[    {        "title": "2401.17861v2.Sharp_pinching_theorems_for_complete_CMC_hypersurfaces_in_the_sphere.pdf",        "content": "arXiv:2401.17861v2  [math.DG]  7 Feb 2024SHARP PINCHING THEOREMS FOR COMPLETE CMC HYPERSURFACES IN\nTHE SPHERE\nLUCIANO MARI ,FERNANDA ROING , ANDANDREAS SAVAS-HALILAJ\nAbstract. In this paper, we prove that every complete, minimally immersed hype rsurface\nf:Mn→Sn+1whose second fundamental form satisfies |A|2≤n, is either totally geodesic\nor (a covering of) a minimal Clifford torus, thereby extending the we ll-known result by\nSimons, Lawson and Chern, do Carmo & Kobayashi from compact to complete Mn. We\nalso extend the corresponding result for hypersurfaces with non vanishing constant mean\ncurvature, due to Alencar & do Carmo, to complete immersed CMC hy persurfaces, under\nthe optimal bound for their umbilicity tensor. Our approach is inspire d by the conformal\nmethod of Fischer-Colbrie and Catino, Mastrolia & Roncoroni.\n1.Introduction\nLetf:Mn→Sn+1,n≥2, be an immersed hypersurface of the unit sphere. According to a\nwell-known result due to Simons [ 24], ifMnis compact, minimal, and the square norm |A|2\nof the second fundamental form Asatisfies|A|2≤n, then either |A| ≡0 or|A|2≡n. If\n|A| ≡0, thenMnis a great sphere and if |A|2≡n, thenMnis a (Riemannian) covering of\nthe Clifford torus given by the natural embedding\nTn(k) =Sk/parenleftbig/radicalbig\nk/n/parenrightbig\n×Sn−k/parenleftbig/radicalbig\n(n−k)/n/parenrightbig\n֒→Sn+1,\nfor somek∈ {1,...,n−1}. The latter characterization is local and was obtained by Chern,\ndo Carmo & Kobayashi [ 10] and, independently, by Lawson [ 16]. This result is known in the\nliterature as Simons’ pinching theorem . This result was extended to compact hypersurfaces\nwith constant mean curvature in the sphere by Alencar & do Carmo [ 1]. More precisely,\nsupposethat f:Mn→Sn+1isacompactimmersed hypersurface with(normalized) constant\nmean curvature H≥0. For short we call such objects CMC hypersurfaces . Denote by g the\nRiemannian metric on Mnand by\nΦ =A−Hg\nthe traceless part of the second fundamental form of the hyper surface, and suppose that\n|Φ|2≤b2(n,H),\nwhereb(n,H) is the positive root of the polynomial\nP(n,H)(x) =x2+n(n−2)/radicalbig\nn(n−1)Hx−n(H2+1). (1.1)\n2000Mathematics Subject Classification. Primary 53C42, 53A10.\nKey words and phrases. CMC and minimal hypersurfaces, Clifford torus, pinching, umbilicity.\nA. Savas-Halilaj is supported by (HFRI) Grant No:14758. L. Mari is s upported by the PRIN 20225J97H5\n“Differential-geometric aspects of manifolds via Global Analysis”.\n12 L. MARI, F. ROING, AND A. SAVAS-HALILAJ\nThen, either |Φ| ≡0 andMnis a sphere or |Φ| ≡b(n,H) andMncovers a minimal Clifford\ntorus or a torus of the form\nSn−1(r)×S1(√\n1−r2)֒→Sn+1,\nof appropriate radius r∈(0,1). This particular example is called H(r)-torus.\nAll the above proofs heavily rely on the strong maximum principle applie d to the inequality\n∆|Φ|2≥ −2|Φ|2P(n,H)(|Φ|)+2|∇Φ|2, (1.2)\nwhereP(n,H)is the polynomial given in ( 1.1); see [1, page 1226]. Indeed, the assumption\n|Φ| ≤b(n,H) implies that P(n,H)(|Φ|)≤0, and therefore ∆ |Φ|2≥0. SinceMnis compact,\n|Φ|2must be constant and thus ∇Φ≡0. The conclusion follows from a careful analysis of\nhypersurfaces with ∇Φ≡0.\nA natural question is whether the same result also holds when the co mpactness of Mnis\nweakened to the assumption that Mnis complete. In this respect, a computation due to\nLeung [17] shows that the Ricci curvature of g satisfies\nRic≥ −n−1\nnP(n,H)(|Φ|)g.\nConsequently, under our assumption it follows that Ric ≥0 on (Mn,g). In dimension\nn= 2, Bishop-Gromov’s theorem implies that M2has quadratic area growth and is therefore\nparabolic, see [ 13,15]. Hence, from ( 1.2) we again obtain that M2is either a sphere or (a\ncovering of) a torus. However, in higher dimensions Ric ≥0 is not enough to guarantee\nthe parabolicity of Mn, and attempts were made to achieve the goal via the Omori-Yau\nmaximum principle at infinity (see [ 2,21] for a thorough investigation of the principle and\nits geometric applications). Although there are partial results, th e problem remains still\nopen. As a matter of fact, one can easily show from ( 1.2) thatMnis a sphere whenever\nsup|Φ|< b(n,H); see for instance [ 31] and [25]. Also, if |Φ(x0)|=b(n,H) for some x0,\nthen the strong maximum principle implies |Φ| ≡b(n,H) and the classification in [ 1] follows.\nConsequently, the main difficulty is to characterize the complete CMC hypersurfaces of the\nunit sphere with\n|Φ|<b(n,H) and sup |Φ|=b(n,H).\nInsuch a case, finding differential inequalities forwhich theOmori-Ya uprincipleyields useful\ninformation seems a hard task.\nFor this reason, we here pursue a different strategy, inspired by t he works of Fischer-Colbrie\n[12] and Catino, Mastrolia & Roncoroni [ 8]. The idea is to conformally change the metric of\nMnby a suitable power of the function\nu=b2(n,H)−|Φ|2>0,\nto show that Mnis compact, from which it readily follows that Mnis a sphere. We obtain:\nTheorem 1.1. Letf:Mn→Sn+1be a complete immersed hypersurface with constant mean\ncurvatureH≥0. Suppose that the square norm |Φ|2of the traceless part of the second\nfundamental form of Mnsatisfies\n|Φ|2≤b2(n,H),\nwhereb(n,H)is the positive root of the polynomial (1.1) (in particular, b(n,0) =√n).CMC HYPERSURFACES 3\nThen, either |Φ| ≡0 (andMnis a totally umbilic sphere )or|Φ| ≡b(n,H). Furthermore,\n|Φ| ≡b(n,H)if and only if:\n(a)H= 0andMncovers a Clifford torus;\n(b)H >0,n≥3andMncovers anH(r)-torus with r2<(n−1)/n;\n(c)H >0,n= 2andMncovers anH(r)-torus with r2/ne}ationslash= (n−1)/n.\nRemark 1.2. Let us make some comments regarding our result.\n(1) In Theorem 1.1we are implicitly assuming that Mnis 2-sided if H/ne}ationslash= 0. On the other\nhand, ifH= 0, thenMnis not assumed to be 2-sided.\n(2) As discussed in [ 1], according to the chosen orientation, the mean curvature of an H(r)-\ntorus is given by either\nH=(n−1)−nr2\nnr√\n1−r2orH=nr2−(n−1)\nnr√\n1−r2.\nThe choice leading to positive His the first one if r2<(n−1)/n, and by direct com-\nputation these H(r)-tori satisfy |Φ| ≡b(n,H). On the other hand, if r2>(n−1)/nthe\nchoice is the second one, but for n≥3 a computation gives |Φ|> b(n,H). Hence, tori\nwith suchrdo not satisfy the assumptions in our theorem. The different behav iour is\ndue to the linear term in P(n,H)and does not occur if n= 2, motivating the presence of\nH(r)-tori with any r2/ne}ationslash= (n−1)/nin the classification.\n(3) Besides [ 8], the method introduced by Fischer-Colbrie in [ 12] was also used in higher\ndimensions by Shen & Ye [ 22], Shen & Zhu [ 23], Cheng [ 9] and Elbert, Nelli & Rosenberg\n[11]. It is worth-mentioning that in all these results it is required that th e manifold Mn\nhas dimension 3, 4 or 5. The absence of a dimensional constraint in ou r theorem was\nsomehow unexpected to us.\n(4) The constant b(n,H) in the theorem is sharp. For example, we infer that the complete\nminimal hypersurfaces constructed by Otsuki [ 19] and do Carmo & Dajczer [ 7] have\nbounded squared norm of the second fundamental form lying in a ra nge containing n;\nnamely\nn(n−1)a2\n0\n1−a2\n0≤ |A|2≤n(n−1)a2\n1\n1−a2\n1,\nwherea0anda1are constants such that\n0<a0<1√n<a1<1;\nfor more details see [ 19, Section 4 & Remark 2, p.162]. Similar results for the norm of the\nsecond fundamental form of a CMC hypersurface with two distinct principal curvatures\ncan be found in [ 3,20].\n(5) For examples of complete CMC hypersurfaces in the sphere Sn+1obtained by gluing\ntechniques, we refer to the paper of Butscher [ 5].\n(6) There are several sphere-type theorems in the literature fo r complete submanifolds in\nspace forms under various pinching conditions on the second funda mental form; see for\nexample [ 4,6,14,26–30]. With the notable exception of [ 14], the compactness either\nis assumed or it directly follows from the assumptions, the Gauss equ ation, and the\nBonnet-Myers theorem.4 L. MARI, F. ROING, AND A. SAVAS-HALILAJ\n2.Proof of the theorem\nLetf:Mn→Sn+1be a complete immersed hypersurface with |Φ| ≤b(n,H).Let us denote\nbyb−<0<b+=b(n,H) the two roots of P(n,H)given in ( 1.1), namely:\nb±=±/radicalBigg\nn2(n−2)2H2\n4n(n−1)+n(H2+1)−n(n−2)H/radicalbig\n2n(n−1).\nFrom|b−|>b+we deduce that, for x∈[0,b+],\nP(n,H)(x) = (x−b+)(x−b−)≤(x−b+)(x+b+) =x2−b2\n+,\nwhence ( 1.2) can be written in the form\n∆|Φ|2≥2(b2−|Φ|2)|Φ|2+2|∇Φ|2≥0, (2.1)\nwhere, hereafter, b=b+=b(n,H). As a consequence, the function\nu.=b2−|Φ|2,\nsatisfies\nu≥0 and ∆ u≤ −2|Φ|2uonM. (2.2)\nWe distinguish two cases:\nCase 1:Ifu(x0) = 0 for some x0∈Mn, by the strong maximum principle u≡0, whence\n|Φ|2≡b2and|∇Φ| ≡0.\nThen the conclusion follows from the results in [ 1, p. 1227-1228] or in [ 16, Theorem 4].\nMore precisely, it is shown that the hypersurface has two distinct p rincipal curvatures, both\nconstants, and that every x∈Mnhas a neighbourhood Ufor whichf(U) is a piece of\neither a Clifford torus Tn(k) or anH(r)-torus (with rin the range stated in the theorem),\naccording to the value of H. We show that f(Mn) is a single such torus. Indeed, any Clifford\norH(r)-torus is the zero set of an appropriate real analytic function on Sn+1; see for instance\n[18, Example 3, p.194]. Fix x∈MnandUas above and let Ψ : Sn+1→Rbe a real analytic\nfunction that vanishes on the torus containing f(U). BeingMnandfreal analytic as well,\nthen Ψ◦fis real analytic and vanishes on U, thus it vanishes on the entire Mn. This shows\nthatf(Mn) is contained in a single torus Σn. Asf:Mn→Σnis a local isometry and Mnis\ncomplete, Ambrose’s Theorem guarantees that fis onto and a Riemannian covering, which\nproves our claim.\nCase 2: Assume now that u >0 onMn. Our goal is to prove that Mnmust be a totally\numbilic sphere. To reach the goal, inspired by [ 8,12] we endow Mnwith the metric\ng =u2βg,\nwhere\nβ=\n\nany number in (0 ,1) ifn= 2,3,\n1\nn−2ifn≥4.(2.3)\nConsider a curve γ: [0,a]→Mparametrized by g-arclength s, and denote by stheg-\narclength of γ. From\n∂s=u−β∂sandds=uβdsCMC HYPERSURFACES 5\nthe length of γin the metric g is\nℓg(γ) =/integraldisplaya\n0uβds.\nWe split the proof into three claims:\nClaim 1: Assume that γis ag-geodesic with non-negative second variation of g-arclength.\nThen, there exist constants t0>1,c0>0depending on n,βsuch that\nc0/integraldisplaya\n0uβψ2ds≤ −2t0/integraldisplaya\n0uβψψssds,∀ψ∈C2\n0([0,a]), (2.4)\nwhereC2\n0([0,a])is the set of functions ψ∈C2([0,a])such thatψ(0) =ψ(a) = 0.\nProof of Claim 1. From the second variation formula, along γwe have\n/integraldisplays(a)\n0/braceleftbig\n(n−1)(ϕs)2−ϕ2Ric(γs,γs)/bracerightbig\nds≥0,∀ϕ∈C2\n0([0,a]),\nor, equivalently,/integraldisplaya\n0/braceleftbig\n(n−1)(ϕs)2−ϕ2Ric(γs,γs)/bracerightbig\nu−βds≥0,∀ϕ∈C2\n0([0,a]). (2.5)\nAs it is shown in [ 11, Appendix, eqn. (14)], along γthe following identity holds:\nRic(γs,γs) = Ric(γs,γs)−β(n−2)(lnu)ss−β∆lnu.\nFrom (2.2),\n∆lnu≤ −2|Φ|2−|∇lnu|2≤ −2|Φ|2−{(lnu)s}2,\nwhence\nRic(γs,γs)≥Ric(γs,γs)+2β|Φ|2−β(n−2)(lnu)ss+β{(lnu)s}2. (2.6)\nLet{e1=γs,e2...,en}be a g-orthonormal frame along γ. From the Gauss equation, we\nhave that the components of the Riemann curvature tensor are g iven by\nRijij= 1−δij+hiihjj−h2\nij, i,j∈ {1,...,n},\nwherehijare the components of the second fundamental form of Mn. The identity can\nequivalently be written in the form\nRijij= 1−δij+/parenleftbig\nΦii+H/parenrightbig\n(Φjj+H)−(Φij+Hδij)2, i,j∈ {1,...,n}.\nUsing the fact that Φ is traceless, we get\nRic(γs,γs) = (n−1)−Φ2\n11+(n−2)HΦ11+(n−1)H2−n/summationdisplay\nj=2Φ2\n1j.\nTo estimate the terms, notice that because tr(Φ) = 0, the Cauchy -Schwarz inequality gives\nΦ2\n11≤(n−1)n/summationdisplay\ni=2Φ2\nii.\nConsequently,\n|Φ|2≥n/summationdisplay\ni=1Φ2\nii+2n/summationdisplay\nj=2Φ2\n1j≥n\nn−1Φ2\n11+2n/summationdisplay\nj=2Φ2\n1j≥n\nn−1n/summationdisplay\nj=1Φ2\n1j.(2.7)6 L. MARI, F. ROING, AND A. SAVAS-HALILAJ\nFixτ∈(0,1] andε>0. Let us estimate the term HΦ11by means of Young inequality and\n(2.7) as follows\nHΦ11=H(1−τ)Φ11+τHΦ11\n≥ −H(1−τ)|Φ|/radicalbigg\nn−1\nn−τH2\n2ε−τεΦ2\n11\n2.\nTherefore,\nRic(γs,γs)≥(n−1)−(n−2)/radicalbigg\nn−1\nn(1−τ)|Φ|H+/parenleftbigg\nn−1−(n−2)τ\n2ε/parenrightbigg\nH2\n−/parenleftbigg\n1+(n−2)τε\n2/parenrightbigg\nΦ2\n11−n/summationdisplay\nj=2Φ2\n1j.(2.8)\nSinceP(n,H)(b) = 0 and |Φ| ≤b, we have\n(n−2)/radicalbigg\nn−1\nnH|Φ| ≤(n−1)(H2+1)−n−1\nn|Φ|2.\nHence,\nRic(γs,γs)≥τ(n−1)+τ/parenleftbigg\nn−1−n−2\n2ε/parenrightbigg\nH2\n+(1−τ)n−1\nn|Φ|2−/parenleftbigg\n1+(n−2)τε\n2/parenrightbigg\nΦ2\n11−n/summationdisplay\nj=2Φ2\n1j(2.9)\nand we conclude that\nRic(γs,γs)≥τ(n−1)+τ/parenleftbigg\nn−1−n−2\n2ε/parenrightbigg\nH2\n+/parenleftbigg\n2β+(1−τ)n−1\nn/parenrightbigg\n|Φ|2−/parenleftbigg\n1+(n−2)τε\n2/parenrightbiggn/summationdisplay\nj=1Φ2\n1j\n−β(n−2)(lnu)ss+β{(lnu)s}2.(2.10)\nBy (2.7),\n/parenleftbigg\n2β+(1−τ)n−1\nn/parenrightbigg\n|Φ|2−/parenleftbigg\n1+(n−2)τε\n2/parenrightbiggn/summationdisplay\nj=1Φ2\n1j\n≥/parenleftbigg2nβ\nn−1−τ−(n−2)τε\n2/parenrightbiggn/summationdisplay\nj=1Φ2\n1j.\nWe shall specify τandεso that\n\n\nn−1−n−2\n2ε≥0,\n2nβ\nn−1−τ−(n−2)τε\n2≥0.CMC HYPERSURFACES 7\nIndeed, it is enough to choose\nε=\n\n1 if n= 2,\nn−2\n2(n−1)ifn≥3,andτsmall enough.\nFor such a choice, setting c0=τ(n−1)>0, inequality ( 2.10) yields\nRic(γs,γs)≥c0−β(n−2)(lnu)ss+β{(lnu)s}2. (2.11)\nReplacing ( 2.11) in (2.5), we obtain\n(n−1)/integraldisplaya\n0(ϕs)2u−βds≥/integraldisplaya\n0ϕ2u−β/parenleftBig\nc0−β(n−2)(lnu)ss+β{(lnu)s}2/parenrightBig\nds.(2.12)\nIntegration by parts gives\n−β/integraldisplaya\n0ϕ2(lnu)ssu−βds=−/integraldisplaya\n0ϕ2(lnuβ)ssu−βds\n= 2β/integraldisplaya\n0ϕϕs(lnu)su−βds−β2/integraldisplaya\n0ϕ2{(lnu)s}2u−βds,\nand plugging into ( 2.12) yields\n(n−1)/integraldisplaya\n0(ϕs)2u−βds≥c0/integraldisplaya\n0ϕ2u−βds (2.13)\n+2β(n−2)/integraldisplaya\n0ϕϕsu−β−1usds+β/parenleftbig\n1−β(n−2)/parenrightbig/integraldisplaya\n0ϕ2u−β−2(us)2ds.\nWe follow ideas by Catino, Mastrolia & Roncoroni [ 8] to treat the integral inequality ( 2.13).\nSet\nϕ=uβψ,withψ∈C2\n0([0,a]).\nThen,\n\nϕs=βuβ−1usψ+uβψs,\nϕϕs=βu2β−1usψ2+u2βψψs,\n(ϕs)2=β2u2β−2(us)2ψ2+u2β(ψs)2+2βu2β−1usψψs.\nThen, (2.13) becomes\n(n−1)/integraldisplaya\n0(ψs)2uβds≥c0/integraldisplaya\n0ψ2uβds (2.14)\n+β(1−β)/integraldisplaya\n0uβ−2(us)2ψ2ds−2β/integraldisplaya\n0uβ−1usψψsds.\nDefine\nI.=/integraldisplaya\n0uβ−1usψψsds.\nIntegration by parts gives\nI=1\nβ/integraldisplaya\n0(uβ)sψψsds=−1\nβ/integraldisplayβ\n0uβ(ψs)2ds−1\nβ/integraldisplaya\n0uβψψssds.8 L. MARI, F. ROING, AND A. SAVAS-HALILAJ\nFor everyt>1 and every ε>0, from Young’s inequality we obtain\n2βI= 2βtI+2β(1−t)I\n=−2t/integraldisplaya\n0uβ(ψs)2ds−2t/integraldisplaya\n0uβψψssds+2β(1−t)/integraldisplaya\n0uβ−1usψψsds\n≤ −2t/integraldisplaya\n0uβ(ψs)2ds−2t/integraldisplaya\n0ψψssuβds\n+β(t−1)ε/integraldisplaya\n0uβ−2(us)2ψ2ds+β(t−1)\nε/integraldisplaya\n0uβ(ψs)2ds.\nChoosing\nε=1−β\nt−1\nwe obtain\n2βI≤ −2t/integraldisplaya\n0uβψψssds+β(1−β)/integraldisplaya\n0ψ2uβ−2(us)2ds\n+β(t−1)2−2t(1−β)\n1−β/integraldisplaya\n0uβ(ψs)2ds. (2.15)\nInserting ( 2.15) into (2.14), we arrive at\n/integraldisplaya\n0uβ/braceleftbig\nc0ψ2−p(n,t,β)(ψs)2+2tψψss/bracerightbig\nds≤0, (2.16)\nwhere\np(n,t,β) =β(t−1)2\n1−β−2(t−1)+(n−3).\nWith our choice of β, we get\np(n,t0,β)≤0 where t0=/braceleftBigg\n1+21−β\nβifn∈ {2,3},\nn−2 ifn≥4.\nNotice that t0>1, whence it is admissible. Then, ( 2.16) becomes ( 2.4). /square\nClaim 2: The manifold Mnis compact.\nProof of Claim 2. Assume by contradiction that Mnis non-compact. We follow [ 12] and\nconstruct the“smallest divergent curve”in ( Mn,g) issuing from a fixed point. For the sake\nof completeness, we include a simplified and a bit more general statem ent:\nLemma 2.1. Let(Mn,g)be a non-compact Remannian manifold. Then, for each o∈M,\nthere exists a divergent curve γ: [0,T)→Mnissuing from owhich is a g-geodesic, minimizes\ntheg-length on any compact subinterval of [0,T)and satisfies the following property:\ngis complete ⇐⇒T=∞.\nProof.The implication ⇒is obvious, since we know that for a complete Riemannian metric\ng every divergent g-geodesic is defined on the entire [0 ,∞). To prove the converse, we shall\nconstruct such a geodesic γ. Consider an exhaustion of Mnby relatively compact, smooth\nopen sets Ω jcontaining o. SinceΩjis a smooth manifold with boundary, there exists aCMC HYPERSURFACES 9\ng-minimizing rectifiable curve γj: [0,Tj]→Ωjjoiningoto∂Ωj, which we parametrize by\ng-arclength. Because γjis ag-geodesic, γj([0,Tj))⊂Ωj, and since Ωj⊂Ωj+1, we have that\n{Tj}is a strictly increasing sequence. Then, up to a subsequence,\nγj→γ: [0,T)→Mn,\nsmoothly on compact sets as j→ ∞. Since each γjminimizes g-length between any pair of\nits points, it follows that γis ag-geodesic that minimizes g-length between any pair of its\npoints as well. Moreover, let σ: [0,Tσ)→Mnbe any other divergent curve, parametrized\nbyg-arclength. Then, for each natural j, lettjbe the first time for which σ(tj)∈∂Ωj. By\nthe minimality of γj, and having fixed S >0, we have for large enough jthat\nTσ=ℓg(σ)≥ℓg(σ|[0,tj])≥ℓg(γj)≥ℓg((γj)[0,S])j→∞→ℓg(γ|[0,S])S→∞→ℓg(γ) =∞.\nWhence,Tσ=∞and thus the Riemannian metric g is complete. ⊛\nPick a smallest divergent curve γin (Mn,g) issuing from a fixed origin. Since ( Mn,g) is\ncomplete, reparametrizing γby g-arclength s, it turns out that γis defined for s∈[0,∞).\nWhence, because of Claim 1 and since γisg-minimizing between any pair of its points, along\nγit holds\nc0/integraldisplaya\n0uβψ2ds≤ −2t0/integraldisplaya\n0uβψψssds,∀a>0 andψ∈C2\n0([0,a]).\nChoosing as test function\nψ(s) = sin/parenleftBigπs\na/parenrightBig\n∈C2\n0([0,a]),\nwe get\n/parenleftBig\nc0−2t0π2\na2/parenrightBig/integraldisplaya\n0sin2/parenleftBigπs\na/parenrightBig\nuβ(γ(s))ds≤0,\nwhich gives a contradiction if\na>π/radicalbigg\n2t0\nc0.\nThis completes the proof of Claim 2. /square\nClaim 3: The Riemannian manifold (Mn,g)is a totally umbilic sphere.\nProof of Claim 3. SinceMnis compact and ∆ |Φ|2≥0, we deduce that |Φ|2is constant.\nInequalityu>0 implies |Φ|2<b2. Thus again from ( 1.2) we get\n(b2−|Φ|2)|Φ|2≤0,\nwhence|Φ|2≡0 andMnis a sphere. /square\nThis completes the proof of the theorem.\nAcknowledgements . The authors would like to thank Aldir Brasil for kindly pointing out\nthe references [ 3,4].10 L. MARI, F. ROING, AND A. SAVAS-HALILAJ\nReferences\n[1] H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature in spheres , Proc. Am. Math.\nSoc.120(1994), 1223-1229.\n[2] L.-J. Al´ ıas, P. Mastrolia, and M. Rigoli, Maximum principles and geometric applications , Springer\nMonogr. Math., Springer, Cham, 2016.\n[3] L.-J. Al´ ıas, S. de Almeida, and A. Brasil, Hypersurfaces with constant mean curvature and two princip al\ncurvatures in Sn+1, An. Acad. Bras. Ciˆ enc. 76(2004), 489-497.\n[4] A.AspertiandE.Costa, Vanishing of homology groups, Ricciestimate for submanifo lds and applications ,\nKodai Math. J. 24(2001), 313-328.\n[5] A. Butscher, Gluing constructions amongst constant mean curvature hype rsurfaces in Sn+1, Ann. Global\nAnal. Geom. 36(2009), 221-274.\n[6] M. Dajczer and Th. 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Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three -manifolds , Invent.\nMath.82(1985), 121-132.\n[13] A. Grigor’yan, Analytic and geometric background of recurrence and non-ex plosion of the Brownian\nmotion on Riemannian manifolds , Bull. Amer. Math. Soc. (N.S.) 36(1999), 135-249.\n[14] Th. Hasanis and Th. Vlachos, Ricci curvature and minimal submanifolds , Pac. J. Math. 197(2001),\n13-24.\n[15] A. Huber, On subharmonic functions and differential geometry in the la rge, Comment. Math. Helv. 32\n(1957), 13-72.\n[16] H.-B. Lawson, Local rigidity theorems for minimal hypersurfaces , Ann. of Math. 89(1969), 187-197.\n[17] P.-F. Leung, An estimate on the Ricci curvature of a submanifold and some a pplications , Proc. Amer.\nMath. Soc. 114(1992), 1051-1061.\n[18] K. Nomizu, Elie Cartan’s work on isoparametric families of hypersurfa ces, Differ. Geom., Proc. Symp.\nPure Math. 27, Part 1, Stanford 1973, 191-200 (1975) 114(1975), 191-200.\n[19] T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature , Am. J. Math. 92\n(1970), 145-173.\n[20] O. Perdomo, CMC hypersurfaces with two principal curvatures , arXiv:2111.01966 (2021), 1-46.\n[21] S. Pigola, M. Rigoli, and A.G. Setti, Maximum principles on Riemannian manifolds and applicatio ns,\nMem. Amer. Math. Soc. 174(2005), no. 822, x+99.\n[22] Y. Shen and R. Ye, On stable minimal surfaces in manifolds of positive bi-Ricc i curvatures , Duke Math.\nJ.85(1996), 109-116.\n[23] Y. Shen and S. Zhu, Rigidity of stable minimal hypersurfaces , Math. Ann. 309(1997), 107-116.\n[24] J. Simons, Minimal varieties in Riemannian manifolds , Ann. of Math. 88(1968), 62-105.\n[25] Th. Vlachos, Complete submanifolds with parallel mean curvature in a sph ere, Glasg.Math. J. 38(1996),\n343-346.\n[26] H.-W. Xu, F. Huang, and E. Zhao, Geometric and differentiable rigidity of submanifolds in sp heres, J.\nMath. Pures Appl. (9) 99(2013), 330-342.\n[27] H.-W. Xu, F. Huang, and E. Zhao, Geometric, topological and differentiable rigidity of subm anifolds in\nspace forms , Geom. Funct. Anal. 23(2013), 1684-1703.CMC HYPERSURFACES 11\n[28] H.-W. Xu and J.-R. Gu, An optimal differentiable sphere theorem for complete manif olds, Math. Res.\nLett.17(2010), 1111-1124.\n[29] H.-W. Xu and J.-R. Gu, The differentiable sphere theorem for manifolds with positi ve Ricci curvature ,\nProc. Am. Math. Soc. 140(2012), 1011-1021.\n[30] H.-W. Xu and L. Tian, A differentiable sphere theorem inspired by rigidity of mini mal submanifolds ,\nPac. J. Math. 254(2011), 499-510.\n[31] P. Yanglian, Pinching theorems of Simons type for complete minimal subma nifolds in the sphere , Proc.\nAm. Math. Soc. 93(1985), 710-712.\nLuciano Mari\nDipartimento di Matematica “Giuseppe Peano”, Universit `a degli Studi di Torino, 10123\nTorino, Italy\ne-mail:luciano.mari@unito.it\nFernanda Roing\nDipartimento di Matematica “Giuseppe Peano”, Universit `a degli Studi di Torino, 10123\nTorino, Italy\ne-mail:fernanda.roing@unito.it\nAndreas Savas-Halilaj\nDepartment of Mathematics, Section of Algebra&Geometry, U niversity of Ioannina, 45110\nIoannina, Greece\nE-mail:ansavas@uoi.gr"    },    {        "title": "2401.17866v1.Making_Sense_of_Knowledge_Intensive_Processes__an_Oil___Gas_Industry_Scenario.pdf",        "content": "  \n Making Sense of K nowledge Intensive \nProcesse s: an Oil & Gas Industry  \nScenario   \n \nAbstract  \nSensemaking is a constant and ongoing process by \nwhich people associate meaning to experiences. It can \nbe an individual process , known as abduction , or a \ngroup process by which people give meaning to \ncollective experiences . The sensemaking of a group is \ninfluenced by the abduction process of each person \nabout the experience. Every collaborative  process \nneeds some level of sensemaking to  show result s. For a \nknowledge –intensive process, sensemaking is central \nand related to most of its tasks. We present findi ngs \nfrom a fieldwork executed in  knowledge -intensive \nprocess from the Oil & Gas  industry . Our findings  \nindicate d that different types of knowledge can be \ncombined to  compose the result  of a sensemaking \nprocess  (e.g. decision, the need for more discussion, \netc.).  This paper presents an initial set of knowledge  \ntypes that can be combined to compose the result of \nthe sensemaking of a collaborative decision -making \nprocess.  We also discuss ideas for using systems \npowered by Artificial Intelligence to support \nsensemaking processes.  \nAuthor Keywords  \nSensemaking;  abduc tion; knowledge -intensive process; \nsocial -technical process; decision -making , artificial \nintelligence .   \n \n \n \n \n \n \n \nThis paper was presented at the Sensemaking in a Senseless World  \nworkshop  during the 2018 ACM CHI Conference on Human Factors in \nComputing Systems.  Juliana Jansen Ferreira  \nVinícius Segura \nAna Fucs  \nVisual Analytics & Comprehension  \nIBM Research, Brazil  \nAvenida Pasteur, 138 - Urca Rio de Janeiro - RJ, 22290 -240 \n{jjansen , vboas , anafucs} @br.ibm.com  \n \nRogério de Paula  \nVisual Analytics & Comprehension  \nIBM Research, Brazil  \nRua Tutóia, 1157, Paraíso São Paulo  - SP, 04007 -005 \nropaula@br.ibm.com  \n  \n ACM Classification Keywords  \nSocial and professional topics → Socio -technical systems ; \nHuman -centered computing → Empirical studies in collaborative and \nsocial computing  \nIntroduction  \nAccording to Weick and Meader ( [7], p. 232)  \nsensemaking is the process of constructing  “moderately \nconsensual def initions  that cohere long enough for  \npeople to be able to infer some idea of what they have,  \nwhat they want, why they can not get it, and why it \nmay not be worth getting in the  first place. ” In \nprofessional practices, it is commonly related to \ndecision -making processes [6, 9], which directly  affect  \nbusiness success . Several practices have challenges \nregarding sensemaking. Software development , for \nexample,  is a very social -technical process which \nsensemaking goes from the early stages ( e.g. \nrequirement elicitation) through software deploy and \nhomologation. The development team among \nthemselves [ 8] and with users are constantly looking to \nreach a common ground about innumerous impasses.  \nFor knowledge -intensive processes, sensemaking \nappear s even during the definition of the process itself. \nFor instances  of these processes , the sensemaking is \ncentral. All people involved in that process need to be  \naligned and have a common understanding about the \nsensemaking process results.  Our sensemaking \nscenario was identified during a fieldwork in an Oil & \nGas company. We collected  the data from 9 interviews \nwith interp reters with different formation s, \nbackgrounds , and industry experience s. Our findings \nunveiled different  types of knowledge that comprise \ntheir sensemaking process. Those types of knowledge \nmay help to identify the people who should  be involved in the sensemaking process to achieve that company’s \ngoal. Our findings also suggest ways of  using Artificial \nIntelligence (AI) powered systems in support of \nsensemaking processes.  \nIn this position paper, we present the related work for \nsensemaking in knowledge -intensive contexts. Then, \nwe describe our fieldwork. After that, we present our \nfinding s from the analysis of fieldwork data. And , in the \nlast section, we discuss the findings, our ongoing \nrelated research , and next steps.  \nRelated work  \nThe s ensemaking  process  has strong social \ncharacteristics  when, for instance, people face \nobstacles and impasses  [2]. In a ttempting to overcome \nthem, people may try to answer  a set of tacit  \nquestions , such as,  what is stopping me , what I can do \nabout it , and where I can find  assistance in choosing \nand taking an action  [1]. In the context of decision -\nmaking, which  may present consequences related to \nhow people surpass those obstacles and impasses , that \nsensemaking may be more sensitive  (e.g. decisions \nregarding people security, large amount of money, \netc.).  \nSchön [ 6] says that sensemaking processes in \nprofessional practice s are motivated by scenarios of \ncomplexity, instability , and uncertainty, combined with \nindeterminacies and value conflicts. Uncertainty and \nthe need to  making decision are characteristics of such \nscenarios [ 9]. The sensemaking of a group is influenced \nby each person’s abduction process about the scenario \nand the decision to be made. Ab duction is the process \nof forming explanatory hypotheses  [4]. It is important \nfor the study of meanings  people assign to any  \n experience  because it des cribes the logic of human \nsense making, from practical  mundane situations to  \nmore elaborate argumentatio n [3].  \nSensemaking is intrinsic to software design and \ndevelopment processes . People need to make sense of \na problem context, then design and build a solution for \nthat problem, which can be, for example, a piece of \nsoftware. The design r ationale tells the story of that \nprocess. From problem to solution, all involved people \nparticip ate in sensemaking so as to attend a client’ \nrequirement . Designers use  different support tools  (e.g. \nwhiteboard , flipcharts, wireframes, storyboards, etc.)  to \nregister a common understa nding [ 8] and communicate  \nthe results of each step of their design sensemaking \nprocess. Software developers externalize their \nsensemaking relative to the users’ problems in different \nartifacts throughout the development process. They \ncreate documents, conceptual models, lines of code, \netc. to communicate their understanding to other \ndevelopers and also to end -users  [3].  \nSoftware development scenario presents  somehow a \nstructured process , but when the scenario is associated \nwith a knowledge -intensive process, the sensemaking \nplays the role of the process itself. A knowledge -\nintensive  process is characterized by  activities that \ncannot be planned in advance , may change  on the fly \nand are driven by the contextual scenario  in which  the \nprocess is embedded. Who should be  involved and who \nshould be  the right person  to execute a particular step  \nare dictated by  the scenario . Plus, the set of users \n \n1 http://wiki.aapg.org/Seismic_interpretation  involved may not  be formally  defined and may be \nidentified  as the process scenario unfolds  [5]. \nOur s ensemaking fieldwork scenario  \nOur fieldwork to discuss sensemaking scenario is a \nknowledge intensive processes related to the  seismic \ninterpretation1 in an Oil & Gas  company. Seismic \ninterpretation is a central process in the E xploration & \nProduction industry and its main goal is  support other \ndecision -making processes by reducing uncertainty . To \nachieve that goal, different people engage in multiple \ninformal interactions and collaboration  sessions , \nembedding biases, decisions, and reputation.  Seismic \ninterpretation  is the process of inferring the geology of \na region at some depth fro m the processed seismic \nsurvey. A seismic survey is a  data set of soundwaves \nrefracted and reflected through Earth’s crust measured \nand recorded respect from a particular area of the \nEarth's surface, to evaluate the subsurface2. Figure 1 \nand Figure 2 show examples of  seismic data line s, \nwhich is  a portion of a seismic survey.  \n \n \n2 http://www.glossary.oilfield.slb.com/Terms/s/  \nseismic_survey.aspx   \n  \nFigure 1. Seismic image example (Netherland - Central Graben  \n– inline 474)3 \nTo perform a seismic interpretation, t he interpreter  \n(which can be a geologist or  a geophysicist ) gets the \nseismic image  and analyzes  it based on a set of other \nknowledge s, from paper s, reference books, knowledge \nfrom previous projects, etc. In our sensemaking \nscenario, we have a fiction al interpreter called Paul. He \nanalyzes  the seismic image and identifies a particular \ngeological formation called salt diapir  (Figure 1). He \nknow s that this kind of formation is frequently related \nto regions that present potential for reservoir of \nhydrocarbonate , which means presence of oil or gas \nthat can be explored and provide revenue.  \nAfter raising the hypothesis of a salt diapir  existence , \nPaul plays with seismic data  using different filters  and \nimage representation ( Figure 2) in different tools  (e.g. \nPetrel4, Paradigm5, etc.) to check if there are more \nvisual evidence s of that formation in the seismic data.  \n \n3 Images taken from the Open Seismic Repository, a free and \npublic data set, of the Netherlands Offshore F3 Bloc k - \nhttps://opendtect.org/osr /  \n4 https://www.software.slb.com/products/petrel  He collect s more evidence to show to his peers in a \nreview meeting.   \n \nFigure 2. Same seismic line as Figure 1 with different \nvisualization characteristic3 \nTo support his salt diapir hypothesis, Paul looks for \npapers relating the Central Graben basin in the North \nSea to the salt diapir geological formation. He  finds \npaper s, such as,  “Geometry and late -stage structural \nevolution of Central Graben salt diapirs, North Sea ” 6 , \nwhich support his hypothesis of salt presence , with  top \nsalt indication in  Figure 3.  \nPaul combines what he discovered in his interpretation \nwith the knowledge he acquired from the experiences \nof previous project s, documents , and his personal \nnotes . He also talks to Gary, another interpreter , who \nworked in another project on the Central Graben  basin \na few years ago, to collect more information about that \nbasin. Once Paul is confident about his salt diapir \n5 http://www.pdgm.com/solutions/seismic -processing -and-\nimaging/  \n6 I Davison, I Alsop, P Birch, et al. 2000. Geometry and late -\nstage structural evolution of Central Graben salt diapirs, North \nSea. Marine and Petroleum Geology 17, 4: 499 –522. \n \n hypothesis, he puts together a report to discuss with \nhis peers. He prepares a presentation  with his \nassessment about investing money on Central Graben \nbasin area , showing all his steps to get to the \nhypothesis  and the  related evidence  of return -of-\ninvestment . \n \nFigure 3. Seismic line interpretation presented in paper about \nsalt diapirs in Netherland - Central Graben6 \nIn the review meeting, there are a group of interpreters \nwith different background s, experience s, and \nformation s (geologist or geophysicist s). Paul presents \nhis findings, illustrating and  sustaining  his hypothesis \nwith a set of evidence s. All interpreter s participating in \nthe review meeting need to get into a common ground \nabout  Paul’s findings in Central Graben basin.  They \nmust decide if the company should invest in that area \nor not. That decision is associated with the spending of \nmillions of dollars. The y want to invest in areas that will \neventually have large revenue, generating company’s \nbusiness value.  We anonymized the fieldwork data to protect our \nparticipant’s identities and any confidential information  \nor data . This did not impact our findings or discussions.  \nFindings related to sensemaking  \nIn the fieldwork, we identified two moments where the \nsensemaking needs to be externalized in some way: 1) \nwhen Paul puts together a report to be reviewed by his \npeers (individual sensemaking), and 2) when all \ninterpreters involved in the review need to decide about \nthe investment in the  investigated area (collective  \nsensemaking ). This scenario in the Oil & Gas  industry \nreveals a significant and distinct system of \ncollaboration. Given the intrinsic risk and uncertainty \nproperties of seismic interpretation , we observed  that \ncombining efforts in a teamwork is a way to construct a \nbetter supported model and, in a simplified vision, \nshare res ponsibility over decisions in an Oil & Gas  \nproject.  \nThroughout the interpretation process, t he \nsensemaking is built with different types of knowledge . \nThe analyzed data from fieldwork allowed us to define \nthree types of knowledge necessary in the process. \nThese knowledges combined construct the substantial \nasset of information used in the seismic interpretation \nprocess . The types of knowledge are:  \nDomain knowledge  – It concerns the information \nsomeone knows about the activity he/she executes in \nthe project. For instance, an interpreter has a domain \nknowledge of seismic image interpretation.  \nContext knowledge  – Some people might have a vast \nexperience in a specific context, such as a country, a \n \n region , a field  or a basin. This knowledge enables some \ninference of geological properties in a project.  \nHistorical knowledge  – In the Oil & Gas  industry it is \nnot unusual  to observe experts that resort to previous \nsimilar projects and apply prior learnings and \nknowledge while creating a new model. The history of \nsomeone’s career in this industry  matters in a very \nsignificant way.  \nPeople with different profiles regarding types and levels \nof knowledge are responsible of building data resources \nfor a project. This data stands for a result of what we \ncall “collaborative knowledge construction” and intends \nto minimize unc ertainties  related to the project . \nParticipants in our fieldwork highlighted this \n“collaborative knowledge construction” : \nIt always happens as a symbiosis. I'm not here working \nstatically. I work with a geologist to have the \ninformation, to know what’s his opinion, what he thinks \nabout my project. 90% of the time the discussion ends \nwith these people: myself, Tom, Mary , and Mike. The \ngeologist is also always present. (Participant #1)  \nWe work very close to a lot of them (interpreters). They \nare geophysicist... petrophysicist too...very close of all! \nWe rely on them all. We integrate. (Participant #2)  \nThe quotes above show the cooperative aspect of the \nseismic interpretation and model creation  process . \nDiscussions, consulting , and feedback are some of the \nsystematically identified routines. We identif ied some \ninteresting characteristics in that sensemaking process  \nof collabora tive knowledge c onstruction  and categorize d \nthem as showed  in Figure 4. As we described, individualities are a significant part of \na team’s construction. As a teammate starts working in \na project, his/her inputs contribute to the workflow of \ndata and expands the project’s resources \n(sensemaking) according to its knowledge type : \ndomain, context , or historical. For instance, a less \nexperienced interpreter ( Figure 4, A) may consult a  \n \nFigure 4. Collaborative Knowledge Construction  \ncolleague with context knowledge ( Figure 4, B) to \ngather more significant data about the basin in which \nhis project is situated. Other colleagues with historical \nknowledge might collaborate ( Figure 4, C, D) \ncontributing to a much more structured information \nnetwork that will grow dynamically, resulting in a lower \nrisk decision making situation.  \n \n We confirm that one  person  might be required or \ndemanded to participate in a team, considering all \naspects of his /her professional profile. A specific \nexperience or past participation in the same or \nanalogue region is a strong reason for a contribution. \nFindings also define d profile needs. Specific experiences \nestablish key players in the process, as we might \nobserve in the excerpts below:  \nWe ask for a participant according to what was found … \ntypes of gases, fluids etc. (Participant 3)  \nAlthough they ( other  team) know how to calculate \nvolume data conversions, they might sometimes not \nhave sensitivity to realize if values are correct \n(Participant #1)  \nI did take part in several studies so I already have \nmany case studies in my head (Participant #4)  \nThe whole  sensemaking  process in our scenario is \nshrouded in uncertainty, so all stages need to be \ncomposed of pieces of e vidence, which must be \ntraceable at any time to support someone else’s \ndecision or next step:  \nWhen I receive the information, if it is not properly \naudited or if we can’t track it, I take a risk using these \nparameters. Tracking the work is very important for us. \nThis will be successively accumulated ( Participant #2)  \nLet’s suppose a colleague has used a cutoff he \nconsidered the best choice  for that project. Then I start \nusing a cutoff that doesn’t correspond at all with the \nfirst one . For instance, if the rock does not have a \nminimum porosity of 5% , it is so close d up that it \ndoesn’t interests us. A petrophysicist  said the cutoff was 5%. So, I realize  with additional data  that we need \na new cutoff of 10% because 5% is not enough. There \nare adjustments to make. If this is not reported  and \ntracked, we lose information and  it has a direct impact \non the final volume we calculate. ( Participant #2)  \nDuring the sensemaking process , the execution  and its \nresults should consistently be reviewed and adjusted \naccording to the accumulated knowledge. Quality check \nis an established practice that  clearly reduces risk along \nthe project.  \nWe always try to set some criteria within a project. If \nit’s wrong it needs to stand consistently wrong so it’s \nmuch easier to get it corrected in the future.  (…) When \nanalyzing uncertainty, we evaluate the volumes and we \nhave 4 to 5 controlling parameters. Our job is always to \nintegrate more data to reduce uncertainty. (Participant \n#2) \nFindings related to sensemaking  \nThe size  of the  Oil & Gas  company  (not a big company)  \ninfluences the close collaborative interaction among \npeople.  It helps the sensemaking process by knowing \nwho to look for if a certain knowledge is needed in a \nproject. Some have experience in different p hases in \nthe process, so those have an overall idea of the \nsensemaking necessary for mak ing a decision  in a \nproject . \nThe most interesting quotes were from people in the \nmiddle of the process, those who rely heavily on other \npeople assessment  (individual sensemaking)  to make \ntheir own evaluation and send it to someone else. For \nthose people, tracing the data is very important.  The  \n sensemaking process register is a huge and crucial \nchallenge.  \nDifferent profiles act on different parts in the process. \nDetails about people ’s profile  (geologist, petrophysicist, \netc.) interfere in the sensemaking process. The \nidentification of key people in a sensemaking process \nmight increase the success of the process result.  \nRemarks  and future work  \nOur fieldwork was executed i n a small Oil & Gas  \ncompany. In that context, people somehow know who \nto look for in specific situations or decisions. Also , due \nto the small company context, people have an idea in \nwhich project colleagues are involved  in that moment . \nHowever, the current context  of a company  does not \ngive any indication about a person previous experience \nin other companies ( historical knowledge ), with \nother geological context s (context knowledge ), or \nother phases in the Oil & Gas  decision -making process \n(domain knowledge ). That knowledge could enrich \nthe sen semaking of future projects by involving people \nwith the necessary knowledge to make better decisions.  \nOur findings indicated the need of tracking resources to \nsupport the sensemaking process for decision -making \nat Oil & Gas  industry. People rely on others \nsensemaking results to build their own. To trust t he \ninput for  their individual sensemaking, people need to \nverify the sensemaking traces of that input.  \nWe are currently studying  how Artificial Intelligence  \n(AI) Powered Systems  could impact the sensemaking \nprocess in a knowledge -intensive context. With the \nrapid development and adoption of AI Powered \nSystems, the humans -machines relationship has becom e increasingly more complex and nuanced . \nTherefore, the sensemaking related to complex \nprocesses  can benefit from AI t o address situation like \nthe usage  of large amounts of data,  make relations \namong data that people cannot do without \ncomputational aid , and so on. \nWe have been working on a context -aware advisor \npowered by artificial intelligence technology for \nsupporting knowledge -intensive processes [ 10]. The \nidea of a context -aware advisor is to associate \nknowledge from different domains, tasks, events, and \ncontacts that are related to a user. A context -aware \nadvisor is an assistant, but it knows different contexts  \nof the user’s life: work, personal, hobbies, routines, \netc. It has the common features of an assistant like \nGoogle Assistant or Siri, but it can set its “mind” to \nmore specific context s and partner up with user to \nexecute a task . For exampl e, a HCI researcher has her  \nown annotated data collected throughout his res earch \nlife. What if she could get a notification about  of a new \npaper that relates the same concepts that she \ncommented on another paper? It would be nice, right?  \nReferences  \n1. Albert M Selvin, Simon J Buckingham Shum, and \nMark Aakhus. 2010. The practice level in \nparticipatory design rationale: studying practitioner \nmoves and choices. Human Technology: An \nInterdisciplinary Journal on Humans in ICT \nEnvironments . \n2. Brenda Dervin. 1998. Sense ‐making theory and \npractice: an overview of user interests in \nknowledge seeking and use. Journal of Knowledge \nManagement 2, 2: 36 –46.  \n3. C.S. de Souza, R.F.G. Cerqueira, L.M. Afonso, \nR.R.M. Brandão, and J.S.J Ferreira. 2016. Software  \n Developers As  Users Semiotic Investigations in \nHuman -centered Software Development. Springer -\nVerlag New York Inc.  \n4. Charles Sanders Peirce. 1974. Collected papers of \nCharles  Sanders Peirce . Harvard University Press.  \n5. Claudio Di Ciccio, Andrea Marrella, and Alessandro \nRusso. 2015. Knowledge -Intensive Processes: \nCharacteristics, Requirements and Analysis of \nContemporary Approaches. Journal on Data \nSemantics 4, 1: 29 –57. \n6. D. A. Schön. 1983. The reflective practitioner: how \nprofessionals think in action. Basic Books, New \nYork. \n7. Karl E Weick and David K Meader. 1993. \nSensemaking and group support systems. Group \nsupport systems: New perspectives : 230 –252. \n8. Marian Petre and André van der Hoek, eds. 2014. \nSoftware designers in action: a human -centric look \nat design work. CRC Press, Taylor & Francis Group, \nBoca Raton.  \n9. Mark Aakhus. 2003. Neither naïve nor critical \nreconstruction: Dispute mediators, impasse, and \nthe design of argumentation. Argumentation  17, 3: \n265–290. \n10. Vinícius Segura, Juliana Jansen Ferreira, Ana Fucs, \nMárcio Moreno Ferreira, Rogério d e Paula, and \nRenato Cerqueira. 2018. CoRgI: Cognitive \nReasoning Interface. Proceedings of 20th \nInternational Conference on Human -Computer \nInteraction  (to appear) . \n \n \n "    },    {        "title": "2401.17893v1.Cosmological_solutions_in_the_Brans_Dicke_theory_via_invariants_of_symmetry_groups.pdf",        "content": "arXiv:2401.17893v1  [gr-qc]  31 Jan 2024Cosmological solutions in the Brans-Dicke theory\nvia invariants of symmetry groups\nE. Ahmadi Azar1, K. Atazadeh2, A. Eghbali3,\nDepartment of Physics, Faculty of Basic Sciences,\nAzarbaijan Shahid Madani University, 53714-161, Tabriz, Iran\nWe proceed to obtain an exact analytical solution of the Bran s-Dicke (BD) equations for the\nspatially flat ( k= 0) Friedmann-Lamaitre-Robertson-Walker (FLRW) cosmologi cal model in both\ncases of the absence and presence of the cosmological consta nt. The solution method that we use to\nsolve the field equations of the BD equations is called the “in variants of symmetry groups method”\n(ISG-method). This method is based on the extended Prelle-S inger (PS) method and it employs\nthe Lie point symmetry, λ-symmetry, and Darboux polynomials (DPs). Indeed, the ISG- method\ntries to provide two independent first-order invariants ass ociated to the one-parameter Lie groups\nof transformations keeping ordinary differential equation s (ODEs) invariant, as solutions. It should\nbe noted for integrable ODEs, the ISG-method guarantees the extraction of these two invariants.\nIn this work for the BD equations in FLRW cosmological model, we find the Lie point symmetries,\nλ-symmetries and DPs, and obtain the basic quantities of the e xtended PS method (which are the\nnull forms and the integrating factors). By making use of the extended PS method we find two\nindependent first-order invariants, in such a way appropria te cosmological solutions from solving\nthese invariants as a system of algebraic equations are simu ltaneously obtained. These solutions\nare wealthy so that they include many known special solution s, such as O’Hanlon-Tupper vacuum\nsolutions, Nariai’s solutions, Brans-Dicke dust solution s, inflationary solutions, and etc.\nKeywords: Cosmological Solutions, Brans-Dicke Theory, Inv ariants of Symmetry Groups\nMethod\nContents\n1 Introduction 2\n2 A short review of the ISG-method 4\n1e.ahmadi.azar@azaruniv.ac.ir\n2atazadeh@azaruniv.ac.ir\n3eghbali978@gmail.com3 FLRW cosmological model in BD theory 8\n3.1 A short review of the theory of BD . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.2 FLRW cosmological model in BD theory . . . . . . . . . . . . . . . . . . . . . 10\n4 Solving the FBD-equation 12\n5 Completing the solutions 16\n6 Special solutions of the FBD-equation 20\n7 Conclusions 23\n1 Introduction\nMach’s ideas strongly inspired Albert Einstein, so he attem pted to include this principle in\nhis theory when formulated the theory of general relativity (GR). During the years 1907-1916\nwhen Einstein was developing his theory of GR, he strongly in fluenced by Mach’s philosophy\nand tried to include this principle in his theory in every pos sible. Finally, he could reflect the\nrelationship between the geometry of space-time and also th e distribution of matter in the\nGR equations. Later, it was shown that the Einstein’s field eq uations (EFEs) can be include\nsolutions in the absence of matter and in empty space (Minkow ski space-time, Taub-NUT\nspace-time, etc.) [ 1]. Therefore, contrary to Mach’s opinion (the stronger form of Mach’s\nprinciple), the GR theory includes the weaker form of Mach’s principle, and hence space-time\nhas an existence independent of matter. Among physicists, t here is a group that accepts\nMach’s opinion and believes that GR, which does not include t he stronger form of Mach’s\nprinciple, should be modified to include it. Since the 70s, eff orts have been made in this\ndirection, among which we can mention the scalar-tensor the ories (STTs) [ 2]. The Brans-\nDicke (BD) theory of gravity (sometimes called the Jordan-Bra ns-Dicke scalar-tenser theory\nof gravity) is the simplest version of the STT, which was deve loped by Pascual Jordan in the\n1950s and later by Brans and Dicke in the early 1960s [ 3,4]. Actually, theories about this\nhad already been proposed years earlier by Jordan, Fierz and Thiery. One of the motivations\nof these authors was to include the stronger form of the Mach’ s principle in gravity. This\nprinciple states that the inertial mass of a body is due to all the rest of the matter in the\nuniverse. For the first time, the idea of varying the Newton’s gravitational constant Gwith\nthe age of the universe was proposed by Dirac [ 5]. His purpose in doing this was to explain the\nweakness of the gravitational force among the fundamental f orces in nature. In the Jordan\ngravity theory, the variation of the constant Gwith space-time was described by a simple\nscalar field. However, scientists such as Fierz and Bondi crit icized this theory, because energy\nwas not conserved in it. In 1961, Brans and Dicke proposed [ 3] a more complete theory of\ngravity incorporating a varying gravitational constant (s ee also [ 4]). In the BD theory, the\nconstant Ghas been replaced by the inverse of a scalar field ϕ. The scalar field ϕplays\nthe role of the G, and Mach’s principle is revived through it. Also, in the BD th eory a\n2dimensionless parameter ω, which measures the coupling of the ϕto matter, is included. In\nthe GR, there is only one tensor field gµνdescribing the geometry of space-time, while in the\nBD theory, in addition to the tensor field gµν, a scalar field ϕis incorporated for reviving\nMach’s principle. In 1980, Batista [ 6] extended the cosmological model obtained by Brans and\nDicke to non-homogeneous models. He considered density ρand pressure pas functions of\nthe variables r,tand showed that for the dust-dominant universe, the same sol utions for a(t)\nandϕare obtained as previously found by Brans and Dicke. In other w ords, the assumption\nof inhomogeneity of dust distribution goes not change the BD s olutions. McIntosh [ 7] showed\nthat with the spatially flat FLRW metric, the scalar field ϕ, the pressure pand its density\nρ, there are solutions for the BD equations. He showed that for s olutions of the BD field\nequations, it must not be assumed that the ϕ,pandρare only functions of the variables on\nwhich the metric gµνdepend. This true not just in the cosmological case but in vac uum and\nother cases as well. The vacuum solutions of BD equations for t he spatially flat FLRW metric\nwere obtained by O’Hanlon and Tupper [ 8], Chauvet [ 9], Cervero [ 10] and Lorentz-Petzold [ 11].\nChauvet also obtained the solutions of the BD equations param etrically in the case k=±1.\nUehara and Kim [ 12] studied the BD theory in the presence of the cosmological con stant\nΛ, in such a way that they obtained solutions for the BD equation s in the presence of the\ncosmological constant for the spatially flat FLRW metric. Th ey also obtained an exponential\nsolution for the radiation universe (p=ρ/3)whenω=−3/2.\nCosmological dynamical equations (DEs) of FLRW model in the framework of GR theory\nand many other models in the framework of extended gravity th eories such as BD theory [ 13],\nf(T)gravity [ 14],f(R)theory [ 15], Rastall theory [ 16], etc., have been solved analytically and\nnumerically by different methods. But, there is no unique solu tion method that is general,\nso that it can be used not only to solve the field equations of BD t heory, but also the field\nequations of any extended gravitational theories. Note tha t the cosmological DEs are coupled\nnon-linear differential equations, which require special m ethods. The PS-method, which was\nlater modified and completed by Chandrasekar et al. [17], is one of these methods. This solu-\ntion method has made great progress in the field of theory of no n-linear differential equations.\nThe ability that this method has shown in solving a number of w ell-known non-linear differen-\ntial equations, such as the 2-dimensional Kepler problem, the modified Emden equation, a nd\nthe Helmholtz oscillator [ 18], is indicative of the fact that the modified extended PS-met hod,\nwhen combined with other symmetry methods, it can be a powerf ul algorithmic solution for\nsolving non-linear DEs in physics, such as cosmology [ 19,20]. We may use this method to\nextract independent invariants from the cosmological DEs i n a mini-super space, and hence\nwe have actually solved the problem, because these invarian ts can be considered as a system\nof algebraic equations. Solving this system of algebraic eq uations is much easier than solving\nthe system of cosmological DEs (which are in the form of non-l inear differential equations). It\nshould be noted that the extended PS-method and combined wit h other symmetry methods\nthat we call the ISG-method, they are not included in other br anches of science, especially\ngravity and cosmology. The remarkable point in this method i s that it is difficult to calculate\nits basic quantities, i.e., null form Sand integrating factor Rin the extended PS-method are\nrelated to the fundamental quantities of other symmetry met hods such as the infinitesimals\n3τandξin the Lie point symmetry and the function λin theλ-symmetries, and DPs in the\nDPs method [ 19–26]. Recently, we have presented the ISG-method in order to sol ve the field\nequations of the spatially flat FLRW cosmological model in th e presence of Λin the context\nof GR [ 27]. Here, we employ this method to obtain the solutions of the fi eld equations of BD\ntheory for spatially flat FLRW cosmological model in both cas es of the absence and presence\nof the cosmological constant. We show that when the coupling constant ωtends to infinity, the\ncosmological solutions of GR theory can be obtained from the analytical solutions of BDEs.\nFor the dust dominated universe, when the cosmological cons tant is present, we conclude that\nthe solutions of Uehara and Kim [ 20] are special cases of the solutions of BDEs. In addition,\nwe show that the case Λ = 0 of the solutions of BDEs include Nariai’s solutions, BD dust\nsolutions, radiation solutions, inflationary solutions, O ’Hanlon-Tupper vacuum solutions and\nfinally general relativity solutions.\nThe paper is organized as follows. In Sec. 2, we briefly describe the ISG-method applicable\nfor second-order ODEs in physics, especially in cosmology. We give the main points of the\nISG-method in this section. In Sec. 3, we first introduce the BD theory of gravity, and then\nextract the DEs of the FLRW cosmological model in this theory . Then, by combination these\nequations, we will reduce them to a second-order ODE called t he Friedmann-Brans-Dicke\nequation (FBD-equation). In Sec. 4, we will apply the ISG-method to solve the FBD-\nequation. This solution enables us to obtain the cosmologic al solutions of the BD theory. In\nSec.5, the solutions of the BDEs are completed so that they are suita ble for cosmological\napplications. In Sec. 6, we will examine the cosmological solutions in special case s. Finally,\nconclusions are reported in Sec. 7.\n2 A short review of the ISG-method\nIn order to introduce the ISG-method [ 27], let us consider a particle of unit mass moveing\nin one-dimensional configuration space Q= (q). A force F=φ(t,q,˙q)∂/∂q is applied to the\nparticle and gives it an acceleration a= ¨q∂/∂q . According to Newton’s second law of motion,\nthe DE of the particle is as follows:\n¨q=φ(t,q,˙q), (2.1)\nwhereφas a function of t,qand˙qis called a force function of the particle. Here, “.” denotes\nthe total derivative operator with respect to the time t, which is defined as\n.=d\ndt=∂\n∂t+ ˙q∂\n∂q+φ(t,q,˙q)∂\n∂˙q. (2.2)\nIn the force function φ(t,q,˙q), timetis the independent variable, and coordinate qis the\ndependent variable. Many physical phenomena have a DE in the form of equation ( 2.1).\nThat is, the dynamics of such phenomena can be imagined as the dynamic of a particle with\na unit mass, which is subject to force F. The ISG-method provides us an algorithmic and\nsystematic solution processes for solving second-order OD Es of the form equation ( 2.1), in\n4which the force function φ(t,q,˙q) =P/Qis a fractional function of polynomials PandQof\nthe variables t,q, and˙qwith coefficients in the set of complex numbers. In this work, w e use\nthe ISG-method to solve second-order non-linear ODEs like ( 2.1), in which the force function\nis as in those of ( 2.1). This method can be developed to solve non-linear ODEs of hi gher\norders and even for system of ODEs of any order, which are not o ur discussion here. The\ngeneral strategy of the ISG-method for solving second-orde r ODE ( 2.1) is to systematically\nand algorithmically extract a certain number (equal to the o rder of the DE, which is two\nhere) of the independent “first integral” (or first-order “in variant”) such as I1(t,q,˙q) =c1and\nI2(t,q,˙q) =c2associated to the DE ( 2.1), whereI1andI2are known functions of the variables\nt,qand˙qandci’s (i=1,2) are constants on the solutions of the DE ( 2.1). When we obtain\nthese two invariants, the solution of the DE ( 2.1) is actually prepared, because one can look\nat the first-order ODEs, I1(t,q,˙q) =c1andI2(t,q,˙q) =c2as a system of algebraic equations\nin terms of variables t,qand constants of motion c1andc2. It is easy to solve this system of\nequations. To solve this system of algebraic equations, it i s enough to remove the variable ˙q\nbetween them. For this purpose, one can obtain from one of the m, for example, I1(t,q,˙q) =c1,\nthe variable ˙qin terms of t,qand the constant of motion c1as follows: ˙q=χ1(t,q;c1)where\nχ1is a known function of t,qandc1. Then, by replacing this function in the second equation\nI2(t,q,˙q) =c2, one obtains the following equation:\nc2=I2/parenleftbig\nt,q,χ1(t,q;c1)/parenrightbig\n:= Σ(t,q;c1), (2.3)\nwhereΣis a known function of variables t,qand constant of motion c1. Equation ( 2.3) can\nbe solved algebraically and obtained q= Π(t;c1,c2), whereΠis a known function of variable\ntand constants of motion c1andc2. In this way, the solution of the problem is complete, and\nq= Π(t;c1,c2)derived by the ISG-method is the general solution of the DE ( 2.1).\nNow, to get into the details of the ISG-method, how can one find two independent invari-\nantsI1(t,q,˙q) =c1andI2(t,q,˙q) =c2? As mentioned before, if the force function φ(t,q,˙q)\nin the DE ( 2.1) is a fractional function of polynomials like PandQof variables t,qand˙q\nwith coefficients in the set of complex numbers as φ(t,q,˙q) =P/Q, then, to find these two\ninvariants, one can use the following theorem.\nTheorem 1. (Duarte’s integral formula) Let¨q=φ(t,q,˙q)be the DE of a particle with\none degree of freedom in the configuration space Q= (q), where the force function φ(t,q,˙q)\nis a fractional function of polynomials PandQof variables t,qand˙q. If this second-order\nODE admits a first integral as I(t,q,˙q) =c, then\nI(t,q,˙q) =r1+r2−/integraldisplay/bracketleftBig\nR+∂\n∂˙q(r1+r2)/bracketrightBig\nd˙q, (2.4)\nwhere\nr1=/integraldisplay\nR(φ+S˙q)dt, r 2=−/integraldisplay/parenleftBig\nRS+∂r1\n∂q/parenrightBig\ndq. (2.5)\n5In equations (2.4)and(2.5),SandR, which satisfy in the PS determining equations:\nD[S] =−φq+Sφ˙q+S2, (2.6)\nD[R] =−R(S+φ˙q), (2.7)\nRq=R˙qS+RS˙q, (2.8)\nare called the “null form” and the “integrating factor” of th e DE (2.1), respectively [21,22,28].\nHere,φq:=∂φ\n∂q,φ˙q:=∂φ\n∂˙qand so on.\nIn order to calculate the functionally independent invaria ntsI1(t,q,˙q) =c1andI2(t,q,˙q) =\nc2associated to the DE ( 2.1), we must first obtain the null form Sand integrating factor R,\nusing the PS determining equations ( 2.6)-(2.8). Solving these differential equations for Sand\nRis difficult, except for certain simple cases. Therefore, to o btain these two basic functions\nin the PS procedure, we must resort to another symmetry metho ds. These methods are Lie\npoint symmetry, λ-symmetry and DPs.\nIf the calculation of functions SandRis not achieved by the PS determining equations,\nthen one must do them indirectly. For this purpose, suppose [ 27]\nΦ:R2→R2,\n(t,q)/ma√sto→(t,q) =Φ(t,q;ǫ) =/parenleftbig\nT(t,q;ǫ),Q(t,q;ǫ)/parenrightbig\n, (2.9)\nwith the transformation equations\nt→¯t=t+ǫτ(t,q)+O(ǫ2),\nq→¯q=q+ǫξ(t,q)+O(ǫ2), ǫ≪1, (2.10)\nbe the group of one-parameter Lie point transformations suc h that the DE ( 2.1) under which\nremains invariant. In equations ( 2.10), functions τandξdefined by\nτ(t,q) =∂T(t,q;ǫ)\n∂ǫ|ǫ=0, (2.11)\nξ(t,q) =∂Q(t,q;ǫ)\n∂ǫ|ǫ=0, (2.12)\nare called infinitesimals of the group of transformations Φ. By these two functions, the\ninfinitesimal generator vector of the group of transformati onsΦis defined as\nX=τ(t,q)∂\n∂t+ξ(t,q)∂\n∂q. (2.13)\nIn order to find the Lie point symmetries associated to the DE ( 2.1), we must first obtain the\ninfinitesimals τandξ. For this purpose, we use the following theorem.\nTheorem 2. (Lie’s invariance condition) Suppose for a dynamical system with one degree\n6of freedom, the second-order ODE H(t,q,˙q,¨q) := ¨q−φ(t,q,˙q) = 0 be the DE (or governing\nequation) of the particle in the configuration space, and let X=τ(t,x)∂t+ξ(t,q)∂qbe the\ninfinitesimal generator of the one-parameter Lie group of tr ansformations (2.9)with transfor-\nmation equations (2.10)acting on the (t,q)-space. The group of transformations Φis admitted\nby the DE (2.1), if and only if the following condition is holds:\nX(2)H(t,q,˙q,¨q)/vextendsingle/vextendsingle/vextendsingle\nH=0= 0, (2.14)\nwhere\nX(2):=τ∂\n∂t+ξ∂\n∂q+(˙ξ−˙q˙τ)∂\n∂˙q+(¨ξ−2¨q˙τ−˙q¨τ)∂\n∂¨q, (2.15)\nis the 2th extended (prolonged) infinitesimal generator of t he vector field X. We note that\nequation (2.14)is called the Lie’s invariance condition which is the fundam ental equation of\nthe Lie symmetry analysis [27,29,30].\nIn the other words, the establishing equation ( 2.14) is a necessary and sufficient condition\nfor the ODE H= 0to admit Φwith infinitesimal generator Xas a one-parameter Lie group\nof transformations. By solving the Lie’s invariance conditi on, which are in fact a system of\nsecond-order partial differential equations for the unknow n functions τandξ, all infinitesimal\ngenerators Xassociated to the ODE H= 0can be found.\nAssume that the ODE H= 0has at least two Lie point symmetries and Φ1andΦ2be\narbitrarily two of them with the infinitesimal generators X1=τ1∂t+ξ1∂qandX2=τ2∂t+ξ2∂q,\nrespectively. By the symmetries X1andX2, we define the characteristics Q1:=ξ1−˙qτ1and\nQ2:=ξ2−˙qτ2, respectively. Similarly, by the characteristics Q1andQ2we define the functions\nλi:=D[Qi]/Qi, (i=1, 2). It can be shown that −λi’s are solutions of the determining equation\n(2.6). Thus, we conclude that the null forms are Si=−λi. Now, by using the Darboux’s\neigenvalue equation4[21,27]:\nD[F] =/braceleftBigg\nφ˙qF if DE is not an explicit function of t,\nKF if DE is an explicit function of t.(2.16)\none can obtain the DPs Fassociated to the DE ( 2.1). It can be shown that the ratio Q/F\nis a solution of the determining equation ( 2.7), and hence R=Q/F. Therefore, two 2-tuples\n(S1,R1)and(S2,R2)are obtained as follow:\n(S1,R1) = (−λ1,Q1\nF),(S2,R2) = (−λ2,Q2\nF). (2.17)\nThus, by substituting each of these 2-tuples (S1,R1)and(S2,R2)in the Duarte’s integral\nformula ( 2.4) one can obtain the first integrals corresponding to each of t hese 2-tuples. Cer-\ntainly, these first integrals are independent of each other. Therefore, they are what we needed\n4In this equation, F’s are DPs associated to the DE ( 2.1), and the eigenvalue K(t,q,˙q)/parenleftbig\nφ˙q(t,q,˙q)/parenrightbig\nis called\nthe cofactor of Fwhen DE is an explicit function of t(when DE is not an explicit function of t) and has degree\nat most 2.\n7to solve the DE ( 2.1). It should be noted if the DE ( 2.1) does not have any Lie point sym-\nmetry that can be used to calculate the null form Sand the integrating factor R, then, one\nmust use its generalized symmetries. The most suitable of th ese symmetries is λ-symmetry.\nSimilar to Lie point symmetry, λ-symmetry has also a symmetry condition. If this symmetry\ncondition is used for the DE ( 2.1), then, the function λ(t,q,˙q)and its infinitesimals τand\nξare obtained. When the functions λ,τandξare known, then the first integral can be\ncalculated by a process which is given in Ref. [ 27].\n3 FLRW cosmological model in BD theory\nThe purpose of this section is to study the FLRW cosmological model in the BD theory.\nBefore proceeding to do this, let us give a short review of the t heory of BD.\n3.1 A short review of the theory of BD\nThe BD theory of gravity is a generalization of Einstein’s the ory of general relativity, which\nwas formulated by Brans and Dicke [ 3,4]. As mentioned earlier, this theory is consistent with\nMach’s principle (the stronger form) and relies less on the i nherent properties of space. In\nthis theory, in addition to the tensor field that describes th e geometry of space-time, a scalar\nfield is also included. Hence, BD theory is also called scalar- tensor theory. To obtain the field\nequations of this theory, let us consider Einstein-Hilbert action in the theory of GR\nIEH=/integraldisplay\nM√−g/parenleftBig\n−R\n16πG+LM/parenrightBig\nd4x. (3.1)\nwhereRis the Ricci scalar associated to the four-dimensional spac e-time manifold Mwith\nthe local coordinates xµ= (x0,x1,x2,x3), andLM=LM(gµν,∂kgµν)is the Lagrangian density\nof the ordinary matter and g:=det gµν. This action in the BD theory and in the “Jordan\nframe” is generalized as follows [ 10,12]:\nIBD=1\n16π/integraldisplay\nM√−g/parenleftBig\nϕR−ω\nϕgµν∇µϕ∇νϕ−V(ϕ)+16πLM/parenrightBig\nd4x, (3.2)\nwhereϕis the BD scalar field, ωis the adjustable BD coupling parameter, V(ϕ)is the\npotential energy of the field ϕand∇µdenotes the covariant derivative operator in the space-\ntimexµ. Moreover, the BD scalar field ϕinversely proportional to the effective gravitational\nconstant Geffby the following relation\nϕ=1\nGeff4+2ω\n3+2ω, (3.3)\nwhereGeffis equal to the constant Gin the limit when ωtends to infinity. In addition, LM\nas the Lagrangian density is minimally coupled to the BD scala r fieldϕ. In analogy to the\nGR theory, the law of conservation of energy-momentum of the ordinary matter ∇µTµν\nM= 0\n8and the definition of the energy-momentum tensor (EMT) of the ordinary matter fields in the\nfour-dimensional space-time is given by\nTµν\nM:=2√−g∂\n∂gµν(√−gLM). (3.4)\nBy varying the action ( 3.2) with respect to the scalar field ϕand metric gµνone can get the\nfollowing equations, respectively [ 31]\n2ω\nϕ/squareϕ+R−ω\nϕ2∇αϕ∇αϕ−dV\ndϕ= 0, (3.5)\nRµν−1\n2gµνR=8π\nϕTMµν+ω\nϕ2(∇µϕ∇νϕ−1\n2gµν∇αϕ∇αϕ)\n+1\nϕ(∇µ∇νϕ−gµν/squareϕ)−V\n2ϕgµν, (3.6)\nwhereRµνis the Ricci tensor and /square:=∇α∇αis the covariant d’Alemberian operator of\nthe metric. Note that equation ( 3.5) is known as the “modified Klein-Gordon equation”, and\nequation ( 3.6) is a generalization of the EFEs Rµν−(1/2)gµνR+Λgµν= 8π G TMµν. Now,\nby contracting on equation ( 3.6) one obtains\nR=−8πTM\nϕ+ω\nϕ2∇αϕ∇αϕ+3/squareϕ\nϕ+2V\nϕ. (3.7)\nIn order to eliminate the Ricci scalar Rfrom ( 3.5) one may insert ( 3.7) into ( 3.5). It then\nresults\n/squareϕ=8π\n3+2ω(8πTM+ϕdV\ndϕ−2V), (3.8)\nwhereTM:=gµνTMµνis the trace of the energy-momentum tensor of the ordinary ma tter\nfields. The equation ( 3.8) together with ( 3.6) form a system of coupled second-order non-\nlinear differential equations which are called the general f orm of the BD equations. Choosing\nthe potential V(ϕ)in the form V(ϕ) = 2Λϕ, the field equations ( 3.6) and ( 3.8) yield the\nfollowing standard BD equations\n/squareϕ=8π\n3+2ωTM−2Λϕ\n3+2ω, (3.9)\nRµν−1\n2gµνR+Λgµν=8π\nϕTMµν+ω\nϕ2(∇µϕ∇νϕ−1\n2gµν∇αϕ∇αϕ)\n+1\nϕ(∇µ∇ν−gµν/squareϕ). (3.10)\nThis system of the field equations, without considering the c osmological constant Λ, were\nobtained for the first time by Brans and Dicke in 1961, and known as the BD equations in the\npresence of the cosmological constant Λ. These equations form the basis of the BD theory. In\n9fact, BD theory with the BD equations as the field equations is an extended gravity theory of\nthe “scalar-tensor” type, in which the potential energy fun ction sets to V(ϕ) = 2Λϕ. In BD\ntheory, although the initial value of the BD coupling paramet er was very small, its present\nvalue exceeds 500 in many cosmological models, such that ω >500is in good agreement\nwith experimental tests and general relativity prediction s (for example, neutron stars, black\nholes, gravitational waves, binary pulsars, etc.) [ 32]. The BD theory of gravity reduces the\nGR theory in the ω→ ∞ limit [ 33,34].\nIn the following, we will use the ISG-method to solve the DEs o f the FLRW cosmological\nmodel in the framework of the BD theory. In fact, we will obtain the cosmological solutions\nof the BD equations ( 3.9) and ( 3.10) by the ISG-method.\n3.2 FLRW cosmological model in BD theory\nIn this subsection, the FLRW cosmological model is briefly st udied in the framework of the\nBD theory. We consider a homogeneous and isotropic universe a s a cosmological model for\nour study. To a good approximation, this universe is describ ed by the FLRW metric. This\nmetric in the coordinates xµ= (t,r,θ,ϕ)is defined as\nds2=−dt2+a2(t)/parenleftBigdr2\n1−kr2+r2dθ2+r2sin2θ dϕ2/parenrightBig\n, (3.11)\nwherea(t)is the scale factor, and the number kcan admits three values of −1,1,0for spaces\nof negative, positive and zero curvature, which represents closed, open and flat universes,\nrespectively [ 35,36]. The Weyl’s principle requires that the constituent mater ial of a homoge-\nneous and isotropic universe is a perfect fluid on large scale s with high accuracy. So, according\nto the Weyl’s principle, the FLRW universe can be considered as a perfect fluid with a good\napproximation. If the energy density of this fluid be ρc2and its pressure p, then its normal\nbarotropic equation of state is\np=wρc2, (3.12)\nwherewis called the perfect fluid state parameter and cis the velocity of light in vacuum.\nThe equation of state ( 3.12) includes most of the interesting states that are important in\ncosmology. For example, for dust: w= 0, for radiation: w= 1/3, for false vacuum: w=−1,\nand for stiff fluid: w= 1[37,38]. In the present work, a system of units is used in which the\nvelocity of light in vacuum is set to unity. Furthermore, the components of the EMT of the\nfluid will be as follows:\nTMµν= (ρ+p)UµUν+pgµν, (3.13)\nwhereUµ= (1,0,0,0)is the co-moving 4-velocity. The trace of this tensor is give n by\nTM= 3p−ρ. (3.14)\n10The non-vanishing components of the BDE ( 3.10) read\n3˙a2\na2−Λ =8π\nϕρ+ω\n2˙ϕ2\nϕ2−3˙a\na˙ϕ\nϕ, (3.15)\n−2¨a\na−˙a2\na2+Λ =8π\nϕp+ω\n2˙ϕ2\nϕ2+2˙a\na˙ϕ\nϕ+¨ϕ\nϕ. (3.16)\nAlso, using equation ( 3.14), one concludes that equation ( 3.9) becomes\n¨ϕ\nϕ+3˙a\na˙ϕ\nϕ=2Λ\n3+2ω+8π\nϕ/parenleftBigρ−3p\n3+2ω/parenrightBig\n. (3.17)\nNote that equation ( 3.17) is not an independent equation in BD theory, because it can be\nobtained by using Bianchi identities ∇µGµν= 0and equations ( 3.15) and ( 3.16). So, we\nhave only two independent equations to determine four unkno wn functions a(t),ϕ(t),ρ(t)\nandp(t). As such this system of equations does not seem to have a uniqu e solution. Hence,\nto find these unknowns uniquely, two equations are needed. On e of these is the equation of\nstate of the universe, ( 3.12). To obtain the second equation we use the following Corolla ry.\nCorollary1. In the spatially flat (k= 0)FLRW cosmological model in the framework of\nthe BD theory, the necessary and sufficient condition for the d eceleration parameter of the\nuniverse q:=−a¨a/˙a2to be a constant, is that a power-law between the cosmic scale factor\na(t)and the Brans-Dicke scalar field ϕ(t)holds in the following form [35–40]:\nϕan=C, (3.18)\nwherenandCare the functions of the parameters w,ωand constants GandΛ, that is,\nC:=C(w,ω,G,Λ)andn:=n(w,ω,G,Λ).\nProof. See Ref. [ 39].\nThe history of this Corollary backs to the Dirac’s hypothesi s that states the constant Gis\na variable quantity and should be related by a power-law rela tion such as G=´Canto the scale\nfactor of the universe, where ´Candnare some constants. Since in the BD theory, the scalar\nfieldϕis proportional to the inverse of the Newton’s gravitationa l constant, that is, ϕ=c0/G,\n(wherec0is the proportionality constant), combining this relation withG=C′anyields the\npower-law in the form ( 3.18) in which C:=c0/C′. By combining equations ( 3.15)-(3.17) with\nequation of state ( 3.12) and power-law relation ( 3.18), we then obtain\n8πρ\nCan=˙a2\na2/parenleftBig6−ωn2−6n\n2/parenrightBig\n−Λ, (3.19)\n8πwρ\nCan= (n−2)¨a\na+˙a2\na2/bracketleftBig2(n−1)−(ω+2)n2\n2/bracketrightBig\n+Λ, (3.20)\n8π\nCan(ρ−3wρ) = (3+2 ω)/bracketleftBig\n−n¨a\na+n(n−2)˙a2\na2/bracketrightBig\n−2Λ. (3.21)\n11The combination of these three differential equations after performing some algebraic calcu-\nlations leads to the following non-linear second-order ODE for the scale factor a(t)\n2(ωn+3)¨a\na+(6+4nω−ωn2)˙a2\na2= 2Λ, ωn +3/ne}ationslash= 0. (3.22)\nThis equation can be rewritten in the following form\n¨a=α˙a2\na+βa, (3.23)\nwhere the new parameters αandβin terms of the old parameters ω,w,nand the constant\nΛare defined by\nα:=−6+4nω−ωn2\n2(ωn+3), β :=Λ\nωn+3. (3.24)\nTherefore, the combination of BD equations with the fluid stat e equation of the universe\nand the power-law relation leads to a second-order ODE for th e cosmic scale factor a(t)in\nthe form ( 3.23). In this work, we call this equation, the “Friedman-Brans-D icke equation”\n(FBD-equation). The FBD-equation alone describes the dynami cs of spatially flat (k= 0)\nFLRW universe in the framework of the BD theory. Equation ( 3.23) can be imagined as the\nequation of motion of a particle with a unit mass in one-dimen sion. By defining the cosmic\nscale factor as the coordinate of this particle, i.e., q(t) :=a(t), equation ( 3.23) is the familiar\nform of Newton’s second law\n¨q=φ(t,q,˙q), (3.25)\nwhere\nφ(t,q,˙q) :=α˙q2\nq+βq, (3.26)\nis the component of the force function acting on the particle , and¨qis the component of its\nacceleration vector. Thus, to analyze the dynamics of cosmo logical model, it is enough to\nsolve only the FBD-equation by the ISG-method. We will do this in the next section.\n4 Solving the FBD-equation\nIn this section we employ the ISG-method to solve the FBD-equa tion (3.25). To this end, we\nuse the following steps:\n(1)-We re-consider spatially flat (k= 0) FLRW cosmological model in the framework of\nBD theory with a dynamical system S2\n1. As mentioned earlier, the governing equation of this\nsystem is ¨q=φ(t,q,˙q), whereφ(t,q,˙q)is the force function of a particle that moves in a\n121-dimensional mini-super space with configuration Q= (q)whereq:=a(t)is the coordinate\nof the particle.\n(2)-A one-parameter Lie group of point transformations Φ(t,q;ε)with the transformation\nequations ( 2.10) must be found (if there is a solution) such that the FBD-equat ion (3.25)\nremains invariant under this group of transformations. For this purpose, by using Lie’s in-\nvariance condition, a set of the PDEs should be extracted for the infinitesimals τandξof\ngroupΦ. By simultaneously solving this system of PDEs, the infinites imalsτandξas func-\ntions oftandqshould be obtained: τ=τ(t,q),ξ=ξ(t,q). For our case, these functions\nare obtained as τ=τ(t,q) =c1,ξ=ξ(t,q) =c2qwherec1andc1are some arbitrary con-\nstants [ 27]. Then, by using these results one must find all Lie point symm etry vectors of the\ntransformations group Φsuch that for our case these vectors are obtained to be X1=∂tand\nX2=q∂qwhich are respectively the infinitesimal generator of the tr anslation group along the\ntimeΦ1(t,q;ε) = (t+ε,q)and the scaling group Φ2(t,q;ε) = (t,eεq)[27].\n(3)-To calculate the first integral corresponding to the DE ¨q=φ(t,q,˙q)of the particle in\nthe configuration space Q= (q), which is equation ( 2.4) of Theorem 1, the null form Sand\nintegrating factor Rshould be obtained by using the determining equations ( 2.6)-(2.8). It is\ndifficult to solve these equations simultaneously, except in special simple cases. Therefore, to\nobtain these two basic functions in the extended PS method, w e must resort to other sym-\nmetry methods. The most important other symmetry methods, t hat can be used here, are:\n(a)Lie point symmetry, (b)λ-symmetry, (c)DPs. Therefore, the functions SandRshould\nbe calculated indirectly by the symmetry methods (a)-(c).\n(4)-Using the Lie point symmetry mentioned in section 2, the characteristic Q(t,q,˙q) :=ξ−˙qτ\nand then λ(t,q,˙q) :=D[Q]/Qmust be defined. It can be shown that −D[Q]/Qis a solution\nof the determining equation ( 2.6), in such a way that S(t,q,˙q) =−λ(t,q,˙q).\n(5)-By using Darboux’s eigenvalue equation ( 2.16) [21], the DP F(t,q,˙q)and the eigen-\nvalueK(t,q,˙q)(φ˙q(t,q,˙q))corresponding to this polynomial should be obtained. It can be\nshown that the ratio Q/F is a general solution of the determining equation ( 2.7). So,\nR(t,q,˙q) =Q/F. In this way, the null form Sand the integrating factor Rare obtained\nindirectly by the Lie point symmetry method and DPs, respect ively, without the need to\nsolve their determining equations.\nLet us turn into the main goal of this section which is nothing but calculating the first\nintegrals I1andI2corresponding to the FBD-equation ( 3.25). Since the FBD-equation ad-\nmits two Lie point symmetries, both λ-symmetries associated to FBD-equation can be only\ncalculated by using the relation λ=D[Q]/Qwithout solving the λ-symmetry condition [ 27].\nTherefore, both λ-symmetries associated to the Lie symmetry vectors X1=∂tandX2=q∂q\n13are, respectively,\nλ1(t,q,˙q) =D[Q1]\nQ1=D[−˙q]\n−˙q=α˙q\nq+βq\n˙q, (4.1)\nλ2(t,q,˙q) =D[Q2]\nQ2=D[q]\nq=˙q\nq. (4.2)\nIt can be easily check that functions −D[Qi]/Qiare solutions of the determining equations\n(2.6). Thus, for the FBD-equation, these functions are nothing bu t two null forms associated\nto the Lie symmetry vectors X1=∂tandX2=q∂q:\nS1(t,q,˙q) =−λ1(t,q,˙q) =−α˙q\nq−βq\n˙q, (4.3)\nS2(t,q,˙q) =−λ2(t,q,˙q) =−˙q\nq. (4.4)\nOn the other hand, the determining equation for DPs of the FBD- equation ¨q=φ(t,q,˙q)is\n(2.16). As mentioned above, Q/Fis a solution of the determining equation ( 2.7) such that\nR=Q/F, and hence by considering S=−D[Q]/Qwe arrive at\nD[Q\nF]+Q\nF/parenleftBig\n−D[Q]\nQ+φ˙q/parenrightBig\n= 0. (4.5)\nFurthermore, the characteristics Q=ξ−˙qτobtained from the λ-symmetry and the DPs F\nobtained from solving of the Darboux eigenvalue equation as sociated to the FBD-equation\ngives us the integrating factor RasR=Q/F. Therefore, by using λ-symmetry and DPs\nwithout solving the determining equations, we can obtain 2-tuple(S,R)as follows (S,R) =\n(−D[Q]/Q,Q/F ). In the ISG-method, the basic quantities of the Lie point sym metry, ex-\ntended PS method, λ-symmetry, and DPs read τ,ξ,λ,F,RandS. These quantities are\nrelated to each other as shown in [ 27].\nNow, we consider one of the Lie symmetries, for example X1=∂t. This vector is the\ninfinitesimal generator of the group of translation time: Φ1(t,q;ε) = (t+ε,q). The charac-\nteristic corresponding to this symmetry vector is Q1=ξ1−˙qτ1=−˙qandλassociated to\nthis characteristic is given by ( 4.1). Therefore, the null form associated to the infinitesimal\ngenerator X1=∂tis obtained from equation ( 4.3). We find the integrating factor R1corre-\nsponding to the null form S1. For this purpose, by using the characteristic Q1=−˙qand by\nconsidering DP5F(D)\n6=q2α, the integrating factor R1reads\nR1=Q1\nF(D)\n6=−˙q\nq2α, (4.6)\n5According to Table 1 of Ref. [ 27], by the characteristics Q1(t,q,˙q) =−˙q,Q2(t,q,˙q) =qand DPs\nF(D)\n1,F(D)\n2,···,F(D)\n6, two null forms Si:=−D[Qi]/Qi,(i= 1,2)and twelve integrating factors Rij:=\nQi/F(D)\nj,(i= 1,2,j= 1,2,...,6)can be defined.\n14and2-tuple(S1,R1)is then obtained to be of the form\n(S1,R1) = (−α˙q\nq−βq\n˙q,−˙q\nq2α). (4.7)\nBy substituting this 2-tuple in the Duarte’s integral formula ( 2.4), and then by calculating\nthe integrals, one can get the first integral associated to th e infinitesimal generator X1=∂t,\ngiving us\nI1(q,˙q) =1\n2˙q2q−2α−β\n2(1−α)q−2α+2=c1, (4.8)\nwherec1is an arbitrary constant. In the same way, the 2-tuple(S2,R2)corresponding to the\nsymmetry vector X2=q∂qcan be obtained as follows:\n(S2,R2) = (−λ2,Q2/F(D)\n1) =/parenleftbig\n−˙q\nq,q\nβ\nα−1q2+ ˙q2/parenrightbig\n. (4.9)\nI2(t,q,˙q) =\n\n(α−1)t+/radicalBig\nα−1\nβtan−1(q\n˙q/radicalBig\nβ\nα−1) =c2,β\nα−1>0,\n(α−1)t+q\n˙q=c2,β\nα−1= 0,\n(α−1)t+/radicalBig\n|α−1\nβ|tanh−1(q\n˙q/radicalBig\n|β\nα−1|) =c2,β\nα−1<0.(4.10)\nAccording to the above results and as earlier mentioned, a se t of twelve members of the null\nforms and integrating factors can be constructed, which we c all the PS set:\nSPS={(Si,Rij)}i=1,2,j=1,...,6={(S1,R11),(S1,R12),(S1,R13),(S1,R14),(S1,R15),\n(S1,R16),(S2,R21),(S2,R22),(S2,R23),(S2,R24),(S2,R25),(S2,R26)},(4.11)\nBy using the formula ( 2.4), a first integral can be constructed with each of the 2-tuple of this\nset. Therefore, for twelve first integrals we have\nIi,ij=r1;ij+r2;ij−/integraldisplay/bracketleftbig\nRij+∂\n∂˙q(r1;ij+r2;ij)/bracketrightbig\nd˙q=ci,ij, i= 1,2;j= 1,...,6,(4.12)\nwhereci,ij’s are some constants on the solutions of the DE ¨q=φ(t,q,˙q). Also, according to\n(2.5),r1;ij’s andr2;ij’s are defined as\nr1;ij(t,q,˙q) =/integraldisplay\nRij(φ+ ˙qSi)dt, (4.13)\nr2;ij(t,q,˙q) =−/integraldisplay\n[RijSi+∂\n∂q(ri;ij)]dq. (4.14)\nThe FBD-equation ¨q=φ(t,q,˙q)admits all members of the PS set ( 4.10) as2-tuple(S,R). The\nmembers of SPSand their corresponding first integrals have been listed in T able 2 in Ref. [ 27].\nAmong the first integrals of Table 2, only two of them, I1,16andI2,21, which we denoted by I1\n15andI2, respectively, are independent and the rest are dependent o n two first integrals I1and\nI2. In the PS set ( 4.11),(S1,R16)and(S2,R21)are actually the 2-tuples(S1,R1),(S2,R2),\nrespectively, and the first integrals associated to these 2-tuples are I1=I1,16andI2=I2,21,\nrespectively.\nAs was mentioned earlier, the infinitesimal generator of the translation group is the sym-\nmetry vector X1=∂t. Therefore, the first integral associated to this symmetry v ector gives\nthe energy of the dynamical system S2\n1. Thus, the invariant I1is the energy of the dynamical\nsystem. Now, consider the independent first integrals ( 4.8) and ( 4.10). We look at these two\nindependent invariants as a system of algebraic equations. To solve this system of equations\nwe must remove the variable ˙qamong the equations of this system. To do this, we find ˙qfrom\nthe equation I2(t,q,˙q) =c2, and then employ I1(q,˙q) =c1. Finally, the general solutions of\nthe FBD-equation, which include all three cases c <0,c= 0andc >0, are worked out\nq(t) =\n\n/parenleftBig\n2c1\n|c|/parenrightBig1\n2(1−α)sinh1\n1−α/bracketleftBig\n(α−1)/radicalbig\n|c|(t+´c2)/bracketrightBig\n, c < 0,\n/parenleftBig\n2c1(α−1)2/parenrightBig1\n2(1−α)/parenleftBig\nt+c′2/parenrightBig1\n1−α, c = 0,\n/parenleftBig\n2c1\nc/parenrightBig1\n2(1−α)sin1\n1−α/bracketleftBig\n(α−1)√c(t+´c2)/bracketrightBig\n, c > 0.(4.15)\nwhereαandβare given by ( 3.24). Moreover,\nc=β\nα−1, c′2:=c2\n1−α. (4.16)\nBy redefinition c′1:= (2c1)1\n2(1−α), the general solutions ( 4.15) can be written as follows:\nq(t) =\n\nc′\n1/parenleftBig−2Λ\ns/parenrightBig−wn+3\ns\nsin2(wn+3)\ns/bracketleftBig1\nwn+3/radicalBig\n−Λs\n2(t+c′\n2)/bracketrightBig\n, Λ<0,\nc′\n1/parenleftBigs\n2(wn+3)/parenrightBig2(wn+3)\ns/parenleftBig\nt+c′\n2/parenrightBig2(wn+3)\ns\n, Λ = 0,\nc′\n1/parenleftBig2Λ\ns/parenrightBig−wn+3\ns\nsinh2(wn+3)\ns/bracketleftBig1\nwn+3/radicalBig\nΛs\n2(t+c′\n2)/bracketrightBig\n, Λ>0.(4.17)\nwheres:= 12+6 nω−ωn2. As we expected, since the FBD-equation is an ODE of the second -\norder, its general solution contains two integration const antsc′1andc′2. These constants\ncan be determined by using the initial conditions of the prob lem, i.e., q(t= 0) = q0and\n˙q(t= 0) = ˙q0.\n5 Completing the solutions\nNow, we show that the parameter nin the power-law equation ( 3.18) with the parameter of\nstatewin equation ( 3.12), are consistent, provided that there is a relationship bet ween them.\n16For this purpose, subtracting both sides of equations ( 3.19) and ( 3.20) we obtain\n8πan\nCρ(1−w) =˙a2\na2/parenleftBig6−ωn2−6n\n2−2(n−1)−(ω+2)n2\n2/parenrightBig\n−(n−2)¨a\na−2Λ.(5.1)\nBy taking two consecutive times of the total derivative with r espect to cosmic time tfrom\nequation ( 3.18) one obtains\n˙a\na=−1\nn˙ϕ\nϕ, (5.2)\n¨a\na=n+1\nn2˙ϕ2\nϕ2−1\nn¨ϕ\nϕ. (5.3)\nBy substituting ( 3.18), (5.2) and ( 5.3) into equation ( 5.1) and after a little simplification we\narrive at\n¨ϕ−3\nn˙ϕ2\nϕ=2n\nn−2Λϕ+8πρ(1−w)n\nn−2. (5.4)\nIn addition, one may insert equations ( 3.12), (3.18) and ( 5.2) into ( 3.17) to get\n¨ϕ−3\nn˙ϕ2\nϕ=2Λϕ\n3+2ω+8π\n3+2ωρ(1−3w). (5.5)\nComparing equations ( 5.5) and ( 5.4) yields the following equation:\nn=\n\n1−3w\nω(w−1)−1,Λ = 0, w∈[−1,1],\n−1\nω+1, Λ/ne}ationslash= 0, w= 0.(5.6)\nIt can be seen that the parameter nin the power-law equation ( 3.18) cannot admit any value,\nbut only those values satisfying in ( 5.6). Thus, one may use ( 5.6) to write the general solution\n(4.17) in terms of the parameters ω,wandΛ, and hence by changing q(t)→a(t)we get\na(t) =\n\nc′1/parenleftBig\n−2Λ(ω+1)2\n(4+3ω)(3+2ω)/parenrightBig−ω+1\n3ω+4sin2(ω+1)\n3ω+4/bracketleftBig/radicalBig\n−(3ω+4\n2ω+3)Λ\n2(t+c′2)/bracketrightBig\n,Λ<0,\nc′1/parenleftBig\n4+3ω(1−w2)\n2[ω(1−w)+1]/parenrightBig2[ω(1−w)+1]\n4+3ω(1−w2)/parenleftbig\nt+c′2/parenrightbig2[ω(1−w)+1]\n4+3ω(1−w2), Λ = 0,\nc′1/parenleftBig\n2Λ(ω+1)2\n(4+3ω)(3+2ω)/parenrightBig−ω+1\n3ω+4sinh2(ω+1)\n3ω+4/bracketleftBig/radicalBig\n(3ω+4\n2ω+3)Λ\n2(t+c′2)/bracketrightBig\n,Λ>0.(5.7)\nAccording to ( 5.6), the above equation is actually the solution of the BD equati ons for the\nscale factor of the dust-dominated universe in both cases Λ<0andΛ>0, while the solution\n17of the case Λ = 0 can be included the universe fulfilling with the dust, radiat ion and false\nvacuum, as well as stiff fluid. Having the cosmic scale factor a(t), one can calculate the BD\nscalar field ϕ(t). Using equations ( 3.18), (5.6) and ( 5.7) and simplifying them we then get\nϕ(t) =\n\nCc′11\nω+1/parenleftBig\n−2Λ(ω+1)2\n(4+3ω)(3+2ω)/parenrightBig−1\n3ω+4sin2\n3ω+4/bracketleftBig/radicalBig\n−(3ω+4\n2ω+3)Λ\n2(t+c′2)/bracketrightBig\n,Λ<0,\nCc′13w−1\nω(w−1)−1/parenleftBig\n4+3ω(1−w2)\n2[ω(1−w)+1]/parenrightBig2(1−3w)\n4+3ω(1−w2)/parenleftbig\nt+c′2/parenrightbig2(1−3w)\n4+3ω(1−w2), Λ = 0,\nCc′11\nω+1/parenleftBig\n2Λ(ω+1)2\n(4+3ω)(3+2ω)/parenrightBig−1\n3ω+4sinh2\n3ω+4/bracketleftBig/radicalBig\n(3ω+4\n2ω+3)Λ\n2(t+c′2)/bracketrightBig\n,Λ>0.(5.8)\nIn order to calculate the energy density of the universe with the equation of state p=ωρ,\nin all three cases Λ<0,Λ = 0 , andΛ>0we use equations ( 3.19), (5.6) and ( 5.7). By\ndoing some algebraic calculations, we get the following sol ution for the energy density of the\nuniverse\nρ(t) =\n\n(−ΛC\n8π)c′1−1\nω+1/parenleftBig\n−2Λ(ω+1)2\n(4+3ω)(3+2ω)/parenrightBig−1\n3ω+4sin−6(ω+1)\n3ω+4/bracketleftBig/radicalBig\n−(3ω+4\n2ω+3)Λ\n2(t+´c2)/bracketrightBig\n,Λ<0,\nM(ω,w)X(ω,w)(t+c′2)N(ω,w), Λ = 0,\n(ΛC\n8π)c′1−1\nω+1/parenleftBig\n2Λ(ω+1)2\n(4+3ω)(3+2ω)/parenrightBig−1\n3ω+4sinh−6(ω+1)\n3ω+4/bracketleftBig/radicalBig\n(3ω+4\n2ω+3)Λ\n2(t+´c2)/bracketrightBig\n,Λ>0,(5.9)\nwhereM,X, andNare the functions of the parameters ωandwwhich are defined as\nM(ω,w) :=C c′13w−1\nω(w−1)−1\n4π/parenleftBig4+3ω(1−w2)\n2[ω(1−w)+1]/parenrightBig2(1−3w)\n4+3ω(1−w2), (5.10)\nX(ω,w) :=6(w−1)2ω2−ω[12(w−1)+(1−3w)(3w−5)]+12−18w\n[4+3ω(1−w2)]2,(5.11)\nN(ω,w) :=2(1−3w)−8−6(1−w2)ω\n4+3ω(1−w2). (5.12)\nAs mentioned earlier, the cases Λ<0andΛ>0of solution ( 5.9) give the energy density for\ndust dominated universe only, while the case Λ = 0 of this solution gives the energy density\nof the perfect fluid-filled universe with equation of state ( 3.12). Accordingly and also using\nthe equation of state p=wρand energy density ( 5.9), the pressure of the universe is worked\n18out to be\np(t) =\n\n0 Λ <0,\nwM(ω,w)X(ω,w)(t+c′2)N(ω,w), Λ = 0, w∈[−1,1]\n0 Λ >0.(5.13)\nAccording to this solution, the pressure of the dust univers e forΛ<0,Λ = 0 andΛ>0is\nzero as we expected. Furthermore, for Λ = 0 , this solution gives the pressure of a universe\nfilled by a perfect fluid. We call the solutions ( 5.7), (5.8), (5.9) and ( 5.13) in this article\nas cosmological solutions of the BD equations. It should be no ted that these solutions were\npreviously obtained in less detail without the use of the con cept of symmetry by one of the\nauthors of this article in [ 41,42]. Note that when the coupling constant ωtends to infinity in\nsolutions ( 5.7), (5.8), (5.9) and ( 5.13), one concludes that they are actually the cosmological\nsolutions of general relativity, which can be obtained from solving EFEs in the presence of\nthe cosmological constant for a universe filled with a perfec t fluid with the equation of state\np=wρ.\nBefore closing this section, let us assume that the perfect flu id, filling the universe, is\nradiation. For w= 1/3andΛ = 0 it follows from ( 5.6) thatn= 0. Then, equations ( 3.19)\nand (3.20) are reduced to\n8πρ\nC= 3˙a2\na2, (5.14)\n8πp\nC=−2¨a\na−˙a2\na2. (5.15)\nNow we compare these equations with the case Λ = 0 of the following Friedmann equations\nin the theory of general relativity\n8πGρ= 3˙a2\na2, (5.16)\n8πGρ=−2¨a\na−˙a2\na2. (5.17)\nIn order to be compatible with general relativity in this spe cial case (for radiation in the\nabsence of the Λ), the value of the constant Cshould be\nC(w=1\n3,ω,G,Λ = 0) =1\nG. (5.18)\nBy using condition ( 5.18), one can obtain the cosmological solutions of the BD equatio ns for\nradiation in the absence of the Λ, giving us\na(t) =c′1/radicalbig\n2(t+c′2), ϕ(t) =1\nG, p(t) =1\n32πGt2, ρ(t) = 3p(t). (5.19)\n19Therefore, when the condition ( 5.18) holds, then not only general relativity and Brans-Dicke\ntheories are compatible with each other, but also the soluti ons of both theories will be the\nsame for radiation. This result is indicative of the fact tha t for a universe full of radiation\nw= 1/3whose metric is FLRW flat (k= 0), in the absence of the cosmological constant Λ,\nthe coupling constant of the BD scalar field ϕwith the gravitational field gµνis at its lowest\npossible, so that the two fields ϕandgµνare completely separate and have no paining with\nother. In other words, the curvature of space-time is very la rge compared to the curvature of\nthe scalar field.\n6 Special solutions of the FBD-equation\nIn order to make sure the correctness of cosmological soluti ons of the BD equations ( 5.7),\n(5.8), (5.9) and ( 5.13), it is necessary to compare our solutions with special solu tions such\nas Nariai’s solutions [ 43], O’Hanlon-Tupper vacuum solutions [ 8] and the inflation solutions\nthat have been obtained before. It can be shown that when w∈[0,1/3]andc′2= 0, our\nsolutions include the Nariai’s solutions. Nariai’s soluti ons in the absence of the Λ, when the\nfluid constituting the universe is dust, include the BD dust so lutions\na(t) =c′1/parenleftBig4+3ω\n2(ω+1)/parenrightBig2(ω+1)\n4+3ω(t+c′2)2(ω+1)\n4+3ω, (6.1)\nϕ(t) =C/parenleftBig\nc′1(4+3ω)\n2(ω+1)/parenrightBig2\n4+3ω(t+c′2)2\n4+3ω, (6.2)\nρ(t) =M(ω,0)X(ω,0) (t+c′2)−6(ω+1)\n4+3ω, (6.3)\np(t) = 0. (6.4)\nwhereω/ne}ationslash=−4/3,−1.To obtain the vacuum solution ( p= 0,ρ= 0), we reconsider equation\n(3.19) that for the vanishing energy density we have\n3−3n−ω\n2n2= 0, (6.5)\nThe solutions of this quadratic equation are:\nn±=−3±/radicalbig\n3(3+2ω)\nω. (6.6)\nThe solutions a(t)andφ(t)obtained for these two values of nare called the “vacuum solutions”\nof the BD equations. By substituting ( 5.6) into equation ( 6.6) one may obtain the parameter\nof the equation of state win terms of the coupling parameter ω, expressing\nw±=−(2ω+3)±(ω+1)/radicalbig\n3(3+2ω)\n±ω/radicalbig\n3(3+2ω). (6.7)\n20For these two values of w±, the cosmological solutions of BD equations in the absence of the\nΛare worked out\na±(t) =a0(1+bt)q±, (6.8)\nϕ±(t) =ϕ0(1+bt)1−3q±, (6.9)\nwhere\nq±=ω+1±/radicalBig\n2ω+3\n3\n3ω+4,\nalso, the constants a0,ϕ0andbare\na0:=a(t= 0) =c′1/parenleftBigc′2\nq±/parenrightBigq±, ϕ 0:=ϕ(t= 0) =1\nGc′11−3q±\nq±/parenleftBigc′2\nq±/parenrightBig1−3q±, b:=1\nc′2,\nwe note that the range of parameter ωisω >−3/2withω/ne}ationslash=−4/3,0. These solutions are\nin agreement with the solutions of those of [ 15]. The special solutions ( 6.8) and ( 6.9) were\nfirst obtained by O’Hanlon and Tupper in 1972 and are usually c alled the vacuum solutions of\nO’Hanlon-Tupper [ 8]. These solutions are completely consistent with Chauvet’ s solutions [ 9].\nAt the end of this section, let us examine the inflation soluti ons of BD equations. For\nthe state parameter w=−1, the universe was passing through the false vacuum era [ 44–46].\nWe expect that the solutions obtained from the BD equations in this particular state to be\ninflationary. To this end, we begin with the case Λ = 0 of the cosmological solutions ( 5.7),\n(5.8) and ( 5.9) forw=−1, which are, respectively, given by\na(t) =c′1/parenleftBig1\nω+1\n2/parenrightBigω+1\n2(t+c′2)ω+1\n2, (6.10)\nϕ(t) =1\nGc′2\nω+1\n2\n1/parenleftBig2\n2ω+1/parenrightBig2\n(t+c′2)2, (6.11)\nρ(t) =1\n8πGc′2\nω+1\n2\n1(2ω+3)(6ω+5)\n(2ω+1)2:=ρ−1. (6.12)\nwhereω/ne}ationslash=−1/2,−3/2,−5/6. The last one shows that the energy density of the universe in\nthe false vacuum era is a constant. Here, we have denoted this constant value by ρ−1. Now,\nit is necessary to express the integration constant c′1in equations ( 6.10) and ( 6.11) in terms\nof cosmic scale factor of the universe at the time of the Big Bang t= 0, in which we denote\nbya0. For this purpose, by putting t= 0in cosmological solution ( 6.10) and according to the\ninitial condition a0=a(t= 0), we then get\nc′2=2ω+1\n2/parenleftBiga0\nc′1/parenrightBig2\n2ω+1. (6.13)\n21By substituting ( 6.12) and ( 6.13) into solutions ( 6.10) and ( 6.11), one can obtain\na(t) =a0(1+χt)ω+1\n2, (6.14)\nϕ(t) =1\nGa2\nω+1\n2\n0(1+χt)2, (6.15)\nwhere\nχ2:=32πGρ−1c′−2\nω+1\n2\n1\n(6ω+5)(2ω+3), (6.16)\nis a constant. The cosmic scale factor ( 6.14) and the BD scalar field ( 6.15) are the inflation\nsolutions of the BD equations in the absence of the cosmologic al constant, respectively. When\nthe coupling parameter ωtends to infinity, for very small times χt≪1, the limits of the\ninflation solutions of the BD equations become\nlim\nω→∞a(t) =a0lim\nω→∞/parenleftBig\n1+1\nω/radicalbigg\n8πGρ−1\n3t/parenrightBigω\n=a0e/radicalBig\n8πGρ−1\n3t, (6.17)\nlim\nω→∞ϕ(t) =1\nG. (6.18)\nIt can be seen that in the limit ω→ ∞, the cosmic scale factor a(t)changes exponentially\nwith cosmic time t, while the BD scalar field ϕ(t)is a constant. Notice that solutions ( 6.17)\nand (6.18) are the same solutions which are obtained from the theory of the GR in the absence\nof the cosmological constant in the case where the equation o f state of the universe is in the\nformp=−ρ−1. For the case that χt≫1, then the inflationary solutions ( 6.14) and ( 6.15)\nare as follows:\na(t) =a0(χt)ω+1\n2, (6.19)\nϕ(t) =ϕ0(χt)2, (6.20)\nwhere\nϕ0=1\nGa2\nω+1\n2\n0. (6.21)\nFor the universe in the false vacuum era (p=−ρ−1)and in the absence of cosmological\nconstant Λ, it follows from ( 5.6) that\nn|w=−1,Λ=0=−2\nω+1\n2. (6.22)\nThen, by substituting ( 6.22) into the power-law equation ( 3.18) and by using the fact that\nC= 1/G, we find\n1\nG=C=ϕ(t)an(t) =ϕ(t= 0)an(t= 0) =ϕ0an\n0=ϕ0a−2\nω+1\n2\n0.\n22which is nothing but equation ( 6.21). This issue can be a confirmation of the correctness of\nthe inflationary solutions obtained by solving the BD equatio ns for the false vacuum universe\nin the absence of the cosmological constant.\n7 Conclusions\nIn this work, we have focused on solving the BD equations for sp atially flat (k= 0)FLRW\nuniverse fulfilling a perfect fluid with the equation of state p=ωρ,(−1≤ω≤1). Using\nequations ( 3.12) and ( 3.18) we have solved the BD equations in both cases of the absence\nand presence of the cosmological constant to obtain functio nsa(t),ϕ(t),ρ(t)andp(t). As\nwas shown, the constant nin equation ( 3.18) and the state equation parameter win (3.12)\nare compatible with each other, provided that the condition (5.6) is held. Equations ( 3.12)\nand (3.18) helped us summarize all BD equations in one equation, ( 3.25), which we called\nFBD-equation. This equation alone could describe the dynami cs of the spatially flat FLRW\ncosmological model in the framework of BD theory. As we have se en, this cosmological model\ncould be imagined as a 1-dimensional dynamical system with the configuration space Q= (a)\nin which the governing equation of particle motion was given byF=φ(t,a,˙a)∂a. As the\nfirst step to solve the FBD-equation we found the correspondin g Lie point symmetries. Then,\nwe showed that the FBD-equation has two independent point sym metries with infinitesimal\ngenerators X1=∂tandX2=a∂a. Using DPs and λ-symmetries, we found two independent\n2-tuple(S1,R1)(equation ( 4.7)) and(S2,R2)(equation ( 4.9)) consisting of null form Sand\nintegrating factor R. Using these independent 2-tuples and applying Duarte’s integral formula\n(2.4), we obtained the two independent invariants ( 4.8) and ( 4.10) which were associated to\nthe infinitesimal generators X1andX2, respectively. We showed that these two independent\ninvariants, each of which was a first-order ODE for the cosmic scale factor a(t)in terms of\ncosmic time t, could be viewed as a system of algebraic equations for the un known functions\na(t)and˙a(t). By eliminating ˙a(t)between the two equations of the system, we obtained the\nanalytical solution ( 5.7) for the cosmological scale factor a(t). By employing this solution in\nthe BD equations, we were able to obtain the BD scalar field ϕ(t), the energy density of the\nuniverse ρ(t)and its pressure p(t), which were given by equations ( 5.8), (5.9), and ( 5.13),\nrespectively. Not only these solutions gave the scale facto ra(t), the BD scalar field ϕ(t), the\nenergy density ρ(t)and pressure of the universe p(t)for the dust, but also they gave us the\nmore general state of a perfect fluid with equation of state p=ωρ,(−1≤ω≤1).\nAs an interesting result, we have shown that when the couplin g parameter ωtends to\ninfinity, the cosmological solutions of GR theory can be obta ined from the analytical solu-\ntions of BD equations ( 5.7), (5.8), (5.9) and ( 5.13); this can be a confirmation of limω→∞\nBD = GR. Note that our cosmological solutions are rich, so that they include many well-\nknown special solutions that were previously obtained by ot her methods. For example, the\nsolutions presented by Uehara and Kim [ 12] are special cases of our solutions, when the cosmo-\nlogical constant Λis present for the dust dominated universe. 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Quantum Gravity, 19(18)\n(2002) 4747-4752.\n27"    },    {        "title": "2401.17943v1.Large_amplitude_traveling_waves_for_the_non_resistive_MHD_system.pdf",        "content": "arXiv:2401.17943v1  [math.AP]  31 Jan 2024LARGE AMPLITUDE TRAVELING WAVES FOR THE NON-RESISTIVE MHD\nSYSTEM\nGENNARO CIAMPA, RICCARDO MONTALTO, AND SHULAMIT TERRACINA\nAbstract. We prove the existence of large amplitude bi-periodic trave ling waves (stationary in a\nmoving frame) of the two-dimensional non-resistive Magnet ohydrodynamics (MHD) system with a\ntraveling wave external force with large velocity speed λ(ω1,ω2) and of amplitude of order O(λ1+)\nwhereλ≫1 is a large parameter. For most values of ω= (ω1,ω2) and for λ≫1 large enough,\nwe construct bi-periodic traveling wave solutions of arbit rarily large amplitude as λ→+∞. More\nprecisely, we show that the velocity field is of order O(λ0+), whereas the magnetic field is close\nto a constant vector as λ→+∞. Due to the presence of small divisors, the proof is based on a\nnonlinear Nash-Moser scheme adapted to construct nonlinea r waves of large amplitude. The main\ndifficulty is that the linearized equation at any approximate solution is an unbounded perturbation\nof large size of a diagonal operator and hence the problem is n ot perturbative. The invertibility\nof the linearized operator is then performed by using tools f rom micro-local analysis and normal\nforms together with a sharp analysis of high and low frequenc y regimes w.r. to the large parameter\nλ≫1. To the best of our knowledge, this is the first result in whic h global in time, large amplitude\nsolutions are constructed for the 2D non-resistive MHD syst em with periodic boundary conditions\nand also the first existence results of large amplitude quasi -periodic solutions for a nonlinear PDE\nin higher space dimension.\nKey words: Fluid Mechanics, Magnetohydrodynamics, Traveling Waves, Normal Forms, Micro-\nlocal analysis.\nMSC 2020: 35Q35, 76W05, 35S50.\nContents\n1. Introduction 2\n1.1. Main result 4\n1.2. Main ideas, novelties and strategy of the proof 6\nRemarks and future perspectives 10\nOutline of the paper 10\nAcknowledgements 10\n2. Functions spaces, norms and linear operators 11\n2.1. Diophantine equations 12\n2.2. Linear operators 12\n2.3. Pseudo-differential operators and norms 13\n2.4. A quantitative Egorov Theorem 18\n3. The linearized operator 25\n4. Decoupling of linearized operator up to a smoothing remai nder 26\n4.1. Decoupling of the first order part 27\n4.2. The iterative step 35\n5. Inversion of the first equation 39\n5.1. Inversion of the operator L(1)\n1 39\n5.2. Reduction to a scalar transport-type operator 42\n6. Normal form reduction and inversion of the transport oper atorP 43\n6.1. Reduction of the transport operator of order one 43\n6.2. Reduction of the lower order terms 45\n6.3. Inversion of the operator P 47\n7. Inversion of the linearized operator L 49\n8. Construction of an approximate solution 50\n12 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\n9. The Nash Moser scheme 51\n10. Measure estimates 55\n11. Proof of Theorems 1.1, 1.2 57\nReferences 59\n1.Introduction\nInthispaperweconsiderthetwodimensionalnon-resistive MHDequationsonthetwo-dimensional\ntorusT2,T:=R/(2πZ)\n\n\n∂tu+(u·∇)u+∇p= ∆u+(b·∇)b+f(t,x),\n∂tb+(u·∇)b= (b·∇)u,\ndivu= 0,divb= 0,(1.1)\nwhere the unknowns are the velocity field of the fluid u:R×T2→R2, the magnetic field\nb:R×T2→R2and the pressure p:R×T2→R, whereas the external force acting on the\nfluidf:R×T2→R2is given. This system describes the behavior of a viscous, el ectrically\nconducting, incompressible fluid, whose resistivity is neg ligible and it is commonly used to model\nstrongly collisional plasmas in astrophysics. We refer to [ 14, 43] for a derivation of the model,\nhowever we recall that the equations are given by a combinati on of the Navies-Stokes equations\nand the Faraday-Maxwell system via Ohm’s law. The equations (1.1) were analyzed by Moffat [46]\nin the context of topological hydrodynamics with the aim to c onstruct magneto-static equilibria\n(and thus also steady solutions of the Euler equations) with arbitrary prescribed topology. In fact,\nthe second equation in (1.1) is “topology preserving” since the integral lines of the magnetic field b\nare transported by the flow of the velocity uas they evolve in time. Indeed, as long as one considers\nsmooth solutions, the flow of uis a diffeomorphism and then the magnetic lines keep their topo logy\nunchanged during the evolution. This is also known in the lit erature as Alfven’s Theorem. The\nambitious program of Moffat was to construct magneto-static e quilibria in the infinite limit (w.r.t.\ntime) of solutions to a topology preserving diffusive equatio n. We point out that Moffat’s original\nproposal did not involve external forces in the system (1.1) , but we refer to [42] for the forced\ncase. This procedure is known as magnetic relaxation and some partial results have been obtained\nin recent years, see [3, 11, 13, 42]. We also point out that Enc iso and Peralta-Salas [18] proved\nthe existence of (infinite energy) stationary solutions of t he Euler equations with vortex lines of\nprescribed topology, however it is an open problem whether s uch solutions arise as limits of the\nsystem (1.1). In contrast, in the resistive case the topolog y of magnetic lines is expected to change\nleading to magnetic reconnection phenomena. Thelatter are of extreme relevance to astrophys icists\nbeing a phenomenon observed virtually everywhere in the uni verse. We refer to [12, 48, 49] and\nreferences therein for an overview of this topic from a numer ical and analytical point of view.\nThemathematical aspectsof theMHDequations havebeenexte nsively studiedinrecent decades,\nespecially withregard to theCauchy problem. Before going i nto thedetails of our result, we provide\na (non-exhaustive) review of the literature on the problem, with a particular focus on the two-\ndimensional case. In the presence of the resistive term ∆ bin the second equation, global existence\nand uniqueness of finite energy weak solutions (in 2D) has bee n proved in [16]. Furthermore, if\nthe initial datum and the external force are smooth, the well -posedness of (1.1) (with the resistive\nterm ∆b) in 2D has been shown in [52]. In contrast, for the three-dime nsional case we only have\nuniquelocalsmooth solutions, see again [16, 52]. For the non-resistive case, the local existence of\nsmooth solutions in 2D has been proved in [40] when the initia l datum and the body force belong\nto the space Hswiths≥3, while the 3D case was treated in [21, 22] at a sharp level of S obolev\nregularity. Moreover, global weak solutions in 2D for the sy stem without the viscous term ∆ u\nin the first equations but the resistive term ∆ bin the second equations have been constructed in\n[30]. This case is less difficult than (1.1). Indeed in (1.1), t he equations for the magnetic field look\nlike the vorticity formulation of the 3D Euler equation (the termb·∇uis similar to the strechingLARGE AMPLITUDE TRAVELING WAVES MHD 3\nterm). Global well-posedness results in the three-dimensi onal (1.1) equations are available under\nsome geometrical constraints on the initial datum. In parti cular, special type of axisymmetric\nsolutions have been constructed for the system (1.1) in [41] . See also [32] for the analogous result\nfor the system with no viscosity ∆ uin the first equations but with resistive term ∆ bin the second\nequation.\nFinally, assuming that the initial datum b0is a small perturbation of a constant steady state,\nthe existence of global-in-time smooth smallsolutions (namely the velocity field is small and the\nmagnetic field is close to a constant steady state) of (1.1) ha s been established in [44, 50]. See also\n[45, 57] for similar results in the three dimensional case. H owever, the existence of global-in-time\nsmooth solutions for (1.1) with generic periodic initial da ta (no perturbation of a constant steady\nstate, big velocity) is still open even in 2D.\nThe aim of this paper is to construct quasi-periodic solutio ns (global-in-time) of large amplitude\nfor the system (1.1) on the bi-dimensional torus T2. We consider the case of bi-periodic traveling\nwave solutions. We shall assume that the external force fis a smooth bi-periodic traveling wave\npropagating in the direction ω= (ω1,ω2)∈R2, with large amplitude λ1+η(0<η≪1,λ≫1) and\nwith large velocity speed λω. More precisely, we take\nf(t,x) =λ1+ηf(x+λωt) where\nf∈ C∞(T2,R2),/integraldisplay\nT2f(x)dx= 0,\nω= (ω1,ω2)∈ OwhereO ⊂R2is a bounded domain ,\nλ≫1 is a large parameter and 0 <η≪1 is a small parameter .(1.2)\nNote thatλ1+ηrepresents the amplitude of the forcing term and λis the size of the frequency\n(velocity speed). We shall prove that for λ≫1 large enough, 0 < η≪1 small enough and for a\nfull measure set of frequencies ω∈ O, the system (1.1) admits traveling wave solutions u(x+λωt),\nb(x+λωt),p(x+λωt) oflarge amplitude of orderO(λ0+) asλ→+∞. This is a small divisors\nproblem, typical in the KAM (Kolmogorov-Arnold-Moser) the ory for PDEs, since the linearized\nequation at the origin has spectrum accumulating to zero for most values of the parameter ω.\nThe major difficulty in this problem is that we look for large quasi-periodic solutions and hence the\nproblem is not perturbative: indeed, even at a formal level, if one tries to construct by perturbation\ntheory the solution (by expanding in decreasing powers of λ), one gets that the size of the expected\nsolution is essentially given by the ratio\nsize of the solution =size of the perturbation\nsize of the frequency=O(λ1+)\nO(λ)=O(λ0+). (1.3)\nOther difficulties are: the equations (1.1) are strongly coup led, this makes quite difficult their\nanalysis; the system (1.1) is a system of PDEs in higher space dimension with derivatives in the\nnonlinearity, for which it is particularly difficult to prove KAM-type results. Indeed the problem\nthat we study is even non trivial for small amplitude solutio ns.\nWe overcome the afore-mentioned difficulties by implementin g a Nash-Moser scheme adapted for\nconstructing solutions of large amplitude. The key difficult y is the inversion of the linearized\noperator at any approximate solution which is the sum of a dia gonal operator plus a large variable\ncoefficients operator of size O(λ0+). In order to invert such an operator we develop normal form\nmethods based on micro-local analysis and pseudo-differenti al calculus which allows to use the\nstabilizing effect of the large velocity speed λω. This requires some sharp analysis in high and low\nfrequency regimes w.r. to the large parameter λcombined with tools from micro-local analysis.\nWe refer to the next section for a more precise description of the main ideas of the proof.\nWe conclude this part of the introduction with the following remarks. Our result is the first one\nin which one shows the existence of large amplitude quasi-periodic solutions for a non-integrable\nPDE with a perturbation (forcing term) of large size. Up to no w large amplitude quasi-periodic\nsolutions have been constructed only for small perturbatio nsof defocusingNLSand KdVequations,\nby exploiting the integrable structures of these two equati ons, see [9], [10]. We also mention that\nKAM and Normal Form techniques have been developed to study t he dynamics of one dimensional4 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nlinear wave and Klein Gordon equations with a time dependent quasi-periodic potential of size\nO(1) and large frequencies (in which the ratio (1.3) is small f orλ≫1 large enough), see [26], [29].\n1.1.Main result. We are interested in the existence of large amplitude traveling wave solutions\nof (1.1). As we have already mentioned, the unknowns are the m agnetic field b:R×T2→R2, the\nvelocity of the fluid u:R×T2→R2and the pressure p:R×T2→R, whereas, the data of the\nproblem are the large parameter λ≫1, the small parameter 0 <η≪1,ω= (ω1,ω2)∈ O ⊂R2\nis the two-dimensional frequency vector in a bounded domain O ⊂R2and the external force fis\na bi-periodic traveling wave of the form (1.2). We then look f or smooth solutions ( u,b,p) of the\nsystem (1.1), having the form\nu(t,x) :=U(x+λωt), U:T2→R2,/integraldisplay\nT2U(x)dx= 0,\nb(t,x) :=b+B(x+λωt),b/\\e}atio\\slash= 0, B:T2→R2,/integraldisplay\nT2B(x)dx= 0,\np(t,x) :=P(x+λωt),/integraldisplay\nT2P(x)dx= 0,\nwith frequency ω= (ω1,ω2)∈ O ⊆R2.(1.4)\nThe main feature of solutions of the form (1.4) is that they look steady in the reference frame\ny1=x1+λω1t,y2=x2+λω2t. This implies that plugging the ansatz (1.4) in the equation s (1.1),\nwe obtain a stationary problem of the form\n\n\nλω·∇U+(U·∇)U+∇P= ∆U+(b+B)·∇B+λ1+ηf,\nλω·∇B+(U·∇)B= (b+B)·∇U,\ndivU= divB= 0(1.5)\nwhereω· ∇=ω1∂x1+ω2∂x2. We look for large amplitude solution (growing as λ→+∞)\n(U,B,P)∈Hs\n0(T2,R2)×Hs\n0(T2,R2)×Hs\n0(T2,R) fors≫0 large enough where\nHs(T2,Rn) :=/braceleftBig\nu(x) =/summationdisplay\nk∈Z2/hatwideu(k)eik·x∈L2(T2,Rn) :/ba∇dblu/ba∇dbls:=/parenleftBig/summationdisplay\nk∈Z2/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2s|/hatwideu(k)|2/parenrightBig1\n2<∞/bracerightBig\n,\nHs\n0(T2,Rn) :=/braceleftBig\nu∈Hs(T2,Rn) :/integraldisplay\nT2u(x)dx= 0/bracerightBig\n.(1.6)\nwhere/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht:= (1 +|k|2)1\n2. As a notation, we often write Hs≡Hs(T2)≡Hs(T2,R) andHs\n0≡\nHs\n0(T2)≡Hs\n0(T2,R). We denote by M(n)the Lebesgue measure on Rn.\nWe shall make the following assumption on the forcing term fand on the average of the magnetic\nfieldbthat guarantees that we construct non trivial solutions.\n•Assumption. Letb∈R2\\ {0}be the average of the magnetic field and let f:=\n(f1,f2)∈ C∞(T2,R2),/integraltext\nT2f(x)dx= 0. Let F:= curl(f) :=∂x1f2−∂x2f1,F(x) =/summationtext\nk∈Z2\\{0}/hatwideF(k)eix·k.\nThere exists k∈Z2\\{0}such that\nb·k/\\e}atio\\slash= 0 and/hatwideF(k)/\\e}atio\\slash= 0.(1.7)\nOur main result is the following.\nTheorem 1.1. Letf∈ C∞(T2,R2)with zero average and b∈R2\\ {0}satisfies (1.7). There\nexist¯S >0,η∈(0,1)such that for any S≥¯S, there exist λ0≡λ0(f,η,S,b)≫1andC1≡\nC1(f,η,S,b), C2≡C2(f,η,S,b)≫1such that for any λ>λ0the following holds. There exists a\nBorel set Oλ⊂ Oof asymptotically full Lebesgue measure, i.e. limλ→+∞M(2)(O \\Oλ) = 0, such\nthat for every ω∈ Oλthere exists/parenleftBig\nU(·;ω),B(·;ω),P(·;ω)/parenrightBig\n∈HS\n0(T2,R2)×HS\n0(T2,R2)×HS\n0(T2),LARGE AMPLITUDE TRAVELING WAVES MHD 5\nU,B,P/\\e}atio\\slash= 0that solves the MHD equations (1.5). Moreover\ninf\nω∈Oλ/parenleftBig\n/ba∇dblU(·;ω)/ba∇dblS+/ba∇dblB(·;ω)/ba∇dblS/parenrightBig\n≥inf\nω∈Oλ/ba∇dblU(·;ω)/ba∇dblS≥C1λη,\nifdiv(f)/\\e}atio\\slash= 0then inf\nω∈Oλ/ba∇dblP(·;ω)/ba∇dblS≥C1λ1+η\nsup\nω∈Oλ/ba∇dblU(·;ω)/ba∇dblS≤C2λ3η,sup\nω∈Oλ/ba∇dblB(·;ω)/ba∇dblS≤C2λ−3η,sup\nω∈Oλ/ba∇dblP(·;ω)/ba∇dblS≤C2λ1+η.(1.8)\nWe now make some comments on the assumption (1.7). This assum ption is verified for instance\nifb∈R2\\{0}is an irrational vector, i.e. b·k/\\e}atio\\slash= 0 for any k∈Z2\\{0}and ifF= curl(f)/\\e}atio\\slash= 0\nis a non trivial forcing term. If bis a rational vector, for instance b= (1,0), then the assumption\n(1.7) holds if (for instance) the forcing term Fsatisfies/hatwideF(k)/\\e}atio\\slash= 0 fork= (k1,0),k1∈Z\\{0}.\nTheorem 1.1 will be obtained by Theorem 1.2 below, dealing wi th the vorticity formulation of the\nsystem (1.10). We define the vorticity and the current densit y as\nΩ :=∂x1U2−∂x2U1, U(x) = (U1(x),U2(x)),\nJ:=∂x1B2−∂x2B1, B(x) = (B1(x),B2(x)).(1.9)\nApplying the curl operator to both the equations in (1.1), we obtain the following system\n\n\nλω·∇Ω+(U·∇)Ω = ∆Ω+( b+B)·∇J+λ1+ηF,\nλω·∇J+(U·∇)J= (b+B)·∇Ω+2H(U,B),\nU=UΩ :=∇⊥(−∆)−1Ω, B=UJ:=∇⊥(−∆)−1J,(1.10)\nwhere we recall that F= curlf=∂x1f2−∂x2f1, the orthogonal gradient is defined as ∇⊥:=\n(∂x2,−∂x1), (−∆)−1denotes the inverse of the Laplacian (i.e. the Fourier multi plier with symbol\n|ξ|−2forξ∈Z2\\{0}), and the bilinear form His defined by\nH(U,B) :=∂x1B·∇U2−∂x2B·∇U1. (1.11)\nWe remark that, once (1.10) is solved, the pressure is recove red from the first equation in (1.5) via\nthe elliptic equation\n∆P=λ1+ηdivf+div[(B·∇)B]−div[(U·∇)U]. (1.12)\nSince/integraltext\nT2Ω(x)dxand/integraltext\nT2J(x)dxare conserved quantities, we shall restrict to the space of z ero\naverage vector fields in x. We look for large amplitude waves of order O(λ0+), therefore it is\nconvenient to rescale the variables\nΩ/mapsto→λδΩ, J/mapsto→λδJ,\nwhere we fix δ:= 3η.(1.13)\nUnder the latter rescaling, the system (1.10) reads as\nF(Ω,J) = 0,\nF(Ω,J) :=\nλω·∇Ω−∆Ω−b·∇J+λδ/bracketleftBig\nU·∇Ω−B·∇J/bracketrightBig\n−λ1−2\n3δF\nλω·∇J−b·∇Ω+λδ/bracketleftBig\n(U·∇)J−B·∇Ω−2H(U,B)/bracketrightBig\n,\nU=UΩ :=∇⊥(−∆)−1Ω, B=UJ:=∇⊥(−∆)−1J(1.14)\nand we look for solutions Ω ,J∈Hs\n0(T2) fors≫1 large enough. The following theorem holds.\nTheorem 1.2. Letf∈ C∞(T2,R2)with zero average and b∈R2\\ {0}satisfies (1.7). There\nexist¯S >0,δ∈(0,1)such that for any S≥¯S, there exist λ0=λ0(f,δ,S,b)≫1andC1≡\nC1(f,δ,S,b), C2≡C2(f,δ,S,b)>0such that for any λ>λ0the following holds. There exists a\nBorel set Oλ⊂ Oof asymptotically full Lebesgue measure, i.e. limλ→∞M(2)(O \\ Oλ) = 0, such6 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nthat for any ω∈ Oλ, there exists (Ω(·;ω),J(·;ω))∈HS\n0(T2)×HS\n0(T2)withΩ,J/\\e}atio\\slash= 0which is a\nzero ofFdefined in (1.14), namely F/parenleftBig\nΩ(·;ω),J(·;ω)/parenrightBig\n= 0for anyω∈ Oλ. Moreover\ninf\nω∈Oλ/parenleftBig\n/ba∇dblΩ(·;ω)/ba∇dblS+/ba∇dblJ(·;ω)/ba∇dblS/parenrightBig\n≥inf\nω∈Oλ/ba∇dblΩ(·;ω)/ba∇dblS≥C1λ−2\n3δ,\nsup\nω∈Oλ/ba∇dblΩ(·;ω)/ba∇dblS≤C2,sup\nω∈Oλ/ba∇dblJ(·;ω)/ba∇dblS≤C2λ−2δ(1.15)\nOur approach is based on a set of techniques and tools (normal forms, microlocal analysis, Nash-\nMoser implicit function theorem, etc.) that are now commonl y referred to as KAM (Kolmogorov-\nArnold-Moser) techniques for PDEs. In recent years, these t echniques have been successfully\ndeveloped to construct nonlinear periodic and quasi-perio dic waves of small amplitude in Fluid\nMechanics. This has been the case of the two-dimensional wat er waves equations, see [1, 8] for\ntime quasi-periodic standing waves, [6, 7, 23] for time quas i-periodic traveling wave solutions\nand [38], [39] for diamond waves in the three-dimensional wa ter waves equations. Moreover,\nwe also mention the recent results on the evolution of vortex -patches for active-scalar equations\n[5, 17, 31, 33, 34, 35, 37, 51]. By considering quasi-periodi c external forces, time quasi-periodic\nsolutions of the forced Euler equations have been construct ed in [2], in [47] for the Navier-Stokes\nequations and in [27] quasi-periodic solutions of 2D Navier -Stokes equations were constructed in\nthe vanishing viscosity limit and it was proved the converge nce to quasi-periodic solutions of the\nforced Euler equation (uniformly and globally in time) as th e viscosity goes to zero.\nWe finally remark that quasi-periodic solutions to the (unfo rced) Euler equations were con-\nstructed also in [15, 19] with a completely different approach (see also the recent extension to\nalmost periodic solutions [28]). In particular, the soluti ons are built by suitably gluing localized\ntraveling profiles, so they are not of KAM type since no small d ivisors are involved.\n1.2.Main ideas, novelties and strategy of the proof. We construct large amplitude bi-\nperiodic solutions of the 2D non-resistive MHD equations (w e work in the vorticity-current formu-\nlation (1.14)) of size O(λδ), 0<δ≪1,λ≫1. Due to the presence of small divisors , the scheme\nof the proof is based on a Nash-Moser iteration scheme in scal e of Sobolev spaces Hs. The main\ndifficulty compared with previous works is that, since we prod uce solutions of large amplitude, the\nproblem is not perturbative. The heart of the proof lies in th e analysis of the linearized operator\nL, associated to (1.10), arising at each step of the Nash-Mose r iteration.\nFor the whole introduction, we use the following notation. G ivenκ,m∈Rwe writeO(λκ|D|m) to\ndenote an operator of size λκand of order m.\nThe linearized operator Lis a variable coefficients perturbation of large size of a diag onal operator,\nnamely it has the form\nL:=D+P,\nD:=/parenleftbigg\nλω·∇−∆ 0\n0λω·∇/parenrightbigg\n,P:=/parenleftbigg\na(x)·∇d(x)·∇\nd(x)·∇a(x)·∇/parenrightbigg\n+/parenleftbigg\nR1R2\nR3R4/parenrightbigg\n,\nwhered(x),a(x) =O(λδ) and\nR1,R2=O(λδ|D|−1)...,R3,R4=O(λδ|D|0).(1.16)\nWe want to show the invertibility of Land provide tame estimates for its inverse L−1. Theformal\nideabehind the procedure is the following. By imposing the stand ard diophantine condition on ω,\ni.e.\n|ω·k| ≥γ\n|k|τ,∀k∈Z2\\{0},0<γ≪1, τ≫0, (1.17)\none has that Dis invertible with loss of derivatives and/ba∇dblD−1/ba∇dblB(Hs+τ,Hs)=O(λ−1). Hence formally\nD−1P=O(λδ−1)≪1 ifδ≪1 is small enough and λ≫1 is large enough. This heuristic\nargument suggests that the fact that the velocity speed is of sizeO(λ) is the key point to enter in a\nperturbative regime. On the other hand, due to the small divi sors, the Neumann series argument\ncannot be implemented directly and hence one needs to implem ent normal form arguments based\non pseudo-differential calculus in order to reduce to a situat ion in which the heuristic argumentLARGE AMPLITUDE TRAVELING WAVES MHD 7\ndescribed above can be made rigorous. As we mentioned alread y before, the major difficulty in this\nprocedure is that the perturbation is large for λ≫1. Now, we shall describe more in details the\nmain ideas and points of such a procedure.\nIn order to invert the operator Lwe have to solve the linear system of 2 equations in 2 unknowns\n(recall (1.16)) of the type\n/braceleftBigg\nL(1)h1+L(2)h2=g1,\nL(3)h1+L(4)h2=g2,\nL(1):=λω·∇−∆+a(x)·∇+R1,\nL(2):=d(x)·∇+R2,L(3):=d(x)·∇+R3,\nL(4):=λω·∇+a(x)·∇+R4.(1.18)\nThe leading idea to solve this system is to try to reduce it to t he invertibility of a scalar transport\ntype operator with large variable coefficients, that can be in verted by extending the strategy devel-\noped in [2], [27] in the framework of small amplitude solutio ns. This reduction can be done since\nL(1)isdissipative , due to the presence of the Laplacian and this avoid the small divisors problem at\nleast in the first equation. On the other hand, we mention that this inversion is anyway subtle since\nthe perturbation has large variable coefficients and require s a splitting in high and low frequencies\nwith respect to the large parameter λ. We can summarize the strategy of the inversion in three\nmain steps that we shall describe separately below:\n(1) decoupling of the equations up to an arbitrarily regular izing remainder;\n(2) inversion of the first equation;\n(3) inversion of the second equation.\nDecoupling of the equations up to an arbitrarily regularizing remainder.\nFirst of all, in Section 4, we implement an iterative procedu re that decouples the equations up to\na remainder of large size but regularizing, namely we use pse udo-differential operators in order to\nconjugate Lto another linear operator of the form\nL1=/parenleftBigg\nL(1)\n1L(2)\n−N\nL(3)\n−NL(4)\n1/parenrightBigg\n, (1.19)\nwith\nL(1)\n1=λω·∇−∆+a(x)·∇+R(1)\n0,\nL(1)\n4=λω·∇+a(x)·∇+R(4)\n0,(1.20)\nwhereR(1)\n0,R(4)\n0=O(λδM|D|0) are operators of order 0 andof size O(λδM) for someM≡M(N)≫\n1 large enough, whereas the remainders L(2)\n−N,L(3)\n−N=O(λδM|D|−N) are smoothing operators of\norder−N(meaningthattheyareboundedlinearoperatorsfrom HsintoHs+N)andofsize O(λδM).\nThe presence of the Laplacian in the first equation is crucial in this procedure, since it makes\npossible to solve homological equations which in turn provides the generator of the transformations\nthat decouple the equations. In order to simplify the exposi tion we shall describe the first step of\nsuch an iterative procedure in which we eliminate from Lin (1.16), the highest order off-diagonal\nterm\nQ1:=/parenleftbigg\n0d(x)·∇\nd(x)·∇0/parenrightbigg\n. (1.21)\nWe look for a map\nΦ := exp(Ψ) ,Ψ :=/parenleftbigg\n0 Op( ψ1(x,ξ))\nOp(ψ2(x,ξ)) 0/parenrightbigg\n,\nin such a way to eliminate-normalize the off-diagonal term Q1. If one computes Φ−1LΦ, by a\ndirect calculation, using pseudo-differential calculus, it turns out that in order to cancel the new8 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\noff-diagonal term of order one, the symbols ψ1(x,ξ) andψ2(x,ξ) has to satisfy\nλω·∇ψ1(x,ξ)+|ξ|2ψ1(x,ξ)+id(x)·ξ= lower order terms ,\nλω·∇ψ2(x,ξ)−|ξ|2ψ2(x,ξ)+id(x)·ξ= lower order terms .(1.22)\nThe two latter equations are solved in the same way, hence let us discuss how to solve the first\none. The main issue is that d(x) =O(λδ)≫1, but one needs estimates on ψ1which are uniformly\nbounded w.r. to λ, in such a way that exp(Ψ) has not bad divergences w.r. to the l arge parameter\nλ. A convenient way to proceed is then to introduce a suitable c ut-off function χλ(ξ) which is\nsupported on |ξ| ≥λ6δ. Then, we split the symbol id(x)·ξas\nid(x)·ξ=iχλ(ξ)d(x)·ξ+i(1−χλ(ξ))d(x)·ξ.\nThe symbol (1 −χλ(ξ))d(x)·ξis smoothing and it is of order −N,N≫1 with estimates\n|(1−χλ(ξ))d(x)·ξ|/lessorsimilarλ6δ(N+1)/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−N,\nand hence we shall solve\nλω·∇ψ1(x,ξ)+|ξ|2ψ1(x,ξ)+iχλ(ξ)d(x)·ξ= 0. (1.23)\nThis equation can be solved by expanding ψ1anddin Fourier series w.r. to xand one obtains that\nψ1(x,ξ) =/summationdisplay\nk∈Z2/hatwideψ1(k,ξ)eik·x, d(x) =/summationdisplay\nk∈Z2/hatwided(k)eik·x,/hatwideψ1(k,ξ) =−iχλ(ξ)/hatwided(k)·ξ\niλω·k+|ξ|2.\nBy the latter formula, one has that ψ1is supported on |ξ| ≥λ6δand (recall that d(x) =O(λδ)),\n|ψ1(x,ξ)|/lessorsimilarλδ|ξ|−1|ξ|1\n2≥λ3δ\n/lessorsimilarλδλ−3δ|ξ|−1\n2/lessorsimilarλ−2δ|ξ|−1\n2.\nWe then find ψ1,ψ2of order1\n2and of size O(λ−2δ)≪1 forλ≫1 large enough. The outcome of\nthis procedure, by estimating all the terms in the expansion ofL(0):= Φ−1LΦ is then given by\nL(0):=D+/parenleftbigg\na(x)·∇0\n0a(x)·∇/parenrightbigg\n+/parenleftBigg\n0O(λ3δ|D|1\n2)\nO(λ3δ|D|1\n2) 0/parenrightBigg\n+O(λ3δ|D|0)+O(λMδ|D|−N),\nM:= 6(N+1).\nWe remark that in L(0), the size of the remainders is O(λ3δ), which is worse than the ones in L\n(which is of order O(λδ)), cf. (1.16). This is due to the presence of the term Op/parenleftBig\nid(x)·ξ/parenleftbig\nψ1−ψ2/parenrightbig/parenrightBig\nwhich is of order O(λ3δ|D|0), see (4.42), which follows from the fact that Op( ψ1−ψ2) is of order\n−1, thanks to a cancellation exploited in Lemma 4.2-( ii) (even ifψ1,ψ2are only of order −1\n2). In\norder to normalize the lower order terms, the procedure is qu ite similar to the one described above\nand the equations to be solved are similar to (1.22), (1.23). The only difference is that the size\nof the remainders to be normalized is O(λ3δ). In order to get uniform estimates w.r. to λof the\nmaps along the iteration, one uses that the cut-off function χλis supported on |ξ| ≥λ6δ. We refer\nto sub-section 4.2 for more details.\nInversion of the first equation and reduction to the second component.\nAfter the previous decoupling procedure, we can invert the fi rst equation (namely the operator\nL(1)\n1in (1.20)) by implementing a Neumann series argument. This i s done in subsection 5.1. The\noperator L(1)\n1is of the form\nL(1)\n1:=Lλ+R(1)\n1,\nLλ:=λω·∇−∆,R(1)\n1=O(λδM|D|).(1.24)\nIn order to invert this operator by Neumann series we need tha tL−1\nλR(1)\n1is of order zero and small\nin size. Clearly\nLλ= diagk/\\e}atio\\slash=0iλω·k+|k|2,LARGE AMPLITUDE TRAVELING WAVES MHD 9\nand hence |iλω·k+|k|2| ≥ |k|2for anyk∈Z2\\{0}, meaning that L−1\nλ∼(−∆)−1gains two space\nderivatives. Thisfact thenbalance thefactthat R(1)\n1is unboundedoforder one. Ontheother hand,\nthis argument is not enough to use Neumann series, since in th is wayL−1\nλR(1)\n1=O(λδM|D|−1),\nwhose size is still large w.r. to λ≫1. Hence, in order to overcome this problem we require a gain\nof only one derivative in the estimate of L−1\nλand we gain also in size, in such a way to compensate\nO(λδM). This is done in Lemma 5.1. More precisely, for some constan tp>0 (depending on the\nconstantτappearing in the small divisors (1.17)), One shows that\nL−1\nλ=O(λ−p|D|−1),\nby analyzing separately the regimes |k| ≥λpand|k| ≤λp. For large frequencies one uses that\n|k|2≥λp|k|and for low frequencies one uses the diophantine condition o nω. We remark that the\ndissipation provided by the laplacian −∆ is used in a crucial way to analyze the high frequency\nregime|k| ≥λp. One then obtains that\nL−1\nλR(1)\n1=O(λ−p+δM|D|0),\nwhich is small and bounded by choosing 0 < δ≪p\nMandλ≫1 large enough. The precise,\nquantitative Neumann series argument is performed in Lemma 5.2 and one shows that ( L(1)\n1)−1=\nO(λ−p|D|−1). The inversion of L(1)\n1then reduces the invertibility of L1in (1.19), (1.20) to the\ninvertibility of a transport type operator with large varia ble coefficients of the form\nP:=λω·∇+a(x)·∇+R0,R0=O(λδM|D|0), (1.25)\nsee (5.21)-(5.23) and Lemma 5.3 in sub-section 5.2.\nInversion of the transport operator P.\nThe inversion of the transport operator Pis done in section 5.2. This is an adaptation of the\nmethods developed in [2], [27] (for small amplitudewaves). We implement a normal form procedure\ntransforming PtoP1where the operator P1is a smoothing perturbation of size λδMof a diagonal\noperator, namely it has the form\nP1=D1+R1,\nD1:= diagk/\\e}atio\\slash=0µ(k), µ(k) =iλω·k+O(λδM), k∈Z2\\{0},\nR1=O(λδM|D|−N).(1.26)\nThis is proved in Propositions 6.1, 6.2 and 6.3. Also in this s teps, it is essential that the size\nof the frequency is of order O(λ). This allows to estimate uniformly in λthe size of the normal\nform transformations that we need in order to transform PinP1. Then in sub-section 6.3, we\ninvertP1(and then P) by a Neumann series argument. By imposing diophantine cond itions on\nthe eigenvalues of D1\n|µ(k)| ≥λγ\n|k|τ, k∈Z2\\{0},\nand by choosing N∼τ, one gets that\nD−1\n1R1=O(λδM−1|D|0),\nwhich is small for δ≪1\nMandλ≫1 large enough. This last step then conclude the whole\ninvertibility procedure of the linearized operator Lwhich is summarized in Section 7. We point\nout that by an appropriate choice of the parameters which ent ers in the procedure, L−1has size\n/ba∇dblL−1/ba∇dblB(Hs+¯σ,Hs)=O(λ−δM) (for some large constant ¯ σ≫1).\nThe nonlinear Nash-Moser scheme. We conclude this explanation of the scheme of the proof\nwith some comments on the nonlinear Nash-Moser scheme imple mented on the map F. The first\nproblem is already in the first approximation of the solution . Indeed (Ω ,J) = (0,0) is not a good\napproximation, since F(0,0) =O(λ1+η−δ) =O(λ1−) is not uniformly bounded w.r. to λ. Hence in\nsection 8, we construct a non trivial approximate solution ( Ωapp,0), with Ω app/\\e}atio\\slash= 0 satisfying the\nproperty that F(Ωapp,0) satisfies estimates which are uniform w.r. to the large par ameterλ. This\nis enough in order to implement a rapidly convergent Nash-Mo ser scheme in Section 9. A crucial10 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\npoint is actually the gain of size O(λ−δM) in the estimate of L−1, described above. This gain is\nactually used in order to obtain that the difference between tw o consecutive approximate solutions\nconverges to zero super exponentially and it is uniformly bo unded w.r. to λ≫1. We point out\nthat the average of the magnetic field b/\\e}atio\\slash= 0 in (1.4) and the forcing term Fhave to satisfy the\nnon-degeneracy condition (1.7) in order to guarantee that t he solutions (Ω ,J) that we produce are\ndifferent from zero, see Proposition 8.1 and Section 11. Indee d, to show that Ω /\\e}atio\\slash= 0, one provides\na lower bound on Ω app(in Proposition 8.1) and then by a perturbative argument one also shows\nthat Ω satisfies the same lower bound since the difference Ω −Ωappis small. The proof that J/\\e}atio\\slash= 0\nis more delicate and it is done by contraddiction. The key poi nt is to shows that b· ∇Ω is not\nidentically zero. This is true (by the assumption (1.7)) for the approximate solution Ω appand it is\nproved also for Ω by a perturbative argument.\nRemarks and future perspectives. We conclude this introduction with some final remarks\nand future perspectives. By using the techniques developed in this paper (with some techinal\nadaptations), one can also prove the same kind of result for t he inviscid, resistive MHD system,\nnamely there is not the viscous term ∆ uin the first equations of (1.1) whereas, there is the resistiv e\nterm ∆bin the second equations of (1.1). As we described above, the p resence of the dissipation\nin one of the two equations is used in a crucial way in the analy sis of the linearized equation\nand in particular it allows to (approximately) decouple the system in a parabolic equation and a\ntrasnport equation. The decoupling procedure for the non re sistive inviscid case (no ∆ uin the\nfirst equation, no ∆ bin the second equation) seems to be quite difficult and require s new ideas.\nA natural extension of our result would be to analyze the gene ral case of quasi periodic solutions\nwith an arbitrary number of frequencies ω= (ω1,...,ω ν). In this case the solutions one look for\nare not stationary in a moving frame, hence space and time are really independent unlike the case\nof diamond waves where the time derivative becomes a space de rivative. A major issue in the\nextension of our strategy seems to be in the reduction proced ure to the scalar transport equation,\nin which one has to invert the dissipative operator ∂t−∆ on the space of time quasi-periodic\nfunctionsu(ωt,x),u:Tν×T2→R. In this inversion then one loses the “structure of dynamica l\nsystem” and this seems to be a non-trivial technical problem in implementing KAM-Normal forms\nmethods for reducing the linearized equation to constant co efficients. Other future perspectives are\nthe analysis of the three dimensional case and the analysis o f the inviscid case.\nOutline of the paper. The paper is organized as follows. In Section 2 we recall the f unctional\nsetting and some general lemmata which will then be used cons istently throughout the paper. In\nSection 3, we compute the linearized operator Lof the vorticity-current system (1.10) and we state\nsome properties of it that will be used in the sequel. We then i mplement the decoupling procedure\nofL(up to smoothing remainders) in Section 4. Section 5 will be d evoted to the inversion of\nthe first equation of the decoupled linear operator, whereas in Section 6 we invert the transport\noperator Pand in Section 7 we conclude the invertibility of the lineari zed operator L. In Section 8,\nwe construct an approximate solution that will be the starti ng point of the Nash-Moser iteration.\nThe convergence of the Nash-Moser scheme and the measure est imates on the set of non-resonant\nfrequencies will be shown in Section 9 and 10 respectively. F inally, the proof of the main theorems\nis provided in Section 11.\nAcknowledgements. G. Ciampa, R. Montalto and S. Terracina are supported by the E RC\nSTARTING GRANT 2021 “Hamiltonian Dynamics, Normal Forms an d Water Waves” (HamDy-\nWWa), ProjectNumber: 101039762. Viewsandopinionsexpres sedarehoweverthoseoftheauthors\nonly and do not necessarily reflect those of the European Unio n or the European Research Council.\nNeither the European Union nor the granting authority can be held responsible for them.\nThe work of the author Riccardo Montalto is also supported by PRIN 2022 “Turbulent effects vs\nStability in Equations from Oceanography” (TESEO), projec t number: 2022HSSYPN.\nRiccardo Montalto and Shulamit Terracina are also supporte d by INDAM-GNFM and Gennaro\nCiampa is supported by INDAM-GNAMPA.LARGE AMPLITUDE TRAVELING WAVES MHD 11\nThe authors warmly thank Luca Franzoi, Renato Luc` a and Mich ela Procesi for many useful dis-\ncussions and comments.\n2.Functions spaces, norms and linear operators\nIn this section we fix the notation and we state some techincal tools concerning function spaces\nand pseudo-differential operators. We consider\ns0>5 that can be arbitrarily large but fixed for the whole paper (2 .1)\nNotations. In the whole paper, the notation A/lessorsimilars,m,αBmeans that A≤C(s,m,α)Bfor some\nconstantC(s,m,α)>0 depending on the Sobolev index s, the constants α,m. Ifs=s0we simply\nwrite/lessorsimilar,/lessorsimilarm,αinstead of/lessorsimilars0,/lessorsimilars0,m,α. We always omit to write the dependence on τ, which is the\nconstant appearing in the non-resonance conditions (see fo r instance (1.17) in the introduction).\nWe denote by Nthe set of the positive integer numbers N={1,2...}andN0:=N∪{0}. Givenf:\nΩ→C, Ω⊆Rdand a multi-index β= (β1,...,β d)∈Nd\n0, we write∂β\nxf=∂β1x1...∂βdxdf(x1,...,x d).\nWe also define the lenght of a multi-index as |β|=β1+...+βd.\nFor anys≥0 we define the scale of Sobolev spaces\nHs(T2,Kn) :=/braceleftBig\nu(x) =/summationdisplay\nk∈Z2/hatwideu(k)eik·x∈L2(T2,Kn) :/ba∇dblu/ba∇dbls:=/parenleftBig/summationdisplay\nk∈Z2/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2s|/hatwideu(k)|2/parenrightBig1\n2<∞/bracerightBig\n,\nHs\n0(T2,Kn) :=/braceleftBig\nu∈Hs(T2,Kn) :/integraldisplay\nT2u(x)dx= 0/bracerightBig\n.(2.2)\nwhereK=C,R. As a notation, we often write Hs≡Hs(T2)≡Hs(T2,K) andHs\n0≡Hs\n0(T2)≡\nHs\n0(T2,K).\nFors>s0one hasHs(T2)⊂ C0(T2) whereC0(T2) is the set of continuous functions on T2, and\nHs(T2) is an algebra and they have the interpolation structure\n/ba∇dbluv/ba∇dbls≤C(s)/ba∇dblu/ba∇dbls/ba∇dblv/ba∇dbls0+C(s0)/ba∇dblu/ba∇dbls0/ba∇dblv/ba∇dbls, (2.3)\nwhereC(s),C(s0) are positive constants independent from u,v.\nWe also denote by L2\n0(T2,Kn), the space of L2functions from T2toKnwith zero average.\nGiven two Banach spaces X1,X2, we denote by B(X1,X2) the space of bounded, linear operators\nfromX1toX2equipped by the standard operator norm /ba∇dbl · /ba∇dblB(X1,X2). IfX1=X2we write\nB(X1)≡ B(X1,X2).\nDefinition 2.1 (Weighted norms) .LetO⊂R2be a bounded subset of R2and let(X,/ba∇dbl·/ba∇dblX)be a\nBanach space. Given γ∈(0,1)and a Lipschitz function u:O→X, we define\n/ba∇dblu/ba∇dblsup\nX:= sup\nω∈O/ba∇dblu(ω)/ba∇dblX,/ba∇dblu/ba∇dblLip\nX:= sup\nω1,ω2∈O\nω1/\\e}atio\\slash=ω2/ba∇dblu(ω1)−u(ω2)/ba∇dblX\n|ω1−ω2|,\n/ba∇dblu/ba∇dblLip(γ)\nX:=/ba∇dblu/ba∇dblsup\nX+γ/ba∇dblu/ba∇dblLip\nX.\nIfX=Hs(T2), we write /ba∇dbl·/ba∇dblsup\ns,/ba∇dbl·/ba∇dblLip\nsinstead of /ba∇dbl·/ba∇dblsup\nHs,/ba∇dbl·/ba∇dblLip\nHsand\n/ba∇dblu/ba∇dblLip(γ)\ns:=/ba∇dblu/ba∇dblsup\ns+γ/ba∇dblu/ba∇dblLip\ns−1. (2.4)\nIfX=R,C, we write |·|sup,|·|Lip,|·|Lip(γ)instead of /ba∇dbl·/ba∇dblsup\nX,/ba∇dbl·/ba∇dblLip\nX,/ba∇dbl·/ba∇dblLip(γ)\nX.\nFor anyN∈Nwe define the smoothing operators (Fourier truncations)\nΠNu(x) :=/summationdisplay\n|k|≤N/hatwideu(k)eik·x,Π⊥\nN= Id−ΠN. (2.5)\nLemma 2.2 (Smoothing estimates) .The operators ΠN,Π⊥\nNsatisfy the smoothing estimates\n/ba∇dblΠNu/ba∇dblLip(γ)\ns≤Na/ba∇dblu/ba∇dblLip(γ)\ns−a,0≤a≤s, (2.6)\n/ba∇dblΠ⊥\nNu/ba∇dblLip(γ)\ns≤N−a/ba∇dblu/ba∇dblLip(γ)\ns+a,∀a≥0. (2.7)12 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nLemma 2.3. For alls≥s0,u,v∈Hs(T2)\n/ba∇dbluv/ba∇dblLip(γ)\ns≤C(s)/ba∇dblu/ba∇dblLip(γ)\ns/ba∇dblv/ba∇dblLip(γ)\ns0+C(s0)/ba∇dblu/ba∇dblLip(γ)\ns0/ba∇dblv/ba∇dblLip(γ)\ns. (2.8)\nMoreover for s1≤s≤s2one has the following interpolation inequality\n/ba∇dblu/ba∇dblLip(γ)\ns/lessorsimilar(/ba∇dblu/ba∇dblLip(γ)\ns1)λ(/ba∇dblu/ba∇dblLip(γ)\ns2)1−λ, λ=s2−s\ns2−s1,1−λ=s−s1\ns2−s1. (2.9)\nAs a consequence, one has that for any a0,b0≥0,p,q>0\n2.1.Diophantine equations. LetO⊂R2be a bounded domain of R2. Givenγ∈(0,1),τ >0,\nwe define the set of Diophantine vectors DC( γ,τ), where\nDC(γ,τ) :=/braceleftbigg\nω∈O:|ω·k| ≥γ\n|k|τ,∀k∈Z2\\{0}/bracerightbigg\n, (2.10)\nthe equation\nω·∇v=u, (2.11)\nwithusatisfying/integraltext\nT2u(x)dx= 0, has the periodic solution\nv(x) := (ω·∇)−1u(x) =/summationdisplay\nk∈Z2\\{0}/hatwideu(k)\niω·keix·k.\nThe following estimates hold (see [9, Lemma 2.2]).\nLemma 2.4. Lets≥s0andω∈DC(γ,τ). Then, for any u∈Hs+τ(T2)withˆu(0) = 0, the linear\nequation (2.11)has a unique solution v:= (ω·∇)−1u∈Hs\n0(T2)withˆv(0) = 0such that\n/ba∇dbl(ω·∇)−1u/ba∇dbls≤Cγ−1/ba∇dblu/ba∇dbls+τ, (2.12)\nIfu=uω∈Hs+2τ+1\n0(T2)is Lipschitz continuous in ω∈O⊆R2, then the solution v=vω∈Hs\n0(T2)\nis Lipschitz continuous in DC(γ,τ)and satisfies the estimate\n/ba∇dbl(ω·∇)−1u/ba∇dblLip(γ)\ns≤Cγ−1/ba∇dblu/ba∇dblLip(γ)\ns+2τ+1. (2.13)\n2.2.Linear operators.\nDefinition 2.5. LetAbe a linear operator acting on L2(T2). We say that an operator Ais real if\nit maps real valued functions into real valued functions.\nLemma 2.6. GivenA1,A2real linear operators, then A1◦A2andA−1\n1are real operators.\nWe represent a real linear operator Racting on (f,g)∈L2(T2,R2) by a matrix\nR/parenleftbiggf\ng/parenrightbigg\n=/parenleftbigg\nA1A2\nA3A4/parenrightbigg/parenleftbiggf\ng/parenrightbigg\n, (2.14)\nwhereA1,A2,A3,A4are real operators acting on the scalar components f,g∈L2(T2). The action\nof an operator A ∈ B(L2(T2)) on a function u∈L2(T2,R2) can be defined as\nAu(x) =/summationdisplay\nk,k′∈Z2ˆAk′\nkˆu(k′)eix·k. (2.15)\nSo, we can identify an operator Awith the matrix ( ˆAk′\nk)k,k′∈Z2.\nDefinition 2.7. LetAbe an operator as in (2.15). We define DAas the operator\nDA:= diagk∈Z2ˆAk\nk,(DA)k′\nk=/braceleftBiggˆAk′\nkk=k′,\n0otherwise.(2.16)\nIn particular, we say that the operator Ais diagonal if DA=A.LARGE AMPLITUDE TRAVELING WAVES MHD 13\n2.3.Pseudo-differential operators and norms.\nDefinition 2.8 (Pseudo-differential operators and symbols) .Letm∈R,s≥0andα∈N0. We\nsay that a function a:T2×R2→Cbelongs to Sm\ns,αif there exists a constant C(s,α)>0such that\nfor anyβ∈N2,|β| ≤αsatisfies the inequality\n/ba∇dbl∂β\nξa(·,ξ)/ba∇dbls≤C(s,α)/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htm−|β|,∀ξ∈R2. (2.17)\nWe say that a linear operator Abelongs to the class OPSm\ns,αif there isa∈ Sm\ns,αsuch that A= Op(a),\nnamely\nAu(x) :=/summationdisplay\nk∈Z2a(x,k)/hatwideu(k)eix·k, u(x) =/summationdisplay\nk∈Z2/hatwideu(k)eix·k.\nWe set\nSm:=/intersectiondisplay\ns≥0\nα∈NSm\ns,α,OPSm:=/intersectiondisplay\ns≥0\nα∈NOPSm\ns,α,\nwhich are the classes of classical symbols and classical pse udo-differential operators of order m. We\nalso denote by S−∞:=∩m∈RSmthe class of smoothing symbols and OPS−∞:=∩m∈ROPSmthe\nclass of smoothing pseudo-differential operators.\nFor a matrix of pseudo-differential operators\nA=/parenleftbigg\nA1A2\nA3A4/parenrightbigg\n, (2.18)\nwe say that A ∈ OPSm\ns,αifAi∈ OPSm\ns,αandA ∈ OPSmifAi∈ OPSm,i= 1,2,3,4.\nWe will say that Ais a block diagonal matrix operator if A2=A3= 0. On the other hand, we\nwill say that that Ais a block off-diagonal matrix operator if A1=A4= 0. If instead an operator\nis diagonal in the sense of the Definition 2.7, it will be speci fied.\nThe following characterization for real pseudo-differentia l operators holds.\nLemma 2.9 (Lemma 2.10 [8]) .LetA= Op(a(x,ξ))be a pseudo-differential operator. Then Ais\nreal if and only if the symbol a(x,ξ) =a(x,−ξ)for any(x,ξ)∈T2×R2.\nWe now introduce a norm which controls the regularity in x, and the decay in ξ, of the symbol\na(x,ξ)∈ Sm\ns,α, together with its derivatives ∂β\nξa(x,ξ)∈ Sm−|β|\ns,α,|β| ≤αin the Sobolev norm /ba∇dbl·/ba∇dbls.\nDefinition 2.10 (Weighted norms) .Letm∈R,s≥0,α∈N0. IfA= Op(a)∈ OPSm\ns,α, we\ndefine the norm\n|A|m,s,α:= sup\n|β|≤αsup\nξ∈R2/ba∇dbl∂β\nξa(·,ξ)/ba∇dbls/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−m+|β|<∞; (2.19)\nIfA=A(ω)depends in a Lipschitz way on a parameter ω∈O⊂R2, we define\n|A|Lip(γ)\nm,s,α:= sup\n|β|≤αsup\nξ∈R2/ba∇dbl∂β\nξa(·,ξ)/ba∇dblLip(γ)\ns/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−m+|β|. (2.20)\nThe norm of a 2×2matrix of pseudo-differential operators in OPSm\ns,αof the form (2.18)is\n|A|Lip(γ)\nm,s,α:= max{|Ai|Lip(γ)\nm,s,α,i= 1,2,3,4}. (2.21)\nThepseudo-differential norm |·|Lip(γ)\nm,s,αsatisfies thefollowing elementary properties: for any s≤s′,\nα≤α′, andm≤m′,\n|·|Lip(γ)\nm,s,α≤ |·|Lip(γ)\nm,s′,α,|·|Lip(γ)\nm,s,α≤ |·|Lip(γ)\nm,s,α′,|·|Lip(γ)\nm′,s,α≤ |·|Lip(γ)\nm,s,α. (2.22)\nFor a Fourier multiplier Op( g(ξ)) of order m, one has\n|Op(g)|Lip(γ)\nm,s,α=|Op(g)|Lip(γ)\nm,0,α≤C(m,α),∀s≥0, (2.23)\nand for a function a(x;ω),\n|Op(a)|Lip(γ)\n0,s,α=|Op(a)|Lip(γ)\n0,s,0/lessorsimilar/ba∇dbla/ba∇dblLip(γ)\ns. (2.24)14 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nComposition. IfA= Op(a)∈ OPSmandB= Op(b)∈ OPSm′, then the composition\nAB:=A◦Bis a pseudo-differential operator with symbol σAB∈ Sm+m′\nσAB(x,ξ) :=/summationdisplay\nk∈Z2a(x,ξ+k)/hatwideb(k,ξ)eix·k=/summationdisplay\nk,k′∈Z2ˆa(k′−k,ξ+k)/hatwideb(k,ξ)eix·k′,(2.25)\nwhere the symbol ˆ·denotes the Fourier coefficient with respect to x. We will also use the notation\na#btoexpressthesymbolofthecomposition. Moreover, forany N≥0,σABadmitstheasymptotic\nexpansion\nσAB(x,ξ) :=/summationdisplay\n|β|≤N−11\ni|β|β!∂β\nξa(x,ξ)∂β\nxb(x,ξ)+rN(x,ξ), (2.26)\nwith remainder rN∈ Sm+m′−N. The remainder rNhas the explicit formula\nrN(x,ξ) :=1\niN(N−1)!/integraldisplay1\n0(1−τ)N−1/summationdisplay\n|β|=N/summationdisplay\nk∈Z2(∂β\nξa)(x,ξ+τk)/hatwidest∂β\nxb(k,ξ)eix·kdτ.(2.27)\nWe recall the following tame estimate for the composition of two pseudo-differential operators.\nLemma 2.11 (Composition of pseudo-differential operators) .Lets≥s0,m,m′∈R,α∈N0.\n(i) LetA= Op(a)∈ OPSm\ns,α,B= Op(b)∈ OPSm′\ns,α. Then the product σab(x,ξ) :=a(x,ξ)b(x,ξ)\nsatisfies for any s≥s0,α∈N0the estimate\n|Op(σab)|Lip(γ)\nm+m′,s,α/lessorsimilars,α|A|Lip(γ)\nm,s,α|B|Lip(γ)\nm′,s0,α+|A|Lip(γ)\nm,s0,α|B|Lip(γ)\nm′,s,α. (2.28)\n(ii) Lets≥s0,α∈N0,A= Op(a)∈ OPSm\ns,α,B= Op(b)∈ OPSm′\ns+|m|+α,α. Then, the\ncomposition operator ABbelongs to OPSm+m′\ns,αand it satisfies the estimate\n|AB|Lip(γ)\nm+m′,s,α/lessorsimilars,m,α|A|Lip(γ)\nm,s,α|B|Lip(γ)\nm′,s0+|m|+α,α+|A|Lip(γ)\nm,s0,α|B|Lip(γ)\nm′,s+|m|+α,α.(2.29)\nMoreover if A= Op(a)∈ OPSm\ns,N,B= Op(b)∈ OPSm′\ns+2N+|m|,0then the remainder\nRN= Op(rN)∈ OPSm+m′−N\ns,0satisfies the estimates\n|RN|Lip(γ)\nm+m′−N,s,0/lessorsimilars,N|A|Lip(γ)\nm,s,N|B|Lip(γ)\nm′,s0+2N+|m|,0+|A|Lip(γ)\nm,s0,N|B|Lip(γ)\nm′,s+|m|,0(2.30)\n(iii) The estimates (2.28)-(2.30)hold verbatim for matrices of pseudo-diffrential operators A ∈\nOPSm\ns,αandB ∈ OPSm′\ns,α.\n(iv) Lets≥s0,α∈N0,A= Op(a)∈ OPSm\ns+|m|+|m′|+α+1,α+1,B= Op(b)∈ OPSm′\ns+|m|+|m′|+α+1,α+1.\nThen, the commutator [A,B] :=AB−BA ∈ OPSm+m′−1\ns,αand it satisfies the estimate\n|[A,B]|Lip(γ)\nm+m′−1,s,α/lessorsimilars,m,m′,α|A|Lip(γ)\nm,s+|m|+|m′|+α+1,α+1|B|Lip(γ)\nm′,s0+|m|+|m′|+α+1,α+1\n+|A|Lip(γ)\nm,s0+|m|+|m′|+α+1,α+1|B|Lip(γ)\nm′,s+|m|+|m′|+α+1,α+1.(2.31)\nProof.See Lemma 2.13 in [8]. /square\nLemma 2.12 (Exponential map) .Lets≥s0,α∈N0,A= Op(a(x,ξ;ω))∈ OPS0\ns+α,α, then\nΦ = exp( A)satisfies the estimate\n|Φ−Id|Lip(γ)\n0,s,α≤ |A|Lip(γ)\n0,s,αexp(C(s,α)|A|Lip(γ)\n0,s0+α,α). (2.32)\nThe same statement holds even for matrix valued symbol A=/parenleftbigg\nA1A2\nA3A4/parenrightbigg\n∈ OPS0\ns+α,α.\nProof.See Lemma 2.13 in [10]. /squareLARGE AMPLITUDE TRAVELING WAVES MHD 15\nWe also define, for 0 ≤n≤N−1,\na#nb:=/summationdisplay\n|β|=n1\nβ!i|β|(∂β\nξa)(∂β\nxb)∈ Sm+m′−n,\na#<Nb:=N−1/summationdisplay\nn=0a#nb∈ Sm+m′, a#≥Nb:=rN:=rN(A,B)∈ Sm+m′−N.(2.33)\nRemark 2.13. We remark that if Op(a),Op(b)are real operators, then Op(a#b),Op(a#n)and\nOp(a#<Nb)are so.\nLemma 2.14. Letm,m′∈R,s≥s0,α∈N0,N∈N,A= Op(a)∈ OPSm\ns,α+N,B= Op(b)∈\nOPSm′\ns+2N+|m|,α. Then the composition operator ABadmits the expansion\nAB= Op(ab)+Op(σN)+RN\nwhere\n|Op(σN)|Lip(γ)\nm+m′−1,s,α/lessorsimilars,α,m,m′|A|Lip(γ)\nm,s,α+N|B|Lip(γ)\nm′,s0+N,α+|A|Lip(γ)\nm,s0,α+N|B|Lip(γ)\nm′,s+N,α\nand the remainder RNsatisfies for any s≥s0, the estimate\n|RN|Lip(γ)\nm+m′−N,s,0/lessorsimilars,N|A|Lip(γ)\nm,s,N|B|Lip(γ)\nm′,s0+2N+|m|,0+|A|Lip(γ)\nm,s0,N|B|Lip(γ)\nm′,s+2N+|m|,0.\nThe same statement holds true also for matrices of pseudo-diff erential operators of the form (2.18)\nA= Op(a)∈ OPSm\ns,α+NandB= Op(b)∈ OPSm′\ns+2N+|m|,α.\nProof.By (2.25), (2.27), we define\nσN(x,ξ) :=/summationdisplay\n1≤|β|≤N−11\ni|β|β!∂β\nξa(x,ξ)∂β\nxb(x,ξ),RN:= Op(rN(x,ξ)).\nThe claimed bound on RNfollows directly by (2.30). Moreover by Lemma 2.11-( i), one obtains\nthat for any 1 ≤ |β| ≤N−1, one gets\n|Op(∂β\nξa∂β\nxb)|Lip(γ)\nm+m′−1,s,α≤ |Op(∂β\nξa∂β\nxb)|Lip(γ)\nm+m′−|β|,s,α\n/lessorsimilars,α|Op(∂β\nξa)|Lip(γ)\nm−|β|,s,α|Op(∂β\nxb)|Lip(γ)\nm′,s0,α\n+|Op(∂β\nξa)|Lip(γ)\nm−|β|,s0,α|Op(∂β\nxb)|Lip(γ)\nm′,s,α\n/lessorsimilars,α|Op(a)|Lip(γ)\nm,s,α+N|Op(b)|Lip(γ)\nm′,s0+N,α\n+|Op(a)|Lip(γ)\nm,s0,α+N|Op(b)|Lip(γ)\nm′,s+N,α.\nThelatterboundimpliesthentheestimateonOp( σN)andtheproofofthelemmaisthenconcluded.\n/square\nGiven two linear operators AandB, we define for any ℓ∈N, the linear operator Adℓ(A)Bas\nfollows\nAd(A)B:= [B,A],\nAdℓ+1(A)B:= [Adℓ(A)B,A], ℓ≥1.\nBy (2.26), (2.27), (2.33) one deduces the expansion for any N≥2\na⋆b:=σAB−σBA=−i{a,b}+/summationdisplay\n2≤|β|≤N−1(a#|β|b−b#|β|a)+rN (2.34)\nwhere\n{a,b}:=∇ξa·∇xb−∇xa·∇ξb,rN=rN(A,B)−rN(B,A). (2.35)\nBy construction,\nOp(a⋆b) = Ad(B)A. (2.36)\nBy iterating Lemma 2.14, one obtains the following lemma.16 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nLemma 2.15. Letm>0,m′∈R,ℓ,N∈N,α∈N0,ℓ≤N,s≥s0. Then there exists a constant\nσ≡σ(N,m,m′)>0large enough such that if A=/parenleftbigg\nA1A2\nA3A4/parenrightbigg\n∈ OPS−m\ns+σ,α+σ,B=/parenleftbigg\nB1B2\nB3B4/parenrightbigg\n∈\nOPSm′\ns+σ,α+σ, one has that\nAdℓ(A)B=Cℓ,N+Rℓ,N,\nwhere\n|Cℓ,N|Lip(γ)\nm′−ℓm,s,α/lessorsimilars,α,N|A|Lip(γ)\nm,s+σ,α+σ/parenleftBig\n|A|Lip(γ)\nm,s0+σ,α+σ/parenrightBigℓ−1\n|B|Lip(γ)\nm′,s0+σ,α+σ\n+/parenleftBig\n|A|Lip(γ)\nm,s0+σ,α+σ/parenrightBigℓ\n|B|Lip(γ)\nm′,s+σ,α+σ,\nand\n|Rℓ,N|Lip(γ)\nm′−mN,s,0/lessorsimilars,N|A|Lip(γ)\nm,s+σ,σ/parenleftBig\n|A|Lip(γ)\nm,s0+σ,σ/parenrightBigℓ−1\n|B|Lip(γ)\nm′,s0+σ,σ\n+/parenleftBig\n|A|Lip(γ)\nm,s0+σ,σ/parenrightBigℓ\n|B|Lip(γ)\nm′,s+σ,σ.\nIn particular AdN(A)Bsatisfies the estimate\n|AdN(A)B|Lip(γ)\nm′−mN,s,0/lessorsimilarm,m′,s,N|A|Lip(γ)\nm,s+σ,σ/parenleftBig\n|A|Lip(γ)\nm,s0+σ,σ/parenrightBigN−1\n|B|Lip(γ)\nm′,s0+σ,σ\n+/parenleftBig\n|A|Lip(γ)\nm,s0+σ,σ/parenrightBigN\n|B|Lip(γ)\nm′,s+σ,σ.(2.37)\nLemma 2.16 (Conjugation by an exponential map) .Letm>0,m′∈R,ℓ,N∈N,α∈N0,s≥s0.\nThen there exists a constant σ≡σ(N,m,m′)>0large enough such that if A=/parenleftbigg\nA1A2\nA3A4/parenrightbigg\n∈\nOPS−m\ns+σ,α+σ,B=/parenleftbigg\nB1B2\nB3B4/parenrightbigg\n∈ OPSm′\ns+σ,α+σ. If|A|Lip(γ)\n−m,s0+σ,α+σ≤Cfor some constant C >0,\nthen one has the following expansion\ne−ABeA=B+CN+RN,\nwhere for any s≥s0, α∈N0,\n|CN|Lip(γ)\nm′−m,s,α/lessorsimilars,α,N|A|Lip(γ)\n−m,s+σ,α+σ|B|Lip(γ)\nm′,s0+σ,α+σ+|A|Lip(γ)\n−m,s0+σ,α+σ|B|Lip(γ)\nm′,s+σ,α+σ,\nand\n|RN|Lip(γ)\nm′−Nm,s,0/lessorsimilars,α,N|A|Lip(γ)\n−m,s+σ,σ|B|Lip(γ)\nm′,s0+σ,σ+|A|Lip(γ)\n−m,s0+σ,σ|B|Lip(γ)\nm′,s+σ,σ.\nProof.By the Lie expansion, one has that\ne−ABeA=B+N−1/summationdisplay\nℓ=1Adℓ(A)B\nℓ!+QN,\nQN:=1\n(N−1)!/integraldisplay1\n0(1−τ)N−1e−τA◦AdN(A)B ◦eτAdτ.(2.38)\nThen by applying Lemma 2.15 in order to expand any termAdℓ(A)B\nℓ!, we then obtain there exists\nσ≡σ(N,m,m′)≫0 large enough such that if |A|Lip(γ)\nm,s0+σ,α+σ/lessorsimilar1, then\nAdℓ(A)B\nℓ!=Cℓ,N+Rℓ,N,\n|Cℓ,N|Lip(γ)\nm′−ℓm,s,α/lessorsimilars,α,N|A|Lip(γ)\nm,s+σ,α+σ|B|Lip(γ)\nm′,s0+σ,α+σ+|A|Lip(γ)\nm,s0+σ,α+σ|B|Lip(γ)\nm′,s+σ,α+σ,\n|Rℓ,N|Lip(γ)\nm′−mN,s,0/lessorsimilars,α,N|A|Lip(γ)\n−m,s+σ,σ|B|Lip(γ)\nm′,s0+σ,σ+|A|Lip(γ)\n−m,s0+σ,σ|B|Lip(γ)\nm′,s+σ,σ.\nMoreover by the estimate (2.12) (applied for α= 0), the estimate (2.37) and by the composition\nestimate (2.29) (still applied for α= 0), one obtains the bound for QNin (2.38)\n|QN|Lip(γ)\nm′−mN,s,0/lessorsimilars,α,N|A|Lip(γ)\n−m,s+σ,σ|B|Lip(γ)\nm′,s0+σ,σ+|A|Lip(γ)\n−m,s0+σ,σ|B|Lip(γ)\nm′,s+σ,σ.LARGE AMPLITUDE TRAVELING WAVES MHD 17\nThe claimed statement then follows by defining\nCN:=N−1/summationdisplay\nℓ=1Cℓ,N,RN:=QN+N−1/summationdisplay\nℓ=1Rℓ,N.\n/square\nGiven an operator A= Op(a(x,ξ))∈ OPSm\ns,α, we define the averaged symbol /a\\}b∇acketle{ta/a\\}b∇acket∇i}htx(ξ) as\n/a\\}b∇acketle{ta/a\\}b∇acket∇i}htx(ξ) :=1\n(2π)d/integraldisplay\nTda(x,ξ)dx. (2.39)\nThe following elementary lemma holds.\nLemma 2.17. LetA= Op(a(x,ξ))∈ OPSm\ns,α,s≥s0,α∈N0. Then\n|Op(/a\\}b∇acketle{ta/a\\}b∇acket∇i}htx)|Lip(γ)\nm,s,α/lessorsimilar|A|Lip(γ)\nm,s0,α.\nMoreover, if Ais real, then one has that /a\\}b∇acketle{ta/a\\}b∇acket∇i}htx(ξ) =/a\\}b∇acketle{ta/a\\}b∇acket∇i}htx(−ξ).\nFollowing the argument used in Lemma 2 .21 of [8] one can prove the following result.\nLemma 2.18 (Action of a pseudo-differential operator ).LetN >0,A= Op(a)∈ S−N\ns,0,\ns≥s0. Then\n/ba∇dblAh/ba∇dblLip(γ)\ns+N/lessorsimilars,m|A|Lip(γ)\n−N,s0,0/ba∇dblh/ba∇dblLip(γ)\ns+|A|Lip(γ)\n−N,s,0/ba∇dblh/ba∇dblLip(γ)\ns0.\nWe now collect some properties of composition operators. We consider a diffeomorphism of the\n2-dimensional torus defined by\ny=x+α(x)⇐⇒x=y+ˇα(y), (2.40)\nwhereαis a small real valued smooth vector function, and the induce d operators\n(Au)(x) :=u(x+α(x)),(A−1u)(y) :=u(y+ˇα(y)). (2.41)\nLemma 2.19. Lets≥s0,α(·;ω)∈Hs,ω∈O⊂R2,/ba∇dblα/ba∇dblLip(γ)\ns0≤δ(s0)small enough. Then, the\ncomposition operator Asatisfies the following tame estimates\n/ba∇dblAu/ba∇dblLip(γ)\ns/lessorsimilars/ba∇dblu/ba∇dblLip(γ)\ns+/ba∇dblα/ba∇dblLip(γ)\ns/ba∇dblu/ba∇dblLip(γ)\ns0, (2.42)\nand the function ˇαdefined by the inverse diffeomorphism satisfies\n/ba∇dblˇα/ba∇dblLip(γ)\ns/lessorsimilars/ba∇dblα/ba∇dblLip(γ)\ns, (2.43)\nand as a consequence\n/ba∇dblA−1u/ba∇dblLip(γ)\ns/lessorsimilars/ba∇dblu/ba∇dblLip(γ)\ns+/ba∇dblα/ba∇dblLip(γ)\ns/ba∇dblu/ba∇dblLip(γ)\ns0. (2.44)\nProof.It follows by the same argument of Lemma 2.30 in [8]. /square\nSince the linearized operator has invariance properties on the space of zero average function we\nintroduce the projections Π 0and Π⊥\n0as follows\nΠ0h:=1\n(2π)2/integraldisplay\nT2h(x)dx,Π⊥\n0:= Id−Π0. (2.45)\nNote that given an even cut-off function η0such that\nη0∈ C∞(R2,R), η0is even,\nη0(ξ) = 1,∀|ξ| ≤1\n2, η0(ξ) = 0∀|ξ| ≥2\n3,(2.46)\none has that\nΠ0= Op(η0(ξ))∈ OPS−∞,\n|Π0|−m,s,α/lessorsimilarm,s,α1,∀m,s≥0, α∈N0,\n|Π⊥\n0|0,s,α/lessorsimilars,α1,∀s≥0, α∈N0.(2.47)\nFinally, one has that Π 0is real since η0is even inξand therefore |Π⊥\n0is so. We state the following\nlemma, see Lemma 4.2 in [27]18 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nLemma 2.20. LetAbe the map in (2.41)and let us define A⊥:= Π⊥\n0AΠ⊥\n0. Then under the same\nassumptions of Lemma 2.19, one has that A⊥:Hs\n0→Hs\n0is invertible and A−1\n⊥= Π⊥\n0A−1Π⊥\n0:\nHs\n0→Hs\n0and ifu(·;ω)∈Hs\n0, then\n/ba∇dblA±1\n⊥u/ba∇dblLip(γ)\ns/lessorsimilars/ba∇dblu/ba∇dblLip(γ)\ns+/ba∇dblα/ba∇dblLip(γ)\ns/ba∇dblu/ba∇dblLip(γ)\ns0.\nWe also prove the following lemma that we shall use in the sequ el.\nLemma 2.21. LetN >0,m∈R,s≥s0. Then there is σ≫0large enough such that if A=\nOp(a)∈ OPSm\ns+σ,0then the operator A⊥:= Π⊥\n0AΠ⊥\n0(recall(2.45)-(2.47)) satisfies A⊥=A+R⊥\nwhereR⊥satisfies the estimate |R⊥|Lip(γ)\n−N,s,0/lessorsimilarN,s|A|Lip(γ)\nm,s+σ,0. Finally, if Ais real, then the operators\nA⊥andR⊥are real.\nProof.One has\nA⊥=A+R⊥,R⊥:=−Π0AΠ⊥\n0−AΠ0,\nhencetheclaimedboundon R⊥followsby (2.47)andthecompositionestimate(2.29). Theo perator\nA⊥is real by composition with Π⊥\n0andR⊥by difference. /square\nWe now mention some elementary properties of the Laplacian o perator−∆ and of its inverse\n(−∆)−1acting on functions with zero average in x:\n−∆u(x) =/summationdisplay\nk/\\e}atio\\slash=0|k|2/hatwideu(k)eix·k,(−∆)−1u(x) =/summationdisplay\nk/\\e}atio\\slash=01\n|k|2/hatwideu(k)eix·k. (2.48)\nBy the properties of the cut off function η0we can identify ( −∆)−1with Op/parenleftBig1−η0(ξ)\n|ξ|2/parenrightBig\nsince the\naction of these two operators on functions with zero average is the same. By recalling Definition\n2.10 one easily checks that\n|−∆|2,s,α/lessorsimilarα1,|(−∆)−1|−2,s,α/lessorsimilarα1. (2.49)\nIn what follows, we denote by U:Hs(T2)→Hs+1(T2) the Biot-Savart operator, which is defined\non zero average functions as follows\nUf(x) :=∇⊥(−∆)−1f(x) =/parenleftbigg\n∂x2(−∆)−1f\n−∂x1(−∆)−1f/parenrightbigg\n. (2.50)\nOne asily verifies that U ∈ OPS−1and satisfies\n|U|−1,s,α/lessorsimilarα1,∀s≥0, α∈N0. (2.51)\nFinally, given f∈ C∞(T2) we define the operator R(f) acting on h∈Hs(T2) as follows\nR(f)h= [∇⊥(−∆)−1h·∇]f=Uh·∇f, (2.52)\nwhich is a pseudo-differential operator of order −1 satisfying\n|R(f)|−1,s,α/lessorsimilarα/ba∇dblf/ba∇dbls+1,∀s≥0, α∈N0. (2.53)\n2.4.A quantitative Egorov Theorem. We also prove a quantitative version of the Egorov\ntheorem. The proof follows word by word the arguments in [4] ( See also Theorem 3 .4 of [25] for\nthe main idea). All the following sums over k1,k2,k3∈N0such thatk1+k2+k3=sare actually\nsums overk1,k2,k3∈N0such thatk1+k2+k3=s,k1+k2≥1.\nTheorem 2.22. (Egorov Theorem). Fixm∈R,M >max{0,−m}. There exists σ:=\nσ(m,M)≫0large enough such that for any S≥s0+σand for any α≥0there exists\nε=ε(S,m,M)≪1small enough such that, if (recall the definition of Ain(2.41))\n/ba∇dblα/ba∇dblLip(γ)\ns0+σ≤ε, (2.54)\nthen given a symbol w(x,ξ)∈ Sm\nS,α+σ, Lipschitz in the variable ω∈O,then the following hold.\nA−1Op(w(x,ξ))A= Op/parenleftBig\nq(x,ξ)/parenrightBig\n+R (2.55)LARGE AMPLITUDE TRAVELING WAVES MHD 19\nwhere, for any s∈[s0,S−σ]and for any α≥0,q∈ Sm\ns,αand satisfies the estimates\n|Op(q)|Lip(γ)\nm,s,α/lessorsimilarm,M,s,α|Op(w)|Lip(γ)\nm,s,α+σ+/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,α+k2+σ/ba∇dblα/ba∇dblLip(γ)\nk3+σ,(2.56)\n|∆12Op(q)|m,s,α/lessorsimilarm,M,s,α|Op(w)|m,s+1,α+σ/ba∇dbl∆12α/ba∇dbls+1+|∆12Op(w)|m,s,α+σ\n+/summationdisplay\nk1+k2+k3=s+1|Op(w)|m,k1,α+k2+σ/ba∇dblα/ba∇dblk3+σ1/ba∇dbl∆12α/ba∇dbls0+1\n+/summationdisplay\nk1+k2+k3=s|∆12Op(w)|m,k1,α+k2+σ/ba∇dblα/ba∇dblk3+σ.(2.57)\nFurthermore the remainder R ∈ B(Hs,Hs+M), for anys∈[s0,S−σ], satisfies the estimate\n/ba∇dblRh/ba∇dblLip(γ)\ns+M/lessorsimilarm,s,MM(s)/ba∇dblh/ba∇dblLip(γ)\ns0+M(s0)/ba∇dblh/ba∇dblLip(γ)\ns (2.58)\nwhere\nM(s) :=/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+σ/ba∇dblα/ba∇dblLip(γ)\nk3+σ\nFinally, if Op(w(x,ξ))is a real operator, then Op(q(x,ξ))andRare a real operators.\nIn the proof we use the following lemma proved in the Appendix of [25, Lemma A.7]. Given a\nsquare matrix A, we denote by ATthe transpose matrix and if Ais invertible we denote by A−T\nthe inverse of the transpose matrix.\nLemma 2.23. There exists σ≫0such that for any S≥s0+σ, ifα∈CS(T2,R2)is a real\nfunction satisfying /ba∇dblα/ba∇dblLip(γ)\ns0+σ<1then, for any symbol w∈ Sm\nS,α,\nAw:=w/parenleftBig\nx+α(x),(Id+∇xα(x))−Tξ/parenrightBig\n(2.59)\nis a symbol in Sm\nS,αsatisfying, for any s∈[s0,S−σ]and for any α≥0,\n|Op(Aw)|Lip(γ)\nm,s,α≤ |Op(w)|Lip(γ)\nm,s,α+C/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,α+k2/ba∇dblα/ba∇dblLip(γ)\nk3+σ,(2.60)\nfor someC=C(s,α)>0. Fors=s0we have the rougher estimate |Op(Aw)|Lip(γ)\nm,s0,α/lessorsimilar|Op(w)|Lip(γ)\nm,s0,α+s0.\nProof of Thm. 2.22. Let us consider for τ∈[0,1] the composition operators\nAτh(x) :=h(x+τα(x)),(Aτ)−1h(y) :=h(y+˘α(τ;y)), (2.61)\nFor anyτ∈[0,1], the conjugated operator\nPτ:=Aτ◦Op(w)◦(Aτ)−1\nsolves the Heisenberg equation\n∂τPτ= [Xτ,Pτ], P0= Op(w), (2.62)\nwhere1\nXτ:=b(τ;x)·∇x= Op(χ), χ:=χ(τ;x,ξ) :=χ(τ;x,ξ) :=ib(τ;x)·ξ,(2.64)\nand\nb(τ;x) = (I+τ∇xα)−1α. (2.65)\nNotice that\n|Op(χ)|Lip(γ)\n1,s,α/lessorsimilars/ba∇dblb/ba∇dblLip(γ)\ns/lessorsimilars/ba∇dblα/ba∇dblLip(γ)\ns+1,∀p≥0. (2.66)\n1By using the definition of Aτin (2.61) and (2.64)-(2.65) one can easily check that\n∂τAτ=XτAτ,A0= Id. (2.63)20 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nFor simplicity we omit the dependence of the symbols on the va riablesω∈O⊂R2. We look for\nan approximate solution of (2.62) of the form2\nQτ:= Op(q(τ;x,ξ)), q=q(τ;x,ξ) =m+M−1/summationdisplay\nk=0qm−k(τ;x,ξ), (2.67)\nwhereqm−kare symbols in Sm−kto be determined iteratively so that\n∂τQτ= [Xτ,Qτ]+Mτ, Q0= Op(w), (2.68)\nwhereMτ= Op(r−M(τ;x,ξ)) with symbol r−M∈ S−M. Passing to the symbols we obtain (recall\n(2.25), (2.67) and (2.34))\n/braceleftBigg\n∂τq(τ;x,ξ) =χ(τ;x,ξ)⋆q(τ;x,ξ)+r−M(τ;x,ξ)\nq(0;x,ξ) =w(x,ξ).(2.69)\nwhere the unknowns are now q(τ;x,ξ),r−M(τ;x,ξ).\nWe expand χ(τ;x,ξ)⋆q(τ;x,ξ) into a sum of symbols with decreasing orders. Using that χis\nlinear inξ∈R2(see (2.64)), (2.26), (2.27), and by the expansion (2.67) (t ogether with the ansatz\nthatqm−k∈Sm−k) we note that\nχ⋆q=χ#q−q#χ=χq+1\ni∇ξχ·∇xq−q#χ\n(2.67)=χq+m+M−1/summationdisplay\nk=01\ni∇ξχ·∇xqm−k−/parenleftBigm+M−1/summationdisplay\nk=0qm−k(τ;x,ξ)/parenrightBig\n#χ\n(2.33)=m+M−1/summationdisplay\nk=01\ni{χ,qm−k}−m+M−1/summationdisplay\nk=0m−k+M/summationdisplay\nn=2qm−k#nχ−m+M−1/summationdisplay\nk=0qm−k#≥m−k+M+1χ.\nBy rearranging the sums we can write\nχ⋆q=−i{χ,qm}/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nordm−ordersfrom −M+1 tom−1/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright\nm+M−1/summationdisplay\nk=1/parenleftbig\n−i{χ,qm−k}+rm−k/parenrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nordm−k−r−M/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nord−M(2.70)\nwhere we defined, denoting w=w(k,h) :=k−h+1,\nrm−k:=−k−1/summationdisplay\nh=0qm−h#wχ(2.33)\n∈ S(m−h)+1−(k−h+1)\ns,α ≡ Sm−k\ns,α, (2.71)\nr−M:=m+M−1/summationdisplay\nk=0qm−k#≥M+m−k+1χ(2.33)\n∈ Sm−k+1−(m−k+1+M)\ns,α ≡ S−M\ns,α. (2.72)\nWe have reduced the problem to finding symbols qm−k∈ Sm−k\ns,α, 0≤k≤m+M−1, which solve\nfork= 0 (recall the form of χin (2.64))\n/braceleftBigg\n∂τqm(τ;x,ξ) ={b(τ;x)ξ,qm(τ;x,ξ)}\nqm(0;x,ξ) =w(x,ξ),(2.73)\nwhile for 1 ≤k≤m+M−1/braceleftBigg\n∂τqm−k(τ;x,ξ) ={b(τ;x)ξ,qm−k(τ;x,ξ)}+rm−k(τ;x,ξ)\nqm−k(0;x,ξ) = 0.(2.74)\nWe note that the symbols rm−kin (2.71) with 1 ≤k≤m+M−1 depend only on qm−hwith\n0≤h<k. This means that equations (2.74) can be solved iteratively .\n2We suppose that m+M≥2 is an integer, otherwise we can replace m+Mwith [m+M]+1.LARGE AMPLITUDE TRAVELING WAVES MHD 21\nOrderm.To solve (2.73), we consider the solutions of the Hamiltonia n system\n\n\nd\ndsx(s) =−b(s;x(s))\nd\ndsξ(s) = (∇xb(s;x(s)))Tξ(s)(x(0),ξ(0)) = (x0,ξ0)∈T2×R2. (2.75)\nOne can note that if qmis a solution of (2.73), then it is constant when evaluated al ong the flow\nof (2.75). In other words setting g(τ) :=qm(τ;x(τ),ξ(τ)) one has thatd\ndτg(τ) = 0 implies that\ng(τ) =g(0), for any τ∈[0,1]. Let us denote by γτ0,τ(x,ξ) the solution of the characteristic system\n(2.75) with initial condition γτ0,τ0= (x,ξ)3. By a standard ODE argument the flow is regular w.r.t.\nxandξ. Then the equation (2.73) has the solution\nqm(τ;x,ξ) =w(γτ,0(x,ξ)) (2.76)\nwhereγτ,0(x,ξ) has the explicit form (recall (2.65))\nγτ,0(x,ξ) =/parenleftbig\nf(τ;x),g(τ;x)ξ/parenrightbig\n, f(τ;x) :=x+τα(x), g(τ;x) := (I+τ∇xα(x))−T.(2.77)\nIndeed\n∂\n∂τ(x+τα(x(τ))) =∂\n∂τx(τ)+α(x(τ))+τ∇xα(x(τ))∂\n∂τx(τ)\n=α(x(τ))−(I+τ∇xα(x(τ)))b(x(τ))\n=α(x(τ))−(I+τ∇xα(x(τ)))(I+τ∇xα(x(τ)))−1α(x(τ)) = 0\nFor the second one, let us start looking at ∇xb\n∂xk[b(τ,x)] =∂xk[(I+τ∇xα(x))−1α(x)]\n=−τ(I+τ∇xα(x))−1(∂xk∇xα)(I+τ∇xα(x))−1α+(I+τ∇xα(x))−1∂xkα\n=−τ(I+τ∇xα(x))−1(∂xk∇xα)[b]+(I+τ∇xα(x))−1∂xkα\nAnd therefore\n(I+τ∇xα(x))∂xkb(τ,x) =∂xkα−τ(∂xk∇xα)[b] (2.78)\n˙ξk(τ)(2.75)=2/summationdisplay\nj=1∂xkbj(x(τ))ξj(τ)(2.77)=2/summationdisplay\nj=1∂xkbj(x(τ))[(I+τ∇xα)Tξ]j\n=∂xkb(x(τ))·(I+τ(∇xα)T)ξ= (I+τ(∇xα)T)∂xkb(x(τ))·ξ\n(2.78)=∂xkα(x(τ))·ξ−τ(∂xk∇xα[b])·ξ=∂xkα(x(τ))·ξ−τ2/summationdisplay\nj,p=1((∂xk∂xpαj)bp)ξj\n=∂xkα(x(τ))·ξ−τ2/summationdisplay\nj=1/parenleftBig2/summationdisplay\np=1∂xk(∇xα)p\njbp/parenrightBig\nξj=∂xkα(x(τ))·ξ−τ∂xk/parenleftbig\n(∇xαb)·ξ/parenrightbig\n(2.79)\nand therefore\n˙ξ(τ) = (∇xα)T[ξ]−τ∇x/parenleftbig\n(∇xαb)·ξ/parenrightbig\n(2.80)\nBy the other hand, if ξ(τ) = (I+τ∇xα(x(τ)))T[ξ] then\nξk(τ) =ξk+τ2/summationdisplay\nj=1∂xkαj(x(τ))ξj (2.81)\nand therefore\n∂τξk(τ) =2/summationdisplay\nj=1∂xkαj(x(τ))ξj+τ2/summationdisplay\nj,p=1((∂xk∂xpαj)∂τxp(τ))ξj (2.82)\nthat, ifξ(τ) solves (2.75), it is equal to (2.79).\nHence by Lemma 2.23 the symbol qm(τ;x,ξ) in (2.76) satisfies (2.56).\n3In other words ( x(τ),ξ(τ)) =γ0,τ(x0,ξ0) and the inverse flow is given by γτ,0(x,ξ) = (x0,ξ0).22 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nOrderm−kwith1≤k≤m+M−1.We now assume inductively that we have already found\nthe appropriate solutions qm−h∈ Sm−hof (2.74) with 0 ≤h < k, for some k≥1. We assume\nmoreover that\n|Op(qm−h)|Lip(γ)\nm−h,s,α/lessorsimilarm,s,α,M/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+σh+α/ba∇dblα/ba∇dblLip(γ)\nk3+σh,1≤h<k, (2.83)\nfor some non decreasing sequence of parameters σhdepending only on |m|andM. Notice that in\n(2.83) the Sobolev norm of αdoes not contain the parameter α. This is due to the fact that χis\nlinear inξ. We now construct the solution qm−kof (2.74) satisfying estimate (2.83) with h=k.\nUsing Lemma 2.14 we deduce (recall also that χis linear in ξandw:=k−h+1)\n|Op(rm−k)|Lip(γ)\nm−k,s,α(2.71)\n/lessorsimilark−1/summationdisplay\nh=0/summationdisplay\n|β|=w1\nh!|(∇β\nξqm−h)(∇β\nxχ)|Lip(γ)\nm−k,s,α\n(2.66),(2.28)\n/lessorsimilarm,s,α|Op(qm)|Lip(γ)\nm,s,α+k+1/ba∇dblα/ba∇dblLip(γ)\ns0+k+2+|Op(qm)|Lip(γ)\nm,s0,α+k+1/ba∇dblα/ba∇dblLip(γ)\ns+k+2\n+k−1/summationdisplay\nh=1|Op(qm−h)|Lip(γ)\nm−h,s,α+w/ba∇dblα/ba∇dblLip(γ)\ns0+1+w+|Op(qm−h)|Lip(γ)\nm−h,s0,α+w/ba∇dblα/ba∇dblLip(γ)\ns+1+w.\n(2.84)\nLet us consider first the second summand, which is the most com plicated: by the inductive as-\nsumption (2.83) with 1 ≤h≤k−1, we deduce\nk−1/summationdisplay\nh=1|Op(qm−h)|Lip(γ)\nm−h,s,α+w/ba∇dblα/ba∇dblLip(γ)\ns0+1+w+|Op(qm−h)|Lip(γ)\nm−h,s0,α+w/ba∇dblα/ba∇dblLip(γ)\ns+1+w\n(2.83)\n/lessorsimilarm,s,α,Mk−1/summationdisplay\nh=0/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+σh+p+w/ba∇dblα/ba∇dblLip(γ)\nk3+σh+w/ba∇dblα/ba∇dblLip(γ)\ns0+1+w\n+k−1/summationdisplay\nh=0/summationdisplay\nk1+k2+k3=s0|Op(w)|Lip(γ)\nm,k1,k2+σh+α+w/ba∇dblα/ba∇dblLip(γ)\nk3+σh+w/ba∇dblα/ba∇dblLip(γ)\ns+1+w\n/lessorsimilarm,s,α,Mk−1/summationdisplay\nh=0/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+σh+α+w/ba∇dblα/ba∇dblLip(γ)\nk3+σh+w\n+k−1/summationdisplay\nh=0/summationdisplay\n1≤k1+k2≤s0|Op(w)|Lip(γ)\nm,k1,k2+σh+α+w/ba∇dblα/ba∇dblLip(γ)\ns−s0+1+w+s0\n/lessorsimilarm,s,α,M∗/summationdisplay\ns|Op(w)|Lip(γ)\nm,k1,k2+/hatwideσk−1+α/ba∇dblα/ba∇dblLip(γ)\nk3+/hatwideσk−1.\nInthefourthandfifthlinewehaveusedthesmallnessassumpt ion(2.54)(assuming σ≥σk−1+1+k),\nand in the last line we have set ˆ σk−1:=σk−1+k+2+s0. In dealing with the first summand (i.e.\nthe third line in (2.71)) we proceed in the same way, only we su bstitute (2.56) instead of (2.83).\nWe conclude that\n|Op(rm−k)|Lip(γ)\nm−k,s,α/lessorsimilarm,s,α,M/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+/hatwideσk−1+α/ba∇dblα/ba∇dblLip(γ)\nk3+/hatwideσk−1.(2.85)\nWe now reason similarly to qm, and solve (2.74) by variation of constants. We define fm−k(τ) :=\nqm−k(τ;x(τ),ξ(τ)) wherex(τ),ξ(τ) are the solution of the Hamiltonian system (2.75). One has\nthat, ifqm−ksolves (2.74), then\nd\ndτfm−k(τ) =rm−k(τ;x(τ),ξ(τ))⇒fm−k(τ) =/integraldisplayτ\n0rm−k(σ;x(σ),ξ(σ))dσ,LARGE AMPLITUDE TRAVELING WAVES MHD 23\nwhere we used that f(0) =qm−k(0,x(0),ξ(0)) = 0. Therefore the solution of (2.74) is\nqm−k(τ;x,ξ) =/integraldisplayτ\n0rm−k(γ0,σγτ,0(x,ξ))dσ.\nWe observe also that\nγ0,σγτ,0(x,ξ) = (˜f,˜gξ), σ,τ ∈[0,1],\nwith\n˜f(σ,τ,x) :=x+τα(x)+˘α(σ,x+τα(x)),˜g(σ,τ,x) := (∇xf(σ,τ,x))−T.\nThus if˜Ar:=r(˜f,˜gξ) we have (recall that τ∈[0,1])\n|Op(qm−k)|Lip(γ)\nm−k,s,α/lessorsimilars,α|Op(˜Arm−k)|Lip(γ)\nm−k,s,α,\n|Op(qm−k)|Lip(γ)\nm−k,s0,α/lessorsimilarα|Op(˜Arm−k)|Lip(γ)\nm−k,s0,α/lessorsimilar|Op(rm−k)|Lip(γ)\nm−k,s0,α+s0,\nand by Lemma 2.23 with α/squigglerightτα(x)+˘α(σ,x+τα(x))4\n|Op(qm−k)|Lip(γ)\nm−k,s,α/lessorsimilars,α|Op(rm−k)|Lip(γ)\nm−k,s,α+/summationdisplay\nk1+k2+k3=s|Op(rm−k)|Lip(γ)\nm−k,k1,α+k2/ba∇dblα/ba∇dblLip(γ)\nk3+/tildewideσ,(2.86)\nwhere/tildewideσdepends only on s0. It remains to prove that actually the symbol qm−ksatisfies the bound\n(2.83) with h=kand for some new σk≥σk−1depending only on |m|andM. By substituting\n(2.85) in the estimate (2.86) we get\n|Op(qm−k)|Lip(γ)\nm−k,s,α/lessorsimilarm,s,α,M/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+/hatwideσk−1+α/ba∇dblα/ba∇dblLip(γ)\nk3+/hatwideσk−1\n+/summationdisplay\nk1+k2+k3=s/parenleftBig/summationdisplay\nk′\n1+k′\n2+k′\n3=k1|Op(w)|Lip(γ)\nm,k′\n1,k′\n2+/hatwideσk−1+α+k2/ba∇dblα/ba∇dblLip(γ)\nk′\n3+/hatwideσk−1/parenrightBig\n/ba∇dblα/ba∇dblLip(γ)\nk3+/tildewideσ.\n(2.87)\nBy Lemma 2.3 we deduce that (if k′\n3/\\e}atio\\slash= 0, otherwise we do nothing)\n/ba∇dblα/ba∇dblLip(γ)\nk′\n3+/hatwideσk−1/ba∇dblα/ba∇dblLip(γ)\nk3+/tildewideσ/lessorsimilars/parenleftBig\n/ba∇dblα/ba∇dblLip(γ)\n/hatwideσk−1/parenrightBigk3\nk3+k′\n3/parenleftBig\n/ba∇dblα/ba∇dblLip(γ)\nk′\n3+k3+/hatwideσk−1/parenrightBigk′\n3\nk3+k′\n3/parenleftBig\n/ba∇dblα/ba∇dblLip(γ)\n/tildewideσ/parenrightBigk′\n3\nk3+k′\n3/parenleftBig\n/ba∇dblα/ba∇dblLip(γ)\nk′\n3+k3+/tildewideσ/parenrightBigk3\nk3+k′\n3\n/lessorsimilars/ba∇dblα/ba∇dblLip(γ)\n/hatwideσk−1+/tildewideσ/ba∇dblα/ba∇dblLip(γ)\nk′\n3+k3+/hatwideσk−1+/tildewideσ(2.54)\n/lessorsimilar/ba∇dblα/ba∇dblLip(γ)\nk′\n3+k3+/hatwideσk−1+/tildewideσ.\nTherefore the (2.87) becomes (by renaming the indexes in the sum)\n|Op(qm−k)|Lip(γ)\nm−k,s,α/lessorsimilarm,s,α,M/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+/hatwideσk−1+α/ba∇dblα/ba∇dblLip(γ)\nk3+/hatwideσk−1\n+/summationdisplay\nk1+k2+k3=s/summationdisplay\nk1+k2+k3=k1|Op(w)|Lip(γ)\nm,k′\n1,k′\n2+/hatwideσk−1+α+k2/ba∇dblα/ba∇dblLip(γ)\nk′\n3+k3+/tildewideσ+/hatwideσk−1\n/lessorsimilarm,s,α,M/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+σk+α/ba∇dblα/ba∇dblLip(γ)\nk3+σk\nfor someσk≥/hatwideσk−1+/tildewideσdepending only on |m|andM. This is the estimate (2.83) with h=k. To\nprove (2.57) we reason as a above, using that ∆ 12Op(q) = Op(∆ 12q) and Leibnitz rule.\n4Notice that the function t(τ,σ,x) :=τα(x)+˘α(σ,x+τα(x)) satisfies, using that y+˘αis the inverse diffeomor-\nphism of x+τα(recall for instance (2.43)) , the estimate /bardblt(τ,σ)/bardblLip(γ)\ns/lessorsimilars/bardblα/bardblLip(γ)\ns+s0uniformly in τ,σ∈[0,1].24 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nLet us now estimate the remainder term. Similarly to (2.84), we have\n|Op(r−M)|Lip(γ)\n−M,s,0(2.72)\n/lessorsimilarm+M−1/summationdisplay\nk=0|Op(rm−k+1+M(qm−k,χ))|Lip(γ)\n−M,s,0\n(2.30)\n/lessorsimilarm,s,Mm+M−1/summationdisplay\nk=0|Op(qm−k)|Lip(γ)\nm−k,s,|m|+1+M|χ|Lip(γ)\n1,s0+3|m|+3+3M,0\n+|Op(qm−k)|Lip(γ)\nm−k,s0,|m|+1+M|χ|Lip(γ)\n1,s+3|m|+3+3M,0\n(2.66)\n/lessorsimilarm,s,Mm+M−1/summationdisplay\nk=0|Op(qm−k)|Lip(γ)\nm−k,s,|m|+1+M/ba∇dblα/ba∇dblLip(γ)\ns0+3|m|+4+3M\n+|Op(qm−k)|Lip(γ)\nm−k,s0,|m|+1+M/ba∇dblα/ba∇dblLip(γ)\ns+3|m|+4+3M.(2.88)\nSubstituting the bound (2.83) we get\n|Op(r−M)|Lip(γ)\n−M,s,0/lessorsimilarm,s,Mm+M−1/summationdisplay\nk=0/parenleftbig/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+σk+|m|+1+M/ba∇dblα/ba∇dblLip(γ)\nk3+σk/parenrightbig\n/ba∇dblα/ba∇dblLip(γ)\ns0+3|m|+4+3M\n+m+M−1/summationdisplay\nk=0/parenleftbig/summationdisplay\nk1+k2+k3=s0|Op(w)|Lip(γ)\nm,k1,k2+σk+|m|+1+M/ba∇dblα/ba∇dblLip(γ)\nk3+σk/parenrightbig\n/ba∇dblα/ba∇dblLip(γ)\ns+3|m|+4+3M\n/lessorsimilarm,s,M/summationdisplay\nk1+k2+k3=s|Op(w)|Lip(γ)\nm,k1,k2+/hatwideσ/ba∇dblα/ba∇dblLip(γ)\nk3+/hatwideσ,\n(2.89)\nfor some/hatwideσ≥σm+M−1+ 3|m|+ 4 + 3M(recall that σkis a non-decreasing sequence in k) and\nprovided that (2.54) holds for some σ≥/hatwideσ. In the last inequality we also used the interpolation\nestimate in Lemma 2.3 following the reasoning used for (2.87 ). We now conclude the construction\nof the solution of (2.62). We set (recall (2.67))\nPτ=Qτ+Rτ, Qτ= Op(q)∈ OPSm.\nBy construction we have that (recall (2.67), (2.69))\n∂τQτ= [Xτ,Qτ]+Mτ,Mτ:= Op(r−M),\nso/braceleftBigg\n∂τRτ= [Xτ,Rτ]−Mτ\nR0= 0.(2.90)\nWe setVτ:= (Aτ)−1◦Rτ◦Aτand we note that V0= (A0)−1◦R0◦A0= Id◦R0◦Id =R0= 0.\nMoreover, by using (2.90), (2.63), we have\n∂τVτ=∂τ((Aτ)−1)◦Rτ◦Aτ+(Aτ)−1◦(∂τRτ)◦Aτ+(Aτ)−1◦Rτ◦(∂τAτ)\n(2.90),(2.63)=−(Aτ)−1◦Xτ◦Rτ◦Aτ\n+(Aτ)−1◦/parenleftbig\n[Xτ,Rτ]−Mτ/parenrightbig\n◦Aτ+(Aτ)−1◦Rτ◦Xτ◦Aτ\n=−(Aτ)−1◦Mτ◦Aτ.\nThereforeVτsolves the problem\n/braceleftBigg\n∂τVτ=−(Aτ)−1◦Mτ◦Aτ\nV0= 0,\nwhose solution is\nVτ=−/integraldisplayτ\n0(At)−1◦Mt◦Atdt.LARGE AMPLITUDE TRAVELING WAVES MHD 25\nWe then deduce that\nRτ=−/integraldisplayτ\n0Aτ◦(At)−1◦Mt◦At◦(Aτ)−1dt.\nBy using (2.8), Lemmata 2.19 and 2.18 and Lemma 2.3 we obtain t he bound (2.58).\nFinally, if Op( w(x,ξ)) is a real operator then by induction one can prove that Op( q(x,ξ)) is real.\nIndeed, Op( qm(τ;x,ξ)) = Op(w(γτ,0(x,ξ))) so it is real and Op( qm−k) is real by (2.4), (2.71) and\nLemma 2.13. Therefore, by difference also Ris a real operator. /square\n3.The linearized operator\nWe will show the existence of bi-periodic traveling waves fo r the MHD system, namely we will\nfind zeroes of the nonlinear map Fdefined in (1.14), i.e.\nF(Ω,J) :=/parenleftbiggλω·∇Ω−∆Ω−b·∇J+λδ[(U·∇)Ω−(B·∇)J]−λ1−2\n3δF\nλω·∇J−b·∇Ω+λδ[(U·∇)J−(B·∇)Ω−2H(Ω,J)]/parenrightbigg\n,\nU:=UΩ, B:=UJ(3.1)\nwhere we recall that Uis the Biot-Savart operator defined in (2.50) and His defined as follows\nH(Ω,J) =H(UΩ,UJ)\n=−∂x1UJ·∇∂x1(−∆)−1Ω−∂x2UJ·∇∂x2(−∆)−1Ω.(3.2)\nThe main point is the inversion of the linearized operator at any (Ω,J) where Ω,J∈C∞(T2), with\nzero average. We shall make the following ansatz that will be used systematically in all our proofs.\nI:= (Ω,J) satisfies /ba∇dblI/ba∇dblLip(γ)\ns0+σ≤C(s0,σ),\nfor some large constants σ, C(σ,s0)≫0.(3.3)\nThe linearized operator at (Ω ,J) is given by L:=DF(Ω,J)\nL(ˆΩ,ˆJ) :=/parenleftbigg(λω·∇−∆)ˆΩ+a(x)·∇ˆΩ+d(x)·∇ˆJ+R1(ˆΩ)+R2(ˆJ)\nλω·∇ˆJ+a(x)·∇ˆJ+d(x)·∇Ω+R3(ˆΩ)+R4(ˆJ)/parenrightbigg\n, (3.4)\nwhere the operators Riare defined as follows (recall the definition (2.52))\nR1[/hatwideΩ] =λδR(Ω)[/hatwideΩ] =λδ(UˆΩ·∇)Ω,\nR2[/hatwideJ] =−λδR(J)[/hatwideJ] =−λδ(UˆJ·∇)J,\nR3[/hatwideΩ] =λδ(UˆΩ·∇)J−2λδH(ˆΩ,J)\n=λδR(J)[/hatwideΩ]+2λδ∂x1UJ·∇∂x1(−∆)−1/hatwideΩ\n+2λδ∂x2UJ·∇∂x2(−∆)−1/hatwideΩ,\nR4[/hatwideJ] =λδ(UˆJ·∇)Ω−2λδH(Ω,ˆJ)\n=λδR(Ω)[/hatwideJ]+2λδ∂x1U/hatwideJ·∇∂x1(−∆)−1Ω\n+2λδ∂x2U/hatwideJ·∇∂x2(−∆)−1Ω,(3.5)\nand the vector fields a(x),d(x) are given by\na(x) :=λδ(UΩ)(x), (3.6)\nd(x) :=−b−λδ(UJ)(x). (3.7)\nBy using that UΩ,UJare zero divergence vector fields and have zero average, a dir ect calculation\nshows that a·∇,d·∇,Ri,i= 1,...,4 leave invariant the space of zero average functions, namel y\n[a·∇,Π⊥\n0] = [d·∇,Π⊥\n0] = 0,[Ri,Π⊥\n0] = 0, i= 1,...,4\nand hence a·∇= Π⊥\n0a·∇Π⊥\n0, d·∇= Π⊥\n0d·∇Π⊥\n0,\nRi= Π⊥\n0RiΠ⊥\n0, i= 1,...,4.\nThe latter properties imply that L= Π⊥\n0LΠ⊥\n0.(3.8)26 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nMoreover, since Uis real then all the operators in Lin (3.4) are real. With these notations, we can\nrewrite the operator Lin the following matrix formulation\nL=/parenleftbigg\nλω·∇−∆+a(x)·∇d(x)·∇\nd(x)·∇ λω·∇+a(x)·∇/parenrightbigg\n+/parenleftbigg\nR1R2\nR3R4/parenrightbigg\n. (3.9)\nThe following lemma holds.\nLemma 3.1. Assume (3.3)withσ= 1. Then for any s≥s0,α∈N0one has\n/ba∇dbla/ba∇dblLip(γ)\ns/lessorsimilarsλδ/ba∇dblI/ba∇dblLip(γ)\ns,/ba∇dbld/ba∇dblLip(γ)\ns/lessorsimilarsλδ(1+/ba∇dblI/ba∇dblLip(γ)\ns),\nR1,R2∈ OPS−1,R3,R4∈ OPS0,\n|R1|Lip(γ)\n−1,s,α,|R2|Lip(γ)\n−1,s,α,|R3|Lip(γ)\n0,s,α,|R4|Lip(γ)\n0,s,α/lessorsimilars,αλδ/ba∇dblI/ba∇dblLip(γ)\ns+1.(3.10)\nLets1≥s0,α∈N0and assume that I1,I2satisfy(3.3)withs1+1instead ofs0+1. Then\n/ba∇dbl∆12a/ba∇dbls1,/ba∇dbl∆12d/ba∇dbls1/lessorsimilars1λδ/ba∇dblI1−I2/ba∇dbls1,\n|∆12R1|−1,s1,α,|∆12R2|−1,s1,α,|∆12R3|0,s1,α,|∆12R4|0,s1,α/lessorsimilars1,αλδ/ba∇dblI1−I2/ba∇dbls1+1.(3.11)\nProof.The lemma follows by the definitions (3.5)-(3.7) and by apply ing Lemma 2.3, and the esti-\nmates (2.23), (2.24), (2.51), (2.53). /square\nWe now state the following lemma which summarize some elemen tary properties of the map F\nthat we shall use in the sequel.\nLemma 3.2. LetFbe the nonlinear map defined in (1.14). We have the following.\n(i)The map Fis invariant on the space of zero average functions. More pre cisely for any\ns≥s0,F:Hs+2\n0×Hs+2\n0→Hs\n0×Hs\n0.\n(ii)Lets≥s0,I(ω)∈ C∞(T2,R2)(with zero average) with /ba∇dblI/ba∇dblLip(γ)\ns0+1≤C0. Then the linearized\noperator L(I) :=DF(I)satisfies the following properties. One has that [L(I),ΠN] =\n[Π⊥\nN,L(I)]and for any a>0,\n/ba∇dbl[L(I),Π⊥\nN][/hatwideI]/ba∇dblLip(γ)\ns0/lessorsimilarsλδN1−a/parenleftBig\n/ba∇dbl/hatwideI/ba∇dblLip(γ)\ns0+a+/ba∇dblI/ba∇dblLip(γ)\ns0+a/ba∇dbl/hatwideI/ba∇dblLip(γ)\ns0+1/parenrightBig\n,\nand\n/ba∇dbl[L(I),ΠN][/hatwideI]/ba∇dblLip(γ)\ns0+a/lessorsimilarsλδN/parenleftBig\n/ba∇dbl/hatwideI/ba∇dblLip(γ)\ns0+a+/ba∇dblI/ba∇dblLip(γ)\ns0+a+1/ba∇dbl/hatwideI/ba∇dblLip(γ)\ns0+1/parenrightBig\n.\n(iii)Lets≥s0,I(ω),/hatwideI(ω)∈Hs+1\n0×Hs+1\n0and let us define\nQ(/hatwideI,/hatwideI) :=F(I+/hatwideI)−F(I)−L(I)[/hatwideI].\nThen, one has\n/ba∇dblQ(/hatwideI,/hatwideI)/ba∇dblLip(γ)\ns/lessorsimilarsλδ/ba∇dbl/hatwideI/ba∇dblLip(γ)\ns+1/ba∇dbl/hatwideI/ba∇dblLip(γ)\ns0+1.\nProof.The fact that if Ihas zero average, then F(I) has zero average follows by explicit intgration\nby parts, using that Ω ,J,U,Bhave zero average and U=UΩ,B=UJare zero divergence vector\nfields. All the claimed bounds follow by explicit calculatio ns, by applying interpolation estimates\n(2.8), the smoothing estimates of Lemma 2.2, The estimates o f Lemma 3.1 and using the trivial\nfact that Π Nand Π⊥\nNcommute with the diagonal operator/parenleftbigg\nλω·∇−∆ 0\n0λω·∇/parenrightbigg\n./square\n4.Decoupling of linearized operator up to a smoothing remaind er\nTheaimofthissectionistodecoupletheequationsfortheve locity andthemagneticfield. Wewill\nimplement an iterative procedure whose goal is to remove the operators in the off-diagonal entries\nofLup to an arbitrary regularizing term of order −N, for someN∈Nfixed. The trickiest part\ncomes from the decoupling of the order 1 which we present in Su bsection 4.1. Then, we illustrate\nthe iterative procedure in Subsection 4.2. We use the notati onFto denote block diagonal matrix\noperators, while we denote with Qblock off-diagonal matrix operators. Lastly, we denote with R\nthe remainders term.LARGE AMPLITUDE TRAVELING WAVES MHD 27\n4.1.Decoupling of the first order part. We look for a transformation which removes the\ntransport term d(x)·∇from the off-diagonal entries of Lup to a remainder of order1\n2. The main\ntechnical obstruction comes from the fact that d(x) =O(λδ),λ≫1 large enough.\nThen, we consider a transformation Ψ ⊥of the form\nΨ⊥:= Π⊥\n0ΨΠ⊥\n0,Ψ :=/parenleftbigg\n0 Ψ1\nΨ20/parenrightbigg\n, (4.1)\nwhere Ψ 1= Op(ψ1(x,ξ)),Ψ2= Op(ψ2(x,ξ)) whereψ1,ψ2∈ S−1\n2have to be determined in order\nto cancel the highest order off diagonal term/parenleftbigg\n0d(x)·∇\nd(x)·∇0/parenrightbigg\n. We split Las\nL=D+F1+Q1+R0,\nD:=/parenleftbigg\nλω·∇−∆ 0\n0λω·∇/parenrightbigg\n,\nF1:=/parenleftbigg\na·∇0\n0a·∇/parenrightbigg\n,\nQ1:=/parenleftbigg\n0d·∇\nd·∇0/parenrightbigg\n,\nR0:=/parenleftbigg\nR1R2\nR3R4/parenrightbigg\n.(4.2)\nNote that Dis a diagonal operator in the sense of Definition 2.7. We compu teL(0):=e−Ψ⊥LeΨ⊥:\nwe apply the Lie expansion and we get that\ne−Ψ⊥DeΨ⊥=D+[D,Ψ⊥]+1\n2[[D,Ψ⊥],Ψ⊥]+R(0)\n1,\ne−Ψ⊥(F1+Q1)eΨ⊥=F1+Q1+[F1,Ψ⊥]+[Q1,Ψ⊥]+R(1)\n2,\nR(0)\n1:=1\n2/integraldisplay1\n0(1−τ)2e−τΨ⊥[[[D,Ψ⊥],Ψ⊥],Ψ⊥]eτΨ⊥dτ,\nR(0)\n2:=1\n2/integraldisplay1\n0(1−τ)2e−τΨ⊥[[F1+Q1,Ψ⊥],Ψ⊥]eτΨ⊥dτ.(4.3)\nThus we have that\nL(0)=D+F1+[D,Ψ⊥]+Q1+1\n2[[D,Ψ⊥],Ψ⊥]+[F1,Ψ⊥]+[Q1,Ψ⊥]\n+R(0)\n1+R(0)\n2+e−Ψ⊥R0eΨ⊥.(4.4)\nWe shall construct Ψ in such a way that\n[D,Ψ⊥]+Q1= an operator of order1\n2.\nMoreover, by exploiting the explicit formulas for ψ1andψ2, we will show that there is a regular-\nization effect for which1\n2[[D,Ψ⊥],Ψ⊥],[Q1,Ψ⊥] are operators of order zero.\n4.1.1.Construction of Ψ.We define an even cut-off function χ∈C∞(R2) such that\n0≤χ≤1, χ(ξ) = 1,∀|ξ| ≥1,\nχ(ξ) = 0,∀|ξ| ≤1\n2,\nand defineχλas\nχλ(ξ) :=χ/parenleftBigξ\nλ6δ/parenrightBig\n. (4.5)\nWe need the following lemmas.28 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nLemma 4.1. Letm∈R,f∈ Smand let us consider the symbol fλ(x,ξ) :=/parenleftbig\n1−χλ(ξ)/parenrightbig\nf(x,ξ).\nThen one has that for any N∈N,fλ∈ S−Nand for any s≥0,α∈N0,\n|Op(fλ)|Lip(γ)\n−N,s,α/lessorsimilars,N,αλ6δ(m+N)|Op(f)|Lip(γ)\nm,s,α.\nProof.By the properties of the cut-off function χλone has that for any K∈N,β∈N2\n|/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htK∂β\nξ(1−χλ(ξ))|/lessorsimilarK,βλ6δ(K−|β|). (4.6)\nHence for any β∈N2,s≥s0, for anyξ∈R2, one has that\n/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htN+|β|/ba∇dbl∂β\nξfλ(·,ξ)/ba∇dbls/lessorsimilars,β/summationdisplay\nβ1+β2=β/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htN+|β||∂β1\nξ(1−χλ)(ξ)|/ba∇dbl∂β2\nξf(·,ξ)/ba∇dbls\n/lessorsimilars,β/summationdisplay\nβ1+β2=β/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htN+|β||∂β1\nξ(1−χλ)(ξ)|/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htm−|β2||Op(f)|m,s,|β2|\n/lessorsimilars,β|Op(f)|m,s,|β|/summationdisplay\nβ1+β2=β/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htN+|β1|+|β2|+m−|β2||∂β1\nξ(1−χλ(ξ))|\n/lessorsimilars,βλ6δ(m+N)|Op(f)|m,s,|β|,\nwhere in the last line we used the property (4.6) and that for β1/\\e}atio\\slash= 0 and the function ∂β1\nξ(1−\nχλ(ξ))/\\e}atio\\slash= 0 as long asλ6δ\n2≤ |ξ| ≤λ6δ. The Lipschitz estimate can be proved similarly. /square\nIn the next lemma we establish some properties of the solutio ns of the homological equations\nthat one has to solve in the reduction procedure of this secti on and the one of section 4.2.\nLemma 4.2. (i). Letω∈DC(γ,τ)anda1,a2∈ Sm. Then there exist ψ1,ψ2∈ Sm−3\n2such that\n/braceleftBigg/parenleftbig\nλω·∇+|ξ|2/parenrightbig\nψ1(x,ξ)+χλ(ξ)a1(x,ξ) = 0,/parenleftbig\nλω·∇−|ξ|2/parenrightbig\nψ2(x,ξ)+χλ(ξ)a2(x,ξ) = 0.(4.7)\nMoreover, for any s≥s0,α∈N0,i= 1,2, one gets that |Op(ψi)|Lip(γ)\nm−3\n2,s,α/lessorsimilars,αλ−3δ|Op(ai)|Lip(γ)\nm,s+τ+1,α.\nFinally, if Op(a1(x,ξ)),Op(a2(x,ξ))are real operators then Op(ψ1(x,ξ)),Op(ψ2(x,ξ))are real op-\nerators.\n(ii)Letω∈DC(γ,τ)and let us assume that a1(x,ξ) =a2(x,ξ) =id(x)·ξwhere the function\ndis defined in (3.7). Then, the two corresponding solutions ψ1,ψ2of(4.7)satisfy the following\nproperty:ψ1−ψ2∈ S−1and\n|Op(ψ1−ψ2)|Lip(γ)\n−1,s,α/lessorsimilars,αλδγ−1/parenleftbig\n1+/ba∇dblI/ba∇dblLip(γ)\ns+2τ+3/parenrightbig\n,∀s≥s0, α∈N0. (4.8)\nProof.Proof of (i).We prove the lemma for ψ1. The corresponding properties of ψ2can be\nproved in a similar way. To shorten notations we write ψ≡ψ1anda≡a1. We expand a(x,ξ) and\nψ(x,ξ) in Fourier series\na(x,ξ) =/summationdisplay\nk∈Z2/hatwidea(k,ξ)eix·k, ψ(x,ξ) =/summationdisplay\nk∈Z2/hatwideψ(k,ξ)eix·k.\nThen we need to determine /hatwideψ(k,ξ),k∈Z2,ξ∈R2in such a way that\n/parenleftBig\niλω·k+|ξ|2/parenrightBig\n/hatwideψ(k,ξ)+χλ(ξ)/hatwidea(k,ξ) = 0,\nand therefore we define\n/hatwideψ(k,ξ) =−χλ(ξ)/hatwidea(k,ξ)\niλω·k+|ξ|2,(k,ξ)∈Z2×R2. (4.9)\nNote that by the definition of the cut-off function χλ, one has that\nχλ(ξ)/\\e}atio\\slash= 0 =⇒ |ξ| ≥1\n2λ6δ,\n∂β\nξχλ(ξ)/\\e}atio\\slash= 0 =⇒ |ξ| ∼λ6δ, β∈N2,(4.10)LARGE AMPLITUDE TRAVELING WAVES MHD 29\nand\n|∂β\nξχλ(ξ)|/lessorsimilarλ−6δ|β|/lessorsimilarβ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−|β|,∀β∈N2,∀ξ∈R2. (4.11)\nA direct computation shows that (use that χand all its derivatives vanish near the origin) for any\nk∈Z2,λ6δ/lessorsimilar|ξ|, one has that\n/vextendsingle/vextendsingle/vextendsingle∂β\nξ/parenleftBig1\niλω·k+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarβ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−2−|β|/lessorsimilarβλ−3δ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β|,∀β∈N2. (4.12)\nHence, the estimates (4.11), (4.12) imply that\n/vextendsingle/vextendsingle/vextendsingle∂β\nξ/parenleftBigχλ(ξ)/hatwidea(k,ξ)\niλω·k+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarβλ−3δ/summationdisplay\nβ1+β2=β/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β1||∂β2\nξ/hatwidea(k,ξ)|,\nwhich implies that\n/ba∇dbl∂β\nξψ(·,ξ)/ba∇dbls/lessorsimilarβλ−3δ/summationdisplay\nβ1+β2=β/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β1|/ba∇dbl∂β2\nξa(·,ξ)/ba∇dbls\n/lessorsimilarβλ−3δ|Op(a)|m,s,|β|/summationdisplay\nβ1+β2=β/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β1|/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htm−|β2|\n/lessorsimilarβλ−3δ|Op(a)|m,s,|β|/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htm−3\n2−|β|.\nThe latter chain of inequalities then implies the bound\n|Op(ψ)|m−3\n2,s,α/lessorsimilarαλ−3δ|Op(a)|m,s,α,∀s≥s0, α∈N0. (4.13)\nNow we compute the Lipschitz norm w.r.t. the variable ω. Letω1,ω2∈DC(γ,τ), we have that\nψ(x,ξ;ω1)−ψ(x,ξ;ω2) =/summationdisplay\nk∈Z2χλ(ξ)/bracketleftbigg/hatwidea(k,ξ;ω2)\niλω2·k+|ξ|2−/hatwidea(k,ξ;ω1)\niλω1·k+|ξ|2/bracketrightbigg\neix·k\n= Φ1(x,ξ)+Φ2(x,ξ),\nΦ1(x,ξ) :=/summationdisplay\nk∈Z2χλ(ξ)\niλω2·k+|ξ|2/parenleftBig\n/hatwidea(k,ξ;ω2)−/hatwidea(k,ξ;ω1)/parenrightBig\neix·k,\nΦ2(x,ξ) :=/summationdisplay\nk∈Z2Γ(k,ξ)/hatwidea(k,ξ;ω1)eix·k,\nΓ(k,ξ) :=χλ(ξ)/bracketleftbiggiλ(ω1−ω2)·k\n(iλω2·k+|ξ|2)(iλω1·k+|ξ|2)/bracketrightbigg\n.(4.14)\nArguing as in (4.13), one gets that for any s≥s0,α∈N0,\nγ|Op(Φ1)|m−3\n2,s,α/lessorsimilars,αλ−3δγ|Op(a)|Lip\nm−3\n2,s,α|ω1−ω2|/lessorsimilars,βλ−3δ|Op(a)|Lip(γ)\nm−3\n2,s,α|ω1−ω2|.(4.15)\nWe then estimate Φ 2. Arguing by induction, one can show that since ω1∈DC(γ,τ), one gets that\nforλ6δ/lessorsimilar|ξ|\n/vextendsingle/vextendsingle/vextendsingle∂β\nξ/parenleftBig1\niλω1·k+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarβ|k|τ\nλγ|ξ||β|∀β∈N2. (4.16)\nTherefore, the estimates (4.11), (4.12), (4.16) imply that for anyξ∈R2,β∈N2,ω1,ω2∈DC(γ,τ),\n|∂β\nξΓ(k,ξ)|/lessorsimilarβλ|k||ω1−ω2|/summationdisplay\nβ1+β2+β3=β|∂β1\nξχλ(ξ)|/vextendsingle/vextendsingle/vextendsingle∂β2\nξ/parenleftBig1\niλω1·k+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂β3\nξ/parenleftBig1\niλω2·k+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilarβλ|k||ω1−ω2|/summationdisplay\nβ1+β2+β3=β/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−|β1|λ−3δ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β2||k|τλ−1γ−1/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−|β3|\n/lessorsimilarα|k|τ+1γ−1λ−3δ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β||ω1−ω2|.\n(4.17)30 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nHence, for any s≥s0,β∈N2, one obtains that\n/ba∇dbl∂β\nξΦ2(·,ξ)/ba∇dbl2\ns/lessorsimilarβ/summationdisplay\nβ1+β2=β/summationdisplay\nk∈Z2/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2s|∂β1\nξΓ(k,ξ)|2|∂β2\nξ/hatwidea(k,ξ;ω1)|2\n(4.17)\n/lessorsimilarβ/parenleftBig\nγ−1λ−3δ|ω1−ω2|/parenrightBig2/summationdisplay\nβ1+β2=β/parenleftbig\n/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β1|/parenrightbig2/summationdisplay\nk∈Z2/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2(s+τ+1)|∂β2\nξ/hatwidea(k,ξ;ω1)|2\n/lessorsimilarβ/parenleftBig\nγ−1λ−3δ|ω1−ω2|/parenrightBig2/summationdisplay\nβ1+β2=β/parenleftbig\n/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β1|/parenrightbig2/ba∇dbl∂β2\nξa(·,ξ;ω1)/ba∇dbl2\ns+τ+1\n/lessorsimilarβ/parenleftBig\nγ−1λ−3δ|ω1−ω2|/parenrightBig2/summationdisplay\nβ1+β2=β/parenleftbig\n/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−3\n2−|β1|/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htm−|β2|/parenrightbig2|Op(a)|2\nm,s+τ+1,|β2|\n/lessorsimilarβ/parenleftBig\nγ−1λ−3δ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htm−3\n2−|β||Op(a)|m,s+τ+1,|β||ω1−ω2|/parenrightBig2\n.\n(4.18)\nHence (4.13), (4.14), (4.15), (4.18) imply the claimed boun d\n|Op(ψ)|Lip(γ)\nm−3/2,s,α/lessorsimilars,αλ−3δ|Op(a)|Lip(γ)\nm,s+τ+1,α,∀s≥s0, α∈N0.\nBy (4.9) and by the fact that χλ(ξ) is even, one can see that Op( ψ1) is a real operator. For ψ2one\ncan reason as above.\nProof of (ii).By the proof of the item ( i), one gets that for any k∈Z2,ξ∈R2\nψ1(x,ξ) =/summationdisplay\nk∈Z2/hatwideψ1(k,ξ)eix·ξ, ψ2(x,ξ) =/summationdisplay\nk∈Z2/hatwideψ2(k,ξ)eix·ξ,\n/hatwideψ1(k,ξ) =−χλ(ξ)i/hatwided(k)·ξ\niλω·k+|ξ|2,/hatwideψ2(k,ξ) =−χλ(ξ)i/hatwided(k)·ξ\niλω·k−|ξ|2,\nand hence\np(x,ξ) :=ψ1(x,ξ)−ψ2(x,ξ) =/summationdisplay\nk∈Z2/hatwidep(k,ξ)eik·ξ,\n/hatwidep(k,ξ) :=/hatwideψ1(k,ξ)−/hatwideψ2(k,ξ) =−2i/hatwided(k)·ξχλ(ξ)|ξ|2\nλ2(ω·k)2+|ξ|4.\nA direct calculation shows that (recall (4.11)) for any β∈N2, one gets\n|∂β\nξ/hatwidep(k,ξ)|/lessorsimilarβ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−1−|β||/hatwided(k)|,\nimplying that for any s≥s0,α∈N0,\n|Op(p)|−1,s,α/lessorsimilars,α/ba∇dbld/ba∇dbls(3.10)\n/lessorsimilars,αλδ(1+/ba∇dblI/ba∇dbls+1). (4.19)\nand for any given ω1,ω2∈DC(γ,τ), one has that\n/hatwidep(k,ξ;ω1)−/hatwidep(k,ξ;ω2) =−2i/hatwided(k;ω1)·ξχλ(ξ)|ξ|2\nλ2(ω1·k)2+|ξ|4+2i/hatwided(k;ω2)·ξχλ(ξ)|ξ|2\nλ2(ω2·k)2+|ξ|4\n= Γ1(k,ξ)+Γ2(k,ξ),\nΓ1(k,ξ) :=−2i[/hatwided(k;ω1)−/hatwided(k;ω2)]·ξχλ(ξ)|ξ|2\nλ2(ω1·k)2+|ξ|4\nΓ2(k,ξ) := 2i/hatwided(k;ω2)·ξχλ(ξ)|ξ|2λ2/bracketleftbig\n(ω1−ω2)·k/bracketrightbig/bracketleftbig\n(ω1+ω2)·k/bracketrightbig\n(λ2(ω1·k)2+|ξ|4)(λ2(ω2·k)2+|ξ|4).\nLetω1,ω2∈DC(γ,τ),λ6δ/lessorsimilar|ξ|. Then, one can easily show by induction that\n/vextendsingle/vextendsingle/vextendsingle∂β\nξ/parenleftBig1\nλ2(ω1·k)2+|ξ|4/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarβ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−4−|β|,∀β∈N2,\n/vextendsingle/vextendsingle/vextendsingle∂β\nξ/parenleftBig1\nλ2(ω2·k)2+|ξ|4/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarβλ−2γ−2|k|2τ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−|β|,∀β∈N2.(4.20)LARGE AMPLITUDE TRAVELING WAVES MHD 31\nIn the first estimate above, one always estimate the denomina tors asλ2(ω1·k)2+|ξ|4≥ |ξ|4whereas\nin the second one, λ2(ω2·k)2+|ξ|4≥λ2(ω2·k)2≥λ2γ2|k|−2τfork∈Z2\\{0}, by using that ω2is\ndiophantine.\nBy using the estimate in (4.20) and the estimate (4.11) on χλ, one proves that\n|∂β\nξΓ1(k,ξ)|/lessorsimilarβ/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−1−|β||/hatwided(k;ω1)−/hatwided(k;ω2)|,∀β∈N2, (4.21)\nand\n|∂β\nξΓ2(k,ξ)|/lessorsimilarβ|k|2(τ+1)γ−2/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht−1−|β||/hatwided(k;ω2)||ω1−ω2|,∀β∈N2. (4.22)\nHence, by the estimates (4.21), (4.22) one deduces that for a nys≥s0,α∈N0\nγ/vextendsingle/vextendsingle/vextendsingleOp/parenleftBig\np(·;ω1)−p(·;ω2)/parenrightBig/vextendsingle/vextendsingle/vextendsingle\n−1,s,α/lessorsimilarαγ/ba∇dbld(·;ω1)−d(·;ω2)/ba∇dbls+γ−1/ba∇dbld/ba∇dblsup\ns+2τ+2|ω1−ω2|\n/lessorsimilars,αγ−1/ba∇dbld/ba∇dblLip(γ)\ns+2τ+2|ω1−ω2|\n(3.10)\n/lessorsimilars,αλδγ−1/parenleftbig\n1+/ba∇dblI/ba∇dblLip(γ)\ns+2τ+3/parenrightbig\n|ω1−ω2|.(4.23)\nThe claimed statement then follows by the bounds (4.19), (4. 23). /square\nWe are now in position to define the operator Ψ.\nLemma 4.3. Assume that (3.3)holds for some σ≫0large enough and let ω∈DC(γ,τ). Then\nthere exist two real operators Ψ1= Op(ψ1(x,ξ)),Ψ2= Op(ψ2(x,ξ))whereψ1,ψ2∈ S−1/2such that\nthe operator Ψ⊥defined as\nΨ⊥= Π⊥\n0ΨΠ⊥\n0,Ψ =/parenleftbigg\n0 Op( ψ1(x,ξ))\nOp(ψ2(x,ξ)) 0/parenrightbigg\n,\nsatisfies the equation\n[D,Ψ⊥]+Q1= Π⊥\n0/parenleftBig\nQΨ+RΨ/parenrightBig\nΠ⊥\n0, (4.24)\nwithΨ∈ OPS−1/2such that\n|Ψ|Lip(γ)\n−1/2,s,α/lessorsimilars1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, α∈N0,\nΨ⊥= Ψ+RΨ\n⊥,\n|RΨ\n⊥|Lip(γ)\n−N,s,0/lessorsimilars,N1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, N∈N,(4.25)\nand the matrix operators QΨ,RΨare off-diagonal, real, and satisfy the estimates\n|QΨ|Lip(γ)\n1/2,s,α/lessorsimilars1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, α∈N0,\n|RΨ|Lip(γ)\n−N,s,0/lessorsimilarN,sλ6δ(N+1)(1+/ba∇dblI/ba∇dblLip(γ)\ns+σ),∀s≥s0.(4.26)\nMoreover let s1≥s0,α∈N0and let us assume that I1,I2satisfy(3.3)withs1+σinstead of\ns0+σ. Then we also have that\n|∆12Ψ|−1/2,s1,α,|∆12QΨ|1/2,s1,α/lessorsimilar/ba∇dblI1−I2/ba∇dbls1+σ, (4.27)\nProof.First of all, we define ψ1andψ2to be, respectively, the solutions of the equation (4.7) wit h\na1anda2given by\na1(x,ξ) =a2(x,ξ) :=id(x)·ξ. (4.28)\nThus, since ai∈ S1and are their associated pseudo-differential operators are r eal, we can apply\nLemma 4.2 and Lemma 3.1 obtaining the existence of ψ1,ψ2∈ S−1/2such that Ψ 1,Ψ2are real and\nsatisfy for some σ≫0 large enough\n|Ψi|Lip(γ)\n−1/2,s,α/lessorsimilars,αλ−3δ/ba∇dbld/ba∇dblLip(γ)\ns+τ+1/lessorsimilars,α1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, α∈N0. (4.29)\nIn particular, it follows that\n|Ψ|Lip(γ)\n−1/2,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, α∈N0. (4.30)\nThe expansion of Ψ ⊥and the estimate of RΨ\n⊥then follows by applying Lemma 2.21.32 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nNow, we consider [ D,Ψ⊥]+Q1. Using that [ D,Π⊥\n0] = 0 and that Q1= Π⊥\n0Q1Π⊥\n0(recall (3.8)), one\nhas that\n[D,Ψ⊥]+Q1= Π⊥\n0/parenleftBig\n[D,Ψ]+Q1/parenrightBig\nΠ⊥\n0. (4.31)\nand by splitting id(x)·ξ=iχλ(ξ)d(x)·ξ+i(1−χλ(ξ))d(x)·ξ,i= 1,2 (recall (4.5)), and by using\nthat by a direct calculation\n−∆◦Op/parenleftBig\nψ1(x,ξ)/parenrightBig\n= Op/parenleftBig\nψ1(x,ξ)|ξ|2−2iξ·∇xψ1(x,ξ)−∆xψ1(x,ξ)/parenrightBig\n,\nOp(ψ2)◦(−∆) = Op/parenleftBig\nψ2(x,ξ)|ξ|2/parenrightBig\n,(4.32)\none obtains that\n[D,Ψ]+Q1\n=/parenleftbigg\n0 [ λω·∇,Op/parenleftbig\nψ1/parenrightbig\n]−∆◦Op/parenleftbig\nψ1/parenrightbig\n[λω·∇,Op/parenleftbig\nψ2/parenrightbig\n]+Op/parenleftbig\nψ2/parenrightbig\n◦∆ 0/parenrightbigg\n+/parenleftbigg\n0 Op/parenleftbig\niχλ(ξ)d(x)·ξ/parenrightbig\nOp/parenleftbig\niχλ(ξ)d(x)·ξ/parenrightbig\n0/parenrightbigg\n+/parenleftbigg\n0 Op/parenleftbig\ni/parenleftbig\n1−χλ(ξ)/parenrightbig\nd(x)·ξ/parenrightbig\nOp/parenleftbig\ni/parenleftbig\n1−χλ(ξ)/parenrightbig\nd(x)·ξ/parenrightbig\n0/parenrightbigg\n(4.32)=/parenleftbigg\n0 Op/parenleftbig/parenleftbig\nλω·∇+|ξ|2/parenrightbig\nψ1(x,ξ)+iχλ(ξ)d(x)·ξ/parenrightbig\nOp/parenleftbig/parenleftbig\nλω·∇−|ξ|2/parenrightbig\nψ2(x,ξ)+iχλ(ξ)d(x)·ξ/parenrightbig\n0/parenrightbigg\n+QΨ+RΨ,\nQΨ:=/parenleftBigg\n0 Op/parenleftBig\n−2iξ·∇xψ1(x,ξ)−∆xψ1(x,ξ)/parenrightBig\n0 0/parenrightBigg\n,\nRΨ:=\n0 Op/parenleftBig\ni/parenleftbig\n1−χλ(ξ)/parenrightbig\nd(x)·ξ/parenrightBig\nOp/parenleftBig\ni/parenleftbig\n1−χλ(ξ)/parenrightbig\nd(x)·ξ/parenrightBig\n0\n.\n(4.33)\nBy the explicit form in (4.33), by the fact that χλ(ξ) is even and by the reality of the symbol ψ1\nand the function d(x), we have that QΨandRΨare real matrix operators.\nBy applying the estimate (4.30), together with Lemma 4.1 and the estimates (3.10) on d(x), one\nobtains that\n|QΨ|1/2,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dbls+σ,∀s≥s0, α∈N0,\n|RΨ|−N,s,0/lessorsimilarN,sλ6δ(N+1)(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0,(4.34)\n(for some constant σ≫0 large enough). Moreover, we apply Lemma 4.2 to get that\n/parenleftbig\nλω·∇+|ξ|2/parenrightbig\nψ1(x,ξ)+iχλ(ξ)d(x)·ξ= 0,\n/parenleftbig\nλω·∇−|ξ|2/parenrightbig\nψ2(x,ξ)+iχλ(ξ)d(x)·ξ= 0,\nand hence we have obtained that\n[D,Ψ]+Q1=QΨ+RΨ\nand this concludes the proof. The estimates (4.27) can be pro ved by similar arguments, by using\nalso the estimates (3.11). /square\nWe now analyze the conjugation of Lby means of the map Φ := exp(Ψ ⊥) (recall (4.1)). This is\nthe content of the following lemma.\nLemma 4.4 (Conjugation of L).LetN∈N,γ∈(0,1),τ >0,λ−δγ−1≤1,ω∈DC(γ,τ). Then\nthere exists σ≡σN≫0such that if (3.3)is fulfilled the following holds. There exists an invertible\nreal map Φsatisfying\nΦ±1:Hs\n0→Hs\n0,|Φ±1|Lip(γ)\n0,s,0/lessorsimilars1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, (4.35)LARGE AMPLITUDE TRAVELING WAVES MHD 33\nsuch that the operator L(0):= Φ−1LΦin(4.4)admits the expansion\nL(0)=D+F1+Π⊥\n0/parenleftBig\nQ(0)\n1/2+R(0)\n0+R(0)\n−N/parenrightBig\nΠ⊥\n0, (4.36)\nwhereF1=/parenleftbigg\na·∇0\n0a·∇/parenrightbigg\nis defined in (4.2),Q(0)\n1/2∈ OPS1/2is a block off-diagonal matrix-\noperator, R(0)\n0∈ OPS0andR(0)\n−Nis an operator of order −Nsatisfying\n|Q(0)\n1/2|Lip(γ)\n1/2,s,α,|R(0)\n0|Lip(γ)\n0,s,α/lessorsimilars,αλ3δ(1+/ba∇dblI/ba∇dblLip(γ)\ns+σ),∀s≥s0, α∈N0,\n|R(0)\n−N|Lip(γ)\n−N,s,0/lessorsimilars,Nλ6δ(N+1)(1+/ba∇dblI/ba∇dblLip(γ)\ns+σ),∀s≥s0.(4.37)\nMoreover let s1≥s0,α∈N0and letI1,I2satisfy(3.3)withs1+σinstead ofs0+σ. Then\n|∆12Φ±1|0,s1,0/lessorsimilar/ba∇dblI1−I2/ba∇dbls1+σ,\n|∆12Q(0)\n1/2|1/2,s1,α,|∆12R(0)\n0|0,s1,α/lessorsimilars1,αλ3δ/ba∇dblI1−I2/ba∇dbls1+σ.(4.38)\nFinally the operators L(0),F1,Q(0)\n1/2,R(0)\n0andR(0)\n−Nare real matrix operators.\nProof.To simplify the notations we write /ba∇dbl·/ba∇dblsinstead of /ba∇dbl·/ba∇dblLip(γ)\nsand|·|m,s,αinstead of |·|Lip(γ)\nm,s,α.\nFirst of all, by the estimates (4.25) and by Lemma 2.12, one im mediately gets the estimates (4.35)\non Φ := exp(Ψ ⊥). Moreover, since Ψ ⊥is real, then also Φ is so. Then, we analyze all the terms\nappearing in the expansions (4.3), (4.4) of L(0)= Φ−1LΦ.\nExpansion of the term1\n2[[D,Ψ⊥],Ψ⊥] + [F1,Ψ⊥] + [Q1,Ψ⊥].By applying Lemma 4.3, one\nobtains that\n1\n2[[D,Ψ⊥],Ψ⊥]+[Q1,Ψ⊥]+[F1,Ψ⊥] = [F1,Ψ⊥]+1\n2[Q1,Ψ⊥]\n+1\n2[Π⊥\n0QΨΠ⊥\n0,Ψ⊥]+1\n2[Π⊥\n0RΨΠ⊥\n0,Ψ⊥].\nBy the estimates (4.25), (4.26) and by applying the composit ion Lemmata 2.11 (to estimate\n[Π⊥\n0RΨΠ⊥\n0,Ψ⊥]), 2.14 (to expand [Π⊥\n0QΨΠ⊥\n0,Ψ⊥]) together with the ansatz (3.3), one easily ob-\ntains that\n1\n2[Π⊥\n0QΨΠ⊥\n0,Ψ⊥] = Π⊥\n0/parenleftbig\nF(1)\n1+R(1)\n−N/parenrightbig\nΠ⊥\n0whereF(1)\n1∈ OPS0has zero off-diagonal entries\n|F(1)\n1|0,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dbls+σ,∀s≥s0, α∈N0,\n|R(1)\n−N|−N,s,0/lessorsimilars,N1+/ba∇dblI/ba∇dbls+σ,∀s≥s0,\n|[Π⊥\n0RΨΠ⊥\n0,Ψ]|−N,s,0/lessorsimilars,Nλ6δ(N+1)(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0.\n(4.39)\nWe analyze in more details the terms [ Q1,Ψ⊥],[F1,Ψ⊥]. By the properties (3.8), [Π⊥\n0,Q1] = 0,\n[Π⊥\n0,F1] = 0 and therefore\n[F1,Ψ⊥] = Π⊥\n0[F1,Ψ]Π⊥\n0,[Q1,Ψ⊥] = Π⊥\n0[Q1,Ψ]Π⊥\n0. (4.40)\nMoreover, byanexplicitcalculation, onehasthat(recalli ngthatΨ 1= Op(ψ1(x,ξ)),Ψ2= Op(ψ2(x,ξ)))\n[Q1,Ψ] =/parenleftbigg\n0d·∇\nd·∇0/parenrightbigg/parenleftbigg\n0 Ψ1\nΨ20/parenrightbigg\n−/parenleftbigg\n0 Ψ1\nΨ20/parenrightbigg/parenleftbigg\n0d·∇\nd·∇0/parenrightbigg\n=/parenleftbigg\n(d·∇)◦Ψ2−Ψ1◦(d·∇) 0\n0 ( d·∇)◦Ψ1−Ψ2◦(d·∇)/parenrightbigg\n.34 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nThen by applying Lemma 2.14 and by using the estimates on m(x) and Ψ 1of Lemmata 3.1, 4.3,\none obtains the expansion\n[Q1,Ψ]\n=\nOp/parenleftBig\nid(x)·ξ/parenleftbig\nψ2(x,ξ)−ψ1(x,ξ)/parenrightbig/parenrightBig\n0\n0 Op/parenleftBig\nid(x)·ξ/parenleftbig\nψ1(x,ξ)−ψ2(x,ξ)/parenrightbig/parenrightBig\n\n+F(2)\n1+R(2)\n−N,\nF(2)\n1∈ OPS−1\n2,|F(2)\n1|−1\n2,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, α∈N0,\n|R(2)\n−N|−N,s,0/lessorsimilars,N1+/ba∇dblI/ba∇dbls+σ,∀s≥s0.(4.41)\nFurthermore, by using Lemmata 2.11-( i), 3.1, 4.2-( ii) one obtains that\nid·ξ/parenleftbig\nψ1−ψ2/parenrightbig\n∈ S0,/vextendsingle/vextendsingle/vextendsingleOp/parenleftBig\nid·ξ/parenleftbig\nψ1−ψ2/parenrightbig/parenrightBig/vextendsingle/vextendsingle/vextendsingle\n0,s,α/lessorsimilars,αλ2δγ−1(1+/ba∇dblI/ba∇dbls+σ)\nλ−δγ−1≤1\n/lessorsimilars,αλ3δ(1+/ba∇dblI/ba∇dbls+σ)∀s≥s0, α∈N0.(4.42)\nMoreover, by applying Lemma 2.14 and by using the estimates o na(x) and Ψ 1,Ψ2of Lemmata\n3.1, 4.3, one obtains the expansion\n[F1,Ψ] =/parenleftbigg\n0 [a·∇,Ψ1]\n[a·∇,Ψ2] 0/parenrightbigg\n=Q(3)\n1+R(3)\n−N,\nQ(3)\n1∈ OPS−1\n2,|Q(3)\n1|−1\n2,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dbls+σ,∀s≥s0, α∈N0,\n|R(3)\n−N|−N,s,0/lessorsimilars,N1+/ba∇dblI/ba∇dbls+σ,∀s≥s0.(4.43)\nHence, by summarizing (4.39), (4.40), (4.41), (4.43), (4.4 2), one gets the final expansion\n1\n2[[D,Ψ⊥],Ψ⊥]+[F1,Ψ⊥]+[Q1,Ψ⊥] = Π⊥\n0/parenleftbig\nS(4)\n1+R(4)\n−N/parenrightbig\nΠ⊥\n0,\nS(4)\n1∈ OPS0,|S(4)\n1|0,s,α/lessorsimilars,αλ3δ(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0, α∈N0,\n|R(4)\n−N|−N,s,0/lessorsimilars,Nλ6δ(N+1)(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0.(4.44)\nAnalysis of the term R(0)\n1+R(0)\n2+e−Ψ⊥R0eΨ⊥in(4.3),(4.4).By using Lemmata 2.14, 2.16,\n3.1, 4.3, the expansions (4.41), (4.43), (4.44), one obtain s that\nR(1)\n1+R(1)\n2+e−Ψ⊥R0eΨ⊥= Π⊥\n0/parenleftbig\nS(5)\n1+R(5)\n−N/parenrightbig\nΠ⊥\n0,\nS(5)\n1∈ OPS0,|S(5)\n1|0,s,α/lessorsimilars,αλ3δ(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0, α∈N0,\n|R(5)\nN|−N,s,0/lessorsimilars,Nλ6δ(N+1)(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0.(4.45)\nThe claimed expansion (4.36) and the claimed bounds (4.37) t hen follows by Lemma 4.3, the\nexpansions (4.44), (4.45) and by defining\nQ(0)\n1\n2:=QΨ,R(0)\n0:=S(4)\n1+S(5)\n1,R(0)\n−N:=RΨ+R(4)\n−N+R(5)\n−N.\nThe bounds (4.38) follows by similar arguments, using also t he estimates (3.11), (4.27). Finally\nwe discuss the algebraic properties of the operators. L(0)is real by composition, F1by definition,\nQ(0)\n1/2by Lemma 4.3. By Remark 2.13 we have that S(4)\n1,S(5)\n1are real and therefore also R(0)\n0and\nfinally by difference RΨ,R(4)\n−N,R(5)\n−Nare real and hence R(0)\n−N. /squareLARGE AMPLITUDE TRAVELING WAVES MHD 35\n4.2.The iterative step. We now describe the iterative argument that will eventually allow us to\ndecouple the equations up to an arbitrary smoothing operato r of order −N.\nLemma 4.5. LetN∈N,γ∈(0,1),τ >0and assume that λ−δγ−1≤1. Then there exists\nσ≡σN≫0large enough such that if (3.3)holds, then the following statements holds for all\nn∈ {0,...,2N+1}. There exists a linear operator\nL(n)=D+F1+Π⊥\n0/parenleftBig\nF(n)\n0+Q(n)\n1−n\n2+R(n)\n−N/parenrightBig\nΠ⊥\n0, (4.46)\ndefined for all ω∈DC(γ,τ), whereF(n)\n0∈ OPS0is a block diagonal matrix-operator, Q(n)\n1−n\n2∈\nOPS1−n\n2is a block off-diagonal matrix-operator, and R(n)\n−Nis an operator of order −Nsatisfying\nthe estimates\n|F(n)\n0|Lip(γ)\n0,s,α,|Q(n)\n1−n\n2|Lip(γ)\n1−n\n2,s,α/lessorsimilars,αλ3δ/parenleftbig\n1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/parenrightbig\n,∀s≥s0, α∈N0,\n|R(n)\n−N|Lip(γ)\n−N,s,0/lessorsimilars,Nλ6δ(N+1)/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0.(4.47)\nForn∈ {1,2,...,2N+1}there exists a real, linear and invertible map Φn−1satisfying\nΦ±1\nn−1:Hs\n0→Hs\n0,|Φ±1\nn−1|Lip(γ)\n0,s,0/lessorsimilars1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0, (4.48)\nsuch that the operator L(n)satisfies the conjugation\nL(n)= Φ−1\nn−1L(n−1)Φn−1.\nLets1≥s0,α∈N0and letI1,I2satisfy(3.3)withs1+σinstead ofs0+σ. Then\n|∆12Φ±1\nn−1|0,s1,0/lessorsimilars1/ba∇dblI1−I2/ba∇dbls1+σ,\n|∆12F(n)\n0|0,s1,α,|∆12Q(n)\n1−n\n2|1−n\n2,s1,α/lessorsimilars1,αλ3δ/ba∇dblI1−I2/ba∇dbls1+σ.(4.49)\nFinally, the opeators L(n),F(n)\n0,Q(n)\n1−n\n2andR(n)\n−Nare real matrix operators.\nProof.To simplify notations we write /ba∇dbl·/ba∇dblsfor/ba∇dbl·/ba∇dblLip(γ)\nsand|·|m,s,αfor|·|Lip(γ)\nm,s,α.\nPROOF OF THE STATEMENT FOR n= 0. The claimed statement follows from Lemma 4.4.\nPROOF OF THE INDUCTION STEP. Let n≥0 and we assume that we have an operator of\nthe form\nL(n)=D+F1+Π⊥\n0/parenleftBig\nF(n)\n0+Q(n)\n1−n\n2+R(n)\n−N/parenrightBig\nΠ⊥\n0, (4.50)\nwhereF(n)\n0,Q(n)\n1−n\n2,R(n)\n−Nsatisfy the properties (4.47). Our goal is to normalize the o ff-diagonal part\nQ(n)\n1−n\n2=\n0 Op/parenleftBig\nqn,1(x,ξ)/parenrightBig\nOp/parenleftBig\nqn,2(x,ξ)/parenrightBig\n0\n∈ OPS1−n\n2. (4.51)\nWe look for\nΨn,⊥:= Π⊥\n0ΨnΠ⊥\n0,Ψn=\n0 Op/parenleftBig\nψn,1(x,ξ)/parenrightBig\nOp/parenleftBig\nψn,2(x,ξ)/parenrightBig\n0\n∈ OPS−n\n2−1,(4.52)36 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nwhere the symbols ψn,1andψn,2have to be determined and we consider Φ n= exp(Ψ n,⊥). We\nanalyze the conjugation L(n+1):= Φ−1\nnL(n)Φn. By Lie expansion one computes\nL(n+1)=D+F1+[D,Ψn,⊥]+Π⊥\n0/parenleftBig\nQ(n)\n1−n\n2+F(n)\n0/parenrightBig\nΠ⊥\n0+S(n)\n1+S(n)\n2+S(n)\n3,\nS(n)\n1:=1\n2/integraldisplay1\n0(1−τ)e−τΨn,⊥[[D,Ψn,⊥],Ψn,⊥]eτΨn,⊥dτ\nS(n)\n2:=/integraldisplay1\n0e−τΨn,⊥/bracketleftBig\nF1+Π⊥\n0F(n)\n0Π⊥\n0+Π⊥\n0Q(n)\n1−n\n2Π⊥\n0,Ψn,⊥/bracketrightBig\neτΨn,⊥dτ\nS(n)\n3:= Φ−1\nnΠ⊥\n0R(n)\n−NΠ⊥\n0Φn.(4.53)\nConstruction of Ψnand expansion of [D,Ψn,⊥]+Π⊥\n0Q(n)\n1−n\n2Π⊥\n0.Since [D,Π⊥\n0] = 0, one has\nthat\n[D,Ψn,⊥]+Π⊥\n0Q(n)\n1−n\n2Π⊥\n0= Π⊥\n0/parenleftBig\n[D,Ψ]+Q(n)\n1−n\n2/parenrightBig\nΠ⊥\n0.\nBy Lemma 4.2 and using the induction estimates (4.47) on Q(n)\n1−n\n2, We choose ψn,i,i= 1,2 so that\n/parenleftbig\nλω·∇+|ξ|2/parenrightbig\nψn,1(x,ξ)+χλ(ξ)qn,1(x,ξ) = 0,\n/parenleftbig\nλω·∇−|ξ|2/parenrightbig\nψn,2(x,ξ)+χλ(ξ)qn,2(x,ξ) = 0,(4.54)\nwithψn,1,ψn,2∈S−n\n2−1satisfy\n|Op(ψn,i)|−n\n2−1,s,α/lessorsimilarαλ−3δ|Op(qn,i)|1−n\n2,s+τ+1,α/lessorsimilars,α1+/ba∇dblI/ba∇dbls+σ,∀s≥s0, α∈N0.(4.55)\nMoreover the latter properties, together with Lemma 2.21 im ply that\nΨn,⊥= Ψn+RΨn\n⊥,\n|RΨn\n⊥|−N,s,0/lessorsimilars,N1+/ba∇dblI/ba∇dbls+σ,∀s≥s0.(4.56)\nFurthermore, by splitting qn,i=χλqn,i+(1−χλ)qn,i,i= 1,2 (recall (4.5)), and by using that by a\ndirect calculation\n−∆◦Op/parenleftBig\nψn,1(x,ξ)/parenrightBig\n= Op/parenleftBig\nψn,1(x,ξ)|ξ|2−2iξ·∇xψn,1(x,ξ)−∆xψn,1(x,ξ)/parenrightBig\n,\nOp(ψn,2)◦(−∆) = Op/parenleftBig\nψn,2(x,ξ)|ξ|2/parenrightBig\n,(4.57)\none obtains that\n[D,Ψn]+Q(n)\n1−n\n2\n=/parenleftbigg\n0 [ λω·∇,Op/parenleftbig\nψn,1/parenrightbig\n]−∆◦Op/parenleftbig\nψn,1/parenrightbig\n[λω·∇,Op/parenleftbig\nψn,2/parenrightbig\n]+Op/parenleftbig\nψn,2/parenrightbig\n◦∆ 0/parenrightbigg\n+/parenleftbigg\n0 Op/parenleftbig\nχλ(ξ)qn,1(x,ξ)/parenrightbig\nOp/parenleftbig\nχλ(ξ)qn,2(x,ξ)/parenrightbig\n0/parenrightbigg\n+/parenleftbigg\n0 Op/parenleftbig/parenleftbig\n1−χλ(ξ)/parenrightbig\nqn,1(x,ξ)/parenrightbig\nOp/parenleftbig/parenleftbig\n1−χλ(ξ)/parenrightbig\nqn,2(x,ξ)/parenrightbig\n0/parenrightbigg\n=/parenleftbigg\n0/parenleftbig\nλω·∇+|ξ|2/parenrightbig\nψn,1(x,ξ)+χλ(ξ)qn,1(x,ξ)/parenleftbig\nλω·∇−|ξ|2/parenrightbig\nψn,2(x,ξ)+χλ(ξ)qn,2(x,ξ) 0/parenrightbigg\n+QΨn+RΨn,\n(4.57)=QΨn+RΨn,\n(4.58)\nQΨn:=/parenleftBigg\n0 Op/parenleftBig\n2iξ·∇xψn,1(x,ξ)−∆xψn,1(x,ξ)/parenrightBig\n0 0/parenrightBigg\n,\nRΨn:=/parenleftbigg\n0 Op(/parenleftbig\n1−χλ(ξ)/parenrightbig\nqn,1(x,ξ))\nOp/parenleftbig/parenleftbig\n1−χλ(ξ)/parenrightbig\nqn,2(x,ξ)/parenrightbig\n0/parenrightbigg\n.(4.59)LARGE AMPLITUDE TRAVELING WAVES MHD 37\nBy using Lemma 4.1, by the estimates (4.54) and by the inducti on estimates (4.47), one easily gets\nthat\nQΨn∈ OPS−n\n2,|QΨn|−n\n2,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dbls+σ,∀s≥s0, α∈N0,\n|RΨn|−N,s,0/lessorsimilars,Nλ6δ(N+1−n\n2)λ3δ(1+/ba∇dblI/ba∇dbls+σ)\n/lessorsimilars,Nλ6δ(N+1\n2)λ3δ(1+/ba∇dblI/ba∇dbls+σ)\n/lessorsimilars,Nλ6δ(N+1)(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0.(4.60)\nThe estimates (4.54), (2.47), together with Lemma 2.12 and t he ansatz (3.3) allows to deduce that\nsup\nτ∈[−1,1]|eτΨn,⊥|0,s,0/lessorsimilars1+/ba∇dblI/ba∇dbls+σ,∀s≥s0, (4.61)\nform which one deduces the claimed bound (4.48) for Φ±1\nn=e±Ψn,⊥.\nSince by induction one has that Q(n)\n1−n\n2, then by using Lemma 4.2 one deduce that Φ nis real and\ntherefore by composition also L(n+1)is so.\nAnalysis of S(n)\n1.By recalling (4.53) and using (4.58), (4.59), (4.60) one wri tes\nS(n)\n1=S(n)\na+S(n)\nb,\nS(n)\na:=1\n2/integraldisplay1\n0(1−τ)e−τΨn,⊥[Π⊥\n0QΨnΠ⊥\n0,Ψn,⊥]eτΨn,⊥dτ,\nS(n)\nb:=1\n2/integraldisplay1\n0(1−τ)e−τΨn,⊥[Π⊥\n0RΨnΠ⊥\n0,Ψn,⊥]eτΨn,⊥dτ.\nClearlyS(n)\n1is invariant on the space of zero average functions, namely S(n)\n1= Π⊥\n0S(n)\n1Π⊥\n0,S(n)\na=\nΠ⊥\n0S(n)\naΠ⊥\n0andS(n)\nb= Π⊥\n0S(n)\nbΠ⊥\n0. By theestimates (4.54), (4.60), (4.61), (2.47) by thecomp osition\nestimate (2.29) (using also the ansatz (3.3)), one gets that the operator S(n)\nbis of order −Nand it\nsatisfies the estimate\n|S(n)\nb|−N,s,0/lessorsimilars,Nλ6δ(N+1)(1+/ba∇dblI/ba∇dbls+σ),∀s≥s0. (4.62)\nMoreover, by (4.54), (4.60), Lemma 2.14 (to expand [Π⊥\n0QΨnΠ⊥\n0,Ψn,⊥]), Lemma 2.16 (to expand\ne−τΨn,⊥[Π⊥\n0QΨnΠ⊥\n0,Ψn,⊥]eτΨn,⊥,τ∈[0,1]) ,using again (3.3)), one obtains the following expansio n\nfor the operator S(n)\na:\nS(n)\na= Π⊥\n0/parenleftBig\nS(n)\nc+S(n)\nd/parenrightBig\nΠ⊥\n0,\nS(n)\nc∈ OPS−n−1,|S(n)\nc|−n−1,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dbls+σ,∀s≥s0,∀α∈N0,\n|S(n)\nd|−N,s,0/lessorsimilars1+/ba∇dblI/ba∇dbls+σ,∀s≥s0.(4.63)\nThis concludes the analysis of the remainder S(n)\n1.\nAnalysis of S(n)\n2.Recall the formula(4.53) for S(n)\n2. By recalling the expressionof F1in(4.2), the\nestimates of a(x) in Lemma 3.1 and by the induction estimates (4.47) on F(n)\n0,Q(n)\n1−n\n2, one obtains\nthat\nF1+F(n)\n0+Q(n)\n1−n\n2∈ OPS1and\n/vextendsingle/vextendsingle/vextendsingleF1+F(n)\n0+Q(n)\n1−n\n2/vextendsingle/vextendsingle/vextendsingle\n1,s,α/lessorsimilars,αλ3δ/parenleftbig\n1+/ba∇dblI/ba∇dbls+σ/parenrightbig\n,∀s≥s0, α∈N0.(4.64)\nHence by (4.64), (4.54), Lemma 2.14 (to expand [Π⊥\n0/parenleftbig\nF1+F(n)\n0+Q(n)\n1−n\n2/parenrightbig\nΠ⊥\n0,Ψn,⊥]), 2.16 (to ex-\npande−τΨn,⊥[Π⊥\n0/parenleftbig\nF1+F(n)\n0+Q(n)\n1−n\n2/parenrightbig\nΠ⊥\n0,Ψn,⊥]eτΨn,⊥), using again (3.3), one obtains the following38 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nexpansion for the operator S(n)\n2:\nS(n)\n2= Π⊥\n0/parenleftBig\nS(n)\ne+S(n)\nf/parenrightBig\nΠ⊥\n0,\nS(n)\ne∈ OPS−n\n2,|S(n)\ne|−n\n2,s,α/lessorsimilars,αλ3δ/parenleftbig\n1+/ba∇dblI/ba∇dbls+σ/parenrightbig\n,∀s≥s0,∀α∈N0,\n|S(n)\nf|−N,s,0/lessorsimilarsλ3δ/parenleftbig\n1+/ba∇dblI/ba∇dbls+σ/parenrightbig\n/lessorsimilarsλ6δ(N+1)/parenleftbig\n1+/ba∇dblI/ba∇dbls+σ/parenrightbig\n,∀s≥s0.(4.65)\nThis concludes the analysis of the remainder S(n)\n2.\nAnalysis of S(n)\n3.By (4.53), by the estimates (4.61), the induction estimate ( 4.47) on R(n)\n−N, and\nby the composition estimate (2.29) (together with the ansat z (3.3)), one deduces the S(n)\n3is an\noperator of order −Nand it satisfies the estimate\n|S(n)\n3|−N,s,0/lessorsimilarsλ6δ(N+1)/parenleftbig\n1+/ba∇dblI/ba∇dbls+σ/parenrightbig\n,∀s≥s0. (4.66)\nFinally, by collecting (4.53), (4.58), (4.59), (4.60), (4. 62), (4.63), (4.65), (4.66) one gets that\nL(n+1)= Φ−1\nnL(n)Φn=D+F1+Π⊥\n0/parenleftBig\nF(n)\n0+R(n+1)\n−n\n2+R(n+1)\n−N/parenrightBig\nΠ⊥\n0,\nR(n+1)\n−n\n2:=QΨn+S(n)\nc+S(n)\ne,\nR(n+1)\n−N:=RΨn+S(n)\nb+S(n)\nd+S(n)\nf+S(n)\n3,\n|R(n+1)\n−n\n2|−n\n2,s,α/lessorsimilars,αλ3δ/parenleftbig\n1+/ba∇dblI/ba∇dbls+σ/parenrightbig\n,∀s≥s0, α∈N0,\n|R(n+1)\n−N|−N,s,0/lessorsimilars,Nλ6δ(N+1)/parenleftbig\n1+/ba∇dblI/ba∇dbls+σ/parenrightbig\n,∀s≥s0.(4.67)\nfor someσ≡σN≫0 large enough. Then the claimed properties (4.47) for L(n+1)then follows\nby (4.67) by splitting R(n+1)\n−n\n2=F(n+1)\n−n\n2+Q(n+1)\n−n\n2whereF(n+1)\n−n\n2is the diagonal part of R(n+1)\n−n\n2and\nQ(n+1)\n−n\n2is the off-diagonal part of R(n+1)\n−n\n2and by defining F(n+1)\n0:=F(n)\n0+F(n+1)\n−n\n2. The claimed\nexpansion (4.46) and the claimed estimates (4.47) at the ste pn+ 1 has then been proved. The\nestimates (4.49) at the step n+1 follows by similar arguments.\nTo prove that F(n+1)\n0,Q(n+1)\n−n\n2are real one can reason as donein Lemma 4.4 by usingtheinduct ive\nhypothesis on F(n)\n0,Q(n)\n1−n\n2and finally R(n+1)\n−n\n2is real by difference. /square\nProposition 4.6. LetN∈N,M:= 6(N+ 1),γ∈(0,1),τ >0,λ−δγ−1≤1,ω∈DC(γ,τ).\nThen there exists σ≡σN≫0such that if (3.3)holds, then there exists a real, invertible map ΦN\nsatisfying\nΦ±1\nN:Hs\n0→Hs\n0,|Φ±1\nN|Lip(γ)\n0,s,0/lessorsimilars,N1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s≥s0 (4.68)\nand such that the linear operator Lin(3.9)transforms as follows:\nL1:=Φ−1\nNLΦN=/parenleftBigg\nL(1)\n1L(2)\n−N\nL(3)\n−NL(4)\n1/parenrightBigg\n,\nL(1)\n1:=λω·∇−∆+a(x)·∇+Π⊥\n0S(1)\n0Π⊥\n0+S(1)\n−N,\nL(1)\n4:=λω·∇+a(x)·∇+Π⊥\n0S(4)\n0Π⊥\n0+S(4)\n−N,(4.69)LARGE AMPLITUDE TRAVELING WAVES MHD 39\nwhereS(1)\n0,S(4)\n0∈ OPS0, whereas the remainders L(2)\n−N,L(3)\n−N,S(1)\n−N,S(4)\n−Nare operators of order −N\nsatisfying\n|S(1)\n0|Lip(γ)\n0,s,α,|S(4)\n0|Lip(γ)\n0,s,α/lessorsimilars,αλ3δ/parenleftbig\n1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/parenrightbig\n,∀s≥s0, α∈N0,\nL(2)\n−N,L(3)\n−N,S(1)\n−N,S(1)\n−N∈ B(Hs\n0, Hs+N\n0),∀s≥s0and\n/ba∇dblL(2)\n−Nh/ba∇dblLip(γ)\ns+N,/ba∇dblL(3)\n−Nh/ba∇dblLip(γ)\ns+N,/ba∇dblS(1)\n−Nh/ba∇dblLip(γ)\ns+N,/ba∇dblS(4)\n−Nh/ba∇dblLip(γ)\ns+N\n/lessorsimilarsλMδ/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightBig\n,∀s≥s0.(4.70)\nLets1≥s0,α∈N0and letI1,I2satisfy(3.3)withs1+σinstead ofs0+σ. Then\n|∆12S(1)\n0|0,s1,α,|∆12S(4)\n0|0,s1,α/lessorsimilars1,αλ3δ/ba∇dblI1−I2/ba∇dbls1+σ (4.71)\nFinally, the operators L1,S(1)\n0,S(4)\n0,L(2)\n−N,L(3)\n−N,S(1)\n−N,S(4)\n−Nare real.\nProof.We apply Lemmata 4.4, 4.5. We set\nΦN:= Φ◦Φ0◦...◦Φ2N.\nThe estimate (4.68) then follows by (4.35), (4.48) and by app lying the composition estimate (2.29)\n(usingalso (3.3)). Moreover, the operator L1≡ L(2N+1)=Φ−1\nNLΦNis of the form (4.69) by setting\nF(2N+1)\n0≡/parenleftBigg\nS(1)\n00\n0S(4)\n0/parenrightBigg\n,Π⊥\n0/parenleftBig\nQ(2N+1)\n−N+R(2N+1)\n−N/parenrightBig\nΠ⊥\n0≡/parenleftBigg\nS(1)\n−NL(2)\n−N\nL(3)\n−NS(4)\n−N/parenrightBigg\n.\nThen by the estimate (4.47) and using also Lemma 2.18 one dedu ces the estimates (4.70). Finally,\nthe estimates (4.71) are a direct consequence of the estimat es (4.49) for ∆ 12F(2N+1)\n0. Since the\nmaps Φ,Φ0,...,Φ2Nare real by Lemmata 4.5 and 4.4, then the map ΦNis real by composition,\ntogether with L(2N+1). To prove the realty of S(1)\n0,S(4)\n0,L(2)\n−N,L(3)\n−N,S(1)\n−N,S(4)\n−None reasons as done\nin the proof of Lemma 4.4. /square\n5.Inversion of the first equation\n5.1.Inversion of the operator L(1)\n1.In order to invert the operator L1, we first invert the\noperator\nL(1)\n1:=λω·∇−∆+R(1)\n1,R(1)\n1:=a(x)·∇+Π⊥\n0S(1)\n0Π⊥\n0+S(1)\n−N, (5.1)\ngiven in (4.69). Note that by Lemma 3.1, by Lemma 2.18 and by th e estimates (4.70), and by the\nansatz (3.3), one obtains that\nR(1)\n1∈ B(Hs+1\n0,Hs\n0),∀s≥s0and\n/ba∇dblR(1)\n1h/ba∇dblLip(γ)\ns/lessorsimilarsλδM/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+1/parenrightBig\n,∀s≥s0.(5.2)\nThen, we write L(1)\n1as\nL(1)\n1:=Lλ+R(1)\n1, Lλ:=λω·∇−∆. (5.3)\nWe have the following.\nLemma 5.1. Letλ >1,γ∈(0,1),0<p<1\nτ+2andω∈DC(γ,τ). Then the operator Lλis\ninvertible and its inverse is a continuous operator L−1\nλ:Hs\n0(T2)→Hs+1\n0(T2)satisfying the bound\n/ba∇dblL−1\nλ/ba∇dblB(Hs\n0,Hs+1\n0)/lessorsimilarγ−1λ−p. (5.4)\nMoreover, one also has the estimate with loss of derivatives\n/ba∇dblL−1\nλ/ba∇dblB(Hs+τ\n0,Hs\n0)/lessorsimilarγ−1λ−1,∀s≥0. (5.5)40 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nProof.We denote by Λ( k)≡Λ(k;ω),ω∈DC(γ,τ), the eigenvalues of the operator Lλ≡Lλ(ω),\nwhich are given by the formula\nΛ(k) :=iλω·k+|k|2, k∈Z2\\{0}. (5.6)\nWe look for a lower bound on the eigenvalues Λ( k). Letp>0. In the high frequency regime, we\nhave that\n|Λ(k)| ≥ |k|2≥λp|k|for|k| ≥λp. (5.7)\nOn the other hand, for low frequencies |k| ≤λpwe use the Diophantine condition, obtaining that\n|Λ(k)| ≥λ|ω·k| ≥λγ\n|k|τ≥γλ1−τpfor|k| ≤λp. (5.8)\nThus, for any h∈Hs\n0(T2), one has that\n/ba∇dblL−1\nλh/ba∇dbl2\ns+1=/summationdisplay\nk/\\e}atio\\slash=0/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2(s+1)|ˆh(k)|2\n|Λ(k)|2\n=/summationdisplay\n0<|k|≤λp/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2(s+1)|ˆh(k)|2\n|Λ(k)|2+/summationdisplay\n|k|≥λp/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2(s+1)|ˆh(k)|2\n|Λ(k)|2\n/lessorsimilarγ−2λ2(τp−1)/summationdisplay\n|k|≤λp/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2(s+1)|ˆh(k)|2\n+λ−2p/summationdisplay\n|k|≥λp/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2(s+1)|ˆh(k)|2\n|k|2\n/lessorsimilar/parenleftBig\nγ−2λ2(τp+p−1)+λ−2p/parenrightBig\n/ba∇dblh/ba∇dbl2\ns/lessorsimilarλ−2pγ−2/ba∇dblh/ba∇dbl2\ns,(5.9)\nsinceγ∈(0,1), 0<p<1\nτ+2, which implies that λτp+p−1/lessorsimilarλ−p. The latter chain of inequalities\nthen implies (5.4). In order to prove (5.5), one simply uses t hatω∈DC(γ,τ), namely\n|Λ(k)| ≥λ|ω·k| ≥λγ\n|k|τ,∀k∈Z2\\{0}\nwhich implies that /ba∇dblL−1\nλh/ba∇dbls/lessorsimilarλ−1γ−1/ba∇dblh/ba∇dbls+τfor anys≥0,h∈Hs+τ\n0.\n/square\nWe now fix the constants pandδas\np:=δ(M+1),0<δ <1\n(M+1)(τ+2),\nwhere we recall that M= 6(N+1).(5.10)\nLemma 5.2. Letγ∈(0,1),τ >0,N∈Nandω∈DC(γ,τ). Then there exists σ≡σN≫0large\nenough such that if (3.3)holds, then for any S >s0+σthere exists ε(S)≪1small enough such\nthat ifλ−δγ−1≤ε(S), then the following holds\n•Invertibility of L(1)\n1with gain of regularity. The operator L(1)\n1is invertible with inverse/parenleftBig\nL(1)\n1/parenrightBig−1\n:Hs\n0(T2)→Hs+1\n0(T2), for anys0≤s≤S−σand it satisfies the estimate\n/vextenddouble/vextenddouble/vextenddouble/parenleftBig\nL(1)\n1/parenrightBig−1\nh/vextenddouble/vextenddouble/vextenddouble\ns+1/lessorsimilarsλ−δM/parenleftBig\n/ba∇dblh/ba∇dbls+/ba∇dblI/ba∇dbls+σ/ba∇dblh/ba∇dbls0/parenrightBig\n,∀s0≤s≤S−σ. (5.11)\n•Lipschitz estimate of (L(1)\n1)−1with loss of derivatives. For anys0≤s≤S−σ, for\nh(·;ω)∈Hs+2τ+1\n0,ω∈DC(γ,τ), the operator (L(1)\n1)−1satisfies the estimate\n/vextenddouble/vextenddouble/vextenddouble/parenleftBig\nL(1)\n1/parenrightBig−1\nh/vextenddouble/vextenddouble/vextenddoubleLip(γ)\ns/lessorsimilarsλ−δM/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+2τ+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+2τ+1/parenrightBig\n. (5.12)LARGE AMPLITUDE TRAVELING WAVES MHD 41\nProof.Proof of (5.11).First of all, by recalling (5.3), we rewrite the operator L(1)\n1as\nL(1)\n1=Lλ/parenleftBig\nId+Pλ/parenrightBig\n,Pλ:=L−1\nλR(1)\n1. (5.13)\nIn order to invert L(1)\n1, we need to invert the operator Id + Pλ. We will make this by Neumann\nseries. Indeed, by the definition of pin (5.10), by the estimate (5.2) and by the estimate (5.4) in\nLemma 5.1, one has that for σ≡σN≫0 large enough, /ba∇dblI/ba∇dbls0+σ/lessorsimilar1 (see (3.3)), for any S≥s0+σ,\nfor anys0+1≤s≤S−σ, one has that\nPλ:Hs\n0→Hs\n0,\n/ba∇dblPλh/ba∇dbls/lessorsimilarλ−δ(M+1)γ−1/ba∇dblR(1)\n1h/ba∇dbls−1\n/lessorsimilarsλ−δ(M+1)γ−1λδM/parenleftBig\n/ba∇dblh/ba∇dbls+/ba∇dblI/ba∇dbls+σ/ba∇dblh/ba∇dbls0/parenrightBig\n,\n/lessorsimilarsλ−δγ−1/parenleftBig\n/ba∇dblh/ba∇dbls+/ba∇dblI/ba∇dbls+σ/ba∇dblh/ba∇dbls0/parenrightBig\n.(5.14)\nThe estimate (5.14) together with the ansatz (3.3) gives the estimate\n/ba∇dblPλh/ba∇dbls0/lessorsimilarλ−δγ−1/ba∇dblh/ba∇dbls0. (5.15)\nBy iterating the estimates (5.14), (5.15), one gets that for any integer n∈N\n/ba∇dblPn\nλh/ba∇dbls≤/parenleftBig\nC(s)λ−δγ−1/parenrightBign/parenleftBig\n/ba∇dblh/ba∇dbls+/ba∇dblI/ba∇dbls+σ/ba∇dblh/ba∇dbls0/parenrightBig\n,∀s0+1≤s≤S−σ,\n/ba∇dblPn\nλh/ba∇dbls0≤/parenleftBig\nC(s0)λ−δγ−1/parenrightBign\n/ba∇dblh/ba∇dbls0,(5.16)\nfor some constants C(s)≫C(s0)≫0 large enough. Hence by expanding (Id + Pλ)−1=/summationtext\nn≥0(−1)nPn\nλby Neumann series and by using the estimates (5.16) one obtai ns that for any\ns0≤s≤S−σ, and by taking λ≫1 large enough in such a way that C(S)λ−δγ−1≪1\n/ba∇dbl(Id+Pλ)−1h/ba∇dbls/lessorsimilars/ba∇dblh/ba∇dbls+/ba∇dblI/ba∇dbls+σ/ba∇dblh/ba∇dbls0,∀s0+1≤s≤S−σ. (5.17)\nBytheestimateofLemma5.1,usingthat p=δ(M+1)andλ−δγ−1≪1, onehasthat /ba∇dblL−1\nλ/ba∇dblB(Hs\n0,Hs+1\n0)/lessorsimilar\nλ−δMand therefore the claimed bound (5.11) on the operator ( L(1)\n1)−1follows by (5.13), (5.17).\nProof of (5.12).Lets0≤s≤S−σ,h(·;ω)∈Hs+2τ+1\n0,ω∈DC(γ,τ). By the estimate (5.5) in\nLemma 5.1 and by the estimate (5.17), one gets that\n/vextenddouble/vextenddouble/vextenddouble/parenleftBig\nL(1)\n1/parenrightBig−1\nh/vextenddouble/vextenddouble/vextenddoublesup\ns/lessorsimilarsλ−1γ−1/parenleftBig\n/ba∇dblh/ba∇dblsup\ns+τ+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblsup\ns0+τ/parenrightBig\n. (5.18)\nNow letω1,ω2∈DC(γ,τ). Since by (5.10), 0 < δ <1\nM+1, forλ−δγ−1≪1, one has that\nλδMγ−1/lessorsimilarλ. Hence, by (5.2), (5.3), one obtains for s≥s0,h∈Hs+1\n0\n/vextenddouble/vextenddouble/vextenddouble/parenleftBig\nL(1)\n1(ω2)−L(1)\n1(ω1)/parenrightBig\nh/vextenddouble/vextenddouble/vextenddouble\ns/lessorsimilarsλ/ba∇dblh/ba∇dbls+1|ω1−ω2|\n+λδMγ−1/parenleftBig\n/ba∇dblh/ba∇dbls+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dbls0+1/parenrightBig\n|ω1−ω2|\n/lessorsimilarsλ/parenleftBig\n/ba∇dblh/ba∇dbls+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dbls0+1/parenrightBig\n|ω1−ω2|.(5.19)\nThen, we write\n(L(1)\n1(ω1))−1h(·;ω1)−(L(1)\n1(ω2))−1h(·;ω2) =G1h(·;ω1)+(L(1)\n1(ω2))−1[h(·;ω1)−h(·;ω2)],\nG1:= (L(1)\n1(ω1))−1−(L(1)\n1(ω2))−1= (L(1)\n1(ω1))−1/parenleftBig\nL(1)\n1(ω2)−L(1)\n1(ω1)/parenrightBig\n(L(1)\n1(ω2))−1.42 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nBy the estimates (5.18), (5.19), one then obtains that for an ys0≤s≤S−σ(use also the ansatz\n(3.3)),\nγ/vextenddouble/vextenddouble/vextenddouble(L(1)\n1(ω1))−1h(·;ω1)−(L(1)\n1(ω2))−1h(·;ω2)/vextenddouble/vextenddouble/vextenddouble\ns\n/lessorsimilarsλ−1γ−1/parenleftBig\n/ba∇dblh/ba∇dblsup\ns+2τ+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblsup\ns0+2τ+1/parenrightBig\n|ω1−ω2|\n+λ−1γ−1/parenleftBig\nγ/ba∇dblh/ba∇dbllip\ns+τ+/ba∇dblI/ba∇dblLip(γ)\ns+σγ/ba∇dblh/ba∇dbllip\ns0+τ/parenrightBig\n|ω1−ω2|\n/lessorsimilarsλ−1γ−1/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+2τ+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+2τ+1/parenrightBig\n|ω1−ω2|.(5.20)\nThe claimed bound (5.12) then follows by (5.18), (5.20), usi ng again that λ−1γ−1/lessorsimilarλ−δM./square\n5.2.Reduction to a scalar transport-type operator. In order to invert the linear operator\nL1in (4.69), we need to solve the system\n/braceleftBigg\nL(1)\n1h1+L(2)\n−Nh2=g1,\nL(3)\n−Nh1+L(4)\n1h2=g2.(5.21)\nBy Lemma 5.2, we have that\nh1= (L(1)\n1)−1g1−(L(1)\n1)−1L(2)\n−Nh2, (5.22)\nand we can replace the latter expression of h1in the second equation of (5.21), by obtaining the\nequation\nP[h2] =g2−L(3)\n−N(L(1)\n1)−1[g1],\nP:=L(4)\n1−L(3)\n−N(L(1)\n1)−1L(2)\n−N.(5.23)\nHence, in order to conclude the invertibility of L1, it is enough to solve the equation (5.23), namely\nto invert the linear operator P. The following lemma holds.\nLemma 5.3. Letγ∈(0,1),τ >0,N≥2τ+1andω∈DC(γ,τ). There exists σ≡σN≫0large\nenough such that if (3.3)holds then the linear operator Phas the form\nP=λω·∇+a(x)·∇+Π⊥\n0QΠ⊥\n0+R (5.24)\nwhereQ ∈ OPS0andRis a smoothing operator of order −Nsatisfying the following properties.\nThe operator Qsatisfies the estimate\n|Q|Lip(γ)\n0,s,α/lessorsimilars,αλδM(1+/ba∇dblI/ba∇dblLip(γ)\ns+σ),∀s≥s0, α∈N0, (5.25)\nand for any S >s0+σthere exists ε(S)≪1small enough such that if λ−δγ−1≤ε(S), then\nR ∈ B(Hs\n0,Hs+N\n0),∀s0≤s≤S−σ,\n/ba∇dblRh/ba∇dblLip(γ)\ns+N/lessorsimilars,NλδM/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightBig\n,∀s0≤s≤S−σ.(5.26)\nLets1≥s0,α∈N0and assume that I1,I2satisfies (3.3)withs1+σinstead ofs0+σ. Then\n|∆12Q|0,s1,α/lessorsimilars1,αλδM/ba∇dblI1−I2/ba∇dbls1+σ. (5.27)\nFinally, the operators P,QandRare real.\nProof.To simplify the notations we write /ba∇dbl · /ba∇dblsinstead of /ba∇dbl · /ba∇dblLip(γ)\ns. By recalling (4.69), (4.70),\n(4.71), (5.23), the estimates (5.25), (5.27) on Q ≡ S(4)\n0immediately follows, hence we estimate the\noperator R:=−L(3)\n−N(L(1)\n1)−1L(2)\n−N+S(4)\n−N. LetS > s0+σ, then ifλ−δγ−1≤ε(S)≪1 is smallLARGE AMPLITUDE TRAVELING WAVES MHD 43\nenough, we can apply the estimate (5.12) in Lemma 5.2 and by us ing the estimates (4.70), Lemma\n2.18 and the ansatz (3.3), one gets\n/ba∇dblL(3)\n−N(L(1)\n1)−1L(2)\n−Nh/ba∇dbls+N≤ /ba∇dblL(3)\n−N(L(1)\n1)−1L(2)\n−Nh/ba∇dbls+2N−2τ−1\n/lessorsimilarsλδM/parenleftBig\n/ba∇dbl(L(1)\n1)−1L(2)\n−Nh/ba∇dbls+N−2τ−1+/ba∇dblI/ba∇dbls+σ/ba∇dbl(L(1)\n1)−1L(2)\n−Nh/ba∇dbls0+N−2τ−1/parenrightBig\n/lessorsimilarsλδMλ−δM/parenleftBig\n/ba∇dblL(2)\n−Nh/ba∇dbls+N+/ba∇dblI/ba∇dbls+σ/ba∇dblL(2)\n−Nh/ba∇dbls0+N/parenrightBig\n/lessorsimilarsλδMλ−δMλδM/parenleftBig\n/ba∇dblh/ba∇dbls+/ba∇dblI/ba∇dbls+σ/ba∇dblh/ba∇dbls0/parenrightBig\n/lessorsimilarsλδM/parenleftBig\n/ba∇dblh/ba∇dbls+/ba∇dblI/ba∇dbls+σ/ba∇dblh/ba∇dbls0/parenrightBig\nwhich imply the claimed bound for R. By Proposition 4.6 and by composition we have that Qand\nRare real and therefore Pis so, since it is a sum of real operators. The proof of the lemm a is then\nconcluded. /square\n6.Normal form reduction and inversion of the transport operat orP\nIn this section we discuss the invertibility of the linear op eratorPin (5.24). The first step to\nbe implemented is to reduce the highest order part to constan t coefficients, namely the transport\noperator\nTλ:=λω·∇+a(x)·∇=λT,T:=ω·∇+aλ(x)·∇,\naλ(x) :=λ−1a(x).(6.1)\nBy (3.6) and Lemma 3.1, one clearly has that aλsatisfies\n/ba∇dblaλ/ba∇dblLip(γ)\ns/lessorsimilarsλδ−1/ba∇dblI/ba∇dblLip(γ)\ns,∀s≥s0,\nand ifs1≥s0,/ba∇dblaλ(I1)−aλ(I2)/ba∇dbls1/lessorsimilars1λδ−1/ba∇dblI1−I2/ba∇dbls1.(6.2)\nWe also recall that the function a(x) has zero average and zero divergence\n/integraldisplay\nT2a(x)dx= 0,div(a) = 0. (6.3)\n6.1.Reduction of the transport operator of order one. We recall the following result from\n[2, Proposition 4.1] (see also [24]).\nProposition 6.1. LetTas in(6.1)and assume (6.3). Then there exists σ≡σ(τ)≫0large\nenough such that if (3.3)holds, forS >s0+σthere exists ε≡ε(S)≪1small enough such that if\nλδ−1γ−1≤ε, (6.4)\nthen the following holds. There exists an invertible diffeomo rphismx∈T2/mapsto→x+α(x;ω)∈T2with\ninversey/mapsto→y+ˇα(y;ω), defined for all ω∈DC(γ,τ), satisfying, for any s0≤s≤S−σ,\n/ba∇dblα/ba∇dblLip(γ)\ns,/ba∇dblˇα/ba∇dblLip(γ)\ns/lessorsimilarsλδ−1γ−1/ba∇dblI/ba∇dblLip(γ)\ns+σ, (6.5)\nsuch that, by defining\nAh(x) :=h(x+α(x)),withA−1h(y) =h(y+ˇα(y)), (6.6)\none gets the conjugation\nA−1(ω·∇+aλ(x)·∇)A=ω·∇.\nMoreover, let s1≥s0and let us assume that I1,I2satisfy(3.3)withs1+σinstead ofs0+σ. Then\n/ba∇dbl∆12α/ba∇dbls1,/ba∇dbl∆12ˇα/ba∇dbls1/lessorsimilars1λδ−1γ−1/ba∇dblI1−I2/ba∇dbls1+σ,\n/ba∇dbl(∆12A)h/ba∇dbls1,/ba∇dbl(∆12A−1)h/ba∇dbls1/lessorsimilars1λδ−1γ−1/ba∇dblI1−I2/ba∇dbls1+σ/ba∇dblh/ba∇dbls1+1.(6.7)\nIn the next lemma we provide the full conjugation of the opera torPby means of the map\nA⊥:= Π⊥\n0AΠ⊥\n0.44 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nProposition 6.2. Letγ∈(0,1),τ >0,N≥2τ+ 1,ω∈DC(γ,τ). There exists σ≡σN≫0\nlarge enough such that for any S > s0+σthere exists ε(S)≪1such that if λ−δγ−1≤ε(S)and\n(3.3)holds then for any ω∈DC(γ,τ), the linear operator Pin(5.24)transforms under the action\nof the map A⊥:= Π⊥\n0AΠ⊥\n0as follows.\nP0:=A−1\n⊥PA⊥=λω·∇+Π⊥\n0Q0Π⊥\n0+R0, (6.8)\nwhereQ0∈ OPS0\nS−σ,αfor anyα∈N0andR0is a smoothing operator of order −Nsatisfying\n|Q0|Lip(γ)\n0,s,α/lessorsimilars,αλδM(1+/ba∇dblI/ba∇dblLip(γ)\ns+σ),∀s0≤s≤S−σ,∀α∈N0,\nR0∈ B(Hs\n0,Hs+N\n0),∀s0≤s≤S−σ,\n/ba∇dblR0h/ba∇dblLip(γ)\ns+N/lessorsimilarsλδM/parenleftbig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightbig\n,∀s0≤s≤S−σ.(6.9)\nLets1≥s0,α∈N0and assume that I1,I2satisfy(3.3)withs1+σinstead ofs0+σ. Then\n|∆12Q0|0,s1,α/lessorsimilars,αλδM/ba∇dblI1−I2/ba∇dbls1+σ. (6.10)\nFinally, the operators P0,Q0andR0are real.\nProof.To simplify notations we write /ba∇dbl · /ba∇dblsfor/ba∇dbl · /ba∇dblLip(γ)\nsand| · |m,s,αfor| · |Lip(γ)\nm,s,α. By (5.10),\n0< δ <1\n2hence forλ≫1 large enough, λδ−1γ−1≤λ−δγ−1≪1 and hence the smallness\ncondition of Proposition 6.1 is verified. By (3.8) [ a(x)·∇,Π⊥\n0] = 0, one has that\nΠ⊥\n0/parenleftbig\nλω·∇+a(x)·∇/parenrightbig\nΠ⊥\n0=λω·∇+a(x)·∇\nand by applying Lemma 2.20 ( A−1\n⊥= Π⊥\n0A−1Π⊥\n0) and Proposition 6.1, one gets that\nA−1\n⊥/parenleftbig\nλω·∇+a(x)·∇/parenrightbig\nA⊥= Π⊥\n0A−1/parenleftbig\nλω·∇+a(x)·∇/parenrightbig\nAΠ⊥\n0=λω·∇,\nA−1\n⊥QA⊥= Π⊥\n0A−1Π⊥\n0QΠ⊥\n0AΠ⊥\n0= Π⊥\n0A−1QAΠ⊥\n0+R(1)\nQ,\nR(1)\nQ:=−Π⊥\n0A−1Π0QΠ⊥\n0AΠ⊥\n0−Π⊥\n0A−1QΠ0AΠ⊥\n0,(6.11)\ntherefore\nA−1\n⊥PA⊥=λω·∇+Π⊥\n0A−1QAΠ⊥\n0+A−1\n⊥RA⊥+R(1)\nQ.\nWe now analyze the conjugation A−1QAofQ= Op(q(x,ξ)). By Theorem 2.22 we have that\nA−1QA= Op(q0(x,ξ))+R(2)\nQ, (6.12)\nand there exists σ≡σN≫µ≫0 such that if (3.3) holds then for any S > s 0+σ, for any\ns0≤s≤S−σ, for anyα∈N0, one has\n|Op(q0)|0,s,α/lessorsimilars,α|Q|0,s,α+µ+/summationdisplay\nk1+k2+k3=s|Q|0,k1,α+k2+µ/ba∇dblα/ba∇dblk3+µ,\n/ba∇dblR(2)\nQh/ba∇dbls+N/lessorsimilars,NM(s)/ba∇dblh/ba∇dbls0+M(s0)/ba∇dblh/ba∇dbls,\nM(s) :=/summationdisplay\nk1+k2+k3=s|Q|0,k1,k2+µ/ba∇dblα/ba∇dblk3+µ(6.13)\nand furthermore, by applying the estimates (5.25), (6.5) on Qandα, one obtains that for k1+\nk2+k3=s\n|Q|0,s,α+µ,|Q|0,s+µ,µ/lessorsimilars,αλδM(1+/ba∇dblI/ba∇dbls+σ),\n|Q|0,k1,α+k2+µ,|Q|0,k1,k2+µ/lessorsimilars,αλδM(1+/ba∇dblI/ba∇dbls1+σ),\n/ba∇dblα/ba∇dbls3+µ/lessorsimilarsλδ−1γ−1/ba∇dblI/ba∇dbls3+σ,\nfor someσ≡σN≫µ≫0 large enough. Hence by the latter estimates and by (6.13), u sing that\nλδ−1γ−1≤λ−δγ−1≪1, one immediately obtains that\n|Op(q0)|0,s,α,M(s)/lessorsimilars,αλδM/parenleftBig\n/ba∇dblI/ba∇dbls+σ+/summationdisplay\nk1+k2+k3=s/ba∇dblI/ba∇dblk1+k2+σ/ba∇dblI/ba∇dblk3+σ/parenrightBig\n.(6.14)LARGE AMPLITUDE TRAVELING WAVES MHD 45\nFinally by the interpolation estimate (2.9) one gets that fo rk1+k2+k3=s\n/ba∇dblI/ba∇dblk1+k2+σ/ba∇dblI/ba∇dblk3+σ≤ /ba∇dblI/ba∇dbls−k1−k2\nsσ/ba∇dblI/ba∇dblk1+k2\ns\ns+σ/ba∇dblI/ba∇dbls−k3\nsσ/ba∇dblI/ba∇dblk3\ns\ns+σk1+k2+k3=s\n≤ /ba∇dblI/ba∇dbl σ/ba∇dblI/ba∇dbls+σ(3.3)\n/lessorsimilar/ba∇dblI/ba∇dbls+σ,\nwhich implies that by (6.13), (6.14), Q0= Op(q0) andR(2)\nQsatisfy the claimed estimate (6.9). By\ntheestimates (5.26), Lemma2.19, theestimates(6.5)on α,˘α,theproperty(2.47)onΠ 0andLemma\n2.18 (using also (3.3) and λδ−1γ−1≤1), one obtains that also the remainders R(1)\nQandA−1\n⊥RA⊥\nsatisfy the claimed bound (6.9). The claimed estimate (6.9) onR0:=R(1)\nQ+R(2)\nQ+A−1\n⊥RA⊥then\nfollows. The claimed bound (6.10) follows by similar argume nts, using also (5.27), (6.7).\nThe operators Q0= Op(q0) andRare real by Theorem 2.22 and therefore Pis so./square\n6.2.Reduction of the lower order terms. In this section we shall prove the following propo-\nsition in which one reduces the operator P0to a diagonal one plus a smoothing remainder.\nProposition 6.3. Letγ∈(0,1),τ >0,N≥2τ+1,ω∈DC(γ,τ). Then there exists σ≡σN≫0\nsuch that for any S >s0+σthere exists ε=ε(S)≪1such that if (3.3)andλ−δγ−1≤ε(S)are\nfullfilled then the following holds. There exists a real, inve rtible map Vsatisfying\nV±1:Hs\n0→Hs\n0,|V±1|Lip(γ)\n0,s,0/lessorsimilars1+/ba∇dblI/ba∇dblLip(γ)\ns+σ,∀s0≤s≤S−σ, (6.15)\nand such that the linear operator P0in(6.8)transforms as follows:\nP1:=V−1P0V=λω·∇+Z+R1, (6.16)\nwhereZ(ω) := diagk∈Z2\\{0}z(k;ω)is a diagonal operator whose eigenvalues satisfy the estima te\nsup\nk∈Z2\\{0}|z(k;·)|Lip(γ)/lessorsimilarλδM, (6.17)\nand the remainder satisfies\nR1∈ B(Hs\n0,Hs+N\n0),∀s0≤s≤S−σ,\n/ba∇dblR1h/ba∇dblLip(γ)\ns+N/lessorsimilarsλδM/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightBig\n,∀s0≤s≤S−σ.(6.18)\nMoreover, let I1,I2satisfy(3.3). Then\nsup\nk∈Z2\\{0}|∆12z(k)|/lessorsimilarλδM/ba∇dblI1−I2/ba∇dbls0+σ.(6.19)\nFinally, the operators P1,ZandR1are real.\nProposition 6.3 follows by the iterative normal form Lemma 6 .5. Before proving this, we prove a\nlemma in which we give study the homological equations requi red in this normal form procedure.\nLemma 6.4. Letγ∈(0,1),τ >0,m∈R,α∈N0,ω∈DC(γ,τ)anda≡a(·,·;ω)∈ Sm\ns+2τ+1,α.\nThen there exists a unique symbol f∈ Sm\ns,αwith zero average in x, that solves the equation\nλω·∇f(x,ξ)+a(x,ξ) =/a\\}b∇acketle{ta/a\\}b∇acket∇i}htx(ξ). (6.20)\nMoreover\n|Op(f)|Lip(γ)\nm,s,α/lessorsimilarλ−1γ−1|Op(a)|Lip(γ)\nm,s+2τ+1,α. (6.21)\nFinally, if Op(a)is real, then Op(f)is real.\nProof.By expanding in Fourier series, one gets that the only soluti on with zero average of the\nequation (6.20) is given by\nf(x,ξ) =−/summationdisplay\nk∈Z2\\{0}/hatwidea(k,ξ)\niλω·keix·kdefined for ω∈DC(γ,τ).\nBy applying Lemma 2.4, one gets that for any β∈N2,|β| ≤α,\n/ba∇dbl∂β\nξf(·,ξ)/ba∇dblLip(γ)\ns/lessorsimilarλ−1γ−1/ba∇dbl∂β\nξa(·,ξ)/ba∇dblLip(γ)\ns+2τ+1.\nThe claimed bound (6.21) then immediately follows by recall ing Definition 2.10. By the hypothesis\nonawe have that Op( f) is a real operator by definition. /square46 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nLemma 6.5. Letα∈N0,γ∈(0,1),τ >0,N≥2τ+ 1,ω∈DC(γ,τ). Then there exists\nσi≡σi(α,τ)≫0,i= 1,2,...,Nlarge enough, σ1<σ2<...<σ Nsuch that for any S >s0+σN,\nthere exists ε(S)≪1small enough such that if (3.3)holds andλ−δγ−1≤ε(S)≪1, then the\nfollowing holds. For any n= 0,...,N, there exists a linear operator P(n)\n0of the form\nP(n)\n0:=λω·∇+Π⊥\n0Z(n)Π⊥\n0+Π⊥\n0Q(n)\n0Π⊥\n0+R(n)\n0, (6.22)\nwhereZ(n)= Op(z(n)(ξ))∈ OPS0\nS−σn,α,Q(n)\n0= Op(q(n)\n0(x,ξ))∈ OPS−n\nS−σn,αandR(n)\n0satisfy for\nanys0≤s≤S−σn, the estimates\n|Z(n)|Lip(γ)\n0,s,α/lessorsimilars,nλδM(1+/ba∇dblI/ba∇dblLip(γ)\ns0+σn),\n|Q(n)\n0|Lip(γ)\n−n,s,α/lessorsimilars,n,αλδM(1+/ba∇dblI/ba∇dblLip(γ)\ns+σn),\nR(n)\n0∈ B(Hs\n0,Hs+N\n0),\n/ba∇dblR(n)\n0h/ba∇dblLip(γ)\ns+N/lessorsimilars,NλδM/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σn/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightBig\n.(6.23)\nThe operators P(n)\n0,Z(n),Q(n)\n0,R(n)\n0are real. For any n= 1,...,Nthere exists a real, invertible\nmapTndefined for any ω∈DC(γ,τ), satisfying\nT±1\nn:Hs\n0→Hs\n0,|T±1\nn|Lip(γ)\n0,s,α/lessorsimilars,n,α1+/ba∇dblI/ba∇dblLip(γ)\ns+σn,∀s0≤s≤S−σn (6.24)\nand\nP(n)\n0=T−1\nn−1P(n−1)\n0Tn−1∀ω∈DC(γ,τ). (6.25)\nLets1≥s0,α∈N0and assume that I1,I2satisfy(3.3)withs1+σninstead ofs0+σ. Then, for\nanyω∈DC(γ,τ),\n|∆12T±1\nn−1|−n,s1,α/lessorsimilars1,n,α/ba∇dblI1−I2/ba∇dbls1+σn,\n|∆12Z(n)|0,s1,α,|∆12Q(n)\n0|−n,s1,α/lessorsimilars1,n,αλδM/ba∇dblI1−I2/ba∇dbls1+σn.(6.26)\nProof.The proof is made by implementing an induction normal form pr ocedure. We describe the\ninduction step. Assume that the claimed statement holds at t he stepn. We look for a transforma-\ntion\nTn:= exp(Mn,⊥),Mn,⊥:= Π⊥\n0MnΠ⊥\n0,Mn= Op(mn(x,ξ))∈ OPS−n\nwhere the symbol mn(x,ξ) has to be determined in order to normalize Q(n)\n0= Op/parenleftbig\nq(n)\n0(x,ξ)/parenrightbig\n∈\nOPS−n. By conjugating the operator P(n)\n0by means of Tn, one obtains that\nP(n+1)\n0: =T−1\nnP(n)\n0Tn\n=λω·∇+Π⊥\n0Z(n)Π⊥\n0+[λω·∇,Π⊥\n0MnΠ⊥\n0]+Π⊥\n0Q(n)\n0Π⊥\n0\n+Q(n+1)\n0+R(n+1)\n0,(6.27)\nwhere\nQ(n+1)\n0:=/integraldisplay1\n0(1−τ)e−τMn,⊥[[λω·∇,Mn,⊥],Mn,⊥]eτMn,⊥dτ\n+/integraldisplay1\n0e−τMn,⊥[Π⊥\n0Z(n)Π⊥\n0+Π⊥\n0Q(n)\n0Π⊥\n0,Mn,⊥]eτMn,⊥dτ,\nR(n+1)\n0:=T−1\nnR(n)\n0Tn.(6.28)\nNote that\nQ(n+1)\n0= Π⊥\n0Q(n+1)\n0Π⊥\n0and [λω·∇,Π⊥\n0] = 0, (6.29)\ntherefore, the term of order −nis then given by\n[λω·∇,Π⊥\n0MnΠ⊥\n0]+Π⊥\n0Q(n)\n0Π⊥\n0= Π⊥\n0/parenleftBig\n[λω·∇,Mn]+Q(n)\n0/parenrightBig\nΠ⊥\n0\n= Π⊥\n0Op/parenleftBig\nλω·∇mn(x,ξ)+q(n)\n0(x,ξ)/parenrightBig\nΠ⊥\n0.LARGE AMPLITUDE TRAVELING WAVES MHD 47\nWe choose the symbol mn(x,ξ) in such a way that\nλω·∇mn(x,ξ)+q(n)\n0(x,ξ) =/a\\}b∇acketle{tq(n)\n0/a\\}b∇acket∇i}htx(ξ),/a\\}b∇acketle{tq(n)\n0/a\\}b∇acket∇i}htx(ξ) :=1\n(2π)2/integraldisplay\nT2q(n)\n0(x,ξ)dx. (6.30)\nThis can be done by applying Lemma 6.4 and using that, since 1 −δM > δ (see (5.10)) and\nλδM−1γ−1≤λ−δγ−1≤1, one gets\n|Mn|Lip(γ)\n−n,s,α/lessorsimilarλ−1γ−1|Q(n)\n0|Lip(γ)\n−n,s+2τ+1,α(6.23)\n/lessorsimilars,n,αλδM−1γ−1(1+/ba∇dblI/ba∇dblLip(γ)\ns+σn+2τ+1)\n/lessorsimilars,n,α1+/ba∇dblI/ba∇dblLip(γ)\ns+σn+2τ+1,∀s0≤s≤S−σn−2τ−1.\nFurthermore, by using also the composition estimate (2.29) and the estimates (2.47) on Π⊥\n0, the\nlatter estimate together with Lemma 2.12 and the ansatz (3.3 ) imply that\n|Mn,⊥|Lip(γ)\n−n,s,α/lessorsimilars,n,α1+/ba∇dblI/ba∇dblLip(γ)\ns+σn+µ(α),∀s0≤s≤S−σn−µ(α),\nsup\nτ∈[−1,1]|eτMn,⊥|Lip(γ)\n0,s,α/lessorsimilars,α1+/ba∇dblI/ba∇dblLip(γ)\ns+σn+µ(α),∀s0≤s≤S−σn−µ(α),(6.31)\nfor some constant µ(α)≫0 large enough. By inductive hypothesis on q(n)\n0and by using Lemma\n6.4 we have that the maps Mn,Mn,⊥andTnare real. Then by composition and by using the\ninductive hypotesis on P(n)\n0one also has that P(n+1)\n0is real. By (6.27), (6.30), one gets that P(n+1)\n0\ntakes the form\nP(n+1)\n0=λω·∇+Π⊥\n0Z(n+1)Π⊥\n0+Π⊥\n0Q(n+1)\n0Π⊥\n0+R(n+1)\n0,\nZ(n+1):=Z(n)+Op/parenleftbig\n/a\\}b∇acketle{tq(n)\n0/a\\}b∇acket∇i}htx(ξ)/parenrightbig\n,\nQ(n+1)\n0=/integraldisplay1\n0(1−τ)e−τMn,⊥[Π⊥\n0Z(n)Π⊥\n0−Π⊥\n0Q(n)\n0Π⊥\n0,Mn,⊥]eτMndτ\n+/integraldisplay1\n0e−τMn,⊥[Π⊥\n0Z(n)Π⊥\n0+Π⊥\n0Q(n)\n0Π⊥\n0,Mn,⊥]eτMn,⊥dτ,\nR(n+1)\n0=T−1\nnR(n)\n0Tn.(6.32)\nThe estimate of Z(n+1)follows by the induction estimates (6.23) on Z(n),Q(n)\n0and by Lemma\n2.17. The estimate of Q(n+1)\n0can be done by using Lemma 2.11, the estimate (6.31), the indu ction\nestimates (6.23), the estimates (2.47), together with the a nsatz (3.3). The estimate of R(n+1)\n0\nfollows by Lemma 2.18, together with the estimates (6.31) an d the induction estimate on R(n)\n0.\nTheestimate (6.26) at thestep n+1can beprovedby similar arguments. Z(n+1)isreal by inductive\nhypothesis and Lemma 2.17, R(n)\n0is real by composition and by difference Q(n)\n0is so./square\nProof of Proposition 6.3. We set\nV ≡ VN:=T0◦T1◦...◦TN−1.\nTheestimate(6.15)thenfollowsby (6.24), byLemma2.11and byusingtheansatz(3.3). ByLemma\n6.5 and by composition one has that Vis real. Moreover the operator P1≡ P(N)\n0=V−1P0Vis\nof the form (6.16) with Z:= Π⊥\n0Z(N)Π⊥\n0andR1:= Π⊥\n0Q(N)\n0Π⊥\n0+R(N)\n0and by composition it is\na real operator. Then ZandR1satisfies the desired properties (6.17), (6.18), (6.19) by a pplying\n(6.23), (6.26) with n=N.\n6.3.Inversion of the operator P.In this section we prove the invertibility of the operator Pin\n(5.24). We define the non-resonant set\nGλ(γ,τ)≡ Gλ(γ,τ;I) :=/braceleftBig\nω∈DC(γ,τ) :|iλω·k+z(k;ω)| ≥λγ\n|k|τ,∀k∈Z2\\{0}/bracerightBig\n.(6.33)48 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nForτ >0, we fix the constants N,M,δas\nN:= 2τ+2, M= 6(N+1) = 6(2τ+3),\n0<δ<1\n(M+1)(τ+2)(6.34)\nwhere we recall (5.10). We prove the following proposition.\nProposition 6.6. Letγ∈(0,1),τ >0andM,δas in(6.34). Then there exists σ≡σ(τ)≫0\nlarge enough such that for any S >s0+σ, there exists ε(S)≪1small enough such that if (3.3)\nholds andλ−δγ−1≤ε(S)≪1, then the following holds. For any ω∈ Gλ(γ,τ), the operator Pis\ninvertible and its inverse P−1is a real operator which satisfies for any s0≤s≤S−σ, the estimate\n/ba∇dblP−1h/ba∇dblLip(γ)\ns/lessorsimilarsλ−1γ−1/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+2τ+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+2τ+1/parenrightBig\n.\nProof.First of all P−1is real since the operator Pis real. Then, since by Propositions 6.2, 6.3,\nP=A⊥VP1V−1A−1\n⊥, we need to invert the operator P1constructed in Proposition 6.3. We write\nP1=D+R1,D:=λω·∇+Z.\nThe first thing that we discuss is the invertibility of D.\nInvertibility of D.We consider the operator D(ω) = diagk/\\e}atio\\slash=0µ(k;ω) whereµ(k;ω) =iλω·k+\nz(k;ω),k∈Z2\\{0}. Ifω∈ Gλ(γ,τ) (see (6.33)), then |µ(k;ω)| ≥λγ\n|k|τ,j∈Z2\\{0}, implying that\nD(ω)−1= diagk/\\e}atio\\slash=01\nµ(k;ω),\nsatisfies the bound\n/ba∇dblD−1h/ba∇dbls/lessorsimilarλ−1γ−1/ba∇dblh/ba∇dbls+τ,∀s≥0, h∈Hs+τ\n0(T2). (6.35)\nMoreover, if ω1,ω2∈ Gλ(γ,τ), one has that\n/vextendsingle/vextendsingle/vextendsingle1\nµ(k;ω1)−1\nµ(k;ω2)/vextendsingle/vextendsingle/vextendsingle=|µ(k;ω1)−µ(k;ω2)|\n|µ(k;ω1)||µ(k;ω2)|\n≤λ−1γ−2|k|2τ+1|ω1−ω2|+λ−2γ−2|k|2τ|z(k;ω1)−z(k;ω2)|\n(6.17)\n/lessorsimilarλ−1γ−2|k|2τ+1|ω1−ω2|+λMδ−2γ−3|k|2τ|ω1−ω2|\nλMδ−1γ−1≪1\n/lessorsimilarλ−1γ−2|k|2τ+1|ω1−ω2|.\nThe latter estimate then implies that\n/ba∇dbl(D(ω1)−1−D(ω2)−1)h/ba∇dbls/lessorsimilarγ−2λ−1/ba∇dblh/ba∇dbls+2τ+1|ω1−ω2|,∀s≥0,∀h∈Hs+2τ+1\n0(T2).(6.36)\nThe estimates (6.35), (6.36) imply that\n/ba∇dblD−1/ba∇dblLip(γ)\nB(Hs+2τ+1\n0,Hs\n0)/lessorsimilarλ−1γ−1,∀s≥0. (6.37)\nInvertibility of P1.Forω∈ Gλ(γ,τ), we write\nP1=D/parenleftbig\nId+F/parenrightbig\n,F:=D−1R1.\nBy the hypotheses of the proposition, we choose the order of r egularization in Proposition 6.3 as\nN= 2τ+2≥2τ+1. By (5.10), since 0 <δ<1\n1+M,λδM−1γ−1≤λ−δγ−1, hence, by the estimates\n(6.18), (6.37), one has that for for any s0≤s≤S−σ,\n/ba∇dblFh/ba∇dblLip(γ)\ns≤C(s)λδM−1γ−1/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightBig\n,\n≤C(s)λ−δγ−1/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightBig\n,\n/ba∇dblFh/ba∇dblLip(γ)\ns0≤C(s0)λδM−1γ−1/ba∇dblh/ba∇dblLip(γ)\ns0≤C(s0)λ−δγ−1/ba∇dblh/ba∇dblLip(γ)\ns0LARGE AMPLITUDE TRAVELING WAVES MHD 49\nforσ≫0 andC(s)≫C(s0)≫1. By iterating the latter estimate, using (3.3), one gets th at for\nanyn∈N,\n/ba∇dblFnh/ba∇dblLip(γ)\ns≤/parenleftBig\nC(s)λ−δγ−1/parenrightBign/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0/parenrightBig\n,∀s0≤s≤S−σ,\n/ba∇dblFnh/ba∇dblLip(γ)\ns0≤/parenleftBig\nC(s0)λ−δγ−1/parenrightBign\n/ba∇dblh/ba∇dblLip(γ)\ns0\nHence for any s0≤s≤S−σ, by using the smallness condition λ−δγ−1≤ε(S)≪1, the operator\nId+Fis invertible by Neumann series and\n/ba∇dbl(Id+F)−1h/ba∇dblLip(γ)\ns/lessorsimilars/ba∇dblh/ba∇dblLip(γ)\ns+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0,∀s0≤s≤S−σ. (6.38)\nThen by the estimates (6.37), (6.38), the operator P−1\n1= (Id+F)−1D−1satisfies the bound\n/ba∇dblP−1\n1h/ba∇dblLip(γ)\ns/lessorsimilarsγ−1λ−1/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+2τ+1+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+2τ+1/parenrightBig\n,∀s0≤s≤S−σ.(6.39)\nEstimate of P−1.By using that P−1=AVP−1\n1V−1A−1, the estimate (6.39), together with the\nestimates (6.5), (6.15) and Lemmata 2.19, 2.18 (to estimate A,V) imply the claimed bound on\nP−1. /square\n7.Inversion of the linearized operator L\nWe now invert the linear operator L1in (4.69). The following proposition holds.\nProposition 7.1. Letγ∈(0,1),τ >0andM,δas in(6.34). Then there exists σ≡σ(τ)≫0\nlarge enough such that for any S >s0+σ, there exists ε(S)≪1small enough such that if (3.3)\nholds andλ−δγ−1≤ε(S)≪1, then the following holds. For any ω∈ Gλ(γ,τ), the operator L1is\ninvertible and its inverse L−1\n1is a real operator and satisfies the estimate\n/ba∇dblL−1\n1h/ba∇dblLip(γ)\ns/lessorsimilarsλ−δM/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+σ+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+σ/parenrightBig\n,∀s0≤s≤S−σ. (7.1)\nProof.To shorten notations we write /ba∇dbl·/ba∇dblsinstead of /ba∇dbl·/ba∇dblLip(γ)\ns. By Proposition 6.6, one gets that\nforω∈ Gλ(γ,τ), the operator Pis invertible and hence the equation (5.23) can be solved by s etting\nh2=P−1[g2]−P−1L(3)\n−N(L(1)\n1)−1[g1].\nBy replacing the latter expression of h2in formula (5.22), one obtains that\nh1= (L(1)\n1)−1g1−(L(1)\n1)−1L(2)\n−NP−1[g2]+(L(1)\n1)−1L(2)\n−NP−1L(3)\n−N(L(1)\n1)−1[g1],\nand hence for any ω∈ Gλ(γ,τ)\nL−1\n1=/parenleftBigg\n(L(1)\n1)−1+(L(1)\n1)−1L(2)\n−NP−1L(3)\n−N(L(1)\n1)−1−(L(1)\n1)−1L(2)\n−NP−1\n−P−1L(3)\n−N(L(1)\n1)−1P−1/parenrightBigg\n.\nBy applying (5.12) in Lemma 5.2 to estimate ( L(1)\n1)−1, Proposition 6.6 to estimate P−1, the esti-\nmates (4.70) on L(2)\n−N,L(3)\n−N, using also (3.3), one obtains that for any s0≤s≤S−σ\n/ba∇dbl(L(1)\n1)−1L(2)\n−NP−1L(3)\n−N(L(1)\n1)−1h/ba∇dbls,/ba∇dbl(L(1)\n1)−1L(2)\n−NP−1h/ba∇dbls,/ba∇dblP−1L(3)\n−N(L(1)\n1)−1h/ba∇dbls\n/lessorsimilarsλ−1γ−1/parenleftBig\n/ba∇dblh/ba∇dbls+σ+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+σ/parenrightBig\n.\nforσ≡σ(τ)≫0. By sing that λ−1γ−1≤λ−δM, since by (5.10), 0 <δ<1\nM, one then obtains the\nclaimed bound on L−1\n1. /square\nWe can finally invert the linearized operator Lin (3.9). The following Proposition holds.\nProposition 7.2. Letγ∈(0,1),τ >0andM,δas in(6.34). Then there exists σ≡σ(τ)≫\n2τ+ 1≫0large enough such that for any S > s0+σ, there exists ε(S)≪1small enough such\nthat if(3.3)holds andλ−δγ−1≤ε(S)≪1, then the following holds. For any ω∈ Gλ(γ,τ), the\noperator Lis invertible and its inverse L−1satisfies for any s0≤s≤S−σ, the estimate\n/ba∇dblL−1h/ba∇dblLip(γ)\ns/lessorsimilarsλ−δM/parenleftBig\n/ba∇dblh/ba∇dblLip(γ)\ns+σ+/ba∇dblI/ba∇dblLip(γ)\ns+σ/ba∇dblh/ba∇dblLip(γ)\ns0+σ/parenrightBig\n. (7.2)50 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nProof.First of all L−1is real since the operator Lis real. Then, by Propositions 4.6, 7.1, for any\nω∈ Gλ(γ,τ), one has that L−1=ΦNL−1\n1Φ−1\nN. Then the claimed bound follows by the estimate\n(4.68), Lemma 2.18 and by the estimate (7.1), using also (3.3 ). /square\n8.Construction of an approximate solution\nIn this section we construct an approximate solution of the n onlinear equation F(Ω,J) = 0\nwhere the nonlinear operator Fis defined in (1.14). This is the starting point to implement t he\nnonlinear Nash-Moser scheme of Section 9. The following pro position holds.\nProposition 8.1 (Approximate solutions). Letλ>1,γ∈(0,1),τ >0,δ∈(0,1)and assume\nthatλ−δ\n3γ−1≤1. Then, there exists Iapp(·;ω) := (Ω app(·;ω),0),Ωapp∈ C∞(T2)with zero average\ndefined for ω∈DC(γ,τ)such that Ωapp/\\e}atio\\slash= 0and for any s≥s0\n/ba∇dblIapp/ba∇dblLip(γ)\ns=/ba∇dblΩapp/ba∇dblLip(γ)\ns/lessorsimilarsλ−2\n3δγ−1,/ba∇dblF(Iapp)/ba∇dblLip(γ)\ns/lessorsimilarsλ−δ\n3γ−2. (8.1)\nMoreover, for any s≥2, one has the following lower bound on the approximate soluti on\ninf\nω∈DC(γ,τ)/ba∇dblIapp(·;ω)/ba∇dbls= inf\nω∈DC(γ,τ)/ba∇dblΩapp(·;ω)/ba∇dbls/greaterorsimilarsλ−2\n3δ. (8.2)\nMoreover, there exists K(f,b)>0such that\n/ba∇dblb·∇Ωapp/ba∇dblL2≥λ−2\n3δK(f,b). (8.3)\nProof.We determine Ω appas the only solution with zero average of the following equat ion\nLλΩapp=λ1−2\n3δF,\nLλ:=λω·∇−∆ = diagk∈Z2\\{0}Λ(k),\nΛ(k)≡Λ(k;ω) =iλω·k+|k|2, k∈Z2\\{0}.(8.4)\nBy using that ω∈DC(γ,τ), one has that\nL−1\nλ= diagk∈Z2\\{0}1\niλω·k+|k|2,\nand\n|Λ(k)|=|iλω·k+|k|2| ≥λ|ω·k| ≥λγ|k|−τ,∀k∈Z2\\{0}. (8.5)\nMoreover if ω1,ω2∈DC(γ,τ),k∈Z2\\{0}, one estimates\n|Λ(k;ω1)−1−Λ(k;ω2)−1| ≤|Λ(k;ω1)−Λ(k;ω2)|\n|Λ(k;ω1)||Λ(k;ω2)|\n(8.5)\n/lessorsimilarλ−1γ−2|k|2τ+1|ω1−ω2|.(8.6)\nThe bounds (8.5), (8.6) easily imply that\n/ba∇dblL−1\nλ/ba∇dblB(Hs+τ\n0,Hs\n0),/ba∇dblL−1\nλ/ba∇dblLip(γ)\nB(Hs+2τ+1\n0,Hs\n0)/lessorsimilarλ−1γ−1,∀s≥0.\nTherefore Ω app:=λ1−2\n3δL−1\nλFsatisfies the bound\n/ba∇dblΩapp/ba∇dblLip(γ)\ns/lessorsimilarλ−2\n3δγ−1/ba∇dblF/ba∇dblLip(γ)\ns+2τ+1/lessorsimilarsλ−2\n3δγ−1,∀s≥0. (8.7)\nClearly, by the Assumption (1.7), the forcing term F/\\e}atio\\slash= 0 and hence one also has that Ω app/\\e}atio\\slash= 0.\nBy (8.4) and by recalling (1.14), one has that\nF(Ωapp,0) =/parenleftbigg\nλδUapp·∇Ωapp\n−b·∇Ωapp/parenrightbigg\n,\nUapp=UΩapp= (−∆)−1∇⊥Ωapp.\nClearly the estimates (8.7) also implies that\n/ba∇dblUapp/ba∇dblLip(γ)\ns/lessorsimilarsλ−2\n3δγ−1,∀s≥0. (8.8)LARGE AMPLITUDE TRAVELING WAVES MHD 51\nHence, the estimates (8.7), (8.8), together with the interp olation estimate (2.8), allow to deduce\nthat\nλδ/ba∇dblUapp·∇Ωapp/ba∇dblLip(γ)\ns/lessorsimilarsλ−δ\n3γ−2,/ba∇dblb·∇Ωapp/ba∇dblLip(γ)\ns/lessorsimilarsλ−2\n3δγ−1,∀s≥0.(8.9)\nSinceλ−δ\n3γ−1≤1,λ>1,γ∈(0,1),δ∈(0,1) one has\nλ−2\n3δγ−1≤λ−δ\n3γ−2\ntherefore the estimates (8.7), (8.9) imply the claimed uppe r bound (8.2) on Ω app,F(Iapp). Now,\nwe prove the lower bound on Ω app. One has that\nΩapp(x) =λ1−2\n3δL−1\nλF=λ1−2\n3δ/summationdisplay\nk∈Z2\\{0}/hatwideF(k)\niλω·k+|k|2eix·k. (8.10)\nNote that for λ>1 large enough, k∈Z2\\{0}, one has the estimate\n|iλω·k+|k|2|/lessorsimilarλ|ω·k|+|k|2/lessorsimilarλ|k|+|k|2/lessorsimilarλ|k|2, (8.11)\ntherefore\n/ba∇dblΩapp/ba∇dbl2\ns=λ2(1−2\n3δ)/summationdisplay\nk∈Z2\\{0}|k|2s\n|iλω·k+|k|2|2|/hatwideF(k)|2\n/greaterorsimilarλ2(1−2\n3δ)λ−2/summationdisplay\nk∈Z2\\{0}|k|2(s−2)|/hatwideF(k)|2\n/greaterorsimilarλ−4\n3δ/ba∇dblF/ba∇dbl2\ns−2/greaterorsimilarsλ−4\n3δ\nwhich then implies the lower bound (8.2). We now prove (8.3). By the assumption (1.7), one has\nthat there exists k∈Z2\\{0}such that\nb·k/\\e}atio\\slash= 0,/hatwideF(k)/\\e}atio\\slash= 0. (8.12)\nHence, by (8.10)\n/ba∇dblb·∇Ωapp/ba∇dblL2/greaterorsimilarλ1−2\n3δ/parenleftBig/summationdisplay\nk∈Z2\\{0}|b·k|2|/hatwideF(k)|2\n|iλω·k+|k|2|2/parenrightBig1\n2\n(8.11)\n/greaterorsimilarλ−2\n3δ/parenleftBig/summationdisplay\nk∈Z2\\{0}|b·k|2|/hatwideF(k)|2\n|k|4/parenrightBig1\n2\n≥Cλ−2\n3δ|b·k|2|/hatwideF(k)|2\n|k|2.\nThe claimed bound then follows by defining K(f,b) :=C|b·k|2|/hatwideF(k)|2\n|k|2>0, by (8.12). The proof\nof the Lemma is then concluded. /square\n9.The Nash Moser scheme\nIn this section we construct solutions of the equation F(I) = 0,I= (Ω,J) in (1.14) by means\nof a Nash Moser nonlinear iteration. We define the constants\nN0>0, Nn:=Nχn\n0, n≥0, N−1:= 1, χ:= 3/2,\nτ >0, M:= 6(2τ+3),0<δ<1\n(M+1)(τ+2),\nµ:= 3σ+3,a:= max{3µ+1,2(2σ+τ+1)},b:=µ+a+1(9.1)\nwhere ¯σ≫1 is given in Proposition 7.2 and we recall (6.34). We denote b y Πnthe orthogonal\nprojector Π Nn(see (2.5)) on the finite dimensional space\nHn:=/braceleftbig\nI ∈L2\n0(T2,R2) :I= ΠnI= (ΠnΩ,ΠnJ)/bracerightbig\n,52 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nand Π⊥\nn:= Id−Πn. The projectors Π n, Π⊥\nnsatisfy the usual smoothing properties (cf. Lemma\n2.2), namely\n/ba∇dblΠnI/ba∇dblLip(γ)\ns+a≤Na\nn/ba∇dblI/ba∇dblLip(γ)\ns,/ba∇dblΠ⊥\nnI/ba∇dblLip(γ)\ns≤N−a\nn/ba∇dblI/ba∇dblLip(γ)\ns+a, s,a≥0. (9.2)\nNotethatProposition8.1providesanapproximatesolution Iappofthefunctionalequation F(I) = 0\nand such that /ba∇dblF(Iapp)/ba∇dblLip(γ)\nsis uniformly bounded with respect to the large parameter λ≫1 for\nanys≥0. The following lemma is a direct consequence of Propositio n 8.1.\nLemma 9.1. (Initialization of the Nash-Moser iteration). Letλ >1,γ∈(0,1),τ >0,\nδ∈(0,1)and assume that λ−δ\n3γ−2≤1.For anys≥0, there exists a constant C(s)>0such that\n/ba∇dblF(Iapp)/ba∇dblLip(γ)\ns,/ba∇dblIapp/ba∇dblLip(γ)\ns≤C(s),\ninf\nω∈DC(γ,τ)/ba∇dblIapp(·;ω)/ba∇dbls= inf\nω∈DC(γ,τ)/ba∇dblΩapp(·;ω)/ba∇dbls≥C(s)λ−2\n3δ, s≥2(9.3)\nuniformly w.r. to λ.\nProposition 9.2. (Nash-Moser) There exist ε∈(0,1),N0>0,λ>0,C∗>0such that if\nλ≥λ, λ−δ\n3γ−2≤ε, (9.4)\nthen the following properties hold for all n≥0.\n(P1)nThere exists a constant C0>0large enough and a sequence (In)n≥0withI0:=Iapp,\nIn−I0:Gn→ Hn−1, satisfying\n/ba∇dblIn/ba∇dblLip(γ)\ns0+σ≤C0. (9.5)\nIfn≥1, the difference hn:=In−In−1satisfies\n/ba∇dblhn/ba∇dblLip(γ)\ns0+σ/lessorsimilarN2σ\nn−1N−a\nn−2λ−δM,forn≥1. (9.6)\nThe sets {Gn}n≥0are defined as follows. If n= 0we define G0:= DC(γ,τ)(recall(2.10)). If\nn≥1, we define\nGn+1:=Gn∩Gλ(γn,τ;In) (9.7)\nwhereγn:=γ(1+2−n)and the sets Gλ(γn,τ;In)are defined in (6.33).\n(P2)nOn the set Gn, one has the estimate\n/ba∇dblF(In)/ba∇dblLip(γ)\ns0≤C∗N−a\nn−1; (9.8)\n(P3)nOn the set Gn, one has the estimate\n/ba∇dblIn/ba∇dblLip(γ)\ns0+b+/ba∇dblF(In)/ba∇dblLip(γ)\ns0+b≤C∗Na\nn−1 (9.9)\nProof.To simplify notations, in this proof we write /ba∇dbl·/ba∇dblsinstead of /ba∇dbl·/ba∇dblLip(γ)\ns.\nProof of (P1,2,3)0. By (9.3), we have /ba∇dblI0/ba∇dbls=/ba∇dblIapp/ba∇dbls≤C(s) and/ba∇dblF(I0)/ba∇dbls=/ba∇dblF(Iapp)/ba∇dbls≤\nC(s) for anys≥0. Then (9.5), (9.8) and (9.9) hold taking1\n2C0,C∗(s0+σ)≥C(s0+σ). In\nparticular, we have\n/ba∇dblI0/ba∇dbls0+¯σ≤1\n2C0≤C0. (9.10)\nAssume that (P1,2,3)nhold for some n≥0, and prove (P1,2,3)n+1.By (P1)n, one\nhas/ba∇dblIn/ba∇dbls0+¯σ≤C0, for someC0≫0 large enough. The assumption (9.4) implies the smallness\nconditionλ−δγ−1≪1 of Proposition 7.2. Indeed γ∈(0,1),δ >δ\n3,λ≫1 implies that λ−δγ−1≤\nλ−δ\n3γ−2≪1. Then Proposition 7.2 applies to the linearized operator\nLn:=DF(In). (9.11)\nThis implies that, for any ω∈ Gn+1, the operator Ln(ω)≡ L(ω,In(ω)) admits a right inverse\nLn(ω)−1satisfying the tame estimates, for any s0≤s≤s0+b+1 (choose S:=s0+b+1),\n/ba∇dblL−1\nnh/ba∇dbls/lessorsimilarsλ−δM/parenleftbig\n/ba∇dblh/ba∇dbls+σ+/ba∇dblIn/ba∇dbls+σ/ba∇dblh/ba∇dbls0+σ/parenrightbig\n, (9.12)LARGE AMPLITUDE TRAVELING WAVES MHD 53\nusing the bound γn=γ(1+2−n)∈[γ,2γ]. Note that since Iapp=I0∈ C∞(T2,R2) satisfies (9.3),\nsinceIn−I0∈ Hn−1, by (9.2) one obtains that for any s,µ≥0,\n/ba∇dblIn/ba∇dbls+µ≤ /ba∇dblIapp/ba∇dbls+µ+/ba∇dblIn−Iapp/ba∇dbls+µ\n/lessorsimilars,µ1+Nµ\nn−1/ba∇dblIn−Iapp/ba∇dbls\n/lessorsimilars,µ1+Nµ\nn−1(/ba∇dblIn/ba∇dbls+/ba∇dblIapp/ba∇dbls)\n/lessorsimilars,µNµ\nn(1+/ba∇dblIn/ba∇dbls).(9.13)\nBy (9.12), (9.5) and (9.13), for s=s0ands=s0+1, one gets that one has\n/ba∇dblL−1\nnh/ba∇dbls0/lessorsimilarλ−δM/ba∇dblh/ba∇dbls0+σ,\n/ba∇dblL−1\nnh/ba∇dbls0+1/lessorsimilarλ−δM/parenleftBig\n/ba∇dblh/ba∇dbls0+σ+1+/ba∇dblIn/ba∇dbls0+σ+1/ba∇dblh/ba∇dbls0+σ/parenrightBig\n/lessorsimilarNnλ−δM/ba∇dblh/ba∇dbls0+σ+1.(9.14)\nWe define the next approximation as\nIn+1:=In+hn+1, hn+1:=−ΠnL−1\nnΠnF(In)∈ Hn, (9.15)\ndefined for any ω∈ Gn+1, and the remainder\nQn:=F(In+1)−F(In)−Lnhn+1.\nWe now estimate hn+1. By the estimates (9.12), (9.14), (9.5), (9.2), (9.1), (9.4 ), (9.13) and (using\nalso thatγ−1=N0≤Nn), we get that\n/ba∇dblhn+1/ba∇dbls0/lessorsimilarλ−δM/ba∇dblΠnF(In)/ba∇dbls0+σ/lessorsimilarN¯σ\nnλ−δM/ba∇dblF(In)/ba∇dbls0\n/ba∇dblhn+1/ba∇dbls0+¯σ/lessorsimilarN¯σ\nn/ba∇dblL−1\nnΠnF(In)/ba∇dbls0/lessorsimilarN2¯σ\nnλ−δM/ba∇dblF(In)/ba∇dbls0,(9.16)\nand\n/ba∇dblhn+1/ba∇dbls0+b/lessorsimilarλ−δM/parenleftBig\n/ba∇dblΠnF(In)/ba∇dbls0+b+σ+/ba∇dblIn/ba∇dbls0+b+σ/ba∇dblΠnF(In)/ba∇dbls0+σ/parenrightBig\n/lessorsimilarλ−δMN2¯σ\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b−2+/ba∇dblIn/ba∇dbls0+b/ba∇dblF(In)/ba∇dbls0/parenrightBig\n/lessorsimilarλ−δMN2¯σ\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+C∗N−a\nn−1/ba∇dblIn/ba∇dbls0+b/parenrightBig\n/lessorsimilarN2¯σ\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightBig\n.(9.17)\nMoreover, (9.15) and the estimate (9.17) imply that\n/ba∇dblIn+1/ba∇dbls0+b≤ /ba∇dblIn/ba∇dbls0+b+/ba∇dblhn+1/ba∇dbls0+b/lessorsimilarN2¯σ\nn/parenleftbig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightbig\n.(9.18)\nNext, we estimate /ba∇dblF(In+1)/ba∇dbls0and/ba∇dblF(In+1)/ba∇dbls0+b. By the definition of hn+1in (9.15), we obtain\nthat, for any ω∈ Gn+1,\nF(In+1) =F(In)+Lnhn+1+Qn\n=F(In)−LnΠnL−1\nnΠnF(In)+Qn\n=F(In)−ΠnLnL−1\nnΠnF(In)+[Ln,Πn]L−1\nnΠnF(In)+Qn\n= Π⊥\nnF(In)+[Ln,Πn]L−1\nnΠnF(In)+Qn\nΠn=Id−Π⊥\nn= Π⊥\nnF(In)+[Π⊥\nn,Ln]L−1\nnΠnF(In)+Qn(9.19)\nWe estimate separately the three terms in (9.19). By (9.2), w e have\n/ba∇dblΠ⊥\nnF(In)/ba∇dbls0≤N−b\nn/ba∇dblF(In)/ba∇dbls0+b,/ba∇dblΠ⊥\nnF(In)/ba∇dbls0+b≤ /ba∇dblF(In)/ba∇dbls0+b. (9.20)54 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nBy (9.5), (9.8), Lemma 3.2-( ii), (9.2), (9.12), (9.4), (9.13), we have\n/ba∇dbl[Π⊥\nn,Ln]L−1\nnΠnF(In)/ba∇dbls0/lessorsimilarλδN1−b\nn/parenleftBig\n/ba∇dblL−1\nnΠnF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/ba∇dblL−1\nnΠnF(In)/ba∇dbls0+1/parenrightBig\n/lessorsimilarλδN1−b\nnλ−δM/parenleftBig\n/ba∇dblΠnF(In)/ba∇dbls0+b+¯σ\n+/ba∇dblIn/ba∇dbls0+b+¯σ/ba∇dblΠnF(In)/ba∇dbls0+¯σ+Nn/ba∇dblΠnF(In)/ba∇dbls0+¯σ+1/parenrightBig\n/lessorsimilarN2−b\nnλ−(M−1)δ/parenleftBig\n/ba∇dblΠnF(In)/ba∇dbls0+b+¯σ\n+/ba∇dblIn/ba∇dbls0+b+¯σ/ba∇dblΠnF(In)/ba∇dbls0+¯σ+/ba∇dblΠnF(In)/ba∇dbls0+¯σ+1/parenrightBig\n/lessorsimilarN2¯σ+3−b\nnλ−(M−1)δ/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+C∗N−a\nn−1/ba∇dblIn/ba∇dbls0+b/parenrightBig\n/lessorsimilarN2¯σ+3−b\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightBig\nand\n/ba∇dbl[Π⊥\nn,Ln]L−1\nnΠnF(In)/ba∇dbls0+b=/ba∇dbl[Ln,Πn]L−1\nnΠnF(In)/ba∇dbls0+b\n/lessorsimilarλδNn/parenleftBig\n/ba∇dblL−1\nnΠnF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/ba∇dblL−1\nnΠnF(In)/ba∇dbls0+1/parenrightBig\n/lessorsimilarNnλ−(M−1)δ/parenleftBig\n/ba∇dblΠnF(In)/ba∇dbls0+b+¯σ+/ba∇dblIn/ba∇dbls0+b+¯σ/ba∇dblΠnF(In)/ba∇dbls0+¯σ/parenrightBig\n+Nnλ−(M−1)δ/ba∇dblIn/ba∇dbls0+b/ba∇dblΠnF(In)/ba∇dbls0+¯σ+1\nM>1,λ≫1\n/lessorsimilarN2¯σ+2\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/ba∇dblF(In)/ba∇dbls0/parenrightBig\n/lessorsimilarN2¯σ+2\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+C∗N−a\nn−1/ba∇dblIn/ba∇dbls0+b/parenrightBig\n/lessorsimilarN2¯σ+2\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightBig\n.\n(9.21)\nBy Lemma 3.2-( iii), (9.2), (9.5), (9.16), (9.17) and (9.4), we have\n/ba∇dblQn/ba∇dbls0/lessorsimilarλδN2\nn/ba∇dblhn+1/ba∇dbl2\ns0/lessorsimilarλδN2¯σ+2\nnλ−2δM/ba∇dblF(In)/ba∇dbl2\ns0\n/lessorsimilarN2¯σ+2\nnλ−(2M−1)δ/ba∇dblF(In)/ba∇dbl2\ns0and\n/ba∇dblQn/ba∇dbls0+b/lessorsimilarλδ/ba∇dblhn+1/ba∇dbls0+b+1/ba∇dblhn+1/ba∇dbls0+1/lessorsimilarN2\nnλδ/ba∇dblhn+1/ba∇dbls0+b/ba∇dblhn+1/ba∇dbls0\n/lessorsimilarN2¯σ+2\nnλδ/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightBig\nN¯σ\nnλ−δM/ba∇dblF(In)/ba∇dbls0\n/lessorsimilarN3¯σ+2\nnλ−(M−1)δ/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightBig\nN−a\nn−1\n/lessorsimilarN3¯σ+2\nn/parenleftBig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightBig(9.22)\nTherefore, by collecting (9.18)-(9.22), we have proved the following inductive inequalities for any\nn≥0 and on the set Gn+1:\n/ba∇dblF(In+1)/ba∇dbls0+b+/ba∇dblIn+1/ba∇dbls0+b/lessorsimilarN¯µ\nn/parenleftbig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightbig\n,\n/ba∇dblF(In+1)/ba∇dbls0/lessorsimilarN¯µ−b\nn/parenleftbig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightbig\n+N¯µ\nnλ−(2M−1)δ/ba∇dblF(In)/ba∇dbl2\ns0,\n¯µ:= 3¯σ+3.(9.23)\nProof of (P1)n+1.By (9.16), (9.8), (9.4) and (9.1), we obtain\n/ba∇dblhn+1/ba∇dbls0+¯σ/lessorsimilarN2¯σ\nnN−a\nn−1λ−δM(9.24)LARGE AMPLITUDE TRAVELING WAVES MHD 55\nwhich proves (9.6) at the step n+1. By (9.10), by the definition of the constants in (9.1) and b y\nthe smallness condition in (9.4), the estimate (9.5) at the s tepn+1 follows since\n/ba∇dblIn+1/ba∇dbls0+¯σ≤ /ba∇dblI0/ba∇dbls0+¯σ+n+1/summationdisplay\nk=1/ba∇dblhk/ba∇dbls0+¯σ\n≤1\n2C0+C∗∞/summationdisplay\nk=1N2σ\nk−1N−a\nk−2λ−δM≤1\n2C0+C1λ−δM≤C0(9.25)\nforλ≫1 large enough.\nProof of (P2)n+1,(P3)n+1.By the induction estimate (9.23) and using the induction hyp othesis\non (P2)n,(P3)n, we obtain that\n/ba∇dblF(In+1)/ba∇dbls0+b+/ba∇dblIn+1/ba∇dbls0+b/lessorsimilarN¯µ\nn/parenleftbig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightbig\n≤CC∗N¯µ\nnNa\nn−1≤C∗Na\nn\nby the choice of the constants in (9.1) and taking N0≫1 large enough. This latter chain of\ninequalities proves ( P3)n+1. Moreover, by (9.23) and by the induction estimates (9.8), ( 9.9), we\nhave\n/ba∇dblF(In+1)/ba∇dbls0/lessorsimilarN¯µ−b\nn/parenleftbig\n/ba∇dblF(In)/ba∇dbls0+b+/ba∇dblIn/ba∇dbls0+b/parenrightbig\n+N¯µ\nnλ−(2M−1)δ/ba∇dblF(In)/ba∇dbl2\ns0\n≤CC∗N¯µ−b\nnNa\nn−1+CC2\n∗N¯µ\nnN−2a\nn−1λ−(2M−1)δ\n≤C∗N−a\nn\nprovided\nNb−¯µ−a\nn≥2C, λ−(2M−1)δ≤N2a\nn−1N−a−¯µ\nn\n2CC∗,∀n≥0.\nThe latter condition is verified by the smallness condition ( 9.4), forN0,λ≫1 large enough and by\nthe choice of the constants in (9.1). The proof of the claimed statement is then concluded. /square\n10.Measure estimates\nIn this section we estimate the measure of the resonant set O \\G∞where the set G∞is defined\nas\nG∞:=∩n≥0Gn (10.1)\nwhere the sets Gnare given in Proposition 9.2. We recall that we denote by M(d)the Lebesgue\nmeasure on Rd. The main result of this section is the following\nProposition 10.1. Let\nτ:= 2. (10.2)\nUnder the same assumption of Proposition 9.2, we have that M(2)(O\\G∞)/lessorsimilarγ.\nThe rest of this section is devoted to the proof of Propositio n 10.1. By the definition (10.1), one\nhas\nO\\G∞⊆(O\\G0)∪/uniondisplay\nn≥0(Gn\\Gn+1). (10.3)\nHence, it is enough to estimate the measure of O\\G0andGn\\Gn+1for anyn≥0. By (9.7), (6.33),\none gets that G0= DC(γ,2), therefore it is standard the fact that\nM(2)(O\\G0)/lessorsimilarγ. (10.4)\nMoreover for n≥0, one has that Gn\\Gn+1can be written as\nGn\\Gn+1=/uniondisplay\nk∈Z2\\{0}Rk(In),\nRk(In) :=/braceleftBig\nω∈ Gn:|iλω·k+z(k;ω,In(ω))|<λγn\n|k|2/bracerightBig\n, k∈Z2\\{0}.(10.5)\nIn the next lemma, we estimate the measure of the resonant set sRk(In),k∈Z2\\{0}defined\nin (10.5).56 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nLemma 10.2. For anyn≥0, one has that\nM(2)(Rk(In))/lessorsimilarγ\n|k|3.\nProof.Fork∈Z2\\{0}, we writez(k;ω)≡z(k;ω,In(ω)), and we set\nφ(ω) :=iλω·k+z(k;ω).\nSincek/\\e}atio\\slash= 0, we write\nω=k\n|k|s+v, v·k= 0,\nand we estimate the measure of the set\nQ:=/braceleftBig\ns:|ψ(s)|<γnλ\n|k|2,k\n|k|s+v∈ Gn/bracerightBig\n,\nψ(s) :=φ/parenleftBigk\n|k|s+v/parenrightBig\n=iλ|k|s+z(k;s),\nz(k;s)≡z/parenleftBig\nk;k\n|k|s+v/parenrightBig\n.\nBy the estimate (6.17) of Proposition 6.3, one has that\n|z(k;s1)−z(k;s2)|/lessorsimilarγ−1λδM|s1−s2|,∀k∈Z2\\{0},\nand therefore\n|ψ(s1)−ψ(s2)| ≥/parenleftBig\nλ|k|−CλδMγ−1/parenrightBig\n|s1−s2| ≥λ|k|\n2|s1−s2|,\nsince, by (9.1), (9.4), one has that λδM−1γ−1≤λ−δ\n3γ−2≪1. This implies that the measure of the\nsetQsatisfies\nM(1)(Q)/lessorsimilarλγn\nλ|k|3/lessorsimilarγ\n|k|3\nand by the Fubini theorem one also gets that the claimed bound on the measure M(2)(Rk(In))./square\nLemma 10.3. Letn≥1,k∈Z2\\{0},|k| ≤Nn−1. ThenRk(In) =∅.\nProof.Letω∈ Gn. We shall prove that\n|iλω·k+z(k;ω,In(ω))| ≥λγn\n|k|2, k∈Z2\\{0},|k| ≤Nn−1, (10.6)\nimplying that Rk(In) =∅. We shall prove the bound (10.6). By applying the estimate (6 .19) in\nProposition 6.3, one has\n|iλω·k+z(k;ω,In(ω))| ≥ |iλω·k+z(k;ω,In−1(ω))|−|z(k;ω,In(ω))−z(k;ω,In−1(ω))|\n≥λγn−1\n|k|2−CλδM/ba∇dblIn−In−1/ba∇dbls0+σ\n(9.6)\n≥λγn−1\n|k|2−CN2σ\nn−1N−a\nn−2≥λγn\n|k|2,\nprovided that\nC|k|2N2σ\nn−1\nNa\nn−2λ(γn−1−γn)≤1,∀0<|k| ≤Nn−1.\nSince|k| ≤Nn−1andγn−1−γn=γ\n2nand 2n/lessorsimilarNn−1, the latter condition reads\nC′N2σ+3\nn−1N−a\nn−2λ−1γ−1≤1.\nfor some constant C′≫0 large enough. This latter condition is satisfied by the smal lness condition\n(9.4) and since a≥2(2σ+τ+1)>χ(2σ+τ+1) =χ(2σ+3) (recall (9.1) and that τ= 2). The\nclaimed bound (10.6) has then been proved. /square\nProposition 10.4. One has M(2)(G0\\G1)/lessorsimilarγand for any n≥1,M(2)(Gn\\Gn+1)/lessorsimilarγN−3\nn−1.LARGE AMPLITUDE TRAVELING WAVES MHD 57\nProof.By (10.5), Lemma 10.2 and by recalling (10.2), one gets\nM(2)(G0\\G1)≤/summationdisplay\nk∈Z2\\{0}M(2)(Rk(I0))/lessorsimilarγ/summationdisplay\nk∈Z2\\{0}1\n|k|3/lessorsimilarγ.\nMoreover, (10.5) and Lemma 10.3 imply that for any n≥1\nGn\\Gn+1⊆/uniondisplay\n|k|≥Nn−1Rk(In)\nand hence, using again Lemma 10.2 and (10.2), one gets\nM(2)(Gn\\Gn+1)≤/summationdisplay\n|k|≥Nn−1M(2)(Rk(In))/lessorsimilarγ/summationdisplay\n|k|≥Nn−11\n|k|3/lessorsimilarγN−3\nn−1.\nThe proof of the lemma is then concluded. /square\nProof of Proposition 10.1 .It follows by the inclusion (10.3), the estimate (10.4), Pro position\n10.4, and the fact that the series/summationtext\nn≥1N−3\nn−1<+∞converges. /square\n11.Proof of Theorems 1.1, 1.2\nProof of Theorem 1.2.\nFixγ:=λ−c, see (9.1), (9.4). Then\nλ−δ\n3γ−2=λ−δ\n3+2c≪1\nby takingλ≫1 large enough, and 0 <c<δ\n6which implies that the smallness condition (9.4) is\nfullfilled. We then set\nOλ:=G∞ (11.1)\nwhere the set G∞is defined in (10.1). By Proposition 10.1, using that γ=λ−c, one has that\nM(2)(O\\Oλ)/lessorsimilarλ−c→0 asλ→+∞.\nMoreover, by Proposition 9.2-( P1)n, using a telescoping argument, for any ω∈ G∞=Oλ, the\nsequence ( In(·;ω))n≥0⊂Hs0+σ\n0(T2,R2) converges to I∞(·;ω)∈Hs0+σ\n0(T2,R2) with respect to the\nnorm/ba∇dbl·/ba∇dblLip(γ)\ns0+σ, and\n/ba∇dblI∞/ba∇dblLip(γ)\ns0+σ≤C,/ba∇dblI∞−In/ba∇dblLip(γ)\ns0+σ/lessorsimilarN2σ\nnN−a\nn−1λ−δM,∀n≥0. (11.2)\nBy Proposition 9.2-( P2)n, for anyω∈ Oλ,F(In)→0 asn→ ∞, therefore, the estimate (11.2)\nimplies that\nF/parenleftBig\nΩ(·;ω),J(·;ω)/parenrightBig\n= 0∀ω∈ Oλwhere\n/parenleftBig\nΩ(·;ω),J(·;ω)/parenrightBig\n:=I∞(·;ω), ω∈ Oλ.\nIt remains only to prove the lower bound on (Ω ,J)\ninf\nω∈Oλ/ba∇dblI∞(·;ω)/ba∇dbls0+σ≥inf\nω∈Oλ/ba∇dblΩ(·;ω)/ba∇dbls0+σ/greaterorsimilarλ−2\n3δ. (11.3)\nIndeed, by the estimate (9.3) and by (11.2) for n= 0 (recall that I0=Iapp= (Ωapp,0)), one has\nthat there is a constant C1>0 such that\n/ba∇dblIapp/ba∇dbls0+σ=/ba∇dblΩapp/ba∇dbls0+σ≥C1λ−2\n3δ,\n/ba∇dblΩ−Ωapp/ba∇dbls0+σ+/ba∇dblJ/ba∇dbls0+σ≤ /ba∇dblI∞−Iapp/ba∇dbls0+σ/lessorsimilarλ−δM/lessorsimilarλ−2δ(11.4)\nforλ≫1 large enough where we recall that by (9.1), (10.2), M≥2. Therefore\n/ba∇dblI∞/ba∇dbls0+σ≥ /ba∇dblΩ/ba∇dbls0+σ≥ /ba∇dblΩapp/ba∇dbls0+σ−/ba∇dblΩ−Ωapp/ba∇dbls0+σ\n/greaterorsimilarλ−2\n3δ−λ−2δ/greaterorsimilarλ−2\n3δ58 G. CIAMPA, R. MONTALTO, AND S. TERRACINA\nforλ≫1 large enough, uniformly w.r. to ω∈ Oλ. The latter estimate then implies the claimed\nbound (11.3) and also that clearly Ω /\\e}atio\\slash= 0. We also prove that J/\\e}atio\\slash= 0. Indeed, assume that by\ncontradiction J= 0. Then F(Ω,0) = 0, implying that the second component in (1.14) becomes\nb·∇Ω≡0 is identically zero. (11.5)\nOn the other hand, by (8.3) and by the estimate (11.4), using t hats0+σ≥1, one has that\n/ba∇dblb·∇Ωapp/ba∇dblL2≥λ−2\n3δK(f,b),\n/ba∇dblb·∇Ω−b·∇Ωapp/ba∇dblL2/lessorsimilar/ba∇dblΩ−Ωapp/ba∇dbl1/lessorsimilar/ba∇dblΩ−Ωapp/ba∇dbls0+σ≤Cλ−2δ\ntherefore\n/ba∇dblb·∇Ω/ba∇dblL2≥ /ba∇dblb·∇Ωapp/ba∇dblL2−/ba∇dblb·∇Ω−b·∇Ωapp/ba∇dblL2\n≥K(f,b)λ−2\n3δ−Cλ−2δ≥K(f,b)\n2λ−2\n3δ\nby takingλ4\n3δ≥K(f,b)\n2C. The latter lower bound then implies that b·∇Ω is not identically zero\nand this is a contraddiction with (11.5). The proof of Theore m 1.2 is then concluded.\nProof of Theorem 1.1. We deduce Theorem 1.1 from Theorem 1.2. Let (Ω( ·;ω),J(·;ω))∈\nHS\n0(T2,R2),S≫0,ω∈ Oλbe the solution of F/parenleftBig\nΩ(·;ω),J(·;ω)/parenrightBig\n= 0 constructed in Theorem 1.2\nand satisfying the estimate (1.15). Then, by (1.10), (1.12) , (1.13), (1.14)\nU:=λδ(−∆)−1∇⊥Ω, B:=λδ(−∆)−1∇⊥J,\nP:= ∆−1/parenleftBig\nλ1+ηdivf+div[(B·∇)B]−div[(U·∇)U]/parenrightBig\n, δ= 3η\nsolves the system (1.5). By the form of the Fourier multiplie r (−∆)−1∇⊥, one clearly has that\nU,B∈HS+1\n0(T2,R2) and\n/ba∇dblU/ba∇dbls+1≃λδ/ba∇dblΩ/ba∇dbls,/ba∇dblB/ba∇dbls+1≃λδ/ba∇dblJ/ba∇dbls,∀0≤s≤S.(11.6)\nHence, the latter estimate, together with the second estima te in (1.15) and the interpolation esti-\nmate (2.8) (and δ= 3η) imply that\n/ba∇dblU/ba∇dblS+1/lessorsimilarSλ3η,/ba∇dblB/ba∇dblS+1/lessorsimilarSλ−3ηand\n/ba∇dblP/ba∇dblS+1/lessorsimilarλ1+η/ba∇dblf/ba∇dblS+/ba∇dblB·∇B/ba∇dblS+/ba∇dblU·∇U/ba∇dblS\n/lessorsimilarSλ1+η+/ba∇dblB/ba∇dbl2\nS+1+/ba∇dblU/ba∇dbl2\nS+1\n/lessorsimilarSλ1+η+λ6η0<η≪1\n/lessorsimilarSλ1+η.\nTheupperboundfor U,B,Pin(1.8) thenfollows. Wenow prove thelower bounds. By thees timate\n(11.6) and by the first estimate in (1.15), one has that\n/ba∇dblU/ba∇dblS+1+/ba∇dblB/ba∇dblS+1≥ /ba∇dblU/ba∇dblS+1≃λδ/ba∇dblΩ/ba∇dblS/greaterorsimilarSλδλ−2\n3δδ=3η\n/greaterorsimilarSλη.\nNow by assumingthat div f/\\e}atio\\slash= 0, usingthat f∈HS\n0(T2,R2), divf∈HS−1\n0(T2,R2), by the estimate\n(2.8), using that /ba∇dblU/ba∇dblS+1/lessorsimilarSλ3η,/ba∇dblB/ba∇dblS+1/lessorsimilarSλ−3η/lessorsimilarS1,\n/ba∇dblP/ba∇dblS+1/greaterorsimilarλ1+η/ba∇dbldivf/ba∇dblS−2−/ba∇dblU·∇U/ba∇dblS−/ba∇dblB·∇B/ba∇dblS\n/greaterorsimilarSλ1+η−/ba∇dblU/ba∇dbl2\nS+1−/ba∇dblB/ba∇dbl2\nS+1\n/greaterorsimilarSλ1+η−λ6ηλ≫1,0<η≪1\n/greaterorsimilarSλ1+η.\nThe proof of the claimed theorem is then concluded.LARGE AMPLITUDE TRAVELING WAVES MHD 59\nReferences\n[1]P. Baldi, M. Berti, E. Haus, R. Montalto :Time quasi-periodic gravity water waves in finite depth . Invent.\nMath.214(2), 739-911 (2018).\n[2]P. Baldi, R. Montalto :Quasi-periodic incompressible Euler flows in 3D . Adv. 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Lay :Introduction to Functional Analysis , 2nd edition, John Wiley & Sons, New York,\nChichester, Brisbane, Toronto, 1980.\n[56]G. Wu:Regularity criteria for the 3D generalized MHD equations in terms of vorticity . Nonlinear Analysis 71,\n4251-4258 (2009).\n[57]L. Xu, P. Zhang :Global Small Solutions to Three-Dimensional Incompressib le Magnetohydrodynamical System .\nSIAM J. Math. Anal. 47, 26-65 (2015).\n[58]V.I. Yudovich. Non-stationary flow of an incompressible liquid. Vychisl. Mat. Mat. Fiz. 3, 1032-1066, 1963.\n(G. Ciampa) Dipartimento di Matematica “Federigo Enriques”, Universi t`a degli Studi di Milano,\nVia Cesare Saldini 50, 20133 Milano, Italy.\nEmail address :gennaro.ciampa@unimi.it\n(R. Montalto) Dipartimento di Matematica “Federigo Enriques”, Universi t`a degli Studi di Milano,\nVia Cesare Saldini 50, 20133 Milano, Italy.\nEmail address :riccardo.montalto@unimi.it\n(S. Terracina) Dipartimento di Matematica “Federigo Enriques”, Universi t`a degli Studi di Milano,\nVia Cesare Saldini 50, 20133 Milano, Italy.\nEmail address :shulamit.terracina@unimi.it"    },    {        "title": "2401.18014v1.Bayesian_regularization_for_flexible_baseline_hazard_functions_in_Cox_survival_models.pdf",        "content": "arXiv:2401.18014v1  [stat.ME]  31 Jan 2024Biometrical Journal 52(2010) 61, zzz–zzz / DOI: 10.1002/bimj.200100000\nBayesian regularization for flexible baseline hazard funct ions in\nCox survival models\nElena L ´azaro∗,1, Carmen Armero1,andDanilo Alvares2\n1Department of Statistics and Operation Research, Universi ty of Valencia, 46100 Burjassot, Spain\n2Department of Statistics, Pontificia Universidad Cat´ olic a de Chile, 7820436 Macul, Chile\nReceived zzz, revised zzz, accepted zzz\nFully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric ap-\nproaches generally provide monotone estimations. Semi-pa rametric choices allow for more flexible patterns\nbut they can suffer from overfitting and instability. Regula rization methods through prior distributions with\ncorrelated structures usually give reasonable answers to t hese types of situations.\nWe discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards spec-\nified by a mixture of piecewise constant functions and by a cub ic B-spline function. For those “semi-\nparametric” proposals, different prior scenarios ranging from prior independence to particular correlated\nstructures are discussed in a real study with micro-virulen ce data and in an extensive simulation scenario\nthat includes different data sample and time axis partition sizes in order to capture risk variations. The pos-\nterior distribution of the parameters was approximated usi ng Markov chain Monte Carlo methods. Model\nselection was performed in accordance with the Deviance Inf ormation Criteria and the Log Pseudo-Marginal\nLikelihood.\nThe results obtained reveal that, in general, Cox models pre sent great robustness in covariate effects and\nsurvival estimates independent of the baseline hazard spec ification. In relation to the “semi-parametric”\nbaseline hazard specification, the B-splines hazard functi on is less dependent on the regularization process\nthan the piecewise specification because it demands a smalle r time axis partition to estimate a similar be-\nhaviour of the risk.\nKey words: Correlated prior process; Cubic B-splines; Piecewise func tions; Survival analysis;\nWeibull distribution.\nSupporting Information for this article is available from t he author or on the WWW under\nhttp://dx.doi.org/10.1022/bimj.XXXXXXX (please delete if not applicable)\n1 Introduction\nThe Cox proportional hazards model (Cox, 1972) is the most po pular regression model in survival analysis.\nIt expresses the hazard function h(t)of the survival time of each individual in the target populat ion as\nthe product of a common baseline hazard function h0(t), which determines the shape of h(t), and an\nexponential regression term which includes the relevant co variates.Baseline hazard misspecification can\nimply a loss of valuable information that is necessary to ful ly report the estimation of the outcomes of\ninterest, such as probabilities or survival curves (Roysto n, 2011). This issue is especially important in\nsurvival studies where h0(t)represents the natural course of a disease or an infection, o r even the control\ngroup when comparing several treatments.\nThe frequentist estimation of the Cox model focuses on the re gression coefficients β, which can be\nobtained without specifying a model for h0(t)by using the partial likelihood methodology (Cox, 1972;\n∗Corresponding author: e-mail: elena.lazaro@uv.es , Phone: +34-665-813-113\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com2 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\nvan Houwelingen and Stijnen, 2014). Frequentist Cox can als o provide a point estimation of h0(t)by\nmeans of the Breslow estimator by plugging the estimate ˆβintoβand point estimations of the survival\nfunction via analogues of the Nelson-Aalen and the Kaplan-M eier estimators (van Houwelingen and Stijnen,\n2014). Uncertainty about these estimates is assessed throu gh confidence intervals which rely on asymp-\ntotics (Andersen and Gill, 1982; Tsiatis, 1981).\nBayesian analysis of the Cox model needs to specify a model fo rh0(t)(Christensen et al., 2011). It\nprovides a natural framework to jointly analyse all the unce rtainties in the statistical modelling, h0(t)\nandβ, by means of its joint posterior distribution. This posteri or contains all the relevant information\nfrom the study and it is usually the starting point for the sub sequent estimation and prediction of the out-\ncomes of interest. In this regard, Bayesian inference, unli ke frequentist statistics, does not generally use\nasymptotic arguments to assess the variability of the estim ates (Ibrahim et al., 2001). Baseline hazard\nfunctions can be defined through parametric or semi-paramet ric approaches. Parametric models give re-\nstricted shapes which do not allow for the presence of irregu lar behaviours (Dellaportas and Smith, 1993;\nKim and Ibrahim, 2000). Semi-parametric choices result in fl exible baseline shapes (Sahu et al., 1997;\nIbrahim et al., 2001) but they may suffer from overfitting and instability (Breiman, 1996). Regularization\nmethods modify the estimation procedures to solve these typ es of problems. Frequentist regularization\nintroduces some changes in the likelihood function. Bayesi an reasoning accounts for this issue through\nprior distributions.\nIn fully Bayesian studies, the joint posterior distributio n is obtained via Bayes’ theorem from the like-\nlihood function and the prior distribution. This is why the p rior can be considered as the element that\nregularizes the likelihood and the reason why the elicitati on of prior distributions is relevant, particularly\nin survival analysis when h0(t)is defined in terms of flexible modelling. The selection of dif ferent base-\nline hazard functions implies different likelihood specifi cations and different prior distributions, which for\na givenh0(t)can range from prior independence to some particular correl ated prior distributions in order\nto avoid overfitting.\nThe prior distribution is a fundamental element of Bayesian methodology that serves as a starting point\nfor any Bayesian study. In general terms, prior distributio ns can be non-informative (or almost) or in-\nformative. Non-informative distributions try to play a neu tral role in the inferential process and give full\nprominence to the data. Informative prior distributions ar e relevant in the statistical procedure, especially\nin studies with little data. In these cases, it is especially important to add sensitivity analyses to the study\nin order to check the robustness of the results with regard to the elicited prior distributions. A non-robust\nprior distribution can be the source of important biases in t he results (Berger et al., 1994; Ibrahim et al.,\n2011).\nRegularization methods originated in mathematical settin gs and were fruitfully and widely dissemi-\nnated to the world of statistics, providing many different a pproaches and concepts (Girosi et al., 1993;\nBenner et al., 2010). All of them share the general and easy id ea of combining the aim of simultaneously\nlooking for a function that is close to the data and also smoot h. The statistical background on the subject,\nBayesian and mainly frequentist, is so extensive that revie wing and understanding the concepts, issues and\nrelationships within each statistical approach is beyond t he scope of this paper (see Bickel (2006) for an\nup-to-date review).\nWe have a twofold objective in this paper: to assess the role o f the specification of h0(t)and to discuss\nthe effect of the Bayesian regularization in the case of semi -parametric modelling of h0(t). We consider\ntwo flexible specifications for h0(t)that allow for multimodal patterns: a mixture of piecewise c onstant\nfunctions (Sahu et al., 1997) and a cubic B-spline function ( Hastie et al., 2009). A Weibull baseline hazard\ndistribution, the usual parametric proposal for h0(t), is also included for comparison purposes. The base-\nline risk functions with which we work in this paper, as well a s the different prior distributions considered,\nare methodological proposals known in Bayesian literature that, as far as we know, have not been compared\nto date. The novelty of our work lies in this comparison, whic h we carry out through different criteria of\ngoodness for the estimated models.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 3\nPiecewise constant functions for h0(t)have a long tradition in Bayesian survival (Kalbfleisch and P rentice,\n1973; Sahu et al., 1997). Relevant proposals that induce cor related structures in the subsequent prior\ndistribution for the coefficients of the piecewise function s are based on discrete time martingale pro-\ncesses, Gamma process priors, and random-walk priors (Ibra him et al., 2001). Cubic B-spline functions\nforh0(t)are far more recent. They come from the world of generalized a dditive models (Hastie et al.,\n2009) and are widely used in spatial and spatio-temporal ana lysis. Their use in survival settings was\nproposed by Cai et al. (2002), Fahrmeir and Hennerfeind (200 3) and Sharef et al. (2010) by means of\nfirst or second random walk smoothness priors with Gaussian e rrors. Other flexible models for base-\nline hazard functions are based on low-rank thin plate linea r splines (Murray et al., 2016), truncated basis\nsplines (Crainiceanu et al., 2005), M-splines (Benner et al ., 1988) or the popular P-splines (P. H. C. Eilers and Durb´ an ,\n2015), particular B-splines with penalties in the frequent ist setting.\nThe remainder of this article is organized as follows. Secti on 2 introduces Weibull, piecewise constant\nand B-spline baseline hazard functions for the Cox model as w ell as the most common prior distributions\nfor these scenarios. Section 3 explores non-penalized freq uentist and Bayesian estimation with piecewise\nconstant and cubic B-spline functions and discusses Bayesi an regularization for h0(t)for a real microbial\nvirulence study. Section 4 explores various simulation sce narios to compare the behaviour of the different\nh0(t)and prior distributions. These last two sections deal with r egularization in the semi-parametric set-\ntings with regard to different partitions of the time axis in which a mixture of piecewise constant and cubic\nB-spline functions are defined. The article ends with some ge neral remarks and conclusions.\n2 Cox proportional hazards model\nLetTibe the random variable that accounts for the observed event t ime for individual i,i= 1,...,n .\nIt is defined as Ti=min(T∗\ni,Ci), the minimum between the true failure time for individual i,T∗\ni, and\nthe right-censoring time, Ci, determined by the end of the study (administrative censori ng). The event\nindicatorδi=I(T∗\ni≤Ci)is 1 if the survival time is observed, and 0 otherwise. We assu me thatT∗\ni\nis a continuous random variable with survival function, Si(t) =P(T∗\ni> t), and hazard function hi(t),\n∀t≥0, which represents the instantaneous rate of occurrence of t he event.\nThe Cox proportional hazards model for T∗\niexpresses the hazard function for individual iin the form\nhi(t|h0,xi,β) =h0(t)exp{x′\niβ}, (1)\nwherexiis a vector of Jcovariates, βis the vector of regression coefficients, and h0(t)is the baseline\nhazard function.\n2.1 Baseline hazard function\nWe discuss three different proposals for h0(t), a Weibull hazard function and two semi-parametric ones,\nnamely a mixture of piecewise constant functions and a cubic B-spline function.\nWeibull function\nThe most popular parametric model for h0(t)is the Weibull distribution, We (α,λ), with shape parameter\nα>0and scaleλ>0, and baseline hazard function\nh0(t|α,λ) =λαtα−1, t>0. (2)\nThis is a traditional model for survival data in biometrical applications. It is highly suitable thanks to\nits computational simplicity, especially in small-sample settings, but it has no flexibility to represent risks\naway from monotonicity (Lee et al., 2016).\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com4 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\nMixture of piecewise constant functions\nPiecewise functions are defined by polynomial functions. Th ey generate a flexible framework for mod-\nelling survival data with a long tradition (Henschel et al., 2009; Ibrahim et al., 2001) in the Bayesian liter-\nature as alternative models to Weibull h0(t). The overall shape of the baseline hazard function does not\nhave to be imposed in advance as is the case with the parametri c models.\nWe assume a finite partition of the time axis with knots c0≤c1≤...≤cK, wherec0= 0, andcKare\nusually taken as the last observed survival or censoring tim e. The hazard function is a mixture of piecewise\nconstant functions defined as\nh0(t|ϕ) =K/summationdisplay\nk=1ϕkI(ck−1,ck](t), t>0, (3)\nwhereϕ= (ϕ1,...,ϕ K),I(ck−1,ck](t)is the indicator function defined as 1 when t∈(ck−1,ck]and 0\notherwise. This baseline hazard function is usually known a s the piecewise constant (PCfrom now on).\nCubic B-spline functions\nWe assume the same finite partition of the time axis as specifie d for the PCbaseline hazard function. The\nspline function for the baseline hazard function is usually defined in logarithmic scale (Murray et al., 2016)\nto accommodate normality and positivity for the subsequent selection of prior distributions. It is defined as\nlog(h0(t|γ)) =K+3/summationdisplay\nk=1γkB(k,4)(t), t>0, (4)\nwhereγ= (γ1,...,γ K+3),{B(k,4)(t),k= 1,...,K+ 3}is a cubic basis of B-splines with boundary\nknotsc0andcKand internal knots ck, k= 1,..,K−1defined recursively by means of the de Boor\nformula (Boor, 1978) as\nB(k,4)(t) =t−τk\nτk+3−τkB(k,3)(t)+τk+4−t\nτk+4−τk+1B(k+1,3)(t), k= 1,...,K+3, (5)\nwhereB(k,1)(t) = 1 ifτk≤t≤τk+1,k= 1,2,...,K and zero otherwise. It is worth noting that the\ndefinition of this B-spline function needs augmentation of t he original knot sequence c= (c0,c1,...,c K)\ntoτ, defined as (Hastie et al., 2009)\nτ1≤...≤τ4≤c0;τj+4=cj, j= 1,2,...,K−1;cK≤τK+4≤...≤τK+7. (6)\nThis modelling strategy is known as a piecewise cubic B-spline function (PSfrom now on). Note that\nfunctions in hazard (3) are B-spline functions of order 1.\n2.2 Bayesian inferential process\nRegularization\nPCandPSbaseline hazard functions can accommodate different shape s depending on the particular char-\nacteristics of the partition of the time axis. This is a relev ant issue with a great amount of research ac-\ntivity: Breslow (1974) considered various failure times as end points of intervals; Kalbfleisch and Prentice\n(1973) supported the theory that the grid should be selected independently of the data; Murray et al. (2016)\nproposed equally-spaced partitions; Henschel et al. (2009 ) fixed the intervals assuming the condition that\nall the intervals contain comparable information, i.e. a si milar number of events; and Lee et al. (2016)\navoided reliance on fixed partitions of the time scale by intr oducing the number of splits as a parameter\nto be estimated. When Kis large, the model has so many parameters that it could suffe r from overfit-\nting problems. On the contrary, choices of Kthat are too small will lead to poor model fitting. When\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 5\nusing a shrinkage or regularization procedure, the effect o f increasing Koften diminishes. Regularization\nprocesses in the Bayesian setting are usually carried out by means of informative prior distributions that\nrestrict the freedom of the parameters.\nThe elicitation of prior distributions for PCandPSbaseline hazard functions includes different prior\ndistribution proposals for the coefficients ϕandγin (3) and (4), respectively. They range from a default\nsituation of prior independence among all the coefficients t o a correlated prior distribution that accounts\nfor shape restrictions in order to avoid overfitting and stro ng irregularities in the estimation process.\nWe consider four prior scenarios for h0(t)defined in terms of a mixture of piecewise constant functions\nbased on different correlation patterns among the coefficie nts associated with the piecewise functions.\nScenario PC1. Independent gamma prior distributions\nπ(ϕk) =Ga(ηk,ψk), k= 1,2,...,K. (7)\nThis is the most flexible and general prior scenario. A common selection isηk=ψk= 0.01.\nScenario PC2. Independent gamma prior distributions\nπ(ϕk) =Ga(w0η0(ck−ck−1),w0(ck−ck−1)), k= 1,...,K. (8)\nAll these marginal prior distributions share the same prior expectation, η0, but the prior variance of each\nϕkis inversely proportional to the corresponding interval le ngth,ck−ck−1. The selection w0= 0.01is\na usual value which provides the prior distribution with a hi gh level of uncertainty. We will assume the ad\nhocproposal by Christensen et al. (2011) for the elicitation of η0that considers η0= 0.69315/˜t, where˜t\nis the median survival time of the reference group.\nScenario PC3. Correlated conditional gamma prior distributions\nπ(ϕk|ϕ1,...,ϕ k−1) =Ga(ηk, ηk/ϕk−1), k= 2,...,K. (9)\nThis prior is based on a discrete-time martingale process (S ahu et al., 1997) which correlates the ϕ’s\nof adjacent intervals with E (ϕk|ϕ1,...,ϕ k−1) =ϕk−1and Var(ϕk|ϕ1,...,ϕ k−1) =ϕ2\nk−1/ηk. The\nparameterηkis very important because it controls the level of smoothnes s, which decreases as ηkreaches\nzero. A common elicitation is ηk= 0.01, k= 2,...,K andπ(ϕ1) =Ga(0.01,0.01).\nScenario PC4. Correlated conditional normal prior distributions for the ϕcoefficients in a logarithmic\nscale\nπ(log(ϕk)|ϕ1,...,ϕ k−1) =N(log(ϕk−1),σ2\nϕ), k= 2,...,K, (10)\nwithπ(log(ϕ1)) = N(0,σ2\nϕ). This is also a proposal based on a discrete-time martingale process. It comes\nfrom the areas of spatial statistics (Banerjee et al., 2014) and Bayesian B-splines (Lang and Brezger, 2004),\nwhere it is better known as a first-order random walk. Correla tion between the log (ϕk)corresponding to\nneighbouring intervals is expressed assuming conditional normal prior distributions.\nNon-informative prior distributions for σ2\nϕhave generally been taken as inverse gamma distributions,\nIG(ν0,ν0), with small values for ν0. However, some research questions the role of these distrib utions for\ndescribing lack of prior information. Gelman (2006) propos ed the use of proper uniforms and half-t dis-\ntributions for the standard deviations as sensible choices , which were understood as reference models to\nbe used as a standard of comparison or a starting point of the i nferential process (Bernardo, 1979). We\nalso considered different prior specifications for the coef ficients of the PSmodelling of baseline hazard\nfunctions that follow the idea of smoothing its level of flexi bility and prevent overfitting. These scenarios\nare not a mere repetition of those considered for PCbaseline hazard functions. They have been chosen\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com6 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\nbecause they are usual proposals in the statistical literat ure regarding cubic B-splines specifications.\nScenario PS1. Independent normal prior distributions\nπ(γk) =N(0,σ2\nk), k= 1,...,K+3. (11)\nThis is the simplest scenario, similar to PC1, in whichγkare considered as independent and normally\ndistributed with a known variance.\nScenario PS2. Hierarchical normal prior distributions\nπ(γk|σ2\nγ) =N(0,σ2\nγ), k= 1,...,K+3, (12)\nwhereσ2\nγis the common variance population. As mentioned previously , a usual choice for the hyperprior\ndistribution for σ2\nγis an inverse gamma distribution or also a proper uniform dis tribution (Gelman, 2006).\nScenario PS3. Correlated conditional normal prior distributions defined as\nπ(γk|γ1,...,γ k−1) =N(γk−1,σ2\nγ), k= 2,...,K+3, (13)\nand based on a first-order Gaussian random walk which involve s an intrinsic Gaussian Markov random\nfield as the conditional joint prior distribution for the spl ine coefficients given σ2\nγ. This proposal comes\nfrom the so-called Bayesian P-splines (Lang and Brezger, 20 04; Fahrmeir and Kneib, 2011). It has been\nwidely used in Bayesian spatial statistics (Banerjee et al. , 2014), where it is usually expressed in terms of\nconditional distributions in the form\nπ(γk|γ−k) =N/parenleftbigg1\n2(γk−1+γk+1),2σ2\nγ/parenrightbigg\n, k= 2,...,K+3, (14)\nwhereγ−kdenotes all spline coefficients except γk. Popular marginal prior distribution choices for σγ\nthat try to be as neutral as possible are Ga (1,0.0005) (Lang and Brezger, 2004) and Ga(0.001, 0.001)\nas a default option in the software BayesX (Belitz et al., 2015). This scenario is analogous to Scenario\nPC4. Consequently, all the discussion regarding the elicitati on of the prior distribution for the variance σ2\nγ\n(precision or standard deviation τγandσγ, respectively) also applies here.\nPosterior distribution\nWe considered a prior independent scenario between the para meters inh0(t)and the regression coefficients\nassociated to covariates. We also reckoned prior independe nce between the regression coefficients within\nanon-informative scenario, with normal distributions centred at zero and a wi de known variance:\nπ(h0,β) =π(h0)π(β) =π(h0)/producttextJ\nj=1N(βj|0,σ2\nj), (15)\nwhereπ(h0)is the prior distribution of all parameters and hyperparame ters inh0(t). The model needs\nto be fed with data D={(ti,δi,xi),i= 1,...,n}, wheretiis the observed survival time for the ith\nindividual,δiis the indicator taking 1 if the event has occurred and 0 other wise, and xiare the subsequent\ncovariates.\nBayes’ theorem combines prior knowledge and experimental i nformation in the posterior distribution\nπ(h0,β| D)∝ L(h0,β)π(h0,β),\nwhereL(h0,β)is the likelihood function of (h0,β)given by Ibrahim et al. (2001) as\nL(h0,β) =n/productdisplay\ni=1h0(ti)δiexp{−H0(ti)}[exp{x′\niβ}]δiexp{exp{x′\niβ}}, (16)\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 7\nwithH0(t) =/integraltextt\n0h0(u)duas the cumulative baseline hazard function.\nIn the case of a Weibull hazard baseline function, the cumula tive baseline hazard function is H0(t) =\nλtα,t>0.When the baseline function is defined via a mixture of piecewi se constant functions, as in (3)\nH0(t) =/summationtextk−1\nm=1ϕm(cm−cm−1)+ϕk(t−ck−1), ck−1≤t<ck, k= 1,...,K.\nThe expression of the cumulative baseline hazard for h0(t)defined in (4) in logarithmic scale in terms\nof cubic B-spline functions needs to take into account some a dditional properties of B-splines (Boor, 1978;\nSherar, 2004). In particular,\n/integraldisplayt\n0K+3/summationdisplay\nk=1γkB(k,4)(u)du=K+4/summationdisplay\nk=1φkB(k,5)(t), (17)\nwithφ1= 0, andφm+1=τm+1−τ5\n4/summationtextm\nj=1γj, m= 1,2,...,K+3.Note that B-splines of order 5 need\nto add two additional nodes to the augmented knot sequence τin (6).\n3 An experiment on microbial virulence\n3.1 Virulence data and modelling\nA dataset involving a virulence assay is taken into account t o explore the baseline hazard specifications\ndiscussed above. The data came from an experiment designed t o assess the effect of the use of a cauliflower\nby-product infusion treatment in Salmonella enterica serovar Typhimurium ( S.Typhimurium) virulence\nbehaviour. S.Typhimurium is one of the most usual serotypes related to sal monellosis outbreaks and\ncauliflower by-product infusion treatment is an alternativ e preservation treatment against it.\nOne and three exposures to the treatment were evaluated. A pa thogen S.Typhimurium ( ST) popula-\ntion non-exposed to the treatment was considered as the cont rol group. The nematode Caenorhabditis\nelegans (C. elegans) was used as a host model to quantify the virulence of the patho gen. STnon-treated\n(ST0),STtreated once ( ST1), and STtreated three times ( ST3) was the source of nutrition of 250 syn-\nchronized young adult nematodes kept in identical environm ental conditions throughout their lifespan (ap-\nproximately three weeks at the most). Virulence for each wor m was defined in terms of their survival time\n(see Sanz-Puig et al. (2017) for more details about the valid ation and special conditions of the study). Most\nof the data were fully observed. Only five survival times were right-censored due to the accidental death\nof the individuals when they were being transferred from one plate to another.\nFigure 1 shows a Kaplan-Meier curve, in days, for each of the STpopulations considered. Individuals\nfed on ST0 (the control group) showed a survival curve that was lower ov er time in relation to the ones\nfed on ST1andST3, with a median survival time of 5.58days versus 8.40and9.24, respectively. The ST1\nandST3 groups exhibit similar trajectories which cross at certain time points, thus confirming a similar\nbehaviour.\nFIGURE 1 AROUND HERE\nVirulence for the i-th worm was modelled by means of the Cox proportional hazard s model\nhi(t|h0,xi,β) =h0(t)exp{β1I1(i)+β3I3(i)}, (18)\nwhereI1(i)andI3(i)are indicator variables for groups ST1 andST3, respectively. It is important to\nhighlight that hi(t| ·) =h0(t)in the case of ST0, which acts as the control group, hi(t| ·) =\nh0(t)exp{β1I1(i)}when it is ST1, andhi(t| ·) =h0(t)exp{β3I3(i)}when the group is ST3.\nWe considered a Weibull model for h0(t)as well asPCandPSbaseline hazard functions based on four\ndifferent partitions of the time axis with number of knots K= 5, 10, 25 and 40. All these partitions were\nchosen following the proposal by Murray et al. (2016) based o n selecting intervals with the same length.\nThe last knot in all PCandPSmodels is 24.50days, which was the longest survival time observed.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com8 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\n3.2 Posterior inferences\nWe carried out all Bayesian survival inferential processes derived from the combination of the generic\nspecifications of the baseline hazard function above with th e different prior scenarios and number of knots\n(K= 5, 10, 25 and 40) for PC andPSmodels. The joint posterior distribution for each model was\napproximated using the JAGS software (Plummer, 2003). For each estimated model, we ran t hree parallel\nchains with 50,000 iterations and a burn-in of 5,000. Chains were also thinned by storing every 5th iteration\nto reduce autocorrelation in the sample. Convergence to the joint posterior distribution was guaranteed\nwith a potential scale reduction factor close to 1 and an effe ctive number of independent simulation draws\ngreater than 100.\n3.3 Model selection, hazard ratios and baseline hazard-sur vival function\nDeviance information criterion (DIC) (Spiegelhalter et al ., 2002) and log pseudo-marginal likelihood (LPML) (Geisse r and Eddy,\n1979) were considered for model selection. DIC measures the information on a model by means of its de-\nviance penalized with regard to its complexity. Additional ly, from the DIC computation we derived the\neffective number of parameters (pD) to evaluate the model co mplexity (Spiegelhalter et al., 2002). LPML\nis based on predictive criteria. It combines, on a logarithm ic scale, the conditional predictive ordinate value\n(CPO) associated with observations of each individual (Gel fand, 1996). Smaller values for DIC are pre-\nferred, while larger LPML values indicate better predictiv e performance. pD is interpreted together with\nDIC, as a complementary criterion.\nAs a rule of thumb, if two models differ in the DIC by more than 3 , the one with the smaller DIC is\npreferred as the best fitting (Spiegelhalter et al., 2002). I n the case of LPML, there is no rule of thumb\nabout how much this difference should be (Bogaerts et al., 20 17). However, the LPML statistics from\ntwo competing models, LPML 1and LPML 2, can be used to compute what has been termed a “pseudo\nBayes factor” (PBF), which roughly indicates which model is superior at predicting the observed data:\nPBF12= exp( LPML 1−LPML 2)(Hanson and Yang, 2007; Branscum and Hanson, 2008; Zhao et al .,\n2014). We interpret the PBF following the guidelines propos ed by Jeffreys (1961) and Kass and Raftery\n(1995); thus, a PBF 12above 3 denotes there is substantial evidence in favour of mo del 1.\nTable 1 shows the DIC, the pD and LPML values of the estimated m odels. Based on the DIC and LPML\nvalues, PSmodels exhibit better behaviour than Weibull or PCspecifications. The Weibull model shows\nthe worst preformance even if showing the lowest complexity (as measured by pD value). An increase\nin the number of knots for PCmodels generally results in a clear improvement in the model ling (from\nK= 5 toK= 25 ), since increasing Kup to 40 does not substantially improve goodness of fit while\nmeaningfully increasing model complexity. Differences in DIC and PSB that are higher than 3 favour\nmodels with K≤25. This fact is more relevant with correlated prior distribut ions, especially for scenario\nPC4.PS(regardless of the number of knots and prior setting) are alw ays the best models, showing no\nrelevant differences between their DIC and LPML (PBF) value s. Thus, PS models with K= 5 show\nsimilar performance to their K= 10 ,K= 25 andK= 40 counterparts. In relation to the pD values,\nthe complexity of the models is clearly influenced by prior sp ecification.PC4andPS3models (above all\nforK= 25 andK= 40) show a prior-induced parameter reduction (the true par ameters (not considering\nhyperparameters) for PC andPSmodels can be estimated as K+2 andK+5, respectively); hence they\nshow an improvement in model complexity with respect to thei r counterparts.\nTABLE 1 AROUND HERE\nBelow we focus on the posterior stability of the posterior di stribution for the hazard ratios as well as the\nbehaviour of the subsequent marginal posterior distributi on for the baseline hazard function, which reflects\nthe natural course of the infection, and the survival functi on.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 9\nHazard ratios\nDiscrepancies between the posterior marginal distributio ns for the regression coefficients and for any of\ntheir corresponding derived quantities, such as hazard rat ios, are a result of the different modelling of\nh0(t). Figure 2 shows the posterior mean and a 95% credible interva l for the hazard ratios of interest\nHRST1, HRST3, HRST1/ST3(computed as π(exp(β1)| D),π(exp(β3)| D)andπ(exp(β1−β3)| D))\nwith regard to the different specifications of the baseline h azard function, prior scenarios and number of\nknots for PCandPSmodels. HR ST1and HR ST3posterior distributions behave in a similar way, with\nvalues below 0 indicating efficacy in bacterium virulence re duction. HR ST1/ST3posterior distributions\nare centred at approximately 1, pointing to similar efficacy for both treatments. We observe great internal\nrobustness in the results of the PSmodels and the PC models. Weibull estimated coefficients are also\nquite similar to those obtained from PCandPSmodels.\nFIGURE 2 AROUND HERE\nBaseline hazard and baseline survival functions\nWe now discuss the posterior distribution for h0(t)and the survival function of the different models in\nthe study. Models with K= 25 knots were selected for PCspecification given that the PC4 (K= 25)\nshowed the best performance based on DIC and LPML. For PSspecification, PS1underK= 40 showed\nthe best performance based on DIC, but it was dismissed since it presented clear signs of overfitting and\ninstability in the baseline hazard value associated to the l ast interval. Thus, models under K= 5 were\nselected because the PS1was the best model according to the two selection scores and i t also shows\nsimilar values of pD to those of its counterparts ( PS2andPS3underK= 5). Figures 3 and 4 are a matrix\nof graphs for illustrating baseline hazard (logarithmic sc ale) and survival functions posterior distributions\nunderWe(row one), PC (K= 25) (row two) and PS(K= 5) (row three) models.\nBaseline hazard estimates are sensitive to their specificat ion and their implicit regularization. The We\nmodel displays an increasing monotone behaviour. PCmodels report a general increasing trend with dif-\nferent ups and downs. They show wider credible intervals in r egions with very little data. The PC4model\nevidences that Bayesian regularization not only smooths th e posterior mean but also reduces the uncer-\ntainty of the estimate. PSmodels present a more flexible baseline hazard than PC’s and a regularization\neffect is mainly observed only in uncertainty estimates. On the contrary, estimates of posterior distribution\nπ(S0(t)| D), which is encapsulated in the unit interval, are robust to ba seline hazard function specification\nand differences between the different modelling proposals are imperceptible.\nFIGURES 3 AND 4 AROUND HERE\n3.4 Frequentist and Bayesian Cox model\nAlthough it is not a main objective of the article, we have per formed a comparison of Bayesian Cox\nmodels against their frequentist counterparts. The compar ison considered the three generic baseline hazard\nspecifications We,PCandPSto be baseline hazard functions based on the four different p artitions of the\ntime axis exploited earlier ( K= 5, 10, 25, and 40). For the PCandPSmodels, we only considered models\nPC1andPS1due to their “non-informative” nature in prior specificatio n. Frequentist Cox with Weibull\nbaseline hazard was estimated through the survreg function of the survival library. Results for the\nCoxPCandPSmodels were obtained by the mexhaz function of the mexhaz library, which uses the\nequivalence between PCmodels and Poisson regression models (Holford, 1980; Laird and Olivier, 1981).\nTABLE 2 AROUND HERE\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com10 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\nTable 2 refers to the estimation of the hazard ratios HR ST1and HR ST3. Bayesian reasoning provides\nthe corresponding posterior mean and 95 %credible interval. Frequentist statistics includes maxim um\nlikelihood estimates and 95% confidence intervals. Both est imation procedures are very stable, with similar\nresults forPCandPSmodels.\n4 Simulation study\nWe continue with the exploration of the impact of the baselin e hazard specification in the whole inferential\nprocess, specifically the posterior estimates of the regres sion coefficients as well as the posterior for the\nhazard and survival function. We conduct three simulation s tudies (based on three different h0(t)defini-\ntions) to assess the performance of the Weibull, PCandPSdefinitions. PC andPSare also discussed\nwith regard to different partitions of the time axis.\n4.1 Simulation scenarios\nThree simulation scenarios were generated from a CPH model w ith different specifications for h0(t)as\ndescribed below.\nScenario 1 . A Weibull distribution with an increasing hazard function (α= 1.5andλ= 0.5).\nScenario 2 . A mixture of five piecewise functions\nh0(t|ϕ) =5/summationdisplay\nk=1ϕkI(ck−1,ck](t), t>0,\nwhereϕ1= 0.5in0< t≤0.2,ϕ2= 2.5in0.2< t≤0.4,ϕ3= 0.5in0.4< t≤0.6,ϕ4= 1 in\n0.6<t≤0.8, andϕ5= 1.5int>0.8.\nScenario 3 . A mixture of two Weibull distributions\nh0(t|α1,α2,λ1,λ2) =λ1α1tα1−1pexp{−λ1tα1}+λ2α2tα2−1(1−p)exp{−λ2tα2}\npexp{−λ1tα1}+(1−p)exp{−λ2tα2}, t>0\nwith shapeα1= 3,α2= 1, scaleλ1=λ2= 0.5, and mixing probability parameter p= 0.2.\nThese scenarios included an indicator covariate with regre ssion coefficient β= 1. Data were assigned\nto each group according to a Bernoulli distribution with pro bability 0.5. We considered right censoring\nat timeCR. It was previously fixed for each scenario from the condition S0(CR) = 0.3for the baseline\nsurvival function. Each scenario was replicated R= 100 times for sample sizes of N= 100 andN= 300 .\nAll the simulated dataset were analysed via each of the state d modellings discussed in Section 2. The\nestimation of the PCandPSmodels was based on two different partitions of the time axis withK= 5 and\n15 knots with intervals of the same length ((Murray et al., 20 16)). The last knot in all models corresponds\nto the previously referred censored time ( CR), which is the longest survival time observed.\n4.2 Generating survival times\nWe follow the inversion method (Bender et al., 2005; Austin, 2012; Crowther and Lambert, 2013) to sim-\nulate survival data for Scenarios 1 and2. This method is based on the relationship between the cumula tive\ndistribution function (CDF) of a survival random variable a nd a standard uniform random variable. It\ncan be directly applied when the subsequent CDF has a closed f orm expression and can be directly in-\nverted and easily implemented with R(R Core Team, 2013) packages simsurv (Brilleman, 2013) and\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 11\nSimSCRPiecewise (Chapple, 2016). The inversion method for Scenario 3 is not directly suitable.\nThe subsequent cumulative hazard function cannot be direct ly inverted and we have used iterative root-\nfinding techniques (Crowther and Lambert, 2013) to solve it. This procedure is implemented for the R\nsoftware (R Core Team, 2013) in the simsurv (Brilleman, 2013) package. Further details of the inver-\nsion method and its corresponding extension to simulate com plex baseline hazard functions are described\nin the supporting information.\n4.3 Posterior inferences\nEach simulation dataset was used to estimate all the surviva l models with all the specifications of h0(t)\nand the different prior scenarios in Section 2. Posterior di stributions were approximated by JAGS soft-\nware (Plummer, 2003) based on three parallel chains with 20, 000 iterations each plus another 2,000 for\nthe burn-in period. Moreover, the chains were additionally thinned by storing every 10th draw to reduce\nautocorrelation in the sequences. Convergence of the chain s to the posterior distribution was guaranteed\nby monitoring in all inferences to ensure that the potential scale reduction factor was close to 1 and the\neffective number of independent simulation draws was great er than 100.\n4.4 Regression coefficients and baseline hazard function\nWe considered R= 100 replicas of each inferential process and, consequently, we constructed 100 ap-\nproximate random samples of the posterior distribution for β. Let{β(1)\n(r),...,β(N)\n(r)}be the approximate\nMCMC sample of size Nof the posterior marginal distribution for βcorresponding to the replica r.\nThe stability of the posterior distribution for the regress ion coefficients were assessed by means of the\nfollowing measures:\n•Bias: Difference between the average of the posterior sample mea ns of the replicas and the true\nregression coefficient, (/summationtextR\nr=1¯β(r)/R)−β, where¯β(r)is the sample mean of the posterior sample\ncorresponding to the replica r.\n•Standard error (SE) : Square root/radicalBig/summationtextR\nr=1s2\n(r)/Rof the average of the posterior variances s2\n(r)of\nthe replicas.\n•Standard deviation (SD) : Standard deviation of the set {¯β(1),...,¯β(R)}that includes the posterior\nsample mean of the regression coefficient of all replicas.\n•Coverage probability (CP) : Proportion of the R= 100 95%credible intervals which contain the\ntrue value of the regression coefficient.\nThe performance of the set of models considered was also eval uated in terms of the posterior baseline\nhazard estimates (logarithmic transformation). For the po sterior sample of each replica we construct an\napproximate posterior sample of the log baseline hazard fun ction at each time, whose average can be used\nas a point estimate of the true baseline hazard at that time. W e then merge the information of all the replicas\nto obtain a global estimation, log (/hatwideh0(t)), by calculating their average. This procedure is also usefu l for\nextracting information about the posterior variability an d constructing, for example, 95 %credible intervals\nfor the posterior of the baseline hazard at each time.\nThe accuracy of the estimation was measured through the diff erence between the posterior estimation\nof the baseline hazard and the true hazard function. A genera l measure that accounts for this difference\nover the time period of the study is the root-mean squared dev iation (RMSD), computed as\nRMSD=/radicalBigg/summationtextM\nm=1[log(/hatwideh0(tm))−log(h0(tm))]2\nM, (19)\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com12 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\na discrete approximation based on the idea of the Riemann sum s to approach an integral. At this point,\nwe would like to note that we have used a wide partition of the t ime axis, with knots spaced at 0.01time\npoints from 0 to the maximum time value of each scenario. This maximum time value is determined by\nthe corresponding censoring time ( CR).\nTABLES 3, 4, 5 AROUND HERE\nTables 3, 4 and 5 display the values of the average, bias, SE, S D and CP (related to βand RMSD (related\nto log(h0(t))) referring to the three simulation scenarios. In relation to theβestimate, the Wemodel is\nvery stable for the three scenarios and the effect of Nis not appreciated. PCandPSmodels approximate\nthe regression coefficient quite well, which is slightly aff ected by the number of knots ( K) and the sample\nsize (N).\nUnder Scenario 1 , theWe models provide the closest fit to the true function with the lo west RMSD\nvalues. PSmodels are generally better than PC’s, which show the worst performance, possibly because of\ntheir non-continuous behaviour. Under Scenario 2 ,PC4models (for N= 100 andN= 300 ) provide the\nclosest fit to the true function with the lowest RMSD values, t hereby underlining the relevance of sensitivity\nto prior scenarios. PSmodels also seem to capture the behaviour of the true functio n, on the whole, showing\nRMSD values lower than the PC1,PC2,PC3 models. The Wemodels present the highest RMSD. Under\nScenario 3 ,PSmodels provide the lowest RMSD values as a general rule. PS3 specification shows the\nlowest values for all Kconfigurations. The Wemodels present higher RMSD estimates in relation to PS’s.\nBetween PC’s,PC4specification improves the RMSD values of its PCcounterparts. For all scenarios,\nthe prior distribution has a strong effect on the baseline ha zard estimation of PCmodels.\nFigures 5, 6 and 7 show the posterior mean of the baseline haza rd function and a 95% credible bound\nfor the best models (based on RMSD criterion) between the thr ee generich0(t)specifications and for both\nNvalues for Scenario 1 ,Scenario 2 andScenario 3 . In general, models under N= 300 present lower\nRMSD values than their N= 100 counterparts as well as more accurate baseline hazard e stimates (95% of\ncredible bounds are narrower).\nFIGURES 5, 6, 7 AROUND HERE\n5 Conclusions\nWe have discussed different proposals for performing a full y time-to-event Bayesian analysis in the con-\ntext of the CPH model via parametric and semi-parametric defi nitions of the baseline hazard function. The\nBayesian methodology allows the baseline hazard functions to be implemented in an easy conceptual way,\neven semi-parametric proposals that are necessary in conte xts in which a certain complexity in the shape of\nthe underlying function is expected. On this matter, we have examined some of the most popular proposals\nin the literature related to the subject: the Weibull distri bution as the most common parametric model,\nand piecewise constant and cubic B-spline baseline hazards as semi-parametric definitions. Flexibility\nand overfitting were discussed within both semi-parametric options with regard to different regularization\nschemes expressed in terms of prior distributions and time a xis partition configurations. These develop-\nments provide a unified framework to conduct a fully Bayesian analysis of complex survival data that will\nsurely encourage more comprehensive analyses, which curre ntly often rely on some versions of the CPH\nmodel without further examination. The flexibility of our ap proach allows for easy subsequent research\non prior sensitivity, different criteria for determining t he axis partition of non-parametric proposals and\nrelationships between covariates and baseline hazard func tions. Additionally, we have also incorporated\na comparison with the frequentist approach to evaluate the p erformance of both methodologies under the\nCPH model.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 13\nThe virulence database in Section 3 illustrates the main goa ls of this paper. All inferential processes\nagree with the conclusions in Sanz-Puig et al. (2017) that th e cauliflower by-product infusion can be an al-\nternative preservation treatment. This fact evidences the robustness (regardless of the h0(t)specification)\nof the Cox model in estimating covariate effects. However, PCmodels show a certain sensitivity to axis\npartition in estimating covariate effects. The outcomes al so highlight the fact that piecewise constant and\nB-splines specifications allow us to capture and introduce ( dealing with different axis partition configura-\ntions) more flexibility in h0(t). However, piecewise constant options exhibit less flexibil ity, thus requiring\na higher number of Kas well as a prior correlation specification to behave in a sim ilar way to B-splines.\nHence, in this illustrative example the PCmodel underlines the efficacy of regularization Bayesian me th-\nods (based on defining correlation by means of prior definitio n) to overcome overfitting and instability in\nbaseline hazard estimation under high Kvalues. In relation to the survival function estimation, th is derived\nquantity shows greater robustness regardless of the baseli ne hazard specification. Both DIC and LPML re-\ninforce the evidence observed in sensitivity analyses in wh ichPSmodels show better behaviour than PC\nmodels irrespective of the number of pre-fixed knots. Freque ntist methods showed similar performance to\nthe Bayesian in the Cox inferential process within a framewo rk of non-regularization in relation to Weibull\nand B-spline specification.\nWe have also exemplified our proposals through different sim ulated data generated by Weibull, piecewise\nconstant and mixtures of Weibull baseline hazard functions . In general, the outcomes indicate that moder-\nate bias can be observed in estimates of the regression coeffi cient for a treatment effect when the baseline\nhazard function specification does not match the origin spec ification. For baseline hazard estimates, we\nappreciate small differences between the true baseline haz ard and their point estimates, and lower RMSD\nvalues have a close relationship with the data-generating m odel. In terms of RMSD estimates the Weibull\nmodel provides the best results with Weibull simulated data , althoughPSmodels also exhibit good be-\nhaviour. In the case of piecewise constant simulated data, t hePC4 model is the best model, although PS\nmodels present a very good behaviour in terms of RMSD values. PS3 models provide the best estimates\nfor the Weibull mixture data. In relation to the performance of the different number of knot configurations\n(K) explored, it is generally noticeable that PC models requir e a higher number of KthanPSmodels\nwithin the same scenario. Thus, the need for regularization becomes more evident under PCmodels. In\nall scenarios, the impact of the database size has generally been evident mainly in the estimation of the\nbaseline hazard function, but has been less evident in the re gression coefficient estimate.\nAlthough in this article we have extolled the potential of Ba yesian inference in dealing with semi-parametric\nspecifications for the baseline hazard in the context of the C PH model, it must be stated that in many set-\ntings a simpler distribution may be suitable. However, usin g a more complex distribution can provide far\nmore realistic inferences in certain situations. Some inte resting issues that are beyond the scope of this pa-\nper deal with introducing uncertainty in the number of knots , including new regularization proposals such\nas penalized complexity priors, carrying out a sensitivity analysis within each scenario and also exploring\nin greater depth the performance of the frequentist approac h under the “semi-parametric” specification of\nthe baseline hazard function.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com14 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\n0 5 10 15 20 250.0 0.2 0.4 0.6 0.8 1.0\nTime (days)Survival probability (%)\nFigure 1 Kaplan-Meier survival curve, in days, for individuals fed o n a)ST0(black), b) ST1(dark gray),\nand c) ST3(gray).0.6 0.8 1.0 1.2\n0.6 0.8 1.0 1.2\n0.6 0.8 1.0 1.2\n0.6 0.8 1.0 1.20.6 0.8 1.0 1.2\n0.6 0.8 1.0 1.2\n0.6 0.8 1.0 1.2\n0.6 0.8 1.0 1.20.6 0.8 1.0 1.2\nWePC1 PC2 PC3 PC4 PS1 PS2 PS3\n(a)K=5\n0.6 0.8 1.0 1.2\nWePC1 PC2 PC3 PC4 PS1 PS2 PS3\n(b)K=10\n0.6 0.8 1.0 1.2\nWePC1 PC2 PC3 PC4 PS1 PS2 PS3\n(c)K=25\n0.6 0.8 1.0 1.2\nWePC1 PC2 PC3 PC4 PS1 PS2 PS3\n(d)K=40\nFigure 2 Posterior mean and 95% credible interval for the hazard rati os, HR ST1(row one), HR ST3(row\ntwo) and HR ST1/ST3(row three), for all survival models under evaluation.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 15−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(a)We−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(b)PC1\n−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(c)PC2\n−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(d)PC3\n−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(e)PC4−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(f)PS1\n−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(g)PS2\n−6 −3 0 3 60 5 10 15 20 25\nTime (days)\n(h)PS3\nFigure 3 Posterior mean and 95% credible interval for the log baselin e hazard function under Weibull\n(row one),PC(row two) and PS(row three) scenarios. PCandPSmodels are estimated with K= 25\nandK= 5knots, respectively.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com16 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions0.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(a)We0.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(b)PC1\n0.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(c)PC2\n0.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(d)PC3\n0.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(e)PC40.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(f)PS1\n0.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(g)PS2\n0.00 0.25 0.50 0.75 1.000 5 10 15 20 25\nTime (days)\n(h)PS3\nFigure 4 Posterior mean and 95% credible interval for the baseline su rvival function under Weibull (row\none),PC(row two) and PS(row three) scenarios. PCandPSmodels are estimated with K= 25 and\nK= 5knots, respectively.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 17−4 −2 0 2\n0.0 0.5 1.0 1.5\nTime\n(a)We. RMSD=0.039\n−4 −2 0 2\n0.0 0.5 1.0 1.5\nTime\n(b)PC1(K= 5). RMSD=0.205\n−4 −2 0 2\n0.0 0.5 1.0 1.5\nTime\n(c)PS2(K= 5). RMSD=0.095−4 −2 0 2\n0.0 0.5 1.0 1.5\nTime\n(d)We. RMSD=0.007\n−4 −2 0 2\n0.0 0.5 1.0 1.5\nTime\n(e)PC3 (K= 15 ). RMSD=0.130\n−4 −2 0 2\n0.0 0.5 1.0 1.5\nTime\n(f)PS1(K= 5). RMSD=0.063\nFigure 5 Average replica pointwise of the posterior approximate mea ns of the log-baseline hazard es-\ntimate (black solid line), average replica of the posterior 95% credible intervals (dark grey area), and\ntrue log-baseline hazard function (grey dash-dotted line) in the simulated Scenario 1 under theWe,PC1\n(K= 5),PS2(K= 5) forN=100 (row 1) and under the We,PC3(K= 15 ),PS1(K= 5) forN=\n300 (row 2).\n−4 −2 0 2\n0.00 0.25 0.50 0.75 1.00\nTime\n(a)We. RMSD=0.626\n−4 −2 0 2\n0.00 0.25 0.50 0.75 1.00\nTime\n(b)PC4 (K= 5). RMSD=0.095\n−4 −2 0 2\n0.00 0.25 0.50 0.75 1.00\nTime\n(c)PS2(K= 15 ). RMSD=0.289−4 −2 0 2\n0.00 0.25 0.50 0.75 1.00\nTime\n(d)We. RMSD=0.626\n−4 −2 0 2\n0.00 0.25 0.50 0.75 1.00\nTime\n(e)PC3 (K= 5). RMSD=0.042\n−4 −2 0 2\n0.00 0.25 0.50 0.75 1.00\nTime\n(f)PS2(K= 15 ). RMSD=0.254\nFigure 6 Average replica pointwise of the posterior approximate mea ns of the log-baseline hazard esti-\nmate (black solid line), average replica of the posterior 95 % credible intervals (grey area), and true log-\nbaseline hazard function (grey dash-dotted line) in the sim ulated Scenario 2 under theWe,PC4(K= 5),\nPS2(K= 15 ) forN= 100 (row 1) and under the We,PC3(K= 5),PS2(K= 15 ) forN= 300 (row\n2).\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com18 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions−4 −2 0 2\n0.0 0.5 1.0 1.5 2.0\nTime\n(a)We. RMSD=0.131\n−4 −2 0 2\n0.0 0.5 1.0 1.5 2.0\nTime\n(b)PC4 (K= 5). RMSD=0.116\n−4 −2 0 2\n0.0 0.5 1.0 1.5 2.0\nTime\n(c)PS3(K= 5). RMSD=0.074−4 −2 0 2\n0.0 0.5 1.0 1.5 2.0\nTime\n(d)We. RMSD=0.132\n−4 −2 0 2\n0.0 0.5 1.0 1.5 2.0\nTime\n(e)PC1 (K= 5). RMSD=0.066\n−4 −2 0 2\n0.0 0.5 1.0 1.5 2.0\nTime\n(f)PS3(K= 5). RMSD=0.043\nFigure 7 Average replica pointwise of the posterior approximate mea ns of the log-baseline hazard esti-\nmate (black solid line), average replica of the posterior 95 % credible intervals (grey area), and true log-\nbaseline hazard function (grey dash-dotted line) in the sim ulated Scenario 3 under theWe,PC4(K= 5),\nPS3(K= 5) forN= 100 (row 1) and under the We,PC1(K= 5),PS3(K= 5) forN= 300 (row 3).\nModel K DIC pD LPML Model K DIC pD LPML\nWe - 4553.309 3.960 -2276.334\nPC1 5 4484.455 7.030 -2241.921 PS1 5 4460.598 9.930 -2230.660\n10 4478.040 12.067 -2238.658 10 4462.866 14.368 -2231.988\n25 4469.406 27.313 -2235.836 25 4462.494 29.007 -2236.958\n40 4488.393 43.036 -2249.157 40 4419.711 42.537 -2230.357\nPC2 5 4484.457 7.030 -2241.917 PS2 5 4460.024 9.537 -2230.207\n10 4478.069 12.081 -2238.661 10 4462.249 13.831 -2231.412\n25 4469.371 27.295 -2236.586 25 4463.873 26.345 -2233.509\n40 4488.417 43.047 -2249.814 40 4463.732 38.084 -2235.947\nPC3 5 4484.439 7.021 -2241.905 PS3 5 4459.578 8.572 -2229.787\n10 4477.979 12.036 -2238.632 10 4458.998 10.467 -2229.443\n25 4469.221 27.219 -2235.719 25 4460.255 13.471 -2230.112\n40 4487.049 42.356 -2245.979 40 4458.403 15.583 -2229.296\nPC4 5 4484.445 7.014 -2241.894\n10 4477.070 11.508 -2238.193\n25 4463.265 22.566 -2231.649\n40 4471.340 29.782 -2235.798\nTable 1 DIC, pD and LPML values for the survival models defined by mean s of Weibull, PCandPS\nspecifications of the baseline hazard function with number o f knotsK= 5, 10, 25, and 40.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 19\nBayesian approach Frequentist approach\nModel K HRST1 HRST3 HRST1 HRST3\nWe – 0.640 (0.533, 0.760) 0.654 (0.546, 0.774) 0.637 (0.534, 0.760) 0.652 (0.546, 0.777)\n5 0.604 (0.503, 0.722) 0.619 (0.515, 0.737) 0.601 (0.503, 0.719) 0.616 (0.515, 0.736)\nPC1 10 0.598 (0.498, 0.712) 0.615 (0.513, 0.732) 0.596 (0.498, 0.713) 0.613 (0.512, 0.733)\n25 0.594 (0.495, 0.707) 0.607 (0.505, 0.723) 0.592 (0.495, 0.708) 0.605 (0.506, 0.723)\n40 0.594 (0.494, 0.708) 0.608 (0.507, 0.725) 0.593 (0.496, 0.709) 0.608 (0.508, 0.727)\nPS15 0.596 (0.496, 0.709) 0.610 (0.508, 0.725) 0.593 (0.495, 0.709) 0.607 (0.508, 0.726)\n10 0.592 (0.493, 0.706) 0.605 (0.505, 0.719) 0.593 (0.495, 0.709) 0.606 (0.506, 0.725)\n25 0.592 (0.493, 0.705) 0.610 (0.509, 0.725) 0.592 (0.495, 0.709) 0.606 (0.507, 0.725)\n40 0.590 (0.491, 0.702) 0.603 (0.501, 0.719) 0.592 (0.495, 0.709) 0.606 (0.507, 0.725)\nTable 2 HRST1and HR ST3: posterior mean and 95% credible interval (Bayesian approa ch), and estimate\nand 95% confidence intervals (Frequentist approach).\nAcknowledgements L´ azaro’s work was supported by a predoctoral FPU fellowshi p (FPU2013/02042) from the\nSpanish Ministry of Education, Culture and Sports. This res earch work was funded by grant MTM2016-77501-P from\nthe Spanish Ministry of Economy and Competitiveness co-fina nced with FEDER funds.\nConflict of Interest\nThe authors have declared no conflict of interest.\nReferences\nAndersen, P. K., and R. D. Gill. 1982. Cox’s regression model for counting processes: A large sample study. Annals of\nStatistics 10: 1100–1120.\nAustin, P. C. 2012. Generating survival times to simulate Co x proportional hazards models with time-varying covari-\nates. Statistics in Medicine 31 (29): 3946–3958.\nBanerjee, S., B. P. Carlin, and Alan E A. E. Gelfand. 2014. Hierarchical modeling and analysis for spatial data . Boca\nRaton, Chapman & Hall Crc Press.\nBelitz, C., A. Brezger, T. Kneib, S. Lang, and N. Umlauf. 2015 . Bayesx software for Bayesian inference in structured\nadditive regression models version 2.0.1. URL http://www. bayesx. org .\nBender, R., T. Augustin, and M. Blettner. 2005. 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Series B (Methodological) .\nBickel, P. J. 2006. Regularization in statistics. Test15: 271–344.\nBogaerts, K., A. Komarek, and E. Lesaffre. 2017. Survival analysis with interval-censored data: A practica l approach\nwith examples in R, SAS, and BUGS . Chapman and Hall/CRC.\nBoor, C. De. 1978. A practical guide to splines . New York, Springer-Verlag.\nBranscum, A., and T. E. Hanson. 2008. Bayesian nonparametri c meta-analysis using Polya tree mixture models. Bio-\nmetrics 64 (3): 825–833.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com20 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\nModel N Kβ log(h0(t))\nAverage Bias SE SD CP RMSD\nWe100 – 1.035 0.035 0.230 0.211 0.97 0.039\n300 – 1.008 0.008 0.132 0.136 0.95 0.007\nPC11005 1.037 0.037 0.233 0.216 0.96 0.205\n15 1.049 0.049 0.234 0.216 0.97 2.158\n3005 1.004 0.004 0.133 0.140 0.95 0.198\n15 1.013 0.013 0.133 0.142 0.95 0.131\nPC21005 1.038 0.038 0.233 0.215 0.96 0.205\n15 1.051 0.051 0.234 0.216 0.97 3.607\n3005 1.004 0.004 0.133 0.140 0.96 0.198\n15 1.013 0.013 0.134 0.141 0.97 0.131\nPC31005 1.037 0.037 0.234 0.216 0.95 0.205\n15 1.050 0.050 0.234 0.216 0.96 1.083\n3005 1.004 0.004 0.133 0.140 0.96 0.198\n15 1.014 0.014 0.134 0.142 0.97 0.130\nPC41005 0.946 -0.054 0.234 0.210 0.97 0.212\n15 0.882 -0.118 0.233 0.203 0.96 0.206\n3005 0.970 -0.030 0.134 0.140 0.96 0.204\n15 0.944 -0.056 0.133 0.139 0.93 0.145\nPS11005 1.031 0.031 0.232 0.211 0.98 0.117\n15 0.996 -0.004 0.228 0.203 0.97 0.205\n3005 1.010 0.010 0.133 0.140 0.95 0.063\n15 0.994 -0.006 0.132 0.137 0.96 0.120\nPS21005 0.925 -0.075 0.231 0.205 0.96 0.095\n15 0.788 -0.212 0.225 0.189 0.88 0.201\n3005 0.967 -0.033 0.133 0.139 0.95 0.064\n15 0.902 -0.098 0.131 0.134 0.86 0.116\nPS31005 1.027 0.027 0.233 0.210 0.97 0.096\n15 1.023 0.023 0.234 0.209 0.97 0.121\n3005 1.007 0.007 0.134 0.140 0.97 0.071\n15 1.005 0.005 0.134 0.140 0.97 0.089\nTable 3 Average, bias, SE, SD and CP of the regression coefficient βand RMSD of the log (h0(t))\ncorresponding to all inferential and replicate processes f or the Scenario 1 simulated data.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.comBiometrical Journal 52(2010) 61 21\nModel N Kβ log(h0(t))\nAverage Bias SE SD CP RMSD\nWe100 – 1.077 0.077 0.234 0.251 0.93 0.626\n300 – 1.074 0.074 0.133 0.163 0.88 0.626\nPC11005 1.018 0.018 0.234 0.232 0.94 0.276\n15 1.018 0.018 0.235 0.229 0.95 7.933\n3005 1.012 0.012 0.133 0.149 0.95 0.058\n15 1.013 0.013 0.134 0.150 0.92 0.889\nPC21005 1.018 0.018 0.234 0.232 0.94 0.760\n15 1.017 0.017 0.235 0.229 0.95 13.085\n3005 1.011 0.011 0.133 0.149 0.94 0.058\n15 1.013 0.013 0.134 0.151 0.92 1.291\nPC31005 1.017 0.017 0.233 0.232 0.94 0.345\n15 1.017 0.017 0.235 0.229 0.95 4.381\n3005 1.012 0.012 0.134 0.149 0.94 0.058\n15 1.013 0.013 0.134 0.150 0.92 0.276\nPC41005 1.001 0.001 0.230 0.226 0.94 0.095\n15 0.973 -0.027 0.228 0.216 0.95 0.202\n3005 1.006 0.006 0.133 0.148 0.94 0.042\n15 0.996 -0.004 0.133 0.147 0.92 0.102\nPS11005 1.012 0.012 0.233 0.225 0.94 0.421\n15 0.992 -0.008 0.231 0.223 0.95 0.402\n3005 1.013 0.013 0.134 0.150 0.92 0.387\n15 1.003 0.003 0.133 0.147 0.94 0.303\nPS21005 1.001 0.001 0.226 0.211 0.96 0.405\n15 0.975 -0.025 0.214 0.190 0.97 0.289\n3005 1.008 0.008 0.132 0.147 0.92 0.386\n15 0.993 -0.007 0.128 0.137 0.94 0.254\nPS31005 1.018 0.018 0.234 0.229 0.94 0.424\n15 1.015 0.015 0.235 0.229 0.94 0.305\n3005 1.014 0.014 0.134 0.151 0.92 0.388\n15 1.012 0.012 0.134 0.150 0.92 0.261\nTable 4 Average, bias, SE, SD and CP of the regression coefficient βand RMSD of the log (h0(t))\ncorresponding to all inferential and replicate processes f or the Scenario 2 simulated data.\n© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.biometrical-journal.com22 Elena L´ azaro et al. : Bayesian regularization for flexible baseline hazard func tions\nModel N Kβ log(h0(t))\nAverage Bias SE SD CP RMSD\nWe100 – 0.955 -0.045 0.230 0.234 0.93 0.131\n300 – 0.960 -0.040 0.131 0.119 0.94 0.132\nPC11005 0.983 -0.017 0.234 0.254 0.93 0.309\n15 0.989 -0.011 0.235 0.254 0.93 4.524\n3005 0.979 -0.021 0.133 0.120 0.95 0.066\n15 0.984 -0.016 0.133 0.121 0.95 0.245\nPC21005 0.985 -0.015 0.234 0.255 0.93 0.831\n15 0.992 -0.008 0.235 0.255 0.93 7.012\n3005 0.980 -0.020 0.133 0.121 0.95 0.066\n15 0.984 -0.016 0.133 0.122 0.96 0.313\nPC31005 0.984 -0.016 0.234 0.254 0.93 0.466\n15 0.991 -0.009 0.235 0.255 0.94 3.962\n3005 0.979 -0.021 0.133 0.120 0.95 0.066\n15 0.984 -0.016 0.133 0.122 0.96 0.102\nPC41005 0.865 -0.135 0.236 0.251 0.88 0.116\n15 0.802 -0.198 0.232 0.240 0.83 0.141\n3005 0.938 -0.062 0.133 0.121 0.94 0.077\n15 0.902 -0.098 0.133 0.118 0.91 0.075\nPS11005 0.978 -0.022 0.232 0.251 0.93 0.136\n15 0.941 -0.059 0.228 0.243 0.93 0.224\n3005 0.980 -0.020 0.133 0.121 0.96 0.053\n15 0.967 -0.033 0.132 0.121 0.94 0.129\nPS21005 0.822 -0.178 0.233 0.252 0.84 0.127\n15 0.675 -0.325 0.223 0.236 0.71 0.235\n3005 0.917 -0.083 0.133 0.123 0.91 0.058\n15 0.844 -0.156 0.132 0.122 0.80 0.114\nPS31005 0.966 -0.034 0.233 0.244 0.92 0.074\n15 0.964 -0.036 0.233 0.242 0.92 0.084\n3005 0.974 -0.026 0.133 0.120 0.95 0.043\n15 0.973 -0.027 0.133 0.119 0.95 0.048\nTable 5 Average, bias, SE, SD and CP of the regression coefficient βand RMSD of the log (h0(t))\ncorresponding to all inferential and replicate processes f or the Scenario 3 simulated data.\n© 2010 WILEY-VCH Verlag GmbH & Co. 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Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Notkestr. 9, 22607 Hamburg, Germany\n2Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia\n3The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany\n4Physics Department and Research Center OPTIMAS, University Kaiserslautern-Landau,\nErwin-Schr¨ odinger Str. 46, 67663 Kaiserslautern, Germany\n(Dated: February 1, 2024)\nWe report on a nonequilibrium quantum phase transition (NQPT) in a hybrid quantum many-\nbody system consisting of a vibrational mode of a damped nanomembrane interacting optomechan-\nically with a cavity, whose output light couples to two internal states of an ultracold Bose gas held\nin an external quasi-one-dimensional box potential. For small effective membrane-atom couplings,\nthe system is in a homogeneous Bose-Einstein condensate (BEC) steady state, with no membrane\ndisplacement. Depending on the transition frequency between the two internal atomic states, ei-\nther one or both internal states are occupied. By increasing the atom-membrane couplings, the\nsystem transitions to a symmetry-broken self-organized BEC phase, which is characterized by a\nconsiderably displaced membrane steady-state and density wave-like BEC profiles. This NQPT can\nbe both discontinuous and continuous for a certain interval of transition frequencies, and is purely\ndiscontinuous outside of it.\nI. INTRODUCTION\nMany physical systems can be classified in terms of\ntheir emergent collective behavior when they undergo\nphase transitions [1–3]. While equilibrium phase tran-\nsitions are fairly well understood, nonequilibrium quan-\ntum phase transitions (NQPTs) are a relatively new and\nexciting field of research [4, 5]. Such phase transitions oc-\ncur in a nonequilibrium steady state of driven quantum\nsystems in contact with an environment, so that their\nenergy is not conserved due to external driving and dis-\nsipation [6–16]. Among other cases, NQPTs have been\nstudied in ultracold atoms in an optical cavity [17–24],\nin driven-dissipative Bose-Einstein condensates [25, 26],\nin microcavity-polariton systems [27–29], and in photon\nBose-Einstein condensates [30–32]. Additionally, dynam-\nical quantum phase transitions and universal scaling have\nbeen observed both in the nonequilibrium dynamics of\nisolated quantum systems after a quench, with time play-\ning the role of the control parameter [33], and recently in\nopen quantum systems, whose dynamics is driven by the\ndissipative contact to an environment [34].\nHybrid atom-optomechanical systems, which combine\noptomechanics with atom optics, represent one recent\npromising platform for studying NQPTs [35–43]. In these\nsystems, a nanomembrane in an optical cavity is coupled\nto the motional [35] or to the internal [36] degrees of free-\ndom of a distant cloud of cold atoms that are trapped\nin the optical lattice of the outcoupled light field. By\n∗milan.radonjic@physik.uni-hamburg.de\n†lmixa@physnet.uni-hamburg.de\n‡axel.pelster@rptu.de\n§michael.thorwart@physik.uni-hamburg.deemploying different cooling mechanisms, such as sym-\npathetic cooling [35, 37–39] and optical feedback cool-\ning [39, 44, 45], the nanomembrane can be cooled down\nalmost to its quantum ground state. Moreover, hybrid\natom-optomechanical systems are a versatile playground\nfor various quantum phenomena, such as indirect quan-\ntum measurement, atom-membrane entanglement and\ncoherent state transfer [46–50].\nThe internal state coupling scheme [36] is realized by\na nanomembrane that is coupled to transitions between\ninternal states of the atoms via translating the phase\nshift of the light, caused by the membrane displacement,\ninto a polarization rotation using a polarizing beam split-\nter. This scheme allows for resonant coupling and miti-\ngates the drawbacks of the motional coupling scheme [35],\nlike strong frequency mismatch between the nanooscil-\nlator and the atomic motion in the optical trap. It has\nbeen utilized for membrane cooling [36, 51], displacement\nsqueezing [52], the realization of a effective negative mass\noscillator [53], and quantum back-action evading mea-\nsurements [54]. In addition, a peculiar NQPT, whose\norder can be controlled by changing a directly accessi-\nble experimental parameter, has recently been proposed\nin this scheme [55]. It has been found that the system\ncan undergo both a first- and a second-order phase tran-\nsition in the same physical setup, by tuning the atomic\ntransition frequency.\nIn this work, we examine NQPTs in a hybrid atom-\noptomechanical system in the internal state coupling\nscheme. We consider an ultracold atomic gas trapped in a\nlarge quasi-1D box, where the lattice potentials generated\nby the light fields have been canceled. Since the mem-\nbrane is naturally damped by its suspension, the atoms\nrelax into a steady state whose features critically depend\non the state of the membrane. For large enough trap-\nping box and weak effective coupling between the atomsarXiv:2401.18015v1  [cond-mat.quant-gas]  31 Jan 20242\nand the nanomembrane, we find that the atomic conden-\nsates are essentially uniform and the membrane is not\ndisplaced. When the atom-membrane couplings become\nstrong enough, the system transitions into a symmetry-\nbroken self-organized phase that exhibits density wave-\nlike condensate profiles and significant nanomembrane\ndisplacement. Depending on the values of the readily\ncontrollable experimental parameters, this NQPT can be\nof either first or second order. We map out the relevant\nparts of the parameter space and study the hallmarks\nof the phase transition. Hence, the rest of the paper\nis organized as follows. In Sec. II we outline the basic\nmean-field theory. In the following Sec. III we examine\nthe homogeneous steady states. Next, in Sec. IV we fo-\ncus on the inhomogeneous steady state and its features.\nWe also address and examine the occurring 1st and 2nd\norder phase transitions. We discuss our results in Sec. V\nand provide an outlook.\nII. MODEL AND EQUATIONS OF MOTION\nThe system under consideration [36] consists of N\nidentical ultracold bosonic atoms held in an external\nquasi-1D trap (see Fig. 1). The relevant atomic states\n{|−⟩,|+⟩,|e⟩}constitute a Λ level scheme, where the two\nlower states are energetically separated by the atomic\ntransition frequency Ωa. An applied σ−circularly polar-\nized laser of frequency ωLdrives the internal transition|+⟩ ↔ | e⟩at a finite large detuning ∆. The passing\nbeam is sent to a polarizing beam splitter (PBS), which\nsplits the circularly polarized light into linearly polarized\nπxandπylight beams in two perpendicular directions.\nIn the vertical arm, a fixed mirror simply reflects light\nwith conserved polarization πx. The horizontal beam en-\nters a low-finesse cavity which hosts a semitransparent\nnanomembrane with the mechanical resonance frequency\nΩm. The lengths of the two arms are made equal when\nthe membrane is undisplaced. In that case, the cavity\nreflects πylight without any relative phase shift. In a\nquasistatic picture, any finite displacement of the mem-\nbrane induces a relative phase shift between the πxand\nπylight beams from the two arms. This relative phase is\nconverted to a polarization rotation after the two beams\nhave passed backwards through the PBS. The emerging\nσ+polarized light then drives the internal atomic tran-\nsition |−⟩ ↔ | e⟩and may induce a two-photon Raman\ntransition between the two lower states of the Λ scheme\nvia the excited state |e⟩. On the other hand, an inter-\nnal transition between the two lower atomic states leads\nto the emission of a σ+polarized photon which alters\nthe radiation pressure on the membrane after reaching\nit, which leads to an effective atom-membrane coupling.\nAn adiabatic elimination of the excited state and the\nlight field results [55] in an effective Hamiltonian involv-\ning the membrane and the low-energy atomic degrees of\nfreedom ( ℏ=c= 1)\nˆHeff= Ωmˆa†ˆa+X\nν=±Z\ndzˆψ†\nν(z)\u0014\nνΩa\n2−ωR∂2\nz+Vν(z) +1\n2X\nν′=±gνν′ˆψ†\nν′(z)ˆψν′(z)\u0015\nˆψν(z)\n−(ˆa+ ˆa†)Z\ndzsin(2z)\u0014λ\n2\u0000ˆψ†\n+(z)ˆψ−(z) +ˆψ†\n−(z)ˆψ+(z)\u0001\n+λexˆψ†\n+(z)ˆψ+(z)\u0015\n, (1)\nwhere the position coordinate has been re-scaled as z→z/ωL. The bosonic operator ˆ adenotes the annihilation\noperator of the excitations of one single mechanical mode of the nanomenbrane, while ˆψν(z) denotes the field operator\nof the atoms in the state ν. The atomic recoil frequency reads ωR=ω2\nL/(2m), the external trapping potentials are\ndenoted by V±(z), while gνν′are atomic state-dependent contact interaction strengths, and the effective membrane-\natom coupling strengths are λandλex. Note that the effective cavity-mediated global atom-atom interaction [56] has\nbeen neglected. Taking into account that the membrane is damped at the rate Γm, we derive the following equations\nof motion for the membrane and the atomic operators\ni∂tˆa=\u0000\nΩm−iΓm\u0001\nˆa−Z\ndzsin(2z)\u0014λ\n2\u0000ˆψ†\n+(z)ˆψ−(z) +ˆψ†\n−(z)ˆψ+(z)\u0001\n+λexˆψ†\n+(z)ˆψ+(z)\u0015\n−ˆξm, (2a)\ni∂tˆψ−(z) =\u0014\n−Ωa\n2−ωR∂2\nz+V−(z) +X\nν=±g−νˆψ†\nν(z)ˆψν(z)\u0015\nˆψ−(z)−λ\n2sin(2z)\u0000\nˆa+ ˆa†\u0001ˆψ+(z), (2b)\ni∂tˆψ+(z) =\u0014Ωa\n2−ωR∂2\nz+V+(z) +X\nν=±g+νˆψ†\nν(z)ˆψν(z)\u0015\nˆψ+(z)−λ\n2sin(2z)\u0000\nˆa+ ˆa†\u0001ˆψ−(z)\n−λexsin(2z)\u0000\nˆa+ ˆa†\u0001ˆψ+(z), (2c)\nwhere we have omitted the time argument for brevity. In accordance with the damping of the membrane mode,3\nFIG. 1. A semitransparent nanomembrane in an optical cav-\nity is coupled to the internal states of a distant atomic ensem-\nble that is trapped in an external quasi-1D trap. The internal\nstates of the atoms constitute a Λ level scheme according to\nthe inset. The figure is adapted from Ref. [55].\nwe have introduced the corresponding bosonic noise op-erator ˆξmthat is characterized by the zero mean fluctua-\ntions⟨ˆξm⟩=⟨ˆξ†\nm⟩= 0, as well as by the auto-correlations\n⟨ˆξm(t)ˆξ†\nm(0)⟩= 2Γm(nm+ 1)δ(t), (3a)\n⟨ˆξ†\nm(t)ˆξm(0)⟩= 2Γmnmδ(t). (3b)\nThe environment occupation number nmdetermines the\nsteady-state occupation of the membrane mode when it\nis affected only by its environment [57].\nWe assume that the atoms are at ultra-low tempera-\nture and that the atom-membrane coupling is such that a\nlarge fraction of atoms condenses. Thus, the system dy-\nnamics can be described by the following coupled mean-\nfield equations of motion\ni∂tα=\u0000\nΩm−iΓm\u0001\nα−Z\ndzsin(2z)\u0014√\nNλ\n2\u0000\nψ∗\n+(z)ψ−(z) +ψ∗\n−(z)ψ+(z)\u0001\n+√\nNλex|ψ+(z)|2\u0015\n, (4a)\ni∂tψ−(z) =\u0014\n−Ωa\n2−ωR∂2\nz+V−(z) +NX\nν=±g−ν|ψν(z)|2\u0015\nψ−(z)−√\nNλ\n2sin(2z)\u0000\nα+α∗\u0001\nψ+(z), (4b)\ni∂tψ+(z) =\u0014Ωa\n2−ωR∂2\nz+V+(z) +NX\nν=±g+ν|ψν(z)|2\u0015\nψ+(z)−√\nNλ\n2sin(2z)\u0000\nα+α∗\u0001\nψ−(z)\n−√\nNλexsin(2z)\u0000\nα+α∗\u0001\nψ+(z), (4c)\nwhere we have neglected all mutual correlations and introduced the complex membrane amplitude α=⟨ˆa⟩/√\nN, and\nthe wave functions of the two condensates ψ±(z) =⟨ˆψ±(z)⟩/√\nN. We assume that the membrane damping is so fast\nthat the membrane almost instantaneously relaxes to its steady state determined by the condensate wave functions\n¯α=1\nΩm−iΓmZ\ndzsin(2z)\u0014√\nNλex|ψ+(z)|2+√\nNλ\n2\u0000\nψ∗\n+(z)ψ−(z) +ψ∗\n−(z)ψ+(z)\u0001\u0015\n, (5)\nso that we obtain the following coupled equations for the two condensates\ni∂tψ−(z) =\u0014\n−Ωa\n2−ωR∂2\nz+V−(z) +NX\nν=±g−ν|ψν(z)|2\u0015\nψ−(z)−√\nNλ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001\nψ+(z), (6a)\ni∂tψ+(z) =\u0014Ωa\n2−ωR∂2\nz+V+(z) +NX\nν=±g+ν|ψν(z)|2\u0015\nψ+(z)−√\nNλ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001\nψ−(z)\n−√\nNλexsin(2z)\u0000\n¯α+ ¯α∗\u0001\nψ+(z). (6b)\nDue to the damping of the membrane, the atoms also eventually reach the minimal-energy steady state, i.e., the\nground steady state (GSS), described by the ansatz ψ±(z, t) =e−iµtψ±(z). It involves the chemical potential µthat\nis fixed by the normalization condition\nZ\ndz\u0000\n|ψ−(z)|2+|ψ+(z)|2\u0001\n= 1. (7)\nIt follows that the GSS solutions of the equations of motion (6) can be obtained either by solving the stationary\nequations\n0 =\u0014\n−Ωa\n2−ωR∂2\nz−µ+V−(z) +NX\nν=±g−ν|ψν(z)|2\u0015\nψ−(z)−√\nNλ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001\nψ+(z), (8a)4\n0 =\u0014Ωa\n2−ωR∂2\nz−µ+V+(z) +NX\nν=±g+ν|ψν(z)|2\u0015\nψ+(z)−√\nNλ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001\nψ−(z)\n−√\nNλexsin(2z)\u0000\n¯α+ ¯α∗\u0001\nψ+(z), (8b)\nor, equivalently, by minimizing the energy functional\nEN[ψ±] =NX\nν=±Z\ndz ψ∗\nν(z)\u0014\nνΩa\n2−ωR∂2\nz+Vν(z) +N\n2X\nν′=±gνν′|ψν′(z)|2\u0015\nψν(z)\n−N2Ωm\nΩ2m+ Γ2m\u001aZ\ndzsin(2z)\u0014λ\n2\u0000\nψ∗\n+(z)ψ−(z) +ψ∗\n−(z)ψ+(z)\u0001\n+λex|ψ+(z)|2\u0015\u001b2\n, (9)\nwith respect to ψ∗\n±(z) and under the normalization constraint (7). This minimization justifies our notion of the GSS.\nThe chemical potential stems from the relation µ=∂EN[ψ±]/∂N and reads\nµ=X\nν=±Z\ndz ψ∗\nν(z)\u0014\nνΩa\n2−ωR∂2\nz+Vν(z) +NX\nν′=±gνν′|ψν′(z)|2\u0015\nψν(z)\n−2NΩm\nΩ2m+ Γ2m\u001aZ\ndzsin(2z)\u0014λ\n2\u0000\nψ∗\n+(z)ψ−(z) +ψ∗\n−(z)ψ+(z)\u0001\n+λex|ψ+(z)|2\u0015\u001b2\n. (10)\nLet us next examine the ground steady state of the system. The external trapping potentials are designed to cancel\nthe lattice potentials due to the driving laser and result in the box-like traps V±(z) with the hard walls at z∈ {z1, z2},\nwhere z1=−π/4−ℓπ,z2=−π/4+ℓπfor some large integer ℓ≳100. That imposes the Dirichlet boundary conditions\nψ±(z1) =ψ±(z2) = 0. According to our assumption ℓ≫1, we can safely neglect any edge effects that significantly\naffect the condensates only within a few healing lengths of the walls. Since the atom-membrane interaction is periodic\nwith the period of π, we can reduce our attention onto the single period [ −π/4,3π/4]. We impose the periodic\nboundary conditions ψ−(−π/4) = ψ−(3π/4),ψ+(−π/4) = ψ+(3π/4) and take V±(z) = 0 henceforward. Note that\nthe spatial dependence of the atom-membrane interaction, ∝sin(2z), is even with respect to the center of the chosen\nperiod. This brings us to the energy per period\n˜E˜N[˜ψ±]≡EN[ψ±]\n2ℓ=˜NX\nν=±Z\nπdz˜ψ∗\nν(z)\u0014\nνΩa\n2−ωR∂2\nz+˜N\n2X\nν′=±gνν′|˜ψν′(z)|2\u0015\n˜ψν(z)\n−˜N2Ωm\nΩ2m+ Γ2m\u001aZ\nπdzsin(2z)\u0014˜λ\n2\u0000˜ψ∗\n+(z)˜ψ−(z) +˜ψ∗\n−(z)˜ψ+(z)\u0001\n+˜λex|˜ψ+(z)|2\u0015\u001b2\n, (11)\nwhere we introduced the rescaled quantities\nN= 2ℓ˜N, ψ±(z) =˜ψ±(z)/√\n2ℓ, λ =˜λ/√\n2ℓ, λex=˜λex/√\n2ℓ. (12)\nThe integrals in Eq. (11) run over one single period of length π. In terms of the above, the membrane amplitude\nbecomes\n¯α=1\nΩm−iΓmZ\nπdzsin(2z)\u0014p\n˜N˜λex|˜ψ+(z)|2+p\n˜N˜λ\n2\u0000˜ψ∗\n+(z)˜ψ−(z) +˜ψ∗\n−(z)˜ψ+(z)\u0001\u0015\n, (13)\nwhile the rescaled wave functions evolve in accordance with\ni∂t˜ψ−(z) =\u0014\n−Ωa\n2−ωR∂2\nz+˜NX\nν=±g−ν|˜ψν(z)|2\u0015\n˜ψ−(z)−p\n˜N˜λ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001˜ψ+(z), (14a)\ni∂t˜ψ+(z) =\u0014Ωa\n2−ωR∂2\nz+˜NX\nν=±g+ν|˜ψν(z)|2\u0015\n˜ψ+(z)−p\n˜N˜λ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001˜ψ−(z)\n−p\n˜N˜λexsin(2z)\u0000\n¯α+ ¯α∗\u0001˜ψ+(z), (14b)\nand are normalized according to\nZ\nπdz\u0000\n|˜ψ−(z)|2+|˜ψ+(z)|2\u0001\n= 1. (15)5\nThe stationary wave function equations, which determine the GSS, turn into\n0 =\u0014\n−Ωa\n2−ωR∂2\nz−µ+˜NX\nν=±g−ν|˜ψν(z)|2\u0015\n˜ψ−(z)−p\n˜N˜λ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001˜ψ+(z), (16a)\n0 =\u0014Ωa\n2−ωR∂2\nz−µ+˜NX\nν=±g+ν|˜ψν(z)|2\u0015\n˜ψ+(z)−p\n˜N˜λ\n2sin(2z)\u0000\n¯α+ ¯α∗\u0001˜ψ−(z)\n−p\n˜N˜λexsin(2z)\u0000\n¯α+ ¯α∗\u0001˜ψ+(z), (16b)\nwhere the chemical potential is given by\nµ=X\nν=±Z\nπdz˜ψ∗\nν(z)\u0014\nνΩa\n2−ωR∂2\nz+˜NX\nν′=±gνν′|˜ψν′(z)|2\u0015\n˜ψν(z)\n−2˜NΩm\nΩ2m+ Γ2m\u001aZ\nπdzsin(2z)\u0014˜λ\n2\u0000˜ψ∗\n+(z)˜ψ−(z) +˜ψ∗\n−(z)˜ψ+(z)\u0001\n+˜λex|˜ψ+(z)|2\u0015\u001b2\n. (17)\nIn the following, we will assume that the rescalings (12)\nhave been performed, and, for the sake of brevity, we will\nomit the tildes. All extensive quantities addressed below\nare therefore per period.\nIII. HOMOGENEOUS GROUND STEADY\nSTATES\nWe first analyze the homogeneous solutions using the\nansatz ψ±(z) =p\nn±/π, where n±are the occupation\nfractions of the two atomic states. In such a case, the\nnormalization condition simply reads n−+n+= 1 and\nthe membrane is not excited, i.e., ¯ α= 0. The stationary\nequations (16) become\n0 =\u0014\n−Ωa\n2−µ+N\nπX\nν=±g−νnν\u0015p\nn−, (18a)\n0 =\u0014Ωa\n2−µ+N\nπX\nν=±g+νnν\u0015p\nn+. (18b)\nWe will be considering the on average repulsive regime\nwhere g−−, g++>0> g+−andg−−g++> g2\n+−. Three\nsolutions are possible, which read\nn−= 1, n+= 0, µ =−Ωa\n2+N\nπg−−,(19a)\nn−= 0, n+= 1, µ =Ωa\n2+N\nπg++,(19b)\nas well as\nn−=g++−g+−+πΩa/N\ng−−+g++−2g+−,\nn+=g−−−g+−−πΩa/N\ng−−+g++−2g+−,µ=Ωa\n2g−−−g++\ng−−+g++−2g+−\n+N\nπg−−g++−g2\n+−\ng−−+g++−2g+−. (19c)\nThe first solution is the lowest in energy when πΩa/N >\ng−−−g+−holds, while for πΩa/N < g+−−g++, the\nsecond one takes over. The third solution exists only if\ng+−−g++< πΩa/N < g−−−g+−is satisfied and has the\nlowest energy of the three. For πΩa/N < (g−−−g++)/2\nwe have n−< n+, while for πΩa/N≥(g−−−g++)/2\nwe get n−≥n+. Thus, it is convenient to introduce the\nreference transition frequency Ωa,0=N(g−−−g++)/(2π)\nand study the behavior of the system as a function of the\nrelative detuning\nδΩa≡Ωa−Ωa,0, (20)\nof the atomic transition frequency from it. For strong\nenough couplings λandλexthe system transitions into a\nself-organized GSS. The membrane becomes excited and\nthe condensates become inhomogeneous. The nature of\nthis phase transition is what we are going to investigate\nnext.\nIV. PHASE TRANSITIONS TO AND FROM\nINHOMOGENEOUS GROUND STEADY STATE\nLet us now explore the self-organized phase. To this\nend, we evolve the equations of motion (14) in imagi-\nnary time starting with a suitable initial state until the\nground steady state is reached. As can be seen from these\nequations, the membrane couples to the condensates via\nthe combination Re[¯ α] sin(2 z). The inhomogeneous GSS\nwave functions have to reach their maximum together\nwith sin(2 z), i.e., in the middle of the chosen period.\nAlso, ψ±(z) have to be even functions with respect to the\nsame point, in the same manner as sin(2 z). In this way,6\nFIG. 2. Spatial profiles of Re[¯ α] sin(2 z) (green) and the\nground steady state condensate wave functions ψ−(z) (red)\nandψ+(z) (blue) over one period for large negative (a) and\nlarge positive (b) relative detuning. Parameters: Ω m=\n100ωR, Γm= 10 ωR,Ng−−=Ng++= 100 ωR,Ng+−=\n−90ωR,√\nNλex= 35ωR, while in (a) δΩa=−40ωR,√\nNλ=\n5ωRand in (b) δΩa= 70ωR,√\nNλ= 70ωR.\nthe atom-membrane interaction term in Eq. (11) con-\ntributes most to the minimization of the energy of the\nsystem. Thus, the combination Re[¯ α] sin(2 z) captures\nthe essence of the membrane’s influence on the atoms,\nboth in terms of its spatial behavior and its relative\nstrength. Figure 2 shows two exemplary cases of spatial\nprofiles of Re[¯ α] sin(2 z), together with the two GSS wave\nfunctions ψ±(z) over the chosen period. In the panel\n(a) we present the case of a negative relative detuning\nδΩa, where the |+⟩state usually has higher occupation.\nThe panel (b) displays the opposite case with the |−⟩\nstate being more populated. As can be seen, larger spa-\ntial variances of the wave functions correspond to larger\nmembrane amplitude. For large negative relative de-\ntunings, the condensate is homogeneous and the atoms\noccupy exclusively the |+⟩state. Likewise, for large posi-\ntiveδΩathe atoms homogeneously condense into the |−⟩\nstate. In between, both atomic states exhibit a macro-\nscopic population, while the system may transition from\nthe homogeneous to the self-organized phase. Such tran-\nsitions occur for sufficiently large couplings to the mem-\nbrane λandλex. Their values determine whether the\ntransitions are continuous or discontinuous.\nIn Fig. 3 we showcase the behavior of Re[¯ α] and the\n-200 -100 0 100 2000.00.20.40.6\n-200 -100 0 100 200-1.0-0.50.00.51.0FIG. 3. Dependence of the real part of the membrane ampli-\ntude Re[¯ α] (a) and the population imbalance d−+≡(N−−\nN+)/N(b) between the |−⟩and the |+⟩state on the relative\ndetuning δΩafor a fixed value of the coupling√\nNλex= 40ωR\nand for three values of the coupling√\nNλ∈ {20,70,120}ωR\n(green, red, blue curves). Other parameters: Ω m= 100 ωR,\nΓm= 10ωR,Ng−−=Ng++= 100 ωR,Ng+−=−90ωR.\npopulation imbalance d−+≡(N−−N+)/Nwhen the\nrelative detuning is scanned across zero, for a fixed value\nof the coupling√\nNλex= 40ωRand for three values of\nthe coupling√\nNλ∈ {20,70,120}ωR(green, red, blue\ncurves). Here, we defined N±as the population per pe-\nriod of the state |±⟩. The green curves depict a case of a\ncontinuous second-order phase transition on the negative\nside, followed by a continuous second-order transition on\nthe positive side as well. This feature is manifest in both\nRe[¯α] and d−+. For a larger value of λ(red curves),\nthe transition on the negative side becomes discontinu-\nous and of first order, while on the positive side it is still\ncontinuous. For even larger value of λ(blue curves), the\nlatter transition also becomes a discontinuous first-order\none.\nIn order to acquire additional insight into the proper-\nties of the system, in Fig. 4 we show the behavior of the\ncondensate spatial profiles ψ±(z) along the three curves\nshown in Fig. 3 as we sweep the detuning from nega-\ntive to positive values. One can visually distinguish the\nhomogeneous and the self-organized phases. The wave\nfunctions ψ±(z) are symmetric and maximal in the cen-\nter of the period. The sharpness of the color gradients\naround the transition points reflects the (dis)continuity of\nthe respective transition. The width of ψ+(z) decreases7\nFIG. 4. Spatial profiles of the condensate wave functions ψ+(z) (left panels) and ψ−(z) (right panels) depending on the relative\ndetuning δΩafor a fixed value of the coupling√\nNλex= 40ωRand for three values of the coupling√\nNλ∈ {20,70,120}ωR(top,\nmiddle, bottom row). The remaining parameters are Ω m= 100 ωR, Γm= 10ωR,Ng−−=Ng++= 100 ωR,Ng+−=−90ωR.\nuntil shortly before the second transition, while the width\nofψ−(z) increases monotonically. Stronger couplings λ\nandλextypically lead to narrower wave function profiles\nψ±(z) due to the membrane-induced localization of the\natoms.\nWe extend the study of the transitions discussed above\nto a range of values of the coupling strength λ. Figure 5\ndisplays the dependence of Re[¯ α] (left panels) and d−+\n(right panels) on δΩaandλfor two values of the coupling√\nNλex= 40ωR(top row) and√\nNλex= 25ωR(bottom\nrow). The horizontal dashed lines in the top left panelmark explicitly those cases shown in Figs. 3 and 4. The\nsingle-colored blue (red) regions in the right panels cor-\nrespond to the homogeneous phase where the atoms con-\ndense into the |+⟩(|−⟩) state. Careful inspection of the\nleft panels reveals that there are blue intermediate re-\ngions where the condensates are homogeneous and both\natomic states have significant population. These appear\nfor√\nNλ≲20ωRand only around |δΩa| ≈50ωRfor\nthe larger coupling λex, as opposed to the entire range\n|δΩa|≲50ωRfor the smaller coupling λex. Therefore,\nto discriminate experimentally between the homogeneous\nand the inhomogeneous phase, one has first to measure8\n0 0.2 0.4 0.6 0.81.0\n-1.0 -0.5 0 0.51.0\n0 0.2 0.4 0.6 0.8\n-1.0 -0.5 0 0.51.0\nFIG. 5. Dependence of the real part of the membrane amplitude Re[¯ α] (left panels) and the population imbalance d−+=\n(N−−N+)/N(right panels) on the relative detuning δΩaand the coupling strength√\nNλ, for√\nNλex= 40 ωR(top row)\nand√\nNλex= 25ωR(bottom row). The horizontal dashed lines in the top left panel mark those cases shown in Figs. 3 and\n4. The black solid lines in the right panels correspond to N−=N+. Other parameters are: Ω m= 100 ωR, Γm= 10 ωR,\nNg−−=Ng++= 100 ωR,Ng+−=−90ωR.\nthe membrane displacement. If it turns out to be essen-\ntially zero, further characterization of the homogeneous\nphase can be done by measuring the population imbal-\nance d−+.\nThe continuous nature of the transitions to the self-\norganized phase is manifest in the gradual changes of\nthe color-coding on both the negative and the positive\ndetuning side for√\nNλ≲40ωRin the former case and\nfor√\nNλ≲110ωRin the latter case. In the top row,\nthe transition on the negative side becomes sharp and\ndiscontinuous within the strip 40 ωR≲√\nNλ≲110ωR,\nwhile the one on the positive side is still continuous.\nAbove that, both transitions become discontinuous. For\nthe smaller coupling λexin the bottom row, the cor-responding coupling strengths are√\nNλ≈110ωRand√\nNλ≈160ωR, respectively.\nFurther understanding can be gained by investigat-\ning the energy behavior. Figure 6 portrays the energy\nper period, given by Eq. (11), as a function of the rel-\native detuning δΩaand the coupling strength√\nNλ, for√\nNλex= 40ωR(top panel) and√\nNλex= 25ωR(bottom\npanel). Noticeably, there are no discontinuities within\nthe energy landscape. However, the energy features gra-\ndient jumps along the curves that correspond to discon-\ntinuous phase transitions observed in Fig. 5. This is il-\nlustrated both by the shading and the non-smoothness\nof the contour lines. The energy is maximal when both\nthe relative detuning and the coupling λare zero. As ex-9\n-100 -80 -60 -40 -20 0\n-125 -100 -75 -50 -25 0\nFIG. 6. Dependence of the energy per period, Eq. (11), on\nthe relative detuning δΩaand the coupling strength√\nNλ,\nfor√\nNλex= 40ωR(top panel) and√\nNλex= 25ωR(bottom\npanel). The black curves represent the contour lines. Used\nparameters: Ω m= 100 ωR, Γm= 10 ωR,Ng−−=Ng++=\n100ωR,Ng+−=−90ωR.\npected, the larger either of the couplings λorλexis, the\nlower is the energy due to the stabilizing atom-membrane\ninteraction. We also observe that for the fixed couplings\nthe energy of the homogeneous phases decreases with in-\ncreasing magnitude of δΩa, while the self-ordered phase\nexhibits a non-monotonic dependence in this respect.\nIt is worth illustrating the asymmetric role played by\nthe couplings λandλex, which is obvious from the equa-\ntions of motion (14). Figure 7 shows Re[¯ α] as a function\nof the two coupling strengths, for δΩa=−120ωR(top\npanel) and δΩa= 120 ωR(bottom panel). For presen-\ntation purposes, and different than in other plots, we\nhave chosen in this context the imbalanced contact in-\n0 0.5 1.0 1.5 2.0\n0 0.5 1.0 1.5 2.0\nFIG. 7. Dependence of the real part of the membrane am-\nplitude Re[¯ α] on the coupling strengths√\nNλand√\nNλex,\nforδΩa=−120ωR(top panel) and δΩa= 120 ωR(bottom\npanel). Other parameters are: Ω m= 100 ωR, Γm= 10 ωR,\nNg−−= 50ωR,Ng++= 200 ωR,Ng+−=−90ωR.\nteraction strengths Ng−−= 50ωRandNg++= 200 ωR.\nThe regions that correspond to the homogeneous phase\nare not symmetric with respect to λandλex. For large\nenough relative detuning, increasing λexleads to a first-\norder transition for small λand potentially to a second-\norder transition for larger λ. Conversely, increasing λ\nleads to a second-order transition for small λexand to a\nfirst-order transition for larger λex.\nV. DISCUSSION AND OUTLOOK\nThe above examples demonstrate that the considered\nsystem is versatile and offers multiple tuning knobs for10\nchanging the order of the phase transition between the\nhomogeneous and the self-organized density wave-like\nstate. These include, but are not limited to, the rela-\ntive detuning δΩaof the atomic transition frequency Ωa\nand the coupling strengths λandλex. The two states\ncan be experimentally distinguished by measuring the\nmembrane displacement. To further characterize the ho-\nmogeneous state, one can resort to the population im-\nbalance d−+. We have not explored the corresponding\nrole of the atomic contact interaction strengths gνν′in\nthis work. They represent, of course, further means of\ncontrolling the behavior of the system. The asymmet-\nric nature of the atom-membrane coupling can either be\nfurther enhanced or effectively suppressed by an appro-\npriate asymmetric choice of gνν′. Depending on the path\ntaken through the experimental parameter space, one can\nencounter either first- or second-order transition hyper-\nsurfaces. A possible future work could involve dynamics\nbeyond the studied nonequilibrium steady state. In such\na case, the system parameters could be dynamically con-\ntrolled along different open or closed paths. In certain\nscenarios, hysteresis should appear and can be studied.\nIt would be interesting to investigate how the above\nmean-field analysis may be amended once quantum fluc-\ntuations have been taken into account. The spatial pro-\nfiles of the two condensates critically determine which\natomic normal modes are coupled to the membrane.\nThose that are coupled should exhibit enhanced occu-\npation and may significantly alter primarily the homoge-\nneous state. Therefore, the order of the observed phase\ntransitions should be reanalyzed. Moreover, quantum\nfluctuations may lead to the emergence of new phases,\nakin to the formation of self-bound quantum droplets in\nBose-Bose mixtures and dipolar Bose gases [58, 59]. In\nthe considered system the membrane may have an essen-tial role to play. Beyond mean-field quantum effects may\nalso reveal the imprint of NQPTs and their order on the\nstatistics of the outcoupled light.\nWe find it important to note the differences between\nthe NQPT reported here and the one studied in Ref. [55].\nFirst, in the cited work it was essential to have a lattice\npotential present for the atoms. This led to exclusively\ndensity wave-like condensate steady states. Second, only\nthe special case of atomic contact interaction strengths\ngνν′=gwas considered. Moreover, the phase transition\nstudied therein would occur even in the absence of atom-\natom interactions within the BEC, i.e., for gνν′= 0. Ob-\nviously, our present scenario requires the overall repulsive\ninteractions.\nIn conclusion, in this paper we have presented a flexible\nhybrid atom-optomechanical experimental platform for\nthe study of both continuous and discontinuous NQPTs.\nImportantly, it is possible to choose between the two by\ntuning readily accessible experimental parameters. The\npresent work lays the foundation for further exploration\nof possible quantum phases of ultracold atomic gases\nalong the lines of self-bound quantum droplets.\nACKNOWLEDGMENTS\nThis work was supported by the Deutsche Forschungs-\ngemeinschaft (DFG, German Research Foundation) via\nthe Collaborative Research Center SFB/TR185 (Project\nNo. 277625399) (A.P.) and via the Research Grant\n274978739 (M.R. and M.T.). We also acknowledge the\nsupport from the DFG Cluster of Excellence “CUI: Ad-\nvanced Imaging of Matter” −EXC 2056 (Project ID\n390715994).\n[1] S. Sachdev, Quantum Phase Transitions , 2nd ed. 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Lotnik´ ow 32/46, 02-668 Warsaw, Poland\n2Astronomy Department, Universidad de Concepci´ on, Barrio Universitario S/N, Concepci´ on 4030000, Chile\n3Laborat´ orio Nacional de Astrof´ ısica, MCTI, Rua dos Estados Unidos, 154, 37504-364 Itajub´ a, MG, Brazil\n4Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotl´ aˇ rsk´ a 2, 611 37 Brno, Czech Republic\nABSTRACT\nWe present a set of new theoretical Fe II templates for bright quasars covering a wavelength range\nof 1000-10000 ˚A, based on the recent atomic database available in the C23.00 version of the photoion-\nization code CLOUDY . We compute a grid of models for a range of incident photon flux and gas density\nand for multiple microturbulence velocities. We analyze the ratios of Fe II emission over a variety of\nwavebands and compare them with observations. Our key results are: (1) Despite the use of the newest\natomic data we still confirm the long-standing problem that the predicted Fe II UV/optical ratio is\nsignificantly larger than that observed in the AGN spectra. (2) The ratio is not significantly affected by\nthe variations in the microturbulence and the metallicity. (3) The turbulence can create an additional\napparent velocity shift of up to 1000 km/s in the spectra. (4) There is no single Fe II template that\ncan fit the observational data covering UV to optical wavelength range. We shortly discuss the most\nlikely effects responsible for the Fe II UV/optical mismatch problem: the assumption of the constant\ncolumn density and the assumption of the isotropic emission implying equal contribution of the bright\nirradiated faces and the dark shielded faces of the clouds.\nKeywords: Line: formation — methods: numerical — galaxies: active —quasars: emission lines;\nRadiative transfer simulations; Photoionization\n1.INTRODUCTION\nThe dominant component of the bright active galactic nuclei (AGN) in the optical/UV band is the continuum\ncomponent coming from the cold optically thick geometrically thin disk (see e.g. Krolik 1999; Karas et al. 2021;\nZajaˇ cek et al. 2023a, for reviews). Locally this component is well described as a power law although at high quality\nand/or in broadband data some curvature of the component is visible (e.g. Capellupo et al. 2015). Superimposed on\nthis continuum component are numerous strong and broad emission lines like H β, Mg II, C IV, and Ly α, depending\non the redshift range covered by the data. These lines are easy to identify in the individual spectra of AGN and quasar\ncomposite spectra (e.g. Francis et al. 1991; Vanden Berk et al. 2001; Telfer et al. 2002; Scott et al. 2004; Ren et al.\n2022). These lines come from the Broad Line Region (BLR) which is located at a fraction of a parsec from the central\nsupermassive black hole (SMBH). This region reprocesses the bulk of the radiation from the central parts very close\nto the SMBH.\nApart from these well-resolved strong individual lines, we have a contribution from transitions in weakly ionized\nheavy elements, predominantly iron. These transitions are so numerous that they form a pseudo-continuum well visible\nin the near-IR (NIR), optical and UV bands. This pseudo-continuum affects considerably the fitting of the strong\nindividual emission lines from BLR. The issue has been identified a long time ago (Sargent 1968; Collin-Souffrin et al.\n1979; Wills et al. 1985, 1980). The important role of Fe II diagnostics has been recognized by Boroson & Green (1992)\nCorresponding author: Ashwani Pandey\nashwanitapan@gmail.com\n∗CNPq FellowarXiv:2401.18052v1  [astro-ph.GA]  31 Jan 20242\nin their studies of the many AGN parameters using the Principal Component Analysis (PCA) approach. Their study\nled to the identification of the key role of the Eigenvector 1 (EV1) in the quasar sequence. The ratio, RFeII, of the\nequivalent width (EW) of the Fe II in the 4434-4684 ˚A wavelength range to the EW of H βplayed a key role in EV1,\nand later it became the foundation of the optical Quasar Main Sequence plots showing the Full Width Half Maximum\n(FWHM) of H βvsRFeII(Sulentic et al. 2000; Shen & Ho 2014; Marziani et al. 2018; Panda et al. 2018, 2019b).\nSince Fe II transitions are so numerous, certain standard templates were developed separately for the optical and for\nthe UV bands which were used for detailed modeling of AGN spectra. Some of the templates were purely observational,\ni.e. created by selecting the object with relatively narrow lines and subtracting the known lines and continuum. The\nresidual was considered as coming from Fe II (e.g. Vestergaard & Wilkes 2001; Marziani et al. 2003; Park et al. 2022).\nOther templates were mostly theoretical, based on computing the transition rates (e.g. Bruhweiler & Verner 2008;\nKovaˇ cevi´ c-Dojˇ cinovi´ c & Popovi´ c 2015). Some templates were a combination of both approaches (Tsuzuki et al. 2006,\nbut also partially Kovaˇ cevi´ c-Dojˇ cinovi´ c & Popovi´ c 2015).\nThe main issue with the empirical or observational templates is that these templates are limited by the signal-to-\nnoise ratio (SNR) and the wavelength coverage of the spectra. There is not yet a single empirical template that can\ncover the entire optical to UV bands simultaneously.\nIn this paper, we use the latest version C23.00 of the photoionization code CLOUDY (Chatzikos et al. 2023), which\ncontains a large database of the Fe II transitions, to develop a new theoretical Fe II template that covers a wide\nwavelength range of 1000-10000 ˚A. This new template will be useful to fit the AGN spectra simultaneously from\noptical to UV bands. Some properties of Fe II emission, like the ratio of the Fe II to Mg II line in UV and the trends\nwith the EV1 were studied by Sarkar et al. (2021); Dias dos Santos et al. (2024) but here we study systematically\nseveral correlations which can also be tested against observational data in the NIR to UV wavebands.\n2.METHOD\nWe use the latest version C23.00 of the photoionization code CLOUDY (Chatzikos et al. 2023). CLOUDY code has\nfour different Fe II atomic data sets reported by Verner et al. (1999), Bautista et al. (2015), Tayal & Zatsarinny (2018)\nand Smyth et al. (2019). A comparison of these four data sets is presented in Sarkar et al. (2021). For our model\ncalculations, we use the Fe II data set of Smyth et al. (2019) which is the default Fe+ dataset in CLOUDY C23.00. The\nSmyth et al. (2019) Fe II data set includes 716 levels extended up to 26.4 eV generating 255974 emission lines.\nWe computed several grids of photoionization models that span five decades in the hydrogen gas density, 9 ≤\nlognH(cm−3)≤14, and the incident hydrogen-ionizing photon flux, 17 ≤log Φ H(cm−2s−1)≤22, with a step size\nof 0.25 dex. We adopted a fixed hydrogen column density of 1024cm−2and set the abundance to the default solar\nabundance, as in Sarkar et al. (2021). For the description of the spectral shape of the incident radiation, we used\nthe standard AGN.SED available in the CLOUDY database, which is similar to the continuum of Mathews & Ferland\n(1987).\nWe generated such models with different values of microturbulence velocity (V turb). We used six values of microtur-\nbulence velocity: 0, 10, 20, 30, 50, and 100 km/s. Such range for the microturbulence velocity was inspired by previous\nworks (Baldwin et al. 2004; Bruhweiler & Verner 2008; Panda et al. 2018). For a certain set of values of log Φ Hand\nlognH, we did not get any Fe II emission. We listed these values in Table 1.\nHaving the set of computed spectra, we constructed several ratios of Fe II emission by integrating the spectra over\nthe selected wavelength ranges. Such ratios are frequently studied in the observed spectra, and we aim to analyze the\ntrends in these ratios as functions of the density and the ionizing flux.\nWe also examine the effect of metallicity on these ratios by varying metallicity as 2, 5, 7, 10 and 20 times the solar\nmetallicity.\n2.1. Definitions of optical and UV Fe II\nSince the Fe II emission is of a broad band character the integrated value depends critically on the adopted range\nof integration. In our computations, we use several such ranges that were frequently considered in the literature.\nTo characterize the broadband emission in the optical range we integrate Fe II from 3500 to 7200 ˚A (V´ eron-Cetty\net al. 2004). Some of the studies concentrate on much narrower ranges. For example, the range from 4434 - 4684\n˚A was introduced to parameterize the Fe II emission close and bluewards of the H βline and to characterize the\nrelative importance of the Fe II to H βemission by the corresponding RFeIIratio (Boroson & Green 1992). This ratio\nwas well studied observationally in many papers (e.g. Osterbrock 1977; Bergeron & Kunth 1984; Shen & Ho 2014;\n´Sniegowska et al. 2018; Mart´ ınez-Aldama et al. 2021; Marziani et al. 2022).3\nTable 1. The set of parameters, for which no Fe II emission was obtained.\nlog Φ HlognH\n21.25 9.00\n21.50 9.00\n21.50 9.25\n21.75 9.00\n21.75 9.25\n21.75 9.50\n22.00 9.00\n22.00 9.25\n22.00 9.50\n22.00 9.75\nNote: This set of parameters is valid for all the turbulence velocities except for zero. In case of no turbulence velocity in\naddition to the upper parameters, Fe II emission was also not obtained for log Φ H(cm−2s−1)= 22 and log nH(cm−3) = 10.\nBroad UV Fe II range can be well characterized by the Fe II emission integrated in the wavelength range from 1250\nto 3090 ˚A (Vestergaard & Wilkes 2001). This range contains the Fe II bump at ∼2500 ˚A seen in the spectra as well\nas in early models of Fe II emission (Wills et al. 1985).\nWe also conduct a systematic analysis of two UV and two optical wavelength ranges that may be used to determine\nthe spectral shape. The UV ranges, 2650-2800 ˚A and 2800-3090 ˚A characterize the Fe II emission on each side of the\nMg II line. The optical ranges are 4434-4684 ˚A and 5150-5350 ˚A corresponding to the Fe II emission at the blue and\nred sides of the H βline, respectively. The Fe II emission on both sides of Mg II and H βforms clear local bumps or\npeaks. Therefore, for convenience, we refer to these regions as blue and red wings (e.g. Sarkar et al. 2021).\n3.RESULTS\n3.1. UV to optical Fe II emission\nWe calculated the ratio of the UV to optical broadband Fe II emission with the integration range specified in\nSection 2.1. The results are shown in Figure 1 using contour plots. The representative value in the central parts of\nthe plot is 0.7 in the log space, i.e. the ratio is typically of the order of 5 in linear space. As already argued by Joly\n(1981) and Wills et al. (1985), a ratio of the order of 1 in the relative intensities of the optical to UV lines implies a\ncolumn density of about 1023to 1024cm−2to balance Fe II line optical depths and the Balmer opacity. This value is\nonly weakly sensitive to the ionization flux or density. However, we see that this ratio rises rapidly when either the\nionization parameter becomes very high or very low, and the cloud density rises. At higher values of the ionization\nparameter, above 1020cm−2s−1, this steep rising wall forms a linear trend in the log nH−log Φ Hdiagram, with the\ncharacteristic position correlated with the density. The ratio at this wall exceeds 100. At low values of the ionization\nflux, below 1018cm−2s−1, the ratio depends mostly on the density, changing from the factor 2.5 in the linear scale\nfor densities below 1010cm−3to values above 100 for densities above ∼3×1013cm−3. Lines are almost vertical\nthere, so for low values of the ionization flux this ratio can serve as a density indicator. At intermediate values of the\nionization flux, above 1018cm−2s−1, the dependence on the two parameters is more complex. Thus, when we do not\nhave an estimate of the ionization parameter in the source, high values of the UV to optical Fe II emission can indicate\neither very high or very low ionization flux. Sarkar et al. (2021) stressed the role of the turbulent velocity in the Fe II\nintensity but the global ratio of the UV to optical is relatively unaffected.\nThe ratio shown in Figure 1 represents the expectations based on the assumption of a single zone responsible for\nboth UV and optical emission, which may not be the case in actual objects. We address this issue in the discussion.\n3.2. Red-to-blue UV Fe II wings ratio\nIn our next plot, we concentrate on the global properties of the UV part of Fe II emission which is important for the\nproper modelling of the Mg II line. The Fe II emission under the Mg II line is never close to zero, even right under the\ncentral parts of the Mg II, at 2800 ˚A, as adopted in the creation of UV Fe II observational templates (e.g. Vestergaard\n& Wilkes 2001). However, due to the overlap of the Fe II and Mg II there, it is much easier to measure the strong4\n10 12 14171819202122Vturb=0 km/s\n0.250.500.75\n1.001.251.25\n1.501.50\n1.751.752.00\n2.252.50\n10 12 14171819202122Vturb=10 km/s\n0.500.751.00\n1.00\n1.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.25\n2.50\n10 12 14171819202122Vturb=20 km/s\n0.500.751.00\n1.001.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.25\n2.50\n10 12 14171819202122Vturb=30 km/s\n0.500.751.00\n1.00\n1.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.252.50\n10 12 14171819202122Vturb=50 km/s\n0.500.751.00\n1.00\n1.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.25\n2.502.50\n10 12 14171819202122Vturb=100 km/s\n0.500.751.00\n1.001.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.25\n2.502.500.00.51.01.52.02.5\n0.00.51.01.52.02.53.0\n0.250.751.251.752.252.75\n0.250.751.251.752.252.75\n0.250.751.251.752.252.75\n0.250.751.251.752.252.75\nlognH (cm3)\nlogH (cm2 s1)\nFigure 1. Contour plots for the logarithmic ratio of UV Fe II to optical Fe II emission for different turbulence velocities.\n10 12 14171819202122Vturb=0 km/s\n0.00\n0.250.25\n0.500.50 0.75\n0.75\n1.00\n1.25\n1.50\n1.75\n2.00\n10 12 14171819202122Vturb=10 km/s\n0.00\n0.250.25\n0.500.50\n0.750.75\n1.001.00\n1.25\n1.50\n1.75\n2.002.25\n10 12 14171819202122Vturb=20 km/s\n0.00\n0.250.25\n0.500.50\n0.750.75\n1.001.00\n1.251.25\n1.50\n1.75\n2.00\n10 12 14171819202122Vturb=30 km/s\n0.0\n0.20.2\n0.40.4\n0.60.6\n0.80.8\n1.01.0\n1.21.21.2\n1.4\n1.61.8\n2.0\n10 12 14171819202122Vturb=50 km/s\n0.0\n0.20.2\n0.40.4\n0.60.6\n0.80.8\n1.01.0\n1.21.2\n1.4 1.61.8\n10 12 14171819202122Vturb=100 km/s\n0.0\n0.20.2\n0.40.4\n0.60.6\n0.80.8\n1.01.0\n1.2 1.41.60.000.250.500.751.001.251.501.752.002.25\n0.00.51.01.52.02.5\n0.00.51.01.52.02.5\n0.00.40.81.21.62.0\n0.00.40.81.21.62.0\n0.00.20.40.60.81.01.21.41.61.8\nlognH (cm3)\nlogH (cm2 s1)\nFigure 2. Contour plots for the ratio of red (2800-3090 ˚A) to blue (2650-2800 ˚A) part of the UV Fe II spectra for different\nturbulence velocities.5\nwings that Fe II forms at both the red and blue sides of Mg II. Proper modeling of the underlying Fe II may affect the\nconclusions about the dynamics of the Mg II emitting region (e.g. Joni´ c et al. 2016; Popovi´ c et al. 2019).\nThe contour plots for the ratio of red (2800-3090 ˚A) to blue (2650-2800 ˚A) wings of UV Fe II are shown in Figure 2.\nThe plot of the ratio shows a complicated structure. We again see a trend similar to the one displayed in Figure 1.\nThe same values of the ratio are found at low-density high ionization flux and high-density low ionization flux. At\nlow ionization flux, the ratio again depends mostly on the density. However, in this case, we additionally observe the\ndependence on the turbulent velocity, as in the central parts the ratio is initially rising, and finally decreasing when\nthe turbulent velocity crosses the value 50 km s−1. Maybe this is connected with the fact that the overall range in the\nmeasured ratio is much smaller for the red to the blue wing (typical range between 0.3 and 1.8) than in the case of\nthe entire UV to optical range covering a broad range of magnitudes.\nThe two wings represent different multiplets in templates of Kovaˇ cevi´ c-Dojˇ cinovi´ c & Popovi´ c (2015), so they may,\nor may not, come from the same region (see Section 4.\n3.3. The effect of the UV Fe II broadening on the measurements of the outflow properties\nIn the analysis of their Sloan Digital Sky Survey (SDSS) sample, Kovaˇ cevi´ c-Dojˇ cinovi´ c & Popovi´ c (2015) noticed\na systematic averaged redshift in the UV Fe II to the optical Fe II. To check if such a shift can be caused just by\nthe shape of the Fe II emission when broadened with the turbulence effect included, we used a specific Fe II template\ncharacterized by the ionization flux, log Φ H(cm−2s−1) = 20 .5, and the density, log nH(cm−3) = 12. We applied the\nGaussian smearing to the template, and we determined the peak wavelength of the blue wing of UV Fe II. The peak\nis usually located close to ∼2750 ˚A , and it is dominated by the multiplets 62 and 63 (see Kovaˇ cevi´ c-Dojˇ cinovi´ c &\nPopovi´ c 2015). However, when we model the entire Fe II emission of the blue wing we see that the position of the peaks\nmoves with the adopted smear velocity. We plot this effect in Figure 3 simply in ˚A units, and in Figure 4 in velocity\nspace. The shift corresponds to the apparent velocity from 100 km s−1up to 300 km s−1when the turbulence is not\nincluded, and the presence of the turbulence affects the apparent direction of the outflow and its amplitude. Apparent\nvelocity shifts up to 1000 km s−1are possible even though no velocity is present in the model which corresponds in\neach case to the rest frame spectrum. In Kovaˇ cevi´ c-Dojˇ cinovi´ c & Popovi´ c (2015) more than half of the sources show a\nkinematic offset above 1000 km s−1. We do not see such large shifts due to the Fe II smoothing but we experimented\nonly with a single template.\n3.4. Red-to-blue optical Fe II wings ratio\nThis part of Fe II emission is important for accurate modelling of the H βzone, as recently stressed by Popovi´ c et al.\n(2023). Proper modelling of the Fe II pseudo-continuum may strongly affect the conclusions about the dynamics of\nthe BLR (like the presence or absence of the very broad line component to H β, and the role of the intermediate line\nregion), and the modelling of the quasar main sequence (Panda et al. 2018, 2019a,b; Marziani et al. 2021).\nWe show contour plots for the ratio of red (5150-5350 ˚A) to blue (4434-4684 ˚A) parts of optical Fe II in Figure 5. This\nratio shows again the non-monotonic dependence of the ionization parameter but interestingly, almost no dependence\non the density. The minimum value of around 1.0 is present at the ionization parameter 1020cm−2s−1, increasing\nwhen we move both up and down in the ionization flux. The dependence on the turbulent velocity is marginally\nnoticeable, particularly for this minimum value. Some traces of dependence on the density are only seen for very low\nionization flux, below 1018cm−2s−1, and only for very high densities. The value of the turbulent velocity affects the\nextension of this region.\n3.5. RFeIIin the optical band\nWhen computing the Fe II emission with CLOUDY we also have, as a by-product, the intensities of other emission\nlines. The ratio RFeIIof the optical Fe II emission to H βplays an exceptional role in the EV1 of the quasar sequence\n(Boroson & Green 1992; Sulentic et al. 2000; Shen & Ho 2014; Marziani et al. 2018; Panda et al. 2018, 2019b; Du\n& Wang 2019; Panda & Marziani 2023). We estimated RFeIIas the ratio of the integrated optical Fe II flux in the\nstandard wavelength range 4434-4684 ˚A to the flux of H βat 4861.32 ˚A. We present the corresponding contour maps\nin Figure 6.\nThese plots show that most of the map area corresponds to the value of RFeIIsmaller than 1. Higher values, above\n3, are only predicted for the intermediate values of the turbulent velocity (20 - 30 km s−1), and only for very high\nvalues of the ionization parameter (log Φ H(cm−2s−1)>21) which supports the earlier findings by Panda et al. (2018)6\n1000 2000 3000 40004849\nVturb=0 km/s\n1000 2000 3000 4000485052\nVturb=10 km/s\n1000 2000 3000 40004648505254\nVturb=20 km/s\n1000 2000 3000 400048505254\nVturb=30 km/s\n1000 2000 3000 40004648505254\nVturb=50 km/s\n1000 2000 3000 400040.042.545.047.5\nVturb=100 km/s\nSmear velocity (km/s)Peak wavelength (2700+ Å)\nFigure 3. Variation of UV Fe II peak wavelength with the smear velocity for log Φ H(cm−2s−1) = 20 .5,log n H(cm−3) = 12,\nand different turbulence velocities.\n1000 2000 3000 4000300\n200\n100\nVturb=0 km/s\n1000 2000 3000 4000200\n0200\nVturb=10 km/s\n1000 2000 3000 4000400\n200\n0200400\nVturb=20 km/s\n1000 2000 3000 4000200\n0200400\nVturb=30 km/s\n1000 2000 3000 4000500\n250\n0250500\nVturb=50 km/s\n1000 2000 3000 40001000\n500\nVturb=100 km/s\nSmear velocity (km/s)Peak (km/s)\nFigure 4. Variation of UV Fe II peak around 2750 ˚A wavelength in velocity space with the smear velocity for log Φ H(cm−2s−1) =\n20.5,log n H(cm−3) = 12, and different turbulence velocities. Here, a negative peak value represents an outflow, while a positive\nvalue denotes an inflow.7\n10 12 14171819202122Vturb=0 km/s\n0.00.30.6\n0.9\n0.90.9\n1.21.21.2\n1.5\n1.8\n2.1\n2.42.4\n2.7\n10 12 14171819202122Vturb=10 km/s\n0.00.3\n0.60.60.6\n0.9\n0.90.9\n1.21.21.2\n1.5\n1.82.1\n2.42.7\n10 12 14171819202122Vturb=20 km/s\n0.00.3\n0.60.6\n0.6\n0.9\n0.90.9\n1.2\n1.21.2\n1.5 1.8\n2.1\n2.4\n2.72.7\n10 12 14171819202122Vturb=30 km/s\n0.00.3\n0.60.6\n0.9\n0.90.9\n1.21.21.2\n1.5\n1.82.1\n2.42.4\n2.7\n10 12 14171819202122Vturb=50 km/s\n0.00.30.6 0.9\n0.90.9\n1.2\n1.21.2\n1.5\n1.8\n2.12.1\n2.42.4\n2.7\n3.0\n10 12 14171819202122Vturb=100 km/s\n0.00.30.6 0.9\n0.90.9\n1.2\n1.21.2\n1.5\n1.8\n2.12.1\n2.42.4\n2.72.7\n3.00.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.61.21.82.43.0\nlognH (cm3)\nlogH (cm2 s1)\nFigure 5. Contour plots for the ratio of red (5150-5350 ˚A) to blue (4434-4684 ˚A) part of the optical Fe II spectra for different\nturbulence velocities.\n10 12 14171819202122Vturb=0 km/s\n0.000.00\n0.150.15\n0.300.30\n0.450.45\n0.600.60\n0.750.75\n0.900.901.051.20\n10 12 14171819202122Vturb=10 km/s\n0.00.0\n0.30.3\n0.60.6\n0.90.9\n1.21.21.5\n1.82.12.4\n2.73.0\n10 12 14171819202122Vturb=20 km/s\n0.00.0\n0.40.4\n0.80.8\n1.21.2\n1.62.02.42.8\n3.23.6\n10 12 14171819202122Vturb=30 km/s\n0.00.0\n0.40.4\n0.80.8\n1.21.21.62.02.4\n2.83.2\n10 12 14171819202122Vturb=50 km/s\n0.00.0\n0.40.4\n0.80.8\n1.21.21.62.0\n2.42.83.2\n10 12 14171819202122Vturb=100 km/s\n0.000.00\n0.250.25\n0.500.50\n0.750.75\n1.001.001.251.501.752.002.250.000.150.300.450.600.750.901.051.201.35\n0.00.61.21.82.43.0\n0.00.81.62.43.24.0\n0.00.81.62.43.24.0\n0.00.40.81.21.62.02.42.83.23.6\n0.00.51.01.52.02.5\nlognH (cm3)\nlogH (cm2 s1)\nFigure 6. Contour plots for R FeIIi.e. the ratio of Fe II 4434-4684 ˚A to H β(λ4861.32) fluxes for different turbulence velocities.8\n0 0:5 1:0 1:5 2:0 2:5 3:0 3:5 4:0\nRFeII0500100015002000NRakshit+20\nShen+11\nFigure 7. Histogram of the RFeIIdistribution in SDSS quasar spectra based on Shen et al. (2011) (blue) and Rakshit et al.\n(2020) (green).\nTable 2. The model predicted ratios of Fe II emission for various abundances, with a fixed microturbulence velocity of 20\nkm/s, in different wavebands considered in this work.\nAbundance (in Z⊙) log Φ H,lognHUV Fe II/optical Fe II UV red/blue wings Optical red/blue wings R FeII\n1 18,10 2.56 2.13 1.81 0.91\n20.5,12 8.98 1.26 0.84 1.09\n5 18,10 2.28 2.05 1.99 1.27\n20.5,12 5.63 1.21 0.75 2.49\n20 18,10 3.03 1.77 1.88 0.99\n20.5,12 4.94 1.03 0.76 4.42\nand Panda (2021). Smaller values of RFecorrespond to the intermediate A-type quasars. The basic classification of\nquasars into class A and B, introduced by Sulentic et al. (2000) was analogous to the older classification of Seyfert\ngalaxies into Narrow Line Seyfert 1 (NLS1) class and other Seyfert 1 galaxies (Osterbrock & Pogge 1985), only the\nlimiting velocity was different (4000 km s−1vs. 2000 km s−1), which reflected the systematic difference in the mass\nof the central black hole. In both cases, one class (NLS1 and type A) of objects corresponded to relatively higher\nEddington rate sources than the other class, which was reflected in line properties (intensities and shapes). Later, the\nspecific ranges of RFewere used to explore full 2-D classification (see e.g., Marziani et al. 2018). A-type quasars with\nRFeII≳1 were named Extreme Population A (xA) sources, and they were found to be applicable as standard candles\nfor cosmology (Marziani & Sulentic 2014; Dultzin et al. 2020; Marziani 2023; Panda & Marziani 2023). Most of the\nxA sources have the values of RFeIIbetween 1 and 2 (Marziani et al. 2018) but ´Sniegowska et al. (2018) found one\nsource with a large ratio. A few more such sources were reported by Shen et al. (2011) but as discussed in ´Sniegowska\net al. (2018), those measurements were not of high quality. Moreover, the sources with RFeII≳2 are reported in the\nnewer catalogs (e.g. Negrete et al. 2018; Rakshit et al. 2020, 2021; Wu & Shen 2022).\nWe, thus, could conclude, from Figure 5, that xA sources require high-density BLR (log nH(cm−3) = 12 −13) and\nvery high radiation flux. However, our templates were composed of solar metallicity. Some authors argued (e.g. Panda\net al. 2019b; ´Sniegowska et al. 2021; Panda 2021; Marziani et al. 2024) that these sources, in addition, require highly\nsuper-solar metallicity.\nWe compared these predictions with the statistical distribution of the RFeIIreported on the basis of SDSS quasar\nstudies. We used two data sets: older massive fits to the SDSS quasars, based on data release 7 (DR 7), were reported\nby Shen et al. (2011). Newer results, based on DR 14, were reported by Rakshit et al. (2020). The results are based\non the spectral decomposition of the Fe II emission integrated within the traditional range 4434-4684 ˚A (Boroson &\nGreen 1992). In both cases, we selected only the sources with reliable spectra decomposition by requiring that the\nrelative error of the FWHM of H βand the RFeIImeasurement are below 20%. The resulting histograms, shown in9\nTable 3. Basic properties of the sources.\nSource RA Dec Redshift log(L 3000[erg/s]) log ( MBH[M⊙])λEdd Reference\nHE 0413-4031 4h15m14s−40◦23′41′′1.39117∗46.74 8.87 1.66 Prince et al. (2023)\nSDSS RM102 14h13m53s+52◦34′44′′0.86114 45.00 7.92 0.329 Shen et al. (2019)\n∗The redshift to this source was recently firmly established from the narrow [OIII] line using the near-IR spectrum (Zajaˇ cek\net al. 2023b).\nFigure 7, consistently show the peak of the RFeIIvalues at about 0.6 in both samples. There are very few objects with\nvalues above 4.0 which we ignored for better clarity of the plot. Such values are present either in the region of higher\ndensity and high ionization flux, above ∼1020cm−2s−1, or low ionization flux and low density (lower left corner of\nplots in the Figure 6). The first location seems more likely as it is smoothly connecting to even larger values of RFeII\nthan the average value, and the observational histogram also smoothly extends to higher values. However, we should\nstress that the observational results are not based on the Fe II emission models presented in this paper. Results of\nboth Shen et al. (2011) and Rakshit et al. (2020) are based on the use of the optical Fe II template from Boroson &\nGreen (1992).\n3.6. The effect of metallicity\nTo investigate the effect of metallicity on Fe II emission, we rebuild our models for abundances equal to 2, 5, 7,\n10, and 20 times solar abundances, with a fixed microturbulence velocity of 20 km/s. We plot contours for the ratios\nestimated in the previous sections in Figures 8-11 for various abundances. The values of these ratios at two arbitrary\npositions are given in Table 2 to quantify their variations with abundances.\nThe UV Fe II to optical Fe II contours, shown in Figure 8, exhibit marginal changes with abundances. The ratio\ndecreases with increasing abundance at (log Φ H,lognH)=(20.5,12), however at no such trend is seen at lower incident\nflux, and lower density (log Φ H,lognH)=(18,10). Verner et al. (2003) investigated the effect of varying abundance on\nthe ratio of UV Fe II to optical Fe II and found that the ratio decreases from 8 to 5.6 when abundance increases from\nsolar to 10 times solar abundance for log Φ H(cm−2s−1) = 20 .5,log n H(cm−3) = 9 .5 and a microturbulence velocity of 1\nkm/s. They concluded that the optical Fe II emission increases more than the UV Fe II emission when the abundance\nincreases which is supported by the findings from Panda et al. (2019a).\nFigure 9 shows the contours for the ratio of UV wings, whereas Figure 10 shows those of optical wings. As seen\nfrom the Figures and Table 2, both ratios are essentially unaffected by the abundance.\nThe contours for R FeIIare plotted in Figure 11 which shows small variations with abundance. The ratio increases\nwith increasing abundance at high ionization flux and high density (log Φ H,lognH)=(20.5,12), but no such trend is\nseen at (log Φ H,lognH)=(18,10).\n3.7. Fe II templates\nFor convenient use in future spectral fitting, we created a new family of Fe II templates. They come from the\nsimulations as described in Section 2, i.e. those are 2646 models covering the range of the local density, ionization flux\nand turbulent velocity. Output files from CLOUDY are usually large, and data fitting does not require full resolution\navailable in these output files since most of AGN have broad Fe components. Therefore, we binned the output spectra\nusing the 2 ˚A bins. We tested whether this does not lead to a considerable problem in data fitting by assuming that\nthe AGN spectrum comes from the medium with the velocity dispersion 2000 km s−1and we compared the Fe II\nspectrum in the 2650 - 2850 ˚A range without any prior binning, and with a bin size of 1, 2 and 5 ˚A. The result is\nshown in Figure 12. Up to the bin size of 2 ˚A there is no net difference in the broadened spectra but the bin size of 5\n˚A seems to imply already a systematic difference, so we adopted the 2 ˚A bin size optimizing between the accuracy and\nthe size of the file. The complete set of Fe II templates for solar metallicity is available through zenodo1and GitHub2\nlinks.\n3.8. Exemplary cases\n1https://zenodo.org/doi/10.5281/zenodo.10532690\n2https://github.com/Ashwani-88/Fe2 template10\n10 12 14171819202122Z=Z\n0.500.751.00\n1.001.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.25\n2.502.75\n10 12 14171819202122Z=2Z\n0.500.751.00\n1.001.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.252.502.75\n10 12 14171819202122Z=5Z\n0.500.751.00\n1.00\n1.251.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.252.502.75\n10 12 14171819202122Z=7Z\n0.500.75\n1.001.00\n1.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.252.502.75\n10 12 14171819202122Z=10Z\n0.500.75\n1.001.00\n1.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.252.50\n2.75\n10 12 14171819202122Z=20Z\n0.50\n0.75\n1.001.001.001.00\n1.251.251.25\n1.501.50\n1.751.75\n2.002.00\n2.252.252.502.750.00.51.01.52.02.53.0\n0.00.51.01.52.02.53.0\n0.250.751.251.752.252.75\n0.250.751.251.752.252.75\n0.250.751.251.752.252.75\n0.250.751.251.752.252.75\nlognH (cm3)\nlogH (cm2 s1)\nFigure 8. Contour plots for the logarithmic ratio of UV Fe II to optical Fe II emission for turbulence velocity V turb= 20\nkm/s and different metallicities. The white diamonds in the plots show that the ratio cannot be estimated because there is no\nFe II emission available for the specified set of parameters.\n10 12 14171819202122Z=Z\n0.00\n0.250.25\n0.500.50\n0.750.75\n1.001.00\n1.251.25\n1.50\n1.75\n2.00\n10 12 14171819202122Z=2Z\n0.00\n0.250.25\n0.500.50\n0.750.75\n1.001.00\n1.251.50\n1.75\n2.00\n10 12 14171819202122Z=5Z\n0.0\n0.20.2\n0.40.4\n0.60.6\n0.80.8\n1.01.01.21.2\n1.41.6\n1.82.0\n10 12 14171819202122Z=7Z\n0.0\n0.20.2\n0.40.4\n0.60.6\n0.80.8\n1.01.0\n1.21.4\n1.61.8\n2.0\n10 12 14171819202122Z=10Z\n0.0\n0.20.2\n0.40.4\n0.60.6\n0.80.8\n1.01.0\n1.21.4\n1.61.8\n2.0\n10 12 14171819202122Z=20Z\n0.0\n0.20.2\n0.40.4\n0.60.6\n0.80.80.8\n1.01.01.0\n1.21.4\n1.61.80.00.51.01.52.02.5\n0.000.250.500.751.001.251.501.752.002.25\n0.00.40.81.21.62.0\n0.00.40.81.21.62.0\n0.00.40.81.21.62.0\n0.00.40.81.21.62.0\nlognH (cm3)\nlogH (cm2 s1)\nFigure 9. Contour plots for the ratio of red (2800-3090 ˚A) to blue (2650-2800 ˚A) part of the UV Fe II spectra for turbulence\nvelocity V turb= 20 km/s and different metallicities.11\n10 12 14171819202122Z=Z\n0.00.3\n0.60.6\n0.6\n0.9\n0.90.9\n1.2\n1.21.2\n1.5 1.8\n2.1\n2.4\n2.72.7\n10 12 14171819202122Z=2Z\n0.00.3\n0.60.60.60.9\n0.90.9\n1.21.21.2\n1.5\n1.82.1\n2.42.4\n2.7 2.7\n10 12 14171819202122Z=5Z\n0.00.3\n0.60.60.6\n0.6\n0.90.90.9\n1.21.2 1.2\n1.2\n1.51.8\n2.12.4\n2.72.7\n10 12 14171819202122Z=7Z\n0.00.3\n0.60.6\n0.90.9 0.9\n1.21.21.2\n1.5\n1.8\n2.1\n2.42.72.7\n3.0\n10 12 14171819202122Z=10Z\n0.00.3\n0.60.6\n0.6\n0.90.9 0.9\n1.21.2 1.2\n1.5\n1.8\n2.1 2.4\n2.72.7\n3.0\n10 12 14171819202122Z=20Z\n0.00.40.4\n0.8\n0.80.8\n1.21.21.2\n1.2\n1.6\n2.02.4\n2.82.8\n3.23.60.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.61.21.82.43.0\n0.00.81.62.43.24.0\nlognH (cm3)\nlogH (cm2 s1)\nFigure 10. Contour plots for the ratio of red (5150-5350 ˚A) to blue (4434-4684 ˚A) part of the optical Fe II spectra for turbulence\nvelocity V turb= 20 km/s and different metallicities.\n10 12 14171819202122Z=Z\n0.00.0\n0.40.4\n0.80.8\n1.21.2\n1.62.02.42.8\n3.23.6\n10 12 14171819202122Z=2Z\n0.00.0\n0.50.5\n1.01.0\n1.52.02.53.03.5\n4.04.5\n10 12 14171819202122Z=5Z\n0.00.0\n0.80.81.6\n2.43.24.04.85.6\n6.47.2\n10 12 14171819202122Z=7Z\n00\n112\n234\n56 7 8\n10 12 14171819202122Z=10Z\n00\n11 234\n56 789\n10 12 14171819202122Z=20Z\n0.00.0\n1.53.04.56.07.5\n9.010.512.00.00.81.62.43.24.0\n012345\n0.01.63.24.86.48.0\n0123456789\n0246810\n03691215\nlognH (cm3)\nlogH (cm2 s1)\nFigure 11. Contour plots for R FeIIi.e. the ratio of Fe II 4434-4684 ˚A to H β(λ4861.32) fluxes for turbulence velocity V turb= 20\nkm/s and different metallicities.12\n2650 2700 2750 2800 2850\nWavelength (Å)0.50.60.70.80.91.0FFeII (a.u.)unbinned\n2A-bin\n5A-bin\n1A-bin\nFigure 12. Plot showing the justification of the choice of 2 ˚A bin size. The larger bin size (5 ˚A) already shows a visible\ndeparture in the smoothed shape from the denser grid.\nWe test how the new Fe II templates compare with the observed AGN spectra. Since we use the SED more\nappropriate for bright quasars to construct the Fe II templates, we selected two exemplary sources of this type namely,\nHE 0413-4031 and SDSS RM102. The basic properties of these sources are mentioned in Table 3. In the case of HE\n0413-4031, only narrow UV band data are available while the spectrum of SDSS RM102 continuously covers the broad\nUV-to-optical band.\n3.8.1. HE 0413-4031\nThis quasar has been observed with the Southern African Large Telescope (SALT) for 13 years with the aim of\nperforming the reverberation mapping of the Mg II line and the UV Fe II (Zajaˇ cek et al. 2020; Prince et al. 2023;\nZajaˇ cek et al. 2023b). The obtained spectra cover a narrow range around the Mg II line but the spectral shape did not\nshow any strong variability apart from the normalization of the spectrum. We thus constructed the mean spectrum of\nthis quasar by combining all 32 available data sets for a better S/N ratio. The shape of the Mg II line in this source\nis particularly simple, well fitted as a single kinematic component of the Lorentzian shape.\nWe fit all the parameters together (slope and normalization of the power law, normalization and the broadening of\nthe Fe II template, the position, normalization and FWHM of the Mg II) since in case of such a short data range we\ndo not have any possibility to find parts of the spectrum unaffected by Fe II and fit first the underlying power law.\nThe narrow spectral range makes the fits not strongly constraining the Fe II templates but the fits are very satisfac-\ntory. The results of the spectral fitting are shown in Figure 13. The best solution corresponds to log Φ H(cm−2s−1) =\n18.25, shown in the middle panel of Figure 13. The best fit FWHM of the Fe II (5400 km s−1) is broader than the\nFWHM of Mg II (4280 km s−1) but it comes with a large error. Higher turbulent velocity (100 km s−1) gives fits with\nsomewhat higher reduced χ2, and pushes the FWHM of Fe II toward even higher values, at the same time increasing\nthe Fe II intensity. This results from considerable degeneracy between the Fe II normalization and the underlying\npower law. A lower value of the turbulent velocity (20 km s−1) gives narrower Fe II lines but the quality of the fit is\nmuch worse.\nFor comparison, we show two other fits, for higher (log Φ H(cm−2s−1) = 19) and lower (log Φ H(cm−2s−1) = 18)\nionization fluxes in the upper and lower panels of Figure 13, respectively. The fit quality is good in all these cases\n(χ2differences ∼1.0), and the value of the EW of the Mg II line is similar (31.2 ˚A, 30.2 ˚A and 28.4 ˚A), but slowly\nrising towards larger ionization fluxes. We calculated also EW(Fe II) but only in the fitted range. EW(Fe II) is more\naffected by the change of the ionizing flux (the corresponding values are 37.9 ˚A, 31.2 ˚A and 15.9 ˚A), and all these\nvalues are larger than achieved with the use of different templates by Zajaˇ cek et al. (2023b). Such a large change is\nrelated to a corresponding change in the slope and normalization of the underlying power law.\nThe chosen fit density (log nH(cm−3) = 12) is consistent with the best fit theoretical template d12-m20-20-5.dat\nselected from the collection of Bruhweiler & Verner (2008) in our previous papers, including Prince et al. (2023).\nHowever, the ionization flux is now lower by more than two orders of magnitude.13\nHE 0413-4031\nHE 0413-4031\nHE 0413-4031\nFigure 13. The exemplary fits to the SALT spectrum with the Fe II template phi19.0-nH12.0-m50.dat, FWHM = 5400 km\ns−1(upper panel), phi18.25-nH12.0-m50.dat, FWHM = 5400 km s−1(middle panel), and phi18.0-nH12.0-m50.dat, FWHM =\n5800 km s−1(lower panel). The middle panel shows the best fit.\nWe compared the requested value for the ionizing flux from Fe II modelling with the expected ionizing flux based on\nSED of this object and the Mg II time delay of 302.6 days for Mg II (Zajaˇ cek et al. 2020) but the obtained ionization\nflux from the SED best fit presented in Figure 9 of Zajaˇ cek et al. (2020) gave only log Φ H(cm−2s−1) = 14 .6, since the\nspectrum appeared to decrease rapidly in the UV band.\nThe narrow available spectral range on one hand guarantees that a range of templates fits well the spectrum but it\nopens a question of how reliable is such a decomposition.14\nTable 4. The ratios of Fe II intensities measured in the rest frame of the object RM102\nwavelength range 1 wavelength range 2 ratio 1 to 2\n2800-3090 ˚A 2650-2800 ˚A 1.939\n5150-5350 ˚A 4434-4684 ˚A 0.713\n2000-3090 ˚A 4000-5350 ˚A 2.151\n3.8.2. SDSS RM102\nAs a second example, we selected the quasar SDSS J141352.99+523444.2 (SDSS RM102) from the SDSS Reverber-\nation Mapping (SDSS-RM) catalogue (Shen et al. 2019) considering a high signal-to-noise, moderate Fe II emitter\n(RFeII>0.5), and a reliable time lag for H βor Mg II λ2800 (maximum Pearson correlation coefficient >0.4). The\nspectrum of the source covers a broad wavelength range (2000-5500 ˚A). This allowed us to cover both the rest-frame\noptical Fe II as well as UV Fe II and to test the feasibility of using the same Fe II template to fit the entire UV-to-optical\nspectral range. The size of its BLR region has been estimated by different methods, in this paper we took the one\nestimated by the Interpolated Cross-Correlation Function (ICCF), τMgII−ICCF = 101 .7+11.6\n+10.3andτHβ−ICCF = 104 .6+20.5\n+18.5\ndays in the observed-frame (Shen et al. 2023). Using the SDSS Science Archive Server (SAS) from the DR163, we\nfound 70 spectroscopic epochs observed during four years. All the spectra were combined in a single one to obtain a\nhigh signal-to-noise spectrum, S/N=30 and 35 at 3000 ˚A and 4000 ˚A, respectively. Due to the presence of spikes around\n5100˚A, it was not possible to estimate the S/N at this wavelength.\nIn the spectral fitting, we considered all the emission lines reported by Vanden Berk et al. (2001) between 2000\nand 5500 ˚A, 31 emission lines in total. We also included a model of the Balmer Continuum with an optical depth of\n1 plus the High-Order Balmer lines (Bernal et al in prep.). The continuum was fitted with a single power law. To\ndecrease the number of free parameters, we fixed the same FWHM for each family of emission lines: Balmer lines,\nHelium transitions, forbidden lines (narrow emission lines), and coronal lines. Flux intensities, FWHM, and shift sum\na total of 63 free parameters. We applied Gaussian smearing to the UV and optical Fe II template and a single flux\nintensity value was taken for the UV and optical wavelengths. In the first attempt, we noticed that it was not possible\nto fit the UV and optical Fe II contribution with the same flux intensity. Thus, we performed a careful fitting of all\nthe other spectral features and what remained can be considered as the Fe II component. Due to the overlapping\nof the emission lines and the Fe II pseudocontinuum in some ranges, the flux intensity of the emission line would\nbe underestimated. So, we considered the semiempirical UV/optical Fe II templates implemented in the fitting code\nPyQSOFit (Guo et al. 2018) to constrain the intensity of the emission lines. As we previously noticed, even with these\nempirical Fe II templates, a different flux intensity value is needed to fit the Fe II contribution. The flux intensity\nfactor has a difference of 3.5 between the UV and the optical Fe II contribution. The final spectral fitting is shown in\nFigure 14. The potential Fe II contribution is shown at a zero level. As a reference, we also plotted the UV (dotted\nline) and optical (dashed-line) Fe II templates implemented in PyQSOFit . There is good agreement in the optical Fe\nII, while slight variations are seen in the UV part.\nThe new spectral fitting indicates a RFeII=1.24, which is approximately twice the value reported by Shen et al.\n(2019). On the other hand, this source shows a moderate Eddington ratio λEdd=0.329 (Shen et al. 2019). The weak\ncontribution of the doublet [OIII] λλ5007,4959 suggests that RM102 has a moderate Eddington ratio and is close to\nthe xA sources (Marziani 2023).\nTo locate the source in the log nH−log Φ Hplane, we measured three ratios: red to blue wings of UV Fe II, red to\nblue wings of optical Fe II, and UV to optical Fe II. The first two ratios were measured in the same wavelength range\nas used to create the general maps in Figure 2 and Figure 5. The last ratio had to be modified since the data cover\nthe wavelength range only up to 2000 ˚A. The measured ratios are given in Table 4. Looking at the plots in Section 3\nabove we could not easily identify the most likely parameters of the successful template. We thus created a new map,\nshown in Figure 15, where all the three ratios given in Table 4 are shown as contour plots, each for all 6 turbulent\nvelocities.\nWe see that the three families of lines in Figure 15 never cross. This means that there is no single template in our\ncollection which could fit the data for the quasar RM102. The blue lines representing the ratio of red to blue wings\n3https://dr16.sdss.org/optical/spectrum/search15\n2000 3000 4000 5000\nrestframe(Å) \n0204060F   (1017 erg s1 cm2 Å1)\nFigure 14. Spectral fitting of the sources RM102. The cyan line corresponds to the best fit of the emission lines (dotted line)\nand the continuum (dashed line). The Fe II pseudocontinuum is shown at the zero level (red line). For reference, we also show\nthe semi-empirical UV (dotted line) and optical (dashed) templates implemented in PyQSOfit in dark red.\n9 10 11 12 13 14171819202122\nt=0t=10\nt=20t=30\nt=0t=10t=20t=30t=0t=0\nt=0t=0t=10\nt=10t=10\nt=20t=20t=20\nt=30t=30t=30\nt=50t=50\nt=50\nt=100t=100\nt=100\nlognH (cm3)\nlogH (cm2 s1)\nFigure 15. Contours corresponding to the ratios observed in the source RM102 of Fe II UV-to-optical ratio (denoted by black\nlines), red to blue wings of UV Fe II (denoted by red lines) and red to blue wings of optical Fe II (denoted by blue lines).\nThe values of microturbulent velocity are labelled on each contour. The three families do not cross, with the optical wing ratio\nimplying much higher ionization than the other two.\nof optical Fe II occupy a narrow range of high ionization flux values. This reflects the fact that the measured ratio\nis low and the corresponding solutions are located in the upper part of the general diagram (see Figure 5). On the\nother hand, the UV wings ratio in general forms a two-family solution for low values (see Figure 2), but for high values\nas the one measured in our source there is only an island located at low ionization flux and low density. The global\nUV to optical ratio (although measured in a slightly different wavelength range than the generic Figure 1) also points\ntoward low density and low ionization flux. The results for densities of the order of 1011−1012cm−3and moderate\nionization fluxes of the order of 1018cm−2s−1are similar to the parameter range concluded for HE 0413-4031, but\nthe optical wing ratio remains unexplained.\nWe estimated the ionization flux from this source at the distance corresponding to the BLR position as given by\nthe H βline delay. However, the SDSS spectrum does not cover the spectral range above 1 Rydberg, needed for this\npurpose. We thus extrapolated the spectrum of RM102 using the AGN SED template from CLOUDY. The observed16\n2000 2500 3000 3500 4000 4500 5000 55000.6\n0.4\n0.2\n0.00.20.40.60.81.0\nWavelength (Å)f (a.u.)\nFigure 16. A comparison of the observed Fe II emission in RM102 (blue points) with the two options for a theoretical template\nfrom our set (see text). The red curve denotes the Fe II flux in the wavelength range 1989-4000 ˚A for log Φ H(cm−2s−1) = 18\nwhile the black curve represents the Fe II emission in the wavelength range 4000-5530 ˚A for log Φ H(cm−2s−1) = 20. The\nmagenta curve is obtained using the second approach.\nspectrum is only marginally bluer in the optical band. Matching the two spectra at UV part we obtained the ionization\nflux of 1 .99×1020cm−2s−1. We also tested another way of extension, using the HST composite spectrum of Zheng\net al. (1997) which covers the rest frame range from 350 ˚A up to 2200 ˚A, and they give the broken power law fit.\nMatching this broken power law to the shortest wavelengths with our spectrum we obtained the ionization flux of\n1.35×1020cm−2s−1. The two values are comparable, they eventually favour the value implied by the red-to-blue\noptical Fe II ratio, although at (rather likely) higher densities than 1010cm−3they are not in full agreement with\nany of the ratios. However, some composites predict a stronger decrease of the quasar far-UV flux towards shorter\nwavelengths (see Cai & Wang 2023, and the references there).\nOverall, Figure 15 implies that a single Fe II template is unlikely to fit well the whole spectrum. In particular, the\nmismatch in the values of the ionization flux is by a factor of 100. The metallicity is not affecting the ratios strongly\nenough. We thus set the density at the likely value of 1011cm−3, and we took two approaches to set the ionizing flux.\nIn the first case, we allowed for a discontinuity in the ionizing flux at 4000 ˚A, where the Fe II emission is very low,\nand we used low ionizing flux in UV band (log Φ H(cm−2s−1) = 18) and high ionizing flux (log Φ H(cm−2s−1) = 20) in\nthe optical band. Separate fitting of the Fe II and optical is usually done when modelling a specific source. The result\nof this approach is shown in Figure 16. In the second approach, we allowed for a linear change of the ionizing flux\nwith the wavelength in the log-log scale. This result is shown in the same plot (magenta line). None of these attempts\nare fully satisfactory. However, the discontinuous model seems to work much better. Still, in both cases, we face a\nconsiderable mismatch between the model and the data in the region 3000 - 3500 ˚A. Semi-empirical models do include\nconsiderable emission there (see Figure 14, dotted line). The spike close to ∼2000 ˚A is most likely a Fe III emission.\nWe did not try a more complex approach in the form of the Locally Optimized Cloud (LOC) model of Baldwin et al.\n(1995) since we do not include Fe III emission which may be important for exact data fitting. We address this issue\nand additional likely effects in the discussion.\n4.DISCUSSION17\nWe used the most recent version of the code CLOUDY C23.00 to construct the grid of theoretical Fe II templates cov-\nering the broad UV to NIR range from 1000 ˚A to 10000 ˚A. The templates represent single-zone emitters, parametrized\nby the local density and ionization flux. We consider a range of turbulent velocities and for a fixed turbulent velocity of\n20 km−1we calculated the results for different metallicities. For the other turbulent velocity the considered metallicity\nis solar.\nThe advantage of the theoretical templates is that they predict the Fe II emission underlying the strong emission\nlines while purely observational templates, modelling first the lines, force low Fe II emission at the position of lines\nlike H βor Mg II. In addition, the construction of the observational templates is thus most successful for the objects\nwith extremely narrow emission lines like I Zw 1 (Vestergaard & Wilkes 2001) or Mrk 493 (Park et al. 2022). Line\nwidths of these spectra imply high Eddington ratio, strong Fe II emission and the incident SED shape not necessarily\nrepresentative for the majority of quasars.\nThe availability of the Fe II templates in the full spectral range will address better the issue of the production of\nthe Fe II in the BLR. In the case of the observational data covering a narrow spectral range, the templates allow us to\nfit the data and obtain an estimate of the density and the ionization flux in the BLR, as we showed in Section 3.8.1.\nHowever, our attempt to use these templates for a quasar spectrum covering both optical and UV ranges was not\nsuccessful (see Section 3.8.2). The problem seems generic since the selected quasar is rather a typical object, and the\nissue of the UV to optical ratio in the models and the data has been present in the literature for many years.\nAlready the early papers asked the question of whether the Fe II emission actually forms as a result of the irradiation\nby the central source or due to mechanical heating. For example, Joly (1981) in her paper argued that either mechanical\nheating is the dominating channel of production, or the clouds must be very optically thick, in which case the irradiation\nflux is thermalized and the mechanism is similar to the mechanical heating. However, those early models were not\nsuitable for column densities larger than 1024cm−2.\nVerner et al. (2003) calculated the predicted ratio of the UV to optical Fe II emission from their model of the Fe II\natomic transitions. The adopted ranges were 2000–3000 ˚A and 4000–6000 ˚A range, the range is comparable although\nnot identical to the one studied in Section 3.8.2. The UV/optical ratio obtained in this paper is similar to what\nwe measure from RM102 in the case of collisional heating, but radiative heating gives much higher values, 8 - 11,\ndepending on the number of transitions. These values were obtained under a turbulent velocity of 1 km s−1. They\nnoted that Fe II optical emission flux is more sensitive to abundance than the Fe II (UV) band. Conversely, the Fe II\n(UV) band is more sensitive to microturbulence than the Fe II optical band.\nBaldwin et al. (2004) investigated the Fe II emission in the UV (2200-2800 ˚A) and optical (4450-4750 and 5080-\n5460 ˚A) ranges. They obtained the observed Fe II UV/optical ratio of 1-2, which is significantly lower than the\nmodel-predicted ratio of 40-50. Using the LOC model, they found that the observed UV Fe II emission requires a\nhigh microturbulence velocity ( ≥100 km/s). They examined collisional heating as an alternate excitation mechanism,\nhowever only the Fe II lines were visible in the UV range, which has the disadvantage of adding another specialised\nregion to the already complicated structure of the inner region of AGN. They noticed that other complex ions, such\nas Ni II, might have created similar patterns despite their lower abundance, but their atomic data was not available.\nAs a result, they favoured a microturbulence-based model for explaining UV Fe II emissions and concluded that the\noptical Fe II emission originates from an entirely different region compared to the UV Fe II emission.\nThe problem of matching the theoretical templates over the full optical to UV spectral range is not likely due to the\nuse of a single-zone approximation, particularly as some of the papers mentioned above used a multi-zone approach.\nThe standard LOC model can in principle represent both the broad range of radii as well as the broad range of densities\nbut may not be actually appropriate to solve the physical problem.\nThe radially very extended emission is unlikely. Kovaˇ cevi´ c-Dojˇ cinovi´ c & Popovi´ c (2015) argued that there is a\nkinematic correlation between the UV and the optical Fe II emissions, although they reported considerable positive\nshift of the UV Fe II with respect to the systemic redshift for the majority of their sources (their Fig. 10). The\nradius-luminosity relation discussed by Zajaˇ cek et al. (2023b) for both spectral ranges looks roughly similar, so the\ntwo regions cannot be considerably shifted. Furthermore, the possibility of measuring the Fe II delay in the optical and\nUV bands contradicts the major collisional origin of the emission, but some contributions cannot be ruled out. The\natomic content of the recent version C23.00 of CLOUDY used in this paper does not seem questionable, and apparently,\nthe systematic improvement in the data did not solve the long-standing issue of UV and optical discrepancy. The\nmetallicity dependence also does not seem to be strong enough to account for the issue.18\nThere are thus two remaining issues which can solve the problem but they were not modelled in this paper since our\nprimary goal was to provide a set of theoretical templates for further use.\nThe first weakness in the simulation is the use of a constant density assumption. Such an approach is widely\nemployed, yet it is not physically warranted. The higher temperature of the illuminated face of the cloud and the\nconsiderably lower temperature of the dark side would create an enormous pressure gradient which could disrupt the\ncloud during the dynamical timescale. Instead, the illuminated cloud must be in pressure equilibrium which also\ntakes into account the radiation pressure (R´ o˙ za´ nska et al. 2006; Baskin et al. 2014). Overall, the bright face has a\nmuch lower density, and the density is rising towards the dark side. Clouds in pressure equilibrium were considered\nin several papers. Some of these computations were done in the context of the warm absorber (R´ o˙ za´ nska et al. 2006;\nAdhikari et al. 2015, 2019) but this medium may be the same as BLR. The change of the density and temperature\nis not like a power law but rather like a rapid transition (see e.g. Fig. 1 of Baskin et al. 2014, or Fig. 3 of Adhikari\net al. 2019), so the LOC model would not reproduce it well, and eventually, a discontinuous jump in density and\nionization parameter might be a better approximation. Computations of clouds under constant pressure are possible\nwithin CLOUDY environment, and this could be possibly done in the future.\nThe second weakness is the assumption of isotropic emission. If the clouds are few and randomly distributed around\nthe central source, then we will see clouds at all angles with respect to their orientation towards the source. The entire\ncontribution summed up is well represented by the isotropic emission. If cloud distribution is flattened, but our line\nof sight is not passing through the cloud region the dark sides may contribute a little stronger to the total emission in\nterms of the effective surface but most of the cloud emission comes from the bright sides. Such expectations were given,\nfor example, by Czerny & Hryniewicz (2011) in the context of the time delay. However, the bright sides contribute\nmore to the isotropic part, therefore overexposure of the cloud’s bright sides may not have a significant impact on Fe\nII emission. However, Ferland et al. (2009) suggested that we predominantly see the dark side of the clouds. They\nconnected this to the Fe II emission in the optical band showing predominantly the redshift (Hu et al. 2008) and\nthus implying the inflow. Also in Panda et al. (2018), in Section 4.3, they discussed the issue that the optical Fe\nII is produced predominantly in the dark side of a cloud, and the dominance of the dark side helped to reproduce\nRFeIIratio without metallicity enhancement. However, the requested geometrical setup is not simple since the line of\nsight does not cross the BLR. It may require a highly ionized absorber, between the black hole and the BLR, almost\n‘invisibly’ shielding the well-irradiated bright sides of clouds.\n5.SUMMARY\nOne of the most prominent characteristic features of the quasar spectrum is the iron emission. In this work, we\nexamine the properties of Fe II emission using CLOUDY version C23.00 and generate new theoretical Fe II templates\nthat can be used to model Fe II emission in the quasar spectra covering the wavelength range 1000-10000 ˚A. We study\nthe impact of microturbulence and metallicity on iron emission and compare our model predictions with observational\ndata. The primary outcomes of our study are outlined below.\n1. The long-standing problem of a mismatch between the expected and observed values of Fe II UV to optical ratio\nunder the single zone approximation could not be resolved by the updated atomic database available in CLOUDY\nC23.00.\n2. Microturbulence has less impact on the UV to optical Fe II emission ratio.\n3. An additional apparent velocity shift of up to 1000 km/s can be added to the spectrum by microturbulence.\n4. The ionisation flux has a more significant impact on the ratio of red-to-blue parts of optical Fe II around H β\nthan does the gas density.\n5. High-density BLR (log nH(cm−3) = 12 - 13) and extremely high radiation fluxes are needed for the highly\naccreting (xA) sources.\n6. The ratios of Fe II emissions are not significantly affected by the metallicity.\n7. The observed UV-to-optical spectra of a quasar cannot be well fitted by a single theoretical Fe II template\ngenerated assuming fixed column density and isotropic cloud emission.19\nThe project was partially supported by the Polish Funding Agency National Science Centre, project\n2017/26/A/ST9/00756 (MAESTRO 9). This project has received funding from the European Research Council\n(ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. [951549]).\nBC and MZ acknowledge the OPUS-LAP/GA ˇCR-LA bilateral project (2021/43/I/ST9/01352/OPUS 22 and GF23-\n04053L). MLM-A acknowledges financial support from Millenium Nucleus NCN19-058 and NCN2023 002 (TITANs).\nSP acknowledges the financial support of the Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq)\nFellowships 164753/2020-6 and 300936/2023-0. 32 spectroscopic observations of HE 0413-4031 used in this paper were\nobtained with the Southern African Large Telescope (SALT). 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I particularly value his advice on mastering both numerical and\nanalyticaltechniques,astrategythatmarkedaturningpointinmyresearchtrajectory.\nHisamiablenatureandcharmhavenotonlymadehimabelovedfigureinthephysics\ncommunitybuthavealsofosteredacollaborativeandinspiringenvironmentforme.\nThe conferences and seminars I attended under his guidance were fertile grounds\nfor the research ideas I am zealously pursuing today.\nI would also like to extend my heartfelt thanks to Prof. Tada, a former assistant\nprofessor in our lab. His support and regular check-ins during the challenging\nCOVID-19 pandemic were a great source of comfort and encouragement. His\nkindness, along with the support of his family, played a crucial role in helping me\nnavigate through this difficult period. Additionally, I am deeply thankful to my\nfriends who provided unwavering support and companionship during these trying\ntimes, making the journey less daunting.\nI am grateful for the financial support provided by the MERIT-WINGS program\nand the JSPS fellowship (DC1), which were pivotal in facilitating my studies.\nOn a personal note, I owe a tremendous debt of gratitude to my family. To my\nparents, my sister, and her husband, whose concern and regular check-ins were\na constant source of reassurance. The delicious meals and moments of joy they\nshared with me have been both a comfort and an inspiration. I extend my deepest\nappreciation to my wife, whose unwavering support, cheer, and love have been my\npillars of strength. Her presence in my life is a blessing I cherish immensely.\nIn summary, my journey through my PhD has been enriched and made possible\nby each of these individuals and their unique contributions to my life, for which I\nam eternally grateful.\niiABSTRACT\nWe propose the integration of an energy-based finite-size scaling methodology\nwith tensor network renormalization (TNR) techniques. TNR, serving as a numer-\nical implementation of real-space renormalization group (RG) methods, provides\na pathway to access the low-lying energy spectrum of various systems. By meld-\ning TNR with conformal perturbation theory, we can effectively calculate running\ncoupling constants. This combined methodology is particularly valuable in prac-\ntical calculations, as it adeptly navigates around the numerical errors commonly\nencountered in TNR applications.\nA primary objective of numerical simulations in the study of lattice models is\noften the precise determination of their phase diagrams. The demarcation of phase\ntransition points within these diagrams often necessitates simulations of systems\nwith very large sizes. For example, accurately identifying the phase boundary of\nthe Ising model through spontaneous magnetization typically requires simulating\nthousands of lattice sites. While TNR is capable of handling large system sizes, it\nis also known to suffer from amplified numerical errors as the size of the system\nincreases.\nIncontrast,ourproposedmethodologyrequiresonlyafewstepsofRG,therebyin-\nducingfewernumericalerrorsandreducingcomputationalcosts. Theenergy-based\nfinite-sizescalingapproachdoesnotrelyonlargesystemsizes,unlikeconventional\nmethodsthatuseobservablessuchasmagnetizationandheatcapacitytodetermine\nphasetransitions. Thisapproachisnotonlymoreefficientbutalsomoreresilientto\nthechallengesposedbythescaleofthesimulations,offeringasignificantadvantage\nin the study of critical phenomena in lattice models.\nAdditionally,wewilldelveintotheoriginsofnumericalerrorsinTNRsimulations\nfromafield-theoreticalperspective. Thisanalysiswillshedlightonhowtheseerrors\nscale with the approximation parameter, denoted as 𝐷. Understanding this scaling\nis critical for accurately estimating and managing errors in simulation results.\nThrough the application of this methodology, we aim to provide a more accurate\nand computationally efficient means of exploring phase transitions and the critical\npropertiesoflatticemodels,enhancingourunderstandingofthesecomplexsystems.\nIn the subsequent discussion, we delve into the tensor structure of fixed points in\nthe context of lattice models. A significant challenge in this area arises from the\niiieffectsoffinitebonddimensions,whichmakethetruefixed-pointtensorpractically\nunattainable through direct numerical methods. To circumvent this issue, we adopt\nan analytical approach, employing conformal mappings to study the fixed-point\ntensors.\nThisanalyticalexplorationleadstoarevealinginsight: thetensorelementsofthe\nfixed-pointtensorcorrespondtothefour-pointfunctionsofprimaryoperatorswithin\nthe framework of conformal field theory (CFT). This correspondence is not just a\ntheoretical conjecture; it is corroborated by empirical observations showing that\ntensors renormalized for finite sizes tend to align with our theoretical predictions.\nThe significance of this finding cannot be overstated. It suggests that the tensor\nrepresentations of fixed points in lattice models embody the universality of non-\ntrivial infrared (IR) physics at the lattice level. Our approach thus is not only a\nnewsolutiontoadecade-oldproblem,butalsobridgesthegapbetweentheabstract\ntheoretical constructs of CFT and the practical, computable structures in lattice\nmodels. By demonstrating this universal behavior, we provide robust support for\nthe concept of universality in critical phenomena, particularly as it manifests in the\nintricate world of lattice models.\nThrough this investigation, we aim to offer a deeper understanding of the funda-\nmental principles underlying critical phenomena, specifically highlighting how the\nuniversal aspects of CFT are reflected in the practical, numerical realm of lattice\nmodel simulations.\nivPUBLISHED CONTENT AND CONTRIBUTIONS\n1A.UedaandM.Oshikawa,“Finite-sizeandfinitebonddimensioneffectsoftensor\nnetwork renormalization”, Phys. Rev. B 108, 024413 (2023).\n2A. Ueda and M. Yamazaki, “Fixed-point tensor is a four-point function”, 10.\n48550/arXiv.2307.02523 (2023).\n3A.UedaandM.Oshikawa,“Tensornetworkrenormalizationstudyonthecrossover\nin classical heisenberg and RP2models in two dimensions”, Phys. Rev. E 106,\n014104 (2022).\n4A. Ueda and M. Oshikawa, “Resolving the Berezinskii-Kosterlitz-Thouless tran-\nsition in the two-dimensional XY model with tensor-network-based level spec-\ntroscopy”, Phys. Rev. B 104, 165132 (2021).\nThe main content of this thesis is the first paper on this list(Phys. Rev. B 108,\n024413 (2023)).\nContributions\n1A.U participated in the conception of the project, coding of the numerical imple-\nmentations, and participated in the writing of the manuscript.\n2A.U participated in the conception of the project, analytical calculations, coding\nofthenumericalimplementations,andparticipatedinthewritingofthemanuscript.\n3A.U participated in the coding of the numerical implementations, analysis of the\nresults, and participated in the writing of the manuscript.\n4A.U participated in the conception of extending level-spectroscopy to visualizing\nRG flows using TNR, coding of the numerical implementations, upgrading level-\nspectroscopy, performing third-order perturbations, and participated in the writing\nof the manuscript.\nvTABLE OF CONTENTS\nAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii\nAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii\nPublished Content and Contributions . . . . . . . . . . . . . . . . . . . . . . v\nTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v\nList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii\nList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi\nChapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\n1.1 Renormalization group theory in statistical mechanics . . . . . . . . 3\n1.2 Review on tensor network renormalization . . . . . . . . . . . . . . 17\nChapter II: Finite-size and finite bond dimension effects of tensor network\nrenormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38\n2.1 Operator product expansion coefficients . . . . . . . . . . . . . . . . 39\n2.2 Precise determination of the transition temperature . . . . . . . . . . 43\n2.3 Renormalization group flow . . . . . . . . . . . . . . . . . . . . . . 49\n2.4 Finite bond-dimension effects . . . . . . . . . . . . . . . . . . . . . 51\nChapter III: Tensor network representation of fixed-point tensors . . . . . . . 57\n3.1 Fixed-point tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 57\n3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59\n3.3 Numerical fixed point tensor . . . . . . . . . . . . . . . . . . . . . . 61\n3.4 Tests on critical lattice models . . . . . . . . . . . . . . . . . . . . . 62\nChapter IV: Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . 67\nBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71\nAppendix A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75\nA.1 Perturbation theory on the finite-size spectrum . . . . . . . . . . . . 75\nA.2 Finite-entanglement scaling of the three-state potts model. . . . . . . 78\nA.3 Supplemental information on the fixed-point tensor . . . . . . . . . . 80\nviLIST OF FIGURES\nNumber Page\n1.1 ThesketchofMigdal-Kadanofftransformation. Itaimstoanalytically\nsee how the coupling 𝐾=𝛽𝐽changes when the system is coarse-\ngrained by a factor of two. . . . . . . . . . . . . . . . . . . . . . . . 6\n1.2 The RG flow of the two-dimensional Ising model following Eq. (1.9). 7\n1.3 A schematic figure of fusing of operators. . . . . . . . . . . . . . . 13\n1.4 A schematic picture of the tensor network renormalization. The\neffectivelocalBoltzmannweightat 𝑛-thRGstep𝑇(𝑛)isdecomposed\nintothetwothree-legtensorsandrecombinedas 𝑇(𝑛+1). Theeffective\nsystem size enlarges by√\n2each RG step. . . . . . . . . . . . . . . . 20\n1.5 An example of singular values of the four-leg tensor 𝑇𝑖𝑗𝑘𝑙. The blue,\norange, and green lines show the decay of 𝑠𝑚for the classical Ising\nmodelatthelow,critical,andhigh-temperatureregimes,respectively\nat𝑑=𝐷=10after six RG steps. . . . . . . . . . . . . . . . . . . . 22\n1.6 TheCDLtensorsafterTRG.Thelocalloop,markedbyaredsquare,\npersists after the TRG steps, indicating that ultraviolet information\nremains even after extensive coarse-graining. . . . . . . . . . . . . . 24\n1.7 The conformal mapping from a plane to a cylinder. The black and\nred dotted circles on the plane correspond to different time slices on\nthe cylinder. As a result, the scale translation indicated by the red\narrow on the 𝑧-plane transforms into a translation in imaginary time\non the𝑤-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n1.8 The scaling dimensions of the critical Ising model that are obtained\nfromthetransfermatrixspectrumofLoop-TNR.Thebluedottedline\nare the theoretical values from the character. . . . . . . . . . . . . . 36\n2.1 The scaling dimension obtained from the first and second leading\neigenvalue of the transfer matrix of the Ising model as 𝑥𝜎(𝐿)=\n1\n2𝜋ln𝜆0\n𝜆1. At the critical temperature, denoted by a red dotted line,\nthevalueisconsistentwiththescalingdimension1\n8regardlessofthe\nsystem sizes. Away from criticality, however, the deviation from1\n8\ngrows as𝐿increases. . . . . . . . . . . . . . . . . . . . . . . . . . . 44\nvii2.2 Example of estimating the transition temperature using Loop-TNR.\nWe set𝑇−=2.66and𝑇+=2.68as an initial estimate. The level-\ncrossing temperature 𝑇∗(𝐿)is linearly fitted to extrapolate the tran-\nsition temperature. The insert shows how we compute 𝑇∗(𝐿)for\nvarious system sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . 48\n2.3 (Left panel) The system size dependence of 𝛿𝑥cmb=𝛿𝑥𝜎+𝛿𝑥𝜖/16\nforℎ=±10−5(purple and green), 𝑇=1.0001𝑇𝑐(red) and𝑇=\n0.9999𝑇𝑐(blue). The purple and green dots are on top of each other,\nand “+\" denotes the data with a negative sign. After removing the\n𝐿−2irrelevantperturbations,thenextleading 𝐿−4perturbationshown\nwithabluedottedlineappears. ThedatawasobtainedviaLoop-TNR\nwith a bond dimension of 𝐷=24, which was deemed sufficient for\nthe finitely-correlated systems being considered. (Right panel) The\nresulting renormalization group flow. Only data after six steps are\nexhibited, where the 𝐿−4perturbations disappear. . . . . . . . . . . . 50\n2.4 Shift|𝛿𝑥𝜎(𝐿)|for the Ising model at 𝑇=𝑇𝑐,ℎ=0computed by\nLoop-TNR with 𝐷=32. There is little finite- 𝐷effect for small\nsystemsizes 𝐿 <256. Theemergentperturbationsof 𝜖and𝜎appear\nat𝐿∼256and𝐿∼104,scalingas𝐿and𝐿15/4. Theinducedgapby\nfinite-𝐷goestowardsconstantat 𝐿 >105asdenotedwiththepurple\ndotted line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52\nviii2.5(𝑎)The scaling of the shift 𝛿𝑥𝜎in TNR of the Ising model at the\ncritical point, for various bond dimensions 𝐷=4,..., 28. The\nverticalaxisisscaledas 𝐿2𝛿𝑥𝜎sothatitisconstantwhen 𝐿≪𝜉(𝐷).\nWhen𝐿≪𝜉(𝐷), the shift is dominated by the emergent relevant\nperturbation 𝜖; this is confirmed by the scaling 𝐿2𝑔𝜖∝𝐿3. The\nhorizontal axis is scaled as 𝐿/𝜉(𝐷), where the correlation length\n𝜉(𝐷)is hypothesized as in Eqs. (2.21) and (2.22). The collapse\nof the data for different bond dimensions is strong evidence of the\nhypothesizedscalingofthecorrelationlength 𝜉(𝐷). Thebluedotted\nlineindicates 𝐿/𝜉(𝐷)=1. Weset𝜉(𝐷)=2.0𝐷𝜅sothat𝐿/𝜉(𝐷)=1\nbecomes the crossover scale between the finite-size scaling regime\nand the finite- 𝐷scaling regime.(𝑏)Similar scaling analysis of\nthe shift𝛿𝑥𝜖in TNR of the three-state Potts model at the critical\npoint, for various bond dimensions 𝐷=16,..., 40with𝜉(𝐷)=\n0.067𝐷𝜅. The scaled shift 𝐿0.8𝛿𝑥𝜖behaves as a constant in the\nfinite-size scaling regime 𝐿/𝜉(𝐷)<1, whereas it scales as 𝐿3.2\nin the finite- 𝐷scaling regime 𝐿/𝜉(𝐷)>1, as expected from the\nCFT analysis (see Appendix A.2 for details). The data for different\nbond dimensions collapse again, giving compelling evidence for the\nscaling of the correlation length (2.21) and (2.22) . . . . . . . . . . 55\n3.1 Thepath-integralrepresentationofthetensorelements (𝑎)𝑆∗\n𝛼𝛽𝛾and\n(𝑏)𝑇∗\n𝛼𝛽𝛾𝛿. Thefixed-pointtensorlivesatthecenterofcylinders,and\nsurroundingcylindersarebravectorsofprimaryfields. SincetheFP\ntensorcorrespondstotheidentityoperator,“insertionofnooperator\"\nisillustratedasemptyspace. Thisidentityoperatorattheoriginin 𝑧\ncoordinate will be mapped to the infinity in 𝑤. . . . . . . . . . . . . 62\n3.2 Estimationof 𝑥𝑆(𝐿)fromLevin-TRG( 𝐷=96)andEvenbly-TNR( 𝐷=\n40). The values of 𝑥(𝐿)from the Ising and three-state Potts model\nconvergetothetheoreticalvalue 𝑥𝑆=𝑒𝜋/4denotedbyablackdotted\nline. We plot 𝑥𝑆=2.23035obtained from Loop-TNR [10] on the\ncritical 9-state clock model [19] with a lime dashed line. The three-\nstatePottsmodelexhibitsadeviationfor 𝐿 >100becausesimulating\nsystems with higher central charges involves larger numerical errors. . 63\nix3.3 The OPEcoefficients of thecritical Isingmodel evaluated bysetting\n𝑥𝑆=𝑒𝜋/4. Theblackdottedlinesdenotethetheoreticalvalues0,0.5,\nand1. Thedatapoints,denotedbyfilledcircles\" ◦\"andcrosses\"+,\"\nare obtained from Levin-TRG( 𝐷=96) and Evenbly-TNR( 𝐷=40),\nrespectively. Relativelylargefinite-sizeeffectshaveuniversalscaling\nas tested in Sec. A.3. . . . . . . . . . . . . . . . . . . . . . . . . . . 64\n3.4 Thefinite-sizeeffectofthefixedpointtensor 𝛿𝑇𝛼𝛽𝛾𝛿≡⟨𝜙𝛼𝜙𝑗𝛽𝜙𝛾𝜙𝛿⟩−\n𝑇𝛼𝛽𝛾𝛿(𝐿)from Levin-TRG( 𝐷=96, red) and Evenbly-TNR( 𝐷=40,\nblue). We plot 𝛿𝑇𝛼𝛽𝛾𝛿of𝜎𝜎𝜎𝜎(“+\"),𝜎𝜎𝜖 1(“★\"), and𝜎𝜎11(“×\")\nwith different colors depending on the algorithm. The difference\nconverges to zero for 𝐿→∞with the power-law ∼𝐿−1/3. . . . . . . 66\n4.1 The RG flow of the classical XY model in two dimensions stands\nas a quintessential example of a topological phase transition. This\nparticular type of RG flow is commonly referred to as the Kosterlitz\nRG flow. The right panel is numerically obtained RG flow in a\nsimilar manner. However, a key distinction lies in the consideration\nof up to third-order perturbations in our computational approach.\nThe deviation in the smaller system size is due to the irrelevant\nperturbations. Further details can be found in Ref. [26]. . . . . . . . 67\n4.2 (Left panel) A schematic picture of the reduced density matrix 𝜌𝐴\nfor a bipartition of the system in the path integral picture. The\nuncontracted legs correspond to the indices of the reduced density\nmatrix. (Right panel) Each of the four quadrants of the space-time\nin the left panel may be replaced by the renormalized tensor in TNR\nwith appropriate boundary conditions. . . . . . . . . . . . . . . . . 69\nA.1 The size dependence of the (a) 𝛿𝑥𝜎and (b)𝛿𝑥𝜖at𝑇=0.999995𝑇𝑐\nand𝑇=1.000005𝑇𝑐. The pink and green dotted lines denote 𝐿−0.8,\n(a)𝐿1.2, and (b)𝐿2.4fittings respectively. For the low-temperature\nphase, the sign of 𝛿𝑥𝜎is negative at 𝐿 > 100. The dip on the left\npanelaround 𝐿∼102correspondstothezeropointofEq.(A.17). (b)\nThefinite-sizeeffecttothe 𝑥𝜖sufferslessfrom 𝑇2\ncyl+¯𝑇2\ncylinamplitude.\nThe scaling of Eq. (A.18) is clearly observed. . . . . . . . . . . . . . 77\nxA.2 36𝛿𝑥𝜎−𝛿𝑥𝜖for the high temperature phase. “+” is used when the\nsignisnegative. Thereddottedlinedenotesthe 𝐿−2fittingwhilethe\nlightgreenoneisjustarelevant 𝐿1.2contributionfrom 𝜖. Loop-TNR\nrotatesthelatticeby𝜋\n4ateachRGstep,andthetiltedsystemisplotted\nwith the blue dots. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79\nA.3 Rescaled 𝛿𝑥𝜎by𝜉(𝐷)=𝐷𝜅atthecriticaltemperature. Theresulting\ndatacollapseontoauniversalfunctionthatisindependentof 𝐿/𝜉(𝐷).\nIf𝐿/𝜉(𝐷)<1,thesystemisintheFSSregion,whileif 𝐿/𝜉(𝐷)≥1,\nitisintheFESregion. IntheFESregion,thescalingofthefirst-order\nandsecond-orderperturbationsareindicatedbyagrayandpinkline,\nrespectively. 𝑥𝜎is computed as an average value of the first and\nsecond excitation energy. . . . . . . . . . . . . . . . . . . . . . . . . 81\nA.4 The contraction of the fixed-point tensors. We obtain 𝑆from TRG\nand combine together to make 𝑆∗and𝑇∗. In this way, 𝑇∗respects\nreflection symmetry along the dotted lines in addition to 𝐶4rotation\nsymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83\nA.5 The finite-size corrections 𝛿𝐶𝛼𝛽𝛾(𝐿)obtained from the numerical\nsimulation of the critical Ising model. The numerical results for\nhigher energy levels 𝛿𝐶𝜖𝜖1(𝐿)and𝛿𝐶1𝜖𝜖(𝐿)suffer from finite- 𝐷\neffects for𝐿 > 100. The scalings of the finite-size corrections are\nnevertheless universal, which is consistent with Table III in Ref. [34] 84\nA.6 TheOPEcoefficientsofthecriticalthree-statePottsmodelevaluated\nby setting𝑥𝑆=𝑒𝜋/4. The black dotted lines denote the theoretical\nvalues0,0.546,and1[50]. Thedatapoints,denotedbyfilledcircles\n\"◦\" and crosses \"+,\" are obtained from Levin-TRG( 𝐷=88) and\nEvenbly-TNR( 𝐷=40), respectively. . . . . . . . . . . . . . . . . . . 86\nxiLIST OF TABLES\nNumber Page\n2.1 The numerically obtained OPE coefficients of the Ising CFT from\nTRG. The bond dimension and the system size are 𝐷=56and\n𝐿=16√\n2(9 RG steps), respectively. . . . . . . . . . . . . . . . . . 40\n2.2 The finite-size scaling dimension of the Ising model. 𝛼is a constant\ndetermined from the second-order perturbation. Since 𝑔𝑇2and𝑔¯𝑇2\ndecay in the same manner, we write them as 𝑔. . . . . . . . . . . . . 49\n3.1 Thecomparisonofthenumerically-obtainedfixed-pointtensor 𝑇𝛼𝛽𝛾𝛿\nat𝐿=2048andtheexactfour-pointfunction ⟨𝜙𝛼(−𝑥𝑇)𝜙𝛽(𝑖𝑥𝑇)𝜙𝛾(𝑥𝑇)𝜙𝛿(−𝑖𝑥𝑇)⟩pl\nof the Ising model with 𝑥𝑇=𝑒𝜋/2/2. . . . . . . . . . . . . . . . . . 65\nA.1 A set of primary operators of the three-state Potts model. . . . . . . . 77\nxiiC h a pte r 1\nINTRODUCTION\nPhase transitions hold a particular fascination in the field of statistical mechanics\ndue to their display of universality. This universality is most notably observed\nin the behavior of systems undergoing continuous phase transitions, which are\ncharacterized by the divergence of derivatives of the partition function. These\ndivergences are quantified by critical exponents, which are key indicators of the\nsystem’s behavior near the critical point.\nWhat makes these critical exponents particularly intriguing is their universality\nacross different physical systems. Despite the diverse nature of these systems,\nthe critical exponents of certain systems tend to exhibit the same values. For\ninstance, a striking example of this universality is seen when comparing the liquid-\nvaportransitioninwaterwiththeferromagnetictoparamagnetictransitioninthree-\ndimensional magnets. Remarkably, these vastly different systems share the same\ncriticalexponents. However,thisconceptofuniversalityinphasetransitionspresents\na somewhat counter-intuitive picture at first glance. Consider the stark differences\nbetween substances like water and magnets: water is composed of hydrogen and\noxygen,whilemagnetscanbemadefrommaterialslikeneodymium. Moreover,the\ntemperaturesatwhichthesesubstancesundergophasetransitionsarevastlydifferent.\nImagine measuring the critical exponents of water in a national laboratory in the\nU.S. and then measuring those for magnets in a makeshift lab in your basement.\nDespitethedifferencesinsubstances,environments,andmethodologies,theresults\nwould be surprisingly consistent.\nThisphenomenonalmostsuggeststhatnaturepossessesaninnateunderstandingof\ntheessenceofphasetransitions. Itisasthoughtheunderlyingprinciplesgoverning\nthese phenomena inherently “ know” to discard irrelevant details, focusing instead\non fundamental aspects that are common across diverse systems. This remarkable\naspect of universality in phase transitions not only challenges our intuitive under-\nstandingbutalsohighlightstheprofoundsimplicityandelegancewithwhichnature\noperates at a fundamental level.\nNature’s method of simplifying complex phenomena can be likened to how we\nperceive images in our daily lives. Consider, for example, the iconic Windows XP\n1wallpaper “Bliss 1.,” which depicts a serene landscape of a green hill and blue sky.\nAt first glance, we see just these broad elements. However, upon closer inspection,\none might notice finer details like yellow flowers dotting the hill. With an even\nmorefocusedview,likethroughamicroscope,onecouldobservebeesaroundthese\nflowers or even delve into the atomic structures of these elements.\nYet,fromanormalviewingdistance,theseminutedetailsareeffectivelyinvisible.\nOurperceptionsimplifiesthescene,focusingonthemostsignificantelementswhile\n“ignoring”thesmaller,lessimpactfulones. Thisisakintohowweapproachphase\ntransitions in statistical mechanics. In the study of phase transitions, we often\nconsiderthethermodynamiclimit,whichimpliesobservingthesystemasifitwere\ninfinitelylarge. Thisperspectiverequiresusto“stepback”andviewthesystemfrom\na great distance, thereby making any finite-sized clusters or features with limited\ncorrelation lengths appear increasingly smaller and less significant.\nThis“steppingback”inobservingphysicalsystemsisanalogoustoobservingthe\n“Bliss” wallpaper from a distance. Just as we see only the broad strokes of the\nhill and sky rather than the minute details, in phase transitions, the focus is on\nmacroscopic properties that emerge when viewing the system in its entirety, from\nafar. Small-scale variations and details become irrelevant at this scale, allowing us\ntodiscerntheuniversalaspectsthatdominatethebehaviorofthesystemasawhole.\nToquantitativelyexplorehowsystemsappeartotransformwhenwe“stepback\"and\nobserve them from a larger scale, the renormalization group (RG) theory becomes\nindispensable. This chapter is dedicated to reviewing RG theory, with a specific\nemphasis on its application in statistical mechanics.\nRG theory provides a framework to understand how the behavior of physical\nsystems changes across different scales. It allows us to systematically “zoom out”\nfrom the microscopic details and observe how the collective properties of a system\nevolve. Thisperspectiveiscrucialforgraspingtheessenceofphasetransitions,asit\nrevealstheunderlyinguniversalitiesthatmanifestwhenmicroscopicdetailsbecome\nless relevant at macroscopic scales.\nFinally,asapreludetodelvingintothedetailedtheoreticalaspectsofRGtheoryand\nits applications in statistical mechanics, it can be beneficial for readers to visualize\nhowtheconceptof“seeingfromadistance\"manifestsinphysicalsystems. Toaidin\n1It was a wallpaper of my first computer. https://en.wikipedia.org/wiki/Bliss_\n(image). They now have a 4K version of it. https://msdesign.blob.core.windows.net/\nwallpapers/Microsoft_Nostalgic_Windows_Wallpaper_4k.jpg\n2this visualization, we recommend viewing a short YouTube video 2that illustrates\nhow it happens in the Ising model.\n1.1 Renormalization group theory in statistical mechanics\nIn this section, we delve into critical aspects of phase transition and RG theory\nwithin the realm of statistical mechanics, using the Ising model as a foundational\nexample. A key element in this context is the partition function, particularly when\nexpressed in the transfer matrix formalism, and its intricate relationship with the\ncorresponding action or Hamiltonian. This conceptual framework, central to our\ndiscussion, paves the way for a natural extension to and incorporation within the\ntensor-network language, which will be explored in the following section.\nWe begin our exploration with the classical one-dimensional Ising model, a fun-\ndamental and illustrative example in the study of magnetism. In this model, local\nspin states are characterized by a binary variable, 𝜎𝑖=±1, where each spin can be\nthought of as a miniature magnet. The values of ±1can be metaphorically equated\ntothenorth(N)andsouth(S)polesofamagnet. Thisanalogyisusefulinvisualiz-\ning how each local spin, akin to a tiny magnet, contributes to the overall magnetic\nproperties of the system. In the one-dimensional Ising model, each spin aligns\nalong a single direction in an array. By considering these local spins collectively,\nwe gain insight into the emergent magnetic behavior of the entire system, where\nthe alignment or randomness of these “mini magnets\" underpins the macroscopic\nproperties observed. Specifically, the system is classified as ferromagnetic if the\naverage spin, calculated as1\n𝑁Í𝑁\n𝑖=1𝜎𝑖, is non-zero. This non-zero average indicates\na net alignment in one direction, characteristic of ferromagnetic ordering. Con-\nversely, if this average equals zero,1\n𝑁Í𝑁\n𝑖=1𝜎𝑖=0, the system is said to be in a\nparamagnetic state. Inthisstate,thespinsareorientedrandomly,resultinginnonet\nmagnetization. This binary representation not only simplifies the analysis but also\nprovidesprofoundinsightsintotheunderlyingmechanismsofmagneticinteractions\nand phase transitions.\nIntherealmofstatisticalmechanics,thebehaviorofspinsintheIsingmodelisgov-\nerned by local Boltzmann weights. The probability of a specific spin configuration\noccurringisquantifiedbytheseBoltzmannweights. Theseweightsaremathemati-\ncally defined as exp[−𝛽𝜖{𝜎𝑖}], where𝛽=1/𝑇represents the inverse temperature,\nand𝜖{𝜎𝑖}denotes the energy associated with a particular spin configuration. Con-\n2https://www.youtube.com/watch?v=MxRddFrEnPc&t=1s\n3sequently, the probability of observing a certain configuration, 𝑝{𝜎𝑖}, is calculated\nusing the formula 𝑝{𝜎𝑖}=exp[−𝛽𝜖{𝜎𝑖}]\n𝑍(𝛽). Here,𝑍(𝛽)=Í\n{𝜎𝑖}exp[−𝛽𝜖{𝜎𝑖}]is the\npartition function, which serves as a normalization factor ensuring that the sum of\nprobabilities over all possible configurations equals one.\nTheenergyoftheIsingmodelinperiodicboundarycondition(PBC)isdefinedas\n𝜖{𝜎𝑖}=−𝐽𝑁−1∑︁\n𝑖=1𝜎𝑖𝜎𝑖+1−𝐽𝜎𝑁𝜎1. (1.1)\nThis definition ensures that the energy is minimized when all spins are aligned in\nthe same direction, making this configuration highly favored at low temperatures.\nIn contrast, at higher temperature regimes, where 𝛽∼0, the probability becomes\nignorantoftheenergyconfiguration. Asaresult,undertheseconditions,theprefer-\nential status of spin alignment dictated by lower energy considerations diminishes.\nInstead, the system tends to favor states that are more ’typical’ or statistically com-\nmon, reflecting a shift from energy-driven order to entropy-driven disorder. These\ntwo distinct phases – the ordered, low-temperature phase and the disordered, high-\ntemperature phase – are typically delineated by a critical point known as the phase\ntransition. Around this point, various singularities emerge in the derivatives of\nthe free energy, which is defined as 𝑓(𝛽)=−1\n𝛽ln𝑍(𝛽). These singularities are\nindicative of drastic changes in the system’s behavior and are a hallmark of phase\ntransitions in statistical mechanics.\nLet us compute the partition function for the one-dimensional Ising model as\ndefined in Eq. (1.1). We denote the partition function for a system of 𝑁spins with\nfixedspinsatbothendsas 𝑍𝑁(𝜎1,𝜎𝑁). Calculatingthisfunctionisstraightforward\nfor a system with only two spins. Let us set 𝐽=1and then, we have 𝑍2(+,+)=\n𝑍2(−,−)=𝑒𝛽,𝑍2(+,−)=𝑍2(−,+)=𝑒−𝛽, where+and−denotes𝜎=1and−1,\nrespectively. Now, consider adding an additional spin at site 𝑖=3. The resulting\npartitionfunctionforthreespinscanbedeterminedbyconsideringcaseswherethe\nsecond and third spins are either aligned or opposite. This yields the following\nequations:\n𝑍3(+,+)=𝑒−𝛽𝑍2(+,−)+𝑒𝛽𝑍2(+,+),\n𝑍3(+,−)=𝑒𝛽𝑍2(+,−)+𝑒−𝛽𝑍2(+,+).\nThe above equations are rewritten using matrix multiplications as\n \n𝑍3(+,+)\n𝑍3(+,−)!\n= \n𝑒𝛽𝑒−𝛽\n𝑒−𝛽𝑒𝛽!  \n𝑍2(+,+)\n𝑍2(+,−)!\n.\n4We can repeat this procedure to obtain the partition function of 𝑁spin systems as\n \n𝑍𝑁(+,+)\n𝑍𝑁(+,−)!\n= \n𝑒𝛽𝑒−𝛽\n𝑒−𝛽𝑒𝛽!  \n𝑍𝑁−1(+,+)\n𝑍𝑁−1(+,−)!\n,\n= \n𝑒𝛽𝑒−𝛽\n𝑒−𝛽𝑒𝛽!𝑁−2 \n𝑍2(+,+)\n𝑍2(+,−)!\n,\n= \n1\n2\u0002\n(2 cosh𝛽)𝑁−1+(2 sinh𝛽)𝑁−1\u0003\n1\n2\u0002\n(2 cosh𝛽)𝑁−1−(2 sinh𝛽)𝑁−1\u0003!\n,(1.2)\nwhere the remaining two boundary conditions, 𝑍𝑁(−,−)and𝑍𝑁(−,+)are respec-\ntively equal to 𝑍𝑁(+,+)and𝑍𝑁(+,−). Thus, the partition function for PBC is\n𝑍𝑁(𝛽)=𝑍𝑁+1(−,−)+𝑍𝑁+1(+,+)\n=(2 cosh𝛽)𝑁+(2 sinh𝛽)𝑁(1.3)\nThisresultcanbefurtherelucidatedbyemployingtheconceptofa transfermatrix .\nThe transfer matrix, denoted as T, is defined by the matrix:\nT= \n𝑒𝛽𝑒−𝛽\n𝑒−𝛽𝑒𝛽!\n. (1.4)\nThismatrixeffectively transfersthepartitionfunctionfrom 𝑍𝑁to𝑍𝑁+1,servingasa\ntooltocalculatethepartitionfunctionforalargersystembasedontheknownresults\nof a smaller system. The eigenvalues of the transfer matrix, T, are particularly\nsignificant as they dictate the thermodynamic properties of the system. In fact, it\ncanbedemonstratedthatthepartitionfunctioncanbesuccinctlyexpressedusing T\nin the following manner:\n𝑍𝑁(𝛽)=TrT𝑁(1.5)\n=1∑︁\n𝑛=0𝜆𝑁\n𝑛, (1.6)\nwhere𝜆0=2 cosh𝛽and𝜆1=2 sinh𝛽are the eigenvalues of T. This elucidates an\nimportant concept: the partition function is essentially the sum of the 𝑁-th powers\nof the eigenvalues of the transfer matrix. This concept is applicable to any spatial\ndimension.\nReal-space renormalization: Migdal-Kadanoff transformaiton\nThetwo-dimensionalIsingmodelpresentsahigherlevelofcomplexitycompared\ntoitsone-dimensionalcounterpart. Toaddressthis,MigdalandKadanoffdeveloped\n5Figure 1.1: The sketch of Migdal-Kadanoff transformation. It aims to analytically\nseehowthecoupling 𝐾=𝛽𝐽changeswhenthesystemiscoarse-grainedbyafactor\nof two.\na methodology to simplify the problem [1, 2]. Kadanoff’s approach focused on\nunderstanding how the effective coupling constant 𝐾=𝛽𝐽evolves with changes\nin scale, rather than attempting to directly calculate the partition function. This is\nschematically illustrated in Figure 1.1, where the method involves tracing out the\ndegrees of freedom at the ⊗sites while retaining the spins at the ◦sites. Notably,\nthespinatthecenterofeachnewsquarelatticeisignored,enablingexacttreatment\nof the system. Kadanoff’s key assumption is that during the coarse-graining of the\nsystem, the effective coupling effectively doubles. This doubling occurs as a result\nof bundling together two interaction bonds from the original lattice. Moreover,\nthere are spins situated at the center of the new bonds, denoted as ⊗. These central\nspins need to be traced out to determine the new coupling constant. Then, the new\ncoupling constant 𝐾′is derived as follows:\n𝐴exp[𝐾′𝜎1𝜎3]=Tr𝜎2exp[2𝐾𝜎1𝜎2+2𝐾𝜎 2𝜎3]\n=2 cosh(2𝐾(𝜎1+𝜎3))\n=2𝜎1𝜎3sinh2(2𝐾)+2 cosh2(2𝐾) (1.7)\nGiven that𝜎2=1, we can express the left-hand side of the equation as:\nexp[𝐾′𝜎1𝜎3]=cosh(𝐾′)+𝜎1𝜎3sinh(𝐾′) (1.8)\n6Figure 1.2: The RG flow of the two-dimensional Ising model following Eq. (1.9).\nThis leads us to the relation between the old and new couplings:\ntanh(𝐾′)=tanh2(2𝐾) (1.9)\nEquation (1.9) is fundamental to understanding how coupling constants transform\nunder scale transformations in the two-dimensional Ising model. It indicates that\nwith each coarse-graining of the lattice by two sites, the coupling constants evolve\naccording to the relation 𝐾′=arctanh(tanh2(2𝐾)). This process of evolution is\ndepicted in Fig. 1.2.\nAt the critical point 𝐾𝑐, marked by a black dotted line, there is no evolution of 𝐾\nsince tanh(𝐾𝑐)=tanh2(2𝐾𝑐)holds true. This point is identified as a fixed-point ,\ncorresponding to the critical temperature. In the high-temperature regime where\n𝐾 <𝐾𝑐, as shown by the red arrows, the effective coupling decreases towards zero\nwith increasing scale. Since 𝐾=0corresponds to the 𝐽→0or𝑇→∞limit, this\nindicates each spin is decoupled, being a phase of complete disorder. Conversely,\nstarting from a low-temperature regime where 𝐾 > 𝐾𝑐,𝐾increases upon coarse-\ngraining, as indicated by the blue arrows. This behavior, aligning with the 𝐽→∞\nor𝑇→0limits, corresponds to a phase characterized by spontaneous symmetry\nbreaking.\nFigure 1.2 effectively illustrates the concept of renormalization group flow (RG\nflow) in the context of the coupling constant 𝐾in the Ising model. As depicted,\n7when the scale increases, the couplings 𝐾that start from values below the critical\npoint(𝐾 < 𝐾𝑐)and above it(𝐾 > 𝐾𝑐)appear to “flow\" towards 𝐾=0and\n𝐾=∞,respectively. Thisdynamicbehaviorofthecouplingconstantunderscaling\ntransformations is a fundamental aspect of RG flow.\nThepoints𝐾=0,𝐾𝑐,and∞areidentifiedasfixed-pointswithinthisflow. These\npoints are unique in that they remain invariant under scale transformations as for-\nmulated in Eq. (1.9). They can be conceptualized as “the terminal stations” of the\nscaletransformationprocess. Whenreachingthesepoints,thesystemhaseffectively\ndiscarded all irrelevant information through an infinite series of scale transforma-\ntions. Drawing a parallel to the ’Bliss’ wallpaper analogy, each step we take back\nfrom the image can be likened to each step of scale transformation in physical sys-\ntems. As we step back, finite-sized clusters within the wallpaper appear smaller\nand smaller. Similarly, in physical systems undergoing scale transformations, the\ncorrelation length, denoted as 𝜉, reduces by half with each coarse-graining step.\nThis process leads the system towards a “simpler theory”, where 𝜉=0, represents\na state devoid of significant correlations at large scales.\nAnotableexceptionarisesatthepointofcriticality,where 𝜉reachesinfinity. This\ninfinitecorrelationlengthisadefiningfeatureofcontinuousphasetransitions,mark-\ning a state where correlations extend across all scales. The fixed-point associated\nwiththiscriticalityisofparticularinterest,asitembodiesthekeyprinciplesofuni-\nversalityandscaleinvariancefundamentaltounderstandingthesephasetransitions.\nDespite the microscopic differences between systems like water and magnets, the\nmacroscopic behaviors of these systems converge to the same fixed-points. This\nconvergence to common fixed-points is what gives rise to the universal properties\nobserved in critical phenomena across different physical systems.\nTo further understand this concept, consider linearizing Eq. (1.9) near the critical\npoint𝐾=𝐾𝑐. This linearization yields:\n(𝐾′−𝐾𝑐)≈1.68(𝐾−𝐾𝑐).\nThis relationship indicates that, after 𝑛-steps of scale transformation, the effective\ncoupling constant increases exponentially as:\n(𝐾′−𝐾𝑐)≈1.68𝑛(𝐾−𝐾𝑐).\nExtending this to a continuous scale transformation 𝑛, we can describe how the\n8deviation𝛿𝐾=𝐾−𝐾𝑐grows with scale:\n𝑑(𝛿𝐾(𝑛))\n𝑑𝑛≈ln(1.68)𝛿𝐾(𝑛). (1.10)\nSince the system size scales as 𝐿=2𝑛, Eq. (1.10) can be reformulated in terms of\nthe logarithmic scale 𝑙=ln(𝐿):\n𝑑(𝛿𝐾(𝑙))\n𝑑𝑙≈ln(1.68)\nln(2)𝛿𝐾(𝑙),\n≈0.75𝛿𝐾(𝑙). (1.11)\nThis formulation represents the RG equation. Phase transitions that belong to the\nsameuniversalityclassarecharacterizedbythesameRGequation,eventhoughthey\nmay have different specific scale transformations as in Eq. (1.9). The coefficient\n∼0.75in this equation is referred to as the RG dimension, which plays a crucial\nrole in determining the critical exponents of the system. It is noteworthy that while\nthe exact RG dimension for the Ising universality class is one as\n𝑑(𝛿𝐾(𝑙))\n𝑑𝑙=𝛿𝐾(𝑙). (1.12)\nNamely, Kadanoff’s approximation yields a close value, demonstrating the effec-\ntiveness of this simplified approach.\nIn the following sections, we will delve deeper into this concept, interpreting\nit through the lens of field theory. This approach will provide a more nuanced\nand comprehensive understanding of the dynamics at play in phase transitions and\ncritical phenomena.\nField theory describing two-dimensional fixed-point\nUniversality at criticality can be exemplified by examining the behavior of the\ncorrelation function in critical systems. In the specific case of the critical two-\ndimensional Ising model, the spin-spin correlation function exhibits a polynomial\ndecay characterized by:\n⟨𝜎(𝑟𝑖)𝜎(𝑟𝑗)⟩∝1\n|𝑟𝑖−𝑟𝑗|1/4, (1.13)\nwhere𝑟𝑖and𝑟𝑗denote the positions of spins. This equation illustrates how, at\ncriticality, the correlation between spins decays in a manner inversely proportional\nto the distance raised to the power of 1/4.\n9Incontrast,forsystemsthatarenotatcriticality,whichpossessafinitecorrelation\nlength𝜉, the correlation function typically exhibits an exponential decay:\n⟨𝜎(𝑟𝑖)𝜎(𝑟𝑗)⟩∝𝑒−|𝑟𝑖−𝑟𝑗|/𝜉.\nIn this scenario, the correlation diminishes exponentially with increasing distance\nbetween spins, governed by the correlation length 𝜉.\nTherefore,thecorrelationfunctioninEq.(1.13)forthecriticalIsingmodelcanbe\nunderstood as the limit where 𝜉→∞. This infinite correlation length at criticality\nis what leads to the power-law decay of the correlation function, distinguishing\nit from the exponential decay observed in systems with finite correlation lengths.\nThis behavior exemplifies the concept of universality at criticality, where different\nsystemsexhibitsimilarlong-rangecorrelationsastheyapproachtheircriticalpoints.\nIn field theory, correlation functions are expressed in terms of operators. For\ninstance, Eq. (1.13) in the field theory framework is represented as:\n⟨ˆ𝜎(𝑟𝑖)ˆ𝜎(𝑟𝑗)⟩=1\n|𝑟𝑖−𝑟𝑗|1/4, (1.14)\nwhere ˆ𝜎denotes the spin operator and the brackets indicate the expectation value\nofinsertingtwosuchspinoperatorsintothevacuumstate. Thisformulationreflects\nhow the correlation between spins decays with distance in the critical Ising model.\nAnothercrucialoperatorinthecriticalIsingmodelistheenergyoperator,denoted\nasˆ𝜖. The correlation function for this operator is given by:\n⟨ˆ𝜖(𝑟𝑖)ˆ𝜖(𝑟𝑗)⟩=1\n|𝑟𝑖−𝑟𝑗|2. (1.15)\nThis equation demonstrates that the correlation of the energy operator also decays\nwith distance, but at a different rate compared to the spin operator.\nOnthelattice,thecorrelationfunctioncorrespondingtotheenergyoperatortakes\nthe form:\n⟨(𝜎(𝑟𝑖)𝜎(𝑟′\n𝑖))(𝜎(𝑟𝑗)𝜎(𝑟′\n𝑗))⟩∝1\n|𝑟𝑖−𝑟𝑗|2, (1.16)\nwhere𝑟′\n𝑖is a neighboring site of 𝑟𝑖. The product 𝜎(𝑟𝑖)𝜎(𝑟′\n𝑖)represents the local\nenergyintheIsingmodel,whichexplainswhy ˆ𝜖isreferredtoastheenergyoperator.\nIn addition to the above operators, there is also the identity operator denoted as 𝐼\nthat represents “inserting nothing.\"\n10The exact values of the exponents in Eqs. (1.14) and (1.15), specifically 1/4\nand2, naturally lead to speculation about an underlying theoretical framework that\nexplainstheseprecisefigures. Indeed,fortwo-dimensionalcriticalsystems,thereis\naprofoundtheoreticalbasisbehindtheseexponents. Theconceptofscaleinvariance,\nwhichisinherenttofixedpoints,extendstoabroaderprincipleknownasconformal\ninvariance in two dimensions. Conformal invariance imposes strict constraints on\noperators and their correlation functions, often enabling the precise determination\nof critical exponents.\nThefieldtheorythatincorporatesthisinvarianceisknownasconformalfieldtheory\n(CFT).ThekeytoCFT’seffectivenessliesinitsexploitationofconformalinvariance,\nwhich significantly enhances the symmetry of the system and, consequently, its\nanalytical traceability.\nIn CFT, each universality class corresponds to a specific CFT. For example, the\nIsinguniversalityclassisdescribedbytheIsingCFT,whichincludesthreeprimary\noperators 3:𝐼,𝜎, and𝜖(hereafter, we will refer to operators without the hat\nnotation). More generally, primary operators in CFT are denoted as Φ𝑖. These\noperators exhibit the following characteristics: The two-point correlation function\nis defined as\n⟨Φ𝑖(𝑟𝑖)Φ𝑗(𝑟𝑗)⟩=𝛿𝑖,𝑗\n|𝑟𝑖−𝑟𝑗|2𝑥𝑖, (1.17)\nwhere𝑥𝑖represents the scaling dimension of the operator Φ𝑖. In the case of the\nIsing CFT, the scaling dimensions are 𝑥𝐼=0,𝑥𝜎=1/8, and𝑥𝜖=1. Similarly, the\nthree-point correlation function is expressed in a universal form:\n⟨Φ𝑖(𝑟𝑖)Φ𝑗(𝑟𝑗)Φ𝑘(𝑟𝑘)⟩=𝐶𝑖𝑗𝑘\n|𝑟𝑖−𝑟𝑗|Δ𝑘\n𝑖𝑗|𝑟𝑗−𝑟𝑘|Δ𝑖\n𝑗𝑘|𝑟𝑘−𝑟𝑖|Δ𝑗\n𝑘𝑖, (1.18)\nwhere𝐶𝑖𝑗𝑘isanoperatorproductexpansion(OPE)coefficient,and Δ𝑘\n𝑖𝑗=𝑥𝑖+𝑥𝑗−𝑥𝑘.\nNotable OPE coefficients for the Ising CFT include:\n𝐶𝐼𝐼𝐼=𝐶𝐼𝜎𝜎=𝐶𝐼𝜖𝜖=1, (1.19)\n𝐶𝜎𝜎𝜖=1\n2. (1.20)\n3Primary operators are a specific class of operators that play an important role in elementary\nexcitations. While we will only discuss this class of operators in this section, there is another class\nnamed descendants that govern higher excited states.\n11It is important to note that the permutation of indices in these coefficients does not\nalter their values, preserving the Z2symmetry 4.\nEmphasizingthephysicalsignificanceofOPEcoefficientsinRGtheoryiscrucial,\nasthesecoefficientsplayapivotalroleinunderstandinghowlocaloperatorsinteract\nand combine, or “fuse,\" within the field theory framework. To grasp this concept,\nconsider an analogy involving a canvas with blue and red dots placed close to each\nother. When viewed from a close distance, these dots are seen as distinct entities.\nHowever, as one steps back, the dots may appear to merge into a single purple dot.\nThisphenomenonofblendingor“fusion\"ofthedotsmirrorshowoperatorsinfield\ntheory can be combined.\nInfieldtheory,thefusionofoperatorsismathematicallyrepresentedbythemixing\nof two local operators situated in close proximity. Let Φ𝑖andΦ𝑗be two such\noperators. The fusion of these operators can be expanded in terms of the local\noperator basis, as illustrated below:\nΦ𝑖(𝑟𝑖)Φ𝑗(𝑟𝑗)≈∑︁\n𝑘𝐶𝑖𝑗𝑘\n|𝑟𝑖−𝑟𝑗|𝑥𝑖+𝑥𝑗−𝑥𝑘Φ𝑘\u0010𝑟𝑖+𝑟𝑗\n2\u0011\n. (1.21)\nIn this equation, the exponents with respect to |𝑟𝑖−𝑟𝑗|are chosen to ensure con-\nsistency with the two-point correlation function, as described in Eq. (1.17). This\nexpansionsignifieshowtwolocaloperatorswhenincloseproximity,caneffectively\ncombine to form a different operator, Φ𝑘, with the OPE coefficients 𝐶𝑖𝑗𝑘dictating\nthe nature and strength of this fusion.\nIn the Ising CFT, the fusion of two spin operators 𝜎results in the formation\nof the energy operator 𝜖. This concept aligns intuitively with the corresponding\nlattice model of the Ising system. Recall that in the lattice model, the individual\nspin operator 𝜎𝑖and the product of adjacent spin operators 𝜎𝑖𝜎𝑖+1correspond to\nthe CFT operators 𝜎and𝜖, respectively. Consequently, when two spin operators\npositioned close to each other on the lattice are multiplied, the resulting interaction\nclosely resembles 𝜎𝑖𝜎𝑖+1, which is the lattice analog of the energy operator 𝜖. This\ninteraction is mirrored in the CFT framework, as evidenced by the non-zero OPE\ncoefficient𝐶𝜎𝜎𝜖, indicating a significant fusion between two 𝜎operators into 𝜖.\nAdditionally, in CFT, the identity operator 𝐼represents the concept of inserting\nno operator into the system. As such, fusing any operator with 𝐼does not result in\n4For a detailed discussion on CFT, readers are referred to Ref. [3]. In this chapter, we aim to\ngive a practical introduction to CFT.\n12Figure 1.3: A schematic figure of fusing of operators.\nanychangetotheoriginaloperator. ThisisreflectedintheOPEcoefficients,where\nfusing an operator with the identity operator maintains the operator unchanged,\nleading to𝐶𝑖𝑖𝐼=1. This property highlights the foundational role of the identity\noperator in maintaining the integrity of the system’s operators during the fusion\nprocess.\nBuilding upon our understanding of OPE coefficients and their role in operator\nfusion, we can discern the rationale behind their appearance in the three-point\ncorrelationfunction,asshowninEq.(1.18). Considerthesituationwherethepoints\n𝑟𝑖and𝑟𝑗are in close proximity to each other. Under this condition, Eq. (1.18) can\nbe approximated as:\n⟨Φ𝑖(𝑟𝑖)Φ𝑗(𝑟𝑗)Φ𝑙(𝑟𝑙)⟩≈𝐶𝑖𝑗𝑙\n|𝑟𝑖−𝑟𝑗|𝑥𝑖+𝑥𝑗−𝑥𝑙1\n|𝑟𝑖−𝑟𝑙|2𝑥𝑙. (1.22)\nThis approximation effectively combines the principles outlined in Eqs. (1.17) and\n(1.18). Specifically, it demonstrates that a three-point correlation function can be\ninterpreted as a two-point function following the fusion of the first two operators,\nΦ𝑖andΦ𝑗.\nThis fusion process results in a single operator, which then interacts with the\nthird operator, Φ𝑙. The corresponding correlation function thus encapsulates this\ninteraction, with the OPE coefficient 𝐶𝑖𝑗𝑘playing a crucial role in quantifying the\nstrength and nature of the fusion between Φ𝑖andΦ𝑗. This concept is graphically\nrepresentedinFig.1.3,wherethefusionofthefirsttwooperatorsbeforeinteracting\nwith the third is illustrated. Through this lens, the three-point function can be\nunderstoodasamanifestationoftheunderlyingfusiondynamicsamongtheoperators\nin the field theory framework.\nIn CFT, conformal mapping plays a crucial role, especially in two-dimensional\ncontexts. In such systems, the physical 𝑥-𝑦plane is effectively represented using a\ncomplexplanewithcoordinates 𝑧=𝑥+𝑖𝑦anditscomplexconjugate ¯𝑧=𝑥−𝑖𝑦. In\n13thisframework,primaryoperatorstransformwhenweperformaconformalmapping\n𝑤=𝑓(𝑧)as:\n˜Φ𝑖(𝑤,¯𝑤)=\u0012𝜕𝑤\n𝜕𝑧\u0013−ℎ𝑖\u0012𝜕¯𝑤\n𝜕¯𝑧\u0013−¯ℎ𝑖\nΦ𝑖(𝑧,¯𝑧), (1.23)\nIn this formula, ℎ𝑖and ¯ℎ𝑖are known as the conformal weights of the operator\nΦ𝑖, determining its scaling dimension 𝑥𝑖=ℎ𝑖+¯ℎ𝑖and conformal spin 𝑠𝑖=ℎ𝑖−¯ℎ𝑖.\nConsiderascaletransformationdescribedby 𝑤=𝑏−1𝑧5. Underthistransformation,\nthe primary operator Φ𝑖behaves as:\n˜Φ𝑖(𝑤,¯𝑤)=𝑏𝑥𝑖Φ𝑖(𝑧,¯𝑧). (1.24)\nThis relationship illustrates how the operator scales with the transformation factor\n𝑏. Setting𝑏=|𝑧|, we can derive the power-law decay of the two-point correlation\nfunction, as observed in Eq. (1.17):\n⟨Φ𝑖(𝑧,¯𝑧)Φ𝑖(0)⟩=|𝑧|−2𝑥𝑖⟨Φ𝑖(1)Φ𝑖(0)⟩. (1.25)\nThis outcome highlights how the decay rate of the two-point function is directly\nlinkedtothescalingdimensionoftheoperator,demonstratingtheprofoundinfluence\nofconformalmappinginCFT.Suchinsightsarepivotalforunderstandingcorrelation\nfunctions in critical phenomena and the symmetry principles that underpin them.\nThe collection of scaling dimensions 𝑥𝑖and OPE coefficients 𝐶𝑖𝑗𝑘, collectively\nreferred to as CFT data, is crucial for a comprehensive understanding of critical\nphenomena. Therefore, the determination of this CFT data, particularly from a\nnumerical perspective, is a fundamental objective in the study of critical systems.\nRenormalization group and CFT\nThe concept of CFT is intrinsically linked to the RG theory. Specifically, CFT\nprovidesaframeworktocalculatehowdeviationsfromcriticalvalues,suchas 𝛿𝐾(𝑙)\nin Eq. (1.12), evolve through scale transformations. When considering Kadanoff’s\nreal-spaceRGapproach,onemightquestionwhytheanalysisisfocusedsolelyonthe\ncouplingconstant 𝐾. Inanexactcoarse-grainingofthemodel,othercouplingcon-\nstants, like the next nearest-neighbor coupling, could emerge. However, Eq. (1.11)\n5𝑏is the factor of scale transformation, where Kadanoff’s RG corresponds to 𝑏=2. This is\nbecause 2𝑐𝑚in𝑧-plane becomes 1𝑐𝑚in𝑤-coordinate.\n14still qualitatively represents the correct RG flow for Ising criticality. So, why is it\nvalid to concentrate only on 𝐾? Field theory answers this by identifying 𝐾as a\nrelevantparameter,whereasothersarenot. Tounderstandthisindetail,Wilsonfirst\nintroduced the concept of considering allpossible perturbations that might arise\nfromthemicroscopicdetailsoftheHamiltonian[4,5]. Deviationssuchas 𝛿𝐾from\nthe scale-invariant action 𝑆∗are expressed by 𝑔𝑗with corresponding operators Φ𝑗.\nHence, the action can be formulated as:\n𝑆=𝑆∗+∫\n𝑑𝑑𝑟∑︁\n𝑗𝑔𝑗Φ𝑗, (1.26)\nwhereΦ𝑗are normalized operators, and the sum of 𝑗goes through all possible\nperturbations. Weareinterestedinhowonlyafewofthesebecomerelevantduring\nthe scale transformation. To do this, we expand the Euclidean action around the\nfixed point as follows:\nTr𝑒−𝑆=Tr𝑒−𝑆∗*\n1−∑︁\n𝑖∫\n𝑑𝑑𝑟𝑔𝑖Φ𝑖+1\n2∑︁\n𝑖,𝑗∫\n𝑑𝑑𝑟𝑖𝑑𝑑𝑟𝑗𝑔𝑖𝑔𝑗Φ𝑖Φ𝑗+···+\n𝑆∗\nUnder scale transformations 𝑟→𝑏𝑟, the second and third terms transform as:\n∑︁\n𝑖∫\n𝑑𝑑𝑟𝑔𝑖Φ𝑖→∑︁\n𝑖∫\n𝑑𝑑𝑟𝑏𝑑−𝑥𝑖𝑔𝑖Φ𝑖,\n1\n2∑︁\n𝑖,𝑗∫\n𝑑𝑑𝑟𝑖𝑑𝑑𝑟𝑗𝑔𝑖𝑔𝑗Φ𝑖Φ𝑗→𝑆𝑑\n2(𝑏−1)∑︁\n𝑖,𝑗∫\n𝑑𝑑𝑟𝑖𝑔𝑖𝑔𝑗𝐶𝑖𝑗𝑘Φ𝑘,\nwhere𝑆𝑑is the surface area of a (𝑑−1)-dimensional sphere, with 𝑆2=2𝜋in\ntwo dimensions. In the analysis of the third term, we integrate over the region\n𝑎 <|𝑟𝑖−𝑟𝑗|< 𝑏𝑎, thereby accounting for the short-range physics. Here, 𝑎\nrepresents the lattice spacing at the current scale. We define 𝑔𝑘such that𝑎is\nnormalizedtounity. Uponapplyinganinfinitesimalscaletransformation 𝑏≈1,the\neffective coupling 𝑔𝑘evolves to𝑔𝑘(𝑏), given by\n𝑔𝑘(𝑏)=𝑏𝑑−𝑥𝑘𝑔𝑘−𝑆𝑑\n2(𝑏−1)∑︁\n𝑖,𝑗𝐶𝑖𝑗𝑘𝑔𝑖𝑔𝑗,\nConsequently, the RG equation up to 1-loop expansion becomes:\n𝑑𝑔𝑘\n𝑑𝑙=𝑏𝑑𝑔𝑘\n𝑑𝑏\n=(𝑑−𝑥𝑘)𝑔𝑘−𝑆𝑑\n2∑︁\n𝑖,𝑗𝐶𝑖𝑗𝑘𝑔𝑖𝑔𝑗, (1.27)\n15To this order, the RG equation (beta function) is universally determined by the\nscaling dimension 𝑥𝑘and the OPE coefficients 𝐶𝑖𝑗𝑘.\nRevisiting the justification for concentrating solely on 𝛿𝐾in Eq. (1.12), it is\nessential to understand the dynamics of operators near criticality, as defined by the\nfixed-point CFT. The Ising CFT, although potentially allowing for infinite kinds of\nperturbations,hasaspecificcriterionforwhichperturbationsaresignificantduring\nscaletransformations. AccordingtoEq.(1.27),therunningcouplingconstantsscale\nas\n𝑔𝑘(𝐿)∝𝐿𝑑−𝑥𝑘,\nand thereby only operators with scaling dimensions 𝑥𝑘< 𝑑remain influential\nthrough these transformations. Such operators are termed “relevant operators.”\nTherationalebehindfocusingonalimitednumberofparametersinRGanalysisis\nrootedinthefactthattypically,onlyafewrelevantoperatorsexistinthefixed-point\nCFT.InthecaseoftheIsingCFT,forinstance,theoperator 𝜖istheprimaryrelevant\noperatorthatpersists 6,whichcorroboratesthevalidityofKadanoff’sapproach. This\nfocus on a few relevant parameters simplifies the RG analysis while still capturing\nthe critical behavior of the system.\nConversely, operators with scaling dimensions 𝑥𝑘> 𝑑, known as “irrelevant op-\nerators,\" tend to diminish and become negligible throughout scale transformations.\nThese operators do not significantly influence the macroscopic properties of the\nsystemandthereforedonotneedtobeconsideredintheRGanalysis. Thisselective\napproach,focusingonlyonrelevantoperatorslike 𝜖intheIsingCFT,iswhatmakes\ncritical phenomena universal across diverse systems.\nTosummarizethissection,theuniversalityobservedincriticalphenomenacanbe\ntracedbacktothesharedRGfixedpointsamongdifferentsystems. TheseRGfixed\npoints,conceptualizedas’terminalstations’inthescaletransformationprocess,are\ncharacterizedbyalimitedsetofparameterscorrespondingtorelevantoperators. In\ncontrast, other microscopic details, which are associated with irrelevant operators,\ndiminish and lose significance through the coarse-graining process.\nThisparadigmhighlightsthefundamentalprinciplethat,atthemacroscopiclevel,\nthe critical behavior of a system is governed not by the myriad of its microscopic\ndetails, but by a select few relevant parameters. These parameters, represented\n6𝜖and𝜎respectively corresponds to the shift of temperature and applying a uniform magnetic\nfield. Under the Z2spin flip symmetry, only 𝜖is allowed.\n16by the relevant operators at the RG fixed point, dictate the universal aspects of the\nsystem’scriticalbehavior. Asaresult,systemswithdifferentmicroscopicstructures\ncanexhibitthesamemacroscopiccriticalphenomena,providedtheyconvergetothe\nsameRGfixedpoint. Thisconvergenceiswhatunderliestheuniversalityofcritical\nphenomena, emphasizing the profound impact of scale and relevant operators in\ndetermining the nature of phase transitions and critical behavior.\nMovingtothenextsection,ourfocusshiftstothemethodologiesforextractinguni-\nversal information from critical lattice models. A particularly promising approach\nis the tensor network formalism. This method can be seen as a generalization of\nthe transfer matrix formalism, which we previously discussed in the context of the\none-dimensional Ising model. When combined with recent advances in computa-\ntional techniques, tensor network formalism enables a more sophisticated form of\nreal-space RG analysis.\nThis enhanced RG approach transcends Kadanoff’s methodology by its ability\nto retain multiple parameters throughout the scale transformation process. This\ncharacteristic aligns more closely with Wilson’s original vision for RG theory,\nprovidingamorecomprehensiveandfaithfulrepresentationofscaletransformations\nin physical systems.\nIn the upcoming section, we will review the fundamental concepts underlying\ntensor network-based RG. This will set the stage for the main chapter, where we\npropose a novel approach to calculate the RG flow using tensor networks. This\napproach not only leverages the strengths of tensor network formalism but also\naddressessomeofthelimitationsofpreviousmethodologies,offeringamorerobust\nand accurate tool for analyzing critical behavior in lattice models.\n1.2 Review on tensor network renormalization\nWhat is a tensor?\nBefore delving into the intricacies of our study, it is essential to establish a fun-\ndamental understanding of what a tensor is. A tensor can be seen as a generalized\nformofvectorsandmatrices. Toillustrate,considerthefollowingexamples,which\n17are referred to as one-leg and two-leg tensors, respectively:\n𝑉=©\n«1\n2\n3ª®®\n¬,\n𝑀= \n𝑎00𝑎01\n𝑎10𝑎11.!\nFor clarity in this thesis, these tensors are denoted as 𝑉𝑖and𝑀𝑖𝑗, indicating their\nrespective number of legs. The indices run as 𝑖=0,1,2for𝑉𝑖and(𝑖,𝑗)=(0,0),\n(0,1),(1,0), and(1,1)for𝑀𝑖𝑗. This notation allows us to interpret the values as\n𝑉𝑖=𝑖+1and𝑀𝑖𝑗=𝑎𝑖𝑗. Thedimensionassociatedwitheachindexisknownasthe\nbond dimension 𝐷. Accordingly, we denote 𝐷=3for𝑉𝑖and𝐷=2for𝑀𝑖𝑗.\nAnother example is the anti-symmetric tensor 𝜖𝑖𝑗𝑘, defined as:\n𝜖𝑖𝑗𝑘= \n1for(𝑖,𝑗,𝑘)=(0,1,2),(1,2,0),(2,0,1),\n−1for(𝑖,𝑗,𝑘)=(2,1,0),(1,0,2),(0,2,1),\n0otherwise.\nThistensorrepresentsa3-legtensorwithabonddimensionof 𝐷=3. Theconcept\nof tensors, as previously introduced, extends to structures with any number of legs,\nknown as𝑛-leg tensors. These multi-dimensional tensors play a pivotal role in\nthe discussions that follow in this thesis. To aid in the comprehension of tensors,\nespecially in more complex scenarios, we often employ a graphical representation.\nInthisvisualdepiction,tensorsareillustratedascircleswithacorrespondingnumber\nof legs emanating from them.\nFor instance, the tensors 𝑉,𝑀, and𝜖, previously defined, would be represented\ngraphically as follows:\n.\nSimilarly, the multiplication of tensors is represented as a contraction of legs.\n.\nThese graphical representations provide an intuitive way to visualize the connec-\ntions between tensors, particularly when dealing with complex tensor networks or\n18operations involving multiple tensors. By utilizing these diagrams, we can more\neasily conceptualize the multi-dimensional relationships and transformations that\ntensors undergo in our analyses.\nTensor network renormalization\nThetensornetworkisanumericaltechniqueusedtorepresentthepartitionfunction\nof statistical models. The partition functions of two-dimensional statistical models\nwith a system size of 𝐿can be expressed through the contraction of 𝐿2tensors 7.\nEach tensor represents a local Boltzmann weight, and its dimensions correspond\nto physical degrees of freedom. For instance, the local tensor of the Ising model\non the square lattice is a 𝐷=2four-leg tensor 𝑇(1)\n𝑖𝑗𝑘𝑙=𝑒𝛽(𝑠𝑖𝑠𝑗+𝑠𝑗𝑠𝑘+𝑠𝑘𝑠𝑙+𝑠𝑙𝑠𝑖), where\n𝑠𝛼=2𝛼−1. The tensor network representation often provides an efficient method\nfor simulating complex systems.\nHowever, the exact contraction of 𝐿2tensors is generally impracticable for larger\nsystemsizesduetotheconstraintsimposedbythehigh-dimensionalHilbertspace 8.\nTNR aims to circumvent this issue by utilizing the principles of renormalization\ngroup theory. During each step of the RG process, 𝑇(𝑛)is coarse-grained to 𝑇(𝑛+1)\nviaaseriesofdecompositionsandrecombinations,asillustratedinFig.1.4. Starting\nfrom the local tensor 𝑇(1), we can simulate a system size of 𝐿=√\n2𝑛after𝑛RG\nsteps. Thecoarse-grainingprocessinTNRinvolvesnumericaltruncation,reducing\nthenumberofdegreesoffreedomwhilepreservingessentialphysics. Consequently,\nTNR facilitates efficient numerical simulation of complex systems.\nTensor Renormalization Group\nLevin and Nave were the pioneers in applying the technique of singular value\ndecomposition (SVD) to the decomposition of tensor networks [6]. SVD offers\na straightforward yet effective method for the tensor decomposition of a four-leg\ntensor𝑇𝑖𝑗𝑘𝑙.\n7Tensors can be understood as a generalization of vectors and matrices, extending into higher\ndimensions and complexities. A tensor characterized by 𝑛indices is referred to as an 𝑛-leg tensor.\nIn this framework, vectors, and matrices are special cases of tensors: a vector is a 1-leg tensor,\npossessing a single index, while a matrix is a 2-leg tensor, defined by two indices.\n8The one-dimensional Ising model was the simplest case, where we could perform exact con-\ntractions\n19Figure1.4: Aschematicpictureofthetensornetworkrenormalization. Theeffective\nlocal Boltzmann weight at 𝑛-th RG step𝑇(𝑛)is decomposed into the two three-leg\ntensors and recombined as 𝑇(𝑛+1). The effective system size enlarges by√\n2each\nRG step.\nThe decomposition of 𝑇𝑖𝑗𝑘𝑙using SVD can be represented as follows:\n𝑇𝑖𝑗𝑘𝑙=𝑑2∑︁\n𝑚,𝑛=1𝑈𝑖𝑗𝑚Σ𝑚𝑛𝑉†\n𝑛𝑘𝑙, (1.28)\nwhere𝑈𝑖𝑗𝑚and𝑉𝑛𝑘𝑙are unitary matrices, and Σ𝑚𝑛=𝛿𝑚,𝑛𝑠𝑚is a diagonal matrix.\nThediagonalelementsof Σarethesingularvaluesthatisnon-negativerealnumbers,\nandSVDeffectivelygeneralizestheconceptofmatrixdiagonalizationtorectangular\nmatrices. In the context of tensor networks, when the bond dimension of 𝑇𝑖𝑗𝑘𝑙is\n𝑑, the summation in Eq. (1.28) runs over 𝑑2terms. This new index 𝑚represents a\nnew bond dimension, which necessitates a truncation process to prevent the bond\ndimensionfrombecomingexcessivelylargeduringsuccessivecoarse-grainingsteps.\nTruncation is thus a crucial aspect of maintaining computational efficiency and\nfeasibility in TRG.\n20Now, we want to truncate the index 𝑚to minimize the Hilbert-Schmidt norm of\nthe difference between the original and truncated tensors as follows:\n(1.29)\nWe truncate the singular values when 𝑑2is larger than the desired bond dimension\n𝐷. LetΣ𝐷\n𝑚𝑛bethetruncatedmatrixthatkeepsonly 𝐷leadingsingularvalues. Then,\nEq. (1.29) is\n(1.30)\n=Tr\u0002\n𝑈†𝑈(Σ2+(Σ𝐷)2−2ΣΣ𝐷)𝑉†𝑉\u0003\n=𝑑2∑︁\n𝑚=1𝑠2\n𝑚+𝐷∑︁\n𝑚=1𝑠2\n𝑚−2𝐷∑︁\n𝑚=1𝑠2\n𝑚\n=𝑑2∑︁\n𝑚=𝐷+1𝑠2\n𝑚 (1.31)\nIn statistical mechanics systems, the singular values 𝑠𝑚typically exhibit an expo-\nnential decay as a function of 𝑚. This characteristic decay pattern implies that,\nas𝑚increases, the singular values become progressively smaller. Consequently,\nwhen evaluating the sum in Eq. (1.31), this exponential decay of 𝑠𝑚ensures that\nthe cumulative sum remains extremely small, particularly when a sufficiently large\nbond dimension 𝐷is considered. Therefore, for practical computations, a large 𝐷\neffectivelycapturesthesignificantcontributionsofthesum,whilethecontributions\nfrom higher values of 𝑚become negligibly small due to this exponential decay.\nFigure 1.5 presents a typical example of singular value distributions in TRG for\nthe two-dimensional classical Ising model. The blue and green lines represent the\nsingular values 𝑠𝑚for the low and high-temperature phases, respectively. In this\ncase, the bond dimension is set to ten, resulting in a total of one hundred singular\nvalues. However, it is observed that for 𝑚 > 10, the values of 𝑠𝑚drop below 10−6,\naffirming the efficacy and validity of this truncation method in these phases.\nConversely,thecriticalphase,depictedbytheorangeline,exhibitsaslowerdecay\nin𝑠𝑚. At the same bond dimension of 𝐷=10, the singular values are still around\n21Figure 1.5: An example of singular values of the four-leg tensor 𝑇𝑖𝑗𝑘𝑙. The blue,\norange, and green lines show the decay of 𝑠𝑚for the classical Ising model at the\nlow,critical,andhigh-temperatureregimes,respectivelyat 𝑑=𝐷=10aftersixRG\nsteps.\n10−2, highlighting a marked difference from the non-critical phases. This slower\ndecayatcriticalityunderscoresacrucialchallengeinTRGcalculations: thedifficulty\nin effectively capturing critical phenomena. Truncation of tensors at criticality can\nlead to systems with finite correlations, potentially introducing significant errors.\nGiven the importance of accurately estimating errors, especially in the context of\ncritical systems, we will delve into a more detailed discussion on this topic in a\nlatersection. ThisanalysisiscrucialforbothunderstandingthelimitationsofTRG\nat criticality and for developing strategies to mitigate error propagation in practical\ncalculations.\nGiventhevalidityoftheSVDtruncationsinstatisticalmechanics,LevinandNave\nproposed the following algorithm:\n22Algorithm 1 Algorithm for TRG\nInput:Initial tensor 𝑇(0)representing local Boltzmann weights\nOutput: Renormalized tensor 𝑇(𝑛)\nTRG 1-step: First, decompose the four-leg tensor 𝑇with bond dimension 𝑑as\nfollows:\n𝑇𝑖𝑗𝑘𝑙=𝑑2∑︁\n𝑚=1𝑈𝑖𝑗𝑚Σ𝑚𝑛𝑉†\n𝑛𝑘𝑙,\n𝑆1\n𝑖𝑗𝑛=𝑚𝑖𝑛(𝑑2,𝐷)∑︁\n𝑚=1𝑈𝑖𝑗𝑚√︁\nΣ𝑚𝑛\n𝑆2\n𝑛𝑘𝑙=𝑚𝑖𝑛(𝑑2,𝐷)∑︁\n𝑚=1√︁\nΣ𝑚𝑛𝑉†\n𝑛𝑘𝑙\n√︁\nΣ𝑚𝑛=𝛿𝑚,𝑛√𝑠𝑚\nRepeatthesameproceduretomake 𝑆3and𝑆4bydoingSVDwithapairofindices\n{𝑗𝑘}and{𝑙𝑖}. Then, contract 𝑆1∼𝑆4inside the red dotted square and rotate by\n45degrees as below:\nfor𝑖=1to𝑛do\nRepeat TRG 1-step\nend for\nreturn𝑇(𝑛)\nThe practical implementation in Python is presented in my GitHub repository\n23Figure 1.6: The CDL tensors after TRG. The local loop, marked by a red square,\npersists after the TRG steps, indicating that ultraviolet information remains even\nafter extensive coarse-graining.\n(https://github.com/dartsushi/Loop-TNR_RGflow/tree/main/TRG_tutorial ).\nApplications to a honeycomb lattice are also discussed in the original paper [6].\nLimitation of TRG\nDespite the successes of TRG in various applications, it is not without its limita-\ntions. AsignificantshortcomingofTRG,asnotedinreferences[6,7],isitsinability\nto accurately reproduce the fixed-points of disorder phases. Levin attributed this\nlimitation to the inherent nature of SVD-based TRG, which tends to produce un-\nphysical fixed-point tensors, referred to as corner double line (CDL) tensors. The\nstructure of CDL tensors is illustrated as follows:\n(1.32)\n24InSVD-typeTRG,thereisanotablechallengeineliminating localentanglement,\nas depicted by the red square in Fig. 1.6. This figure demonstrates that ultraviolet\ninformation persists throughout successive coarse-graining steps. Consequently,\nthesedegreesoffreedomconsumevaluablebonddimensions,leadingtosuboptimal\napproximations after multiple RG steps.\nThis inherent limitation of TRG underscores the need for more sophisticated\ncoarse-graining algorithms that transcend the local tensor approximation presented\nin Eq. (1.29). Such advanced algorithms are encompassed under the umbrella of\nTNR, which aims to address these specific challenges and improve the accuracy of\ncoarse-graining in complex systems.\nTensor network renormalization\nThe field has witnessed the development of numerous algorithms aimed at over-\ncoming the CDL problem, a known limitation in TRG applications [8–13]. While\nthese algorithms vary in their technical details, they generally embrace two core\nprinciples: employing a larger unit-cell for optimization processes and effectively\nfiltering out local entanglement. Let us focus on Loop-TNR [10], known to be one\nof the best ones for two-dimensional TNR.\nOptimization with a larger unit-cell\nLoop-TNR aims to perform the approximation of the following:\n. (1.33)\nContrastingwiththeapproachdescribedinEq.(1.29),TNRfocusesonatwo-by-two\nunitcellcomposedoftwodistincttensors. Inthisprocess,an8-legtensor(asshown\non the left side) is approximated through the contraction of eight 3-leg tensors,\ndenoted as𝑆1∼𝑆8(as depicted on the right side).\nThismethodologyiscentraltotheconceptofTNR:ratherthanoptimizingindivid-\nual tensors, the focus is on optimizing contracted tensor networks. This approach\nallows for a more nuanced and effective handling of complex tensor structures,\nparticularly in addressing the limitations of traditional SVD methods in TRG. By\n25optimizing the entire network of tensors, TNR provides a more robust framework\nfor accurately capturing the intricate interactions within these systems.\nIn TRG,𝑆1∼𝑆8are constructed through SVD as explained in the algorithm. As\nthosetensorsfromSVDareagoodenoughapproximationoflocaltensors,weadopt\nthemasinitialtensorsofoptimizations. LettheleftandrightsidesofEq.(1.33)be\n|Ψ𝐴⟩and|Ψ𝐵⟩. Then, the cost function is rewritten in the following forms.\n||Ψ𝐴⟩−|Ψ𝐵⟩|2=⟨Ψ𝐴|Ψ𝐴⟩+⟨Ψ𝐵|Ψ𝐵⟩−⟨Ψ𝐴|Ψ𝐵⟩−⟨Ψ𝐵|Ψ𝐴⟩, (1.34)\n.(1.35)\nNow,wewanttooptimize 𝑆1∼𝑆8tominimizethecostfunction. Forconvenience,\nwe define𝐶,𝑁𝑖,𝑊𝑖, and𝑊†\n𝑖as followings:\n.\nThis allows to rewrite Eq. (1.35) as\n𝑓({𝑆𝑖})=||Ψ𝐴⟩−|Ψ𝐵⟩|2\n=𝐶+(𝑆𝑖)†𝑁𝑖𝑆𝑖−𝑊†\n𝑖𝑆𝑖−(𝑆𝑖)†𝑊𝑖. (1.36)\nThis function is quadratic if we fix every tensor except for 𝑆𝑖. In this case, the\nminimum can be found by solving𝛿𝑓({𝑆𝑖})\n𝛿𝑆𝑖=0. It is straightforward to check that it\nis equivalent to solving the following linear equation:\n𝑁𝑖𝑆𝑖=𝑊𝑖. (1.37)\nEntanglement filtering\n26AnotherimportantingredientofTNRisentanglementfiltering. Thisisaprocedure\nto discard the local degrees of freedom under the existence of CDL tensors. When\nthetensorhastheformoftheCDLtensorsshowninEq.(1.32),thecontractedfour\ntensors can be expressed as the following:\n,\nwhere(𝑇1,𝑇2,𝑇3,𝑇4)=(𝑇𝐴,𝑇𝐵,𝑇𝐴,𝑇𝐵), and the red loop corresponds to the local\nloop in Fig. 1.6. In scenarios where the red loop encompasses 𝑛dimensions, each\ntensorwithinthisredloopisassociatedwitharank- 𝑛diagonalmatrix,asillustrated\nin the following diagram:\n. (1.38)\nHere,each𝜆𝑖isarrangedindescendingorderbasedonabsolutevalues. Theprimary\nobjectiveinentanglementfilteringistoeffectivelycompressthismatrixtoarank-1\nconfiguration. Thiscompressionisachievedbyconstructingaprojectorbetween 𝑇𝑖\nand𝑇𝑖+1thattargetsthesubspacecorrespondingto 𝜆1,thelargestsingularvalue. To\ndothis,weuseaQRdecomposition. Considertheprocedureofinsertingaprojector\nbetween tensors 𝑇4and𝑇1in a TNR setup, where we denote 𝑇𝑖+4=𝑇𝑖for cyclic\nconsistency. Thefirststepinvolvesplacingarank- 𝑑identitymatrix,denotedas 𝐿[1]\n1\n, to the left of 𝑇1. Subsequently, we apply QR decomposition to the tensor product\nof𝐿[1]\n1and𝑇1, resulting in:\n𝐿[1]\n1𝑇1=˜𝑇1𝐿[2]\n1, (1.39)\nwhere ˜𝑇1isanorthogonalmatrixand 𝐿[2]\n1isanuppertriangularmatrix. Thenextstep\ninvolves normalizing 𝐿[2]\n1appropriately and repeating a similar QR decomposition\nprocess with 𝐿[2]\n1and𝑇2, and then proceeding with 𝐿[3]\n1and𝑇3. This iterative\nprocess is continued until convergence is achieved, resulting in the final projector\n𝐿[∞]\n1(The convergence is checked when 𝐿1comes back to between 𝑇4and𝑇1.\nDuring this process, 𝐿accumulates the matrix in Eq. (1.38) to end up having\nlim\n𝑚→∞©\n«𝜆𝑚\n10··· 0\n0𝜆𝑚\n2··· ···\n··· ··· ··· 0\n0··· 0𝜆𝑚\n𝑛ª®®®®®\n¬∝©\n«1 0··· 0\n0 0··· ···\n··· ··· ··· 0\n0··· 0 0ª®®®®®\n¬.\n27Werepeatthesamethingtotheleftstartingfrom 𝑅[1]\n4and𝑇4toobtain𝑅[∞]\n4. Finally,\nwe obtain the projectors using SVD as follows:\n𝐿[∞]\n1𝑅[∞]\n4=𝑈41Λ41𝑉†\n41,\n𝑃4𝑅=𝑅∞\n4𝑉411√Λ41, (1.40)\n𝑃1𝐿=1√Λ41𝑈†\n41𝐿∞\n1, (1.41)\nObtaining all projectors in hand, we redefine 𝑇𝐴/𝐵by contracting four projectors\naround as below:\n.\nFormoredetails,readersshallconsulttheoriginalpaper[10]. However,itisimpor-\ntanttonotethatwecanreducetheCDLloopstructurethroughthisprocedure. Loop-\nTNR’s algorithm is summarized in Algorithm. 2. The practical implementation\nin Python is in my GitHub repository( https://github.com/dartsushi/Loop-\nTNR_RGflow ).\n28Algorithm 2 Algorithm for Loop-TNR\nInput:Initial tensor 𝑇(0)\n𝐴/𝐵representing local Boltzmann weights\nOutput: Renormalized tensor 𝑇(𝑛)\n𝐴/𝐵\nLoop-TNR 1-step:\n1. Entanglement filtering\n2. Decomposition of 𝑇𝐴/𝐵into𝑆1∼𝑆8using SVD.\n3. Optimize 𝑆𝑖using from 1 to 8.\n𝑁𝑖𝑇𝑖=𝑊𝑖. (1.42)\n4. Repeat 3 until you reach the desired accuracy.\n5. Combine(𝑆1,𝑆4,𝑆5,𝑆8)and(𝑆2,𝑆3,𝑆6,𝑆7)to make new 𝑇𝐴/𝐵respectively.\nfor𝑖=1to𝑛do\nRepeat Loop-TNR 1-step\nend for\nreturn𝑇(𝑛)\n𝐴/𝐵\nLimitation of TNR\nWhile TNR effectively removes CDL tensors, another significant limitation arises\nfrom computational constraints, often termed finite bond dimension effects. For\nexample, the computational cost of Loop-TNR scales as 𝑂(𝐷6), limiting practical\ncomputationsonatypicaldesktopcomputertobonddimensionsuptoapproximately\n𝐷≈40. Although TNR achieves exactness in the limit of 𝐷=∞, the necessity\nof using a finite bond dimension inevitably introduces numerical errors. Prior to\nour research, the methodology for estimating these numerical errors was not well-\nestablished. Addressing this gap, we will delve into the strategies for estimating\nnumerical errors in the context of tensor-network based methodologies in Chapter\n2.\n29CFT data from TRG/TNR\nUptothispoint,wehavediscussedtheimplementationofTRGandTNRmethods\nin the context of real-space RG analysis. A key aspect of these methods is their\nability to reveal the properties of fixed points in critical lattice models. In practice,\nwhenTRGandTNRareappliedtosimulatecriticallatticemodels,therenormalized\ntensor𝑇(𝑛)is observed to converge rapidly to a specific tensor, denoted as 𝑇∗. This\nconvergence behavior is indicative of the system approaching a fixed point in its\nparameter space.\nThe tensor𝑇∗, aptly referred to as a fixed-point tensor, is believed to encapsulate\nthe properties associated with the fixed-point of the system. Notably, Gu and Wen\nhaveproposedmethodologiestocalculatethescalingdimensionsdirectlyfromthis\nfixed-point tensor [7]. In our review, we will adopt a slightly different notation to\nfacilitate a more seamless integration with the main content of our discussion 9.\nThepivotaloutcomeofGuandWen’sresearchconcernstheanalysisofthefixed-\npointtensorforasquarelattice,whichisafour-leggedtensormirroringthestructure\nof the original lattice. A critical step in their methodology involves contracting the\nlegsalongtheverticalaxisofthistensor. Thiscontractionprocessleadstoamatrix,\nfrom which the scaling dimensions can be inferred based on the eigenvalues. The\nprocess can be represented as:\n∑︁\n𝑛𝑇𝛼𝑛𝛽𝑛=\n,(1.43)\nIn this formulation, the largest eigenvalue is normalized to one, aligning with the\nscaling dimension of the identity operator, 𝑥𝐼=0. Subsequently, the scaling\ndimensions associated with other operators can be derived from the ratio of the\neigenvalues, denoted as 𝜆𝑛, using the equation:\n𝑥𝑛=1\n2𝜋ln𝜆0\n𝜆𝑛. (1.44)\nHere, the eigenvalues 𝜆𝑛are arranged in descending order. This methodology\noffers a robust means to extract scaling dimensions from the fixed-point tensor,\nproviding a valuable tool for analyzing the critical properties of the system. By\n9While our explanation is fundamentally equivalent to that of Gu and Wen, we approach the\ntopic using the energy basis, in contrast to their use of the character basis of CFTs.\n30utilizingtheeigenvalueratios,onecaneffectivelydeterminethescalingdimensions\ncorresponding to various operators, thereby gaining deeper insights into the nature\nof the critical points in the lattice model.\nTounderstandwhyscalingdimensionsarereflectedintheeigenvaluesofrenormal-\nizedtensors,letusrevisitthenatureofrenormalizedtensorsinTRG/TNRschemes,\nas discussed in the previous section. In these schemes, each RG step corresponds\nto a scale transformation with a factor of 𝑏=√\n2. Consequently, after 𝑛steps of\ncoarse-graining, the renormalized tensor effectively represents the tensor networks\nof a system whose size has been scaled to 𝐿=√\n2𝑛. This process can be visualized\nschematically as shown below:\n(1.45)\nInthis𝐿×𝐿tensornetwork,contractingtheverticalindicestransformsthenetwork\nintoacylindricalshape. Theeigenvaluesoftheresultantmatrixareessentiallythe 𝐿\nrepetitionsofthecolumn-to-columntransfermatrix,whichIwillhenceforthreferto\nsimplyasthe’transfermatrix’. Inthecontextofclassical-quantumcorrespondence,\na single layer of the transfer matrix equates to a translation in imaginary time. In\none-dimensionalquantumsystems,thistranslationcorrespondstotheHamiltonian.\nConsequently, there exists a relationship between the eigenvalues of the transfer\nmatrix and the spectrum of the one-dimensional Hamiltonian, 𝐸𝑛(𝐿), as follows:\n𝜆𝑛=𝑒−𝐿𝐸𝑛(𝐿), (1.46)\nwhere the𝐸𝑛values are arranged in ascending order, such that 𝐸0represents the\ngroundstate. Thisperspectiveenablesthesubsequentchapterstoanalyzethetransfer\nmatrix spectrum within the framework of Hamiltonian formalism.\nAt criticality, the system’s conformal invariance enables the computation of the\nenergy spectrum, 𝐸𝑛(𝐿). Consider an infinitesimal transformation defined as 𝑟′\n𝜇=\n𝑟𝜇+𝜖𝜇. Within this framework, the variation of the action can be articulated as:\n𝛿𝑆=−∫𝑑2𝑟\n2𝜋𝑇𝜇𝜈(𝑟)𝜕𝜇𝜖𝜈(𝑟), (1.47)\n31where𝑇𝜇𝜈denotes the stress tensor. In two-dimensional systems at criticality, con-\nformal invariance is a key characteristic. This invariance includes transformations\nsuch as translations, rotations, dilatations, and special conformal transformations.\nAsaresult,conformaltransformationsinthesesystemscanberepresentedthrough\nanalyticfunctionsonthecomplexplane,usingthetransformation 𝑧→𝑤(𝑧),where\n𝑧=𝑥+𝑖𝑦and¯𝑧=𝑥−𝑖𝑦.\nThegeneratorsoftheseconformaltransformationsareexpressedas 𝑇(𝑧)=1\n4(𝑇𝑥𝑥−\n𝑇𝑦𝑦−2𝑖𝑇𝑥𝑦)and ¯𝑇(¯𝑧)=1\n4(𝑇𝑥𝑥−𝑇𝑦𝑦+2𝑖𝑇𝑥𝑦). The variation of an operator 𝐴\nunderconformaltransformationsiseffectivelyencapsulatedbytheWard-Takahashi\nidentity:\n𝛿𝜖𝐴(𝑧,¯𝑧)=∮\n𝐶𝑧𝑑𝜁\n2𝜋𝑖𝜖(𝜁)𝑇(𝜁)𝐴(𝑧,¯𝑧). (1.48)\nThis equation implies that the variation of 𝐴(𝑧,¯𝑧)can be computed by applying\nthe product of 𝜖and the stress tensor 𝑇to𝐴, followed by performing a contour\nintegration around the point 𝑧. To facilitate this calculation, it is useful to express\nthe stress tensor through Laurent expansions:\n𝑇(𝑧)=∑︁\n𝑛∈Z𝑧−𝑛−2𝐿𝑛,¯𝑇(¯𝑧)=∑︁\n𝑛∈Z¯𝑧−𝑛−2¯𝐿𝑛.\nHere,𝐿𝑛and¯𝐿𝑛serve as the generators of conformal transformations and obey the\nVirasoro algebra, a cornerstone of CFT. The Virasoro algebra is given by:\n[𝐿𝑚,𝐿𝑛]=(𝑚−𝑛)𝐿𝑚+𝑛+𝑐\n12(𝑚3−𝑚)𝛿𝑚+𝑛,0, (1.49)\nwhere𝑐representsthecentralcharge,afundamentalcharacteristicoftheCFT.The\ncentral charge is a critical parameter that helps classify the universality class of\nthe theory. It provides an intuitive measure of the number of bosonic excitations\npresent: for instance, a theory with 𝑛decoupled free bosons has a central charge of\n𝑐=𝑛, while a theory with 𝑛decoupled fermions has 𝑐=𝑛/2.\nIn this framework using the stress tensor, the dilatation ˆ𝐷is defined as following:\nˆ𝐷=∮𝑑𝑧\n2𝜋𝑖𝑧𝑇(𝑧)+∮𝑑¯𝑧\n2𝜋𝑖¯𝑧𝑇(¯𝑧), (1.50)\n=𝐿0+¯𝐿0. (1.51)\nThe eigenvalues of 𝐿0and ¯𝐿0are conformal weights denoted as (ℎ,¯ℎ), and the\nscaling dimension 𝑥=ℎ+¯ℎbecomes indeed the eigenvalue of the dilatation.\n32Figure 1.7: The conformal mapping from a plane to a cylinder. The black and red\ndotted circles on the plane correspond to different time slices on the cylinder. As\na result, the scale translation indicated by the red arrow on the 𝑧-plane transforms\ninto a translation in imaginary time on the 𝑤-axis.\nThe intriguing aspect of dilatation in two-dimensional conformal field theory is\nthatitsgenerator, ˆ𝐷,canbereinterpretedasthegeneratoroftranslationinimaginary\ntime, denoted as ˆ𝐻𝑃, in a (1+1)-dimensional context. This relationship becomes\nevidentwhenconsideringtheconformalmapping 𝑤=𝐿\n2𝜋ln𝑧,asdepictedinFig.1.7.\nIn this mapping, the original 𝑧-plane is transformed onto a cylinder with circum-\nference𝐿. Thecylindercanbeviewedasaquantumsystemwithsize 𝐿andperiodic\nboundary conditions in one dimension. The translation in scale, represented by the\nred arrow on the left panel of Fig. 1.7, corresponds to a translation in time on the\ncylinder. Sincetheenergydensityinaquantumsystemisthediagonal 𝑡component\nofthestresstensor,itfollowsthattheenergyofafinitequantumsystemcanbecal-\nculated using ˆ𝐷. Conversely, if we know how the stress tensor 𝑇transforms under\nconformaltransformations,wecandeducethesystem’senergy. Thistransformation\nis known and is given by:\n𝑇(𝑧)=\u0012𝑑𝑤\n𝑑𝑧\u00132\n˜𝑇(𝑤)+𝑐\n12{𝑤,𝑧}, (1.52)\nwhere{𝑤,𝑧}is the Schwarzian derivative.\nThe Hamiltonian on the cylinder is then expressed as:\n𝐻𝑃=∫𝐿\n0𝑑𝑥\n2𝜋(𝑇cyl(𝑤)+𝑇cyl(𝑤))\n=2𝜋\n𝐿(𝐿0+𝐿0−𝑐\n12), (1.53)\n33yieldingtheenergy 𝐸𝑛=2𝜋\n𝐿(𝑥𝑛−𝑐\n12)[14,15]. Therefore,thescalingdimension/the\nenergy spectrum can be calculated from the energy spectrum of the transfer matrix\nin𝑦direction𝜆𝑖=𝑒−2𝜋(𝑥𝑖−𝑐\n12), being consistent with Eq. (1.46).\nIn lattice models, it is crucial to consider the contribution of bulk energy when\nanalyzingtheenergyspectrum. Thisconsiderationleadstothefollowingexpressions\nfor the energy spectrum:\n𝐸𝑛−𝐸0=2𝜋\n𝐿𝑥𝑛, (1.54)\n𝐸0=𝜖0𝐿−𝜋𝑐\n6𝐿. (1.55)\nHowever, a challenge arises in determining the central charge 𝑐due to the lack\nof a sufficient number of equations to separate the contribution of the bulk energy.\nTo address this, one can utilize the partition function from the previous RG step,\ndenoted as𝑍(𝑛−1):\n𝑍(𝑛−1)=Tr𝑥𝑖exp\u0010\n−2𝜋\u0010\n𝑥𝑖−𝑐\n12\u0011\n−𝜖0𝑏2𝑛−2\u0011\n. (1.56)\nFor the fixed-point tensor, it is reasonable to assume that both 𝑐and𝜖0remain\nconstant. Under this assumption, the central charge can be determined as follows:\n𝑐=6\n𝜋1\n𝑏2−1\u0010\n𝑏2ln𝜆(𝑛−1)\n0−ln𝜆(𝑛)\n0\u0011\n. (1.57)\nA widely used formula for calculating the central charge is given by [7]:\n𝑐=6\n𝜋\"\n𝑏2\n𝑏2−1\u0012\nln𝑍(𝑛−1)−ln𝑍(𝑛)\n𝑏2\u0013\n+ln𝜆(𝑛)\n0\n𝑍(𝑛)#\n. (1.58)\nIn the critical case, Eqs.(1.57) and (1.58) are equivalent since𝜆(𝑛)\n0\n𝑍(𝑛)=𝜆(𝑛−1)\n0\n𝑍(𝑛−1). How-\never,Eq.(1.58)maybecomeunstablewhenthesystemsizesurpassesthecorrelation\nlength. Inthisthesis,wehavecalculatedtheeffectivecentralchargeusingEq.(1.58).\nThroughthismethodology,weareabletoextractthecentralcharge,akeyparam-\neter in conformal field theory that characterizes the universality class of the model.\nThis approach enables a deeper understanding of the critical properties of lattice\nmodels,particularlyinthecontextoftensornetworkrenormalizationandfinite-size\nscaling theory.\n34How to read CFT dictionaries: Character\nTodeterminetheapplicabilityofaspecificConformalFieldTheory(CFT)toyour\nmodels,oneeffectiveapproachistoutilizetheconceptofthe’character’,atoolthat\nallowsyoutodeciphertheenergyspectrumofthemodel. Bycomparingtheenergy\nspectrumofyourmodelwithknownresultsinCFTliterature,youcanascertainthe\nuniversality class of the model in question. This method offers a practical solution\nto the often esoteric nature of CFT literature, facilitating its application to specific\nmodels.\nIn the previous section, we discussed how the partition function, excluding bulk\nenergy contributions, is expressed as:\n𝑍(𝐿,𝐿)=Tr exph\n−2𝜋(𝐿0+¯𝐿0−𝑐\n12)i\n, (1.59)\nwheretheeigenvalueof (𝐿0+¯𝐿0)correspondstothescalingdimension 𝑥𝑖. Thekey\ntakeaway is that the trace in Eq. (1.59) is calculated using the transfer matrix basis,\nmeaning the eigenvalues of the transfer matrix in the 𝑦-direction are predicted to\nbe𝑒−2𝜋(𝑥𝑖−𝑐\n12). For the partition function of a rectangular shape, such as 𝑍(𝐿,2𝐿),\nthe eigenvalue becomes 𝑒−4𝜋(𝑥𝑖−𝑐\n12), reflecting the squaring of eigenvalues due to\nthe double transfer distance. Moreover, the spectrum of 𝑍(2𝐿,2𝐿)is identical to\nthat of𝑍(𝐿,𝐿), and𝑍(2𝐿,4𝐿)mirrors𝑍(𝐿,2𝐿). This uniformity results from the\nscale-invariance inherent in CFT, with the shape ratio𝐿𝑦\n𝐿𝑥being the critical factor.\nTherefore, the partition function is often represented in CFT literature as:\n𝑍(𝑞)=Tr𝑞𝐿0−𝑐\n24¯𝑞¯𝐿0−𝑐\n24, (1.60)\nwhere𝜏=𝑖𝐿𝑦\n𝐿𝑥,𝑞=𝑒2𝜋𝑖𝜏, and ¯𝑞=𝑒−2𝜋𝑖𝜏∗. Here,𝜏is known as the modular\nparameter. Defining this parameter allows for the generalization of the partition\nfunction concept to parallelograms, as illustrated below:\n35Figure1.8: ThescalingdimensionsofthecriticalIsingmodelthatareobtainedfrom\nthe transfer matrix spectrum of Loop-TNR. The blue dotted line are the theoretical\nvalues from the character.\nWithperiodicboundaryconditionsonbothedges,thissetupequatestothepartition\nfunction on a torus, a phenomenon termed modular invariance. This principle\nimposes significant constraints on CFT. In the context of transfer matrix spectra,\nconsideringthespectrumonaparallelogramisinsightful,asEq.(1.60)encapsulates\ninformation about the conformal spin. This arises from the phase acquired by the\noperator due to momentum when the real part of 𝜏is non-zero. As depicted in\nFig. 1.7,this shift onthe plane correspondsto the additionalphase acquired during\noperator rotation around the origin. Here, we specifically address the case where\n𝜏=𝑖. For a broader understanding encompassing more generic cases, readers are\nencouraged to consult comprehensive CFT literature [3].\n\"An essential aspect to understand is that Eq.(1.60) possesses a universal form,\nand its specifics are precisely known for certain universality classes. For instance,\nthe partition function of the Ising CFT is expressed as:\n𝑍(𝑞)=|𝜒0(𝑞)|2+\f\f\f𝜒1\n16(𝑞)\f\f\f2\n+\f\f\f𝜒1\n2(𝑞)\f\f\f2\n. (1.61)\nHere,𝜒0(𝑞),𝜒1\n16(𝑞), and𝜒1\n2(𝑞)correspond to the excitations of the 𝐼,𝜎, and𝜖\nfamilies, respectively, and are known as characters. These characters often con-\ntain known quantities, allowing for the prediction of low-lying energies or scaling\n36dimensions. For the Ising model, the characters are as follows:\n𝑞𝑐/24𝜒0(𝑞)=1+𝑞2+𝑞3+2𝑞4+2𝑞5+···, (1.62)\n𝑞𝑐/24𝜒1\n16(𝑞)=𝑞1/16(1+𝑞+𝑞2+2𝑞3+2𝑞4+3𝑞5+···),(1.63)\n𝑞𝑐/24𝜒1\n2(𝑞)=𝑞1/2(1+𝑞+𝑞2+𝑞3+2𝑞4+2𝑞5+···). (1.64)\nBysubstitutingEqs.(1.62-1.64)intoEq.(1.60)andcomparingitwithEq.(1.59),we\ncan identify low-lying scaling dimensions from the exponents of 𝑞. For the Ising\nmodel, we obtain 𝑍(𝑞)=𝑞−𝑐/12(1+𝑞1/8+𝑞1+2𝑞9/8+4𝑞2+···), leading to the\nscaling dimensions 𝑥0,𝑥1,𝑥2,𝑥3,𝑥4,𝑥5,···=0,1\n8,1,9\n8,9\n8,2,···. This aligns with\nthe numerical results of the critical Ising model, as demonstrated in Fig. 1.8.\nIn a similar vein, by comparing the character of a potential universality class\nfrom CFT literature with your numerical data, you can accurately determine the\nuniversality class of an unknown phase transition in lattice models.\n37C h a pte r 2\nFINITE-SIZE AND FINITE BOND DIMENSION EFFECTS OF\nTENSOR NETWORK RENORMALIZATION\nIn this chapter, we introduce a comprehensive procedure aimed at extracting the\nrunning coupling constants denoted as 𝑔𝑛(𝐿)of the underlying field theory for\nclassical statistical models on two-dimensional lattices. This approach synergizes\nTNR with the finite-size scaling principles of CFT. Our methodology extends Gu\nand Wen’s analysis, originally focused on the transfer matrix spectrum of critical\nsystems, to encompass off-critical systems.\nIn systems away from criticality, the spectral properties exhibit a departure from\nscale invariance during the RG steps, deviating from the universal values charac-\nteristic of CFT. We propose that these deviations are indicative of the RG flow.\nBy meticulously analyzing these deviations, we can calculate the running coupling\nconstants with extremely high precision at each scale. This process enables us to\ntrack the evolution of these coupling constants through successive scales, thereby\noffering a detailed visualization of the RG flow. To demonstrate the efficacy of our\napproach,weapplyittoclassicallatticemodelssuchastheIsingandthree-statePotts\nmodels. This concept is also extended to determine the transition with extremely\nhigh precision.\nFurthermore, we explore the potential of utilizing the eigenvectors of the transfer\nmatrix to compute another critical component of CFT data: the OPE coefficients.\nThis advancement in our methodology allows us to derive a complete set of CFT\ndata from the TRG/TNR scheme. The ability to obtain both the running coupling\nconstantsandOPEcoefficientsmarksasignificantstepforwardinourunderstanding\nof these models, bridging the gap between tensor network approaches and the rich\ntheoretical framework of CFT.\nFinally,utilizingournewmethodology,werevealthelimitationsduetofinitebond\ndimension𝐷on TNR applied to critical systems. We find that a finite correlation\nlength is induced by the finite bond dimension in TNR, and it can be attributed\nto an emergent relevant perturbation that respects the symmetries of the system.\nThe correlation length shows the same power-law dependence on 𝐷as the \"finite\nentanglementscaling\"oftheMatrixProductStates. Usingthis,wecanestimatethe\n38errors arising from TRG/TNR scheme, which was unclear before.\nThe following sections mainly discuss the Ising and three-state Potts models on\nthe square lattice. The energy (classical Hamiltonian) of the Ising and three-state\nPotts models are\nE𝐼𝑠𝑖𝑛𝑔=−∑︁\n⟨𝑖,𝑗⟩𝜎𝑖𝜎𝑗−ℎ∑︁\n𝑖𝜎𝑖, (2.1)\nE𝑃𝑜𝑡𝑡𝑠=−∑︁\n⟨𝑖,𝑗⟩𝛿𝑠𝑖,𝑠𝑗, (2.2)\nwhere𝜎𝑖=±1(Ising) and𝑠𝑖=0,1,2(three-state Potts). The first terms and ℎrep-\nresentthenearest-neighborinteractionsandtheuniformmagneticfield. Employing\nthetemperature 𝑇,theBoltzmannweightisdefinedas 𝑒−E/𝑇,wherewesettheBoltz-\nmann constant to unity. Our primary focus in this chapter is the Ising model, while\nadetaileddiscussionofthethree-statePottsmodelisprovidedintheappendix. The\nIsing model reaches its critical point at (𝑇,ℎ)=(𝑇𝑐,0), where𝑇𝑐=2/ln(1+√\n2).\nAt this criticality, physical quantities like the spin-spin correlation function are\ngoverned by the Ising CFT, which comprises three primary operators: the identity\noperator𝐼, magnetic operator 𝜎, and energy operator 𝜖.\nIn the context of the lattice model, a shift from the critical temperature and the\napplication of a magnetic field correspond to the perturbative insertion of 𝜖and\n𝜎into the effective Hamiltonian. As a result, 𝜎is odd in the Z2spin-flip, while\n𝐼and𝜖are even. Given the operator structure of the CFT, certain quantities are\nconsequently fixed.\n2.1 Operator product expansion coefficients\nOperator product expansion is another fundamental concept in field theory and\nstatistical mechanics as explained in the previous chapter [16, 17]. Since OPE\ncoefficients determine the structure of the field theory, their computation is quite\nimportant. Numerical computation of OPE coefficients [18, 19] has not been so\nstraightforwardcomparedtothatofscalingdimensions. Here,wepresentasimpler\nway to compute them, which is applicable to TRG [6], HOTRG [20], and Loop-\nTNR [10].\nThe renormalized tensor 𝑇(𝑛)contracted in 𝑥-direction is a transfer matrix in the\n𝑦-direction. While the eigenvalues of the transfer matrix correspond to the energy\nor scaling dimension of the primary operators, the eigenvectors thereof are the\nwavefunctions of the corresponding “primary states” |𝜓𝑛(𝐿)⟩. This is graphically\n39𝜓𝛼𝜓𝛽,𝜓𝛾𝐶𝛼𝛽𝛾 22𝑥𝛽+2𝑥𝛾−𝑥𝛼𝐴𝛼𝛽𝛾/𝐴𝐼𝐼𝐼\n𝐼 𝜎,𝜎 1 0.8938\n𝜎 𝜎,𝐼 1 0.9473\n𝐼 𝜖,𝜖 1 0.9966\n𝜖 𝜖,𝐼 1 0.9968\n𝜖 𝜎,𝜎 0.5 0.5007\n𝜎 𝜎,𝜖 0.5 0.2705\nTable 2.1: The numerically obtained OPE coefficients of the Ising CFT from TRG.\nThe bond dimension and the system size are 𝐷=56and𝐿=16√\n2(9 RG steps),\nrespectively.\nrepresented below.\nNote that the tensor has been rotated for ease of viewing. We do not change the\ncontracted index. Likewise, we can compute the wavefunctions of the system size\n2𝐿as depicted below.\n|𝜓𝑛(𝐿)⟩and|𝜓𝑛(2𝐿)⟩are one-leg and two-leg tensors, respectively. Thus, we\nproposeanovelmethodforcalculatingOPEcoefficients,utilizingthecontractionof\neigenstatesderivedfromthetransfermatrixoftherenormalizedtensor. Specifically,\nthis method involves computing the overlaps between the states |𝜓𝛼(2𝐿)⟩and the\ntensor product|𝜓𝛽(𝐿)⟩⊗|𝜓𝛾(𝐿)⟩by contracting their respective indices. This\ncomputed quantity is directly proportional to the OPE coefficients. This approach\nalignswiththediscussionontheoverlapofquantumwavefunctionsinRef.[21–23].\nIn CFT, the overlap of wavefunctions, denoted as ⟨𝜓𝛼(2𝐿)|𝜓𝛽(𝐿)𝜓𝛾(𝐿)⟩, is pro-\nportionaltothe’pantsdiagram’ofpathintegrals. Thisrelationshipcanbeelucidated\nthrough a review of how eigenstates |𝜓𝑛(𝐿)⟩are expressed in CFT.\n40For the ground state, the process begins with a random initial state |𝜓𝑖𝑛𝑖⟩, which\nundergoes imaginary time evolution:\n|𝜓0⟩∝lim\n𝜏→∞𝑒−𝜏𝐻|𝜓𝑖𝑛𝑖⟩, (2.3)\nwhere the dominance of the smallest eigenvalue of the Hamiltonian after sufficient\nimaginary time evolution ensures the ground state is attained. To obtain primary\nstates corresponding to excited states, initial eigenstates are prepared, with state-\noperator correspondence in CFT allowing the creation of states by inserting the\ncorresponding operator into the vacuum. On a cylinder, these operators acquire\na prefactor\u0010\n2𝜋\n𝐿\u0011−𝑥𝑛, derived from Eq. (1.24). Additionally, an exponential factor\n𝑒2𝜋\n𝐿𝑥𝑛𝜏should be placed to ensure normalization relative to the ground state as in\nEq. (1.53) 1. Consequently, the eigenstate at the 𝜏=0slice is given by:\n|𝜓𝑛(𝐿)⟩=lim\n𝜏→−∞𝑒−2𝜋\n𝐿𝑥𝑛𝜏\u00122𝜋\n𝐿\u0013−𝑥𝑛\n𝜓𝑐𝑦𝑙\n𝑛(𝜏)|𝐼𝑐𝑦𝑙⟩, (2.4)\n=𝜓𝑛(−∞)|𝐼𝑐𝑦𝑙⟩.\nThisformulationappliesto |𝜓𝛽(𝐿)⟩and|𝜓𝛾(𝐿)⟩. Similarly,|𝜓𝛼(2𝐿)⟩isconstructed\nfrom the infinite future:\n|𝜓𝛼(2𝐿)⟩=lim\n𝜏→∞𝑒−𝜋\n𝐿𝑥𝛼𝜏\u0010𝜋\n𝐿\u0011−𝑥𝛼𝜓𝑐𝑦𝑙\n𝛼(𝜏)|𝐼𝑐𝑦𝑙⟩, (2.5)\n=𝜓𝛼(∞)|𝐼𝑐𝑦𝑙⟩.\nIn this setup, these three vectors meet at the 𝜏=0slice, forming the basis for the\n’pants diagram’ path integral representation. (It does look like a pair of pants!)\n1Inessence,Eq.(1.53)states 𝐸𝑛−𝐸0=2𝜋\n𝐿𝑥𝑛. Weimplementthefactorinadvancetocompensate\n𝑒−𝜏𝐻=𝑒−2𝜋\n𝐿𝑥𝑛𝜏.\n41Inthiscontext,theoverlapoftheeigenstatescanbeconceptualizedasathree-point\nfunction on the ’pants’ manifold:\n𝐴𝛼𝛽𝛾\n𝐴𝐼𝐼𝐼=⟨𝜓𝛼(∞)𝜓𝛽(−∞)𝜓𝛾(−∞)⟩𝑝𝑎𝑛𝑡𝑠, (2.6)\nwhere𝐴𝛼𝛽𝛾representstheoverlap ⟨𝜓𝛼(2𝐿)|𝜓𝛽(𝐿)𝜓𝛾(𝐿)⟩. Tocomputethisquantity,\na conformal mapping from the ’pants’ manifold to a plane is required. This type\nof mapping, common in string theory calculations for string interactions, is well-\nunderstood. The conformal mapping, known as the Mandelstam mapping [24], is\ndefined as:\n𝑧=𝐿\n2𝜋[ln(𝑤−𝑖)+ln(𝑤+𝑖)−2 ln(𝑤)], (2.7)\nwherethepoints 𝑧=−∞and𝑧=∞correspondto 𝑤=±𝑖and𝑤=0,respectively 2.\nThis mapping, which effectively stitches three cylinders to a plane after opening\nthem,callsfortheadditionalfactor (𝑑𝑤\n𝑑𝑧)𝑥𝑛tothecorrelationfunctionasinEq.(1.24).\nThus, Eq. (2.6) is transformed to:\n𝐴𝛼𝛽𝛾\n𝐴𝐼𝐼𝐼=|𝐽𝛼|𝑥𝛼|𝐽𝛽|𝑥𝛽|𝐽𝛾|𝑥𝛾⟨𝜓𝛼(0)𝜓𝛽(𝑖)𝜓𝛾(−𝑖)⟩𝑝𝑙𝑎𝑛𝑒, (2.8)\nwhere the three-point function is now evaluated on the plane. The prefactor 𝐽is\nderived from a combination of Eqs. (2.4-2.5) and the (𝑑𝑤\n𝑑𝑧)𝑥𝑛factor:\n|𝐽𝛼|=\f\f\f\flim\n𝑧→∞𝑒−𝜋\n𝐿𝑧\u0010𝜋\n𝐿\u0011−1\u0012𝑑𝑤\n𝑑𝑧\u0013\f\f\f\f\n𝑤→0, (2.9)\n|𝐽𝛽|=\f\f\f\f\flim\n𝑧→−∞𝑒−2𝜋\n𝐿𝑧\u00122𝜋\n𝐿\u0013−1\u0012𝑑𝑤\n𝑑𝑧\u0013\f\f\f\f\f\n𝑤→𝑖, (2.10)\n|𝐽𝛾|=\f\f\f\f\flim\n𝑧→−∞𝑒−2𝜋\n𝐿𝑧\u00122𝜋\n𝐿\u0013−1\u0012𝑑𝑤\n𝑑𝑧\u0013\f\f\f\f\f\n𝑤→−𝑖. (2.11)\nUpon evaluation, it is straightforward to verify that |𝐽𝛼|=1and|𝐽𝛽|=|𝐽𝛾|=1/2,\nand⟨𝜓𝛼(0)𝜓𝛽(𝑖)𝜓𝛾(−𝑖)⟩𝑝𝑙𝑎𝑛𝑒=2𝑥𝛼−𝑥𝛽−𝑥𝛾𝐶𝛼𝛽𝛾. Consequently, the relationship\nbetween the OPE coefficient and the overlap becomes:\n𝐴𝛼𝛽𝛾\n𝐴𝐼𝐼𝐼=2𝑥𝛼−2𝑥𝛽−2𝑥𝛾𝐶𝛼𝛽𝛾, (2.12)\n2The coefficients of the logarithmic of 𝑤terms correspond to the length of the string, whereas\nits sign is negative for the states in the infinite future. The sum of the prefactors should be zero so\nthat the total length of the strings from the infinite past is equal to that of the infinite future.\n42illustrating how the OPE coefficients are intimately connected to the eigenstate\noverlaps within the ’pants’ manifold framework. (For more generic cases, readers\nshall consult Ref. [21–23].)\nIn most cases, the identity operator denoted as 𝐼, corresponds to the ground state\nor equivalently the leading eigenvector. Thus, the OPE coefficients 𝐶𝛼𝛽𝛾can be\ncomputedfromtheratiooftheoverlap 𝐴𝛼𝛽𝛾and𝐴𝐼𝐼𝐼,giventhescalingdimensions\nfrom the transfer matrix. We benchmark our method by the critical Ising model.\nTable.2.1showsthenumericallyobtainedOPEcoefficientsbyTRG[6]at 𝐿=16√\n2\nand𝐷=56. Naturally, there are finite-size corrections to Eq. (2.12). Since\nEq. (2.12) is exact in the thermodynamic limit, using a very large system size 𝐿\nmightappeardesirable. However,aswewilldiscusslaterinSec.2.4,correctionsdue\nto the finite bond-dimension effect appear for system sizes larger than a correlation\nlength𝜉(𝐷)3. AsreportedinRef.[23],thefinite-sizeeffectsaresignificantfor 𝐶𝜎𝜎𝜖\nand𝐶𝜖𝜖𝐼. Nevertheless,evenwiththemoderatesize 𝐿=16√\n2,theobtainedvalues\n𝐶𝐼𝜖𝜖=0.9966and𝐶𝜖𝜎𝜎=0.5007are rather close to exact CFT results. While we\ntestedourmethodbythesimplestalgorithm,LevinandNave’sTRG,themethodfor\ncalculatingOPEisstraightforwardlyapplicabletootherTRGandTNRalgorithms,\nsuch as HOTRG [20].\n2.2 Precise determination of the transition temperature\nAswehavementionedearlier,theratiosofthetransfermatrixspectrumrepresent\nthe scaling dimension as𝜆𝑛\n𝜆0=exp(−2𝜋𝑥𝑛)at criticality after sufficient coarse-\ngraining. However,therescaledenergylevelsofalatticemodelgenerallydependon\nthesystemsize 𝐿,astheeffectiveHamiltonianofthesystemcontainsperturbations\ntotheCFT.Thisletsusdefineageneralizedconceptofthescalingdimension,which\ndepends on the system size. We denote it as “rescaled energy\" defined as\n𝜆𝑛(𝐿)\n𝜆0(𝐿)=exp(−2𝜋𝑥𝑛(𝐿)). (2.13)\nFigure2.1exhibits“therescaledenergy\"ofthefirstexcitedstateoftheIsingmodel,\ncorresponding to 𝑥𝜎. At the critical temperature, which we denote with a red\ndotted line, this scaling dimension consistently aligns with the expected value of\n1\n8, a characteristic feature of the Ising universality class in two dimensions. This\nconsistency is observed regardless of the variations in system size 𝐿.\n3Thiseffectisevenstrongerandnon-trivialforTRGduetotheCDLtensorsasdiscussedinthe\nprevious chapter.\n43Figure 2.1: The scaling dimension obtained from the first and second leading\neigenvalue of the transfer matrix of the Ising model as 𝑥𝜎(𝐿)=1\n2𝜋ln𝜆0\n𝜆1. At the\ncritical temperature, denoted by a red dotted line, the value is consistent with the\nscalingdimension1\n8regardlessofthesystemsizes. Awayfromcriticality,however,\nthe deviation from1\n8grows as𝐿increases.\nHowever, a notable shift in behavior occurs when the system deviates from the\ncritical temperature. This observed shift in the behavior of the rescaled energy\n𝑥𝜎(𝐿)is a key aspect of critical phenomena. In off-critical scenarios, 𝑥𝜎(𝐿)starts\nto diverge from its critical value of1\n8. Notably, as the system size 𝐿increases, this\ndeviationbecomesmoresignificant. Thistrendisnotjustasimpleanomaly;rather,\nitsignifiestheevolutionoftherunningcouplingconstants,denotedas 𝑔𝑛(𝐿),inthe\nsystem.\nThis relationship between the deviation in scaling dimensions and the running\ncoupling constants is deeply rooted in the theoretical framework combining pertur-\nbation theory with CFT. The perturbation theory, when applied within the context\nof CFT, provides a robust explanation for this phenomenon. It elucidates how the\nchanges in the system’s parameters, as it moves away from criticality, influence the\nrunning coupling constants and consequently, the scaling dimensions.\nFor a detailed exploration of this relationship and the underlying theoretical prin-\nciples, readers are directed to Sec. A.1 in the appendix. Here, we only use Cardy’s\nresults[14,15]. Therescaledenergylevelsinafinite-sizeperturbedCFTaregiven\n44as\n𝑥𝑛(𝐿)=𝑥𝑛+2𝜋∑︁\n𝑗𝐶𝑛𝑛𝑗𝑔𝑗(𝐿), (2.14)\nwhere𝑔𝑗(𝐿)scales as∝𝐿2−𝑥𝑗4. Comparing Eq. (2.13) from TNR and Eq. (2.14)\nfrom the conformal perturbation theory, we can obtain the running coupling con-\nstants𝑔𝑗(𝐿)at each scale from the finite-size effect 𝛿𝑥𝑛(𝐿)=𝑥𝑛(𝐿)−𝑥𝑛.\nAn immediate and practical application of our observations is the precise deter-\nmination of critical points. This approach, often referred to as ’level spectroscopy,’\nwas originally developed to address the complexities of the Berezinskii-Kosterlitz-\nThouless(BKT)transition,particularlynotedforitschallengesinstandardfinite-size\nscalinganalysis. Initially,thistechniquewasappliedtoquantumspinsystemsinone\ndimension, as demonstrated by Nomura in 1994 [25]. More recently we extended\nfor classical statistical systems in two dimensions using TNR [26].\nThecoreprincipleoflevelspectroscopyisthecarefulanalysisoftheenergylevels\noreigenvalues,particularlyhowtheyshiftandevolveasthesystemapproachesand\nmoves away from criticality. While this technique was conceived in the context of\nthe BKT transition, its basic concept is broadly applicable to more conventional\ntypes of critical phenomena, such as those observed in the Ising model.\nTheRGfixed-pointforthetwo-dimensionalIsingmodelhastworelevantoperators,\ntheenergydensity 𝜖andthemagnetizationdensity 𝜎. Thecouplingconstant 𝑔𝜖for\n𝜖isproportionaltothedeviationofthetemperaturefromthecriticalpoint,andalso\nscaled∼𝐿in the small coupling limit 𝑔𝜖≪1because𝑥𝜖=1. Thus\n𝑔𝜖(𝐿)∼𝛼(𝑇−𝑇𝑐)𝐿, (2.15)\nwhen𝑔𝜖(𝐿)≪ 1. Likewise, the coupling 𝑔𝜎is proportional to the magnetic field\nℎand scaled∼𝐿15/8because𝑥𝜎=1/8. When determining the critical point, we\nfocus on the critical temperature with zero magnetic fields, where 𝑔𝜎=0.\nAlthough the Ising critical phenomena are mostly described by the two relevant\ncouplingconstants 𝑔𝜖and𝑔𝜎,moreaccuratedescriptioncanbeobtainedbyinclud-\ning irrelevant perturbations. Including the leading irrelevant operators, namely the\n4The second term in Eq. (2.14) is the first-order perturbation term. In CFT, the unperturbed\neigenstates and perturbations correspond to the primary states and Φ𝑗. In this framework, the\ncorrections to the energy are expressed as ⟨𝑛|𝑔𝑗Φ𝑗|𝑛⟩, which yields the OPE coefficients 𝐶𝑛𝑛𝑗\n45irrelevant operators with the smallest scaling dimension permitted by the symme-\ntries, the effective Hamiltonian of the Ising model is described as following:\n𝐻=𝐻∗\n𝐼𝑠𝑖𝑛𝑔+∫𝐿\n0𝑑𝑥[𝑔𝜎𝜎(𝑥)+𝑔𝜖𝜖(𝑥)\n+𝑔𝑇2𝑇2\ncyl(𝑥)+𝑔¯𝑇2¯𝑇2\ncyl(𝑥)], (2.16)\nwhere𝑇cyland ¯𝑇cylare the holomorphic and anti-holomorphic parts of stress tensor\non a cylinder [15]. The holomorphic part 𝑇cylof the stress tensor on a cylinder is\nrelated to that on the infinite plane 𝑇𝑧𝑧(𝑧)via the conformal mapping 𝑧=𝑒2𝜋𝑤/𝐿,\nwhere𝑤=𝜏+𝑖𝑥and0≤𝑥 < 𝐿. More explicitly, 𝑇𝑧𝑧(𝑧)transforms as\n𝑇cyl(𝑤)=\u00122𝜋\n𝐿\u00132\u0010\n𝑧2𝑇𝑧𝑧(𝑧)−𝑐\n24\u0011\n. (2.17)\nThis leads to\n𝑇cyl(𝑥)=2𝜋\n𝐿 ∞∑︁\n𝑛=−∞𝐿𝑛𝑒2𝜋𝑖𝑥/𝐿−𝑐\n24!\n, (2.18)\nwhere𝑐isthecentralchargecharacterizingtheCFT,and 𝐿𝑛’saregeneratorsofthe\nVirasoro algebra defined by\n𝑇𝑧𝑧(𝑧)=∞∑︁\n𝑛=−∞𝐿𝑛\n𝑧𝑛+2, (2.19)\nintermsoftheholomorphicpart 𝑇𝑧𝑧oftheenergy-momentumtensorontheinfinite\nplane. Inserting the above 𝑇cyland integrating over 0≤𝑥 < 𝐿with an appropriate\nregularization, the 𝑔𝑇2-term of the perturbation is given as [27]\n∫\n𝑑𝑥𝑇2\n𝑐𝑦𝑙(𝑥)=𝐿2\n0−𝑐+2\n12𝐿0+2∞∑︁\n𝑛=1𝐿−𝑛𝐿𝑛+𝑐(22+5𝑐)\n2880\nOnly the first and second terms affect the energy levels, and the contributions\nto𝑥𝜎(𝐿)and𝑥𝜖(𝐿)are calculated to be −7\n768𝑔𝑇2and7\n48𝑔𝑇2respectively. The\ncomputation of the contributions from ¯𝑇2is exactly the same, and we denote their\nsum as𝑔. These operators are the leading irrelevant operators for the Ising model\nonthesquarelattice. Animportantaspecttoconsideristheoriginandimplications\nof the squared terms of the stress tensor on the cylinder, 𝑇2\ncyland ¯𝑇2\ncyl. These\nterms possess conformal spins of +4and−4, respectively. The presence of these\nconformal spins is significant because they lead to the breaking of continuous\nrotational symmetry, which is the breaking of Lorentz invariance in Minkowski\n46space-time. However,inthecontextofasquarelattice,whichonlypossessesdiscrete\n𝐶4rotational symmetry, the inclusion of these terms is permissible. The presence\nof𝑇2\ncyland ¯𝑇2\ncylservestoadjustthesymmetryofthecontinuumtheorytomatchthat\nof the discrete lattice model. Essentially, these ’irrelevant’ operators play a crucial\nroleinaligningthetheoreticalmodel’ssymmetrywiththeinherentsymmetryofthe\nsquare lattice. This aspect of conformal spin becomes particularly evident during\nodd-numberedRGsteps. Atthesestages,thelatticeundergoesa45-degreerotation,\nleadingtoasignchangeintheseoperators,asindicatedbythefactor (𝑒𝑖𝜋/4)4=−1.\nThisrotation-inducedsignchangehasobservableconsequences. Forinstance,when\nexamining finite-size corrections at criticality, we notice an alternating sign in the\ncorrections to the scaling dimension, 𝛿𝑥𝜎, at each RG step.\nIncluding the contributions from relevant perturbations, the resulting finite-size\ncorrections to 𝑥𝜎(𝐿)and𝑥𝜖(𝐿)are shown in Table. 2.2 5. While the exact critical\npoint is known for the Ising model on the square lattice, let us demonstrate the\ndetermination of the critical point from the TNR spectrum without using prior\nknowledgeofthecriticalpoint(bututilizingtheCFTdata,assumingthatweidentify\ntheuniversalityclass). Sinceweareinterestedinthecriticalpointatzeromagnetic\nfields, we can set 𝑔𝜎∝ℎ=0. The simplest way to determine the critical point\nis to look at the lowest rescaled energy level 𝑥𝜎(𝐿)in the lowest order of the\nrelevant coupling constant 𝑔𝜖, ignoring the irrelevant perturbation 𝑔. Within this\napproximation, the shift 𝛿𝑥𝜎(𝐿)=𝑥𝜎(𝐿)−𝑥𝜎vanishes at the critical point 𝑇=\n𝑇𝑐where𝑔𝜖=0. Away from the critical point, 𝛿𝑥𝜎(𝐿)is non-zero and grows\nproportionally to 𝐿because𝑔𝜖(𝐿)scales as𝐿. Because of this, we can identify\nthe critical point with the temperature where 𝛿𝑥𝜎(𝐿)=0is observed in the TNR\nspectrum. However, this estimate suffers from the corrections due to the leading\nirrelevant perturbations 𝑇2\ncyland ¯𝑇2\ncyl. Since they have scaling dimension 4, the\ncorrespondingcouplingconstantisrenormalizedas 𝑔∝𝐿−2. Thisleadstoanerror\nof𝑂(𝐿−2)in the naive estimate of the critical point using 𝛿𝑥𝜎(𝐿)=0.\nWe can improve the accuracy by removing the effects of the leading irrelevant\nperturbation 𝑔. This can be done by combining the shifts of the rescaled energy\nlevels𝛿𝑥𝜎(𝐿)and𝛿𝑥𝜖(𝐿)following Table. 2.2 as\n𝛿𝑥cmb≡𝛿𝑥𝜎(𝐿)+1\n16𝛿𝑥𝜖(𝐿)\n=𝜋𝑔𝜖+(𝛼𝜎\n𝜎+1\n16𝛼𝜎\n𝜖)𝑔2\n𝜎+(𝛼𝜖\n𝜎+1\n16𝛼𝜖\n𝜖)𝑔2\n𝜖. (2.20)\n5As𝑇2\ncyland ¯𝑇2\ncylare not primary operators, we need to pay special attention.\n47Figure2.2: ExampleofestimatingthetransitiontemperatureusingLoop-TNR.We\nset𝑇−=2.66and𝑇+=2.68as an initial estimate. The level-crossing temperature\n𝑇∗(𝐿)is linearly fitted to extrapolate the transition temperature. The insert shows\nhow we compute 𝑇∗(𝐿)for various system sizes.\nNotethatthefirst-ordercorrectionintheirrelevantcoupling 𝑔iscanceledout. Now\nwe can identify the critical point by finding the temperature for which 𝛿𝑥cmb∝\n𝑔𝜖(𝐿)=0. Having eliminated the effects of the leading irrelevant perturbation\n𝑇2\ncyl,¯𝑇2\ncyl, the dominant error is now caused by the next-leading irrelevant operator\nwith scaling dimension 6and thus should be scaled as 𝐿−4.\nIn practice, the determination of the critical point can be efficiently implemented\nas follows. First, we pick up one temperature from each phase: 𝑇+> 𝑇𝑐and\n𝑇−<𝑇𝑐, and calculate the combined shift 𝛿𝑥cmbat these temperatures. The phase\nofthesystemcanbeconfirmedbyobservingthegrowthof 𝛿𝑥cmbasthesystemsize\nincreasesbecauseitincreases/decreasesifthesystemisinthehigh-temperature/low-\ntemperature phase (if the initial choice of the temperature turns out to be wrong,\nchange the temperature and restart the process). Next, linear interpolations of the\ncombined shift between the two temperatures 𝑇±are made, and the crossing of\nthe lines for system sizes 𝐿and√\n2𝐿is found, as shown in the insert of Fig. 2.2.\nWe denote the temperature where the two lines cross as 𝑇∗(𝐿). Because of the\nsecond-order contribution 𝑂(𝑔𝜖2)in Eq. (2.20), the crossing temperature 𝑇∗(𝐿)\nobtained by the linearinterpolation deviates from the true critical point 𝑇𝑐as\n48model operator Rescaled energy level\nIsing model 𝑥𝜎(𝐿)1\n8+𝛼𝜎\n𝜎𝑔2\n𝜎+𝜋𝑔𝜖+𝛼𝜖\n𝜎𝑔2\n𝜖−7\n768𝜋𝑔\n𝑥𝜖(𝐿) 1+𝛼𝜎\n𝜖𝑔2\n𝜎+𝛼𝜖\n𝜖𝑔2\n𝜖+7\n48𝜋𝑔\nTable 2.2: The finite-size scaling dimension of the Ising model. 𝛼is a constant\ndetermined from the second-order perturbation. Since 𝑔𝑇2and𝑔¯𝑇2decay in the\nsame manner, we write them as 𝑔.\n𝑇∗(𝐿)−𝑇𝑐∝𝑔𝜖∝𝐿, when𝑔𝜖≪16. The critical point 𝑇𝑐is estimated by fitting\n𝑇∗(𝐿)byalinearfunctionof 𝐿as𝑇∗(𝐿)∼𝑇𝑐+const.𝐿. Whilethe“extrapolation”\nto𝐿=0used here might look unusual, this procedure is done to remove the\neffect of the nonlinearity due to 𝑂(𝑔𝜖2)in Eq. (2.20), and the condition 𝛿𝑥cmb=0\nitself is accurate for 𝑇𝑐up to the error of 𝑂(𝐿−4)due to the next-leading irrelevant\nperturbations. An example of the estimate of 𝑇𝑐with the above procedure with\nthe choice of the temperatures 𝑇+=2.68and𝑇−=2.66and with system sizes\n16≤𝐿 < 64is depicted in Fig. 2.2. The final estimate of the critical point is\n𝑇est\n𝑐=2.269177. Remarkably, even with the choice of two temperatures differ by\n10−2and the relatively low bond-dimension 𝐷=20, the estimated critical point is\nquiteaccurate: 𝑇est\n𝑐−𝑇𝑐=−8.11×10−6. Thisisthankstothesuppressionoftheerror\nto𝑂(𝐿−4)by eliminating the contributions from the leading irrelevant operators.\nOnce the critical point is estimated with good accuracy with this procedure, the\naccuracy can be further improved by choosing 𝑇±closer to the estimated critical\ntemperature and then applying the same procedure.\n2.3 Renormalization group flow\nThe comparison between the TNR spectrum in Eq. (2.13) and the conformal per-\nturbationtheoryinEq.(2.14)canalsobeusedtoextractrunningcouplingconstants\nand their scale dependence, enabling visualization of the RG flow. This analysis\nwill be particularly useful in investigating the effects of finite bond dimensions in\ndetail, a topic we plan to explore comprehensively in Sec. 2.4.\nFor the Ising model, the extraction of running coupling constants is based on\nobserving shifts in the rescaled energy levels, as detailed in Table 2.2. It is also\nbeneficial to consider the combined shift as described in Eq. (2.20). Given that 𝑔𝜎\nand𝑔𝜖aresmallnearthecriticality,wesimplifyourcalculationsbyneglecting 𝑔2\n𝜖for\n6It is proportional to 𝐿2−𝑥thermal, where𝑥thermalis the scaling dimension of the thermal operator.\n49Figure 2.3: (Left panel) The system size dependence of 𝛿𝑥cmb=𝛿𝑥𝜎+𝛿𝑥𝜖/16for\nℎ=±10−5(purple and green), 𝑇=1.0001𝑇𝑐(red) and𝑇=0.9999𝑇𝑐(blue). The\npurple and green dots are on top of each other, and “ +\" denotes the data with a\nnegative sign. After removing the 𝐿−2irrelevant perturbations, the next leading\n𝐿−4perturbation shown with a blue dotted line appears. The data was obtained\nvia Loop-TNR with a bond dimension of 𝐷=24, which was deemed sufficient\nfor the finitely-correlated systems being considered. (Right panel) The resulting\nrenormalization group flow. Only data after six steps are exhibited, where the 𝐿−4\nperturbations disappear.\nℎ=0. Consequently, we redefine two relevant coupling constants for convenience:\n𝑔𝑡=𝜋𝑔𝜖and𝑔ℎ=√︃\n(𝛼𝜎𝜎+1\n16𝛼𝜎𝜖)𝑔𝜎. In this way, the combined shift Eq. (2.20)\nsimplygives 𝑔𝑡whenℎ=0and𝑔ℎ2when𝑇=𝑇𝑐,inthelowestorderof 𝑔𝑡,𝑔ℎ. Using\nthese relations, we can read off the relevant coupling constants 𝑔𝑡or𝑔ℎfrom the\nTNRdata,asshowninFig.2.3( 𝑏). Aswehavediscussedintheprevioussubsection,\ntheeffectsoftheleadingirrelevantperturbations 𝑇2\ncyl,¯𝑇2\ncylwithscalingdimension 4\nareeliminatedinthecombinedshiftEq.(2.20),andthusthefinite-sizecorrectionis\nnowof𝑂(𝐿−4),duetothenext-leadingirrelevantoperatorswithscalingdimension\n6. This𝑂(𝐿−4)scaling is indeed observed in Fig. 2.3 near the critical point for\nsmall system size 𝐿when relevant perturbations are still negligible. Since it is safe\nto say that these contributions disappear after five RG steps, we can conclude that\nthe origin of 𝑔𝑡and𝑔ℎare purely from 𝜖and𝜎after six steps.\nThe right panel illustrates the scale-dependence of the coupling constants 𝑔𝑡and\n𝑔ℎ. ItisnothingbuttheRGflowoftheIsingcriticalpoint,andweconcludethatwe\nsucceed in calculating the RG flow of the celebrated Ising fixed-point.\n50There is one thing to note on the left panel of Fig. 2.3. While the combined\nshift (2.20), which is an estimator for |𝑔ℎ|2, scales as𝐿3.75at𝐿 < 103, it starts to\nflatten and scales as 𝐿at𝐿 > 103. This behavior has a rather simple origin. Since\nthe magnetic perturbation is relevant, the system has a finite correlation length or\nequivalently, a non-zero gap Δ. This implies that the rescaled energy levels are\nproportional to 𝐿for sufficiently large system size 𝐿≫Δ−1. As a consequence,\nthe shift Eq. (2.20) also grows proportionally to 𝐿. In this regime, the conformal\nperturbationtheorybreaksdown(higher-ordercontributionsareimportant),andwe\nnolongeridentifytheshiftEq.(2.20)with |𝑔ℎ|2. Thisshouldbedistinguishedfrom\nthe𝐿-linear behavior of the combined shift Eq. (2.20) observed for 𝐿 > 10with\nℎ=0and𝑇≠𝑇𝑐, which corresponds to the renormalization of 𝑔𝑡∝𝐿because of\n𝑥𝜖=1. The𝐿-linear behavior due to the gap is observed in the non-perturbative\nregime𝛿𝑥𝜖,𝜎≫𝑥𝜖,𝜎, whereas the 𝐿-linear behavior due to the scaling is observed\nin the perturbative regime 𝛿𝑥𝜖,𝜎≪𝑥𝜖,𝜎.\n2.4 Finite bond-dimension effects\nLet us examine the impacts of a finite bond-dimension 𝐷on TNR from the per-\nspective of our method. In any computation that employs tensor networks, it is\nnecessary to restrict the bond dimension to a finite value 𝐷due to the increasing\nstorage requirements and computational costs associated with larger bond dimen-\nsions. Thefinitenessofthebonddimensioninevitablyleadstoalossofinformation\nineachstepofrenormalizationafteracertainnumberofiterations. AlthoughTNR\ncannominallyhandlearbitrarylargesystems,andtheTNR-typecalculationsareof-\ntenusedtostudyextremelylargesystems,wehavetobecarefulaboutthelimitations\ndue to the finite bond dimension.\nThelimitationofthefinitebonddimension 𝐷onthematrixproductstate(MPS)is\ncharacterizedbythefinite(maximum)correlationlength 𝜉(𝐷)oftheMPS[28–30].\nThe correlation length of MPS is known to obey the scaling law\n𝜉(𝐷)∼𝐷𝜅, (2.21)\n𝜅=6\n𝑐(1+√︃\n12\n𝑐). (2.22)\nWhile the TNR-type calculation of two-dimensional statistical systems appears\nrather different from the MPS applied to one-dimensional quantum systems, the\nemergenceofthefinitecorrelationlength 𝜉(𝐷)obeyingthesimilarscalinglaw(2.21)\nwasreportedinRef.[31]foraHOTRGcalculationofthecriticalIsingmodelintwo\n51Figure 2.4: Shift|𝛿𝑥𝜎(𝐿)|for the Ising model at 𝑇=𝑇𝑐,ℎ=0computed by Loop-\nTNR with𝐷=32. There is little finite- 𝐷effect for small system sizes 𝐿 < 256.\nThe emergent perturbations of 𝜖and𝜎appear at𝐿∼256and𝐿∼104, scaling\nas𝐿and𝐿15/4. The induced gap by finite- 𝐷goes towards constant at 𝐿 > 105as\ndenoted with the purple dotted line.\ndimensions. Theexponent 𝜅fortheIsingmodelwasestimatedtobeapproximately\n2,whichisclosetotheMPSexponent(2.22) 𝜅=2.03425...fortheIsingCFTwith\ncentral charge 𝑐=1/2. A similar emergence of the finite correlation length 𝜉(𝐷)\nwas also reported in our TNR finite-size scaling study of the two-dimensional XY\nmodel [26], with the MPS exponent (2.22) for 𝑐=1.\nIn the following, using our TNR finite-size scaling methodology, we will demon-\nstrate that the emergence of the finite correlation length due to the finite bond\ndimension in TNR can be attributed to an emergent relevant perturbation. Further-\nmore, we present evidences for the scaling (2.21) with the MPS exponent (2.22) in\nTNR of Ising and three-state Potts models.\nEmergent relevant perturbation\nIfafinitecorrelationlengthemergesintheTNR,itwouldbenaturaltoidentifythe\nrenormalizedtensorwithaHamiltonianforthesystemawayfromthecriticalpoint,\n52that is, an RG fixed-point (CFT) Hamiltonian perturbed with relevant operators\n𝐻FB(𝐷)=𝐻∗\nCFT+∑︁\n𝑖∫𝐿\n0𝑑𝑥𝑔𝑖(𝐷,𝐿)Φ𝑖(𝑥,𝐷), (2.23)\nwhere𝐻𝐹𝐵istheeffectiveHamiltonianofthefinite- 𝐷systemandΦ𝑖(𝑥,𝐷)arethe\nscaling operators representing the perturbations. In this view, we expect relevant\nperturbations to emerge in order to mimic the finite correlation length imposed by\nthe finite bond dimension.\nTo demonstrate the emergence of the relevant perturbation, we investigate the\nsystem-size dependence of the shift in the rescaled energy levels 𝛿𝑥𝜎. In Fig. 2.4,\nwe show the absolute value of the shift |𝛿𝑥𝜎|as a function of the system size 𝐿\nusedincalculatingthetransfermatrixspectruminTNRexactlyatthecriticalpoint\nℎ=0,𝑇=𝑇𝑐. The conformal perturbation theory in Eq. (2.14) implies that the\nshift𝑥𝜎contains contributions from the irrelevant perturbations. Since the leading\nirrelevant operators at the critical points are 𝑇2\ncyland ¯𝑇2\ncylwith scaling dimension\n4, we expect𝛿𝑥𝜎(𝐿)decays as𝐿−2. (This is to be contrasted with Eq. (2.20) and\nFig.2.3,inwhichthecontributionsfrom 𝑇2\ncyland ¯𝑇2\ncylareeliminated.) Theexpected\n𝐿−2behaviorintheshift 𝛿𝑥𝜎(𝐿)isindeedobservedforsmallsystemsizes 𝐿 <256.\nFor larger system sizes, however, |𝛿𝑥𝜎(𝐿)|starts to increase, deviating from the\nconformal perturbation theory scaling 𝐿−2. We identify the finite bond-dimension\n𝐷effects as the origin of this deviation. More remarkably, we can observe a clear\nscaling behavior of the deviation. That is, the shift |𝛿𝑥𝜎(𝐿)|scales with the system\nsizes as𝐿and𝐿15/4for256< 𝐿 < 104and104< 𝐿, respectively. Compared with\nthe off-critical cases in Fig. 2.3, we realize that these scalings are identical to those\ninduced by the thermal and magnetic perturbations. In other words, the relevant\nperturbations emerge in the TNR calculation.\nLet us first discuss the 𝐿15/4scaling of the shift, observed for 𝐿 > 104. This\ncan be understood as the effect of an emerging magnetic perturbation ℎbecause its\nsecondorderperturbationscalesas 𝑔2\n𝜎∝𝐿15/4. Althoughthemagneticperturbation\nℎis forbidden by the Z2spin-flip symmetry, the symmetry could be broken by the\nlimitations in the machine precision. Once the spin-flip symmetry is broken, the\nmagnetic field ℎ, which is a relevant perturbation, is effectively generated. Even if\ntheeffectivemagneticfield ℎisextremelysmall,itwillbeenhancedateachRGstep\nand eventually dominate the system at sufficiently large length scales. This is what\nwe observe for 𝐿 > 104. This phenomenon should be related to machine precision\n53andnotintrinsictothealgorithm. Ifweareinterestedina Z2symmetricsystem,we\ncan impose the symmetry at each step of TNR in order to avoid this effect.\nIncontrast,the 𝐿scalingobservedfor 256< 𝐿 < 104ismoreintrinsic. Themost\nrelevantperturbationallowedunderthe Z2symmetrytothecriticalIsingfixed-point\nisthethermaloperator. Thus,weexpectthatthefinitebonddimensioneffectcanbe\nmimickedbythethermalperturbation 𝜖tothefixed-pointHamiltonian 𝐻∗\nCFT. Ifthis\nisthecase,theeffectivecoefficient 𝑔𝜖growsproportionallyto 𝐿asthesystemsize 𝐿\nisincreased,becausethethermaloperator 𝜖hasthescalingdimension 1. According\nto Eq. (2.14), this will lead to a correction proportional to 𝐿in the rescaled energy\nlevel𝛿𝑥𝜎(𝐿). This is indeed supported by the numerical result shown in Fig. 2.4.\nIngeneral,thefinite- 𝐷effectinTNRwouldbedescribedintermsoftheemergence\nof relevant perturbation(s) to the fixed-point Hamiltonian, which induces the finite\ncorrelation length 𝜉(𝐷). In addition to the emergence of the relevant operator 𝜖in\nthecriticalIsingmodeldiscussedabove,asimilaremergenceoftherelevantoperator\nisobservedinthecriticalthree-statePottsmodel,asdemonstratedinAppendixA.2.\nScaling of the emergent correlation length\nNowletusdemonstratethatthefinitecorrelationlength 𝜉(𝐷)inducedbythefinite\nbonddimension 𝐷inTNRobeysthesamescaling(2.21)and(2.22)asintheMPS,\nas suggested in Refs. [26, 31].\nIn Fig. 2.5, we demonstrate the scaling of the correlation length induced by the\nfinite bond dimension in TNR of the critical Ising and the three-state Potts models.\nIn Fig. 2.5(a), we plot the shift 𝛿𝑥𝜎in the Ising model obtained by the TNR of\nthe Ising model at the critical point, which was also studied in Fig. 2.4, with the\nseveral different bond dimensions 𝐷=4,..., 28. Here, we rescaled the vertical\naxis as𝐿2𝛿𝑥𝜎so that the constant behavior is observed for system size smaller\nthanthecorrelationlength,wheretheleadingirrelevantperturbation(whichcauses\n𝛿𝑥𝜎∝𝐿−2) is dominant. The deviation from the constant at larger system sizes\n𝐿can be attributed to the emergent relevant perturbation 𝜖induced by the finite\nbond dimension 𝐷, as discussed in the previous subsection. This is confirmed\nby the𝐿3scaling (𝐿2times𝛿𝑥𝜎∝𝑔𝜖∝𝐿). Most importantly, the horizontal\naxis is the rescaled system size 𝐿/𝜉(𝐷)using the hypothesized correlation length\n𝜉(𝐷)=𝑎𝐷𝜅given by Eqs. (2.21) and (2.22). The collapse of the data for different\nbond dimensions strongly supports our hypothesis on the correlation length. Note\nthat we roughly fit the prefactor 𝑎so that the cross-over occurs at 𝐿=𝜉(𝐷).\n54Figure 2.5:(𝑎)The scaling of the shift 𝛿𝑥𝜎in TNR of the Ising model at the\ncritical point, for various bond dimensions 𝐷=4,..., 28. The vertical axis is\nscaled as𝐿2𝛿𝑥𝜎so that it is constant when 𝐿≪𝜉(𝐷). When𝐿≪𝜉(𝐷), the\nshift is dominated by the emergent relevant perturbation 𝜖; this is confirmed by the\nscaling𝐿2𝑔𝜖∝𝐿3. The horizontal axis is scaled as 𝐿/𝜉(𝐷), where the correlation\nlength𝜉(𝐷)is hypothesized as in Eqs. (2.21) and (2.22). The collapse of the\ndata for different bond dimensions is strong evidence of the hypothesized scaling\nof the correlation length 𝜉(𝐷). The blue dotted line indicates 𝐿/𝜉(𝐷)=1. We\nset𝜉(𝐷)=2.0𝐷𝜅so that𝐿/𝜉(𝐷)=1becomes the crossover scale between the\nfinite-size scaling regime and the finite- 𝐷scaling regime.(𝑏)Similar scaling\nanalysis of the shift 𝛿𝑥𝜖in TNR of the three-state Potts model at the critical point,\nfor various bond dimensions 𝐷=16,..., 40with𝜉(𝐷)=0.067𝐷𝜅. The scaled\nshift𝐿0.8𝛿𝑥𝜖behaves as a constant in the finite-size scaling regime 𝐿/𝜉(𝐷)<1,\nwhereas it scales as 𝐿3.2in the finite- 𝐷scaling regime 𝐿/𝜉(𝐷)>1, as expected\nfrom the CFT analysis (see Appendix A.2 for details). The data for different\nbond dimensions collapse again, giving compelling evidence for the scaling of the\ncorrelation length (2.21) and (2.22)\nInordertoconfirmthefinite- 𝐷scalingofthecorrelationlengthanditsuniversality,\nwehavealsostudiedthethree-statePottsmodelatthecriticalpoint. Asanexample,\nin Fig. 2.5(b), we plot the shift of the rescaled energy level corresponding to the\nenergy operator 𝜖in the three-state Potts model. For this shift 𝛿𝑥𝜖, the contribution\nfromtheleadingirrelevantoperatoris ∼𝐿−4/5,andthedominantcontributionfrom\nthe emergent relevant perturbation 𝜖is expected to be proportional to 𝑔𝜖2∝𝐿12/5.\n(See Appendix A.2 for details). We rescaled the vertical axis as 𝐿0.8𝛿𝜖so that it is\nconstantinthefinite-sizescalingregime 𝐿 <𝜉(𝐷). Thehorizontalaxisisagainthe\n55rescaledsystemsize 𝐿/𝜉(𝐷),withthecorrelationlength 𝜉(𝐷)definedinEqs.(2.21)\nand (2.22) with the central charge 𝑐=4/5for the three-state Potts model. The data\nfordifferentbonddimensionsagainshowacollapse,providingcompellingevidence\nfor our hypothesis on the correlation length scaling. For 𝐿/𝜉(𝐷)>1, the data fits\nwell the expected behavior 𝐿0.8×𝑔𝜖2∝𝐿0.8×𝐿2.4=𝐿3.2.\n56C h a pte r 3\nTENSOR NETWORK REPRESENTATION OF FIXED-POINT\nTENSORS\nTNR, while a powerful tool in the study of critical phenomena, encounters lim-\nitations when dealing with systems at criticality. A key challenge arises from the\nemergence of finite correlation lengths, which inherently restrict the effectiveness\nof TNR in capturing the true nature of the critical point. This limitation is com-\npounded by the practical inability to simulate an infinite bond dimension ( 𝐷=∞),\neffectively creating a sort of ’no-go theorem’ for TNR in accurately obtaining the\ntrue fixed-point tensor.\nFixed-point tensors, being invariant under RG transformations, are expected to\nexhibitcertaindistinctproperties. RecognizingtheconstraintsofTNRatcriticality,\nwe propose in this chapter an analytical approach aimed at uncovering the tensor\nelements of these fixed-point tensors. This method is designed to complement the\nRG techniques employed in tensor networks.\nOur approach leverages analytical methods to probe deeper into the structure of\nfixed-point tensors, enabling us to bypass some of the limitations posed by finite\ncorrelation lengths and finite bond dimensions. By analytically determining the\ntensor elements of fixed-point tensors, we aim to provide a more comprehensive\nunderstanding of the behavior of systems at criticality. This method not only\nenhances our ability to study critical phenomena using tensor networks but also\ncontributes to a more nuanced understanding of the universal properties associated\nwith RG fixed points.\n3.1 Fixed-point tensor\nTo simulate two-dimensional statistical models, we use the tensor network meth-\nods, where the local Boltzmann weight is represented as a four-legged tensor 𝑇(0).\nWeobtainthetransfermatrixinthe 𝑦-directionifwecontract 𝐿copiesofthefour-leg\ntensorsalongacircleinthe 𝑥-direction;weobtainthepartitionfunction 𝑍(𝐿,𝑇(0))\nif we contract 𝐿×𝐿copies along the torus in the 𝑥,𝑦-directions. In practical sim-\nulations, the exact contraction of two-dimensional tensor networks is notoriously\nchallenging, often proving to be exponentially hard. To address this, we consider a\n57tensor RG map that effectively coarse-grains the local tensors, as illustrated below:\nIntheinitialRGstep,weapplytheRGmaptotheoriginalBoltzmanntensor,denoted\nas𝑇(0), to yield𝑇(1). Subsequently, 𝑇(1)is used to generate the next tensor in the\nsequence. ThisRGmapisdesignedtoensurethattherenormalizedtensorremainsa\nclose approximation of theoriginal tensor group, while selectively discarding local\nentanglement. While the specifics of the technical implementation vary depending\non the chosen algorithms, it is established that 𝑇(𝑛)converges to a universal tensor,\n𝑇∗,atcriticalpoints. Thisuniversallyconvergenttensor, 𝑇∗,isreferredtoastheFP\ntensor. ItssignificanceliesinitscloseassociationwiththeRGfixed-point,reflecting\ntheunderlyingprinciplesofscaleinvarianceanduniversalityintherenormalization\ngroup theory.\nIf the original tensor 𝑇(0)hasD4symmetry(reflection and 𝜋/2rotation),𝑇∗also\nrespectsit. ThisallowsthedecompositionoftheFPtensorintoapairoftwoidentical\nthree-leg tensors 𝑆∗:\n(3.1)\nThe FP tensor 𝑇∗has gauge degrees of freedom that change the basis of each leg.\nThe insertion of the gauge transformation (unitary operators) does not change the\nspectral property of the FP tensor. In the following, we fix the gauge so that each\nindexoftheFPtensorislabeledbytheeigenstatesoftheHamiltonian 𝐿0+¯𝐿0ona\ncylinder,where 𝐿𝑛(¯𝐿𝑛)arethestandardgeneratorsoftheleft-moving(right-moving)\nVirasoro algebras. By the state-operator correspondence, we can label these states\nbyasetofoperators 𝜙𝛼,amongwhichwewillfindtheidentityoperator 𝜙1withthe\nlowest scaling dimension 1. In tensor-network representations, the projector to this\n1Note that the label 𝛼refers to both the primaries and the descendants of the Virasoro algebra.\n58basis can be found by diagonalizing the transfer matrix as follows [7]:\n(3.2)\nIn the following, we choose the states 𝛼,𝛽,...to be primary operators.\n3.2 Main results\nLet us now state the main results of this chapter. First, the three-leg tensor 𝑆∗\nis proportional to the three-point functions of the FP CFT on the complex plane\ndenoted as pl:\n𝑆∗\n𝛼𝛽𝛾\n𝑆∗\n111=⟨𝜙𝛼(−𝑥𝑆)𝜙𝛽(𝑖𝑥𝑆)𝜙𝛾(0)⟩pl. (3.3)\nSecond,thefour-legFPtensordeterminesthefour-pointfunctionsoftheFPCFTas\n𝑇∗\n𝛼𝛽𝛾𝛿\n𝑇∗\n1111=⟨𝜙𝛼(−𝑥𝑇)𝜙𝛽(𝑖𝑥𝑇)𝜙𝛾(𝑥𝑇)𝜙𝛿(−𝑖𝑥𝑇)⟩pl. (3.4)\nTheseequalitiesholdwhenwechoosethevalues 𝑥𝑆=𝑒𝜋/4and𝑥𝑇=𝑒𝜋/2/2.𝑥𝑆and\n𝑥𝑇are just numbers. Do not confuse them with scaling dimensions.\nWe can now reproduce the fulldefining data for the FP CFT. Recall that we can\nextract the scaling dimensions 𝑥𝛼operators from Eq. (3.2). The remaining data is\nthe OPE coefficients 𝐶𝛼𝛽𝛾of the operators 𝜙𝛼, which can be extracted by applying\na conformal transformation to Eq. (3.3):\n𝑆∗\n𝛼𝛽𝛾\n𝑆∗\n111=𝐶𝛼𝛽𝛾\n𝑥𝑥𝛽+𝑥𝛾−𝑥𝛼\n𝑆𝑥𝑥𝛾+𝑥𝛼−𝑥𝛽\n𝑆(√\n2𝑥𝑆)𝑥𝛼+𝑥𝛽−𝑥𝛾,\n=2𝑥𝛾𝐶𝛼𝛽𝛾\n(√\n2𝑥𝑆)𝑥𝛼+𝑥𝛽+𝑥𝛾. (3.5)\nEquation(3.1)representstheequivalenceoftwodifferentdecompositions( 𝑠-and\n𝑡-channels) of the four-point function into a pair of three-point functions, i.e. the\ncelebrated crossing relation of the CFT.\n59To better understand Eqs. (3.3-3.4), we apply conformal transformations to the\ntwo equations to obtain\n𝑆∗\n𝛼𝛽𝛾\n𝑆∗\n111=𝑒−𝜋\n4(𝑥𝛼+𝑥𝛽+𝑥𝛾)⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(0)⟩pl, (3.6)\n𝑇∗\n𝛼𝛽𝛾𝛿\n𝑇∗\n1111=\u0012𝑒𝜋\n2\n2\u0013−𝑥tot\n⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(1)𝜙𝛿(−𝑖)⟩pl, (3.7)\nwhere𝑥tot≡𝑥𝛼+𝑥𝛽+𝑥𝛾+𝑥𝛿.\nEquations(3.6-3.7)naturallyarisefromthefollowingarguments. Oncewefixthe\nbasisfortheFPtensor,eachindexcorrespondstothestatesofCFT.Thus,thetensor\nelements of the FP tensor are the coefficients of each basis:\n𝑇∗=𝑇∗\n𝛼𝛽𝛾𝛿|𝜙𝛼⟩|𝜙𝛽⟩|𝜙𝛾⟩|𝜙𝛿⟩ (3.8)\nOn the other hand, the FP tensor itself is a lattice representation of the identity\noperator 1. InRef.[18,32,33],theyconfirmedthatlocalscale-transformationcould\nbe realized using the FP tensors.\nThescaletransformationofafour-legtensor,comprisingtheFPtensor(coloredblue)\nandisometry(coloredorange),resultsinprimaryoperatorsemergingaseigenstates.\nNotably, the scale-invariant FP tensor corresponds to 𝑥𝛼=0. This specific cor-\nrespondence is significant, equating the FP tensor to the identity operator 2. This\nobservationleadsustoaconceptualizationwheretheelementsofthetensorcanbe\nexpressedasanoverlapbetweenthefour-legidentityoperatorandfourone-legpri-\nmary operators as 𝑇∗\n𝛼𝛽𝛾𝛿=⟨𝜙𝛼𝜙𝛽𝜙𝛾𝜙𝛿|𝜙4−𝑙𝑒𝑔\n1⟩. The same argument can be applied\nto the three-leg FP tensor 𝑆∗. To calculate these values, we employ a technique\nsimilartotheonedescribedinthereferencedliterature,specificallyinRef.[21,34].\nFirst,utilizingstate-operatorcorrespondence,thenormalizedwavefunctionofthe\nfirst index of 𝑆∗, for instance, is created by inserting 𝜙𝛼in the future infinity of the\n2unitary CFTs have ground states corresponding to the identity operator\n60cylinder as follows:\n|𝜙1⟩=\u00122𝜋\n𝐿\u0013−𝑥𝛼\nlim\n𝑧→∞𝑒2𝜋𝑧𝑥𝛼/𝐿𝜙𝛼(∞)|𝐼cyl⟩,\nwhere|𝐼cyl⟩represents the ground state corresponding to the identity operator.\nSubsequently,theFPtensors 𝑆∗and𝑇∗canbeexpressedbythepathintegralonthe\nmanifoldsΣ𝑆andΣ𝑇, respectively, as illustrated in Fig. 3.1. Then, the FP-tensor\nelements are\n𝑆∗\n𝛼𝛽𝛾\n𝑆∗\n111=⟨𝜙𝛼(∞)𝜙𝛽(𝑖∞)𝜙𝛾(−(1+𝑖)∞)⟩Σ𝑆, (3.9)\n𝑇∗\n𝛼𝛽𝛾𝛿\n𝑇∗\n1111=⟨𝜙𝛼(−∞)𝜙𝛽(𝑖∞)𝜙𝛾(∞)𝜙𝛿(−𝑖∞)⟩Σ𝑇. (3.10)\nΣ𝑆andΣ𝑇canbemappedtothecomplexplane 𝑤byusing(generalized)Mandelstam\nmapping [24, 35],\n𝑧𝑆=𝐿\n2𝜋[−ln(𝑤−𝑖)−𝑖ln(𝑤+1)+(1+𝑖)ln𝑤], (3.11)\n𝑧𝑇=𝐿\n2𝜋\u0014\nln\u0012𝑤+𝑖\n𝑤−𝑖\u0013\n+𝑖ln\u0012𝑤−1\n𝑤+1\u0013\u0015\n. (3.12)\nEach operator in the 𝑧-coordinate transforms accordingly as\n𝑆∗\n𝛼𝛽𝛾\n𝑆∗\n111=⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(0)⟩plÖ\n𝑛∈(𝛼,𝛽,𝛾)|𝐽𝑛|𝑥𝑛,\n𝑇∗\n𝛼𝛽𝛾𝛿\n𝑇∗\n1111=⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(1)𝜙𝛿(−𝑖)⟩plÖ\n𝑛∈(𝛼,𝛽,𝛾,𝛿)|𝐽𝑛|𝑥𝑛,\nwhere|𝐽𝑛|=|\u0010\n2𝜋\n𝐿\u0011−1\nlim𝑧→𝜁∞𝑒2𝜋𝑧𝜁∗/(𝐿|𝜁|)|𝑤′(𝑧)|, and𝜁∞is the coordinate of the\nindexintheoriginalmanifold. Theresulting |𝐽𝑛|are𝑒−𝜋/4and2𝑒−𝜋/2,respectively,\nbeing consistent with Eqs. (3.6-3.7). Detailed calculations are presented in the\nappendix.\n3.3 Numerical fixed point tensor\nLet us provide numerical confirmations of our main results using Levin’s tensor\nrenormalization group (TRG) [6] and Evenbly’s TNR [10]. TRG and TNR are\nnumerical techniques that calculate effective 𝐿×𝐿tensor networks. In our study,\nour interest lies in computing those of large system sizes to obtain a tensor that is\nas close as possible to the FP tensor. However, performing an exact contraction is\n61Figure 3.1: The path-integral representation of the tensor elements (𝑎)𝑆∗\n𝛼𝛽𝛾and\n(𝑏)𝑇∗\n𝛼𝛽𝛾𝛿. The fixed-point tensor lives at the center of cylinders, and surrounding\ncylinders are bra vectors of primary fields. Since the FP tensor corresponds to\nthe identity operator, “insertion of no operator\" is illustrated as empty space. This\nidentity operator at the origin in 𝑧coordinate will be mapped to the infinity in 𝑤.\nexponentiallydifficult,promptingustofocusonextractinglow-lyingspectralprop-\nerties. TRG/TNR seeks to circumvent this issue by employing the principles of the\nrenormalizationgrouptheory. Eachcoarse-grainingstepentailsdecompositionsand\nrecombinations. Truncation,parameterizedbythebonddimension 𝐷,isperformed\nto maintain the tractability of numerical computation. However, it is important to\nnote that this scheme is considered exactwhen𝐷=∞, and thus, employing larger\n𝐷improvesthenumericalaccuracy. Additionally,weimposespecial 𝐷4symmetry\nin TRG. The details can be found in the appendix. It is crucial to acknowledge\nthat the TRG method is known to exhibit instabilities, primarily due to its inherent\nlimitations in eliminating certain types of local entanglement. In contrast, TNR,\nwhich includesa local entanglementfiltering process,typically demonstrates supe-\nrior performance in extracting infra-red information. This enhanced capability of\nTNR is attributed to its more effective handling of local entanglement, making it a\nmore robust approach for studying systems at criticality.\n3.4 Tests on critical lattice models\nLet us first test the value 𝑥𝑆=𝑒𝜋/4in Eq. (3.6), by computing 𝑥𝑆from the critical\nIsing and three-state Potts models. Given Eq. (3.6), we can numerically compute\nthe OPE coefficients 𝐶𝛼𝛽𝛾from Eq. (3.5). We define 𝑥𝑆(𝐿)by solving Eq. (3.5) to\nbe\n𝑥𝑆(𝐿)≡1√\n2\u00122𝑥𝛾𝐶𝛼𝛽𝛾\n𝑆𝛼𝛽𝛾(𝐿)\u00131/(𝑥𝛼+𝑥𝛽+𝑥𝛾)\n. (3.13)\nEach model has a primary operator 𝜖, called the energy and the thermal operator,\nrespectively. Since 𝐶𝜖𝜖1=1,𝑥𝑆(𝐿)can be computed from the finite-size three-leg\n62Figure 3.2: Estimation of 𝑥𝑆(𝐿)from Levin-TRG( 𝐷=96) and Evenbly-TNR( 𝐷=\n40). The values of 𝑥(𝐿)from the Ising and three-state Potts model converge to the\ntheoretical value 𝑥𝑆=𝑒𝜋/4denoted by a black dotted line. We plot 𝑥𝑆=2.23035\nobtained from Loop-TNR [10] on the critical 9-state clock model [19] with a lime\ndashed line. The three-state Potts model exhibits a deviation for 𝐿 > 100because\nsimulating systems with higher central charges involves larger numerical errors.\ntensor𝑆𝜖𝜖1(𝐿).\nFigure 3.2 shows the value of 𝑥𝑆(𝐿)obtained from TRG and TNR at the bond\ndimension𝐷=96and𝐷=40, respectively. The numerically derived 𝑥𝑆(𝐿)’s for\nboth models converge to the theoretical value of 𝑒𝜋/4. The noticeable increase in\namplitude for the three-state Potts model by TRG at 𝐿 > 102is attributed to the\neffectofthefinitebonddimensionandtheremaininglocalentanglement. Itisworth\nnotingthatourvaluefor 𝑥𝑆deviatesslightlyfromthevalue 𝑥𝑆=2.23035 3obtained\nin a previous study on the 9-state clock model [19]. We speculate that this minor\ndeviation is due to the finite bond-dimension effect because higher central charges\nlead to more pronounced numerical errors [36]. For the system size 𝐿=2048and\nbond dimension 𝐷=96, we ascertain 𝑥𝑆=2.193257for the Ising model, a value\nremarkably close to 𝑒𝜋/4=2.193280. Once we are certain of the value 𝑥𝑆=𝑒𝜋/4,\nwe can verify Eq. (3.6) for all the OPE coefficients, which are computed from the\n3Their paper showed that the three-leg FP tensor 𝑆∗had the same structure as a three-point\nfunction by numerical experiments. In this process, they treat 𝑥𝑆as a fitting parameter to reproduce\nknown OPE coefficients. Our results show that they are indeed three-point functions and 𝑥𝑆is a\nuniversal number and not a fitting parameter.\n63Figure 3.3: The OPE coefficients of the critical Ising model evaluated by setting\n𝑥𝑆=𝑒𝜋/4. The black dotted lines denote the theoretical values 0, 0.5, and 1. The\ndata points, denoted by filled circles \" ◦\" and crosses \"+,\" are obtained from Levin-\nTRG(𝐷=96) and Evenbly-TNR( 𝐷=40), respectively. Relatively large finite-size\neffects have universal scaling as tested in Sec. A.3.\nthree-leg tensor 𝑆as\n𝐶𝛼𝛽𝛾(𝐿)=(√\n2𝑒𝜋/4)𝑥𝛼+𝑥𝛽+𝑥𝛾2−𝑥𝛾𝑆𝛼𝛽𝛾(𝐿). (3.14)\nThe results for the critical Ising model are exhibited in Fig. 3.3. The obtained\nOPE coefficients are consistent with our theory with the finite-size effects of ex-\npectedscaling. Thefinite-sizeeffectoriginatesfromthetwistoperatoratthebranch\npoints [21, 34], whose scaling is universal. The detailed analysis is discussed in\nthe appendix. The same plot for the critical three-state Potts model is shown in the\nsupplemental information in the appendix. While it has less accuracy due to the\nstronger finite bond dimension effect for higher central charges, the result is still\nconsistent with the expected OPE coefficients.\nWe next computed four-point tensors 𝑇𝛼𝛽𝛾𝛿and compared with the theoretical\nvalues from Eq. (3.7), where the explicit forms of the four-point functions of the\ncriticalIsingmodelarelistedinthesupplementalinformationintheappendix. The\nresultisconsistentuptotwodigitsformosttensorelements,asshowninTable3.1.\nTheexceptionsare 𝑇𝜎𝜎𝜎𝜎and𝑇𝜎𝜎11,whosenumericalvaluesdeviateapproximately\n64Table 3.1: The comparison of the numerically-obtained fixed-point tensor 𝑇𝛼𝛽𝛾𝛿at\n𝐿=2048and the exact four-point function ⟨𝜙𝛼(−𝑥𝑇)𝜙𝛽(𝑖𝑥𝑇)𝜙𝛾(𝑥𝑇)𝜙𝛿(−𝑖𝑥𝑇)⟩plof\nthe Ising model with 𝑥𝑇=𝑒𝜋/2/2.\n𝛼𝛽𝛾𝛿𝑇𝛼𝛽𝛾𝛿(𝐿=2048)⟨𝜙𝛼𝜙𝛽𝜙𝛾𝜙𝛿⟩\n1111 1 1\n𝜎𝜎𝜎𝜎 0.610 0.645\n𝜎𝜎𝜖𝜖 0.0714 0.0716\n𝜎𝜖𝜎𝜖 0.000 0\n𝜖𝜖𝜖𝜖 0.0168 0.0168\n𝜎𝜎𝜖 10.0618 0.0765\n𝜎𝜖𝜎 10.133 0.140\n𝜎𝜎𝜎 10.000 0\n𝜖𝜖𝜖10.001 0\n𝜎𝜎110.708 0.736\n𝜎1𝜎10.639 0.675\n𝜖𝜖110.0863 0.0864\n𝜖1𝜖10.0439 0.0432\n𝜖𝜎110.000 0\n5% from the theoretical values. As for 𝑇𝜎𝜎𝜖 1, the deviation is almost 24% 4. This\ndiscrepancy,however,canbeattributedtofinite-sizeeffectsandbecomesnegligible\nfor infinite system sizes. To illustrate this, we define the finite-size deviation as (do\nnot confuse with temperature)\n𝛿𝑇𝛼𝛽𝛾𝛿≡𝑇∗\n𝛼𝛽𝛾𝛿−𝑇𝛼𝛽𝛾𝛿(𝐿).\nFigure 3.4 presents the values of 𝛿𝑇𝜎𝜎𝜎𝜎(𝐿),𝛿𝑇𝜎𝜎𝜖 1(𝐿), and𝛿𝑇𝜎𝜎11(𝐿)obtained\nfrom TRG calculations. A clear power-law decay with respect to the system size is\nobserved,supportingtheclaimthatthelargedeviationsforthoseelementsarefinite-\nsizeeffects. However,itisworthmentioningthattheexponentcloselyapproximates\n∼𝐿−1/3,hintingattheexistenceofanunderlyingtheorythatmightaccountforthis.\n4The TNR scheme has a similar performance at the same length-scale as seen in Fig. 3.4.\n65Figure3.4: Thefinite-sizeeffectofthefixedpointtensor 𝛿𝑇𝛼𝛽𝛾𝛿≡⟨𝜙𝛼𝜙𝑗𝛽𝜙𝛾𝜙𝛿⟩−\n𝑇𝛼𝛽𝛾𝛿(𝐿)from Levin-TRG( 𝐷=96, red) and Evenbly-TNR( 𝐷=40, blue). We plot\n𝛿𝑇𝛼𝛽𝛾𝛿of𝜎𝜎𝜎𝜎(“+\"),𝜎𝜎𝜖 1(“★\"),and𝜎𝜎11(“×\")withdifferentcolorsdepending\non the algorithm. The difference converges to zero for 𝐿→∞with the power-law\n∼𝐿−1/3.\n66C h a pte r 4\nCONCLUSION AND DISCUSSION\nIn the first part of Chapter 2, we discussed a method for computing the coupling\nconstantsusingrenormalizedtensorsbasedonthefinite-sizescalingtheoryofCFT.\nByplottingtheresultingvaluesateachscale,wewereabletovisualizetheRGflow,\nandweconfirmedthatthetheoreticalRGflows,asshowninFig.2.3,areconsistent\nwith the Ising models. Our methodology has undergone further validation through\nits application to the three-state Potts model and the XY model, as detailed in the\nappendixandinRef.[26],respectively. InthecaseoftheXYmodel,particularcare\nisrequiredintheperturbativecalculationsduetothemarginalnatureoftherunning\ncouplingconstantsunderconsideration. Despitethesecomplexities,wesuccessfully\ndemonstratethattheRGflowintheXYmodelalignswiththeKosterlitzRGflow[37],\nas confirmed by our inclusion of third-order perturbative effects. This finding is\ndepictedinFig.4.1,offeringavisualrepresentationoftheKosterlitzRGflowinthe\ncontext of the XY model.\nFigure 4.1: The RG flow of the classical XY model in two dimensions stands as\na quintessential example of a topological phase transition. This particular type\nof RG flow is commonly referred to as the Kosterlitz RG flow. The right panel is\nnumericallyobtainedRGflowinasimilarmanner. However,akeydistinctionliesin\nthe consideration of up to third-order perturbations in our computational approach.\nThe deviation in the smaller system size is due to the irrelevant perturbations.\nFurther details can be found in Ref. [26].\n67Ourmethodhastheadvantageofbeingabletoextractbothultravioletandinfrared\ninformation, making it a valuable tool for investigating gapped and crossover sys-\ntems. We applied this idea to reveal the asymptotic freedom of the Heisenberg and\nexotic cross-over behavior of RP2models in Ref. [38].\nIn the second part of the chapter, applying the methodology developed in the\nfirst part, we explored the impact of finite bond-dimension 𝐷on the RG flow.\nThe finiteness of the bond-dimension results in a finite correlation length 𝜉(𝐷), or\nequivalently in a non-zero gap in the energy spectrum of the corresponding one-\ndimensional quantum system. We find that this gap formation can be attributed\nto the emergence of a relevant perturbation enforced by the finite bond dimension.\nThis is demonstrated by the RG flow of the emergent relevant coupling.\nThe finite-size scaling of TNR shows a crossover at 𝐿∼𝜉(𝐷), above which the\nsystem is governed by the finite correlation length. The correlation length 𝜉(𝐷)\ninducedbythefinitebonddimensioninTNRshowsthesamescaling(2.21),(2.22)\nas the correlation length of MPS. While such scaling in TNR was suggested earlier\nin Refs. [26], in this thesis, we presented more convincing evidence.\nAlthough we do not have a mathematical proof for the scaling of 𝜉(𝐷)in TNR\nat this point, it may be natural from the following point of view. Besides the\nconstruction of the transfer matrix by contracting horizontal legs, the renormalized\ntensorobtainedinTNRcangivethecornertransfermatrixbycontractingtheupper\nand left legs. The same finite- 𝐷scaling (2.21), (2.22) as in MPS was observed\nin corner transfer matrix renormalization group (CTMRG) [39–41]. Moreover, the\nentanglement spectrum for the half-bipartition of the system of length 2𝐿can be\nrelatedtoacontractionoffourrenormalizedtensorsoflinearsize 𝐿[42],asshown\nin Fig. 4.2. These relations are suggestive of the identical scaling of 𝜉(𝐷)in MPS,\nCTMRG, and TNR as we have observed.\nOur study highlights the importance of considering the impact of the finite bond\ndimensionintheTNR-typeapproach. Inparticular,adirectstudyofthethermody-\nnamic limit with TNR would be prone to errors due to the finite correlation length\n𝜉(𝐷)imposed by the finite bond dimension. As a resolution of this problem, we\nhave demonstrated that accurate data for the thermodynamic limit can be extracted\nby finite-size scaling of TNR spectra obtained for system sizes smaller than 𝜉(𝐷),\ncombined with conformal field theory. Even with this limitation, the tractable sys-\ntemsizeisgreatlyincreasedfrom ∼log𝐷withexactdiagonalizationto 𝜉(𝐷)∼𝐷𝜅\nin TNR.\n68Figure 4.2: (Left panel) A schematic picture of the reduced density matrix 𝜌𝐴\nfor a bipartition of the system in the path integral picture. The uncontracted legs\ncorrespond to the indices of the reduced density matrix. (Right panel) Each of the\nfourquadrantsofthespace-timeintheleftpanelmaybereplacedbytherenormalized\ntensor in TNR with appropriate boundary conditions.\nMeanwhile, the unattainability of true critical fixed-point tensors in numerical\nsimulations does not prohibit their calculation by analytical approach. In Chapter\n3,wehavesuccessfullycomputedexplicitelementsofthecriticalfixed-pointtensor\nby field theory, which we have identified as the four-point function in CFT. This\nachievement enables us to directly extract the OPE coefficients from these tensor\nelements. When combined with the scaling dimensions derived from the transfer\nmatrix, this approach allows us to determine a complete set of CFT data for any\ncritical unitary lattice model.\nOne of the key strengths of our method for extracting OPE coefficients is its\nsimplicityandeffectivefinite-sizescalingproperties. Thesecharacteristicsmakeour\napproachparticularlyusefulforextrapolatingother,morecomplexOPEcoefficients.\nAsanexampleofthismethod’sapplication,wehavecomputedtheOPEcoefficients\nfor the three-state Potts model, detailed in Sec. A.3.\nDeterminingtheinfraredCFTsforlatticemodelsstandsasacornerstonechallenge\nin theoretical physics. This problem has garnered widespread attention across var-\nious domains of physics, including high energy physics, condensed matter physics,\nand statistical physics, due to its fundamental nature and broad implications.\nIn our research, we have introduced a novel solution to this long-standing is-\nsue, employing a synergistic blend of analytical and numerical techniques. This\napproach not only addresses the immediate problem at hand but also opens new\navenuesforexplorationandinquiryintherealmsofformalconformalfieldtheories\n69and numerical tensor networks. Our work, therefore, not only contributes to the\nresolution of a decade-old problem but also poses many intriguing questions for\nfuture research, potentially leading to significant advancements in both theoretical\nand applied physics\nLookingtothefuture,exploringthetensorelementscorrespondingtodescendants\nwithin the CFT framework presents an exciting avenue for further research. This\nexplorationcoulddeepenourunderstandingoftheintricatestructureswithinlattice\nmodels. Infact,followingourwork,Ref.[43]hasexamineddescendantsinasimilar\ncontext, indicating the growing interest and potential in this area of study.\nUltimately, our approach, which combines the exact treatment of RG and fixed-\npoint tensors in tensor networks, could significantly contribute to a more profound\nunderstanding of universality in lattice models. 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A 13, 1013–1030 (1998).\n74A p p e n d i x A\nAPPENDIX\nA.1 Perturbation theory on the finite-size spectrum\nIn the realm of effective field theory, particularly for lattice models or condensed\nmattersystems,thetheoryoftenencompassesvariousperturbationstotheConformal\nField Theory (CFT). Let us consider the effective Hamiltonian as follows:\nˆ𝐻=ˆ𝐻∗+∑︁\n𝑗𝑔𝑗∫\n𝑑𝑣ˆΦ𝑗(𝑣). (A.1)\nIn this formulation, 𝐻∗represents the Hamiltonian of the unperturbed CFT, and\nthe additional terms correspond to various perturbations, each characterized by a\ncoupling constant 𝑔𝑗and a field operator ˆΦ𝑗(𝑣). When all these perturbations are\ndeemed irrelevant in the RG context, the behavior of the system asymptotically\napproaches that described by the CFT in the large-distance or low-energy limit.\nHowever,itiscrucialtonotethattheseirrelevantperturbationscanstillsignificantly\ninfluencethefinite-sizespectrumofthesystem. Thisaspectwasextensivelystudied\nby Cardy, who delved into the effects of such perturbations on the finite-size spec-\ntrum[14]. Hisworkprovidesadeeperunderstandingofhowminordeviationsfrom\nthe idealized CFT can have measurable impacts in finite systems.\nNow,weapplythestandardRayleigh-Schrödingerperturbationtheorytoourgen-\neral Hamiltonian framework:\nH=H0+𝑉, (A.2)\nwhere𝑉is treated as a perturbation. The unperturbed eigenstates are defined as\nH0|𝑛(0)⟩=𝐸(0)\n𝑛|𝑛(0)⟩, (A.3)\nand the perturbative expansion of the energy eigenvalue is given by\n𝐸𝑛=𝐸(0)\n𝑛+𝜖𝐸(1)\n𝑛+𝜖2𝐸(2)\n𝑛, (A.4)\nwhere the first-order correction is\n𝐸(1)\n𝑛=⟨𝑛(0)|𝑉|𝑛(0)⟩. (A.5)\n75In this context, the unperturbed eigenstate corresponds to a primary state of CFT:\n|𝑛(0)⟩=|Φ𝑛⟩, (A.6)\nand the perturbation 𝑉is represented as\n𝑉=∑︁\n𝑗𝑔𝑗∫\n𝑑𝑣ˆΦ𝑗(𝑣), (A.7)\nUsing conformal mapping for the three-point correlation function, we obtain\n⟨Φ𝑖|ˆΦ𝑗(𝑣)|Φ𝑘⟩=𝐶𝑖𝑗𝑘\u00122𝜋\n𝐿\u0013𝑥𝑗\n𝑒2𝜋𝑖(𝑠𝑖−𝑠𝑘)𝑣/𝐿. (A.8)\nThus, for the matrix element of 𝑉, we have\n⟨𝑙(0)|𝑉|𝑛(0)⟩=∑︁\n𝑗𝑔𝑗∫𝐿\n0𝑑𝑣⟨Φ𝑙|ˆΦ𝑗(𝑣)|Φ𝑛⟩ (A.9)\n=𝛿𝑠𝑙,𝑠𝑛∑︁\n𝑗𝑔𝑗𝐶𝑛𝑗𝑙𝐿\u00122𝜋\n𝐿\u0013𝑥𝑗\n(A.10)\n=𝛿𝑠𝑙,𝑠𝑛2𝜋∑︁\n𝑗𝑔𝑗𝐶𝑛𝑗𝑙\u00122𝜋\n𝐿\u0013𝑥𝑗−1\n. (A.11)\nHence, for𝑙=𝑛, the first-order energy correction is\n𝐸(1)\n𝑛=2𝜋∑︁\n𝑗𝑔𝑗𝐶𝑛𝑛𝑗\u00122𝜋\n𝐿\u0013𝑥𝑗−1\n. (A.12)\nTherefore, to the first order in perturbation theory, the energy difference relative\nto the ground state is\n𝐸𝑛−𝐸0=2𝜋\n𝐿 \n𝑥𝑛+2𝜋∑︁\n𝑗𝑔𝑗𝐶𝑛𝑛𝑗\u00122𝜋\n𝐿\u0013𝑥𝑗−2!\n. (A.13)\nWhen the perturbations are irrelevant ( 𝑥𝑗>2), they give subleading corrections\nto the CFT scaling (A.13). In contrast, relevant perturbations with 𝑥𝑗<2eventu-\nally dominates the CFT scaling for a sufficiently large system size 𝐿, signalling a\nbreakdownoftheperturbationtheory. Nevertheless,theperturbationtheorycanbe\nusefulforsmallperturbations(atthemicroscopicscale)whenthesystemsizeisnot\nvery large. When 𝑥𝑗=2, the perturbation is marginal.\nThe formula (A.13) may be interpreted in an alternative way. We can apply the\nscale transformation so that the length of the system becomes 2𝜋. The scaled gap\n76Symbol Dimension Meaning\n𝐼 0 identity\n𝜖2\n5thermal op.\n𝜎1\n15spin\n𝑋7\n5\n𝑌 3\n𝑍2\n3\nTable A.1: A set of primary operators of the three-state Potts model.\nFigure A.1: The size dependence of the (a) 𝛿𝑥𝜎and (b)𝛿𝑥𝜖at𝑇=0.999995𝑇𝑐and\n𝑇=1.000005𝑇𝑐. Thepinkandgreendottedlinesdenote 𝐿−0.8,(a)𝐿1.2,and(b)𝐿2.4\nfittings respectively. For the low-temperature phase, the sign of 𝛿𝑥𝜎is negative at\n𝐿 > 100. The dip on the left panel around 𝐿∼102corresponds to the zero point\nof Eq. (A.17). (b) The finite-size effect to the 𝑥𝜖suffers less from 𝑇2\ncyl+¯𝑇2\ncylin\namplitude. The scaling of Eq. (A.18) is clearly observed.\nis𝐿(𝐸𝑛−𝐸0)/(2𝜋). Applying the perturbation theory to the system on the ring of\nradius 1,\n𝐿\n2𝜋(𝐸𝑛−𝐸0)=𝑥𝑛+2𝜋∑︁\n𝑗𝑔𝑗(𝐿)𝐶𝑛𝑛𝑗. (A.14)\nUsing the scale-dependent coupling constant\n𝑔𝑗(𝐿)=\u00122𝜋\n𝐿\u0013𝑥𝑗−2\n𝑔𝑗, (A.15)\nwhere𝑔𝑗is the “bare” value of the coupling constant that we find in Eq. (A.13).\n77A.2 Finite-entanglement scaling of the three-state potts model.\nDefinition of the model and its CFT\nWe can further verify the emergence of relevant perturbations by applying it to\nthe three-state Potts model. It is a natural extension of the Ising model to the Z3\nsymmetry, and the Hamiltonian is\n𝐻=−∑︁\n⟨𝑖,𝑗⟩𝛿𝑠𝑖,𝑠𝑗, (A.16)\nwhere𝑠𝑖takes 0, 1, and−1. It has a phase transition of Z3symmetry breaking at\n𝑇𝑐=1/log(1+√\n3). The critical theory of the three-state Potts model is another\ntype of the minimal model M(6,5)with𝑐=4\n5[3, 44]. A set of primary operators\nare shown in Table. A.1.\nAsopposedtotheIsingmodel,thereareoff-diagonaloperatorsas Φ2\n5,7\n5,Φ7\n5,2\n5and\nΦ3,0,Φ0,3(currents).\nLet us first examine the RG flow in a gapped system. Similar to the Ising model,\nthe phase transition is identified by spontaneous symmetry breaking. The high-\ntemperature phase is a trivial phase, whereas the low-temperature region is Z3\nsymmetry breaking phase. Thus, the fixed-point tensor is a stacking of three states\nwith their Z3charge 0,−1, and 1.\nConstruction of the effective Hamiltonian\nThe RG flow can be seen by investigating the scaling dimensions. For instance,\nwe can take the spin operator 𝜎= Φ 1\n15,1\n15and plot the value of 𝛿𝑥𝜎=𝑥𝜎(𝐿)−\n2\n15. Similarly, as in the Ising model, there is competition between irrelevant and\nrelevant operators: 𝑋=Φ 7\n5,7\n5and𝜖=Φ 2\n5,2\n5. The thermal operator separates the Z3\nsymmetry-breakingphasefromthetrivialone. Thefinite-sizecorrectionsof 𝑋and𝜖\nto𝑥𝜎are𝐿−0.8and𝐿1.2,respectively. Thefusionrulesare 𝜎×𝜎=1+𝜖+𝜎+𝑋+𝑌+𝑍,\n𝜖×𝜖=1+𝑋, and𝜖×𝜎=𝜎+𝑍. Hence,𝛿𝑥𝜎has the following form:\n𝛿𝑥𝜎=2𝜋𝐶𝜎𝜎𝑋𝑔𝑋\u0012𝐿\n2𝜋\u0013−0.8\n+2𝜋𝐶𝜎𝜎𝜖𝑔𝜖\u0012𝐿\n2𝜋\u00131.2\n. (A.17)\nOn the other hand, the perturbation of 𝜖appears as a second-order term for 𝛿𝑥𝜖\nbecause the fusion rule says 𝜖×𝜖=1+𝑋. Consequently, 𝛿𝑥𝜖can be computed as\n𝛿𝑥𝜖=2𝜋𝐶𝜖𝜖𝑋𝑔𝑋\u0012𝐿\n2𝜋\u0013−0.8\n+𝛼𝑔2\n𝜖\u0012𝐿\n2𝜋\u00132.4\n, (A.18)\nwhere𝛼is a constant determined from the second-order calculation.\n78FigureA.2: 36𝛿𝑥𝜎−𝛿𝑥𝜖forthehightemperaturephase. “+”isusedwhenthesign\nis negative. The red dotted line denotes the 𝐿−2fitting while the light green one is\njust a relevant 𝐿1.2contribution from 𝜖. Loop-TNR rotates the lattice by𝜋\n4at each\nRG step, and the tilted system is plotted with the blue dots.\nFigure. A.1 shows the computed 𝛿𝑥𝜎by TNR. As expected, it exhibits the com-\npetition between irrelevant and relevant operators. The sign of 𝑔𝜖is the opposite\nbetween two phases, which is a manifest indication of the RG flow in the opposite\ndirection due to the thermal operator. 𝑥𝜎has doubly degenerate states with Z3\ncharge±1. In the low-temperature phase, these two states flow to 𝑥𝜎(𝐿)→ 0, and\nthe fixed-point tensor becomes three-fold degenerate. As for the irrelevant pertur-\nbation, there seems to be a discrepancy between 𝛿𝑥𝜎in Fig. A.1 and Eq. (A.17).\nThe data points are scattered for small system sizes and not precisely on the fitting\nlines. Thisisduetotheleadingirrelevantoperatorwehavenotconsidered. Wecan\nidentify it as 𝑇2\ncyl+¯𝑇2\ncylas followings. Just as we did in the left panel of Fig. 2.3,\nthe contributions from 𝑔𝑋can be eliminated by combining 𝛿𝑥𝜎and𝛿𝑥𝜖. The OPE\ncoefficients for the three-state Potts model are known, and the ratio of the two OPE\ncoefficients is 𝐶𝜖𝜖𝑋/𝐶𝜎𝜎𝑋=36[45–49]. Thus, the origin of the \"scattering\" shall\nbe observed by plotting 36𝛿𝑥𝜎−𝛿𝑥𝜖.\nFigure. A.2 displays the result for the high-temperature phase. It is now obvious\nthat the scattering of Fig. A.1 comes from the 𝐿−2perturbation denoted with the\nred dotted line. Also, it has a conformal spin 𝑠because it flips a sign at each step\n79and𝑠≡4(mod 8) 1. As a result, we can conclude the irrelevant operator has the\nconformal weights as (ℎ,¯ℎ)=(4,0)and(0,4), which are𝑇2\ncyland ¯𝑇2\ncyl. Finally, the\neffectiveHamiltonianofthecriticalthree-statePottsmodelonthesquarelatticecan\nbe constructed as\n𝐻=𝐻∗\n𝑃𝑜𝑡𝑡𝑠+∫𝐿\n0𝑑𝑥h\n𝑔𝑋Φ7\n5,7\n5(𝑥)+𝑔𝑇(𝑇2\ncyl+¯𝑇2\ncyl)i\n. (A.19)\nFinite-Entanglement scaling\nAt the critical temperature of the Ising model, the finite- 𝐷effect proves to be a\nperturbation from the thermal operator. Let us verify it for the critical three-state\nPottsmodel. Duetotheirrelevantperturbationsfrom 𝑇2\ncyl+¯𝑇2\ncyl,thefinite-𝐷effects\nare clearer for 𝑥𝜖(𝐿)as seen in Fig. A.1( 𝑏). This is shown in Fig. 2.5 of the main\ntext. Here, we demonstrate that 𝛿𝑥𝜎(𝐿)also shows the universal behavior with\n𝐿/𝜉(𝐷). Figure. A.3 shows the rescaled correction to 𝛿𝑥𝜎(𝐿). For𝐿 > 𝜉(𝐷), the\nperturbation grows as 𝐿2denoted by a gray line, which means that the emergent\nperturbationscalesas 𝐿1.2. ComparedwithEq.(A.17),itisclearthattheemergent\nperturbationisfromthethermaloperator. However,asthesystemsizeincreases,the\nsecond-orderperturbationbecomespredominantasshownwithapinkline. As 𝜖is\nthemostrelevantoperatorthatispermittedbysymmetry,itsupportsourconjecture\nstated in the main text.\nA.3 Supplemental information on the fixed-point tensor\nConformal mapping of S\nThethree-legtensor 𝑆∗\n𝛼𝛽𝛾representsthethree-sidedthermofielddoublestate[21]\ncorresponding to the geometry in Fig. 1(a) in the main text. This manifold Σ𝑆is\nmapped to the plane by a conformal mapping\n𝑧=𝐿\n2𝜋[−ln(𝑤−𝑖)−𝑖ln(𝑤+1)+(1+𝑖)ln𝑤], (A.20)\nwhichmapsthethreepointsin Σ𝑆,(𝑧1,𝑧2,𝑧3)=(∞,𝑖∞,−(1+𝑖)∞),to(𝑤1,𝑤2,𝑤3)=\n(𝑖,−1,0). Then, the tensor element is\n𝑆∗\n𝛼𝛽𝛾\n𝑆∗\n111=|𝐽1|𝑥𝛼|𝐽2|𝑥𝛽|𝐽3|𝑥𝛾⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(0)⟩p𝑙, (A.21)\n1For each iteration, the lattice rotates by 45 degrees, and it corresponds to the conformal\ntransformation 𝑤=𝑒𝑖𝜋\n4𝑧on a complex plane. As the irrelevant perturbations 𝑇2\ncyland ¯𝑇2\ncylhave a\nconformal spin 4and−4, they get an additional factor (𝑒𝑖𝜋\n4)4=(𝑒𝑖𝜋\n4)−4=−1for an odd number of\nsteps. We can see this by plotting the data from even steps (original) and odd steps (tilt) separately.\n80Figure A.3: Rescaled 𝛿𝑥𝜎by𝜉(𝐷)=𝐷𝜅at the critical temperature. The resulting\ndatacollapseontoauniversalfunctionthatisindependentof 𝐿/𝜉(𝐷). If𝐿/𝜉(𝐷)<\n1, the system is in the FSS region, while if 𝐿/𝜉(𝐷)≥1, it is in the FES region.\nIn the FES region, the scaling of the first-order and second-order perturbations are\nindicated by a gray and pink line, respectively. 𝑥𝜎is computed as an average value\nof the first and second excitation energy.\nwhere𝐽𝑖is the Jacobian of the conformal mapping (A.20). The initial states are\n|𝜙1⟩=\u00122𝜋\n𝐿\u0013−𝑥𝛼\nlim\n𝑧→∞𝑒2𝜋𝑧𝑥𝛼/𝐿𝜙𝛼(𝑧)|𝐼c𝑦𝑙⟩,\n|𝜙2⟩=\u00122𝜋\n𝐿\u0013−𝑥𝛽\nlim\n𝑧→𝑖∞𝑒−𝑖2𝜋𝑧𝑥𝛽/𝐿𝜙𝛽(𝑧)|𝐼c𝑦𝑙⟩,\n|𝜙3⟩= √\n2𝜋\n𝐿!−𝑥𝛾\n(A.22)\nlim\n𝑧→(−𝑖−1)∞𝑒(𝑖−1)√\n22𝜋√\n2𝐿𝑧𝑥𝛾𝜙𝛾(𝑧)|𝐼c𝑦𝑙⟩.\nThe Jacobian can be computed as\n|𝐽1|=\f\f\f\f\f\u00122𝜋\n𝐿\u0013−1\nlim\n𝑧→∞𝑒2𝜋𝑧/𝐿𝑤′(𝑧)\f\f\f\f\f\n=\f\f\f\f\f\u00122𝜋\n𝐿\u0013−1\nlim\n𝑤→𝑖𝑒2𝜋𝑧/𝐿\u0012𝑑𝑧\n𝑑𝑤\u0013−1\f\f\f\f\f. (A.23)\n81Using Eq. (10) in the main text, the first and second term is\n𝑒2𝜋𝑧/𝐿=exp[ln𝑤\n𝑤−𝑖+𝑖ln𝑤\n𝑤+1], (A.24)\n𝑑𝑧\n𝑑𝑤=𝐿\n2𝜋\u0014\n−1\n𝑤−𝑖−𝑖\n𝑤+1+(1+𝑖)\n𝑤\u0015\n. (A.25)\nSubstituting these into Eq. (A.23),\n|𝐽1|=\f\f\f\flim\n𝑤→𝑖𝑤\n𝑤−𝑖exph\n𝑖ln𝑤\n𝑤+1i\u0012\u0014\n−1\n𝑤−𝑖−𝑖\n𝑤+1+(1+𝑖)\n𝑤\u0015\u0013−1\f\f\f\f\n=\f\f\f\fexp\u0012\n𝑖ln𝑖\n1+𝑖\u0013\f\f\f\f\n=𝑒−𝜋/4. (A.26)\nIn the same way, we can show |𝐽2|=|𝐽3|=𝑒−𝜋/4. Thus, the 3-leg tensor is\n𝑆∗\n𝛼𝛽𝛾=𝑒−𝜋\n4(𝑥𝛼+𝑥𝛽+𝑥𝛾)⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(0)⟩p𝑙. (A.27)\nConformal mapping of T\nThe conformal mapping from the four-sided thermofield double state is\n𝑧=𝐿\n2𝜋[−ln(𝑤−𝑖)+log(𝑤+𝑖)−𝑖ln(𝑤+1)+𝑖ln(𝑤−1)]\n=𝐿\n2𝜋\u0014\nln\u0012𝑤+𝑖\n𝑤−𝑖\u0013\n+𝑖ln\u0012𝑤−1\n𝑤+1\u0013\u0015\n. (A.28)\nTo compute the Jacobian, we compute\n𝑒2𝜋𝑧/𝐿=exp\u0014\nln𝑤+𝑖\n𝑤−𝑖+𝑖ln𝑤−1\n𝑤+1\u0015\n, (A.29)\n𝑑𝑧\n𝑑𝑤=𝐿\n2𝜋\u0014\n−1\n𝑤−𝑖+1\n𝑤+𝑖−𝑖\n𝑤+1+𝑖\n𝑤−1\u0015\n. (A.30)\nThe Jacobian is then computed similarly as before:\n|𝐽1|−1=lim\n𝑤→𝑖\f\f\f\f𝑒−2𝜋𝑧/𝐿\u0014\n−1\n𝑤−𝑖+1\n𝑤+𝑖−𝑖\n𝑤+1+𝑖\n𝑤−1\u0015\f\f\f\f\n=𝑒𝜋/2\n2. (A.31)\nThe four-point function thus transforms as\n𝑇∗\n𝛼𝛽𝛾𝛿\n𝑇∗\n1111=|𝐽1|𝑥𝛼|𝐽2|𝑥𝛽|𝐽3|𝑥𝛾|𝐽4|𝑥𝛿⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(1)𝜙𝛿(−𝑖)⟩p𝑙,\n=\u0012𝑒𝜋\n2\n2\u0013−𝑥t𝑜𝑡\n⟨𝜙𝛼(−1)𝜙𝛽(𝑖)𝜙𝛾(1)𝜙𝛿(−𝑖)⟩p𝑙. (A.32)\n82Figure A.4: The contraction of the fixed-point tensors. We obtain 𝑆from TRG and\ncombine together to make 𝑆∗and𝑇∗. In this way, 𝑇∗respects reflection symmetry\nalong the dotted lines in addition to 𝐶4rotation symmetry.\n𝐷4-symmetric TRG\nWe use the TRG scheme which aligns closely with the original paper’s method-\nology [6]. In principle, singular-value decomposition (SVD) of the four-leg tensor\nshouldyieldtwoidenticalsymmetrictensors,giventhe 𝐷4symmetryoftheoriginal\ntensor. However,numericalerrorssometimesmakethesetwotensorsnon-identical.\nTo mitigate this, we consistently select one of the three-leg tensors and supplement\nthe other with its reflection. By adopting this approach, the fixed-point tensors,\ndepicted in Fig. A.4, maintain the 𝐷4symmetry at every RG step by construction.\nFour-point function of the critical Ising model\nHere,welistthefour-pointfunctionoftheIsingmodel. Giventhefourcoordinates\n𝑧𝑖anditscross-ratio 𝑥≡(𝑧12𝑧34)/(𝑧13𝑧24),thefour-pointfunctionsoftheIsingCFT\nare\n⟨𝜖4⟩=\f\f\f\f\f\fÖ\n1≤𝑖<𝑗≤4𝑧−1\n3\n𝑖𝑗1−𝑥+𝑥2\n𝑥2\n3(1−𝑥)2\n3\f\f\f\f\f\f2\n,\n⟨𝜎2𝜖2⟩=\f\f\f\f\f\u0014\n𝑧1\n4\n12𝑧−5\n8\n34(𝑧13𝑧24𝑧14𝑧23)−3\n16\u00151−𝑥\n2\n𝑥3\n8(1−𝑥)5\n16\f\f\f\f\f2\n,\n⟨𝜎4⟩=|𝑧13𝑧24|−1/4|1+√\n1−𝑥|+|1−√\n1−𝑥|\n2|𝑥|1\n4|1−𝑥|1\n4.\n83Figure A.5: The finite-size corrections 𝛿𝐶𝛼𝛽𝛾(𝐿)obtained from the numerical\nsimulation of the critical Ising model. The numerical results for higher energy\nlevels𝛿𝐶𝜖𝜖1(𝐿)and𝛿𝐶1𝜖𝜖(𝐿)sufferfromfinite- 𝐷effectsfor𝐿 >100. Thescalings\nof the finite-size corrections are nevertheless universal, which is consistent with\nTable III in Ref. [34]\nThefunctionsaboveareusedtoevaluatetheanalyticFPtensorelementsinthemain\ntext.\nUniversal finite-size corrections\nHere, we discuss the finite-size corrections to Eq. (13) in the main text. The\nfinite-size corrections of the OPE coefficients are defined as\n𝛿𝐶𝛼𝛽𝛾(𝐿)=|𝐶𝛼𝛽𝛾−𝐶𝛼𝛽𝛾(𝐿)|, (A.33)\nwhere𝐶𝛼𝛽𝛾(𝐿)is defined in Eq. (13) in the main text. We found that 𝛿𝐶𝛼𝛽𝛾(𝐿)\nexhibits a universal power-law decay as\n𝛿𝐶𝛼𝛽𝛾(𝐿)∼𝐿−𝑝𝛼𝛽𝛾. (A.34)\nOurnumericalresultssuggest 𝑝𝛼𝛽𝛾=1/2for(𝛼,𝛽,𝛾)=(1,1,𝜖),(1,𝜖,1),(𝜖,𝜖,𝜖),\n(1,𝜎,𝜎),(𝜎,𝜖,𝜎),(𝜎,𝜎, 1),and(𝜎,𝜎,𝜖), and𝑝𝛼𝛽𝛾=2for(𝛼,𝛽,𝛾)=(𝜖,𝜖,1)\nand(1,𝜖,𝜖)as shown in Fig. A.5. Similar universal scalings were discussed in\nRef. [34], where they considered the overlap of critical wavefunctions 𝐴𝛼𝛽𝛾=\n⟨𝜙3∗\n𝛾|𝜙1\n𝛼𝜙2\n𝛽⟩. Thethreewavefunctionsaredefinedonaringwithacircumferenceof\n84𝐿1,𝐿2, and𝐿3=𝐿1+𝐿2, respectively, and the lower indices are the label of the\ncorresponding primary states. Ref. [34] found the overlap of wavefunctions to be\n𝐴𝛼𝛽𝛾\n𝐴111∼\u0012𝐿3\n𝐿1\u0013𝐿1\n𝐿3\u0012𝐿3\n𝐿2\u0013𝐿2\n𝐿3−𝐿3\n𝐿1𝛼−𝐿3\n𝐿2𝛽+𝛾\n𝐶𝛼𝛽𝛾+ ˜𝐴(𝑝)\n𝛼𝛽𝛾𝐿−𝑝𝛼𝛽𝛾\n3,(A.35)\nwhere𝑝𝛼𝛽𝛾is the leading finite-size correction and ˜𝐴(𝑝)\n𝛼𝛽𝛾is a prefactor that is\nindependent of 𝐿3.\nOur scaling exponents 𝑝𝛼𝛽𝛾in Eq. (A.34) coincide with those from the previous\nworkinEq.(A.35)forallfusionchannels(seeTableIIIofRef.[34]). Thisuniversal\nscaling can be explained by considering rings 1 and 2 as an orbifold theory. The\nscaling𝑝𝛼𝛽𝛾=1/2is then attributed to the difference in the scaling dimensions\nof the orbifold theory, which is 𝑥𝜖/2=1/2. (See Ref. [34] for details.) Similarly,\nwe conjecture that the universal scaling for 𝛿𝑇𝛼𝛽𝛾𝛿∼𝐿−1/3can be understood by\nconsidering the three of four legs to be an orbifold theory.\nThe three-State Potts model\nHere,wepresenttheOPEcoefficientsobtainedfromnumericalsimulationsofthe\nclassicalcriticalthree-statePottsmodel. Thelow-lyingprimarystatesofthismodel\nare the identity operator \"1\", the two spin operators \" 𝜎,\" and the thermal operator\n\"𝜖,\" whose scaling dimensions are 0, 2/15, and 4/5. The non-trivial coefficients is\n𝐶𝜎𝜎𝜖=0.546[50]. Figure.A.6exhibitsthenumericalresultsfromLevin-TRGand\nEvenbly-TNR as the Ising model in the main text. TRG/TNR schemes, generally\nspeaking, have finite- 𝐷effects for larger system sizes, and this effect is larger in\nhigher central charges. Since the central charge 𝑐=0.8of the three-state Potts\nmodel is larger than 𝑐=0.5of the Ising model, these numerical errors manifest\nin the data plots. In particular, the TRG data is unstable due to CDL tensors and\nquickly diverts from the theoretical values. However, Evenbly-TNR’s results still\nconverge to the correct values.\n85FigureA.6: TheOPEcoefficientsofthecriticalthree-statePottsmodelevaluatedby\nsetting𝑥𝑆=𝑒𝜋/4. Theblackdottedlinesdenotethetheoreticalvalues0,0.546,and\n1 [50]. The data points, denoted by filled circles \" ◦\" and crosses \"+,\" are obtained\nfrom Levin-TRG( 𝐷=88) and Evenbly-TNR( 𝐷=40), respectively.\n86"    },    {        "title": "2401.18076v3.Searching_for_scalar_field_dark_matter_with_LIGO.pdf",        "content": "Searching for scalar field dark matter with LIGO\nAlexandre S. Göttel1,†, Aldo Ejlli2, Kanioar Karan2, Sander M.\nVermeulen3, Lorenzo Aiello4,5, Vivien Raymond1, and Hartmut Grote1\n1Gravity Exploration Institute, Cardiff University, Cardiff CF24 3AA, United Kingdom\n2Max-Planck-Institute for Gravitational Physics and Leibniz University Hannover, Callinstr. 38, 30167 Hannover, Germany\n3California Institute of Technology, Department of Physics, Pasadena, California 91125, USA\n4Università di Roma Tor Vergata, I-00133 Roma, Italy\n5INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy and\n†gottela@cardiff.ac.uk\n(Dated: March 11, 2024)\nWe report on a direct search for scalar field dark matter using data from LIGO’s third observing\nrun. We analyse the coupling of size oscillations of the interferometer’s beamsplitter and arm test\nmasses that may be caused by scalar field dark matter. Using new efficient search methods to\nmaximise sensitivity for signatures of such oscillations, we set new upper limits for the coupling\nconstants of scalar field dark matter as a function of its mass, which improve upon bounds from\nprevious direct searches by several orders of magnitude in a frequency band from 10 Hzto180 Hz.\nLaser interferometers, with their extreme sensitivity to\nminute length changes, have revolutionized astronomy\nwith a wide range of gravitational-wave (GW) detections\nover the last years [1]. Due to their capabilities at or\nbeyond quantum limits, GW detectors can also be used\ndirectly in the search for new physics, without the media-\ntion of gravitational waves, for example in the search for\ndark matter (DM). Several ideas have been put forward\nas to how different candidates of DM can directly cou-\nple to GW detectors, ranging from scalar field DM [2–4]\nto dark photon DM [5, 6], and to clumpy DM coupling\ngravitationally or through an additional Yukawa force [7].\nScalar field DM influences objects by accelerating them\nwhen a field gradient exists, and by expanding them (and\naltering their refractive index) without net acceleration.\nUpper limits for scalar field DM have been set using data\nfromtheGEO600GWdetector[8], theFermilabHolome-\nter [9], and LIGO [10]. The scalar field searches of the\nGEO600 and Holometer instruments used the dominant\nexpansion (and refractive index) effect in those instru-\nments, while the work of [10] used the acceleration effect.\nLikewise, upper limits on Dark photon DM, which is in\nsimplified terms represented by a vector field causing ob-\njects to accelerate, have been set using data from the first\n(O1) and third (O3) observing runs of the LIGO detec-\ntors [11, 12].\nInthiswork, weanalysethe expansion effectofscalarfield\nDM on dual-recycled Fabry-Pérot Michelson interferom-\neters such as LIGO [13], Virgo [14], or KAGRA [15]. We\nalso develop enhanced spectral search techniques that are\noptimised to search efficiently at a lower frequency ( i.e.\nmass) range of DM, and apply these to search for DMsignatures using data of the third observing run of the\ntwo LIGO observatories. Not finding viable candidates,\nwe set new upper limits that surpass existing bounds by\nup to three orders of magnitude in a frequency band from\n10 Hzto180 Hz, and are competitive up to 5000 Hz.\nExpected dark matter signal\nThe astronomically-inferred mass density associated with\nDMmaybeattributedtoanundiscoveredscalarfieldwith\nahighoccupationnumber. Modelsofweakly-coupledlow-\nmass( ≪1 eV)scalarfieldspredictthatsufficientparticles\ncouldbeproducedintheearlyuniversethroughavacuum\nmisalignment mechanism and manifest as the observed\nDM. This scalar field DM would behave as a coherently\noscillating classical field [2, 16]:\nϕ(t,⃗ r) =ϕ0cos\u0010\nωϕt−⃗kϕ·⃗ r\u0011\n, (1)\nwhere ωϕ=mϕc2/ℏis the angular Compton frequency,\n⃗kϕ=mϕ⃗ vobs/ℏis the wave vector, mϕis the mass of the\nfield, and ⃗ vobsthe velocity relative to the observer. The\namplitudeofthefieldcanbesetas ϕ0=ℏ√2ρlocal/(mϕc),\nunder the assumption that this scalar field constitutes the\nlocal DM density ρlocal[17].\nMoreover, dynamical models of scalar field DM pre-\ndict that such matter would be trapped and virialised in\ngravitationalpotentials,leadingtoaMaxwell-Boltzmann-\nlike distribution of velocities ⃗ vobsrelative to an observer.\nAs non-zero velocities produce a Doppler-shift of the ob-\nserved DM field frequency, this virialisation results in the\nDM field having a finite coherence time or, equivalently, a\nspread in observed frequency (linewidth) [5, 18]. The ex-\npectedlinewidthis ∆ωobs/ωobs∼10−6forDMtrappedinarXiv:2401.18076v3  [astro-ph.CO]  8 Mar 20242\nthe galactic gravity potential, as in the standard galactic\nDM halo model. Similarly, the observed frequency ωobs\nshifts from ωϕby a∝v2\nobsterm, negligible in our analysis.\nScalar field DM could couple to the fields of the Stan-\ndard Model (SM) in various ways. These couplings are\nmodelled by the addition of a parameterised interaction\nterm to the SM Lagrangian [19, 20]. In this paper, we\nconsider linear interaction terms with the electromagnetic\nfield tensor Fµνand the electron rest mass me:\nLint⊃ϕ\nΛγFµνFµν\n4−ϕ\nΛeme¯ψeψe, (2)\nwhere ψe,¯ψearetheSMelectronfieldanditsDiracconju-\ngate, respectively, and Λγ,Λeparameterise the coupling.\nThey can also be expressed in terms of the dimensionless\nparameters deanddme, with de,m e=MPl/(√\n4πΛγ,e),\nwhere MPlis the Planck mass. The terms in Eq. (2)\ncause effective changes of the fine structure constant α\nand the effective rest mass me[16, 21]. These changes in\nturn modify the lattice spacing and electronic modes of\nsolids, driving modulations of size land refractive index\nn:\nδl\nl=−\u0012δα\nα+δme\nme\u0013\n, (3)\nδn\nn=−5·10−3\u0012\n2δα\nα+δme\nme\u0013\n, (4)\nwhere δxdenotes a change of the parameter x:\nx→x+δx. Eqs. 3, 4 hold in the adiabatic limit, which\napplies for solids with a mechanical resonance frequency\nmuch higher than the driving frequency ωϕ[3, 22, 23].\nIn the LIGO interferometers, light from a laser source im-\npinges on a beamsplitter (BS) and splits into two orthog-\nonal arms, each containing a Fabry-Pérot cavity (com-\nprised of two mirrors, referred to as test masses) to in-\ncrease the effective optical path length and optical power\nof the arms. A sketch of this optical layout can be seen\nin Fig. 1. While all components of the interferometers\ncan be affected by DM, the BS has been identified as a\ndominant coupling element for scalar field dark matter, as\nargued in [3]. This is because the “splitting” effect occurs\non one surface of the BS and not at its centre of mass,\nsee Fig. 1. This results in DM causing a path length dif-\nference between the arms.\nFor a BS of thickness tBand index of refraction n, one\nexpects from Eqs. (1) and (3) a length change to be pro-\nduced in the LIGO interferometers [3]:\nδ(Lx−Ly)≈\u00121\nΛγ+1\nΛe\u0013\n·n tBℏ√2ρlocal\nmϕc,(5)where δ(Lx−Ly)is the optical path difference between\nboth arms and we have neglected the contribution of the\nrefractive index changes to the signal, as it is more than\ntwo orders of magnitude smaller than that of the size\nchanges. In this work, for the first time, we also take\ninto account the contribution of the four arm test masses,\nwhich predominantly produce a signal by changing the\noptical path lengths within the arms. While this effect\nmostly cancels out if the test masses have identical thick-\nnesses, as pointed out in [3], we find that the real small\nthickness differences between LIGO’s test masses lead to\nnon-negligible additions to the BS-induced coupling.\nLengthfluctuations,suchasthosecausedbytheBSand\nthe test mass couplings, are transduced by the optical in-\nterferometric setup into signals on the photodetector (see\nFig. 1). This conversion can be represented by so-called\ntransfer functions , which describe how the interferometer\nresponds to signals of different frequencies. In LIGO, the\nphotodetector signal IPD(ω)is calibrated to GW-induced\nstrain h(ω)according to:\nh(ω) =IPD(ω)\nLTGW(ω)eiϕGW, (6)\nwhere Lis the arm length of the interferometer ( ≈4 km),\nand T GWis the optical transfer function from GW-\ninduced strain (with phase ϕGW) to photodetector signal.\nHowever, to search for the expansion effect of scalar field\ndark matter, we are interested in a different type of strain\nthat corresponds to thickness changes of the optical com-\nFIG. 1. Simplified optical layout of a LIGO-type inter-\nferometer. A beamsplitter BS splits a laser beam into\ntwo long arms that contain Fabry-Pérot cavities com-\nposed of test masses ITMX/Y and ETMX/Y, respec-\ntively. The interferometric output is read by a photode-\ntector PD.3\nponents of the interferometer, as described by Eq. (5) for\nthe beamsplitter. To also take into account the effect of\nthe arm test masses, we define:\ntM= (tETMY +tITMY )−(tETMX +tITMX ),(7)\nwhere tETMY,tITMY,tETMX, and tITMXrepresent the\nmirror thicknesses for the End Test Mass (ETM) and In-\nput Test Mass (ITM) in the Y- and X- interferometer\narms, respectively. The thickness variations of the op-\ntics under the effect of DM, i.e.the DM-induced strain\nsDM(ω), can then be expressed as:\nsDM(ω) =IPD(ω)\n|n tBTBeiϕB+tMTMeiϕM|,(8)\nwhere T Band T Mare the transfer functions correspond-\ning to the beamsplitter and test mass effects, respectively,\nandϕBandϕMare the phases of those transfer functions.\nSince we have access to the GW-induced strain h(ω)only\n(and not IPD(ω)), we express the DM-induced strain as:\nsDM(ω) =h(ω)·\f\f\f\fLTGWeiϕGW\nn tBTBeiϕB+tMTMeiϕM\f\f\f\f.(9)\nFinally, we can express h(ω)in relation to the coupling\nconstants ΛγandΛe:\nh(ω)·Acal(ω)≈\u00121\nΛγ+1\nΛe\u0013\n·\u0012ℏ√2ρlocal\nmϕc\u0013\n,(10)\nwhere\nAcal=TGW·L/(n tBTB)p\n1 + 2 ψcos(ϕB−ϕM) +ψ2;ψ=tMTM\nn tBTB.\nWe derive the required GW and DM transfer functions\nconsidering both the BS and test mass effects using a\nsimulation-based approach. While the underlying prin-\nciples of the transfer function calculation are rigorously\nunderstood analytically, see [24], this approach is bet-\nter suited for handling the many complexities specific to\nindividual optical setups. We achieve this using the Fi-\nnesse [25] software package, which was designed to model\noptical-interferometric systems in the frequency domain\nand has been widely corroborated experimentally. We\nnote that the simulation also takes into account any ef-\nfects stemming from the light travel time, as have been\npointed out in [6]. We also considered the phase differ-\nence between end test masses that is caused by the finite\nde Broglie wavelength of the DM field. This increases the\ntotal transfer function’s magnitude by about 5%at5 kHzwhen averaged over Earth’s sky coverage. Given its neg-\nligible impact on our results below, when compared to\nstatistical uncertainty, we disregard this effect here.\nThe obtained transfer functions, as well as individual re-\nsults for BS and test mass effects are shown in Fig. 2 for\nthe LIGO Livingston (LLO) and Hanford (LHO) obser-\nvatories, respectively. The most relevant optical data for\nthis simulation are listed in Table I. As can be seen from\nLHO LLO\nThickness Transm. Thickness Transm.\n(mm) (%) (mm) (%)\nBS 60.41 50 59.88 50\nITMX 199.763 1.5 199.960 1.48\nETMX 199.846 3.9e-4 199.245 7.1e-4\nITMY 199.904 1.5 199.290 1.48\nETMY 199.792 3.8e-4 199.954 7.6e-4\nTABLE I. Relevant thickness and transmission val-\nues [26] of relevant optical components in LHO and\nLLO, as used for our transfer function simulations. See\nFig. 1 for an overview of the different components and\ntheir meaning.\nFIG. 2. Simulated transfer functions as a function of\nfrequency. a)TGWfor both interefometers, b) and c):\nTMfor the test-mass effect (dashed line) and TSfor the\nBS effect (dot-dashed line), see text, and their in-phase\ncombination (dotted line) for LLO and LHO, respec-\ntively.\nthe middle and bottom panels in Fig. 2, the influence of4\nthe BS in both detectors becomes dominant at 10 Hzto\n20 Hz. The differences between the detectors is caused by\nsmall differences in test mass thicknesses.\nLogarithmic spectral analysis\nGiven the frequency-dependence of the expected DM sig-\nnal, with ∆ωobs/ωobs∼10−6, maximising signal to noise\nratio in our analysis implies a bin spacing in frequency\nspace with the same constant width-to-frequency ratio as\nthat expected from the signal [8, 27]. The dataset used\nin this paper is from LIGO’s third observing run [1]. We\nuse 40 segments of data that are at least 28 hin length,\nto ensure integration over at least one coherence time at\nour lower frequency bound ( 10 Hz), with a total of about\n1500 hof data sampled at 16 kHz. The calculation of Dis-\ncrete Fourier Transforms (DFT) we thus require presents\na unique technical challenge, as it explores unprecedented\nfrequencies (below 50 Hz) for this kind of analysis. It\nfor example needs to use amounts of data ( O(100GB))\nexceeding typical memory capacities. More importantly,\nthe cost of this calculation scales as O(N2), where N\nis the number of data points, leading to a prohibitively\nexpensive regime. While methods exist to accelerate log-\narithmic DFT calculations [28, 29], none reach the Fast\nFourierTransforms’(FFT)speed,andnoexistingpackage\nsatisfies this analysis’ requirements (including memory).\nThe use of logarithmic bin spacing precludes the use of\nthe FFT due to the resultant frequency-dependent terms\nand variable data points in the DFT calculations. How-\never, we observed that this frequency-dependence was rel-\natively weak, allowing us to implement small approxima-\ntions to modify the DFT. This adjustment allowed for\nthe effective use of FFTs, significantly enhancing compu-\ntational efficiency. For details about this calculation and\nthe aforementioned software requirements see [30]. Over-\nall, we achieve a speed-up factor of O(104)with negligible\nimpact on the results.\nSince the effect of DM on the detector cannot be\n“turned off”, it is necessary to build a background model\nthat is resistant to the influence of existing peaks in the\ndata (see [18]). This is done, after having calculated the\nPSD, by implementing a recursive procedure making use\nofsplinefitssimilartothatusedbyLIGOcalibration[31].\nIn each iteration, bins containing identified peaks are re-\nmoved and the fits are repeated on the “cleaned” data,\nuntil the solutions converge. The method was validated\nby varying the degrees of freedom of the aforementioned\napproximation.\nWe find empirically that the residuals of the background\nmodel with respect to the log of the observed PSD arewell described by a skew-normal distribution. While the\nparameters of said distribution vary over frequency, this\nvariationissmall: thefollowinganalysisisthusperformed\nin chunks of 10,000 frequency bins in which the distribu-\ntion parameters can safely be viewed as constant. We\nuse a likelihood-based analysis in order to combine data\nfrom different segments (in time) and from the different\ninterferometers. The likelihood is defined as:\nL(µ,⃗θ,˜⃗θ) =X\nseg,ifoX\nj=0logfseg\u0010\ngj,seg,ifo (µ,⃗θ),˜⃗θ\u0011\n,(11)\nwhere the sums are over data segments and frequency, re-\nspectively, µis proportional to the amplitude of the DM\npeak, fsegis a skew normal distribution with parameters\nheldby˜⃗θ, thesubscript jdenotesthefrequencybinindex,\nseg the data segment, and ifo the corresponding interfer-\nometer. gseg,iforepresents the residuals between the data\nand the expected background, with a term to allow for\nDM effects:\ngj,seg,ifo = log Yseg−log\u0000\nebkgseg+µ·βifo\u0001\n,(12)\nwhere Y(ω)is the PSD data based on h(ω),bkg(ω)refers\nto the fitted background shape in log space, and βifo(ω)\nis an interferometer-specific calibration term based on\nEq. (10). Frequency-dependence throughout the equation\nis left implicit for simplicity. The parameter µ= Λ−2\niwas\nchosen because, as can be seen in Eq. (10), it is not possi-\nble to differentiate between a non-zero Λ−1\nγorΛ−1\ne.Λ−1\ni\ncan thus be interpreted as being either one of the two cou-\npling constants, under the assumption that the other is\nnull. Although the proximity of the interferometers could\nallow us to exploit coherence effects, the abundance of\nnon-coincident data segments in our dataset prompted us\nto disregard this method, as we estimated that it would\nlead to only a roughly 10% improvement in our results.\nIn this framework, finding a DM signal is thus equivalent\nto rejecting the hypothesis that µ= 0. Additionally, in\norder to correctly make use of the fact that physically,\nµcannot be negative, we use the profile-likelihood-ratio\nbased test statistic q0, as described in eq. (12) in [32],\nand corresponding asymptotic methods, to search for a\npositive signal.\nConversely, in order to calculate the upper limit on Λ−1\ni,\nthe complementary test-statistic ˜qµas described in eq.\n(16) in [32], was used. This test-statistic correctly ac-\ncounts for cases where the value of µmaximizing the like-\nlihood is greater than the hypothesized value, ensuring\nthat upward noise fluctuations are not considered as less\ncompatible with a given upper limit.5\nThis approach enabled the search for DM signals by\nidentifying local excesses in the q0value across different\nfrequency bins. A 5σthreshold, corrected for the look-\nelsewhere-effect , resulted in 349 candidates, which was re-\nduced to 159 by associating neighbouring over-threshold\nbins to single candidates. Finally, since the reconstructed\namplitude of DM should not vary much over time, the\nconsistency of the results was further probed with a t=5\nthreshold on a student-t test comparing results from dif-\nferentsegmentcombinations. Afinalcutontheremaining\n42candidateswasthensetonrequiringthatbothinterfer-\nometers have results that are significantly different from\nzero. Faced with the lack of surviving candidates, our\nupper limits can be seen in the context of other measure-\nments in Fig. 3 for Λ−1\neandΛ−1\nγ, respectively.\nThese results assume a local dark matter density ρCDM =\na)\nb)\nFIG. 3. Upper limit on Λ−1\ni(95% C.L.) as a function of\nfrequency. a) and b) depict our results in the context\nof other experimental results on Λe,Λγ, respectively.\nOur results are shown by the thick blue line, constraints\nfrom direct experimental searches for DM [8–10, 33–43]\nare shown in thin grey, and constraints from searches\nfor ‘fifth forces’ [44, 45] are depicted by the dashed red\nlines. Our results were smoothed for visual purposes.0.4GeV/cm3(as in [46] for the standard smooth DM halo\nmodel). Models in which DM forms a relaxion halo [47,\n48] predict local DM overdensities of up to ρRH/ρCDM≤\n1016[49]. Our results impose significantly more stringent\nconstraints on the coupling constants for higher assumed\nvalues of the DM density ρA> ρ CDM: the constraint\nbecomes more stringent by a factor (ρA/ρCDM)1/2(see\nEq. 5).\nOur limits represent a several order of magnitude im-\nprovement on other direct searches in a band from 10 Hz\nto180 Hz(roughly 5×10−14eVto1×10−12eV). The\nmain limiting factor being detector noise, we expect those\nresults to be improved greatly in future LIGO runs and\nwith future gravitational wave detectors. We emphasise\nin particular that the results could also be improved dras-\ntically by increasing mirror thickness differences in the\ninterferometer arms, for which this study paves the way.\nAcknowledgements\nThe authors are grateful for support from the Sci-\nence and Technology Facilities Council (STFC), grants\nST/T006331/1 and ST/W006456/1 for the Quantum\nTechnologiesforFundamentalPhysicsprogram, aswellas\nST/I006285/1, and ST/L000946/1, and the Leverhulme\nTrust, grant RPG-2019-022. This work was supported in\npart by Oracle Cloud credits and related resources pro-\nvided by the Oracle Corporation. This research has made\nuse of data or software obtained from the Gravitational\nWave Open Science Center (gwosc.org), a service of the\nLIGO Scientific Collaboration, the Virgo Collaboration,\nandKAGRA.Thismaterialisbaseduponworksupported\nby NSF’s LIGO Laboratory which is a major facility fully\nfunded by the National Science Foundation, as well as\nthe Science and Technology Facilities Council (STFC)\nof the United Kingdom, the Max-Planck-Society (MPS),\nand the State of Niedersachsen/Germany for support of\nthe construction of Advanced LIGO and construction and\noperation of the GEO600 detector. Additional support\nfor Advanced LIGO was provided by the Australian Re-\nsearch Council. Virgo is funded, through the European\nGravitational Observatory (EGO), by the French Cen-\ntre National de Recherche Scientifique (CNRS), the Ital-\nian Istituto Nazionale di Fisica Nucleare (INFN) and the\nDutch Nikhef, with contributions by institutions from\nBelgium, Germany, Greece, Hungary, Ireland, Japan,\nMonaco, Poland, Portugal, Spain. KAGRA is supported\nby Ministry of Education, Culture, Sports, Science and\nTechnology (MEXT), Japan Society for the Promotion of6\nScience (JSPS) in Japan; National Research Foundation\n(NRF)andMinistryofScienceandICT(MSIT)inKorea;\nAcademia Sinica (AS) and National Science and Technol-\nogy Council (NSTC) in Taiwan. This document has been\nassigned LIGO document number LIGO-P2400010.\n[1] R. Abbott et al. (LVK Collaboration), ApJS 267, 29\n(2023).\n[2] Y. Stadnik and V. Flambaum, Phys. Rev. Lett. 115,\n201301 (2015).\n[3] H. Grote and Y. Stadnik, Phys. Rev. Res. 1, 033187\n(2019).\n[4] S. Morisaki and T. Suyama, Phys. Rev. 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This leads to a reinte rpretation of the\nStrong CP problem as arising from a spontaneously broken instanto nic symmetry in\nQCD. We discuss how known solutions to the problem are unified in this f ramework\nand explore some, so far unsuccessful , attempts to find new solutions.Contents\n1 Introduction and summary 2\n1.1 Defining a ( −1)-form U(1) global symmetry 3\n1.2 Spontaneous breaking of a ( −1)-form U(1) symmetry 6\n1.3 Application to the Strong CP problem 8\n1.4 Outline 9\n2 Gauge theories as spontaneously broken phases 9\n2.1 4dMaxwell theory 10\n2.2 3dMaxwell theory 12\n2.3 2dMaxwell theory 13\n2.4 The Schwinger model 15\n2.5 The massive Schwinger model 17\n3 A different look at the Strong CP problem 18\n3.1 Low energy effective theory in 4d Yang-Mills theory and QCD 18\n3.2 SSB of the ( −1)-form U(1) symmetry in 4d Yang-Mills theory and QCD 20\n3.3 Reformulation of the Strong CP problem 21\n4 Solutions to the Strong CP problem and its analogues 22\n4.1 Solving the problem by gauging with an axion 22\n4.2 Solving the problem by gauging with massless fermions 23\n4.3 Solving the problem with non-compact symmetries 24\n4.4 Failing to solve the problem with explicit breaking 25\n4.5 Solving the problem with gauged reflection symmetries 28\n5 Outlook 29\nA The 2dAbelian-Higgs model 31\nB The CPN−1model 32\nC A SymTFT for 2dMaxwell theory. 33\n– 1 –1 Introduction and summary\nSymmetries are extremely useful for understanding quantum the ories. In quantum\nfield theory, symmetries have traditionally been taken to act on loca l operators and\nto obey a group law multiplication, but recent years have seen many g eneralizations,\nstarting with p-form generalized global symmetries [ 1,2]. The subsequent literature\nis too large to comprehensively cite here; instead, we refer reader s to several recent\npedagogical reviews [ 3–9]. Generalized symmetries have proved to be useful for char-\nacterizing phase transitions, strong dynamics, and other nonper turbative aspects of\nquantum field theory. Quantum gravity, in contrast to quantum fie ld theory, is be-\nlieved to lack such symmetries (though approximate symmetries are common). The\nabsence of global symmetries has powerful implications, for examp le requiring a com-\nplete spectrum of charged objects in quantum gravity [ 10–14]. Our goal in this paper is\nto argue that an apparently degenerate case of generalized symm etry, namely the case\nof (−1)-form U(1) global symmetry, is a useful concept that provides a unifying lan-\nguage for discussing many interesting dynamical phenomena in quan tum field theory.\nAlthough this degenerate case has received relatively little attentio n in the literature,\nit has previously made an appearance in [ 15–19], and our discussion will build on ideas\nintroduced therein.1\nThe case of p-form invertible global symmetries in d-dimensional quantum field\ntheory, with 0 ≤p≤d−2, is well-established. A p-form global symmetry with group\nGis associated with a family of topological operators U(g,Σ), known as symmetry\noperators, labeled by a group element g∈Gand a closed (i.e., compact and without\nboundary) ( d−p−1)-manifold Σ. These operators are topological, in the sense that\ncorrelation functions are invariant under deformations of Σ provid ed that Σ does not\ncross another operator insertion when deformed. The symmetry operatorU(g,Σ) acts\nonp-dimensional charged operators living on p-manifolds that are linked by Σ. Al-\nthough the symmetry charge in general is defined on a ( d−p−1)-manifold, in many\ncases it is localized to the ( p+1)-dimensional worldvolumes of massive charged objects\ncreatedby thecharged p-dimensional operators. Inthespecial caseofa U(1)symmetry,\nthe symmetry operators take the form\nU(eiα,Σ) = exp/parenleftbigg\niα/integraldisplay\nΣ⋆jp+1/parenrightbigg\n, (1.1)\nwherejp+1is a conserved ( p+1)-form current. In other words, jp+1is co-closed:\nd⋆jp+1= 0. (1.2)\n1Other recent discussions of ( −1)-form global symmetries, with less overlap with our current focu s,\nappear in [ 20,21]. (d−1)-form global symmetry is another interesting special case; see [22].\n– 2 –The topological nature of the operator follows from this local cons ervation equation.\n(In some literature, the ( d−p−1)-form operator Jd−p−1=⋆jp+1is referred to as the\nconserved current, and it is closed rather than co-closed. Here w e follow the classic\nconvention in which an ordinary conserved current for a symmetry acting on local\noperators is a 1-form.) For a U(1) symmetry, the charge Q=/integraltext\nΣ⋆jp+1is an integer,\nwhich is equivalent to saying that the right-hand side of ( 1.1) is invariant under α/mapsto→\nα+2π, anecessary conditionfortheoperatortobeawell-defined funct ionofeiα∈U(1).\nEvery symmetry is associated with topological operators, even if it is a traditional\nNoether symmetry that is conserved only on the equations of motio n. The topolog-\nical nature of the operator is a statement about correlation func tions in the theory.\nHowever, in special cases the symmetry itself is topological in natur e. For example, in\nMaxwell theory, the magnetic flux1\n2π/integraltext\nΣF∈Zis an integer topological invariant for\nany U(1) bundle and any closed 2-manifold Σ, without the need to use equations of\nmotion. In such cases, the path integral decomposes into topolog ical sectors, and the\nsymmetry operator insertion is identical for all field configurations in a given sector.\nWhether a symmetry is topological in this sense can depend on the du ality frame in\nwhich one works, so it is not a physical invariant. On the other hand, many ordinary\nsymmetries are not topological in anyduality frame. In this paper, we focus on sym-\nmetries that are topological in the strong sense, in the duality fram e in which we define\nthe theory.\nAp-form U(1) global symmetry (without an ’t Hooft anomaly) can be co upled to\na background ( p+ 1)-form gauge field (U(1) connection) Ap+1by adding a coupling\nAp+1∧⋆jp+1. We can gauge the symmetry by making Ap+1dynamical, i.e., by summing\nover U(1) bundles with connection Ap+1in the path integral (and generally including\na kinetic term for Ap+1, which will typically be generated by loops). In this case,\nMaxwell’s equation1\ne2d⋆Fp+2=⋆jp+1indicates that the would-be co-closed current has\nbecome co-exact, and as a result the symmetry operators becom e trivial.\n1.1 Defining a (−1)-formU(1)global symmetry\nThe special case p=−1 ofp-form global symmetry is somewhat degenerate, in a few\nsenses. At first glance, one may reasonably be skeptical that it is a useful notion at all.\nA standard p-form global symmetry acts on p-dimensional charged operators. There\nis, apparently, no such thing as a ( −1)-dimensional operator, so a ( −1)-form symmetry\nwould appear to have nothing to act on. On the other hand, a p-form symmetry is\nalso associated to dynamical charged objects with a ( p+1)-dimensional worldvolume,\nand wedohave a notion of a dynamical object with a 0-dimensional worldvolume ,\nnamely, an instanton. (For example, it is common to speak interchan geably of D( −1)-\nbranes or D-instantons in Type IIB string theory [ 23].) A related concern is that the\n– 3 –symmetry operators U(g,Σ) for a ( −1)-form symmetry are associated with closed d-\ndimensional spacetime manifolds Σ; in other words, they are integra ted over the entire\nspacetime. In this case, the question of whether the operator’s c orrelation functions\nare topological is not obviously meaningful, because we cannot locally deform Σ while\nkeeping the spacetime background of our theory fixed.2Similarly, in the case of a\ncontinuous symmetry, a ( −1)-form symmetry is associated with a conserved 0-form\ncurrent, d⋆j0= 0. However, this condition is trivial, because ⋆j0is a top form in the\ntheory. Thus, everyscalar operator in the theory defines, in some sense, a ( −1)-form\nglobal symmetry, which threatens to render the concept vacuou s. There may still be\nsome merit to this concept, even in the extremely general case, wh ere the absence of\nsuch (−1)-form global symmetries has been identified with the longstanding claim that\nthere are no free parameters in quantum gravity [ 18].\nIn this paper, we focus on the case of ( −1)-form U(1) global symmetries, which\nretain enough structure to be a useful concept [ 15–17,19]. The U(1) case is associated\nwith integer charges,/integraltext\nΣ⋆j0∈Z. Correspondingly, these theories can be coupled to a\nbackground axion field, i.e., a compact scalar θ(x)∼=θ(x) +2π, which we think of as\na 0-form U(1) gauge field. The gauge redundancies of θare simplyθ(x)/mapsto→θ(x)+2πn\nforn∈Z. These are the analogues of “large” or winding gauge transformat ionsAp/mapsto→\nAp+2πnωpwith [ωp]∈Hp(M,Z) for ap-form gauge field; the axion has no analogue\nof the local gauge transformations Ap/mapsto→Ap+dλp−1.3\nWewill followthepragmaticapproachoftakingthepossibility tocouple atheoryto\nabackgroundaxionfieldasourworking definition ofa(−1)-formU(1)globalsymmetry:\nDefinition. We say that a theory has a ( −1)-formU(1)global symmetry when it\ncontains an operator j0that can be consistently linearly coupled to a background field\nθ(x) taking values in a circle ( θ∼=θ+2π),\ne−SE/mapsto→e−SEexp/parenleftbigg\ni/integraldisplay\nMθ(x)⋆j0(x)/parenrightbigg\n. (1.3)\nWereferto j0(x)asthe(−1)-formU(1)symmetrycurrent and(forthecaseoforientable\nspacetime manifolds M) we refer to/integraltext\nM⋆j0(x)∈Zas the (−1)-form symmetry charge.\nThe canonical example, and the case of greatest relevance to par ticle physics, is a\n4dgauge theory with\n⋆j0=1\n8π2tr(F∧F), (1.4)\n2There may be useful perspectives in which the deformation occurs in configuration space, or in\nsome type of auxiliary extra dimension. We will not make use of such pe rspectives in this paper.\n3See for instance [ 24].\n– 4 –for which the ( −1)-form symmetry charge is the instanton number of a gauge field\nconfiguration. We will point out a number of other examples as we go a long.\nA few comments about our definition are in order. We require that th e theory\ncan be defined on arbitrary orientable spacetime manifolds Mfor arbitrary θ(x) back-\ngrounds, which in general are bundles over spacetime—i.e., they adm it configurations\nin whichθ(x) winds around a cycle.4In some cases, turning on other background fields\ncan clash with turning on general θbackgrounds; in that case, we say that there is a\nmixed ’t Hooft anomaly involving the ( −1)-form symmetry, or an anomaly in the space\nof coupling constants, as discussed extensively in [ 15,16].\nWe have referred to orientable spacetime manifolds because in a the ory with an\norientation-reversing spacetime symmetry like parity (by which we m ean reflection of\nan odd number of spatial dimensions) or time reversal, there are ad ditional subtleties.\nSuch theories may be defined on non-orientable manifolds (see, e.g., [26–28] for recent\ndiscussions). In this case, the quantity that we can integrate ove rMis a pseudoform\nor twisted form, i.e., one that transforms with an extra minus sign un der parity. If\n⋆j0is an ordinary form, then θ(x) must be a pseudoscalar in order for ( 1.3) to make\nsense. Our definition is valid in that case, but the charge/integraltext\nM⋆j0(x) is not defined on\narbitrary spacetime backgrounds, and in particular it does not mak e sense to couple\nthe theory to a constant θ-term on a general background. This is as expected: such a\nterm violates parity.\nThe definition of a ( −1)-form U(1) global symmetry that we have chosen is useful,\nbecausetheorieswiththispropertyhavemanyfeaturesincommon withtheorieswith p-\nform U(1) global symmetries for higher p. For example, the symmetry can be gauged.\nIn our context, we can do this by making the field θ(x) dynamical, summing over\nθfield configurations in the path integral. In some theories we can also gauge the\nsymmetry by introducing massless chiral fermions [ 19]. In theses cases, as in ordinary\nelectromagnetism, gaugingthesymmetry renders thecurrent co -exact instead ofmerely\nco-closed. The central point of this paper is that the analogy also e xtends to the notion\nof spontaneous breaking of the global symmetry, and correspon dingly, to higgsing (and\ndual confinement) when the symmetry is gauged.\nForanordinaryU(1)globalsymmetry, itispossibletogaugeasubgr oupZk⊂U(1).\nThis operation also extends to the ( −1)-form case, where it corresponds to summing\n4One might wonder if a weaker notion of ( −1)-form symmetry is of interest, in which a theory need\nonly admit a coupling to a constant background θterm. An interesting candidate is discussed in [ 25]:\ntheCP1sigma model in 3d has a topological invariant characterized by π3(CP1)∼=Z, which one might\nexpect can be coupled to a constant θ, but this topological invariant is not given by an integral of a\nlocal term. It turns out that the theory is only consistent with the choicesθ= 0 and θ=π. We do\nnot know any theory admitting a coupling to generic constant θbut not to a background axion.\n– 5 –over only field configurations with topological charge a multiple of k[17,29–32].\n1.2 Spontaneous breaking of a (−1)-form U(1) symmetry\nThe spontaneous breaking of p-form global symmetries for p≥0 has been exten-\nsively discussed in the literature [ 2,33–35]. A standard diagnostic for breaking of\nan ordinary 0-form symmetry is that a charged operator obtains a vacuum expec-\ntation value. When the symmetry is continuous, we also find massless , propagating\nNambu-Goldstone bosons that nonlinearly realize the symmetry. Th is picture extends\nto higher-form symmetries: for example, a Wilson loop generally has a n expectation\nvalue that obeys a perimeter law or an area law. In the case of a perim eter law, a\ncounterterm in the definition of the Wilson loop can cancel the perime ter dependence,\nleaving behind a constant expectation value even for arbitrarily larg e loops. This is\nthe case of spontaneous breaking of a 1-form global symmetry, a nd the photon can be\nviewed as a massless Nambu-Goldstone mode. For a confining theory with an area law\nfor the Wilson loop, on the other hand, the expectation value decay s for large loops,\nand the 1-form global symmetry is considered to be unbroken.\nSuch diagnostics cannot be extended to the case of ( −1)-form global symmetries,\nbecause there is no ( −1)-dimensional charged operator that can obtain a vacuum ex-\npectation value. Similarly, there is no possibility of a propagating Namb u-Goldstone\nboson created by a ( −1)-formfield nonlinearly realizing the symmetry. Nonetheless, we\nwill argue that there is a useful notion of spontaneous symmetry b reaking for a ( −1)-\nform U(1) symmetry, and even a sense in which there is an emergent Nambu-Goldstone\nfieldin the infrared (though not a propagating boson).\nWe propose that a useful diagnostic of spontaneous symmetry br eaking for a ( −1)-\nform global symmetry is that the vacuum energy for the theory in Minkowski space\ndepends on the value of a constant θbackground . In particular, one order parameter\nfor such spontaneous symmetry breaking is the topological susceptibility , defined as\nX=−i/integraldisplay\nddx/an}bracketle{tT{j0(x)j0(0)}/an}bracketri}htconn.=∂\n∂θ/an}bracketle{tj0/an}bracketri}ht=∂2\n∂θ2V(θ), (1.5)\nwhere, conn .denotes the connected two-point function. One suggestive link be tween\nthis expression and familiar cases of spontaneous symmetry break ing is that of the\nKogut-Susskind pole [ 36,37], which we briefly review here. Specifically, in many the-\nories, the ( −1)-form topological charge density j0is a total derivative of a (gauge-\ndependent) quantity vµ(x),\nj0(x) =∂µvµ(x). (1.6)\n– 6 –In this context, a nonzero value of Xsignals the existence of a pole in the two-point\nfunction of vµ(x):\nX= lim\nq→0−iqµqν/integraldisplay\nddxeiq·x/an}bracketle{tT{vµ(x)vν(0)}/an}bracketri}htconn., (1.7)\nwhich implies that\nlim\nq→0/integraldisplay\nddxeiq·x/an}bracketle{tT{vµ(x)vν(0)}/an}bracketri}htconn.= iqµqν\nq2X\nq2. (1.8)\nThat is, the topological susceptibility is the residue of a pole at q2= 0 in a (gauge-\ndependent) two-point function. Because of the gauge dependen ce, this pole does not\nsignal the existence of a propagating particle, but it does relate to important long-\ndistance correlations in the theory [ 37].\nAn interesting perspective on the Kogut-Susskind pole is that it sign als that the\ninfrared theory has a description in terms of an emergent ( d−1)-form gauge field [ 38–\n41]:\n⋆j0→IRdCd−1. (1.9)\nA (d−1)-form gauge field has no propagating degrees of freedom, but t here can be\ndomain walls carrying a gauge charge under it, and such fields prove u seful in various\napplications (see, e.g., [ 42–45]).\nWe would like to propose a reinterpretation of the emergent ( d−1)-form gauge\ntheory. Ingeneral, spontaneousbreakingofa p-formU(1)globalsymmetryisassociated\nwith the emergence of a p-formNambu-Goldstone gauge field in the IR. Inthe standard\ncase of a 0-form symmetry, we think of this simply as a compact boso n, but as we have\nargued, such bosons can also be thought of as 0-form gauge fields . Ap-form gauge field\nhas a complementary description in terms of a magnetic dual ( d−p−2)-form gauge\nfield, whose field strength is the Hodge dual of the original field stre ngth:\n⋆dap∼dbd−p−2. (1.10)\nIt is unclear what it would mean to seek a ( −1)-form gauge field emerging in the IR\ndescription of a spontaneously broken ( −1)-form global symmetry, but the magnetic\ndual makes perfect sense: it should be a ( d−1)-form gauge field, precisely as in ( 1.9).\nThus, we argue that the Kogut-Susskind pole can be thought of as signaling that the\nIR theory admits an emergent description in terms of a ( d−1)-form Nambu-Goldstone\ngauge field. There is no Nambu-Goldstone boson, because there are no propagating\ndegrees of freedom; nonetheless, the Nambu-Goldstone fieldcan be useful.\nOneexample oftheutility ofsuch adescription ariseswhen wegauget he(−1)-form\nglobal symmetry. When we gauge a spontaneously broken p-form global symmetry, the\n– 7 –resulting theory is in the Higgs phase. We can summarize the phenome non of higgsing\nand its magnetic dual, confinement, by saying that in this phase elect rically charged\nworldvolumes haveboundaries (they can end on a vacuum insertion) and magnetically\ncharged worldvolumes areboundaries (they are confined by a higher-dimensional ob-\nject). Manyoftheconsequences ofhiggsinghaveanalogueswhen wecoupleadynamical\naxion field to a spontaneously broken ( −1)-form global symmetry. To name a few:\n•The gauge field acquires a mass. For the axion, this is apparent: the re is a\npotentialV(θ) and the axion mass is proportional to Xat the minimum of the\npotential.\n•Electric charges are screened. In the axion case, the electrically c harged objects\nare instantons. We can think of the local operator eiθ(x)as the analogue of a\nWilson line: it inserts a static instanton configuration at a point. The e ffects of\nthis insertion in correlation functions fall off at long distances, beca use the axion\nis massive.\n•Magnetic charges are confined. In the axion case, these are vort ices, codimension-\ntwo objects in spacetime around which the axion field winds. They are charged\nunder the gauge field Bd−2dual toθ. This field is eaten by the emergent ( d−1)-\nform gauge field via a Stueckelberg structure of the form |dBd−2−Cd−1|2[41].\nEquivalently, axion vortices are the boundaries of domain walls, which carry\ncharge under Cd−1.\nWe consider this set of parallels to be a strong argument that our de finition of sponta-\nneous breaking of a ( −1)-form global symmetry is a useful one, allowing us to success-\nfully apply intuition from more standard cases in a different context.\n1.3 Application to the Strong CP problem\nThe language that we have introduced above provides a useful fra mework for thinking\nabout the Strong CP problem. The Strong CP problem is the puzzle th at the Standard\nModel admits a CP-violating term of the form1\n8π2¯θ/integraltext\ntr(G∧G),5but experiment finds\nthat this term is extraordinarily small, |¯θ|/lessorsimilar10−10[46,47]. This cries out for some\nexplanation in terms of symmetries or dynamics. Because the CKM ph ase in the quark\nmixing matrix has been measured to be an O(1) number, the simplest answer that our\nuniverse respects CP is not a viable one. A number of solutions to this puzzle have\nbeen proposed over the years.\n5The physical quantity ¯θin fact is a linear combination of the coefficient of tr( G∧G) and the phase\nof the determinant of the quark mass matrix; here we assume we ha ve rephased the quarks to move\nthe physical quantity entirely into the gluonic term.\n– 8 –From our perspective, the Strong CP problem is closely related to th e existence of\na spontaneously broken ( −1)-form U(1) global symmetry of the Standard Model, with\ncharge the QCD instanton number. The symmetry itself allows us to t urn on a ¯θterm\n(in the absence of an additional symmetry like CP, which would forbid a constantθ(x)\nbackground field on generic spacetimes). The spontaneous break ing of the symmetry\nallows¯θto affect physical observables like the neutron EDM. This suggests that a\nuseful strategy for solving the Strong CP problem is to seek mecha nisms for eliminating\nthis global symmetry. A global symmetry can be eliminated by effects that explicitly\nbreak the symmetry, or by gauging. As already noted in [ 19], two different classic\nsolutions to the Strong CP problem, the QCD axion and the massless u p quark, can\nbe understood as different ways of gauging the ( −1)-form global symmetry. Explicitly\nbreaking the symmetry is more challenging, since the underlying char ge is topological.\nNonetheless, there are physical mechanisms that can break such symmetries. We will\ndiscuss some of these mechanisms, and see that for the most part they do not offer a\nsatisfactory resolution of the Strong CP problem. A final, classic se t of solutions to\nthe Strong CP problem rely on the spontaneous breaking of an orien tation-reversing\nspacetime symmetry (parity or CP). These mechanisms, again, are linked to the fate\nof the (−1)-form symmetry, since the topological charge is not defined on t he non-\norientable spacetime backgrounds that are allowed in such theories .\n1.4 Outline\nThe remainder of this text is structured as follows. In section 2we present a discussion\non how generic abelian gauge theories in the Coulomb phase can be und erstood as\ndescribing spontaneously broken higher form symmetries. We argu e that this still\nholds in 2d, where Maxwell theory realizes a spontaneously broken ( −1)-form U(1)\nsymmetry. We examine several deformations of this theory and pr opose universal\nfeatures of spontaneously broken ( −1)-form symmetries. In section 3we argue that\nthe instantonic symmetry of SU( N) Yang Mills and QCD is spontaneously broken and\nlink this fact with the Strong CP problem. In section 4we explore solutions to the\nStrong CP problem from this point of view. We list some open questions and provide\nan outlook in section 5.\n2 Gauge theories as spontaneously broken phases\nStandard lore holds that abelian gauge theories in the Coulomb phase describe spon-\ntaneously broken higher form symmetries. The lore further specifi es that the Nambu-\nGoldstone bosons realizing the spontaneously broken symmetries n onlinearly are the\n– 9 –photons themselves.6This section aims to show that this lore holds even in 2 dtheories,\nwhere the higher-form symmetry is a ( −1)-form symmetry. To gain some intuition we\nreview Maxwell’s theory in 4 dand 3d, making our way to the two-dimensional world.\nThen we describe abelian gauge theories in 2 dand give explicit realizations of the con-\ncepts introduced in Sec. 1. As we will describe, 2 dgauge theories have instantons that\nare charged under a ( −1)-form symmetry that is spontaneously broken.\n2.14dMaxwell theory\nFree electromagnetism is a theory of a U(1) gauge field Awith field strength F= dA\nand the following action,\nS=/integraldisplay\n−1\n2e2F∧⋆F. (2.1)\nThe equation of motion for the gauge field is d ⋆F= 0. This equation signals the\nexistence of a conserved 2-form current Je=1\ne2Fthat generates a 1-form U(1)(1)\ne\nsymmetry. The topological symmetry operator can be construct ed by exponentiation\nof the current,\nUα(Σ2) = eiα\ne2/integraltext\nΣ2⋆F. (2.2)\nThis symmetry operatorcounts theelectric charge inside Σ 2andactsby linking onnon-\ndynamical probe electric charges dubbed Wilson lines. If massless ele ctrically charged\ndynamical matter, such as the electron, is added to this theory, t he electric charge of\nWilson lines is screened and the U(1)(1)\nesymmetry is explicitly broken. Provided that\nthe gauge group is U(1), electric charge is quantized and α∈[0,2π), as befits a U(1)(1)\ne\nsymmetry. The gauge field obeys a topological constraint, the Bian chi identity d F= 0.\nAs before, this equation signals the existence of a conserved 2-fo rm magnetic current\nJm=⋆Fthat generates a U(1)(1)\nmsymmetry. Exponentiation of the current yields the\nsymmetry operators ˜Uα(Σ2) that measure the magnetic charge inside of Σ 2. If the\ngauge group is U(1),/contintegraltextF\n2π∈Z, which is a topological invariant labeling gauge bundles\nby their monopole number. The topological nature of the magnetic s ymmetry makes\nexplicitly breaking it a non-perturbative statement in the action in Eq uation (2.1).\nIn other words, no modification of the Lagrangian, no matter how d rastic it is, can\nexplicitly break this symmetry as long as Ais a U(1) gauge field.\nA 1-form symmetry is generated by codimension 2 topological opera tors. There\nis no invariant way of defining an action of an operator of such dimens ionality on\nlocal operators. This implies that a local operator can’t transform under a 1-form\nsymmetry. This is not true for local operators that are not gauge invariant, which\n6This also holds for the compact scalar, which we understand as a gau ge boson for a ( −1)-form\nU(1) symmetry. It nonlinearly realizes a spontaneously broken 0-f orm symmetry.\n– 10 –do not correspond to physical observables. In fact the action of the electric 1-form\nsymmetry can be encoded as a shift of the gauge field by a closed but not quantized\n1-formΛ 1,A→A+Λ1,/contintegraltext\nΛ1∈[0,2π). Note that if Λ 1was quantized this shift would\ncorrespond to a largegaugetransformation. The gaugeinvariant operator transforming\nunder the symmetry is the Wilson line, defined as Wq(γ) = eiq/integraltext\nγAfor some integer q\nand its transformation rules follow from those of A. This is one of the nice features of\ngauge fields: they allow for the description of line operators in terms of local (but not\ngauge invariant) ones. A further important lesson follows from the transformation of\nA; it realizes the 1-form symmetry non-linearly. This is a familiar proper ty of Nambu-\nGoldstone bosons φin phases with spontaneously broken U(1) 0-form symmetries.\nGiven a symmetry transformation with compact parameter c∈[0,2π) the Nambu-\nGoldstone boson shifts as φ→φ+c. The lesson that follows from this observation is\nthatAis the Nambu-Goldstone boson of a spontaneously broken U(1)(1)\nesymmetry [ 2].\nThis heuristic observation can be made precise by computing the exp ectation value\nof an object charged under U(1)(1)\ne, a Wilson line. It obeys a perimeter law in the\nCoulomb phase, signaling the spontaneous breaking of the symmetr y. The Goldstone\ntheorem implies that, in a phase with a spontaneously broken U(1) 0- form symmetry,\nthe conserved current creates Nambu-Goldstone bosons from t he vacuum, which prop-\nagate a massless excitation. In the case at hand the conserved cu rrent creates a 1-form\nNambu-Goldstone boson, the photon [ 2],\n/an}bracketle{t0|Je,µν(x)|λ,p/an}bracketri}ht= (λµpν−λνpµ)eipx. (2.3)\nThe equation of motion and the Bianchi identity are exchanged unde r1\n2πF↔1\ne2⋆F.\nOne can introduce a magnetic photon ˜Aby defining a Hodge dual field strength1\n2π˜F=\n1\ne2Fand a dual coupling ˜ e= 2πe−1and the action remains invariant. In the electric\nframe an ’t Hooft line is defined as a boundary condition for the gauge field along a\n1-dimensional locus. In the dual frame however it can be defined in t erms of the dual\ngauge field Hq(γ) = eiq/integraltext\nγ˜A. If one substitutes U(1)(1)\newith U(1)(1)\nm,Awith˜Aand the\nWilson lines with ’t Hooft lines all the discussion regarding the spontane ous breaking\nof the symmetry remains unchanged. It follows that both U(1)(1)\neand U(1)(1)\nmare\nspontaneously broken in the Coulomb phase giving rise to a single Namb u-Goldstone\nboson in either the electric or magnetic frame.\nIt is interesting to note that the photon remains exactly massless e ven if both\nsymmetries are explicitly broken at some scale by adding fundamenta l matter and\ndynamical monopoles. Indeed this is plausibly what happens in our univ erse. Since\nlocal operators may not carry 1-form charge, no relevant (or irr elevant) couplings can\nspoil the emergent 1-formsymmetry, making it exact at low energie s. This fact protects\nthe masslessness of the photon. For related references see for instance [ 35,48–54].\n– 11 –2.23dMaxwell theory\nThe symmetries of 3 delectromagnetism follow a similar pattern to its 4 dcounterpart.\nThere is a U(1)(1)\ne1-form symmetry under which Wilson lines are charged. In this\ncase, however, the magnetic symmetry is 0-form U(1)(0)\nm. Magnetic charge is sourced\nby local operators called monopole operators which are defined by e xcising a point of\nspacetime and prescribing a boundary condition for Asourcing magnetic flux. Due\nto the change in dimensionality the gauge field Ais dual to a compact scalar field\nσ. In terms of this field the monopole operator is defined as Mp(x) = eipσ(x). In\nthe 3dworld a continuous 1-form symmetry can’t spontaneously break an d give rise to\nNambu-Goldstonemodes, aresultwhichfollowsfromageneralization oftheHohenberg-\nMermin-Wagner-Coleman theorem [ 2,33,55,56]. This implies that the electric 1-\nform symmetry can’t be spontaneously broken in 3 d. This result is made apparent by\nintroducing dynamical monopoles which give rise to a confining force b etween electric\nparticles [ 57]. The Wilson line then follows an area law, which in the large area limit\nvanishes, and the U(1)(1)\nesymmetry remains unbroken. This means that the 3 dphoton\ncan’t be understood as the Nambu-Goldstone boson for the spont aneous breaking of\nthe electric symmetry. This result follows from monopole proliferatio n, which in turn\nimplies that the vacuum is magnetically charged and the magnetic 0-fo rm symmetry\nis spontaneously broken. Furthermore, explicit computation in the magnetic frame\nshows that the matrix element between the magnetic current Jm,µ= (⋆F)µand the\ndual photon is,\n/an}bracketle{t0|Jm,µ(x)|p/an}bracketri}ht=pµeipx. (2.4)\nThe lesson is that 3 delectromagnetism can be understood as in the magnetic Coulomb\nphase and describes a spontaneously broken U(1)(0)symmetry.\nAs already mentioned, addition of magnetic monopoles leads to their p roliferation\nand the onset of confinement of electric charges. A further effec t of this proliferation\nis to give the photon a mass exponentially small in the monopole action Smon. This is\nunderstood by noticing that once dynamical monopoles are included , the magnetic 0-\nformsymmetryisonlyemergentatenergiesbelow Smon. Anemergent0-formsymmetry,\nunlike an emergent 1-form symmetry, is not enough to protect the masslessness of the\nphoton. This was beautifully exemplified by Polyakov in [ 57]. He studied how the\nphoton, when embedded in SU(2) through adjoint Higgsing gets a ma ss from vortices.\nThe magnetic symmetry is absent in the SU(2) theory and, corresp ondingly, the U(1)\nphoton is massive.\n– 12 –2.32dMaxwell theory\nWe arenow ready to tackle the wacky two dimensional world. In2 dthe Maxwell theory\nadmits aθ-term, which we omitted in the 4 dcase. It will play a starring role in our\ndiscussion, so let us spell it out,\nS=/integraldisplay\n−1\n2e2F∧⋆F+1\n2π/integraldisplay\nθF. (2.5)\nThe equation of motion is unchanged, d ⋆F= 0, and gives rise to a conserved 2-\nform current Je. The Bianchi identity is more subtle than in the higher dimensional\ncounterparts. It still reads d F= 0 but it is a tautological equation, since every top\nform is closed. As in higher dimensions, the first Chern class of a U(1) gauge bundle\nin 2 dimensions is quantized/contintegraltext\nF= 2πZ. This is what allows for the introduction of\ntheθ-term in the first place. We can use this fact to identify 2 πF=⋆j0as a magnetic\n(−1)-form U(1) current and the θ-term as the coupling to a background gauge field\nfor it,7following our definition in Equation ( 1.3). In our terminology the symmetry\nof 2dMaxwell theory is then U(1)(1)\ne×U(1)(−1)\nm.8Furthermore, in analogy with the\nhigher dimensional counterparts, it is natural to expect the ( −1)-form symmetry to be\nspontaneously broken. In the following we explore this possibility in de tail. The field\nstrength in 2 dhas a single component F01, and the action can be rewritten.\nS=/integraldisplay\nd2x/bracketleftbigg1\n2e2F2\n01+1\n2πθF01/bracketrightbigg\n. (2.6)\nIt is useful to quantize the theory by choosing the space manifold t o be a circle S1\nof radius R. By a suitable choice of gauge A0= 0 one can argue that only the zero\nmode survives and the theory can be rewritten in terms of an angula r variableφ(t) =/integraltext2πR\n0dxA1(x,t). The angular nature follows from the large gauge transformation s ofA\nwinding along the circle, which become φ→φ+2π. In terms of φthe action becomes,\nS=/integraldisplay\ndt/bracketleftbigg1\n4πe2R˙φ2+θ\n2π˙φ/bracketrightbigg\n. (2.7)\nThis is just the action for a particle in a circle in the presence of a magn etic field. The\nsystem can be quantized and has energy eigenstates ψl= eilφwith energy,\nEl=πe2R/parenleftbigg\nl−θ\n2π/parenrightbigg2\n. (2.8)\n7Given that θis a background gauge field, the meaning of the transformations θ→θ+ 2πis\nclear, they are just the large gauge transformations of the back ground gauge field. These large gauge\ntransformations shift the scalar background gauge field by a close d but not exact 0-form: a constant.\nIn the (−1)-form symmetry case this is all there is, since small gauge transf ormations, described by\nshifts by an exact 0-form, are trivially zero.\n8A SymTFT realization of this symmetry can be found in appendix C.\n– 13 –The ground state is ψ0, which we denote |0/an}bracketri}ht=|ψ0/an}bracketri}ht. The system in state ψlis charac-\nterized by a constant electric field,\nF01=e2/parenleftbigg\nl−θ\n2π/parenrightbigg\n. (2.9)\nThe ground state is the state with lowest electric field,\n/an}bracketle{tF01/an}bracketri}ht0=−e2θ\n2π. (2.10)\nThis result is hardly surprising since the classical equations of motion forA0,A1in\neq. (2.6) are∂0F01=∂1F01, whose only solution is a constant electric field. This also\nagrees with the photon not propagating any degree of freedom in 2 d. In free Maxwell\ntheory there are no electric particles but one can consider adding h eavy particles, or\nWilson lines, to probe the theory. On the circle we must add at least tw o, of opposite\ncharge and separated by a distance L. Regardless of the chosen state, the electric field\nbetween them will increase by one unit, giving rise to an energy that g rows linearly\nwithL. This shows that, even if the photon does not propagate any degr ee of freedom\nthere is a long range force that confines probe charges classically.\nClassic confinement implies that large Wilson loops obey an area law and t he elec-\ntric 1-form symmetry is not spontaneously broken. A similar check is not available for\nthe magnetic ( −1)-form symmetry. We would need a charged operator that is analo -\ngous to the ’t Hooftloopin 4 dor the monopoleoperator in 3 dbut such a thing does not\nseem to exist. As discussed in Sec. 1, we propose instead that the spontaneous breaking\nof the (−1)-form symmetry is diagnosed by an explicit dependence of the vac uum en-\nergyV(θ) on the value of the background θ. The leading measure of such dependence\nis the topological vacuum susceptibility X=∂2\n∂θ2V(θ). Given the topological density\n⋆F, the topological susceptibility can be rewritten as,\nX=−i1\n4π2/integraldisplay\nd2x/an}bracketle{tT(⋆F(X)⋆F(0))/an}bracketri}htconn.=1\n2π∂\n∂θ/an}bracketle{t⋆F/an}bracketri}ht=e2\n4π2, (2.11)\nwhich is nonzero in the present case, signaling spontaneous breakin g of the ( −1)-form\nsymmetry. So far we have linked the spontaneous breaking of the ( −1)-form symmetry\nwith a physical dependence on its gauge background. A further mo tivation for this\ndefinition is the relation between a non-vanishing Xand the appearance of a double\npole at zero momentum in the 2-point function of the photon. By con sidering the\ngauge-dependent two point function /an}bracketle{tAµ(x)Aν(y)/an}bracketri}ht, one can show that it is written in\nterms of a propagator G(q2) that satisfies,\nlim\nq2→0q2G(q2)∼ X. (2.12)\n– 14 –AlthoughG(q2) is gauge dependent, this limit matches the manifestly gauge invarian t\nquantity ( 2.11) and so the pole at q2= 0 is independent of the gauge [ 37]. While\nin higher dimensions the Kogut-Susskind pole is somewhat mysterious , there is no\nmystery in the abelian two dimensional case where the role of the non -propagating\nphoton field is well understood. In particular, it is responsible for th e long-range force\nthat confines probe particles. In this sense, the gauge field Aµmediates a long range\nforce thanks to a pole in its “propagator,” in complete analogy with Ma xwell theory in\nhigher dimensions. We conclude that the non-vanishing of the topolo gical susceptibility\nsignals the spontaneous breaking for the ( −1)-form symmetry giving rise to a Nambu-\nGoldstone fieldthat creates a long range force, even if it does not propagate.\nAfter explicitly establishing the connection between free 2 dMaxwell theory and\nthe (−1)-form symmetries introduced in sec. 1, in the following we explore the fate of\nthese universal features in theories with fermions, both massless and massive. We have\nalso considered two other 2 dmodels displaying interesting low energy dynamics that\ncan be understood in terms of the magnetic ( −1)-form symmetry but, for deference to\nthe exhausted reader, those discussions are left to the appendic es. In appendix Awe\nreview the 2 dAbelian-Higgs model while in appendix Bwe review the CPN−1model.\n2.4 The Schwinger model\nIf we couple the 2 dU(1) gauge theory with a massless Dirac fermion we obtain the\nSchwinger model. The action is,\nS=/integraldisplay\n−1\n2e2F∧⋆F+1\n2πθF+i¯ψ/Dψ. (2.13)\nExplicit computation of the equation of motion shows that the electr ic current is no\nlongerconserved. Ontheotherhandthereisstilla θ-termconsistent withourdefinition\nof (−1)-form symmetry. There is also a would-be chiral U(1) symmetry t hat is ABJ\nanomalous. The symmetry of this theory is just U(1)(−1)\nm.\nThe gauge coupling is dimensionful in two dimensions and the theory is s trongly\ncoupled in the IR. Nonetheless, it is simple enough for Schwinger to be able to solve\nit explicitly using the operator formalism [ 58]. If you are not Schwinger, a simpler\napproach was pioneered by Coleman [ 59] taking advantage of 2 dbosonization. This\nduality states that a strongly coupled fermion can be exchanged wit h a weakly coupled\nboson, provided that adictionaryisused. The fermiontheoryconfi nes classically atlow\nenergies but it is equivalently described by a theory of a free scalar w hich only couples\nto the gauge field through a topological term. In the present case the bosonized version\nof the theory takes the following form,\nS′=/integraldisplay\n−1\n2e2F∧⋆F+1\n2π(θ+φ)F+1\n8π(dφ)2. (2.14)\n– 15 –One could naively think that there is still a ground state electric field g iven by,\nF01=−e2\n2π(θ+φ). (2.15)\nHowever, we can redefine φ→φ−θto absorbθ, signaling that θis unphysical. In fact,\nthis was already apparent in the original formulation of the theory, which has an ABJ\nanomaly that allows θto be absorbed in a chiral rotation. The bottom line is that the\nelectric field in the ground state, which was proportional to θin the free Maxwell case,\ncan now relax to a vanishing value.\n/an}bracketle{t0|(⋆F)|0/an}bracketri}ht= 0. (2.16)\nIn more detail, in 2 dthe ABJ anomaly is computed by the vacuum polarization dia-\ngram. The vacuum polarizes and the electric field is screened. We high light that this\nscreening is not mediated by Schwinger pair production since the elec tric field to be\nscreened is fractional in units of the charges of the massless ferm ions.9Consequently,\nthe topological susceptibility Xvanishes and the Schwinger model does not sponta-\nneously break the ( −1)-form symmetry. By virtue of eq. ( 2.12) the vanishing of the\ntopological susceptibility implies that the pole disappears from the ga uge 2-point func-\ntion, signaling that the Nambu-Goldstone fieldhas been lifted and we are no longer in\nthe Coulomb phase. Hence we also expect the long-range force bet ween probe particles\nto vanish.\nIndeed the polarized vacuum can also screen the electric field sourc ed by probe\nparticles and the long-range force is well known to be absent in the S chwinger model\n[58,62]. We see that all our expectations regarding a theory that sponta neously breaks\na (−1)-form symmetry are negated in this model.\nWe finish our discussion by noting that the Schwinger model is equivale nt, through\nthe bosonization dictionary, to a theory where the ( −1)-form symmetry has been\ngauged. In eq. ( 2.14) the (−1)-form symmetry current J0=⋆Fhas been coupled\nto a dynamical gauge field φ(a compact scalar), which is the canonical way of gauging\nsymmetries associated to conserved currents. From this point of view, it is very natural\nthat the Schwinger model cannot possibly realize a spontaneously b roken (−1)-form\nsymmetry, since it has been gauged! This observation will be useful when we leverage\nthe knowledge gained from this toy model to understand QCD and th e Strong CP\nproblem.\n9One can also argue for this by noticing that the constant electric fie ld becomes F01=−e2\n2πφ, which\ngives a non-zero energy. Another way to argue for the field relaxin g to zero is by integrating out\nF=dAfrom2.14. One finds a quadratic potential for φ, which naturally relaxes to zero setting\nF01= 0. For related discussions see for instance [ 60,61].\n– 16 –2.5 The massive Schwinger model\nA further twist can be made by adding a mass to the Dirac fermion in th e Schwinger\nmodel. In the massless Schwinger model the vanishing of the topolog ical susceptibility\nwas intimately tied with the ABJ anomaly, which is absent in this case due to the\nfermionshavingamass. Forthisreason, weexpectthismodeltore alizeaspontaneously\nbroken (−1)-form global symmetry. If the fermion is massive enough m2≫e2, we\nrecover free Maxwell theory in the IR and our expectation is trivially satisfied. A more\ninteresting question is what happens in the opposite case of m2≪e2. Consider the\nfollowing action of a massive Dirac fermion coupled to a U(1) gauge field ,\nS=/integraldisplay\n−1\n2e2F∧⋆F+1\n2πθF+i¯ψ/Dψ−im¯ψψ. (2.17)\nIn them2≪e2regime the theory is strongly coupled in the IR. Luckily, Coleman\ntaught us how to solve it using the bosonization dictionary. The boso nized theory is\n[59,63],\nS′=/integraldisplay\n−1\n2F∧⋆F+1\n2π(θ+φ)F+1\n8π|dφ|2+m\nπǫcosφ. (2.18)\nThe only difference with the bosonized action of the massless Schwing er model is the\nlast term in Equation ( 2.18) which is absent in Equation ( 2.14). This term obstructs\nthe absorption of θby aφfield redefinition, so we expect θto be physical and to give\nrise to a vacuum electric field. As discussed in [ 59,63], this is precisely what happens.\nThere is a non-zero electric field in the vacuum, which can’t be screen ed in this case,\n/an}bracketle{t0|(⋆F)|0/an}bracketri}ht=e2\n2π(θ+φ). (2.19)\nFrom Equation ( 2.11) it follows that there is a non-zero topological susceptibility and,\nconsequently a zero momentum pole in the 2-point function of the ga uge field. Further-\nmore, this theory displays a long-range force between probe part icles. The bottom line\nis by now clear, the magnetic ( −1)-form symmetry, which was gauged in the massless\ncase, is now ungauged and spontaneously broken, giving rise to a Na mbu-Goldstone\nfieldand a long-range force.\nA difference with the pure Maxwell case is that the long range force b etween two\nparticles vanishes if the difference of their charges is a multiple of 2. T he reason is that\na Schwinger pair of massive fermions may nucleate, screening the ele ctric field created\nby the particles. We learn that the long range characteristic of spo ntaneously broken\n(−1)-form symmetries may be dynamically screened but will be present for improperly\nquantized probes. Asimilar phenomenon happens withthevacuumele ctric field, giving\na physical explanation for the periodicity of θin this model.\n– 17 –Anexplicit, numerical computation of the topologicalsusceptibility w as carried out\nin the recent work [ 61] by using a tensor network approach in the lattice, where it was\nindeed found to be non-vanishing.\n3 A different look at the Strong CP problem\nWhile 2 dimensions are fun, they are somewhat detached from the hig h energy physics\nof our universe. In this section we present two 4 dgauge theories whose low energy dy-\nnamics are governed by a spontaneously broken ( −1)-form symmetry. Namely, SU( N)\nYang-Mills theory and QCD. We will see how the low energy dynamics in th ese theo-\nries have similarities with the physics of 2 dMaxwell theory, particularly in the large N\nlimit of Yang-Mills. In the last part of this section we use this insight to r eformulate\nthe Strong CP problem as a consequence of the spontaneous brea king of a ( −1)-form\nsymmetry.\n3.1 Low energy effective theory in 4d Yang-Mills theory and QC D\nFor concreteness, let us consider SU( N) Yang-Mills theory with no light matter, which\nserves us as a toy model for QCD. The action is,\nS=/integraldisplay\ntr/parenleftbigg\n−1\ng2F∧⋆F+θ\n8π2F∧F/parenrightbigg\n. (3.1)\nAnimportantpropertyofSU( N)YMtheory, isthatthe θ-termleadstophysical effects.\nNote that this property lies at the core of the Strong CP problem. S uch a physical\ndependence on θis probed by the topological susceptibility of the vacuum Xwhich can\nbe defined as [ 64],\nX ≡/parenleftbiggd2E\ndθ2/parenrightbigg\n= lim\nq→0−i/parenleftbigg1\n16π2/parenrightbigg2/integraldisplay\nd4xeiqx/an}bracketle{t0|T(tr(Fµν˜Fµν(x))tr(Fρσ˜Fρσ(0)))|0/an}bracketri}ht.\n(3.2)\nThe low energy dynamics of 4 dYang-Mills theory is strongly coupled and notoriously\ndifficult to study. Nonetheless, the non-vanishing of the topologica l susceptibility for\ngenericθgives us important hints about the vacuum structure. As noticed b y L¨ uscher\nin [37], the 2-point function above can be recast in terms of a 2-point fun ction of the\nChern-Simons 1-form current K1, which is the Hodge dual of the Chern Simons 3-form\nK1=⋆C3, at vanishing momentum. Using the fact that ∂µKµ=1\n16π2tr(Fµν˜Fµν) one\ncan rewrite eq. ( 3.2) as,\nX= lim\nq→0−iqµqν/integraldisplay\neiqx/an}bracketle{t0|TKµ(x)Kν(0)|0/an}bracketri}htd4x. (3.3)\n– 18 –As discussed in the introduction, the non-vanishing of the topologic al susceptibility\nimplies that the two point function has a pole at q2= 0. Given that the 2-point\nfunction in question is not gauge invariant, one may be wary that this pole may be\nunphysical. Luscher argued that the pole remains in any gauge and, as we will explain,\nits physical implications are profound. The existence of this pole at z ero momentum\nsignals the appearance of a massless mode for C3in the IR. Since we believe that\nYang-Mills theory is otherwise gapped, it is natural to expect the va cuum structure of\nSU(N) Yang-Mills theory to be captured by an effective theory of a massle ss 3-form\ngauge field C3. In fact, this observation should hold for QCD as well, since Xis also\nnonzero in that case. For related discussions in Yang-Mills and QCD se e [38,65–67].\nThe precise form of the Lagrangian describing this effective theory will depend on\nstrongly coupled dynamics and is, in general, not available to us. In th e large N limit\nmatters are simpler, as discussed in [ 38,39,65,66]. At small momenta only terms\nwith less than two derivatives will be relevant. One can assume that a kinetic term is\ngenerated and all other 2-derivative terms are in fact suppresse d in the large N limit.\nIn this limit the effective theory takes the following form,\nL=−1\n2XF4∧⋆F4+1\n2πθF4. (3.4)\nWhereF4= dC3is anabelian4-formfield strength that should not be confused with F,\nthe 2-form non-abelian field strength. This action describes a 3-fo rm gauge field in 4 d\nwith a topological coupling or θ-term. This theory is reminiscent to electromagnetism\nin 2d, see2.3. In fact, the physics of both theories is very similar, as explored in, for\ninstance [ 68]. Like its 2 dcounterpart, a 3-form gauge field in 4 ddoes not propagate\nand the different vacua are characterised by a constant electric fi eld,\n/an}bracketle{t⋆F4/an}bracketri}htl= (θ+2πl)X. (3.5)\nThe energy density of these vacua is\nEl(θ) =1\n2(θ+2πl)2X. (3.6)\nThe true vacuum, or ground state, is selected by minimizing the expr ession above. One\nthen finds a non-zero electric field in the ground state,\n/an}bracketle{t⋆F4/an}bracketri}ht0=Xθ. (3.7)\nReassuringly these results match the expectations that one infer s from holography [ 69].\nAwayfromthelarge Nlimit, thepreciseformoftheactionforsuchafieldisunknown.10\n10Its form may be determined in some approximations such as the dilute instanton gas; see [ 41].\n– 19 –In general the Lagrangian will take the following form,\nL=−1\n2|F4|2+1\n2πθF4+K(F4), (3.8)\nwhereK(F4) denotes higher order contributions with F4. For instance, in QCD θand\nK(F4) will depend explicitly on the quark masses. Regardless of the precis e form of the\nLagrangian the equations of motion still admit a constant solution fo r⋆F4such that\nthe physical picture remains unchanged.\n3.2 SSB of the (−1)-formU(1)symmetry in 4d Yang-Mills theory and QCD\nAn important property of any theory with a Lagrangian of the form 3.8is that it has a\nmagnetic ( −1)-form U(1) symmetry. The existence of this symmetry follows fr om the\nBianchi identity of the C3gauge field and the quantization of/contintegraltext\nF4. From the Biachi\nidentity we identify the conserved current as,\nj0=⋆F4, (3.9)\nwhich we can couple to a background gauge field θwhich is periodic. This symmetry\nin the IR effective theory is already present in the UV of both SU( N) Yang-Mills and\nQCD: it is a Chern-Weil symmetry with conserved current [ 19],\n⋆j0=1\n8π2tr(F∧F), (3.10)\nwhere now Fis the non-abelian field strength. This symmetry is sometimes called th e\ninstantonic symmetry, as it measures the instanton number. The n on-vanishing of the\ntopologicalsusceptibility inSU( N)YMandQCDimpliesthatthephysics dependsnon-\ntrivially on the value of the background field θ. Following our discussion in section 2.3\nwe take this dependence to signal the spontaneous breaking of th e (−1)-form U(1)\nsymmetry. Finally, we identify the 3-form gauge field C3as the Nambu-Goldstone field\nof the spontaneously broken ( −1)-form symmetry.\nA further way of arguing that the ( −1)-form symmetry is spontaneously broken is\nby considering its gauging. It can be explicitly gauged by introducing a kinetic term\nfor the gauge field θ(x) and summing over it in the path integral. This is equivalent to\ncoupling the Chern-Weil current to an axion. Importantly, the axio n has a non-trivial\npotential arising from the θdependence of the vacuum energy density, i.e., the non-\nvanishing X. This potential endows the axion with a non-zero mass, signaling tha t the\ngauged(−1)-form symmetry is spontaneously broken giving rise to a Higgs mec hanism.\nFurthermore, an electric Higgs phase is dual to magnetic confineme nt. This can be\nexplicitly checked in this case by replacing θ(x) by its Hodge dual 2-form gauge field\n– 20 –dθ∼⋆dB2. The 3-form field C3is no longer massless as it picks up a mass from a\nStueckelberg-like mass term of the form,\n|dB2−C3|2, (3.11)\nwhich implies that the would-be Nambu-Goldstone field C3is eaten by the gauge field\nB2. Asimilar Stueckelberg-like massisobtainedin, e.g., thedualdescript ion ofgauging\nthe magnetic 1-form U(1) symmetry. From 3.11it follows that, to preserve gauge\ninvariance, axion strings must be attached to C3domain walls. These domain walls\nhave a finite tension, giving rise to a confining force between strings . We conclude\nthat axionic strings are confined, in agreement with the gauge ( −1)-form symmetry\nbeing spontaneously broken and Higgsed. Note that, were Xto vanish, the effective\nfield theory would not have a massless 3-form gauge field which we hav e associated\nwith the spontaneous breaking of the ( −1)-form U(1) symmetry. Furthermore, in the\ngauged ( −1)-form symmetry theory, the axion would remain massless and the (−1)-\nform symmetry unhiggsed. These two facts, together with similar c onsiderations in\nsection2lead us to propose Xas an order parameter for the spontaneous breaking of\nthe (−1)-form symmetry.\nSpontaneous Breaking of a ( −1)-form U(1) symmetry.\nA (−1)-form U(1) symmetry with background gauge field θis spontaneously\nbroken if the vacuum energy in Minkowski space Vdepends on θ. An order\nparameter for such spontaneous breaking is the topological susc eptibility:\nX=∂2\n∂θ2V(θ). (3.12)\n3.3 Reformulation of the Strong CP problem\nAn important consequence of the non-vanishing of Xin QCD is that physical observ-\nables can depend on ¯θ=θ+ Arg(det M), where Mis the quark mass matrix. An\nexample of such observable is the neutron electric dipole moment (nE DM). Experi-\nmental measurements of the nEDM place stringent constraints,\n|¯θ|/lessorsimilar10−10. (3.13)\nIn the following we will refer to ¯θasθ. That is, we take θto be the physical parameter.\nGiven that θis a free angular parameter of the quantum theory, its experiment al\nvalue is unnaturally small, giving rise to the Strong CP problem. We have argued\nthat the non-vanishing of Xand, more generally, a partition function that depends\n– 21 –onθsignal the spontaneous breaking of the ( −1)-form U(1) Chern-Weil symmetry of\nQCD. More broadly, a theory with a spontaneously broken ( −1)-form symmetry has a\nphysical dependence on a circle valued background field θ. Ifθis measured to be too\nsmall, a naturalness problem arises. The Strong CP problem is the QCD avatar of this\nnaturalness problem. We can now extract a necessary condition fo r the QCD Strong\nCP problem to arise.\nA necessary condition for the Strong CP problem in QCD.\nA necessary condition for Quantum Chromodynamics to have a Stro ng CP prob-\nlem is that the global ( −1)-form U(1) symmetry is spontaneously broken.\nIn the next section we will use this necessary condition for the Stro ng CP problem\nto arise in QCD to provide a new perspective on the problem and its solu tions.\n4 Solutions to the Strong CP problem and its analogues\nWe have argued that the Strong CP problem is intimately tied with a spo ntaneously\nbroken (−1)-form U(1) symmetry that arises in the Standard Model. If the p hysics\nassociated with this spontaneous breaking is prevented in some way , the Strong CP\nproblem should be solved. This problem has direct analogues in various other theories\nwith spontaneously broken, global ( −1)-form symmetries, some of which are easier to\nanalyze because they are low-dimensional, as we have discussed abo ve. In this section,\nwe discuss various solutions to the Strong CP problem from this pers pective.\n4.1 Solving the problem by gauging with an axion\nThe classic Peccei-Quinn-Weinberg-Wilczek solution to the Strong CP problem [ 70–73]\nmay be thought of as gaugingthe (−1)-form global U(1) symmetry with a dynamical\naxion field θ(x). The existence of a ( −1)-form global U(1) symmetry means that our\ntheory can be consistently coupled to a background axion field θ(x); we now simply\nsum over all such possible backgrounds in the path integral.\nIn the analogue problem in 2 dMaxwell theory, we have already introduced the\nrelevant action in ( 2.14), where the field φplays the role of the dynamical axion. Such\na coupling explicitly removes the physical dependence on θby polarizing the vacuum\nand screening the constant electric field. In this case, the original 0-form U(1) gauge\nsymmetry is Higgsed. One can see this explicitly by dualizing φto˜φ. The resulting\nkinetic term is ∼ |d˜φ−A|2, which shows that Ais made massive by a Stueckelberg\nmechanism. However, the ( −1)-form U(1) gauge symmetry is alsohiggsed, eliminating\n– 22 –the Kogut-Susskind pole. Wecan see this by dualizing thefield streng thF= dAto a 0-\nform integer field strength n, which acquires a Stueckelberg-type “kinetic term” |n−φ|2\nthat can also be interpreted as a potential that makes the gauge fi eldφfor the (−1)-\nformsymmetry massive. Ingeneral, we expect higgsing ofa ( −1)-formgaugesymmetry\ntocorrespond toconfinement ofaxionvortices bydomainwalls. Int he(1+1)dcase, the\ndomainwallsaresimplyparticleschargedunder A, whiletheoperatorei˜φinsertsastatic\nvortex at a point in spacetime. Because ˜φshifts under Agauge transformations, such\na vortex musthave an attached domain wall. This is the expected dual confinement\nphenomenon. Higher-dimensional analogues of this have been exte nsively discussed in\nthe literature on inflation [ 44,45].\nFor the Strong CP problem in QCD, the relevant coupling takes the fo llowing form:\nS⊂1\n8π2/integraldisplay\nθ(x)tr(G∧G). (4.1)\nThe Vafa-Witten theorem [ 74] ensures that the axion potential generated by QCD\ndynamics sets the effective low-energy θangle to zero. As in the 2 dcase, this term\ndescribes the coupling of the ( −1)-form symmetry current j0=1\n8π2⋆tr(G∧G) to a\ndynamical gauge field, the axion. The net effect is that the ( −1)-form symmetry is\ngauged. The axion equation of motion then shows that the instanto n number current\nbecomes co-exact,\nf2d⋆dθ=1\n8π2tr(G∧G). (4.2)\nThis exactness conditionisequivalent to gauging: anexact current integrates tozero on\nany closed manifold, implying that there are no charged operators t hat may link with\nthe symmetry operators ei/contintegraltext⋆j. There are thus no objects charged under a symmetry\ngenerated by an exact current. This is the chief property of a gau ge symmetry in gauge\ntheory. As discussed in §3.2, the gauged ( −1)-form symmetry is in a higgsed phase,\nwhich is reflected in the confinement of magnetically charged object s (axionic strings)\nby axion domain walls.\n4.2 Solving the problem by gauging with massless fermions\nA second canonical solution to the Strong CP problem is to postulate a chiral massless\nfermion. In the case of 2 dMaxwell theory the resulting theory is the Schwinger model,\nwhose action we wrote in ( 2.13). Such a massless chiral fermion comes with an ABJ\nanomaly for the chiral symmetry that allows the θangle to be rotated away, making\nit an unphysical parameter. This effect is particularly explicit in the 2 dcase thanks to\nthe 2dbosonization by which the Schwinger model is equivalent to the boson ic theory\nin eq. (2.14), where the θis absorbed by a redefinition of the compact scalar field. It\n– 23 –is now clear what is happening in terms of the ( −1)-form symmetry. The ( −1)-form\nsymmetry hasbeen gauged, making its spontaneous breaking innoc uous. It followsthat\nthe ground state electric field is screened by a polarized vacuum and the Strong CP\nproblem is avoided. In 2 dimensions it is clear that these two solutions t o the Strong\nCP problem are really the same. Furthermore, the lesson that addin g massless fermions\ngauges (−1)-formsymmetries holds more generally. Forinstance, in Yang-Mills , adding\na massless fermion produces an ABJ anomaly for the chiral current Jc,\nd⋆Jc=1\n8π2tr(F∧F). (4.3)\nThis equation implies that ⋆j0=1\n8π2F∧Fis (globally) exact, and hence (as explained\nin [19]) the (−1)-form symmetry is gauged. As in Sec. 4.1, this gauged ( −1)-form\nsymmetry is in a Higgs phase. In this case, although there is no elemen tary axion\nfield, there is still a magnetic confinement phenomenon. The confine d vortices are the\nboundaries of η′domain walls, which have chiral excitations carrying baryon number,\nas described in [ 75].\n4.3 Solving the problem with non-compact symmetries\nAn alternative solution to the CP problem was proposed in [ 76]. As discussed there,\nif one considers a 2 dabelian gauge theory with gauge group Rinstead of U(1), the\nanalogue of the Strong CP problem is immediately solved. The reason f or this to work\nis most easily understood in terms of the ( −1)-form symmetry, which we recall, is the\nmagnetic symmetry of the U(1) gauge theory in 2 d. As is well known, for the gauge\ngroup Rthe would-be magnetic symmetry operators act trivially.11In more detail,\nthere is a topological constraint that/integraltext\nMF= 0 for any closed 2-manifold M, so a/integraltext\nMθF\nterm for constantθdoes not affect the physics. In particular, the vacuum energy is\nindependent ofconstant θandsowewouldnotsaythatthetheoryspontaneouslybreaks\na (−1)-form symmetry. Physically, a background electric field on a non- compact space\ncan be screened by combinations of particles with mutually irrational electric charges.\nAn analogous 4 dsetup to this 2 dtheory requires modifying the instanton sum by\ncoupling to a topological theory (TQFT) with a non-compact 3-form gauge field [ 32].\n11Equivalently one maysaythat an Rgaugetheoryis obtained fromthe U(1) theoryby performinga\ntopologicalgaugingofthe U(1) magnetic symmetry. This gaugingise nforced bycouplingthe magnetic\ncurrent to a non-dynamical (i.e. with no kinetic term) U(1) gauge fie ld with only flat connections and\nsumming over it. In the present case this auxiliary field is a flat compac t scalar. From this point\nof view, the Strong CP problem is avoided also in this case by gauging th e (−1)-form symmetry. A\nSymTFT discussion of this model and the topological gauging can be f ound in appendix C. We thank\nAndrea Antinucci for comments on this point and for careful expla nation of his recent paper [ 77].\n– 24 –As in the 2dtheory however, a dynamical mechanism, i.e., adding mutually irration ally\ncharged domain walls, is needed to relax θand fully solve the Strong CP problem.\nFor the Strong CP problem of the Standard Model, [ 76] also proposed a related\nmechanism, relying on a non-compact axion fielda(x), which has couplings\n/integraldisplay1\n8π2[ξHa(x)tr(GH∧GH)+ξa(x)tr(G∧G)]. (4.4)\nHereG(x) is the usual Standard Model gluon field strength, while GHis the field\nstrength of a hidden Yang-Mills group that confines at a much higher scale. This\nconfinement generates a potential with a set of minima for a(x). IfξHandξare\nmutuallyirrational, thentheinfinitesetofminimaofthe GH-generatedpotentialallows\nthe effective theta term of QCD to scan over a dense discretuum of values, some of\nwhich will be very small. One then must invoke a cosmological argument for why\nwe find ourselves in a universe with such a small value. In our language , this model\nhas gauged a ( −1)-form Rglobal symmetry, which is an irrational combination of two\n(−1)-form U(1) global symmetries. This is only possible with an axion field that is\nnon-compact.\nThe common feature of the models of [ 32,76] is the introduction of non-compact\ngauge fields (either an ordinary gauge field, or an axion, or a three- form field). This\ncan enable novel solutions of CP problems in quantum field theory, bu t we expect that\nsuch models do not have consistent UV completions in quantum gravit y (see, e.g., [ 11]).\n4.4 Failing to solve the problem with explicit breaking\nAs we have discussed, standard solutions to the Strong CP problem rely on gauging the\n(−1)-form global symmetry. One might wonder if, instead, we could sim ply break the\nsymmetry explicitly. In the following we discuss a couple of strategies that implement\nthis idea but that ultimately fail to solve the problem.\n4.4.1 Explicit breaking via gauging and mixed anomalies\nFirst, itisoftenthecasethatwecanbreakasymmetry bygauginga different symmetry\nwith which it has a mixed anomaly.12One can attempt this strategy for solving the\n2dMaxwell theory analogue of the Strong CP problem, as follows. A well known fact\nabout Maxwell theory in any number of dimensions is that it has a U(1)(1)\ne×U(1)(d−3)\nm\nsymmetry under which Wilson lines and ’t Hooft operators are charge d. There is an\nobstruction to gauging these two symmetries at the same time, or a mixed ’t Hooft\n12There are many exceptions to this statement when gauging gives ris e to a 2-group structure or\nnon-invertible symmetries, see for instance [ 78–82]. Here we restrict to anomalies that break the\nungauged symmetry.\n– 25 –anomaly, that can be succinctly encapsulated in terms of the backg round gauge fields\nBeandBmby its anomaly polynomial,\nA=1\n2πdBe∧dBm. (4.5)\nIn the 2dcase these facts also hold, with the background field for the magne tic sym-\nmetry being θitself. In this case the anomaly polynomial is,\nA2d=1\n2πdBe∧dθ. (4.6)\nIt follows that an easy way of breaking the magnetic ( −1)-form symmetry is to gauge\nthe electric symmetry. The gauging is implemented by coupling the elec tric symmetry\nto a background gauge field Be, adding suitable local counterterms and summing over\nthe background field configurations in the path integral. The result ing action is13\nS=/integraldisplay\n−1\n2e2(F−Be)∧⋆(F−Be)+1\n2πθ(F−Be). (4.7)\nA first observation is that the kinetic term for the gauge field becom es a Stueckelberg-\nlike coupling and the U(1) gauge field is “eaten” by Be. This mass term removes the\npole from the photon 2-point function, destroying the long range f orce. Thus, the\ninfrared physics of the theory is trivial, and X= 0. Whether one considers this to be\na solution of the 2d CP problem or not is perhaps a matter of semantic s: the problem\nis gone, but so is all of the physics, since the photon is gapped.\nEven this pyrrhic victory is lost in the case of the actual Strong CP p roblem for\nQCD. The analogue would be to gauge a U(1) 3-form symmetry that h as an anomaly\npolynomial of the form\nA4d=1\n2πdB4∧dθ, (4.8)\nwithB4a background gauge field for the 3-form symmetry. However, QCD has no\nsuch 3-form symmetry! The 3-form gauge field associated with the Kogut-Susskind\npole emerges in the IR, rather than existing as a fundamental UV fie ld. There is no\nelectric 3-form global symmetry associated with it that we can gaug e.\n4.4.2 Explicit breaking in the UV\nIt ispossible that the( −1)-formU(1) globalsymmetry associated withQCD instantons\nis explicitly broken in the UV. Because the symmetry is topological, we e xpect that this\nwill occur only when thegaugegroup itself issomehow modified in theUV . Anexample\n13A similar computation has recently appeared in section 4.1.2 of [ 83].\n– 26 –was discussed in [ 19]. Suppose that the Standard Model gauge group is embedded in\nSU(5) (and let us ignore fermions for the moment, which slightly comp licate the story\nwithout changing the punchline). The UV theory has a single ( −1)-form Chern-Weil\nsymmetry with current\nj0SU(5)=1\n8π2⋆tr(FSU(5)×FSU(5)). (4.9)\nIn the IR however, there are three such Chern-Weil symmetries, one for each field\nstrength. Clearly, two linear combinations thereof are emergent in the low energy\ntheory, and explicitly broken in the UV. One could then ask: could the UV breaking of\nan emergent ( −1)-form symmetry be sufficient to remove the Nambu-Goldstone po le\nand solve the Strong CP problem? From one point of view, the answer should be no,\nas it would be a dramatic failure of decoupling if physics at the GUT scale could set\nthe topological susceptibility computed in the infrared limit of QCD to z ero. From a\ndifferent point of view, however, one may have the intuition that Nam bu-Goldstone\npoles are fragile and easily removed by UV effects. Thus, it is worth dis cussing this\npoint in more detail.\nA well-known fact about Nambu-Goldstone bosons parametrizing th e degenerate\nvacua of a spontaneously broken 0-form symmetry is that, if the s ymmetry is not\nexact in the UV, they get a small mass [ 84]. Consider for instance a symmetry that\nis explicitly broken by some irrelevant coupling with a characteristic sc ale ΛUV. If\nthe emergent symmetry is spontaneously broken at a scale Λ IR, the pseudo-Goldstone\nboson will typically have a mass scaling like ∼(ΛIR/ΛUV)p, wherepis some power\ndepending on the specific details.\nA surprising feature of Nambu-Goldstone bosons for spontaneou sly broken 1-form\nsymmetries is that their masslessness remains protected even if th e symmetry is only\napproximate in the sense above. In other words, 1-form symmetr ies (and higher) are\nexact emergent symmetries [ 54]. An example of this fact is the electromagnetic field\nthat we observe in nature. At low energies there are two U(1) symm etries, electric and\nmagnetic. At high energies these two symmetries are explicitly broke n by the presence\nof electric fermions and, presumably, magnetic monopoles. Those t wo symmetries are\nspontaneously broken and the Nambu-Goldstone boson is the phot on, which is exactly\nmassless despite the explicit breaking in the UV. More detailed example s were given\nin [54]. A difference with 0-form symmetries is that no local operator can b e charged\nunder a 1-form symmetry, which implies that emergent 1-form symm etries are exact\nin perturbation theory. The standard lore is that this helps in keepin g the photon\nmassless.\nFor the case of ( −1)-form symmetries one can pose a similar question: If the ( −1)-\nformsymmetry isemergent intheIR, isthemassless natureofthee mergent gaugefield,\n– 27 –i.e., the pole, protected? In other words, are emergent ( −1)-form symmetries exact?\nThis question is relevant because the pole is behind all the features o f spontaneously\nbroken (−1)-form symmetries and, in particular, if there is no pole, there is no vacuum\nelectric field and, thus, no Strong CP problem. If ( −1)-form symmetries behave like\n0-form symmetries it should be possible to lift the pole by changing the UV physics in\nsuch a way that the ( −1)-form symmetry is explicitly broken.\nAgain, it is useful to consider the case of Maxwell theory in various d imensions.\nIn 3d, where the photon is dual to a compact scalar, Maxwell theory has an electric\n1-form symmetry and a magnetic 0-form symmetry. The magnetic 0 -form symmetry\ncan be explicitly broken in the UV by embedding U(1) gauge theory in SU (2) gauge\ntheory, higgsed by a real adjoint scalar field. This theory admits a f amous semiclassical\nanalysis of confinement due to Polyakov [ 57], in which magnetic monopoles (which are\ninstantons in 3 d) produce an exponentially small mass for the dual photon. Thus, in\nthis case, the would-be Nambu-Goldstone boson is removed by the e xplicit breaking of\na 0-form symmetry.\nOn the other hand, in 2 dMaxwell theory, the magnetic symmetry is a ( −1)-form\nsymmetry. Again, the magnetic symmetry can be explicitly broken by embedding the\nU(1) gauge theory in SU(2); no axion coupling that UV completes1\n2πθFis possible\nin that theory, because the SU(2) field strength is not gauge invar iant. There are no\ninstanton effects in this theory, and the photon should remain mass less. Thus, in this\ncase we expect that thepole associated with the spontaneous bre aking of the ( −1)-form\nsymmetry is still present in the IR, despite the explicit breaking of th e symmetry in\nthe UV. We expect a similar behavior in the case of SU(5) breaking into the SM gauge\ngroup in 4d, and leave a detailed investigation for a future study.\nWe expect that the lesson here generalizes: a Nambu-Goldstone po le can be re-\nmoved only in the case where the Nambu-Goldstone is protected by a 0-form symmetry\nthatisexplicitlybrokenintheUV.TheKogut-Susskindpoleisassociat edwithanemer-\ngent (d−1)-formgauge field, and can be protected by either a ( d−1)-formsymmetry or\na (−1)-form symmetry, and hence its existence is robust against UV sy mmetry break-\ning ind >1 spacetime dimensions. Thus, embedding the Standard Model in a GU T\nshould not have any impact on the Strong CP problem.\n4.5 Solving the problem with gauged reflection symmetries\nAside fromthe axion, the most well-studied solutionto the Strong CP problem assumes\na fundamental spacetime reflection symmetry, which is either a gen eralized parity sym-\nmetry [85] or CP symmetry [ 86–89]. Here we will refer to the latter case, generally\nknown as Nelson-Barr models, though our remarks will apply more br oadly. Theories\nwith a spacetime reflection symmetry can be defined on non-orienta ble manifolds. On\n– 28 –such a space,1\n8π2tr(F∧F) is not defined, because F∧Fis an ordinary differential\nform, but only pseudo-forms can be integrated without a choice of orientation. Thus,\nthe instanton number is not well-defined (although a topological inva riant valued in Z2\nsurvives). However, by our definition, these theories still have a ( −1)-form U(1) global\nsymmetry, becausetheycanbeconsistentlycoupledtoabackgro undpseudoscalar axion\nfieldθ(x), which transforms with a minus sign under spacetime reflections. F urther-\nmore, the symmetry is still spontaneously broken, because on Minkowski space we can\nstill turn on an arbitrary constant ¯θterm and evaluate a nonzero topological suscep-\ntibility (3.2). However, the only constant θ(x) backgrounds that can be defined on an\narbitrary space are¯θ= 0 and ¯θ=π. Thus, the reflection symmetry requires that the\ntheory be defined with one of these two special ¯θterms, and the Strong CP problem\ncould, in principle, be solved.\nThe difficulty begins when we recall that the world in which we live is not CP\nsymmetric, and indeed the CP-violating phase in the CKM matrix is an O(1) number.\nThus, if we live in a universe with an underlying CP symmetry, the symme try must be\nspontaneously broken, and (at least as measured by the CKM phas e) badly so. Below\nthe scale of CP breaking, we should match the fundamental theory onto a theory\nwithout CP, and such a theory in principle admits an arbitrary consta nt¯θterm. The\nlow-energy value of ¯θneed not be one of the special values ¯θ= 0 orπdefining the\ntheory in the ultraviolet, because integrating out massive particles that couple to CP-\nbreaking can generate effective contributions to ¯θin the IR. Nelson-Barr models are\nengineered so that such effects are small, whereas the CKM phase is large. It is difficult\ntogiveapurelysymmetry-based explanationofhowtheywork, with outdelvingintothe\ndetailed structure of the quark mass matrices, which must be enfo rced with additional\n(model-dependent) gauge symmetries.\n5 Outlook\nIn this work we have extended the notion of spontaneous breaking to (−1)-form U(1)\nsymmetries and started the exploration of its applications. We finish this text with\nsome open questions.\n•We have provided a useful working definition of a ( −1)-form U(1) symmetry and\nits spontaneous breaking, but it would be useful to put ( −1)-form symmetries\nin general on a more similar footing to other p-form symmetries in QFT. For\nexample, the SymTFT approach could be a useful way to formulate ( −1)-form\nsymmetries. Itwouldalsobeinterestingtogainabetterunderstan dingofwhether\n(−1)-form Rsymmetries are a useful concept in QFT.\n– 29 –•We have explored several solutions to the Strong CP problem which, broadly\nspeaking, aim at removing the Nambu-Goldstone fieldby either gauging or ex-\nplicitly breaking the ( −1)-form symmetry. It turns out that gauging has been\nextensively covered in the literature. On the other hand, it seems t o us that ex-\nplicit breaking is still poorly understood and we hope to study it furth er in future\nwork. Besides, theexplicit breaking ofthe( −1)-formsymmetry bymonopoleshas\nnot been thoroughly investigated except in the case of U(1) gauge theory [19]. It\nwould be interesting to examine the case of more general gauge gro ups, including\nGrand Unified Theories.\n•A fundamental ingredient in our understanding of spontaneous br eaking of con-\ntinuous (higher form) symmetries is the Goldstone Theorem. While we have\nestablished the presence of a pole in the 2-point function of a Nambu -Goldstone\nfieldwhenever a ( −1)-form U(1) symmetry is spontaneously broken, we have\nnot been able to explicitly prove a theorem that follows the reasoning of the\nusual Goldstone theorem. Difficulties arise because the symmetry o perator fills\nthe entire spacetime and cannot be deformed, and because there are no (−1)-\ndimensional charged operators that can obtain vacuum expectat ion values. A\nbetter understanding of the formalism of ( −1)-form symmetries may overcome\nthese obstacles and provide a more direct Goldstone Theorem.\n•While most of the solutions to the Strong CP-problem that are discus sed in\nour paper are concentrated on lifting the ( −1)-form U(1) symmetry, Nelson-Barr\nmodels are conceptually different. Unlike the other solutions that lea d to no\ndependence of the vacuum energy on θ, Nelson-Barr models are constructed such\nthat the value of θis small. It remains an open question to understand a generic\nsymmetry-based explanation for such models.\n•Axion-like fields play a role in several solutions to longstanding natura lness prob-\nlems in particle physics. A prime example is the hierarchy problem, which sees\npotential mitigation through the relaxion model [ 90,91]. This model introduces\nan axion-like field having a coupling with the Higgs mass term. Another e xample\nis the axion monodromy framework for inflation [ 43]. In these models the axion\ncouplings appear to violate the axionperiodicity but it is restored by m onodromy,\nmuch as in the 2 dMaxwell example we discussed in Sec. 2.3. Understanding the\ninterplay between monodromy and SSB of ( −1)-form symmetries in such models\nwould be interesting.\nWe hope that this novel case of spontaneous symmetry breaking w ill prove a useful\nunifying tool for physical phenomena.\n– 30 –Acknowledgements\nWe wish to thank Andrea Antinucci, Riccardo Argurio, Diego Delmastr o, Ben Heiden-\nreich, Ohad Mamroud, Jake McNamara, Miguel Montero, Tom Rudeliu s, John Stout,\nHo Tat Lam, Angel Uranga, and Irene Valenzuela for insightful con versations. The\nwork of DA is supported by the U.S. Department of Energy (DOE) un der Award DE-\nSC0015845. The work of EGV has been partially supported by Marga tita Salas award\nCA1/RSUE/2021-00738 and by MIUR PRIN Grant 2020KR4KN2 “Str ing Theory as\na bridge between Gauge Theories and Quantum Gravity”. MR is suppo rted by the\nDOE Grant DE-SC0013607. MS is supported by JSPS KAKENHI Grant Numbers\nJP22J00537. EGV also thanks the Simons Center for Geometry and Physics and its\nSummer Physics Workshop for kind hospitality where part of this wor k was completed.\nMS would like to thank Fermilab theory division for their hospitality.\nA The 2dAbelian-Higgs model\nA nice exposition of this model, which we follow, may be found in [ 92,93]. Consider\nthe action,\nS=/integraldisplay\nd2x1\n2e2F2\n01+θ\n2πF01+|Dφ|2−m2|φ|2−λ\n2|φ|4. (A.1)\nAs already mentioned, in 2 dthe gauge coupling is dimensionful and the theory is\nstrongly coupled in the IR. Hence, the regime |m2|/lessorsimilare2will be complicated to solve.\nConsider instead |m2|≫e2. There are then two regimes to consider,\n•m2≫e2: In this case the gauge symmetry is not spontaneously broken and the\ntheoryisjustelectrodynamicswithaheavyscalarmeson. Thebeha viourissimilar\nto the massive Schwinger model. In particular, there is a vacuum elec tric field, a\nnon-zero topological susceptibility and a long range constant forc e mediated by\nthe photon. We conclude that there is a magnetic ( −1)-form symmetry in the IR\nwhich is spontaneously broken.\n•m2≪ −e2: In this case the gauge symmetry is spontaneously broken by the\ncondensationofthescalarfield. Thenaiveexpectationisthatthep hotonbecomes\nmassive, the long-range force is screened, the topological susce ptibility vanishes\nand there is no electric field in the vacuum. We therefore expect tha t the (−1)-\nform symmetry is not spontaneously broken. It turns out that th is expectation\nis wrong. Due to non-perturbative effects mediated by instantons (which are\nvortices in 2 dimensions), the gauge symmetry is restored, there is a long-range\nforce between probe particles and there is a vacuum electric field wh ich depends\n– 31 –linearly with θ. In fact the physics is the same as in the m2≫e2regime but all\neffects are exponentially suppressed. We learn that the ( −1)-form symmetry is,\ncontrary to expectation, realized in the IR and spontaneously bro ken!\nB The CPN−1model\nThis model has been extensively discussed in the literature, we follow [94,95]. The\nCPN−1model can be defined by the following Euclidean action,\nS=/integraldisplay\nd2x1\n2e2|F|2+iθ\n2πF+N/summationdisplay\na=1|Dφa|2+λ\n2/parenleftBiggN/summationdisplay\na=1|φa|2−v2/parenrightBigg2\n,(B.1)\nwhichdescribesasetof NmassivecomplexscalarfieldswithanSU( N)globalsymmetry\nand coupled to a U(1) gauge field. Classically we expect the scalar pot ential to be\nminimized, spontaneously breaking the SU( N) symmetry to SU( N−1)×U(1),\n|φa|2=v2. (B.2)\nThus, classically, the low energy is described by N−1 massless scalar fields (Goldstone\nbosons) with target space,\nCPN−1=SU(N)\nSU(N−1)×U(1). (B.3)\nThis is of course in contradiction with the MWC theorem and it is well kno wn that\nstrong dynamics radically change the low energy physics of this theo ry. We will not\nreview the computations but merely state the result. It turns out that the low energy\ndynamics is that of Nmassive scalar fields coupled to a U(1) gauge field, whose dy-\nnamics is emergent in the IR14. The mass is given by a dynamically generated scale\nΛCPN−1analog to Λ QCD. The resulting dynamics are pretty much the same as the ones\nof the abelian-Higgs model in the unbroken phase:\n•There is an electric field in the vacuum. In the large N limit it was compute d in\n[37,96],\n/an}bracketle{tF01/an}bracketri}ht ∼Λ2\nCPN−1\nNθ+O(1/N). (B.4)\n•It follows that the topological susceptibility, which is the order para meter for the\n(−1)-form symmetry SSB, takes a non-zero value,\nX ∼Λ2\nCPN−1\nN. (B.5)\n14In fact, if one starts with no kinetic term for the UV gauge field a non zero kinetic term is dynam-\nically generated in the IR. In this sense, the U(1) dynamics are emer gent.\n– 32 –•There is a long range force between fractionally quantized probe pa rticles. Inte-\nger quantized particles don’t experience such force because they are screened by\nSchwinger pair production.\n•The spectrum is composed of mesons. As the θangle is dialed from 0 to 2 πaφφ⋆\npair is created and the spectrum undergoes a flow.\n•Interestingly all these phenomena are not exponentially suppress ed as befits an\ninstanton effect, signaling that one can’t hope to explain them using s emiclassical\ntechniques.\n•We conclude that there is a ( −1)-form symmetry which is spontaneously broken\nand all the expected features are present.\nNote that this model has many of the salient features of QCD. It is w ell-known that\nit hasθ-vacua, a dynamically generated scale and instantons that are insu fficient to\nexplain the low energy physics. We learn now that it shares a further feature with\nQCD, namely an ( −1)-form symmetry which is spontaneously broken.\nC A SymTFT for 2dMaxwell theory.\nThe SymTFT [ 97–99] of a given ddimensional Quantum Field Theory Tdis ad+ 1\nTQFT that encodes the (categorical) symmetry of Tdand of all other theories that can\nbe obtained from Tdby a topological manipulation. The SymTFT is placed on a ( d+1)\nslab with two boundaries. On one of them the boundary condition is no t topological\nand the local degrees of freedom live. On the other boundary a top ological boundary\ncondition is imposed that prescribes the symmetry of Tdonce the slab is collapsed to\nrecoverTd. Different topological boundary conditions encode the symmetry o f theories\nobtained from Tdby a topological manipulation.\nThis construction has recently been extended to abelian continuou s symmetries\nin [77,100]. In this appendix we present the SymTFT for 2 dabelian gauge theories,\nwhich is an application of [ 77]. This construction puts ( −1)-form U(1) symmetries in\nthe same footing as more familiar symmetries and also clarifies the rela tion between\nthe (−1)-form symmetries of abelian gauge theories with different global f orms of the\ngauge group.\nConsider a 3 dBF theory with action,\nS=1\n2π/integraldisplay\nφdb2, (C.1)\n– 33 –where both φandb2areRgauge fields15. This means that they don’t have any large\ngauge transformations and charged operators with arbitrary re al coefficients are al-\nlowed. In this case there are local operators and surfaces,\nUα(x) = eiαφ(x),˜Uβ(Σ2) = eiβ/integraltext\nΣ2b2, (C.2)\nwhereα,βare real-valued. The braiding between these operators is,\n/an}bracketle{tUα(x)˜Uβ(Σ2)/an}bracketri}ht=e2παβ·Link(x,Σ2). (C.3)\nThis SymTFT encodes the symmetry of continuous free abelian gaug e theories in 2 d.\nDifferent topological boundary conditions correspond to a choice o f mutually transpar-\nent bulk operators that can terminate on the boundary, i.e. their b raiding is trivial.\nDifferent boundary conditions give different symmetries on the boun dary. Here we\nmention those corresponding to the global forms that we have enc ountered in the main\ntext16.\n•DirichletBoundaryConditions(DBC’s)for b2allowtheWilsonsurfaces ˜Uβ(Σ2) =\neiβ/contintegraltext\nΣ2b2to endonthe boundary, giving rise toa R1-formsymmetry. Thetopolog-\nical symmetry operators on the boundary are Uα(x). For the variational problem\nto be well posed φmust obey Neumann Boundary Conditions (NBC’s), which\nimply that it must be summed over in the 2 dtheory. This sum explicitly imple-\nments the topological gauging mentioned in footnote 11. The total symmetry of\nthe 2dtheory is an R1-form symmetry, which is the symmetry of the R2dgauge\ntheory that we met in section 4.3.\n•Mixed boundary conditions are allowed. In particular, one may impose NBC’s\nfor the Zpiece of both fields and DBC’s for the remaining U(1) ≃R/Zpieces\n[100]. These boundary conditions allow operators ˜Uβ(Σ2) withβ∈Zto end on\ntheboundary. The endpoints becomethe chargedlines under a U(1 )(1)symmetry.\nThe remaining operators ˜Uβ(Σ2) withβ∈U(1) can be placed on the boundary\nand correspond to symmetry operators generating a U(1)(−1)symmetry. The\noperatorsUα(x) = eiαφ(x),α∈Zcan’t end onthe boundarybecause they arezero-\ndimensional, so the U(1)(−1)symmetry does not have charged operators . Finally,\nthe operators Uα(x) = eiαφ(x),α∈U(1) are topological on the boundary and\ngenerate the U(1)(1)symmetry. We conclude that the total symmetry is,\nU(1)(1)×U(1)(−1), (C.4)\n15For the case of φthis means that it is a non-compact scalar field.\n16The different boundary conditions can be expressed in terms of a va riational problem that must\nbe well-defined, see [ 82,100] for a similar discussion. We refrain from going into such details and\nmerely state the results here.\n– 34 –which matches the symmetry of the U(1) gauge theory discussed in section2.3.\nThe SymTFT of the gauge theories with electric matter can be similarly realized by\nturningφinto a compact scalar. Through this exercise we see that ( −1)-form symme-\ntries are very similar to more usual symmetries, at least from the Sy mTFT point of\nview. We plan to return to these considerations in more generality an d depth in future\nwork.\n– 35 –References\n[1] A. Kapustin and N. Seiberg, “Coupling a QFT to a TQFT and Du ality,”\nJHEP04(2014) 001 ,arXiv:1401.0740 [hep-th] .\n[2] D. Gaiotto, A. Kapustin, N. Seiberg, and B. 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Teleman, “Topological sy mmetry in quantum field\ntheory,”arXiv:2209.07471 [hep-th] .\n[100] T. D. Brennan and Z. Sun, “A SymTFT for Continuous Symme tries,”\narXiv:2401.06128 [hep-th] .\n– 42 –"    },    {        "title": "2402.00125v1.Migration_of_low_mass_planets_in_inviscid_disks__the_effect_of_radiation_transport_on_the_dynamical_corotation_torque.pdf",        "content": "MNRAS 000, 1–11 (2023) Preprint 2 February 2024 Compiled using MNRAS L ATEX style file v3.0\nMigration of low-mass planets in inviscid disks:\nthe effect of radiation transport on the dynamical corotation torque\nAlexandros Ziampras1⋆, Richard P. Nelson1, Sijme-Jan Paardekooper1,2\n1Astronomy Unit, School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK\n2Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2600 AA Delft, The Netherlands\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nLow-mass planets migrate in the type-I regime. In the inviscid limit, the contrast between the vortensity trapped inside the\nplanet’s corotating region and the background disk vortensity leads to a dynamical corotation torque, which is thought to slow\ndown inward migration. We investigate the e ffect of radiative cooling on low-mass planet migration using inviscid 2D hydro-\ndynamical simulations. We find that cooling induces a baroclinic forcing on material U-turning near the planet, resulting in\nvortensity growth in the corotating region, which in turn weakens the dynamical corotation torque and leads to 2–3 ×faster in-\nward migration. This mechanism is most e fficient when cooling acts on a timescale similar to the U-turn time of material inside\nthe corotating region, but is nonetheless relevant for a substantial radial range in a typical disk ( R∼5–50 au). As the planet\nmigrates inwards, the contrast between the vortensity inside and outside the corotating region increases and partially regulates\nthe effect of baroclinic forcing. As a secondary e ffect, we show that radiative damping can further weaken the vortensity barrier\ncreated by the planet’s spiral shocks, supporting inward migration. Finally, we highlight that a self-consistent treatment of ra-\ndiative di ffusion as opposed to local cooling is critical in order to avoid overestimating the vortensity growth and the resulting\nmigration rate.\nKey words: planet–disc interactions — accretion discs — hydrodynamics — methods: numerical\n1 INTRODUCTION\nMore than 5500 exoplanets have been discovered so far1, and\ntheir population shows remarkable diversity in orbital parameters,\nmasses, and multiplicity. The idea that these planets formed in cir-\ncumstellar disks is supported by both the direct observation of plan-\nets embedded in the disk around PDS 70 with VLT (Keppler et al.\n2018; Ha ffert et al. 2019), as well as kinematic signatures in ob-\nservations of gas emission with ALMA (e.g., Teague et al. 2018).\nUnderstanding how planets formed and evolved to their current state\nis a key goal of modern astrophysics.\nThe final orbital configuration of a planet is directly a ffected by its\ninteraction with the disk it is embedded in. The disk exerts a torque\non the planet, which causes it to migrate through the disk (Goldreich\n& Tremaine 1979; Ward 1997a; Tanaka et al. 2002). The migration\nrate depends on the disk’s properties as well as on the planet’s mass,\nwith low-mass planets migrating in the rapid type-I regime (Ward\n1997a). For a recent review, see Paardekooper et al. (2022).\nModeling planet–disk interaction over long timescales is not an\neasy task. The population synthesis approach (e.g., Mordasini et al.\n2009a,b) and N-body modeling (e.g., Coleman & Nelson 2014;\nIzidoro et al. 2017) are computationally inexpensive and cover a\nwide range of parameters in a relatively short time, but rely on\nanalytical prescriptions for the torques acting on the embedded\n⋆E-mail: a.ziampras@qmul.ac.uk\n1https://exoplanetarchive.ipac.caltech.edu/planet(s). Hydrodynamical simulations, on the other hand, are sig-\nnificantly more computationally restrictive, but allow a full treat-\nment of planet–disk interaction that yields more accurate results.\nHydrodynamical modeling is favorable for massive-enough plan-\nets, where planet–disk interaction becomes nonlinear (Rafikov 2002)\nand gap opening can occur (Crida et al. 2006). It is also crucial\nfor low-viscosity disks, where the corotation torque can desaturate\n(Paardekooper et al. 2011) and dynamical corotation torques can\ncome into play (McNally et al. 2017), as well as for non-isothermal\ndisks, where thermal e ffects can even reverse the direction of mi-\ngration (Kley & Crida 2008; Pierens 2015). In light of the recent\nparadigm shift towards low-turbulence disks, where accretion is\nthought to be powered by MHD-driven winds (Bai & Stone 2013),\nand the constraints on the radiative properties of protoplanetary disks\nthrough their dust distribution (Birnstiel et al. 2018), the need for un-\nderstanding how radiative e ffects can a ffect planet migration in the\nnearly inviscid limit is more relevant than ever.\nA simplified thermodynamics model that is often employed in\nsimulations of planet–disk interactions is the locally isothermal\nequation of state, which assumes instant cooling of the gas towards\nan equilibrium temperature. However, recent models (Miranda &\nRafikov 2019, 2020a; Ziampras et al. 2020), based on observations\n(e.g., Andrews et al. 2018; Öberg et al. 2021), highlight the impor-\ntance of a more realistic treatment of thermodynamics and therefore\nradiative e ffects, even in regions where cooling occurs on dynami-\ncal time scales. Further work has showcased the impact of thermal\ndiffusion (Ziampras et al. 2023; Miranda & Rafikov 2020b) in disk\n©2023 The AuthorsarXiv:2402.00125v1  [astro-ph.EP]  31 Jan 20242 A. Ziampras et al.\nthermodynamics. It therefore becomes clear that an investigation of\nhow radiation transport and in particular thermal di ffusion influence\nplanet migration in the marginally optically thin 20–40 au range is\nnecessary.\nIn this study, we carry out inviscid radiation hydrodynamics sim-\nulations of planet–disk interaction, with the focus being the e ffect\nof radiation transport in the type-I regime of planet migration. Our\ngoal is to investigate the source of torques related to radiative e ffects\nusing a realistic prescription of radiation transport in 2D, extend-\ning works that have focused on viscous models (e.g., Kley & Crida\n2008; Pierens 2015) but also providing an explanation on the origin\nof these “radiative” torques and constraining where in the disk they\nmight be important.\nWe describe our physical and numerical setup in Sect. 2. We\npresent our results in Sect. 3, and discuss them in Sect. 4. We then\nsummarize our findings in Sect. 5.\n2 PHYSICS AND NUMERICS\nIn this section we lay out our physical and numerical framework. We\nlist the vertically integrated hydrodynamical equations, introduce the\nsources of cooling used in our models, and describe our numerical\nsetup.\n2.1 Physics\nWe consider a disk of ideal gas with adiabatic index γ=7/5 and\nmean molecular weight µ=2.35 around a star with mass M⋆and\nluminosity L⋆. The inviscid Navier–Stokes equations then read\n∂Σ\n∂t+u·∇Σ =−Σ∇·u, (1a)\n∂u\n∂t+(u·∇)u=−1\nΣ∇P−∇(Φ⋆+ Φ p), (1b)\n∂e\n∂t+u·∇e=−γe∇·u+Q, (1c)\nwhere Σ,uandPdenote the surface density, velocity vector, and\npressure of the gas, and the internal energy density is given by the\nideal gas law as e=P/(γ−1). The stellar potential at distance R\nisΦ⋆=−GM⋆/R, where G is the gravitational constant. We can\nalso define the gas isothermal sound speed cs=√P/Σ =p\nRT/µ,\nwhere Tis the gas temperature and Ris the ideal gas constant. The\npressure scale height is then H=cs/ΩK, where ΩK=p\nGM⋆/R3is\nthe Keplerian angular velocity, and the disk aspect ratio is h=H/R.\nThe planetary potential follows a Plummer-like prescription with\na smoothing length ϵ=0.6Hpsimilar to Müller & Kley (2012) to\naccount for the vertical disk stratification:\nΦp=−GMp√\nd2+ϵ2,d=R−Rp. (2)\nFinally, Qcontains any additional radiative terms. In our models\nthe disk is heated by stellar irradiation following the passive, irradi-\nated disk model of Menou & Goodman (2004)\nQirr=2L⋆\n4πR2(1−ε)θ\nτeff, θ =Rdh\ndR, (3)\nwhereθis the flaring angle with θ=2h/7 (as T∝R−3/7, see Chiang\n& Goldreich 1997), εis the disk albedo (here ε=1/2), andτeffis an\neffective optical depth following Hubeny (1990)\nτeff=3τR\n8+√\n3\n4+1\n4τP. (4)Here,τR,P=1\n2κR,PΣis the optical depth due to the Rosseland and\nPlanck mean opacities κRandκP. We assume κR=κP=κ, following\nthe opacity model of Lin & Papaloizou (1985).\nWe then treat thermal cooling through the disk surfaces as\nQcool=−2σSBT4\nτeff, (5)\nand in-plane radiation transport with a flux-limited di ffusion ap-\nproach (FLD, Levermore & Pomraning 1981)\nQrad=√\n2πH∇· \nλ4σSB\nκRρmid∇T4!\n, ρ mid=1√\n2πΣ\nH, (6)\nwith the flux limiter λfollowing Kley (1989). The balance between\nQirr,Qcool, and Qradresults in a temperature profile T∝R−3/7.\n2.2 Vortensity growth due to cooling\nIn this study we often use the vertical component of the gas vorten-\nsity, defined in 2D as ϖ=∇×u/Σ·ˆz, to interpret our results. Taking\nthe curl of Eq. (1b) and dividing by Σyields the vortensity equation,\nwhich, for a 2D flow with uz=0, reads\n∂ϖ\n∂t+(u·∇)ϖ=∇Σ×∇P\nΣ3=S, (7)\nwhereS∝∇ Σ×∇Pis a vortensity source term due to baroclinic\nforcing. For a barotropic flow ( P=P(Σ)),ϖis conserved along\nstreamlines.\nLet us now consider a gas parcel in the planet’s corotating region.\nIn the planet’s corotating frame this parcel follows a streamline with\na horseshoe pattern (see e.g., Kley & Nelson 2012), performing a\nU-turn once behind and once ahead of the planet before completing\na closed loop. We can now analyze what happens to the vortensity of\nthe gas parcel during the U-turn at R=Rp, along the radial direction\n(see schematic in Fig. 1).\n2.2.1 Locally isothermal limit\nIn the limit where cooling happens on timescales much shorter than\nthe orbital timescale Ω−1\nK, we can assume that T(R)∝Rqis set by a\nbalance between the radiative terms listed in Sect. 2.1, and U-turning\nmaterial is expected to drive vortensity growth for q,0 (Casoli\n& Masset 2009). The vortensity of gas U-turning behind the planet\n(ϖb) will change according to\nSiso\nb=P\nTΣ3∇Σb×∇T|p=−Pb\nTbΣ3\nb ∂Σ\n∂ϕ∂T\n∂R!\f\f\f\f\f\fb=−qPb\nRpΣ3\nb∂Σ\n∂ϕ\f\f\f\f\f\nb,(8)\nwhere a subscript ‘p’ denotes the value at the planet’s radial loca-\ntion. Assuming a dense, circular envelope around the planet such\nthat∂ϕΣ|b>0 and q<0, this implies that Siso\nb>0. In other words,\nϖwill increase as material U-turns behind the planet. In a similar\nway, the vortensity of gas U-turning ahead of the planet ( ϖa) will\nchange as\nSiso\na=−qPa\nRpΣ3\na∂Σ\n∂ϕ\f\f\f\f\f\na, (9)\nresulting in a decrease in ϖsince∂ϕΣ|a<0.\nFinally, assuming that the U-turns are symmetric such that ∂ϕΣ|b=\n−∂ϕΣ|aandxa=xbforx∈[P,Σ,T], we have thatSiso\na+Siso\nb=\n0. In other words, while ϖwill increase (decrease) as material U-\nturns behind (ahead of) the planet, the total change along a closed\nhorseshoe loop is zero and ϖis ultimately conserved.\nMNRAS 000, 1–11 (2023)Planet migration in radiative disks 3\n1.0\n 0.5\n 0.0 0.5 1.0\n-1.00.01.0(RRp)/xh\n0.2\n 0.1\n 0.0 0.1 0.2\n( p)/\n-1.00.01.0(RRp)/xh\nFigure 1. Gas streamlines around an embedded planet (marked with a black\ndot) in the{R,ϕ,z=0}plane. Material just outside the planet’s corotating\nregion (CR) feels a kick as it shears past the planet (green curves). Inside the\nCR, material follows closed orbits as it U-turns behind (orange) and ahead\n(blue) of the planet. Purple orbits mark the L4 and L5 Lagrange points. The\nbottom panel is a zoom-in of the top between the dashed lines, showing that\nmaterial U-turning closer to the planet librates closer to the edge of the CR.\n2.2.2 Adiabatic limit\nWhen cooling is ine fficient, the parcel contracts (expands) adiabati-\ncally as it approaches (recedes from) the planet. This process is re-\nversible and entropy is conserved along the streamline. Each stream-\nline therefore retains its own entropy. Defining the entropy function\nK=P/Σγ, with Σ∝Rs, we initially have that K(R)∝Rξ, with\nξ=q+(1−γ)s. Assuming, without loss of generality, that ξ <0,\na high-entropy streamline at R<RpU-turning behind the planet\nwill advect into a low-entropy radial zone at R>Rp, resulting in\nan entropy discontinuity near R≈Rpand a nonzeroSb. The low-\nentropy streamline will, of course, also U-turn ahead of the planet\ninto the high-entropy zone at R<Rp, resulting inSa=−S band\na netSadb=0. Furthermore, after a few libration timescales (see\nEq. (14)), entropy will be completely mixed in the horseshoe region\nand therefore\nSadb∝∇Σ×∇P∝∇Σ×∇K=0. (10)\nIn our models, ξ≈−0.03 and therefore vortensity growth in adia-\nbatic models is negligible. We do, however, explore di fferent values\nofξin Appendix B.\n2.2.3 Finite cooling timescale\nWhen cooling acts on a finite timescale, this calculation becomes\ncomplicated. We can however show that the two source terms ahead\nof and behind the planet ( Sa,Sb) are asymmetric such that vortensity\nalways increases along a pair of streamlines U-turning in opposite\ndirections. We provide a non-rigorous derivation in Appendix A as\na proof of concept.2.3 Dynamical corotation torque\nIn a su fficiently laminar disk, where mixing between the planet’s\nhorseshoe region and the background disk is negligible, a migrating\nplanet will experience a dynamical corotation torque (McNally et al.\n2017)\nΓh=2π \n1−ϖ(Rp)\nϖh!\nΣpR2\npxhΩp dRp\ndt−uR!\n, (11)\nwhere the subscript ‘p’ denotes quantities at the planet’s radial posi-\ntion and xhis an estimate of the half-width of the corotating region\nin the 2D approximation following Paardekooper et al. (2010)\nxh=1.1\nγ1/4 0.4\nϵ/H!1/4s\nMp\nhM⋆Rp. (12)\nIn Eq. (11), ϖ(Rp) denotes the vortensity of the background disk\natRpandϖhrefers to the characteristic vortensity enclosed in the\nplanet’s horseshoe region, which would correspond to the vortensity\nof the initial radial location of the planet ( R0) ifϖis conserved inside\nthe corotation region.\nProvided that the background disk radial velocity uRis small\ncompared to the planet’s migration speed, this dynamical corota-\ntion torque is positive for a planet migrating inwards and becomes\nstronger as ϖ(Rp)/ϖ hincreases (assuming Σ∝R−3/2or shal-\nlower), eventually stalling the planet (Paardekooper 2014). Based\non the discussion in Sects. 2.2.2–2.2.3, a vortensity source term\nin the horseshoe region would cause ϖhto evolve as ϖh(t)≈\nϖ(R0)+Rt\n0Srad(t′) dt′, weakening the dynamical corotation torque\nand speeding up inward migration, possibly preventing the planet\nfrom stalling.\n2.4 Numerics\nWe solve the equations Eq. (1) using the Godunov-scheme numer-\nical hydrodynamics code PLUTO v.4.4 (Mignone et al. 2007). We\nuse a cylindrical grid {R,ϕ}with NR×Nϕ=805×3141 cells, loga-\nrithmically spaced in Rand uniformly spaced in ϕ. Our grid covers\nthe radial extent R∈[0.4,2]R0with R0=30 au, andϕ∈[0,2π].\nOur initial conditions assume power-law profiles for ΣandTwith\nΣ0= Σ ref R\nR0!−1\n, T0=Tref R\nR0!−3/7\n, (13)\nwithΣref=56.7 g/cm2andTref=21 K, or equivalently h0=0.05,\nobtained by a balance between QirrandQcool(see Eqs. (3) & (5)).\nWith these parameters, our chosen resolution yields 25 cells per\nscale height at R=R0. The disk surface density corresponds to\n1700 g/cm2at 1 au. The chosen set of parameters also results in a\ncooling timescale tcool≈1.6Ω−1\nKatR=R0(Ziampras et al. 2023,\nsee also Eqs. (15) & (16)).\nThe velocity field is initialized with a Keplerian profile corrected\nfor pressure support, and the planet is placed at Rp=R0with Mp=\n2×10−5M⋆≈6.7M⊕. The disk boundaries are periodic in azimuth\nand closed in the radial direction, and we apply the wave-damping\nprescription of de Val-Borro et al. (2006) in the radial zones R<\n0.525 and R>1.525 with a damping timescale of 0.1 orbits at the\nrespective boundary.\nThe planet is integrated using an adaptation of the N-body module\nby Thun & Kley (2018), where the correction due to the neglect of\ndisk self-gravity by Baruteau & Masset (2008b) is applied to mod-\nels where the planet is allowed to migrate. The indirect term by the\nplanet and star orbiting their common center of mass is included in\nMNRAS 000, 1–11 (2023)4 A. Ziampras et al.\nall models, but the star feels the gravitational influence of the disk\nonly in models where the planet can migrate (Crida et al. 2022). The\nplanet grows to its final mass over 10 orbits using the formula in de\nVal-Borro et al. (2006).\nIn all runs we use the FARGO algorithm (Masset 2000), imple-\nmented in PLUTO by Mignone et al. (2012). We use the hllc Rie-\nmann solver (Toro et al. 1994) with the VAN_LEER limiter (Van Leer\n1974), 2nd-order RK2timestepping and the 3rd-order WENO3 recon-\nstruction (Yamaleev & Carpenter 2009).\nWe run two sets of models, where the planet is either fixed or\nallowed to migrate through the disk. In each set we carry out the\nfollowing models:\nradiative : realistic disk with Q=Qirr+Qcool+Qrad,\nadiabatic : no radiative terms in Eq. (1c) (i.e., Q=0),\nlocally isothermal : Eq. (1c) is not solved; T=T0(R).\nThese models are sometimes tagged “ rad”, “adb”, and “ iso” in\nour plots. For consistency, we always use orange, blue, and green\nrespectively to refer to each model in all plots.\n3 RESULTS\nIn this section we present the results of our numerical models. We\nfirst focus on the vortensity evolution in models with fixed planets,\nbefore moving on to models where the planet is allowed to migrate.\nWe then investigate how di fferent cooling regimes a ffect the vorten-\nsity evolution.\n3.1 Fixed planet\nWe first compare the vortensity evolution in our models with a fixed\nplanet. Even though there is initially a radial vortensity gradient\nthrough the disk ( ϖ0(R)≈ΩK/2), the vortensity is mixed in the\nplanet’s corotating region. This process happens over the libration\ntimescale\nτlib=8πRp\n3Ωpxh. (14)\nAfter a few τlib,ϖis flat inside the corotating region and the corota-\ntion torque vanishes.\nIn Fig. 2 we compare the vortensity evolution in our fiducial mod-\nels with a fixed planet after 1 τlib. In all models ϖundergoes phase\nmixing and is roughly constant in the corotating region after a few\nlibration timescales. In the radiative model, however, a vortensity\nexcess is visible (in red) at R≈Rp±xh, indicating that vorten-\nsity is not conserved. We plot the azimuthally averaged vortensity in\nFig. 3, where we show that the excess grows with time and a ffects\nthe vortensity profile in the entire corotating region.\nThis excess, driven by cooling and unrelated to vortensity growth\ndue to the Rossby Wave Instability (Lovelace et al. 1999), is due to\nthe source termSin Eq. (7), which acts to increase ϖas material\nU-turns near the planet (see Sect. 2.2). To verify this, we plot Sfor\nall models at t=τlibin Fig. 4. Our findings agree with the expec-\ntations in Sect. 2.2: Sis zero in the adiabatic model, nonzero but\nantisymmetric about the planet in the locally isothermal model such\nthat it averages to zero, and asymmetric in the radiative model such\nthatS>0 in the corotating region. As a result, ϖin the corotating\nregion only increases in models with cooling, and in particular on\nstreamlines that U-turn very close to the planet. Since these stream-\nlines librate at the edges of the corotating region, the excess piles up\natR≈Rp±xh(see Figs. 2, 3).Given that the dynamical corotation torque for an inwardly mi-\ngrating planet is proportional to ϖp/ϖ h−1, whereϖpis the\nbackground disk vortensity at the planet’s location and ϖhis the\ncharacteristic vortensity trapped in the corotating region (McNally\net al. 2017), the vortensity growth we found in our radiative mod-\nels should result in faster inward migration. We investigate this in\nSect. 3.2.\n3.2 Migrating planet\nWe now repeat the models in Sect. 3.1, allowing the planet to migrate\nafter 10 orbits. The migration tracks for these models are shown in\nFig. 5.\nInitially, the planet migrates inwards faster in the radiative model\nthan in the adiabatic. This can be attributed to a combination of a\nweaker dynamical corotation torque due to the vortensity growth\nmechanism we described in Sects. 2.2 & 3.1, as well as an e ffec-\ntively smaller γdue to cooling (Paardekooper et al. 2010).\nAfter 200 orbits, migration in the radiative model slows down to\nmatch the adiabatic model. This becomes clear when comparing the\nmigration timescales τmig=a/˙a, which become equal for the ra-\ndiative and adiabatic models for t∼200–600 orbits. This can be\ninterpreted as the result of a combination of e ffects. For one, the dy-\nnamical corotation torque scales with the planet’s migration speed,\nsee Eq. (11), therefore partially slowing the planet down. In addi-\ntion, the vortensity excess via baroclinic forcing becomes less im-\nportant as the planet migrates inwards, because the contrast to the\nlocal vortensity ( ϖp/ϖ h) increases faster than the vortensity input\nvia baroclinic forcing in the corotating region when the planet mi-\ngrates quickly.\nTo support this argument we show that vortensity growth is not as\nefficient in the case of a migrating planet. In part, this happens be-\ncause the mechanism relies on material U-turning near the planet\nrepeatedly. As the planet migrates inwards, however, material U-\nturning behind the planet is no longer trapped in the corotating re-\ngion. Instead, the latter is now a tadpole-shaped zone in front of the\nplanet (Masset & Papaloizou 2003; Papaloizou et al. 2007), and the\nvortensity-rich streamlines behind the planet are simply left behind\nthe planet’s wake, unable to sustain the feedback loop that led to sub-\nstantial vortensity growth in the fixed case in Sect. 3.1. We illustrate\nthis picture in Fig. 6.\nHowever, after t∼600 orbits, the planet speeds up in the ra-\ndiative model, migrating roughly 3 ×and 2×faster compared to\nthe adiabatic and locally isothermal models, respectively. We in-\nterpret this as a combination of two e ffects: a rapidly migrating\nplanet will not capture as many U-turning streamlines, weakening\nvortensity growth due to cooling. This results in the planet slow-\ning down over time as the vortensity contrast to the background\ndisk increases and the dynamical corotation torque becomes more\npositive ( Γh∝dRp/dt), which is why the radiative model briefly\nmatches the adiabatic migration timescale at t∼200 orbits. By\nslowing down, however, the planet can capture more vortensity-rich\nU-turning streamlines, enhancing ϖhand weakening the dynami-\ncal corotation torque, causing the planet to speed up once again.\nThe two e ffects seemingly reach a balance after ∼1000 orbits, with\nthe planet migrating inwards without slowing down. We neverthe-\nless stress that this is only a qualitative explanation, and as such we\ncannot rule out the possibility of runaway migration in the radiative\nmodel (e.g., Pierens 2015).\nMNRAS 000, 1–11 (2023)Planet migration in radiative disks 5\n0.04\n 0.00 0.040.00.51.01.52.0( p)\nloc. iso: 65 orb.\n0.04\n 0.00 0.04adiabatic: 65 orb.\n0.04\n 0.00 0.04radiative: 65 orb.\n0.04\n 0.00 0.04radiative: 130 orb.\n0.04\n0.02\n0.000.020.04\n( p)/p\n(RRp)/Rp\nFigure 2. Two-dimensional heatmaps showing the vortensity ϖin the planet’s corotating region (CR). The first three panels from left to right compare di fferent\nmodels at t=τlib(see Eq. (14)), showing that ϖis being mixed throughout in the CR but an excess develops in the radiative run. The rightmost panel shows\nthat the excess grows with time in the radiative model. Vertical dashed lines mark the edges of the CR at ±xh(see Eq. (12)).\n0.04\n 0.00 0.04\n(RRp)/Rp\n0.03\n0.000.030.060.090.12( p)/p\nadb: 2lib\niso: 2lib\nrad: 2lib\nrad: 4lib\nFigure 3. Azimuthally averaged perturbed vortensity in the planet’s corotat-\ning region (CR) for the models in Fig. 2. The vortensity excess in the radiative\nmodel grows with time and a ffects the entire CR.\n3.3 Di fferent cooling timescales\nIn the previous sections we showed how cooling can speed up inward\nmigration. We also showed that in the limit of both ine fficient and\nrapid cooling (adiabatic and isothermal, respectively), this e ffect is\nabsent. We therefore expect that the e ffect depends on the cooling\ntimescaleβcool=tcoolΩK, and is maximized for a critical value.\nTo measure this dependency of vortensity growth on βcool, we run\na set of radiative models where we vary the reference surface den-\nsityΣrefto controlβcool. For these models, the planet is kept fixed at\nR0. We compute the cooling timescale following the prescription in\nMiranda & Rafikov (2020b)\nβ−1\ncool=β−1\nsurf+β−1\nmid, (15)\nwhereβsurfandβmidare the cooling timescales for surface and in-\n0.04\n0.000.04\nadiabatic\n0.04\n0.000.04(RRp)/Rp\nloc. iso\n0.04\n 0.02\n 0.00 0.02 0.04\n( p)/\n0.04\n0.000.04\nradiative\n0.75\n 0.50\n 0.25\n 0.00 0.25 0.50 0.75100×[pp]\nFigure 4. The baroclinic forcing term Sin Eq. (7), which leads to vortensity\ngrowth. This term is zero throughout the corotating region in the adiabatic\nmodel, averages to zero in the locally isothermal model, but is asymmetric in\nthe radiative model such that vortensity grows over time. Black curves mark\nthe position of the planetary spirals following Rafikov (2002).\nplane cooling and are given by (Ziampras et al. 2023)\nβsurf=ΣcvT\n|Qcool|ΩK,cv=R\nµ(γ−1)\nβmid=ΩK\nη \nH2+l2\nrad\n3!\n, η =16σSBT3\n3κRρ2\nmidcv,lrad=1\nκPρmid.(16)\nMNRAS 000, 1–11 (2023)6 A. Ziampras et al.\n0.60.70.80.91.0ap [30 au]adiabatic\nloc. iso\nradiative\n cooling\nLindblad\n0 1000 2000 3000 4000\ntime [orbits at 30 au]12mig=a/a [Myr]\n0.00 0.16 0.33 0.49 0.66time [Myr]\nFigure 5. Migration tracks for the models with migrating planets. The radia-\ntive model (orange) initially migrates at the locally isothermal rate (green)\nfort≲200P0, then matches the adiabatic rate (blue) until t∼600P0, and\nafterwards diverges significantly as the planet speeds up. This is also evident\nin the migration timescales τmig(bottom), which match between the radiative\nand adiabatic models for t∼200–600 P0, but then show that the planet mi-\ngrates roughly 2–3 ×as fast in the radiative model compared to the adiabatic.\nThe dashed curve marks the migration rate without corotation torques. The\npurple curve (discussed in detail in Sect. 3.4) shows a model where thermal\ndiffusion was omitted, resulting in unphysically rapid migration at the rate\ngiven by Lindblad torques only. Given that we use a wave damping zone for\nR<0.53R0, we exclude data for ap<0.55R0. The damping zone interface\nis also the reason why migration slows down in all models for ap≲0.6R0\n.\nWe then compute the quantity Ssimilar to Fig. 4 after 20 plane-\ntary orbits and integrate it over the corotating region for each model\nto obtain a proxy for the vortensity growth rate. We plot the results\nin Fig. 7 (orange curve), where we show that this integral peaks at\nβcool≈4.3 and approaches zero in the adiabatic and locally isother-\nmal limits.\nThe peak agrees very well with the estimate of the U-turn\ntimescaleβU-turn≈hτlibΩK≈3.65 in Baruteau & Masset (2008a),\nhighlighted with a vertical line in Fig. 7. This suggests that the\nvortensity growth rate is maximized when the cooling timescale is\ncomparable to the U-turn timescale of material in the corotating re-\ngion. In other words, the e ffect of cooling is strongest when it hap-\npens over a timescale similar to the time over which U-turning ma-\nterial interacts dynamically with the planet.\n3.4 The e ffect of thermal di ffusion\nOur radiative models discussed above include a treatment of ther-\nmal di ffusion along the disk plane through the flux-limited di ffusion\n(FLD) approximation. Thermal di ffusion serves to smooth out tem-\nperature gradients in the disk, and therefore could a ffect the vorten-\nsity growth rate.\nA more simplistic (but less accurate) approach would be to im-\nplement radiation transport as a local cooling term in the energy\nequation, similar to the method in Miranda & Rafikov (2020b) (see\n0.00 0.050.00.51.01.52.0( p)/\nadiabaticRp=28.50 au\n0.00 0.05radiative\n0.03\n 0.00 0.03/0\n(RRp)/Rp\nFigure 6. Similar to Fig. 2, but for the models with migrating planets. The\ncorotating region is now tadpole-shaped, with a vortensity excess around it\n(resembling a flame) in the radiative model. The vortensity-rich streamlines\nU-turning behind the planet are no longer confined to the corotating region,\nas the planet has drifted inwards by the time they would have reached the\nplanet from the front.\n101\n100101102103104\ncooling timescale cool\n108\n107\n106\n105\nxhdV[R2\nppp]\nadiabatic isothermalradiative\nlocal cooling\nUturn\n1 2 5 10 20 50R [au]\nFigure 7. The baroclinic source term Sin Eq. (7), integrated over the corotat-\ning region, as a function of the cooling timescale βcoolor the corresponding\ndistance Rfor our models. The function peaks where βcool≈βU-turn (vertical\nline), or R∼10–35 au for our setup. The orange curve shows the results for\nour radiative models, while the purple curve shows models without thermal\ndiffusion (discussed in Sect. 3.4).\nMNRAS 000, 1–11 (2023)Planet migration in radiative disks 7\nZiampras et al. 2023, for a comparison). This approach, however,\nmisses the di ffusive component of in-plane radiation transport. To\ninvestigate the di fference, we run a set of models where we replace\nQirr+Qcool+Qradin Eq. (6) with a local cooling term Qrelaxgiven by\n(e.g., Gammie 2001)\nQrelax=−4ΣcvT−T0\nβΩK, (17)\nwhere cvis the heat capacity at constant volume and β=βcoolis\nthe cooling timescale for both surface and in-plane cooling (see\nEqs. (15) & (16)). The factor of 4 is a correction following Zi-\nampras et al. (2023), obtained by linearizing Qcoolin Eq. (5). We\nnote that Dullemond et al. (2022) cite a correction of 4 +bwith\nb=dlogκ\ndlog T≈2 instead, but this applies only in the optically thin\nlimit where τeff∝τ−1∝κ−1(see Eq. (4)). In the optically thick\nlimit one would instead use 4 −bsinceτeff∝τ, but since we have\nτ∼2⇒τeff∼1 atR=R0in our models we ignore b.\nIn Fig. 5 we also plot the migration tracks for a run with local, β\ncooling. We find that the planet migrates much faster when thermal\ndiffusion is omitted, leading to drastically faster migration. In addi-\ntion, we compute the integrated baroclinic term as a function of βcool\nsimilar to Sect. 3.3 and include it in Fig. 7 (purple curve). While the\nfunction also peaks at βcool≈βU-turn, it consistently overestimates the\nvortensity growth rate compared to our radiative models where ther-\nmal di ffusion is included. This suggests that, while cooling results\nin vortensity growth due to baroclinic forcing and therefore faster\ninward migration, thermal di ffusion acts to partially suppress this\neffect.\n3.5 Radiative damping of spiral shocks\nWhile the focus of this study has been how radiative cooling can in-\nduce vortensity growth via baroclinic e ffects, it is worth highlighting\nthat an additional source of vortensity exists in the planet’s vicin-\nity. The planet’s spiral arms will steepen into shocks as they propa-\ngate through the disk (Rafikov 2002), resulting in a vortensity excess\n(Lin & Papaloizou 2010). This creates a vortensity pileup about ±xsh\naway from the planet, where xshis the shock distance following Zhu\net al. (2015) (see also Goodman & Rafikov 2001)\nxsh≈0.93 γ+1\n12/5Mp\nMth!−2/5\nH, Mth=2c3\ns\n3GΩp≈h3\npM⋆. (18)\nHere, Mthis the thermal mass at the location of the planet (Goodman\n& Rafikov 2001).\nHowever, radiative cooling is expected to weaken the spiral\nshocks (Miranda & Rafikov 2020a; Zhang & Zhu 2020; Ziampras\net al. 2023), possibly resulting in weaker vortensity growth. This\nwould imply that, while a planet in an adiabatic or isothermal model\nwould also need to overcome the dynamical corotation torque due to\nthe vortensity “barrier” created by its own spiral shocks, this e ffect\nis mitigated or even absent in a radiative model. The result would\nbe faster inward migration when β∼1, where the damping e ffect of\nradiative cooling on spiral shocks is maximized (Miranda & Rafikov\n2020a; Ziampras et al. 2023). This condition is met in our models,\nas we haveβ∼1.5 atR=30 au (see Fig. 7).\nFigure 8 shows a comparison of the azimuthally averaged per-\nturbed vortensity for our fiducial models similar to Fig. 3, but cover-\ning a wider radial range and normalized to the Keplerian vortensity\nto highlight any excess. Here, it becomes clear that radiative cooling\ninterferes with the damping of spiral shocks, resulting in a vorten-\nsity excess that is weaker and closer to the planet. Finally, in Fig. 9\n0.3\n 0.2\n 0.1\n 0.0 0.1 0.2 0.3\n(RRp)/Rp\n0.02\n0.000.020.040.06/0\nadb\niso\nradFigure 8. Azimuthally averaged perturbed vortensity for static planet models\natt=2τlib, similar to Fig. 3 but normalized to the Keplerian vortensity.\nColored bands mark the location where spiral shocks deposit vortensity in\nthe disk. The radiative model shows a weaker vortensity excess due to spiral\nshocks, closer to the planet.\n0.20.61.01.41.8( p)/\nadiabaticRp=25.50 au\nradiative\n0.1\n 0.00.20.61.01.41.8( p)/\nloc. iso\n0.1\n 0.0local cooling\n0.1\n 0.0 0.1/0\n(RRp)/Rp\nFigure 9. Similar to Fig. 6, with a focus on the vortensity profile ahead of\n(inside) the planet’s orbit. Models with radiative cooling (right) show practi-\ncally no vortensity excess due to spiral shocks.\nwe show a heatmap of the perturbed vortensity for models with mi-\ngrating planets, where we find that the excess due to spiral shocks is\npractically absent in models with radiative cooling.\nWe expect that this e ffect is of secondary importance compared\nto the baroclinic forcing we discuss in this paper. Nevertheless, we\nstress that this can further enhance the e ffect of radiative cooling\non inward migration such that baroclinic forcing in the horseshoe\nMNRAS 000, 1–11 (2023)8 A. Ziampras et al.\nregion (β∼βU-turn∼10) and radiative damping of shocks ( β∼1)\ncan operate e fficiently together for β∼1–10. Such an investigation\nis beyond the scope of this paper, but could be the subject of future\nwork.\n4 DISCUSSION\nIn this section we discuss our findings in the context of previous\nstudies and more realistic physics. We also highlight the relevance\nof our results in planet migration.\n4.1 Previous radiative models\nLega et al. (2014) showed that radiative cooling results in the “cold\nfinger” e ffect in the corotating region, as material is colder and\ndenser after a U-turn compared to an adiabatic case. This would\nformally result in vortensity generation due to the e ffect discussed\nin this paper, but only if the cooling timescale is right (see Fig. 7).\nPierens (2015) also investigated the e ffect of radiation transport on\nplanet migration, but did not report on the mechanism described\nhere. Yun et al. (2022) similarly carried out 3D radiative hydrody-\nnamic simulations of planet migration, showcasing the “cold finger”\neffect but not finding evidence of cooling-induced vortensity growth.\nThis is likely due to the fact that all of these studies probed the in-\nner 1–5 au, where the cooling timescale is βcool∼103–104≫βU-turn,\nsuch that the e ffect we describe was too ine fficient (see Fig. 7). For\na solar-type star, we expect cooling-induced vortensity growth to be\nmost e fficient at R∼10–35 au (see Fig. 7).\n4.2 More realistic physics\nIn this paper we aimed to isolate the e ffect of radiative cooling on\nthe migration rate of low-mass planets. In reality, however, many\nmore processes that we did not include and which can directly or\nindirectly a ffect migration could come into play.\nFor example, accretion heating can give rise to thermal torques\n(Benítez-Llambay et al. 2015; Masset 2017, who also included ther-\nmal di ffusion) or even result in vortex formation (Cummins et al.\n2022) that could a ffect migration (Lega et al. 2021). The Hall e ffect\nin non-ideal MHD can lead to a torque in the disk midplane, which\nstrongly a ffects migration (McNally et al. 2017, 2018). In addition,\nthe torque by an MHD-driven wind could reverse the direction of\nmigration (Kimmig et al. 2020). Finally, including the vertical di-\nrection can give rise to buoyancy-related torques (Zhu et al. 2012;\nMcNally et al. 2020) as well as introducing a vertical dependence of\nthe MHD wind-driven torques (Wa fflard-Fernandez & Lesur 2023).\nWe expect that a more realistic picture should include all such\neffects, but also note that a treatment of cooling—and in particular\nradiative di ffusion—would likely regulate their contribution to the\ntotal torque. This could be an interesting topic for future work.\n4.3 Sustaining, retriggering, or exaggerating the e ffect\nWhile the dynamical corotation torque will not fully halt inward mi-\ngration, it can slow the planet down enough for it to open a partial\ngap and slowly transition into the much slower type-II migration\n(Ward 1997b). Alternatively, migration can halt as the planet ap-\nproaches the inner edge of its gap and the (negative) outer Lindblad\ntorque is suppressed (though see McNally et al. 2019, for a discus-\nsion on the e ffect of vortices).\nIn both cases the corotating region will span the full disk azimuth,trapping trailing vortensity-rich streamlines and retriggering rapid\ninward migration if the planet can break the stall (e.g., McNally et al.\n2019). This could lead to a sustained faster inward migration, or\na series of migration stalls and restarts. Investigating this scenario\ncould be the subject of future work.\nIn the case of a more traditionally viscous disk, one or more zero-\ntorque regions (“migration traps”) could exist in the disk, such as\nnear icelines where the background disk density and temperature\nprofiles change (e.g., Coleman & Nelson 2016). In such a scenario,\nthe (static, outward) corotation torque remains unsaturated and bal-\nances against the (inward) Lindblad torque, allowing the planet to\nremain stationary. Here, the cooling-induced vortensity growth we\ndiscuss could result in the planet migrating either inwards or out-\nwards (depending on the accretion velocity of the disk, see Eq. (11))\nat a rate that will strongly depend on the disk properties (Pierens\n2015).\nOur analysis in Sect. 3.4 further showed that thermal di ffusion\nplays a critical role in the vortensity generation and subsequent\ntorque by smoothing temperature gradients. It is therefore important\nto treat radiation transport appropriately (using e.g., FLD instead of\na local cooling prescription for the in-plane cooling component) to\navoid exaggerating the related torque.\nIn the more realistic scenario of a planet migrating through the\ndisk as it grows, we expect that the e ffect discussed in this work will\nbe less important during the earlier stages of migration, when the\nplanet is still small and the corotating region is not yet fully devel-\noped. However, as the planet grows, the corotating region widens,\nand the cooling-induced vortensity growth becomes stronger, its im-\npact on migration will quickly become relevant. Finding a critical\nmass beyond which this happens is beyond the scope of our work,\nhowever.\nFinally, as we have discussed, it becomes apparent that holding\nthe planet in a fixed radial location would both hide the contribution\nof this e ffect to the total torque, as it mainly a ffects the dynami-\ncal corotation torque, but also artificially speed up inward migration\nonce the planet is released due to the pileup of vortensity in the coro-\ntating region. For this reason, we recommend that numerical models\nof planet migration allow the planet to migrate as soon as possible\nafter the start of the simulation.\n4.4 Connecting to population synthesis models\nAs discussed in Sect. 1, we expect that dynamical corotation\ntorques will be important for low- to moderate-mass planets in low-\nturbulence disks. In order to investigate its e ffects on planet pop-\nulations, however, a recipe for the dynamical corotation torque is\nrequired. McNally et al. (2018) provided such a recipe for a planet\nmigrating in a disk where the Hall e ffect drives a Lorentz accelera-\ntion on the disk, resulting in a background radial velocity uRand a\nvortensity source term on the planet given by\ndϖh\ndt=−ϖ2\nh\"1\nΣRd\ndR\u0010\nRaϕ\u0011#\n, aϕ=√GM⋆R\n2R2uR. (19)\nThe vortensity source term Sand the associated torque that we dis-\ncuss in this paper is not related to magnetic fields as in the work of\nMcNally et al. (2018), but it would be useful to parametrize Sas\na function of planet and disk properties such that its e ffects can be\nincluded in population synthesis models. This will be the focus of\nfollowup work.\nAt the same time, it is important to note that our models focus\non disks with a single planet. In a multiple-planet configuration, on\nMNRAS 000, 1–11 (2023)Planet migration in radiative disks 9\ntop of planet–planet interactions, it is possible that the vortensity-\nrich streamlines left behind by the innermost, inwardly-migrating\nplanet can interfere with the migration of planets further out by in-\ncreasing the local disk vortensity above its nominal Keplerian value.\nThis could enhance the dynamical corotation torque on the remain-\ning planets, slowing them down.\n5 SUMMARY\nWe carried out numerical simulations of planet–disk interaction in\ninviscid disks, and investigated the e ffect of cooling on the vorten-\nsity evolution in the planet’s corotating region. We summarize our\nfindings below.\nWe found that radiative cooling interferes with the otherwise adi-\nabatic compression and expansion of material in the planet’s coro-\ntating region as it U-turns near the planet. The result is a net increase\nin vortensity in the corotating region, which weakens the stalling ef-\nfect of the dynamical corotation torque (McNally et al. 2017) and\nsubstantially speeds up inward migration. We identified the reason\nfor this vortensity growth as baroclinic forcing along streamlines U-\nturning very close to the planet, causing the vortensity excess to pile\nup at the edges of the corotating region.\nWe then compared models of migrating planets with and without\nradiative cooling, and found that this baroclinic forcing results in\nslightly faster inward migration for the first 200 orbits. As the planet\ncontinues to migrate inwards, a competition between the dynamical\ncorotation torque becoming stronger for a rapidly-migrating planet\nand baroclinic forcing enhancing vortensity growth for a slowly-\nmigrating planet prevents the planet from stalling and instead allows\nit to sustain a migration rate 2–3 ×faster than the isothermal or adia-\nbatic rates. We also found that the vortensity growth rate (and there-\nfore inward migration rate) is maximized when the cooling timescale\nis comparable to the U-turn timescale of material in the corotating\nregion, making this mechanism relevant in a quite broad radial range\nin the disk ( R∼5–50 au).\nFurthermore, we showed that thermal di ffusion acts to suppress\nvortensity growth due to baroclinic forcing. We therefore conclude\nthat the e ffect of radiative cooling on inward migration is twofold:\nit acts to speed up inward migration via vortensity growth, but this\ngrowth is partly suppressed by thermal di ffusion.\nFinally, we discussed how radiative damping of spiral shocks can\nfurther enhance the e ffect of cooling on inward migration by weak-\nening the vortensity barrier created by the planet’s own spiral shocks.\nWhile this e ffect is of secondary importance compared to the baro-\nclinic forcing we discuss in this paper, we expect that it will be rel-\nevant in the radial range where β∼0.1–10 (Miranda & Rafikov\n2020a; Ziampras et al. 2023), and note that the two e ffects could\noperate e fficiently together for β∼1–10.\nOur results highlight the importance of radiative e ffects in planet–\ndisk interaction, and in particular the role of thermal di ffusion.\nAn important take-home message is that type-I migration in low-\nturbulence, radiative disks might be more challenging to understand\nthan previously thought, and mechanisms that can naturally lead to\noutward migration (e.g., magnetic fields, planetary accretion lumi-\nnosity) will become necessary from a modeling perspective in order\nto explain the population of low-mass planets at R∼1–10 au. We\nalso expect that the e ffect of thermal di ffusion will be even more\ncentral when additional thermal e ffects are considered.ACKNOWLEDGEMENTS\nAZ would like to thank Roman Rafikov, Kees Dullemond and\nJosh Brown for their suggestions and helpful discussions. This re-\nsearch utilized Queen Mary’s Apocrita HPC facility, supported by\nQMUL Research-IT (http: //doi.org /10.5281 /zenodo.438045). This\nwork was performed using the DiRAC Data Intensive service at Le-\nicester, operated by the University of Leicester IT Services, which\nforms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk).\nThe equipment was funded by BEIS capital funding via STFC cap-\nital grants ST /K000373 /1 and ST /R002363 /1 and STFC DiRAC\nOperations grant ST /R001014 /1. DiRAC is part of the National\ne-Infrastructure. AZ and RPN are supported by STFC grant\nST/P000592 /1, and RPN is supported by the Leverhulme Trust\nthrough grant RPG-2018-418. 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K., Carpenter M. H., 2009, Journal of Computational Physics,\n228, 3025\nYun H.-G., Kim W.-T., Bae J., Han C., 2022, ApJ, 938, 102\nZhang S., Zhu Z., 2020, MNRAS, 493, 2287\nZhu Z., Stone J. M., Rafikov R. R., 2012, ApJ, 758, L42\nZhu Z., Dong R., Stone J. M., Rafikov R. R., 2015, ApJ, 813, 88\nZiampras A., Kley W., Dullemond C. P., 2020, A&A, 637, A50\nZiampras A., Nelson R. P., Rafikov R. R., 2023, arXiv e-prints, p.\narXiv:2305.14415\nde Val-Borro M., et al., 2006, MNRAS, 370, 529\nAPPENDIX A: VORTENSITY GROWTH WITH COOLING\nWe follow a similar approach to Sect. 2.2, where we look at a gas\nparcel U-turning ahead of or behind the planet. We can rewrite the\nsource termSin Eq. (7) using the ideal gas law as\nS=P\nTΣ3∇Σ×∇T=P\nTΣ3 ∂Σ\n∂R∂T\n∂ϕ−∂Σ\n∂ϕ∂T\n∂R!\n. (A1)Similar to Sect. 2.2, we now assume a hot and dense circular en-\nvelope around the planet such that ∂ϕΣ|b=−∂ϕΣ|a=σϕand\n∂ϕT|b=−∂ϕT|a=τϕ. We also assume that the gas parcel in a disk\nwith cooling contracts and heats up in the same way that it would in\nan adiabatic disk, except that the expansion and cooling phases are\ndifferent. In other words, σadb\nϕ=σcool\nϕ=σϕandτadb\nϕ=τcool\nϕ=τϕ,\nwhere the superscripts denote the adiabatic and cooling cases.\nIn the adiabatic limit, S=0 and therefore\nSadb\na=Pa\nTaΣ3\na \n−∂Σ\n∂R\f\f\f\f\fadb\naτϕ+∂T\n∂R\f\f\f\f\fadb\naσϕ!\n=0, (A2)\nSadb\nb=Pb\nTbΣ3\nb ∂Σ\n∂R\f\f\f\f\fadb\nbτϕ−∂T\n∂R\f\f\f\f\fadb\nbσϕ!\n=0. (A3)\nIn the case with cooling, we assume that thermal emission and\nin-plane cooling happen alongside adiabatic expansion. The heated\ngas parcel will then cool faster (and expand less) than it would have\nadiabatically. We then write that\n∂T\n∂R\f\f\f\f\fcool\n≈∂T\n∂R\f\f\f\f\fadb\n−|δT|\nδR,∂Σ\n∂R≈∂Σ\n∂R\f\f\f\f\fadb\n+|δΣ|\nδR. (A4)\nWe then obtain ahead of the planet ( δR<0)\nScool\na≈Pa\nTaΣ3\na \n−∂Σ\n∂R\f\f\f\f\fcool\naτϕ+∂T\n∂R\f\f\f\f\fcool\naσϕ!\n≈1\n|δR|Pa\nTaΣ3\na\u0010\n|δΣ|τϕ+|δT|σϕ\u0011\n,(A5)\nand behind the planet ( δR>0)\nScool\nb≈1\n|δR|Pb\nTbΣ3\nb\u0010\n|δΣ|τϕ+|δT|σϕ\u0011\n. (A6)\nAs we have defined τϕandσϕto be positive, we find that Scool>0\nduring a U-turn both ahead of and behind the planet. In other words,\nvortensity will always increase along streamlines during a U-turn.\nWe note, however, that this is a crude approach as, in principle, the\nassumption that σadb\nϕ=σcool\nϕandτadb\nϕ=τcool\nϕwill not hold. We also\ndid not take into account the presence of a background temperature\nand density gradient, which would further complicate this calcula-\ntion. We can nevertheless expect that, in any case, the two terms in\nEqs. (A5) & (A6) will sum to a positive value such that there is a net\nvortensity growth.\nAPPENDIX B: VORTENSITY GROWTH IN ADIABATIC\nMODELS WITH A RADIAL ENTROPY GRADIENT\nIn Sect. 2.2.2, we argued that a radial entropy gradient K∝Rξ\nshould result in a vortensity source term about the planet’s azimuthal\nlocation. This term should however average to zero for each pair of\nstreamlines U-turning ahead of and behind the planet, and only exist\nuntil the horseshoe is completely phase-mixed to a constant entropy\nstate.\nIn Fig. B1 we show the vortensity source term similar to Fig. 4\natt=20 orbits for three adiabatic models with di fferent surface\ndensity radial profiles Σ∝Rs, where s∈{−2,−1,0}. We find that the\nsource term is zero throughout the corotating region for our fiducial\nmodel with s=−1 andξ≈0, but nonzero for the models with\nξ,0, where the source term is antisymmetric about the planet such\nthat it averages to zero. This is consistent with our expectations in\nSect. 2.2.2.\nWe further run a set of models with di fferent surface density ex-\nponents sto investigate the long-term evolution of Sadbfor di fferent\nMNRAS 000, 1–11 (2023)Planet migration in radiative disks 11\n0.04\n0.000.04\ns=2, =0.37\n0.04\n0.000.04(RRp)/Rp\ns=1, =0.03\n0.04\n 0.02\n 0.00 0.02 0.04\n( p)/\n0.04\n0.000.04\ns=0, =0.43\n0.75\n 0.50\n 0.25\n 0.00 0.25 0.50 0.75100×[pp]\nFigure B1. The baroclinic forcing term Sin Eq. (7), similar to Fig. 4, for\nadiabatic models with di fferent radial entropy exponents ξ, controlled by the\nsurface density exponent s.\nvalues ofξ. We compute the integrated vortensity source term inside\nthe horseshoe similar to Sect. 3.3 and Fig. 7, and plot the results as a\nfunction of sand time in Fig. B2. In line with our expectations, the\nsource term is very small (about 100 ×smaller than a typical value\nin a radiative model) and further decreases with time.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\n0.4\n 0.2\n 0.0 0.2 0.4\nentropy exponent  (KR)\n1.0\n0.5\n0.00.5106×xhdV[R2\nppp]\n0.3lib\n1lib\n2lib\n2.0\n 1.5\n 1.0\n 0.5\n 0.0surface density exponent s (Rs)\nFigure B2. The baroclinic source term Sin Eq. (7), integrated over the coro-\ntating region similar to Fig. 7, as a function of the radial entropy exponent ξ\nand time. This term is very small compared to radiative models and decreases\nwith time, consistent with our expectations in Sect. 2.2.2.\nMNRAS 000, 1–11 (2023)"    },    {        "title": "2402.00133v2.On_the_Constant_Depth_Circuit_Complexity_of_Generating_Quasigroups.pdf",        "content": "arXiv:2402.00133v2  [cs.CC]  14 Feb 2024On the Constant-Depth Circuit Complexity of Generating\nQuasigroups∗\nNathaniel A. Collins1, Joshua A. Grochow2,3, Michael Levet4, and Armin Weiß5\n1Department of Mathematics, Colorado State University\n2Department of Computer Science, University of Colorado Boulder\n3Department of Mathematics, University of Colorado Boulder\n4Department of Computer Science, College of Charleston\n5Universit¨ at Stuttgart, FMI\nFebruary 15, 2024\nAbstract\nWe investigate the constant-depth circuit complexity of th eIsomorphism Problem ,Minimum Gen-\nerating Set Problem (MGS), andSub(quasi)group Membership Problem (Membership ) for\ngroups and quasigroups (=Latin squares), given as input in t erms of their multiplication (Cayley) tables.\nDespite decades of research on these problems, lower bounds for these problems even against depth-2\nACcircuits remain unknown. Perhaps surprisingly, Chattopad hyay, Tor´ an, and Wagner (FSTTCS 2010;\nACM Trans. Comput. Theory , 2013) showed that Quasigroup Isomorphism could be solved by AC\ncircuits of depth O(loglogn) usingO(log2n) nondeterministic bits, a class we denote ∃log2nFOLL. We\nnarrow this gap by improving the upper bound for many of these problems to quasiAC0, thus decreasing\nthe depth to constant.\nIn particular, we show that Membership can be solved in NTIME(polylog( n)) and use this to prove\nthe following:\n•MGSfor quasigroups belongs to ∃log2n∀lognNTIME(polylog( n))⊆quasiAC0. Papadimitriou and\nYannakakis ( J. Comput. Syst. Sci. , 1996) conjectured that this problem was ∃log2nP-complete; our\nresults refute a version of that conjecture for completenes s underquasiAC0reductions uncondition-\nally, and under polylog-space reductions assuming EXP/negationslash=PSPACE.\nIt furthermore implies that this problem is not hard for any c lass containing Parity. The analogous\nresults concerning Parity were known for Quasigroup Isomorphism (Chattopadhyay, Tor´ an, &\nWagner, ibid.) andSubgroup Membership (Fleischer, Theory Comput. 2022), though not for\nMGS.\n•MGSfor groups belongs to AC1(L). OurAC1(L) boundimproves on the previous, veryrecent, upper\nbound of P(Lucchini & Thakkar, J. Algebra , 2024). Our quasiAC0upper bound is incomparable to\nP, but has similar consequences to the above result for quasig roups.\n•Quasigroup Isomorphism belongs to ∃log2nAC0(DTISP(polylog( n),log(n)))⊆quasiAC0. As a\nconsequence of this result and previously known AC0reductions, this implies the same upper bound\nfor theIsomorphism Problems for: Steiner triple systems, pseudo-STS graphs, Latin Square\nIsotopy , Latin square graphs, and Steiner ( t,t+ 1)-designs. This improves upon the previous\nupper bound for these problems, which was ∃log2nL∩ ∃log2nFOLL⊆quasiFOLL (Chattopadhyay,\nTor´ an, & Wagner, ibid.; Levet,Australas. J. Combin. 2023).\n•As a strong contrast, we show that MGS for arbitrary magmas is NP-complete.\nOur results suggest that understanding the constant-depth circuit complexity may be key to resolving\nthe complexity of problems concerning (quasi)groups in the multiplication table model.\n∗JAG and ML were partially supported by JAG’s NSF CAREER award CISE-204775 during this work.1 Introduction\nTheGroup Isomorphism (GpI) problem is a central problem in computational complexity and compu ter\nalgebra. When the groups are given as input by their multiplication (a.k .a. Cayley) tables, the problem\nreduces to Graph Isomorphism (GI), and because the best-known runtimes for the two are quite clos e\n(nO(logn)[Mil78]1vs.nO(log2n)[Bab16]2), the former stands as a key bottleneck towards further improv e-\nments in the latter.\nDespite this, GpIseems quite a bit easierthan GI. For example, Tarjan’s nlogn+O(1)algorithm for groups\n[Mil78] can now be given as an exercise to undergraduates: every gr oup is generated by at most ⌈log2|G|⌉\nelements, so the algorithm is to try all possible/parenleftbign\nlogn/parenrightbig\n≤nlogngenerating sets, and for each, check in nO(1)\ntime whether the map of generating sets extends to an isomorphism . In contrast, the quasi-polynomial time\nalgorithm for graphs was a tour de force that built on decades of cu tting-edge research into algorithms and\nthe structure of permutation groups. Nonetheless, it remains un known whether the problem for groups is\nactually easier than that for graphs, or even whether both proble ms are in P!\nUsing a finer notion of reduction, Chattopadhyay, Tor´ an, and Wa gner [CTW13] proved that there was no\nAC0reduction from GItoGpI. This gave the first (and still only known) unconditional evidence th at there\nissome formal sense (namely, the AC0sense) in which GpIreally is easier than GI. The key to their result\nwas that the generator-enumerationtechnique described above can be implemented by non-deterministically\nguessing log2nbits (describing the log ngenerators, each of log nbits), and then verifying an isomorphism\nby a non-deterministic circuit of depth only O(loglogn), a class we denote ∃log2nFOLL. Observe that\n∃log2nFOLL⊆quasiFOLL (by trying all 2O(log2n)settings of the non-deterministic bits in parallel), which\ncannot compute Parity [Raz93, Smo87, CTW13]. As GIisDET-hard [Tor04]—and hence can compute\nParity—there can be no AC0reduction from GItoGpI.\nSuch a low-depth circuit was quite surprising, although that surpris e is perhaps tempered by the use of\nnon-determinism. Nonetheless, it raises the question:\nIs it possible that Group Isomorphism is inAC0?\nThe authors would be shocked if the answer were “yes,” and yet we d o not even have results showing that\nGroup Isomorphism cannot be computed by polynomial-size circuits of (!) depth 2. Indee d, the upper\nbound of ∃log2nFOLLrules out most existing lower bound techniques against AC0, as most such techniques\nalso yield similar lower bounds against ∃log2nFOLL.\nIn this paper, we aim to close the gap between AC0and∃log2nFOLLin the complexity of Group Iso-\nmorphism and related problems. Our goal is to obtain constant-depth circuit s of quasipolynomial size, a\nnatural benchmark in circuit complexity [Bar92]. We leave it for futur e work to focus on reducing the size\nof the circuits. Our first main result along these lines is:\nTheorem A. (Quasi)Group Isomorphism can be solved in quasiAC0.\n(We discuss quasigroups more below.)\nRemark 1.1. We in fact get a more precise bound of ∃log2nAC0(DTISP(polylog( n),log(n))) (Theorem 4.1\ngives an even more specific bound). This more precise bound is notab le because it is contained in both\nquasiAC0and∃log2nFOLL∩∃log2nL, the latter thus improving on [CTW13]. We get similarly precise bounds\nwith complicated-looking complexity classes for the other problems m entioned in the introduction, but we\nomit the precise bounds here for readability.\nThe prior best results in terms of depth complexity all had super-co nstant depth (with quasipolynomial\nsize):∃log2nSC2by Tang [Tan13] yields (by a standard simulation argument) circuits o f depth log2n, while\n∃log2nL∩∃log2nFOLL[CTW13] has depth loglog n.\n1Miller [Mil78] credits Tarjan for nlogn+O(1).\n2Babai [Bab16] proved quasi-polynomial time, and the expone nt of the exponent was analyzed and improved by Helfgott\n[HBD17]\n1Minimum generating set. Another very natural problem in computational algebra is the Min Gen-\nerating Set (MGS) problem. Given a group, this problem asks to find a generating set o f the smallest\npossible size. Given that many algorithms on groups depend on the siz e of a generating set, finding a mini-\nmum generating set has the potential to be a widely applicable subrou tine. Despite this, the MGSproblem\nfor groups was shown to be in Pby Lucchini & Thakkar very recently [LT24]. We improve their complex ity\nbound to:\nTheorem B. Min Generating Set for groups can be solved in quasiAC0and inAC1(L)(O(logn)-depth,\nunbounded fan-in circuits with a logspace oracle).\nWenotethat, although quasiAC0isincomparableto Pbecauseofthequasi-polynomialsize(whereas AC1(L)⊆\nP), the key we are focusing on here is reducing the depth.\nFor nilpotent groups (widely believed to be the hardest cases of GpI), if we only wish to compute the\nminimum numberofgenerators,wecanfurtherimprovethiscomplexityto L∩AC0(NTISP(polylog( n),log(n)))\n(Proposition 7.3).\nWhile our AC1(L) bound above is essentially a careful complexity analysis of the polyn omial-time algo-\nrithm of Lucchini & Thakkar [LT24], the quasiAC0upper bound is in fact a consequence of our next, more\ngeneral result for quasigroups, which involves some new ingredients.\nEnter quasigroups. Quasigroups can be defined in (at least) two equivalent ways: (1) an algebra whose\nmultiplication table is a Latin square,3or (2) a group-like algebra that need not have an identity nor be\nassociative, but in which left and right division are uniquely defined, th at is, for all a,b, there are unique x\nandysuch that ax=bandya=b.\nIn the paper in which they introduced log2(n)-bounded nondeterminism, Papadimitriou and Yannakakis\nshowed that for arbitrary magmas4, testing whether the magma has log ngenerators was in fact complete\nfor∃log2nP, and conjectured:\nConjecture (Papadimitriou & Yannakakis [PY96, p. 169]) .Min Generating Set for Quasigroups is\n∃log2nP-complete.\nThey explicitly did notconjecture the same for MGSforgroups, writing:\n“We conjecture that this result [ ∃log2nP-completeness] also holds for the more structured MIN-\nIMUM GENERATOR SET OF A QUASIGROUP problem. In contrast, QUAS IGROUP ISO-\nMORPHISM was recently shown to be in DSPACE(log2n) [Wol94]. Notice that the corre-\nsponding problems for groups were known to be in DSPACE(log2n) [LSZ77].”—Papadimitriou\n& Yannakakis [PY96, p. 169]\nWe thus turn our attention to the analogous problems for quasigro ups:MGSfor quasigroups, Quasigroup\nIsomorphism ,andthekeysubroutine, Sub-quasigroup Membership . Wenotethatthe ∃log2nFOLLupper\nbound of Chattopadhyay, Tor´ an, and Wagner [CTW13] actually ap plies toQuasigroup Isomorphism and\nnot just GpI; we perform a careful analysis of their algorithm to put Quasigroup Isomorphism into\nquasiAC0as well.\nTheorem C. Min Generating Set for Quasigroups is inquasiAC0∩DSPACE(log2n).\nTo the best of our knowledge, MGS for Quasigroups has not been studied from the complexity-\ntheoretic viewpoint previously. While a DSPACE(log2n) upper bound for MGSforgroupsfollows from\n[Tan13, AT06], as far as we know it remained open for quasigroups pr ior to our work.\nAs with prior results on Quasigroup Isomorphism andGroup Isomorphism [CTW13], and other\nisomorphism problems such as Latin Square Isotopy andLatin Square Graph Isomorphism [Lev23],\nThm. C shows that Paritydoes not reduce to MGS for Quasigroups , thus ruling out most known lower\nbound methods that might be used to prove that MGS for Quasigroups is not in AC0. We also observe\na similar bound for MGS for Groups using Fleischer’s technique [Fle22].\n3A Latin square is an n×nmatrix where for each row and each column, the elements of [ n] appear exactly once\n4A magma is a set Mtogether with a function M×M→Mthat need not satisfy any additional axioms.\n2Papadimitriou and Yannakakis did not specify the type of reduction u sed in their conjecture, though\ntheir∃log2nP-completeness result for Log Generating Set of a Magma works in both logspace and AC0\n(under a suitable input encoding). Ourtwo upper bounds rule out (u nconditionally in one case, conditionally\nin the other) such reductions for MGS for Quasigroups :\nCorollary C. The conjecture of Papadimitriou & Yannakakis [PY96, p. 169] is false under quasiAC0re-\nductions. It is also false under polylog-space reductions a ssuming EXP/\\⌉}atio\\slash=PSPACE.\nIn strong contrast, we show that MGS for Magmas isNP-complete (Thm. 7.13).\nA key ingredient in our proof of Thm. C is an improvement in the complex ity of another central problem\nin computational algebra: the Sub-quasigroup Membership problem ( Membership ,5for short):\nTheorem D. Membership for quasigroups is in NTIME(polylog( n))⊆quasiAC0.\nMembership forgroupsis well-known to belong to L, by reducing to the connectivity problem on the\nappropriate Cayley graph (cf. [BM91, Rei08]), but as Lsits in between AC0andAC1, this is not low enough\ndepth for us.\nAdditional results. We also obtain a number of additional new results on related problems , some of\nwhich we highlight here:\n•By known AC0reductions (see, e.g., Levet [Lev23] for details), our quasiAC0analysis of Chattopadhyay,\nTor´ an, and Wagner’s algorithm for Quasigroup Isomorphism yields the same upper bound for the\nisomorphism problems for Steiner triple systems, pseudo-STS grap hs, Latin square graphs, and Steiner\n(t,t+1)-designs, as well as Latin Square Isotopy . Prior to our work, Quasigroup Isomorphism\nwas not known to be solvable using quasiACcircuits of depth o(loglogn).\n•Group Isomorphism for simple groups or for groups from a dense set Υ of orders can be solved\ninAC0(DTISP(polylog( n),log(n)))⊆quasiAC0∩L∩FOLL. For groups in a dense set of orders, this\nimproves the parallel complexity compared to the original result of D ietrich & Wilson [DW22]. As in\ntheir paper, note that Υ omits large prime powers. Thus, we essent ially have that for groups that are\nnotp-groups, Group Isomorphism belongs to a propersubclass of DET. This evidence fits with the\nwidely-believed idea that p-groups are a bottleneck case for Group Isomorphism .\n•Abelian Group Isomorphism is in∀loglognMAC0(DTISP(polylog( n),log(n))). The key novelties\nhere are (1) a new observation that allows us to reduce the number of co-nondeterministic bits from\nlogn(as in [GL23]) down to loglog n, and (2) using an AC0(DTISP(polylog( n),log(n))) circuit for order\nfinding, rather than FOLLas in [CTW13].\n•Membership for nilpotent groups is in NTISP(polylog( n),log(n))⊆FOLL∩quasiAC0.\n1.1 Methods\nSeveralof our results involvecareful analysis of the low-levelcirc uit complexity ofextant algorithms, showing\nthat they in fact lie in smaller complexity classes than previously known . One important ingredient here is\nthat we use simultaneous time- and space-restricted computation s. This not only facilitates several proofs\nand gives better complexity bounds, but also gives rise to new algorit hms such as for Membership for\nnilpotent groups, which previously was not known to be in FOLL.\nOnesuchinstanceisinourimprovedboundfororder-findingandexp onentiationinasemigroup(Lem.3.1).\nThe previous proof [BKLM01] (still state of the art 23 years later) used a then-novel and clever “double-\nbarrelled” recursive approach to compute these in FOLL. In contrast, our proof uses standard repeated\ndoubling, noting that it can be done in DTISP(polylog( n),log(n))⊆FOLL∩quasiAC0, recovering their result\nwith standard tools and reducing the depth. We use this improved bo und on order-finding to improve the\ncomplexity of isomorphism testing of Abelian groups (Thm. 5.1), simple groups (Cor. 3.3), and groups of\nalmost all orders (Thm. 6.1).\n5In the literature, the analogous problem for groups is somet imes called Cayley Group Membership orCGM, to highlight\nthat it is in the Cayley table model.\n3For a few of our results, however, we need to develop new tools to w ork with quasigroups. In partic-\nular, for the quasiAC0upper bound on MGSfor quasigroups, we cannot directly adapt the technique of\nChattopadhyay, Tor´ an, and Wagner. Indeed, their analysis of t heir algorithm already seems tight to us in\nterms of having depth Θ(loglog n). The first key is Thm. D, putting Membership for quasigroups into\nNTIME(polylog( n)). To do this, we replace the use of a cube generating sequences f rom [CTW13] with\nsomething that is nearly as good for the purposes of MGS: we extend the Babai–Szemer´ edi Reachability\nLemma [BS84, Thm. 3.1] from groups (its original setting) to get str aight-line programs in quasigroups.\nDivision in quasigroups is somewhat nuanced, e.g., despite the fact th at for any a,b, there exists a unique x\nsuch that ax=b, this does not necessarily mean that there is an element “ a−1” such that x=a−1b, because\nof the lack of associativity. Our proof is thus a careful adaptation of the technique of Babai & Szemer´ edi,\nwith a few quasigroup twists that result in a slightly worse, but still su fficient, bound.\n1.2 Prior work\nIsomorphism testing. The best known runtime bound for GpIisn(1/4)logp(n)+O(1)-time due to Rosem-\nbaum [Ros13] (see [LGR16, Sec. 2.2]), though this tells us little about p arallel complexity. In addition\nto Tarjan’s result mentioned above [Mil78], Lipton, Snyder, & Zalcste in [LSZ77] independently observed\nthat if a group is d-generated, then we can decide isomorphism by considering all poss ibled-element sub-\nsets. This is the generator enumeration procedure. Using the fact that every group admits a generating\nset of size ≤logp(n) (where pis the smallest prime dividing n), Tarjan obtained a bound of nlogp(n)+O(1)-\ntime for Group Isomorphism , while Lipton, Snyder, & Zalcstein [LSZ77] obtained a stronger boun d of\nDSPACE(log2n). Miller [Mil78] extended Tarjan’s observation to the setting of quas igroups. There has been\nsubsequent work on improving the parallel complexity of generator enumeration for quasigroups, resulting\nin bounds of ∃log2nAC1(AC1circuits that additionally receive O(log2n) non-deterministic bits, denoted by\nother authors as β2AC1) due to Wolf6[Wol94], ∃log2nSAC1due to Wagner [Wag10], and ∃log2nL∩∃log2nFOLL\ndue to Chattopadhyay, Tor´ an, & Wagner [CTW13]. In the special case of groups, generator enumeration is\nalso known to belong to ∃log2nSC2[Tan13]. There has been considerable work on polynomial-time, isomor -\nphism tests for several families of groups, as well as more recent w ork onNCisomorphism tests—we refer\nto recent works [GQ17, DW22, GL23] for a survey. We are not awar e of work on isomorphism testing for\nspecific families of quasigroups that are not groups.\nMin Generating Set. As every (quasi)group has a generating set of size ≤ ⌈logn⌉,MGSadmits an\nnlog(n)+O(1)-time solution for (quasi)groups. In the case of groups, Arvind & T or´ an [AT06] improved the\ncomplexity to DSPACE(log2n). They also gave a polynomial-time algorithm in the special case of nilpo tent\ngroups. Tang further improved the general bound for MGSfor groups to ∃log2nSC2[Tan13]. We observe\nthat Wolf’s technique for placing Quasigroup Isomorphism intoDSPACE(log2n) also suffices to get MGS\nfor Quasigroups into the same class. Very recently, Das & Thakkar [DT23] improved the algorithmic\nupper bound for MGSin the setting of groups to n(1/4)log(n)+O(1). A month later, Lucchini & Thakkar\n[LT24] placed MGSfor groups into P. Prior to [LT24], MGS for Groups was considered comparable to\nGroup Isomorphism in terms of difficulty. Our AC1(L) bound (Thm. C) further closes the gap between\nMembership Testing in groups and MGS for Groups , and in particular suggests that MGSis of\ncomparable difficulty to Membership for groups rather than GpI. Note that Membership for groups has\nlong been known to belong to L[BM91, Rei08].\n2 Preliminaries\n2.1 Algebra and Combinatorics\nGraph Theory . Astrongly regular graph with parameters ( n,k,λ,µ) is a simple, undirected k-regular, n-\nvertex graph G(V,E) where any two adjacent vertices share λneighbors, and any two non-adjacent vertices\nshareµneighbors. The complement of a strongly regular graph is also stron gly regular, with parameters\n6Wolf actually claims a bound of ∃log2nNC2; however, he uses NC1circuits to multiply two elements of a quasigroup rather\nthanAC0circuits.\n4(n,n−k−1,n−2−2k+µ,n−2k+λ).\nAmagmaMis an algebraic structure together with a binary operation ·:M×M→M. We will frequently\nconsider subclasses of magmas, such as groups, quasigroups, an d semigroups.\nQuasigroups . Aquasigroup consists of a set Gand a binary operation ⋆:G×G→Gsatisfying the\nfollowing. For every a,b∈G, there exist unique x,ysuch that a⋆x=bandy⋆a=b. We write x=a\\band\ny=b/a, i.e.,a⋆(a\\b) =band (b/a)⋆a=b. When the multiplication operation is understood, we simply\nwriteaxfora⋆x. Asub-quasigroup of a quasigroup is a subset that itself is a quasigroup. This means it is\nclosed under the multiplication as well as under left and right quotient s. Given X⊆G, the sub-quasigroup\ngenerated by Xis denoted as /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht. It is the smallest sub-quasigroup containing X.\nUnless otherwise stated, all quasigroups are assumed to be finite a nd represented using their Cayley\n(multiplication) tables.\nAs quasigroups are non-associative, the parenthesization of a giv en expression may impact the resulting\nvalue. For a sequence S:= (s0,s1,...,s k) and parenthesization Pfrom a quasigroup, define:\nCube(S) ={P(s0se1\n1···sek\nk) :e1,...,e k∈ {0,1}}.\nWe say that Sis acube generating sequence if each element gin the quasigroup can be written as\ng=P(s0se1\n1···sek\nk), fore1,...,e k∈ {0,1}. Here,s0\niindicates that siis not being considered in the product.\nFor every parenthesization, every quasigroup is known to admit a c ube generating sequence of size O(logn)\n[CTW13].\nALatin square of ordernis ann×nmatrixLwhere each cell Lij∈[n], and each element of [ n] appears\nexactly once in a given row or a given column. Latin squares are precis ely the Cayley tables correspondingto\nquasigroups. We willabusenotationbyreferringtoaquasigroupan ditsmultiplication tableinterchangeably.\nAnisotopyof Latin squares L1andL2is an ordered triple ( α,β,γ), where α,β,γ:L1→L2are bijections\nsatisfying the following: whenever ab=cinL1, we have that α(a)β(b) =γ(c) inL2. Alternatively, we may\nviewαas a permutation of the rows of L1,βas a permutation of the rows of L1, andγas a permutation of\nthe values in the table. Here, L1andL2are isotopic precisely if (i) the ( i,j) entry of L1is the (α(i),β(j))\nentry of L2, and (ii) xis the (i,j) entry of L1if and only if γ(x) is the ( α(i),β(j)) entry of L2.\nFor a given Latin square Lof order n, we associate a Latin square graph G(L) that has n2vertices; one\nfor each triple ( a,b,c) that satisfies ab=c. Two vertices ( a,b,c) and (x,y,z) are adjacent in G(L) precisely\nifa=x,b=y, orc=z. Miller showed that two Latin square graphs G1andG2are isomorphic if and only\nif the corresponding Latin squares, L1andL2, aremain class isotopic ; that is, if L1andL2can be placed\ninto compatible normal forms that correspond to isomorphic quasig roups [Mil78]. A Latin square graph on\nn2vertices is a strongly regular graph with parameters ( n2,3(n−1),n,6). Conversely, a strongly regular\ngraph with these same parameters ( n2,3(n−1),n,6) is called a pseudo-Latin square graph . Bruck showed\nthat forn >23, a pseudo-Latin square graph is a Latin square graph [Bru63].\nAlbert showed that a quasigroup Qis isotopic to a group Gif and only if Qis isomorphic to G. In\ngeneral, isotopic quasigroups need not be isomorphic [Alb43].\nGroup Theory . For a standard reference, see [Rob82]. All groups are assumed t o be finite. For a group\nG,d(G) denotes the minimum size of a generating set for G. TheFrattini subgroup Φ(G) is the set of\nnon-generators of G. IfGis ap-group, the Burnside Basis Theorem (see [Rob82, Theorem 5.3.2]) pr ovides\nthat (i)G=Gp[G,G], (ii)G/Φ(G)∼=(Z/pZ)d(G), and (iii) if Sgenerates ( Z/pZ)d(G), then any lift of S\ngenerates G. Achief series ofGis an ascending chain ( Ni)k\ni=0of normal subgroups of G, whereN0= 1,\nNk=G, and each Ni+1/Ni(i= 0,...,k−1) is minimal normal in G/Ni. Forg,h∈G, thecommutator\n[g,h] :=ghg−1h−1. Thecommutator subgroup [G,G] =/a\\}⌊ra⌋k⌉tl⌉{t{[g,h] :g,h∈G}/a\\}⌊ra⌋k⌉tri}ht.\nAlgorithmic Problems. A multiplication table of a magma G={g1,...,g n}of order nis an array Mof\nlengthn2where each entry M[i+(j−1)n] fori,j∈ {1,...,n}contains the binary representation of ksuch\nthatgigj=gk. In the following all magmas are given as their multiplication tables.\nWe will consider the following algorithmic problems. The Quasigroup Isomorphism problem takes\nas input two quasigroups Q1,Q2given by their multiplication tables, and asks if there is an isomorphism\nϕ:Q1∼=Q2. TheMembership problem for groups takes as input a group Ggiven by its multiplication\n5table, a set S⊆G, and an element x∈G, and asks if x∈ /a\\}⌊ra⌋k⌉tl⌉{tS/a\\}⌊ra⌋k⌉tri}ht. We may define the Membership problem\nanalogously when the input is a semigroup or quasigroup, and /a\\}⌊ra⌋k⌉tl⌉{tS/a\\}⌊ra⌋k⌉tri}htis considered as the sub-semigroup or\nsub-quasigroup, respectively.\nTheMinimum Generating Set (MGS) problem takes as input a magma Mgiven by its multiplication\ntable and asks for a generating set S⊆Mwhere|S|is minimum. At some point we will also consider the\ndecision variant ofMGS: here we additionally give an integer kin the input and the question is whether\nthere exists a generating set of cardinality at most k.\nWe will be primarily interested in Minimum Generating Set andMembership in the setting of\n(quasi)groups, with a few excursions to semigroups and magmas. N ote that Membership for groups is\nknown to be in L∩quasiAC0[BM91, Rei08, Fle22]. Here the containment in Lfollows from the deep result by\nReingold [Rei08] that symmetric logspace (nondeterministic logspace where each transition is also allowed\nto be applied backward) coincides with L.\n2.2 Computational Complexity\nWe assume that the reader is familiar with standard complexity classe s such as L,NL,NP, andEXP. For\na standard reference on circuit complexity, see [Vol99]. We consider Boolean circuits using the AND,OR,\nNOT, andMajority, whereMajority(x1,...,x n) = 1 if and only if ≥n/2 of the inputs are 1. Otherwise,\nMajority(x1,...,x n) = 0. In this paper, we will consider DLOGTIME -uniform circuit families ( Cn)n∈N. For\nthis, one encodes the gates of each circuit Cnby bit strings of length O(logn). Then the circuit family\n(Cn)n≥0is called DLOGTIME -uniform if (i) there exists a deterministic Turing machine that computes for\na given gate u∈ {0,1}∗ofCn(|u| ∈O(logn)) in time O(logn) the type of gate u, where the types are\nx1,...,x n,NOT,AND,OR,Majority, or oracle gates, and (ii) there exists a deterministic Turing machine\nthat decides for two given gates u,v∈ {0,1}∗ofCn(|u|,|v| ∈O(logn)) and a binary encoded integer iwith\nO(logn) many bits in time O(logn) whether uis thei-th input gate for v.\nDefinition 2.1. Fixk≥0. We say that a language Lbelongs to (uniform) NCkif there exist a (uniform)\nfamily of circuits ( Cn)n∈Nover the AND,OR,NOTgates such that the following hold:\n•TheANDandORgates take exactly 2 inputs. That is, they have fan-in 2.\n•Cnhas depth O(logkn) and uses (has size) nO(1)gates. Here, the implicit constants in the circuit\ndepth and size depend only on L.\n•x∈Lif and only if C|x|(x) = 1.\nThe complexity class ACkis defined analogously as NCk, except that the AND,ORgates are permitted to\nhave unbounded fan-in. That is, a single ANDgate can compute an arbitrary conjunction, and a single\nORgate can compute an arbitrary disjunction. The class SACkis defined analogously, in which the OR\ngates have unbounded fan-in but the ANDgates must have fan-in 2. The complexity class TCkis defined\nanalogously as ACk, except that our circuits are now permitted Majority gates of unbounded fan-in. We also\nallow circuits to compute functions by using multiple output gates.\nFurthermore, for a language Lthe class ACk(L), apart from Boolean gates, also allows oracle gates for L\n(an oracle gate outputs 1 if and only if its input is in L). IfK∈ACk(L), thenKis said to be ACkTuring\nreducible to L. Finally, for some complexity class CdenoteACk(C) to be the set of decision problems that\nareACk-Turing reducible to problems in C. Be aware that here we follow the notation of [Vol99], which is\ndifferent from [Wag10, GL23] (where ACk(C) is used to denote composition of functions).\nFor every k, the following containments are well-known:\nNCk⊆SACk⊆ACk⊆TCk⊆NCk+1.\nIn the case of k= 0, we have that:\nNC0/subset⋉oteqlAC0/subset⋉oteqlTC0⊆NC1⊆L⊆NL⊆SAC1⊆AC1.\nWe note that functions that are NC0-computable can only depend on a bounded number of input bits. Thu s,\nNC0is unable to compute the ANDfunction. It is a classical result that AC0is unable to compute Parity\n6[Smo87]. The containment TC0⊆NC1(and hence, TCk⊆NCk+1) follows from the fact that NC1can\nsimulate the Majority gate.\nWe will crucially use the following observation throughout the paper.\nObservation 2.2. The product of O(logn)-many integers each of O(logn)bits can be computed in AC0.\nProof.By [HAM02], the iterated multiplication of k k-bit integers is in TC0. In our setting, k=m∈\nO(logn). Thus, each Majority gate can be implemented using an ACcircuit of constant depth and size\n2O(logn)= poly(n). The result now follows.\nFurther circuit classes. The complexity class MAC0is the set of languages decidable by constant-depth\nuniform circuit families with a polynomial number of AND,OR, andNOTgates, and a single Majority gate\nat the output. The class MAC0was introduced (but not so named) in [ABFR91], where it was shown th at\nMAC0/subset⋉oteqlTC0. This class was subsequently given the name MAC0in [JKS02].\nThe complexity class FOLLis the set of languages decidable by uniform ACcircuit families of depth\nO(loglogn) and polynomial size. It is known that AC0/subset⋉oteqlFOLL/subset⋉oteqlAC1, and it is open as to whether FOLL\nis contained in NL[BKLM01].\nWe will be particularly interested in NCcircuits of quasipolynomial size (i.e., 2O(logkn)for some constant\nk). For a circuit class C ⊆NC, the analogous class permitting a quasipolynomial number of gates is denoted\nquasiC. We will focus specifically on quasiAC0. Here, we stress that our results for quasiAC0will be stated for\nthe nonuniform setting. Note that DLOGTIME uniformity does not make sense for quasiAC0, as we cannot\nencode gate indices using O(logn) bits. Nonetheless, there exist suitable notions of uniformity for quasiAC0\n[Bar92, FGSTT20].\nBounded nondeterminism. For a complexity class C, we define ∃loginCto be the set of languages Lsuch\nthat there exists an L′∈ Csuch that x∈Lif and only if there exists yof length at most O(logi|x|) such\nthat (x,y)∈L′. Similarly, define ∀loginCto be the set of languages Lsuch that there exists an L′∈ Csuch\nthatx∈Lif and only if for all yof length at most O(logi|x|), (x,y)∈L′. For any i≥0 and any c≥0, both\n∃loginFOLLand∀loginFOLLare contained in quasiFOLL , and so cannot compute Parity[CTW13, Smo87].\nNote that ∀lognC ∪∃lognC ⊆AC0(C).\nTime and space-restricted Turing machines. When considering complexity classes defined by Turing\nmachines with a time bound t(n)∈o(n), we use Turing machine with random access and a separate addres s\n(orindex) tape. After writing an address, the machine can go to a query sta te reading the symbol from the\ninput at the location specified by the address tape.\nFor functions t(n),s(n)∈Ω(logn), the classes DTISP(t(n),s(n)) andNTISP(t(n),s(n)) are defined by\ndeterministic (resp. nondeterministic) t(n) time and s(n) space bounded Turing machines. Note that there\nmust be one Turing machine that simultaneously satisfies the time and space bound. For details we refer to\n[Vol99, Section 2.6]. For further reading on the connection to quasiAC0, we refer to [Bar92, FGSTT20].\nRemark 2.3. Because DTISP(polylog( n),log(n)) is such a small class, when we say some function is com-\nputable in DTISP(polylog( n),log(n)), there is some ambiguity as to what this might mean (e.g., is there a\nseparate output tape? Does it output one bit at a time?). Through out this paper, where relevant (e.g.,\nLemma 3.1), it means that at the end of the computation, the result is written on the work tape of the\nTuring machine (and, hence, also is subject to the space bound).\nFact 2.4. NTISP(polylog( n),log(n))⊆NTIME(polylog( n))⊆quasiAC0.\nProof sketch. Take the OR over all 2polylog( n)possible computation histories, of the AC0circuit that verifies\na computation history (the latter as in the proof of the Cook–Levin Theorem).\nLemma 2.5. NTISP(polylog( n),log(n))⊆FOLL.\nProof.This is essentially the proof of Savitch’s Theorem: A configuration of a Turing machine consist of the\ncurrent state, the work and index tape content, but notthe content of the input tape. For configurations\n7α,βdefine the Reach predicate as follows:\nReach(α,β,0)⇐⇒βis reachable from αin at most one computation step\nReach(α,β,i)⇐⇒ ∃γ:/parenleftbig\nReach(α,γ,i−1)∧Reach(γ,β,i−1)/parenrightbig\nfori≥1. Thus, Reach( α,β,i) holds if and only if βcan be reached from αin at most 2icomputation steps.\nSince the running time is bounded by logknfor some k, by letting Start( w) denote the initial configuration\nfor the input w∈Σ∗and Accept the accepting configuration of M(which can be assumed to be unique after\na suitable manipulation of M), we have\nReach(Start( w),Accept,loglogkn)⇐⇒w∈L.\nNow, it remains to observe that the inductive definition of the Reach predicate can be evaluated in FOLL\nsince the recursiondepth is O(loglogn), each recursion step is clearly in AC0and there are only polynomially\nmany configurations (because of the space bound of log n).\nBy the very definition we have DTISP(polylog( n),log(n))⊆LandNTISP(polylog( n),log(n))⊆NL.\nThus, we obtain\nFact 2.6.\n•AC0(DTISP(polylog( n),log(n)))⊆L∩FOLL∩quasiAC0and\n•AC0(NTISP(polylog( n),log(n)))⊆NL∩FOLL∩quasiAC0.\nDisjunctive truth-table reductions. We finally recall the notion of a disjunctive truth-table reduction.\nAgain let L1,L2be languages. We say that L1isdisjunctive truth-table (dtt) reducible to L2, denoted\nL1≤dttL2, if there exists a function gmapping a string xto a tuple of strings ( y1,...,y k) such that x∈L1\nif and only if there is some i∈ {1,...,k}such that yi∈L2. Whengis computable in a complexity class C,\nwe call this a C-dtt reduction, and write L1≤C\ndttL2. For any class C,C-dtt reductions are intermediate in\nstrength between C-many-one reductions and C-Turing reductions.\nFact 2.7. ∃log2nAC0(C)is closed under ≤AC0\ndttreductions for any class C.\nProof.LetL∈ ∃log2nAC0(C), and suppose L′≤AC0\ndttviag. We show that L′is also in ∃log2nAC0(C). Let\nV∈AC0(C) be a predicate such that x∈L⇐⇒[∃log2|x|w]V(x,w) for all strings x. Then we have\nx∈L′iff∃i∈ {1,...,k},∃w∈ {0,1}log2n:V(g(x)i,w), where g(x)idenotes the i-th string output by\ng(x) = (y1,...,y k). Since gis, in particular, polynomial size, we have k≤poly(|x|), so we have the bit-\nlengthoftheindex iisatmost O(log|x|). Finally, since AC0(C) denotestheoracleclass, it isclosedunder AC0\nreductions, and thus the function ( x,w,i)/ma√sto→V(g(x)i,w) is inAC0(C), and therefore L′∈ ∃log2nAC0(C).\n3 Order Finding and Applications\nIn this section, we consider the parallel complexity of order finding. We begin with the following lemma.\nLemma 3.1. The following problem is in DTISP(polylog( n),log(n)): On input of a multiplication table of\na semi-group S, an element s∈S, and a unary or binary number k∈Nwithk≤ |S|, compute sk(NB:\nRemark 2.3).\nProof.Ifkis given in unary, we first compute its binary representation using a b inary search (note that we\ncan write it on the work tape as is uses at most ⌈log|S|⌉bits). We identify the semigroup elements with the\nnatural numbers 0 ,...,|S|−1. Now, we compute skusing the standard fast exponentiation algorithm. Note\nthat the multiplication of two semigroup elements can be done in DTIME(logn) as we only need to write\ndown two log nbit numbers on the address tape (if the multiplication table is not padd ed up to a power of\ntwo, this is still in DTISP(polylog( n),log(n)) because we need to multiply two log nbit numbers to compute\nthe index in the multiplication table).\nIt is well-known that the fast exponentiation algorithm needs only O(logk) elementary multiplications\nandO(logk+logn) space; hence, the lemma follows.\n8Note that Lem. 3.1 together with Lem. 2.5 gives a new proof that the problem of computing a power sk\nin a semigroup can be done in FOLL. This approach seems easier and more general than the double-ba rrelled\nrecursive approach in [BKLM01].\nLem. 3.1 also yields the following immediate corollary:\nCorollary 3.2. On input of a multiplication table of a group G, an element g∈G, andk∈N, we may\ndecide whether ord(g) =kin∀lognDTISP(polylog( n),log(n)). The same applies if, instead of k, another\ngroup element h∈Gis given with the question whether ord(g) = ord(h).\nProof.First check whether gk= 1 using Lem. 3.1. If yes (and k >1), we use O(logn) universally quantified\nco-nondeterministic bits to verify that for all 1 ≤i < kthatgi/\\⌉}atio\\slash= 1 using again Lem. 3.1. For the second form\nof input observe that ord( g) = ord(h) if and only if for all i≤ |G|we have gi= 1 if and only if hi= 1.\n3.1 Application to isomorphism testing\nUsing Cor. 3.2, we can improve the upper bound for isomorphism test ing of finite simple groups. Previously,\nthis problem was known to be in L[Tan13] and FOLL[GL23]. We obtain the following improved bound.\nCorollary 3.3. LetGbe a finite simple group and Hbe arbitrary. We can decide isomorphism between G\nandHinAC0(DTISP(polylog( n),log(n))).\nProof.AsGis a finite simple group, Gis determined up to isomorphism by (i) |G|, and (ii) the set of orders\nof elements of G: spec(G) :={ord(g) :g∈G}[VGM09]. We may check whether |G|=|H|inAC0. By\nCor. 3.2, we may compute and compare spec( G) and spec( H) inAC0(DTISP(polylog( n),log(n))). The result\nnow follows.\nIn light of Cor. 3.2, we obtain an improved bound for testing whether a group Gis nilpotent. Testing\nfor nilpotency was previously known to be in L∩FOLL[BKLM01].\nCorollary 3.4. LetGbe a finite group given by its multiplication table. We may dec ide whether Gis\nnilpotent in AC0(DTISP(polylog( n),log(n))).\nProof.A finite group Gis nilpotent if and only if it is the direct product of its Sylow subgroups. Thus, for\neach prime pdividing |G|, we identify the elements Xp:={g: ord(g) is a power of p}and then test whether\nXpforms a group. By Cor. 3.2, we can identify XpinAC0(DTISP(polylog( n),log(n))). Verifying the group\naxioms is AC0-computable. The result follows.\n3.2 Application to membership testing\nA group Ghas the log npower basis property (as defined in [BKLM01, Fle22]) if for every subset X⊆G\neveryg∈ /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}htcan be written as g=ge1\n1···gemmwithm≤lognand suitable gi∈Xandei∈Z.\nObservation 3.5. Membership testingfor semigroups withthe lognpower basis property is in NTISP(polylog( n),log(n)).\nProof sketch. This follows by guessing suitable exponents and using Lem. 3.1 to comp ute the respective\npowers.\nThisobservationallowsustoimproveseveralresultsfrom[BKLM01 ,Fle22]from FOLLtoNTISP(polylog( n),log(n)).\nCorollary 3.6. Membership for commutative semigroups is in NTISP(polylog( n),log(n)).\nCorollary 3.7. Letdbe aconstant. Membership for solvable groups of class boundedby dis inNTISP(polylog( n),log(n)).\nProof.LetXbe the input set for Membership andGthe input group. Theorem 3.5 in [BKLM01] is proved\nby showing that /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht=XdwhereX0=Xand for some suitable constant C\nXi={xe1\n1···xemm|m≤Clogn,xj∈Xi−1,ej∈Zforj∈ {1,...,m}}.\nNow, as in Observation 3.5, we can guess an element of XiinNTISP(polylog( n),log(n)) given that we\nhavetheelementsof Xi−1. Asdisaconstant, wecanguessthe elementsof Xi−1inNTISP(polylog( n),log(n))\nby induction. Thus, we can decide membership in /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht=XdinNTISP(polylog( n),log(n)).\n9For nilpotent groups, we can do even better and show NTISP(polylog( n),log(n)) even without a bound\non the nilpotency class—thus, improving considerably over [BKLM01 ]. This also underlines that the class\nNTISP(polylog( n),log(n)) is useful not only because it provides a better complexity bound t hanFOLL, but\nit also facilitates some proofs.\nTheorem 3.8. Membership for nilpotent groups is in NTISP(polylog( n),log(n)).\nProof.LetXbe the input set for Membership andGthe input group. Write n=|G|. Note that the\nnilpotency class of Gis at most log n. Define C={[x1,...,x ℓ]|xi∈X,ℓ≤logn}where [x1,...,x ℓ] is\ndefined inductively by [ x1,...,x ℓ] = [[x1,...,x ℓ−1],xℓ] forℓ≥3.\nWe claim that there is a subset C′={c1,...,c m} ⊆Cwithm≤lognsuch that every g∈ /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}htcan be\nwritten as g=ce1\n1···cemmwithei∈Z.\nLet Γmdenote the m-th term of the lower central series of /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht(meaning that Γ 0=/a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}htand Γ m+1=\n[Γm,/a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht]). Then Γ mis generated by Cm={[x1,...,x k]|xi∈X,k≥m}(e.g., [CMZ17, Lemma 2.6]).\nObserve that, although kis unbounded, the terms with k >lognare trivial because log nis a bound on the\nnilpotency class. We obtain the desired set C′by first choosing a minimal generating set for the Abelian\ngroup Γ 0/Γ1, then a minimal generating set of Γ 1/Γ2and so on (meaning that ( c1,...,c m) is a so-called\npolycyclic generating sequence of/a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht). Note that mis bounded by log nsince/a\\}⌊ra⌋k⌉tl⌉{tci,...,c m/a\\}⌊ra⌋k⌉tri}htis a proper\nsubgroup of /a\\}⌊ra⌋k⌉tl⌉{tci−1,...,c m/a\\}⌊ra⌋k⌉tri}htfor alli.\nNext, let us show that we can guess an element of C′inNTISP(polylog( n),log(n)). This is not difficult:\nwe start by guessing x1and seta=x1. Then, as long as we guess to continue, we guess xiand compute\na:= [a,xi]. Asℓ(in the definition of C) is bounded by log n, we can guess any element of C′in this way in\npolylogarithmic time. Moreover, note that at any point we only need t o store a constant number of group\nelements.\nOnce we have guessed an element of C′, we can guess a power of it as in Observation 3.5 and then guess\nthe next element of C′and so on. As m≤logn, this gives an NTISP(polylog( n),log(n)) algorithm.\n4 Quasigroup Isomorphism\nIn this section, we establish the following.\nTheorem 4.1. Quasigroup Isomorphism belongs to ∃log2n∀logn∃lognDTISP(polylog( n),log(n)).\nNote that we have ∃log2n∀logn∃lognDTISP(polylog( n),log(n))⊆ ∃log2nAC0(DTISP(polylog( n),log(n)))⊆\nquasiAC0∩∃log2nL∩∃log2nFOLL.\nThis statement is inspired by [DEP+22] where a similar problem is shown to be in the third level of the\npolynomial-time hierarchy using the same approach. Our proof follow s closely the algorithm for [CTW13,\nTheorem 3.4].\nProof of Thm. 4.1 . LetGandHbe quasigroups given as their multiplication tables. We assume that\nthe elements of the quasigroups are indexed by integers 1 ,...,|G|. If|G| /\\⌉}atio\\slash=|H|(this can be tested in\nDTIME(logn) by a standard binary search), we know that GandHare not isomorphic. Otherwise, let us\nwriten=|G|andk=⌈2logn⌉+1.\nThe basic idea is to guess cube generating sequences ( g0,...,g k) and (h0,...,h k) forGandHand verify\nthat the map gi/ma√sto→hiinduces an isomorphism between GandH. Hence, we start by guessing cube gener-\nating sequences ( g0,...,g k) and (h0,...,h k) with respect to the parenthesization Pwhere the elements are\nevaluated left-to-right (so P(g1g2g3) = (g1g2)g3), withgi∈G,hi∈H. This amounts to guessing k·log(n)∈\nO(log2n)manybits(thus, ∃log2n). Now,weneedtoverifytwopointsin ∀logn∃lognDTISP(polylog( n),log(n)):\n•that these sequences are actually cube generating sequences,\n•thatgi/ma√sto→hiinduces an isomorphism.\nLet us describe the first point for G(forHthis follows exactly the same way): we universally verify for\nevery element g∈G(which can be encoded using O(logn) bits, hence, ∀logn) that we can existentially guess\n10a sequence ( e1,...,e k)∈ {0,1}k(i.e.∃logn) such that g=P(g0ge1\n1···gek\nk). We can compute this product in\nDTISP(log3n,logn) by multiplying from left to right.7Each multiplication can be done in time O(log2n)\nbecause we simply need to compute i+j·nfor two addresses i,jof quasigroup elements, write the result\non the index tape and then read the corresponding product of gro up elements from the multiplication table.\nMoreover, note that for this procedure we only need to store one intermediate result on the working tape at\nany time. Thus, computing the product P(g0ge1\n1···gek\nk) can be done in time O(log3n) and space O(logn)\nand it can be checked whether the result is g.\nTocheck the secondpoint, by [CTW13], we need to verifyuniversallyt hat for all ( c1,...,c k), (d1,...,d k),\n(e1,...,e k)∈ {0,1}k(hence,∀logn) whether\nP(g0gc1\n1···gck\nk)·P(g0gd1\n1···gdk\nk) =P(g0ge1\n1···gek\nk).\nAgain the products can be computed in time O(polylog n) and space O(logn) as outlined above. Now, it\nremains to observe that we can combine the two ∀lognblocks into one ∀lognblock because the check whether\ngi/ma√sto→hiinduces an isomorphism does not depend on the existentially guessed bits in the first check.\nWe obtain several corollaries.\nCorollary 4.2. Isomorphism testing of two Steiner triple systems is in ∃log2nAC0(DTISP(polylog( n),log(n))).\nProof.Given a Steiner triple system, we may write down a quasigroup Qin the following manner. For each\nblock{a,b,c}in the Steiner triple system, we include the products ab=c,ba=c,ac=b,ca=b,bc=a,cb=\na. Thus, we may write down the multiplication table for QinAC0. The Steiner triple system determines Q\nup to isomorphism. The result now follows.\nBose [Bos63] previously showed that pseudo-STS graphs with >67 vertices are STS graphs. Now given a\nblock-incidence graph from a Steiner 2-design with bounded block siz e, we can recover the underlying design\ninAC0[Lev23, Proposition 4.7]. This yields the following.\nCorollary 4.3. Isomorphism testing of pseudo-STS graphs belongs to ∃log2nAC0(DTISP(polylog( n),log(n))).\nAs isomorphism testing of Steiner ( t,t+ 1)-designs is ∃log2nAC0-reducible to isomorphism testing of\nSteiner triple systems [BW13] (cf., [Lev23, Corollary 4.11]), we obtain the following corollary.\nCorollary 4.4. Isomorphism testing of Steiner (t,t+1)-designs is in ∃log2nAC0(DTISP(polylog( n),log(n))).\nGiven two Latin square graphs G1,G2, we may recover corresponding Latin squares L1,L2inAC0(cf.\n[Lev23, Lemma 3.9]). Now G1∼=G2if and only if L1andL2are main class isotopic [Mil78, Lemma 3]. We\nmay decide whether L1,L2aremain classisotopic using an AC0-computable disjunctive truth-table reduction\ntoQuasigroup Isomorphism (cf. [Lev23, Remark1.6]). As ∃log2nAC0(DTISP(polylog( n),log(n))) is closed\nunderAC0-computable dtt reductions (Fact 2.7), we get the following corollar y.\nCorollary 4.5. Isomorphism testing of Latin square graphs belongs to ∃log2nAC0(DTISP(polylog( n),log(n))).\nRemark 4.6. Latin square graphs are one of the four families of strongly regular graphs under Neumaier’s\nclassification[Neu79] (the other families being line graphsofSteiner 2 -designs, conference graphs, and graphs\nwhoseeigenvaluessatisfy the clawbound). Levet [Lev23] previou slyestablished an upper bound of ∃log2nAC0\nforisomorphismtestingofconferencegraphs,whichisastronger upperboundthanweobtainforLatinsquare\ngraphs. In contrast, the best known algorithmic runtime for isomo rphism testing of conference graphs is\nn2log(n)+O(1)due to Babai [Bab80], whereas isomorphism testing of Latin square g raphs is known to admit\nannlog(n)+O(1)-time solution [Mil78].\nWe note that the reduction outlined in [Mil78, Theorem 2] and [Lev23, R emark 1.6] in fact allow us to\ndetermine whether two quasigroup are isotopic, and not just main c lass isotopic. Thus, we have an AC0-\ncomputable disjunctive truth-table reduction from Latin Square Isotopy toQuasigroup Isomorphism .\nWe now obtain the following corollary.\nCorollary 4.7. Latin Square Isotopy belongs to ∃log2nAC0(DTISP(polylog( n),log(n))).\n7Here, we could impose the additional requirement that the le ngth of each row/column of the multiplication table is padde d\nup to a power of two in order to get a bound of ∃log2n∀logn∃lognDTISP(log2n,logn))\n115 Abelian Group Isomorphism\nIn this section, we consider isomorphism testing of Abelian groups. O ur main result in this regard is:\nTheorem 5.1. LetGbe an Abelian group, and let Hbe arbitrary. We can decide isomorphism between G\nandHin∀loglognMAC0(DTISP(polylog( n),log(n))).\nChattopadhyay, Tor´ an, and Wagner [CTW13] established a TC0(FOLL) upper bound on this problem.\nGrochow & Levet [GL23, Theorem 5] gave a tighter analysis of their a lgorithm, placing it in the sub-class\n∀lognMAC0(FOLL).8We note that Chattopadhyay, Tor´ an, & Wagner also established a n upper bound of L\nfor this problem, which is incomparable to the result of Grochow & Lev et (ibid.). We improve upon both\nthese bounds by (i) showing that O(loglogn) non-deterministic bits suffice instead of O(logn) bits, and\n(ii) using an AC0(DTISP(polylog( n),log(n))) circuit for order finding rather than an FOLLcircuit. We note\nthat while ∀lognMAC0(FOLL) is contained in TC0(FOLL), it is open whether this containment is strict. In\ncontrast, Cor. 5.3 shows that our new bound of ∀loglognMAC0(DTISP(polylog( n),log(n))) is a class that is\nin factstrictlycontained in L∩TC0(FOLL).\nProof of Thm. 5.1 . Following the strategy of [GL23, Theorem 7.15], we show that non-isomorphism can be\ndecided in the same class but with existentially quantified non-determ inistic bits.\nWe may check in AC0whether a group is Abelian. So if His not Abelian, we can decide in AC0that\nG/\\⌉}atio\\slash∼=H. So suppose now that His Abelian. By the Fundamental Theorem of Finite Abelian Groups, Gand\nHare isomorphic if and only if their multiset of orders are the same. In p articular, if G/\\⌉}atio\\slash∼=H, then there\nexists a prime power pesuch that there are more elements of order peinGthan inH. We first identify the\norder of each element, which is AC0(DTISP(polylog( n),log(n)))-computable by Lem. 3.1.\nWe will show how to nondeterministically guess and check the prime pow erpesuch that Ghas more\nelementsoforder pethanHdoes. Let n=pe1\n1···peℓ\nℓbe the primefactorizationof n. We havethat ℓ≤log2n,\nand the number of distinct prime powers dividing nise1+···+eℓ≤log2n. Nondeterministically guess a\npair (i,e) with 1 ≤i≤ ⌊log2n⌋and 1≤e≤ ⌊log2n⌋. We treat this pair as representing the prime power pe\ni\nat which we will test that Ghas more elements of that order than H. Because both iandeare bounded in\nmagnitude by log2n, the number of bits guessed is at most log2log2n. So we may effectively guess peusing\nO(loglogn) bits to specify p(implicitly, i.e., by its index i) and to specify e(explicitly, i.e., in its binary\nexpansion).\nHowever, to identify elements of order pe, we will need the number peexplicitly, in its full binary\nexpansion. First we show that we can get peexplicitly if we can get pexplicitly. Once we have pin its binary\nexpansion, the function ( p,e)/ma√sto→peis a function of O(logn) bits, so can be computed by an AC0circuit of\nsize 2O(logn)=nO(1)(see Obs. 2.2). Thus all that remains is to get pexplicitly.\nBecause all the numbers involved are only O(logn) bits, any arithmetic function of these numbers can\nbe computed in AC0, in particular, testing if an O(logn)-bit number is prime, testing if one O(logn)-bit\nnumber divides another. So, in parallel, for all numbers x= 2,3,...,n/2, anAC0checks which ones are\nprime and divide n. Now, consider the surviving primes as a list of length O(logn), consisting of numbers\neach ofO(logn) bits. Now for each pair of primes pj,ph, we define an indicator X(j,h) = 1⇐⇒pj> ph.\nAspj,phare representable using O(logn) bits, we may compute X(j,ℓ) inAC0. Now as the number of\nprimesℓ≤log2(n), we may in AC0find a prime jwith:\nℓ/summationdisplay\nh=1X(j,h) =i.\nThe result now follows.\nAs with many of our other results, we show that this class is restrict ive enough that it cannot compute\nParity. To do this, we appeal to the following theorem of Barrington & Stra ubing:\n8Grochow & Levet consider ∀lognMAC0◦FOLL, where◦denotes composition (see [GL23] for a precise formulation) . We note\nthat asAC0◦FOLL=FOLL=AC0(FOLL), we have that ∀lognMAC0◦FOLL=∀lognMAC0(FOLL). Thus, Thm. 5.1 improves\nupon the previous bound of ∀lognMAC0(FOLL) obtained by Grochow & Levet.\n12Theorem 5.2 (Barrington & Straubing [BS94, Thm. 7]) .Letk >1. AnyTCcircuit family of constant\ndepth, size 2no(1), and with at most no(1)Majority gates cannot compute the MOD kfunction.\nCorollary 5.3. Letk >1, and let Qo(logn)be any finite sequence (of O(1)length) of alternating ∃and∀\nquantifiers, where the total number of bits quantified over is o(logn). Then\nModk/∈Qo(logn)MAC0(DTISP(polylog( n),log(n))).\nProof.LetL∈Qo(logn)MAC0(DTISP(polylog( n),log(n))). ByFact2.4, DTISP(polylog( n),log(n))⊆quasiAC0,\nsoMAC0(DTISP(polylog( n),log(n)))⊆quasiMAC0, that is, quasi-polynomial size circuits of constant depth\nwith a single Majority gate at the output. Thus L∈Qo(logn)quasiMAC0.\nLetCbe thequasiMAC0circuit such that x∈L⇐⇒(Qy)C(x,y), where |y|< o(log|x|), for all strings\nx. There are 2o(logn)=no(1)possible choices for y; letCy(x) =C(x,y), where Cydenotes the circuit Cwith\nthe second inputs fixed to the string y.\nNow, to compute L, the quantifiers Qo(logn)can be replaced by a constant-depth circuit (whose depth is\nequal to the number of quantifier alternations), and whose total size is 2o(logn)=no(1), where at each leaf of\nthis circuit, we put the corresponding circuit Cy. The resulting circuit is a quasiTC0circuit with only no(1)\nMajority gates, hence by Theorem 5.2, cannot compute Parity.\n6 Isomorphism for Groups of Almost All Orders\nDietrich & Wilson [DW22] previously established that there exists a den se set Υ⊆Nsuch that if n∈Υ and\nG1,G2are magmas of order ngiven by their multiplication tables, we can (i) decide if G1,G2are groups,\nand (ii) if so, decide whether G1∼=G2in timeO(n2log2n), which is quasi-linear time relative to the size of\nthe multiplication table.\nIn this section, we establish the following.\nTheorem 6.1. Letn∈Υ, and let G1,G2be groups of order n. We can decide isomorphism between G1\nandG2inAC0(DTISP(polylog( n),log(n))).\nNote that verifying the group axioms is AC0-computable.\nRemark 6.2. Theorem 6.1 provides that for almost all orders, Group Isomorphism belongs to\nAC0(DTISP(polylog( n),log(n))), which is contained within L∩FOLL/subset⋉oteqlPand cannot compute Parity.\nWhile it is known that Group Isomorphism belongs to complexity classes such as ∃log2nL∩∃log2nFOLL\n[CTW13] and quasiAC0(Theorem 4.1) that cannot compute Parity, membership within P—let alone a\nsubclass of Pthat cannot compute Parity—is a longstanding open problem.\nProof of Theorem 6.1. Dietrich & Wilson showed [DW22, Theorem 2.5] that if Gis a group of order n∈Υ,\nthenG=H⋉B, where:\n•Bis a cyclic group of order p1···pℓ, where for each i∈[ℓ],pi>loglognandpiis the maximum power\nofpidividing n.\n•|H|= (logn)polyloglog n; and in particular, if a prime divisor pofnsatisfiesp≤loglogn, thenpdivides\n|H|.\nAsG1,G2aregivenbytheirmultiplicationtables, wemayin AC0compute(i)the primedivisors p1,...,p k\nofn, and (ii) determine whether, for each i∈[k],piis the maximal power of pidividing n. Furthermore, in\nAC0, we may write down ⌊loglogn⌋and test whether pi>⌊loglogn⌋.\nFix a group Gof order n. We will first discuss how to decompose G=H⋉B, as prescribed by [DW22,\nTheorem 2.5]. Without loss of generality, suppose that p1,...,p ℓ(ℓ≤k) are the unique primes where pi(i∈\n[ℓ]) divides nonly once and pi>loglogn. Now as each pican be representedas a string of length ≤ ⌈log(n)⌉,\nwe may in AC0compute p:=p1···pℓ(Obs. 2.2). Using Lem. 3.1, we may in AC0(DTISP(polylog( n),log(n)))\nidentify an element g∈Gof order p.\nNow inAC0, we may write down the multiplication table for Hj∼=Gj/Bj. As|Hj| ≤(logn)polyloglog n,\nthere are poly( n) possible generating sequences for Hjof length at most log |Hj|. By [BS84] it actually\n13suffices to consider cube generating sequences for Hj. Now given cube generating sequences xjforHj,\nwe may by the proof of Theorem 4.1 decide whether x1/ma√sto→x2extends to an isomorphism of H1andH2\nin∀logn∃lognDTISP(polylog( n),log(n))⊆AC0(DTISP(polylog( n),log(n))). As there are only poly( n) such\ngenerating sequences to consider, we may decide whether H1∼=H2inAC0(DTISP(polylog( n),log(n))).\nSuppose H1∼=H2,B1∼=B2, and gcd( |Bj|,|Hj|) = 1 for j= 1,2. We have by the Schur–Zassenhaus\nTheorem that Gj=Hj⋉θjBj(j= 1,2). By Taunt’s Lemma [Tau55], it remains to test whether the action s\nθ1andθ2are equivalent in the following sense. We first note that, as Bjis Abelian, for any two elements\nh1,h2ofGjbelonging to the same coset of Gj/Bjand any element b∈Bj, thath1bh−1\n1=h2bh−1\n2. Thus,\ngiven an isomorphism α:H1∼=H2(whereα(h1) =h2is represented using arbitrary coset representatives\nofh1,h2), we may by Lemma 3.1, search in AC0(DTISP(polylog( n),log(n))) for generators b1∈B1,b2∈B2\nand write down the isomorphism β:B1∼=B2induced by the map b1/ma√sto→b2. We may now check in AC0\nwhether for all h1∈H1and allb∈Bjwhether\nβ(h1bh−1\n1) =α(h1)β(b)α(h−1\n1).\nHere, we abuse notation, where h1bh−1\n1is evaluated using the coset representatives h1∈h1andh−1\n1∈h−1\n1.\nThe result now follows.\n7 Minimum Generating Set\nIn this section, we consider the Minimum Generating Set (MGS) problem for quasigroups, as well as\narbitrary magmas.\n7.1 MGS for Groups in AC1(L)\nIn this section, we establish the following.\nTheorem 7.1. MGSfor groups belongs to AC1(L).\nWe begin with the following lemma.\nLemma 7.2. LetGbe a group. We can compute a chief series for GinAC1(L).\nProof.We will first show how to compute the minimal normal subgroups N1,...,N ℓofG. We proceed as\nfollows. We first note that the normal closure ncl( x) is the subgroup generated by {gxg−1:g∈G}. Now we\nmay write down the elements of {gxg−1:g∈G}inAC0, and then compute ncl( x) inLusing a membership\ntest. Now in L, we may identify the minimal (with respect to inclusion) subgroups am ongst those obtained.\nGivenN1,...,N ℓ, we may easily in Lcompute/producttextk\ni=1Nifor each k≤ℓ. In particular, we may compute\nSoc(G) inL. We claim that/producttextk\ni=1Ni, fork= 1,...,ℓ, is in fact a chief series of Soc( G) (which will then\nfit into a chief series for G). To see this, we have that/producttextk\ni=1Niis normal in/parenleftBig/producttextk\ni=1Ni/parenrightBig\n×Nk+1and that\n(/producttextk+1\ni=1Ni)/(/producttextk\ni=1Ni)∼=Nk+1is a normal subgroup of G//producttextk\ni=1Ni. By the Lattice Isomorphism Theorem,\n(/producttextk+1\ni=1Ni)/(/producttextk\ni=1Ni) is in fact minimal normal in G//producttextk\ni=1Ni.\nWe iterate on this process starting from G/Soc(G). Note that, as we have computed Soc( G) from the\nprevious paragraph, we may write down the cosets for G/Soc(G) inAC0. Furthermore, given a subgroup\nH≤G/Soc(G), we may write down the elements of HSoc(G) inAC0. By the above, the minimal normal\nsubgroups of a group are computable in L. As there are at most log nterms in a chief series, we may compute\na chief series for GinAC1(L), as desired. (Recall that we use this notation to mean an AC1circuit with\noracle gates calling a Loracle, not function composition such as AC1◦L.)\nWe now prove the Theorem 7.1.\nProof of Theorem 7.1. By Lem. 7.2, we can compute a chief series for GinAC1(L). So let N1⊳···⊳ Nk\nbe a chief series SofG. Lucchini and Thakkar [LT24] showed that minimum generating sets ofG/Ni+1\nhave specific structure depending on whether or not Ni+1/Niis Abelian. We proceed inductively down S\nstarting from Nk−1. AsG/Nk−1is a finite simple group, and hence at most 2-generated, we can write all\n14/parenleftbign\n2/parenrightbig\npossible generating sets in parallel with a single AC0circuit and test whether each generates the group\nwith a membership test. This can be done in L.\nFixi < k. Suppose we are given a minimum generating sequence g1,...,g d∈GforG/Ni. We will\nconstruct a minimum generating sequence for G/Ni−1as follows. We consider the following cases:\n•Case 1: Suppose that N=Ni/Ni−1is Abelian. By [LM94, Theorem 4], we have three cases:\n– Case 1a: G/Ni−1=/a\\}⌊ra⌋k⌉tl⌉{tg1,···,gd/a\\}⌊ra⌋k⌉tri}ht. We may test whether this holds in Lusing a membership test.\n– Case 1b: We have G/Ni−1=/a\\}⌊ra⌋k⌉tl⌉{tg1,···,gi,gjn,gj+1,···,gd/a\\}⌊ra⌋k⌉tri}htfor some j∈[d] and some n∈N.\nThere are at most d·(|N| −1) generating sets to consider in this case and we can test each of\nthem inL.\n– Case 1c: If neither Cases 1a or 1b hold, then we necessarily have that G/Ni−1=/a\\}⌊ra⌋k⌉tl⌉{tg1,···,gd,x/a\\}⌊ra⌋k⌉tri}ht\nfor some any non-identity element x∈N.\nNote that there are at most d·|N|generating sets to consider, we may construct a minimum generatin g\nset forG/Ni−1inLusing a membership test.\n•Case 2: Suppose instead that N=Ni/Ni−1is non-Abelian. Then by [LT24, Corollary 13] we have\nthe following cases:\n– Case 2a: G/Ni−1=/a\\}⌊ra⌋k⌉tl⌉{tg1,···,gd/a\\}⌊ra⌋k⌉tri}ht. We may test whether this holds in Lusing a membership test.\n– Case 2b: If Case 2a does not hold, then we have the following case. Let ηG(N) denote the\nthe number of factors in a chief series with order |N|. Letu= max{d,2}andt= min{u,⌈8\n5+\nlog|N|ηG(N)⌉}. Thenthereexist n1,...,n t∈Ni−1suchthat G/Ni−1=/a\\}⌊ra⌋k⌉tl⌉{tg1n1,···,gtnt,gt+1,···,gd/a\\}⌊ra⌋k⌉tri}ht.\nBy [LT24, Corollary 13], there are at most |N|⌈8\n5+log|N|ηG(N)⌉generating sets of this form. We\nmay write down these generating sets in parallel with a single AC0circuit and test whether each\ngenerate G/Ni−1inLusing a membership test.\nDescendingalongthe chiefseriesinthis fashion, wecompute quotien tsNi/Ni−1andcomputeagenerating\nset forG/Ni−1. The algorithm terminates when we’ve computed a generating set fo rG/N0=G. Since a\nchief series has O(logn) terms, this algorithm requires O(logn) iterations and each iteration is computable\ninL. Hence, we have an algorithm for MGSinAC1(L).\nImproving upon the AC1(L) bound on MGSfor groups appears daunting. It is thus natural to inquire as\nto families of groups where MGSis solvable in complexity classes contained within AC1(L). To this end, we\nexamine the class of nilpotent groups. Arvind & Tor´ an previously es tablished a polynomial-time algorithm\nfor nilpotent groups [AT06, Theorem 7]. We improve their bound as fo llows.\nProposition 7.3. For a nilpotent group G, we can compute d(G)inL∩AC0(NTISP(polylog( n),log(n))).\nProof.LetGbe our input group. Recall that a finite nilpotent group is the direct p roduct of its Sylow\nsubgroups (which by the Sylow theorems, implies that for a given prim epdividing |G|, the Sylow p-subgroup\nofGis unique). We can, in AC0(DTISP(polylog( n),log(n))) (using Cor. 3.2), decide if Gis nilpotent; and if\nso, compute its Sylow subgroups. So we write G=P1×P2×···×Pℓ, where each Piis the Sylow subgroup\nofGcorresponding to the prime pi. Arvind & Tor´ an (see the proof of [AT06, Theorem 7]) established t hat\nd(G) = max 1≤i≤ℓd(Pi). Thus, it suffices to compute d(Pi) for each i∈[ℓ].\nThe Burnside Basis Theorem provides that Φ( P) =Pp[P,P]. We may compute Ppin\nL∩AC0(DTISP(polylog( n),log(n))) (the latter using Cor. 3.2). We now turn to computing [ P,P]. Us-\ning a membership test, we can compute [ P,P] inL. By [Ser97, I. §4 Exercise 5], every element in [ P,P]\nis the product of at most log |P|commutators. Therefore, we can also decide membership in [ P,P] in\nNTISP(polylog( n),log(n)), andsowecanwritedowntheelementsof[ P,P]inAC0(NTISP(polylog( n),log(n))).\nThus, we may compute Φ( P) inL∩AC0(NTISP(polylog( n),log(n))). Given Φ( P), we may compute\n|P/Φ(P)|inAC0. Thus, we may recover d(P) from|P/Φ(P)|inAC0, by iterated multiplication of the\nprime divisor pof|P|. As the length of the encoding of pis at most log |P|and we are multiplying pby\nitself log |P|times, iterated multiplication is AC0-computable. Thus, in total, we may compute d(P) in\nL∩AC0(NTISP(polylog( n),log(n))). It follows that for an arbitrary nilpotent group G, we may compute\nd(G) inL∩AC0(NTISP(polylog( n),log(n))).\n15Remark 7.4. While Prop. 7.3 allows us to compute d(G) for a nilpotent group G, the algorithm is non-\nconstructive. It is not clear how to find such a generating set in L. We can, however, provide such a\ngenerating set in AC1(NTISP(polylog( n),log(n))). Note that this bound is incomparable to AC1(L). We\noutline the algorithm here.\nThe Burnside Basis Theorem provides that for a nilpotent group G, (i) every generating set of Gprojects\nto a generating set of G/Φ(G), and (ii) for every generating set SofG/Φ(G),everylift ofSis a generating\nset ofG. Furthermore, every minimum generating set of Gcan be obtained from the Sylow subgroups in\nthe following manner. Write G=P1× ··· ×Pℓ, wherePiis the Sylow pi-subgroup of G. Suppose that\nPi=/a\\}⌊ra⌋k⌉tl⌉{tgi1,...,g ik/a\\}⌊ra⌋k⌉tri}ht(where we may have gij= 1 for certain values of j). Write gj=/producttextℓ\ni=1gij. As in the proof\nof [AT06, Theorem 7] we obtain G=/a\\}⌊ra⌋k⌉tl⌉{tg1,...,g k/a\\}⌊ra⌋k⌉tri}ht.\nGiven generating sets for P1,...,P ℓ, we may in L∩AC0(NTISP(polylog( n),log(n))) recover a generating\nset forG. Thus, it suffices to compute a minimum generating set P/Φ(P)∼=(Z/pZ)d(P), where Pis\nap-group. Note that we may handle each Sylow subgroup of Gin parallel. To compute a minimum\ngenerating set of P/Φ(P), we use the generator enumeration strategy. As P/Φ(P) is Abelian, we may check\ninAC0(NTISP(polylog( n),log(n))) Cor. 3.6whether a set ofelements generatesthe group. As d(P)≤log|P|,\nwe have log |P|steps where each step is AC0(NTISP(polylog( n),log(n)))-computable. Thus, we may compute\na minimum generating set for PinAC1(NTISP(polylog( n),log(n))), as desired.\n7.2 MGS for Quasigroups\nInthissection, weconsiderthe Minimum Generating Set problemforquasigroups. Ourgoalistoestablish\nthe following.\nTheorem 7.5. ForMGSfor quasigroups,\na. The decision version belongs to ∃log2n∀lognNTIME(polylog( n));\nb. The decision version belongs to ∃log2nSAC1⊆DSPACE(log2n);9and\nc. The search version belongs to quasiAC0.\nIn the paper in which they introduced (polylog-)limited nondeterminis m, Papadimitriou and Yannakakis\nconjectured that MGS for quasigroups was ∃log2nP-complete [PY96, after Thm. 7]. While they did not\nspecify the type of reductions used, it may be natural to consider polynomial-time many-one reductions.\nTheorem 7.5 refutes two versions of their conjecture under othe r kinds of reductions, that are incomparable\nto polynomial-time many-one reductions: quasiAC0reductions unconditionally and polylog-space reductions\nconditionally. We note that their other ∃log2nP-completeness result in the same section produces a reduction\nthat in fact can be done in logspace and (with a suitable, but natural, encoding of the gates in a circuit)\nalso inAC0, so our result rules out any such reduction for MGS. (We also note: assuming EXP/\\⌉}atio\\slash=PSPACE,\nshowing that this problem is complete under polynomial-time reduction s would give a separation between\npoly-time and log-space reductions, an open problem akin to P/\\⌉}atio\\slash=L.)\nCorollary 7.6. MGSfor quasigroups and Quasigroup Isomorphism are\na. not∃log2nP-complete under quasiAC0Turing reductions.\nb. not∃log2nP-complete under polylog-space Turing reductions unless EXP=PSPACE.\nProof.(a) Thm. 7.5 (a) for MGS, resp. Thm. 4.1 for Quasigroup Isomorphism , place both problems into\nclasses that are contained in quasiAC0by Fact 2.4. The result then follows by noting that quasiAC0is closed\nunderquasiAC0Turing reductions, Parity∈P⊆ ∃log2nP, butParity /∈quasiAC0.\n(b) Both MGSfor quasigroups and Quasigroup Isomorphism are inDSPACE(log2n) by Thm. 7.5 (b),\nresp. [CTW13]. The closure of DSPACE(log2n) under poly-log space reductions is contained in polyL=/uniontext\nk≥0DSPACE(logkn). If either of these two quasigroup problems were complete for ∃log2nPunder polylog-\nspace Turing reductions, we would get ∃log2nP⊆polyL. Under the latter assumption, by a straightforward\npadding argument, we now show that EXP=PSPACE.\n9Wolf [Wol94] showed the containment ∃log2nSAC1⊆DSPACE (log2n).\n16LetL∈EXP; letkbe such that L∈DTIME(2nk+k). Define Lpad={(x,12|x|k+k) :x∈L}. By\nconstruction, Lpad∈P. Let us use Nto denote the size of the input to Lpad, that is, N= 2nk+k+n. By\nassumption, we thus have Lpad∈polyL. Suppose ℓis such that Lpad∈DSPACE(logℓN). We now give a\nPSPACE algorithm for L. In order to stay within polynomial space, we cannot write out the p adding 12nk+k\nexplicitly. What we do instead is simulate the DSPACE(logℓN) algorithm for Lpadas follows. Whenever\nthe head on the input tape would move off the xand into the padding, we keep track of its index into\nthe padding, and the simulation responds as though the tape head were reading a 1. When the tape head\nmoves right the index increases by 1, when it moves left it decreases by 1, and if the index is zero and the\ntape head moves left, then we move the tape head onto the right en d of the string x. The index itself is a\nnumber between 0 and 2nk+k, so can be stored using only O(nk) bits. The remainder of the Lpadalgorithm\nuses only O(logℓN) =O(nkℓ) additional space, thus this entire algorithm uses only a polynomial a mount of\nspace, so L∈PSPACE, and thus EXP=PSPACE.\nNow we return to establishing the main result of this section, Thm. 7.5 . To establish Thm. 7.5 (a) and\n(c), we will crucially leverage the Membership for quasigroups problem. To this end, we will first establish\nthe following.\nTheorem 7.7. Membership for quasigroups belongs to NTIME(polylog( n)).\nRemark 7.8. A natural approach to Thm. 7.5 (b) is also to attempt to use Thm. 7.7 . For comparison\nto the approach we take instead, we note that a careful analysis o f the proof for Thm. 7.7 only yields a\nDSPACE(log3n) bound for MGSfor quasigroups.\nThm. 7.7 immediately yields the following corollary.\nCorollary 7.9. For quasigroups, Membership andMGSare not hard under AC0-reductions for any com-\nplexity class containing Parity.\nThe proofs of Theorem 7.7 and Theorem 7.5 rely crucially on the followin g adaption of the Babai–\nSzemer´ edi Reachability Lemma [BS84, Theorem 3.1] to quasigroups . We first generalize the notion of\na straight-line program for groups [BS84] to SLPs for quasigroups . LetXbe a set of generators for a\nquasigroup G. We call a sequence of elements g1,...,g ℓ∈Gastraight-line program (SLP for short) if each\ngi(i∈[ℓ]) either belongs to X, or is of the form or gjgk,gj\\gk, orgj/gkfor some j,k < i(wheregj\\gk, resp.\ngj/gk, denotes the quasigroup division as defined in Section 2.1). An SLP is s aid tocompute orgenerate a\nsetS(or an element g) ifS⊆ {g1,...,g ℓ}(resp.g∈ {g1,...,g ℓ}).\nLemma 7.10 (Reachability Lemma for quasigroups) .LetGbe a finite quasigroup and let Xbe a set of\ngenerators for G. For each g∈G, there exists a straight-line program over Xgenerating gwhich has length\nO(log2|G|).\nProof.We follow the same strategy as in the proof of [BS84, Theorem 3.1], bu t there are some subtle, yet\ncrucial, modifications due to the fact that quasigroups are non-as sociative and need not posses an identity\nelement. We will inductively construct a sequence z0,z1,...,z t(similar to a cube generating sequence).\nFor any sequence of elements z1,...,z k, letP(z1z2···zk) denote the left-to-right parenthesization, e.g.,\nP(z1z2z3) = (z1z2)z3. For some initial segment z0,z1,...,z idefine the cube\nK(i) ={P(z0ze1\n1···zei\ni) :e1,...,e i∈ {0,1}},\nwhereej= 0 denotes omitting zjfrom the product (since there need not be an identity element).\nDefineL(i) =K(i)\\K(i) ={g\\h:g,h∈K(i)}. We will construct a sequence z0,z1,...,z tsuch that\nt≤ ⌈log2(n)⌉andL(t) =G. Moreover, we derive a bound on the straight-line cost c(i) for{z0,z1,...,z i}\n(1≤i≤t), which is defined as the length of the shortest SLP generating {z0,z1,...,z i}.\nWe take z0as an arbitrary element from X. Hence, K(0) ={z0}, and soc(0) = 1. Next, let us construct\nK(i+1)from K(i). IfL(i)/\\⌉}atio\\slash=G, we setzi+1to be the element z′/∈L(i) that minimizes c(i+1)−c(i). We first\nclaim that |K(i+1)|= 2·|K(i)|. Note that K(i+1) =K(i)∪K(i)zi+1by definition. As right-multiplication\nby a fixed element is a bijection in a quasi-group, it suffices to show tha tK(i)∩K(i)zi+1=∅. So, suppose\n17that there exists some a∈K(i)∩K(i)zi+1. Then a=gzi+1for some g∈K(i). Hence, zi+1=g\\a,\ncontradicting zi+1/\\⌉}atio\\slash∈L(i), since both aandgare inK(i).\nIt now follows that |K(i)|= 2iand, hence, t≤ ⌈log2(|G|)⌉andL(t) =G. Moreover, for every g∈Gwe\nobtain an SLP of length at most c(t)+2t+1: write g=a\\bfora,b∈K(t) and start with the SLP computing\n{z0,z1,...,z t}. We obtain an SLP for a=z0ze1\n1···zet\ntby adding a new element for each z0ze1\n1···zei\niwith\nei= 1 (1≤i≤t, thus, at most tnew elements). Likewise, we get an SLP for b. Adding a last element\ng=a\\b, we obtain an SLP computing g.\nIt remains to bound the straight-line cost c(i) of{z0,z1,...,z i}. Here, we claim that that c(i+1)−c(i)≤\n4i+3 (note that this is slightly worse than the bound c(i+1)−c(i)≤2i+1 for groups [BS84]). It follows\nfrom this claim that c(i)∈O(i2). We will now turn to proving our claim.\nProof of Claim. IfG/\\⌉}atio\\slash=L(i), then either X/\\⌉}atio\\slash⊆L(i) orL(i) is not a sub-quasigroup. Hence, we have one of\nthe following cases:\n•Case 1: Suppose that there is some g∈XL(i). In this case, there is an SLP of length one for g.\nThus, given an SLP for {z0,...,z i}at costc(i), we may construct an SLP for {z0,...,z i,g}of length\nc(i)+1 and we obtain c(i+1)−c(i)≤1 in this case.\n•Case 2: Suppose there exist g,h∈L(i) with one of gh,g/h, org\\h/\\⌉}atio\\slash∈L(i). For simplicity, suppose\ngh/\\⌉}atio\\slash∈L(i).The argument is identical for g/handg\\h. As above, given a straight-line program to\ncompute {z0,z1,...,z i}, we may construct straight-line programs for gandheach of additional length\n2i+1. This yields a straight-line program for ghof total length at most c(i)+4i+3, and shows that\nc(i+1)−c(i)≤4i+3.\nThe result now follows.\nForprovingTheorem7.7 wefollowessentiallythe ideasof[Fle22] (thou gh weavoidintroducingthe notion\nof Cayley circuits). Fleischer obtained a quasiAC0bound for Membership for groups by then showing that\nthe Cayley circuits for this problem can be simulated by a quasiAC0circuit. We will instead directly analyze\nthe straight-line programs using an NTIME(polylog( n)) (sequential) algorithm.\nProof of Theorem 7.7. To decide whether g∈ /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht, we guess a sequence of elements g1,...,g ℓ∈Gwith\nℓ∈O(log2|G|) andgℓ=gand in a next step verify whether this, indeed, is a straight-line prog ram. Clearly,\nthe conditions ( gi(i∈[ℓ]) either belongs to X, or is of the form gjgk,gj\\gk, orgj/gkfor some j,k < i) can\nbe checked in time polynomial in the length of the straight-line progra m and, thus, in time polylogarithmic\nin the input length.\nThe proof of Thm. 7.5 (a) and (c) below is by describing a reduction fr omMGStoMembership . To\nestablish Thm. 7.5 (b), we use the following result of Wagner:\nLemma 7.11 ([Wag10, Theorem 10.2.1]) .LetQbe a quasigroup, and let X⊆Q. We can compute /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}htin\nSAC1.\nWe now have all the ingredients we need to prove all three parts of T hm. 7.5.\nProof of Theorem 7.5. (a) LetGdenote the input quasigroup (of order n). First, observe that every quasi-\ngroup has a generating set of size ≤ ⌈logn⌉[Mil78]. Therefore, we start by guessing a subset X⊆Gof size\nat most≤ ⌈logn⌉(resp. the size bound given in the input). For the decision version, w e useO(log2n) exis-\ntentially quantified non-deterministic bits ( ∃log2n) to guess a generating sequence. To find a minimum-sized\ngenerating sequence, we can enumerate all possible generating se quences in quasiAC0. In the next step, we\nverify whether Xactually generates G. This is done by checking for all g∈G(universally verifying O(logn)\nbits,∀logn) whether g∈ /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}ht, which can be done in NTIME(polylog( n))⊆quasiAC0by Theorem 7.7. This\nconcludes the proof of the bound (a) for the decision variant.\n(c) If we want to compute the minimum generating set, we have to ch eck whether it is actually of smallest\npossible size. To do this, in a last step, we use the same technique as a bove to check that all Y⊆Gwith\n|Y|<|X|do not generate G.\n(b) Every quasigroup has a generating set of size ≤ ⌈logn⌉. Existentially guess the generating set using\nlog2nbits, then use Lem. 7.11 to compute /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}htinSAC1. With a log-depth tree of bounded fan-in AND\ngates, we then check that /a\\}⌊ra⌋k⌉tl⌉{tX/a\\}⌊ra⌋k⌉tri}htis indeed the whole quasigroup.\n18Remark 7.12. We may similarly reduce Quasigroup Isomorphism toMembership for quasigroups.\nThis formalizes the intuition that membership testing is an essential s ubroutine for isomorphism testing\nandMGS. In particular, in the setting of quasiAC0, we have that isomorphism testing and MGSreduce\ntoMembership . This latter consequence might seem surprising, as in the setting of groups,Membership\nbelongs to L, whileMGSbelongs to AC1(L) (Thm. 7.1) and it is a longstanding open problem whether\nGroup Isomorphism is inP.\n7.3 MGS for Magmas\nIn this section, we establish the following.\nTheorem 7.13. The decision variant of Minimum Generating Set for magmas is NP-complete.\nThisNP-completeness result helps explain the use of Integer Linear Progr amming in practical heuristic\nalgorithms for the search version of this problem, e.g., [JMV23].\nFor the closely related problem they called Log Generators —given the multiplication table of a binary\nfunction (=magma) of order n, decide whether it has a generating set of size ≤ ⌈log2n⌉—Papadimitriou\nand Yannakakis proved that Log Generators of Magmas is∃log2nP-complete under polynomial-time\nreductions [PY96, Thm. 7]. Our proof uses a generating set of size r oughly√n, wherenis the order of\nthe magma; this is analogous to the situation that Log-Clique is in∃log2nP, while finding cliques of size\nΘ(√n) in ann-vertex graph is NP-complete.\nProof.It is straightforward to see that the problem is in NPby simply guessing a suitable generating set.\nTo show NP-hardness we reduce 3SATtoMinimum Generating Set for magmas. Let F=/logicalandtextm\nj=1Cjwith\nvariables X1,...,X nbe an instance of 3SAT. Our magma Mconsists of the following elements:\n•for each variable Xi, two elements Xi,Xi,\n•for each clause Cjan element Cj,\n•for each 1 ≤j≤k≤man element Tj,k, and\n•a trash element 0.\nWe useTas an abbreviation for T1,m.\nWe define the multiplication as follows:\nCjX=Tj,jif the literal Xappears in Cj\nTXi=Xi,\nTXi=Xi,\nTj,kTk+1,ℓ=Tj,ℓ.\nall other products are defined as 0.\nNote that any generating set for Mmust include all the Cj, since without them, there is no way to\ngenerate them from any other elements. Similarly, any generating s et must include, for each i= 1,...,n, at\nleast one of XiorXi, since again, without them, there is no way to generate those from any other elements.\nWhenFis satisfiable, we claim that Mcan be generated by precisely n+melements. Namely, include\nallCjin the generating set. Fix a satisfying assignment ϕtoF. Ifϕ(Xi) = 1, then include Xiin the\ngenerating set, and if ϕ(Xi) = 0, include Xiin the generating set. Since ϕis a satisfying assignment, for\neachj, there is a literal Xin our generating set that appears in Cj, and from those two we can generate\nCjX=Tj,j. Next,Tj,jTj+1,j+1=Tj,j+1, and by induction we can generate all Tj,k. In particular, we can\ngenerate T=T1,m, and then using Tand our literal generators, we can generate the remaining literals.\nConversely, suppose Mis generated by n+melements; we will show that Fis satisfiable. As argued\nabove, those elements must consist of {Cj:j∈[m]}together with precisely one literal corresponding to each\nvariable. Since the final defining relation can only produce elements Tj,ℓwithjstrictly less than ℓ, the only\nway to generate Tj,jfrom our generating set is using the first relation. Namely, at least o ne of the literals in\nour generating set must appear in Cj. But then, reversing the construction of the previous paragrap h, the\nliterals in our generating set give a satisfying assignment to F.\n19As with most NP-complete decision versionsof optimization problems, we expect tha t theexactversion—\ngiven a magma Mand an integer k, decide whether the minimum generating set has size exactly k—is\nDP-complete, but we leave that as a (minor) open question.\n8 Conclusion\nThe biggest open question about constant-depth complexity on alg ebras given by multiplication tables is,\nin our opinion, still whether or not Group Isomorphism is inAC0in the Cayley table model. 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Springer Berlin, Heidelberg, 1 edition, 1997.\ndoi:10.1007/978-3-642-59141-9 .\n[Smo87] Roman Smolensky. Algebraic methods in the theory of lower b ounds for boolean circuit\ncomplexity. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium\non Theory of Computing, 1987, New York, New York, USA , pages 77–82. ACM, 1987.\ndoi:10.1145/28395.28404 .\n22[Tan13] Bangsheng Tang. Towards Understanding Satisfiability, Group Isomorphism\nand Their Connections . PhD thesis, Tsinghua University, 2013. URL:\nhttp://papakonstantinou.org/periklis/pdfs/bangsheng _thesis.pdf .\n[Tau55] D. R. Taunt. Remarks on the isomorphism problem in theories of construction of finite\ngroups. Mathematical Proceedings of the Cambridge Philosophical S ociety, 51(1):16–24, 1955.\ndoi:10.1017/S030500410002987X .\n[Tor04] Jacobo Tor´ an. On the hardness of graph isomorphism. SIAM J. Comput. , 33(5):1093–1108,\n2004.doi:10.1137/S009753970241096X .\n[VGM09] Valery Vasil’ev, Maria Grechkoseeva, and V. Mazurov. Char acterization of the fi-\nnite simple groups by spectrum and order. Algebra and Logic , 48:385–409, 12 2009.\ndoi:10.1007/s10469-009-9074-9 .\n[Vol99] Heribert Vollmer. Introduction to Circuit Complexity - A Uniform Approach . Texts in Theoret-\nical Computer Science. An EATCS Series. Springer, 1999. doi:10.1007/978-3-662-03927-4 .\n[Wag10] F. Wagner. On the complexity of isomorphism testing for restricted\nclasses of graphs . PhD thesis, Universit¨ at Ulm, 2010. URL:\nhttps://oparu.uni-ulm.de/xmlui/bitstream/handle/123 456789/3923/vts_7264_10267.pdf .\n[Wol94] Marty J. Wolf. Nondeterministic circuits, space complexity an d quasigroups. Theoretical Com-\nputer Science , 125(2):295–313, 1994. doi:10.1016/0304-3975(92)00014-I .\n23"    },    {        "title": "2402.00134v2.Investigations_of__Λ__states_with_spin_parity___frac_3__2____pm__.pdf",        "content": "arXiv:2402.00134v2  [hep-ph]  4 Apr 2024Investigations of Λstates with spin-parity3\n2±\nK. Azizi∗\nDepartment of Physics, University of Tehran, North Karegar Avenue, Tehran 14395-547, Iran\nDepartment of Physics, Doˇ gu¸ s University, Dudullu- ¨Umraniye, 34775 Istanbul, Turkey and\nSchool of Particles and Accelerators, Institute for Resear ch in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehra n, Iran\nY. Sarac†\nElectrical and Electronics Engineering Department, Atili m University, 06836 Ankara, Turkey\nH. Sundu‡\nDepartment of Physics Engineering, Istanbul Medeniyet Uni versity, 34700 Istanbul, Turkey\n(Dated: April 5, 2024)\nThe present study provides spectroscopic investigations o f spin-3\n2Λ baryons with both positive\nand negative parities. The analysis mainly focuses on three states, namely 1 P, 2P, and 2S, and\ncorresponding masses are calculated using the QCD sum rule m ethod. To implement the method,\nwe apply two types of interpolating currents with octet and s inglet quantum numbers and compare\nthe corresponding results with the reported masses of exper imentally observed states. From the\ncomparisons, it is extracted that the results of interpolat ing current with octet quantum numbers\nare in good agreement with the experimentally measured mass es. The masses obtained with this\ninterpolating current are m= 1513.64±8.76 MeV for 1 Pstate with JP=3\n2−,m′= 1687.91±\n0.31 MeV for 2 Pstate with JP=3\n2−and ˜m= 1882.37±11.95 MeV for 2 Sstate with JP=3\n2+and\nthey are consistent with the experimental masses of Λ(1520) , Λ(1690) and Λ(1890), respectively,\nwhich confirm their spin-parity quantum numbers. Besides, w e calculate the corresponding current\ncoupling constants, which are utilized as inputs in the calc ulations of different form factors defining\nthe widths of the states under study.\nI. INTRODUCTION\nThe advances in experimental researchareashave broughtmany new observations ofboth conventionaland noncon-\nventional hadrons. As a result of the improved experimental tech niques and analyses, the excited states of light and\nheavy baryons have been observed with progressively higher confi dence levels, and our understanding of the strong\ninteraction has been enhanced via the investigations conducted to comprehend the properties of these observed states.\nThe discrepancy between the numbers of the observed excited nu cleon and other resonance states and the expectation\nof the quark model keeps the subject hot and collects interest in t heir investigation. Consequently, comprehending\nthe spectroscopic characteristics, substructures, and intera ctions of the present or newly observed states is crucial for\ngaining a deeper understanding of both the strong interaction and the absent resonances.\nThe studies on the hyperon resonances and their excited states a re of importance to enhance our understanding of\nthese states. The need for the investigation of these states is au gmented by our poor knowledge of their properties\ncompared to those of non-strange baryons such as Nand ∆ resonances. The presence of squark with a mass\nheavier than the mass of uanddquarks and lighter than that of candbquarks puts these states in an interesting\nplace among the other baryons. The excited states of baryons ha ve been among the recent focus of investigations\nas a result of new observations pointing out possible excited states of heavy and light baryons [ 1–4]. Positive and\nnegative parity excitations of these baryons have been investigat ed in several works using different methods, such\nas the quark model [ 5–17], lattice QCD [ 18–20], basis lattice front quantization approach [ 21], covariant three-body\nFadeev approach [ 22], using Dyson-Schwinger and Bethe-Salpeter equations [ 23] and QCD sum rules method [ 24–\n32]. To advance our understanding of particle physics and the nonper turbative regime of QCD, understanding the\nstructure and interaction mechanisms of both heavy baryons and baryons with light quark content have been of\ninterest. Looking at the light baryons in Particle Data Group (PDG) [ 33], it is seen that there exist fifteen N\nresonances, eight ∆ resonances, ten Λ resonances, eight Σ reso nances, three Ξ resonances, and one Ω resonance with\n4-star status. However, the quark model predicts more states in these energy regions [ 5], whose reason may be the\neffective degrees of freedom of the model. These indicate a need fo r deeper investigation of the present and missing\n∗kazem.azizi@ut.ac.ir ; Corresponding author\n†yasemin.sarac@atilim.edu.tr\n‡hayriyesundu.pamuk@medeniyet.edu.tr2\nlight baryon resonances to understand their nature, structure , and interactions. With these motivations, these states\nwere investigated via different approaches [ 5,34,35].\nAmong these light baryons are the Λ hyperon and its excited states . These states have been investigated in the\nlast decades via the data coming from K−pinvariant mass spectra of hyperons [ 36–38], partial wave analyses of K−p\nreactions [ 39] and photo-production of Λ(1405) from JLab [ 40,41]. The recent couple channel analyses resulted in\nmore hyperons [ 42–49]. Besides experimental analyses, the spectrum of the Λ baryons w as investigated via various\ntheoretical models, such as the quark model [ 5,50–59] and lattice QCD [ 60–62]. The QCD sum rule method was\napplied to study the mass of Λ baryons [ 63–65]. The masses of the excited states of light strange baryons, includ ing\nΛ baryon, were studied in Ref. [ 66] using the Regge phenomenology with quasi-linear Regge trajector ies. In Ref. [ 57],\nit was pointed out that Λ(1405), Λ(1520), Λ(1670), and Λ(1690) are possible mixtures of the p-wave q2sstates and\ngroundq3¯sqpentaquark states. In Ref. [ 67], 71 Λ∗’s were predicted by lattice QCD calculations. Previous analyses\nwith the constituent quark models provided similar predictions [ 5,10]. All these put the baryon states containing\nstrange quarks among the focus of experimental and theoretica l researches for either the investigation of missing\nhyperon states or better understanding the observed ones. Be sides, these studies improve our understanding of the\nnonperturbativeregimeofQCD.Theirinvestigationsprovideuswith abetterunderstandingoftheirintrinsicstructure\nand the degrees of freedom in a baryon and help us improve our unde rstanding of confinement mechanisms as well.\nWith this motivation, baryon states with strange quarks have been the subject of investigations in many experimental\nfacilities around the world to investigate their spectroscopy and int erection mechanisms (for some recent experimental\ninvestigations, see, for instance, the Refs. [ 66,68–73]).\nIn this work, our main purpose is to study the Λ states with spin J=3\n2. With this aim, we make spectrum analyses\nfor Λ state with this spin and investigate not only the ground state b ut also the corresponding first orbital and first\nradial excitations. Among Λ states, our analyses with the consider ed spin-parity may supply relevant information\nabout the states Λ(1520) with spin-parity JP=3\n2−, Λ(1690) with spin-parity JP=3\n2−, and Λ(1890) with spin-\nparityJP=3\n2+which have the same spin-parity quantum numbers with the consider ed states in the present work.\nSuch analyses require a nonperturbative method, and in this work, we apply the QCD sum rule method [ 74–76] as\nit is among the successful nonperturbative methods with predictio ns exhibiting good consistency with experimental\nobservations. To apply the method, the main ingredient is a proper in terpolating current describing the considered\nstate, which is composed of the quark fields according to the valenc e quark content of the state. We investigate the\nmasses and current coupling constants with two types of interpola ting currents with octet or flavor-singlet quantum\nnumbers [ 65]. The comparison of the results with the present experimental mas s values helps us understand the\nstructure of these states, gain information about the interactio n mechanisms of the quarks in the low energy domain\nof the QCD, and may also contribute to future investigations of miss ing states.\nThe outline of the present work is as follows: In next section, the de tails of the QCD sum rule calculations for the\nmasses and current coupling constants of all considered states a re given. The numerical analyses for the results are\nattained in Sec. III. SecIVis devoted to the summary and conclusion.\nII. THE QCD SUM RULE FOR THE ΛSTATES\nAmong the efficient approaches for elucidating the structure and p roperties of a particular resonance is determining\nits mass by assigning a proper structure to the resonance. The co mparison of the computed mass results with\nexperimental findings significantly helps us understand their natur e and substructures. With this angle, our purpose\nin the present work is to calculate the masses for the Λ states with s pin-3\n2and both negative and positive parities. To\naccomplish this task, we choose proper interpolating fields comprise d of the quark fields consistent with the valence\nquark content and quantum numbers of the Λ states. To this end, we choose two types of interpolating currents with\noctet and flavor-singlet quantum numbers given as [ 65]:\nηo\nµ=/radicalbigg\n1\n6ǫabc[2(uaTCσκδdb)σκδγµsc+(uaTCσκδsb)σκδγµdc−(daTCσκδsb)σκδγµuc],\nηs\nµ=/radicalbigg\n1\n3ǫabc[(uaTCσκδdb)σκδγµsc−(uaTCσκδsb)σκδγµdc+(daTCσκδsb)σκδγµuc]. (1)\nIn these currents, Cdenotes charge conjugation operator; a, b, cindices are used to represent the color indices of the\nquark fields u, dands, ando(s) inηo(s)denotes octet (singlet)-type current. These two currents are used in the\nfollowing correlation function to obtain the mass sum rules:\nΠµν(q) =i/integraldisplay\nd4xeiq·x/an}bracketle{t0|T {ηo(s)\nµ(x)¯ηo(s)\nν(0)}|0/an}bracketri}ht, (2)3\nwhereTrepresents the time ordering operator.\nThe calculation of the correlation function, Eq. ( 2), is performed in two ways called the Hadronic and the QCD\nsides, respectively. The results obtained from these two sides are matched via a dispersion relation to obtain the\nQCD sum rules for the physical parameters under quest. These re sults contain various Lorenz structures, and in\nthe matching procedure, one equates the coefficients of one of th ese Lorentz structures. The contributions coming\nfrom the higher states and continuum are suppressed by the applic ation of the Borel transformation and continuum\nsubtraction operations to both sides.\nIn the hadronic side, the correlator is calculated treating the inter polating currents as creation and annihilation\noperators, which create or annihilate the considered states from the vacuum. To proceed in the calculation a complete\nset of hadronicstates with same quantum numbers carried by the in terpolatingcurrents is inserted into the correlation\nfunction. Then, the integral over four- xis performed giving the following result:\nΠHad\nµν(q) =/an}bracketle{t0|ηo(s)\nµ|Λ(q,s)/an}bracketri}ht/an}bracketle{tΛ(q,s)|¯ηo(s)\nν|0/an}bracketri}ht\nm2−q2+/an}bracketle{t0|ηo(s)\nµ|Λ′(q,s)/an}bracketri}ht/an}bracketle{tΛ′(q,s)|¯ηo(s)\nν|0/an}bracketri}ht\nm′2−q2+/an}bracketle{t0|ηo(s)\nµ|˜Λ(q,s)/an}bracketri}ht/an}bracketle{t˜Λ(q,s)|¯ηo(s)\nν|0/an}bracketri}ht\n˜m2−q2\n+···, (3)\nwhere the explicitly presented terms denote the contributions of n egative parity spin-3\n2ground state as well as its\nexcitations with negative and positive parities, respectively, and ···shows the contributions of higher resonances and\ncontinuum. |Λ(q,s)/an}bracketri}ht,|Λ′(q,s)/an}bracketri}htand|˜Λ(q,s)/an}bracketri}htcorrespond to their one-particle states with respective masses m,m′and\n˜m. The matrix elements in Eq. ( 3) are defined in terms of current coupling constants and spin-vect ors,uµ, in Rarita\nSchwinger representation as:\n/an}bracketle{t0|ηo(s)\nµ|Λ(q,s)/an}bracketri}ht=λγ5uµ(q,s),\n/an}bracketle{t0|ηo(s)\nµ|Λ′(q,s)/an}bracketri}ht=λ′γ5uµ(q,s),\n/an}bracketle{t0|ηo(s)\nµ|˜Λ(q,s)/an}bracketri}ht=˜λuµ(q,s). (4)\nOnce these matrix elements are used in the Eq. ( 3), the summation over spin is required to proceed, which has the\nfollowing form:\n/summationdisplay\nsuµ(q,s)¯uν(q,s) =−(/ne}ationslashq+m)/bracketleftBig\ngµν−1\n3γµγν−2qµqν\n3m2+qµγν−qνγµ\n3m/bracketrightBig\n. (5)\nAt this point we need to mention that the interpolating currents tha t we use couple also to spin-1\n2positive and\nnegative parity states with their corresponding matrix elements giv en as:\n/an}bracketle{t0|ηo(s)\nµ|1\n2+\n(q)/an}bracketri}ht=A1\n2+(γµ+4qµ\nm1\n2+)γ5u(q,s), (6)\nand\n/an}bracketle{t0|ηo(s)\nµ|1\n2−\n(q)/an}bracketri}ht=A1\n2−(γµ+4qµ\nm1\n2−)u(q,s), (7)\nrespectively. From these matrix elements, it can be seen that the c oefficients of the Lorentz structures containing γµ\nandqµget contributions from spin-1\n2states also. To avoid these contributions and to select the ones on ly coming\nfrom the spin-3\n2states, we make a proper choice of Lorentz structure giving mere spin-3\n2states’ contribution. Using\nthe above relations, the result of hadronic side becomes\nΠHad\nµν(q) =λ2\nq2−m2(/ne}ationslashq−m)/bracketleftBig\ngµν−1\n3γµγν−2qµqν\n3m2−qµγν−qνγµ\n3m/bracketrightBig\n+λ′2\nq2−m′2(/ne}ationslashq−m′)/bracketleftBig\ngµν−1\n3γµγν−2qµqν\n3m′2−qµγν−qνγµ\n3m′/bracketrightBig\n+˜λ2\nq2−˜m2(/ne}ationslashq+ ˜m)/bracketleftBig\ngµν−1\n3γµγν−2qµqν\n3˜m2+qµγν−qνγµ\n3˜m/bracketrightBig\n+···. (8)\nAs it is seen, this expression contain many Lorentz structures, ho wever, only gµνand/ne}ationslashqgµνstructures are free from\nspin-1\n2pollution and give contributions only to spin-3\n2states. In principle, as the standard application of QCD sum4\nrule method, both of these structures can be selected to predict the mass of the desired states. However, in our case,\nboth of these structures lead to roughly the same results. To this end, we consider the structure gµνin our analyses\nto extract the physical parameters of the states under study. Hence, we have\nΠHad\nµν(q) =−λ2\nq2−m2mgµν−λ′2\nq2−m′2m′gµν+˜λ2\nq2−˜m2˜mgµν+other structures+ ···. (9)\nAfter the Borel transformation with respect to −q2, which is applied to suppress the contribution coming from higher\nstates and continuum, the final result becomes\n/hatwideBΠHad\nµν(q) =λ2e−m2\nM2mgµν+λ′2e−m′2\nM2m′gµν−˜λ2e−˜m2\nM2˜mgµν+other structures+ ···. (10)\nAs for the QCD side of the calculation, the operator product expan sion (OPE) is applied, and the correlation\nfunction is computed using the interpolating currents explicitly in Eq. (2). This gives a result in terms of quark fields,\nand, applying Wick’s theorem, possible contractions between the qu ark fields are obtained. So, the results turn into\nthe ones given in terms of the quark propagators. The correspon ding result for the octet current is provided here to\nexemplify the form of the results:\nΠQCD\nµν(q) =i/integraldisplay\nd4xeiq·x1\n6ǫabcǫa′b′c′σκδγµ/braceleftBig\n4Scc′\ns(x)γνσκ′δ′Tr[Sbb′\nd(x)σκ′δ′˜Saa′\nu(x)σκδ]\n−2Scb′\ns(x)σκ′δ′˜Saa′\nu(x)σκδSbc′\nd(x)γνσκ′δ′−2Scb′\ns(x)σκ′δ′˜Sba′\nd(x)σκδSac′\nu(x)γνσκ′δ′\n−2Scb′\nd(x)σκ′δ′˜Saa′\nu(x)σκδSbc′\ns(x)γνσκ′δ′+Scc′\nd(x)γνσκ′δ′Tr[Sbb′\ns(x)σκ′δ′˜Saa′\nu(x)σκδ]\n+Sca′\nd(x)σκ′δ′˜Sbb′\ns(x)σκδSac′\nu(x)γνσκ′δ′−2Sca′\nu(x)σκ′δ′˜Sab′\nd(x)σκδSbc′\ns(x)γνσκ′δ′\n+Sca′\nu(x)σκ′δ′˜Sbb′\ns(x)σκδSac′\nd(x)γνσκ′δ′+Scc′\nu(x)γνσκ′δ′Tr[Sbb′\ns(x)σκ′δ′˜Saa′\nd(x)σκδ]/bracerightBig\n,(11)\nwhere˜Saa\nqrepresents CSaa′T\nqC. The quark propagator for light quark has the following form in xspace:\nSab\nq(x) =ix /\n2π2x4δab−mq\n4π2x2δab−/an}bracketle{tqq/an}bracketri}ht\n12/parenleftBig\n1−imq\n4x //parenrightBig\nδab−x2\n192m2\n0/an}bracketle{tqq/an}bracketri}ht/parenleftBig\n1−imq\n6x //parenrightBig\nδab−igsGθη\nab\n32π2x2/bracketleftBig\nx /σθη+σθηx //bracketrightBig\n−x /x2g2\ns\n7776/an}bracketle{tqq/an}bracketri}ht2δab−x4/an}bracketle{tqq/an}bracketri}ht/an}bracketle{tg2\nsG2/an}bracketri}ht\n27648δab+mq\n32π2[ln(−x2Λ2\n4)+2γE]gsGθη\nabσθη. (12)\nIn the light quark propagator, Eq. ( 12),γE≃0.577 is the Euler constant and Λ is the QCD scale parameter. The\ncalculation of this side is straightforward using this propagator in th e correlator. After the applications of the Fourier\nand Borel transformations as well as continuum subtraction, the result takes the following form:\n/hatwideBΠQCD(s0,M2) =/integraldisplays0\n(mu+md+ms)2e−s\nM2ρ(s)ds+Γ(M2), (13)\nwhereρ(s) and Γ(M2) are the results obtained considering the coefficient of the struct uregµνas in the hadronic side\nands0is the continuum threshold parameter. Note that we apply the quar k-hadron duality assumption to omit the\nsuppressedcontributions ofthe higherstates and continuum fro m the hadronicside with their equivalent contributions\nfrom the QCD side above the threshold s0.\nThe match of the coefficients of the selected structure from both the hadronic and QCD sides gives the following\nQCD sum rule:\nmλ2e−m2\nM2+m′λ′2e−m′2\nM2−˜m˜λ2e−˜m2\nM2=/hatwideBΠQCD(s0,M2). (14)\nTo extract masses and current coupling constants from Eq. ( 14), we consider each state one by one and follow such\nan approach: First, we calculate the mass for the ground state fr om the first term in the left-hand side of Eq. ( 14),\nkeeping others in continuum, namely the ground state + continuum f rame. To get the mass of the ground state, after\ntaking the derivative of Eq. ( 14) with respect to −1\nM2, we divide this by Eq. ( 14) itself, i.e.:\nm2=d\nd(−1\nM2)/hatwideBΠQCD(s0,M2)\n/hatwideBΠQCD(s0,M2). (15)5\nThe mass predicted in this way is applied together with Eq. ( 14), and the current coupling constant for this state is\nattained as\nλ2=em2\nM2/hatwideBΠQCD(s0,M2). (16)\nThe mass and current coupling constants for the excited states a re obtained similarly: To get results for the first\nradial excitation, the first two terms in Eq. ( 14) are considered, and the remaining term is treated to be inside the\ncontinuum, which means ground state + first radial (2 P) excitation + continuum frame is applied using the results\nfor the ground state as input. And finally, the same treatment is ap plied for the next excited state, namely the 2 S\nstate; that is, ground state + first radial excitation + 2 S+ continuum frame is now taken into account to get the\nphysical parameters for the 2 Sstate. The numerical analyses of the results for these states ar e given in the next\nsection.\nIII. NUMERICAL ANALYSES\nThe QCD sum rules, attained for masses and current coupling const ants in the previous section, are numerically\nanalyzed in this section with the necessary input parameters. Some of these parameters are given in Table I. The\nParameters Values\nmu 2.16+0.49\n−0.26MeV [33]\nmd 4.67+0.48\n−0.17MeV [33]\nms 93.4+8.6\n−3.4MeV [33]\n/angbracketleft¯qq/angbracketright(1GeV) (−0.24±0.01)3GeV3[77]\n/angbracketleft¯ss/angbracketright 0.8/angbracketleft¯qq/angbracketright[77]\nm2\n0 (0.8±0.1) GeV2[77]\n/angbracketleftqgsσGq/angbracketright m2\n0/angbracketleft¯qq/angbracketright\n/angbracketleftαs\nπG2/angbracketright(0.012±0.004) GeV4[78]\nTABLE I. Some input parameters required for the numerical an alyses.\nparameters present in Table Iare necessary but not the only required parameters. Besides, th e results contain\ntwo more auxiliary parameters: the Borel parameter M2and the threshold parameter s0. To be able to fix these\nparameters, the QCD sum rules have some prescriptions that are s tandard for the method. These are the weak\ndependence of the results on these auxiliary parameters, the con vergence of the OPE, and pole dominance. By OPE\nconvergence, we mean: The perturbative part exceeds the nonp erturbative contributions and the higher the dimension\nof the nonperturbative operator, the lower its contribution. Pole dominance guarantees the dominant contributions\nof the considered resonances compared to the higher states and continuum.\nIndeed, the Borel parameter is fixed by the dominance of the cons idered first three resonances over the higher states\nand continuum, which sets its upper limit, as well as the OPE converge nce, which sets its lower limit. The Borel\nmass intervals obtained considering these restrictions are presen ted in Table II. In this table, we also present the\nworking intervals of the threshold parameters fixed by the ground state + 2 Pexcitation + 2 Sexcitation + continuum\napproach, as explained in the previous section. First, considering t he relation of the threshold parameter to the\nenergy of the next excited state, we fix the threshold parameter by ground state + continuum frame and get the\nmass and corresponding current coupling constant for the groun d state, whose values are also present in Table II.\nThese results are used as inputs in the calculations of the mass and c urrent coupling constant for the next excited\nstate. To make this calculation, now the ground state + 2 Pexcitation + continuum frame is taken into account, and\nsimilar to the previous calculation, the interval for the new thresho ld parameter is determined. With this threshold\nparameter interval, again, the mass and current coupling constan t for this excited state are obtained and presented\nin Table II. Finally, the mass and current coupling constant for the next excit ed state are obtained from ground state\n+ 2Pexcitation + 2 Sexcitation + continuum frame, and the threshold parameter interv al for this new consideration\nis decided in a similar way. With this new threshold interval, the results f or mass and current coupling constant in\nthis case are also obtained and given in Table II. The errors of the results are presented in Table II, as well, and these\nerrors are due to the uncertainties present in the input paramete rs and the determination of the working intervals of\nthe auxiliary parameters.\nIt is instructive to find the first three resonance’ contribution (F TRC) at average values of the Borel parameter and6\nResults for octet current\nΛ state M2(GeV2)s0(GeV2)Mass (MeV) Current Coupling Constant (GeV3)\nΛ(JP=3\n2−)(1P)3.0−4.02.7−2.91513.64±8.76 (3.35±0.15)×10−2\nΛ(JP=3\n2−)(2P)3.0−4.03.1−3.31687.91±0.31 (2.05±0.27)×10−2\nΛ(JP=3\n2+)(2S)3.0−4.03.5−3.71882.37±11.95 (2.08±0.30)×10−2\nResults for singlet current\nΛ state M2(GeV2)s0(GeV2)Mass (MeV) Current Coupling Constant (GeV3)\nΛ(JP=3\n2−)(1P)3.0−4.02.7−2.91470.44±16.08 (4.85±0.20)×10−2\nΛ(JP=3\n2−)(2P)3.0−4.03.1−3.31732.56±11.55 (2.90±0.39)×10−2\nΛ(JP=3\n2+)(2S)3.0−4.03.5−3.71843.47±11.21 (3.00±0.41)×10−2\nTABLE II. The working windows of the Borel masses and thresho ld values and the mass and current coupling constant results\nobtained using the octet and singlet currents for the Λ state s with spin-3\n2and different parities\ncontinuum threshold. It is defined as\nFTRC =/hatwideBΠQCD(s0,M2)\n/hatwideBΠQCD(∞,M2). (17)\nBy using the average values of the auxiliary parameters, we find FTR C = 0.91 indicating that the main contribution\nin the QCD side comes from the first three resonances and the highe r states contribute with only 9%. This well\nsatisfies the corresponding requirement of the method. To depict the behavior and stability of the results in the\nworking intervals of the auxiliary parameters, we present the figur es1,2,3,4,5, and6for the masses and current\ncoupling constants of all the considered states. From these figur es, we see good stability of the results with respect to\nthe auxiliary parameters. The mild variations of the results with resp ect to the variations of these parameters, seen\nin these figures, are reflected in the errors of the predictions pre sented in Table IIas mentioned before.\n●●●●●●●●●●●■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆\n●s0=2.9GeV2\n■s0=2.8GeV2\n◆s0=2.7GeV2\n3.03.23.43.63.84.01.01.21.41.61.82.0\nM2(GeV2)m(GeV)\n●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n●M2=4.0GeV2\n■M2=3.5GeV2\n◆M2=3.0GeV2\n2.702.752.802.852.901.01.21.41.61.82.0\ns0(GeV2)m(GeV)\nFIG. 1. Left:The dependence of the mass of the ground state Λ on M2at different values of s0.Right:The dependence of\nthe mass of the ground state Λ on s0at different values of M2.\nIV. SUMMARY AND CONCLUSION\nIn the present work, among the light hyperon states, we focused on the Λ state with spin-parity quantum numbers\nJ=3\n2−and its excitations with negative and positive parities. In the analyse s, we especially targeted to account\nfor the Λ(1520) through its spectroscopic property, together with to see whether any experimentally observed state\npossessing the identical quantum numbers can be explained via the r esults attained for the excited states. To analyze\nthese states, we applied the QCD sum rule method with two possible cu rrents that may define these states with\noctet and singlet quantum numbers. From the analyses, we got the mass values of the three lowest lying states.\nThe results were obtained using the current with octet quantum nu mbers as: m= 1513.64±8.76 MeV for the 1 P7\n●●●●●●●● ● ●●■■■■■■■■ ■ ■■◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆●s0=2.9GeV2\n■s0=2.8GeV2\n◆s0=2.7GeV2\n3.03.23.43.63.84.0123456\nM2(GeV2)λ×10-2(GeV3)\n●●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆●M2=4.0GeV2\n■M2=3.5GeV2\n◆M2=3.0GeV2\n2.702.752.802.852.90123456\ns0(GeV2)λ×10-2(GeV3)\nFIG. 2.Left:The dependence of the current coupling constant of the groun d state Λ on M2at different values of s0.Right:\nThe dependence of the current coupling constant of the groun d state Λ on s0at different values of M2.\n●●●●●●●●●●● ■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆\n●s0=3.3GeV2\n■s0=3.2GeV2\n◆s0=3.1GeV2\n3.03.23.43.63.84.01.21.41.61.82.02.2\nM2(GeV2)m'(GeV)\n●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n●M2=4.0GeV2\n■M2=3.5GeV2\n◆M2=3.0GeV2\n3.103.153.203.253.301.21.41.61.82.02.2\ns0(GeV2)m'(GeV)\nFIG. 3.Left:The dependenceof the mass of the (2 P) exited state of Λ on M2at different values of s0.Right:The dependence\nof the mass of the (2 P) exited state of Λ on s0at different values of M2.\n●●●●●●●●●●●\n■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆●s0=3.3GeV2\n■s0=3.2GeV2\n◆s0=3.1GeV2\n3.03.23.43.63 \u0000 \u00014.0012\n345\nM2(GeV2)λ'×10-2(GeV3)\n●●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆●M2=4.0GeV2\n■M2=3.5GeV2\n◆M2=3.0GeV2\n3.103.153.203.253.30012345\ns0(GeV2)λ'×10-2(GeV3)\nFIG. 4. Left:The dependence of the curent coupling constant of the (2 P) exited state of Λ on M2at different values of s0.\nRight:The dependence of the curent coupling constant of the (2 P) exited state of Λ on s0at different values of M2.\nstate with JP=3\n2−,m′= 1687.91±0.31 MeV for the 2 Pstate with JP=3\n2−and ˜m= 1882.37±11.95 MeV\nfor the 2 Sstate with JP=3\n2+. The mass values using current with singlet quantum numbers were o btained as:\nm= 1470.44±16.08 MeV for the 1 Pstate with JP=3\n2−,m′= 1732.56±11.55 MeV for the 2 Pstate with JP=3\n2−\nand ˜m= 1843.47±11.21MeV for the 2 Sstate with JP=3\n2+. In other works present in the literature, the masses for8\n●●●●●●●●●●●■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆\n●s0=\u0002 \u0003 \u0004GeV2\n■s0=3.6GeV2\n◆s0=3.5GeV2\n3.03.23.43.6 \u0005 \u0006 \u00074.01.51.61.71.81.92.02.12.2\nM2(GeV2)m˜(GeV)\n●●●●●●●●●●●●●●●●●●●●● ■■■■ ■ ■ ■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆◆◆◆◆◆◆◆\n●M2=4.0GeV2\n■M2=3.5GeV2\n◆M2=3.0GeV2\n3.503.553.603.65\b \t \n \u000b1.51.61.71.81.92.02.12.2\ns0(GeV2)m˜(GeV)\nFIG. 5.Left:The dependence of the mass of the (2 S) exited state of Λ on M2at different values of s0.Right:The dependence\nof the mass of the (2 S) exited state of Λ on s0at different values of M2.\n●●●●●●●●●●●\n■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆●s0=3.7GeV2\n■s0=3.6GeV2\n◆s0=3.5GeV2\n3.03.23.43.6 \f \r \u000e4.0012\n\f45\nM2(GeV2)λ˜\n×10-2(GeV\n\u000f)\n●●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆●M2=4.0GeV2\n■M2=3.5GeV2\n◆M2=3.0GeV2\n3.503.553.603.65\u0010 \u0011 \u0012 \u0013012\n\u001045\ns0(GeV2)λ˜\n×10-2(GeV\n\u0014)\nFIG. 6. Left:The dependence of the curent coupling constant of the (2 S) exited state of Λ on M2at different values of s0.\nRight:The dependence of the curent coupling constant of the (2 S) exited state of Λ on s0at different values of M2.\nthe lowest lying Λ states with spin-3\n2consistent with the ones we search for in the present work were pr edicted. For\ncompleteness and comparison with our predictions, we present the results of some of these works here. In Ref. [ 7], the\nmasses for the lowest lying P-waveΛ states with spin-3\n2were given as m= 1490MeV and m= 1690MeV. In Ref. [ 10],\nthe masses for the same negative parity lowest lying states were ob tained as m= 1545 MeV and m= 1645 MeV\nfor3\n2−states, and the mass was obtained as m= 1900 MeV for JP=3\n2+positive parity state. The spectral values\nwere predicted in Ref. [ 11] asm= 1498 MeV and m= 1629 MeV for3\n2−states and m= 1855 MeV for JP=3\n2+\nstate. In Ref. [ 50] the masses were given as m= 1490 MeV, and m= 1690 MeV for3\n2−states. In Ref. [ 54] the\nmasses for spin-parity3\n2−states were attained as m= 1508 MeV, and m= 1662 MeV, and for spin-parity3\n2+as\nm= 1823 MeV. The masses were given as m= 1549 MeV and m= 1693 MeV for3\n2−states and m= 1854 MeV\nforJP=3\n2+state in Ref. [ 55]. In Ref. [ 59] the mass predictions for JP= 3/2−state corresponding to Λ(1520)\nwere calculated as m= 1534 MeV and m= 1544 MeV, the masses for JP= 3/2−state given for Λ(1690) were\npredicted to be m= 1819 MeV and m= 1841 MeV and the masses for JP= 3/2+state for Λ(1890) were predicted\nasm= 1769 MeV and m= 1789 MeV. The mass values were given in Ref. [ 56] asm= 1431 MeV, m= 1650 MeV,\nandm= 1896 MeV for Λ(1520), Λ(1690), and Λ(1890) spin-3\n2states, respectively. The mass for the 1 Pspin-3\n2state\nwas given in Ref. [ 66] asm= 1551.23±0.43 MeV.\nCompared to the experimental findings, our predictions obtained f rom the current with octet quantum numbers\nare consistent with the masses of Λ(1520), Λ(1690) and Λ(1890) states possessing mass values and quantum numbers\nmΛ(1520)≈1519 MeV and JP=3\n2−,mΛ(1690)≈1690 MeV and JP=3\n2−, andmΛ(1890)≈1890 MeV and JP=3\n2+,\nrespectively [ 33]. Given the consistency of these results with experimental observ ations, it is also pertinent to compare\nthem with the predictions of other studies. The mass result obtaine d using the interpolating current with octet\nquantum number in the present work for the 2 Pstate is consistent with that of Ref. [ 7] given for the excited state,9\nPresent Work\n(Results of\noctet current)Present Work\n(Results of\nsinglet current )[7][10][11][50][54][55][56][59][66]Exp.\n[33]\nm(JP=3\n2−)1513.64±8.761470.44±16.0814901545149814901508154914311534\n15441551.23±0.431519\nm′(JP=3\n2−)1687.91±0.311732.56±11.5516901645162916901662169316501819\n1841- 1690\n˜m(JP=3\n2+)1882.37±11.951843.47±11.21-19001855-1823185418961769\n1789- 1890\nTABLE III. The mass results of the present work and that of diff erent works and experimental masses for the Λ states with\nspin-parity quantum numbers JP=3\n2±in units of MeV.\nhowever, the 1 Pstate is larger than their corresponding result. Compared to the r esults of Ref. [ 10], our value for\nthe 1Pstate is smaller, and the 2 Pstate is larger than their corresponding predictions. In Ref. [ 11], the reported\nresult for Λ(1520) state is close to our prediction, but their predic tions for Λ(1690) and Λ(1890) are smaller than ours.\nThe results given in Ref. [ 50] are close to those of the present work obtained for 2 Pand 2Scases, and their result\ncorresponding to the 1 Pcase is smaller than our prediction. While the result we obtained for Λ( 1520) is consistent\nwith that of Ref. [ 54], the results obtained for the other two states in that work are sm aller than the predictions of\nthe present work. The predictions for the mass of negative parity states in Ref. [ 55] are larger, on the other hand,\nthe positive parity state is smaller than our corresponding predictio ns. In Ref. [ 59], the mass predictions given for\nΛ(1520) and Λ(1690) states are larger, and that for Λ(1890) is s maller than ours. The reported masses for Λ(1520)\nand Λ(1690) in Ref. [ 56] are smaller than our corresponding results, while the result given f or Λ(1890) is in agreement\nwithin the errors with our prediction. To provide a more explicit illustra tion of the aforementioned comparison, we\npresent the Table IIIlisting the results from other methods, experiment, and the result s obtained from our analyses.\nAs is seen, though there are various studies in literature with altern ative methods about the Λ baryon with spin3\n2such as Λ(1520), Λ(1690), and Λ(1890), the discrepancies amon g the results necessitate more such works to provide\nfurther information over these particles. Studying the propertie s of these states helps us better identify these states,\nunderstand their nature, and interactions with other particles. T hese studies may contribute to the understanding of\nthe underlying mechanisms of the interactions of these particles wit h others and their role in various particle physics\nprocesses. 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However, for unimodular gravity to be considered a viable theory of\ngravity, one has to show that it has a well-posed initial value formulation. Working in vacuum, we\napply Dirac’s algorithm to find all the constraints of the theory. Then we prove that, for initial data\ncompatible with these constraints, the evolution is well posed. Finally, we find sufficient conditions\nfor a matter action to preserve the well-posedness of the initial value problem of unimodular gravity.\nAs a corollary, we argue that the “unimodular” restriction on the spacetime volume element can be\nsatisfied by a suitable choice of the lapse function.\nI. INTRODUCTION\nGeneral relativity (GR) is nowadays accepted as the\ntheory of gravity. However, GR is not problem-free. In\nparticular, it requires a cosmological constant to describe\nthe universe at cosmological scales whose measured value\ndeparts, by many orders of magnitude, from the value\nestimated by considering vacuum state contributions [1].\nUnimodular gravity (UG) is a modified theory of gravity\nwhere the cosmological constant arises as an integration\nconstant, which is independent of vacuum state contri-\nbutions.\nOne important property of any fundamental physical\ntheory is its ability to predict a system’s evolution from\ninitial data. Given some initial data, perhaps subject\nto some constraints, this evolution ought to be unique.\nHowever, for some theories, these properties are not\nenough; the evolution must also be continuous and causal\nin the following sense: we expect that small perturbations\non initial data should produce small changes in the solu-\ntions, where the notion of “smallness” is given by certain\nSobolev norms [2]. In other words, we require the solu-\ntions to depend continuously on the initial data to avoid\nlosing predictability given that initial data can only be\nmeasured with finite precision. Also, changes in initial\ndata supported in a given spacetime region should only\naffect its causal future (and past). This notion is rele-\nvant in relativistic theories for consistency with space-\ntime causal structure. A theory in which the evolution is\nunique, continuous, causal in the above described sense\nis said to have a well-posed initial value formulation.\nA very important feature of GR is that it has a well-\nposed initial value formulation [3]. This is not a trivial re-\nsult since the metric, which is a dynamical field, contains\nthe causal information. Nevertheless, the well-posedness\nof the initial problem of GR can be shown by writing the\nequation of motion, via a judicious choice of coordinates,\nin a form where one can prove that the above-mentioned\nproperties hold.\n∗bonder@nucleares.unam.mxIt is worth mentioning that, besides GR, there is a\nrelatively small set of modified gravity theories for which\nproofs of well-posed initial value formulation are known.\nExamples of modified gravity theories where such results\nhave been obtained include scalar-tensor theories [4], k-\nessence theory [5], Einstein-æther theory [6], Horndeski\ntheories [7, 8], and a 4-derivative scalar-tensor theory [9].\nStill, to the best of our knowledge, there are no previous\nproofs of a proper initial value formulation for theories\nwith nondynamical tensors.\nThe goal of this work is to show that UG has a well-\nposed initial value formulation. We must stress that the\nproof we present is not a simple application of the GR\ntechniques, since the UG constraints are different from\nthose of GR. In this sense, this work introduces methods\nthat may be used when investigating the initial value\nformulation of other modified gravity theories.\nWe structured the paper as follows: in Sec. II, to have a\nself-contained paper, we introduce the UG theory and the\nmathematical tools to study a relativistic theory of grav-\nity in terms of 3-dimensional geometrical objects. The\nmost important part of this paper is the identification\nof the evolution and constraint equations of the theory,\nwhich is presented in Sec. III. In this section, we also\nperform the constraint analysis for UG and we carry out\nthe analysis of the evolution equations, relying on the\nwell-known BSSN formulation. Finally, we present our\nconclusions in Sec. IV.\nII. PRELIMINARIES\nA. Unimodular Gravity\nHistorically, UG dates back to Einstein [10] and Pauli\n[11] who were interested in possible interplays between\ngravity and elementary particles (for historical remarks\nsee Ref. 12). Yet, the framework that is close to what is\npresented here emerged some fifty years later [13] when\nthe theory was studied in the context of field theory [14–\n16]. In 2011, attention was drawn into UG with the ob-\nservation that the energy associated with the vacuum\nstate does not gravitate (in a semiclassical framework),arXiv:2402.00141v2  [gr-qc]  21 Feb 20242\nbypassing the cosmological constant problem [17]. This\nclaim, however, is not free of criticisms [18].\nRecent works rekindled the interest in UG. For ex-\nample, it has been shown that energy nonconservation\navoids some incompatible features with Quantum Me-\nchanics and could give rise to an effective cosmological\nconstant that has an adequate sign and size [19, 20].\nAlso, cosmological diffusion models in the UG framework\naffect the value of the Hubble constant [21], among other\ninteresting features [22].\nHere, we work on a 4-dimensional spacetime M\nequipped with the pseudo-Riemannian metric gab(we fol-\nlow the notation and conventions of Ref. 2 and, in par-\nticular, pairs of indexes in between parenthesis/brackets\nstand for its symmetric/antisymmetric part with a 1 /2\nfactor). Moreover, spacetime is assumed to be globally\nhyperbolic, which allows us to foliate Mby constant time\n(Cauchy) hypersurfaces Σ t.\nThere are several ways to introduce UG [23]; the UG\naction we consider here is\nS\u0002\ngab, λ,Φ\u0003\n=1\n2κZ\nd4x\b√−gR+λ\u0000√−g−f\u0001\t\n+SM\u0002\ngab,Φ\u0003\n, (1)\nwhere gabis the inverse of gab,λis a scalar field that\nacts as a Lagrangian multiplier, κis the gravitational\ncoupling constant, Ris the curvature scalar associated\nwith the metric-compatible and torsion-free derivative\n∇a. Moreover, gis the determinant of gab, and fis a\nnondynamical positive scalar density, i.e., a real function\nthat transforms under coordinate transformations mim-\nicking√−g. Also, S Mis the matter action, which takes\nthe form\nSM\u0002\ngab,Φ\u0003\n=Z\nd4x√−gLM\u0000\ngab,Φ\u0001\n, (2)\nwhere LM\u0000\ngab,Φ\u0001\nis the matter Lagrangian and Φ col-\nlectively describes all matter fields.\nWe want to emphasize that the only difference of the\nUG action when compared with that of GR is the pres-\nence of the term with the Lagrange multiplier. This term\nfixes the differential spacetime volume element√−gd4x\nto coincide with fd4x, which is only consistent if f >0,\nwhich is something we assume.\nAn arbitrary variation of the action (1) has the form\nδS =Z\nd4x\u001a√−g\u00141\n2κ(Gab−1\n2gabλ)−1\n2Tab\u0015\nδgab\n+1\n2κ\u0000√−g−f\u0001\nδλ+δLM\nδΦδΦ\u001b\n, (3)\nwhere we omit the boundary terms, something we\ndo throughout the paper, and we define the energy-\nmomentum tensor as\nTab:=−2√−gδ(LM√−g)\nδgab. (4)Hence, the metric equation of motion is\nRab−1\n2Rgab−1\n2λgab=κTab, (5)\nNotice that λenters this last equation as a cosmological\nconstant, even though at this point there is no reason for\nit to be constant. Furthermore, the equation of motion\nassociated with λyields the “unimodular constraint”\n√−g=f. (6)\nOf course, there are also matter field equations that can\nbe written generically as\nδSM\nδΦ= 0. (7)\nThe trace of Eq. (5) takes the form\nR+ 2λ+κT= 0, (8)\nwhere T:=gabTabis the trace of the energy-momentum\ntensor. Introducing this trace into Eq. (5) yields\nEab:=Rab−1\n4Rgab−κ\u0012\nTab−1\n4Tgab\u0013\n= 0,(9)\nwhich is explicitly traceless, namely,\nEabgab= 0. (10)\nOn the other hand, the divergence of Eq. (5) produces\nκ∇aTab=−1\n2∇bλ, (11)\nwhere we use the Bianchi identity. If ∇aTab= 0, λbe-\ncomes a constant, which acts as the cosmological con-\nstant. However, UG allows for more general matter so-\nlutions where ∇aTabis not necessarily zero, opening the\ndoor to interesting phenomenological applications [24].\nThe main difference between GR and UG is the the-\nories’ symmetries and conservation laws. As it is well\nknown, GR is invariant under all diffeomorphisms, which\nin turn implies that ∇aTab= 0. This is not the case in\nUG where the nondynamical function f, which does not\ntransform under (active) diffeomorphisms, by assump-\ntion, partially breaks invariance under diffeomorphisms\n[25]. To show this, we first consider theory in vacuum,\nnamely, S M= 0. The variation of the vacuum UG action\nwith respect to a diffeomorphism associated with ξais\ngiven by Eq. (3) with\nδgab=£ξgab=−2∇(aξb), (12)\nδλ=£ξλ=ξa∇aλ, (13)\nwhere £vis the Lie derivative along va. Then, the UG\naction variation with respect to a diffeomorphism takes\nthe form\nδS =Z\nd4x\u001a\u00141\nκ(−Gab+1\n2gabλ)\u0015\n∇aξb\n+1\n2κ\u0000√−g−f\u0001\nξa∇aλ\u001b\n. (14)3\nAfter we integrate by parts (with the appropriate volume\nform), this variation can be written as\nδS =1\n2κZ\nd4x√−gλ∇a\u0012\nξaf√−g\u0013\n, (15)\nwhere we use the Bianchi identity. On shell, Eq. (6) is\nvalid, and thus,\nδS =1\n2κZ\nd4x√−gλ∇aξa. (16)\nHence, the vacuum action is only invariant under diffeo-\nmorphisms associated with divergence-free vector fields,\nnamely, vector fields such that ∇aξa= 0. This restricted\nset of diffeomorphisms goes by the name of volume-\npreserving diffeomorphisms.\nTo obtain the matter conservation law associated with\nvolume-preserving diffeomorphisms, we notice that a\ndivergence-free vector field ξacan be written in terms of\nageneric antisymmetric tensor αabasξa=ϵabcd∇bαcd,\nwhere ϵabcdis the volume form associated with gab. Thus,\nthe on-shell variation of S Mwith respect to a volume-\npreserving diffeomorphism can be written as\nδSM=Z\nd4x√−gTabϵbcde∇a∇cαde\n=Z\nd4x√−gαdeϵbcde∇c∇aTab, (17)\nwhere, in the last step, we integrate by parts twice.\nTherefore, the matter action is invariant under all\nvolume-preserving diffeomorphisms if\n∇[b∇aTc]a= 0. (18)\nThis last equation is the UG matter conservation law,\nwhich is more general than that of GR. What is more,\nusing the Poincar´ e Lemma [26] and under the hypothe-\nsis that Mis simply connected, which we assume, this\nconservation law implies that there exists a scalar Qsuch\nthat\n∇aTab=∇bQ. (19)\nWhen Qis constant, we recover the GR matter conserva-\ntion law. However, in UG, Qcan be arbitrary. We turn\nto present the framework to describe gravity theories in\nterms of an evolving 3-dimensional geometry.\nB. Space and time decomposition\nUnlike other physical theories fields propagate on a\nfixed spacetime, in relativistic theories of gravity, space-\ntime is a priori unknown and the goal is to deduce its\ngeometry from initial data. In this section, we discuss\nfundamental aspects of the 3 + 1-formalism we use to\ndescribe these theories by evolving geometrical objects,\nclosely following Ref. 2, chapter 10.2.\nFigure 1. A spacetime diagram illustrating the definition of\nthe field ta, the shift vector, Na, and the lapse function, N.\nRecall that we work under the assumption that Mcan\nbe foliated by Cauchy surfaces, Σ t, parameterized by a\nglobal time function, t. We further assume that the nor-\nmal vector nato Σ tis timelike and is normalized accord-\ning to gabnanb=−1. On each Σ t, the spacetime metric\ninduces a Riemannian metric\nhab=gab+nanb. (20)\nThe inverse of this metric is hab=gab+nanb, where gabis\nthe only metric used throughout the text to raise/lower\nindices. Additionally, habna= 0 and ha\nbis a projector\nonto Σ t.\nConsider an arbitrary spacetime vector field va. We\ncan express this field as\nva=v⊥na+va\n∥, (21)\nwhere va\n∥is the part parallel to Σ t. When va=va\n∥, we\ncan think of va, restricted on Σ t, as a vector field on Σ t.\nMore generally, a spacetime tensor τa1···ak\nb1···blis said to be\ntangent to Σ tif\nτa1···ak\nb1···bl=ha1\nc1···hakckhd1\nb1···hdl\nblτc1···ck\nd1···dl. (22)\nLettabe a timelike vector field in Mdefined by\nta∇at= 1. (23)\nThis vector field identifies points on infinitesimally close\nhypersurfaces of constant t, providing a flow of time that\nis used for the evolution. We can decompose tainto its\nnormal and tangential parts as\nta=Nna+Na, (24)\nwhere Nis the lapse function and Na, which is tangen-\ntial, is the shift vector. Relevantly, the fact that tamust\nbe timelike implies that N > 0. Broadly speaking, N\ngives the rate of change of physical time as compared\nwith t. On the other hand, Natells us how the coordi-\nnates are transported from Σ tto Σ t+dt. Fig. 1 illustrates\nthis construction.4\nWe define the derivative operator on Σ t,Da, by\nDeτa1···ak\nb1···bl=ha1\nc1···hakckhd1\nb1···hdl\nblhf\ne∇fτc1···ck\nd1···dl,(25)\nwhere τa1···ak\nb1···blis a tangential tensor. Importantly, one can\nreadily show that this is the only (torsionless) derivative\noperator such that Dahbc= 0 [2]. Moreover, the time\nderivative of any tangential tensor is defined by\n˙τa1···ak\nb1···bl=ha1\nc1···hakckhd1\nb1···hdl\nbl£tτc1···ck\nd1···dl,(26)\nwhich is, by construction, a tangential tensor.\nOne of the dynamical variables we use to describe the\nevolution of spacetime’s geometry is hab; thus, we need\nto calculate its time derivative. We define\nKab:=hc\na∇cnb, (27)\nwhich can be shown to be symmetric. The tensor Kab\nis known as the extrinsic curvature and it describes the\nembedding of Σ tinM. If we use the time derivative on\nhab, we obtain\n˙hab= 2NKab+ 2D(aNb), (28)\nshowing that Kabis related to the time derivative of hab.\nGiven that the equations of motion for UG are second\norder, it is natural to propose that the appropriate initial\ndata for UG should be given by habandKab.\nTo end this subsection, we state some expressions that\nrelate the Riemann tensor associated with gab,Rd\nabc,\nwith objects in Σ t. One can show that the purely tangen-\ntial projection of the Riemann tensor satisfies a Gauss-\nCodazzi relation [2]:\nRd\nabcha\nehb\nfhc\nghj\nd=(3)Rj\nefg+KegKj\nf−KfgKj\ne,(29)\nwhere(3)Rd\nabcis the Riemann tensor associated with hab.\nIn addition,(3)Raband(3)Rrespectively represent the\n3-dimensional Ricci tensor and curvature scalar. Other\nprojections of the Riemann tensor are\nRd\nabcha\nehb\nfhc\ngnd=DeKfg−DfKeg, (30)\nRd\nabcnahb\nehc\nfnd=hb\nehc\nfna∇aKbc−aeaf\n−D(eaf)+Ka\neKaf,(31)\nwhere aa:=nb∇bnais a tangential vector field. Some\nuseful relations concerning the projections and the trace\nof the Ricci tensor are given by\nRcdhc\nahd\nb=(3)Rab+KKab−2Kc\naKcb+N−1˙Kab\n−N−1NcDcKab−2N−1Kc(aDb)Nc\n−aaab−D(aab), (32)\nRcdncnd=−N−1hcd˙Kcd+N−1NcDcK+acac\n+2N−1KcdDcNd+KcdKcd+Dcac,\n(33)\nRbcnbhc\na=DbKb\na−DaK, (34)\nR=(3)R+K2−3KcdKcd+ 2N−1hcd˙Kcd\n−2N−1NcDcK−4N−1KcdDcNd\n−2acac−2Dcac, (35)where K:=Ka\na. Also, we can show that\n˙Kef=Nhb\nehc\nfna∇aKbc+NaDaKef\n+2Ka(eDf)Na+ 2NKa\neKaf. (36)\nWith this result, Eq. (31) can be written as\nRd\nabcnahb\nehc\nfnd=N−1˙Kef−N−1NaDaKef\n−2N−1Ka(eDf)Na−Ka\neKaf\n−aeaf−D(eaf). (37)\nThese are all the technical results we require. In the\nnext section, we classify the UG equations of motion as\nevolution or constraints, and study the Cauchy problem\nfor UG.\nIII. INITIAL VALUE PROBLEM\nThe first task when studying the initial value prob-\nlem of a theory is to identify the constraints. In this\nsection, we classify the UG field equations into evolu-\ntion equations and constraints. When working with a\ngeometrical gravity theory with second-order equations\nof motion, an evolution equation, by definition, contains\ntime derivatives of the extrinsic curvature, which can be\nthought of as second-time derivatives of hab. Conversely,\nan equation with no time derivatives of Kabis a con-\nstraint, which must be imposed on the initial data, and,\nfor consistency, must be kept valid under evolution. No-\ntice that we cannot perform the initial value study for\ngeneric matter fields without specifying the matter ac-\ntion. Thus, in what follows, vacuum UG is considered;\nwe present the sufficient conditions for the matter action\nto respect the well-posedness of UG in Subsec. III C.\nIn most parts of this section, we consider Tab= 0.\nThus, we define E(v)\nabas the tensor that, when it vanishes,\ngives the metric equations of motion, Eq. (9), in the case\nwhere Tab= 0. Importantly, in this part of our study,\nE(v)\nabisnotassumed to be zero throughout M. Instead, a\nweaker assumption is considered: the constraints are only\nvalid on the initial data hypersurface, while the evolution\nequations are satisfied all over M. Interestingly, we find\nthat dynamical consistency implies E(v)\nab= 0 in M.\nFrom the Bianchi identity, we can show that\n∇aE(v)\nab=1\n4∇bR. (38)\nRecall that, since we do not assume that E(v)\nabvanishes\nthroughout M, we cannot claim that its divergence also\nvanishes, and thus, we cannot argue, at this stage, that\nRis constant. Also, from Eq. (10), we can show that\nhabE(v)\nab=E(v)\nabnanb. (39)\nThis last equation allows us to identify the 3-dimensional\ntrace of tangential-tangential projection with the normal-\nnormal projection, rendering the latter redundant; in5\nwhat follows, we omit the normal-normal projection.\nThus, there are only nine independent components of\nE(v)\nab= 0. These components, together with the unimod-\nular constraint, amount to ten equations, which coincides\nwith the number of independent equations of GR.\nWe separate E(v)\nab= 0 into its tangential-tangential\nprojection and its normal-tangential projection, which\nare respectively given by\n0 =(3)Rab+KKab−2Kc\naKcb+N−1˙Kab−aaab\n−N−1NcDcKab−2N−1Kc(aDb)Nc−D(aab)\n−1\n4\u0002(3)R+K2−3KcdKcd+ 2N−1hcd˙Kcd\n−2N−1NcDcK−4N−1KcdDcNd−2acac\n−2Dcac\u0003\nhab, (40)\n0 = DbKb\na−DaK, (41)\nwhere we use Eqs. (32) and (34). The tangential-\ntangential projection of the field equation, Eq. (40), has\ntime derivatives of Kab. Therefore, it is an evolution\nequation. On the other hand, the normal-tangential pro-\njection, Eq. (41), does not contain time derivatives of\nKab, and it is thus a constraint, which is reminiscent of\nthe GR momentum constraint.\nNotice that the unimodular constraint, Eq. (6), is a\nconstraint in the sense that it does not have time deriva-\ntives of Kab. Still, it is not a relation that can be im-\nposed on the initial data and that is automatically satis-\nfied throughout spacetime by the evolution. This is be-\ncause the values of f, which is nondynamical, are given\na priori all over M. However, we will show that the uni-\nmodular constraint can be satisfied by choosing the lapse\nfunction.\nIn summary, in terms of components, six of the field\nequations of vacuum UG, the tangent-tangent projec-\ntions, are evolution equations, while the remaining three,\nthe normal-tangential projections, are constraints. In the\nnext subsection, we analyze the evolution of the con-\nstraints.\nA. Constraint equations\nWe now study if the constraint, Eq. (41), is maintained\nunder evolution. In other words, given initial data on Σ t\nthat satisfies Eq. (41), we must check if the fields ob-\ntained by evolving this initial data still satisfy the con-\nstraint. This can be done by proving that the constraints\non the initial data hypersurface have zero time derivative.\nWithout loss of generality, we take the initial value hy-\npersurface to be Σ 0.\nWe define the tangential tensors\nEab:=hc\nahd\nbE(v)\ncd, (42)\nCa:=DbKb\na−DaK. (43)\nLetA1be the following set of assumptions: the evolution\nequations, Eab= 0, are valid throughout M, and theconstraint, Ca= 0, is only valid on Σ 0. What remains to\ncheck is if, under A1,˙Ca= 0 on Σ 0. From the definition\nof the time derivative, we readily obtain\n˙Ca=Nhd\nane∇eCd+NeDeCa\n+NCeKe\na+CeDaNe. (44)\nGiven that Ca= 0 = DaCbon Σ 0, Eq. (44), when re-\nstricted to Σ 0, takes the form\n˙Ca\f\f\f\nΣ0=Nhb\nanc∇cCb, (45)\nwhich is notautomatically zero under A1.\nLet\nSa:=hb\nanc∇cCb. (46)\nClearly, the time derivative of Cavanishes if Sa= 0. We\ncan show that, under A1,Sa=−DaR/4 on Σ 0:\nProof. We can verify that\nhd\na∇b\u0010\nE(v)\nbchc\nd\u0011\n=DbEab+Eabab−Sa−KCa,(47)\nwhere we use K=∇cnc. Now, using Eq. (38), we obtain\nhd\na∇b\u0010\nE(v)\nbchc\nd\u0011\n=1\n4DaR+Kb\naCb−aaE, (48)\nwhere\nE:=Eabhab\n=1\n4(3)R+1\n4K2+1\n4KabKab−1\n2N−1hab˙Kab\n+1\n2aaaa+1\n2N−1NcDcK+N−1KabDaNb\n+1\n2Daaa. (49)\nCombining Eqs. (47) and (48) produces\nSa=−1\n4DaR−Kb\naCb−KCa+Eabab\n+aaE+DbEab. (50)\nHence, on Σ 0and assuming A1,\nSa|Σ0=−1\n4DaR|Σ0. (51)\nAccording to Dirac’s method [27], it is necessary to\npromote Sa= 0 as a constraint; in Dirac’s terminology,\nSa= 0 is a “secondary constraint.” This, of course, im-\nplies that the 4-dimensional curvature scalar, R, must\nbe constant throughout Σ 0. Namely, R= 4Λ 0, where\nDaΛ0= 0. Notice that we must consider Ras a short-\nhand notation for the right-hand side of Eq. (35), which\nonly contains tangential objects. Also, as the notation6\nsuggests, Λ 0will end up playing the role of the cosmolog-\nical constant. However, at this stage, we can only claim\nthat Λ 0is constant along Σ 0and there is noreason to\nassume that ˙Λ0= 0; dynamical consistency fixes Λ 0to\nbe constant throughout spacetime, as we show next.\nWe now prove that, under A1,˙R= 0 on Σ 0:\nProof. Using the fact that hb\na−nanbis the identity tensor\nin spacetime, we can verify that\nE(v)\nabnb=Ca−naE. (52)\nThe divergence of this last equation produces\n∇a(E(v)\nabnb) =∇aCa−KE −na∇aE. (53)\nAlternatively, we can calculate ∇a(E(v)\nabnb) using the\nLeibniz rule and Eq. (38), producing\n∇a\u0010\nE(v)\nabnb\u0011\n=1\n4na∇aR+E(v)\nabKab−Caaa.(54)\nWhen we compare Eqs. (53) and (54), we obtain\nDaCa+2Caaa−KE−na∇aE=1\n4na∇aR+EabKab,(55)\nwhere we use ∇aCa=DaCa+Caaa. Hence, on Σ 0and\nassuming A1,\nna∇aR|Σ0= 0. (56)\nThis result, together with the fact that DaR= 0 on Σ 0,\nimplies that ˙R= 0 on Σ 0.\nThe lesson from the last proof is that, under A1, Λ0is\nconstant throughout M. Recall that A1is a significantly\nweaker set of assumptions than assuming that E(v)\nab= 0\nthroughout M. We now need to check if ˙Sa= 0 on\nΣ0, assuming A1andSa|Σ0= 0; we refer to this set of\nassumptions by A2. Clearly, under A2, the vanishing of\n˙Sais equivalent to\nhb\nanc∇cSb\f\f\nΣ0= 0, (57)\nwhich we show to hold:\nProof. Using Eq. (50), we get\nhb\nanc∇cSb=−1\n4hb\nanc∇cDbR−Cdhb\nanc∇cKd\nb\n−Kb\naSb−Canb∇bK−KSa\n+adhb\nanc∇cEbd+Eabnc∇cab+aanb∇bE\n+Ehb\nanc∇cab+hb\nanc∇cDdEbd, (58)\nwhere we write nc∇cKabandna∇aKin terms of R=\n4Λ0and other tangential objects using Eqs. (35) and (36).\nNotice that, except for the first term, all the terms in\nEq. (58) are proportional to Ca,Sa,E,Eabor derivativesofEandEab, all of which vanish under A2. Thus, we\nneed to focus on\nhb\nanc∇cDbR=hb\nanc∇c(hd\nb∇dR)\n=hb\na(nc∇chd\nb)∇dR+hb\nanc∇c∇bR\n=aanb∇bR+Da(nb∇bR)−Kb\naDbR,\n(59)\nwhere we repeatedly use the definition of the tangen-\ntial derivative, the Leibniz rule, and the fact that co-\nvariant derivatives acting on scalars commute; this is a\nconsequence of the torsion-free hypothesis (a study of an\nunimodular theory with torsion is presented in Ref. 28).\nWhen we evaluate on Σ 0, the first and last terms in\nEq. (59) can be seen to vanish using R= 4Λ 0. More-\nover, if we take the tangential derivative of Eq. (55), we\ncan write the second term in terms of objects that also\nvanish under A2. With all this, we can show that, under\nA2,hb\nanc∇cSb= 0 on Σ 0.\nRelevantly, the secondary constraint Sa= 0 can be\nwritten in a more familiar way. Starting from R= 4Λ 0,\nwhich, under A2, is equivalent to Sa= 0, and using E= 0\nand Eqs. (35) and (49), it is possible to obtain\n4Λ0= 2(3)R+ 2K2−2KcdKcd, (60)\nwhich has the form of the Hamiltonian constraint of GR\nwith a cosmological constant. However, in vacuum UG,\nEq. (60) appeared for dynamical consistency and not as\na (primary) constraint. The main conclusion of this sub-\nsection is that, for consistency, initial data in vacuum\nUG must be subject to Eqs. (41) and (60). In the next\nsubsection, we verify that the evolution predicted in vac-\nuum UG is unique, continuous, and causal in the above-\ndescribed sense.\nB. Evolution equations\nIn this subsection, we study the evolution equations,\nnamely, the tangential-tangential projection of the vac-\nuum field equations, and the unimodular constraint. We\nhave taken habandKabas the dynamic variables for vac-\nuum UG. However, an approach that better adapts to our\nneeds is the BSSN formulation, developed for GR initially\nby Shibata and Nakamura [29] and later by Baumgarte\nand Shapiro [30]. The BSSN formulation consists of sepa-\nrating the conformal factor for the spatial metric and the\ntrace of the extrinsic curvature, and studying their evo-\nlution separately. Also, we decompose the tensors into\ntheir components on the foliation hypersurfaces, which\nare denoted with Latin indexes i, j, k, . . . . Also, for sim-\nplicity, we take Na= 0, a condition that can be trivially\nrelaxed.\nLet˜hijbe such that\nhij=ψ4˜hij, (61)7\nwhere\nψ=h1/12, (62)\nso that the determinant of ˜hijis˜h= 1 (hthe determinant\nassociated with hij). Notice that\n˜hij=ψ4hij=h1/3hij. (63)\nis the inverse of ˜hij. Moreover, let Aijbe the symmetric\ntensor field defined by\nAij:=Kij−1\n3Khij. (64)\nClearly Aij˜hij= 0, namely, Aijis the traceless part of\nthe extrinsic curvature.\nIn the BSSN formulation the dynamical varialbes are\nϕ,K,˜hab,˜Aaband˜Γa, which are given by\nϕ:= ln ψ=1\n12lnh, (65a)\nK=hijKij, (65b)\n˜hij= e−4ϕhij, (65c)\n˜Aij= e−4ϕAij, (65d)\n˜Γi:=˜hjk˜Γi\njk. (65e)\nThe first expression is a redefinition of the conformal fac-\ntor, the second is the trace of the extrinsic curvature, the\nthird is the conformal transformation given in Eq. (61),\nthe fourth is a rescaling of the traceless part of the extrin-\nsic curvature, and finally, ˜Γiare the conformal connection\nfunctions, where\n˜Γi\njk=1\n2˜hil\u0010\n−∂l˜hjk+∂j˜hkl+∂k˜hlj\u0011\n, (66)\nare the Christoffel symbols associated with ˜hij. Further-\nmore, we can show that Eq. (65e) is equivalent to\n˜Γi=−∂j˜hji. (67)\nWhat remains to be done is to rewrite the constraint and\nevolution equations in terms of the BSSN variables; this\nallows us to argue when comparing with the GR evolution\nequations, that vacuum UG has a well-posed initial value\nformulation.\nUsing Eq. (60) we can cast the evolution equation,\nEq. (40), as\n0 =(3)Rab+KKab−2Kc\naKcb+N−1˙Kab\n−N−1NcDcKab−2N−1Kc(aDb)Nc\n−N−1DaDbN−Λ0hab, (68)\nwhich coincides with the evolution equation of GR with\na cosmological constant. Thus, the system of evolutionequations of vacuum UG, in BSSN variables, takes the\nsame form of the corresponding GR equations, namely,\n∂t˜hij= 2N˜Aij, (69a)\n∂tϕ=1\n6NK, (69b)\n∂t˜Aij= e−4ϕ\u0010\nDiDjN−N(3)Rij\u0011TL\n−\nN\u0010\nK˜Aij−2˜Ak\ni˜Akj\u0011\n, (69c)\n∂tK=DkDkN−N\u0012\n˜Aij˜Aij+1\n3K2−Λ0\u0013\n,(69d)\n∂t˜Γi=−2N\u0012\n˜Γi\njk˜Ajk+ 6˜Aij∂jϕ−2\n3˜hij∂jK\u0013\n+2˜Aij∂jN. (69e)\nGiven that GR with a cosmological constant is strongly\nhyperbolic [31], then, it follows from our analysis that the\nevolution equations of vacuum UG are also strongly hy-\nperbolic, and thus, this theory has a well-posed evolution\nin the sense of being unique, continuous, and causal.\nWe turn now to discuss the unimodular constraint.\nUsing the well-known expression√−g=N√\nhand\nEq. (65a), we get√−g=Ne6ϕ. On the other hand,\nthe unimodular constraint is√−g=f. By direct com-\nparison, we can conclude that the unimodular constraint\nis satisfied as long as\nN=fe−6ϕ. (70)\nNotably, both sides of Eq. (70) must be positive. More-\nover, the fact that the unimodular constraint can be\nsolved by simply choosing the lapse function is compati-\nble with the result of Ref. [32] where it is noted that one\ncomponent of the metric is sufficient to solve the unimod-\nular constraint.\nWith this analysis, we shown that vacuum UG has an\nevolution that is unique, continuous, and causal. How-\never, there are cases where the initial data must be given\non different charts covering Σ 0. In this case, a “gluing”\nof the different “evolutions” must be made. Fortunately,\nthis procedure can be carried out in the same manner as\nin GR [2], since this procedure relies on making coordi-\nnate transformations, which are available in UG. Like-\nwise, there is no obstruction to applying the GR proof\n[3] that shows that there is a maximal evolution of the\ninitial data. Therefore, we can conclude that vacuum UG\nhas a well-posed initial value formulation which gives rise\nto a maximal evolution of the initial data. In the next\nsubsection, we discuss UG with matter.\nC. Initial value formulation with matter\nThe initial value formulation of a UG theory with mat-\nter can only be studied once the relevant matter action\nis given. Still, we provide a set of sufficient conditions to8\nhave a well-posed initial value formulation for a UG the-\nory with a generic matter action. Of course, the methods\nwe present can be used for a particular matter action.\nThe metric field equation is equivalent to Eab= 0, as\nin Eq. (9). Again, we do notassume that Eabvanishes\ninM, but only a weaker assumption: the constraints\nvanish on Σ 0and the evolution equations vanish in M.\nImportantly, we can check that\n∇aEab=1\n4∇b(R−4κQ+κT). (71)\nwhere we use Eq. (19).\nThe tangential-tangential projection and normal-\ntangential projection of Eab= 0 are respectively given\nby\n0 =Eab−κ\u0012\nTcdhc\nahd\nb−1\n4Thab\u0013\n, (72)\n0 = Ca−κTbcnbhc\na, (73)\nwhich are tangential tensors. Recall that, since Eabgab=\n0, the normal-normal projection of Eab= 0 can be ob-\ntained from Eq. (72).\nWe know, from the vacuum UG analysis, that Eq. (72)\nis an evolution equation. On the other hand, the normal-\ntangential projection, Eq. (73), is a constraint provided\nthat Tbcnbhc\nacan be written in such a way that it does\nnot contain second time derivatives of habor the matter\nfields. Assuming this is the case, we find the conditions\nnecessary for the constraint to remain valid under evolu-\ntion. This analysis can be done as in Subsec. III A, the\nmain difference is that now 4Λ 0=R−4κQ+κTis con-\nstant, as expected from Eq. (71). The conclusion is that,\nto have dynamical consistency, the initial data must be\nsubject to\nDbKb\na−DaK=κTbcnbhc\na, (74)\n(3)R+K2−KabKab= 2( κTabnanb+ Λ 0+κQ),(75)\nwhere, to write Eq. (75) in this form, we take the trace of\nEq. (72) and we use that T=Tabhab−Tabnanb. Equation\n(75) looks like the Hamiltonian constraint with matter\nand a cosmologican constant but it depends on Q, which,\nrecall, is closely related to ∇aTab; this is an important\ndifference when comparing with GR.\nWe can study the well-posedness of the evolution equa-\ntions with matter in a simple way. Mimicking the analysis\nof vacuum UG in terms of BSSN variables, we can show\nthat the UG evolution equations have the same form as\nthose of GR with matter and a cosmological constant,\nbut in this case Λ 0+κTabnanb+κQplays the role of the\ncosmological constant. Now, it is known that GR with\nmatter and a cosmological constant has a well-posed ini-\ntial value formulation, in a weak hyperbolic sense [33, 34],\nprovided that Tabonly depends on the fields and its\nfirst derivatives and that the matter equations of mo-\ntion, Eq. (7), are also well-posed by themselves. To prove\nthis result one can use the fact that the evolution equa-\ntions are quasilinear, diagonal, second-order hyperbolic(QDSOH), and apply Leray’s theorem [35], as done in\nRef. 2. We can thus conclude that UG has a well-posed\ninitial value formulation as long as neither TabnorQ,\nwhich also appears in the dynamical equations, have sec-\nond derivatives of the fields, and the matter field equa-\ntions are well posed. Observe that this restriction on the\nenergy-momentum tensor is enough for Eq. (73) to be\na constraint. Of course, these are sufficient conditions;\nthere could be cases that do not meet this hypothesis and\nstill have a well-posed initial value formulation.\nIV. CONCLUSIONS\nOne of the most important properties of any theory\nis its capacity to make predictions out of initial data.\nTherefore, if UG is to be considered as a viable physical\ntheory, it needs to have a well-posed initial value formu-\nlation. Moreover, to utilize this theory, we need to find\nall the constraints. Here, we show that UG is well-posed\nand we find all the constraints. For the latter, we fol-\nlowed the well-known Dirac method; the result is that\nthere are primary constraints, Ca= 0, and secondary\nconstraints, Sa= 0. Then, we used these constraints in\nthe evolution equations to cast them in the form of those\nin GR, which have a well-posed initial value formulation,\ncompleting the proposed analysis.\nRemarkably, the unimodular constraint does not be-\nhave as a constraint in the sense that it is not imposed\non the initial data and preserved under evolution. This\nfeature ought to be present in any theory with nondy-\nnamical fields, like the gravitational sector of the Stan-\ndard Model Extension [36]. In the present case, the uni-\nmodular constraint can always be satisfied by a choice of\nthe lapse function, shedding some light on the role of the\nUG nondynamical function.\nWe also found interesting that the equivalence between\nGR and vacuum UG, in the context of an initial value\nproblem, is not a priori obvious even when Qis con-\nstant. Interestingly, only by requiring dynamical consis-\ntency does a spacetime constant emerge (e.g., R= 4Λ 0in\nvacuum) that can be used to show this equivalence. Still,\nUG can be a suitable test theory to perform numerical\ncomputation of modified geometrical gravity theories. In\naddition, the results we obtain suggest that one can run\nGR numerical calculations using only the traceless part\nof the Einstein equations. Of course, the relevant case\nwhen looking for new physical phenomena is to allow for\n∇aTab̸= 0.\nFinally, UG has more constraints than the 4 con-\nstraints of GR. This enhanced number should be related\nto the reduced symmetries of UG [37], but a detailed\ncounting of constraints is not direct [38]. For that, we\nwould need to use Hamiltonian methods, which have been\nused to study the problem of time in UG [39, 40]. Un-\nfortunately, other studies of theories with nondynamical\nfields [41, 42] suggest that a full Hamiltonian analysis\nshould not be straightforward.9\nACKNOWLEDGMENTS\nWe acknowledge getting valuable feedback from M.\nChernicoff, U. Nucamendi, N. Ortiz, M. Salgado, O. Sar-\nbach, and D. Sudarsky. This project used financial sup-\nport from CONAHCyT FORDECYT-PRONACES grant\n140630 and UNAM DGAPA-PAPIIT grant IN101724.\n[1] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).\n[2] R. M. Wald, General Relativity (Chicago Univ. Press,\n1984).\n[3] Y. Choquet-Bruhat and R. Geroch, Comm. Math. Phys.\n14, 329 (1969).\n[4] M. Salgado, Class. Quantum Grav. 23, 4719 (2006).\n[5] A. D. Rendall, Class. Quantum Grav. 23, 1557 (2006).\n[6] O. Sarbach, E. Barausse, and J. A. Preciado-L´ opez,\nClass. Quantum Grav. 36, 165007 (2019).\n[7] A. D. Kov´ acs, Phys. Rev. D 100, 024005 (2019).\n[8] A. D. Kov´ acs and H. S. Reall, Phys. Rev. D 101, 124003\n(2020).\n[9] L. A. Sal´ o, K. Clough, and P. 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Bonder, in Proceedings of the Ninth Meeting on CPT\nand Lorentz Symmetry (World Scientific, 2023) p. 176."    },    {        "title": "2402.00146v2.Collapse_and_expansion_kinetics_of_a_single_polyelectrolyte_chain_with_hydrodynamic_interactions.pdf",        "content": "Collapse and expansion kinetics of a single polyelectrolyte chain with\nhydrodynamic interactions\nJiaxing Yuan1and Tine Curk2,a)\n1)Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba,\nMeguro-ku, Tokyo 153-8904, Japan\n2)Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland 21218,\nUSA\nWe investigate the collapse and expansion dynamics of a linear polyelectrolyte (PE) with hydrodynamic\ninteractions. Using dissipative particle dynamics with a bead-spring PE model, long-range electrostatics\nand explicit ions we examine how the timescales of collapse tcoland expansion texpdepend on the chain\nlength N, and obtain scaling relationships tcol∼Nαandtexp∼Nβ. For neutral polymers, we derive values\nofα= 0.94±0.01 and β= 1.97±0.10. Interestingly, the introduction of electrostatic interaction markedly\nshifts αtoα≈1.4±0.1 for salt concentrations within c= 10−4M to 10−2M. A reduction in ion-to-monomer\nsize ratio noticeably reduces α. On the other hand, the expansion scaling remains approximately constant,\nβ≈2, regardless of salt concentration or ion size considered. We find β > α for all conditions considered,\nimplying that expansion is always slower than collapse in the limit of long polymers. This asymmetry is\nexplained by distinct kinetic pathways of collapse and expansion processes.\nPolyelectrolytes (PE) are charged polymers that con-\nstitute a class of materials with profound significance in\nbiological systems (such as proteins, DNA, and RNA)\nand diverse industrial applications.1,2A substantial body\nof theoretical and computational research has been ded-\nicated to unraveling the structural attributes of these\npolymers1,3and gaining insights into their expansion-\ncollapse transition.4–6In particular, the non-equilibrium\nkinetics governing the PE expansion-collapse transition\nholds vital implications for elucidating the folding dy-\nnamics of biomolecules7–9and for designing responsive\nsmart materials.10,11\nThe non-equilibrium phase ordering kinetics of soft\nmaterials can be significantly influenced by hydrody-\nnamic interactions (HI) induced by solvent flow.12–16\nFor a neutral polymer in a solvent a collapse can be\ninduced by a change in the solvent quality, in which\ncase HI were shown to accelerate the collapse kinetics\ncompared to Brownian dynamics (BD) simulations.17–20\nStarting from the fully expanded chain Kikuchi et al.\nprovided an analytical prediction asserting that the col-\nlapse timescale tcolfollows a power-law relationship,\ntcol∼N4/3, with Nrepresenting the chain length.19\nGuo et al. employed dissipative particle dynamics (DPD)\nsimulations and reported a different scaling behavior,\ntcol∼N0.98±0.09, if the initial condition is an equilibrated\npolymer.\nHowever, it is not clear how the presence of electro-\nstatic interactions impacts the scaling relationship of\nPE collapse. The full hydrodynamic simulation of the\nexpansion–collapse kinetics of charged PE is complex due\nto the necessity of properly accounting for long-range and\nmany-body electrostatic interactions as well as HI. Such\ninvestigations are relatively rare with only one recent\na)Electronic mail: tcurk@jhu.educomputational study21delving into PE collapse kinetics\nusing fluid particle dynamics method,22which revealed\nthat HI significantly speeds up PE collapse but did not\nexplore how HI changes the scaling behavior. Moreover,\nthe dependence of expansion timescale texp, the reverse\nprocess of collapse, on the chain length Nhas to our\nknowledge not been investigated.\nIn this work, we conduct DPD simulations of a coarse-\ngrained (CG) bead-spring PE23,24to investigate how the\ncollapse time tcoland expansion time texpscales with\nthe chain length N. Our CG model comprises an an-\nionic PE, monovalent counterions, and additional mono-\nvalent salt at concentration ccontained in a cubic three-\ndimensional periodic box of size L. The PE is represented\nas a bead-spring chain23,24composed of Nmonomers,\neach carrying a charge of −e. The PE monomers and\nions are treated as spherical particles with diameter σ\nandσs, respectively. We primarily focus on the case of\nσ=σs, but we also explore the impact of a smaller\nion size ( σs= 0.5σ). The particles interact through\na standard 12-6 Lennard-Jones (LJ) potential charac-\nterized by an energy coupling constant εand a cutoff\ndistance rcut. We choose rcut= 3σto represent the\nshort-range hydrophobic attraction between monomers,\nwhereas rcut= 21/6(σ+σs)/2 and rcut= 21/6σsare\nset for purely repulsive monomer–ion and ion–ion LJ\ninteraction. The electrostatic coupling is controlled by\nthe Bjerrum length lBgiven by lB=e2/(4πkBTεsol)\nwhere εsolrepresents the permittivity of the solvent, kB\nis Boltzmann’s constant, and Tis the absolute tem-\nperature. We set σ=lB= 0.72 nm to represent a\ntypical monomer size and electrostatic coupling in an\naqueous electrolyte at room temperature.24,25Neighbor-\ning monomers along a chain are connected through har-\nmonic potential Ubond(rij) = ( K/2)(rij−R0)2with\nspring constant K= 400 kBT/σ2and bond length R0=\n21/6σ, where rijrepresents the center-to-center distance\nbetween monomer iandj. Electrostatic interactions\n1arXiv:2402.00146v2  [cond-mat.soft]  4 Apr 2024are calculated using the particle-particle-particle-mesh\n(PPPM) algorithm with a relative force accuracy of\n10−3.26,27Each simulation contains a single polymer\nchain in a box size of L= 120 σforN≤100 and\nL= 180 σforN= 200, which is sufficiently large to\navoid polymer–polymer interaction through periodic im-\nages (see Table S1 and Fig. S1 in Supplemental Informa-\ntion (SI) for details).\nTo account for hydrodynamic interactions, we employ\nthe computational method based on dissipative particle\ndynamics (DPD)28that couples the polymers to the DPD\nsolvent (DPDS).29The monomers and ions are immersed\nin a DPD solvent characterized by parameters typical\nfor an aqueous solution: particle density ρ= 3r−3\nc, cut-\noff distance rc=λ= 0.646 nm, friction parameter\nγ= 4.5kBTτ/r2\nc, with τ=λp\nm/(kBT) the standard\nmolecular dynamics simulation time unit, mthe mass of\nthe particles, and interaction prefactor aij= 78kBT; see\nRef. 29 for more details). The coupling between the so-\nlute particles (monomers and ions) and the DPD solvent\nis achieved by friction parameter γs= 5γand cutoff dis-\ntance rs=rc. This setting captures the hydrodynamic\ninteractions and yields the typical diffusion constant of\nions and monomers D≈0.07λ2/τ≈1.1 nm2/ns (τ≈\n0.027ns). The system is evolved using the velocity-Verlet\nintegrator with a time step of ∆ t= 0.005τ. When plot-\nting the results, we scale the time in terms of the Brow-\nnian time for a free particle τBD=σ2/(24D)≈0.74τ.\nWe first perform equilibrium simulations to prepare\ninitial equilibrated configurations of the polymer. When\ninvestigating the collapse process, we initially employ\npurely repulsive LJ interactions between the polymer\nsegments ( rcut= 21/6σandε= 3kBT) while incor-\nporating full electrostatic interactions. To initiate the\ncollapse, we introduce an attractive LJ interaction be-\ntween the monomers by setting rcut= 3σand keeping\nε= 3kBT. This attractive interaction is activated in-\nstantaneously, leading to a sudden quench that drives\nthe polymer collapse. Conversely, when studying expan-\nsion, we follow the opposite protocol. The equilibrium\nstructures are generated using attractive LJ interactions\n(rcut= 3σ) with ε= 3kBT. Subsequently, we change the\nmonomer–monomer LJ interaction to a purely repulsive\n(rcut= 21/6σ) to investigate the expansion kinetics.\nUsing this setup, we examine the collapse and expan-\nsion dynamics of both neutral polymer and charged PE\nat different salt concentrations c= 10−4M (Debye length\nlD≈30.4nm), 10−3M (lD≈9.61nm), and 10−2M\n(lD≈3.04nm). The temporal change of Rgas a function\nof elapsed time tfor the collapse and expansion of a single\nneutral polymer and a charged PE at salt concentration\nc= 10−3M are shown in Fig. 1 and Fig. 2, respectively.\nThe raw data of PE collapse and expansion at different\nsalt concentrations c= 10−4M and c= 10−2M are pro-\nvided in Fig. S2 in SI. Comparing PE with a neutral poly-\nmer, we observe that the electrostatic repulsion between\nmonomers significantly slows the collapse kinetics, while\nexpansion is accelerated. This effect becomes particu-larly important when the salt concentration is low since\nelectrostatic repulsion is not effectively screened. More-\nover, the equilibrium configurations depend sensitively\non the electrostatic repulsion and salt concentrations c:\nneutral polymer forms a globule, but the PE forms an\nelongated rod or a bead-spring configuration with larger\nRgat lower c, in agreement with previous studies.30,31\nThis change in equilibrium Rgsignificantly affects the\ncollapse and expansion timescales in addition to direct\nelectrostatic interactions.\nFor these simulations (Fig. 2) we employed the stan-\ndard choice with salt ions of the same size as the\nmonomers, σs=σ. However, ion size can impact ion\nbinding, as it determines the closest distance between\nions and monomers. In Fig. 3, we explore the influ-\nence of ion size. It is evident that a smaller ion size\n(σs= 0.5σ) yields more compact structures in the col-\nlapsed PE due to stronger electrostatic attraction be-\ntween ions and monomers, consistent with previous sim-\nulations of PE.31–33Compared to PE collapse where ions\nand monomers have identical sizes (Fig. S1(b)), smaller\nion size substantially accelerates the collapse since elec-\ntrostatic repulsion between monomers is screened more\neffectively. Moreover, smaller ions can more easily incor-\nporate into the globular polymer thus yielding compact\nequilibrium globular structures that are similar to neu-\ntral polymer configurations (compare configurations in\nFig. 3b with Fig. S1(d)and Fig. 1b)\nBased on the time evolution of Rgin Fig. 1–Fig. 3, we\ncalculate how the collapse and expansion timescale de-\npend on the chain length N. The collapse and expansion\ntimescale tcolandtexpare defined by the time t∗required\nfor the change in the radius of gyration Rg(t) to reach a\nfraction fcof the maximum change,19\nRg(t∗) =Rg,init−fc(Rg,init−Rg,final), (1)\nwhere Rg,initandRg,finalare, respectively, the initial and\nthe final equilibrated values of Rg. In the present work,\nwe set fc= 0.9 and fit the obtained timescales to a power-\nlaw scaling ( tcol∼Nαandtexp∼Nβ) using the non-\nlinear Levenberg-Marquardt algorithm34that takes into\naccount the error bars when fitting the data. We employ\nthe implementation of this algorithm in Gnuplot.35For\nthe collapse process, we conduct 15 independent simula-\ntions. Conversely, for the expansion process, owing to the\nabsence of driving force and larger variance, we conduct\na range of 25 to 35 independent runs. This enables us\nto acquire high-resolution data concerning Rg(t) and its\nassociated timescale. Note the choice of fc= 0.9 ensures\nthe extracted timescales are not affected by the slow local\narrangements of the collapsed globule in the late stage.19\nThe scaling behaviors of collapse and expansion\ntimescales are summarized in Fig. 4. For a neutral\npolymer we obtain α= 0.94±0.01 (Fig. 4(a)) and\nβ= 1.97±0.10 (Fig. 4(e)). This agrees with previ-\nous reported simulation results36of neutral polymer col-\nlapse starting from an initially equilibrated conformation,\ntcol∼N0.98±0.09.\n2(a)neutral polymer collapse(b)neutral polymer expansion\n110100100010000051015t [τBD]Rg [σ]N=10N=20N=40N=100N=200110100100010000100000051015t [τBD]Rg [σ]\nFIG. 1. The temporal change of Rgas a function of elapsed time tfor a single neutral polymer of different chain lengths N\nundergoing (a) collapse and (b) expansion. The dashed lines show the equilibrium collapsed (a) and expanded (b) Rg. The\ninsets in (b) show the equilibrium collapsed and expanded polymer configurations for N= 100. Error bars denote standard\nerrors obtained from 10 independent simulations. Error bars in (a) are smaller than the symbol size.\n110100100010000110100t [BD]Rg [](a)c=10-3 M(b)c=10-3 M\n110100100010000110100t [BD]Rg []\nFIG. 2. The temporal change of Rgfor a single PE of various chain lengths Nundergoing collapse (a) and expansion (b) at salt\nconcentration c= 10−3M. The insets in (b) show the representative equilibrated collapsed and expanded PE configurations for\nN= 100. Salt ions are not shown for clarity. The monomer size and ion size are identical ( σs=σ). Error bars show standard\nerrors, for most points they are smaller than the symbol size.\n(a)c=10-2 M (small ion size)(b)c=10-2 M (small ion size)\n1101001000100000.1110100t [BD]Rg []\n1101001000100000.1110100t [τBD]Rg [σ]N=20N=40N=100N=200N=10\nFIG. 3. The temporal change of Rgfor a single PE undergoing (a) collapse and (b) expansion at c= 10−2Mand smaller ion\nsize ( σs= 0.5σ). The insets in (b) show the collapsed and expanded PE for N= 100.\n3Notably, we find the collapse scaling for PE is α≈\n1.4±0.1 (Fig. 4b–d) for different salt concentrations, indi-\ncating that the incorporation of electrostatic interactions\nnot only shifts the absolute values of timescales but also\nmarkedly changes the scaling behavior compared to the\nneutral polymer case. We speculate that α≈1.4±0.1\nscaling may be a result of the initial state resembling a\nfully expanded polymer with Rg∼Ndue to electrostatic\nrepulsion between segments. Consequently, α≈1.4±0.1\nis close to the theoretical prediction of α= 4/3 for\nneutral polymer collapse starting from a fully expanded\nstate.19We speculate that scaling could change to lower\nvalues of αfor very long polymers N > 200 and high salt\nconcentration c > 10−2M. Conversely, the expansion\nscaling does not change appreciably with the introduc-\ntion of electrostatics and we find β≈2 (Fig. 4f–h) for\nthe charged PE under different salt concentrations.\nIn the case of smaller ion-to-monomer size ratio,\nσs/σ= 0.5, we obtain α= 1.15±0.04 at c= 10−2M\n(Fig. 4a), which is closer to the corresponding values\nfor neutral polymers, while the expansion scaling re-\nmains unaffected within statistical accuracy of our re-\nsults, β= 1.94±0.09 (Fig. 4e). We attribute the reduced\nαto enhanced screening and the ability of small ions to be\nincorporated into the globule, enabling a charge-neutral\nglobule configuration. In contrast, with larger ion-to-\nmonomer size ratio ( σs/σ= 1), the ions cannot incorpo-\nrate into a collapsed globule, leading to more expanded,\npearl-necklace-like equilibrium structures. These find-\nings suggests that the collapse scaling is sensitive to the\nshort-range ion–polymer interaction and whether ions\ncan incorporate in, or are excluded from, the collapsed\npolymer globule. We anticipate that the scaling exponent\nαvaries continuously with the size ratio σs/σand that by\nfurther decreasing the size ratio, the strengthened screen-\ning would result in αcloser to that of a neutral polymer\n(α≈1).\nInterestingly, we find β > α for all conditions inves-\ntigated, implying that expansion is always slower than\ncollapse for long polymers. This intriguing observation\ncan be understood by considering the kinetic pathways\nfor collapse and expansion (Fig. 5a). The pathway is\nquantified by a Rg–ncparametric plot where ncis the\nnumber of contacts per monomer defined as the average\nnumber of neighbors within a distance 1 .5σ(Fig. 5b).\nNotably, we find that a closed circle emerges confirming\nthat collapse and expansion follow distinct kinetic path-\nways. The collapse behavior resembles a shrinking pro-\ncess along the backbone where small pearls form whose\nhydrodynamic drag is proportional to the pearl diameter\n(N1/3) and in addition HI induce directional flow acceler-\nating the merging of local pearls.19,21Conversely, during\nexpansion the coil expands isotropically where the total\ndrag scales as the sum over monomers ( N) and there is no\ndirectional flow along the backbone, resulting in a longer\ntimescale compared to the collapse.\nFurthermore, while collapse scaling is substantially af-\nfected by electrostatics and ion–polymer interaction, theexpansion scaling remains β≈2 regardless of polymer\ncharge, salt concentration and ion size (Fig. 4e–h). This\nsuggests that the expansion scaling is governed by the\ndiffusive expansion of the polymer and electrostatic re-\npulsion does not appear to noticeably alter this scaling,\neven at very low salt concentrations. We analyze the\nRg–ncplot for the collapse and expansion of PE under\nsalt concentrations of c= 10−4M (Fig. 5c, top) and\nc= 10−2M (Fig. 5d, top). For expansion, we observe\na two-step process: initially, the coil experiences slight\nswelling, during which the number of contacts per par-\nticlencis significantly reduced. Subsequently, diffusion-\ndriven expansion occurs, constituting the rate-limiting\nstage, which is only weakly influenced by electrostatics\ndue to the relatively small monomer contacts nc. Con-\nversely, nccontinues to rise during the entire collapse\nprocess, which underscores the critical role of electro-\nstatics on the collapse. The structural analysis of radial\ndensity distribution for polymer and charged PE during\ncollapse and expansion collectively support the above sce-\nnario (Fig. S3 in SI). These findings also align with the\nevolution of electrostatic energy Vele(Fig. 5c–d, bottom)\nwhere Velereaches saturation much earlier in the expan-\nsion compared to the collapse. Note Veleis defined for\nthe entire system encompassing monomers, counterions,\nand salt ions and its value depends on the relative num-\nber of ions to monomers in the system. Veleis positive\nin the low-salt regime, which is dominated by monomer–\nmonomer repulsion, but it becomes negative in the high\nsalt case where electrostatics is dominated by ion–ion at-\ntractive interactions. The meaningful aspect lies in the\ntemporal change of Veleduring the collapse/expansion\nprocess, which is related to the conformational change\nof the PE. The different kinetic pathways of collapse and\nexpansion can explain why the collapse scaling αis sensi-\ntive to electrostatics, while the expansion scaling, β≈2,\nis not.\nFollowing previous works on polymer collapse,17–21,37\nwe immediately quench the system to initiate collapse\nor expansion. For a typical real PE with chain length\nN≈104, based on the predicted scaling relationships\ntcol∼Nαandtexp∼Nβ, we infer that the timescales\nfor collapse and expansion under the salt concentration of\nc= 10−2M are tcol≈13µsandtexp≈1350µs. Thus, our\nwork should be relevant to experimental conditions where\nthe rapid quenching into a new regime can be achieved\nwithin microseconds.\nTo summarize, we systematically examined the col-\nlapse and expansion timescales of neutral polymers and\ncharged polyelectrolytes with long-range electrostatics,\nexplicit salt ions and hydrodynamic interactions. Our\nfindings illustrate that the inclusion of electrostatic in-\nteractions alters the scaling behaviors tcol∼Nαof col-\nlapse, while it does not noticeably affect the expansion\nscaling texp∼Nβ. Specifically, in neutral polymer sys-\ntems, we obtain α= 0.94±0.01, consistent with the\nprevious prediction36andβ= 1.97±0.10. The inclusion\nof electrostatic interactions significantly shifts the value\n41101001000101102103104\nNtcol [τBD]c=10-2mol/Lslope:1.38±0.07(a)(b)\n1101001000101102103104\nNtcol [τBD]neutral slope:0.94±0.01slope:1.15±0.04c=10-2mol/L(small ion)\n1101001000101102103104\nNtcol [τBD]c=10-3mol/Lslope:1.42±0.081101001000101102103104\nNtcol [τBD]slope:1.38±0.07c=10-4mol/L(c)(d)1101001000101102103104\nNtexp [τBD]c=10-2mol/Lslope:2.09±0.05(e)(f)\n1101001000101102103104105\nNtexp [τBD]neutralc=10-2mol/L(small ion)slope:1.97±0.10slope:1.94±0.09(g)(h)\n1101001000101102103104\nNtexp [τBD]c=10-3mol/L\nslope:2.14±0.071101001000101102103104\nNtexp [τBD]c=10-4mol/Lslope:1.91±0.05collapseexpansionFIG. 4. Collapse and expansion timescales of charged polymer and neutral polymer calculated by applying Eq. (1) to data\nin Figs. 1–3. The exponents for collapse αand expansion βare obtained by fitting the data to tcol∝Nαandtexp∝Nβ\nrespectively.\n110100100010000-2.0-1.5-1.0-0.50.0\nt [BD]Vele [kBT]collapseexpansion(b)neutral polymer\n(d)charged PE (N=100; c=10-2M)(c)charged PE (N=100; c=10-4M)\n(a)\ncollapse kinetic pathwayexpansion kinetic pathway02468100246810\nRg []ncexpansioncollapse\n101520250246\nRg []ncexpansioncollapse510152002468\nRg []ncexpansioncollapse\n1101001000100000.00.51.01.52.0\nt [BD]Vele [kBT]collapseexpansion\nFIG. 5. Characterization of kinetic pathways in collapse and expansion processes for a polymer and PE with N= 100. (a)\nSnapshots from simulations depicting polymer collapse and expansion kinetic pathways for a neutral polymer. (b) Relationship\nbetween the radius of gyration Rgand the number of contacts per monomer ncduring collapse and expansion of a neutral\npolymer. (c)–(d) Top panels: The same analysis as in panel (b) conducted for the collapse and expansion of a charged PE at\nsalt concentrations of c= 10−4M and c= 10−2M. Bottom panels: The temporal change of the average electrostatic energy\nper charged particle (i.e., PE and salt ions), denoted as Vele, for the collapse and expansion of a PE at salt concentrations of\nc= 10−4M (bottom left) and c= 10−2M (bottom right).\n5of collapse scaling to α≈1.4. Interestingly, changes\nin salt concentration ranging from 10−4M to 10−2M do\nnot have a substantial impact on the scaling relationships\nof both collapse and expansion timescales. However, re-\nducing ion-to-monomer size ratio changes the scaling to\nα≈1.15 at c= 10−2M, which we mainly attribute to\nthe ability of small ions to incorporate into the collapsed\npolymer globule. This highlights that non-electrostatic\nshort-range ion–polymer interaction can substantially in-\nfluence the collapse scaling. The expansion scaling, how-\never, remains constant with β≈2 regardless of the\ncharge, salt concentration or ion size. We find that β > α\nfor all conditions considered implying that expansion is\nalways slower than collapse for long polymers. We eluci-\ndate these observations by contrasting the distinct kinetic\npathways of collapse and expansion (Fig. 5), highlighting\nthat these two processes are not simply opposites of each\nother. Our study provides fundamental insights into the\ncollapse and expansion kinetics of linear polyelectrolytes,\nuseful for rationalizing the dynamics of polyelectrolytes\nin solution and understanding the folding and unfolding\nprocess of charged biopolymers.\nI. 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Janke, “Kinetics\nof polymer collapse: Effect of temperature on cluster\ngrowth and aging,” Soft Matter 13, 1276–1290 (2017).\n7"    },    {        "title": "2402.00189v1.The_clique_number_of_the_exact_distance__t__power_graph__complexity_and_eigenvalue_bounds.pdf",        "content": "arXiv:2402.00189v1  [math.CO]  31 Jan 2024The clique number of the exact distance t-power\ngraph: complexity and eigenvalue bounds\nAida Abiad∗Afrouz Jabal Ameli†Luuk Reijnders‡\nAbstract\nThe exact distance t-power of a graph G,G[♯t], is a graph which has the same\nvertex set as G, with two vertices adjacent in G[♯t]if and only if they are at distance\nexactlytin the original graph G. We study the clique number of this graph, also\nknown as the t-equidistant number. We show that it is NP-hard to determine the\nt-equidistant number of a graph, and that in fact, it is NP-har d to approximate it\nwithinaconstant factor. Wealsoinvestigate howthe t-equidistantnumberrelates to\nanother distance-based graph parameter; the t-independencenumber. In particular,\nwe show how large the gap between both parameters can be. The h ardness results\nmotivate deriving eigenvalue bounds, which compare well ag ainst a known general\nbound. In addition, the tightness of the proposed eigenvalu e bounds is studied.\nKeywords: Exact distance graphpower; Clique number; Complexity; Eigenvalue bounds\n1 Introduction\nA setSof vertices of a graph Gis at-equidistant set ifd(u,v) =tfor allu,v∈S,\nwhereu/\\e}atio\\slash=v. A set S⊆V(G) is anequidistant set if there exists t∈Nsuch that\nSis at-equidistant set; we will say that a t-equidistant set Shasdiameter t, which\nwe will denote by D(S) =t. Thet-equidistant number of a graph Gis the number of\nvertices in a largest t-equidistant set of Gand theequidistant number eq(G) is defined\nby eq(G) = max{eqt(G) : 1≤t≤D(G)}. Astrict equidistant set Sis an equidistant set\nsuch that if Sinduces a clique then |S|= 1; the strict equidistant number eq′(G) is the\nsize of a largest strict equidistant set in G.\nFor a positive integer t, thepower graph ofG, denoted by Gt, is a graph which has\nthe same vertex set as G, with two vertices adjacent in Gtif and only if they are at\n∗a.abiad.monge@tue.nl , Department of Mathematics and Computer Science, Eindhoven Un iversity\nof Technology, The Netherlands\n†a.jabal.ameli@tue.nl , Department of Mathematics and Computer Science, Eindhoven Un iversity\nof Technology, The Netherlands\n‡l.e.r.m.reijnders@student.tue.nl , Department of Mathematics and Computer Science, Eind-\nhoven University of Technology, The Netherlands\n1distance at least tin the original graph G. A similar, but less studied, notion is the exact\ndistance t-powerof a graph G. This is denoted by G[♯t], and is a graph which has the\nsame vertex set as G, with two vertices adjacent in G[♯t]if and only if they are at distance\nexactlytin the original graph G. One of the first explicit definitions of this concept is\ndue to Simi´ c [25], who studied graphs which had an exact distance po wer isomorphic to\ntheir line graph. Other work includes the investigation of graphs tha t are isomorphic to\ntheir exact square [4] and the structure of exact distance power s of products of graphs\n[6]. Also, the notion of exact distance powers often appears in the c ontext of the theory\nof sparse graphs, see [22, Section 11.9]. The main focus in earlier inve stigations of exact\ndistance graphs was on their chromatic number, see for instance [ 16, 23, 11, 19]. Note\nthat a connected graph Ghasχeq(G) = 1 if and only if it is a clique.\nUsing the exact distance- tpower graph we have an alternative definition of the t-\nequidistant number: eqt(G) =ω(G[♯t]). In this work we focus on investigating the clique\nnumber of the exact distance t-power of a graph, a parameter that has been investigated\nbefore. Instances of it are the work by Foucaud et al. [12], who st udied the problem of\nfinding the maximum possible clique number among exact distance t-powers of graphs\nof given maximum degree. Haemers [13] studied a similar parameter re lated to the t-\nequidistant number.\nIn this paper we show that the problem of determining the t-equidistant number is\nNP-hard, and also that it is impossible to approximate the t-equidistant number with a\nconstant factor unless P=NP. We also investigate the relation between the equidistant\nnumber and the well studied t-independence number of a graph G, which is the maximum\nsize of a set of vertices in Gat pairwise distance greater than t. In particular, we focus\non understanding how large the gap between both parameters can be. The hardness\nresults motivate deriving eigenvalue bounds, since these can be com puted in polynomial\ntime. Using the results on the parameters’ relationship and eigenva lue interlacing, we\nderive several eigenvalue bounds on the t-equidistant number. We show that, for several\ninstances, the obtained bounds compare favorably to previously k nown bounds. Finally,\nwe study the tightness of the proposed eigenvalue bounds.\nSome applications of the parameter eqt(G)\nThe parameter of interest, eqt(G), has a direct application in the field of Information\nTheory. Let Aq={0,...,q−1}for an integer q≥2. A non-empty subset C⊂An\nqis\ncalled a q-ary code. If|C|=Nanddis the minimum Hamming distance in C, then\nCis called a ( n,d,N)q-Hamming code . ByAq(n,d) we denote the largest N∈Nfor\nwhich there exists a ( n,d,N)q-Hamming code. If the Hamming distance between any\ncodewords in Cis exactly d, then the code Cis calledequidistant . A quantity of interest\nin the study of q-ary codes is Aq(n,d), which is the largest N∈Nfor which there exists\na (n,d,N)q-Hamming code. A similar quantity exists for equidistant codes specifi cally,\nBq(n,d) is the largest N∈Nfor which there exists an equidistant ( n,d,N)q-Hamming\ncode. Much like the ( t−1)-independence number of the hypercube graph Qnis equal\nA2(n,t)[3],thet-equidistantnumberof Qnisequalto B2(n,t). Hence, thespectralbounds\n2on the equidistant number presented here can be applied in the cont ext of equidistant\ncodes. Equidistant codes have been extensively studied, see for in stance [5, 21], and their\ninterest is a consequence of their application in random network cod ing [18].\nAnotherapplicationisto equidistant subspace codes ,i.e., subspace codeswiththeprop-\nerty that the intersection of any pair of codewords has the same d imension. Equidistant\nsubspace codes were shown to have relevant applications in distribu ted storage in [24].\n2 Complexity of eqt(G)\nWe investigate the complexity of the t-equidistant number, and show that it is NP-hard\nto determine eqt. We do this by constructing auxiliary graphs where edges in the origin al\ngraphs become paths of length t. We remark that the gadget used for this reduction for\nodd values of tresembles the gadget used for the NP-hardness proof of ω(Gt) by Lin and\nSkiena [20, Theorem 5.2]. Finally, we show that it is NP-hard to approxim ate eqtwithin\na constant factor.\nWe begin this section by exploiting some properties of a well-studied fa mily of graphs\nknown as split graphs. Such properties will be useful in particular fo r our reduction when\ntis even.\nDefinition 1 (Split graph) .A graphG= (V,E)is called splitif there exists a partition\nof the vertices into subsets V1,V2such that V1is a clique and V2is an independent set.\nSplit graphs were defined and characterized by F¨ oldes and Hammer [8, 10]. These\ngraphs were also introduced independently by Tyshkevich and Cher nyak [7] under the\nnamepolar graphs .\nLemma 2. Whether or not a graph Gis split can be determined in polynomial time.\nFurthermore if Gis split, we can find a partition of V(G)into a clique and independent\nset in polynomial time.\nProof.Letd1≥d2...≥dnbe the degree sequence of Gand letv1,v2,...,vnbe a labeling\nof vertices such that vihas degree di. Hammer and Simeone [9] showed that a graph is a\nsplit iff for largest index ksuch that dk≥k−1, we have:\nk(k−1)−k/summationdisplay\ni=1di+n/summationdisplay\ni=k+1di= 0.\nFurthermore the authors showed that if Gis split, then (/uniontextk\ni=1vi,/uniontextn\nj=k+1vj) is a partition\nofGinto a clique and an independent set. Note that the above-mentione d process can\nclearly be done in polynomial time.\nTheorem 3. LetG= (V,E)be a graph. For any fixed integer t≥1finding a maximum\nt-equidistant number of Gis an NP-hard problem.\n3Proof.IfGis split, then determining the clique number of Gcan be done in polynomial\ntime. Let ( V1,V2) be a partition of the vertices of Gsuch that V1is a clique and V2is\nan independent set. Note that we can obtain such a partition (if it ex ists) in polynomial\ntime using Lemma 2. Note that as V2is an independent set of nodes any clique of Gcan\ncontain at most one vertex in V2. Furthermore, we already know V1is a clique. Hence if\nV1∪{v}is a clique for any v∈V2then this is the maximum clique and otherwise V1itself\nis the maximum clique of G.\nNow assume Gis not split. We will give a reduction of determining the clique number\nofGto determining the t-equidistant number of a graph H. We split into two cases, t\neven and todd.\n•tis odd: Construct a graph H= (V′,E′) fromGby replacing every edge e∈E\nby a path of length t, that is add t−1 additional vertices into every edge. We\nhenceforth call the vertices in Voriginal vertices, and the vertices in V′\\Vdummy\nvertices. Note that the distance of original vertices uandvisdinGthen their\ndistance is k×tinH.\nWe now want to show that Ghas a clique of size mif and only if Hhas at-\nequidistant set of size m. The first implication follows directly from the definition\nofH. For the second, note that:\n(i) IfU⊂Vis at-equidistant set in H, thenUis a clique in G.\n(ii) IfU⊂V′is at-equidistant set in H, thenUeither consists entirely of original\nvertices, or entirely ofdummy vertices. This is due to the fact that the distance\nof a dummy vertex from an original vertex can not be a multiple of t.\nMeaning we only need to consider the additional case when U⊂V′\\Vis at-\nequidistant set in H. Form= 1 or 2 this is trivial, hence assume |U|=m≥3.\nNow, consider a t-equidistant set in H, for which U⊂V′\\Vand|U|= 3. Consider\ntwo vertices u1,u2∈U. By construction, they must lie on two distinct paths corre-\nsponding to two distinct edges in Gthat share an endpoint, say ( v1,v2),(v1,v3)∈E.\nNow,\nt=d(u1,u2) =dH(u1,v1)+dH(u2,v1) =⇒dH(u1,v1)/\\e}atio\\slash=dH(u2,v1).\nThis is because tis odd. Hence, for the third vertex u3∈Uit cannot be on a path\ncorresponding to an edge incident to v1, as it could never be at distance tfrom both\nu1andu2in this case. Instead, it must be on the path corresponding to the e dge\n(v2,v3). For this to occur, v1v2v3must be a triangle in G, meaning we have a clique\nof size 3 in G.\nFinally, assume in the above that m≥4. Then there exists a fourth vertex u4∈U,\nhowever, just as u3, this vertex must lie on the path corresponding to the edge\n(v2,v3). But then dH(u3,u4)< twhich is a contradiction and hence this case\ncannot occur.\nThusω(G) = eqt(H) and hence we can find ω(G) by computing eqt(H).\n4•tis even: Just as in the odd case, construct a graph H= (V′,E′) fromGby\nreplacing every edge e∈Eby a path of length t, that is add t−1 additional\nvertices into every edge. However, now we additionally connect all t he ”central”\ndummy vertices, i.e. the vertices in the center of the paths of lengt htreplacing\nthe edges. We now want to show that Ghas a clique of size mif and only if H\nhas at-equidistant set of size m. The first implication still follows directly from the\ndefinition of H. For the second we use a few observations:\n(i)U⊆Vis at-equidistant set in Hif and only if Uis a clique in G.\n(ii) Any t-equidistant set U⊆V′inHcan at most have 1 dummy vertex.\n(iii) For any clique U⊆VinG, there exists an edge e∈Esuch that ehas no\nendpoint in U.\nThese are true because:\n(i) Follows directly from the construction of H.\n(ii) Ifv∈V′\\Vis a dummy vertex, then it will be at distance at mostt\n2−1 from\na central dummy vertex. But since this holds for all dummy vertices , and all\ncentral dummy vertices form a clique, we conclude any two dummy ve rtices\nare within distance (t\n2−1)+(t\n2−1)+1 = t−1 from each other, and hence\nno two dummy vertices could be in the same t-equidistant set of H.\n(iii) By assumption, Gis not split and thus if Uis a clique in G, thenV\\Uis not\nan independent set in G. In other words, there exist two vertices u,v∈V\\U\nsuch that ( u,v)∈E.\nFrom assertions (i) and (ii) we find ω(G) = eqt(H) orω(G)+1 = eqt(H). Assertion\n(iii) tells us we are always in the latter case. Indeed, let Ube a clique in Gof size\nω(G), thenUis at-equidistant set in H. Now consider the edge ( v1,v2)∈Esuch\nthat{v1,v2}∩U=∅, andinparticular consider its corresponding dummy vertices in\nH. Denote by wthe dummy vertex in Hfor which d(v1,w) = 1 and d(v2,w) =t−1.\nNow, any vertex in Uwill be at distance exactlyt\n2to some central dummy vertex in\nH, andhence at distancet\n2+1tothecentral dummy vertex corresponding to e. This\nin turn means they are all at distance exactly tfrom the dummy vertex w. Thus\nU∪{w}is at-equidistant set in Hof sizeω(G)+1 and hence ω(G)+1 = eqt(H).\nThis means we can exactly determine ω(G) by computing eqt(H), as we wanted.\nNow since ωis NP-hard, eqtmust be NP-hard as well.\nInfact, inthenextresultweshowthatitisimpossibletoapproximate thet-equidistant\nnumber with a constant factor unless P=NP.\nTheorem 4. For every fixed integer tandǫ≥0, there exists no constant approximation\nfor computing the t-equidistant number of a graph H, wherenis the number of vertices\nofH, unlessP=NP.\n5Proof.For this purpose we show that if such an approximation algorithm exis t then there\nexists a constant approximation for the maximum clique problem wher ekis the number\nof vertices of the input graph. Now to prove use we use a similar argu ment to the proof\nof Theorem 3. Let Gbe the input graph of a maximum clique instance. We first use\nLemma 2 to check whether Gis split or no. If Gis split then we can find its maximum\nclique in polynomial time. Hence we assume that Gis not split. Assume there exists\nac-approximation for eqtfor some constant c >1. Now we use the same reduction as\ndiscussed in the proof of Theorem 3. We distinguish the following case s:\n•tis odd:Note that our reduction constructs an instance HofGsuch that the size\nofHis polynomial in terms of size of G. Furthermore for every feasible solution\nofGwe can compute in polynomial time a feasible solution with the same size\nforHand vice versa. Therefore given a c-approximation for H, we can obtain a\nc-approximation for Gas well.\n•tis even: Again our reduction constructs an instance HofGsuch that the size\nofHis polynomial in terms of size of G. However now for every feasible solution\nofHwe can compute in polynomial time a feasible solution that is at most one\nunit smaller for G. Furthermore we know that ω(G) = eqt(H) + 1. Now we can\nuse thec-approximation algorithm for maximum t-equidistant set to compute a t-\nequidistant set Sof size eqt(H)/c. Then we can use this to find a clique Cof size\nat least eqt(H)/c−1 inG. Observe that eqt(H)/c−1 = (ω(G)+ 1)/c−1. Now\nasGis not split it has at least one edge. Therefore we can compute a clique of size\nmax{2,(ω(G) + 1)/c−1}> ω(G)/2c. Thus this leads to a 2 c-approximation for\nmaximum clique problem.\nHowever under the assumption of P/\\e}atio\\slash=NPthere exists no constant approximation for\nmaximum clique problem ([27]), hence the claim.\n3 The relation between the equidistant number and\nthet-independence number\nA setUof vertices of a graph Gis called a t-independence set ifd(u,v)> tfor all\nu,v∈U. Thet-independence number of a graph G, denoted αt, is the size of the\nlargestt-independence set of G. Clearly, the t-equidistant number is closely related to the\n(t−1)-independence number, but with the underlying set having strict er requirements.\nThe following lemmas make this connection more precise.\nThe following lemmas investigate the relation between the above para meters.\nLemma 5. [26] For any graph Gwith clique number ω(G)we have ω(G) = eq1(G)≤\neq(G)≤max{ω(G),α(G)}. For graphs with diameter two we have the equality eq(G) =\nmax{ω(G),α(G)}.\nA straight-forward extension of Lemma 5, using αt(G), is as follows.\n6Lemma 6. LetGbe a graph. Then, at least one of the following holds\n1.eq(G) = eqi(G), for some i∈[1,t],\n2.eq(G)≤αt(G).\nLemma 7. LetGbe a graph. Then for t,t∗∈N+witht < t∗we haveαt(G)≥eqt∗(G).\nProof.IfUis at∗-equidistant set, then all its vertices are at pairwise distance at lea stt,\nmeaning it is also a t-independent set. Hence αt(G)≥eqt∗(G).\nLemma 8. LetGbe a graph. If t≥D(G), thenαt−1(G) = eqt(G).\nProof.The case t > D(G) is trivial, as in this case no two vertices are far enough apart\nto form a ( t−1)-independent set or t-equidistant set, and hence αt−1(G) = 1 = eqt(G).\nNext ist=D(G).αt−1(G)≥eqt(G) follows from Lemma 7. To prove the converse, let\nUbe a (t−1)-independent set. Then for all u,v∈Uwe haved(u,v)> t−1, additionally,\nsincet=D(G), we have d(u,v)≤D(G) =t. Thusd(u,v) =tfor allu,v∈U, and thus\nUis at-equidistant set. Since this holds for arbitrary U, we conclude αt−1(G)≤eqt(G),\nand thus αt−1(G) = eqt(G).\nLemma 9. LetGbe a graph. Then there exists a t∗∈N+such that for all t < t∗we\nhaveαt(G)≥eq(G).\nProof.By definition, there is a t∗∈N+such that eq( G) = eqt∗(G). The desired inequality\nfollows from applying Lemma 7.\nBy way of the above lemmas, we are able to apply eigenvalue bounds fo r the indepen-\ndence number, such as the ones seen in [1] and its optimization from [2], to the equidistant\nnumber. We will investigate this further in Section 4.\nThanks to Lemma 7, we have an easy way to apply bounds on αt−1to eqt. However, a\nquick look at the two definitions suggests that αt−1can in general be significantly larger\nthan eqt. Thus, next we will investigate how this gap behaves, and we will also lo ok at\nthe extremal cases, i.e. the graphs of given order for which αt−1−eqtis maximal.\nFirst, we define the main quantity we are interested in:\nDefinition 10. Letk≥1,t≥2, we denote the maximum distance between αt−1andeqt\nover all connected graphs of order kby:\nAE(k,t) = max{αt−1(G)−eqt(G) :Ghas order k}.\nIn particular, we will focus on studying\nlim\nk→∞AE(k,t)\nk. (1)\nThefollowingresultshowsthat(1)isentirelydeterminedbyhowfast thet-independence\nnumber grows relative to the amount of vertices.\n7Theorem 11.\nlim\nk→∞AE(k,t)\nk= lim\nk→∞max/braceleftbiggαt−1(G)\nk:Gis of order k/bracerightbigg\n.\nProof.The inequality\nlim\nk→∞AE(k,t)\nk≤lim\nk→∞max/braceleftbiggαt−1(G)\nk:Gis of order k/bracerightbigg\n.\nfollows trivially from αt−1−eqt≤αt−1.\nFor the other inequality, fix some t∈N+. Now choose the graphs G(n,t) such that\nG(n,t) hasnvertices, and\nαt−1(G(n,t)) = max {αt−1(G) :Gis of order n}.\nNext, define the graphs H(m,n,t) by connecting mcopies of G(n,t) in a line, by way of\npathsof length t. This graphhas m(n+t)−tvertices, forthe mcopies of G(n,t) that have\nnvertices, and the m−1 paths of length tconnecting them. Since none of these copies are\nwithin distance t, we can take the union of the largest t-independence sets of the copies of\nG(n,t) as at-independence set of H(m,n,t). Hence, αt−1(H(m,n,t))≥m·αt−1(G(n,t)).\nFurthermore, note that eqt(H(m,n,t)) will be bounded by some constant Cn,twhich\ndoes not depend on m. This is because for any arbitrary vertex v∈V(H(m,n,t)), the\namount of vertices within distance t(and subsequently those at exactly t) does not grow\nas you add more copies of G(n,t) in the way described here.\nNow, fix an nand observe\nlim\nk→∞AE(k,t)\nk≥lim\nm→∞αt−1(H(m,n,t))−eqt(H(m,n,t))\n|V(H(m,n,t))|≥lim\nm→∞m·αt−1(G(n,t))−Cn,t\nm(n+t)−t\n=αt−1(G(n,t))\nn+t\nSince this holds for arbitrary n, it must also hold in the limit.\nlim\nk→∞AE(k,t)\nk≥lim\nn→∞αt−1(G(n,t))\nn+t= lim\nn→∞αt−1(G(n,t))\nn\n= lim\nn→∞max/braceleftbiggαt−1(G)\nn:Gis of order n/bracerightbigg\nwhere this last equality is by definition of G(n,t).\nIn view of Theorem 11, it would be of interest to know how\nlim\nk→∞max/braceleftbiggαt(G)\nk:Gis of order k/bracerightbigg\n(2)\nbehaves. For t= 1, the classical independence number, we know this limit is equal to 1 ,\ntake for instance the star graphs Sn. For larger thowever, this remains an open problem.\n84 Eigenvalue bounds for the ( t-)equidistant number\nWe have seen in Section 2 that the t-equidistant number is hard to compute. Thus, it\nmakes sense to derive bounds for this parameter using eigenvalues , since these can be\ncomputed in polynomial time. In particular, we will investigate how kno wn bounds on\nαtandωcan be used to derive bounds on eqtand eq. In addition, we will introduce a\ncompletely different bound using the distance matrix spectra (Theo rem 22). We will also\nshow graph classes that attain equality for our bounds and show th at our bounds compare\nfavorably to a previous bound on the t-equidistant number.\n4.1 Bounds on eqt\nAs a point of comparison, we will use the following bound on eqt, which is a straight\nforward generalization of a bound on eq2by Foucaud et al. [12, Theorem 2.1].\nProposition 12. LetGbe a graph with maximum degree ∆. Then\neqt(G)≤∆(∆−1)t−1+1.\nProof.For some vertex v∈V, consider the maximum amount of vertices that can be at\ndistance exactly tfromv, this is ∆(∆ −1)t−1. Hence, the size of any t-equidistant set is\nat most ∆(∆ −1)t−1+1.\nAswehaveseeninSection3, wecanrelatetheequidistant number to theindependence\nnumberandthecliquenumber. Therearetwomainwaysofdoingthis. WecanuseLemma\n7 and we can use Lemma 5.\nFirst, we will look at the bounds we can obtain on eqt+1using bounds on αt; and we\nwill do it applying Lemma 7.\nProposition 13 (Inertial-type bound) .LetG= (V,E)be a graph with adjacency eigen-\nvaluesλ1≥λ2≥ ··· ≥ λnand adjacency matrix A. Letp∈Rt[x]with corresponding\nparameters W(p) := max u∈V{(p(A))uu}andw(p) := min u∈V{(p(A))uu}. Then, the (t+1)-\nequidistant number of Gsatisfies the bound\neqt+1(G)≤min{|{i:p(λi)≥w(p)}|,|{i:p(λi)≤W(p)}|}.\nProof.Use Lemma 7, and then bound αtwith [1, Theorem 3.1].\nThe implementation of the Inertial-type bound from Proposition 13 u sing Mixed In-\nteger Linear Programming (MILP) is analogous to the MILP for the s ame bound for αt\n[2]. More details can be found in Section 5.1 of the Appendix.\nProposition14 (Ratio-typebound) .Lett≥1andletGbe a regular graphwith nvertices\nand adjacency eigenvalues λ1≥λ2≥ ··· ≥ λnand adjacency matrix A. Letp∈Rt[x]\nwith corresponding parameters W(p) := max u∈V{(p(A))uu}andλ(p) := min i∈[2,n]{p(λi)},\nand assume p(λ1)> λ(p). Then\neqt+1(G)≤nW(p)−λ(p)\np(λ1)−λ(p).\n9Proof.Use Lemma 7, and then bound αtwith [1, Theorem 3.2].\nMore details on the MILP implementation of this Ratio-type bound can be found in\nSection 5.2 from the Appendix.\nFortheoriginalRatio-typeboundon αt[1, Theorem3.2], thebest choiceofpolynomial\nis known for α2[1, Corollary 3.3] and α3[17]. These polynomials are also the best choice\nfor bounding eq3and eq4respectively. This follows because in Proposition 14 we directly\napply the bound from αt. If a different polynomial were to give a better bound on eqt+1,\nthen it must automatically also give a better bound on αt−1, a contradiction. Thus we\nobtain the following two straight forward corollaries.\nCorollary 15 (Best Ratio-type bound, t= 3).LetGbe ak-regular graph with nvertices\nwith distinct adjacency eigenvalues k=θ0> θ1>···> θdwithd≥2. Letθibe the\nlargest eigenvalue such that θi≤ −1. Then, using p(x) =x2−(θi+θi−1)xin Proposition\n14 gives the following bound:\neq3(G)≤nθ0+θiθi−1\n(θ0−θi)(θ0−θi−1).\nThis is the best possible bound for t= 3that can be obtained from Proposition 14.\nCorollary 16 (Best Ratio-type bound, t= 4).LetGbe ak-regular graph with nvertices\nwith distinct adjacency eigenvalues k=θ0> θ1>···> θd, withd≥3. Letθsbe\nthe largest eigenvalue such that θs≤ −θ2\n0+θ0θd−∆3\nθ0(θd+1), where∆3= max u∈V{(A3)uu}. Let\nb=−(θs+θs−1+θd)andc=θdθs+θdθs−1+θsθs−1. Then, using p(x) =x3+bx2+cx\nin Proposition 14 gives the following bound:\neq4(G)≤n∆3−θ0(θs+θs−1+θd)−θsθs−1θd\n(θ0−θs)(θ0−θs−1)(θ0−θd).\nThis is the best possible bound for t= 4that can be obtained from Proposition 14.\nIn the Appendix we will illustrate the tightness of the Ratio-type bou nd from Propo-\nsition 14.\nNext we show that the bounds from Corollaries 15 and 16 are tight fo r certain John-\nson graphs. A Johnson graph , denoted J(n,k), is a graph where the vertices represent\nthek-element subsets of a set of size n. Two vertices are adjacent if and only if their\ncorresponding subsets differ by exactly 1 element. The graph J(n,k) has/parenleftbign\nk/parenrightbig\nvertices and\neigenvalues {(k−j)(n−k−j)−j:j∈[0,min{k,n−k}]}.\nProposition 17. The bound from Corollary 15 is tight for the Johnson graphs J(n,3),\nn >6, and the bound from Corollary 16 is tight for the Johnson grap hsJ(n,4),n >8.\nProof.First, we will determine the exact value of eqk(J(n,k)) (note that we use eqkhere\ninsteadofeqttohighlightthefactthatitmatchesthe kfromJ(n,k)). Usingthe k-element\nsubset representation of the vertices, we find a set U⊂V(J(n,k)) is ak-equidistant set\nif and only if the subsets are disjoint. From this we find eqk(J(n,k)) =/floorleftbign\nk/floorrightbig\n.\n10Nowwewillshowourboundsmeetthisexactvalue. For J(n,3)wehavetheeigenvalues\n(θ0,θ1,θ2,θ3) = (3n−9,2n−9,n−7,−3). Since n >6, we find θ3is thelargest eigenvalues\nless than −1. By plugging these values into the bound from Corollary 15 we obtain\neq3(J(n,3))≤n\n3,\nwhich of course can be rounded down to/floorleftbign\n3/floorrightbig\n, since eq3is an integer.\nNow we move on to the case eq4(J(n,4)). Here we have eigenvalues ( θ0,θ1,θ2,θ3,θ4) =\n(4n−16,3n−16,2n−8,n−10,−4). Further, we have ∆ 3=k(n−k)(n−2), which can\nbe obtained by a simple counting argument. For Corollary 16 we find θsis the largest\neigenvalues less than n−6. Since n >8, we find θs=θ3=n−10. By plugging these\nvalues into the bound from Corollary 16 we obtain\neq4(J(n,4))≤n\n4,\nwhich can be rounded down to/floorleftbign\n4/floorrightbig\n, since eq4is an integer.\nWe can also make use of the relation eqt(G) =ω(G[♯t]) to extend a bound on the clique\nnumber by Haemers [15, Theorem 3.5].\nProposition 18. [15, Theorem 3.5] Let Gbe a regular graph with adjacency eigenvalues\nλ1≥ ··· ≥λn. Then the clique number of Gsatisfies:\nω(G)≤n1+λ2\nn−λ1+λ2.\nProposition 19. LetGbe a graph such that its exact distance t-power,G[♯t]is regular.\nLetG[♯t]have adjacency eigenvalues β1≥ ··· ≥βn. Then the t-equidistant number of G\nsatisfies\neqt(G)≤n1+β2\nn−β1+β2.\nProof.Apply Proposition 18 to G[♯t], then use the fact that eqt(G) =ω(G[♯t]).\nAnother parameter that Haemers [13] looked at is Φ( G). LetA,B⊂V(G), thenA\nandBare called disconnected if there is no edge between AandB. The quantity Φ( G)\nis defined as the maximum of/radicalbig\n|A||B|whereA,Bare disjoint and disconnected. We can\nrelate Φ( G) to thet-equidistant number and apply [13, Theorem 2.4] to obtain a bound\non eqt(G). First, we state the original bound on Φ( G):\nProposition 20. [13, Theorem 2.4] Let Gbe a graph with Laplacian eigenvalues µ1≤\n··· ≤µn, then\nΦ(G)≤n\n2/parenleftbigg\n1−µ2\nµn/parenrightbigg\n.\n11Proposition 21. LetGbe a graph, and let µ1≤ ··· ≤µnbe the Laplacian eigenvalues\nofG[♯t]. Then\neqt(G)≤n/parenleftbigg\n1−µ2\nµn/parenrightbigg\n+1.\nProof.LetUbe at-equidistant set of size eqt(G). Then, in G[♯t], any two vertices in U\nwill not be adjacent. Hence, if we split UintoU1,U2, such that |Ui| ≥eqt(G)−1\n2, then\nU1,U2are disconnected, and hence\neqt(G)−1\n2≤/radicalbig\n|U1||U2| ≤Φ(G[♯t]).\nFinally, apply Proposition 20 to G[♯t]to obtain the desired bound.\nIt is of interest to investigate how well these bounds induced from t he different pa-\nrameters perform in practice, especially in view of the results from S ection 3 which tell us\nthe gap between αt−1and eqtcan grow very large. For this reason, it is worth considering\nan alternative and more direct approach to derive eigenvalue bound s. Instead of using\ninterlacing on the adjacency matrix (as we did to derive Propositions 13 and 14), in the\nfollowing result (and its corollaries in the next section) we apply the int erlacing method to\nthe distance matrix of a graph. This makes sense, as the adjacenc y eigenvalue interlacing\nmay not fully capture the properties of an equidistant set, wherea s the distance matrix\nmight. While eigenvalue interlacing using the adjacency matrix is a widely used tool, the\ndistance matrix is much less studied when it comes to using eigenvalue in terlacing for\nbounding graph parameters.\nTheorem 22. LetGbe a graph, and let Dbe its distance matrix, with eigenvalues ˜λ1≥\n··· ≥˜λn. Then the t-equidistant number satisfies:\neqt(G)≤/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −t/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1.\nProof.Assume eqt≥2, else the result is trivial. Let Ube at-equidistant set of size eqt.\nThen, we can label the vertices in such a way that Dhas principal eqt×eqtsubmatrix:\nB=t(J−I),\nwhereJis the all 1 matrix and Ithe identity matrix, both of appropriate size. Bhas\nspectrum {(t(eqt−1))[1],(−t)[eqt−1]}. Cauchy interlacing then tells us:\n˜λn−eqt+i≤ −tfori∈[2,eqt].\nHence, we must have at least eqt−1 eigenvalues less than −t, meaning\neqt≤/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −t/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1.\n124.2 Bounds on eq\nUsing the bound on eqtfrom Theorem 22, we can obtain a bound on eq, captured in the\nfollowing corollary.\nCorollary 23. LetGbe a graph, and let Dbe its distance matrix, with eigenvalues\n˜λ1≥ ··· ≥˜λn. Then the equidistant number satisfies:\neq(G)≤/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −1/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1.\nProof.By Theorem 22 we have\neqt(G)≤/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −t/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1≤/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −1/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1\nfor allt. Hence, eq( G) will also be bounded by the above.\nAninteresting consequence ofTheorem 22, isthatifwe knowthebo undfromTheorem\n22 is tight for some t∗, then we can reduce the number of values for tthat need to be\nconsidered when determining eq. This is illustrated in the following resu lt.\nCorollary 24. LetGbe a graph, and let Dbe its distance matrix, with eigenvalues\n˜λ1≥ ··· ≥˜λn. Then, if there exists a t∗∈[1,D(G)]such that\neqt∗=/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −t∗/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1,\nthen\neq(G) = max{eqt(G) : 1≤t≤t∗}.\nProof.Fort > t∗, we have\neqt≤/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −t/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1≤/vextendsingle/vextendsingle/vextendsingle/braceleftBig\ni:˜λi≤ −t∗/bracerightBig/vextendsingle/vextendsingle/vextendsingle+1 = eqt∗.\nHence\neq(G) = max{eqt(G) : 1≤t≤D(G)}= max{eqt(G) : 1≤t≤t∗}.\nFor obtaining the next bound on eq we will use Lemma 5 combined with a b ound on\nαand a bound on ω.\nTheorem 25 (Ratio bound, unpublished, see e.g. [14]) .LetGbe a regular graph with n\nvertices and adjacency eigenvalues λ1≥ ··· ≥ λn. Then the independence number of G\nsatisfies:\nα(G)≤n−λn\nλ1−λn.\nFor bounding ωwe will use the earlier stated result by Haemers (Proposition 18).\nCorollary 26. LetGbe a regular graph with adjacency eigenvalues λ1≥ ··· ≥λn. Then\nthe equidistant number of Gsatisfies:\neq(G)≤n·max/braceleftbigg−λn\nλ1−λn,1+λ2\nn−λ1+λ2/bracerightbigg\n.\nProof.Use Lemma 5, and then bound αandωwith Theorem 25 and Proposition 18,\nrespectively.\n134.3 Bounds performance\nFinally, using a computational approach, we investigate the tightne ss of our bounds, how\nthey compare with each other and with the previous bound from Pro position 12. The\nresults can be found in the Appendix 5.3.\nIn Tables 1 and 2, the performance of the distance spectrum boun d on eqt(Theorem\n22) is compared to the performance of the induced adjacency spe ctrum bounds from αt\n(Propositions 13 and 14) and the induced spectral bounds from ωand Φ (Propositions 19\nand 21) for t= 2,3, respectively. In Table 3 the performance of the distance spect rum\nbound on eq (Corollary 23) is compared to the performance of the in duced adjacency\nspectrum bound from αandω(Corollary 26). These bounds were computed using Sage-\nMath.\nIn these three tables, a graph name is in boldface when at least one o f the new bounds\nis tight. Note that while Proposition 21 is only tight for trivial cases in T ables 1 and 2,\nwe have found several smaller graphs for which Proposition 21 gives a non-trivial tight\nbound.\nAcknowledgements\nThe authors thank James Tuite for bringing the equidistant number to their attention,\nand Sjanne Zeijlemaker for her help with the MILP from Section 5.2. A ida Abiad is\nsupported by the Dutch Research Council through the grant VI.V idi.213.085.\nReferences\n[1] A. Abiad, G. Coutinho, and M. Fiol. On the k-independence number of graphs.\nDiscrete Mathematics 342(10):2875–2885, 2019.\n[2] A. Abiad, G. Coutinho, M. A. Fiol, B. D. Nogueira, S. Zeijlemaker. O ptimization\nof eigenvalue bounds for the independence and chromatic number o f graph powers.\nDiscrete Mathematics 345(3):112706, 2022.\n[3] A. Abiad, A.P. Khramova, A. Ravagnani. 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Split graphs, in: Proceedings of the 8th South-Eastern Con-\nference on Combinatorics, Graph Theory and Computing , 1977, pp. 311–315.\n[11] F. Foucaud, H. Hocquard, S. Mishra, N. Narayanan, R. Naser asr,´E. Sopena, P. Val-\nicov. Exact square coloring of subcubic planar graphs. Discrete Applied Mathematics\n293:74-89, 2021.\n[12] F. Foucaud, S. Mishra, N. Narayanan, R. Naserasr, P. Valicov . Cliques in exact\ndistance powers of graphs of given maximum degree. Procedia Computer Science\n195:427–436, 2021.\n[13] W.H. Haemers Disconnected Vertex Sets and Equidistant Code P airs.The Electronic\nJournal of Combinatorics 4(1):#R7, 1997\n[14] W.H. Haemers. Hoffman’s ratio bound. Linear Algebra and its Applications 617:215-\n219, 2021.\n[15] W.H. Haemers. Interlacing eigenvalues and graphs. Linear Algebra and its Applica-\ntions226-228:593–616, 1995.\n[16] J.vandenHeuvel, H.A.Kierstead, D.A.Quiroz.Chromaticnumbe rsofexactdistance\ngraphs.Journal of Combinatorial Theory, Series B 134:143–163, 2018.\n[17] L.C. Kavi, M. Newman. The optimal bound on the 3-independence number obtain-\nable from a polynomial-type method. Discrete Mathematics 346(7):113471, 2023.\n[18] R. Koetter and F.R. Kschischang. Coding for errors and erasu res in random network\ncoding.IEEE Transactions on Information Theory 54:3579–3591, 2008.\n[19] X.H. La. 2-distance coloring of sparse graphs. PhD thesis Unive rsit´ e de Montpellier,\n2022.\n[20] Y.-L. Lin, S. Skiena. Algorithms for Square Roots of Graphs. SIAM Journal on\nDiscrete Mathematics 8(1):99–118, 1995.\n[21] J.H. van Lint. A Theorem on Equidistant Codes. Discrete Math 6:353–358, 1973.\n[22] J. Neˇ setˇ ril, P. Ossona de Mendez. Sparsity - Graphs, Struc tures, and Algorithms.\nSpringer, 2012.\n15[23] D.A.Quiroz.Colouringexact distancegraphsofchordal graph s.Dicrete Mathematics\n343(5):111769, 2020.\n[24] N. Raviv, T. Etzion, Distributed storage systems based on equ idistant subspace\ncodes. arXiv:1406.6170\n[25] S.K. Simi´ c. Graph equations for line graphs and n-distance gra phs.Publications de\nl’Institut de Math´ ematiques de Beograd 33(47):203–216, 1983.\n[26] J.Tuit, G.Erskine, V.Taranchuk, C.Tompkins, N.Salia.Equidis tancesetsingraphs.\nIn preparation (private communication), 2024.\n[27] D. Zuckerman. Linear Degree Extractors and the Inapproxim ability of Max Clique\nand Chromatic Number. Theory of Computing 3:103–128, 2007.\n165 Appendix\n5.1 MILP for the Inertial-type bound (Proposition 13)\nBelow is an MILP implementation by Abiad et al. [2] for the Inertial-typ e bound on αt,\nwhich can also be used for finding the best polynomial in Proposition 13 .\nLetG= (V,E) have adjacency matrix Ahaving distinct eigenvalues θ0>···> θd\nwith multiplicities m= (m0,...,m d). Letu∈V, and let p(x) =atxt+···+a0,b=\n(b0,...,b d)∈ {0,1}d+1,M∈Rlarge, and ε∈Rsmall, then we have the following MILP:\nvariables: ( a0...,at),(b0,...,b d)\nparameters: t,M,ε\ninput: For a regular graph G, its adjacency matrix A,\nits spectrum {θm0\n0,...,θmd\nd},a vertex u∈V\noutput: ( a0...,at), the coefficients of a polynomial p\nminimize m⊤b\nsubject tot/summationdisplay\ni=0ai(Ai)vv≥0, v∈V\\{u}\nt/summationdisplay\ni=0ai(Ai)uu= 0\nt/summationdisplay\ni=0aiθi\nj−Mbj+ε≤0, j= 0,...,d\nb∈ {0,1}d+1(3)\nImplementing such an MILP for all u∈Vwill find the best pforG. Note that when the\ngraph is walk-regular, this MILP only needs to be run once.\n175.2 MILP for the Ratio-type bound (Proposition 14)\nThe following MILP can be used to compute the optimal polynomial of P roposition 14.\nLetG= (V,E) have adjacency matrix Aand distinct eigenvalues θ0>···> θd. Then,\nfor some u∈Vandl∈[1,d] we have the MILP:\nvariables: ( a0...,at)\nparameters: t\ninput: For a graph G, its adjacency matrix Aand its spectrum {θ0,...,θ d}.\nA vertex u∈V.Anl∈[1,d]\noutput: ( a0...,at), the coefficients of a polynomial p\nmaximizet/summationdisplay\ni=0aiθi\n0−t/summationdisplay\ni=0aiθi\nl\nsubject tot/summationdisplay\ni=0ai((Ai)vv−(Ai)uu)≤0, v∈V\\{u}\nt/summationdisplay\ni=0ai((Ai)uu−θi\nl) = 1\nt/summationdisplay\ni=0ai(θi\n0−θi\nj)>0, j∈[1,d]\nt/summationdisplay\ni=0ai(θi\nj−θi\nl)≥0, j∈[1,d](4)\nSolving this for all u∈Vandl∈[1,d] yields the best polynomial.\n185.3 The performance of the eigenvalue bounds from Section 4\nGraph Proposition 12 Proposition 13 Proposition 14 Proposi tion 19 Proposition 21 Theorem 22 eq2\nBalaban 10-cage 7 36 35 7 10 37 3\nFrucht graph 7 6 5 - 41 4 3\nMeredith Graph 13 45 34 6 9 40 5\nMoebius-Kantor Graph 7 8 8 11 36 9 4\nBidiakis cube 7 8 5 - 42 4 3\nGosset Graph 703 8 5 7 27 8 4\nBalaban 11-cage 7 59 54 6 6 52 3\nMoser spindle 13 3 - - 54 3 2\nGray graph 7 35 27 7 12 25 3\nNauru Graph 7 14 12 8 27 16 3\nBlanusa First Snark Graph 7 8 8 5 29 7 4\nGrotzsch graph 21 5 - - 47 6 5\nPappus Graph 7 11 9 10 36 7 3\nBlanusa Second Snark Graph 7 8 7 4 27 6 3\nHall-Janko graph 1261 37 10 10 18 37 10\nPoussin Graph 31 6 - - 38 4 4\nBrinkmann graph 13 9 8 8 43 10 5\nHarborth Graph 13 20 20 - 17 15 4\nPerkel Graph 31 20 19 7 21 19 6\nBrouwer-Haemers 381 20 21 21 24 61 15\nHarries Graph 7 35 35 7 9 31 3\nPetersen graph 7 4 4 4 43 6 4\nBucky Ball 7 30 27 5 10 25 3\nHarries-Wong graph 7 35 35 7 9 31 3\nRobertson Graph 13 8 7 9 49 10 7\nHeawood graph 7 7 7 14 36 8 7\nSchl¨ afli graph 241 7 3 3 24 7 3\nHerschel graph 13 7 - - 39 6 6\nShrikhande graph 31 7 4 4 36 7 4\nHigman-Sims graph 463 22 26 26 24 78 22\nHoffman Graph 13 11 8 - 36 8 8\nSousselier Graph 21 6 - - 45 7 5\nClebsch graph 21 5 6 6 36 11 5\nHoffman-Singleton graph 43 21 15 15 36 29 15\nSylvester Graph 21 17 13 9 29 17 5\nCoxeter Graph 7 13 12 4 16 9 3\nHolt graph 13 10 10 5 27 11 4\nSzekeres Snark Graph 7 24 23 5 11 17 4\nDesargues Graph 7 10 10 9 29 6 4\nHorton Graph 7 50 48 7 7 40 4\nThomsen graph 7 5 3 6 36 5 3\nTietze Graph 7 6 5 - 41 6 4\nDouble star snark 7 15 13 5 17 10 3\nKrackhardt Kite Graph 31 4 - - 49 4 3\nDurer graph 7 7 5 - 39 4 3\nKlein 3-regular Graph 7 29 25 4 9 16 3\nTruncated Tetrahedron 7 6 4 3 29 4 3\nDyck graph 7 16 16 8 18 17 3\nKlein 7-regular Graph 43 9 6 6 30 9 6\nTutte 12-Cage 7 77 63 7 5 71 3\nEllingham-Horton 54-graph 7 29 27 7 12 24 4\nTutte-Coxeter graph 7 20 15 8 22 19 3\nEllingham-Horton 78-graph 7 40 39 7 9 30 4\nLjubljana graph 7 63 56 7 6 43 3\nErrera graph 31 7 - - 35 5 4\nWagner Graph 7 3 3 3 45 5 3\nF26A Graph 7 13 13 8 23 7 3\nM22 Graph 241 21 21 21 26 56 21\nFlower Snark 7 10 9 5 27 8 4\nMarkstroem Graph 7 10 10 - 20 7 3\nWells graph 21 13 12 10 36 9 5\nFolkman Graph 13 15 10 - 36 10 10\nWiener-Araya Graph 13 19 - - 17 14 4\nFoster Graph 7 50 45 7 7 42 3\nMcGee graph 7 12 11 5 22 10 3\nHexahedron 7 4 4 8 36 4 4\nDodecahedron 7 11 8 3 22 8 3\nOctahedron 13 4 2 2 24 4 2\nIcosahedron 21 4 3 4 34 4 3\nTable 1: Comparison of eq2bounds for Sage named graphs. Graph names are in bold\nwhen one of the new bounds is tight.\n19Graph Proposition 12 Proposition 13 Proposition 14 Proposi tion 19 Proposition 21 Theorem 22 eq3\nBalaban 10-cage 13 19 17 5 19 28 2\nFrucht graph 13 3 3 - 39 4 3\nMeredith Graph 37 10 14 - 16 15 2\nMoebius-Kantor Graph 13 6 4 3 38 5 2\nBidiakis cube 13 4 3 - 36 4 2\nGosset Graph 18253 8 2 2 3 8 2\nBalaban 11-cage 13 39 27 5 9 52 2\nMoser spindle 37 2 - 1 1 2 1\nGray graph 13 19 11 5 24 7 2\nNauru Graph 13 8 6 4 46 7 2\nBlanusa First Snark Graph 13 4 4 - 31 5 3\nGrotzsch graph 81 1 - 1 1 6 1\nPappus Graph 13 7 3 3 40 7 2\nBlanusa Second Snark Graph 13 4 4 - 29 5 4\nHall-Janko graph 44101 1 1 1 1 37 1\nPoussin Graph 151 4 - - 28 4 2\nBrinkmann graph 37 6 3 3 21 8 3\nHarborth Graph 37 13 10 - 21 12 3\nPerkel Graph 151 18 5 7 16 19 5\nBrouwer-Haemers 7221 1 1 1 1 61 1\nHarries Graph 13 18 17 5 19 27 2\nPetersen graph 13 1 1 1 1 6 1\nBucky Ball 13 16 14 5 11 15 3\nHarries-Wong graph 13 18 17 5 19 27 2\nRobertson Graph 37 5 3 3 15 8 3\nHeawood graph 13 2 2 2 34 8 2\nSchl¨ afli graph 3601 1 1 1 1 7 1\nHerschel graph 37 3 - - 28 4 2\nShrikhande graph 151 1 1 1 1 7 1\nHigman-Sims graph 9703 1 1 1 1 78 1\nHoffman Graph 37 5 2 3 31 5 2\nSousselier Graph 81 5 - - 36 6 3\nClebsch graph 81 1 1 1 1 11 1\nHoffman-Singleton graph 253 1 1 1 1 29 1\nSylvester Graph 81 10 6 6 14 17 6\nCoxeter Graph 13 7 7 7 22 9 7\nHolt graph 37 7 4 4 25 7 3\nSzekeres Snark Graph 13 13 12 - 15 13 2\nDesargues Graph 13 6 5 3 36 5 2\nHorton Graph 13 30 24 - 13 32 2\nThomsen graph 13 1 1 1 1 1 1\nTietze Graph 13 4 3 - 36 6 3\nDouble star snark 13 9 7 6 21 10 4\nKrackhardt Kite Graph 151 4 - - 43 3 2\nDurer graph 13 3 2 - 30 4 2\nKlein 3-regular Graph 13 19 13 6 13 16 3\nTruncated Tetrahedron 13 4 3 3 31 4 3\nDyck graph 13 8 8 4 33 7 2\nKlein 7-regular Graph 253 9 3 3 9 9 3\nTutte 12-Cage 13 44 28 5 10 71 2\nEllingham-Horton 54-graph 13 20 13 - 20 16 2\nTutte-Coxeter graph 13 10 6 4 50 10 2\nEllingham-Horton 78-graph 13 27 19 - 16 26 2\nLjubljana graph 13 35 27 5 11 43 2\nErrera graph 151 4 - - 30 4 2\nWagner Graph 13 1 2 1 1 3 1\nF26A Graph 13 7 6 3 41 7 2\nM22 Graph 3601 1 1 1 1 56 1\nFlower Snark 13 7 5 - 31 6 3\nMarkstroem Graph 13 7 6 - 25 6 3\nWells graph 81 9 3 3 13 9 2\nFolkman Graph 37 5 3 4 36 5 2\nWiener-Araya Graph 37 12 - - 21 10 3\nFoster Graph 13 23 22 5 14 14 2\nMcGee graph 13 7 6 7 36 9 5\nHexahedron 13 2 2 2 18 4 2\nDodecahedron 13 4 4 4 21 4 4\nOctahedron 37 1 1 1 1 1 1\nIcosahedron 81 4 2 2 12 4 2\nTable 2: Comparison of eq3bounds for Sage named graphs. Graph names are in bold\nwhen one of the new bounds is tight.\n20Graph Corollary 23 Corollary 26 eq\nBalaban 10-cage 37 35 9\nFrucht graph 6 5 3\nMeredith Graph 46 34 5\nMoebius-Kantor Graph 9 8 4\nBidiakis cube 7 5 3\nGosset Graph 8 14 7\nBalaban 11-cage 60 54 7\nMoser spindle 5 - 3\nGray graph 31 27 6\nNauru Graph 16 12 4\nBlanusa First Snark Graph 8 8 4\nGrotzsch graph 8 - 5\nPappus Graph 8 9 3\nBlanusa Second Snark Graph 10 7 4\nHall-Janko graph 37 10 10\nPoussin Graph 7 - 4\nBrinkmann graph 12 8 5\nHarborth Graph 24 20 4\nPerkel Graph 19 19 6\nBrouwer-Haemers 61 21 15\nHarries Graph 31 35 10\nPetersen graph 6 4 4\nBucky Ball 28 27 3\nHarries-Wong Graph 31 35 9\nRobertson Graph 11 7 7\nHeawood Graph 8 7 7\nSchl¨ afli Graph 7 9 6\nHerschel graph 6 - 6\nShrikhande Graph 7 4 4\nHigman-Sims Graph 78 26 22\nSims-Gewirtz Graph 36 16 16\nChvatal Graph 9 5 4\nHoffman Graph 9 8 8\nSousselier Graph 7 - 5\nClebsch Graph 11 6 5\nHoffman-Singleton Graph 29 15 15\nSylvester Graph 26 13 6\nCoxeter Graph 16 12 7\nHolt Graph 11 10 4\nSzekeres Snark Graph 21 23 5\nDesargues Graph 6 10 4\nHorton Graph 46 48 4\nThomsen graph 5 3 3\nDejter Graph 8 56 8\nKittell Graph 10 23 4\nTietze Graph 8 5 4\nDouble star snark 14 13 4\nKrackhardt Kite Graph 6 - 4\nDurer Graph 5 5 3\nKlein 3-regular Graph 29 25 7\nTruncated Tetrahedron 9 4 3\nDyck Graph 17 16 4\nKlein 7-regular Graph 9 6 6\nTutte 12-Cage 71 63 9\nEllingham-Horton 54-Graph 26 27 4\nTutte-Coxeter Graph 19 15 5\nEllingham-Horton 78-Graph 37 39 5\nLjubljana Graph 50 56 8\nErrera Graph 9 - 4\nWagner Graph 6 3 3\nF26A Graph 13 13 3\nM22 Graph 56 21 21\nFlower Snark 12 9 4\nMarkstroem Graph 12 10 3\nWells Graph 9 12 5\nFolkman Graph 11 10 10\nWiener-Araya Graph 19 - 4\nFoster Graph 42 45 5\nMcGee Graph 12 11 5\nFranklin Graph 7 6 6\nHexahedron 4 4 4\nDodecahedron 8 8 4\nOctahedron 4 3 3\nIcosahedron 4 4 3\nTable 3: Comparison of eq bounds for Sage named graphs. Graph na mes are in bold\nwhen one of the new bounds is tight.\n21"    },    {        "title": "2402.00215v1.Schrödinger_Operators_with_Potentials_Generated_by_Hyperbolic_Transformations__II__Large_Deviations_and_Anderson_Localization.pdf",        "content": "arXiv:2402.00215v1  [math.SP]  31 Jan 2024SCHR¨ODINGER OPERATORS WITH POTENTIALS\nGENERATED BY HYPERBOLIC TRANSFORMATIONS:\nII. LARGE DEVIATIONS AND ANDERSON LOCALIZATION\nARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nAbstract. We consider discrete one-dimensional Schr¨ odinger operat ors\nwhose potentials are generated by H¨ older continuous sampl ing along the orbits\nof a uniformly hyperbolic transformation. For any ergodic m easure satisfying\na suitable bounded distortion property, we establish a unif orm large deviation\nestimate in a large energy region provided that the sampling function is locally\nconstant or has small supremum norm. We also prove full spect ral Anderson\nlocalization for the operators in question.\nContents\n1. Introduction 1\n2. The Setting and the Main Results 3\n2.1. The Base Dynamical System 3\n2.2. Measures With the Bounded Distortion Property 4\n2.3. The Associated Operator Family 6\n2.4. The Main Results 7\n3. Large Deviation Estimates – Proof of Theorem 2.10 10\n3.1. Stable and Unstable Holonomies 10\n3.2. Reduction of the Uniform LDT 11\n3.3. Proof of Theorem 3.6 14\n4. Establishing Localization – Proof of Theorem 2.12 22\n4.1. Bounding the Green Function 23\n4.2. Elimination of Double Resonances 30\n5. Applications – Proof of Corollaries 2.14 and 2.15 34\nReferences 35\n1.Introduction\nIt is a well-known phenomenon that the one-dimensional Anderson m odel is\nlocalized throughout the spectrum. Given a compactly supported p robability mea-\nsureνonR, the Anderson model is given by the discrete Schr¨ odinger operat or on\nℓ2(Z), acting as\n(1) [ Hωψ](n) =ψ(n+1)+ψ(n−1)+Vω(n)ψ(n),\nD. D. was supported in part by NSF grants DMS–1700131 and DMS– 2054752.\nZ. Z. was supported in part by NSF grant DMS–1764154.\n12 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nwhere the random potential Vω(n) is given by independent random variables,\neach distributed according to ν. Formally this can be modeled by consider-\ning the compact product Ω = (supp ν)Z, the shift transformation T: Ω→Ω,\n[Tω](n) =ω(n+1), and the T-ergodic measure µ=νZon Ω, as the potential can\nthen be written in the form of a dynamically defined potential in the se nse of [DF],\n(2) Vω(n) =f(Tnω),\nwhere the sampling function f: Ω→Ris given by the evaluation at the origin,\n(3) f(ω) =ω(0).\nThe localization statement for the operator family {Hω}ω∈Ωtypically takes two\nforms. Spectral localization means that for µ-almost every ω∈Ω, the operator\nHωhas pure point spectrum with exponentially decaying eigenfunctions . Dynami-\ncal localization means that the unitary group associated with Hωhas exponential\noff-diagonal decay relative to the standard orthonormal basis {δn}n∈Zofℓ2(Z) uni-\nformly in time, either in an almost sure sense or in expectation with res pect toµ.\nSpectral localization and dynamical localization are not equivalent fo r general\noperators, but they both hold for the Anderson model. Proving th ese localization\nstatements for the Anderson model is most difficult in the case of a s ingular single-\nsite distribution, for example the Bernoulli case, where supp νhas cardinality 2.\nIn fact it is true that the Bernoulli case is the most difficult case to ha ndle, as\nany localization proof that covers the Bernoulli case will also cover t he general\ncase. We refer the reader to [CKM] for the first proof of spectra l localization in\nthe Bernoulli case and to [BDF+, GZ, GK, JZ] for recent treatment s of this model,\nwhich establish both spectral and dynamical localization and are simp ler and more\nconceptual (in that they rely on one-dimensional tools, rather th an verifying the\ninput necessary to run a multi-scale analysis).\nGiven this state of affairs, one might say that the one-dimensional A nderson\nmodel is completely understood. Under the hood, however, all the proofs rely\ncrucially on the independence of the potential values, and hence on ly apply to\nthe special choice (3) of the sampling function f. From the perspective of the\ngeneral theory of ergodic Schr¨ odinger operators in ℓ2(Z), [DF], it is natural to ask\nwhether localization extends to more general choices of f. Indeed, this should even\nbe expected to be the case, based on the heuristics of the situatio n. Alas, prior\nto the present work, such an extension was well outside the scope of the existing\napproaches. One could arguethat it is the independence ofthe und erlyingentries of\nωthat is responsible for localization phenomena, as long as the sampling function is\nstill sufficiently “local” in the sense that the values of f(ω) should be most heavily\ninfluenced by the values of ωnfornnot too large. Of course this will be true for any\ncontinuous sampling function f: Ω→R, but upon closer inspection, quantitative\ncontinuity properties of fbecome relevant, as many exotic spectral phenomena\nturn out to occur for a generic continuous sampling function.\nAs a very specific example of interest, consider the case of locally co nstantf∈\nC(Ω,R), which are those for which f(ω) only depends on the values of ωnin a fixed\nfinite range of n’s. The localization problem for the associated operators {Hω}ω∈Ω\nwas previously inaccessible, but it will be fully covered by our work.\nThe approach to proving localization developed in the current series of papers\napplies to a much larger class of models. This is relevant as this class inc ludes\nadditionalspecific examplesofinterest that areformallydistinct fr om the AndersonANDERSON LOCALIZATION 3\nmodel, but share the feature that is crucial to our method, namely the fact that\nthe underlying ergodic base dynamical system (Ω ,T,µ) is a subshift of finite type\nwith the ergodic measure having the bounded distortion property. We will give\nprecise definitions in Section 2 but mention here that this allows us to e stablish\na localization result throughout the entire spectrum for Arnold’s ca t map. For\nthis particular example, there was only the previous work [BS] that h ad to exclude\nenergy intervals of positive length from consideration.\nThe general structure of our localization proof follows the outline o f previous\nworks (e.g., [B, BG, BS]):\n(i) Establish the positivity of the Lyapunov exponent, L(E), for sufficiently\nmany energies (i.e., the exceptional set {E:L(E) = 0}should be shown to\nbe discrete).\n(ii) Prove large deviation estimates for the Lyapunov exponent.\n(iii) Eliminate double resonances.\nItem (i) was addressed in part I of this series [ADZ]. In the present follow-up\nwork we address items (ii) and (iii).\nTo summarize, together with part I of the series, [ADZ], the presen t paper suc-\nceeds in establishing spectral and dynamical localization for Schr¨ o dinger operators\nthat were expected to share these features with the Anderson m odel, but were\nstructurally so different from it that previous methods were insuffic ient to make\nprogress towards results of this kind.\nWe give precise definitions and statements of our results in Section 2 . As we can\nquotepositivityresultsforthe Lyapunovexponentfrom[ADZ], wen eed toestablish\nlarge deviation estimates and eliminate double resonances, which the n completes\nthe proof of localization. We do the former in Section 3 and the latter in Section 4.\nFinally, we discuss applications to concrete examples in Section 5.\n2.The Setting and the Main Results\n2.1.The Base Dynamical System. LetA={1,2,...,ℓ}withℓ≥2beequipped\nwith the discrete topology. Consider the product space AZ, whose topology is\ngenerated by the cylinder sets, which are the sets of the form\n[n;j0,···,jk] ={ω∈ AZ:ωn+i=ji,0≤i≤k}\nwithn∈Zandj0,...,j k∈ A. The topology is metrizable and for definiteness we\nfix the following metric on AZ. Set\nN(ω,˜ω) = max{N≥0 :ωn= ˜ωnfor all|n|<N},\nand equip AZwith the metric ddefined by\nd(ω,˜ω) =e−N(ω,˜ω).\nWe consider the left shift T:AZ→ AZdefined by ( Tω)n=ωn+1forω∈ AZand\nn∈Z. Let Orb(ω) ={Tnω:n∈Z}be the orbit of ωunder the dynamics T.\nDefinition2.1. LetΩ⊆ AZbeasubshiftoffinitetypeandconsiderthetopological\ndynamical system (Ω ,T).\nWe say that a finite word j0j1...jk, whereji∈ {1,...,ℓ}for 0≤i≤k, is\nadmissible if it occurs in some ω∈Ω, that is, there are ω∈Ω andn∈Zsuch that\nωn+i=jifor all 0≤i≤k.4 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nThelocal stable set of a pointω∈Ω is defined by\nWs\nloc(ω) ={˜ω∈Ω :ωn= ˜ωnforn≥0}\nand thelocal unstable set ofωis defined by\nWu\nloc(ω) ={˜ω∈Ω :ωn= ˜ωnforn≤0}.\nA set is called s-locally saturated (resp.,u-locally saturated ) if it is a union of\nlocal stable (resp., local unstable) sets of the form above.\nFor eachj∈ Aand each pair of points ω,˜ω∈[0;j], we denote the unique point\ninWu\nloc(ω)∩Ws\nloc(˜ω) byω∧˜ω.\n2.2.Measures With the Bounded Distortion Property. Let the subshift Ω\nbe equipped with the Borel σ-algebra and let µbe a probability measure on Ω that\nis ergodic with respect to T. We define\nΩ+={(ωn)n≥0:ω∈Ω}\nΩ−={(ωn)n≤0:ω∈Ω}\nto be the spaces of one-sided right and left infinite sequences, res pectively, associ-\nated with Ω. Metrics for Ω±can be defined in a way similar to the definition of\nthe metric for AZabove. Abusing notation slightly, we still let ddenote their met-\nrics. Letπ+be the projection from Ω to Ω+andµ+=π+\n∗(µ) be the pushforward\nmeasure of µon Ω+. Similarly, we let π−be the projection to Ω−andµ−be the\npushforward measure on Ω−. LetT+be the left shift operator on Ω+andT−be\nthe right shift on Ω−. Forn≥0, we let [n;j0,...,j k]+denote the cylinder sets\nin Ω+; forn≤ −k, we let [n;j0,...,j k]−denote the cylinder sets in Ω−. Letω±\ndenote points in Ω±, respectively.\nFor simplicity, for each 1 ≤j≤ℓ, we setµj=µ|[0;j]and Ω j= Ω∩[0;j].\nSimilarly, we set µ±\nj=µ±|[0;j]±and Ω±\nj= Ω±∩[0;j]±, respectively.\nNote that we do not have Ω = Ω−×Ω+. However, for each 1 ≤j≤ℓ, we have\na natural homeomorphism:\nP: Ωj→Ω−\nj×Ω+\njwhereP(ω) = (π−ω,π+ω).\nThus, abusing the notation a bit, we may just write Ω j= Ω−\nj×Ω+\nj. Moreover, we\nhave for all ω∈Ω,\n(4) ( π+)−1(π+ω) =Ws\nloc(ω),(π−)−1(π−ω) =Wu\nloc(ω).\nDefinition 2.2. We sayµhas alocal product structure if there is a ψ: Ω→R+\nsuch that for each 1 ≤j≤ℓ,ψ∈L1(Ωj,µ−\nj×µ+\nj) and\n(5) dµj=ψ·d(µ+\nj×µ−\nj).\nIfµhasalocalproductstructure, thenitstopologicalsupportsupp µisasubshift\nof finite type (see, e.g., [BGV, Lemma 1.2]) and hence, without loss of g enerality,\nwe will assume throughout that the measure µhas full support in Ω. For results\nconcerning positivity of the LE, a local product structure is usually sufficient, see\ne.g. [ADZ]. However, for further results such as large deviation est imates, we will\nneed the measure µto obey a quantitative version of local product structure, which\nis defined as follows.ANDERSON LOCALIZATION 5\nDefinition 2.3. We say that µsatisfies the bounded distortion property if there\nisC≥1 such that for all cylinders [ n;j0,...,j k] and [l;,i0, ...,j m] in Ω, where\nl>n+kand [n;j0,...,j k]∩[l;,i0,...,is]/\\⌉}atio\\slash=∅, we have\n(6) C−1≤µ([n;j0,...,j k]∩[l;i0,...,is])\nµ([n;j0,...,j k])·µ([l;i0,...,is])≤C.\nThe bounded distortion property implies local product structure; see, for exam-\nple, [ADZ, Section 2.1]. Equilibrium states of H¨ older continuous poten tials have\nthe bounded distortion property; compare, for example, [ADZ, Le mma 3.4]. It is\na standard result that if an invariant measure µof aC2transitive Anosov diffeo-\nmorphism (or a C2transitive, uniformly expanding map) is absolutely continuous\nwith respect to the volume measure, then it is an equilibrium state of a H¨ older\ncontinuous potential; see, for example, [B]. In particular, it include s hyperbolic\nendomorphisms on the d-dimensional torus, for all d≥1, such as the doubling map\nonR/Zor Arnold’s cat map on ( R/Z)2, where the ergodic measure is simply taken\nto be the Lebesgue measure . It is alsowell-known that Markovchain s with Markov\nmeasures have locally constant potentials, which are clearly H¨ older continuous.\nRecall that (Ω ,T) istopologically mixing if for any pair of nonempty open sets\nU,V⊂Ω, there is an N≥1 such that Tn(U)∩V/\\⌉}atio\\slash=∅for alln≥N. Following\n[ADZ, Section 2.1.4], we can assume without loss of generality that (Ω ,T) is topo-\nlogically mixing in Section 3. Concretely, by the spectral decompositio n theorem\nfor hyperbolic basic sets, we can decompose Ω as Ω =/unionsqtexts\nl=1Ω(l)for somes≥1\nand for closed subsets Ω(l), so that the following holds true: T(Ω(l)) = Ω(l+1)for\n1≤l<s,T(Ω(s)) = (Ω(1)), andTs|Ω(l)is a topologically mixing subshift of finite\ntype for each 1 ≤l≤s. Moreover, the normalized restriction µ(l)ofµto Ω(l)\nis aTs-invariant ergodic, fully supported probability measure with local pr oduct\nstructure or bounded distortion property, provided the same pr operty is true for µ\non Ω.\nFor a cocycle map A: Ω→SL(2,R) with/bar⌈blA/bar⌈bl∞≤Γ, we consider As: Ω(l)→\nSL(2,R)asAs(ω) (see(9)belowforthedefinition of As), whichmaybeconsidereda\ncocycle map defined overthe base dynamics Ts: Ω(l)→Ω(l). Clearly,L(As,µ(l))>\n0 for some 1 ≤l≤simplies that L(A,µ)>0. Moreover, large deviation estimates\non each (Ts,Ω(l),µ(l)) imply large deviation estimates on ( T,Ω,µ) as follows. For\nn≥1, we may write n=sk+rfor some 0 ≤r<s. Then for each ω∈Ω,\n/bar⌈blAr(Tksω)/bar⌈bl−1·/bar⌈blAks(ω)/bar⌈bl ≤ /bar⌈blAn(ω)/bar⌈bl ≤ /bar⌈blAr(Tksω)/bar⌈bl·/bar⌈blAks(ω)/bar⌈bl.\nHence for each ε>0, there is a n0=n0(ε,Γ) such that for all n≥n0, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnlog/bar⌈blAn(ω)/bar⌈bl−1\nkslog/bar⌈blAks(ω)/bar⌈bl/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε\n2.\nIn particular, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnlog/bar⌈blAn(ω)/bar⌈bl−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε⇒/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nklog/bar⌈blAks(ω)/bar⌈bl−sL(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>s\n2ε,\nwhich implies for all n≥n0that\n/braceleftbigg\nω:/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnlog/bar⌈blAn(ω)/bar⌈bl−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n⊂s/uniondisplay\nl=1/braceleftbigg\nω∈Ω(l):/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nklog/bar⌈blAks(ω)/bar⌈bl−sL(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>s\n2ε/bracerightbigg\n.6 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nExponential decay of the measure of the set at the right side abov e is clearly a\ndirect consequence of the LDT of ( Ts,As) on each (Ts,Ω(l),µ(l)).\nThe main consequence of topological mixing that we will use is the follow ing.\nThere isr0∈Z+so that for all [ k;j0,...,j n]⊂Ω and all [l;i0,...,im]∈Ω, where\nl−(k+n)≥r0, we have\n(7) [ k;j0,...,j n]∩[l;i0,...,im]/\\⌉}atio\\slash=∅.\n(7) could be an easy consequence of the specification property of such systems; see,\nfor example, [ADZ, Proposition 2.7]. In Section 3, for our (Ω ,T,µ), we letr0be a\nnumber satisfying (7), which will be used in the proof of Lemma 3.7.\n2.3.The Associated Operator Family. In this paper, we are mainly con-\ncerned with the Anderson localization phenomenon for one-dimensio nal discrete\nSchr¨ odinger operators Hωinℓ2(Z) acting by\n(8) [ Hωu](n) =u(n+1)+u(n−1)+Vω(n)u(n).\nHere we assume Ω to be any compact metric space, T: Ω→Ω a homeomorphism,\nandf: Ω→Rcontinuous. We consider potentials Vω:Z→Rdefined by\nVω(n) =f(Tnω) forω∈Ω andn∈Z. Let\nσ(Hω) ={E:Hω−Edoes not have a bounded inverse }\nbe the spectrum of Hω. Choose a T-ergodic measure µ. Then there is a com-\npact set Σ such that σ(Hω) = Σ for µ-a.e.ω. We say that Hωhas (spectral)\nAnderson localization if it has pure point spectrum with exponentially decaying\neigenfunctions.\nA continuous map A: Ω→SL(2,R) gives rise to the cocycle ( T,A) : Ω×R2→\nΩ×R2, (ω,v)/ma√sto→(Tω,A(ω)v). Forn∈Z, we let (T,A)n= (Tn,An). In particular,\nwe have\n(9) An(ω) =\n\nA(Tn−1ω)···A(ω), n≥1;\nI2, n = 0;\n[A−n(Tnω)]−1, n ≤ −1,\nwhereI2is the identity matrix. The Lyapunov exponent is given by\nL(A,µ) = lim\nn→∞1\nn/integraldisplay\nlog/bar⌈blAn(ω)/bar⌈bldµ(ω)\n= inf\nn≥11\nn/integraldisplay\nlog/bar⌈blAn(ω)/bar⌈bldµ(ω)≥0.\nBy Kingman’s subaddive ergodic theorem, we have\nlim\nn→∞1\nnlog/bar⌈blAn(ω)/bar⌈bl=L(A,µ)\nforµ-a.e.ω. By linearity and invertibility of each A(ω), we can projectivize the\nsecond component and consider ( T,A) : Ω×RP1→Ω×RP1.\nSpectral properties of the operators Hωcan be investigated in terms of the\nbehavior of the solutions to the difference equation\n(10) u(n+1)+u(n−1)+Vω(n)u(n) =Eu(n), n∈Z,ANDERSON LOCALIZATION 7\nwithE∈R. These solutions in turn can be described with the help of the\nSchr¨ odingercocycle ( T,AE) with the cocycle map AE: Ω→SL(2,R) being defined\nas\nAE(ω) =A(E−f)(ω) :=/parenleftbiggE−f(ω)−1\n1 0/parenrightbigg\n,\nwhere we often leave the dependence on f: Ω→Rimplicit as it will be fixed\nmost of the time. Such cocycles describe the transfer matrices as sociated with\nSchr¨ odinger operators with dynamically defined potentials. Specifi cally,usolves\n(10) if and only if\n/parenleftbigg\nu(n)\nu(n−1)/parenrightbigg\n=AE\nn(ω)/parenleftbigg\nu(0)\nu(−1)/parenrightbigg\n, n∈Z.\nWe setL(E) =L(AE,µ) and let Zf={E:L(E) = 0}.\n2.4.The Main Results. As mentioned in the introduction, there are two key\ningredients to a localization proof – positivity of the Lyapunov expon ent and large\ndeviation estimates. Let us formulate precisely the two statement s that are needed.\nDefinition 2.4. LetI⊂R. We sayAEhas PLE on Iif\ninf\nE∈IL(E)>0.\nWe sayAEhas ULD on Iif for every ε>0, there are constants C,c>0, depending\nonly onεandf, such that it for all E∈Iandn≥1, we have\nµ/braceleftbig\nω∈Ω :/vextendsingle/vextendsingle1\nnlog/bar⌈blAE\nn(ω)/bar⌈bl−L(E)/vextendsingle/vextendsingle>ε/bracerightbig\n<Ce−cn.\nSince PLE may be quoted from [ADZ], our main tasks will be to establish U LD\nand then to deduce the desired localization results from PLE and ULD . In this\npaper, we shall show ULD for the following two classes of sampling fun ctions. They\narebothsubsetsof C(Ω,R) andtheyrequirequantitativecontrolonthedependence\nof the function of entries of the input sequence that are far away from the origin.\nThe first class is the following, where there is in fact no dependence o n entries\nthat are sufficiently far away from the origin.\nDefinition 2.5. We say that f: Ω→Rislocally constant if there exists an\nn0∈Z+such that for each ω∈Ω,f(ω) depends only on the cylinder set\n[−n0;ω−n0,...,ω n0]. In other words, fis constant on each such cylinder set. Let\nus denote by LC the set of all locally constant f: Ω→R.\nRemark 2.6. (a) Clearly, any locally constant fis Lipschitz continuous and hence\nLC⊂Cα(Ω,R) for any 0 <α≤1, which is defined below.\n(b) We can similarly define locally constant cocycles A: Ω→SL(2,R) and, abusing\nnotation slightly, denote by LC the set of all such cocycles as well. Cle arly,f∈LC\nimplies that AE∈LC for allE.\nThe second class consists of sufficiently small H¨ older continuous fu nctionsf:\nΩ→R. We write Cα(Ω,R), 0< α≤1 for the space of real-valued α-H¨ older\ncontinuous functions. That is,\nsup\nω/ne}ationslash=ω′|f(ω)−f(ω′)|\nd(ω,ω′)α<∞.\nLikewise, we can define the space of α-H¨ older continuous cocycles Cα(Ω,SL(2,R)).8 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nDefinition 2.7. We say that A∈Cα(Ω,SL(2,R)) isfiber bunched if there exists\nN≥1 such that for every ω∈Ω, we have\n(11) /bar⌈blAN(ω)/bar⌈bl2<eαN.\nEquivalently, there is θ<αsuch that /bar⌈blAN(ω)/bar⌈bl2<eθNfor everyω∈Ω.\nRemark 2.8. As explained in [ADZ, Remark 7.5], AEis fiber bunched for all\nE∈[−2−λ0,2+λ0]⊃Σf, provided /bar⌈blf/bar⌈bl∞<λ0:=(eα\n2−1)2\n9.\nHence, we make the following definition:\nDefinition 2.9. We sayf∈Cα(Ω,R) isglobally fiber bunched if/bar⌈blf/bar⌈bl∞<λ0and\ndenote by SH the set of all such f.\nSupposeThas a fixed point on Ω and µis aT-ergodic measure that has a local\nproduct structure. Then the following holds true if f∈LC∪SH. Let Ffbe the set\nofenergieswhere AEhasansu-state(seeSection3.1belowforadetaileddescription\nofsu-states). By [ADZ, Proposition 5.9], Zf⊂ Ffand by [ADZ, Lemma 5.11], Ff\nis finite. By [BBB, Theorem 2.8], L(E) is continuous on R. Then for any compact\nintervalJand anyη>0, we have PLE on\n(12) Jη:=J\\B(Ff,η),\nwhich consists of a finite number of connected compact intervals. H ere,B(Ff,η)\ndenotes the open η-neighborhood of Ff.\nThe first main result of the present paper addresses the ULD prop erty:\nTheorem 2.10. LetΩbe a subshift of finite type and µbe aT-ergodic measure\nthat has the bounded distortion property. Assume that Thas a fixed point. Let\nf∈LC∪SHbe nonconstant. Then there is a connected compact interval J⊃Σf\nsuch thatAEsatisfies ULDonJηfor allη>0.\nRemark 2.11. Large deviation estimates in similar contexts were previously ob-\ntained by several authors, see for example [DKP, GS, PP]. In partic ular, [DKP]\nproved a local uniform version, which enabled them to obtain H¨ older continuity\nof the Lyapunov exponent. However, they all assumed certain ty pical conditions\nof the cocycles that were first introduced by [BGV, BV], while in the co ntext of\nTheorem 2.10, we do not necessarily have these typical conditions f orAE,E∈Jη.\nIndeed, typical conditions were used in [BGV, BV] to prove positivity of the Lya-\npunov exponent (or simplicity of the Lyapunov spectrum in case of h igher dimen-\nsional cocycles), whereas the proof of positivity of the Lyapunov exponent on Jηin\n[ADZ] does not use any perturbation argument. What [ADZ] uses ins tead is a cer-\ntain analyticity argument together with certain aspects of inverse spectral theory.\nIn fact, Theorem 2.10 is stronger than the previous results in the s ense that the\ntypical conditions of [BGV, BV] imply the uniqueness of the u-state, which is the\ncondition needed to prove our ULD. Although we do not pursue a loca l uniform\nversion of LDT in the space of Cα(Ω,SL(2,R)), our proof does imply that without\nany extra work since uniformity is a direct consequence of the uniqu eness of the u-\nstate. We refer the reader to Section 3.1 for a more detailed descr iption. Moreover,\nour techniques, which are different from those of the previous wor ks, can be further\ndeveloped to obtain stronger results concerning ULD which we shall explore in the\nthird paper of this series.ANDERSON LOCALIZATION 9\nOur second main result deduces localization from PLE and ULD:\nTheorem 2.12. LetΩbe a subshift of finite type and µbe aT-ergodic measure that\nsatisfies the bounded distortion property. Let f∈Cα(Ω,R). LetI⊂Ron which\nwe have PLEandULD. Then,Hωhas spectral localization on Iforµ-almost every\nω∈Ω.\nRemark 2.13. In fact, our proof implies exponential dynamical localization of Hω\nonIforµ-a.e.ωwhich is known to be stronger than the spectral localization. More\nconcretely, for any µ-a.e.ω, allε >0, and 0< β < γ = infE∈IL(E), there is a\nconstant/tildewideC=/tildewideCω,β,ε>0 such that\n(13) sup\nt∈R|/a\\}brack⌉tl⌉{tδn,e−itHωPI,ωδm/a\\}brack⌉tri}ht| ≤/tildewideCeǫ|m|e−β|n−m|\nfor allm,n∈Z, wherePI,ωdenotes the spectral projection of HωtoI.\nCorollaries 2.14 and 2.15 below are direct consequences of Theorems 2.10\nand 2.12. We state them separately to facilitate reference to one o f them.\nCorollary 2.14. Let(Ω,µ)be as in Theorem 2.10. If f∈LCis nonconstant, then\nforµ-almost every ω∈Ω,Hωhas full spectral localization.\nCorollary 2.15. Let(Ω,µ)be as in Theorem 2.10. If f∈SHis nonconstant, then\nforµ-almost every ω∈Ω,Hωhas full spectral localization.\nIn the same spirit, for ease of reference, let us formulate the follo wing sample\napplications to specific base dynamics.\nCorollary 2.16. ConsiderT: (R/Z)m→(R/Z)mwhereTis the doubling map\nifm= 1or Arnold’s cat map if m= 2. Letµbe the Lebesgue measure. Let\nf: (R/Z)m→Rbe H¨ older continuous and nonconstant, and consider the pot entials\ngiven byVω(n) =λf(Tnω). Then there is a λ0>0such that for all 0< λ≤λ0,\nHωhas full spectral localization for µ-a.e.ω∈(R/Z)m.\nCorollary 2.17. Let(Ω,T,µ)be a Markov chain with a Markov measure µ. Sup-\nposeThas a fixed point and f∈LCis nonconstant. Consider the potentials given\nbyVω(n) =λf(Tnω). Then for all λ>0,Hωhas full spectral localization for µ-a.e.\nω.\nRemark 2.18. We wish to point out that according to Definition 2.9 in Corol-\nlary 2.15,f∈SH simply means f∈Cα(Ω,R) and/bar⌈blf/bar⌈bl∞≤λ0, where\nλ0=(eα\n2−1)2\n9.\nBut in Corollary 2.16 the λ0might be different due to the coding process that\nconverts the smooth hyperbolic systems to their corresponding s ubshifts of finite\ntype. Take the doubling map for example, by [ADZ, Remark 7.4] we may choose\nλ0to be\nλ0=(2α\n2−1)2\n9.\nIn any case, λ0is not too small if αis not small.\nRemark 2.19. Corollary 2.17 covers in particular our initial motivating example\nfrom the introduction: for the Bernoulli shift with the Bernoulli mea sure, any\nnonconstant locally constant sampling function will produce an oper ator family10 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nthat is almost surely spectrally localized throughout the spectrum. One should\nremark, though, that dynamical localization will only hold away from a discrete set\nof exceptional energies. There are examples where such energies are indeed present\nand may produce transport at a rate that is understood; compar e [ADZ, DT1,\nDT2, JS, JSS] for relevant discussions of this phenomenon. We also emphasize that\nCorollary 2.16 fully covers the main results of [BS] and fills in the energy intervals\non which [BS] was inconclusive regarding the localization statement.\n3.Large Deviation Estimates – Proof of Theorem 2.10\nIn this section, we prove Theorem 2.10. Thus, we focus on sampling f unctions\nf∈LC orf∈SH. For simplicity, we denote the projectivized action of B∈\nSL(2,R) onRP1byB·v,v∈RP1.\n3.1.Stable and Unstable Holonomies. Let us recall the following central con-\ncept.\nDefinition 3.1. Astable holonomy hsfor a continuous A: Ω→SL(2,R) is a\nfamily of matrices\n{hs\nω,ω′∈SL(2,R) :ω∈Ω,ω′∈Ws\nloc(ω)},\nsuch that\n(i)hs\nω′,ω′′hs\nω,ω′=hs\nω,ω′′andhs\nω,ω= id,\n(ii)A(ω′)hs\nω,ω′=hs\nTω,Tω′A(ω),\n(iii) (ω,ω′)/ma√sto→hs\nω,ω′is uniformly continuous for all ω∈Ω and allω′∈Wu\nloc(ω).\nAnunstable holonomy {hu\nω,ω′:ω∈Ω,ω′∈Wu\nloc(ω)}forAis a stable holonomy for\nA−1overT−1.\nIfAis locally constant or fiber bunched, then it is a standard result (see e.g.\nthe proof of [ADZ, Lemma 4.1]) that the stable and unstable holonomie s can be\ndefined as\n(14)Hs\nω,ω′= lim\nn→∞An(ω′)−1An(ω), Hu\nω,ω′= lim\nn→∞A−n(ω′)−1A−n(ω)\nforω,ω′in the same stable and unstable sets, respectively. Moreover, the con-\nvergence is uniform for all ω∈Ω and allω′∈Ws\nloc(ω) and allω′∈Wu\nloc(ω),\nrespectively. The properties (i)–(iii) for Hs\nω,ω′,Hu\nω,ω′follow directly from the con-\nstruction. Holonomies that arise from (14) are called canonical holonomies ofA.\nDefinition 3.2. Suppose we are given a ( T,A)-invariant probability measure m\non Ω×RP1that projects to µin the first component. A disintegration ofmalong\nthe fibers is a measurable family {mω:ω∈Ω}of conditional probability measures\nonRP1such thatm=/integraltext\nmωdµ(ω), that is,\nm(D) =/integraldisplay\nΩmω({z∈RP1: (ω,z)∈D})dµ(ω)\nfor each measurable set D⊂Ω×RP1.\nBy Rokhlin’s disintegration theorem, such a disintegration exists. Mo reover,\n{˜mω:ω∈Ω}is another disintegration of mif and only if mω= ˜mωforµ-almost\neveryω∈Ω. By a straightforward calculation one checks that {A(ω)∗mω:ω∈Ω}\nis a disintegration of ( T,A)∗m. In particular, the facts above imply that mis\n(T,A)-invariant if and only if A(ω)∗mω=mTωforµ-almost every ω∈Ω.ANDERSON LOCALIZATION 11\nSuch a measure mwill be called an s-state(resp., au-state) if it is in addition\ninvariant under the stable (resp., unstable) holonomies. That is, th e disintegration\n{mω:ω∈Ω}satisfies (hs\nω,ω′)∗mω=mω′forµ-almost every ω∈Ω and every\nω′∈Ws\nloc(ω) (resp., (hu\nω,ω′)∗mω=mω′forµ-almost every ω∈Ω and every ω′∈\nWu\nloc(ω)). In this case, we say that {mω}iss-invariant (resp.,u-invariant ). A\nmeasure that is both an s-state and a u-state is called an su-state.\nDefinition 3.3. We say that a function defined on Ω only depends on the future\n(resp.,past) if it is constant on every local stable (resp., unstable) set.\nWe can use the stable or unstable holonomy to reduce Aso that it is constant\non local unstable or stable sets as follows. For each 1 ≤j≤ℓ, fix a choice of\nω(j)∈[0;j]. We define ϕ(ω) =ω∧ω(ω0), which depends only on the past. We\ndefine a new cocycle map as follows.\n˜A−1(ω) :=Hu\nT−1ω,ϕ(T−1ω)·A−1(ω)·Hu\nϕ(ω),ω.\nIt is clear that ( T−1,˜A−1) is conjugate to ( T−1,A−1) via the unstable holonomy.\nBy property (iii), ˜A−1is continuous. By conditions (i) and (ii), we have that\n˜A−1(ω) =Hu\nT−1ω,ϕ(T−1ω)·A−1(ω)·Hu\nϕ(ω),ω\n=Hu\nT−1ω,ϕ(T−1ω)·Hu\nT−1ϕ(ω),T−1ω·A(ϕ(ω))\n=Hu\nT−1φ(ω),ϕ(T−1ω)·A(ϕ(ω)),\nwhich implies that ˜A(ω) depends only on the past. Similary, we can conjugate\n(T,A) to (T,¯A) via the stable holonomy so that ¯Adepends only the future.\nThe mapsHu,E\nω,ω′are continuous and uniformly bounded for all Ein any compact\nset. It is then straightforward to see that large deviation estimat es as stated in\nDefinition 2.4 are preserved under such a conjugacy. Hence, from now on, we may\nwithout loss of generality assume that AEdepends only the past. In particular, for\nsuch matrices, we have\nHu\nω,ω′=I2\nfor allω∈Ω and allω′∈Wu\nloc(ω).\nIn the context of Theorem 2.10, f∈LC orf∈SH. Hence, AEis fiber bunched\nthroughout some connected compact interval Jcontaining the almost sure spec-\ntrum Σ f. So we may let Hs,E\nω,ω′andHu,E\nω,ω′be their canonical stable and unstable\nholonomies, respectively.\n3.2.Reduction of the Uniform LDT. LetFE: Ω×RP1→Ω×RP1be given\nbyFE(ω,v) = (Tω,AE(ω)·v). For a continuous function ϕ: Ω×RP1, we let\nSE\nn(ϕ)(ω,v) =/summationtextn−1\nk=0ϕ((FE)k(ω,v)) be its Birkhoff sum with respect to the dy-\nnamicsFE. Thus a measure νon Ω×RP1is (T,AE)-invariant if and only if\n(FE)∗ν=ν. Throughout this section, we may sometimes leave the dependence of\nAE,FE, andHs(u),EonEimplicit if it is clear from the context.\nLemma 3.4. For everyE∈Jη,AEhas a unique u-state. Moreover, the unique\nu-state is continuous in Ewith respect to the weak*-topology.\nProof.LetE∈Jηbe arbitrary. By the definition of Jηin (12) and the discussion\nsurrounding (12), we have L(E)>0. Thus, by Oseledec’s multiplicative ergodic\ntheorem,Ahas stable and unstable directions for µ-almost every ω. We consider\nthe Dirac measures δs(ω) andδu(ω) onRP1, whereδs(ω) andδu(ω) are Dirac12 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nmasses concentrated on the stable and unstable directions, resp ectively. We can\ndefine twoFE-invariant probability measures msandmson Ω×RP1as follows:\nms=/integraldisplay\nδs(ω)dµ(ω) andmu=/integraldisplay\nδu(ω)dµ(ω).\nThe invariance of msandmufollows from the ( T,A)-invariance of the stable\nand unstable directions, respectively. Property (ii) of Definition 3.1 implies\nAn(ω′)Hs\nω,ω′=Hs\nTnω,Tnω′An(ω),\nwhich in turn implies that the length of the vector An(ω′)Hs\nω,ω′vdecays exponen-\ntially for any vector v∈suppδs(ω). ThusHs\nω,ω′δs(ω) =δs(ω′), which implies that\nmsis ans-state (with disintegration {δs(ω)}ω∈Ω). Applying the same argument\nto (T−1,A−1) and the unstable holonomies Hu\nω,ω′, we obtain that muis au-state\n(with disintegration {δu(ω)}ω∈Ω). Moreover, it is not difficult to see that any F-\ninvariant measure mon Ω×RP1is a convex combination of muandms; see, for\nexample, [BBB].\nWe claim that muis the unique u-state ofA. Indeed, suppose there is a u-state\nm/\\⌉}atio\\slash=mu. Thenm=tmu+(1−t)msfor some 0 ≤t<1. Thus\nms=1\n1−tm+t\n1−tmu,\nwhich implies msis au-state, hence an su-state, ofAE. Thus,E∈ Ff, which\ncontradicts our choice of E.\nTo show the continuity of muinE, we letmu,Edenote the unique u-state ofAE\nfor eachE∈Jη. Let{En}be a convergent sequence in Jηand lim\nn→∞En=E0∈Jη.\nTo show that mu,Enconverges to mu,E0in the weak*-topology, it suffices to show\nthat the limit of any convergent subsequence is mu,E0. Without loss of generality,\nwe may just assume that {mu,En}is convergent with the limit denoted by m; we\nneed to show that m=mu,E0. Note that\nAEn(ω)−AE0(ω) =/parenleftbiggEn−E00\n0 0/parenrightbigg\n,\nwhich clearly implies\nlim\nn→∞/bar⌈blAEn−AE0/bar⌈bl∞= 0.\nNote by our assumption, Hu,E\nω,ω′=I2for all parameters. Thus, by a special case of\n[BBB, Lemma 4.3], it follows that mis au-state with respect to ( T,AE0). By the\nuniqueness of the u-state, we have m=mu,E0, as desired. /square\nBy the definition of a local unstable set, we may think of Wu\nlocas a map on Ω−:\nWu\nloc(ω−) ={ω∈Ω :π−(ω) =ω−}.\nThus we may decompose Ω as a disjoint union of local unstable sets as follows:\nΩ =/unionsqdisplay\nω−∈Ω−Wu\nloc(ω−).\nAdisintegration ofµalong the local unstable sets is a measurable family\n{µu\nω−: a probablity measure on Wu\nloc(ω−)}ω−∈Ω−ANDERSON LOCALIZATION 13\nsuch thatµ=/integraltext\nΩ−µu\nω−dµ−(ω−). In other words,\n(15) µ(D) =/integraldisplay\nΩ−µu\nω−/parenleftbig\n{ω∈D:π−(ω) =ω−}/parenrightbig\ndµ−(ω−)\nfor each measurable set D⊂Ω. Again by Rokhlin’s disintegration theorem, such a\ndisintegration exists. For v∈RP1, we letvbe a unit vector in the direction of v.\nThen we have:\nTheorem 3.5. For everyε >0, there exist C,c >0such that uniformly for all\n(ω−,v)∈Ω−×RP1and allE∈Jη, we have\nµu\nω−/braceleftbigg\nω∈Wu\nloc(ω−) :/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnlog/bar⌈blAE\nn(ω)v/bar⌈bl−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n<Ce−cn.\nTheorem 2.10 is an easy consequence of Theorem 3.5 as follows.\nProof of Theorem 2.10. By Theorem 3.5 and the disintegration (15), we clearly\nhave\nµ/braceleftbigg\nω∈Ω :/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnlog/bar⌈blAE\nn(ω)v/bar⌈bl−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n<Ce−cn\nuniformly for all E∈Jη. Choosing v= (0,1) and (1,0), it then follows from\na standard computation (see, e.g., the proof of [BDF+, Theorem 3.1 ]) that the\nestimate above implies\nµ/braceleftbigg\nω∈Ω :/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnlog/bar⌈blAE\nn(ω)/bar⌈bl−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n<Ce−cn\nuniformly for all E∈Jη, as desired. /square\nOn the other hand, Theorem 3.5 follows from the next theorem. Rec all\nthat for a continuous function ϕon Ω×RP1, we denote by SE\nn(ϕ)(ω,v) =/summationtextn−1\nk=0ϕ((FE)k(ω,v)) its Birkhoff sum with respect to the dynamics FE.\nTheorem 3.6. For everyϕ∈Cα(Ω×RP1,R),ε >0, andE0∈Jη, there exist\nC,c,r> 0such that uniformly for all (ω−,v)∈Ω−×RP1and allE∈(E0−r,E0+\nr)∩Jη, we have that\nµu\nω−/braceleftbigg\nω∈Wu\nloc(ω−) :/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnSE\nn(ϕ)(ω,v)−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n<Ce−cn.\nTheorem 3.5 follows from Theorem 3.6:\nProof of Theorem 3.5. LetϕE(ω,v) = log/bar⌈blAE(ω)v/bar⌈bl. It is a direct computation to\nsee thatϕE∈Cα(Ω×RP1) sincef∈Cα(Ω,R). Applying Theorem 3.6 to ϕE0and\nthe givenε>0, we obtain some positive constants r0,c,Csuch that\nµu\nω−/braceleftbigg\nω∈Wu\nloc(ω−) :/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnSE\nn(ϕE0)(ω,v)−/integraldisplay\nϕE0dmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε\n3/bracerightbigg\n<Ce−cn.\nuniformly for all ( ω−,v)∈Ω−×RP1and allE∈(E0−r0,E0+r0)∩Jη.\nFor anyE0∈Jη, let{En}be a sequence converging to E0. Then it is clear that\nlim\nn→∞/bar⌈blϕEn−ϕE0/bar⌈bl∞= 0,14 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nwhere the supremum norm is taken over Ω ×RP1. Thus for the given ε>0, there\nexistsr1>0 such that for all Ewith|E−E0|<r1, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nnSE\nn(ϕE)−1\nnSE\nn(ϕE0)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n0<ε\n3and\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nϕEdmu,E−/integraldisplay\nϕE0dmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε\n3.\nSettingr= min{r0,r1}and combining the estimates above, we obtain\nµu\nω−/braceleftbigg\nω∈Wu\nloc(ω−) :/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnSE\nn(ϕE)(ω,v)−/integraldisplay\nϕEdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n<Ce−cn,\nuniformly for all ( ω−,v)∈Ω−×RP1and allE∈(E0−r,E0+r)∩Jη.\nBy compactness of Jη, and changing C,cif necessary, we get the uniformity in\nE∈Jη. This yields the desired conclusion of Theorem 3.5. Indeed, we just n eed\nto note that\n1\nnSE\nn(ϕE)(ω,v) =1\nnn−1/summationdisplay\nk=0ϕE(Tkω,AE\nk(ω)v)\n=1\nnn−1/summationdisplay\nk=0log/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAE(Tkω)AE\nk(ω)v\n/bar⌈blAE\nk(ω)v/bar⌈bl)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n=1\nnlog/bar⌈blAE\nn(ω)v/bar⌈bl,\nand by Birkhorff ergodic theorem, we have\n/integraldisplay\nϕEdmu,E=L(E).\nThis concludes the proof. /square\n3.3.Proof of Theorem 3.6. In this subsection, we shall prove Theorem 3.6. Let\nC,cbe some universal constants with 0 < c <1< C. We also let ω−,j∈Ω−\njfor\neachj∈ {1,...,ℓ}.\nLemma 3.7. Consider any ω−,j∈Ω−\njand letνube a probability measure on\nWu\nloc(ω−,j)such that\n(16) C−1≤d((π+)∗νu)\ndµ+\nj≤C.\nThen, we have\n(17)1\nnn−1/summationdisplay\nk=0Tk\n∗νu→µ\nin the weak- ∗topology as n→ ∞, uniformly in ω−,j,1≤j≤ℓ, andνu.\nProof.Letνj=µ−\nj×(π+)∗νube a measure on [0; j]. Letν=1\nµ−\nj(Ω−\nj)νj, which is\na probability measure. By (16), we have\nC−1≤d(µ−\nj×(π+)∗νu)\nd(µ−\nj×µ+\nj)≤C.ANDERSON LOCALIZATION 15\nBy the bounded distortion property of µ, it is straightforward to see (see, e.g.,\n[ADZ, Section 2.1.2]) that\nC−1≤dµj\nd(µ−\nj×µ+\nj)≤C.\nCombining the two estimates above, we obtain\nC−1≤dνj\ndµj≤C,\nwhich implies that for all n≥1, we have\n(18) C−1≤d/parenleftbig/summationtextn−1\nk=0Tk\n∗νj/parenrightbig\nd/parenleftbig/summationtextn−1\nk=0Tk∗µj/parenrightbig≤C.\nBy(7), foreverycylinderset[ i;j0,···,js], wehaveT−k([i;j0,···,js])∩[0;j]/\\⌉}atio\\slash=∅\nfor allksufficiently large. In particular, we can consider such k’s so thatk+i>0.\nBy the bounded distortion property, we then have\n(Tk\n∗µj)([n;j0,···,js]) =µ(T−k([i;j0,···,js]∩[0;j])\n=µ([i+k;j0,···,js]∩[0;j])\n≥C−1µ([i+k;j0,···,js])·µ([0;j])\n≥C−1µ([i+k;j0,···,js]).\nThus byT-invariance of µ, for every cylinder set [ i;j0,···,js], we have for each\nlargenthat\nC−1≤1\nn/summationtextn−1\nk=0(Tk\n∗µj)([i;j0,···,js])\nµ([i;j0,···,js])≤C.\nThus by (18), the estimates above hold true if we replace µjbyν=1\nµ−\nj(Ω−\nj)νj. This\nfurther implies that if ν′is a limit point of the sequence {1\nn/summationtextn−1\nk=0(Tk\n∗ν)}n, then we\nhave\n(19) C−1≤dν′\ndµ≤C.\nSince Ω is a compact metric space, by the Stone-Weierstrass theor em the set\nC(Ω,R) is separable, which in turn implies that the weak- ∗topology on the set of\nBorel probability measures on Ω is metrizable. For instance, such a m etric may be\nchosen as\nρ(ν,ν′) :=∞/summationdisplay\ns=12−s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nfsdν−/integraldisplay\nfsdν′/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhere{fs:s∈Z+}is a dense subset of the set {f∈C(Ω,R) :/bar⌈blf/bar⌈bl∞≤1}.\nNote that by the Banach-Alaoglu theorem, the weak- ∗topology on the set of Borel\nprobability measures on any topological space is compact. Combining the two facts\nabove, we see that the the weak- ∗topology on the set of Borel probability measures\non Ω is sequentially compact.\nNow letν(n)=1\nn/summationtextn−1\nk=0Tk\n∗νfor anyn∈Z+. By the discussion above and (19),\nthe sequence {ν(n)}has a weak- ∗accumulation point ν(0)that is equivalent to µ.\nIt is clear that ν(0)is alsoT-invariant. Set g=dν(0)\ndµ, which isT-invariant since\nbothν(0)andµareT-invairant. By ergodicity of µ, there is a constant csuch that16 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\ng(ω) =cforµ-a.e.ω. Since both ν(0)andµare probability measures, we must\nhavec= 1, which implies ν(0)=µ.\nThus, by uniqueness of the accumulation point, we have ν(n)→µasn→ ∞in\nthe weak- ∗topology. Applying uniqueness of the limit and metrizability of the set\nof Borel probability measures, we have that the convergence is un iform inν. That\nis, the convergence is uniform in the choices of νusatisfying (16) and the choices\nofω−,j. Indeed, assume for the sake of contradiction that the converg ence is not\nuniform. Then there is ε0>0 and a strictly increasing sequence {nl}l≥1inZ+\nwith the following property. For every l, there isν(l) satisfying (19) such that\nρ(ν(nl)(l),µ)≥ε0,\nwhereν(n)(l) :=1\nn/summationtextn−1\nk=0Tk\n∗ν(l). Without loss of generality, we may assume\n{ν(nl)(l)}l≥1is convergent and the limit is /tildewideν. Thus/tildewideνsatisfies (19) and ρ(/tildewideν,µ)≥ε0.\nMoreover,/tildewideνisT-invariant. Indeed,\nρ(T∗ν(nl)(l),ν(nl)(l)) =1\nnl∞/summationdisplay\ns=12−s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglenl−1/summationdisplay\nk=0/integraldisplay\nfsd(Tk+1\n∗ν(l))−/integraldisplay\nfsd(Tk\n∗ν(l))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=1\nnl∞/summationdisplay\ns=12−s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nfsd(Tnl∗ν(l))−/integraldisplay\nfsd(ν(l))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤2\nnl,\nwhich implies\nT∗/tildewideν= lim\nl→∞T∗ν(nl)(l) = lim\nl→∞ν(nl)(l) =/tildewideν.\nThus the same argument above showing ν(0)=µimplies that/tildewideν=µ, which contra-\ndictsρ(/tildewideν,µ)≥ε0. In summary, so far we have obtained the estimate (17) except\nthat we only did so for ν. The final step is to replace νbyνu. To this end, it\nsuffices to prove the following claim.\nIf we letνu,(n)=1\nn/summationtextn−1\nk=0Tk\n∗νu, then for any continuous ϕ, we claim that\nlim\nn→∞/parenleftbigg/integraldisplay\nϕdνu,(n)−/integraldisplay\nϕdν(n)/parenrightbigg\n= 0,\nwhere the convergence is also uniform in ω−,jandνu. Indeed, we have\n/integraldisplay\nϕdνu,(n)−/integraldisplay\nϕdν(n)=1\nnn−1/summationdisplay\nk=1/parenleftBigg/integraldisplay\nϕd(Tk\n∗νu)−1\nµ−\nj(Ω−\nj)/integraldisplay\nϕd(Tk\n∗νj)/parenrightBigg\n,ANDERSON LOCALIZATION 17\nwhich implies that it suffices to show/integraltext\nϕd(Tk\n∗νu)−1\nµ−\nj(Ω−\nj)/integraltext\nϕd(Tk\n∗νj)→0, uni-\nformly inω−andνu. To this end, we have/integraldisplay\nϕd(Tk\n∗νu)−1\nµ−\nj(Ω−\nj)/integraldisplay\nϕd(Tk\n∗νj)\n=/integraldisplay\nϕ◦Tkdνu−1\nµ−\nj(Ω−\nj)/integraldisplay\nϕ◦Tkdνj\n=/integraldisplay\nϕ◦Tkdνu−1\nµ−\nj(Ω−\nj)/integraldisplay\nϕ◦Tkd/parenleftbig\nµ−\nj×(π+)∗νu/parenrightbig\n=/integraldisplay\nWu\nloc(ω−,j)ϕ◦Tkdνu−/integraldisplay\nΩ+\nj/parenleftBigg\n1\nµ−\nj(Ω−\nj)/integraldisplay\nΩ−\njϕ◦Tk(ω−,ω+)dµ−\nj(ω−)/parenrightBigg\nd/parenleftbig\n(π+)∗νu/parenrightbig\n(ω+)\n=/integraldisplay\nΩ+\nj/parenleftBigg\nϕ◦Tk(ω−,j,ω+)−1\nµ−\nj(Ω−\nj)/integraldisplay\nΩ−\njϕ◦Tk(ω−,ω+)dµ−\nj(ω−)/parenrightBigg\nd/parenleftbig\n(π+)∗νu/parenrightbig\n(ω+).\nClearly, we have/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕ◦Tk(ω−,j,ω+)−1\nµ−\nj(Ω−\nj)/integraldisplay\nΩ−\njϕ◦Tk(ω−,ω+)dµ−\nj(ω−)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nµ−\nj(Ω−\nj)/integraldisplay\nΩ−\njϕ◦Tk(ω−,j,ω+)−ϕ◦Tk(ω−,ω+)dµ−\nj(ω−)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤1\nµ−\nj(Ω−\nj)/integraldisplay\nΩ−\nj/vextendsingle/vextendsingleϕ◦Tk(ω−,j,ω+)−ϕ◦Tk(ω−,ω+)/vextendsingle/vextendsingledµ−\nj(ω−),\nwhich tends to 0 uniformly in ω+ask→ ∞by uniform continuity of ϕand the\nfactd(Tk(ω−,j,ω+),Tk(ω−,ω+))→0 ask→ ∞. The estimates above imply that/integraltext\nϕd(Tk\n∗νu)−1\nµ−\nj(Ω−\nj)/integraltext\nϕd(Tk\n∗νj)→0, uniformly in ω−,jandνu, as desired. /square\nLemma 3.8. LetE∈Jηandω−,j∈Ω−\nj. Supposemis a probability measure on\nWu\nloc(ω−,j)×RP1, whose projection νutoWu\nloc(ω−,j)satisfies the assumptions of\nLemma 3.7. Then, we have\n1\nnn−1/summationdisplay\nk=0(FE)k\n∗m→mu,E\nin the weak- ∗topology, uniformly in ω−,j,E∈Jη, and such choices of m.\nProof.By compactness of Ω ×RP1and the same reasoning as in the proof of\nLemma 3.7, we see that weak- ∗accumulation points of1\nn/summationtextn−1\nk=0(FE)k\n∗mexist.\nConsider such an accumulation point and denote it by /tildewidem. Without loss of gen-\nerality, we may just assume that1\nn/summationtextn−1\nk=0(FE)k\n∗mconverges to /tildewidem. Clearly,/tildewidemis\ninvariantunder FEand it projects to an accumulation point of1\nn/summationtextn−1\nk=0Tk\n∗νuin the\nfirstcomponent. Byourassumptionon νuandLemma3.7,1\nn/summationtextn−1\nk=0Tk\n∗νuconverges\ntoµasn→ ∞. That is,/tildewidemprojects to µin the first coordinate.\nNext, we show that any disintegration {/tildewidemω}ω∈Ωof/tildewidemis invariant under the\nunstable holonomy. By our assumption that Adepends only on the past, this is\nequivalent to having /tildewidemω=/tildewidemω′forωandω′in the same local unstable set. To\nthis end, we let /tildewidem−= (π−×id)∗(/tildewidem), which is a measure on Ω−×RP1. First, we18 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nnote thatAEnaturally descends to a map on Ω−since it depends only on the past.\nThat is,AE(ω) =˜AE(π−ω) for some map ˜AE: Ω−→SL(2,R). Abusing notation\nslightly, we still let AEdenote the map on Ω−. LetFE\n−denote the projectivized\naction of (T−,(AE)−1) on Ω−×RP1whereT−is the right shift on Ω−. Then we\nnaturally have\n(π−×id)◦(T,AE)−1= (T−,(AE)−1)◦(π−×id),\nwhich implies ( π−×id)◦(FE)−1=FE\n−◦(π−×id). We claim that /tildewidem−is invariant\nunderFE\n−. Indeed,\n(FE\n−)∗/tildewidem−= (FE\n−)∗(π−×id)∗(/tildewidem)\n= (FE\n−◦(π−×id))∗/parenleftbig\nlim\nn→∞1\nnn−1/summationdisplay\nk=0(FE)k\n∗m/parenrightbig\n= ((π−×id)◦(FE)−1)∗/parenleftbig\nlim\nn→∞1\nnn−1/summationdisplay\nk=0(FE)k\n∗m/parenrightbig\n= (π−×id)∗/parenleftbig\nlim\nn→∞1\nnn−1/summationdisplay\nk=0(FE)k−1\n∗m/parenrightbig\n= (π−×id)∗/tildewidem\n=/tildewidem−.\nLet{/tildewidem−\nω−}be a disintegration of /tildewidem−. ByFE\n−-invariance, we then have\n(AE(T−ω−)−1)∗/tildewidem−\nω−=/tildewidem−\nT−ω−,\nor equivalently,\n(20) ( AE\n−1(ω−))∗/tildewidem−\nω−=/tildewidem−\nT−ω−.\nByaspecialcaseof[AV, Lemma3.4], thedisintegration {/tildewidemω}of/tildewidemcanberecovered\nfrom the disintegration {/tildewidem−\nω−}of/tildewidem−via\n/tildewidemω= lim\nn→∞(AE\n−n(π−(Tnω)))∗/tildewidem−\nπ−(Tnω)\nforµ-a.e.ω. But (20) and the fact T−◦π−=π−◦T−1imply that\n(AE\n−n(π−(Tnω)))∗/tildewidem−\nπ−(Tnω)=/tildewidem−\nπ−ω\nfor alln≥1. Thus we have\n/tildewidemω=/tildewidem−\nπ−ω,\nwhich stays constant for all ω′∈Wu\nloc(π−ω). This concludes the proof that /tildewidem\nis au-state. Therefore, by uniqueness of the u-state, it must be equal to mu,E.\nUniform convergence follows again from uniqueness of the limit as in th e proof of\nLemma 3.7. /square\nThe final piece needed to prove Theorem 3.6 is the following large devia tion\nestimate for martingale difference sequences; see, for example, [ AEP, BS].\nLemma 3.9. Let{dn}n≥1be a martingale difference sequence adapted to some\nfiltration. That is, {dn}n≥1is a sequence of variables such that\n(21) E(dn+1|Fn) = 0,ANDERSON LOCALIZATION 19\nwhereFnis theσ-algebra generated by d1,...,d n. Then for every ε>0, there is a\nc>0such that for all N≥1,\n(22) P\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\n1dn/vextendsingle/vextendsingle/vextendsingle/vextendsingle>εN1\n2/parenleftBiggN/summationdisplay\n1/bar⌈bldn/bar⌈bl2\n∞/parenrightBigg1\n2\n<e−cε2N.\nRemark 3.10. Note that if there is asuch that\n(23) sup\nn≥1/bar⌈bldn/bar⌈bl∞<a,\nthen (22) becomes\n(24) P/parenleftBig/vextendsingle/vextendsingle/vextendsingle1\nNN/summationdisplay\n1dn/vextendsingle/vextendsingle/vextendsingle>ε/parenrightBig\n≤e−cε2\n2a2n,\nwhich is alreadyknown; compare, for example, [Az] or [LV]. In fact, cmay be taken\nto be 1 in (24).\nWe are now ready to prove Theorem 3.6.\nProof of Theorem 3.6. By the bounded distortion property of µ, there exists C≥1\nsuch that for each 1 ≤j≤ℓandµ−-almost every ω−,j∈Ω−\nj,\n(25) C−1≤d(π+\n∗µu\nω−,j)\ndµ+\nj≤C.\nSince the estimates on a zero µ−-measure subset of Ω−are negligible when we\npass the estimates to µvia (15), we may without loss of generality assume that\n(25) holds uniformly for all ω−,j∈Ω−\nj. Thusµu\nω−,jsatisfies the assumption of\nLemma 3.7. Choose any v∈RP1and letmbe the lift of µu\nω−,jtoWu\nloc(ω−,j)×{v}.\nThenmsatisfies the assumption of Lemma 3.8. Since π+T=T+π+, we have\nπ+\n∗T∗µu\nω−,j= (T+)∗π+\n∗µu\nω−,jwhich implies\nC−1≤d(π+\n∗T∗µu\nω−,j)\nd((T+)∗µ+\nj)≤C.\nByT+-invariance of µ+, forjiadmissible we have (( T+)∗µ+\nj)|[0;i]+=µ+\ni. Hence,\nthe above estimate implies that1\nµu\nω−,j([0:ji])T∗µu\nω−,j|[0;i]is a probability measure\nthat satisfies (25), hence the assumption of Lemma 3.7. Note that\nFE\n∗m|[0;i]×RP1= (FE|[0;j,i]×RP1)∗m= (FE|(Wu\nloc(w−,j)∩[0;j,i])×RP1)∗m\nis the lift of T∗µu\nω−,j|[0:i]to (T(Wu\nloc(ω−,j))∩[0 :i])×{AE(ω)v}, whereAE(ω)vis\na constant since ω∈Wu\nloc(ω−,j)∩[0;j,i] andAEdepends only on the past. Thus\n1\nµu\nω−,j([0 :ji])FE\n∗m|[0:i]×RP1=1\nm([0 :j,i]×RP1)FE\n∗m|[0:i]×RP1\nsatisfies the assumptions of Lemma 3.8 as well. By induction, if ji1···isis admis-\nsible, then\n1\nµu\nω−,j([0 :j,i1,...,is])Ts\n∗µu\nω−,j|[0;i1,...,is]\nsatisfies the assumption of Lemma 3.7. Likewise,\n(FE)s\n∗m|[0;i1,...,is]=/parenleftbig\n(FE)s|Wu\nloc(w−,j)∩[0;j,i1,...,is]/parenrightbig\n∗m20 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nis the lift of Ts\n∗µu\nω−,j|[0:i1,...,is]toTs(Wu\nloc(ω−,j))∩[0 :i1,...,is]×{AE\ns(ω)v}, where\nagainAE\ns(ω)vstays constant since ω∈Wu\nloc(ω−,j)∩[0;j,i1,...,is−1,is]. Hence,\n1\nm([0 :j,i1,...,is]×RP1)(FE)s\n∗m|[0:i1,...,is]\nsatisfies the assumption of Lemma 3.8. Now for each ω∈Wu\nloc(ω−,j), we define\nDi(ω) :=/parenleftbig\n[0;ω0,...,ω i]∩Wu\nloc(ω−,j)/parenrightbig\n×RP1.\nNote that for every integrable function ψ, we have\n/integraldisplay\nDi−1(Tω)ψ(˜ω)d(FE)i\n∗m(˜ω) =/integraldisplay\nDi(ω)ψ◦(FE)i(˜ω)dm(˜ω),\nwhere ˜ωis the variable of the integrand, which should not be confused with ω. We\nleave ˜ωimplicit wheneveris it clear from the context. Consider a H¨ older cont inuous\nfunctionϕ∈Cα(Ω×RP1,R). By Lemma 3.8 and the facts described above, given\nε>0, there isN≥1 such that for every i≥0 and every ω∈Ω,\n(26)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nm(Di(ω))/integraldisplay\nDi(ω)1\nNSN(ϕ◦(FE)i)dm−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε\n4.\nLet us rewrite this as follows. Define Yi:Wu\nloc(ω−,j)→Rby\nYi(ω) =1\nm(Di(ω))/integraldisplay\nDi(ω)Si(ϕ)dm,\nwhich depends only on ω0,...,ω i. LetBibe theσ-algebra generated by Y0,...,Y i,\nwhich is basically generated by the cylinder sets [0; n0,...,n i]. In particular, the\nconditional expectation of Yi+Nrespect to Biis\nE/parenleftBig\nYi+N/vextendsingle/vextendsingle/vextendsingleBi/parenrightBig\n=1\nm(Di(ω))/summationdisplay\n˜ω∈Di(ω)m(Di+N(˜ω))Yi+N(˜ω)\n=1\nm(Di(ω))/summationdisplay\n˜ω∈Di(ω)/integraldisplay\nDi+N(˜ω)Si+N(ϕ)dm (27)\n=1\nm(Di(ω))/integraldisplay\nDi(ω)Si+N(ϕ)dm.\nClearly,E/parenleftBig\nYi/vextendsingle/vextendsingle/vextendsingleBi/parenrightBig\n=Yi. Thus the estimate (26) reads\n(28)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nNE/parenleftBig\nYi+N−Yi/vextendsingle/vextendsingle/vextendsingleBi/parenrightBig\n−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε\n4.\nIf we define\nXn=YnN−n/summationdisplay\nk=1E/parenleftbig\nYkN−Y(k−1)N|B(k−1)N/parenrightbig\n,\nit follows that {Xn}is a martingale, that is, (21) holds for {Xn}. Indeed,\nXn+1−Xn=Y(n+1)N−YnN−E/parenleftBig\nY(n+1)N−YnN/vextendsingle/vextendsingle/vextendsingleBnN/parenrightBig\n=Y(n+1)N−E/parenleftBig\nY(n+1)N/vextendsingle/vextendsingle/vextendsingleBnN/parenrightBig\n. (29)ANDERSON LOCALIZATION 21\nSince theσ-algebra Fngenerated by X1,...,X nis precisely BnN, the above equa-\ntion implies that for all n≥1,\nE/parenleftBig\nXn+1−Xn/vextendsingle/vextendsingle/vextendsingleFn/parenrightBig\n= 0.\nWe claim that (23) also holds true for {Xn}because of the H¨ older continuity of ϕ.\nIndeed, it is clear that |X1|=|YN−E(YN−Y0/vextendsingle/vextendsingle/vextendsingleB0)|=|YN−Y0−E(YN/vextendsingle/vextendsingle/vextendsingleB0)|<CN.\nBy (27) and (29), Xn+1−Xnmay be rewritten as\n1\nm(D(n+1)N(ω))/integraldisplay\nD(n+1)N(ω)S(n+1)N(ϕ)dm−1\nm(DnN(ω))/integraldisplay\nDnN(ω)S(n+1)N(ϕ)dm.\nNote thatAE\ni(ω′)vis independent of ω′for allω′∈[0;ω0,...,ω nN]∩Wu\nloc(ω−,j)\nand all 0 ≤i≤nN. So we may denote the projection of such AE\ni(ω′)vinRP1by\nvi. Now for any ω′,ω′′∈[0;ω0,...,ω nN]∩Wu\nloc(ω−,j), we have\n/vextendsingle/vextendsingleS(n+1)N(ϕ)(ω′,v)−S(n+1)N(ϕ)(ω′′,v)/vextendsingle/vextendsingle\n≤\nnN−1/summationdisplay\ni=0+(n+1)N−1/summationdisplay\nnN\n/vextendsingle/vextendsingle/vextendsingleϕ/parenleftbig\n(FE)i(ω′,v)/parenrightbig\n−ϕ/parenleftbig\n(FE)i(ω′′,v)/parenrightbig/vextendsingle/vextendsingle/vextendsingle\n≤nN−1/summationdisplay\ni=0/vextendsingle/vextendsingle/vextendsingleϕ/parenleftbig\n(FE)i(ω′,v)/parenrightbig\n−ϕ/parenleftbig\n(FE)i(ω′′,v)/parenrightbig/vextendsingle/vextendsingle/vextendsingle+CN\n≤nN−1/summationdisplay\ni=0/vextendsingle/vextendsingleϕ/parenleftbig\nTiω′,vi/parenrightbig\n−ϕ/parenleftbig\nTiω′′,vi/parenrightbig/vextendsingle/vextendsingle+CN\n≤CnN−1/summationdisplay\ni=0d(Tiω′,Tiω′′)α+CN\n≤CnN−1/summationdisplay\ni=0e−α(nN−i)+CN\n≤C(e−α+N),\nwhereCmay vary but is independent of ω,ω−,j,v, orE. Leta=C(e−α+N).\nThis implies that\n|X1|<aand|Xn+1−Xn|<afor alln≥1.\nThus, it follows from Lemma 3.9 that for every δ>0 and alln≥1, we have\n(30) µu\nω−,j/braceleftbigg\nω:/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnXn/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δ/bracerightbigg\n=P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnXn/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δ/parenrightbigg\n<e−δ2\n2a2n.\nSupposeω∈Wu\nloc(ω−,j) satisfies\n(31)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNSnN(ϕ)(ω)−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε.\nThen by the same argument above bounding |Xn+1−Xn|, we have for all ω′∈\n[0;ω0,...,ω nN]∩Wu\nloc(ω−,j) that\n|SnN(φ)(ω,v)−SnN(φ)(ω′,v)|<C.22 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nCombining (31) and the estimate above, we obtain for all n≥N0that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNYnN(ω)−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε\n2,\nwhereN0depends only on ϕandε. By the estimate above and (28), we have for\nallωsatisfying (31) that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNXn/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNYnN−1\nnn/summationdisplay\nk=11\nNE/parenleftBig\nYkN−Y(k−1)N/vextendsingle/vextendsingle/vextendsingleB(k−1)N/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNYnN−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\nnn/summationdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nNE/parenleftbig\nYkN−Y(k−1)N|B(k−1)N/parenrightbig\n−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n>ε\n4.\nApplying (30) to δ=εN\n4, we obtain for all n≥N0,\nµu\nω−,j/braceleftbigg\nω:/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNSnN(ϕ)(ω)−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n≤µu\nω−,j/braceleftbigg\nω:/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnXn/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥Nε\n4/bracerightbigg\n≤e−Nε2\n16a2nN.\nAbsorbingthe statement for 1 ≤n≤N0into some constant C=C(ϕ,ε), we obtain\nthe desired estimate\nµu\nω−,j/braceleftbigg\nω:/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNSnN(ϕ)(ω)−/integraldisplay\nϕdmu,E/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n<Ce−Nε2\n16a2nN\nalong the sequence {nN}with all the constants depending only on ϕandε.\nReplacing the sequence {nN}aboveby {nN+l}forl= 1,...,N−1 and running\nthe same argument, we obtain the estimate (24) for all n. /square\n4.Establishing Localization – Proof of Theorem 2.12\nIn this section, we derive localization from PLE and ULD, and hence pr ove\nTheorem2.12. Theoverallstructureoftheprooffollowstheoutlin eofthetreatment\nof the Bernoulli Anderson case in [BDF+]. However, many of the key s teps in the\nproof given in [BDF+] require the independence of the potential valu es and hence\nwon’t work in the same way in our far more general setting. Consequ ently, we will\ngive full details whenever the absence of independence requires a n ew argument,\nwhile referring the reader to [BDF+] for details regarding the steps that extend\nwithout changes from the Bernoulli Anderson case to our setting.\nFor some of the key lemmas we have to use techniques from [ADZ, BS]. For ex-\nample, Lemma 4.3 generalizes [BS, Lemma 8.1], while the proof of Propos ition 4.6,\nthe elimination of double resonances, combines various different tec hniques from\n[ADZ, Section 3], [BS, Section 9], and [BDF+, Section 6].ANDERSON LOCALIZATION 23\n4.1.Bounding the Green Function. In the proof we shall use the H¨ older conti-\nnuity of the Lyapunove exponent, which is a direct consequence of PLE and ULD;\nsee, for example, [BDF+, Theorem 4.1]. We formulate it as follows. The re exist\nconstantsC >0,β >0 depending on fandIsuch that\n(32) |L(E)−L(E′)| ≤C|E−E′|β\nfor allE,E′∈I. First, we define\ngn(ω,E) =1\nnlog/bar⌈blAE\nn(ω)/bar⌈bl.\nLet Γ = supω∈Ω,E∈I/bar⌈blAE(ω)/bar⌈bl. Sincef∈Cα(Ω,R), a direct computation shows\nthat\n(33) |gn(ω′,E′)−gn(ω,E)|<Γn−1/parenleftbig\nCd(ω,ω′)α+|E′−E|/parenrightbig\nuniformly for all E∈I. Then we have:\nLemma 4.1. For everyε>0, there is a n0=n0(ε)such that\nµ/braceleftBig\nω:/vextendsingle/vextendsingle/vextendsingleL(E)−1\nrr−1/summationdisplay\ns=0gn(Tns+s0ω,E)/vextendsingle/vextendsingle/vextendsingle>ε/bracerightBig\n≤e−cε2\nn2r\nfor allE∈I,r≥Cn\nε,s0∈Zand alln≥n0.\nUsing the techniques of [ADZ, Section 3], we can reduce gn(·,E) to a family of\nfunctions in Cα\n2(Ω+,R). We briefly introduce the reduction process since we need\na bit more information than what is discussed in [ADZ, Section 3]. Follow ing the\ndiscussion after Definition 3.3, for each 1 ≤j≤ℓ, we fix a choice ω(j)∈[0;j] and\nconsiderϕ(ω) =ω(ω0)∧ω∈Wu\nloc(ω(ω0))∩Ws\nloc(ω). For each n, we construct a\nfamily of functions hn(ω,E) via\n(34) hs\nn(ω,E) :=∞/summationdisplay\nk=0/bracketleftbig\ngn(Tkω,E)−gn(Tkϕ(ω),E)/bracketrightbig\n.\nThen\n(35) g+\nn(ω,E) :=gn(ω,E)+hn(Tω,E)−hn(ω,E)\nis constant on every local stable set Ws\nloc(ω). By the estimate of [ADZ, Section 3],\nwe see that/vextendsingle/vextendsingleh(ω,E)−h(ω′,E)/vextendsingle/vextendsingle<C′Γn−1d(ω,ω′)α\n2,\nwhereC′=C\n1−e−α\n2andCis from (33). In particular, we may just say that\n(36) |g+\nn(ω,E)−g+\nn(ω′,E)|<CΓn−1d(ω,ω′)α\n2.\nMoreover, choosing N=⌈logΓ\nαn⌉, we see that\n|hs\nn(ω)| ≤N/summationdisplay\nk=0/vextendsingle/vextendsinglegn(Tkω)−gn(Tkϕ(ω))/vextendsingle/vextendsingle+/vextendsingle/vextendsingle∞/summationdisplay\nN/vextendsingle/vextendsinglegn(Tkω)−gn(Tkϕ(ω))/vextendsingle/vextendsingle\n≤2ΓN+Ce−αNΓn(37)\n≤CΓN\n≤Cn,24 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nwhich implies that\n(38) sup\nE∈I/bar⌈blg+\nn(·,E)/bar⌈bl∞≤Cn.\nThen, following the strategy of [ADZ, Section 3], Lemma 4.1 will be a con sequence\nof the following lemma:\nLemma 4.2. For everyε>0, there is a n0=n0(ε)such that\nµ+/braceleftBig\nω+:/vextendsingle/vextendsingle/vextendsingleL(E)−1\nrr−1/summationdisplay\ns=0g+\nn(Tns\n+ω+,E)/vextendsingle/vextendsingle/vextendsingle>ε/bracerightBig\n≤e−cε2\nn2r\nfor allE∈I,r∈Z+, and alln≥n0.\nThe proof of Lemma 4.2 relies on Lemma 4.3 below, whose proof again us es\nLemma 3.9.\nLemma 4.3. LetK >1and assume that Fisα-H¨ older continuous on Ω+with\n/bar⌈blF/bar⌈bl∞<1and|F(ω)−F(ω′)|<Kd(ω,ω′)α. Then for all ε>0and allr,n≥1,\nwe have\nµ+/braceleftBigg\nω+:/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0F(Tns\n+ω+)−/integraldisplay\nFdµ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε/bracerightBigg\n<exp/parenleftbigg\n−cε2n2\nlog2(Kε−1)r/parenrightbigg\n.\nProof.Forl= (l1,...,ln), where the finite word l1···lnis admissible, we instead\njust say that lis admissible. Let |l|denotes the number of entries in l. For an\nadmissiblel, we set Ω+\nl:= [0;l]+.\nFix anyn≥1 and let Bibe theσ-algebra generated by the cylinder sets Ω+\nl,\nwhere|l|=ni. Denote by Eithe conditional expectation operator with respect to\nBi. Hence,\nEi[F](ω+) =/summationdisplay\nl1\nµ+(Ω+\nl)/integraldisplay\nΩ+\nlFdµ+·χΩ+\nl(ω+),\nwhere the sum is taken over all admissible l. Note that E0[F] =/integraltext\nFdµ+and we\nlet ∆iF=Ei[F]−Ei−1[F] for alli≥1. Clearly, we only need to consider ε >0\nsmall. Thus we may choose an integer Nsuch that\nlog10K\nε<αnN < 2log10K\nε.\nIn particular, we have\n(39) Ke−αnN<ε\n10and1\nN>cn\nlogK\nε.\nBy H¨ older continuity of F, we have\n(40) /bar⌈blF−EN[F]/bar⌈bl∞<Ke−nNα<ε\n10.\nDenoteEN[F] byFNfor simplicity. Let i∧j= max{i,j}. Note that\nFN=/integraldisplay\nFdµ++N/summationdisplay\ni=0∆i(F).ANDERSON LOCALIZATION 25\nThus, forr≥1\n1\nrr−1/summationdisplay\ns=0FN(Tns\n+ω+) =/integraldisplay\nFdµ+\n+1\nr∆1F (=d1)\n+1\nr[∆1F(Tn\n+ω+)+∆2F(ω+)] (= d2)\n+···+1\nri∧N−1/summationdisplay\nk=(i−r)∧1∆kF(Tn(i−k)\n+ω+)+··· (=di)\n+1\nr[∆N−1F(Tn(r−1)\n+ω+)+∆NF(Tn(r−2)\n+ω+)] (=dN+r−2)\n+1\nr∆NF(Tn(r−1)\n+ω+) (= dN+r−1).\nIt is clear that {di}i≥1is a martingale sequence adapted to BisinceEi−1(di) = 0.\nMoreover, since /bar⌈blF/bar⌈bl∞≤1, it is straightforward to see that for r≤N:\n/bar⌈bldi/bar⌈bl∞≤\n\n2i\nr,1≤i<r\n2, r ≤i≤N\n2N+r−i\nr, N <i ≤N+r−1,\nwhich implies\nN+r/summationdisplay\ni=1/bar⌈bldi/bar⌈bl2\n∞≤8r−1/summationdisplay\ni=1i2\nr2+4(N−r+1)≤4N≤4N2\nr.\nIfr≥N, we have\n/bar⌈bldi/bar⌈bl∞≤\n\n2i\nr,1≤i<N\n2N\nr, N ≤i≤r\n2N+r−i\nr, r<i≤N+r−1,\nwhich implies\nN+r/summationdisplay\ni=1/bar⌈bldi/bar⌈bl2\n∞≤8N−1/summationdisplay\ni=1i2\nr2+4N2\nr2(r−N+1)≤4N2\nr.\nIn all cases, we have\n(41)/parenleftBiggN+r/summationdisplay\ni=1/bar⌈bldi/bar⌈bl2\n∞/parenrightBigg1\n2\n≤2N√r.\nBy (22), we have for all δ >0 andm∈Z+that\nP\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglem/summationdisplay\n1di/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δm1\n2/parenleftBiggm/summationdisplay\n1/bar⌈bldi/bar⌈bl2\n∞/parenrightBigg1\n2\n<e−cδ2m.26 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nBy settingm=r+Nandδm1\n2/parenleftbig/summationtextm\n1/bar⌈bldi/bar⌈bl2\n∞/parenrightbig1\n2=ε\n10, and using (41), we obtain\nµ+/braceleftBigg\nω+:/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0FN(Tns\n+ω+)−/integraldisplay\nFdµ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε\n10/bracerightBigg\n=P\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglem/summationdisplay\n1di/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δm1\n2/parenleftBiggm/summationdisplay\n1/bar⌈bldi/bar⌈bl2\n∞/parenrightBigg1\n2\n\n<exp/parenleftbigg\n−cε2\n/summationtextm\n1/bar⌈bldi/bar⌈bl2∞/parenrightbigg\n≤exp/parenleftbigg\n−cε2\nN2r/parenrightbigg\n,\nand hence, in view of (39) and (40), we have\nµ+/braceleftBigg\nω+:/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0F(Tns\n+ω+)−/integraldisplay\nFdµ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightBigg\n<exp/parenleftbigg\n−cε2n2\nlog2(ε−1K)r/parenrightbigg\n,\nwhich is the desired estimate. /square\nNow we are ready to prove Lemma 4.2.\nProof of Lemma 4.2. By ULD on Iand the relation between gn(·,E) andg+\nn(·,E),\nfor everyε>0, there are c,C >0 such that\nµ+/braceleftbig\nω+:/vextendsingle/vextendsingleg+\nn(ω+,E)−L(E)/vextendsingle/vextendsingle>ǫ/bracerightbig\n<Ce−cn,\nuniformly for all E∈I, which implies the existence of some n1(ε) so that for all\nn≥n1(ε),/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\ng+\nn(ω+,E)dµ+−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε\n10.\nBy (36) and (38), we may apply Lemma 4.3 to F=g+\nn\nCnandK=CΓnto obtain\nsomen2(ε) so that for all n≥n2(ε),\nµ+/braceleftBigg\nω+:/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0g+\nn(Tns\n+ω+,E)−/integraldisplay\ng+\nndµ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε\n10/bracerightBigg\n<e−cε2\nn2r.\nCombining the two estimates above, we obtain\nµ+/braceleftBigg\nω+:/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0g+\nn(Tns\n+ω+,E)−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightBigg\n<e−cε2\nn2r,\nuniformly for all E∈I,r≥1, andn≥n0(ε) = max{n1(ε),n2(ε)}. /square\nProof of Lemma 4.1. This proof is similar to the proof of [ADZ, Theorem 3.1]. We\ncopy a part of it here since we need the explicit lower bound on r.\nLet\nB+\nr(ε) :=/braceleftbigg\nω+∈Ω+:/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0g+\nn(Tns\n+ω+,E)−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n.\nIn view of (35) and (37), for r≥Cn\nε, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0gn(Tnsω,E)−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε⇒/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0g+\nn(Tns\n+ω+,E)−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε\n2,ANDERSON LOCALIZATION 27\nwhich implies for all n≥n0(ε),r≥Cn\nε, and allE∈Ithat\nµ/braceleftbigg\nω∈Ω :/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrr−1/summationdisplay\ns=0gn(Tnsω,E)−L(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ε/bracerightbigg\n≤µ[(π+)−1B+\nn(ε/2)]\n=µ+(B+\nn(ε/2))\n<e−cε2\nn2r.\nFinally, the ability to add the shift by s0in the statement of Lemma 4.1 is a\nconsequence of the T-invariance of µ. /square\nProposition 4.4. For anyε∈(0,1), there exists a subset Ω′= Ω′(ε)⊂Ωof full\nµ-measure such that the following statement holds true. For e veryε∈(0,1)and\neveryω∈Ω′(ε), there exists/tildewiden0=/tildewiden0(ω,ε)such that\n(42)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL(E)−1\nn4n4−1/summationdisplay\ns=0gn/parenleftbig\nTs0+ns(ω),E/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε\nfor alln,s0∈Zwithn≥max/parenleftbig\n/tildewiden0,(log(|s0|+1))2/3/parenrightbig\nand allE∈I.\nProof.For a given large enough n(we will determine largeness later), we first\nconsider sets Bn,s0=Bn,s0(ε) where (42) fails to hold:\nBn,s0:=\n\nω: sup\nE∈I/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL(E)−1\nn4n4−1/summationdisplay\ns=0gn(Ts0+nsω,E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥ε\n\n.\nLetκ=|I|and given 0 <δ≤κ/2, define the na¨ ıve grid\nI0:= [I∩(δZ)].\nIt is straightforward to check that I0isδ-dense inIin the sense that\n(43) I⊆/uniondisplay\nt∈I0[t−δ,t+δ].\nMoreover, we may estimate the cardinality of I0via\n#I0≤κ\nδ+1≤2κ\nδ.\nNow, fixing 0 <ε<1 and taking δ=ε(3Γn)−1, we may produce a finite set I0⊂I,\nwhich isε(3Γn)−1-dense inIin the sense of (43), with cardinality bounded above\nby\n(44) # I0≤6κΓn\nε.\nIf necessary, enlarge nto ensure that\n(45) C/parenleftBigε\n6Γn/parenrightBigβ\n<ε\n3,\nwhereCandβare from (32). Then (32) and (33) yield\nBn,s0⊂/uniondisplay\nE∈I0\n\nω:/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL(E)−1\nn4n4−1/summationdisplay\ns=0gn/parenleftbig\nTs0+nsω,E/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥ε\n3\n\n.28 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nConsequently, by taking nlarge enough that n≥max{n0(ε/3),Cε−1/3}and (45)\nholds, and using T-invariance of µ, we obtain\n(46) µ(Bn,s0)≤6κΓn\nεe−cε2n2\nby Lemma 4.1. Now, with\nBn:=/uniondisplay\n|s0|≤en3/2Bn,s0,\nit is clearthat µ(Bn)≤e−cεn2fornlarge, so Ω′:= Ω\\limsupBnsatisfiesµ(Ω′) = 1\nby the Borel–CantelliLemma. Naturally, for each ω∈Ω′, we can find/tildewiden0=/tildewiden0(ω,ε)\nlarge enough that ω /∈ Bnwhenevern≥/tildewiden0. In other words,\nω /∈ Bn,s0whenevern≥/tildewiden0(ω,ε) and|s0| ≤en3/2.\nChanging the order of nands0, the statement above implies\nω /∈ Bn,s0whenevers0∈Zandn≥max/parenleftBig\n/tildewiden0,(log(|s0|+1))2/3/parenrightBig\n.\nBy the definition of Bn,s0, we obtain the statement of the proposition. /square\nWe are now in a position to estimate the finite-volume Green functions . Before\nstating the estimate, we fix some notation. Let Λ = [ a,b]∩Zbe a finite subinterval\nofZ, and denote by PΛ:ℓ2(Z)→ℓ2(Λ) the canonical projection. We denote the\nrestriction of Hωto Λ by\nHω,Λ:=PΛHωP∗\nΛ.\nFor anyE /∈σ(Hω,Λ), define\nGE\nω,Λ:= (Hω,Λ−E)−1\nto be the resolvent operator associated to Hω,Λ. LikeHω,Λ,GE\nω,Λhas a represen-\ntation as a finite matrix; denote its matrix elements by GE\nω,Λ(m,n), that is,\nGE\nω,Λ(m,n) :=/angbracketleftbig\nδm,GE\nω,Λδn/angbracketrightbig\n, n,m∈Λ.\nAdditionally, for N∈Z+, let us define Hω,N:=Hω,[0,N−1]to be the restriction of\nHωto the box Λ N:= [0,N−1]∩Z.We will likewise use the same notation for the\nassociated resolvent GE\nω,N.\nUsing Cramer’s rule, we know that\n(47) GE\nω,N(j,k) =det[Hω,j−E]det[HTk+1ω,N−k−1−E]\ndet[Hω,N−E]\nforany0 ≤j≤k≤N−1andE /∈σ(Hω,N), whereoneinterpretsdet[ Hω,0−E] = 1.\nAnother relation that will be important in what follows is\nAE\nN(ω) =/bracketleftbiggdet(E−Hω,N)−det(E−HTω,N−1)\ndet(E−Hω,N−1)−det(E−HTω,N−2)/bracketrightbigg\n, N≥2. (48)\nThis is a standard fact, which one may prove inductively.\nIn particular, since the norm of a matrix majorizes the absolute valu e of any of\nits entries, we obtain\n(49)/vextendsingle/vextendsingleGE\nω,N(j,k)/vextendsingle/vextendsingle≤/bar⌈blAE\nj(ω)/bar⌈bl/bar⌈blAE\nN−k(Tkω)/bar⌈bl\n|det[Hω,N−E]|ANDERSON LOCALIZATION 29\nfor all 0≤j≤k≤N−1 by combining (47) and (48). Thus, it is straightforward\nto transform estimates on transfer matrices into estimates on Gr een functions of\ntruncations of Hω; to complete our goal of estimating Green functions, we will use\nProposition 4.4 to estimate transfer matrix norms, and then apply ( 49).\nCorollary 4.5. Givenε∈(0,1)andω∈Ω′(ε), there exists /tildewiden1=/tildewiden1(ω,ε)so that\nthe following statements hold true. For all E∈I, we have\n(50)1\nnlog/vextenddouble/vextenddoubleAE\nn(Ts0ω)/vextenddouble/vextenddouble≤L(E)+2ε\nwhenevern,s0∈Zsatisfyn≥max/parenleftbig\n/tildewiden1,log2(|s0|+1)/parenrightbig\n.\nMoreover, for all n,s0∈Zwithn≥ε−1max/parenleftbig\n/tildewiden1,2log2(|s0|+1)/parenrightbig\n, we have\n(51)/vextendsingle/vextendsingleGE\nTs0ω,n(j,k)/vextendsingle/vextendsingle≤exp[(n−|j−k|)L(E)+C0εn]\n|det[HTs0ω,n−E]|\nfor allE∈I\\σ(HTs0ω,n)and allj,k∈[0,n), whereC0is a constant that only\ndepends on fandα.\nProof.Fixε∈(0,1),ω∈Ω′(ε), andE∈I. Asusual,ourestimatesareindependent\nof the energy, so we suppress Efrom the notation. Choose /tildewiden1∈Z+large enough\nthat\n(52) /tildewiden1≥max/parenleftbig\n/tildewiden0(ω,ε)3,4ε−1,154/parenrightbig\n,and20logΓ\n/tildewiden1/4\n1−5<ε.\nGivenn,s0∈Zwithn≥max/parenleftbig\n/tildewiden1,log2(|s0|+1)/parenrightbig\n, we want to apply Proposition4.4\nwithm=⌈n1/4⌉. Notice that 0 ≤m4−n≤5m3. Thus, by submultiplicativity of\nthe matrix norm and unimodularity of the transfer matrices, we hav e\n(53) /bar⌈blAn(Ts0ω)/bar⌈bl ≤Γ5m3m3−1/productdisplay\ns=0/bar⌈blAm(Ts0+smω)/bar⌈bl.\nBy our choice of /tildewiden1, we havem≥/tildewiden0andm≥(log(|s0|+1))2/3. Thus, combining\n(53) with Proposition 4.4 and (52), a direct computation shows that\n1\nnlog/bar⌈blAn(Ts0ω)/bar⌈bl ≤5m3logΓ\nn+m4\nn(L+ε)\n≤L+10logΓ\nm−5+m\nm−5ε\n≤L+2ε,(54)\nwhich yields (50).\nNow suppose n≥ε−1max/parenleftbig\n/tildewiden1,2log2(|s0|+1))/parenrightbig\nand puth:=⌈εn⌉. Forj≥0,\nwe have\n/bar⌈blAj(Ts0ω)/bar⌈bl ≤/vextenddouble/vextenddoubleAj+h(Ts0−hω)/vextenddouble/vextenddouble/vextenddouble/vextenddouble[Ah(Ts0−hω)]−1/vextenddouble/vextenddouble.\nClearlyj≥0 andh≥/tildewiden1. Moreover, by our choice of /tildewiden1and the relation between\nnands0, a direct computation shows h≥log2(|s0−h|+1). Thus we can apply\n(54) to estimate the norms on the right hand side and obtain\n(55) /bar⌈blAj(Ts0ω)/bar⌈bl ≤e(j+2h)L(E)+2ε(j+2h)≤ejL(E)+C0εn,\nwhereC0is a constant that depends only on /tildewideµ.\nNaturally, one can also apply the analysis above to estimate the tran sfer matrix\nAn−k(Ts0+kω) withj≤k≤n−1 as well. Using (52) and the relationships among30 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nh,s0andk, a direct computation shows that log2(|s0−k+h|+1)≤h. Thus, (54)\nyields\n/bar⌈blAn−k(Ts0+kω)/bar⌈bl ≤/vextenddouble/vextenddoubleAn−k+h(Ts0+k−hω)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbig\nAh(Ts0+k−hω)/bracketrightbig−1/vextenddouble/vextenddouble/vextenddouble\n≤exp[(n−k)L(E)+C0εn]. (56)\nCombining equations (55) and (56) with the observation (49), one s ees that for a\nsuitable choice of C0,\n|GE\nTs0ω,n(j,k)| ≤/bar⌈blAj(Ts0ω)/bar⌈bl/bar⌈blAn−k(Ts0+kω)/bar⌈bl\n|det[HTs0ω,n−E)]|\n≤exp[(n−|j−k|)L(E)+C0εn]\n|det[HTs0ω,n−E)]|\nfor allE∈I\\σ(HTs0ω,n) and all 0 ≤j≤k <n. The case j≥kfollows because\nHis self-adjoint and Eis real. /square\n4.2.Elimination of Double Resonances. ForN∈Z+, we define\nN:=/floorleftbig\nNlogN/floorrightbig\n=/floorleftBig\ne(logN)2/floorrightBig\n,\nwhich is a super-polynomially and subexponentially growing function of N.\nWe now introduce the set of double resonances. Given ε >0 andN∈Z+, let\nDN=DN(ε) denote the set of all those ω∈Ω such that\n(57) /bar⌈blGE\nTsω,[−N1,N2]/bar⌈bl ≥eK2\nand\n(58)1\nmlog/bar⌈blAm(Ts+rω,E)/bar⌈bl ≤L(E)−ε\nfor some choice of s∈Z,K≥max(N,log2(|s|+ 1)), 0 ≤N1,N2≤K9,E∈I,\nK10≤r≤K, andm∈ {K,2K}.\nProposition 4.6. For all0<ε<1, there exist constants C >0and/tildewideη >0such\nthat\nµ(DN(ε))≤Ce−/tildewideηN\nfor allN∈Z+.\nTo prove Proposition 4.6, we first need the following lemma from [ADZ]. R ecall\nthat for an admissible l= (l1,...,ln), we have defined at the begining of proof of\nLemma 4.3 that Ω+\nl:= [0;l]+. We further define\n(59) µ+\nl=1\nµ+/parenleftbig\nΩ+\nl/parenrightbig(T|l|\n+)∗µ+/vextendsingle/vextendsingle\nΩ+\nl.\nIn other words, µ+\nlis the normalized push-forward of µ+under the injective map\nT|l|\n+: Ω+\nl→Ω+. By definition, we have\n/integraldisplay\nΩ+fdµ+\nl=1\nµ+(Ω+\nl)/integraldisplay\nΩ+\nlf◦T|l|\n+dµ+.\nIf we viewT−|l|\n+as a map from T|l|\n+(Ω+\nl) to Ω+\nl, we obtain\n(60) µ+(Ω+\nl)/integraldisplay\nT|l|\n+(Ω+\nl)f◦T−|l|\n+dµ+\nl=/integraldisplay\nΩ+\nlfdµ+.ANDERSON LOCALIZATION 31\nNote thatµ+\nlis concentrated on T|l|\n+(Ω+\nl). Then we have [ADZ, Lemma 3.3]; stated\nas Lemma 4.7 below. We point out that although in the statement of Le mma 3.3,\n[ADZ] assumes the topological mixing property, it is not needed since the proof\nthere did not use it.\nLemma 4.7. Consider a one-sided subshift of finite type (Ω+,T+,µ+), whereµ+\nhas the bounded distortion property. There exists a C≥1so that, uniformly for\nall admissible l, we have\n(61)dµ+\nl\ndµ+(ω+)≤Cforµ-a.e.ω+,\nwheredµ+\nl\ndµ+is the Radon-Nikodym derivative of µ+\nlwith respect to µ+. In particular,\nwe have for all nonnegative measurable functions fand all admissible l,\n(62)/integraldisplay\nfdµ+\nl≤C/integraldisplay\nfdµ+.\nProof of Proposition 4.6. Define auxiliary “bad sets” for fixed sandK:\nDK,s={ω: (57),(58) are satisfied for some choice of E,N1,N2,r,mas above}.\nFixε∈(0,1) and begin by noticing that\n(63) DK,s⊂/uniondisplay\nK10≤r≤K/uniondisplay\n0≤N1,N2≤K9/tildewideD1(N1,N2,r,s)∪/tildewideD2(N1,N2,r,s),\nwhere/tildewideDj(N1,N2,r,s) denotes the collection of all ω∈Ω for which there exists\nE∈Isuch that (57) and (58) hold with m=jK.\nWe will estimate µ(/tildewideD1). The estimates for /tildewideD2are completely analogous. To that\nend, suppose ω∈/tildewideD1(N1,N2,r,s), that is, (57) and (58) hold for some E∈I. By\nthe spectral theorem, there exists E0∈σ(HTsω,[−N1,N2]) with\n(64) |E−E0| ≤/vextenddouble/vextenddouble/vextenddoubleGE\nTsω,[−N1,N2]/vextenddouble/vextenddouble/vextenddouble−1\n≤e−K2.\nOn the other hand, choosing Klarge enough that CΓKe−αK2≤ε\n6andCe−αβK2≤\nε\n6(whereC,βare from (32) and (33) and the αin the second inequality will be\nused later), we get\ngK(Ts+rω,E0)≤gK(Ts+rω,E)+ε\n6\n≤L(E)−ε+ε\n6(65)\n≤L(E0)−2ε\n3,\nwhere we have used (33) in the first line, (58) in the second line, and ( 32) in the\nfinal line. Thus, when Kis large enough, we get\n(66)/tildewideD1(N1,N2,r,s)⊆ˆD1(N1,N2,r,s) for allN1,N2,r,ands,\nwhereˆD1=ˆD1(N1,N2,r,s) denotes the set of all ω∈Ω such that\ngK(Ts+rω,E0)≤L(E0)−ε\n232 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nfor someE0∈σ/parenleftbig\nHTsω,[−N1,N2]/parenrightbig\n. To estimate the measure of ˆD1, byT-invariance,\nwe may instead consider\nTs(ˆD1) =/uniondisplay\nE0∈σ/parenleftbig\nHω,[−N1,N2]/parenrightbig/braceleftbig\nω:gK(Trω,E0)≤L(E0)−ε\n2/bracerightbig\n.\nDivideΩintoΩ =/uniontext\nlΩl, wherelisadmissiblewith |l|= 2K2+1andΩ l= [−K2:l].\nFix someω(l)∈Ωl. Then for all ω∈Ωl, we haved(ω,ω(l))≤e−K2. Thus for each\nE0∈σ/parenleftbig\nHω,[−N1,N2]/parenrightbig\n, there is a E′∈σ/parenleftbig\nHω(l),[−N1,N2]/parenrightbig\nsuch that\n|E′−E0| ≤ /bar⌈blHω,[−N1,N2]−Hω(l),[−N1,N2]/bar⌈bl<Ce−αK2.\nRunning the estimate (65) again, we obtain that for ω∈Ωl\ngK(Trω,E0)<L(E0)−2\n3ε⇒gK(Trω,E′)<L(E′)−ε\n3.\nIn other words, we have\n(67)Ts(ˆD1)⊂/uniondisplay\nl/uniondisplay\nE′∈σ/parenleftbig\nHω(l),[−N1,N2]/parenrightbig/braceleftBig\nω∈Ωl:gK(Trω,E′)≤L(E′)−ε\n3/bracerightBig\n.\nLetSK(E,ε) ={ω:gK(ω,E)≤L(E)−ε}. Then the set on the right-hand side of\n(67) becomes\nΩl/intersectiondisplay\nT−r/parenleftbig\nSK(E′,ε/3)/parenrightbig\n.\nToestimate the measureofthe set above, wedefine the following s-locallysaturated\nset,\n˜S(E) =/uniondisplay\nω∈T−K2SK(E,ε/3)Ws\nloc(ω).\nFor anyω′∈Ws\nloc(ω), we haved(TK2ω′,TK2ω)≤e−K2, which implies\n|gK(TK2ω,E)−gK(TK2ω′,E)| ≤CΓKe−αK2<ε\n6.\nFor anyω′∈˜S(E),ω′∈Ws\nloc(ω) for someω∈T−K2(SK(E,ε/3)). Hence,\ng(TK2ω′,E)≤g(TK2ω,E)+ε\n6\n≤L(E)−ε\n3+ε\n6\n=L(E)−ε\n6,\nwhich implies\nT−K2SK(E,ε/3)⊂˜S(E)⊂T−K2SK(E,ε/6).\nClearly,\nT−K2/parenleftBig\nΩl/intersectiondisplay\nT−rSK(E′,ε/3)/parenrightBig\n⊂T−K2/parenleftBig\nΩl/intersectiondisplay\nT−r+K2˜S(E′)/parenrightBig\n=/parenleftBig\nT−K2Ωl/parenrightBig/intersectiondisplay/parenleftBig\nT−r˜SK(E′)/parenrightBig\n,\nwhich iss-locally saturated. Let\n˜S+(E) =π+(˜S(E)) and Ω+\nl:=π+/parenleftbig\nT−K2Ωl/parenrightbig\n= [0;l]+.ANDERSON LOCALIZATION 33\nIt is straightforward to see that for any s-locally saturated set S, we haveµ(S) =\nµ+(π+S) andπ+T−nS=T−n\n+π+S, which yields\nµ/parenleftBig/parenleftbig\nT−K2Ωl/parenrightbig/intersectiondisplay/parenleftbig\nT−r˜SK(E′)/parenrightbig/parenrightBig\n=µ+π+/parenleftBig/parenleftbig\nT−K2Ωl/parenrightbig/intersectiondisplay/parenleftbig\nT−r˜SK(E′)/parenrightbig/parenrightBig\n≤µ+/parenleftBig\nπ+/parenleftbig\nT−K2Ωl/parenrightbig/intersectiondisplay\nπ+/parenleftbig\nT−r˜SK(E′)/parenrightbig/parenrightBig\n=µ+/parenleftBig\nΩ+\nl/intersectiondisplay\nT−r\n+/parenleftbig˜S+\nK(E′)/parenrightbig/parenrightBig\n.\nBy Lemma 4.7, the three estimates above, T- andT+-invariance of µandµ+\nrespectively, and the fact that r≥K10>2K2+1, we obtain\nµ/parenleftbig\nΩl/intersectiondisplay\nT−r/parenleftbig\nSK(E′,ε/3)/parenrightbig/parenrightbig\n=µ/parenleftBig\nT−K2/parenleftbig\nΩl/intersectiondisplay\nT−rSK(E′,ε/3)/parenrightbig/parenrightBig\n≤µ+/parenleftBig\nΩ+\nl/intersectiondisplay\nT−r\n+/parenleftbig˜S+(E′)/parenrightbig/parenrightBig\n=/integraldisplay\nΩ+\nlχT−r\n+(˜S+(E′))dµ+\n=/integraldisplay\nΩ+\nlχ˜S+(E′)◦Tr\n+dµ+\n=µ+(Ω+\nl)/integraldisplay\nΩχ˜S+(E′)◦Tr−2K2−1\n+dµ+\nl\n≤Cµ+(Ω+\nl)/integraldisplay\nΩχ˜S+(E′)◦Tr−2K2−1\n+dµ+\n=Cµ+(Ω+\nl)/integraldisplay\nΩχ˜S+(E′)dµ+\n=Cµ+(Ω+\nl)·µ+/parenleftbig˜S+(E′)/parenrightbig\n=Cµ(Ωl)·µ/parenleftbig˜S(E′)/parenrightbig\n≤Cµ(Ωl)µ(T−K2SK(E′,ε/6))\n=Cµ(Ωl)µ(SK(E′,ε/6))\n≤Cµ(Ωl)e−cK,\nwhere the last line follows from ULD. Then, we have\nµ(ˆD1) =µ(TsˆD1)\n≤C/summationdisplay\nlµ(Ωl)K9e−cK\n≤CK9e−cK\n≤Ce−η1K.\nThus, we obtain µ(/tildewideD1(N1,N2,r,s))≤Ce−η1Kby applying (66).\nApplying similar reasoning to /tildewideD2, one can estimate µ(/tildewideD2(N1,N2,r,s))≤\nCe−η2K.34 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\nPutting everything together yields\nµ(DK,s)≤/summationdisplay\n0≤N1,N2≤K9/summationdisplay\nK10≤r≤K/parenleftBig\nµ(/tildewideD1(N1,N2,r))+µ(/tildewideD2(N1,N2,r))/parenrightBig\n≤CK18Ke−η3K\n≤Ce−2/tildewideηK\nfor some suitable choice of /tildewideη. Changing the order of Kands, we have\nDN=/uniondisplay\ns∈Z/uniondisplay\nK≥max{N,log2(|s|+1)}DK,s⊆/uniondisplay\nK≥N/uniondisplay\n|s|≤e√\nKDK,s.\nThen, the estimates above yield\nµ(DN)≤/summationdisplay\nK≥N(2e√\nK+1)Ce−2/tildewideηK≤Ce−/tildewideηN\nfor large enough N. Adjusting the constants to account for small Nconcludes the\nproof. /square\nOnce we have Corollary 4.5 and Proposition 4.6, we can then mimic the pr o-\ncess after the proof of Proposition 6.1 of [BDF+, Section 6] to prov e (13), hence\nTheorem 2.12.\nRemark 4.8. An analogous proof can be given for half-line operators associated\nwith non-invertible maps. The changes are relatively simple. In partic ular, mod-\nulo these modifications, the arguments presented in this paper will e stablish the\nlocalization result for the doubling map model stated in Corollary 2.16.\n5.Applications – Proof of Corollaries 2.14 and 2.15\nThe two corollaries can be proved together. In both cases, we may letJ⊂R\nbe a compact interval so that it contains the almost sure spectrum Σ andAEis\neither locally constant or fiber bunched over J. LetFfbe the finite set and Jη,\nη >0, be the finite union of compact intervals as described before the s tatement\nof Theorem 2.10. In the same description, we know that PLE holds tr ue onJη,\nand by Theorem 2.10, ULD holds true on Jη. Hence, by Theorem 2.12 Hωhas\nexponential dynamical localization on Jηforµ-a.e.ω.\nThus for every n∈Z+, there is a full measure set Ω(n)⊂Ω so thatHωhas\nspectral localization for all ω∈Ω(n)onJ1\nn. Also, for every E∈R, the set\nΩE:={ω:Eis not an eigenvalue of Hω}\nhas full measure. This is an easy consequence of Oseledec’s theore m or it follows\ndirectly from [P]. Now let\nΩ∗=\n/intersectiondisplay\nn∈Z+Ω(n)\n∩\n/intersectiondisplay\nE∈FfΩE\n.\nThen it is clear that µ(Ω∗) = 1. Moreover, for each ω∈Ω∗,Hωhas pure point\nspectrumon R\\Ff, exponentiallydecayingeigenfunctionsforalleigenvalues z∈R\\\nFf, andFfcontains no eigenvalue of Hω. HenceHωexhibits Anderson localization\nfor eachω∈Ω∗, concluding the proof.ANDERSON LOCALIZATION 35\nFinally, Corollary 2.16 is a special case of Corollary 2.15, again noting th e brief\ndiscussion of half-line localization in Remark 4.8, while Corollary 2.17 is a sp ecial\ncase of Corollary 2.14.\nReferences\n[AEP] N. Alon, P. Erd¨ os, J. Spencer, The Probabilistic Method , John Wiley & Sons, New-York,\n1992.\n[ADZ] A. Avila, D. Damanik, Z. Zhang, Schr¨ odinger operator s with potentials generated by\nhyperbolic transformations: I. Positivity of the Lyapunov exponent, Invent. Math. 231\n(2023), 851–927.\n[AV] A. Avila, M. Viana, Extremal Lyapunov exponents: an inv ariance principle and applica-\ntions,Invent. Math. 181(2010), 115–189.\n[Az] K. Azuma, Weighted sums of certain dependent random var iables,Tˆ ohoku Math. Journ.\n19(1967), 357–367.\n[BBB] L. Backes, A. Brown, C. Butler, Continuity of Lyapunov exponents for cocycles with\ninvariant holonomies, J. Mod. Dyn. 12(2018), 223–260.\n[B] J. Bourgain, Green’s Function Estimates for Lattice Schr¨ odinger Opera tors and Appli-\ncations, Annals of Mathematics Studies 158, Princeton University Press, Princeton, NJ,\n2005.\n[BGV] C. Bonatti, X. G´ omez-Mont, M. Viana, G´ en´ ericit´ e d ’exposants de Lyapunov non-nuls\npour des produits d´ eterministes de matrices, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire\n20(2003), 579–624.\n[BV] C. Bonatti, M. Viana, Lyapunov exponents with multipli city 1 for deterministic products\nof matrices, Ergodic Theory Dynam. Systems 24(2004), 1295–1330.\n[BG] J.Bourgain, M.Goldstein, On nonperturbative localiz ation with quasi-periodicpotential,\nAnn. of Math. 152(2000), 835–879.\n[BS] J. Bourgain, W. Schlag, Anderson localization for Schr ¨ odinger operators on Zwith\nstrongly mixing potentials, Commun. Math. Phys. 215(2000), 143–175.\n[B] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomo rphisms, Lec-\nture Notes in Mathematics 470, Springer-Verlag, 1975.\n[BDF+] V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. Vanden Boom, F. Wang, Z. Zhang,\nLocalization for the one-dimensional Anderson model via po sitivity and large deviations\nfor the Lyapunov exponent, Trans. Amer. Math. Soc. 372(2019), 3619–3667.\n[CKM] R. Carmona, A. Klein, F. Martinelli, Anderson localiz ation for Bernoulli and other sin-\ngular potentials, Commun. Math. Phys. 108(1987), 41–66.\n[DF] D. Damanik, J. Fillman, One-Dimensional Ergodic Schr¨ odinger Operators – I. Gen-\neral Theory , Graduate Studies in Mathematics 221, American Mathematical Society,\nProvidence, RI, 2022.\n[DT1] D. Damanik, S. Tcheremchantsev, Power-law bounds on t ransfer matrices and quantum\ndynamics in one dimension, Commun. Math. Phys. 236(2003), 513–534.\n[DT2] D. Damanik, S. Tcheremchantsev, Upper bounds in quant um dynamics, J. Amer. Math.\nSoc.20(2007), 799–827.\n[DKP] P. Duarte, S. Klein, M. Poletti, H¨ older continuity of the Lyapunov exponents of linear\ncocycles over hyperbolic maps, Math. Z. 302(2022), 2285–2325.\n[GZ] L. Ge, X. Zhao, Exponential dynamical localization in e xpectation for the one dimen-\nsional Anderson model, J. Spectr. Theory 10(2020), 887–904.\n[GK] A. Gorodetski, V. Kleptsyn, Parametric Furstenberg th eorem on random products of\nSL(2,R) matrices, Adv. Math. 378(2021), Paper No. 107522, 81 pp.\n[GS] S. Gou¨ ezel, L. Stoyanov, Quantitative Pesin theory fo r Anosov diffeomorphisms and\nflows,Ergodic Theory Dynam. Systems 39(2019), 159–200.\n[JS] S. Jitomirskaya, H. Schulz-Baldes, Upper bounds on wav epacket spreading for random\nJacobi matrices, Commun. Math. Phys. 273(2007), 601–618.\n[JSS] S. Jitomirskaya, H. Schulz-Baldes, G. Stolz, Delocal ization in random polymer models,\nComm. Math. Phys. 233(2003), 27–48.\n[JZ] S. Jitomirskaya, X. Zhu, Large deviations of the Lyapun ov exponent and localization for\nthe 1D Anderson model, Commun. Math. Phys. 370(2019), 311–324.36 ARTUR AVILA, DAVID DAMANIK, AND ZHENGHE ZHANG\n[LV] E. Lesigne, D. Voln´ y, Large deviations for martingale s,Stochastic Process. Appl. 96\n(2001), 143–159.\n[PP] K. Park, M. Piraino, Transfer operators and limit laws f or typical cocycles Commun.\nMath. Phys. 389(2022), 1475–1523.\n[P] L. Pastur, Spectral properties of disordered systems in the one-body approximation,\nCommun. Math. Phys. 75(1980), 179–196.\n[V] M. Viana, Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov\nexponents, Ann. of Math. 167(2008), 643–680.\nInstitut f ¨ur Mathematik, Universit ¨at Z¨urich, Winterthurerstrasse 190, 8057 Z ¨urich,\nSwitzerland and IMPA, Estrada D. Castorina 110, Jardim Bot ˆanico, 22460-320 Rio de\nJaneiro, Brazil\nEmail address :artur@math.sunysb.edu\nDepartment of Mathematics, Rice University, Houston, TX 7700 5, USA\nEmail address :damanik@rice.edu\nDepartment of Mathematics, University of California, River side, CA-92521, USA\nEmail address :zhenghe.zhang@ucr.edu"    },    {        "title": "2402.00252v1.Reformulating_polarized_radiative_transfer___I__A_consistent_formalism_allowing_non_local_Magnus_solutions.pdf",        "content": "Astronomy &Astrophysics manuscript no. aanda ©ESO 2024\nFebruary 2, 2024\nReformulating polarized radiative transfer\nI. A consistent formalism allowing non-local Magnus solutions\nE.S. Carlin1, S. Blanes2, F. Casas3\n1Instituto de Astrofísica de Canarias (IAC), E-38205, La Laguna, Tenerife, Spain\ne-mail: ecarlin@iac.es\n2Instituto de Matemática Multidisciplinar, Universitat Politècnica de Valencia, E-46022 Valencia, Spain\n3Departament de Matemàtiques, Universitat Jaume I, E-12071 Castellón, Spain\nReceived January 1, 2024; accepted XXXX\nABSTRACT\nThe physical diagnosis of the solar atmosphere is achieved by solving the polarized radiative transfer problem for plasmas in Non-\nLocal Thermodynamical Equilibrium (NLTE). This complex scenario poses theoretical challenges for integrating the radiative transfer\nequation (RTE) e fficiently and demands better numerical methods for synthesis and inversion of polarization spectra. Indeed, the\ncurrent theory and methods are limited to constant propagation matrices, thus imposing local solutions. To spark significant advances,\nwe propose a formalism that reformulates the polarized transfer problem. Namely, this paper lays the foundations to achieve a non-\nlocal geometrical integration of the RTE based on the Magnus expansion. First, we revisit the statement of the problem and its general\nsolutions in Jones and Stokes formalisms to clarify some inconsistencies in the literature, and to unify and update the nomenclature.\nLooking at both formalisms as equivalent representations of the Lorentz /Poincaré group of rotations, we interpret the RTE in terms\nof Lie group theory to remark the suitability of the Magnus expansion for obtaining accurate non-local solutions. We then present a\ndetailed algebraic characterization of the propagation matrix involving a new generalized Lorentz matrix. Combining this, our \"basic\nevolution\" theorem, and the Magnus expansion, we reformulate the homogenous solution to the RTE in Stokes formalism. Thus,\nwe obtain the first compact analytical evolution operator that supports arbitrary spatial variations of the propagation matrix to first\norder in the Magnus expansion, paving the road to higher orders. Finally, we also reformulate the corresponding inhomogeneous\nsolution as an exact equivalent homogeneous system, which is then solved analytically with the Magnus expansion. This gives the\nfirst e fficient and consistent formal solution of the RTE that furthermore is non-local, natively accurate (prevents any numerical break\ndown of the group structure of the solution), and that separates the integration from the formal solution. Such disruptive formulation\nleads to a new whole family of numerical radiative transfer methods and suggests accelerating NLTE syntheses and inversions with\nnon-local radiative transfer. With minor cosmetic changes, our results are valid for other universal physical problems sharing the\nLorentz /Poincaré algebra of the RTE and special relativity (e.g., motion of charges in inhomogeneous electromagnetic fields).\nKey words. Sun: atmosphere – radiative transfer – polarization. ......\n1. Introduction\nAstrophysics studies the distant universe through light. There-\nfore, it is of upmost importance to have a precise understanding\nof how light is a ffected by di fferent physical processes. In the\nframework of Maxwell’s electromagnetic wave theory, a light\nbeam is fully characterized by the spectra of intensity and of po-\nlarization, as described either by the Stokes /Müller formalism\nor equivalently by the Jones formalism (Landi Degl’Innocenti &\nLandolfi 2004; Stenflo 1994). Currently, the measurement and\ninterpretation of Stokes spectropolarimetry in atomic and molec-\nular spectral lines is the preferred and best known way of di-\nagnosing astrophysical plasmas. Namely, the Stokes vector is a\npractical and powerful way to describe any partially (i. e. natu-\nrally) polarized light beam by providing four vector components\nquantifying the number of photons and their direction of oscilla-\ntion in the reference frame of the light beam and using common\n(intensity) units (Born & Wolf 1980).\nHaving a way of quantifying polarization, the next step is\nto describe its transference through a plasma with the polarized\nradiative transfer equation (RTE). The best scenario to investi-\ngate this key astrophysical problem is the near solar atmosphere.In such a context with spatial resolution, radiative transfer is\na fundamental part of the Non-Local Thermodynamical Equi-\nlibrium (NLTE) problem, in which radiation emitted at a given\npoint modifies, via radiative transfer, the physical state of other\ndistant points in the solar atmosphere. The polarized RTE con-\ntains a four-dimensional propagation matrix quantifying the mi-\ncroscopic physics through optical coe fficients that depend on an-\ngle, wavelength, and distance along the ray. This makes the in-\ntegration of the RTE the most frequent and complex operation\ncarried out in NLTE iterative schemes, specially when consider-\ning multidimensional solar models. One of the motivations for\nthe present paper is to accelerate the NLTE calculations by solv-\ning the RTE with more consistent, robust, and accurate methods.\nBut this implies to beat a more fundamental goal: solving the\nRTE consistently in spatially resolved solar models without as-\nsuming constant local properties . Let us see why this is the next\nfrontier in polarized radiative transfer.\nAfter Unno (1956) pioneered the first derivation of the RTE\naccounting for magnetic fields, Rachkovsky (1967) completed\nthe equation and gave a first analytical solution valid for spa-\ntially homogeneous (Milne-Eddington) atmospheres with con-\nstant propagation matrix. Later, van Ballegooijen (1985) and\nArticle number, page 1 of 22arXiv:2402.00252v1  [astro-ph.SR]  1 Feb 2024A&A proofs: manuscript no. aanda\nLandi Deglinnocenti & Landi Deglinnocenti (1985) provided\nspecific evolution operators solving the homogeneous RTE with\nconstant spatial properties. However, spatial variations are not\nnegligible in the solar atmosphere. Soon it was known that the\nnumerical solutions of the RTE improved when subdividing the\natmosphere into numerous layers (e.g., Rees et al. 1989; Bellot\nRubio et al. 1998) and the need for models with spatial varia-\ntions was confirmed by numerical inversions of observed solar\nStokes profiles (Collados et al. 1994; Del Toro Iniesta & Ruiz\nCobo 1996).\nAt the beginning, the inhomogeneous character of the solar\natmosphere was mainly attributed to the large-scale stratification\nin density, temperature, and magnetic field. However, the evolu-\ntion of telescopes and simulations has also shown omnipresent\nsmall-scale variations. In particular, synthetic MHD solar mod-\nels started to support large inhomogeneity due to macroscopic\nmotions, especially in the dynamic chromosphere (e.g., Carlsson\n& Stein 1997). Indeed, plasma velocity gradients in sound /shock\nwaves are main sources of solar small-scale variations and dis-\ncontinuities, having a large and specific impact in the polariza-\ntion via radiative transfer e ffects (Carlin et al. 2013), either due\nto their modulation of radiation field anisotropy (Carlin & Asen-\nsio Ramos 2015) and /or due to a relatively new phenomenon that\nwe call dynamic dichroism (Carlin 2019).\nIn order to cope with spatial atmospheric variations in re-\nsolved atmospheres with formalisms based on constant proper-\nties, strategies such as \"ray characteristics\" (e.g., Kunasz & Auer\n1988) and several integration numerical methods have been de-\nveloped. Some of the most representative are: the piecewise con-\nstant Evolution Operator method (van Ballegooijen 1985; Landi\nDeglinnocenti & Landi Deglinnocenti 1985, hereafter vB85 and\nLD85), the “DELO” methods (Rees et al. 1989) of linear, semi-\nparabolic, and parabolic kind (see Janett et al. 2017, and refer-\nences therein); and the order-3 DELO-Bezier (De la Cruz Ro-\ndríguez & Piskunov 2013) and Hermite (Bellot Rubio et al.\n1998) methods. Numerically, these schemes are considered ad-\nvantageous against older standards, such as the diagonalization\nof the propagation matrix (Šidlichovsky,1976) or the Runge-\nKutta scheme, which demands a too large number of grid points\ncontrolled by the eigenvalues of the propagation matrix (Landi\nDegl’Innocenti 1976).\nHence, despite nowadays we know that both large and small\nscale spatial variations are needed to represent stellar atmo-\nspheres, all existing numerical methods for solving the polarized\nRTE in solar physics and astrophysics remain limited to constant\nspatial properties. The bottleneck is in a theory based on an evo-\nlution operator constrained to constant propagation matrix. Thus,\nin order to approach the limit of constant properties in every nu-\nmerical cell, the current methods demand to sequentially solve\nthe RTE many times along every ray. This process scales with\nthe resolution of the models, becoming a main source of compu-\ntational cost. In addition, the approach cannot be exact because\nthe limit of constant properties is never fully achieved due to the\nfact that the radiative transfer depends on several atmospheric\nphysical quantities with di fferent sources of gradients and scales\nof variation. Furthermore, it cannot be achieved simultaneously\nand equally for all wavelengths (e.g.around a spectral line) be-\ncause the integration step of any numerical cell change with\nwavelength (e.g. because opacity does it). This fact is reinforced\nby the sensitivity of the optical depth step to Doppler shifts and\nopacity variations in the atmospheres.\nAnother issue that is never mentioned is that current methods\npropagate the physical solution along the light beam by imposing\nan emissivity that does change continuously while the propaga-tion matrix is locally constant, which is inconsistent. Physically,\nit is incorrect because emissivity cannot change if the propaga-\ntion matrix is constant, but it also corrupts indirectly the mathe-\nmatical meaning of the RTE by breaking its Lie group structure.\nIn other words, current methods are actually not solving the tar-\nget physical problem, but a di fferent one.\nFor reasons like these, it should be expected that the numer-\nical properties of current methods are limited by construction:\nthey are not designed for spatial variations. Obviously, all this\nhas negative consequences in the more general NLTE problem.\nIn the work starting here, we clarify these limitations while mov-\ning to the real case in which arbitrary spatial variations can be\nconsistently accounted in the RTE. Until now, the only attempt to\nsolve this general problem was made by Semel & López Ariste\n(1999) and López Ariste & Semel (1999), hereafter SLA99 and\nLAS99. Considering the group structure of the RTE, they applied\nthe Wei & Norman (1963) method to formally pose a non-local\nevolution operator based on products of exponentials. However,\ntheir formulation does not provide a closed formula for the ra-\ndiative transfer solution and demands to solve several di fferential\nequations for the optical coe fficients of the RTE, such that they\ndid not materialize it numerically arguing that it would not be\neconomic. Without discarding their results, which will be com-\npared with ours a posteriori, we present a di fferent approach.\nHere we must mention the key problem of commutativity:\nwhen an atmosphere varies spatially, its propagation matrices\ndo not generally commute between points, which complicates\nan exact solution. SLA99 saw this as a limitation, arguing that,\nwithout commuting layers, the solution requires constant propa-\ngation matrices and forces a discretization of the atmosphere. In\nour work, we assume a discretization and arbitrary spatial vari-\nations as requirements. Consequently, we fully embrace non-\ncommutativity as part of the problem, posing the foundations\nto incorporate it in the theory. To do this, we use the Magnus\nexpansion (Magnus 1954), an approach that SLA99 explicitly\nintended to avoid due to its complexity. Among its several re-\nmarkable aspects (e.g., Blanes et al. 2009), this object preserves\nthe algebraic group structure and intrinsic properties of the exact\nsolution after truncation in the algebra of a Lie group. As will\nbe seen later, its group elements are neatly defined with a single\nexponential, thus avoiding products of exponentials, di fferential\nformulations, and (inexact) perturbative developments.\nThis paper has to be understood as the first of a series where\nwe reformulate the radiative transfer problem with the Magnus\nexpansion. Here, we present the general theory of our formalism:\n1) In Section 2, we re-derive the RTE and its general so-\nlutions both in Stokes and Jones formalisms, using a common\nframework and nomenclature and stating some relations that are\nunclear in the literature. This introductory step is also necessary\nto motivate the future extension of our approach to the Jones\nformalism, where the solution and the Magnus expansion could\nadopt an advantageous analytical form.\n2) In section 3, we set a minimal framework to work with\nLie group rotations, showing the limitations of current evolution\noperators, and introducing the Magnus solution and our Basic\nEvolution theorem to solve the exact exponential of any integral\nof the propagation matrix.\n3) Section 4 combines some aspects of Lie group representa-\ntion and multidimensional rotations, leading to a detailed charac-\nterization of the propagation vector and the propagation matrix.\n5) In Section 5, we calculate our compact analytical evolu-\ntion operator using the Magnus solution to first order.\n6) Finally, Section 6 reformulates and calculates the inhomo-\ngeneous formal solution in Stokes formalism.\nArticle number, page 2 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\n2. A brief description of natural polarized light\nElectromagnetic waves travel along the direction of propagation\nk=E×B, as the electric field oscillates and rotates in the plane\nΠ⊥k. We adopt a right-handed reference system ( ˆe1,ˆe2,k) with\ntwo perpendicular unit vectors ˆe1andˆe2contained in Π. Natural\nlight is then described as an incoherent superposition of waves in\na wave packet constrained to a small range of wavenumbers ∆k\naround k, hence having a finite but small (quasi-monochromatic)\nfrequency bandwidth ∆ω≈c|∆k|and an angular spread in the\nsolid angle ∆Ω≈|∆k|/k. Integrating in δkall waves in the range\n∆k, the total electric oscillation at any point P in Πis:\nE(P,t)=Reh\n(E1(P,t)ˆe1+E2(P,t)ˆe2)e±i(k·rP−ωt)i\n, (1)\nwhere\nEi(P,t)=Z\nd3(δk) n(δk) Ei(k+δk) e±i(ϕi+δk·rP−δωt)(2)\nare the total complex electric field amplitudes of the wave\npacket along each reference axis, while Ei(k+δk) andϕiare\nthe corresponding real electric amplitudes and phases for every\nsingle wave with wavenumber k′=k+δkand angular frequency\nω+δω=c|k′|. Finally, n(δk) is the number density of waves in\nthe three-dimensional wavenumber space.\nTwo signs are possible in the above exponentials. Following\nLandi Degl’Innocenti & Landolfi (2004), we choose as conven-\ntion the positive sign (i.e., negative temporal exponent).\nWith products of the electric amplitudes given by Eq. (2)\nwe can define the average polarization tensor J i j. From it, we\nalso define the Stokes vector I=(I,Q,U,V)⊺, which quantifies\nthe observable properties of the average polarization ellipse pro-\nduced by the total electric field vector oscillating in Π. These\ntwo quantities1are related as ( uis a constant converting from\nsquared electric field to intensity units):\nJ=u· \n<E∗\n1E1> <E∗\n1E2>\n<E∗\n2E1> <E∗\n2E2>!\n=1\n2· \nI+Q U−iV\nU+iV I−Q!\n(3a)\nI=J11+J22, Q=J11−J22,\nU=J12+J21, V=i(J12−J21), (3b)\nThus, a light beam propagating along khas a total number of\nphotons quantified by Stokes I, a linear polarization defined by\nboth Stokes Q and U along two reference directions in Π, and\na circular polarization given by Stokes V , which is the di ffer-\nence between right-handed and left-handed circularly polarized\nphotons using kas reference direction. Hence, Stokes V quan-\ntifies the rotation of the electric field vector around k. By con-\nvention, it is defined positive /negative when that field is seen\nrotating clockwise /counter-clockwise from the observer (right-\nhanded /left-handed circular polarization).\nA consequence of considering a realistic superposition of\nquasi-monochromatic waves to quantify polarization is that the\nideal physical relation I2=Q2+U2+V2, holding for a\nmonochromatic wave, is substituted by the constrain:\nI2>Q2+U2+V2(|J|,0). (4)\nEquation (4) defines the Poincaré sphere by mapping the Stokes\nvector with points in the three-dimensional space Q/I,U/I,V/I\n1The average character of JandScomes from averaging (with the\noperator<...> ) the bilinear products of electric amplitudes in Eq. (3a)\nover time scales much larger than the period of the waves.(Shurcli ff2013). The polarization degree is the radius of such\nsphere p=p\nQ2+U2+V2/I<1, meaning that the wave\npacket has become partially polarized because there are photons\nthat do not contribute to Q, U or V .\n2.1. The evolution equation for polarized radiation\nWe now consider a general atmosphere with di fferent complex\nrefractive indices nαalong each perpendicular spatial direction\nαin an atmosphere reference frame. From Eq.(1), one can then\nderive the spatial evolution of the transverse components of the\nelectric field for a stationary quasi-monochromatic wave in our\nbasis ( ˆe1,ˆe2,k) as a function of the optical properties of the\nmedium, and from there the components Ji jin Eq.(3a). Assum-\ning the physical conditions associated to astrophysical plasmas\n(natural light in linear optics regime2), and calling sthe geomet-\nrical distance along the ray k, the result is (Landi Degl’Innocenti\n& Landolfi 2004, Sect. 5.2) :\ndJi j\nds=−2X\nk=1(P∗\nikJk j+PjkJik),(i,j=1,2) (5)\nwhere the propagation tensor is3Pjk=−ikgα\njknαwith real and\nimaginary parts ¯Pjk+i˘Pjk. However, in astrophysics we use opti-\ncal coe fficients instead of the propagation tensor. We have found\nthe following succint relations holding between them:\nηI=P∗\n11+P∗\n22, η Q+iρQ=P∗\n11−P∗\n22, (6a)\nηU+iρU=P∗\n12+P∗\n21, η V+iρV=i(P∗\n12−P∗\n21), (6b)\nwhere we shall call propagation vector to that with components:\nak=ηk+iρk, k=1,2,3 (7)\nHereafter, k=0,1,2,3 refer to quantities associated to Stokes\nvector components I,Q,U,V. Thus, the above coe fficients de-\nscribe total absorption ( η0), dichroism ( η1,2,3) and anomalous\ndispersion ( ρ1,2,3) for the corresponding Stokes parameter. Since\nlight propagates through an anisotropic plasma while oscillating\ntransversally, the complex components of the electric field, ε1\nandε2, find di fferent refraction indexes along the transfer, which\nexplains the meaning of each optical coe fficient.4.\nThe RTE is completed adding to Eq.(5) a term with emis-\nsivity coe fficientsϵ0,1,2,3quantifying the sources of photons. The\nemissivity and the seven propagation coe fficients are involved\nfunctions of wavelength, propagation angle, and other parame-\nters defining the micro-physics of the atmosphere (temperature,\nvelocity, density, and magnetic field). We shall assume them al-\nready calculated for every wavelengh and spatial point along a\ngiven ray of light.\n2This means that the dielectric properties of the medium does not de-\npend on the amplitude of the electric field, and that the dielectric con-\nstants of the medium are close to one.\n3This summation changes from the basis of the atmosphere ( ˆuα) to that\nof the electric field ( ˆej), with coe fficients gα\njkquantifying the geometry\nof propagation.\n4Dichroism is explained as the preference of the plasma to absorbe\nphotons oscillating along a given direction (selective absorption of po-\nlarization states) and it is connected with a di fferential attenuation be-\ntween the modulus of ε1andε2. Anomalous dispersion ( ρQ,U,V) is the\nability of the plasma element to dephase ε1andε2, which couples polar-\nization states, inducing mutual conversions as they propagate. Finally,\nthe absorption η0quantifies the total number of photons absorbed per\nunit distance in any polarization state (intensity), and it is related to\nvariations in the total modulus of the refraction index. Emissivity quan-\ntifies the opposite process in which the atoms re-emit energy previously\nabsorbed from collisional and radiative processes.\nArticle number, page 3 of 22A&A proofs: manuscript no. aanda\n2.2. Radiative transfer equation in the Jones formalism\nDeveloping Eq.(5), adding the source term, and considering the\nrelations in Eqs.(3b) and (6), we obtain the RTE in terms of 2x2\nmatrices (Jones formalism; see Shurcli ff2013):\nd\ndsJ=E−(˜KJ+J˜K+), (8)\nwhere the emissivity term is\nE=1\n2· \nϵI+ϵQϵU−iϵV\nϵU+iϵVϵI−ϵQ!\n, (9)\nthe 2x2 propagation matrix is:\n˜K=1\n2· \nηI+aQaU−iaV\naU+iaVηI−aQ!\n. (10)\nandJwas written in terms of the Stokes parameters in Eq.(3a).\nThese equations di ffer from those in van Ballegooijen (1985)\n(also Kalkofen 1987) in some aspects. First of all, we use a more\nconvenient, modern and general nomenclature. Namely, we do\nnot constrain the equations to the particular geometry of a strat-\nified atmosphere, we use geometrical distance along the optical\npath instead of optical depth as the evolution parameter, and we\ndo not assume yet any particular expressions for the optical co-\nefficients, allowing to represent any general physical situation\nto describe the plasma element (e.g. to consider scattering po-\nlarization, Hanle physics, etc.). Secondly, we use the definition\nak=ηk+iρkin Eq. (7), instead of the frequency profiles associ-\nated toηk−iρk, to obtain compact and cleaner expressions. This\nalso allows us to present similar treatments for both Stokes and\nJones formalisms and thus compare them using similar quan-\ntities agreeing with the monograph of Landi Degl’Innocenti &\nLandolfi (2004).\nFinally, with respect to vB85’s, our expressions have non-\nobvious sign di fferences in all ϵV,aV, and Stokes V, which de-\nmands an explanation. The convention of signs mentioned after\nEq. (2) for the temporal exponentials of the electric field sets\nthe handedness (sign) of Stokes V: inverting the convention is\nequivalent to complex conjugation in the polarization tensor and\ntherefore in Stokes V , as obvious from Eqs. (3). Despite vB85’s\nspecifies our same (right-handed) basis for the electric field, that\nauthor does not specify a sign convention. A direct calculation\nshows that the sign di fferences comes from that issue, and that\nour choice leads to the same 2 ×2 system of matrices if akis\ndefined as we did. In summary, one convention corresponds to\nour equations, while the other one uses instead ak=ηk−iρkand\nnegative signs for the exponentials in Eqs. (1) and (2). These lat-\nter choice would imply doing J→J∗,iV↔− iV,iaV↔− iaV,\nandiϵV↔− iϵVin our Jones formulation.\n2.3. Radiative transfer equation in Stokes formalism\nCarrying out the matrix products in Eq.(8) and separating real\nand imaginary parts, we obtain the polarized RTE in Stokes’\nformalism. It is a system of four coupled first-order, ordinary\ndifferential equations whose homogeneous /inhomogeneous term\ncontains the propagation /emissivity matrix /vector (e.g., Landi\nDegl’Innocenti & Landolfi 2004, chapter 8):\nd\ndsI=ϵ−KI, (11)with ϵthe emissivity vector and Kthe 4x4 propagation matrix:\nd\ndsI\nQ\nU\nV=ϵI\nϵQ\nϵU\nϵV−ηIηQηUηV\nηQηIρV−ρU\nηU−ρVηIρQ\nηVρU−ρQηII\nQ\nU\nV.(12)\nThe Stokes and Jones formalisms are equivalent in that they de-\nscribe the same physical problem (e.g., Sanchez Almeida 1992).\nThe fact that one can derive the Stokes RTE from Jones’ proves\nthis. Historically, the solar and astrophysical community has pre-\nferred the Stokes formalism. The reason to include them both in\nthis first paper is to facilitate a posterior comparison from the\nstandpoint of our formulation, because the algebraic properties\nof the Jones 2×2 matrices could imply advantages when working\nwith the Magnus expansion.\n2.4. Generic solution in Jones formalism\nIn order to solve the RTE, one first consider its homogeneous\npart, i.e. the RTE without emissivity, and then use it to solve\nthe inhomogeneous part. For the Jones formalism, we follow a\nderivation similar to that of vB85, but adapting it to our general\nnomenclature and adding small improvements. First, we assume\nan homogeneous solution to Eq. (8) as the following variation of\na certain auxiliar Jaalong the trajectory parameter s:\nJ(s)=O(s,s0)Ja(s)O†(s,s0), (13)\nwhere5the evolution operator O≡O(s,s0;˘K) is a 2×2 complex\nmatrix fulfilling:\ndO\nds=−˘K(s)O, \n⇒dO†\nds=−O†˘K†(s)!\n, (14)\nDifferentiating Eq. (13) with respect to s, substituting Jand its\nderivative in Eq. (8), and removing terms cancelling each other,\nwe obtain:\ndJa(s)\nds=O−1E(s) [O−1]†, (15)\nwhose integral\nJa(s)=Ja(s0)+Zs\ns0ds′O−1(s′,s0)E(s′) [O−1]†(s′,s0), (16)\nis then inserted into Eq. (13). Now, we improve the result of\nvB85 by specifying the boundary term and combining evolution\noperators to the left and to the right into a total evolution operator\nOT. Thus, we obtain our generic inhomogenous solution in Jones\nformalism:\nJ(s)=J(s0)+Zs\ns0ds′OT(s,s′)E(s′)O†\nT(s,s′), (17)\nwhere\nOT(s,s′)=O(s,s0)·O−1(s′,s0). (18)\nand where the first term is the boundary contribution, which in\nanalogy with Eq.(13), has been called\nJ(s0)=O(s,s0)Ja(s0)O†(s,s0). (19)\nVB85 did not specify the boundary term because applied the so-\nlution to an optically thick atmosphere as a whole piece, in which\n5A†denote the conjugate transpose or Hermitian adjoint of A.\nArticle number, page 4 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\ncase the boundary term cancels out due to the physical properties\nof the boundaries (zero illumination for incoming rays and op-\ntically thick atmosphere for emerging rays). But in practice, the\natmosphere is discretized and the integrals are applied sequen-\ntially to every cell of a reduced physical domain that is optically\nthin. Hence, we need to retain and specify the changing bound-\nary contribution for each step. Note that in Eq. (17) the total solu-\ntion is the direct addition of an integral to the previous solution\nat the boundary, without multiplying the latter by an evolution\noperator as in the Eq. (23) of the Stokes formalism.\n2.5. Generic solution in Stokes formalism\nFrom Eq.(11), the homogenous RTE in Stokes formalism is\nposed as the initial value problem:\ndI(s)\nds=−K(s)I(s); I(s0)=I0, (20)\nIntegrating in an interval ∆s=s−s0, its homogeneous formal\nsolution is given by an evolution operator O(s,s0;K) applied to\nthe initial value\nI(s)=O(s,s0;K)·I(s0), (21)\nwith Os\ns0≡O(s,s0;K) being solution to6\nd\ndsOs\ns0=−K(s)Os\ns0, Os0s0=1 (22)\nOnce the evolution operator is known, the general formal inho-\nmogenous solution to the RTE is posed\nI(s)=O(s,s0)I(s0)+Zs\ns0ds′O(s,s′)ϵ(s′). (23)\nThe suitability of this solution is physically and numerically de-\ntermined by the specific expressions adopted for the evolution\noperator and for the above integral. In the following sections, we\nwill both reformulate the evolution operator and the inhomoge-\nneous solution to develop better ways of solving the RTE.\n3. Foundations for reformulating the radiative\ntransfer solution with the Magnus expansion\nIn astrophysics, the term evolution operator has been used to re-\nfer both to a very general concept and to very particular meth-\nods, which is confusing. The concept was originally known in\nmatrix theory as the matricant (Gantmacher 1959), and refers to\nthe matrix advancing the solution of a di fferential equation along\na trajectory of integration. For instance, in Stokes formalism, Eq.\n(21) defines the evolution operator as the 4 ×4 real matrix that,\nwhen multiplied by the Stokes vector at s0, gives the Stokes vec-\ntor solution at point s. Therefore it fully characterizes the final\nsolution, both physically and numerically.\nInstead, the methods associated to the evolution operator\nwere introduced in solar physics by vB85 (see also Kalkofen\n2009) and later by LD85 to approximate explicit analytical evo-\nlution operators for the RTE in the Jones and Stokes formalism,\nrespectively. Essentially, these methods consists in conveniently\ndecomposing a constant propagation matrix to approximate its\nexponential. Thereafter, a numerical representation of it can be\n6This is seen derivating the homogenous equation with respect to s(or\ns0) and using Eq.(21)implemented to obtain the inhomogeneous solution for the trans-\nfer between s0andsfrom Eqs. (23) or (17). Thus, to solve them\nwe need an explicit expression for the evolution operator (homo-\ngeneous solution), and a way of solving the integrals (inhomo-\ngeneous solution).\nThe numerical approximations to the evolution operator are\napparently easier than the analytical ones in their formulation,\nwhich made them dominate in our field. However, an analyt-\nical approach as the one we are going to develop can lead to\nmore accurate numerical implementations because part of the\nresult has been already calculated analytically and exactly. Fur-\nthemore, it allows to study the theoretical dependence of the so-\nlution on some parameters. The key to develop a suitable analyti-\ncal solution that can later be translated into a powerful numerical\nmethod is to respect the Lie group structure of the RTE, which\npreserves the qualitative properties of the exact solution. As part\nof the foundations supporting our work with the Magnus expan-\nsion, we present now a brief conceptual framework to interpret\nthe evolution of the solution to the RTE from the point of view\nof Lie groups7.\nIn particular, a Lie group is a group of elements with a main\ngroup operation mapping its elements smoothly to form a di ffer-\nentiable manifold M(topological condition), but also fulfilling\nan algebraic condition: the result of combining the elements of\nMtogether with the operation of commutation stays in M. The\ninterest in Lie groups for solving di fferential equations is that be-\ning di fferentiable, hence analytical, there exists tangents to them\nat any point p∈M . A tangent vector at pcan be defined by dif-\nferentiating a smooth parametric curve γ(s) such thatγ(0)=p\n7In general, a group Gis a set of elements Ai∈G that together with\na binary operation fulfills the axioms of closure ( A1·A2=A3∈G),\nassociativity ( A·(B·C)=(A·B)·C), and existence of neutral, identity,\nand inverse ( A·A−1=1). We shall only consider Lie groups of invert-\nible (hence square) N×Nmatrices with the ordinary product as group\noperation and the commutator as the Lie bracket.\nFig. 1. Solution to the RTE evolving in the ordinary 3D space of the ray\n(orange) and in the 4D space of the Poincaré group (red). A local solu-\ntion (white circles) is an element of the Lie group and an exponential\nmap of vector fields (blue) associated to the corresponding Lie algebra.\nArticle number, page 5 of 22A&A proofs: manuscript no. aanda\nFig. 2. Examples of basic vector fields (arrows) and corresponding flow solutions (red lines) to the homogeneous RTE for di fferent wavelengths.\nLeft: rotational flow due to magneto-optical e ffects for constant Kin plane Q-U. The trajectory, which should be circular, deviates to describe a\nspiral due to the accumulated inaccuracy error in the standard integration method employed here. Middle: hyperbolic flow due to dichroism in I-V\nplane for constant K. Right: combination of vector fields and evolution of solution in QUV space for non-constant K.\n(e.g., Bonfiglioli & Fulci 2011):\nv=dγ(s−s0)\nds\f\f\f\f\fs=s0(24)\nAs illustrated in Fig. 1, the set of all possible tangent vectors\natpforms the linear vector spaceT(the tangent space of the\ngroup). If locally (in the neighourghood of each p) we associate\nγ(s) with the evolution of the solution to a linear ODE on M,\nthen the infinitesimal advance of the solution at poccurs in a di-\nrection contained in its tangent space. Hence, instead of working\nin the non-linear manifold M, Lie groups o ffer the mathematical\nfoundation for solving the ODE in a linear vector space while\npreserving the local structure of the group.\nThe key to do this is that the elements in Mcan also be ob-\ntained by mapping (e.g., exponentiating) those of the so-called\nLie algebra gof the group. The algebra is defined as the com-\nbination of the tangent vector space around the group identity\nelement with the operation of commutation. In that way, the al-\ngebraic structure of a Lie group is captured by its Lie algebra, a\nsimpler object (since it is a vector space).\nTo particularize these basic ideas, consider the homogeneous\nRTE and its formal solution in Stokes formalism given by Eqs.\n(20) and (21) with system matrix A=−K:\nI′(s)=A(s)I(s); I(s0)=I0, (25)\nI(s)=O(s,s0;A)I0. (26)\nAs the l.h.s. of the RTE is a first derivative with respect to s,A\ncan be seen as a vector field tangent to the s-parametric stream-\nline followed by the system when transiting from I(s0) toI(s) in\nM, while the evolution operator O(s,s0;A) is called the flow of\nAbecause it advances the solution following A. The matrix A\nrepresents a vector field (a.k.a. infinitesimal generator ) when is\nseen as the derivative of the flow at s=s0:\nA=dO(s,s0;A)\nds\f\f\f\f\fs=s0. (27)\nThis operation represents a local coordinate map between the el-\nements of the group and its tangent linear space Tformed by\nthe set of all possible vector fields A. Namely, one can see that\nsuch a coordinate map is possible at the identity of the group,\ni.e. where O(s,s0;A)=1ins=s0. In general, this condition is\nsatisfied if the evolution operator is an exponential map. Then,as commutators of system matrices at points nearby the identity\nelement ([ A1,A2]=A1A2−A2A1) represent the total deriva-\ntive of the vector field Aing, their exponentiation is equivalent\nto infinitesimal translations in the group. A more specific repre-\nsentation of the evolution of the solution to the RTE in terms of\nvector fields and flows is presented in Fig. 2(see caption).\nIn the rest of this section we make the evolution operator\nmore explicit. As it is formally the same in both the Stokes and\nJones formalisms, we shall focus in the former.\n3.1. The universal solution to the evolution operator is a\nseries\nIn general, the evolution operator solving the homogeneous RTE\nin Eq. (25) is given by the V olterra (1887)’s integral equation\nO(s,s0;A)=1+Zs\ns0dtA(t)O(t,s0,A),\n(28)\nafter applying the fundamental theorem of calculus to Eq.(26)\nwith O(s0,s0)=1. A Picard iteration on Eq. (28) can start with\nO(t,s0,A)=0 and be continued up to infinity. Thus, an expres-\nsion for O(s,s0) was first obtained by Peano (1888) (also Peano\n(1890) or translation in Kannenberg (2000)) and further studied\nby Baker (1902, 1905). It is the funny Peano-Baker series:\nO(s,s0;A)=1+HA+HAHA+HAHAHA+...(29)\nwhere H Ais the result of applying the integral operator Hfor in-\ntegrating the matrix A=(a)i jin∆s, i.e. (H A)ij(s)=Rs\ns0dt a i j(t).\nEach term of the series can be recursively written in terms of\nmatrix components:\na(1)\ni j=Zs\ns0dt a i j(t), a(2)\ni j=Zs\ns0dtdX\nn=1ain(t)a(1)\nn j(t), ...\nor directly as:\nO(s,s0;A)=1+Zs\ns0A(t)dt+Zs\ns0A(t1)Zs\ns0A(t2)dt2dt1+...(30)\nWhen A=−Kis constant, Eq. (30) is easily reduced to:\nO(s)=1+A(s−s0)+...+An(s−s0)n\nn!+...=e−K(s−s0)(31)\nArticle number, page 6 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\nThe Peano-Baker series is unique and converges absolutely\nand uniformly in every closed interval where A(s) is continu-\nous and bounded (Ince 1956; Gantmacher 1959). This solution is\nemployed in di fferent contexts, where is often known as the Neu-\nmann series or the Dyson perturbative solution (Dyson 1949).\nThe problem with this series is that, when truncated (either nu-\nmerically or analytically), it stops evolving in the Lie group of\nthe RTE, no longer preserving the algebraic properties of the ex-\nact solution (see e.g. Blanes et al. 2009).\n3.2. The limitations of assuming constant K\nLet us see now how the Peano-Baker evolution operator has been\napproximated to solve the radiative transfer problem in astro-\nphysics. Landi Degl’Innocenti & Landolfi (2004) starts rewriting\nEq. (30) in Stokes formalism. Noting the minus sign acompany-\ningK, one has ( Step-1 ):\nO(s,s0)=1+∞X\nn=1(−1)nZs\ns0ds1Zs1\ns0ds2···\n···Zsn−1\ns0dsnK(s1)K(s2)···K(sn) (32)\nMaking the integration regions independent on the integration\nvariables , one obtains the so-called \"time-ordered\" exponential\nin terms of the (time-)ordered8or chronological product of op-\neratorsP(Step-2 ):\nO(s,s0)=1+∞X\nn=1(−1)n\nn!Zs\ns0ds1Zs\ns0ds2···\n···Zs\ns0dsnP{K(s1)K(s2)···K(sn)}=P\be−Rs\ns0dtK(t)\t(33)\nHere, the Dyson chronological operator Pis such that\nP{K(s1)K(s2)···K(sn)}=K(sj1)K(sj2)···K(sjn)\nsj1≤sj2≤···≤ sjn (34)\nThe ordered exponential can only become an ordinary exponen-\ntial when the products inside the integral commute among all of\nthem ([ K(si),K(sj)]=0), thus setting us free to reorder them. In\nthat case, the products inside the integral become ordinary prod-\nucts of identical integrals and the expression becomes the Taylor\nexpansion of the exponential of such repeated integral( Step-3 ):\nO(s,s0)=1+∞X\nn=1(−1)n\nn!\u0014Zs\ns0dtK(t)\u0015n\n=e−Rs\ns0dtK(t)(35)\nAs it is imposible that all propagation matrices commute among\nthem for all rays inside a realistic solar atmosphere, the only\nplausible assumption to achieve commutation is that the propa-\ngation matrix Kis constant along the whole integration path of\ngeometrical length ∆s=s−s0. But this was already implicit\nin the previous step and any further assumption on the variation\nofK(e.g. to design a numerical method) would be inconsistent.\nThe remaining step now is actually part of assuming a constant\nK, just extracting Kout of the integral to obtain ( Step-4 ):\nO(s,s0)=e−K·(s−s0)(36)\nUntil now, this has been the starting expression to solve any po-\nlarized radiative transfer problem. When calculated numerically,\n8In our case, it is a space-ordered product of operators.it is truncated, and hence it is equivalent to the Peano-Baker se-\nries, with the same truncation problems as Eqs. (30) and (31).\nIn addition, this evolution operator limits the consistency of cur-\nrent approaches to numerical cells with constant properties (lo-\ncal methods of solution). Here, a fully consistent inhomogeneous\nsolution should imply both Kand the emissivity to be constant,\nsimplifying Eq. (23) to the poor approximation\nI(s)=e−K(s−s0)I0+e−Ks\u0014Zs\ns0eKtdt\u0015\nϵ=\n=e−K(s−s0)I0+(1−e−K(s−s0))K−1ϵ (37)\nWe see then that current formal solutions and methods for the\nRTE are physically and mathematically inconsistent because\nthey start from an expression obtained with Kconstant while\nlater assuming that its exponential and the emissivity vary in-\nside the cell when a numerical method is defined. Furthermore,\na simple inspection of the physical dependences driving emis-\nsivity and the propagation matrix shows that one cannot change\nif the other is constant. Thus, the numerical approximations im-\nposed by common radiative transfer methods to pose and calcu-\nlate Eqs. (36) and (23) break the inhomogeneous group structure,\ni.e. their solution does not evolve in its true physical algebraic\ngroup, rendering them both numerically and analytically inac-\ncurate. This shows that the numerical properties of a radiative\ntransfer method should be quantified with a reference solution\nbased on an exact method, i.e. one that respects the evolution in\nthe Lie group to a su fficiently high level of accuracy. To allow\nthis, we need an exact evolution operator: the Magnus solution.\n3.3. Magnus solution: an exact Lie-friendly evolution operator\nInstead of solving Eq. (22) in the Lie group with Eq. (30), Haus-\ndorff(1906) did it generically in the Lie algebra ( g). His general\nsolution was an exponential evolution operator9O=eΩ(s), with\nΩ(s)∈g. From (here A=−Kto avoid confusion with the sign)\ndO\nds=AO⇒\u0012deΩ(s)\nds\u0013\ne−Ω(s)=A, (38)\nhe derived the non-linear di fferential Hausdor ff’s equation for Ω:\ndΩ\nds=∞X\nn=0Bn\nn![A,Ω[n]]. (39)\nTheBnare the Bernoulli numbers10and\n[A,Ω[n]]=[···[[A,nz            }|            {\nΩ],Ω]···,Ω], (40)\nis the adjoint of Ωwritten as right-nested commutators with\n[A,Ω[0]]=A. The infinite recursive series in Eq. (39) is analytic\nin allCexcept in the points P={2πni,n∈Z>0}. Iteration on\nEq. (39) led Magnus (1954) to the solution Ω(s)=P∞\nk=1Ωk(s)\n9AsΩ(s0)=0⇒O(s0)=1, and O(s) is continuous and invertible\naround s=s0. Then, the inverse evolution operator is O−1=e−Ω(s).\n10The first values are Bn=1,−1/2,1/6,0,−1/30,0,1/42,0,−1/30,...\nwith Bn=0 for any odd n di fferent than 1 (Abramowitz & Stegun 1972).\nArticle number, page 7 of 22A&A proofs: manuscript no. aanda\nwith the Magnus expansion being such that Ω(0)=0and\nΩ1(s)=Zs\ns0d1A1,\nΩ2(s)=−1\n2Zs\ns0\u0014Zs1\ns0d2A2,A1\u0015\nd1,\nΩ3(s)=1\n4Zs\ns0\u0014Zs1\ns0\u0014Zs2\ns0A3d3,A2\u0015\nd2,A1\u0015\nd1,\nΩ4(s)=1\n12Zs\ns0\u0014Zs1\ns0A2d2,\u0014Zs1\ns0A2d2,A1\u0015\u0015\nd1,\nΩ5(s)=−1\n8Zs\ns0\u0014Zs1\ns0\u0014Zs2\ns0\u0014Zs3\ns0A4d4,A3\u0015\nd3,A2\u0015\nd2,A1\u0015\nd1,\nΩ6(s)=−1\n24Zs\ns0\u0014Zs1\ns0\u0014Zs2\ns0A3d3,\u0014Zs2\ns0A3d3,A2\u0015\u0015\nd2,A1\u0015\nd1,\n...\n(41)\nwith dn=dsnandAn=A(sn). This series converges in the\nneighborhood of eΩ(s0)if the di fferences between any two eigen-\nvalues of Ω(s0) is not in P. In general, di fferent terms Ωncom-\npose a certain order kof the expansion. Any of such terms con-\ntains kpropagation matrices, a multivariate integral with knested\nintegrals, and k−1 nested commutators. For n≥3, more than\none term Ωn(s) is always necessary to fully account for any given\norder of the expansion. Thus, conveniently, in this paper we shall\nonly consider Ω1andΩ2, because they alone fully quantify the\norder 1 and 2 of the expansion respectively, something that does\nnot change in any other formulation of the Magnus expansion.\nIt should be clear that the order we are talking about is that of\nthe Magnus series, not the one of the corresponding numerical\nmethods to be developed to solve the integrals and the RTE.\nA key point here is that, if a truncated Magnus expansion is\nused in approximation of Ω, then the subsequent Magnus meth-\nods shall yet preserve peculiar geometric properties of the solu-\ntion (e.g., preservation of a norm). As the Magnus solution, i.e.\nthe homogeneous solution to the RTE, is in the Lie group, and\nany commutator of A=−Kis in the associated algebra g, then a\nsuitable numerical discretization can also stay on the manifold.\nBy the properties of the integrals, the Magnus expansion also\naccomplishes Ω(s0,s1)+Ω(s1,s2)=Ω(s0,s2) andΩ(s,s0)=\n−Ω(s0,s). Hence, the Magnus evolution operator allows seri-\nalization and preserves a symmetric homogeneous evolution.\nNamely, calling Osj\nsi≡O(si,sj),:\nOs2s0=Os2s1Os1s0, (42a)\nOs0s0=1=Os\ns0·[Os\ns0]−1, [Os\ns0]−1=Os0s, (42b)\nThis could in principle allow using the same homogeneous\nsolution for both outgoing and incoming rays along a same di-\nrection in the solar atmosphere, which might be e fficient when\napplying our methods to the full NLTE problem.\nA deeper analysis of the Magnus expansion is not necessary\nnow (see e.g., Blanes et al. 2009). However, we should point out\nthat there are many ways of formulating it. Interestingly, we havefound that the terms in (41) can be neatly written as\nΩ1(s)=Zs\ns0d1A(s1),\nΩ2(s)=1\n2Zs\ns0\u0014\nA(s1),Ω1(s1)\u0015\nds1,\nΩ3(s)=1\n2Zs\ns0\u0014\nA(s1),Ω2(s1)\u0015\nds1,\nΩ4(s)=1\n12Zs\ns0\u0014\u0014\nA(s1),Ω1(s1)\u0015\n,Ω1(s1)\u0015\nds1,\nΩ5(s)=1\n2Zs\ns0\u0014\nA(s1),Ω3(s1)\u0015\nds1,\nΩ6(s)=1\n2Zs\ns0\u0014\nA(s1),Ω4(s1)\u0015\nds1,\n...(43)\nWith this way of presentation, we point out that the Magnus\nexpansion is a peculiar object that, as a result of the Hausdor ff\niteration, seems to exhibit a certain fractal formal character: its\nterms can be written as functions of lower-order terms chang-\ning at smaller integration scales11. Despite such a character is\npurely formal, it arises our interest in investigating the Magnus\nexpansion from the point of view of fractality.\n3.4. The Basic Evolution (BE) theorem\nThe simplest evolution operator resulting from the Magnus ex-\npansion includes only the first term of the expansion after trun-\ncation of higher-order terms12. A Magnus solution based on at\nleast that first term would be significantly better than one based\non Eq.(36), because the presence of the integral in the Magnus\noperator consistently preserves memory of the evolution, i.e. of\nthe variation of the propagation matrix along the ray. To specify\nthis operator we need to calculate the matrix exponential of in-\ntegrals of the propagation matrix, for which we have derived the\nfollowing theorem.\nConsider the decomposition of an arbitrary matrix N=f# »n·# »σin terms of a real constant f, a vector# »n=(n1,..., nd) with\nmodule n, and a vector# »σ=(σ1,...,σ d) of basis matrices of an\nalgebra g. Our theorem states that if these matrices fulfill\nσi·σj=δi j1+c·ϵi jk·σk, (c=constant) (44)\nthen the exact exponential of the integral of the matrix N is given\nby (see demo in Appendix):\ne±R\nds f·# »n(s)·# »σ=ch(f b)1±sh(f b)# »u·# »σ, (45)\nwhere# »uis the unitary vector resulting of integrating# »nalong\nthe trajectory, such that:\n# »u=# »b\nb=(b1,b2,···,bd)\n[b2\n1+b2\n2+···+b2\nd]1/2(46a)\nbk=Z\nds n k(s) (46b)\n11The formal form of any term always results of substituting a matrix\nAof a term of previous orders by the simplest elemental commutator in\nΩ2. In that sense, the structure of terms could be seen as fractal.\n12This is not the same as assuming constant propagation matrices,\nwhich would also cancel higher-order terms but reducing the first term\nto the common oversimplification given by Eq.(36)\nArticle number, page 8 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\nThe particularization of this general solution to the case of con-\nstant N(i.e., constant nk) gives bk=s·nk,# »u=# »n/nandb=s·n.\nWe identify our solution (45) with a rotor, an element of Clif-\nford geometric algebra. Geometric algebra allows to associate\nalgebraic transformations with space properties (e.g., volumes)\nand to extend the treatment of rotations and the concept of vec-\ntor to an arbitrary number of dimensions without the need of\nimaginary numbers (Hestenes 2003). One of the insights that we\nobtain from identifying Eq. (45) with a rotor is that the evolu-\ntion of the homogeneous solution behaves as a rotation around a\ndirection defined by the vector# »u.\n4. Algebraic representations for the polarization\nApplying the BE theorem to the polarized RTE, one can explain\nthe evolution of the homogeneous solution of the RTE as rota-\ntions in a Lie algebra around the identity of a group. It will be\nthen useful now to consider the basic groups and algebraic struc-\ntures capable of representing rotations (e.g., Hall 2015). Groups\ncan be represented by mapping their elements onto linear op-\nerators (e.g., matrices) acting on some vector space V. But it\nis possible to represent the same group elements using di ffer-\nent kinds of operators, as we do for instance when expressing\nthe RTE in Stokes or Jones formalisms. To easily connect this\nwith rotations, we start considering the simplest example of an\northonormal basis β={(1,0),(0,1)}forV=R2and the coun-\nterclockwise rotation Rθby an angle θ. Applying trigonometry,\nthe basis vectors transform as:\nRθ(1,0)=(cosθ,sinθ), (47a)\nRθ(0,1)=(−sinθ,cosθ). (47b)\nAnd using the transformed vectors as matrix columns we can\nconstruct a parameterized matrix representation of Rθforβ:\nU= \ncosθ−sinθ\nsinθcosθ!\n←→ cosθ+isinθ=eiθ, (48)\nwhere we have also considered that any complex number a+ib\n(with a,b∈R) can also be represented as a real 2 ×2 matrix13\na+ib←→ \na−b\nb a!\n. (49)\nThus, rotations of a vector v=(v1,v2) in a plane can be alge-\nbraically quantified both as Uvor as ( v1+iv2)·eiθ. From here, ro-\ntations in any number of dimensions can be seen as linear super-\npositions of two-dimensional rotations embedded into the higher\ndimensions (any nD rotation is a superposition of 2D rotations\nin planes formed by each couple of dimensions in the space).\n4.1. Rotations in 3D and 4D\nThe group of rotations in ndimensions is the special orthogo-\nnal group SO( n), which leads to the ordinary vector represen-\ntation inRn, whose operators are orthogonal ( UTU=1)Un×n\nmatrices with|U|=1. The corresponding algebra so(n) is com-\nposed by skew-symmetric matrices. Hence, in vector representa-\ntion, three-dimensional rotations can be represented with SO(3).\nHere the Lie algebra so(3) can be expressed either as the Eu-\nclidean space R3with the vector product as the Lie bracket, or\n13This is a ring isomorphism from the field of complex numbers to the\nring of these matrices.as the set of all skew-symmetric matrices with the matrix com-\nmutator as the Lie bracket. Thus the elements of their algebras\nare linked by the map:\n# »n=(n1,n2,n3)←→ N=0−n3n2\nn3 0−n1\n−n2n1 0, (50)\nsuch that a rotation of a vector# »vin a plane with normal vector# »n\nis# »n×# »vorN# »v. An important extension of the group of rotations\nto four dimensions is the Lorentz group SO(1,3), whose elements\nsatisfy\nUTJU=J, (51)\nwith J=diag{1,−1,−1,−1}. Its Lie algebra so(1,3) is deter-\nmined by computing the tangent vectors to parametric curves\ns7→U(s) on SO(1,3) through the identity 1. Hence, di fferen-\ntiating Eq.(51) with U(0)=1, the elements H≡U′(0) of the\nalgebra satisfy:\nHTJ+JH=0, (52)\nwhich implies that JHandHare symmetric in the first dimen-\nsion and skew-symmetric in the others:\nso(1,3)=( 0u\nuTB!\n|u∈R3,BT=−B)\n. (53)\nThis condition is satisfied by the propagation matrix in Stokes\nformalism, and Eq. (51) can then be used to check whether a\nsolution to the RTE belong to its corresponding Lorentz group.\nAn alternative way of representing n-dimensional rotations\nis the special unitary group SU(n), which is the set of all n×n\ncomplex unitary14matrices with unit determinant and whose al-\ngebra su(n) is composed by skew-Hermitian traceless matrices.\nNamely, three-dimensional rotations can be represented in SU(2)\n(spinorial representation). This group uses traceless Hermitian\nmatrices in a vector space H, which maps the vector representa-\ntion in R3through the correspondence:\n# »n=(n1,n2,n3)←→ N= \nn1 n2−in3\nn2+in3−n1!\n∈H, (54)\nThus, from a right-handed base in R3, we construct one in H:\nˆe1=(1,0,0)←→σ1= \n1 0\n0−1!\n, (55a)\nˆe2=(0,1,0)←→σ2= \n0 1\n1 0!\n, (55b)\nˆe3=(0,0,1)←→σ3= \n0−i\ni0!\n, (55c)\nwhereσiare the Pauli matrices15. They satisfy:\nσiσj=δi j1+iϵi jkσk, (56a)\n[σi,σj]=2iϵi jkσk, (56b)\nwithϵi jkthe Levi-Civita permutation symbol16. Defining the\nPauli vector# »σ=(σ1,σ2,σ3), any element of the algebra su(2)\n14A matrix is unitary when A†A=AA†=1.\n15These definitions only di ffer from those used in quantum mechanics\nin the labels: our basis is defined such that the Pauli matrix with complex\nunit is associated to the third axis of the right-handed cartesian base.\n16It is+1/−1 if the number of permutations of ( i,j,k) to obtain (1 ,2,3)\nis odd /even, or 0 if two indices are equal.\nArticle number, page 9 of 22A&A proofs: manuscript no. aanda\nFig. 3. Geometry associated to a±anda2\n±.Right panel : deduction of the triangle r-q-h assuming one of its corners at λ1. To obtain the figure: 1)\nAsa+fits the side of the romboid, we associate it with the distance from η0to the point λ1+iλ3(i.e., as if the origin were at η0); 2) We obtain\nαthrough the area of the romboid A♢=2|ˆa||˜a|=|q|=hsin(2α); 3) As h2=q2+r2and sin(2α)=q/h, we associate h, q, and r with the sides\nof a rectangular triangle with a corner in λ1; 4) We identify auxiliary triangles with sides ˆd,˜dand ˆD,˜D, finding: ˆd+i˜d=(√\nh−1)·(|ˆa|+i|˜a|)\nand ˆD+i˜D=√\nh·(|ˆa|+i|˜a|). 5) Finally, we identify a vector a2\n±=r+iqenclosed in the diagonal of the rectangle of sides r-q-h with origin in\nO′=ˆO′+i˜O′=λ1−reiα. The distance C=OO′is then: C=[(ˆO′)2+(˜O′)2]1/2=[λ2\n1−2rcos(α)λ1+r2]1/2.Left panel :a2\n+anda2\n−when their\norigin O′is chosen at λ2. .\n(i.e., the matrices in Eq.(54)) can be written as N=# »n·# »σ. Ex-\nponentiating and applying the BE theorem (Section 3.4) with\nconstant# »n, one obtains the elements of the group SU(2):\nes# »n·# »σ=ch(θ)1+sh(θ)# »u·# »σ= (57a)\n= \nch(θ)+sh(θ)u1sh(θ) (u2−i u3)\nsh(θ) (u2+i u3) ch(θ)−sh(θ)u1!\n, (57b)\nwith normalized (unitary) vector# »u=# »n/n=(u1,u2,u3), and\na parameter θ=s·n. The latter matrix describes the evolu-\ntion of group elements as hyperbolic rotations by an angle θ\nin a plane with normal unit vector# »uembedded in R3. Thus,\nrotations in R3can be seen as compositions of three indepen-\ndent two-dimensional rotations, one per plane perpendicular to\na cartesian basis axes17. To describe polarized light in Jones for-\nmalism, we work with the special linear group SL(2 ,C), com-\nposed by 2×2 complex matrices with unit determinant. In other\nwords, the Jones formalism is the result of considering the SU(2)\ngroup elements of Eq.(57) with a complex, instead of real, vector# »u. We shall see that in the case of the polarization, the role of# »u\nis played by the (normalized) propagation vector.\n17Pure rotations around such axes would be\nes# »n1·# »σ= \neθ0\n0e−θ!\n(θ=s·n1)\nes# »n2·# »σ= \nch(θ) sh(θ)\nsh(θ) ch(θ)!\n(θ=s·n2)\nes# »n3·# »σ= \nch(θ)−ish(θ)\nish(θ) ch(θ)!\n(θ=s·n3).4.2. The propagation vector\nFor notational compactness, the propagation vector# »a, defined\nin Eq.(7), and its complex conjugate shall be denoted with +and\n−symbols, respectively:\n# »a±=# »η±i# »ρ=(η1±iρ1,η2±iρ2,η3±iρ3). (59)\nThe module aof the complex vectors# »a+and# »a−is a real num-\nber with:\na2=# »a+·# »a−=X\nkaka∗\nk=X\nk|ak|2=η2+ρ2, (60a)\na4=(η2−ρ2)2+(2ηρ)2, (60b)\nwhereη2=# »η2andρ2=# »ρ2. Instead, the vector modules of# »a+\nand# »a−are the complex numbers a+anda−, fulfilling\na2\n±=# »a±·# »a±=r±iq⇒ a±=ˆa±i˜a, (61)\nwith (see Appendix A):\nˆa=ˆs h+r\n2!1/2\n,˜a=˜s h−r\n2!1/2\n(62a)\nh=[r2+q2]1/2=a−·a+=ˆa2+˜a2(∈R) (62b)\nr=ˆa2−˜a2=η2−ρ2(62c)\nq=2ˆa˜a=2(# »η·# »ρ)=2ηρcos(θ) (62d)\nˆs˜s=sign(# »η·# »ρ) (62e)\nHere the signs ˆ sand ˜sare only constrained to Eq. (62e) (see\nAppendix A). Additional relations are:\n|ˆa|4+|˜a|4=h2+r2\n2, (63a)\na2\n±=r±iq=a±2# »ρ·#»a±. (63b)\nArticle number, page 10 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\nFig. 4. Mandelbrot fractal arising from iterating the square of the propa-\ngation vector C+a2\n±shifted for di fferent values of C. The iteration only\nconverges in the white region.\nNote that the real and imaginary parts of a±define the four eigen-\nvalues of the propagation matrix λ1,2,3,4={η0±|ˆa|, η0±i|˜a|},\nsuch that they form an equilateral romboid of side√\nhin the\ncomplex plane. This and the above algebraic relationships can\nbe translated into the geometry of Fig. (3). As explained there,\nthe distances between the eigenvalues of Kcan be related with\nthe propagation vector and its square, given by randqin Eq.\n(61). By choosing a reference system defined by the distance\nC=OO′and inclined with respect to the original real axis by\nan angleα, the projections randqofa2\n+can be made coincident\nwithλ1and be related to the romboid sides. Once the relative\nsize and inclination between r, q, and h has been obtained, we\ncould arbitrarily move and rotate their triangle to a di fferent ori-\nginO′, as shown in the second panel of Fig.(3). Interestingly,\nas the values of randqcompose the squared complex number\na2\n±, they form the Mandelbrot fractal (Mandelbrot 1982) when\n(a2\n±)n+1=(a2\n±)n+Cis iterated for all values of Cin the com-\nplex plane (see Fig. 4). We suspect a possible application of the\nMandelbrot fractal to understand certain aspects of the Magnus\nexpansion (e.g., its convergence properties, or an eventual rela-\ntion among its terms, rotations, and the powers of a±).\n4.3. Algebraic characterization of the propagation matrix\nIn this section, we present the engine of our formalism, a bat-\ntery of decompositions and calculations that allows to under-\nstand the algebra of the propagation matrix and operate con-\nveniently within it. Semel & López Ariste (1999) and López\nAriste & Semel (1999) investigated the RTE in Stokes formal-\nism and found important to investigate the algebraic structure of\nthe polarized RTE. One of their insights was that the Stokes 4-\nvector can be seen as an element of a Minkowski-like R1,3space\n(with Iplaying the role of the temporal dimension) with a metric\nhaving the norm18∥I∥≤I2−Q2−U2−V2. Thus, the RTE\nquantifies all possible infinitesimal transformations between two\nnearby points lying inside the “Stokes cone of light”. Such trans-\n18In the solar atmosphere this norm is close to one because intensity\nand all its related quantities ( I,ηI,ρI) are normally much larger than the\npolarization counterparts because the light is only partially polarized.formations are divided in 11 kinds composing a Poincaré group\n(the inhomogeneous generalization of a Lorentz group) in which\ndilatations are allowed. The inhomogeneous character is given\nby the inhomogeneous term in the RTE, i.e. ϵ, which quantifies\ntranslations along each of the four dimensions. Dilatations ap-\npear due to the diagonal of the absorption matrix in Eq. (12):\nK(s)=η0(s)1+ˆL(s), (64)\nwhich reduces /amplifies the number of reemitted photons\nthrough indiscriminated positive /negative absorption of photons\nin any polarization states as of η0=η(A)\n0−η(S)\n0is dominated by\nordinary absorption ( η(A)\n0as in the solar atmosphere, or by stim-\nulated emission ( η(S)\n0).\nThe remaining (i.e. the homogeneous) part of the Poincaré\ngroup is given by the Lorentz group19SO(1,3), whose elements\nUcan be expressed as exponentials of Lorentz matrices ˆL, ac-\ncomplishing (see Section 4.1):\nUTJU=J, (65a)\nˆLTJ+JˆL=0. (65b)\nwith J=diag(1,−1,−1,−1). The Lorentz matrix ˆLis com-\nposed of the 6 infinitesimal generators of rotations of the Lorentz\ngroup, i.e. the three 4 ×4 matrices ˜G1,˜G2,˜G3generating ordinary\n3D rotations (anomalous dispersion) in the QUV space, plus the\nthree 4×4 matrices ˆG1,ˆG2,ˆG3producing hyperbolic rotations\n(i.e., rotations involving the first dimension) due to dichroism\n(Lorentz boosts in special relativity theory).\nFrom here, our characterization of the propagation matrix\ndistinguishes four algebraic levels associated to decompositions\nin: 1) individual Lorentz generators; 2) base matrices; 3) hy-\nperbolic and circular subspaces; and 4) Lorentz matrices. The\nLorentz generators in the first level are encoded in ˆLas20:\nˆG1+ˆG2+ˆG3+˜G1+˜G2+˜G3=\n=0 1 1 1\n1 0 1−1\n1−1 0 1\n1 1−1 0. (66)\nHere the geometrical symbols point out the only elements of\nthe corresponding generator matrices that are not zero. They de-\nscribe rotations in every subspace and they are all orthogonal\namong them. For all k, we find:\nˆG2\nk−˜G2\nk=1 (67a)\nˆGk·˜Gk=0 (67b)\nˆG2\nk=1k⇒ ˜G2\nk=1k−1, (67c)\nwhere\n1k=1 0 0 0\n0δ1,k0 0\n0 0δ2,k0\n0 0 0 δ3,k. (68)\n19Actually, the group representing the polarization in Stokes formalism\nis SO+(1,3) because, as I≥0, one only needs the positive part of the\n\"Stokes cone of light\" to describe polarization.\n20Our notation emphasizes that they are real and imaginary part of a\nsame complex matrix.\nArticle number, page 11 of 22A&A proofs: manuscript no. aanda\nEquivalently, all the above implies ( ˆGk±i˜Gk)2=1. We also\nintroduce a set of dual generators:\nˆD1+ˆD2+ˆD3+˜D1+˜D2+˜D3=\n=0 1−1 1\n−1 0 1 1\n1 1 0 1\n−1 1 1 0. (69)\nAs before, every dual generator ˆDkor˜Dkis a 4×4 matrix whose\nonly entries di fferent than zero are those pointed out with the\ncorresponding geometrical symbol. Thus, the anticommutators\nof the original generators can be neatly expressed as:\n[ˆGi,ˆGj]+=2·δi j1i+|ϵi jk|˜Dk(∀i,j) (70a)\n[˜Gi,˜Gj]+=2(1i−1)+|ϵi jk|˜Dk(∀i,j) (70b)\n[ˆGi,˜Gj]+=ϵi jk(−1)kˆDk(∀i,j) (70c)\nAnd their ordinary commutators are:\n[ˆGi,˜Gj]=−ϵi jkˆGk(∀i,j) (71a)\n[ˆGi,ˆGj]=ϵi jk˜Gk(∀i,j) (71b)\n[˜Gi,˜Gj]=−ϵi jk˜Gk(∀i,j) (71c)\nInstead of infinitesimal generators, sometimes it is more con-\nvenient to use denser matrices. Thus, our second algebraic level\nconsiders a decomposition in larger subspaces. Namely, we de-\nfine vectors of hyperbolic and ordinary 4 ×4 rotation matrices\n# »ˆG=(ˆG1,ˆG2,ˆG3) (72a)\n# »˜G=(˜G1,˜G2,˜G3), (72b)\nsuch that any Lorentz matrix ˆL(si)≡ˆLi=Hi+Riat a point i\ncan be neatly decomposed into a hyperbolic rotation Hi=# »ηi·# »ˆG\nand an ordinary rotation Ri=# »ρi·# »˜G:\nˆLi=0η1,iη2,iη3,i\nη1,i0 0 0\nη2,i0 0 0\nη3,i0 0 0+0 0 0 0\n0 0 ρ3,i−ρ2,i\n0−ρ3,i 0ρ1,i\n0ρ2,i−ρ1,i 0.\n(73)\nRemarkably, any power n∈ZofHandRcan be reduced to\nH2n=η2(n−1)H2(74a)\nH2n+1=η2nH (74b)\nR2n=(−1)n−1ρ2(n−1)R2(74c)\nR2n+1=(−1)nρ2nR, (74d)\nwhich allows to use power series to calculate exactly any func-\ntion of HorRin terms of only H,R,H2,R2, where H2,R2are\neasily obtained with\nH2=C(# »η), (75a)\nR2=C(# »ρ)−ρ21, (75b)andCis the generic correlation matrix defined as:\nC(# »u)=u20 0 0\n0 u2\n1u1u2u1u3\n0u1u2u2\n2u2u3\n0u1u3u2u3u2\n3. (76)\nFor the crossed products, we find interesting that\nHRH =RHR =(HR)n=(RH)n=0(n∈Z), (77)\nwhile the basic product RHis\n−QT=RH=0 0 0 0\n(# »η×# »ρ)10 0 0\n(# »η×# »ρ)20 0 0\n(# »η×# »ρ)30 0 0, (78)\nforQ=HR. Here the oriented surface vector# »S=# »η×# »ρhas\nas module the area of the paralellogram enclosed by# »ηand# »ρ.\nFurthermore, HandRdo not commute:\n[Hi,Rj]=Q+QT=−(# »ηi×# »ρj)·# »ˆG (H−like) (79a)\n[Hi,Hj]=(# »ηi×# »ηj)·# »˜G (R−like) (79b)\n[Ri,Rj]=−(# »ρi×# »ρj)·# »˜G (R−like). (79c)\nWe also obtain nested commutators that are useful to describe\nthe algebra of the Magnus expansion:\n[...[[H1,R1],R2]...,Rn]=\n=(−1)n(...((# »η1×# »ρ1)×# »ρ2)×...×# »ρn)·# »ˆG (H−like) (80a)\n[...[[H0,H1],H2]...,Hn]=\n=(...((# »η0×# »η1)×# »η2)×...×# »ηn)·# »˜G (R−like) (80b)\n[...[[R0,R1],R2]...,Rn]=\n=(−1)n(...((# »ρ0×# »ρ1)×# »ρ2)×...×# »ρn)·# »˜G (R−like) (80c)\nFinally, we need the anti-commutator [ Hi,Rj]+. Despite being\nskew-symmetric in the hyperbolic space, it cannot be neatly ex-\npressed in terms of vector of matrices with the generators that\nwe have defined, due to an alterning sign:\n[Hi,Rj]+=X\nk(# »ηi×# »ρj)k(−1)kˆDk= (81a)\n=0−S1−S2−S3\nS1 0 0 0\nS2 0 0 0\nS3 0 0 0. (81b)\nThe third algebraic level to consider involves a decomposition in\nbasis matrices. When the vector field of any d-dimensional Lie\nalgebra gis expressed in terms of basis matrices Bi(i=1,..., d),\nthe algebra is characterized by its structure constants ci jk:\n[Bi,Bj]=X\nkci jkBk(∈g)\nWe define our basis matrices in terms of generators as:\nBk=ˆGk−i˜Gk, (k=1,2,3) (82a)\n# »B=(B1,B2,B3), (82b)\nArticle number, page 12 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\nsatisfying\nBiBj=δi j1+iϵi jkBk=[Bi,Bj]++[Bi,Bj]\n2(∀i,j) (83a)\nB∗\niB∗\nj=δi j1−iϵi jkB∗\nk=[B∗\ni,B∗\nj]++[B∗\ni,B∗\nj]\n2(∀i,j) (83b)\nBk·B∗\nk=1+2˜G2\nk=21k−1(∀k) (83c)\n[Bi,B∗\nj]=[B∗\ni,Bj]=0 (∀i,j) (83d)\n[Bi,Bj]=2iϵi jkBk(∀i,j) (83e)\n[B∗\ni,B∗\nj]=−2iϵi jkB∗\nk(∀i,j) (83f)\n[Bi,Bj]+=2δi j1(∀i,j) (83g)\n[Bi,B∗\nj]+=2|ϵi jk|˜Dk+2iϵi jk(−1)kˆDk(∀i,j) (83h)\nThis decomposition leads to a fourth algebraic level involving\nLorentz matrices and the propagation vector. Here, we propose\nthe existence of a generalized (complex) Lorentz matrix:\nL=# »a# »B=ˆL+i˜L. (84)\nIts real part is the ordinary Lorentz matrix ˆLin Eq. (64). How-\never, the imaginary part is a new dual Lorentz matrix containing\nthe same physical information as ˆLbut reorganized such that\n˜L=(# »η·# »ρ)ˆL−1(85a)\nˆL=L+L∗\n2=# »η·# »ˆG+# »ρ·# »˜G (85b)\ni˜L=L−L∗\n2=i(# »ρ·# »ˆG−# »η·# »˜G) (85c)\n[L,L∗]=[˜L,ˆL]=0. (85d)\nThese expressions can be verified with a direct calculation. They\nshow that both Lorentz matrices can be decomposed not only\nin terms of non-commuting Lorentz generators but also in terms\nofLandL∗, which always commute in a same spatial point. The\ninverse of the Lorentz matrix in Eq. (85a) does not exist when the\ndeterminant|ˆL|=0. The determinant of a matrix is the product\nof all its eigenvalues, which for ˆLandKare given by the sets\nλ′\n1,2,3,4={±|ˆa|,±i|˜a|} (86a)\nλ1,2,3,4={η0±|ˆa|, η0±i|˜a|}, (86b)\ncontaining the components of a±(see Eq.61). Multiplying all\nvalues in each set, we obtain the determinants:\n|ˆL|=|˜L|=−p2(87a)\n|K|=η2\n0[η2\n0−r]+|ˆL|. (87b)\nwith p=# »η·# »ρandr=η2−ρ2(see Eqs. 62). Hence, ˆL−1\ndoes not exist when p=0, i.e. when ηorρare zero21, or when# »η⊥# »ρ. This omnipresent quantity p=# »η·# »ρis actually de-\ntermining whether the algebraic systems represented by ˆLand\nˆL−1have a unique solution. A null determinant implies eigen-\nvalues with larger degeneracy because one of the matrix rows or\ncolumns is not linearly independent, which reduces the dimen-\nsionality of the problem (the volumen enclosed by the column\nvectors collapses along at least one dimension).\n21The wavelength dependence and symmetry properties of# »ηor# »ρ\nmakes them zero in a small set of wavelengths (e.g.# »ρ=0 at the line\ncenter of the atomic transition considered).Several useful relationships arise from relating the basis and\nthe Lorentz matrices. Considering the normalized vector\n# »u=# »a\na+=a−·# »a\nh=(ˆa−i˜a)·# »a\nh(88)\nit is direct to show that:\n# »u# »B=a−·L\nh(89a)\n# »u∗# »B∗=a+·L∗\nh(89b)\n# »u# »B+# »u∗# »B∗\n2=ˆaˆL+˜a˜L\nh(89c)\n# »u# »B−# »u∗# »B∗\n2=iˆa˜L−˜aˆL\nh(89d)\n(# »u# »B)·(# »u∗# »B∗)=L·L∗\nh. (89e)\nAnd using the commutation rules obtained, we discover that:\nL2=a2\n+1; (L∗)2=a2\n−1 (90a)\nˆL2−˜L2=r·1 (90b)\n[˜L,ˆL]+\n2=ˆL·˜L=i(L2−(L∗)2)\n4=p·1 (90c)\n[L,L∗]+\n2=L·L∗=ˆL2+˜L2=2K2−4η0K+(2η2\n0−r)1(90d)\nTo obtain the last equality22of Eq.(90d), we made use of Eq.\n(90b) and of ˆL2=(K−η01)2=K2−2η0K+η2\n01.\nUsing the latter expressions, the powers of ˆLcan be recur-\nsively calculated as functions of ˆL,ˆL2, and ˜L:\nˆL3=rˆL+p˜L (91a)\nˆL4=rˆL2+p21 (91b)\nˆL5=(r2+p2)ˆL+rp˜L (91c)\nˆL6=(r2+p2)ˆL2+rp21 (91d)\nˆL7=(r2+2p2)rˆL+(r2+p2)p˜L (91e)\n...\nForn≥2 and starting from an odd power of the kind ˆL2n−1=\nαˆL+β˜L, the rules to obtain the next even and odd powers are:\nˆL2n=αˆL2+pβ1 (92a)\nˆL2n+1=(rα+pβ)ˆL+pα˜L (92b)\nAnd from previous relations, we obtain:\nˆL2+˜L2=2ˆL2−r1 (93a)\nˆL3+˜L3=(p+r)ˆL+(p−r)˜L (93b)\nˆL4+˜L4=(2p2+r2)1 (93c)\n...\nWe shall need to have an insight into the specific structure\nofˆL2andˆL3. The latter (and any odd power of ˆL) can be neatly\ndecomposed by subspaces with ( H+R)3=H3+R3+[H2,R]++\n[R2,H]++(HRH +RHR ), which using Eqs. (74a) and (77) gives:\nˆL3=[η2# »η+(# »S×# »ρ)]·# »ˆG−[ρ2# »ρ+(# »S×# »η)]·# »˜G, (94)\n22Note that while L2and (L∗)2are diagonal, LandL∗are not: there can\nbe an infinite set of nondiagonal square roots to a square matrix.\nArticle number, page 13 of 22A&A proofs: manuscript no. aanda\nForˆL2, we wrote that ˆL2=K2−2η0K+η2\n0. To obtain a more\ndirect expression, using Eqs. (75a) and (81), the structure of ˆL2\ncan be calculated as ( H+R)2=H2+R2+[H,R]+:\nˆL2=(U(a)−ρ21)+# »d# »˜D+X\nkSk(−1)kˆDk, (95)\nwith U(a)=diag{a2,a2\n1,a2\n2,a2\n3}(akgiven in Eq.60a), dk=ηiηj+\nρiρj(i,j,k) and# »S=# »η×# »ρ. This shows that the structure of\nˆL2(and hence of any even power of ˆL) cannot be expressed only\nin terms of 1,# »ˆD, and# »˜D(due to U(a) and (−1)k).\nTo obtain a neater expression reflecting the structure of ˆL2,\nwe combine Eqs. (93a) and (90d) to see that ˆL2=1\n2(LL∗+r1).\nDeveloping the summations appearing in L·L∗, we reach:\nL·L∗=2X\nk|ak|21k−a21+X\nk=1,2,3(gkDk+g∗\nkD∗\nk),\nwhere Dk=˜Dk+i(−1)kˆDkare basis matrices obtained from dual\ngenerators and gk=aia∗\njϵi jkhas coe fficients cyclically ordered\nto giveϵi jk=1. Writting them as vectors, we finally see that\nˆL2=\u0012\nU(a)−ρ21\u0013\n+ℜe{# »g·# »D}. (96)\n4.4. Brief characterization of the Jones propagation matrix\nAlthough we shall focus in the Stokes formalism, in this brief\nsection we present a minimal characterization of the propagation\nmatrix in Jones formalism, with the aim of motivating the study\nof Magnus solutions in it.\nWe have said that the Jones formalism uses SL(2 ,C) to rep-\nresent the four-dimensional rotations associated to the evolution\nof the polarization. Now the mapping between a point in R1,3\nand the matrix vector space is sometimes called spinor map23.\nThen, the correspondence between the Stokes vector emissivity\nand the Jones emissivity matrix is:\n# »ϵ=(ϵ0,ϵ1,ϵ2,ϵ3)←→ E= \nϵ0+ϵ1ϵ2−iϵ3\nϵ2+iϵ3ϵ0−ϵ1!\n, (97)\nwhere we can separate diagonal and non-diagonal parts using\nPauli matrices (Eqs.(55)) and the auxiliar vector# »e=(ϵ1,ϵ2,ϵ3):\nE=1\n2[ϵ01+# »e# »σ], (98)\nThe map between the full propagation vector (associated to\nStokes I,Q,U,V) and the Jones propagation matrix is:\n# »α=(η0,a1,a2,a3)←→ ˘K= \nη0+a1a2−ia3\na2+ia3η0−a1!\n, (99)\nwith a1,a2,a3∈C. This matrix can also be decomposed as:\n˘K=1\n2[η01+# »a·# »σ]=1\n2[η01+a+X], with (100a)\nX=1\na+· \na1 a2−ia3\na2+ia3−a1!\n=# »u·# »σ, (100b)\n23With this map one could define# »σas a four vector of Pauli matri-\nces withσ0=1, but we keep the vector with only three components\nbecause it is more useful for separating diagonal and non-diagonal ma-\ntrices in the calculations.with normalized propagation vector# »u=# »a/a+. Note that\nX2=1andX=X−1, which makes the calculation of the\npowers of ˘Kand its functions a simple task. However, while\nthe Pauli matrices and the Jones emissivity matrix are Hermi-\ntian (σk=σ†\nk,E=E†),Xis not, because the akcomponents in\nits complex entries are themselves complex:\nX†=1\na−· \na∗\n1a∗\n2−ia∗\n3a∗\n2+ia∗\n3−a∗\n1!\n=# »u∗·# »σ. (101)\nThe eigenvalues of Xand ˘Kareλ′\n±=±1 andλ±=(η0±a+)/2,\nrespectively. This implies the determinant |˘K|=(η2\n0−a2\n+)/4.\nIn the spirit of what we did in the analysis of the Stokes\npropagation matrix, we can calculate several quantities that ap-\npear when operating with the Jones propagation matrix. Calling\nz:=# »a a−=ˆz+i˜z, we define and calculate:\nˆY:=X+X†\n2=ˆz\nh# »σ (102a)\n˜Y:=X−X†\n2=˜z\nh# »σ (102b)\n˜Y·ˆY=[X,X†]\n4=−(# »η×# »ρ)·# »σ\nh(102c)\n(a+X)·(a−X†)=a·1−2(# »η×# »ρ)·# »σ (102d)\nˆX:=a+X+a−X†\n2= \nηQηU−iηV\nηU+iηV−ηQ!\n=# »η·# »σ(102e)\n˜X:=a+X−a−X†\n2=i \nρQρU−iρV\nρU+iρV−ρQ!\n=i# »ρ·# »σ\n(102f)\nˆX·(−i˜X)=(# »η·# »σ)(# »ρ·# »σ)=(# »η·# »ρ)1−i(# »η×# »ρ)·# »σ.(102g)\nThe vector products can be substituted by commutators with:\n−(# »η×# »ρ)·# »σ=i\n2[# »η·# »σ,# »ρ·# »σ]=[ˆX,˜X]\n2. (103)\nSuch vector products give a vector perpendicular to the plane\ndefined by# »ηand# »ρ, and its mutiplication by# »σgives a ma-\ntrix describing three-dimensional rotations around such a nor-\nmal vector (see Section 4.1). The above quantities are simpler\nand more compact than their counterparts in Stokes formalism.\nThey also gives a better insight because are easy to interpret in\nterms of rotors and base Pauli matrices. Furthermore, the simple\ndecomposition in Eq. (100) allows to obtain the corresponding\nMagnus evolution operator in Jones formalism by just applying\nEq. (45)(BE theorem) with# »uthe normalized propagation vector.\nComparing with the Stokes formalism, the only step that could\nparallel its complexity in the Jones formalism seems to be the\ninhomogeneous solution in Eq. (17), which involves a double\nmultiplication by the evolution operator. These reasons justify\nour interest in using the Magnus expansion to reformulate the\nradiative transfer problem also in Jones formalism, as we do in\nthe following sections for the Stokes formalism.\nArticle number, page 14 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\n5. Reformulation of the homogeneous solution in\nthe Lorentz group (Stokes formalism)\n5.1. Derivation of the evolution operator for non-constant K\nWe can finally derive an analytical evolution operator that allows\nfor arbitrary variations of K. We start truncating the Magnus ex-\npansion in Eq. (41) to retain its first term. The algebraic analy-\nsis performed in Section (4.3) allows to calculate it explicitly in\nmore than one way. Our approach begins decomposing K(s) in\nterms of generalized Lorentz matrices with Eqs. (64) and (85b):\nK(s)=η0(s)1+L(s)+L∗(s)\n2. (104)\nThe commutation among LandL∗allows to divide the Magnus\nevolution operator in three exponentials:\nO(s)=e−Rs\ns0dtK(t)=\n=e−Rs\ns0dtη0(t)·e−1\n2Rs\ns0dt# »a(t)# »B·e−1\n2Rs\ns0dt# »a∗(t)# »B∗\n(105)\nThis step is not obvious because the exponents contain integrals\ncombining all points along the ray, so let us be more specific.\nCalling# »b=Rs\ns0dt# »a(t), the two latter exponents are matrices\nthat can be written together as\n−1\n2[# »B# »b+# »B∗# »b∗]\nbecause# »Bis a vector of basis (hence constant) matrices. To di-\nvide in two exponentials, any component of# »B# »bmust commute\nwith any other of# »B∗# »b∗, which occurs due to Eq. (83d) because\n[Bibi,B∗\njb∗\nj]=(biBi)(b∗\njB∗\nj)−(b∗\njB∗\nj)(biBi)=\n=bib∗\njBiB∗\nj−bib∗\njB∗\njBi=bib∗\nj[Bi,B∗\nj]=0,(106)\nThen, we can apply the BE theorem (Sec. 3.4) to each matrix\nexponential24in Eq. (105), obtaining\nO(s)=e−η′\n0·[ch(b) 1−sh(b)# »u# »B]·[ch(b∗)1−sh(b∗)# »u∗# »B∗],\n(107)\nwith band# »ucasting a+and# »a/a+in Sec. 4.2, but built from\nintegrated coe fficientsη′\nk(k=0,1,2,3) andρ′\nk(k=1,2,3):\n# »u=# »b\nb=(b1,b2,b3)\n[b2\n1+b2\n2+b2\n3]1/2=(ˆb−i˜b)\nh(b1,b2,b3) (108a)\nbk(s)=Zs\ns0ak(t)dt=Zs\ns0ηk(t)dt+iZs\ns0ρk(t)dt=η′\nk+iρ′\nk\n(108b)\nˆb2=(h+r)/2 (108c)\n˜b2=(h−r)/2 (108d)\nh=[r2+q2]1/2(=ˆb2+˜b2) (108e)\nr=(# »η′)2−(# »ρ′)2(=ˆb2−˜b2) (108f)\nq=2(# »η′·# »ρ′) (=2ˆb˜b) (108g)\nThis move allows to express everything in terms of the seven\nbasic integrals bkas elementary coe fficients. Next, we perform\nthe products in Eq. (107), identify matrix terms with Eqs. (89),\n24The conditions of the theorem are fulfilled because the Bkmatrices\nform a basis of the Lorentz algebra that accomplishes with Eq. (83a).and apply Appendix C to separate the real ( ˆb) and imaginary ( ˜b)\nparts of bandb∗. This leads to:\nO(s)=e−η′\n0·n\nc01−1\nhh\nc1ˆL+c2˜L+c3(ˆL2+˜L2)io\n, (109)\nwith\nc0,3=ch(2ˆb)±cos(2 ˜b)\n2\nc1,2=sh(2ˆb)∓isin(2 ˜b)\n2. (110a)\nFinally, substituting Eq. (93) for ˆL2+˜L2and operating, we find\nO(s)=e−η′\n0·n\nf01+f1aˆL+f1b˜L+f2ˆL2o\n, (111)\nwith scalar functions25\nf0=˜b2ch(2ˆb)+ˆb2cos(2 ˜b)\nˆb2+˜b2;f1a=−\"ˆbsh(2ˆb)+˜bsin(2 ˜b)\nˆb2+˜b2#\nf2=ch(2ˆb)−cos(2 ˜b)\nˆb2+˜b2;f1b=ˆbsin(2 ˜b)−˜bsh(2ˆb)\nˆb2+˜b2.\n(112)\nAlternative expressions for the evolution operator can be ob-\ntained using Eqs. (91) to remove the ˜Lin Eq.(111), writting it\nas function of any odd power of ˆL. For instance, substituting it\nin terms of ˆLandˆL3with Eq. (91a), we obtain:\nO(s)=e−η′\n0·n\nf01+f1ˆL+f2ˆL2+f3ˆL3o\n, (113)\nwhere the new functions are just\nf1=−\"˜b2sh(2ˆb)+ˆb2sin(2 ˜b)\nˆb2+˜b2#\nf3=−\"sh(2ˆb)\nˆb+sin(2 ˜b)\n˜b.#\n(114)\nThe two expressions for the evolution operator can be simpli-\nfied further by identifying subspaces of constant infinitesimal\ngenerators(# »ˆG,# »˜G), which allows a convenient integration of the\nevolution operator in the next sections. Namely, dividing the ma-\ntrices in Eq. (111) in subspaces, operating, and regrouping, we\nfind the remarkable expression:\nO(s)=e−η′\n0·n˚K+f2ˆL2o\n, (115)\nwhose new matrix ˚K=f01+# »ˆα·# »ˆG+# »˜α·# »˜Ghas same structure\nasKbut with a new propagation vector\n# »α=# »ˆα+i# »˜α=−[sh(2 ˆb)+isin(2 ˜b)]# »u, (116)\nbeing# »ustill given by Eq.(108) and ˆL2by Eq.(95).\nOur three expressions for the evolution operator, Eqs. (111),\n(113), and (115), are fully equivalent. They all are more general\nand significantly simpler than the one given by LD85. They are\ngeneral because they preserve memory of the spatial variations\n(of any arbitrary K) encoded as ray path integrals of optical co-\nefficients, as defined in Eq.(108b). They express everything in\n25An alternative set of relative parameters β=r/q,γ=[1+β2]1/2,\nδ2=(˜b/ˆb)2=(γ−β)(γ+β)−1, andε=(1+δ2)−1, could be defined to\nexpress or calculate f0,f1a,f1b,f2succintely in terms of εand (1−ε).\nArticle number, page 15 of 22A&A proofs: manuscript no. aanda\nterms of the seven basic scalar integrals in bk, which seems to be\nthe minimal and most e fficient integration possible to solve the\nRTE for non-constant properties. Thus, by separating the inte-\ngration from the algebraic calculation of the evolution operator,\nwe expect to obtain more general and e fficient numerical solu-\ntions for the RTE.\nWe remark that the Lorentz matrices in our expressions also\ncontain those integrated coe fficients (although the matrices are\nstill called as in previous sections). Indeed, the relative simplic-\nity of our expressions comes from writing them in terms of the\n(integrated) Lorentz matrix ˆL=K′(s)−η′\n0(s)1=# »η′·# »ˆG+# »ρ′·# »˜G.\nThis is also true for the dual Lorentz matrix ˜Lin Eq. (111)\nbecause it has the same algebraic structure as ˆL. Namely, re-\ncalling Eqs.(85), ˜Lis not only proportional to ˆL−1but also\n˜L=# »ρ′·# »ˆG−# »η′·# »˜G, hence both ˜Land ˆLare effortlessly built\nfrom each other by exchanging η′\nk↔ρ′\nkandρ′\nk↔−η′\nk.\nConsidering also that we know the specific elements of ˆL2\nby Eq.(95), or that it can be cheaply calculated multiplying ˆL\nby itself, we see that our solution for the exponential evolu-\ntion operator is optimal. In essence, it is just a matter of adding\nup two composed matrices ( ˆL2and ˚K), half of them contain-\ning redundant information (due to their symmetries). Hence, our\ncalculation of the matrix exponential is not only analytical, but\nalso seems ideal for numerical applications, because involves the\nminimal number of operations to solve exactly an evolution op-\nerator accounting for arbitrary spatial variations of K.\nOur evolution operator in Eq. (115) reveals other insight: ˆL2\nis the only term whose algebra di ffers from that of Kin the ho-\nmogeneous solution, making explicit that ˆL2is the only alge-\nbraic di fference between the Lorentz group (where the evolution\noperator belongs) and its Lie algebra, where Kbelongs. To ver-\nify that our result belongs to the Lorentz group, one could use\nEq. (51). Particularizing our expressions to the case of constant\npropagation matrix, one recovers the result of LD85.\n5.2. Higher order terms of the Magnus expansion\nThe BE theorem allows to calculate the exponential of the Mag-\nnus expansion including only its first order term Ω1. Here we\ngive a preliminar exploration of our Magnus solution when Ω2\nis included. Decomposing with Eqs. (85b) and (85d), the Ω2for\nthe RTE becomes:\nΩ2(s)=−1\n2Zs\ns0d1Zs1\ns0d2\u0014\nK(s1)K(s2)−K(s2)K(s1)\u0015\n=\n(117a)\n=−1\n2Zs\ns0d1Zs1\ns0d2\u0014\nˆL(s1),ˆL(s2)\u0015\n. (117b)\nThe latter commutator can be worked out with the relation\n[# »v1·# »B,# »v2·# »B]=2i(# »v1×# »v2)·# »B, (118)\nbeing# »Bour vector of basis matrices. Thus, we obtain (calling\ngenerically XitoX(si))\n[ˆL1,ˆL2]=i\n2\u0014\n(# »a1×# »a2)# »B−(# »a∗\n1×# »a∗\n2)# »B∗\u0015\n=ℜe(i\n2# »a1x2·# »B)\n=\n=\u0014\n(# »η1×# »η2)−(# »ρ1×# »ρ2)\u0015# »˜G−\u0014\n(# »η1×# »ρ2)+(# »ρ1×# »η2)\u0015# »ˆG,\n(119)with# »a1x2=# »a(s1)×# »a(s2). Then,\nΩ2(s)=\u0014Zs\ns0ds1(# »η1×# »ρ′\n1)+(# »ρ1×# »η′\n1)\n2\u0015# »ˆG+\n+\u0014Zs\ns0ds1(# »ρ1×# »ρ′\n1)−(# »η1×# »η′\n1)\n2\u0015# »˜G. (120)\nwith\n# »η′\n1=# »η′(s1)=Zs1\ns0ds2# »η(s2) (121a)\n# »ρ′\n1=# »ρ′(s1)=Zs1\ns0ds2# »ρ(s2). (121b)\nComparison of this result with the di fferent decompositions\nof the Lorentz matrix (Section 4.3) shows explicitly that the ma-\ntrix structure of Ω2is that of the Lorentz matrix, as expected in\nthis Lie algebra. As the same is true for all terms of the Magnus\nexpansion, their addition to Ω1will give an exponential evolu-\ntion operator of the same form as Eq.(115), but with f0,f2and# »αincreasingly complicated by the presence of nested integrals\nof nested vector products, in a similar fashion to Eq. (120). It\nis a matter of ongoing numerical investigation to find out how\nto calculate such integrals e fficiently to avoid significant penalty\nwhen extending our methods to higher orders of the expansion.\n6. Reformulation of the inhomogeneous problem\nwith Magnus in Stokes formalism\nThe general evolution operator derived analytically in the previ-\nous section can be directly inserted into Eq. (23) to obtain a new\nfamily of numerical methods based on the Magnus expansion.\nSuch methods would already represent a fundamental improve-\nment to solve the RTE. However, there are certain issues that\nmotivate an alternative formulation of the inhomogeneous prob-\nlem too. The inhomogeneous RTE:\nI′(s)=A(s)I(s)+ϵ(s) ; I(0)=I0 (122)\nwith A(s)=−K(s) can be solved with Eq. (23). In the most\ngeneral case, the evolution operators in it are substituted by the\nfull Magnus exponential. In this paper we shall start considering\nEq. (123) as a result of truncating the Magnus expansion to first\norder, in consistency with our previous section, and yet allowing\nfor variations of the K(s) matrix in A(s). Then:\nI(s)=eRs\n0A(τ)dτI0+Zs\n0eRs\ntA(τ)dτϵ(t)dt. (123)\nOnce we study and test our methods for this case, we will study\nhigher orders. The problem here is that the calculation of the\ninhomogeneous integral with the nested integral in the inner ex-\nponent is costly and of di fficult evaluation because we need two\ndifferent sets of quadrature points for nested integrals changing\nin different intervals. To solve this problem, we reformulate it.\n6.1. Reformulating the case with constant properties\nForAconstant, the typical formal inhomogeneous solution given\nby Eq.(23) can be considered local because is restricted to small\nregions /cells of constant properties. As we anticipated, it is also\ninconsistent and innacurate, because it breaks the inhomoge-\nneous group structure when Ais constant but ϵis not. In order\nArticle number, page 16 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\nto introduce our method of solution for the general case, we first\nconsider the simplest consistent system, given by Eq. (123) with\nboth Aandϵconstant. Its solution can be written in terms of a\nmatrix function ϕ1(sA), both as\nI(s)=esAI0+Zs\n0e(s−t)Aϵdt=esAI0+sϕ1(sA)ϵ (124)\nor as\nI(s)=esAI0+(esA−1)A−1ϵ=esA(I0+A−1ϵ)−A−1ϵ=\n=I0+sϕ1(sA)(AI0+ϵ), (125)\nwhere we define ϕ1(sA) in several ways:\nϕ1(sA)=1\nsZs\n0e(s−t)Adt=1\nsZs\n0etAdt= (126a)\n=∞X\nn=0(sA)n\n(n+1)!= (126b)\n=(esA−1)(sA)−1=\u0012\n≡ex−1\nx\u0013\n(126c)\n=Z1\n0ezsAdz. (126d)\nEq. (126a) shows the relation of ϕ1(sA) with Eq. (125), while\nEq. (126b) shows its relation with the exponential\nesA=∞X\nn=0(sA)n\nn!.\n(127)\nAlternatively, Eq. (126c) gives an ine fficient way of calculating\nϕ1(sA), while Eq. (126d) gives an e fficient integral definition in\nterms of a parameter z.\nOur method of solution is based on the realization that the\nEq. (123) with both Aandϵconstant is equivalent to a 5 ×5\nhomogeneous system\nd\nds \nI(s)\nIt!\n= \nAϵ\n0⊺0! \nI(s)\nIt!\n(128)\nwith I(0)=I0,It(0)=1,Ita mere auxiliar value, and whose new\n5×5 propagation matrix A5contains A≡A4×4. Correspond-\ningly, the solution in Eq. (124) can be seen to be equivalent to\n(see also Eq. (26))\nI5(s)=esA5I5,0= \nesAsϕ1(sA)ϵ\n0⊺0!\nI5,0 (129)\nwith I5,0=I5(0)=(I0,1)⊺. Eq. (129) tells us that the inho-\nmogeneous solution for A5constant can be written as a 5 ×5\nevolution operator containing both the corresponding 4 ×4 evo-\nlution operator and a special product involving ϕ1. Let us now\napply this to the general case.\n6.2. Reformulating the general case with variable properties\nThe method that we propose consists in extending by one the\ndimension of the inhomogeneous problem to convert it in a five-\ndimensional homogeneous one, and thus solve it with the Mag-\nnus expansion. Namely, Eq. (123) is equivalent to a 5 ×5 homo-\ngeneous system\nd\nds \nI(s)\nIt!\n= A(s)ϵ(s)\n0T0! \nI(s)\nIt!\n(130)with I5,0=I5(0)=(I0,1)⊺, and where again we call I5(s) to the\nsolution vector of unknowns and A5(s) to the new propagation\nmatrix containing the original A(s)=−K(s). Being homoge-\nneous, this system can be solved applying the Magnus expan-\nsion toA5(s). We do this considering only Ω1in the Magnus\nexpansion, to keep consistency with the evolution operator that\nwe derived in the previous section. Thus, we have to calculate:\nO5(s)=eRs\ns0A5(t)dt=e¯A5=∞X\nn=0¯An\n5\nn!. (131)\nwhere the overbars mean integration hereafter:\n¯A5= ¯A ¯ϵ\n0⊺0!\n(132a)\n¯A=−Zs\ns0K(t)dt (132b)\n¯ϵ=Zs\ns0ϵ(t)dt=(ϵ′\n0,ϵ′\n1,ϵ′\n2,ϵ′\n3)⊺(132c)\nThe integral of Kcontains the seven ray-path scalar integrals of\nEq.(108),η′\nk(k=0,1,2,3) andρ′\nk(k=1,2,3), while ¯ϵcontains\nthe four equivalent scalar integrals ϵ′\nk(k=0,1,2,3).\nAs the powers of ¯A5show the simple general form\n¯An\n5= ¯An¯An−1¯ϵ\n0⊺0!\n, (133)\nall terms in Eq. (131) can be readily resummed to obtain\nO5(s)=e¯A\"P∞\nn=0¯An\n(n+1)!#\n¯ϵ\n0⊺1= \ne¯Aϕ1(¯A)¯ϵ\n0⊺1!\n(134a)\n=15+ \nϕ1(¯A)¯Aϕ1(¯A)¯ϵ\n0⊺0!\n, (134b)\nInserting these two equivalent expressions into the general solu-\ntion I5(s)=O5(s)I5,0, and taking the 4 ×4 subspace, we find:\nI(s)=e¯AI0+ϕ1(¯A)¯ϵ= (135a)\n=I0+ϕ1(¯A)(¯AI0+¯ϵ), (135b)\nwhich reduces to Eqs. (124) and (125) when both A(i.e.,−K)\nandϵare constant.\nWhat has just happened in Eqs. (135)? The inhomogeneous\nformal integral that has always characterized the solution to the\nradiative transfer problem and its numerical methods has van-\nished. Eq. (135) substitutes it by the product of an integrated\nemissivity vector and a special function of an integrated prop-\nagation matrix. We can explain this saying that, by solving the\n4×4 inhomogeneous problem as the Magnus solution to a 5 ×5\nhomogeneous problem, we are solving the translations given by\nthe emissivity term in the Poincaré space of the solution as if they\nwere rotations quantified by the new algebra of the 5 ×5 Mag-\nnus expansion. Thus, Eq. (135) gives a radically di fferent way of\nsolving the radiative transfer problem (see next subsection 6.4).\nLet us also comment on Eq. (134a), which shows that the al-\ngebra of the 5D propagation matrix is such that the 4D subspace\ncontaining the homogeneous solution (i.e. the evolution opera-\ntor) is always independent on the inhomogeneous part of the sys-\ntem containing the special function ϕ1and the emissivity. This\nimplies that if we extend the Magnus expansion to higher orders,\nArticle number, page 17 of 22A&A proofs: manuscript no. aanda\nwe can always continue using the corresponding evolution oper-\nator because the only thing that changes is how the function ϕ1\nis combined with the emisivity. For instance, if we add Ω2to\nthe Magnus expansion we would be adding a term of the kind of\nEq.(120), with the commutator :\n[A5(s1),A5(s2)]=A5(s1)A5(s2)−A 5(s2),A5(s1)=\n= \n[A(s1),A(s2)]A(s1)ϵ(s2)−A(s2)ϵ(s1)\n0⊺0!\n.(136)\nThis shows that the 4 ×4 subspace is preserved, containing a\ncommutator similar to that with A5. Thus, after integrating and\nexponentiating, the new solution will have the new evolution op-\nerator (now including Ω2) and a composition of matrix-vector\nproducts between ϕ1and integrals of Aandϵ. We will have to\nfind the right balance between more general theoretical descrip-\ntion (higher order in Magnus) and more e fficient computational\nrepresentation (higher numerical order of integration and fastest\ncalculation).\n6.3. Calculation of ϕ1: integrating the evolution operator\nTaking now Eq. (135a) as an e fficient general expression, we\nmake it fully explicit by calculating ϕ1(¯A)=ϕ1(−¯K). Namely,\nintegrating along a parameter zwith Eq. (126d)\nϕ1(−¯K)=Z1\n0ez¯A(s)dz=Z1\n0e−Rs\ns0zK(t)dtdz=Z1\n0O(t,z)dz,\n(137)\nwe see that ϕ1(−¯K) is obtained integrating our evolution opera-\ntor in Eq. (115) after assigning the parameter zto every matrix\ncomponent of K, i.e. to every integrated optical coe fficientη′and\nρ′in Eqs. (108c). Guided by Eq. (107), and propagating zto all\nthe subsequent quantities, we see that the only place where the\npropagated parameter zdoes not cancel out is inside the trigono-\nmetrical expressions, because any other quantity depending on\nη′andρ′(i. e. the normalized vector# »u, and bk,ˆb, or˜b) has a\ndenominator that always cancels out the zdependence. We make\nthis clear by specifying all the subspaces in Eq. (115) for the\nevolution operator and adding the parameter z where it remains:\nO(s,z)=e−η′\n0z·(˜b2ch(2ˆbz)+ˆb2cos(2 ˜bz)\nˆb2+˜b21+\n−ℜen\n[sh(2 ˆbz)+isin(2 ˜bz)]# »uo\n·# »ˆG+\n−ℑmn\n[sh(2 ˆbz)+isin(2 ˜bz)]# »uo\n·# »˜G+\n+ch(2ˆbz)−cos(2 ˜bz)\nˆb2+˜b2\u0012\nU(a)−ρ21+ℜe{# »g·# »D}\u0013)\n,(138)with U(a)=diag{a2,a2\n1,a2\n2,a2\n3},gk=aia∗\njϵi jk, and akgiven in\nEq.(60a). Hence, the only integrals that appear are\nZ1\n0e−η′\n0zch(2ˆbz)dz=e−η′\n0[−η′\n0ch(2ˆb)−2ˆbsh(2ˆb)]+η′\n0\nη′\n02−(2ˆb)2\n(139a)\nZ1\n0e−η′\n0zcos(2 ˜bz)dz=e−η′\n0[−η′\n0cos(2 ˜b)+2˜bsin(2 ˜b)]+η′\n0\nη′\n02+(2˜b)2\n(139b)\nZ1\n0e−η′\n0zsh(2ˆbz)dz=e−η′\n0[−η′\n0sh(2ˆb)−2ˆbch(2ˆb)]+2ˆb\nη′\n02−(2ˆb)2\n(139c)\nZ1\n0e−η′\n0zsin(2 ˜bz)dz=e−η′\n0[−η′\n0sin(2 ˜b)−2˜bcos(2 ˜b)]+2˜b\nη′\n02+(2˜b)2\n(139d)\nOnce substituted the integrals and the vector# »u(see Eq.108a):\n# »u=# »b\nb=(ˆb−i˜b)\nh(# »η′+i# »ρ′), (140)\nwe rearrange and identify terms. First, we arrange things to iden-\ntify the trigonometric functions:\nˆc=ch(2ˆb)\nη′\n0ˆd; ˜c=cos(2 ˜b)\nη′\n0˜d(141a)\nˆs=sh(2ˆb)\n2ˆbˆd; ˜s=sin(2 ˜b)\n2˜b˜d, (141b)\ncontaining well-behaving sync functions\nlim\nˆb→0sh(2ˆb)\n2ˆb=lim\n˜b→0sin(2 ˜b)\n2˜b=1 (142)\nand\nˆd=η′\n02−(2ˆb)2;˜d=η′\n02+(2˜b)2. (143)\nRecombining subspaces for# »ˆGand# »˜G, and keeping terms in 1\nandˆL2, we identify the Lorentz matrices ˆL,˜LandˆL2. Thus, after\na cumbersome calculation, we obtain:\nϕ1(−¯K)=ϕA+ϕB=\n=η′\n02e−η′\n0\nˆb2+˜b2·(\nn11+n2ˆL+n3˜L+n4ˆL2)\n+\n+4η′\n0\nˆd˜d·(\u0012η′\n0\n2−r\u0013\n1−η′\n0\n2ˆL+nr˜L+ˆL2)\n, (144)\nwith nr=2ˆb˜b\nη′\n0and\nn1=(˜b2ˆc+ˆb2˜c)+n2\nr(ˆs−˜s) (145a)\nn2=2\nη′\n0h\n(ˆs+ˆc)ˆb2+(˜s+˜c)˜b2i\n(145b)\nn3=nrh\n(ˆs+ˆc)−(˜s+˜c)i\n(145c)\nn4=4(ˆb2ˆs+˜b2˜s)−(ˆc−˜c) (145d)\nFor the sake of clarity, we have divided Eq. (144) into two matri-\ncesϕAandϕB, showing di fferent external dependence on η′\n0, but\nArticle number, page 18 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\nFig. 5. Research prospects.\nboth belonging to the Lorent group. I.e., their algebraic structure\nmatch that of the evolution operator, thus depending on the same\nbasic Lorentz matrices that were already built from integrated\noptical coe fficients, but multiplied by simple scalar functions.\nWith this explicit analytical calculation, the computational cost\nof building our formal solution in Eq. (135a), which furthermore\nonly contains two matrix-vector products, should be minimal.\nNote also that, once the order of the Magnus expansion is set,\nour whole development of the evolution operator and of the in-\nhomogeneous solution with ϕ1has been mathematically exact.\n6.4. Additional comments and prospects\nIn summary, we have reformulated the radiative transfer prob-\nlem to obtain consistent integral solutions based on the Magnus\nexpansion. Starting by truncating the expansion to first order, we\nhave arrived to Eq. (135)\nI(s)=e−¯KI0+ϕ1(−¯K)¯ϵ,\nwith the exponential evolution operator given explicitly by Eq.\n(115), ϕ1(−¯K) given by Eq. (144), and ¯Kand¯ϵcontaining the\neleven basic scalar ray-path integrals of the optical coe fficients.\nBy having an exact analytical formula for both a consistent in-\nhomogeneous solution and an evolution operator considering ar-\nbitrary variations of K, we now can (see Fig. 5):\n–1) Find consistent solutions respecting the Lie group struc-\nture at any step of the calculations, thus solving accurately\nthe exact physical problem, instead of a similar one. The ob-\nvious prospect here is to extend our formulation to include\nhigher order terms of the Magnus expansion26.\n–2) Define a new family of numerical methods implementing\nEq. (135a). These methods should be advantageous from the\nstart, simply because they are based on a more accurate, com-\npact, and e fficient representation of the physical and mathe-\nmatical problem. However, one of the numerical aspects that\nshould be investigated is how non-local is our theory for the\norder of the Magnus expansion considered. A key to investi-\ngate this is the convergence of the Magnus expansion, which\n26Remind that in general the order of the Magnus expansion is not the\norder of the numerical methods to solve it.could set a limiting range that depends on our specific ap-\nplication. Our research and numerical developments are on-\ngoing and will be available to the astrophysical community\nwhen ready.\n–3) Understand polarized radiative transfer in depth. As our\nformal solution is analytical and non-local (not restricted\nto relatively small regions or cells of constant properties),\nthe general long-range solution to the RTE can be stud-\nied theoretically as a function of its parameters, thus tran-\nscending previous approximate radiative transfer models.\nFor instance, now it would be possible to explain without\nstrong approximations the joint action of magneto-optical\nand dichroic e ffects in certain wavelengths of the polariza-\ntion profiles, something that could only be done indepen-\ndently for every mechanism (as made for dichroism in Carlin\n2019).\n–4) Incorporate a defined geometry in the radiative trans-\nfer problem analytically, thus building analytical models of\ncomplex astrophysical objects (e.g., a whole star) that can\nhowever be taylored to the physical information available.\nThus, we can study the physics of these objects with polar-\nization fingerprints transparently, without radiative transfer\nitself being a source of error. This is possible because the\nmain input for our equations, the eleven ray-path scalar in-\ntegrals of the optical coe fficients, cannot only be provided\nby realistic atmosphere models but also by exact prescribed\nfunctional variations of the atmosphere.\n–5) Reformulate the methods of solution for the NLTE prob-\nlem with polarization to make them non-local. This novel\nidea seems very interesting. By considering variations of K,\nboth our evolution operator and the final inhomogeneous so-\nlution are non-markovian in space , i.e. they preserve mem-\nory of the atmospheric physics in previous locations along\nthe ray27. This memory can be used, at least in principle, to\naccelerate the NLTE interation between the RTE and the rate\nequations in the theory of scattering polarization. Of interest\nhere is that our theory separates the integration of the optical\n27This problem is equivalent to considering partial redistribution ef-\nfects in the derivation of the equations defining the NLTE radiative\ntransfer problem. However, in that case the non-markovian character\narises from integrating along time the Schrödinger equation describing\nthe evolution of matter-radiation interaction.\nArticle number, page 19 of 22A&A proofs: manuscript no. aanda\ncoefficients from the algebraic construction of the formal so-\nlution, which permits to compute the latter only once directly\nat the end of the ray (or at any intermediate location), and re-\nduces the numerical task to the simplest integration possible,\ni.e. to the scalar integration of the eleven basic optical coef-\nficients of the problem.\n–6) Solve general physical problems with a similar alge-\nbraic structure to that of the RTE. With small cosmetic\nchanges (reinterpreting the physical quantities), our equa-\ntions also give novel solutions for other homogeneous and\ninhomogeneous problems of universal interest that share the\nLorentz /Poincaré group structure, again allowing arbitrary\nvariations of their system matrix. Well-known examples are\nthe motion of masses at relativistic speeds, and the calcu-\nlation of Lorentz forces on charged particles moving in an\nelectromagnetic field.\n7. Conclusions\nWe have laid the foundations to solve the polarized radiative\ntransfer problem in a new, e fficient, and fully consistent way\nusing the Magnus expansion. First, we presented a succint red-\nerivation of the radiative tranfer equations (RTE) in Stokes and\nJones formalisms using a common nomenclature and consistent\nphysical conventions for the phase of the electric field and the\nsigns of Stokes V . We believe that now the equations of both\nformalism can be used and compared on a more clear basis.\nTruncating the Magnus expansion to first order, and applying\nour BE theorem (Sec. 3.4), we have obtained three equivalent an-\nalytical expressions of a new and more general evolution opera-\ntor that allows for arbitrary spatial variations of the propagation\nmatrix (Sec. 5.1). Our Basic Evolution theorem calculates ana-\nlytically the exact exponential of any integral of the propagation\nmatrix, illustrating the possibility of introducing the scalar inte-\ngrals of the optical coe fficients inside the terms of our solutions.\nThis ends up being equivalent to separate the integration from\nthe composition of the formal solution. Furthermore, it allows\nto calculate the solution at the end of a ray without necessarily\ndoing matrix calculations or solving the formal solution at ev-\nery intermediate point. Developing the order two of the Magnus\nexpansion, we have shown some prospects for generalizing the\nevolution operator to higher orders of the expansion.\nAs an insight, we interpret the evolution of the solution to\nthe RTE as complex multidimensional rotations in the Lie alge-\nbra of the Lorentz /Poincaré group. Such rotations are quantified\nby rotors whose direction and strength of rotation are given by\nthe normalized propagation vector. As shown by the compact\nformulation of our evolution operator, such a rotational evolu-\ntion is neatly expressed at the level of the Lie group with a linear\ncombination of our generalized Lorentz matrix and the square of\nthe ordinary Lorentz matrix (the only factor in the solution that\ndoes not belong to the Lie algebra). Here we remark that our cal-\nculations were made easier by our algebraic characterization of\nthe propagation matrix, including its generators, powers, com-\nmutators, and the di fferent ways in which we can decompose it\nto operate (Sec. 4.3).\nFinally, by increasing the dimensionality of the 4 ×4 prob-\nlem to 5×5, we have reformulated the inhomogeneous solution\nas an homogeneous one, which is again solved with the Magnus\nexpansion to first order. Thus, the typical formal integral of pre-\nvious formulations is substituted by an integral function of our\nevolution operator. Calculating it analytically and exactly, we ar-\nrive to an explicit radiative transfer solution whose inputs are thesimple ray-path scalar integrals of the eleven optical coe fficient\nof the radiative transfer problem.\nFor several reasons, our approach is potentially disruptive for\nsolar spectropolarimetry. First of all, it is the first one that fully\nsolves the RTE consistently, explicitly, and allowing for arbitrary\nray-path variations of the propagation matrix. Secondly, it poses\nthe integration of the RTE in a radically new way. Instead of inte-\ngrating an standard formal solution to solve the inhomogeneous\nproblem many times along every ray, our theory is built in such a\nway that in principle allows to integrate directly and exactly the\nscalar optical coe fficients of the problem for the whole ray and,\nonly then, substitute the result in an analytical expression with\nminimal algebraic complexity.\nFurthermore, by including a Magnus evolution operator and\na Magnus inhomogeneous solution, our results natively pre-\nserves the full Lie group structure of the problem. On one hand,\nthis means that the evolution of the solution follows its correct\ngeometrical space in the algebra. In other words, we should now\nbe able to integrate over large distances along the ray with a sig-\nnificantly smaller departure from the right solution. On the other\nhand, in principle we could use our new evolution operator to\ncalculate more general response functions in the NLTE problem.\nThese two key aspects suggests a reformulation of the NLTE\nproblem with polarization that could allow to solve this non-\nlocal problem in a non-local way for the first time. This is critical\nfor accelarating inversions of Stokes profiles, on which solar di-\nagnosis is based. Besides, our preliminar calculations (Sec. 4.4)\nsuggests to investigate whether is advantageous to reformulate\nour Magnus-based solutions in Jones formalism, whose algebra\ninvolves simpler 2 ×2 matrices. In that case, a reformulation of\nthe NLTE problem in Jones formalism could be advisable.\nUsing the Magnus expansion, our formulation allows to nat-\nurally generalize our evolution operator for higher orders of ac-\ncuracy even before considering to increase the numerical order,\nsomething that previous methods were unable to do. We specu-\nlate that if the new terms of such generalization could be ideally\nresummed into tractable scalar functions, we could be opening\nthe door to solve the radiative transfer problem in the most gen-\neral, e fficient, and exact way possible.\nGiven the algebraic similarity with other physical universal\nproblems with Lorentz /Poincaré algebra, our results are applica-\nble to them after a mere reinterpretation of the physical quanti-\nties.\nAcknowledgements. E.S.C. acknowledges financial support from the Spanish\nMinistry of Science and Innovation (MICINN) through the Spanish State Re-\nsearch Agency, under Severo Ochoa Centres of Excellence Programme 2020-\n2023 (CEX2019-000920-S). Part of his work has been funded by Ministe-\nrio de Ciencia e Innovación (Spain) through project PID2022-136585NB-C21,\nMCIN /AEI/10.13039 /501100011033 /FEDER, UE, and also by Generalitat Va-\nlenciana (Spain) through project CIAICO /2021/180.\nReferences\nAbramowitz, M. & Stegun, I. A. 1972, Handbook of Mathematical Functions\n(New York: Dover)\nBaker, H. 1902, Proc. London Math. Soc., 35, 334\nBaker, H. F. 1905, Proceedings of the London Mathematical Society, s2-2, 293\nBellot Rubio, L. R., Ruiz Cobo, B., & Collados, M. 1998, ApJ, 506, 805\nBlanes, S., Casas, F., Oteo, J., & Ros, J. 2009, Physics Reports, 470, 151\nBonfiglioli, A. & Fulci, R. 2011, Topics in Noncommutative Algebra: The Theo-\nrem of Campbell, Baker, Hausdor ffand Dynkin, Lecture Notes in Mathemat-\nics (Springer Berlin Heidelberg)\nBorn, M. & Wolf, E. 1980, Principles of Optics: Electromagnetic Theory of Prop-\nagation of Light (Oxford: Pergamon Press)\nCarlin, E. S. 2019, A&A, 627, A47\nCarlin, E. S. & Asensio Ramos, A. 2015, The Astrophysical Journal, 801, 16\nArticle number, page 20 of 22E.S. Carlin, S. Blanes, F. Casas: Reformulating polarized radiative transfer\nCarlin, E. S., Asensio Ramos, A., & Trujillo Bueno, J. 2013, ApJ, 764, 40\nCarlsson, M. & Stein, R. F. 1997, ApJ, 481, 500\nCollados, M., Martínez Pillet, V ., Ruiz Cobo, B., del Toro Iniesta, J. C., &\nVázquez, M. 1994, A&A, 291, 622\nDe la Cruz Rodríguez, J. & Piskunov, N. 2013, ApJ, 764, 33\nDel Toro Iniesta, J. 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Transfer, 39, 67\nLandi Degl’Innocenti, E. 1976, A&A, 25, 379\nLandi Deglinnocenti, E. & Landi Deglinnocenti, M. 1985, Sol. Phys., 97, 239\nLandi Degl’Innocenti, E. & Landolfi, M. 2004, Polarization in Spectral Lines\n(Kluwer Academic Publishers)\nLópez Ariste, A. & Semel, M. 1999, A&A, 350, 1089\nMagnus, W. 1954, Communications on Pure and Applied Mathematics, 7, 649\nMandelbrot, B. B. 1982, The Fractal Geometry of Nature (San Francisco: W. H.\nFreeman)\nPeano. 1890, Mathematische Annalen, 37, 182\nPeano, G. 1888, Calcolo geometrico secondo l’ Ausdehnhungslehre di H. Grass-\nman precedoto dalle operazioni della logica deduttiva. (Bocca, Torino)\nRachkovsky, D. N. 1967, Izvestiya Ordena Trudovogo Krasnogo Znameni\nKrymskoj Astrofizicheskoj Observatorii, 37, 56\nRees, D. E., Durrant, C. J., & Murphy, G. A. 1989, ApJ, 339, 1093\nSanchez Almeida, J. 1992, Sol. Phys., 137, 1\nSemel, M. & López Ariste, A. 1999, A&A, 342, 201\nShurcli ff, W. A. 2013, in Polarized Light (Harvard University Press)\nStenflo, J. O. 1994, Solar Magnetic Fields. Polarized Radiation Diagnostics\n(Dordrecht: Kluwer Academic Publishers)\nUnno, W. 1956, PASJ, 8, 108\nvan Ballegooijen, A. A. 1985, in NASA Conference Publication, V ol. 2374,\nNASA Conference Publication, 322–334\nWei, J. & Norman, E. 1963, Journal of Mathematical Physics, 4, 575\nArticle number, page 21 of 22A&A proofs: manuscript no. aanda\nAppendix A: Calculation of real and imaginary\nparts of the propagation module\nIf# »a=# »η+i# »ρ=(η1+iρ1,η2+iρ2,η3+iρ3), the modules of# »a(≡# »a+) and# »a∗(≡# »a−) are complex numbers that we call a+\nanda−, respectively. To calculate them we state:\na±=ˆa±i˜a (ˆa,˜a∈R),\na2\n±=# »η2−# »ρ2±i2# »η·# »ρ=ˆa2−˜a2±i2ˆa˜a\nwhere we identify the real numbers\nr=# »η2−# »ρ2=ˆa2−˜a2\nq=2# »η·# »ρ=2ˆa˜a,\nh=[r2+q2]1/2=a−·a+=ˆa2+˜a2\nSubstituting ˆ a=q/(2˜a) inr=ˆa2−˜a2we obtain 4˜ a4+4r˜a2−q2=\n0 and find the solutions:\nˆa=± ±h+r\n2!1/2\n,˜a=± ±h−r\n2!1/2\n.\nThe signs acompanying hmust be chosen positive for ˆ a, ˜a∈R.\nRegarding the outer signs of the roots, one is tempted to choose\nthe sign of ˆ apositive because ˆ aworks as an attenuation fac-\ntor. Landi Deglinnocenti & Landi Deglinnocenti (1985) solved\na similar problem to this before us, imposing positive sign for\na proportional quantity equivalent to our ˆ a(which they called\nα). However such imposition is unnecesary to solve the above\nequations and would be wrong in a general case in which stim-\nulated emission could dominate. Note that the total absorption\ncoefficientη0±|ˆa|>0, appearing e.g. in the eigenvalues of the\npropagation matrix or in the expressions of the evolution opera-\ntor, is always positive with η0>0 and|ˆa|< η 0when stimulated\nemission does not dominate. We then avoid to impose the outer\nsigns, labelling them as ˆ sand ˜s:\nˆa=ˆs h+r\n2!1/2\n, ˜a=˜s h−r\n2!1/2\n.\nThe signs are however constrained by the angle θbetween# »ηand# »ρin the QUV space, because ˆ s·˜s=sign(ˆa·˜a)=sign(# »η·# »ρ)=\nsign(cosθ). Thus, when# »η⊥# »ρthe signs are undetermined, but\nthen the expressions involucrating them will become indepen-\ndent on them.\nAppendix B: Proof of the Basic Evolution theorem\nThe evolution theorem gives the solution to the exponential of\nthe integral of an arbitrary matrix N. To demonstrate the theo-\nrem, we start decomposing the matrix as N=f# »n·# »σ, with fan\narbitrary constant,# »n=(n1,..., nd) a vector of components, and# »σ=(σ1,...,σ d) a vector of basis matrices with algebra gthat\nmust fulfill:\nσk·σℓ=δkℓ1+h·ϵkℓm·σm, (h=ct.) (B.1)\nThis condition implies both σ2\nk=1and vanishing anticommuta-\ntors [σk,σℓ]+=σkσℓ+σℓσk=0 for all k,ℓ. The following\nstep is now to calculate the generic integral of N:\nI=Z\ndsN(s)=Z\nds f# »n(s)·# »σ=fdX\nk=1bkσk=f# »b·# »σ\n(B.2a)where we defined the integrated vector# »b=(b1,..., bd), having\nmodule band components:\nbk=Z\nnk(s)ds (B.3)\nCondition (B.1) simplify calculation of I2as:\nI2\nf2=dX\nk=1bkσk2\n=X\nk=ℓb2\nkσ2\nk+X\nk,ℓbkbℓ[σk,σℓ]+=b21(B.4a)\nHence, the even and odd powers of the integral are I2k=(f b)2k1\nandI2k+1=I2k·I=(f b)2k+1# »u·# »σ, in terms of the unitary vector\n# »u=# »b\nb. (B.5)\nThus, the exponential of Ican be finally calculated as:\ne±R\ndsN(s)=∞X\nk=0(±I)k\nk!=∞X\nk=0I2k\n(2k)!±∞X\nk=0I2k+1\n(2k+1)!=\n=∞X\nk=0(f b)2k\n(2k)!1±∞X\nk=0(f b)2k+1\n(2k+1)!# »u·# »σ=\n=ch(f b)1±sh(f b)U, (B.6a)\nwhere the matrix U=# »u·# »σ, and band# »uare given by Eqs. (B.3)\nand (B.5).\nAppendix C: Trigonometrical expressions for the\ncoefficients of the analytical EO\nThe following four coe fficients with complex argument x=ˆx+i˜x\nare developed to find several equivalent convenient expressions:\nc0=cosh( x) cosh( x∗)=cosh2( ˆx)+sinh2(i˜x)=\n=cosh2( ˆx)−sin2( ˜x)=\n=cosh(2 ˆ x)+cos(2 ˜ x)\n2\nc1=cosh( x) sinh( x∗)=sinh(2 ˆ x)−isin(2 ˜ x)\n2\nc2=cosh( x∗) sinh( x)=sinh(2 ˆ x)+isin(2 ˜ x)\n2\nc3=sinh( x) sinh( x∗)=cosh2( ˆx)−cosh2(i˜x)=\n=cosh2( ˆx)−cos2( ˜x)\n=cosh(2 ˆ x)−cos(2 ˜ x)\n2. (C.1a)\nAs trigonometrical expressions for double angle miminimizes\nnumerical error for small arguments, we choose:\nc0,3=cosh(2 ˆ x)±cos(2 ˜ x)\n2\nc1,2=sinh(2 ˆ x)∓isin(2 ˜ x)\n2. (C.2a)\nArticle number, page 22 of 22"    },    {        "title": "2402.00355v1.Adaptive_Primal_Dual_Method_for_Safe_Reinforcement_Learning.pdf",        "content": "Adaptive Primal-Dual Method for Safe Reinforcement Learning\nWeiqin Chen\nRensselaer Polytechnic Institute\nTroy, NY, USA\nchenw18@rpi.eduJames Onyejizu\nRensselaer Polytechnic Institute\nTroy, NY, USA\nonyejj@rpi.eduLong Vu\nIBM T.J. Watson Research Center\nYorktown Heights, NY, USA\nlhvu@us.ibm.com\nLan Hoang\nIBM Research\nDaresbury, Warrington, UK\nlan.hoang@ibm.comDharmashankar Subramanian\nIBM T.J. Watson Research Center\nYorktown Heights, NY, USA\ndharmash@us.ibm.comKoushik Kar\nRensselaer Polytechnic Institute\nTroy, NY, USA\nkark@rpi.edu\nSandipan Mishra\nRensselaer Polytechnic Institute\nTroy, NY, USA\nmishrs2@rpi.eduSantiago Paternain\nRensselaer Polytechnic Institute\nTroy, NY, USA\npaters@rpi.edu\nABSTRACT\nPrimal-dual methods have a natural application in Safe Reinforce-\nment Learning (SRL), posed as a constrained policy optimization\nproblem. In practice however, applying primal-dual methods to\nSRL is challenging, due to the inter-dependency of the learning\nrate (LR) and Lagrangian multipliers (dual variables) each time an\nembedded unconstrained RL problem is solved. In this paper, we\npropose, analyze and evaluate adaptive primal-dual (APD) methods\nfor SRL, where two adaptive LRs are adjusted to the Lagrangian\nmultipliers so as to optimize the policy in each iteration. We theo-\nretically establish the convergence, optimality and feasibility of the\nAPD algorithm. Finally, we conduct numerical evaluation of the\npractical APD algorithm with four well-known environments in\nBullet-Safey-Gym employing two state-of-the-art SRL algorithms:\nPPO-Lagrangian and DDPG-Lagrangian. All experiments show that\nthe practical APD algorithm outperforms (or achieves comparable\nperformance) and attains more stable training than the constant\nLR cases. Additionally, we substantiate the robustness of selecting\nthe two adaptive LRs by empirical evidence.\nKEYWORDS\nSafe Reinforcement Learning; Adaptive Primal-Dual; Adaptive Learn-\ning Rates\nACM Reference Format:\nWeiqin Chen, James Onyejizu, Long Vu, Lan Hoang, Dharmashankar Sub-\nramanian, Koushik Kar, Sandipan Mishra, and Santiago Paternain. 2024.\nAdaptive Primal-Dual Method for Safe Reinforcement Learning. In Proc.\nof the 23rd International Conference on Autonomous Agents and Multiagent\nSystems (AAMAS 2024), Auckland, New Zealand, May 6 – 10, 2024 , IFAAMAS,\n13 pages.\n1 INTRODUCTION\nReinforcement learning (RL) has a rich history of solving a wide\nrange of decision-making problems. Recently, RL has succeeded in\nProc. of the 23rd International Conference on Autonomous Agents and Multiagent Systems\n(AAMAS 2024), N. Alechina, V. Dignum, M. Dastani, J.S. Sichman (eds.), May 6 – 10, 2024,\nAuckland, New Zealand .©2024 International Foundation for Autonomous Agents\nand Multiagent Systems (www.ifaamas.org). This work is licenced under the Creative\nCommons Attribution 4.0 International (CC-BY 4.0) licence.training large language models such as ChatGPT [ 1], playing video\ngames at superhuman level [ 2–4], mastering Go [ 5,6], and manip-\nulating robotics [ 7,8]. RL problems, in general, are formulated as\nMarkov Decision Processes (MDPs). In this work, we are interested\nin conditions where the underlying dynamics are unknown, the\noptimal policy thus needs to be learned from data (samples). The\ngoal for an agent is to explore the environment so that it is able to\nmaximize the expected cumulative reward.\nNevertheless, classical RL techniques might lead to risky/unsafe\nactions [ 9–11], if they are only concerned with the reward. There-\nfore, safety constitutes a foundational aspect in realistic domains\nor physical entities. Specifically, in the realm of robot navigation,\nensuring collision avoidance [ 12,13] is essential for their proper\nfunctioning, and to ensure the preservation of human safety in the\nvicinity. Taking into account the safety requirements motivates the\ndevelopment of policy optimization under safety guarantees [ 14–\n16].\nA common approach is to employ the framework of Constrained\nMDPs (CMDPs) [ 17] where auxiliary costs (analogous to reward)\nare considered in the constraints. This framework has gained wide-\nspread adoption for inducing safe behaviors [ 16,18–22]. We briefly\nintroduce these work below.\n1.1 Related Work\nThe state-of-the-art algorithms for solving CMDPs commonly in-\nclude two types of methods: primal methods and primal-dual meth-\nods.\n1.1.1 Primal Methods. [18] develops a constrained policy opti-\nmization (CPO) algorithm for SRL that searches the feasible policy\nwithin the confines of the trust region while guaranteeing a mono-\ntonic performance improvement as well as constraint satisfaction.\nProjection-based constrained policy optimization (PCPO) [ 19] em-\nploys TRPO [ 23], the cutting-edge unconstrained RL algorithm first,\nand then projects the policy back into the feasible set. Nevertheless,\nboth CPO and PCPO suffer from the approximation error and ex-\npensive computation of the inversion of high-dimensional Hessian\nmatrices. To address these issues, [ 20] proposes first-order con-\nstrained optimization in policy space (FOCOPS), which optimizesarXiv:2402.00355v1  [cs.LG]  1 Feb 2024the constrained policy in the non-parametric space, and subse-\nquently derives the first-order gradients of the ℓ2loss function\nin the parameterization space. Another alternative is the penal-\nized proximal policy optimization (P3O) [ 24], which solves the\nconstrained policy iteration based on the minimization of an equiv-\nalent unconstrained problem. However, both FOCOPS and P3O\nintroduce more auxiliary parameters that need to be learned.\n1.1.2 Primal-Dual Methods. Lagrangian-based primal-dual algo-\nrithms like primal-dual optimization (PDO) [ 16] and reward con-\nstrained policy optimization (RCPO) [ 21] have succeeded in solv-\ning CMDP optimization problems. Nevertheless, the convergence\nguarantees of these algorithms are limited to either local (locally\noptimal policies or stationary-point) [ 16,21,25] or asymptotic sce-\nnarios [ 26]. [22] establishes the analysis on the duality gap for\nCMDPs within the policy space and provides a provably dual de-\nscent algorithm under the assumption of access to a non-convex\noptimization oracle. However, obtaining the solution to a primal\nnon-convex problem remains an issue, thus lacking global con-\nvergence guarantees. To tackle these challenges, [ 27] proposes a\nnatural policy gradient primal-dual (NPG-PD) method that achieves\nnon-asymptotic global convergence with sublinear rates regarding\nboth the optimality gap and the constraint violation. [ 28] provides\nthe first provably efficient online policy optimization algorithm–\noptimistic primal-dual proximal policy optimization (OPDOP) and\nestablishes the bounds on the regret and constraint violation. [ 29]\nproposes a safe primal-dual (SPG) algorithm that is used to solve a\nCMDP problem with probabilistic safety constraints. Nonetheless,\nachieving satisfactory performance with the existing primal-dual\nmethods remains challenging, mainly due to their sensitivity to\nhyper-parameters such as learning rates.\n1.2 Main Contribution\nThis paper addresses a core challenge in applying primal-dual meth-\nods to SRL problems, the inter-dependence of the primal Learning\nRate (LR) and Lagrangian Multiplier (LM) (dual variable) param-\neters in the primal-dual method. We provide analytical expres-\nsions (bounds) of the amount of progress made in each step of\nthe primal-dual algorithm, based on which we develop two adap-\ntive LR choices that optimize these bounds. The two LR choices\nhave an inverse-linear and inverse-quadratic dependence on the\nLM, and are incorporated into the proposed adaptive primal-dual\n(APD) algorithm. We provide theoretical analyses of the conver-\ngence, return optimality, and constraint feasibility of the APD algo-\nrithm. Finally, we numerically evaluate the practical version of APD\n(PAPD) algorithm using four environments in the Bullet-Safety-\nGym[ 30], and compare the performance of PAPD with constant-\nLR solutions using two state-of-the-art constrained RL methods:\nPPO-Lagrangian (PPOL) and DDPG-Lagrangian (DDPGL). We also\nnumerically demonstrate the robustness of PAPD algorithm with\nrespect to certain key parameter choices.\n2 SAFE REINFORCEMENT LEARNING\nMDPs are defined by a tuple ( 𝑆,𝐴,𝑅, P,U,𝛾) [31], where𝑆is the\nstate space, 𝐴is the action space, 𝑅:𝑆×𝐴×𝑆→Ris the reward\nfunction describing the quality of the decision. For any ˆ𝑆⊂𝑆,𝑠𝑡∈\n𝑆,𝑎𝑡∈𝐴,𝑡∈ {0,1,···},P(𝑠𝑡+1∈ˆ𝑆|𝑠𝑡,𝑎𝑡)(i.e., the probabilityof𝑠𝑡+1being in ˆ𝑆given𝑠𝑡and𝑎𝑡) is the transition probability\ndescribing the dynamics of the system, U(ˆ𝑆):=P(𝑠∈ˆ𝑆)is the\nstarting state distribution, and 𝛾is the discount factor. Note that a\ntable of notations is provided in the supplementary material, which\nserves to facilitate tracking and enhance comprehension for the\nreaders.\nConsider a parameterization space Θand a probability density\nfunction𝑃(·). Given𝜃∈Θ, a parameterized stationary policy\n𝜋𝜃:𝑆→𝑃(𝐴)maps states to probability distributions over the set\nof actions, and 𝜋𝜃(𝑎|𝑠)indicates the probability density of drawing\naction𝑎∈𝐴in the corresponding state 𝑠∈𝑆. Common parameter-\nizations include neural networks (NN) and Radial Basis Functions\n(RBFs). In this work, we are particularly interested in situations\nwhere the state transition distributions are unknown, and thus the\npolicies need to be computed through interacting with the environ-\nment.\nIn the context of MDPs, the objective is to find an optimal pa-\nrameter that maximizes the expected discounted return\n𝐽𝑅(𝜋𝜃)=E𝜏∼𝜋𝜃\"∞∑︁\n𝑡=0𝛾𝑡𝑅(𝑠𝑡,𝑎𝑡,𝑠𝑡+1)#\n, (1)\nwhere𝜏=(𝑠0,𝑎0,𝑠1,𝑎1,···) denotes a sample trajectory. Given\nfixed initial state distribution 𝑠0∼U and transition distribution\n𝑠𝑡+1∼P(·|𝑠𝑡,𝑎𝑡), let us define a shorthand 𝜏∼𝜋𝜃indicating that\nthe distribution over trajectories depends on the policy through\n𝑎𝑡∼𝜋𝜃(·|𝑠𝑡).\nCMDPs [ 17] impose additional constraints on the allowable poli-\ncies. More concretely, auxiliary cost functions 𝐶𝑖:𝑆×𝐴×𝑆→\nR,𝑖=1,2,···,𝑚are introduced. We make the following assump-\ntion about the costs.\nAssumption 1. Consider𝐵>0and𝐶=[𝐶1,···,𝐶𝑚]𝑇, assume\nwe have bounded costs such that ||𝐶||≤𝐵.\nAnalogous to the expected discounted return, define the expected\ndiscounted cost as\n𝐽𝐶𝑖(𝜋𝜃)=E𝜏∼𝜋𝜃\"∞∑︁\n𝑡=0𝛾𝑡𝐶𝑖(𝑠𝑡,𝑎𝑡,𝑠𝑡+1)#\n. (2)\nDenote by𝑑𝑖,𝑖=1,2,···,𝑚the desired threshold for the ex-\npected discounted cost. Accordingly, SRL problems can be formu-\nlated as\n𝜃∗∈arg max\n𝜃∈Θ𝐽𝑅(𝜋𝜃)\ns.t.𝐽𝐶𝑖(𝜋𝜃)≤𝑑𝑖,𝑖=1,2,···,𝑚. (3)\nThe SRL problem posed in (3)can be solved by applying gradient-\nbased methods on a regularized objective function [ 32] or using\nprimal-dual methods [ 16,29] to achieve local optimal solutions. The\nperformance of the first type of method can heavily depend on the\nchoice of the regularization parameter, and this choice is problem\ndependent for which there is no easy method. In general, for a given\nchoice of regularization parameter, the first type of method may\nnot produce optimal results. In this work, therefore, we focus on\nthe primal-dual method, where the regularization parameter is the\nLM (dual variable) that is progressively updated. We introduce this\nmethod in the next section and motivate the need for an adaptive\nLR in the primal update step.(a) Return\n (b) Cost\nFigure 1: Learning curves of PPOL over five independent\nruns with fixed LM values of 1 and 5. The horizontal axis\nrepresents time steps. Cost limit d=10(black dashed line) in\nall experiments. LR = 0.0006 outperforms LR = 0.0003 at LM\n=1 (red curve is infeasible), while the opposite holds when\nLM = 5.\n3 ADAPTIVE PRIMAL-DUAL ALGORITHM\n3.1 Motivation\nPrimal-dual methods rely on iterative training of the policy param-\neters𝜃to minimize the Lagrangian of (3):\nL(𝜃,𝜆)¤=−𝐽𝑅(𝜋𝜃)+𝜆𝑇(𝐽𝐶(𝜋𝜃)−d), (4)\nwhere𝐽𝐶(·)=[𝐽𝐶1(·),𝐽𝐶2(·),···,𝐽𝐶𝑚(·)]𝑇,d=[𝑑1,𝑑2,···,𝑑𝑚]𝑇,\nand𝜆∈R𝑚is the LM.\nAs stated in Section 1, despite the impressive capabilities demon-\nstrated by state-of-the-art algorithms like PDO [ 16] and RCPO [ 21]\nin addressing a wide range of SRL problems, these primal-dual like\nalgorithms still encounter challenges of selecting an appropriate LR\nfor the embedded RL problem during the training process. Indeed,\nit is natural to posit that employing a constant LR throughout the\ntraining process might not be optimal, given that the LM under-\ngoes continuous changes. We illustrate this through a numerical\nexample in Figure 1. Figure 1 presents learning curves (both Return\nand Cost) for the PPOL method over five independent runs using\nfixed LM values of 1 and 5. We use the SafetyCarRun-v0 environ-\nment from the Bullet-Safety-Gym [ 30] for this test. Notably, as LM\ntransitions from 1 to 5, the LR needs adjustment from 0.0006 to\n0.0003 for optimal results (in terms of return and cost). This means,\nmaintaining a consistent LR across varying LMs might lead to sub-\noptimal results. Specifically, deploying an optimized LR tailored for\nan LM value of 5 (green curve) in a scenario meant for LM value\nof 1 (red curve) and vice versa leads to compromised performance.\nTherefore, accounting for the dependence between LM and LR is\ncrucial for achieving good performance in SRL.\n3.2 Adaptive Learning Rate\nA naive approach to addressing the above challenge is to solve the\nprimal problem in (4)for different values of 𝜆(LM), each with its\noptimized LR. However, this approach does not scale when the LM\nis multi-dimensional (i.e., there are multiple safety constraints) or\nhas a large range. Primal-dual algorithms offer a practical way tofind the LM (dual variable) 𝜆∗that maximizes the dual function\n𝑑(𝜆)=min\n𝜃∈R𝑑−𝐽𝑅(𝜋𝜃)+𝜆𝑇(𝐽𝐶(𝜋𝜃)−d). (5)\nConsequently the optimal policy (primal variable) 𝜃∗is computed\nthrough iterative coordinated updates of the primal and dual vari-\nables\n\u001a𝜃𝑘+1=𝜃𝑘−𝜂𝑘∇𝜃L(𝜃𝑘,𝜆𝑘), (6)\n𝜆𝑘+1=[𝜆𝑘+𝜁𝑔(𝜃𝑘+1)]+, (7)\nwhere𝑔(·)is the constraint function, i.e.,\n𝑔(𝜃𝑘+1)¤=𝐽𝐶(𝜋𝜃𝑘+1)−d, (8)\nand𝜂𝑘,𝜁denote the primal and dual LRs, respectively. Note the\ndependence of the primal LR 𝜂𝑘on iteration 𝑘in (6); this is because\nin our adaptive primal-dual algorithm, described next, 𝜂𝑘depends\non the LM𝜆𝑘.\nWe have reached the phase where we articulate the convergence\nof (7). This is formally established in the following theorem.\nTheorem 1. Consider the dual function 𝑑(·)defined in (5), the con-\nstraint function 𝑔(·)and cost limit din(8). Let𝜆∗∈arg max𝜆∈R𝑚+𝑑(𝜆)\nand define𝐷∗=𝑑(𝜆∗). Let𝜃𝑘and𝜆𝑘be the sequences generated\nby(6)and (7). Denote by 𝜖𝑘the primal error of updating the La-\ngrangian given 𝜆𝑘, i.e.,𝜖𝑘=L(𝜃𝑘+1,𝜆𝑘)−min𝜃L(𝜃,𝜆𝑘). Define\n𝜆best=arg max𝜆∈{𝜆𝑘}𝐾\n𝑘=0𝑑(𝜆), it holds that\n0≤𝐷∗−𝑑(𝜆best)≤∥𝜆0−𝜆∗∥2\n2𝜁𝐾+𝜁(𝐵+(1−𝛾)∥d∥)2\n2(1−𝛾)2\n+1\n𝐾𝐾−1∑︁\n𝑘=0𝜖𝑘. (9)\nProof. The proof follows the standard stochastic gradient descent\nanalysis. See the supplementary material for details. □\nNote that the first two terms on the right-hand side of (9)depend\nexclusively on the problem and on the dual LR. The last term,\nhowever, depends on the primal error 𝜖𝑘. Our goal is to minimize\nthis error through an adaptive selection of the primal LR 𝜂𝑘. We\nresort to two different analyses to bound these errors (see Lemma 1).\nFrom these, we derive two adaptive LRs. Before stating Lemma 1,\nwe require the following assumptions.\nAssumption 2. Assume the gradients of the objective and cost\nfunctions in (3)are Lipschitz continuous with constants 𝐿𝑅,𝐿𝐶1,···,𝐿𝐶𝑚.\nLet𝐿𝐶=\u0002\n𝐿𝐶1,···,𝐿𝐶𝑚\u0003𝑇. This implies that the gradient of La-\ngrangian (4)is Lipschitz on 𝜃\n∥∇𝜃L(𝜃1,𝜆)−∇𝜃L(𝜃2,𝜆)∥≤𝐿(𝜆)∥𝜃1−𝜃2∥. (10)\nwhere𝐿(𝜆)=𝐿𝑅+𝜆𝑇𝐿𝐶.\nAssumption 3. Assume the Lagrangian function L(𝜃,𝜆)is𝜇-\nstrongly convex on 𝜃. This is,L(𝜃1,𝜆𝑘)≥L(𝜃2,𝜆𝑘)+∇𝜃L(𝜃2,𝜆𝑘)𝑇\n(𝜃1−𝜃2)+𝜇/2∥𝜃1−𝜃2∥2.\nLemma 1. Assume the Lagrangian function L(𝜃𝑘,𝜆𝑘)is𝐿′-Lipschitz\ncontinuous. Define 𝛿𝑘=\r\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r\r2\n. Then, the following boundshold for𝜖𝑘given the assumptions that the term inside the square root\nis non-negative.\n𝜖𝑘≤𝐿′√︄\n2\n𝜇\u0012\n𝐿(𝜆)𝛿𝑘+\u0012\n𝐿(𝜆)\u0010\n𝜂1\n𝑘\u00112\n−𝜂1\n𝑘\u0013\n∥∇𝜃L(𝜃𝑘,𝜆𝑘)∥2\u0013\n,\n(11)\n𝜖𝑘≤𝐿′√︄\u0012\n1+\u0010\n𝜂2\n𝑘\u00112\n𝐿(𝜆)2−𝜂2\n𝑘𝜇\u0013\n𝛿𝑘. (12)\nProof. We start the proof by employing the 𝐿′-Lipschitz continuity\nof the Lagrangian to bound 𝜖𝑘\n𝜖𝑘=L(𝜃𝑘+1,𝜆𝑘)−L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\n≤𝐿′\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r. (13)\nHence, to bound 𝜖𝑘, one can instead investigate\r\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r\r.\nWe first focus on (11). It follows from the strong convexity of\nthe Lagrangian with respect to 𝜃that\nL(𝜃𝑘+1,𝜆𝑘)≥L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\n+∇𝜃L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011𝑇\u0010\n𝜃𝑘+1−𝜃∗\n𝑘+1\u0011\n+𝜇\n2\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2. (14)\nSince∇𝜃L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\n=0the previous inequality reduces to\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤2\n𝜇\u0010\nL(𝜃𝑘+1,𝜆𝑘)−L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\u0011\n.(15)\nWe apply the Taylor expansion on both terms on the right-hand\nside of the previous expression around 𝜃𝑘\nL(𝜃𝑘+1,𝜆𝑘)=L(𝜃𝑘,𝜆𝑘)+∇𝜃L(𝜃𝑐,𝜆𝑘)𝑇(𝜃𝑘+1−𝜃𝑘), (16)\nwhere𝜃𝑐=𝜉𝜃𝑘+(1−𝜉)𝜃𝑘+1, 𝜉∈[0,1], and\nL(𝜃∗\n𝑘+1,𝜆𝑘)=L(𝜃𝑘,𝜆𝑘)+∇𝜃L(𝜃′\n𝑐,𝜆𝑘)𝑇\u0010\n𝜃∗\n𝑘+1−𝜃𝑘\u0011\n, (17)\nwhere𝜃′𝑐=𝜉′𝜃𝑘+(1−𝜉′)𝜃∗\n𝑘+1, 𝜉′∈[0,1]. Substituting (16)and\n(17) into (15) reduces to\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤2\n𝜇\u0010\n∇𝜃L(𝜃′\n𝑐,𝜆𝑘)𝑇(𝜃𝑘−𝜃∗\n𝑘+1) (18)\n+∇𝜃L(𝜃𝑐,𝜆𝑘)𝑇(𝜃𝑘+1−𝜃𝑘)\u0011\n.\nBy virtue of∇𝜃L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\n=0(18) reduces to\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤2\n𝜇\u0012\u0010\n∇𝜃L\u0000𝜃′\n𝑐,𝜆𝑘\u0001−∇𝜃L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\u0011𝑇\n\u0010\n𝜃𝑘−𝜃∗\n𝑘+1\u0011\n+∇𝜃L(𝜃𝑐,𝜆𝑘)𝑇(𝜃𝑘+1−𝜃𝑘)\u0011\n.\n(19)\nwhere the last inequality follows from the Lipschitz continuity of\nthe gradient of Lagrangian (see Assumption 2).\nBy adding and subtracting ∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇(𝜃𝑘+1−𝜃𝑘)we obtain\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤2\n𝜇\u0010\n𝐿(𝜆)\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2+\n(∇𝜃L(𝜃𝑐,𝜆𝑘)−∇𝜃L(𝜃𝑘,𝜆𝑘))𝑇(𝜃𝑘+1−𝜃𝑘)\n+∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇(𝜃𝑘+1−𝜃𝑘)\u0011\n. (20)Employing Lipschitz continuity of the gradient of Lagrangian yields\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤2\n𝜇\u0010\n𝐿(𝜆)\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2+𝐿(𝜆)∥𝜃𝑘+1−𝜃𝑘∥2\n+∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇(𝜃𝑘+1−𝜃𝑘)\u0011\n. (21)\nBy definition of 𝛿𝑘the previous inequality can be rewritten as\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤2\n𝜇\u0012\n𝐿(𝜆)𝛿𝑘+𝐿(𝜆)\u0010\n𝜂1\n𝑘\u00112\n∥∇𝜃L(𝜃𝑘,𝜆𝑘)∥2\n−𝜂1\n𝑘∥∇𝜃L(𝜃𝑘,𝜆𝑘)∥2\u0011\n. (22)\nTaking the square root on the previous inequality and combining\nwith (13) completes the proof of (11).\nWe then turn our attention to proving (12). Note that\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2=\r\r𝜃𝑘−𝜂2\n𝑘∇𝜃L(𝜃𝑘,𝜆𝑘)−𝜃∗\n𝑘+1\r\r2\n=\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2+\u0010\n𝜂2\n𝑘\u00112\n∥∇𝜃L(𝜃𝑘,𝜆𝑘)∥2\n−2𝜂2\n𝑘∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇\u0010\n𝜃𝑘−𝜃∗\n𝑘+1\u0011\n. (23)\nBy∇𝜃L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\n=0the previous equation is equivalent to\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2=\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2\n+\u0010\n𝜂2\n𝑘\u00112\r\r\r∇𝜃L(𝜃𝑘,𝜆𝑘)−∇𝜃L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\r\r\r2\n−2𝜂2\n𝑘∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇\u0010\n𝜃𝑘−𝜃∗\n𝑘+1\u0011\n. (24)\nAssumption 2 indicates that the gradient of Lagrangian function is\n𝐿(𝜆)-Lipschitz continuous, and we thus obtain\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2\n+\u0010\n𝜂2\n𝑘\u00112\n𝐿(𝜆)2\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2\n−2𝜂2\n𝑘∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇\u0010\n𝜃𝑘−𝜃∗\n𝑘+1\u0011\n.(25)\nIn addition, the strong convexity of the Lagrangian with respect\nto𝜃shows that\n∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇\u0010\n𝜃∗\n𝑘+1−𝜃𝑘\u0011\n≤L\u0010\n𝜃∗\n𝑘+1,𝜆𝑘\u0011\n−L(𝜃𝑘,𝜆𝑘)\n−𝜇\n2\r\r𝜃∗\n𝑘+1−𝜃𝑘\r\r2\n≤−𝜇\n2\r\r𝜃∗\n𝑘+1−𝜃𝑘\r\r2, (26)\nwhere the last inequality follows from the fact that 𝜃∗\n𝑘+1is the\nminimizer of the Lagrangian. Subsequently, combining (25)and\n(26) yields\n\r\r𝜃𝑘+1−𝜃∗\n𝑘+1\r\r2≤\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2−𝜂2\n𝑘𝜇\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2\n+\u0010\n𝜂2\n𝑘\u00112\n𝐿(𝜆)2\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2\n=\u0012\n1+\u0010\n𝜂2\n𝑘\u00112\n𝐿(𝜆)2−𝜂2\n𝑘𝜇\u0013\r\r𝜃𝑘−𝜃∗\n𝑘+1\r\r2.(27)\nTaking the square root and combining with (13)completes the proof\nof the result. □\nBy virtue of Assumptions 2, 3 and Lemma 1, we are able to derive\ntwo optimal LRs that minimize the two bounds in (11)and(12). We\nformalize this claim in the following theorem.Theorem 2. Let Assumption 2 and Assumption 3 hold. Then, the\nlearning rates 𝜂1\n𝑘=1\n2𝐿(𝜆)and𝜂2\n𝑘=𝜇\n2𝐿(𝜆)2minimize the bounds in\n(11) and (12), respectively. Denote by 𝜖1\n𝑘and𝜖2\n𝑘the bounds on the\nprimal error using 𝜂1\n𝑘and𝜂2\n𝑘. Consider the 𝐿′-Lipschitz continuity of\nthe LagrangianL(𝜃𝑘,𝜆𝑘)and𝛿𝑘in Lemma 1. Then, it holds that\n𝜖1\n𝑘≤𝐿′√︄\n2𝛿𝑘\n𝜇\u0012\n𝐿(𝜆)−𝜇2\n16𝐿(𝜆)\u0013\n, (28)\n𝜖2\n𝑘≤𝐿′√︄\n𝛿𝑘\u0012\n1−𝜇2\n4𝐿(𝜆)2\u0013\n. (29)\nProof. We start by establishing two LRs 𝜂1\n𝑘and𝜂2\n𝑘. As observed\nin(11), the right-hand side is convex with respect to 𝜂1\n𝑘since\n𝐿′,𝐿(𝜆),𝛿𝑘are independent of 𝜂1\n𝑘. To minimize it one can com-\npute the derivative and make it zero. Then the minimizer of the\nright-hand side of (11) is given by\n𝜂1\n𝑘=1\n2𝐿(𝜆). (30)\nLikewise, the right-hand side of (12)is convex with respect to\n𝜂2\n𝑘, and the optimal LR (minimizer) takes the form of\n𝜂2\n𝑘=𝜇\n2𝐿(𝜆)2. (31)\nHaving established 𝜂1\n𝑘and𝜂2\n𝑘, we are now in the stage of proving\n𝜖1\n𝑘and𝜖2\n𝑘. For𝜖1\n𝑘, substituting 𝜂1\n𝑘into(11)yields the tightest upper\nbound, i.e.,\n𝜖1\n𝑘≤𝐿′√︄\n2\n𝜇\u0012\n𝐿(𝜆)𝛿𝑘−∥∇𝜃L(𝜃𝑘,𝜆𝑘)∥2\n4𝐿(𝜆)\u0013\n. (32)\nStrong convexity indicates that\n−∇𝜃L(𝜃𝑘,𝜆𝑘)𝑇\u0010\n𝜃∗\n𝑘+1−𝜃𝑘\u0011\n≥𝜇\n2\r\r𝜃∗\n𝑘+1−𝜃𝑘\r\r2. (33)\nSquaring both sides of the previous inequality, using the Cauchy-\nSchwartz inequality and the definition of 𝛿𝑘it follows that\n∥∇𝜃L(𝜃𝑘,𝜆𝑘)∥2≥𝜇2\n4\r\r𝜃∗\n𝑘+1−𝜃𝑘\r\r2=𝜇2\n4𝛿𝑘. (34)\nCombining the previous inequality with (32) yields\n𝜖1\n𝑘≤𝐿′√︄\n2𝛿𝑘\n𝜇\u0012\n𝐿(𝜆)−𝜇2\n16𝐿(𝜆)\u0013\n. (35)\nOur focus now shifts towards the derivation of 𝜖2\n𝑘. Substituting\n𝜂2\n𝑘into (12) yields the tightest upper bound on 𝜖2\n𝑘\n𝜖2\n𝑘≤𝐿′√︄\n𝛿𝑘\u0012\n1−𝜇2\n4𝐿(𝜆)2\u0013\n. (36)\nThese complete the proof of Theorem 2. □\nNotice that both LRs proposed by Theorem 2 demonstrate an in-\nverse relationship with respect to the LM, which is also empirically\nvalidated by our preliminary observations in Figure 1. Moreover, 𝜂1\n𝑘\nhas an inverse-linear dependence on LM while 𝜂2\n𝑘has an inverse-\nquadratic dependence on LM. We therefore term them InvLin and\nInvQua , respectively. It is important to point out that we assume\nthat the Lagrangian is strongly convex. This is generally not the\ncase for RL problems. However, one can assume local convexityAlgorithm 1 Adaptive Primal-Dual (APD)\n1:Input :𝜃0,𝜆0,𝜁,𝐿𝑅,𝐿𝐶,𝜇(optional)\n2:for𝑘=0,1,···,do\n3: Choose primal LR from (30) or (31)\n4: Update primal variable (policy parameter) as in (6)\n5: Update dual variable (LM) as in (7)\n6:end for\nAlgorithm 2 Practical Adaptive Primal-Dual (PAPD)\n1:Input :𝜃0,𝐻1(𝐻′\n1),𝐻2(𝐻′\n2),𝐾𝑃,𝐾𝐼,𝐾𝐷\n2:for𝑘=0,1,···,do\n3: Choose primal LR from (39)\n4: Update primal variable (policy parameter) as in (6)\n5: Update dual variable (LM) as in (40)\n6:end for\naround a local minimum. We chose this stronger assumption for\nsimplicity in the exposition.\nHaving established InvLin andInvQua as well as corresponding\nbounds𝜖1\n𝑘and𝜖2\n𝑘, we are able to propose an adaptive primal-dual\n(APD) algorithm, which is summarized under Algorithm 1. Notice\nthat Theorem 1 indicating the (approximate) convergence of the\nLM also holds for the APD algorithm. Furthermore, with 𝜖1\n𝑘and𝜖2\n𝑘,\nwe can further establish the guarantee of proximity to the primal\noptimum𝐽𝑅(𝜋𝜃∗). The following theorem addresses this aspect\nexplicitly.\nTheorem 3. Consider𝜃∗and𝐽𝑅(𝜋𝜃)in(3). Let the hypotheses of\nTheorem 1 hold, and consider the sequence of the LM generated by\nAlgorithm 1. Then, it holds that\nlim inf\n𝐾→∞1\n𝐾𝐾−1∑︁\n𝑘=0𝐽𝑅(𝜋𝜃𝑘+1)≥𝐽𝑅(𝜋𝜃∗)−1\n𝐾 𝐾−1∑︁\n𝑘=0𝜖𝑘!\n−𝜁(𝐵+(1−𝛾)∥d∥)2\n2(1−𝛾)2. (37)\nProof. See the supplemental material. □\nNotice that𝜖𝑘in(37)can be bounded by (28)or(29), depending\nonInvLin orInvQua is selected. Theorem 3 demonstrates that the\nlimit inferior of the average of the sequence derived by Algorithm 1\napproximates well the value of 𝐽𝑅(𝜋𝜃∗). In principle, the limit supe-\nrior has the potential to be significantly larger than 𝐽𝑅(𝜋𝜃∗), which\nwould lead to constraint violations. In the following result, we prove\nthat this is not the case, i.e., the sequence generated by Algorithm 1\nis feasible on average.\nTheorem 4. Let hypotheses of Theorem 1 hold. It holds that\nlim sup\n𝐾→∞1\n𝐾𝐾−1∑︁\n𝑘=0𝐽𝐶𝑖(𝜋𝜃𝑘)≤𝑑𝑖,𝑖=1,2,···,𝑚. (38)\nProof. See the supplemental material. □\nDespite the theoretical guarantees on APD (Algorithm 1) re-\ngarding the convergence, optimality, and feasibility, estimating the\nLipschitz constant 𝐿(𝜆)and the strongly convex constant 𝜇in the\nLRs (30)and (31)is, in general, computationally expensive. Wethus consider a practical version of adaptive LRs. Under the single\nconstraint, the adaptive LRs in (30) and (31) can be written as\n𝜂1\n𝑘=𝐻1/(𝜆𝑘+𝐻2), 𝜂2\n𝑘=𝐻′\n1/(𝜆𝑘+𝐻′\n2)2, (39)\nwhere𝐻1,𝐻2,𝐻′\n1,𝐻′\n2are hyper-parameters.\nNote that the update on the policy parameter is updated by run-\nning a step of any RL algorithm with the adaptive LRs in (39). In\nparticular, we consider state-of-the-art algorithms such as PPOL\nand DDPGL. However, the dual variables frequently exhibit tenden-\ncies to overshoot and oscillate, thereby hindering performance. To\naddress these challenges, we adopt the Proportional Integral Deriv-\native (PID) Lagrangian strategies as introduced by [ 33]. Drawing\ninspiration from the feedback control, the LM is updated as\n𝜆𝑘=\u0010\n𝐾𝑃(𝐽𝑘\n𝐶−𝑑)+𝐾𝐼𝐼𝑘+𝐾𝐷(𝐽𝑘\n𝐶−𝐽𝑘−1\n𝐶)\u0011\n+, (40)\nwhere𝐼𝑘is recursively defined as 𝐼𝑘=(𝐼𝑘−1+𝐽𝑘\n𝐶−𝑑)+and sub-\nscript+indicates projection onto the non-negative orthant. The\nthree hyper-parameters 𝐾𝑃,𝐾𝐼, and𝐾𝐷represent the proportional,\nintegral, and derivative gains. In particular, selecting 𝐾𝑃=𝐾𝐷=0\nthe update reduces to gradient descent on the dual domain.\nWith the practical adaptive LRs ( InvLin andInvQua ) defined as in\n(39), as well as the dual update rule (40)using PID-Lagrangian, the\npractical version of APD (PAPD) algorithm, which we implement\nand evaluate in the subsequent section, is described in Algorithm 2.\n4 EXPERIMENTS\nThe experimental details are deferred to the supplementary mate-\nrial.\n4.1 Environment\nTo validate our findings, we consider the Bullet-Safety-Gym envi-\nronments [ 30] and the Fast Safe Reinforcement Learning (FSRL)\nframework [ 34] in this work. The Bullet-Safety-Gym is a platform\ndesigned to train and evaluate safety features in constrained RL\nscenarios. Meanwhile, the FSRL library offers structured modules\nfor implementing SRL algorithms including PPOL and DDPGL.\nTable 1: Running time (seconds) using PPOL for all experiments.\nEach case contains five independent runs.\nEnvironment BallRun CarRun BallCircle CarCircle\nInvLin 128.1±5.2159.0±5.6 223.8±5.6 861.0±28.2\nInvQua 128.1±2.7 161.9±14.5 228.2±11.1 993.2±33.8\nLR=0.0001 127.1±1.5 162.0±13.8 234.4±2.6 984.5±56.1\nLR=0.00025 125.8±4.3 164.0±18.4 230.3±9.1 961.9±61.1\nLR=0.0005 124.4±3.9 170.9±17.7 222.8±13.6 908.8±35.1\nLR=0.001 120.5±6.3163.6±22.7214.7±14.5868.9±27.9\n4.2 Results\nWe compare Algorithm 2 with the constant LR primal-dual algo-\nrithm in four environments of Bullet-Safety-Gym: SafetyBallRun-v0 ,\nSafetyCarRun-v0 ,SafetyBallCircle-v0 ,SafetyCarCircle-v0 (described\nin the supplementary material). For a fair comparison, we maintainuniformity in all parameters and hyper-parameters (except for the\nLR) across each case (see the supplementary material for details\nof the hyper-parameters). While Algorithm 2 relies on five hyper-\nparameters (other than 𝜃0), its performance is fairly robust to the\nchoice of these parameter values. We employ the default values\nof𝐾𝑃,𝐾𝐼,𝐾𝐷in FSRL across all experiments. In addition, we nu-\nmerically substantiate the robustness of the algorithm performance\nagainst variations in 𝐻1(𝐻′\n1),𝐻2(𝐻′\n2)values in Section 4.3.\nFigure 2 depicts the training curves of return, cost, and LR using\nPPOL over five random seeds. The solid line illustrates the mean and\nthe shaded area depicts the minimum and maximum values across\nseeds. More specifically, the smallest constant LR (LR = 0.0001) has\nthe worst performance in all four environments. Despite the stable\ntraining process, this LR makes significant sacrifices in both aspects\nof convergence rate and optimal value (return). To some extent,\nthe above issue can be alleviated by using a larger LR. The purple\ncurves in Figure 2 show a better performance achieved by LR =\n0.00025. Nevertheless, the performance is not maintained as one\nkeeps increasing LR. Indeed, the experiments in SafetyBallCircle-v0\npresent that the training processes become more unstable (larger\nvariance) and/or converge to a worse solution when LR = 0.0005\nis selected. Ultimately, if LR is continuously raised until it reaches\n0.001, all experiments exhibit either a significant fluctuation in re-\nturn and cost, or in some cases, an even worse average performance\ncompared to using a LR of 0.0005.\nOn the contrary, PAPD with InvLin andInvQua outperform all\nconstant-LR cases in SafetyBallCircle-v0 andSafetyCarCircle-v0 , and\nachieve comparable performance in terms of return and cost with\nthe best constant-LR trials in SafetyBallRun-v0 andSafetyCarRun-v0 .\nThis stems from the common criterion in the optimization literature,\nwhere the solutions with higher returns but violating constraints (in-\nfeasible) are regarded as having “inferior performance”. Moreover,\nwe present the running time employing PPOL for all experiments,\nas detailed in Table 1, where BallRun ,CarRun ,BallCircle ,CarCir-\ncleserve as succinct abbreviations for the corresponding environ-\nments, namely SafetyBallRun-v0 ,SafetyCarRun-v0 ,SafetyBallCircle-\nv0,SafetyCarCircle-v0 . Table 1 unveils the absence of substantial\ndifference in running time between our PAPD algorithm and the\nconstant-LR baselines. In addition, it is noteworthy that we use\nthe same hyper-parameters 𝐻1=0.001,𝐻2=3inInvLin and same\n𝐻′\n1=0.015,𝐻′\n2=6inInvQua across all experiments.\nWe also apply DDPGL to update policy parameters in the PAPD\nalgorithm. Similar to what we observe in Figure 2, InvLin andIn-\nvQua employing DDPGL surpasses or matches the best perfor-\nmance of all constant-LR cases. Given the limited space, the results\nof DDPGL experiments are meticulously detailed and analyzed\nin the supplementary material. Likewise, we maintain uniformity\nin hyper-parameter values throughout all conducted experiments.\nThis consistency, despite the varied experimental scenarios, speaks\nto the robustness of our PAPD approach. Further evidence sup-\nporting this robustness statement will be discussed in the next\nsubsection.\n4.3 Robustness Verification\nHaving observed the sensitivity of experimental results with respect\nto the constant-LR, one might naturally find themselves intriguedReturn- SafetyBallRun-v0\n Return- SafetyCarRun-v0\n Return- SafetyBallCircle-v0\n Return- SafetyCarCircle-v0\nCost- SafetyBallRun-v0\n Cost- SafetyCarRun-v0\n Cost- SafetyBallCircle-v0\n Cost- SafetyCarCircle-v0\nLR-SafetyBallRun-v0\n LR-SafetyCarRun-v0\n LR-SafetyBallCircle-v0\n LR-SafetyCarCircle-v0\nFigure 2: Learning curves for PPOL over four environments with five independent runs. In all figures, the horizontal axis is the\nnumber of time step. The solid line illustrates the mean and the shaded area depicts the maximum and the minimum. In all\nexperiments, 𝐻1=0.001,𝐻2=3forInvLin ,𝐻′\n1=0.015,𝐻′\n2=6forInvQua , and cost limit d =10(black dashed line).\nby the sensitivity exhibited by 𝐻1(𝐻′\n1),𝐻2(𝐻′\n2). As shown in (39),\n𝐻1(𝐻′\n1)plays a similar role to constant-LR in a natural way. There-\nfore, a compelling point of interest would involve their comparison\nwith the constant-LR. On the other hand, 𝐻2(𝐻′\n2)is added to the LM\n𝜆𝑘. Indeed, for large values of 𝜆𝑘,𝐻2(𝐻′\n2)becomes more negligible,\nwhereas for small values of 𝜆𝑘, the LR will focus on 𝐻2(𝐻′\n2)itself.\nThe LM describes the level of complexity involved in solving the\nproblem, which indicates that 𝐻2(𝐻′\n2)generally depends on the\nproblems/tasks. In this subsection, we substantiate the robustness\nof𝐻1(𝐻′\n1)and𝐻2(𝐻′\n2), respectively. The results are summarized in\nTables 2, 3 and 4.\nTable 2 summarizes the experimental results of PPOL algorithm\ninSafetyCarRun-v0 , where each case contains five independent runs.\nMore concretely, the parameters in the first block are the same as\nin Figure 2, where constant LR (0.00025) has the most comparable\nperformance with InvLin (𝐻1=0.001) and InvQua (𝐻′\n1=0.015), andwe thus select it for robustness verification. In the next three blocks,\nthe constant LR, 𝐻1, and𝐻′\n1are reduced by the same proportion,\nand we also increase the three parameters by the same scale in the\nlast four blocks. These 8 blocks contain a wide range of constant\nLR,𝐻1,𝐻′\n1, and thus enable us to fairly compare their sensitivity.\nThe goal of problem (3)is to maximize the expected return while\nsatisfying the constraint, i.e., cost<10in our experiments. With\nthis in mind, Table 2 shows that InvLin achieves the best perfor-\nmance in the first, second, and fifth blocks, while InvQua stands\nout as the epitome of exceptional performance in the rest of blocks.\nThe fifth block is the specialty, where all three cases are infeasible.\nIn this case, InvLin is selected to be the best due to the smallest\nconstraint violations and almost the largest return. To summarize,\nour proposed InvLin andInvQua outperform the constant LR in a\nwide range of values.Table 2: Robustness verification for 𝐻1/𝐻′\n1using SafetyCarRun-v0 and PPOL. Each\ncase contains five independent runs. In all experiments, 𝐻2=3/𝐻′\n2=6are fixed as in\nFigure 2. The first block showcases three baselines: constant LR = 0.00025 (Baseline1),\nInvLin𝐻1=0.001(Baseline2), InvQua𝐻′\n1=0.015(Baseline3), as selected in Figure 2. In\nthe rest of blocks, constant LR, 𝐻1,𝐻′\n1are decreased/increased by the same proportion.\nParameter Return Cost Parameter Return Cost\nBaseline1 538.10±8.71 12.48±10.96 Baseline1×0.2 421.26±40.82 9.64±4.90\nBaseline2 537.59±11.40 9.98±5.98 Baseline2×0.2 502.48±27.43 8.80±5.15\nBaseline3 545.22±4.93 13.40±4.78 Baseline3×0.2 513.93±7.35 11.50±4.55\nBaseline1×0.32 492.65±15.78 7.66±5.34 Baseline1×0.4 510.41±13.62 9.22±6.14\nBaseline2×0.32 511.41±22.70 11.46±7.22 Baseline2×0.4 526.35±4.30 18.42±8.38\nBaseline3×0.32 509.91±29.01 8.94±5.74 Baseline3×0.4 519.52±11.61 8.04±5.66\nBaseline1×1.6 545.26±4.50 15.22±7.48 Baseline1×2.4 539.62±19.01 10.02±10.87\nBaseline2×1.6 548.04±8.59 14.32±7.21 Baseline2×2.4 528.23±28.56 4.74±5.68\nBaseline3×1.6 548.45±7.52 15.96±11.93 Baseline3×2.4 542.24±8.92 8.28±8.27\nBaseline1×3.2 536.08±12.57 6.42±8.25 Baseline1×4.0 529.97±15.72 1.26±1.08\nBaseline2×3.2 544.97±8.28 10.70±5.69 Baseline2×4.0 533.70±19.98 9.56±10.51\nBaseline3×3.2 538.08±11.93 5.66±6.96 Baseline3×4.0 534.07±20.06 7.70±11.52\nTable 3: Summary of Table 2\nParamter Best Performance Overall Return Overall Cost\nConstant LR 536.08/6.42 514.17±41.45 8.99±4.15\nInvLin -𝐻1 537.59/9.98 529.10±15.74 10.99±4.03\nInvQua -𝐻′\n1542.24/8.28 531.43±14.92 9.94±3.41\nTable 4: Robustness verification for 𝐻2/𝐻′\n2using SafetyCarRun-\nv0and PPOL. Each case contains five independent runs. In all\nexperiments, 𝐻1=0.001/𝐻′\n1=0.015are fixed as in Figure 2. The\ntwo baselines, denoted as BS1andBS2, correspond to InvLin\n𝐻2=3andInvQua𝐻′\n2=6as selected in Figure 2. Starting from\nthe first block, 𝐻2/𝐻′\n2are increased by 10%.\n𝐻2/𝐻′\n2Return Cost 𝐻2/𝐻′\n2Return Cost\nBS1×0.8 543.05±10.21 12.40±8.66 BS1×0.9 531.73±12.61 5.80±6.15\nBS2×0.8 545.09±3.82 6.56±5.05 BS2×0.9 543.58±6.84 8.04±5.26\nBS1×1.1 538.27±5.70 8.86±6.90 BS1×1.2 531.27±8.97 7.90±6.14\nBS2×1.1 547.12±2.44 13.28±2.49 BS2×1.2 537.37±7.68 9.50±4.40\nWe further analyze Table 2 from the perspective of all blocks.\nIndeed, Table 3 extracts the best performance of three parameters\nacross all blocks. All of them attain feasible solutions, with InvQua\nemerging as the optimal choice by achieving return =542.24,cost=\n8.28. On the other hand, we collect the mean values of the return\nand the cost in each block of Table 2, and compute their mean and\nstandard deviation across all eight blocks. These are shown in the\nlast two columns: “Overall Return” and “Overall Cost”. Notice that\nthe best “overall performance”, i.e., the best overall return-cost pair\ngoes to InvQua again. In addition, InvLin andInvQua yield smaller\nstandard deviation than the constant LR case in terms of both return\nand cost.Furthermore, Table 4 tests the robustness of 𝐻2(𝐻′\n2). We also use\nthe same𝐻1(𝐻′\n1),𝐻2(𝐻′\n2)as in Figure 2 as our baselines. Then, we\nincrease𝐻2(𝐻′\n2)by10%starting from the first block in Table 4, while\nkeeping𝐻1(𝐻′\n1)constant. Note that we maintain the satisfactory\nperformance in three of the blocks and encounter small loss in\nthe worst case while changing 𝐻2(𝐻′\n2). Moreover, InvQua shows\nlarger sensitivity to 𝐻′\n2, which could be naturally explained by its\nquadratic format.\nIn summary, we are able to employ the same 𝐻1(𝐻′\n1),𝐻2(𝐻′\n2)in\nfour different Bullet-Safety-Gym environments for both PPOL and\nDDPGL algorithms. In addition, Tables 2 and 3 indicate that InvLin\nandInvQua outperform the constant LR baselines in a wide range\nof𝐻1(𝐻′\n1)that are selected, and Table 4 validates the robustness of\n𝐻2(𝐻′\n2)as well. Therefore, these numerical results substantiate the\nrobustness of our proposed PAPD algorithm.\n5 CONCLUDING REMARKS\nIn this work, we propose the Adaptive Primal-Dual (APD) algorithm\nand its practical version (PAPD) that leverage adaptive learning\nrates for safe reinforcement learning (SRL). Theoretically, we pro-\nvide the analyses of the APD algorithm in terms of the convergence,\noptimality and feasibility. 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PMLR, 2020.\n[34] Zuxin Liu, Zijian Guo, Haohong Lin, Yihang Yao, Jiacheng Zhu, Zhepeng Cen,\nHanjiang Hu, Wenhao Yu, Tingnan Zhang, Jie Tan, et al. Datasets and benchmarks\nfor offline safe reinforcement learning. arXiv preprint arXiv:2306.09303 , 2023.\n[35] Miguel Calvo-Fullana, Santiago Paternain, Luiz FO Chamon, and Alejandro\nRibeiro. State augmented constrained reinforcement learning: Overcoming the\nlimitations of learning with rewards. arXiv preprint arXiv:2102.11941 , 2021.\n[36] Alla Dita Raza Choudary and Constantin P Niculescu. Real analysis on intervals .\nSpringer, 2014.\n[37] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov.\nProximal policy optimization algorithms. arXiv preprint arXiv:1707.06347 , 2017.6 SUPPLEMENTARY MATERIAL\nTo make this work self-contained and easy to read, the supplemen-\ntary material is organized as follows. In the first subsection, we\nshowcase a table of notations for the reader’s effortless tracking.\nIn Sections 6.2 - 6.4, we provide the detailed proofs of theorems\nclaimed in the main body of paper. In Section 6.5, we present the de-\ntails of experiments including environment, methodology, results,\nand implementation.\n6.1 Table of Notations\nWe present a table of notations, as outlined in Table 5, which serves\nto facilitate tracking and enhance comprehension for the readers.\nTable 5: Table of Notations\nNotation Definition Notation Definition\n𝑆 state space 𝐴 action space\n𝑠 state 𝑎 action\n𝑅 reward function 𝐶 cost function\nP transition probability 𝛾 discount factor\nU initial state distribution 𝜋 policy\nΘ parameterization space 𝜏 trajectory\n𝜃 primal variable 𝜆 dual variable\n𝐽𝑅 expected return 𝐽𝐶 expected cost\n𝐵 cost bound L Lagrangian\nLM Lagrangian multiplier LR learning rate\nd constraint threshold 𝑑 dual function\n𝜂𝑘 primal stepsize 𝜁 dual stepsize\n𝜆∗optimal dual variable 𝐷∗dual optimum\n𝑔 constraint function 𝜖𝑘 Lagrangian error\n𝜇 strongly convex constant 𝐿 Lipschitz constant\n𝐻1(𝐻2) practical LR constant 𝐾𝑃 proportional gain\n𝐾𝐼 integral gain 𝐾𝐷 derivative gain\n6.2 Proof of Theorem 1\nWe start by recalling the dual update (7)\n𝜆𝑘+1=[𝜆𝑘+𝜁𝑔(𝜃𝑘+1)]+. (41)\nThen, we obtain\n||𝜆𝑘+1−𝜆∗||2=||[𝜆𝑘+𝜁𝑔(𝜃𝑘+1)]+−𝜆∗||2\n≤||𝜆𝑘−𝜆∗+𝜁𝑔(𝜃𝑘+1)||2, (42)\nwhere the last inequality follows from the non-expansiveness of\nthe projection.\nExpanding the norm square yields\n||𝜆𝑘+1−𝜆∗||2≤||𝜆𝑘−𝜆∗||2+𝜁2||𝑔(𝜃𝑘+1)||2\n+2𝜁(𝜆𝑘−𝜆∗)𝑇𝑔(𝜃𝑘+1). (43)\nCombining Assumption 1 and (2) yields ||𝐽𝐶||≤𝐵/(1−𝛾). By the\ndefinition of 𝑔(𝜃𝑘+1)in (8) and the triangle inequality we obtain\n||𝑔(𝜃𝑘+1)||2≤(||𝐽𝐶||+|| d||)2≤(𝐵\n1−𝛾+||d||)2. (44)Then we have\n||𝜆𝑘+1−𝜆∗||2≤||𝜆𝑘−𝜆∗||2+𝜁2(𝐵\n1−𝛾+||d||)2\n+2𝜁(𝜆𝑘−𝜆∗)𝑇𝑔(𝜃𝑘+1). (45)\nTo proceed, we rely on the following technical lemma.\nLemma 2. Consider the dual function 𝑑(·)defined in (5).∀𝜆1,𝜆2∈\nR𝑚, denote by𝜃∗(𝜆1)and𝜃∗(𝜆2)the optimizer of the Lagrangian\nwith respect to 𝜆1and𝜆2. Suppose that there exists a 𝜖>0and a\n𝜃†(𝜆1)such that\nL(𝜃†(𝜆1),𝜆1)−L(𝜃∗(𝜆1),𝜆1)=𝜖. (46)\nThen, it holds that\n𝑑(𝜆2)≤𝑑(𝜆1)+(𝜆2−𝜆1)𝑇𝑔(𝜃†(𝜆1))+𝜖. (47)\nProof. We proceed by recalling the definition of the dual function\n𝑑(𝜆1)and𝑑(𝜆2)and computing their difference\n𝑑(𝜆1)−𝑑(𝜆2)=L(𝜃∗(𝜆1),𝜆1)−L(𝜃∗(𝜆2),𝜆2), (48)\nBy virtue of the optimality of 𝜃∗(𝜆2), (48) can be rewritten as\n𝑑(𝜆1)−𝑑(𝜆2)≥L(𝜃∗(𝜆1),𝜆1)−L(𝜃†(𝜆1),𝜆2). (49)\nSubstituting (46) into the previous inequality yields\n𝑑(𝜆1)−𝑑(𝜆2)≥L(𝜃†(𝜆1),𝜆1)−L(𝜃†(𝜆1),𝜆2)−𝜖. (50)\nBy expanding the Lagrangian the previous inequality reduces to\n𝑑(𝜆1)−𝑑(𝜆2)≥−𝐽𝑅(𝜋𝜃†(𝜆1))+𝜆𝑇\n1𝑔(𝜃†(𝜆1))−\n\u0010\n−𝐽𝑅(𝜋𝜃†(𝜆1))+𝜆𝑇\n2𝑔(𝜃†(𝜆1))\u0011\n−𝜖\n=(𝜆1−𝜆2)𝑇𝑔(𝜃†(𝜆1))−𝜖. (51)\nReordering (51) completes the proof of Lemma 2. □\nEmploy Lemma 2 with 𝜆∗and𝜆𝑘yields\n𝑑(𝜆∗)≤𝑑(𝜆𝑘)+(𝜆∗−𝜆𝑘)𝑇𝑔(𝜃𝑘+1)+𝜖𝑘, (52)\nwhere𝑔(𝜃𝑘+1)=𝑔(𝜃†(𝜆𝑘))and𝜖𝑘denotes the primal error of\nupdating the Lagrangian, as stated in Theorem 1.\nCombing (45) and (52) yields\n||𝜆𝑘+1−𝜆∗||2≤||𝜆𝑘−𝜆∗||2+𝜁2(𝐵\n1−𝛾+||d||)2\n+2𝜁(𝑑(𝜆𝑘)−𝐷∗+𝜖𝑘). (53)\nwhere𝐷∗=𝑑(𝜆∗).Unrolling the previous inequality starting from\n𝜆𝐾to𝜆0yields\n0≤||𝜆𝐾−𝜆∗||2≤||𝜆0−𝜆∗||2+𝐾𝜁2(𝐵\n1−𝛾+||d||)2\n+2𝜁(𝐾−1∑︁\n𝑘=0𝑑(𝜆𝑘)−𝐾𝐷∗+𝐾−1∑︁\n𝑘=0𝜖𝑘). (54)\nReordering terms in the previous expression yields\n𝐷∗−1\n𝐾𝐾−1∑︁\n𝑘=0𝑑(𝜆𝑘)≤||𝜆0−𝜆∗||2\n2𝜁𝐾+𝜁(𝐵+(1−𝛾)||d||)2\n2(1−𝛾)2\n+1\n𝐾𝐾−1∑︁\n𝑘=0𝜖𝑘. (55)Note that𝜆best∈arg max𝜆∈{𝜆𝑘}𝐾\n𝑘=0𝑑(𝜆), it holds that\n0≤𝐷∗−𝑑(𝜆best)≤||𝜆0−𝜆∗||2\n2𝜁𝐾+𝜁(𝐵+(1−𝛾)||d||)2\n2(1−𝛾)2\n+1\n𝐾𝐾−1∑︁\n𝑘=0𝜖𝑘. (56)\nwhere the leftmost inequality follows from the definition of the\ndual function. This completes the proof of Theorem 1.\n6.3 Proof of Theorem 3\nFor∀𝑘>0, the following chain of inequalities holds due to the fact\nthat the dual function is a lower bound on the value of the primal\nand it is concave\n−𝐽𝑅(𝜋𝜃∗)≥𝑑 \n1\n𝐾𝐾−1∑︁\n𝑘=0𝜆𝑘!\n≥1\n𝐾𝐾−1∑︁\n𝑘=0𝑑(𝜆𝑘). (57)\nSince the primal variable 𝜃𝑘+1is updated by the APD algo-\nrithm (Algorithm 1) with primal error 𝜖𝑘and by definition 𝑑(𝜆𝑘)=\nL(𝜃∗(𝜆𝑘),𝜆𝑘), we obtain\n𝑑(𝜆𝑘)=L(𝜃𝑘+1,𝜆𝑘)−𝜖𝑘\n=−𝐽𝑅(𝜋𝜃𝑘+1)+𝜆𝑇\n𝑘𝑔(𝜃𝑘+1)−𝜖𝑘. (58)\nSubstituting the previous equation into (57) yields\n−𝐽𝑅(𝜋𝜃∗)≥1\n𝐾𝐾−1∑︁\n𝑘=0\u0010\n−𝐽𝑅(𝜋𝜃𝑘+1)+𝜆𝑇\n𝑘𝑔(𝜃𝑘+1)−𝜖𝑘\u0011\n.(59)\nBy reordering terms the previous inequality reduces to\n1\n𝐾𝐾−1∑︁\n𝑘=0𝐽𝑅(𝜋𝜃𝑘+1)≥𝐽𝑅(𝜋𝜃∗)+1\n𝐾𝐾−1∑︁\n𝑘=0𝜆𝑇\n𝑘𝑔(𝜃𝑘+1)\n−1\n𝐾𝐾−1∑︁\n𝑘=0𝜖𝑘. (60)\nBy virtue of Lemma 1 in [ 35] and the fact that ||𝑔(𝜃𝑘+1)||2≤\n(𝐵/(1−𝛾)+||d||)2as described in (44), we obtain\n1\n𝐾𝐾−1∑︁\n𝑘=0𝜆𝑇\n𝑘𝑔(𝜃𝑘+1)≥−𝜁(𝐵+(1−𝛾)||d||)2\n2(1−𝛾)2−||𝜆0||2\n2𝜁𝐾. (61)\nCombining (60) and (61) yields\n1\n𝐾𝐾−1∑︁\n𝑘=0𝐽𝑅(𝜋𝜃𝑘+1)≥𝐽𝑅(𝜋𝜃∗)−1\n𝐾𝐾−1∑︁\n𝑘=0𝜖𝑘\n−𝜁(𝐵+(1−𝛾)||d||)2\n2(1−𝛾)2−||𝜆0||2\n2𝜁𝐾.(62)\nThe proof is completed by taking the limit inferior in both sides of\nthe previous inequality.\n6.4 Proof of Theorem 4\nConsider𝑔(·)as defined in (8). By virtue of the non-expansiveness\nof the projection, (7) can be re-written as\n𝜆𝑘+1≥𝜆𝑘+𝜁𝑔(𝜃𝑘+1). (63)Note that𝜆𝑘,𝜆𝑘+1,𝑔(𝜃𝑘+1)are𝑚-dimensional vectors, and the in-\nequality holds for every entry of the vectors, i.e.,\n𝜆(𝑖)\n𝑘+1≥𝜆(𝑖)\n𝑘+𝜁𝑔(𝜃𝑘+1)(𝑖), 𝑖=1,2,···,𝑚. (64)\nUnrolling the previous inequality recursively starting from 𝜆(𝑖)\n𝐾to\n𝜆(𝑖)\n0yields\n𝜆(𝑖)\n𝐾≥𝜆(𝑖)\n0+𝜁𝐾−1∑︁\n𝑘=0𝑔(𝜃𝑘+1)(𝑖), 𝑖=1,2,···,𝑚. (65)\nRearranging the previous inequality and dividing by 𝜁𝐾on both\nsides results in\n1\n𝐾𝐾−1∑︁\n𝑘=0𝑔(𝜃𝑘+1)(𝑖)≤𝜆(𝑖)\n𝐾−𝜆(𝑖)\n0\n𝜁𝐾, 𝑖=1,2,···,𝑚. (66)\nTo proceed, we rely on the following technical lemma.\nLemma 3. Assume that there exists a strictly feasible policy 𝜋˜𝜃\nsuch that for 𝑖=1,2,···,𝑚,∃𝐶𝑖>0so that𝑔(𝜋˜𝜃)𝑖≤−𝐶𝑖and\n𝐽𝑅(𝜋˜𝜃)is bounded. Then, it holds that\nlim sup\n𝐾→∞𝜆(𝑖)\n𝐾/𝐾=0,𝑖=1,2,···,𝑚. (67)\nProof. Let𝐶=[𝐶1,𝐶2,···,𝐶𝑚]𝑇. By definition of the dual function\nin (5),𝑑(𝜆)can be upper bounded as\n𝑑(𝜆)≤−𝐽𝑅(𝜋˜𝜃)+𝜆𝑇𝑔(𝜋˜𝜃)≤−𝐽𝑅(𝜋˜𝜃)−𝜆𝑇𝐶. (68)\nThe previous inequality is equivalent to\n𝜆𝑇𝐶≤−𝐽𝑅(𝜋˜𝜃)−𝑑(𝜆),∀𝜆∈R𝑚. (69)\nSince𝜆(𝑖)≥0,𝐶𝑖>0(𝑖=1,2,···,𝑚), then we obtain\n𝜆(𝑖)𝐶𝑖≤𝜆𝑇𝐶≤−𝐽𝑅(𝜋˜𝜃)−𝑑(𝜆),𝑖=1,2,···,𝑚. (70)\nThe previous inequality indicates that\n𝜆(𝑖)≤−𝐽𝑅(𝜋˜𝜃)−𝑑(𝜆)\n𝐶𝑖,𝑖=1,2,···,𝑚. (71)\nSince𝑑(𝜆∗)i.e.,𝐷∗and𝐽𝑅(𝜋˜𝜃)are bounded, (71)shows that 𝜆∗\nis bounded as well. On the other hand, the definition of the dual\nfunction yields\n𝑑 \n1\n𝐾𝐾−1∑︁\n𝑘=0𝜆(𝑖)\n𝑘!\n≤𝐷∗. (72)\nCombining the previous inequality and (71)further reveals that\n1/𝐾Í𝐾−1\n𝑘=0𝜆(𝑖)\n𝑘is bounded. Taking the limit superior yields\nlim sup\n𝐾→∞1\n𝐾𝐾−1∑︁\n𝑘=0𝜆(𝑖)\n𝑘<∞. (73)\nBy virtue of the Stolz–Cesàro Theorem [36], it holds that\nlim sup\n𝐾→∞𝜆(𝑖)\n𝐾<∞. (74)\nUltimately, dividing by 1/𝐾completes the proof of Lemma 3. □\nSince𝜆(𝑖)\n0,𝜁in(66)are bounded constants, by virtue of Lemma 3\nthe limit superior of the right-hand side of (66)is zero. Then, the\ndefinition of 𝑔(·)completes the proof of Theorem 4.(a) Ball\n (b) Car\n (c) Run\n (d) Circle\nFigure 3: Agents and Tasks from [30]: (a) The Ball agent; (b) The Car agent; (c) The Run task; (d) The Circle task.\nReturn- SafetyBallRun-v0\n Return- SafetyCarRun-v0\n Return- SafetyBallCircle-v0\n Return- SafetyCarCircle-v0\nCost- SafetyBallRun-v0\n Cost- SafetyCarRun-v0\n Cost- SafetyBallCircle-v0\n Cost- SafetyCarCircle-v0\nLR-SafetyBallRun-v0\n LR-SafetyCarRun-v0\n LR-SafetyBallCircle-v0\n LR-SafetyCarCircle-v0\nFigure 4: Learning curves for DDPGL over four environments with five independent runs. In all figures, the horizontal axis is\nthe number of time step. The solid line illustrates the mean and the shaded area depicts the maximum and the minimum. In all\nexperiments, 𝐻1=0.003,𝐻2=4.5forInvLin ,𝐻′\n1=0.045,𝐻′\n2=7.5forInvQua , and cost limit d =10(black dashed line).\n6.5 Additional Experimental Details\n6.5.1 Environment. We consider the tasks Run and Circle using\nthe agents Ball and Car from the Bullet-Safety-Gym [ 30] as shownin Figure 3. The experiments are performed on a workstation with\nan NVIDIA RTX 3070 GPU, 32GB memory, and an Intel Core i7-\n10750H clocked at 2.60 GHz. In the two tasks, both the Ball and\nCar agents utilize states that encompass their position, linear, andTable 6: Summary of Experimental Parameters\nParameters PPOL DDPGL\nCost Limit 10 10\nNumber of Hidden Layers 2 2\nHidden Layer Size 128 128\nStep Per Epoch 10000 10000\nDiscount Factor 0.99 0.97\n𝐾𝑃,𝐾𝐼,𝐾𝐷 (0.05, 0.0005, 0.1) (0.05, 0.0005, 0.1)\nBatch Size 256 256\nGAE Lambda 0.95 N/A\nTarget KL 0.02 N/A\nClip Ratio 0.2 N/A\nSoft Update Ratio N/A 0.05\nExploration Noise N/A 0.1\nangular velocities. The Ball agent’s action is determined by a two-\ndimensional external force, while the Car agent’s action is defined\nby its target velocity and steering angle.\nIn the Run task, the objective is to maintain a constant speed\nwhile keeping its position within the boundaries illustrated in Fig-\nure 3(c). To induce this behavior the reward and the cost with\nrespect to the agent’s current state ( 𝑠𝑡) are defined as:\n𝑟(𝑠𝑡)=\r\r𝒑𝒕−1−𝒈\r\r2−\r\r𝒑𝒕−𝒈\r\r2+𝑟robot(𝑠𝑡),\n𝑐(𝑠𝑡)=1\u0010\n|𝑝𝑡\n𝑦|>𝑦lim\u0011\n+1\u0010\r\r𝒗𝒕\r\r2>𝑣lim\u0011\n,(75)\nwhere𝑟robot(𝑠𝑡)specifies the unique reward for various robots,\n𝒑𝒕=h\n𝑝𝑡𝑥,𝑝𝑡𝑦i\ndefines the position of the agent at time step 𝑡,𝒈=\n\u0002\n𝑔𝑥,𝑔𝑦\u0003represents the position of a fictitious target, 𝑦limdefines\nthe safety region, 𝒗𝒕=h\n𝑣𝑡𝑥,𝑣𝑡𝑦i\ndenotes the agent’s velocity at time\n𝑡, and𝑣limdenotes the speed limit.\nIn the Circle task, the agent earns rewards for moving in a circular\npath but must remain within a confined area that has a smaller\nradius than that of the target circle. The reward and cost functions\nfor this task are delineated as follows:\n𝑟(𝑠𝑡)=−𝑝𝑡𝑦𝑣𝑡𝑥+𝑝𝑡𝑥𝑣𝑡𝑦\n1+|\r\r𝒑𝒕∥2−𝑜\f\f+𝑟robot(𝑠𝑡),\n𝑐(𝑠𝑡)=1\u0000|𝑝𝑡\n𝑥|>𝑥lim\u0001,(76)\nwhere𝑜represents the radius of the circle and 𝑥limdelineates the\nboundaries of the safety region.\n6.5.2 Methodology. To underscore our algorithm’s adaptability\nand independence from specific methods, we employ two state-\nof-the-art SRL methods: PPOL and DDPGL to update the policy\nparameter𝜃. Before proceeding, let us define the advantage func-\ntions for reward and cost as follows\n𝐴𝜋𝜃\n𝑅(𝑠,𝑎)=𝑄𝜋𝜃\n𝑅(𝑠,𝑎)−𝑉𝜋𝜃\n𝑅(𝑠), (77)\n𝐴𝜋𝜃\n𝐶(𝑠,𝑎)=𝑄𝜋𝜃\n𝐶(𝑠,𝑎)−𝑉𝜋𝜃\n𝐶(𝑠), (78)where𝑉𝜋𝜃\n𝑅(𝑠)and𝑄𝜋𝜃\n𝑅(𝑠,𝑎)symbolize the state-value function\nand the action-value function for reward, respectively. 𝑉𝜋𝜃\n𝐶(𝑠)and\n𝑄𝜋𝜃\n𝐶(𝑠,𝑎)play similar roles for cost.\nDenote byℓ𝑝𝑝𝑜andℓ𝑑𝑑𝑝𝑔 the objectives of PPO and DDPG. Ac-\ncording to [37], ℓ𝑝𝑝𝑜is structured as\nℓppo=min \n𝜋𝜃(𝑎|𝑠)\n𝜋old\n𝜃(𝑎|𝑠)𝐴𝜋old\n𝜃\n𝑅(𝑠,𝑎),\nclip \n𝜋𝜃(𝑎|𝑠)\n𝜋old\n𝜃(𝑎|𝑠),1−𝜖,1+𝜖!\n𝐴𝜋old\n𝜃\n𝑅(𝑠,𝑎)!\n,(79)\nwhere𝜋𝜃(𝑎|𝑠)and𝜋old\n𝜃(𝑎|𝑠)characterize the current and old policy,\nrespectively. 𝜖is a hyper-parameter ensuring the new policy does\nnot deviate too much from the old policy. Then, the loss of the\nPPOL is formally defined as\nℓ𝑝𝑝𝑜𝑙=1\n1+𝜆\u0012\nℓ𝑝𝑝𝑜−𝜆𝐴𝜋old\n𝜃\n𝐶(𝑠,𝑎)\u0013\n. (80)\nIn an analogous fashion, the objective of DDPGL is given by\nℓ𝑑𝑑𝑝𝑔𝑙 =1\n1+𝜆\u0012\nℓ𝑑𝑑𝑝𝑔−𝜆𝐴𝜋old\n𝜃\n𝐶(𝑠,𝑎)\u0013\n. (81)\n6.5.3 Results. Figure 4 displays the training curves of return, cost,\nand LR where we apply DDPGL to update policy parameters in\nthe PAPD algorithm. Analogous to our observation from Figure 2,\nInvLin andInvQua surpasses or matches the best performance of all\nconstant-LR cases. The similar trends on performance are observed\nwhen we change the LR. Furthermore, Figure 4 illustrates that\nDDPGL displays a heightened sensitivity to the LR when compared\nto PPOL. Indeed, there is no constant-LR that can effectively operate\nacross all environments, which underscores the significance of\nemploying our PAPD approach. As in the PPOL experiments, we\nmaintain consistent hyper-parameter values across all tests: 𝐻1=\n0.003,𝐻2=4.5forInvLin and𝐻′\n1=0.045,𝐻′\n2=7.5forInvQua .\n6.5.4 Implementation (Codes). All experiments are implemented\nvia the existing framework FSRL [ 34], exactly as is with the default\nparameter settings (see Table 6) and the sole change consisting\nof the learning rates. The FSRL is available at https://github.com/\nliuzuxin/FSRL."    },    {        "title": "2402.00378v1.On_the_Minimum_Depth_of_Circuits_with_Linear_Number_of_Wires_Encoding_Good_Codes.pdf",        "content": "On the Minimum Depth of Circuits with Linear Number of Wires\nEncoding Good Codes\nAndrew Drucker∗Yuan Li†\nAbstract\nLetSd(n) denote the minimum number of wires of a depth- d(unbounded fan-in) circuit\nencoding an error-correcting code C:{0,1}n→ {0,1}32nwith distance at least 4 n. G´ al,\nHansen, Kouck´ y, Pudl´ ak, and Viola [IEEE Trans. Inform. Theory 59(10), 2013] proved that\nSd(n) = Θ d(λd(n)·n) for any fixed d≥3. By improving their construction and analysis, we\nprove Sd(n) =O(λd(n)·n). Letting d=α(n), a version of the inverse Ackermann function, we\nobtain circuits of linear size. This depth α(n) is the minimum possible to within an additive\nconstant 2; we credit the nearly-matching depth lower bound to G´ al et al., since it directly follows\ntheir method (although not explicitly claimed or fully verified in that work), and is obtained by\nmaking some constants explicit in a graph-theoretic lemma of Pudl´ ak [Combinatorica, 14(2),\n1994], extending it to super-constant depths.\nWe also study a subclass of MDS codes C:Fn→Fmcharacterized by the Hamming-distance\nrelation dist( C(x), C(y))≥m−dist(x, y) + 1 for any distinct x, y∈Fn. (For linear codes this\nis equivalent to the generator matrix being totally invertible.) We call these superconcentrator-\ninduced codes , and we show their tight connection with superconcentrators. Specifically, we\nobserve that any linear or nonlinear circuit encoding a superconcentrator-induced code must be\na superconcentrator graph, and any superconcentrator graph can be converted to a linear circuit,\nover a sufficiently large field (exponential in the size of the graph), encoding a superconcentrator-\ninduced code.\nKeywords: error-correcting codes, circuit complexity, superconcentrator.\n1 Introduction\nUnderstanding the computational complexity of encoding error-correcting codes is an important\ntask in theoretical computer science. Complexity measures of interest include time, space, and\nparallelism. Error-correcting codes are indispensable as a tool in computer science. Highly ef-\nficient encoding algorithms (or circuits) are desirable in settings studied by theorists including\nzero-knowledge proofs [Gol+21], circuit lower bounds [CT19], data structures for error-correcting\ncodes [Vio19], pairwise-independent hashing [Ish+08], and secret sharing [DI14]. Besides that, the\nexistence of error-correcting codes with efficient encoding circuits sheds light on the designing of\npractical error-correcting codes.\n∗Independent. Email:andy.drucker@gmail.com.\n†School of Computer Science, Fudan University. Email: yuan li@fudan.edu.cn. An earlier version of this work\nappeared as Chapter 2 of the author’s Ph.D. thesis [Li17]. Part of this work was done while the authors were affiliated\nwith the University of Chicago Computer Science Dept.\n1arXiv:2402.00378v1  [cs.CC]  1 Feb 2024We consider codes with constant rate and constant relative distance, which are called asymptot-\nically good error-correcting codes or good codes for short. The complexity of encoding good codes\nhas been studied before. Bazzi and Mitter [BM05] proved that branching programs with linear\ntime and sublinear space cannot encode good codes. By using the sensitivity bounds [Bop97], one\ncan prove that AC0circuits cannot encode good codes; Lovett and Viola proved that AC0circuits\ncannot sample good codes [LV11]; Beck, Impagliazzo and Lovett [BIL12] strengthened the result.\nDobrushin, Gelfand and Pinsker [DGP73] proved that there exist linear-size circuits encoding\ngood codes. Sipser and Spielman [Spi96; SS96] explicitly constructed good codes that are encod-\nable by bounded fanin circuits of depth O(logn) and size O(n), and decodable by circuits of size\nO(nlogn). For bounded fan-in, the depth O(logn) is obviously optimal. Henceforth, unless other-\nwise stated, we consider circuits with unbounded fan-in, where the size is measured by the number\nofwires instead of gates.\nG´ al, Hansen, Kouck´ y, Pudl´ ak, and Viola [Gal+13] investigated the circuit complexity of en-\ncoding good codes. G´ al et al. constructed circuits recursively and probabilistically, with clever\nrecursive composition ideas, which resemble the construction of superconcentrators in [Dol+83].\nThey also proved size lower bounds for bounded depth, by showing that any circuit encoding good\ncodes must satisfy some superconcentrator-like properties; the lower bound follows from the size\nbounds for a variant of bounded-depth superconcentrators studied by Pudl´ ak [Pud94]. Their con-\nstruction’s wire upper bounds are of form Od(n·λd(n)) (in our notation1) and their lower bounds\nare of form Ω d(n·λd(n)), matching up to a multiplicative constant cdfor constant values d. They\nalso proved that there exist Od(n)-size O(logλd(n))-depth circuits encoding good codes. Here λd(n)\nare slowly growing inverse Ackermann-type functions, e.g., λ2(n) = Θ(log n),λ3(n) = Θ(log log n),\nλ4(n) = Θ(log∗n).\nDruk and Ishai [DI14] proposed a randomized construction of good codes meeting the Gilbert-\nVarshamov bound, which can be encoded by linear-size logarithmic-depth circuits (with bounded\nfan-in). Their construction is based on linear-time computable pairwise independent hash functions\n[Ish+08].\nChen and Tell [CT19] constructed explicit circuits of depth dencoding linear codes with constant\nrelative distance and code rate Ω\u0010\n1\nlogn\u0011\nusing n1+2−Ω(d)wires, for every d≥4. They used these\nexplicit circuits to prove bootstrapping results for threshold circuits.\n1.1 Background and results\nTo encode good error-correcting codes, linear size is obviously required. It is natural to ask, what\nis the minimum depth required to encode goods using a linear number of wires? This question is\naddressed, but not fully answered, by the work of G´ al et al. [Gal+13].\nWe show that one can non-explicitly encode error-correcting codes with constant rate and\nconstant relative distance using O(n) wires, with depth at most α(n), for sufficiently large n. Here,\nα(n) is a version of the inverse Ackermann function. This is nearly optimal, by a lower bound of\nα(n)−2 that we credit to G´ al et al. [Gal+13] as discussed below. Our new upper bound states:\nTheorem 1.1. (Upper bound) Let r∈(0,1) and δ∈(0,1\n2) such that r <1−h(δ). For sufficiently\nlarge nand for any d≥3, not necessarily a constant, there exists a linear circuit C:{0,1}n→\n1Our definition of λd(n) follows Raz and Shpilka [RS03]. It is slightly different from G´ al et al. ’s. In [Gal+13], the\nfunction λi(n) is actually λ2i(n) in our notation.\n2{0,1}⌊n\nr⌋of size Or,δ(λd(n)·n) and depth dthat encodes an error-correcting code with relative\ndistance ≥δ. In particular, when d=α(n), the circuit is of size Or,δ(n).\nOur upper bound improves the construction and the analysis by G´ al et al. [Gal+13]. Specifically,\nletSd(n) denote the minimum size of a depth- dlinear circuit encoding a code C:{0,1}n→ {0,1}32n\nwith distance 4 n. G´ al et al. proved that Sd(n) =Od(λd(n)·n), where the hidden constant in\nOd(λd(n)·n) grows rapidly as an Ackermann-type function (if one follows their arguments and\nexpand the computation straightforwardly in Lemma 26 in [Gal+13].) They also proved that, for\nanyfixed m, when the depth d=O(log(λm(n))),Sd(n) =Om(n). Their upper bound is strong, but\nsuboptimal. Our main technical contribution is to prove Sd(n) =O(λd(n)·n), where the hidden\nconstant is an absolute constant. Explaining how this improvement is obtained is a bit technical,\nand defeered to the next subsection.\nTurning to the lower bounds: we credit the lower bound in the result below to G´ al et al.\n(although it was not explicitly claimed or fully verified in that work), since it is directly obtainable\nby their size lower-bound method and the tool of Pudl´ ak [Pud94] on which it relies, when that tool\nis straightforwardly extended to super-constant depth.2\nTheorem 1.2. (Lower bound) [Gal+13] Let ρ∈(0,1) and δ∈(0,1\n2), and let constant c >0. Let\nCn:{0,1}n→ {0,1}⌊n/ρ⌋be a family of circuits of size at most cnthat encode error-correcting\ncodes with relative distance ≥δ. Arbitrary Boolean-function gates of unrestricted fanin are allowed\ninCn. Ifnis sufficiently large, i.e., n≥N(r, δ, c ), the depth of the circuit Cnis at least α(n)−2.\nThe proof for Theorem 1.2 closely follows [Gal+13] and is an application of a graph-theoretic\nargument in the spirit of [Val77; Dol+83; Pud94; RS03]. In detail, we use Pudl´ ak’s size lower bounds\n[Pud94] on “densely regular” graphs, and rely on the connection between good codes and densely\nregular graphs by G´ al et al. [Gal+13]. Pudl´ ak’s bound was originally proved for bounded depth;\nto apply it to unbounded depth, we explicitly determine the hidden constants by directly following\nPudl´ ak’s work, and verify that their decay at higher unbounded depths is moderate enough to allow\nthe lower-bound method to give superlinear bounds up to depth α(n)−3. Even after our work,\nthe precise asymptotic complexity of encoding good codes remains an open question for din the\nrange [ ω(1),α(n) - 3].\nStepping back to a higher-level view, the strategy of the graph-theoretic lower-bound arguments\nin the cited and related works is as follows:\n•Prove any circuit computing the target function must satisfy some superconcentrator-like\nconnection properties;\n•Prove any graph satisfying the above connection properties must have many edges;\n•Therefore, the circuit must have many wires.\nValiant [Val75; Val76; Val77] first articulated this kind of argument, and proposed the definition\nof superconcentrators. Somewhat surprisingly, Valiant showed that linear-size superconcentrators\nexist. As a result, one cannot prove superlinear size bounds using this argument (when the depth is\n2The Ω( ·) notation in the circuit size lower bound of G´ al et al. , for example, Theorem 1 in [Gal+13], involves\nan implicit constant which decays with the depth d, as can be suitable for constant depths; similarly for the tool\nof Pudl´ ak, Theorem 3.(ii) in [Pud94], on which it relies. For general super-constant depths, more explicit work is\nrequired to verify the decay is not too rapid.\n3unbounded). Dolev, Dwork, Pippenger, and Wigderson [Dol+83] proved Ω( λd(n)·n) lower bounds\nfor bounded-depth (weak) superconcentrators, which implies circuit lower bounds for functions sat-\nisfying weak-superconcentrator properties. Pudl´ ak [Pud94] generalized Dolev et al. ’s lower bounds\nby proposing the definition of densely regular graphs , and proved lower bounds for bounded-depth\ndensely regular graphs, which implies circuit lower bounds for functions satisfying densely regular\nproperty, including shifters, parity shifters, and Vandermonde matrices. Raz and Shpilka [RS03]\nstrengthened the aforementioned superconcentrator lower bounds by proving a powerful graph-\ntheoretic lemma, and applied it to prove superlinear lower bounds for matrix multiplication. (This\npowerful lemma can reprove all the above lower bounds.) G´ al et al. [Gal+13] proved that any\ncircuits encoding good error-correcting codes must be densely regular. They combined this with\nPudl´ ak’s lower bound on densely regular graphs [Pud94] to obtain Ω( λd(n)·n) size bounds for\ndepth- dcircuits encoding good codes.\nAll the circuit lower bounds mentioned above apply even to the powerful model of arbitrary-gate\ncircuits, that is,\n•each gate has unbounded fanin,\n•a gate with fanin scan compute any function from {0,1}sto{0,1},\n•circuit size is measured as the number of wires.\nIn this “arbitrary-gates” model, any function from {0,1}nto{0,1}mcan be computed by a circuit\nof size mn.\nIt is known that any circuits encoding good codes must satisfy some superconcentrator-like\nconnection properties [Spi96], [Gal+13]. Our other result is a theorem in the reverse direction in the\nalgebraic setting over large finite fields. Motivated by this connection, we study superconcentrator-\ninduced codes (Definition 5.1), a subclass of maximum distance separable (MDS) codes [LX04;\nGRS12], and observe its tight connection with superconcentrators.\nTheorem 1.3. (Informal) Given any ( n, m)-superconcentrator, one can convert it to a linear\narithmetic circuit encoding a code C:Fn→Fmsuch that\ndist(C(x), C(y))≥m−dist(x, y) + 1 ∀x̸=y∈Fn(1)\nby replacing each vertex with an addition gate and assigning the coefficient for each edge uniformly\nat random over a sufficiently large finite field (where d2Ω(n+m)suffices, and dis the depth of the\nsuperconcentrator).\nWe also observe that any arithmetic circuit, linear or nonlinear, encoding a code C:Fn→Fm\nsatisfying (1), viewed as a graph, must be a superconcentrator.\nThe proof of Theorem 1.3 relates the connectivity properties with the rank of a matrix, and\nuses the Schwartz-Zippel lemma to estimate the rank of certain submatrices; these techniques are\nwidely used, for example, in [CKL13; Lov18]. In addition, the idea of assigning uniform random\ncoefficients (in a finite field) to edges, to form linear circuits, has appeared before in e.g. network\ncoding [Ahl+00; LYC03]. The question we study is akin to a higher-depth version of the GM-MDS\ntype questions about matrices with restricted support [Lov18].\nObserve that any code satisfying the distance inequality (1) is a good code. The existence of\ndepth- dsize-O(λd(n)·n) superconcentrators [Dol+83; AP94], for any d≥3, immediately implies\n4the existence of depth- d(linear) arithmetic circuits of size O(λd(n)·n) encoding good codes over\nlarge finite field .\nIn a subsequent work [Li23], inspired by this connection and using similar techniques, the second\nauthor proved that any ( n, m)-superconcentrator can compute the shares of an ( n, m) linear thresh-\nold secret sharing scheme. In other words, any ( n, m)-superconcentrator-induced code induces an\n(n, m) linear threshold secret sharing scheme. Results in [Li23] can be viewed as an application of\nsuperconcentrator-induced codes.\n1.2 Circuit construction techniques\nIn terms of techniques, G´ al et al. proposed the notion of range detectors (Definition 3.6). They use\nprobabilistic methods to prove certain linear-size depth-1 range detectors exist and then compose\nthe range detectors recursively to construct circuits encoding good codes. We improve the recursive\nconstruction by tuning the parameters and changing the way range detectors are composed (in the\ninductive step), which leads to an absolute constant in the size bound O(λd(n)·n), eliminating\nthe dependency on d. By utilizing a property of the inverse Ackermann function, we immediately\nobtain circuits of depth α(n) and size O(n), avoiding a size-reduction argument.\nLet us give an overview of the construction, highlighting the difference between G´ al et al. ’s.\nFirst, we distill some notions to facilitate the construction, each of which is either explicit or\npresent in essence in [Gal+13].\n•Partial good code (i.e., Definition 3.1), or PGC for short: when the weight of an input is\nwithin a certain range, a PGC outputs a codeword whose relative weight is above an absolute\nconstant.\n•Condenser: a condenser function reduces the number of inputs while preserving the minimum\nabsolute weight. Using probabilistic arguments, G´ al et al. proved that certain unbalanced\nunique-neighbor expanders exist, and converted the expanders to linear-size condensers, that\nis,\u0000\nn,⌊n\nr⌋, s,n\nr1.5, s\u0001\n-range detectors. We reuse their condensers.\n•Amplifier: these amplify the number of inputs while preserving the relative distance . G´ al et\nal.uses a probabilistic argument and applies the Chernoff bound to show linear-size amplifiers\nexist (e.g., in the proof of Theorem 22 and Lemma 26). We choose to decouple the argument\nby using the disperser graphs given by [RT00], choosing the coefficients at random, and then\napplying a union bound.\n•Rate booster: these do not change the number of inputs, but can boost the rate close to\nthe Gilbert-Varshamov bound. Lemma 17 in [Gal+13] embodies a rate booster, although a\nnot fully optimized one; G´ al et al. also state3that a tighter analysis would approach the\nGilbert-Varshamov bound. Using a probabilistic argument similar to that with the amplifiers\nand to the argument in [Gal+13], we provide a verification of this claim.\n•Composition Lemma: this tool allows us to combine hpartial good codes into a larger one,\nat the cost of increasing the depth by 1, and the size by O(n). Furthermore, if each partial\ngood code has bounded output fanin O(1), then one can collapse the last layer at the cost\n3In the conference version (COCOON 2023), we mistakenly stated that such rate-boosting is not discussed by G´ al\net al. (based on non-journal versions of the paper), but in fact it is discussed in Section 4 of the journal version.\n5of increasing the size by O(hn). The composition idea was implicit in G´ al et al. ’s (e.g., in\nTheorem 22 and Lemma 26 in [Gal+13]).\nWith the these concepts, the task is to construct the circuits using these building blocks. Our\nanalysis framework is also inspired by [Dol+83]; we show the minimal size of a depth- d(n, r, s )-\nPGC satisfies a similar recursion as that of partial superconcentrators . For the base case when the\ndepth is small (i.e., d≤4), we completely reuse G´ al et al. ’s construction. In the key inductive step,\nG´ alet al. construct a depth-( d+ 2) circuits encoding an\u0000\nn,n\nr, n\u0001\n-PGC by composing a number of,\nsayh, PGCs. Assuming each smaller PGC is of size O(n), the total size would be O(hn). They let\n(\nk1 = min {r,⌈m3/4⌉}\nki+1=λd(ki)3,(2)\nwhere his the least integer such that khis below a particular universal constant. Each inner part is\nan\u0010j\nn\nλd(ki)2k\n,n\nki,n\nλd(ki)2\u0011\n-PGC of depth dand size Od(n), topped by a condenser, appended by an\nampilifier. Then they argue that h= 2λd+2(n) +Od(1), and thus the total size of the depth-( d+ 2)\ncircuit becomes Od+2(λd+2(n)·n). They did not mention the constant in Od(1); if one works out\nthe constant straightforwardly, it is a very fast-growing constant depending on d.\nOur remedy is to choose the parameters and compose the PGCs differently in the inductive\nconstruction. To obtain a depth-( d+ 2) circuit encoding an\u0000\nn,n\nr, n\u0001\n-PGC, we compose at most\nλd+2(n) PGCs that are of the parameter\u0012\nn,n\nA(i)\nk−1(c0),n\nA(i−1)\nk−1(c0)\u0013\n, and of size O(n), where 1 ≤i≤\nλd+2(n). In addition, we ensure that each PGC has bounded output fanin O(1), which allows us to\ncollapse the last layer in the composition lemma without blowing up the size. Let r=A(i−1)\nk−1(c0). To\nconstruct a depth-( d+ 2)\u0010\nn,n\nAk−1(r),n\nr\u0011\n-PGC, we cannot reduce it to a depth- dPGC directly, due\nto the parameter restriction in the condenser. (Recall that the linear-size condenser constructed\nis an\u0000\nn,⌊n\nr⌋, s,n\nr1.5, s\u0001\n-range detector, not an\u0000\nn,⌊n\nr⌋, s,n\nr, s\u0001\n-range detector.) We work around\nthis by composing two PGCs: an\u0010\nn,n\nA(k−1,r),n\n4r2\u0011\n-PGC, and an\u0000\nn,n\n4r2,n\nr\u0001\n-PGC, where each is\nof linear size, and have bounded output fanin O(1). The former can be constructed by a linear-\nsize condenser, an\u0010\u0004n\n2r\u0005\n,n\nA(k−1,r),n\n2r\u0011\n-PGC of depth d, and an amplifier; the latter we construct\ndirectly by a simple argument. This composition is inspired by the superconcentrator construction\nby Dolev et al. [Dol+83], who constructed superconcentrators of depth dand size O(λd(n)·n).\nIndeed, the encoding circuits have to resemble the superconcentrators, since it was known that the\ncircuits must be a (weak) superconcentrators.\nThe above sketch leads to a construction of depth- dsize-O(λd(n)·n) circuits encoding good\ncodes, where dis not necessarily a constant; by letting d=α(n), one immediately gets depth-\nα(n) size- O(n) circuits. This improves the analysis, getting rid of an additive O(logα(n)) factor\nin the depth (compared with Corollary 32 in [Gal+13], or Corollary 1.1 in [Dol+83]). But a tight\nanalysis to prove this requires one small new ingredient compared to prior works. Specifically, we\nobserve that, if λd(n)≤d, then λd+2(n) =O(1) (See Proposition A.2 (ii) for details.) The above\nobservation can also improve the depth bound on the linear-size superconcentrators in [Dol+83]\nfrom α(n) +O(logα(n)) to α(n) (without change in the construction). In contrast, G´ al et al.\ndeployed a different strategy to achieve linear size. Having constructed an encoding circuit of depth\nd=O(1) and size Od(λd(n)·n), they then apply a size-reduction transformation O(logλd(n)) times,\n6a transformation which reduces a size- minstance to a size m/2 instance with additional cost O(m)\nwires. In this way they obtain a circuit of depth 2 d+O(logλd(n)) and size Od(n).\nWe verify our codes can obtain any constant rate and constant relative distance within the\nGilbert-Varshamov bound, by using a disperser graph at the bottom layer and collapsing that layer\nafterward (for linear circuits). A similar rate-boosting by probabilistic construction is described\nin [Gal+13]; see their (quantitatively somewhat weaker) Lemma 17 and Corollary 18 and the\ncomments following, stating an improved analysis is possible.\n2 Inverse Ackermann functions\nDefinition 2.1. (Definition 2.3 in [RS03]) For a function f, define f(i)to be the composition of f\nwith itself itimes. For a function f:N→Nsuch that f(n)< nfor all n >0, define\nf∗(n) = min {i:f(i)(n)≤1}.\nLet\nλ1(n) = ⌊√n⌋,\nλ2(n) = ⌈logn⌉,\nλd(n) = λ∗\nd−2(n).\nAsdgets larger, λd(n) becomes extremely slowly growing, for example, λ3(n) = Θ(log log n),\nλ4(n) = Θ(log∗n),λ5(n) = Θ(log∗n), etc.\nWe define a version of the inverse Ackermann function as follows.\nDefinition 2.2 (Inverse Ackermann Function) .For any positive integer n, let\nα(n) = min {even d:λd(n)≤6}.\nThere are different variants of the inverse Ackermann function; they differ by at most a multi-\nplicative constant factor.\nWe need the definition of the Ackermann function. We put some relevant properties in Appendix\nA.\nDefinition 2.3. (Ackermann function [Tar75; Dol+83]) Define\n\n\nA(0, j) = 2 j, forj≥1\nA(i,1) = 2 , fori≥1\nA(i, j) =A(i−1, A(i, j−1)),fori≥1, j≥2.(3)\nThe Ackermann function grows rapidly. From the definition, one can easily verify that A(0, i) =\n2i,A(1, i) = 2iandA(2, i) = 22...2\n|{z}\nicopies of 2.For notational convenience, we often write A(i, j) as\nAi(j).\n73 Upper bound\nIn this section, we prove Theorem 1.1. That is, we non-explicitly construct, for any rate r∈(0,1)\nand relative distance δ∈(0,1\n2) satisfying r <1−h(δ), circuits encoding error-correcting codes\nC:{0,1}n→ {0,1}⌊n/r⌋with relative distance δ, where the circuit is of size Or,δ(λd(n)·n) and\ndepth d. In particular, when the depth d=α(n), the circuit is of size Or,δ(n).\nFirst, we construct circuits encoding codes C:{0,1}n→ {0,1}32nwith relative distance1\n8,\nwhere the constants 32 and1\n8are picked for convenience. Then, we verify that one can boost the\nrate and the distance to achieve the Gilbert-Varshamov bound (without increasing the depth of\nthe circuit).\nNote that random linear codes achieve the Gilbert-Varshamov bound. However, circuits encod-\ning random linear codes have size O(n2). In contrast, our circuits have size O(n). Our circuits\nconsist of XOR gates only; we call these linear circuits hereafter. We point out that the construc-\ntion generalizes to any finite field, where XOR gates are replaced by addition gates (over that finite\nfield).\nRecall that let Sd(n) denote the minimum size of a depth- dlinear circuit encoding a code\nC:{0,1}n→ {0,1}32nwith distance 4 n.\nDefinition 3.1. (Partial good code) C:{0,1}n→ {0,1}32nis called ( n, r, s )-partial good code , or\n(n, r, s )-PGC for short, if for all x∈ {0,1}nwith wt( x)∈[r, s], we have wt( C(x))≥4n.\nDenote by Sd(n, r, s ) the minimum size of a linear circuit of depth- dthat encodes an ( n, r, s )-\nPGC, where r, sare real numbers.\nDefinition 3.2. [Sip88; CW89] Bipartite graph G= (V1= [n], V2= [m], E) is a ( k, ϵ)-disperser\ngraph if for any X⊆V1with|X| ≥k,|Γ(X)| ≥(1−ϵ)m.\nWe rely on the following probabilistic construction of dispersers.\nTheorem 3.3. (Theorem 1.10 in [RT00]) For every n≥k >1,m≥1, and ϵ >0, there exists a\n(k, ϵ)-disperser graph G= (V1= [n], V2= [m], E) with left degree\nD=\u00181\nϵ\u0010\nln\u0010n\nk\u0011\n+ 1\u0011\n+m\nk\u0012\nln\u00121\nϵ\u0013\n+ 1\u0013\u0019\n.\nAprobabilistic (linear) circuit is a linear circuit with random coefficients. That is, each XOR\ngate computes c1x1+c2x2+. . .+cmxm, where x1, x2, . . . , x mare incoming wires, coefficients\nc1, c2, . . . , c n∈ {0,1}are chosen independently and uniformly at random. A probabilistic circuit\ncan be viewed as a distribution of linear circuits. Our work in what follows relies on the following\nsimple observation: if one of the incoming wires is nonzero, the output is uniformly distributed in\n{0,1}.\nLemma 3.4. (Rate booster) Let δ∈(0,32), and let c >1,γ∈(0,1\n2) be such that h(γ)<1−1\nc.\nFor sufficiently large n≥N(c, γ), there exists a probabilistic circuit C:{0,1}32n→ {0,1}⌊cn⌋of\ndepth 1 and size Oδ,c,γ(n) such that\nPr\nC[wt(C(x))≥γ· ⌊cn⌋]>1−2−n\nfor any x∈ {0,1}32nwith wt( x)≥δn. Moreover, the output gates have bounded fanin Oδ,c,γ(1).\n8Figure 1: Gate in a probabilistic circuit\nProof. Letc′=⌊cn⌋\nn. Let nbe sufficiently large such that h(γ)<1−1\nc′.\nBy Theorem 3.3, there exists a ( δn, ϵ)-disperser graph G′= (V1= [32 n], V′\n2= [(c′+ 1)n], E),\nwhere ϵ >0 is small constant to be determined. After removing nvertices in V′\n2= [(c′+ 1)n] with\nthe largest degree, we are left with a bipartite graph G= (V1= [32 n], V2= [c′n], E) such that\n|Γ(X)| ≥c′n−ϵ(c′+ 1)n=c′n\u0010\n1−ϵ−ϵ\nc′\u0011\n(4)\nfor all X⊆V1of size at least δn. Moreover, the degree of the vertices in V2is bounded by 32 D,\nwhere D=l\n1\nϵ(ln(32\nδ) + 1) +c′+1\nδ(ln(1\nϵ) + 1)m\n=Oδ,c,ϵ(1).\nWe transform bipartite graph Ginto a probabilistic circuit by identifying vertices in V1with 32 n\ninputs, denoted by x1, x2, . . . , x 32n, and replacing each vertex in V2by an XOR gate with random\ncoefficients. That is, if vertex j∈V2is connected to i1, i2, . . . , i k∈V1, then vertex jcomputes an\noutput\nyj=cj,i1xi1+cj,i2xi2+. . .+cj,ikxik(mod 2) ,\nwhere cj,i1, cj,i2, . . . , c j,ik∈ {0,1}are chosen independently and uniformly at random.\nFor any x∈ {0,1}32nwith wt( x)≥δn, letX= supp( x). By the expansion property (4), we\nhave|Γ(X)| ≥c′n(1−ϵ′), where ϵ′= (1 +1\nc′)ϵ. For every j∈Γ(X), vertex jis incident to at least\none vertex in X; pick an arbitrary neighbor, denoted by i∈X, and leave coefficient ci,junfixed.\nIn other words, we fix all the coefficients except those |Γ(X)|coefficients (that are incident to X).\nObserve that C(x), restricted to Γ( X), is uniformly distributed in {0,1}|Γ(X)|. Thus,\nPr\nC[wt(C(x))< γc′n]≤\u0000⌈c′n(1−ϵ′)⌉\n0\u0001\n+\u0000⌈c′n(1−ϵ′)⌉\n1\u0001\n+. . .+\u0000⌈c′n(1−ϵ′)⌉\n⌊γc′n⌋\u0001\n2⌈c′n(1−ϵ′)⌉\n≤2−c′n(1−ϵ′)(1−h(γ\n1−ϵ′)). (5)\nRecall that ϵ′= (1 +1\nc′)ϵ. Let ϵ >0 be sufficiently small such that c′(1−ϵ′)(1−h(γ\n1−ϵ′))>1,\nthe probability (5) would be less than 2−n. (Such ϵ′>0 clearly exists, since h(γ)<1−1\nc′⇔\nc′(1−h(γ))>1).\nNext, a “composition lemma” allows us to combine several partial good codes into a larger one.\nThe lemma is already implicit in [Gal+13].\n9Lemma 3.5. (Composition Lemma) For any n, d≥1,ℓ≥2, and any 1 ≤r1≤r2≤. . . r ℓ+1≤n,\nassume there exist linear circuits of size siand depth dencoding some ( n, ri, ri+1)-PGC, i=\n1,2, . . . , ℓ . We have\nSd+1(n, r1, rℓ+1)≤ℓX\ni=1si+O(n). (6)\nMoreover, if the output gates of the linear circuits encoding ( n, ri, ri+1)-PGCs have bounded\nfan-in D, for all i= 1,2, . . . , ℓ , then we have\nSd(n, r1, rℓ+1)≤ℓX\ni=1si+O(Dℓn). (7)\nIn addition, the fanin of the output gates (of the depth- dcircuit encoding the ( n, r1, rℓ+1)-PGC) is\nbounded by O(ℓD).\nProof. Denote the linear circuit encoding an ( n, ri, ri+1)-PGC by Ci,i= 1,2, . . . , ℓ . Recall that\neach Ci:{0,1}n→ {0,1}32nhas 32 noutputs, denoted by w(i)\n1, w(i)\n2, . . . , w(i)\n32n.\nCreate 32 ngates, denoted by y1, y2, . . . , y 32n. Let\nyj=ℓX\ni=1c(i)\njw(i)\nj(mod 2) ,\nwhere coefficients c(i)\nj∈ {0,1}are chosen independently and uniformly at random. See Figure 2 for\nillustration.\nFigure 2: Partial good codes composition\nFigure 2 is a bit misleading. Instead of creating 32 nnew gates y1, y2, . . . , y 32n, we merge\nw(1)\nj, w(2)\nj, . . . , w(ℓ)\nj, for each j= 1,2, . . . , 32n. Fixing j, independently for each i∈ {1,2, . . . , ℓ},\nwith probability1\n2, merge w(i)\njwith yj, and with probability1\n2, do nothing. As such, the depth of\nthe circuit is still d, and the size is at mostPℓ\ni=1si.\nLet us analyze the weight of y= (y1, y2, . . . , y 32n)∈ {0,1}32n. For any x∈ {0,1}nwith\nwt(x)∈[r1, rℓ+1], there exists ksuch that wt( x)∈[rk, rk+1]. By the definition of PGC, Ck(x) has\nweight at least 4 n. Fix all coefficients c(i)\njexcept for those i=kandj∈supp( Ck(x)), one can\n10easily see that {yj:j∈supp( Ck(x))}is uniformly distributed in {0,1}|supp( Ck(x))|. Thus, for any\nfixed x,\nPrh\nwt(y)<n\n2i\n≤\u00004n\n0\u0001\n+\u00004n\n1\u0001\n+. . .+\u00004n\n⌊n\n2⌋\u0001\n24n\n≤2−4n(1−h(1\n8))<2−1.8n.\nTaking a union bound over x, we know there exists an assignment of coefficients such that the\nweight of the output is always ≥n\n2.\nFinally, we boost the distance from1\n2nto 4nusing a rate booster in Lemma 3.4. Specifically,\napply Lemma 3.4 with δ=1\n2, c= 32, γ=1\n8. Adding the rate booster increases the depth by 1, and\nincreases the size by O(n).\nFor the “moreover” part, observe that the outputs in the rate booster have bounded fan-in\nO(1). As in the use of rate boosting in [Gal+13], we collapse the last layer, where, originally, the\nlast layer has bounded fan-in O(1), and the second last layer has fan-in ≤ℓD. As such, O(n) wires\nin the rate booster will extend to O(Dℓn) wires, which proves (7). Furthermore, the output gates\nhave fanin O(ℓD).\nDefinition 3.6. [Gal+13] An ( m, n, ℓ, k, r, s )-range detector is a linear circuit that has minputs, n\noutputs, and on any input of Hamming weight between ℓandk, it outputs a string with Hamming\nweight between rands. We omit the last parameter if s=n.\nLemma 3.7. (Condenser, Lemma 23 in [Gal+13]) There exists a constant c0≥6 such that\nfor any c0≤r≤nand 1 ≤s≤n\nr1.5, where nis an integer, r, sare reals, there exists an\u0000\nn,⌊n\nr⌋, s,n\nr1.5, s\u0001\n-range detector of depth 1 and size 6 n.\nThe following lemma says there exist linear size “amplifiers” that amplify the number of outputs\nwhile preserving the relative distance.\nLemma 3.8 (Amplifier) .For any positive integers n, m≥3n, there exists an ( n, m,n\n8, n,m\n8)-range\ndetector computable by a depth-1 linear circuit of size O(m). Moreover, the fanin of the outputs\nis bounded by an absolute constant.\nProof. By Theorem 3.3, there exists a bipartite graph G= (V1= [n], V2= [m], E) with O(m) edges\nsuch that for every X⊆V1of size at leastn\n8,|Γ(X)| ≥9\n10·m. In addition, the vertices in V2have\nbounded degrees. (For example, applying Theorem 3.3 with n=m,m= 2m,k=n\n8, and ϵ=1\n20,\nwe have a disperser with left degree O(m\nn) and right average degree O(1). After removing mright\nvertices with largest degree, we have the desired graph.)\nConvert graph Ginto a probabilistic circuit C:{0,1}n→ {0,1}mas follows:\n•View the vertices in V1as inputs, denoted by xu, for each u∈V1.\n•Replace the vertices in V2by XOR gates. View them as outputs, denoted by yv, for each\nv∈V2.\n•For each edge ( u, v)∈E, choose a coefficient cu,v∈ {0,1}independently at random. Let the\noutput gate vcompute the function\nyv=X\n(u,v)∈Ecu,vxu(mod 2) .\n11For any x∈ {0,1}nwith wt( x)≥n\n8, by the expansion property of graph G,|Γ(supp( x))| ≥9\n10·m.\nIn other words, there exist at least ⌈9\n10·m⌉outputs, which are incident to at least one nonzero\ninput. So, C(x), restricted to Γ(supp( x)), is uniformly distributed in {0,1}|Γ(supp( x))|. Thus,\nPr[wt( C(x))<m\n8]≤\u0000⌈9\n10·m⌉\n0\u0001\n+\u0000⌈9\n10·m⌉\n1\u0001\n+. . .+\u0000⌈9\n10·m⌉\n⌊m\n8⌋\u0001\n2⌈9\n10·m⌉\n≤2−(1−h(5\n36))·9\n10m\n<2−0.37m<2−n.\nFinally, taking a union bound over all x, we conclude there exists a linear circuit that computes an\n(n, m,n\n8, n,m\n8)-range detector.\nThe following lemma says, by putting a condenser at the top and an amplifier at the bottom,\none can reduce the size of the problem (of encoding partial good codes).\nLemma 3.9. (Reduction Lemma) Let c0be the constant in Lemma 3.7. For any r∈[c0, n] and\n1≤s≤t≤n\nr1.5,\nSd(n, s, t )≤Sd−2\u0010jn\nrk\n, s,n\nr\u0011\n+O(n). (8)\nIn addition, the output gates encoding ( n, s, t )-PGC have bounded fanin O(1).\nProof. Construct a depth- dlinear circuit that encodes an ( n, s, t )-PGC by concatenating three\ncircuits:\n•(Condenser) The top is an ( n,⌊n\nr⌋, s, t, s )-range detector of size O(n) and depth 1, whose\nexistence is proved by Lemma 3.7.\n•(Partial good code) The middle is an ( ⌊n\nr⌋, s,⌊n\nr⌋)-PGC of depth d−2.\n•(Amplifier) The bottom is an (32 ⌊n\nr⌋,32n,4⌊n\nr⌋,32⌊n\nr⌋,4n)-range detector of depth 1 and size\nO(n), whose existence is proved by Lemma 3.8. Moreover, the output gates have bounded\nfanin O(1).\nThe output gates of the overall circuit have bounded fanin O(1), because the output gates of the\namplifier have bounded fanin O(1).\nThe following bounds are tight for fixed depth d. To avoid redoing the same work, we will use\nit for the case d= 4 at the base step in our main construction. Our technical contribution is to\nremove the dependence on d, i.e., we prove Sd=O(λd(n)·n) for any d, not necessarily fixed, where\nthe hidden constant is an absolute constant.\nLemma 3.10. (Lemma 28 in [Gal+13]) Fix an even constant d≥4. For any 1 ≤r≤n,\nSd\u0010\nn,n\nr, n\u0011\n=Od(λd(r)·n),\nwhere the constant in Od(λd(r)·n) only depends on d.\nLemma 3.11. (Lemma 27 in [Gal+13]) For any 1 ≤r≤n,\nS2\u0010\nn,n\nr, n\u0011\n=O(nlog2r).\n12We will need the following handy result in our recursive construction.\nLemma 3.12. For any 1 ≤r≤n,\nS4\u0010\nn,n\n4r2,n\nr\u0011\n=O(n),\nwhere the constant in O(n) is an absolute constant. Moreover, the output gates of the linear circuits\nhave bounded fanin O(1), which is an absolute constant.\nProof. We apply Lemma 3.9 with r=⌊√r⌋,s=n\n4r2,t=n\nr. Thus,\nS4\u0010\nn,n\n4r2,n\nr\u0011\n≤S2\u0012\u0016n\n⌊√r⌋\u0017\n,n\n4r2,n\n⌊√r⌋\u0013\n+O(n)\n≤O\u0012n\n⌊√r⌋log2(4r2)\u0013\n+O(n) By Lemma 3.11\n=O(n).\nThe output gates have bounded fanin O(1) by Lemma 3.9.\nThe following theorem is our main construction, which is proved by an induction on the depth.\nTheorem 3.13. Letc0be the constant in Lemma 3.7. There exist absolute constants c, D > 0\nsuch that the following holds.\nFor any c0≤r≤n, and for any k≥3, we have\nS2k\u0012\nn,n\nA(k−1, r),n\nr\u0013\n≤2cn. (9)\nMoreover, the output gates of the linear circuits encoding\u0010\nn,n\nA(k−1,r),n\nr\u0011\n-PGC have bounded fanin\nD.\nFor any 2 ≤r≤nand for any k≥2,\nS2k\u0010\nn,n\nr, n\u0011\n≤3cλ2k(r)·n. (10)\n(But the circuit encoding\u0000\nn,n\nr, n\u0001\n-PGC does not necessarily have bounded output fanin.)\nProof. (of Theorem 3.13) The overall proof is an induction on k, as illustrated by Figure 3.\nWe pick constants D, csuch that\nD≥2c2max( D1, D2)\nand\nc≥max\u0000\nc1D+ (log c0)2c3,2(2c1D1+c4+c5), c6\u0001\n.\nRelevant constants used in the proof are summarized in Table 1 on page 15.\nThe base step is (10) for k= 2, which is true by Lemma 3.10.\nFor the induction step, we shall prove\n•(9) implies (10) for the same k, and\n13Figure 3: Inductive proof of Theorem 3.13\n•(10) with k−1 implies (9) with k.\nFirst, let us prove (9) implies (10) for the same k, where k≥3. By the composition lemma\n(i.e., Lemma 3.5),\nS2k\u0010\nn,n\nr, n\u0011\n≤S2k\u0012\nn,n\nc0, n\u0013\n+hX\ni=2S2k \nn,n\nA(i)\nk−1(c0),n\nA(i−1)\nk−1(c0)!\n+c1Dhn ,\nwhere his the minimum integer such that A(h)\nk−1(c0)≥r(and hence A(h−1)\nk−1(c0)< r). Note that\nA(h)\nk−1(c0)≥A(h)\nk−1(1) = A(k, h) and A(h−1)\nk−1(c0)≥A(k, h−1). Since A(k, h−1)≤A(h−1)\nk−1(c0)< r, by\nProposition A.3, we have h≤λ2k(r). By Lemma 3.11, S2k\u0010\nn,n\nc0, n\u0011\n≤c3n(logc0)2, and the circuit\nencoding\u0010\nn,n\nc0, n\u0011\n-PGC has output fanin 1, which can be done by adding an extra layer to the\ndepth-2 circuit given by Lemma 3.11. By induction hypothesis, S2k\u0012\nn,n\nA(i)\nk−1(c0),n\nA(i−1)\nk−1(c0)\u0013\n≤2cn,\nand the fanin of the outputs in the circut encoding\u0012\nn,n\nA(i)\nk−1(c0),n\nA(i−1)\nk−1(c0)\u0013\n-PGC is at most D.\nThus\nS2k\u0010\nn,n\nr, n\u0011\n≤c3n(logc0)2+ (λ2k(r)−1)·2cn+c1Dλ2k(r)n\n≤2cλ2k(r)n+λ2k(r)n(c1D+ (log c2)2c3)\n≤3cλ2k(r)·n,\n14c0 the constant in Lemma 3.7 and 3.9\nc1the constant in (7) in Lemma 3.5, i.e.,\nSd(n, r1, rℓ+1)≤Pℓ\ni=1si+c2Dℓn.\nc2Lemma 3.5, the fanin of the output gates (of the depth- dcircuit\nencoding the ( n, r1, rℓ+1)-PGC) is bounded by c2ℓD\nc3 Lemma 3.11, S2\u0000\nn,n\nr, n\u0001\n≤c3nlog2r\nc4the constant in (8) in Lemma 3.9, i.e.,\nSd(n, s, t )≤Sd−2\u0000\u0004n\nr\u0005\n, s,n\nr\u0001\n+c4n\nc5the constant in Lemma 3.12, i.e.,\nS4\u0000\nn,n\n4r2,n\nr\u0001\n≤c5n\nc6when d= 4, the constant in Od(λd(r)·n) in Lemma 3.10, i.e.,\nS4\u0000\nn,n\nr, n\u0001\n≤c6λ4(r)·n\nD1 Lemma 3.9, output gates have fanin at most D1\nD2 Lemma 3.12, output gates have fanin at most D2\nTable 1: Absolute constants used in the proof.\nsince c≥c1D+ (log c2)2c3.\nNow, we prove (10) with k−1 implies (9) with k. That is, assuming S2(k−1)\u0000\nn,n\nr, n\u0001\n≤\n3cλ2(k−1)(r)·nfor any 2 ≤r≤n, we will prove S2k\u0010\nn,n\nA(k−1,r),n\nr\u0011\n≤2cnfor any r∈[c0, n], and\nthe output gates of the linear circuit encoding S2k\u0010\nn,n\nA(k−1,r),n\nr\u0011\n-PGC have bounded fanin ≤D.\nNote that r≥c0by our condition. Applying the reduction lemma (i.e., Lemma 3.9) with\nr′= 2r, we have\nS2k\u0012\nn,n\nA(k−1, r),n\n4r2\u0013\n≤S2(k−1)\u0012jn\n2rk\n,n\nA(k−1, r),n\n2r\u0013\n+c4n\n≤3c·n\n2r·λ2(k−1)\u0012A(k−1, r)\n2r\u0013\n+c4n By induction hypothesis\n≤3c\n2·n+c4n,\nwhere the last step λ2(k−1)\u0010\nA(k−1,r)\n2r\u0011\n≤λ2(k−1)(A(k−1, r)) = ris by Proposition A.3. By the\nreduction lemma (i.e., Lemma 3.9), the output gates of the linear circuit encoding\u0010\nn,n\nA(k−1,r),n\n4r2\u0011\n-\nPGC have bounded fanin D1.\nApplying the composition lemma (i.e., Lemma 3.5), we have\nS2k\u0012\nn,n\nA(k−1, r),n\nr\u0013\n≤S2k\u0012\nn,n\nA(k−1, r),n\n4r2\u0013\n+S2k\u0010\nn,n\n4r2,n\nr\u0011\n+ 2c1D1n\n≤3c\n2·n+c4n+c5n+ 2c1D1n\n≤2cn ,\n15where the second last step is by Lemma 3.12, and the last step is because c≥2(2c1D1+c4+c5).\nWe claim the output gates of the overall linear circuit encoding\u0010\nn,n\nA(k−1,r),n\nr\u0011\n-PGC have fanin\nat most 2 c2max( D1, D2), because\n•The output gates of the circuit encoding\u0010\nn,n\nA(k−1,r),n\n4r2\u0011\n-PGC have bounded fanin D1by the\nreduction lemma (i.e., Lemma 3.9). (Notice that the circuit encoding\u0010\u0004n\n2r\u0005\n,n\nA(k−1,r),n\n2r\u0011\n-\nPGC does not have bounded outputs fanin, which lies in the inner part in the reduction\nlemma.)\n•The output gates of the depth-2 kcircuit encoding\u0000\nn,n\n4r2,n\nr\u0001\n-PGC have bounded fanin D2\nby Lemma 3.12.\n•In the composition lemma (i.e., Lemma 3.5), we compose 2 circuits, encoding\u0010\nn,n\nA(k−1,r),n\n4r2\u0011\n-\nPGC and\u0000\nn,n\n4r2,n\nr\u0001\n-PGC respectively. So the fanin of the outputs of the overall circuit is\nbounded by 2 c2max( D1, D2),as desired.\nBy taking r=nin Theorem 3.13, we immediately have\nCorollary 3.14. For any n,\nSα(n)(n) = O(n).\nWe have constructed a linear circuit of size O(n) and depth α(n) encoding a code C:{0,1}n→\n{0,1}32nwith relative distance1\n8. By putting a rate booster at the bottom, one can achieve any\nconstant rate and relative distance within the Gilbert-Varshamov bound.\nProof. (of Theorem 1.2) By Corollary 3.14, there exists a linear circuit of size O(λd(n)·n) and\ndepth dencoding a code C:{0,1}n→ {0,1}32nwith relative distance1\n8.\nBy Lemma 3.4 with c=1\nr,γ=δ,δ= 4, we know there exists a probabilistic circuit D:\n{0,1}32n→ {0,1}⌊n/r⌋of depth 1 and size Or,δ(n) such that\nPr\nDh\nwt(D(x))≥δ·jn\nrki\n>1−2−n\nfor any x∈ {0,1}32nwith wt( x)≥4n. Applying a union bound over all C(x), where 0 ̸=x∈ {0,1}n,\nwe claim there exists a (deterministic) linear circuit Dof depth 1 and size Or,δ(n) such that\nwt(D(C(x)))≥δ· ⌊n\nr⌋for all nonzero x∈ {0,1}n. In addition, the output gates in circuit Dhave\nbounded fanin Or,δ(1).\nNote that the size of the circuit D(C(x)) is O(n) +Or,δ(n) =Or,δ(n) and the depth of the\ncircuit is d+ 1. Notice that the output gates have bounded fanin Or,δ(1). So collapsing the last\nlayer (for the linear circuit) will increase the size by a multiplicative factor Or,δ(1). As such, we\nobtain a circuit of depth dand size Or,δ(n).\nIn particular, by letting d=α(n), we obtain a circuit of depth α(n) and size Or,δ(n).\n164 Depth lower bound\nDefinition 4.1. (Densely regular graph [Pud94]) Let Gbe a directed acyclic graph with ninputs\nandnoutputs. Let 0 < ϵ, δ and 0 ≤µ≤1. We say Gis (ϵ, δ, µ )-densely regular if for every\nk∈[µn, n ], there are probability distributions XandYonk-element subsets of inputs and outputs\nrespectively, such that for every i∈[n],\nPr\nX∈X[i∈X]≤k\nδn, Pr\nY∈Y[i∈Y]≤k\nδn,\nand the expected number of vertex-disjoint paths from XtoYis at least ϵkfor randomly chosen\nX∈ XandY∈ Y.\nDenote by D(n, d, ϵ, δ, η ) the minimal size of a ( ϵ, δ, µ )-densely regular layered directed acyclic\ngraph with ninputs and noutputs and depth d.\nTheorem 4.2. (Theorem 3 in [Pud94]) Let ϵ, δ > 0. For every d≥3, and every r≤n,\nD\u0012\nn, d, ϵ, δ,1\nr\u0013\n= Ω d,ϵ,δ(nλd(r)).\nTo apply the above lower bounds when dis not fixed, we need to figure out the hidden constant\nthat depends on d, ϵ, δ .\nTheorem 4.3. Letϵ, δ > 0. For every d≥3, and every r≤n,\nD\u0012\nn, d, ϵ, δ,1\nr\u0013\n≥Ω\u0010\n2−d/2ϵδ2λd(r)n\u0011\n.\nThe constant in Ω(2−d/2ϵδ2λd(r)n) is an absolute constant. The proof of Theorem 4.3 is almost\nthe same as Theorem 4.2, which is in Appendix B.\nCorollary 4.4. (Corollary 15 in [Gal+13]) Let 0 < ρ, δ < 1 be constants and Cbe a circuit\nencoding an error-correcting code {0,1}ρn→ {0,1}nwith relative distance at least δ. If we extend\nthe underlying graph with (1 −ρ)ndummy inputs, then its underlying graph is ( ρδ, ρ,1\nn)-densely\nregular.\nThe proof of Theorem 1.2 readily follows from Theorem 4.3 and Corollary 4.4.\nProof. (of Theorem 1.2) By Corollary 4.4, the underlying graph Gof the circuit Cn, is\u0010\nρ′δ, ρ′,1\n⌊n/ρ⌋\u0011\n-\ndensely regular with ⌊n/ρ⌋inputs and ⌊n/ρ⌋outputs, by adding dummy inputs, where ρ′=n\n⌊n/ρ⌋>\n1\n2ρ.\nNotice that graph Gis not necessarily layered. We can always make Glayered, by increasing\nthe size by a factor of d, where dis the depth of G. By slightly abusing notations, we use Gto\ndenote a layered ( ρ′δ, ρ′,1\n⌊n/ρ⌋)-densely regular graph of depth d, where |E(G)| ≤cdn.\nOn the other hand, by Theorem 4.3, we have\n|E(G)| ≥ Ω\u0010\n2−d/2δρ′3λd(⌊n/ρ⌋)⌊n/ρ⌋\u0011\n.\n17Combining with |E(G)| ≤cdn, we have\nλd(⌊n/ρ⌋)≤O\u00121\nδρ′2cd2d/2\u0013\n=Oρ,δ,c\u0010\nd2d/2\u0011\n. (11)\nTherefore,\nλd+2(n)≤λd+2(⌊n/ρ⌋)\n=λ∗\nd(⌊n/ρ⌋)\n≤λ∗\nd(λd(⌊n/ρ⌋)) + 1 By Definition 2.1\n≤λ∗\nd(Oρ,δ,c(1)·d2d/2) + 1 By (11)\n≤λ∗\nd(λd(Oρ,δ,c(1)·d2d/2)) + 2 By Definition 2.1\n≤λ∗\nd(d) + 2 By Proposition A.4\n≤6. By Proposition A.2\nSo, we conclude d+ 2≥α(n), i.e., d≥α(n)−2.\nWe would like to point out that, alternatively, one can use a powerful lemma by Raz and Shipilka\n[RS03], to prove the depth lower bound (i.e., Theorem 1.2).\n5 Superconcentrator-induced codes\nIt is already known that circuits for encoding error-correcting codes must satisfy some superconcentrator-\nlike connectivity properties. For example, Speilman [Spi96] observed that any circuits encoding\ncodes from {0,1}nto{0,1}mwith distance δmmust have δnvertex-disjoint paths connecting any\nchosen δninputs to any set of (1 −δ)moutputs; using matroid theory, G´ al et al. [Gal+13] proved\nthat, for any k≤n, for any k-element subset of inputs X, taking a random k-element subset of\noutputs Y, the expected number of vertex-disjoint paths from XtoYis at least δk.\nWe observe a connection in the reverse direction by showing that anysuperconcentrator graph,\nconverted to an arithmetic circuit over a sufficiently large field, can encode a good code. (Recall that\na directed acyclic graph G= (V, E) with minputs and noutputs is an ( m, n)-superconcentrator , if\nfor any equal-size inputs X⊆[m] and outputs Y⊆[n], the number of vertex-disjoint paths from\nXtoYis|X|.) Furthermore, the code C:Fn→Fm(encoded by the above circuits) satisfies\na distance criterion, stronger than MDS (maximum distance separable) codes, captured by the\nfollowing definition.\nDefinition 5.1. (Superconcentrator-induced code) C:Fn→Fmis asuperconcentrator-induced\ncode if\ndist(C(x), C(y))≥m−dist(x, y) + 1\nfor any distinct x, y∈Fn.\nFor a linear code C:Fn→Fm, it is well known that Cis an MDS code, i.e., Csatisfies the\nSingleton bound\ndist(C(x), C(y))≥m−n+ 1∀x̸=y∈Fn,\nif and only if any ncolumns of its generator matrix are linearly independent. Similarly, we show\nthat a linear code is a superconcentrator-induced code if and only if every square submatrix of its\ngenerator matrix is nonsingular. (Such matrices are sometimes called totally invertible matrices .)\n18Lemma 5.2. Linear code C:Fn→Fmis a superconcentrator-induced code if and only if every\nsquare submatrix of its generator matrix is nonsingular.\nProof. For the “if” direction, assume for contradiction that there exists a nonzero x∈Fnsuch that\nwt(C(x))≤m−wt(x). Let X= supp( x)⊆[n], and let Ybe any subset of [ m]\\supp( C(x)) of\nsize|X|. Denote the generator matrix by M∈Fn×m. By the definition of the generator matrix,\nwe have C(x) =xM. Thus,\nC(x)|Y=xM[n],Y\n=x|XMX,Y+x|[n]\\XM[n]\\X,Y\n=x|XMX,Y since x|[n]\\X= 0,\nwhere MX,Ydenotes the |X|×|Y|submatrix indexed by rows Xand columns Y. Since matrix MX,Y\nis nonsingular, and x|Xis nonzero, we claim C(x)|Yis nonzero. This contradicts the definition of\nY.\nFor the “only if” direction, we prove the contraposition. That is, if there exists a singular square\nsubmatrix, then Cis not a superconcentrator-induced code. Suppose there exist equal-size X⊆[n]\nandY⊆[m] such that MX,Yis singular. Since rank MX,Y<|X|, there exists a nonzero x′∈F|X|\nsuch that x′MX,Y= 0. Extend x′to a vector x∈Fnby adding zeros on the coordinates outside of\nX, that is, x′=x|supp( x). As such, we have\nC(x)|Y= (xM)|Y\n=xM[n],Y\n=x|XMX,Y+x|[n]\\XM[n]\\X,Y\n=x|XMX,Y since x[n]\\Xis zero\n=x′MX,Y\n= 0,\nwhich implies that wt( C(x))≤m−|Y|=m−wt(x). That is, Cis not a superconcentrator-induced\ncode.\nWe justify the naming “superconcentrator-induced codes” by proving\n•Any (unrestricted) arithmetic circuit encoding a superconcentrator-induced code, viewed as\na graph, must be a superconcentrator.\n•Any superconcentrator can be converted to an arithmetic circuit encoding a superconcentrator-\ninduced code, by replacing each vertex with an addition gate over a sufficiently large finite\nfield, and choosing the coefficients uniformly at random. (The field size is exponential in the\nsize of the graph.)\nWe consider the following unrestricted arithmetic circuit model encoding a superconcentrator-\ninduced code C:Fn→Fm:\n•LetFbe a finite field.\n•There are ninputs, denoted by x1, x2, . . . , x n∈F; there are moutputs, denoted by y1, y2, . . . , y m∈\nF.\n19•Gates have unbounded fanin and fanout. The size of the circuit is defined to be the number\nof wires, instead of gates.\n•Each internal and output gate can compute anyfunction from Fs→F, where sis the fanin\nof the gate (unbounded).\nIn this model, any single output function from FntoFcan be computed by a single gate. It makes\nsense to consider the circuit complexity of multiple output functions.\nLemma 5.3. Any unrestricted arithmetic circuit encoding a superconcentrator-induced code must\nbe a superconcentrator.\nProof. LetC:Fn→Fmbe an unrestricted arithmetic circuit encoding a superconcentrator-induced\ncode. Let Gdenote the underlying directed acyclic graph with ninputs and moutputs.\nSuppose for contradiction that Gis not an ( n, m)-superconcentrator, that is, there exist equal-\nsize inputs S⊆[n] and outputs T⊆[m] such that the number of vertex-disjoint paths from StoT\nis less than |S|. By Menger’s theorem, there exists a cut vertex set, denoted by V, whose removal\ndisconnects SandTand|V|<|S|.\nFix the inputs outside of Sarbitrarily, while leave the inputs in Sundetermined. So, the\noutputs in Tare completely determined by the gates in V, the “cut vertices” between SandT.\nSince |V|<|S|, the outputs in Tcan take at most |F||V|<|F||S|values. By the pigeonhole\nprinciple, there exist x, x′∈Fnsuch that\n•x|S̸=x′|S,\n•x|[n]\\S=x′|[n]\\S, and\n•C(x)|T=C(x′)|T.\nTherefore, dist( C(x), C(x′))≤m−|T|=m−|S|, and dist( x, x′)≤ |S|, which implies dist( C(x), C(x′))≤\nm−dist(x, x′). This contradicts the definition of superconcentrator-induced codes.\nIn the reverse direction, we prove the following.\nLemma 5.4. LetGby an ( n, m)-superconcentrator. Let CG:Fn→Fmbe an arithmetic circuit\nby replacing each vertex in Gwith an addition gate, and choosing the coefficient on each edge\nuniformly and random (over the finite field F). With probability at least 1 −Pn\ni=1\u0000n\ni\u0001\u0000m\ni\u0001di\n|F|, CG\nencodes a superconcentrator-induced code.\nProof. Denote the inputs by x1, x2, . . . , x n, and outputs by y1, y2, . . . , y m. For each edge e∈E(G),\nassign a random coefficient re. Let Mbe the n×mgenerator matrix of the linear code CG. One\ncan see that\nMi,j=X\npath pfrom xitoyjY\ne∈E(p)re,\nwhere penumerates allpaths from xitoyj. View Mi,jas a (multilinear) polynomial in F[{re}e∈E(G)].\nNote that the degree of Mi,jis exactly the length of the longest path from xitoyj.\nClaim 5.5. For any equal-size subsets X⊆[n] and Y⊆[m],\nPr[det( MX,Y) = 0] ≤d|X|\n|F|,\nwhere dis the depth of the superconcentrator.\n20Proof. (of Claim 5.5) Note that det( MX,Y) is a polynomial of degree at most d|X|, since each entry\nis of degree at most d. By the definition of superconcentrator, there are |X|vertex-disjoint paths\nconnecting XandY. Setting reto 1 for all edges eon these |X|vertex-disjoint paths, and leaving\nall others 0, the polynomial det( MX,Y) evaluates to ±1, which implies that det( MX,Y) is a nonzero\npolynomial. The claim follows from the Schwarz-Zipple lemma.\nBy Lemma 5.2, CGencodes a superconcentrator-induced code if and only if all square submatrix\nofMis nonsingular. Taking a union bound over all equal-size X⊆[n] and Y⊆[m], we have\nPr[CGencodes a superconcentrator-induced code] ≥1−nX\ni=1\u0012n\ni\u0013\u0012m\ni\u0013di\n|F|.\n6 Conclusion\nIn this work, we determine, up to an additive constant 2, the minimum depth required to encode\nasymptotically good error-correcting codes, i.e., codes with constant rate and constant relative\ndistance, using a linear number of wires. The minimum depth is between α(n)−2 and α(n),\nwhere α(n) is a version of the inverse Ackermann function. The upper bound is met by certain\nbinary codes we construct (building on G´ al et al. [Gal+13] with a few new ingredients) for any\nconstant rate and constant relative distance within the Gilbert-Varshamov bounds. The lower\nbound applies to any constant rate and constant relative distance. We credit the lower bound\nto G´ al et al. [Gal+13], although not explicitly claimed or fully verified in [Gal+13]; because our\ncontribution is a routine checking of details.\nValiant articulated graph-theoretic arguments for proving circuit lower bounds [Val75; Val76;\nVal77]. Since then, there have been fruitful results along this line. We show a result in the reverse\ndirection, that is, we prove that any superconcentrator (after being converted to an arithmetic\ncircuit) can encode a good code over a sufficiently large field (exponential in the size of the super-\nconcentrator graph).\nReferences\n[Ahl+00] Rudolf Ahlswede, Ning Cai, S-YR Li, and Raymond W Yeung. “Network information\nflow”. 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In: Journal\nof Computer and System Sciences 13.3 (1976), pp. 278–285.\n[Val77] Leslie G. Valiant. “Graph-theoretic arguments in low-level complexity”. In: Mathe-\nmatical Foundations of Computer Science 1977, 6th Symposium, Tatranska Lomnica,\nCzechoslovakia, September 5-9, 1977, Proceedings . 1977, pp. 162–176.\n[Vio19] Emanuele Viola. “Lower bounds for data structures with space close to maximum imply\ncircuit lower bounds”. In: Theory of Computing 15.1 (2019), pp. 1–9.\nA Properties of the inverse Ackermann-type functions\nProposition A.1. (Claim 2.4 in [RS03])\n(i). Each λi(n) is a monotone function tending to infinity with n.\n(ii). For i≥2,λ2i+1(n)≤λ2i(n)≤2λ2i+1(n).\n(iii). For i≥2 and n≥27,λi(n)≤ ⌊pn\n2⌋.\n23The following proposition is given as Proposition 2 and 3 in the subsequent paper [Li23], but\nwas developed first in the present work.\nProposition A.2. (i) For any d≥1, and for all n≥4,\nλd(n)≤n−2.\n(ii) For any d≥1,\nλd(d)≤4.\nProof. (i) We prove by induction on d. When d= 1 or 2, it is easy to verify that λd(n)≤n−2.\nAssuming λd(n)≤n−2 for all n≥4, let us prove that λd+2(n)≤n−2. By the definition of\nλd+2(n), we have\nλd+2(n) = λ∗\nd(n)≤λd(n)≤n−2.\n(ii) We prove by induction on d. When d≤3, it is easy to verify by straightforward calculation.\nAssuming λd(d)≤4 is true for d, where d≥2, let us prove λd+2(d+ 2)≤4. We have\nλd+2(d+ 2) = λ∗\nd(d+ 2)\n=λ∗\nd(λd(d+ 2)) + 1\n≤λ∗\nd(d) + 1 Proposition A.2 (i)\n=λ∗\nd(λd(d)) + 2\n≤λ∗\nd(4) + 2 By induction hypothesis\n≤max{λ∗\n2(4), λ∗\n3(4)}+ 2\n= 4.\nProposition A.3. For any i, d≥1,\n(1).A(i+ 1, d) =A(d)\ni(1).\n(2).λ2i(A(i, d)) =d.\n(3).λ2i(A(i, d) + 1) = d+ 1.\n(4).A(i, d) = max {n:λ2i(n)≤d}.\n(5).A(i, λ2i(d))≥d.\nProof. (1). By Definition 2.3,\nA(i+ 1, d) = A(i, A(i+ 1, d−1))\n=A(i, A(i, A(i+ 1, d−2)))\n=A(i, A(i, . . .|{z}\nd−1 times, A(i+ 1,1). . .))\n=A(d)\ni(1),\n24where we write A(i, j) asAi(j).\n(2). Prove by induction on i. For the base case i= 1, we have A(1, d) = 2d, and λ2(n) =⌈log2n⌉.\nIt is obvious that the λ2(A(1, d)) =dholds.\nFor the induction step, assuming the conclusion holds for i, let us prove it for i+ 1.\nλ2(i+1)(A(i+ 1, d))\n=λ∗\n2i\u0010\nA(d)\ni(1)\u0011\n=λ∗\n2i\u0010\nA(d−1)\ni(1)\u0011\n+ 1 By induction hypothesis\n=λ∗\n2i\u0010\nA(d−2)\ni(1)\u0011\n+ 2 By induction hypothesis\n...\n=d.\n(3). Prove by induction on i. For the base case i= 1, we have A(1, d) = 2d, and λ2(n) =⌈log2n⌉.\nIt is obvious that the λ2(A(1, d) + 1) = d+ 1 holds.\nFor the induction step, assuming the conclusion holds for i, let us prove it for i+ 1.\nλ2(i+1)(A(i+ 1, d) + 1)\n=λ∗\n2i\u0010\nA(d)\ni(1) + 1\u0011\n=λ∗\n2i\u0010\nA\u0010\ni, A(d−1)\ni(1)\u0011\n+ 1\u0011\n=λ∗\n2i\u0010\nA(d−1)\ni(1) + 1\u0011\n+ 1 By induction hypothesis\n=λ∗\n2i\u0010\nA(d−2)\ni(1) + 1\u0011\n+ 2 By induction hypothesis\n...\n=λ∗\n2i(1 + 1) + d\n=d+ 1.\n(4) follows from (2) and (3).\n(5) follows from (4).\nProposition A.4. Letc >0 be a constant. When dis large enough, λd(c·d2d/2)≤d.\nProof. Ifdis even, i.e., d= 2i, by Proposition A.3, it suffices to prove\nc(2i)2i≤Ai(2i). (12)\nNotice that A1(2i) = 22iandAi+1(n)≥Ai(n). Clearly, (12) is true when iis large enough, and\nthis sufficiently large number only depends on c.\nIfdis odd, i.e., d= 2i+ 1, it suffices to prove λ2i(c·d2d/2)≤d, asλ2i+1(n)≤λ2i(n) by\nProposition A.1. By Proposition A.3, it suffices to prove\nc(2i+ 1)2i+1\n2≤Ai(2i+ 1),\nwhich is true because A1(2i+ 1) = 22i+1.\n25B Proof of Theorem 4.3\nLemma B.1. (Lemma 8 in [Pud94]) For any 1 ≤r≤n, and for any ϵ, δ∈(0,1], we have\nD\u0012\nn,1, ϵ, δ,1\nr\u0013\n≥ϵδ2nr.\nLemma B.2. (Lemma 5 in [Pud94]) Suppose f(0) = f(1) = 0 and f(n)≤ ⌊√n⌋for every n >1.\nFor every n≥0, we have\n(i)f∗(n)≤f(n)≤ ⌊√n⌋,\n(ii)f(i)(n)\nf(i+1)(n)≥f(i+1)(n) for every i≥1 provided the denominator is not 0,\n(iii)f(i)(n)≥f∗(n)\n2for every i≤f∗(n)\n2.\nIndeed, (iii) can be strengthened.\nLemma B.3. Suppose f(n)≤ ⌊√n⌋. For every n≥1 and every i≤f∗(n)\n2, we have f(i)(n)≥f∗(n).\nProof. Let us assume f∗(n)≥2. Otherwise, f∗(n)≤1, that is, f(n)≤1. Note that i≤f∗(n)/2≤\n1/2, and thus i= 0. So f(0)(n) =n≥f∗(n) = 1 always holds.\nBy the definition of f∗(n) and f(n)≤ ⌊√n⌋, we have\nf(f∗(n))(n) = 1 ,\nf(f∗(n)−1)(n)≥2,\nf(f∗(n)−2)(n)≥22,\nf(f∗(n)−3)(n)≥24,\n. . .\nf(f∗(n)−j)(n)≥22j−1\nfor any 1 ≤j≤f∗(n).\nWhen i≤f∗(n)\n2,f∗(n)−i≥f∗(n)\n2. Thus,\nf(i)(n)≥22f∗(n)−i−1\n≥22f∗(n)\n2−1\n≥f∗(n),\nwhere the last step is because 22x\n2−1≥xfor all real x≥4. (If f∗(n) = 2, then f(1)(n)≥2. If\nf∗(n) = 3, then f(1)(n)≥4.)\nThe following lemma is proved by Pudl´ ak [Pud94]. To apply it for super-constant depth, we\nexplicitly determine an appropriate β(in terms of αandϵ, δ). We reproduce the proof here.\n26Lemma B.4. (Lemma 9 in [Pud94]) Let f(n)≤ ⌊√n⌋for every n. For every positive reals α, β, ϵ\nsuch that if\n∀n∀r≤n D\u0012\nn, d,ϵ\n2, δ,1\nr\u0013\n≥αnf(r), (13)\nthen\n∀n∀r≤n D\u0012\nn, d+ 2, ϵ, δ,1\nr\u0013\n≥βnf∗(r), (14)\nwhere β= min\bϵδ\n27, α\t\n.\nProof. LetGbe a depth-( d+2) layered graph with ninputs and noutputs which is (ϵ\n2, δ,1\nr)-densely\nregular. Let Vi,i= 0,1, . . . , d + 2, be the ith level of G, where V0are inputs and Vd+2are outputs.\nLet\nA0={v∈V1∪Vd+1: deg( v)> f(r)},\nand\nAi=n\nv∈V1∪Vd+1: deg( v)∈(f(i+1)(r), f(i)(r)]o\nfori= 1,2, . . . , f∗(r).\nClaim B.5. For every i, where 1 ≤i≤f∗(r)\n2−3, at least one of the following inequalities holds:\n|A0∪ ··· ∪ Ai−1| ≥ϵ\n4·n\nf(i+1)(r); (15)\n|{(u, v) : (u, v) incident with Ai∪Ai+1∪Ai+2}| ≥ϵδ\n4n; (16)\n|{(u, v) : (u, v) not incident with A0∪ ··· ∪ Ai+2}| ≥ αnf(i+2)(r)\nf(i+3)(r)n. (17)\nProof. (of Claim B.5) The proof is reproduced from [Pud94]. For any i∈[1,f∗(r)\n2−3], suppose\nneither (15) nor (16) is true, we prove (17).\nLetk∈h\nn\nf(i+1)(r), ni\nbe an arbitrary integer. Let X,Ybe probability distributions on k-element\nsubsets of inputs and outputs respectively satisfying the condition in Definition 4.1. Let X∈ X,\nY∈ Ybe random subsets of inputs and outputs of size k.\nBy the definition of densely regular graphs, there are at least ϵkvertex-disjoint paths connecting\nthem. Out of these, there are at mostϵ\n4·n\nf(i+1)(r)≤ϵ\n4·kpaths passing through A0∪ ··· ∪ Ai−1\nby non (15). On average , there are at mostϵ\n4·kpaths passing through Ai∪Ai+1∪Ai+2, since\nthe expected number of vertices of X∪Ydirectly connected with with Ai∪Ai+1∪Ai+2is at\nmostk\nδn·ϵδ\n4n=ϵ\n4·k. So, on average, the number of vertex-disjoint paths from XtoYavoiding\nA0∪ ··· ∪ Ai+2is at leastϵ\n2·k.\nWe transform Ginto a depth- dgraph as follows:\n•Omit the vertices in V1andVd+1.\n•For each pair of edges ( u, v),(v, w), where v∈Ai+3∪Ai+4∪ ··· , add the edge ( v, w).\n27As such, the size of the new graph, denoted by G′, is at most f(i+3)(r) times larger than the size\nofG, since f(i+3)(r) is an upper bound on the degrees of vertices in Ai+3∪Ai+4∪ ··· . We have\nshown that G′is (ϵ\n2, δ,1\nf(i+1)(r))-densely regular. By the assumption of the lemma, G′is of size at\nleast\nD\u0012\nn, d,ϵ\n2, δ,1\nf(i+1)(r)\u0013\n≥αnf(f(i+1)(r)) = αnf(i+2)(r),\nwhich implies that the number of edges not incident with A0∪ ··· ∪ Ai+2is at least\n|E(G′)|\nf(i+3)(r)≥αnf(i+2)(r)\nf(i+3)(r)n,\nwhere the last step is by Lemma B.2 (ii).\nConsider the three cases according to Claim B.5.\nCase 1: Forsome i, where 1 ≤i≤f∗(r)\n2−3, (15) holds. Since each vertex in A0∪. . .∪Ai−1\nhas degree at least f(i)(r), we have\n|E(G)| ≥ϵ\n4·n\nf(i+1)(r)·f(i)(r)\n≥ϵ\n4·n·f(i+1)(r) By Lemma B.2 (ii)\n≥ϵ\n4·n·f∗(r) By Lemma B.3 .\nCase 2: Foralli, where 1 ≤i≤f∗(r)\n2−3, (16) holds. If f∗(r)≥54, we have\n|E(G)| ≥1\n3·\u0012f∗(r)\n2−3\u0013\n·ϵδ\n4n\n≥1\n3·4\n9f∗(r)·ϵδ\n4n\n=1\n27f∗(r)ϵδn.\nIff∗(r)<54, we still have |E(G)| ≥1\n27·f∗(r)ϵδnby Claim B.6.\nClaim B.6. LetGbe a layered graph with ninputs and noutputs and of depth at least 2, which\nis (ϵ, δ,1\nr)-densely regular. Then |E(G)| ≥2ϵδn.\nProof. (of the claim) Let k∈[n\nr, n]. Denote the inputs of GbyI(G), and the outputs of Gby\nO(G). Let ∆ G(X, Y) denote the number of vertex-disjoint paths between XandY.\nBy the definition of densely regular graphs, there are nonempty sets of k-element subsets X ∈\u0000I(G)\nk\u0001\nandY ∈\u0000O(G)\nk\u0001\nsuch that for every i∈I(G), for every j∈O(G), Pr X∈X[i∈X]≤\nk\nδnand Pr Y∈Y[j∈Y]≤k\nδn, andE[∆G(X, Y)]≥ϵk. On the other hand,\nE\nX∈X\nY∈Y[∆G(X, Y)]≤E\nX∈X[X\nx∈X1deg(x)>0]\n≤X\nx∈I(G) and deg( x)>0Pr\nx∈X[x∈ X]\n≤k\nδnX\nx∈I(G)deg(x).\n28Putting them together, we haveP\nx∈I(G)deg(x)≥ϵδn. Similarly, we can proveP\ny∈O(G)deg(y)≥\nϵδn. Since Gis a layered graph of depth at least 2, we have |E(G)| ≥2ϵδn.\nCase 3: Forsome i, where 1 ≤i≤f∗(r)\n2−3, (17) holds. So we have\n|E(G)| ≥αnf(i+2)(r)\nf(i+3)(r)n≥αf(i+3)(r)≥αnf∗(r).\nCombining 3 cases, we have β= min\bϵ\n4,ϵδ\n27, α\t\n= min\bϵδ\n27, α\t\n.\nWe are ready to prove Theorem 4.3.\nProof. (of Theorem 4.3) Since λ2i(n) = Θ( λ2i+1(n)) for i≥2, it suffices to prove\nD\u0012\nn,2d+ 1, ϵ, δ,1\nr\u0013\n≥min\u001a1\n27,δ\n2\u001b\n2−dϵδnλ 2d+1(r). (18)\nThe base case for d= 0 is proved in Lemma B.1. Assume (18) is true for d, let us prove it for\nd+ 1. By induction hypothesis, we have\nD\u0012\nn,2d+ 1,ϵ\n2, δ,1\nr\u0013\n≥min\u001a1\n27,δ\n2\u001b\n2−(d+1)ϵδnλ 2d+1(r)\nholds for any r≤n. By Lemma B.4, we have D(n,2d+ 3, ϵ, δ,1\nr)≥βnλ∗\n2d+1(r) =βnλ 2d+3(r),\nwhere\nβ= min\u001aϵδ\n27,min\u001a1\n27,δ\n2\u001b\n2−(d+1)ϵδ\u001b\n= min\u001a1\n27,δ\n2\u001b\n2−(d+1)ϵδ,\nwhich completes the induction step.\n29"    },    {        "title": "2402.00387v1.Superconductivity_in___textit_a___MoGe_thin_films__effect_of_phase_fluctuations_with_decreasing_thickness_and_study_of_vortex_dynamics_in_presence_of_low_frequency_ac_excitation.pdf",        "content": "Superconductivity in a-MoGe thin\nfilms: effect of phase fluctuations\nwith decreasing thickness and study\nof vortex dynamics in presence of\nlow-frequency ac excitation\nA Thesis\nSubmitted to the\nTata Institute of Fundamental Research, Mumbai\nfor the Degree of Doctor of Philosophy\nin Physics\nby\nSoumyajit Mandal\nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nMumbai, India\nJuly, 2023\nFinal Version Submitted in January, 2024arXiv:2402.00387v1  [cond-mat.supr-con]  1 Feb 202426-1-2024To Maa, Baba and BhaiAcknowledgement\nFirst and foremost, I express my deepest gratitude to my supervisor, Prof. Pratap Ray-\nchaudhuri, for his unwavering guidance and immense support throughout my Ph.D. jour-\nney. Without his invaluable mentorship, this work would not have been possible. His\nperseverance and enthusiasm for exploring new ideas have been a constant source of in-\nspiration for me. I will forever cherish the numerous insightful discussions we had over\nthe years.\nI extend heartfelt appreciation to Prof. Lara Benfatto for her invaluable theoretical\ncollaborations and valuable input, which enriched the quality of this research.\nI am grateful to Prof. Vikram Tripathi and Prof. Arnab Bhattacharya for dedicating\ntheir time and expertise as members of the thesis monitoring committee, offering valuable\ninsights that greatly contributed to the refinement of this work.\nMy sincere thanks go to my esteemed colleagues, both past and present. Dr. Somesh\nChandra Ganguli, Dr. Rini Ganguly, Dr. Harkirat Singh, Dr. Aditya Narayan Roy\nChoudhury, Dr. Pradnya Parab, Dr. Ruchi Tomar, Dr. Indranil Roy, Dr. Surajit Dutta,\nJohn Jesudasan, Vivas Bagwe, Somak Basistha, Rishabh Duhan, Srijita Das, Pritam\nDas, Arghya Dutta, Sulagna Dutta, Sitakanta Behera and Dr. Subhamita Sengupta,\nyour collaboration, unwavering support, and camaraderie have been instrumental in the\nsuccess of this research.\nI am grateful to Subash Pai, Supriya, and the team at Excel Instruments for their\nprompt and indispensable technical support. My heartfelt thanks also go to Atul Raut\nand the TIFR Central Workshop for their invaluable technical assistance. I am deeply\nappreciative of Ganesh Jangam and Prof. P. L. Paulose for their assistance with SQUID\nmagnetization measurements. I extend my thanks to Bhagyashree Chalke, and Rudhir\nBapat for characterizing the samples using EDX and SEM, and Jayesh B. Parmar for\ntheir invaluable help in TEM sample-surface preparations. Moreover, I extend my deepest\nappreciation to the Low Temperature Facility team of TIFR: Srinivasan, Vijay Arolkar,\nviiBosco, Arvind, Jaison, Purao, and Nikhil for their unfaltering support in providing liquid\nhelium and nitrogen, without which these experiments would not have been possible.\nI am grateful for the invaluable friendship and camaraderie extended to me by my\nfellow researchers at TIFR. The constant support of Triparno, Nairit, Aritra, Manibrata,\nChandradoy, Sabir, Saikat, Amrita and other seniors, as well as Anupam, Rabisankar,\nAnirban, Soumen, Sawani, Sanat, Anindya, Dibysankar, Souvik, Deepjyoti, Sunil, Arnab,\nand other juniors, Sudip, Indranil, Pratap, Sandip, Soham, Saikat, Soumen, Anurag,\nHaritha, Arusha, Kishor, Vishal, Dibyendu, Sumeru, Arkadipta, Sazedur, Suman, and\nother wonderful friends and batch-mates have been a delightful part of this voyage. Dur-\ning challenging times towards the conclusion of my tenure, I would like to acknowledge\nthe unwavering support from Mintu, Sanat, Aman, Anirban Bala, Anindya, Dibyasankar,\nand many others. Beyond the TIFR community, I have been fortunate to have friends\nlike Madhumita, Semanti, Srijoyee, Shreya, and Debamitra during various phases of my\nPh.D. journey. My sincere appreciation goes to Aditi for being a companion in sharing\nour daily life experiences — some enthralling, others mere moments of shared frustration.\nI consider myself fortunate to have forged meaningful friendships throughout my aca-\ndemic journey, spanning from school to bachelors and masters. While it is impossible\nto name everyone, I want to express my gratitude to friends like Arunava, Subir, Di-\npanjan, Aritra, and others from school. In my college days, I cherish the camaraderie\nof Poulab, Bodhisattwa, Snehashish, Rahul, Jui, Amartya, Rajesh, Supriyo, Durgesh,\nSantu, Shirsendu, and many others. Each of these friendships has enriched my life in\nunique ways.\nIt would be remiss of me not to acknowledge the impact of my fellow Duffianz from\nScottish Church College. Mustak, Subhaskar, Ankur, Mithun, Soumyadeep, Subhranil,\nSuman, Sumit – you all are truly incredible. Our circle of friends was a rare haven where\nI could freely share both the frustrations and joys of my life without any hesitation.\nI am deeply grateful for the continuous encouragement and unwavering support I\nhave received from my family members, including dida, dadu, masi, mama, mami, and\nmy younger cousins, throughout my academic journey.\nLastly, I would like to acknowledge my beloved family - baba, maa, and bhai for their\nunconditional love, unwavering encouragement, and unyielding support throughout this\nacademic journey. Their belief in me has been my greatest motivation, and I am forever\nindebted to them for shaping the person I have become today.\nviiiContents\nAbstract xiii\nList of Publications xv\nList of Figures xix\nList of Tables xxi\nPreamble xxiii\nStatement of Joint Work xxv\n1 Introduction 1\n1.1 Basics of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 2\n1.1.1 Zero resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n1.1.2 Perfect diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.1.3 Critical field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.1.4 London equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.1.5 Pippard’s Coherence length . . . . . . . . . . . . . . . . . . . . . 4\n1.1.6 Type-I and Type-II superconductors . . . . . . . . . . . . . . . . 5\n1.2 BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n1.2.1 Cooper pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n1.2.2 BCS ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n1.2.3 Elementary excitations and BCS gap variation at finite T . . . . . 9\n1.2.4 BCS density of states . . . . . . . . . . . . . . . . . . . . . . . . . 10\n1.2.5 Broadening of coherence peaks: Dynes parameter . . . . . . . . . 11\nixCONTENTS\n1.2.6 Measurement of superconducting gap by normal metal-to-superconductor\ntunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n1.2.7 Temperature variation of penetration depth . . . . . . . . . . . . 14\n1.3 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n1.3.1 Order parameter and GL free energy . . . . . . . . . . . . . . . . 15\n1.3.2 Ginzburg-Landau equations . . . . . . . . . . . . . . . . . . . . . 16\n1.3.3 GL coherence length . . . . . . . . . . . . . . . . . . . . . . . . . 16\n1.3.4 GL parameter and Classification of superconductors . . . . . . . . 17\n1.3.5 Fluxoid quantization . . . . . . . . . . . . . . . . . . . . . . . . . 17\n1.3.6 Little-Parks experiment . . . . . . . . . . . . . . . . . . . . . . . 18\n1.3.7 Linearized GL equation and Hc2. . . . . . . . . . . . . . . . . . . 19\n1.3.8 Abrikosov vortex state . . . . . . . . . . . . . . . . . . . . . . . . 19\n1.4 Effect of phase fluctuations: the concept of phase stiffness . . . . . . . . 20\n1.5 Effect of disorder on superconductivity . . . . . . . . . . . . . . . . . . . 21\n1.5.1 Beyond Anderson’s hypothesis . . . . . . . . . . . . . . . . . . . . 22\n1.5.2 Fermionic mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 22\n1.5.3 Bosonic mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n1.5.4 Distingushing features in the Fermionic and Bosonic mechanisms . 24\n1.5.5 Fermionic to Bosonic transition . . . . . . . . . . . . . . . . . . . 25\n1.6 Thermally activated flux creep . . . . . . . . . . . . . . . . . . . . . . . . 25\n1.7 Effect of ac excitation on vortex lattice . . . . . . . . . . . . . . . . . . . 26\n1.7.1 Gittleman-Rosenblum model . . . . . . . . . . . . . . . . . . . . . 26\n1.7.2 Campbell penetration depth . . . . . . . . . . . . . . . . . . . . . 28\n1.7.3 Effect of thermally activated flux flow . . . . . . . . . . . . . . . . 29\n1.7.4 Universal expression for vortex resistivity . . . . . . . . . . . . . . 32\n2 Experimental methods 33\n2.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n2.1.1 Pulsed laser deposition . . . . . . . . . . . . . . . . . . . . . . . . 33\n2.1.2 Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35\n2.1.3 Thermal evaporation . . . . . . . . . . . . . . . . . . . . . . . . . 36\n2.1.4 Masks used in deposition . . . . . . . . . . . . . . . . . . . . . . . 37\n2.2 Thickness measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 38\n2.3 Low-temperature measurements . . . . . . . . . . . . . . . . . . . . . . . 38\n2.3.14He VTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n2.3.2 Janis3He VTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n2.3.3 Homemade3He VTI inside existing4He VTI . . . . . . . . . . . . 40\n2.4 Measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 41\n2.4.1 Two-coil mutual inductance measurements . . . . . . . . . . . . . 41\nxCONTENTS\n2.4.2 Broadband microwave spectroscopy using Corbino geometry . . . 47\n2.4.3 Scanning tunneling microscopy/ spectroscopy (STM/S) . . . . . . 52\n2.4.4 Magneto-transport setup . . . . . . . . . . . . . . . . . . . . . . . 55\n2.4.5 Setup for simultaneous two-coil mutual inductance and magneto-\ntransport measurements . . . . . . . . . . . . . . . . . . . . . . . 55\n2.4.6 SQUID-VSM setup . . . . . . . . . . . . . . . . . . . . . . . . . . 56\n3 Our model system: amorphous Molybdenum Germanium thin films 59\n3.1 Sample details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59\n3.1.1 Target preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 60\n3.2 Hexatic phase in a-MoGe . . . . . . . . . . . . . . . . . . . . . . . . . . 62\n3.3 Weakly pinned vortex lattice in a-MoGe . . . . . . . . . . . . . . . . . . 64\n3.3.1 Extremely small M-H loop . . . . . . . . . . . . . . . . . . . . . . 64\n3.3.2 ZFC and FC data . . . . . . . . . . . . . . . . . . . . . . . . . . . 64\n3.4 Signatures of Peak effect in a-MoGe . . . . . . . . . . . . . . . . . . . . . 65\n3.4.1 Peak in JcandρTAFF . . . . . . . . . . . . . . . . . . . . . . . . . 65\n3.4.2 Hint of peak-like feature in broadband microwave spectroscopy . . 67\n3.4.3 Peak effect in the movement of vortices using scanning tunneling\nspectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68\n3.4.4 Unusual peak-like feature in low-field magnetization data . . . . . 70\n3.5 a-MoGe grown on anodic alumina template . . . . . . . . . . . . . . . . 70\n4 The role of phase fluctuations in ultrathin a-MoGe films: Crossover\nfrom Fermionic to Bosonic regime 75\n4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75\n4.2 Theoretical motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76\n4.2.1 Fermionic and Bosonic mechanism . . . . . . . . . . . . . . . . . 76\n4.2.2 Reason for choosing a-MoGe thin film: popular candidate for Fermionic\nroute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77\n4.3 Samples used in this work . . . . . . . . . . . . . . . . . . . . . . . . . . 77\n4.4 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77\n4.4.1 Transport data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77\n4.4.2 Scanning tunneling spectroscopy data . . . . . . . . . . . . . . . . 79\n4.4.3 Penetration depth data . . . . . . . . . . . . . . . . . . . . . . . . 84\n4.5 Effect of phase fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 86\n4.6 Comparison of TcandT∆. . . . . . . . . . . . . . . . . . . . . . . . . . . 88\n4.7 Interpretation of penetration depth data from “Mean-field” perspective . 89\n4.8 Comparison of energy scales . . . . . . . . . . . . . . . . . . . . . . . . . 90\n4.9 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 92\nxiCONTENTS\n5 Study of vortex dynamics in a-MoGe using low-frequency two-coil mu-\ntual inductance measurements: extracting vortex parameters 93\n5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93\n5.2 Theoretical background: Coffey-Clem Model . . . . . . . . . . . . . . . . 94\n5.3 Sample used in this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 95\n5.4 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95\n5.5 Applicability of Coffey-Clem equation: linearity of ac excitation amplitude 99\n5.6 Temperature dependence of vortex parameters . . . . . . . . . . . . . . . 100\n5.6.1 Tvariation of η. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100\n5.6.2 Tvariation of U. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101\n5.6.3 Tvariation of αL. . . . . . . . . . . . . . . . . . . . . . . . . . . 103\n5.6.4 Correction to UandαLdue to thermal fluctuations . . . . . . . . 103\n5.7 Fitting with experimental λ−2data . . . . . . . . . . . . . . . . . . . . . 104\n5.8 Field dependence of vortex lattice parameters . . . . . . . . . . . . . . . 104\n5.8.1 Field variation of U0. . . . . . . . . . . . . . . . . . . . . . . . . 105\n5.8.2 Field variation of αL0. . . . . . . . . . . . . . . . . . . . . . . . . 106\n5.8.3 Field variation of T0. . . . . . . . . . . . . . . . . . . . . . . . . 107\n5.9 Inconsistency with δ−2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108\n5.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109\n5.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110\n6 Summary and outlook 113\n6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113\n6.2 Future problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114\nAppendices 117\nAppendix A BKT transition in 2.2-nm-thick MoGe film 119\nAppendix B Fitting of experimental data with Coffey-Clem equation 123\nB.1 Fitting with trial temperature variations of UandαL. . . . . . . . . . . 124\nB.2 Best fit parameters and error bar . . . . . . . . . . . . . . . . . . . . . . 125\nBibliography 129\nxiiAbstract\nWe have studied the evolution of superconductivity in amorphous Molybdenum Germa-\nnium ( a-MoGe) thin films. The work can be broken down into two parts. In the first part,\nwe investigate the effect of decreasing thickness on the suppression of superconductivity\nina-MoGe thin films. Thick a-MoGe thin film is a typical type-II superconductor and\nfollows the conventional Bardeen-Cooper-Schrieffer (BCS) equation. Conventionally, it is\nbelieved that decreasing thickness will decrease the effective attractive pairing interaction\nbecause of the gradual loss of screening which holds true for large thicknesses in a-MoGe.\nBut for lower thicknesses, a new mechanism comes into the picture where superfluid den-\nsity is suppressed making the superconductor vulnerable to phase fluctuations. This is\nknown as the Bosonic mechanism, where superconductivity is destroyed due to the loss of\nphase coherence of the superconducting state even though the attractive pairing ampli-\ntude remains finite above the transition temperature. The study unveils that the impact\nof phase fluctuations is significantly more potent than anticipated. Specifically, these fluc-\ntuations suppress the critical temperature ( Tc) of the thinnest films, where both superfluid\ndensity and resistivity vanish, well below the predicted conventional Kosterlitz-Thouless-\nBerezinskii transition. In the second part, we explore the electromagnetic response of\nvortices in a-MoGe thin film by low-frequency two-coil mutual inductance technique.\nPenetration depth measured from the two-coil technique was earlier used to determine\nsuperfluid density. However in the present work, by analyzing the in-field penetration\ndepth data with the help of a mean-field model proposed by Coffey and Clem, we have\ndemonstrated a procedure of extraction of vortex parameters such as pinning restoring\nforce constant or Labusch parameter, vortex lattice drag coefficient and pinning poten-\ntial barrier for the thermally activated motion of vortices. The temperature variation of\nvortex parameters suggests the dominant effect of thermal fluctuations.\nxiiiList of Publications\nPublications related to this thesis\n•Soumyajit Mandal , Surajit Dutta, Somak Basistha, Indranil Roy, John Jesu-\ndasan, Vivas Bagwe, Lara Benfatto, Arumugam Thamizhavel, and Pratap Ray-\nchaudhuri; Destruction of superconductivity through phase fluctuations in ultrathin\na-MoGe films ;Physical Review B ,102, 060501(R) (2020).\n•Soumyajit Mandal , Somak Basistha, John Jesudasan, Vivas Bagwe, and Pratap\nRaychaudhuri; Study of vortex dynamics in an a-MoGe thin film using low-frequency\ntwo-coil mutual inductance measurements ;Superconductor Science and Technology ,\n36, 014004 (2023).\nOther publications\n•Indranil Roy, Surajit Dutta, Aditya N. Roy Choudhury, Somak Basistha, Ilaria\nMaccari, Soumyajit Mandal , John Jesudasan, Vivas Bagwe, Claudio Castellani,\nLara Benfatto, and Pratap Raychaudhuri; Melting of the vortex lattice through\nintermediate hexatic fluid in an a-MoGe thin film ;Physical Review Letters ,122,\n047001 (2019).\n•Surajit Dutta, Indranil Roy, Soumyajit Mandal , John Jesudasan, Vivas Bagwe,\nand Pratap Raychaudhuri; Extreme sensitivity of the vortex state in a-MoGe films\nto radio-frequency electromagnetic perturbation ;Physical Review B ,100, 214518\n(2019).\nxv•Surajit Dutta, Indranil Roy, Somak Basistha, Soumyajit Mandal , John Jesu-\ndasan, Vivas Bagwe and Pratap Raychaudhuri; Collective flux pinning in hexatic\nvortex fluid in a-MoGe thin filmn ;Journal of Physics: Condensed Matter ,32,\n075601 (2020).\n•Somak Basistha, Vivas Bagwe, John Jesudasan, Gorakhnath Chaurasiya, Soumya-\njit Mandal , Surajit Dutta, and Pratap Raychaudhuri; Growth and Character-\nization of amorphous Molybdenum Germanium (a-MoGe) and amorphous Rhe-\nnium Zirconium (a-Re 6Zr) superconducting thin films using Pulsed Laser Deposition\n(PLD) technique ;Society for Materials Chemistry Bulletin ,11(3), 150 - 159 (Dec\n2020).\n•Somak Basistha, Soumyajit Mandal , John Jesudasan, Vivas Bagwe, and Pratap\nRaychaudhuri; Low frequency electrodynamics in the mixed state of supercon-\nducting NbN and a-MoGe films using two-coil mutual inductance technique ;\narXiv:2311.03752. (submitted)\nxviList of Figures\n1.1 Universal BCS superconducting gap variation . . . . . . . . . . . . . . . 9\n1.2 Density of states in superconducting and normal state . . . . . . . . . . . 11\n1.3 Variation of normalized density of states with different Dynes parameter\nvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n1.4 Temperature variation of simulated normalized differential conductance\nspectra for normal-to-superconductor tunneling. . . . . . . . . . . . . . . 13\n1.5 Comparison of the temperature variation of λ−2\nLin the clean and dirty limit. 15\n1.6 Variation of normalized vortex relaxation time and flux-creep factor as a\nfunction of normalized potential barrier . . . . . . . . . . . . . . . . . . . 31\n2.1 Pulsed laser deposition setup . . . . . . . . . . . . . . . . . . . . . . . . . 34\n2.2 Sputtering setup for thin film deposition . . . . . . . . . . . . . . . . . . 35\n2.3 Thermal evaporation setup . . . . . . . . . . . . . . . . . . . . . . . . . . 36\n2.4 Different masks used for thin film deposition . . . . . . . . . . . . . . . . 37\n2.5 Instrument assembly for low-temperature measurements . . . . . . . . . 38\n2.6 The4He and3He VTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n2.7 Schematic of the two-coil circuit . . . . . . . . . . . . . . . . . . . . . . . 41\n2.8 The two-coil mutual inductance probe head . . . . . . . . . . . . . . . . 42\n2.9 The two-coil mutual inductance setup . . . . . . . . . . . . . . . . . . . . 43\n2.10 Comparison of simulated radial distribution of field and induced current\ndensity due to quadrupole and dipole coils . . . . . . . . . . . . . . . . . 47\n2.11 Schematic of the broadband microwave circuit . . . . . . . . . . . . . . . 48\n2.12 The broadband microwave insert . . . . . . . . . . . . . . . . . . . . . . 49\n2.13 Schematic of scanning tunneling microscope and typical conductance spec-\ntrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52\n2.14 Typical vortex image and Delaunay triangulation . . . . . . . . . . . . . 53\nxviiLIST OF FIGURES\n2.15 Tracking the movement of vortices by STM . . . . . . . . . . . . . . . . . 54\n2.16 Hall-bar geometry for four-probe magneto-transport setup . . . . . . . . 55\n2.17 Setup for simultaneous two-coil mutual inductance and magneto-transport\nmeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56\n2.18 SQUID-VSM setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57\n3.1 Variation of TcandkFlfora-MoGe thin films of different thicknesses. . . 60\n3.2 Commercial and arc-melted MoGe target . . . . . . . . . . . . . . . . . . 61\n3.3 Representative STM images of vortices in vortex solid (VS), hexatic vortex\nfluid (HVF), and isotropic vortex liquid (IVL) in a 20- nm-thick a-MoGe\nfilm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63\n3.4 Phase space boundaries in a 20-nm-thick a-MoGe film . . . . . . . . . . . 63\n3.5 M-H loop in 20- nm-thick a-MoGe films . . . . . . . . . . . . . . . . . . . 64\n3.6 ZFC and FC data in a 20-nm-thick a-MoGe film . . . . . . . . . . . . . . 65\n3.7 Peak effect in JcandρTAFF in a 20-nm-thick a-MoGe film . . . . . . . . 66\n3.8 A hint of peak effect in broadband microwave measurements in amorphous\n20-nm-thick MoGe film . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67\n3.9 Histogram of the number of vortices with different displacements as a frac-\ntion of vortex lattice constant at 2 Kfor different fields in amorphous\n20-nm-thick MoGe film . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68\n3.10 RMS displacement of vortices as a fraction of vortex lattice constant at 2\nKfor different fields in amorphous 20- nm-thick MoGe film . . . . . . . . 69\n3.11 Signature of peak effect at extremely low fields in SQUID-VSM measure-\nments in amorphous 20- nm-thick MoGe film . . . . . . . . . . . . . . . . 70\n3.12 SEM and AFM images of a MoGe sample grown on anodic alumina template 71\n3.13 Vortex matching effect in amorphous MoGe film . . . . . . . . . . . . . . 72\n4.1 Sheet resistance RsvsTfora-MoGe films with different thicknesses, and\nvariation of normal state sheet resistance and resistivity as a function of\ninverse thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78\n4.2 Normal state RsandTcs as a function of film thickness and the Finkel’stein\nfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79\n4.3 Representative tunneling G(V) vsVspectra and normalization for Altshuler-\nAronov-type electron-electron interactions . . . . . . . . . . . . . . . . . 80\n4.4 Normalized tunneling spectra and zero-bias conductance maps for 11-, 4 .5-\nand 2- nm-thick films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81\n4.5 Normalized tunneling spectra for different thicknesses and temperature\nvariation of ∆ and Γ extracted from the spectra . . . . . . . . . . . . . . 83\n4.6 ∆(0) /kBTcand ∆(0) as a function of Tc, and variation of Tcand gap-\nvanishing temperature T∆with film thickness . . . . . . . . . . . . . . . 84\nxviiiLIST OF FIGURES\n4.7 λ−2as a function of temperature for films with different thicknesses . . . 85\n4.8 Comparison of λ−2(T→0) with BCS estimate . . . . . . . . . . . . . . . 86\n4.9 Temperature variation of λ−2for 2.2-nm-thick sample highlighting quadratic\n(T2) dependence at very low temperatures . . . . . . . . . . . . . . . . . 87\n4.10 λ−2as a function of temperature for films with different thicknesses with\nBCS variation taking TDelta as transition . . . . . . . . . . . . . . . . . . 89\n4.11 Comparison of Jsand ∆ as a function of thickness d. . . . . . . . . . . . 91\n5.1 M−Hloop and comparison of ZFC and FC data for weakly-pinned 20-\nnm-thick a-MoGe film . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95\n5.2 Temperature variation of experimentally measured M′andM′′at different\nfields with the corresponding λ−2andδ−2. . . . . . . . . . . . . . . . . . 96\n5.3 “ λ−2-vanishing” temperature points ( T∗) in the H−Tphase boundary . 97\n5.4 “Magnetic field variation of λ−2andδ−2at 2K. . . . . . . . . . . . . . 98\n5.5 Simulated radial distribution of field on the sample due to primary quadrupole\ncoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99\n5.6 Hc2as a function of temperature . . . . . . . . . . . . . . . . . . . . . . 100\n5.7 Temperature variation of λ−2with fit from Coffey-Clem equation . . . . 105\n5.8 Field dependence of U0/2kB. . . . . . . . . . . . . . . . . . . . . . . . . 105\n5.9 Field dependence of αL0. . . . . . . . . . . . . . . . . . . . . . . . . . . 106\n5.10 Field dependence of T0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108\n5.11 Experimental δ−2vsTwith fit to the imaginary part of the Coffey-Clem\nequation with best-fit parameters from λ−2vsTfits . . . . . . . . . . . . 108\n5.12 Comparison of superfluid stiffness J2K\nsdirectly calculated from λ−2at 2K\nand the in-field transition temperature T∗\nc. . . . . . . . . . . . . . . . . . 109\nA.1 λ−2vsTfor 2.2-nm-thick sample with dirty BCS and BKT fits . . . . . 120\nB.1 λ−2vsTwith trial temperature variations of αLandU. . . . . . . . . . . 123\nB.2 χ2as a function of T0,αL0andU0/2kBrespectively for four representative\nfields: 2, 5, 10 and 30 kOe. . . . . . . . . . . . . . . . . . . . . . . . . . . 125\nB.3 Field dependence of T0,αL0andU0/2kBvalues at the χ2minima points. 126\nxixList of Tables\n1.1 The expressions of flux-creep factor ϵeffand vortex relaxation time τeff\nfor different models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\n4.1 Comparison of Tc,T∆and Γ (in K) for two thinnest samples of thickness\nd(innm) as curated from Fig 4.5 and Fig. 4.6. . . . . . . . . . . . . . . 88\nB.1 Comparison of Coffey-Clem fits using different trial temperature variations\nofUandαL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124\nxxiPreamble\nThis thesis explores the evolution of superconductivity in amorphous Molybdenum Ger-\nmanium ( a-MoGe) thin films, focusing on the effect of phase fluctuations on the sup-\npression of superconductivity with decreasing thickness. Additionally, it investigates the\nelectromagnetic response of vortices in these films, extracting vortex parameters and\nexamining the impact of thermal fluctuations.\nThe layout of the thesis is as follows:\n•Chapter 1 discusses the basic concepts of superconductivity, including BCS and\nGinzburg-Landau theory. Additionally, necessary theoretical concepts and litera-\nture surveys required for the ensuing work are also explored and presented.\n•Chapter 2 focuses on the discussion of sample preparation and measurement tech-\nniques.\n•Chapter 3 describes the sample details and characterization of the sample. Addi-\ntionally, it includes a discussion of some recent results that have not been covered\nin the later chapters.\n•Chapter 4 delves into the effect of decreasing thickness in a-MoGe thin films and\nhow phase fluctuations affect the superconductivity at lower thicknesses.\n•Chapter 5 examines the vortex dynamics in a-MoGe in the presence of low-frequency\nsmall ac excitation and demonstrates the extraction of vortex parameters.\n•Chapter 6 provides a summary of our work and presents an outlook for future\nstudies.\nxxiiiStatement of Joint Work\nThe experiments reported in this thesis were conducted in the Department of Condensed\nMatter Physics and Materials Science at the Tata Institute of Fundamental Research\nunder the guidance of Prof. Pratap Raychaudhuri. A significant portion of the results\npresented here has already been published in refereed journals.\nWhile I have conducted the majority of the experiments and analyses presented in this\nthesis, I have also included some collaborative work carried out by others. The details of\nthese collaborations are as follows:\n•Thea-MoGe thin films for two-coil mutual inductance measurements were grown\nin collaboration with Somak Basistha and John Jesudasan. The MoGe target for\nsample deposition in Pulsed Laser Deposition (PLD) was grown in collaboration\nwith Prof. Arumugam Thamizhavel.\n•In the investigation of the effect of phase fluctuations on ultrathin a-MoGe films, I\nconducted the penetration depth measurements and analyses. Scanning Tunneling\nSpectroscopy (STS) measurements and analyses were performed by Surajit Dutta\nand Indranil Roy. Transport measurements were carried out by Surajit Dutta and\nSomak Basistha. Theoretical inputs were provided by Prof. Lara Benfatto.\n•For the study of vortex dynamics in a-MoGe, I collaborated with Somak Basistha\nfor the penetration depth measurements and analyses.\n•SQUID-VSM measurements on a-MoGe were performed with assistance from Ganesh\nJangam.\n•Broadband microwave measurements on a-MoGe films were conducted in collabo-\nration with Somak Basistha.\n•The study of the movement of vortices in a-MoGe using STS and subsequent anal-\nyses were done in collaboration with Indranil Roy.\nxxvCHAPTER1\nIntroduction\nSuperconductivity, discovered by Heike Kamerlingh Onnes1,2in 1911, has been a subject\nof intense research for over a century. While the practical applications of supercon-\nductivity have been a driving force behind extensive research, understanding its funda-\nmental aspects has also been crucial. In the 1950s, Bardeen, Cooper, and Schrieffer3\n( BCS ) revolutionized our understanding of superconductivity, providing a microscopic\nexplanation for the phenomenon and paving the way for practical applications. In an-\nother major breakthrough, Vitaly Ginzburg and Lev Landau introduced a macroscopic\nframework within Landau’s general theory4of second-order phase transitions, known as\nGinzburg Landau theory5,6( GL ), which further deepened our comprehension of super-\nconducting materials and their behavior near the critical temperature. The discovery of\nhigh-temperature superconductors by Bednorz and M¨ uller7further revitalized the field,\nalthough the basic microscopic mechanism in these new superconductors remains an open\nquestion to date. However, their overall phenomenology follows the patterns observed in\nconventional superconductors. Hence, a common strategy is often adopted to explore\nseveral high-temperature phenomena, including the presence of a pseudogap phase8,9,\nquantum fluctuations of vortices10, and phase fluctuations11in conventional supercon-\nductors with low superfluid density, in order to uncover similarities and draw connections\nbetween the behavior of high-temperature superconductors and that of traditional super-\nconductors.\nThe superconducting state is primarily dependent on two phenomena: (i) weak at-\ntractive interaction between a pair of electrons with opposite momenta and opposite\nspins forming a spin-zero Boson-like object called Cooper pairs and (ii) condensation\n11.1. BASICS OF SUPERCONDUCTIVITY\nof all such Cooper pairs into a phase-coherent macroscopic quantum state. The extent\nof phase coherence depends on the phase rigidity of the superconducting state and it\ngives rise to two of the most defining properties of the superconductor: zero resistance\nand perfect diamagnetism. The response to the external magnetic field further classifies\nthe superconductors- type-I, and type-II. Type-I superconductors behave like a perfect\ndiamagnet in the presence of an external field until it reaches a critical field ( Hc) char-\nacteristic of each material. On the other hand, at low fields type-II superconductors are\nin perfect diamagnetic or Meissner state up to the lower critical field Hc1, above which\nquantized flux lines ( vortices ) penetrate the sample before going to the normal state\nabove the upper critical field Hc2.\nWe have studied the evolution of superconductivity in amorphous Molybdenum Ger-\nmanium ( a-MoGe ) thin films. The work can be broken down into two parts. In the\nfirst work, we have studied the effect of disorder on the superconductivity in the absence\nof a magnetic field in a-MoGe thin films and the interplay of disorder and phase fluctu-\nations on the superconducting state. In the second work, we have explored the response\nof vortices to small ac excitation ( kHz frequency range ) in the weakly-pinned a-MoGe\nfilm.\nIn this chapter, I shall first introduce the basic theoretical understanding and the\nphenomenology of superconductivity including BCS and GL theory. Then I shall provide\na brief literature review of the effect of disorder on superconductivity as an introduction\nfor the first work ( Chapter 4 ). Also, I shall briefly discuss the current understanding of\nthe effect of ac excitation on the vortex lattice as an introduction to the second work\n( Chapter 5 ).\n1.1 Basics of Superconductivity\nThe two fundamental properties distinguishing a superconductor from a normal metal\nare its zero resistance and perfect diamagnetism.\n1.1.1 Zero resistance\nIt was first observed by Kamerlingh Onnes1,2in 1911 that the electrical resistance of\nvarious metals such as mercury, tin, and lead drops below measurable limits below a\ncertain critical temperature, Tc, characteristic of the corresponding material. Later this\nphenomenon was observed in a lot of other materials. This property is most successfully\ndemonstrated by the persistent current in superconducting loops which has the charac-\nteristic decay time of more than 105years12. This perfect conductivity has made the\nsuperconductors the potential candidates for applications in the transmission of high\ncurrent and high field magnets.\n2CHAPTER 1. INTRODUCTION\n1.1.2 Perfect diamagnetism\nThe other defining property of a superconductor, discovered by Meissner and Oschsen-\nfeld13in 1933, is perfect diamagnetism. When a superconductor is cooled below its\ncharacteristic transition temperature Tc, it repels the external magnetic field irrespective\nof its magnetic history, whether the field was applied before or after cooling. In contrast,\na perfect conductor would trap the flux inside if they are cooled in the field. This perfect\ndiamagnetism in superconductors is known as the Meissner effect.\n1.1.3 Critical field\nWhile superconductors exhibit perfect diamagnetism in weak magnetic fields, it is im-\nportant to note that their superconductivity can be disrupted by strong magnetic fields.\nWhen the magnetic field surpasses a certain threshold known as the critical field, Hc,\nmost elemental superconductors lose their diamagnetic properties and start behaving like\nnormal metals. It reflects in the increase in the free energy of the superconducting state\nin the presence of an external magnetic field, also known as condensation energy, ϵcond:\nϵcond=fs(Hc)−fs(0) =1\n2µ0H2\nc. (1.1)\nHcis a parameter dependent on the superconducting material. Experimental findings\nhave revealed that the temperature dependence of the critical fields in different super-\nconductors can be approximated by the following empirical relation14,\nHc(T) =Hc(0)\u0002\n1−(T/T c)2\u0003\n. (1.2)\n1.1.4 London equation\nStrictly speaking, the perfect diamagnetism mentioned above is only valid for bulk sam-\nples. For finite-size superconductors, the magnetic field does penetrate the superconduc-\ntor to a finite distance from the surface, known as London penetration depth ( λL). Fritz\nand Heinz London15in 1935 came up with a phenomenological model to explain the two\nbasic electrodynamic properties of superconductors mentioned above. They proposed\nthat the microscopic electric and magnetic fields of the superconducting state follow the\nfollowing two equations:\nE=∂(ΛJs)\n∂t, (1.3)\nB=∇×A=−∇×(ΛJs), (1.4)\nHere, Λ = me/nse2is a phenomenological parameter ( meis the effective electron mass,\nµ0is vacuum permeability, nsis superfluid density and eis electron charge ), which is\n31.1. BASICS OF SUPERCONDUCTIVITY\nrelated to the London penetration depth by the following relation:\nλL=rme\nµ0nse2=s\nΛ\nµ0(1.5)\nEq. 1.3 pertains to the acceleration of electrons, rather than their ability to sustain\nvelocity in the presence of resistance caused by electric fields. As such, it signifies the\nperfect conductivity exhibited by superconductors, where electrons can move without\nhindrance. Eq. 1.4 combined with the Maxwell’s equation, ∇×B=µ0Js, gives rise to\nthe following equation:\n∇2B=B\nλ2\nL(1.6)\nLet us write Eq. 1.6 in a simple case where a superconductor is placed in a uniform\napplied magnetic field Ba=B0ˆz. Hence, the magnetic field outside the surface of the\nsuperconductor is, B(x= 0) = B0ˆzwhere B(x) is the magnetic induction inside the\nsuperconductor and x is the distance from the surface. Thus Eq. 1.6 is simplified as:\n∂2B(x)\n∂x2=B(x)\nλ2\nL(1.7)\nSolution to Eq. 1.7 is given by:\nB(x) =B0e−x/λL(1.8)\nThis means that magnetic induction Bdecays exponentially inside the superconductor\nwith the characteristic length λL. Additionally, it follows from Eq. 1.4 that the current\nalso decays exponentially inside the superconductor.\n1.1.5 Pippard’s Coherence length\nIn 1953, Pippard16introduced a crucial concept regarding superconducting electrons. He\nargued that the density of superconducting electrons in space cannot abruptly change\nto zero. The shortest length scale, over which the superconducting electron density can\nbe modified to transition from a superconductor to a normal metal, is known as the\ncoherence length, denoted by ξ. However, the presence of impurities in the material\ncan reduce this coherence length. For impure materials or alloys, ξis given by ( ξ0l)1/2,\nwhere ξ0is the coherence length of the pure material and lis the electronic mean free\npath of the material. Pippard employed the position-momentum uncertainty argument\nto demonstrate that ξ0∼aℏvF\nkBTc, where ais a constant of order 1, vFis the Fermi velocity,\nℏis the reduced Planck constant, kBis the Boltzmann constant, and Tcis the transition\ntemperature.\n4CHAPTER 1. INTRODUCTION\n1.1.6 Type-I and Type-II superconductors\nThe response to the external magnetic field classifies the superconductors into two cate-\ngories - type-I and type-II. Type-I superconductors exhibit perfect diamagnetism (Meiss-\nner effect) up to a critical magnetic field, Hc, beyond which they transition to a normal\nstate. Except for Nb, elemental superconductors fall in the type-I class of superconduc-\ntors.\nOn the other hand, in type-II superconductors, which include nearly all supercon-\nducting alloys and compounds, the magnetic response to the applied field is relatively\ncomplex. It shows perfect diamagnetism up to a lower characteristic field, Hc1, where\nthe magnetic flux lines do not penetrate the superconductor. However, when the applied\nmagnetic field exceeds Hc1, quantized magnetic flux lines start to penetrate inside the\nsuperconductor, giving rise to a vortex state where superconducting regions coexist with\nnormal regions, and the flux lines pass through the normal regions. With further increase\nin the magnetic field beyond a second characteristic critical field, Hc2, the superconduc-\ntivity is completely destroyed, and the material transitions to a normal state. Hc2is\nknown as the upper critical field.\n1.2 BCS theory\nThe Bardeen-Cooper-Schrieffer ( BCS ) theory of superconductivity revolutionized our\nunderstanding of the behavior of electrons in materials at low temperatures. The theory,\nproposed in 1957 by John Bardeen, Leon Cooper, and John Robert Schrieffer3, provided\na comprehensive explanation for the emergence of superconductivity, the phenomenon in\nwhich certain materials can conduct electricity with zero resistance at very low tempera-\ntures. According to the theory, even a weak attractive interaction between the electrons\ncan lead to the formation of a macroscopic quantum state, in which many electrons be-\nhave as a single entity. This state arises due to the interaction between electrons and\nthe crystal lattice and is characterized by the presence of an energy gap in the density of\nstates of electrons. The BCS theory has since been confirmed by numerous experiments\nand has led to many important technological applications, such as magnetic resonance\nimaging ( MRI ) and particle accelerators.\nIn this section, I shall give a brief overview of this elegant theory and its novelty to\ndescribe different properties of superconductors.\n1.2.1 Cooper pairs\nIn 1950, Fr¨ ohlich17suggested that electrons can experience an attractive interaction me-\ndiated by phonons, contrary to the commonly held belief that electrons always repel each\n51.2. BCS THEORY\nother. This occurs because when an electron travels through the lattice, it creates a\npositive charge imbalance that attracts other electrons.\nSix years later, Cooper18showed that even a weak attractive interaction can cause two\nelectrons with opposite momenta and spin to form a bound pair known as a Cooper pair.\nTo investigate the properties of a superconducting state, Cooper considered a two-particle\nwave function for the Cooper pair, assuming that the two electrons did not interact\nwith the rest of the Fermi sea. For simplicity, he assumed a momentum-independent\nattractive potential ( −V). By solving Schrodinger’s equation, Cooper was able to obtain\nthe eigenvalues of the wave function for this state as the following:\nE≈2EF−2ℏωDe−2/N(0)V, (1.9)\nwhere N(0) represents the density of states ( DOS ) at the Fermi sea for electrons of\none spin, while EFis the Fermi energy and ωDis the Debye frequency of the material.\nEq. 1.9 indicates that even if the net attractive potential Vis very small, the eigenvalues\nof the wave function are negative with respect to the Fermi energy, resulting in the\nformation of bound states of electrons with kinetic energy greater than EF. This analysis\nassumes that the coupling between the electrons and the lattice is weak, i.e., N(0)V≪1\n( weak-coupling approximation )12.\n1.2.2 BCS ground state\nCooper’s18approach of considering a single pair of electrons was not sufficient to describe\nreal-world systems. Dealing with multiple Cooper pairs in a state was a challenging task.\nTo tackle this issue, Bardeen, Cooper, and Schrieffer3proposed a novel expression for the\nground state of a superconductor:\n|ΨBCS⟩=Y\nk(uk+vkeiϕkc†\nk↑c†\n−k↓)|0⟩, (1.10)\nwhere |0⟩is the vacuum state in the absence of any particle. c†\nk↑(c−k↓) is the creation\n(annihilation) operator of an electron of momentum k(−k) and spin up (down). The\nprobability of having a bound pair of electrons ( Cooper pair ) is v2\nkand not having is\nu2\nk, with the normalization condition: u2\nk+v2\nk= 1. Here, ukandukare chosen to be real\nwhile the phase part is absorbed in ϕk, which is the phase of each Cooper pair. The use\nof a wavefunction expressed as a product of individual wavefunctions shown in Eq. 1.10,\nrather than a many-body wavefunction composed of Slater determinants of momentum\neigenfunctions, is only justified when dealing with a large number of particles.\nWe first consider the ground state of the superconductor. We can determine the\n6CHAPTER 1. INTRODUCTION\ncoefficients uk,vk, and ϕkby minimizing the ground state energy Eby variational method,\nδE=δ⟨ΨBCS|Hpair|ΨBCS⟩= 0, (1.11)\nwhere Hpairis the BCS pairing Hamiltonian denoted by,\nHpair=X\nk,σ(ϵk−µ)nk,σ+X\nk,lVk,lc†\nk↑c†\n−k↓c−l↓cl↑\n=X\nk,σξkc†\nk,σck,σ+1\n2 X\nk,lVk,lc†\nk↑c†\n−k↓c−l↓cl↑+X\nk,lVl,kc†\nl↑c†\n−l↓c−k↓ck↑!\n.(1.12)\nThe first term in the right-hand side of Eq. 1.12 corresponds to the kinetic energy of\nnoninteracting electrons, where ϵkis the kinetic energy of a single electron, µis the\nchemical potential and nk,σis called particle number operator defined by nk,σ=c†\nk,σck,σ\nwith momentum kand spin σ. The chemical potential µ( or Fermi energy EF) is\nincluded in Eq. 1.12 under the framework of grand canonical ensemble, where the mean\nnumber of particles is kept fixed instead of the total number of particles, N. This choice of\nthe ensemble is justified in the large Nlimit, where fractional uncertainty in Nbecomes\nnegligible. In the second equality, ξkis defined as the kinetic energy of a single electron\nwith respect to Fermi level ( EForµ),ξk=ϵk−µ.\nOn the other hand, the second term in the right-hand side of Eq. 1.12 pertains to the\npairing interaction due to second-order phonon scattering. In the second equality, the\nscattering term has been split between both the scattering possibilities for the combina-\ntion of momenta kandl, taking into account the symmetry of the term. Additionally,\nfollowing Cooper’s approach18, BCS adopted a simplified form for the electron-phonon\nscattering matrix element Vk,las follows,\nVk,l=\n\n−Vfor|ξk|,|ξl| ≤ℏωD\n0 otherwise .(1.13)\nBy utilizing Eq. 1.10, 1.12, and 1.13, we can calculate the expectation value of energy\nat the ground state as follows,\nE=⟨ΨBCS|Hpair|ΨBCS⟩\n= 2X\nkξkv2\nk+1\n2X\nk,l(−V)ukvkulvl\u0002\ne−i(ϕk−ϕl)+ei(ϕk−ϕl)\u0003\n.(1.14)\n71.2. BCS THEORY\nNow, to minimize Ein Eq. 1.14 by varying ϕk, we obtain,\n∂E\n∂ϕk=1\n2X\nk,l(−V)ukvkulvl\u0002\n−ie−i(ϕk−ϕl)+iei(ϕk−ϕl)\u0003\n= 0\n⇒VX\nk,lukvkulvl\u0014ei(ϕk−ϕl)−ei(ϕk−ϕl)\n2i\u0015\n= 0\n⇒VX\nk,lukvkulvlsin (ϕk−ϕl) = 0 .(1.15)\nNow, excluding a trivial solution for uk(ul) orvk(vl) for the ground state energy, we get\nto the condition,\nsin (ϕk−ϕl) = 0\n⇒ϕk=ϕl.(1.16)\nThus, ϕkis independent of momentum k, indicating that the phase ϕis the same for\nall the Cooper pairs, implying phase coherence in the superconducting ground state.\nHowever, this condition is only true for T= 0. At finite temperatures, the effects of\nphase fluctuations come into play, although they have a negligible impact on conventional\nsuperconductors.\nSimilar to ϕk,ukandvkcan also be determined by minimizing the ground state\nenergy as follows12,19,\nv2\nk=1\n2\u0012\n1−ξk\nEk\u0013\n=1\n2\"\n1−ξk\n(∆2+ξk)1/2#\n, (1.17)\nand\nu2\nk=1\n2\u0012\n1 +ξk\nEk\u0013\n= 1−v2\nk. (1.18)\nOnce we get the values of the coefficients ukandvk, both |ΨBCS⟩and ground state\nenergy can be determined. It is found that the energy difference between the ground\nstate superconducting state and normal state is given by the following,\nϵcond=fn(0)−fs(0) =1\n2N(0)∆2(0), (1.19)\nwhere fnandfsare the free energy per unit volume at normal and superconducting states\nrespectively and ∆(0) is the superconducting gap which is the manifestation of the pairing\nenergy in the electronic spectrum. This difference is equivalent to the condensation energy\nϵcondstated in Eq. 1.1. ∆(0) at ground state ( or zero temperature ) is determined in the\n8CHAPTER 1. INTRODUCTION\nBCS theory by the following expression12:\n∆(0) = 2 ℏωDe−1/N(0)V(1.20)\n0.00.20.40.60.81.00.00.20.40.60.81.0Δ(Τ) / Δ(0)T\n / TcΔ(0) = 1.76 kBTc\nFigure 1.1: BCS variation of normalized ∆ as a function of T/T c(universal BCS gap variation)\nobtained by numerically solving Eq. 1.22 under weak-coupling approximation12(N(0)V≈0.3).\n1.2.3 Elementary excitations and BCS gap variation at finite T\nAt finite temperatures, the breaking of Cooper pairs gives rise to electron and hole-like\nquasi-particles, which are the elementary excitations in superconductors. While the vari-\national method can be used to deal with excited states, a more elegant and sophisticated\napproach is the Bogoliubov-Valatin20–22self-consistent method. The energy of the quasi-\nparticles is given by12,19,\nEk= (ξ2\nk+ ∆2)1/2. (1.21)\nBy solving the Bogoliubov-Valatin self-consistent equation, we get the following equation:\n1\nN(0)V=ZℏωD\n0tanhβ\n2(ξ2+ ∆2)1/2\n(ξ2+ ∆2)1/2dξ. (1.22)\nThe temperature variation of superconducting gap ∆ can be obtained by numerically\nsolving Eq. 1.22 ( shown in Fig. 1.1 ). According to the BCS theory, ∆( T) or the pairing\nenergy of the Cooper pairs goes to zero at transition temperature Tc. Hence, at Tcthe\nEq. 1.22 translates into the expression:\nkBTc= 1.13ℏωDe−1/N(0)V. (1.23)\n91.2. BCS THEORY\nNow, Eq. 1.23 combined with Eq. 1.20 gives rise to the BCS prediction about the relation\nbetween ∆ and Tc:\n∆(0)\nTc= 1.764. (1.24)\nThis signifies that the superconducting gap vanishes at Tcaccording to BCS theory, which\nis true for conventional superconductors.\nHowever, it is worth noting that BCS did not consider the effect of phase fluctuations\nat finite temperatures. To date, a comprehensive microscopic theory that can effectively\naddress the entire problem, accounting for both quasi-particle excitations and phase fluc-\ntuations on equal terms, remains elusive. Nevertheless, the effect of phase fluctuations\ncan be formulated in the framework of a phase-only Hamiltonian, which will be discussed\nin Section 1.4.\n1.2.4 BCS density of states\nThe quasi-particle excitation is Fermionic in nature and the quasi-particles behave like\nnormal electrons. Also, the Bogoliubov creation operator which creates the quasi-particles\nis in one-to-one correspondence with the creation operator c∗\nkfor normal electrons. Hence,\nwe can get the superconducting DOS, Ns(E) by equating with the normal DOS, Nn(ξn),\nNs(E)dE=Nn(ξn)dξn. (1.25)\nSince we are considering ξnonly few meV around the Fermi level, we can assume nor-\nmal DOS to be constant in that range so that Nn(ξn)≈Nn(0). Hence, we get the\nsuperconducting DOS by the following relation,\nNs(E)\nNn(0)=dξ\ndE=\n\nE\n(E2−∆2)1/2(E >∆)\n0 ( E <∆)(1.26)\nThe comparison of superconducting and normal density of states (DOS) is illustrated in\nFig. 1.2(a) ( showing only the positive half of the quadrant ). At E= ∆, as also evident\nfrom Eq. 1.26, the superconducting DOS diverges. In Fig. 1.2(b), the superconducting\nDOS is plotted for different temperatures. At temperatures very close to zero, it diverges\nin proximity to the superconducting gap at zero temperature, ∆(0). These points of di-\nvergence at E=±∆ are known as coherence peaks. With an increase in temperature, the\ncoherence peak shifts to lower values of ∆, identifiable from the temperature-dependent\ngap values curated in Fig. 1.1. Beyond Tc, it becomes similar to the normal DOS.\n10CHAPTER 1. INTRODUCTION\n01 2 3 4 01234Ns(E) / Nn(0)E\n ( meV ) T → 0 \nT = 0.9Tc \nT = 0.99Tc \nT > Tc(\nb)\n01 2 3 4 01234Ns(E) / Nn(0)E\n / ΔSuperconductingN\normal  E/(E2 - D2)1/2, E > D \n          0, E < D(\na)\nFigure 1.2: (a) The comparison of the BCS density of states (DOS) (red line) with the normal\nmetal DOS (blue line). At E= ∆ (indicated by the vertical dotted line), the superconducting\nDOS diverges. Both DOS are normalized with respect to the DOS of the normal metal at\nE=EF.(b)The variation of the normalized superconducting DOS is shown at intermediate\ntemperatures ( Tint), where the superconducting DOS diverges at E= ∆( Tint) - derived from\nthe corresponding temperature profile of ∆ in Fig. 1.1. All the plots were simulated using\nEq. 1.26.\n1.2.5 Broadening of coherence peaks: Dynes parameter\nIn the case of disordered superconductors, the coherence peaks in the superconducting\nDOS are comparatively broader than expected from BCS theory. To address this broad-\nening, Dynes et al.23proposed to incorporate a phenomenological parameter Γ in the\nBCS DOS such that Eq. 1.26 becomes,\nNs(E) =Re\nE+iΓq\n(E+iΓ)2−∆2\n. (1.27)\nThe parameter Γ, named after Dynes, is integral to our discussion. The modified density\nof states ( DOS ), as defined in Eq. 1.27, has proven effective in accurately modeling\ntunneling data in both zero-field24–29environments and in the presence of a magnetic\nfield30–32. In fact, the Dynes formula for the density of states ( DOS ) has been ap-\nplied to a wide variety of superconductors, encompassing tunneling spectra in bulk33and\nthin film24,27,29superconductors, superconducting nanoparticles34, two-band supercon-\nductors like MgB 235, and novel superconductors such as CaC 636,37. Additionally, it has\nbeen utilized in photoemission studies of the superconducting h-ZrRuP38and the filled\nskutterudite superconductor LaRu 4P1239.\nDynes et al.23,33initially interpreted the observed broadening as arising from the\nfinite lifetime of quasi-particles at the superconducting gap edge. However, this inter-\npretation has faced criticism40,41on the basis that the parameter is not directly derived\nfrom a microscopic pair-breaking mechanism and alternative forms of the single-particle\n111.2. BCS THEORY\nsuperconducting density of states ( DOS ) have been proposed, each with its own set of\nmicroscopic justifications. Rather than introducing the parameter Γ in Eq. 1.26, Mitrovi´ c\net al.40considered the gap to be complex. On the other hand, Dentelski et al.41employed\na theory of free electrons coupled to superconducting fluctuations with finite-range cor-\nrelations to fit the tunneling spectra. Nevertheless, in recent years, some researchers42,43\nhave proposed microscopic interpretations for the utilization of Γ. Interestingly, for ad-\ndressing the broadening in the presence of magnetic scatterers, Herman et al.42utilized\nthe following elegant transformation suggested by Mikhailovsky et al.44,\n∆′(E) =E∆\nE+iΓ, (1.28)\nwhich, upon substituting ∆ →∆′(E), transforms Eq. 1.26 into Eq. 1.27.\n-4- 20 2 4 01234Ns(E) / Nn(0)E\n ( meV )Γ ( meV ) \n0 \n0.1 \n0.5 \n1 \n2T ® 0\nFigure 1.3: Normalized density of states spectra ( Ns/NnvsE) are presented for various Dynes\nparameter values (Γ), simulated using Eq. 1.27 as T→0. For Γ = 0, a sharp divergence occurs\natE=±∆(0), where ∆(0) = 1 .76kBTc= 1.0626 meV with Tc= 7 K. In the intermediate\nregion, Nsis zero for Γ = 0, as indicated by Eq. 1.26. As we increase Γ, the coherence peaks\nflatten, leading to a finite density of states (DOS) in the intermediate gap region.\nDyne broadening can be attributed to various factors, encompassing spatial inhomo-\ngeneity within the electron system43and the averaging effect over the anisotropic gap\ndistribution in a superconductor26. Another significant contributor to the broadening of\nquasiparticle peaks in Ns(E) stems from pair-breaking scattering, limiting the lifetime\nof quasiparticle states43. Cooper pair-breaking can occur due to diverse factors, such as\nexternal magnetic fields, current, rotation, spin exchange, hyperfine fields, and magnetic\nimpurities12. Thermal phase fluctuations represent another such pair-breaking mecha-\nnism45, holding particular relevance for our system ( refer to Chapter 4 for further\ndetails ). Abrikosov and Gor’kov46initially demonstrated the effect of pair-breaking on\nthe depression of Tcand the modification of BCS density of states in the presence of\nmagnetic impurities. Later, Maki47–50and de Gennes et al.51–53extended these results\nto describe the effects of other pair-breaking perturbations, i.e., those that destroy the\n12CHAPTER 1. INTRODUCTION\ntime-reversal degeneracy of the paired states.\n1.2.6 Measurement of superconducting gap by normal metal-\nto-superconductor tunneling\nAccording to the BCS theory, an energy gap, ∆, which is equivalent to the pairing energy,\nemerges in the electronic density of states ( DOS ) during the transition from the normal\nto the superconducting state. In 1960, Ivar Giaever54–56conducted direct measurements\nof ∆ for superconductors like Al, Pb, and others. He employed tunneling experiments\nin Superconductor-Insulator-Normal ( SIN ) tunnel junctions for this purpose. I shall\nbriefly explain the basics of normal-to-superconductor tunneling.\nTunneling is a quantum mechanical phenomenon that allows particles to pass through\na potential barrier with a non-zero transmission probability, which is forbidden in classical\nphysics. When a normal metal and a superconductor are brought close to each other while\nmaintaining a thin insulating barrier between them, electron tunneling occurs and reveals\nthe density of states of the superconductor.\n-4-20 2 4 012345T\nc = 7 KGN(V)V\n ( mV )  T ( K ) \n0.1 \n1 \n2 \n3 \n4 \n5 \n6 \n6.5 \n7 \n7.5Δ(0) = 1.76kBTc\nFigure 1.4: Normalized differential conductance ( GN(V) vs V) spectra for normal-to-\nsuperconductor tunneling for different temperatures simulated from Eq. 1.30 assuming ∆(0) =\n1.76kBTcwhere Tc= 7K.\nIn the absence of an externally applied voltage, the Fermi levels of the normal metal\nand superconductor align themselves at the same energy in accordance with thermo-\ndynamic principles, resulting in a single chemical potential for the combined system.\nHowever, due to the presence of a single-particle energy gap ( ∆ ) in the superconductor\nacross the Fermi level, tunneling from the normal metal to the superconductor at the\nsame energy is prohibited. Therefore, in the absence of an external bias, no steady-state\ntunneling current is observed.\nTunneling becomes possible only when the applied bias voltage ( V) exceeds the\nenergy gap ( ∆ ), specifically when eV > ∆. By observing the tunneling current under\n131.2. BCS THEORY\napplied bias, the superconducting energy gap can be determined accurately.\nThe tunneling current from the normal to the superconducting side takes the form:\nIns=A|T|2Nn(0)Z∞\n−∞Ns(E)[f(E)−f(E+eV)]dE, (1.29)\nwhere Tis the tunneling matrix element, fis the Fermi-Dirac distribution, NnandNs\nare the electronic density of states of normal and superconducting state respectively.\nThe differential conductance of such SIN tunnel junctions, as a function of the bias\nvoltage, V, can be accurately described using Fermi’s golden rule:\nGns=dIns\ndV=GnnZ∞\n−∞Ns(E)\nNn(0)\u0014\n−∂f(E+eV)\n∂(eV)\u0015\ndE, (1.30)\nwhere Gnnis the differential conductance of the junction above the transition temperature\nwhen the superconductor becomes normal. Now at the limit T= 0, ∂f(E+eV)/∂(eV)\nbecomes a δfunction at E=eVand the differential conductance becomes,\nGns|T=0=GnnNs(e|V|)\nNn(0)=Gnne|V|\n(e|V|2−∆2)1/2. (1.31)\nWe employ this normal-to-superconductor tunneling phenomenon to directly measure\nthe superconducting DOS with the help of a scanning tunneling microscope where the\n“insulator” in the superconductor-insulator-normal metal junction is the vacuum between\nthe STM tip (metal) and the sample (superconductor).\n1.2.7 Temperature variation of penetration depth\nIn the clean limit ( l > ξ 0), the BCS variation of penetration depth ( λL) with temper-\nature is as follows12,\nλ−2\nL(T)\nλ−2\nL(0)\f\f\f\f\nClean=\u0014\n1−2Z∞\n∆\u0012\n−∂f\n∂E\u0013E\n(E2−∆2)1/2dE\u0015\n, (1.32)\nwhere λ−2\nL(0) in the clean limit is determined from Eq. 1.5 where nsbecomes the total\ncarrier density nin the limit T→0,\nλ−2\nL,Clean (0) =µ0ne2\nme. (1.33)\nOn the other hand, the variation of λLIn the dirty limit ( l < ξ 0) is given by57,\nλ−2\nL(T)\nλ−2\nL(0)\f\f\f\f\nDirty=∆(T)\n∆(0)tanh∆(T)\n2kBT, (1.34)\n14CHAPTER 1. INTRODUCTION\nwhere the zero-temperature dirty limit BCS estimate of λLis given by the following\nrelation58,\nλ−2\nL,Dirty (0) =πµ0∆(0)σn\nℏ, (1.35)\nwhich can be estimated from the sum rule for the suppression of superfluid density in the\npresence of disorder59,60(σnis the normal state conductivity ). It should be noted that\na comprehensive expression of the temperature variation of λLas a function of disorder\nhas been derived in Ref. 60.\n0.00.20.40.60.81.00.00.20.40.60.81.0λ−2 L\n(Τ) / λ−2L\n (0)T\n/Tc Clean limit \nDirty limit\nFigure 1.5: Temperature variation of λ−2\nLin the clean and dirty limit simulated from Eq. 1.32\nand 1.34 respectively.\n1.3 Ginzburg-Landau theory\nThe microscopic BCS theory offers excellent explanations for various characteristics of\nsuperconductors, such as nuclear relaxation, energy gap, and elementary excitations, as-\nsuming a constant energy gap throughout space. However, in the case of inhomogeneous\nsuperconductors where the gap varies spatially and involves fluctuations, the BCS theory\nbecomes highly complex. In such scenarios, an intriguing theoretical framework pro-\nposed by Ginzburg and Landau5,6,61( GL ) in 1950, predating the development of the\nBCS theory, presents an elegant depiction of superconductivity near Tc. Although this\ntheory originated as a phenomenological assumption, Gor’kov62demonstrated that the\nGinzburg-Landau theory can be deduced from the BCS theory.\n1.3.1 Order parameter and GL free energy\nTaking a cue from the general theory on second-order phase transition by Landau4,\nGinzburg and Landau proposed a pseudo-wavefunction in the form of a complex order\nparameter, ψ=|ψ|eiϕassociated with the superconducting state. Local superconducting\n151.3. GINZBURG-LANDAU THEORY\nelectron density, ns(r), is assumed to be related to the order parameter in the following\nway:\nns(r) =|ψ(r)|2(1.36)\nAssuming small ψand slow variation in space, free energy density fcan be expanded in\na power series of |ψ|2and|∇ψ|2as follows12:\nf=fn0+α|ψ|2+β\n2|ψ|4+1\n2m∗\f\f\f\f\u0012ℏ\ni∇−e∗A\u0013\nψ\f\f\f\f2\n+1\n2µ0H2, (1.37)\nwhere fn0is the normal state free energy, αandβare phenomenological coefficients, m∗\nande∗are the effective mass and effective charge of a pair of electrons respectively, Ais\nthe vector potential.\nSimilar to the Landau theory of phase transitions4, temperature variation of αandβ\nacross the transition is taken as:\nα=α0(T/T c−1), (1.38)\nand,\nβ=β0. (1.39)\nwhere α0, β0>0\n1.3.2 Ginzburg-Landau equations\nBy minimizing the free energy expression in Eq. 1.37 by applying variational method, we\nget the celebrated Ginzburg-Landau equations:\nαψ+β|ψ|2ψ+1\n2m∗\u0012ℏ\ni∇−e∗A\u00132\nψ= 0, (1.40)\nand\nJ=e∗\nm∗|ψ|2(ℏ∇ϕ−e∗A) =e∗|ψ|2vs, (1.41)\nwhere Jis the supercurrent density and vsis interpreted as the superelectron velocity\ngiven by:\nvs=1\nm∗(ℏ∇ϕ−e∗A) (1.42)\n1.3.3 GL coherence length\nIn zero field, the first GL equation 1.40 can be written in one dimension as follows,\nℏ2\n2m∗|α|d2f\ndx2+f−f3= 0, (1.43)\n16CHAPTER 1. INTRODUCTION\nwhere f=ψ/ψ∞is a normalized wave function with ψ∞=−α/β > 0. A new character-\nistic length scale ξGLcan be defined as evident in Eq. 1.43:\nξ2\nGL(T) =ℏ2\n2m∗|α|=Φ0\n2√\n2Hc(T)λL(T). (1.44)\nξGLis called Ginzburg-Landau coherence length which is the characteristic length scale\nover which a small disturbance of ψwith respect to ψ∞will decay spatially.\n1.3.4 GL parameter and Classification of superconductors\nGinzburg-Landau parameter is defined as the ratio of two characteristic lengths of the\nsystem, namely, London penetration depth ( λL) and GL coherence length ( ξGL):\nκ=λL(T)\nξGL(T). (1.45)\nIn their original paper5,6, Ginzburg and Landau showed by numerical integration that\nthere is a crossover from positive to negative surface energy at κ= 1/√\n2. But, the\nenormity of this crossover was later realized in the work of Abrikosov’s seminal work63–65,\nwhere he demonstrated that the presence of negative surface energy leads to the division\nof flux-bearing normal regions, eventually reaching a quantum limit where each quantum\nof flux Φ 0=h/2epenetrates the sample as an individual flux tube. This new class of\nsuperconductors was named “superconductors of the second group” by Abrikosov and\nlater known as type-II superconductors. We have already introduced the type-I and\ntype-II superconductors in section 1.1.6 which are classified on the basis of the following\nrelation:\nκ\n\n<1√\n2for type-I\n>1√\n2for type-II .(1.46)\n1.3.5 Fluxoid quantization\nWhile investigating the state of a multiply connected superconductor in the presence of a\nmagnetic field, F. London66put forward the concept of the fluxoid associated with each\nhole ( or normal region ) passing through the superconductor. Fluxoid Φ′is defined by,\nΦ′= Φ + µ0λ2I\nJs·dl= Φ +m∗\ne∗I\nvs·dl, (1.47)\n171.3. GINZBURG-LANDAU THEORY\nwhere m∗ande∗are twice the mass mand charge eof electron respectively, and Φ is the\nordinary magnetic flux passing through the integration loop expressed as,\nΦ =ZZ\nSB·dS=I\nA·dl. (1.48)\nIn the framework of GL theory, it is possible to explain the quantization of fluxoids.\nUsing the Bohr-Sommerfeld quantum condition, it can be shown that:\nΦ′=1\ne∗I\n(m∗vs+e∗A)·ds=1\ne∗I\np·ds=nh\ne∗=nΦ0, (1.49)\nwhere Φ 0=h/2e= 2.07×10−15Wbis the flux quantum ( his the Planck constant ).\n1.3.6 Little-Parks experiment\nLittle and Parks67,68conducted a remarkable experiment to demonstrate the quantization\nof the fluxoid. They utilized a thin-walled superconducting cylinder placed in an axial\nmagnetic field and observed shifts ∆ Tc(H) in the critical temperature corresponding to\nthe magnetic flux enclosed by the cylinder.\nFor a thin-walled cylinder of radius Rin the presence of applied field H, flux Φ is\ngiven by,\nΦ =πR2H. (1.50)\nNow, Putting Eq. 1.50 and the fluxoid quantization relation Φ′=nΦ0into Eq. 1.47, we\nget the expression of supercurrent density,\nvs=ℏ\nm∗R\u0012\nn−Φ\nΦ0\u0013\n. (1.51)\nAlso, by minimising GL equation (Eq. 1.37) for a uniform current density in a thin\ncylinder, we get,\n|ψ|2=ψ2\n∞\"\n1−\u0012ξGLm∗vs\nℏ\u00132#\n, (1.52)\nwhere ψ∞=−α/β > 0. Now, at the transition |ψ|2= 0, which leads us to the condition,\n1\nξ2\nGL=\u0012m∗vs\nℏ\u00132\n=1\nR2\u0012\nn−Φ\nΦ0\u00132\n. (1.53)\nUsing the temperature variation of ξGLin Eq. 1.53, we get the following form of fractional\ndepression of Tc,\n∆Tc(H)\nTc∝ξ2\nGL(0)\nR2\u0012\nn−Φ\nΦ0\u00132\n(1.54)\n18CHAPTER 1. INTRODUCTION\nIt is evident from Eq. 1.54 that the suppression of Tcis maximum when\u0010\nn−Φ\nΦ0\u0011\n=1\n2.\nOn the other hand, when\u0010\nn−Φ\nΦ0\u0011\n= 0, vs= 0. As a result, |ψ|2=ψ2\n∞andTcis\nenhanced. For other nonzero values of vs,Tcis less.\n1.3.7 Linearized GL equation and Hc2\nIfψreduces by the application of magnetic field such that ψ≪ψ∞, the|ψ|4term in\n1.40 can be dropped and we get the linearized GL equation12:\n\u0012∇\ni−2πA\nΦ0\u00132\nψ=−2m∗α\nℏ2ψ=ψ\nξ2\nGL(T)(1.55)\nIf we solve Eq. 1.55 for a bulk sample with magnetic field along z-axis with the choice of\nvector potential being Ay=Hx, we can get the expression of the maxima of field H as,\nHc2=Φ0\n2πξ2\nGL(T), (1.56)\nwhere the maxima is the upper critical field ( Hc2) introduced before ( Section 1.1 ).\n1.3.8 Abrikosov vortex state\nGeneral solution to the linearized GL equation 1.55 for a bulk type-II SC was given by\nAbrikosov64as follows:\nψ(x, y) =∞X\nn=−∞Cneinkye−1\n2κ2(x−nk\nκ2)2, (1.57)\nwhere the coefficients kandCnneed to be adjusted to minimize the Gibbs free energy\nsubject to the periodicity condition: Cn+N=Cn. Now, for finding the configuration with\nminimum energy, Abrikosov coined a new parameter,\nβA=⟨ψ4⟩\n⟨ψ2⟩2. (1.58)\nThe configuration with minimum βAwill have the minimum energy. It was calculated by\nAbrikosov64that the minimum energy configuration was a square lattice with βA= 1.18\nand lattice constant ( a□) as follows:\na□=\u0012Φ0\nB\u00131/2\n(1.59)\nBut, later it was found to be incorrect. Kleiner et al.69showed that it is a triangular (\nor hexagonal ) lattice that possesses the minimum energy with βA= 1.16 and has the\n191.4. EFFECT OF PHASE FLUCTUATIONS: THE CONCEPT OF PHASE\nSTIFFNESS\nnearest neighbor distance ( a∆):\na∆=\u00124\n3\u00131/4\n×\u0012Φ0\nB\u00131/2\n= 1.075×\u0012Φ0\nB\u00131/2\n(1.60)\n1.4 Effect of phase fluctuations: the concept of phase\nstiffness\nThe superconducting ground state, characterized by the complex order parameter ψ=\n∆eiϕ, can be destroyed by two types of excitations: (i) quasi-particle excitations formed\ndue to the breaking of Cooper pairs and (ii) phase fluctuations destroying the global\nphase coherence. The energy scale associated with the quasi-particle excitations is the\npairing energy or superconducting energy gap ∆, while the extent of phase fluctuations\nis dictated by another energy scale called phase stiffness Js, which is defined by the\nenergy required to apply a phase twist of unit magnitude over a unit length. For conven-\ntional bulk superconductors, the phase of the superconducting wave function is rigid, i.e.,\nJs≫∆; hence ∆ is the defining energy scale of the superconducting transition which is\nwell explained by the Bardeen-Cooper-Schrieffer3( BCS ) and Eliashberg70mean-field\ntheories, where phase fluctuations were not considered. However, this hierarchy of Jsand\n∆ can be altered in certain situations, which include (i) strongly correlated unconven-\ntional superconductors such as underdoped high- Tccuprates or organic superconductors,\nand (ii) thin films of extremely disordered conventional superconductors71such as NbN,\nInO x, TiN, MoGe, and SrTiO 3/LaAlO 3interfacial superconductors. In such situations,\nthe superfluid density becomes very low and the superconducting state can get destroyed\nby strong phase fluctuations even when the pairing energy ∆ remains finite. Up until\nnow, a comprehensive microscopic theory that can effectively address the entire problem,\nencompassing both quasi-particle excitations and phase fluctuations on equal footing,\nremains elusive. However, if we consider only the phase degree of freedom, the phase\nstiffness Jscan be defined in the framework of GL theory as follows.\nWhen there is no magnetic field, GL equation ( Eq. 1.41 ) for the supercurrent density\nreduces to:\nJ=ℏe∗\nm∗|ψ|2∇ϕ, (1.61)\nwhere we have taken the gauge choice A= 0. If we identify |ψ|2with ns/2,nsbeing the\nsuperfluid density in the unit of electrons per unit volume, Jis given by,\nJ=ns\n2e∗vs, (1.62)\n20CHAPTER 1. INTRODUCTION\nwithvsexpressed as,\nvs=ℏ\nm∗∇ϕ. (1.63)\nNow, the superfluid kinetic energy density is given by,\n1\n2\u0010ns\n2\u0011\n(m∗v2\ns) =1\n2ℏ2ns\n2m∗|∇ϕ|2=1\n2ℏ2ns\n4m|∇ϕ|2, (1.64)\nwhere effective mass of Cooper pair m∗is replaced by the mass of an electron msuch\nthat m∗= 2m. Now, considering only the phase degree of freedom, the Hamiltonian of\nthe system is expressed as,\nH=1\n2ℏ2ns\n4mZ\n|∇ϕ|2d3r. (1.65)\nIt is evident from Eq. 1.65 that the pre-factor in the integralℏ2ns\n4mis a measure of the\nresilience of the superconductor against the phase fluctuations. In order to convert this\nquantity in the characteristic energy scale, we multiply a suitable length a, and define\nthe superfluid stiffness ( Js) expressed as,\nJs=ℏ2nsa\n4m=ℏ2a\n4µ0λ2\nLe2, (1.66)\nwhere the characteristic length scale afor phase fluctuations depends on the relation:\na≈min(d, ξ);dandξbeing the thickness and coherence length respectively. In the\nsecond equality, nsis replaced by London penetration depth λLby the relation 1.5.\n1.5 Effect of disorder on superconductivity\nApart from the search for room-temperature superconductors, another area of research\nin the field of superconductivity focuses on understanding how external parameters, such\nas magnetic fields, disorder, dimensionality, or changes in composition, can affect the\nsuperconducting state. In this section, we shall specifically discuss the impact of the\ndisorder on the state of superconductivity, particularly in the context of superconducting\nthin films.\nDisorder, arising from impurities, lattice imperfections, or external perturbations,\nintroduces spatial and temporal variations that significantly impact the behavior of su-\nperconductors. As the disorder level increases, the transition temperature ( Tc) gradually\ndecreases, eventually leading to a nonsuperconducting ground state. However, even in\nthe absence of the global superconducting ground state, superconducting correlations\ncontinue to considerably influence the electronic properties. These correlations give rise\nto several intriguing phenomena, such as finite high-frequency superfluid stiffness above\nTc72, a giant magnetoresistance peak73–77in strongly disordered superconducting films,\n211.5. EFFECT OF DISORDER ON SUPERCONDUCTIVITY\nthe persistence of magnetic flux quantization even after being driven into an insulating\nstate observed in strongly disordered Bi films78, and the persistence of a pronounced pseu-\ndogap79–81at temperatures significantly higher than Tc. This interplay between disorder\nand superconductivity offers an intricate and captivating avenue for condensed matter\nphysics research.\nThe research on the effect of disorder on superconducting thin films was pioneered by\nShalnikov82, followed by Buckel and Hilsch83,84. The first comprehensive and systematic\nstudy of the suppression of Tcwith decreasing thickness of the films was done by Strongin\net al.85,86.\n1.5.1 Beyond Anderson’s hypothesis\nDebate on how the disorder affects superconductivity dates back to the late fifties when\nAnderson57,87predicted that in an s-wave superconductor attractive interaction forming\nthe Cooper pairs would remain unaffected by the presence of non-magnetic impurities.\nHe argued that the Cooper pairs are formed by time-reversed eigenstates, whose density\nof states should not be strongly perturbed by the disorder. This hypothesis, also known\nas “Anderson’s theorem”, was also demonstrated in the random potential scattering up\nto the leading order by Gor’kov62. This was then loosely interpreted that Tcwould\nalso not be strongly sensitive to disorder. But, subsequent experiments showed that\nthe above is only true in the limit of weak disorder: In the presence of increasingly\nstrong disorder, Tcgets gradually suppressed85,86, and eventually, the material is driven\ninto a non-superconducting state at a critical disorder. The seminal work by Haviland\net al.88on the SI transition inspired a lot of experimental research89–92, where Tcof\na thin superconducting film vanishes at some critical value of the dimensionless sheet\nconductance: g=σℏ/e2. In fact, the suppression of Tcwith an increase in disorder can\nhappen from two origins, namely Fermionic and Bosonic mechanisms71,93–98.\n1.5.2 Fermionic mechanism\nIn the Fermionic route, suppression of the mean-field transition temperature occurs due to\nthe loss of effective screening with an increase in disorder scattering. Due to less screening,\nthe repulsion among the oppositely charged electrons gets dominant over the attractive\npairing interaction, and at a critical disorder, superconductivity is destroyed. There have\nbeen several efforts to capture the effect of Coulomb interaction on the suppression of\nTcwith increasing disorder. Ovchinnikov99calculated the shift in Tcfor thick films (\nthickness d≫l, where lis the mean free path ) taking into account the electromagnetic\nfluctuations, whose main contribution was related to the scalar potential or Coulomb in-\nteraction. However, interest in the interplay of disorder and electron-electron interaction\ntruly blossomed when Altshuler and Aronov100–102, in 1979, showed that Coulomb inter-\n22CHAPTER 1. INTRODUCTION\naction along with the impurity scattering creates a dip in the single-particle DOS at the\nFermi level. By applying the diagrammatic technique, Maekawa et al.103,104and Takagi\net al.105calculated the lowering of transition temperature with the increasing disorder:\nlnTc\nTc0=−1\n3gle2\n2π2ℏR□\u0012\nlnh\nkBTcτ\u00133\n, (1.67)\nwhere Tc0is the transition temperature of the sample at the bulk limit, R□is the sheet\nresistance of the film, τis the mean-free-path time and glis a constant dependent on the\nelectron-electron interaction and has the value 1 /2 for the screened Coulomb potential.\nThe above analysis illustrates an abnormal rise in the magnitude of the small momentum\ntransfer component of the Coulomb interaction within the effective amplitude of the\nelectron-electron interaction in the Cooper channel. This escalation of electron repulsion,\nresulting from the interplay between the Coulomb interaction and disorder, serves as\nthe primary cause for the reduction in Tcwithin homogeneous films. Finkel’stein106–108\nimproved Eq. 1.67 by employing normalization-group analysis as follows:\nTc\nTc0=eγ 1\nγ+r\n4−pr\n2\n1\nγ+r\n4+pr\n2!1/√\n2r\n, (1.68)\nwhere r=e2\n2π2ℏR□and the adjustable parameter γ= lnh\nkBTc0τ,τbeing the transit time.\nThis Eq. 1.68 by Finkel’stein is considered a benchmark for the Fermionic route in the\ncase of homogeneous thin films.\nThere has been another work by Anderson et al.109where a possible mechanism of\nthe degradation of Tchas been proposed in the case of high- Tcsuperconductors taking\ninto account the effect of Coulomb interaction. Based on the scaling theory developed by\nAbrahams et al.110, they showed that disorder increases the Coulomb pseudopotential µ∗\nor the effective Coulomb repulsion between the elections in the Cooper pair. They came\nout with the following expression for the variation of Tc109:\nTc=ωD\n1.45e−[1.04(1+ λ)\nλ−µ∗(1+0.62λ)], (1.69)\nwhere ωDis the Debye frequency, and λis the electron-phonon coupling.\n1.5.3 Bosonic mechanism\nOn the other hand, in the Bosonic mechanism111,112disorder scattering diminishes super-\nfluid density ( ns) and hence superfluid phase stiffness Js(∼ns/m∗). As a result, the\nphase coherent superconducting state gets vulnerable to phase fluctuations and eventu-\nally can get destroyed due to strong phase fluctuations even when the pairing amplitude\nremains finite. Experimentally, this manifests as a persistence of the superconducting\n231.5. EFFECT OF DISORDER ON SUPERCONDUCTIVITY\ngap in the electronic excitation spectrum, even after the global superconducting state is\ndestroyed. Now, if we warm up the system from the superconducting state, it becomes\nresistive in DC transport measurements due to thermally excited phase fluctuations way\nbefore the superconducting gap ∆( T) in the quasiparticle density of states decreases\ntoward zero. Thus, in a certain temperature range, the state of superconductivity (\nindicated by the gap ) and “normality” ( DC resistance ) coexist. This state can be qual-\nitatively visualized as being composed of phase incoherent Cooper pairs. This regime is\nreferred to as “pseudogapped”80,81,113,114state.\nThe bosonic scenario tends to be realized in superconductors with low electron density\ndue to their weaker shielding and relatively lower rigidity when it comes to phase changes.\nConsequently, the significance of phase fluctuations is amplified in such cases11,115,116.\nThough the Bosonic formalism was developed primarily for disordered homogeneous thin\nfilms93, granular superconductors are naturally suited for this scenario in the BCS frame-\nwork. Interestingly, some signatures of the Bosonic mechanism were observed in granular\n2D systems117,118even before the concepts of SI transition came into the picture.\n1.5.4 Distingushing features in the Fermionic and Bosonic mech-\nanisms\nMagneto-transport measurements and subsequent scaling analysis have traditionally been\nemployed to differentiate between Fermionic and Bosonic mechanisms93. However, rely-\ning solely on the success of scaling and the identification of universality classes based on\ncritical exponents may not fully elucidate the microscopic physics underlying the tran-\nsition or the nature of the insulating state. In fact, the lack of universality in scaling\nexponents and the large variation of critical resistances across different systems have\nundermined the confidence in arguments based solely on the scaling analysis in recent\nyears.\nHowever, the distinction between the two scenarios can be conveniently formulated\nusing the complex order parameter, ψ= ∆eiϕ, where |ψ|is the superconducting en-\nergy gap ∆ and ϕis the phase of the condensate. When fluctuations are present, the\nsuperconducting state is preserved if the correlator95,\nG(r) =⟨ψ(r)ψ(0)⟩, (1.70)\nremains non-vanishing in the limit |r| → ∞ , i.e., G(|r| → ∞ )̸= 0. In the case of\nthe Fermionic route119–121, ∆ vanishes at the transition and the phase becomes mean-\ningless automatically. Thus, certain aspects of the BCS theory, such as the complete\ndisappearance of the superconducting gap precisely at the transition temperature, re-\nmain intact. On the other hand, G(r) can vanish due to the phase fluctuations in the\n24CHAPTER 1. INTRODUCTION\nBosonic scenario80,81,113,114even if the modulus of the order parameter remains non-zero,\nwhich manifests in the form of a superconducting gap or pseudogap above Tc. Recent\nadvancements in local spectroscopy80,81,113,114,122,123have provided experimental evidence\nfor the disorder-induced granularity of superconductivity and the existence of preformed\nCooper pairs above Tc.\n1.5.5 Fermionic to Bosonic transition\nRecent studies indicate that the aforementioned classification may have been overly sim-\nplified. In the vicinity of the critical resistance, the interplay between quantum phase fluc-\ntuations, localization effects, and disorder-induced spatial variations in electronic proper-\nties disrupts the conventional distinction between amplitude- and phase-driven pathways.\nThis disruption gives rise to novel scenarios in which these two scenarios become inter-\ntwined71. In fact, to explain the absence of universal critical resistance as originally\npredicted in the Bosonic scenario, Yazdani et al.90proposed the possibility of a parallel\nFermionic conductance channel at finite temperatures, an idea that is further supported\nby Gantmakher et al.124.\nExperimentally, the significant advancements in recent times can be largely attributed\nto the utilization of low-temperature scanning tunneling spectroscopy80,81,113,114,122,123,\nscanning squid microscopy125, and high-frequency electrodynamic response measure-\nments59,126–128. Using these techniques, it has been possible to observe the emergent\nelectronic granularity in homogeneously disordered thin films and the presence of a pseu-\ndogap for preformed Cooper pairs. These discoveries have led to the unexpected break-\ndown of the frequently mentioned Fermionic and Bosonic dichotomy and necessitate a\nnew microscopic description of superconductivity under conditions of strong disorder. It\nappears to be a common feature in many superconductors such as TiN, NbN and MoGe\nthat the same system can follow the Fermionic route at moderate disorder and crossover\nto a Bosonic scenario at a stronger disorder127,129–131. It is therefore interesting to inves-\ntigate whether a system can follow the Fermionic route all the way to the disorder level\nwhere the superconducting ground state is completely destroyed.\n1.6 Thermally activated flux creep\nAnderson and Kim132,133demonstrated that at finite temperatures, flux lines have the\nability to jump from one pinning point to another, surpassing the pinning barrier Udue\nto the driving force of the current and flux-density gradient. This phenomenon, known\nas flux creep, manifests in two ways: firstly, it gives rise to measurable resistive voltages,\nand secondly, it leads to gradual changes in trapped magnetic fields. In the absence\nof current, the flux line has an equal probability of moving in either direction relative\n251.7. EFFECT OF AC EXCITATION ON VORTEX LATTICE\nto the pinning potential. However, in the presence of a transport current or flux-density\ngradient, the tilted potential grants priority to downhill creep over uphill movement. The\ntiny resistivity resulting from this thermally activated flux creep ( TAFF ) is expressed\nas follows,\nρTAFF =ρffe−U\nkBT (1.71)\n1.7 Effect of ac excitation on vortex lattice\nIn a clean superconductor, the interaction between the vortices arranges them in a trian-\ngular lattice known as the Abrikosov63–65,134vortex lattice ( VL ). However, the inevitable\npresence of crystalline defects in solids acts as a random pinning potential for the vor-\ntices. If these vortices are made to oscillate under the influence of an oscillatory current or\nmagnetic field, their motion is governed by the following competing forces135: i) Lorentz\nforce due to the external current density driving the motion, ii) restoring force due to the\ncombined effect of pinning by crystalline defects and repulsion from neighboring vortices,\nand iii) the dissipative viscous drag of the vortices. In addition, at finite temperature\nthermal activation can cause the vortices to spontaneously jump over the pinning barrier\nresulting in thermally activated flux flow ( TAFF ), which produces a small resistance\neven for external current much below the critical current density ( Jc).\n1.7.1 Gittleman-Rosenblum model\nTo model the above problem, Gittleman and Rosenblum136( GR ) considered the follow-\ning simplified picture for a single vortex(neglecting vortex mass term137,138and TAFF ),\nwhere the displacement uof the vortex due to small ac excitation follows the following\nequation of motion ( force per unit length of the vortex ):\nη˙u+αLu= Φ 0Jac׈n, (1.72)\nwhere Jacis the external alternating current density on the vortex containing one flux\nquantum Φ 0(ˆnbeing the unit vector along the vortex ), ηis the viscous drag coefficient\non the vortex in the absence of pinning and flux creep, and the restoring pinning force\nconstant αLis called the Labusch139–141parameter. Even though this model does not\nexplicitly invoke the interaction between vortices, it nevertheless captures the dynamics\nof a vortex solid where the pinning acts collectively over a typical length scale over which\nthe vortices maintain their positional order, namely the Larkin length. In this case, the\nresultant pinning parameters have to be interpreted in a mean-field sense135, where they\nincorporate both the effect of interactions and the pinning potential. As we will show in\nchapter 5, it can also be applied in certain vortex fluid states as long as the thermally\n26CHAPTER 1. INTRODUCTION\nactivated motion of the vortices is very slow compared to the excitation frequency. αL\ndetermines the shielding response of the superconductor in the vortex state. Assuming\nharmonic solution ( ∼eiωt) in the equation of motion ( Eq. 1.72 ), we get,\nu= Φ 0Jac׈n\nαL+iωη(1.73)\nThe resultant umodifies the London equation12in the following way135,142:\nA=−µ0λ2\nLJac+u×B=−µ0\u0012\nλ2\nL+Φ0B\nµ0(αL+iωη)\u0013\nJac\n=−µ0λ2\neffJac.(1.74)\nWe see that Eq. 1.74 has a form similar to the usual London equation, where the London\npenetration depth ( λL) is replaced by the effective complex penetration depth ( λeff),\nλeff=s\nλ2\nL+Φ0B\nµ0(αL+iωη)(1.75)\nEq. 1.75 captures the effect of small ac excitation on vortices in addition to the London\npenetration depth, where λeffis the effective ac screening length. λeffcan be written in\nthe following way,\nλ2\neff=λ2\nL+λ2\nvortex =λ2\nL+λ2\nC(1 +iωτ0)−1(1.76)\nHere, τ0=η/α Lis the vortex relaxation time without any vortex creep and λC=p\nBΦ0/µ0αLis the Campbell penetration depth which determines how much the mag-\nnetic field penetrates the superconductors when you apply small ac excitation in the\npresence of vortices. We shall discuss about λCin detail in section 1.7.2.\nNow, looking at Eq. 1.76 we observe two frequency regimes: The imaginary or the\ndissipative part becomes significant at high enough frequency ( flux-flow regime, ωτ0>1\n); while as we decrease frequency ( Campbell regime, ωτ0>1 ), real part dominates and\nvortex contribution is given by λC.\nλ2\neffin Eq. 1.76 can also be rewritten in terms of vortex resistivity, ρv, as:\nλ2\neff=λ2\nL−iρv(ω)\nµ0ω, (1.77)\nwhere τ0=η/α Lis the vortex relaxation time and ρvis the complex vortex resistivity\nexpressed as:\nρv(ω) =ρffiωτ0\n1 +iωτ0, (1.78)\n271.7. EFFECT OF AC EXCITATION ON VORTEX LATTICE\nwhere ρffis the dc flux flow resistivity ( ρff) given by the relation143,144:\nρff=Bϕ0\nη=B\nBc2ρn, (1.79)\nρnbeing the normal state resistivity. We shall see in the later sections, how Eq. 1.78\nmodifies in the presence of TAFF.\n1.7.2 Campbell penetration depth\nCampbell145,146penetration depth is not a penetration depth in the true sense. It is\nrather an ac screening response of the superconductor to small amplitude ac excitation in\nthe presence of vortices. When an external ac field is applied, it at first interacts with the\nvortices on the surface generating an oscillation. This oscillation then propagates towards\nthe centre of the sample through the vortices resulting in a change in screening response.\nThis vortex contribution to the screening response is called Campbell penetration depth,\nnamed after Campbell, who first tried to estimate the contribution via a simple calculation\nwhich we shall briefly discuss as follows.\nLet us consider square vortex lattice for the sake of simplicity in our calculation147,\nwhere the positions of a certain vortex are given by xanduis the displacement of that\nvortex due to external ac excitation. In the absence of excitation, the distance between\nconsecutive vortices is given the vortex lattice constant, a0=p\nΦ0/B0, where B0is the\nunperturbed magnetic induction. In the presence of excitation, the inter-vortex distance\nbecomes,\na(x) =a0+u(x+a0)−u(x)\n=a0\u0014\n1 +u(x+a0)−u(x)\na0\u0015\n≈a0\u0012\n1 +du\ndx\u0013\n.(1.80)\nHence the changed magnetic induction is given by ( assuming one flux quantum per\nvortex ):\nB(x) =Φ0\na0a(x)=Φ0\na2\n0(1 +du\ndx)≈B0(1−du\ndx). (1.81)\nInvoking the Maxwell equation, µ0J=∇×B, with B=B(x)ˆz, we get,\nµ0Jy=−∂B(x)\n∂x=B0d2u\ndx2(1.82)\n28CHAPTER 1. INTRODUCTION\nHence, Lorentz force per unit volume on vortices is given by,\nFL=JB0=B2\n0\nµ0d2u\ndx2(1.83)\nAgain, the Lorentz force is balanced by the pinning force, which can be written in terms\nof the Labusch parameter αL. In single vortex approximation, pinning force per unit\nvolume is given by fp=−αLuand the areal density of vortices is nA=B0/Φ0. Here,\npinning force per unit volume is given by,\nFp=nAfp=−αLB0\nΦ0u (1.84)\nAt the steady state,\nFL+Fp= 0. (1.85)\nNow, putting the expressions from Eqs. 1.83 and 1.84, we get,\nB0Φ0\nµ0αLd2u\ndx2≡λ2\nCd2u\ndx2=u. (1.86)\nThis λC=p\nB0Φ0/µ0αLis the Campbell penetration depth we discussed in Eq. 1.76.\n1.7.3 Effect of thermally activated flux flow\nSo far, we have not considered the thermal motion of the vortices in GR model which is\napplicable at higher frequencies. However, at extremely low frequency ( ωτ0≪1 ), the\nthermally activated flux jumps among different metastable states of VL become impor-\ntant which could not be captured in GR model. As it is difficult to map the distribution of\nrandom pinning potential barriers ( due to non-uniformity between different metastable\nstate minima ), a typical single activation energy value135,148U ( dependent on both tem-\nperature and field ) is usually assumed for the sample under study, in the same spirit that\nthe single αLvalue was considered before. It is the factor ( U/k BT) that determines the\nstrength of thermally activated motion in the improved models independently proposed\nby Brandt149and Coffey-Clem150. Physically, TAFF relaxes the restoring pinning force\nover large time scales and is therefore important when measurements are performed at\nvery low frequencies.\n1.7.3.1 Brandt model\nBrandt149introduced the thermal creep by the thermal relaxation of Labusch parameter,\nαL(t). He argued that after a sudden step-like displacement of the vortex lattice, the\n291.7. EFFECT OF AC EXCITATION ON VORTEX LATTICE\npinning restoring force should decay exponentially in time ( t):\nαL(t) =αLe−t/τ, (1.87)\nwhere modified vortex relaxation time τ≫τ0,τ0being the vortex relaxation time without\nthe thermal creep ( Eq. 1.76 ). Brandt suggested the form of τas,\nτ=τ0eU/kBT(1.88)\nIn Fourier space, Eq. 1.87 becomes149,151–154,\nαL(ω) =αL\n1−i/ωτ. (1.89)\nBrandt also gave an alternative derivation155,156of the modified expression of αL( Eq.\n1.89 ) by taking into account the contribution from both elastic velocity due to αLand\nplastic velocity due to η. Effective velocity is given by,\nv=vel+vpl=˙Feff\nαL+Feff\nη=\u0012iω\nαL+1\nη\u0013\nFeff, (1.90)\nwhere Feff=Fext−iωηis the effective driving force density. We have used the simplified\nrelations for a sinusoidal external force of frequency ω:˙Feff=iωF eff, ˙v=iωv. Now,\nputting the expressions in Eq. 1.90, we get:\nFeff=Fext−iωη=αL\n1−i/ωτu, (1.91)\nwhere τ=η/αandv=iωu. Here, we have got the same expression of αLas in Eq. 1.89.\n1.7.3.2 Coffey-Clem model\nCoffey and Clem150,157–162addressed the problem of TAFF in a more elaborate manner.\nTo account for the thermally activated flux motion over the pinning barriers, a random\nLangevin force which depends on the pinning barrier height U is added to the right-\nhand side of Eq. 1.72. The solution to the above problem is similar to that of a particle\nundergoing Brownian motion in a periodic potential. In their model, the expression of\nvortex resistivity ( ρTAFF\nv ) gets modified to the following form163:\nρTAFF\nv (ω) =ρffϵ+iωτ\n1 +iωτ(1.92)\n30CHAPTER 1. INTRODUCTION\n02 4 6 8 10-510-410-310-210-1100ε ( ≡ 1 / Ι20\n(ν) )ν\n ( ≡ U/2kBT ) C(\nb)\n05 1 01 52 00.00.20.40.60.81.01.2τ/τ0ν\n ( ≡ U/2kBT )(a)\nFigure 1.6: Variation of (a)the normalized vortex relaxation time ( τ/τ0= (I2\n0(ν)−\n1)/(I1(ν)I0(ν))) and (b)the flux-creep factor ( ϵ= 1/I2\n0(ν)) as a function of the normalized\npotential barrier ( ν=U/2kBT) according to the Coffey-Clem model150.\nwhere τis the vortex relaxation time in the presence of TAFF given by:\nτ=τ0I2\n0(ν)−1\nI1(ν)I0(ν), (1.93)\nandϵis called the flux creep factor (0 < ϵ < 1) defined by,\nϵ=1\nI2\n0(ν). (1.94)\nHere, Ipis the modified Bessel’s function of the first kind of order p, and ν=U/2kBT;\nUbeing the pinning barrier relevant for the thermally activated vortex motion. The\nvariations of τ/τ0andϵwith ν≡U/2kBTare plotted in Fig. 1.6.\nIf one looks at Eq. 1.92 in zero-frequency limit,\nρTAFF\nv (ω→0)\nρff≈ϵ, (1.95)\nwhich implies that the flux-creep factor ϵquantifies the extent to which thermal activation\nfacilitates the approach of the electric field, generated by vortex motion, towards the value\nit would have reached in the absence of pinning.\nHence the Eq. 1.77 is modified to the following form:\nλ2\neff=λ2\nL−iρff\nµ0ωϵ+iωτ\n1 +iωτ. (1.96)\nAtT→0,ϵ≈0 and τ≈τ0, i.e., the effect of thermal creep becomes negligible. Hence\nEq. 1.96 reduces to Eq. 1.75. At high temperatures where ϵ≈1, vortices behave as if\nthere are no pinning centers.\n311.7. EFFECT OF AC EXCITATION ON VORTEX LATTICE\nFurthermore, the normal fluid skin depth, δnfintroduces an additional correction such\nthat Eq. 1.75 is modified as follows:\nλ2\neff,CC =\u0012\nλ2\nL−iρTAFF\nv\nµ0ω\u0013\u001e \n1 + 2 iλ2\nL\nδ2\nnf!\n, (1.97)\nwhere δnffollows the phenomenological variation150of the form:\nδ2\nnf(t, h) =(2ρn\nµ0ω)\n1−f(t, h)(1.98)\nwhich is complementary to the λLvariation considering they come from normal and\nsuperconducting electrons respectively in the framework of the two-fluid model:\nλ2\nL(t, h) =λ2\nL\nf(t, h), (1.99)\nwhere f(t, h) = (1 −t4)(1−h) with t=T/T candh=H/H c2;TcandHc2being the\nzero-field superconducting transition temperature and upper critical field respectively.\nConsequently, the vortex state is characterized by the minimal set of three parameters:\nαL, η, and U. We shall use this Coffey-Clem equation to fit the experimental in-field\npenetration depth data in Chapter 5.\n1.7.4 Universal expression for vortex resistivity\nPompeo and Silva163came up with a universal expression for the vortex resistivity ρv:\nρv(ω) =ρffϵeff+iωτeff\n1 +iωτeff. (1.100)\nAll the models discussed above can be written in terms of Eq. 1.100, where the parameters\nfor various models are written in Table 1.1.\nTable 1.1: The expressions of flux-creep factor ϵeffand vortex relaxation time τefffor\ndifferent models.\nModel ϵeff τeff Defined parameters\nGittleman and Rosenblum1360 τ0 τ0=η\nαL\nBrandt149τ0\nτ0+τBτ0τB\nτ0+τBτB=τ0eU/kBT\nCoffey and Clem1501\nI2\n0(ν)τ0I2\n0(ν)−1\nI0(ν)I1(ν)ν=U\n2kBT\n32CHAPTER2\nExperimental methods\n2.1 Sample preparation\nThe majority of our samples comprise amorphous Molybdenum Germanium ( a-MoGe )\nthin films of different thicknesses, grown by pulsed laser deposition ( PLD ) technique.\nAdditionally, a few epitaxial Niobium Nitride ( NbN ) thin films were deposited using\nthe reactive DC magnetron sputtering method. Thermal evaporation was used to deposit\nsilver ( Ag ) on the sample as a contact pad for Corbino geometry in the broadband\nmicrowave setup.\n2.1.1 Pulsed laser deposition\nMoGe thin films were deposited using the pulsed laser deposition164,165( PLD ) tech-\nnique. During the PLD process, an arc-melted Mo 70Ge30target was ablated by a KrF\n248nmexcimer laser ( COHERENT COMPex 205 ) while keeping the substrate at\nroom temperature. Both surface-oxidized Si substrates ( with an oxide layer thickness\nof approximately 200 nm) and (100) oriented MgO substrates were used based on the\nexperimental requirements. The deposition was performed at a base pressure of ∼10−7\nTorr , achieved through the use of high vacuum rotary and turbo pumps. For metallic\nthin films, a relatively high energy density of the incident laser is necessary to maintain\nthe stoichiometry of the amorphous film close to that of the target, compared to oxide\nthin films. The laser was focused onto a small spot on the target, and laser pulses were\nbombarded at a repetition rate of 10 Hz, with an effective energy density of around 240\n332.1. SAMPLE PREPARATION\nmJ/mm2per pulse. The growth rate was ∼1nmper 100 pulses. Films with different\nthicknesses were obtained by varying the number of laser shots.\n(a)(b)Quartz\nLaser \nbeamManipulator \nfor loading \nsample\nMotor for rotating \ntarget carousel\nFigure 2.1: (a) Schematic of the pulsed laser deposition (PLD) technique (adapted from\nRef. 166). (inset) A bright plume emanating from the MoGe target ablated by laser pulses and\ngetting deposited on the sample holder placed in front of it (Photo courtesy: Arghya Dutta).\n(b)Top view of the PLD setup (Photo courtesy: Dr. Subhamita Sengupta).\n2.1.1.1 Optimizing the growth parameters\nThe thickness of the thin film is determined by several parameters: (i) the number of laser\nshots, (ii) laser energy, and (iii) the size of the focused laser spot on the target. The size\nand position of the laser spot can be controlled by adjusting the position of a converging\nlens in the laser path. In order to achieve better target ablation, our approach focused\non using a smaller laser spot, which corresponds to higher energy density, rather than a\nlarger spot size, as typically used in standard PLD practice. To determine the spot size,\nwe scanned the size of the spot on thermal paper while changing the position of the lens.\nUsually, laser ablation is performed by focusing the demagnified image of the aperture\nonto the target. However, in our setup, we focused on the focal point of the lens to\nconcentrate higher laser intensity and achieve greater ablation on the metal-like surface\nof the MoGe target. This approach is not commonly used because it can produce side\nlobes, leading to inhomogeneity in the stoichiometry of the deposited film. Nevertheless,\nour observations showed that the laser intensity at the side lobe was not sufficient to\ncause ablation on the MoGe surface. Consequently, we optimized the position of the lens\nto achieve a smaller spot size.\n34CHAPTER 2. EXPERIMENTAL METHODS\nAr+ionsSubstrate \nholder \nwith \nheater\nSubstratePlasmaTarget\nGate valve\nTurbo + rotary pumpingSputter gunTop view\n(a)(b)\nFigure 2.2: (a) Top view of the Sputtering chamber. (inset) Plasma plume ejecting from the\nNb target and getting deposited on the sample holder placed in front of it. (Photo courtesy:\nArghya Dutta/ John Jesudasan). (b)Schematic of the sputtering setup.\n2.1.2 Sputtering\nThe NbN thin films were grown using reactive DC magnetron sputtering. The sputtering\nsystem, manufactured by Excel Instruments, featured four ports for loading the sputtering\ngun or viewing inside the chamber. Additionally, it had a top port for mounting the\nsubstrate and smaller ports for the sputtering gas inlet, venting, pressure gauge, and\nother purposes.\nTo achieve epitaxial NbN films, a high-purity Nb metal ( 99 .999% ) from Kurt-Lesker\nwas mounted on the sputtering gun, while single crystal MgO substrates with a lattice\nconstant of 4 .212˚A were utilized. The MgO substrates were cleaned using trichloroethy-\nlene ( TCE ) and securely attached to the sample holder using high-temperature silver\npaste. Sample patterning was performed using shadow masking.\nThe chamber was evacuated using a Pfeiffer turbo pump, and the substrate temper-\nature was elevated to 600 °Cusing an embedded nichrome wire heater. Two mass flow\ncontrollers were employed to precisely control the partial pressures of Ar and N 2gases\nwhile maintaining a total pressure of 5 ×10−3Torr during all depositions. Following the\ngas introduction, a negative high voltage was applied using an Aplab high-voltage power\nsupply to initiate plasma discharge. The sputtering power was regulated by adjusting the\ncurrent passing through the plasma. Pre-sputtering was carried out for approximately 3\nminutes prior to the actual deposition.\nThe properties of NbN thin films strongly rely on deposition conditions such as sput-\n352.1. SAMPLE PREPARATION\ntering power, Ar/N 2gas ratio, and deposition time, all of which were systematically\nvaried in this study to obtain the desired film properties.\n(a)(b)\n(c)Molybdenum \nBoatTungsten Filament\nAg as target\nCorbino mask\nMagnetSample\nFigure 2.3: (a) Thermal evaporation chamber. (b)Tungsten filament and Molybdenum\nboat used for thermally evaporating targets such as Silver (Ag). (c)Magnet assembly for the\ndeposition of Corbino-shaped contact pad. (inset) Schematic of the side view of the assembly.\n(Photo courtesy: Arghya Dutta).\n2.1.3 Thermal evaporation\nThermal evaporation is a popular technique for depositing thin films onto solid substrates.\nIt involves heating a material in a vacuum chamber until it vaporizes and then allowing\nthe vaporized atoms or molecules to condense onto a substrate, forming a thin film. The\nmaterial to be evaporated, known as the target material, is heated using resistive heating,\nand the substrate is maintained at a lower temperature to facilitate condensation and\nfilm growth.\nIn the evaporation chamber, the target is usually kept on a Molybdenum boat or\ninside a Tungsten filament ( as shown in Fig. 2.3(b) ) while the sample is placed just\nabove it. For patterning the contact pad in the Corbino form, the sample is kept on top\nof an assembly of two button magnets such that the annular mask remains attached while\nthe deposition is going on ( Fig. 2.3(c) ).\nFor the deposition of silver ( Ag ), a small piece of Ag wire, with a diameter of\n1mm, is placed inside the filament. Prior to the deposition process, the chamber is\n36CHAPTER 2. EXPERIMENTAL METHODS\nsubjected to a low-pressure vacuum of approximately 10−5mbar . During the deposition,\na high current of 5 A is applied to the filament, heating it to a red-hot state, causing the\nAg to evaporate. The evaporated Ag then gets deposited onto the substrate, which is\nmaintained at a temperature of 150 °C.\n(a) (b)\n(c)\nFigure 2.4: Deposited thin films and different stainless steel masks used for thin film deposi-\ntion: (a)8-mm-diameter a-MoGe thin film deposited on MgO substrate with the stainless steel\nmask, (b)8-mm-diameter a-MoGe thin film grown in surface-oxidized Si substrate with strip\nlines for measuring the thickness with the mask, and (c)deposited hall-bar-patterned a-MoGe\nthin film with the mask. (Photo courtesy: Dr. Subhamita Sengupta).\n2.1.4 Masks used in deposition\nFor two-coil measurements, the sample was deposited in the form of an 8- mm-diameter\ncircle using a shadow stainless steel mask on 10 ×10mm2substrate as illustrated in\nFig. 2.4(a), while Fig. 2.4(b) shows the mask with two additional strip lines in the middle\nused for measuring thickness on the 8- mm-diameter sample. In the case of transport\nmeasurements, hall-bar-shaped geometry was patterned on samples for better sensitivity\n( Fig. 2.4(c) ). For both two-coil and transport measurements, samples were capped with\na 2-nm-thick Si layer to prevent surface oxidation.\nFor the broadband microwave measurements, thin films grown on the substrate with-\nout any mask, were placed in the evaporator chamber for silver ( Ag ) deposition using\nthe annular mask to form the Corbino geometry ( Fig. 2.3(c) ).\n372.2. THICKNESS MEASUREMENTS\n2.2 Thickness measurements\nThe thickness of all films was measured using an Ambios XP2 Stylus profilometer. Here,\nthe metal tip scratches through the sample to find any change in the topography. Thick-\nness measurements were carried out at various positions of the sample and the mean\nvalue was taken as the film thickness ( d). For d≥10nm, the thickness of the film\nwas directly measured using the stylus profilometer whereas for thinner samples it was\nestimated from the number of laser pulses using two films with d≥10nmgrown before\nand after the actual run for calibration.\n2.3 Low-temperature measurements\nAll our measurements were conducted in our lab using various cryogenic setups to achieve\nextremely low temperatures. Fig. 2.5 shows the measurement assembly comprising the\n4He cryostat and the measuring instruments. Below, we provide brief descriptions of the\ncryogenic setups extensively utilized for our experiments.\nVector \nnetwork\nAnalyzer\nTemperature \ncontrollerMeasurement \nprobe\n4He cryostatCurrent \nsource and \nvoltmeter\nLock -In \nAmplifierInterfacing \nwork station\nMagnet power \nsupply\nFigure 2.5: The Instrument assembly for low-temperature measurements. (Photo courtesy:\nDr. Subhamita Sengupta).\n38CHAPTER 2. EXPERIMENTAL METHODS\n(a)(c)Sorb\n1K \npot\n3He \npot\nMeasurement \nprobeTurbo \npumping \nlineRotary line \nfor pumping \n4He\nCapillary \ntube for \ninserting \n4He\ninto sample \nspace\n(b)Turbo \npumping \nIVCConnected \nto 3He \nVessel\nFigure 2.6: (a)4He VTI. (c)Homemade3He VTI design to sit inside the existing4He VTI\n(c)Janis3He VTI. (Photo courtesy: Dr. Subhamita Sengupta).\n2.3.14He VTI\nAll the in-field two-coil mutual inductance ( Chapter 5 ) and broadband microwave mea-\nsurements above ∼1.7Kwere performed using a4He cryostat equipped with a NbTi-\nNb3Sn superconducting magnet capable of reaching a maximum field of 11 T( Cryomag-\nnetics Inc. ). To control the temperature, a custom-made variable temperature insert\n( VTI ) built by Excel Instruments was employed. The VTI consists of two concentric\nthin-walled stainless steel tubes separated by an evacuated annular space. The outer\ntube is immersed in liquid helium, while the inner tube accommodates the sample insert.\nLiquid helium is introduced into the sample space through a capillary from the helium\nbucket. The flow of4He into the sample space is controlled by adjusting the temperature\nof the capillary with the help of a heater attached to it.\n2.3.2 Janis3He VTI\nTo conduct measurements at temperatures below 2 K, we utilized a custom-made3He in-\nsert provided by Janis Research Co. This insert was specifically designed to fit into our 11\nTNbTi-Nb 3Sn superconducting magnet cryostat. Zero-field two-coil mutual inductance\nmeasurements in Chapter 4 were conducted using the3He VTI.\nAt the start of the cooling procedure, the sorption pump temperature is set to approx-\nimately 45 K, and the 1 Kpot is cooled down to around 1.6 K by continuously pumping\nover liquid4He, through a capillary immersed in the4He liquid bath. The3He, in thermal\ncontact with the 1K pot, is allowed to condense inside the pot. Subsequently, the heater\nfor the sorption pump is deactivated, and it cools down naturally due to the flow of liquid\n4He in narrow tubes surrounding it. As the sorption pump cools, it adsorbs3He vapor,\n392.3. LOW-TEMPERATURE MEASUREMENTS\nwhich further cools the liquid in the pot down to 300 mK. Although the entire setup\noperates under vacuum, the sample holder maintains thermal contact with the3He pot\nthrough a solid cold finger made of copper.\nThe lowest temperature achieved in this3He VTI was ∼330−350mK. The tem-\nperature control of the3He pot is achieved through a combination of regulating the\ntemperature of the sorption pump, which determines the rate of pumping over the liquid\n3He and applying a small amount of heater power directly at the3He pot.\n2.3.3 Homemade3He VTI inside existing4He VTI\nSome preliminary two-coil measurements were carried out in a custom-made3He VTI\nmanufactured by Excel Instruments, which is designed to sit inside the sample space\nof our existing4He VTI. The3He VTI consists of two annular stainless steel cylinders,\nwith the lower part made of copper for better heat conduction. The space between\nthese two cylinders, known as the inner vacuum chamber ( IVC ), can be pumped by a\nturbomolecular pump.\nNow, the cooling process is carried out in two steps. In the first step, the sample\nspace in the4He VTI is cooled down to ∼1.2Kusing the same procedure mentioned in\nSection 2.3.1. At the same time, the3He slowly cools via conduction through the copper\npart, aided by a small amount of4He gas ( exchange gas ) injected into the IVC. Once\nthe system reaches equilibrium at 1 .2K, the3He sample space is isolated by removing\nthe exchange gas from the IVC with the help of the turbo pump.\nThe second step of cooling involves pumping out the3He gas. Initially, the sample\nspace is pumped by a turbo to ∼10−6Torr . Then, the3He gas is gradually allowed\ninto the sample space from a separate container known as the3He dump vessel. With\nthe temperature at 1 .2K, the system is left for some time to allow all the3He gas to\ncondense into the3He pot. To further decrease the temperature, the condensed3He needs\nto be pumped similar to the process used for the evaporative cooling of4He in the4He\nVTI. For this purpose, we utilize a cryopump, which consists of a long pipe filled with\nactivated charcoal. The cryopump, previously kept at room temperature, is then inserted\ninto a helium Dewar to cool the charcoal and start adsorbing all the3He gas in contact\nwith the sample. As a result, the system is further cooled by evaporative cooling. With\nthis homebuilt setup, we were able to achieve the lowest temperature of 700 mK.\nSince the sample is in physical contact with the3He in this system, it is necessary to\nrecover the3He from the sample space and return it to the dump before the sample can\nbe taken out at the end of a measurement. For this purpose, the cryopump is used as\nbefore to evacuate the3He gas in contact with the sample. The cryostat is then isolated,\nand the cryopump is taken out of the Dewar to allow the3He gas to desorb and return\nto the dump. This process is carried out 2 −3 times before the sample is removed from\n40CHAPTER 2. EXPERIMENTAL METHODS\nthe cryostat.\nThe main advantage of this system is its greater cooling power since the sample is\ndirectly immersed in the liquid3He. Additionally, achieving thermal equilibrium is easier\nin this dip system compared to a dry vacuum system like the Janis3He system discussed\nin Section 2.3.2.\n2.4 Measurement techniques\n2.4.1 Two-coil mutual inductance measurements\nFor both the works presented Chapters 4 and 5, we have primarily used a two-coil mutual\ninductance setup. In this section, I shall provide a detailed discussion of this setup. Mag-\nnetic field decays exponentially in a superconductor over a characteristic length known\nas the penetration depth ( λ). Using the two-coil mutual inductance setup, we measure\nthe absolute value of λfrom the screening response of the superconductor.\nPassive\nvoltage -to-current \nconverter\n 220 𝛀 10 𝛀\n2 mm\nSine \nout\nRef inBlaze X\nInput\nInputSR830SR865A\nPick -up coilDrive \ncoil\nQuadrupolar Quadru -\npoleDipolar \nDipoleThin \nfilmSubs -\ntrate(a)\n(b)\nFigure 2.7: (a) Schematic of the two-coil assembly: quadrupole as drive coil (top) and dipole\nas pick-up coil (bottom) with sample sandwiched in between. The coil wire diameter (50 µm)\nis drawn bigger than the actual for clarity. (b)Schematic of the electronic circuit for the\ntwo-coil mutual inductance measurement. The SR865A Lock-in amplifier sends a voltage V\nat a frequency of kHz to the V-Iconverter, connected in series with the primary drive coil,\nand measures the pick-up voltage in the secondary coil. Simultaneously, the SR830 Lock-in\namplifier, locked at the same frequency, measures the voltage drop across a 10 Ω resistance that\nis in series with the drive coil.\n412.4. MEASUREMENT TECHNIQUES\n(a) (b)\nMacor\nhousingShapal\nhousingTwo -coil Probe \nhead \ncompatible for \n4He VTITwo -coil Probe \nhead \ncompatible for \n3He VTI\nDrive \ncoil\nPick -up \ncoilSample\nFigure 2.8: (a) The two-coil mutual inductance probe head made of Macor compatible with\n4He VTI. (b)The two-coil mutual inductance probe head machined from Shapal compatible\nwith3He VTI. (Photo courtesy: Dr. Subhamita Sengupta).\n2.4.1.1 Instrument and measurement details\nWe sandwich the superconductor between the primary quadrupolar coil and the secondary\ndipolar coil. In the case of4He VTI, Both the coils are wound on 2- mm-diameter bobbins\nmade out of Delrin ( Fig. 2.8(a) ). The entire coil assembly is kept inside a Macor housing\nsurrounded by a heater-can made of copper ( Cu ) in a helium ( He ) gas environment,\nwhich provides excellent thermal equilibrium. However, achieving thermal equilibrium in\n3He VTI is slightly difficult, where the whole coil assembly is kept within ∼10−6vacuum.\nTo overcome the challenge, both the coil bobbins and the outer housing are made of\nShapal, which offers higher thermal conductivity compared to Macor ( Fig. 2.8(b) ).\nThe quadrupolar primary coil comprises 15 clockwise turns wound in the one half of\nthe coil and 15 anticlockwise turns in the other half of the coil, while the dipole secondary\ncoil has 120 turns wound in 4 layers. 50 µmdiameter copper wire is used for both coils.\nThe complex mutual inductance ( M=M′+iM′′) between the two coils is measured by\npassing a small ac excitation current ( Iac) through the primary coil and measuring the\nresulting in-phase and out-of-phase components of the induced voltage, VinandVout, in\nthe secondary using a lock-in amplifier,\nM′(M′′) =Vout(Vin)\nωIac, (2.1)\nwhere ωis the excitation frequency. The schematic of the electronic circuit used in the\n42CHAPTER 2. EXPERIMENTAL METHODS\nmeasurement is shown in Fig. 2.7. For the voltage-to-current ( V−I) converter employed\nto send the ac excitation current, we used a simple circuit with resistances in series instead\nof the usual V−Iconverter comprising OPAMPs. The reason for this choice was that\nwe conducted preliminary measurements at high frequencies ( MHz ) using the SR865A\nLock-in amplifier, where the common OPAMPs ceased to work effectively. To address this\nissue, we attempted this passive V−Iconverter consisting only of resistances in series\nand observed that in the low-temperature range, the coil resistance changed negligibly,\nresulting in very little current variation. As a result, we were able to maintain a practically\nconstant current source in the low-temperature range ( below 10 K). We have been using\nthis simple V−Iconverter for low frequencies ( kHz ) as well.\n8 mmBNC \nconnectors\n10-pin \nconnector for \ntemperature \ncontroller \nAfter \nopening\ncopper \ncanSecondary \ndipole coil Primary \nquadrupole\ncoil (a)(b)\n(c)\n(e)Macor\nhousing MoGe grown \non MgO\n(d)\nFigure 2.9: (a) The two-coil mutual inductance insert compatible with4He VTI. (b)The con-\nnector assembly on the upper part of the insert consists of BNC connectors used to connect the\ndrive and pick-up coils to the lock-in amplifiers, along with a 10-pin connector for temperature\nmeasurements. (c)8-mm-diameter a-MoGe film grown on (100) oriented MgO substrate. (d)\nThe two-coil probe, enclosed within a macor housing, is displayed. (e)The two-coil assembly:\nquadrupole as drive coil (top) and dipole as pick-up coil (bottom) with sample sandwiched in\nbetween. (f)The side view of the two-coil assembly shows the MoGe film, deposited on MgO,\npositioned facing the secondary coil. (Photo courtesy: Dr. Subhamita Sengupta and Somak\nBasistha).\nThe use of a quadrupolar coil as the primary ensures fast radial decay of the ac\nmagnetic field on the film which minimizes edge effects on the measurements ( see Sec-\ntion 2.4.1.3 ). The degree of shielding between the primary and the secondary is governed\nbyλof the superconductor, while skin depth ( δ) captures the loss in the system, which\neventually gets reflected in M′andM′′respectively. One can represent both λandδin\n432.4. MEASUREMENT TECHNIQUES\na single complex form known as complex penetration depth ( λeff) such that,\nλ−2\neff=λ−2+iδ−2. (2.2)\nHowever, extracting λ−2\nefffrom M is not straightforward because M′andM′′both depend\nonλandδ. For this, we numerically solve the coupled Maxwell and London equations for\nthe geometry of the coils and samples using finite element analysis and create a lookup\ntable M=M′+iM′′for a range of values of λandδ.λeffis then extracted by comparing\nthe experimentally measured value of M with the corresponding value in the lookup table.\nIn Section 2.4.1.2, we shall give an overview of how we extract λandδfrom experimental\nM.\nThough this method is primarily applied to zero-field measurements, it can also be\napplied in the presence of a magnetic field.\n2.4.1.2 Analysis\nIn this section, we shall discuss the general scheme to formulate the lookup table between\n(M′, M′′) and ( λ, δ). We have followed a numerical scheme suggested by Turneaure et\nal.167,168, which takes into account the finite radius of the films.\nWe start with the mutual inductance expression between the two coils given by,\nM=1\nIdnpX\nj=1I\nAp\nj·dl, (2.3)\nwhere npis the number of loops in the pickup coil, Ap\njis the total vector potential on the\njthturn of the pickup coil. Ap\njconsists of two contributions: (i) Ad→p\njdue to the drive\ncurrent Id( denoted by Iacin Eq. 2.1 ) and (ii) Afilm→p\nj due to the induced current on\nthe superconducting film generated by excitation magnetic field due to Idin drive coil.\nNow, utilizing the azimuthal symmetry of the coaxial coil-sample assembly in our setup,\nEq. 2.3 simplifies as follows,\nM=2π\nIdnpX\nj=1rp\njAp\nj\n=2π\nIdnpX\nj=1rp\njh\nAd→p\nj+Afilm→p\nji\n,(2.4)\nwhere Ap\njis the ϕ-component of the total vector potential at the radius of jthloop of the\npickup coil, rp\nj.\nNow, the vector potential due to the current in the drive coil is a standard problem\n44CHAPTER 2. EXPERIMENTAL METHODS\nof vector potential between two concentric rings and can be expressed as169,\nAd→p\nj=µ0Id\n4πndX\ni4rd\ni\nmij\u0014(2−k2\nij)K(kij)−2E(kij)\nk2\nij\u0015\n, (2.5)\nwith\nmij=q\n(rd\ni+rp\nj)2+D2\nij, (2.6)\nand, E(kij) and K(kij) are the complete elliptic integrals with argument,\nk2\nij=4rd\nirp\nj\n(rd\ni+rp\nj)2+D2\nij, (2.7)\nwhere rd\njis the radius of the ithloop in the drive coil and Dijis the distance between ith\ndrive loop and jthpickup loop. The summation in Eq. 2.5 runs on the loops of the drive\ncoil,ndbeing the number of drive loops.\nIn order to determine the vector potential due to the screening currents in the film,\nwe need to know the current density across the film, Jfilm. Now, in the local limit, vector\npotential in the film is related to the current density according to London’s equation as\nfollows,\nJfilm(r) =−Afilm(r)\nµ0λ2. (2.8)\nAlso, the vector potential in the film can be expressed in the following self-consistent\nmanner,\nAfilm(r) =Ad(r) +µ0\n4πZ\nd3r′Jfilm(r′)\n|r−r′|, (2.9)\nwhere Ad(r) is the contribution at rcaused by the drive coil and r′is the film coordinate.\nAd(r) =−µ0λ2Jfilm(r)−µ0\n4πZ\nd3r′Jfilm(r′)\n|r−r′|. (2.10)\nNow, for simplification in the calculation, the z-integral is replaced by a multiplicative\nfactor,\ndeff=λtanh\u0012d\nλ\u0013\n, (2.11)\nwhich is a good approximation to the exact solution for a stripline geometry. Also, after\ndoing the ϕ-integration, we express Eq. 2.10 in terms of single ρ-integration,\nAd(ρ) =−µ0λ2Jfilm(ρ)−µ0\n2πdeffZ\ndρ′Jfilm(ρ′)ρ+ρ′\nρ\n×\u0014\u0012\n1−k2\n2\u0013\nK(k)−E(k)\u0015\n,(2.12)\n452.4. MEASUREMENT TECHNIQUES\nwhere E(k) and K(k) are the complete elliptic integrals, with the argument kdefined as,\nk2=4ρρ′\n(ρ+ρ′)2. (2.13)\nTo numerically solve Eq. 2.12, we divide the circular film into N number of concentric\nannular rings. Thus, the integration in Eq. 2.12 gets converted to a matrix equation,\nAd\ni(ρi) =−µ0λ2Jfilm\ni(ρi)−µ0\n2πdeffX\nj∆ρjJfilm\njρi+ρj\nρi\n×\u0014\u0012\n1−k2\n2\u0013\nK(k)−E(k)\u0015\n,(2.14)\nwhere ρiis the radius of the ithannular ring. Now, Eq. 2.14 can be expressed as,\naijcj=bi, (2.15)\nwhere\naij=\u0014λ2\ndeff+∆R\n2π\u0012\nln16πρj\n∆R−2\u0013\u0015\nδij\n+∆R\n2πρi+ρj\nρi\u0014\u0012\n1−k2\n2\u0013\nK(k)−E(k)\u0015\n(1−δij),(2.16)\nbi=Ad\ni\nµ0, (2.17)\nci=−deffJfilm\ni, (2.18)\nwhere the first term in aijcorresponds to the self-term of the loop from the self-mutual\ninductance calculated by Gilchrist and Brandt170.\nNow, by numerically solving Eq. 2.15, we can calculate the current density in ith\nannular ring, which can be used to calculate Afilm→p. Hence, for a given λ2\nω=λ−2+iδ−2,\nMcan be determined from Eq. 2.4.\nWe generally take a 100 ×100 sets of ( λ, δ) and calculate the corresponding 100 ×100\nmatrix of ( M′, M′′) values.\n2.4.1.3 Advantage of using quadrupole as drive coil\nDespite producing a weaker signal compared to the dipolar coil, we opted to use a\nquadrupolar coil as the drive coil. In Fig. 2.10, we present a comparison of the simu-\nlated field and induced current density in the film for our setup geometry using both the\nquadrupolar and dipolar coils. It is evident that the field corresponding to the quadrupo-\nlar coil diminishes significantly at the boundary of the film ( radius r= 4mm), while a\n46CHAPTER 2. EXPERIMENTAL METHODS\n01 2 3 4 0123I\n = 5 mAJ ( ×108 A/m2 )r\n ( mm ) Quadrupole \nDipole(\nb)0\n2040J\n ( ×108 A/m2 )\n01 2 3 4 0102030I\n = 5 mAhac ( mOe )r\n ( mm ) Quadrupole \nDipole0\n200400h\nac ( mOe )(\na)\nFigure 2.10: Comparison of simulated radial distribution of (a)magnetic field and (b)\ninduced current density due to quadrupolar and dipolar coils. In the simulation, the following\nparameters were considered: thickness d= 21 nm,λ= 570 nm, and δ=∞.\nsubstantial field is still present in the case of the dipolar coil ( Fig. 2.10(a) ). Additionally,\nthe induced current density on the superconducting film due to the quadrupole nearly\nbecomes zero at the film’s edge, whereas that induced by the dipole exhibits a finite value\nwith a sharp peak at the edge ( Fig. 2.10(b) ). Consequently, using the quadrupolar coil\nrenders the edge effects negligible, simplifying our calculation of the penetration depth.\n2.4.2 Broadband microwave spectroscopy using Corbino geom-\netry\nResonant-cavity techniques are commonly used for highly sensitive and stable conduc-\ntivity measurements at microwave frequencies. However, these techniques are limited\nto discrete frequencies determined by the resonant modes of the cavity, providing in-\nformation only at specific frequencies. In contrast, broadband microwave spectroscopy\nserves as a complementary technique, allowing for the measurement of the electrodynamic\nresponse of materials across a wide frequency range, typically spanning from MHz to\nGHz . This enables researchers to obtain continuous information on the behavior of ma-\nterials as a function of frequency, providing a more comprehensive understanding of their\nelectromagnetic properties. By combining resonant-cavity techniques with broadband\nmicrowave spectroscopy, researchers can gain insights into the behavior of materials at\ndifferent frequencies, leading to a more detailed characterization of their electromagnetic\nproperties and potential applications.\n2.4.2.1 Instrument and measurement details\nOur laboratory has a custom-built setup for broadband microwave spectroscopy, following\nthe method proposed by J. C. Booth171. We utilize a vector network analyzer( VNA ) from\nRhode and Schwarz, which is capable of generating microwave signals in the frequency\n472.4. MEASUREMENT TECHNIQUES\nVector Network Analyser\nSS coaxial cable\nMicrowave probeTemperature controller\nSource meter for dc \nresistance measurements\nCryostatCommercial microwave cable\nFigure 2.11: Schematic of the electronic circuit of the broadband microwave setup. The\nmicrowave signal at a frequency of GHz range is transmitted through both the commercial mi-\ncrowave cable and the SS-coaxial cable from the vector network analyzer (VNA), enabling the\nmeasurement of the complex reflection coefficient. The Lakeshore controller is utilized to mon-\nitor the temperature, while simultaneous two-probe DC resistance measurement is conducted\nusing the source meter connected via the VNA.\nrange of 10 MHz to 20 GHz and measuring the complex reflection coefficient. The\nVNA is connected to a flexible microwave cable with a characteristic impedance of 50 Ω.\nFor low-temperature measurements, we extend the setup with a 50 cmlong 0 .181” SS-\ncoaxial cable, connected via an SMA connector to the commercially available microwave\ncable. The sample under investigation is positioned at the lower end of the SS-coaxial\ncable, acting as a terminator and reflecting back the microwave signal toward the VNA. A\nzoomed-in illustration of the lower end of the SS-coaxial cable, attached to the microwave\nprobe, is shown in Fig. 2.12.\nThe microwave probe in our setup features a female SMA connector, with a spring-\nloaded Kita pin made of beryllium copper inserted at its center. This design ensures\nreliable electrical contact even at low temperatures, accommodating the thermal contrac-\ntion of the cable and connector assembly. The sample is positioned on a Teflon base,\nwhich can be adjusted in height using spring and screw arrangements from below. The\nhousing of the probe is constructed with gold-plated copper for improved performance.\nIn order to fit the sample within the probe, the sample dimensions are maintained at\napproximately 5 mm×5mm. To ensure better electrical contact with the SMA con-\nnector and center pin, a silver ( Ag ) contact pad with a thickness of approximately 200\nnmis deposited on the thin film in a Corbino geometry. This is achieved by thermally\n48CHAPTER 2. EXPERIMENTAL METHODS\nCenter pin\nTeflon \nbase\nCorbino -shaped sample10-pin \nconnector for \ntemperature \ncontroller SMA \nconnector \nfor VNA \na1 a2\nAg contact \npads (a)(b)\n(c)(d)\n(e)\nOuter \nconductor\nSample\nScrew to adjust \nsample stage\nFigure 2.12: (a) The broadband microwave insert compatible with4He VTI. (b)Connector\nassembly on the upper part of the insert comprising SMA connector to send microwave signal\nfrom VNA and 10-pin connector to measure the temperature. (c)The microwave probe, which\nis connected to the lower part of the SS coaxial cable, is displayed. It is enclosed within a\nhousing made of gold-plated copper. At the center of the SMA connector, a spring-loaded pin\nis attached to ensure good electrical contact with the sample, while the sample is placed on\na Teflon base. (d)Corbino-shaped sample with silver deposited as a contact pad. (e)Probe\nhead without the sample. (Photo courtesy: Dr. Subhamita Sengupta and Somak Basistha).\nevaporating Ag using an annular mask with inner and outer diameters of 1 .5mmand 4\nmm, respectively. The Corbino geometry allows the self-field generated by the current\nflowing through the film to be parallel to the surface of the film, eliminating edge effects.\nTemperature is monitored using Lakeshore 336 controller, and at each temperature, the\nfrequency is swept across the entire available range, and the corresponding complex re-\nflection coefficient is measured using a vector network analyzer ( VNA ). Additionally, the\ntwo-probe DC electrical resistance is measured at the same temperature using a Keithley\n2400 source meter via the bias port on the back side of the VNA.\nIfSmeasis the measured complex reflection coefficient, then sample impedance, ZLis\ngiven as,\nZL=Z0\u00121 +Smeas\n1−Smeas\u0013\n. (2.19)\nHere, Z0is the characteristic impedance of the transmission line which is 50Ω in our case.\nNow, the complex conductivity, σis obtained as,\nσ=ln (a2\na1)\n2πdZ L, (2.20)\n492.4. MEASUREMENT TECHNIQUES\nwhere a2anda1are outer and inner radius of the sample and dis the film thickness.\nWe can then find out the complex penetration depth λωusing the relation,\nλ−2\nω=iµ0ωσ. (2.21)\n2.4.2.2 Error correction\nIn microwave measurements, ensuring accurate measurement of the reflection coefficient\nbecomes particularly challenging at lower temperatures and higher frequencies due to sig-\nnal attenuation along transmission cables and unwanted reflections from improper joints.\nTo overcome these systematic errors, precise calibration of the probe is essential to extract\nonly the sample contribution. The calibration process involves establishing a reference\nplane, which, in this case, is the plane of the Corbino probe. Any contribution from\nthe other side of this plane is considered the sample contribution, as it is experimentally\nimpossible to separate the contact resistance between the Corbino probe and the sample\nfrom the sample impedance.\nThere are three main types of errors expressed by a model. The first one is Directivity\n(ED), which arises from reflected signals due to improper soldering or joints. The second\nerror is Reflection tracking ( ER), which occurs due to the damping of the signal along\nthe coaxial cable. The third error is Source mismatch ( ES), which happens when\nthe reflected signal from the sample gets reflected again from improper junctions and\nadds to the incoming signal. To mitigate these error contributions, precise calibration\nis conducted using three known standards: the short standard, the open standard, and\nthe load standard. The reflection coefficient for a short terminator is −1, while for an\nopen terminator, it is 1. As for the load, its reflection coefficient value depends on its\nimpedance, as specified by Eq. 2.19. There are commercially available calibrators for\nroom temperature, but they are not suitable for low-temperature calibration. Therefore,\nalternative methods need to be employed for low-temperature calibration171–174:\n2.4.2.3 Short standard\nA thick Al/Au film or the superconducting spectra of an ordered thick superconductor\nat a temperature far below Tccan be used as a short standard.\n2.4.2.4 Open standard\nThe sample stage is left empty, and the Teflon base of the sample stage with a thickness\nof 1.5mmacts as the open standard.\n50CHAPTER 2. EXPERIMENTAL METHODS\n2.4.2.5 Load standard\nMetallic NiCr films have proven to be reliable load standards due to their temperature-\nindependent resistivity over a wide temperature range, spanning from a few Kelvin to\nroom temperature. This characteristic makes them excellent candidates for load cali-\nbration purposes, with typical resistance values falling within the range of 20 −30 Ω.\nTraditionally, these NiCr films are thermally evaporated onto a glass substrate to serve\nas calibration standards.\nHowever, in our recent experiments, we adopted an alternative approach. Instead of\nemploying three distinct choices for load standards, we utilized a superconducting short\nstandard along with two load standards. These load standards were based on the spectra\nof the actual sample above its critical temperature ( Tc) at two different temperatures.\nThe advantage of this approach was that it enabled us to measure two of the three\ncalibrators during the same thermal cycle as the actual sample. By doing so, we could\nreduce potential errors associated with variations in heating and cooling cycles within\nthe experimental setup. This improvement in the calibration process contributes to more\naccurate and reliable measurements.\nBelow are the expressions for the three error terms concerning the three calibration\nstandards:\nED(ω) =M1(M2−M3)A2A3+M2(M3−M1)A3A1+M3(M1−M2)A1A2\n(M1−M2)A1A2+ (M2−M3)A2A3+ (M3−M1)A3A1,(2.22)\nER(ω) =(M1−M2)(M2−M3)(M3−M1)(A1−A2)(A2−A3)(A3−A1)\n[(M1−M2)A1A2+ (M2−M3)A2A3+ (M3−M1)A3A1]2(2.23)\nES(ω) =M1(A2−A3) +M2(A3−A1) +M3(A1−A2)\n(M1−M2)A1A2+ (M2−M3)A2A3+ (M3−M1)A3A1(2.24)\nThe expressions provided above utilize Mito represent the measured reflection coefficient\nandAito represent the actual reflection coefficient for the ithcalibrator. Once the\nerror terms are established as a function of frequency at different temperatures, the\ncorrected reflection coefficient, denoted as Scorr, can be computed for each frequency and\ntemperature using the given formula,\nScorr=Smeas−ED\nER+ES(Smeas−ED). (2.25)\nSubsequently, the obtained Scorrcan be utilized in Eqs. 2.19, 2.20 and 2.21 to derive the\nelectrodynamic properties of the sample in the frequency domain.\n512.4. MEASUREMENT TECHNIQUES\n2.4.3 Scanning tunneling microscopy/ spectroscopy (STM/S)\nAll Scanning Tunneling Spectroscopy/Microscopy ( STS/M ) measurements on the a-\nMoGe thin films were conducted using a custom-built3He STM capable of operating at\ntemperatures as low as 350 mKand equipped with a superconducting magnet of 90 kOe.\nThe STM is connected to a sample preparation chamber, which includes a dc magnetron\nsputter gun, Ar ion milling, and a recently installed Ferrovac GmbH ion pump. Data\nacquisition and basic analysis are performed using a Rev 9 controller from RHK Tech.,\nwhich serves both functions. A Pt-Ir tip ( 300 µmdiameter ) is utilized for scanning the\nsample, and it is mounted on a molybdenum sample holder. The tip is attached to a\npiezoelectric tube that can be controlled in ±X,±Y, and Zdirections. The piezo-tube is\npositioned on a coarse positioner, controlled by the sixth channel of Attocube controller\nANC300, operating at 30 Vand 250 Hzmode, utilizing the raising part of the pulse.\nVibration isolation is achieved by floating the cryostat on four air-cushion legs, further\ndamped by a piezo-controlled active vibration isolator from Newport Smart Table.\n-4 -3 -2 -1 0 1 2 3 40.40.60.81.0\n  GN(V)\nV ( mV )(b)Superconducting \nDOSNormal DOS\n(a)\nFigure 2.13: (a) Schematic of scanning tunneling microscope. (b)Representative normalized\ntunneling conductance GN(V) vsVspectrum.\nThe operation of STS/M is based on the phenomenon of tunneling, as explained in\nsection 1.2.6. In terms of the instrument, the tunneling current falls within the range of\npico-nano-amperes, which is then pre-amplified by a factor of 109and converted to digital\nsignals using an analog-to-digital converter for acquisition by the computer. The feedback\nloop, depending on the set current for the measurement, controls the distance between\nthe tip and the surface of the sample to maintain a constant current mode of operation,\nenabling scanning over rough surfaces. Specific parameters for different purposes are\ndetermined through trial and error, such as setting the bias at 10 mVand the set current\nat 50 pAfor topography imaging, for instance.\n52CHAPTER 2. EXPERIMENTAL METHODS\n2.4.3.1 Bias spectroscopy\nIn addition to topography data, an interesting type of data that can be obtained using\nSTS/M is bias spectra, specifically the differential conductance ( dI/dV orGns) as a\nfunction of bias. To acquire this data, we apply an alternating voltage of dV= 150 µV\nand∼2.677kHz to our bias voltage and momentarily pause the feedback loop to record\nthedIusing the built-in lock-in amplifier. This allows us to vary the bias from −V( e.g.,\n−6mV) to + V( e.g., +6 mV), and obtain the dI/dV vs.Vspectra. To normalize\nthe data, we divide the full spectra by Gns(V≫∆) so that GnsandGnn( normalized\ndifferential conductance ) become unity at high bias.\nUsing the Gns(V) vsVdata, one can extract information such as the BCS gap ( ∆ )\nby fitting the Gns(V) curve using Eq. 1.30. However, the BCS density of states ( DOS ) is\nslightly modified to incorporate a Γ factor , accounting for non-thermal sources of broad-\nening. The modified form of the equation is:\nNs(E)\nN(0)=\n\nRe\u0012\n|E|+iΓ√\n(|E|+iΓ)2−∆2\u0013\n(E >∆)\n0 ( E <∆)(2.26)\nAn experimental Gns(V) obtained from a 20- nm-thick a-MoGe film is shown in Figure,\nand it can be fitted using equation 5 to obtain ∆ = 1 .3meV and Γ = 0 .2meV at 450\nmK. It is possible to perform the same experiment and fitting for different temperatures\nto extract ∆( T) values, which can be further fitted using the universal BCS curve.\n(a) (b)\nFigure 2.14: (a) Representative vortex image obtained using scanning tunneling spectroscopy\n(450 mK, 10kOe, 6-nm-thick a-MoGe film). (b)Representative Delaunay triangulated vortex\nimage (450 mK, 55 kOe, 21-nm-thick a-MoGe film). Color codes are used to differentiate\nvortices based on the number of nearest neighbors or coordination numbers: Black - 6, Red -\n5, White - 7, Magenta - 4, and Yellow - 8.\n532.4. MEASUREMENT TECHNIQUES\n2.4.3.2 Vortex imaging\nBy utilizing the fact that a vortex core behaves as a normal metallic region, where the\nsuperconducting gap and coherence peaks are suppressed ( as shown in Fig. 2.13(b) ),\nwe set our bias voltage ( Vpeak) at the location of the coherence peak and record the\ndifferential conductance across the surface. This results in a G(V=Vpeak) map, as\ndepicted in Fig. 2.14, where vortices are visualized as minima due to their lower value of\nG(V=Vpeak) compared to the superconducting regions.\n2.4.3.3 Delaunay Triangulation\nOnce the positions of the vortices are determined, all the nearest neighbor points are\njoined by the Delaunay triangulation method such that the circumcircle of any triangle\nmade of three nearest neighbor points consists of no other point. Topological defects are\nthen identified from the lattice points having coordination numbers larger or smaller than\n6 ( Colour code: black - 6, red - 5, white - 7, magenta - 4, and yellow - 8 ).\n(a) (b)\nFigure 2.15: (a) Shift in vortex positions tracked through two representative consecutive\nvortex images at the same position: Red – initial position, Blue – after 1 hour and 20 minutes\n(450 mK, 55kOe, 21-nm-thick a-MoGe film). (b)Classification of Vortices according to their\nshifted positions after one hour and 20 minutes: color codes mentioned in the plot.\n2.4.3.4 Vortex Movement Tracking\nTo track how much the vortices are moving with time, we scanned successive vortex\nimages at the intervals of 1 hour and 20 minutes and compared the positions of the\nvortices in those images to see how much they moved. Each vortex was then classified\naccording to its displacement being a fraction of the theoretical vortex lattice constant\n(a∆) given by:\na∆=\u00124\n3\u00131/4\n×\u0012Φ0\nB\u00131/2\n= 1.075×\u0012Φ0\nB\u00131/2\n(2.27)\n54CHAPTER 2. EXPERIMENTAL METHODS\n2.4.4 Magneto-transport setup\nTransport measurements were done by a standard four-probe technique using a current\nsource and nanovoltmeter. To improve the sensitivity, samples were patterned in the\nshape of a hall bar ( Fig. 2.16 ). The bridge between the two voltage probes was 0 .3mm\nwide ( w) and 1 .3mmlong ( L).\nV\nI\n(a) (b)\nFigure 2.16: (a) Schematic of Hall-bar geometry with four-probe contacts for current (I) and\nvoltage (V) measurements. (b)Sample with four-probe contacts placed in the measurement\nsetup. (Photo courtesy: Dr. Subhamita Sengupta).\nThe protocol for making contacts for the magneto-resistance measurements for MoGe\nsamples is described briefly. To avoid excessive heating in the case of direct soldering at\nthe sample surface, contacts are made in two steps. The sample is attached to a slightly\nbigger glass slide. At first, thin gold wires are attached to the voltage and current contact\npads using two-component epoxy from EPOTEK. The other end of the gold wire is glued\nby epoxy on a small glass slide glued to the sample. This glass end is then connected to\nthe PCB board via a thin copper wire.\n2.4.5 Setup for simultaneous two-coil mutual inductance and\nmagneto-transport measurements\nIn this setup, we have made additional contacts to measure magneto-resistance alongside\nthe two-coil mutual inductance measurements. To create contacts for the resistance\nmeasurements, we deposited the thin film using a shadow mask, as shown in Fig. 2.17(a),\nin a manner that allows the current to flow through a small bridge at the center of the\n552.4. MEASUREMENT TECHNIQUES\nV+\nV-\nI-I+\nV+\nV-\n(a) (b) (c)\nFigure 2.17: (a) Stainless steel mask for depositing thin films for simultaneous two-coil mutual\ninductance and magneto-transport measurements. (b)Contacts for current (I) and voltage (V)\nmeasurements. (c)Sample with four-probe contacts placed inside the two-coil measurement\nsetup.\n8-mm-diameter sample. The sample dimensions are kept the same as our two-coil setup,\nexcept for the small bridge, aiming to closely mimic the two-coil environments.\n2.4.6 SQUID-VSM setup\nIn Chapter 5, we conducted characterization of 20- nm-thick a-MoGe films grown on both\n(100) oriented MgO and surface-oxidized Si substrates using a commercial Superconduct-\ning Quantum Interference Device-Vibrating Sample Magnetometer( SQUID-VSM ) setup.\nThe SQUID-VSM combines two crucial components: a SQUID, an extremely sensitive\nmagnetometer that operates based on the quantum interference of superconducting cur-\nrents, and a vibrating sample magnetometer, which enables precise control of the sample’s\nmagnetic orientation.\nThe SQUID measures the magnetic flux generated by the sample, while the vibrat-\ning sample setup applies a small oscillating magnetic field to the sample, facilitating the\nmeasurement of its magnetic response as a function of field strength and temperature.\nThis remarkable technique excels at detecting even the tiniest magnetic signals, rendering\nit invaluable for studying superconductors, magnetic materials, and other systems with\ndistinctive magnetic properties. The SQUID-VSM measurements yield valuable insights\ninto various aspects of magnetic behavior, including the determination of critical tem-\nperatures, characterization of superconducting and magnetic phase transitions, analysis\nof hysteresis phenomena, and quantification of magnetic moments.\nFor our experiments, we employed a SQUID-VSM system manufactured by Quantum\nDesign. This high-quality instrument allows measuring samples within a temperature\nrange of 300 Kto 2Kwhile subjecting them to magnetic fields of up to 70 kOe. To ensure\nthat the sample is aligned perpendicular to the external magnetic field, it is sandwiched\nbetween two empty plastic capsules ( Fig. 2.18(c) ) inside a straw. The straw, containing\n56CHAPTER 2. EXPERIMENTAL METHODS\nsampleCapsule to \nalign the \nsample \nvertically\n(a) (b) (c)xyzH\n8 mmSecond derivative coil\nFigure 2.18: (a) The assembly of the SQUID-VSM setup. (b)The schematic illustrates the\npickup coil geometry in the SQUID-VSM systems used, with the direction of current indicated\nby brown arrows. The vertical orange arrows approximate the length and direction of sample\nmovement during measurement. (Adapted from Ref. 175). (c)Sample is aligned perpendicular\nto the applied magnetic field by sandwiching between two plastic capsules.\nthe sample, is securely attached to a carbon-fiber rod, which is then inserted into a helium\nflow cryostat. Finally, the sample assembly is positioned between two pick-up coils to\nfacilitate the magnetic measurements. During the measurements, we applied a constant\nrate of vibration to the sample ( with a peak amplitude of 3 mm and a frequency of\n40Hz), allowing us to capture the resulting change in flux using the SQUID, and thus\nenabling the determination of the sample’s magnetic moment.\n57CHAPTER3\nOur model system: amorphous Molybdenum Germanium thin films\n3.1 Sample details\nOur samples consist of amorphous Molybdenum Germanium ( a-MoGe ) thin films of dif-\nferent thicknesses. The amorphous phase can be considered the most disordered state of a\nsolid. Similar to their crystalline counterparts, lattice vibrations or phonons also occur in\namorphous solids. An amorphous solid can be thought of as the limit where the unit cell\nbecomes infinitely large176, and the Brillouin zone reduces to just the point k= 0. No-\ntably, the electron-phonon coupling is found to be stronger for amorphous superconduc-\ntors, with the superconducting gap ( ∆(0) ) far exceeding the Bardeen-Cooper-Schrieffer\n( BCS ) ratio ( ∆(0) /kBTc∼1.76 ), and in some cases, they exhibit Tcvalues higher\nthan their crystalline counterparts177,178. Thanks to their exceptionally short electronic\nmean free paths, these amorphous materials serve as excellent model systems for explor-\ning the impact of disorder on superconductivity78,107,108,126,177,179–183. The emergence of\nnovel vortex phases in these amorphous superconductors has become an area of grow-\ning interest184–188. Additionally, the low pinning strength of amorphous superconductors\nsuggests a highly soft vortex lattice, making them ideal candidates for investigating the\nBerezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young ( BKTHNY ) phase transition189.\nThe a-MoGe samples were grown by pulsed laser deposition ( PLD ) technique, on\nsurface-oxidized Si and (100) oriented MgO substrates. Zero-field superconducting tran-\nsition temperatures ( Tc) varied from 7 .5Kto 0.7Kfor thicknesses 40 nmto 1.8nm.\nDuring the PLD run, an arc-melted Mo 70Ge30target was ablated using a 248 nmexcimer\nlaser keeping the substrate at room temperature. Laser pulses of comparatively high en-\n593.1. SAMPLE DETAILS\nergy density ( ∼240mJ/mm2per pulse ) were bombarded to maintain the stoichiometry\nof the amorphous film close to the stoichiometry of the target. The growth rate was ∼1\nnm/100 pulse.\nThe disorder in thin amorphous MoGe films is tuned by changing thickness or dimen-\nsionality. We can quantify disorder by Ioffe Regel parameter, kFlas,\nkFl=ℏ(3π2)2/3\n[nH(25K)]1/3ρ(295K)e2, (3.1)\nwhere kFis the Fermi wave vector, lis the mean free path, nH(25K) is the Hall carrier\ndensity at 25 K,ρ(295K) is the resistivity at 295 Kandeis the electron charge.\n48 1216202.02.42.83.23.6kF ld\n ( nm )(b)\n05 1 01 52 002468d\n ( nm )Tc ( K )(\na)\nFigure 3.1: Variation of (a)Tc(connected red circles) and (b)kFl(connected green triangles)\nfora-MoGe thin films of different thicknesses.\n3.1.1 Target preparation\n3.1.1.1 Commercial target\nWe started with commercial MoGe targets purchased from Superconductor Materials\nInc.. But the problem with commercial ones is that there are impurities including oxy-\ngen residue as observed in EDAX data. Commercial targets are typically prepared using\npress-and-sinter powder metallurgy190, where powdered metal components are mixed and\ncompacted to a desired shape before being subjected to sintering or high-temperature\nheating. During the powdering procedure of the metal components, oxides may inad-\nvertently mix in when the molten metal is passed through high-pressure water, resulting\nin the formation of powder via the water atomization process. The presence of higher\noxygen residue or impurities does not significantly affect the Tc, but it appears to en-\nhance pinning, resulting in stronger pinning forces. As we are dealing with extremely low\npinned samples, it is preferable to have the least amount of impurities.\n60CHAPTER 3. OUR MODEL SYSTEM: A-MOGE THIN FILMS\n1 kOe\n(b) (d)(c) (a)\n-0.01     0     0.01\n1/nm\nFigure 3.2: (a) Commercial MoGe target purchased from Superconductor Materials Inc. and\n(b)arc-melted MoGe target prepared in Prof. Arumugam Thamizhavel’s lab (picture courtesy:\nDr. Indranil Roy). (c)-(d)(left) Representative vortex images (conductance map ∆ Grecorded\nat a fixed bias, V= 1.52mVfor 1 kOe at 2 K) of 20- nm-thick MoGe films prepared from the\ncommercial and arc-melted target respectively. The black dots represent vortices connected by\nblack lines to illustrate the Delaunay triangulation. The topological defects are indicated by\nthe red, green, magenta, and yellow points, corresponding to 5, 7, 4, and 8-fold coordination,\nrespectively. While topological defects are present in the former image, there is no defect in\nthe visible area for the latter. (right) 2D Fourier transforms (FT) of the corresponding images\non the left, with a scale bar in nm−1. FT image for the film made from the commercial target\ndiffuses to a ring, while that from the arc-melted target shows six spots. Vortex images were\ntaken by Dr. Indranil Roy.\n3.1.1.2 Home-made target\nSo, we turned to a homemade target which was prepared in Prof. Thamizhavel’s lab using\na tetra-arc furnace. At first, a commercially available Molybdenum rod was taken along\nwith small pieces of Germanium. The components were taken in stoichiometric amounts\n(Mo:Ge≡3 : 1 ) and placed on a copper hearth. This hearth is water-cooled and plays\nthe role of an anode in the arc-melting procedure employed to make the MoGe target\nbuttons. The hearth along with the starting materials are loaded onto a chamber in the\ntetra-arc furnace. The furnace is equipped with a combination of high vacuum rotary\nand turbo pumps, which bring down the pressure inside the chamber to ∼10−6mbar .\n613.2. HEXATIC PHASE IN A-MOGE\nThe chamber was flushed three times with gaseous Argon to drive out any impurities\nthat might have been left. The arcs, made of Tungsten, acting as cathodes, are ignited\nby sending a current of ∼50A. When the temperature of the chamber rises to ∼3000\nK, the materials kept on the hearth start melting. The melting was performed in an Ar\natmosphere with pressure slightly less than 1 atm. The chamber is heated for about 10\nminutes and then rapidly cooled to achieve an amorphous material. The end product\ntakes a dome shape, with a concave upper surface. The alloy obtained is then flipped\nupside down, placed on the hearth, and re-melted again. The process is carried out a\ntotal of four times to ensure homogeneity. Two such targets were made of a diameter of\naround 18 mmand a thickness of around 3 .4−3.5mm. Subsequently, the MoGe was\nsliced by wire cutting and then the top surface was polished with fine emery paper to\nobtain a flat, smooth ablating surface which helps to ascertain a fixed plume direction\nduring the laser shots. No oxygen residue was found in EDAX for films synthesized from\na homegrown MoGe target.\nIt is evident from Fig. 3.2 that the MoGe thin films produced from the commercial\ntarget exhibit stronger pinning, manifesting in the form of topological defects in the\nvortex image. In contrast, the films made from the arc-melted target show no defects in\nthe visible area of the vortex image, implying significantly lower pinning.\n3.2 Hexatic phase in a-MoGe\nMelting of 3D VL can happen by increasing temperature or magnetic field, similar to\nthe melting of any crystalline solid via a first-order phase transition through the “Lin-\ndemann191criterion” where the lattice vibration amplitude exceeds a certain fraction of\nthe lattice constant. However, for 2D solid melting can happen through an alternate\nroute via two continuous phase transitions mediated by topological defects as predicted\nby Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young ( BKTHNY )189,192–194theory.\nIn fact, this two-step melting was observed for 2D vortex lattice of weakly pinned a-MoGe\nthin film188. In the first step of melting, thermally excited free dislocations proliferate in\nthe VL, forming an intermediate state between crystalline solid and liquid, which is called\nHexatic fluid when the solid ( e.g., VL ) has hexagonal symmetry. Then in the second\nstep, dislocations dissociate into isolated disclinations producing isotropic liquid. Hexatic\nvortex fluid ( HVF ) differs from regular fluid in that it has short-range positional order\nlike in isotropic vortex liquid ( IVL ), but retains the quasi-long-range orientational order\nof vortex solid ( VS ). In the HVF state, vortices at low temperatures move extremely\nslowly, and hence no stark change in pinning properties was observed in the VS-HVF\ntransition during the magneto-transport measurements195ina-MoGe thin films. More-\nover, VL was found to be extremely sensitive to small electromagnetic perturbations in\nthe HVF state in a-MoGe196.\n62CHAPTER 3. OUR MODEL SYSTEM: A-MOGE THIN FILMS\n(a) (b) (c)\nFigure 3.3: Representative STM images of vortices in (a)vortex solid (VS), (b)hexatic\nvortex fluid (HVF), and (c)isotropic vortex liquid (IVL) in 20- nm-thick a-MoGe film (adapted\nfrom Ref. 195).\nWe note that although the TAFF characteristics in the I-V curves undergo nontrivial\nmodifications across the transition from vortex solid to hexatic vortex fluid188, the flux\nflow regime of the VL is not significantly altered. This is because the movement of the\ndislocations, which provides the small mobility of vortices in the TAFF regime, is much\nslower compared to the vortex velocity in the flux flow regime. Therefore, we can apply\nthe general arguments of collective pinning developed by Larkin and Ovchinnikov ( LO )\nto understand the extrapolated Jcobtained from the I-V curve in the flux flow regime.\n23 4 5 6 7 110100H ( kOe )T\n ( K ) VS-HVF \nHVF-IVL \nHc2\nFigure 3.4: Phase space boundaries in a 20-nm-thick a-MoGe fim: VS-HVF (connected black\nsquares), HVF-IVL (connected red circles) and Hc2(connected blue triangles) respectively (VS:\nVortex solid, HVF: Hexatic vortex fluid and IVL: Isotropic vortex liquid) (adapted from Ref.\n188).\nIn 2-D, the LO theory incorporates two fundamental elements: (i) the existence of\nLarkin domains with short-range order in a VL lacking long-range order and (ii) a finite\n633.3. WEAKLY PINNED VORTEX LATTICE IN A-MOGE\nshear modulus C66of the vortex lattice. In the hexatic state, the characteristic length scale\nRcrepresents the typical separation between two dislocations. On the other hand, in the\nhexatic state, the modulus of the orientational stiffness, known as the Frank constant κ,\nassumes the same role as C66a2\n∆( where a∆is the vortex lattice constant ) does in a vortex\nsolid197. Formally, κis the coefficient of the term1\n2|∇θ|2in the free energy density, where\nθrepresents the bond angle193. Therefore, by establishing these two formal associations,\nwe can employ the LO arguments to comprehend the magnetic field variation of Jc, which\nwe shall discuss in Section 3.4.1.\n-200-1000 1 002 00-6-4-202-\n40-2002040-6-4-20M ( ×10 − 4 emu )H\n ( Oe )Substrate:(\n100) oriented M\ngO(b)\n-200-1000 1 002 00-6-303-\n6-3036-6-303M ( ×10 − 4 emu )H\n ( Oe )Substrate:s\nurface-oxidised S\ni(a)\nFigure 3.5: M-H loop for 20- nm-thick a-MoGe films grown on (a)surface-oxidized Si and\n(b)(100) oriented MgO.\n3.3 Weakly pinned vortex lattice in a-MoGe\n3.3.1 Extremely small M-H loop\nThe weakly pinned nature of the 20- nm-thick a-MoGe thin film is reflected in the very\nsmall M-H loop measured using the SQUID-VSM technique ( Fig. 3.5 ). We have done\nthe magnetization measurements on both surface-oxidized Si ( Fig. 3.5(a) ) and (100)\noriented MgO ( Fig. 3.5(b) ). It is observed that the MH-loop is bigger in the case of the\nMgO substrate than the Si substrate, suggesting a slightly stronger pin in the former.\n3.3.2 ZFC and FC data\nThe extremely small M-H loop in Fig. 3.5(a) is corroborated by the field-cooled ( FC )\nand zero-field-cooled ( ZFC ) data ( Fig. 3.6(a) ) measured on the same sample grown\non surface-oxidized Si. In the presence of pinning, the vortex lattice ( VL ) can exhibit\nmultiple metastable states with varying degrees of order. As pinning increases, the sus-\nceptibility in the ZFC and FC states begins to display distinct responses, representing\n64CHAPTER 3. OUR MODEL SYSTEM: A-MOGE THIN FILMS\n24 6 8 -0.5-0.4-0.3-0.2-0.10.0 \n500 Oe (ZFC) \n500 Oe (FC) \n1 kOe (ZFC) \n1 kOe (FC) \n3 kOe (ZFC) \n3 kOe (FC)χ / χ0T\n ( K )(b)\n24 6 8 -3-2-10M ( ×10 − 4 emu )T\n ( K ) 0.5 Oe (ZFC)  \n0.5 Oe (FC) \n2 Oe (ZFC) \n2 Oe (FC) \n3 Oe (ZFC) \n3 Oe (FC) \n5 Oe (ZFC) \n5 Oe (FC)(a)\nFigure 3.6: (a) Temperature variation of Magnetization ( M) in the zero-field-cooled (ZFC)\nand field-cooled (FC) states at 0 .5Oe, 2Oe, 3Oe, and 5 Oerespectively measured on a 20-\nnm-thick a-MoGe film using SQUID-VSM technique. There is a significant gap between the\nZFC and FC states at 0 .5Oe, which gradually decreases as the magnetic field is increased.\nEventually, the ZFC and FC curves nearly converge for fields at or above 5 Oe.(b)The\nnormalized real part of the ac susceptibility, χ, as a function of temperature for the FC and\nZFC states at 500 Oe, 1kOeand 3 kOe(χ0is the zero-field susceptibility at 1 .5K) measured\nusing the two-coil mutual inductance setup on a 20- nm-thick a-MoGe film (adapted from Ref.\n188). The FC and ZFC curves at each field overlap with each other showing the very weakly\npinned nature of the VL. The density of points for the FC curves has been reduced for clarity.\ntwo limiting cases. It is observed in Fig. 3.6(a) that with increasing field, the difference\nbetween ZFC and FC diminishes and they nearly become identical above 5 Oe, signifying\nextremely weak pinning.\nWeak pinning nature was also evident in previous ac susceptibility measurements,\nwhere the FC and ZFC data on similar samples188showed a negligible difference for\nfields above 500 Oe, indicating very weak pinning in the sample ( Fig. 3.6(b) ).\n3.4 Signatures of Peak effect in a-MoGe\n3.4.1 Peak in Jcand ρTAFF\nOne of the most fascinating findings of our magneto-transport measurements on 20- nm-\nthick a-MoGe films, is the presence of shallow minima in ρTAFF , which is associated with\nthe “peak effect” in critical current density Jc. It is a common feature in many weakly\npinned type-II superconductors, where at the boundary between a Bragg glass and a\nvortex glass198, a dip in ρis accompanied by a corresponding peak in JcforJ≳Jc.\nHowever, in a Bragg glass and the vortex glass, where ρTAFF is extremely small, the peak\neffect is typically not detected in measurements with J≪Jc. In contrast, in our study,\nwe observe the peak effect at the boundary between two vortex fluid states, where ρTAFF\nhas a finite value. Therefore, our first objective is to investigate the internal consistency\n653.4. SIGNATURES OF PEAK EFFECT IN A-MOGE\nbetween the observed variations in JcandρTAFF .\nFigure 3.7: Field variation of Jc(connected blue circles) and ρTAFF (connected black trian-\ngles) at 2 Kfor a 20- nm-thick MoGe film. Here, three distinct phases of vortex lattice: vortex\nsolid (VS), hexatic vortex fluid (HVF), and isotropic vortex liquid (IVL) are shown in orange,\nwhite, and green color area respectively (Adapted from Ref. 195).\nAs depicted in Fig. 3.7, a shallow minimum is observed in ρTAFF at the boundary\nbetween the hexatic vortex fluid and the isotropic vortex liquid. We demonstrate that\nthis phenomenon arises due to the non-monotonic variation of the flux flow critical cur-\nrent density, Jc, which can be explained by the theory of weak collective pinning. The\ncorrelation between the peak in Jcand the minima in ρTAFF is evident; both occur at the\nsame magnetic field. Therefore, the reduction in ρTAFF reflects the increase in Jc, where\nenhanced collective pinning influences the vortex lattice and reduces thermally activated\nflux motion in the TAFF region, resulting in an effective decrease in linear resistivity.\nWe will now elucidate the origin of the maximum in the flux flow critical current, Jc.\nIn the case of a well-ordered vortex lattice with an area of R2\nc, the pinning force exerted\non the vortex lattice can be expressed as ( ⟨f2⟩N)1/2, where frepresents the elementary\npinning force and Ndenotes the total number of random pinning centers within this area.\nThus, the volume pinning force density is given by,\nFp=(⟨f2⟩N)1/2\nR2\ncd=(⟨f2⟩nA)1/2\ndRc, (3.2)\nwhere nAis the areal density of pinning centers and dis the thickness of the film. Equating\nFpwith the Lorentz force density we obtain the flux flow critical current density,\nJcB=(⟨f2⟩nA)1/2\ndRc. (3.3)\n66CHAPTER 3. OUR MODEL SYSTEM: A-MOGE THIN FILMS\nDepending on the nature of pinning, the variation of ffollows the general form199:\n⟨f2⟩nA∝bn(1−b)2, (3.4)\nwhere b=B/B c2. Here, n= 1 corresponds to pinning due to dislocation loops200and\nn= 3 due to impurities and vacancies201respectively. We have observed that n= 1\ncorrectly describes the pinning in a similar 20- nm-thick MoGe sample195. Thus from\nEq. 3.3 we obtain,\nJcB∝b1/2(1−b)\nRc⇒Jc∝b−1/2(1−b)\nRc. (3.5)\nWe can now gain a qualitative understanding of the variation of Jcbased on Eq. 3.5. As\nthe magnetic field increases within the hexatic vortex fluid, the density of dislocations\ngradually rises, resulting in a decrease in Rc. Near the phase transition between the\nhexatic fluid and the isotropic liquid, the rapid decline in Rccan offset the slow decrease\ninJcfrom the numerator, leading to an increase in Jc. Generally, this argument remains\nvalid until Rcreaches a limiting value around Rc∼a∆, where a∆represents the vortex\nlattice constant. Beyond this limiting value of Rc,Jcwill decrease due to the reduction\nof (⟨f2⟩nA)1/2.\n0204060801000.1110100RS ( Ω )H\n ( kOe )  0.1  GHz \n 0.2  GHz \n 0.3  GHz \n 0.4  GHz \n 0.5  GHz \n 0.7  GHz \n 1.5  GHz \n 3  GHz \n 5  GHz \n 6  GHz3 K \n 4-probe DC \n 2-probe DC\nFigure 3.8: A hint of peak effect in the field dependence of surface resistance Rsfor dif-\nferent frequencies at 3 Kin amorphous 20- nm-thick MoGe film using broadband microwave\nspectroscopy. DC 2-probe measurement was done simultaneously using a current source meter,\nbut not sensitive enough to capture the effect (shown as the pink line in the same plot). For\ncomparison, 4-probe DC resistance, measured separately using hall-bar geometry, is also plotted\nin the same plot (green line).\n3.4.2 Hint of peak-like feature in broadband microwave spec-\ntroscopy\nFigure 3.8 illustrates the surface resistance Rsplotted against the magnetic field at 3 Kfor\nvarious frequencies obtained through broadband microwave spectroscopy using Corbino\n673.4. SIGNATURES OF PEAK EFFECT IN A-MOGE\ngeometry. Notably, a subtle plateau-like feature resembling the peak effect is observed\natGHz frequencies. Bhangale and co-workers202–205demonstrated the occurrence of\nthe peak effect in the temperature or magnetic field dependence of microwave surface\nresistivity during the order-disorder transition of the flux-line lattice in DyBa 2Cu3O7−δ\nand YBa 2Cu3O7−δ. They proposed a plausible explanation for this effect within the\nframework of collective pinning, attributing it to the non-monotonic behavior of the\nLabusch parameter associated with the peak in the critical current Jc. This explanation is\nalso applicable to the HVF-IVL transition in a-MoGe. However, at 3 K, the peak feature\nis not very prominent, which further diminishes at higher frequencies. For comparison,\nthe 4-probe DC surface resistance, measured on a similar a-MoGe film with a thickness\nof 20 nm, is plotted on the same graph. Additionally, the 2-probe DC surface resistance,\nmeasured simultaneously with microwave spectroscopy, is included, although it is not\nsensitive enough to capture the peak effect.\n10 kOe 25 kOe 40 kOe\n55 kOe 70 kOe 85 kOe\nFigure 3.9: Histogram of the number of vortices with different displacements as a fraction\nof vortex lattice constant ( a∆) at 2 Kfor different fields in an amorphous 20- nm-thick MoGe\nfilm. Bars in each histogram are normalized by the total number of vortices so that the sum of\nall the pillars is unity.\n3.4.3 Peak effect in the movement of vortices using scanning\ntunneling spectroscopy\nIn order to examine the dynamics of vortices in the field region exhibiting the peak\neffect, we monitored their motion using a scanning tunneling microscope at intervals\n68CHAPTER 3. OUR MODEL SYSTEM: A-MOGE THIN FILMS\nof one hour and 20 minutes. Fig. 3.9 illustrates a histogram presenting the number of\nvortices exhibiting various displacements, expressed as a fraction of the vortex lattice\nconstant a∆, after one hour and 20 minutes. This analysis was conducted through three\nconsecutive scans at a specific temperature ( 2 Kin Fig. 3.9 ) across various magnetic\nfields.\nUpon examining the first distribution at 10 kOe, it is evident that the average dis-\nplacement of the vortices is minimal. This is because, at low fields, the influence of\nvortex-vortex interaction is insignificant compared to the strength of pinning. As the\nfield is increased, the movement of vortices intensifies due to enhanced vortex-vortex re-\npulsion as more vortices come into proximity with each other ( resulting in a decrease in\nthe vortex lattice constant ), as observed at 25 kOe.\nWith further increases in the field, the motion of vortices appears to be impeded, indi-\ncating that pinning forces become increasingly dominant over vortex-vortex interaction.\nThis can be explained by the fact that as the field approaches Hc2, the energy associated\nwith vortex-vortex interaction decreases rapidly, following ( H−Hc2)2, while the energy\nrelated to vortex-impurity interaction varies as ( H−Hc2)206. Consequently, the former\ndiminishes more rapidly than the latter207. As a result, vortices tend to pass through\nthe pinning centers as vortex-impurity interaction starts to prevail over vortex-vortex\ninteraction.\nFinally, at 55 kOe, the pinning potential becomes exceptionally strong, leading to\nthe majority of the vortices becoming pinned and resulting in a highly rigid vortex lat-\ntice. Beyond 55 kOe, the average displacement of the vortices begins to increase again,\nindicating that the vortex lattice is approaching the threshold of melting.\n02 04 06 08 00.150.200.250.300.35 \n < r 2 >1/2 / aΔH\n ( kOe )2 K\nFigure 3.10: RMS displacement of vortices (in the unit of inter-vortex distance, a∆) at 2 K\nfor different fields in amorphous 20- nm-thick MoGe film, calculated from Fig. 3.9.\nTo consolidate the findings from the histograms presented in Fig.3.9, we have plotted\nthe root-mean-square ( RMS ) displacement across all the vortices in Fig.3.10 as a function\n693.5. A-MOGE GROWN ON ANODIC ALUMINA TEMPLATE\nof the magnetic field. As anticipated, this plot exhibits a dip at 55 kOe. This phenomenon\nserves as a microscopic manifestation of the “Peak Effect”.\n24 6 8 -1.2-0.8-0.40.0M ( ×10 − 4 emu )T\n ( K )      FC data \n 1.5 Oe \n 2 Oe \n 2.5 Oe \n 3 Oe(b)\n24 6 8 -3-2-10M ( ×10 − 4 emu )T\n ( K )   ZFC data \n 1.5 Oe \n 2 Oe \n 2.5 Oe \n 3 Oe(a)\nFigure 3.11: Signature of peak effect in both (a)zero-field-cooled (ZFC) and (b)field-cooled\n(FC) states at extremely low fields in SQUID-VSM measurements in an amorphous 20- nm-thick\nMoGe film.\n3.4.4 Unusual peak-like feature in low-field magnetization data\nIn Fig. 3.11, we present an unusual peak-like signature observed around Hc1in a 20- nm-\nthick a-MoGe film grown on an oxidized Si substrate. As depicted in Fig. 3.11, both the\nzero-field-cooled ( ZFC ) and field-cooled ( FC ) states exhibit a dip in the magnetization\ndata. However, this characteristic is only sustained within a narrow range of magnetic\nfield values ( 2 - 2 .5Oe) before leveling off above 3 Oe. We do not yet have a complete\nunderstanding of the origin of this “peak effect” and needs to be explored more.\n3.5 a-MoGe grown on anodic alumina template\nThis section delves into the impact of periodic pinning centers on the vortex lattice in\na-MoGe. Instead of using surface-oxidized Si or MgO as substrates, we have deposited\nthea-MoGe thin film on the background of a nanoporous anodic alumina membrane (\nAAM ) template, which acts as an array of pinning centers inside the superconductor.\nAs demonstrated in Section 1.3.6, the presence of a physical hole or void in the supercon-\nductor results in oscillatory behavior of the order parameter and related thermodynamic\nquantities with the magnetic field, known as the Little-Parks effect67,68. Furthermore, it\nhas been revealed208–210that in an array of voids, or an antidot array, the cancellation\nof circulating supercurrents around each antidot can lead to a Little-Parks-like quantum\ninterference ( QI ) effect.\n70CHAPTER 3. OUR MODEL SYSTEM: A-MOGE THIN FILMS\nIn an antidot array, each vortex experiences two distinct forces: the confining potential\nof the antidot, which remains nearly constant with the magnetic field, and the field-\ndependent confining potential arising from the repulsive interaction with surrounding\nvortices. At matching fields, where the number of vortices in each antidot is an integer,\nthe vortices become tightly confined within a cage formed by neighboring vortices. This\nphenomenon, known as the vortex matching effect ( VME ), gives rise to a pronounced\noscillatory response of the superconductor to the magnetic field. The VME manifests as\noscillations in magnetoresistance close to the superconducting transition temperature\n(Tc), exhibiting a period corresponding to the first matching field ( Hm). In addition,\nthere are noticeable periodic variations with the magnetic field in all dynamic quantities\ninfluenced by the motion of flux lines under external driving, including the magnetic\nshielding response and the critical current.\n1 μm ×1 μm\n0 100 200 300 40005101520Depth ( nm )\nScanning Distance ( nm )(c)(a)\n(b)\nFigure 3.12: (a) Scanning electron microscopy (SEM) images of a representative MoGe\nsample grown on an AAM template (film thickness = 55 nmandTc= 6.1K). (b) AFM scan\nof the same sample. ( c) Depth profile of the sample surface along the line shown in ( b), passing\nthrough multiple pores on the template. AFM scan was performed with help from Mr. Vivas\nBagwe.\nThe samples utilized in this study comprise 3- mm-diameter MoGe films grown on\na nanoporous anodic alumina membrane ( AAM ) using pulsed laser deposition ( PLD ).\n713.5. A-MOGE GROWN ON ANODIC ALUMINA TEMPLATE\nThe AAM template, with a thickness of 50 µm, pore diameter of approximately 35\nnm, and inter-hole distance of around 100 nm, is affixed onto a Si substrate with GE\nVarnis prior to deposition. In Fig. 3.12(a), a scanning electron micrograph ( SEM ) of a\nrepresentative sample ( 55 nmMoGe grown on AAM ) is presented. Fig. 3.12(b) depicts\nthe surface scan of the sample using an atomic force microscope ( AFM ). The depth\nprofile, shown in Fig. 3.12(c), corresponds to a representative line traversing multiple\npores seen in Fig. 3.12(b). The depth profile reveals that the pores are filled up to 10 nm\nafter deposition of around 50 - 60 nmof MoGe.\n-10- 50 5 1 03040506070T\n = 4.4 KM' ( nH )H\n ( kOe ) 0.5 mA \n5 mA \n10 mA \n15 mA(d)5\n5 nm\n-10- 50 5 1 020406080M' ( nH )H\n ( kOe ) 3.23 K        4.4 K \n5.8 K          6 KI\nac = 0.5 mA(b)5 5 nm\n-10- 50 5 1 030405060T\n = 3.9 KM' ( nH )H\n( kOe ) 0.5 mA \n5 mA \n7.5 mA \n10 mA \n11 mA \n12 mA \n13 mA \n14 mA \n15 mA(c)1\n5 nm\n-10- 50 5 1 020406080M' ( nH )H\n ( kOe ) 5.1 K           5.3 K         5.4 K \n3.9 K  4.4 K  4.7 K  5 K0.5 mA( a)1\n5 nm\nFigure 3.13: (a) -(b)Variation of screening response M′as a function of magnetic field Hat\ndifferent temperatures with Iac∼0.5mAin MoGe deposited on AAM template for thicknesses\naround 15 nmand 55 nmrespectively (with Tcaround 5 .3Kand 6 .1Krespectively). VMEs\nmanifest as minima in M′at matching fields. (c)-(d)M′−Hfor different oscillation amplitudes\nIacat 3.9Kand 4 .4Kfor 15- nm- and 55- nm-thick samples respectively.\nWe begin by examining the field variation of the screening response, represented by\n(M′−Hplots ), at different temperatures using a small excitation current ( ac amplitude\nIac= 0.5mA), for two distinct samples: (i) a 15- nm-thick MoGe film deposited on AAM,\nas shown in Fig. 3.13(a), with a Tcof approximately 5 .3K; and (ii) a 55- nm-thick MoGe\nfilm deposited on AAM, as shown in Fig. 3.13(b), with a Tcof around 6 .1K. For\nboth samples, the matching effect is very prominent around near Tc. Here, only one\n72CHAPTER 3. OUR MODEL SYSTEM: A-MOGE THIN FILMS\nmatching field is found at around 2 kOe, consistent with the expected matching field,\nHM= 2Φ 0/√\n3a2, Φ0represents the flux quantum and adenotes the average lattice\nconstant of the antidot array. This is probably because at high fields, vortices start\nmoving, causing an increasing M′response. As a result, oscillations are suppressed by\nthis background response.\nThe diminishing effect of decreasing temperature on the matching effect can be ex-\nplained as follows. The magnitude of the oscillation observed in the screening response is\ndetermined by the ratio210,211FI/F, where (i) FIrepresents the repulsive force between\nvortices, which reduces the compressibility of the vortex lattice when each vortex is sur-\nrounded by neighboring vortices, and (ii) Fis the restoring force on a vortex due to the\nconfining potential created by the surrounding superconductor. For the antidot array211,\nthe ratio FI/F∼ξGL/a. As the temperature decreases, ξGLfollows the usual GL relation,\nξGL∝(1−T/T c)−1/2, resulting in less pronounced oscillations. From Fig. 3.13(a) and\n(b), it is evident that oscillation nearly vanishes at 3 .9Kand 4 .4Krespectively.\nWe now explore the effect of VME on the excitation amplitude Iac. In Fig. 3.13(c)-(d),\nwe look at the magnetic field evolution of M′for different Iacat the lowest temperature\nwhere the oscillations start to diminish. It is evident that the oscillations get enhanced as\nIacincrease. This is consistent with the transition of the vortex lattice from a Mott-like\nto a metal-like state, previously observed210in a superconducting NbN film comprising\na periodic array of holes.\n73CHAPTER4\nThe role of phase fluctuations in ultrathin a-MoGe films:\nCrossover from Fermionic to Bosonic regime\n4.1 Introduction\nIn the case of conventional superconductors, superconductivity is well explained by the\ncelebrated Bardeen-Cooper-Schrieffer3( BCS ) theory, which states that the supercon-\nducting state is primarily dependent on two phenomena: (i) weak attractive interaction\nbetween a pair of electrons with opposite momenta and opposite spins forming a spin-\nzero Boson-like object called Cooper pairs and (ii) condensation of all such Cooper pairs\ninto a phase-coherent macroscopic quantum state. The superconducting state is given by\nΨ = ∆ eiφ. But things get changed when the disorder is introduced.\nDisorder, stemming from impurities, lattice imperfections, or external perturbations,\nintroduces spatial and temporal variations that can have a substantial impact on the\nbehavior of superconductors. With an increase in disorder level, the transition tem-\nperature ( Tc) gradually decreases, eventually leading to a nonsuperconducting ground\nstate. However, even after the destruction of the global superconducting ground state,\nsuperconducting correlations continue to significantly influence the electronic properties.\nThese correlations manifest through several intriguing phenomena: finite high-frequency\nsuperfluid stiffness above Tc72, giant magnetoresistance peak73–77in strongly disordered\nsuperconducting films, the persistence of magnetic flux quantization even after being\ndriven into an insulating state observed in strongly disordered Bi films78, and the persis-\ntence of pronounced pseudogap79–81at temperatures significantly higher than Tc. This\n754.2. THEORETICAL MOTIVATION\ninterplay between disorder and superconductivity presents an intricate and captivating\navenue of study in condensed matter physics. In Section 1.5, a detailed discussion has\nbeen provided regarding the influence of the disorder on superconductivity. I shall how-\never touch upon a few points in this chapter for the sake of maintaining continuity.\n4.2 Theoretical motivation\nDebate on how the disorder affects superconductivity dates back to the late fifties when\nAnderson87predicted that, in an s-wave superconductor, attractive interaction forming\nthe Cooper pairs would remain unaffected by the presence of non-magnetic impurities.\nThis was then interpreted that Tcwould also not be strongly sensitive to disorder. But,\nsubsequent experiments showed that the above is only true in the limit of weak disorder:\nIn the presence of increasingly strong disorder, Tcgets gradually suppressed86,92, and\neventually, the material is driven into a non-superconducting state at a critical disorder.\nIn fact, the suppression of Tcwith an increase in disorder can happen from two origins,\nnamely Fermionic and Bosonic mechanisms93–98.\n4.2.1 Fermionic and Bosonic mechanism\nIn the Fermionic route103,104,107–109, the repulsion among the oppositely charged elec-\ntrons becomes dominant over the attractive pairing interaction due to the loss of effective\nscreening with an increase in disorder, and thus the mean-field transition temperature is\nsuppressed, and at a critical disorder superconductivity is destroyed. On the other hand,\nin the Bosonic mechanism111,112, disorder scattering diminishes superfluid density ( ns)\nand hence superfluid phase stiffness Js(∼ns/m∗). As a result, the phase coherent\nsuperconducting state gets vulnerable to phase fluctuations11,115and eventually can get\ndestroyed due to strong phase fluctuations even when the pairing amplitude remains fi-\nnite212–214. Experimentally, this manifests as a persistence of the superconducting gap in\nthe electronic excitation spectrum, known as the pseudogap, even after the global super-\nconducting state is destroyed80,81,113,114. However, recent studies indicate that these two\nmechanisms need not be exclusive: The interplay between quantum phase fluctuations,\nlocalization effects, and disorder-induced spatial variations in electronic properties dis-\nrupts the conventional distinction between amplitude- and phase-driven pathways. The\nsame system can follow the Fermionic route at moderate disorder and crossover to a\nBosonic scenario at a stronger disorder127,129,130. It is therefore interesting to investigate\nwhether a system can follow the Fermionic route all the way to the disorder level where\nthe superconducting ground state is completely destroyed.\n76CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\n4.2.2 Reason for choosing a-MoGe thin film: popular candidate\nfor Fermionic route\nSeveral amorphous films of conventional superconductors including our sample a-MoGe\nhave long been observed to follow the Fermionic route towards the destruction of super-\nconductivity95,96,98,215. However, the above evidence was primarily through the transport\nmeasurements, where the variation of sheet resistance ( Rs) with Tcis consistent with the\nFermionic theory proposed by Finkel’stein107,108. Nevertheless, the presence of quantum\nphase fluctuations at very low thicknesses has also been recognized90. Our motivation\nwas to see if a-MoGe still shows Fermionic behaviour even if we increase the disorder or\nif there is a transition to Bosonic regime at stronger disorder.\n4.3 Samples used in this work\nThea-MoGe thin films used in this study were grown on oxidized silicon substrates using\npulsed laser deposition technique ablating a homegrown Mo 70Ge30arc-melted target.\nFilm thickness ( d) was varied between 20 nmto 1.8nmwith Tcvarying between 7 Kto\n1.8Krespectively. For d >10nm, the thickness of the film was directly measured using\na stylus profilometer whereas for thinner samples it was estimated from the number of\nlaser pulses using two films with d >10nmgrown before and after the actual run for\ncalibration.\n4.4 Experimental data\nIn this work, we have explored a-MoGe films of varying thickness using a combination of\nlow-temperature penetration depth ( λ), scanning tunneling spectroscopy ( STS ), and\nmagneto-transport measurements. Magneto-transport and penetration depth measure-\nments were performed in3He cryostats operating down to 300 mK. STS measurements\nwere performed using a home-built scanning tunneling microscope ( STM ) operating\ndown to 450 mKand in magnetic fields up to 90 kOe.\n4.4.1 Transport data\nAt first, we present the results from transport measurements on samples of varying thick-\nnesses. We have done all the measurements on standard hall-bar geometry designed by\nshadowing masks for better sensitivity. Fig. 4.1(a) shows the sheet resistance, Rs, as a\nfunction of temperature for samples with different thicknesses. Rsis defined as follows:\nRs=ρ\nd=Rw\nl, (4.1)\n774.4. EXPERIMENTAL DATA\n0.00 .20 .40 .60.00.51.01.51\n/d ( nm-1 )R9Ks\n ( kΩ )(b)1\n.01.52.02.53.0ρ\n9Κ ( µΩ-m )\n02 4 6 8 0.00.40.81.21.6T\n ( K ) 16nm    8nm  \n4nm      3nm  \n2nm      1.5nm Rs ( kΩ )(a)\nFigure 4.1: (a) RsvsTfora-MoGe films with different thicknesses. (b)Variation of R9K\ns\n(black squares) and ρ9K\nN(red circles) as a function of 1 /d, where dis the thickness of the film.\nThe blue line shows the linear R9K\ns−1/dvariation for thickness down to 4 nm.\nwhere ρis the resistivity, dis the thickness, Ris the resistance, wis the width, and lis\nthe length of the sample. As Ris proportional to the length and inversely proportional\nto the width of the sample geometry, it is evident from Eq. 4.1 that Rsis independent\nof the area of the film ( i.e., l×w), but not the thickness of the film. Hence, it is an\nideal parameter to characterize resistive behavior and quantify disorder in amorphous 2D\nthin films of varying thicknesses92. In our sample a-MoGe, similar to other amorphous\nsuperconducting thin films, thickness is used as the tuning parameter for modulating the\nlevel of disorder86,215,216. In the case of amorphous superconductors such as a-MoGe thin\nfilms, defects in the substrate affect the films more as we go to lower thickness, whereas\nthe effect of the substrate is screened for thicker samples. Hence, the inverse of thickness\nhas been conventionally taken as a measure of disorder in amorphous superconducting\nthin films86. In Fig. 4.1(b), variation of normal state sheet resistance ( taken as the sheet\nresistance at 9 K,R9K\ns) and normal state resistivity ( taken as the resistivity at 9 K,\nρ9K\nN) are plotted as a function of the inverse of the thickness ( 1 /d). For d >4nm,R9K\ns\nvaries linearly with 1 /d, showing that the increase in the sheet resistance is primarily a\ngeometric effect ( Fig. 4.1(b) ). Below this thickness, R9K\nsfollows an upward trend and\nthe corresponding resistivity ( ρ9K) shows an increase from approximately 1 .5µΩ−m\nto 2.6µΩ−m.\nWe define Tcas the temperature where the resistance becomes <0.05% of the normal\nstate value. Fig. 4.2(a) shows the variation of the normal state sheet resistance ( taken\nas the sheet resistance at 9 K, R9K\ns) and Tcas a function of film thickness. With\ndecreasing thickness, Tcdecreases whereas Rsincreases, but remains well below the\nquantum resistance, h/(4e2) = 6 .45kΩ ( where eis the electron charge and his the\nPlanck constant ).\nTo compare with the previous results in the literature, we have plotted the Tcas a\nfunction of the corresponding normal state values of Rsat 9Kfor all the samples which\n78CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\n01 0002 00002468 \nExpt. data \nFinkel'stein fittingTc ( K )R\n9Ks\n ( Ω )γ = 8.25 (\nb)\n05 1 01 52 00.51.01.5d\n ( nm )R9Ks\n ( kΩ )(\na)0\n246T\nc ( K )\nFigure 4.2: (a) R9K\ns(black squares) and Tc(blue circles) as a function of film thickness ( d);\nthe solid lines are guides to the eye. (b)TcvsR9K\ns(black circles) for a-MoGe thin films of\ndifferent thicknesses with fit to Finkel’stein model (red line).\nfit well with the Finkel’stein107,108model which considered the suppression of Tcaccording\nto Fermionic route,\nTc\nTc0=eγ\"1\nγ+r\n4−pr\n2\n1\nγ+r\n4+pr\n2#1/√\n2r\n, (4.2)\nwhere Tc0is the Tcof the bulk sample, r=e2\n2π2ℏR9K\nsand the adjustable parameter\nγ= lnh\nkBTc0τ,τbeing the transit time. Our data fit with almost the same value as\nin Refs. 107 and 215. But this result alone does not validate that it is following the\nFermionic route. Moreover, it is not known whether the Bosonic route will also have\na similar variation of Tcsince it has not been explored in detail yet. To investigate it\nfurther, we go to STS and penetration depth measurements.\n4.4.2 Scanning tunneling spectroscopy data\nWe next concentrate on the STS measurements. The tunneling conductance ( G(V)≡\ndI\ndV\f\f\nVvsV) spectra were recorded over a 32 ×32 grid over 200 nm×200nmarea at\neach temperature. Fig. 4.3(a) and (b) show the average spectra at different temperatures\nfor representative samples ( here 20- nm- and 2- nm-thick a-MoGe films respectively ).\nAt low temperatures, the spectra for all samples have the characteristic features of a\nsuperconductor: a depression in G(V) at low bias corresponding to the superconducting\nenergy gap, ∆, and the presence of coherence peaks at the gap edge. In addition, for\nthe 2- nm-thick sample, we observe a broad V-shaped, nearly temperature-independent\nbackground. This feature, also observed in other disordered superconductors27,122,217,\nis attributed to the Altshuler-Aronov type electron-electron interactions in disordered\nmetals218, which is discussed in Section 4.4.2.1.\n794.4. EXPERIMENTAL DATA\n-6-303 6 0.60.81.0GN (V)V\n ( mV ) 0.45 K \n0.75 K \n1.15 K \n1.72 K \n2.1 K 2.35 K \n2.7 K \n3.1 K \n3.85 K(d)2  nm\n02 4 6 3.54.04.50\n12340.000.010.02G ( 10-8 S )V\n ( mV ) G =A ´ Va(\nc)3.85 K2 nmΔA / ANT\n ( K )\n-6-303 6 234G ( 10-8 S )V\n ( mV ) 0.45 K \n0.75 K \n1.15 K \n1.72 K \n2.1 K 2.35 K \n2.7 K \n3.1 K \n3.85 K \n4.4 K2 nm( b)\n-4-202 4 0369G ( 10-8 S )V\n ( mV ) 0.45 K \n0.97 K \n1.55 K \n2.15 K \n2.65 K 3.75 K \n4.45 K \n5.30 K \n6.60 K \n7.60 K20 nm( a)\nFigure 4.3: (a), (b) Representative tunneling spectra G(V) vsVfor 20- nm- and 2- nm-thick\nfilms respectively at different temperatures using scanning tunneling microscope. (c)The fit of\nthe spectra for the 2- nm-thick sample at 3 .85KtoG(V) =A×Vαexpected from Altshuler-\nAronov-type electron-electron interactions, where A= 3.84×10−8andα= 0.076; the deviation\nat very low bias is due to thermal smearing at finite temperature. (inset) The fractional change\nof total spectral weight with respect to normal state. (d)The spectra for 2 nmafter normalizing\nwith the spectra at 4 .4K.\n4.4.2.1 The Altshuler-Aronov Anomaly and Normalization of the tunneling\nspectra\nApart from the low bias feature associated with the superconductivity, the G(V)−V\ntunneling spectra contain a broad V-shaped nearly temperature-independent background\nthat extends up to high bias. With an increase in temperature at some temperature,\nthe low bias feature disappears; above this temperature, the spectra overlap with each\nother but the V-shape remains. For the d= 2 nmsample, this temperature is 3 .85\nK( Fig. 4.3(b) ). We observe that the V-shape can be fitted with a functional form\nG(V) =AVα, consistent with the predicted dependence from extensions of Altshuler-\nAronov-like interactions218beyond the perturbative limit ( Fig. 4.3(c) ). In order to\nremove the V-shaped background and keep only the feature related to superconductivity,\n80CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\nwe divide all the spectra with the spectrum at 4 .4K( Fig. 4.3(d) ). One consistency\ncheck of this procedure is to verify that the normalized spectra, GN(V), conserves the total\nspectral weight, i.e., lim\nV≫∆/eRV\n−VGN(V′)dV′between the normal and the superconducting\nstate. Defining the quantity,\nA(T) =Z6mV\n−6mVGN(V′, T)dV′, (4.3)\nwe plot in the inset of Fig. 4.3(c),\n∆A\nA=A(T)−AN\nAN, (4.4)\nwhere ANis the value of the integral in the normal state, i.e., for GN(V) = 1. We observe\nthat ∆ A/A < 0.02 which is zero within the limits of resolution of our measurement.\n24 6 -4-2024V ( mV )T\n ( K )0.10.50.91GN (V)1\n1 nm050100150R\ns ( W ) \n11 n\nm2\n4 6 -4-2024V ( mV )T\n ( K )0.20.50.91GN (V)0\n.00.20.4R\ns ( kW )(b) \n4.5 \nnm0\n.61.21.82.43.0-4-2024V ( mV )T\n ( K )0.50.70.91GN (V)( c) \n2n\nm   4.5 nm0\n.00.51.0R\ns ( kW )501001502002001\n501\n005\n0 x\n ( nm )y ( nm )0\n.10.40.60.9GN (0)(a)5\n01001502002001\n501\n005\n0 x\n ( nm )y ( nm )0\n.10.40.60.9GN (0)5\n01001502002001\n501\n005\n0 0\n.10.40.60.9GN (0)x\n ( nm )y ( nm ) \n2 nm(d)(\ne)(\nf)\nFigure 4.4: (a)-(c) Intensity plot of normalized spectra GN(V) as a function of bias voltage\n(V) and temperature for three films with thickness 11 nm, 4.5nm, 2nmrespectively. (d)-\n(f)Spatial maps of zero-bias conductance (ZBC), GN(0), at 450 mK for 11, 4 .5, and 2 nm\nrespectively.\n814.4. EXPERIMENTAL DATA\nTo extract the superconducting contribution alone, we calculated the normalized spec-\ntra,GN(V) vsV, by dividing it with the spectra obtained at high temperatures where\nthe low bias feature associated with superconducting pairing disappears. In Fig. 4.4(a)-\n(c), the similar temperature dependence of GN(V) spectra at different thicknesses are\nrepresented as intensity plots along with Rsmeasured on the same samples ( hall-bar-\npatterned after taking out of STM ). We can see that for thicknesses 11 nmor higher, the\nsuperconducting gap vanishes very close to the transition from the transport measure-\nments consistent with the expectation from Bardeen-Cooper-Schrieffer theory12. But for\nthe sample with 4 .5nmthickness, we can see a small hint of a pseudogap, where a soft\ngap in the tunneling spectra extends approximately 0 .5Kabove Tc. For the 2- nm-thick\nsample, pseudogap exists until almost double the Tc(∼1.8K). We define the pseudogap\nvanishing temperature, T∆, as the temperature where GN(0)≈0.95. We observe that\neven though the coherence peaks are greatly suppressed compared to the BCS estimate,\npseudogaps continue to exist up to a temperature much above Tc∼1.8Ktill about\n3.1K. This is consistent with theories where the pseudogap arises from the destruction\nof global phase coherence state214, but inconsistent with theories that ascribe the pseu-\ndogap to parity gap219. It is also pertinent to note that we observe significant spectral\nweight inside the gap, which is similar to earlier observations in strongly disordered in\nsitu grown122,123NbN ( different from ex situ grown80,81TiN and InO xwhich do not show\nstates inside the gap ), even though its origin is not completely understood41,220.\nThe zero bias conductance ( GN(0) ) maps obtained at 450 mK ( Fig. 4.4(d)-(f) )\nreveal that the superconducting state becomes progressively inhomogeneous as we move\nto lower thicknesses. This nanoscale inhomogeneity113,122,123,129,221in the thinnest sample\n(d= 2nm) is also evident in the histogram in the inset of Fig. 4.6(a), where the root\nmean square width of the distribution of GN(0) values increases significantly when d= 2\nnm. It should be noted that the surface morphology remains homogeneous down to the\nthinnest sample.\n4.4.2.2 Fitting of the normalized tunneling spectra and temperature varia-\ntion of superconducting energy gap\nThe tunneling conductance between a normal-metal tip and a superconductor is theoret-\nically described by the equation12,\nG(V)∝1\nRNZ∞\n−∞Ns(E)∂f(E+eV)\n∂EdE, (4.5)\nwhere\nNs(E) =Re\nE+iΓq\n(E+iΓ)2−∆2\n (4.6)\n82CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\n-4- 20 2 4 0.60.81.01.2-4- 20 2 4 0.30.60.91.2-4- 20 2 4 0.30.60.91.21.5-4- 20 2 4 0.40.81.21.60\n1 2 3 0.20.40.60.802 4 6 0.30.60.902 4 6 8 0.40.81.202 4 6 8 0.40.81.2GN(V)V\n ( mV ) 0.45 K \n0.75 K \n1.15 K \n1.72 K 2.1 K \n2.35 K \n2.7 K \n3.1 K \n3.85 K(a)(\nb)(\nc)2\n nm 0.45 K \n0.68 K \n1 K \n1.57 K \n2.1 K \n2.65 KGN(V)V\n ( mV ) 3.4 K \n3.9 K \n4.34 K \n4.72 K \n5.34 K \n5.92 K4.5 nmGN(V)V\n ( mV ) 3.7 K \n4.38 K \n4.9 K \n5.4 K \n5.9 K \n6.5 K 0.45 K \n0.76 K \n1.15 K \n1.72 K \n2.2 K \n2.63 K11 nmGN(V)V\n ( mV ) 0.45 K \n0.97 K \n1.55 K \n2.1 K \n2.65 K 3.75 K \n4.45 K \n5.30 K \n6.60 K \n7.60 K(\nd)21 nmΔ\n, Γ ( meV )T\n ( K )2 nm0\n.00.51.0R\ns ( kΩ  )Δ, Γ ( meV )T\n ( K )4.5 nm0\n.00.20.4R\ns ( kΩ )Δ, Γ ( meV )T\n ( K )11 nm0\n4080120R\ns ( Ω )Δ, Γ ( meV ) T\n ( K )(e)(\nf)(\ng)(\nh)21 nm0\n204060R\ns ( Ω )\nFigure 4.5: (a)-(d) The normalized GN(V)−Vspectra along with the fits with the BCS+\nΓ model for 4 samples with different thicknesses: 21, 11, 4 .5, and 2 nmrespectively. (e)-(h)\nTemperature variation of the corresponding ∆ and Γ extracted from the fits; the temperature\nvariation of the sheet resistances are shown in the same panels. The blue lines in (e)and(f)\nshow the BCS fit to the temperature variation of ∆. The solid grey lines in (g)and(h)are\nguides to the eye; for these two thicknesses pseudogaps are observed above Tc.\nis the single-particle density of states in the superconductor. When Γ = 0, this expression\nreduces to the usual BCS expression ( Eq. 1.30 ). Γ is a phenomenological parameter that\nis incorporated in the density of states to account for possible broadening in the presence\n834.4. EXPERIMENTAL DATA\n24 6 0.81.01.21.4Δ(0) ( meV )T\nc ( K )0\n5 1 01 52 02345Δ(0)/kBTc, Δ(0)/kBTΔd\n ( nm )(b)\n05 1 01 52 0234567 \nTc \nTDTc, TD ( K ) d\n ( nm )\nGN (0)ZBC count0 .020.0210\n.034(a)\nFigure 4.6: (a) Variation of TcandT∆(connected black squares and red circles respectively)\nwith respect to film thickness d. Here, Tcis the transition temperature where the resistance\nbecomes <0.05% of the normal state value, while T∆is the gap-vanishing temperature where\nGN(0)≈0.95. (inset) Histogram of the ZBC for samples of different thicknesses; the root mean\nsquare width of each ZBC distribution is written next to each histogram. (b)∆(0)/kBTcand\n∆(0)/kBT∆(connected black squares and red circles respectively) as a function of film thickness\nd. (inset) Variation of ∆(0) with Tc(connected blue triangles).\nof disorder23( see Section 1.2.5 for detailed discussion on Γ ). Fig. 4.5(a)-(d) show the fit\nof the normalized GN(V)−Vspectra for 21-,11-,4 .5-, and 2- nm-thick films respectively,\nwhile the temperature variation of ∆ and Γ for the corresponding samples are shown in\n4.5(e)-(h). The temperature variations of Rsare also shown in the right panels.\nT∆has been plotted in Fig. 4.6(a) along with the Tcfrom transport measurements.\nIt is observed that they start to deviate below 11 nm. After extracting ∆ at 450\nmK( denoted by ∆(0) ) using BCS + Γ model, we observe that ∆(0) /(kBTc) increases\nrapidly for samples below 5 nmthickness ( Fig. 4.6(b) ) reaching a value of 5 at 2 nm.\nIn contrast, the ratio ∆(0) /(kBT∆) hovers between 2 and 2 .5, as expected for a strong-\ncoupling type-II superconductor. The emergence of a pseudogap between TcandT∆,\ncoupled with the anomalously large value of ∆(0) /(kBTc) signals a departure from the\nBCS ( or Fermionic ) scenario113,129,131fora-MoGe films with a thickness below 5 nm.\nTo summarize, the STM data reveal three key findings: Firstly, the gap ∆( T) for the\ntwo thinnest films appears to extrapolate to zero at a temperature T∆well above the\nonset of dc resistivity at Tc. Secondly, a consistent low-temperature gap ratio is observed\nacross all four films, regardless of thickness ( both thick and thin ), ∆ /(kBT∆)≈2.\nThirdly, there is evidence pointing to a mechanism responsible for broadening the peak\nin the density of states for quasiparticles.\n4.4.3 Penetration depth data\nWe now examine the penetration depth ( λ) data. The temperature dependence of λ−2\nfor different thicknesses ( d) is shown in Fig. 4.7(a). As evident from Fig. 4.7(a), for\n84CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\nlarger thicknesses ( d >5nm)λ−2vsTfollows the dirty limit BCS equation12( 1.34 ):\nλ−2(T)\nλ−2(0)=∆(T)\n∆(0)tanh\u0014∆(T)\n2kBT\u0015\n, (4.7)\nwhere ∆(0) is constrained within 10% of the value obtained from tunneling measurements\n0.51 .01 .52 .00.00.20.40.60.80\n.51.01.50.40.60.8 Measured  \nLinear fitλ−2 ( µm-2 )T\n ( K )2.2 nm(b)λ-2 ( µm-2 )T\n ( K ) BCS fit\n02 4 6 8 01234 20 nm \n12 nm \n 5 nm \n2.2 nmλ-2 ( µm-2 )T\n ( K )´2.5(a)\nFigure 4.7: (a) λ−2as a function of temperature for films with different thicknesses ( d). For\n2.2nm,λ−2is multiplied by 2 .5 for clarity. Solid lines represent the temperature variation\nexpected from the dirty-limit BCS theory. (b)Temperature variation of λ−2for 2.2-nm-thick\nsample (blue circles); the green line is fit to the linear T region. (inset) λ−2vsTfor 2.2nm\nzoomed at low-temperature region along with the dirty-limit BCS fit (cyan line).\nat 450 mKand ∆( T)/∆(0) is assumed to have BCS temperature dependence for a weak-\ncoupling s-wave superconductor ( Eq. 1.22 ). But as the thickness is decreased, λ−2\ndeviates from the BCS behavior as evident in the 2 .2nmsample data ( zoomed in the\ninset of Fig. 4.7(b) ). As seen from Fig. 4.7(b), λ−2vsTfor the most disordered sample\n( 2.2nm) slowly varies at low temperature and crosses over to a linear variation before\ndecreasing rapidly close to transition.\nAt low temperatures, λ−2saturates towards a constant value for all samples. We now\nanalyze the thickness variation of λ−2(T→0). With a decrease in thickness, λ−2(T→0)\nprogressively decreases by more than an order of magnitude. Within BCS theory, with an\nincrease in disorder, superfluid density, ns(≡m∗/µ0e2λ2, where m∗is the effective mass\nandµ0is the vacuum permeability ) gets suppressed from the electronic carrier density,\nn, due to an increase in electron scattering. In the dirty limit, this is captured by the\nrelation,\nλ−2\nBCS(0) =πµ0∆(T→0)\nℏρN, (4.8)\nwhere ℏis the reduced Planck constant, and ρNis the resistivity in the normal state ( at\n10K). This suppression is evident in Fig. 4.8(a), where the experimental ns(T→0) and\nns,BCS (0) are observed to be suppressed by four orders of magnitude from the total carrier\ndensity in the normal state ( at 25 K) measured from Hall measurements222. Moreover,\n854.5. EFFECT OF PHASE FLUCTUATIONS\nwe plotλ−2(T→0)\nλ−2\nBCS(0)for different thicknesses to compare the experimental saturated zero\ntemperature value with the dirty BCS estimate ( Fig. 4.8(b) ). Here, we observe that the\nratio is nearly 1 within the error bar for higher thicknesses, but measured λ−2(T→0)\nfalls significantly below the disorder-suppressed BCS value below d∼5nm, reaching a\nvalue∼0.55 for 2 .2-nm-thick sample. To explain this additional suppression in the 2 .2\nnmdata, one has to invoke the effect phase fluctuation in the sample, which was not\nconsidered in the dirty BCS fit.\n05 101520250.40.60.81.0λ-2(T→ 0) / λ-2B\nCS(0)d\n ( nm )(b)\n05 1 01 52 0102510261027102810291030n ( m-3 )d\n ( nm ) ns(T→ 0) \nns,BCS(0) \nn at 25 K(a)\nFigure 4.8: (a) Comparison of the thickness dependence of λ−2\nexpt(T→0) and λ−2\nBCS(0)\nconverted to superfluid density (connected red circles and blue squares respectively). For com-\nparison total carrier density obtained by Hall measurements at 25 Khas been added (connected\ngreen triangles). (b)λ−2\nexpt(T→0)/λ−2\nBCS(0) for different thicknesses, with error bars set at 20%\nof the corresponding y-axis values.\n4.5 Effect of phase fluctuations\nSince the suppression of λ−2(0) and the linear- Tvariation are consistent with the expecta-\ntions from quantum and classical longitudinal phase fluctuations223,224, we now attempt a\ncomparison of these features with theory225,226. For the 2 .2-nm-thick sample, we estimate\nξ∼8nmfrom Hc2, such that it is in the 2D limit. Now, the resilience of a supercon-\nductor against phase fluctuations is given by the superfluid stiffness ( Js), which for a\ntwo-dimensional superconductor ( d < ξ ) is expressed as,\nJs=ℏ2nsd\n4m∗. (4.9)\nA rough estimate of the suppression of nsdue to quantum phase fluctuations can be\nobtained by considering two energy scales227: The Coulomb energy, Ec=e2\n2ϵ0ϵBξand\nJs(T→0), where ϵ0is the vacuum permeability and ϵBis the background dielectric\nconstant. Using the carrier density measured from Hall effect measurements n= 4.63×\n86CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\n1029m−3and the plasma frequency228Ωp= 1.625×1016Hz, we estimate ϵB=e2n\nϵ0mΩ2p∼\n5.6. Here, the suppression of the superfluid density due to quantum phase fluctuations is\ngiven by,\nns\nn0\ns=e−⟨(∆θ)2⟩/4, (4.10)\nwhere\n⟨(∆θ)2⟩ ≈1\n5πs\nEc\nJs(0), (4.11)\nandn0\nsis the bare superfluid density in the absence of phase fluctuations. We obtain\nns\nn0s∼0.78. This value would get further reduced in a disordered system if the local\nsuperfluid density is spatially inhomogeneous229–232. Though it is difficult to quantify\nthis effect in our film, from the large spatial variation of GN(0) in the 2- nm-thick films\n( Fig. 4.4(f) - (h) ), we believe that this effect could be substantial. This additional\nsuppression of the superfluid density due to inhomogeneity reflects in the emergence of a\nfinite-frequency absorption that is expected to occur at relatively low energies, i.e., below\nthe 2∆ threshold for quasiparticle absorption in dirty superconductors233,234. This effect\nhas been observed in NbN and InO xfilms128,235. Such low energy dissipative mode has a\nfeedback effect on quantum phase fluctuations such that the effect of quantum corrections\ngets reduced and the quantum to classical cross-over shifts to a lower temperature225.\nConsidering these uncertainties at the moment we can only state that the observed value\nofλ−2(300mK)\nλ−2\nBCS(0)∼0.55 falls in the correct ballpark.\n0.51 .01 .52 .00.00.20.40.60.80\n.10.20.30.700.720.740.762.2 nm \nµ T \nµ T 2λ−2 ( µm−2 )T\n ( K )λ−2 ( µm−2 )T\n2 ( K2 )\nFigure 4.9: Temperature variation of λ−2for 2.2-nm-thick sample highlighting quadratic\n(T2) dependence at very low temperatures which becomes linear ( T) at around 0 .5K. (inset)\nExpanded view of T2dependence of λ−2at low temperatures.\nUpon further scrutiny of the λ−2variation in the low-temperature range, particularly\nbelow the linear regime, we observe a shift towards quadratic ( T2) behavior in the 2 .2-\nnm-thick sample ( see Fig. 4.9 ). This T2trend at lower temperatures aligns with findings\ndocumented in various reports in the literature236–241. In our disordered a-MoGe thin\n874.6. COMPARISON OF TcAND T∆\nfilm, the rationale behind this quadratic variation can be elucidated by considering the\nrole of dissipation. Theoretical predictions242and experimental observations in d-wave\nsuperconductors suggest the presence of low-energy dissipation, as evidenced by high-\nfrequency conductivity measurements243,244. Recent studies on amorphous InO x235and\nstrongly disordered NbN113films indicate that low-energy dissipation may also manifest\nin disordered s-wave superconductors. Although the precise origin of this dissipation\nremains unclear, quantum phase fluctuations contribute to the T2temperature depletion\nfor the superfluid density ( ns∝1/λ2) in the presence of dissipation113,227. This is\nexpressed as ns/ns0= 1−DT2at low temperatures, where Dis directly proportional to\nthe dissipation. For the 2 .2-nm-thick sample with a critical temperature of approximately\n1.8 K, the T2variation becomes distinctly evident below 500 mK, while a linear variation\npredominates beyond this temperature threshold.\n4.6 Comparison of Tcand T∆\nIn exploring the impact of phase fluctuations, we turn our attention to the comparison\nbetween the critical temperature ( Tc) and the superconducting gap “vanishing” temper-\nature ( T∆). We have already observed in Fig. 4.6(a) that TcandT∆are almost identical\nfor higher thicknesses and only start to deviate below 11 nm. Interestingly, across all\nfilm thicknesses, both thick and thin, we consistently observe a low-temperature gap ra-\ntio of ∆(0) /(kBT∆)≈2 ( Fig. 4.6(b) ), implying that T∆corresponds to the expected\nmean-field transition.\nTable 4.1: Comparison of Tc,T∆and Γ (in K) for two thinnest samples of thickness d(innm)\nas curated from Fig 4.5 and Fig. 4.6.\nd T cT∆(TBulk\nc−T∆) Γ Γ /(TBulk\nc−T∆)\n2 1 .8 4.4 2 .6 6 .1 2 .3\n4.5 5 5 .6 1 .4 2 .9 2 .1\nNotably, the difference between the bulk critical temperature ( TBulk\nc) of the thickest\nfilm ( 7 Kfor 21 nm) and the measured T∆of thinner films corresponds closely to half\nof the tunneling “pair-breaking” parameter Γ, as evident in Table 4.1. This correlation,\nwhile not conclusive, hints at a consistent pairing interaction in all films, capable of\nyielding a Tcof 7K. However, the expected “mean-field” transition is suppressed by pair-\nbreaking mechanisms such as electron-electron scattering or thermal phase fluctuation45.\nThis aligns with the idea that the phenomenological broadening parameter Γ embodies\nthe influence of the pair-breaking mechanism within the system.\nCollectively, these findings strongly suggest that T∆essentially represents the “mean-\nfield” superconducting transition temperature ( T∆≈Tc0), diminished in thinner films\n88CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\ndue to Fermionic effects. The presence of Bosonic effects, potentially linked to phase\nfluctuations, further reduces the observed Tcbelow T∆.\n4.7 Interpretation of penetration depth data from\n“Mean-field” perspective\nIn this section, we shall attempt to compare and interpret the measured penetration\ndepth data from the “mean-field” viewpoint in light of scanning tunneling spectroscopy\n( STS ) data. It is revealed in the STS data ( Section 4.4.2 ) that the “gap”, which signi-\nfies the suppression of the low-energy quasi-particle density of states, disappears at T∆.\nAs discussed in Section 4.6, T∆is essentially the expected “mean-field” superconducting\ntransition temperature. Mean-field theories, which do not consider phase fluctuations,\nsuggest that the superfluid density also disappears at T∆. These theories encompass\nBCS3, the Abrikosov-Gor’kov theory46, and the Eliashberg theory70. The Abrikosov-\nGor’kov46theory incorporates elastic pair-breaking, revealing that pair-breaking smears\nthe BCS peak in the quasiparticle density of states. On the other hand, the Eliash-\nberg theory70, which includes inelastic electron-phonon scattering, indicates a low-energy\nsuppression in the density of states, but not a true gap, reminiscent of the Dynes phe-\nnomenological formula23.\n02 4 6 8 01234 \n20 nm \n12 nm \n 5 nm \n2.2 nmλ-2 ( µm-2 )T\n ( K )\nFigure 4.10: λ−2as a function of temperature for films with different thicknesses. Solid lines\nrepresent the temperature variation expected from the dirty-limit BCS theory with respect to\nT∆. For 2 .2nmquadratic fit has been used (cyan line). For 2 .2 and 5 nm, BKT lines are drawn\n(cyan and magenta dots respectively).\nNow, in Fig. 4.10, we juxtapose the temperature dependence of the measured super-\nfluid density ( ∝1/λ2(T) ) with a “mean-field” superfluid density that diminishes at\nT∆. For the thick samples, we apply the dirty-BCS temperature dependence of λ−2(T)\n( Eq. 4.7 ) to fit the data. However, in the case of the thinner samples ( d <5nm),\n894.8. COMPARISON OF ENERGY SCALES\nwhere ∆( T) deviates from the BCS equation ( refer to Fig. 4.5 ), the use of the BCS func-\ntional form is not justified. Instead, we adopt a mean-field approach for the 2 .2-nm-thick\nsample, assuming a simple quadratic form for superfluid density:\nλ−2(T)\nλ−2(0)\f\f\f\f\nmeanfield= 1−(T/T ∆)2. (4.12)\nThis Eq. 4.12 is consistent with Fig. 4.9, where the quadratic temperature ( T2) depen-\ndence is observed at low temperatures before transitioning to linearity in the presence of\nthermal phase fluctuations.\nThe fits in Fig.4.10 are distinct from those in Fig.4.7 in that we now fix the transition\natT∆instead of finding the best-fit Tc. This approach provides a “mean-field” perspective\non the temperature variation of superfluid density. Fig. 4.10 highlights the temperatures\nbetween TcandT∆, known as the “pseudogap” region, where presumably some Bosonic\nmechanism, such as thermal phase fluctuations, triggers the onset of DC resistance.\nTo further emphasize the anomalously strong phase fluctuations, we have incorporated\nthe universal BKT245–247line into Fig. 4.10 for the two thinnest films, given by,\nJs=2\nπT\n⇒1\nλ2=8µ0e2\nπℏ2d×T,(4.13)\nwhere Jsis the superfluid stiffness ( defined in Eq. 4.9 ) and dis the thickness of the thin\nfilm. The intersection of the BKT line with the “mean-field” λ−2(T) visually indicates\nthe region where conventional phase fluctuation theory predicts the disappearance of su-\nperfluid density248,249. It is observed in Fig. 4.10 that the superfluid density for both 2 .2\nand 5 nmfalls well before the possible intersection point mentioned above. Consequently,\nit becomes evident that phase fluctuations suppress superfluid density much more vigor-\nously than anticipated. Furthermore, the spatial inhomogeneity in the local superfluid\ndensity within the thinnest films could amplify the overall suppression of the superfluid\ndensity. This results in a reduction of the critical temperature ( Tc) in the thinnest\nfilms, where both superfluid density and resistivity vanish, well below the anticipated\noccurrence of the familiar Kosterlitz-Thouless-Berezinskii transition.\n4.8 Comparison of energy scales\nAt last, to reconcile the penetration depth measurements with the emergence of the\npseudogap, we discuss the energy scales from the above two measurements: superfluid\nstiffness Js(0) from the penetration depth measurements and superconducting gap or\npairing energy ∆(0) from STS measurements. ∆(0) is the energy scale important for the\n90CHAPTER 4. FERMIONIC TO BOSONIC CROSSOVER IN ULTRATHIN A-MOGE\nsuperconducting transition in conventional superconductors according to the BCS theory\nwhere the superconducting state is resilient to phase fluctuations. On the contrary, for\nsuperconductors vulnerable to phase fluctuations, the energy scale corresponding to Js(0),\nwhich is the measure of the resilience of a superconductor against phase fluctuations,\nplays an important role. Thus, in general, the superconducting Tcis determined by the\nlower of these two energy scales11. When ∆(0) ≪Js(0), the superconducting transition\ntemperature is given by226,Tc∼∆(0)/(AkB), where A∼2 for a-MoGe. On the other\nhand, when Js(0)≪∆(0), thermal phase fluctuations play a dominant role, where\nthe superconducting state gets destroyed due to thermal phase fluctuations at Tc∼\nJs(0)/(BkB) ( B is a constant of the order of unity ) even if the pairing amplitude\nremains finite up to higher temperatures.\n05 1 01 52 010100JS(0), Δ(0) ( K )d\n ( nm ) JS(0) \nΔ(0)\nFigure 4.11: Comparison of Superfluid stiffness Js(0) and pairing energy ∆(0) in the limit\nT→0 as a function of thickness d.\nIt is observed in Fig. 4.11 that Js(0) is an order of magnitude larger than ∆(0) for\nlarger thicknesses ( d > 5nm). But Js(0) drops sharply below 5 nmand becomes\ncomparable to ∆(0) around d∼2nm. Thus, around this thickness, we expect phase\nfluctuation to dominate the superconducting properties. Indeed, in the corresponding\nSTS intensity maps ( Fig. 4.4(d)-(e) ), a soft gap appears in STS for d∼5nm, while\npseudogap persists up to almost double of Tcford∼2nm. The observation of the\npseudogap suggests that at this thickness the pairing amplitude remains finite even when\nthe global phase coherent state has been destroyed by phase fluctuations. In addition,\nsince the film is in the 2D limit, the measured temperature dependence of the superfluid\nstiffness near Tcturns out to be consistent with a Berezinskii-Kosterlitz-Thouless246,247,249\n( BKT ) jump smeared by disorder-induced inhomogeneity250,251( see Appendix A ).\n914.9. SUMMARY AND CONCLUSION\n4.9 Summary and conclusion\nIn summary, we have shown that the suppression of superconductivity in a-MoGe with\ndecreasing film thickness has two regimes. At moderate disorder, Tcdecreases but ∆ /kBTc\nshows only a small increase ( Fig. 4.6(a) ) and R9K\nsvsTc( Fig. 4.2(b) ) follows the\nFinkel’stein model both consistent with the Fermionic scenario. But, at stronger disorder,\nthe system moves from Fermionic to Bosonic regime, where the pairing amplitude remains\nfinite and manifests in the form of pseudogap even after the superconducting state is\ndestroyed by phase fluctuations. This evolution of the superconducting state is consistent\nwith earlier observations on disordered NbN, TiN, and InO x80,81,113, where the eventual\ndestruction of superconductivity through the Bosonic route is well established.\nIn this context, we would like to note that some early tunneling measurements per-\nformed at low temperatures on thin superconducting amorphous films ( Bi, Pb ) ap-\npeared consistent with the Fermionic scenario119,120. However, these measurements were\nperformed on planar tunnel junctions where the influence of the normal-metal electrode\non the superconductor through the relatively low resistance tunnel barrier and the pos-\nsibility of chemical mixing at the interface are difficult to completely rule out. Also, it\nshould be approached with caution to draw the conclusion that the disappearance of the\ngap in the insulating regime implies the absence of Cooper pairs, because vanishing of the\ngap in superconducting tunneling can also arise from other factors, such as pair breaking\ncaused by disorder252.\nOn the other hand, the observation of the Bosonic scenario in a-MoGe, which was\nwidely believed to follow the Fermionic route, raises the important question on whether a\ndisordered superconductor can follow the Fermionic route all the way to the destruction of\nsuperconductivity or whether all superconductors become Bosonic at sufficiently strong\ndisorder. We conclude with the following conjecture, which needs to be addressed in\nfuture theoretical and experimental studies.\nConjecture. In the limit of extreme disorder, the suppression of the superconductivity\nin all superconductors will follow the Bosonic mechanism, regardless of whether it follows\nthe Fermionic or Bosonic route at moderate disorder.\n92CHAPTER5\nStudy of vortex dynamics in a-MoGe using low-frequency two-coil\nmutual inductance measurements: extracting vortex parameters\n5.1 Introduction\nUnderstanding the dynamics of vortices in type-II superconductors is of paramount im-\nportance both from a fundamental standpoint and for the practical application of these\nmaterials253–258. In type-II superconductors, when we apply a magnetic field larger than\nthe lower critical field Hc1, quantized flux lines ( vortices ) penetrate the sample. In a\nclean superconductor, the interaction between the vortices arranges them in a triangu-\nlar lattice known as the Abrikosov63–65,134vortex lattice ( VL ). However, the inevitable\npresence of crystalline defects in solids acts as a random pinning potential for the vortices.\nIf these vortices are made to oscillate under the influence of an oscillatory current or\nmagnetic field, their motion is governed by the following competing forces135: i) Lorentz\nforce due to the external current density driving the motion, ii) restoring force due to the\ncombined effect of pinning by crystalline defects and repulsion from neighboring vortices,\nand iii) the dissipative viscous drag of the vortices. In addition, at finite temperature\nthermal activation can cause the vortices to spontaneously jump over the pinning barrier\nresulting in thermally activated flux flow ( TAFF ), which produces a small resistance\neven for external current much below the critical current density ( Jc).\nIn this work, we have studied the dynamics of vortices in the presence of small ac\nexcitation ( 0 .05mAand 30 kHz for our two-coil167,168,210,211,259–261measurements ) in\nweakly-pinned 20- nm-thick a-MoGe film, where the ac magnetic shielding response is\n935.2. THEORETICAL BACKGROUND: COFFEY-CLEM MODEL\nprimarily governed by the dynamics of vortices. The vortex contribution can be formally\naccounted for through an effective penetration depth ( the Campbell145,146penetration\ndepth or λC) which operates in addition to the London15penetration depth ( λL).\nTo analyze the experimental temperature-dependent variation of the penetration depth\ndata, we utilized mean-field models to describe the motion of vortices under small ac\nexcitation.\n5.2 Theoretical background: Coffey-Clem Model\nTo interpret the extracted penetration depth data from the two-coil mutual induc-\ntance measurements, we have made use of a mean-field model proposed by Coffey and\nClem150,157–162for the motion of vortices in response to small ac excitation. This model\nwas discussed in detail in section 1.7.3.2. For continuation in this chapter, I shall briefly\nstate the relevant part needed for our analysis. According to this model, the equation of\nmotion of a single vortex ( force per unit length ) is given by:\nη˙u+αLu= Φ 0Jac׈n+F(U,T), (5.1)\nwhere Jacis the external alternating current density on the vortex containing one flux\nquantum Φ 0(ˆnbeing the unit vector along the vortex ), ηis the viscous drag coefficient on\nthe vortex, αLis the restoring pinning force constant called the Labusch parameter139–141,\nandF(U,T)is a random Langevin ( thermal ) force depending on the pinning barrier U\nfor the thermally activated flux flow ( TAFF ) of the vortices. Coffey and Clem150,157–162\n( CC ) solved Eq. 5.1 and derived the following equation for ac penetration depth:\nλ2\nac=\u0012\nλ2\nL−iρTAFF\nv\nµ0ω\u0013\u001e \n1 + 2 iλ2\nL\nδ2\nnf!\n, (5.2)\nwhere normal fluid correction is given by the normal fluid skin depth ( δnf) and the\nvortex contribution is expressed in the vortex resistivity ( ρTAFF\nv ) term given by,\nρTAFF\nv =ρffϵ+iωτ\n1 +iωτ, (5.3)\nwhere τis the vortex relaxation time in the presence of TAFF expressed as,\nτ=η\nαLI2\n0(ν)\nI1(ν)I0(ν), (5.4)\nandϵis the vortex creep parameter given by,\nϵ=1\nI2\n0(ν). (5.5)\n94CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\nHere, Ipis the modified Bessel’s function of the first kind of order p. The effect of TAFF\nenters in the argument of Ipthrough the parameter νdefined as,\nν=U\n2kBT. (5.6)\n5.3 Sample used in this work\nOur sample consists of a 20- nm-thick ( d) weakly-pinned amorphous Molybdenum Ger-\nmanium ( a-MoGe ) thin film, grown on (100) oriented MgO substrate by pulsed laser\ndeposition ( PLD ) technique, ablating an arc-melted Mo 70Ge30target. The zero field\nsuperconducting transition temperature ( Tc) for the 20- nm-thick samples was ∼7K\n( similar to the samples in ref. 188,195 ).\n2345678-0.5-0.4-0.3-0.2-0.10.0 \n500 Oe (ZFC) \n500 Oe (FC) \n1 kOe (ZFC) \n1 kOe (FC) \n3 kOe (ZFC) \n3 kOe (FC)χ / χ0T\n ( K )(b)\n-200-1000 1 002 00-6-4-202-\n40-2002040-6-4-20M ( 10 − 4 emu )H\n ( Oe )(a)\nFigure 5.1: (a) M−Hloop for a 20- nm-thick a-MoGe film grown on (100) oriented MgO\nsubstrate measured using the SQUID-VSM technique. The loop has been enlarged in the inset.\n(b)Normalized real part of the ac susceptibility, χ, as a function of temperature for the FC\nand ZFC state at 500 Oe, 1kOe and 3 kOe (χ0is the zero-field susceptibility at 1 .5K) on a\nsimilar sample (adapted from Ref. 188). The FC and ZFC curves at each field overlap with\neach other showing the very weakly pinned nature of the VL. The density of points for the FC\ncurves has been reduced for clarity.\nThe weakly pinned nature of the 20- nm-thick a-MoGe thin film is reflected in the very\nsmall M-H loop measured using the SQUID-VSM technique ( Fig. 5.1(a) ). Weak pinning\nnature was also evident in the previous ac susceptibility measurements ( Fig. 5.1(b) ),\nwhere the field-cooled ( FC ) and zero-field-cooled ( ZFC ) data on similar samples188\nshowed a negligible difference for fields above 500 Oe.\n5.4 Experimental data\nIn Fig. 5.2(a) and (b), we plot the temperature variation of M′andM′′for different fields\nmeasured using the two-coil technique. From the data, the inverse square of the penetra-\n955.4. EXPERIMENTAL DATA\n108109101010112\n3 4 5 6 7 1081091010 H ( kOe ) \n1 \n2 \n3.5 \n5 \n7 \n10 \n15 \n20 \n30 \n40 \n50 \n60λ−2 ( m−2 )(c)δ−2 ( m−2 )T\n ( K )(d)\n304050607080901002\n3 4 5 6 7 -10-8-6-4-20M' ( nH )H ( kOe ) \n1 \n2 \n3.5 \n5 \n7 \n10 \n15 \n20 \n30 \n40 \n50 \n60(a)M'' ( nH )T\n ( K )(b)\nFigure 5.2: (a), (b) Temperature variation of experimentally measured M′andM′′respec-\ntively at the different fields (1 −60kOe);(c), (d) Corresponding λ−2andδ−2as a function of\ntemperature respectively extracted using finite element analysis described in Section 2.4.1.\ntion depth λ−2and the skin depth δ−2data were extracted using the process mentioned\nin Section 2.4.1 and shown in Fig. 5.2(c) and (d) respectively. The ac penetration depth\ncan be broken down in terms of λandδas,\nλ−2\nac=λ−2\neff=λ−2+iδ−2. (5.7)\nEq. 5.7 can also be expressed in terms of the effective electrical conductivity of the film\nas follows,\nσeff=1\niµ0ωλ2\neff=1\nµ0ωδ2+1\niµ0ωλ2. (5.8)\nAlternatively, the effective impedance of the film is,\nρeff=iµ0ωλ2\neff=iµ0ω(λ2\nL+λ2\nvortex), (5.9)\nIt is evident that the electrical impedances of both the superfluid, denoted by iµ0ωλ2\nL,\nand the vortices, represented by iµ0ωλ2\nvortex, are in series ( also see Eq. 1.76 ). Conse-\nquently, their impedances add, in contrast to their conductivities. It should be noted\nthat Eq. 5.9 is just the simplified version of Eq. 5.2 barring the normal-fluid correction.\nNow, to gain a sense of the values, let us tabulate the experimental details for our\nsample: film surface area = π×42= 50.3mm2, film thickness d= 20 nm, normal state\nresistivity ρn=Rnd= 1.55µΩ−m, normal state sheet resistance Rn= 77.5 Ω, zero-field\nsuperconducting transition temperature Tc= 7.05K, zero-temperature upper critical\n96CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\nfieldHc2(0) = 11 .5T= 115 kOe, GL coherence length ξ(0) = 5 .35nm, inverse square of\nthe London penetration depth λ−2\nL≡λ−2(T= 0, B= 0) = 3 .55µm−2, Pearl penetration\ndepth λP≡2λ2\nL/d≈28µm. For vortex-creating dc perpendicular fields ( B) from 1\nkOeto 60 kOe, the intervortex distance ( a∆= 1.075(Φ 0/B)1/2) decreases from 155 nm\nto 20 nm.\nFor the dc fields studied herein, the impedance of the superfluid is negligible, being less\nthan 10% of the impedance of vortices. Hence, the statement above that the experimental\nparameters in Fig. 5.2 refer only to vortices. So, the dominant contribution to σeffcomes\nfrom the vortices, i.e., σeff≈σvortex.\nNow, coming back to Fig. 5.2, we first concentrate on λ−2. We observe that λ−2(T→\n0) decreases as a function of field signifying an increase in the Campbell penetration depth.\nWith an increase in temperature, λ−2smoothly starts decreasing at low temperatures\nbefore it drops rapidly eventually going below the resolution limit.\n23 4 5 6 7 110100H ( kOe )T\n ( K ) VS-HVF \nHVF-IVL \nHc2 \nT*\nFigure 5.3: “λ−2-vanishing” temperature points ( T∗) were noted for all the fields from Fig. 5.2,\nwhere λ−2(T > T∗) drops below the resolution limit 108m−2and plotted (Cyan connected\npentagons) in the H-T parameter space along with earlier phase space boundaries188: VS-to-\nHVF (connected black squares), HVF-to-IVL (connected red circles) and Hc2(connected blue\ntriangles) respectively (VS: Vortex solid, HVF: Hexatic vortex fluid and IVL: Isotropic vortex\nliquid).\nFrom earlier measurements188, we know that the vortex state in a 20- nm-thick a-\nMoGe film undergoes two transitions: From a vortex solid ( VS ) at low fields and\ntemperatures to a hexatic vortex fluid ( HVF ), and then from a hexatic vortex fluid\n( HVF ) to an isotropic vortex liquid ( IVL ). In Fig. 5.3, we plot the temperature\nT∗for every field at which λ−2(T > T∗) drops below our resolution limit of 108m−2.\nIn the same graph, we plot the locus of the transitions from VS to HVF and HVF to\nIVL obtained earlier188from the scanning tunneling spectroscopy and magneto-transport\nmeasurements. At low temperatures, T∗(H) is just below the HVF-IVL boundary188\nshowing that the screening response vanishes in the IVL. However, we do not see any\nsignature of the VS-HVF transition in the temperature variation of λ−2. This is due to the\n975.4. EXPERIMENTAL DATA\nfact that at low temperatures the mobility of the vortices in the HVF is extremely slow195,\nand hence the ac pinning response in HVF is practically indistinguishable from the vortex\nsolid. This is also consistent with earlier magneto-transport measurements where no\nsharp change in pinning properties was observed across the VS-HVF transition at low\ntemperatures. In fact, it was shown earlier195that the Larkin-Ovchinnikov262collective\npinning model, originally developed to explain vortex pinning in an imperfect vortex\nsolid is largely applicable in the HVF state as well, at temperatures well below Tc. At\nhigher temperatures, T∗bends towards the VS-HVF boundary. This can be understood\nfrom the fact that the mobility of the vortices increases with increasing temperatures\ndue to thermal fluctuations and now, even in the HVF, the vortices experience very little\nrestoring force due to pinning, thereby destroying the shielding response.\nComing to δ−2( Fig. 5.2(d) ), we observe that for fields above 10 kOe,δ−2shows\na single dissipative peak close to T∗as expected from fluctuations close to a transition.\nHowever, at low fields, in addition to this dissipative peak, we observe an increase in δ−2\nwith a decrease in temperatures. This increase in δ−2at low temperatures indicates the\npresence of additional dissipative modes at low temperatures which is not accounted for\nby the theoretical framework used here.\n0.00 .51 .01 .52 .00.000.020.040.060.08 \nδ−2 (2 K) \n∝ 1/B2δ−22\nΚ ( µm−2 )B\n ( T )(b)\n0.00 .51 .01 .52 .00.000.050.100.15 \nλ−2 (2 K) \n∝ 1/Bλ−22\nΚ ( µm−2 )B\n ( T )(a)\nFigure 5.4: (a) ,(b)Magnetic field variation of λ−2andδ−2values at 2 K(black circles and\nblack triangles respectively) extracted from Fig. 5.2(c) and (d) respectively. They are observed\nto follow: λ−2\n2K=kλ(1/B) (red line) and δ−2\n2K=kδ(1/B2) (blue line), where the proportionality\nconstants are kλ= 1.7×1010T/m2andkδ= 1.1×109T2/m2respectively. The magnetic field\n(B) values are in Tesla (T).\nNow, We shall explore the magnetic field variation of low-temperature values of λ−2\nandδ−2. Theoretically, we expect both of them to follow 1 /Bvariation as the vortex\nimpedance is proportional to the density of vortices, hence to the applied field:\nρvortex =1\nσvortex=\u00121\nµ0ωδ2+1\niµ0ωλ2\u0013−1\n∝B (5.10)\n98CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\nFig. 5.4(a) shows that for H < 20kOe λ−2∝1/Bfor 2 Kas expected. On the contrary,\nδ−2should be vanishingly small at low T, but it is not, and it is proportional to ∝1/B2\nas shown in Fig. 5.4(b), not as expected. This anomaly should be investigated in future\nstudies.\n01 2 3 4 0.00.10.20.30.4hac ( mOe )r\n ( mm ) Field due to drive coilI\nac = 0.05 mA, f = 30 kHz\nFigure 5.5: Simulated radial distribution of the magnetic field on the sample due to ac\nexcitation ( Iac= 0.05mA,f= 30 kHz) at the primary quadrupole coil.\n5.5 Applicability of Coffey-Clem equation: linearity\nof ac excitation amplitude\nIn order to quantitatively analyse the data, we fit the data with Eq. 5.2. First, we note\nthat in order for the harmonic approximation ( Eq. 5.1 ) to hold, the ac excitation needs\nto satisfy the following condition135:\nhac≪µ0JcλC= (µ0JcBrf)1/2≡hp. (5.11)\nUsing previously published data195on magneto-transport measurements on a similar film,\nJcB∼2×107−3×108N/m3within our field of interest ( 1 −60kOe) and rf∼ξGL∼5.5\nnm, where ξGLis the Ginzburg-Landau ( GL ) coherence length. Using the above values\nwe estimate hp∼3−14Oewhich is several orders of magnitudes larger than our hac\n( Fig. 5.5 ). Furthermore, from the normal state resistivity of the film, ρN∼1.55µΩ−m\nwe estimate δnf(0)∼3.5mm. Since λL(0)∼587nm, we obtain,\nλ2\nL\nδ2\nnf∼5×10−8. (5.12)\n995.6. TEMPERATURE DEPENDENCE OF VORTEX PARAMETERS\nWe can therefore neglect this term in Eq. 5.2 and set the denominator to unity. The\nsimplified equation now reads as follows:\nλ−2\nac=λ−2+iδ−2=\u0012\nλ2\nL−i1\nµ0ωBΦ0\nηϵ+iωτ\n1 +iωτ\u0013−1\n(5.13)\n5.6 Temperature dependence of vortex parameters\nWe have done the subsequent analysis on λ−2data only. Now, we attempt to fit the\ntemperature variation of the real part of experimental λ−2\nacdata with the Eq. 5.13, where\nwe also have to make some assumptions on the temperature dependence of the vortex pa-\nrameters: η,αL, and U. In the following section, I shall discuss some of the existing forms\nof temperature variations in literature and then state the form used in our subsequent\nanalysis.\n23 4 5 6 7 0255075100 \nExp. Hc2 \nHc20[(1-t2)/(1+t2)]0.66Hc2 ( kOe )T\n ( K )Hc2(0) = 115 kOet\n = T/Tc, Tc = 7.05 K\nFigure 5.6: Temperature dependence of Hc2(kOe) adapted from ref. 188 (black circles),\ndefined by the criterion: ρ(Hc2)∼0.95ρn, along with the fit: Hc2(0)[(1−t2)/(1 +t2)]0.66(green\nline), with Hc2(0) = 115 kOe,t=T/T c,Tc= 7.05K. Temperature variation of ηwas taken to\nbe the same as this form of Hc2.\n5.6.1 Tvariation of η\nThe temperature dependence of ηis assumed to follow the variation of Hc2according to\nthe Bardeen-Stephen formula143,144:\nη=µ0Φ0Hc2\nρn. (5.14)\nHc2was earlier188measured from isotherms in magneto-transport experiments and de-\nfined from the criterion, ρ(Hc2)∼0.95ρn. As shown in Fig. 5.6, Hc2follows the empirical\nvariation188:Hc2∝[(1−t2)/(1 +t2)]0.66, where t=T/T c,Tcbeing the zero-field super-\n100CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\nconducting transition temperature. Therefore, variation of ηis taken as:\nη=η0\u00121−t2\n1 +t2\u00130.66\n, (5.15)\nwhere η0=µ0Φ0Hc2(0)\nρn. Our sample ( amorphous MoGe thin film ) being an s-wave\nsuperconductor, η0is assumed to be field-independent263. For our sample, η0≈1.55×\n10−8Ns/m2( using Bc2≡µ0Hc2(0) = 11 .5Tandρn= 1.5−1.55µΩ−m).\n5.6.2 Tvariation of U\n5.6.2.1 By scaling approach\nThere have been various temperature variations of Usuggested in the literature. While\nPalstra et al.264,265used a temperature-independent Uto analyse the arrhenius plots\nin YBa 2Cu3O7and Bi 2.2Sr2Ca0.8Cu2O8 +δ, there were other predictions for temperature\nvariations for U. Let us start with the scaling approach adopted by Yeshurun and Mal-\nozemoff266( YM ). They took cues from the Anderson-Kim model132,133and predicted\nthe temperature dependence for different scenarios. I shall discuss the possible forms\nfor the temperature dependence of Ufollowing their arguments. It is to be noted that\nonly the temperature dependence terms are included in the following forms, where the\nconstant coefficients are omitted for simplification. We shall make use of the well-known\nempirical temperature variation of Hcandξgiven by12,14,\nHc(t)∼(1−t2), (5.16)\nand\nξ(t)∼\u00121 +t2\n1−t2\u00131/2\n, (5.17)\nwhere t=T/T c,Tcbeing the superconducting transition temperature.\nYM started with the form of Uproposed by Anderson and Kim132,133,\nU∼H2\ncξ3, (5.18)\nwhere Hcis the thermodynamic critical field and ξis the coherence length. Using the\nempirical temperature variation of Hcandξ, we get the form,\nU(t)∼(1−t2)1/2(1 +t2)1/2. (5.19)\nEq. 5.19 can be approximated in the vicinity of transition temperature ( T→Tc≡t→1)\nin the following form266:\nU(t)∼(1−t)1/2. (5.20)\n1015.6. TEMPERATURE DEPENDENCE OF VORTEX PARAMETERS\nAnother form of Uwas suggested by YM for high fields, specifically when more\nand more flux lines penetrate the sample. When the vortex lattice constant a0=\n1.075(Φ 0/B)1/2becomes significantly smaller than the penetration depth λ, especially\nin the case of high- κsuperconductors, we can expect a crossover in the pinning behavior\ndue to the collective effects. This transition was indeed observed at Hcross= 0.2Hc2in\nthe critical current measurements on Y Ba 2Cu307267, which corresponds to a0≈6ξ266.\nHowever, it should be noted that Hcrossmay vary depending on the system. Beyond this\nthreshold field Hcross,Uis expected to be constrained by a0in two dimensions, while\nξremains the minimum characteristic length in the third dimension. Accordingly, YM\nproposed the following form:\nU∼H2\nca2\n0ξ. (5.21)\nInserting the previous Temperature dependencies of Hcandξand, using a2\n0∼1/H, we\nget the form268,269:\nU(t, H)∼(1−t2)3/2(1 +t2)1/2\nH∼(1−t2)(1−t4)1/2\nH. (5.22)\nAtt→1, Eq. 5.22 becomes:\nU(t, H)∼(1−t)3/2\nH. (5.23)\nTheUexpression in Eq. 5.23 has been used extensively in the literature266,269–271. YM266\nwas able to qualitatively explain the “quasi de Almeida - Thouless”272–276or the irre-\nversibility line, (1 −t)∝H2/3by substituting the form of Ufrom Eq. 5.23 into the the\ncritical current expression.\n5.6.2.2 Using collective pinning model\nThe functional dependence of Ucan also be estimated in the framework of the collective\npinning model. It is to be noted that the 20- nm-thick MoGe film has been earlier shown\nto be in 2D limit following the condition277:J1/2\nc≪0.2B1/4d−1from the critical current\n(Jc) measurements on a similar sample195(dis the thickness of the thin film ). It\nis also expected because the thickness of the thin film ( ∼20nm) is much less than\nthe characteristic bending length of the vortices278( few µm). Therefore, in the 2D\nscenario, the characteristic length along the vortex direction, previously represented by\nthe coherence length in 3D, is now substituted by the film thickness d.\nNow, according to the 2D collective pinning theory, the activation barrier Ugoverning\nthe thermal creep rate can be estimated by the shearing elastic energy of the vortex lattice\ndue to the small displacement ( u) of a single vortex as10,279,\nU∼C66\u0012ξ\nRc\u00132\nVc, (5.24)\n102CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\nwhere the displacement uhas been taken as the core dimension, ξ, and C66,RcandVc\nare the shear modulus, transverse Larkin length, and collective pinning volume of the\nVL respectively. In the case of the 2D collective pinning model, Vc∼R2\ncdand hence\nU∼C66ξ2d. Using the well known variation of C66from the literature156,262,280,281,\nC66∼µ0H2\nch(1−h)2, the temperature variation of Ubecomes,\nU(t)∼H2\nc(t)ξ2(t). (5.25)\nNow, Using the known empirical dependence of Hcandξ12,14( Eqs. 5.16 and 5.17 ), one\nobtains,\nU(t, H) =U0(H)(1−t2)(1 + t2). (5.26)\nAs an illustration of the magnitude of U, consider a 3- nm-thick NbN film, which was\nreported to be measured at U0≈20K282using transport.\n5.6.3 Tvariation of αL\nThe restoring force term αLuin Eq. 5.1 can be represented as the derivative of pinning\npotential U, which means that the spring constant αLcan be expressed as∂2U\n∂u2, i.e.,\nαLdepends on the curvature of the pinning potential at the bottom. Though the exact\nshape and curvature of Uare difficult to predict, an estimate of αLcan be found from the\nenergy consideration of the core of the vortex when subjected to the small displacement\nconsidered in Eq. 5.1. If we equate the total condensation energy in the displaced core\nwith the stored elastic energy: µ0H2\nc(πξ2d) =1\n2αLξ2d, we get the temperature variation\nofαLas283,284:\nαL(t)∼H2\nc(t)∼(1−t2)2. (5.27)\nA further modification was suggested by Belk et al.285taking into consideration the effect\nof quasi-particle scattering due to pinning centers formed by point defects286,287:\nαL(t)∼(1−t2)2\n(1 +t)4. (5.28)\nValues of αL288,289were reported to be approximately 3 ×105N/m2for YBa 2Cu3O7-x,\nmeasured using the parallel plate resonator ( PPR ) technique.\n5.6.4 Correction to Uand αLdue to thermal fluctuations\nA significant modification of the functional forms was suggested by Feigel’man et al.290\nand Koshelev et al.291considering the effect of thermal fluctuations. They suggested\nthat the smearing of the pinning potential due to thermal fluctuations would result in\nthe exponential decay of αLandU. Experimentally, such exponential decay was indeed\n1035.7. FITTING WITH EXPERIMENTAL λ−2DATA\nobserved for YBa 2Cu3O7and Bi 2Sr2CaCu 2O8thin films288,289,292for temperatures lower\nthan 0 .8Tc. Since the two-dimensional vortex state in thin a-MoGe films is extremely\nsensitive to small perturbations196, it is expected that the vortex lattice would also be\nsusceptible to thermal fluctuations.\n5.7 Fitting with experimental λ−2data\nOur attempts to fit the data with the above temperature dependence for UandαL\nestimated from the 2D collective pinning model ( Eqs. 5.26 and 5.27 respectively ) resulted\nin poor fit, particularly at low magnetic fields. On the other hand, a much better fit over\nthe entire field range was obtained by assuming temperature dependence of the form:\nαL∼αL0e−\u0010\nT\nT0\u0011\n, (5.29)\nU∼U0e−\u0010\nT\nT0\u0011\n. (5.30)\nHere, T0is the characteristic temperature dependent on the effect of thermal fluctuations.\nFor comparison, fits using both the temperature variations on a representative field 5 kOe\nare shown in Fig. 5.7(a). We also tried other combinations of temperature variations of\nUandαL, but the fitting was worse than the above two combinations ( see Appendix B.1\nfor further details ). However, from Fig. 5.7(a), it is clearly evident that the dominant\neffect on temperature dependence comes from the smearing of pinning potential due to\nthermal fluctuations. Nevertheless, a small deviation is observed at higher temperatures\nclose to the knee region above which λ−2drops rapidly. This is however unsurprising since\nthis is close to the boundary where the harmonic approximation assumed in Eq. 5.1 will\nstart to break down. For 50 and 60 kOe, the temperature range of data is too small to\nperform a reliable best fit, but the qualitative variation is captured by using extrapolated\nparameters from lower fields.\nIn Fig. 5.7(b), we have shown the fits ( black lines ) to the experimental λ−2data\naccording to the real part of Eq. 5.13 where we have taken the exponential dependence of\nαLandU, where the fitting parameters are αL0,U0andT0. Further details on the fitting\nprocedure and finding the best-fit procedure are shown in Appendix B.1 and B.2.\n5.8 Field dependence of vortex lattice parameters\nAs we assumed the temperature dependence of the parameters in Eq. 5.13 as explained\nabove, the field dependence of the vortex parameters was reflected in the fitted coefficients\nU0,αL0, and T0as shown in Fig. 5.8, 5.9 and 5.10 respectively. In the following sections,\nI shall discuss some existing field variations from the literature for perspective, before\n104CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\n23 4 5 6 7 10810910101011H ( kOe ) \n1 \n2 \n3.5 \n5 \n7 \n10 \n15 \n20 \n30 \n40 \n50 \n60λ−2 ( m−2 )T\n ( K )(b)\n23 4 5 6 10810910101011λ−2 ( m−2 )T\n ( K )5 kOe(a)\nFigure 5.7: (a) Experimental λ−2vsTdata at a representative field (5 kOe) (black circles),\nalong with the fit to Eq. 5.13 using (i) αL=αL0e−T/T 0, U=U0e−T/T 0(red line) and (ii)\nαL=αL0(1−t2)2, U=U0(1−t2)(1 + t2) (blue line). (b)λ−2vsTat different fields along with\nthe fit to Eq. 5.13 with the best-fit parameters following the method discussed in Section B.2.\ndiscussing our data.\n5.8.1 Field variation of U0\nIn literature, there have been several examples of power-law decay in the field variation of\nUsuch as H−β,βdepending on the materials and the choice of the temperature depen-\ndence model. If indeed Uis following the Eq. 5.22 or 5.23, βis supposed to be 1. Zeldov\net al.269found β≈0.95 while Kim et al.268gotβ≈0.73 in YBa 2Cu3O7−δconsidering\nEq. 5.22 as the temperature dependence. Palstra et al.264found that β≈1\n2forH∥and\nβ≈1\n6forH⊥considering the temperature-independent Ufor Bi 2.2Sr2Ca0.8Cu2O8+δ(H∥\nandH⊥are the field parallel and perpendicular to a−bplane ).\n01020304005001000U0/2kB ( K )H\n ( kOe ) ∝ Η − 0.82\nFigure 5.8: U0/2kBvsH(black circles) which follows power law decay (red line): U0∝H−0.82\nOn the other hand, logarithmic dependence was suggested by Feigel’man et al.293\narising from the interaction of dislocations in the vortex lattice. However, after the\n1055.8. FIELD DEPENDENCE OF VORTEX LATTICE PARAMETERS\nformation of a free dislocation pair, the potential barrier for moving the dislocations is\nquite less than the energy to form the pair. So, the defining contribution comes from the\nformation of dislocation pairs given by294–296:U0(H)≈U′\n0ln (H′\n0/H), where U′\n0=Φ2\n0d\n64µ0π2λ2\nandH′\n0≈Hc2. The derived expression is an approximation, where some factors of the\norder of unity were neglected in the argument.\nHowever, as seen in Fig. 5.8, U0follows the power decay law: H−0.82which is consistent\nwith the earlier studies297ona-MoGe, though our exponent is larger than the earlier value\nof 0.66, calculated from magneto-transport measurements.\n0.00 .10 .20 .3108109 \nαL0 ξ(B/Φ0) \n∝ h0.5(1-h)/dRcFp ( N/m3 )h\n ( ≡ H/Hc2(0) )(b)\n0102030404060801000\n10203040500.51.01.5αL0 ( N/m2 )H\n ( kOe )(a)Jc ( 108 A/m2 )H\n ( kOe )\nFigure 5.9: (a) Field dependence of αL0(connected red circles). (inset) Critical current\ndensity ( Jc) as a function of field (blue connected circles) measured on a similar sample from\nref. 195. (b)Pinning force density Fp(=αL0ξB/Φ0) vs normalized field h(=H/H c2(0)) (black\ncircles) along with the theoretical field variations proportional to h0.5(1−h)/dR c(blue line)\n(semi-log scale). Field variation of Rcis taken from ref. 195 and d= 20 nmis the film thickness.\nError bars for the parameters were determined following the protocol explained in Section B.2.\n5.8.2 Field variation of αL0\nFig. 5.9(a) demonstrates the non-monotonic behavior of αL0. Initially, it decreases with\nthe field and reaches a shallow minimum at 15 kOe before starting to increase again.\nThis minimum closely resembles the pattern observed in the field-dependent variation\nofJcmeasured on similar samples195. The similarity can be understood based on the\ntheory of collective pinning. Within Larkin-Ovchinnikov262theory of collective pinning,\nthe pinning force on a vortex ( per unit length ) for a displacement uis given by,\nFΦ0\np=αL0u≈Φ0\nB\u0000\n⟨f2\nαL0⟩nA\u00011/2\ndRc(5.31)\nwhere fαL0is the elementary restoring pinning force and nAis the areal density of pinning\ncentres. FΦ0pis governed by two counteracting effects: the variation of fαL0andRcwith\nmagnetic field. With increase in field, fαL0andRccontrol the behaviour of αL0making\n106CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\nit non-monotonic with field. Depending on the nature of pinning, the variation of fαL0\nfollows the general form199:\n⟨f2\nαL0⟩nA∝Chn(1−h)2, (5.32)\nwhich suggests that,\nFΦ0\np∝hn\n2−1(1−h)\ndRc, (5.33)\nwhere n= 1 corresponds to pinning due to dislocation loops200andn= 3 due to\nimpurities and vacancies201respectively. We have earlier observed that n= 1 correctly\ndescribes the pinning in a similar sample195. Thus, at low fields the initial decrease in\nαL0comes from the dominant numerator because of a decrease in elementary pinning\nforce with an increasing field. However, above a certain field, the decrease in Rc195with\nincreasing fields starts to dominate, thus increasing αL0( Fig. 5.9 ). Similar field variation\ninJcfrom the magneto-resistance measurements was observed on a similar sample195\n( inset of Fig. 5.9(a) ) as a precursor of the peak effect observed in the sample. We\ncould not capture the peak at higher fields since the shielding response goes below our\nresolution limit. To further analyse the data quantitatively, we express Eq. 5.31 in terms\nof pinning force density ( Fp):\nFp≈αL0ξ\u0012B\nΦ0\u0013\n∝hn/2(1−h)\ndRc, (5.34)\nwhere displacement of a vortex is assumed to be of the order of coherence length ξ. Using\nthe field variation of Rcobtained from STS images in Ref. 195, in Fig. 5.9(b), we plot\nthe variation of Fpas a function of the normalized field h(≡H/H c2(0) ) from Eq. 5.34\nwhere the proportionality constant is taken as an adjustable parameter. The qualitative\nagreement is indeed very good when we consider that Rchas been measured on a different\nsample.\n5.8.3 Field variation of T0\nFinally, in Fig. 5.10, we observe that T0increases from 1 .5−5Kup to 20 kOe, and then\nexhibits a gentle decrease in the same range of fields where we observe the increase in\nαL0. Even though we do not have a model to explain this variation at the moment, we\nbelieve that the initial increase is related to the increase in rigidity of the VL with the\nincreasing field as the vortex lattice is squeezed. The decrease in the high field is more\ndifficult to understand. However, in this range, the error bar on the extracted value of T0\nis large ( due to the small range of temperature considered for the fit ) and it is difficult\nat the moment to assess the significance of this decrease. The variation of T0with the\nfield needs to be explored further in the future.\n1075.9. INCONSISTENCY WITH δ−2\n01 02 03 04 012345T0 ( K )H\n ( kOe )\nFigure 5.10: Field variation of the characteristic temperature T0(Used in the the temperature\ndependence: αL(U) =αL0(U0)e−T/T 0(connected blue circles).\n5.9 Inconsistency with δ−2\nIn Fig. 5.11, we have plotted the corresponding imaginary part of Eq. 5.13 with the same\nparameters used for fitting λ−2. At high fields, the simulated curve qualitatively captures\nthe dissipation peak observed close to T∗. However, at low fields, we observe an additional\nincrease at low temperatures which is not captured within this model. This increase\nsignals some additional mode of dissipation present in the system. One possibility is that\nat low fields the oscillatory field excites additional internal modes within a Larkin domain\nin the soft lattice that is not accounted for in the mean-field description considered in\nEq. 5.1.\n23 4 5 6 7 10810910101011δ−2 ( m−2 )T\n ( K )H ( kOe ) \n1 \n2 \n3.5 \n5 \n7 \n10 \n15 \n20 \n30 \n40 \n50 \n60\nFigure 5.11: Experimental δ−2vsTdata with fit to the imaginary part of Eq. 5.13 with\nbest-fit parameters used in Fig. 5.7.\n108CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\n5.10 Discussion\nThis work was inspired by two influential papers298,299published around a decade ago.\nOne of the fascinating results in the phenomenology of superconductors is that the super-\nfluid density is proportional to the inverse square of the London penetration depth, which\nis a measurable quantity. Using the Ginzburg-Landau theory5,6, it is possible to connect\nthe superfluid density to an energy scale called the superfluid stiffness, which measures\nthe ability of the superconductor to withstand phase fluctuations. This means that by\nmeasuring the penetration depth, it is possible to accurately determine the superfluid\nstiffness, providing an elegant solution.\n01020304050600.11T*c\n, J2Ks\n ( K )H\n ( kOe ) T*c\n (H) \nJ2Ks\nFigure 5.12: (Connected green circles) Field variation of superfluid stiffness J2K\nsdirectly\ncalculated from in-field λ−2values at 2 K. For comparison, the in-field transition temperatures\n(T∗\nc) are also plotted (connected red squares), i.e., where λ−2falls below the resolution limit\n(108m−2) in Fig. 5.2(c).\nHowever, this approach has limitations and should not be used in the presence of a\nmagnetic field, when vortices nucleate within the system. In such cases, the measured\npenetration depth will be influenced by the motion of pinned vortices, which is not related\nto the superfluid density or superfluid stiffness. In fact, A. M. Campbell145,146had already\ndiscovered this phenomenon in the 1960s while studying the shielding response of pinned\nvortices. Nonetheless, the above-mentioned papers298,299disregarded this and treated the\nvortex lattice in a coarse-grained manner, such that the impact of vortex motion on bulk\nmeasurements get nullified across a sizable sample. This allowed for the extension of the\nGinzburg-Landau relation in the presence of a magnetic field and the direct estimation\nof the superfluid stiffness from the ac penetration depth. Consequently, it was concluded\nthat the superfluid stiffness becomes zero at a specific critical magnetic field. To check\nthe consistency of this argument, we have directly calculated J2K\nsusing the well-known\nformula: Js=ℏ2d\n4µ0e2λ2, where dis the film thickness, and λ−2\n2Kis taken from Fig. 5.2(c).\nIn Fig. 5.12, we present a comparison of this calculated J2K\nswith the in-field transition\ntemperature ( T∗\nc) obtained from Fig. 5.2. The plot clearly indicates that with an increase\n1095.11. CONCLUSION\nin the magnetic field, J2K\nsdecreases rapidly while T∗\ncslowly decreases but remains at least\nan order of magnitude greater than the former. However, as discussed in Section 4.8, Js\nshould either be much greater than Tcor of the same order, depending on the relative\nimportance of phase fluctuations. But, it can never be an order of magnitude less than\nTc, as observed in Fig. 5.12. This discrepancy implies that it was not correct, in the first\nplace, to directly connect the superfluid stiffness with the penetration depth data in the\npresence of the magnetic field.\nThe confusion arises from the fact that we typically measure the penetration depth\nusing a finite-frequency screening response for practical reasons. However, in a zero mag-\nnetic field, the measurement is equivalent to what would be measured with a dc field,\nnamely the London15penetration depth. In contrast, the situation is entirely different\nin the presence of a magnetic field. The dc field almost entirely penetrates the sample in\nthe form of vortices. So, if we do a dc magnetization ( M) measurement, measured M\nwill be zero, indicating an almost infinite coarse-averaged penetration depth. However,\nthis is not what we actually measure. The measured penetration depth corresponds to\nthe ac screening response, which results from the fact that the vortices are unable to\nrespond to small changes in the magnetic field due to pinning. This quantity, known as\nthe Campbell145,146penetration depth, is not necessarily linked to the superfluid stiffness\nin the same way as the London penetration depth, and therefore cannot be used inter-\nchangeably. While it is possible to calculate an energy scale using the zero-field formula\nfor superfluid stiffness, doing so would not be physically meaningful.\nThe purpose of this work is to support our point of view by analyzing in-field pene-\ntration depth data in detail and demonstrating that the variation in penetration depth\nwith a magnetic field can be explained quantitatively by the electrodynamic response of\npinned vortices along with the London penetration depth.\n5.11 Conclusion\nIn this work, we have demonstrated a method of extracting vortex parameters from pen-\netration depth measurements using the low-frequency mutual inductance technique. We\nfitted the measured λ−2vsTplots at different fields with a model developed by Coffey\nand Clem, where the vortex parameters were determined as fitted parameters. The accu-\nracy of the extracted parameter values depends on their correct temperature dependence\nmodels. Though the CC model has been extensively used ( with some approximations ) to\nextract vortex parameters from the microwave vortex resistivity measurements163,288,300,\nit has not been used very much for low-frequency ac susceptibility measurements bar-\nring a few301,302. Our data fitted well for exponential temperature variations of both the\nLabusch parameter and activation potential barrier, suggesting that the dominant effect\nof temperature comes from the smearing of pinning potential due to thermal fluctuations.\n110CHAPTER 5. STUDY OF VORTEX DYNAMICS IN A-MOGE\nHowever, using the CC model we could not fully capture the variation of the skin depth\nsignifying the loss in the system which suggests that there might be additional modes of\ndissipation present in the system beyond the model presented here. One limitation of our\napproach is that we have to pre-assume the temperature dependence of some parameters\nin our analysis. If the chosen temperature dependence is not accurate for a material, the\nshape of the fitted curve does not reproduce the experimental data accurately. If one\nknows the precise temperature dependence of the vortex parameters for the concerned\nmaterial, the estimate of the final result only gets better.\n111CHAPTER6\nSummary and outlook\n6.1 Summary\nIn this thesis, we have focused on the evolution of superconductivity in amorphous Molyb-\ndenum Germanium ( a-MoGe ) thin films and the extraction of vortex parameters using\nthe low-frequency mutual inductance technique. Our research has provided valuable\ninsights into the behavior of superconductors and their electromagnetic response, con-\ntributing to the understanding of superconductivity and its underlying mechanisms.\nFirstly, we investigated the effect of decreasing film thickness on the suppression of\nsuperconductivity in a-MoGe thin films. We observed two distinct regimes: a Fermionic\nregime at moderate disorder and a Bosonic regime at stronger disorder. In the Fermionic\nregime, the critical temperature ( Tc) decreased with decreasing film thickness, while the\nratio of the superconducting gap ( ∆ ) to the temperature ( kBTc) showed only a small\nincrease. The behavior of the sheet resistance at the normal state as a function of Tc\nfollowed the Finkel’stein model, consistent with the Fermionic scenario. However, in the\nBosonic regime, the system transitioned to a phase-fluctuation-dominated state. Even\nafter the superconducting state was destroyed by phase fluctuations, the finite pairing\namplitude manifested as a pseudogap. Our observations aligned with earlier studies\non disordered superconductors, suggesting the universality of the Bosonic route in the\neventual destruction of superconductivity.\nSecondly, we presented a method for extracting vortex parameters using the low-\nfrequency mutual inductance technique. By analyzing the temperature dependence of\nthe measured penetration depth data, we successfully determined vortex parameters such\n1136.2. FUTURE PROBLEMS\nas the Labusch parameter, vortex lattice drag coefficient, and pinning potential barrier.\nOur analysis revealed the dominant effect of thermal fluctuations on the vortex behavior.\nHowever, the Coffey-Clem ( CC ) model used in our analysis had limitations in fully\ncapturing the variation of the skin depth, indicating the presence of additional dissipa-\ntion modes in the system. Nonetheless, our method provided valuable insights into the\nbehavior of vortices in a-MoGe thin films and their interaction with thermal fluctuations.\nIn conclusion, our research sheds light on the evolution of superconductivity in a-\nMoGe thin films and the characteristics of vortices. This thesis aims to enhance our\nunderstanding of the suppression of superconductivity with decreasing film thickness\n( or increasing disorder ) in MoGe thin films through various measurements, with a par-\nticular emphasis on tunneling measurements of the superconducting gap and two-coil\nmeasurements of superfluid density, conducted on films of varying thicknesses. While\nextensive two-coil measurements on the same material already exist303, our study in-\ntroduces tunneling ( STM ) data to enhance the interpretation of superfluid density.\nThe work reveals that the effect of phase fluctuations is much stronger than expected,\ni.e., they suppress the critical temperature ( Tc) of the thinnest films ( where super-\nfluid density and resistivity vanish ) well below the predicted occurrence of the familiar\nKosterlitz-Thouless-Berezinskii transition. Additionally, spatial inhomogeneity of the lo-\ncal superfluid density in the thinnest films might accentuate the suppression of the total\nsuperfluid density. We have observed the transition from the Fermionic to the Bosonic\nregime in the suppression of superconductivity, highlighting the importance of disorder\nand phase fluctuations in determining the behavior of superconductors. Furthermore, our\nmethod of extracting vortex parameters using the low-frequency mutual inductance tech-\nnique has provided a means to investigate the dynamics of vortices in superconducting\nsystems.\n6.2 Future problems\n1. It would be interesting to explore the universality of the Bosonic mechanism in\nother disordered superconductors and investigate exceptions to the Fermionic route\nof superconductivity destruction.\n2. Refining the analysis methods for extracting vortex parameters and investigating\nadditional dissipation modes will greatly enhance our understanding of supercon-\nducting behavior. It is evident that the simple Bardeen-Stephen model falls short\nof fully capturing the flux flow viscosity, necessitating modifications to account\nfor additional dissipation mechanisms. One such mechanism is the contribution of\nshear viscosity arising from the relative motion of vortices. By incorporating these\nadditional factors into our analysis, we can gain a more comprehensive picture of\n114CHAPTER 6. SUMMARY AND OUTLOOK\nthe intricate dynamics and interactions of vortices in superconducting systems.\n3. A hint of a peak-like feature in the field variation of microwave surface resistance\nin the GHz range was observed during the transition from the Hexatic fluid to the\nisotropic fluid state. This finding aligns with the results obtained from magneto-\ntransport, scanning tunneling microscopy ( STM ), and penetration depth measure-\nments, thereby emphasizing the need for a detailed exploration of this phenomenon.\nBy conducting further investigations, we can gain a deeper understanding of the\nunderlying mechanisms and dynamics associated with this peak-like feature.\n4. An unusual peak-like feature was observed in the extremely weakly pinned a-MoGe\nsample at a very low field near Hc1. Despite extensive investigation, a complete\nunderstanding of this phenomenon remains elusive, necessitating further exploration\nin future studies. By delving deeper into the nature and origin of this peak-like\nfeature, we can unravel the underlying mechanisms and gain valuable insights into\nthe unique behavior exhibited by the system under these specific conditions.\n5. We can explore the magnetization ( M) data of MoGe samples with varying thick-\nnesses to systematically understand the M−Hloops and, consequently, the under-\nlying pinning strengths. By analyzing the M−Hcurves, we can obtain valuable\ninsights into the behavior and response of the samples to external magnetic fields.\nThis systematic investigation will provide a comprehensive understanding of the\npinning mechanisms and their influence on the superconducting properties of the\nMoGe samples. Such knowledge is crucial for optimizing the design and perfor-\nmance of superconducting materials and devices.\n115Appendices\n117APPENDIXA\nBKT transition in 2.2-nm-thick MoGe film\nEven for our thinnest film ( ∼2.2nm), the low-temperature stiffness Js(T→0)≈\n10 K ( Fig. 4.11 ), calculated from Eq. 4.9, is still large enough that the Berezinskii-\nKosterlitz-Thouless245–247( BKT ) temperature TBKT is very close to the BCS mean-\nfield temperature ( TBCS). In addition, the sample inhomogeneity emerging at strong\ndisorder is expected to smear out the BKT signatures, for example, the universal jump\nof the superfluid stiffness231, as already observed experimentally in thin films250,251, of\nNbN. To quantify these effects, we analyzed the temperature dependence of the superfluid\nstiffness in the thinnest film within the same scheme described in Ref. 250,251.\nBoth quasiparticle excitations and phase fluctuations ( longitudinal and transverse )\ncontribute to the depletion of Jstowards zero. The usual approach used to analyze the\nBKT effect is to include all the other excitations except vortices in a phenomenologi-\ncal ansatz JBCS\ns which serves as starting point for the RG equations. Here we extract\nJBCS\nsfrom a phenomenological fit of the superfluid stiffness with a dirty-limit BCS-like\nexpression,\nJBCS\ns(T)\nJBCS\ns(0)=∆(T)\n∆(0)tanh∆(T)\n2kBT, (A.1)\nby using ∆(0) /TBCSas a free parameter. The resulting JBCS\ns(T) here acts as a mean-field\nansatz for the renormalization group ( RG ) analysis192,245–247,249.\nHowever, to account for the sample inhomogeneity observed by STM we will follow the\nsame procedure suggested in Ref. 250,251. We will assume that the local BCS stiffness\nJi\ns,BCS is distributed according to a given probability density P(Ji\ns), and that the local\nBCS transition temperature Ti\nBCSscale accordingly. Then by solving numerically the RG\n1190.00 .51 .01 .52 .00.00.20.40.60.80\n.81.21.62.00.00.20.40.60.8 2.2 nm \nBCS fit \nBKT fitλ−2 ( µm−2 )T\n ( K ) BCS fit \nBKT fit \nBKT lineλ−2 ( µm−2 )T\n ( K )σ = 0Figure A.1: Temperature variation of λ−2(blue circles) for the 2 .2-nm-thick sample along with\nthe fits from dirty-limit BCS formula (green line) and BKT theory (cyan line); The inset shows\nthe corresponding BCS (green line) and BKT (cyan line) variation for the hypothetical case of\nzero disorder, i.e., σ= 0, where σis the standard deviation in the Gaussian distribution A.3.\nThe universal BKT line248,249is shown in orange.\nequations, we can compute the local stiffness Ji\ns(T). The overall superfluid stiffness is\nthen computed phenomenologically as an average value Jav\ns:\nJav\ns(T) =X\niP(Ji\ns)Ji\ns(T). (A.2)\nFor the sake of concreteness, we will use for P(Ji\ns) a Gaussian distribution centered around\nJ0\ns:\nP(Ji\ns) =1√\n2πσe−(Ji\ns−J0\ns)2/2σ2. (A.3)\nWhen all the stiffness Ji\ns(T) are different from zero, as is the case at low temperatures,\nthe average stiffness will be centered around the center of the Gaussian distribution\n(Eq. A.3) so that it will coincide with J0\ns(T). However, by approaching TBKT defined\nby the average J0\ns(T), not all the patches make the transition at the same temperature\nso that the BKT jump is rounded and Jav\nsremains finite above the average TBKT. This\nleads to a rather symmetric smearing of the superfluid-density jump with respect to the\nabrupt downturn observed for the clean case, which is progressively more pronounced\nfor increasing σ/J0\ns(T= 0). It is worth noting that such a phenomenological approach\naccounts rather well for experiments in thin films on NbN250,251, and it has been recently\nvalidated theoretically by Monte Carlo simulations within an inhomogeneous 2D XY\nmodel in the presence of correlated disorder231.\nIn Fig. A.1, we show the result of the above fitting procedure for the 2 .2-nm-thick a-\nMoGe film. The curve labeled BCS represents the BCS fit of the average stiffness based on\nEq. A.1 above. From the fit, we obtain the values TBCS= 1.84Kand ∆(0) /(kBTBCS) =\n1.9, that is consistent with the STM estimate. As mentioned above, the BCS fit cannot\n120APPENDIX A. BKT TRANSITION IN 2 .2-nm-THICK MOGE FILM\ncapture the linear depletion at low temperatures, but it captures rather well the overall\nsuppression due to quasiparticle excitations up to a temperature T∼1.5K. Here,\nthe BKT curve starts to deviate from the BCS one due to the effect of bound vortex-\nantivortex pairs, as is evident in the inset where we show the homogeneous case ( σ= 0).\nThe abrupt jump is however smeared out in the presence of inhomogeneity, as shown\nin the main panel, where the BKT line corresponds to the average procedure encoded\nin Eq. A.2 for σ/J0\ns= 0.05. The overall fit reproduces reasonably well the experimental\ntrend, with a global transition temperature Tc≈1.8 K. Overall, we can conclude that the\nBKT analysis further supports the conclusion that our thinnest sample is in the 2D limit.\nHowever, the manifestation of BKT effects is partly blurred out by the inhomogeneity,\nwhich unavoidably comes along with the increase of effective disorder in the thinnest\nsamples.\n121APPENDIXB\nFitting of experimental data with Coffey-Clem equation\nWe analysed the experimental λ−2vsTdata ( Fig. 5.7 ) primarily on Mathematica.\nWe have used the full expression in Eq. 5.2 to fit the data. As I discussed earlier,\nwe had to choose the temperature dependence of the vortex parameters η,αL, and U\nsuitable for our system. As we did not know the precise temperature dependence of\nthe parameters, we had to follow a trial-and-error procedure. To check whether the\nchosen temperature variations are correct, we varied the parameters with the help of the\n“Manipulate” function in Mathematica and found the estimated fit to the experimental\ndata. Finally, we used a detailed scheme to find the exact best-fit parameters using the\nχ2method ( Section B.2 ).\n23 4 5 6 7 10810910101011(b)H ( kOe ) \n1 \n2 \n3.5 \n5 \n7 \n10 \n15 \n20 \n30 \n40 \n50 \n60λ−2 ( m−2 )T\n ( K )Trial 2\n23 4 5 6 7 10810910101011(\na)H ( kOe ) \n1 \n2 \n3.5 \n5 \n7 \n10 \n15 \n20 \n30 \n40 \n50 \n60λ−2 ( m−2 )T\n ( K )Trial 1\nFigure B.1: λ−2vsTwith the trial temperature variations of αLandU:(a)αL∼(1−t2)2\nandU∼(1−t2)(1−t4)1/2;(b)αL∼(1−t2)2andU∼(1−t2)(1 + t2), where t=T/T c.\n123B.1. FITTING WITH TRIAL TEMPERATURE VARIATIONS OF UAND αL\nB.1 Fitting with trial temperature variations of U\nand αL\nIn this section, I shall discuss some of the trial temperature variations we have consid-\nered in order to find a suitable function. For ηvariation, we have assumed the simple\nvariation mimicking the temperature variation of Hc2. This might not be true for the\nlow-temperature part of δ−2vsTdata as discussed in section 5.9.\nForUandαL, at first, we have taken the temperature variation from Eqs. 5.27 and\n5.22 and tried to fit ( Trial 1 in Fig. B.1(a) ). But, as seen in Fig. B.1, the fit is not\nthat good, especially at low fields. To improve on the fitting, we tried other functions.\nWe found a better fitting ( Trial 2 in Fig. B.1(b) ) for the variation of Uin Eq. 5.26\nwhile keeping the same variation of αLas in Eq. 5.27. It is evident in Fig. B.1 that the\nfit around the transition is better for trial 2. However, in low fields, the increase in λ−2\ncould not be captured in either of the above trial variations. Comparisons for the fits\nwith the trial variations are tabulated in the following table.\nTable B.1: Comparison of Coffey-Clem fits using different trial temperature variations of U\nandαL.\nTrialT dependence of UandαL(Eq.\nno.)Fig.\nNo.Remarks\n1.•U∼(1−t2)(1−t4)1/2(5.22)\n•αL∼(1−t2)2(5.27)B.1(a)•Fits well for high-field ( >10kOe)\ndata.\n•For low-field data, provides a good\nfit except at the transition and at\nlowT.\n2.•U∼(1−t2)(1 + t2) (5.26)\n•αL∼(1−t2)2(5.27)B.1(b)•Fit for high-field data is similar to\ntrial 1.\n•For low-field data, the fit is more\naccurate at the transition, but it\nfails to capture the behavior at\nlowTsimilar to trial 1.\n3.•U∼e−T/T 0(5.30)\n•αL∼e−T/T 0(5.29)5.7(b)Fits reasonably well over the entire\nrange of T(2−6K) and field (1 −60\nkOe).\n124APPENDIX B. FITTING OF EXPERIMENTAL DATA WITH COFFEY-CLEM\nEQUATION\n2.02 .22 .41.01.52.06\n5707580851.01.52.04\n005 006 001.01.52.02\n.62.83.03.20.30.40.50.60.75\n0556065700.30.40.50.60.72\n002402800.30.40.50.60.73\n.43.63.84.04.20.160.180.200.224\n446485052540.160.180.200.221\n101201301401500.160.180.200.223\n.54.04.55.05.56.00.090.100.115\n25660640.090.100.115\n0556065700.090.100.11χ2 ( ×108 m−2 )T\n0 ( K )2 kOe(a)χ\n2 ( ×108 m−2 )α\nL0 ( N/m2 )2 kOe(b)χ\n2 ( ×108 m−2 )U\n0/2kB ( K )2 kOe(c)χ2 ( ×108 m−2 )T\n0 ( K )5 kOe(d)χ\n2 ( ×108 m−2 )α\nL0 ( N/m2 )5 kOe(e)χ\n2 ( ×108 m−2 )U\n0/2kB ( K )5 kOe(f)χ2 ( ×108 m−2 )T\n0 ( K )10 kOe(g)χ\n2 ( ×108 m−2 )α\nL0 ( N/m2 )10 kOe(h)χ\n2 ( ×108 m−2 )U\n0/2kB ( K )10 kOe(i)χ2 ( ×108 m−2 )T\n0 ( K )30 kOe(j)χ\n2 ( ×108 m−2 )α\nL0 ( N/m2 )30 kOe(k)χ\n2 ( ×108 m−2 )U\n0/2kB ( K )30 kOe(l)\nFigure B.2: χ2as a function of T0,αL0andU0/2kBrespectively for four representative fields:\n(a)-(c) 2kOe;(d)-(f) 5kOe;(g)-(i) 10kOeand(j)-(l) 30kOe. The horizontal black line in\neach of the plots is at 1.05 times the minimum χ2value: the spread of parameter values below\nthe line has been chosen as the error bar of that parameter for the given field.\nB.2 Best fit parameters and error bar\nIn this section, I shall elaborate on the scheme we have followed to find out the best-fit\nvalues of the vortex parameters- U0,αL0, and T0. The goodness of fit is calculated by the\nformula:\nχ2=1\nNP(y−yfit)2\nyfit, (B.1)\n125B.2. BEST FIT PARAMETERS AND ERROR BAR\nwhere yis the experimental λ−2data in Fig. 5.7 and yfitis calculated using the right-hand\nside of Eq. 5.13; the normalization factor 1 /Nwas to account for the fact that different\ndatasets had a different number of points, N. Here, the free parameters are αL0,T0and\nU0/2kB. To carry out the error analysis, we used the following protocol: One particular\nparameter was fixed and the other two were varied freely so as to have the best fit using\nthe “FindFit” function in Mathematica, and the corresponding χ2was calculated. In this\nprocedure, the best fit was determined by the set of values where χ2has the minimum\nvalue. The results are shown in Fig. B.2(a) - (l) for 4 representative fields ( 2 ,5,10,and\n30kOe).\n123454\n060801000\n1 02 03 04 005001000T0 ( K )χ\n2 minimised by \nvarying T0 \nvarying αL0 \nvarying U0/2kB(a)αL0 ( N/m2 )χ2 minimised by \nvarying T0 \nvarying αL0 \nvarying U0/2kB(b)U0/2kB ( K )H\n ( kOe )χ2 minimised by \nvarying T0 \nvarying αL0 \nvarying U0/2kB(c)\nFigure B.3: (a)-(c) Field dependence of T0,αL0andU0/2kBvalues at the χ2minima points\nrespectively. In each plot, minimization was done by varying each parameter individually:\nminima of the χ2vsT0plots are given by connected black squares, minima of the χ2vsαL0\nplots by connected red circles and minima of the χ2vsU0/2kBplots by connected blue triangles.\nIn principle, all three curves should give the same value of χ2at the minimum. The\n126APPENDIX B. FITTING OF EXPERIMENTAL DATA WITH COFFEY-CLEM\nEQUATION\nslight difference is owing to the fact that the in-built “FindFit” function minimizes the\nleast-square error instead of χ2. Nevertheless, the extracted parameter values from the\nminima of either of these three curves are very close to each other and are shown in\nFig. B.3. The set of parameters ( αL0,T0andU0/2kB) corresponding to the lowest\nof the three minima are plotted in Fig. B.3. Here we did not include parameters for\n50 and 60 kOe since the temperature range of data is too small to perform a reliable\nbest fit. 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Publisher: American Physical Society.\n152"    },    {        "title": "2402.00401v1.Higgs_boson_pair_production_and_decay_at_NLO_in_QCD__the__b_bar_b_γγ__final_state.pdf",        "content": "Prepared for submission to JHEP\nHiggs boson pair production and decay at NLO in\nQCD: the b¯bγγfinal state\nHai Tao Li,aZong-Guo Si,aJian Wang,a,bXiao Zhang,aDan Zhaoa\naSchool of Physics Shandong University, Jinan, Shandong 250100, China\nbCenter for High Energy Physics, Peking University, Beijing 100871, China\nAbstract: The Higgs boson pair production at the LHC provides a probe to the Higgs\nboson self-coupling. The higher-order QCD corrections in this process are sizable and\nmust be taken into account in comparison with data. Due to the small cross section, it\nis necessary to consider at least one of the Higgs bosons decaying to bottom quarks. The\nQCD corrections to the decay processes would also be important in such cases. We present\na full calculation of the total and differential cross sections for the b¯bγγfinal state with\nnext-to-leading order (NLO) QCD corrections. After applying typical kinematic cuts in the\nfinal state, we find that QCD NLO corrections in the decay decrease the LO result by 19%\nand reduce the scale uncertainties by a factor of two. The QCD corrections to the invariant\nmass mjjγγdistribution, the transverse momentum spectra of the leading bottom quark jet\nand photon are significant and can not be approximated by a constant factor.arXiv:2402.00401v1  [hep-ph]  1 Feb 2024Contents\n1 Introduction 1\n2 Calculation framework 2\n3 Numerical results 4\n4 Conclusion 8\nAH→b¯bdecay 11\nBH→γγdecay 12\n1 Introduction\nThe discovery of the Higgs boson more than ten years ago [1, 2] marks the great success of\nthe standard model (SM), and inaugurates a new era in particle physics. The presence of\na Higgs field is essential to explain the masses of the WandZgauge bosons via the Higgs\nmechanism [3–6]. Until now, the properties of this Higgs boson, including its mass and\nwidth [7–9], spin and parity [10, 11], as well as various production and decay rates [12, 13],\nhave been measured, and the results are in agreement with the SM predictions.\nA precise understanding of the trilinear Higgs self-coupling λHHHis crucial for unrav-\neling the mechanism of electroweak symmetry breaking and offering insights into vacuum\nstability [14]. The self-interaction of the Higgs boson contributes to higher-order elec-\ntroweak corrections in both the production and decay processes of a single Higgs boson,\nand thus the Higgs self-coupling gets constrained by precisely measuring the cross sections\nof the single Higgs processes [15–20]. However, this method relies on the assumption that\nthere is no modification of the other Higgs couplings [21], and the dependence appears as a\nloop effect. Therefore, it is more informative to measure the Higgs boson pair productions,\nwhich are sensitive to the trilinear Higgs self-coupling at leading order (LO).\nAt the LHC, the gluon fusion is the dominant production channel, in which the Higgs\nbosons are emitted from a top quark loop. The cross section of this process has been\ncalculated up to QCD N3LOin the infinite large top quark mass limit [22–26] and up to\nQCD NLO with finite mtdependence [27–30]. In the mt→ ∞limit, the NLO corrections\nare significant, increasing the LO results by 87%, and the NNLO corrections provide an\nadditional 33%enhancement. The N3LOcorrections amount to about 6.4%of the LO\ncross sections. With finite top quark mass, the NLO QCD corrections increase the LO\ncross section by 66%. Partial mteffects at NNLO in QCD and the electroweak corrections\nat NLO have been studied in [31–34] and [35–39], respectively. Soft gluon resummation\neffects have been taken into account in [40–42].\n– 1 –The CMS collaboration has searched for the Higgs pair productions in the bbγγ,bbbb,\nbbττ,bbZZand multilepton final states, giving a limit of 3.4 times the cross section pre-\ndicted by the SM at 95%confidence level (CL), which is converted to the allowed range\nfor the self-coupling −1.24< κ λ(=λHHH/λSM\nHHH)<6.49at95%CL [13]. The ATLAS\ncollaboration has measured the Higgs pair productions in the bbγγ,bbττ, and bbbbfinal\nstates, and set an upper limit of 2.4 times SM predictions. Combined with the single Higgs\nboson analyses, this is transformed to the constraint −0.4< κλ<6.3at95%CL [19]. The\nrange would be significantly narrowed at the high-luminosity LHC [43].\nIn obtaining the above limits, precise predictions for the kinematic distributions of the\nHHevents, especially the mHHdistribution, play an important role [44]. So far, only the\nhigher-order QCD corrections in the production processes have been studied. It is not clear\nwhether the kinematic distributions would change after considering the QCD corrections in\nthe decays of the Higgs bosons. Our aim in this paper is to address this issue. Because the\ncrosssectionoftheHiggsbosonpairproductionisalreadyverysmall, itisrequiredtosearch\nfor the signal events with at least one of the Higgs bosons decaying to two bottom quarks.\nWe focus on the HH→b¯bγγfinal state in the present work. We find that the NLO QCD\ncorrection in the decay could reduce the cross section for the process pp→HH→b¯bγγ\nsignificantly and that the impact on the kinematic distributions can not be described by a\nconstant.\nThispaperisorganizedasfollows. Insection2, weintroducethecalculationframework.\nThenumericalresultsforthetotalcrosssectionsandkinematicaldistributionsarepresented\nin section 3. The conclusion is given in section 4.\n2 Calculation framework\nWearegoingtocalculatethefullQCDcorrectionsto gg→HHproductionwiththe HH→\nb¯bγγdecay at the LHC. The interference between the QCD corrections in the production\nand decay processes starts from NNLO and is suppressed by a factor of ΓH/mH∼10−5\ncompared to the resonance contribution, in which the Higgs bosons are taken to be on-shell.\nIn this paper, we consider only the O(αs)correction and therefore neglect the interference.\nIn the narrow-width approximation, the cross section is written as\nσpro+dec =σpro1\nΓH11\nΓH2Γdec (2.1)\nwhere σprodenotes the production cross section of gg→H1H2andΓH1= ΓH2= ΓHis the\ntotal width of the Higgs boson. The double decay width is defined by\nΓdec≡Z\ndΓH1Z\ndΓH2FJ (2.2)\nwith FJthe measurement function which embodies the cut information in event selection.\nFor a specific decay mode, i.e., H1→X1andH2→X2, the cross section is expressed as\nσpro+dec(X 1,X2)=σpro1\nΓH1→X11\nΓH2→X2Γdec(X 1,X2)×R(H1→X1)R(H2→X2)(2.3)\n– 2 –where we have introduced R(Hi→Xi)as the branching ratio for the Higgs decay into the\nfinal state Xi.\nWe focus on the higher-order QCD corrections to the production and decay to a specific\nfinal state X1+X2. The cross sections and decay widths can be expanded in a series of\nthe strong coupling constant αs, e.g.,\nσ=σ(0)+X\nn=1αn\nsσ(n). (2.4)\nAnd therefore, we have\nσ(n)\npro+dec(X 1,X2)=σpro1\nΓH1→X11\nΓH2→X2Γdec(X 1,X2)\f\f\f\f\nexpanded to αns\n×R(H1→X1)R(H2→X2). (2.5)\nWe have taken the branching ratios as constants in our calculation, and the most precise\nvalues given in refs. [45–47] are adopted.\nAs a first step, we incorporate only the NLO QCD corrections in this study. The O(αs)\ncorrection is decomposed into two parts,\nσ(1)\npro+dec(b ¯b,γγ)=σpro(1)\npro+dec(b ¯b,γγ)+σdec(1)\npro+dec(b ¯b,γγ)(2.6)\nwith\nσpro(1)\npro+dec(b ¯b,γγ)=σ(1)\npro1\nΓ(0)\nH1→b¯b1\nΓ(0)\nH2→γγΓ(0)\ndec(b ¯b,γγ)×R(H1→b¯b)R(H2→γγ)(2.7)\nand\nσdec(1)\npro+dec(b ¯b,γγ)=σ(0)\npro1\nΓ(0)\nH1→b¯b1\nΓ(0)\nH2→γγΓ(1)\ndec(b ¯b,γγ)×R(H1→b¯b)R(H2→γγ)\n−σ(0)\nproΓ(1)\nH1→b¯b\u0010\nΓ(0)\nH1→b¯b\u001121\nΓ(0)\nH2→γγΓ(0)\ndec(b ¯b,γγ)×R(H1→b¯b)R(H2→γγ)\n−σ(0)\npro1\nΓ(0)\nH1→b¯bΓ(1)\nH2→γγ\u0010\nΓ(0)\nH2→γγ\u00112Γ(0)\ndec(b ¯b,γγ)×R(H1→b¯b)R(H2→γγ).(2.8)\nIf no cuts are applied and the full phase space of the final state particles is integrated over,\nσdec(1)\npro+dec(b ¯b,γγ)is vanishing. In practice, kinematic cuts have to be imposed to select the\nevents. In this case, σdec(1)\npro+dec(b ¯b,γγ)may provide a significant correction. It is our purpose\nto investigate the contribution of this part.\nBecause there is no real correction from H→γγ+gat NLO in QCD due to color\nconservation, the decay H→γγreceives only virtual corrections. Then the third line in\neq.(2.8) could cancel with part of the first line that is given by\nZ\ndΓ(0)\nH1→b¯b×Z\ndΓ(1)\nH2→γγ(2.9)\n– 3 –since they have the same kinematics. As a consequence,\nσdec(1)\npro+dec(b ¯b,γγ)=σ(0)\npro1\nΓ(0)\nH1→b¯b1\nΓ(0)\nH2→γγΓ(0)\ndec(b ¯b,γγ)\nR\ndΓ(1)\nH1→b¯bFJ\nR\ndΓ(0)\nH1→b¯bFJ−Γ(1)\nH1→b¯b\nΓ(0)\nH1→b¯b\n\n×R(H1→b¯b)R(H2→γγ). (2.10)\nWe have written the first term in the bracket in differential form because it is subject to\nthe kinematic cuts.\nThe above eqs.(2.6,2.7,2.10) represent the master formulae of our calculation. The\nO(αs)correction to the production process σ(1)\nprohas been computed in [28] with full top\nquark mass dependence. We have checked the results independently using our code, al-\nthough the two-loop gg→HHamplitude is calculated via the grid implemented in\nPOWHEG BOX [48]. The one-loop gg→HHgand other similar 2→3partonic am-\nplitudes are generated by the automatic tool OpenLoops [49, 50] with the scalar integrals\nevaluated by Collier/OneLoop [51, 52]. All the tree-level and one-loop amplitudes of the\ndecay process H→b¯b+Xare calculated employing the FeynArts [53, 54] and FeynCalc\n[55–57] packages. The analytical results are collected in Appendix A. The final-state bot-\ntom quarks have been taken to be massless while the Yukawa coupling of the bottom quark\nis kept finite. The result of the squared amplitude for the decay H→γγis recorded in\nAppendix B. The infrared divergences in the virtual and real corrections are subtracted\nusing the dipole method [58, 59]. In constructing the analytical subtraction terms, we need\nthe one-loop amplitude of gg→HHwhich is evaluated with the aid of QCDLoop [60].\n3 Numerical results\nIn numerical calculation, the SM input parameters are set to be\nGF= 1.1663787 ×10−5GeV−2, m W= 80.399 GeV ,\nmH= 125 GeV , m t= 173 GeV , m b(mb) = 4 .18 GeV ,\nR(H→bb) = 0 .5824, R (H→γγ) = 2 .27×10−3. (3.1)\nWe choose the parton distribution function (PDF) set PDF4LHC15_nlo_100_pdfas [61].\nThe strong coupling is evaluated using the function associated with this PDF set. The\ndefault values of the factorization scale µFand renormalization scale µRare set as mHH/2.\nIn the calculation of the real corrections to pp→HH, we have applied a phase space cut\npcut\nT∼0.1GeV on the Higgs pair system to ensure numerical stability. Below this cut,\nthe contributions from the squared amplitude and the differential subtraction terms cancel\nagainst each other and could be neglected. We have checked that the total cross sections\nare insensitive to this cut when it is chosen from 0.001 GeV to 1 GeV. A kinematic cut\nscut\nij∼10−5GeV2is applied to the final-state partons when considering the higher-order\nQCD corrections to H→b¯bdue to the same reason.\nBefore we show any numerical results, it is necessary to elaborate on the importance\nof including decay processes in the theoretical calculation. One crucial reason is that all\n– 4 –400\n6008001000120014001600180000.20.40.63−10×) [fb/20 GeV] γγjj(dmHH/dmσd )γγ→bb)H(→H(→pp \n=13.6 TeV s ∞→t m \nHH m \nγγjj m500\n1000 1500 ) [GeV]\nγγjj(mHHm11.051.1Ratio\n0\n10020030040050060070080000.513−10×) [fb/20 GeV] jjT\n(dpHT\n/dpσd )γγ→bb)H(→H(→pp \n=13.6 TeV s ∞→t m \nHT\n p \njjT\n p0\n200 400 600 800 ) [GeV]\njjT\n (pHT\np11.011.021.03RatioFigure 1 : Comparison of the kinematic distributions for the intermediate and\nreconstructed Higgs bosons in the heavy top quark mass limit at LO. The lower panels\nshow the ratio of the distributions for the intermediate Higgs bosons over those for the\nreconstructed Higgs bosons. The ratio rises sharply for mjjγγlarger than 1200 GeV and\ncan not be shown properly in the plot. The value is about 2.5 at mjjγγ= 1800GeV. In\nthe right plot, the ratio is about 7 at pjj\nT=800 GeV.\nthe events in experiments are collected under kinematic cuts. Therefore it is the cross\nsection with cuts that is measured. In experiments, the cuts can only be applied to the\nstable particles, i.e., the jets, photons, and leptons. One has to take into account the decay\nprocesses of unstable particles to fully simulate the events.\nForillustration, wecomparethekinematicdistributionsoftheintermediateHiggsboson\nresonancesandthereconstructedHiggsbosonsfromthefinal-statejetsandphotonsinfigure\n1. The intermediate Higgs boson appears in the final state of the production process but\ndecaystobottomquarksorphotonsimmediately. Itsmomentumisknowninthetheoretical\ncalculation but can not measured directly. In contrast, the momentum of the reconstructed\nHiggs boson is defined as the sum of the momenta of the two bottom jets, denoted by j,\nor the two photons. The final-state (anti-)bottom quark and other partons are clustered to\njets using the anti- ktalgorithm [62, 63] with a separation parameter R= 0.4. The angular\ndistancebetweenphotonsandthe b-jetsshouldbelargerthan0.4. Fromfigure1, weseethat\nthe reconstructed invariant mass spectrum mjjγγis generally lower than the intermediate\nHiggs boson invariant mass spectrum mHHby about 2%∼9%atmHH(mjjγγ)<600GeV.\nThis is mainly because the phase space with Rjγ< Rhas been cut out. The difference\nbecomes smaller with the increasing of mHH(mjjγγ), because the b-jets and photons tend\nto fly away from each other for larger mHH(mjjγγ). The single Higgs transverse momentum\ndistribution is lowered by about 2%∼3%in the reconstructed kinematics for pjj\nT<200\nGeV, and the deviation between the distributions using the reconstructed and intermediate\nHiggs boson becomes smaller as pjj\nT(pH\nT)increases from 100 GeV to 600 GeV due to the\nsame reason in the discussion of mHH.\n– 5 –An obvious deviation can be found in the region of mHH>1250GeV or pH\nT>620GeV,\nwhere the reconstructed distribution drops quickly. The reason is that the bottom quark\npair from the Higgs decay is highly boosted so that they become collinear with each other\nand are subject to the requirement Rjj> R. The threshold value of pH\nTcan be estimated\nby\n2mH\nR(3.2)\nwhich is the minimum of the momentum pH=mH/R/p\nz(1−z)of the Higgs boson\ndecaying to a collinear bottom (anti-)quark pair with a separation angle of Rand the\nmomentum fraction zcarried by one bottom quark. One can not observe such an interesting\nfeature without considering the decays of the Higgs bosons.\nIn practice, the particles with very low transverse momentum or very large rapidity are\nnot detected. To calculate cross sections with decays, we apply the following cuts that are\nsimilar to those used in the analysis of the CMS collaboration [64],\npj\nT≥25 GeV , pγ\nT≥25 GeV ,|ηj| ≤2.5,|ηγ| ≤2.5,\n90 GeV ≤mjj≤190 GeV , R jj,jγ,γγ ≥0.4. (3.3)\nThese cuts are rather loose because we want to keep more signals after cuts.\nwithout decays with decays but no cuts with decays and cuts\nLOdecδNLOdecLOdecδNLOdec\nLOpro\n∞ 17.07+31%\n−22%0.02257+31%\n−22%0 0.01257+30%\n−22%−0.00175+42%\n−28%\nLOpro\nmt19.85+28%\n−21%0.02624+28%\n−20%0 0.01395+27%\n−20%−0.00261+39%\n−27%\nδNLOpro\n∞ 14.86+6%\n−7%0.01964+6%\n−7%− 0.01064+6%\n−7%−\nδNLOpro\nmt13.08+4%\n−8%0.01729+4%\n−8%− 0.00914+4%\n−8%−\nFull NLO result\nNLO ∞ 31.93+18%\n−15%0.04221+18%\n−15%0.02146+15%\n−14%\nNLO mt 32.93+14%\n−13%0.04354+14%\n−13%0.02047+10%\n−11%\nTable 1: Total cross sections (in fb) of the Higgs boson pair production and decays\ngg→HH→b¯bγγat the 14 TeV LHC. The PDF set is PDF4LHC15 _nlo_100_pdfas.\nThe relative uncertainties are obtained by taking the seven-point scale variations around\nthe default values. The subscripts mtand∞indicate that the results are calculated with\nfullmtdependence or in the large mtlimit, respectively. The notation δNLOdenotes the\nO(αs)correction while the NLO results include both LO and δNLOcontributions. The\nsymbol ‘ −’ represents the higher-order corrections that we neglected in this work.\nIn table 1 we show the total cross sections with or without decays or cuts. The second\ncolumn gives the total cross section without decays. We have performed the calculations\nup to NLO in QCD both with full mtdependence and in the infinitely large mtlimit. Our\nresults coincide with the previous computations taken with the infinite mt[26] and finite mt\n[28] setup if the same PDF sets and SM input parameters are adopted. The LO production\n– 6 –cross section with infinite mtis smaller than that with finite mtby14%. But it receives\nlarger QCD corrections, e.g., 87%in the infinite mtlimit compared to 66%with full mt\ndependence. As a result, the difference between the two schemes is only 3%at NLO. We\nhave estimated the scale uncertainty by varying the factorization µFand renormalization\nµRscales independently by a factor of two, excluding the cases when the ratio of the two\nscales µF/µR= 4,1/4. We see that the scale uncertainty is reduced almost by a factor of\ntwo after including the QCD corrections.\nThe third column shows the total cross sections with decays but without cuts. The\nresults with LO decays agree with the production cross sections multiplied by a product of\nthe branching ratios, which is about 0.13%. The cross sections with the LO production and\nO(αs)decays are vanishing, as expected from eq. (2.10). These features serve as strong\nchecks of the validity of our numerical program.\nThe cross sections with kinematics cuts (3.3) on the final-state jets and photons are\nlisted in the fourth column. The results of the LO production and LO decay after the cuts\nare56%and53%of those without cuts in the infinite and finite mtcases, respectively. The\nNLO corrections in the production suffer from similar suppression. The NLO corrections in\nthe decay decrease the LO cross section by 19%(14%) for finite (infinite) mt. This effect\nis much larger than the N3LOQCD correction, which is about +6%[26].\nThis cut rate and the NLO corrections in the decay can be understood by inspecting\nthe transverse momentum distribution for the subleading b-jet, which is shown in figure 2.\nAs indicated by the LO result, the transverse momentum spectrum has a peak around 40\nGeV. Therefore a large portion of the events would be neglected under a typical cut pj\nT≥25\nGeV on the jet. The NLO corrections can be divided into two parts, i.e., the virtual and\nreal corrections with associated subtraction terms, denoted by δNLO V+IandδNLO R−Sin\nthe figure, respectively. The former has the same distribution as the LO because they are\nproportional to each other with a factor independent of the kinematics; see eq. (A.5) in the\nappendix. However, the latter gives a positive and negative contribution for pT<30GeV\nandpT>30GeV, respectively. The requirement of a cut pj\nT>25GeV leads to a significant\ncancellation between the two corrections. As a result, the first term in the bracket in eq.\n(2.10) is small and the dominant contribution comes from the second term there, which is\naround −20%.\nIn figure 3, we show the influence of the QCD corrections in the decay process on\nsome kinematic distributions. We see that the QCD corrections are most significant in\nthe peak region of the invariant mass mjjγγdistribution, which can be as large as −25%.\nThe corrections become smaller when mjjγγmoves away from the peak region. The QCD\ncorrection to the transverse momentum of the leading b-jet is also prominent in the peak\nregion. The impact of the QCD correction on the transverse momentum of the subleading\nb-jet is important in the small pTregion; it can reduce the LO cross section by −28%. The\nrapidity distribution receives a correction in the range from −21%to−18%.\nInfigure4, wepresentthekinematicdistributionsincludingQCDNLOcorrectionsboth\nin the production and decay processes. The NLO corrections in the production increase\nthemjjγγdistribution by 54%−82%in the region of mjjγγ<600GeV, which contributes\nthe dominant part of the cross section. After including the NLO corrections in the decay\n– 7 –0\n50 100 150 200  [GeV] \nj2T\np00.20.43−10× [fb/GeV] j2T\n/dpσd )γγ→bb)H(→H(→pp \nt=14 TeV  Finite m s LO \n \ndecR-S\nNLOδ  \ndecV+I\nNLOδ Figure 2 : The transverse momentum distribution of the subleading b-jet. The black line\nrepresents the LO result. The red and blue lines denote the NLO virtual and real\ncorrections with corresponding subtraction terms, respectively.\nprocesses, the distribution is enhanced further for mjjγγ<350GeV but decreased for\nmjjγγ>350GeV. A similar pattern appears in the transverse momentum distribution of\nthe leading b-jet. The NLO corrections in the production improve the LO results by a\nfactor of 1.63−1.77, while the NLO corrections in the decay increase and decrease the\ncross section for pj1\nT<80GeV and pj1\nT>80GeV, respectively. The transverse momentum\nspectrum of the leading photon is increased by 1.62−1.78, considering only the QCD\ncorrections in the production. This improvement reduces to 1.43−1.56after taking the\nQCD corrections in the decay into account as well. The higher-order QCD corrections to\nthe rapidity distribution of the leading b-jet are stable except for the region with very large\nrapidity.\n4 Conclusion\nThe Higgs boson pair production is an important process at the LHC that can be used\nto measure the Higgs boson self-couplings. A precise theoretical prediction on its cross\nsection enables an accurate and reliable constraint extracted from the present and future\ndata. Though there has been great progress in the calculations of higher-order QCD or\nelectroweak corrections to the production processes, the QCD corrections in the Higgs\nboson decays are rarely included. In this work, we present a full calculation of the QCD\nNLO corrections to the process pp→HH→b¯bγγ. The importance of including the decay\nis reflected by the difference between the kinematic distributions of the intermediate and\nthe reconstructed Higgs boson. After adopting loose kinematic cuts, we have found that\nthe QCD corrections in the decay decrease the LO result by 19%. This impact is much\nmore significant than the N3LOQCD corrections in the production. We also show the QCD\ncorrections to various kinematic distributions, such as the invariant mass of the b-jets and\nphotons, the transverse momentum and rapidity distributions of the b-jets, and find that\nthe QCD corrections in the decay process are significant, and do not change the kinematic\ndistributions simply by a constant multiplier.\n– 8 –400\n6008001000120000.513−10× [fb/20 GeV] γγjj/dmσd )γγ→bb)H(→H(→pp \nt=13.6 TeV Finite m s LO \n  \ndecNLOδ 500\n1000  [GeV]γγjjm0.2−00.2Ratio(a)\n100\n2003004005006000123−10×  [fb/20 GeV] j1T\n/dpσd )γγ→bb)H(→H(→ppt\n=13.6 TeV Finite m s LO \n  \ndecNLOδ 200\n400 600    [GeV]j1T\np0.2−00.20.4Ratio (b)\n40\n60801001201401601802000123−10×  [fb/10 GeV] j2T\n/dpσd )γγ→bb)H(→H(→ppt\n=13.6 TeV Finite m s LO \n \ndecNLOδ 50\n100 150 200  [GeV]\nj2T\np0.3−0.2−0.1−Ratio\n(c)\n2.5\n−2−1.5−1−0.5−00.511.522.500.53−10×  [fb/0.15] j1η/dσd )γγ→bb)H(→H(→pp \nt=13.6 TeV Finite m s LO   \ndecNLOδ 2\n−1−012j1\nη0.3−0.2−0.1−Ratio (d)\nFigure 3 : Impact of the QCD NLO corrections in the H→b¯bdecay process on the\nkinematic distributions after the cuts (3.3). The production cross section is calculated\nwith finite mt. The plot (a)shows the invariant mass of the two b-jets and two photons.\nThe plots (b)and(c)present the transverse momenta of the leading and subleading b-jets,\nrespectively. The rapidity distribution of the leading b-jet is shown in the plot (d). The\nratio in the lower panel in each plot illustrates the magnitude of O(αs)corrections in the\ndecay process compared to the LO result.\nAcknowledgments\nThis work was supported in part by the National Science Foundation of China under grant\nNos. 12275156, 12321005, 12375076 and the Taishan Scholar Foundation of Shandong\n– 9 –400\n6008001000120000.511.523−10×  [fb/20 GeV] γγjj/dmσd )γγ→bb)H(→H(→pp \nt=13.6 TeV Finite m s LO \n  \nproNLOδ LO+ \ndecNLOδ+proNLOδ LO+500\n1000  [GeV]γγjjm11.52K-factor(a)\n100\n200300400500600012343−10×  [fb/20 GeV] j1T\n/dpσd )γγ→bb)H(→H(→ppt\n=13.6 TeV Finite m s LO \n  \nproNLOδ LO+ \ndecNLOδ+proNLOδ LO+200\n400 600   [GeV]j1T\np1.52K-factor (b)\n100\n200300400500600012343−10×  [fb/20 GeV] 1γT\n/dpσd )γγ→bb)H(→H(→ppt\n=13.6 TeV Finite m s LO \n  \nproNLOδ LO+ \ndecNLOδ+proNLOδ LO+200\n400 600   [GeV]1γT\np1.41.61.8K-factor\n(c)\n2.5\n−2−1.5−1−0.5−00.511.522.500.511.53−10×  [fb/0.15] j1η/dσd )γγ→bb)H(→H(→pp \nt=13.6 TeV Finite m s LO \n  \nproNLOδ LO+ \ndecNLOδ+proNLOδ LO+2\n−1−012j1η1.41.51.61.7K-factor (d)\nFigure 4 : Kinematic distributions of the process pp→HH→b¯bγγafter the cuts (3.3).\nThe production cross section is calculated with finite mt. The plot (a)shows the invariant\nmass of the two b-jets and two photons. The plots (b)and(c)present the transverse\nmomenta of the leading b-jet and photon, respectively. The rapidity distribution of the\nleading b-jet is shown in the plot (d). The K-factor is defined as the ratio of the NLO\nresults over the LO ones.\nprovince (tsqn201909011).\n– 10 –AH→b¯bdecay\nThe Higgs boson to bottom quarks decay is the dominant channel. The QCD NLOcor-\nrections were calculated forty years ago [65–68]. The electroweak NLO corrections were\nobtained around the same time [69–72]. When the final-state bottom quark is taken to\nbe massless, the inclusive and differential cross sections have been computed up to N4LO\n[73–78] and N3LO [79–81], respectively. The differential cross sections with massive bottom\nquarks have been calculated at NNLO in QCD [82–84], and the NNLO analytical inclusive\nresult was studied in [85–87]. In addition, the combined QCD and electroweak corrections\nhave been computed in [88], and the parton shower effect is considered in [89, 90].\nIn our calculation, we consider massless bottom quarks but keep the Yukawa coupling\nybfinite. The LO squared amplitude is given by\n\f\fMtree(H→b¯b)\f\f2= 2√\n2Ncm2\nHm2\nb(µ)GF, (A.1)\nwhere we have replaced ybby23/4mb(µ)G1/2\nF. Setting mH= 125 GeV , Nc= 3, mb(mH) =\n2.8 GeV , GF= 1.1663787 ×10−5GeV−2and performing phase space integration, we obtain\nΓLO\nH→bb= 1.93 MeV. The NLO QCD corrections to H→b¯binclude contributions from both\nreal and virtual gluon effects. For the real corrections, we need the tree-level amplitude\nsquared of the process H(p)→b(p1) +¯b(p2) +g(p3),\n\f\fMreal(H→b¯bg)\f\f2=8παsCF\u0000\ns2\n12+m4\nH\u0001\ns13s23m2\nH|Mtree|2(A.2)\nwith sij≡(pi+pj)2. In the soft or collinear limit of the gluon’s momentum, i.e., s13→0\nors23→0, the integration of this amplitude squared becomes divergent. In numerical\ncalculation, we subtract these divergences using dipole counter-terms which have the same\ninfrared structure but are easy to integrate analytically.\nThe virtual loop corrections in this process are easy to compute, and the results contain\nbothultravioletandinfrareddivergences, whichareregularizedbyworkingin d-dimensional\nspace-time with d= 4−2ϵ. Adopting the MSand on-shell schemes for the Yukawa coupling\nand the bottom quark wave function, respectively, the contribution from the counter-terms\nis given by\nMCT=−3αsCF\n4πϵMtree, (A.3)\nwhich would cancel the ultraviolet divergences in the loop corrections. The sum of the loop\ncorrections and counter-terms is\n2Re [Mloop+CT · M∗\ntree]\n=αsCF\n2πeγEϵ\nΓ(1−ϵ)\u0014\n−2\nϵ2−1\nϵ\u0012\n3 + 2 logµ2\nR\nm2\nH\u0013\n−log2\u0012µ2\nR\nm2\nH\u0013\n+π2−2\u0015\n|Mtree|2.(A.4)\nThe remaining infrared divergences cancel with the integrated dipole subtraction terms. At\nthe end, the finite result of the virtual correction reads\n2Re [Mloop+CT · M∗\ntree] +I(ϵ)× |M tree|2=αsCF\n2π\u0012\n3 logµ2\nR\nm2\nH+ 8\u0013\n|Mtree|2,(A.5)\n– 11 –where the operator I(ϵ)is the integrated dipole subtraction term [58, 59]. After performing\nthe phase space integration, our result agrees with those in [73, 75, 80, 91].\nBH→γγdecay\nThe Higgs boson to a photon pair decay is a clean channel in discovering the Higgs boson.\nThe QCD NLO corrections to this decay channel have been calculated in the heavy top\nquark limit [92–94] and with full quark mass dependence [95–99]. Higher-order results are\nalso available [100–102]. Since the NLO QCD corrections do not contribute to the final\nresult, as discussed around eq.(2.9), we consider the Higgs boson decay to a photon pair at\nLO. 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Villegas-Blas¶\nFebruary 2, 2024\nAbstract\nWe consider a modulated magnetic field, B(t) =B0+εf(ωt), perpendicular to a\nfixed plane, where B0is constant, ε>0 andfa periodic function on the torus Tn.\nOur aim is to study classical and quantum dynamics for the correspo nding Landau\nHamiltonian. It turns out that the results depend strongly on the c hosen gauge.\nFor the Landau gauge the position observable is unbounded for ”alm ost all” non\nresonant frequencies ω. On the contrary, for the symmetric gauge we obtain that,\nfor ”almost all” non resonant frequencies ω, the Landau Hamiltonian is reducible to\na two dimensional harmonic oscillatorand thus gives rise to bounded d ynamics. The\nproofs use KAM algorithms for the classical dynamics. Quantum app lications are\ngiven. In particular, the Floquet spectrum is absolutely continuous in the Landau\ngauge while it is discrete, of finite multiplicity, in symmetric gauge.\nThank you, Thomas, for sharing your enthusiasm\nand your joy of playing with mathematics.\n1 Introduction and main results\nIn this paper we study the dynamics of time dependent perturb ations of the Schr¨ odinger\nequation\ni∂tψ=HA(t)ψ+V(t)ψ , (1.1)\n∗Dipartimento di Matematica Federigo Enriques, Universit` a degli Studi di Milano, Via Saldini 50,\nI-20133 Milano.\nEmail:dario.bambusi@unimi.it\n†Laboratoire de Math´ ematiques Jean Leray, Universit´ e de N antes, 2 rue de la Houssini` ere BP 92208,\n44322 Nantes Cedex 3\nEmail:benoit.grebert@univ-nantes.fr\n‡International School for Advanced Studies (SISSA), Via Bon omea 265, 34136, Trieste, Italy\nEmail:amaspero@sissa.it\n§Laboratoire de Math´ ematiques Jean Leray, Universit´ e de N antes, 2 rue de la Houssini` ere BP 92208,\n44322 Nantes Cedex 3\nEmail:didier.robert@univ-nantes.fr\n¶Universidad Nacional Autonoma de M´ exico, Instituto de Mat em´ aticas, Unidad Cuernavaca\nEmail:villegas@matcuer.unam.mx\n1whereHA(t) is the magnetic Schr¨ odinger operator in L2(R3):\nHA(t) :=/summationdisplay\n1≤j≤3/parenleftbig\nDxj−Aj(t,x)/parenrightbig2, Dx:= i−1∂\n∂x, (1.2)\nwithA(t,x) = (A1(t,x),A2(t,x),A3(t,x)) a time dependent vector potential and finally\nV(t,x) is a time dependent scalar potential. We recall that the ele ctric field is given\nby/vectorE(t,x) =−∂A\n∂t(t,x)− ∇xV(t,x) and the magnetic field by /vectorB(t,x) =∇x∧A(t,x).\nWe shall assume that the magnetic field has a fixed direction or thogonal to the plane\n{e1,e2}. Choosing A3(t,x)≡0 thenA1(t,x) andA2(t,x) depend only on ( t,x1,x2) and\nit is enough to consider the 2D magnetic Hamiltonian\nHA(t) = (Dx1−A1(t,x))2+(Dx2−A2(t,x))2\nas an operator in L2(R2).\nAnimportantparticularcaseistheconstant(inposition)m agneticfield B(t) = (0,0,−B(t)),\nfor which\nB(t) =∂x1A2−∂x2A1. (1.3)\nThis is usually studied using either the symmetric Gauge or t he Landau Gauge, namely\n-the symmetric gauge :A1(t,x) =B(t)\n2x2, A2(t,x) =−B(t)\n2x1;\n-the Landau gauge: A1(t,x) =B(t)x2, A2(t,x) = 0.\nIn this paper we consider the case when Bslightly fluctuates around a fix value B0>0:\nB(t) =B0+εf(ωt) whereω∈Rnis a frequency vector, fis a periodic function real\nanalytic on the torus Tnandε>0 is a small parameter.\nMathematically the source of most of the interesting featur es of the Landau Hamil-\ntonian rests in the fact that when ε= 0 the Hamiltonian is degenerate, in the sense that\nit is equivalent (unitary equivalent in the quantum case, ca nonically equivalent in the\nclassical case) to the Hamiltonian of a 1-d harmonic oscilla tor. As a result the quantum\nspectrum of the system is composed just by essential spectru m and coincides with the\nset{λj= 2B0(j+1/2)|j∈N}.\nThe caseε/\\e}atio\\slash= 0 will be discussed in the two different gauges:\n(i) the Landau gauge HL(t) = (Dx1−B(t)x2)2+D2\nx2;\n(ii) the symmetric gauge HsL(t) = (Dx1−B(t)x2\n2)2+(Dx2+B(t)x1\n2)2.\nIt turns out that both the main part of the Hamiltonian and the time dependent pertur-\nbation are quadratic polynomials in the position and the mom entum variables, and this\nallows to study the problems (both classical and quantum) us ing the ideas of [3], namely\nby using classical KAM theory to conjugate the Hamiltonian t o a suitable normal form\nwhose dynamics is easy to study.\nThe results depend drastically of the choice of the gauge:\n•In case (i), provided ωis non resonant, a condition which is fulfilled in a set\nof asymptotically full measure, we get that for ε/\\e}atio\\slash= 0 the position observable is\n2unbounded as t→ ∞as well for the classical motion and the quantum motion. It\nmay be surprising that for a dynamical system a non-resonanc e condition generates\nan instability1.\nAs a consequence, in the quantum side, we prove that the Floqu et spectrum is\nabsolutely continuous.\n•In case (ii) we prove that for ωin a set of asymptotically full measure, the dynam-\nics is reducible to a harmonic oscillator with two degrees of freedom, hence with\nbounded dynamics.\nAs a consequence, in the quantum side, we prove that the Floqu et spectrum is\ndiscrete with finite multiplicity.\nNotice that HL(t) andHsL(t) are gauge equivalent modulo a quadratic scalar po-\ntential (see section 1.3). Therefore the two models are not p hysically equivalent: in the\ntwo cases we have the same magnetic field but not the same elect ric field.\n1.1 Main result in the Landau gauge\nWe consider first the Landau gauge, namely\nHL(t) = (Dx1−B(t)x2)2+D2\nx2, B(t)>0, (1.4)\nwithB(t) :=B0+εf(ωt),fis real analytic on the torus Tn,ˆf(0) = 0 and ω∈[0,2π)n:=\nD0. Here and below we denote by ˆf(k) thek-th Fourier coefficients of f,\nˆf(k) := (2π)−n/integraldisplay\nTnf(θ)e−ik·θdθ .\nWe decompose the Hamiltonian HL(t) in (1.4) as\nHL(t) =HL+RL(ωt)\nwhere\nHL= (Dx1−B0x2)2+D2\nx2,\nRL(ωt) =−2εf(ωt)x2(Dx1−B0x2)+ε2f(ωt)2x2\n2.\nWe denote by hL(t) the corresponding classical Hamiltonian:\nhL(t,x,p) = (p1−B(t)x2)2+p2\n2=hL(x,p)+rL(ωt,x,p). (1.5)\n1But the phenomenon is similar of that encoded in Theorem 3.3 o f [3], in which the non resonance\ncondition is used to eliminate from the Hamiltonian as many t erms as possible, so that one remains only\nwith the terms actually generating the instability.\n3We introduce now complex coordinates in which the classical Hamiltonian hLhas the\nform of a degenerate 2-dimensional Harmonic oscillator. Fi rst introduce the symplectic\nvariables\nQ1=−1\nB0(p1−B0x2), P1=p2\nand\nQ2=−1\nB0(p2−B0x1), P2=p1.\nIn these variables we have\nhL=B2\n0Q2\n1+P2\n1.\nThen we introduce the complex variables\nz1=B0Q1+iP1√2B0, z2=B0Q2+iP2√2B0,\nfulfilling idzi∧dzi=dQi∧dPi,i= 1,2. In these variables\nhL= 2B0|z1|2. (1.6)\nThe link with the initial coordinates ( x,p)∈R4is given by ( z1,z2) =τ0(x,p) whereτ0\nis the linear symplectic transformation τ0:R4→C2such that\nz1=B0√2B0/parenleftbiggB0x2−p1\nB0/parenrightbigg\n+ip2√2B0\nz2=B0√2B0/parenleftbiggB0x1−p2\nB0/parenrightbigg\n+ip1√2B0. (1.7)\nIn order to state the first result we need to define a constant cωwhich is only defined\nforωfulfilling a nonresonance condition. To this end we prelimin ary restrict the set of\nthe allowed frequencies.\nDefinition 1.1. The set D0⊂[0,2π]nis the set of the frequencies ωs.t. there exist\npositiveγ,τs.t.\n|ω·k+2B0| ≥γ\n1+|k|τ,∀k∈Zn(1.8)\n|ω·k| ≥γ\n|k|τ,∀k∈Zn\\{0}. (1.9)\nRemark that such a set has full measure in [0 ,2π]n.\nWe now introduce\ncω:=/integraldisplay\nTngω(θ)2dθ (1.10)\nwhere the function\ngω(θ) :=−1√2B0/summationdisplay\nk/\\e}atio\\slash=0ω·k\nω·k+2B0ˆf(k)eik·θ,\n4is well defined for ω∈D0(recall that fis real analytic).\nFurthermore, in the following we will say that two polynomia ls areO(ǫ) close from\neach other, if their coefficients are O(ǫ) close from each other.\nOur first result is the following:\nTheorem 1.2. There exists ε0>0such that for |ε|<ε0, there exist\n•an asymptotically full measure set of frequencies Cε⊂D0satisfying:\nlim\nε→0meas(D0\\Cε) = 0;\n•a linear, symplectic change of variable τ, depending on ω,t,ε, which is close to the\nidentity, namely τ=I+O(ε)uniform in the other parameters,\nsuch that, for ω∈ Cε, the time quasi-periodic Hamiltonian hL(t,x,p)in(1.5)is conju-\ngated to the constant coefficient quadratic Hamiltonian\n(hL(t)◦τ−1\n0)◦τ=b(ε)|z1|2+c(ε)(z2−z2)2(1.11)\nwhere\nb(ε) = 2B0+O(ε2), c(ε) =cωε2+O(ε4) (1.12)\nandcωis given by (1.10).\nIn the new coordinates ( z1,z2) the motion is easily computed:\nz1(t) = e−ib(ε)tz1(0),\nℑz2(t) =ℑz2(0),ℜz2(t) =−4c(ε)ℑz2(0)t+ℜz2(0).\nLetuscomeback totheoriginalcoordinates ( x,p). FirstweremarkthattheHamiltonian\nat r.h.s. of (1.11) takes the form\nb(ε)\n2B0hL(x,p)+αc(ǫ)p2\n1\nwithhLthe original Landau Hamiltonian and α/\\e}atio\\slash= 0 a numerical constant. As a con-\nsequence, the corresponding dynamics is just given by the st andard circular motion of\na particle in a magnetic field with a slightly different frequen cy and with superimposed\na uniform motion in the direction of x1. This uniform motion is the new effect which\ngives rise to the growth of the solution. Actually this descr iption holds in the system\nof coordinates introduced by the KAM procedure. In the true o riginal coordinates this\nmotion is slightely deformed, so that it has superimposed a s mall oscillation. Precisely\nthe motion for the quadratic Hamiltonian hL(t) is a linear flow Φ B(t) where we have\n/parenleftbiggx(t)\np(t)/parenrightbigg\n= ΦB(t)/parenleftbiggx(0)\np(0)/parenrightbigg\n.\nwhere Φ B(t) is a real symplectic matrix. Then we have:\n5Corollary 1.3. Forω∈ Cεwe have, with α/\\e}atio\\slash= 0,\nx1(t) =x1(0)+αc(ε)p1(0)t+1\nB(t)(p2(t)−p2(0))+aε,ω(t)·x(0)+bε,ω(t)·p(0).(1.13)\nMoreover, modulo an error term of the form E˜a,˜b(t) = ˜a(t)·x(0)+˜b(t)·p(0), such that\nuniformly for t∈R,ω∈ Cε, we have |˜a(t)|+|˜b(t)|=O(ε),\np2(t) =/radicalbig\n2B0ℑ(z1(t)), z1(t) = e−2ibtz1(0) (1.14)\np1(t) =p1(0)\nx2(t) =p1(0)\nB(t)+√\n2B3/2\n0\nB(t)2ℜz1(t). (1.15)\nIn particular, if both cωandp1(0) are not zero, then the classical flow is not bounded\nas soon asε/\\e}atio\\slash= 0 is small enough.\nRemark 1.4. Clearly, in view of (1.10),cω/\\e}atio\\slash= 0holds forωin a set of asymptotically\nfull measure and for fin a set of codimension 1 (e.g. in L2). For instance, by a simple\ncalculus one has that if f(θ) = sin(θ)(and thusn= 1) thencω/\\e}atio\\slash= 0as soon asω/\\e}atio\\slash= 2B0.\nFrom the result on the classical evolution of x1(t), we get a direct application to\nthe large time evolution of the quantum position observable ˆx1(t). Let us explicit our\nnotations: we denote by ˆ xjand ˆpj,j= 1,2, the position and momentum operator\nˆxjψ(x) =xjψ(x),ˆpjψ=1\ni∂\n∂xjψ, x= (x1,x2), p= (p1,p2), ψ∈ H1\nosc.\nForr≥0,Hr\noscis the weighted Sobolev space associated with the harmonic o scillator\nH0:= ˆp2+ ˆx2,\nHr\nosc={ψ∈L2(R2) :Hr/2\n0ψ∈L2(R2)},\nendowed with the norm /ba∇dblψ/ba∇dblr=/ba∇dblHr/2\n0ψ/ba∇dblL2(R2).\nRecall that we are working here with polynomials classical H amiltonians of degree at\nmost 2 so the correspondence classical-quantum is exact. Th at means\n/parenleftbiggˆx(t)\nˆp(t)/parenrightbigg\n= ΦB(t)/parenleftbiggˆx\nˆp/parenrightbigg\n,\nwhere (ˆx(t),ˆp(t)) is the solution of the Heisenberg equation.\nOur first corollary regards the existence of solutions of the quantum Landau Hamil-\ntonian undergoing unbounded growth of Sobolev norms.\nCorollary 1.5. Letω∈ Cε. We have, in the operator sense,\nˆx1(t) = ˆx1+αcωε2tˆp1+(1+ε4t)(Aε,ω(t)·ˆx+Bε,ω(t)·ˆp),\n6whereα/\\e}atio\\slash= 0andcωis given by (1.10), and|Aε,ω(t)|+|Bε,ω(t)|=O(1). In particular, if\ncω/\\e}atio\\slash= 0, there exist µ>0,K≥0such that for any ψ∈ H1\noscwe have\n/ba∇dblˆx1(t)ψ/ba∇dbl0≥αcωtε2/ba∇dblDx1ψ/ba∇dbl0−K(1+ε4t)/ba∇dblψ/ba∇dbl1. (1.16)\nIn particular, if εis sufficiently small, then /ba∇dblˆx1(t)ψ/ba∇dbl0ր+∞astր+∞.\nWe have also a lower bound for the quantum average of the time e volution of the position\nobservable, for ψ∈ H1/2\nosc:\n|/a\\}b∇acketle{tψ,ˆx1(t)ψ/a\\}b∇acket∇i}ht| ≥αcωtε2|/a\\}b∇acketle{tψ,Dx1ψ/a\\}b∇acket∇i}ht|−K(1+ε4t)/ba∇dblψ/ba∇dbl1/2. (1.17)\nOur second corollary regards the Floquet spectrum of the tim e quasi-periodic Hamil-\ntonianHL(t).\nCorollary 1.6. The quantum dynamics Uε,ω(t,0)ofHL(t)is conjugated to the quantum\ndynamics e−it˜HL,ε,∞of the stationary Hamiltonian ˜HL,ε,∞:=b(ε)(D2\n˜x1+ ˜x2\n1)+c(ε)D2\n˜x2.\nMoreover as far as b(ε)>0andc(ε)>0, the spectrum σ(HL,ε,∞)of˜HL,ε,∞is absolutely\ncontinuous, with threesholds at the Landau levels,\nσ(HL,ε,∞) =/uniondisplay\nj≥0[b(ε)(j+1/2),+∞[.\nProof.Denote by hL,ε,∞:=b(ε)|z1|2+c(ε)(z2−z2)2the stationary classical Hamilto-\nnian to which hL(t) is conjugated, see (1.11). In the real coordinates ( x,p) we have\nhL,ε,∞(x,p) =b(ε)\n2(p2\n2+x2)2+c(ε)\n2(x1−p2)2. By the symplectic change of coordinates\n˜x1=x1−p2,˜p1=p1, ˜x2=x2−p1,˜p2=p2, we have\n˜hL,ε,∞(˜x,˜p) =b(ε)(˜p2\n2+ ˜x2)2+c(ε)˜p2\n1.\nThe first part of the Corollary follows.\nForthesecondpartnoticethatwehavethefollowingfamilyo fgeneralized eigenfunctions:\nΨj,ξ(x1,x2) =ψj(x1)eix2ξsuch that we have\n˜HL,ε,∞Ψj,ξ= (b(ε)(j+1/2)+c(ε)ξ2)Ψj,ξ.\nHence we get a description of the spectrum of ˜HL,ε,∞.\n1.2 Main result in symmetric gauge\nWe still consider a magnetic Schr¨ odinger operator with a ti me-quasiperiodic magnetic\nfield, i.e.B(t) =B0+εf(ωt) withfreal analytic on the torus Tn, andˆf(0) = 0, but\nnow in the symmetric gauge, namely\nHsL(t) =/parenleftbigg\nDx1−1\n2B(t)x2/parenrightbigg2\n+/parenleftbigg\nDx2+1\n2B(t)x1/parenrightbigg2\n, (1.18)\n7and the frequency vector ωin the set of nonresonant frequencies D0of Definition 1.1.\nWe can write\nHsL(t) =HsL+RsL(ωt)\nwhere\nHsL=/parenleftbigg\nDx1−1\n2B0x2/parenrightbigg2\n+/parenleftbigg\nDx2+1\n2B0x1/parenrightbigg2\n,\nRsL(ωt) =εf(ωt)/parenleftbigg\nx1(Dx2+B0\n2x1)−x2(Dx1−B0\n2x2)/parenrightbigg\n+1\n4ε2f(ωt)2(x2\n1+x2\n2).\nWe denote by hsL(t) the corresponding classical Hamiltonian:\nhsL(t) =/parenleftbig\np1−B(t)\n2x2/parenrightbig2+/parenleftbig\np2+B(t)\n2x1/parenrightbig2=hsL+rsL(ωt). (1.19)\nWe introduce the symplectic variables\np′\n1=x1, x′\n2=x2x′\n1=−p1, p′\n2=p2\nand in these variables hsLreads\nhsL=/parenleftbigg\nx′\n1+B0\n2x′\n2/parenrightbigg2\n+/parenleftbigg\np′\n2+B0\n2p′\n1/parenrightbigg2\n.\nThen in the new symplectic variables ( y1,y2,η1,η2) defined by\ny1=1√B0x′\n1+√B0\n2x′\n2, η1=√B0\n2p′\n1+1√B0p′\n2\nand\ny2=1√B0x′\n1−√B0\n2x′\n2, η2=√B0\n2p′\n1−1√B0p′\n2\nwe obtain that hsLis thedegenerate two-dimensional Harmonic oscillator\nhsL=B0(y2\n1+η2\n1) = 2B0|z1|2\nwherez1=y1+iη1√\n2and similarly z2=y2+iη2√\n2. We denote by τ0the linear symplectic\ntransformation from R4toC2defined by ( z1,z2) =τ0(x,p). We note that in the complex\nvariables the symplectic form reads:\ndy1∧dη1+dy2∧dη2= i(dz1∧dz1+dz2∧dz2).\nIn order to state the next result we introduce\ndω=/integraldisplay\nTnh(θ)2dθ (1.20)\nwhere\nh(θ) =−1√2B0/summationdisplay\nk/\\e}atio\\slash=0ω·k+2iB0\nω·k+2B0ˆf(k)eik·θ. (1.21)\n8Remark 1.7. The number dωis well defined and real for ω∈D0.\nOur next result shows that, in the symmetric gauge, the pertu rbed classical Hamil-\ntonianhsL(t) in (1.19) is conjugated to a two dimensional Harmonic oscil lator which is\nnon-degenerate provided the constant dω/\\e}atio\\slash= 0.\nTheorem 1.8. There exists ε0>0such that for |ε|<ε0, there exist\n•an asymptotically full measure set of frequencies Cε∈D0satisfying:\nlim\nε→0meas(D0\\Cε) = 0;\n•a linear, symplectic change of variable τ, depending on ω,t,ε, which is close to the\nidentity, namely τ=I+O(ε)uniform in the other parameters,\nsuch that, for ω∈ Cε, and provided dωin(1.20)does not vanish,\n(hsL(t)◦τ−1\n0)◦τ=b(ε)|z1|2+d(ε)|z2|2(1.22)\nwith\nb(ε) = 2B0+O(ε2), d(ε) =dωε2+O(ε4)\nanddωis given by (1.20)and does not vanish for ω∈ Cε.\nIn the new coordinates the motion of (1.22) is easily compute d:\nz1(t) = e−ib(ε)tz1(0), z 2(t) = e−id(ε)tz2(0).\nInparticular, inthesymmetricgauge, providedtheconstan tdωin(1.20)doesnot vanish,\nall the trajectories are bounded, contrary to what happens i n the Landau gauge.\nAlso in this case we are able to describe the quantum flow, whic h in this case is\nuniformly bounded in any Sobolev space Hr\nosc.\nLetUε,ω(t,s) be the quantum propagator defined by the Hamiltonian HsL(t) in (1.18).\nSo we have i ∂tUε,ω(t,s) =HsL(t)Uε,ω(t,s),Uε,ω(s,s) =I.\nCorollary 1.9. There exists ε0>0such that for |ε| ≤ε0, for anyr >0there exist\n0<cr≤Crsuch that if ω∈ Cεwe have for any ψ0∈ Hr\nosc,\ncr/ba∇dblψ0/ba∇dblr≤ /ba∇dblUε,ω(t,0)ψ0/ba∇dblr≤Cr/ba∇dblψ0/ba∇dblr,∀t∈R. (1.23)\nProof.We follow the proof given in [3], Corollary 1.3. We shall give here only the main\nsteps.\nFor simpler notation we assume that B0= 1. Let\nHsL,ε,∞=b(ε)\n2((ˆp1−ˆx2)2+ ˆp2\n2)+d(ε)\n2((ˆp2−ˆx1)2+ ˆp2\n1),\nandZ1= (ˆp1−ˆx2)2+ ˆp2\n2,Z2= (ˆp2−ˆx1)2+ ˆp2\n1. Notice that Zj= ˆz∗\njˆzj. Moreover\n[Z1,Z2] = 0. So [Hε,∞,Z1+Z2] = 0. ButZ1+Z2is a non degenerate harmonic oscillator\n9hence e−itHε,∞ψ0satisfies the estimates (1.23). Then as in [3], from the class ical KAM\nconstruction there exists\nχε,ω(t,x,p) =/parenleftbiggx\np/parenrightbigg\n·Sε,ω(t)/parenleftbiggx\np/parenrightbigg\n,\nwhereSε,ω(t) is a symmetric matrix with uniformly bounded entries. Let Uε,ω(t) =\neiεˆχε,ω(t)we have\nUε,ω(t,0) =U∗\nε,ω(t)e−itHε,∞Uε,ω(t).\nUsing that ([3], Theorem 2.7), uniformly in ( t,ε,ω), we have\n˜cr/ba∇dblψ/ba∇dblr≤ /ba∇dblU∗\nε,ω(t)ψ/ba∇dblr≤˜Cr/ba∇dblψ/ba∇dblr,\nwe get (1.23).\nCorollary 1.10. The symmetric Landau Hamiltonian HsL(t)is reducible to a stationary\nHamiltonian HsL,ε,∞with a discrete spectrum with all eigenvalues of finite multi plicities\nas farb(ε)>0,d(ε)/\\e}atio\\slash= 0. Moreover, up to a linear symplectic transformation, HsL,ε,∞\nis combination of two 1-d harmonic oscillators\nHsL,ε,∞=b(ε)(D2\nx1+x2\n1)+d(ε)(D2\nx2+x2\n2).\nProof.Keeping the notations of the proof of Corollary 1.9, since Z1+Z2has compact\ninverse, it has pure point spectrum and there exists a basis o f eigenfunctions, which,\nsince [Z1,Z2] = 0 can be choose in such a way that they are also eigenfunctio ns of\nbothZ1andZ2. Thus, if one denotes by λi,j=j+ 1/2,j∈Nthe eigenvalues of\nZi, one has that HsL,ε,∞is diagonal in the same basis, with eigenvalues µj1,j2(ε) =\nb(ε)(j1+1/2)+d(ε)(j2+1/2). So the symmetric LandauHamiltonian HsL(t) is reducible\ntoastationaryHamiltonian HsL,ε,∞withadiscretespectrumwithalleigenvalues offinite\nmultiplicities as far b(ε)>0,d(ε)/\\e}atio\\slash= 0.\n1.3 About the change of gauge\nThe fact that one has a completely different behaviour in the ca se of the Landau gauge\nand in the case of the symmetric gauge could seem surprising a t first sight, however we\nremark that they correspond to different physical situations . Indeed, the electric and\nthe magnetic field are given by\nB=∇×A(x,t) and E=−∂A\n∂t(x,t)−∇U(x,t). (1.24)\nThus, in the time dependent case, they differ for the case of the Landau gauge and the\ncase of the symmetric gauge. As it is well known, there exists a gauge transformation\nwhich allows to pass from one gauge to the other by keeping the same electromagnetic\nfields. For example the electric and magnetic fields can be cho sen as\nU(x,t) = 0, A(x,t) =B(t)(x2,0,0)\n10or as\nU′(x,t) =−B′(t)\n2x1x2, A(x,t) :=B(t)\n2(x2,−x1,0).\nThe first corresponds to the Landau gauge, while the second co rresponds to the sym-\nmetric gauge plus a scalar potential.\n1.4 Related literature\nMost of the literature about the Landau Hamiltonian regards the asymptotic behavior\nof the perturbed spectrum under time independent perturbations, for example scalar\npotentials in different classes. These works ensure conditio ns on the perturbation so that\nthe perturbed spectrum is asymptotically localized around the Landau levels {2B0(j+\n1\n2)}j∈N, a fact which is not trivial due to the infinite multiplicity o f these. We mention\nfor example the works [31, 30, 28, 20, 21, 15] and reference th erein.\nThe case of time dependent perturbations, such as (1.1), is less studied. We mention\n[33, 34, 9] which prove the existence of the quantum flow, and [ 26] giving time upper\nbounds on the dynamics. The present paper aims to prove finer p roperties about the\nquantum dynamics, and in particular investigates the dicot omy of “existence of solu-\ntions with unbounded trajectories” vs “all trajectories ar e bounded”. This question has\nreceived, in the last decade, a lot of attention.\nIn case of linear, time dependent Schr¨ odinger equations, s uch as (1.1), the first result\nabout existence of solutions with unbounded paths is due to B ourgain [6] on the torus.\nRecently, several works have considered non-degenerate Ha rmonic oscillators on Rdand\nconstructed time dependent perturbations in the form of pse udodifferential operators\n[8, 23], polynomial functions [3, 17, 19, 18], or classical p otentials [11, 32], that create\nsolutions with unbounded trajectories. We also cite the rec ent results [24, 25] which\nprove that generic, time periodic, pseudodifferential pertu rbations provoke instability\nphenomena.\nOn the opposite side, many works prove that, when the perturb ation is small in size\nand quasi-periodic in time with a non resonant frequency ω, all trajectories are bounded\nin Sobolev spaces. There results are based on KAM reducibili ty methods ensuring that\nthe linear propagator has operatorial norm (in Sobolev spac es) bounded uniformly in\ntime, in the same spirit of our Corollary 1.9. This is the case , in great generality, for\nsystems in 1-spatial dimensions: limiting ourselves to res ults considering perturbations\nof the Harmonic oscillator on R, we cite [7, 35, 14, 2, 1]. In a higher dimensional setting,\nsuch as theone of equation (1.1), there are few KAM reducibil ity results. We cite [10, 29]\nfor the Schr¨ odinger on Td, [13, 3] for the Harmonic oscillators on Rd, [27] for the wave\nonTd, and [12, 5] for transport equations on Td.\n112 Proofs of main Theorems\n2.1 Proof of Theorem 1.2\nWe prefer to work in the extended phase space in which we add th e anglesθ∈Tn\nas new variables and their conjugated momenta I∈Rn. So our phase space is now\nTn×Rn×C4∋(θ,I,z1,z2), withC2considered as a real vector space. The symplectic\nform isdI∧dθ+ idz∧dzand the Hamiltonian equations of a Hamiltonian function\nh(θ,I,z1,z2) are\n˙I=−∂h\n∂θ,˙θ=∂h\n∂I,˙z1=−i∂h\n∂z1,˙z2=−i∂h\n∂z2. (2.1)\nIn this framework theHamiltonian equation associated with theclassical timedependent\nHamiltonian function hLin (1.5) is equivalent to the autonomous Hamiltonian system\n(2.1) with\nh=h0+r1+r2\nand\nh0=ω·I+2B0|z1|2, (2.2)\nr1=ε(z1+z1)(z1+z1−i(z2−z2))f(θ), (2.3)\nr2=ε2\n2B0(z1+z1−i(z2−z2))2f(θ)2. (2.4)\nThe proof of Theorem 1.2 follows a KAM strategy: we want to eli minate the angles in h\nby canonical changes of variables. Thiscanonical changes o f variables will beconstructed\nas time 1 flows, Φ1\nχ, of some Hamiltonian χ. We begin by computing explicitly the first\ntwo KAM steps and then we will bein position to apply a KAM theo rem with symmetry,\nnamely Theorem 3.4. First we construct the first change of var iables and we begin by\nsolving a so called homological equation.\nLemma 2.1. Let\nχ1:=iε/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbiggz2\n1\nω·k+4B0+z2\n1\nω·k−4B0+2z1z1\nω·k/parenrightbigg\n+ε(z2−z2)/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbiggz1\nω·k+2B0+z1\nω·k−2B0/parenrightbigg\n,\nthenχ1solves the following homological equation:\n{χ1,h0}+r1= 0. (2.5)\nProof.First recall that\n{F,G}:=n/summationdisplay\nj=1∂F\n∂θj∂G\n∂Ij−∂F\n∂Ij∂G\n∂θj+i/summationdisplay\nj=1,2∂F\n∂zj∂G\n∂zj−∂G\n∂zj∂F\n∂zj. (2.6)\n12Then we introduce\nN(θ,z1) =/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbiggz1\nω·k+2B0+z1\nω·k−2B0/parenrightbigg\nand\nM(θ,z1) =/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbiggz2\n1\nω·k+4B0+z2\n1\nω·k−4B0+2z1z1\nω·k/parenrightbigg\nin such way we have\nχ1= iεM+ε(z2−z2)N.\nSo, sinceh0in (2.2) doesn’t depend on z2, we get\n{χ1,h0}= iε{M,h0}+ε(z2−z2){N,h0}.\nThen we compute\n{N,h0}= i/summationdisplay\nk∈Zn\\{0}(k·ω)ˆf(k)eik·θ/parenleftbiggz1\nω·k+2B0+z1\nω·k−2B0/parenrightbigg\n+i/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbigg2B0z1\nω·k+2B0+−2B0z1\nω·k−2B0/parenrightbigg\n= i(z1+z1)f(θ),\nand\n{M,h0}= i/summationdisplay\nk∈Zn\\{0}(k·ω)ˆf(k)eik·θ/parenleftbiggz2\n1\nω·k+4B0+z2\n1\nω·k−4B0+2z1z1\nω·k/parenrightbigg\n+i/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbigg4B0z2\n1\nω·k+4B0+−4B0z2\n1\nω·k−4B0/parenrightbigg\n= i(z1+z1)2f(θ),\nfrom which (2.5) follows.\nThen since,\n{χ1,h0}+r1= 0,{χ1,{χ1,h0}}=−{χ1,r1},\nwe get\nh◦Φ1\nχ1=h+{χ1,h}+1\n2{χ1,{χ1,h0}}+O(ε3) =h0+1\n2{χ1,r1}+r2+O(ε3).(2.7)\n13Next we wish to compute explicitly {χ1,r1}. In view of the expressions of χ1andr1, we\nfirst compute\n{M,(z1+z1)2}= 2(z1+z1){M,z1+z1},\n{M,(z1+z1)(z2−z2)}= (z2−z2){M,z1+z1},\n{M,z1+z1}= 2i/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbiggz1\nω·k+4B0−z1\nω·k−4B0+z1−z1\nω·k/parenrightbigg\n=−8iB0/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbiggz1\nω·k(ω·k+4B0)+z1\nω·k(ω·k−4B0)/parenrightbigg\nand\n{N,(z1+z1)2}= 2(z1+z1){N,z1+z1},\n{N,(z1+z1)(z2−z2)}= (z2−z2){N,z1+z1},\n{N,z1+z1}=−4iB0/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ\n(ω·k)2−4B2\n0.\nTherefore\n{χ1,r1}={iεM+ε(z2−z2)N,ε(z1+z1)(z1+z1−i(z2−z2))f(θ)}\n= 2iε2f(θ)(z1+z1){M,(z1+z1)}+ε2(z2−z2)f(θ){M,(z1+z1)}\n+2ε2f(θ)(z2−z2)(z1+z1){N,z1+z1}−iε2(z2−z2)2f(θ){N,z1+z1}\n=ε2f(θ)(2i(z1+z1)+(z2−z2)){M,(z1+z1)}\n−iε2f(θ)(z2−z2)(2i(z1+z1)+(z2−z2)){N,z1+z1}\n=ε2f(θ)(2i(z1+z1)+(z2−z2))({M,(z1+z1)}−i(z2−z2){N,z1+z1})\n=−4iε2B0f(θ)(2i(z1+z1)+(z2−z2))×\n×/summationdisplay\nk∈Zn\\{0}ˆf(k)eik·θ/parenleftbigg2z1\nω·k(ω·k+4B0)+2z1\nω·k(ω·k−4B0)−iz2−z2\n(ω·k)2−4B2\n0/parenrightbigg\n.\n(2.8)\nAt the nextKAM step we remove from1\n2{χ1,r1}+r2all theε2terms except resonant\nmonomials, i.e. in our case, |z1|2and (z2−z2)2. In view of the expressions (2.8), (2.4),\nwe thus obtain\nh2=h◦Φ1\nχ1◦Φ1\nχ2=h0+aωε2|z1|2+cωε2(z2−z2)2+O(ε3) (2.9)\n14where\ncω=−/summationdisplay\nk∈Zn\\{0}ˆf(k)ˆf(−k)/parenleftbigg2B0\n(ω·k)2−4B2\n0/parenrightbigg\n−1\n2B0/a\\}b∇acketle{tf2/a\\}b∇acket∇i}ht\n=−/summationdisplay\nk∈Zn\\{0}ˆf(k)ˆf(−k)/parenleftbigg2B0\n(ω·k)2−4B2\n0+1\n2B0/parenrightbigg\n=−1\n2B0/summationdisplay\nk∈Zn\\{0}ˆf(k)ˆf(−k)(ω·k)2\n(ω·k)2−4B2\n0\nas stated in (1.10) while\naω= 8B0/summationdisplay\nk∈Zn\\{0}ˆf(k)ˆf(−k)/parenleftbigg1\n(ω·k)((ω·k)+4B0)+1\n(ω·k)((ω·k)−4B0)/parenrightbigg\n+1\nB0/a\\}b∇acketle{tf2/a\\}b∇acket∇i}ht\n=/summationdisplay\nk∈Zn\\{0}ˆf(k)ˆf(−k)/parenleftbigg16B0\n(ω·k)2−16B2\n0+1\nB0/parenrightbigg\n=1\nB0/summationdisplay\nk∈Zn\\{0}ˆf(k)ˆf(−k)(ω·k)2\n(ω·k)2−16B2\n0.\nTo end the proof of Theorem 1.2 we just have to iterate this KAM step and to prove\nthe convergence of such a process (concretely we will apply T heorem 3.4). In particular\nwe will check that we only remain with resonant monomials, i. e. in our case, |z1|2and\n(z2−z2)2.\nThe details of the KAM iteration are given section 3. In fact, the proof is quite standard\nexcept that the normal form h0is degenerate (the frequency associated to z2is null) and\nthe perturbation r=r1+r2commutes with z2−z2.\nWe stress out that the difference between the estimate (3.12) i n Theorem 3.4 and the\nestimate (1.12) in Theorem 1.2 is due to the specific form of rin (2.3) and the fact that\nˆf(0) = 0.\n2.2 Proof of Theorem 1.8\nWe still work in the same framework as in section 2.1, i.e. in t he extended phase space in\nwhich we add the angles θ∈Tnas new variables and their conjugated momenta I∈Rn.\nSo our phase space is still Tn×Rn×C2∋(θ,I,z1,z2) (see (2.1)).\nIn this framework the Hamiltonian equation associated with the classical time depen-\ndent Hamiltonian function (1.19) hsL(t) is equivalent to the autonomous Hamiltonian\nsystem (2.1) with h=h0+rwhere\nh0=ω·I+2B0|z1|2\n15andr=rsL(θ). To compute explicitly this term, recall that, in the coord inates intro-\nduced in section 1.2,\nx2=x′\n2=1√B0(y1−y2) =1√2B0(z1+z1−(z2+z2)),\nx1=p′\n1=1√B0(η1+η2) =1\ni√2B0(z1−z1+(z2−z2)),\np1−B0\n2x2=−x′\n1−B0\n2x′\n2=−/radicalbig\nB0y1=−√B0√\n2(z1+z1),\np2+B0\n2x1=p2−B0\n2ξ1=/radicalbig\nB0η1=√B0\ni√\n2(z1−z1).\nThereforer(θ) reads\nr(θ) =εf(θ)(y1(y1−y2)+η1(η1+η2))+ε2\n4B0f(θ)2((y1−y2)2+(η1+η2)2)\n=ε\n2f(θ)((z1+z1)(z1+z1−(z2+z2))−(z1−z1)(z1−z1+(z2−z2)))\n+ε2\n8B0f(θ)2((z1+z1−(z2+z2))2−(z1−z1+(z2−z2))2)\n=εr1+ε2r2.\nWe follow the same strategy than in the previous section and w e are interested by the\nquadratic terms in z2,z2after the second KAM step. At the first step (at order ε)\nwe don’t have such term (in r1) and thus we eliminate all the terms of order εby a\nsymplectic change of variables Φ1\nχ1whereχ1is the solution of the homological equation.\n{χ1,hsL}=−r1.\nSince\nr1=−εf(θ)(z1z2+z1z2)+ quadratic terms in z1,z1\nwe take (with ν= 2B0)\nχ1=−iε/summationdisplay\nk/\\e}atio\\slash=0ˆf(k)eik·θ/parenleftbiggz2z1\nk·ω+ν+z2z1\nk·ω−ν/parenrightbigg\n+ quadratic terms in z1,z1.\nThen, using (2.7), the quadratic terms in z2,z2after the second KAM step come from\nthe quadratic terms in z2,z2in1\n2{χ1,r1}and inr2.\nFromr2we get\nε2f(θ)2\n4B0z2z2\nand from1\n2{χ1,r1}we get\n−1\n2ε2f(θ)/summationdisplay\nk/\\e}atio\\slash=0ˆf(k)eik·θ/parenleftbiggz2z2\nk·ω+ν−z2z2\nk·ω−ν/parenrightbigg\n.\n16Now the second KAM step will eliminate all the eik·θz2z2fork/\\e}atio\\slash= 0 and therefore we\nobtain like in (2.9)\nh2=h◦Φ1\nχ1◦Φ1\nχ2=ω·I+(2B0+aε2)|z1|2+dωε2|z2|2+O(ε3) (2.10)\nwhere\ndω=1\n4B0/summationdisplay\nℓˆf(ℓ)ˆf(−ℓ)+/summationdisplay\nℓ/\\e}atio\\slash=0ˆf(ℓ)ˆf(−ℓ)2B0\n(ℓ·ω)2−4B2\n0\n=1\n4B0/summationdisplay\nℓ/\\e}atio\\slash=0ˆf(ℓ)ˆf(−ℓ)(ℓ·ω)2+4B2\n0\n(ℓ·ω)2−4B2\n0\nas stated in (1.20).\nThen, ifdω/\\e}atio\\slash= 0,h2appears as a perturbation of the non degenerate Hamiltonian (2B0+\naε2)|z1|2+dωε2|z2|2and we can apply Theorem 3.5.\n3 Proofs of two reducibility Theorems\nIn this section we prove the two reducibility theorems that w e need to conclude the\nproofs of Theorem 1.2 and Theorem 1.8.\nFirst, it is usefull to change the notation in order to make cl ear that the variable z\nis not necessarily the complex conjugate of z: we only have to define the concept of real\nsubmanifold of the phase space and it depends on the variable s we use (see Definition\n3.1 and Remark 3.3 below for more details).\nSo, first of all we define\nξj:=zj, ηj:=zj, j= 1,2. (3.1)\nDefinition 3.1. For(ξ,η)∈C4, define the involution\n(ξ,η)/ma√sto→I(ξ,η) = (η,ξ), (3.2)\nthe states (ξ,η)s.t.(ξ,η) =I(ξ,η)will be said to be real.\nRemark 3.2. The Hamiltonians we are dealing with are real when (ξ,η)is real.\nIn order to deal with the Hamiltonian (2.9) whose expansion u p to order ε2only\ndepends on z1,z1andz2−z2, it is useful to introduce the following canonical change of\nvariables\nξ′\n2:=ξ2−η2, η′\n2:=η2. (3.3)\nIn these variables (and omitting primes) (2.9) reads\nh2=ω·I+(2B0+aωε2)ξ1η1+cωε2ξ2\n2+O(ε3). (3.4)\n17Remark 3.3. in these variables the involution Itakes the form (omitting the primes)\n(ξ1,η1,ξ2,η2)/ma√sto→I(ξ1,η1,ξ2,η2)≡(η1,ξ1,ξ2,ξ2−η2) (3.5)\nand the real submanilfold reads: η1=ξ1,ξ2is real and 2ℜ(η2) =ξ2.\nIn all the situations we will encounter the Hamiltonian is ind ependent of η2, therefore\nthe fact that the reality condition becomes more complicate will be completely irrelevant.\nOf course, in the following a Hamiltonian expressed in the va riables(3.3)will be said\nto be real if it takes real values for real (ξ,η), i.e. for (ξ,η)which are fixed points with\nrespect to the involution (3.5).\nThe first theorem we will prove concerns quasi-periodic in ti me Hamiltonians of the\nform\nhε(t,ξ,η) =ν1ξ1η1+εq(ωt,ξ1,η1,ξ2) (3.6)\nwhereqis a polynomial in ( ξ1,η1,ξ2) homogeneous of degree 2 with coefficients that\ndepend quasi periodically on time. The important point is th atqdoes not depend on\nη2.\nIn the following we denote, for σ>0,Tn\nσ:={x+iy:x∈Tn, y∈Rn,|y| ≤σ}.\nTheorem 3.4. Assume that ν1>0and that Tn×C4∋(θ,ξ)/ma√sto→q(θ,ξ)∈Cis a\npolynomial in (ξ1,η1,ξ2)homogeneous of degree 2, independent of η2, with coefficients\nreal analytic in θ∈Tn\nσfor someσ>0(i.e. real when (ξ,η)is real and θ∈Tn)\nThen there exists ε∗>0andC >0, such that for |ε|<ε∗:\n- there exists a set Eε⊂(0,2π]nwithmeas((0,2π]n\\Eε)≤Cε1\n9,\n- for anyω∈ Eε, there exists an analytic map θ/ma√sto→Aω(θ)∈sp(2), such that the change\nof coordinates\n(ξ′,η′) =eAω(ωt)(ξ,η) (3.7)\nconjugates the Hamiltonian equations of (3.6)to the Hamiltonian equations of a homo-\ngeneous polynomial\nh∞(ξ,η) =ν1(ε)ξ1η1+c(ε)ξ2\n2\nwith\n|ν1(ε)−ν1| ≤Cε,|c(ε)| ≤ε. (3.8)\nFinallyAωisε-close to zero and eAω(ωt)leaves invariant the space of real states.\nThe second reducibility theorem deals with Hamiltonians of the form (in the original\nvariables (3.1))\nhε(t,ξ,η) =ν1ξ1η1+ν2ξ2η2+ε3q(ωt,ξ1,η1,ξ2,η2) (3.9)\nin which one has that the frequencies depend on ω∈Dandεand fulfill\nc≤ |ν1(ω)| ≤C, cε2≤ |ν2(ω)| ≤Cε2,|∂ων1|,|∂ων2| ≤Cε2, (3.10)\nwith some positive c,C. Soit is designed to deal with Hamiltonian (2.10). Thediffere nce\nwith the standard KAM context is that the second frequency is of orderε2while the\n18perturbation is of order ε3. The following theorem says that the standard conclusion\nstill holds true, i.e. that the inhomogeneous Hamiltonian s ystem associated with (3.9)\nis reducible for almost all values of ω:\nTheorem 3.5. Assume (3.10)and that Tn×C4∋(θ,ξ,η)/ma√sto→q(θ,ξ,η)∈Cis a polyno-\nmial in(ξ1,η1,ξ2,η2)homogeneous of degree 2, with coefficients analytic in θ∈Tn\nσfor\nsomeσ>0and taking real values when θ∈Tnandηis the complex conjugate of ξ.\nThen there exists ε∗>0andC >0, such that for |ε|<ε∗:\n- there exists a set Eε⊂(0,2π]nwithmeas((0,2π]n\\Eε)≤Cε1\n9,\n- for anyω∈ Eε, there exists an analytic map θ/ma√sto→Aω(θ)∈sp(2)such that the change\nof coordinates\n(ξ′,η′) =eAω(ωt)(ξ,η) (3.11)\nconjugates the Hamiltonian equations of (3.6)to the Hamiltonian equations of a homo-\ngeneous polynomial\nh∞(ξ,η) = ˜ν1(ε)ξ1η1+ ˜ν2(ε)ξ2η2\nwith\n|˜νj(ε)−νj| ≤Cε3, j= 1,2. (3.12)\nFinallyAωisε-close to zero.\nIn the remainder of this section we give the details of the pro of of Theorem 3.4 while\nwe only point out the small changes needed to prove Theorem 3. 5.\nIn fact, the proof of Theorem 3.4 is very standard, but since w e’re dealing with a\ndegenerate Hamiltonian h0, we have to be a little careful. The fact that the perturbatio n\nqis independent of η2is crucial here. If not, the Poisson bracket {h,χ}could generate\nnew quadratic terms in ( ξ2,η2) and the iteration could diverge.\n3.1 General strategy\nThe canonical change of variables is constructed applying a KAM strategy to the Hamil-\ntonian\nh(y,θ,ξ,η) =ω·y+ν0ξ1η1+εq(ωt,ξ1,η1,ξ2)\nin the extended phase space Rn×Tn×C2endowed with the standard symplectic form\ndy∧dθ+idξ∧dη.\nWe will say that the Hamiltonian his innormal form if it reads\nh(y,θ,ξ,η) =ω·y+aξ1η1+cξ2\n2=ω·y+N(ξ) (3.13)\nwhereaandcare real constants (independent of θ).\nLetq≡qωbe a polynomial Hamiltonian homogeneous of degree 2. We writ e\nq(θ,ξ,ξ) =/summationdisplay\nα,βqα,β(θ)ξαηβ\n(3.14)\n19where the coefficients qα,β(θ) are analytic functions of θ∈Tn\nσ, we used the standard\nnotation\nξαηβ:=ξα1\n1ξα2\n2ηβ1\n1ηβ2\n2\nand according to the independence on η2and the fact that this must be a quadratic\npolynomial one has the restrictions\nβ2= 0,|α|+|β|= 2. (3.15)\nThe size of such polynomial function depending analyticall y onθ∈Tn\nσandC1on\nω∈D⊆(0,2π]nwill be controlled by the norm\n[q]σ,D:= sup\n|ℑθ|<σ, ω∈D\nα,β, j=0,1|∂j\nωqα,β(θ)|\nand we denote by Q(σ,D) the corresponding class of Hamiltonians of the form (3.14)\nwhose norm [ ·]σ,Dis finite.\nLet us assume that [ q]σ,D=O(ε). We search for χ≡χω∈Q(σ,D) withχ=O(ε) such\nthat its time-one flow Φ χ≡Φt=1\nχtransforms the Hamiltonian h+qinto\n(h+q(θ))◦Φχ=h++q+(θ),∀ω∈D+, (3.16)\nwhereh+=ω·y+N+(ξ) is a new normal form, ε-close toh, and the new perturbation\nq+∈ Q(σ+,D+) is of size2O(ε3\n2), andD+⊂Dis an open set εα-close to Dfor some\nα>0. As a consequence of the Hamiltonian structure we have that\n(h+q(θ))◦Φχ=h+{h,χ}+q(θ)+O(ε3\n2).\nSo to achieve the goal above we should solve the homological equation :\n{h,χ}=h+−h−q(θ)+O(ε3\n2), ω∈D+. (3.17)\nRepeating iteratively the same procedure with h+instead ofh, we will construct a\nchange of variable Φ such that\n(h+q(θ))◦Φ =h∞, ω∈D∞\nwithh∞=ω·y+N(ξ,η) in normal form and D∞aεα-close subset of D. Note that\nwe will be forced to solve the homological equation, not only for the original normal\nformh0=ω·y+νξ1η1, but for more general normal form Hamiltonians (3.13) with\nN(ξ) =aξ1η1+cξ2\n2close toN0=νξ1η1. To control this closeness we define a norm on\nN=aξ1η1+cξ2\n2:\n/ba∇dblN/ba∇dbl:= max(|a|,|c|).\nThe key remark is that ∀q∈ Q(σ,D) one has/braceleftbig\nξ2;q/bracerightbig\n≡0.\n2Formally we could expect q+to be of size O(ε2) but the small divisors and the change of analyticity\ndomain will led to O(ε3\n2).\n203.2 Homological equation\nProposition 3.6. LetD⊂D0. LetD∋ω/ma√sto→N(ω)be aC1mapping that verifies\n/vextenddouble/vextenddouble∂j\nω(N(ω)−N0)/vextenddouble/vextenddouble≤min(1,ν0)\n4(3.18)\nforj= 0,1andω∈D. Leth=ω·y+N(ξ),q∈ Q(σ,D),κ>0andK≥1.\nThen there exists an open subset D′=D′(κ,K)⊂D, satisfying\nmeas(D\\D′)≤4d2K2nκ, (3.19)\nand there exist χ,r∈ ∩0≤σ′<σQ(σ′,D′)and˜Nin normal form such that for all ω∈D′\n{h,χ}+q=˜N+r . (3.20)\nFurthermore for all 0≤σ′<σ\n[r]σ′,D′≤Ce−1\n2(σ−σ′)K\n(σ−σ′)n[q]σ,D, (3.21)\n[χ]σ′,D′≤C\nκ2(σ−σ′)n[q]σ,D, (3.22)\n/ba∇dbl∂j\nω˜N(ω)/ba∇dbl ≤[q]σ,Dj= 0,1,∀ω∈D. (3.23)\nThe constant Cdepends on n.\nProof.As usual we consider the ”homological operator” L:={h;.}and decompose the\nspaceQ(σ,D) on the basis of its eigenfunctions. Such a basis is given by t he monomials\nξαηβeikθ,\nwhereαandβare subject to the restrictions (3.15). The corresponding e igenvalues are\ni(ν1(α1−β1)+ω·k), (3.24)\nwhile Ker( L) =span/braceleftbig\nξ2\n2,ξ1η1/bracerightbig\n. So, decomposing qas in (3.14), and expand the coeffi-\ncients in Fourier series:\nqα,β(θ) =/summationdisplay\nk∈Znˆqα.β(k)eikθ,\nthen one is lead to define\nχα,β(θ) =/summationdisplay\n|k|≤Kˆqα.β(k)\ni(ν1(α1−β1)+ω·k)eikθ, (3.25)\nwhere, forα= (1,0),β= (1,0) and forα= (0,0) andβ= (2,0) the sum is restricted to\nk/\\e}atio\\slash= 0. We also define\n˜N:= ˆq1,0,1,0(0)ξ1η1+ ˆq0,2,0,0(0)ξ2\n2, (3.26)\nr(ξ,η,θ) :=/summationdisplay\n|k|>K,α,βˆqα,β(k)ξαηβeikθ, (3.27)\n21so that the homological equation is satisfied. Still it remai ns to prove the estimates of\nthe various terms. We give explicitly the estimate of χ. To this end we have to control\nthe small denominators (3.24) under the restriction\n|α1−β1|+|k| /\\e}atio\\slash= 0,|k| ≤K , α 1+β1≤2. (3.28)\nWe define D′to be the set for which the above small denominators are bigge r thanκ.\nIn order to estimate its measure we recall the following clas sical lemma:\nLemma 3.7. Letf: [0,1]/ma√sto→Rbe aC1-map satisfying |f′(x)| ≥δfor allx∈[0,1]and\nletκ>0. Then\nmeas{x∈[0,1]| |f(x)| ≤κ} ≤κ\nδ.\nSince|∂ω(k·ω)(k\n|k|))|=|k| ≥1, we get, using condition (3.18),\n|∂ω(k·ω+ν1(α1−β1))(k\n|k|))| ≥1/2. (3.29)\nUsing (3.29) and Lemma 3.7, we conclude that for any fixed k\n|k·ω+ν1(α1−β1)|>κ , (3.30)\noutside a set Fk,α,βof measure ≤2d2κ(the casek= 0 being evident) so that if Fis the\nunion ofFk,α,βfor|k|<Kand asαandβvary, we have\nmeas(F)≤4K2nd2κ . (3.31)\nThus, defining D′≡D′(κ,K) =D\\F, we get for all ω∈D′, 0≤σ′<σandθ∈Tn\nσ′\n|χα,β(θ,ω)| ≤C\nκ(σ−σ′)nsup\n|ℑθ|<σ|qα,β(θ)|.\nThe estimates for the derivatives with respect to ωare obtained by differentiating the\ndefinition of χ(for more details see for instance [16]).\n3.3 Iterative lemma\nTheorem 3.4 is proved by an iterative KAM procedure. We begin with the initial Hamil-\ntonianh0+q0where\nh0(y,θ,ξ,η) =ω·y+ν0ξ1η1, (3.32)\nandq0=εq∈ Q(σ0,D0),D0= [ε,2π]n. Then we construct iteratively the change of\nvariablesθm, the normal form hm=ω·y+νmξ1η1+cm(ξ2−η2)2and the perturbation\nqm∈ Q(σm,Dm) as follows: assume that the construction is done up to step m≥0 then\n(i) usingProposition3.6 weconstruct χm+1(ω,θ),˜Nm(ω),rm+1(ω,θ)andDm+1⊂Dm\nsuch that\n{h,χm+1}=˜Nm−qm+rm+1, ω∈Dm+1, θ∈Tn\nσm+1(3.33)\nwhere 0<σm+1<σmhas to be chosen later;\n22(ii) we define hm+1=ω·y+Nm+1by\nNm+1=Nm+˜Nm, (3.34)\nand\nqm+1=rm+1+/integraldisplay1\n0{(1−t)(hm+1−hm+rm+1)+tqm,χm+1}◦Φt\nχm+1dt.(3.35)\nFor any regular Hamiltonian fwe have, using the Taylor expansion of g(t) =f◦Φt\nχm+1\nbetweent= 0 andt= 1\nf◦Φχm+1=f+{f,χm+1}+/integraldisplay1\n0(1−t){{f,χm+1},χm+1}◦Φt\nχm+1dt.\nTherefore we get for ω∈Dm+1\n(hm+qm)◦Φχm+1=hm+1+qm+1.\nFollowing the general scheme above we have for all m≥0\n(h0+q0)◦Φ1\nχ1◦···◦Φ1\nχm=hm+qm.\nAt stepmthe Fourier series are truncated at order Kmand the small divisors are\ncontrolled by κm. Now we specify the choice of all the parameters for m≥0 in term of\nεmwhich will control [ qm]Dm,σm.\nFirst we define ε0=ε,σ0=σand form≥1 we choose\nσm−1−σm=C∗σ0m−2,\nKm=2(σm−1−σm)−1lnε−1\nm−1,\nκm=ε1\n8\nm−1,\nwhere (C∗)−1= 2/summationtext\nj≥11\nj2.\nLemma 3.8. There exists ε∗>0depending on d,nsuch that, for |ε| ≤ε∗and\nεm=ε(3/2)m, m≥0,\nwe have the following:\nFor allm≥1there exist an open set Dm⊂Dm−1functionsχm,qm∈ Q(Dm,σm)and\nNmin normal form such that\n(i) The mapping\nΦm(·,ω,θ) = Φ1\nχm:C2→C2, ω∈Dm, θ∈Tσm (3.36)\nis a linear isomorphism, C1inω∈Dm, analytic in θ∈Tn\nσm, linking the Hamilto-\nnian at step m−1and the Hamiltonian at step m, i.e.\n(hm−1+qm−1)◦Φm=hm+qm,∀ω∈Dm.\n23(ii) we have the estimates\nmeas(Dm−1\\Dm)≤ε1\n9\nm−1, (3.37)\n/ba∇dbl∂j\nω(Nm(ω)−Nm−1(ω))/ba∇dbl ≤εm−1, j= 0,1, ω∈Dm, (3.38)\n[qm]σm,Dm≤εm, (3.39)\n/ba∇dblΦm(·,ω,θ)−Id/ba∇dblL(C2)≤ε1\n2\nm−1,forθ∈Tn\nσm, ω∈Dm.(3.40)\nProof.At step 1,h0=ω·y+ν0ξ1η1and thus hypothesis (3.18) is trivially satisfied and\nwe can apply Proposition 3.6 to construct χ1,N1,r1andD1such that for ω∈D1\n{h0,χ1}+q0=N1−N0+r1.\nThen, using (3.19), we have\nmeas(D\\D1)≤CK2n\n1κ1≤ε1\n9\n0\nforε=ε0small enough. Using (3.22) we have for ε0small enough\n[χ1]σ1,D1≤C1\nκ2\n1(σ0−σ1)nε0≤ε1\n2\n0.\nSimilarly using (3.21), (3.23) we have\n/ba∇dblN1−N0/ba∇dbl ≤ε0,\nand\n[r1]σ1,D1≤Cε2\n0\n(σ0−σ1)n≤ε3\n2\n0\nforε=ε0small enough.\nIn particular we deduce /ba∇dblΦ1(·,ω,θ)−Id/ba∇dblL(C2)≤ε1\n2\n0.Thus using (3.35) we get for ε0\nsmall enough\n[q1]σ1,D1≤ε3/2\n0=ε1.\nNow assume that we have verified Lemma 3.8 up to step m. We want to perform\nthe stepm+1. We have hm=ω·y+Nmand since\n/ba∇dblNm−N0/ba∇dbl ≤ /ba∇dblNm−Nm−1/ba∇dbl+···+/ba∇dblN1−N0/ba∇dbl ≤m−1/summationdisplay\nj=0εj≤2ε0,\nhypothesis (3.18) is satisfied and we can apply Proposition 3 .6 to construct Dm+1,χm+1\nandqm+1. Estimates (3.37)-(3.40) at step m+ 1 are proved as we have proved the\ncorresponding estimates at step 1.\n243.4 Transition to the limit and proof of Theorem 3.4\nLet\nEε:=∩m≥0Dm.\nIn view of (3.37), this is a Borel set satisfying\nmeas(D\\Eε)≤/summationdisplay\nm≥0ε1\n9m≤2ε1\n9\n0.\nLet us denote Ψ N(·,ω,θ) = Φ1(·,ω,θ)◦ ··· ◦ΦN(·,ω,θ). Due to (3.40) it satisfies for\nM≤Nand forω∈ Eε,θ∈Tn\nσ/2\n/ba∇dblΨN(·,ω,θ)−ΨM(·,ω,θ)/ba∇dblL(C2)≤N/summationdisplay\nm=Mε1\n2m≤2ε1\n2\nM.\nTherefore (Ψ N(·,ω,θ))Nis a Cauchy sequence in L(C2). Thus when N→ ∞the maps\nΨN(·,ω,θ) converge to a limit mapping Ψ ∞(·,ω,θ)∈ L(C2).Furthermore since the\nconvergence is uniform on ω∈ Eεandθ∈Tn\nσ/2, (ω,θ)→Ψ1\n∞(·,ω,θ) is analytic in θand\nlipschitzian in ω. Moreover,\n/ba∇dblΨ∞(·,ω,θ)−Id/ba∇dblL(C2)≤ε1\n2\n0. (3.41)\nBy construction, the map Ψ m(·,ω,ωt) transforms the original Hamiltonian\nhε(t,ξ,η) =N0(ξ,η)+εq(ωt,ξ,η), N0(ξ,η) =ν0ξ1η1\ninto\nHm(t,ξ,η) =Nm(ξ,η)+qm(ωt,ξ,η).\nWhenm→ ∞, by (3.39) we get qm→0 and by (3.38) we get Nm→Nwhere\nN≡N(ω) =N0++∞/summationdisplay\nk=1˜Nk=:ν(ω,ε)ξ1η1+c(ω,ε)(ξ2−η2). (3.42)\nFurther for all ω∈ Eεwe have using (3.38)\n/ba∇dblN(ω)−N0/ba∇dbl ≤∞/summationdisplay\nm=0εm≤2ε.\nLet us denote Ψ ∞(θ) = Ψ1\n∞(·,ω,θ). Denoting the limiting Hamiltonian h∞(ξ,η) =\nN∞(ξ,η) we have\nhε(θ,Ψ∞(θ)(ξ,η)) =h∞(ξ,η), θ∈T,(ξ,η)∈C2d, ω∈Dε.\n25Finally we show that the linear symplectomorphism Ψ ∞can be written as (3.7). To\nbegin with, write each Hamiltonian χmconstructed in the KAM iteration as\nχm(θ,ξ,η) =1\n2/parenleftbiggξ\nη/parenrightbigg\n·EcBm(θ)/parenleftbiggξ\nη/parenrightbigg\n, Ec:=/bracketleftbigg0−i\ni 0/bracketrightbigg\n, (3.43)\nwhereBm(θ) is a skew-adjoint matrix of dimension 4 ×4 of sizeεm. Then Ψ mhas the\nform\nΨm(θ,ξ,η) =eBm(θ)(ξ,η). (3.44)\nThe following lemma is proved analogously to Lemma 3.5 in [3] .\nLemma 3.9. There exists a sequence of Hamiltonian matrices Al(θ)such that\nΨ1◦...◦Ψl(θ,ξ,η) =eAl(θ)(ξ,η)∀ξ∈C2. (3.45)\nFurthermore, there exist a Hamiltonian matrix Aω(θ)such that\nlim\nl→+∞eAl(θ)=eA∞(θ),sup\n|Imθ|≤σ/2/ba∇dblAω(θ)/ba∇dbl ≤Cǫ , (3.46)\nand for each θ∈Tn,\nΨ(θ,ξ,η) =eAω(θ)(ξ,η)∀ξ∈C2.\nThis concludes the proof of Theorem 3.4.\n3.5 Changes for proving Theorem 3.5.\nThe main change needed for the proof of Theorem 3.5 rests in Pr oposition 3.6. Indeed\none has to assume the r.h.s. of (3.18) to be smaller than a smal l constant times ε2and\nthe conclusion changes in the fact that at the denominator of the r.h.s. (3.22) instead\nofκ2, one gets\nmin/braceleftbig\nε2,κ2/bracerightbig\n. (3.47)\nIn the next lines we are going to prove this version of Proposi tion 3.6.\nIndeed, the proof of Proposition 3.6 goes excatly in the same way, except that now\nthe eigenvalue (3.24) are substituted by\ni(ν1(α1−β1)+ν2(α2−β2)+ω·k) (3.48)\nwith the only selection rule\n|α−β|+|k| /\\e}atio\\slash= 0,and|k| ≤K .\nThe estimates of the small divisors are obtained exactly in t he same way as far as\n|α1−β1|+|k| /\\e}atio\\slash= 0,\nhowever when such a quantity vanishes, the modulus of the sma ll divisor becomes\n|2ν2| ≥cε2\n26(remarkthatinthiscasethenormalformcontains span {ξ1η1}). Concerningcontribution\nof the terms with this denominator to the derivative with res pect toω, remark that the\ninvolved terms are\nξ2\n2\n2iν2,−η2\n2\n2iν2\n(multiplied by a constant). Thus the derivative with respec t to, sayωiof the coefficient\nof one of these terms gives, by (3.10),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2ν2\n2∂ν2\n∂ωi/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cε−2,\nwhich proves the claimed statement.\nThen also the iterative lemma changes. Actually, only few fir st steps change, for\nwhich one has at the first step\nmin/braceleftbig\nε2,κ2\nm/bracerightbig\n=ε2;\nit is easy to see that with the choices just before the stateme nt of Lemma 3.8, after\na finite number of steps one has min/braceleftbig\nε2,κ2\nm/bracerightbig\n=κ2\nm, and therefore after this step the\niteration can be repeated exactly in the same way as in the pre vious case.\nFunding. D.BambusiandA.Masperoaresupportedby PRIN2020(2020XB3 EFL001)\n“Hamiltonian and dispersive PDEs”, PRIN 2022 (2022HSSYPN) ”TESEO - Turbulent\nEffects vs Stability in Equations from Oceanography”.\nReferences\n[1] D. Bambusi. 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Phys. ,\n277(2):459–496, 2008.\n29"    },    {        "title": "2402.00439v1.Degradation_Analysis_of_Perovskite_Solar_Cells_via_Short_Circuit_Impedance_Spectroscopy__A_case_study_on_NiOx_passivation.pdf",        "content": "1 \n Degradation Analysis of Perovskite Solar Cells via Short -\nCircuit Impedance Spectroscopy :  A case study on NiO x \npassivation  \nOsbel Almora,1,* Pilar López -Varo,2,* Renán Escalante,3 John Mohanraj,4 Luis F Marsal,1 \nSelina Olthof,4 Juan A. Anta3,* \n \n1 Departament d'Enginyeria Electrònica Elèctrica i Automàtica, Universitat Rovira i Virgili, \n43007 Tarragona, Spain  \n2 Institut Photovoltaïque d'Ile -de-France (IPVF), 91120 Palaiseau, France  \n3 Department of Physical, Chemical and Natural Systems, Universidad Pablo de Olavide, \nSevilla 41013, Spain  \n4 Department of Chemistry, University of Cologne, Greinstrasse 4−6, Cologne 50939, Germany  \n* osbel.almora@ urv.cat , pilar.lopez -varo@ipvf.fr , jaantmon@upo.es   \nAbstract  \nPerovskite solar cells (PSCs) continue to be the “front runner” technology among emerging \nphotovoltaic devices in terms of power conversion efficiency and application versatility. \nHowever, not only the stability but also the understanding of their ionic -electronic transport \nmechanisms continues to be challenging. In this work, the case study of NiO x-based inverted \nPSCs and the effect of different interface passivating treatments on the device performance are  \napproached. Our experiments include impedance spectroscopy (IS) measurements in shor t-\ncircuit under different illumination intensities and operational stability tests under constant \nillumination intensit y. It is found that certain surface treatments lead to more stable \nperformance . However,  protic anion donors can induce , both, an initial  performance decrease \nand a subsequent reactivity during light exposure which apparently improves the cells  \nperformance . Our d rift-diffusion simulations suggest that the modification of the interface with \nthe hole transport material may have decrease the conductivity, as well as  the ion and electron \nmobilities at the perovskite and the NiO x, respectively. Importantly, capacitance and resistance \nare shown to peak maximum and minimum values, respectively, around specific ranges of \nmobile ion concentration.  Our results introduce a general route for characterization of \ndegradation paths in PSCs via IS in short -circuit.  \nTOC figure: \nResistancePower conversion efficiency\nDegradation pathCapacitance\nMobile ion concentrationShort Circuit  2 \n  \n1. Introduction  \nThe optimal optoelectronic properties of metal -halide perovskites have gained major \nattention of the semiconductor device  research community during the last decade resulting in \nunprecedented progress in several fields, such as photovoltaics,[1] light emitting diodes,[2] \nlasers[3] and ionizing radiation detectors.[4] Particularly, single junction perovskite solar cells \n(PSCs) have been reported with record power conversion efficiency ( PCE ) values >25% with \nrelatively low -cost solution -based fabrications methods.  These devices are based on the \nsuperposition of sequenced thin film layers where the perovskite absorber is sandwiched \nbetween the electron and hole selective layers. In addition,  the compatibility with different \nsubstrates have pr oduced a broad range of versatile applications, for instance in \ntransparent/semitransparent and flexible photovoltaics. However, the understanding of the \nworking mechanisms of these devices is still limited . Particularly , the long -term operational \nstabilit y issues in  PSCs,  which remain a key limiting factor for upscaling and industrial \ndeployment , are still in early phases of elucidation .[5]  \nThe long-term degradation of thin film devices , such as PSCs, is a complex process  that \ndepends on several mechanism s, parameters and constitut ing elements (see Figure S1  in Section \nS1, supporting information ). Most typically, instability originates from  chemical re actions \ncreating oxidati on and/or unintended products between the pristine materials and reactant \nleftovers of each layer,[6, 7] the interfaces [8] and the air (e.g. humidity ) in operational \nconditions.[9, 10] Moreover , reactivity can be catal yzed and/or triggered by mechanical,[9] \nthermal[9-11] and bias[11-14] stress es as well as photon interaction.[5, 9, 15]  In addition, besides \nreactivity, the material s crystal structure of each layer and the interfaces can be modified, \ncreating undesired defects that reduce photon absorption and enhance charge carrier \nrecombination.[16] Among these  non-reactive sources of defect s one can f ind temperature stress \nand the migration of various species inside the layer due to either diffusion, illumination or \nelectrical stressing .[13, 14, 17, 18] Notably, not only species from different layers present within the  \ndevice may migrate increasing the leakage currents of the cell,[19] but also the ion migration of \nintrinsic halide vacancies and other charged defects[20, 21] have been demonstrated in halide \nperovskites.[22] Therefore, assessing the individual contribution of each mechanism and element \nin the device is challenging, which motivates the design of experiments where one can neglect \nsome of the degradation agents.  In this context , the use of advanced characterizat ion techniques 3 \n (beyond the routine measurement of current density -voltage ( 𝐽−𝑉)  curves) and support by \nnumerical device simulation is particularly useful .  \nThe device structure of PSCs includes a sequence of thin film layers where the intrinsic \nabsorber perovskite is sandwiched between the electron and hole transport layers, ETL and \nHTL, respectively. The conductivity of the ETL and the HTL are n - and p -type,  respectively, \nwhich define p -i-n or n -i-p structures, depending on which of the transport materials comes first \nin the direction of the incident light path. Particularly, p -i-n structures , also known as inverted \nPSCs, have reported efficiencies over 25%[23, 24] and are attractive due to their low-temperature  \nfabrication methods ,[25] and compatibility with existing industrial techniques .[26] An example of \nthis structure is schemed in Figure 1a, where non -stoichiometric nickel oxide (NiO x) is chosen \nas inorganic HTL and the organic semiconductor films comprised  of C 60 and bathocuproine \n(BCP) serve as ETL and hole blocking layers, r espectively.  The use of organic ETLs in inverted \nPSCs has been suggested  because of  the facile low-temperature synthesis with orthogonal \nsolvents and purification  methods resulting in optimal performance with relatively low \ninstability.[27] In addition, the fact that the p -i-n structure places the ETL behind the absorber \nperovskite in the direction of the light path reduces the radiative an d temperature stress es on the \norganic HTL . On the other hand, the use of NiO x is proposed considering the already \ndemonstrated PCE  values over 24% .[28, 29] \nThe direct contact between as -prepared NiO x and perovskite has proven to be problematic \nfor solar cell efficiency and stability, as for example summarized in the recent review by Cai et \nal.[30]. To circumvent this issue, t he NiO x/perovskite interface has been optimized via several \nmaterial s and fabrication methods. For instance, t he energy level alignment between NiO x and \nthe perovskite has been  optimized with the introduction o f inorganic extrinsic doping  such as \nthe case of  Wang et al.[31] who sited  Ag via a sol -gel method  and Yi et al.[32] who fabricated \nnanopatterned Zn:NiO x with an advantageous 1D nanosca le architecture and synergistic \nsubstitutional Zn doping. Alternatively, Hu et al.[33] used an organic  doping by including 4 -tert-\nbutylpyridine (tBP) as additive in the NiO x precursor solution , whereas Kang et al.[34] used 4 -\niodophenylboronic acids to modify the NiO x/perovskite layer interface, believed to be affected \nby the intrinsic defects (Ni vacancies) in the NiO x film and the I vacancies at the buried interface \nof the perovskite. Similarly, high perform ance devices were reported by Wang et al.,[29] who \nutilized a multi -fluorine organic molecule 6FPPY, to manage the buried interface of NiOx -\nbased p -i-n PSC. Notably, Pu et  al.[28] introduced a poly[bis(4phenyl)(2,4,6 -\ntrimethylphenyl)amine] (PTAA) interlayer between the NiO x and the perovskite, resulting in 4 \n high efficiency and stability. Furthermore, Shen et al.[35] propose d that the hydrophilic chain of \nthe amphipathic molecule Triton X100  can coordinate as a Lewis additive with the Ni3+ on the \nNiO x surface, passivating interfacial defects and hindering the detrimental reactions at the \nNiO x/perovskite interface.  \nFrom the device operation point of view, the degradation of a solar cell is common ly \nassociated with the decrease of the 𝑃𝐶𝐸 , as measured from the current density -voltage ( 𝐽−𝑉) \ncurve under standard 1 sun illumination  incident power density ( 𝑃𝑖𝑛). The 𝑃𝐶𝐸  value can be \nexpressed in terms of the complementary performance parameters as  \n𝑃𝐶𝐸 =𝑉𝑜𝑐𝐽𝑠𝑐𝐹𝐹 \n𝑃𝑖𝑛 (1) \nwhere 𝑉𝑜𝑐 is the open -circuit,  𝐽𝑠𝑐 is the short -circuit current , and FF the fill factor. Accordingly, \na decrease  in PCE  can be found in terms of the decrease  of one or up to three performance \nparameters (see Figure S2 ) in overlapping . In practice, 𝐹𝐹 and 𝑉𝑜𝑐 modifications are more likely \nto be connected since these parameters relate to the near-flat-band condition  where charge \ncarrier recombination rate approaches that  of the photogeneration.  Note that under illumination  \nintensities close to and above 1 sun standard , high -performance devices  are typically  expected \nto achieve a maximum power point  (MPP) voltage 𝑉𝑚𝑝𝑝<𝑉𝑜𝑐 close to the built -in voltage ( 𝑉𝑏𝑖). \nIn contrast, the lower the illumination intensity, the farther the values of 𝑉𝑚𝑝𝑝 and 𝑉𝑜𝑐 from 𝑉𝑏𝑖. \nAround flat -band (V=𝑉𝑏𝑖), the energy diagram is expected as shown by the  simulation in Figure \n1b for the device structure in Figure 1a. This is a regime where the diffusion current is similar \nto, if not larger than, drift current due to the  low electric field  (see Figure 1d), resulting in large  \ngradients in the charge carrier density profiles (see Figure 1e). Diffusio n-related l ong-term \nrelaxation s may overlap with degradation processes , hindering dedicated investigations \nregarding device stability .  Moreover, the illumination intensity is not expected to produce a big \nchange in  the energy diagram around flat-band  (dashed lines close to solid lines in Figure 1) \nbecause the slower electrical response hinders  the separation between injected and \nphotogenerated exce ss charge carriers , which recombine mostly radiatively . In addition , the \ndual electronic -ionic conductivity of metal halide perovskites not only hinders the evaluation of \nthe MPP, which defines the 𝐹𝐹, but also  complicates the transport by adding field s creening by \nions to a regime where diffusion transport would otherwise be more dominant.[4, 36] The field \nscreening effect can be seen in Figure 1d towards the interfaces of the perovskite  and the mobile \nions could distribute along the entire perovskite bulk, as shown in Figure 1e. Accordingly, it 5 \n may be convenient to explore another biasing regime for studying stability.  \nAt short -circuit ( 𝑉=0 V) , the electric field in the perovskite bulk is significantly higher and \nsensitive to the illumination intensity (see Figure 1d). Under illumination,  this is a regime where \nthe drift current and nonradiative recombination towards the interf aces are predominant due to \nthe band leaning (see Figure 1c) and the lower concentrations of charge carriers located within \nthe perovskite near the transport selective layers (see Figure 1f).  In addition, the short -circuit \nand reverse bias regimes are commonly used for the characterization of the shunt resistance, \nwhich is a parameter typically affected during long -term operation. Interestingly, PSCs have \nbeen found to exhibit linear  photo -shunt  resistance  (see Table S1 )[37] and photo -capacitive[38] \nincrease  at short -circuit , which are properties characterized via impedance spectroscopy (IS) . \nNotably, the use of IS analyses at short -circuit is preferrable  due to the higher linearity of the \nsignal  response , compared to near flat -band situation . Moreover,  unlike t he IS studies in quasi -\nopen -circuit condition,[39, 40] little attention have been paid in the literature to the IS spectra in \nshort -circuit  for PSCs.  Table S2  in the supporting information summarizes th e comparison \nbetween short -circuit  and near flat -band regime s, from which one can assess the suitability of \neach condition for designing experiments.  \n-10123\n100 200 300 4000204060\n100 200 300 40010141015101610171018\n100 200 300 400Short-circuit\n(a) EC  EV  EFn  EFpEnergy (eV)(b)Flat-band\n(c)\na)BCP\nMAPbI3Ag\nNiOxC60\nFTO\nGlass\nIlluminationAbsolute electric field (kV cm-1)\nPosition (nm) V=Vbi=1.2 V  (flat-band)\n V=0 V (short-circuit)(d) n  p  a  cCharge density (cm-3)\nPosition (x)(e) (f)\nPosition (x)\n \nFigure 1. Simulated steady -state electrical response of the reference PSCs with the structure \nin (a) in terms of energy diagrams ( b, c), electric field (d) and charge density profiles ( e, f). \nThe depicted conditions include flat -band (b, e) , short -circuit ( e, f), 1 sun illumi nation (solid \nlines) and dark (dashed lines) . In (b, c), 𝐸𝐶 and 𝐸𝑉 are the conduction band minimum and \nvalence band maximum  energy levels, respectively; and 𝐸𝐹𝑛 and 𝐸𝐹𝑝 are the quasi -Fermi \nlevels for electrons and holes, respectively. In (d), 𝑉𝑏𝑖 is the built -in voltage . In (e, f), the \ndensities are 𝑛, 𝑝, 𝑎, and 𝑐 for electrons, holes, mobile anions and mobile cations, \nrespectively. Setfos -Fluxim[41] was used for these simulations  using the parameters enclosed 6 \n in Table S 9. \nIn this work, the stability of passivated -NiO x PSCs is analyzed with focus on the HTL -\nperovskite interface and the evolution of electrical properties. In a separate publication these \npassivating surface treatments of NiOx have been studied with respect  to film formation, \ninterface composition, as well as solar cell initial performance and stability. [Ref]  In contrast, \nthe focus here is set on the modeling and understanding of fundamental transport roperties. \nExperimentally, the current evolution over ti me under high power condition was sensed for \nseveral passivation methods targeting the interface between the NiO x and MAPbI 3 absorber \nlayer. Moreover,  impedance spectra were measured over time and under different illumination \nintensities  at short -circuit  in each case . The  results were contrasted with equivalent circuit \nmodeling and numerical simulations on Setfos Fluxim[41] and Driftfusion[42] transport equation \nsolvers.   Our results correlate different modification in the electron and mobile ion \nconcentrations and mobilities with the degradation observed in the samples.  \n2. Methods  \nThe fabrication details for the studied devices with structure \nFTO/NiO x/MAPbI 3/C60/BCP/Ag (see Figure 1a) can be found in our simultaneous article [ref]. \nAn adapted version is presented in Section S2  (supporting information)  along with illustrative  \ninitial PCE  characterization  (see Figure S4  and Table S3 ).    For the passivation, the selected \nmaterials distribute into three categories attending to the po tential conductivity effect , as \ndiscussed in more detail in reference [ref]. Firstly, PbI 2 as cation donor which is a metallic salt \nwith divalent Pb cation that may potentially occupy Ni2+ vacant sites, and corresponding anions \nto interact with high valent  (> 2+)  Ni centers. Secondly, 1 -Iodobutane (C 4H9I, also labeled here \nas iodobutane ) and 1 -Phenylethylamine (C 8H11N, also labeled here as amine ) neutral bases, \nwhich are Lewis bases, i.e. neutral organic molecules with a capacity to donate electron  pairs to \nthe Ni atoms with higher formal charge (>2+) . Thirdly, HI and MAI protic anion donors , i.e. \nBronsted acids, which are  organic molecules that could donate protons (or accept electron pairs ) \nto O atoms with increased formal charge due to Ni2+ vacancies (O -defect sites) and anions to \ninteract with Ni>2+.  \nThe IS data were measured with an Autolab PGSTAT302N potentiostat including a \nFRA32M unit and a kit Autolab Optical Bench from MetroOhm. Th e samples were illuminated \nwith a white LED (CREE XM -L3 U4 on star PCB, XMLDWT -U40E1) at different steady -state \nillumination intensities, then the short -circuit  current was stabilized before applying 15 mV of 7 \n alternating current (AC) mode perturbation . The  details for the IS characterization under \ndifferent illumination intensities are in Section S3 . The stability assessment was carried out \nunder 0. 2 sun equivalent illumination intensity, as measured with a calibrated reference cell \n91150 -112/PVM 164 from Newport.  The samples were under continuous N 2 flux, at room \ntemperature, and were kept in operation at a forward bias close MPP under illumination with \nsystematic switch ing to short -circuit  for IS  measurements  (see Figure S16 ). Section S4  provides \nfurther detail s on the operational stability test procedure and current and IS data.  \nThe equivalent circuit modeling was carried out with ZView from Scribners. Details of \neach equivalent circuits and fitting parameters can be found in Sections S3 and S4. SETFOS  \nFluxim[41] software and MATLABS’s Driftfusion[42] code were used f or the numerical  \nsimulatio ns of the time -dependent solutions of the transport equations with electronic and \nmobile ion charge carriers . The corresponding details can be found in Sections S5 and S6, in \nthe supporting information.   \n3. Results and discussion  \nThe different NiOx  surface treatments employed in the various samples in this study \nresulted in two main behaviours, as seen in the solar cell characteristics presented  in Figure S4. \nOn the one hand, the NiO x surface passivation with 1-Iodobutane, 1 -Phenylethylamine and PbI 2 \nimproved the performance mostly due to a slight increase of the photocurrent. The resultant \nPCE values ranged around 13% for the un -passivated reference sample (NiO x Ref.) and nearly \n15% for the passivated samples , in line with state -of-the-art pure MAPb I3 NiO x-based PSCs .[43, \n44] This suggests that the treatment with  these passivation agents could reduce electrical losses \ndue to charge carrier recombination at the interface. Nevertheless, these passivation processes \ncould also modify the morphology and optoelectronic properties of the perovskite layers  \nresulting in fu rther effects such as the reduction of optical losses due to interference and the \ndecrease of bulk recombination in the perovskite. On the other han d, the MAI and HI passivated \nsamples showed a significant PCE reduction to less than 6% due to a remarkable decrease of \nthe fill factor and the photocurrent.  The absence of rectifying behavior in these samples indicate \nmajor modifications of the charge carrier density profiles, large parasitic resistive effects and \nlarge recombination rates.  \nFor the understandi ng of these two different behaviors, complementary experiments were \nperformed with a focus on the short -circuit  condition, the different illumination intensities and \nthe IS analysis. This is presented in Section 3.1 where the equivalent circuit modeling is used 8 \n to estimate resistive, capacitive and characteristic response times. Subsequently, the operational \nstability and the evolution of the IS spec tra over time under illumination is presented in Section \n3.2.  \n Different illumination intensities  in short -circuit . \nThe impedance spectra under different illumination intensities were measured in short -\ncircuit condition for representative samples of the studied device set with different passivation \ntreatment on the NiO x interface with the perovskite , as described in Section S3 of the supporting \ninformation and summarized in Figure 2. The characteristic two -arcs spectra in the impedance \nNyquist plots are present in th e reference and optimized samples (amine, iodobutane, PbI 2), as \nindicated in Figure 2a. Typically, the high - and low -frequency semicircles relate to el ectronic \nand ionic -electronic resistance ( R)-capacitance ( C) contributions, respectively. In the \ncapacitance Bode plot representation, two plateaus can be identified  (see Figure 2b), where the \nnearly constant values toward high -frequency (Hf, >10 kHz) are related with the geometrical \ndielectric capacitance[39] and the capacitance step-like increase toward low -frequency (Lf, <10 \nHz) has been associated with electrode polarization and mobile ion accumulation at the \ninterfaces.[40]  However the HI and MAI samples not only demonstrate  smaller resistances in \nSC, but als o show three arcs in the impedance Nyquist representation (see Figure 2a) and a \ngradual capacitance increase in the capacitance Bode plot (see Figure 2b). Notably, the higher \nthe illumination intensity, the clearer the definition of the three arcs  in the Nyquist plot . Then, \na transition from a “two (larger)” to a “three (smaller)” arcs situation complicates the “high -\nfrequency electronic” versus “low -frequency  ionic” interpretation of the IS spectra.  \nThe equivalent circuit models used for parameterization of the IS spectra are shown in \nFigure 2c,d for the two- and three -𝑅𝐶 constants, respectively. Particularly , for the reference \nand optimized samples two resistors and two capacitors were fitted for the high - and low -\nfrequency features of the spectra as RHf, RLf, CHf and CLf, respectively.  In contrast, f or the MAI \nand HI treated samples an additional set of medium -frequency resistor ( RMf) and capacitor ( CMf) \nwere considered. The resultant values for the resistances as a function of the short -circuit current \nunder different i llumination intensities are summarized in Figure 2e. First, in general, all the \nsamples follow a similar trend where a saturation is observed towards dark condition and a \ndecrease 𝑅∝𝐽𝑠𝑐−1 follows as the illumination intensity increases. This is a common trend in \nphotovoltaic solar cells (see Table S1) due to the photoconductivity increase upon charge carrier \ngeneration.  Second, the low -frequency resistance is always higher than that of the high -\nfrequency resistance . This may suggest that in short -circuit  condition the ion migration and/or 9 \n ion-related charge carrier transport is significantly hindered in comparison to the faster “pure ” \nelectronic response.  Third, the higher the short -circuit resistance the higher the device \noperational performance. This behavior  can be directly related with the shunt resistance (Rsh) of \nthe samples which mostly affects the fill factor and , ultimately , the photovoltage. Interestingly, \nFigure 2e highlights one of the advantages of the IS over the 𝐽−𝑉 curve for the estimation of \nthe Rsh. Not only hyster esis issues are avoided but also smaller resistance contributions are \nidentified, which would not be measured  in series connected added resistances with direct \ncurrent (DC) mode methods. In addition, the fact that low -frequency resistance s saturate in dark  \nsuggests that ion migration is occurring in the grain boundaries and surface defects , in addition \nto the bulk perovskite .[45]  \nThe low -frequency capacitance linear  increase  as a function of short -circuit current 𝐶𝐿𝑓∝\n𝐽𝑠𝑐 for different illumination intensit ies is illustrated  in Figure 2f, whereas the constant high -\nfrequency can be found in Section S7. This increase of low -frequency capacitance in short \ncircuit is a characteristic feature of PSCs[38] due to mobile ion accumulation toward the \ninterfaces (see  Figure 1f). Interestingly, the reference and surface -treated (amine, iodobutane, \nPbI 2) samples show a threshold illumination value for 𝐽𝑠𝑐≈10-5 A·cm-2 which indicates a \ntransition between a nearly constant behavior and the linear  trend. In contrast, the HI and MAI \nsamples show a gradual increase of CLf without an apparent baseline capacitance in the explored \nrange of illumination intensity.  Furthermore, by coupling the capacitors and the correspondi ng \ndischarge resistors, the characteristic short -circuit  charge carrier response times 𝜏=𝑅𝐶 can be \naccessed, as presented in Figure 2g for the high-frequency times. The low -frequency , and \nslower,  response times can be found in Section S3 showing nearly linear or slightly increasing \ntrends with the augment of illumination intensity. On the other hand, the fastest high -frequency \nresponse times evol ve from saturated maximum values up to linear decreasing trends as the \nillumination intensity is augmented. This can be interpreted as a transition of main charge carrier \nrecombination mechanism between non -radiative and radiative, when dark and under \nillumination , respectively. Notably, the higher the response time the higher the operational \nperformance of the samples in terms of PCE.   10 \n \n1 1k 1M100n10µ\n1µ 1m1k1M\n1µ 1m100n10µ10-710-610-510-410-30.010.11101001000\n1µ 1m1m10 250 500 7500250500750\n(d)(a)\n NiOx Ref. \n Iodobutane\n HI-Z'' (W cm2)\nZ' (W cm2)(c)\nDarkC (F cm-2)\nf (Hz)(b)\nLight \nLf\nHfR (W cm2)\nJsc (A cm-2)(e)\nC (F cm-2)\nJsc (A cm-2)(f)\nCHf\nRMfRLfCLf\nRHfCMf\nCHf\nRHfRLfCLf\nt (s)\nJsc (A cm-2)(g) \nFigure 2. Experimental selected IS spectra under different illumination intensities in short -\ncircuit  condition ( V=0V) for illustrative samples  with different passivation procedures , as \nindicated, and equivalent circuit parameterization. I n (a) there are impedance Nyquist plots \nunder 0. 2 sun equivalent LED light and (b) shows the capacitance Bode plots  in perspective \nwith the dark spectra (darker dots) . Solid lines in (a) and (b) are the fitting to the equivalent \ncircuits in (c) and (d), in each case. The corresponding (e) resistance, (f)  capacitance (low-\nfrequency) and (g) a characteristic response time  (high frequency) are plotted in the lower \npanel with darker and lighter dot fill colours for high and low frequency related components, \nrespectively. The lines in (e -g) are empirical fitti ngs introduced in Section S3.  \nThe experimental evidence shown in Figure 2 and Section S3 with IS spectra over a range \nof illumination intensities hig hlights the paramount importance of the HTL/perovskite interface \nfor the device operation. Nevertheless, several aspects  require further understanding. For \ninstance , the role of 1 -iodobutane, 1 -phenylethylamine and PbI 2 treatments appears to be \npassivating surface defects and improving the charge extraction by hindering recombination. \nHowever, it is unclear how the MAI and HI treatments deteriorate the rectifying barrier , \nreducing  the overall resistance and showing the c haracteristic “three arcs ” in the impedance \nNyquist plots. Furthermore , the fact that a large low -frequency impedance arc evolves into two \nsmaller arcs in the Nyquist plot could be related with  ion migration and the modification of the  \nnature or concentrat ion of the mobile ions and electrons in the perovskite  [ref]. \n Constant illumination over operational time   \nThe operational stability of the samples was studied over time under constant illumination \nintensity , at room temperature , and with nitrogen flux cir culation . The stability tests procedure 11 \n is schemed and described in detail in Section S4 . The sample was exposed to constant 0.2 sun \nwhite LED equivalent illumination intensity , next a voltage Vhpp close to the MPP was applied \nand the current was recorded for measuring the output power ( Pout). Subsequently, the biasing \nwas switched to short -circuit  and IS spectra were acquired in each iteration of the loop. Th is \nprocess was continued until the sampl es broke , the duration ranged from 20 h to 50 h . The \noverall Pout performance of the  devices  is depicted in Figure 3a, where two main trends are \nobser ved. For the reference and surface -optimized (amine, iodobutane, PbI 2) samples, a rather \nstable Pout with a slight increase is detected. On the other hand, the HI - and MAI -treated cells \nexperienced a large increase of Pout with an apparent extrapolated saturation around 300% of  \nthe initial value. Nevertheless, despite the final  tripled  Pout values for the HI and MAI samples, \nthe performance of the reference and optimized cells was still 30% to 50% better, as summarized \nin Figure 3b and Figure S17. When comparing final versus initial Pout values  in this plot , the \ncloser the sample is to the “x=y” diagonal (dashed line in Figure 3b), the higher the stability. \nAlternatively, considering the different initial performance and test duration of each sample, \none can also use the concept of effective or overall  degradation rate[1] \n𝐷𝑅=𝑃𝑜𝑢𝑡 ,𝑓𝑖𝑛𝑎𝑙 −𝑃𝑜𝑢𝑡 ,𝑖𝑛𝑖𝑡𝑖𝑎𝑙   \n𝑡𝑆𝑇𝐷 (2) \nwhere 𝑃𝑜𝑢𝑡 ,𝑖𝑛𝑖𝑡𝑖𝑎𝑙  and 𝑃𝑜𝑢𝑡 ,𝑓𝑖𝑛𝑎𝑙  are the initial and final output powers for the stability test of \nduration 𝑡𝑆𝑇𝐷. In the definition of equation (2), the smaller the value of DR the better . Moreover, \ndespite the  typical decrease of per formance would entail DR<0, the report of DR>0 during the \nfirst 200 hours of operation under constant illumination is well -known in the literature,[1] and \ntypically precedes a subsequent performance decrease period with DR<0. In Figure 3b (right \naxis) the studied samples show DR>0 with the highest and lowest values for the HI and the PbI 2. \nsample s, respectively. Interestingly, the reference sample resulted in a DR val ue close to that of \nthe PbI 2-treated device . This latter behavior mismatch es that of  previous MPP tracking \nexperiments [ref] where the reference device was more  instab le. Nevertheless , in that study  [ref] \nthere were not  intermittent short -circuit periods for current stabilization and IS measurements. \nTherefore, one can hypothesize  that the short -circuit and near -flat-band hinders and favors the \ndegradation of the untreated sample, respectively; and the opposite can be tr ue for the PbI 2-\ntreated samples.        \nThe IS spectra in short -circuit  condition were obtained along the above -described \noperational stability tests and the results are depicted in Figure 3c-e (see also Section S4). \nAnalogously to the previous discussion, two main trends were observed. For the reference and 12 \n optimized samples, a relatively stable evolution of the IS spectra was found. Figure 3c zooms \nthe high -frequency region in the impedance Nyquist plots for the reference sample, which \nslightly decrease the high -frequency resistance over time durin g the degradation test. This \nhighlights the accuracy  of the IS experiments, even though the main contribution  to resistance \nand device performance comes from the larger low -frequency range of the spectra. In Figure \n3d (see also Figure S18), the complete Nyquist plots for the PbI 2-treated sample illustrate the \napparent constancy of the device over time of degradation test. At th e same time , the spectra \nare also shown similar ly unchanged in the impedance and capacitance Bode plots in figures S19 \nand S20, respectively. This steadiness of the reference and optimized samples in short -circuit  \nsuggest s the stability of the device s shun t resistance which agrees  with the slight increase of the \noutput power shown in Figure 3a. Furthermore, it is implie d that any significant improvement \nof the device operation performance should be mostly related with the modification of optical \nabsorption and/or the improvement of the transport properties which affect s the near -flat-band \nregime.   \nThe HI and MAI cells show an unstable behavior in the IS spectra over time under \nconstant illumination, as in Figure 3e-f (see also Figure S18-20). Not only the impedance is \nsmalle r in these samples with respect to the re ference  and optimized ones, but also a clear \nincrease is observed. The longer the operation time under constant illumination, the higher the \nimpedance of the sample  which correlate s with the performance improvement in Figure 3a. \nInterestingly, the three -similar -arcs feature in the Nyquist plot evolve s into the more typical \ntwo-arcs shape where the high frequency arc is significantly smaller than that of the low -\nfrequency  range. Once more, the qualitative change of the low frequency part of the IS spectra \nsuggests that the nature and/or quantity of the mobile ion properties may be modified by the \naction o f the current flow of current under continuous illumination.  \n0 50 100100200300 Ref.\n Amine\n Iodobutane\n PbI2\n MAI\n HINormalized output power (%)\nNormalized stability test time (%)(a)\n   \n MAI\n HI Ref.\n Amine\n Iodobutane\n PbI2\n0.5 1.0 1.5 2.00.51.01.52.02.53.0Final output power (mW cm-2)\nInitial output power (mW cm-2)(b)\n102030405060\nDegradation rate ( mWcm-2h-1)13 \n \n0.0 0.5 1.0 1.50.00.51.01.5\nAged-Z´´ (kW cm2)\nZ´ (kW cm2)Ref.(c)\nFresh  \n0 10 20 30 40010203040\n0.0 0.5 1.00.00.51.0-Z'' (k W cm2)\nZ' (kW cm2)PbI2(d)\n-Z'' (k W cm2)\nZ' (kW cm2)\n0 100 200 300 4000100200300400\nFresh-Z'' (W cm2)\nZ' (W cm2)HI\nAged(e)\n1 1k 1M100101102103AgedMAI-Z'' (W cm2)\nf (Hz)(f)\nFresh\n \n \nFigure 3. Experimental stability test under 0.2 sun white LED equivalent illumination \nintensity. The normalized time evolution of output power in (a) is summarized in (b) with the  \nfinal values (left axis, filled symbols) and degradation r ates (right axis, open symbols)  as a \nfunction of the initial output power  for each sample, as indicated . Illustrative impedance \nspectra in short -circuit under constant illumination over operation time are in Nyquist \nrepresentation in (c-e) and Bode plots ( f) for different samples, as indicated . Dots and lines \nare the experimental data and simulated spectra from fitting to equivalent circuit models in \nFigure S 5, respectively. The dashed line in (b) indicate s the x=y coordinates. In (c -f), the \nlighter the colour the longer the operational time  in the IS spectra ; and the inset in (d) \nmagnifies  the high frequency part of the spectra .  \nThe instability of the MAI and HI sample indicates the presence of intrinsic reactivity , \nwhich could be understood in terms o f the creation of a non -stoichiometric excess of MA, I or \nH that modifies the local composition of the perovskite in contact with the NiO x interface . It \nmay appear counter -intuitive that these  excess reagents contribute to performance improvement \nrather th an further  degradation. However, one must bear in mind that intrinsic reactivit y \ntransform s the material composition and properties regardless of whether it contributes to or \nhinders the final device operational performance. In addition, the chemical “soft ness” of the \nperovskite under biasing is herein highlighted since only 0.2 sun white LED equivalent \nillumination intensit y with room temperatures (<40°C) have been sufficient  to catalyze  the 14 \n modification of the samples , when compared with previous studies  [ref] on unbiased samples \ncomprising MAPbI 3 on treated NiO x films .   \nEvidence that the use of non -stoichiometric formulations using volatile precursors of \nPSCs modifies the operational stability of the PSC[46] which  create an excess of mobile ions \nalready exists in the literature. For instance, Lammar et al.[18] found that the excess of a protic \nprecursor FAI, linked to a larger concentration of mobile ionic defects, also  leads to an initial \nincrease of the PCE under illumination, due to ionic redistribution. HI and MAI are not only \nmore acidic but also smaller than the rest of the studied additives and, consequently, more likely \nto infiltrate in the perovskite lattice and  create ionic defects  [ref].  \n  Drift -diffusion simulations  \nThe experiments described in sections 3.1 and 3.2 illustrate the effects of using different \ntreatments to the surface of the NiO x HTL before subsequent deposition of the absorber \nperovskite layer. Two main trends are found. On the one hand, 1-iodobutane , 1-\nphenylethylamine  and PbI 2 treatments result in similar, if not better, device performance and \nstability when compared with the reference untreated samples. On the other hand, MAI - and \nHI-treated s amples not only show very low fill factors in the J-V curves and anomalous three -\narcs low impedance IS spectra in the Nyquist plots, but also behave unstable under constant \nillumination in operation over time.  These observations rise several questions on the nature of \nthe material modifications and the corresponding implications for the transport properties. \nAccordingly, this section present s a series of numerical simulation s which are  firstly validate d \nby qualitatively reproducing the experimental behavio rs and subsequently explore the possible \norigins and consequences of the observed  phenomena. The simulation parameters and detailed \ndescription of the simulated spectra can be found in Section S5 and S6 for the simulations made \nwith SETFOS -Fluxim  software[41] and MATLAB ’s Driftfusion  code.[42] The use of two \nsimulators not only provide further validation to the a nalyses but also allows to assess the impact \nof the different  boundary conditions to solve similar problems.   \nThe initial current -voltage curves under standard 1 sun illumination were qualitatively \nfitted to match the DC electrical response before simulation of the IS spectra. This is shown in  \nFigure 4a,b (see also  Sections S5.1 and S6.1 ) within a range of mobile ion concentration \nbetween 1015 and 1019 cm-3, which include s the minimum defect concentration measurable with \nIS methods and the effective density of states at the bands .[47] Notably, in order to reproduce the \n𝐽−𝑉 response from the HI - and MAI -treated samples , three main parameters were modified 15 \n with respect to those used to simulate the reference and the optimized samples.  First, a change \nof absorptivity was considered  by reducing the charge carrier generation rate from Gref=3.3×1021 \ncm-3 to GHI = 2.2×1021 cm-3, mostly accounting for the 𝐽𝑠𝑐 difference between the reference and \nthe HI samples, respectively . Second, an increase of series resistance from 2.5 up to 25 Ω ·cm2 \nwas taken for the HI sample that directly diminishes the 𝐹𝐹 from around 70% , for the reference \nand optimized cells , to less than 40% for the MAI and HI devices. Third, the mobility of electron \nand hole charge carriers was reduced at the NiO x HTL in the HI sample simulations to a tenth \nof those used in the reference cell. The modification of this para meter can be correlated with \nthe modification of the HTL during the surface treatment, resulting in  the decrease  of the 𝐹𝐹 in \nthe 𝐽−𝑉 characteristic. Moreover, the modification of the electron and hole mobilities in the \nperovskite layer has also been f ound to alter the shape of the 𝐽−𝑉 curve accounting for some \nof the above -described  lessening in the 𝐽𝑠𝑐 and the 𝐹𝐹.  \nThe increase of mobile ion concentration (𝑁𝑖𝑜𝑛) decreases the 𝐽𝑠𝑐 of the simulated 𝐽−𝑉 \ncurves in Figure 4a,b, and the 𝐹𝐹 also appears to be inversely proportional to 𝑁𝑖𝑜𝑛. The reason \nfor this can be associated to the electric field screening  due to ion space charge regions toward \nthe electrode. The higher the 𝑁𝑖𝑜𝑛 the smaller the effective built -in field of the junction, which \ndiminishes the drift current producing  photocurrent around the short -circuit condition  (see \nFigure S24).[18] On the other hand, the closer to the flat -band condition, which for these samples \napproaches the open -circuit voltage ( 𝑉𝑜𝑐), the smaller the effect of 𝑁𝑖𝑜𝑛 resulting in very similar \ncurrents for the same bias. Around flat -band condition , the diffusion current  takes over as the \nmain transport mechanism, which is no longer that dependent on the electric field shielding \neffect of the mobile ions.   Notably, the curves of Figure 4a,b can be considered quasi -steady -\nstate solutions since little hysteresis effect was obtained in the simulations (see Section S6.1)  \nand the broad range of 𝑁𝑖𝑜𝑛 values that reproduce similar  𝐽−𝑉 shapes suggest that further \nevidence should be considered.  \nThe simulated IS spectra in short -circuit under constant 0.2 sun illumination intensity are \npresented in Figure 4c,d for the reference and HI samples, respectively. A common feature for \nboth device types  is that the impedance does not behave smoothly with the increase of 𝑁𝑖𝑜𝑛. \nThe highest impedance arcs in the Nyquist plot representation are obtained for  the lowest and \nhighest values of 𝑁𝑖𝑜𝑛 whereas  a minimum impedance is found for intermediate va lues. For \n𝑁𝑖𝑜𝑛<1015 cm-3 the electric field screening is not sufficient to shield the built -in field and the \nconduction is hindered. On the other hand, for 𝑁𝑖𝑜𝑛>1018 cm-3 the excess mobile ions behave \nlike a doping concentration that no longer shield but maximize the built -in field. In an 16 \n intermediate range 1015<𝑁𝑖𝑜𝑛<1018 cm-3 the charge density profile of mobile ions is modified, \nwhich favors diffusion current and thus conductivity. This increase of diffusion current \nenhances the associated low  frequency capacitance, which behaves opposite to the impedance: \nminimum capacitance values are found for the lowest and the highest 𝑁𝑖𝑜𝑛 values (see Section \nS6). \nFor the IS spectra of the reference sample  in Figure 4c, our simulations suggest mobile \nion concentrations in the range 1 017<𝑁𝑖𝑜𝑛<1018 cm-3. This is concluded from the proportion \nbetween the low -frequency and high -frequency arcs of the Nyquist plots. Figure 2d illustrate s \nthe experimental behavior of the IS spectra after equivalent circuit parameterization resulting in  \n𝑅𝐿𝑓/𝑅𝐻𝑓>10. Therefore , the simulations with a low -frequency semicircle smaller than that of \nthe high frequency range in the Nyquist plot would not relate to the studied samples.   Moreover, \nthe IS spectra of the HI sample in Figure 4d is complicated by the appearance  of the \ncharacteristic three -arcs feature in the Nyquist plot representation. Particularly, the closest \nqualitative resemblance between simulations and the experiments (see Figure 2e) was obtained \nfor 𝑁𝑖𝑜𝑛~ 1016 cm-3. This may suggest that the absolute concentration of mobile ions had \ndecreased in the  HI and MAI  samples, with respect to the reference sample.  However, when \nconsidering the stability data in Figure 3 and the IS spectra in Figure 4d, one can correlate the \nreactivity over time of operation under constant illumination with the increase of concentration \nof mobile ions in the perovskite layer. More importantly, an increase of the mobile ion mobi lity \nis inferred from the following simulations, which may account as a major contribution.  \nA narrow range of mobile ion concentrations has been found key to reproduce the \nexperimental trends. However, the multi -dependance of the simulated 𝐽−𝑉 curves and  IS \nspectra on several parameters should be considered in detail. For instance, in sections S5.2 and \nS6.2, our simulations of IS spectra as a function of illumination intensity nicely reproduce the \n𝑅𝐿𝑓∝𝑃𝑖𝑛−1 and 𝐶𝐿𝑓∝𝑃𝑖𝑛 trends which collapse un til 𝑅𝑠ℎ limits the transport in the \nexperiments (see Figure 2 and Section S3). Yet, spurious results with 𝑅𝐿𝑓/𝑅𝐻𝑓<1 can be found \nwhen simulating IS spectra with low concentration of mobile ions and shunt resistance. \nPurposely, the effect of shunt resistance is analyzed in sections S5.3 and S6.3 where an \nexclusively resistive effect is identified. Shunt resistance do not modify the capacitance of the \nsample but instead alters the 𝐹𝐹 of the 𝐽−𝑉 curves and the 𝑅𝐿𝑓/𝑅𝐻𝑓 ratio of the IS spectra for \nintermediate ranges of ion concentrations. Interestingly, the mobile ion concentration and shunt \nresistance modify 𝑅𝐿𝑓 and 𝑅𝐻𝑓 simultaneously, whereas the ion mobility (𝜇𝑖𝑜𝑛) of the \nperovskite only changes the low -frequency part of the spectra (see section S5.4).[48]  17 \n  \n0.0 0.2 0.4 0.6 0.8 1.0 1.205101520\nSimulation with Nion (cm-3) \n  1015\n  1018\n   1019 \nExperiment \n  -J (mA cm-2)\nV (V)(a)Reference\n0.0 0.2 0.4 0.6 0.8 1.0 1.2051015\nSimulation for Nion (cm-3)\n   1015\n   1018\n   1019\nExperiment\n -J (mA cm-2)\nV (V)(b)HI\n0 25 50 750255075Nion (cm-3)\n 1019 \n 1018 \n 1017 \n 1016 \n 1015 -Z'' (kW cm2)\nZ' (kW cm2)(c)\n0 1 201-Z'' (kW cm2)\nZ' (kW cm2)\n0 100 200 300 4000100200300400 -Z'' (W cm2)\nZ' (W cm2)(d)Nion (cm-3)\n 1019 \n 1018 \n 1017 \n 2´1016 \n 1.5´1016 \n 1016 \n 1015 0 100 200 300050\n \nFigure 4. Simulated (a,b) current -voltage under 1 sun illumination intensity and (c, d) \nimpedance spectra in short -circuit under 0.2 sun illumination intensity for (a,c) the reference \nand (b,d) the HI devices considering different mobile ion concentrations, as indicated.  The \nsimulation were done with MATLAB’s Driftfusion[42] code  with details as described in \nSection S6 . \nCharge carrier recombination is also closely linked with the mobile ion concentration and \nthe electrical response of the samples . However, care must be taken with the model employed \nfor the simu lations. For instance, the Setfos -software by Fluxim utilizes the recombination 18 \n velocity at the interface between the NiO x HTL and the perovskite. With this model, Section \nS5.5 shows an impedance decrease with the augment of the recombination velocity with a more \nsignificant modification of the low -frequency part of the spectrum. Since a large contribution \nto the current thro ughout the device in short -circuit is due to recombination, the higher the \nsurface recombination velocity  (𝑣𝑠), the higher the leakage current and the smaller the \nimpedance. Alternatively, the MATLAB’s code Driftfusion  simulates the interfaces with \nnarro w interlayers where a surface recombination lifetime 𝜏𝑠 is set, as in section S6.5. Our \nsimulations with this approach show a nalogous reduction of the impedance with the decrease \nof the 𝜏𝑠, but a  clear  dependency on 𝑅𝑠ℎ and 𝑁𝑖𝑜𝑛 is also identifie d. For instance, the capacitance \nalways peaks a maximum for intermediate values of 𝑁𝑖𝑜𝑛, regardless of 𝑅𝑠ℎ or 𝜏𝑠. Similarly, \nwhen 𝑅𝑠ℎ is sufficiently small that controls the transport in SC, the total impedance only varies \nthe 𝑅𝐿𝑓/𝑅𝐻𝑓 ratio and behaves steady  regardless of 𝑁𝑖𝑜𝑛 or 𝜏𝑠. Moreover , for high 𝑅𝑠ℎ and all \nthe calculated interface 𝜏𝑠 values,  the impedance reaches a minimum for intermediate values of \n𝑁𝑖𝑜𝑛 which often includes spurious 𝑅𝐿𝑓/𝑅𝐻𝑓>1 ratios . Furthermore, decreasing b ulk \nrecombination lifetime  (𝜏𝑏) in the perovskite layer similarly reduces the impedance , as \npresented in Section S6.6.  However, unlike 𝜏𝑠 whose decrease slightly declines 𝐽𝑠𝑐 and 𝐹𝐹, the \nreduction of 𝜏𝑏 implies a large lessening of 𝐽𝑠𝑐, 𝐹𝐹 and even the 𝑉𝑜𝑐. From all these simulations, \ncapacitance and resistance are shown to peak maximum and minimum values, respectively, \naround specific threshold mobile ion concentration as schemed in Figure 5c. In this figure, we \nalso highlight the inverse relation between the 𝑃𝐶𝐸  and 𝑁𝑖𝑜𝑛: the higher the mobile ion density, \nthe smaller the device efficiency.  Accordingly,  and assuming constant ion mobility, degradation \nroutes with decrease or increase of device performance can be outlined in terms of the resistance \nand capacitance in short -circuit condition , (see arrows in Figure 5c).  \n \n10-1100101102103104105106101102103104105-Z'' (W·cm2)\nf (Hz) 1015\n 1016\n 1018\n 1019Nion (cm-3) for t = 10-6,  10-9,  10-10s (a)\n10-110010110210310410510610-710-6C (F·cm-2)\nf (Hz) 1019\n 1018\n 1016\n 1015Nion (cm-3) for t = 10-6,  10-9,  10-10s (b)19 \n \nResistancePower conversion efficiencyCapacitance\nMobile ion concentration(c)\nShort Circuit \nFigure 5. Simulated impedance spectra in short -circuit  for different concentrations of mobile \nions ( Nion= 1015, 1016, 1018, 1019 cm-3) and trap recombination lifetime values at the interface \n(τ = 10-6, 10-9, 10-10 s) considering a shunt resistance of Rsh = 1 MΩ  in imaginary impedance \n(a) and ( b) capacitance Bode plots . In (c) there is a schematic on the se trends for the reference \nand optimized devices including degradation paths with PCE increase (purple open arrows) \nand decrease (blue filled arrows) , assuming consta nt ion mobility .  \n \nThe effects of the electron  and hole  mobilit ies (𝜇𝑒,ℎ) in the perovskite were also simulated, \nas displayed in Section S6.7, and no simple trend is observed. Instead, a multi -dependent \nrelationship is found w hereby  the concentration of mobile ions increases o r decreases the \nimpedance and the capacitance in different ranges. Remarkably, the perovskite electronic \nmobility has a high impact on the 𝐽𝑠𝑐 and 𝐹𝐹 of the 𝐽−𝑉 curve, although mostly negligible on \nthe 𝑉𝑜𝑐. A similar behavior is also found for the hole mobility (𝜇ℎ) at the  HTL  in terms of the \nvariated relation of the impedance and capacitance dependence on the mobility and the \nconcentration of mo bile ions. In contrast, the change of mobility of the NiO x layer (see Section \nS6.8) is mostly affecting the 𝐹𝐹, whereas the 𝐽𝑠𝑐 and the 𝑉𝑜𝑐 are mostly independent.   Last but \nnot least, the effect of the perovskite dielectric constant (𝜀) was also investiga ted in section S6.9. \nIt is shown that the increase of the dielectric permittivity of the absorber layer reduces the \nimpedance and increases the capacitance in the high -frequency range where the geometrical \ncontribution is dominant.  \nThe multiple r elations between simulation parameters and the behaviors of the IS spectra \nin short -circuit condition and the 𝐽−𝑉 curves under illumination are summarized in Table 1 \nwith a comprehensible “arrow -based” scheme. It is highlighted, that while some parameters are \nstraightforwardly related, some others present complicated dependencies that hinder the \ninterpretation of the experimental data in terms of more simplistic model s.   20 \n   \nTable 1. Summary  of different effects on the impedance spectra in short -circuit and \ncurrent -voltage characteristic due to the modification of simulation parameters . The up/down \narrows indicate direct or inverse proportionality, respectively. Several arrows indicate intens ity \nof dependence when in the same direction or complex multi -dependent relation when in \ndifferent directions.  \nParameter  IS spectra in SC  𝐽−𝑉 characteristic  Sections in \nSI Impedance  Capacitance  𝐽𝑠𝑐 𝐹𝐹 𝑉𝑜𝑐 \n𝑁𝑖𝑜𝑛 ↓↑ ↑↓ ↓ ↓ ↓ S5.1, S6.1  \n𝜇𝑖𝑜𝑛 ↑ -    S5.4 \n𝑃𝑖𝑛 ↓ ↑ ↑ ↑ ↑ S5.2, S6.2  \n𝑅𝑠ℎ ↑ - - ↑ ↑ S5.3, S6.3  \n𝑣𝑠 ↑↑↓ ↑↑↓ ↑ ↑↑ - S5.5 \n𝜏𝑠 ↓ ↓ ↓ ↓ - S6.5 \n𝜏𝑏 ↓ ↓ ↓ ↓ ↓ S6.6 \n𝜇𝑒,ℎ \n(perovskite)  ↓↓↑ ↑↑↓ ↑↑ ↑ - S6.7 \n𝜀 (perovskite)  ↓ ↑    S6.9 \n𝜇ℎ (HTL)  ↑↑↓ ↑↑↓ - ↓ - S6.8 \n4. Conclusions  \nThis work summarizes  an optoelectronic characterization of perovskite solar cells with \ndifferent passivation treatments on the interface between the perovskite absorber material and \nthe NiO x hole transport layer. From the initial assessment of the device performance and the \nsubsequent operational stability test two main trends where identified. The untreated reference \nsample and th ose utilizing 1-iodobutane, 1 -phenylethylamine  and PbI2 as passivators resulted \nin optimal and rather stable performances. In contrast, the MAI - and HI-treaded samples not \nonly resulted in low fill factor and photocurrent but also evidences substantial instability with \nan apparent improvement of the output power up to three times the initial value during the \noperational stability test.  \nThe use of impedance spectroscopy analysis in short -circuit condition under different \nillumination intensities and over time during operational stability test have been introduced, \ndiscussed and carried out as a resourceful procedure for understanding the electrical response \nof solar cells. Several practical and theoretical advantages of this approach have been discussed \nover the more common open -circuit/flat -band  and/or MPP studies . The experimental spectra \nshowed that the untreated and optimized devices behave in a more typical way where two main \n𝑅𝐶 constants produce two arcs in the impedance Nyquist representation. By using equivalent \ncircuit modelling the resistances for the low and high frequency ranges of spectra can be 21 \n identified as 𝑅𝐿𝑓≫𝑅𝐻𝑓. Characterist ically, the MAI and HI treated samples evidenced up to \nthree 𝑅𝐶 constant with an extra middle frequency feature that apparently merged with the low \nfrequency part of the spectra by reduction of the illumination intensity or exposure time during \nthe opera tional stability test.  Notably, the continuous switching between the operational \nforward bias and the short -circuit conditions was found to break the cells, which could be \nexplored as a reliability test protocol in future works.  \nOur drift -diffusion simul ations were able to qualitatively reproduce most of the \nexperimental behaviors for the current -voltage curves and the impedance spectra. The explored \nsimulation parameters suggest that the HI and MAI treatments on the NiO x HTL reduced the \ndevice absorptanc e, which could be associated with an interface morphology change that \nincreases photon reflectivity and electrical  series resistance . At the same time, the apparently \ndamaged HTL experienced a decrease in the electronic mobility . Moreover, the ionic mobili ty \nand concentrations of the perovskite were also found to be modified with respect to the values \nin the reference device . This work not only presents a comprehensible analysis on our study \ncase, but also explore a wide range of simulation parameters and i ts simultaneous effects on the \nJ-V curve and the IS spectra. These simulations results can be used to further comprehend the \nworking mechanism of the perovskite solar cells, their ionic properties and stability.  \n \nAuthor s Information  \nCorresponding Authors  \nOsbel Almora — Department of Electronic, Electric and Automatic Engineering, Universitat \nRovira i Virgili, 43007 Tarragona, Spain; https://orcid.org/0000 -0002 -2523 -0203  ; Email: \nosbel.almora@urv.cat   \nPilar López -Varo — Institut Photovoltaïque d'Île -de-France (IPVF), 91120 Palaiseau, France; \nhttps://orcid.org/0000 -0002 -2170 -1581  ; Email  : pilar.lopez -varo@ipvf.fr  \nJuan A. Anta — Department of Physical, Chemical and Natural Systems, Universidad Pablo de \nOlavide, Sevilla 41013, Spain;  https: //orcid.org/0000 -0002 -8002 -0313 ; Email: \njaantmon@upo.es   \nAuthors  \nRenán Escalante — Department of Physical, Chemical and Natural Systems, Universidad Pablo \nde Olavide, Sevilla 41013, Spain; https://orcid.org/0000 -0002 -5100 -5448 ; Email: \nraescqui1@upo.es   \nLluis F. Marsal — Department of Electronic, Electric and Automatic Engineering, Universitat \nRovira i Virgili, 43007 Tarragona,  Spain; https://orcid.org/0000 -0002 -5976 -1408 ; Email: \nlluis.marsal@urv.cat   22 \n Notes  \nThe authors declare no competing financial interests.  \nAcknowledgment  \nS. Mohamed acknowledges the financial support from Programa Martí i Franquès. M. Ramírez -\nComo acknowledges the financial support from Diputació de Tarragona under Grant \n2021CM14 and 2022PGR -DIPTA -URV04. This work was further supported by th e Spanish \nMinisterio de Ciencia e Innovación (MICINN/FEDER) under Grants PDI2021 -128342OB -I00 \nand RTI2018 -094040 -B-I00, by the Agency for Management of University and Research Grants \n(AGAUR) ref. 2017 -SGR -1527, and from the Catalan Institution for Research  and Advanced \nStudies (ICREA) under the ICREA Academia Award. O.A. thanks the National Research \nAgency (Agencia Estatal de Investigación) of Spain for the Juan de la Cierva 2021 grant  \n(FJC2021 -046887 -I). P.LV. thanks the French Government in the frame of t he program of \ninvestment for the future (Programme d’Investissement d’Avenir – ANR -IEED -002-01). J.M. \nand S.O. thank the Ministry of Economic Affairs Innovation, Digitalization and Energy of the \nState of North Rhine -Westphalia for funding under the grant S CALEUP (SOLAR -ERA.NET \nCofund 2, id: 32).  \n \nReferences  \n[1] Osbel Almora, Carlos I. 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Anta, \nIdentification of recombination losses and charge collection efficiency in a \nperovskite solar cell by comparing impedance response to a drift -diffusion model, \nNanoscale  2020 , 12, 17385, https://doi.org/10.1039/D0NR03058A  \n s1 \n Supporting information:  \nDegradation Analysis of Perovskite Solar Cells  via Short -\nCircuit Impedance Spectroscopy:  A case study on NiO x \npassivation  \nOsbel Almora,1,* Pilar López -Varo,2,* Renán Escalante,3 John Mohanraj,4 Luis F Marsal,1 \nSelina Olthof,4 Juan A. Anta3,* \n1 Departament d'Enginyeria Electrònica Elèctrica i Automàtica, Universitat Rovira i \nVirgili, 43007 Tarragona, Spain  \n2 Institut Photovoltaïque d'Ile -de-France (IPVF), 91120 Palaiseau, France  \n3 Department of P hysical, Chemical and Natural Systems, Universidad Pablo de Olavide, \nSevilla 41013, Spain  \n4 Department of Chemistry, University of Cologne, Greinstrasse 4−6, Cologne 50939, \nGermany  \n* osbel.almora@urv.cat , pilar.lopez -varo@ipvf.fr , jaantmon@upo.es   \n \n  s2 \n S1. Introduction  \nMaterials & \nExcess \nReagents\nInterfaces Air(a)\nReactivity\nElectric \nField \nStress\nPhoton \nInteractionThermo -\ncatalysisElectro -\ncatalysis\nPhoto -\nelectric \ndegradationPhoto -\nthermal \ndegradationTemperature\nStress(b)\nPhoto -\ncatalysis\n \nFigure S1. Schemed interconnection of different degradation (a) elements and  (b) \nmechanisms . \n \n(a) (b) (c)\n(d)\n  \n  \n \n  \n \n  \n \n \n  \n (g)\n  \n    (f) (e)\n \nFigure S 2. Schemed different types of degradation effect s on the current -voltage curve: sole \nor significantly dominant decrease of the (a) short -circuit current density  ( 𝑠𝑐), (b) fill factor \n(𝐹𝐹) and , (c) open -circuit voltage (  𝑜𝑐); and combined reduction of (d)  𝑠𝑐+𝐹𝐹, (e)  𝑜𝑐+\n𝐹𝐹, (f)  𝑜𝑐+ 𝑠𝑐+𝐹𝐹, and (f)  𝑜𝑐+ 𝑠𝑐. \n \n \n s3 \n Table S1. Dependency of shunt  resistance  (𝑅𝑠ℎ) and/or shor -circuit resistance as a \nfunction of illumination intensity ( 𝑃𝑖𝑛). \nAbsorber material  Dependency  Ref. Comment  \nc-Si 𝑅𝑠𝑐∝𝑃𝑖𝑛−1.13 [1]  \na-Si:H 𝑅𝑠𝑐∝ 𝑠𝑐−1 [2] AM1.5G spectrum  \nSi 𝑅𝑠ℎ=𝑅0 𝑅1(𝑃𝑖𝑛\n𝑃0)1.08\n [3] Polycrystalline silicon ; 𝑅0 , 𝑅1 and 𝑃0 are independent of 𝑃𝑖𝑛 \nc-Si 𝑅𝑠ℎ=𝑅0\n1+𝑃0\n𝑃𝑖𝑛 [4] 𝑅0 and 𝑃0 are independent of 𝑃𝑖𝑛  \nCdTe  𝑅𝑠ℎ∝𝑃𝑖𝑛−0.86 [5]  \nPBDB -T:F-M 𝑅𝑝∝𝑃𝑖𝑛−1 [6]  \n \nTable S2. Comparative summary between flat -band and short -circuit condition s in \nperovskite solar cells.  \nProperties  Flat-band (bias close to)  Short -circuit  \nPCE related parameters  Voc, FF, Vmpp Jsc \nLight effect on energy diagram and charge density profile  Low High  \nDrift versus diffusion contributions  Diffusion>Drift  Drift >>Diffusion  \nRecombination mechanism  Radiative >>Nonradiative  Nonradiative >>Radiative  \nMobile ion distribution in the perovskite  Bulk  Mostly towards interface  \nElectrical response time  Slower  Faster  \nCurrent linearity  Lower  Higher  \nParasitic resistance relation  Rs Rsh \n \n  s4 \n S2. Device fabrication  and initial performance characterization.  \nThe description below is adapted from our simultaneous work [ref]. \n Material  acquisition  \nFor MAPbI 3 layer deposition, PbI 2 and MA I were purchased from TCI Deutschland \nGmbH. The dry solvents dimethylformamide ( ≥ 99.9%), chlorobenzene, ethanol (99.9%), \nisopropanol (Honeywell 99.9%), and ethanolamine were purchased from Sigma -Aldrich and \nused as received. Nickel(II) acetate tetrahydrat e, HI (57 wt%), Li -TFSI, 1 -phenylethylamine, \n1,2,-ethanedithiol, 1 -iodobutane,  C 60 and bathocuproine were also purchased from Sigma -\nAldrich or Merck KGaA. The FTO -coated glass substrates (TEC 10, S2002S1) used for NiO x \ndeposition as well as solar cells preparation were procured from Ossila BV, The Netherlands.  \n Preparation of NiO x thin films  \nFirst, 400 mg of nickel ( II) aetate tetrahydrate and 97 L of ethanolamine were mixed in \n3.2 mL of ethanol, resulting in 0.5 M so lution . This solution  was stirred for at 2 h ours \n(minimum)  at room temperature to obtain a transparent dark blue -greenish colored solution. \nSubsequently, the solution was filtered with a 0.2 m PTFE filter, and then 100 L of it was \nspin-coated on clean FT O (TEC 10) substrates at 3000 rpm for 30 s. Immediately after, the \nsubstrates were annealed at 100 °C for 5 mi nutes i nside a N 2 glovebox and then at 450 °C for \n30 min in a furnace. Once the substrates were cooled to room temperature, they were transferred \ninto a N 2 filled  glovebox. Prior depositing any further layer on, the substrates were heated to \n100 °C for 10 min.  \n Preparation of MAPbI 3 thin films  \nMAPbI 3 films for degradation studies and solar cells were prepared by following the \nconventional antisolvent method. For degradation studies, 1 M of MAI and PbI 2 were dissolved \nin dimethylformamide by stirring the solution for at least 2 h at 50 °C inside a N 2 filled glovebox, \nand the solution was filtered with a 0.2 m PTFE filter. This was used as a stock solution to \nprepare 0.5 M, 0.375 M, 0.25 and 0.1 M MAPbI 3 solutions by diluting with appropriate amount \nof dimethylformamide. For film preparation, 80 L of M APbI 3 solution of desired concentration \nwas spin coated on substrates at 3000 rpm for 50 s. During this process, after 8 s, 200 L of \nchlorobenzene was dripped continuously onto the substrates. The substrates were then annealed s5 \n at 80 °C for 1 h on a hot pl ate inside a glovebox.   \n Deposition of surface passivation materials on NiO x \nIn separate vials, 0.1 M solutions of Li -TFSI, 1 -phenylethanol amine, 1 -iodobutane, HI, \nMAI and 1,2 -ethanedithiol were prepared in isopropanol and dimethylformamide was used as \na solvent for 0.1 M solution of PbI 2. These solutions were stirred at room temperature and \nfiltered with 0.2 m PTFE filter. At that time, 80 L of the filtered solution of desired \npassivating material was spin coated on FTO/N iOx substrates at 3000 rpm for 50 s and then \nannealed at 100 °C for 15 min on a hotplate inside a N 2 filled glove box. Importantly, the \nsubstrates were then washed with a copious amount of the respective solvent (isopropanol or \ndimethylformamide) and dried  with a N 2 gun to remove any unbound material from the NiO x \nsurface. These substrates were used as such for further analysis or to prepare further samples by \ndepositing MAPbI 3 films.  \n Fabrication of perovskite solar cells  \nAll solar cells were prepared by us ing etched FTO substrates with a20 -30 nm thick NiO x \nhole transport layer prepared from 0.2 M nickel  (II) acetate tetrahydrate and ethanolamine \nmixture in ethanol. On NiO x layer, passivating materials (0.1 M solution) and MAPbI 3 (from 1 \nM solution, 350 -400 nm thick) were deposited by following the procedures described above. \nThe substrates were then moved to an evaporation chamber, where C 60 electron transport layer \n(30 nm) and bathocuproine (8 nm) were evaporated. Finally, the devices were completed with \nAg cathode (100 nm) deposition. Each substrate contains 7 devices with an active area of 0.0785 \ncm2, defined by the deposited Ag cathode  (see Figure S 3).  \n \nFigure S 3. Photography of the studied samples  with an active area of 0.0785 cm2.  \n \ns6 \n \n Device performa nce \n0.0 0.2 0.4 0.6 0.8 1.005101520-J (mA cm-2)\nV (V) Ref.\n HI\n MAI\n PbI2\n 1-Phenylethylamine\n 1-Iodobutane(a)\n0.0 0.2 0.4 0.6 0.8 1.0051015 Ref.\n 1-Phenylethylamine\n 1-Iodobutane\n PbI2\n HI\n MAI-P (mW cm-2)\nV (V)(b)\n \nFigure S 4. Representative as-fabricated current density -voltage (   ) curves under \nstandard 1 sun illumination . In each case, the best two pixels per substrate are shown for two \nbias scan sweep directions at a scan rate of 70 mV/s.  \n \n \nTable S3. Summary of best performance samples in terms of power conversion efficiency  from \n    curves under standard 1 sun illumination.   \nSample  Reverse scan  Forward scan  \nVoc \n(V) Jsc \n(mA·cm-2) FF \n(%) PCE \n(%) Voc \n(V) Jsc \n(mA·cm -2) FF \n(%) PCE \n(%) \nRef. 1.04 20.0 64 13.4 1.01 19.6 68 13.4 \nPhenylethyl amine  1.04 21.5 62 13.8 1.02 20.7 68 14.4 \nIodobutane  1.04 21.8 60 13.6 1.02 20.8 67 14.1 \nPbI 2 1.03 21.2 64 14.1 1.01 20.0 73 14.7 \nHI 0.96 11.9 35 4.0 0.96 11.8 38 4.3 \nMAI  0.98 14.7 29 4.2 0.99 15.3 34 5.2 \n  s7 \n S3. Impedance spectra in short -circuit at different illumination intensities  \n Experimental Procedure and Equivalent Circuit Modelling  \nThe impedance spectroscopy was measured  with an Autolab PGSTAT302N  potentiostat \nincluding a FRA32M unit and a kit Autolab Optical Bench  from MetroOhm. The samples were \nilluminated with a white LED at different steady -state illumination intensities, then the short -\ncircuit (SC) condition was applied ( V=0V) , allowing a dire ct current (DC) density Jsc. Upon \nthese DC condition the 15  mV perturbation was applied for measuring the impedance \nspectroscopy (IS) as a function of the Jsc for each illumination intensity. Subsequently, the IS \nspectra were fitted to the equivalent circu it (EC) model of Figure S5.  \nCHf\nRHfRLfRsCLf LsCHf\nRMfRLfRsCLf\nLs RHfCMf(a) (b)\n \nFigure S5. Equivalent circuit s used for numerical model of impedance spectra. Here Ls and Rs \nare series inductor and resistor , respectively;  RHf , RMf and RLf stand for high, medium  and low \nfrequency resistors, respectively , and CHf, CMf and CLf stand for high, medium  and low \nfrequency capacitors , respectively . \nThe resistance parameters at SC extracted from the EC model were subsequently \nparameterized as a function of Jsc, following the empirical equation  \n𝑅𝑠𝑐=𝑅𝑠+𝑅𝑠ℎ0\n(1+ 𝑠𝑐\n 𝜎) (S1) \nwhere 𝑅𝑠 is the series resistance;  𝑅𝑠ℎ0, the equilibrium (dark) shunt resistance;  and  𝜎 is \nthe short -circuit current density corresponding to the threshold illumination intensity whose \ncharge carrier concentration at SC increase s photoconductivity , thus decreas ing the resistance. \nNotably,  𝑅𝑠𝑐≅𝑅𝑠ℎ/2 for  𝑠𝑐= 𝜎.  \nThe capacitance paramet ers at SC extracted from the EC model were subsequently \nparameterized as a function of Jsc, following the empirical equation  s8 \n 𝐶𝑠𝑐=𝐶0(1+( 𝑠𝑐\n 𝜀)𝑝\n) (S2) \nwhere 𝐶0 is the equilibrium  (dark) SC capacitance ; 𝑝 is a dimensionless power parameter, \nand  𝜀 is the short -circuit current density corresponding to the threshold illumination intensity \nwhose charge carrier concentration at SC enables a capacitive regime transition from a purely \ndielectric response (geom etric or space -charge capacitance) to a presumably mobile ions -related \nbehavior . Notably,  𝐶𝑠𝑐≅2𝑅𝑠ℎ for  𝑠𝑐= 𝜀 and dielectric contributions which are independent \nof the light intensity would require significantly large threshold currents (  𝜀→∞⇒𝐶𝑠𝑐=𝐶0) \nThe characteristic response times for each mechanism can be extracted via the estimation \nof the correspondin g peak maxima across the Bode plots of the imaginary part of the impedance, \nand/or by considering the resistor -capacitor coupling ( 𝑅∙𝐶 product) assumed in the EC model. \nEither way, the behaviour  of the response time s follows the empirical equation   \n𝜏𝑠𝑐=𝜏0\n(1+ 𝑠𝑐\n 𝜎)(1+( 𝑠𝑐\n 𝜀)𝑝\n) (S3) \nwhere 𝜏0≈𝑅𝑠ℎ0𝐶0 is the  characteristic  equilibrium (dark) response time constant . \nImportantly, Equation (S3) behaves constant for 𝑝=1 in the high illumination intensity limit \n( 𝑠𝑐→∞⇒𝜏𝑠𝑐=𝜏0 𝜎/ 𝜀); provided equal threshold currents and/or in the low illumination \nintensity limit (  𝜎= 𝜀∨ 𝑠𝑐→0⇒𝜏𝑠𝑐=𝜏0). \n s9 \n \n Experimental impedance spectroscopy data and equivalent circuit \nfittings  for different illumination intensities  \n NiO xRef \n10-210-110010110210310410510610710-710-610-5\nDark C (F cm-2)\nf (Hz)(a) NiOxRef\nLight \n \n10-210-110010110210310410510610710-1100101102103104105106107Z'' (W cm2)\nf (Hz)(b) NiOxRef\nDark \nLight  \n0.0 1.0M 2.0M0.01.0M\nLight-Z'' (W cm2)\nZ' (W cm2)(c) NiOxRef\nDark\n \nFigure S6. Impedance spectra for the NiOxRef  Sample in SC under different DC illumination \nintensities in (a) capacitance and (b) imaginary Bode, and (c) impedance Nyquist \nrepresentations. In each case, the dots indicate experimental data, the lines are the numerical \nsimulations via EC model (see  Figure S5), and darker and lighter colours  indicate low and \nhigh illumination intensities (and Jsc values), respectively.  \n \n  s10 \n \n10-710-610-510-410-3100101102103104105106 Rs (NiOxRef)   RHf (NiOxRef)   RLf (NiOxRef)R (W cm2)\nJsc (A cm-2)(a)\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(s)\nRsh 1.89884 ± 0.01177\nJr 10 ± 749.42415\nReduced Chi-Sqr 0.0012\nR-Square (COD) -0.15404\nAdj. R-Square -0.28227\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(1)\nRsh 52887.00758 ± 4814.74629\nJr 2E-5 ± 0\nReduced Chi-Sqr 1.11319E8\nR-Square (COD) 0.80626\nAdj. R-Square 0.80626\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(2)\nRsh 800000 ± 72596.63993\nJr 8.2E-5 ± 0\nReduced Chi-Sqr 3.2687E10\nR-Square (COD) 0.78146\nAdj. R-Square 0.78146 \n10-710-610-510-410-310-810-710-610-5 CHf (NiOxRef)   CLf (NiOxRef)C (F cm-2)\nJsc (A cm-2)(b)  \n  \nFigure S7. Results for (a) resistance, (b) capacitance, and (c) characteristic times extracted \nfrom the EC numerical modelling  of the impedance s pectra in SC of the NiO x Re.f sample \nunder different DC illumination intensities. In each case, the dots indicate fitted parameters \nfrom the EC model (see Figure S5), and the lines  are the numerical simulations to the \nanalytical trends of equations (S1), (S2), and (S3) for (a), (b), and (c), respectively.  \n \nTable S4. Parameterization of photovoltage trend from the EC modelling  following Equations \n(S1), (S2), (S3) and from the IS spectra of the sample NiO x Ref. in SC .  \nParameters  𝑅𝑠  𝑅𝐻𝑓  𝑅𝐿𝑓  𝐶𝐻𝑓  𝐶𝐿𝑓  𝜏𝐻𝑓  𝜏𝐿𝑓  \n𝑅𝑠 (Ω⸱cm2) 1.90 — — — — — — \n𝑅𝑠ℎ (Ω⸱cm2) — 58053  8E5 — — — — \n 𝜎 (A cm-2) — 2E-5 8.2E-5 — — 8e-5 3.6E-4 \n𝐶0 (F⸱cm-2) — — — 1.38E -7 3.93E -8 — — \n 𝜀 (A cm-2) — — — — 2.42E -5 — 5E-5 \np (a.u.)  — — — — 1.35 — 1.35 \n𝜏0 (s) — — — — — 0.0472  0.03 \n 𝑝ℎ (V) — — — — — — — \n 𝑉 (A cm-2) — — — — — — — \n \n \n10-710-610-510-410-310-210-1100 tHf (NiOxRef)   tLf (NiOxRef)\nt (s)\nJsc (A cm-2)Model tscJsc_light (User)\nEquationt0*(1+(Jsc/Jc)^p)/\n(1+Jsc/Jr)\nPlot t1\nt0 0.12 ± 0.01517\nJc 1 ± 0\nJr 8E-5 ± 6.83784E-5\nReduced Chi-Sqr 0.00107\nR-Square (COD) 0.77736\nAdj. R-Square 0.68194\nModel tscJsc_light (User)\nEquationt0*(1+(Jsc/Jc)^p)/\n(1+Jsc/Jr)\nPlot t2\nt0 0.02937 ± 0.01377\nJc 5E-5 ± 2.45664E-5\nJr 3.66904E-4 ± 1.51336E-4\np 1.35 ± 0\nReduced Chi-Sqr 0.00127\nR-Square (COD) 0.98205\nAdj. R-Square 0.97757(c)s11 \n \n Amine  \n10-210-110010110210310410510610710-710-610-510-4\nDark C (F cm-2)\nf (Hz)(a) Amine\nLight \n \n10-210-110010110210310410510610710-1100101102103104105106107Z'' (W cm2)\nf (Hz)(b) Amine\nDark \nLight  \n0.0 500.0k 1.0M 1.5M0.0500.0k1.0M\nLight-Z'' (W cm2)\nZ' (W cm2)(c)Amine\nDark\n \nFigure S8. Impedance spectra for the Amine  Sample in SC under different DC illumination \nintensities in (a) capacitance and (b) imaginary Bode, and (c) impedance Nyquist \nrepresentations. In each case, the dots indicate experimental data, the lines are the numerical \nsimulations via EC model (see Figure S5), and darker and lighter colors indicate low and \nhigh illumination intensities (and Jsc values), respectively.  \n \n  s12 \n \n10-710-610-510-410-3100101102103104105106 Rs (Amine)   RHf (Amine)   RLf (Amine)R (W cm2)\nJsc (A cm-2)(a)\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(s)\nRsh 1.62932 ± 0.02698\nJr 10 ± 1079.40746\nReduced Chi-Sqr 0.00637\nR-Square (COD) -0.06477\nAdj. R-Square -0.18308\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(1)\nRsh 7188.79871 ± 261.14916\nJr 2E-4 ± 0\nReduced Chi-Sqr 457792.96477\nR-Square (COD) 0.94517\nAdj. R-Square 0.94517 \n10-710-610-510-410-310-810-710-610-5 CHf (Amine)   CLf (Amine)C (F cm-2)\nJsc (A cm-2)(b)  \n \nFigure S9. Results for (a) resistance, (b) capacitance, and (c) characteristic times extracted \nfrom the EC numerical modelling  of the impedance spectra of the Amine  sample in SC under \ndifferent DC illumination intensities. In each case, the dots indicate fitted parameters from \nthe EC model (see Figure S5), and the lines are the numerical simulations to the analytical \ntrends of equations (S1), (S2), and (S3) for (a), (b), and (c), respectively.  \n \n \nTable S5. Parameterization of photovoltage trend from the EC modeling following Equations \n(S1), (S2), (S3) and from the IS spectra of the sample Amine  in SC .  \nParameters  𝑅𝑠  𝑅𝐻𝑓  𝑅𝐿𝑓  𝐶𝐻𝑓  𝐶𝐿𝑓  𝜏𝐻𝑓  𝜏𝐿𝑓  \n𝑅𝑠 (Ω⸱cm2) 1.61 — — — — — — \n𝑅𝑠ℎ (Ω⸱cm2) — 6959  1.28E6  — — — — \n 𝜎 (A cm-2) — 2E-4 8.2E-5 — — 8.2e-5 3E-6 \n𝐶0 (F⸱cm-2) — — — 8.7e-8 8E-8 — — \n 𝜀 (A cm-2) — — — — 5E-5 — 1.3E-5 \np (a.u.)  — — — — 1.5 — 1.65 \n𝜏0 (s) — — — — — 0.12 0.17 \n 𝑝ℎ (V) — — — — — — — \n 𝑉 (A cm-2) — — — — — — — \n10-710-610-510-410-310-210-1100 tHf (Amine)   tLf (Amine)\nt (s)\nJsc (A cm-2)(c)\nModel tscJsc_light (User)\nEquationt0*(1+(Jsc/Jc)^p)/\n(1+Jsc/Jr)\nPlot t2\nt0 0.17 ± 0.51317\nJc 1.3E-5 ± 8.85815E-5\nJr 3E-6 ± 3.84685E-5\np 1.65 ± 0.42066\nReduced Chi-Sqr 0.37425\nR-Square (COD) -0.72784\nAdj. R-Square -1.37578s13 \n  \n Iodobutane  \n10-210-110010110210310410510610710-710-610-510-4\nDark C (F cm-2)\nf (Hz)(a) Iodobutane\nLight \n10-210-110010110210310410510610710-1100101102103104105106107108Z'' (W cm2)\nf (Hz)(b) Iodobutane\nDark \nLight \n0.0 3.0M0.03.0M6.0M9.0M\n0 50k 100k050k100k\nLight-Z'' (W cm2)\nZ' (W cm2)(c)\nDark\nLight-Z'' (W cm2)\nZ' (W cm2)Iodobutane\nDark\n \nFigure S10. Impedance spectra for the Iodobutane  Sample in SC under different DC \nillumination intensities in (a) capacitance and (b) imaginary Bode, and (c) impedance Nyquist \nrepresentations. In each case, the dots indicate experimental data, the lines are the numerical \nsimulations via EC model (see Figure S5), and darker and lighter colors indicate low and \nhigh illumination intensities (and Jsc values), respectively.  \n \n  s14 \n \n10-710-610-510-410-3100101102103104105106107108 Rs (Iodobutane)   RHf (Iodobutane)   RLf (Iodobutane)R (W cm2)\nJsc (A cm-2)(a)\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(s)\nRsh 1.62531 ± 0.04226\nJr 10 ± 1919.87081\nReduced Chi-Sqr 0.01723\nR-Square (COD) -0.01438\nAdj. R-Square -0.11581\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(1)\nRsh 21227.30234 ± 2689.3319\nJr 5.2E-5 ± 5.39427E-5\nReduced Chi-Sqr 4.27479E7\nR-Square (COD) 0.65273\nAdj. R-Square 0.62116 \n10-710-610-510-410-310-810-710-610-5 CHf (Iodobutane)   CLf (Iodobutane)C (F cm-2)\nJsc (A cm-2)(b)  \n  \n \nFigure S11. Results for (a) resistance, (b) capacitance , and (c) characteristic times   extracted \nfrom the EC numerical modelling  of the impedance spectra of the Iodobutane  sample in SC \nunder differen t DC illumination intensities. In each case, the dots indicate fitted parameters \nfrom the EC model (see Figure S5), and the lines are the numerical simulations to the \nanalytical trends of equations (S1), (S2), and (S3) for (a), (b), and (c), respectively.  \n \nTable S6. Parameterization of photovoltage trend from the EC mode lling following Equations \n(S1), (S2), (S3) and from the IS spectra of the sample Iodobutane  in SC .  \nParameters  𝑅𝑠  𝑅𝐻𝑓  𝑅𝐿𝑓  𝐶𝐻𝑓  𝐶𝐿𝑓  𝜏𝐻𝑓  𝜏𝐿𝑓  \n𝑅𝑠 (Ω⸱cm2) 1.62 — — — — — — \n𝑅𝑠ℎ (Ω⸱cm2) — 21227  8.62E7  — — — — \n 𝜎 (A cm-2) — 5.2E-5 1E-6 — — 1E-6 3E-6 \n𝐶0 (F⸱cm-2) — — — 1.46E -7 5E-8 — — \n 𝜀 (A cm-2) — — — — 1.22E-5 — 2E-5 \np (a.u.)  — — — — 1.02 — 1.06 \n𝜏0 (s) — — — — — 12.1 2.6 \n 𝑝ℎ (V) — — — — — — — \n 𝑉 (A cm-2) — — — — — — — \n10-710-610-510-410-310-310-210-1100101 tHf (Iodobutane)   tLf (Iodobutane)\nt (s)\nJsc (A cm-2)(c)\nModel tscJsc_light (User)\nEquationt0*(1+(Jsc/Jc)^p)/\n(1+Jsc/Jr)\nPlot t2\nt0 2.6 ± 1.05585\nJc 2E-5 ± 1.23684E-4\nJr 3E-6 ± 9.354E-6\np 1.06121 ± 0.97111\nReduced Chi-Sqr 2.38356\nR-Square (COD) 0.41336\nAdj. R-Square 0.21781s15 \n \n HI \n10-210-110010110210310410510610710-710-610-510-4\nDark C (F cm-2)\nf (Hz)(a) HI\nLight \n \n10-210-110010110210310410510610710-1100101102103104105106107Z'' (W cm2)\nf (Hz)(b) HI\nDark \nLight  \n0 500k 1M0250k500k\nLight-Z'' (W cm2)\nZ' (W cm2)(c)HI\nDark\n0 2k01k\nLight-Z'' (W cm2)\nZ' (W cm2)(c)HI\nDark\n \n \nFigure S12. Impedance spectra for the HI Sample in SC under different DC illumination \nintensities in (a) capacitance and (b) imaginary Bode, and (c) impedance Nyquist \nrepresenta tions. In each case, the dots indicate experimental data, the lines are the numerical \nsimulations via EC model (see Figure S5), and darker and lighter colors indicate low and \nhigh illumination intensities (and Jsc values), respectively.  \n \n  s16 \n \n10-710-610-510-410-310-710-610-510-4  CHf (HI) \n CMf (HI) \n CLf (HI)C (F cm-2)\nJsc (A cm-2)(b) \n10-710-610-510-410-310-410-310-210-1 tHf (HI)  tMf (HI)  tLf (HI)\nt (s)\nJsc (A cm-2)(c)\n \nFigure S13. Results for (a) resistance, (b) capacitance, and (c) characteristic times  extracted \nfrom the EC numerical modelling  of the impedance spectra of the HI sample in SC under \ndifferent DC illumination intensities. In each case, the dots indicate fitted parameters from \nthe EC model (see Figure S5), and the lines are the n umerical simulations to the analytical \ntrends of equations (S1), (S2), and (S3) for (a), (b), and (c), respectively.  \n \nTable S7. Parameterization of photovoltage trend from the EC modelling  following Equations \n(S1), (S2), (S3) and from the IS spectra of the sample HI in SC .  \nParameters  𝑅𝑠  𝑅𝐻𝑓  𝑅𝑀𝑓  𝑅𝐿𝑓  𝐶𝐻𝑓  𝐶𝑀𝑓 𝐶𝐿𝑓  𝜏𝐻𝑓  𝜏𝑀𝑓  𝜏𝐿𝑓  \n𝑅𝑠 (Ω⸱cm2) 1.31 — — — — — — — — — \n𝑅𝑠ℎ \n(Ω⸱cm2) — 17342  21043  1.79E6  — — — — — — \n 𝜎 (A cm-2) — 2.82E -\n6 2.48E -\n6 3.19E -7 — — — 4E-7 4E-7 1.04E -\n5 \n𝐶0 (F⸱cm-2) — — — — 9.49E -\n8 1E-7 2.45E -\n8 — — — \n 𝜀 (A cm-2) — — — — — 3.23E -\n4 4E-7 — — 1.13E -\n4 \np (a.u.)  — — — — — 1.5 1.00 — — 1.86 \n𝜏0 (s) — — — — — — — 0.175  0.234  0.0586  \n 𝑝ℎ (V) — — — — — — — — — — \n 𝑉 (A cm-2) — — — — — — — — — — \n10-910-810-710-610-510-410-310-210-1100101102103104105106107 Rs (HI)  RHf (HI)  RMf (HI)  RLf (HI)R (W cm2)\nJsc (A cm-2)(a)s17 \n \n PbI2  \n10-210-110010110210310410510610710-810-710-610-510-4\nDark C (F cm-2)\nf (Hz)(a) PbI2\nLight \n \n10-210-110010110210310410510610710-1100101102103104105106107108Z'' (W cm2)\nf (Hz)(b) PbI2\nDark \nLight  \n0 2M 4M02M4M\nLight-Z'' (W cm2)\nZ' (W cm2)(c)\nPbI2\nDark\n \n \n \nFigure S14. Impedance spectra for the PbI2  Sample in SC under different DC illumination \nintensities in (a) capacitance and (b) imaginary Bode, and (c) impedance Nyquist \nrepresentations. In each case, the dots indicate experimental data, th e lines are the numerical \nsimulations via EC model (see Figure S5), and darker and lighter colors indicate low and \nhigh illumination intensities (and Jsc values), respectively.  \n \n  s18 \n  \n10-710-610-510-410-3100101102103104105106107108 Rs (IPbI2)   RHf (PbI2)   RLf (PbI2)R (W cm2)\nJsc (A cm-2)(a)\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(s)\nRsh 1.37812 ± 0.00815\nJr 10 ± 731.04349\nReduced Chi-Sqr 6.39308E-4\nR-Square (COD) -0.02811\nAdj. R-Square -0.13092\nModel RJsc_light (User)\nEquation Rsh/(1+Jsc/Jr)\nPlot R\\-(1)\nRsh 11412.57726 ± 1113.36453\nJr 1E-4 ± 7.66024E-5\nReduced Chi-Sqr 8.5231E6\nR-Square (COD) 0.7003\nAdj. R-Square 0.67305\n \n10-710-610-510-410-310-810-710-610-5 CHf (PbI2)   CLf (PbI2)C (F cm-2)\nJsc (A cm-2)(b)  \n10-710-610-510-410-310-310-210-1100101 tHf (PbI2)   tLf (PbI2)\nt (s)\nJsc (A cm-2)(c)\nModel tscJsc_light (User)\nEquationt0*(1+(Jsc/Jc)^p)/\n(1+Jsc/Jr)\nPlot t2\nt0 0.68512 ± 0.25292\nJc 4.97029E-6 ± 3.95463E-5\nJr 3.81929E-6 ± 3.17469E-5\np 1.07555 ± 0.14267\nReduced Chi-Sqr 0.12558\nR-Square (COD) 0.06441\nAdj. R-Square -0.24745\n  \nFigure S15. Results for (a) resistance, (b) capacitance, (c) characteristic times and (d) internal \nphotovoltages extracted from the EC numerical mode lling of the impedance spectra of the HI \nsample in SC under different DC i llumination intensities. In each case, the dots indicate fitted \nparameters from the EC model (see Figure S5), and the lines are the numerical simulations \nto the analytical trends of equations (S1), (S2), and (S3) for (a), (b), and (c), respectively.  \n \nTable S8. Parameterization of photovoltage trend from the EC modelling  following Equations \n(S1), (S2), (S3) and from the IS spectra of the sa mple HI in SC .  \nParameters  𝑅𝑠  𝑅𝐻𝑓  𝑅𝑀𝑓  𝑅𝐿𝑓  𝐶𝐻𝑓  𝐶𝐿𝑓  𝜏𝐻𝑓  𝜏𝐿𝑓  \n𝑅𝑠 (Ω⸱cm2) 1.38 — — — — — — — \n𝑅𝑠ℎ (Ω⸱cm2) — 1734\n2 11412  8.19E6  — — — — \n 𝜎 (A cm-2) — 2.82E\n-6 1E-4 1.84E -5 — — 1.5E -\n5 1.04E -5 \n𝐶0 (F⸱cm-2) — — — — 9.64E -8 7E-8 — — \n 𝜀 (A cm-2) — — — — — 1.09E -5 — 1.13E -4 \np (a.u.)  — — — — — 1.00 — 1.86 \n𝜏0 (s) — — — — — — 1.00 0.0586  \n 𝑝ℎ (V) — — — — — — — — \n 𝑉 (A cm-2) — — — — — — — — \n s19 \n S4. Stability test in short -circuit  under 0.2 sun white LED equivalent  \nillumination intensity  \nVhppisset\nΔtddelay\nSC (V=0) issetΔtscstabilization\ndelay @SCIS ismeasured\n \nFigure S 16. Scheme of the degradation test loop.  The devices were initially set under \nillumination to a high-power point ( Vhpp), close to that of the starting maximum -power point \n(MPP) without correcting hysteresis effect. Next, a delay time ( Δtd) for degradation was \nwaited while the illumination and the bias Vhpp were constant . Subsequently a biasing change \nfrom near -to-MPP to short -circuit (V=0V) is set. Another delay time ( Δtsc) is considered \nallowing the short circuit current to  stabilize . Afterwards , the impedance  spectrum (IS) is \nmeasured at short -circuit (SC). The loop is repeated when a new Vhpp is set.  For the correction \nof the Vhpp value , an empirical rule was implemented. Provided the initial difference ΔV=Voc,0-\nVmpp between the open circuit voltage and the MPP voltage, the open circuit voltage of the nth \ncycle is measure after the IS and then the nth high-power voltage value is set as : Vhpp,n =Voc,n-\n1- ΔV \n \n s20 \n \n High power condition over time  \n0 10 20 30 40 5080100120140160180200220240260280300-J (%)\nt (h) Iodobutane\n Amine\n PbI2\n NiOxref\n MAI\n HI\n  \n0 10 20 30 40 50100150200250300-P (%)\nt (h) Iodobutane\n Amine\n PbI2\n NiOxref\n MAI\n HI  \n0 10 20 30 40 500.51.01.52.02.5-J (mA cm-2)\nTime (h) Iodobutane\n Amine\n PbI2\n NiOxref\n MAI\n HI\n0 10 20 30 40 500.51.01.52.0-P (mA cm-2)\nTime (h) Iodobutane\n Amine\n NiOxref\n MAI\n HI\n \nFigure S17. Stability test at high power point under 0.2 sun white LED equivalent \nillumination .  \n s21 \n \n Impedance spectroscopy in short -circuit over time  \n0 10k 20k 30k 40k010k20k -Z'' (W cm2)\nZ' (W cm2)NiOxRef1 (13%)\n0 3k 6k03k -Z'' (W cm2)\nZ' (W cm2)iodobutane1 (14%)\n0 5k 10k 15k05k10k -Z'' (W cm2)\nZ' (W cm2)Amine 1 (14%)\n \n0 2k 4k 6k 8k 10k02k-Z'' (W cm2)\nZ' (W cm2)MAI1 (5%)\n0.0 500.0 1.0k 1.5k 2.0k 2.5k 3.0k0500-Z'' (W cm2)\nZ' (W cm2)Hi1 (4%)\n  \nFigure S18. Impedance  spectra in Nyquist plot representation in short -circuit under 0.2 sun \nwhite LED equivalent over time  for different sample surface treatment, as indicated. Dots and \nlines are the experimental data and equivalent circuit model fittings, respectively. The li ghter \n(yellower) the colors the longer the time. The same spectra are shown in Figure S20 and \nFigure S19 \n \n  s22 \n \n10-210-1100101102103104105106107100101102103104105\nNiOxRef1 (13%)-Z'' (W cm2)\nFrequency (Hz)\n10-210-1100101102103104105106107100101102103iodobutane1 (14%)-Z'' (W cm2)\nFrequency (Hz) \n10-210-1100101102103104105106107100101102103104Amine 1 (14%)-Z'' (W cm2)\nFrequency (Hz)\n \n10-210-1100101102103104105106107100101102103104\nPbI2-1 (13%)-Z'' (W cm2)\nFrequency (Hz)  \n10-210-1100101102103104105106107100101102103Hi1 (4%)-Z'' (W cm2)\nFrequency (Hz)\n \nFigure S19. Bode plots of imaginary part of impedance spectra over time in short -circuit \ncondition under 0.2 sun white LED equivalent illumination. The dots are the experimental \ndata and lines are the fittings to the equivalent circuit models in Figure S5. The same spectra \nare shown in Figure S18 and Figure S20. \n \n s23 \n \n10-210-110010110210310410510610710-710-610-5\nNiOxRef1 (13%)C (F cm-2)\nFrequency (Hz) \n10-210-110010110210310410510610710-710-610-5\niodobutane1 (14%)C (F cm-2)\nFrequency (Hz)  \n10-210-110010110210310410510610710-710-610-5\nAmine 1 (14%)C (F cm-2)\nFrequency (Hz)\n \n10-210-110010110210310410510610710-710-610-5 PbI2-1 (13%)C (F cm-2)\nFrequency (Hz)  \n10-210-110010110210310410510610710-710-610-510-4\nMAI (5%)C (F cm-2)\nFrequency (Hz)\n \n10-210-110010110210310410510610710-710-610-510-4HIC (F cm-2)\nFrequency (Hz)  \nFigure S20. Bode plots of capacitance spectra over time in short -circuit condition under 0.2 \nsun white LED equivalent illumination for different sample surface treatment, as indicated. \nThe dots are the experimental data and lines are the fittings to the equivalent cir cuit models \nin Figure S5. The same spectra are shown in Figure S18 and Figure S19 \n \n \n s24 \n \n0 10 20 30 40 507075808590CHf (F cm-2)\nt (h) HI (4%)\n MAI (5%)\n Ref (13%)\n Amine (14%)\n PbI2 (13%)\n IodoButane (14%) \n0 10 20 30 40 50102103104105106107108CLf (F cm-2)\nt (h) HI (4%, Mf)\n HI (4%, Lf1)\n HI (4%, Lf2)\n MAI (5%, Mf)\n MAI (5%, Lf)\n Ref (13%)\n PbI2 (13%)\n Amine (14%)\n IodoButane (14%)  \n0 10 20 30 40 50100110Normalized CHf (%)\nt (h) HI (4%)\n MAI (5%)\n Ref (13%)\n PbI2 (13%)\n Amine (14%)\n IodoButane (14%)\n \n0 10 20 30 40 5010100Normalized CLf (%)\nt (h) HI (4%, Mf)\n HI (4%, Lf)\n MAI (5%, Mf)\n MAI (5%, Lf)\n PbI2 (13%)\n Ref (13%)\n Amine (14%)\n IodoButane (14%)  \nFigure S21. Capacitance evolution over time during the stability test in short -circuit \ncondition  under 0.2 sun white LED equivalent illumination. The values are the result of \nnumerical fittings of the spe ctra in Figure S20 to the equivalent circuit model s in Figure S5. \n \n  s25 \n   \n0 5 10 15 20 25 30 35 40 45 5011.52Rs (W cm2)\nt (h) HI (4%)\n MAI (5%)\n Ref (13%)\n PbI2 (13%)\n Amine (14%)\n IodoButane (14%)  \n0 10 20 30 40 5080100120140Normalized Rs (%)\nt (h) HI (4%)\n MAI (5%)\n Ref (13%)\n PbI2 (13%)\n Amine (14%)\n IodoButane (14%)  \n0 5 10 15 20 25 30 35 40 45 50102103RHf (W cm2)\nt (h) HI (4%)\n MAI (5%)\n PbI2 (13%)\n Ref (13%)\n Amine (14%)\n IodoButane (14%)\n \n0 10 20 30 40 50100150200250300350400Normalized RHf (%)\nt (h) HI (4%)\n MAI (5%)\n Ref (13%)\n PbI2 (13%)\n Amine (14%)\n IodoButane (14%)  \n0 10 20 30 40 5010100100010000 RLf (W cm2)\nt (h) MAI (5%, Mf)\n MAI (5%, Lf)\n HI (4%, Mf)\n HI (4%, Lf1)\n HI (4%, Lf2)\n Ref (13%)\n PbI2 (13%)\n Amine (14%)\n IodoButane (14%)\n \n0 10 20 30 40 50100100010000Normalized RLf (%)\nt (h) HI (4%, Mf)\n HI (4%, Lf)\n MAI (5%, Mf)\n MAI (5%, Lf)\n Ref (13%)\n PbI2 (13%)\n Amine (14%)\n IodoButane (14%)  \nFigure S22. Resistance  evolution over time during the stability test in short -circuit condition  \nunder 0.2 sun white LED equivalent illumination for different samples, as indicated. The \nvalues are the result of numerical fittings of the spectra in Figure S18 to the equivalent circuit \nmodel s in Figure S5. \n \n  s26 \n S5. SEFTOS -FLUXIM[7] simulations  \nTable S9. Simulation parameters for S etfos -Fluxim  \nParameter  Value  Unit  \nActive layer (perovskite)  \nDielectric constant  24.1   \nThickness  400 nm \nBimolecular recomb ination p re-factor  9.40E -10 cm3/s \nLangevin efficiency  3.25E -04  \nTrap density  1.00E+16  cm3 \nElectron /hole  pseudo -lifetime  5.00E -09 s \nElectron / hole  capture rates  2.00E -08 cm3/s \nElectron /hole  Diffusion coefficient  5.00E -05 m2/s \nElectron /hole  mobility  1.92E+01  cm2/sV \nHole conductor (NiOx)  \nDielectric constant  3  \nThickness   30 nm \nHole Diffusion coefficient  2.60E -05 cm2/s \nHole Mobility  1.00E -03 cm2/sV \nDoping density (acceptor)  1.40E+20  cm-3 \nElectron conductor (C 60) \nDielectric constant  3.9  \nThickness  30 nm \nElectron Diffusion coefficient  2.31E -05 cm2/s \nElectron Mobility  8.90E -04 cm2/sV \nDoping density (donors)  1.50E+18  cm-3 \nExternal parameters  \nShunt resistance  1E6 Ω cm2 \n \n \n  s27 \n \n Qualitative initial simulation of current -voltage and impedance \nspectroscopy spectra  \n \n(a)  (b)  \n(c)  \nFigure S23.  Initial settings for Setfos simulator: (a) l ayer structure , (9) energy diagram s and \n(c) base simulated current density -voltage curves in dark and under illuminatio n. \ns28 \n  \n(a) (b)  \nFigure S24.  Simulated electrical potential (Psi) as a function of the distance (x) between \nelectrodes over the external voltage for mobile ion concentrations of (a) 1015 cm-3 and (b) \n1018 cm-3. The simulation parameters are in Table S9. \n \n  \nNion=1021 m-3 Nion=1024 m-3 s29 \n  \n (a)   \n(b) (c)  \nFigure S25.  Simulated impedance spectra in short -circuit under 0.2 sun illumination \nintensity with mobile ion concentrations of 1018 cm-3 in (a) impedance Nyquist plot, (b) \ncapacitance Bode plot and (c) impedance Bode plots . Further simulation parameters are in \nTable S9. \n \n \n \nNion = 1024\n m-3\n Nion = 1024\n m-3\n \nNion = 1024\n m-3\n s30 \n \n Incident  illumination intensity  effects  \n(a)  (b)  \n(c)  (d)   \nFigure S26.  Simulated impedance spectra in short -circuit under different illumination \nintensit ies (in units of sun) with mobile ion concentrations of (a, b) 1018 cm-3 and (c, d) 1015 \ncm-3 in (a, c) impedance Nyquist plot, and (b, d) imaginary part of impedance Bode plot . \nFurther simulation parameters are in Table S9. \n \n \n \n \nNion = 1024\n m-3\n \nNion = 1024\n m-3\n \nNion = 1021\n m-3\n Nion = 1021 m-3 s31 \n \n0 2 4 6 802468        Nion (cm-3)\n 1012\n 1015\n 1018-Z'' (W cm2)\nZ' (W cm2)(a) \n0.001 1 10 100103104105106(b)         Nion (cm-3)\n 1012\n 1015\n 1018R (W cm2)\nPin (mW cm-2)0\n \n4E-46E-48E-40.0010.00120.00140.00160.0018 1 10 10010-710-610-510-4(c)C (F cm-2)\nPin (mW cm-2) CHf (Nion=1012 cm-3)\n CHf (Nion=1015 cm-3)\n CHf (Nion=1018 cm-3)\n CLf (Nion=1018 cm-3)\n0  \nFigure S27.  Numerical simulation of impedance spectra from the reference sample in \nshort -circuit under different illumination intensities and for dif ferent mobile ion \nconcentrations. In (a) there are the Nyquist plots for 1 sun illumination intensity and (b) and \n(c) show the resultant low frequency resistances and capacitances, respectively, from \nequivalent circuit modeling. The dots in (a) are Setfos -Fluxim simulations and the lines are \nZview equivalent circuit fittings. The solid lines in (b) and (c) are allometric fittings with \npower -1 and +1, respectively.   \n \n s32 \n \n Shunt resistance effects  \n(a)  (b)  \n(c)  (d)  \n(e) (f)  \nFigure S 28.  Simulated (a, b) current -voltage curves and (c -f) impedance spectra in short -\ncircuit condition under 0.2 sun illumination intensity  for different values of shunt resistance, \nas indicated with mobile ion concentrations of (a, c, e) 1015 cm-3 and ( b, d, f) 1018 cm-3 in (c, \nd) impedance Nyquist plot, and ( e, f) impedance imaginary part Bode plot s. Further \nsimulation parameters are in Table S9. \n \nNion = 1021 m-3 \nNion = 1024 m-3 \nNion = 1021 m-3 \nNion = 1021 m-3 \nNion = 1024 m-3 Nion = 1024 m-3 s33 \n  \n Ion mobility effects  \n(a)  \n(b)  \nFigure S 29.  Simulated impedance spectra in short -circuit under 0.2 sun illumination for a \nmobile ion concentration of 1018 cm-3 with mobile ion mobility of 10-8 cm/s  (solid line) and \n10-10 cm/s  (dashed line). The results are shown in (a) impedance Nyquist plot, and ( b) \nimpedance imaginary part Bode plot s. The simulation parameters are in Table S9. \n  \ns34 \n \n  Interfacial recombination effect  \n(a)  \n(b)  (c)  \nFigure S 30.  Simulated impedance spectra in short -circuit under 0.2 sun illumination for a \nmobile ion concentration of 1018 cm-3 with an interfac e (c) recombination velocity of 105 cm/s  \n(solid line) and without interface recombination (dashed line). The results are sho wn in (a) \nimpedance Nyquist plot, and ( b) impedance imaginary part Bode plot s. The simulation \nparameters are in Table S9. \n \n  \ns35 \n \n HTL (NiOx) doping effect  \n(a) (b)  \n \n \n \n(b) (c)  \nFigure S 31.  Simulated electrostatic potential as a function of external voltage for for \ndifferent acceptor concentrations at the hole transport layer (HTL), i.e. the NiOx , with (a) \nNA=1.4×1020 cm-3 and Nion=1015 cm-3, (b) NA=1.4×1018 cm-3 and Nion=1015 cm-3, (c) \nNA=1.4×1020 cm-3 and Nion=1018 cm-3 and (d) NA=1.4×1018 cm-3 and Nion=1018 cm-3. Further \nsimulation parameters are in Table S9. \n \n \n  \nNA=1.4×1020 cm-3 \nNion=1021 m-3 \n NA=1.4×1018 cm-3 \nNion=1021 m-3 \n \nNA=1.4×1018 cm-3 \nNion=1024 m-3 \n NA=1.4×1020 cm-3 \nNion=1024 m-3 \n s36 \n S6. Matlabs’s Driftfusion[8] simulations  \nTable S 10: Simulation parameters used in this work for the input sheet of Driftfusion.[8]  \nLayer type  Contact  Layer  Interface  Active  Interface  Layer  Contact  \nSack  Anode NiOx  Interface 1  Perovskite  Interface 2  C60 Cath ode \nThickness (cm)  -- 2E-6 1E-6 4E-5 1E-6 3E-6 -- \nElectron affinity (eV)  -- -1.85   -3.8  -4.1 -- \nIonization potential (eV)  -- -5.4  -5.4  -5.8 -- \nWork function (eV)  -5.2 -5.2  -4.6  -4.25  -4.2 \nTrap energy level (eV)  -- -3.65   -4.6  -4.95  -- \nEffective density of states at the conduction band (cm-3) -- 1E19   1E19   1E20 -- \nEffective density of states at the valence band (cm-3) -- 1E19   1E19   1E20 -- \nEquilibrium concentration of cations (cm-3) -- 0 0 1E16 0 0 -- \nEquilibrium concentration of anions (cm-3) -- 0 0 1E16 0 0 -- \nMaximum concentration of cations (cm-3) -- 0 0 1.21E22  0 0 -- \nMaximum concentration of anions (cm-3) -- 0 0 1.21E22  0 0 -- \nCharge carrier mobility of electrons (cm2V-1s-1) -- 0.001  2  0.001 -- \nCharge carrier mobility of holes (cm2V-1s-1) -- 0.001  2  0.001 -- \nCharge carrier mobility of cations (cm2V-1s-1) -- 0 0 1E-8 0 0 -- \nCharge carrier mobility of anions (cm2V-1s-1) -- 0 0 1E-10 0 0 -- \nDielectric constant  -- 3  64  3 -- \nGeneration rate (s-1cm-3) -- 0 0 2e21 \n(1 sun)  0 0 -- \nBand -to-band radiative recombination rate (s-1cm3) -- 6.3E-\n11 0 4.8E-11 0 6.8E -11 -- \nShockley -Read -Hall non -radiative recombination \nlifetime for electrons (s)  -- 1E-6 1E-10 0.8E-7 1E-10 1E-6 -- \nShockley -Read -Hall non -radiative recombination \nlifetime for holes (s)  -- 1E-6 1E-10 0.8E-7 1E-10 1E-6 -- \nRecombination velocity for electrons (cm·s-1) 1E7 --  --  -- 1E7 \nRecombination velocity for holes (cm·s-1) 1E7 --  --  -- 1E7 \nShunt Resistance (Ωcm2) 1E6       \n \n  s37 \n \n Qualitative initial simulation of c urrent -voltage and impedance \nspectroscopy spectra  \n0.0 0.2 0.4 0.6 0.8 1.0 1.205101520\nNion (cm-3) in R and F\n   1015\n   1018\n   1019 \nExeriment in R and F \n  -J (mA cm-2)\nV (V)(a)\n  \n10-210-1100101102103104105106107100101102103104-Z'' (W cm2)\nf (Hz)Nion = 1018 cm-3Nion = 1019 cm-3(c)\n10-210-110010110210310410510610710-710-610-5C (F cm-2)\nf (Hz)Nion = 1019 cm-3Nion = 1018 cm-3(d)\n \n0 10k 20k 30k 40k010k20k-Z´´ (W cm2)\nZ' (W cm2)Nion = 1019 cm-3\nNion = 1018 cm-3(e)\n \nFigure S32.  (a) The experimental current -voltage curve of the reference structure  (dots) and \nthe numerical simulations (lines) considering different ion concentrations  and voltage sweep \ndirections . (b) Energy band diagram of the simulated structure: NiO x/perovskite/C 60 \nconsidering Nion=1015 cm-3, 0.2 sun equivalent illumination intensity, and short -circuit \ncondition. The corresponding simulated impedance spectra are shown the Bode plots of (c) \nimaginary part of impedance and (d) capacitance and the (e) impedance Nyquist plot. The \nsimulation paramet ers for the current -voltage curve include Vbi=1V, G=33×1020 cm-3s-1, \nRsh=1 M Ω·cm2, in addition to those parameters in  Table S 10. \n \n(b) s38 \n \n0.0 0.2 0.4 0.6 0.8 1.0 1.2051015\nNion (cm-3) in R and F\n   1015\n   1018\n   1019\nExeriment in R and F \n  -J (mA cm-2)\nV (V)(a)\n0 5k 10k 15k05k10k15k\n 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 2*1016 cm-3\n 1.5*1016 cm-3\n 1016 cm-3\n 1016 cm-3-Z'' (W cm2)\nZ' (W cm2)(b)\n0 200 4000200-Z'' (W cm2)\nZ' (W cm2) \n10-210-110010110210310410510610710-710-610-510-410-3 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 2*1016 cm-3\n 1.5*1016 cm-3\n 1016 cm-3\n 1015 cm-3C (F cm-2)\nf (Hz)(c)\n \n10-210-110010110210310410510610710-1100101102103104105Z'' (W cm2)\nf (Hz) 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 2*1016 cm-3\n 1.5*1016 cm-3\n 1016 cm-3\n 1015 cm-3(d)  \nFigure S33. Simulations for the the MAI and HI samples.  The current -voltage curves in (a) \ncompare t he experimental data of (dots) and the numerical simulations (lines) considering \ndifferent ion concentrations  and voltage sweep directions (solid and dashed lines for forward \nand backward directions) . The simulated impedance spectra with the characteristic three arcs \nare shown in (b) impedance Nyquist plot and the Bode plots of (a) capacitance and (d) \nimaginary parts of impedance. The simulation parameters for the current -voltage curve \ninclude scan rate o f 70 mV·s-1, Vbi=1V, µ= 10-4 cm2V-1s-1, G=4×1020 cm-3s-1, Rsh=1 M Ω·cm2, \nin addition to those parameters in Table S 10. \n s39 \n \n Incident illumination  intensity effects  \n0 5 10 15 20 2505101520\n0 1 2 301\n Nion (cm-3)\n 1012\n 1016\n 1019-Z'' (k W·cm2)\nZ' (kW·cm2)1 sun (a)\n-Z'' (k W·cm2)\nZ' (kW·cm2)\n0 40 80 120 160020406080-Z'' (k W·cm2)\nZ (kW·cm2)(b) Pin= 10 mW cm-2\n \n0 100 200 300 400 500050100150200250300 -Z'' (k W·cm2)\nZ (kW·cm2)(c) Pin= 1 mW cm-2\n0 200 400 600 8000100200300400500 -Z'' (k W·cm2)\nZ (kW·cm2)(d) Pin=0.1 mW cm-2\n \n0.0 0.2 0.4 0.6 0.8 1.00.00.5Nion=1012 cm-3-Z'' (M W·cm2)\nZ (MW·cm2)(e)\n0 10 200510\n0.0 0.2 0.4 0.60.00.2-Z'' (M W·cm2)\nZ (MW·cm2)(f) Nion=1016 cm-3\n0 10 20 30010\n \n0.0 0.2 0.40.00.2-Z'' (M W·cm2)\nZ (MW·cm2)Nion=1019 cm-3(g)\n0.0 0.5 1.00.00.51.0\n \nFigure S34. Simulated effect of the light intensity on the IS Nyquist plot.  The initial \ngeneration rate G = 2·1021 cm-3 is reduced by a factor of 1, 0.1, 0.01, 0.001 in (a -d), \nrespectively, for Nion= 1012 (blue), 1016 (green), 1019 (lime) cm-3. The same data is reordered \nfor each ion concentration in (e -g), where darker (brown) line colours indicate lower \nillumination inten sities .  The simulated s hunt resistance was 1 M Ω. \n \n s40 \n \n0 200 400 600 800 1k0200400600Nion (cm-3)\n 1012\n 1016\n 1019-Z'' (W·cm2)\nZ (W·cm2)(a)1 sun for \n0 200 400 600 800 1k0200400600(b)\n-Z'' (W·cm2)\nZ (W·cm2)Nion (cm-3)\n 1012\n 1016\n 10190.1 sun for  \n0 200 400 600 800 1k0200400600(c)\n-Z'' (W·cm2)\nZ (W·cm2)0.01 sun for Nion (cm-3)\n 1012\n 1016\n 1019\n0 200 400 600 800 1k0200400600(d)\n-Z'' (W·cm2)\nZ (W·cm2)Nion (cm-3)\n 1012\n 1016\n 10190.001 sun for \n \n0 200 400 600 800 1k0200400600for 1 sun ´\n 1\n 0.1\n 0.001\n 0.0001-Z'' (W·cm2)\nZ (W·cm2)(e)Nion=1012 cm-3\n0 200 400 600 800 1k0200400600for 1 sun ´\n 1\n 0.1\n 0.001\n 0.0001-Z'' (W·cm2)\nZ (W·cm2)(f)Nion=1016 cm-3\n \n0 200 400 600 800 1k0200400600for 1 sun ´\n 1\n 0.1\n 0.001\n 0.0001-Z'' (W·cm2)\nZ (W·cm2)(g)Nion=1019 cm-3\n \nFigure S 35. Effect of the light intensity (the initial generation rate is reduced by a factor of 1, \n0.1, 0.01, 0.001) on the Nyquist plot. At lower light intensity higher is the impact of mobile \nions on the total impedance of the system. Simulated ion concentrations are: Nion= 1012(blue), \n1016(green), 1019(red) cm-3.  The simulation parameters include s hunt resistance Rsh=1 kΩ, \nbesides those in the Table S 10. \n s41 \n  \n \n \n \n \n \n10-210-110010110210310410510610710-710-610-5for 1 sun\n 1012\n 1016\n 1019\nfor 0.001 sun\n 1012\n 1016\n 1019C (F·cm-2)\nf (Hz)1 sun\n0.001 sunNion (cm-3)(a)\n10-210-1100101102103104105106107100101102103104105106107(b)\n-Z'' (F·cm2)\nf (Hz)1 sun0.001 sunfor 1 sun\n 1012\n 1016\n 1019\nfor 0.001 sun\n 1012\n 1016\n 1019Nion (cm-3)\n \n10-210-110010110210310410510610710-710-610-5C (F·cm-2)\nf (Hz)1 sun\n0.001 sun(c)\n10-210-110010110210310410510610710-1100101102103-Z'' (F·cm2)\nf (Hz)1 sun\n0.001 sun(d)\n \nFigure S36. Simulated IS  spectra with the effect of the light intensity (the initial generation \nrate is reduced by a factor of 1, 0.001) on the capacitance and imaginary -frequency plot  for \ndifferent values of mobile ion concentration ( Nion= 1012(blue),  1016(green),  1019(red) c m-3) \nand shunt resistance ( 1 MΩ and 1 k Ω in top and bottom, respectively ). Further simulation \nparameters are in Table S 10. \n \n \n \n \n \n s42 \n  \n Shunt resistance eff ect \n0 20k 40k 60k020k40k -Z'' (W·cm2)\nZ' (W cm2) 1015\n 1016\n 1.2´1016\n 2´1016\n 1017\n 1018\n 1019Nion (cm-3) \nfor Rsh =1 M W cm2(a)\n0 2k 4k 6k 8k02k4k6kNion (cm-3) \nfor Rsh =10 k W cm2-Z'' (W·cm2)\nZ (W·cm2) 1015\n 1016\n 1.2´1016\n 2´1016\n 1017\n 1018\n 1019(b)\n \n0 20 40 60 80 100020406080 -Z'' (W·cm2)\nZ' (W cm2)Nion (cm-3) \nfor Rsh =100 W cm2(c) 1015\n 1016\n 1.2´1016\n 2´1016\n 1017\n 1018\n 1019\n \n10-110010110210310410510610-710-610-5\n    1015\n     1016\n     1.2´1016\n     2´1016\n     1017\n     1018\n     1019C (F/cm2)\nf (Hz)                      Nion (cm-3) for \nRsh (W)= 1M,      1k,     100(d)\n                      Nion (cm-3) for \nRsh (W)= 1M,      10k,     100\n10-110010110210310410510610-310-210-1100101102103104\n    1015\n     1016\n     1.2´1016\n     2´1016\n     1017\n     1018\n     1019-Z'' (W cm-2)\nf (Hz)(e)\n \n \nFigure S37. Simulated impedance spectra fo r the reference sample  considering different \nshunt resistance and ion concentration s, as indicated . The physical simulation parameters are \nin the Table S 10. \n \n \n  s43 \n \n0.0 0.2 0.4 0.6 0.8 1.0-3-2-10\nNion (cm-3) for\nRsh= 104, 106 Wcm2\n   1015\n   1016\n  1019J (mA cm-2)\nV (V)(a) \n0.0 0.2 0.4 0.6 0.8 1.010-510-410-310-210-1100\nNion (cm-3) for\nRsh= 104, 106 Wcm2\n   1015\n   1016\n  1019J (mA cm-2)\nV (V)(b)\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.010.1110100\nNion (cm-3) for\nRsh= 104, 106 Wcm2\n   1015\n   1016\n  1019R (kW cm2)\nV (V)(c)\n \n0.0 0.2 0.4 0.6 0.8 1.0 1.2100101102103104105106107\nNion (cm-3) for\nRsh= 104, 106 Wcm2\n   1015\n   1016\n  1019R (W cm2)\nV (V)(b)  \n \nFigure S38. Simulated current -voltage curves (a) under 0.2 sun illumination intensity and (b) \nin dark, with DC resistances (in c, d, respectively) for the reference sample considering \ndifferent shunt resistance and  mobile  ion concentration s, as indicated . The physical \nsimulation parameters are in the Table S 10. \n \n \n \n \n \n \n \n s44 \n  \n0 2k 4k 6k02k4k6k\n 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 5*1016 cm-3\n 2*1016 cm-3\n 1016 cm-3\n 1015 cm-3-Z'' (W cm2)\nZ' (W cm2)(b)\n0 1k 2k 01k -Z'' (W cm2)\nZ' (W cm2)\n10-210-110010110210310410510610710-710-610-510-4 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 2*1016 cm-3\n 1016 cm-3\n 1015 cm-3C (F cm-2)\nf (Hz)(c)\n10-1100101102103104105106101102103104Z'' (W cm2)\nf (Hz) 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 2*1016 cm-3\n 1016 cm-3\n 1015 cm-3(c)\n \nFigure S39. Simulated impedance spectra considering different high shunt resistance  (1 \nMΩ) ion concentrations  in (a) impedance Nyquist representation, (b) capacitance Bode Plot \nand (c) imaginary part of impedance versus frequency . The physical parameters are in the \nTable S 10. \n \n  s45 \n \n0 250 500 7500200400 1015\n 1016\n 3´1016\n 5´1016\n 1017\n 1018\n 1019-Z'' (W·cm2)\nZ' (W·cm2) \n10-210-110010110210310410510610710-710-610-510-4C (F·cm-2)\nFrequency (Hz)\n \n10-210-1100101102103104105106107100101102-Z'' (F·cm2)\nFrequency (Hz)\n \nFigure S40. Simulated impedance spectra for the HI or MAI samples with c ombined effect \nof low shunt resistance  (1 kΩ cm2), positive/negative ion mobility (10-6/10-8 cm2V-1s-1), low \nelectron/hole mobility (0.2 cm2V-1s-1) and high light intensity G 0 = 2·1021 cm-3s-1. Simulated \nion concentrations are: Nion= 1015, 1016, 3×1016, 5×1016, 1018, 1019 cm-3.   \n s46 \n \n0 250 500 750 10000200400600 1015\n 1016\n 3´1016\n 5´1016\n 1017\n 1018\n 1019-Z'' (W·cm2)\nZ' (W·cm2) \n10-210-110010110210310410510610710-710-6C (F·cm-2)\nFrequency (Hz)\n10-210-110010110210310410510610710-1100101102103-Z'' (F·cm2)\nFrequency (Hz)\n \nFigure S41. Simulated impedance spectra for the HI or MAI samples with lower generation \nrate than those of Figure S40. At low light intensity , or upon reduction of optical absorption , \nthe lower the shunt resistance the more negligible the three -arcs effects within the explored \nperturbation frequency for high concentration of mobile ions and electron mobility.  Low \nshunt resistance : 1 kΩ cm2, positive/negative ion mobility : 10-6/10-8 cm2V-1s-1, electron/hole \nmobility 2 cm2V-1s-1 and low light intensity G 0 = 4·1020 cm-3s-1. Simulated ion concentrations \nare: Nion= 1015, 1016, 3×1016, 5×1016, 1018, 1019 cm-3.   \n s47 \n \n0 250 500 750 10000200400600 1015\n 1016\n 3´1016\n 5´1016\n 1017\n 1018\n 1019-Z'' (W·cm2)\nZ' (W·cm2) \n10-210-110010110210310410510610710-710-610-5C (F·cm-2)\nFrequency (Hz)\n \n10-210-110010110210310410510610710-1100101102-Z'' (F·cm2)\nFrequency (Hz)\n \nFigure S42. Simulated impedance spectra for the HI of MAI sample with c ombined effect of \nlow shunt resistance (1 kΩcm2), ion mobility (positive/negative ion mobility: 10-6/10-8 cm2V-\n1s-1), electron/hole m obility (2 cm2V-1s-1) and low light intensity G 0 = 2·1021 cm-3s-1. \nSimulated ion concentrations are: Nion= 1015, 1016, 3×1016, 5×1016, 1018, 1019 cm-3.   \n \n \n s48 \n \n Mobile ion concentration (in the perovskite) effect s \n \n0 25k 50k 75k025k50k75k 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 1016 cm-3\n 1015 cm-3-Z'' (W cm2)\nZ' (W cm2)(d)\n10-110010110210310410510610-710-610-5 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 1016 cm-3\n 1015 cm-3C (F cm-2)\nf (Hz)(e)\n \n \n10-1100101102103104105106101102103104Z'' (W cm2)\nf (Hz) 1019 cm-3\n 1018 cm-3\n 1017 cm-3\n 1016 cm-3\n 1015 cm-3(f)  \nFigure S43. Schemed impedance spectra with effect s due to  different mobile ion \nconcentration s (a-c) and corresponding numerical simulations  (d-f). The plots show (d) \nNyquist re presentation , (e) capacitance vs frequency, and (f) imaginary part of impedance vs \nfrequency . Further simulation  parameters are in Table S 10. \n \ns49 \n \n0 10 20 30 40 50 60 700102030405060\n0 1 2 3 40123-Z'' (kW cm2)\nZ' (kW cm2)Nion (cm-3)\n 1015\n 1016\n 1017\n 1018\n 2´1018\n 5´1018\n 1019(a)\n-Z'' (kW cm2)\nZ' (kW cm2)\n10-210-110010110210310410510610710-810-710-610-5   Nion (cm-3)\n 1015\n 1016\n 1017\n 1018\n 2´1018\n 5´1018\n 1019C (F cm-2)\nf (Hz)(b)\n10-1100101102103104105106101102103104-Z'' (W cm2)\nf (Hz)Nion (cm-3)\n 1015\n 1016\n 1017\n 1018\n 2´1018\n 5´1018\n 1019(c) \nFigure S44. Simulated impedance spectra in (a) Nyquist  plot and Bode plots of (b) \ncapacitance and (c)  imaginary part of impedance for different perovskite mobile ion \nconcentrations, as indicated. The simulation  parameters include Rsh = 106 Ω·cm2; R s = 2.5 \nΩ·cm2; εr=100; G = 4×1020 cm-3s-1, besides those  in the Table S 10. \n \n s50 \n \n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.0-J (mA cm-2)\nV (V) Nion=1015 cm-3\n Nion=1016 cm-3\n Nion=1017 cm-3\n Nion=1018 cm-3\n Nion=1019 cm-3(a)\n0.0 0.2 0.4 0.6 0.8 1.00.0000.0010.0020.003-J (mA cm-2)\nV (V) Nion=1015 cm-3\n Nion=1016 cm-3\n Nion=1017 cm-3\n Nion=1018 cm-3\n Nion=1019 cm-3(b) \nFigure S45. Simulated current -voltage curve s under 20% of 1 sun illumination intensity for  \n(a) the reference sample   and (b) the HI/MAI samples considering different values of mobile \nion concentration, as indicated.  The simulation parameters include scan rate of 70 mV s-1, \ngeneration rate of 4×1020 cm-3s-1, in addition to those values in Table S 10. Notably shunt \nresistance is Rsh=1 M Ω·cm2 and 1k Ω·cm2, and electron mobility is µe=2 and 0.2 cm2V-1s-1 in \n(a) and (b), respectively.   \n \n  s51 \n \n Interface (perovskite/HTL) charge carrier recombination effects  \n0 250 500 750 10000200400600800\n(Rsh= 1kW cm2) 1015\n 1016\n 1018\n 1019-Z'' (W·cm2)\nZ (W·cm2)Nion (cm-3) for t = 10-6, 10-9, 10-10s (a)\n \n0 50k 100k 150k 200k050k100k150k\n(Rsh= 1M W cm2)-Z'' (W·cm2)\nZ' (W·cm2) 1015\n 1016\n 1018\n 1019Nion (cm-3) for t = 10-6, 10-9, 10-10s (b)  \n \n10-110010110210310410510610-710-6\n(Rsh= 1kW cm2)Nion (cm-3) for t = 10-6,  10-9,  10-10s \n 1019\n 1018\n 1016\n 1015C (F·cm-2)\nf (Hz)(c)\n \n10-110010110210310410510610-710-6\n(Rsh= 1M W cm2)C (F·cm-2)\nf (Hz) 1019\n 1018\n 1016\n 1015Nion (cm-3) for t = 10-6,  10-9,  10-10s (d)  \n10-110010110210310410510610-1100101102-Z'' (W·cm2)\nf (Hz) 1015\n 1016\n 1018\n 1019Nion (cm-3) for t= 10-6,  10-9,  10-10s (e)\n(Rsh= 1kW cm2)\n \n10-1100101102103104105106101102103104105\n(Rsh= 1M W cm2)-Z'' (W·cm2)\nf (Hz) 1015\n 1016\n 1018\n 1019Nion (cm-3) for t = 10-6,  10-9,  10-10s (f)  \n \nFigure S 46. Simulated impedance spectra in SC for different concentrations of mobile ions \n(Nion= 1015, 1016, 1018, 1019 cm-3) and trap  recombination lifetime values at the interface ( τ = \n10-6, 10-9, 10-10 s) considering a shunt resistance of  Rsh = 1 kΩ and 1 MΩ, for the left and right \npanels, respectively. Further simulation parameters are in the Table S 10.  \n \n \n s52 \n \n0.0 0.2 0.4 0.6 0.8 1.005101520\nt  for \nNion =1015, 1019cm-3\n         10-6 s\n         10-8 s\n         10-10 s-J (mA cm-2)\nV (V)(a) \n0.0 0.2 0.4 0.6 0.8 1.00123\nt  for \nNion =1015, 1019cm-3\n         10-6 s\n         10-8 s\n         10-10 s-J (mA cm-2)\nV (V)(b)\n \nFigure S 47. Simulated current -voltage curve  under (a) 1 and (b) 0.2 sun illumination for the \nreference sample considering different values of mobile ion concentration and trap \nrecombination lifetime at the interface between the perovskite and the NiO x transport layer, \nas indicated.  The simulation parameters include scan rate of 70 mV s-1, generation rate of \n2×1021 cm-3s-1, in addition to those values in Table S 10. \n s53 \n \n Bulk (of the perovskite) charge carrier recombination effects.  \n0 20k 40k 60k020k40k60k\n0 1k 2k 3k01k2k-Z'' (W·cm2)\nZ (W·cm2)   10-6\n   10-7\n   10-8\n   10-9t (s) for Nion = \n1015cm-3, 1018cm-3 (a)\n-Z'' (W·cm2)\nZ (W·cm2)\n0 20k 40k 60k020k40k60k\n0 1k 2k 3k01k2k-Z'' (W·cm2)\nZ (W·cm2)   10-6\n   10-7\n   10-8\n   10-9t (s) for Nion = \n1016cm-3, 1019cm-3 (b)\n-Z'' (W·cm2)\nZ (W·cm2)\n \n10-110010110210310410510610-710-610-510-4\n    10-6\n    10-7\n    10-8\n    10-9C (F·cm-2)\nf (Hz)t (s) for Nion= 1015 ,  1016 ,     1018 ,    1019 cm-3(c)\n10-1100101102103104105106100101102103104-Z'' (F·cm2)\nf (Hz)(d)\n    10-6\n    10-7\n    10-8\n    10-9t (s) for Nion= 1015 ,  1016 ,     1018 ,    1019 cm-3\n \nFigure S 48. Simulated impedance spectra in short -circuit for de reference sample with \ndifferent concentration  of mobile ions and bulk trap recombination lifetime values , as \nindicated .  The physical simulation parameters are in the Table S 10. \n \n \n \n \n \n s54 \n \n0.0 0.2 0.4 0.6 0.8 1.0 1.205101520\n   10-6\n   10-8\n   10-9-J (A cm-2)\nV (V)t for Nion= 1015, 1019 cm-3 (a) \n0.0 0.2 0.4 0.6 0.8 1.0 1.20123-J (mA cm-2)\nV (V)   10-6\n   10-8\n   10-9t for Nion= 1015, 1019 cm-3 (b)\n \nFigure S 49. Simulated current -voltage curve  under (a) 1 and (b) 0.2 sun illumination for the \nreference sample considering different values of mobile ion concentration and trap \nrecombination lifetime τ within the perovskite bulk, as indicat ed. The simulation parameters \ninclude scan rate of 70 mV s-1, generation rate of 4×1020 cm-3s-1, in addition to those values \nin Table S 10. \n \n  s55 \n \n Electron/hole mobility (of the perovskite) effect s \n0 50 100 150 200050100150200\n0 2 4024-Z'' (W·cm2)\nZ (W·cm2)e,h (cm2V-1s-1) \nfor Nion=1015 cm-3\n 0.01 \n 0.1 \n 1 \n 10\n 0.01\n 0.1\n 1\n 10(a)\n-Z'' (W·cm2)\nZ (W·cm2)for Nion=1019 cm-3\n \n0 50 100 150 200050100150200\n0 2 4024-Z'' (W·cm2)\nZ (W·cm2)e,h (cm2V-1s-1) \nfor Nion=1016 cm-3\n 0.01 \n 0.1 \n 1 \n 10\n 0.01\n 0.1\n 1\n 10(a)\n-Z'' (W·cm2)\nZ (W·cm2)for Nion=1019 cm-3  \n10-110010110210310410510610-810-710-610-510-4\nfor Nion=1018 cm-3e,h (cm2V-1s-1) for Nion=1015 cm-3\n 0.01\n 0.1\n 1\n 10 0.01 \n 0.1 \n 1 \n 10C (F·cm-2)\nf (Hz)(b)\n10-110010110210310410510610-810-710-610-510-4C (F·cm-2)\nf (Hz)(b)e,h (cm2V-1s-1) for Nion=1016 cm-3\n 0.01 \n 0.1 \n 1 \n 10\n for Nion=1019 cm-3\n 0.01\n 0.1\n 1\n 10\n \n10-1100101102103104105106100101102103104105-Z'' (W·cm2)\nf (Hz)for Nion=1018 cm-3e,h (cm2V-1s-1) for Nion=1015 cm-3\n 0.01\n 0.1\n 1\n 10 0.01 \n 0.1 \n 1 \n 10(c)\n \n10-1100101102103104105106100101102103104105-Z'' (W·cm2)\nf (Hz)for Nion=1019 cm-3e,h (cm2V-1s-1) for Nion=1016 cm-3\n 0.01\n 0.1\n 1\n 10 0.01 \n 0.1 \n 1 \n 10(c)  \nFigure S 50. Simulated impedance spectra  in (a) Nyquist and Bode plots of (b) capacitance and \n(c) imaginary part of impedance considering different mobile ion concentration and \nelectron/hole mobilities in the perovskite, as indicated. The simulation parameters include Rsh= \n1 MΩ , in addition to those parameters in the Table S 10. \n s56 \n \n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-2.5-2.0-1.5-1.0-0.50.0\n (cm2V-1s-1) for\nNion= 1015, 1016, 1019cm-3\n      0.01\n      0.1\n     1\n     10J (mA cm-2)\nV (V) \nFigure S51. Simulated current -voltage curves  for the reference sample under 0.2 sun \nillumination intensity considering different mobile ion concentration and electron and hole \nmobilities in the perovskite bulk , as indicated. Further simula tion parameters can be found \nin in the Table S 10. \n \n  s57 \n \n Hole mobility effects at the NiO x hole transport layer   \n0 250 500 750 10000200400600800Nion (cm-3) for h= 10-3,   10-5,   10-6 cm2V-1s-1-Z'' (W·cm2)\nZ' (W·cm2)(Rsh= 1kW cm2) 1015\n 1016\n 1018\n 1019(a)\n \n0 20k 40k 60k020k40k60k\n0.0 1.0k 2.0k 3.0k0.01.0k-Z'' (W·cm2)\nZ' (W·cm2) 1015\n 1016\n 1018\n 1019Nion (cm-3) for  h= 10-3, 10-5, 10-6 cm2V-1s-1\n(Rsh= 1M W cm2)(b)\n-Z'' (W·cm2)\nZ' (W·cm2)\n10-110010110210310410510610-710-610-5C (F·cm-2)\nf (Hz)(c)\n 1015\n 1016\n 1018\n 1019Nion (cm-3) for h= 10-3,  10-5,  10-6 cm2V-1s-1\n(Rsh= 1kW cm2)\n \n10-110010110210310410510610-710-610-5C (F·cm-2)\nf (Hz) 1015\n 1016\n 1018\n 1019Nion (cm-3) for  h= 10-3, 10-5, 10-6 cm2V-1s-1\n(Rsh= 1M W cm2)(d)  \n10-110010110210310410510610-1100101102-Z'' (W·cm2)\nf (Hz) 1015\n 1016\n 1018\n 1019Nion (cm-3) for h= 10-3,   10-5,   10-6 cm2V-1s-1\n(Rsh= 1kW cm2)(e)\n \n10-1100101102103104105106100101102103104-Z'' (W·cm2)\nf (Hz) 1015\n 1016\n 1018\n 1019Nion (cm-3) for  h= 10-3, 10-5, 10-6 cm2V-1s-1\n(Rsh= 1M W cm2)(f)  \n \nFigure S 52. Simulated impedance spectra in SC for different concentrations of mobile ions \n(Nion= 1015, 1016, 1018, 1019 cm-3), shunt resistance ( Rsh = 1 kΩ and 1 MΩ) and hole mobility \nat the hole transport material ( µh= 10-3, 10-4, 10-5, 10-6 cm2/Vs), as indicated.  The simulation \nparameters are in the Table S 10. \n s58 \n \n0.0 0.2 0.4 0.6 0.8 1.005101520\nNion=1015 cm-3\n h= 10-3 cm2/Vs\n h= 10-4 cm2/Vs\n h= 10-5 cm2/Vs\nNion=1019 cm-3\n h= 10-3 cm2/Vs\n h= 10-4 cm2/Vs\n h= 10-5 cm2/Vs-J (mA cm-2)\nV (V)(a) \n0.0 0.2 0.4 0.6 0.8 1.00123\nNion=1015 cm-3\n h= 10-2 cm2/Vs\n h= 10-3 cm2/Vs\n h= 10-4 cm2/Vs\n h= 10-5 cm2/Vs\nNion=1019 cm-3\n h= 10-2 cm2/Vs\n h= 10-3 cm2/Vs\n h= 10-4 cm2/Vs\n h= 10-5 cm2/Vs-J (mA cm-2)\nV (V)(b)\n \nFigure S 53. Simulated current -voltage curve  under (a) 1 and (b) 0.2 sun illumination for the \nreference sample in SC considering different values of mobile ion concentration and hole \nmobility at the NiO x transport layer, as indicated. The simulation parameters include scan rate \nof 70 mV s-1, generation rate of 2×1021 cm-3s-1, shunt resistance of 1M Ωcm2, in addition to \nthose values in Table S 10. \n \n s59 \n \n Dielectric permittivity (of the perovskite)  effect s \n0 50 100050100for er=100\n 1015\n 1016\n 1017\n 1018\n 1019for er=20\n 1015\n 1016\n 1017\n 1018\n 1019-Z'' (k W cm2)\nZ' (kW cm2)Nion (cm-3) (a)\n \n0 2 4 6 802468-Z'' (k W cm2)\nZ' (kW cm2)(a)  \n \n \n10-110010110210310410510610-710-610-5 For er=20\n 1015\n 1016\n 1017\n 1018\n 1019For er=100\n 1015\n 1016\n 1017\n 1018\n 1019C (F·cm-2)\nf (Hz)Nion (cm-3)(b)\n \n10-1100101102103104105106101102103104105- Z'' (W·cm2)\nf (Hz)For er=100\n 1015\n 1016\n 1017\n 1018\n 1019For er=20\n 1015\n 1016\n 1017\n 1018\n 1019Nion (cm-3)(c)  \n \n \n \nFigure S54. Simulated impedance spectra  in (a) Nyquist and Bode plots of (b) capacitance \nand (c) imaginary part or impedance for the reference sample considering different mobile \nion concentrations and  dielectric permittivity of the perovskite , as indicated . The lower the \nmobile ion concentration the stronger the influence of the d ielectric permittivity.  The \nsimulation parameters  include R sh=1 M Ω cm2, besides those in Table S 10. \n \n \n \n \n \n \n s60 \n Acknowledgment   \nS. Mohamed acknowledges the financial support from Programa Martí i Franquès. M. Ramírez -\nComo acknowledges the financial support from Diputació de Tarrago na under Grant \n2021CM14 and 2022PGR -DIPTA -URV04. This work was further supported by the Spanish \nMinisterio de Ciencia e Innovación (MICINN/FEDER) under Grants PDI2021 -128342OB -I00 \nand RTI2018 -094040 -B-I00, by the Agency for Management of University and Res earch Grants \n(AGAUR) ref. 2017 -SGR -1527, and from the Catalan Institution for Research and Advanced \nStudies (ICREA) under the ICREA Academia Award . O.A. thanks the National Research \nAgency (Agencia Estatal de Investigación ) of Spain for the Juan de la Cierva  2021 grant  \n(FJC2021 -046887 -I). P.LV. thanks the French Government in the frame of the program of \ninvestment for the future (Programme d’I nvestissement d’Avenir – ANR -IEED -002-01). J.M. \nand S.O. thank the Ministry of Economic Affairs Innovation, Digitalization and Energy of the \nState of North Rhine -Westphalia for funding under the grant SCALEUP (SOLAR -ERA.NET \nCofund 2, id: 32) . \n \n \n \nReferences  \n[1] Firoz Khan, Seong -Ho Baek, Jae Hyun Kim, Intensity dependency of \nphotovoltaic cell parameters under high illumination conditions: An analysis, \nAppl. Energy  2014 , 133, 356, https://doi.org/10.1016/j.apenergy.2014.07.107  \n[2] S. N. Agbo, T. Merdzhanova, U. Rau, O. Astakhov, Illumination intensity \nand spectrum -dependent performance of thin -film silicon single and multijunction \nsolar cells, Sol. Energy Mater. Sol. Cel ls 2017 , 159, 427, \nhttps://doi.org/10.1016/j.solmat.2016.09.039  \n[3] M. Chegaar, A. Hamzaoui, A. Namoda, P. Petit, M. Aillerie, A. Herguth, \nEffect of Illumination Intensity on Solar Cells Paramete rs, Energy Procedia  2013 , \n36, 722, https://doi.org/10.1016/j.egypro.2013.07.084  \n[4] Firoz Khan, S. N. Singh, M. Husain, Effect of illumination intensity on cell \nparameters of a silicon solar cell , Sol. Energy Mater. Sol. Cells  2010 , 94, 1473, \nhttps://doi.org/10.1016/j.solmat.2010.03.018  \n[5] Kai Shen, Qiang Li, Dezhao Wang, Ruilong Yang, Yi Deng, Ming -Jer Jeng, \nDeliang Wang, CdTe solar ce ll performance under low -intensity light irradiance, \nSol. Energy Mater. Sol. Cells  2016 , 144, 472, \nhttps://doi.org/10.1016/j.solmat.2015.09.043  \n[6] Dana Lübke, Paula Hartnagel, Markus Hülsbeck, Thomas Kirchartz, \nUnderstanding the Thickness and Light -Intensity Dependent Performance of \nGreen -Solvent Processed Organic Solar Cells, ACS Materials Au  2023 , 3, 215, \nhttps://doi.org/10.1021/acsmaterialsau.2c00070  s61 \n [7] SETFOS: Simulation Software for Organic and Perovskite Solar Cells and \nLEDs, https://www.fluxim.com/setfos -intro , accessed: 15.11.202 3. \n[8] Philip Calado, Ilario Gelmetti, Benjamin Hilton, Mohammed Azzouzi, \nJenny Nelson, Piers R. F. Barnes, Driftfusion: an open source code for simulating \nordered semiconductor devices with mixed ionic -electronic conducting materials \nin one dimension, J. Comput. Electron.  2022 , 21, 960, \nhttps://doi.org/10.1007/s10825 -021-01827 -z \n "    },    {        "title": "2402.00460v1.Octet_baryon_and_heavy_meson_interactions_in_chiral_effective_field_theory.pdf",        "content": "arXiv:2402.00460v1  [hep-ph]  1 Feb 2024Octet baryon and heavy meson interactions in\nchiral effective field theory\nBo-Lin Huang∗1, Bo Wang†2,3,4, and Shi-Lin Zhu‡5\n1School of Physical Science and Technology, Inner Mongolia U niversity, Hohhot 010021, China\n2College of Physics Science &Technology, Hebei University, Baoding 071002, China\n3Hebei Key Laboratory of High-precision Computation and App lication of Quantum Field Theory,\nBaoding, 071002, China\n4Hebei Research Center of the Basic Discipline for Computati onal Physics, Baoding, 071002, China\n5School of Physics and Center of High Energy Physics, Peking U niversity, Beijing 100871, China\nFebruary 2, 2024\nAbstract\nWe calculate the effective potentials of the octet baryon and heavy meson systems using\nthe chiral effective field theory up to the next-to-leading or der. We consider the contact\nterms, one-pseudoscalar-meson and two-pseudoscalar-mes on exchange contributions, facil-\nitating a comprehensive analysis of the short-, mid-, and lo ng-range interactions in these\nsystems. The low energy constants (LECs) are correlated wit h those of the ¯NNinteraction\nusing a quark-level Lagrangian approach. We also incorpora te the decuplet baryon contribu-\ntions in the loop diagrams. Our research provides new insigh ts into several near-threshold\ncharmed baryons [e.g., Λ c(2940), Ξ c(3055), and Ω c(3188), etc.] around 3 GeV from the\nhadronic molecular perspective. We also identify several m olecular states, designated as\nΞc, within the mass range of 3100 −3500 MeV. Further measurements of their resonance\nparameters and decay patterns in experiments will help to di scriminate the conventional\nbaryon and hadronic molecule explanations for these near-t hreshold states.\nKeywords: Chiral effective field theory, Octet baryon and heavy meson in teraction, Bound\nstates\n1 Introduction\nInvestigations of the baryon-meson interactions provide significa nt insights into quantum\nchromodynamics (QCD) at hadronic scales, forming a critical found ation for advancing hadron\nspectroscopy. The quark model, known for its efficacy in demystify ing hadron spectra [ 1],\nfaces substantial challenges in incorporating some near-thresho ld states such as X(3872) [2]\nandDs0(2317) [3–5] within its theoretical framework [ 6–12]. These complications are similarly\nevident in charmed baryons, exemplified by Λ c(2940)and Ξ c(3055), which mirrorthe perplexities\nencountered with X(3872) and Ds0(2317).\nThe charmed baryon Λ c(2940)+was first reported in 2007 by the BABAR Collaboration [ 13].\nIt wasidentified in the D0pinvariantmassspectrum. The Λ c(2940)+wasanisosinglet, evidenced\nby the absence of any signals in the D+pfinal state. This discovery was later corroborated by\nthe Belle experiment, which uncovered the decay mode of Λ c(2940)+→Σ0,++\ncπ+,−[14]. Further\nadvancements were made in 2017 when the LHCb Collaboration shed lig ht on the quantum\nnumbersJPof Λc(2940)+, suggesting JP= 3/2−as the most likely spin-parity assignment [ 15].\nThere are two primary hypotheses on the internal structure of Λ c(2940): either a traditional\ncharmed baryon or a D∗Nmolecular state. The latter proposition arises primarily due to the\nchallenge of categorizing Λ c(2940) as a 2 Pstate in the charmed baryon spectroscopy. Its mass\n∗blhuang@imu.edu.cn\n†wangbo@hbu.edu.cn\n‡zhusl@pku.edu.cn\n1falls short by approximately 60 −100 MeV compared to the quark model predictions [ 16–19].\nThe molecular state hypothesis gained traction considering that Λ c(2940) is situated roughly\n6 MeV below the D∗0pthreshold. The authors of ref. [ 20] analyzed its decay patterns and\nsuggested Λ c(2940) as a 1 /2−molecular state. He et al.[21] examined the D∗Ninteraction via\nthe one-boson-exchange model and endorsed the view of Λ c(2940) as a D∗Nbound state with\nI(JP) = 0(1/2+) or 0(3/2−). Ortega et al.[22] delved into the D∗Nmolecular framework using\nthe constituent quark model and deduced a binding solution in the iso scalarJP= 3/2−channel.\nThis molecular picture is further supported by the analyses of the s trong and radiative decays of\nΛc(2940) in refs. [ 23,24]. Moreover, a QCD sum rule study [ 25] posits that Λ c(2940) does not\nmanifest as a compact state. This perspective gains additional bac king from recent calculations\nwithin the chiral quark model [ 26]. For a more in-depth exploration and diverse viewpoints on\nthe nature and characteristics of Λ c(2940), one might consult the comprehensive reviews and\nrelated studies in refs. [ 27–49].\nBaryons composed of one up or down quark, one strange quark, a nd one charmed quark are\nclassified as Ξ cbaryons [ 1]. The landscape of charmed strange baryons has witnessed signifi-\ncant enrichment in recent years with the discovery of several high ly excited states. Notably, the\nΞc(2980) and Ξ c(3080) baryons were first identified by the Belle Collaboration [ 50] and subse-\nquently confirmed by the BABAR Collaboration [ 51]. Additionally, the BABAR Collaboration\nobserved the Ξ c(3055)+and Ξ c(3123)+baryons [ 51]. However, only the Ξ c(3055)+was later\nconfirmed by the Belle Collaboration [ 52]. The Belle Collaboration [ 53] reported three charmed\nstrange baryons. The Ξ c(3055)0was first observed in the Λ D0decay mode, while the Ξ c(3055)+\nand Ξc(3080)+were initially detected in the Λ D+decay mode. A notable aspect of these exper-\niments is that the JP(total angular momentum and parity) quantum numbers for most o f the\nobserved charmed baryons have not been determined.\nSince its discovery, the Ξ c(3055) baryon has been the subject of extensive study. Liu et al.\nproposed that the Ξ c(3055) is a D-wave state based on their analysis of its strong decay [ 54].\nVarious studies have suggested different possibilities for the JPof the Ξ c(3055), including 3 /2+,\n5/2+, and 7/2+, as detailed in refs. [ 18,55–61]. Additionally, Ye et al.proposed in refs. [ 62,63]\nthat the Ξ c(3055) is not a P-wave excited Ξ cbaryon, but rather a 2 S-wave state in the3P0\nmodel. Contrary to these views, a molecular state description of th e Ξc(3055) was provided in\nref. [64], suggesting its JPas 1/2−. Given these diverse perspectives, it is clear that further\nresearch is essential to fully comprehend the nature of the Ξ c(3055) baryon.\nThe explorationof the octet baryonand heavy meson interactions is fundamental to resolving\nthe mysteries associated with these charmed baryons. Moreover , a deep understanding of the\noctet baryon and heavy meson interactions is key to investigating t heD-mesic nuclei [ 65,66] as\nwell as the characteristics of charmed mesons in nuclear environme nts [67,68]. An alternative\napproach,themeson-exchangemodel[ 69], hasbeeneffectivelyutilizedbytheJ¨ ulichgroup[ 70–72]\nfor the development of the DNandD∗Ninteraction framework.\nThe conceptual focus of modern nuclear force theory has shifte d from the boson-exchange\nmodel to an approach more steeped in the foundational work of We inberg [73,74] and mainly\nshaped within the frameworkofeffective field theory. Particularly, the chiraleffective field theory\nhas been widely and effectively utilized to examine the nucleon-nucleon (NN) interaction with\nsignificant success [ 75–80]. This theory has also shown considerable promise in the investigation\nof systems incorporating heavy flavors, as exemplified in various st udies [81–90] (see ref. [ 12]\nfor a recent review). The utility of the approach becomes apparen t in its diverse application:\nprojectingBB∗andB∗B∗bound states [ 83], deciphering the newly discovered pentaquarks [ 85],\nand extrapolating the Σ cNpotential from the lattice QCD data to the physical pion mass [ 86]\nbeing some prominent examples.\nIn ref. [91], we examined the D(∗)Ninteractions through chiral effective field theory. We\nextend our earlier research in this work. The findings from refs. [ 92–102] suggest that employing\nthe SU(3) framework for such calculations yields plausible prediction s. This current study aims\nto further explorethe interactionsbetween octet baryonsand h eavymesons up to next-to-leading\norder within the chiral effective field theory. Our approach is compr ehensive, incorporating the\nlong-, mid-, and short-range interactions. Additionally, we include t he contributions of the\ndecuplet baryons, considering them as intermediate states in the lo ops. Utilizing chiral effective\nfield theory, we not only calculate the effective potentials between t he octet baryons and heavy\nmesons but also investigate the possible bound states.\nThis paper is structured as follows. Section 2introduces the chiral Lagrangians. Section 3\ncovers Feynman diagrams and expressions of the effective potent ials. Section 4presents and\n2discusses our findings. The final section provides a brief summary. The Appendix includes the\nparameters for the potentials.\n2 Chiral Lagrangian\nOur calculation of the octet baryon and heavy meson interaction is b ased on the SU(3)\neffective chiral Lagrangian in the heavy hadron formulation,\nLeff=LBφ+LBφT+LHφ+LBH. (1)\nThe traceless hermitian 3 ×3 matricesBandφinclude the octet baryon fields ( N,Λ,Σ,Ξ) and\nthe light pseudoscalar Goldstone boson fields ( φ=π,K,¯K,η), respectively. The Trepresents\nthe decuplet baryon fields (∆ ,Σ∗,Ξ∗,Ω). TheHis the superfield for the charmed mesons. The\nlowest-order SU(3) chiral Lagrangian for the octet baryon and lig ht meson interaction take the\nform [103]\nL(1)\nBφ= tr(i¯B[v·D,B])+2Dtr(¯BSµ{uµ,B})+2Ftr(¯BSµ[uµ,B]), (2)\nwhere tr( ···) represents the trace in flavor space, vµ= (1,0,0,0) is the heavy meson velocity,\nDµdenotes the covariant derivative\n[Dµ,B] =∂µB+[Γµ,B], (3)\nandSµis the covariant spin operator. In practice one works with /vector σ(the Pauli spin matrices)\nSµ= (0,/vector σ\n2). (4)\nThe chiral connection Γµ= [ξ†,∂µξ]/2 and the axial vector quantity uµ=i{ξ†,∂µξ}/2 contain\nan even and odd number of meson fields, respectively. The SU(3) ma trixU=ξ2= exp(iφ/f)\ncollects the light pseudoscalarGoldstone boson fields. The paramet erfis the pseudoscalardecay\nconstant in the chiral limit. The axial vector coupling constants DandFcan be determined\nby fitting the semi-leptonic decays of the octet baryons [ 104]. The lowest-order SU(3) chiral\nLagrangian involving the decuplet baryon takes the form [ 105]\nL(1)\nBφT=−¯Tµ(iv·D−δB)Tµ+C(¯TµuµB+¯BuµTµ)+2H¯TµS·uTµ, (5)\nwhereδBdenotes the mass difference between the decuplet and octet bary on in the chiral limit.\nThe chiral covariant derivative with the decuplet baryon field reads\niDµTν\nabc=i∂µTν\nabc+(Γµ)d\naTν\ndbc+(Γµ)d\nbTν\nadc+(Γµ)d\ncTν\nabd, (6)\nwith\nTabc=\n∆++∆+\n√\n3Σ∗+\n√\n3\n∆+\n√\n3∆0\n√\n3Σ∗0\n√\n6\nΣ∗+\n√\n3Σ∗0\n√\n6Ξ∗0\n√\n3\n\n∆+\n√\n3∆0\n√\n3Σ∗0\n√\n6\n∆0\n√\n3∆−Σ∗−\n√\n3\nΣ∗0\n√\n6Σ∗−\n√\n3Ξ∗−\n√\n3\n\nΣ∗+\n√\n3Σ∗0\n√\n6Ξ∗0\n√\n3\nΣ∗0\n√\n6Σ∗−\n√\n3Ξ∗−\n√\n3\nΞ∗0\n√\n3Ξ∗−\n√\n3Ω−\n.(7)\nIn this representation, one can assign any particular permutation of indicesa,b,cto denote the\nrow, column, and sub-matrix. The decuplet remains completely symm etric after rearranging the\nflavor indices.\nThe lowest-order chiral Lagrangian for the heavy mesons reads [ 106,107]\nL(1)\nHφ=−/angbracketleftbig\n(iv·∂H)¯H/angbracketrightbig\n+/angbracketleftbig\nH(iv·Γ)¯H/angbracketrightbig\n−1\n8δD/angbracketleftbig\nHσµν¯Hσµν/angbracketrightbig\n+g/angbracketleftbig\nHuµγµγ5¯H/angbracketrightbig\n,(8)\nwhere/an}bracketle{t···/an}bracketri}htrepresents the trace in spinor space and δDdenotes the mass difference between the\nheavy pseudoscalar and vector meson. The doublet of the ground state heavy mesons reads\nH=1+/v\n2(P∗\nµγµ+iPγ5),¯H=γ0H†γ0= (P∗†\nµγµ+iP†γ5)1+/v\n2, (9)\n3Figure 1: The leading order Feynman diagrams O(ǫ0) in the calculation of the octet baryon ( B)\nand the heavy meson ( D/D∗) effective potentials. The thick, thin, double-thin, and dashed lines\ndenote the B,D,D∗, andφfields, respectively.\nFigure 2: The two-pseudoscalar-meson-exchange diagrams for t heBDsystem at O(ǫ2). These\ndiagrams are categorized into the football diagram (F11), triangle diagrams (T11,T12,T13), box\ndiagrams (B11,B12), and crossed box diagrams (R11,R12). The dec uplet fields are represented\nby heavy-thick lines within the loops. The notations for other elemen ts remain consistent with\nthose presented in Fig. 1.\nP= (D0,D+,D+\ns), P∗\nµ= (D0∗,D+∗,D+∗\ns)µ. (10)\nAt last, we construct the leading order contact Lagrangian to des cribe the short distance\ninteraction between the octet baryon and heavy meson,\nL(0)\nBH=Datr(¯BB)/angbracketleftbig\nH¯H/angbracketrightbig\n+Dbtr(¯Bγµγ5B)/angbracketleftbig\nHγµγ5¯H/angbracketrightbig\n+Eatr(¯BλaB)/angbracketleftbig\nHλa¯H/angbracketrightbig\n+Ebtr(¯Bγµγ5λaB)/angbracketleftbig\nHγµγ5λa¯H/angbracketrightbig\n, (11)\nwhereDaandDbare two low energy constants that contribute to the central pot ential and the\nspin-spin interaction, respectively. On the otherhand, EaandEbareassociated with the isospin-\nisospin interaction, influencing the central potential and spin-spin interaction, respectively. With\nthe quark model, we can determine their values using the nucleon-an tinucleon interaction as\ninputs, as done in the Appendix of ref. [ 91]. Theλarepresents the conventional Gell-Mann\nmatrix.\n3 Expressions of the effective potentials\nIn the context of heavy hadron chiral perturbation theory, the scattering amplitudes for the\nBDandBD∗systems can be systematically expanded in a series. These expansio ns are based\non small dimensionless parameters, denoted as ǫ={q/Λχ,mφ/Λχ,δ/Λχ}. Here, Λ χ≃1GeV\nrepresents the scale of chiral symmetry breaking. The paramete rqrefers to the momentum of\nGoldstone bosons or the residual momentum of heavy hadrons, mφindicates the masses of the\nlight pseudoscalar mesons, and δsignifies the mass gap between the decuplet and octet baryons\nor between the heavy pseudoscalar and vector mesons. This powe r counting scheme is often\ndenoted as the small scale expansion (SSE).\nWe present the explicit expressions of the effective potentials for t heBDandBD∗systems at\nO(ǫ0). The corresponding Feynman diagrams are shown in Fig. 1, which contain the contact and\none-pseudoscalar-meson exchange diagrams. The one-pseudos calar-meson exchange diagram for\n4Figure 3: The two-pseudoscalar-meson-exchange diagrams for t heBD∗system at O(ǫ2). The\nnotations remain consistent with those depicted in Fig. 2.\ntheBDsystem vanishes because the three pseudoscalar meson vertex is forbidden. Then, the\npotentials read\nV(X11)\nBD=Da+α(X11)Ea, (12)\nV(X21)\nBD∗=Da+Db+α(X21)(Ea+Ebσ·T), (13)\nV(H21)\nBD∗=α(H21)\nπg\nf2π(σ·q)(T·q)\nq2+m2π+α(H21)\nηg\nf2η(σ·q)(T·q)\nq2+m2η, (14)\nwheretheparameters α(X11),α(X21),α(H21)\nπ, andα(H21)\nη, whichincorporatecrucialisospinfactors,\nare detailed in the Appendix. The operators σandTare connected to the spin operators of\nthe spin-1\n2baryon and spin-1 meson, being represented as (1\n2σ) and (−T) respectively. This\nrelationship is elaborated in detail in ref. [ 85]. The Breit approximation formula,\nV(q) =−M(q)√2M12M22M32M4(15)\nis employed to establish a connection between the scattering amplitu deM(q) and the effective\npotential V(q) in momentum space, where M1,...,4are the masses of the scattering particles.\nThe two-pseudoscalar-meson-exchangediagrams for the BDsystem at order O(ǫ2) are shown\nin Fig.2. The effective potentials from these diagrams read\nV(F11)\nBD=α(F11)\nππ1\nf4πJF\n22(mπ,mπ)+α(F11)\nKK1\nf4\nKJF\n22(mK,mK), (16)\nV(T11)\nBD=/braceleftigα(T11)\nππ\nf4π,α(T11)\nKK\nf4\nK/bracerightig\ng2/bracketleftig\n(d−1)JT\n34−(JT\n24+JT\n33)q2/bracketrightig\n{(mπ,mπ,−δD),(mK,mK,−δD)},\n(17)\n5V(T12)\nBD=/braceleftigα(T12)\nππ\nf4π,α(T12)\nKK\nf4\nK/bracerightig\nC2/bracketleftig\n(2−d)JT\n34−2−d\nd−1(JT\n24+JT\n33)q2/bracketrightig\n{(mπ,mπ,−δB),(mK,mK,−δB)},\n(18)\nV(T13)\nBD=/braceleftigα(T13)\nππ\nf4π,α(T13)\nKK\nf4\nK/bracerightig/bracketleftig\n(d−1)JT\n34−(JT\n24+JT\n33)q2/bracketrightig\n{(mπ,mπ),(mK,mK)},(19)\nV(B11)\nBD=/braceleftigα(B11)\nππ\nf4π,α(B11)\nKK\nf4\nK,α(B11)\nηη\nf4η,α(B11)\nπη\nf2πf2η,α(B11)\nηπ\nf2ηf2π/bracerightig\ng2/braceleftig\n(d2−1)JB\n41−[JB\n21+2(d+1)(JB\n31+JB\n42)]q2\n+(JB\n22+2JB\n32+JB\n43)q4/bracerightig\n{(mπ,mπ,−δD),(mK,mK,−δD),(mη,mη,−δD),\n(mπ,mη,−δD),(mη,mπ,−δD)}, (20)\nV(B12)\nBD=/braceleftigα(B12)\nππ\nf4π,α(B12)\nKK\nf4\nK,α(B12)\nηη\nf4η,α(B12)\nπη\nf2πf2η,α(B12)\nηπ\nf2ηf2π/bracerightig\nC2g2d−2\nd−1/braceleftig\n(JB\n22+2JB\n32+JB\n43)q4+(d2−1)JB\n41\n−[JB\n21+2(d+1)(JB\n31+JB\n42)]q2/bracerightig\n{(mπ,mπ,−δB,−δD),(mK,mK,−δB,−δD),\n(mη,mη,−δB,−δD),(mπ,mη,−δB,−δD),(mη,mπ,−δB,−δD)}, (21)\nV(R11)\nBD=V(B11)\nBD|JB\nij→JR\nij,α(B11)\n∗∗→α(R11)\n∗∗, (22)\nV(R12)\nBD=V(B12)\nBD|JB\nij→JR\nij,α(B12)\n∗∗→α(R12)\n∗∗, (23)\nwhere the parameters α(∗∗)\n∗∗are detailed in the Appendix. The loop functions JF\nij,JT\nij,JB\nij, and\nJR\nijare given in refs. [ 83–85]. Nevertheless, due to the varying masses ofthe internal pseudo scalar\nmeson fields in the SU(3) framework, we have employed the formulas ¯∆ =m2\n1+(m2\n2−m2\n1)x+\nq2x(x−1)−iǫandA=m2\n1−ω2+(m2\n2−m2\n1)x+q2x(x−1)−iǫ(wherem1andm2denote the\nmasses of the internal pseudoscalar meson fields), as substitute s for the equivalent expressions\noutlined in ref. [ 85]. In addition, we takethe scale λ= 4πfπin the loopfunctions. The parameter\nddenotes the dimension introduced in the dimensional regularization. Note that, to generate\nthe effective potential, one must subtract the two-particle reduc ible contributions from the box\ndiagrams. A detailed deduction, based on the principal value integra l method, is provided in\nAppendix B of ref. [ 85].\nThe two-pseudoscalar-meson-exchange diagrams for the BD∗system at order O(ǫ2) are de-\npicted in Fig. 3. The effective potentials arising from these diagrams are expresse d as follows:\nV(F21)\nBD∗=α(F21)\nππ1\nf4πJF\n22(mπ,mπ)+α(F21)\nKK1\nf4\nKJF\n22(mK,mK), (24)\nV(T21)\nBD∗=/braceleftigα(T21)\nππ\nf4π,α(T21)\nKK\nf4\nK/bracerightig\ng2(d−3)(d−2)\nd−1/bracketleftig\n(d−1)JT\n34−(JT\n24+JT\n33)q2/bracketrightig\n{(mπ,mπ),(mK,mK)},\n(25)\nV(T22)\nBD∗=/braceleftigα(T22)\nππ\nf4π,α(T22)\nKK\nf4\nK/bracerightig\ng2/bracketleftig\nJT\n34−1\nd−1(JT\n24+JT\n33)q2/bracketrightig\n{(mπ,mπ,δD),(mK,mK,δD)},(26)\nV(T23)\nBD∗=/braceleftigα(T23)\nππ\nf4π,α(T23)\nKK\nf4\nK/bracerightig/bracketleftig\n(d−1)JT\n34−(JT\n24+JT\n33)q2/bracketrightig\n{(mπ,mπ),(mK,mK)},(27)\nV(T24)\nBD∗=/braceleftigα(T24)\nππ\nf4π,α(T24)\nKK\nf4\nK/bracerightig/bracketleftig\n(2−d)JT\n34−2−d\nd−1(JT\n24+JT\n33)q2/bracketrightig\n{(mπ,mπ,−δB),(mK,mK,−δB)},\n(28)\n6V(B21)\nBD∗=/braceleftigα(B21)\nππ\nf4π,α(B21)\nKK\nf4\nK,α(B21)\nηη\nf4η,α(B21)\nπη\nf2πf2η,α(B21)\nηπ\nf2ηf2π/bracerightig\ng2(d−3)(d−2)\nd−1/braceleftig\n(d2−1)JB\n41\n−/bracketleftig/parenleftig1\nd−2σ·T+1/parenrightig\nJB\n21+2(d+1)(JB\n31+JB\n42)/bracketrightig\nq2+(JB\n22+2JB\n32+JB\n43)q4/bracerightig\n{(mπ,mπ),\n(mK,mK),(mη,mη),(mπ,mη),(mη,mπ)}, (29)\nV(B22)\nBD∗=/braceleftigα(B22)\nππ\nf4π,α(B22)\nKK\nf4\nK,α(B22)\nηη\nf4η,α(B22)\nπη\nf2πf2η,α(B22)\nηπ\nf2ηf2π/bracerightigg2\nd−1/braceleftig\n(d2−1)JB\n41+(JB\n22+2JB\n32+JB\n43)q4\n−[(σ·T+1)JB\n21+2(d+1)(JB\n31+JB\n42)]q2/bracerightig\n{(mπ,mπ,δD),(mK,mK,δD),\n(mη,mη,δD),(mπ,mη,δD),(mη,mπ,δD)}, (30)\nV(B23)\nBD∗=/braceleftigα(B23)\nππ\nf4π,α(B23)\nKK\nf4\nK,α(B23)\nηη\nf4η,α(B23)\nπη\nf2πf2η,α(B23)\nηπ\nf2ηf2π/bracerightig\ng2C2(d−3)(d−2)2\n(d−1)2/braceleftig\n(d2−1)JB\n41+(JB\n22+\n2JB\n32+JB\n43)q4+/bracketleftigσ·T\n(d−2)2JB\n21−JB\n21−2(d+1)(JB\n31+JB\n42)/bracketrightig\nq2/bracerightig\n{(mπ,mπ,−δB),\n(mK,mK,−δB),(mη,mη,−δB),(mπ,mη,−δB),(mη,mπ,−δB)}, (31)\nV(B24)\nBD∗=/braceleftigα(B24)\nππ\nf4π,α(B24)\nKK\nf4\nK,α(B24)\nηη\nf4η,α(B24)\nπη\nf2πf2η,α(B24)\nηπ\nf2ηf2π/bracerightig\ng2C2d−2\n(d−1)2/braceleftig\n(JB\n22+2JB\n32+JB\n43)q4\n+(d2−1)JB\n41+/bracketleftig/parenleftigσ·T\nd−2−1/parenrightig\nJB\n21−2(d+1)(JB\n31+JB\n42)/bracketrightig\nq2/bracerightig\n{(mπ,mπ,−δB,δD),\n(mK,mK,−δB,δD),(mη,mη,−δB,δD),(mπ,mη,−δB,δD),(mη,mπ,−δB,δD)},(32)\nV(R21)\nBD∗=V(B21)\nBD∗|JB\nij→JR\nij,α(B21)\n∗∗→α(R21)\n∗∗,σ·T→−σ·T, (33)\nV(R22)\nBD∗=V(B22)\nBD∗|JB\nij→JR\nij,α(B22)\n∗∗→α(R22)\n∗∗,σ·T→−σ·T, (34)\nV(R23)\nBD∗=V(B23)\nBD∗|JB\nij→JR\nij,α(B23)\n∗∗→α(R23)\n∗∗,σ·T→−σ·T, (35)\nV(R24)\nBD∗=V(B24)\nBD∗|JB\nij→JR\nij,α(B24)\n∗∗→α(R24)\n∗∗,σ·T→−σ·T, (36)\nwhere the parameters α(∗∗)\n∗∗are also detailed in the Appendix. In the above expressions, all\nphysical quantities are defined in ddimensions, such as S2= (d−1)/4 andqiqj= 1/(d−1)q2δij\nfor theSwave.\nAt the next-to-leading order, in addition to the previously mentione d two-pseudoscalar-\nmeson-exchange potentials, there exist one-loop corrections pe rtaining to the one-pseudoscalar-\nmeson-exchangeandcontact terms. These one-loopcorrection scan be incorporatedby substitut-\ning the physical values of relevant parameters such as the coupling s, decay constant, and masses\nof light-mesons, baryons, and heavy-mesons. Furthermore, we also have the corrections from\nthe subleading contact Lagrangians involving some unrenormalized O(ǫ2) LECs. In contrast to\nthe nucleon-nucleon case, the current unavailability of baryon and heavy-meson scattering data\nimpedes the fitting of these O(ǫ2) LECs. Consequently, we refrain from including their contri-\nbutions and focus solely on the leading-order contact terms. Once the lattice QCD simulations\nprovide access to the baryon and heavy meson scattering phase s hifts, revisiting these systems\nwhile incorporating higher-order contributions would be an intriguing and worthwhile avenue for\nexploration.\nUtilizingthe momentum-spacepotentials V(q)derivedabove, weproceedwiththe subsequent\nFourier transformation to acquire the effective potential V(r) in coordinate space,\nV(r) =/integraldisplayd3q\n(2π)3eiq·rV(q)F(q). (37)\n7In this context, the introduction of a regulator function F(q) becomes essential to suppress\nthe high momentum contributions. For this purpose, we opt for the Gaussian form F(q) =\nexp/parenleftbig\n−q2n/Λ2n/parenrightbig\n, previously utilized in studies related to the NNandN¯Nsystems [ 108–110].\nTheTaylorexpansionofthisregulatorfunctioncanbeexpresseda sF(q) = 1−q2n/Λ2n+···. The\nchoice of the power nis crucial to ensure that the contributions from the cutoff-induce d terms,\nV(q)O(q2n/Λ2n), lie beyond the chiral order at which the analysis is conducted. Alth ough our\ncalculations primarily focus on the leading and subleading contribution s, choosing n= 1 would\nsuffice. However, for consistency with the estimation of the LECs in theN¯Nsystem [110], where\nn= 3 was adopted, we also select n= 3. For the cutoff parameter Λ, we can make adjustments\nwithin the rangeof 400 −600MeV, guided by insights from alternative analyses[ 86,87,110–113].\nNotably, due to δD> mπ, certain diagrams in Fig. 3, specifically (T22) and (B22), would\nyield imaginary components, influencing the width of the associated b ound state. If one were\nto solve the Lippmann-Schwinger equation, the presence of these imaginary parts would alter\nthe pole position within the Riemann sheet. In this work, we opt for so lving the Schr¨ odinger\nequation, with aprimaryfocus onthe binding energies. Consequent ly, the imaginarycomponents\nare disregarded in our calculations.\n4 Results and discussion\nTo derive the numerical results, the LECs ( Da,Db,Ea,Eb) need to be determined. Fol-\nlowing the methodology proposed in ref. [ 86,87,91], we estimate the LECs through the de-\nvelopment of a contact Lagrangian at the quark level. This involves t he coupling constants\ncsandct. The relationships ( Da= 3cs,Db=−ct,Ea= 3cs,Eb=−5ct) were deduced by\naligning the ND∗potentials from the hadron and quark levels in the SU(2) framework , as\ndetailed in ref. [ 91]. Similarly, for the SU(3) framework, the relationships are establish ed as\n(Da= 2cs,Db= 2ct/3,Ea= 3cs,Eb=−5ct). The values of csandctare derived from the\nN¯Ninteraction, as detailed in refs. [ 91,110]. In this work, we adopt cs=−8.1GeV−2and\nct= 0.65GeV−2, as sourced from the Appendix of ref. [ 91].\nFurthermore,weusetheaveragevaluesfortheisospinmultiplet ma ssesofmesonsandbaryons\nfrom PDG [ 1]:mπ= 137.27MeV,mK= 495.64MeV,mη= 547.86MeV,MD= 1867.24MeV,\nMDs= 1968.34MeV,MD∗= 2008.55MeV,MDs∗= 2112.20MeV,MN= 938.92MeV,MΣ=\n1193.15MeV,MΞ= 1318.28MeV,MΛ= 1115.68MeV,M∆= 1232.00MeV,MΣ∗= 1384.57MeV,\nMΞ∗= 1533.40MeV,MΩ= 1672.45MeV. We also consider the average values for the mass dif-\nferences:δB= 314.10MeV and δD= 142.59MeV. The physical values of the pseudoscalar decay\nconstants, as reported in PDG, are used: fπ= 92.07MeV,fK= 110.03MeV, and fη= 1.2fπ.\nThe axial vector coupling constant gAhas been determined to be approximately 1.27, based\non lattice quantum chromodynamics calculations [ 114] and measurements in the decay of free\nneutrons [ 115]. Accordingly, we adopt D= 0.80 andF= 0.47 as their physical values. These\nparameters are crucial for obtaining accurate numerical results in our theoretical framework.\nIn our analysis, we adopt C= 1.2 based on the study of strong and electromagnetic decays of\ndecuplet baryons, as detailed in ref. [ 116]. The coupling constant g=−0.59 is determined from\nthe decay width of D∗+, as reported in refs. [ 117,118]. The negative sign for gis derived using\ninsights from the quark model. Regarding the cutoff parameter Λ, an extensive discussion is\npresented in ref. [ 91]. The findings from this reference suggest that a cutoff value of Λ= 0.4GeV\nis suitable for accurately describing bound states. On the other ha nd, a higher cutoff value,\nsuch as Λ= 0.6GeV, is considered highly unnatural within the context of our calcu lation.\nConsequently, for the purpose of making predictions, we opt for Λ= 0.4GeV. This choice is\ninformed by the need to ensure consistency and accuracy in our th eoretical framework and the\ncorresponding results.\n4.1D(∗)\n(s)Nsystems\nWe first analyze the D-meson and nucleon systems. The effective potentials of each poss ible\nIJPconfiguration are shown in Fig. 4. For the [DN]I=0\nJ=1/2system, the O(ǫ0) contact and O(ǫ2)\ntwo-meson-exchange potentials are both attractive. However, the attraction of the two-meson-\nexchange potentials is relatively weak. The dominant source of the a ttractive potential arises\nfrom the contact interaction. For the [ DN]I=1\nJ=1/2and [DsN]I=1/2\nJ=1/2systems, the O(ǫ0) contact\ninteractionsvanish inourcalculation, andthe totalpotentials stem fromthe two-meson-exchange\n801\n2-1D N system\n0123456-100-80-60-40-200\nr(fm)V(r) (MeV)\n11\n2-1D N system\n0123456-30-25-20-15-10-50\nr(fm)V(r) (MeV)\n1\n21\n2-1DSN system\n0 1 2 3 4 5 6-6-4-20\nr(fm)V(r) (MeV)\n01\n2-1D*N system\n0123456-100-500\nr(fm)V(r) (MeV)\n11\n2-1D*N system\n0123456-10-505\nr(fm)V(r) (MeV)\n1\n21\n2-1DS*N system\n0 1 2 3 4 5 6-4-2024\nr(fm)V(r) (MeV)\n03\n2-1D*N system\n0123456-80-60-40-20020\nr(fm)V(r) (MeV)\n13\n2-1D*N system\n0 1 2 3 4 5 6-6-4-20246\nr(fm)V(r) (MeV)\n1\n23\n2-1DS*N system\n0 1 2 3 4 5 6-7-6-5-4-3-2-10\nr(fm)V(r) (MeV)\nFigure 4: The effective potentials of the D(∗)\n(s)Nsystems. Their quantum numbers ( IJP) are\nindicated in each subfigure. The red-dashed, brown-dotted, gre en-dotdashed, blue-solid lines de-\nnote the contact, one-meson-exchange, two-meson-exchang e, and their total potentials in SU(3)\nframework, respectively. Meanwhile, the pink-dashed, orange-d otted, cyan-dotdashed, purple-\nsolid lines present the contact, one-meson-exchange, two-meso n-exchange, and their total poten-\ntials in SU(2) framework, respectively.\ncontributions. Notably, the potentials in the two channels exhibit co nsiderably shallower depths\nwhencomparedtothe[ DN]I=0\nJ=1/2channel,withaparticularemphasisonthe[ DsN]I=1/2\nJ=1/2channel.\nIn the [D∗N]I=0\nJ=1/2system, the contactinteraction exhibits an attractivepotential, while both\nthe one-meson-exchangeand two-meson-exchangepotentials a rerepulsive. Despite this, the total\npotential of the system maintains approximatelyhalf the depth com pared to that observed in the\n[DN]I=0\nJ=1/2channel. For the channel [ D∗N]I=0\nJ=3/2, the behavior of its potentials is intriguing. It\nis noteworthy that the contributions from one-meson-exchange and two-meson-exchange nearly\noffset each other. Consequently, the total potential is predomin antly determined by the contact\nterm, reaching a maximum depth of −80 MeV. In the [ D∗N]I=1\nJ=1/2, [D∗N]I=1\nJ=3/2, [D∗\nsN]I=1/2\nJ=1/2,\nand [D∗\nsN]I=1/2\nJ=3/2systems, the contact, one-meson-exchange, and two-meson- exchange potentials\nexhibit weakly attractive or weakly repulsive behavior. Consequent ly, their total potentials are\nalso either weakly attractive or weakly repulsive.\nIn the [DN]I=0,1\nJ=1/2and [D∗N]I=0,1\nJ=1/2,3/2channels, we present the potentials arising from the\nSU(2) framework, where the η- andK-exchange contributions are excluded. In both the SU(2)\nand SU(3) frameworks, the contact and one-meson potentials ex hibit nearly identical character-\nistics, while a small distinction emerges in the two-meson potentials. T he total potentials from\nthe SU(3) and SU(2) frameworks are not significantly different. Th erefore, our calculation in\nSU(3) framework is reasonable.\nWeobtainthebindingsolutionsinthe [ DN]I=0\nJ=1/2, [D∗N]I=0\nJ=1/2, and[D∗N]I=0\nJ=3/2systems, and\nthe results are detailed in Table 1. Notably, the binding energy values for the SU(2) framework\nexhibit slightly deviations from those reported in ref. [ 91]. This discrepancy primarily arises\nfrom our selection of different constants, particularly a smaller cou pling constants for the ∆ Nπ\nvertex. A key point of interest is the [ D∗N]I=0\nJ=3/2channel. The mass of this bound state in the\nSU(3) framework closely aligns with the Λ c(2940) mass measurements.\n91\n21\n2-1DΣsystem\n0123456-80-60-40-200\nr(fm)V(r) (MeV)\n3\n21\n2-1DΣsystem\n0123456-4-3-2-10\nr(fm)V(r) (MeV)\n11\n2-1DSΣsystem\n0123456-30-20-100\nr(fm)V(r) (MeV)\n1\n21\n2-1D*Σsystem\n0123456-80-60-40-20020\nr(fm)V(r) (MeV)\n3\n21\n2-1D*Σsystem\n0123456-6-4-2024\nr(fm)V(r) (MeV)\n11\n2-1DS*Σsystem\n0123456-10-505\nr(fm)V(r) (MeV)\n1\n23\n2-1D*Σsystem\n0123456-60-40-200\nr(fm)V(r) (MeV)\n3\n23\n2-1D*Σsystem\n0123456-4-2024\nr(fm)V(r) (MeV)\n13\n2-1DS*Σsystem\n0123456-15-10-50\nr(fm)V(r) (MeV)\nFigure 5: The effective potentials of the D(∗)\n(s)Σ systems. The notation is the same as in Fig. 4.\n4.2D(∗)\n(s)Σsystems\nWe then examine the D-meson and Σ-baryonsystems. The effective potentials correspo nding\nto each possible IJPconfiguration are depicted in Fig. 5. The features of DΣ are very similar to\nthose ofDN, with the distinction that the effective potential of [ DsΣ]I=1\nJ=1/2is deeper than that\nof [DΣ]I=3/2\nJ=1/2, while the effective potential of [ DsN]I=1/2\nJ=1/2is shallower than that of [ DN]I=1\nJ=1/2.\nThis discrepancy is anticipated, given the presence of an attractiv es¯squark pair in [ DsΣ]I=1\nJ=1/2.\nDespite the effective potential of [ DΣ]I=1/2\nJ=1/2being slightly shallower than that of [ DN]I=0\nJ=1/2, it\nremains sufficiently deep to facilitate the formation of a bound state .\nThe characteristics of the D∗Σ system show a striking similarity to those of the D∗Nsys-\ntem. For the low-isospin channels, [ D∗Σ]I=1/2\nJ=1/2,3/2, we observe significantly deep attractive po-\ntentials. Conversely, the high-isospin channels, [ D∗Σ]I=3/2\nJ=1/2,3/2, as well as the strange channels\n[D∗\nsΣ]I=1\nJ=1/2,3/2, arecharacterizedbyshallowerattractivepotentials. Inthecas eofthe[D∗Σ]I=1/2\nJ=1/2\nsystem, there is an attractive contact interaction, although bot h one-meson-exchange and two-\nmeson-exchange potentials are repulsive. This particular trait mirr ors that of the [ D∗N]I=0\nJ=1/2\nTable 1: The bound states forthe D(∗)\n(s)Nsystems in SU(2) and SU(3)\nframeworks. The ∆ E,M, and/radicalbig\n/an}bracketle{tr2/an}bracketri}htdenote the binding energy, the\nmass of the bound state, and the root mean square radius, respe c-\ntively.\nSU(3)/SU(2) [ DN]I=0\nJ=1/2[D∗N]I=0\nJ=1/2[D∗N]I=0\nJ=3/2\n∆E(MeV) −12.9/−10.5−1.3/−2.9−6.9/−9.3\nM(MeV) 2793 .2/2795.6 2946.2/2944.6 2940.5/2938.1\n/radicalbig\n/an}bracketle{tr2/an}bracketri}ht(fm) 1 .7/1.8 4 .1/3.0 2 .0/1.9\n1001\n2-1DΞsystem\n0123456-60-50-40-30-20-100\nr(fm)V(r) (MeV)\n11\n2-1DΞsystem\n0123456-4-3-2-10\nr(fm)V(r) (MeV)\n1\n21\n2-1DSΞsystem\n0123456-60-40-200\nr(fm)V(r) (MeV)\n01\n2-1D*Ξsystem\n0123456-25-20-15-10-505\nr(fm)V(r) (MeV)\n11\n2-1D*Ξsystem\n0123456-1012345\nr(fm)V(r) (MeV)\n1\n21\n2-1DS*Ξsystem\n0123456-50-40-30-20-100\nr(fm)V(r) (MeV)\n03\n2-1D*Ξsystem\n0123456-20-15-10-505\nr(fm)V(r) (MeV)\n13\n2-1D*Ξsystem\n0123456-5-4-3-2-10\nr(fm)V(r) (MeV)\n1\n23\n2-1DS*Ξsystem\n0123456-40-30-20-100\nr(fm)V(r) (MeV)\nFigure 6: The effective potentials of the D(∗)\n(s)Ξ systems. The notation is the same as in Fig. 4.\nsystem. In contrast, in the [ D∗Σ]I=1/2\nJ=3/2system, the influence of the two-meson-exchange is negli-\ngible, diverging from the characteristics of the [ D∗N]I=0\nJ=3/2system. Nevertheless, the attractive\nnature of the low-isospin channels is sufficient to facilitate the forma tion of bound states.\nWealsoobtainbinding solutionsin the[ DΣ]I=1/2\nJ=1/2, [D∗Σ]I=1/2\nJ=1/2, and[D∗Σ]I=1/2\nJ=3/2systems. The\nnumerical results are presented in Table 2. It is evident that the bound states from D∗Σ exhibit\nsimilarities to those from D∗N. The channel with low spin, [ D∗Σ]I=1/2\nJ=1/2, has a much shallower\nbinding energy compared to the channel with high spin, [ D∗Σ]I=1/2\nJ=3/2. In analogy to interpreting\nΛc(2940) as a molecular state of D∗N, we may propose to interpret the Ξ c(3196) as a bound\nstate ofD∗Σ withIJP=1\n23\n2−or1\n21\n2−. It is noteworthy that the molecular states also emerge\nin the [DΣ]I=1/2\nJ=1/2and [D∗Σ]I=1/2\nJ=3/2systems from the calculations in ref. [ 119]. Additionally, the\nΞc(3055) may be interpreted as the bound state of [ DΣ]I=1/2\nJ=1/2system considering their nearly\nidentical mass.\n4.3D(∗)\n(s)Ξsystems\nWe also investigate the systems composed of a Dmeson and a Ξ baryon. The effective poten-\ntials for each possible IJPconfiguration are illustrated in Fig. 6. A notable distinction emerges\nTable 2: The bound states for the D(∗)\n(s)Σ systems in SU(3)\nframework. The notation is the same as in Table 1.\nSU(3) [ DΣ]I=1/2\nJ=1/2[D∗Σ]I=1/2\nJ=1/2[D∗Σ]I=1/2\nJ=3/2\n∆E(MeV) −7.5 −3.6 −6.2\nM(MeV) 3052 .9 3198 .1 3195 .5\n/radicalbig\n/an}bracketle{tr2/an}bracketri}ht(fm) 1 .9 2 .5 2 .0\n111\n21\n2-1DΛsystem\n0123456-30-25-20-15-10-50\nr(fm)V(r) (MeV)\n01\n2-1DSΛsystem\n0123456-40-30-20-100\nr(fm)V(r) (MeV)\n1\n21\n2-1D*Λsystem\n0123456-5-4-3-2-10\nr(fm)V(r) (MeV)\n1\n23\n2-1D*Λsystem\n0123456-12-10-8-6-4-20\nr(fm)V(r) (MeV)\n01\n2-1DS*Λsystem\n0123456-30-20-100\nr(fm)V(r) (MeV)\n03\n2-1DS*Λsystem\n0123456-30-25-20-15-10-50\nr(fm)V(r) (MeV)\nFigure 7: The effective potentials of the D(∗)\n(s)Λ systems. The notation is the same as in Fig. 4.\nwhen comparing the characteristics of the DΞ systems to those of DN. Specifically, for the\n[DΞ]I=0\nJ=1/2and [DΞ]I=1\nJ=1/2systems, the O(ǫ0) contact interactions vanish in our calculation, since\nthe total potentials are derived solely from the two-meson-excha nge contributions. Conversely,\nin the [DsΞ]I=1/2\nJ=1/2system, the O(ǫ0) contact interaction is significant. This distinction leads\nto the emergence of sufficiently attractive potentials in the [ DΞ]I=0\nJ=1/2and [DsΞ]I=1/2\nJ=1/2systems,\npotentially indicating the formation of bound states. Meanwhile, a mo destly attractive potential\nis observed in the [ DΞ]I=1\nJ=1/2system.\nTheD∗Ξ system exhibits distinct characteristics compared to the D∗Nsystem. In the low-\nisospin channels, specifically [ D∗Ξ]I=0\nJ=1/2,3/2, we detect attractive potentials. However, these\npotentials are not sufficiently deep to facilitate the formation of bou nd states. Conversely, in the\nhigh-isospin channels, [ D∗Ξ]I=1\nJ=1/2,3/2, we observe a mixture of shallower repulsive and attractive\npotentials. With regard to the [ D∗\nsΞ]I=1/2\nJ=1/2,3/2systems, both the one-meson-exchange and two-\nmeson-exchange potentials appear relatively weak. Notably, the p rimary source of the attractive\npotential in these cases stem from the contact interactions. Des pite the total potentials are not\nprofoundly deep, they are sufficiently strong to potentially lead to t he formation of bound states.\nWe obtain binding solutions in the [ DΞ]I=0\nJ=1/2, [DsΞ]I=1/2\nJ=1/2, [D∗\nsΞ]I=1/2\nJ=1/2, and [D∗\nsΞ]I=1/2\nJ=3/2sys-\ntems, with comprehensive results detailed in Table 3. Notably, the bound state associated with\nthe [D∗\nsΞ]I=1/2\nJ=1/2,3/2system may correspond to the Ξ c(3429). Furthermore, the Ξ c(3279) may be\ninterpreted as the bound state of the [ DsΞ]I=1/2\nJ=1/2. Intriguingly, its mass closely aligns with that\nof the bound state in the [ D∗Σ]I=1/2\nJ=1/2system, and they share identical quantum numbers. These\ntwo channels may be considered as the coupled-channels. Additiona lly, the bound state in the\n[DΞ]I=0\nJ=1/2system may correspond to the Ω c(3188) observed by LHCb [ 120]. Our result for this\nchannel is in agreement with ref. [ 121], where a bound state was identified. However, it contrasts\nwith the findings in refs. [ 119,122], in which a 2 SΩc-baryon and a virtual state were reported,\nrespectively.\nTable 3: The bound states for the D(∗)\n(s)Ξ systems in SU(3) framework.\nThe notation is the same as in Table 1.\nSU(3) [ DΞ]I=0\nJ=1/2[DsΞ]I=1/2\nJ=1/2[D∗\nsΞ]I=1/2\nJ=1/2[D∗\nsΞ]I=1/2\nJ=3/2\n∆E(MeV) −4.1 −7.7 −1.7 −0.4\nM(MeV) 3181 .4 3279 .0 3428 .8 3430 .1\n/radicalbig\n/an}bracketle{tr2/an}bracketri}ht(fm) 2 .3 1 .9 3 .2 5 .7\n124.4D(∗)\n(s)Λsystems\nFinally, we explore the D-meson and Λ-baryon systems. For each possible IJPconfiguration\nwithinthesesystems,weillustratetheeffectivepotentialsinFig. 7. Comparedwiththepreviously\ndiscussed systems, it is noteworthy that all the O(ǫ0) contact interactions are nonvanishing here.\nHowever, these interactionsare not sufficiently attractiveto for m bound states, even when taking\ninto account both the one-meson-exchange and two-meson-exc hange contributions. Particularly\nin the [DsΛ]I=0\nJ=1/2system, we observe a significantly attractive contact potential c oupled with\nan attractive two-meson-exchange potential, results in the most significant potential observed\nwithin the D(∗)\n(s)Λ systems. It is crucial to emphasize that no binding solution exists in these\nsystems.\n5 Summary\nIn this work, we have comprehensively calculated the effective pote ntials of the octet baryon\nandD-meson systems up to the next-to-leading order using the chiral e ffective field theory. Our\napproachencompassesthe long-rangeone-pseudoscalar-meso n-exchangepotential, the mid-range\ntwo-pseudoscalar-meson-exchangepotential, and the short-r ange contact interactions. Addition-\nally, we have incorporated the decuplet baryons as the intermediat e states within the loop dia-\ngrams. The involvedlow energyconstants arejudiciously estimated based on the N¯Ninteraction\nwith help of the quark model.\nIn ourinvestigationofthe D(∗)\n(s)Nsystems, wehavealmostobtained attractivepotentials in all\nchannels, with the exception of the [ D∗\nsN]I=1/2\nJ=1/2channel. Significantly, bound states emerge only\nin the [DN]I=0\nJ=1/2, [D∗N]I=0\nJ=1/2, and [D∗N]I=0\nJ=3/2systems, owing to the sufficient depth of their\nattractive potentials. Notably, the mass of the bound state in the [D∗N]I=0\nJ=3/2system exhibits a\nremarkable correspondence with the mass of the Λ c(2940).\nIn our examination of the D(∗)\n(s)Σ systems, we observe that their characteristics closely mirror\nthose of the D(∗)\n(s)Nsystems. We have obtained attractive potentials in all channels. Sp ecifically,\nbound states appear in the [ DΣ]I=1/2\nJ=1/2, [D∗Σ]I=1/2\nJ=1/2, and [D∗Σ]I=1/2\nJ=3/2systems. The bound state\nin the [DΣ]I=1/2\nJ=1/2system notably aligns in mass with Ξ c(3055), suggesting its interpretation as\na molecular state. The bound state corresponding to D∗Σ leads to the molecular explanation\nof the Ξ c(3196). These two cases parallel the interpretation of Λ c(2940) as a molecular state of\nD∗N.\nIn our analysis of the D(∗)\n(s)Ξ systems, we observe distinct characteristics compared to the\nD(∗)\n(s)Nsystems. While a repulsive potential is identified in the [ D∗Ξ]I=1\nJ=1/2channel, attractive\npotentials are present in the other channels. Significantly deep pot entials, capable of generating\nbound states, are found in the [ DΞ]I=0\nJ=1/2, [DsΞ]I=1/2\nJ=1/2, [D∗\nsΞ]I=1/2\nJ=1/2, [D∗\nsΞ]I=1/2\nJ=3/2systems. The\nbound state in the [ DΞ]I=0\nJ=1/2system may correspond to the Ω c(3188). Moreover, the Ξ c(3279)\nand Ξ c(3429) may be interpreted as the molecular states of [ DsΞ]I=1/2\nJ=1/2and [D∗\nsΞ]I=1/2\nJ=1/2,3/2,\nrespectively.\nIn our study of the D(∗)\n(s)Λ systems, attractive potentials are identified in all channels. Neve r-\ntheless, these attractive potentials are not sufficiently deep to fa cilitate the formation of bound\nstates in any of these systems.\nTo summarize, our investigations provide new insights into the Λ c(2940), Ξ c(3055), and\nΩc(3188) states from the hadronic molecular perspective. We have a lso identified specific molec-\nular states, designated as Ξ c, within the mass range of 3100 −3500 MeV. The further experi-\nmental measurements of their masses, widths, spin-parity quant um numbers and decay patterns\nwill help to discriminate the conventional baryon and hadronic molecu le explanations for these\nnear-threshold states. We look forward with great anticipation to future experimental valida-\ntion of the possible existence of the singly heavy molecular baryon st ates. Such findings would\nsignificantlyenhance ourcomprehensionofthe molecularstate dyn amics in heavyquarksystems.\n13Acknowledgments\nThis work is supported by the National Natural Science Foundation of China under Grants\nNo. 11975033, No. 12070131001, No. 12147127, No. 12105072, and No. 12305090. B. Wang\nwas also supported by the Youth Funds of Hebei Province (No. A20 21201027) and the Start-\nup Funds for Young Talents of Hebei University (No. 52110022102 1). We thank Zi-Yang Lin\n(Peking University), Lu Meng (Ruhr-Universit¨ at Bochum) for ver y helpful discussions.\nA Parameters for the potentials\nThis appendix details the parameters for the potentials, as shown in TablesA.1toA.2below.\n14Table A.1: Parameters for the potentials of the BDorBD∗systems.\nSystems ND(∗)(I= 1) ND(∗)(I= 0) ND(∗)\nS(I=1\n2) ΣD(∗)(I=3\n2) ΣD(∗)(I=1\n2) ΣD(∗)\nS(I= 1) Ξ D(∗)(I= 1) Ξ D(∗)(I= 0) Ξ D(∗)\nS(I=1\n2) ΛD(∗)(I=1\n2) ΛD(∗)\nS(I= 0)\nα(X11)/α(X21)−2\n310\n3−2\n3−2\n37\n3−2\n3−2\n3−2\n34\n3−1\n32\n3\nα(H21)πD+F\n4−3\n4(D+F) 0F\n2−F 0F−D\n43(D−F)\n40 0 0\nα(H21)η1\n12(D−3F)1\n12(D−3F)1\n6(3F−D) −D\n6−D\n6D\n31\n12(D+3F)1\n12(D+3F)1\n6(−D−3F)D\n6−D\n3\nα(F11)ππ/α(F21)ππ1\n4−3\n401\n2−1 01\n4−3\n40 0 0\nα(F11)KK/α(F21)KK −3\n16−9\n1611\n163\n16−3\n81\n163\n80 −5\n80 0\nα(T11)ππ1\n4−3\n401\n2−1 01\n4−3\n40 0 0\nα(T11)KK−1\n4−3\n43\n41\n4−1\n201\n20 −3\n40 0\nα(T12)ππ/α(T24)ππ1\n6−1\n20 −1\n241\n120 −1\n121\n40 0 0\nα(T12)KK/α(T24)KK1\n120 −1\n8−1\n12−5\n241\n4−1\n243\n8−1\n81\n8−1\n4\nα(T13)ππ/α(T23)ππ1\n4(D+F)2−3\n4(D+F)201\n6/parenleftbig\nD2+3F2/parenrightbig\n−D2\n3−F201\n4(D−F)2−3\n4(D−F)20 0 0\nα(T13)KK/α(T23)KK−1\n4(D−F)2−1\n12(D+3F)2 1\n12/parenleftbig\n5D2−6DF+9F2/parenrightbig1\n4(D−F)21\n2/parenleftbig\n−D2−DF−F2/parenrightbig\nDF1\n6/parenleftbig\nD2+3F2/parenrightbig1\n3D(D+3F)1\n12/parenleftbig\n−5D2−6DF−9F2/parenrightbigDF\n2−DF\nα(B11)ππ1\n16(D+F)2 9\n16(D+F)20F2\n4D2\n4+F201\n16(D−F)2 9\n16(D−F)20D2\n40\nα(B11)KK1\n4(D−F)2 1\n12(D+3F)20 03\n8(D+F)2 1\n4(D−F)20 01\n12/parenleftbig\n5D2+6DF+9F2/parenrightbig1\n24(D−3F)21\n12(D+3F)2\nα(B11)ηη1\n144(D−3F)2 1\n144(D−3F)2 1\n36(D−3F)2 D2\n36D2\n36D2\n91\n144(D+3F)2 1\n144(D+3F)2 1\n36(D+3F)2 D2\n36D2\n9\nα(B11)πη/α(B11)ηπ1\n48(D−3F)(D+F)−1\n16(D−3F)(D+F) 0 −DF\n12DF\n60 −1\n48(D−F)(D+3F)1\n16(D−F)(D+3F) 0 0 0\nα(B12)ππ1\n30 01\n481\n1201\n483\n1603\n160\nα(B12)KK1\n120 0 01\n81\n301\n21\n81\n80\nα(B12)ηη 0 0 01\n481\n481\n121\n481\n481\n120 0\nα(B12)πη/α(B12)ηπ 0 0 0 −1\n481\n240 −1\n481\n160 0 0\nα(R11)ππ5\n16(D+F)2−3\n16(D+F)20D2\n6+3F2\n4−D2\n1205\n16(D−F)2−3\n16(D−F)20D2\n40\nα(R11)KK0 01\n12/parenleftbig\n5D2−6DF+9F2/parenrightbig1\n4(D−F)2−1\n8(D−F)2 1\n4(D+F)2 1\n6/parenleftbig\nD2+3F2/parenrightbig1\n3D(D+3F) 01\n24(D+3F)21\n12(D−3F)2\nα(R11)ηη1\n144(D−3F)2 1\n144(D−3F)2 1\n36(D−3F)2 D2\n36D2\n36D2\n91\n144(D+3F)2 1\n144(D+3F)2 1\n36(D+3F)2 D2\n36D2\n9\nα(R11)πη/α(R11)ηπ1\n48(D−3F)(D+F)−1\n16(D−3F)(D+F) 0 −DF\n12DF\n60 −1\n48(D−F)(D+3F)1\n16(D−F)(D+3F) 0 0 0\nα(R12)ππ1\n61\n201\n160 05\n48−1\n1603\n160\nα(R12)KK0 01\n81\n121\n31\n121\n241\n81\n401\n4\nα(R12)ηη 0 0 01\n481\n481\n121\n481\n481\n120 0\nα(R12)πη/α(R12)ηπ 0 0 0 −1\n481\n240 −1\n481\n160 0 0\nα(T21)ππ1\n4−3\n401\n2−1 01\n4−3\n40 0 0\nα(T21)KK−1\n4−3\n43\n41\n4−1\n201\n20 −3\n40 0\nα(T22)ππ1\n4−3\n401\n2−1 01\n4−3\n40 0 0\nα(T22)KK −1\n4−3\n43\n41\n4−1\n201\n20 −3\n40 0\nα(B21)ππ1\n16(D+F)2 9\n16(D+F)20F2\n4D2\n4+F201\n16(D−F)2 9\n16(D−F)20D2\n40\n15Table A.2: Continue.\nSystems ND(∗)(I= 1) ND(∗)(I= 0) ND(∗)\nS(I=1\n2) ΣD(∗)(I=3\n2) ΣD(∗)(I=1\n2) ΣD(∗)\nS(I= 1) Ξ D(∗)(I= 1) Ξ D(∗)(I= 0) Ξ D(∗)\nS(I=1\n2) ΛD(∗)(I=1\n2) ΛD(∗)\nS(I= 0)\nα(B21)KK1\n4(D−F)2 1\n12(D+3F)20 03\n8(D+F)2 1\n4(D−F)20 01\n12/parenleftbig\n5D2+6DF+9F2/parenrightbig1\n24(D−3F)21\n12(D+3F)2\nα(B21)ηη1\n144(D−3F)2 1\n144(D−3F)2 1\n36(D−3F)2 D2\n36D2\n36D2\n91\n144(D+3F)2 1\n144(D+3F)2 1\n36(D+3F)2 D2\n36D2\n9\nα(B21)πη/α(B21)ηπ1\n48(D−3F)(D+F)−1\n16(D−3F)(D+F) 0 −DF\n12DF\n60 −1\n48(D−F)(D+3F)1\n16(D−F)(D+3F) 0 0 0\nα(B22)ππ1\n16(D+F)2 9\n16(D+F)20F2\n4D2\n4+F201\n16(D−F)2 9\n16(D−F)20D2\n40\nα(B22)KK1\n4(D−F)2 1\n12(D+3F)20 03\n8(D+F)2 1\n4(D−F)20 01\n12/parenleftbig\n5D2+6DF+9F2/parenrightbig1\n24(D−3F)21\n12(D+3F)2\nα(B22)ηη1\n144(D−3F)2 1\n144(D−3F)2 1\n36(D−3F)2 D2\n36D2\n36D2\n91\n144(D+3F)2 1\n144(D+3F)2 1\n36(D+3F)2 D2\n36D2\n9\nα(B22)πη/α(B22)ηπ1\n48(D−3F)(D+F)−1\n16(D−3F)(D+F) 0 −DF\n12DF\n60 −1\n48(D−F)(D+3F)1\n16(D−F)(D+3F) 0 0 0\nα(B23)ππ1\n30 01\n481\n1201\n483\n1603\n160\nα(B23)KK1\n120 0 01\n81\n301\n21\n81\n80\nα(B23)ηη 0 0 01\n481\n481\n121\n481\n481\n120 0\nα(B23)πη/α(B23)ηπ 0 0 01\n48−1\n2401\n48−1\n160 0 0\nα(B24)ππ1\n30 01\n481\n1201\n483\n1603\n160\nα(B24)KK1\n120 0 01\n81\n301\n21\n81\n80\nα(B24)ηη 0 0 01\n481\n481\n121\n481\n481\n120 0\nα(B24)πη/α(B24)ηπ 0 0 0 −1\n481\n240 −1\n481\n160 0 0\nα(R21)ππ5\n16(D+F)2−3\n16(D+F)20D2\n6+3F2\n4−D2\n1205\n16(D−F)2−3\n16(D−F)20D2\n40\nα(R21)KK 0 01\n12/parenleftbig\n5D2−6DF+9F2/parenrightbig1\n4(D−F)2−1\n8(D−F)21\n4(D+F)2 1\n6/parenleftbig\nD2+3F2/parenrightbig1\n3D(D+3F) 01\n24(D+3F)21\n12(D−3F)2\nα(R21)ηη1\n144(D−3F)2 1\n144(D−3F)2 1\n36(D−3F)2 D2\n36D2\n36D2\n91\n144(D+3F)2 1\n144(D+3F)2 1\n36(D+3F)2 D2\n36D2\n9\nα(R21)πη/α(R21)ηπ1\n48(D−3F)(D+F)−1\n16(D−3F)(D+F) 0 −DF\n12DF\n60 −1\n48(D−F)(D+3F)1\n16(D−F)(D+3F) 0 0 0\nα(R22)ππ5\n16(D+F)2−3\n16(D+F)20D2\n6+3F2\n4−D2\n1205\n16(D−F)2−3\n16(D−F)20D2\n40\nα(R22)KK 0 01\n12/parenleftbig\n5D2−6DF+9F2/parenrightbig1\n4(D−F)2−1\n8(D−F)21\n4(D+F)2 1\n6/parenleftbig\nD2+3F2/parenrightbig1\n3D(D+3F) 01\n24(D+3F)21\n12(D−3F)2\nα(R22)ηη1\n144(D−3F)2 1\n144(D−3F)2 1\n36(D−3F)2 D2\n36D2\n36D2\n91\n144(D+3F)2 1\n144(D+3F)2 1\n36(D+3F)2 D2\n36D2\n9\nα(R22)πη/α(R22)ηπ1\n48(D−3F)(D+F)−1\n16(D−3F)(D+F) 0 −DF\n12DF\n60 −1\n48(D−F)(D+3F)1\n16(D−F)(D+3F) 0 0 0\nα(R23)ππ1\n61\n201\n160 05\n48−1\n1603\n160\nα(R23)KK 0 01\n81\n121\n31\n121\n241\n81\n401\n4\nα(R23)ηη 0 0 01\n481\n481\n121\n481\n481\n120 0\nα(R23)πη/α(R23)ηπ 0 0 01\n48−1\n2401\n48−1\n160 0 0\nα(R24)ππ1\n61\n201\n160 05\n48−1\n1603\n160\nα(R24)KK 0 01\n81\n121\n31\n121\n241\n81\n401\n4\nα(R24)ηη 0 0 01\n481\n481\n121\n481\n481\n120 0\nα(R24)πη/α(R24)ηπ 0 0 0 −1\n481\n240 −1\n481\n160 0 0\n16References\n[1]Particle Data Group Collaboration, R. 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Taniguchi,3Diptiman Sen,2and Anindya Das†1\n1Department of Physics, Indian Institute of Science, Bangalore 560012, India\n2Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India\n3National Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan\nThis letter presents a non-local study on the electric field tunable edge transport in an hBN-\nencapsulated dual-gated Bernal stacked (ABA) trilayer graphene across various displacement fields\n(D) and temperatures ( T). Our measurements revealed that the non-local resistance ( RNL) sur-\npassed the expected classical ohmic contribution by a factor of at least two orders of magnitude.\nThrough scaling analysis, we found that the non-local resistance scales linearly with the local resis-\ntance ( RL) only when the Dexceeds a critical value of ∼0.2V/nm. Additionally, we observed that\nthe scaling exponent remains constant at unity for temperatures below the bulk-band gap energy\nthreshold ( T <25K). Further, the value of RNLdecreases in a linear fashion as the channel length\n(L) increases. These experimental findings provide evidence for edge-mediated charge transport in\nABA trilayer graphene under the influence of a finite displacement field. Furthermore, our theo-\nretical calculations support these results by demonstrating the emergence of dispersive edge modes\nwithin the bulk-band gap energy range when a sufficient displacement field is applied.\nThe emergence of gapless edge modes at the physical\nboundariesofatwo-dimensionalsystemisoneofthemost\nfascinating phenomena in condensed matter physics.\nUsually, these edge modes are related to the bulk topo-\nlogicalorderofthesystem[1–6]andplayasignificantrole\nin electronic transport. Some notable examples are the\nhelical edge modes in Z2topological insulators [7–10],\nchiral quantum Hall edge modes [11, 12], valley-helical\nedge modes in graphene [13–17], kink states[13, 18], and\nso on. These edge modes are also believed to be key in-\ngredients for the observation of the electric field-induced\nmagnetism [19, 20], valley-dependent transport [16, 17],\nand half-metallic behavior in graphene or multilayer\ngraphene [19, 20].\nRecently, trilayer graphene (TLG) has emerged as\na novel two-dimensional material, where several elec-\ntronicphases, forexample, spin-polarizedhalf-metal[21],\nspin and valley polarized quarter metal [21], supercon-\nductivity [22], correlated Chern insulators and ferro-\nmagnetism [23], have been realized experimentally. To\ncompletely understand the electronic properties of these\nphases, it is essential to study both bulk and edge trans-\nport. Usually, in the absence of a perpendicular displace-\nment field, the band structure of Bernal stacked TLG is\ndescribed by a set of linear and quadratic bands, which\nare similar to the low-energy bands of single and bilayer\ngraphene, respectively, as shown in Fig. 1 (b). However,\nunder the application of the large displacement field, the\ninterplay of layer asymmetry and trigonal warping leads\nto the formation of new sets of Dirac cones, as shown in\nFig. 1 (c). In addition to modification in the bulk-band\nstructure, the application of displacement field also in-\nduces a non-trivial valley Hall state, where the energy\ngap at the emergent Dirac points is filled by chiral edge\nmodes which propagate in opposite directions between\ntwo valleys [24]. Although the emergence of the new sets\nof the Dirac cones has been experimentally observed ina quantum capacitance measurement [25], an experimen-\ntal manifestation of the predicted edge modes [24] is still\nlacking.\nNon-local transport measurements have been widely\nused to study the unconventional transport mechanism\nintwo-dimensionalsystemslikethedetectionofbulkspin\nand valley transports [26–32]. Along with that, the non-\nlocal resistance measurement is believed to be an im-\nportant tool to probe the edge states in the topological\ninsulator [33–35] and has been widely used to explore\nthe edge transport mechanism in several electronic sys-\ntems [36, 37] including the twisted bilayer system [38].\nIn this work, we have carried out non-local resistance\nmeasurements in an hBN-encapsulated dual-gated ABA\nTLG device. The measured non-local resistance was\nfound to be at least two orders of magnitude larger than\nthe classical ohmic contribution. More importantly, the\nnon-local resistance RNLscales linearly with the local\nresistance RL, suggesting that the charge transport is\nedge-mediated [34, 35, 39–42]. The scaling exponent α\n(RNL∝Rα\nL)was found to be close to 1beyond a crit-\nical displacement field D. Below the critical field, the\nedge states are not dispersive and do not contribute sig-\nnificantly to non-local transport. Similarly, for temper-\natures ( T) smaller than the scale of the band gap, α\nremains close to 1. On further increasing the tempera-\nture, αstarts deviating from 1due to the contribution\nof bulk transport. Moreover, we have measured the RNL\nat different distances between the injected and measured\nprobes, and found that the value of RNLdecreases in a\nlinear fashion as the distance Lbetween the probes in-\ncreases. Thisisexpectedfortheedge-mediatedtransport\nas described in [42]. To further establish our findings, we\nperform a theoretical calculation of the edge-mode dis-\npersion for different displacement fields, and found that\nthe edge modes become dispersive only above a critical\ndisplacement field, which is consistent with our experi-arXiv:2402.00461v1  [cond-mat.mes-hall]  1 Feb 20242\nV\nBGRxy\nISD\nV\nTG(a)                                                           (d)\n0.5 1 1.5 2 2.5 3\nVTG (V)2468B (T)\n-1012345\ndRxy / dVTG (a.u.) (e)\n(b)                (c)LW\nh/4e2\nh/6e2\nh/10e2\nh/12e2\nh/14e2\nh/22e2 h/18e2Rxy (k )\n0.5 1 1.5 2 2.5\nVTG(V)24682.10T\n2.20T\n2.30T\n2.40T\n2.50T\n2.60T\n2.70T\n2.80T\n2.90T\n3.00T\n3.10T\n3.20T\n3.30T\nFigure 1: (a)Schematic of the device configuration: Bernal stacked trilayer graphene (TLG) is encapsulated between two hBN\nsubstrates and is gated by a graphite back gate ( VBG) and a metal top gate ( VTG).(b)and(c), Band dispersions of Bernal\nstacked TLG at zero and finite ( ∆ = 200 meV) displacement field, respectively. (d)Rxyresponse of the device as a function\nofVTGfor several values of the magnetic field. (e)dRxy\ndVTGis plotted as a function of VTGand magnetic field. The observation\nof the crossing points between the different Landau levels, depicted as discontinuities in the QH plateau structure (blue strips)\nalong the white dashed lines, confirms the ABA character of the TLG.\nmental findings.\nFor the non-local resistance measurement, we fabri-\ncated a hBN-encapsulated Bernal stacked TLG device\nusing the standard dry transfer technique with a high\nmobility of ∼300000cm2V−1s−1. Our device is gated by\na graphite back gate and a metal top gate. The details\nof the device fabrication is described in the Supplemen-\ntal Material[43]. Fig. 1 (a) shows the schematic of the\ndevice structure. The electrical resistance was measured\nusing the standard low-frequency lock-in technique. Be-\nfore discussing details of the non-local measurement, we\nfirst discuss the quantum Hall response of the device,\nwhich establishes the Bernal stacked trilayer character\nof the graphene. Fig. 1 (d) shows a plot of Rxyas a\nfunction of the top gate voltage VTG. The various colour\ntraces correspond to different values of the magnetic field\nas shown in the legend. One can see that at a low mag-\nnetic field, well-developed robust quantum Hall plateaus\nappear at h/6e2,h/10e2,h/14e2,···, which are the char-\nacteristic plateaus of TLG [55–59]. We also observe other\nsymmetry broken intermediate plateaus, suggesting the\nhigh quality of the device. To further confirm the Bernal\nstacked trilayer nature of the graphene, in Fig. 1 (e),\nwe plot a two-dimensional color map of dRxy/dVTGas\na function of the magnetic field (B)and the top gate\nvoltage VTG. The crossing between the Landau levels\nof the monolayer and bilayer-like bands, whose energies\nscale differently with the magnetic field, can be seen as\ndiscontinuities in the QH plateau structure (blue strips)\nalong the white dashed lines in Fig. 1 (e). The positions\nof the crossing points are similar to the earlier experi-\nmental observations suggesting the ABA character of the\nTLG [57, 60, 61].\nThe dual gate architecture of the device allows us to\ntune the carrier density nand the displacement field D\nindependently. Fig. 2 (a) shows a color map of the localresistance (RL)as a function of the displacement field D\nand total density n. The resistance at the Dirac point at\na higher displacement fields do not change significantly,\nsuggesting a small band gap of the ABA TLG consis-\ntent with the earlier observations [58, 62, 63]. Similarly,\nFig. 2 (b) shows a color map of the non-local resistance\nRNL, (defined as VNL/I) as a function of the displace-\nment field Dand total density n. In Fig. 2 (c), we plot\nthe line cuts of RL(red) and RNL(black) as a function\nof the total density at D=−0.4V/nm. RNL(black)\ncurve is multiplied by 20 to show it on the same resis-\ntance scale axis. To rule out the origin of the ohmic\ncontribution due to a classical diffusion of charge trans-\nport, we calculate the ohmic contribution using the equa-\ntionRNL=WRL\nπLexp(−πL/W ), [26, 30, 31, 64, 65] with\nL= 4µm and W= 1.8µm. The measured RNLis two\norders of magnitude larger than the theoretically calcu-\nlated ohmic contribution, suggesting a non-trivial origin\nof the observed RNL.\nMotivated by the earlier non-local resistance measure-\nments in graphene/hBN superlattice and gapped bilayer\ngraphene devices, we perform a scaling analysis of RNL\nagainst RL. We look for a simple scaling relation RNL∝\nRα\nLto determine the value of α. We plot lnRNLversus\nlnRLin Fig. 2 (d) as a function of Dfor different val-\nues of nfrom−3.2to5.6×1014m−2, and in Fig. 2 (e)\nas a function of nfor different values of Dfrom−0.25\nto−0.50V/nm. The data points for these plots are ex-\ntracted along the vertical dashed black arrows and the\nhorizontal dashed yellow arrows shown in Figs. 2 (a,b),\nrespectively. The scaling analysis of both Figs. 2 (d) and\n(e)showthatlinearfittingoftheplot lnRNLversus lnRL\ngives a slope equal to one ( α≈1).\nTo further investigate the linear scaling of RNLwith\nRL, we extract the thermal activation gap by measuring\nthe temperature dependence of the local and non-local3\n(d) (e)\n6.8 7 7.2 7.4 7.63.23.43.63.84.04.2\nln(RL)ln(RNL)\nn = -3.2 X 1014 m-2n = 5.6 X 1014 m-2\nn = -1.2 X 1014 m-2n = 1.3 X 1014 m-2n = 4.0 X 1014 m-2\nRNL  RL\n1.52.53.54.5\n5 7 8 6D = -0.50 V/nm\nD = -0.45 V/nm\nD = -0.40 V/nm\nD = -0.35 V/nm\nD = -0.30 V/nm\nD = -0.25 V/nmRNL  RLln(RNL)\nln(RL)(f)\nEg,NL ≈ 1.1Eg,L\n1.4 1.6 1.8 2.01.71.92.12.3\nEg, L (meV)Eg,NL (meV)\nI+\nI-V+\nV-\nRNL(Ω)\n Rxx(kΩ)I+ I-\nV+V-(a) (b)\n-0.9 -0.6 0.3 0 0.3 0.6 0.90.00.51.01.52.02.5(c)\nn (1012 cm-2)R (k )RL\nRohmRNL\n× 20× 20\nRch× 20\nFigure 2: Color map of RL(a)andRNL(b)as a function of total carrier density nand the displacement field D. (c) The line\ncuts of RL(red) and RNL(black) are plotted with density at D=−0.4V/nm. The RNLis multiplied by a factor of 20. The\nmagenta and green curves (both multiplied by 20) represent the theoretically expected non-local contributions from the charge\naccumulation (near the edge) and classical ohomic one, respectively. ( d,e) Log-log plots of RNLwith RL. Open circles are\nextracted from Figs. 2 (a,b) for different values of n(along vertical black arrows) near the Dirac point ( d) and ( e) for different\nvalues of D(along horizontal yellow arrows). The solid lines correspond to the linear fitting of the data points with slope ∼1.\n(f)The activation gap extracted from RNLis plotted versus the gap extracted from RL. The red line is the linear fit to these\ndata with slope ∼1.1. Different filled circles correspond to the gap extracted at Dranging from −0.29to−0.50V/nm. Error\nbars correspond to the standard deviation associated with the slope of the linear fit.\nresistances. As shown in figure S6 in [43], RNLalso fol-\nlows an activated behavior at high temperatures similar\nto the RL. The temperature dependence of RLin the\nactivation transport regime is proportional to eEg/kBT.\nIf the non-local resistance follows the scaling relation\nRNL∝Rα\nL, then its temperature dependence will be\nproportional to eαEg/kBT. As a result, the activation gap\nextracted from the non-local resistance (Eg,NL)should be\nαtimes of the gap obtained from local resistance (Eg,L),\ni.e.,Eg,NL=αEg,L. In Fig. 2 (f), we have plotted the ac-\ntivation gap extracted from non-local resistance against\nthe gap extracted from local resistance. The filled circles\ncorrespond to the gaps extracted at displacement fields\nranging from -0.29 to -0.50 V/nm. The red line is the\nlinear fit of these data points with slope ∼1.1, again es-\ntablishing the scaling exponent αclose to 1. Note that\nthough the scaling analysis in Fig. 2 is limited (the range\nofRL) due to the small band gap opening in ABA TLG\n(as seen in Fig. 2 (f)), further scaling analysis for various\ndisplacement fields, temperatures, and channel lengths,\nwhich will be discussed in the next section, shows that\nlinear scaling is robust for ABA TLG.\nThe linear scaling of RNLwith RLresembles the edge-mediated charge transport by the helical edge modes ob-\nserved earlier in several two-dimensional electronic sys-\ntems [34, 35, 39–41, 66]. Thus, we attribute the lin-\near scaling of RNLwith RLto an edge-mediated charge\ntransport in TLG. To strengthen the claim of our find-\nings, we study the effect of the displacement field, tem-\nperature and separation between the probes as described\nbelow. It can be seen from Fig. 3 (a) that only above a\ncritical |D|≳0.2V/nm the αbecome close to 1. Fig. 3\n(b) shows how αvaries with TatD=−0.45V/nm, and\ncanbeseenthat T≳25Kαstartsdeviatingfrom 1. This\nis consistent with the edge-mediated charge transport in\nABA TLG. Above 25K, which corresponds to an energy\nscale similar to the band gap opened in ABA TLG, the\nbulk states start contributing to the transport, and α\ndeviates from 1. Fig. 3 (c) shows RNLfor different L,\nand it can be seen from the inset that RNLdecreases lin-\nearly with L, which is in accordance with edge-mediated\ntransport as reported for quantum spin Hall phase and\ntopological insulators [34, 35, 39–42]. Note that similar\nresults (critical D) are obtained for positive displacement\nfields. However, due to the limited range of Don the\npositive side, as seen in Fig. 2 (b), we have presented the4\n(a) (b)\n(c)\n0.1 0.2 0.3 0.4 0.5\n|D|  (V/nm)0.81.01.21.4 Edge \ntransportT = 5.2K\n(d)\nEdge transport\nT < E\ng\n10 20 30 40 50\nT (K)0.91.11.3D = -0.45 V/nm\n1.01.2)\n-0.50 -0.25 0 0.25 0.50\nn (1012 cm-2)20406080\nL = 4 m\nL = 8 m\nL = 12 mRNL (4 8L ( m)\n04080\n \nRNL ()12\nE(meV)20\n0\n∆ = 20 meVE(meV)20\n0\n∆ = 40 meV\nFigure 3: (a)The scaling exponent αplotted as a func-\ntion of DatT= 5.2K. For |D|<0.2V/nm, αdeviates\nsignificantly from unity. (b)αplotted as a function of T\nforD=−0.45V/nm. For T≳25K αstarts deviating from\nunity. (c)RNLis plotted with density for channel lengths of\nL= 4µm (black), 8 µm (blue) and 12 µm (red). Inset: The\npeak value of RNL(blue circles) plotted for different L. The\nred line is a linear fit. (d)The dispersion of TLG, along with\nthe zigzag edge, is plotted for ∆ = 20meV (top) and ∆ = 40\nmeV (bottom). The green (red) curves correspond to modes\non the right (left) edge of the system.\ndata for negative D. We have repeated the experiment in\ndifferent thermal cycles, which is summarized in section-\n8 in [43], and shows similar results with scaling exponent\n1.\nTo understand edge-mediated non-local charge trans-\nport, we now perform theoretical calculations to confirm\nthe presence of edge states in TLG in the presence of the\ndesired displacement field. We consider a potential drop\nof2∆between the top and bottom layer of TLG due\nto applying the external displacement field. The magni-\ntude ∆is related to the experimentally applied displace-\nment field Dvia the relation 2∆ =−(d⊥\nϵTLG×D)e, where\nd⊥= 0.67nm is the separation between the top and\nbottom layers of TLG, ϵTLGis the dielectric constant\nof the TLG, and eis the electronic charge. Considering\nthe electric field, the Hamiltonian for this system has a\nform described in detail in [43]. The presence of ∆opens\nup a gap in the bulk dispersion but, more interestingly,\nalso gives rise to six new Dirac points each around the\nDirac points KandK′points. These play a major role\nin hosting the edge states in this system.\nThe Hamiltonian is time-reversal symmetric, implying\nthat the total Hall conductivity σxysummed over all the\nvalleysmustbezero. Butifwelookataparticularvalley,\nthe system has a non-zero σV\nxy. We estimate this quan-\ntity by numerically calculating the valley Chern number\nusing the method of Fukui et al [67] in the discretizedBrillouin zone close to a Dirac point (See [43] for details).\nAlthough the total Chern number summed over valleys is\nzero, the valley Chern number CVequals 2.5 for all the\nvalues of ∆relevant to the corresponding experimental\nvalues of D. This implies that there is a non-zero val-\nley Hall conductivity of σV\nxy=−2.5(e2/ℏ), which agrees\nwith the theoretically predicted value in Ref. [24]. The\nnon-zero valley Chern number suggests that there is a\npossibility of having edge modes in the system. How-\never, the edge modes would not be robust to perturba-\ntions since the counter-propagating modes from Kand\nK′valley can hybridize.\nIn TLG, for ∆from 20meV to 50meV, we find that\nfor a zig-zag edge configuration, the edges host gapless\nmodes in the bulk gap. The method used for determin-\ning these edge modes is given in [43]. Figs. 3 (d) show\nplots for ∆ = 20 meV and ∆ = 40 meV with the edge\nmodes in green (red) for the right (left) edge of the sys-\ntem. Since they are present in the bulk gap, they partici-\npate in transport along the edges. However, we note that\nthese edge modes are not protected from backscattering.\nHence, there can be intervalley scattering between the\nstates, and a simple dissipative model for edge transport\ncan mimic this and explain the linear scaling between\nlocal and non-local resistances as described using a re-\nsistor network circuit model in Ref. [42]. Further, the\ncircuit model (equation S20 of Ref. [42]) as explained in\nsection-4 in [43] captures linear decay of the non-local\nresistance with the channel length ( L) as seen in our ex-\nperiment (the inset of Fig. 3c). We would like to point\nout that this is unlike the experimental results for bilayer\ngraphene [42], where the bulk valley Hall effect domi-\nnates and gives a cubic relation between RNLandRL\nas reported in Refs. [31] and [42]. From our theoretical\ncalculation, we also find that for small values of the dis-\nplacement field below 20meV (See Fig. S10 in [43]), the\nedge modes are approximately flat and non-dispersive,\nthus not contributing to the non-local charge transport\nsignificantly. This is consistent with the experimental\nresults, where we find that as a function of the displace-\nment field, αdeviates from one for values of |D|below\n0.15V/nm ( ∆<20 meV) as shown in Fig. 3(b). This\nis also consistent with Zibrov et al. [25], where at the\nsimilar displacement field, the Fermi surface undergoes a\nLifshitz transition from one electron pocket to multiple\nisolated Dirac cones.\nIn general, the non-local signal can originate from\nmainly three different sources: (i) classical contribution,\n(ii) a new kind of topological effect - bulk valley Hall ef-\nfect [30, 31, 42] or (iii) edge transport due to either topo-\nlogical [24, 68, 69] or non-topological (charge accumula-\ntion) [70] edge modes. These three mechanisms have also\nbeen highlighted in Refs. [71, 72]. Although the non-local\nmeasurement is not a smoking gun to distinguish its ori-\ngin, estimating the non-local contributions from the dif-\nferent sources can help to find its dominant contribution.5\nAsshowninFig.2(c)bythegreensolidline, theclassical\nohmic one is ruled out, and similarly, as mentioned be-\nfore, the linear scaling between RNLwith RLin specific\nparameter spaces of temperature and displacement field,\nrule out the bulk valley Hall effect. Now, the question\nis whether the observed edge transport in our experi-\nment originated from a topological or non-topological ef-\nfect. To figure it out, we estimate the contribution from\nthe non-topological charge accumulation effect ( Rch)[70]\nand shown by the solid magenta line in Fig. 2 (c) (de-\ntail in section-7 in [43]), which is one order of magnitude\nsmaller than the measured non-local signal (solid black\nline in Fig. 2 (c)). Further, the linear decay of RNLwith\nL(Fig. 3 (c)) rules out the charge accumulation contri-\nbution, which would have scaled exponentially with the\nlength[70] (see section-7 in [43]). Thus, the dominant\ncontribution to our non-local signal presumably comes\nfrom the dispersive edge modes of TLG as predicted in\nRef. [24] and shown by our theoretical calculation (be-\nyond a critical displacement field). Our findings are in\nsharp contrast to Ref[70] on a non-aligned single-layer\ngraphene device, where the dominant contribution to the\nnon-local signal was the charge accumulation effect [70],\nand is expected due to the absence of dispersive edge\nmodes [73].\nIn conclusion, the consistent linear scaling of non-local\nresistance across temperature variations, displacement\nfield changes, and a threefold variation in channel length\ncorresponds to the dispersive edge mode transport in cor-\nrelation with our theoretical calculations.\nS.K.S. thanks, Adhip Agarwala, Priya Tiwari, and\nManbendra Kuiri, for the useful discussions. S.K.S. ac-\nknowledges PMRF, Ministry of Education, India for fi-\nnancial support. D.S. thanks SERB, India for support\nthrough project JBR/2020/000043. 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B 54, 17954\n(1996)\n.Supplemental Material for “Electric field tunable edge transport in\nBernal stacked trilayer graphene”\nSaurabh Kumar Srivastav1*, Adithi Udupa2, K. Watanabe3, T. Taniguchi3, Diptiman Sen1,2, Anindya Das1†\n1Department of Physics, Indian Institute of Science, Bangalore 560012, India\n2Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India\n3National Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan\n*ssaurabh@iisc.ac.in\n†anindya@iisc.ac.in\n1SM section-1: Device fabrication and basic characterization.\nWe fabricated a dual gated hBN encapsulated stacked trilayer graphene (TLG) device for the non-local\nresistance measurement using the standard dry transfer technique1. The process involved the mechanical\nexfoliation of graphite and bulk hBN crystal on the SiO2/Si wafer to obtain TLG and thin hBN. TLG was\nfirst identified under an optical microscope and was later confirmed via Raman spectroscopy. As shown in\nSupplemental Fig. S1 (a), we observe two characteristic Raman peaks, which appear around ∼1580 cm−1\n(‘G’) and ∼2960 cm−1(‘2D’) and belong to the “graphene” family. Furthermore. the peak intensity ratio\nand the spectral decomposition of the ‘2D’ peak into six Lorentzian (Supplemental Fig. S1 (b)) confirms the\n(ABA) trilayer nature of the graphene flake2–4. As discussed in the main manuscript, the ABA character of\nthe used TLG flake was further confirmed in quantum Hall measurements.\nThe fabrication of the hBN/TLG/hBN/graphite involves the following process. First, a top hBN was\npicked up at a temperature of 90◦Cusing a poly-bisphenol-A-carbonate-coated polydimethylsiloxane block\nmounted on a glass slide attached to the tip of a micro-manipulator. The picked-up hBN flake was aligned\nover the TLG flake, which was picked up at a temperature of 90◦C. Following a similar procedure, we picked\nup the bottom hBN and graphite. The resulting heterostructure (hBN/TLG/hBN/graphite) was dropped down\non top of an oxidized silicon wafer ( p++doped silicon with SiO2) at a temperature of 180◦C. This final\nstack was cleaned in chloroform (CHCl3) followed by acetone and isopropyl alcohol (IPA). The next step\ninvolved electron-beam lithography (EBL) to define the contact region. Poly-methyl-methacrylate (PMMA)\nwas coated on the resulting heterostructure. The contact regions were defined using EBL. The edge contacts\nwere achieved by reactive ion etching (a mixture of CHF3and O2gas was used with a flow rate of 40 and\n4 sccm, respectively, at 25◦Cwith RF power of 60 W), where the etching time was optimized such that the\nbottom hBN is not etched completely. This is done to isolate the metallic contacts from the bottom graphite\nthat will serve as the back gate. Next, thermal deposition of Cr/Pd/Au (5/15/60 nm) was performed to make\nthe contacts in an evaporator chamber having a base pressure of ∼1×10−7to2×10−7mbar and followed\nby lift-off procedure in acetone and IPA. We again coat the PMMA on the resulting device and define the\ntop gate area using EBL. Finally, we did a thermal deposition of Cr/Au(5/60 nm) followed by the lift-off\nprocedure. The optical image of the final device is shown in Supplemental Fig. S2 (a).\nAfter fabricating the device, we first study the zero-field back gate response of the device, as shown\nin Supplemental Fig. S2 (b). The measured resistance data is fitted with the equation5–9\nR=RC+L\nWeµ/radicalbigg\nn2\n0+/parenleftig\nCBG(VBG−VDP)\ne/parenrightig2, (S1)\nwhere RC,L,W,µ, and eare, respectively, the contact resistance, length, width, mobility, and electron\ncharge. The carrier concentration of the channel is given byCBG(VBG−VDP)\newithCBGandVDPbeing\nthe capacitance per unit area of the bottom graphite gate, and the voltage at the charge neutrality point,\n20.00.10.2  0.30.4  0.5\n2550 2600 2650 2700 2750 2800\nRaman Shift (cm-1)Intensity (a.u.)2D peak\nCumulative\nLorentzianfit\n0.00.20.4  0.60.8  1.0 Intensity (a.u.)\n1600 2000 2400 2800\nRaman Shift (cm-1)G\n2DABA-TLG\nABA-TLG(a)                                                              (b)Fig. S 1: Raman Spectroscopy of Bernal stacked (ABA) trilayer graphene: (a) Raman spectrum of\nthe used “graphene” flake. (b)Zoomed Raman spectra around 2D peak. It is fitted well with the six\ncharacteristic Lorentzian peaks of Bernal stacked (ABA) trilayer graphene.\n-1.5 -1 -0.5 0 0.5 1 1.5\nVBG- VDP (V)0.00.51.01.52.02.5R (k ) µ ~ 300,000 cm2 V-1s-1(a)                                                                               (b)\nISDRxx\nBGTG\nFig. S 2: optical image and basic device characterization at zero magnetic fields: (a) Optical image of\nthe dual gated hBN encapsulated Bernal stacked trilayer graphene. (b)The four-probe resistance (measure-\nment configuration is shown in (a)) is plotted as a function of back gate voltage at T= 5.2K. Open circles\nshow the experimental data and the red curve is the fit of data in accordance with Eq. (S1). From this fit,\nextracted mobility was found to be ∼300,000 cm2V−1s−1.\nrespectively, and n0is the charge inhomogeneity. The extracted mobility from the fitting was found to be ∼\n300,000 cm2V−1s−1, indicating the high quality of the device.\n3SM section-2: Measurement scheme of non-local resistance detection and absence of spurious non-\nlocal voltage\nDetecting the true non-local resistance requires a careful design of the measurement setup. One of the most\ncommon artifacts that can give rise to spurious non-local resistance is the current leakage to the voltage\nterminals due to the finite input impedance between the input and ground terminals. The origin and the\nmagnitude of this spurious non-local resistance can be simply understood with the basic schematic shown\nin Fig. S3 (a). The magnitude of the spurious non-local voltage for the given schematic will be\nVs\nNL,amp =VCRamp\nRamp+RA−VCRamp\nRamp+RB≈(RB−RA)VC\nRamp, (S2)\nwhere VCis the potential of the point C, and RA(RB) is the net resistance from point C to the input of\nvoltage metal lead terminal A(B). So any difference between RAandRBwill, in principle, contribute to\nthis spurious non-local voltage signal. Since the typical values of RAandRB, should be of the order of\nmagnitude of the sample resistance Rsample , the magnitude of this spurious non-local resistance will be\nRs\nNL,amp ≈(RB−RA)Rsample\nRamp. (S3)\nIt is evident from Eq. S3 that using the high input impedance voltage amplifier Rampwill significantly re-\nduce the magnitude of this spurious non-local resistance signal. In our experiment, the non-local voltage\nwas measured with a Stanford Research Systems SR830 lock-in amplifier after being amplified with a com-\nmercial differential voltage amplifier SR560 ( Ramp= 100 MΩ) (see Fig. S3 (b)). Since the typical sample\nresistance in our device was found to be ∼2.5kΩ, the maximum magnitude of this spurious non-local\nresistance will be of the order of 0.1Ωor less, which is at least two orders of magnitudes smaller than the\nobserved non-local resistance in our device.\n4+_Lock-in \nAmplifier100 M Ω\n100 M ΩIIS\nIDA\nB+_ VNLRamp\nRampdifferential\nAmplifier\nCRA\nRB(a) (b)\nSR 560 AmplifierFig. S 3: Non-local measurement schematic to eliminate spurious signal from the current leakage:\n(a)A simple circuit model of the non-local resistance measurement set-up. This simple model is used to\nestimate the upper bound on the spurious non-local signal due to the current leakage arising from the finite\ninput impedance of the voltage pre-amplifier. RA(RB) is the total resistance between the point Cand the\nmetal lead A(B).(b)Schematic of the non-local voltage measurement with SR 560 voltage pre-amplifier.\nSM section-3: Absence of heating effect for observed non-local resistance\nTo rule out the heating effect10as the origin of the observed non-local signal, we measure the current\ndependence of the non-local resistance. Fig. S4 shows the plot of non-local resistance with density at 20,\n50, and 100 nA of the excitation current. The measured non-local resistance was the same for three different\nexcitation currents, confirming the absence of a heating effect as the origin of the observed signal.\n5-0.8 -0.4 0 0.4 0.8\nn (1012 cm-2)20406080RNL ()20 nA\n50 nA\n100 nAFig. S 4: Absence of heating effect for observed non-local resistance: The non-local resistance is plotted\nfor three different excitation currents of 20, 50, and 100 nA. As the plot shows, the measured signal is\nindependent of the excitation current, ruling out the heating effect as the origin of the observed non-local\nresistance.\nSM section-4: Length dependence of the non-local resistance\nIn Fig. S5 (b), we plot the gate response of the non-local resistance for three different channel lengths ( L:\ndistance between the injection and detection terminals) of 4 (black), 8 (blue) and 12 µm (red) as can be\nseen from the optical image of the device (Fig. S2a). A schematic for the measurement configuration is\nshown in Fig. S5 (a). As the channel length is increased, the non-local resistance reduces significantly. To\nunderstand the functional behaviour of this decrease with the sample length, we plot the peak values of the\nnon-local resistance as a function of the length, as shown in Fig. S5(c) and the data points are best captured\nby a linear fitting. The linear decrease of the measured non-local signal with the length is in accordance\nwith edge-mediated transport. The length dependence of the non-local resistance has also been observed in\nthe quantum spin Hall phase and topological insulator samples, where the charge transport was dominated\nmainly by the helical edge modes at the boundary of the sample11–16. As discussed later in the theoretical\nsection, there are edge modes in ABA TLG. The edge states are counter-propagating originating from the\nKandK′valleys. Since the system is non-topological and only has a non-zero valley Hall conductance,\nthe counter-propagating modes are not protected by backscattering into each other. Thus, this transport can\nbe modeled by a dissipative transport along the edges. Such a circuit model has been studied in Ref. 16, and\nthe non-local resistance based on the resistor network is given by16\nRNL∝4l1l2\nW(L+l1+l2)RL, (S4)\nwhere l1is the distance to the current probe from the nearest end of the sample, l2is the distance to the\nvoltage probe from the other end, Lis the distance between the current and voltage probes, and Wis the\n6(b) (c)\n10 12\nL ( m)020406080\n RNL ()\n-0.50 -0.25 0 0.25 0.50\nn (1012 cm-2)20406080\nL = 4 m\nL = 8 \nL = 12 RNL ()\n4 6 8(a)\n(R19,46)(R19,28)1 2 3\n9 7 65 10 WL l2l1I+\nI-V+\nV-4\n8(R19,37)m\nmFig. S 5: Length dependence of non-local resistance: (a) Schematic of the non-local resistance measure-\nment for three different channel lengths. The current is injected between contact 1 and 9, and the non-local\nvoltage is measured between contacts 2 and 8 (3 and 7 / 4 and 6) for 4µm(8µm/12µm) of channel length.\nl1is the distance to the current probe from the nearest end of the sample, l2is the distance to the voltage\nprobe from the other end, Lis the distance between the current and voltage probes, and Wis the width of\nthe device. (b)The non-local resistance is plotted for three different channel lengths of L= 4µmm (black,\nR19,28), 8µm(blue, R19,37), and 12 µm(red,R19,46). With increasing channel length L, the non-local re-\nsistance reduces significantly. The first two subscripts of R19,28correspond to the current injection contacts,\nwhile the last two correspond to the voltage probes. (c)The peak value of the non-local resistances shown\nin(b)is plotted as a function of the length L. The red line is the linear fit to these data points. The linear\ndecrease in the non-local resistance with sample length is consistent with edge-mediated non-local charge\ntransport.\nwidth of the device. This simple resistor model captures the α= 1 scaling relations between the local\nand non-local resistances. In our measurement configuration, as shown in Fig. S5 (a), the current injection\ncontacts are always fixed; hence l1is constant. In the denominator of the Eqn. S4, the term W(L+l1+l2)\nis constant because as we change the voltage probes, Lincreases but l2decreases by the same amount. In\nthe numerator, we can replace l2byL′−L, where L′is the distance from the current injection contact to the\nother end of the sample, which is fixed in our case. Hence, RNLin Eqn. S4 will be proportional to the length\nL(RNL∝(L′−L)). This simple model clearly explains the linear length dependence of our non-local\nresistance. However, we would like to point out that the scaling relation α= 1 seen in our experiment\nis quite different than the experimental results obtained by measurements done on bilayer graphene. In the\ncase of bilayer graphene, despite the presence of zig-zag edge modes, the bulk conduction dominates, giving\na cubic relation between RNLandRLas reported in Refs. 16 and 17.\n70.02 0.04 0.06 0.08 0.1 0.12\n1/T (K-1)6.66.87.07.27.47.6ln(RL,max)D  = -0.29 V/nm\nEg = 1.52 meV \n0.05 0.1 0.152.42.62.83.03.23.43.6ln(RNL,max)D  = -0.29 V/nm\nEg = 1.74 meV (a) (b)\n1/T (K-1)-0.8 -0.4 00.4 0.8n(10 12 cm-2)\n0.51.01.52.0\n8 K\n50 KR (k )\n102030\n8 K\n50 K-0.8 -0.4 0 0.4 0.8n(10 12 cm-2)R ( )Fig. S 6: Temperature dependence of local and non-local resistances: (a/b) The peak resistance values\nof the local/non-local resistance ( lnRL,max,lnRNL, max) are plotted as a function of 1/Tat a displacement\nfield value of D=−0.29V/nm. Open circles represent the experimental data points, while the solid line is\nthe linear fit to these data points to extract the thermal activation gap. The activation gap extracted from the\nlocal (non-local) resistance was 1.52 (1.74) meV . Insets show plots of local (RL)and non-local resistance\n(RNL)as a function of the carrier density at D=−0.29V/nm. Different colour traces in (a,b) correspond\nto different values of the temperature ranging from 8 to 50 K.\nSM section-5: Thermal activation gap from the temperature dependence of local and non-local resis-\ntances\nTo extract the activation gap from the temperature dependence of the local and non-local resistances, we\nplot the lnRL,maxandlnRL,max(peak resistance value at Dirac point) versus 1/T. In Fig. S6 (a), lnRL,max\nis plotted versus 1/TatD=−0.29V/nm. The open circles represent the experimental data points, and\nthe solid line is the linear fit to the data points to extract the activation gap. The activation gap extracted\nfrom the local resistance was found to be 1.52 meV . Similarly, to determine the activation gap from the\nnon-local resistance, lnRNL, maxis plotted versus 1/TatD=−0.29V/nm in Fig. S6 (b). Here also, the\nopen circles represent the experimental data points, and the solid line is the linear fit to the data points to\nextract the activation gap. The activation gap from the non-local resistance measurement was found to be\n1.74 meV . The insets of Figs. S6 (a) and (b) show plots of RLandRNLas a function of the carrier density at\nD=−0.29V/nm, respectively. Different colour traces correspond to different temperature values ranging\nfrom 8 to 50 K. To extract the displacement field dependence of scaling exponent α, we perform a similar\nmeasurement at several values of the displacement field.\n8SM section-6: Bulk and Edge mode dispersion of Bernal stacked (ABA) trilayer graphene at finite ∆\nThe non-local and local resistance have a scaling relation given by RNL∼Rα\nL. The measurements show\nthatα= 1for the range of displacement field from 0.2V/nm to 0.5V/nm. The value of αstays close to 1\nup to a temperature of around 25Kat which the temperature is close to the band gap. These observations\nstrongly suggest that the transport is dominated in this sample by edge modes. In this section, we show\ntheoretically that for trilayer graphene with displacement fields consistent with the experiments, the valley\nChern number is non-zero with a large value of 2.5for a given valley and spin. This suggests the presence\nof edge modes in the system, although they are not topologically protected. We show that the system does\nhost zig-zag edge modes for the displacement fields of our interest. A simple resistor circuit model then\nexplains the linear relation between the non-local and local resistance measurements.\nBulk properties of trilayer graphene in the presence of a perpendicular electric field: Considering\nABA stacked trilayer graphene with |Ai⟩and|Bi⟩representing the Bloch functions on the ithlayer, the\nHamiltonian in the basis |A1⟩,|B1⟩,|A2⟩,|B2⟩,|A3⟩and|B3⟩has the following form close to the Dirac\npoint18,\nH(p) =\nH0V W\nV†H′\n0V†\nW V H 0\n,\nwhere H0=/parenleftigg\n0v0p∗\nv0p δ/parenrightigg\n,H′\n0=/parenleftigg\n∆′v0p∗\nv0p 0/parenrightigg\n,V=/parenleftigg\n−v4p∗v3p\nγ1−v4p∗/parenrightigg\n, and W=/parenleftigg\nγ2/2 0\n0γ5/2/parenrightigg\n,\nwithp=px+ipybeing the momentum as measured from the Dirac point. The different band velocities\nare given by vi= 3aγi/2ℏ, where the γi’s are the couplings as described below, and a∼2.46Å is the\nlattice constant, i.e., the distance between the two nearest Aatoms in the same layer. The coupling γ0comes\nfrom hopping between nearest-neighbour sites within a layer. The parameters γ1,γ3andγ4are the nearest\ninterlayer couplings in this model. γ1comes from coupling between the dimer atoms ( B1↔A2,A2↔\nB3). Since this is a vertical coupling (see Fig. S7), the corresponding matrix element in the Hamiltonian\nis independent of the momentum p. The parameter γ3describes the couplings between the non-dimer\norbitals ( A1↔B2,B2↔A3) and γ4between dimer and non-dimer atoms ( A1↔A2,B1↔B2,\nA2↔A3andB2↔B3). These couplings have an in-plane component (Fig. S7)) analogous to the intra-\nlayer nearest-neighbour hopping and therefore come with a momentum-dependent factor in the Hamiltonian.\nThe couplings between the first and third layers have strengths γ2between the Asublattices and γ5between\ntheBsublattices. The parameter δis the on-site energy asymmetry coming from the sites which are involved\nin the γ1couplings and the sites which are not.\n9Fig. S 7: (a) Lattice structure of ABA stacked trilayer graphene. The first and the third layer sit exactly on\ntop of each other, whereas the second layer has the atoms in the Asublattice right below the atoms in the B\nsublattice on the first layer. Since these are positioned vertically above each other, the orbitals B1-A2and\nsimilarly A2-B3are called dimer atoms. (b) Side view of the lattice with the intra- and interlayer couplings\nmarked.\nAdditionally, in the presence of the displacement field, the Hamiltonian becomes\nH(p) =\nH0+U V W\nV†H′\n0 V†\nW V H 0−U\n,where U= ∆/parenleftigg\n1 0\n0 1/parenrightigg\n. (S5)\nWe can block diagonalize the Hamiltonian by re-defining the basis as\n1√\n2(|A1⟩ − |A3⟩),|B2⟩,1√\n2(|B1⟩+|B3⟩),1√\n2(|B1⟩ − |B3⟩),|A2⟩,1√\n2(|A1⟩+|A3⟩). (S6)\nIn this basis, the TLG Hamiltonian takes the form18,\nH=/parenleftigg\nH◦D−\nD+H•/parenrightigg\n, (S7)\nwhere H◦(H•) is written in the first (second) three bases of Eq. (S6),\n10H◦=\n−γ2/2 0 0\n0 0 −√\n2v4p\n0−√\n2v4p∗γ5/2 + ∆′\n, (S8)\nH•=\n−γ5/2 +δ 0 0\n0 δ −√\n2v4p\n0 −√\n2v4p∗γ2/2\n, (S9)\nD+=\nv0p 0 ∆\n0 v0p∗√\n2γ1\n∆√\n2v3pv0p∗\n,and D−=D†\n+. (S10)\nInD±,2∆is the potential drop between the top and bottom layers of TLG due to the application of\nthe external displacement field via local gating. It is related to the experimentally applied displacement field\nDvia the relation 2∆ = −(d⊥\nϵTLG×D)e, where d⊥= 0.67nm is the distance between the top and bottom\nlayers of TLG, ϵTLG is the dielectric constant of the material, and eis the electronic charge.\nWe now plot the dispersion obtained for this model using the values of the parameters given in the\nliterature as follows: γ0= 3.16eV ,γ1= 0.39eV ,γ2=−0.02eV ,γ3= 0.32eV ,γ4= 0.044eV ,γ5= 0.038\neV ,δ= 0.05eV , and ∆ = 0 .2eV . From Fig. S9, we can clearly see that the bulk dispersion of TLG contains\na single Dirac cone and six other off-centred Dirac cones, which are emerging due to the interplay between\nthe trigonal warping and the layer asymmetry term ∆. The emergent new Dirac cones in the bulk band\ndispersion were observed experimentally in quantum capacitance measurements of a high-quality, Bernal\nstacked TLG sample19.\nValley Chern number calculation for trilayer graphene The Hamiltonian in Eq. (S7) is time-reversal\nsymmetric. Thus the total Hall conductivity σxysummed over the different valleys must be zero. But if we\nlook at a particular valley, we find that this system has a non-zero σV\nxy. We estimate this quantity using the\nmethod of Fukui et al.20. For a system with nbands and |ψ(k)⟩being the normalized wave functions in the\nnthband, the total Chern number is given by\nC=/summationdisplay\nn,occ1\n2π/integraldisplay\nBZdk(Fn)z, (S11)\n11where the summation runs over all the occupied bands. The quantity (Fn)zis called the Berry curvature and\nis equal to the curl of the Berry connection An= (An\nx,An\ny). These quantities are given by\nAn=−i/parenleftig\n⟨ψn(k)|∂kx|ψn(k)⟩,⟨ψn(k)|∂ky|ψn(k)⟩/parenrightig\n, (S12)\nand therefore\n(Fn)z= (∇ ×A)z=−i/parenleftig∂\n∂kxAn\ny−∂\n∂kyAn\nx/parenrightig\n. (S13)\nWe are interested in evaluating the valley Chern number numerically. Instead of integrating over the entire\nBrillouin zone, we take a discretized segment of the (kx, ky)space around one of the Dirac points ( Kor\nK′) about which we would like to calculate σV\nxy. We ensure that this space contains the Dirac point and the\nsix-off centred Dirac points of the trilayer graphene. Since most of the contribution to the Berry curvature\n(Fig. S8 (b)) comes from these points, this is a valid region to work with. We then divide this region into\nsmaller plaquettes (see Fig. S8 (a)) and use Fukui’s method to calculate σV\nxyas follows. For the nthband,\nwe define a quantity\nUn\ndk(k) =⟨ψn(k)|ψn(k+dk⟩), (S14)\nwhere dktakes us to the next point in the chosen plaquette. The quantity Fn(k)is now written in terms of\nUn(k)as\nFn(k) =−ilog/parenleftig\nUn\ndkx(k)Un\ndky(k+dkx)Un−1\ndkx(k+dky)Un−1\ndky(k)/parenrightig\n. (S15)\nTo understand this quantity better, consider the grid shown in Fig. S8 (a). Let the wave functions at ( kx, ky),\n(k+dkx, ky), (k+dkx, ky+dky) and ( k, ky+dky) be denoted by ψ1,ψ2,ψ3andψ4respectively as shown.\nThe quantity Fnfor this (kx, ky)is then given by\nFn(k) =−ilog(⟨ψ1|ψ2⟩⟨ψ2|ψ3⟩⟨ψ3|ψ4⟩⟨ψ4|ψ1⟩), (S16)\nand the valley Chern number will then be\nCV\nn=1\n2π/summationdisplay\nkFn(k). (S17)\nThe total valley Chern number is obtained by summing over all the occupied bands, namely, the valence\nbands, since the Fermi level lies in the band gap. Due to time-reversal symmetry, we find that CK=CK′\n.\nWe find numerically that the valley Chern number CVfor a particular valley has a finite value for all values\nof∆. The value of CVatKpoint changes from 2.5to−0.5close to ∆ = 0 .27eV as shown in Fig. S8\n(b). This matches with the theoretical estimate of σV\nxyin Ref. 18. Our region of interest lies is the range\nof∆from 0.02eV to 0.05eV where we have a finite CVof2.5giving σV\nxy=−(e2/ℏ)CV= 2.5(e2/ℏ).\nThis suggests that there is a possibility of having edge modes in the system. However, they would not be\nrobust to perturbations since the counter-propagating modes from KandK′valley can hybridize. In the\nnext section, we will explicitly look for edge modes in trilayer graphene within the range of ∆of interest.\n12xy/uni007C/uni03C84/uni27E9(kx,ky+dky)/uni007C/uni03C83/uni27E9/uni007C/uni03C82/uni27E9/uni007C/uni03C81/uni27E9(kx+dkx,ky+dky)(kx+dkx,ky)(kx,ky)\n-0.50.50\nvpy//uni03B31\nvpx//uni03B31-1.601.6×10−384030-3-5vpx//uni03B31-0.50.5\n-1.61.60vpy//uni03B31/uni0394</uni0394c\n/uni0394>/uni0394c\n-0.50.50\nvpy//uni03B31\nvpx//uni03B31-1.601.6×10−384030-3-5vpx//uni03B31-0.50.5\n-1.61.60vpy//uni03B31/uni0394</uni0394c\n/uni0394>/uni0394c(a)(b)(c)(d)Fig. S 8: (a) The discretized mesh grid of the kspace used to evaluate the Chern number using Fukui’s\nmethod. The quantity Fn(k)as defined in Eq. (S16) is then summed over all these plaquettes in the chosen\nregion of kspace. The region is chosen such that all the contribution coming from the Berry curvature lie\nwithin this region. As we can see from in Figure (b), the contribution from the klying outside this region\nis zero. (b) Surface plot of the Berry curvature for the third band (i.e., the valence band) of the system for\n∆ = 0 .05eV , which is less than the critical field. We see that the contribution to the Berry curvature rapidly\ngoes to zero away from the six off-center Dirac points. (c) Surface plot of the Berry curvature of the valence\nband for ∆ = 0 .7eV . Once again, the maximum contribution comes from the six off-centered Dirac points.\nPresence of edge modes in trilayer graphene at finite ∆:As presented above, the experimental data\nsuggest linear scaling between the local and non-local resistance. This indicates the presence of edge modes\nin the system, as will be discussed further in SM Sec. 7. We will first investigate theoretically if trilayer\ngraphene can indeed host edge modes at the values of ∆of our interest. We implement the method described\nby T. Morimoto et al. in their work18. We consider a semi-infinite system with a zig-zag boundary along\nthex-direction. The edge modes on the left and right edges are given by the regions y > 0andy < 0,\nrespectively. We note that the presence of armchair edge dilutes the contribution from edge transport as\nit enhances valley mixing in the presence of atomic-size scatterers. Since the system is periodic in the\nx-direction, pxis a good quantum number. However along the y-direction, we need to replace pyby the\ncorresponding operator ˆpy=−iℏ∂\n∂y. The Hamiltonian can now be divided into parts dependent on and\nindependent of px, asˆH=Aˆpy+B(px). In the basis defined in Eq. (S6), these matrices have the form\nA=\n0 0 0 −iv0 0 0\n0 0 −i√\n2v40 iv0−i√\n2v3\n0i√\n2v4 0 0 0 iv0\niv0 0 0 0 0 0\n0−iv0 0 0 0 −i√\n2v4\n0i√\n2v3−iv0 0i√\n2v4 0\n,\n13B=\n−γ2/2 0 0 v0px 0 ∆\n0 0 −√\n2v4px 0 v0px√\n2v3px\n0−√\n2v4pxγ5/2 + ∆′∆√\n2γ1 v0px\nv0px 0 ∆ −γ5/2 + ∆′0 0\n0 v0px√\n2γ1 0 ∆′−√\n2v4px\n∆√\n2v3px v0px 0 −√\n2v4px γ2/2\n.\nThe Schrödinger equation ˆHψ=ϵψthen gives\n∂\n∂yψ=iA−1(ϵ−B)ψ=Cψ (S18)\nwhere Cis an eigenvalue of ∂/∂y . For a given pxwith an energy lying within the bulk gap, we look for a\nwave function of the form exp(py,ny)un, where py,nandunare the complex eigenvalues and eigenvectors\nof Eq. (S18). For this 6×6Hamiltonian, we have six eigenvalues for every pair of values (px, ϵ). Out of\nthese, three eigenvalues have Re(py)>0and the other three have Re(py)<0. For modes at the right\nedge, lying in the region y <0, we choose the eigenvalues for which Re(py)>0so that our wave function\ndecays into the system as we go away from the edge. For modes at the left edge, we take the eigenvectors\nwhich have Re(py)<0. The six-component general wave function for either region thus looks like\nψ=3/summationdisplay\nn=1CnUn,where Un= exp( py,ny)un. (S19)\nFrom Fig. S10 (a), we can see that the atoms from the Bsublattice form the right edge. This puts a condition\non the wave function of the atoms on the Asublattice and their components in the wave function are set to\nzero. Thus we have |A1⟩= 0,|A2⟩= 0 and|A3⟩= 0. This implies that in the re-defined basis, the wave\nfunction in Eq. (S19) must satisfy the constraint\n\nU1\n1U1\n2U1\n3\nU5\n1U5\n2U5\n3\nU6\n1U6\n2U6\n3\n\nC1\nC2\nC3\n=\n0\n0\n0\n, (S20)\nwhere Uj\nnis the jthcomponent of the ntheigenvector and is a function of pxandϵ. For a given px, we look\nfor values of ϵwithin the band gap such that the determinant of the matrix involving Uj\nnis zero. Similarly,\nwe calculate the allowed values of ϵfor the left edge. In this case, we consider eigenvalues with Re(py)<0\nwith the boundary condition |B1⟩= 0,|B2⟩= 0and|B3⟩= 0.\n14vpx/1(a)                                                    (b)                                                          (c)\n-0.4 0 0.4\nvpx/1-0.400.4vpy/1\n-0.2-0.10\nE (eV)Fig. S 9: (a) Three-dimensional plot of the dispersion of TLG for ∆ = 0 .2eV close to the Dirac point. (b)\nContour plot for the same quantity showing the six off-centred Dirac points around the Kpoint marked with\nblack circles. (c) The bulk dispersion for ∆ = 0 .2eV near the Fermi level as a function of pxalong lines\npassing through the six Dirac points.\nThe displacement field used in the experimental measurements goes from D= 0.25V/nm to 0.5\nV/nm. The potential difference between the top and the bottom layer is 2∆ = −(d⊥\nϵTLG×D)e, where d⊥nm\nis the distance between the top and bottom layers of TLG, ϵTLG is the dielectric constant of the material,\nandeis the electronic charge. Substituting the literature values of d⊥= 0.67nm and ϵ0= 4, we find that\nthe corresponding range of ∆goes from 20meV to 40meV . In Figs. S10 (b) and (c), we have plotted the left\n(red) and right (green) edge modes for ∆ = 10 meV and ∆ = 40 meV . For values of ∆less than 20meV ,\nwe notice that these edge modes become less dispersive and might not effectively contribute to transport.\nThis supports the experimental data where the exponent αdeviates from 1 for smaller values of D.\n15y<0y>0y (periodic)x\nRe(py)>0Re(py)<0\n(a)(b)(c)\nΔ=10meVΔ=40meVFig. S 10: (a) Zig-zag edge for trilayer graphene. The yellow-coloured atoms B1, B2andB3in the region\ny <0form the right edge. The blue circled atoms A1, A2andA3in the region y >0form the left edge.\n(b) The dispersion of trilayer graphene, along with the energies of the zig-zag edge modes, are plotted. The\ngreen (red) curves correspond to modes on the right (left) edge. This plot is for ∆ = 10 meV . (c) The same\nplot shows edge modes within the band gap for ∆ = 40 meV .\nSM section-7: Various origins for non-local signal and their contributions\nIn this section, we discuss the few mechanisms that are believed to be possible origins of non-local signals\nmeasured in graphene-based devices. (1) Classical ohmic contribution which can be calculated with the\nwell-known formula RNL=WRL\nπLexp(−πL/W ),17, 21, 22where Lis the separation between the current\nand voltage probe and Wis width of the device. (2) The non-local signal originating from the bulk valley\ncurrent due to the Berry curvature hot spots. This mechanism has been used to explain the findings reported\nin graphene/hBN superlattice21and gapped bilayer graphene16, 17devices. (3) The edge-mediated non-local\ncharge transport due to topological18, 23, 24or non-topological (charge accumulation at the physical edge of\nthe sample)25edge modes. These three mechanisms have also been highlighted in the recent articles26, 27.\nThe classical ohmic contribution can be estimated using the well-known equation mentioned above\nand is ruled out in our experiment as its contribution is orders of magnitude smaller than the observed\nnon-local signal. The second mechanism (bulk valley current) is under debate. Theoretical work reported in\nRef. 28 pointed out that non-local resistance peaks cannot be taken as a signature of the anomalous velocities\nproduced by an electric field and a non-vanishing Berry curvature. A crucial point leading to this conclusion\nwas that the use of a semi-classical theory needs to be clarified because, in graphene/hBN superlattice and\ngapped bilayer graphene experiments, the Fermi level was tuned to lie in an electronic gap, where transport\noccurs in the tunnelling regime and a semi-classical formalism should not be used.\nNow we discuss the third mechanism (edge-mediated transport), which is believed to be the possible\n16origin of the observed non-local signal in graphene devices as supported by several theoretical articles23, 24.\nHowever, it should be noted that these theoretical articles do not consider the charge accumulation effect\n(non-topological edge) as the origin of the non-local signal, as reported for the non-aligned single-layer\ngraphene device25.\nNon-topological (charge accumulation) origin of non-local resistance: The magnitude of the non-\nlocal resistance due to the charge accumulation at the physical boundary of the sample can be calculated\nusing the equation25\nRNL=WR L\nπLexp/parenleftbigg\n−πL\nλ/parenrightbigg\n, (S21)\nwith\nλ=W/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalbt1 +2η/parenleftbigg\n1 +π2\n4/parenleftbig1+η(µ2B2+1)\n1+η/parenrightbig2/parenrightbigg\n1 + 2 η1+η(µ2B2+1)\n1+η. (S22)\nwhere η, µandBare the charge accumulation factor, mobility, and magnetic field, respectively.\nAs shown in Figure 2 (c) of the main manuscript, the estimated value of the non-local resistance due\nto the charge accumulation factor of η= 0.7(this is the typical charge accumulation factor reported in\nRef. 25), Rch, is one order of magnitude smaller than the observed non-local signal in our device. Further,\nthe linear decay of RNLwithL(Fig. S5 (c)) rules out the charge accumulation contribution, which would\nhave scaled exponentially with the length as seen in eqn. S2125. The magnitude and its length dependence\nprovide us with enough grounds to firmly believe that the significant contribution to the non-local signal\nobserved in our devices is due to the dispersive edge modes predicted to exist in Bernal stacked trilayer\ngraphene under the application of a sufficient displacement field18and is not due to the charge accumulation\neffect25.\nThe contribution from topological edge modes and non-topological edge modes (due to charge accu-\nmulation) can co-exist in an edge-mediated transport mechanism. However, in the work of Aharon-Steinberg\net al25, the single-layer graphene sample was not aligned with hBN. Naively, one does not expect the pres-\nence of dispersive edge modes in single-layer graphene devices. Some edge modes can exist along the\nzig-zag edge of single-layer graphene29. However, these edge modes are flat (non-dispersive) and do not\ncontribute to the transport. This is probably why charge accumulation was the dominant source of non-local\ntransport reported in Re. 25. As pointed out in Refs. 23, 24, the alignment with hBN (as in Ref. 21) helps\nthese edge modes to gain a dispersion effectively, and then they start contributing to the charge transport.\nThis shortcoming is further discussed in Ref. 27.\n17SM section-8: Thermal cycle\nTo strengthen the robustness and reproducibility of our data shown in the main manuscript, we perform\nsimilar non-local measurements in a different thermal cycle of the device. A thermal cycle is meant to expose\nthe device from the low temperature to ambient temperature and atmospheric environment and further cool\ndown the device to low temperature from the room temperature in a different cool-down process. A thermal\ncycle changes the disorder configurations and impurity concentration in graphene devices and, in principle,\ncan be treated as an entirely new device from the disorder configuration perspective. The results of the\nlocal and non-local resistance measurements at T= 1.8K temperature after the second cooldown are\nsummarized in Fig. S 11. Fig. S 11 (a, b) shows the color map of the local resistance (RL)and non-local\nresistance (RNL)as a function of the displacement field D, and total density n. In Fig. S 11 (c), we plot\nlnRNLversus lnRLas a function of nfor different values of Dfrom−0.20to−0.50V/nm. The data\npoints for these plots are extracted along the horizontal dashed yellow arrows shown in Fig. S 11 (a,b). The\ninset of Fig. S 11 (c) shows lnRNLversus lnRLas a function of Dfor the different values of nclose to the\nDirac point. The data points for the inset plot are extracted along the vertical dashed black arrows shown in\nFig. S 11 (a,b). The scaling analysis of Fig. S 11 (c) and its inset show that linear fitting of the plot lnRNL\nversus lnRLgives a slope equal to one ( α≈1). Fig. S 11 (d) shows the scaling exponent (α)as a function\nof the D. Scaling exponent (α)remains close to one only above a critical |D|≳0.2V/nm, similar to the\nplot shown in Fig. 3(a) of the main manuscript obtained during the first cooldown of the device. The data\nin Fig. S 11 shows the robustness and reproducibility of the experimental results in two different cooldown\nprocesses.\n18RNL(Ω)Rxx(kΩ)(a) (b)\n(c)I+ I-\nV+V-I+\nI-V+\nV-\n1.52.53.54.57.0 7.4 7.8\n3.03.43.84.2\nln(RL)ln(RNL)\nln(RNL)ln(RL)\n5 7 8 6D = - 0.50 V/nmD = - 0.45 V/nmD = - 0.40 V/nmD = - 0.35 V/nmD = - 0.20 V/nm\nD = - 0.25 V/nm\nD = - 0.30 V/nm\n0.1 0.2 0.3 0.4 0.5\n|D| (V/nm)0.81.01.21.4\nEdge \ntransportT = 1.8 K(d)\nRNL  RLRNL  RLFig. S 11: Non-local charge transport in different thermal cycles of the device Color map of RL(a)and\nRNL(b)as a function of total carrier density nand the displacement field DatT= 1.8K.(c)Log-log plots\nofRNLwithRL. Open circles are extracted from Figs. (a, b) for different values of D(along horizontal\nyellow arrows). The solid lines correspond to the linear fitting of the data points with slope ∼1. Inset:\nLog-log plots of RNLwithRL. Open circles are extracted from Figs. (a, b) for different values of n(along\nvertical black arrows) near the Dirac point. (d)The scaling exponent αplotted as a function of DatT= 1.8\nK. For |D|<0.2V/nm, αdeviates significantly from unity.\nReferences\n1. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617\n(2013).\n2. Malard, L., Pimenta, M. A., Dresselhaus, G. & Dresselhaus, M. Raman spectroscopy in graphene.\nPhysics Reports 473, 51–87 (2009).\n193. Lui, C. H. et al. Imaging stacking order in few-layer graphene. Nano Letters 11, 164–169 (2011).\n4. Cong, C. et al. Raman characterization of aba-and abc-stacked trilayer graphene. ACS Nano 5, 8760–\n8768 (2011).\n5. Venugopal, A. et al. Effective mobility of single-layer graphene transistors as a function of channel\ndimensions. Journal of Applied Physics 109, 104511 (2011).\n6. Kumar, C. et al. Localization physics in graphene moiré superlattices. Phys. Rev. B 98, 155408 (2018).\n7. Kumar, C., Srivastav, S. K. & Das, A. Equilibration of quantum hall edges in symmetry-broken bilayer\ngraphene. Phys. Rev. B 98, 155421 (2018).\n8. Kuiri, M. et al. Enhanced electron-phonon coupling in doubly aligned hexagonal boron nitride bilayer\ngraphene heterostructure. Phys. Rev. B 103, 115419 (2021).\n9. Tiwari, P., Srivastav, S. K. & Bid, A. Electric-field-tunable valley zeeman effect in bilayer graphene\nheterostructures: Realization of the spin-orbit valve effect. Phys. Rev. Lett. 126, 096801 (2021).\n10. Renard, J., Studer, M. & Folk, J. A. Origins of nonlocality near the neutrality point in graphene. Phys.\nRev. Lett. 112, 116601 (2014).\n11. Nichele, F. et al. Insulating state and giant nonlocal response in an InAs/GaSb quantum well in the\nquantum hall regime. Phys. Rev. Lett. 112, 036802 (2014).\n12. Gusev, G. M. et al. Transport in disordered two-dimensional topological insulators. Phys. Rev. B 84,\n121302 (2011).\n13. Knez, I. et al. Observation of edge transport in the disordered regime of topologically insulating\nInAs/GaSb quantum wells. Phys. Rev. Lett. 112, 026602 (2014).\n14. Du, L., Knez, I., Sullivan, G. & Du, R.-R. Robust helical edge transport in gated InAs/GaSb bilayers.\nPhys. Rev. Lett. 114, 096802 (2015).\n15. Knez, I., Du, R.-R. & Sullivan, G. Evidence for helical edge modes in inverted InAs/GaSb quantum\nwells. Phys. Rev. Lett. 107, 136603 (2011).\n16. Shimazaki, Y . et al. Generation and detection of pure valley current by electrically induced berry\ncurvature in bilayer graphene. Nature Physics 11, 1032–1036 (2015).\n17. Sui, M. et al. Gate-tunable topological valley transport in bilayer graphene. Nature Physics 11, 1027–\n1031 (2015).\n18. Morimoto, T. & Koshino, M. Gate-induced dirac cones in multilayer graphenes. Physical Review B 87,\n085424 (2013).\n2019. Zibrov, A. A. et al. Emergent dirac gullies and gully-symmetry-breaking quantum hall states in a b a\ntrilayer graphene. Physical Review Letters 121, 167601 (2018).\n20. Takahiro Fukui, Y . H. & Suzuki, H. Chern numbers in discretized brillouin zone: Efficient method of\ncomputing (spin) hall conductance. Journal of the Physical Society of Japan 74, 1674–1677 (2005).\n21. Gorbachev, R. et al. Detecting topological currents in graphene superlattices. Science 346, 448–451\n(2014).\n22. Balakrishnan, J., Koon, G. K. W., Jaiswal, M., Neto, A. C. & Özyilmaz, B. Colossal enhancement of\nspin–orbit coupling in weakly hydrogenated graphene. Nature Physics 9, 284–287 (2013).\n23. Brown, R., Walet, N. R. & Guinea, F. Edge modes and nonlocal conductance in graphene superlattices.\nPhysical Review Letters 120, 026802 (2018).\n24. Marmolejo-Tejada, J. et al. Deciphering the origin of nonlocal resistance in multiterminal graphene on\nhexagonal-boron-nitride with ab initio quantum transport: fermi surface edge currents rather than fermi\nsea topological valley currents. Journal of Physics: Materials 1, 015006 (2018).\n25. Aharon-Steinberg, A. et al. Long-range nontopological edge currents in charge-neutral graphene. Na-\nture593, 528–534 (2021).\n26. Roche, S., Power, S. R., Nikoli ´c, B. K., García, J. H. & Jauho, A.-P. Have mysterious topological valley\ncurrents been observed in graphene superlattices? Journal of Physics: Materials 5, 021001 (2022).\n27. Torres, L. F. & Valenzuela, S. O. A valley of opportunities. Physics World 34, 43 (2021).\n28. Kirczenow, G. Valley currents and nonlocal resistances of graphene nanostructures with broken inver-\nsion symmetry from the perspective of scattering theory. Physical Review B 92, 125425 (2015).\n29. Nakada, K., Fujita, M., Dresselhaus, G. & Dresselhaus, M. S. Edge state in graphene ribbons: Nanome-\nter size effect and edge shape dependence. Phys. Rev. B 54, 17954–17961 (1996).\n2129"    },    {        "title": "2402.00473v1.Detecting_Extragalactic_Axion_like_Dark_Matter_with_Polarization_Measurements_of_Fast_Radio_Bursts.pdf",        "content": "Detecting Extragalactic Axion-like Dark Matter with Polarization Measurements of\nFast Radio Bursts\nBao Wang ,1, 2,∗Xuan Yang,1, 2,∗Jun-Jie Wei ,1, 2,†Song-Bo Zhang,1,‡and Xue-Feng Wu1, 2,§\n1Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China\n2School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei 230026, China\nAxions or axion-like particles (ALPs) are one of the most promising candidates of dark matter\n(DM). A prevalent method to detect axion-like DM is to seek the periodic oscillation feature in\nthe polarization angles of linearly polarized light emitted from astrophysical sources. In this work,\nwe use the time-resolved polarization measurements of the hyperactive repeating fast radio burst,\nFRB 20220912A, detected by the Five-hundred-meter Aperture Spherical radio Telescope (FAST)\nto search for extragalactic axion-like DM for the first time. Given a DM density profile of FRB\n20220912A’s host, we obtain upper limits on the ALP-photon coupling constant of gaγ<(2.9×\n10−11−1.1×10−9) GeV−1for the ALP masses ma∼(1.4×10−21−5.2×10−20) eV. Persistent\npolarimetric observations with FAST would further improve the constraints. We prove that FRBs\noffer an alternative way to detect axion-like DM on extragalactic distance scales, complementary to\nother galactic DM probes.\nI. INTRODUCTION\nNowadays, one of the most important tasks that re-\nmain unresolved in modern physics is the detection of\ndark matter (DM) particles, and numerous candidates of\nDM have been proposed. Axion-like particles (ALPs), a\ntype of ultralight bosons, have emerged as the most pre-\nvailing candidates in the search for DM [1–3]. Thanks\nto the special properties of ultralight scalar particles\n(ma∼10−22eV), they can provide a natural solution\nto the challenges encountered in small-scale structures of\nthe Universe [4, 5]. A lot of strategies for finding axion-\nlike DM have been explored, including photons from ax-\nion conversion [6–9], nuclear magnetic resonance [10, 11],\nperiodic oscillations of linearly polarized light [12–15],\nand other many terrestrial or astronomical experiments\n[16, 17].\nAmong the various axion-like DM detection methods,\ndetecting periodic oscillations of polarized light has been\nregarded as a promising approach in astrophysics [12–\n15, 18–20]. When light propagates through the ALP\nfield, photons would interact with ALPs, and the interac-\ntion term is Laγ=1\n4gaγaFµν˜Fµν, where gaγrepresents\nthe coupling strength between the axion field ( a) and\nthe electromagnetic field ( Fµν). The interaction leads\nto modifications in the dispersion relations [21, 22]. The\nleft- and right-handed circular polarization modes of light\nexperience opposite corrections due to their different dis-\npersion relations. This effect is known as cosmic bire-\nfringence, resulting in changes in the polarization angles\n(PAs) of the light. Therefore, in the presence of an ALP\nfield, if the light is linearly polarized, its PAs will oscil-\nlate with the ALP field, with an amplitude proportional\n∗These two authors contributed equally to this work.\n†Corresponding author: jjwei@pmo.ac.cn\n‡Corresponding author: sbzhang@pmo.ac.cn\n§Corresponding author: xfwu@pmo.ac.cntogaγ. So far, ALP-photon couplings have been pro-\nvided increasingly stringent constraints from a multitude\nof astrophysical observations [12–15, 18–20].\nFast radio bursts (FRBs) are brief and intense radio\ntransients that originate from cosmological distances [23–\n25]. They have been powerful astrophysical laboratories\nfor studying cosmology and fundamental physics [26–34].\nFRBs also have the potential to play an important role\nin the detection of DM, such as constraining millicharged\nDM from timing of FRBs [19], bounding the dark pho-\nton DM parameter space with the time delays of pho-\ntons from FRBs [35], probing compact DM with FRB\nmicrostructure [36] or with strongly lensed FRBs [37–40],\nand constraining primordial black hole DM with lensed\nFRBs [41]. More recently, Gao et al. [42] proposed a\nfuture-feasible method to hunt axion DM with gravita-\ntionally lensed FRBs, indicating the potential of FRBs\nfor searching for axion DM.\nTo date, no studies have utilized real polarization ob-\nservations of repeating FRBs to directly constrain axion-\nlike DM. The repetition pattern of FRBs enables us to\nmonitor their polarization properties long-term to detect\naxion-like DM on extragalactic distance scales, comple-\nmentary to other galactic DM probes. One intriguing\nsample for such a study is FRB 20220912A, an active\nrepeating source with highly linear polarization [43, 44]\nand its local environment is very clean [45]. Its long-term\npolarization observations from the Five-hundred-meter\nAperture Spherical radio Telescope (FAST) [44] provide\na remarkable opportunity to detect extragalactic axion-\nlike DM through searching for a periodic oscillation in\nthe PAs.\nIn this work, we constrain the ALP-photon coupling\nconstant gaγusing polarization data of FRBs for the first\ntime. We analyze the polarization angle variations of lin-\nearly polarized emission from FRB 20220912A observed\nby FAST. All currently available observations from Oc-\ntober 28th, 2022 to December 5th, 2022 are adopted for\nour study. The observational time coverage of ∼38 daysarXiv:2402.00473v1  [astro-ph.HE]  1 Feb 20242\nwith a cadence of ∼1 day is sensitive to ALPs with mass\nranging from 1 .4×10−21eV to 5 .2×10−20eV. By es-\ntimating the changes in linear PAs, we can place upper\nlimits on gaγdirectly. Finally, we also predict further\nconstraints from continued polarization observations of\nFRB 20220912A in the future.\nII. MODELS\nA. ALP-photon Coupling\nALPs can be described as a scalar field a(x, t) with\nmass ma, where xis the spatial coordinates and tis the\ntime. The ALP field can interact with the electromag-\nnetic field, and its dynamics can be captured by the La-\ngrangian terms [46]:\nL=−1\n4FµνFµν+1\n2∂µa∂µa−1\n2m2\naa2+1\n4gaγaFµν˜Fµν,\n(1)\nwhere Fµνdenotes the electromagnetic field tensor,\n˜Fµν=1\n2ϵµνλσFλσis the dual of Fµν, and gaγrepre-\nsents the ALP-photon coupling constant which charac-\nterizes the strength of interaction. This coupling leads\nto a modification in the dispersion relation:\nω±≃k±1\n2gaγnµ∂µa, (2)\nwhere nµis null tangent vector of light, kis the wave vec-\ntor1, and the frequency ω±corresponds to two circular\npolarization states. When two vertically polarized elec-\ntromagnetic waves of these two states propagate, a phase\nshift occurs between them due to the disparity in their\nphase velocities. This phase shift leads to the rotation\nof the polarization plane, known as cosmic birefringence.\nSpecifically, the frequency difference between the two po-\nlarization components is ∆ ω=ω+−ω−=gaγnµ∂µa. If\nwaves emitted from the source at position x1at time t1\nare received by an observer at position x2and time t2,\nthe ALP-induced PA shift is then expressed as\nϕ=1\n2Z\nC∆ωds=1\n2gaγ[a(x2, t2)−a(x1, t1)],(3)\nwhere Cis the propagation path of waves. The equation\nof motion for the ALP field is given by the Klein-Gordon\nequation. When we neglect the backaction term, the so-\nlution simplifies and exhibits as an oscillating form:\na(x, t) =a0(x) sin ( mat+δ), (4)\nwhere δis the position-dependent phase. a0(x) is the\namplitude that relates to the energy density of the ALP\nfield ρ(or equivalently the energy density of DM, if the\n1The natural unit system ℏ=c=1.dominant DM is assumed to be made up of ALPs), i.e.,\nρ=1\n2maa2\n0. The oscillation period of the ALP field\nis given by T= 2π/m a, which depends on the ALP\nmass. If the energy density of the ALP field at the ob-\nserver is much lower than the one at the source (i.e.,\na(x2, t2)≪a(x1, t1)), Eq. (3) can be converted to an\noscillatory expression,\nϕ(t) =−√\n2\n2gaγm−1\naρ1\n2sin\u0012\n2πt\nT′+δ\u0013\n, (5)\nwhere T′=T(1 + z) is the observed period on Earth,\ntaking into account of the cosmic expansion. Eq. (5)\ndescribes that the PAs have the periodic oscillation char-\nacteristic when linearly polarized waves are coupled with\nALPs.\nB. DM Density Profile\nOutside the solitonic cores of galaxies, the DM density\ndistribution ρ(r) can be approximately described by the\ngeneralized Navarro-Frenk-White (NFW) profile [47]:\nρ(r) =ρ0\n(r/rs)β(1 +r/rs)3−β, (6)\nwhere ris the distance from the galaxy center, ρ0is the\ncharacteristic density, rsis the scale radius, and βis the\npower-law index. Also, ρ(r)∝r−βwhen r≪rsand\nρ(r)∝rβ−3when r≫rs. For the case of β= 1, Eq. (6)\nis reduced to the original NFW profile [48]. In principle,\nthese coefficients ( ρ0,rs, and β) can be determined by\nfitting the rotation curves of galaxies. The physical ori-\ngins of FRBs are still unknown, but some of them have\nbeen localized in extragalactic host galaxies. Once we\nhave enough observational information about the FRB\nhost galaxy, we can estimate the DM density ρin the\nvicinity of the FRB source.\nIII. OBSERVATIONS OF FRB 20220912A\nA. The Host Galaxy\nThe Deep Synoptic Array localised the repeater FRB\n20220912A to a host galaxy, PSO J347.2702+48.7066, at\nredshift z= 0.0771 [43]. The host galaxy has a stellar\nmass of approximately 1010M⊙, a star-formation rate of\n≳0.1M⊙yr−1, and an effective radius of 2 .2 kpc. Gordon\net al. [49] compared the optical host luminosities of re-\npeating and nonrepeating FRBs across redshift, and de-\nfined a demarcation at luminosity 109L⊙below which a\nhost can be classified as a dwarf galaxy. FRB 20220912A\nsits just above the borderline at ≈1.1×109L⊙(see Fig. 9\nof [49]), suggesting that its host may be a dwarf galaxy.\nSince they have higher fractions of DM compared to\nmore massive systems, dwarf galaxies are deemed as ideal3\n50\n050obs [deg]\n59880 59885 59890 59895 59900 59905 59910 59915 59920\nTime [MJD]50\n050100(t) [deg]\nFigure 1. Linear polarization measurements of the bursts from FRB 20220912A detected by FAST. The upper and lower\npanels denote the observed PAs ϕobsand the ALP-induced PA shifts ϕ(t), respectively. Each burst is depicted by a yellow\npoint, the green points represent the daily medians, and the light green shaded area encompasses the 1 σconfidence range. The\nALP-induced PA shifts are estimated through ϕ(t) =ϕobs− ⟨ϕbkg⟩, where ⟨ϕbkg⟩=−24.95◦(see section IV).\nsystems to probe the DM density profile [50]. How-\never, we currently lack rotation curve observations of the\nhost of FRB 20220912A to investigate its DM density\ndistribution. Here we use the DM density profile of a\ndwarf galaxy, NGC 4451 (with a similar stellar mass of\n∼1010M⊙and a similar radius of ∼2.2 kpc [50]), as\na reference. Note that the differences in DM density\nprofiles between NGC 4451 and FRB 20220912A’s host\nhave negligible effects within the precision range of our\nstudy. Based on the stellar rotation curve, Cooke et al.\n[50] determined the coefficients of the generalized NFW\nprofile (Eq. (6)) for NGC 4451, i.e., ρ0= 0.41M⊙pc−3,\nrs= 2.2 kpc, and β= 0. Furthermore, a recent mil-\nliarcsecond localization of FRB 20220912A shows that its\ntransverse offset from the host galaxy center is r≈0.8\nkpc [51]. With this information, an estimate on the DM\ndensity at the location of FRB 20220912A from Eq. (6)\nisρ∼0.16M⊙pc−3. This value is much larger than the\nDM density near our Earth, which is ∼0.01M⊙pc−3\nestimated by the Galactic NFW profile [52].\nB. The Polarization Angles\nAfter the initial discovery of FRB 20220912A, subse-\nquent observations from multiple telescopes have consis-\ntently detected a large number of bursts from this spe-\ncific source [44, 45, 53–62]. It is noteworthy that FRB\n20220912A was monitored by FAST for a period of sev-\neral dozen days, during which a total of 1076 bursts were\nrecorded [44]. Most of these bursts exhibit nearly 100%\nlinear polarization. The rotation measure (RM) of FRB\n20220912A is very close to 0 and did not show any varia-tion during the FAST observation period, indicating that\nFRB 20220912A is located in a relatively clean local envi-\nronment [44, 45]. The non-variable RM also means that\nthe PAs of FRB 20220912A are relatively stable.\nAs described in section II, when a linearly polarized\nlight propagates in an external ALP field, the correspond-\ning PAs would have a time-dependent change due to the\nALP-photon coupling effect. By analyzing the PA vari-\nations, we thus can constrain the ALP-photon coupling\nstrength, thereby facilitating the potential detection of\naxion-like DM. Here, the PAs of FRB 20220912A are\nprocessed from the raw data of FAST. The central fre-\nquency, bandwidth, number of frequency channels, and\nsampling time for the raw data were 1.25 GHz, 0.5 GHz,\n4096, and 49.152 µs, respectively. We use the GPU ac-\ncelerated transient detection software HEIMDALL2and\nprocess the data on FAST’s high-performance-computer\nfacilities. The dispersion measure range that we search\nis between 200 and 250 pc cm−3. The signal-to-noise ra-\ntio threshold and maximum boxcar are set as 6.5 and\n512, respectively. The RM is searched from −2000 to\n2000 rad m−2in steps of 1 rad m−2using the rmfit pro-\ngram [63]. After correcting the data with the best-fitted\nRMs, we derive the position angles of the linearly polar-\nized component. During a total of 9.2 hours of obser-\nvations between October 28th, 2022 and December 5th,\n2022 (corresponding to MJD 59880 and MJD 59918), we\nobtain 700 bursts with RM measurements. Fig. 1 dis-\nplays PA measurements of FRB 20220912A as a func-\ntion of time. One can see from this plot that PAs vary\n2https://sourceforge.net/projects/heimdall-astro4\ngreatly within a day, but they are relatively stable on\nthe timescale of months. Our focus is primarily on the\nPA variations in the order of days, so we calculate the\nmedian value of the PA in each day. In this case, the\nminimum time interval is 1 day, while the total observa-\ntional time is 38 days. According to the theory presented\nin section II, the lower and upper limits of the ALP mass\nmadepend on the total observational time and the min-\nimum time interval, respectively. That is, the sensitive\nALP mass mafalls within the range of 1 .4×10−21eV to\n5.2×10−20eV.\nIV. CONSTRAINTS ON THE ALP-PHOTON\nCOUPLING\nVariations in PAs of FRBs are complex and puzzling\n[64–66]. The prevailing understanding is that these varia-\ntions are mainly attributed to the significant fluctuations\nin the magnetic fields surrounding FRB sources. How-\never, if the axion-like DM exits in the host galaxy and\nenvelops the FRB sources, the observed PA ( ϕobs) is ex-\npected to be composed of two components: one from the\nPA contribution of the astrophysical background (e.g.,\nthe magnetic field), ϕbkg, and the other one is the ALP-\ninduced PA shift, ϕ(t), i.e.,\nϕobs=ϕbkg+ϕ(t)\n≃ ⟨ϕbkg⟩+ ∆ϕbkg+ϕ(t),(7)\nwhere ⟨ϕbkg⟩represents the mean value of the PA caused\nby the background magnetic field and ∆ ϕbkgcorresponds\nto the PA fluctuation arising from the changes of the\nmagnetic field. Since ∆ ϕbkgis unpredictable, we assume\nthat the observed PA fluctuations are attributed to the\nALP-photon coupling effect, i.e., ϕobs=⟨ϕbkg⟩+ϕ(t),\nwhich will conduct conservative upper limits on the ALP-\nphoton coupling constant gaγfor different ALP mass.\nIn this work, we employ two analysis methods to search\nfor the ALP-photon coupling effect. The first way is to es-\ntimate the standard deviation (∆ ϕ) of the ALP-induced\nPA shift ϕ(t). Given the randomness of the value of the\nphase δ(see Eq. (5)), we use the standard deviation of\nϕ(t) to characterize its oscillation amplitude [14]. This\nyields\n∆ϕ≡p\n⟨ϕ2(t)⟩\n≃1.96◦\u0012ρ\n0.16M⊙pc−3\u00131\n2\n×\u0010ma\n10−21eV\u0011−1\u0012gaγ\n10−11GeV−1\u0013\n.(8)\nThe mean value of the observed daily median PAs is\n⟨ϕMed⟩=−24.95◦±4.16◦. Here the mean −24.95◦is re-\ngarded as the mean background ⟨ϕbkg⟩, and the 1 σstan-\ndard deviation 4 .16◦is regarded as ∆ ϕ. From Eq. (8), we\ncan see that ∆ ϕis proportional to gaγfor a given ALP\nmass ma. With the ALP mass ranging from 1 .4×10−21eV to 5 .2×10−20eV, the corresponding upper limits on\ngaγcan be obtained.\nAdditionally, the Markov Chain Monte Carlo (MCMC)\nfitting method is also applied for our analyses. Firstly,\nwe construct a grid in the parameter space of ( ma,δ).\nFor a set of fixed ALP mass mafrom 1 .4×10−21eV\nto 5.2×10−20eV and fixed phase δin the range of [0,\n2π], we perform a fitting procedure to simultaneously\noptimize the ALP-photon coupling constant gaγand the\nmean background ⟨ϕbkg⟩. This process involves minimiz-\ning the χ2value by adjusting the two parameters gaγ\nand⟨ϕbkg⟩with the MCMC technology. The standard\nχ2statistic is given by\nχ2=NX\ni=1\u0014ϕobs,i−ϕ(ti)− ⟨ϕbkg⟩\nσi\u00152\n, (9)\nwhere ϕobs,iandσiare the observed PA and its error.\nThen we average over gaγfor the same ma, which is\nequivalent to marginalizing over the nuisance parameter\nδ.\nThe resulting constraints on gaγfrom both the two\nmethods are shown in Fig. 2, along with other con-\nstraints from different astrophysical sources. For the\nfirst method, we obtain an upper limit of gaγ<2.1×\n10−11GeV−1for an ALP mass ma∼10−21eV. For\nthe second one, the obtained upper limit is gaγ<2.7×\n10−10GeV−1for the same ALP mass. The worse result\nobtained in the second method is probably due to the\nlarge daily fluctuations of PAs after using the complete\ndata in MCMC, which can be regarded as a pessimistic\nestimation. It should be emphasized that these results\nare the first constraints on the ALP-photon coupling de-\nrived from FRB polarization measurements. Finally, we\nalso forecast a future constraint. If the polarization ob-\nservations of FRB 20220912A last for up to one year, the\nlimit would be further improved by one order of magni-\ntude, yielding gaγ<3.0×10−12GeV−1.\nRecently, Gao et al. [42] proposed gravitationally\nlensed FRBs as probes for hunting Galactic axion DM.\nThey predicted that, with a single lensed FRB, the\nALP-photon coupling could be constrained to be gaγ<\n7.3×10−11GeV−1for an axion mass ma∼10−21eV.\nThis forecast limit is similar to our result of gaγ<\n2.1×10−11GeV−1from real FRB polarization observa-\ntions. Furthermore, as shown in Fig. 2, our result is\nslightly better than the constraint from the CERN Ax-\nion Solar Telescope (CAST) ( gaγ<6.6×10−11GeV−1)\n[8] and comparable with the constraint from the super-\nnovae observed by the Fermi Large Area Telescope (LAT)\n(gaγ<2.6×10−11GeV−1) [17]. In contrast to the\ngalactic probes, such as pulsars [14], black hole Sgr A∗\n[15], and protoplanetary disk [18], our method can detect\nALPs on kiloparsec scales, which highlights the potential\nof FRBs for detecting extragalactic axion-like DM.5\n1023\n1022\n1021\n1020\n1019\n1018\nma [eV]1014\n1013\n1012\n1011\n1010\n109\n108\n107\nga[GeV1]\nSN-Fermi\nSN 1987ACAST\nSuper star clusters\nPulsarActive galaxies\nSgr A*FRB (This Work)\nFigure 2. The resulting constraints on the ALP-photon coupling constant gaγfor different ALP mass, obtained using the\npolarization measurements of FRB 2022012A (green shaded area). The green straight and irregular lines correspond to the\nupper limits of gaγderived from the standard deviation method and the MCMC method, respectively. The green dashed line\nrepresents the future constraints from continued polarization observations of FRB 2022012A up to one year. Other constraints\nfrom different astrophysical sources are also displayed for comparisons, including the CAST experiment (yellow shaded area)\n[8], the X-ray observation of super star clusters (blue solid line) [16], the Fermi-LAT observation of supernovae (red dot-dashed\nline) [17], the supernova SN1987A (gray dot-dashed line) [7], the VLBA polarization observations of jets from active galaxies\n(orange shaded area) [12], the polarized pulsar light (gray shaded area) [14], and the polarization measurements of the black\nhole Sgr A∗(red solid line) [15]. Please refer to this linkafor a more complete summary.\nahttps://cajohare.github.io/AxionLimits/docs/ap.html\nV. SUMMARY\nLinearly polarized light emitted from FRBs can inter-\nact with the ALPs, producing a periodic oscillation of\nthe PAs. Searching for this periodic oscillation feature\nin FRB polarization measurements can thus help us to\ndetect the axion-like DM. In this work, we attempted to\ndetect extragalactic ALPs in the host galaxies of the hy-\nperactive repeating FRBs for the first time. With the\nstate-of-the-art observations the PAs of the bursts from\nthe repeating source FRB 20220912A taken by FAST,\nwe obtained upper limits on the ALP-photon coupling\nconstant of gaγ<(2.9×10−11−1.1×10−9) GeV−1for\nthe ALP masses ma∼(1.4×10−21−5.2×10−20) eV.\nIf the polarization observations of FRB 20220912A are\nexpected to continue for one year, the gaγlimit would\nbe improved to be gaγ<3.0×10−12GeV−1for an ALP\nmass ma∼1.4×10−22eV. Extragalactic FRBs provide\nan alternative way to detect axion-like DM on kiloparsec\nscales, complementary to other galactic DM probes.ACKNOWLEDGMENTS\nBao Wang thanks Lei Lei for helpful discussions on\ndark matter. This work is partially supported by the\nNational SKA Program of China (2022SKA0130100),\nthe National Natural Science Foundation of China\n(grant Nos. 12373053, 12321003, and 12041306), the\nKey Research Program of Frontier Sciences (grant No.\nZDBS-LY-7014) of Chinese Academy of Sciences, Inter-\nnational Partnership Program of Chinese Academy of\nSciences for Grand Challenges (114332KYSB20210018),\nthe CAS Project for Young Scientists in Basic Research\n(grant No. YSBR-063), the CAS Organizational Scien-\ntific Research Platform for National Major Scientific and\nTechnological Infrastructure: Cosmic Transients with\nFAST, and the Natural Science Foundation of Jiangsu\nProvince (grant No. BK20221562).6\n[1] J. Preskill, M. B. Wise, and F. Wilczek, Physics Letters\nB120, 127 (1983).\n[2] D. J. E. Marsh, Physics Reports 643, 1 (2016),\narXiv:1510.07633 [astro-ph.CO].\n[3] K. Choi, S. H. Im, and C. S. 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We present four sets of boundary value problems for\na pure stress formulation of three-dimensional solids, and in two dimensions for plane stress and plane strain.\nThe associated governing equations are derived by modifications and combinations of the Beltrami-Michell\nequations and the Navier-Cauchy equations. The corresponding variational forms of dimension d∈ {2,3}\nallow to approximate the stress tensor directly, without any presupposed potential stress functions, and are\nshown to be well-posed in H1⊗Sym(d) in the framework of functional analysis via the Lax-Milgram theorem,\nmaking their finite element implementation using C0-continuous elements straightforward. Further, in the\nfinite element setting we provide a treatment for constant and piece-wise constant body forces via distri-\nbutions. The operators and differential identities in this work are provided in modern tensor notation and\nrely on exact sequences, making the resulting equations and differential relations directly comprehensible.\nFinally, numerical benchmarks for convergence as well as spectral analysis are used to test the limits and\nidentify viable use-cases of the equations.\nKey words: Beltrami-Michell equations, stress formulation, linear elasticity, numerical analysis, finite ele-\nment method.\n1 Introduction\nThe classical formulation of linear elasticity is per the Navier-Cauchy equations, where the displacement field\nuis the main unknown. This formulation has been implemented countless times in finite element frameworks\ndue to its simplicity, its straight-forward incorporation into the variational framework and the corresponding\nproofs of robustness in the compressible regime. However, as is well-known, the formulation is unstable for\nincompressible materials ν= 0.5, since the Saint Venant-Kirchhoff material tensor tends to infinity C→ ∞\nin its components related to the trace of the strain tensor tr ε. A remedy for the latter phenomenon was\nintroduced via the Hellinger-Reissner principle, which reformulated linear elasticity as a mixed problem, making\nuse of the compliance tensor A. The compliance tensor, being the inverse of the material tensor A=C−1,\nremains finite for incompressible materials. Thus, a two-field variational formulation of linear elasticity in both\ndisplacements and stresses emerged, where the displacement field is set to be discontinuous u∈[L2]dand\nthe stress tensor is symmetric and normal-continuous σ∈Hsym(Div). Subsequently, various finite element\napproaches were introduced for the mixed formulation [7, 12, 65]. However, the consistent solution relying\non conforming finite elements based on commuting Hilbert space complexes [8, 13, 62] remained impractical,\nsince the elements required for Hsym(Div)-discretisations did not allow for a reference-to-physical mapping of\ntheir base functions or were simply computationally too expensive. In fact, an efficient mapping procedure in\ntwo-dimensions was only recently introduced in [61] and implemented in the open source software for finite\n1Corresponding author: Adam Sky, Institute of Computational Engineering and Sciences, Department of Engineering, Faculty\nof Science, Technology and Medicine, University of Luxembourg, 6 Avenue de la Fonte, L-4362 Esch-sur-Alzette, Luxembourg,\nemail: adam.sky@uni.lu\n2Andreas Zilian, Institute of Computational Engineering and Sciences, Department of Engineering, Faculty of Science, Tech-\nnology and Medicine, University of Luxembourg, 6 Avenue de la Fonte, L-4362 Esch-sur-Alzette, Luxembourg, email: an-\ndreas.zilian@uni.lu\n1arXiv:2402.00480v2  [math.NA]  17 Mar 2024element computations NGSolve [54, 55] for the Hu-Zhang [25–27] element HZ ⊂ Hsym(Div), which satisfies\nthe commutative property in the elasticity complex [15, 16, 45, 46]. We mention that an alternative mapping\nby combination of mapped base functions was recently introduced in [11] for the Arnold-Winther element\nAW ⊂ Hsym(Div) [6, 10], being the first conforming element for Hsym(Div) in the elasticity complex. An\nalternative novel method called Tangential-Displacement-Normal-Normal-Stresses (TDNNS) [47–50] alleviated\nthe need for conforming Hsym(Div)-elements by applying the Hellan-Herrmann-Johnson principle, for which\nthe two-dimensional Hellan-Herrmann-Johnson [9, 39] elements HHJ ⊂∼H(divDiv) were available and the\nthree-dimensional Pechstein-Sch¨ oberl elements PS⊂∼H(divDiv) were introduced. This innovative approach\nrelied on an unconventional interpretation of the contraction between the divergence of the stress tensor and\nthe displacement field as the dual product ⟨Divσ,u⟩H−1(div)×H(curl), where the stress field is defined as σ∈\nH(divDiv), such that its divergence is in H−1(div) which is dual to H(curl), in which the displacement lives.\nThe newly introduced finite elements for the stress field σin this formulation are symmetric normal-normal\ncontinuous tensor fields, such that the contra-variant Piola transformation can be readily applied in order to\nmap base functions from the reference to the physical element. For the displacement field u∈H(curl) the\nformulation makes use of the classical N´ ed´ elec elements N ⊂ H(curl) [58–60,63,64].\nIn a mixed formulation, the relation of the stress field to the displacement is given by the compliance tensor\nsym D u=Aσ. This relation can also be understood as the requirement that σis to be related through the\ncompliance tensor to some compatible tensor field Aσ∈ker(Inc). This interpretation gives rise to the Beltrami-\nMichell equations [14,41,42], which replace the compliance relation with the compatibility relation Inc( Aσ) = 0,\nalso known as the Saint-Venant compatibility condition. A core characteristic of these equations is that they\nare formulated purely in stresses without any additional fields. Thus, they represent the stress analogue of the\ndisplacement formulation, and proof of the equivalence of the two formulations can be found in [24] per the\nuniqueness of the solution. Despite this desirable feature, the equations have found only limited numerical usage,\nspecifically using finite differences [1,2,66] or recently via Fourier series [4], and in fact, application of the theory\nis almost exclusively analytical. Possibly, this is due to the fact that the equations are ill-suited for standard\nfinite element computations. Indeed, in [17] the authors prove well-posedness of the variational formulation of\nthe Beltrami-Michell equations for a discontinuous and compatible stress field σ∈[L2⊗Sym(3)] ∩ker(Inc)\nwith Inc σ∈H−2⊗Sym( d), for the which the construction of conforming finite subspaces is not trivial. As a\ncuriosity, we mention here that one other special case of a pure stress formulation can be derived specifically\nfor the eigenvalue problem of linear elasticity [28], not yielding the Beltrami-Michell equations.\nThe Beltrami-Michell equations pose several difficulties for a straight-forward implementation in a standard\nfinite element framework. Firstly, the three-dimensional equations are asymmetric, making the use of efficient\nsolvers for symmetric positive definite problems inapplicable. Secondly, as we show in this work, for a mixed\nDirichlet-Neumann boundary the associated variational form is unstable. Thirdly, the two-dimensional Beltrami-\nMichell equations for plane stress and plane strain characterise only the mean stress in the plane, making them\nincomplete in the sense that one cannot solve for the full stress tensor, but rather only its trace tr σ. We note\nthat for the three-dimensional case the problem of asymmetry in the strong form was first noticed and alleviated\nin [51]. In spite of the non-existent finite element framework for the Beltrami-Michell equations, the latter have\nbeen employed successfully in multiple applications [31, 40, 43]. Often, not the Beltrami-Michell equations\nthemselves but rather their special-case descendants, the Beltrami, Maxwell, Morera and Airy [3, 20, 29, 52]\nstress equations, were used in order to analytically investigate the flow of stresses in a medium. Otherwise,\nthe Beltrami-Michell equations in their original form have been used in the theory of mixtures [36], in order\nto describe coupled thermal-hydraulic-mechanical processes [1, 2, 66], and in the prediction of stresses in a\nreservoir [4].\nIn this work we derive the Beltrami-Michell equations in modern tensor notation while relying on the concept\nof exact sequences for our argumentation, thus making the involved operators and differential relations clear.\nWe symmetrise the three-dimensional equations, and combine a weaker version of planar static equilibrium with\nthe two-dimensional Beltrami-Michell equations to introduce a novel full planar field problem for stresses in two-\ndimensions. By variational calculus we introduce the complete boundary value problems and their variational\nforms, for which we subsequently prove well-posedness on the Hilbert space H1\n0⊗Sym( d) via the Lax-Milgram\ntheorem and several original lemmas. Finally, we investigate the numerical stability of the formulations via\nconvergence estimates and spectral analysis, which leads us to present novel stabilised forms of the equations for\nmixed Dirichlet-Neumann boundary conditions. The novel equations introduced in this paper are summarised\nin Appendix A.\n2The numerical benchmarks in this work are computed using the open source software NGSolve1, such that\nwe rely on three- and two-dimensional conforming elements Lp(V)⊂H1(V), and Lp(A)⊂H1(A), constructed\nvia hierarchical Legendre polynomials [67].\n1.1 Notation\nThe following notation is used throughout this work. Exception to these rules are made clear in the precise\ncontext.\n•Vectors are defined as bold lower-case letters v,ξ\n•Second order tensors are denoted with bold capital letters T\n•Fourth-order tensors are designated by the blackboard-bold format A\n•We denote the Cartesian basis as {e1,e2,e3}\n•Summation over indices follows the standard rule of repeating indices. Latin indices represent summation\nover the full dimension, whereas Greek indices define summation over the co-dimension.\n•The angle-brackets are used to define scalar products of arbitrary dimensions ⟨a,b⟩=aibi,⟨A,B⟩=\nAijBij\n•The matrix product is used to indicate all partial-contractions between a higher-order and a lower-order\ntensor Av=Aijvjei,AB=AijklBklei⊗ej\n•The second-order identity tensor is defined via 1, such that 1v=v. Analogously, the fourth-order identity\ntensor Jyields JT=T\n•Subsequently, we define various differential operators based on the Nabla-operator ∇=∂iei, which is\ndefined with respect to the dimension of the domain\n•Volumes, surfaces and curves of the physical domain are identified via V,Aands, respectively.\n•Tangent and normal vectors on the physical domain are designated by tandn, respectively.\nWe define the constant space of symmetric second order tensors as\nSym( d) ={T∈Rd×d|T=TT}. (1.1)\nIts counterpart is the space of skew-symmetric tensors\nso(d) ={T∈Rd×d|T=−TT}. (1.2)\nFurther, for our variational formulations we introduce the following Hilbert spaces and their respective norms\nL2(V) ={u:V→R| ∥u∥L2<∞}, ∥u∥2\nL2=Z\nVu2dV , (1.3a)\nH1(V) ={u∈L2(V)| ∇u∈[L2(V)]d}, ∥u∥2\nH1=∥u∥2\nL2+∥∇u∥2\nL2. (1.3b)\nHilbert spaces with vanishing traces are marked with a zero-subscript, for example H1\n0(A). Scalar products\npertaining to the Hilbert spaces are indicated by a subscript on the angle-brackets\n⟨u, v⟩L2=Z\nV⟨u, v⟩dV , (1.4)\nwhere the domain is clear from context.\nIn the following we define various differential operators derived from the ∇-operator.\n•The left-gradient is given via ∇, such that ∇λ=∇ ⊗λ\n1www.ngsolve.org\n3•The right-gradient is defied for vectors and higher order tensors via D, such that D v=v⊗ ∇\n•The Hessian is given as the composite hess λ= D∇λ\n•We define the vectorial divergence as div v=⟨∇,v⟩\n•The tensor divergence is given by Div T=T∇, implying a single contraction and acting row-wise\n•The Laplacian is given via ∆ λ= div∇λfor scalars and as ∆ T= Div D Tfor higher order tensors\n•The vectorial curl operator reads curl v=∇ ×v\n•For tensors the operator is given by Curl T=−T× ∇, acting row-wise\n•The composite incompatibility operator reads Inc T= curl Curl T=−∇ × T× ∇\n•In two dimensions the vectorial curl-operator induces the scalar operator ∇⊥λ=R∇λand vectorial\noperator rot v= div( Rv), with R=e1⊗e2−e2⊗e1\n•Analogously, the tensorial Curl-operator induces the vectorial operator D⊥v= (Dv)RTand the tensorial\noperator Rot T= Div( TRT), acting row-wise\n•Consequently, the composite incompatibility operator reduces to rot Rot Tfor tensors and the Airy-\noperator airy λ= D⊥∇⊥λfor scalars\nLastly, in this work we employ the standard algebraic operators\nsymT=1\n2(T+TT), skwT=1\n2(T−TT), trT=⟨T, 1⟩,\nvolT=1\nd(trT) 1, devT=T−volT, 1,T∈Rd×d, (1.5)\ndefining the symmetry, skew-symmetry, trace, volumetric and deviatoric operators on tensors. We generalise\nthe trace operator to higher order tensors with the notation\nTrT=Tijkei⟨ej⊗ek, 1⟩=Tijjei∈R, T∈Rd×d×d. (1.6)\nFor skew-symmetric tensors we also introduce the operators\na×v= (Anti a)v= [axl(Anti a)]×v∈R, (1.7)\nwhere (Anti a)∈so(3)⊂R3×3maps the vector a∈R3to its corresponding skew-symmetric tensor for an\nequivalent form of the cross product given through a single contraction, and axl(Anti a) =a∈R3is the inverse\noperator extracting the axial vector.\n2 Linear elasticity\nIn this section we briefly recap the Navier-Cauchy equations of linear elasticity. These equations are subsequently\nused in the derivation of the Beltrami-Michell equations.\n2.1 Static equilibrium\nThe energy functional of linear elasticity with the linear strain measure ε= sym D uis given by\nI(u) =Z\nV1\n2⟨sym D u,Csym D u⟩ − ⟨u,f⟩dV−Z\nAN⟨u,µ⟩dA→ min w.r.t. u, (2.1)\nwhere u:V⊂R3→R3is the displacement field, C∈R3×3×3×3is the tensor of material constants, f:V⊂\nR3→R3represents the body forces, and µ:AN⊂R2→R3are the tractions on the Neumann boundary AN,\nsee Fig. 2.1. For a linear isotropic Saint Venant-Kirchhoff material the elasticity tensor reads\n4xyn\nf µ VAD\nAN\nFigure 2.1: The domain V⊂R3with Dirichlet AD⊂R2and Neumann AN⊂R2boundaries under internal\nbody forces fand tractions µwith the outer normal vector n.\nσ=Csym D u, C=E\n1 +ν\u0014\nJ+ν\n1−2ν1⊗ 1\u0015\n, (2.2)\nwhere {E, ν}are Young’s modulus and the Poisson ratio, 1:R3→R3is the second order identity tensor, and\nJ:R3×3→R3×3is the fourth order identity tensor. Variation of the energy functional with respect to the\ndisplacement field yields\nZ\nV⟨sym D δu,Csym D u⟩dV=Z\nV⟨δu,f⟩dV+Z\nAN⟨δu,µ⟩dA ∀δu∈[C∞(V)]3. (2.3)\nApplying partial integration leads to\nZ\n∂V⟨δu,(Csym D u)n⟩dA−Z\nV⟨δu,Div(Csym D u)⟩dV=Z\nV⟨δu,f⟩dV+Z\nAN⟨δu,µ⟩dA∀δu∈[C∞(V)]3,\n(2.4)\nfrom which we extract the boundary value problem of linear elasticity in three dimensions\n−Div(Csym D u) =f in V , (2.5a)\n(Csym D u)n=µ on AN, (2.5b)\nu=eu on AD, (2.5c)\nby splitting the boundary between Dirichlet and Neumann ∂V=AD∪ANwith AD∩AN=∅.\n2.2 Planar static equilibrium\nIn the planar form of linear elasticity it is assumed that the displacement vector is a function solely of the\nx−y-plane u=u(x, y) with relevant components only in the x- and y-directions u:A⊂R2→R2. Under the\nassumption of plane stress σi3=σ3i= 0, the material tensor is adapted to\nσ=Csym D u, C=E\n1−ν2[(1−ν)J+ν 1⊗ 1]∈R2×2×2×2, (2.6)\nwhere now J:R2×2×2×2→R2×2×2×2and 1:R2×2→R2×2. In the case of plane strain εi3=ε3i= 0 with\nε= sym D u, the material tensor has the same form as for the three-dimensional case, reduced to its planar\ncomponents C∈R2×2×2×2. Since in both cases the quadratic form ⟨sym D u,σ⟩=⟨sym D u,Csym D u⟩does\nnot produce energy for the out-of-plane components, the tensor fields of the strain and stress can be reduced\nto sym D u:A⊂R2→Sym(2) and σ:A⊂R2→Sym(2), where Sym(2) is the space of constant symmetric\nsecond order tensors in R2×2. Further, the form of the boundary value problem remains the same as in three\n5RM(V),→[C∞(V)]3sym DC∞(V)⊗Sym(3)IncC∞(V)⊗Sym(3)Div[C∞(V)]3\nFigure 2.2: The three-dimensional elasticity sequence. The space of rigid body motions is the kernel of the\nsymmetrised gradient operator ker(sym D) = RM(V), and the range of the latter is the kernel of the incompat-\nibility operator sym D[ C∞(V)]3= ker(Inc). The range of the incompatibility operator is exactly the kernel of\nthe divergence operator Inc[ C∞(V)⊗Sym(3)] = ker(Div) for symmetric tensors, and the range of the divergence\noperator is a surjection onto the last space in the sequence Div[ C∞(V)⊗Sym(3)] = [ C∞(V)]3.\ndimensions\n−Div(Csym D u) =f in A , (2.7a)\n(Csym D u)n=µ on sN, (2.7b)\nu=eu on sD, (2.7c)\nwith an adjustment of the domain and boundary to ∂A=sD∪sN, and the body forces and tractions to\nf:A⊂R2→R2andµ:sN⊂R→R2.\n2.3 The Beltrami-Michell equations\nThe Beltrami-Michell equations reformulate isotropic linear elasticity purely in stresses. In order to derive the\nBeltrami-Michell equations we introduce the compliance tensor for the inverse stress-strain relation\nsym D u=Aσ, A=C−1=1\nE[(1 + ν)J−ν 1⊗ 1]. (2.8)\nWith the strong form of equilibrium and constitutive relation in strain-form\n−Divσ=f in V , (2.9a)\nAσ−sym D u= 0 in V , (2.9b)\nthe two-field problem of displacement and stresses as per the Hellinger-Reissner principle [61] is obtained. In\norder to construct a problem purely in σ, one starts from the Saint-Venant compatibility condition\nIncε= curl Curl ε= 0 ∀ε∈sym D[ C∞(V)]3, (2.10)\nwhich asserts that the strain tensor εis the symmetric part of the gradient of some vector field on contractible\ndomains by the exact sequence property of the elasticity complex [15, 16, 45, 46], see Fig. 2.2. Note that the\ncompatibility condition does not identify the underlying vector field uniquely since Inc(sym D u) = Inc[sym D( u+\nv)] = 0 for any pair {u,v} ∈[C∞(V)]3[5]. Consequently, the equation of static equilibrium is paramount to\ndetermine a unique solution field in the domain V. The field equations are now given by\n−Divσ=f in V , (2.11a)\nInc(Aσ) = 0 in V . (2.11b)\nThe three-dimensional compliance relation Aσcan be expanded into\nAσ=1\nE[(1 + ν)σ−(νtrσ) 1]. (2.12)\nSince the incompatibility equation is zero on the right-hand-side, for a medium composed of a single isotropic\nmaterial we can multiply the equation with Young’s modulus Eand divide it by (1 + ν) before inserting the\ncompliance relation\nE\n1 +νInc(Aσ) = Inc σ−ν\n1 +νInc[(tr σ) 1] = 0 . (2.13)\n6R,→C∞(V)∇[C∞(V)]3curl[C∞(V)]3 divC∞(V)\nFigure 2.3: The three-dimensional de Rham sequence. The space of constants is the kernel of the gradient\noperator R= ker( ∇), and the range of the latter is the kernel of the curl operator ∇C∞(V) = ker(curl). The\nrange of the curl operator is exactly the kernel of the divergence operator curl[ C∞(V)]3= ker(div), of which\nthe range is a surjection onto the last space div[ C∞(V)]3=C∞(V).\nThe application of the incompatibility operator [35] on a symmetric tensor can be expressed as\nIncσ= 2 sym(D Div σ)−∆σ−hess(tr σ) + [∆ tr σ−div Div σ] 1, (2.14)\nas per the Schaefer-Kr¨ oner formula [30,53]. For a volumetric tensor such as (tr σ) 1the result simplifies [32] to\nInc[(tr σ) 1] = (∆ tr σ) 1−hess(tr σ). (2.15)\nInserting the latter two identities into Eq. (2.13) yields\n−∆σ−1\n1 +νhess(tr σ) +1\n1 +ν(∆ trσ) 1−(div Div σ) 1+ 2 sym D Div σ= 0, (2.16)\nwhere we can now directly insert the static equilibrium equation Eq. (2.11a) to find\n−∆σ−1\n1 +νhess(tr σ) +1\n1 +ν(∆ trσ) 1= 2 sym D f−(divf) 1. (2.17)\nThe equation can be further simplified by observing that the first invariant of the vanishing field Inc( Aσ) = 0\nmust also vanish tr[Inc( Aσ)] = 0. With\ntr(Inc σ) = ∆ tr σ−div Div σ, (2.18a)\ntr(Inc[(tr σ) 1]) = 2∆ tr σ, (2.18b)\nthe first invariant leads to\n(1−ν)∆ trσ−(1 +ν) div Div σ= 0. (2.19)\nWith the latter identity at hand we can eliminate ∆ tr σin Eq. (2.17) and retrieve the classical form of the\nBeltrami-Michell field equation\n−∆σ−1\n1 +νhess(tr σ) = 2 sym D f+ν\n1−ν(divf) 1 in V . (2.20)\nClearly, the Beltrami-Michell equations introduce the problem of finding meaningful stress fields for certain\nforce fields f. In fact, we can characterise cases for which F(f) = 2 sym D f+ν/(1−ν)(divf) 1vanishes, leaving\nno right-hand side F(f) = 0.\nRemark 2.1 (Vanishing right-hand side)\nLet the body forces fbe in the polynomial space of rigid body motions, there holds\nF(f) = 0 ∀f∈RM(V) =R3⊕R3×x, x=xe1+ye2+ze3, (2.21)\nasserting that the right-hand side of the Beltrami-Michell equation vanishes. This is clear since the overlap of\nthe kernel of the divergence operator ker(div) = range(curl) and the symmetrised gradient operator ker(sym D) =\nRM(V)is the space of rigid body motions ker(div) ∩ker(sym D) = RM(V). The respective characterisations\nare given by the exact de Rham sequence [18,44] in Fig. 2.3 and the elasticity sequence in Fig. 2.2.\n7RM(A),→[C∞(A)]2sym DC∞(A)⊗Sym(2)rot RotC∞(A)\nP1(A),→C∞(A)airyC∞(A)⊗Sym(2)Div[C∞(A)]2\nFigure 2.4: Two planar elasticity sequences derived from the reduction of the incompatibility operator.\nFor two-dimensional tensors, the incompatibility operator turns into the rot Rot-operator and for scalars\ninto the airy operator. In the first sequence the two-dimensional space of rigid body motions is the ker-\nnel of the sym D-operator RM(A) = ker(sym D), of which the range is the kernel of the rot Rot-operator\nsym D[ C∞(A)]2= ker(rot Rot). Finally, the rot Rot-operator yields a surjection onto the last space in the se-\nquence rot Rot[ C∞(A)⊗Sym(2)] = C∞(A). In the second sequence the kernel of the airy-operator is given by\nthe linear polynomial space P1(A) = ker(airy), and the airy-operator maps the kernel the divergence operator\nfor symmetric tensors airy C∞(A) = ker(Div). Lastly, the divergence yields a surjection onto the last space in\nthe sequence Div[ C∞(A)⊗Sym(2)] = [ C∞(A)]2.\n2.4 The Beltrami stress function\nUnder the assumption of vanishing body forces in the domain f= 0, the stress field σmust clearly belong to\nthe null space ker(Div), thus satisfying static equilibrium a priori Div σ= 0. By the exact elasticity sequence\nthere holds\n∀σ∈[C∞(V)⊗Sym(3)] ∩ker(Div) ∃Σ∈C∞(V)⊗Sym(3) : σ= Inc Σ, (2.22)\non contractible domains as per Fig. 2.2. Consequently, we can insert the tensorial potential Inc Σinto the\nBeltrami-Michell equations to find\n−∆[∆Σ−2 sym D div Σ−(∆ trΣ−div Div Σ) 1]−1\n1 +νhess(ν∆ trΣ+ div Div Σ) = 0 , (2.23)\nwhich is a fourth-order partial differential equation with Σbeing the Beltrami stress function.\nRemark 2.2 (Specialised stress functions)\nWe note that the Maxwell stress function Σ= Σ 1e1⊗e1+Σ 2e2⊗e2+Σ 3e3⊗e3and the Morera stress function\nΣ= Σ 12e1⊗e2+ Σ 13e1⊗e3+ Σ 23e2⊗e3are specialised forms of the general Beltrami stress function [52],\nlimiting the number of independent terms in the Beltrami stress tensor.\n2.5 The planar Beltrami-Michell equations\nIn two dimensions the compatibility condition is given by the operator rot Rot ε= 0, and the equilibrium\nequation is simply adapted to a two-dimensional plane\n−Divσ=f in A , (2.24)\nwhere σ:A→R2×2andf:A→R2. The out-of-plane strain can be retrieved with the classical relation ε33=\n−(ν/E) trσ. The case of plane stress follows analogously to the three-dimensional case since the compliance\nrelation is the same, adjusted to two-dimensions A= (1/E)[(1 + ν)J−ν 1⊗ 1]∈R2×2×2×2. For a symmetric\ntensor there holds\nrot Rot ε= ∆ tr ε−div Div ε= 0 ∀ε∈sym D[ C∞(A)]2, (2.25)\nas per the two-dimensional elasticity complex, Fig. 2.4. Applying the latter to Aσyields\nE\n1 +νrot Rot( Aσ) =1\n1 +ν∆ trσ−div Div σ= 0, (2.26)\nsince rot Rot[(tr σ) 1] = ∆ tr σ. Now inserting Eq. (2.24) results in the two-dimensional field equations\n−1\n1 +ν∆ trσ= div f in A . (2.27)\n8R,→C∞(A)∇⊥\n[C∞(A)]2 divC∞(A)\nFigure 2.5: One of two possible two-dimensional de Rham sequences derived by the reduction of the curl operator\nto two dimensions. The space of constants is the kernel of the rotated gradient operator R= ker( ∇⊥), and the\nrange of the latter is the kernel of the divergence operator ∇⊥C∞(A) = ker(div). The range of the divergence\noperator is a surjection onto the last space div[ C∞(A)]2=C∞(A).\nIn the case of plane strain one assumes zero out-of-plane strains, such that the compliance equation reads\nε=Aσ=1 +ν\nE[σ−(νtrσ) 1], A=1 +ν\nE[J−ν 1⊗ 1], (2.28)\nfor the in-plane components of the stain tensor. The out-of-plane stress can be recovered via σ33=λtr(sym D u) =\nλdivu,with the Lam´ e constant λ=Eν/[(1 + ν)(1−2ν)], but does not produce energy. Consequently, we find\nfor the compatibility\nE\n1 +νrot Rot( Aσ) = (1 −ν)∆ trσ−div Div σ= 0, (2.29)\nwhich by incorporating static equilibrium results in the field equation\n−(1−ν)∆ trσ= div f in A . (2.30)\nRemark 2.3 (Vanishing planar right-hand side)\nFor both plane stress and plane strain, the right-hand side of the equations is given by the divergence of the body\nforces divf. Consequently, solenoidal force fields f=∇⊥fare not captured by the equations since ker(div) =\nrange( ∇⊥)in two dimensions by the exact de Rham complex as per Fig. 2.5.\n2.6 The Airy function\nIn the special case where in the two-dimensional Beltrami-Michell equations no body forces occur in the domain\nA, the field equations simplify to\nDivσ= 0 in A , (2.31a)\n∆ trσ= 0 in A . (2.31b)\nBy the exact two-dimensional elasticity complex [10,61] there holds\n∃ς∈C∞(A) : σ= airy ς∀σ∈[C∞(A)⊗Sym(2)] ∩ker(Div) , (2.32)\non contractible domains as per Fig. 2.4, where the airy operator reads\nairyς= D⊥∇⊥ς . (2.33)\nConsequently, we can insert the scalar potential ςinto the two-dimensional Beltrami-Michell equations to find\nthe biharmonic field equation\n∆∆ς= 0 in A , (2.34)\nsince tr(airy ς) = ∆ ς. In this context ςis called the Airy-stress function.\n3 Symmetrised stress formulation of isotropic linear elasticity\nThe Beltrami-Michell equation in its classical form in Eq. (2.20) does not lend itself to the construction of\na symmetric bilinear form. The reason lies in the occurrence of the Hessian of the trace of the stress tensor\n9hess(tr σ), which does not yield a symmetric bilinear form after testing and partial integration. Fortunately,\nthe field equation of static equilibrium can be reformulated into\n−1\n1 +ν(div Div σ) 1=1\n1 +ν(divf) 1, (3.1)\nby taking the divergence of both sides and multiplying with the constant tensor [1 /(1 +ν)] 1. Now, adding this\nzero-sum term to Eq. (2.20) yields\n−∆σ−1\n1 +ν[hess(tr σ) + (div Div σ) 1] = 2 sym D f+1 +ν2\n1−ν2(divf) 1, (3.2)\nwhich does lend itself to a symmetric bilinear form. We apply the test function τ∈C∞(V)⊗Sym(3) to find\nZ\nV⟨Dτ,Dσ⟩+1\n1 +ν[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] dV=Z\nV2⟨τ,sym D f⟩+1 +ν2\n1−ν2⟨trτ,divf⟩dV\n+Z\n∂V⟨τ,(Dσ)n⟩+1\n1 +ν[⟨τ,(∇trσ)⊗n⟩+ (trτ)⟨Divσ,n⟩] dA ∀τ∈C∞(V)⊗Sym(3) ,(3.3)\nby partial integration. Thus, we extract the Neumann boundary term\nκ= (Dσ)n+1\n1 +ν[(∇trσ)⊗n+⟨Divσ,n⟩ 1], (3.4)\nby splitting the boundary between Dirichlet and Neumann ∂V=AD∪AN, such that AD∩AN=∅.\nDefinition 3.1 (The pure stress boundary value problem for solids I)\nThe field equations and boundary conditions of the novel three-dimensional stress boundary value problem\nread\n−∆σ−1\n1 +ν[hess(tr σ) + (div Div σ) 1] = 2 sym D f+1 +ν2\n1−ν2(divf) 1 in V , (3.5a)\n(Dσ)n+1\n1 +ν[∇trσ⊗n− ⟨f,n⟩ 1] =κ on AN, (3.5b)\nσ=eσ on AD. (3.5c)\nwhere κis a mixed measure of equilibrium and compatibility on the Neumann boundary where we applied\nDivσ=−f.\nWe observe that continuous constant force fields f∈R3over the domain are captured by the Dirichlet and\npartially by the Neumann boundary conditions. We can now state the variational form.\nDefinition 3.2 (Variational form of the pure stress problem I)\nThe weak formulation of the boundary value problem of linear elasticity written purely in stresses reads\nZ\nV⟨Dτ,Dσ⟩+1\n1 +ν[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] dV=Z\nAN⟨τ,κ⟩dA (3.6)\n+Z\nV2⟨τ,sym D f⟩+1 +ν2\n1−ν2⟨trτ,divf⟩dV ∀τ∈C∞\nAD(V)⊗Sym(3) ,\nand yields a fully symmetric left-hand side. Here τis assumed to be compatible with the Dirichlet boundary.\nClearly, it can be derived from the variation of a functional.\n10Definition 3.3 (Variational functional in pure stresses I)\nThe weak formulation of the new boundary value problem can directly constructed as the variation of the\nfunctional\nI(σ) =Z\nV1\n2∥Dσ∥2+1\n1 +ν⟨Divσ,∇trσ⟩ −2⟨σ,sym D f⟩ −1 +ν2\n1−ν2⟨trσ,divf⟩dV−Z\nAN⟨σ,κ⟩dA ,\n(3.7)\nwith respect to the stress tensor σ.\n3.1 Existence and uniqueness\nFrom the weak form in Eq. (3.6) we extract the bilinear and linear forms\na(τ,σ) =Z\nV⟨Dτ,Dσ⟩+χ[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] dV , (3.8a)\nl(τ) =Z\nV2⟨τ,sym D f⟩+1 +ν2\n1−ν2⟨trτ,divf⟩dV , (3.8b)\nwith the material parameter χ= 1/(1 +ν)∈[2/3,1], given by the Poisson ratio ν∈[0,1/2]. In order to show\nexistence and uniqueness we first derive some preliminary results.\nLemma 3.1 (Upper bound of ∥Dσ∥L2)\nLetσ∈H1\n0(V)⊗Sym(3) for all ϵ >0there holds\n∥Dσ∥2\nL2≤max\u001a\n2 +ϵ,1 +1\nϵ\u001b\n(∥Divσ∥2\nL2+∥∇trσ∥2\nL2+∥sym Curl σ∥2\nL2)∀σ∈H1\n0(V)⊗Sym(3) .(3.9)\nProof. Starting with Eq. (2.14), we reformulate the differential identity to\n∆σ= 2 sym(D Div σ)−hess(tr σ) + [∆ tr σ−div Div σ] 1−Incσ. (3.10)\nNow, testing the equation with −σ∈H1\n0(V)⊗Sym(3) and integrating by parts yields\n∥Dσ∥2\nL2= 2∥Divσ∥2\nL2−2⟨Divσ,∇trσ⟩L2+∥∇trσ∥2\nL2+⟨σ,Incσ⟩L2∀σ∈H1\n0(V)⊗Sym(3) ,(3.11)\nin the distributional sense. For the incompatibility of the stress tensor Inc σ= curl Curl σwe observe\n⟨σ,curl Curl σ⟩L2=⟨curlσ,Curlσ⟩L2=⟨(Curlσ)T,Curlσ⟩L2=∥sym Curl σ∥2\nL2− ∥skw Curl σ∥2\nL2,(3.12)\nwhere we algebraically decomposed (Curl σ)Tand Curl σinto their symmetric and skew-symmetric parts. Using\nthe latter we find the upper bound\n∥Dσ∥2\nL2= 2∥Divσ∥2\nL2−2⟨Divσ,∇trσ⟩L2+∥∇trσ∥2\nL2+∥sym Curl σ∥2\nL2− ∥skw Curl σ∥2\nL2\nY\n≤(2 +ϵ)∥Divσ∥2\nL2+\u0012\n1 +1\nϵ\u0013\n∥∇trσ∥2\nL2+∥sym Curl σ∥2\nL2− ∥skw Curl σ∥2\nL2 (3.13)\n≤max\u001a\n2 +ϵ,1 +1\nϵ\u001b\n(∥Divσ∥2\nL2+∥∇trσ∥2\nL2+∥sym Curl σ∥2\nL2)∀σ∈H1\n0(V)⊗Sym(3) ,\nwhere we applied Young’s inequality2and dropped the negative term −∥skw Curl σ∥2\nL2.\nThe properties of the sym Curl-operator along with its corresponding bounds can be found in [22,23,32–34,\n37,38,62].\nWe also require an upper bound for the skew symmetric curl of the stress tensor.\n2Young’s inequality: ⟨x, y⟩ ≤∥x∥2\n2ϵ+ϵ∥y∥2\n2∀ϵ >0.\n11Lemma 3.2 (Upper bound of ∥skw Curl σ∥)\nLetσ∈H1\n0(V)×Sym(3) , then there holds the upper bound\n∥skw Curl σ∥2\nL2≤1 +ϵ\n2∥Divσ∥2\nL2+1 +ϵ\n2ϵ∥∇trσ∥2\nL2∀σ∈H1(V)⊗Sym(3) , (3.14)\nfor all constant ϵ >0.\nProof. The norm of the skew-symmetric part of the Curl of the stress tensor can be reformulated into\n∥skw Curl σ∥L2= sup\nτ∈L2⊗R3×3⟨τ,skw Curl σ⟩L2\n∥τ∥L2\n= sup\nτ∈L2⊗R3×3⟨Curl(skw τ),σ⟩H−1×H1\n∥τ∥L2\n= sup\nτ∈L2⊗R3×3⟨(div axl skw τ) 1−(D axl skw τ)T,σ⟩H−1×H1\n∥τ∥L2\n= sup\nτ∈L2⊗R3×3⟨(div axl skw τ) 1−sym(D axl skw τ),σ⟩H−1×H1\n∥τ∥L2\n= sup\nτ∈L2⊗R3×3⟨axl skw τ,Divσ− ∇trσ⟩L2\n∥τ∥L2\n= sup\nτ∈L2⊗R3×3⟨τ,Anti(Div σ− ∇trσ)⟩L2\n2∥τ∥L2\n=1\n2∥Anti(Div σ− ∇trσ)∥L2\n=1√\n2∥Divσ− ∇trσ∥L2∀σ∈H1\n0(V)⊗Sym(3) , (3.15)\nwhere H−1(V) is the dual space of H1\n0(V) and⟨·,·⟩H−1×H1is the dual product. In the derivation we used [32]\nthe algebraic identity Curl A= (div axl A) 1−(D axl A)Tand the relation ⟨axlA,axlA⟩= (1/2)⟨A,A⟩for\nsome skew-symmetric tensor A, as well as the symmetry of σ. In fact, by the estimate we derive the following\nalgebraic identity\nskw Curl σ=1\n2Anti(Div σ− ∇trσ)∀σ∈C∞(V)⊗Sym(3) , (3.16)\nwhich holds in general for symmetric tensors σ=σT. Employing the latter we find\n∥skw Curl σ∥2\nL2=1\n2∥Divσ− ∇trσ∥2\nL2\nT\n≤1\n2(∥Divσ∥L2+∥∇trσ∥L2)2\n=1\n2(∥Divσ∥2\nL2+ 2∥Divσ∥L2∥∇trσ∥L2+∥∇trσ∥2\nL2)\nY\n≤1 +ϵ\n2∥Divσ∥2\nL2+1 +ϵ\n2ϵ∥∇trσ∥2\nL2∀σ∈H1(V)⊗Sym(3) , (3.17)\nwhere we used the triangle3inequality and Young’s inequality.\nWith the latter bounds at hand we can show well-posedness of the three-dimensional problem.\nTheorem 3.1 (Existence and uniqueness in three dimensions)\nLet the Poisson ratio be ν∈[0,1/2]such that χ= 1/(1 +ν)∈[2/3,1], then the problem\na(τ,σ) =l(τ)∀τ∈X0(V), (3.18)\n3Triangle inequality: ∥x+y∥ ≤ ∥ x∥+∥y∥.\n12where the bilinear and linear forms are according to Eq. (6.10) , and the space X 0(V)is given by\nX0(V) =H1\n0(V)⊗Sym(3) , ∥σ∥2\nX=∥σ∥2\nH1=∥σ∥2\nL2+∥Dσ∥2\nL2, (3.19)\nhas a unique solution σ∈X0(V)for every right-hand side with the stability estimate\n∥σ∥X≤1\nβ∥l∥X′, (3.20)\nwhere β=β(ν)>0is the coercivity constant and ∥·∥X′is the dual norm with respect to X 0(V).\nProof. The proof follows by the Lax-Milgram theorem, where we neglect Neumann boundary conditions. The\ncontinuity of the linear form is obvious. For the bilinear form we first observe that there holds\n∥∇trσ∥L2=∥Tr(∇ ⊗σ)∥L2CS\n≤ ∥Tr∥∗∥∇ ⊗σ∥L2=√\n3∥Dσ∥L2,\n∥Divσ∥L2=∥Tr Dσ∥L2CS\n≤ ∥Tr∥∗∥Dσ∥L2=√\n3∥Dσ∥L2, (3.21)\nwhich follows from the Cauchy-Schwarz inequality4with the operator norm ∥Tr∥∗=∥ 1∥=√\n3. Thus, we find\na(τ,σ) =⟨Dτ,Dσ⟩L2+χ[⟨Divτ,∇trσ⟩L2+⟨∇trτ,Divσ⟩L2]\nCS\n≤ ∥Dτ∥L2∥Dσ∥L2+χ(∥Divτ∥L2∥∇trσ∥L2+∥∇trτ∥L2∥Divσ∥L2)\n∗\n≤ ∥Dτ∥L2∥Dσ∥L2+ 6χ∥Dτ∥L2∥Dσ∥L2\n≤(1 + 6 χ)∥τ∥X∥σ∥X ∀τ,σ∈X(V), (3.22)\nwhere we applied the Cauchy-Schwarz inequality and the operator bounds. In order to show coercivity we\nreformulate Eq. (3.11) into\n2⟨Divσ,∇trσ⟩L2= 2∥Divσ∥2\nL2− ∥Dσ∥2\nL2+∥∇trσ∥2\nL2+∥sym Curl σ∥2\nL2− ∥skw Curl σ∥2\nL2∀σ∈X0(V).\n(3.23)\nNow, for the bilinear form we find\na(σ,σ) =∥Dσ∥2\nL2+ 2χ⟨Divσ,∇trσ⟩L2\n= (1−χ)∥Dσ∥2\nL2+ 2χ∥Divσ∥2\nL2+χ∥∇trσ∥2\nL2+χ∥sym Curl σ∥2\nL2−χ∥skw Curl σ∥2\nL2\n∗\n≥(1−χ)∥Dσ∥2\nL2+(3−ϵ)χ\n2∥Divσ∥2\nL2+(ϵ−1)χ\n2ϵ∥∇trσ∥2\nL2+χ∥sym Curl σ∥2\nL2\n≥(1−χ)∥Dσ∥2\nL2+χmin\u001a3−ϵ\n2,1,ϵ−1\n2ϵ\u001b\n(∥Divσ∥2\nL2+∥sym Curl σ∥2\nL2+∥∇trσ∥2\nL2)\n⋆\n≥(1−χ)∥Dσ∥2\nL2+χ\n3min\u001a3−ϵ\n2,1,ϵ−1\n2ϵ\u001b\n∥Dσ∥2\nL2\nPF\n≥1\n1 +c2\nF\u0014\n(1−χ) +χ\n3min\u001a3−ϵ\n2,1,ϵ−1\n2ϵ\u001b\u0015\n∥σ∥2\nX ∀σ∈X0(V), (3.24)\nwhere we used the upper bound of ∥skw Curl σ∥2\nL2from Lemma 3.2 in ∗, the upper bound of ∥Dσ∥2\nL2from\nLemma 3.1 in ⋆with ϵ= 1, and finally applied the Poincar´ e-Friedrich inequality5. The inequality holds for the\ncomplete range of χ∈[2/3,1] with 1 ≤ϵ≤3, since even for χ= 1 we can set 1 < ϵ < 3, such that the gradient\nof the stress tensor ∥Dσ∥2\nL2does not vanish.\n4Cauchy-Schwarz inequality: ⟨x, y⟩ ≤ ∥ x∥∥y∥.\n5Poincar´ e-Friedrich inequality: ∃cF>0 :∥x∥L2≤cF∥∇x∥L2∀x∈H1\n0(V),such that cFdepends only on the domain\nV⊆R3and its boundary ∂V.\n13Corollary 3.1 (Well-posedness of the three-dimensional Beltrami-Michell equations)\nFrom the proof of Theorem 3.1 we automatically get that the variational formulation of the original Beltrami-\nMichell equations in Eq. (2.20) is also well-posed for σ∈X0(V). The bilinear form is simply\na(τ,σ) =Z\nV⟨Dτ,Dσ⟩+χ⟨Divτ,∇trσ⟩dV , (3.25)\nsuch that continuity and coercivity follow the same lines of our proof simply with χinstead of 2χ. However, this\nbilinear form is non-symmetric, making computational approaches to its solution far less efficient.\n4 Novel stress formulation of planar isotropic linear elasticity\nFrom Eq. (2.27) and Eq. (2.30) it is clear that the planar Beltrami-Michell equations characterise only the mean\nstress σm= (1/2) trσ, related to the first invariant of the stress tensor. However, we can derive from static\nequilibrium\n−ψsym(D Div σ) =ψsym D f, (4.1)\nwith some constant ψ >0. Now, multiplying either Eq. (2.27) or Eq. (2.30) with the two-dimensional identity\ntensor 1and adding the symmetrised gradient of static equilibrium to it yields the field equations\n−ψsym(D Div σ)−1\n1 +ν(∆ trσ) 1=ψsym D f+ (div f) 1, (4.2a)\n−ψsym(D Div σ)−[(1−ν)∆ trσ] 1=ψsym D f+ (div f) 1. (4.2b)\nThe right-hand side now clearly vanishes for body forces fin the space of rigid body motions\nRM(A) =R2⊕Rx= ker(sym D) ∩ker(div) ⊂ker(div) ,R=\u00140 1\n−1 0\u0015\n,x=xe1+xe2, (4.3)\nanalogously to the three-dimensional case. By respectively defining the constant χ= 1/(1 +ν) orχ= 1−νfor\nplane stress or plane strain, we can denote either problem with the single equation\n−ψsym(D Div σ)−(χ∆ trσ) 1=ψsym D f+ (div f) 1. (4.4)\nWe apply the test function τ∈C∞(A)⊗Sym(2) and integrate by parts to find\nZ\nAψ⟨Divτ,Divσ⟩+χ⟨∇trτ,∇trσ⟩dA\n=Z\n∂Aψ⟨τ,Divσ⊗n⟩+χtrτ⟨∇trσ,n⟩ds (4.5)\n+Z\nAψ⟨τ,sym D f⟩+⟨trτ,divf⟩dA ∀τ∈C∞(A)⊗Sym(2) .\nWe split the boundary between Dirichlet and Neumann ∂A=sD∪sNwith sD∩sN=∅to finally derive the\nnew equations and boundary conditions, where we apply Div σ=−ffor the Neumann boundary.\nDefinition 4.1 (Pure stress planar boundary value problem I)\nLetχ= 1/(1 +ν)orχ= 1−νfor plane stress or plane strain, respectively, the complete boundary value\nproblem reads\n−ψsym(D Div σ)−(χ∆ trσ) 1=ψsym D f+ (div f) 1 in A , (4.6a)\nχ⟨∇trσ,n⟩ 1−ψ(f⊗n) =κ on sN, (4.6b)\nσ=eσ on sD. (4.6c)\nWith the boundary conditions at hand, we can construct the variational form of Definition 4.1.\n14Definition 4.2 (Pure stress planar variational form I)\nThe variational form reads\nZ\nAψ⟨Divτ,Divσ⟩+χ⟨∇trτ,∇trσ⟩dA=Z\nAψ⟨τ,sym D f⟩+⟨trτ,divf⟩dA (4.7)\n+Z\nsN⟨τ,κ⟩ds∀τ∈C∞\nsD(A)⊗Sym(2) ,\nwithψ >0andχ= 1/(1 +ν)orχ= 1−ν, for plane-stress or plane stain, respectively.\nThe latter can clearly be derived from a variation functional.\nDefinition 4.3 (Planar variation functional in pure stresses I)\nThe weak formulation of the new boundary value problem can directly constructed as the variation of the\nfunctional\nI(σ) =Z\nAψ\n2∥Divσ∥2+χ\n2∥∇trσ∥2−ψ⟨τ,sym D f⟩ − ⟨trτ,divf⟩dA−Z\nsN⟨σ,τ⟩ds , (4.8)\nwithψ >0andχ= 1/(1 +ν)orχ= 1−ν, for plane-stress or plane stain, respectively.\nIn the planar case we have σ:A⊂R2→Sym(2), such that the stress tensor is a function of the x−y-plane\nσ=σ(x, y). Consequently, we find\nIncbσ= (∆ tr bσ−div Div bσ)e3⊗e3, (4.9)\nwherebσ∈C∞(A)⊗Sym(3) is the three dimensional insertion of the two-dimensional tensor σ∈C∞(A)⊗\nSym(2) into a three-dimensional tensor bσαβ=σαβwith zeroes in the out-of-plane direction bσi3=bσ3i= 0. Due\nto the latter Eq. (2.16) reduces to\n∆σ= 2 sym(D Div σ)−hess(tr σ) + [∆ tr σ−div Div σ] 1, (4.10)\nbeing a new identity for the Laplacian of a symmetric two-dimensional tensor field. We can reformulate the\nidentity into\n−sym(D Div σ)−(∆ trσ) 1= sym(D Div σ)−∆σ−hess(tr σ)−(div Div σ) 1. (4.11)\nNow, let ψ=χ, then we find for the left-hand side of Eq. (4.4)\n−χsym(D Div σ)−(χ∆ trσ) 1=χ[sym(D Div σ)−∆σ−hess(tr σ)−(div Div σ) 1]\n=χ[−sym D f−∆σ−hess(tr σ)−(div Div σ) 1]. (4.12)\nThus, we derive a new equivalent form for the planar equations\n−∆σ−hess(tr σ)−(div Div σ) 1= 2 sym D f+1\nχ(divf) 1, (4.13)\nby dividing the entire equation by χ. Clearly, the new form is extremely similar to our symmetrised formulation\nin Eq. (3.2) of the three-dimensional problem. However, it leads to a more involved bilinear form.\n4.1 Existence and uniqueness in the planar case\nWe extract the bilinear and linear forms of the planar problem from Eq. (4.7)\na(τ,σ) =Z\nAψ⟨Divτ,Divσ⟩+χ⟨∇trτ,∇trσ⟩dA , (4.14a)\nl(τ) =Z\nAψ⟨τ,sym D f⟩+⟨trτ,divf⟩dA . (4.14b)\nAs a preparatory step we introduce a new norm equivalence.\n15Lemma 4.1 (Norm equivalence for ∥Dσ∥L2)\nLetσ∈H1\n0(A)⊗Sym(2) , then there holds\n∃c1, c2>0 : c1(∥Divσ∥2\nL2+∥∇trσ∥2\nL2)≤ ∥Dσ∥2\nL2≤c2(∥Divσ∥2\nL2+∥∇trσ∥2\nL2)∀σ∈H1\n0(A)⊗Sym(2) ,\n(4.15)\nwith\nc1= min\u001a\n2−ϵ,1−1\nϵ\u001b\n, c 2= max\u001a\n2 +ϵ,1 +1\nϵ\u001b\n, (4.16)\nwhere ϵ >0.\nProof. Starting with Eq. (4.10), we test with −σ∈H1\n0(A)⊗Sym(2) and integrate by parts to find\n∥Dσ∥2\nL2= 2∥Divσ∥2\nL2−2⟨Divσ,∇trσ⟩L2+∥∇trσ∥2\nL2∀σ∈H1\n0(A)⊗Sym(2) . (4.17)\nNow, applying Young’s inequality to find an upper bound yields\n∥Dσ∥2\nL2Y\n≤(2 +ϵ)∥Divσ∥2\nL2+\u0012\n1 +1\nϵ\u0013\n∥∇trσ∥2\nL2\n≤max\u001a\n2 +ϵ,1 +1\nϵ\u001b\n(∥Divσ∥2\nL2+∥∇trσ∥2\nL2)∀σ∈H1\n0(A)⊗Sym(2) , (4.18)\nwhich is satisfied for any ϵ >0. Alternatively, we find the lower bound via the negative Young’s inequality\n∥Dσ∥2\nL2Y\n≥(2−ϵ)∥Divσ∥2\nL2+\u0012\n1−1\nϵ\u0013\n∥∇trσ∥2\nL2\n≥min\u001a\n2−ϵ,1−1\nϵ\u001b\n(∥Divσ∥2\nL2+∥∇trσ∥2\nL2)∀σ∈H1\n0(A)⊗Sym(2) , (4.19)\nwhich holds for 1 < ϵ < 2.\nCorollary 4.1 (Norm equivalence for ∥σ∥H1)\nFrom Lemma 4.1 we automatically derive\nc1(∥Divσ∥2\nL2+∥∇trσ∥2\nL2)≤ ∥σ∥2\nH1≤c2(1 +c2\nF)(∥Divσ∥2\nL2+∥∇trσ∥2\nL2)∀σ∈H1\n0(A)⊗Sym(2) ,(4.20)\nby using ∥Dσ∥2\nL2≤ ∥σ∥2\nL2+∥Dσ∥2\nL2and the Poincar´ e-Friedrich inequality.\nWith the norm equivalence at hand, we can directly show well-posedness.\nTheorem 4.1 (Well-posedness of the planar forms)\nLet the Poisson ratio be ν∈[0,1/2]such that χ∈[2/3,1]orχ∈[1/2,1]andψ >0, then the problem\na(τ,σ) =l(τ)∀τ∈Y0(A) =H1\n0(A)⊗Sym(2) , (4.21)\nwhere the bilinear and linear forms are given in Eq. (4.14) , has a unique solution σ∈Y0(A)for every right-hand\nside with the stability estimate\n∥σ∥Y≤1\nβ∥f∥Y′, (4.22)\nwhere β=β(ψ, χ)>0is the coercivity constant and ∥·∥Y′is the dual norm with respect to Y 0(A).\nProof. We assume a vanishing Neumann boundary. The continuity of the linear form is obvious. For the bilinear\nform we first observe\n∥∇trσ∥L2=∥Tr(∇ ⊗σ)∥L2CS\n≤ ∥Tr∥∗∥∇ ⊗σ∥L2=√\n2∥Dσ∥L2,\n∥Divσ∥L2=∥Tr Dσ∥L2CS\n≤ ∥Tr∥∗∥Dσ∥L2=√\n2∥Dσ∥L2, (4.23)\n16which follows by the Cauchy-Schwarz inequality with ∥Tr∥∗=∥ 1∥=√\n2 since 1∈R2×2in the two-dimensional\ncase. Now, the continuity of bilinear form is given due to\na(τ,σ) =ψ⟨Divτ,Divσ⟩L2+χ⟨∇trτ,∇trσ⟩L2\nCS\n≤ψ∥Divτ∥L2∥Divσ∥L2+χ∥∇trτ∥L2∥∇trσ∥L2\n∗\n≤2ψ∥Dτ∥L2∥Dσ∥L2+ 2χ∥Dτ∥L2∥Dσ∥L2\n≤2(ψ+χ)∥Dτ∥Y∥Dσ∥Y ∀σ∈Y0(A), (4.24)\nwhere we used the Cauchy-Schwarz inequality and Eq. (4.23) in ∗. For the coercivity we find\na(σ,σ) =ψ∥Divσ∥2\nL2+χ∥∇trσ∥2\nL2\n≥min{ψ, χ}(∥Divσ∥2\nL2+∥∇trσ∥2\nL2)\n∗\n≥1\n3min{ψ, χ}∥Dσ∥2\nL2\nPF\n≥1\n3(1 + c2\nF)min{ψ, χ}∥σ∥2\nY ∀σ∈Y0(A), (4.25)\nwhere we used the norm equivalence from Lemma 4.1 with ϵ= 1 in ∗, and then applied the Poincar´ e-Friedrich\ninequality.\nRemark 4.1 (Vanishing ψ-constant)\nNote that the proof fails for ψ= 0, emphasising that the original planar Beltrami-Michell equation is insufficient\nin order to fully characterise the stress tensor σ.\nCorollary 4.2 (Well-posedness of the alternative planar form)\nForψ=χthe bilinear form given by partial integration of Eq. (4.13)\na(τ,σ) =Z\nA⟨Dτ,Dσ⟩+⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩dA , (4.26)\nis coercive. The proof follows the same lines as in the proof of Theorem 4.1, since for coercivity we have\na(σ,σ) =∥Dσ∥2\nL2+ 2⟨Divσ,∇trσ⟩L2= 2∥Divσ∥2\nL2+∥∇trσ∥2\nL2∀σ∈Y0(A), (4.27)\ndue to Eq. (4.17) .\n5 Discrete variational formulations\nThe well-posedness of the three-dimensional and planar formulations is proven by the Lax-Milgram theorem,\nsuch that any conforming discretisation is automatically well-posed. Consequently, we directly obtain the\nquasi-best approximation via Cea’s lemma.\nTheorem 5.1 (Quasi-best approximation)\nLetσ∈X(V)orσ∈Y(A)be the exact solution and σh∈Xh(V)⊂X(V)orσh∈Yh(V)⊂Y(V)be the\napproximate solution, respectively, then there holds\n∥σ−σh∥X≤α\nβinf\nτ∈Xh∥σ−τ∥X, ∥σ−σh∥Y≤α\nβinf\nτ∈Yh∥σ−τ∥Y, (5.1)\nwhere αandβare the continuity and coercivity constants, respectively.\nSince Cea’s lemma is satisfied, the a priori error estimates follow from standard interpolation errors.\n17Theorem 5.2 (Convergence estimates)\nLetσ∈X(V)orσ∈Y(A)be the exact solution and σh∈Xh(V)⊂X(V)orσh∈Yh(V)⊂Y(V)be its\napproximation, respectively, there holds the a priori error estimate\n∥σ−σh∥X≤chp|σ|Hp+1, ∥σ−σh∥Y≤chp|σ|Hp+1, (5.2)\nwhere the constant cis independent of the element size and the exact solution c̸=c(h,σ), and | · |Hp+1is the\nSobolev semi-norm.\nProof. The proof follows directly by replacing τ∈Xh(V) orτ∈Yh(V) in Theorem 5.1 with the interpolation\nΠp\ngσ∈Xh(V) or Πp\ngσ∈Yh(V), respectively, and classical interpolation error estimates [18,67].\nWe conclude our numerical treatment with an observation with respect to discontinuous body forces under\nthe assertion that C0-continuous Lagrange-type elements are employed in the discretisation.\nObservation 5.1 (Discontinuous forces)\nIn both the three-dimensional problem and the planar problems we can understand scalar products with derivatives\noffin the form of distributions\n⟨τ,sym D f⟩T=X\nT∈T⟨τ,f⊗n⟩∂T− ⟨Divτ,f⟩T, (5.3a)\n⟨trτ,divf⟩T=X\nT∈T(trτ)⟨f,n⟩∂T− ⟨∇ trτ,f⟩T, (5.3b)\nwhere Trepresents the triangulation, Tis an element in the mesh and ∂Tis its respective boundary. The term\non the boundary of the elements vanishes if the jump in the force field does not occur directly on the interface\nbetween two elements, since τis continuous. Consequently, the distributive scalar products read\n⟨τ,sym D f⟩T=⟨τ,f⊗n⟩∂N−X\nT∈T⟨Divτ,f⟩T,⟨trτ,divf⟩T=⟨f,n⟩∂N−X\nT∈T⟨∇trτ,f⟩T, (5.4)\nIn practice, these terms are simply Lebesgue integrals over the domain, and over the correspond-\ning Neumann boundary ∂N, since no jump occurs there .\nWith Observation 5.1 we can state the discrete weak forms as\nZ\nV⟨Dτ,Dσ⟩+1\n1 +ν[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] dV= 2⟨τ,sym D f⟩T+1 +ν2\n1−ν2⟨trτ,divf⟩T (5.5)\n+Z\nAN⟨τ,κ⟩dA ∀τ∈Xh(V),\nfor the three-dimensional problem, and\nZ\nAψ⟨Divτ,Divσ⟩+χ⟨∇trτ,∇trσ⟩dA=ψ⟨τ,sym D f⟩T+⟨trτ,divf⟩T+Z\nsN⟨τ,κ⟩ds∀τ∈Yh(A),\n(5.6)\nfor the planar problems of plane stress and plane strain.\n6 Numerics of three-dimensional solids\nIn the following, relative errors are measured using the Lebesgue norm ∥eσ−σh∥L2/∥eσ∥L2, where eσrepresents\nthe exact solution and σhis the finite element approximation.\n18(a)\n (b)\n (c)\nFigure 6.1: Depiction of the displacement field eu(a) with the corresponding von Mises σv(b) and mean stresses\nσm(c). The stresses are depicted on the middle-plane of the cube.\n6.1 Stress convergence\nWe start with the three-dimensional problem. Let the domain be the axis-symmetric cube V= [−1,1]3, with a\ncomplete Dirichlet boundary AD=∂V, we construct an artificial analytical solution by defining the material\nconstants E= 200, ν= 0.25 and the displacement field\neu=1\n2[(x5+y5)e1+ (y5+z5)e2+ (z5+x5)e3], (6.1)\nfrom which we retrieve the stress tensor and the forces\neσ=Asym Deu= 200\n3x4+y4+z4y4x4\ny4x4+ 3y4+z4z4\nx4z4x4+y4+ 3z4\n,f=−Diveσ= 800\n−3x3−y3\n−3y3−z3\n−x3−3z3\n,\n(6.2)\nby using the compliance relation and static equilibrium. Further, from the stress tensor we extract the von\nMiseseσv=p\n3/2∥devσ∥and mean stresses eσm= (1/3) trσ, which are both important invariants of the stress\ntensor in design. The displacement field along with the von Mises stress and the mean stress are depicted in\nFig. 6.1. For comparison we compute the results of our newly introduced stress based formulation σhand the\nstress field retrieved by post-processing in the displacement based formulation σ(uh). Convergence towards the\nexact solution is studied on structured meshes of uniform hexahedra of three polynomial orders p∈ {1,2,3}.\nFrom the the results in Fig. 6.2 we observe that the stress formulation converges in the optimal rate in all three\nmeasurements and yields comparable results to the displacement-based formulation, thus validating the method\nfor a complete Dirichlet boundary.\n6.2 Operator spectrum\nThe discrete three-dimensional variational problem in Eq. (5.5) can be written in operator form as\nfind σh∈Xh(V) s.t. Ahσh=f in [ Xh(V)]′, (6.3)\ngiving rise to the discrete operator Ah, which maps an element of Xh(V) to the dual space [ Xh(V)]′. By our\nproof, the operator is coercive under the very sharp restriction of a complete Dirichlet boundary AD=∂Vif a\nconforming finite element space is employed Xh\n0(V)⊂X0(V). However, often the boundary is not completely\nDirichlet, such that Neumann boundary conditions are imposed. Thus, it is necessary to determine the behaviour\nof the operator for a mixed boundary. In the finite case this measure is given by the spectrum of the operator,\nwhich is characterised by its eigenvalues. Zero eigenvalues correspond with a positive semi-definite operator,\nwhereas negative eigenvalues imply an indefinite operator, and correspondingly a non-coercive problem. In the\nfollowing we count the number of negative and zero eigenvalues in relation to the full spectrum (all eigenvalues),\nwhich corresponds to the total number of degrees of freedom. We employ the cubical domain V= [−1,1]3\n1910210310410510−310−210−1100\nDegrees of freedom∥eσ−σh∥L2/∥eσ∥L2\nσh.1\nσh.2\nσh.3\nσ(uh.1)\nσ(uh.2)\nσ(uh.3)O(h4)O(h1)\n(a)10210310410510−310−210−1100\nDegrees of freedom∥eσv−σh\nv∥L2/∥eσv∥L2\nσh.1\nσh.2\nσh.3\nσ(uh.1)\nσ(uh.2)\nσ(uh.3)O(h4)O(h1)\n(b)10210310410510−310−210−1100\nDegrees of freedom∥eσm−σh\nm∥L2/∥eσm∥L2\nσh.1\nσh.2\nσh.3\nσ(uh.1)\nσ(uh.2)\nσ(uh.3)O(h4)O(h1)\n(c)\nFigure 6.2: Convergence slopes for ∥σ∥L2(a),∥σv∥L2(b), and ∥σm∥L2(c), for polynomial orders p∈ {1,2,3}.\n0 0.13 0.25 0.38 0.501\nνNegative eigenvalues [%]∂V=AN\n∂V=AN∪AD\n∂V=AD\n(a)xyz\n(b)\n (c)\nFigure 6.3: Percent of negative eigenvalues out of the total eigenvalues for varying Poisson ratios (a). Depiction\nof the Neumann boundary in the mixed domain (b) and the 33= 27-hexahedral mesh (c).\nand consider a uniform hexahedral mesh with 33= 27-elements of cubic polynomial order corresponding to\n6000-degrees of freedom. We examine the three boundary definitions\n∂V=AN, ∂V =AN∪AD, ∂V =AD, (6.4)\nwhere the mixed boundary is defined such that ANis composed of the left x=−1, back, y=−1 and top z= 1\nsurfaces of the cube, and the Dirichlet boundary is AD=∂V\\AN, see Fig. 6.3. Further, we vary the Poisson\nratio within its admissible bounds ν∈ {0,0.125,0.25,0.375,0.5}. From the results of the spectral analysis in\nFig. 6.3 we observe that a domain with a Neumann boundary may introduce negative eigenvalues, depending on\nthe Poisson ratio. Interestingly, for incompressible materials ν= 0.5, no negative eigenvalues arise. For a total\nNeumann boundary we observe exactly 6-zero eigenvalues irrespective of the Poisson ratio, which is consistent\nwith the displacement formulation for which (in 3D) a total of 6 rigid body motions have to be eliminated for\nthe positive definiteness of the operator [21].\nObservation 6.1 (Discrete coercivity of form I)\nGiven the numerical results of the eigenvalues of the discrete operator Ahwe surmise that for a complete\nDirichlet boundary AD=∂Vor an incompressible material ν= 0.5with a prescription of eσon at least one\nnode|AD|>0the discrete variational problem is coercive.\n6.3 Stabilised formulation\nIn order to introduce a coercive form of the equations also for a mixed boundary conditions we add the zero-sum\nterm derived from static equilibrium\n−ωsym D Div σ=ωsym D f, ω ≥0. (6.5)\n20to the strong form in Definition 3.1. The corresponding variational form and boundary term are given by testing\nwithτ∈C∞(V)⊗Sym(3) and partial integration\n−ωZ\nV⟨τ,sym D Div σ⟩dV=ωZ\nV⟨Divτ,Divσ⟩dV−ωZ\n∂V⟨τ,Divσ⊗n⟩. (6.6)\nThus, we recover the Neumann boundary term by splitting the boundary between Dirichlet and Neumann\n∂V=AD∪AN, allowing us to state the modified boundary value problem.\nDefinition 6.1 (The pure stress boundary value problem for solids II)\nThe field equations and boundary conditions of the stabilised symmetric three-dimensional stress boundary\nvalue problem read\n−∆σ−1\n1 +ν[hess(tr σ) + (div Div σ) 1]−ωsym D Div σ= (2 + ω) sym D f+1 +ν2\n1−ν2(divf) 1inV ,\n(6.7a)\n(Dσ)n+1\n1 +ν[∇trσ⊗n− ⟨f,n⟩ 1]−ω(f⊗n) =κ onAN,\n(6.7b)\nσ=eσ onAD,\n(6.7c)\nFurther, we retrieve the corresponding variational form.\nDefinition 6.2 (Variational form of the pure stress problem II)\nThe weak formulation of the stabilised boundary value problem of linear elasticity written purely in stresses\nreads\nZ\nV⟨Dτ,Dσ⟩+1\n1 +ν[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] +ω⟨Divτ,Divσ⟩dV=Z\nAN⟨τ,κ⟩dA\n+Z\nV(2 +ω)⟨τ,sym D f⟩+1 +ν2\n1−ν2⟨trτ,divf⟩dV ∀τ∈C∞\nAD(V)⊗Sym(3) , (6.8)\nand yields a fully symmetric left-hand side. Here τis assumed to be compatible with the Dirichlet boundary.\nLastly, the stabilised variational form can be derived from a variation functional.\nDefinition 6.3 (Variation functional in pure stresses II)\nThe weak formulation of the new boundary value problem can directly constructed as the variation of the\nstabilised functional\nI(σ) =Z\nV1\n2∥Dσ∥2+1\n1 +ν⟨Divσ,∇trσ⟩+ω\n2∥Divσ∥2−(2 +ω)⟨σ,sym D f⟩\n−1 +ν2\n1−ν2⟨trσ,divf⟩dV−Z\nAN⟨σ,κ⟩dA , (6.9)\nwith respect to the stress tensor σ.\nWe can now state the well-posedness of the stabilised problem.\nTheorem 6.1 (Stabilised bilinear form)\nLet the bilinear and linear forms read\na(τ,σ) =Z\nV⟨Dτ,Dσ⟩+χ[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] +ω⟨Divτ,Divσ⟩dV , (6.10a)\nl(τ) =Z\nV(2 +ω)⟨τ,sym D f⟩+1 +ν2\n1−ν2⟨trτ,divf⟩dV , (6.10b)\n21withω≥3χ2, then the variational problem with a non-vanishing Dirichlet boundary |AD|>0, reading\na(τ,σ) =l(τ)∀τ∈X(V), (6.11)\nhas a unique solution σ∈X(V)for every right-hand side with the stability estimate\n∥σ∥X≤1\nβ∥l∥X′, (6.12)\nwhere β=β(ν, ω)>0.\nProof. Continuity follows analogously to the proof of Theorem 3.1 with the continuity constant α= 1+9 χ. For\ncoercivity we observe\na(σ,σ) =∥Dσ∥2\nL2+ 2χ⟨Divσ,∇trσ⟩L2+ω∥Divσ∥2\nY\n≥ ∥Dσ∥2\nL2+ (ω−ϵχ)∥Divσ∥2\nL2−χ\nϵ∥∇trσ∥2\nL2\n∗\n≥\u0012\n1−3χ\nϵ\u0013\n∥Dσ∥2\nL2+ (ω−ϵχ)∥Divσ∥2\nL2\nPF\n≥1\n1 +c2\nF\u0012\n1−3χ\nϵ\u0013\n∥σ∥2\nX ∀σ∈X(V), (6.13)\nwhere we used Young’s inequality, the bound of ∥∇trσ∥2\nL2≤3∥Dσ∥2\nL2, and the Poincar´ e-Friedrich inequality.\nThe proof holds if ϵ >3χ, which is possible for the choice ω >3χ2.\nFor the discrete variational problem the coercivity condition can be relaxed to a(σh,σh)>0, implying a\npositive-definite matrix. Thus, we can inspect ωin the discrete case by investigating the eigenvalues of the\nnewly induced operator Ahof the stabilised bilinear form, characterising its spectrum. We apply the same\ntest as in the previous benchmark with a complete Neumann boundary AN=∂Vto find that for ω=χno\nnegative eigenvalue arise. However, specifically and only for ν= 0 we find 10 zero eigenvalues instead of 6,\nwhich indicates a positive semi-definite operator. For ω= 1.01χall negative eigenvalues vanish and we always\nretrieve exactly 6 zero eigenvalues for all possible Poisson ratios ν∈[0,0.5]. We thus conclude that the discrete\nvariational problem is coercive for ω > χ .\nDefinition 6.4 (Stabilised discrete variational problem for solids)\nThe stabilised discrete variational problem reads\nZ\nV⟨Dτ,Dσ⟩+1\n1 +ν[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] +ω⟨Divτ,Divσ⟩dV\n=Z\nAN⟨τ,κ⟩dA+ (2 + ω)⟨τ,sym D f⟩T+1 +ν2\n1−ν2⟨trτ,divf⟩T ∀τ∈Xh(V), (6.14)\nand is coercive for any choice ω > χ .\nIn order to compare the formulations we repeat the benchmark from Section 6.1 using the three-sided\nNeumann boundary definition from Fig. 6.3 (b) for the stabilised formulation with ω= 1.01χand for the non-\nstabilised formulation over ν∈ {0,0.25,0.5}. The convergence curves depicted in Fig. 6.4 clearly demonstrate\nthat the initial formulation without the stabilisation term leads to fluctuating results unless an incompressible\nmaterial is used ν= 0.499≈0.5. In contrast, the stabilised formulation is robust regardless of the Poisson ratio.\nWe note that the latter results are reproduced when using the Cholesky, PARDISO, or UMFPACK solvers in\nNGSolve.\n7 Numerics of the planar problems\n7.1 Characterisation of ψ\nIn order to characterise appropriate values for the dimensionless parametre ψwe examine plane stress and\nplane strain with varying Poisson ratios and Young’s modulus set to E= 200. The domain is defined as\n2210210310410510−310−210−1100\nDegrees of freedom∥eσ−σh∥L2/∥eσ∥L2\nσh.1\nσh.2\nσh.3\nσh.1\ns\nσh.2\ns\nσh.3\nsO(h4)\n(a)10210310410510−310−210−1100\nDegrees of freedom∥eσ−σh∥L2/∥eσ∥L2\nσh.1\nσh.2\nσh.3\nσh.1\ns\nσh.2\ns\nσh.3\nsO(h4)\n(b)10210310410510−310−210−1100\nDegrees of freedom∥eσ−σh∥L2/∥eσ∥L2\nσh.1\nσh.2\nσh.3\nσh.1\ns\nσh.2\ns\nσh.3\nsO(h4)\n(c)\nFigure 6.4: Convergence slopes of the non-stabilised σand stabilised σsforms with p∈ {1,2,3}forν= 0 (a),\nν= 0.25 (b) and ν= 0.499≈0.5 (c).\nA= [−3,3]×[−1,1] with a complete Dirichlet boundary sD=∂A. We guarantee highly accurate approximations\nby relying on a structured mesh with 1200 uniform quadrilateral elements of cubic polynomial order, yielding\n33123 degrees of freedom. We note beforehand that standard problems such as simple shear, pure extension and\nbending, lead to constant stress fields for the first two and a linear stress field in the case of bending, and as a\nresult, the formulation captures these exactly, making them invalid for the characterisation of ψ. Consequently,\nwe construct the following artificial analytical solutions\neus=1\n10sinh(x)e2,eub=1\n10[sin(x)e1+ sin( y)e2],eup=1\n10(sin[π(x+y)]e1+ sin[ π(x+y)]e2),(7.1)\nwhich lead to a pure shear field, bi-axial tension, and periodicity. We retrieve the stresses and forces via the\nconstitutive equation eσ=Csym Deuand static equilibrium f=−Diveσ. We use eσto enforce the Dirichlet\nboundary conditions and compute error estimates. The fields and the resulting stresses for plane stress with\nν= 0.25 are depicted in Fig. 7.1. The errors with respect to the exact solution are explored in Fig. 7.2\nfor the cases of plane stress and plane strain. The value of ψis defined with respect to χusing multiples of\nk∈ {1e−3,1e−2, . . . , 1e+2,1e+3}viaψ=kχ, and the Poisson ration is varied as ν∈ {0,0.125,0.25,0.375,0.5}.\nIn the case of plane strain we use ν= 0.499≈0.5 to derive the analytical solution for a quasi-incompressible\nmaterial. We observe that up to ψ=χand even for ψ= 1 with a negligibly worse result, the error remains\npractically the same. At the same time, the solver fails for ψ= 0. Consequently, it appears that any not too\nsmall or too large value leads to accurate results. Clearly, a simple and valid choice is therefore ψ=χ, which\nallows to derive the simplest form of the equations by diving by χ.\n7.2 Operator spectrum and stabilisation\nAnalogously to Section 6.2 for the three-dimensional case, we introduce the operator form of the two-dimensional\nvariational problem in Eq. (5.6) as\nfind σ∈Yh(A) s.t. Ahσ=f in [ Yh(A)]′, (7.2)\nand investigate its eigenvalues on the domain A= [−1,1]2with a complete Neumann boundary sN=∂A. The\nfinite element mesh is given by 32= 9-quadrilateral elements of cubic polynomial order, yielding 300-degrees of\nfreedom.\nFrom our computations we observe that irrespective of the value the Poisson ratio νorψ > 0 take, we\nalways retrieve 9-zero eigenvalues and no negative eigenvalues for both plane stress and plane strain, implying\na positive semi-definite operator. However, from the displacement-based formulation we expect only 3-zero\neigenvalues, accounting for rigid body motions in the plane. Observe that for ψ=χthe resulting field equation\nis equivalent to Eq. (4.13), allowing us to reformulate the boundary value problem.\n23(a)\n (b)\n (c)\nFigure 7.1: The displacement field eusand the corresponding stress field σ12, where the remaining stress com-\nponents vanish (a). The bi-axial tension field eubalong with the stresses σ11andσ22, such that the shear stress\nvanishes (b). The periodic field eupalong with σ11=σ22andσ12(c). For all the visualisations we set ν= 0.25.\n10−310−210−110010110210310−710−610−5\nk-value∥eσ−σh∥L2/∥eσ∥L2ν= 0\nν= 0.125\nν= 0.25\nν= 0.375\nν= 0.5\n(a)10−310−210−110010110210310−710−610−5\nk-value∥eσ−σh∥L2/∥eσ∥L2ν= 0\nν= 0.125\nν= 0.25\nν= 0.375\nν= 0.5\n(b)10−310−210−110010110210310−710−610−5\nk-value∥eσ−σh∥L2/∥eσ∥L2\nν= 0\nν= 0.125\nν= 0.25\nν= 0.375\nν= 0.5\n(c)\n10−310−210−110010110210310−710−610−5\nk-value∥eσ−σh∥L2/∥eσ∥L2ν= 0\nν= 0.125\nν= 0.25\nν= 0.375\nν≈0.5\n(d)10−310−210−110010110210310−710−610−5\nk-value∥eσ−σh∥L2/∥eσ∥L2ν= 0\nν= 0.125\nν= 0.25\nν= 0.375\nν≈0.5\n(e)10−310−210−110010110210310−710−610−5\nk-value∥eσ−σh∥L2/∥eσ∥L2\nν= 0\nν= 0.125\nν= 0.25\nν= 0.375\nν≈0.5\n(f)\nFigure 7.2: Respective errors for plane stress (a)-(c) and plane strain (d)-(e) relative to ψ=kχfor the cases of\nsheareus, bi-axial tension eub, and periodic displacements eup.\n24Definition 7.1 (Pure stress planar boundary value problem II)\nLetχ= 1/(1 +ν)orχ= 1−νfor plane stress or plane strain, respectively, the complete boundary value\nproblem reads\n−∆σ−hess(tr σ)−(div Div σ) 1= 2 sym D f+1\nχ(divf) 1, in A , (7.3a)\n(Dσ)n+∇trσ⊗n− ⟨f,n⟩ 1=κ on sN, (7.3b)\nσ=eσ on sD. (7.3c)\nThe variational form of Definition 7.1 follows by testing and partial integration, and is well-posed due to\nCorollary 4.2.\nDefinition 7.2 (Pure stress planar variational form II)\nThe variational form reads\nZ\nA⟨Dτ,Dσ⟩+⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩dA=Z\nsN⟨τ,κ⟩ds\n+Z\nA2⟨τ,sym D f⟩+1\nχ⟨trτ,divf⟩dA ∀τ∈C∞\nsD(A)⊗Sym(2) , (7.4)\nwithχ= 1/(1 +ν)orχ= 1−ν, for plane-stress or plane stain, respectively.\nThe latter can clearly be derived from a functional.\nDefinition 7.3 (Planar variational functional in pure stresses II)\nThe weak formulation of the new boundary value problem can directly constructed as the variation of the\nfunctional\nI(σ) =Z\nA1\n2∥Dσ∥2+⟨Divσ,∇trσ⟩ −2⟨σ,sym D f⟩ −1\nχ⟨trσ,divf⟩dA−Z\nsN⟨σ,κ⟩ds ,\nwithχ= 1/(1 +ν)orχ= 1−ν, for plane-stress or plane stain, respectively.\nLastly, the discrete form can be derived by using distributions of the derivatives of the body forces.\nDefinition 7.4 (Planar discrete variational form)\nThe discrete variational form reads\nZ\nA⟨Dτ,Dσ⟩+⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩dA (7.5)\n= 2⟨τ,sym D f⟩T+1\nχ⟨trτ,divf⟩T+Z\nsN⟨τ,κ⟩ds∀τ∈Yh(V),\nwithχ= 1/(1 +ν)orχ= 1−ν, for plane-stress or plane stain, respectively.\nNow, repeating the benchmark with the new formulation reveals that the associated discrete operator bAhalways\nyields 3-zero eigenvalues, being a sound result considering the analogy to the displacement-based formulation.\nWe apply to both formulations the benchmark from Section 7.1 for a complete Dirichlet sD=∂Aand mixed\nDirichlet Neumann boundary ∂A=sD∪sN, using the periodic displacement field eupfor the generation of\nappropriate boundary conditions. We test for both plane stress σσand plane strain σεwith ν= 0.25 using\ncubic quadrilateral finite elements. From the convergence rates in Fig. 7.3 we observe that for a complete\nDirichlet boundary both formulations are stable and converge optimally. However, for a mixed Dirichlet-\nNeumann boundary the initial formulation barely converges at all at the suboptimal rate O(√\nh). In contrast,\n2510310410510−510−510−410−310−210−1\nDegrees of freedom∥eσ−σh∥L2/∥eσ∥L2σh\nσ\nσh\nε\nσh\nσs\nσh\nεs\nO(h4)\n(a)10310410510−510−510−410−310−210−1\nDegrees of freedom∥eσ−σh∥L2/∥eσ∥L2\nσh\nσ\nσh\nε\nσh\nσs\nσh\nεsO(h4)O(√\nh)\n(b)sD\nsNA\n(c)\nFigure 7.3: Convergence of non-stabilised forms for plane stress σh\nσand plane strain σh\nεversus the respective\nstabilised forms denoted with σh\nσsandσh\nεs. In (a) the complete boundary is Dirichlet sD=∂A, whereas in (b)\nthe boundary is mixed between Dirichlet and Neumann ∂A=sD∪sNas per the depiction in (c).\nthe stabilised formulation is robust and yields optimal convergence irrespective of the boundary condition in\nuse.\n8 Conclusions and outlook\nThis work introduces new formulations of isotropic linear elasticity in terms of the stress tensor. The for-\nmulations are based on the Beltrami-Michell equations, which in the three-dimensional case we symmetrised\nand stabilised. In the two-dimensional cases of plane stress and plane strain, the Beltrami-Michell equations\ncharacterise only the mean stress and are thus insufficient for a variational formulation of the full stress tensor.\nConsequently, we introduced a new boundary value problem in two-dimensions by adding to the Beltrami-\nMichell equations a weaker form of static equilibrium scaled with a constant, which we later characterised\nnumerically. In addition, for all the presented variational formulations we proposed the treatment of constant\nand piece-wise constant body forces via distributions.\nBoth, the modified Beltrami-Michell equations in three dimensions as well as the newly introduced equa-\ntions for two-dimensions, were proven to be well-posed on a domain with a complete Dirichlet boundary, which\nwas verified by numerical benchmarks yielding optimal convergence. In the case of mixed Dirichlet-Neumann\nboundary conditions stabilisation is required in both three- and two dimensions. In three-dimensions we in-\ntroduced stabilisation by adding a weighted zero-sum term of a weaker form of static equilibrium into the\nboundary value problem. We investigated the constant weight ωof the stabilised form both analytically and\nnumerically, concluding that the result ω > χ guarantees stability, which we verified by spectral analysis and\nconvergence tests. In the two-dimensional case we derived an alternative boundary value problem via a novel\nidentity of the Laplacian of a symmetric second order tensor. The formulation was shown to be robust via\nspectral and convergence analysis irrespective of the split of the boundary between Dirichlet and Neumann.\nThe newly introduced equations are summarised in Appendix A. Considering our observations on the numerical\nbehaviour of the variational forms, we recommend to only use the stabilised versions in order to guarantee\nrobust computations.\nDespite being classical instruments for the (analytical) determination of stresses in mechanics, the Beltrami-\nMichell equations were not considered a suitable starting point for approximate solutions to the stress state,\ne.g. using finite element computations. As shown in this work, this is likely due to the inadequacy of the\nclassical form for a variational framework. Thus, the newly introduced equations and variational forms in this\npaper represent the foundation for future numerical application of pure stress formulations of linear elasticity,\nfor example for thermal-hydraulic-mechanical processes [1,2,66]. A clear advantage of the numerical framework\nis its flexibility with respect to the geometry of the domain and variation of the material constants within\nit. Specifically in material characterisation [3] and in the theory of mixtures [36] we expect the equations\nto be beneficial, considering the intensive research effort currently being invested into the reconstruction of\ncomputational representations of such materials for numerical simulations [19,56,57].\n26The equations presented in this manuscript are restricted to the case of isotropic linear elasticity. 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Computational Mechanics 68(1), 1–24 (2021)\n[65] Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numerische Mathematik 53(5), 513–538 (1988)\n[66] Wang, S., Wu, Y.S.: Theoretical analysis and semi-analytical formulation for capturing the coupled thermal-hydraulic-\nmechanical process using the stress formulation. Journal of Petroleum Science and Engineering 208, 109752 (2022)\n[67] Zaglmayr, S.: High order finite element methods for electromagnetic field computation. Ph.D. thesis, Johannes Kepler\nUniversit¨ at Linz (2006). URL https://www.numerik.math.tugraz.at/ ~zaglmayr/pub/szthesis.pdf\nA Glossary of stress equations\nDefinition A.1 (Pure stress boundary value problem for solids I)\nThe field equations and boundary conditions of the novel three-dimensional stress boundary value problem\nread\n−∆σ−1\n1 +ν[hess(tr σ) + (div Div σ) 1] = 2 sym D f+1 +ν2\n1−ν2(divf) 1 in V , (A.1a)\n(Dσ)n+1\n1 +ν[∇trσ⊗n− ⟨f,n⟩ 1] =κ on AN, (A.1b)\nσ=eσ on AD. (A.1c)\nwhere κis a mixed measure of equilibrium and compatibility on the Neumann boundary. The problem yields\na stable variational form the case of a complete Dirichlet boundary condition, or a mixed Dirichlet-Neumann\nboundary provided the material is incompressible ν= 0.5.\nDefinition A.2 (Discrete variational problem for solids I)\nThe corresponding discrete variational problem reads\nZ\nV⟨Dτ,Dσ⟩+1\n1 +ν[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] dV\n=Z\nAN⟨τ,κ⟩dA+ 2⟨τ,sym D f⟩T+1 +ν2\n1−ν2⟨trτ,divf⟩T ∀τ∈Xh(V), (A.2)\nand is stable either for a complete Dirichlet boundary AD=∂Vor a mixed boundary assuming incompre-\nhensibility ν= 0.5.\n29Definition A.3 (Pure stress boundary value problem for solids II)\nThe field equations and boundary conditions of the stabilised symmetric three-dimensional stress boundary\nvalue problem read\n−∆σ−1\n1 +ν[hess(tr σ) + (div Div σ) 1]−ωsym D Div σ= (2 + ω) sym D f+1 +ν2\n1−ν2(divf) 1inV ,\n(A.3a)\n(Dσ)n+1\n1 +ν[∇trσ⊗n− ⟨f,n⟩ 1]−ω(f⊗n) =κ onAN,\n(A.3b)\nσ=eσ onAD.\n(A.3c)\nDefinition A.4 (Stabilised discrete variational problem for solids II)\nThe stabilised discrete variational problem reads\nZ\nV⟨Dτ,Dσ⟩+1\n1 +ν[⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩] +ω⟨Divτ,Divσ⟩dV\n=Z\nAN⟨τ,κ⟩dA+ (2 + ω)⟨τ,sym D f⟩T+1 +ν2\n1−ν2⟨trτ,divf⟩T ∀τ∈Xh(V), (A.4)\nand is coercive for any choice ω > χ even for a mixed Dirichlet and Neumann boundary irrespective of the\nvalue of the Poisson ratio.\nDefinition A.5 (Pure stress planar boundary value problem I)\nLetχ= 1/(1 + ν)orχ= 1−νfor plane stress or plane strain, respectively, and ψ > 0, the complete\nboundary value problem reads\n−ψsym(D Div σ)−(χ∆ trσ) 1=ψsym D f+ (div f) 1 in A , (A.5a)\nχ⟨∇trσ,n⟩ 1−ψ(f⊗n) =κ on sN, (A.5b)\nσ=eσ on sD. (A.5c)\nThe boundary value problem leads to an optimally convergent discrete formulation only for a complete\nDirichlet boundary.\nDefinition A.6 (Pure stress planar discrete variational form I)\nThe variational form reads\nZ\nAψ⟨Divτ,Divσ⟩+χ⟨∇trτ,∇trσ⟩dA=ψ⟨τ,sym D f⟩T+⟨trτ,divf⟩T (A.6)\n+Z\nsN⟨τ,κ⟩ds∀τ∈Yh(A),\nwithψ >0andχ= 1/(1+ν)orχ= 1−ν, for plane-stress or plane stain, respectively. This form converges\noptimally only for a complete Dirichlet boundary sD=∂A.\n30Definition A.7 (Pure stress planar boundary value problem II)\nLetχ= 1/(1 +ν)orχ= 1−νfor plane stress or plane strain, respectively, the complete boundary value\nproblem reads\n−∆σ−hess(tr σ)−(div Div σ) 1= 2 sym D f+1\nχ(divf) 1, in A , (A.7a)\n(Dσ)n+∇trσ⊗n− ⟨f,n⟩ 1=κ on sN, (A.7b)\nσ=eσ on sD. (A.7c)\nDefinition A.8 (Pure stress planar discrete variational form II)\nThe discrete variational form reads\nZ\nA⟨Dτ,Dσ⟩+⟨Divτ,∇trσ⟩+⟨∇trτ,Divσ⟩dA (A.8)\n= 2⟨τ,sym D f⟩T+1\nχ⟨trτ,divf⟩T+Z\nsN⟨τ,κ⟩ds∀τ∈Yh(V),\nwith χ= 1/(1 + ν)orχ= 1−ν, for plane-stress or plane stain, respectively. The form yields optimal\nconvergence also for a mixed Dirichlet and Neumann boundary.\nB Mathematical formulae\nIn this work we introduced the following original three-dimensional and two-dimensional identities\nskw Curl σ=1\n2Anti(Div σ− ∇trσ)∀σ∈C∞(V)⊗Sym(3) , (B.1a)\n∆σ= 2 sym(D Div σ)−hess(tr σ) + [∆ tr σ−div Div σ] 1 ∀σ∈C∞(A)⊗Sym(2) , (B.1b)\nwhich were used in subsequent lemmas. Further, we made use of the following classical three-dimensional results\nIncσ= 2 sym(D Div σ)−∆σ−hess(tr σ) + [∆ tr σ−div Div σ] 1, (B.2a)\ntr(Inc σ) = ∆ tr σ−div Div σ, (B.2b)\ntr(Inc[(tr σ) 1]) = 2∆ tr σ, (B.2c)\nInc[(tr σ) 1] = (∆ tr σ) 1−hess(tr σ), (B.2d)\nCurl(skw σ) = (div axl skw σ) 1−(D axl skw σ)T, (B.2e)\nwhose thorough derivation can be found in [32].\nC An observation on planar linear elasticity\nIn following we characterise the curl of the stress tensor in planar linear elasticity. We start by introducing the\nplanar Navier-Cauchy equations.\nC.1 The planar Navier-Cauchy equations\nStarting with the boundary value problem of linear elasticity, we can derive the planar Navier-Cauchy equations\nby inserting the constitutive law for plane stress or plane stain, respectively. In the case of plane stress σi3=\nσ3i= 0, the constitutive equation reads\nσ=Cε=E\n1−ν2[(1−ν)ε+ (νtrε) 1] =E\n1−ν2[(1−ν) sym D u+ (νtr sym D u) 1]. (C.1)\n31Now, recalling the identities\nDiv sym D u=1\n2(∆u+∇divu), tr sym D u= tr D u= div u, Div[(div u) 1] =∇divu, (C.2)\nwe find the equations\n−DivCε=−E\n2(1 + ν)∆u−E\n2(1−ν)∇divu=f in A , (C.3a)\n\u0014E\n1 +νsym D u+Eν\n1−ν2(divu) 1\u0015\nn=µ on sN, (C.3b)\nu=eu on sD, (C.3c)\nbeing the Navier-Cauchy equations of plane stress.\nIn the case of plane strain the constitutive equation reads\nσ=Cε=E\n1 +ν\u0014\nε+ν\n1−2ν(trε) 1\u0015\n=E\n1 +ν\u0014\nsym D u+ν\n1−2ν(tr sym D u) 1\u0015\n, (C.4)\nleading to the equations\n−DivCε=−E\n2(1 + ν)∆u−E\n2(1 + ν)(1−2ν)∇divu=f in A , (C.5a)\n\u0014E\n1 +νsym D u+Eν\n(1 +ν)(1−2ν)(divu) 1\u0015\nn=µ on sN, (C.5b)\nu=eu on sD, (C.5c)\nbeing the Navier-Cauchy equations of plane strain.\nC.2 The curl of the stress tensor\nWe note that the skew-symmetric part of the gradient of the stress tensor is proportional to the curl of the\nstress tensor Rot σ. In order to find an identity concerning Rot σ, we observe\nRot sym Div u= Rot(D u)T=1\n2R∆u−1\n2R∇divu=1\n2(R∆u− ∇⊥divu), (C.6)\nRot[(tr sym D u) 1] = Rot[(div u) 1] =−R∇divu=−∇⊥divu. (C.7)\nNow, the curl of the the stress tensor given by the constitutive equation of plane stress reads\nRotσ= Rot Csym D u=E\n2(1 + ν)(R∆u− ∇⊥divu)−Eν\n1−ν2∇⊥divu\n=E\n2(1 + ν)R∆u−E\n2(1−ν)∇⊥divu. (C.8)\nBy rotating the Navier-Cauchy equation of plane stress we derive\nE\n2(1 + ν)R∆u=−E\n2(1−ν)∇⊥divu−Rf, (C.9)\nleading to\nRotσ=−E\n1−ν∇⊥divu−Rf. (C.10)\nWe can express\n∇⊥divu=R∇divu=RDiv[(tr sym D u) 1] =RDiv[(tr[ Aσ]) 1] =R∇tr(Aσ) =∇⊥tr(Aσ), (C.11)\n32which by using the compliance relation of plane stress reads\n∇⊥divu=∇⊥tr(Aσ) =1\nE∇⊥[(1 + ν) trσ−2νtrσ] =1−ν\nE∇⊥trσ. (C.12)\nThus, we find for the curl of the stress tensor\nRotσ+∇⊥trσ=−Rf, (C.13)\nbeing an alternative form of equilibrium for planar problems.\nFollowing the same procedure for plane strain we observe\nRotσ=E\n2(1 + ν)R∆u−E\n2(1 + ν)(1−2ν)∇⊥divu. (C.14)\nThe rotated Navier-Cauchy equation of plane strain yields\nE\n2(1 + ν)R∆u=−E\n2(1 + ν)(1−2ν)∇⊥divu−Rf, (C.15)\nleading to\nRotσ=−E\n(1 +ν)(1−2ν)∇⊥divu−Rf. (C.16)\nWe compute\n∇⊥divu=∇⊥tr(Aσ) =1 +ν\nE∇⊥[trσ−2νtrσ] =(1 +ν)(1−2ν)\nE∇⊥trσ, (C.17)\nwhere we relied on the plane strain constitutive equation. Thus, we obtain the same relation as for plane stress\nin Eq. (C.13).\nConsequently, we find the new differential identities\nRotσ+∇⊥trσ=RDivσ, Divσ=∇trσ−RRotσ. (C.18)\nrelating the divergence of a symmetric two-dimensional second order tensor σ=σTto its curl.\n33"    },    {        "title": "2402.00484v1.Extreme_value_statistics_of_nerve_transmission_delay.pdf",        "content": "Extreme value statistics of nerve transmission delay\nSatori Tsuzuki∗\nResearch Center for Advanced Science and Technology,\nThe University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8904, Japan\nNerve transmission delay is an important topic in neuroscience. Spike signals fired or received\nat the dendrites of a neuron travel from the axon to the presynaptic cell. The spike signal triggers\na chemical reaction at the synapse, wherein a presynaptic cell transfers neurotransmitters to the\npostsynaptic cell, and regenerates electrical signals by a chemical reaction process through ion chan-\nnels and transmits it to neighboring neurons. In the context of describing the complex physiological\nreaction process as a stochastic process, this study aimed to show that the distribution of the max-\nimum time interval of spike signals follows extreme order statistics. By considering the statistical\nvariance in the time constant of the Leaky Integrate-and-Fire model, which is a deterministic time\nevolution model of spike signals, we enabled randomness in the time interval of spike signals. When\nthe time constant follows an exponential distribution function, the time interval of the spike signal\nalso follows an exponential distribution. In this case, our theory and simulations confirmed that\nthe histogram of the maximum time interval follows the Gumbel distribution, which is one of the\nthree types of extreme value statistics. We also confirmed that the histogram of the maximum time\ninterval follows a Fr´ echet distribution when the time interval of the spike signal follows a Pareto\ndistribution. These findings confirm that nerve transmission delay can be described using extreme\nvalue statistics and could, therefore, be used as a new indicator for transmission delay.\nI. INTRODUCTION\nThe neurological system connects multitudes of neu-\nrons, and spike signaling between two neurons is a com-\nplex physiological process. A schematic of a single neuron\nis illustrated in Fig. 1(a) and a summary of the mecha-\nnism of nerve transmission is as follows [1–3]. Each neu-\nron has a physiological parallel resistor–capacitor circuit,\nin which K+ and Na+ ion channels and K+ leak chan-\nnels maintain the membrane potential at resting poten-\ntial in a non-stimulation state. However, an influx of ex-\nternal current causes the depolarization of neurons, pro-\nducing a sudden and sharp increase in potential, called\na spike. During the spike, the membrane potential ex-\nceeds a threshold value, followed by repolarization and\nhyperpolarization, and, subsequently, a period of prepa-\nration for the next spike generation. During this time,\nthe Na+ ion channels enter a temporal inactive state,\ncalled the refractory period, which prevents neurotrans-\nmission backflow and ensures that the generated spike\nsignal is transmitted in one direction in the neuron. The\nspike signal is then transmitted to the synapse, i.e., the\nedge of the neuron. Each synapse is composed of a pair of\nseparated presynaptic and postsynaptic cells. Transmis-\nsion begins with the release and diffusion of neurotrans-\nmitters into the synaptic cleft in the presynaptic cell,\nfollowed by reception of the transmitters by receptors in\nthe membranes of the postsynaptic cell, and then by ion\nchannels in the postsynaptic membrane that convert the\ntransmitters into electrical signals which are transmit-\nted to the neighboring neurons. In this way, a neural\ntransmission between two neurons (hereafter referred to\nas a “nerve transmission unit” or “NTU”) occurs in a\nfinite amount of time. The time taken by an NTU fluc-\n∗Email: tsuzukisatori@g.ecc.u-tokyo.ac.jptuates around its average value because of physiological\nand chemical reactions of several minute particles such\nas transport of K+ and Na+ ions through the membrane\nor numerous neurotransmitters at the synapse. Thus,\nthe time required for signal transmission at the NTU\ncan be described as a statistical variable rather than a\ndeterministic constant [4–7]. In contrast, the time be-\ntween two spike signals corresponds to a waiting time\nin the transmission processes of the entire nervous sys-\ntem called a chain set of NTUs. Hereinafter, We refer to\nthis waiting time as the “nerve transmission period” or\n”NTP,” defined as the total transmission time taken by\na spike signal to pass through the dendrites and axons\n(hereinafter referred to as a nerve part or NP) and the\n“synaptic delay” as defined in the literature [4].\nTo date, several studies have reported that a delay\nor disruption of spike signals can cause diseases such as\nschizophrenia [8, 9]. Nevertheless, investigations on the\ndetailed relationship of NTP with these nerve diseases\nare limited. Elucidating the statistical behavior of NTUs\nmight be one of the methods to gain deeper insights into\nthe behavioral aspects of NTUs. Estimating the fre-\nquency distribution of maximum delays and the upper\nlimits from the set of time-series data for NTPs can allow\nfor the metrics and scale parameters of the distribution\nto become new indicators for evaluating the statistical\nproperties of the nervous system. We can quantitatively\nevaluate the impact of the disease on the nervous system\nby comparing the degree to which such statistical met-\nrics change from their normal state before and after the\ndisease. Observation of NTPs over several days can help\nderive a theoretical histogram such that the maximum\ndistribution of NTPs asymptotes, and the parameters of\nthe theoretical histogram can also be used as a new math-\nematical tool for evaluating neurological dysfunction.\nNotably , neural transmission in NTU can be expressed\nby a directed graph consisting of two nodes. For clarity,arXiv:2402.00484v1  [q-bio.NC]  1 Feb 20242\nFIG. 1. Schematic representation of a systematic description of a single neuron.\nwe explain the NT system using the classical Petri net\nmodel [10–12], which is an established representation for\nstate transition diagrams. Below, we have explained the\noutline of the Petri net model with reference to Fig. 1(b).\nFirst, the components of a system or their states are de-\npicted by a circular symbol called “place” and an event is\ndepicted by a bar-shaped symbol called “transition.” The\ncorrelation between the components of place and transi-\ntion can be expressed by a directed arrow called “arc.”\nThe objects processed on the system are represented by\na black filled circle called a “token. ” The flow of objects\nis traced on the Petri net diagram by these tokens. The\nfirst place in (b) corresponds to part of a neuron consist-\ning of NP and a presynaptic cell in (a), and the second\nplace in (b) corresponds to a postsynaptic cell in (a),\nwith unidirectional transmissions from the presynaptic\nto the postsynaptic cell. Additionally, the “transition”\nshows all the events such as spike generation, release,\ndiffusion, reception, and conversion of neurotransmitters\ninto electrical signals. The “token” shown as a black cir-cle in (b) indicates the spike signals flow at the NP or\nneurotransmitters at the synapse. The weights w1and\nw2represent the condition that can fire the transition\non each arc; and these weights usually represent the cri-\nteria of firing. When the number of tokens exceeds the\nweights, the tokens are ready to pass through the arc.\nIn NT problems, we regard arcs as always transmittable\nstates as long as we ignore such exceptional dysfunction\nphenomena as synaptic fatigue [13], where neurotrans-\nmitters in the synapse become scarce and transmission is\naborted. Hence, we can ignore the conditions for w1and\nw2in the first approximation and set them to 1. A com-\nparison of (a) and (b) clearly shows that we can interpret\nNTU as a two-state transition problem.\nThe mechanism of NTUs has been studied by both\nmodeling and experimentation. Hodgkin and Huxley pre-\nsented the deterministic time evolution equations of an\nequivalent circuit model describing membrane potential\nchanges in a neuron [14–19]. We will refer to this model\nas the Hodgkin–Huxley model or HHM. The HHM de-3\nscribes the detailed mechanism of NTU as a physiological\nparallel resistor–capacitor circuit mentioned at the begin-\nning of this section. Meanwhile, the Leaky Integrate-and-\nFire model (LIFM) [20–25] is a simplified version of the\nHHM which focuses on the timing of spike generations\nand the frequencies, without the detailed generation pro-\ncesses to reduce the number of variables. LIFM is more\ncomputationally friendly than HHM and is well suited\nfor large-scale numerical simulations [24, 25]. HHM or\nLIFM can be combined with the exponential synapse\nmodels [26–28] that express a decay of the electric current\nat the synapse, representing the behavior of the synap-\ntic delays. Here, we formulated the NTP as a random\nvariable (to be later discussed). Discrete elapsed time Ti\n(i= 1,2,···, n) (used as a random variable sequence)\nwas defined as the time at which a spike signal transmis-\nsion is just converted to electrical signals. Additionally,\nthe random variable sequence of the time difference is de-\nfined as Tiasdi:=Ti−Ti−1(i= 1,2,···, n), and dmax\nis defined as the maximum of di(i= 1,2,···, n) for\neach measurement. In this paper, for sufficiently large m\nmeasurements, a probability density distribution Dcan\nbe obtained using extreme value theory (EVT) [29–32],\nto which the histogram (frequency distribution) of dmax\nasymptotes. In this case, Dderived from EVT can be\nused as a new indicator for objectively evaluating the\nmaximum distribution of NTPs.\nImportantly, transmission in a single NTU can be as-\nsumed to follow a Poisson distribution for the following\nreasons. Consider a single NTU in a nerve chain con-\nnecting multiple NTUs that is long enough to ignore the\neffect of the edges of the chain, based on which, we can\nassume an open boundary condition. Because of the re-\nfractory period, the postsynaptic cell in the NTU does\nnot receive other signals during each signal transmission.\nTherefore, we can assume each event to be independent,\nwhere each “event” is defined as the transmission process\nfrom the generation of a spike signal at a neuron to its\nreception by receptors, and its conversion into electrical\nsignals at a postsynaptic cell. As a thought experiment,\nconsider observing the transmission process in the NTU\nfor a finite time T. Because of physiological homeostasis,\nit is reasonable to consider transmission flow in a steady\nstate. Additionally, in a zero-order approximation, the\ntransmission rate can be assumed to be invariant during\nthe observation. The invariant transmission rate and in-\ndependence of each event imply that the probability of\nevents occurring at any time interval ( ti, ti+t) are inde-\npendent of the events that occurred before Ti. Therefore,\nthe system is memoryless. The reason for stating “in\nthe zero-order approximation” is that at this stage, we\nignore synaptic fatigue [13], where neurotransmitters in\nthe synapse become scarce and transmission is aborted.\nConsider discrete time ∆ tas a small time of the same\norder as di. The probability of an event occurring more\nthan once during ∆ tis negligible because diis the inter-\nval at which an event occurs. Accordingly, this system\nhas the property of event scarcity. From the above, the\nsystem has the following conditions for being considereda Poisson distribution: stationarity, independence, mem-\norylessness, and scarcity. Thus, Tican be a sequence of\nrandom variables following a Poisson process. Recall that\ndiis a sequence of random variables for the time interval\nTi−Ti−1(i= 1,2,···, n). If a sequence of random vari-\nables follows a Poisson process, the difference sequence of\nrandom variables is known to follow an exponential dis-\ntribution [33]. Therefore, dican be expected to follow an\nexponential distribution. In reality, several experiments\nhave shown that the frequency distribution of synaptic\ndelays follows an exponential function, strongly support-\ning that Tifollows a Poisson process and a sequence of\nrandom variables for that time interval difollows an ex-\nponential distribution [34–36].\nIf a random variable sequence Xi(i= 1,2,···, n) fol-\nlows an exponential or Pareto distribution, it is known\nthat the distribution of the maximum order statistic of\nXiis asymptotic to the Gumbel or Fr´ echet distribution\nof EVT, respectively [29–32] (see the Methods section for\nthe mathematical description of EVT). It follows that the\nasymptotic distribution of the maximum order statistic\nof the random variable of time interval dican be deter-\nmined if difollows an exponential or Pareto distribution.\nAs mentioned earlier, several experiments have confirmed\nthat NTPs follow an exponential distribution [34–36];\ntherefore, the distribution of the maximum values of di\nobtained by solving the time evolution of HHM or LIFM\nis expected to follow the Gumbel distribution. This study\ndemonstrated this principle in a single neuron simulation.\nTo the best of our knowledge, this study is the first to ap-\nply EVT to the nervous system. Beginning with the theo-\nretical study on the statistical distribution that the max-\nimum values follow, EVT is established with the Trinity\nTheorem; this shows that the distribution following such\na maximum, if it exists, belongs to only one attraction\ndomain of the three distributions: the Gumbel, Fr´ echet,\nand Weibull distributions [29–32]. In the related studies,\nEVT has been used in various fields, such as predicting\nthe maximum amount of precipitation [37–39], the max-\nimum age limit in a region, and the maximum speed at\nwhich an athlete can run a marathon [40–42]. However,\nthere are not so many examples of the application of EVT\nto traffic transportation systems, even if we look at other\nrelated fields. In the cross fields of physics and aeronau-\ntics, we have previously considered the spot assignment\nproblem in airport ground traffic from a physical point\nof view [43]. Focusing on a roadway consisting of a sin-\ngle lane with a parking lane, we solved the problem of a\nvehicle moving along the roadway in one direction, stop-\nping at one of the randomly selected spots in an adjacent\nparking lane, and starting again after a certain period of\ntime using a stochastic model known as the total asym-\nmetric simple exclusive process (TASEP) [44–46]. In our\nprevious study [43], we examined the stochastic process\nof a vehicle stopping at a spot in a parking lane while\nmoving along a road. We found that this scenario follows\nthe so-called generation-death process and that to obtain\nthe frequency distribution of spot usage, it is necessary\nto consider the order statistics of spot usage. Accord-4\nFIG. 2. Schematic view of the extreme order statistics\ningly, the distribution of spot usage was described using\nan extreme value statistical distribution [43]. This is an\nexample of the application of EVT to stochastic processes\nin the spatial direction. In contrast to the previous study,\nwe applied EVT to the problem of delays in neural spike\nsignals that follow a Poisson process in the time direction.\nOnce the asymptotic distribution of the histogram of the\nmaximum value of NTPs in nerve transmission is iden-\ntified, the upper bounds and their probability of occur-\nrence can be predicted. This could be helpful in clinical\npractice. We could use electroencephalography (EEG) to\nmeasure the delays of NTPs and use EVT to estimate theupper bounds of their distribution. This can serve as a\nstepping stone for treatment methods to develop preven-\ntive measures. In addition, extreme value statistics can\nbe used for the derivation of the general extreme value\ndistribution (GEVD) [47–49], encompassing these three\ndistribution types. Because GEVD is adaptable and is\nsuitable for fitting measurement data, studies that use\nGEVD for the analysis of real data are becoming popu-\nlar. Semiempirical approaches that apply GEVD to the\nmeasured distributions and estimate the model param-\neters of GEVD by probabilistic maximum likelihood or\nsimilar methods have dominated research on spatiotem-5\nporal data [50–53]. In particular, in areas related to neu-\nroscience, a study reported on fitting experimental data\nof the velocities for dynein and kinesin in neurons with\nGEVD [54]. In contrast to these data analysis studies,\nour study aimed to demonstrate that the histogram of\nmaximum time intervals of neural spike signals follows\nEVT both theoretically and by simulation.\nFinally, HHM and LIFM are deterministic time evo-\nlution models; however, the actual spike signal time in-\nterval is confirmed to follow an exponential distribution.\nTherefore, it is necessary to modify these deterministic\ntime evolution models to stochastic time evolution mod-\nels. This study aims to reproduce the stochastic proper-\nties in the LIFM model by giving a stochastic fluctuation\nto the time constant τ, which determines the decay rate\nof the spike signal (see the Methods section). In brief,\nevery time a spike occurs, τis updated according to a\ncertain probability distribution. Because the time depen-\ndence of the voltage drop fluctuates slightly with each\nsignal in real neurotransmission processes, making the\ntime constant a random variable is a more accurate rep-\nresentation of real physical phenomena. Therefore, this\nis a reasonable method to convert a deterministic time\nevolution model to a stochastic time evolution model. In\nsummary, we demonstrated our theory that NTPs follow\nextreme order statistics in a nerve transmission simula-\ntion for a single NTU using stochastic LIFM combined\nwith an exponential synapse model.\nII. METHODS\nSingle neuron models. The first detailed mathe-\nmatical model of the transmission process of a single neu-\nron was the HHM formulated in 1952 by Dr. Hodgkin\nand Dr.Huxley, based on their experimental results of ac-\ntion potentials in squid axons [14–19]. In that study, each\nneuron was shown to be a parallel resistance–capacitor\ncircuit with K+ ion channels, Na+ ion channels, and K+\nleak channels, and the spike signal was shown to prop-\nagate with a waveform of finite width as a solution to\nthe time-dependent differential equation of the electri-\ncal circuit. The HHM model can accurately reproduce\nthe neurotransmission process including the waveform of\nthe spike signal. However, the HHM model has some\npractical drawbacks, such as the fact that it retains four\nvariables per neuron and requires a relatively accurate\ncomputational method because of the complexity of the\nequations. In physiological experiments, only the tim-\ning of the spike signal is often of interest, and not the\nwaveform of the signal. Accordingly, LIFM is an intrin-\nsic model that simplifies HHM by focusing only on the\nmembrane potential shift and reducing the number of\nvariables to one. The time evolution equation for the\nmembrane potential in LIFM is described as follows [20–\n25]:\nτd\ndtV(t) =−(V(t)−Vrest) +RIext,\nV(t)> θ⇒s(t) = 1 , V(t)←Vreset,V(0) = Vinit. (1)\nwhere τis the time constant, Vrestis the resting poten-\ntial,V(t) is the membrane potential at time t,Ris the\nmembrane resistance, Iextis the constant external cur-\nrent, and θis the threshold for spike firing. Vresetis the\nreset potential, and Vinitis the initial value of the mem-\nbrane potential. The membrane potential is the equilib-\nrium point at Vrestand asymptotes to Vrest+Rlextat a\nrate of τ. In addition, s(t) is a discrete function that is\n1 if a spike is fired at time tand zero otherwise. When\nthe membrane potential Vexceeds θ, we set s(t) = 1 to\nexpress the spike was fired at that time, simultaneously\nresetting the membrane potential VtoVreset. According\nto Eq. (1), the change and resetting of the membrane po-\ntential are repeated, and the spike signal travels through\nthe neuron as a periodic signal.\nIn contrast to the original LIFM shown in Eq. (1), we\ntransformed Eq. (1), a deterministic differential equation,\ninto a stochastic differential equation by replacing the\ntime constant τbyξ(µ, σ) where ξ(µ, σ) is the indepen-\ndent identity distribution given by two parameters τand\nσ, corresponding to the location and rate parameters for\nan exponential distribution and shape and location pa-\nrameters for the Pareto distribution, respectively. We\ncomputed the variation of the membrane potential ac-\ncording to Eq. (1) after replacing the time constant τ\nwith ξ(µ, σ). We assumed that the current spike signal is\ntransmitted to the presynaptic cell when the threshold θ\nis reached, and then we calculated the change in current\nat the synapse using an exponential decay model, ex-\npressed as Is(t) =Is(0)e−t/τs, where Is(t) is the current\nat synapse at time tandτsindicates the time constant.\nTo consider the time required for the chemical reaction\nprocess at the synapse, i.e., synaptic delay, the mem-\nbrane potential is reset when both conditions are met:\nthe membrane potential exceeds the threshold θand the\nsynaptic decay is below a time constant. Thus, the con-\ndition V(t)> θfor resetting the membrane potential in\nEq. (1) is modified to V(t)> θ∩Is(t)< ω, where ωcorre-\nsponds to Is(0)e−1. Accordingly, we calculate both spike\nsignal transmission and synaptic delay which makes the\ndescription closer to the actual nerve transmission com-\npared with Eq. (1). Nevertheless, in practice, the effect of\nsynaptic delay is negligible in a single neuron simulation\nin one direction. This can be explained as follows. In the\ncase of multiple neurons, the current Isat the synapse\nfeeds back to the variation of membrane potential as Iext\nin some cases; however, in a single neuron problem, the\nsignal flows in one direction. In addition, it immediately\ndecays compared with the variation in membrane voltage.\nTherefore, Isdoes not affect the behavior of the entire\nsystem in our case because we focused on the distribu-\ntion of maximum time intervals when the synaptic time\nintervals follow an exponential distribution. In summary,\nthe modified version of Eq. (1) is described as follows:\nξ(µ, σ)d\ndtV(t) =−(V(t)−Vrest) +RIext,\nV(t)> θ∩I(t)< ω⇒s(t) = 1 , V(t)←Vreset,6\nV(0) = Vinit. (2)\nExtreme value theory (EVT). The maximum or-\nder statistic of the independent identical random variable\nis defined as Xj(j= 1,2,···, n) asZn. In the mathe-\nmatical form, this definition can be expressed as follows:\nZn:= max {X1, X2,···, Xn}= max Xj(j= 1,2,···, n).\nUnder this definition, EVT discusses the asymptotic dis-\ntribution of Znfor sufficient measurements of Zn. Fig-\nure 2 schematically explains the details of extreme value\nstatistics. Suppose we measured mdata sets of Znin the\nexample in Fig. 2, Znisd4for case 1, d2for case 2, and\nd1for case m. Ifmis sufficiently large, we can obtain\nthe histogram (frequency distribution) of Zn. In partic-\nular, if we find kcounts of data such that Znbecomes a\nvalue of Y, there are mCkpossible combinations in which\nwe observe Yforktimes among mmeasurements. It is\nnecessary to consider all of them to derive the probabil-\nity density function (PDF) for Zn. PDF is obtained as\nthe first derivative of the cumulative distribution func-\ntion (CDF); EVT begins by deriving the CDF of Zn.\nNow consider the case in which the independent identi-\ncal variable Xihas population distribution F. The CDF\nofZnis expressed as P(Zn≤x), where Pis the proba-\nbility and xis a continuous point. From the definition of\nZn, we can derive the following relationships:\nP(Zn≤x) = P\u0000\nmax\n1≤j≤nXj≤x\u0001\n=P(Xj≤x, j= 1,2,···, n)\n=nY\nj=1P(Xj≤x)\n=Fn(x). (3)\nEquation (3) shows that the CDF of Znis equal to the\npopulation distribution Fto the power of n; thus, we fo-\ncus on the distribution where Fn(x) converges. Fisher\nand Tippett [29] mathematically proved the following\nequivalence relation with respect to the convergence of\nFn(x):\nF∈ D(G)⇔lim\nn→∞Fn(anx+bn) = G(x),\nan>0, bn∈R. (4)\nHere, Gis a continuous distribution that is neither de-\ngenerate nor divergent. F∈ D(G) represents that F\nbelongs to a domain of attraction of G. Equation (4)\nstates the following: if F∈ D(G), i.e., if the population\ndistribution Fof independent identical random variable\nXj(1≤j≤n) belongs to the attraction domain of G,\nwhich is the asymptotic distribution of the maximum or-\nder statistic Znand is neither degenerate nor divergent,\nanandbnexist for which Fn(anx+bn) converges in G\nfor sufficiently large n. Conversely, if we can find a pair\nofanandbnwith which Fn(anx+bn) converges to G,\nthen Fbelongs to the attraction domain of G. Equa-\ntion (4) also implies that the linear transformation of G\nwith respect to scale anand location bnis permitted to\navoid degenerate or divergence. This can be understood\nby employing ¯ x=anx+bnand rewriting Eq. (4) for ¯ x.The Trinity Theorem by Fisher and Tippett [29],\nFr´ echet [30], and Gnedenko [31] proved that there are\nonly three types of extreme distributions that satisfy Eq.\n(4): Gumbel, Fr´ echet, and Weibull distributions. The\ntheorem also proved that any population distribution F\nasymptotic to one of these three extreme distributions\nifF∈ D(G). The CDFs of the Gumbel, Fr´ echet, and\nWeibull distributions are described as follows:\nGumbel : G(x):= exp[ −exp(−x)], x∈R, (5)\nFr´ echet : G(x):= exp( −x−α), x≥0, α > 0,(6)\nWeibull : G(x):= exp[ −(−x)α], x≤0, α≥0.(7)\nIfXj(j= 1,2,···, n) follows an exponential distribu-\ntion, Fis expressed as F(x) = 1−exp(−x),x∈R. In\nthis case, the convergence of Fn(x) for sufficiently large\nncan be identified by selecting anandbnto be 1 and\nlog(n) as follows [43]:\nlim\nn→∞Fn(anx+bn)\n= lim\nn→∞(\n1 +−n[1−F(anx+bn)]\nn)n\n= lim\nn→∞(\n1 +−exp(−x)\nn)n\n= exp[ −exp(−x)], x∈R. (8)\nTherefore, the maximum order statistic Znof the ran-\ndom variable Xj(1≤j≤n) converges to the Gum-\nbel distribution if Xjfollows an exponential distribution.\nMeanwhile, if Xj(1≤j≤n) follows the Pareto distri-\nbution, Fis expressed as F(x) = 1−1/xα, where α >0\nandx≥1. In this case, using the derivation similar to\nthat in exponential distribution, Fn(anx+bn) converges\nto the Fr´ echet distribution by selecting anandbnto be\nn1/αand 0, respectively.\nlim\nn→∞Fn(anx+bn) = exp( −x−α). (9)\nIn summary, Znconverges to the Gumbel distribution if\nXj(1≤j≤n) follows an exponential distribution and\nto the Fr´ echet distribution if Xj(1≤j≤n) follows the\nPareto distribution.\nIII. ANALYSIS\nWe performed stochastic LIFM simulations using\nEq. (2) in two different cases in which ξ(µ, σ) follows\nan exponential distribution or a Pareto distribution. In\nour tests, we set ( Vrest, Vreset, θ, R, I ext) to be (-65 mV,\n-65 mV, -55 mV, 1.0 MOhm, 12 nA) by reference to the\nliterature [55–57]. We set the small time ∆ tand the\nnumber of time steps nto 1 ms and 1000 iterative steps;\n1000 ms of physical time was computed in each simula-\ntion. We used the inverse transformation method [58, 59]\nto generate a time interval di(i= 1,2,···, n) random se-\nquence of spike signals following exponential and Pareto7\ndistributions. In the inverse transformation method, the\nrandom sequence di(i= 1,2,···, n) is given by di=\nξ(µ, σ) = (−1.0/σ)log(U) +µfor exponential distribu-\ntions, where Urepresents a uniform distribution. In this\nformula, parameter σcorresponds to the rate parameter\nof the exponential function, which determines the scale\nof the distribution. Parameter µshifts the distribution\nin the positive direction to asymptotes to µ; this param-\neter corresponds to parameter τin Eq. (1). Accordingly,\nwe set µto be 20 ms and set σto be 5 ms. In con-\ntrast, for the Pareto distribution, the random sequence\ndi(i= 1,2,···, n) is given by di=ξ(µ, σ) =µ/(U)1.0/σ.\nIn this model, the location and attenuation rate are given\nby parameters µandσ, respectively. We set parameter\nµto be 20 ms. On the other hand, we performed simu-\nlations in two different cases of σ= 7.5 and σ= 20.\nFigure 3 shows the simulation results in the case\nthat ξ(µ, σ) followed an exponential distribution. First,\nFig. 3(a) evidently shows that ξ(µ, σ) followed an expo-\nnential distribution. Figure 3(b) shows the resulting time\ninterval of the spike signals obtained by solving Eq. (2)\nwith randomly updated ξwhen each spike signal was\ngeenerated. This confirmed that the time interval of gen-\nerated spike signals followed an exponential distribution.\nFigure 3(c) shows the variation of current I(which cor-\nresponds to Isin Methods section) at the synapse and\nmembrane voltage V. Once the voltages reached 55 mV,\nthey triggered the generation of a spike signal, which is\nimmediately transmitted to the synapse, where the cur-\nrent attenuates with the exponential decay model. As\nmentioned earlier, in a single neuron problem, the signal\nflows in one direction, and the current Isimmediately de-\ncays compared with the variation in membrane voltage.\nAs a result, Isdoes not affect the behavior of the entire\nsystem. Nonetheless, we introduced this simplification\nbecause we were interested in the distribution of maxi-\nmum time intervals if the synaptic time intervals followed\nan exponential distribution. Meanwhile, the circle dots\nin Fig. 3(d) show the histogram of the maximum time\nintervals measured in 100000 trials. Each trial iterated\n1000 time steps for the time evolution of Eq. (2). The\nhistogram was normalized so that the distribution sum-\nmation was 1. Notably, the obtained histogram corrob-\norated that it followed the Gumbel distribution, which\nwas consistent with EVT. Note that for the values of R-\nsquared, the coefficient of determination was greater than\n0.99 in all the cases. Accordingly, we demonstrated that\nin single neuron problems, the histogram of maximum\ntime intervals followed the Gumbel distribution when the\ntime intervals followed an exponential distribution.\nFigure 4 shows the simulation results in a case that\nξ(µ, σ) followed a Pareto distribution in small and large\ncases of σ= 7.5 or σ= 20. As reported in the ex-\nperiments on transmission delays, the distribution of\nNTPs primarily followed exponential distributions [34–\n36]. However, in some cases, we observed that the NTP\ndistributions exhibited sharper peaks and long tails sim-\nilar to the Pareto distribution. In response to these ob-\nservations, we investigated the histogram of the maxi-mum NTPs for the Pareto distribution. Figures 4(a)-(d)\nshows the results for σ= 20. First, (a) displays an ex-\nample snapshot of spike signals for reference, (b) shows\nthe obtained distribution of time intervals and fitting re-\nsults using a Pareto distribution. The results indicated\nthat the measured distribution followed the Pareto distri-\nbution. Because the maximum and minimum values are\ngreatly separated in the Pareto distribution, we evidently\nshowed the logarithmic scale of (b) in (c) to confirm the\nexistence of the long tail, which is a trademark of the\nPareto distribution. Figure 4(d) shows a histogram of the\nmaximum time intervals measured in one million trials.\nEach trial iterated 1000 time steps for the time evolution\nof Eq. (2). The histogram was normalized so that the\ndistribution summation was 1. The value of R-squared,\nthe coefficient of determination, was greater than 0.99 in\nthis case. Notably, the obtained histogram followed the\nFrechet distribution, which is consistent with EVT.\nThe Figs. 4(e)-(h) shows the results for σ= 7.5. Fig-\nure 4 (e) displays an example snapshot of the spike sig-\nnals. Figures 4 (f) and (g) shows the distribution of time\nintervals on normal and logarithmic scales, respectively.\nUnlike the case of σ= 20, a drastically delayed case of\nd > 430 ms for σ= 7.5 was observed. Even in this\nextreme case, a Pareto distribution fitted the measured\ndistribution of time intervals while maintaining its char-\nacteristic high peak and long tails. Figure 4(h) shows\na histogram of the maximum time intervals measured in\none million trials. Each trial iterated 1000 time steps for\nthe time evolution of Eq. (2). The histogram was nor-\nmalized so that the distribution summation was 1. No-\ntably, the obtained histogram followed the Frechet distri-\nbution, which is consistent with the theory. Accordingly,\nwe reported that in synaptic problems, the histogram of\nmaximum time intervals followed the Fr´ echet distribution\nwhen the time intervals followed a Pareto distribution.\nIV. CONCLUSION\nThis study described the nerve transmission process,\nwhich is a complex physiological reaction process, as a\nstochastic process. The study showed that the distri-\nbution of the maximum time interval of spike signals\nfollowed an extreme order statistics. By introducing the\nstatistical variance in the time constant of the Leaky\nIntegrate-and-Fire model, which is a deterministic time\nevolution model of spike signals, we allowed randomness\nto emerge in the time interval of spike signals. We\nconfirmed that the time interval of the spike signal\nfollowed an exponential distribution if the time constant\nfollowed an exponential distribution function. In this\ncase, our theory and simulations confirmed that the\nhistogram of the maximum time interval followed the\nGumbel distribution, which is one of the three types of\nextreme value statistics. In addition, we confirmed that\nthe histogram of the maximum time interval follows\na Fr´ echet distribution when the time interval of the\nspike signal followed a Pareto distribution. Our results8\nFIG. 3. Simulation results when ξ(µ, σ) follows an exponential distribution: (a) distribution of ξ(µ, σ), (b) distribution of the\ntime interval, (c) example snapshot of spike signals, and (d) obtained Gumbel distribution.\nshowed that neurotransmission delays can be described\nby extreme value statistics. In this study, we used the\nclassical LIFM model. Although several improved LIFM\nmodels have been presented previously, however, as\nlong as the elapsed time of spike signal transmission\nfollows a Poisson process, the results from the improved\nmodels with excellent waveform reproducibility follow\nthe same patterns as that presented in this study. We\nhave successfully demonstrated that the distribution of\nextreme values is a new indicator of nerve transmissionproblems.\nACKNOWLEDGMENT\nThe author would like to thank Editage\n(www.editage.jp) for the English language editing.\nDATA AVAILABILITY STATEMENT\nNo new data were created or analysed in this study.\n[1] A. E. Pereda, Electrical synapses and their functional in-\nteractions with chemical synapses, Nature Reviews Neu-\nroscience 15, 250 (2014).\n[2] C. Henley, Foundations of neuroscience ([sn], 2021).\n[3] Z. Xiang, C. Tang, C. Chang, and G. Liu, A new view-\npoint and model of neural signal generation and trans-\nmission: Signal transmission on unmyelinated neurons,Nano Research 14, 590 (2021).\n[4] B. Katz and R. Miledi, The measurement of synaptic de-\nlay, and the time course of acetylcholine release at the\nneuromuscular junction, Proceedings of the Royal So-\nciety of London. Series B. Biological Sciences 161, 483\n(1965).9\nFIG. 4. 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Schouder,1, 3and Henrik Stapelfeldt1,b)\n1)Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C,\nDenmark\n2)Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,\nDenmark\n3)LIDYL, CNRS, CEA, Universit´ e Paris-Saclay, 91191 Gif-sur-Yvette, France\n(Dated: 2 February 2024)\nAlkali trimers, Ak3, located on the surface of He nanodroplets are triply ionized following multiphoton\nabsorption from an intense femtosecond laser pulse leading to fragmentation into three correlated Ak+ions.\nCombining the information from three-fold covariance analysis of the emission direction of the fragment\nions and from their kinetic energy distributions P(Ekin), we find that Na3,K3, and Rb3have an equilateral\ntriangular structure, corresponding to that of the lowest-lying quartet state4A′\n2, and determine the equilibrium\nbond distance Req(Na3) = 4.65 ±0.15˚A,Req(K3) = 5.03 ±0.18˚A, and Req(Rb3) = 5.45 ±0.22˚A. For K3and\nRb3these values agree well with existing theoretical calculations, while for Na3the value is 0.2–0.3 ˚A larger\nthan the existing theoretical results. The discrepancy is ascribed to a minor internuclear motion of Na3\nduring the ionization process. Also, we determine the distribution of internuclear distances P(R) under the\nassumption of fixed bond angles. The results are compared to the square of the internuclear wave function\n|Ψ(R)|2.\nWhen molecules are multiply ionized, nowadays almost\nalways achieved by exposure to an intense femtosecond\nlaser pulse, they typically break apart into positively\ncharged fragment ions. This process, termed Coulomb\nexplosion1,2, provides information about the structure of\nthe molecule at the instant the laser pulse arrives because\nthe molecular structure is imprinted on the emission di-\nrections and the kinetic energies of the ion fragments, i.e.,\non accessible experimental observables. As such, laser-\ninduced Coulomb explosion has been applied in a large\nnumber of works to explore both the static structure of\nmolecules in stationary states and the time-dependent\nstructure of molecules undergoing intramolecular motions\ninitiated by a femtosecond pump pulse. Examples include\nidentification of structural isomers3–5, determination of\nthe absolute configuration of chiral molecules6,7, proton\nmigration8, torsion9,10, dissociation11–18and roaming19.\nThe structure determination of a parent molecule from\nthe measured momentum vectors of the fragment ions\nrelies on the assumption that their interaction is given\nby Coulomb repulsion. While this assumption is suffi-\nciently good for some of the examples listed above, non-\nCoulombic effects set a limitation on the accuracy of\nreconstructed structures. For instance, in the case of\nCoulomb explosion of an atomic dimer, such as Ar2, non-\nCoulombic effects cause the one-to-one correspondence\nbetween the initial internuclear distance and the kinetic\nenergy of the fragment ions, valid in the Coulomb approx-\nimation, to disappear20unless all valence electrons are\nstripped. This was typically not the case for the many past\nstudies employing intense visible or near-infrared laser\na)These authors contributed equally to the work.\nb)henriks@chem.au.dkpulses although recent results indicate that ultrashort\nx-ray pulses from free-electron lasers are better suited\nfor this purpose21. Here we study trimers of alkali-metal\natoms in which case only three, relatively loosely bound\nelectrons need to be removed to empty the valence shell.\nThis appears as a favorable situation for accurate struc-\nture determination via laser-induced Coulomb explosion.\nThe objective of our work is to explore if this is indeed\nthe case.\nPrevious studies have shown that alkali trimers can be\nformed at the surface of helium nanodroplets22–24. Em-\nploying vibrationally-resolved absorption spectroscopy of\nelectronically excited states, it was found that the trimers\nare mainly, if not exclusively, created in the lowest-lying\nquartet state,4A′\n2. According to theory25,26, the trimers\nin this state have an equilateral triangular structure with\nD3hsymmetry [see Fig. 1(a) for Na3]. The spectroscopic\nstudies also showed that the trimers are populated only in\nthe vibrational ground state, indicating that the trimers\nequilibrate to the 0.4 K temperature of the droplets27.\nThis finding is consistent with results from magnetic cir-\ncular dicroism experiments28,29. Despite the many ex-\nperimental investigations, a direct determination of the\ninternuclear distances of the alkali trimers has, to our\nknowledge, not been achieved. In the present work, we\ndemonstrate that Coulomb explosion, triggered by an\nintense femtosecond pulse, identifies that Na3,K3, and\nRb3are formed in the4A′\n2state and allows determination\nof their equilibrium bond lengths. As such the studies\nhere are related to previous work on Coulomb explosion\nimaging of trimers composed of noble gas atoms30–33.\nOur method is based on triple ionization of al-\nkali trimers via multiphoton absorption as illustrated\nin Fig. 1(c) for Na3in the4A′\n2state. As a result of the\nionization, the vibrational wave function of Na3is pro-arXiv:2402.00587v1  [physics.atm-clus]  1 Feb 20242\njected onto the repulsive potential curve of Na3+\n3. There\nis only one such curve because Na3+\n3has a closed-shell\nelectron structure and this curve is well approximated by\nthe Coulomb energy available between the three charges,\nVcoul(R) = 3/ Rin atomic units, where Ris the inter-\nnuclear distance in the equilateral triangular structure.\nUpon release of this potential energy, the trimer will\nfragment into three Na+ions, each receiving a kinetic\nenergy,\nEkin=VCoul(R)/3 = 14 .4 eV/R[˚A]. (1)\nWe measured EkinforNa3,K3, and Rb3and in each case\nobserved a peak close to the value expected for Coulomb\nexplosion of the trimers in the4A′\n2state. This interpre-\ntation is supported by the relative emission directions of\nthe fragment ions, localized at 120◦and 240◦, i.e., consis-\ntent with the equilateral triangular structure of trimers\nin the4A′\n2state. Furthermore, via Eq. 1 we determine\nthe distribution of internuclear separations P(R) by a\ntransformation of probabilities, using a standard Jaco-\nbian transformation.\nThe experimental setup has been described previ-\nously35,36. Thus, no elaborate details are given here.\nTo create a continuous beam of helium droplets, a pre-\ncooled gas of high purity (5.0) helium is expanded into\na vacuum chamber at a stagnation pressure of 25 bars\nthrough a 5- µm-diameter nozzle. The nozzle is cooled to\nTnozzle = 10.5–15 K, thereby producing droplets with a\nmean size of ∼5000–18000 He atoms37. After formation,\nthe droplets pass through a pickup cell containing a gas of\nalkali atoms, see Fig. 2. The vapor pressure in the pickup\ncell is adjusted such that some of the helium droplets\npick up three Akatoms. The three atoms can then form\na trimer on the surface of the droplet25,38,39. Hereafter,\nthe doped droplets enter a velocity map imaging (VMI)\nspectrometer inside which a pulsed ( τFWHM ≈50 fs) lin-\nearly polarized, focused laser beam intersects the droplet\nbeam. The intensity and central wavelength of the probe\npulses are specified in Fig. 4. The VMI spectrometer\nprojects the Ak+ions created by the laser pulses onto a\n2D imaging detector monitored by a CCD camera. The\ndetector is gated such that it only detects ions with a\nspecific mass-to-charge ratio (23Na+,39K+,85Rb+, and\n133Cs+). The camera records ion hits from 10 laser shots\nin each frame.\nFigure 3(a1)–(d1) presents the 2D velocity images\nrecorded for various alkali ions. Each image consists\nof tens of thousands of frames stacked into one final im-\nage. Three radial stripes and a center with no signal\nis present in all images. The lacking signal is due to\na metal disk mounted on support rods in front of the\ndetector20,40. The metal disk screens the detector from\nunwanted low-speed Ak+ions created during ionization\nof droplets doped with only one Akatom or ionization of\nisolated effusive Akatoms present inside the VMI spec-\ntrometer. To improve the visual contrast of the outer\nfeatures of the images, the centers in Fig. 3(a1), (b1), and\n(d1) have been cut digitally to remove the remaining few\n4A'21\n2 31\n2 3r = 3.31 Å\nr = 3.31 Å\nR = 4.33 Å  R = 4.33 Å \nR' = 3.89 Å  R = 4.33 Å\n 60° 60°60°80°\n50° 50°\nθ2,3 = 120°θ\n1,2\n= 120°\nθ1,3 = 120°\nθ1,3 = 121°θ\n1,2 = 121°\nθ2,3 = 118°(a) (b)\n(c)Na34A'2 Na32B2FIG. 1. (a),(b) Sketch of the structure of Na3in the4A′\n2\nstate25(a) and in the2B2state34(b). The arrows indicate\nthe emission angles of the Na+fragments upon Coulomb\nexplosion induced by triple ionization. The Na+ion fragments\nare labelled 1, 2 and 3 for reference, and θindicates the angles\nbetween the emitted fragments. (c) Potential energy curve for\nthe4A′\n2state of Na325extrapolated to 15 ˚A and the Coulomb\npotential curve representing Na3+\n3. The red curve displays\nthe square of the wave function of the vibrational ground\nstate in the4A′\n2potential. The vertical black arrows illustrate\nthe multiphoton triple ionization process leading to Coulomb\nexplosion.\nAk+ions that passed by the edge of the metal disk. Out-\nside the screened center, three radially separated channels\ncan be observed in the images. The innermost channel,\nlying between the annotated solid and dashed circles, has\npreviously been identified as Ak+fragments from double\nionization of Ak2in the 13Σ+\nustate, while the channel\nbetween the dashed and dot-dashed circles were identified\nasAk+fragments from double ionization of Ak2in the\n11Σ+\ngstate35,36. In this work, we focus on the last channel\noutside the dot-dashed rings.\nTo investigate the Ak+ions in the outermost channel,\nwe calculate the three-fold covariance41image for the Ak+\nions located outside the dot-dashed rings in a 2D velocity\nion image. Three-fold covariance has previously been used\nin gas-phase experiments to identify correlations between\nthree ionic fragments42–44. A useful representation is\nachieved by transforming the coordinate system of the\nmeasurements into the recoil frame of one of the Ak+\nfragments. This Ak+fragment is designated the ’reference3\nPickup cell (Ak) He droplet\nbeam Skimmer\nProbe\npulse2D imaging\ndetector\nAk+ Ak+ Ak+\nx\nyz\nE\nFIG. 2. Sketch of the core parts of the experiment. Alkali\ntrimers Ak3are formed on helium droplets after capture of\ngas-phase alkali atoms in a pickup cell. The trimers are then\ntriply ionized and the Ak+fragment ions recorded with a 2D\nimaging detector.\nion’ and the covariance between the reference ion and the\nother identical Ak+fragments in the outermost channel\nwithin the same camera frame is calculated, forming a\ncovariance image in the recoil frame of the reference ion.\nThis is done for all ion events in the channel. Finally, all\nthe covariance images are rotated such that the reference\nions lie along one common vector and summed together.\nThe result is a recoil frame covariance image where the\ncovariances with two partner ions are plotted relative to\nthe reference ions, which now all lie along one direction\n(vertical). The covariance images obtained for Na3,K3,\nRb3, and Cs3are displayed in Fig. 3(a2)–(d2).\nEach covariance image shows two areas with a signifi-\ncantly enhanced covariance. To gain further insight, we\nintegrate the covariance images along the radial axis and\nobtain the angular distribution P(θ) of the covariance,\nsee Fig. 3(a3)–(d3). We then determine the central posi-\ntions of the two major peaks in P(θ), which correspond to\nthe two areas in the covariance images. The peak positions\n[see annotations on Fig. 3(a3)–(d3)] show that the emis-\nsion direction of the reference Ak+ion is correlated with\ntwo departing Ak+ions at 120◦and 240◦, correspond-\ning to an emission angle of 120◦between neighbouring\nfragments. This is in agreement with the equilateral struc-\nture of the4A′\n2state trimers, as illustrated for Na3on\nfigure Fig. 1(a). A classical trajectory calculation reveals,\nhowever, that Coulomb explosion via triple ionization\nofAk3in the ground state2B2, an obtuse triangular\nstructure with C2vsymmetry, will eject Ak+ions with\nemission angles very similar to those observed for the\n4A′\n2state. In the case of Na3[see Fig. 1(b)] and Cs3,\nthe emission angle between neighbouring ion fragments\nwill be 121◦and 118◦, while the angles will be 121◦and\n117◦forK3andRb3. We are unable to resolve such small\ndifferences in the emission angles, due to the 9◦FWHM of\nthe peaks in Fig. 3(b3) and the 11◦FWHM of the peaks in\nFig. 3(a3), (c3), and (d3). Thus, the three-fold covariance\nimages are consistent with Ak3in either the2B2or the\n4A′\n2state, but they cannot by themselves distinguish the\ntwo states.\nTo obtain more information about the observed tri-\nangular structures, we determine the kinetic energy dis-tributions P(Ekin) of the Ak+ions. For this task we\nemploy an Abel inversion algorithm45to retrieve the ra-\ndial velocity distributions from the ion images. These\ndistributions are then converted to P(Ekin) by applying\nan energy calibration for the VMI spectrometer and then\nfinally a standard Jacobian transformation20. Figure 4\ndisplays P(Ekin) for all Ak+. For Na+,K+, and Rb+\nthe distribution contains three significant peaks, of which\ntwo can be assigned as Ak+ions produced from Coulomb\nexplosion of Ak2in either the 11Σ+\ngor the 13Σ+\nustate,\nas demonstrated previously35,36. These peaks are labeled\ncorrespondingly in Fig. 4. Here our interest is the peaks\nat higher energies, which were not identified before.\nForNa+, see Fig. 4(a), the unidentified peak is cen-\ntered at 3.10 eV (determined by a Gaussian fit). Coulomb\nexplosion of Na3in the4A′\n2state at the equilibrium dis-\ntance, Req= 4.41 ˚A25would produce three Na+fragments\nwithEkin= 3.3 eV. In comparison, Coulomb explosion of\nNa3in the2B2state, would result in one Na+ion with\nEkin= 4.3 eV [fragment 1, see Fig. 1(b)] and two Na+\nions with Ekin= 4.0 eV (fragment 2 and 3). Therefore, we\ninterpret the peak at 3.10 eV as ions formed by Coulomb\nexplosion of Na3in the4A′\n2state. Similarly, the center of\nthe high-energy peaks, marked by vertical dashed lines,\nfor the K+(2.86 eV) and Rb+(2.64 eV) ions in Fig. 4(b)\nand (c) match well with the kinetic energy each fragment\nion receives upon Coulomb explosion of trimers in the\n4A′\n2state and, importantly, do not match the energy of\n3.5 and 3.2 eV (3.2 and 3.0 eV) fragment ions would get\nfrom Coulomb explosion of K3(Rb3) in the2B2state. As\nsuch, we also conclude that K3andRb3are produced in\nthe4A′\n2state46.\nFrom the central position of the peaks in P(Ekin), we\ndetermine the equilibrium internuclear distance Reqfor\nthe trimers in the identified4A′\n2state via Eq. 1. Ta-\nble I provides a summary of the identified peaks and the\ncorresponding equilibrium bond distances, alongside theo-\nretical values for Reqpreviously reported in the literature.\nThe uncertainty on the peak positions in P(Ekin) results\nfrom a combination of the energy resolution of the VMI\nspectrometer, the slightly asymmetric shape of the peaks\nand, for K3andRb3, the presence of different isotopo-\nlogues47. For K3andRb3the determined Reqis within\n0.1˚A of the theoretical values, while for Na3the mea-\nsured Reqis 0.2–0.3 ˚A larger than the theoretical values.\nThe reason for this discrepancy is, we believe, a minor\ninternuclear motion on potential curves in Na3,Na+\n3or\ninNa2+\n3transiently excited during the laser pulse. Its 50\nfs duration is long enough that such internuclear motion\ncan occur at least for Na3, which is the lightest of the\ntrimers. A similar shift of the equilibrium internuclear\ndistance was also observed for Coulomb explosion studies\nof Na2on helium droplets whereas Reqmeasured for K2\nand Rb2were very close to the theoretical values36.\nWe also conducted the experiment with cesium. In\nthis case, an unidentified broad structure composed of\ntwo individual peaks is observed in P(Ekin) with peaks\nat 2.36 eV and 2.57 eV (determined by Gaussian fits),4\nFIG. 3. (a1)–(d1) 2D velocity images of Ak+ions. The annotated white rings mark the regions pertaining to the alkali dimers\nand to the alkali trimers, the latter being the ions outside the dot-dashed ring, see the text. The annotated white arrows in the\nbottom right corners of each panel show the polarization axis of the laser pulses. For the data in (a1) and (c1) TNozzle was 11 K,\nwhile for (b1) and (d1) it was 12 K. (a2)–(d2) Three-fold covariance images for the ions from Ak3in 2D velocity map images.\nFor improved visual contrast, negative values have been set to zero. (a3)–(d3) The angular distribution P(θ) for the covariance\nimages in (a2)–(d2). The central positions of the two peaks are annotated next to the peaks. The FWHM of the peaks are 9◦\nfor b3) and 11◦for a3), c3), and d3).\nsee Fig. 4(d). Three-fold covariance analysis of the asso-\nciated ions in the VMI image, see Fig. 3(d1)–(d3), shows\nthat these ions originate from Cs3. To understand the\norigin of the two peaks, we varied the mean droplet size\nby adjusting the nozzle temperature to Tnozzle = 10.5 K\nand 15 K. The measured kinetic energy distributions are\ndepicted in the inset in Fig. 4(d), with the distribution\nforTnozzle = 15 K scaled by a factor of 10. We observed\nthat the peak at 2.36 eV was dominating for the 15 K\nmeasurement, which indicate that those ions should come\nfrom Coulomb exploded Cs3in the high-spin state4A′\n2, as\nthe high-spin trimers are better preserved than low-spin\ntrimers on helium droplets, especially on small ones. The\nreleased energy of 2.36 eV is, however, a little lower than\nthe expected kinetic energy release Ekin= 2.59 eV for\nCoulomb explosion of Cs3in the4A′\n2state starting from\nthe equilibrium distance Req= 5.56 ˚A48. In fact, the ex-\npected energy release matches the other peak observed at\n2.57 eV. For the 10.5 K measurement we observe that the\npeak at 2.57 eV begin to dominate, which is unexpectedsince the peak containing ions from Coulomb explosion\nof the high-spin state, whichever of the two peaks that is,\nshould dominate at all droplet sizes. Finally, Coulomb\nexplosion of Cs3at the equilibrium distance of the2B2\nstate51would produce Cs+fragments with energies of 3.0\nand 2.8 eV, which does not match the observed peaks.\nThus, the observations do not currently enable us to\nunambiguously determine the quantum states for Cs3.\nInspired by recent results on laser-induced Coulomb\nexplosion of alkali dimers36, we now discuss the possi-\nbility of determining the distribution of internuclear dis-\ntances P(R) from P(Ekin). In the dimer case, Ris the\nonly internuclear (vibrational) coordinate. Therefore, it\nis straightforward to determine P(R) from P(Ekin) via\nEkin= 7.2eV/R[˚A], the equivalent of Eq. 1, using the\nappropriate transformation of probability distributions.\nIn the trimer case, the situation is more complex because\nthere are three vibrational degrees of freedom, i.e., the\npairwise distances between the three nuclei, R1,R2, and\nR3are not necessarily the same. As such, the internuclear5\n0.00.51.0\n0.00.51.0\n0.00.51.0\n2.0 2.5 3.000.8\n0.5 1.0 1.5 2.0 2.5 3.0 3.50.00.51.0(a)\n(b)\n(c)\n(d)(e)Na+\n:400nm\nI:1.5x1013W/cm2\nTnozzle:11K\nK+\n:400nm\nI:3x1014W/cm2\nTnozzle:12K\nRb+\n:800nm\nI:3x1013W/cm2\nTnozzle:11K\nCs+\n:1300nm\nI:1x1013W/cm2\nTnozzle:12KTnozzle:10.5KTnozzle:15K3+\nu\n1+\ng\n3+\nu\n1+\ng\n3+\nu\n1+\ng\n3+\nu\n1+\ng\n4A/prime\n2\n4A/prime\n2\n4A/prime\n23.10 eV\n2.86 eV\n2.64 eV\n2.36 eV\n2.57 eV\nEkin (eV)P(Ekin) (arb. units)\nFIG. 4. (a)–(d) Kinetic energy spectra for Na+,K+,\nRb+, and Cs+ions extracted from the 2D velocity images\nshown in Fig. 3(a1)–(d1). (e) Close-up view of the Cs+\nP(Ekin) peak structure associated with trimers, measured\nwithTNozzle = 10.5 K (black dotted line) and TNozzle = 15 K\n(red dot-dashed line, scaled by a factor of 10). The vertical\ndashed lines mark the center of the peaks ascribed to trimers.\n4 5 60 0.5 1.0\n4564567\nR (Å)P(R) (arb. units)Na3K3Rb3\nFIG. 5. P(R) distributions for Na3,K3, and Rb3in the4A′\n2\nstate, determined by transformation of the measured P(Ekin)\nand a flat background subtraction. The red lines show the\ntheoretical |Ψ(R)|2distributions for the isolated4A′\n2state.TABLE I. Central position of the peak in P(Ekin) pertaining\nto Coulomb explosion of Ak3in the4A′\n2state. Theoretical\nvalues from the literature are provided for Req.\nP(Ekin) peak (eV) Req(˚A)\nNa3 3.10±0.10 4.65 ±0.15\nNa3Theory – 4.33a, 4.41b\nK3 2.86±0.10 5.03 ±0.18\nK3Theory – 4.95a, 5.05c\nRb3 2.64±0.10 5.45 ±0.22\nRb3Theory – 5.35a, 5.50c, 5.31d, 5.45e\naFrom Ref. 48.\nbFrom Ref. 25.\ncFrom Ref. 26.\ndFrom Ref. 49.\neFrom Ref. 50.\nwave function depends on three variables and can, for\ninstance, be expressed as Ψ( R1, R2, R3). Measurement\nof the kinetic energy of the Ak+fragment ions does not\nimmediately enable determination of |Ψ(R1, R2, R3)|2as\na function of three variables. Therefore, we make the sim-\nplifying assumption that the trimer retains its equilateral\ntriangular shape, i.e., that it is only the symmetric-stretch\nvibration that is important for the zero-point vibrations.\nThis assumption, which was adopted in previous work\non laser-induced Coulomb explosion of neon and argon\ntrimers30, means that the wave function can be written as\nΨ(R). Thus, we can apply the procedure from the dimer\ncase to determine P(R) using Eq. 1 to connect Ekinand\nR.\nThe results for Na3,K3, and Rb3are depicted in Fig. 5\ntogether with |Ψ(R)|2calculated by solving the stationary\nvibrational one-dimensional Schr¨ odinger equation for iso-\nlated Na3,K3, and Rb3in the4A′\n2state using potentials\nfrom the literature25,49,52. The FWHM of the theoretical\n|Ψ(R)|2are 0.3 ˚A for Na3, 0.3 ˚A for K3, and 0.2 ˚A for\nRb3and the FWHM of the measured P(R) are about 2–3\ntimes larger. This is very similar to that observed for the\nalkali dimers and we believe the factors causing the larger\nwidths of P(R) compared to those of |Ψ(R)|2are the\nsame as for the dimer case, i.e., in particular the energy\nresolution of the VMI spectrometer and the distortion\nof the trajectories of the recoiling Ak+ions due to their\ninteraction with the droplet surface36.\nIn summary, we showed that Coulomb explosion of\nalkali trimers, triggered by laser-induced triple ionization,\nproduces three Ak+fragment ions that recoil with a pair-\nwise angle of 120◦. While this observation is consistent\nwith trimers in both the2B2and the4A′\n2state, the ki-\nnetic energy of the fragment ions unambiguously show\nthat Na3,K3, and Rb3are populated in the4A′\n2state\nwhere the structure is that of an equilateral triangle. The\nquantum-state-sensitivity through measurement of Ekin\nof the fragment ions is similar to that demonstrated for6\nalkali dimers where the distinction is between the 13Σ+\nu\nstate and the 11Σ+\ngstate35,53. Furthermore, the measured\nEkinof the Ak+ions enabled the first experimental deter-\nmination of the equilibrium bond distance for Na3,K3,\nandRb3in the4A′\n2state. We expect that the accuracy\nof the bond distances can be improved by employing a\nVMI spectrometer with a higher energy resolution and,\nforK3andRb3, by selecting individual isotopologues\nthrough coincident filtering of the recoil ions53. One\nfuture application of the results presented here, is to ex-\ncite vibrational wave packets in the alkali trimers with a\nfemtosecond pump pulse and follow their evolution with\ntimed Coulomb explosion. For instance, it should be\npossible to excite the symmetric-stretch vibration in the\n4A′\n2state by stimulated Raman transitions54–56. Tracking\nthe internuclear distance with timed Coulomb explosion\ncould be an informative way to explore if the vibration\nstays localized in the initial mode or if it couples to other\nmodes through intramolecular vibrational redistribution.\nACKNOWLEDGMENTS\nWe acknowledge valuable discussions with James Picker-\ning. We also thank Jan Thøgersen for carefully maintain-\ning the laser system. H.S. acknowledges support from The\nVillum Foundation through Villum Investigator Grant No.\n25886.\n1T. Yatsuhashi and N. Nakashima, J. Photochem. Photobiol. C\n34, 52 (2018).\n2C. A. Schouder, A. S. Chatterley, J. D. Pickering, and\nH. Stapelfeldt, Annu. Rev. Phys. Chem. 73, 323 (2022).\n3U. Ablikim, C. Bomme, H. Xiong, E. Savelyev, R. Obaid,\nB. Kaderiya, S. Augustin, K. Schnorr, I. Dumitriu, T. Os-\nipov, R. Bilodeau, D. Kilcoyne, V. Kumarappan, A. Rudenko,\nN. Berrah, and D. Rolles, Sci. Rep. 6, srep38202 (2016).\n4M. Burt, K. Amini, J. W. L. Lee, L. Christiansen, R. R. Johansen,\nY. 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Phys.\nLett. 683, 234 (2017)."    },    {        "title": "2402.00641v1.Testing_side_channel_security_of_cryptographic_implementations_against_future_microarchitectures.pdf",        "content": "Testing side-channel security of cryptographic implementations\nagainst future microarchitectures\nGilles Barthe∗\nMax Planck Institute\nfor Security and Privacy (MPI-SP)\nBochum, Germany\ngilles.barthe@mpi-sp.orgMarcel Böhme\nMax Planck Institute\nfor Security and Privacy (MPI-SP)\nBochum, Germany\nmarcel.boehme@mpi-sp.orgSunjay Cauligi\nMax Planck Institute\nfor Security and Privacy (MPI-SP)\nBochum, Germany\nsunjay.cauligi@mpi-sp.org\nChitchanok Chuengsatiansup\nThe University of Melbourne\nMelbourne, Australia\nc.chuengsatiansup@unimelb.edu.auDaniel Genkin\nGeorgiaTech\nAtlanta, United States\ngenkin@gatech.eduMarco Guarnieri\nIMDEA Software Institute\nMadrid, Spain\nmarco.guarnieri@imdea.org\nDavid Mateos Romero\nIMDEA Software Institute\nMadrid, Spain\ndavid.mateos.romero@gmail.comPeter Schwabe†\nMax Planck Institute\nfor Security and Privacy (MPI-SP)\nBochum, Germany\npeter@cryptojedi.orgDavid Wu\nUniversity of Adelaide\nAdelaide, Australia\ndavid.wu@adelaide.edu.au\nYuval Yarom\nRuhr University Bochum\nBochum, Germany\nyuval.yarom@rub.de\nABSTRACT\nHow will future microarchitectures impact the security of exist-\ning cryptographic implementations? As we cannot keep reducing\nthe size of transistors, chip vendors have started developing new\nmicroarchitectural optimizations to speed up computation. A re-\ncent study (Sanchez Vicarte et al., ISCA 2021) suggests that these\noptimizations might open the Pandora’s box of microarchitectural\nattacks. However, there is little guidance on how to evaluate the\nsecurity impact of future optimization proposals.\nTo help chip vendors explore the impact of microarchitectural\noptimizations on cryptographic implementations, we develop (i) an\nexpressive domain-specific language, called LmSpec , that allows\nthem to specify the leakage model for the given optimization and\n(ii) a testing framework, called LmTest , to automatically detect leaks\nunder the specified leakage model within the given implementation.\nUsing this framework, we conduct an empirical study of 18 proposed\nmicroarchitectural optimizations on 25 implementations of eight\ncryptographic primitives in five popular libraries. We find that every\nimplementation would contain secret-dependent leaks, sometimes\nsufficient to recover a victim’s secret key, if these optimizations were\nrealized. Ironically, some leaks are possible only because of coding id-\nioms used to prevent leaks under the standard constant-time model.\n∗Also with IMDEA Software Institute.\n†Also with Radboud University.1 INTRODUCTION\nAs reducing the size of transistors is increasingly more challenging\ndue to physical limits, chip vendors are looking into microarchitec-\ntural optimizations as an alternative to further speed up computa-\ntions. These optimizations exploit the spatial and temporal locality\nexhibited by programs to predict future behavior and perform some\nof the computation in advance. Thus, the processor can cache or\nprefetch data in anticipation of a near-future use, or it can predict\nfuture instruction flow and even computation results in an effort to\nbetter exploit the processor’s inherent parallelism.\nHowever, a recent study [ 60] suggests that this surge of new\nmicroarchitectures opens the Pandora’s box of microarchitectural\nattacks [23], which exploit the side effects of microarchitectural\noptimizations on a program’s execution time to compromise the\nconfidentiality of an otherwise secure computation. Although the\nimpact of existing optimizations, like caching [ 65,79] or specu-\nlative execution [ 35], is well understood and captured by secure\nprogramming guidelines [ 5,13], there is little guidance on how to\nevaluate the security impact of possible future microarchitectures.\nAs a first step in this direction, Sanchez Vicarte et al . [60] per-\nformed a systematic review of recent optimizations proposed by\nthe computer architecture community and suggested that many\nof those would likely leak secret information in unexpected ways.\nHowever, the authors left open how exactly these optimizations\nwould impact existing cryptographic implementations and whether\nexisting mitigations, like leakage-resistent coding idioms [ 9,10],\nwould help in preventing leaks.\nTo substantiate this prognosis, in this paper we develop a frame-\nwork for assessing the side-channel guarantees of programs againstarXiv:2402.00641v1  [cs.CR]  1 Feb 2024G. Barthe, M. Böhme, S. Cauligi, C. Chuengsatiansup, D. Genkin, M. Guarnieri, D. Mateos Romero, P. Schwabe, D. Wu, Y. Yarom\n(future) microarchitectural optimizations. Our framework consists\nof (1) LmSpec , an expressive domain-specific language that allows\nchip vendors and software developers to specify the leakage model\nassociated with a given optimization in terms of leakage traces, and\n(2)LmTest , a testing approach that generates random inputs and au-\ntomatically detects secret-dependent leaks for the specified LmSpec\nleakage models within a given cryptographic implementation. We\nuse this framework to (3) perform a large scale study of the impact\nof the optimization proposals studied in [ 60] on the side-channel\nguarantees of mainstream cryptographic implementations. Next,\nwe overview these contributions in more detail.\nLmSpec language: We develop the LmSpec language for speci-\nfying leakage models at the ISA-level. Following the formalism of\nGuarnieri et al. [ 27], we define an LmSpec leakage model as (a) a\nleakage clause (§3), which specifies what observations are leaked\nduring the execution of a program, and (b) a prediction clause (§4),\nwhich specifies the prediction mechanisms supported by the mi-\ncroarchitecture and what their effects are. Hence, LmSpec models\nmap each program execution to a leakage trace capturing the leaked\ninformation. Importantly, models specified in LmSpec areexecutable\nfor the x86 ISA (§5.1). That is, given an initial program state, we\ncan execute an arbitrary x86 binary and derive the leakage trace\ngenerated by the LmSpec model. To achieve this, LmSpec models\nare automatically translated into handlers for the Unicorn CPU em-\nulator [ 1] capturing the relevant information to generate leakage\ntraces for a program execution.\nLmTest testing tool: We develop the LmTest testing framework\nfor detecting leaks in x86 programs with respect to a given Lm-\nSpec leakage model (§5). LmTest takes as input the binary of the\nprogram under test, an entry point in the program together with\nlabels for each program input (indicating whether the parameter is\npublic or secret), and the LmSpec model. To detect leaks, LmTest\nadopts a relational testing approach [ 50]: (1) it randomly generates\ntest cases consisting of pairs of initial program states that are low-\nequivalent , i.e., that differ only in the value of secret inputs, (2) it\nthen executes the program on both inputs in a test case to derive\nthe leakage traces according to the LmSpec model, and (3) it finally\nchecks for differences in the two leakage traces in a test case. A\ntest case demonstrating different leakage traces indicates a leak of\nsecret information. We remark that differently from other random\nrelational testing approaches [ 28,50], which target fixed leakage\nmodels, LmTest is parametric in the LmSpec leakage model, thereby\nallowing one to study the security implications of any optimization\nproposal whose leakage profile can be represented in LmSpec .\nCase study: To evaluate our framework and to assess the risks\nthat proposed optimizations pose to current software, we report on\na large-scale empirical study of the impact of microarchitectural op-\ntimizations on popular cryptographic libraries (§6). We consider 18\noptimizations, under six different execution models, and 25 crypto-\ngraphic implementations selected from five popular cryptographic\nlibraries, including libsodium and rust-crypto. We find that every\nimplementation would contain secret-dependent leaks if these op-\ntimizations were realized. In some cases, an optimization-induced\nleak would be sufficient to recover a victim’s secret key directly\nfrom the leakage trace (e.g., for the X25519 implementation in lib-\nsodium [ 40]). Ironically, some leaks are possible only because ofcoding idioms, such as constant-time swap or bit-masking, used to\nprevent leaks under the standard constant-time model.\nSummary of contributions: In summary, the paper makes the\nfollowing contributions: (1) the LmSpec language for rapidly pro-\ntotyping leakage models (§3–4), (2) the LmTest testing tool for\ndetecting secret-dependent leaks in programs against an arbitrary\nLmSpec model (§5), and (3) a large-scale case study analyzing the\nside-channel guarantees of mainstream cryptographic implementa-\ntions against recent microarchitectural proposals (§6).\nArtifacts: The implementation of LmTest andLmSpec , together\nwith all leakage and prediction clauses from our case study as\nwell as scripts to reproduce all our results are available at https:\n//github.com/hw-sw-contracts/leakage-model-testing.\n2 OVERVIEW\nHere, we illustrate the key components of our approach. We start\nby illustrating how leakage (§2.1) and speculation (§2.2) can be\nmodeled using the LmSpec language. Next, we show how LmSpec ’s\nexecutable semantics can be used to derive leakage traces (§2.3).\nWe then overview our notion of side-channel security (§2.4), and\nwe conclude by illustrating how the LmTest testing tool can be\nused to detect leaks in programs given an LmSpec model (§2.5).\n2.1 Modeling leaks with LmSpec\nInLmSpec , leaks are modeled by specifying leakage clauses (de-\nscribed in §3), which describe what an attacker might observe\nthrough side-channels during program execution. As an example,\nFigure 1 depicts the leakage clause formalizing the constant-time\nmodel in LmSpec . We start by defining and naming the clause with\nthedefleakage statement (line 1). Next, we model the three usual\nrequirements on constant-time programs as handlers (lines 2–7).\nEach handler consists of a guard, which specifies the operation\nbeing handled and the names of its arguments, and a body that\ncomputes leakage observations.\nThe first handler deals with memory load operations (lines 2–\n3); it assigns the target address to a new variable called addr and\nthe target size (in bytes) to a new variable called sz. The body of\nthe handler is the expression #(\"load\" addr) ; it produces a tuple\nconsisting of the string “load” and the target address passed to the\nhandler. The second handler deals with memory store operations\n(lines 4–5) and is defined similarly.\nThe last handler (lines 6–7) targets control-flow instructions\n(represented as jump s inLmSpec ). The handler assigns the jump\ntarget address to addr , and the result of the jump condition to\nvariable n. For unconditional jumps, nis simply assigned True . The\nbody of the handler (line 7) returns a tuple consisting of the string\n“jump” and the final jump target, which we calculate by evaluating\nthe if-then-else expression. If nis true, then the jump will be taken,\nso we leak the instruction target addr . Otherwise, we leak the\naddress of the next instruction calculated using the special context\nvariables &pcand&insn , where &pcholds the current instruction’s\naddress and &insn.size contains its size.\nWe describe other examples of leakage clauses in §3.Testing side-channel security of cryptographic implementations against future microarchitectures\n1 (defleakage ConstantTime []\n2 (on[(load [addr ]_sz)\n3 #(\"load\" addr )]\n4 [(store [addr ]_sz := val )\n5 #(\"store\" addr )]\n6 [(jump addr : n )\n7 #(\"jump\" (if n addr (+&pc&insn.size )))]))\nFigure 1: Modeling constant-time requirements in LmSpec\n1 (defpredictor branchSpec []\n2 (on[(jump addr : n )\n3 (when (& insn.group CS_GRP_JUMP )\n4 [(\"PC\" (if n(+&pc&insn.size )addr ))])]))\nFigure 2: Modeling branch speculation in LmSpec\n2.2 Modeling speculation with LmSpec\nBy default, LmSpec assumes that programs are executed following\nthe instruction set architectural semantics. However, processors\nachieve significant performance improvements by speculating on\nthe values of intermediate computations and continuing the ex-\necution based on these predictions. Speculation does not affect\nthe correctness of the computation, because the processor always\nchecks the correctness of its guesses and squashes all speculative\nexecution steps in case of a misprediction. However, the microarchi-\ntectural effects of such instructions are not reversed; the resulting\ninformation leakage can be exploited to recover secrets [35].\nTo model the effects of new leakage models under speculation,\nLmSpec also allows modeling the effects of speculatively executed\ninstructions. For this, LmSpec relies on prediction clauses (described\nin §4) describing (a) which instructions might trigger speculation\nand (b) how the predicted values (for control or data flows) are\ncomputed. As an example, Figure 2 depicts a prediction clause\ncapturing the effects of speculating over branch instructions [ 35].\nPrediction clauses are specified using defpredictor statements\n(line 1) and, similarly to leakage clauses, they are defined using\nhandlers. Here, however, a handler’s body models the set of pre-\ndicted values used when speculatively executing instructions. For\ninstance, the branchSpec clause consists of a single handler (line 2–\n4) that deals with control-flow instructions jump with target addr\nand condition n. The handler uses a when block to test whether the\ncurrent instruction is a conditional branch (line 3).\nImportantly, the prediction clause only specifies the predicted\nvalues; it does not specify how speculative instructions are squashed\nor under which conditions the processor checks for misspecula-\ntion. These aspects are automatically handled by LmSpec , which\nwill speculatively explore all wrong predictions produced by the\nprediction clause following the always mispredict strategy [ 26]. In\nour example, the handler computes the mispredicted target in line 4\n(where \"PC\" indicates a control-flow prediction). LmSpec , then, will\nuse the target computed by the handler to speculatively explore,\nfor a fixed number of steps, the mispredicted branch.\nIn §4, we describe in detail how prediction clauses work and\nmodel in LmSpec different forms of speculative execution.2.3 Generating leakage traces\nTesting for leakage with respect to a given LmSpec model requires\nderiving leakage traces , following the model’s leakage and predic-\ntion clauses, directly from program executions. For this, we imple-\nmented an executable version of LmSpec on top of the Unicorn CPU\nemulator [ 1], which allows simulating architectural executions.\nGiven an LmSpec model, we automatically compile leakage and\nexecution clauses into Unicorn event hooks . These hooks are used\nto instrument program execution whenever user-defined architec-\ntural events are triggered. For leakage clauses, LmSpec handlers are\ntranslated into hooks monitoring events and generating leakage\nobservations. For instance, the load andstore handlers from Fig-\nure 1 are translated into hooks intercepting memory requests and\nrecording the address as part of the leakage trace. For prediction\nclauses, LmSpec handlers are translated into hooks that instrument\nprogram execution to explore speculative paths in addition to the\narchitectural execution. As an example, the clause from Figure 2\nis translated into a hook that starts the speculative execution of\nthe mispredicted branch. Thus, the executable version of LmSpec\nallows us to directly derive the leakage traces associated with each\nprogram execution for arbitrary x86 binaries. We provide further\ndetails on LmSpec ’s executable version in §5.1.\n2.4 Specifying side-channel security\nAs standard, we formalize side-channel security as non-interference\nwith respect to a leakage model [ 5]. Informally, a program is secure\nif its leakage traces do not leak secrets. In more detail, the notion\nof security is parametrized by a labeled interface for the program\nunder consideration. The interface declares, for each program argu-\nment, its security level (secret or public) and additional constraints\nsuch as size. This interface defines a notion of valid input , and a\nnotion of low-equivalence between two valid inputs: informally, two\ninputs are low-equivalent if they only differ in their secrets. Then,\naprogram is non-interfering iff executing the program on pairs of\nlow-equivalent valid inputs yield equal leakage traces.\n2.5 Detecting leaks with LmTest\nFollowing our formalization of side-channel security, security viola-\ntions are pairs of program executions that (i) yield different leakage\ntraces; and (ii) start from valid low-equivalent inputs, i.e., program\ninputs that only differ in their secrets.\nTo automatically discover such violations, we developed the Lm-\nTest testing tool, whose workflow is depicted in Figure 3. LmTest\ntakes as input: (a) the binary of the program under test; (b) an en-\ntry point in a cryptographic library, with a labeled interface ; and\n(c) an LmSpec leakage model. LmTest starts by generating random\ntest cases, each one consisting of a pair of valid low-equivalent\ninputs. For generating a test case, LmTest first generates a random\ninput satisfying the constraints of the labeled interface and then\nre-randomizes all secret bytes to generate a public-equivalent com-\nparison input. Next, LmTest derives the leakage traces associated\nwith each input in a test case using LmSpec ’s executable semantics.\nFinally, LmTest compares the traces in a test case and reports all\ncases where traces differ, i.e., indicating secret-dependent leaks.G. Barthe, M. Böhme, S. Cauligi, C. Chuengsatiansup, D. Genkin, M. Guarnieri, D. Mateos Romero, P. Schwabe, D. Wu, Y. Yarom\nProgram\nLabeled\ninterfaceInput\ngeneratorUnicorn\nengineEngine controller Prediction clause\nTracerLeakage clause\nCheckerDetected\nviolations input\npairshooks predictions\nhooks leakage\ntraces\nFigure 3: Diagram showing LmTest ’s internal workflow. A user provides as input a leakage clause and possibly a prediction\nclause, as well as the program to test. The labeled interface specifies the format of inputs and which inputs are secret.\ninitializer-pair ::::name expr\nuop-binding :\n(read:::reg)\f\f(write:::reg:=::val)\f\f(expr (::op::val *))\f\f(addr:::base +::::index *:::scale +::off)\f\f(load [:::addr]::sz)\f\f(store [:::addr]::sz:=::val)\f\f(jump::::addr ::n)\nleakage-def :\n(defleakage::::::::::LeakageModel [initializer-pair*]\n(on [ uop-binding body obs]*))\npredictor-def :\n(defpredictor::::::::::PredictorModel [initializer-pair*]\n(on [ uop-binding body preds ]*))\nFigure 4: Syntax of LmSpec . Words and symbols in\nfixed-width are verbatim keywords, words in italics are syn-\ntactic forms either of Hy or defined here, and names with\n::::::::underwave can be replaced by any valid identifier. Syntactic\ngroups with a square hat are either optional?or repeatable*.\n3 MODELING LEAKAGE IN LMSPEC\nHere, we first introduce the LmSpec domain specific language with a\nfocus on leakage clauses (§3.1); prediction clauses are the focus of §4.\nNext, we illustrate the core features of LmSpec by modeling three\nmicroarchitectural optimization proposals from [ 60]: silent store\nsuppression (§3.2), register file compression (§3.3), and computation\nreuse (§3.4).\n3.1 The LmSpec language\nWe developed the LmSpec domain-specific language to specify leak-\nage models. LmSpec is implemented in Hy [ 32], a Lisp-like language\nthat compiles to Python bytecode. This provides our models with\nthe full expressiveness of Python as well as access to the data struc-\ntures from the Python standard library, allowing users to explore\ndifferent leakage models with minimal overhead.\nThe syntax of leakage models is given in Figure 4. An LmSpec\nleakage model consists of a leakage clause (defined by defleakage\nstatements, explained below) and a prediction clause (defined by\ndefpredictor statements, explained in §4).\nEach leakage clause is defined by a name, a list of state variables\nwith their initial values, and a list of handlers. The state variablesVar. Description\n&insn information about the current instruction\n&mem current mapping of addresses to bytes\n&regs current mapping of registers to values\n&pc current value of the program counter\nFigure 5: LmSpec ’s context variables\nare used to capture stateful leakage clauses, as we show in our\nmodel of the computation reuse optimization in §3.4. Their initial\nvalues can be defined using arbitrary Python expressions.\nLeakage handlers, instead, are used to generate sequences of\nleakage observations based on specific events, which we call micro-\noperations, happening during program execution. LmSpec con-\nsider seven different micro-operations :read ,write ,expr ,addr ,load,\nstore , and jump . The read andwrite micro-operations refer to\nread and write operations on registers, whereas the expr micro-\noperation denotes computations. There is also a dedicated addr\nmicro-operation for the computation of memory addresses. Next,\nload andstore micro-operations refer to memory reads and writes.\nFinally, jump micro-operations refer to (potentially conditional)\ncontrol-flow changes, i.e., changes to the program counter.\nAleakage handler consists of a guard and a body. The guard of a\nleakage handler uop-binding is an operation with bindings for its\narguments. In contrast, the body of a leakage handler ( body obs) is a\nPython expression that evaluates to a tuple of values representing\nleakage observations, or to None if no observation is leaked. Note\nthat bodies can update state variables, and can freely use special\ncontext variables, such as those in Figure 5.\nNext, we illustrate how LmSpec captures concisely different kinds\nof leaks. In §6, we describe variants of these leakage clauses as well\nas further clauses used in our case study.\n3.2 Silent store suppression\nSilent store suppression [33,37,60] is an optimization that suppresses\na write-back to memory when the associated store operation does\nnot change the value at that memory location. This could result\nin timing leaks, e.g., due to lowering the pressure on the CPU’s\npipeline. Note that silent stores are considered, for instance, in\nthe RISC-V architecture [ 72] and have been observed (on specific\nvalues) in some Intel CPUs [18].Testing side-channel security of cryptographic implementations against future microarchitectures\nThe leakage clause capturing these leaks emits an observation if\nthe value about to be stored matches the value already in memory\nat the target address. This is formalized in LmSpec as follows:\n1(defleakage SilentStore []\n2 (on[(store [addr ]_sz := val )\n3 (when ( = val (&mem.read addr sz ))\n4 #(\"ss\" addr val ))]))\nThe guard of the handler is a store expression (line 2). It binds the\nvariable addr to the target address, szto the value size, and valto\nthe target value to be written. The final part of the handler’s body\n(line 4) returns a triple consisting of a logging string (here “ss” for\n“Silent Store”), the current address, and the stored value. The first\npart of the handler’s body is more interesting. It is a conditional\nwhen block (line 3) that uses the special &mem context variable to\nretrieve values from memory. In this case, the method &mem.read\nis called with the target address addr and size szto get the current\nvalue in memory that the store operation would overwrite. The\nwhen block compares this result to the store target value val, and\ngenerates an observation if these values are equal, i.e., a necessary\ncondition for store suppression.\n3.3 Register file compression\nRegister file compression [8,64] is an optimization that maps multiple\nlogical (architectural) registers to the same physical (microarchi-\ntectural) register file entry when they hold the same value. This\nallows processors to perform more computations in parallel. Its\ncorresponding leakage model emits an observation whenever a\nregister is updated with a value that is already stored in another\nregister. This is captured by the following LmSpec leakage clause:\n1(defleakage RegisterFileCompression []\n2 (on[(write reg := val )\n3 (when ( and (in reg X86_64_GPRS )\n4 (exists reg_i X86_64_GPRS\n5 :where (!= reg_i reg )\n6 (= val (&regs.read reg_i ))))\n7 #(\"rfc\" reg val ))]))\nThe guard of the handler (line 2) is a write operation. It binds reg\nwith the target register name and valwith the target value. The\nbody of the handler (lines 3–7) checks if regis a general purpose\nregister—as opposed to segment registers or other special registers—\nusing the predicate (in reg X86_64_GPRS) (line 3) and scans for\nanother general purpose register that holds the value val(lines\n4–6). For this, it uses the special context variable &regs to query\nthe logical registers by name, and the built-in exists function\nto iterate over all general purpose register names (skipping the\nregister currently being written). If any other register already holds\nthe target value, then the handler exposes the target register and\nthe shared value in the leakage observation (line 7).\n3.4 Computation reuse\nComputation reuse [62] is an optimization that caches results of\nrecent arithmetic instructions in a hardware memoization table.\nComputation reuse thus avoids re-executing any instruction whose\nresults are already present in the table. Its corresponding leakage\nmodel emits an observation if a computation is performed twice.\nThis is captured by the following LmSpec leakage clause:1 (defleakage ComputationReuse [memo (OrderedDict )]\n2 (on[(expr (op#* vals ))\n3 (when (in op CACHING_OPS )\n4 (if (in vals (.get memo &pc#()))\n5 #(\"cr\" op #* vs )\n6 (update memo &pc vals )))]))\nWe first explain the initializer. To mimic a memoization table and\nto have it persist between calls to our handler, we use a state vari-\nable—these are declared within the brackets following the model\nname in a defleakage definition as a sequence of names and initial\nvalues. In line 1, we declare a state variable called memo ; we initial-\nizememo to an empty OrderedDict , which is a map data structure\nfrom Python’s standard library that allows (re)ordering its keys,\nallowing us to simulate a simple ( 𝑛-way) LRU cache of operand\nvalues, indexed by instruction address.1\nNext, we turn to the guard (line 2). The form expr is used to\nmodel general arithmetic instructions. It uses Hy’s list-assignment\nsyntax to write (expr (op #* vals)) , where opis assigned the\ninstruction mnemonic corresponding to the operation and vals is\nassigned the list of operand values.\nWe now turn to the handler’s body (lines 3–6). It uses a predi-\ncate (in op CACHING_OPS) to check if the current instruction is\nincluded in a set of common arithmetic instructions that we are\nexplicitly caching (line 3). Then, on line 4, it retrieves the cache way\ncorresponding to the current instruction from memo , indexed by the\ncurrent instruction address ( &pc). If there is no such mapping yet,\nit returns the empty collection literal #(). Then, if the operand list\nvals exists as an entry within the retrieved way, the body returns\nthe tag cr, the operation and its operands as leakage observation.\nOtherwise, the body calls a function update to update the memo\ncache with the current instruction address and operands.\n4 MODELING SPECULATION IN LMSPEC\nIn this section, we first show how speculation can be modeled in\nLmSpec using prediction clauses (§4.1). These clauses allow users to\nexplore the interactions between speculative execution and leakage\nclauses. We then illustrate LmSpec ’s flexibility by giving examples\nof control-flow (§4.2) and data-flow speculation (§4.3).\n4.1 Prediction clauses in LmSpec\nLmSpec supports prediction clauses that can be used to model spec-\nulative execution. These clauses allow users to specify the possible\npredictions for every operation. For simplicity, we do not require\nusers to specify when the processor checks for misspeculation or\nhow misspeculated instructions are squashed; these checks are\nbuilt-in into LmSpec ’s semantics. In particular, LmSpec adopts the\nconservative always mispredict approach for capturing the effects\nof speculative execution [26]. Whenever the execution reaches an\ninstruction that can result in control flow or data prediction (as\nindicated by the prediction clauses), LmSpec ’s semantics (1) exe-\ncutes the prediction clause to retrieve all predictions, (2) explores\n1The choice of 𝑛is left open, as we are not modeling any known processor design. We\nexpect processor designers and library developers to tailor leakage models to their\nown needs.G. Barthe, M. Böhme, S. Cauligi, C. Chuengsatiansup, D. Genkin, M. Guarnieri, D. Mateos Romero, P. Schwabe, D. Wu, Y. Yarom\nall executions associated with wrong predictions,2before (3) finally\nproceeding along the path of correct execution, i.e, the architectural\npath. These design choices allow us to drastically reduce the user\neffort for specifying speculative execution in LmSpec .\nThe syntax of prediction clauses is given in Figure 4. Prediction\nclauses consist of a name, a list of state variables with their initial\nvalues, and a list of prediction handlers. Prediction handlers are\nvery similar to leakage handlers, and support flexible and generic\nspeculation. This contrasts with prior tools [ 26,50], which rely\non hard-coded rules for identifying prediction points and possi-\nble mispredictions. The core difference is that prediction handlers\noutput a list of predicted values for control-flow or data predic-\ntions. A control-flow prediction consists of the \"PC\" keyword and\nthe address of the next predicted instruction. In contrast, a data\nflow prediction consists of a keyword (either \"MEM\" indicating that\nwe are predicting a memory value or \"REG\" indicating that we\nare predicting a register value) and a prediction, which can be a\ntriple (address size value) for memory predictions or a pair\n(registerId value) for register predictions. These predictions\nare then used by LmSpec when exploring speculative paths, i.e.,\nwhenever LmSpec starts a new speculative path, it first selects one\nof the outstanding predictions and applies it to the program state\nbefore starting the speculative execution.\nIn the rest of this section, we show examples of how control-flow\n(§4.2) and data speculation (§4.3) can be implemented in LmSpec .\nIn §6, we describe other variants of the prediction clauses that we\nused in our case study.\n4.2 Control-flow speculation\nWe already presented a first example of control-flow speculation,\ni.e., misprediction of conditional branch instructions, in §2.2. Here,\nwe present two other examples: straight-line speculation and spec-\nulation using a return stack buffer.\nStraight-line speculation: Some AMD processors can, under cer-\ntain circumstances, execute the instructions following anybranch\ninstruction [ 77], including unconditional branches. This is captured\nby the following prediction clause:\n1 (defpredictor StraightLineSpec []\n2 (on[(jump addr : n )\n3 [(\"PC\" (+&pc&insn.size ))]]))\nThe handler always adds a prediction for the instructions following\nthe branch (line 3). In certain cases, e.g., when a conditional branch\nis not taken, the following instruction is the correct next address. To\nprevent treating this as a misprediction, LmSpec always removes the\ncorrect outcome from the list of possible mispredictions returned\nby the handler.\nReturn stack buffer: To speculate over return instructions, pro-\ncessors employ an auxiliary data structure called the return stack\nbuffer (RSB), which stores the return addresses of recent call in-\nstructions and uses them as prediction for the actual return address.\nWe model RSB speculation (where the RSB is implemented using a\ncircular buffer) with the following prediction clause:\n2The predictions generated by the prediction clause might contain the correct value.\nTo avoid treating it as a misprediction, LmSpec always removes the correct value from\nthe clause’s result.1 (setv RSB_SIZE 16)\n2 (defpredictor RSBCircular [stack (*[0] RSB_SIZE )\n3 idx 0]\n4 (on[(jump addr : n )\n5 (cond\n6 (&insn.group CS_GRP_CALL )\n7 (do (assoc stack idx\n8 (+&pc&insn.size ))\n9 (setv idx (%\n10 (&insn.group CS_GRP_RET )\n11 (do (setv idx (%\n12 [(\"PC\" (get stack idx ))]))]))\nThe code first defines a 16-entry buffer used for the RSB (line 1).\nWhen handling jump operations, the code checks if these are calls\nor returns (lines 6 and 10). In the case of calls, the code pushes\nthe return address into the stack (lines 7–9). For return instruction,\ninstead, the code pops an entry from the stack using it as a predicted\ndestination (lines 11–12).\n4.3 Data speculation\nSo far, we have shown how control-flow speculation can be formal-\nized in LmSpec . Here, we show an example of data speculation. In\nparticular, we model speculation over store bypasses (exploited in\nSpectre-STL attacks [ 30]), in which the processor speculates (possi-\nbly incorrectly) that an older store does not conflict with a younger\nload thereby accessing stale data. This can be modeled with the\nfollowing prediction clause:\n1(defpredictor StoreBypassSpec [buf (deque :maxlen SIZE )]\n2 (on[(load [addr ]_sz)\n3 (lfor [waddr wsz val ]buf\n4 :if (=[addr sz ] [waddr wsz ])\n5 (\"MEM\" (addr sz val )))]\n6 [(store [addr ]_sz := val )\n7 (.append buf [addr sz (&mem.read addr sz )])]))\nThe clause keeps track of recent store instructions and the contents\nthey overwrite using the bufstate variable, which is a buffer of size\nSIZE (defined in line 1). In particular, when executing a store micro-\noperation, the old overwritten value (extracted by the (&mem.read\naddr sz) expression) is appended to buf(lines 6–7). In contrast,\nwhen executing a load micro-operation, the code compares its\naddress and size with those in the store buffer, returning the old\nvalue as a potential prediction (lines 2–5).\n5 TESTING FOR LEAKS\nIn this section, we introduce our approach for automatically testing\nthe side-channel guarantees of programs against leaks captured by\nLmSpec models. We first describe our implementation of LmSpec ’s\nexecutable semantics (§5.1) on top of the Unicorn CPU emulator [ 1],\nwhich enables us to derive leakage traces for arbitrary x86 program\nexecutions. We then proceed to describe our testing approach (§5.2),\nwhich we implement in the LmTest testing tool.\n5.1 Generating leakage traces for LmSpec\nTo study the security implications associated with a given LmSpec\nmodel for real-world cryptographic implementations, we need an\nautomated way of deriving leakage traces directly from program\nexecutions. To address this, we implemented an executable versionTesting side-channel security of cryptographic implementations against future microarchitectures\nofLmSpec on top of the Unicorn emulator [ 1], which allows simu-\nlating x86 programs architecturally. As mentioned in §2.3, we do so\nby translating leakage and prediction clauses in LmSpec into event\nhooks in Unicorn. Event hooks allow injecting arbitrary instrumen-\ntation code that is automatically executed by the Unicorn emulator\nwhenever specific events happen during program execution. Our\nimplementation is inspired by the Revizor tool [ 50], which imple-\nments fixed leakage models on top of Unicorn using event hooks.\nLeakage clauses: We compile each LmSpec leakage clause with 𝑛\nhandlers into 𝑛different event hooks that monitor the execution of\nthe current instruction and record the associated leakage observa-\ntions. In particular, handlers of load andstore micro-operations\nare compiled into memory hooks, which are executed whenever\nthe emulator executes a memory request (as a result of an instruc-\ntion). Handlers for all other micro-operations, instead, are compiled\ninto instruction hooks, which are executed whenever the emulator\nfetches a new instruction. The body of a leakage handler, which\nrecords the leakage observation, is a Hy expression. We directly\ncompile it into Python (using the Hy backend [32]) as the body of\nthe event hook.\nPrediction clauses: By default, Unicorn emulates instructions\nfollowing the x86 architectural semantics. To capture the effects\nof speculatively executed instructions, we extended Unicorn with\nanalways mispredict speculation model [ 26,50]. First, we compile\nLmSpec prediction clauses into event hooks. These hooks, however,\ncompute a list of predictions, rather than observations (as do the\nhooks associated with leakage clauses). Following [ 50], whenever\nthe emulator executes an instruction that triggers a hook associated\nwith a prediction clause, it (1) takes a checkpoint of the computation\nstate, (2) retrieves the predicted values computed by the hook (and,\nif present, removes the correct value from the list of predictions)\nand (3) continues the (speculative) simulation based on one of the\npredictions. When speculative execution terminates,3the emulator\nrestores the computation state, using the previously taken check-\npoint, and either explores another speculative path (if there are\nother predictions) or restarts the architectural simulation.\nContext variables: At every point during the simulation, the\nvalues of LmSpec context variables (see Figure 5) are derived by\ninspecting the emulator’s state. For instance, a call to &mem.read\nis translated into a call to the Unicorn’s API for reading memory.\nSimilarly, the &insn structure is initialized by retrieving the current\ninstruction from the emulator’s state, disassembling it using the\nCapstone library [17], and extracting the necessary information.\nDeriving traces: Given a program, an initial program state, and\nanLmSpec model, the corresponding leakage trace is obtained\nby simulating the program execution using the Unicorn emulator\nextended with the event hooks obtained from the LmSpec model.\nIn particular, the hooks associated with the prediction clause will\ntrigger speculative execution whereas those associated with the\nleakage clause will track the leakage observations during execution.\n3Following the always mispredict model [ 26], this happens in one of the following\ncases: (1) the predefined speculation window is exhausted, (2) the execution encounters\na speculation barrier, e.g., an lfence instruction, or (3) the program terminates.5.2 LmTest testing tool\nHere, we describe our testing approach, implemented in the LmTest\ntool, for detecting leaks in a program given an LmSpec model.\nAs anticipated in §2.5, we characterize side-channel security as\na non-interference property [ 5]. In particular, we say that program\n𝑃is secure under LmSpec model LMif for all pairs of initial program\nstates𝑠,𝑠′, if𝑠and𝑠′only differ in their secrets, then the leakage\ntraces (according to the model LM) associated with 𝑃’s execution\non states𝑠and𝑠′must be the same. Hence, a leak in program 𝑃\nunder model LMconsists of a violation of this non-interference\nproperty, that is, a pair of low-equivalent program states that result\nin different leakage traces.\nAlgorithm 1 LmTest testing approach\nRequire: Program𝑃, labeled interface 𝐼,LmSpec model LM, num-\nber of test cases 𝑁\n1:𝑠1,...,𝑠 𝑁←genInitConfs(𝐼,𝑁) ⊲generate seed states\n2:for𝑠∈{𝑠1,...,𝑠 𝑁}do\n3:𝑠′←mutate(𝑠,𝐼) ⊲mutate state\n4:𝑡←getTrace(𝑃,𝑠,LM) ⊲collect traces\n5:𝑡′←getTrace(𝑃,𝑠′,LM)\n6: if𝑡≠𝑡′then ⊲Trace comparison\n7: return Violation detected ⟨𝑠,𝑠′⟩\n8:return No violation detected\nTo detect leaks, we use a relational random testing approach\ninspired by Revizor [ 50] and ct-fuzz [ 28].LmTest approach for\ndetecting leaks is summarized in Algorithm 1. We start by generat-\ning𝑁randomly selected initial states following the user-provided\nlabelled interface 𝐼(line 1). For each state 𝑠, we then generate a\nlow-equivalent state 𝑠′(line 3). Next, we derive the leakage traces\nassociated with 𝑠and𝑠′by simulating the program under test us-\ning the Unicorn emulator extended with the LmSpec model LMas\ndescribed in §5.1 (lines 4–5). Finally, we compare the generated\ntraces (lines 6–7): any difference in the traces is caused by a secret-\ndependent leak (since 𝑠and𝑠′are low-equivalent). If we detect a\ndifference, we report it (line 7). Otherwise, we continue the testing\nand move to the next test case.\nBefore concluding, we provide further details on the labelled\ninterface and our test generation strategy.\nLabelled interfaces: The labelled interface provides LmTest with\na description of the program inputs. For each input, the interface de-\nscribes (1) whether it is stored in a register or in memory, (2) its size\nin bytes, (3) whether it is public or secret, and (4) (optionally) its lo-\ncation in memory. This interface is provided by the user and allows\nLmTest to generate the test cases, which, as already mentioned,\nare pairs of low-equivalent initial states.\nGenerating states: To generate a random initial state, we instanti-\nate each input with a sequence of randomly generated bytes of the\nappropriate length, as specified in the labelled interface. This simple\nstrategy (which might not work for structured data) is sufficient\nto generate valid initial states for all cryptographic implementa-\ntions we tested in §6, whose inputs consist of, e.g., secret keys or\nmessages.\nMutating states: To mutate a random state 𝑠, we replace its se-\ncret inputs (as indicated by the interface) with sequences of freshG. Barthe, M. Böhme, S. Cauligi, C. Chuengsatiansup, D. Genkin, M. Guarnieri, D. Mateos Romero, P. Schwabe, D. Wu, Y. Yarom\nrandomly generated bytes of the appropriate length. This ensures\nthat the mutated state 𝑠′is low-equivalent to the original state 𝑠.\n6 EVALUATION AND CASE STUDY\nIn this section, we study the impact of microarchitectural optimiza-\ntions on the side-channel security of widely used cryptographic\nalgorithms. As part of this case study, we identify three core re-\nsearch questions, which we address in the following sections:\nRQ1 Does LmSpec provide an expressive and concise framework\nfor specifying leakage models? (§6.1)\nRQ2 Are real-world cryptographic libraries secure under the dif-\nferent models and can LmTest detect leaks in them? (§6.2)\nRQ3 Can the leaks be exploited? (§6.3)\n6.1 RQ1: Expressiveness of LmSpec\nWe use LmSpec to model 18 leakage clauses (§6.1.1) and six predic-\ntion clauses (§6.1.2) that we implemented in LmSpec , which com-\nbined result in 108 LmSpec leakage models. Full implementations\ninLmSpec of all our clauses are given in Appendices A–B.\n6.1.1 Leakage clauses. We implemented in LmSpec 18 different\nleakage clauses capturing the leaks introduced by eight different\nclasses of microarchitectural optimizations, from the classic con-\nstant time model (capturing cache-related leaks), to complex op-\ntimizations like cache compression [ 66] and prefetching [ 68] as\nwell as security-critical optimization proposals [60]. Implemented\nclauses include:\nConstant-time (CT): We implemented the baseline model (de-\nnoted by CT) shown in Figure 1.\nSilent stores (SS): We implemented the baseline model (denoted\nby SS) shown in §3.2, and two variants. The SSI0 variant restricts\nobservations only to silent stores where the value being written\nis all-zeroes [ 18]. The SSI variant produces an observation only\non silent stores involving memory locations that have already\nbeen initialized by the program—the latter uses initialization\nhandlers, an LmSpec feature not presented in the paper.\nRegister file compression (RFC): We implemented the baseline\nmodel (denoted by RFC) shown in §3.3, and two variants. The\nRFC0 variant only compresses registers that are all-zero [ 64].\nThe NRFC variant only compresses narrow values [ 19]: registers\nwhose values are less than 16 bits are compressed into the same\nphysical register. For the latter, our model checks whether the\nvalue being written to a register is under 16 bits and emits an\nobservation whenever another register also stores a value that is\nunder 16 bits.\nComputation simplification (CS): We implemented three mod-\nels. Two models (denoted respectively by CS and CST) capture\nsimplification strategies proposed by Atoofian and Baniasadi\n[6], which simplify arithmetic and logical operations on values\nlike 0and 1. The third, CSN, captures the effects of simplifying\nmultiplication instructions for narrow operands (under 32 bits).\nOperand packing (OP): Operand packing [ 11] compresses multi-\nple in-flight instructions (of the same type) with narrow operands\ninto a single “compressed” instruction that is forwarded to the\nexecution units. We model it by tracking the latest 𝑛expr micro-\noperations during program execution, checking whether some\nof these micro-operations can be compressed (i.e., they all haveoperands that are less than 16 bits), and producing an observation\nif this is the case.\nComputation reuse (CR): Computation reuse optimizations [ 62]\ncache recent computation results and avoid re-executing compu-\ntations whenever their results are cached. We implemented two\nmodels capturing the effects of reuse optimizations from [ 62].\nThe first model (denoted CR and presented in §3.4) captures\nleaks introduced by computation reuse over arithmetic opera-\ntions. The second model (denoted CRA) additionally captures\nleaks by computation reuse on address calculations in address\nmicro-operations.\nCacheline compression (CC): Cache compression optimizations\naims at compressing cache lines, thereby increasing the amount\nof data that can be stored in caches. We implemented models\ncapturing the effects of two compression strategies: Frequence\nPattern Compression (FPC) [ 2] and Base-Delta-Immidiate com-\npression (BDI) [ 56]. The former compresses several common data\npatterns whereas the latter compresses narrow ranges of values.\nIn a nutshell, both models monitor the execution of memory oper-\nations and produce an observation whenever the corresponding\nmemory request might result in compression according to the\ngiven strategy.\nPrefetching (PF): Prefetchers aim at loading memory blocks into\nthe cache hierarchy before these blocks are requested by instruc-\ntions. We implemented LmSpec models capturing the effects of\nthree different prefetching strategies: (1) next-line prefetching\n(PFNL) [ 7], which prefetches the next memory block for any\nload operation, (2) stream prefetching [ 59] (PFS), which detects\nwhether the program is accessing addresses at a regular stride and\nprefetches further memory blocks along the stride, and (3) data-\ndependent prefetcher (PFDD) based on behavior observed on the\nApple M1 chip [ 68]. All these models (1) monitor the execution\nofload micro-operations, (2) check whether further memory\nblocks need to be prefetched according to the corresponding\nstrategy, and (3) append the prefetched memory addresses to the\nleakage trace.\n6.1.2 Prediction clauses. We implemented in LmSpec six different\nprediction clauses, the default sequential prediction clause (de-\nnoted by S eqand represented in LmSpec by the absence of any\ndefpredictor statement), and five additional clauses, capturing\ndifferent speculation mechanisms. In particular, our models cover\nallspeculation mechanisms that have been formalized in the litera-\nture [20]. The implemented models are:\nConditional branch speculation (P ht):The prediction clause\nfor this model, presented in Figure 2, captures speculating over\nbranch instructions following the so-called “always mispredicts”\nsemantics [26].\nStraight-line speculation (S ls):We implemented an LmSpec\nmodel, illustrated in §4.2, that capturess the effects of straight-\nline speculation implemented in some AMD cores [ 77]. The model\nalways speculates for a fixed number of steps beyond any jump\nmicro-operation .\nStore bypass speculation (S tl):TheLmSpec model illustrated\nin §4.3 captures the effects of speculation over store bypasses [ 30].Testing side-channel security of cryptographic implementations against future microarchitectures\nFollowing [ 20], our model speculatively ignores issued store micro-\noperations for a fixed number of steps.\nReturn address speculation (R sb):We implemented two clauses\nmodeling speculation over return instructions. Both models employ\na return stack buffer to determine the speculation target, but they\ndiffer in how they handle buffer under- and over-flows. One model,\ndenoted R sb◦and presented in §4.2, uses a circular buffer that\nwraps around on over- or underflows. The other, denoted as R sb⊥,\nis inspired by the return speculation in [ 20]. It simply drops its\noldest entry on overflow and halts (refuses to speculate further) on\nunderflow.\n6.1.3 Assessment. In terms of expressiveness, we observe that our\nleakage models span a large class of microarchitectural leaks. In\nparticular, our leakage clauses cover both standard models like\nconstant-time as well as advanced optimizations (cache compres-\nsion and prefetching) and examples from all optimization proposals\nstudied in [ 60]. Similarly, our prediction clauses cover multiple\ndifferent speculation mechanisms and all speculation mechanisms\nused in state-of-the-art tools [ 20,50]. To the best of our knowl-\nedge, this is the largest library of leakage models (108 models) to\nbe implemented in a single language/tool.\nRegarding the conciseness of our LmSpec implementations, we\nobserve that most of the leakage and prediction clauses are imple-\nmented in a few lines of LmSpec . For instance, the prediction clauses\nare all implemented in less than 10 lines of LmSpec code each. We\nremark that LmSpec allows directly leveraging Python’s standard\nlibrary and data structures, which greatly simplifies implementing\nmore complex models (like computation reuse or cache compres-\nsion). Overall, all the aforementioned leakage and prediction clauses\ncan be implemented in about 500 lines of code, including Python\ncode for handling data structures like the computation reuse cache\nand C code for the implementation of cache compression strategies.\n6.2 RQ2: Robustness of cryptographic libraries\nHere, we report the results of our analysis of the security of sev-\neral real-world cryptographic libraries against the leakage models\nfrom §6.1, which we conducted using LmTest . In the following, we\nfirst present the test subjects (§6.2.1) and the overall experimental\nsetup (§6.2.2). We conclude by presenting our results (§6.2.3).\n6.2.1 Test subjects. We focus on eight commonly used crypto-\ngraphic algorithms that span a variety of use cases for crypto-\ngraphic software: AES and SHA512 are widely used primitives for\nencryption and hashing, respectively; poly1305 and salsa20 are the\ndefault primitives used for authenticated encryption in the popu-\nlar cryptographic library libsodium; and ed25519 and x25519 are\nelliptic-curve primitives. We also analyze the HMAC and stream-\nXOR constructions to see any leakage effects from higher-level\ncryptographic operations.\nWe test the implementations of these algorithms from five differ-\nent cryptographic libraries: libsodium [ 40], a popular cryptographic\nlibrary written in C and designed for ease-of-use and safe defaults;\ncryptlib [ 14] and libnettle [ 49] as alternative libraries also written\nin C, to compare results between different implementations; Rust\nCrypto [ 55] to examine how the high level safety guarantees pro-\nvided by Rust [ 43] may affect leakage results; and libjade [ 22], alibrary written in the Jasmin secure assembly language [ 4] and\nverified to be constant-time.\nWe compiled each library with their respective default settings\nand compilers.4Since some libraries only implement a subset of\nthe studied algorithms, this results in 25 different cryptographic\nimplementations. For each implementation, we manually created a\nwrapper function that (1) runs the algorithm implementation, and\n(2) annotates the algorithm inputs as secret or public for LmTest .\n6.2.2 Experimental setup. For each of the test subjects, we use Lm-\nTest to test their security against all the leakage models from §6.1.\nSpecifically, for each test subject trgand leakage model 𝑀, we use\nLmTest to (a) generate 100 test cases, where each test case consists\nof a pair of low-equivalent initial states, (b) run trgfor all test cases\nand collect the leakage traces generated by the model 𝑀, and (c)\nanalyze the leakage traces for leaks (i.e., checking if the traces for\na single test case are different). For each test subject and leakage\nmodel, we impose a total timeout of 240 minutes and a per-test-case\ntimeout of 30 minutes, and stop the testing whenever one of the\ntimeouts expires. We ran our testing campaign on an Intel Xeon\nGold 6132 running Ubuntu 22.04.2 LTS.\n6.2.3 Assessment. Table 1 summarizes the results of our testing\ncampaign. The overall finding is that allthe analyzed implementa-\ntions leak. This confirms that the optimization proposals studied\nin [60], if implemented, could potentially introduce security issues\nin modern cryptographic implementations. The testing campaign\nalso highlights several important observations:\n•For the majority of leakage clauses and implementations, LmTest\ncan already detect leaks under the S eqprediction clause.\n•There are several cases, however, where LmTest can only detect\nleaks under the speculative prediction clauses (e.g., computation\nreuse optimizations in libnettle, rust-crypto, and libjade). This in-\ndicates that, similarly to what happens in Spectre attacks [ 35], the\ninterplay between speculative execution and microarchitectural\noptimizations can weaken the security of implementations.\n•The use of memory-safe languages like Rust does not seem to\nsignificantly improve the security against the new leakage models.\nIn most cases, the implementations from the rust-crypto library\nare as leaky as the corresponding C implementations of other\nlibraries.\n•Constant-time programming does not prevent leaks under these\nnew leakage models. Despite all analyzed implementations except\nAES-CBC5begin constant-time (and libjade being provably so),\nLmTest still detects leaks in them. As we show in §6.3, specific\nconstant-time idioms, such as constant-time swaps, introduce\nleaks under several of the studied leakage clauses and result in\nleaks that allow recovering secret keys directly from leakage\ntraces.\n6.3 RQ3: Exploitability of leaks\nTo illustrate the security relevance of the leaks identified by Lm-\nTest , we performed an in-depth analysis of our findings against\n4We compiled libsodium v1.0.18 with gcc v11.4.0, cryptlib v3.4.6 with clang v14.0.0,\nlibnettle v3.8 with clang v14.0.0, Rust Crypto using sha2 v1.10, salsa20 v0.10.2, poly1305\nv0.8.0, and x25519-delak v2.0.0 all with cargo 1.73.0-nightly, and libjade v2023.05-1\nwith jasminc v2023.06.0.\n5libnettle and cryptlib implement AES using non-constant-time lookup tables.G. Barthe, M. Böhme, S. Cauligi, C. Chuengsatiansup, D. Genkin, M. Guarnieri, D. Mateos Romero, P. Schwabe, D. Wu, Y. Yarom\nTable 1: Results of our testing campaign. Filled-in diamonds qdenote that LmTest detected a secret-dependent leak for a\nspecific configuration (library, algorithm, leakage clause, prediction clause), while outlined diamonds ♦denote that LmTest\ndetected no leaks during testing. Crosses ×indicate that LmTest timed out before completing the testing. To aid readability, we\nvisually combine results for (library, algorithm, leakage clause) triples if they are the same for all prediction clauses.\nCT SS RFC CS OP CR CC PF\nlibsodium·· ·i·i0· · 0·n· ·t·n·· · afpc bdi·nl·s·dd\nsalsa20♦♦qq♦ ♦q q♦qqq♦ ♦q q q q♦qq♦ ♦ ♦♦ q q q q q♦♦qq♦ ♦♦♦qq♦ ♦♦♦q♦ ♦ ♦\npoly1305♦♦qq♦ ♦q q q q q q q q ♦ q q q q q♦♦qq♦ ♦♦♦qq♦ ♦♦\nsha512♦qqq♦ ♦q q q q q q q q ♦×q q q q q♦♦×q♦ ♦♦×♦\nhmac♦q×♦ ♦ ♦q×q×q×q×q×q×q×♦×♦×q×q×q×q× q×♦×♦×♦×\ned25519♦q×q♦ ♦q×q×q×q×q×q×q×q×♦♦×q♦ ♦q×q×q×q× q×♦q×q♦ ♦♦q×q♦ ♦♦♦×q♦ ♦\nx25519♦qqq♦ ♦q q q q q q q q ♦ q q q q q♦qqq♦ ♦♦q♦q♦ ♦q\nstream-xor♦qqq♦ ♦q q♦qqq♦ ♦q q q q♦qq♦ ♦ ♦♦ q q q q q♦qqq♦ ♦♦♦qq♦ ♦♦\ncryptlib\naes-cbc qq q q q q q q q ♦ q q q q q q q q\nsha512♦qqq♦ ♦q q q q q q q q ♦ q q q q q♦qqq♦ ♦♦qqq♦ ♦q♦qqqq\nlibnettle\naes-cbc qq q q q q q q q ♦ q q q q q q qqq♦qqq\nsalsa20♦♦q♦ ♦ ♦q♦qqq♦ ♦♦qqq♦ ♦q q q q♦♦q♦ ♦ ♦♦ q♦qqq♦ ♦♦qqq♦ ♦q q ♦ ♦ ♦\nsha512♦qqq♦ ♦q q q q q q q q ♦×q q q q q♦♦×q♦ ♦♦♦×q♦ ♦q♦×qqq\nhmac♦♦q♦ ♦ ♦q q q q q q q♦♦q♦ ♦ ♦♦ q♦qqq♦ ♦♦qqq♦ ♦q q ♦ ♦ ♦\ned25519 ♦×q×q×q×q×q×q×q×q×♦×q×q×q×q× q×♦×♦×q×\nx25519♦×××q♦q×q×q×q×q×q×q×q×♦×q×q×q×q×♦×××q♦♦×××q♦♦×♦×\nrust-crypto\nsalsa20 ♦q q♦qqq♦ ♦q q q q♦×♦ q♦qqq♦ ♦♦qqq♦ ♦q q ♦ ♦×♦\npoly1305♦♦×q♦ ♦q×q×q×q×q×q×q×q×♦♦×q♦ ♦q×q×q×q× q×♦♦×q♦ ♦♦♦×q♦ ♦♦×\nsha512♦♦qq♦ ♦q q q q q q q q ♦ q q q q q♦♦ ♦q♦ ♦♦♦ ♦q♦ ♦♦\nx25519 ♦×q×q×q×q×q×q×q×q×♦×q×q×q×q× q×♦×♦×♦×\nstream-xor♦♦ ♦q♦ ♦q q♦q♦q♦ ♦q q q q♦ ♦ q♦qqq♦q♦qqq♦ ♦q q♦♦ ♦q♦ ♦♦ ♦\nlibjade\nsalsa20♦♦qq♦ ♦q♦qqq♦ ♦♦qqq♦ ♦q q q q♦♦q♦ ♦ ♦♦ q♦qqq♦ ♦♦qqq♦ ♦q q♦♦qq♦ ♦♦♦q♦ ♦ ♦♦♦q♦ ♦ ♦\npoly1305♦♦q♦ ♦ ♦q♦qq♦ ♦ ♦♦qq♦ ♦ ♦q q q q q q q q q q q ♦ ♦ q\nsha512♦♦qq♦ ♦q q q q q q q q ♦ q♦qqq♦ ♦♦qqq♦ ♦q q♦♦qq♦ ♦♦♦qq♦ ♦♦♦q♦ ♦ ♦\nx25519 ♦×q×q×♦×q×q×q×q×q×♦×q×q×q×q×♦×♦×♦×♦×\nstream-xor♦♦qq♦ ♦q♦qqq♦ ♦♦qqq♦ ♦q q q q♦ ♦ q♦qqq♦ ♦♦qqq♦ ♦q q♦♦qq♦ ♦♦♦q♦ ♦ ♦ ♦\nqqqqqq results for prediction clauses:〈Seq〉 〈Pht〉 〈Sls〉\n〈Stl〉 〈Rsb⊥〉 〈Rsb◦〉\nq/♦/×leak found / not found / time out\nq/♦ leak found / not found for all prediction clauses\nq×/♦×leaks found / not found for all prediction clauses;\none or more speculative clauses timed out\nthe libsodium implementation of the X25519 key exchange algo-\nrithm. Our analysis highlights that in several cases we can recover\na victim’s secret key directly from the leakage traces resulting from\ndifferent leakage models. We present three examples focusing on\nleakage models associated with register file compression, compu-\ntation simplification, and silent store optimizations. All examples\nexploit leaks caused by the constant-time swap implementation\nintroduced as part of the X25519 constant-time protections.6.3.1 X25519 compare-and-swap. Figure 6 depicts the function\nshowing a simplified version of the compare-and-swap algorithm\nused in X25519. The two curve points to be swapped, fandg, are\nencoded as 5-element uint64_t arrays. The secret bit bcontrols\nwhether the elements should be swapped. Since the function is\ncalled for each bit of the secret key, learning bat each iteration\nallows an attacker to recover the secret key.\nThe conditional swap is implemented without using conditional\nbranches. First, the condition bit bis expanded into 64-bit maskTesting side-channel security of cryptographic implementations against future microarchitectures\n1 fe25519_cswap (fe25519_limb f [5], fe25519_limb g [5],\n2 /*secret*/ bool b) {\n3 mask = (-(int64_t) b);\n4\n5 x[0..5] = f[0..5] ^ g[0..5];\n6\n7 x[0..5] &= mask ;\n8\n9 f[0..5] = f[0..5] ^ x[0..5];\n10 g[0..5] = g[0..5] ^ x[0..5];\n11 }\nFigure 6: X25519 constant-time swap from libsodium, con-\ndensed for brevity. The variables f,g, and xare each 5-\nelement uint64_t arrays.\n(line 3). The two curve points fandgare then loaded from memory\nand xor’ed together into x(line 5), which is then masked (line 7).\nThe result is xor’ed again with each structure (lines 9–10); if b\nwas 0, then the mask will also be 0 and thus xtoo will become 0;\nperforming the xors will leave each structure unchanged. On the\nother hand, if bwas 1, then the mask will be 0xfff...ff andx\nwill remain unchanged. Since xis already the xor of each structure,\nxor’ing it back into each structure will serve to swap the values.\nFinally, the curve points fandgare written back to memory.\nUnder the constant-time leakage model, this implementation has\nno secret-dependent leaks, as it contains no branches nor secret-\ndependent memory accesses. Unfortunately, it still leaks under\nseveral of the leakage models from §6.1, as we show next.\n6.3.2 Register compression ( RFC0 ,RFCN ).The value of mask is\nderived from the secret bit band is then used to mask x(line 7).\nWhenever the mask is 0, the resulting operations will also produce 0;\nin the RFC models, we will observe this as a cluster of compressions\nto the zero register, which are recorded in the leakage traces. This\nallows us to determine whether the original condition bit was 0\nor 1 in each loop iteration.\n6.3.3 Computation simplification ( CST).After the temporary value\nxhas been masked (line 7), it is xor’ed back against fandgrespec-\ntively to perform the swap (lines 9–10). Whenever the mask is0,\nthexwill also be 0. However, the xor’ing 0with any value leaves\nit unchanged; thus in the CST model, we will observe any such\nxor operations as being simplified, which is recorded as part of the\nleakage trace. Again, this allows us to recover the condition bit b.\n6.3.4 Silent stores ( SS).After the values of fandgare possibly-\nswapped, the two points are written back to memory. Whenever\nthe values are not swapped, the memory writes on lines 9–10 do\nnot modify the memory. In the SSmodel, these stores will be sup-\npressed and produce observations in the leakage traces, which let\nus recover b.\n6.3.5 Assessment. Our analysis highlights that, for several leak-\nage models, the leaks detected by LmTest can be exploited to re-\ncover a secret key from the leakage traces. In particular, the leaks\nare directly caused by the constant-time implementation of the\ncompare-and-swap algorithm introduced to prevent side-channel\nleaks under the standard constant-time model. We also remark that\nthe exploited leaks are already present under the 𝑆eqprediction\nclause, i.e., they do not result from speculative instructions.7 DISCUSSION\nScope of the models: The goal of the leakage models presented\nin this paper is to capture the core aspects associated with the\nstudied microarchitectural optimizations while (a) enabling testing\nof real-world cryptographic implementations and (b) illustrating\nthe expressiveness of the LmSpec language. As a result, our models\nsimplify many aspects of modern CPUs, which might influence\ntheir faithfulness to any specific hardware. Note also that some of\nour models are associated with optimization proposals rather than\nconcrete implementations; different microarchitectural implemen-\ntations of the same proposal, therefore, might result in different\nleakage profiles.\nScope of the results in §6: Our investigation highlighted that\noptimization proposals can potentially compromise the security of\ncurrent cryptographic implementations, despite the use of constant-\ntime programming. Lifting the results of our testing campaign to\nreal-world CPUs, however, is only possible to the extent that our\nleakage models precisely capture the microarchitectural informa-\ntion flows in these CPUs. Even for the optimizations currently im-\nplemented by modern processors (e.g., prefetching), LmTest results\nmight incorrectly classify programs either as secure or insecure\ndue to mismatches between actual and modeled leaks.\nRegarding the exploitability analysis in §6.3, an important caveat\nis that our analysis only refers to leakage models, given that the\nstudied optimizations are (to the best of our knowledge) not yet im-\nplemented in modern CPUs. Even considering a CPU implementing\nthe target optimizations, lifting our analysis to a full-blown attack\nmight be challenging for two reasons. First, our models abstract\naway from many detailed aspects of modern CPU microarchitec-\ntures and, thus, might not faithfully capture all leaks happening\non a specific CPU. Second, our analysis assumes that every leak-\nage observation is immediately visible to an attacker. However, in\na practical setting, the attacker will not have access to such pre-\ncise observations. Instead, they will only be able to observe (noisy)\nmeasurements, e.g., variations in program execution time.\nScope of LmSpec andLmTest :We see LmSpec andLmTest as\na way for both hardware vendors and cryptographic developers\nalike to easily study the security implications of microarchitectural\noptimization proposals. This will enable identifying potential leaks\nduring the development of new microarchitectural optimizations\nbefore their implementation in silicon, thereby enabling program-\nmers to develop program-level countermeasures early on.\n8 RELATED WORK\nComparison with the Pandora works: Here, we review [ 60],\nwhich is a direct inspiration for our work, and a recent follow-up\npaper [21].\nSanchez Vicarte et al. [ 60] conduct a systematic review of the\nsecurity implications of microarchitectural optimizations and per-\nform an in-depth analysis of seven classes of microarchitectural\noptimizations. Their work provides semi-formal descriptions of the\nleakage models associated to several optimizations (e.g., operand\npacking and computation reuse). Their descriptions are based on the\nnotion of microarchitectural leakage descriptors (MLDs). However,\nthese descriptors are informal and often incomplete. For instance,G. Barthe, M. Böhme, S. Cauligi, C. Chuengsatiansup, D. Genkin, M. Guarnieri, D. Mateos Romero, P. Schwabe, D. Wu, Y. Yarom\nthe MLD for computation reuse does not describe how the reuse\nbuffer is updated throughout execution. Our models, instead, cap-\nture the salient aspects of specific optimization proposals and have\nexecutable implementations. For instance, CRandCRA capture the\nkey aspects of computation reuse from Sodani and Sohi [62].\nA main novelty of our work is a framework for evaluating the se-\ncurity implications of arbitrary microarchitectural proposals against\nreal-world cryptographic implentations by (a) modeling them as Lm-\nSpec leakage models, and (b) using LmTest to automatically detect\nleaks through random testing. Additionally, we consider the interac-\ntions between the leakage models and speculative execution, which\nare not explored systematically in [ 60]. Our evaluation in §6 consid-\ners all optimization classes from [60] and our results confirm that,\nif implemented, such microarchitectural optimizations might com-\npromise the security of existing cryptographic implementations.\nSanchez Vicarte et al. also explore the relevance of security leaks\nfor two classes of optimizations: silent stores and data-dependent\nprefetching. Concretely, they present two proof-of-concept attacks\nexploiting silent stores (against Bitslice AES128) and data-dependent\nprefetching (against Ebpf). Their attack for silent-stores is imple-\nmented on top of Gem5 [ 42] and allows to recover the secret key\nused by the Bitslice AES128 encryption algorithm. The attack relies\non (1) secret-dependent information being copied to the stack, and\n(2) the attacker being able to call the encryption algorithm repeat-\nedly with attacker-controlled data (to trigger silent stores). The\nleak exploited in this attack is similar to several SSleaks found by\nLmTest , e.g., those detected in libsodium’s Salsa20 implementation.\nOur analysis of x25519 corroborates their findings that these leaks\ncould be exploited in idealized scenarios where the attacker is able\nto observe fully the leakage described by the LmSpec model.\nIn a follow up work [ 21], Flanders et al. develop a program\nrewriting approach for hardening cryptographic implementations\nagainst two of Pandora’s leakages: silent stores and computation\nsimplification. Their approach shows that it is possible to protect\nimplementations against some of the Pandora leakages. However,\nit incurs a significant performance overhead due to its generality.\nOur work does not consider mitigations. However, it would be\ninteresting to explore in the future how our approach could be\nleveraged for validating algorithm-specific mitigations.\nAttacks related to the studied leakage models: Here, we review\nexisting attacks targeting leaks related with the leakage models\nstudied in this paper.\nCiphertext attacks [ 38,39] exploit the memory encryption in\nAMD SEV-SNP which employs a tweakable encryption mode where\na ciphertext depends on a plaintext and a physical address. When-\never a store operation at location 𝑛happens, an attacker can infer\nwhether the new value is different from the old value at that location\nby observing changes in the ciphertext at 𝑛(ciphertexts are different\niff plaintexts are different). This is exactly the same leakage model\nas for silent stores (§3.2). Hence, mitigations against ciphertext\nattacks [16, 75] should be effective also against silent store leaks.\nFinally, the Augury attack [ 68] exploits the 1-level pointer chas-\ning prefetcher implemented in M1 processors, whereas the Safe-\ncracker attack [ 66] exploits data compression schemes in caches to\ninfer the content of cache lines. The leakage clauses for prefetching(PFDD ) and cache compression ( CC) used in our case study are\ninspired by the leaks exploited in [66, 68].\nAttacks outside our leakage models: Several recent works have\nexploited leakage through power consumption of the CPU, either\ndirectly [ 41] or indirectly through the impact of the power con-\nsumption on the CPU heat and frequency [ 63,70,71]. As these\nattacks do not observe the microarchitecture directly, it is not clear\nwhether they can be modeled in LmSpec .\nRAMBleed [ 36] exploits the Rowhammer effect [ 34] to leak data\nfrom memory. The attack leaks data at rest, and it is not affected by\nexecution models. Thus it is less compatible with our framework.\nSimilarly, attacks that exploit data compression [ 3,61,69] rely on\nvulnerabilities in software, which are out of scope for LmSpec .\nFormal models and analysis: Many formal models capturing\ntiming leaks at microarchitectural level have been proposed. Ini-\ntially, researchers proposed models capturing leaks associated with\n“constant-time” [ 5,46], e.g., by instrumenting a program’s semantics\nto produce leakage traces exposing memory accesses and control-\nflow. More recently, researchers have proposed models capturing\nleaks associated with speculatively executed instructions. Some\nmodels [ 20,26,27,47,54] extend program-level semantics with\ndedicated observations to capture microarchitectural side effects\n(like cache accesses) and capture the effects of speculatively exe-\ncuted instructions at high level by allowing the program semantics\nto explore mispredicted paths for a fixed number of steps [ 26]. Other\nmodels [ 13,25,27,44,67] rely on more complex models that explic-\nitly capture components like pipeline stages, caches, and branch\npredictors.\nTheLmSpec language provides a way of rapidly prototyping and\nformalizing these leakage models. As a proof of LmSpec ’s expres-\nsiveness, we used it to capture a large class of leakage models. In\nparticular, beyond the constant-time model (Figure 1), we success-\nfully implemented in LmSpec (a) leakage clauses capturing the leaks\ninduced by all optimization classes studied by Sanchez Vicarte et al .\n[60], and (b) speculative models capturing speculation over branch,\nstore, and return instructions as well as straight-line speculation.\nWe remark that LmSpec ’s design took inspiration from prior work:\n(1) the notions of leakage and prediction clauses is inspired by the\nmodels from [ 27], (2) the modeling of speculation is inspired by the\nalways mispredict speculative semantics from [ 26], and (3) its exe-\ncutable implementation on top of the Unicorn emulator is inspired\nby the Revizor testing tool [50].\nTesting for leaks: Here, we review relevant prior work on detect-\ning leaks using testing-based approaches.\nThere are several approaches for detecting leaks in programs\nagainst specific leakage models. For instance, CtFuzz [28] and\nCtGrind [15] detect leaks against the constant-time model (see\nFigure 1 for its LmSpec encoding). In particular, CtFuzz constructs\na self-composition of the program under test with itself, which is\nthen fuzzed for violations (i.e., by inspecting the traces produced by\nself-composed program) using the Aflfuzzer. This is different from\nLmTest , which executes the program under test on individual in-\nputs and compares pairs of traces. In contrast, CtGrind [15] allows\nchecking violations of constant-time on top of ValGrind [ 48] using\ntaint-tracking. Finally, SpecFuzz [52] detects speculative bound\ncheck bypasses (BCB) against an always-mispredict speculationTesting side-channel security of cryptographic implementations against future microarchitectures\nmodel capturing speculation over branch instruction (i.e., the model\nin Figure 2). Differently from LmTest , however, all these approaches\nare tied to specific leakage models.\nRather than relying on a leakage model, dudect [58] detects\nside-channel leaks in programs by directly performing hardware\nmeasurements. Therefore, dudect can detect actual leaks against\ncommercial processors. Finally, Microwalk [ 74,76] employs binary\ninstrumentation to collect leakage log from functions, detecting\nleakage when logs are affected by change in secret variables.\nTesting has also been used to automatically discover leaks in\nprocessors rather than specific programs. For instance, tools like\nScam-V [ 12,47] and Revizor [ 29,50,51] can be used to detect leaks\nin commercial processors against a leakage model used as a spec-\nification. Other approaches [ 24,45,73], instead, detect leaks by\nanalyzing hardware measurements without the help of a formal\nleakage model. Finally, tools like SpecDoctor [ 31], SIGFuzz [ 57],\nand AutoCC [ 53] can test processor designs for leaks and they are\napplicable in the pre-silicon phase. We remark, however, that all\nthese tools differ in scope from LmTest :LmTest detect leaks in\nprograms, whereas these tools detect leaks in the CPU under test.\nFinally, Pensieve [ 78] is a framework for evaluating the security\nof early-stage microarchitectural defenses against Spectre attacks.\nIt allows hardware developers to (a) specify a given countermeasure\non top of an out-of-order processor model, and (b) find counterex-\namples to speculative non-interference [ 26] using model check-\ning. Thus, Pensieve aims at finding problems in hardware coun-\ntermeasures. In contrast, our approach aims that evaluating the\nimplications of microarchitectural proposals on the side-channel\nguarantees of programs (even beyond speculative execution).\n9 CONCLUSION\nIn the future, chip vendors are expected to implement new and more\naggressive microarchitectural optimizations to speed up computa-\ntion. Given the extended development time of hardware mitigations,\nand the performance cost and often slow adoption rate of software\ncountermeasures, it is critical that the security analysis of these new\noptimizations happens early on during their development, ideally\nbefore their availability in commercial processors.\nTo enable this early-stage security analysis, we proposed a frame-\nwork (consisting of the LmSpec language and the LmTest testing\ntool) for evaluating the side-channel guarantees of programs against\n(future) microarchitectural optimizations. With our framework, we\nperformed a large-scale study of the implications of a large class of\noptimizations, recently identified as security-critical, on the secu-\nrity of mainstream cryptographic libraries. Our results confirmed\nthat these optimizations, if implemented, would compromise the\nsecurity of all analyzed libraries.\nACKNOWLEDGMENTS\nWe would like to thank Boris Köpf for his feedback on earlier\nversions of this paper.\nThis work was partially supported by the Spanish Ministry of\nScience and Innovation under the project TED2021-132464B-I00\nPRODIGY; the Spanish Ministry of Science and Innovation under\nthe Ramón y Cajal grant RYC2021-032614-I; the Spanish Ministryof Science and Innovation under the project PID2022-142290OB-\nI00 ESPADA; the Australian Research Council Discovery Project\nDP210102670; the Deutsche Forschungsgemeinschaft (DFG, Ger-\nman Research Foundation) under Germany’s Excellence Strategy\n- EXC 2092 CASA - 390781972; the Air Force Office of Scientific\nResearch (AFOSR) under award number FA9550-20-1-0425 the De-\nfense Advanced Research Projects Agency (DARPA) under contract\nnumber W912CG-23-C-0022 the National Science Foundation (NSF)\nunder grant number CNS-1954712 and gifts from Intel, Qualcomm,\nand Cisco.\nREFERENCES\n[1] [n. d.]. 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In USENIX Security . 719–732.\nA ADDITIONAL LEAKAGE CLAUSES\nHere, we provide the LmSpec modelling of the missing leakage\nclauses from §6.1.1.\nA.1 SSI\n1 (defleakage SilentStoreInitializedOnly [initialized\n(set)] ↩→\n2 (on-start [model input ]\n3 (.update initialized (sfor addr input.mem-initialized\n(- model.STACK addr )))) ↩→\n4 (on[(store [addr ]_sz := val )\n5 (let [ addrs (range addr (+ addr sz ))\n6 was-init (.issuperset initialized addrs )]\n7 (.update initialized addrs )\n8 (when ( and was-init\n9 (= val (&mem.read addr sz )))\n10 #(\"ss\" addr val )))]))\nA.2 SSI0\n1 (defleakage SilentStore0InitializedOnly [initialized\n(set)] ↩→\n2 (on-start [model input ]\n3 (.update initialized (sfor addr input.mem-initialized\n(- model.STACK addr )))) ↩→\n4 (on[(store [addr ]_sz := val )\n5 (let [ addrs (range addr (+ addr sz ))\n6 was-init (.issuperset initialized addrs )]\n7 (.update initialized addrs )\n8 (when ( and was-init\n9 (=0val (&mem.read addr sz )))\n10 #(\"ss\" addr val )))]))A.3 RFC0\n1 (defleakage RegisterFileCompression0 []\n2 (on[(write reg := val )\n3 (when ( and (in reg X86_64_GPRS )\n4 (= val 0)\n5 (exists reg_i X86_64_GPRS\n6 :where (!= reg_i reg )\n7 (= val (&regs.read reg_i ))))\n8 #(\"rfc\" reg val ))]))\nA.4 RFCN\n1 (setv NARROW_RFC_LIMIT (<<1 16))\n2 (defleakage NarrowRegisterFileCompression []\n3 (on[(write reg := val )\n4 (when ( and (in reg X86_64_GPRS )\n5 (< val NARROW_RFC_LIMIT )\n6 (exists reg_i X86_64_GPRS\n7 :where (!= reg_i reg )\n8 (<(&regs.read reg_i )NARROW_RFC_LIMIT )))\n9 #(\"rfc\" reg ))]))\nA.5 CS\n1 (setv ST-ADD #{\"add\" \"shl\" \"sal\" \"shr\" \"sar\" }\n2 ST-SUB #{\"sub\" }\n3 ST-MUL #{\"mul\" \"imul\" }\n4 ST-DIV #{\"div\" \"idiv\" }\n5 ST-AND #{\"and\" \"or\" }\n6 ST-XOR #{\"xor\" }\n7 )\n8 (defleakage SemiTrivialComputationSimplification []\n9 (on[(expr (op v1 v2 ))\n10 (when ( is-any-of op\n11 ST-ADD :if (or(= v1 0) ( = v2 0))\n12 ST_SUB :if (or(= v2 0) ( = v1 v2 ))\n13 ST-MUL :if (or(= v1 0) ( = v2 0)\n14 (= v1 1) ( = v2 1))\n15 ST-DIV :if (or(= v1 0) ( = v2 1) ( = v1 v2 ))\n16 ST-AND :if (or(= v1 0) ( = v2 0)\n17 (= v1 ALL1 ) (= v2 ALL1 )\n18 (= v1 v2 ))\n19 ST-XOR :if (or(= v1 0) ( = v2 0)) ; XXX should\nthis also have (= v1 v2)? ↩→\n20 )\n21 ;; XXX rotations?\n22 #(\"cs\" op v1 v2 ))]))\nA.6 CST\n1 (setv ALL1 (-(<<1 64) 1))\n2 (setv T-MUL #{\"mul\" \"imul\" \"and\" }\n3 T-OR #{\"or\" }\n4 T-DIV #{\"div\" \"idiv\" \"shl\" \"sal\" \"shr\" \"sar\" })\n5 (defleakage TrivialComputationSimplification []\n6 (on[(expr (op v1 v2 ))\n7 (when ( is-any-of op T-MUL :if (or(= v1 0) ( = v2\n0)) ↩→\n8 T-OR :if (or(= v1 ALL1 ) (= v2\nALL1 )) ↩→\n9 T-DIV :if (= v1 0))\n10 #(\"cs\" op v1 v2 ))]))G. Barthe, M. Böhme, S. Cauligi, C. Chuengsatiansup, D. Genkin, M. Guarnieri, D. Mateos Romero, P. Schwabe, D. Wu, Y. Yarom\nA.7 CSN\n1 (setv NARROW_CS_LIMIT (<<1 32))\n2 (defleakage NarrowComputationSimplification []\n3 (on[(expr (op v1 v2 ))\n4 (when ( and (in op ST-MUL )\n5 (< v1 NARROW_CS_LIMIT )\n6 (< v2 NARROW_CS_LIMIT ))\n7 #(\"cs\" op ))]))\nA.8 OP\n1 (setv OP_CTX_SIZE 200)\n2 (defleakage OperandPacking [ctx (deque )]\n3 (on[(expr (op v1 v2 ))\n4 (when ( and (< v1 16) ( < v2 16))\n5 (while ( and ctx (>=(-&tick (. ctx [0][0]))\nOP_CTX_SIZE )) ↩→\n6 (.popleft ctx ))\n7 (for [[ i[tick_i op_i ]] ( enumerate ctx )]\n8 (when ( = op_i op )\n9 (del ( . ctx [i]))\n10 (return #( \"op\" op_i op )))\n11 (else\n12 (.append ctx #(&tick op )))))]))\nA.9 CRA\n1 (defleakage ComputationReuseWithAddresses [ctx\n(OrderedDict ) ↩→\n2 ctx-addrs (OrderedDict )\n3 ctx-loads (OrderedDict )]\n4 (on[(expr (op#* vs ))\n5 (when (in op CACHEING_OPS )\n6 (if (in vs(.get ctx &pc#()))\n7 (update ctx &pc vs )))]\n8 [(addr base + index * scale + off )\n9 (if (in #( base index scale off ) (.get ctx-addrs &pc\n#())) ↩→\n10 (update ctx-addrs &pc[base index scale off ]))]\n11 (if (in addr (.get ctx-loads &pc#()))\n12 (update ctx-loads &pc addr ))]))\nA.10 CC−FPC\n1 (defleakage FPCCacheCompression []\n2 (on[(load [addr ]_sz)\n3 (let [ block-addr (<<(>> addr CACHELINE_BITS )\nCACHELINE_BITS ) ↩→\n4 block-data (&mem.read-bytes block-addr\nCACHELINE_SIZE )] ↩→\n5 #(\"cc\" (fpc-size block-data )))]\n6 [(store [addr ]_sz := val )\n7 (let [ block-addr (<<(>> addr CACHELINE_BITS )\nCACHELINE_BITS ) ↩→\n8 block-data (&mem.read-bytes block-addr\nCACHELINE_SIZE ) ↩→\n9 offset (%\n10 (write-into block-data offset sz val )\n11 #(\"cc\" (fpc-size block-data )))]))\nA.11 CC−BDI\n1 (defleakage BDICacheCompression []\n2 (on[(load [addr ]_sz)3 (let [ block-addr (<<(>> addr CACHELINE_BITS )\nCACHELINE_BITS ) ↩→\n4 block-data (&mem.read-bytes block-addr\nCACHELINE_SIZE )] ↩→\n5 #(\"cc\" (bdi-size block-data )))]\n6 [(store [addr ]_sz := val )\n7 (let [ block-addr (<<(>> addr CACHELINE_BITS )\nCACHELINE_BITS ) ↩→\n8 block-data (&mem.read-bytes block-addr\nCACHELINE_SIZE ) ↩→\n9 offset (%\n10 (write-into block-data offset sz val )\n11 #(\"cc\" (bdi-size block-data )))]))\nA.12 PFNL\n1 (setv POINTER_SIZE 8)\n2 (setv CACHELINE_BITS 6)\n3 (setv PAGE_BITS 12)\n4 (defleakage NextLinePrefetch []\n5 (on[(load [addr ]_sz)\n6 (let [ cache-index (>> addr CACHELINE_BITS )]\n7 #(\"pf\" (+ cache-index 1)))]))\nA.13 PFS\n1 (setv PF_HITS 3)\n2 (defn diff [ns]\n3 (setv [ ns1 ns2 ] (tee ns ))\n4 (next ns2 )\n5 (lfor [n m] (zip ns1 ns2 ) (- m n )))\n6 (defn direction-of? [page-hits ]\n7 (when ( <(len page-hits )page-hits.maxlen )\n8 (return 0))\n9 (setv diffs (diff page-hits ))\n10 (cond ( forall n diffs (> n 0)) 1\n11 (forall n diffs (< n 0)) -1\n12 True 0))\n13 (defleakage StreamPrefetch [all-page-hits (ddict #%\n14 (on[(load [addr ]_sz)\n15 (let [ cache-index (>> addr CACHELINE_BITS )\n16 page-index (>> addr PAGE_BITS )\n17 page-hits (get all-page-hits page-index )\n18 _(when (not-in cache-index page-hits )\n(.append page-hits cache-index )) ↩→\n19 stream-dir (direction-of? page-hits )\n20 next-cache-index (+ stream-dir cache-index )]\n21 (when ( and stream-dir\n22 (=(>> next-cache-index (- PAGE_BITS\nCACHELINE_BITS ))page-index )) ↩→\n23 #(\"pf\" next-cache-index )))]))\nA.14 PFDD\n1 (setv M1PF_SIZE 20)\n2 (setv M1PF_PREFETCH 5) ; we prefetch 5 elements\n3 (defleakage M1Prefetch [initialized (set)\n4 accesses (deque :maxlen M1PF_SIZE )\n5 marks (deque :maxlen PF_HITS )]\n6 (on-start [model input ]\n7 (.update initialized (sfor addr\ninput.mem-initialized (- model.STACK addr )))) ↩→\n8 (on[(store [addr ]_sz := val )Testing side-channel security of cryptographic implementations against future microarchitectures\n9 (.update initialized (range addr (+ addr sz )))]\n10 [(load [addr ]_sz)\n11 (let [ val (&mem.read addr sz )]\n12\n13 (setv stride 0)\n14 (for [[ addr_i val_i ] (reversed accesses )]\n15 (when ( = val_i addr )\n16 (.append marks addr_i )\n17 (let [ diffs (set (diff marks ))]\n18 (when ( =(len diffs ) 1)\n19 (setv [ stride ]diffs )))\n20 (break)))\n21\n22 (.append accesses #(addr val ))\n23\n24 (when stride\n25 (setv last-aop (. marks [-1])\n26 fetches [])\n27 (for [ i(range M1PF_PREFETCH )]\n28 (let [ aop-el (+ last-aop (* i stride ))]\n29 (when (in aop-el initialized )\n30 (.extend fetches [aop-el (&mem.read\naop-el POINTER_SIZE )])))) ↩→\n31 #(\"pf\" #* fetches )))]))\nB ADDITIONAL PREDICTION CLAUSES\nHere, we provide the LmSpec modelling of the missing prediction\nclauses from §6.1.2.\nB.1 R sb⊥\n1(setv RSB_SIZE 16)\n2(defpredictor RSBCircular [stack (*[0] RSB_SIZE )\n3 idx 0]\n4;; RSB that drops oldest entry on overflow\n5;; and halts on underflow\n6(on[(jump addr : n )\n7 (cond\n8 (&insn.group CS_GRP_CALL )\n9 (.append stack (+&pc&insn.size ))\n10 (&insn.group CS_GRP_RET )\n11 (if stack [(.pop stack )] [ HALT ]))]))"    },    {        "title": "2402.00642v1.Variants_of_the_Erdős_distinct_sums_problem_and_variance_method.pdf",        "content": "arXiv:2402.00642v1  [math.CO]  1 Feb 2024VARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM\nAND VARIANCE METHOD\nSIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nAbstract. Let Σ = {a1,...,a n}be a set of positive integers with a1< ... < a n\nsuch that all 2nsubset sums are pairwise distinct. A famous conjecture of Er d˝ os\nstates that an> C·2nfor some constant C, while the best result known to date\nis of the form an> C·2n/√n. In this paper, we propose a generalization of the\nErd˝ os distinct sum problem that is in the same spirit as thos e of the Davenport\nand the Erd˝ os-Ginzburg-Ziv constants recently introduce d in [7] and in [6]. More\nprecisely, we require that the non-zero evaluations of the m-th degree symmetric\npolynomial are all distinct over the subsequences of Σ whose size is at most λn,\nfor a given λ∈(0,1], considering Σ as a sequence in Zkwith each coordinate of\neachaiin [0,M]. IfFλ,ndenotes the family of subsets of [1 ,n] whose size is at\nmostλn, our main result is that, for each k,m,andλ, there exists an explicit\nconstant Ck,m,λsuch that\nM≥Ck,m,λ(1+o(1))|Fλ,n|1\nmk\nn1−1\n2m.\n1.Introduction\nFor any n≥1, consider sets {a1,...,a n}of positive integers with a1< ... < a n\nwhose subset sums are all distinct. A famous conjecture, due to P aul Erd˝ os, is that\nan≥C·2nfor some constant C >0. Using the variance method, Erd˝ os and Moser\n[11] (see also [1] and [14]) were able to prove that\nan≥1/4·n−1/2·2n.\nNo advances have been made so far in removing the term n−1/2from this lowerbound,\nbut there have been several improvements on the constant fact or, including the work\nof Dubroff, Fox, and Xu [12], Guy [13], Elkies [10], Bae [4], and Aliev [3]. In pa rticular,\nthe best currently known lower bound states that\nan≥/radicalbigg\n2\nπ1√n2n(1+o(1)).\nTwo simple proofs of this result, first obtained unpublished by Elkies a nd Gleason,\nare presented in [12]. In the other direction, the best-known cons truction is due to\nBohman [5] (see also [15]), who showed that there exist arbitrarily lar ge such sets with\nan≤0.22002·2n.\nSeveral variations on the problem have appeared during the years , such as [2] and\n[8].\nFollowing the notation of [8], we denote by Fλ,nthe family of all subsets of [1 ,n]\nhaving size at most λn, whereλ∈(0,1] is a given constant. The following problem\nwas introduced in [8].\n2010Mathematics Subject Classification. 05D40, 11B13.\nKey words and phrases. Erd˝ os distinct-sums problem, variance method.\n12 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nProblem 1.1. For every positive integer n, find the least positive M=M(n)such\nthat there exists a sequence Σ = (a1,...,a n)of elements of Zkwithai∈[0,M]kfor\neveryi(i.e. with every aihaving non-negative coordinates) such that for all distinc t\nA1,A2∈ Fλ,nwe have that:/summationdisplay\ni∈A1ai/ne}ationslash=/summationdisplay\ni∈A2ai.\nIn this paper, we propose two progressive variants of the Erd˝ os distinct sum prob-\nlem. The first one, inspired by the recent works [7] and in [6] on Dave nport and the\nErd˝ os-Ginzburg-Ziv constants, is to require that the non-zero evaluations of the m-th\ndegree symmetric polynomial are all distinct over the sub-sequenc es of Σ.\nGiven a sequence of integers Σ = ( a1,...,a n) and a subset A⊆[1,n], we define the\nm-th (degree) evaluation\nem\nΣ(A) =/summationdisplay\n{i1,...,im}⊆A\ni1<...<i mai1···aim,\nwhere we adopt the convention that em\nΣ(A) = 0 if|A|< m.\nProblem 1.2.1For every positive integer n, find the least positive M=M(n)such\nthat there exists a sequence Σ = (a1,...,a n)of integers with ai∈[0,M]for every i\nsuch that for all distinct A1,A2⊆[1,n]of size at least mwe have that:\nem\nΣ(A1)/ne}ationslash=em\nΣ(A2).\nA sequence as in Problem 1.2 will be called M-bounded m-th evaluation distinct .\nThe same terminology applies to sequences in Zksuch that every entry belongs to\n[0,M]kand that the evaluation vectors are all distinct.\nBy computing the variance of the random variable that associates t o a randomly\nchosen set A⊆[1,n] the evaluation em\nΣ(A), one finds, surprisingly (these evaluations\ncan not be seen as the values assumed by the sum of independent ra ndom variables),\nthat the terms of higher degree disappear, and hence the varianc e method provides a\nnontrivial bound on M. In order to understand the cause of this behavior, we then\ninvestigate a generalization of Problem 1.2 that is in the same spirit as P roblem 1.1.\nWe consider here a sequence Σ in Zkand we require that the vector-valued evaluations\nofem\nΣare all distinct.\nProblem 1.3. For every positive integer n, find the least positive M=M(n)such\nthat there exists an M-bounded sequence Σ = (a1,...,a n)inZksuch that, for all\ndistinctA1,A2∈ Fλ,nof size at least m,\nem\nΣ(A1)/ne}ationslash=em\nΣ(A2).\nFor this very general problem, a symmetry argument allows us to sh ow that the\nterms of higher degree disappear, explaining therefore the behav ior of Problems 1.1\nand 1.2. Our result on Problem 1.3 also improves, in the case m= 1 and λ <1/2,\nTheorem 2.5 of [8], that left this case open.\nThe paper is organized as follows. Section 2 is devoted to providing a d irect lower\nbound on the values of Min Problem 1.2. Using the variance method, we provide a\nnontrivial bound on M; in particular we prove that:\nM > C m·2n\nm/n1−1\n2m.\n1This problem has been presented at the international confer ence “Eurocomb 2023”, see [9].VARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM AND VARIANCE 3\nIn Section 3 we obtain, combining a symmetry argument with the varia nce method, a\nnontrivial bound on Mfor Problem 1.3. In particular, we prove that:\nM≥Ck,m,λ(1+o(1))|Fλ,n|1\nmk\nn1−1\n2m\nwhereCk,m,λis an explicit constant.\nFinally, in Section 4, we derive a probabilistic upper bound on the const antMof\nProblem 1.3 and we present a direct construction to improve this bou nd for Problem\n1.2.\n2.Lower Bounds via Variance Method: Problem 1.2\nIn Problem 1.2, a first lower bound to the value of Mcan be provided using the\npigeonhole principle. Indeed, since the number of non-zero evaluat ions ofem\nΣis 2n−/summationtextm−1\ni=0/parenleftbign\ni/parenrightbig\n= (1 +o(1))2n, these evaluations are spaced at least by one, and each of\nthese is smaller than em\nΣ([1,n])≤/parenleftbign\nm/parenrightbig\nMm≤nm/(m!)Mm, it follows that:\nM > D m·2n\nm/n.\nIn this section we show, using the variance method (see [1], [11] or [1 3]), that it is\npossible to improve this lower bound. Note that the lower bound we will find in this\nproof holds also when considering sequences of real numbers. In t his case, we consider\na sequence Σ to be m-th evaluation distinct whenever |em\nΣ(A1)−em\nΣ(A2)| ≥1 for any\ndistinctA1,A2⊆[1,n] of size at least m.\nTheorem 2.1. LetΣ = (a1,...,a n)be anm-th evaluation distinct sequence in Z\n(resp.R) that is M-bounded. Then\nM >21−1\nm((m−1)!)1\nm\n31\n2m2n\nm\nn1−1\n2m(1+o(1)).\nProof.Let Σ = ( a1,...,a n) be anm-th evaluation distinct sequence of integers (resp.\nrealnumbers). Pickasubset Auniformly atrandomfrom2[1,n]anddefine therealran-\ndom variable X=em\nΣ(A). We denote by µ:=E[X] andσ2:=E[X2]−µ2respectively\nthe expected value and the variance of the random variable X.\nWe have that:\nµ=/summationdisplay\nA⊆[1,n]:|A|≥mem\nΣ(A)\n2n.\nTheterm ai1...aimappearsintheevaluation em\nΣ(A)preciselywhen Acontains {i1,...,\nim}; this happens for 2n−msubsets of [1 ,n]. Therefore, we have that:\nµ=2n−m\n2n/summationdisplay\ni1<i2<...<i m\ni1,...,im∈[1,n]ai1...aim=em\nΣ([1,n])\n2m.\nBy definition of variance, we have that:\n2nσ2=/summationdisplay\nA⊆[1,n](em\nΣ(A)−µ)2=/summationdisplay\nA⊆[1,n]\n/summationdisplay\ni1<i2<...<i m\ni1,...,im∈Aai1...aim−/summationdisplay\ni1<i2<...<i m\ni1,...,im∈[1,n]ai1...aim\n2m\n2\n.\nThanks to the symmetry of em\nΣ, there exist coefficients C0,...,C msuch that the\nlatter sum can be written as follows:4 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\n(1)C0/summationdisplay\ni1<i2<...<i 2m\ni1,...,i2m∈[1,n]ai1...ai2m+C1/summationdisplay\ni1<i2<...<i 2m−1\ni1,...,i2m−1∈[1,n]/summationdisplay\nℓ∈[1,2m−1]ai1ai2...a2\niℓ...ai2m−1+\n+...+Cm/summationdisplay\ni1<i2<...<i m\ni1,...,im∈[1,n]a2\ni1...a2\nim.\nWe now claim that:\n(a)C0= 0;\n(b)C1= 2n−2m/parenleftbig2m−2\nm−1/parenrightbig\n;\n(c)Ck=O(2n) for every k∈ {2,...,m}.\n(a) The coefficient of ai1...ai2mis/parenleftbig2m\nm/parenrightbig\ntimes that obtained by taking the term\nai1...aimfrom the first ( em\nΣ(A)−µ) in the product and aim+1...ai2mfrom the sec-\nond one. Here we note that we can choose the monomial 1 ·ai1...aimin the first\n(em\nΣ(A)−µ) and 1·aim+1...ai2min the second one from any set Athat contains\n{i1,...,i2m}, i.e. in 2n−2mpossible ways. Similarly, we can choose the monomial\n2−m·ai1...aimin the first ( em\nΣ(A)−µ) and 2−m·aim+1...ai2min the second one from\nany set, i.e. in 2npossible ways. Finally, we can choose the monomial 1 ·ai1...aimin\nthe first ( em\nΣ(A)−µ) and 2−m·aim+1...ai2min (em\nΣ(A)−µ) in the second one from\nany set that contains {im+1,...,i2m}, i.e. in 2n−mpossible ways. Thus, we have that\nC0/parenleftbig2m\nm/parenrightbig= 2n−2m−2·2n−m·2−m+2n·(2−m)2= 0.\n(b) The coefficient of a2\ni1...ai2...ai2m−1is/parenleftbig2m−2\nm−1/parenrightbig\ntimes that obtained taking the term\nai1...aimfrom the first ( em\nΣ(A)−µ) in the product and ai1aim+1...ai2m−1from the\nsecond one. Symmetrically, the same is true foreveryterm ai1...a2\niℓ...ai2m−1. Hence,\narguing as in ( a), we get:\nC1/parenleftbig2m−2\nm−1/parenrightbig= 2n−2m+1−2·2n−m·2−m+2n·(2−m)2= 2n−2m.\n(c)Thecoefficientof a2\ni1...a2\nikaik+1...ai2m−kis/parenleftbig2m−2k\nm−k/parenrightbig\ntimesthatobtainedtakingthe\ntermai1...aimfrom the first ( em\nΣ(A)−µ) in the product and ai1...aikaim+1...ai2m−k\nfrom the second one. Again by symmetry and reasoning as in ( a), we get:\nCk/parenleftbig2m−2k\nm−k/parenrightbig= 2n−2m+k−2·2n−m·2−m+2n·(2−m)2=O(2n).\nSumming up, we can rewrite (1) as\n(2) 2nσ2=C1/summationdisplay\ni1<i2<...<i 2m−1\ni1,...,i2m−1∈[1,n]/summationdisplay\nℓ∈[1,2m−1]ai1ai2...a2\niℓ...ai2m−1+\n+O(2n)\nm/summationdisplay\nk=2/summationdisplay\ni1<i2<...<i 2m−k\ni1,...,i2m−k∈[1,n]/summationdisplay\nℓ1<...<ℓ k\nℓ1,...,ℓk∈[1,2m−k]ai1ai2...a2\niℓ1...a2\niℓk...ai2m−k\n.\nIn (2), for each value of k, we have a sum of/parenleftbign\n2m−k/parenrightbig\n·/parenleftbig2m−k\nk/parenrightbig\n<n2m−k\n(2m−2k)!k!terms.\nSinceMis larger than any element of the sequence, we get:\n2nσ2<n2m−1\n(2m−2)!/parenleftbigg2m−2\nm−1/parenrightbigg\n2n−2mM2m+O(2nn2m−2)M2m(3)VARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM AND VARIANCE 5\n= (1+o(1))·/parenleftbiggn2m−1\n((m−1)!)22n−2mM2m/parenrightbigg\n.\nOn the other hand, for |A| ≥m, the evaluations em\nΣ(A) are all different and spaced\natleastbyone,andhencewehavethat( em\nΣ(A)−µ)2assumesatleast1\n2(2n−/summationtextm−1\ni=0/parenleftbign\ni/parenrightbig\n)\ndifferent values. Since the sum/summationtext\nA⊆[1,n](em\nΣ(A)−µ)2is minimized when the values\nare around µand are spaced by one, we obtain the lower bound:\n(4)1+o(1)\n1223n= 21\n2(2n−/summationtextm−1\ni=0(n\ni))/summationdisplay\ni=0i2≤2nσ2.\nTo conclude the proof, it is enough to compare (3) and (4). /square\n3.Lower Bounds via Variance Method: A Symmetry Argument\nAlso in Problem 1.3 one can provide a first lower bound on Musing the pigeonhole\nprinciple. Indeed, the number of non-zero evaluations of em\nΣis|Fλ,n|−/summationtextm−1\ni=0/parenleftbign\ni/parenrightbig\n=\n(1 +o(1))|Fλ,n|, these evaluations are spaced at least by one, and each of these is\nsmaller in each coordinate than/parenleftbigλn\nm/parenrightbig\nMm≤(λn)m\nm!Mm. It follows that:\nM > D k,m,λ(1+o(1))|Fλ,n|1/(km)\nn.\nIn this section we show how, using the variance method, one can impr ove the denom-\ninatornin this lower bound. First of all, we consider the 1-dimensional sequen ce\n(1,1,...,1) and we upper-bound its variance. Then with a symmetry argumen t (see\n(17) below), we reduce the general case to this one.\nIn the next three lemmas, we let Σ be the sequence of nintegers (1 ,...,1). For a\nsubsetAchosen uniformly at random from Fλ,n, we define the real random variable\n¯X=em\nΣ(A). Notice that if |A|=i, thenem\nΣ(A) =/parenleftbigi\nm/parenrightbig\n. Finally, we let ¯ µ:=E[¯X] and\n¯σ2:=E[¯X2]−¯µ2. Notice that:\n(5) |Fλ,n|¯µ=λn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbiggi\nm/parenrightbigg\n.\nWe will also use the fact that, if/parenleftbigz\nm/parenrightbig\nis seen as a polynomial in the variable z, then:\n(6)/parenleftbiggz\nm/parenrightbigg\n=zm−(/summationtextm−1\ni=1i)zm−1+o(zm−1)\nm!.\nLemma 3.1. Ifnis large enough and λ <1/2we have that:\n¯σ2≤√n(λn)2m−2(1+o(1))\n((m−1)!)2\nProof.Using (5) and the definition of variance we get:\n|Fλ,n|¯σ2=/summationdisplay\nA∈Fλ,n(em\nΣ(A)−¯µ)2=λn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg\n−/parenleftbigg\n¯µ−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg/parenrightbigg2\n=/parenleftiggλn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2/parenrightigg\n−/parenleftbigg\n¯µ−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2\n|Fλ,n|\n<λn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2\n. (7)6 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nNow using (6), together with the fact that i < λn, we get that if |i−λn|<4√nthen:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=4√nm(λn)m−1(1+o(1))\nm!=4√n(λn)m−1(1+o(1))\n(m−1)!,\nand hence (7) becomes\n|Fλ,n|¯σ2<λn−4√n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2\n+λn/summationdisplay\ni=λn−4√n+1/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2\n<λn−4√n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2\n+√n(λn)2m−2(1+o(1))\n((m−1)!)2|Fλ,n|. (8)\nNext, we notice that:\nλn−4√n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2\n< n2mλn/summationdisplay\ni=4√n/parenleftbiggn\ni−4√n/parenrightbigg\n=n2mλn/summationdisplay\ni=4√n/parenleftbiggn\ni/parenrightbiggi(i−1)···(i−4√n+2)(i−4√n+1)\n(n−i+1)(n−i+2)···(n−(i−4√n)).\nSince for i,j≤λnandλ <1/2 we have thati−j\n(n+1)−(i−j)≤λn\nn+1−λn<λ\n1−λ=:γ <1,\nit follows from the above computation that:\nλn−4√n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggλn\nm/parenrightbigg/parenrightbigg2\n< n2m|Fλ,n|(γ)4√n< n2m−2|Fλ,n|,\nwhere in the second inequality we used the fact that for nlarge enough ( γ)4√n<1\nn2.\nCombining this with (8) we get that:\n¯σ2<√n(λn)2m−2(1+o(1))\n((m−1)!)2.\n/square\nLemma 3.2. Ifnis large enough and λ >1/2we have that:\n¯σ2≤n2m−1(1+o(1))\n(m!)2\nProof.Analogously to the proof of Lemma 3.1 we have:\n|Fλ,n|¯σ2=/summationdisplay\nA∈Fλ,n(em\nΣ(A)−¯µ)2=λn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggn/2\nm/parenrightbigg\n−/parenleftbigg\n¯µ−/parenleftbiggn/2\nm/parenrightbigg/parenrightbigg/parenrightbigg2\n=/parenleftiggλn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggn/2\nm/parenrightbigg/parenrightbigg2/parenrightigg\n−/parenleftbigg\n¯µ−/parenleftbiggn/2\nm/parenrightbigg/parenrightbigg2\n|Fλ,n|\n<λn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggn/2\nm/parenrightbigg/parenrightbigg2\n. (9)\nNow using (6), together with the fact that i≤n, we get:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggn/2\nm/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|i−n/2|/parenleftbigg(n/2)m−1+(n/2)m−2i+...+im−1+o(nm−1)\nm!/parenrightbigg\n<|i−n/2|/parenleftbiggnm−1((1/2)m−1+(1/2)m−2+...+1+o(1))\nm!/parenrightbiggVARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM AND VARIANCE 7\n<|i−n/2|/parenleftbigg2nm−1(1+o(1))\nm!/parenrightbigg\n,\nand hence by (9) we deduce:\n(10) |Fλ,n|¯σ2</parenleftiggλn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(i−n/2)2/parenrightigg/parenleftbigg2nm−1(1+o(1))\nm!/parenrightbigg2\n.\nNow notice that:\nλn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(i−n/2)2<n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(i−n/2)2,\nand hence setting A:=/summationtextn\ni=0/parenleftbign\ni/parenrightbig\niandB:=/summationtextn\ni=0/parenleftbign\ni/parenrightbig\ni2we have:\nn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(i−n/2)2=−nA+B+n2\n4n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n=−nA+B+n22n−2.\nSince\nA=n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\ni=nn/summationdisplay\ni=0/parenleftbiggn−1\ni−1/parenrightbigg\n=n2n−1\nand\nB=n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\ni2=nn/summationdisplay\ni=0/parenleftbiggn−1\ni−1/parenrightbigg\ni=nn/summationdisplay\ni=0/parenleftbiggn−1\ni−1/parenrightbigg\n(i−1+1) =\nn(n−1)n/summationdisplay\ni=0/parenleftbiggn−2\ni−2/parenrightbigg\n+nn/summationdisplay\ni=0/parenleftbiggn−1\ni−1/parenrightbigg\n=n(n−1)2n−2+n2n−1=n(n+1)2n−2,\nit follows that\n−nA+B+n22n−2=n2n−2.\nTherefore, we can deduce from (10) that:\n|Fλ,n|¯σ2< n2n−2/parenleftbigg2nm−1(1+o(1))\nm!/parenrightbigg2\n.\nSince for λ >1/2 we have |Fλ,n|= 2n(1+o(1)), the claim follows. /square\nLemma 3.3. Ifnis large enough and λ= 1/2we have that:\n¯σ2≤n2m−1(1+o(1))\n22m((m−1)!)2\nProof.Arguing as at the beginning of the proof of Lemma 3.2, we get to:\n(11) |Fλ,n|¯σ2<⌊n/2⌋/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg/parenleftbigg/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggn/2\nm/parenrightbigg/parenrightbigg2\n.\nNow using (6) together with the fact that i≤n/2, we get that:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggi\nm/parenrightbigg\n−/parenleftbiggn/2\nm/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|i−n/2|/parenleftbigg(n/2)m−1+(n/2)m−2i+...+im−1+o(nm−1)\nm!/parenrightbigg\n<|i−n/2|/parenleftbiggnm−1(1+o(1))\n2m−1(m−1)!/parenrightbigg\n.\nHence from (11) we get:\n(12) |Fλ,n|¯σ2<\n⌊n/2⌋/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(i−n/2)2\n/parenleftbiggnm−1(1+o(1))\n2m−1(m−1)!/parenrightbigg2\n.8 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nNext, notice that:\n⌊n/2⌋/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(i−n/2)2=1\n2n/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(i−n/2)2=n2n−3.\nIt follows from (12) that:\n|F1/2,n|¯σ2< n2n−3/parenleftbiggnm−1(1+o(1))\n2m−1(m−1)!/parenrightbigg2\n.\nSince for λ= 1/2 we have |F1/2,n|= 2n−1(1+o(1)), the claim follows. /square\nSumming up Lemmas 3.1, 3.2 and 3.3, we get the following corollary.\nCorollary 3.4. LetΣbe the sequence of nintegers(1,...,1). Pick a subset Auni-\nformly at random from Fλ,nand define the real random variable ¯X=em\nΣ(A). Setting\n¯µ:=E[¯X]and¯σ2:=E[¯X2]−¯µ2, for a large enough nwe have that:\n¯σ2≤\n\n√n(λn)2m−2(1+o(1))\n((m−1)!)2ifλ <1/2\nn2m−1(1+o(1))\n22m((m−1)!)2ifλ= 1/2\nn2m−1(1+o(1))\n(m!)2ifλ >1/2.\nNext, we are going to use Corollary 3.4 to deduce upper bounds on th e variance for\na generic k-dimensional sequence.\nLemma 3.5. LetΣ = (a1,...,a n)be a sequence in Zkthat isM-bounded and such\nthat, for all distinct A1,A2∈ Fλ,nof size at least m,\nem\nΣ(A1)/ne}ationslash=em\nΣ(A2).\nPick a subset Auniformly at random from Fλ,nand define the real random variable\n¯X=em\nΣ(A). Setting µ:=E[¯X]andσ2:=E[¯X2]−¯µ2, for a large enough nwe have\nthat:\nσ2≤\n\n2k(λ2m−1+λ2m)n2m−1(1+o(1))\n((m−1)!)2M2mifλ <1/2\nk(22m+1λ2m−1+22m+1λ2m+1)n2m−1(1+o(1))\n22m((m−1)!)2M2mifλ= 1/2\nk(2m2λ2m−1+2m2λ2m+1)n2m−1(1+o(1))\n(m!)2M2mifλ >1/2.\nProof.We start from the case k= 1. Notice that\n(13) µ=/summationtext\nA∈Fλ,n/summationtext\ni1<i2<...<i m\ni1,...,im∈Aai1...aim\n|Fλ,n|.\nBy definition of variance, we have that:\n|Fλ,n|σ2=/summationdisplay\nA∈Fλ,n(em\nΣ(A)−µ)2=/summationdisplay\nA∈Fλ,n\n/summationdisplay\ni1<i2<...<i m\ni1,...,im∈Aai1...aim\n2\n−µ2|Fλ,n|.\nLetC0,...,C mbe such that:VARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM AND VARIANCE 9\n(14)/summationdisplay\nA∈Fλ,n\n/summationdisplay\ni1<i2<...<i m\ni1,...,im∈Aai1...aim\n2\n=C0/summationdisplay\ni1<i2<...<i 2m\ni1,...,i2m∈[1,n]ai1...ai2m+\n+C1/summationdisplay\ni1<i2<...<i 2m−1\ni1,...,i2m−1∈[1,n]/summationdisplay\nℓ∈[1,2m−1]ai1ai2...a2\niℓ...ai2m−1+...+Cm/summationdisplay\ni1<i2<...<i m\ni1,...,im∈[1,n]a2\ni1...a2\nim,\nand letD0,...,D mbe such that:\n(15) µ2|Fλ,n|=D0/summationdisplay\ni1<i2<...<i 2m\ni1,...,i2m∈[1,n]ai1...ai2m+\n+D1/summationdisplay\ni1<i2<...<i 2m−1\ni1,...,i2m−1∈[1,n]/summationdisplay\nℓ∈[1,2m−1]ai1ai2...a2\niℓ...ai2m−1+...+Dm/summationdisplay\ni1<i2<...<i m\ni1,...,im∈[1,n]a2\ni1...a2\nim.\nWe now claim that:\n(a)|C1| ≤/parenleftbig2m−2\nm−1/parenrightbig\nλ2m−1|Fλ,n|;\n(b)|D1| ≤/parenleftbig2m−2\nm−1/parenrightbig\nλ2m|Fλ,n|;\n(c)Cj=O(|Fλ,n|) =Djfor every j∈ {2,...,m}.\n(a) The coefficient of a2\ni1...ai2...ai2m−1in (14) is/parenleftbig2m−2\nm−1/parenrightbig\ntimes that obtained taking\nthe term ai1...aimfrom the first em\nΣ(A) in the product and ai1aim+1...ai2m−1from\nthe second one. Symmetrically, the same is true for every term ai1...a2\niℓ...ai2m−1.\nSince a set Acontaining i1,...,i2m−1contains |A| −(2m−1) further elements, we\nget:\nC1/parenleftbig2m−2\nm−1/parenrightbig=λn/summationdisplay\ni=2m−1/parenleftbiggn−(2m−1)\ni−(2m−1)/parenrightbigg\n=λn/summationdisplay\ni=2m−1(n−(2m−1))!\n(i−(2m−1))!(n−i)!.\nSincei≤λnandλ≤1, we have that:\n(n−(2m−1))!\n(i−(2m−1))!(n−i)!=n!\ni!(n−i)!i(i−1)···(i−2m+2)\nn(n−1)···(n−2m+2)≤/parenleftbiggn\ni/parenrightbigg\nλ2m−1.\nIt follows that:\nC1/parenleftbig2m−2\nm−1/parenrightbig≤λn/summationdisplay\ni=2m−1/parenleftbiggn\ni/parenrightbigg\nλ2m−1≤ |Fλ,n|λ2m−1.\n(b)Arguingasinpoint( a)andusing(13),weseethatthecoefficientof a2\ni1...ai2...ai2m−1\ninµ2is/parenleftbig2m−2\nm−1/parenrightbig\ntimes that obtained taking the term ai1...aimfrom the first µin the\nproduct and ai1aim+1...ai2m−1from the second one. Symmetrically, the same is true\nfor everyterm ai1...a2\niℓ...ai2m−1. Since a set Acontaining i1,...,imcontains |A|−m\nfurther elements, the term ai1...aimappears/summationtextλn\ni=m/parenleftbign−m\ni−m/parenrightbig\ntimes in the sum. Since\n/summationtextλn\ni=m(n−m\ni−m)\n|Fλ,n|≤λm, proceeding as in point ( a) we get:\nD1\n|Fλ,n|/parenleftbig2m−2\nm−1/parenrightbig≤λ2m.\n(c) The coefficient of a2\ni1...a2\nijaij+1...ai2m−jin (14) is/parenleftbig2m−2j\nm−j/parenrightbig\ntimes that obtained\ntakingtheterm ai1...aimfromthefirst em\nΣ(A)intheproductand ai1...aijaim+1...ai2m−j10 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nfrom the second one. Chosen these 2 m−jterms, we need to add |A|−(2m−j) terms\nto complete A. Reasoning as in ( a), we get:\nCj/parenleftbig2m−2j\nm−j/parenrightbig=λn/summationdisplay\ni=2m−1/parenleftbiggn−(2m−j)\ni−(2m−j)/parenrightbigg\n=O(|Fλ,n|).\nArguing analogously, one sees that the coefficient of a2\ni1...a2\nijaij+1...ai2m−jin (15) is\nDj/parenleftbig2m−2j\nm−j/parenrightbig=O(|Fλ,n|).\nClaim (c) allows us to write:\n(16) |Fλ,n|σ2= (C0−D0)/summationdisplay\ni1<i2<...<i 2m\ni1,...,i2m∈[1,n]ai1...ai2m+\n+(C1−D1)/summationdisplay\ni1<i2<...<i 2m−1\ni1,...,i2m−1∈[1,n]/summationdisplay\nℓ∈[1,2m−1]ai1ai2...a2\niℓ...ai2m−1+\n+O(|Fλ,n|)\nm/summationdisplay\nj=2/summationdisplay\ni1<i2<...<i 2m−j\ni1,...,i2m−j∈[1,n]/summationdisplay\nℓ1<...<ℓ j\nℓ1,...,ℓj∈[1,2m−j]ai1ai2...a2\niℓ1...a2\niℓj...ai2m−j\n.\nNow we note that if each aiis 1, then Corollary 3.4 provides us an upper bound on\nσ2. Hence, treating (16) as a polynomial expression in the ai’s and evaluating it at\n(a1,...,a n) = (1,...,1), one obtains:\n(C0−D0)/parenleftbiggn\n2m/parenrightbigg\n+(C1−D1)/parenleftbiggn\n2m−1/parenrightbigg\n(2m−1)+O(|Fλ,n|n2m−2)≤L,\nwhereLis the upper bound provided by Corollary 3.4. From now on, we assume\nλ <1/2. The remaining two cases are treated in an analogous way.\nForλ <1/2 we have that L=√n(λn)2m−2(1+o(1))\n((m−1)!)2|Fλ,n|, and hence\n(17) (C0−D0)/parenleftbiggn\n2m/parenrightbigg\n+(C1−D1)(2m−1)/parenleftbiggn\n2m−1/parenrightbigg\n+O(|Fλ,n|√nn2m−2) = 0.\nNow let us go back to (16), and assume that ( a1,...,a n) isM-bounded. Then we\nhave:\n|Fλ,n|σ2≤ |C0−D0|/summationdisplay\ni1<i2<...<i 2m\ni1,...,i2m∈[1,n]ai1...ai2m+\n+|C1−D1|/summationdisplay\ni1<i2<...<i 2m−1\ni1,...,i2m−1∈[1,n]/summationdisplay\nℓ∈[1,2m−1]ai1ai2...a2\niℓ...ai2m−1+\n+O(|Fλ,n|)\nm/summationdisplay\nj=2/summationdisplay\ni1<i2<...<i 2m−j\ni1,...,i2m−j∈[1,n]/summationdisplay\nℓ1<...<ℓ j\nℓ1,...,ℓj∈[1,2m−j]ai1ai2...a2\niℓ1...a2\niℓj...ai2m−j\n≤\n|C0−D0|/parenleftbiggn\n2m/parenrightbigg\nM2m+|C1−D1|(2m−1)/parenleftbiggn\n2m−1/parenrightbigg\nM2m+O(|Fλ,n|n2m−2)M2m.VARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM AND VARIANCE 11\nCombining this with (17) we get:\n|Fλ,n|σ2≤2|C1−D1|(2m−1)/parenleftbiggn\n2m−1/parenrightbigg\nM2m+O(|Fλ,n|√nn2m−2)M2m,\nand using claims ( a) and (b), we deduce that:\n|Fλ,n|σ2≤2(λ2m−1+λ2m)/parenleftbigg2m−2\nm−1/parenrightbigg\n(2m−1)/parenleftbiggn\n2m−1/parenrightbigg\n(1+o(1))|Fλ,n|M2m\n≤2(λ2m−1+λ2m)n2m−1(1+o(1))\n((m−1)!)2|Fλ,n|M2m.\nFinally, if the the entries of Σ belong to Zkfork≥2, one can just argue compo-\nnentwise, obtaining\n|Fλ,n|σ2≤2k(λ2m−1+λ2m)n2m−1(1+o(1))\n((m−1)!)2|Fλ,n|M2m,\nnamely the claim. /square\nTheorem 3.6. LetΣ = (a1,...,a n)be anM-bounded sequence in Zksuch that, for\nall distinct A1,A2∈ Fλ,nof size at least m,\nem\nΣ(A1)/ne}ationslash=em\nΣ(A2).\nThen\nM2m≥\n\n/parenleftbigg((m−1)!)2Γ(k/2+1)2/k\n2(λ2m−1+λ2m)π(k+2)/parenrightbigg(1+o(1))|Fλ,n|2/k\nn2m−1ifλ <1/2\n/parenleftbigg22m((m−1)!)2Γ(k/2+1)2/k\n(22m+1λ2m−1+22m+1λ2m+1)π(k+2)/parenrightbigg(1+o(1))|Fλ,n|2/k\nn2m−1ifλ= 1/2\n/parenleftbigg(m!)2Γ(k/2+1)2/k\n(2m2λ2m−1+2m2λ2m+1)π(k+2)/parenrightbigg(1+o(1))|Fλ,n|2/k\nn2m−1ifλ >1/2.\nProof.To prove the claim, we combine Lemma 3.5 with a lower bound on |Fλ,n|σ2.\nTheideahereis thatthe minimalvalue of |Fλ,n|σ2=/summationtext\nA∈Fλ,n|em\nΣ(A)−µ|2isobtained\nwhen the em\nΣ(A)’s are packed as close as possible around µ. In other words, they have\nto fill the k-dimensional ball of volume |Fλ,n|−/summationtextm−1\ni=0/parenleftbign\ni/parenrightbig\n= (1+o(1))|Fλ,n|centered\ninµ. The radius Rof such ball is given by:\nR=Γ(k/2+1)1/k\n√π(1+o(1))|Fλ,n|1/k.\nThen, following the computations of [8, p. 176], we obtain:\n|Fλ,n|σ2≥R2+kkπk/2\nΓ(k/2+1)(k+2)(1+o(1))\n=kΓ(k/2+1)2/k\nπ(k+2)(1+o(1))|Fλ,n|2/k+1. (18)\nTo conclude, it is enough to compare (18) with Lemma 3.5. /square\nRemark 3.7. Theorem 3.6 can be generalized to sequences ΣinRksuch that for all\ndistinctA1,A2∈ Fλ,nwe have |em\nΣ(A1)−em\nΣ(A2)| ≥1, at the cost of dividing the\nlower bounds by a factor of 4. Here we give a sketch of the proof. The key idea is that/summationtext\nA∈Fλ,n|em\nΣ(A)−µ|2is lower bounded by/summationtext|Fλ,n|\ni=1|zi|2wherez1= 0andzi∈/parenleftbig1\n2Z/parenrightbigk\nfori≥2. This can be proved by noticing that each real number inside t he ball centered\nat0of radius R:= max i|zi|is contained in two balls of radius 1/2centered at some12 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nzi,zj. On the other hand, since |em\nΣ(Ai)−em\nΣ(Aj)| ≥1two balls of radius 1/2centered\ninem\nΣ(Ai)andem\nΣ(Aj)cannot intersect unless they coincide, and therefore for ev ery\nR′≤Rthe ball centered at 0of radius R′contains more zi’s thanem\nΣ(Ai)−µ’s.\nThe lower bound immediately follows. Now/summationtext|Fλ,n|\ni=1|zi|2is exactly what we compute\nin(18), divided by a factor of 4.\nRemark 3.8. Theorem 3.6 provides a lower bound of the following form:\nM≥Ck,m,λn1/(2m)(1+o(1))|Fλ,n|1/(mk)\nn,\nimproving by a factor of n1/(2m)the trivial pigeonhole lower bound. Since the bound\nholds for any λ, we also improve Theorem 2.5 of [8], where the case λ <1/2was left\nopen.\nOn the other hand, if λ=k= 1, the bound of Theorem 3.6 becomes\nM≥C1,m,12n\nm\nn1−1\n2m(1+o(1))\nwhich has the same trend but a worse constant than the result o f Theorem 2.1 which\nis of the form,\nM≥Cm2n\nm\nn1−1\n2m(1+o(1))\nwhere\nC1,m,1=(m!)1/m\n2m/radicalbig\n3(4m2+1)<21−1\nm((m−1)!)1\nm\n31\n2m=Cm.\nIndeed Corollary 3.4 and (17)explain why the terms of higher degree can be replaced\nbut a careful computation of the coefficients (which we are abl e to do only in some\nparticular cases) allows us to further improve the bound.\n4.Upper bounds\nIn this section, we provide an upper bound to the value of Min Problem 1.3 using\nthe probabilistic method. In particular, we show that M≤4nh(λ)/k+o(n), where\nh(λ) =−λlog2λ−(1−λ)log2(1−λ)\ndenotes the binary entropy function. Then, using a direct constr uction (that we pro-\nvide for λ=k= 1 but can be generalized to λ≤1 andk≥1), we prove that\nM≤2n+o(n). It can be shown computationally that h(λ)<1/2 for every λ∈[0,0.11];\nthis means that when k= 1 and λ∈[0,0.11] the probabilistic method improves the\nresult obtained using our direct construction.\nIn order to proceed, we first need to recall the following celebrate d lemma [16, 17].\nLemma 4.1 (Schwartz-Zippel Lemma) .LetFbe a field and P∈F[x1,x2,...,x t]be\na non-zero polynomial of degree d. Consider a finite subset A⊆F. Ifr1,r2,...,r tare\npicked uniformly at random from A, then\nPr[P(r1,r2,...,r t) = 0]≤d\n|A|.\nTheorem 4.2. Letλbe a real number in [0,1/2)and let\nCλ,k=k/radicaligg\n4h(λ)τλ\nτλandτλ=/ceilingleftbigg1\n4h(λ)−1/ceilingrightbigg\n.\nThen there exists a k-dimensional sequence Σ = (a1,...,a n)of/parenleftbig\nCλ,k·m·4h(λ)n/k/parenrightbig\n-\nbounded elements such that, for all distinct A1,A2∈ Fλ,nof size at least m,\nem\nΣ(A1)/ne}ationslash=em\nΣ(A2).VARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM AND VARIANCE 13\nProof.We choose, uniformly at random, a sequence Σ′with entries in [1 ,M]kand\nlengthn′(where the values of Mandn′will be specified later). Let Xbe the random\nvariable that represents the numbers of pairs of elements of Fλ,n′such that em\nΣ′(A1) =\nem\nΣ′(A2).\nThen we need to estimate the following expected value\nE[X] =E(|{{A1,A2}:em\nΣ′(A1) =em\nΣ′(A2),A1,A2∈ Fλ,n′,|A1| ≥m,|A2| ≥m}|)\n=/summationdisplay\n{A1,A2}:A1,A2∈Fλ,n′,|A1|≥m,|A2|≥mPr[em\nΣ′(A1) =em\nΣ′(A2)].\nSince each of the kcomponents of em\nΣ′(A1)−em\nΣ′(A2) is a polynomial of degree mand\neach element’s component in Σ′is picked independently and uniformly at random in\n[1,M], then by Lemma 4.1 we have that\nPr[em\nΣ′(A1)−em\nΣ′(A2) = 0]≤/parenleftigm\nM/parenrightigk\n.\nThen we have that:\nE[X]≤/parenleftigm\nM/parenrightigk\n|{{A1,A2}:A1,A2∈ Fλ,n′,|A1| ≥m,|A2| ≥m}|.\nThanks to the well-known inequality/summationtextℓ\ni=0/parenleftbigr\ni/parenrightbig\n≤2h(ℓ/r)rforℓ < r/2, we obtain:\n(19) E[X]</parenleftigm\nM/parenrightigk\n·\nλn′/summationdisplay\ni=m/parenleftbiggn′\ni/parenrightbigg\n2\n≤/parenleftigm\nM/parenrightigk\n22h(λ)n′.\nThis means that if ( m/M)k22h(λ)n′≤t, there exists a sequence Σ′= (a1,...,a n′)\nof elements in Zkwith at most tpairs{A1,A2}that have the same evaluation and\nsatisfy the assumptions. Hence, we can remove telements from Σ′and obtain a new\nsequence Σ = ( a1,...,a n), withn=n′−telements, such that em\nΣ(A1)/ne}ationslash=em\nΣ(A2) for\nall distinct A1,A2∈ Fλ,n. Thanks to (19), we deduce that a sequence Σ as in the\nclaim exists whenever\n(20) M≥k/radicalbigg\n4h(λ)t\nt·m·4h(λ)n/k.\nOne can check that the function gλ(t):=4h(λ)t\ntis strictly convex for t >0 and\nthat the minimum integer γfor which gλ(γ+ 1)≥gλ(γ) is equal to τλ. Therefore\nt=τλ=/ceilingleftig\n1\n4h(λ)−1/ceilingrightig\nis the best choice to optimize (20). /square\nUsingthe samemethodofTheorem4.2for λ= 1, onecaneasilyobtainthe following\ntheorem.\nTheorem 4.3. There exists a sequence of nelements of Zkthat ism-th evaluation\ndistinct and M-bounded such that\nM≤m·4n/k.\nNow we provide a direct construction of a sequence Σ that is m-th evaluation\ndistinct. We start by constructing such sequence on the real num bers and then we\nadapt the idea over the integers.\nLemma 4.4. Letǫbe a positive real number and let m≥2be an integer. For every\nnlarge enough the sequence of real numbers Σ = (a1,a2,...,a n), where\nai= (2+ǫ)n−2i−1fori= 1,2,...,n,\nism-th evaluation distinct.14 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nProof.Suppose by contradiction there exists two distinct subsets B,C⊆[1,n] with\n|B|,|C| ≥msuch that\n(21) |em\nΣ(B)−em\nΣ(C)|<1.\nFor an arbitrary subset S⊆[1,n] with|S| ≥m, by definition we have:\n(22) em\nΣ(S) =m/summationdisplay\nj=0(−1)j(2+ǫ)(m−j)n/parenleftbigg|S|−j\nm−j/parenrightbigg/summationdisplay\n{i1,i2,...,ij}⊆S\ni1<i2<...<i j2i1+i2+...+ij−j.\nWefirstshowthatInequality(21)implies |B|=|C|. Supposewithoutlossofgenerality\nthat|B|>|C|. Then (22) implies that:\n(23)em\nΣ(B)−em\nΣ(C) = (2+ ǫ)mn/bracketleftbigg/parenleftbigg|B|\nm/parenrightbigg\n−/parenleftbigg|C|\nm/parenrightbigg/bracketrightbigg\n+m/summationdisplay\nj=1(−1)j(2+ǫ)(m−j)n\n\n/parenleftbigg|B|−j\nm−j/parenrightbigg/summationdisplay\n{i1,i2,...,ij}⊆B\ni1<i2<...<i j2i1+i2+...+ij−j−/parenleftbigg|C|−j\nm−j/parenrightbigg/summationdisplay\n{i1,i2,...,ij}⊆C\ni1<i2<...<i j2i1+i2+...+ij−j\n.\nNow it can be seen that each term in the first summation of (23) is of o rder\nO/parenleftig\nnm(2+ǫ)(m−j)n2jn/parenrightig\n,\nforj= 1,2,...,mandn→ ∞. Hence, asymptotically in n, we can rewrite (23) as\nem\nΣ(B)−em\nΣ(C) = (2+ ǫ)mn/bracketleftbigg/parenleftbigg|B|\nm/parenrightbigg\n−/parenleftbigg|C|\nm/parenrightbigg/bracketrightbigg\n(1+o(1)),\nsinceǫ >0. This clearly contradicts (21), and hence we must have |B|=|C|.\nNext, let tbe an integer such that |B|=|C|=tand letB:={b1,b2,...,b t}and\nC:={c1,c2,...,c t}, wherebi,ci∈[1,n] for every i∈[1,t]. Then we have:\n(24)|em\nΣ(B)−em\nΣ(C)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2+ǫ)(m−1)n/parenleftbiggt−1\nm−1/parenrightbigg/parenleftiggt/summationdisplay\ni=12bi−1−2ci−1/parenrightigg\n+m/summationdisplay\nj=2(−1)j−1\n(2+ǫ)(m−j)n/parenleftbiggt−j\nm−j/parenrightbigg\n/summationdisplay\n1≤i1<i2<...<i j≤t2bi1+bi2+...+bij−j−2ci1+ci2+...+cij−j\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nTo conclude the proof, we need to lower bound (24). Since B/ne}ationslash=C, it is easy to see\nthat /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/summationdisplay\ni=12bi−1−2ci−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥1\nand since each term in the summation over jin (24) is, as n→ ∞, of order\nO/parenleftig\nnm(2+ǫ)(m−j)n2jn/parenrightig\n,\nwe obtain the following lower bound:\n|em\nΣ(B)−em\nΣ(C)|>/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2+ǫ)(m−1)n/parenleftbiggt−1\nm−1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1+o(1)).\nThe claim now follows from the fact that the right-hand side of the ab ove inequality\nis greater than 1 for sufficiently large n’s. /squareVARIANTS OF THE ERD ˝OS DISTINCT SUMS PROBLEM AND VARIANCE 15\nNext, we adapt the idea of Lemma 4.4 to work over the integers, tak ing integer\nparts.\nLemma 4.5. Letǫbe a positive real number and let m≥2be an integer. Then for\neverynlarge enough the sequence Σ = (a1,a2,...,a n), where\nai=/floorleftbig\n(2+ǫ)n−2i−1/floorrightbig\nfori= 1,2,...,n,\nism-evaluation distinct.\nProof.Writeai= (2 +ǫ)n−2i−1−δ, where δis the fractional part of (2 + ǫ)n.\nProceeding as in the proof of Lemma 4.4, we suppose by contradictio n that there exist\ntwo distinct subsets B,C⊆[1,n] such that\n(25) |em\nΣ(B)−em\nΣ(C)|= 0.\nWe first show that (25) implies |B|=|C|. Suppose without loss of generality that\n|B|>|C|. Then we have that\n(26)em\nΣ(B)−em\nΣ(C) = (2+ ǫ)mn/bracketleftbigg/parenleftbigg|B|\nm/parenrightbigg\n−/parenleftbigg|C|\nm/parenrightbigg/bracketrightbigg\n+m/summationdisplay\nj=1(−1)j(2+ǫ)(m−j)n\n/bracketleftigg/parenleftbigg|B|−j\nm−j/parenrightbigg/summationdisplay\n{i1,i2,...,ij}⊆B\ni1<i2<...<i j(2i1−1+δ)···(2ij−1+δ)−\n/parenleftbigg|C|−j\nm−j/parenrightbigg/summationdisplay\n{i1,i2,...,ij}⊆C\ni1<i2<...<i j(2i1−1+δ)···(2ij−1+δ)/bracketrightigg\n.\nWe observe that for fixed i1,i2,...,ijand fornlarge enough we have that\n(2i1−1+δ)···(2ij−1+δ)≤(2n+1)j≤e2jn,\nsinceδ≤1 andj≤m≤n. It follows that each term in the first summation of (26)\nis of order\nO/parenleftig\nnm(2+ǫ)(m−j)n2jn/parenrightig\n,\nforj= 1,2,...,mandn→ ∞. Hence, asymptotically in n, we can rewrite (26) as\nem\nΣ(B)−em\nΣ(C) = (2+ ǫ)mn/bracketleftbigg/parenleftbigg|B|\nm/parenrightbigg\n−/parenleftbigg|C|\nm/parenrightbigg/bracketrightbigg\n(1+o(1)),\nsinceǫ >0. This clearly contradicts (25), and hence we must have |B|=|C|.\nNow, as done in Lemma 4.4, let tbe an integer such that |B|=|C|=tand let\nB:={b1,b2,...,b t}andC:={c1,c2,...,c t}, wherebi,ci∈[1,n] for every i∈[1,t].\nThen we have:\n(27)|em\nΣ(B)−em\nΣ(C)|=\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2+ǫ)(m−1)n/parenleftbiggt−1\nm−1/parenrightbigg/parenleftiggt/summationdisplay\ni=1/parenleftbig\n2bi−1+δ/parenrightbig\n−/parenleftbig\n2ci−1+δ/parenrightbig/parenrightigg\n+m/summationdisplay\nj=2(−1)j−1(2+ǫ)(m−j)n\n/parenleftbiggt−j\nm−j/parenrightbigg\n/summationdisplay\n1≤i1<i2<...<i j≤t(2bi1−1+δ)···(2bij−1+δ)−(2ci1−1+δ)···(2cij−1+δ)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.16 SIMONE COSTA, STEFANO DELLA FIORE, ANDREA FERRAGUTI\nSinceB/ne}ationslash=C, we have that/vextendsingle/vextendsingle/vextendsingle/summationtextt\ni=12bi−1−2ci−1/vextendsingle/vextendsingle/vextendsingle≥1. On the other hand, each\nterm in the summation over jin (27) is, as n→ ∞, of order O/parenleftbig\nnm(2+ǫ)(m−j)n2jn/parenrightbig\n.\nConsequently we obtain the following lower bound:\n|em\nΣ(B)−em\nΣ(C)|>/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2+ǫ)(m−1)n/parenleftbiggt−1\nm−1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1+o(1)).\nOnce again, the right-hand side of the above inequality is greaterth an 1 for sufficiently\nlargen’s, and the proof is complete. /square\nAs a consequence of Corollary 4.5, we obtain the following theorem.\nTheorem 4.6. There exists a sequence Σ = (a1,a2,...,a n)ofnintegers that is m-\nevaluation distinct and M-bounded such that\nM≤2n(1+o(1)),\nforn→ ∞.\nProof.For every h∈N≥1letǫh= 2/h, and let Σ n,h:= (a1,a2,...,a n) withai=\n⌊(2 +ǫh)n−2i−1⌋for every i≥1. By Corollary 4.5, there exists a positive integer\nnhsuch that for n≥nhthe sequence Σ n,his (2+ǫh)n-bounded and m-th evaluation\ndistinct. Since ǫh= 2/h, we have that Σ n,his (2n+n/hlog2e)-bounded. We may\nassume, without loss of generality, that the sequence {nh}h∈Nis increasing in h. Now\nthe sequences of integers Σ nh,harem-evaluation distinct and M-bounded where\nM≤2nh+1\nhnhlog2e.\nLettingh→ ∞, one obtains the claim. /square\nRemark 4.7. We have stated Theorem 4.6 for λ=k= 1but, clearly, the same bound\nholds also for any integer k≥1and any positive real λ≤1. Similarly, the result of\nTheorem 4.3 holds also for λ≤1. Here, assuming k= 1, the bound given in Theorem\n4.2 improves that of Theorem 4.6 for every λ∈[0,0.11]and for every λ <1/2when\nk= 2. If instead k≥3, then the best upper bound is that of Theorem 4.3 when λ >1/2\nand that of Theorem 4.2 otherwise.\nAcknowledgements\nThe first and the third authors were partially supported by INdAM– GNSAGA.\nReferences\n[1] N. Alon and J. H. Spencer, The probabilistic method, 4th e d. Wiley, Hoboken, NJ, 2016.\n[2] M. Axenovich, Y. Caro, R. 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Lunnon, Integer sets with distinct subset sums, Ma th. Compute, 50 (1988) 297-320.\n[16] J. T. Schwartz, Fast probabilistic algorithms for veri fication of polynomial identities. Journal of\nthe ACM (JACM) , 27.4 (1980), 701–717.\n[17] R. Zippel. Probabilistic algorithms for sparse polyno mials.Symbolic and Algebraic Computation:\nEUROSM’79, An International Symposium on Symbolic and Alge braic Manipulation , 1979."    },    {        "title": "2402.00669v1.Global_solutions_of_Euler_Maxwell_equations_with_dissipation.pdf",        "content": "arXiv:2402.00669v1  [math.AP]  1 Feb 2024GLOBAL SOLUTIONS OF EULER-MAXWELL EQUATIONS WITH\nDISSIPATION\nB. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\nAbstract. We consider the Cauchy problem for a damped Euler-Maxwell sy stem with no\nionic background. Forsmooth enoughdatasatisfyingsuitab le so-called dispersive conditions,\nwe establish the global in time existence and uniqueness of a strong solution that decays\nuniformly in time. Our method is inspired bythe works of D. Se rre and M. Grassin dedicated\nto the compressible Euler system.\nKeywords: compressible Euler system, Maxwell, global solution, diss ipation, decay.\nAMS subject classification: 35Q30, 76N10.\n1.Introduction\nIn recent years various proposal of including viscousdissipative terms in Maxwell’s equa-\ntions have been made (see [3, 32, 38, 41]) in order to accommod ate propagation in lossy\nmaterials.\nCompared to standard Maxwell’s equations in lossless media , Maxwell’s equations in a\nmediumwith conductor losses have numerousapplications su ch as high-temperatureplasmas,\nCPU electronic field or perfectly matched layers [3]. In [33, 41, 42] this model leads to\nconstruction of efficient numerical methods for simulating e lectromagnetic dissipation. The\npossible derivation of such system from kinetic models, nam ely from the two species Vlasov-\nBoltzmann-Maxwell system, we refer to the work of Jang and Ma smoudi, [22]. Also the\nhydrodynamics limits of the Boltzmann equation, see e.g. [1 9, 27].\nIn order to couple a dissipative Maxwell’s system to a moving medium we consider the\ndissipative 3D Euler-Maxwell system appearing naturally in plasma phys ics as a coupling\nbetween hydrodynamics and electromagnetism (see the deriv ation in e.g. [9]). The goal of\nthe present paper is to study the associated Cauchy problem f or this system.\nInfact, anumberofworkssolveglobally theCauchyproblemf ortheisentropic(ornonisen-\ntropic) 3D Euler-Maxwell system, when a non-vanishing ioni c background is present. Among\nthem, one can quote [12, 14, 35, 34, 43, 44]. However, all of th ese works deal with a strictly\npositive ionic background, a crucial condition in order to g et good estimates. Hereafter, we\nare interested in the degenerate situation where one neglec ts ionic density and vacuum may\nappear. Our aim is to establish the existence of a class of glo bal solutions in that situation\nassuming some dissipation in the momentum and electro-magn etic equations (see [34]). Re-\ncall that a similar issue has been investigated in the simple r situation of the compressible\nEuler system in a series of papers [19, 39, 17] by D. Serre and M . Grassin. There, it is proved\nthat for ‘well-prepared’ data, the Cauchy problem for the co mpressible Euler system admits\na unique global smooth solution.\nIn our recent works [5, 10, 11], we pointed out that Grassin-S erre strategy was efficient\nto prove global existence results for the the Euler-Helmhol tz, Euler-Poisson or Euler-Riesz\nsystems. Our goal here is to adapt that strategy to the Euler- Maxwell system.\nAfter normalization, the system of equations to be studied f or the fluid density ̺=̺(t,x),\nthe charge density /tildewide̺=/tildewide̺(t,x) (see [18]), the velocity field u=u(t,x), the electric field E(t,x)\n12 B. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\nand the magnetic field B(t,x) as functions of the time tand the Eulerian spatial coordinate\nx∈R3reads:\n(1.1) ∂t̺+divx(̺u) = 0,\n(1.2) ∂t(̺u)+div x(̺u⊗u)+∇xΠ(̺) =−̺E+J×B,\n(1.3) ∂tE−curlxB=−J−α1E,\n(1.4) ∂tB+curlxE=−α2B,\n(1.5) div xB= 0,\n(1.6) div xE=−/tildewide̺,\nwith initial data\n(1.7) ( ̺,/tildewide̺,u,E,B )(0,x) = (̺0,/tildewide̺0,u0,E0,B0)(x),\nwhereα1andα2are given real positive parameters, Π( ρ) =Aργis the barotropic pressure\nwithA >0 and the adiabatic exponent γ >1,and the electric current Jis given by Ohm’s\nlaw:\n(1.8) J=−̺u.\nIt is known that one can discard equations (1.5) and (1.6) pro vided that they are satisfied by\nthe data, namely\n(1.9) div xB0= 0,\n(1.10) div xE0=−/tildewide̺0.\nFinally we see that ̺and/tildewide̺are related by the compatibility relation\n(1.11) ∂t(̺−/tildewide̺)−α1/tildewide̺= 0.\nThese conditions being assumed, the reduced problem under s tudy in the following is (1.1)–\n(1.4) supplemented with initial conditions (1.7).\nNeglecting ionic background introduces the classical diffic ulty of vacuum as first pointed\nout by Kato [23] whensymmetrizing thesystem for solving it. Despite this, thecorresponding\nCauchy problem for Euler-Poisson with vacuum for strong sol utions was solved locally in time\nin the eighties by various authors, among them: Makino [28], Makino-Ukai [31], Makino-\nPerthame [30], Gamblin [13], B´ ezard [4], Braun and Karp [6] (see also [29] for a clear survey).\nIt is well known in this context that existence results are ex pected to be only local in time\neven for small data [8] (see blow-up results of Chemin [7] (3D case) or Makino and Perthame\n[30] (1D spherically symmetric case)). However, in [19, 17, 39], D. Serre and M. Grassin\npointed out that under a suitable ‘dispersive’ spectral con dition on the initial velocity that\nwill be specified in the next section, and a smallness hypothe sis on the initial density, the\ncompressible Euler system (that is (1.1)–(1.2) with E≡B≡0) admits a unique global\nsmooth solution.\nMore recently we have shown [5, 10, 11] that the Serre-Grassi n global existence result\nextends to the compressible Euler system coupled with the Po isson or Helmholtz equations.\nOur goal here is to get a similar result for the whole (dissipa tive) Euler-Maxwell system\n(1.1)–(1.4).GLOBAL SOLUTIONS FOR DAMPED EULER-MAXWELL EQUATIONS 3\nTherest of the paper is structured as follows. In the next sec tion, we state our main results\nand give some insights on our strategy. In Section 3, we estab lish decay estimates in Sobolev\nspaces first for a multi-dimensional Burgers-Maxwell syste m (which will provide us with an\napproximate solution for our system) and, next, for the comp ressible Euler equation coupled\nwith the Maxwell system. Section 4 is devoted to the proofs of the main global existence\nresults, then we show the uniqueness of the solution. For the reader’s convenience, some\ntechnical results like, in particular, first and second orde r commutator estimates are recalled\nin the appendix.\nNotation: Throughout the paper, Cdenotes a harmless ‘constant’ that may change from\nline to line, and we use sometimes A/lessorsimilarBto mean that A≤CB.The notation A≈Bis\nused if both A/lessorsimilarBandB/lessorsimilarA.\nFinally, we shall denote by ˙HsandHsthe homogeneous and nonhomogeneous Sobolev\nspaces of order sonR3,and byWk,p(withk∈Nandp∈[1,∞]) the set of Lpfunctions on\nR3,with derivatives up to order kinLp.\n2.Main results\nLet us introduce the symmetrization introduced by T. Makino in [28], setting\n(2.12) ρ:=2√Aγ\nγ−1̺γ−1\n2.\nAfter that change of unknown, System (1.1)-(1.4) rewrites\n(2.13)\n\n(∂t+u·∇)ρ+γ−1\n2ρdivu= 0,\n(∂t+u·∇)u+γ−1\n2ρ∇ρ=−(E+u×B),\n∂tE−curlxB+α1E=̺u,\n∂tB+curlxE+α2B= 0.\nWe consider the following generalized Burgers equation\n(2.14) ∂tv+v·∇xv= curlxv×v,\ncomplemented with the damping-free Maxwell System\n(2.15)/braceleftBigg∂tE−curlxB+α1E= 0,divxE= 0,\n∂tB+curlxE+α2B= 0,divxB= 0,\nwith initial conditions\n(2.16) v(0,x) =v0(x),\nand\n(2.17) ( E,B)(0,x) = (E0,B0)(x).\nThe impetus in considering such system is that, as we shall sh ow later, (2.14) is a good\napproximation of (1.2) provided that the initial density of the fluid and the initial electro-\nmagnetic field are small enough. This can bedone by extending the observation in [15, 19] for\nthe compressible Euler system and under suitable spectral c onditions on the initial data. It is\nthus natural to expect that system (2.14)–(2.15) is a good ap proximation of (2.13) provided\nthat the initial density and the electromagnetic field are sm all.\nThe main strategy will be to justify this heuristics in order to construct a class of global\nsolutions to our Euler-Maxwell system.4 B. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\nWe consider the following function space for the analysis of the approximate equation\n(2.14):\nEs:=/braceleftbig\nz∈ C(R3;R3), Dz∈L∞andD2z∈Hs−2/bracerightbig\n·\nWe now present the existence of a classical solution to equat ion (2.14) given some spectral\nassumption on the data.\nProposition 2.1. Letv0be inEswiths >5/2and satisfy:\n(H0)there exists ε >0such that for any x∈R3,dist(Sp(Dv0(x)),R−)≥ε,\nwhereSpAdenotes the spectrum of the matrix A. Then(2.14)supplemented with (2.16)has\na classical solution vonR+×R3such that\nD2v∈ Cj/parenleftbig\nR+;Hs−2−j(R3)/parenrightbig\nforj= 0,1.\nMoreover, Dv∈ Cb(R+×R3)and we have for any t≥0andx∈R3,\n(2.18) Dv(t,x) =1\n1+tId +1\n(1+t)2K(t,x),\nfor some function K∈ Cb(R+×R3;R3×R3)that also satisfies\n(2.19) /ba∇dblK(t,·)/ba∇dbl˙Hσ≤Kσ(1+t)1\n2−σ,\nfor all0< σ≤s−1. Moreover, there exists a constant C >0such that vsatisfies the\nfollowing estimates\n/ba∇dblDv(t)/ba∇dblL∞≤C\n1+t, (2.20)\n/ba∇dblv(t,·)/ba∇dbl˙Hσ≤C(1+t)1\n2−σ, (2.21)\n/ba∇dblD2v(t)/ba∇dblL∞≤C\n(1+t)3, (2.22)\nwhere0< σ≤s−1.\nWe mention that the proposition above has been established — in the case where there\nright-hand side of (2.14) is absent — in [15, 19] for integer r egularity exponents, while X.\nBlanc, et al. [5] provided the proof for realexponents.\nConcerningthedamping-freeelectromagnetic systemwecan establishanexponentialdecay\nof its solution as shown in the lemma below.\nLemma 2.1. Assume that B0,E0∈L2. Let(E,B)be the unique solution of the Cauchy\nproblem (2.15)-(2.17). ThenE,B∈L2andEsatisfies the estimate\n(2.23) /ba∇dblE/ba∇dbl2\nL2+/ba∇dblB/ba∇dbl2\nL2≤C2\n0e−αt,\nwithC2\n0=1\n2(/ba∇dblE0/ba∇dbl2\nL2+/ba∇dblB0/ba∇dbl2\nL2)andα= min{α1,α2}.\nProof.Multiplying the first equation (2.15) by Eand the second one by B, we get the energy\nequality\n1\n2d\ndt1\n2(B2+E2)+α1E2+α2B2+div(E×B) = 0.\nIntegrating on [0 ,t]×R3we get\n/integraldisplay\nR31\n2B2dx+1\n2/integraldisplay\nR3E2dx+/integraldisplayt\n0/integraldisplay\nR3/parenleftbig\nα1E2(s)+α2B2(s)/parenrightbig\ndx ds=/integraldisplay\nR3/parenleftbigg1\n2B2\n0+1\n2E2\n0/parenrightbigg\ndx,GLOBAL SOLUTIONS FOR DAMPED EULER-MAXWELL EQUATIONS 5\nwhich gives the inequality\n1\n2/integraldisplay\nR3E2dx+α/integraldisplayt\n0/integraldisplay\nR3(E2(s)+B2(s))dx ds≤C2\n0.\nUsing Gronwall’s inequality we get (2.23). /square\nGoing back to system (1.1)-(1.8), we can derive an analogous L2bound for the actual\nelectric field.\nLemma 2.2. LetT >0be arbitrary. Assume that√̺0u0,B0,E0∈L2and̺∈Lγ. Let\n(̺,u,E,B )be a solution of the Cauchy problem (1.1)-(1.8).\nThenE∈L2and satisfies the estimate\n(2.24) /ba∇dblE/ba∇dbl2\nL2+/ba∇dblB/ba∇dbl2\nL2≤C2\n1e−αt,\nwithC2\n1=/ba∇dbl√̺0u0/ba∇dbl2\nL2+Aγ\nγ−1/ba∇dbl̺0/ba∇dblγ\nLγ+1\n2(/ba∇dblE0/ba∇dbl2\nL2+/ba∇dblB0/ba∇dbl2\nL2).\nProof.Multiplying (1.2) by u, (1.3) by Eand (1.4) by B, we get the energy equality\nd\ndt/parenleftbigg1\n2̺u2+Π\nγ−1/parenrightbigg\n+1\n2d\ndt(E2+B2)+α1E2+α2B2\n+div/parenleftbigg/parenleftbigg\n̺u2+γΠ\nγ−1/parenrightbigg\nu+E×B/parenrightbigg\n= 0.\nIntegrating on [0 ,t]×R3we get\n/integraldisplay\nR3/parenleftbigg1\n2̺u2+Π\nγ−1/parenrightbigg\ndx+1\n2/integraldisplay\nR3(E2+B2)dx+/integraldisplayt\n0/integraldisplay\nR3/parenleftbig\nα1E2(s)+α2B2(s)/parenrightbig\ndx ds\n=/integraldisplay\nR3/parenleftbigg1\n2̺0u2\n0+Π0\nγ−1+1\n2B2\n0+1\n2E2\n0/parenrightbigg\ndx,\nwhich gives the inequality\n1\n2/integraldisplay\nR3(E2+B2)dx+α/integraldisplayt\n0/integraldisplay\nR3(E2(s)+B2(s))dx ds≤C2\n1.\nUsing Gronwall’s inequality we obtain (2.24). /square\nOur main result is proving the following global existence an d uniqueness of solution to\nSystem (1.1)-(1.4).\nTheorem 2.1. Suppose that either 1< γ <5/3and5/2< s <3/2+2/(γ−1)or5/2< s <\n+∞ifγ= 1+2/kfor some integer k.\nAssume that the initial data (ρ0,u0,E0,B0)satisfy:\n•(H1)there exists v0inEs+1satisfying (H0)and such that u0−v0is small in Hs;\n•(H2)̺γ−1\n2\n0is small enough in Hs.\n•(H3)E0andB0are small enough in Hs.\n•(H4)Conditions (1.9),(1.10)andB0= curlxu0are satisfied.\nIf(v,E,B)is the global solution of (2.14)–(2.15)with initial data (2.16)–(2.17)from Proposi-\ntion 2.1 and Lemma 2.1, then there exists a unique global solu tion(̺,u,E,B )to(1.1)–(1.7),\nsuch that/parenleftbig\n̺γ−1\n2,u−v,E−E,B−B/parenrightbig\n∈ C/parenleftbig\nR+;Hs/parenrightbig\n.6 B. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\nAs a consequence of the theorem above, we can show that the con structed global solution\nof (1.1)–(1.7) satisfies a decay estimate towards the soluti on of (2.14)–(2.15).\nTheorem 2.2. Let all the assumptions of Theorem 2.1 be in force. Then, for all σin[0,s],\nthe solution (̺,u,E,B )constructed therein satisfies\n/vextenddouble/vextenddouble/vextenddouble̺γ−1\n2,u−v,E−E,B−B/vextenddouble/vextenddouble/vextenddouble˙Hσ≤Cσ(1+t)3/2−σ−min{1,3/2(γ−1)},\nwhereCσdepends only on the initial data, on γand onσ.\n3.Decay estimates in Sobolev spaces\nIn this section we prove a prioridecay estimates in Sobolev spaces which play a fundamen-\ntal role in the proof of our global existence result. To be spe cific, we first establish a decay\nestimate for the generalized Burgers equation (2.14). Seco ndly, we compare the solutions to\n(2.13) and to (2.14) giving us the estimate promised in Theor em 2.2.\n3.1.Decay estimates for the generalized Burgers equation. The purpose of this part\nis to prove Proposition 2.1 for any real regularity exponent s >5/2.\n1. Using the identity v·∇xv=∇xv2\n2+curlxv×vin equation (2.14) we get\n∂tv+∇xv2\n2= 0.\nBy letting Xbe the flow of v, we see that the matrix valued function A: (t,y)/ma√sto→\nDv(t,X(t,y)) satisfies the Ricatti equation\nA′+A2= 0, A|t=0=Dv0.\nFrom Hypothesis ( H0),one can deduce that v(t,y) is defined for all t≥0 andy∈R3,and\nthat\nDv(t,X(t,y)) = (Id + tDv0(y))−1Dv0(y) with X(t,y) =y+tv0(y).\nTherefore, denoting Xt:y/ma√sto→X(t,y),we get (2.18), that is\nDv(t,x) =Id\n1+t+K(t,x)\n(1+t)2(3.25)\nwhereK(t,x) := (1+ t)(Id +tDv0(X−1\nt(x)))−1(Dv0(X−1\nt(x))−Id). From this, we can also\nfind the divergence of the velocity field vby taking the trace of the tensor Dv\n(3.26) div v(t,y+tv0(y)) =d\n1+t+TrK(t,y+tv0(y))\n(1+t)2·\nFurthermore, Hypothesis ( H0) implies that\n(3.27) /ba∇dbl(Id +tDv0)−1/ba∇dblL∞/lessorsimilar(1+εt)−1,\nandKis thus bounded on R+×R3.\n2. For the proof of (2.22) we refer to [17].\n3. We proceed with the proof of (2.19) in the case σ∈]0,1[. The case of higher order\nregularity exponents may be done by taking advantage of the e xplicit formula for partial\nderivatives of /tildewideKtthat has been derived by M. Grassin in [17, p. 1404]. The same m ethod as\nin the case σ∈]0,1[, which we shall shortly show, has then to be applied to each term of the\nformula. The details are left to the reader.GLOBAL SOLUTIONS FOR DAMPED EULER-MAXWELL EQUATIONS 7\nTo bound /tildewideKt:= (1+t)−1K(t,·) in˙Hσ,we use the following characterization of Sobolev\nnorms by finite differences:\n/ba∇dbl/tildewideKt/ba∇dbl2\n˙Hσ≈/integraldisplay\nR3/integraldisplay\nR3|/tildewideKt(y)−/tildewideKt(x)|2\n|y−x|3+2σdxdy.\nWe can thus write /tildewideKt(y)−/tildewideKt(x) =I1\nt(x,y)+I2\nt(x,y) where\nI1\nt(x,y) = (Id+ tDv0(X−1\nt(y)))−1/parenleftbig\nDv0(X−1\nt(y))−Dv0(X−1\nt(x))/parenrightbig\n,\nI2\nt(x,y) =t(Id+tDv0(X−1\nt(y)))−1/parenleftbig\nDv0(X−1\nt(x))−Dv0(X−1\nt(y))/parenrightbig\n×(Id+tDv0(X−1\nt(x)))−1(Dv0(X−1\nt(x))−Id).\nWe see, thanks to (3.27) and to the change of variable x′=X−1\nt(x) andy′=X−1\nt(y),that\n/integraldisplay\nR3/integraldisplay\nR3|I1\nt(x,y)|2\n|y−x|3+2σdxdy≤C\n(1+εt)2/integraldisplay\nR3/integraldisplay\nR3|Dv0(y′)−Dv0(x′)|2\n|Xt(y′)−Xt(x′)|3+2σJXt(x′)JXt(y′)dx′dy′.\nFurthermore, we infer from (3.27) that\n|y′−x′|=|X−1\nt(Xt(y′))−X−1\nt(Xt(x′))|\n≤ /ba∇dblDX−1\nt/ba∇dblL∞|Xt(y′)−Xt(x′)| ≤C\n1+εt|Xt(y′)−Xt(x′)|.(3.28)\nTherefore, using the fact that /ba∇dblJXt/ba∇dblL∞≤C(1+εt)3and (3.28), we get\n(3.29)/integraldisplay\nR3/integraldisplay\nR3|I1\nt(x,y)|2\n|y−x|3+2σdxdy≤C(1+εt)1−2σ/integraldisplay\nR3/integraldisplay\nR3|Dv0(y′)−Dv0(x′)|2\n|y′−x′|3+2σdxdy.\nSimilarly, (3.27) and the change of variable x′=X−1\nt(x) andy′=X−1\nt(y) imply that\n/integraldisplay\nR3/integraldisplay\nR3|I2\nt(x,y)|2\n|y−x|d+2σdxdy≤Ct2\n(1+εt)4/integraldisplay\nR3/integraldisplay\nR3|Dv0(y′)−Dv0(x′)|2\n|Xt(y′)−Xt(x′)|3+2σJXt(x′)JXt(y′)dx′dy′,\nand we thus also have (3.29) for I2\nt.As a conclusion, using the characterization of /ba∇dblDv0/ba∇dbl˙Hσ\nby finite difference, we get\n/ba∇dbl/tildewideKt/ba∇dbl˙Hσ≤C(1+εt)1\n2−σ/ba∇dblDv0/ba∇dbl˙Hσ,\nwhich gives the desired estimate for σ∈]0,1[. /square\n3.2.Sobolev estimates for System (2.13).Let (v,E,B),be the solution of the Burgers-\nMaxwell system given by Proposition 2.1 and Lemma 2.1. Consi der a sufficiently smooth\nsolution ( ρ,u,E,B ) of (2.13) on [0 ,T]×R3,and setw:=u−v,e:=E−Eandb:=B−B.\nThen, (ρ,w,e,b) satisfies:\n(3.30)\n\n(∂t+w·∇)ρ+γ−1\n2ρdivw+v·∇ρ+γ−1\n2ρdivv= 0,\n(∂t+w·∇)w+γ−1\n2ρ∇ρ+v·∇w+w·∇v\n=−E−u×B+v×curlxv,\n∂te−curlxb=/parenleftBig\nγ−1\n2√Aγ/parenrightBig2\nγ−1ρ2\nγ−1u−α1e,\n∂tb+curlxe=−α2b.\nOf course, we have the compatibility conditions\n(3.31) div xe=−/tildewide̺and div xb= 0.8 B. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\nOur aim is to prove decay estimates in ˙Hσfor (3.30) for all 0 ≤σ≤s.Clearly, arguing by\ninterpolation, it suffices to consider the border cases σ= 0 and σ=s.\n1. Let us start with σ= 0.Taking the L2scalar product of the first equation of (3.30)\nwithρgives\n1\n2d\ndt/ba∇dblρ/ba∇dbl2\nL2−1\n2/integraldisplay\nR3ρ2divwdx+γ−1\n2/integraldisplay\nR3ρ2divwdx\n−1\n2/integraldisplay\nR3ρ2divvdx+γ−1\n2/integraldisplay\nR3ρ2divvdx= 0.(3.32)\nIn order to bound the magnetic term, we first remark, as observ ed by Germain and Mas-\nmoudi in [14] that the quantity Z:=B−curlxu(seen as a vector-field) is conserved by the\nflow ofu. Therefore, according to Hypothesis ( H4), we have\n(3.33) B= curlxu.\nUsing this observation, we see that the Lorentz contributio n in the right hand side of the\nmomentum equation — after taking L2product with w— reduces to\n−/integraldisplay\nR3(E+v×curlxw)·w dx.\nTaking now the L2scalar product of the second equation of (3.30) with wand using (3.33)\ngives\n1\n2d\ndt/ba∇dblw/ba∇dbl2\nL2−γ−1\n4/integraldisplay\nR3ρ2divwdx−1\n2/integraldisplay\nR3|w|2divwdx+/integraldisplay\nR3w Dv wdx\n=−/integraldisplay\nR3E·w dx+1\n2/integraldisplay\nR3|w|2divwdx.−/integraldisplay\nR3w(v×curlxw)dx.\nUsing the identity1\n2∇w2=w×curlxw+w·∇wwe further get\n1\n2d\ndt/ba∇dblw/ba∇dbl2\nL2−γ−1\n4/integraldisplay\nR3ρ2divwdx−1\n2/integraldisplay\nR3|w|2divwdx+1\n2/integraldisplay\nR3w Dv wdx\n=−/integraldisplay\nR3E·w dx−1\n2/integraldisplay\nR3|w|2divvdx.(3.34)\nIn the same stroke, taking the L2scalar product of the last two equations of (3.30) with ( e,b)\ngives\n(3.35)1\n2d\ndt/parenleftbig\n/ba∇dble/ba∇dbl2\nL2+/ba∇dblb/ba∇dbl2\nL2/parenrightbig\n+α1/ba∇dble/ba∇dbl2\nL2+α2/ba∇dblb/ba∇dbl2\nL2=/parenleftbiggγ−1\n2√Aγ/parenrightbigg2\nγ−1/integraldisplay\nR3ρ2\nγ−1e·u dx.\nLet us compute several contributions in (3.34) and (3.35). F rom (2.18) and (3.26), one gets/integraldisplay\nR3w Dv wdx =1\n1+t/ba∇dblw/ba∇dbl2\nL2+1\n(1+t)2/integraldisplay\nR3w(x)K(t,x)w(x)dx,\nand /integraldisplay\nR3|w|2divvdx=3\n1+t/ba∇dblw/ba∇dbl2\nL2+1\n(1+t)2/integraldisplay\nR3TrK(t,x)|w|2dx.\nSecondly,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3E·w dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbig\n/ba∇dble/ba∇dblL2+/ba∇dblE/ba∇dblL2/parenrightbig\n/ba∇dblw/ba∇dblL2,\nand finally/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3ρ2\nγ−1e·u dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dblρ2\nγ−1/ba∇dblL∞/ba∇dble/ba∇dblL2(/ba∇dblw/ba∇dblL2+/ba∇dblv/ba∇dblL2)GLOBAL SOLUTIONS FOR DAMPED EULER-MAXWELL EQUATIONS 9\nLet us set now\n(3.36) cγ:= min/braceleftbigg\n1,3γ−1\n2/bracerightbigg\n−3\n2·\nFrom (3.32), Cauchy-Schwarz inequality and Proposition 2. 1, we deduce that, denoting by\nMa bound of K,\n1\n2d\ndt/ba∇dbl(ρ,w,e,b)/ba∇dbl2\nL2+cγ\n1+t/ba∇dbl(ρ,w,e,b)/ba∇dbl2\nL2\n/lessorsimilarMmax{5/2,3/2|γ−2|}\n(1+t)2/ba∇dbl(ρ,w,e,b)/ba∇dbl2\nL2+/ba∇dbldivw/ba∇dblL∞/ba∇dbl(ρ,w,e,b)/ba∇dbl2\nL2\n(3.37) + C|ρ2\nγ−1/ba∇dblL∞/ba∇dble/ba∇dblL2(/ba∇dblw/ba∇dblL2+/ba∇dblv/ba∇dblL2)+/parenleftbig\n/ba∇dble/ba∇dblL2+/ba∇dblE/ba∇dblL2/parenrightbig\n/ba∇dblw/ba∇dblL2.\n2. The case σ >0.In the sequel, we will use freely the following estimates pro ved in\n[2, 5] (see related results in [24][25]), which we shall also refer to sometimes as Kato-Ponce\nestimates.\nLemma 3.1. •Ifs >0,then we have:\n/ba∇dbl[v,˙Λs]u/ba∇dblL2/lessorsimilar/ba∇dblv/ba∇dbl˙Hs/ba∇dblu/ba∇dblL∞+/ba∇dbl∇v/ba∇dblL∞/ba∇dblu/ba∇dbl˙Hs−1.\n•Ifs >1,then we have:\n/ba∇dbl[v,˙Λs]u−s∇v·˙Λs−2∇u/ba∇dblL2/lessorsimilar/ba∇dblv/ba∇dbl˙Hs/ba∇dblu/ba∇dblL∞+/ba∇dbl∇2v/ba∇dblL∞/ba∇dblu/ba∇dbl˙Hs−2.\nIn order to prove Sobolev estimates, we introduce the homoge neous fractional differential\noperator ˙Λsdefined by F(˙Λsf)(ξ) :=|ξ|sFf(ξ) and observe that ρs:=˙Λsρ, ws:=˙Λsw,\nEs:=˙ΛsE,es:=˙Λseandbs:=˙Λsbsatisfy (with the usual summation convention over\nrepeated indices)\n(∂t+w·∇)ρs+γ−1\n2ρdivws+v·∇ρs−s∂jvk˙Λ−2∂2\njkρs+γ−1\n2˙Λs(ρdivv)\n=˙R1\ns+˙R2\ns+˙R3\ns, (3.38)\n(∂t+w·∇)ws+γ−1\n2ρ∇ρs+v·∇ws−s∂jvk˙Λ−2∂2\njkws+˙Λs(w·∇v),\n+Es+ws×curlxw+vs×curlxw+ws×curlxv=˙R4\ns+˙R5\ns+˙R6\ns+˙R7\ns, (3.39)\n(3.40) ∂tes−curlxbs+α1es=˙R8\ns,\n(3.41) ∂tbs+curlxes+α2bs= 0,\nwith\n˙R1\ns:= [w,˙Λs]∇ρ, ˙R5\ns:=γ−1\n2[ρ,˙Λs]∇ρ,\n˙R2\ns:=γ−1\n2[ρ,˙Λs]divw, ˙R6\ns:= [v,˙Λs]∇w−s∂jvk˙Λ−2∂2\njkws,\n˙R3\ns:= [v,˙Λs]∇ρ−s∂jvk˙Λ−2∂2\njkρs,˙R7\ns:=˙Λs(v×curlxv−u×curlxu)\n−vs×curlxv+us×curlxu,\n˙R4\ns:= [w,˙Λs]∇w, ˙R8\ns:=/parenleftBig\nγ−1\n2√Aγ/parenrightBig2\nγ−1˙Λs(ρ2\nγ−1u).\nThe definitions of ˙R3\nsand˙R6\nsare motivated by the fact that, according to the classical th eory\nof pseudo-differential operators, we expect to have\n[˙Λs,v]·∇z=1\ni/braceleftbig\n|ξ|s,v(x)/bracerightbig\n(D)∇z+remainder .10 B. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\nComputing the Poisson bracket in the right-hand side yields\n1\ni/braceleftbig\n|ξ|s,v(x)/bracerightbig\n(D) =−s∂jv˙Λs−2∂j.\nNow, taking advantage of (3.25), we get\n−∂jvk˙Λ−2∂2\njkz=1\n1+tz−Kkj\n(1+t)2˙Λ−2∂2\njkz,\nand using (3.26) yields\n˙Λs(ρdivv) =d\n1+tρs+1\n(1+t)2˙Λs(ρTrK)\nand˙Λs(w·∇v) =1\n1+tws+1\n(1+t)2˙Λs(K·w).\nHence, taking the L2inner product of (3.38) (3.39) (3.40) and (3.41) respective ly with\n(ρs,ws),denoting cγ,s:=cγ+s,andusingthefactthat /ba∇dblEs·ws/ba∇dblL1≤/parenleftbig\n/ba∇dbles/ba∇dblL2+/ba∇dblEs/ba∇dblL2/parenrightbig\n/ba∇dblws/ba∇dblL2,\nwe end up with\n1\n2d\ndt/ba∇dbl(ρs,ws,es,bs)/ba∇dbl2\nL2+cγ,s\n1+t/ba∇dbl(ρs,ws,es,bs)/ba∇dbl2\nL2+α(/ba∇dbles/ba∇dbl2\nL2+/ba∇dblbs/ba∇dbl2\nL2)\n/lessorsimilarγ−1\n2/ba∇dbl∇ρ/ba∇dblL∞/ba∇dblρs/ba∇dblL2/ba∇dblws/ba∇dblL2+sM\n(1+t)2/ba∇dbl(ρs,ws,es,bs)/ba∇dbl2\nL2\n+1\n(1+t)2/parenleftbiggγ−1\n2/ba∇dbl˙Λs(ρTrK)/ba∇dblL2/ba∇dblρs/ba∇dblL2+/ba∇dbl˙Λs(K·w)/ba∇dblL2/ba∇dblws/ba∇dblL2/parenrightbigg\n+/parenleftbig\n/ba∇dbles/ba∇dblL2+/ba∇dblEs/ba∇dblL2+/ba∇dblv/ba∇dbl˙Hs/ba∇dblcurlxw/ba∇dblL∞/parenrightbig\n/ba∇dblws/ba∇dblL2+3/summationdisplay\nj=1/ba∇dbl˙Rj\ns/ba∇dblL2/ba∇dblρs/ba∇dblL2\n+7/summationdisplay\nj=4/ba∇dbl˙Rj\ns/ba∇dblL2/ba∇dblws/ba∇dblL2+/ba∇dbl˙R8\ns/ba∇dblL2/ba∇dbles/ba∇dblL2.(3.42)\nThe terms ˙R1\ns,˙R2\ns,˙R4\ns,˙R5\nsand˙R7\nsmay be treated according to Lemma 3.1 which gives us\n/ba∇dbl˙R1\ns/ba∇dblL2/lessorsimilar/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbl∇w/ba∇dbl˙Hs−1+/ba∇dbl∇w/ba∇dblL∞/ba∇dblρ/ba∇dbl˙Hs,\n/ba∇dbl˙R4\ns/ba∇dblL2/lessorsimilar/ba∇dbl∇w/ba∇dblL∞/ba∇dblw/ba∇dbl˙Hs,\n/ba∇dbl˙R2\ns/ba∇dblL2/lessorsimilar/ba∇dbldivw/ba∇dblL∞/ba∇dblρ/ba∇dbl˙Hs+/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbldivw/ba∇dbl˙Hs−1,\n/ba∇dbl˙R5\ns/ba∇dblL2/lessorsimilar/ba∇dbl∇ρ/ba∇dblL∞/ba∇dblρ/ba∇dbl˙Hs,\nThe (more involved) terms ˙R3\nsand˙R6\nsmay be also handled thanks to Lemma 3.1. We get\n/ba∇dbl˙R3\ns/ba∇dblL2/lessorsimilar/ba∇dbl∇ρ/ba∇dblL∞/ba∇dblv/ba∇dbl˙Hs+/ba∇dbl∇2v/ba∇dblL∞/ba∇dbl∇ρ/ba∇dbl˙Hs−2,\n/ba∇dbl˙R6\ns/ba∇dblL2/lessorsimilar/ba∇dbl∇w/ba∇dblL∞/ba∇dblv/ba∇dbl˙Hs+/ba∇dbl∇2v/ba∇dblL∞/ba∇dbl∇w/ba∇dbl˙Hs−2.\nWe see also that\n/ba∇dbl˙R7\ns/ba∇dblL2/lessorsimilar/ba∇dbl˙Λs(curlxw×w)−curlxw×ws/ba∇dblL2+/ba∇dbl˙Λs(curlxv×w)−curlxv×ws/ba∇dblL2\n+/ba∇dbl˙Λs(curlxw×v)−curlxw×vs/ba∇dblL2.\nTherefore, Lemma 3.1 gives us\n/ba∇dbl˙R7\ns/ba∇dblL2/lessorsimilar/ba∇dbl∇v/ba∇dblL∞/ba∇dblcurlxw/ba∇dbl˙Hs−1+/ba∇dblcurlxw/ba∇dblL∞/ba∇dblv/ba∇dbl˙Hs+/ba∇dbl∇w/ba∇dblL∞/ba∇dblcurlxv/ba∇dbl˙Hs−1\n+/ba∇dblcurlxv/ba∇dblL∞/ba∇dblw/ba∇dbl˙Hs+/ba∇dbl∇w/ba∇dblL∞/ba∇dblcurlxw/ba∇dbl˙Hs−1+/ba∇dblcurlxw/ba∇dblL∞/ba∇dblw/ba∇dbl˙Hs.GLOBAL SOLUTIONS FOR DAMPED EULER-MAXWELL EQUATIONS 11\nTo bound ˙R8\nswe first observe that — due to [5, Lemma A.2] —\n/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble˙Hσ/lessorsimilar/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl˙Hσfor 0< σ <1\n2+2\nγ−1. (3.43)\nUsing Kato-Ponce estimate we get now\n/vextenddouble/vextenddouble/vextenddouble[v,˙Λs]ρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilar/ba∇dbl∇v/ba∇dblL∞/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble˙Hs−1+/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble\nL∞/ba∇dblv/ba∇dbl˙Hs.\nTherefore/vextenddouble/vextenddouble/vextenddouble˙Λs/parenleftBig\nρ2\nγ−1v/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilar/vextenddouble/vextenddouble/vextenddouble[v,˙Λs]ρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/vextenddoublev˙Λs/parenleftBig\nρ2\nγ−1/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/ba∇dbl∇v/ba∇dblL∞/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble˙Hs−1+/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble\nL∞/ba∇dblv/ba∇dbl˙Hs+/ba∇dblv/ba∇dblL∞/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble˙Hs\nAnalogously\n/vextenddouble/vextenddouble/vextenddouble˙Λs/parenleftBig\nρ2\nγ−1w/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilar/vextenddouble/vextenddouble/vextenddouble[w,˙Λs]ρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/vextenddoublew˙Λs/parenleftBig\nρ2\nγ−1/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/ba∇dbl∇w/ba∇dblL∞/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble˙Hs−1+/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble\nL∞/ba∇dblw/ba∇dbl˙Hs+/ba∇dblw/ba∇dblL∞/vextenddouble/vextenddouble/vextenddoubleρ2\nγ−1/vextenddouble/vextenddouble/vextenddouble˙Hs.\nTherefore, with reinforcement of (3.43), we have\n/ba∇dbl˙R8\ns/ba∇dblL2/lessorsimilar/ba∇dbl∇v/ba∇dblL∞/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl˙Hs−1+/ba∇dblρ/ba∇dbl2\nγ−1\nL∞/ba∇dblv/ba∇dbl˙Hs+/ba∇dblv/ba∇dblL∞/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl˙Hs\n+/ba∇dbl∇w/ba∇dblL∞/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl˙Hs−1+/ba∇dblρ/ba∇dbl2\nγ−1\nL∞/ba∇dblw/ba∇dbl˙Hs+/ba∇dblw/ba∇dblL∞/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl˙Hs.\nUsing similar arguments as above we get the following estime s\n/ba∇dbl˙Λs(ρTrK)/ba∇dblL2/lessorsimilar/ba∇dbl∇ρ/ba∇dblL∞/ba∇dblTrK/ba∇dbl˙Hs−1+/ba∇dblTrK/ba∇dblL∞/ba∇dblρ/ba∇dbl˙Hs+/ba∇dblρ/ba∇dblL∞/ba∇dblTrK/ba∇dbl˙Hs\nand/ba∇dbl˙Λs(K·w)/ba∇dblL2/lessorsimilar/ba∇dbl∇w/ba∇dblL∞/ba∇dblK/ba∇dbl˙Hs−1+/ba∇dblK/ba∇dblL∞/ba∇dblw/ba∇dbl˙Hs+/ba∇dblw/ba∇dblL∞/ba∇dblK/ba∇dbl˙Hs.\nPlugging all the above estimates in (3.42) and using Proposi tion 2.1 , we end up with\n1\n2d\ndt/ba∇dbl(ρ,w,e,b)/ba∇dbl2\n˙Hs+cγ,s\n1+t/ba∇dbl(ρ,w,e,b)/ba∇dbl2\n˙Hs\n/lessorsimilar1\n(1+t)2/ba∇dbl(ρ,w,e,b)/ba∇dbl2\n˙Hs+(/ba∇dbles/ba∇dblL2+/ba∇dblEs/ba∇dblL2)/ba∇dbl(ρ,w,e,b)/ba∇dbl˙Hs\n+/ba∇dbl(∇ρ,∇w)/ba∇dblL∞/ba∇dbl(ρ,w,e,b)/ba∇dbl2\n˙Hs+1\n(1+t)s+1\n2/ba∇dbl(∇ρ,∇w)/ba∇dblL∞/ba∇dbl(ρ,w,e,b)/ba∇dbl˙Hs\n+1\n(1+t)/ba∇dbl(ρ,w,e,b)/ba∇dbl2\n˙Hs+1\n(1+t)s+3\n2/ba∇dbl(ρ,w)/ba∇dblL∞/ba∇dbl(ρ,w,e,b)/ba∇dbl˙Hs\n+1\n(1+t)s−1\n2/ba∇dbl(∇ρ,∇w)/ba∇dblL∞/ba∇dbl(ρ,w,e,b)/ba∇dbl˙Hs+1\n(1+t)3/ba∇dbl(ρ,w)/ba∇dbl˙Hs−1/ba∇dbl(ρ,w,e,b)/ba∇dbl˙Hs\n+/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dbl(ρ,w,e,b)/ba∇dbl2\n˙Hs+1\n(1+t)s−1\n2/ba∇dblρ/ba∇dbl2\nγ−1\nL∞/ba∇dbl(ρ,w,e,b)/ba∇dbl˙Hs\n+1\n(1+t)/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl2\n˙Hs+/ba∇dbl∇w/ba∇dblL∞/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl˙Hs−1/ba∇dblρ/ba∇dbl˙Hs\n+/ba∇dblρ/ba∇dbl2\nγ−1\nL∞/ba∇dblρ/ba∇dbl˙Hs/ba∇dblw/ba∇dbl˙Hs+/ba∇dblw/ba∇dblL∞/ba∇dblρ/ba∇dbl2\nγ−1−1\nL∞/ba∇dblρ/ba∇dbl2\n˙Hs(3.44)\nLet us introduce the notation\n˙Xσ:=/ba∇dbl(ρ,w,e,b)/ba∇dbl˙HσandXσ:=/radicalBig\n˙X2\n0+˙X2σ≈ /ba∇dbl(ρ,w,e,b)/ba∇dblHσforσ≥0.12 B. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\nOur aim is to bound the right-hand side of (3.37) and (3.44) in terms of ˙X0and˙Xsonly.\nArguing by interpolation, we get first:\n/ba∇dbl(ρ,w)/ba∇dblL∞/lessorsimilar˙X1−3\n2s\n0˙X3\n2ss, (3.45)\n/ba∇dbl(Dρ,Dw)/ba∇dblL∞/lessorsimilar˙X1−5\n2s\n0˙X5\n2ss, (3.46)\n/ba∇dbl(ρ,w)/ba∇dbl˙Hs−1/lessorsimilar˙X1\ns\n0˙X1−1\nss. (3.47)\nThen, plugging these inequalities and those of Proposition 2.1 in (3.37) and (3.44), together\nwith decay properties given by Lemma 2.1 and 2.2 yields\nd\ndt˙X0+cγ\n1+t˙X0/lessorsimilar˙X0\n(1+t)2+˙X2−5\n2s\n0˙X5\n2ss+˙X(1−3\n2s)2\nγ−1+1\n0˙X3\n2s2\nγ−1s,\nand\nd\ndt˙Xs+cγ,s\n1+t˙Xs/lessorsimilar˙Xs\n(1+t)2+˙X1−5\n2s\n0˙X5\n2ss˙Xs+˙X1−5\n2s\n0˙X5\n2ss\n(1+t)s+1\n2+˙Xs\n(1+t)\n+˙X1−3\n2s\n0˙X3\n2ss\n(1+t)s+3\n2+˙X1−5\n2s\n0˙X5\n2ss\n(1+t)s−1\n2+˙X1\ns\n0˙X1−1\nss\n(1+t)3+/parenleftbigg\n˙X1−3\n2s\n0˙X3\n2ss/parenrightbigg2\nγ−1−1\n˙Xs\n+/parenleftbig˙X1−3\n2s\n0˙X3\n2ss/parenrightbig2\nγ−1\n(1+t)s−1\n2+/parenleftbig˙X1−3\n2s\n0˙X3\n2ss/parenrightbig2\nγ−1−1˙Xs\n(1+t)+˙X1−3\n2s\n0˙X1+3\n2ss/parenleftbig˙X1−3\n2s\n0˙X3\n2ss/parenrightbig2\nγ−1−1.\nLetabe an arbitrary positive number. Performing the change of un known\n˙Ys= (1+t)cγ,s−a˙Xs,\nwe observe that\nd\ndt˙Ys+a\n1+t˙Ys= (1+t)cγ,s−a/parenleftbiggd\ndt˙Xs+cγ,s\n1+t˙Xs/parenrightbigg\n.\nTherefore introducing the notation Yσ:=/radicalBig\n˙Y2\n0+˙Y2σfor anyσ∈[0,s], we get\nd\ndtYs+a\n1+tYs≤C\nYs\n(1+t)2+Y2\ns\n(1+t)Γ0+Y2\nγ−1−1\ns\n(1+t)Γ1+Y2\nγ−1s\n(1+t)Γ2\n(3.48) +Y2\nγ−1−1\ns\n(1+t)Γ3+Y2\nγ−1+1\ns\n(1+t)Γ4\n,\nwith Γ 0=cγ−a+ 5/2, Γ1= (cγ−a+ 3/2)(2\nγ−1−1), Γ2= (cγ−a+ 3/2)(2\nγ−1−1),\nΓ3= (cγ−a+3/2)(2\nγ−1) and Γ 4= (cγ−a+3/2)(2\nγ−1−1).\nDenoting the new unknown Z(t) := (1+ t)ae−Ct\n1+tYs(t), inequality (3.48) gives\nd\ndtZ≤CeCt\n1+tZ2\n(1+t)Γ0+a+Ce(m−1)Ct\n1+tZm\n(1+t)Γ1+am−1\n+CemCt\n1+tZm+1\n(1+t)Γ2+am−1+Ce(m+1)Ct\n1+tZm+2\n(1+t)Γ4+am−1,GLOBAL SOLUTIONS FOR DAMPED EULER-MAXWELL EQUATIONS 13\nwherem=2\nγ−1−1.\nIn order to prove that t→Z(t) is bounded for any t, we use the same bootstrap argument\nas in [5].\nSuppose indeed that Z0:=Z(0) and assume that\n(3.49) Z(t)≤2Z0on [0,T].\nThen (3.48) implies that\nd\ndtZ≤CeC(2Z0)2\n(1+t)Γ0+a+Ce(m−1)C(2Z0)m\n(1+t)Γ1+am−1\n+CemC(2Z0)m+1\n(1+t)Γ2+am−1+Ce(m+1)C(2Z0)m+2\n(1+t)Γ4+am−1.\nSuppose that\n(3.50) Γ 0+a >1,Γ1+am−1>1,Γ2+am−1>1 and Γ 4+am−1.\nHence, integrating in time, we discover that on [0 ,T],we have\nZ(t)≤Z0+CeC\nΓ0+a−1(2Z0)2/parenleftbig\n1−(1+t)1−Γ0−a/parenrightbig\n+Ce(m−1)C(2Z0)m+1\nΓ1+am−2/parenleftbig\n1−(1+t)2−Γ1−am/parenrightbig\n+CemC(2Z0)m\nΓ1+am−2/parenleftbig\n1−(1+t)2−Γ2−am/parenrightbig\n+Ce(m+1)C(2Z0)m+2\nΓ4+am−2/parenleftbig\n1−(1+t)2−Γ2−am/parenrightbig\nLet us discard the obvious case Z0= 0.Then, if Z0is so small as to satisfy\n(3.51)4CeCZ0\nΓ0+a−1+Ce(m−1)C2m+1Zm\n0\nΓ1+am−2+CemC2mZm−1\n0\nΓ2+am−2+Ce(m+1)C2m+2Zm+1\n0\nΓ4+am−2≤1,\nthe above inequality ensures that we actually have Z(t)<2Z0on [0,T].\nTherefore the supremum of T >0 satisfying (3.49) has to be infinite.\nEventually we get, provided /ba∇dbl(ρ0,w0,e0,b0)/ba∇dblHsis small enough:\n(3.52)/radicalBig\n(1+t)2s/ba∇dbl(ρ,w,e,b)/ba∇dbl2\n˙Hs+/ba∇dbl(ρ,w,e,b)/ba∇dbl2\nL2≤2eCt\n1+t\n(1+t)cγ/ba∇dbl(ρ0,w0,e0,b0)/ba∇dblHs.\nLet us emphasize that in order to derive (3.51), we need to sat isfy constraints (3.50).\nThe first condition reduces to γ >1 and the remaining conditions are equivalent to\n2+/parenleftbigg2\nγ−1−1/parenrightbigg/parenleftbigg\na−cγ−3\n2/parenrightbigg\n< a/parenleftbigg2\nγ−1−1/parenrightbigg\n,\nthat is to say\nmin/parenleftbigg\n1,3\n2(γ−1)/parenrightbigg/parenleftbigg2\nγ−1−1/parenrightbigg\n>2.\nThat latter inequality is equivalent to γ <5/3.\nKeeping in mind the previous constraint (3.43) one can concl ude that (3.52) holds true\nwhenever\n(3.53) 1 < γ <5\n3and5\n2< s <1\n2+2\nγ−1·14 B. DUCOMET, ˇS. NEˇCASOV ´A, AND J. S. H. SIMON\n4.Proving Theorem 2.1\nAs pointed out in the introduction, a number of works have bee n dedicated to the well-\nposedness issue for the Euler-Maxwell system. However, non e of them considered data like\nours. For the reader’s convenience, we here sketch the proof of the global existence in the\nfunctional setting of Theorem 2.1, then establish uniquene ss by means of a classical energy\nmethod.\n4.1.Existence. Here we are given ( ρ0,u0,E0,B0) fulfilling the assumptions of Theorem 2.1.\nOur goal is to prove the existence of a global-in-time soluti on (ρ,u,E,B ) for System (2.13)\nor, equivalently, denoting w:=u−v,of (ρ,w,e,b) for System (3.30). The idea is to use the\ncut-off function χwith range [0 ,1], support in the ball B(0,2) and value 1 on B(0,1), and to\napproximate (3.30) as follows:\n\n\n(∂t+u·∇)ρ+d/tildewideγρ\n1+t+/tildewideγρTrKn\n(1+t)2+/tildewideγρdivw= 0,\n(∂t+u·∇)w+w\n1+t+w·Kn\n(1+t)2+/tildewideγρ∇ρ=−χ(n−1(ρE+J×B)),\ndte−curlxb=−χ(n−1J),\ndtb+curlxe= 0,\n(ρ,w,e,b)|t=0= (ρn\n0,un\n0,en\n0,bn\n0),\nwhere/tildewideγ=γ−1\n2, withKn:=χ(n−1·)K, ρn\n0:=χ(n−1·)ρ0,un\n0:=χ(n−1·)u0,en\n0:=χ(n−1·)e0\nandbn\n0:=χ(n−1·)b0.\nSince the initial data as well as Knare in the Sobolev space Hswiths >1+d/2,a tiny\nmodification of the standard theory of symmetric hyperbolic systems allows to prove that\nthere exists a unique maximal solution ( ρn,wn,en,bn) inC([0,Tn);Hs)∩C1([0,Tn);Hs−1) to\nthe above system.\nThe computations of the previous step may be repeated on [0 ,Tn) and one gets, with\nobvious notation, for some absolute constant C,\nYn\ns(t)≤CeCt\n1+t\n(1+t)1+d/tildewideγYs(0) for all 0 ≤t < Tn.\nThis in particular provides a control on /ba∇dbl∇ρn,∇wn,∇en,∇wn/ba∇dblL∞so that the classical blow-\nup criterion for hyperbolic systems allows to conclude that Tn= +∞.\nFrom that point, classical functional analysis arguments a llow to pass to the limit (up\nto subsequence) and to conclude that ( ρn,wn,en,bn) converges to some solution ( ρ,w,e,b)\nof (3.30) corresponding to data ( ρ0,w0,e0,b0).Of course, that solution fulfills (3.52), and\nlooking at the definition of Ysallows to get the required decay estimates.\nThis completes the proof of the existence part of Theorem 2.1 , and of the decay estimates.\n4.2.Uniqueness. Consider two solutions ( ρ1,w1,e1,b1) and (ρ2,w2,e2,b2) of (3.30) corre-\nsponding to the same data and having the properties of regula rity listed in Theorem 2.1.GLOBAL SOLUTIONS FOR DAMPED EULER-MAXWELL EQUATIONS 15\nThen, (δρ,δw,δe,δb ) := (ρ2−ρ1,w2−w1,e2−e1,b2−b1) fulfills:\n\n\n(∂t+w2·∇)δρ+/tildewideγρ2divδw+v·∇δρ+/tildewideγδρdivv=−δw·∇ρ1−/tildewideγδρdivw1,\n(∂t+w2·∇)δw+/tildewideγρ2∇δρ+v·∇δw\n=−δ(E+u×B−v×curlxv),\n∂tδe−curlxδb+α1e=δ/parenleftBig\nρ2\nγ−1u/parenrightBig\n,\n∂tδb+curlxδe+α2b= 0.\nHence, differentiating with respect to xjyields\n\n\n(∂t+(v+w2)·∇)∂jδρ+/tildewideγρ2div∂jδw=−∂jw2·∇δρ−/tildewideγ∂jρ2divδw−∂jv·∇δρ\n−/tildewideγ∂jδρdivv−/tildewideγδρ∂jdivv−∂jδw·∇ρ1−δw·∇∂jρ1−/tildewideγδρdiv∂jw1−/tildewideγ∂jδρdivw1,\n(∂t+(v+w2)·∇)∂jδw+/tildewideγρ2∇∂jδρ=−∂jw2·∇δw−/tildewideγ∂jρ2∇δρ−∂jv·∇δw\n−∂jδw·∇v−δw·∇∂jv−∂jδw·∇w1−δw·∇∂jw1−/tildewideγ∂jδρ∇ρ1−/tildewideγδρ∇∂jρ1\n−∂jδe−∂jδ(u×B)+∂jδ(v×curlxv),\n∂t∂jδe−curlx∂jδb+α1∂je=∂jδ/parenleftBig\nρ2\nγ−1u/parenrightBig\n,\n∂t∂jδb+curlx∂jδe+α2∂jb= 0.\nHence, applying an energy method and arguing exactly as for t he proof of uniqueness in the\nprevious section, we get:\nd\ndt/ba∇dbl(∇δρ,∇δw,∇δe,∇δe)/ba∇dbl2\nL2/lessorsimilar/parenleftBig\n/ba∇dbl(∇ρ1,∇ρ2,∇u1,∇u2,∇v,∇e1,∇e2,∇b1,∇b2)/ba∇dblL∞\n+/ba∇dbl(∇2ρ1,∇2ρ2,∇2u1,∇2u2,∇2v,∇2e1,∇2e2,∇2b1,∇2b2)/ba∇dblLd/parenrightBig\n×/ba∇dbl(∇δρ,∇δw,∇δe,∇δb)/ba∇dbl2\nL2.\nRecall that ∇vis bounded and that ∇2vis inHs−1withs >1+d/2,and thus in Ld.\nOf course, as previously ∇ρi,∇wi,∇ei,∇biare inL∞and∇2ρi,∇2wi,∇2wi,∇2ei,∇2bi\nare inLd, fori= 1,2. Hence Gronwall lemma ensures that ( ∇δρ,∇δw,∇δe,∇δb)≡0 on\n[0,T]×R3,which, owing to the fact that δρis inLqeventually implies that δρ≡0.Plugging\nthat information in the equation of δw,one can then conclude that δw≡0.Finally one sees\nin the same stroke that δe=δb= 0.\nThis completes the proof of the theorem. /square\nAcknowledgments: The first author warmly thanks D. Serre for a fruitful corresp on-\ndence on Euler-Maxwell equations. ˇS.N. and J.S. acknowledge the supports the Praemium\nAcademiae of ˇS. Moreover, ˇS.N. acknowledge the supports of the Czech Science Foundati on\n(GAˇCR)throughprojectsproject22-01591S. TheInstituteofMa thematics, CASissupported\nby RVO:67985840.\nReferences\n[1] H. Bahouri, J.-Y. Chemin and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equatio ns.\nGrundlehren der mathematischen Wissenschaften, 343, Springer (2011).\n[2] S. Benzoni-Gavage, R. Danchin and S. Descombes. 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Anal. ,44:2002–2017, 2012.\nBernard Ducomet\nUniversit´ e Paris-Est\nLAMA (UMR 8050), UPEMLV, UPEC, CNRS\n61 Avenue du G´ en´ eral de Gaulle, F-94010 Cr´ eteil, France\nE-mail: bernard.ducomet@u-pec.fr\nˇS´ arka Neˇ casov´ a\nInstitute of Mathematics of the Academy of Sciences of the Cz ech Republic\nˇZitna 25, 115 67 Praha 1, Czech Republic\nE-mail: matus@math.cas.cz\nJohn Sebastian H. Simon\nJohann Radon Institute for Computation and Applied Mathema tics (RICAM)\nAustrian Academy of Sciences\nAltenberger Strasse 69, 4040 Linz, Austria\nE-mail: john.simon@ricam.oeaw.ac.at, jhsimon1729@gmai l.com"    },    {        "title": "2402.00714v1.Neutral_carbon_in_diffuse_interstellar_medium__abundance_matching_with_H2_for_DLAs_at_high_redshifts.pdf",        "content": "MNRAS 000, 1–11 (2023) Preprint 2 February 2024 Compiled using MNRAS L ATEX style file v3.0\nNeutral carbon in diffuse interstellar medium: abundance matching with\nH2for DLAs at high redshifts.\nS. A. Balashev,1★, D. N. Kosenko,1\n1Ioffe Institute, Polyteknicheskaya 26, 194021 Saint-Petersburg, Russia\nAccepted 2023. Received 2023; in original form 2023\nABSTRACT\nWe present the study of C i/H2relative abundance in the diffuse cold neutral medium. Using the chemical and thermal balance\nmodel we calculated the dependence of C i/H2on the main parameters of the medium: hydrogen number density, metallicity,\nstrength of the UV field, and cosmic ray ionization rate (CRIR). We show that observed relative C iand H 2column densities\nin damped Lyman alpha systems (DLAs) at high redshifts can be reproduced within our model assuming the typically expected\nconditionsinthediffusecoldneutralmedium(CNM).Usingtheadditionalobservedinformationtheonmetallicity,H icolumn\ndensity,andexcitationofC ifine-structurelevels,aswellastemperatureweestimatedforawiderangemetallicitiesintheCNM\nat high redshifts that CRIRs to be in the range from ∼10−16tofew×10−15s−1, hydrogen number densities to be in range\n∼10−103cm−3,andUVfieldinrangefrom 10−2tofew×102ofMathisfield.Weargue,thatsincetheobservedquantitiesused\ninthisworkarequitehomogeneousandmuchlessaffectedbytheradiativetransfereffects(incomparisonwithe.g.dissociation\nof HD and UV pumping of H 2rotational levels) our estimates are quite robust against the assumption of the exact geometrical\nmodel of the cloud and local sources of the UV field.\nKey words: ISM: abundances – ISM: cosmic rays – Cosmology: observations\n1 INTRODUCTION\nCarbon is one of the most important elements in the neutral phase\nofthelocalinterstellarmedium(ISM).Indeed,atmetallicitiesclose\nto solar, its singly ionized form, C ii, provides the principal source\nof the electrons (that determine the chemistry in the medium) and\nis one of the dominant cooling processes in diffuse regions, that de-\nterminestheformationofthecoldphaseofneutralmedium(CNM).\nInturn,intranslucentanddensemolecularregionscarbonismainly\nin Ciand/or CO, which also dominate the cooling of the medium.\nTherefore,itisessentialtostudycarboninCNMofISMbothobser-\nvationally and theoretically.\nFrom observations, the most robust way to constrain the abun-\ndance and ionization state of carbon in CNM is an absorption line\nspectroscopy. However, while C iis typically a sub-dominant form\nof carbon, regarding measured column densities, its abundance can\nbe much more confidently constrained than C ii. Additionally, C i\ninabsorptionalwaysobservedpopulatingthreefine-structurelevels,\nwhichprovideanimportantdiagnosticsofthephysicalconditionsin\nthe medium (see e.g. Jenkins & Tripp 2001; Silva & Viegas 2002;\nSrianand et al. 2005; Noterdaeme et al. 2007; Balashev et al. 2011;\nJorgenson et al. 2010; Klimenko & Balashev 2020; Kosenko et al.\n2023). C iiabundance (Jenkins & Tripp 2011; Roman-Duval et al.\n2019)andexcitationofitsfine-structurelevel(Balashevetal.2022)\ncan also be used for constraints on the physical state of the gas.\nHowever, the usage of C iiis much more limited than C idue to the\n★E-mail: s.balashev@gmail.comavailability of less number of transitions and saturation of the ab-\nsorptionlines,althoughtheincidencerateofC iiismuchhigherthan\nCi. In turn, while rotational excitation of CO can provide indepen-\ndentdiagnosticsofthephysicalconditions(Klimenkoetal.inprep),\nobservations of CO are much more limited than C iiand C i, since\nthe cross-section of the associated dense medium is quite small.\nThe C ichemistry in diffuse ISM was comprehensively consid-\nered theoretically by e.g. Liszt 2011; Wolfire et al. 2008. However,\nthesestudiesconsideredtheobservationsobtainedinourgalaxy,i.e.\nexamined only the local ISM. In turn, to date, there is a quite large\nnumber of observations of C iobtained in high-redshift DLAs, i.e.\nremote galaxies. These observations are mostly associated with the\nlow-metallicity gas ( 𝑍≲0.3), where one can expect the changes in\nthe thermal (see e.g. Bialy & Sternberg 2019; Balashev et al. 2022)\nandionizationbalance(seee.g.Balashev&Kosenko2020;Balashev\net al. 2021), which is important for the chemistry of C i. An impor-\ntantconstituentofthesechangesarecosmicrays,whichstarttobea\ndominantheatingsourceofcoldISMat 𝑍≲0.3(Bialy&Sternberg\n2019)andgovernitsionizationstate.ThereforeC iisexpectedtobe\nsensitive to the cosmic rays ionization rate (CRIR) and hence can\nprovide a way to measure it.\nPreviously, some indirect methods were also used to estimate the\ncosmicrayionizationratebasedontheobservationsoftheabundance\nof a several molecules that are sensitive to the degree of ionization\nof the medium. Most of the molecules used, such as H+\n3,OH+,\nH2O+,HD(see e.g. Hollenbach et al. 2012; Indriolo & McCall\n2012; Indriolo et al. 2015; Kosenko et al. 2021), represent diffuse\nand translucent molecular gas. These methods show that there is a\n©2023 The AuthorsarXiv:2402.00714v1  [astro-ph.GA]  1 Feb 20242\nlargespreadinthemeasuredCRIR:from 10−17to∼few×10−15s−1.\nSuch a scatter can be caused both by the natural inhomogeneity of\nthe CRIR in the interstellar medium, associated with the locality of\nsourcesandtheeffectsofcosmicraypropagation,andbysystematic\neffects in the methods of estimation used. Therefore independent\nestimations of CRIR are highly valuable.\nIn this work, we present an independent formalism to calculate\nCiabundance in the diffuse ISM (see Sect. 2) and its application\nonthesampleofC imeasurementsobtainedinthehigh-redshiftH 2-\nbearing DLAs. In particular, in comparison with previous studies\n(e.g.Wolfireetal.2008;Liszt2011)wefocusonthelowmetallicity\ngas,howitaffectstheC iabundance,andtheimportanceofthecos-\nmic rays in C iproduction (see Sect. 4.1). We developed the method\n(presented in Sect. 4.2) for estimating the main physical conditions\nin the medium, based on measured C iand molecular hydrogen, H 2,\ncolumn densities, the population of the C ifine structure levels, and\nkinetic temperature (derived through the ortho-para ratio of H 2). A\nnewimportantfeatureofthisworkthatdiscriminatesitfromourpre-\nviousstudies(e.g.Klimenko&Balashev2020;Kosenkoetal.2023)\nis the consideration of C iabundance and constraints on CRIR that\nCiprovides coupled with explicit calculations of the thermal state\nof the gas. We applied this method to a sample of known C i/H2-\nbearingDLAsathighredshiftsandcompareditwithsomeprevious\nestimates (see Sect. 4.3).\n2 MODEL\nIn this Section we describe a formalism that was used to calculate\nrelative C i/H2abundances, thermal state, and the excitation of C i\nfine-structurelevels.Inthefollowing,duringthemodelling,wepaid\nattention to the following quantities: H iand H 2column densities,\nmetallicity, the ortho-para ratio of H 2(providing the measurements\nof the kinetic temperature), and the column densities on three C i\nfine-structurelevels.Thelatterprovidethemeasurementofthetotal\ncolumn density of C iand hence the relative abundance of C ito\nH2. These quantities from the observations will be used to derive\nconstraints on the main physical parameters of the model.\n2.1 Model global parameters\nWeconsideredahomogeneous(specifiedbythetotalhydrogennum-\nber density𝑛tot\nH) medium with metallicity 𝑍(denoted by default in\ndecimal units) that is exposed by the UV field, 𝜒(normalized to\nMathisfield,Mathisetal.1983)andcosmicrayswithionizationrate\n𝜁–theprimaryionizationrateperhydrogenatom(measuredin s−1).\nThese are the main global parameters of interest.\nForeachconsideredspecies Xinthemediumwedenotethenum-\nberandcolumndensityas 𝑛Xand𝑁X,respectively1.Thesequantities\nevidently are related through the integration along the line of sight,\n𝑁X=∫\n𝑛X𝑑𝑙. The column densities, are also observed quantities,\nandhencearethesubjectofthemodelcalculation.Forconvenience,\nin the following text, we will use H 2column density, 𝑁H2, as a\nnatural unit of measure of the cloud thickness, instead of physical\nlength. The column density of the species Xcan be calculated as\n𝑁X=∫\n𝑛X(𝑁H2)𝑑𝑙\n𝑑𝑁H2𝑑𝑁H2=∫𝑛X\n𝑛H2𝑑𝑁H2.\n1In the following, we assume that the number and column densities are\nmeasured in units cm−3and cm−2, respectively.2.2 Metal and dust abundance\nThethermalbalancedependsonthemetalanddustabundances.The\ndust abundance was specified by the dust-to-gas ratio, DTG. The\nobservationssuggestthatitisscaledwithmetallicityasapowerlaw\nor broken power law. However, the measured dispersion of DTG is\nquite large, therefore for simplicity, we assume that\n[DTG]≡DTG\nDTG⊙=𝑍𝛽, (1)\nwhere DTG⊙is the dust-to-gas ratio at solar abundance and 𝛽is\ntypically constrained between 1 and 3. While this dependence is\nsimplistic,aswasmentionedbyBalashevetal.2022,itmatcheswith\nthe observations from Rémy-Ruyer et al. (2014). Finally, when we\nused the model calculations we took into account the dispersion of\nDTG-metallicity dependence.\nThe abundance of the elements in the gas phase (which is also\nnecessary to properly calculate the cooling rate) can be determined\nusing the metallicity and DTG ratio as\n[X/H]≡X/H\nX/H⊙=𝑍(1−𝑑X·[DTG]), (2)\nwhere X/H⊙is a solar photospheric abundance of some species\ndenoted by𝑋(we use values taken from the Asplund et al. 2009),\nand𝑑Xis a depletion of species at solar metallicity. For the latter,\nweusedthefollowingvalues0.53,0.41,and0.0forcarbon,oxygen,\nandargon,respectively.Wedidnotapplydepletionforotherspecies,\nsince they participate in neither thermal balance nor basic chemical\nreactions, considered in this work.\nWealsoconsideredPolycyclicAromaticHydrocarbons(PAH)sep-\narately, while, we assumed that PAH abundance is linearly scaled\nwithDTGratio,i.e.thedistributionofthegrainsizedoesnotchange\nwith metallicity. In principle it doesn’t have to be true, since the\nproduction of the large grains can be non-linearly dependent on the\nsmallgrainabundance,aswellasthelargegrainsdestructed/formed\ninto/from smaller grains, which may change their relative fractions.\nWhile other authors Wolfire et al. (2003, 2008); Bialy & Sternberg\n(2019) used artificially added 𝜙PAHcoefficient to rescale the colli-\nsionalcoefficientsofthePAHmeasuredinthelaboratory,wedidnot\nconsider this factor, i.e. used 𝜙PAH=1.\n2.3 H 2/HI abundance\nThe abundances of H 2and H idetermine the ionization state and\nchemistry of thecloud and are hence important forthe thermal state\nandchemicalabundancesofthespecies.Inprinciple,tocalculatethe\nrelative H 2and H iabundances, and the H i/H2transition one needs\nto calculate full radiative transfer in H 2Lyman and Werner bands.\nHowever, this is a very time-consuming task and here to be able to\nefficientlycalculatethemodels,weusedtheanalyticaldescriptionof\nHi/H2transitionproposedrecentlybySternbergetal.(2014);Bialy\n& Sternberg (2016)2. Following this formalism, the number density\nofH2as a function of H2column density, 𝑁H2can be written as\n𝑛H2=𝑛tot\nH\n𝛼𝑆H2(𝑁H2)𝑒−𝜎𝑔(𝑁H+2𝑁H2)+2, (3)\nwhere𝑁His the atomic hydrogen column density, which can be\nderivedasafunctionoftheH 2columndensity(seeBialy&Sternberg\n2We also applied it in our recent studies of HD/H2andOH/H2in Balashev\n& Kosenko 2020; Balashev et al. 2021.\nMNRAS 000, 1–11 (2023)Ciin diffuse ISM 3\n2016),𝑆H2is a self-shielding function, 𝜎g≈1.9×10−21[DTG](in\ncm2) is the dust Lyman-Werner photon absorption cross-section per\nhydrogenatom,and 𝛼istheratiooftheratesoffreespace H2photo-\ndissociation and H2formation on the dust grains. In turn, atomic\nhydrogen density is simply defined as 𝑛H=𝑛tot\nH−2𝑛H2.\n2.4 Chemical abundances\nForthepurposeofthiswork,weconsideredasampleofspecies,that\nare abundant in diffuse clouds and affect the thermal and ionization\nbalance(hencehaveanimpactonthechemistryandexcitationofC i).\nThe major species of the hydrogen network are H,H2,H+,H+\n2,H+\n3.\nFortheoxygenmolecularnetworkweconsidered O,O+,OH+,H2O+,\nH3O+,H2O, and OH. We also calculated He,He+,HeH+,Ar,Ar+,\nandArH+that impact the oxygen molecular chemistry and electron\n(denotedby𝑒)abundance.Inthecarbonnetwork,weconsideredonly\nCandC+and did not append the network of C-bearing molecular\nspecies. This is reasonable since the model will be applied only for\ndiffuseclouds,beforetheonsetofCO,wheretheabundancesofany\nother C-bearing molecules are very low and affect neither carbon\nchemistry nor thermal balance. We also explicitly considered PAH\nspecies at three ionization states: PAH0,PAH+, and PAH−, since\nthey affect ion recombination, such as H+andC+.\nForHandH2we used analytical description of the abundances\n(see Sect. 2.3). We also used the fixed He,O, and Arabundance (as\ndescribed in Sect. 2.2, except Hefor which we used value 0.085 of\nhydrogen abundance, independent from metallicity), since they are\ndominantionizationformincolddiffusemedium.Forotherspecies,\nwe calculated the chemical abundances by solving the matrix equa-\ntions of the chemical network. We used the same reaction rates as\nprovided by Balashev et al. 2021 mostly compiled from UMIST\ndatabase McElroy et al. 2013. For reaction rates of PAHs we used\ncompilation by Wolfire et al. 2008. We additionally considered the\ndependenceofphotoreactionsonthedilutionoftheUVfield,withco-\nefficientstakenfromtherecentcompilationby(Heaysetal.2017).To\nsolve chemical equations we additionally assume that the total PAH\nabundance is 𝑛PAH+𝑛PAH−+𝑛PAH+≡𝑛tot\nPAH=6×10−7[DTG]ntot\nH\nand𝑛C+𝑛C+≡𝑛tot\nC=[C/H]ntot\nH.\n2.5 Excitation of the fine-structure levels\nTheexcitationoffine-structurelevelsisimportantforthecalculation\nofthecoolingfunctionsandthecolumndensitiesofthefine-structure\nlevels, that was used to compare with observed ones. We calculated\ntheexcitationoffine-structurelevelsofC ii,CiandO iusingstandard\nmatrix calculations (e.g. Silva & Viegas 2002), assuming excitation\nby collisions, radiative excitation by CMB and UV pumping, and\nde-excitation by spontaneous transitions. We considered collisions\nwith the most abundant species in diffuse ISM: H,H2(separating\northo and para), He(if available), and electrons. The collisional\ncoefficients for C+were taken from (Barinovs et al. 2005; Flower &\nLaunay 1977; Keenan et al. 1986); for Cfrom (Abrahamsson et al.\n2007; Schroder et al. 1991; Staemmler & Flower 1991; Johnson\net al. 1987); for Ofrom (Abrahamsson et al. 2007; Jaquet et al.\n1992;Monteiro&Flower1987;Belletal.1998).Weconsideredthe\nexcitation by Cosmic Microwave Background with the temperature\nsetas 2.725(1+𝑧)Kattheparticularredshiftofeachstudiedabsorber.\nThisexcitationcanbeimportantonlyforC i(whichhasthesmallest\nseparationofthefine-structurelevels)andredshift( 𝑧∼2−4)while\nprovides subdominant contribution to the population of C ilevels,\nthat is typically observed.2.6 Thermal balance\nThe temperature of the medium was obtained using the thermal\nbalancebetweenthecoolingandheatingprocessesinISM.Wecon-\nsidered the cooling by Ly 𝛼(see e.g. Spitzer 1978), recombination\n(weusedcoefficientfromBialy&Sternberg2019),andC ii,Ci,and\nOifine-structureemission.Thelatterwascalculatedusingexcitation\nofthefine-structurelevels(seeSect.2.5)andassumingtheoptically\nthin regime. The main heating sources considered in this work are\nthe photoelectric heating on the dust grains and cosmic rays, which\ndepend on the physical parameters and electron abundance. We fol-\nlowthestandardformalismtoobtaintheheatingrates,mostrecently\ndescribedinBialy&Sternberg2019.However,weadditionallytake\ninto account the dependence of the photoelectric heating on the ex-\ntinction of the UV field, using scaling of the rate by a dilution of\nthe UV photons, calculated using the typical extinction curve (Fitz-\npatrick&Massa2007)with 𝐸𝐵−𝑉=𝑁tot\nH/5.6×1021[DTG].Wedid\nnotconsiderheatingbyX-raybackground,sinceitislikelyimportant\nonly in the case of the presence of a strong X-ray emitting source.\nWealsodidnottakeintoaccounttheturbulencedissipationheating.\nWhile as was mentioned in (Balashev et al. 2022) it can be impor-\ntant at low metallicities and even can be comparable with cosmic\nray heating, the intermittent nature of the turbulence heating, may\nindicatethattheheatingcanbeverylocal(seediscussioninKlessen\n&Glover2016),whichcomplicatesthecalculations.Wealsodidnot\nconsiderheatingandcoolingby H2,sinceitisonlyimportanteither\nat low metallicity gas or at dense compact regions exposed by large\nUV field (see e.g. Bialy & Sternberg 2019), while we considered\nmostly diffuse ISM.\n2.7 Solution\nToobtainthecolumndensitiesofthespeciesforeachsetofthephys-\nical parameters, defined in Sect. 2.1 and 2.2, we used the following\nroutine. We divided the cloud of some length (specified by 𝑁H2) by\na number of layers, with 0.1 dex step starting from log𝑁H2=14.\nFor each layer, we first determine 𝑛Hand𝑛H2using the analytical\nformalismofH i/H2transition(seeSect.2.3).Withthesevaluesand\nalsofixedvaluesof He,O,and Arabundance(seeSect.2.4)wecal-\nculatethecloudstructurelayerbylayer(startingfromthefirst)using\niterative routine. At each iteration we solved a chemistry network to\nget abundances, 𝑛Xand𝑁X, of other species in consideration (see\nSect. 2.4) together with the thermal balance to get temperature (see\nSect. 2.6) and excitation of fine-structure levels (see Sect. 2.5). The\niterations were stopped when next step does not change the derived\nvalues by more than 0.5 percent for each quantity, which set by a\ntrade-off between accuracy and computational time. In most layers,\nseveral iterations were enough starting from the values, obtained at\nthe previous layer. The typical calculation time of one model was\nabout a few seconds per one core on the standard laptop, which we\nfoundquiteenoughforefficientsamplingofthemodelspresentedin\nSect. 4.2.\n3 DATA SAMPLE\nTo compare the model presented in the previous section with obser-\nvations,wecompiledobservedC iandH2columndensitiesfromall\nknown H 2-bearing DLAs detected at high redshift (𝑧≳2)towards\nquasar sightlines. We note that there are a few additional detected\nsystems at high-redshifts presented in (Noterdaeme et al. 2018) that\nMNRAS 000, 1–11 (2023)4\nhaveonlytotalC iandH 2columndensities,i.e.thereisnoinforma-\ntiononC ifine-structureexcitation, 𝑇01temperature,andmetallicity.\nWealsodidnotconsiderso-calledproximateH 2-bearingDLAs(e.g.\nNoterdaemeetal.2019,2023),whicharetypicallydefinedassystems\nthat have redshift close to the quasar redshift. i.e. Δ𝑉≲3000 km/s.\nThese systems can be significantly impacted by the AGN emission\n(e.g. Balashev et al. 2020) or even be produced in the AGN out-\nflowinggas(e.g.Noterdaemeetal.2021),thereforetheymaybenot\nrepresentative for the diffuse medium, and require a separate study.\nThe compiled sample is presented in Table 1. In the case where\nseveral H 2/Ci-bearing components were detected along the line of\nsight,weprovideexplicitlytheindividualH 2/Cicomponentswithin\nDLAs, since the described modelling allows us to consider them\nindependently.\n4 RESULTS\nIn this section, we present the results of the modelling and its appli-\ncation to the observational data.\nAn example of the calculated temperature, abundances, and exci-\ntationofC ifine-structurelevels,togetherwithvaluesofthechemical\nreaction and cooling/heating rates are shown in Fig. 1 for the model\nwith𝑛tot\nH=100 cm−3,𝜒=1,𝜁=10−15s−1, and𝑍=0.3. One\ncan see that profiles of the species behave well as expected – where\nthe main feature is H i/H2transition (see panel (b)in Fig. 1), which\naffected the thermal balance (panel (e)), temperature (panel (a)),\nionization (panels(b)) and chemical structure (panel (g),(h)) and\nat the less amount PAHabundances (panel (d)). It is important to\nnotice, that the temperature, C iabundance, and relative population\nofCifine-structurelevelsaremoreorlessconstantanddonotshow\nstrongvariationsthroughoutthecloud.Thereforethesequantitiesare\nlikelytohavelittlebiasregardingthegeometricalmodelofthecloud,\nsince they are little coupled with the radiative transfer. This will be\nimportant for the discussion of the results in the following subsec-\ntions. It is interesting to note that there is a slight increase of the\nkinetic temperature in the H 2dominated region in comparison with\natomic hydrogen dominated region, due to lower particle density in\nH2dominated region and lower collisional excitation rate of excited\nCiifine-structure level by H 2, than by atomic hydrogen.\n4.1 Variation of C i/H2abundance\nWeinvestigatethedependenceofC i/H2relativeabundancesoneach\nphysical parameter in the model. Since our calculations were based\non one-sided radiation field models (inherited from formalism by\nSternberg et al. 2014), to compare model results with observational\ndata,hereandinthefollowingtextwecalculatedthecolumndensities\nof each species at the half of specified H2column density and then\nmultiplied by two, i.e. 𝑁obs\nX=2𝑁X(𝑁H2/2). This approximately\nemulates the plain parallel cloud of the thickness 𝑁H2exposed by\nUV field on both sides3.\nInFigure2weplotthecomparisonoftheobservedandcalculated\ndependence of C iandH2column densities. We plot all the systems\nfrom the parent sample, presented in Sect. 3, while we note that for\nseveral systems, where the column densities on high fine-structure\n3Itdoesnotexactlycorrespondaforementionedcase,sincethereisadditional\nUVradiationcomingfromtheoppositeside,butsinceobserved 𝑁H2column\ndensities that we will consider are relatively high this will not introduce\nsignificant bias on the calculated quantities.were not measured, the total C icolumn densities can be slightly\nhigher,butnotmorethan ≈30percent. ItisevidenttoexpectthatC i\nabundancemaydependonthemetallicity(atleastcarbonabundance\nalmost linearly scaled with it), and since the observed metallicities\nat high redshifts systems span about two orders from log𝑍∼−1.5\nto0.5,wedividedthesampleinthefourmetallicitybins ≈0.4sizes\n(in dex) and calculated theoretical 𝑁obs\nCi(𝑁H2)profiles for the mean\nvalues in each bin.\nAswementionedabove,C iabundancestronglydependsonthree\nglobal parameters of the model: the number density, 𝑛tot\nH, CRIR,𝜁,\nandUVfieldstrength, 𝜒.Toshowhowvariationsoftheseparameters\naffectC i/H2relativeabundanceweconstructedthebasemodelwith\n𝑛tot\nH=100cm−3,𝜁=10−15s−1and𝜒=1and varied indepen-\ndently each of the parameters within two dexes, that correspond to\nthe typical measured variations, see Klimenko & Balashev (2020);\nKosenko et al. (2021). The resulting profiles of C i/H2abundances\nare shown in Figure 2. One can see that except a few points at inter-\nmediatemetallicities ⟨log𝑍⟩∼−0.8and−0.4observeddatacanbe\nreproduced within chosen ranges of the physical parameters.\nOnecanseethatatdifferentmetallicitiesC i/H2abundanceissen-\nsitive to different parameters. At high metallicities, 𝑍≳0.5, it is\nmostlysensitivetothevariationoftheUVfieldandnumberdensity.\nThisissimplybecausetheUVfielddirectlyscalestheC iabundance\nby photoionization process. However, at low metallicities, 𝑍≲0.2\n(actually corresponding to the most DLAs at high redshifts), C i/H2\nabundancealsobecomesquitesensitivetoCRIR,sinceatthesemetal-\nlicities, C iirecombination rate depends on the electron abundance,\nwhich in turn depends on the hydrogen ionization fraction, which is\nsensitivetoCRIR.Oncemetallicityapproachessolarvalue,theelec-\ntrondensitiesstarttobedeterminedsolelybythecarbonabundance\nandhenceC i/H2becomesalittlesensitiveonCRIR(thebottomleft\npanel in Figure 2). Interestingly, at low metallicities, C i/H2abun-\ndance becomes almost insensitive to the UV flux (the top central\npanel in Figure 2), at the typically observed values in the range be-\ntween 0.1 and 10 of the Mathis field. One can additionally note,\nthat at intermediate H 2column densities, i.e. log𝑁H2≲20, Ci/H2\nabundance have opposite dependence on the number density at low\nand high metallicities, i.e. decreases and increases with the number\ndensity increase, respectively.\n4.2 Constraints on the physical parameters\nAsweshowedinprevioussectionabundanceC i/H2,Cifinestructure\nexcitation and ortho-para ratio of H 2depend on physical conditions\n(𝑛tot\nH,𝜒,𝜁) one can constrain the latter using observed quantities\ntogetherwithmetallicitymeasurement.Importantly,aswediscussed\nin Sect. 4, aforementioned quantities (that we will use for compari-\nson with observations) do not drastically vary along the cloud, and\nhence these estimates are most probably less biased by unknown\ngeometry of the absorbing region than most other species do, e.g.\nHD or O-bearing molecules. Since the dependency on the physical\nparameters likely has a relatively complex shape in the parameter\nspace, and our model is quite a little time-consuming to constrain\nthese parameters we followed the Bayesian approach to sample the\nposteriorprobabilitiesfunctionofthe 𝑛tot\nH,𝜒,𝜁usingaffineinvariant\nMonte-CarloMarkovChainsampler(Goodman&Weare2010).The\nlikelihood function was constructed by comparison of model values\nwith observations of the following quantities: (i) the kinetic temper-\nature,measuredbycolumndensitiesof 𝐽=0,1levelsofH 2;(ii)the\ncolumndensitiesoffinestructurelevelsofC i(thisincludestherela-\ntiveexcitationandtotalC icolumndensity)(iii)thetotalH icolumn\nMNRAS 000, 1–11 (2023)Ciin diffuse ISM 5\nTable 1.Observational data on C i/H2-bearing absorption systems at high redshifts (𝑧≳2)used in this work\nQSO 𝑧abs log𝑁HI log𝑁H2 log𝑁CIj0 log𝑁CIj1 log𝑁CIj2 log𝑍 𝑇 01[K] References\nJ0000+0048 2.52546 20.80±0.10 20.43±0.02 16.10±0.08 15.54±0.14 14.67±0.11 0.46±0.45 52±2 (1)\nB0027−1836 2.40183 21.75±0.10 17.30±0.07 12.25+0.09\n−0.15<12.27 -−1.63±0.10 134+41\n−25(2, 3)\nJ0812+3208 2.626443 21.35±0.10 19.93±0.04 13.30±0.23 13.02±0.03 12.47±0.05−0.81±0.10 48±2(4, 5)\nJ0812+3208 2.626276 ’-’ 18.82±0.37 12.70±0.02 12.32±0.04<12.39 ’-’ 50+44\n−16(4, 5)\nJ0812+3208 2.625808 ’-’ 15.98+0.29\n−0.23†12.13±0.05 11.68±0.16 11.37±0.27 ’-’ 147+13\n−11(4, 5)\nJ0816+1446 3.287420 22.00±0.20 18.62+0.21\n−0.1713.43±0.01 13.24±0.02 12.47±0.07−1.10±0.10 69+10\n−8(6)\nJ0843+0221‡2.7865 21.82±0.11 21.21+0.02\n−0.0213.53+0.04\n−0.0213.61+0.02\n−0.0213.34+0.04\n−0.04−1.52+0.08\n−0.10123+9\n−8(7)\nJ0906+0548 2.567181 20.13±0.01 18.88±0.02 13.50+0.08\n−0.0613.61+0.03\n−0.0312.89+0.07\n−0.08−0.16+0.05\n−0.08116+26\n−4(8)\nJ0946+1216 2.606406 21.15±0.02 19.96+0.01\n−0.0214.10+0.05\n−0.0513.89+0.03\n−0.0413.28+0.05\n−0.04−0.46±0.02 131+10\n−14(8)\nJ0946+1216 2.607083 ’-’ 17.26+0.31\n−0.0813.22+0.05\n−0.0613.35+0.03\n−0.0312.83+0.10\n−0.18’-’>90 (8)\nJ1146+0743 2.839459 21.54±0.01 18.76+0.01\n−0.0113.28+0.05\n−0.0413.10+0.09\n−0.1012.93+0.12\n−0.18−0.57±0.02 91+3\n−2(8)\nJ1146+0743 2.841629 ’-’ 17.94+0.11\n−0.1313.17+0.10\n−0.1113.07+0.05\n−0.0812.85+0.12\n−0.11’-’ 57+25\n−16(8)\nQ1232+0815 2.33771 20.90±0.09 19.57±0.11 13.87±0.05 13.56±0.04 12.82±0.07−1.32±0.12 67+20\n−12(9, 10)\nJ1237+0647 2.689550 20.00±0.15 19.20+0.13\n−0.1214.67±0.04 14.46±0.03 13.64±0.02 0.34±0.12 108+84\n−33(11)\nJ1237+0647 2.68801 ’-’ 16.28±0.10 13.24±0.04 12.80±0.03 - ’-’ 108+84\n−33(11)\nJ1237+0647 2.68868 ’-’ 17.62+0.08\n−0.1112.84±0.03 12.70±0.05 - ’-’ 108+84\n−33(11)\nJ1311+2225 3.091410 20.75±0.10 17.87+0.37\n−0.3313.17+0.11\n−0.1113.28+0.10\n−0.1112.00+0.5\n−0.4−0.34+0.13\n−0.1463+109\n−27(12, 13)\nJ1311+2225 3.091535 ’-’ 19.52+0.02\n−0.0213.81+0.73\n−0.2713.45+0.11\n−0.0812.86+0.07\n−0.11’-’ 94+11\n−5(12, 13)\nJ1311+2225 3.091735 ’-’ 18.25+0.22\n−0.3913.48+0.05\n−0.0513.42+0.07\n−0.0813.14+0.04\n−0.09’-’ 97+108\n−46(12, 13)\nJ1311+2225 3.091858 ’-’ 18.57+0.05\n−0.0913.32+0.31\n−0.2013.40+0.16\n−0.0912.81+0.10\n−0.11’-’ 85+25\n−13(12, 13)\nJ1439+1117‡2.41837 20.10+0.10\n−0.0819.38+0.10\n−0.1014.26+0.02\n−0.0214.02+0.02\n−0.0213.10+0.02\n−0.020.16±0.11 107+33\n−20(14, 15)\nJ1513+0352 2.463622 21.82±0.02 21.31+0.01\n−0.0114.82+0.18\n−0.1814.60+0.06\n−0.0614.03+0.06\n−0.06−0.84±0.23 92+5\n−6(16)\nJ2100−0641 3.09145 21.05±0.15 18.76±0.03 12.57±0.03 12.31±0.08 -−1.21±0.15 159+44\n−29(4, 17, 18)\nJ2123−0050 2.059300 19.18±0.15 17.94±0.01 13.73±0.02 13.42±0.03 12.43±0.02−0.19±0.15 139+9\n−8(12, 19, 20)\nJ2123−0050 2.059550 ’-’ 15.16±0.02 12.91±0.03 12.74±0.10 12.16±0.11 ’-’ 650+250\n−140(12, 19, 20)\nJ2140−0321 2.339900 22.40±0.10 20.13±0.07 13.03±0.04 13.20±0.04 13.02±0.05−1.05±0.13 75+12\n−9(21)\nB2318−1107 1.98888 20.68±0.03 15.49±0.03 12.63+0.02\n−0.0212.30+0.04\n−0.04-−0.85±0.06 188+34\n−25(2, 3)\nJ2340−0053 2.054170 20.53±0.15 15.99±0.04 12.27+0.03\n−0.0411.97+0.08\n−0.1310.23+0.44\n−0.22−0.74±0.16 181+77\n−25(4, 13)\nJ2340−0053 2.054291 ’-’ 15.24±0.05 12.17+0.04\n−0.0412.29+0.05\n−0.0512.06+0.07\n−0.09’-’ 400+590\n−170(4, 13)\nJ2340−0053 2.054528 ’-’ 17.11±0.12 13.51+0.04\n−0.0313.06+0.01\n−0.0112.00+0.40\n−0.20’-’ 147+253\n−54(4, 13)\nJ2340−0053 2.054610 ’-’ 18.27±0.06 13.20+0.05\n−0.0412.67+0.02\n−0.0312.20+0.50\n−0.50’-’ 183+97\n−50(4, 13)\nJ2340−0053 2.054723 ’-’ 18.14±0.04 13.29+0.01\n−0.0112.97+0.01\n−0.0112.19+0.06\n−0.05’-’ 175+63\n−31(4, 13)\nJ2340−0053 2.054995 ’-’ 16.43±0.03 12.34+0.05\n−0.0511.72+0.15\n−0.2210.72+0.31\n−0.53’-’ 280+145\n−59(4, 13)\nJ2340−0053 2.055140 ’-’ 17.44+0.04\n−0.0512.42+0.05\n−0.0212.18+0.06\n−0.0710.35+0.27\n−0.27’-’ 175+59\n−33(4, 13)\nJ2347−0051 2.587969 20.47±0.01 19.44±0.01 14.28+0.36\n−0.3713.70+0.11\n−0.0813.18+0.11\n−0.16−0.82+0.05\n−0.0376+2\n−2(8)\nReferences:(1)Noterdaemeetal.(2017);(2)Noterdaemeetal.(2007);(3)Noterdaemeetal.(2008a);(4)Jorgensonetal.(2010);(5)Balashevetal.(2010);\n(6) Guimarães et al. (2012); (7) Balashev et al. (2017); (8) Balashev et al. (2019); (9) Ivanchik et al. (2010); (10) Balashev et al. (2011); (11) Noterdaeme\netal.(2010);(12)Noterdaemeetal.(2018);(13)Kosenkoetal.(2021);(14)Srianandetal.(2008);(15)Noterdaemeetal.(2008b);(16)Ranjanetal.(2018);\n(17) Balashev et al. (2015); (18) Ivanchik et al. (2015); (19) Klimenko et al. (2016); (20) Milutinovic et al. (2010); (21) Noterdaeme et al. (2015);\n†independent analysis\n‡for these DLAs we used the total column densities, since 𝐽=0,1H2lines are not resolved, and hence we have no independent measurements in several\nvelocity components resolved in other species (see Balashev et al. 2017; Srianand et al. 2008).\n∗there is a typo in Table 2 of Balashev et al. 2019, with the correct value provided in Table A.8 of that paper.\ndensity associated with DLA. The latter was used as an upper limit\nonlysincewedonotknowhowmuchoftheobservedH iattributedto\ntheC i-bearingcomponent,butwefounditusefulsinceitcanshrink\nthe posterior distribution functions. We considered H 2column den-\nsity as a model parameter since it determines the thickness of the\ncloud in the model. However, we take into account the uncertainty\nofthemeasurementsof 𝑁(H2)usingaGaussianprior,whichiswell\nwithintheBayessianapproach.Additionally,themetallicitywasalso\nconsideredasamodelparameterwithGaussianpriorcorresponding\nto the measured interval estimate for each system. Since the C ifor-\nmation is quite sensitive to the dust-to-gas ratio, we considered theDTGratioasanindependentparameterduringthemodelling,while\nadding a prior to follow dependence with metallicity with 𝛽=−1.5\n(see Eq. 1) and additional 0.5 dex dispersion, to take into account\ntheobservedscatterinDTG-metallicityrelation.Fortheothermodel\nparameters we used flat priors in log space, that conditionally emu-\nlateawidedistribution,intheabsenceofotherphysicallymotivated\npriors. In the following, to report the point and interval estimates\non the model parameters we used maximum a posterior probability\nestimate and the highest posterior density 0.683 credible interval,\nrespectively.\nAnexampleoftheresultofthemodellingoftheH 2/Ciabsorption\nMNRAS 000, 1–11 (2023)6\n15 16 17 18 19 20 2150607080Temperature [K](a)\n15 16 17 18 19 20 21−4−3−2−1012logn[cm−3](b)\nH\nH2\nH+C+\nCe\nHe+\n15 16 17 18 19 20 21−1.5−1.0−0.50.0lognC(J=j)/nC(J=0)(c)\nC(J=1)\nC(J=2)\n15 16 17 18 19 20 21−8−7−6logn[cm−3](d)\nPAH\nPAH−\nPAH+\n15 16 17 18 19 20 21\nlogNH2[cm−2]−26−25−24logL,G[ergs−1cm−3](e)\nPhotoelectric\nCosmicray\nC+O\nC\nRecomb.\n15 16 17 18 19 20 21\nlogNH2[cm−2]−16.0−15.5−15.0−14.5−14.0logR[cm−3s−1](f)\nC+ + e\nC+ + PAH\nC+ + PAH−\nC + hν\n15 16 17 18 19 20 21\nlogNH2[cm−2]−14.5−14.0−13.5−13.0−12.5logR[cm−3s−1](g)\nPAH + e\nPAH− + H+\nPAH− + C+\nPAH− + hν\n15 16 17 18 19 20 21\nlogNH2[cm−2]−19−18−17−16−15−14−13−12logR[cm−3s−1](h)\nH+ + e\nH+\n2 + e\nH+\n3 + e\nC+ + e\nPAH + e\nPAH+ + e\nHe+ + eC + hν\nPAH + hν\nPAH− + hν\nH + c.r.\nH2 + c.r.\nHe + c.r.\nFigure 1. The results of the calculation of the cloud structure with 𝑛tot\nH=100 cm−3,𝜒=1,𝜁=10−15s−1, and𝑍=0.3. The panels show profiles of: (a)\ntemperature;(b)abundancesofthemainelements; (c)numberdensitiesonthe Cfine-structurelevels; (d)PAHadundances;(e)therateoftheheating(dashed\nlines)andcooling(solidlines); (f,g,and h)therateofreactionsoftheformation(solidlines)anddestruction(dashedlines)for C,PAH−,and𝑒,respectively.\nsystem (at z=2.626 towards Q0812+3208) using described method\nis shown in Fig. 3 using the corner plot python package (Foreman-\nMackey 2016). As we see in this particular system, we were able\ntoreproducetheobservationalquantitiesandobtainreasonablecon-\nstraintsonthemodelparameters.Inthisparticularsystem,wefound\nthat UV field, CRIR, and number density are relatively enhanced\nin comparison with typically found values for diffuse clouds. For\nthe latter, it is consistent with the observed high excitation of C i\nfine-structure levels.\n4.3 High redshift sample\nWe applied the procedure described in the previous Section to the\nCi/H2-bearing DLAs at high redshifts, 𝑧≳2, detected so far. The\nparent sample was described in Sect. 3 and shown in Table 1. We\nselected a subsample of systems, where C icolumn density on each\nfine-structurelevelwasmeasured,tomaketheanalysishomogeneous\nsince excitation of C ifine-structure levels provides important con-\nstraints on the model parameters. The described modelling allows\nus to consider the individual H 2/Cibearing components of DLAs,\nunless they have appropriate column density measurements of H 2,\nCifine-structurelevels,and 𝑇01temperature.However,sincewecan\nnot determine the fraction of the H iassociated with each compo-\nnent,weusedanassumptionthatthemetallicitiesofthecomponents\nwithin DLA are the same and equal to the DLA average.\nThe results of the determinations of the physical parameters are\nsummarizedinTable2.WefoundthatestimatesonCRIRarelocated\nwithin the range from 10−16tofew×10−15s−1, while the UV\nestimates show a wider range of values. The number densities are\nfound to be in the range of 𝑛tot\nH=10−1000 cm−3(note that this is\nthetotaldensityofhydrogennuclei,sotheactualnumberdensitycan\nbe≈two times lower than 𝑛tot\nH), which is consistent with the values,\nexpected for the cold neutral medium.\nInFig.4weshowtherelativedependencebetween obtainedvaluesofCRIRandUVfluxes.Itseemsthatthereisnoevidentcorrelation\nbetween these parameters unless one considers some fraction of\nthe data. We note, that since these parameters compete on their\ncontribution to the thermal balance (through cosmic ray heating or\nphotoelectricheating)insomeabsorptionsystemswefoundthatthey\nare degenerated. However, usage of the C iabundance can relax this\ndegeneration, since it provides independent constraints on CRIR.\nAdditionally, we did not confirm the quadratic dependence of the\nCRIR on UV flux, which was previously reported by Kosenko et al.\n2021 for the diffuse ISM, based on the modelling of HD/H 2relative\nabundance coupled with C ifine-structure excitation. In comparison\nwiththeaforementioned values,weobtainedrelativelyhighervalues\nCRIR on average, while the significant fraction of the points around\nlog𝜁∼−15andlog𝜒∼1areconsistentwitheachother.Thehigher\nvaluesoftheCRIRthatwefoundfromC imodellinglikelyarosefrom\nthe matching of C i/H2abundance, since the C iabundance depends\non the recombination rate, which is tightly related to the electron\nabundance (directly at low metallicities, or indirectly through the\ndependence on the PAH−abundance). However, we note that the\nconstraints based on C iare more robust, regarding the radiative\ntransfer issues than HD/H 2-based measurements. Indeed, D i/HD\ntransition which is important to the total HD abundance is typically\nsharp(seee.g.Balashev&Kosenko2020),andhencecanbesensitive\ntotheexactgeometricalmodelofthecloud,sincetheirregularshape,\nnon-uniformity, and patchy structure may allow deeper penetration\nof the dissociating UV radiation in the medium and making HD\nabundancelessthanincaseofuniformmodel.Inthatsense,onecan\nconsiderthatHD/H 2likelyprovidesalowerlimitontheCRIR,since\nhigher ionization fraction4(and hence higher CRIR) is necessary\nif Di/HD transition is shifted deeper into the cloud. Conversely,\n4As discussed in Balashev & Kosenko 2020, HD is mainly formed through\nthereactionof H2andD+,whereabundanceofthelatteristightlylinkedwith\nMNRAS 000, 1–11 (2023)Ciin diffuse ISM 7\n151617181920211213141516logNobs\nCI\n⟨logZ⟩=-1.20⟨logZ⟩=-1.20⟨logZ⟩=-1.20ζ=10−16\nζ=10−15\nζ=10−14\n151617181920211213141516\n⟨logZ⟩=-1.20⟨logZ⟩=-1.20⟨logZ⟩=-1.20ntot\nH=3\nntot\nH=30\nntot\nH=300\n151617181920211213141516\n⟨logZ⟩=-1.20⟨logZ⟩=-1.20⟨logZ⟩=-1.20χ=0.1\nχ=1\nχ=10\n151617181920211213141516logNobs\nCI\n⟨logZ⟩=-0.80⟨logZ⟩=-0.80⟨logZ⟩=-0.80\nζ=10−16\nζ=10−15\nζ=10−14\n151617181920211213141516\n⟨logZ⟩=-0.80⟨logZ⟩=-0.80⟨logZ⟩=-0.80\nntot\nH=3\nntot\nH=30\nntot\nH=300\n151617181920211213141516\n⟨logZ⟩=-0.80⟨logZ⟩=-0.80⟨logZ⟩=-0.80\nχ=0.1\nχ=1\nχ=10\n151617181920211213141516logNobs\nCI\n⟨logZ⟩=-0.40⟨logZ⟩=-0.40⟨logZ⟩=-0.40\nζ=10−16\nζ=10−15\nζ=10−14\n151617181920211213141516\n⟨logZ⟩=-0.40⟨logZ⟩=-0.40⟨logZ⟩=-0.40\nntot\nH=3\nntot\nH=30\nntot\nH=300\n151617181920211213141516\n⟨logZ⟩=-0.40⟨logZ⟩=-0.40⟨logZ⟩=-0.40\nχ=0.1\nχ=1\nχ=10\n15161718192021\nlogNH21213141516logNobs\nCI\n⟨logZ⟩=0.00⟨logZ⟩=0.00⟨logZ⟩=0.00\nζ=10−16\nζ=10−15\nζ=10−14\n15161718192021\nlogNH21213141516\n⟨logZ⟩=0.00⟨logZ⟩=0.00⟨logZ⟩=0.00\nntot\nH=3\nntot\nH=30\nntot\nH=300\n15161718192021\nlogNH21213141516\n⟨logZ⟩=0.00⟨logZ⟩=0.00⟨logZ⟩=0.00\nχ=0.1\nχ=1\nχ=10\nFigure 2. The dependence of the column densities of C ion H 2. The colored points represent observed abundances in high-redshift DLAs (Balashev et al.\n2019;Klimenko &Balashev 2020;Kosenko etal. 2021,and referencestherein) binnedby themetallicity, wheresystems ineach binshown inrows, withmean\nmetallicity within the bin provided in the right bottom corner of the panel. The calculated dependencies using the described formalism are presented by solid,\ndashed, and dotted lines, that correspond to the variation on one of the parameters from the base model with 𝑛tot\nH=100cm−3,𝜁=10−15s−1and𝜒=1. The\narrow in the middle panel at the second row marked the position of J0812 +3208, whose detailed analysis is presented in Figure. 3\nif we assume higher CRIR (and higher ionization fraction) D i/HD\ntransition occurs at a lower depth in the cloud. In turn, as we note\nabove, all, C i/H2abundance, C ifine-structure excitation and the\ntemperatures used in our study are quite homogeneous within the\nH+abundance, and hence the HD formation rate scales with the hydrogen\nionization fraction in the medium and CRIR.mediumanddonotreveallargevariationsthatcansignificantlybias\nresults.\nAswediscussedabove,theionizationandthethermalstatedepend\nonthedustabundance,sincethehighmetallicities, log𝑍≳−0.5,the\nphotoelectricheatingisthedominantheatingsourceandtheelectron\nandH+recombinationaretightlycoupledwiththePAHabundances.\nIn turn, at low metallicities, log𝑍≲−0.5, dust (as it scales with\nMNRAS 000, 1–11 (2023)8\nFigure3. Posteriordistributionfunctionsoftheconstrainedmodelparameters 𝑛tot,𝜒,𝜁,andDTG(alongwiththedistributionsofthemeasuredquantities)for\ntheH 2/Ciabsorptionsystematz=2.626towardsQ0812+3208.Thebluelinesandstripesshowthepointandintervalestimatefromtheobservations.Thekinetic\ntemperature,andcolumndensityofC ifine-structurelevelswereusedtocalculatealikelihoodfunction,whilethetotalH 2columndensityandmetallicitywere\nused as model parameters, with priors corresponding to the measured interval estimate.\nmetallicity) is less important for both the ionization and thermal\nstate and they are mostly governed by cosmic rays. Additionally, at\nlow metallicities, the electron production rate becomes insensitive\nto metal abundance, since it is dominated by hydrogen ionization,\nrather than carbon photoionization as at high metallicity. Since our\nsampleprobesalargerangeofmetallicities,wehaveanopportunityto\nexplorethismetallicitydependence.InFig.5weplotthedependence\nof the derived CRIR on the metallicity. One can see that our resultsdo not show any significant correlation with metallicity, which may\nindicatethattheresultsarerobustagainstartificiallyinducedeffects\nregardingtheonsetofthecosmicraydominanceatlowmetallicities.\nHowever, in principle, this cannot be separated from the possible\ndependence of the physical conditions on the metallicities, which is\nexpected since different metallicities attributed to different galaxies\nandtheirevolutionarystates.Indeeditisknown,thatthemetallicity\nof the DLAs correlates with the mass of the galaxy and its halo.\nMNRAS 000, 1–11 (2023)Ciin diffuse ISM 9\nQSO 𝑧abs log𝑛tot\nH[cm−3]log𝜒[Mathis]log𝜁[s−1]\nJ0000+0048 2.525458 2.0+0.5\n−0.4<−0.4<−14.8\nJ0812+3208 2.626443 2.5+0.1\n−0.11.2+0.2\n−0.2−15.3+0.2\n−0.2\nJ0812+3208 2.625808 1.2+0.4\n−0.41.5+0.4\n−0.4−16.2+0.6\n−0.5\nJ0816+1446 3.287420 2.1+0.3\n−0.2<−1.6−15.2+0.2\n−0.2\nJ0906+0548 2.569181 2.1+0.1\n−0.10.6+0.2\n−0.3−15.0+1.1\n−1.1\nJ0946+1216 2.606406 2.1+0.1\n−0.2−0.1+0.7\n−0.3−14.5+0.1\n−0.2\nJ0946+1216 2.607083 1.9+0.5\n−0.4<0.7−13.9+0.2\n−0.2\nJ1146+0743 2.839459 2.1+0.2\n−0.10.5+0.4\n−0.8−14.4+0.2\n−0.3\nJ1146+0743 2.841629 >2.7 1.4+0.5\n−0.5<−13.4\nQ1232+0815 2.337710 2.3+0.1\n−0.1−2.3+0.4\n−0.4−15.6+0.2\n−0.2\nJ1237+0647 2.689550 1.9+0.1\n−0.1−0.6+0.3\n−0.5<−15.3\nJ1311+2225 3.091410 >1.7 0.8+0.9\n−0.9<−13.5\nJ1311+2225 3.091535 2.0+0.2\n−0.30.7+0.3\n−0.6<−14.3\nJ1311+2225 3.091735 >1.8 0.8+0.4\n−0.5<−13.7\nJ1311+2225 3.091858 2.3+0.2\n−0.40.7+0.4\n−0.7−14.6+0.7\n−1.9\nJ1439+1118 2.418370 1.8+0.3\n−0.2−0.1+0.2\n−0.2−15.1+0.5\n−0.8\nJ1513+0352 2.463622 2.5+0.1\n−0.2−0.4+1.1\n−0.5−14.8+0.3\n−0.3\nJ2123-0050 2.059300 1.5+0.1\n−0.1<−0.0<−14.5\nJ2123-0050 2.059550 1.8+0.4\n−0.3−0.0+0.2\n−0.5−14.3+0.6\n−0.9\nJ2140-0321 2.339900 2.9+0.3\n−0.21.6+0.3\n−0.5−14.4+0.1\n−0.2\nJ2340-0053 2.054170 1.2+0.4\n−0.40.9+0.4\n−0.4<−16.2\nJ2340-0053 2.054291 >1.9<1.9−14.1+0.3\n−0.3\nJ2340-0053 2.054528 1.5+0.7\n−0.4<−0.0−15.0+0.4\n−1.4\nJ2340-0053 2.054610 1.2+0.2\n−0.2<−0.1−15.6+0.1\n−0.1\nJ2340-0053 2.054723 1.6+0.2\n−0.1<0.3−14.8+0.1\n−0.1\nJ2340-0053 2.054995 1.0+0.5\n−0.7<0.9<−16.7\nJ2340-0053 2.055140 1.5+0.4\n−0.30.8+0.4\n−0.4<−16.0\nJ2347-0051 2.587969 2.4+0.4\n−0.4−1.2+0.7\n−0.6−14.8+0.4\n−0.3\nTable 2.The constraints on the physical conditions in high redshift C i/H2-\nbearingDLAsobtainedfromthejointmodellingofC i/H2/Hicolumndensi-\nties, excitation of C ifine-structure, and lower H 2rotational levels.\n−2 −1 0 1 2\nlogχ[Mathis]−16−15−14logζ[s−1]\nCI based \n(this work)\nHD based \n(Kosenko+2020)\nFigure 4. The estimations of cosmic ray ionization rate, 𝜁, and UV field,\n𝜒in diffuse CNM at high redshift H 2-bearing DLAs. The red circles and\ngreen squares represent the results obtained based on C i/H2(this work)\nand HD/H 2(from Kosenko et al. 2021) abundances, respectively. The gray\nsymbols depicted the upper limits.\n−1.5 −1.0 −0.5 0.0 0.5\nlogZ−18−17−16−15−14logζ[s−1]\nFigure 5. The estimates of cosmic ray ionization rate, 𝜁, as a function of\naverage metallicity in high redshift H 2-bearing DLAs. The color code is the\nsame as in Fig. 4.\nThe previous constraints based on HD/H 2abundance also do not\nreveal a strong correlation, although as we discussed above, have\nsystematically lower values of the derived CRIR.\nInFig.6weplotthedependenceofthederivedUVfluxonthehy-\ndrogennumberdensity.Onecanseeagain,thatthereisnostrongcor-\nrelationbetweenthesequantities.However,onecannoticethatwhile\nUV field estimates are found to be consistent, we derive systemati-\ncallyhighervaluesofthehydrogennumberdensityinourstudy,than\nwere previously obtained using only excitation of C ifine-structure\nlevels and kinetic temperature (note that values found by Kosenko\net al. 2021 generally agree with what was obtained by Klimenko &\nBalashev 2020). The main discrepancy can arise from the fact that\nwe also used C iabundance measurements and C iformation rate\nscales with number density, and hence the higher number density\nis favored to reproduce the observed relatively high C i/H2ratios.\nAdditionally, we note, the results obtained by Kosenko et al. 2021;\nKlimenko & Balashev 2020 based on the modelling by MEUDON\nPDR code, during which they used a fixed CRIR to minimize the\nnumber of the model variables. As we discussed above, the CRIR\nmay significantly impact the thermal and chemical balance at low\nmetallicitiesandthereforecanelevatethecharacteristicnumberden-\nsity of the CNM, since it affects the position of CNM branch of the\nneutral phase diagram (see e.g. Bialy & Sternberg 2019; Balashev\net al. 2022). Additionally, as was noted by Slava Klimenko (private\ncommunication, Klimenko et al. in prep), the previous version of\nMEUDON PDR codethatwasusedbyKlimenko&Balashev2020;\nKosenko et al. 2021 had incorrect treatment of CMB field, which\nintensity was a factor of 2 higher than it should be. Higher values\nof the CMB field, will increase the contribution of CMB to the ex-\ncitation of C ilevels at high redshifts, where the CMB temperature\nis enhanced by (1+z) factor, hence lowering the number density es-\ntimates. The detailed comparison of C i/H2and HD/H 2modelling\ncertainly deservesan additional study.However, takinginto account\nthe aforementioned issues, we suppose that the results presented in\nthat study based on the C i/H2abundance are more robust, than the\nprevious ones (Klimenko & Balashev 2020; Kosenko et al. 2021)\nbased on analysis with MEUDON PDR code.\nMNRAS 000, 1–11 (2023)10\n0.5 1.0 1.5 2.0 2.5 3.0 3.5\nlogntot\nH[cm−3]−3−2−1012logχ[Mathis]\nCI based (this work)\nHD based (Kosenko+2021)\nFigure 6. The estimates of UV field, 𝜒, and hydrogen number density, 𝑛tot\nH,\nindiffuseCNMathighredshiftH 2-bearingDLAs.Thecolorcodeisthesame\nas in Fig. 4.\n5 CONCLUSIONS\nWe present the modelling of the relative abundance of C i/H2in the\ndiffuse cold neutral medium using the solution of the H i/H2tran-\nsition, chemical network (with an explicit treatment of the PAH),\nand excitation of fine-structure levels coupled with the thermal bal-\nance.ThisallowsustoaccuratelydescribethedependenceofC i/H2\non the global physical parameters, metallicity, number density, UV\nfield, and cosmic ray ionization rate. We show that typically es-\ntimated values of these parameters can reproduce the observation\nfrom C i/H2-bearing DLA systems at high redshifts. These systems\nmostly represent the low-metallicity gas, for which the cosmic rays\nstart to dominate the heating and ionization state of the medium,\nand hence severely affects the chemistry of the gas. We used C i/H2\nabundances, the population of C ifine structure, and H 2rotational\nlevelsmeasuredinDLAstogetconstraintsonglobalphysicalparam-\neters.WefoundforthesampleofallknownC i/H2-bearingDLAsat\nhighredshiftsthatCRIRisintherange ∼10−16tofew×10−15s−1,\nhydrogen number density is in the range ∼10−103cm−3, and UV\nfieldisintherange 10−2tofew×102ofMathisfield.Wedidnotfind\nany strong correlation between obtained parameters. We also com-\npare measured values with previous constraints obtained using only\nexcitation of C iand H 2levels, and HD/H 2relative abundance. 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J., 2008,\nApJ, 680, 384\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–11 (2023)"    },    {        "title": "2402.00766v1.Benchmarking_Multipartite_Entanglement_Generation_with_Graph_States.pdf",        "content": "1\nBenchmarking Multipartite Entanglement Generation\nwith Graph States\nRen´e Zander, Colin Kai-Uwe Becker\nFraunhofer Institute for Open Communication Systems (FOKUS)\nrene.zander@fokus.fraunhofer.de, colin.kai-uwe.becker@fokus.fraunhofer.de\nAbstract —As quantum computing technology slowly matures and\nthe number of available qubits on a QPU gradually increases,\ninterest in assessing the capabilities of quantum computing\nhardware in a scalable manner is growing. One of the key\nproperties for quantum computing is the ability to generate mul-\ntipartite entangled states. In this paper, aspects of benchmarking\nentanglement generation capabilities of noisy intermediate-scale\nquantum (NISQ) devices are discussed based on the preparation\nof graph states and the verification of entanglement in the\nprepared states. Thereby, we use entanglement witnesses that are\nspecifically suited for a scalable experiment design. This choice\nof entanglement witnesses can detect A) bipartite entanglement\nand B) genuine multipartite entanglement for graph states with\nconstant two measurement settings if the prepared graph state\nis based on a 2-colorable graph, e.g., a square grid graph or\none of its subgraphs. With this, we experimentally verify that a\nfully bipartite entangled state can be prepared on a 127-qubit\nIBM Quantum superconducting QPU, and genuine multipartite\nentanglement can be detected for states of up to 23qubits with\nquantum readout error mitigation.\nIndex Terms —Quantum computing, Benchmarking, Entangle-\nment, Entanglement Witnesses, Graph States\nI. I NTRODUCTION\nExperiments for verifying entanglement generation capabilities\nof gate-based quantum computers gained traction in the recent\nyears in line with the availability of QPUs with an increasing\nnumber of qubits, which is evident from various published\nresults. These include showing genuine multipartite entangle-\nment for a 27-qubit GHZ state [ 1], bipartite entanglement for\na65-qubit graph state [ 2], genuine multipartite entanglement\non51qubits [ 3] and most recently analyzing bipartite and\nmultipartite entanglement for up to 433qubits [ 4]. Furthermore,\ndefining a benchmarking protocol for assessing entanglement\ngeneration capabilities [ 5] using the volumetric benchmarking\nframework [6] was explored.\nGraph states are well-known and researched due to their\nrelevance as a universal resource state for measurement-based\nquantum computing [ 7], and as graph codes in quantum\ncryptography applications and quantum error correction [ 8].\nRecently, graph states were used as encoding scheme for\nan equivalence checking algorithm for comparing bit-strings\nefficiently on gate-based quantum computers [ 9]. The main\nfocus of this paper lies in establishing graph states as a\nmeans for efficiently benchmarking and comparing aspects of\nentanglement generation capabilities for gate-based quantum\ncomputing architectures. For this, bipartite and genuine multi-\npartite entanglement is detected for graph states based on a 2-colorable graph by using different entanglement witnesses in a\nscalable experiment design that requires only two measurement\nsettings. The corresponding measurement results are already\nsufficient to verify bipartite and multipartite entanglement not\nonly for the entire graph state but also for states that correspond\nto connected subgroups of qubits. This is done by evaluating\nexpectation values for different entanglement witnesses that\nare solely dependent on the obtained measurement results. This\nparticular choice of entanglement witnesses thereby provides\na new approach adding to the methods from previous publi-\ncations. We demonstrate our approach through experiments\non three 127-qubit IBM Quantum QPUs. In particular, we\nexperimentally verify that a fully bipartite entangled state\ncan be prepared for 127 qubits, and genuine multipartite\nentanglement can be detected for states of up to 23qubits with\nquantum readout error mitigation. The proposed experiments\nare aimed to be used as a benchmark for gate-based QPUs.\nThe results provide fair performance comparisons for hardware\narchitectures if the chosen graph states can be natively prepared\non each QPU without limitations imposed by the respective\nqubit topologies.\nThe paper is structured as follows. An introduction to the struc-\nture of entanglement for multipartite systems and entanglement\nwitnesses is provided in Section (II). In Section (III), we start\nwith a brief overview about graph states and the stabilizer\nformalism, and discuss the belonging entanglement witnesses.\nHere, we also provide novel insights into the analysis of the\nstructure of entanglement for subgroups of connected qubits.\nThe experiment design, the obtained results, as well as aspects\nof benchmarking are discussed in Section (IV). Finally, we\nconclude with an outlook based on the experiment results in\nSection (V). The Appendix (VI) contains detailed information\nabout the algorithmic implementation used for conducting the\nexperiments as well as additional mathematical proofs.\nII. E NTANGLEMENT\nIn the following, we briefly discuss the structure of entangle-\nment of multipartite systems as well as entanglement witnesses.\nLetMbe a set of qubits and m=|M|. An m-qubit mixed\nstateρisseparable if it can be written as probabilistic mixture\nof separable pure states with respect to a fixed bipartition A, B\nof the set Mof qubits. That is,\nρ=X\nkpkρA\nk⊗ρB\nk (1)arXiv:2402.00766v1  [quant-ph]  1 Feb 2024where ρA\nkandρB\nkare pure states of the subsystems A\nandB, respectively. The coefficients pkdefine a probability\ndistribution, that is, they are positive and sum up to one. Denote\nthe set of separable states by S. A mixed state is bipartite\nentangled if it is not separable with respect to all bipartitions\nof qubits of the system.\nAnm-qubit mixed state ρisfully-separable if it can be written\nas probabilistic mixture of fully-separable pure states, that is,\nρ=X\nkpkρ(1)\nk⊗ ··· ⊗ ρ(m)\nk(2)\nwhere ρ(i)\nkare pure states of qubit i. Denote the set of fully-\nseparable states by Sf. A mixed state is entangled if it is not\nfully separable.\nAnm-qubit mixed state ρisbiseparable if it can be written\nas convex mixture of separable states, that is,\nρ=X\nkqkρk (3)\nwhere ρkare separable states and may have different bipar-\ntitions. The coefficients qkdefine a probability distribution.\nDenote the set of biseparable states by Sb. A mixed state is\ngenuinely multipartite entangled (GME) if it is not biseparable.\nClearly, Sf⊂S⊂Sb.\nA. Entanglement witnesses\nEntanglement witnesses can serve as a helpful tool for\nexperimentally demonstrating the presence of entanglement\nin a quantum state. An entanglement witness for genuine\nmultipartite entanglement is an operator Wthat has non-\nnegative expectation on all biseparable states, i.e.,\ntr(Wρ)≥0,for all ρ∈Sb, (4)\nand a negative expectation on at least one entangled state ˜ρ /∈\nSb. That is, measuring a negative expectation for Wverifies\nthat a state is genuinely multipartite entangled. Similarly, one\ndefines entanglement witnesses for detecting, e.g., non-full-\nseparability (entanglement) or non-separability with respect to\ncertain bipartitions of the set of qubits.\nIn an experimental setting, the prepared state deviates from\nthe target state due to the presence of noise in state-of-the-art\nNISQ devices. For a given pure state |ψ⟩, the projector-based\nwitness [10]\nW=cI− |ψ⟩⟨ψ| (5)\ncan detect genuine multipartite entanglement. Here, cis the\nsmallest constant such that tr(Wρ)≥0for all biseparable\nstates ρ∈Sb. It can be computed via Schmidt decomposition\n[11]. The witness (5) detects a state ρas genuinely multipartite\nentangled if the fidelity between |ψ⟩andρis greater than c,\ni.e.,F(|ψ⟩, ρ)> c.\nIn general, the number of measurement settings required to\nevaluate the operator (5) grows exponentially with the number\nof qubits. For a scalable experiment design it is desirable\nto construct entanglement witnesses that require only a fewmeasurement settings. For example, for m-qubit graph states\nthere are witnesses of the form [10]:\nW=c0I−mX\nk=1ckSk (6)\nwhere ckare constants and the operators Skare tensor\nproducts of Pauli matrices. Then each Skrequires only one\nmeasurement setting, so that Wcan be evaluated with at most\nmmeasurement settings.\nIII. G RAPH STATES\nWe provide a brief introduction to graph states [8], [12].\nLetG= (V, E)be a graph where Vis the set of vertices and\nEis the set of edges. Denote the number of vertices |V|by\nn. A graph state |G⟩ ∈(C2)⊗ncan be associated as follows:\nvertices represent qubits initialized in the |+⟩= (|0⟩+|1⟩)/√\n2\nstate, and edges e= (i, j)represent the controlled Z-operation\nCZ(i,j)acting on qubits iandj. Recall that\nCZ(i,j)=πi\n+⊗Ij+πi\n−⊗σj\nz (7)\nwhere πi\n±= (I±σi\nz)/2are the projectors onto the eigenspaces\nof the operator σi\nzfor eigenvalues +1and−1, respectively.\nThat is, the graph state |G⟩is defined as\n|G⟩=Y\n(i,j)∈ECZ(i,j)|+⟩⊗n. (8)\nDefine the operators\nSi=σi\nxY\nj∈Niσj\nz, i= 1, . . . , n, (9)\nwhere Ni={j∈V|(i, j)∈E}is the set of neighbors of\nthe vertex i. The operators Sicommute and generate a set of\nso-called stabilizer operators that consists of 2nelements,\nS=(nY\ni=1Sxi\ni|x∈ {0,1}n)\n. (10)\nThe graph state |G⟩is the unique state that is an eigenstate to\neigenvalue +1for all Si, that is,\nSi|G⟩=|G⟩,for all i= 1, . . . , n. (11)\nThe projector on |G⟩can be written as product of so-called\nstabilizer projectors (I+Si)/2onto the eigenspace of Siwith\neigenvalue +1, that is,\n|G⟩⟨G|=Y\ni∈VI+Si\n2. (12)\nFor a graph G= (V, E)and a subset U⊂V, we define the\nstabilizer projector of the subset Uas\nP(U, G) =Y\ni∈UI+Si\n2. (13)\nIn particular, |G⟩⟨G|=P(V, G).\n2A. Entanglement witnesses for bipartite entanglement\nHere, we discuss how to detect bipartite entanglement in a\nprepared quantum state ρwith target quantum state |G⟩.\nThe following entanglement witness can be used to detect\nnon-separability. It is known that the same witness can be used\nto rule out full separability [ 10], [8]. Here, we generalize this\nresult to the weaker assumption of separability.\nProposition III.1. LetG= (V, E)be a graph and (i, j)∈E.\nThe operator Wijcan witness non-separability (entanglement),\nWij=I−Si−Sj (14)\nwith⟨Wij⟩ ≥0for all states ρ∈(C2)⊗nthat are separable\nwith respect to any bipartition A, B withi∈Aandj∈B.\nProof. This is a reformulation of the necessary condition for\nseparability given in Proposition (VI.4).\nLetG= (V, E)andρ∈(C2)⊗nbe a density operator with\nqubits V. For an edge e= (i, j)we define its weight w(e) =\n⟨Wij⟩. Let G′= (V, E′)withE′={e∈E|w(e)<0}be\nthe subgraph of Gwith all edges deleted that have non-negative\nweight. Then the connected components of G′correspond to\nbipartite entangled subsets of qubits.\nB. Entanglement witnesses for multipartite entanglement\nIn the following, we discuss how to detect multipartite\nentanglement in a prepared quantum state ρwith target\nquantum state |G⟩.\nFor any graph state |G⟩the projector-based witness\nW(G) =1\n2I− |G⟩⟨G| (15)\ncan detect genuine multipartite entanglement, with ⟨W(G)⟩ ≥\n0for all biseparable states. This follows from the fact that for\nany graph state the fidelity between |G⟩and any biseparable\nstateρis upper bounded by 1/2[13]. It was also shown that\nthis bound is tight in the sense that there is a biseparable state\nρsuch that tr(|G⟩⟨G|ρ) = 1 /2.\nTo measure the projector |G⟩⟨G|, one considers the expansion\n|G⟩⟨G|=1\n2nX\nS∈SS (16)\nwhich is a weighted sum of all 2nstabilizer operators.\nTherefore, the number of stabilizer measurements grows\nexponentially in the number of qubits and is practically feasible\nonly for small systems.\nWith Lemma (VI.1) we obtain entanglement witnesses that\nmay require fewer measurements [ 13]. LetV={V1, . . . , V k}\nbe a partition of the vertex set Vinto disjoint subsets. Then\nthe operator\nW(V, G) =\u0012\nk−1\n2\u0013\nI−kX\nl=1P(Vl, G) (17)\ncan detect genuine multipartite entanglement, with\n⟨W(V, G)⟩ ≥0for all biseparable states. Typical choices\nof the partition Vare the following: if V={V}, we\nobtain the projector-based witness W({V}, G) =W(G). IfV={{i}}i∈V, we obtain (up to a factor of 1/2) the stabilizer\nsum witness [10], [5]\nWs(G) = (n−1)I−nX\nl=1Sl. (18)\nEach stabilizer Slis a tensor product of Pauli matrices and\nhence requires only one measurement setting. Then Ws(G)\ncan be evaluated with at most nmeasurement settings.\nLastly, a map c:V→C, where Cis the set of colors, is\naproper vertex coloring if any two vertices that have the\nsame color are not connected by an edge. A graph is called k-\ncolorable if it has a coloring with |C|=kcolors. The minimal\nnumber kof colors is called the chromatic number χof the\ngraph. A coloring induces a partition Vc={Vc}c∈Cof the\nvertex set Vinto disjoint subsets such that any two vertices in\nthe same subset are not connected by an edge. The case where\nthe partition Vcorresponds to such a proper vertex coloring\nwas investigated for GHZ and 1-D cluster states [ 10] and\nsubsequently proposed as a systematic method for construction\nof entanglement witnesses for graph states [ 13]. We denote\nthecoloring-based witness byWc(G) =W({Vc}c∈C, G). In\nthis case, the expectation of each projector P(Vc, G)can be\ncomputed with one measurement setting ⊗i∈VcXi⊗j∈V\\VcZj.\nThen the computation of the expectation of Wc(G)requires\nonly|C|measurement settings. In particular, for graph states\ncorresponding to 2-colorable graphs, e.g., 1-D and 2-D\ncluster states, entanglement witnesses that require only two\nmeasurement settings can be found.\nIn experiments, the prepared state ρdiffers from the target\ngraph state |G⟩ ∈(C2)⊗ndue to the presence of noise. The\nwhite noise tolerance is commonly used as indicator of the\nrobustness of a witness [ 12]. It is defined as follows. For a\nstate ˜ρand a witness W, consider the state\nρ(p) = (1 −p)˜ρ+pI/2n, (19)\nforp∈[0,1], that is, ρ(p)is a stochastic mixture of the\nstate ˜ρand the maximally mixed state. Then the white noise\ntolerance is the maximal ptolsuch that ρ(p)is detected by\nthe witness W, i.e., tr(Wρ(p))<0for all p∈[0, ptol). For\nthe witnesses W(V, G), for some partition of the vertex set\nV={V1, . . . , V k}, we have 1/k≥ptol>1/(2k)[13]. In fact,\nfor certain graph states, e.g., 1-D and 2-D cluster states, one\ncan construct witnesses such that their white noise tolerance\napproaches one as the number of qubits increases. That is,\nunder the presence of white noise the fidelity between the\nprepared state ρand the target graph state |G⟩can decrease\nexponentially with the number of qubits, but the state ρis\nstill genuinely multipartite entangled and can be detected by\na witness [ 12]. Yet, as these witnesses are an augmentation\nof the projector-based witness W(G), the number of local\nmeasurement settings grows exponentially with the number of\nqubits.\nFinally, a partition eV={eV1, . . . ,eVl}is a refinement of the\npartition V={V1, . . . , V k}if for all i∈ {1, . . . , l}there is a\nj∈ {1, . . . , k }such that eVi⊂Vj. Then with Lemma (VI.1),\nwe see that\nW(eV, G)≥W(V, G). (20)\n3Here, for Hermitian operators A, B , we write A≥Bindicating\nthat(A−B)is positive semidefinite. In particular, the witness\nW(eV, G)has a lower white noise tolerance. In Appendix\n(VI-A ), it is shown that considering witnesses corresponding\nto refinements of a partition Vcan still be useful, specifically\nin the context of quantum readout error mitigation.\nC. Entanglement witnesses for subgraphs\nGiven sampled measurement results corresponding to a pre-\npared state ρwith target graph state |G⟩, we discuss how to\nobtain information on the ability of the QPU to generate\nmultipartite entangled states that correspond to subgraphs\nG′⊂G.\nLetG= (V, E)be a graph and G′= (V′, E′)⊂Gbe a\nsubgraph with E′={(i, j)∈E|i, j∈V′}. That is, G′is\nthe subgraph induced by the subset of vertices V′⊂V. Let\nthe neighborhood N(V′)⊂Vbe the subset of all vertices\ninG\\G′that are adjacent to at least one vertex in G′. Let\neE⊂Ebe the subset of edges that connect G′withG\\G′.\nThis is illustrated in Figure (1).\nConsider the stabilizer projectors\nP(V′, G) =Y\nv∈V′1\n2(I+Sv), (21)\nP(V′, G′) =Y\nv∈V′1\n2(I+S′\nv) (22)\nwhere SvandS′\nvare the stabilizers of |G⟩and|G′⟩, respec-\ntively. Clearly, P(V′, G′) =|G′⟩⟨G′|. The operator P(V′, G′)\nis obtained from P(V′, G)by replacing all Pauli operators\nacting on the qubits in G\\G′with identities.\nOne could aim to compute the expectation of the projector-\nbased witness W({V′}, G′)with respect to the graph state\n|G⟩. For example, consider the 1-D cluster state ρ=|G⟩⟨G|\non four qubits defined by the graph G= (V, E)with V=\n{0,1,2,3},E={(0,1),(1,2),(2,3)}, and let G′= (V′, E′)\nwithV′={1,2},E′={(1,2)}be a subgraph. In this case,\na straightforward computation shows that the reduced density\noperator ρ{1,2}of the subsystem of qubits {1,2}is given by\nρ{1,2}= tr{0,3}ρ= (1/4)I2⊗I2. That is, the reduced state\nρ{1,2}is separable! Accordingly, we have ⟨W(V′, G′)⟩ ≥0.\nMore generally, given an entangled state ρon qubits V, the\nreduced state ρV′with respect to the subset of qubits V′⊂V\nmight not be entangled.\nInstead, we propose measuring the projector P(V′, G)to\nobtain information on the ability of the QPU to generate\na multipartite entangled state that corresponds to the subgraph\nG′⊂G. An interpretation of P(V′, G)is given by the\nfollowing result.\nLemma III.2. The following identity holds:\nP(V′, G) =Y\n(i,j)∈eECZ(i,j)|G′⟩⟨G′| ⊗IY\n(i,j)∈eECZ(i,j).\n(23)\nProof. Recall that |G′⟩⟨G′|=Q\nv∈V′(I+S′\nv)/2. Let (i, j)∈\neE. The unitary CZ(i,j)commutes with all stabilizer operatorsS′\nvforv /∈ {i, j}. There is exactly one vertex v∈V′such\nthatv∈ {i, j}. Without loss of generality, let v=i. Then\nusing (7) and the relation σxπ±=π∓σx, we find\nCZ(i,j)S′\niCZ(i,j)= CZ(i,j)(πi\n−⊗Ij+πi\n+⊗σj\nz)S′\ni=σj\nzS′\ni.\n(24)\nThe last equation follows from a straightforward calculation\nusing the identities π2\n±=π±,π+π−=π−π+= 0 andπ++\nπ−=Ifor the projectors π±. Then the claim follows by\ninduction on the set of edges eE.\nProposition III.3. The following identity holds:\ntr(P(V′, G)ρ) = tr( |G′⟩⟨G′|ˆρV′) (25)\nwhere\nˆρ=Y\n(i,j)∈eECZ(i,j)ρY\n(i,j)∈eECZ(i,j).\nProof. With equation (23) and the cyclic property of the trace\nwe find tr(P(V′, G)ρ) = tr( |G′⟩⟨G′| ⊗Iˆρ). Then the claim\nfollows from the properties of the partial trace.\nThat is, the state ˆρis obtained from the state ρby applying\ncontrolled- Zoperations to all pairs of qubits (i, j)∈eE. If\nρ=|G⟩⟨G|, this amounts to removing all edges connecting\nG′withG\\G′. Then the resulting graph state ˆρ=|G′⟩⟨G′|⊗\n|G\\G′⟩⟨G\\G′|is a product of the graph states for the two\nsubgraphs. In this case, we have tr(|G′⟩⟨G′|ˆρ) = 1 .\nIn the language of entanglement witnesses this can be stated\nas follows:\nProposition III.4. LetV′be a partition of the vertex set V′.\nThe operator W(V′, G)as defined in (17) can detect genuine\nmultipartite entanglement, with ⟨W(V′, G)⟩ ≥0for all states\nρon the system Vsuch that ˆρV′is biseparable.\nProof. ForV′={V′}this is a consequence of Proposition\n(III.3) and the fact that W=1\n2I−|G′⟩⟨G′|is an entanglement\nwitness for |G′⟩. Then the claim for arbitrary partitions follows\nfrom Lemma (VI.1).\nLetG1, G2⊂Gbe disjoint subgraphs such that G1∪G2is\nconnected, and let V1,V2be partitions of the set of vertices\nV1, V2of the subgraphs G1, G2, respectively. Then we have\nW(V1∪ V2, G) =1\n2I+W(V1, G) +W(V2, G). (26)\nIn experiments, the state ˆρV′differs from |G′⟩⟨G′|due to the\npresence of noise. In particular, it is also affected by non-local\nnoise acting on qubits in the neighborhood N(V′). Therefore,\nwe assume that ⟨W(V′, G)⟩ρ≥ ⟨W(G′)⟩ρ′where ρandρ′are\nthe prepared states corresponding to the graph states |G⟩and\n|G′⟩, respectively. In this sense, evaluating an entanglement\nwitness W(V′, G)on the prepared state ρyields information\non the ability of the QPU to prepare the graph state |G′⟩.\n41 2 3\n4 5 6\n7 8 91\n1\n1\nFig. 1: The 2-D cluster graph Cl3×4. Consider the subgraph\nG′= (V′, E′)withV′={1,2,4,5}and\nE′={(1,2),(1,4),(2,5),(4,5)}. Then N(V′) ={3,6,7,8}\nand, as indicated by the red lines,\neE={(2,3),(4,7),(5,6),(5,8)}.\nIV. E XPERIMENTS\nA. Experiment Design\n1) State preparation: We prepare the native graph state\nG= (V, E), i.e., the graph state corresponding to the graph\ndefined by the coupling map of the device, on the 127-qubit\nIBM Quantum superconducting devices ibm_brisbane ,\nibm_sherbrooke andibm_cusco . All three devices have\nthe same so-called heavy-hex layout, as shown in Figure (2).\nAll qubits are prepared in the |+⟩= (|0⟩+|1⟩)/√\n2state\nby applying a Hadamard gate to their initial |0⟩state. Then\nthe controlled- Zgates corresponding to the edges are applied\nin three layers. Within each layer, the controlled- Zgates are\nexecuted in parallel.\n62\n7267 68\n99 10044 45\n108\n1128 9\n40 410\n14 17\n30\n10191\n98\n11075\n9042\n96 9718 19\n5510\n49\n106 10747 48 43\n73\n8538 3920\n33\n103 10480 81 7666\n86\n1188216\n26\n94\n11912 13\n77 7822 23\n50\n105\n114 11546 51\n8324\n79 84\n11625\n56 57\n93\n11721\n53\n58\n8731 27\n71324\n15\n113111\n123 12428\n34\n64 65\n12054365 6\n60 61\n881 2\n12629\n89\n12170\n74\n1223\n63\n1099537\n52\n59 69\n12535\n10211\n927\nFig. 2: A visualization of the heavy-hex layout of the\n127-qubit IBM devices. The edge coloring (red, green, blue)\ncorresponds to three layers of controlled- Zgates. Within each\nlayer, the controlled- Zgates can be executed in parallel. The\nshape of the nodes (round, rectangular) corresponds to a\nvertex coloring of the graph.\n2) Measurements: Since the heavy-hex graph is 2-colorable,\nwe measure the prepared graph state in two measurement\nsettings ⊗i∈V1Xi⊗j∈V2Zjand⊗i∈V2Xi⊗j∈V1Zj, where\n{V1, V2}is the partition of the vertex set Vcorresponding tothe coloring as indicated in Figure (2). For each measurement\nsetting, N= 30000 shots are executed. In the following,\nthese measurement results are used to calculate expectations\nof stabilizers projectors and with this, entanglement witnesses\nfor bipartite and multipartite entanglement.\n3) QREM: Quantum readout error mitigation aims to correct\nmeasurement errors by a classical post-processing of the\nmeasurement outcomes [ 14], [15]. Measurement noise for\na system of Mqubits can be characterized classically by the\nrelation\npnoisy=A·pideal (27)\nwhere pnoisy is the 2M-dimensional probability vector de-\nscribing the distribution of the measurement outcomes in\nthe presence of measurement errors, and pideal is the 2M-\ndimensional probability vector describing the distribution of\nmeasurement outcomes in the absence of measurement errors\n(but still including, e.g., gate errors), and Ais a2M×2M-\ndimensional stochastic matrix. The entry Aijis the probability\nof observing the outcome i∈ {0, . . . , 2M−1}provided that\nthe ideal outcome is j∈ {0, . . . , 2M−1}. Then equation\n(27) can be solved for pideal. Note that the result is not\nnecessarily a probability distribution but a quasiprobability\ndistribution: it may contain negative values but still sums up to\none. This quasiprobability distribution can be used to compute\nan unbiased estimate for the expectation of an observable [ 14].\nIn the tensor product noise model [ 14], we assume that the\nnoise acts independently on each qubit, i.e.,\nA=M−1M\nk=0A(k). (28)\nHere, A(k)is the calibration matrix for qubit kin the\ncomputational basis, defined as\nA(k)= \n1−P(k)\n0,1 P(k)\n1,0\nP(k)\n0,1 1−P(k)\n1,0!\n(29)\nwhere P(k)\ni,jis the probability of measuring qubit kin state\ni∈ {0,1}if the prepared state is j∈ {0,1}. The error rates\np(k)\ni,jare obtained from O(M)calibration circuits.\nIn general, this error mitigation method scales only to a small\nnumber of qubits M. However, it can be utilized for large\nsystems when expectations of m-local observables (for a small\nnumber m) are computed. Recall that an observable is m-local\nif it can be decomposed asP\nlOlwhere each term Olis a\nHermitian operator acting on at most mqubits. In this case,\nthe expectation for each observable Olcan be computed from\nthe marginal distribution with respect to at most mqubits. The\n2m×2m-dimensional calibration matrices for mitigating the\nmarginal distributions are the tensor products of the calibration\nmatrices for the respective qubits. We apply this method to\ncalculate mitigated expectations of stabilizer projectors and\nthereby entanglement witnesses. In particular, this approach\nis suitable for evaluating the stabilizer sum witness for graph\nstates if the belonging graph has a low maximum vertex degree,\ne.g., for heavy-hex graphs. It may occur that the readout\nerror mitigation yields non-physical values ⟨P(U, G)⟩>1.\nTherefore, we cap the expectations of stabilizer projectors at\n51. Details on the implementation of evaluating entanglement\nwitnesses with the described readout error mitigation method\nare given in Appendix (VI-A).\nB. Results\nWe evaluate bipartite entanglement witnesses, and multipartite\nentanglement witnesses for subgraphs.\n1) Bipartite entanglement: We compute expectations of the\nbipartite entanglement witnesses\nWij=I−Si−Sj (30)\nfor all edges e= (i, j)in the graph G. Negative expectations\n⟨Wij⟩<0show that the system is not separable with respect\nto the pair of qubits iandj. That is, there is no bipartition\nA, B withi∈A,j∈Bof the set of qubits Vsuch that the\nprepared state is separable with respect to the bipartition A, B .\nThe connected subgraphs induced by the edges with negative\nexpectations correspond to bipartite entangled regions of the\ndevice.\nThe results are illustrated in Figures (3), (4) and (5)\nfor the devices ibm_brisbane ,ibm_sherbrooke and\nibm_cusco , respectively. Notably, for ibm_brisbane full\n127-qubit bipartite entanglement can be detected when QREM\nis applied.\nFinally, note that similar results on bipartite entanglement were\npresented for the (now retired) devices ibmq_rochester\n(52qubits) and ibmq_manhattan (65qubits) [ 2], and\nmost recently also for ibm_washington (127qubits) and\nibm_seattle (433 qubits) [ 4]. Information on bipartite\nentanglement was obtained by performing full quantum state\ntomography (QST) on every pair of connected qubits and their\nnearest neighbors, and then computing the negativity between\nevery pair of connected qubits. In general, QST on nqubits\nrequires 3nmeasurement settings. If QST is performed for\neach pair of connected qubits, the total number of measurement\nsettings scales linearly in the number of these pairs. As shown\nrecently, this scaling can be reduced to a constant factor by\nperforming QST in parallel [ 4]. In contrast, in this work\nwe show that bipartite entanglement can be characterized by\nmeasuring the prepared graph state in only two measurement\nsettings (for 2-colorable graphs) and calculating the bipartite\nentanglement witnesses (30).\n2) Multipartite entanglement: We compute expectations of\nmultipartite entanglement witnesses with respect to subgraphs\nG′= (V′, E′)⊂G. Denote the number of vertices |V′|by\nn′. The following entanglement witnesses can be evaluated\nwith only two measurement settings:\n(i) the stabilizer sum witnesses (SSW)\nWs(G′, G) = (n′−1)I−n′X\nl=1Sl. (31)\nThe main advantage in utilizing this witness is that it can be\ncomputed efficiently by summing up the previously calculated\nexpectations of the stabilizers for the qubits in V′. However,\nit comes with a theoretical disadvantage of having the lowest\nwhite noise tolerance ptol= 1/n′among all witnesses of the\nform (17).Here, the SSW is utilized as follows.\nFirst, we evaluate the SSW for all subgraphs G′⊂Gthat are\nisomorphic to the 1-D cluster graph Cln, for n= 2, . . . , 30.\nIf⟨Ws(G′, G)⟩<0for a subgraph G′, this indicates that the\ngraph state |G′⟩can be prepared on the device and verified\nas GME. In the following, we say that the state |G′⟩can be\nverified as GME. For each number of qubits n, we identify the\nsubgraph G∗\nn⊂Gthat minimizes the expectation ⟨Ws(G′, G)⟩\nover all subgraphs G′that are isomorphic to Cln. Expectations\nare calculated with and without QREM.\nThe results are illustrated in Figure (6). For the devices\nibm_brisbane ,ibm_sherbrooke andibm_cusco , the\nresults indicate that a 23-qubit, 21-qubit and 21-qubit 1-D\ncluster state can be verified as genuinely multipartite entangled\nwhen QREM is applied, respectively.\nSecondly, we calculate the SSW for all 12-qubit heavy-hex\nunit cells in the graph. Expectations are calculated with and\nwithout QREM.\nThe results are illustrated in Figures (3), (4) and (5).\nFor the devices ibm_brisbane ,ibm_sherbrooke and\nibm_cusco , the results indicate that 8,4and1heavy-hex\nunit cells can be verified as genuinely multipartite entangled\nwhen QREM is applied, respectively.\nNotably, the size of the largest 1-D cluster state that can be\nverified as GME is similar for all three devices. In contrast,\nthere is a remarkable difference in the number of heavy-hex unit\ncells that can be verified as GME, e.g., 8foribm_brisbane\nand1foribm_cusco . This shows that the ability to generate\nmultipartite entangled states is spread more evenly across\nthe device for ibm_brisbane . Therefore, for applications\nthat require a larger number of qubits one would expect that\nibm_brisbane yields better results. In general, evaluating\nmultipartite entanglement witnesses for different types of\nsubgraphs can lead to more expressive results that can be\ninterpreted in the context of practical applications. For example,\nfor simulations of a Heisenberg model on a 1-D lattice, the\nsize of the largest 1-D cluster state verified as GME could be\na suitable metric. When considering, e.g., a Kagome lattice,\nthe size of the largest heavy-hex subgraph verified as GME\ncould be a suitable metric.\n(ii) The coloring-based witness (CBW)\nWc(G′, G) =3\n2I−P(V′\n1, G)−P(V′\n2, G) (32)\nwhere V′={V′\n1, V′\n2}is the partition of the set of vertices\nV′ofG′induced by the coloring of the graph. This witness\nhas a higher white noise tolerance ptol>1/4. However, as\nthe operator (32) cannot be decomposed as a sum of m-local\nobservables for a fixed mindependent of the number of qubits\nn′, its expectation cannot be efficiently computed with the\nQREM described in Section ( IV-A 3). This can be remedied by\nconsidering a refinement of (32), that is, the operator W(eV′, G)\n(17) for a refinement eV′of the partition V′. For 1-D cluster\nstates, the refinement is chosen by subdividing the state in\ngroups of 5connected qubits and ≤5qubits in the remaining\ngroup. This construction is ambiguous: depending on the order\nof the qubits it yields two different refinements of the CBW.\n6Therefore, we choose the minimum of both evaluated witnesses.\nCompared to the SSW such a refinement of the CBW still has\na higher white noise tolerance, e.g., ptol= 1.54/n′ifn′is a\nmultiple of 5(Appendix (VI-B)).\nHere, the CBW is utilized as follows. We evaluate the CBW\nfor the subgraphs isomorphic to Cln, forn= 2, . . . , 30, that\nminimize the SSW without QREM. Furthermore, we evaluate\nthe refinement of the CBW for the subgraphs isomorphic to\nCln, forn= 2, . . . , 30, that minimize the SSW with QREM.\nThe results are illustrated in Figure (6). For the devices\nibm_brisbane ,ibm_sherbrooke andibm_cusco , the\nresults indicate that a 9-qubit, 11-qubit and 9-qubit 1-D cluster\nstate can be verified as genuinely multipartite entangled without\nQREM, respectively. This is comparable to the results for the\nSSW. Notably, the difference between the expectations of the\nSSW and CBW increases with the number of qubits.\nWhen QREM is applied, the results indicate that a 25-\nqubit, 27-qubit and 27-qubit 1-D cluster state can be verified\nas genuinely multipartite entangled for ibm_brisbane ,\nibm_sherbrooke andibm_cusco , respectively.\nForibm_brisbane , the expectations of the SSW and the\nrefinement of the CBW are almost identical independent of the\nnumber of qubits, despite its higher white noise tolerance. This\nindicates that the white noise tolerance is not a sufficient metric\nto assess the robustness of an entanglement witness under\nrealistic experimental conditions. An avenue for future research\ncould be the investigation of the robustness of entanglement\nwitnesses under more realistic noise models. Also note that\nin some cases the expectation of the CBW is larger than the\nexpectation of the SSW. On first sight, this seems contradictory\nto the fact that Ws(G′, G)≥Wc(G′, G), as the SSW is a\nrefinement of the CBW. Yet, this translates to a similar relation\nfor the expectations, i.e., ⟨Ws(G′, G)⟩ ≥ ⟨Wc(G′, G)⟩, only if\nthey are evaluated for probability distributions. With QREM we\nobtain quasiprobability distributions that may include negative\nprobabilities.\nForibm_sherbrooke andibm_cusco , the expectations\nof the SSW and CBW are comparable, yet, the expectations\nof the CBW are consistently lower. At this point, it is not\nsufficiently investigated if this is a reliable result. This differ-\nence could very well be attributed to the QREM method: for\nthe evaluation of witnesses we cap expectations of projectors\nP(U) = P(U, G)at1. For the SSW, the projectors are\nevaluated for each qubit separately. For the refinement of\nthe CBW, the projectors correspond to subsets of 2or3qubits.\nThen, for example, if one considers a projector for a subset\nof2qubits P({i, j})with⟨P({i})⟩>1(before capping)\nand⟨P({j})⟩<1, it may occur that ⟨P({i, j})⟩ ≥ 1.\nAfter capping we have ⟨P({i})⟩= 1,⟨P({j})⟩<1and\n⟨P({i, j})⟩= 1, so that in this case the CBW is lower than\nthe SSW. This phenomenon should be further investigated, e.g,\nby comparing the findings for different readout error mitigation\nmethods.\nC. Benchmarking\nIn the following, the experiments are considered with regard\nto various aspects of benchmarking [ 16] such as scalability,(a)ibm_brisbane (no QREM)\n0 1 2 3 4 5 6 7 8 9 10 11 12 13\n14 15 16 17\n18 19 20 21 22 23 24 25 26 27 28 29 30 31 32\n33 34 35 36\n37 38 39 40 41 42 43 44 45 46 47 48 49 50 51\n52 53 54 55\n56 57 58 59 60 61 62 63 64 65 66 67 68 69 70\n71 72 73 74\n75 76 77 78 79 80 81 82 83 84 85 86 87 88 89\n90 91 92 93\n94 95 96 97 98 99 100 101 102 103 104 105 106 107 108\n109 110 111 112\n113 114 115 116 117 118 119 120 121 122 123 124 125 126\n(b)ibm_brisbane (QREM)\n0 1 2 3 4 5 6 7 8 9 10 11 12 13\n14 15 16 17\n18 19 20 21 22 23 24 25 26 27 28 29 30 31 32\n33 34 35 36\n37 38 39 40 41 42 43 44 45 46 47 48 49 50 51\n52 53 54 55\n56 57 58 59 60 61 62 63 64 65 66 67 68 69 70\n71 72 73 74\n75 76 77 78 79 80 81 82 83 84 85 86 87 88 89\n90 91 92 93\n94 95 96 97 98 99 100 101 102 103 104 105 106 107 108\n109 110 111 112\n113 114 115 116 117 118 119 120 121 122 123 124 125 126\n1.00\n 0.75\n 0.50\n 0.25\n 0.00 0.25 0.50 0.75 1.00Expectation\nFig. 3: A visualization of entanglement in the native graph\nstate on the 127-qubit ibm_brisbane device. The color of\neach node irepresents the expectation of the stabilizer −Si.\nThe color of each egde (i, j)represents the expectation of the\nentanglement witness Wij. Thick (red) edges indicate that the\nsystem is non-separable with respect to the pair of qubits\n(i, j)with95% confidence, and thin (blue) edges indicate that\nnon-separability was not detected with confidence. The\nconnected subgraphs induced by the thick (red) edges\ncorrespond to bipartite entangled regions of the device. The\nlargest bipartite entangled regions consist of (a) 125qubits,\nand (b) 127qubits. The color of the hexagons represent the\nexpectation (capped at 1) of the stabilizer sum witness (31)\nfor the corresponding heavy-hex unit-cell. A red boundary of\nsuch a hexagon indicates that GME was detected with 95%\nconfidence, and a blue boundary indicates that GME was not\ndetected with confidence. When QREM is applied, 8\nheavy-hex unit cells are detected as GME.\n7(a)ibm_sherbrooke (no QREM)\n0 1 2 3 4 5 6 7 8 9 10 11 12 13\n14 15 16 17\n18 19 20 21 22 23 24 25 26 27 28 29 30 31 32\n33 34 35 36\n37 38 39 40 41 42 43 44 45 46 47 48 49 50 51\n52 53 54 55\n56 57 58 59 60 61 62 63 64 65 66 67 68 69 70\n71 72 73 74\n75 76 77 78 79 80 81 82 83 84 85 86 87 88 89\n90 91 92 93\n94 95 96 97 98 99 100 101 102 103 104 105 106 107 108\n109 110 111 112\n113 114 115 116 117 118 119 120 121 122 123 124 125 126\n(b)ibm_sherbrooke (QREM)\n0 1 2 3 4 5 6 7 8 9 10 11 12 13\n14 15 16 17\n18 19 20 21 22 23 24 25 26 27 28 29 30 31 32\n33 34 35 36\n37 38 39 40 41 42 43 44 45 46 47 48 49 50 51\n52 53 54 55\n56 57 58 59 60 61 62 63 64 65 66 67 68 69 70\n71 72 73 74\n75 76 77 78 79 80 81 82 83 84 85 86 87 88 89\n90 91 92 93\n94 95 96 97 98 99 100 101 102 103 104 105 106 107 108\n109 110 111 112\n113 114 115 116 117 118 119 120 121 122 123 124 125 126\n1.00\n 0.75\n 0.50\n 0.25\n 0.00 0.25 0.50 0.75 1.00Expectation\nFig. 4: A visualization of entanglement in the native graph\nstate on the 127-qubit ibm_sherbrooke device. The\nlargest bipartite entangled regions consist of (a) 122qubits,\nand (b) 125qubits. When QREM is applied, 4heavy-hex unit\ncells are detected as GME.\nverifiability and comparability.\n1) Architecture-specific benchmarks: By performing bench-\nmarks with graph states that correspond to the native qubit\ntopology of the QPU under test, computational overhead for\nclassical preprocessing with circuit optimization and qubit\nrouting is reduced without introducing additional SWAP gates.\nFurthermore, the execution of CZ gates can be straightfor-\nwardly parallelized with respect to the qubit topology as\nis shown in Figure (2). For 2-colorable graphs, only two\nmeasurement settings - independent of the number of qubits\n- are needed. Prominent examples apart from IBM Quantum\ndevices that have 2-colorable coupling graphs are shown in(a)ibm_cusco (no QREM)\n0 1 2 3 4 5 6 7 8 9 10 11 12 13\n14 15 16 17\n18 19 20 21 22 23 24 25 26 27 28 29 30 31 32\n33 34 35 36\n37 38 39 40 41 42 43 44 45 46 47 48 49 50 51\n52 53 54 55\n56 57 58 59 60 61 62 63 64 65 66 67 68 69 70\n71 72 73 74\n75 76 77 78 79 80 81 82 83 84 85 86 87 88 89\n90 91 92 93\n94 95 96 97 98 99 100 101 102 103 104 105 106 107 108\n109 110 111 112\n113 114 115 116 117 118 119 120 121 122 123 124 125 126\n(b)ibm_cusco (QREM)\n0 1 2 3 4 5 6 7 8 9 10 11 12 13\n14 15 16 17\n18 19 20 21 22 23 24 25 26 27 28 29 30 31 32\n33 34 35 36\n37 38 39 40 41 42 43 44 45 46 47 48 49 50 51\n52 53 54 55\n56 57 58 59 60 61 62 63 64 65 66 67 68 69 70\n71 72 73 74\n75 76 77 78 79 80 81 82 83 84 85 86 87 88 89\n90 91 92 93\n94 95 96 97 98 99 100 101 102 103 104 105 106 107 108\n109 110 111 112\n113 114 115 116 117 118 119 120 121 122 123 124 125 126\n1.00\n 0.75\n 0.50\n 0.25\n 0.00 0.25 0.50 0.75 1.00Expectation\nFig. 5: A visualization of entanglement in the native graph\nstate on the 127-qubit ibm_cusco device. The largest\nbipartite entangled regions consist of (a) 87qubits, and (b)\n103qubits. When QREM is applied, 1heavy-hex unit cell is\ndetected as GME.\nFigure (7). Note that especially for ion trap based QPUs, all-\nto-all connectivity can be achieved in the NISQ era. With\nthis, every graph state can be natively implemented without\nintroducing additional SWAP gates.\nThe whole QPU can be benchmarked for A) bipartite entangle-\nment so that regions of connected qubits that are bipartite\nentangled are found. Ideally, all benchmarked qubits are\nbipartite entangled such as shown in (2(b)). With the same\nmeasurement results, the QPU can be benchmarked for B)\ngenuine multipartite entanglement so that regions of connected\nqubits that are genuinely multipartite entangled are found. That\nis, the graph state induced by such a subset of qubits can be\n8(a)ibm_brisbane\n0 5 10 15 20 25 30\nNumber of qubits0.5\n0.00.51.01.5Expectation of witnessstabilizer sum witness (no qrem)\nstabilizer sum witness (qrem)\ncoloring-based witness (no qrem)\ncoloring-based witness (qrem) (b)ibm_sherbrooke\n0 5 10 15 20 25 30\nNumber of qubits0.5\n0.00.51.01.5Expectation of witnessstabilizer sum witness (no qrem)\nstabilizer sum witness (qrem)\ncoloring-based witness (no qrem)\ncoloring-based witness (qrem)\n(c)ibm_cusco\n0 5 10 15 20 25 30\nNumber of qubits0.5\n0.00.51.01.52.02.53.0Expectation of witnessstabilizer sum witness (no qrem)\nstabilizer sum witness (qrem)\ncoloring-based witness (no qrem)\ncoloring-based witness (qrem)\nFig. 6: Minimal expectations of the SSW (up to a factor of 1/2) over all subgraphs that are isomorphic to the 1-D cluster\ngraph Cln, forn= 2, . . . , 30. Expectations are computed without QREM (black) and with QREM (blue). Expectations of the\nCBW (green) and a refinement of the CBW (red) for the subgraphs that minimize the SSW without QREM and with QREM,\nrespectively. GME can be verified if expectations are less than zero. The 95% confidence intervals are computed with\nbootstrapping methods. Note that due to the large sample size N= 30000 error bars are barely visible.\nprepared on the QPU and verified as GME. Realistically, for\nNISQ devices, these subsets correspond to smaller subgraphs\nsuch as shown for the heavy-hex unit cells in Figure (2(b)) and\nthe subgraphs isomorphic to 1-D cluster states in Figure (6).\nBased on our observations, we advise using the stabilizer\nsum witness for performing the benchmarks as it can be\nevaluated efficiently in a scalable manner also with readout\nerror mitigation. The coloring-based witness is more costly to\nevaluate (especially with readout error mitigation) and has not\nshown a significant advantage in detecting GME.\nWith our method, the capability of generating entangled states\nbased on natively implementable graph states can be assessed\nand compared to the results from different suitable architectures.\nIf the comparison is done between hardware platforms where\none platform can only implement the graph state by usingSWAP operations, the comparison is not straightforward\nanymore. If the results of said benchmarks would be worse on\nthis platform, it is not clear if this can only be explained with\nthe CNOT gate overhead introduced by the additional SWAP\ngates, or if the device would also perform worse independent of\nthis gate overhead. Only if such a device performs better despite\nan additional SWAP overhead, the results can be interpreted\ncomparatively with other devices in the sense that it performs\nbetter in said entanglement generation tasks. Hence, we advise\nusing this as an architecture-specific benchmark.\n2) Architecture-independent benchmarks: Multipartite entan-\nglement generation for 1-D cluster states can be benchmarked\non every hardware topology, hence generating the longest chain\nof qubits that exhibits GME can be seen as an architecture-\nindependent benchmark. The context is important here: if\n9Fig. 7: The coupling map for the Rigetti Aspen (top) and\nGoogle Sycamore (bottom) device is 2-colorable.\nentanglement for a 1-D cluster state is verified as part of\na larger experiment that probes an overarching graph state,\nthen these results are not necessarily comparable to just\nverifying entanglement for such a cluster state, mainly due to\nincreased (non-local) noise from the additional gate executions\nsurrounding this subgraph.\nV. O UTLOOK\nIn summary, we discussed a scalable method for benchmarking\nthe entanglement generation capabilities of NISQ devices using\nentanglement witnesses. This method was tested on different\nIBM QPUs for analyzing bipartite entanglement over all qubits\nand the ability to generate GME for 1-D cluster states and\nheavy-hex unit cells. In addition, we discussed the implications\nof using this method as a benchmark. Finally, based on the\nresults presented, we list potential further approaches that can\nbe pursued in future research endeavors.\n1) Benchmarking different QPUs : Since the developed\nmethod can be executed efficiently on several NISQ de-\nvices, performing additional benchmarks on these QPUs\nwill provide insightful data. Based on this, comparisons\nbetween different devices with the criteria discussed in\nSection (IV-C) can be drawn.\n2) Maintaining Entanglement : Measuring the duration for\nwhich verified bipartite entanglement for a graph state\nor GME for a subet of qubits can be maintained on\na QPU under test is an interesting extension for the\nproposed benchmark. For this, the experiments can be\naugmented by delayed measurements with an incremental\nincrease in delay time in order to obtain time-dependentdata. Similar experiments were performed based on\ndifferent entanglement verification criteria [ 4] and can\nbe compared with the presented method.\n3) Parallel Circuit Execution : The verification of entangle-\nment in specific subgraphs could potentially be used\nfor the evaluation of parallelization possibilities on a\nQPU. The simultaneous execution of multiple spatially\nseparated quantum circuits on a single QPU (often called\nmulti programming) is an active field of research and\nseveral compilers that perform such parallel scheduling\ntasks were proposed such as palloq [17],QuMC [18]\nandQuCloud/QuCloud+ [19]. The analysis of regions\non a QPU that show good entanglement generation\ncapabilities could be used for more efficient scheduling\nimplementations. A possible choice for such regions are\ngiven by the heavy-hex unit cells that can be verified as\nGME in Section (IV).\nACKNOWLEDGMENTS\nThis work was funded by the Federal Ministry for Economic\nAffairs and Climate Action (German: Bundesministerium f ¨ur\nWirtschaft und Klimaschutz) under the project funding number\n01MQ22007A. The authors are responsible for the content of\nthis publication.\nVI. A PPENDIX\nA. Evaluation of entanglement witnesses\nSubsequently, we discuss the main aspects of implementing\nthe evaluation of entanglement witnesses and readout error\nmitigation (IV-A3).\nIn our setting, we consider a graph G= (V, E)and a partition\nV={V1, . . . , V k}of the set of vertices Vcorresponding\nto a vertex coloring of Gwith kcolors (here, k= 2). The\nmeasurement results for the prepared state ρwith respect\nto the graph state |G⟩are given by a set of probability\ndistributions ∆ ={∆1, . . . , ∆k}. Each probability distribution\n∆l={(x, p(x))|x∈ {0,1}n, p(x)̸= 0}contains measure-\nment results in the measurement setting ⊗i∈VlXi⊗j∈V\\VlZj.\nFor readout error mitigation we utilize the calibration matrices\n(A(i))i∈V. Then entanglement witnesses are evaluated as\nfollows.\n1) Coloring-based witness: Algorithm (1) computes the\ncoloring-based witness for a subgraph G′= (U, E′). More\nspecifically, let U={U1, . . . , U k}be the partition of U\ninduced by the coloring of the graph G, i.e., we have\nUl=Vl∩U. Then we compute the expectation of the operator\nW(U, G) =\u0012\nk−1\n2\u0013\nI−kX\nl=1P(Ul, G). (33)\nNote that each stabilizer projector P(Ul, G)acts non-trivially\non all qubits in Uland their neighbors N(Ul)in the graph\nG. Then the QREM as described in Section ( IV-A 3) scales\nexponentially in the number of these qubits. Hence, it can\nonly be applied for small subsets of qubits. For larger subsets\nof qubits, one can compute a (refinement of the) coloring-\nbased witness with QREM by subdividing the set of qubits\n10Algorithm 1 witness\nInput: A graph G= (V, E), a partition VofV, distributions\n∆V, a subset of qubits U⊂V, calibration matrices\n(A(i))i∈V, and a Boolean qrem .\nOutput: Expectation of witness W=⟨W(U, G)⟩.\n∆ = ∆ V\nΣ ={} ▷Stabilizers\nforjinUdo\nΣ[j] =Sj\nU={} ▷Partition U={U1, . . . , U k}\nforl= 1 tokdo\nU[l] =Vl∩U\nifqrem then ▷Readout error mitigation\n∆,Σ = qrem( G,∆V,Σ,(A(i))i∈V,U)\nW=k−1\n2\nforl= 1 tokdo\nifU[l]̸=∅then\nP= 0 ▷Evaluate projector P(Ul)\nfor(x, p(x))in∆[l]do\ntemp = 1 ▷Evaluate ⟨x|P(Ul)|x⟩\nforiinU[l]do\ntemp =temp·1 +⟨x|Σ[i]|x⟩\n2\nP=P+p(x)·temp\nW=W−min(P,1) ▷Cap projector at 1\nelse\nW=W−1 ▷ P(∅) = 1\nreturn: W\nThis algorithm computes the expectation of the coloring-based\nwitness (33) for a subset Uof qubits, and the expectation of\nthe stabilizer SiifU={i}, fori∈V. For this, we compute\nthe expectation of the stabilizer projectors for each color. The\nexpectation of the projector P(Ul) =P(Ul, G)is calculated\nfrom the distribution ∆lof measurement outcomes in\nmeasurement setting ⊕i∈Vl⊕j∈V\\VlZj. This corresponds to\na change of basis such that in this basis the stabilizers Si, for\ni∈Vl, are a product of Iand Pauli- Zoperators. Thus, the\nprojector P(Ul)is a product of diagonal operators. Therefore,\ncalculating its expectation on a computational basis state |x⟩\nis accomplished by taking the product of the expectations of\nthe diagonal operators (I+Si)/2.\nUinto smaller subsets and computing the witnesses for each\nsubset. For example, for a subdivision U=A∪Bconsider\nthe partitions A={U1∩A, . . . , U l∩A}andB={U1∩\nB, . . . , U l∩B}of the sets AandB, respectively. Then the\npartition A ∪ B is a refinement of the partition Uof the set\nU=A∪B. With equations (20) and (26), we find\nW(U, G)≤W(A ∪ B , G) =I/2 +W(A, G) +W(B, G).\n(34)\nThe witnesses W(A, G)andW(B, G)act non-trivially on a\nsmaller number of qubits.Algorithm 2 qrem\nInput: A graph G, distributions ∆V, stabilizers Σ, calibra-\ntion matrices (A(i))i∈V, and a partition UofU.\nOutput: Mitigated distributions ∆Uand reduced stabilizers\nΣU.\n∆U={}\nΣU={}\nforl= 1 tokdo\nifU[l]̸=∅then\nπ= sort( Ul∪N(Ul))\nforiinUldo ▷Reduced stabilizers\nΣU[i] = reduce(Σ[ i], π)\n∆U[l] = marginal(∆ V[l], π)\n∆U[l] = mitigate(∆ U[l],(A(π1), . . . , A(πk)))\nreturn: ∆U,ΣU\nThis algorithm computes the reduced stabilizers and the\nmitigated (marginal) distribution with respect to a subset Uof\nqubits.\n2) Stabilizer sum witness: If the set Uconsists of exactly one\nqubit, i.e., U={i}, fori∈V, equation (33) simplifies to\nW(U, G) =1\n2−P({i}, G) =−Si\n2. (35)\nHere, we use that P(∅, G) = 1 . Thus, we can apply Algorithm\n(1) to compute the expectations of all stabilizers Si, fori∈V.\nIn our experiments, each such stabilizer acts non-trivially only\non at most 4qubits since the maximum vertex degree of a\nheavy-hex graph is 3. Therefore, readout error mitigation as\ndescribed in Section ( IV-A 3) can be utilized. The (mitigated)\nexpectations of the stabilizers can further be used to calculate\nstabilizer sum witnesses for subgraphs.\n3) QREM: The quantum readout error mitigation described\nin Section ( IV-A 3) is implemented as shown in Algorithm (2).\nFor this, we assume that the following functions are given:\n•sort:Input: set of integers. Output: sorted list of integers.\n•reduce :Input: stabilizer, (sorted) list of positions. Out-\nput: reduced stabilizers with respect to the given positions.\nFor example, for a stabilizer Z7X8Z9and positions\nπ= (7,8,9), the reduced stabilizer is Z0X1Z2.\n•marginal :Input: distribution, (sorted) list of positions.\nOutput: marginal distributions with respect to the given\npositions.\n•mitigate :Input: distribution, list of calibration matrices.\nOutput: mitigated distribution.\nB. White noise tolerance\nFrom the definition of the white noise tolerance for a graph\nstate|G⟩and a witness Wwe find\nptol=\u0012\n1−tr(W)\n2ntr(W|G⟩⟨G|)\u0013−1\n. (36)\n11For a witness Wof the form (5) we have tr(W|G⟩⟨G|) =\n−1/2. It remains to calculate\ntr(W) = 2n\u0012\nk−1\n2\u0013\n−2nkX\nl=12−nl(37)\nwhere nl=|Vl|is the number of qubits in each vertex set of\nthe partition V, and we use that tr(P(Vl, G)) = 2n−nl. Then\nwe obtain\nptol=1\n2 \nk−kX\nl=12−nl!−1\n>1\n2k. (38)\nIn particular, for the stabilizer sum witness, i.e., V={{i}}i∈V,\nwe find ptol= 1/n.\nIn Section ( IV-B ), we consider a refinement of the coloring-\nbased witness for 1-D cluster states. This refinement is obtained\nby subdividing a state in groups of 5connected qubits and\n≤5qubits in the remaining group. This corresponds to a\npartition V={V0, . . . , V ⌈n/5⌉−1}, where V2i, V2i+1, fori=\n0, . . . ,⌈n/5⌉ −1, are the sets of qubits of the i-th group for\neach color. Then we have k= 2⌈n/5⌉. Ifnis a multiple of\n5, we can assume that |V2i|= 2 and|V2i+1|= 3. In this case,\nwe have\nptol=1\n2\u00122n\n5−n\n5\u00121\n4+1\n8\u0013\u0013−1\n=20\n13n≈1.54n−1.(39)\nThe values of c(n) =n·ptol, forn= 1, . . . , 24, are shown in\nTable (I).\nn 1 2 3 4 5 6 7 8\nc(n) 1.0 1.0 1.2 1.33 1.54 1.41 1.33 1.39\nn 9 10 11 12 13 14 15 16\nc(n) 1.44 1.54 1.47 1.41 1.44 1.47 1.54 1.49\nn 17 18 19 20 21 22 23 24\nc(n) 1.45 1.47 1.49 1.54 1.5 1.47 1.48 1.5\nn 25 26 27 28 29 30\nc(n) 1.54 1.51 1.48 1.49 1.51 1.54\nTABLE I: The factors c(n) =n·ptol, forn= 1, . . .30.\nC. Properties of projectors\nFor Hermitian operators A, B on a finite-dimensional Hilbert\nspaceH, we use the notation A≥Bindicating that (A−B)\nis positive semidefinite. The following result was shown in\n[13] (Proof of Proposition 2).\nLemma VI.1. LetP1, . . . , P kbe commuting Hermitian op-\nerators on a finite-dimensional Hilbert space Hwith all\neigenvalues in {0,1}. Then we have\nkY\nl=1Pl≥kX\nl=1Pl−(k−1)I. (40)D. A necessary condition for separability\nWe prove a necessary condition for separability that can be used\nto construct entanglement witnesses for bipartite entanglement.\nRemark VI.2. We write σifori= 0,1,2,3, where σ0=I\nis the identity, and σ1=σx,σ2=σy,σ3=σzare the three\nPauli matrices. Consider the Hilbert space H= (C2)⊗n. We\nwrite i= (i1, . . . , i n)for a multi-index. The set of matrices\nEi=σi1⊗···⊗ σinis orthogonal with respect to the Hilbert-\nSchmidt inner product, that is, (Ei, E j) = tr( E†\niEj) = 2nδij,\nand forms a basis of the real vector space of Hermitian\nmatrices in H. Then any density operator may be represented\nas\nρ=1\n2n\nI+X\ni̸=0λiσi1⊗ ··· ⊗ σid\n (41)\nwhere λiare real numbers. Equation (41) does not include\nthe non-negativity condition.\nProposition VI.3. LetS, S′be Pauli product operators of the\nform:\nS=Y\nm∈Mσ(m), S′=Y\nm∈Mσ′(m) (42)\nwhere σ(m), σ′(m)∈ {I, σx, σy, σz}are Pauli operators\nacting on qubit m. LetA, B be a partition of M. Suppose that\nthe state ρ∈(C2)⊗|M|is separable with respect to A, B , that\nis,ρ=P\nkpkρA\nk⊗ρB\nk. Consider the Pauli product operators\nSK=Y\nm∈M∩Kσ(m), S′\nK=Y\nm∈M∩Kσ′(m), (43)\nforK∈ {A, B}. The operators SKandS′\nKare obtained\nfrom SandS′, respectively, by replacing all Pauli operators\nacting on qubits in M\\Kto identities. If the anti-commutation\nrelations\n{SK, S′\nK}= 0,forK∈ {A, B}, (44)\nare satisfied, then we have\n⟨S⟩+⟨S′⟩ ≤1. (45)\nProof. We consider the case ρ=ρA⊗ρB. Then we have\n⟨S⟩+⟨S′⟩=⟨SA⟩⟨SB⟩+⟨S′\nA⟩⟨S′\nB⟩. (46)\nDefine the Hermitian operators\nOK(θ) = cos( θ)SK+ sin( θ)S′\nK, (47)\nforK∈ {A, B}. The operators OK(θ)have all eigenvalues\nin{−1,1}: we have\n(OK(θ))2=(cos( θ))2S2\nK+ cos( θ) sin(θ)(SKS′\nK+S′\nKSK)\n+ (sin( θ))2S′2\nK=I.\n(48)\nThe Pauli products SK, S′\nKare of the form Ei=σi1⊗···⊗ σin,\nfor some indexes i=iK,i′\nK. Then with (41) we may write\nρK=1\n2|K|\nI+rKOK(θK) +X\ni̸=0,iK,i′\nKλK\niσi0⊗ ··· ⊗ σi|K|\n,\n(49)\n12for real numbers λK\niandrK≥0,θK∈[0,2π). Then with\n(46) we find\n⟨S⟩+⟨S′⟩=rArB(cos(θA) cos( θB) + sin( θA) sin(θB))\n=rArBcos(θA−θB)≤rArB.\n(50)\nIn the following we show that rK≤1forK∈ {A, B}:\non the one hand we have tr(ρKOK(θK)) =rK(cos2(θK) +\nsin2(θK)) = rK. On the other hand the Hermitian operator\nOK(θK)has a spectral decompositionP\nlλl|vl⟩⟨vl|, where\nλlare the real eigenvalues of OK(θK)and the vectors |vl⟩\nform an orthonormal basis. Then we have:\ntr(ρKOK(θK)) =X\nlλltr(ρK|vl⟩⟨vl|)≤max\nlλl= 1.\n(51)\nHere, we used that the density operator ρKis positive and\nhas trace equal to one. This finishes the proof for the case\nρ=ρA⊗ρB. Then the claim follows from the linearity of\nexpectations.\nAs a special case, we obtain the following necessary condition\nfor separability in the context of graph states. This is a\ngeneralization of a similar condition for full separability [ 10],\n[8].\nProposition VI.4. LetG= (V, E)be a graph and (i, j)∈E.\nConsider the stabilizer operators\nSi=σi\nxY\nl∈Niσl\nz, S j=σj\nxY\nl∈Njσl\nz. (52)\nLetA, B be a partition of Vwithi∈Aandj∈B. Suppose\nthat the state ρ∈(C2)⊗nis separable with respect to A, B ,\nthat is, ρ=P\nkpkρA\nk⊗ρB\nk. Then we have\n⟨Si⟩+⟨Sj⟩ ≤1. (53)\nProof. With S=SiandS′=Sjwe find that\nSA=σi\nxY\nl∈Ni∩Aσl\nz, S′\nA=σi\nzY\nl∈Nj\\{i}∩Aσl\nz,(54)\nSB=σj\nzY\nl∈Ni\\{j}∩Bσl\nz, S′\nB=σj\nxY\nl∈Nj∩Bσl\nz. (55)\nClearly, the anti-commutation relations (44) are satisfied. Then\nthe claim follows from Proposition (VI.3).\nREFERENCES\n[1]G. J. Mooney, G. A. White, C. D. 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Yuan et al. , “Detecting multipartite entanglement\nstructure with minimal resources,” npj Quantum Information , vol. 5,\nno. 1, p. 83, 2019.\n[14] S. Bravyi, S. Sheldon, A. Kandala et al. , “Mitigating measurement errors\nin multiqubit experiments,” Physical Review A , vol. 103, p. 042605,\n2021.\n[15] P. D. Nation, H. Kang, N. Sundaresan et al. , “Scalable mitigation of\nmeasurement errors on quantum computers,” PRX Quantum , vol. 2, p.\n040326, 2021.\n[16] C. K.-U. Becker, N. Tcholtchev, I.-D. Gheorghe-Pop et al. , “Towards a\nQuantum Benchmark Suite with Standardized KPIs,” in 2022 IEEE 19th\nInternational Conference on Software Architecture Companion (ICSA-C) ,\n2022, pp. 160–163.\n[17] Y . Ohkura, T. Satoh, and R. Van Meter, “Simultaneous Execution of\nQuantum Circuits on Current and Near-Future NISQ Systems,” IEEE\nTransactions on Quantum Engineering , vol. 3, pp. 1–10, 2022.\n[18] S. Niu and A. Todri-Sanial, “Enabling Multi-programming Mechanism\nfor Quantum Computing in the NISQ Era,” Quantum , vol. 7, p. 925,\n2023.\n[19] L. Liu and X. Dou, “QuCloud+: A Holistic Qubit Mapping Scheme\nfor Single/Multi-programming on 2D/3D NISQ Quantum Computers,”\narXiv preprint arXiv:2207.14483 , 2022.\n13"    },    {        "title": "2402.00785v2.Apparent_Dark_Matter_Inspired_by_Einstein_Equation_of_State.pdf",        "content": "arXiv:2402.00785v2  [gr-qc]  5 Feb 2024Apparent Dark Matter Inspired by Einstein Equation of State\nKimet Jusufi1,∗and Ahmad Sheykhi2,†\n1Physics Department, State University of Tetovo, Ilinden St reet nn, 1200, Tetovo, North Macedonia\n2Physics Department and Biruni Observatory, College of Scie nce, Shiraz University, Shiraz 71454, Iran\nThe purpose of this article is twofold. First, by means of Pad manabhan’s proposal on the emergence nature\nof gravity, we recover the ΛCDM model and the effect of the dark matter in the context of co smology. Toward\nthis goal, we use the key idea of Padmanabhan that states cosm ic space emerges as the cosmic time progress and\nlink the emergence of space to the difference between the num ber of degrees of freedom on the boundary and in\nthe bulk. Interestingly enough, we show that the effect of th e cold dark matter in the cosmological setup can be\nunderstood by assuming an interaction between the numbers o f degrees of freedom in the bulk. In the second\npart, we follow the Jacobson’s argument and obtain the modifi ed Einstein field equations with additional dark\nmatter component emerging due to the interaction term betwe en dark energy and baryonic matter related by\nΩDM,0=/radicalbig2αΩM,0ΩDE,0, where αis a coupling constant. Finally, a correspondence with Yuka wa cosmology\nis pointed out, and the role of massive gravitons as a possibi lity in explaining the nature of the dark sector as\nwell as the theoretical origin of the Modified Newtonian Dyna mics (MOND) are addressed. We speculate that\nthe interaction coupling αfundamentally measures the entanglement between the gravi tons and matter fields\nand there exists a fundamental limitation in measuring the g ravitons wavelength.\nI. INTRODUCTION\nAfter discovery of the black holes thermodynamics in\n1970’s, physicists have been speculating that there should be\na correspondence between the laws of gravity and the laws\nof thermodynamics. In 1995, Jacobson [ 1] disclosed that the\nEinstein’s field equations of gravity can be derived by ap-\nplying the first law of thermodynamics, δQ=TδS, on the\nboundary of spacetime, where δQis the energy flux across\nthe boundary, TandSare, temperature and entropy associated\nwith the boundary, respectively. This derivation is of grea t\nimportance and reveals that the first law of thermodynamics,\nwhen is applied to a spacetime as a thermodynamic system at\nlarge scales, can be translated to the field equations of grav ity.\nFollowing Jacobson, a lot of works have been done to address\nthe correspondence between the first law of thermodynamics\nand the field equations of gravity in different setups [ 2–13].\nAccording to the ΛCDM cosmological model (which\nstands as a prominent and widely accepted framework in de-\nscribing the evolution of the Universe), our Universe com-\nprises three essential constituents: cold dark matter, dar k en-\nergy, and baryonic matter [ 14]. Cold dark matter, a substance\nthat eludes electromagnetic interaction and manifests sol ely\nthrough its gravitational influence, offers a potential sol ution\nto many problems, including the observed flat rotation curve s\nof spiral galaxies [ 15–20]. In recent years, dark matter has\nbeen widely believed to be made of particles beyond the stan-\ndard model, which describes baryonic matter. However, to-\nday, such an elusive dark matter particle remains undetecte d.\nOn the other hand, the cosmological constant gives the dark\nenergy, which emerges as a manifestation of energy permeat-\ning every corner of the Universe, frequently associated wit h\nvacuum energy [ 21].\nAs an alternative to dark matter, in addressing the flat rota-\ntion curves, the Modified Newtonian Dynamics (MOND) the-\n∗kimet.jusufi@unite.edu.mk\n†asheykhi@shirazu.ac.irory was formulated by Milgrom [ 22–24] (see also [ 25–31]).\nIt was argued that the theoretical origin of the MOND theory\ncan be understood from Debye entropic gravity perspective\n[32,33]. The idea that gravity is not a fundamental interac-\ntion and can be regarded as an entropic force caused by the\nchanges in the information associated with the positions of\nmaterial bodies, was proposed by Verlinde [ 34]. Verlinde’s\nderivation of Newton’s law of gravitation at the very least o f-\nfers a strong analogy with a well understood statistical mec h-\nanism. Therefore, this derivation opens a new window to un-\nderstand gravity from the first principles. The study on the e n-\ntropic force has raised a lot of attention recently (see [ 35–40]\nand references therein). In the framework of mimetic gravit y,\nthe MOND-like acceleration was recovered [ 41]. According\nto the entropic force scenario, dark matter represents an ap -\nparent manifestation, a consequence of baryonic matter [ 42].\nUsing the entropic force, modified Friedmann equations have\nbeen obtained recently [ 43–45].\nVerlinde’s proposal on the entropic nature of gravity opene d\na new window on the origin of gravity, however, it assumes the\nbackground geometry as a preexist structure. An important\nquestion is how one can consider the spacetime as an emergent\nstructure? Padmanabhan [ 46] was the first who answered this\nquestion and argued that the spatial expansion of our univer se\ncan be regarded as the consequence of emergence of space and\nthe cosmic space is emergent as the cosmic time progresses .\nBy calculating the difference between the surface degrees o f\nfreedom and the bulk degrees of freedom in a region of space\nand equating the result with the volume change of the space,\nhe derived the Friedmann equations describing the evolutio n\nof the universe [ 46]. Following Padmanabhan, a lot of works\nhave been carried out to investigate the emergence nature of\ngravity and various aspects/extension of this theory have b een\naddressed in the literatures [ 47–54].\nIn this paper, we would like to employ a novel idea and\naddress the dark matter problem in the context of Padmanab-\nhan’s proposal of emergence gravity. This leads to modifica-\ntion of Einstein equations of general relativity, too. This pa-\nper is structured as follows. In the next section we recover t he2\nΛCDM model by assuming an interaction between baryonic\nmatter and dark energy in the bulk. In section III, Following\nJacobson [ 1], we recover the modified Einstein’s field equa-\ntion with an additional energy-momentum term which can be\nregarded as the apparent dark matter. In section IV , we estab -\nlish a correspondence with Yukawa cosmology and explain\nthe role of the massive gravitons as a possible candidate to\nexplain the nature of the dark sector and the emergence of\nMOND. We finish with final remarks in section V .\nII. RECOVERING ΛCDM IN PADMANABHAN’S\nEMERGENT UNIVERSE\nTo explain why our Universe is expanding, Padmanabhan\nused the holographic principle and the equipartition law of\nenergy [ 46]. According to Padmanabhan the basic law gov-\nerning the emergence of space must relate the emergence of\nspace to the difference between degrees of freedom on the\nsurface and in the bulk, namely (Nsur−Nbulk). He proposed\nrelation [ 46]\ndV\ndt=L2\nP(Nsur−Nbulk), (1)\nwhere Vandtare, respectively, the volume and the cosmic\ntime in Planck units. More generally, it is expected (∆V/∆t)\nto be some function of (Nsur−Nbulk)which vanishes when the\nlatter does (note also that L2\nP=¯hG/c3). In the present paper,\nwe would like to modify the above relation as\ndV\ndt=L2\nP/bracketleftBig\nNsur−/parenleftBig\nNbulk+Nint\nbulk/parenrightBig/bracketrightBig\n, (2)\nwhere we assume there is an additional term for the degrees of\nfreedom in the bulk denoted by Nint\nbulk, which originates from\nthe interaction between matter/energy components in the bu lk.\nThe origin of this interaction term will be discussed later o n.\nSuch a universe obeys the holographic principle where Nsur,\nthe number of degrees of freedom on the spherical surface of\nHubble radius H−1, given by [ 46]\nNsur=4π\nL2\nPH2. (3)\nHere Nbulkis the effective number of degrees of freedom\nwhich obeys the equipartition law of energy contained insid e\nthe universe [ 46]\nNbulk=|E|\n(1/2)kBT. (4)\nWe shall assume a multi-fluid matter inside the universe with\nproper Komar energy density\nE=∑\ni(ρi+3pi)V, (5)\nand hence we can write the equipartition law of energy as\nNbulk=−∑\niεi2(ρi+3pi)V\nkBT, (6)along with\ndV\ndt=L2\nP/bracketleftBig\nNsur−/parenleftBig\nεiNbulk+εintNint\nbulk/parenrightBig/bracketrightBig\n, (7)\nwhere, in our definition, εi=−1 stands for baryonic mat-\nter and radiation and εi=1 for dark energy [in order to en-\nsure that Nbulk>0; for matter and radiation ρM\ni+3pM\ni>0\nandρR\ni+3pR\ni>0, and ρDE+3pDE<0 for dark energy, re-\nspectively]. Furthermore, we take εint=−1, meaning that\nNint\nbulk>0, which should behave as baryonic matter for reasons\nthat we will explain below.\nThe volume of the universe and the temperature associated\nwith the horizon are given by\nV=4π\n3H3,T=H\n2π. (8)\nThe novel idea is to consider an interaction term between\nbaryonic matter and dark energy in the bulk. As a result,\nwe propose there is an additional degrees of freedom which\ncomes from the interaction term and assume, for pure dimen-\nsional reasons, it can be written as\nNint\nbulk=−εint/radicalBig\nαNM\nbulkNDE\nbulk, (9)\nwhere αis a coupling constant, NM\nbulkandNDE\nbulkare the degrees\nof freedom of baryonic matter and dark energy in the bulk,\nrespectively. In the next section, we will give further phys ical\narguments about α. Since pM=0 for baryonic matter, and\npDE=−ρDEfor dark energy fluid, then, for the interaction\nterm, we get\nNint\nbulk=/radicalbig\n2αρMρDE2V\nkBT. (10)\nLet us now obtain the standard Friedmann equation. Using\nEqs. ( 7), (3), (6), (8), (10) and pi=ωiρi, we get\ndV\ndt=4πL2\nP/bracketleftBig1\nL2\nPH2+∑\niρi(1+3ωi)4π\n3H4\n+/radicalbig\n2αρMρDE4π\n3H4/bracketrightBig\n, (11)\nwhere we have set kB=1. From the last equation, we obtain\n¨a\na=−4πL2\nP\n3/bracketleftBigg\n∑\niρi(1+3ωi)+/radicalbig\n2αρMρDE/bracketrightBigg\n. (12)\nIn the second term, we have an evolving matter given by\nρM=ρM,0(1+z)3and dark energy ρDE=ρDE,0, where it is\nnatural to identify this term as the dark matter for reasons w e\nshall argue below. Let us define the following equation as an\nevolving dark matter\nρDM\n(1+z)3/2≡/radicalbig\n2αρMρDE, (13)\nwith using ρM=ρM,0(1+z)3andρDE=ρDE,0, leads to\nρDM=ρDM,0a−3, (14)3\nwhere 1+z=a−1andρDM,0≡/radicalbig2αρM,0ρDE,0. Note that\nassuming dark matter is pressureless, then Eq. ( 14) is also\nsolution of the continuity equation ˙ρDM+3HρDM=0, as ex-\npected and justifies ansatz ( 13).\nIf we multiply both sides of Eq. ( 12) by 2 a˙a, and using the\ncontinuity equation\n˙ρi+3H(1+ωi)ρi=0, (15)\nwhere H=˙a/ais the Hubble parameter and ρi=\nρi,0a−3(1+ωi), assuming several matter fluids with a constant\nequation of state parameters ωi. We then arrive at\nd(˙a2)=8πL2\nP\n3d/parenleftBigg\n∑\niρi,0a−1−3ωi+ρDM,0a−1−3ωDM/parenrightBigg\n,(16)\nwhere ΩDM=0. In the l.h.s, we get a constant of integration\nkas follows\n˙a2+k=8πL2\nP\n3/integraldisplay\nd/parenleftBigg\n∑\niρi,0a−1−3ωi+ρDM,0a−1/parenrightBigg\n.(17)\nBy solving the integral in the r.h.s, we get\n/parenleftbigg˙a\na/parenrightbigg2\n=8πL2\nP\n3/parenleftBig\nρR,0a−4+ρM,0a−3+ρDM,0a−3\n+ρcura2+ρDE,0a0/parenrightBig\n, (18)\nwhere we have also included the spatial curvature density\nρcur≡3k\n8πL2\nPa2. (19)\nIf we use the critical density\nρcrit=3H2\n0\n8πL2\nP, (20)\nwe get the equation in terms of the density parameters or in\nterms of redshift zas follows\nE2(z)=ΩR,0(1+z)4+Ωtot\nM,0(1+z)3+Ωcurv(1+z)2+ΩDE,0,\n(21)\nwhere E(z)=H/H0, and\nΩDM,0=/radicalbig\n2αΩM,0ΩDE,0, (22)\nalong with\nΩtot\nM,0=ΩM,0+/radicalbig\n2αΩM,0ΩDE,0=ΩM,0+ΩDM,0.(23)\nThat suggests that dark matter can evolve due to the expansio n\nof the Universe, which is related to the fact that the baryoni c\nmatter evolves, and the coupling parameter is a function of z,\nas we pointed out. As a special case, if we consider a flat uni-\nverse with k=0 for the late-time Universe, which is consis-\ntent with the result obtained in [ 55]. We shall elaborate more\nabout this connection in this work. However, it is importantto see from these equations that dark matter is only an appar-\nent effect due to the interaction between the dark energy and\nmatter. That is the key distinction as compared to the ΛCDM\nmodel, whose respective Hubble parameter as a function of\nthe redshift for a flat space ( Ωcurv=0) is given by (written in\nour notation)\nH(z)=H0/radicalBig\nΩR,0(1+z)4+Ωtot\nM,0(1+z)3+ΩDE,0,(24)\nthe last expression is obtained from Eq. ( 21) and it perfectly\nmatches the ΛCDM model.\nIII. EINSTEIN FIELD EQUATIONS WITH APPARENT\nDARK MATTER\nIn this section, we will closely follow the seminal paper\nby Jacobson [ 1]. Specifically, Jacobson derived the Einstein’s\nfield equations of gravity through the application of the firs t\nlaw of thermodynamics and one can view the Einstein’s field\nequation as an equation of state. In the current study, we\naim to extend this framework, incorporating the influences o f\nbaryonic matter, dark energy, and an additional term stem-\nming from the interaction between baryonic matter and dark\nenergy. First, we need to consider the heat flow across the\nhorizon attributed to the energy carried by the matter field,\ncharacterized by an energy-momentum tensor TM\nabrepresent-\ning baryonic matter\nδQM=−κ/integraldisplay\nHξTM\nabkakbdξdS. (25)\nIn this equation with Swe represent the area of the horizon\nH. The vector kacorresponds to the tangent vector of the\nhorizon generators, further κis known as the surface gravity\nand, finally, ξis an appropriate affine parameter. We also\nintroduce a dark energy term contribution\nδQDE=−κ/integraldisplay\nHξTDE\nabkakbdξdS. (26)\nBuilding upon the concept introduced in the preceding sec-\ntion, which serves as the fundamental idea in this paper, we\nincorporate an interaction term between baryonic matter an d\ndark energy. Consequently, we anticipate the emergence of a\ncontribution term\nδQint.=−κ/integraldisplay\nHξTint\nabkakbdξdS, (27)\nwhere we have introduced an interaction term\nTint\nab=uaub/radicalbig\n2αρM,0ρDE,0a−3, (28)\nwhere αis an interaction parameter between dark energy and\nbaryonic matter, ais the scale factor, here 4-velocity of the\nprefect fluid with gabuaub=−1. For reasons that will be clar-\nified later on we shall identify the interaction term as the da rk\nmatter term, i.e. TDM\nab≡Tint\nab. We get\nTDM\nab=uaub/radicalbig\n2αρM,0ρDE,0a−3≡ρDMuaub, (29)4\nAccording to the Raychaudhuri equation, for the change of\nthe horizon area we have\nδS=/integraldisplay\nHθdξdS=−/integraldisplay\nHξRabkakbdξdS, (30)\nin which θis known as the expansion/contraction scalar. Fi-\nnally, for the first law of thermodynamics we obtain\nδ(QM+QDE+Qint)=TdS, (31)\nwhereSis the entropy and we note that Tis the temperature\nand it reads T=κ/2π. Using the Bekenstein-Hawking rela-\ntion between the entropy and the horizon area S=S/4, and\nwith the help of the above relations we get\n/integraldisplay\nHξ/parenleftbig\nTM\nab+TDE\nab+Tint\nab/parenrightbig\nkakbdξdS=1\n8πG/integraldisplay\nHξRabkakbdξdS.\n(32)\nFrom the last equation it is evident that one has\n1\n8πG/parenleftBig\nRabkakb/parenrightBig\n=/parenleftbig\nTM\nab+TDE\nab+Tint\nab/parenrightbig\nkakb, (33)\nwhich is valid for all null vector ka. The last equation can be\nfurther rewritten as\n8πGTtot\nab=Rab+ζgab, (34)\nwith ζbeing some function. In addition we have defined the\ntotal energy-momentum tensor to be\nTtot\nab=TM\nab+TDE\nab+TDM\nab. (35)\nIf we apply the covariant derivative to and we further utiliz e\nthe contracted Bianchi identity, we obtain:\n∇b(Rab+ζgab)=0⇒− ∇b/parenleftbiggR\n2/parenrightbigg\n=∂bζ. (36)\nWe have also assumed ∇bTtot\nab=0. From the last equation it\nis easy to see that we get for the function ξ\nξ=−/parenleftbiggR\n2/parenrightbigg\n+C. (37)\nIn our case we can fix the constant Cto zero. But we further\ndefine\nTDE\nab=−Λ\n8πGgab. (38)\nIn this way we obtain the Einstein field equations as a thermo-\ndynamics equation of state with dark matter\nRab−1\n2Rgab+Λgab=8πG/parenleftbig\nTM\nab+TDM\nab/parenrightbig\n. (39)\nOne should keep in mind that the dark matter is only an appar-\nent effect and obtained via the interaction of dark energy an d\nmatter. With this equation in hand, we can apply to find the\ncosmological model of the universe. Let us now use the Ein-\nstein equation with dark matter and extend our discussion ofthe cosmological setup. Assuming the background spacetime\nto be spatially homogeneous and isotropic, which is given by\nthe Friedmann-Robertson-Walker (FRW) metric\nds2=−dt2+a2/bracketleftbiggdr2\n1−kr2+r2(dθ2+sin2θdφ2)/bracketrightbigg\n,(40)\nwhere we can further use R=a(t)r,x0=t,x1=r, the two\ndimensional metric hµν. Here kdenotes the curvature of\nspace with k=0,1,−1 corresponding to flat, closed, and open\nuniverses, respectively. The dynamical apparent horizon, a\nmarginally trapped surface with vanishing expansion, is de ter-\nmined by the relation hµν(∂µR)(∂νR)=0. A simple calcula-\ntion gives the apparent horizon radius for the FRW universe\nR=ar=1//radicalBig\nH2+k/a2. (41)\nFor the matter source in the FRW universe, we shall assume a\nperfect fluid described by the stress-energy tensor\nTµν=(ρi+pi)uµuν+pigµν. (42)\nthis leads to the continuity equation have assumed several\nmatter fluids with a constant equation of state parameters ωi\nand continuity equations we have the expression for densiti es\nρi=ρi0a−3(1+ωi)(this includes baryonic matter ωi=0, radi-\nation ωi=1/3, dark matter ωi=0 and dark energy ωi=−1).\nwith H=˙a/abeing the Hubble parameter. If we rewrite the\nEinstein equations as\nRab=8πG/parenleftbigg\nTtot\nab−1\n2gabTtot/parenrightbigg\n. (43)\nFrom the Einstein field equations, we can obtain the Fried-\nmann’s equation, namely\n¨a\na=−/parenleftbigg4πG\n3/parenrightbigg\n∑\ni(ρi+3pi). (44)\nwhere we have the expression for densities ρi=ρi0a−3(1+ωi).\nEq. ( 44) becomes\n¨a\na=−/parenleftbigg4πG\n3/parenrightbigg\n∑\ni(1+3ωi)ρi0a−3(1+ωi). (45)\nNext, by multiplying 2 ˙ aaon both sides of Eq. ( 45), and by\nintegrating we obtain\n/parenleftbigg˙a\na/parenrightbigg2\n+k\na2=8πG\n3∑\niρi0a−3(1+ωi)(46)\nor\nH2+k\na2=8πG\n3∑\niρi0a−3(1+ωi)(47)\nWe further assume, the energy-momentum tensor of each\ncomponent is in the form of the perfect fluid,\nTi\nab=(ρi+pi)uaub+pigab (48)5\nwhere i=DM,DE,Manduais the 4-velocity of the prefect\nfluid with gabuaub=−1. For the pressureless dark matter with\npDM=0, we have\nTabDM=ρDMuaub. (49)\nComparing with the equation\nTabDM=uaub/radicalbig\n2αρM,0ρDE,0a−3. (50)\nEquating Eqs. ( 49) and ( 50) we finally get\nρDM=/radicalbig\n2αρM,0ρDE,0(1+z)3. (51)\nUsing the continuity equation for the matter, we have ρM=\nρM,0(1+z)3, while for the cosmological constant (DE) the\nenergy density is constant, namely ρDE=ρ0\nDE, in the general\ncase we define\nρDM=/radicalBig\n2αρ0\nMρ0\nDE(1+z)3, (52)\nor\nρDM=ρ0\nDM(1+z)3, (53)\nthat it we obtain the standard Friedmann equation\n/parenleftbigg˙a\na/parenrightbigg2\n=8πG\n3/parenleftBig\nρR,0a−4+ρM,0a−3+ρDM,0a−3\n+ρcurva2+ρDE,0a0/parenrightBig\n, (54)\nIn addition, we have the spatial curvature density\nρcur=3k\n8πGa2. (55)\nIf we use the critical density\nρcrit=3H2\n0\n8πG, (56)\nwe get the equation in terms of the density parameters or in\nterms of redshift zas follows\nE2(z)=ΩR,0(1+z)4+Ωtot\nM,0(1+z)3+Ωcurv(1+z)2+ΩDE,0.\n(57)\nwhere Ωtot\nM,0=ΩM,0+/radicalbig\n2αΩM,0ΩDE,0. This result coincides\nwith Eq. (21) in the last section.\nIV . CORRESPONDENCE WITH YUKA WA COSMOLOGY\nAND RECOVERING MOND\nA. Correspondence with Yukawa cosmology\nIn this section, we would like to elaborate more on the na-\nture of dark energy. Einstein famously introduced the cos-\nmological constant into the field equation Gµν+Λgµν=\n8πGTµν/c4to explain the expansion of the Universe. Dark\nenergy is commonly linked to represent the energy densityof the vacuum [ 56]; hence, it is just a constant. Specifically,\nthe energy density can be calculated by integrating the vac-\nuum fluctuation energies using ρvac\nDE≃Λ4\ncutwhere we inte-\ngrate up to a certain ultraviolet momentum cutoff kmax, we\ngetρvac\nDE∼¯hk4\nmax[56]. That is the origin of the famous dis-\ncrepancy, namely, when one compares the theoretical value\nwith the observed value, that gives a huge discrepancy of the\norder∼10120. Speculation regarding a potential symmetry\nleads to a vanishing vacuum energy. The problem surround-\ning the cosmological constant remains unsolved and continu es\nto challenge researchers. Certainly, alternative concept s have\nbeen proposed to elucidate the cosmological constant. In th is\ncontext, we will present an argument for the involvement of\nmassive gravitons in clarifying the nature of the cosmologi cal\nconstant. Moreover, we will draw attention to a correlation\nwith Yukawa cosmology, the subject of recent investigation\nin [55,57]. At large distances, one can obtain a Yukawa-like\npotential\nΦ=−GMm\nr(1+αexp(−r/λ)), (58)\nwith the wavelength of massive graviton reads [ 58]\nλ=¯h\nmgc. (59)\nIt is interesting to note that the Yukawa potential has been o b-\ntained in modified theories of gravity such as the f(R)gravity\n[59,60]). It was shown that the dark matter can be viewed\nas an apparent effect, namely, dark matter is obtained via th e\nlong-range force modification in terms of the following equa -\ntion [ 55,57]. In particular, one can obtain effectively the\nΛCDM using [ 55,57]\nΩΛCDM\nDM,0=/radicalBig\n2ΩΛCDM\nM,0ΩΛCDM\nDE,0, (60)\nwhere\nΩΛCDM\nDE,0∼c2α\nλ2H2\n0. (61)\nWe can therefore make the correspondence if we define the\ndark energy density in the Padmanbhan universe as\nΩPadmanbhan\nDE,0∼c2\nλ2H2\n0. (62)\nIn other words, there is an equivalence with the Yukawa cos-\nmology [ 55] with two parameters: λthat specifies the dark\nenergy density and it is related to the gravitons mass and α\nthat specifies the interaction between matter and dark energ y.\nWe can further write the energy density of the cosmological\nconstant contribution in terms of the graviton mass. Since w e\nfound a correspondence with Yukawa cosmology, one can see\nthat the parameter αappears in the modified Newton’s poten-\ntial and, therefore, Newton’s law in large distances. The qu est\nfor a viable theory grounded in a healthy and well-defined ac-\ntion that can integrate the concept of massive gravitons re-\nmains an ongoing endeavour that has yet to reach a conclu-\nsive resolution. the parameter αplays a crucial role in al-\ntering Newton’s law. Yet, an intriguing question emerges re -\ngarding the fundamental origin of this parameter. We shall6\npresent compelling arguments linking the modified entropy t o\nadjusted gravity, with αemerging as a consequence of en-\ntropy modification. In some deep sense, αcould potentially\nbe influenced by the entanglement entropy arising from the\nintricate interplay between baryonic matter and the fluctua -\ntions in the gravitational field (gravitons). One can alread y\nsee a hint for this if we use ΩΛCDM\nΛ,0=ρDE,0/ρcrit, where\nρcrit=3H2\n0/(8πG)andρDE,0=Λc2/(8πG)one can obtain\nthe cosmological constant in terms of α, as follows\nΛ∼3α\nλ2. (63)\nIn this way, we get for the energy density of dark energy\nρDE∼3αc2\n8πGλ2, (64)\nwhich has the form of a holographic dark energy expression\nfor the cosmological constant. This signals that αfundamen-\ntally measures the entanglement between the gravitons and\nmatter fields.\nB. Recovering MOND\nIn this section, we would like to point out that the parameter\nαthat modifies the Newton’s law in the large distances, can\nexplain theoretical origin of the MOND theory. To see this,\nlet us rewrite Eq. ( 12) as follows\n¨R\nR=−4πG\n3/bracketleftBigg\n∑\niρi(1+3ωi)+/radicalbig\n2αρMρDE/bracketrightBigg\n, (65)\nwhere we have restored the Newton’s constant G. This equa-\ntion can also be written as\nm¨R=−Gm\nR2/bracketleftBigg\n4πR3\n3∑\niρi(1+3ωi)+4πR3\n3/radicalbig\n2αρMρDE/bracketrightBigg\n.\n(66)\nUsing the fact that the Komar mass [similar to Komar energy\nwith pi=ωiρi] reads\nM=4πR3\n3∑\niρi(1+3ωi), (67)\nand if we consider only matter [ωi=0] and dark energy ωi=\n−1] (we shall neglect radiation contribution), we get\nm¨R=−Gm\nR2/bracketleftbigg4πR3\n3(ρM−2ρDE)+4πR3\n3/radicalbig\n2αρMρDE/bracketrightbigg\n.\n(68)\nFor Newton’s law in cosmological scales, we therefore get\n/vectorF=m/vectora=−/bracketleftbiggG m(M−MDE)\nR2+G m\nR2V/radicalbig\n2αρMρDE/bracketrightbigg\n/vectorr0,\n(69)\nwhere/vectorr0is some unit vector, and M=4πR3ρM/3 (although in\ngeneral Mdepends on R) along with MDE=4πR3(2ρDE)/3.On cosmological scales, the dark energy mass MDEdomi-\nnates over baryonic mass M, implying a negative sign which\nmeans an expanding universe. Conversely, on galactic scale s,\nthe baryonic mass dominates and the total force is toward the\ngalactic center. Nevertheless, the presence of the second t erm\nwhich depends on α, yields an additional contribution from\nthe long-range gravity modification; such a term can be at-\ntributed to the dark matter having a mass\nMDM=4πR3\n3/radicalbig\n2αρMρDE, (70)\nwe can restore Newton’s law of gravity\n/vectorF=−Gm(M−MDE+MDM)\nR2/vectorr0. (71)\nprovided there is an extra mass term due to the dark matter.\nHowever, there is another way of rewriting Eq. ( 66) as follows\na=G(M−MDE)\nR2+/radicalBigg\nα/parenleftbiggG M\nR2/parenrightbigg/parenleftbiggG M DE\nR2/parenrightbigg\n(72)\nLet us define the acceleration due to the matter and the dark\nenergy enclosed in that region of volume Vas\naM=G M\nR2and aDE=G M DE\nR2, (73)\nwe obtain the total acceleration\na=aM−aDE+√αaMaDE. (74)\nThe last equation is the acceleration in MOND theory.\n•Cosmological scales\nIn cosmological scales, as we pointed out the dark energy\ndominates over baryonic matter [in general it also should de -\npend in the scale distance M=M(R)], and for cosmological\nscales R∼1026m,aM→0, or aM<<aDEand, from the lat\nequation we get a=−aDE. It is clear that the negative sign\nexplains the expanding universe due to dark energy. One can\nfurther check that\naDE=G M DE\nR2∼4πGρDER\n3, (75)\nwhere for the dark energy density we have\nρcosmology\nDE∼3αc2\n8πGλ2\ncosmology, (76)\nhence we get\naDE∼αc2R\n2λ2\ncosmology∼10−10m/s2. (77)\nThe last result gives the acceleration of the universe. Here\nwe note that the value of λcosmology is of the order of the ob-\nservable radius of the universe, i.e., λcosmology∼1026m (see\nfrom observation constrains reported in [ 55,57]). In addi-\ntion, we have used α∼0.41 along with R∼1026m, and con-\nsequently the energy density of dark energy can be approxi-\nmated to ρDE∼7×10−27kg/m3.7\n•Galactic scales\nLet us now elaborate the case of galaxies where Ris of pc\nor kpc order. However, near the galactic center in general\nthe mass term is a function of distance, i.e., M(R). In other\nwords, in the galactic center, we know that baryoinic matter\ndominates compared to the dark energy ( aM>>aDE), hence\nwe get\na=aM+√αaMaDE. (78)\nFurther let us define a0, by writing\na0=αaDE, (79)\nto get a MOND-like form for the acceleration\na=aM+√aMa0. (80)\nThe interesting part here is when one studies the large dis-\ntances, say when Ris of pc or kpc order. Hence let us take for\nthe outer part of the galaxy R∼1017m, we can try to compute\na0which gives\na0=αaDE∼4πα2GρDER\n3∼10−20m/s2! (81)\nWe have expected for a0, as MOND predicts, a0∼\n10−10m/s2. To address this inconsistency, let’s initially high-\nlight that the energy density of dark energy on galactic scal es\nis given by\nρgalaxy\nDE∼3αc2\n8πGλ2\ngalaxy, (82)\nand rewrite a0in terms of the energy density as follows\na0∼α2c2\n2λ2\ngalaxy. (83)\nAs we shall elaborate the energy density of dark energy de-\npends on the scale of measurements (observations) due its\ndependence on λ. In some sense there exists a fundamental\nlimitations on measuring the dark energy density due to the\nuncertainty principle when we measure λ. To see this more\nclearly, let us go back in Eq. (59) and after we set λ=∆x,\nand∆p=mgc, for graviton we get the momentum-position\nuncertainty relation\n∆x∆p∼¯h. (84)\nNow lets see the implications of this relation. When\nλcosmology∼1026m., i.e. in cosmological scales, we have\nmore uncertainty in position λcosmology=∆xcosmology∼1026\nm, but more precision on the momentum ∆pcosmology, namely\n∆pcosmology∼¯h\n∆xcosmology∼10−60N·s. (85)\nThis simply means that when we apply the uncertainty rela-\ntion for the graviton entirety of the universe, our knowledg eof position becomes more uncertain, but we gain greater prec i-\nsion in measuring momentum. Conversely, when focusing on\ngalactic scales, the uncertainty in position decreases, ap prox-\nimately λgalaxy=∆xgalaxy∼1019m. However, this reduction\nin position uncertainty is accompanied by an increase in mo-\nmentum uncertainty, denoted as ∆pgalaxy, specifically we can\nsee this from\n∆pgalaxy∼¯h\n∆xgalaxy∼10−53N·s. (86)\nIn order to obtain a consistent result, we need R/greaterorsimilar1015m,\nnamely\na0∼α2c2R\n2λ2\ngalaxy∼10−10m/s2. (87)\nAs we increase the scale of observations R,λwill increase\nand there is no sense to speak about R>λ. Our findings align\nwith recent observations related to the applicability of Ne w-\nton’s law over substantial distances, specifically employi ng\nwide binary systems as exemplified by Hernandez and Chae\n[61,62]. Consequently, it is evident that there are inherent\nconstraints in accurately measuring the graviton waveleng th\nwithin the realm of cosmology. As we elaborated, the incon-\nsistency that emerges when comparing analyses conducted on\ngalactic and cosmological scales, can be explained through\nthe prism of uncertainty relations.\nV . FINAL REMARKS AND CONCLUSIONS\nThe central and novel result we found in this paper is that\nby adopting the emergence perspective of gravity, one can in -\nterpret the dark matter as an effective/apparent phenomeno n.\nIn this viewpoint the dark matter appears as a result of inter ac-\ntion between dark energy and baryonic matter which modifies\nthe numbers of degrees of freedom in bulk. This approach\nleads naturally to derive the ΛCDM model of cosmology. The\nsecond important result is that, we applied the Jacobson’s a p-\nproach by taking into account an interaction term between\nbaryonic matter and dark energy which plays the role of dark\nmatter. We thus constructed the modified Einstein field equa-\ntions as an equation of state. The Einstein field equations\nobtained in this method, contains energy momentum tensor\nof apparent dark matter as well as dark energy. Both meth-\nods are consistent and we confirm how dark matter naturally\nemerges by including an interaction term between dark energ y\nand baryonic matter. Finally, we obtained the ΛCDM model\nin cosmology and established a correspondence with Yukawa\ncosmology. We then explored the relation and the role of mas-\nsive graviton in the dark sector. For the energy density of\ndark energy we obtained a holographic dark energy expres-\nsion which implies that the interaction term αfundamentally\nmeasures the entanglement between the gravitons and matter\nfields. 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Sci., Wesleyan University, Middletown, USA\n2Department of CSSE, Concordia University, Montreal, Canada\nFebruary 2, 2024\nAbstract\nWe study the truck-drone cooperative delivery problem in a setting where a single truck\ncarrying a drone travels at constant speed on a straight-line trajectory/street. Delivery to\nclients located in the plane and not on the truck’s trajectory is performed by the drone, which\nhas limited carrying capacity and flying range, and whose battery can be recharged when on the\ntruck. We show that the problem of maximizing the number of deliveries is strongly NP-hard\neven in this simple setting. We present a 2-approximation algorithm for the problem, and an\noptimal algorithm for a non-trivial family of instances.\n1 Introduction\nThe use of unmanned aerial vehicles or drones for last-mile delivery in the logistics industry has\nreceived considerable attention in business and academic communities, see for example [1, 3, 15, 9].\nDrones have been shown in a recent analysis [13] to have significantly less lifecycle costs, and faster\ndelivery time compared to diesel or electric trucks in urban, suburban, and rural settings, and\nhave less harmful emissions compared to diesel trucks. The potential applications where drone\ndelivery could make a big impact include contactless delivery, return of unsatisfactory goods, rural\nor hard-to-access delivery and delivery in disaster relief scenarios.\nIn this paper we consider a system in which the delivery of physical items to clients located in\nthe plane is done by two cooperating mobile agents having different but complementary properties.\nThe first mobile agent, called the drone can move in any direction but it can travel only a limited\ndistance, called its flying range , before it needs to recharge its battery. Furthermore, it has limited\ncarrying capacity. The second mobile agent, called the truck can travel only along a fixed trajectory,\ncalled a street but its battery/fuel is not only sufficient to follow the street as long as necessary,\nbut it is also equipped with a charging facility where the drone can recharge whenever it reaches\nthe truck. Furthermore, it can carry all items that are to be delivered to the clients.\nThe delivery of items to clients is done as follows. All items to be delivered are preloaded on\nthe truck at the warehouse. The truck then moves along the street at a fixed speed and it delivers\nitems to any client who is located on its trajectory. The delivery of an item to a client who is not\nlocated on the trajectory of the truck must be carried out by the drone. At an appropriate time,\nthe drone flies from the truck with the item to be delivered to the given client, drops the item\nthere, and then flies back to the still-moving truck. There it can recharge, pick up another item,\nand make the next delivery, and so on. Clearly the same set-up can also be used to pick up items\nrather than deliver them. For ease of exposition, we always talk about item delivery in this paper.\nGiven a set of delivery locations and the parameters of the agents, i.e., the trajectory and the\nspeed of the truck, the flying range of the drone and its speed, we want to compute a feasible\n1arXiv:2402.00829v1  [cs.DS]  1 Feb 2024schedule of deliveries that maximizes the number of deliveries made. Such a schedule specifies the\norder in which the deliveries to clients are done by the drone, and for each delivery it gives the\ntime the drone leaves the truck. Clearly, to be feasible, the schedule should ensure that for each\ndelivery, the drone can fly to the delivery location and back to the still-moving truck while having\ntravelled distance at most its flying range, and arrive at the truck in time to start its next delivery.\n1.1 Related work\nThe algorithmic study of truck-drone cooperative delivery problems was initiated by Murray and\nChu [12] and Mathew et al. [11] where the problem of a single truck being helped by a single\ndrone to deliver packages to customers is studied. Since then there has been a great deal of\nwork (Murray and Chu’s paper has received more than 1000 citations) on different versions of\nwhat is variously referred to as Truck-Drone Cooperative Delivery, Drone-Aided Delivery or Last-\nMile Delivery problems. Variations considered include multiple trucks, multiple drones, drone-only\ndelivery, mixed truck-drone delivery, etc. We refer the reader to recent surveys for more details\n[4, 3, 9, 15, 16].\nIn the above work, the problem is most often modelled using a weighted directed graph with\ncustomers as nodes, streets and drone flight paths as edges, etc. Under these circumstances the\nproblems become versions of the Travelling Salesperson Problem or the Vehicle Routing Problem.\nAs such they are all easily seen to be NP-hard in general and are solved by adapting known exact\n(e.g., Mixed Integer Linear Programming) or heuristic (e.g., greedy) techniques. For specialized\ndomains some variants can be shown to be polynomial time, e.g. on trees [2].\nIn most of the previous research it is assumed that the points at which a truck and drone can\nrendezvous are part of the input (e.g., customer locations, depots) and that the truck or drone stops\nat the rendezvous point to wait for the other to arrive. More recent work [7, 8, 10, 14] has focused\non the case where the rendezvous can occur “en route” as the truck is moving and the rendezvous\npoints are to be determined by the algorithm, as is the case with our study. In these papers, the\nproblems studied are again generalized versions of TSP or VRP and are attacked via adaptations\nof known exact or heuristic techniques. Here we restrict ourselves to the simplest version of the\nproblem with one truck and one drone, where the truck travels at a constant speed along a single\nstreet. Surprisingly, even in this case, as shown in Section 3, the problem is strongly NP-hard.\nAll of the above work is concentrated on minimizing either the total delivery time or total energy\nrequirements (or some combination of both) to deliver all of the packages to all of the customers.\nTo the best of our knowledge we are the first to consider the problem of maximizing the number\nof clients that are satisfied in the en route model.\n1.2 Our Truck-Drone Model\nWe define the truck-drone delivery problem more formally as follows. We assume that the delivery\npoints as well as the trajectories of the truck and the drone, are set in the 2-dimensional Cartesian\nplane. Without loss of generality, we assume the warehouse is located at [0 ,0], and the truck starts\nfully loaded with all items to be delivered at the warehouse at time 0, and subsequently moves\nright on the x-axis with constant speed 1. Note that this allows us to measure the elapsed time by\nthe distance of the truck from the origin.\nThe speed of the drone is denoted by vand it is assumed that vis a constant that is greater\nthan 1. The flying range of the drone is given by the value R, and is defined as the maximum\ndistance that the drone can fly on a full battery without needing to be recharged. We assume that\nthe time to recharge the drone’s battery, and to pick up an item from the truck, or to drop off an\nitem at its delivery location are negligible compared to the delivery times, and thus are equal to\n20. Therefore, any time the drone leaves the truck it can fly its full range Rbefore returning to the\ntruck.\nWe are given a multi-set D={d1, d2, . . . , d n}of delivery points in the plane where the deliveries\nare to be made. The truck delivers any item whose delivery point is located is on its trajectory, we\nassume that this can be done with negligible delay. Thus we assume below that none of the points\ninDis located on the trajectory of the truck, i.e., on the positive x-axis.\nWe now define a feasible delivery schedule for the truck-drone delivery problem.\nDefinition 1. Given an instance I= (v, R, D )of the truck-drone problem, where D={d1, d2, . . . , d n},\nwe define a schedule SIto be an ordered list of delivery points to which deliveries are made, and\nthe start time of each delivery, i.e.,\nSI= ((di1, s1),(di2, s2), . . . , (dim, sm)), m≤n\nwhere mis called the length of the schedule and for 1≤j≤mthe drone makes a delivery to dijby\nleaving the truck at point [sj,0]. The schedule is feasible , ifs1≥0, and for each j,1≤j≤m−1,\nthe drone can reach dijwhen leaving the truck at position [sj,0]and return to the truck at or before\n[sj+1,0].\nSchedule SIis called optimal if there is no schedule that is longer than SI, that is, makes more\ndeliveries than SI.\nGiven an instance I= (v, R, D ) of the truck-drone problem, where vandRare the speed and\nthe range of the drone respectively, and Dis the set of delivery points, the goal of the truck-drone\ndelivery problem is to find an optimal delivery schedule.\n4 3\n224 5\n0 1 4.5 5 7 11.5d\nd1\n2\nsd3d4\nFigure 1: Instance I= (2,10,{d1, d2, d3, d4}), and its schedule SI= ((1 , d1),(5, d2),(7, d3)). The\ntrajectory of the drone is in blue, that of the truck in red. The blue numbers give the distances,\nthe black numbers show the time sequence\n.\nFigure 1 shows an example of a truck-drone problem and of a feasible schedule.\n1.3 Our results\nIn Section 3, we show that even for the ostensibly simple case of a single truck travelling on a straight\nline, and a single drone, the truck-drone delivery problem is strongly NP-hard. In particular, we\nshow that given an instance Iof the truck-drone problem and an integer k, it is strongly NP-hard\n[5] to decide whether there is a schedule SIof length k.\n3In Section 4, we describe a greedy algorithm Agand show that it computes a 2-approximation of\nan optimal schedule in O(n2) time. The factor of 2 is shown to be tight for this algorithm. Finally,\nin Section 5, we define a proper family of instances. Roughly speaking, in such instances, the\ndelivery points do not have the same or “nearly” the same x-coordinates, where “nearly” depends\non the difference in their y-coordinates. In particular, the greater the difference in the y-coordinates\nof the points, the greater is the difference in their x-coordinates in proper instances. We then give\nanO(n3) algorithm that calculates an optimal schedule for any proper instance.\n2 Preliminary Observations\nWe say that a point d= [x, y] isreachable by the drone from position [ s,0] if the drone can leave\nthe truck at [ s,0], fly to point dand fly back to the truck with the total distance travelled at most\nits flying range R. First we examine some geometric properties of points in the plane that are\nreachable from [ s,0] by the drone flying with speed vand having flying range R.\nSuppose the drone leaves the truck at position [ s,0], makes a delivery at d= [x, y] and returns\nto the truck using its full range R. To fly range Rthe drone needs time t=R/vand at that time\nthe truck is at position [ s+R/v, 0]. Therefore, the drone can make a delivery at point d= [x, y] if\n[s;0]\nh\ns+R\nv;0i\nd= [x; y]\nE\nm\nM\nFigure 2: In Ellipse Eshown above, the speed of the drone is not much higher than that of the\ntruck. When the speed of the drone increases, the distance between the foci decreases, and the\nellipse becomes closer to a circle.\nthe total distance it flew satisfies the equation\n|[s,0],[x, y]|+|[x, y],[s+R/v, 0]|=R\nClearly all such points dreachable by the drone from [ s,0] using its full flying range lie on ellipse E\n(see Figure 2) with left focus [ s,0] and right focus [ s+R/v, 0]. Furthermore, the major radius , i.e.\nthe length of the semi-major axis of the ellipse is M=R\n2, and minor radius , i.e. the length of its\nsemi-minor axis ism=R\n2v√\nv2−1. Next, considering also the delivery points that can be reached\nby the drone by flying distance < R, we conclude that all points reachable from [ s,0] by the drone\nwithin its flying range are located on or inside the ellipse E.\nAssuming that the ellipse Eis centered at [0 ,0], its left focus [ s,0] = [ −R\n2v,0], and its right\nfocus is [R\n2v,0], and M,mare the major, minor radii as specified above. The equation of the ellipse\nis:\nx2\nM2+y2\nm2= 1 (1)\nClearly, delivery to point d= [x, y] isfeasible only if −m≤y≤m, i.e., all delivery points should\nbe located in a band of width 2 mcentered along the x-axis.\nAssume a delivery point dis on the right half of ellipse E, and the drone makes a delivery to\ndstarting from the truck at point [ s′,0] between the foci of the ellipse E. Since the distance from\n4[s′,0] to dis shorter than the distance from the left focus [ s,0] of Etod, the drone can reach the\ndelivery point d, flying for distance < R. However, the drone when leaving the truck at point [ s,0]\narrives at dearlier than when staying on the truck and leaving for donly later at point [ s′,0], and\ntherefore it also returns to the truck earlier. Thus when using flying distance less than Rthe drone\nreturns to the truck later as shown in Figure 3.\n[s,0] [s’,0]d E\n[s+R/v,0]\nFigure 3: The red lines show the delivery with the full range R, the green lines show the delivery\nwith range less than R.\nThis leads to the next observation:\nObservation 1. Consider a delivery point din the right half of the ellipse E. To make a delivery\ntodflying less that the full range R, the drone must start the delivery at a point to the right of\nthe left focus of Eand the drone returns to the truck to the right of the right focus of E. Starting\npoints for the drone to the left of the left focus are not feasible.\nA symmetric observation holds about delivery points on the left half of E.\nWe now determine for each delivery point an interval on the trajectory of the truck describing\nfeasible departure points for the drone to make a delivery to point d. Given a delivery point d,\nletE1andE2be the ellipses with major radius Mand minor radius m, such that their foci are\nlocated on the x-axis, with E1containing don its right half, while E2contains don its left half.\nLetfi1, fi2be the foci of Eifori∈ {1,2}(see Figure 4). The following observation now follows\nfrom Observation 1 above.\nObservation 2. Focus f11is the point of the earliest start for a delivery to d, and focus f12is the\npoint of the earliest return to the truck from a delivery to d. Focus f21is the point of the latest start\nfor a delivery to dthat can meet the truck, and Focus f22is the latest return to the truck from any\ndelivery to d. Feasible start points for delivery to dlie between f11andf21, with the corresponding\nreturn to the truck occurring between f12andf22.\nIn the rest of this paper, given a delivery point dwe denote its earliest start time as es(d) and\nthe corresponding earliest return as er(d), the latest start time of dasls(d), and the corresponding\nlatest return back to the truck as lr(d),\nNotice that for any delivery point dwe have\ner(d)−es(d) =lr(d)−ls(d) =R/v\nthe distance between the foci of E.\nGiven a point d= [x, y], we can calculate the values es(d),ls(d) as follows. Imagine a horizontal\nline passing through d. It intersects the ellipse Ecentered at 0 at two points [ −x′, y] and [ x′, y].\nAccording to Equation (1), we have ( x′)2/M2+y2/m= 1. Therefore, x′=Mq\n(1−y2\nm2). Now,\n5E1\nE2\nf11\nf21\nf12\nf22\ndFigure 4: The red lines show the earliest delivery to d, the blue lines show the latest delivery to\nd. A delivery to dcould be scheduled to start at a point between f11andf21.\nimagine sliding the ellipse Ealong the x-axis. When Etouches dfor the first time, we obtain E1\nhaving travelled distance x−x′. Similarly, when Etouches dfor the last time, we obtain E2having\ntravelled distance x+x′. Thus, we have:\nObservation 3. Ford= [x, y]\nes(d) =x−R\n2v−x′, and er(d) =es(d) +R/v,\nls(d) =x−R\n2v+x′, and lr(d) =ls(d) +R/v.\nThe next lemma gives the return point of the drone to the truck after a delivery to a delivery\npoint d= [x, y], starting from the truck at a position [ s,0].\nLemma 1. Suppose we wish to make a delivery to a delivery point d= [x, y]using the drone,\nstarting from the truck at position [s,0], and returning to the truck at position [ret,0].\n1. If es(d)≤s≤ls(d).\nret = ret( s, d, v ) :=s+s+av−x+p\nb2−s(v2−1)(b+s+av−x)\nv2−1(2)\nwhere a=p\ny2+ (s−x)2,b=sv2+av−x.\n2. If s < es (d), then ret(s, d, v ) =er(d).\n3. If s > ls (d), then delivery is impossible, thus we set ret(s, d, v ) =∞.\nProof. To see (1), observe that the total distance travelled by the drone is d1=|[s,0],[x, y]|+\n|[x, y],[ret,0]|=a+p\n(ret−x)2+y2, which the drone travels in time d1/v. At the same time the\ntruck travels the distance d2= ret−s. Thus we have the equation\na+p\n(x−ret)2+y2=v(ret−s)\n(x−ret)2+y2= (v(ret−s)−a)2\nx2−2xret + ret2+y2=v2(ret2−2sret +s2)−2av(ret−s) +y2+ (s2−2sx+x2)\nret2−2xret = v2ret2−2v2sret−2avret +v2s2+ 2avs+s2−2sx\n0 = ( v2−1) ret2−2(sv2+av−x) ret + s(v2s+s+ 2av−2x)\n0 = ( v2−1) ret2−2bret +s(b+s+av−x),\n6and by solving the quadratic equation for ret > swe have\nret =b+p\nb2−s(v2−1)(b+s+av−x)\nv2−1\n=s+s+av−x+p\nb2−s(v2−1)(b+s+av−x)\nv2−1,\nas needed.\nFor (2), note that if s < es (d) then the drone remains on the truck until position [ es(d),0] is\nreached and then it starts a delivery from position [ es(d),0], since by Observation 1, this gives the\nearliest time the drone can start from the truck for a delivery to point d. Thus for any such sthe\ndrone returns to the truck at position [ er(d),0].\nFinally, (3) follows from Observation 2.\nForswhere es(d)≤s≤ls(d) and a delivery point d= [x, y], we call ret( s, d, v )−stheround-\ntrip flight time todfrom [ s,0]. It can be seen from Formula 2 that the round-trip flight time is not a\nlinear function in s, which makes a calculation of a schedule for a given instance of the truck-drone\nproblem more complicated.\nObservation 4. For a delivery point d= [x, y]and a point [s,0]between es(d)andls(d), the\nround-trip flight time ret(s, d, v )−sreaches the maximal value R/v ats=es(d), it decreases until\ns=x(1−1/√\nv2−1)and then increases until s=ls(d)where it again reaches the maximal value\nR/v.\nLemma 2. Letd= [x, y]andd′= [x′, y′]be two delivery points, and suppose there are valid\ndrone trajectories from [s,0]todreturning at [r,0]and from [s′,0]tod′returning at [r′,0]. If\ns′< s < r ≤r′, then there is also a valid drone trajectory from [s′,0]todreturning at a point\nbefore [r,0].\nProof. LetR1be the length of the drone trajectory from [ s′,0] tod′and then to [ r′,0], and similarly,\nletR2be the length of the drone trajectory from [ s,0] to dand then to [ r,0]. Then R1/vandR2/v\nrespectively are the distances from [ s′,0] to [ r′,0] and from [ s,0] to [ r,0]. Since s′< s < r ≤r′, it\nfollows that R2< R 1. Now consider the ellipse E1with parameters ( R1, v) with [ s′,0] as its left\nfocus. Then d′is on the right half of E1, and [ r′,0] must be its right focus. Similarly, let E2be the\nellipse with parameters ( R2, v) with [ s,0] and [ r,0] as its left and right foci respectively, and with d\non the right half of the ellipse. Since s′< s < r ≤r′, the ellipse E2is completely contained in E1,\nand the point dis in the interior of the ellipse E1. It follows that there is a valid drone trajectory\ntodstarting at [ s′,0]. Furthermore, since the drone reaches dearlier if it starts at [ s′,0] than if\nit stayed on the truck until [ s,0] and then flew to d, it must also return to the truck earlier than\n[r,0].\nIn the truck-drone instance that we use in the proof of strong NP-hardness in Section 3, many\nof the delivery points are located on the yaxis. For these points we can simplify the expression\nused to define function ret( s, d, v )−s, and this simplified expression is used to obtain upper and\nlower bounds on ret( s, d, v )−s.\nLemma 3. Fors≥0and a delivery point d= [0, y]withv/4≤y≤v/2we have\n2y\nv<ret(s, d, v )−s <2y\nv+1 + 4 s2+s\nv2−1.\n7Proof. Let ∆ s:= ret( s, d, v )−s. Then the distance travelled by the drone isp\ns2+y2+p\n(s+ ∆s)2+y2.\nSince the drone travels at speed v, the time taken by the drone is then\np\ns2+ 2y2+p\n(s+ ∆s)2\nv.\nDuring the delivery, the truck travels distance ∆ sat speed 1 taking the time ∆ s. Equating the\ntwo times we get:p\ns2+y2+p\n(s+ ∆s)2+y2\nv= ∆s.\nSolving for ∆ s, we obtain:\n∆s=2(vp\ns2+y2+s)\nv2−1.\nFrom this expression we immediately obtain the lower bound on ∆ susing s≥0:\n∆s≥2vy\nv2−1≥2y\nv.\nNext observe thatp\ns2+y2≤y+s2\n2y.Plugging this inequality into the expression for ∆ swe obtain:\n∆s≤2(v(y+s2/2y) +s)\nv2−1=2vy+v\nys2+ 2s\nv2−1=2\u0000\n1−1\nv2\u0001\nvy+ 21\nv2vy+v\nys2+ 2s\nv2−1\n=2y\nv+2y\nv+v\nys2+s\nv2−1≤2y\nv+1 + 4 s2+s\nv2−1,\nwhere in the last inequality we used the fact that v/4≤y≤v/2.\n3 Strong NP-hardness\nIn this section we prove that the following decision problem is strongly NP-hard:\nProblem 1 (Schedule Length problem) .Given an instance Iof the truck-drone problem, and an\ninteger p, is there a schedule SIof length p(that is, SImakes pdeliveries)?\nWe show below that there is a polynomial reduction from the well known 3 -Partition problem [5]\nto the Schedule Length problem. Recall that in the 3-Partition problem we are given a multi-set\nof integers Y={y1≤y2≤ ··· ≤ yn}, where n= 3k. Let T=Pn\ni=1yi/k. The 3-Partition problem\nasks if there is a partition of Yintoktriples, such that the sum of elements in each triple is equal\ntoT. The 3-Partition problem is strongly NP-hard [5].\nTheorem 1. The Schedule Length problem is strongly NP-hard.\nProof. We prove the theorem by exhibiting a reduction from a 3-Partition instance Y={y1, y2, . . . , y n}\nto an instance Iof the Schedule Length problem. We use the notation for the 3-Partition instance\nYintroduced immediately prior to the statement of the theorem. We assume that nis sufficiently\nlarge; the values in Yare bounded from above by a polynomial in n, so that nc< T≤nc+1for a\nsufficiently large constant c.\nWe now define the corresponding instance Iof the Schedule Length problem as follows. The\nspeed of the drone is set to v=Tand the flying range of the drone is set to R= 4T. Then the\nminor radius of the ellipse corresponding to the speed and range of the drone is m= 2√\nT2−1.\nFor this proof, we depart from our convention of the truck starting at [0 ,0] and instead specify\nthe starting position of the truck as [2 ,0] (this does not affect the complexity of the problem, but\n8makes some of the formulas nicer). The set of delivery points Dis partitioned into three subsets\ncalled A, B andC, that are defined below:\nA={[0, y1],[0, y2], . . . , [0, yn]}is a set of delivery points located on the y-axis and correspond to\nthe inputs to the 3-Partition problem.\nB={[6 +ϵ(n), m],[2(6 + ϵ(n)), m], . . . , [(k−1)(6 + ϵ(n)), m]}and\nC={[k(6 +ϵ(n)), m],[k(6 +ϵ(n)) + 4 , m], . . . , [k(6 +ϵ(n)) + 4 T, m]}\nare sets of delivery points that are located at distance mfrom the x-axis and ϵ(n)∈(0,1) is a\nfunction of nto be specified later.\nObserve that each delivery point in B∪Ccan be reached by the drone from exactly one location\non the x-axis, and the drone must fly its full range R= 4Tto make the delivery and return to the\ntruck, and therefore, each such delivery takes time R/v= 4. See Figure 5 for an illustration of the\ninstance Iproduced by the reduction, as well as the unique feasible drone trajectories for delivery\npoints in BandC.\nIn total there are n+ (k−1) +T+ 1 delivery points, and we set p=n+ (k−1) +T+ 1 in the\nSchedule Length problem instance. In other words, this instance asks whether there is a schedule\nthat delivers to allthe delivery points. Observe that the number of points and their coordinates\nare all bounded by a polynomial in n, so the reduction runs in polynomial time.\n2p\nT2\u00001\nT\nT=2\nT=4\n[0;y1]\n[0;y2]\n[0;yn]\nA\n[2;0]\nB\nC\n4\n4\n4\n4\n4\n2 +\u000f(n)\n2 +\u000f(n)\nFigure 5: Illustration of the Schedule Length instance Ioutput by the reduction from the 3-\npartition problem. The unique feasible drone trajectories for delivery points in BandCare also\nshown.\nWe claim that the instance Yto the 3-Partition problem is a yes-instance if and only if I\nis a yes-instance to the Schedule Length problem. It is clear that since the flying range of the\ndrone equals 4 T, no deliveries to points in Acan be scheduled after the deliveries to points in C\nare made. Thus a valid schedule delivering to all the points must schedule deliveries to Ain the\nintervals between deliveries to points in B. There are ksuch intervals, and each interval is of length\n2 +ϵ(n). We claim that at most three points [0 , yi1],[0, yi2],[0, yi3] can be scheduled within such\nan interval and if only if yi1+yi2+yi3≤T. Establishing this claim would finish the proof of the\ntheorem.\nAssume we have three integers yi1, yi2, yi3such that yi1+yi2+yi3≤Tand the truck with the\ndrone on it is at position [ i(6 +ϵ(n) + 2, m] for 0 ≤i≤k−1. By the upper bound on the delivery\ntime in Lemma 3 and observing that i < k =n/3, the total time for the three consecutive deliveries\n9started at [ i(6 +ϵ(n) + 2) , m] is at most\n2(yi1+yi2+yi3)\nT+ 31 + 4( k(6 +ϵ(n)))2+ (k(6 +ϵ(n)))\nT2−1≤2 +2n2(6 +ϵ(n))2\nT2−1= 2 + O(n2/T2).\n(3)\nThus, the deliveries to [0 , yi1],[0, yi2],[0, yi3] can be completed before the delivery to [( i+ 1)(6 +\nϵ(n) + 2, m] is scheduled, provided that O(n2/T2) =O(n−2c+2)≤ϵ(n).\nAssume we have three integers yi1, yi2, yi3such that yi1+yi2+yi3> Tand the truck with the\ndrone on it is at position [ i(6 +e(n) + 2, m] for 0 ≤i≤k−1. By the lower bound on the delivery\ntime in Lemma 3, the total time for the three consecutive deliveries started at [ i(6 +ϵ(n) + 2, m]\nis at least 2( yi1+yi2+yi3)/T > 2 + 1 /Tand they cannot be completed before the delivery to\n[(i+ 1)(6 + ϵ(n) + 2, m] is scheduled, provided that the term 1 /T≥n−c−1exceeds ϵ(n).\nIt is left to notice that because we can take nandcsufficiently large, we can find ϵ(n) satisfying:\nO\u00121\nn2c−2\u0013\n< ϵ(n)<1\nnc+1.\nFor example, one could take c= 4 and ϵ(n) = 1 /n6. This completes the proof of strong NP-\nhardness.\n4 A Greedy Approximation Algorithm\nIn this section we describe a greedy scheduling algorithm for the truck-drone problem. Our al-\ngorithm, which we call Ag, assigns deliveries to the drone as the truck moves from left to right\nstarting from the initial position of the truck at [0 ,0]. When the truck with the drone is at position\n[s,0], our greedy algorithm schedules a delivery to point dwhich, from among all feasible delivery\npoints, minimizes the round-trip flight time from [ s,0], i.e., which gives the earliest possible return\nfor the drone to the truck. Notice that the delivery point which minimizes the round-trip flight\ntime from [ s,0] is not necessarily the delivery point that is at the shortest distance from [ s,0]. For\nexample, in Figure 8, the point d1is closer than d2to [s,0]. Thus one needs to use the function\ndefined by Formula 2 to calculate which delivery point requires the shortest time to return to the\ntruck. We then update sto be this shortest return time. If there are no feasible delivery points,\nthen si set to the earliest time any of the remaining points can be reached after the current time.\nd1\nd2\nd3\nd4\nd5\nd1\nd2\nd3\nd4\nd5\nFigure 6: The black arrows, red arrows show the travel of the drone according to an optimal,\ngreedy schedule, respectively, for an instance I= (0,4,8, D) with Dcontaining five delivery points\n{d1, d2, . . . , d 5}. The black crosses and red crosses on the x-axis indicate the return points of OPT\nandAg, respectively.\nAlgorithm 1 gives the pseudocode for Ag. It is straightforward to see that Algorithm Agcan be\nimplemented in O(n2) time, since a single evaluation of ret takes constant time. Figure 6 gives an\nexample of the trajectories of the drone according to an optimal schedule and that of the schedule\ncalculated by Ag.\n10Algorithm 1 Greedy Approximation Algorithm Agto Compute Feasible Delivery Schedule\nRequire: Instance I= (v, R, D ) where D={d1, d2, . . . , d n}, is a list of delivery points.\nEnsure: SIis a feasible schedule of deliveries.\n1:SI←L← ∅\n2:s←0\n3: ▷For each delivery point di, calculate es(di) and ls(di) and insert triple into L.\n4:fori= 1. . . ndo\n5: ifs≤ls(di)then\n6: x.es=es(di)\n7: x.ls=ls(di)\n8: x.d=di\n9: Insert( L, x)\n10: end if\n11:end for\n12:Sort( L, key =es)\n13:while L̸=∅do\n14: x←first (L)\n15:\n16: ifs < x.es then\n17: s←x.es ▷ If no feasible delivery point, move sforward.\n18: end if\n19: ▷Find feasible delivery point which minimizes the return time to truck.\n20: rmin← ∞\n21: while x̸=NIL ands≥x.esdo\n22: r←ret(s, v, x.d i)\n23: ifr < r minthen\n24: rmin←r\n25: save←x\n26: end if\n27: x←next(L)\n28: end while\n29: ▷Insert next delivery point into schedule, update sand list L\n30: Insert ( SI,(save.d, s ))\n31: s←rmin\n32: forx∈Ldo\n33: ifx.lr < s then\n34: Delete ( L, x)\n35: end if\n36: end for\n37:end while\nIn the next theorem we compare the size of the schedule calculated by Algorithm Agwith\nrespect to an optimal algorithm.\nTheorem 2. Given an instance I= (v, R, D )of the truck-drone delivery problem, let Sgbe the\nschedule produced by the algorithm Agand let SOPT be an optimal schedule. Then\n|SOPT| ≤2|Sg|\nProof. LetD={d1, d2, . . . , d n}and let\nSOPT= ((di1, s1),(di2, s2), . . . , (dip, sp)),and\n11Sg= ((dj1, s′\n1),(dj2, s′\n2), . . . , (djq, s′\nq)),where q≤p≤n.\nWe give a function Fthat maps delivery points in SOPT to points in Sg. For every k, with\n1≤k≤p, define rkto be the return time of the drone for the kthdelivery in SOPT and similarly\nfor every k, with 1 ≤k≤q, define r′\nkto be the return time of the drone for the kthdelivery in Sg.\nDefine Qkto be the set of delivery points in Sgwhose return to the truck in the greedy schedule\noccurs during the flight time of the drone to deliver the kthitem in SOPT. That is,\nQk={djℓ:r′\nℓ∈(sk, rk]}\nIfQk̸=∅, define last(Qk) to be the element of Qkwith the latest return according to the greedy\nschedule.\nNow define Pkto be the set of delivery points in Sgwhose start time in the greedy schedule is\nbefore the start time of the kthdelivery in the optimal schedule, but whose return to the truck in\nthe greedy schedule occurs between the return from the kthdelivery in the optimal schedule and\nthe start of the ( k+ 1)stdelivery. That is:\nPk={djℓ:s′\nℓ≤skandrk< r′\nℓ≤sk+1}\nIfPk̸=∅, note that it can have only one element, denote it as pk.\nWe are now ready to define the function F. For all k∈ {1, . . . , p }\nF(dik) =\n\nlast(Qk) if Qk̸=∅ (4a)\npk ifQk=∅andPk̸=∅ (4b)\ndik otherwise (4c)\nWe give an example to illustrate function Fusing an instance shown in Figure 6. In that case\nthe optimal schedule makes 5 deliveries in order to ( d1, d2, d3, d4, d5) and greedy schedule contains\n3 deliveries ( d3, d4, d5) listed in order, omitting the starting times. For this case the function Fis\nas follows:\nF(d1) =d3,F(d2) =d4,F(d3) =d3,F(d5) =d5, andF(d5) =d5.\nFirst we prove that Clauses 4a, 4b, and 4c define a valid function on L′, that is, every delivery\npoint in the optimal schedule is mapped to a delivery point in the greedy schedule. Since QkandPk\nonly contain delivery points in the greedy schedule, the only case to consider is that Qk=Pk=∅\nandF(dik) =dikanddikis not part of the schedule Sgof the greedy algorithm Ag.\nLetℓbe the largest integer such that s′\nℓ≤sk. By assumption djℓ̸=dik. Since Qk=Pk=∅,\neither r′\nℓ> sk+1orr′\nℓ≤sk. Ifr′\nℓ≤sk(see Figure 7(a)), consider the ( ℓ+ 1)stdelivery by the\ngreedy algorithm. We know that s′\nℓ+1> skand since Qk=∅, it must be that r′\nℓ+1> rk. Thus for\nits (ℓ+ 1)stdelivery, the greedy heuristic should have chosen to deliver to dikrather than to djℓ+1,\na contradiction.\nTherefore it must be that r′\nℓ> sk+1. But then, using Lemma 2, there is a valid trajectory for\nthe drone flying to dikstarting at s′\nℓwith an earlier return time that is at most rk≤sk+1< r′\nℓ\n(see Figure 7(b)). Thus for its ℓthdelivery, the greedy heuristic should have chosen to deliver to\ndikrather than to djℓ, a contradiction. Thus dikmust be part of the greedy schedule, and Fis a\nvalid function mapping the delivery points in SOPTto the delivery points in Sg.\nFinally, we claim that Fmaps at most two delivery points in SOPTto one delivery point in Sg.\nFirst, since the half-closed intervals ( s′\n1, r′\n1],(s′\n2, r′\n2], . . . , (s′\nk′, r′\nk′] are all disjoint, and the half-closed\nintervals ( s1, r1],(s2, r2], . . . , (sk′, rk] are also all disjoint, and any return point [ r′,0] can satisfy at\nmost one of Clauses 4a and 4b, it follows that distinct elements in SOPT are mapped to distinct\nelements of Sgby those two clauses. Second, clearly distinct elements in SOPT are mapped to\n12dik\ndik+1\ndj`\ndj`+1\n(a)\ns0\n`\ns0\n`+1\nsk\nsk+1\nrk\nrk+1\nr0\n`\nr0\n`+1\ndik\ndik+1\ndj`\n(b)\ns0\n`\nsk\nsk+1\nrk\nrk+1\nr0\n`Figure 7: This figure illustrates the two cases in the proof that the function Fin Theorem 2 is well\ndefined. Red lines show deliveries in a presumed greedy schedule, and black lines indicate deliveries\nin a presumed optimal schedule.The dashed line represents the xaxis, with distances from the\norigin marked.\ndistinct elements of Sgby Clause 4c. Therefore, the only kind of ”collision” that can occur is that\nF(dik) is mapped to djℓby Clause 4a or Clause 4b and F(diq) is mapped to djℓby Clause 4c. This\nproves our claim that Fmaps at most two delivery points in SOPTto one delivery point in Sg.\nWe conclude that the schedule Sgcreated by Agcontains at least ⌈p/2⌉elements, as desired.\nThat is, Agis a 2-approximation algorithm.\nThe approximation ratio of 2 is tight. To see this, consider the instance given in Figure 8.\nFor this instance the schedule computed by the greedy algorithm contains exactly one half of the\ndelivery points, while an optimal schedule makes deliveries to all points. Thus, the approximation\nfactor of 2 in Theorem 2 cannot be improved.\nd\nsd d\nd2d4d65 3 1\nFigure 8: Approximation factor of 2 is sharp: in this instance, d1, d3, d5, . . .are located at the\nmaximal reach of the drone, and reachable only from the left focus of the corresponding ellipse\n(dashed green). An optimal schedule, shown in black, contains all points in order d1, d2, d3, d4, . . ..\nThe greedy algorithm, can immediately schedule a delivery to d2, but not to d1. After scheduling\na delivery to d2a delivery to d1is not feasible any more, and this scheduling, shown in red, is\nrepeated, resulting in the schedule d2, d4, d6, . . ..\n5 Optimal algorithm for a restricted set of inputs\nAs seen in the proof of strong NP-hardness in Section 3, having many delivery points with the same\nx-coordinate creates a decision problem: should a delivery to a point [0 , y] be scheduled prior to\nor after the truck reaches [0 ,0]. These decisions make the truck-drone problem NP-hard. In this\nsection, we specify a family of instances called proper instances in which the delivery points do\nnot have the same or “nearly” the same x-coordinates, where “nearly” depends on the difference in\ntheir y-coordinates. In particular, the greater the difference in the y-coordinates of the points, the\n13greater is the difference in their x-coordinates in proper instances. We show that there is O(n3)\nalgorithm to compute an optimal schedule for proper instances.\nDefinition 2. LetI= (v, R, D )be an instance of the truck-drone delivery problem where D=\n{d1, d2, . . . , d n}. We say Iis a proper instance if:\n•for every i, j∈ {1, . . . , n }, with i̸=j, the delivery point djis not contained in the triangle\n[es(di),0], di,[lr(di),0], and\n•the set of closed intervals {[es(d1), ls(d1)],[es(d2), ls(d2)], . . . , [es(dn), ls(dn)]}form a proper\ninterval graph [6], i.e., no interval in the set is a subset of another interval in the set.\nFigure 9 shows an example of a proper instance. The definition of a proper instance implies\nthat the delivery points have pairwise different x-coordinates and clearly, not many of them can\nreside in a narrow vertical band.\nThe lemma below implies that for a proper instance with D={d1, d2, . . . , d n}, the intervals\n[es(d1), ls(d1)],[es(d2), ls(d2)], . . . , [es(dn), ls(dn)] are ordered by the x-coordinates of the corre-\nsponding points in D.\nLemma 4. LetI= (v, R, D )be a proper instance of the truck-drone delivery problem with D=\n{d1, d2, . . . , d n}. Let di= [xi, yi]anddj= [xj, yj]be two points of Dwith xi< xj. Then either\nls(di)< es(dj), ores(di)< es(dj)≤ls(di)< ls(dj)\nProof. Ifyj< yithen ls(dj)> ls(di). Since interval [ es(dj), ls(dj)] cannot contain [ es(di), ls(di)],\neither ls(di)< es(dj), or es(di)< es(dj)≤ls(di)< ls(dj).\nIfyj> yithen es(dj)> es (di). Since interval [ es(di), ls(di)] cannot contain [ es(dj), ls(dj)],\neither ls(di)< es(dj), or es(di)< es(dj)≤ls(di)< ls(dj).\nGiven an instance I= (v, R, D ), we can verify if Iis a proper instance in O(n2) time by\nchecking each pair of intervals for non-containment, and each triangle for the non-inclusion of other\npoints of D.\nFigure 9: An example of a proper instance I= (v, R, D ) position for v= 3, and r= 12. For each\ndelivery point the red segment points to the corresponding edandldpoints on the line, and the\ngreen segment points to the corresponding la. The three topmost points are at the limit of the\nreach of the drone.\nThe following lemma is used to show that for proper instances we can restrict our attention to\nschedules in which the subsequent deliveries are ordered by the x-coordinates of delivery points,\nand in which the trajectories of the drone are non-crossing. See Figure 10 for an illustration.\nLemma 5. LetI= (v, R, D )be a proper instance of the truck-drone problem. Assume that there\nis a feasible schedule for this instance in which a delivery to, say d2= [x2, y2]immediately precedes\nthat to d1= [x1, y1], with x1< x 2. Then\n14d1= [x1; y1]\nd2= [x2; y2]\ns2\nr2=s1\nr1\n(i)\nd1= [x1; y1]\nd2= [x2; y2]\ns2\nr2=s1\nr1\n(ii)\nr0\n1\nr0\n2Figure 10: (i) The crossing trajectories to d1andd2are shown in black. (ii) The non-crossing\ntrajectories, shorter in total, are in red.\n1. The trajectories of the drone to d1andd2must cross.\n2. By swapping the order of deliveries to d1andd2the total time of the two deliveries cannot\nincrease, and thus swapping the two deliveries maintains the feasibility of the schedule, i.e.,\ncrossings of two consecutive trajectories can be avoided.\nd2= [x2; y2]\ns2\nr2\nx2\n(a)\ns1\nr2\nd1= [x1; y1]\nx1\n(b)\nFigure 11: Illustration for the proof of (1) in Lemma 5. d1= [x1, y1] and d2= [x2, y2] with x1< x 2\nand delivery to d2occurring before the delivery to d1. Figure (a) illustrates the case of y1≤y2\nand the shaded region demonstrates locations of d1which result in a crossing trajectory. Figure\n(b) illustrates the case of y1≥y2and the shaded region demonstrates locations of d2which result\nin a crossing trajectory.\nProof. To see (1), let si, ridenote the start and return times to delivery point difori∈ {1,2}.\nAssume for contradiction that delivery trajectories do not cross. If y1≤y2, then d1lies inside the\ntriangle [ r2,0], d2,[x2,0]. This triangle is clearly contained in [ es(d2),0], d2,[lr(d2),0] contradicting\nDbeing proper. If y1≥y2then d2lies inside the triangle [ x1,0], d1,[s1,0], which is contained inside\n[es(d1),0], d1,[lr(d1),0]. This also contradicts Dbeing proper. See Figure 11.\nNext, we show (2). By Observation 1 we can assume that the delivery to d1starts immediately\nat time r2, i.e., s1=r2and terminates at time r1. Clearly, in this case es(d2)< ls(d1) and thus,\nby Lemma 4, es(d1)< es(d2)≤s2< r 2≤ls(d1)< ls(d2). Thus, a delivery to d1can be started\nat time s2, and a delivery to d2can be started at time r2or later. It remains to show that the\nreversal in the delivery order can terminate latest at time r1.\nSuppose first that delivery to d1, when started at time s2takes at most as much time as a\ndelivery to d2at time s2, see Figure 10 (ii). In the paragraph below, we use sito denote the point\n[si,0] and similarly rito denote the point [ ri,0]. Consider the quadrilateral s2,d1,d2,r1shown in\nblue. Since our instance is a proper instance, the triangle s2,d1,r1doesn’t contain d2and thus\nthis quadrilateral is convex. By the triangular inequality the sum |s2, d1|+|d2, r1|of the lengths\n15of two opposite sides of the quadrilateral is strictly less than the sum of the length of its diagonals\n|d1, r1|+|s2, d2|. Therefore,\n|s2, d1|+|d1, r2|+|r2, d2|+|d2, r1|<|s2, d2|+|d2, r2|+|r2, d1|+|d1, r1|\nand the path s2, d1, r2, d2, r1is shorter than the trajectory s2, d2, r2, d1, r1. However, the path\ns2, d1, r2, d2, r1isnot necessarily a valid drone trajectory if the delivery to d1from s2takes less\ntime than the delivery to d2from s2. Then, when delivering to d1first, the drone returns to the\ntruck at point r′\n2located strictly between s2andr2. But then the path s2, d1, r′\n2, d2, r1is even\nshorter than path s2, d1, r2, d2, r1. Thus, when starting the delivery to d2atr′\n2, the drone returns\nto the truck at a point r′\n1to the left of r1, which improves the total delivery time to d1andd2.\nd1= [x1; y1]\nd2= [x2; y2]\ns2\nr2=s1\nr1\n(i)\n(ii)\nd1= [x1; y1]\nd2= [x2; y2]\ns2\nr2=s1\nr1\nFigure 12: Reversing the directions of deliveries and the movement of the truck in (ii) converts a\nconfiguration of the second case to the first case\nNow suppose instead that delivery to d1, when started from s2, takes more time then the\ndelivery to d2from s2, as for example on Figure 12 (i). By the shape of the function ret( s, d, v )),\nsee Observation 4, and since es(d1)< es(d2) and ls(d1)< ls(d2), a delivery to d1, when started\nfrom s1also takes more time than the delivery to d2from s1. Consider the configuration on Figure\n12(ii) in which we reverse the movement of the drone and of the truck. Then we reduced this to\nthe previous case and a delivery from s1first to d2and then to d1is shorter, and by reversing this\nonce more we obtain that the delivery from s2first to d1and then to d2is shorter.\nA proper instance is guaranteed to have an optimal schedule with non-crossing trajectories.\nHowever not all optimal schedules give non-crossing trajectories. Indeed there are non-proper\ninstances where crossing of trajectories is required in any optimal schedule as demonstrated in\nFigure 13.\nsd1d3\nd2\nFigure 13: An instance of the problem where any optimal schedule must contain crossing trajec-\ntories. Points d1andd3are at the maximum reach of the drone. When scheduling a delivery first\ntod1then a delivery is possible either to d2or to d3, but not to both.\n16Definition 3. LetI= (v, R, D )be an instance of the truck-drone delivery problem where D=\n{d1, d2, . . . , d n}. We call schedule\nSI= ((di1, s1),(di2, s2), . . . , (dim, sm), m≤n\nmonotone if the x-coordinate of dijis strictly less than the x-coordinate of dij+1for every 1≤j≤\nm−1,\nIn the next theorem we show that there always exists a monotone schedule with the optimal\nsubstructure property for proper instances.\nTheorem 3. LetI= (v, R, D )be a proper instance of the truck-drone delivery problem with\nD={d1, d2, . . . , d n}. Assume that the points in Dare listed according to increasing x-coordinate.\nThen there is an optimal schedule SI= ((di1, s1),(di2.s2), . . . , (dim, sm),m≤nfor this instance\nwith the following properties:\n1.SIis monotone.\n2. For every j≤m, the initial part ((di1, s1),(di2, s2), . . . , (dij, sj))ofSIminimizes the delivery\ncompletion time for any subset of {d1, d2, . . . , d ij}of size j.\nProof. Assume I= (v, R, D ) is a given proper instance of the truck-drone delivery problem and\nletSIbe an optimal schedule for it. By a repeated application of Lemma 5 we can swap any two\nconsecutive deliveries that don’t respect the order of x-coordinates of points, as in the bubble sort,\nwhile maintaining the schedule optimal. This eventually produces a monotone schedule of the same\n(optimal) length, proving (1).\nTo show (2), assume that for some j≤m, there is a subset of jpoints {di′\n1, di′\n2, . . . , d′\nij}of the\nset{d1, d2, . . . , d ij}for which there is a schedule (( d′\ni1, s′\n1),(d′\ni2, s′\n2), . . . , (d′\nij, s′\nj)) with s′\nj< sjand\nwhich minimizes the delivery completion time for any subset of {d1, d2, . . . , d ij}of size j. Then\nby concatenating (( d′\ni1, s′\n1),(d′\ni2, s′\n2), . . . , (d′\nij, s′\nj)) with (( dij+1, sj+1), . . . , (dim, sm), we get a valid\nschedule. In this manner, repeating the process starting with j=imand decreasing appropriately\nthe value of jwe can get a schedule for Ithat is optimal, monotone, and satisfies the property 2\nof the theorem.\nWe use Theorem 3 to describe a dynamic programming algorithm that finds an optimal schedule\nfor proper instances.\nTheorem 4. There is an O(n3)algorithm that calculates an optimal schedule for any proper\ninstance I= (v, R, D )of the truck-drone delivery problem.\nProof. Assume the delivery points in Dare listed in the order of their xcoordinates. Define\nT(i, j) to be the earliest delivery completion time for the truck and the drone to perform exactly i\ndeliveries from among d1, d2, . . . , d jwhere djmust be included in the schedule. If such a schedule is\nnot possible, we define T(i, j) =∞. We can compute T(i, j) using dynamic programming as follows.\nWe clearly have T(1, j) = ret( s, dj, v) for the base case of i= 1 (see Lemma 1 for the definition of\nret) where [ s,0] is the starting position of the truck. For i≥2, we have T(i, j) = min j′<jret(T(i−\n1, j′), dj, v). This recursive formula immediately follows from the optimal substructure property\nstated in Theorem 3: a schedule resulting in the earliest completion time of making iout of the\nfirstjdeliveries where djis included consists of delivering to i−1 out of the first j′< jdelivery\npoints (with earliest completion time T(i−1, j′)) followed by earliest delivery completion to dj.\nNote that defining ret( s′, v, d) =∞when s′> ls(d) and T(i, j) =∞when delivery is impossible\ncorrectly works with the recursive computation of T.\n17Having computed T, we can find the maximum number of deliveries that can completed in a\nvalid schedule by taking the maximum msuch that T(m, j)̸=∞for some j. By recording for each\n(i, j) pair which choice of j′resulted in the table entry T(i, j), we can reconstruct the schedule\nitself using standard backtracking techniques.\nThe running time is dominated by computing the table T(i, j). It has O(n2) entries and each\nentry can be computed in time O(n), since a single evaluation of ret takes constant time. The\noverall runtime is then O(n3).\n6 Discussion\nWe have shown that even in the simple case of a single drone with a single truck travelling in a\nstraight line, the problem of coordinating their efforts to maximize the number of deliveries made\nis hard. Our work raises a number of different questions. We show that a greedy strategy achieves\na 2-approximation. Is a better approximation possible? In particular, is the problem APX-hard\nor might there be a PTAS for it? Our implementation of the greedy strategy runs in O(n2) time.\nIs a better running time for the algorithm possible by taking advantage of the structure of the\nintervals created by the drone paths? The set of proper instances includes those where the y-\ncoordinate is fixed. Could this be expanded to include points with a limited number of different\ny-coordinates? More generally, is there a ”natural” setting in which the problem becomes fixed-\nparameter tractable? Finally, many variations on the problem are worth pursuing. Rather than\nmaximizing the number of deliveries made with a given speed or drone range, one could consider\nthe dual problems of minimizing the speed or range required to complete all deliveries. Versions\nwith multiple drones and/or trucks, larger capacity drones, etc. are also of interest.\nReferences\n[1] A. Cornell, B. Kloss, and R. Riedel. Drones take to the sky, potentially disrupting last-mile de-\nlivery. https://www.mckinsey.com/industries/aerospace-and-defense/our-insights/future-air-\nmobility-blog/drones-take-to-the-sky-potentially-disrupting-last-mile-delivery , 2023.\n[2] Thomas Erlebach, Kelin Luo, and Frits C. R. Spieksma. Package delivery using drones with\nrestricted movement areas. In Sang Won Bae and Heejin Park, editors, 33rd International Sym-\nposium on Algorithms and Computation, ISAAC 2022, December 19-21, 2022, Seoul, Korea ,\nvolume 248 of LIPIcs , pages 49:1–49:16. Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik,\n2022.\n[3] H. Eskandaripour and E. Boldsaikhan. Last-mile drone delivery: Past, present, and future.\nSpecial Issue The Applications of Drones in Logistics , 2023.\n[4] J´ ulia C Freitas, Puca Huachi V Penna, and T´ ulio AM Toffolo. Exact and heuristic approaches\nto truck–drone delivery problems. EURO Journal on Transportation and Logistics , 12:100094,\n2023.\n[5] M. R. Garey and D. S. Johnson. Computers and Intractability; A Guide to the Theory of\nNP-Completeness . W. H. Freeman & Co., New York, NY, USA, 1990.\n[6] M. C. Golumbic. Interval graphs and related topics. Discrete Mathematics , 55:113–121, 1985.\n[7] Arindam Khanda, Federico Cor` o, and Sajal K Das. Drone-truck cooperated delivery under\ntime varying dynamics. In Proceedings of the 2022 Workshop on Advanced tools, programming\n18languages, and PLatforms for Implementing and Evaluating algorithms for Distributed systems ,\npages 24–29, 2022.\n[8] Hongqi Li, Jun Chen, Feilong Wang, and Yibin Zhao. Truck and drone routing problem with\nsynchronization on arcs. Naval Research Logistics (NRL) , 69(6):884–901, 2022.\n[9] Giusy Macrina, Luigi Di Puglia Pugliese, Francesca Guerriero, and Gilbert Laporte. Drone-\naided routing: A literature review. Transportation Research Part C: Emerging Technologies ,\n120:102762, 2020.\n[10] Adriano Masone, Stefan Poikonen, and Bruce L Golden. The multivisit drone routing prob-\nlem with edge launches: An iterative approach with discrete and continuous improvements.\nNetworks , 80(2):193–215, 2022.\n[11] Neil Mathew, Stephen L Smith, and Steven L Waslander. Optimal path planning in coopera-\ntive heterogeneous multi-robot delivery systems. In Algorithmic Foundations of Robotics XI:\nSelected Contributions of the Eleventh International Workshop on the Algorithmic Foundations\nof Robotics , pages 407–423. Springer, 2015.\n[12] Chase C Murray and Amanda G Chu. The flying sidekick traveling salesman problem: Op-\ntimization of drone-assisted parcel delivery. Transportation Research Part C: Emerging Tech-\nnologies , 54:86–109, 2015.\n[13] Aishwarya Raghunatha, Emma Lindkvist, Patrik Thollander, Erika Hansson, and Greta Jon-\nsson. Critical assessment of emissions, costs, and time for last-mile goods delivery by drones\nversus trucks. Scientific Reports , 13(1):11814, 2023.\n[14] Teena Thomas, Sharan Srinivas, and Chandrasekharan Rajendran. Collaborative truck multi-\ndrone delivery system considering drone scheduling and en route operations. Annals of Oper-\nations Research , pages 1–47, 2023.\n[15] Li. X., J. Tupayachi, A. Sharmin, and M. Ferguson. Drone-aided delivery methods, challenge,\nand the future: A methodological review. Drones , 7:191, 2023.\n[16] Ruowei Zhang, Lihua Dou, Bin Xin, Chen Chen, Fang Deng, and Jie Chen. A review on the\ntruck and drone cooperative delivery problem. Unmanned Systems , pages 1–25, 2023.\n19"    },    {        "title": "2402.00850v2.Constant_Degree_Direct_Product_Testers_with_Small_Soundness.pdf",        "content": "Constant Degree Direct Product Testers with Small Soundness\nMitali Bafna* Noam Lifshitz†Dor Minzer‡\nAbstract\nLetXbe ad-dimensional simplicial complex. A function F:X(k)→ {0,1}kis said to be a direct\nproduct function if there exists a function f:X(1)→ {0,1}such that F(σ) = ( f(σ1), . . . , f (σk))for\neachk-face σ. In an effort to simplify components of the PCP theorem, Goldreich and Safra [GS00]\nintroduced the problem of direct product testing, which asks whether one can test if F:X(k)→ {0,1}k\nis correlated with a direct product function by querying Fon only 2inputs. Dinur and Kaufman [DK17]\nconjectured that there exist bounded degree complexes with a direct product test in the small sound-\nness regime. We resolve their conjecture by showing that for all δ >0, there exists a family of high-\ndimensional expanders with degree Oδ(1)and a 2-query direct product tester with soundness δ.\nWe use the characterization given by [BM23a] and independently by [DD23a], who showed that\nsome form of non-Abelian coboundary expansion (which they called “Unique-Games coboundary ex-\npansion”) is a necessary and sufficient condition for a complex to admit such direct product testers. Our\nmain technical contribution is a general technique for showing coboundary expansion of complexes with\ncoefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders con-\nstructed by [CL23] satisfy the conditions of [BM23a], thus admitting a 2-query direct product tester with\nsmall soundness.\n1 Introduction\nThe primary goal of this paper is to construct direct product testers with constant degree. Earlier works by a\nsubset of the authors [BM23a] and by Dinur and Dikstein [DD23a] have shown that there are coboundary-\ntype properties that are sufficient for a complex to have in order to admit such a direct product tester. The\nmain contribution of the current paper is to establish that the simplicial complex of [CL23] has the sufficient\ncondition presented in [BM23a], implying that it admits direct product testers with small soundness.\n1.1 Direct Product Testers\nThe goal in direct product testing is to encode the values of a function f: [n]→ {0,1}via a table Fwhich\nis testable. By that, we mean that there is an efficient, randomized tester that makes queries to F, performs\na test on the received values, and decides to accept or reject. The tester should always accept if Fis indeed\na valid encoding of a function f; this property is often referred to as the completeness of the test. In the\nconverse direction, if the tester accepts Fwith probability at least s >0, then Fmust be close (in the case s\nis close to 1) or correlated (in the case sis close to 0) to a valid encoding of some function f. This property\nis often referred to as the soundness of the tester. The smallest sfor which the last property holds is called\n*Department of Mathematics, Massachusetts Institute of Technology. mitali.bafna@gmail.com\n†Einstein institute of Mathematics, Hebrew University. noamlifshitz@gmail.com . Supported by the Israel Science\nFoundation (grant no. 1980/22).\n‡Department of Mathematics, Massachusetts Institute of Technology. dminzer@mit.edu . Supported by NSF CCF award\n2227876 and NSF CAREER award 2239160.\n1arXiv:2402.00850v2  [cs.CC]  7 Feb 2024the soundness parameter of the encoding, and the number of queries the algorithm makes is called the query\ncomplexity of the algorithm. In this paper, we will restrict our attention to direct product testers with 2\nqueries, which is the smallest query complexity one can hope for.\nThe study of direct product testing and the importance of its parameters such as completeness, soundness\nand query complexity, originally comes from the work of Goldreich and Safra [GS00]. Their motivation\nwas to simplify components in the proof of the PCP theorem [FGL91, AS98, ALM+98], and the parameters\nof the direct product tester correspond exactly to the parameters of the PCP verifier. Another motivation\nfor the study of direct product testing comes from hardness amplification, particularly in the style of the\nparallel repetition theorem [Raz98, Hol09, Rao11, DS14a, BG15]. Indeed, direct product testers with small\nsoundness can be viewed as a combinatorial analog of parallel repetition theorems, and these can sometimes\nbe turned into proper parallel repetition theorems [IKW09, DM11]. We refer the reader to [DK17, BM23a]\nfor further discussion on the role of direct product testing in theoretical computer science.\nThe most natural direct product encoding of a function f: [n]→ {0,1}is given by the Johnson scheme.\nHere, for k∈Nthought of as a constant, one may encode fvia the table F:\u0000[n]\nk\u0001\n→ {0,1}kasF[A] =\nf|A.1The Johnson encoding scheme also admits a corresponding natural direct product tester:\n1. Sample I⊆[n]of size t.\n2. Independently sample A, A′⊆[n]of size kcontaining I.\n3. Query F[A]andF[A′]and check that they agree on the coordinates of I.\nThe above direct product tester has received significant attention, as well as its variations. By now, all\nof these are well understood in the entire range of parameters t, see for example [DG08, IKW09, DS14b,\nBKM23]. In particular, it is well known that the tester has vanishing soundness in the setting that t≈√\nk\nand for sufficiently large k.2\nThe primary disadvantage with the Johnson encoding scheme and the corresponding direct product tester\nis its size. Indeed, the size of the encoding Fof a function fisnk, which is polynomially large in the size of\nf. Additionally, if one wishes the soundness of the Johnson direct product tester to be small, say δ, one must\ntakekto be sufficiently large, hence getting a large blow-up in the encoding size. For some applications,\nthis size blow-up is too costly. For instance, in applications in PCPs, the blow-up introduced by the use of\nparallel repetition theorem is often what dominates the blow-up in the instance size reductions produce.\n1.2 Direct Product Testers via High Dimensional Expanders\nIn the quest for more efficient ways to amplify hardness (often called derandomized hardness amplification),\nDinur and Kaufman [DK17] suggested high dimensional expanders as a sparse object that may facilitate\ndirect product testers. To describe their result, we first take a quick detour to present several basic notions\nfrom the field of high dimensional expansion, abbreviated HDX henceforth.\nAd-dimensional simplicial complex X= (X(0), . . . , X (d))with vertex set X(1) = [ n]is a downwards\nclosed collection of subsets of [n]. We follow the convention that X(0) = {∅}, and for each i >1the set\nofi-faces X(i)is a collection of subsets of X(1)of size i. The size of Xis the total number of faces in X.\nThe degree of a vertex v∈X(1)is the number of faces containing it, and the degree of Xis the maximum\ndegree over all v∈X(1).\n1We think of [n]as being order in a canonical way, thus for a set Aof size kcontaining i1< . . . < i kwe define F[A] =\n(f(i1), . . . , f (ik)).\n2In fact, in that case it is known that the soundness is exponentially small in t.\n2Definition 1.1. For a d-dimensional simplicial complex X= (X(0), X(1), . . . , X (d)),0⩽i⩽d−2and\nI∈X(i), the link of Iis the (d−i)-dimensional complex XIwhose faces are given as\nXI(j−i) ={J\\I|J∈X(j), J⊇I}.\nFor a d-dimensional complex X= (X(0), X(1), . . . , X (d))andI∈Xof size at most d−2, the graph\nunderlying the link of Iis the graph whose vertices are XI(1)and whose edges are XI(2). We associate\nwithXa collection of distributions over faces. The distribution µdis the uniform distribution over X(d),\nand for each i < d the distribution µiis a distribution over X(i)which results by picking D∼µd, and then\ntaking I⊆Dof size iuniformly. For convenience, we encourage the reader to think of µias the uniform\ndistribution over X(i)(though often times this is not the case, the fact that µiis not actually the uniform\ndistribution is rarely an issue).\nDefinition 1.2. We say a d-dimensional simplicial complex Xis aγone-sided local spectral expander if for\nevery I∈Xof size at most d−2, the second eigenvalue of the normalized adjacency matrix of the graph\n(XI(1), XI(2)) is at most γ.\nWith this definition, the result of Dinur and Kaufman [DK17] asserts that if Xis aγone-sided local\nspectral expander for a sufficiently small γ, then Xadmits a 2-query direct product tester with soundness s=\n1−ε. Their direct product tester is a direct analog of the Johnson direct product tester. It is parameterized by\nk∈Nwhich is much smaller than d, and proceeds as follows. Given an assignment F:X(k)→ {0,1}d,\nthe tester proceeds as follows:\n1. Sample D∼µd.\n2. Sample I⊆Dof size√\nkuniformly.\n3. Sample I⊆A, A′⊆Dindependently.\n4. Query F[A]andF[A′]and check that they agree on the coordinates of I.\nThe result of Dinur and Kaufman [DK17] asserts that provided that γis small enough, if Fpasses the above\ntester with probability 1−ε, then there exists f:X(1)→ {0,1}such that with probability 1−O(ε)over\nthe choice of A∼µkwe have that F[A] =f|A. We refer to the above tester as the canonical direct product\ntester of X.\nThe main open problem left from the work [DK17] is whether there are high-dimensional expanders\nthat facilitate direct product testers in the low soundness regime. With regards to this, the works [BM23a,\nDD23a] both show that spectral expansion is insufficient. Namely, there are high-dimensional expanders\nwith arbitrarily good local spectral expansion for which the above natural direct product tester fails in the\nlow-soundness regime. These two works then went on to seek additional properties of high-dimensional\nexpanders that will imply that the canonical direct product tester has small soundness.\n1.3 Main Result\nThe main result of this paper is that there exist families of high-dimensional expanders for which the canon-\nical direct product tester has small soundness. In fact, the complexes we prove this for are variants of\nthe complexes constructed by Chapman and Lubotzky [CL23, Section 5] for sufficiently large parame-\ntersp, n∈N, which are explicit and efficiently computable. The relevance of variants of the complexes\nof [CL23] was communicated to us in [DDL23].\n3Theorem 1.3. For all δ, γ > 0, for sufficiently large d, there is ∆∈Nsuch that the following holds. There\nare infinitely many nfor which there is a simplicial complex Xsuch that:\n1. The complex Xhasnvertices and is a γtwo-sided spectral expander.\n2. The canonical direct product tester for Xhas soundness δ.\n3. The complex Xhas degree at most ∆.\nIt has been brought to our attention that Dikstein, Dinur, and Lubotzky have concurrently and indepen-\ndently established Theorem 1.3.\n1.4 UG Coboundary Expansion\nThe starting point of the proof of Theorem 1.3 is the characterization of [BM23a] of high-dimensional\nexpanders for which the canonical tester has small soundness. Towards this end, we begin by defining\nnon-Abelian affine unique games over graphs.\nDefinition 1.4. An instance of Affine Unique Games Ψ = ( G,Π)over the symmetric group Smconsists of a\ngraph G= (V, E)and a collection of permutations, one for each ordered edge, Π ={πu,v}(u,v)∈E, where\nπu,v∈Smandπu,v=π−1\nv,u. An assignment to Ψis a function A:V→Sm, and we denote by val(A)the\nfraction of constraints satisfied by A, that is,\nval(A) = Pr\n(u,v)∼E[A(u) =πu,vA(v)].\nWe denote by viol (A)the fraction of constraints violated by A, that is, viol (A) = 1−val(A). The value of\nthe instance Ψis defined as val(Ψ) = max A:V→Smval(A).\nWe refer to the above instances as affine Unique Games because they can be thought of as systems\nof equations of the form A(u)A(v)−1=πu,voverSm, wherein the goal is to find an Sm-labeling of the\nvertices that satisfies as many of the equations as possible.\nDefinition 1.5. LetG= (V, E)be a graph, equipped with a distribution Dover triangles in G. We say that\na UG instance Φ = ( G,{πu,v}(u,v)∈E(G))is(1−δ)-triangle consistent if:\nPr\n(u,v,w )∼D[πu,vπv,wπw,u=id]⩾1−δ.\nWe let incons (Φ)denote the fraction of triangles that are inconsistent.\nNote that if we have a graph G= (V, E)and an assignment A:V→Sm, then defining πu,v=\nA(u)A(v)−1gives a fully triangle consistent instance of affine Unique-Games. The property of UG cobound-\nary expansion asserts, roughly speaking, that the only instances that are highly triangle consistent over cer-\ntain graphs associated with a complex Xarise in this way.\nDefinition 1.6. LetXbe a d-dimensional simplicial complex, and let r⩽d/3. We define the graph\nGr[X]whose vertex set is X(r), and whose set of edges Er[X]consists of pairs of vertices (u, v)such\nthatu∪v∈X(2r). The distribution over triangles associated with this graph is the distribution where\nwe pick a face from X(3r)according to the measure µ3r, and then split it randomly as u∪v∪wwhere\nu, v, w ∈X(r).\n4With the definition of the graph Gr[X], we may now define the notion of UG coboundary expansion.\nDefinition 1.7. We say that a d-dimensional simplicial complex Xis an (m, r, ξ, c )UG coboundary ex-\npander if for all t⩽rand for all functions f:Et[X]→Smthat are (1−ξ)-consistent on triangles, there\nisg:X(t)→Smsuch that\nPr\nu∪v∼µ2t\u0002\nπ(u, v) =g(u)g(v)−1\u0003\n⩾1−c.\nThe notion of UG coboundary expansion can be seen as a non-Abelian version of the usual notion of\ncoboundary expansion. For r= 1this definition appeared in earlier works of [DM19, GK23]; for larger rit\nis not clear what is the correct way to generalize the notion of coboundary expansion, as it is often stated for\nAbelian groups. Our notion is motivated by looking at the graph as a constraint satisfaction instance. In this\npaper, this will be the only notion of coboundary expansion discussed, and as such we will often abbreviate\nit as refer to it simply as coboundary expansion.\nThe main result of [BM23a] asserts that a high-dimensional expander Xwith sufficiently strong UG\ncoboundary parameters has a canonical direct product tester with small soundness; see [BM23a, Theorem\n1.9, Theorem B.1]. For future reference, we give here the version we use in our application:\nTheorem 1.8. There exists c >0such that for all ε, δ > 0there is ηand sufficiently large m, r∈N, such\nthat for sufficiently large k, sufficiently large dand sufficiently small γ >0the following holds. Suppose\nthatXis ad-dimensional simplicial complex such that:\n1. Aγone-sided local spectral expander.\n2. An (m, r, 2−r/log log log r, c)UG coboundary expander.\nIfF:X(k)→ {0,1}kpasses the canonical direct product tester on Xwith probability at least δ, then there\nexists f:X(1)→ {0,1}such that\nPr\nA∼µk[∆(F[A], f|A)⩽εk]⩾η.\nUsing Theorem 1.8, it suffices to construct a complex which is simultaneously a strong spectral expander,\nas well as a UG coboundary expander for sufficiently good parameters. With regard to this, we mention the\nwork of [DD23b] that presents a technique that allows one to prove coboundary-type properties with bounds\nthat are independent of the dimension of the complex. We also mention the recent work [DD23c] which\nshows coboundary-type properties for general buildings with for up to δ= 2−O(r)inconsistent triangles,\nand an improved bound of δ= 2−O(√r)for the spherical building of type A.\nStudying spherical buildings and proving coboundary expansion results for them is a central part of our\nproof as well, however the result of [DD23c] is insufficient as far as we know. The spherical buildings we\nhave to study are spherical buildings of type Cn, and for them we have to be able to handle Unique-Games\ninstances with δ= 2−o(r)fraction of inconsistent triangles. Indeed, the main technical contribution is a\nfairly general technique for proving that a simplicial complex is a UG coboundary expander.\n1.5 Our Techniques\nThe rest of this introductory section is devoted to an overview of the proof of Theorem 1.3, and we start\nwith the following definition.\n5Definition 1.9. We say that a graph is a (C(G), β(G))-coboundary expander over Smif for all δ∈[0,1]\nand all UG instances ΨoverSm, with incons (Ψ)⩽δ, there exists an assignment A:V→Smwith\nviolΨ(A)⩽C(G)δ+β(G). When β(G)is0, we say that a graph is a C(G)-coboundary expander over\nSm.\nThe additive error β(G)is necessary in our proofs, but we encourage the reader to ignore it and think\nof it as 0for the purposes of this section. Fix a simplicial complex X; we will eventually take Xto be\nChapman Lubotzky complex with an appropriate choice of parameteres [CL23]. The main components of\nour proof proceed as follows:\n1.Local to global for cosystolic expansion: we use an idea of [EK16, DD23b] who show that to prove\nthat a simplicial complex is a cosystolic expander, it suffices to show that its links are coboundary\nexpanders. Here and throughout, we say that a graph G= (V, E)is aC-cosystolic expander if\nfor every Unique-Games instances over Gwith at most δfraction of inconsistent triangles, we may\nmodify the constraints of Gin at most Cδfraction of the edges and get that all triangles are fully\nconsistent. We use this idea to show that if the links of Xare coboundary expanders, then Xitself is\na cosystolic expander with similar parameters.\n2.Vanishing Cohomology of X:inspired by the first item, one is motivated to ask the question of what\nis the constraint structure of fully triangle consistent Unique Games instances on the graphs associated\nwithX. Here, we use the fact, communicated to us by Dikstein, Dinur and Lubotzky [DDL23], that\nthe parameters of the Chapman–Lubotzky [CL23] can be chosen appropriately so that the complex\nhas vanishing cohomology over Sm.\n3.Coboundary expansion of the links of X:Together, the first two items imply that to prove that Xis\na coboundary expander, it suffices to prove that the links of Xare coboundary expanders. Indeed, in\nthat case, if Ψis a(1−δ)-consistent Unique-Games instance over Sm, by the first item we can modify\nthe constraints of Ψon at most Cδfraction of edges for C= 2o(r)to get an instance Ψ′which is fully\ntriangle consistent. Invoking the second item, we conclude a structural result about the constraints of\nΨ′, which automatically implies a similar result for Ψ.\nProving that the links of Xhave sufficiently good coboundary expansion consists of the bulk of our\neffort in this paper. We do so by an inductive argument on the parameter r.\nIn the rest of this section, we elaborate on the third item above as we consider it to be the main contribution\nof this paper. We begin by noting that for our complex X, the links are either isomorphic to the spherical\nbuildings of type Cnor to product of two such spherical buildings. Given a prime pand a dimension d, the\nspherical building of type Arefers to the complex whose vertex set is the set of all nontrivial subspaces of\nFd\np, and whose k-faces are k-flags of subspaces, namely {A1, . . . , A k}where A1⊆A2⊆. . .⊆Ak.\nThe spherical buildings with type Care similar, except that the vertices only consist of isotropic sub-\nspaces with respect to a symplectic form. Here, the dimension dis an even number 2n, and one defines a\nsymplectic form ω:F2n\np×F2n\np→Fpby\nω(x, y) =nX\ni=1xiyi+n−xi+nyi.\nA subspace V⊆F2n\npis then called isotropic ifω(x, y) = 0 for all x, y∈V. With this in mind, the spherical\nbuilding of type Ccertainly has a nice structure. However, as we are aiming for an inductive approach which\n6will necessitate for us to look at various “restrictions” of this complex, it turns out to be convenient to think\nabout this complex more abstractly as a measure µovernQ\ni=1Xi, where Xiis the collection of all isotropic\nsubspaces of Fd\npof dimension i. Namely, µis just the uniform distribution over top dimensional faces of X.\nThe most crucial feature of µthat makes it easier to work with is the fact that it is an ε-product measure\nin the sense of [GLL22], where ε=O(1/√p). Informally, an ε-product measure is one in which one can\nperform the usual type of discrete Fourier analysis as in product domains such as the Boolean hypercube\n{0,1}n, or product-like domains as the Johnson scheme\u0000[n]\nk\u0001\nwith constant k. Additionally, ε-productness\nis a property that is preserved under conditioning, making it friendly for inductive processes. We defer a\nprecise definition of ε-product measures to Section 2. Below, we will focus on the measure µitself, but we\nremark that our arguments have to also work with restrictions of µ(which they do; the notations become\nsomewhat more complicated, and hence we omit this discussion).\n1.5.1 Moving to Tripartite Instances and ε-product Distributions\nSet-up: moving to the tripartite problem: consider the n-dimensional spherical building of type C\ncomplex Xas above, and let Ψbe a Unique-Games instance over its rfaces which is (1−δ)-triangle\nconsistent. Sample R1, R2, R3⊆[n]independently of size r, and consider the Unique-Games instance Ψ′\ninduced by Ψon the tripartite graph T(R1, R2, R3). Here and throughout, the tripartite weighted graph\nT(R1, R2, R3)is the graph whose faces are r-faces that are subsets of R1, R2andR3, and whose edges\nand constraints are induced by Ψ. It can be shown via standard arguments that the fraction of inconsistent\ntriangles in Ψ′is(1 +o(1))δ. Furthermore, using an idea from [DD23b], we show that it suffices to prove\nthat the instance Ψ′has a good cosystolic coefficient for a good fraction of choices of R1, R2, R3. Roughly\nspeaking, if we show that with probability at least p, the induced instance Ψ′has cosystolic coefficient C,\nthen the instance Ψwill have a cosystolic coefficient at most O(C/p).\nOur argument will not show a strong enough cosystolic coefficient for every choice of R1, R2, R3, and\nit depends on two features of them:\n1. Separatedness: we would like the elements in R1∪R2∪R3to be as far apart as possible form each\nother. Note that the expected magnitude of an element in the union is Θ(n), and as we are choosing\n3rof them uniformly, we expect them to be roughly Θ(n/r)apart. Ideally, we would have liked any\ntwo distinct elements from R1∪R2∪R3to be Θ(n/r)apart, however the probability for that (over\nthe choice of R1, R2, R3) is2−Θ(r), which is too small for our purposes as we are shooting for a 2o(r)\ncoboundary constant. We thus settle for weaker separatedness, and for our argument it suffices to have\nR1, R2, R3to be Θ(n/r2)-separated (which happens with probability Θ(1) ).\n2. Well spread: consider an interval I⊆[d], sayI={s, s+ 1, . . . , s +L−1}, and think of Las being\nof the order n/r0.99. Note that in expectation, each one of R1, R2, R3contains Lr\nnelement from the\ninterval. We say that R1is well spread if for every interval Iof length Lwe have that R1∩Ihas\nsize roughly Lr\nn, and say that R1, R2, R3is well spread if each one of R1,R2andR3is well spread.\nWe would like R1, R2, R3to be well spread, and a standard application of concentration bounds show\nthat the probability a randomly chosen R1, R2, R3is well spread with probability 1−o(1).\nFor the rest of this section we fix R1, R2, R3that are well spread and Θ(n/r2)separated.\nMoving to the language of ε-product distributions: now that we are considering tripartite instances Ψ′\noverR1, R2, R3, it will be useful to think of the measure µoverQ\ni∈R1∪R2∪R3Xithat underlies this complex;\n7namely, this is the distribution µ3rrestricted to triangles in Ψ′. This measure can be proved to be an ε-\nproduct measure for ε=ε(p)which is a vanishing function of the field size p; for our intents this means\nthatεcan be guaranteed to be as small as we wish compared to the dimension of the complex and r. We\ndenote by Cr,r,r(µ)the coboundary constant of the tripartite graph underlying Ψ′. With these notations, our\nprimary objective now is to prove that Cr,r,r(µ) = 2o(r). More generally, for 1⩽t⩽r, we define\nCt,t,t(µ) = max\nR′\n1⊆R1,|R′\n1|=t\nR′\n2⊆R2,|R′\n2|=t\nR′\n3⊆R3,|R′\n3|=tmax\naCoboundary constant of T(R′\n1, R′\n2, R′\n3|x(R1∪R2∪R3)\\(R′\n1∪R′\n2∪R′\n3)=a),\nwhere the graph T(R′\n1, R′\n2, R′\n3|x(R1∪R2∪R3)\\(R′\n1∪R′\n2∪R′\n3)=a)is the induced subgraph of Ton the vertices\nthat agree with the restriction x(R1∪R2∪R3)\\(R′\n1∪R′\n2∪R′\n3)=a. The main benefit of ε-product measures is\nthat it provides a clean abstraction of he spectral properties of our tripartite graphs which is suitable for\ninduction.\n1.5.2 An Inductive Approach to Coboundary Expansion: the Base Case\nTo get some intuition, we first consider the case that r= 1. In that case, the underlying graph of Ψ′is a\ntripartite inclusion graph induced by 3distinct dimensions, say i < j < ℓ , which we know to be Θ(n/r2)\nseparated. Inspecting this graph, one can show that the diameter of this graph is at most\nO\u0012\nmax\u0012j\n|i−j|,ℓ\n|ℓ−j|\u0013\u0013\n=O(r2),\nwhich suggests that the graph is very well connected. In this case, we use the cones method [Gro10, LMM16,\nKM22, KM19, KO19]. The cones method is a standard combinatorial “local correction” technique allowing\none to deduce global assignments from good local consistency. The method requires us to construct a set of\ncanonical paths between vertices which will be used for “propagation”, as well as triangulations of cycles\nthat are formed by an edge (V, W )in the graph and the two canonical constructed paths from some vertex\nU, using only a small number of triangles.3To make this possible, we have to construct the set of canonical\npaths in a rather careful manner. With these paths, we are able to break each formed cycle between U, V, W\ninto a small collection of 8-cycles (as well as some triangles), which we then triangulate via careful case\nanalysis.\nIn our context, using the cones method we can show that the coboundary constant of the graph under-\nlying Gis polynomial in its diameter. In particular, this implies that C1,1,1(µ)⩽rO(1). We remark that\nworking with well-separated indices is a useful idea if one wishes to prove results that are independent of the\ndimension of the complex, first introduced by [DD23b]. For us, it will be important that the above applies\nnot only to the measure µitself, but rather to any restriction of µleaving at least 3coordinates alive.\n1.5.3 An Inductive Approach to Coboundary Expansion: the Extended Base Case\nThe inductive base case above for C1,1,1(µ)is insufficient for our purposes; indeed, using it and our inductive\nstep yields a bound of 2˜O(r)on the coboundary expansion of T(R1, R2, R3).4To go beyond this bound we\n3The cones method also requires sufficient good transitive symmetry from the complex, which holds for free in the case of the\nspherical buildings we study.\n4We remark that with additional work, this bound may be improved to 2O(r), but we do not know how to break this barrier using\nonly the base case for r= 1.\n8identify another base case, that we explain next, for which we can also directly bound the coboundary\nconstant.\nLetkbe a parameter to be thought of as a small power of r, and consider subsets S1, S2, S3⊆R1∪\nR2∪R3with the following properties:\n1. Equal sizes: we have that |S1|=|S2|=|S3|= 3kand|Si∩Rj|=kfor any i, j∈ {1,2,3}.\n2. Well ordered: we have that max s1∈S1s1<mins2∈S2s2⩽max s′\n2∈S2s′\n2<mins3∈S3s3. In words, all\nelements of S1are smaller than all elements of S2, and all elements of S2are smaller than all elements\nofS3.\nOur extended base case gives good bounds for the coboundary constant of the graph T(S1, S2, S3;µ). To get\nsome intuition, note that when we attempt to construct paths and triangulations in the graph T(S1, S2, S3),\nincidences between vertices depend only on partial information on them. For example, if U∈supp(µS1)\nandV∈supp(µS2)are vertices, then the fact of whether (U, V)is an edge or not depends only on the\nsubspace in Ucorresponding to dimension max s1∈S1s1and the subspace in Vcorresponding to dimension\nmins2∈S2s2. Thus, incidences in this graph depend only on the 4parameters in the second condition above,\nand it thus stands to reason one can extend the triangulation from the case r= 1to the current case.\nLuckily, it turns out there is a surprisingly clean way to go about proving a statement along these lines\nthat only uses the base case r= 1in a black box manner. Indeed, letting i= max s1∈S1s1,j= min s2∈S2s2,\nj′= max s2∈S2s2andℓ= min s3∈S3s3, we show, using an argument similar to the inductive step explained\nbelow, that\nC(T(S1, S2, S3;µ))⩽C(T({i},{j},{ℓ};µ)·max( C1, C2, C3), (1)\nwhere\nC1= max\naC(T(S1\\ {i}, S2, S3;µ|Xi=a)), C 2= max\nbC(T(S1, S2\\ {j}, S3;µ|Xj=b)),\nC3= max\ncC(T(S1, S2, S3\\ {ℓ};µ|Xℓ=c)).\nThe first term on the right hand side of (1) can be bounded by the base case r= 1above. The second term,\nnamely max( C1, C2, C3), is more interesting, and we focus on C1for concreteness. Thinking of µas the\nuniform distribution over top faces of the type C spherical building, one observes that once we condition on\nthe vertex of dimension iin a face, the vertices of dimension strictly less than iand the vertices of dimension\nmore than iform a product structure. In particular, since all elements of S1\\ {i}are smaller than iand all\nelements of S2, S3are bigger than it, we have that for ν=µ|Xi=ait holds that\nν(S1\\{i})∪S2=νS1\\{i}×νS2, ν(S1\\{i})∪S3=νS1\\{i}×νS3.\nThus, the tripartite graph T(S1\\ {i}, S2, S3;ν)is composed of the bipartite graph between S2andS3, and\nthe graph between S1\\ {i}and them is complete. This trivializes the task of constructing triangulations,\nand indeed we show that the coboundary constant C1is dominated by the diameter of the bipartite graph\nbetween S2andS3, which is easily seen to be O(r2)using the separatedness.\nOverall, the extended base case allows us to argue that C(S1, S2, S3;µ)⩽rO(1)even when the sets S1,\nS2andS3are large, so long as they are “well ordered”. The ability to find well ordered subsets is precisely\nthe reason we require R1, R2, R3to be well spread.\nRemark 1.10. We remark that the additional feature of µwe use here is that restricting a coordinate of it\nbreaks µinto a proper product distribution of 2distributions. While this property is easy to see directly for\n9type A and C spherical buildings discussed herein, it is in fact more general for any spherical building. This\nis best seen by looking at the Coxeter diagrams that correspond to a given spherical building, and inspecting\nthat removing a vertex from them (which corresponds to the operation of restriction/ taking a link) either\nturns a type into an easier-to-handle type, or else it disconnects the diagram, in which case the product\nstructure as discussed above emerges.\n1.5.4 The Inductive Approach: the Inductive Step\nArmed with the base case, we are now ready to describe an inductive approach to bound Ct(µ) =Ct,t,t(µ).\nRestrictions and finding good local solutions: Suppose that we have R′\n1⊆R1,R′\n2⊆R2andR′\n3⊆R3\nof size tduring our induction, and choose disjoint well ordered subsets S1, S2, S3ofR′\n1∪R′\n2∪R′\n3that each\ncontains kelements from each one of R1, R2andR3. An easy argument shows that for appropriate choice\nof parameters, so long as t⩾r0.99we are able to find such S1, S2, S3, and we fix such sets henceforth. We\nlater explain how to handle the case t < r0.99.\nChoose restrictions (a1, a2, a3)∼µS1∪S2∪S3. We may consider 3distinct induced graphs on T=\nT(R′\n1, R′\n2, R′\n3)corresponding to these restrictions, which are T1=T(R′\n1\\S1, R′\n2\\S1, R′\n3\\S1|µ|xS1=\na1),T2=T(R′\n1\\S2, R′\n2\\S2, R′\n3\\S2|µ|xS2=a2)andT3=T(R′\n1\\S3, R′\n2\\S3, R′\n3\\S3|µ|xS3=a3).\nEach one of these graphs refers to the induced graph on vertices that agree with the restriction a1,a2anda3\nrespectively, and we may consider the induced affine Unique-Games instance Ψ′\na1Ψ′\na2andΨ′\na3on them. Let\nεa1,εa2andεa3denote the fraction of inconsistent triangles in Ψ′\na1,Ψ′\na2andΨ′\na3respectively, and choose\nXa1Xa2andXa3to assignments that satisfy the maximum fraction of constraints. Thus, by definition we\nget that\nviol(Xa1;T1)⩽Ct−kεa1, viol(Xa2;T2)⩽Ct−kεa2, viol(Xa3;T3)⩽Ct−kεa3.\nSince we are working with affine instances of Unique-Games, once we have one good solution we may\napply affine shifts to it to get a collection of good solutions; in our case, for each π∈Smwe may define\nLa1[π] =Xa1π, and have that La1consists of solutions to Ψa1each satisfying all but Ct−kεa1of the\nconstraints. Similarly, we may define La2andLa3.\nRelating good local solutions: consider now the instance induced by Ψon the graph\nGa1,a2=T(R′\n1, R′\n2, R′\n3|xS1∪S2= (a1, a2)).\nDenote by viol (Xa1;Ga1,a2),viol(Xa2;Ga1,a2)the fraction of constraints violated by Xa1, Xa2inGa1,a2\nrespectively. From the point of view of the restriction S1, a1, this is a randomly chosen induced sub-instance\nofT1, and hence we expect that viol (Xa1;Ga1,a2)≲viol(Xa1), and analogously viol (Xa2;Ga1,a2)≲\nviol(Xa2). Using the ε-productness of µ, the graph Ga1,a2has second singular value bounded away from 1,\nand therefore any two good solutions to an Affine Unique-Games over it must be the same up to an affine\nshift (we remark that this is an idea whose origin is algorithms for solving affine Unique-Games over some\nspecial classes of graphs [BBK+21, BM23b]). Thus, we may find a shift πa1,a2∈Smthat nearly forms a\nmatching between the lists La1, La2. More precisely, we are able to show that\nE\na1,a2\u0014\nPr\nu∈Ga1,a2[Xa1(u)̸=πa1,a2Xa2(u)]\u0015\n≲E\na1,a2[viol(Xa1;Ga1,a2) +viol(Xa2;Ga1,a2)]≲Ct−kδ.\n10Using our observations so far, we may consider the tripartite graph over restrictions, whose triangles\ncorrespond to a triplet of restrictions a1, a2, a3that are valid under µ. We consider an Affine Unique-Games\ninstance Ψrestrict on it, that has the constraint π−1\na1,a2on the edge (a1, a2), and the goal is to choose labels\nfromSmto each vertex so as to satisfy as many of the constraints as possible. Using our arguments so far, we\ncan argue that the fraction of inconsistent triangles is at most η=O(Ct−kδ), and as the underlying graph\nofΨrestrict isT(S1, S2, S3)we are able to conclude that we may find a solution to Ψrestrict that satisfies\nall but O(CkCt−kδ)of constraints. Going in this route, one can conclude the recursion Ct≲CkCt−k,\nwhich ultimately gives Ct⩽2O(tlogr). This is, of course, insufficient, and the reason for why we require\nthe extended base case. Using it, we have that C(T(S1, S2, S3))⩽rO(1)giving us the recursion Ct⩽\nrO(1)Ct−kso long as t⩾r0.99. Iterating, this gives\nCr⩽rr/kCr0.99⩽rr/k2O(r0.99logr)⩽2O(r1−c)\nfor some absolute constant c >0.\nTo complete the overall picture, we now explain how we lift good solutions to Ψrestrict to a good solution\nofΨ′. Suppose that Ais an assignment to Ψrestrict satisfying 1−ηfraction of constraints, and let Ube a\nvertex in Ψ′, sayU∈supp(µR′\n1)without loss of generality. The idea is to ask the opinion of a random vertex\ninS1that is consistent with U, and assign Uaccordingly. More precisely, we sample a1∼µ|xR′\n1=U,\nand assign B[U] =A[a1](U). Indeed, using standard spectral arguments, we show that for typical U, the\nvalue of A[a1](U)fora1chosen in this way is almost fixed, and furthermore that the assignment Bsatisfies\nall but ηfraction of constraints in Ψ′.\n2 Preliminaries\nNotations: We use standard big- Onotations: we denote A=O(B)orA≲BifA⩽c·Bfor some\nabsolute constant c >0. Similarly, we denote A= Ω( B)orA≳BifA⩾cBfor some absolute constant\nc >0. We also denote k≪dto denote the fact that dis taken to be sufficiently large compared to any\nfunction of k. For a distribution µoverX1×. . .×Xrand a subset S⊆[r], we denote by µSthe marginal\ndistribution of µon the coordinates of S. We denote by supp(µ)the support of µ, and for a subset S⊆[r]\nand an assignment V∈Q\ni∈SXiwe denote by µ|XS=Vthe distribution of X∼µconditioned on\nXS=V. IfAis a finite set and i⩽|A|, the notation B⊆iAmeans that we sample a subset of size iofA\nuniformly.\n2.1 Graphs Associated with Distributions and ε-product Distributions\nLetµoverX1×. . .×Xr. The following definition describes bipartite graphs that can be associated with µ:\nDefinition 2.1. Letµbe a distribution on X1×. . .×Xr. Let L, R⊆[r]be two disjoint non-empty sets.\nLetA(L, R;µ)be the bipartite graph produced as follows: the vertices are supp(µL)and supp(µR)and\nto sample an edge we choose a sample Xfrom µ, and output (XL, XR). We will let AL,Rdenote the\ncorresponding operator AL,R:L2(XL, µL)→L2(XR, µR), with AL,Rf(v) =Ew∼µL|XR=v[f(w)].\nSimilarly, one can create tripartite graphs from µ:\nDefinition 2.2. Letµbe a distribution on X1×. . .×Xr. LetS1, S2, S3⊆[r]be three disjoint non-empty\nsets. Let T(S1, S2, S3;µ)be the tripartite graph produced as follows: the vertices are supp(µS1),supp(µS2)\n11andsupp(µS3), and to sample an edge we choose a sample Xfromµ, and with probability 1/3each output\n(XSi, XSj)fori̸=j, i, j∈[3].\nNote that when one of the Si’s is∅, sayS1, the graph T(∅, S2, S3;µ)denotes the tripartite graph with\none vertex ∅in its first part, supp(µS2|a),supp(µS3|a)in its second and third parts, and we sample an edge\nby sampling X∼µand and outputting the pairs (∅, XS2),(∅, XS3)or(XS2, XS3)with equal probability.\nWhen more of the Si’s are∅we can similarly create these tripartite graphs.\nWe now discuss the definition of ε-product distributions from [GLL22]. We begin by defining ε-\npseudorandom distributions.\nDefinition 2.3. We say that a distribution DoverX1×X2isε-pseudorandom if the second largest singular\nvalue of A{1},{2}is at most ε.\nNext, we define the notion of having ε-pseudorandom skeletons.\nDefinition 2.4. We say that a distribution DoverY1×. . .×Ythasε-pseudorandom skeletons if for all\ni̸=j∈[t], the marginal distribution D{i,j}isε-pseudorandom.\nWe are now ready to define the notion of ε-product distributions.\nDefinition 2.5. We say that µis an ε-product distribution over X1×. . .×Xrif for all S⊆[r]of size at\nmostr−2and all V∈supp(µS), the conditional distribution µ|XS=Vhasε-pseudorandom skeletons.\nMany properties of ε-product distributions were established in [GLL22], and we will require a few of\nthem. In particular, we will need [GLL22, Lemma 3.3], which asserts that if L, R⊆[r]are disjoint and µis\nanε-product distribution, then the second singular values of the bipartite graphs A(L, R;µ)are small.\nLemma 2.6. Letµbe an ε-product distribution over X1×. . .×Xrand let L, R⊆[r]be two disjoint sets.\nThen the second largest singular value of AL,RandAR,Lis at most poly( r)ε.\n2.2 Properties of Expanders\nWe need the following well known version of the expander mixing lemma for bipartite graphs.\nLemma 2.7. LetG= (U, V, E )be a bipartite graph in which the second singular value of the normalized\nadjacency matrix is at most λ. Then for all A⊂UandB⊂Vwe have that\n\f\f\f\fPr\n(u,v)∈E[u∈A, v∈B]−µ(A)µ(B)\f\f\f\f⩽λp\nµ(A)(1−µ(A))µ(B)(1−µ(B)).\nWe also use the following standard sampling property of bipartite expanders.\nLemma 2.8. LetG= (U, V, E )be a weighted bipartite graph with second singular value at most λ. Let\nB⊂Uand set T={v∈V|Pru∼v[u∈B]> ε+ Pr[ B]}. Then Pr[T]⩽λ2δ/ε2.\n2.3 Properties of Local Spectral Expanders\nRecall that we associated with each d-dimensional simplicial complex Xa sequence of measures {µk}1⩽k⩽d,\nwhere µkis a probability measure over X(k). Note that for all 0⩽t⩽r⩽d, a sample according to µt\ncan be drawn by first sampling R∼µr, and then sampling T⊆tRuniformly. The converse is also true:\na sample from µrcan be drawn by first sampling T∼µt, and then sampling Rfrom µrconditioned on\ncontaining T. These observations give rise to the standard “up” and “down” operators, which we present\nnext. We only mention a few of their properties that are necessary for our arguments, and refer the reader\nto [DDFH18] for a more comprehensive exposition.\n12Definition 2.9. The operator Ui+1\niis a map from L2(X(i);µi)toL2(X(i+ 1); µi+1)defined as\nUi+1\nif(u) =E\nv⊂iu\u0002\nf(v)\u0003\nfor all u∈X(i+1). Forj⩾k+1, we define Uj\nkvia composition of up operators: Uj\nk=Uj\nj−1◦. . .◦Uk+1\nk.\nDefinition 2.10. The operator Di+1\niis a map from L2(X(i+ 1); µi+1)toL2(X(i);µi)defined as\nDi+1\nif(u) = E\nv⊇i+1u\u0002\nf(v)\u0003\nfor all u∈X(i). Forj⩾k+1, we define Dj\nkvia composition of down operators: Dj\nk=Dk+1\nk◦. . .◦Dj\nj−1.\nAbusing notations, we use the notations Uj\nk, Dj\nkto denote the operators, as well as the real valued\nmatrices associated with them. A key property of the down and up operators is that they are adjoint:\nClaim 2.11. For all k⩽j⩽d,Uj\nkandDj\nkare adjoint operators: for all functions f:X(k)→Rand\ng:X(j)→Rit holds that ⟨Uj\nkf, g⟩=⟨f, Dj\nkg⟩.\nWe need the following result due to [Opp18] known as the trickling-down theorem that uses the eigen-\nvalues of links at X(d−2)to show that Xis a one-sided local spectral expander.\nTheorem 2.12. LetXbe ad-dimensional simplicial complex such that the 1-skeleton of every link (includ-\ning the empty one) is connected and for all I∈X(d−2), the 1-skeleton of Ihas second eigenvalue at most\nλ. Then Xis aλ\n1−(d−1)λ-one-sided local spectral expander.\nWe need the following lemma regarding the second eigenvalue of the down-up walks Uj\nkDj\nkonX(j)\n(j⩾k), that can be found in [AL20].\nLemma 2.13. Let(X, µ)be a d-dimensional γone-sided local spectral expander. For all i⩽dand\nα∈(1/i,1), the largest singular value of Ui\nαiandDi\nαiis at most√α+ poly( i)γ. Thus the down-up\nrandom walk Ui\nαiDi\nαionX(i)has second largest singular value at most α+ poly( i)γ.\n2.4 Spherical Buildings of Type A\nAnalyzing the Chapman-Lubotzky complex will require us to study its links. In this section and in the next\none, we present the spherical buildings of type A and of type C, which are morally the graphs we will end\nup needing to study.\nDefinition 2.14. The spherical building of type A over Fd+1\nqis ad-dimensional complex denoted by SBA\nd(Fq)\nwith the set of maximal faces:\n{(V1, . . . , V d) :V1⊂. . .⊂Vd, Vi⊂iFd+1\nq}.\nThed-faces are equipped with the uniform distribution which we also denote by SBA\nd(Fq).\nThe following is a well-known fact, but we give the proof here for completeness.\nLemma 2.15. The distribution SBA\nd(Fq)is aO(1/√q)-product distribution.\n13Proof. Letµ=SBA\nd(Fq). Take any set S⊆[d],|S|⩽r−2and a valid restriction of it, say V. We\nneed to show that µ|XS=Vhas1/q-pseudorandom skeletons. Consider two coordinates i̸=j∈[d]\\S\nand let Ai,jbe the bipartite graph/normalized adjacency operator corresponding to µ{i,j}|XS=V. If there\nexists k∈Ssuch that i < k < j , then Ai,jhas second largest singular value 0. So let us consider the\ncase where there is no such coordinate kbetween iandj. Let i′be the largest coordinate in Sthat is less\nthaniandj′be the smallest coordinate in Sthat is greater than j(i′< i < j < j′). Then the bipartite\ngraph Ai,jis isomorphic to the weighted inclusion graph between i−i′andj−i′-dimensional subspaces of\nFj′−i′\nq. It is well-known (see for example [BCN12, GM16]) that this graph has second largest singular value\n≲1/p\nqj−i⩽1/√q. This shows that µ|XS=Vhas1/√q-pseudorandom skeletons for all SandV, thus\nshowing that µis a1/√q-product distribution.\n2.5 Spherical Buildings of Type C\nNext, we present the spherical buildings of type C. Like the spherical buildings of type A, type Cspherical\nbuildings are too defined using subspaces. However, we only consider subspaces that are isotropic with\nrespect to a symplectic form .\nDefinition 2.16. A symplectic bilinear form is a mapping ω:F2n×F2n→Fis a map which bi-linear,\nanti-symmetric – ω(v, w) =−ω(w, v),∀v, w∈F2nand non-degenerate – ω(u, v) = 0 for all vimplies\nu= 0.\nIn this paper we fix the symplectic form:\nω=\u00120In\n−In0\u0013\n,\nwhich gives the bi-linear form ω(v, w) =Pn\ni=1viwn+i−wivn+i. One can check that ω(·,·)is a valid\nsymplectic bilinear form.\nDefinition 2.17. A subspace V⊆F2d\nqis called isotropic if ω(v, w) = 0 for all v, w∈V.\nDefinition 2.18. The symplectic group Sp (2d, F)is defined as the set of 2d×2dmatrices Mover a field\nFthat preserve the symplectic form ω. This group is characterized by matrices in GL 2d(F)that satisfy\nMTωM=ω, where ωis the matrix representation of the symplectic form.\nProperties of SP 2d(F):For an invertible matrix Mand subspace V⊆F2d, letM◦Vdenote the subspace\nspan(Mv)v∈V. Then the group SP 2d(F)has the following properties:\n1. IfVis at-dimensional isotropic subspace of F2d\nq, then M◦Vis also a t-dimensional isotropic subspace\nofF2d\nq.\n2. Furthermore, SP 2d(F)acts transitively on the set of t-dimensional isotropic subspaces for all t⩽d.\nDefinition 2.19. The spherical buildings of type C over F2d\nqis ad-dimensional complex denoted by SBC\nd(Fq)\nwith the set of maximal faces:\n{(V1, . . . , V d) :V1⊂. . .⊂Vd, Viis an isotropic subspace of dimension i}.\nThed-faces are equipped with the uniform distribution which we also denote by SBC\nd(Fq).\n14The following lemma asserts that the natural distribution associated with the spherical building of type\nCisε-product for a small ε.\nLemma 2.20. Letd, q∈Nwithq⩾poly( d). Then µ=SBC\nd(Fq)is aO(1/√q)-product distribution.\nProof. Fixε=O(1/√q). One can check that proving µis aλ-product distribution is equivalent to showing\nthatSBC\nd(Fq)denoted by Xis aλ-one-sided local spectral expander, and we focus on the latter task. To\ndo so, it suffices to show that the 1-skeleton of all links I∈X(d−2)have second eigenvalue at most ε.\nOnce we show that, the trickling-down theorem, Theorem 2.12, implies that Xis aε\n1−dε≲εone-sided\nlocal spectral expander.\nFix a (d−2)-sized link I, and say that it fixes the set of coordinates S⊂[d]of size d−2to the isotropic\nsubspaces Va, a∈S, that form a valid inclusion chain. Let i < j∈[d]be the two unfixed coordinates and\nletDbe the resulting conditional distribution on X{i,j}. The second largest eigenvalue of the 1-skeleton of\nIis equal to the second largest singular value of A({i},{j};D), since the 1-skeleton is a bipartite graph.\nWe bound the latter using case analysis.\nCoordinates i, jare not consecutive: In this case there exists k∈Ssuch that i < k < j . So we know\nthatDis a product distribution and A({i},{j};D)is the complete bipartite graph, hence the corresponding\nsecond largest singular value is 0.\nCoordinates i, jare consecutive but not equal to (d−1, d):In this case, j=i+ 1and the coordinates\ni+2andi−1belong to S. Then the resulting bipartite graph is over the set of isotropic subspaces contained\nwithin Vi+1and containing Vi−1. Since Vi+1is an isotropic subspace and every subspace within an isotropic\nsubspace is isotropic, we get that A({i},{j};D)is isomorphic to the bipartite weighted inclusion graph\nbetween 1and2-dimensional subspaces of F3\nq. This has largest singular value ⩽εas we saw in Lemma 2.15.\nCoordinates (i, j) = ( d−1, d):In this case the coordinate (d−2)belongs to S. The set of d−1and\nd-dimensional isotropic subspaces containing Vd−2is in one-to-one correspondence with the set of 1and\n2-dimensional subspaces respectively, that are symplectically orthogonal to Vd−2and are not contained in\nVd−2. Taking a quotient by Vd−2we get that this is the set of 1and2-dimensional isotropic subspaces of\nF4\nq. Therefore A({i},{j};D)is isomorphic to the bipartite inclusion graph between 1and2-dimensional\nisotropic subspaces of F4\nq. Using [BCN12, Theorem 9.4.3], we get that this graph has second largest singular\nvalue≲εas required.\n3 A Local to Global Theorem for Coboundary Expansion\nThe primary goal of this section is to present an inductive approach to prove upper bounds on the coboundary\nconstants corresponding to the level rfaces of a simplicial complex X. Roughly speaking, starting with an\ninitial assumption regarding the coboundary constant of Xon constant levels, we show how to lift it to a\nreasonable bound on higher levels.\n3.1 Tools\n3.1.1 Basic Notions and Properties of the Tripartite Graph T(R1, R2, R3)\nIn this section, we present some of the tools and notions that are necessary for our proof. Throughout this\nsection, we fix a set of indices I={i1, . . . , i 3r}and an ε-product distribution µoverQ\ni∈IXi. Here and\n15throughout, εshould be thought of as very small compared to all other parameters, and we encourage the\nreader to think of ε= 0at first reading.\nWe begin by formally defining the notion of coboundary expansion (with additive error) for measures.\nDefinition 3.1. Letµbe a measure overQ\ni∈IXiand let 0⩽r1, r2, r3⩽r∈Nbe integers. We say that µ\nis a(Cr1,r2,r3, βr1,r2,r3)-coboundary expander over Smif for all sets Sof size at most 3r−(r1+r2+r3),\nfor all restrictions a∈supp(µS), and for all disjoint R1, R2, R3⊆I\\Sof sizes r1, r2, r3respectively, the\ntripartite graph T(R1, R2, R3;µ|XS=a)with respect to the distribution µ∪Ri|(XS=a)over triangles, is\na(Cr1,r2,r3, βr1,r2,r3)-coboundary expander over Smas per Definition 1.9.\nWhen βr1,r2,r3= 0, we simply say that µis aCr1,r2,r3-coboundary expander over Sm.\nDefinition 3.1 will be of central interest to us, and to use it we must develop some tools to investi-\ngate coboundary expansion in the tripartite graphs T(R1, R2, R3;µ). We begin with the following claim,\nasserting that this graph always has second singular value bounded away from 1.\nClaim 3.2. Letµbe an ε-product distribution over X1×. . .×Xr, and let S1, S2, S3⊆[r]be three disjoint\nnon-empty sets. Let Tbe the normalized adjacency operator of the graph T(S1, S2, S3;µ). Then the second\nlargest singular value of Tis at most 1/2 + poly( r)ε.\nProof. LetTbe the normalized adjacency operator of T(S1, S2, S3;µ)and let fbe the second eigenvector\nofTwithE[f] = 0 and∥f∥2= 1. We will now bound ∥Tf∥2.\nFor any i= 1,2,3, letfibe the function frestricted to XSithought of as an element in L2(XSi;µSi);\nwe also denote ai=E[fi] =|Ex∼µSi[f(x)]|. We have that for all x∈S1,\nTf(x) =1\n2AS2,S1f(x) +1\n2AS3,S1f(x),\nand similarly for x∈S2orS3. Similarly, we denote by (Tf)ithe restriction of TftoXSi. With these\nnotations, we have that\n∥(Tf)1∥2⩽1\n2∥AS2,S1f2∥2+1\n2∥AS3,S1f3∥2⩽1\n2(a2+ poly( r)ε) +1\n2(a3+ poly( r)ε),\nwhere in the last transition we used Claim 2.6 to bound the second singular value of AS2,S1andAS3,S1to\nbound ∥AS2,S1f2−a2∥2=∥AS2,S1(f2−a2)∥2⩽poly(r)ε. Squaring and simplifying gives us that\n∥(Tf)1∥2\n2⩽1\n4(a2+a3)2+ poly( r)ε.\nAsa1+a2+a3= 3E[f] = 0 , we conclude that ∥(Tf)1∥2\n2⩽1\n4a2\n1+poly(r)ε, and similarly ∥(Tf)2∥2\n2⩽\n1\n4a2\n2+poly(r)εand∥(Tf)3∥2\n2⩽1\n4a2\n3+poly(r)ε. Multiplying by 1/3and summing up we get\n∥Tf∥2\n2=E\ni∈[3][∥(Tf)i∥2\n2]⩽1\n12(a2\n1+a2\n2+a2\n3) + poly( r)ε\n=1\n12(E[f1]2+E[f2]2+E[f3]2) + poly( r)ε\n⩽1\n12(E[f2\n1] +E[f2\n2]2+E[f2\n3]) + poly( r)ε\n=1\n4∥f∥2\n2+ poly( r)ε,\nwhich is at most1\n4+poly(r)εas∥f∥2= 1. Taking a square root gives that ∥Tf∥2⩽1/2 + poly( r)ε.\n163.1.2 Almost Uniqueness of Good Solutions to Affine UG Instances\nWe begin by defining shifts of assignments to Affine Unique-Games instances G.\nDefinition 3.3. Given an affine Unique-Games instance Φwith alphabet Sm, an assignment Ato it and\nπ∈Sm, we denote by A◦πdenote the assignment that (A◦π)(v) =X(v)π.\nSuppose Φis an instance of affine Unique-Games as in Definition 1.4, and suppose that Ais an assign-\nment to it. It can easily be seen that val(A) = val( A◦π)for every π∈Sm. Thus, if Asatisfies many of\nthe constraints of Ψ, then so does A◦π. The following claim says that if the underlying constraint graph is\nan expander, then all good assumptions are essentially shifts of one good assignment X. This idea appeared\nin [BBK+21] in the context of the Abelian version of affine Unique-Games, but the proof in the non-Abelian\ncase is essentially the same and we give it for completeness.\nClaim 3.4. LetΦ = ( G,Π, Sm)be a UG instance on Gwhich is an expander graph with second eigenvalue\nλ, and let X, X′∈SV(G)\nm be two solutions to Φ. Then there exists a permutation π∈Smsuch that\nPr\nv∈G[X(v)̸=X′(v)π]⩽viol(X) +viol(X′)\n1−λ.\nProof. FixXandX′as in the statement of the claim, and partition V(G) =∪π∈SmCπwhere for each\nπ∈Smwe define Cπ={u∈G:X(v) =X′(v)π}. Note that if an edge (u, v)∈Gis satisfied by both X\nandX′, then its endpoints lie in the same part Cπ. Indeed, suppose that X(v) =X′(v)π, then\nX(u) =πu,vX(v) =πu,vX′(v)π=X′(u)π.\nThus, if (u, v)goes across distinct parts of the partition {Cπ}π∈Sm, then it must be violated either by Xor\nbyX′.\nFor a set of vertices Sletµ(S)denote the measure of SinV(G)and for a set of edges Tletµ(T)denote\nthe measure of TinE(G). Let E(S,S)denote the set of edges that cross between Sand its complement.\nBy (the easy direction of) Cheeger’s inequality, for each π∈Smwe have that\n1\n2µ(E(Cπ,Cπ))⩾(1−λ)µ(Cπ)(1−µ(Cπ)).\nSumming this over π∈Smwe get,\n1\n2X\nπ∈Smµ(E(Cπ,Cπ))⩾(1−λ)(1−X\nπµ(Cπ)2).\nNote that each edge that crosses between distinct parts of {Cπ}π∈Smis counted twice on the left side, and\nby our earlier observation it must be violated either by Xor by X′, and so1\n2µ(E(Cπ,Cπ))⩽viol(X) +\nviol(X′). Plugging this and simplifying gives that\nX\nπµ(Cπ)2⩾1−viol(X) +viol(X′)\n1−λ,\nand asP\nπ∈Smµ(Cπ)) = 1 , it follows that there exists π∈Smsuch that µ(Cπ)⩾1−viol(X)+viol(X′)\n1−λ.\n173.2 Exponential Bound on the Coboundary Constant via Lopsided Induction\nOur bounds on the coboundary constants Cr1,r2,r3will always follow an inductive strategy. All of our\ninductive arguments will be of similar spirit, though they differ in some technical aspects that tailor them\nfor different uses. This section is devoted for the most simplistic of these inductive approaches, for which\nwe have two utilities. First, it will be useful for us to handle r1, r2, r3that are relatively small (e.g. r0.99).\nSecondly, it will be useful for us when we extend the base case into the extended base case as described the\nintroduction. In this case, we have the following result:\nLemma 3.5. There exists an absolute constant K > 0such that the following holds. Suppose that µis an\nε-product measure with ε⩽r−Kwhich is a (C1,1,1(µ), β1,1,1(µ))-coboundary expander over Sm. Then for\nallk1, k2, k3∈Nsuch that k1+k2+k3⩽r,µis a(Ck1,k2,k3(µ), βk1,k2,k3(µ))-coboundary expander over\nSmwith\nCk1,k2,k3(µ)⩽O(C1,1,1(µ))k1+k2+k3, β k1,k2,k3(µ)⩽O(C1,1,1(µ))k1+k2+k3β1,1,1(µ).\nProof. Fix any three pairwise disjoint sets R1, R2, R3with sizes 0< r 1, r2, r3∈Nsuch thatPri⩽\nr, indices j1∈R1, j2∈R2andj3∈R3, a set S⊂[d]\\SRi, and a restriction A0∈supp(µS).\nLetDdenote µ|XS=A0. For any restriction a1∈supp(Dj1), letGa1denote the induced subgraph\nT(R1\\j1, R2, R3;D|Xj1=a1)and similarly define the graphs Ga2, Ga3for every a2∈supp(Dj2), a3∈\nsupp(Dj3). Then we will show that,\nC(T(R1, R2, R3;D))⩽C(T({j1},{j2},{j3};D))·max\ni∈[3]\nai∈supp(Dji)(C(Gai)). (2)\nβ(T(R1, R2, R3;D))⩽C(T({j1},{j2},{j3};D))·max\ni∈[3]\nai∈supp(Dji)(β(Gai)) +β(T({j1},{j2},{j3};D)).\n(3)\nAdditionally, when either of the ri’s are 0we show that C(T(R1, R2, R3;D))⩽1, as well as that\nβ(T(R1, R2, R3;D)) = 0 . This along with the equations above is enough to conclude the lemma as we\nshow in the end of the proof.\nTo prove (2) and (3) fix any affine UG instance ΦonT(R1, R2, R3;D)which has δ-fraction of incon-\nsistent triangles.\nSetting up lists on restrictions: For every ai∈supp(Dji)letXai∈SV(Gai)\nm be an assignment on V(Gai)\nwith maximum value, and let Lji→aibe an ordered list of solutions indexed by permutations in Sm, where\nLSi→ai[π] =Xai◦π.\nSetting up permutations between restrictions: For every two restrictions of different parts, say aofj1\nandbofj2where (a, b)is a valid restriction (that is, where (a, b)∈supp(D{j1,j2})), letGa,bbe the induced\nsubgraph T(R1, R2, R3;D|(Xj1=a, X j2=b)). Then by Claim 3.4 there exists a permutation πa,bthat\nsatisfies the following:\nPr\nv∼Ga,b[Xa(v)̸=Xb(v)πa,b]⩽viol(Xa;Ga,b) +viol(Xb;Ga,b)\n1−λ(Ga,b)≲viol(Xa;Ga,b) +viol(Xb;Ga,b),(4)\n18where we used Claim 3.2 to bound λ(Ga,b)⩽1/2 + poly( r)ε⩽0.51. Let\nBad(a, b) ={v∈Ga,b|Xa(v)̸=Xb(v)πa,b}.\nFor any restriction bas above, let Lb◦πdenote the list where Lb◦π[π′] =Lb[ππ′] =Xb◦ππ′. With these\nnotations and the definition of Bad (a, b), we get that La|Ga,b∩Bad(a,b)=Lb◦πa,b|Ga,b∩Bad(a,b).\nCounting bad triangles: For any restriction (a1, a2, a3)∈supp(D{j1,j2,j3})letGa1,a2,a3denote the graph\nT(R1\\j1, R2\\j2, R3\\j3;D|(a1, a2, a3)). Clearly, we have that,\nE\na3∼Dj3|(a1,a2)[µGa1,a2,a3(Bad(a1, a2))] = µGa1,a2(Bad(a1, a2)).\nFixa1, a2. Using Markov’s inequality and (4) it follows that\nPr\na3∼Dj3|(a1,a2)\u0014\nµGa1,a2,a3(Bad(a1, a2))>1\n100\u0015\n≲viol(Xa1;Ga1,a2) +viol(Xa2;Ga1,a2). (5)\nWe say a restriction (a1, a2, a3)is bad if µGa1,a2,a3(Bad(ai, ak))>1/100for some i̸=k. By (5) we get\nE\n(a1,a2,a3)∼D∪ji[I((a1, a2, a3)is bad )]\n⩽X\ni∈[3]E\n(aℓ,ak)∼Djℓ∪jk\nℓ,k̸=iE\nai∼Dji|(aℓ,ak)[I(µGa1,a2,a3(Bad(aℓ, ak))>1\n100)]\n≲X\ni∈[3]E\n(aℓ,ak)∼Djℓ∪jk\nℓ,k̸=i[viol(Xaℓ;Gaℓ,ak) +viol(Xak;Gaℓ,ak)]\n≲X\ni∈[3]E\nai∼Dji[viol(Xai;Gai)], (6)\nwhere in the first inequality we used the union bound, the second inequality we used (5). By definition we\nhave that viol (Xa1;Ga1)⩽C(Ga1)εa1+β(Ga1), where εa1is the fraction of violated triangles in Ga1.\nTherefore,\n(6)≲X\ni∈[3]E\nai∼Dji[C(Gai)εai+β(Gai)]≲ max\ni∈[3]\nai∈supp(Dji)(C(Gai))δ+ max\ni∈[3]\nai∈supp(Dji)(β(Gai)) := δ′.\nCreating a UG instance on graph over restrictions: LetΨbe the following UG instance on H=\nT({j1},{j2},{j3};D): each edge (ai, ak)has the constraint π−1\nai,ak. That is, we want to find a solution A\nthat maximizes the fraction of edges satisfying A(ai) =π−1\nai,akA(ak). Note that a triangle (a1, a2, a3)is\nconsistent if π−1\na3,a1π−1\na1,a2π−1\na2,a3=id or equivalently if πa2,a3πa1,a2πa3,a1=id.\nFirst note that if a triangle (a1, a2, a3)is not bad, then it is consistent. To see this, note that in this case,\nby the union bound we have µGa1,a2,a3(Bad(a1, a2)∪Bad(a2, a3)∪Bad(a1, a3))⩽0.03, which gives that\nµGa1,a2,a3(Good (a1, a2, a3))⩾0.97>0, where the latter set is defined as those vertices in Ga1,a2,a3that\ndon’t belong to any of Bad (ai, ak). We have that:\nLa1|Good(a1,a2,a3)=La2◦πa1,a2|Good(a1,a2,a3)\n19La2|Good(a1,a2,a3)=La3◦πa2,a3|Good(a1,a2,a3)\nLa3|Good(a1,a2,a3)=La1◦πa3,a1|Good(a1,a2,a3),\nwhich implies that:\nLa3|Good(a1,a2,a3)=La3◦πa2,a3◦πa1,a2◦πa3,a1|Good(a1,a2,a3)=La3◦πa2,a3πa1,a2πa3,a1|Good(a1,a2,a3).\nTaking any v∈Good (a1, a2, a3)we get that, La3[id][v] =La3◦πa2,a3πa1,a2πa3,a1[id][v], where the former\nisXa3(v)and the latter is Xa3(v)πa2,a3πa1,a2πa3,a1, which implies πa2,a3πa1,a2πa3,a1=id.\nUsing the coboundary expansion of the graph T({j1},{j2},{j3};D)we get that there exists a UG\nsolution to the vertices violating at most (C(H)δ′+β(H))-fraction of the edges. Call this solution A.\nLifting the solution: We will now use Ato create a highly satisfying solution BtoG=T(R1, R2, R3;D).\nTo each vertex u∈Ri, define B(u) =Lu|ji[A(u|ji)][u]. We will now upper bound the fraction of edges\nthatBviolates for Φ.\nConsider an edge (ui, uk)∈Gbetween parts Ri, Rk, and let (ai, ak)denote the edge (ui|ji, uk|jk)in\nthe restriction graph H. Let Good (G)be the set of edges where the following three events hold: (1) A\nsatisfies the edge (ai, ak), i.e. A(ai) =π−1\nai,akA(ak), (2)Xaisatisfies the edge (ui, uk), i.e. Xai(ui) =\nπui,ukXai(uk)and (3) ukdoes not belong to Bad (ai, ak), i.e. Xai(uk) =Xak(uk)πai,ak. It is easy to see\nthatBsatisfies every good edge, that is, B(ui) =πui,ukB(uk). Indeed, using the events (1), (2) and (3), we\nhave that,\nB(ui) =Lai[A(ai)][ui] =Xai(ui)A(ai)\n=Xai(ui)π−1\nai,akA(ak) by (1)\n=πui,ukXai(uk)π−1\nai,akA(ak) by (2)\n=πui,ukXak(uk)πai,akπ−1\nai,akA(ak) by (3)\n=πui,ukB(uk).\nSo it suffices to upper bound the fraction of edges that are not in Good (G), which we denote by Bad (G).\nSampling an edge, we can upper bound the probability it doesn’t satisfy at least one of the events. For (1),\nPr\n(ui,uk)∈G[A(ai)̸=π−1\nai,akA(ak)] = viol(A)≲C(H)δ′+β(H).\nFor (2),\nPr\n(ui,uk)∈G[Xai(ui)̸=πui,ukXai(uk)] = E\ni∼[3]\nai∼DjiE\n(u1,u2,u3)∼D|(Xji=ai)\nk∼[3]\\i[I(Xai(ui)̸=πui,ukXai(uk))]\n≲E\ni∼[3]\nai∼Dji[viol(Xai;Gai)]≲δ′.\nFor (3), using (4) we have\nPr\n(ui,uk)∈G[uk̸∈Bad(ai, ak)]≲ E\n(ai,ak)∼Dji∪jkE\n(ui,uk,uℓ)∼D|(ai,ak)\nr∼{i,k}[I(ur̸∈Bad(ai, ak))]\n≲ E\n(ai,ak)∼Dji∪jk[viol(Xai;Gai,ak) +viol(Xak;Gai,ak)]\n20≲E\nai∼Dji[viol(Xai;Gai)]\n≲δ′.\nAdding up these probabilities we get,\nPr\n(ui,uk)∈G[(ui, uk)∈Bad(G)]≲δ′+C(H)δ′+β(H)\n≲\nC(H) max\ni∈[3]\nai∈supp(Dji)(C(Gai))\nδ+\nC(H) max\ni∈[3]\nai∈supp(Dji)(β(Gai)) +β(H)\n,\nwhich completes the proof of (2) and (3).\nProving base cases when ri’s are 0: Without loss of generality assume that |R1|= 0. Recall that the\ngraph G=T(R1, R2, R3;D)has one vertex ∅in its first part that is connected to all the vertices supp(µR2)\nandsupp(µR3)in the second and third parts named P2, P3. We can show that C(G)⩽1via the following\nalgorithm to get a satisfying solution to Φ. We assign the identity permutation to ∅, and then propagate this\nsolution to all the vertices in parts P2andP3. Now note that by definition all the edges between ∅andP2\norP3are satisfied. For an edge (u, v)between P2, P3, it is violated only if the triangle (∅, u, v)is violated\nwhich happens only with probability δ. Therefore we get a solution violating at most δ/3-fraction of the\nedges of G, which shows that C(G)⩽1. The same argument holds when more of the Ri’s have size 0.\nConcluding the induction: Firstly note that (2), (3) imply that:\nCr1,r2,r3(µ)≲C1,1,1(µ)C′, β r1,r2,r3(µ)≲C1,1,1(µ)β′+β1,1,1(µ) (7)\nwhere we take C′= max( Cr1−1,r2,r3(µ), Cr1,r2−1,r3(µ), Cr1,r2,r3−1(µ))and analogously pick β′to be\nβ′= max( βr1−1,r2,r3(µ), βr1,r2−1,r3(µ), βr1,r2,r3−1(µ)).\nTo prove the final statement we can then use induction onP3\ni=1ri. The base case of induction is when\neitherPri= 3 and each ri= 1, or one of the ri’s is0. In the first case Cr1,r2,r3(µ)⩽C1,1,1(µ)⩽\nO(C1,1,1(µ))3,βr1,r2,r3(µ)⩽β1,1,1(µ)⩽O(C1,1,1(µ))3β1,1,1(µ)and in the second case Cr1,r2,r3⩽1⩽\nO(C1,1,1(µ))Pri, βr1,r2,r3= 0⩽C1,1,1(µ)Priβ1,1,1(µ)too. Therefore now let us assume the lemma\nstatement holds for all (r1, r2, r3)withPri⩽tand let us prove it for (r′\n1, r′\n2, r′\n3)withPr′\ni=t+ 1and\neachr′\ni>0. Applying (7) on Cr′\n1,r′\n2,r′\n3(µ)we get that,\nCr′\n1,r′\n2,r′\n3(µ)⩽O(C1,1,1(µ))C′⩽O(C1,1,1(µ))O(C1,1,1(µ))r′\n1+r′\n2+r′\n3−1=O(C1,1,1(µ))r′\n1+r′\n2+r′\n3,\nwhere in the second inequality we used the inductive bound on Cr′\n1−1,r′\n2,r′\n3(µ), Cr′\n1,r′\n2−1,r′\n3(µ), Cr′\n1,r′\n2,r′\n3−1(µ).\nA similar argument for βr′\n1,r′\n2,r′\n3(µ)completes the induction thus proving the lemma.\n3.3 Subexponential Bounds via Non-Lopsided Induction\nThroughout this section we fix k=r0.01, three r-sized pairwise disjoint sets R1, R2, R3⊂[d],I=S\ni∈[3]RiandµoverQ\ni∈IXithat satisfy the following assumptions:\nAssumption 1. The following conditions hold:\n211. The measure µis anε-product distribution with ε⩽2−r12.\n2. For all k⩽r,S⊆Iof size at most 3r−3kand restrictions a0∈supp(µS), for all k-sized pairwise\ndisjoint sets A, B, C ofI\\S, such that\nmax\na∈Aa <min\nb∈Bb⩽max\nb∈Bb <min\nc∈Cc,\nthe graph T(A, B, C ;µ|XS=a0)is a(poly( r),2−r12)-coboundary expander over Sm.\n3. For each interval Ij=n\njdk10\nr, . . . ,(j+1)dk10\nro\nand for each i∈[3]we have that\nk10−k6⩽|Ri∩Ij|⩽k10+k6.\nIn words, the number of elements in Riin the interval Ijis roughly k10(which is the number of points\na typical interval of that length has).\nThe main result of this section is the following lemma, asserting that if µsatisfies Assumption 1, then\nthe corresponding tripartite graph is a coboundary expander with good parameters. More precisely:\nLemma 3.6. LetR1, R2, R3⊆and let µbe a probability measure overQ\ni∈IXisatisfying Assumption 1.\nThen the graph T(R1, R2, R3;µ)is a(poly( r)r/k,2−Ω(r12))-coboundary expander over Sm.\nProof. LetIj⊂[d]denote the intervaln\njdk10\nr, . . . ,(j+1)dk10\nro\n, and write |Ri∩Ij|=k10+ci,jwhere\nci,j∈[−k6, k6]; note thatP\njci,j= 0. We will use induction to prove the lemma. Throughout the\ninduction, we will have three sets R′\n1, R′\n2, R′\n3, satisfying:\n1.R′\ni⊆Ri.\n2. For all i, j,|R′\ni∩Ij|=rj+ci,jfor some rj⩽k10.\n3. There exist at least three intervals with rj⩾k9, sayIj1, Ij2, Ij3, with j1< j2< j3.\nInitially, we will take R′\n1=R1,R′\n2=R2,R′\n3=R3, and later steps will take subsets of these. Note that in\nparticular (2) above implies that all R′\niare equal in size. Henceforth, we fix such R′\n1, R′\n2, R′\n3. Note that by\nthe second and third items, we may find three 3k-sized disjoint sets S1, S2, S3satisfying that Si⊂Ijifor\ni= 1,2,3and furthermore for all j∈[3],|Si∩(R′\nj∩Iji)|=k, and we fix such S1, S2, S3henceforth.\nConsider a distribution D=µ|XB=B0, where B⊆I\\S\ni∈[3]R′\niandB0∈supp(µB). The bulk of\nthe argument will be devoted to proving that\nC(T(R′\n1, R′\n2, R′\n3;D))⩽poly( r)·max\ni∈[3],\nai∈supp(DSi)(C(T(R′\n1\\Si, R′\n2\\Si, R′\n3\\Si;D|XSi=ai))),(8)\nβ(T(R′\n1, R′\n2, R′\n3;D))⩽poly( r)·max\ni∈[3],\nai∈supp(DSi)(β(T(R′\n1\\Si, R′\n2\\Si, R′\n3\\Si;D|XSi=ai))).(9)\nOnce we establish these two inequalities, it is straightforward to conclude the lemma, and we do so in the\nend of the proof. Towards proving (8) and (9), fix any UG instance ΦonG=T(R′\n1, R′\n2, R′\n3;D)with\nδ-fraction of inconsistent triangles.\n22Setting up lists on restrictions: For every Siand every restriction a∈supp(DSi), letGabe the subgraph\nT(R′\n1\\Si, R′\n2\\Si, R′\n3\\Si;D|XSi=a). Let Xa∈SV(Ga)\nm be an assignment on V(Ga)with maximum\nvalue, and let Labe an ordered list of solutions indexed by permutations in Sm, where La[π] =Xa◦π.\nSetting up permutations between restrictions: For every i̸=j, and every restriction aofSiandbof\nSjwhere (a, b)∈supp(DSi∪Sj), letGa,bbe the induced subgraph T(R′\n1, R′\n2, R′\n3;D|(XSi=a, X Sj=b)).\nLet viol (Xa;Ga,b)denote the fraction of edges in Ga,bthat are violated by the assignment Xa. Then by\nClaim 3.4 there exists a permutation πa,bsatisfying that\nPr\nv∼Ga,b[Xa(v)̸=Xb(v)πa,b]⩽viol(Xa;Ga,b) +viol(Xb;Ga,b)\n1−λ(Ga,b)≲viol(Xa;Ga,b) +viol(Xb;Ga,b),(10)\nwhere we used Claim 3.2 to bound λ(Ga,b)by1/2 + poly( r)ε <0.51. Let\nBad(a, b) ={v∈Ga,b|Xa(v)̸=Xb(v)πa,b}.\nFor any restriction bas above, let Lb◦πdenote the list where Lb◦π[π′] =Lb[ππ′] =Xb◦ππ′. With these\nnotations and the definition of Bad (a, b), we get that La|Ga,b∩Bad(a,b)=Lb◦πa,b|Ga,b∩Bad(a,b).\nCounting bad triangles: LetS=S\niSi. For any restrictions a1, a2, a3ofS1, S2, S3letGa1,a2,a3denote\nthe graph T(R′\n1\\S, R′\n2\\S, R′\n3\\S;D|(a1, a2, a3))when (a1, a2, a3)is a valid restriction. Clearly, we have\nthats\nE\na3∼DS3|(a1,a2)[µGa1,a2,a3(Bad(a1, a2))] = µGa1,a2(Bad(a1, a2)).\nFixa1, a2. Using Markov’s inequality and (10) we get\nPr\na3∼DS3|(a1,a2)\u0014\nµGa1,a2,a3(Bad(a1, a2))>1\n100\u0015\n≲viol(Xa1;Ga1,a2) +viol(Xa2;Ga1,a2). (11)\nWe say a restriction (a1, a2, a3)is bad if µGa1,a2,a3(Bad(ai, aj))>1/100for some i̸=j. By (11) we\nget that,\nE\n(a1,a2,a3)∼D∪Si[I((a1, a2, a3)is bad )]\n⩽X\ni∈[3]E\n(aj,ak)∼DSj∪Sk\nj,k̸=iE\nai∼DSi|(aj,ak)[I(µGa1,a2,a3(Bad(aj, ak))>1\n100)]\n≲X\ni∈[3]E\n(aj,ak)∼µSj∪Sk\nj,k̸=i[viol(Xaj;Gaj,ak) +viol(Xak;Gaj,ak)]\n=X\ni∈[3]E\nai∼DSi[viol(Xai;Gai)], (12)\nwhere in the first inequality we used the union bound and in the second inequality we used (11). By definition\nwe have that viol (Xai;Gai)⩽C(Gai)εai+β(Gai), where εaiis the fraction of violated triangles in Gai.\nTherefore,\n(12)⩽X\ni∈[3]E\nai∼DSi[C(Gai)εai+β(Gai)]≲ max\ni∈[3]\nai∈supp(DSi)(C(Gai)δ+β(Gai)) := δ′′.\n23Creating a UG instance on graph over restrictions: LetΨbe the following UG instance on H=\nT(S1, S2, S3;D): each edge (ai, aj)has the constraint π−1\nai,aj. That is, we want to find a solution Athat\nmaximizes the fraction of edges satisfying A(ai) =π−1\nai,ajA(aj). Note that a triangle (a1, a2, a3)is consis-\ntent if π−1\na3,a1π−1\na1,a2π−1\na2,a3=id or equivalently if πa2,a3πa1,a2πa3,a1=id.\nFirst note that if a triangle (a1, a2, a3)is not bad, then it is consistent. The proof is the same as that in\nLemma 3.5 hence we omit it here. Using the coboundary expansion of H, we get that there exists a UG\nsolution Ato the vertices violating at most C(H)δ′′+β(H)-fraction of the edges.\nSince Si⊂Ijifor all i, we get that max a∈S1a <minb∈S2bandmax b∈S2b <minc∈S3c. Therefore by\nAssumption 1, C(H)⩽poly( r)andβ(H)⩽2−r12. Therefore viol (A)⩽poly( r)δ′′+ 2−r12:=δ′.\nLifting the solution: We will now use Ato create a highly satisfying solution BtoG=T(R′\n1, R′\n2, R′\n3;D).\nFor every vertex v∈supp(µR′\ni)and restriction a∈supp(DS1|(XR′\ni=v)), letga(v)denote the permutation\nLa[A(a)][v]. We will choose a randomized assignment as follows: to every vertex u∈supp(µR′\ni), choose a\nrandom s∼ DS1|(XR′\ni=u)and assign B(u) =gs(u). We will now upper bound the expected fraction of\nedges that Bviolates for Φ.\nConsider an edge (ui, uj)∈Gbetween parts R′\ni, R′\nj. This edge is satisfied if there exists s′∈\nsupp(DS1|(XR′\ni=ui, XR′\nj=uj))such that, (1) B(ui) =gs′(ui), (2) B(uj) =gs′(uj)and (3) The\nassignment (gs′(ui), gs′(uj))satisfies the edge (ui, uj)or equivalently (ui, uj)is satisfied by the assign-\nment Xs′. To evaluate the probability there is such s′, we sample s′∼ DS1|(XR′\ni=ui, XR′\nj=uj)and\nconsider each one of the events, starting with event (1). For it, the probability it doesn’t hold is at most\nE\nBE\n(ui,uj)∼GE\ns′∼DS1|(XR′\ni=ui,XR′\nj=uj)[I(B(ui)̸=gs′(ui))] = E\nu∈GE\ns,s′∼DS1|u[I(gs′(u)̸=gs(u))].\nTo calculate this, let us first calculate a bound on:\nE\nu∈GE\ni̸=j∈[3]\n(s,s′)∼DSi∪Sj|u[I(gs′(ui)̸=gs(ui))].\nFix a vertex u∈R′\nifor some i. It is easy to check that an if an edge (s, s′)∈T(S1, S2, S3;D|u)is satisfied\nbyAandu̸∈Bad(s, s′), then gs(u) =gs′(u). Thus,\nE\nu∈GE\ni̸=j∈[3]\n(s,s′)∼DSi∪Sj|u[I(gs′(u)̸=gs(u))]\n⩽E\nu∈GE\ni̸=j∈[3]\n(s,s′)∼DSi∪Sj|u[I((s, s′)not satisfied by A)] +E\nu∈GE\ni̸=j∈[3]\n(s,s′)∼DSi∪Sj|u[I(u∈Bad(s, s′))]\n=viol(A) + E\n(s,s′)∼E(T(S1,S2,S3;D))\nu∈Gs,s′[I(u∈Bad(s, s′))]\n=δ′+ 2 E\nu∈T(S1,S2,S3;D)[viol(Xu)]\n≲δ′.\nWe are ready to bound the probability that event (1) does not happen. Towards this end, for each vertex\nudefine pu:= Pr(s,s′)∼DS1∪S2|u[gs′(u)̸=gs(u)], so that the above inequality translates to Eu∈G[pu]≲δ′.\n24Letp′\nu= Prs,s′∼DS1|u[gs′(u)̸=gs(u)]. By Lemma 2.6, the second largest singular value of the bipartite\ngraph A(S1, S2;D|u)is at most ε·poly( r)⩽0.01for all u, so by the easy direction of Cheeger’s inequality\nwe get that, p′\nu⩽O(pu), which gives us that,\nE\nu∈GE\ns,s′∼DS1|u[I(gs′(u)̸=gs(u))]≲δ′,\nthus bounding the probability of event (1). One can check that the probability of event (2) is the same as\nevent (1), hence let us proceed to event (3). For that we get,\nE\n(ui,uj)∼GE\ns∼DS1|(XR′\ni=ui,XR′\nj=uj)[I((ui, uj)∈viol(Xs))]≲δ′.\nWe see that sampling an edge (ui, uj),sands′fails to satisfy at least one of the events (1), (2) and (3) with\nprobability at most O(δ′). Thus, with probability at most O(δ′)over the choice of (ui, uj)ands, there is\nnos′like that and otherwise we get that s′satisfies all of (1), (2) and (3). This shows that in expectation\nthe assignment Bthat we get violates at most O(δ′)⩽poly( r) maxi∈[3]ai∈supp(DSi)(C(Gai)δ+β(Gai))\nfraction of the edges of G, which completes the proof of (8), (9).\nConcluding the induction: Let us now use (8), (9) to prove the lemma. For any sets R′\n1, R′\n2, R′\n3satisfying\nthe first two items in our assumption about R′\ni’s above with |R′\ni|=ℓkand distribution Das above we will\nshow by induction that C(T(R′\n1, R′\n2, R′\n3;D))⩽poly( r)ℓ·poly( r)r/kandβ(T(R′\n1, R′\n2, R′\n3;D))⩽2−Ω(r12).\nLet us prove the base case first. When ℓ= 1, we can use Lemma 3.5 to bound C(T(R′\n1, R′\n2, R′\n3;D))by\nO(C1,1,1(µ))3k⩽poly( r)k⩽poly( r)r/k, since C1,1,1(µ)⩽poly( r)by Assumption 1. Our next base case\nis when the sets R′\ni’s do not satisfy the third item in the assumptions about R′\ni’s. That is, for all but at most\ntwo intervals Ij,|R′\ni∩Ij|⩽k9+ci,j⩽k9+k6. This implies that |R′\ni|⩽(r/k10)·(k9+k6)+2·(k10+k6)≲\nr/k. In this case, again using Lemma 3.5 we get that C(T(R′\n1, R′\n2, R′\n3;D))⩽poly( r)r/kas required.\nLet us now show the inductive step. Assume that we have proved the statement for ℓand let us prove it\nfor sets R′\ni’s satisfying assumptions (1), (2) and (3) with |R′\ni|= (ℓ+ 1)k. Applying (8) we get that there are\n3k-sized Si’s so that,\nC(T(R′\n1, R′\n2, R′\n3;D))⩽poly( r)·max\ni∈[3],\nai∈supp(DSi)(C(T(R′\n1\\Si, R′\n2\\Si, R′\n3\\Si;D|XSi=ai))).\nOne can now check that the sets (R′\n1\\S1, R′\n2\\S1, R′\n3\\S1)satisfy assumptions (1) and (2), therefore\nC(T(R′\n1\\S1, R′\n2\\S1, R′\n3\\S1;D|a1))⩽poly( r)ℓ·poly( r)r/kby the induction hypothesis. The same\nholds for S2andS3, thus giving us that, C(T(R′\n1, R′\n2, R′\n3;D))⩽poly( r)ℓ+1poly( r)r/kas required. This\nimplies that for ℓ=r/k, we get that C(T(R1, R2, R3;µ))⩽poly( r)r/k.\nApplying a similar argument on βgives that β(T(R1, R2, R3;µ))⩽2−Ω(r12).\n4 Base Case and Extended Base Case for Spherical Buildings\nIn this section we establish bounds on the coboundary constants that serve as the base case presented in this\nsection. The discussion will be specialized to spherical buildings of type A, type C (for technical reasons,\nin the end we will also have to discuss tensors of two types, but this will be straightforward).\nTo establish the base case of our induction we will use the cones method [Gro10, LMM16, KM22,\nKM19, KO19]. The cones method allows one to prove coboundary expansion properties for a graph given\nan efficient way triangulate certain cycles in our graphs of interest.\n25Definition 4.1. A triangulation of a cycle Cin a graph Gis given by a set of vertices SC⊆V(G)such that\nthe induced subgraph on SC∪V(C)breaks into a union of triangles, where any two distinct triangles are\neither disjoint or share an edge.\nNotations: throughout this section, for subspaces A, B , we will often use the notation (A, B)to denote\nthe subspace A+B. Similarly, for a vector vand a subspace A, we will denote by (v, A)the subspace\nspan({v}) +A.\n4.1 The Base Case for Spherical Buildings of Type A\nLet Gr d(k1, k2, k3)denote the tripartite graph whose vertices are k1, k2, k3-dimensional subspaces of Fd\nqand\nan edge is sampled by sampling a random chain V1⊂V2⊂V3and choosing (Vi, Vj), i̸=j∼[3]. We drop\nthe subscript dwhen clear from context.\nLemma 4.2. For all m∈N,k1< k2< k3integers, let K= max( ⌈k2\nk3−k2⌉,⌈k2\nk2−k1⌉). Then Gr d(k1, k2, k3)\nis an (O(K2),poly( K)/q)-coboundary expander over Sm.\nWe break the proof into two cases according to which term larger, k2−k1ork3−k2. Let us first discuss\nthe case when k3−k2⩽k2−k1and then later we discuss how to modify the proof in the other case.\nFor simplicity of notation we will assume that k1, k2, k3are all multiples of k3−k2, so set k=k3−k2,\nt=k3/kandt′=k1/k. The same proof with some slight modifications works when this is not the case,\nhence we omit those details.\nLetGdenote the graph Gr d(k1, k2, k3)andGidenote the vertices of dimension ki. First fix an arbi-\ntrary vertex U∈G3and an arbitrary basis for it: u1, . . . , u k3. Let U(1)denote the set of first kvectors,\n{u1, . . . , u k},U(2)the second set of kvectors and so on upto U(t). We now fix a set of paths from UtoV\nfor most V∈G. LetBbe a set of “block decompositions” for each subspace V∈G, i.e.Bassigns V∈Gi\nthe blocks, B(V) = (V(1), . . . , V (t))withV(i)=U(i)for all i⩽t−ki/kandV= (V>t−ki/k).\n4.1.1 Set of Good Vertices and Edges with respect to Bwhen k3−k2⩽k2−k1\n1. Let Good i(B)⊆Gibe the set of V∈Githat satisfy for all i∈[t],dim(V(1), . . . , V (i), U>i) =k3. In\nparticular taking i=tthis implies that dim(U⩽t−ki/k, V) =k3.\n2. For all b < a∈[3]let Good ab(B)⊆E(Ga, Gb)be the set of edges (V, W )withW⊂Vthat satisfy\n(a) Both V, W∈S\na′∈[3]Good a′(B).\n(b) For all j < i∈[t],dim(V⩽j, W(j+1), . . . , W (i), U>i) =k3.\nClaim 4.3. LetV1, V2⊆Fd\nqbed1, d2dimensional subspaces respectively, and suppose that d3+d1⩽d2.\nThen\nPr\nV3⊆d3V2[dim( V3+V1)< d3+d1]⩽1\nq−1.\nProof. LetU=V1∩V2and write V=U+V′\n1withV′\n1∩V2={0}. Then:\nPr\nV3⊂d3V2[V3∩U={0}]⩾d3−1Y\ni=0\u0012qd2−qd1+i\nqd2\u0013\n⩾1−1\nq−1.\nIfV3∩U={0}thenV3∩V1={0}too, hence dim(V3+V1) =d3+d1thus completing the proof.\n26Lemma 4.4. There exists a block decomposition Bsuch that for all b < a∈[3]:\nPr\nV∼Ga[V∈Good a(B)]⩾1−O\u0012t\nq\u0013\nPr\nV∼Ga\nW⊂bV[(V, W )∈Good ab(B)]⩾1−poly( t)\nq.\nProof. Let us consider a random block decomposition B, that for each subspace V∈Ga, picks a sequence\nof random subspaces:\nV(t−ka/k+1)⊂k(V(t−ka/k+1), V(t−ka/k+1))⊂2k. . .⊂ka−kV,\nand sets V(i)=U(i)for all i⩽t−ka/k. Let us prove that with high probability V∈Good a. Fix some\ni∈[t]and let F⊂[t]be the set {1, . . . , t −ka/k} ∪ {i+ 1, . . . , t}. For a set S⊂[t], letUSdenote the\nsubspace span (U(i)|i∈S). Then using Claim 4.3 we get,\nPr\nV∼Ga,B[dim( V⩽i, U>i) =k3] = Pr\nV∼Ga,B[dim( VF, UF) =k3]⩾1−1\nq−1,\nwhere we used that VFis distributed as a uniformly random k(t− |F|)-dimensional subspace. Taking a\nunion bound over i∈[t]we get that,\nPr\nV∼Ga,B[V∈Good a(B)]⩾1−t\nq−1, (13)\nestablishing the first item. For the second item, fix b < a∈[3],j < i∈[t], and let\nF= [t−ka/k]∪ {i+ 1, . . . , t} ∪([t−kb/k]∩ {j+ 1, . . . , i}), R 1= [j]\\F, R 2={j+ 1, . . . , i} \\F.\nWe have:\nPr\nW⊂k1V,B[dim( V⩽j, W(j+1), . . . , W (i), U>i) =k3]\n= Pr\nW⊂V,B[dim( VR1, WR2, UF) =k3]\n= Pr\nV,B[dim( VR1, UF) = (|R1|+|F|)k]\n·Pr\nW⊂k2V,B[dim( VR1, WR2, UF) =k3|dim(VR1, UF) = (|R1|+|F|)k]. (14)\nThe first term is at least 1−1\nq−1by Claim 4.3, and we next lower bound the second term. By symmetry,\nit suffices to bound it for a fixed VR1which satisfies dim(VR1, UF) = (|R1|+|F|)k. We will first show\nthat the distribution over WR2conditioned on VR1isO(1/q)-close to uniform. Indeed, by Claim 4.3 with\nprobability 1−O(1/q)we have that W∩T={0}. Conditioned on this, WR2is a uniformly random |R2|k-\ndimensional subspace amongst |R2|k-dimensional subspaces intersecting VR1at{0}. The latter distribution\nisO(1/q)-close to a uniformly random |R2|k-dimensional subspace, as by Claim 4.3 the probability a ran-\ndom|R2|k-dimensional subspace intersects VR1at{0}is1−O(1/q). When WR2is chosen to be a uniformly\nrandom |R2|k-dimensional subspace we can easily bound the probability that dim(VR1, WR2, UF) =k3us-\ning Claim 4.3. Therefore we get,\nPr\nW⊂k2V,B[dim( VR1, WR2, UF) =k3|dim(VR1, UF) = (|R1|+|F|)k]⩾1−O(1/q).\n27Plugging this back into (14) and taking a union bound over all j < i∈[t]we get,\nPr\n(V,W)∼E(Ga,Gb),B[(V, W )∈Good ab(B)]⩾1−poly( t)\nq. (15)\nAn averaging argument on (13) and (15) now gives us that there is a Bfor which most vertices and edges\nare good as required.\nHenceforth fix a block decomposition Bsatisfying the conclusions of Lemma 4.4.\n4.1.2 Constructing the Paths when k3−k2⩽k2−k1\nIn this section, fixing U, we construct collections of canonical paths between Uand good vertices in our\ngraph. For each V∈S\ni∈[3]Good i(B), this is achieved using the block decomposition of Vas follows:\nP(U, V) =(U(1), . . . , U (t))→(U(2), . . . , U (t))→(V(1), U(2), . . . , U (t))\n→(V(1), U(3), . . . , U (t))→(V(1), V(2), U(3), . . . , U (t))→. . .\n→(V(1), . . . , V (t−1), U(t))→(V(1), . . . , V (t−1))→(V(1), . . . , V (t))→V,\nwhere we omit the last step if V∈G3since V= (V(1), . . . , V (t)). Note that since V∈Good i, this path\nalternates between vertices from S3andS2. When V∈G1orG2, the subspace Uappears more than once\non the path, and the path from UtoVcould be shortened. We use this longer path instead to keep things\nnotationally simpler, since now all paths are of length either 2tor2t+ 1.\n4.1.3 Triangulating Cycles when k3−k2⩽k2−k1\nHaving constructed paths from Uto all good vertices, we notice that if (W, V )is an edge in the graph, then\ntogether with the paths from Uwe have a cycle C(U, V, W ). In the following lemma, we show how to\ntriangulate this cycle using a small number of triangles.\nLemma 4.5. When k3−k2⩽k2−k1, for every edge (V, W )∈S\nb<a∈[3]Good ab(B), the cycle C(U, V, W ) =\nUP(U,V)− − − − → V→WP(W,U)− − − − − → Uhas a triangulation of size O((k3\nk3−k2)2).\nProof. To make notation simpler we will give a triangulation T(U, V, W )of the cycle with some repeating\nvertices. We put in edges between two vertices that are the same, and label them with the identity permu-\ntation. These are called “equality edges”. This introduces triangles that might have at least two identical\nvertices, but it is easy to see that such a triangle is consistent, hence we can use these triangles essentially\nfor free.\nTiling by 8-cycles when V∈G3andW∈G2:LetW′= (W(1), . . . , W (t))withW(1)=U(1),W=\n(W⩾2), andV= (V(1), . . . , V (t)). To get a triangulation we first create paths between the (2i)thvertex on the\npathP(U, V)and the (2i)thvertex on P(U, W )for all i∈ {1, . . . , t}. Recall that P2i(U, V) = (V⩽i, U>i)\nandP2i(U, W ) = ( W⩽i, U>i). For i∈[t]we take the obvious path Ri(U, V, W )between P2i(U, W )and\nP2i(U, V)that flips a block of Wto a block of Vone at a time:\nRi(U, V, W ) :=P2i(U, W )→(W(2), . . . , W (i), U>i)→(V(1), W(2), . . . , W (i), U>i)→. . .→P2i(U, V),\n28that alternates between k3andk2dimensional vertices. Here we used the fact that (V, W )∈Good 32(B)\nwhich implies that the intermediate odd vertices – (V⩽j, W(j+1), . . . , W (i), U>i), on the path Riabove are\nof dimension k3.\nLet us henceforth drop the notation U, V, W inRi(U, V, W )since U, V, W are fixed. First note that by\ncreating the paths Ri, we have broken the original cycle C(U, V, W )intoO(t)cycles of the form:\nCi=P2i(U, W )Ri\n− →P2i(U, V)→P2i+1(U, V)→P2i+2(U, V)Ri+1\n− − − → P2i+2(U, W )→P2i+1(U, W )\n→P2i(U, W ),\nfori∈[t−1]and\nCt=W′= (W(1), . . . , W (t))Rt\n− →V= (V(1), . . . , V (t))→W→V.\nWe will tile each Ciby 8-cycles and triangles, starting with i∈[t−1]. For all j∈[1, i+ 1], letRi\nj\ndenote the (2j−1)thvertex on the path Ri, that is,\nRi\nj= (V<j, W(j), . . . , W (i), U>i).\nIt is easy to see that for all j∈[1, i+ 1], the vertices Ri\njandRi+1\njare connected via a path of length two:\nRi\nj→(V<j, W(j), . . . , W (i), U⩾i+2, . . . , U (t))→Ri+1\nj.\nAs for the last part of the cycle, the vertex P2i+1(U, V), occurs as the middle vertex in all the following\nlength 2 paths: 1) Ri\ni+1=P2i(U, V)→Ri+1\ni+1, 2)P2i(U, V)→P2i+2(U, V)andRi+1\ni+1→Ri+1\ni+2. Thus using\nequality edges between these vertices we can check that the whole cycle has been broken into O(t)8-cycles\nandO(1)triangles (see figure below).\nLet us now tile the cycle Ct. We have that the first three vertices of RtareW′,(W⩾2)and(V(1), W⩾2)\nwhich are all connected to W. Additionally including (V(1), W⩾2), every other odd vertex on Rt, i.e. for all\nj∈[2, t+ 1],Rt\nj= (V<j, W(j), . . . , W (t))is contained (as a subspace) in Vsince both WandV′are in V.\n29Since the dimensions match it means that these are all equal to Vand therefore we can put in equality edges\nbetween them. Similarly every intermediate vertex ( k2-dimensional) on Rtis contained in V. This means\nthatCtis broken into O(t)triangles. So overall C(U, V, W )has been broken into O(t2)8-cycles and O(t)\ntriangles.\nTiling by 8-cycles when V∈G3andW∈G1:LetW′= (W(1), . . . , W (t)), where W= (W>t−k1/k+1)⊂\nV. Now we create the same paths Riof length 2i+ 1between P2i(U, W )→P2i(U, V)for all i∈[t], and\neach vertex on these paths is in S3orS2because (V, W )∈Good 31(B). This breaks the cycle into the cycles\nCi, fori∈[t]. The tiling of Ci, i∈[t−1]proceeds identical to the first case, therefore let us discuss the\ntiling of the last cycle:\nCt=W′= (W(1), . . . , W (t))Rt\n− →V= (V(1), . . . , V (t))→W→W′.\nAs in the case above, we have that the first 2(t−k1/k) + 1 vertices of Rtare all connected to W, since\nthey are of the form (V<j, U(j), . . . , U (t−k1/k), W). After that all subsequent odd vertices on Rt, i.e. for all\nj∈[t−k1/k+ 1, t],Rt\nj= (V<j, W(j), . . . , W (t))are connected to V. This means that Ctis broken into\nO(t)triangles. So overall C(U, V, W )has been broken into O(t2)8-cycles and O(t)triangles.\n(V, W )is an edge between G2andG1:We can show that C(U, V, W )in this case too can be broken into\nO(t2)8-cycles and O(t)triangles. The proof for the cycles Cifori∈[t−1]is the same, so we only discuss\nthe tiling of Ct. We have that,\nCt=W′= (W(1), . . . , W (t))Rt\n− →V′= (V(1), . . . , V (t))→V→W→W′,\nwhere W= (W>t−k1/k), V= (V>t−k2/k). Again we have that the first 2(t−k1/k) + 1 vertices of\nRtare all connected to W, since they are of the form (V<j, U(j), . . . , U (t−k1/k), W). Then the vertex\n(V⩽t−k1/k, W)is additionally connected to V′, V. After that all subsequent vertices on Rt, i.e. for all\nj∈[t−k1/k+ 1, t−1],Rt\nj= (V⩽j, W(j), . . . , W (t))and the intermediate vertices in G1are all connected\ntoV′, by the same reasoning as the above two cases. This means that Ctis broken into O(t)triangles.\nTriangulating the 8-cycles: Having shown that the cycle C(U, V, W )can always be tiled by at most O(t2)\n8-cycles and O(t)triangles, it suffices to show that each one of the resulting 8-cycles can be triangulated in-\ndividually. Towards this end, we first notice that the 8-cycles we formed consist of edges between subspaces\nof dimension k3andk2. Thus, to triangulate them we will have to use auxiliary vertices of dimension k1.\nNote that each 8-cycle in the above tiling is of the following form for some 1⩽j < i∈[t]:\n(W(j), U(i), X)→(W(j), X)→(W(j), W(i), X)→(W(i), X)\n→(V(j), W(i), X)→(V(j), X)→(V(j), U(i), X)→(U(i), X)→(W(j), U(i), X),\nwithX= (V⩽j−1, W(j+1), . . . , W (i−1), U(>i)). We know that dim(X) =k2−k= 2k2−k3and since\nk2−k1⩾k3−k2,dim(X)⩾k1. This implies that there is some k1-dimensional subspace Y⊆Xthat is\ncontained within all the vertices of this 8-cycle. Putting this vertex in the middle of the 8-cycle completes\nthe triangulation of this cycle.\nSize of Triangulation: In total we used O((k3/k3−k2)2)8-cycles, each of which used O(1)triangles\nthus completing the proof.\n304.1.4 The Case that k3−k2> k2−k1\nWe now move on to discuss the case that k2−k1⩽k3−k2and let k=k2−k1, t=k2/khenceforth.\nWe will need to use a different set of paths, and towards this end we fix an arbitrary vertex U∈G2and an\narbitrary basis for it: u1, . . . , u k2. Let U(1)denote the set of first kvectors, {u1, . . . , u k},U(2)the second\nset of kvectors and so on up to U(t). We now fix a set of paths from UtoVfor most V∈G. We will\nchoose a block decomposition Bfor each subspace V∈G, namely:\n1. For V∈G2,B(V) = (V(1), . . . , V (t)) =V.\n2. For every V∈G1, letV′= (V(1), . . . , V (t))forV(1)=U(1)andV= (V⩾2).\n3. For V∈G3, the decomposition Bfirst associates with Vak2-dimensional subspace V′⊆Valong\nwith its block decomposition V′= (V(1), . . . , V (t)).\nAs before, we will need the block decomposition Bto satisfy several genericness properties, that we explain\nnext.\nSet of Good Vertices and Edges with respect to Bwhen k3−k2> k2−k1:\n1. Let Good i(B)⊆Gibe the set of V∈Githat satisfy for all i∈[t],dim(V(1), . . . , V (i), U>i) =k2.\n2. For all b < a∈[3]let Good ab(B)⊆E(Ga, Gb)be the set of edges (V, W )withW⊂bVthat satisfy,\n(a) Both V, W∈S\na∈[3]Good a(B).\n(b) For all j < i∈[t],dim(V⩽j, W(j+1), . . . , W (i), U>i) =k2.\nAnalogously to Lemma 4.4 one can show that there is a Bfor which 1−poly( t)/q-fraction of vertices\nand edges are good, and we fix such Bhenceforth.\n4.1.5 Constructions the Paths when k3−k2> k2−k1\nFor a vertex V∈S\ni∈[3]Good i(B), we first construct a path P(U, V′), where V′= (V(1), . . . , V (t)), by\nflipping a block of Uto a block of V′one at a time. One can check that this path alternates between k2and\nk1-dimensional vertices since Vis good. If V∈G2, then V′=Vand we are done, else in the last step we\ngo from V′→V.\n4.1.6 Triangulating Cycles when k3−k2> k2−k1\nHaving formed the paths P(U, V), we now note that if (V, W )is an edge, then P(U, V),(V, W ), P(U, W )\nform a cycle. We call this cycle C(U, V, W )as before, and show that it can be triangulated using a small\nnumber of triangles.\nLemma 4.6. Suppose that k2−k1⩽k3−k2, and let (V, W )∈S\nb<a∈[3]Good ab(B)be an edge. Then\ncycle C(U, V, W ) =UP(U,V)− − − − → V→WP(W,U)− − − − − → Uhas a triangulation of size O((k2\nk2−k1)2).\nProof. We only give a proof sketch here since the details are exactly the same as Lemma 4.5. To create\nthe paths Riwe take the obvious path between P2i(U, W )andP2i(U, V)by flipping one block at a time,\nalternating between k2andk1-dimensional vertices (instead of k2, k3-dimensional ones). Then putting in\n31the length two paths between Ri\njandRi+1\nj(that are two k2-dimensional vertices, that can be connected\nusing one k1-dimensional vertex) we break the cycle into O(t2) 8-cycles and O(t)triangles.\nNote that each 8-cycle in the above tiling is of the following form for some 1⩽j < i∈[t]:\n(W(j), U(i), X)→(W(j), X)→(W(j), W(i), X)→(W(i), X)\n→(V(j), W(i), X)→(V(j), X)→(V(j), U(i), X)→(U(i), X)→(W(j), U(i), X),\nwithX= (V⩽j−1, W(j+1), . . . , W (i−1), U(>i)), with dim(X) =k1−k= 2k1−k2. Since k3−k2⩾k2−k1,\nwe get that k3⩾k2+k. Therefore to tile this cycle we can use the vertices: X0= (V(j), W(j), X, Y 0), where\nY0is chosen so that dim(X0) =k2,X1= (X0, U(i), Y1), andX2= (X0, W(i), Y2)where Y1, Y2are chosen\nso that dim(X1) = dim( X2) =k3. This is possible since dim(X0, W(i)),dim(X0, U(i))⩽k2+k⩽k3.\nThe figure below shows that after adding these in the cycle breaks into triangles.\nAs for the size of the triangulation: we had O(t2)8-cycles, each tiled by O(1)triangles, which gives an\noverall triangulation size of O((k2/k2−k1)2)as required.\n4.1.7 Proof of Lemma 4.2\nGiven the paths P(U, V)and triangulations T(U, V, W )for every good edge (V, W ), it is easy to complete\nthe proof of Lemma 4.2 using the fact that GL d(Fq)acts transitively on the triangles of Gr (k1, k2, k3).\nFor each invertible linear transformation L∈GLd(Fq), and a subspace VletL(V)denote the sub-\nspace span (Lv|v∈V)and let L−1(V)denote the subspace Wsuch that L(W) = V. For every\nV∈S\ni∈[3]Good i(B)letPL(L(U), L(V))denote the path from L(U)→L(V)where at the ith-step\nwe have the vertex L(Pi(U, V)). It is easy to see that this is a valid path from L(U)toL(V). For a tri-\nangle ∆letL(∆) denote the triangle whose vertices are L(Ui),∀Ui∈∆. In fact for every good edge\n(V, W )we can let TL(L(U), L(V), L(W))be the triangulation of the cycle L(U)PL(L(U),L(V))− − − − − − − − − → L(V)→\nL(W)PL(L(W),L(U))− − − − − − − − − − → L(U)where a triangle in this triangulation is given by L(∆) for∆∈T(U, V, W ).\nAgain it is easy to see that this is a valid triangulation of the cycle for every edge (V, W )∈S\nb<aGood ab(B)\nwith the same size as T(U, V, W ).\nWe have the following randomized algorithm to get a highly satisfying UG solution to an arbitrary UG\ninstance Φ.\n32Algorithm 1 (Φ = ( Gr(k1, k2, k3),Π)).\nInput: UG instance Φon Gr (k1, k2, k3).\nOutput: A function f:V(Gr(k1, k2, k3))→Sm.\n1. Choose a random linear transformation L∈GLd(Fq)and set f(L(U)) = id.\n2. For each subspace V∈ ∪i∈[3]Good i(B), assign fL(L(V))the label obtained by propagating the\nlabel of L(U)toL(V)via the path PL(L(U), L(V)), chosen appropriately according to whether\nk3−k2ork2−k1is larger.\n3. For every subspace V /∈ ∪i∈[3]Good i(B)choose an arbitrary label for L(V).\nWe now complete the proof of Lemma 4.2 via the following lemma:\nLemma 4.7. LetΦbe any UG instance over Smwith incons (Φ) = δ. Then in expectation over L∼\nGLd(Fq), the algorithm violates at most O(K2)δ+ poly( K)/q-fraction of edges, where\nK= max\u0012\u0018k2\nk3−k2\u0019\n,\u0018k2\nk2−k1\u0019\u0013\n.\nProof. Suppose the propagation algorithm chooses a linear transformation L. LetfL:V(Gr(k1, k2, k3))→\nSmdenote the assignment outputted by Algorithm 2 in this case. Let Edenote E(Gr(k1, k2, k3))and let\nGood (E) =S\nb<aGood ab(B). For every (V, W )∈Good (E), the edge (L(V), L(W))is satisfied by fL\nif the cycle L(U)PL(L(U),L(V))− − − − − − − − − → L(V)→L(W)PL(L(W),L(U))− − − − − − − − − − → L(U)is consistent. Furthermore this\nis true if every triangle in TL(L(U), L(V), L(W))is consistent. Recall that this is the set of triangles\nL(∆),∆∈T(U, V, W ). So we get that,\nviol(fL)⩽ Pr\n(V,W)∼E[(V, W )/∈Good (E)] + E\n(V,W)∼Good(E)[I(∃∆∈T(L(U), L(V), L(W))∩incons (Φ)]\n⩽poly( K)\nq+ max\n(V,W)∈Good(E)(|T(U, V, W )|) E\n(V,W)∼Good(E)E\n∆∈T(U,V,W )[I(L(∆)∈incons (Φ)]\n⩽poly( K)\nq+O(K2) E\n(V,W)∼Good(E)E\n∆∈T(U,V,W ))[I(L(∆)∈incons (Φ)],\nwhere in the second inequality we used Lemma 4.4 and the last one we used Lemmas 4.5 and 4.6 to bound\nthe size of the triangulation. Now taking an expectation over L∼GLd(Fq)we get:\nE\nL[viol(fL)]⩽poly( K)\nq+O(K2) E\n(V,W)∼Good(E)E\n∆∈T(U,V,W ))E\nL[I(L(∆)∈incons (Φ)]\n⩽poly( K)\nq+O(K2)δ,\nwhich completes the proof.\n4.2 The Base Case for Spherical Buildings of Type C\nLetµbe the uniform distribution over chains of isotropic subspaces (V1⊂. . . V d)ofF2d\nq, with dim(Vi) =i.\nFork1, k2, k3⩽dletSd(k1, k2, k3)denote the tripartite graph T({k1},{k2},{k3};µ), where we drop the\nsubscript dwhen clear from context. We will prove that:\n33Lemma 4.8. For all m∈N,0< k1, k2, k3⩽d, letK= max( ⌈k2\nk3−k2⌉,⌈k2\nk2−k1⌉). Then S(k1, k2, k3)is an\n(O(K2),poly( K)/q)-coboundary expander over Sm.\nThe proof of this statement will follow closely along the lines of Lemma 4.2.\n4.2.1 Auxiliary Claims\nClaim 4.9. Fort < k < d ∈N, given isotropic subspaces W1, W2⊆F2d\nqwithdim(W1) =d0−kand\ndim(W2) =d0, there exists an isotropic subspace W⊆W2satisfying dim(W) =k, andW∩(W1∩W2) =\n{0}such that W+W1is ad0-dimensional isotropic subspace.\nProof. Define U=W1∩W2and denote d1= dim( U). We can write W1=U+W′\n1andW2=U+W′\n2\nwithW′\n1∩W′\n2={0}. Let Vbe the subspace of vectors vsatisfying ω(v, w′) = 0 for all w′∈W′\n1. Note\nthatVhas dimension 2d−dim(W′\n1) = 2 d−d0+k+d1. Also\ndim(V∩W′\n2) = dim( V) + dim( W′\n2)−dim(V+W′\n2)⩾(2d−d0+k+d1) + (d0−d1)−2d=k.\nTakeW⊆V∩W′\n2of dimension k. Since W⊆W′\n2we get that Wis isotropic, and as W⊆Vwe get that\nW+W′\n1is isotropic. Next, note that W∩W1∩W2⊆W⊆W′\n2∩U={0}. Finally, now that\ndim(W+W1) =k+ (d0−k)−dim(W∩W1) =d0,\nasW∩W1⊆W∩W1∩W′\n2⊆W∩W1∩W2={0}.\nFor an isotropic subspace U, let Symp (U)denote the subspace that is symplectically orthogonal to U, i.e.\nfor all w∈Symp (U), u∈U, ω(w, u) = 0 . We note the standard facts that dim( Symp (U)) = 2 d−dim(U)\nand Symp (U⊕V) = Symp (U)∩Symp (V). We also have that dim(U∩Symp (W)) = dim( U)−\ndim(W)+dim( Symp (U)∩W). In particular, when dim(U) = dim( W)we get that dim(U∩Symp (W)) =\ndim( Symp (U)∩W).\nClaim 4.10. The following facts are true for sufficiently large q:\n1. Fix a subspace U, let1⩽i⩽dand choose an isotropic subspace Wof dimension iuniformly at\nrandom. Then:\n(a) With probability at least 1−4\nqwe have that\ndim(U∩Symp (W)) = max(0 ,dim(U)−dim(W)).\n(b) With probability at least 1−4\nqwe have that\ndim( Symp (U)∩W) = max(0 ,dim(W)−dim(U)).\n2. Choosing maximal isotropic subspaces U, W ofF2d\nqrandomly, we have that U∩W={0}with\nprobability at least 1−4\nq.\nProof. We prove each item separately, and we begin with the first item. We start with (a), and assume that\ndim(U)⩾i; otherwise we replace Uby a subspace of dimension ithat contains it. Choose w1, . . . , w i\nuniformly conditioned on wjbeing symplectically orthogonal to w1, . . . , w j−1for all j. Define the vectors\nw′\n1, . . . , w′\nibywj(s) =−wj(s+d)fors⩽d, and wj(s) =wj(s−d)fors > d . Thus, the space\n34U∩Symp ({w1, . . . , w i})corresponds to all vectors in Uthat are orthogonal to w′\n1, . . . , w′\ni. Note that for\neach1⩽j⩽iandα1, . . . , α j−1∈Fq, the marginal distribution of w′\nj+j−1P\nℓ=1αℓw′\nℓis uniform, and hence\nit is orthogonal to Uwith probability at most q(2d−dim(U))−2d=q−dim(U). It follows that no vector in\nspan({w′\n1, . . . , w′\ni})is orthogonal to Uwith probability at least\n1−iX\nj=1qj−1q−dim(U)⩾1−2\nq(i−1)−dim(U)⩾1−2\nq,\nwhere we used the fact that dim(U)⩾i. Next, we note that the probability that w1, . . . , w iis linearly\nindependent is at least\n1−iX\nj=1q2j\nq2d⩾1−2q2i\nq2d,\nin which case w′\n1, . . . , w′\niare linearly independent too. It follows from the union bound that with probability\nat least 1−3\nq, the set w1, . . . , w′\niis linearly independent and no vector in it is orthogonal to U, in which case\nthe subspace in Uof vectors orthogonal to w1, . . . , w′\nihas dimension dim(U)−i, and (a) follows.\nWe move on to proving (b). Note that\ndim( Symp (U)∩W) = dim( Symp (U)) + dim( W)−dim( Symp (U)⊕dim(W))\n= dim( Symp (U)) + dim( W)−2d+ dim( U∩Symp (W))\n=i−dim(U) + dim( U∩Symp (W)),\nand the result follows from (a).\nThe second item follows immediately from the first item with i=d.\nClaim 4.11. LetA, B be such that dim(A) = dim( B) =k. Then there exists a randomized algorithm to\nchoose A′⊆k−1A, b∈Bsuch that A′+span({b})is an isotropic subspace and for all d0-dimensional C\nwhere (A+C)and(B+C)are two d0+k-dimensional isotropic subspaces,\nPr\nA′,b[dim( A′+span({b}) +C) =d0+k]⩾1−O\u00121\nq\u0013\n.\nProof. We know that A∩C=B∩C={0}, and there are two cases:\n1. If there is no b∈Bthat is isotropic to all of A, then choose A′⊂k−1Aarbitrarily, and let b∈Bso\nthatA′+span({b})is ak-dimensional isotropic subspace by Claim 4.9. In this case, we know that\nB∩(A+C) ={0}, because every vector in A+Cis in Symp (A). Therefore b /∈A′+C, which\ngives that,\ndim(A′+span({b}) +C) = dim( A′+C) + dim( span({b})) = dim( A′) + dim( C) + 1 = d0+k,\nas required.\n2. If there is some non-zero b0∈B∩Symp (A), then choose b=b0and choose a uniformly random\nA′⊂k−1A. Note that A′+span({b})is an isotropic subspace. Furthermore dim(( C+span({b0}))∩\nA)⩽1since dim(C∩A) = 0 . Ifdim(( C+span({b0}))∩A) = 0 , then dim(C+span({b0})+A′) =\nd0+k, as required. Else, (C+span({b0}))∩A=span({v})for some v̸= 0; by Claim 4.3 we know\nthatA′∩span({v}) ={0}with probability 1−O(1/q), in which case (C+span({b0}))∩A′=∅\nanddim(C+span({b0}) +A) =d0+kas required.\n354.2.2 Constructing Paths\nIn this section, we set up paths between vertices in the spherical building that will be convenient for us to\ntriangulate. Let Sidenote the vertices of dimension ki, and let k=k2−k1. For simplicity of notation\nwe will assume that k2is a multiple of kand that k1⩾k(and so k1is also a multiple of k). The same\nproof with some slight modifications works when this is not the case, and in Section 4.2.6 we explain the\nnecessary modifications.\nFix an arbitrary vertex U∈S2, and pick an arbitrary basis for U:{u1, . . . , u k2}. LetU(1)denote the set\nof first kvectors, {u1, . . . , u k},U(2)the second set of vectors and so on up to U(t)fort=k2/k. Note that\nk1=k(t−1). Define:\n1. Good 1⊆S1to be the set of subspaces Vthat satisfy the following: V∩U={0}, and for all\ni∈[t−1],dim(V∩Symp (U>i)) = ( i−1)kanddim( Symp (V)∩Symp (U>i)) = 2 d−k1−k2+ik.\n2. Good 2⊆S2to be the set of subspaces Vthat satisfy the following: V∩U={0}, and for all\ni∈[t−1],dim(V∩Symp (U>i)) =ik.\n3. Good 3⊆S3to be the set of subspaces Vthat satisfy the following: V∩U={0}anddim(V∩\nSymp (U)) =k3−k2.\nLemma 4.12. For all i∈[3]:\nPr\nV∼Si[V∈Good i]⩾1−O\u0012t\nq\u0013\n.\nProof. Immediate from Claim 4.10 and the union bound.\nSetting up Paths from U:We will now be interested in constructing paths from Uto other vertices V\nin the graph. Towards this end, for fixed UandVwe will associate with Va vertex V′. In the case\nthatdim(V) =k2, we will take V′=V, and otherwise V′will be an appropriately chosen subspace or\nsuperspace of V.\n1. For a vertex Vof dimension k2, using Claim 4.9 on VandU⩾2we find V(1)⊆kVsuch that\n(V(1), U(2), . . . , U (t))∈S2. Applying this claim iteratively, we find V(2)such that\n(V(1), V(2), U(3), . . . , U (t))∈S2,\nand so on. Then consider the following path from U→Vwhich flips a block of Uto a block of V\none at a time:\nP(U, V) =(U(1), . . . , U (t))→U⩾2→(V(1), U⩾2)→(V(1), U⩾3)→(V(1), V(2), U⩾3)→. . .\n→(V<t, U(t))→V<t→V.\nNote that this path alternates between vertices of S2andS1. We set V′=V.\n2. For a vertex Vof dimension k3, we know that dim(V∩Symp (U)) =k3−k2, and thus we may choose\nV′⊂k2Vsuch that V′∩Symp (U) ={0}. Consider its block decomposition V′= (V(1), . . . , V (t))\nsuch that for all i⩽t,(V(1), . . . , V (i), U(i+1), . . . , U (t))∈S2. As shown above, we know that such a\ndecomposition is possible using Claim 4.9 iteratively. Then consider the following path from U→V\nwhich flips a block of Uto a block of Vone at a time similar to the above:\nP(U, V) =(U(1), . . . , U (t))→U⩾2→(V(1), U⩾2)→(V(1), U⩾3)→(V(1), V(2), U⩾3)→. . .\n→V′→V.\n363. We only give a path from UtoV∈Good 1. For such a V, we first find V(1)of dimension ksuch that\n(V(1), V)and(V(1), U⩾2)are in S2andV(1)∩U(1)= 0. Such a subspace exists because\ndim( Symp (U⩾2)∩Symp (V)) = 2 d−k1−k2+k⩾2(d−k2) + 2k⩾2k.\nBy Claim 4.10 picking a random k-dimensional isotropic subspace V(1)from the intersection will\nsatisfy that V(1)∩Symp (U(1)) ={0}, since Symp (V)∩Symp (U)is a co-dimension ksubspace of\nSymp (U⩾2)∩Symp (V). Additionally both (V(1), V)and(V(1), U⩾2)are isotropic. Since V∈Good 1,\nSymp (V)∩U⩾2=V∩Symp (U⩾2) ={0}, which gives us that V(1)∩V=V(1)∩U⩾2={0}implying\nthat(V(1), V)and(V(1), U⩾2)are in S2.\nNow set V′= (V(1), V)and consider the a path from U→V. This path is the result of the above\nprocess for k2dimensional vertices applied on V′, where we already picked V(1), followed by a final\nstep from V′toV.\nP(U, V) =U→U⩾2→(V(1), U⩾2)→(V(1), U⩾3)→(V(1), V(2), U⩾3)→. . .\n→(V(1), . . . , V (t))→V.\nNote that this path too alternates between vertices from S2andS1.\nThe block decomposition above satisfies the following properties:\nClaim 4.13. FixUand take V∈S\ni∈[3]Good i, letV′be the associated vertex with Vand write V′=\n(V(1), . . . , V (t))the block decomposition chosen by the path from Uas above. Then for all i∈[t],V(i)∩\nSymp (Ui) ={0}.\nProof. We split the proof to cases.\nThe case that dim(V) =k2:in that case, by construction and definition of Good 2we have that for i⩾2\nit holds that V∩Symp (U⩾i) =V⩽i−1. By this equality for i+ 1instead of i, it follows that each v∈V(i)\nis symplectic to U⩽i+1, and we conclude that\nV(i)∩Symp (Ui) =V(i)∩Symp (U⩾i) =V(i)∩V∩Symp (U⩾i) =V(i)∩V⩽i−1={0}.\nThe case that dim(V) =k3.In that case, we picked V′⊆Vof dimension k2such that V′∩Symp (U) =\n{0}, and the proof is exactly as in the previous case.\nThe case that dim(V) =k1:by construction V(1)∩Symp (U1) ={0}. For i⩾2, the fact that V(i)∩\nSymp (Ui) ={0}follows from the same argument above, as the path from UtoV′is constructed in the\nsame way.\n4.2.3 Constructing the Triangulations: Handling Special 8-cycles\nWith the paths P(U, V), we would like to construct triangulations of cycles that they form. Towards this\nend, we will have to triangulate 8-cycles of a special form as in the following lemma:\n37Lemma 4.14. Letk⩽k1< k 2< k 3⩽dwithk3⩾k2+k, letA, B, C, D bek-dimensional isotropic\nsubspaces, let Xbe ak2−2k-dimensional isotropic subspace and let X1, X2⊆Xbe of dimension k1−k.\nSuppose that C∩Symp (D) =Symp (C)∩D={0}. Then the following 8-cycle, whose odd vertices are in\nS2and even ones are in S1,\nC8= (A, C, X )→(A, X 1)→(A, D, X )→(D, X 2)→(B, D, X )\n→(B, X 1)→(B, C, X )→(C, X 2)→(A, C, X ),\nhas a triangulation of size O(1).\nProof. We break the proof into two cases. We will use the fact that d⩾k3⩾k2+k.\nCase 1– A⊆Symp (B):This implies that B⊆Symp (A)too. Let Z= (A, B, X )anddim(Z) =k2−ℓ\nfor some ℓ∈[0, k]. Let Vbe a subspace such that: V⊆Symp (Z)andV∩Z={0}. We get that\ndim(V)⩾2d−2(k2−ℓ)⩾2k+2ℓ. LetG1⊆Vbe such that G1⊆Symp (C)∩VandG2⊆V∩Symp (D).\nWe get that dim(G1),dim(G2)⩾dim(V)−k, therefore:\ndim(G1∩G2)⩾dim(V)−2k⩾2ℓ.\nPick an arbitrary ℓ-dimensional isotropic subspace Y⊂G1∩G2and add in the isotropic subspace Z0=\n(Z, Y). AsY⊂Vit follows that Y∩Z={0}, and therefore dim(Z0) =k2. Then add in the vertices Z1=\n(Z, Y, C, Y 1)andZ2= (Z, Y, D, Y 2), where Y1, Y2are chosen such that Z1, Z2arek3-dimensional isotropic\nsubspaces. This is possible since (Z, Y, C ),(Z, Y, D )are subspaces of dimension ⩽dim(Z, Y) +k⩽k3\nand they are isotropic since Y⊂G1∩G2. The figure below shows that adding in Z0, Z1, Z2breaks the\n8-cycle into triangles.\nCase 2 – Amay not be in Symp (B):Write A=A1+A2, where A1⊂Symp (B)andA2∩Symp (B) = 0 .\nSimilarly write B=B1+B2, where B1⊂Symp (A)andB2∩Symp (A) = 0 . We know that dim(A1) =\ndim(B1), and say they are equal to k−ℓfor some ℓ∈[0, k]; thus dim(A2) = dim( B2) =ℓ. We know that\ndim(A1, B1, X)⩽2(k−ℓ) +k2−2k=k2−2ℓ, and so we write dim(A1, B1, X) =k2−2ℓ−zfor some\nz⩾0. Let us find a subspace V′that satisfies:\nV′⊆Symp (A2)∩Symp (B2)∩Symp (C)∩Symp (D)∩Symp (A1, B1, X), V′∩(A1, B1, X) ={0},\n38By dimension counting we get,\ndim(V′)⩾2d−dim(A2)−dim(B2)−dim(C)−dim(D)−2 dim( A1, B1, X)\n= 2d−2ℓ−2k−2(k2−2ℓ−z)\n= 2(d−k2−k) + 2( ℓ+z)\n⩾2(ℓ+z).\nWe can now pick any (ℓ+z)-dimensional isotropic subspace Vinside V′. Also fix an arbitrary ℓ-dimensional\nsubspace eVinside V.\nBelow we elaborate on the vertices we add inside the cycle. It is easy to check that by construction each\nof these vertices are isotropic subspaces, therefore we will only check that they are of the correct dimensions\n(k1, k2ork3). Let A′= (B1,eV).\n1.(A′, X1): By construction V∩(B1, X1) ={0}andeV⊆V, hence (B1, X1)∩eV={0}and by the\ndimension formula we get\ndim(A′, X1) = dim( B1, X1)+dim( eV) = dim( B1)+dim( X1)+dim( eV) =k−ℓ+k1−k+ℓ=k1.\n2.(A, B 1, V, X ): This is equal to (A2, A1, B1, X, V ). We know that A2∩(A1, B1, V, X ) ={0},\nsince the latter is symplectically orthogonal to BandA2∩Symp (B) ={0}. Also, by construction\nV∩(A1, B1, X) ={0}, and applying the dimension formula twice we get that:\ndim(A, B 1, V, X ) = dim( A2) + dim( A1, B1, X) + dim( V) =ℓ+k2−2ℓ−z+ℓ+z=k2.\n3.(A, B 1, V, C, X, Y 1)where Y1is chosen so that the whole subspace is k3-dimensional. This is possible\nsince:\ndim(A, B 1, V, C, X ) = dim( A2) + dim( C) + dim( A1, B1, X) + dim( V) =k2+k⩽k3,\nwhere we used that A2∩(A1, B1, V, C ) ={0}since the latter is symplectically orthogonal to Band\nA2∩Symp (B) ={0}, and further C∩(A1, B1, V, X ), since the latter is symplectically orthogonal\ntoDandC∩Symp (D) ={0}by assumption.\n4.(A′, C, X ): This equals (B1,eV , C, X )and like the above, we get\ndim(A′, C, X ) = dim( C) + dim( B1) + dim( X) + dim( eV) =k2.\n5.(A, B 1, V, D, X, Y 2)where Y2is chosen so that the whole subspace is k3-dimensional, where the\nproof that this is possible is the same as the third item above.\n6.(A′, D, X ): This is a k2-dimensional isotropic subspace like the fourth item above.\nThe figure below shows that this breaks the cycle into triangles and another 8-cycle corresponding to the\nsubspaces A′, B, C, D, X, X 1, X2.\n39Now note that A′= (B1,eV)satisfies A′⊆Symp (B). Therefore we can triangulate the remaining 8-\ncycle (in blue above) using Case 1, which gives a triangulation of the original cycle using O(1)triangles.\n4.2.4 Triangulating General Cycles\nLemma 4.15. For every edge (V, W )∈S(k1, k2, k3), where V, W∈S3\ni=1Good i, the cycle C(U, V, W ) =\nUP(U,V)− − − − → V→WP(W,U)− − − − − → Uhas a triangulation of size O(K2), where K= max\u0010\nk2\nk2−k1,k2\nk3−k2\u0011\n.\nProof. Without loss of generality we assume that dim(V)>dim(W), meaning that W⊂V. First, we\nbreak the cycle C(U, V, W )intoO(t2)cycles of length 8andO(t)triangles. This step is slightly different\ndepending on which type of edge we have, and we proceed by case analysis.\nTiling by 8-cycles when V∈S3, W∈S2:LetV′⊂Vbe the vertex chosen by the path from U, with\nV′= (V(1), . . . , V (t))being the corresponding block decomposition. Let (W(1), . . . , W (t))be the block\ndecomposition of W. For all 0⩽i < j⩽twe can check that:\n(V(1), . . . , V (i), W(i+1), . . . , W (j), U(j+1), . . . , U (t))∈S2. (16)\nIndeed, fix i, j. First note that this vertex is an isotropic subspace because (1) we know that (V⩽i, U>i)∈S2\nsoV⩽iis symplectically orthogonal to U>j, (2)(W⩽j, U>j)∈S2implying (W(i+1), . . . , W (j))is sym-\nplectically orthogonal to U>j, and (3) W, V′⊂Vimplying that (W(i+1), . . . , W (j)), V⩽iare symplectically\northogonal. Therefore all vectors in the subspace in (16) are symplectically orthogonal to each other.\nSecond, we check that the dimension of the space in (16) is tk. Since W∈Good 2,dim(W∩\nSymp (U>i)) = dim( W)−dim(U>i) =ik. Since dim(W⩽i) =ik, and by construction W⩽iis sym-\nplectic to U>i, it follows that W∩Symp (U>i) =W⩽i, and so nothing in W>iis symplectically orthogonal\ntoU>i. Thus, as V⩽i⊂Symp (U>i), we conclude that V⩽i∩W>i={0}. We also argue that\n(V⩽i, W(i+1), . . . , W (j))∩(U>j) ={0}.\nIndeed, if v+w∈(V⩽i, W(i+1))andw̸= 0, then vis symplectic to U>jandwis not, so v+wis not\nsymplectic to U>jand hence v+w̸∈U>j. Ifw= 0, then the claim follows as V⩽i∩U>j={0}fori⩽j.\nCombining everything, we get from the dimension formula that\ndim(V⩽i, W(i+1), . . . , W (j), U>j) = dim( V⩽i, W(i+1), . . . , W (j)) + dim( U>j)\n40= dim( V⩽i) + dim( W(i+1), . . . , W (j)) + dim( U>j)\n=tk.\nRecall that P2i(U, V) = ( V⩽i, U>i)and same for W. We will now create paths Riof length 2i+ 1\nbetween P2i(U, W )andP2i(U, V)for all i∈[t]using the intermediate vertices from (16). These paths are\nconstructed as:\nRi:=P2i(U, W )→(W(2), . . . , W (i), U>i)→(V(1), W(2), . . . , W (i), U>i)→. . . →P2i(U, V).\nNote that this is a valid path because the odd vertices (starting from P2i(U, W )) are in S2by (16), which\nalso implies that the even vertices are in S1. By creating the paths Ri, we have broken the original cycle\nC(U, V, W )intoO(t)cycles of the form:\nCi=P2i(U, W )Ri\n− →P2i(U, V)→P2i+1(U, V)→P2i+2(U, V)Ri+1\n− − − → P2i+2(U, W )→P2i+1(U, W )\n→P2i(U, W ),\nfori∈[1, t−1]and\nCt=WRt\n− →V′= (V(1), . . . , V (t))→V→W.\nWe will tile each Ciby 8-cycles and triangles, starting with i∈[t−1]. For all j∈[1, i+ 1], letRi\nj\ndenote the (2j−1)thvertex on the path Ri, that is,\nRi\nj= (V<j, W(j), . . . , W (i), U>i).\nIt is easy to see that for all j∈[1, i+ 1], the vertices Ri\njandRi+1\njare connected via a path of length two:\nRi\nj→(V<j, W(j), . . . , W (i), U⩾i+2, . . . , U (t))→Ri+1\nj.\nAdditionally the middle vertex in the length 2 path is equal to P2i+1(U, V)in the path from Ri\ni+1=\nP2i(U, V)→Ri+1\ni+1,P2i(U, V)→P2i+2(U, V)andRi+1\ni+1→Ri+1\ni+2, thus showing that the whole cycle\nhas been broken into O(t)8-cycles and O(1)triangles.\nLet us now tile the cycle Ct. We have that for all j∈[1, t+1],Rt\nj= (V<j, W(j), . . . , W (t))is contained\ninVsince both WandV′are in V. This means that Ctis broken into O(t)triangles. So overall C(U, V, W )\nhas been broken into O(t2)8-cycles and O(t)triangles.\nTiling by 8-cycles when V∈S3, W∈S1:LetV′⊂Vbe the vertex chosen by the path from U,\nwith V′= (V(1), . . . , V (t))being the corresponding block decomposition. Let W(1)be the additional\nvertex used in the path from UtoW, with the block decomposition W′= (W(1), . . . , W (t)), where\nW= (W(2), . . . , W (t))⊂V. For all 0⩽i < j ∈[t], using the same argument as in (16) we have\nthat\n(V(1), . . . , V (i), W(i+1), . . . , W (j), U(j+1), . . . , U (t))∈S2. (17)\nNow we create the same paths Riof length 2i+ 1between P2i(U, W )→P2i(U, V)for all i∈[t], and\neach vertex on these paths is in S2orS1because of (17). This breaks the cycle into the cycles Ci, fori∈[t].\nThe tiling of Ci, i∈[t−1]proceeds identical to the first case, therefore let us discuss the tiling of the last\ncycle:\nCt=W′= (W(1), . . . , W (t))Rt\n− →V′= (V(1), . . . , V (t))→V→W→W′.\n41We have that the first three vertices of RtareW′,(W⩾2)and(V(1), W⩾2)which are connected to W. Addi-\ntionally including (V(1), W⩾2), every other vertex on Rt, i.e. for all j∈[2, t+1],Rt\nj= (V<j, W(j), . . . , W (t))\nand the intermediate vertices in S1, is contained in Vsince both WandV′are in V. This means that Ctis\nbroken into O(t)triangles. So overall C(U, V, W )has been broken into O(t2)8-cycles and O(t)triangles.\nTiling by 8-cycles when V∈S2, W∈S1:We can show that C(U, V, W )in this case too can be broken\nintoO(t2)8-cycles and O(t)triangles. The proof for the cycles Cifori∈[t−1]is the same, so we only\ndiscuss the tiling of Ct. We have that,\nCt=W′= (W(1), . . . , W (t))Rt\n− →V= (V(1), . . . , V (t))→W→W′,\nwhere W= (W(2), . . . , W (t)). The first and second vertex on Rt,W′, Ware connected to W, and then\none can check that every vertex on Rtexcept for the first one, is connected to V. This breaks CtintoO(t)\ntriangles.\nTriangulating the 8-cycles: Having shown that the cycle C(U, V, W )can always be tiled by at most O(t2)\n8-cycles and O(t)triangles, it suffices to show that each one of the resulting 8-cycles can be triangulated\nindividually. Towards this end, we first notice that the 8-cycles we formed consist of edges between isotropic\nsubspaces of dimension k1and isotropic subspaces of dimension k2. Thus, to triangulate them we will have\nto use auxiliary vertices of dimension k3. Intuitively, the difference k2−k1measures how “different”\nadjacent vertices in the cycle are, and it stands to reason that the closer these vertices are, the easier time we\nwill have triangulating it. In the proof below we handle two cases separately: the case k2−k1⩽k3−k2,\nand the case that k2−k1> k3−k2, and we begin with the former easier case.\nTriangulating the 8-cycles when k3−k2⩾k2−k1:Note that each 8-cycle in the above tiling is of the\nfollowing form for some 1⩽j < i∈[t]:\n(W(j), U(i), X)→(W(j), X)→(W(j), W(i), X)→(W(i), X)\n→(V(j), W(i), X)→(V(j), X)→(V(j), U(i), X)→(U(i), X)→(W(j), U(i), X), (18)\nwithX= (V⩽j−1, W(j+1), . . . , W (i−1), U(>i)). Let A=W(j), B=V(j), C=U(i), D=W(i). By\nClaim 4.13, C∩Symp (D) ={0}. Furthermore each of these blocks are of size k=k2−k1and our\nassumption reads that k3⩾k2+k. This cycle therefore satisfies the assumptions of Lemma 4.14 and can\nthus be triangulated with O(1)triangles.\nTriangulating the 8-cycles when k3−k2⩽k2−k1:This case is more difficult, and we first start with an\n8-cycle of the form we created, and transform it into an 8-cycle that can be triangulated using Lemma 4.14.\nFix some i0< j0∈[t]consider an 8-cycle as in (18), namely:\nC8= (A, C, X )→(A, X)→(A, D, X )→(D, X )→(B, D, X )\n→(B, X )→(B, C, X )→(C, X)→(A, C, X ),\nwhere A, B, C, D, X are defined appropriately as in the above paragraph. We have the property that C∩\nSymp (D) =D∩Symp (C) ={0}.\nPicture the 8-cycle as a square, with the k2-dimensional vertices at the 4 corners. We will construct\ntwo “horizontal” paths: (A, C, X )→(A, C, D )and(B, C, X )→(B, D, X )and two vertical paths:\n42(A, C, X )→(B, C, X )and(A, C, X )→(B, D, X ). To do so first we apply the randomized algorithm\nfrom Claim 4.11 on the subspaces C1:=C, D :=D1to get C2⊂k−1C1, d1∈Dsuch that (d1, C2)is an\nisotropic subspace. We can then write D=D2+d1, such that D2∩span({d1}) =∅, and apply Claim 4.11\nagain to get d2∈D2, C3⊂k−2C2such that (d2, C3)is an isotropic subspace and in fact (d1, d2, C3)is also\nisotropic. Similarly applying the claim ktimes we get a sequence of subspaces:\nC,(d1, C2),(d1, d2, C3), . . . , (d1, . . . , d k−1, Ck), D.\nWe know that Ck⊂1Ck−1⊂2. . . C 2⊂1C. Similarly we create the following sequence between AandB:\nA,(b1, A2),(b1, b2, A2), . . . , (b1, . . . , b k−1, Ak), B.\nWe will show that for all i, j∈[k]:\nPr\na,b,c,d[dim( b⩽i, Ai, d⩽j, Cj, X) = 2 k]⩾1−poly( k)\nq. (19)\nThis is because:\nPr\na,b[dim( b⩽i, Ai, X) =k] =iY\ni′=1Pr\na,b[dim( b⩽i′−1, bi′, Ai′, X) =k|dim(b⩽i′−1, Ai′−1, X) =k]\n⩾\u0012\n1−O\u00121\nq\u0013\u0013i\n⩾1−O\u0012k\nq\u0013\n,\nwhere for each term in the product we used Claim 4.11 with the subspaces C= (b⩽i′−1, X),A=Ai′−1\nandB=Bi′−1. To prove (19) we use the same trick again:\nPr\na,b,c,d[dim( b⩽i, Ai, d⩽j, Cj, X) = 2 k]\n= Pr\na,b[dim( b⩽i, Ai, X) =k] Pr\na,b,c,d[dim( b⩽i, Ai, d⩽j, Cj, X) = 2 k|dim(b⩽i, Ai, X) =k]\n⩾\u0012\n1−O\u0012k\nq\u0013\u0013 jY\nj′=1Pr\na,b,c,d[dim( d⩽j′−1, b⩽i, Ai, dj′, Cj′, X) = 2 k|dim(d⩽j′−1, Cj′−1, b⩽i, Ai, X) = 2 k]\n⩾1−O\u0012poly( k)\nq\u0013\n,\nwhere for each term in the product we used Claim 4.11 with the subspaces C= (d⩽j′−1, b⩽i, Ai, X),\nA=Cj′−1andB=Dj′−1, which proves (19). Since q > poly( k)we can now union bound over all\nchoices of i, j∈[k]to get that there exists a choice of a, b, c, d ’s such that for all i, j∈[k]:\ndim(b⩽i, Ai, d⩽j, Cj, X) = 2 k. (20)\nHenceforth fix such a choice of a, b, c, d ’s.\nDenote k′=k3−k2and write k=t′k′+r′, with r′< k′andt′⩾1. For all i∈[0, t′], set\nPi(C, D) = (d⩽ik′, Cik′+1)andPt′+1(C, D) =D. Then we have the path\nR1= (A, C, X )→(A, X)→(A, P 1(C, D), X)→. . .\n→(A, X)→(A, P t′(C, D), X)→(A, X)→(A, D, X )\n43that alternates between S2andS1by (20). Similarly for all i∈[0, t′], letPi(A, B)denote the subspace,\n(b⩽ik′, Aik′+1)andPt′+1(A, B) =B. We get the following parallel path between (B, C, X )and(B, D, X ):\nR2= (B, C, X )→(B, X )→(B, P 1(C, D), X)→. . .\n→(B, X )→(B, P t′(C, D), X)→(B, X )→(B, D, X ).\nWe now create horizontal paths between (A, X, P i(C, D))and(B, X, P i(C, D))for all 0⩽i⩽t′+ 1:\nHi= (A, P i(C, D), X)→(Pi(C, D), X)→(P1(A, B), Pi(C, D), X)→. . .\n→(Pi(C, D), X)→(Pt′(A, B), Pi(C, D), X)→(Pi(C, D), X)→(B, P i(C, D), X),\nwhere every vertex is either in S2orS1by (20). Let Hi\njdenote the vertex (Pj(A, B), Pi(C, D), X)– this\nis connected to Hi+1\njvia the vertex (Pj(A, B), X)∈S1. Therefore we have broken the whole cycle into\nmultiple triangles on the periphery and a grid of 8-cycles of the form:\n(Pj(A, B), Pi(C, D), X)→(Pj(A, B), X)→(Pj(A, B), Pi+1(C, D), X)\n→(Pi+1(C, D), X)→(Pj+1(A, B), Pi+1(C, D), X)→(Pj+1(A, B), X)\n→(Pj+1(A, B), Pi(C, D), X)→(Pi(C, D), X)→(Pj(A, B), Pi(C, D), X),\nfori, j∈[t′]. To rewrite this cycle in a more convenient form, let Ajk′+1=A(j+1)k′+1+A′, for\nsome k′-dimensional A′satisfying A′∩A(j+1)k′+1={0}. Let B′= (bjk′+1, . . . , b (j+1)k′), and YAB=\n(b⩽jk′, A(j+1)k′+1). In these notations we have that\nPj(A, B) = (RA, YAB)Pj+1(A, B) = (bjk′+1, . . . , b (j+1)k′, YAB).\nLetD′= (dik′+1, . . . , d (i+1)k′), so that Dik′+1=D(i+1)k′+1+D′. Let C′=Symp (D(i+1)k′+1)∩\nCik′+1, and YCD= (d⩽ik′, C(i+1)k′+1). We will now show that C′∩Symp (D′) = 0 anddim(C′) =\nk′. First note that for all x∈[t′],Cx∩Symp (Dx) = Symp (Cx)∩Dx={0}. This is because\nD=span(d⩽x−1) +DxandCx⊂Symp (d⩽x−1)therefore if any vector v∈Cxwas symplectically\northogonal to Dx, then vwould be symplectically orthogonal to Dand we would get a contradiction to\nC∩Symp (D) ={0}; thus Cx∩Symp (Dx) ={0}, and the proof for Symp (Cx)∩Dx={0}is analogous.\nSince C(i+1)k′+1∩Symp (D(i+1)k′+1) ={0}, we get that C(i+1)k′+1∩C′= 0. By dimension counting we\nknow that dim(C′)⩾k′, which gives us the decomposition:\nCik′+1=C(i+1)k′+1+C′,\nanddim(C′) =k′. Since Cik′+1∩Symp (Dik′+1) ={0}andC(i+1)k′+1⊂Symp (D′), we conclude that\nC′∩Symp (D′) ={0}. Finally we get that,\nPi(C, D) = (C′, YCD)Pi+1(C, D) = (dik′+1, . . . , d (i+1)k′, YCD).\nLetting Y= (YAB, YCD), we can rewrite the cycle as:\n(A′, C′, Y)→(A′, YAB)→(A′, D′, Y)→(D′, YCD)→(B′, D′, Y)\n→(B′, YAB)→(B′, C′, YAB)→(C′, YCD)→(A′, C′, Y).\nWe have shown that each of A′, B′, C′, D′is of size k′=k3−k2, with C′∩Symp (D′) = 0 andk3⩾k2+k′.\nThus the cycle satisfies the assumptions of Lemma 4.14 and can be triangulated in O(1)triangles.\n44Size of triangulation: Since we showed a tiling of C(U, V, W )byO(t2)8-cycles, and a tiling of each\n8-cycle by O(⌈k2−k1/k3−k2⌉2)triangles, C(U, V, W )has a triangulation of size O(K2)as required.\n4.2.5 Using Triangulations for Coboundary Expansion\nGiven the triangulations from Lemma 4.15 and as is standard in applications of the cones method, we will\nuse the property that SP 2d(Fq)acts transitively on the triangles of S(k1, k2, k3)to complete the proof of\nLemma 4.8.\nFor a matrix M∈SP2d(F), and a subspace Vrecall that M(V)denotes the subspace span (Mv|\nv∈V). Let M−1(V)denote the subspace Wsuch that M(W) =V. Given a path P(U, V)from U\ntoV, letPM(M(U), M(V))denote the path from M(U)toM(V)where at the ith-step we have the\nvertex M(Pi(U, V)). It is easy to see that this is a valid path from M(U)toM(V)ifV∈S\ni∈[3]Good i.\nFor a triangle ∆letM(∆) denote the triangle whose vertices are M(Ui),∀Ui∈∆. In fact we can let\nTM(M(U), M(V), M(W))be the triangulation of the cycle\nM(U)PM(M(U),M(V))− − − − − − − − − − − → M(V)→M(W)PM(M(W),M(U))− − − − − − − − − − − → M(U)\nwhere a triangle in this triangulation is given by M(∆) for∆∈T(U, V, W ). Again it is easy to see that\nthis is a valid triangulation of the cycle.\nWe have the following randomized algorithm to get a good solution to an arbitrary UG instance Φ.\nAlgorithm 2 (Φ = ( S(k1, k2, k3),Π)).\nInput: UG instance ΦonS(k1, k2, k3).\nOutput: A function f:V(S(k1, k2, k3))→Sm.\n1. Choose a random linear transformation M∼SP2d(Fq)and set f(M(U)) = id.\n2. For each subspace V∈S\niGood i, assign M(V)the label obtained by propagating the label of\nM(U)toM(V)via the path PM(M(U), M(V)). For V /∈S\niGood i, assign an arbitrary label\ntoM(V).\nWe now complete the proof of Lemma 4.8 via the following lemma:\nLemma 4.16. LetΦbe any UG instance over Smwith incons (Φ) = δ. Then in expectation over M∼\nSP2d(Fq), the algorithm violates at most O(K2)δ+ poly( K)/q-fraction of edges where\nK= max\u0012\u0018k2\nk3−k2\u0019\n,\u0018k2\nk2−k1\u0019\u0013\n.\nProof. Suppose the propagation algorithm chooses a linear transformation M. Consider the assignment\nfM:V(S(k1, k2, k3))→Smthat Algorithm 2 outputs in this case. For V, W ∈S\niGood i, an edge\n(M(V), M(W))is satisfied if the cycle M(U)PM(M(U),M(V))− − − − − − − − − − − → M(V)→M(W)PM(M(W),M(U))− − − − − − − − − − − →\nM(U)is consistent, i.e. the permutations on the edges product to id. Furthermore this is true if every triangle\ninTM(M(U), M(V), M(W))is consistent. Recall that this is the set M(∆)as∆ranges over T(U, V, W ).\nLetEdenote E(S(k1, k2, k3))and Good (E)denote the set of edges (V, W )forV, W∈S\niGood i. We get\nthat,\nviol(fM)⩽Pr\n(V,W)∼E[(V, W )/∈Good (E)] + E\n(V,W)∼Good(E)[I(∃∆∈TM(M(U), M(V), M(W))∩incons (Φ)]\n45⩽O\u0012K\nq\u0013\n+ max\n(V,W)∈Good(E)(|T(U, V, W )|) E\n(V,W)∼Good(E)E\n∆∈T(U,V,W )[I(M(∆)∈incons (Φ)]\n⩽O\u0012K\nq\u0013\n+O(K2) E\n(V,W)∼Good(E)E\n∆∈T(U,V,W ))[I(M(∆)∈incons (Φ)],\nwhere in the second inequality we used Lemma 4.12 and the last one we used Lemmas 4.15 to bound the\nsize of the triangulation.\nRecall that Macts transitively on the set of t-dimensional isotropic subspaces, therefore the distribution\nM(∆)overM∼SP2d(Fq)is uniform over the triangles in S(k1, k2, k3). Using this fact, we can now take\nan expectation over M∼SP2d(Fq)for the above equation to get:\nE[viol(fM)]⩽O\u0012K\nq\u0013\n+O(K2) E\n(V,W)∼Good(E)E\n∆∈T(U,V,W ))E\nM[I(M(∆)∈incons (Φ)]\n⩽O\u0012K\nq\u0013\n+O(K2)δ,\nwhich completes the proof.\n4.2.6 Modifications in Triangulations in Edge Cases\nIn this section, we explain the necessary modifications to the triangulation argument in the case that k2and\nk1are not multiples of k=k2−k1. Let k2= (t−1)k+afor some 2⩽t∈Nanda∈ {0, . . . , k −1}.\nThen k1= (t−2)k+a. Fix a vertex U∈S2and let U= (U(1), . . . , U (t))where each block has dimension\nk, except for the last one which has dimension a. We can define the setS\niGood iaccordingly which gives\nthe following set of paths from U.\nSet of Paths from U:The paths from Uare the same for all but the second to last step. For every Vwe\nwill associate with Va vertex V′. In the case that dim(V) =k2, we will take V′=V, and otherwise V′will\nbe an appropriately chosen subspace or superspace of V. We will also associate with Vana-dimensional\nsubspace V′\n(t−1)which is a subset of V(t−1)that is obtained while creating a path from UtoV.\n1. For a vertex Vof dimension k2, using Claim 4.9 on VandU⩾2we find V(1)⊆kVsuch that\n(V(1), U(2), . . . , U (t))∈S2. Applying this claim iteratively, we find V(2)such that\n(V(1), V(2), U(3), . . . , U (t))∈S2,\nand so on. Let V′\n(t−1)be a random a-dimensional subspace of V(t−1). Then consider the following\npath from U→Vwhich flips a block of Uto a block of Vone at a time:\nP(U, V) =(U(1), . . . , U (t))→U⩾2→(V(1), U⩾2)→(V(1), U⩾3)→(V(1), V(2), U⩾3)→. . .\n→(V<t, U(t))→(V<t−1, V′\n(t−1))→V.\nNote that this path alternates between vertices of S2andS1. We set V′=V.\n2. We do the same for vertices in Good 1,Good 3. The paths remain the same for all but the step where\nwe put in the vertex (V<t−1, V′\n(t−1))and one can show that these are valid paths alternating between\nS2andS1.\n46Triangulating the cycles C(U, V, W ):Let us focus on the case where (V, W )is an edge for V∈\nGood 2, W∈Good 1. The other two cases follow analogously. We create the same paths RifromP2i(U, W )\ntoP2i(U, V). The only path that is slightly different is the path Rt:\nRt:=P2i(U, W )→(W(2), . . . , W (t))→(V(1), W(2), . . . , W (t))→. . .→(V<t, W(t))→(V<t−1, V′\n(t−1))\n→P2i(U, V).\nOne can now tile each cycle Cicreated using 8-cycles and triangles in an identical manner when i∈\n[1, t−2]∪ {t}. Let us discuss the tiling of Ct−1. Putting in horizontal length two paths between Rt−1\njand\nRt\njwe can break Ct−1into 8-cycles of the form:\n(W(1), U(t), X, Y )→(W(1), X)→(W(1), W(t), X, Y )→(W(t), X, Y )\n→(V(1), W(t), X, Y )→(V(1), X)→(V(1), U(t), X, Y )→(U(t), X, Y )→(W(1), U(t), X, Y ),(21)\nwithX= (W(2), . . . , W (t−2), W′\n(t−1))andY=W′′\n(t−1)), where W′′\n(t−1)+W′\n(t−1)=W(t−1)andW′′\n(t−1)∩\nW′\n(t−1)={0}. The only difference in this 8-cycle from the one in (18) is that the vertices (W(1), U(t), X, Y )\nand(W(1), W(t), X, Y )have an intersection (W(1), X, Y )which is larger than the k1-dimensional subspace\n(W(1), X), and the same holds for the vertices (V(1), U(t), X, Y )and(V(1), W(t), X, Y ). Intuitively this is\nonly easier to tile than (18) where every two adjacent points on the square intersect in a k1-dimensional\nsubspace. Indeed this holds, and in both the cases k3−k2⩾k2−k1andk3−k2⩽k2−k1we can use the\nsame strategy to tile this 8-cycle with triangles.\nGiven this tiling of C(U, V, W ), the rest of the proof remains the same and we omit the details.\n4.3 The Extended Base Cases\nIn this section we use Lemmas 4.2 and 4.8 to show that the spherical buildings of type A and C, and their\ntensor products, satisfy the Assumptions 1. That is, we will prove a bound on the coboundary constant of\ntripartite graphs T(A, B, C ;µ|XS=A0), where max a∈Aa <minb∈Bbandmax b∈Bb <minc∈Cc, and µ\nis the uniform distribution over the maximal faces of one of these complexes.\n4.3.1 The Extended Base Case for Restrictions of Type A and Type C\nTo establish Assumption 1 we proceed in two steps. First, we prove an auxiliary lemma handling the case\nthatµhas a product structure in the sense that µS1∪S2=µS1×µS2andµS1∪S3=µS1×µS3. Let diam (G)\ndenote the diameter of a graph G.\nLemma 4.17. Letµbe a distribution overQ\ni∈[d]Xi, and let Gbe a group acting onQXisuch that for all\ng∈Git holds that g(Xi) =Xiandg(supp(µ)) = supp(µ), as well as for all X∈supp(µ), the distribution\noverg(X)for a uniformly chosen g∈Gisµ. Then for all pairwise disjoint sets S1, S2, S3⊂[d]that satisfy\nµS1∪S2=µS1×µS2andµS1∪S3=µS1×µS3, we get that\nC(T(S1, S2, S3;µ))≲diam(A(S2, S3;µ)).\nProof. We first construct a set of paths and triangulations for propagation. Let Pidenote the set of vertices\ncorresponding to the set SiinH=T(S1, S2, S3;µ). Since µS1,S2andµS1,S3are both product distributions,\nwe have an edge between U, W for all U∈P1andW∈P3, as well as between any U∈P1andV∈P2.\nFix two vertices U0∈P1, V0∈P2. For every V∈P2fix the path P(U0, V) =U0→Vand similarly for\n47W∈P3fix the path P(U0, W) =U0→W. For any U∈P1fix the path P(U0, U) =U0→V0→U.\nNow for an edge (V, W )between P2andP3the cycle (U0, V, W )is already a triangle, therefore has a\ntriangulation of size 1. For an edge (U, V)we need to tile the cycle U0→V0→U→V→U0. To\ndo so we use a path Rof length diam (A(S2, S3;µ))from V0→Vthat alternates between vertices of P2\nandP3. Every vertex in this path is connected to both UandU0therefore this cycle has a triangulation of\nsize≲diam(A(S2, S3;µ)). We can follow the same strategy for triangulating the cycle U0→W→U→\nV0→U0corresponding to the edge (V, W )– we build a path Rof length diam (A(S2, S3;µ))from V0to\nW, and given that every vertex in the path is connected to both U0andUthis gives us a triangulation of size\n≲diam(A(S2, S3;µ)). We can now use the fact that the triangles in Hare transitive under the action of G\nto get that the coboundary constant of His≲diam(A(S2, S3;µ)). The proof of this fact is the same as that\nof Lemma 4.7, wherein the group Gwas GL d(Fq), and therefore we omit the details.\nArmed with Lemma 4.17, we now prove that the restrictions of type A spherical building give tripartite\ngraphs that are coboundary expanders.\nLemma 4.18. Letµ=SBA\nd(Fq). For all t⩽d/3,S⊆Iof size at most d−3tand restrictions A0∈\nsupp(µS), for all t-sized pairwise disjoint sets A, B, C ofI\\S, with i= max a∈Aa, j= min b∈Bb, j′=\nmax b∈Bbandk= min c∈Ccsatisfying i < j ⩽j′< k , the graph T(A, B, C ;µ|XS=A0)is a\n(poly( K),poly( K)/q)-coboundary expander over Sm, forK= max\u0010\nk\nj−i,k\nk−j′\u0011\n.\nProof. In this proof we will apply Lemma 4.17 multiple times. To do so, we will use the fact that for\nallS⊆[d], A0∈supp(µS), the subgroup Hof GL d(Fq)that fixes A0, has the property that for all\nX∈supp(D), the distribution h(X), h∼His equal to DforD=µ|XS=A0.\nWe first consider the case of t= 1. Fix a set S⊂[d]of size at most d−3,A0∈supp(µS)and consider\ncoordinates i, j, k∈[d]\\Swithi < j < k . LetDdenote the distribution µ|XS=A0. We divide the proof\ninto two cases:\nThere exists ℓ∈Ssuch that ℓ∈(i, j)orℓ∈(j, k):Let us assume that there is ℓ∈Swithℓ∈(i, j).\nThen note that D{i},{j}=D{i}× D{j}andD{i},{k}=D{i}× D{k}. Therefore we can apply Lemma 4.17\nto get that C(T({i},{j},{k};D))≲diam(A({j},{k};D)), which we know is at most O(k/(k−j)).\nThe same proof works when ℓ∈(j, k)to give a coboundary constant O(j/j−i).\nThere is no ℓ∈Swith ℓ∈(i, k):In this case, let d1∈Sbe the largest index smaller than iand let\nd2∈Sbe the smallest index larger than k. We know that Gis isomorphic to Gr (i−d1, j−d1, k−\nd1)over the ambient space Fd2−d1q . Therefore we can use Lemma 4.2 to conclude that our graph is a\n(poly( K′),poly( K′)/q)-coboundary expander for\nK′= max\u0012j−d1\n(j−d1)−(i−d1),j−d1\n(k−d1)−(j−d1)\u0013\n⩽max\u0012k\nk−j,k\nj−i\u0013\n⩽K,\nas required.\nWe now move on to the case that t >1. Namely fix a set S⊂[d]of size at most d−3t,A0∈µS\nand consider three pairwise disjoint t-sized sets A, B, C ⊂[d]\\Swithi < j < j′< k. LetDdenote the\ndistribution µ|XS=A0. Using the argument in Lemma 3.5, and more specifically (2), we get that\nC(T(A, B, C ;D))⩽C(T({i},{j},{k};D))·\n48max\na∈supp(Di)\nb∈supp(Dj)\nc∈supp(Dk)(C(T(A\\ {i}, B, C ;D|a)), C(T(A, B\\ {j}, C;D|b)), C(T(A, B, C \\ {k};D|c)))\nThe first term above is at most Kusing the case t= 1 from above. As for the second term, consider for\ninstance C(T(A\\ {i}, B, C ;D|a), and consider D′=D|aandA′=A\\ {i}. Note that (D′)A′∪B=\n(D′)A′×(D′)Band(D′)A′∪C= (D′)A′×(D′)C, so applying Lemma 4.17 we get that the second term\nabove is at most diam (A(S2, S3;D′)). Note that the diameter of the graph A(S2, S3;D′)is at most a constant\ntimes the diameter of A({j′},{k};D′), which is easily seen to be at most O\u0010\nk\nk−j′\u0011\n⩽O(K). Combining\nthe two bounds, we conclude that C(T(A, B, C ;D))≲K2. Similarly using (3) It is easy to check that the\nadditive error βis at most poly( K)/q.\nNext, we handle restrictions of type C spherical buildings.\nLemma 4.19. Letµ=SBC\nd(Fq). For all t⩽d/3,S⊆Iof size at most d−3tand restrictions A0∈\nsupp(µS), for all t-sized pairwise disjoint sets A, B, C ofI\\S, with i= max a∈Aa, j= min b∈Bb, j′=\nmax b∈Bbandk= min c∈Ccsatisfying i < j ⩽j′< k , the graph T(A, B, C ;µ|XS=A0)is a\n(poly( K),poly( K)/q)-coboundary expander over Sm, forK= max\u0010\nk\nj−i,k\nk−j′\u0011\n.\nProof. The proof is very similar to the proof of Lemma 4.18, and we will need to use the following facts\n(which were already used in Lemma 2.20):\n1. Let Vi′⊂Vk′be two i′andk′-dimensional isotropic subspaces. For i′< i < j < k < k′, the\ntripartite graph over i, jandk-dimensional isotropic subspaces that are contained in Vk′and contain\nVi′is isomorphic to Gr k′−i′(i−i′, j−i′, k−i′)as defined in Section 4.1.\n2. Let Vi′be some i′-dimensional isotropic subspace. For i′< i < j < k , the tripartite graph over i, j\nandk-dimensional isotropic subspaces that contain Vi′, is isomorphic to Sd−i′(i−i′, j−i′, k−i′)as\ndefined in Section 4.2.\nWe will again apply Lemma 4.17 and to do so, we will use the fact that for all S⊆[d], A0∈supp(µS),\nthe subgroup Hof SP 2d(Fq)that fixes A0, has the property that for all X∈supp(D), the distribution\nh(X), h∼His equal to DforD=µ|XS=A0.\nWith these facts in mind, we begin with the proof in the case that t= 1. LetD=µ|XS=A0. We now\nprove that G=T({i},{j},{k}};D)is a(poly( K),poly( K)/q)-coboundary expander, and there are a few\ncases to consider depending on the set of coordinates Sthat we restricted.\nThere exists ℓ∈Ssuch that ℓ∈(i, j)orℓ∈(j, k):Let us assume that there is ℓ∈Swithℓ∈(i, j).\nThen note that D{i},{j}=D{i}× D{j}andD{i},{k}=D{i}× D{k}. Therefore we can apply Lemma 4.17\nto get that C(T({i},{j},{k};D))≲diam(A({j},{k};D)), which we can check is at most O(k/(k−j)).\nThe same proof works when ℓ∈(j, k)to give a coboundary constant O(j/j−i).\nThere is no ℓ∈Swith ℓ∈(i, k)and there is k′∈S, k′> k:In this case, let k′∈Sbe the smallest\nindex larger than kand let i′∈Sbe the largest index smaller than i(set it to 0if no such index). By\nthe above, Gis isomorphic to Gr k′−i′(i−i′, j−i′, k−i′)which is a (poly( K),poly( K)/q)-coboundary\nexpander by Lemma 4.2.\n49There is no ℓ∈Swithℓ∈(i, k)and no k′∈S, k′> k:In this case, again let k′∈Sbe the smallest\nindex larger than kand let i′∈Sbe the largest index smaller than i(i′= 0 if no such index). By Fact\n2 above, we know that Gis isomorphic to Sd−i′(i−i′, j−i′, k−i′)which is a (poly( K),poly( K)/q)-\ncoboundary expander by Lemma 4.8.\nWe now move on to handle t >1. Using the argument in Lemma 3.5, and more specifically (2), we get\nthat\nC(T(A, B, C ;D))⩽C(T({i},{j},{k};D))·\nmax\na∈supp(Di)\nb∈supp(Dj)\nc∈supp(Dk)(C(T(A\\ {i}, B, C ;D|a)), C(T(A, B\\ {j}, C;D|b)), C(T(A, B, C \\ {k};D|c)))\nThe first term above is at most Kusing the case t= 1 from above. As for the second term, consider for\ninstance C(T(A\\{i}, B, C ;D|a), and consider D′=D|aandA′=A\\{i}. Note that (D′)A′∪B= (D′)A′×\n(D′)Band(D′)A′∪C= (D′)A′×(D′)C, so applying Lemma 4.17 we get that the second term above is at\nmost diam (A(S2, S3;D′))≲K. Combining the two bounds, we conclude that C(T(A, B, C ;D))≲K2.\nSimilarly using (3) It is easy to check that the additive error βis at most poly( K)/q.\n4.3.2 The Extended Base Case for Tensors of Type A and C\nWe now extend the base case from the previous section to a base case regarding tensors of type A and type\nC complexes.\nDefinition 4.20. LetX= (X(0), X(1), . . . , X (d1))andY= (Y(0), Y(1), . . . , Y (d2))be simplicial\ncomplexes. We define the tensored simplicial complex X⊗Yas the complex (Z(0), . . . , Z (d1+d2))where\nfor each 1⩽ℓ⩽d1+d2we have\nZ(ℓ) ={A∪B|A∈X, B∈Y,|A|+|B|=ℓ}.\nWe note that if we let µ, ν be the uniform distributions on top dimensional faces of X, Y respectively,\nthen the distribution µ⊗νis uniform over the top dimensional faces of X⊗Y. Thus, it will often be\nconvenient for us to discuss tensors of complexes using the language of associated probability distributions.\nLemma 4.21. Letµ1, µ2be two distributions over the domainsQ\ni∈[d1]XiandQd\ni=d1+1Xi, with µ1being\neither SBA\nd1(Fq)orSBC\nd1(Fq)andµ2being either SBA\nd−d1(Fq)orSBC\nd−d1(Fq). Letµ=µ1×µ2,t⩽d/3,\nS⊆[d]be of size at most d−3t,A0∈supp(µS)a restriction of S, and let A, B, C ⊆[d]\\Sbe of size t\nsuch that\ni= max\na∈Aa < j = min\nb∈Bb⩽j′= max\nb∈Bb < k = min\nc∈Cc.\nThen the graph G=T(A, B, C ;µ|XS=A0)is a(poly( K),poly( K)/q)-coboundary expander over Sm\nforK= max\u0010\nk\nk−j′,k\nj−i\u0011\n.\nProof. LetG1=GLd1(Fq)or SP 2d1(Fq)depending on whether µ1isSBA\nd1(Fq)orSBC\nd1(Fq), and similarly\ndefine G2as GL d−d1(Fq)or SP 2(d−d1)(Fq). The set supp(µ)is transitive under the action of G1×G2and\nmoreover for all S⊆[d], A0supp(µS), the subgroup HofG1×G2that fixes A0, has the property that for\nallX∈supp(D), the distribution h(X), h∼His equal to DforD=µ|XS=A0.\nGiven this fact, we begin with the case that t= 1. Fix a set S⊂[d]of size at most d−3,A0∈\nsupp(µS)and consider coordinates i, j, k∈[d]\\Swithi < j < k . LetD=µ|XS=A0,I1= [d1]and\nI2={d1+ 1, . . . , d }. We divide the proof into 4 cases:\n50The case that i∈I1andj, k∈I2:First note that D{i,j}andD{i,k}are both product distributions. Hence\nwe can apply Lemma 4.17 to get that the coboundary constant of Gis≲diam(A({j},{k};D))≲K.\nThe case that i, j∈I1andk∈I2:This case is similar to the above and we get a bound of O(j/(j−i))≲\nKon the coboundary constant.\nAll three coordinates in I2:Depending on whether µ1isSBA\nd1orSBC\nd1we can use Lemma 4.18 or 4.19\nto get that Gis a(poly( K),2−Ω(r12))-coboundary expander.\nAll three coordinates in I1:This case follows analogously to the third case above.\nWe now move on to the case that t >1, and the argument here is the same as in Lemma 4.18 and 4.19.\nUsing the argument in Lemma 3.5, and more specifically (2), we get that\nC(T(A, B, C ;D))⩽C(T({i},{j},{k};D))·\nmax\na∈supp(Di)\nb∈supp(Dj)\nc∈supp(Dk)(C(T(A\\ {i}, B, C ;D|a)), C(T(A, B\\ {j}, C;D|b)), C(T(A, B, C \\ {k};D|c)))\nThe first term above is at most Kusing the case t= 1 from above. As for the second term, consider for\ninstance C(T(A\\{i}, B, C ;D|a), and consider D′=D|aandA′=A\\{i}. Note that (D′)A′∪B= (D′)A′×\n(D′)Band(D′)A′∪C= (D′)A′×(D′)C, so applying Lemma 4.17 we get that the second term above is at\nmost diam (A(S2, S3;D′))≲K. Combining the two bounds, we conclude that C(T(A, B, C ;D))≲K2.\nSimilarly using (3) It is easy to check that the additive error βis at most poly( K)/q.\n5 UG coboundary expansion of Spherical Buildings\nWe can now use the base case of induction along with the local to global lemma to get that the spherical\nbuildings of type A and C are UG coboundary expander.\nClaim 5.1. There is a constant C > 0such that if α(r)is a function of rsuch that α(r)⩾6, and suppose\nthatd⩾10rα(r). Then\nPr\ni1,...,ir∼[d]\u0014\nmin\nj̸=k∈[r]|ij−ik|⩾d\nrα(r)\u0015\n⩾2−r/α(r)C.\nProof. Denote r′=rα(r)for simplicity. We will show that the event holds even if we pick elements from\n[d]with repetitions, which is clearly stronger. To do so, we first choose a1, then calculate the probability\nthata2lies outside the d/r′interval around a1, then further calculate the probability that a3lies outside the\nd/r′interval around a1anda2and so on. Formally it suffices to lower bound:\nPr\nai∼[d]\u0014\nmin\na∈{a1,...,ai−1}|a−ai|⩾d\nr′\f\f\fa1, . . . a i−1\u0015\n,\nfor all i∈[k].\n51Let us start by bounding the first expression, and let us fix an i. There are a total of at most (i−1)d\nr′\nelements that ared\nr′-close to some ai′for1⩽i′⩽i−1. Thus, the probability aiis chosen among them is\nat mosti−1\nr′, so the first expression is at least 1−i−1\nr′.\nAs the probability of the event we are interested in the product of the expressions over i= 1, . . . , r , we\nget that\nPr\na1,...,ar∼[d]\u0014\nmin\na̸=b∈∪ai|a−b|⩾d\nr′\u0015\n⩾Y\ni∈[r]\u0012\n1−i−1\nr′\u0013\n⩾\u0010\n1−r\nr′\u0011r\n⩾2−O(r/α(r)).\nDefinition 5.2. Letr⩽3dandDbe a distribution overQ\ni∈[d]Xi. We define a\u0000d\nr\u0001\n-partite graph Gr(D)as\nfollows: for every S⊂r[d], let the part PScontain the vertices supp(DS)and to sample an edge we sample\nS, S′⊂r[d]withS∩S′=∅,X∼ D and output (XS, XS′). We associate Gr(D)with the distribution on\ntriangles that samples disjoint sets S1, S2, S3⊂r[d],X∼ D and outputs (XS1, XS2, XS3).\nThe next lemma shows that the graph Gr(D)is a coboundary expander over Smwith strong parameters\nifDis any restriction of a spherical building of type A, of type C or a tensor of them.\nLemma 5.3. Letm, r, d, q ∈Nwithr≪d≪qand let µbe the distribution SBA\nd(Fq),SBC\nd(Fq)\nor one of their tensor products. Then for all S⊂⩽r[d]andA0∈supp(µS),Gr(µ|XS=A0)is a\n(2O(r0.99logr),2−Ω(r12))-coboundary expander over Sm.\nProof. Let us fix µto be the distribution corresponding to the spherical building of type C. The proof when\nµequals SBA\nd(Fq)is identical except that we use Lemmas 2.15 and 4.18 for type A instead of Lemmas 2.20\nand 4.19 for type C that are used below. Similarly, in the case that µis one of the various possible tensor\nproducts, we use the fact that µis an O(1/√q)-product distribution (being the product of two O(1/√q)-\nproduct distributions), and Lemma 4.21 instead of Lemma 4.19.\nConsider D=µ|(XS=A0)forS⊂[d]of size at most randA0∈supp(µS). Let Φ = ( Gr(D),Π)\nbe a UG instance over Smwith incons (Φ) = δ. LetSbe the set of tuples (S1, S2, S3)of subsets of [d]\\S\nof size rthat are pairwise disjoint, and let S′⊆ S be the set of tuples (S1, S2, S3)∈ S where ∪Siis\nΩ(d/r2)-separated and satisfies the third item in Assumption 1. Using Claim 5.1 we have\nPr\n(S1,S2,S3)∈S[(S1, S2, S3)∈ S′]⩾Ω(1)− Pr\n(S1,S2,S3)∈S[∃j∈[r0.9], i∈[3]such that ||Si∩Ij|−r0.1|⩾r0.06].\n(22)\nBy the union bound, symmetry and the fact that r⩽√\ndwe have that\nPr\n(S1,S2,S3)∈S[∃j∈[r0.9], i∈[3]such that ||Si∩Ij| −r0.1|⩾r0.06]≲rPr\nS1[||S1∩I1| −r0.1|⩾r0.06].\nWe bound the latter probability using the multiplicative Chernoff bound (that holds for sampling without\nreplacement too):\nPr\nS1⊂r[d][||S1∩I1| −r0.1|⩾r0.06]⩽2−Ω(r0.1(r0.06/r)2)= 2−Ω(r0.02).\nOverall, plugging in these estimates into (22) we get that\nPr\n(S1,S2,S3)∈S[(S1, S2, S3)∈ S′]≳1.\n52LetTbe the set of S4⊂r[d]\\SwithS4∩(∪i∈[3]Si) =∅, and note that\nE\n(S1,S2,S3)∼S′E\n(a1,a2,a3)∼µ∪Si[(a1, a2, a3)∈incons (Φ)]≲δ, (23)\nE\n(S1,S2,S3)∼S′E\nS4∼TE\n(a1,a2,a4)∼µS1∪S2∪S4[(a1, a2, a4)∈incons (Φ)]≲δ, (24)\nand\nE\n(S1,S2,S3)∼S′E\nS4,S5∼T:\nS4∩S5=∅E\n(a1,a4,a5)∼µS1∪S4∪S5[(a1, a4, a5)∈incons (Φ)]≲δ, (25)\nwhere we used the fact that the expectations of the same event under (S1, S2, S3)∼ S is equal to δ. Using\nMarkov’s inequality we can henceforth fix a tuple (S1, S2, S3)∈ S′where both the above expectations are\nbounded by O(δ). LetI=∪i∈[3]Si. We verify that DIandIsatisfy the Assumptions 1. Indeed,\n1. By Lemma 2.20, the measure Dis an ε-product distribution with ε≲1/√qwhich can be made at\nmost 2−r12by taking qlarge.\n2. The second item in Assumption 1 follows from Lemma 4.19.\n3. The third item holds by the definition of S′.\nThis means that we may apply Lemma 3.6, and indeed we do so.\nSolving ΦonH=T(S1, S2, S3;DI):SinceDIandIsatisfy Assumption 1 we can apply Lemma 3.6 to\nget that His an(2O(r0.99logr),2−Ω(r12))-coboundary expander over Sm. In particular if Φ|His the restricted\nUG instance on Hthen there exists a solution Ato the vertices of Hwith:\nviol(A)⩽2O(r0.99logr)incons (Φ|H) + 2−Ω(r12))⩽2O(r0.99logr)δ+ 2−Ω(r12).\nLifting the solution: We will now use Ato create a highly satisfying solution BtoG=Gr(D). This\nproof is similar to the lifting proof from Lemma 3.6, but we give the full proof here for completeness. For\nevery S4∈ T , every vertex u∈supp(DS4)and every restriction s∈supp(DSi|XS4=u), i∈[3], let\ngs(v)denote the permutation π(v, s)A(s). We will choose a randomized assignment as follows: for every\nsetS4∈ T, every vertex u∈supp(DS4), choose a random s∼ DS1|(XS4=u)and assign B(u) =gs(u).\nWe will now upper bound the expected fraction of edges that Bviolates for Φ.\nConsider an edge (u, v)∈Gbetween two parts S4, S5∈ T (with S4∩S4=∅). This edge is satisfied if\nthere exists s′∈supp(DS1|(XS4=u, X S5=v))such that, (1) B(u) =gs′(u), (2)B(v) =gs′(v)and (3)\nThe triangle (s′, u, v)is consistent.\nTo evaluate the probability there is such s′, we sample s′∼ DS1|(XS4=u, X S5=v)and consider\neach one of the events, starting with event (1). For it, the probability it doesn’t hold is at most\nE\nBE\n(u,v)∼DS4∪S5E\ns′∼DS1|(XS4=u,XS5=v)[I(B(u)̸=gs′(u))] = E\nu∼DS4E\ns,s′∼DS1|u[I(gs′(u)̸=gs(u))].\nTo calculate this, let us first calculate a bound on:\nE\nu∼DS4E\n(s,s′)∼DS1∪S2|u[I(gs(u)̸=gs′(u))].\n53It is easy to check that an if an edge (s, s′)∈T(S1, S2, S3;D|u)is satisfied by A, and the triangle (s, s′, u)\nis consistent then gs(u) =gs′(u). Thus,\nE\nu∈DS4E\n(s,s′)∼DS1∪S2|u[I(gs′(u)̸=gs(u))]\n⩽E\nu∈GE\n(s,s′)∼DS1∪S2|u[I((s, s′)not satisfied by A)] + E\nu∈DS4E\n(s,s′)∼DS1∪S2|u[I((s, s′, u)∈incons (Φ))]\n≲viol(A) + E\n(s,s′,u)∼DS1∪S2∪S4[I((s, s′u)∈incons (Φ))] := δ′(S4).\nWe are ready to bound the probability that event (1) does not happen. Towards this end, for each vertex\nu∈supp(DS4)define pu:= Pr(s,s′)∼DS1∪S2|u[gs′(u)̸=gs(u)], so that the above inequality translates to\nEu∼DS4[pu] =δ′(S4). Let p′\nu= Prs,s′∼DS1|u[gs′(u)̸=gs(u)]. By Lemma 2.6, the second largest singular\nvalue of the bipartite graph A(S1, S2;D|u)is at most ε·poly( r)⩽0.01for all u, so by the easy direction\nof Cheeger’s inequality we get that, p′\nu⩽O(pu), which gives us that,\nE\nu∼DS4E\ns,s′∼DS1|u[I(gs′(u)̸=gs(u))]≲δ′(S4),\nthus bounding the probability of event (1). One can check that the probability of event (2) is the same as\nevent (1), hence let us proceed to event (3). For that we get,\nE\n(u,v)∼DS4∪S5E\ns′∼DS1|(XS1=u,XS2=v)[I((s′, u, v)∈incons (Φ))] = E\n(s′,u,v)∼DS1∪S4∪S5[I((s′, u, v)∈incons (Φ))].\nAdding up the probabilities that events (1),(2) or (3) do not happen and taking an expectation over S4, S5∼\nTwithS4∩S5=∅we get that,\nE\nBE\nS4,S5∼T:\nS4∩S5E\n(u,v)∼DS4∪S5E\ns′∼DS1|(u,v)[I[events (1),(2), or (3) don’t hold ]]≲2O(r0.99logr)δ+ 2−Ω(r12):=δ′,\nwhere we used (24) and (25).\nWe see that sampling sets S4, S5∼ T and an edge (u, v)∼ DS4∪S5, the restrictions su, svchosen by\nthe randomized assignment Bands′fails to satisfy at least one of the events (1), (2) and (3) with probability\n≲δ′. Thus, with probability at most O(δ′)over the choice of S4, S5,(u, v),su, sv, there is no s′that satisfies\nall events and otherwise we get that s′satisfies all of (1), (2) and (3). This shows that in expectation the\nassignment Bthat we get violates ≲δ′-fraction of the edges of Gthat are between disjoint S4, S5∈ T .\nNoting that this is a 1−o(1)-fraction of all the edges of G(since r≪d) completes the proof.\n6 Construction of Sparse UG Coboundary Expander\nWe show that the complex from [CL23] is a sufficiently strong UG coboundary expander. As an immediate\ncorollary of [BM23a] we get that the Chapman-Lubotzky complex admits direct product testers.\n6.1 Vanishing Cohomology for G1[X]overSm\nThe main goal of this section is to present the Chapman-Lubotzky complex. In particular, we need the state-\nment that for all m∈N, for an appropriate choice of parameters, this complex has vanishing 1-cohomology.\nThis statement was communicated to us by Dikstein, Dinur and Lubotzky [DDL23], and below we give an\nalternative proof.\n54Definition 6.1. We say a graph Ghas vanishing cohomology with respect to Smif the following holds. Let\nΦbe a Unique-Games instance on GoverSmin which every triangle in Gis consistent. Then there exists\na solution S∈SV(G)\nm that satisfies all the constraints of Φ. A complex Xhas vanishing 1-cohomology over\nSmifG1[X]has vanishing cohomology over Sm.\nWe note that the above definition is equivalent to the standard topological definition of vanishing 1-\ncohomology. Therein, given a function f:E→Hdefined on the edges where His some finite group, one\ndefines the coboundary map ∂fon triangles via\n∂f(u, v, w ) =f(u, v)f(v, w)f(w, u).\nWith these notations, we care about functions fsuch that ∂f≡id. If f(u, v) =g(u)g(v)−1for all edges\n(u, v)∈Efor some g:V→H, then ∂f≡id clearly. The first cohomology group of Gwith coefficients\ninHis defined via\nH1(G, H) ={f|∂f≡id}\\{f| ∃g:V→H, f(u, v) =g(u)g(v)−1},\nand note that the fact that Ghas vanishing cohomology over Smwith respect to the definition above is\nequivalent to H1(G, H) = 0 . We will use this language henceforth in this section.\nFirst, we need the following general lemma that relates the cohomology of a complex to the homomor-\nphisms of groups.\nLemma 6.2. LetGbe a group acting transitively on a contractible topological space X, letΓbe a discrete\nsubgroup of Gthat acts simply on X, and let Hbe a finite group. Then H1(Γ\\X, H )∼=Hom(Γ , H).\nProof. We may identify functions on Γ\\Xwith functions on Xthat are invariant under Γ. For a cycle\nxlet us write [x]for its equivalence class modulo the boundaries. Given [f]∈H1(Γ\\X, H )we ob-\ntain that ∂(f) = id. Since Xis contractible, we may find a function gonXwith∂(g) =f.Without\nloss of generality g(x) = id for some x∈X. We now set φ(γ) = g(γ(x))for each γ∈Γ. We\nassert that φis a homomorphism. Indeed, for each γinΓchoose a path from γxtox. By hypothe-\nsis,f=∂(g)is invariant under left-multiplication by Γ. Therefore, the value of ∂(g)on the path is\nφ(γ) =g(γx)g(x)−1=g(γτx)g(τ(x))−1=φ(γτ)φ(τ)−1.This yields that φis indeed a homomorphism.\nTo show that the homomorphism is well defined we need to show that if gisγ-invariant, then φ= 1, which\nof course holds. To construct the homomorphism in the reverse direction we choose a representative xfor\neach coset Γxand define a function gonXby setting its value on γxto be φ(γ).We then obtain back\nan element [∂(g)]∈H1(Γ\\X, H ).It is easy to verify that the maps that we defined are inverses of each\nother.\nRecall that a quarternion algebra over Fis a field extension F[i, j, k ]withi2=a, j2=b, k2=cand\nk=ij=−jifora, b, c ∈F. LetQpbe the p-adic rationals and Q∞beR. Let νbe either a prime por\ninfinity. A quarternion algebra DoverQis said to be split at νifD⊗Qpis isomorphic to the algebra of 2×2\nmatrices over Qν. Otherwise, it is a division ring and it is said to be ramified or unsplit. Given a quarternion\nalgebra Dthere exists a natural involution τsending each of i, j, k to its negation. For a matrix Awith\ncoordinates in Dlet us write A∗= (τ(aji)).Then the group SU(n, D)consists of all the matrices with\ncoordinates in D, such that A∗A=I. The first result from number theory used by Chapman and Lubotzky\nis the following.\nFact 6.3. For every p0there exits a quarternion algebra that is ramified (non-split) only over p0and∞.\n55The Chapman-Lubotzky high dimensional expanders consist of Γ\\B, where Γis a lattice and Bis the\nBruhat-Tits building of type ˜CnoverQp. The lattice Γis given by taking a lattice inside the set SU(n, D),\nwhere Dis a quarternion algebra splitting over p0,∞for some p0,∞. In our case we choose the prime p0to\nbe larger than m. The lattice is obtained as follows: we first embed SU(n, D)diagonally inside the productQ\nν̸=p,∞SU(n, D⊗Qν). We then intersect it with a product of the compact groups Kν, where each Kν\ncan be an arbitrary compact open subgroup, which we choose to be SP2n(Zν)for each ν̸=p0, and when\nν=p0,Kνcan be taken to be a pro p0-group, which means that all its finite quotients have order which is\na power of p0:\nFact 6.4. The subgroup SU(n, D)⊗Qpcontains a compact open pro- pgroup for each p.\nFollowing Chapman-Lubotzky we set K=Q\nν̸=p,∞KνandΓ =K∩SU(n, D).\nFinally, using the congruence subgroup property and the strong approximation theorem, Chapman–\nLubotzky showed that Kis the profinite completion of Γ, which means that every homomorphism from\nγto a finite group can be extended uniquely into K. The final number theoretic facts that we need is\nthe following. Recall that a discrete subgroup L⩽Gis said to be a lattice if there exists a G-invariant\nprobability measure on L\\G.\nFact 6.5. Γis a lattice inside SP2n(Qp).\nWe are now ready to state the main lemma of this section.\nLemma 6.6. For all m∈N, choosing nandpto be sufficiently lage, the Chapman-Lubotzky Γ\\Bhas\nvanishing cohomology. Namely, H1(Γ/B, S m) = 0 .\nThe rest of this section is devoted to the proof of Lemma 6.6. First, we note that Lemma 6.2, to show\nthatH1(Γ\\B, S m) = 0 it suffices to show that Hom(Γ , Sm) = 0 . By the above, it is sufficient to establish\nthatHom( Kν, Sm) = 0 for each ν̸=p,∞.\nWhen ν=p0, this follows from the fact that Kνis a pro p0-group and p0> m . This implies that its\nquotients are p-groups and therefore cannot be embedded inside Smwhen m < p .\nForν̸=p0this follows from the fact that the minimal dimension of a sub-representation of SP2n(Zp)\ngoes to infinity with n, and so if nis sufficiently large Hom(SP 2n(Zp), Sm) = 0 as each such permutation\nrepresentation would give rise to a complex representation of dimension m, which would therefore have to\nbe trivial. The lower bound on the dimension can be easily deduced by adapting the method due to Howe\nand Gurevich of U-rank, and we give the details below.5\nLemma 6.7. The minimal dimension of a non-trivial representation of SP2n(Zp)tends to infinity as a\nfunction of nuniformly over all p.\nProof. Letχ= tr◦ρbe the character of a nontrivial representation. Let Bbe the Abelian group of\nsymmetric matrices over Zpand let G= GL n(Zp),which acts on the group BviaAB=BtAB. The group\nSP2n(Zp). Recall that the group SP2n(Zp)conists of the matrices that preserves a symplectic form. They\nare generated by the matrices of the form\u0012I A\nO I\u0013\n, where Ais symmetric matrix with entries in Zp, those\nmatrices of the form\u0012C O\nO\u0000\nCt\u0001−1\u0013\n, where Cis inGLn(Zp)together with the element w=\u0012O I\n−I O\u0013\n.\n5A stronger form of the lemma below will appear in a future paper of Evra, Kindler, Lifshitz, and Pirani that will extend the\ntheory of U-rank over p-adic groups.\n56Let us denote the first group of elements by Band the second group of elements G.Then the product\nBGis a semi direct product B⋊Gand it known as the Siegel parabolic subgroup.\nNow the normal subgroup that Bgenerates can easily be seen to be all of SP2n(Zp).This shows that\nthe restriction of ρtoBis also nontrivial. Indeed, if the restriction of ρtoBis trivial this would imply that\nρon all the conjugates of Band therefore also on the normal subgroup that it generates.\nTherefore, the restriction of χtoBis a a non-constant function. Let us denote the restriction by f. Then\nsince fis invariant under the conjugation action of GonBwe have f(BtAB) =f(A)for all B∈G. This\nimplies that for each character χXin the Pontryagin dual of Ball its equivalence classes χBXBtalso lies in\nthe support of f. Now the dimension of χis equal to f(1)and standard representation theory implies that\nthe Fourier coefficients of fare nonnegative integers.\nWe may therefore lower bound the dimension χ(1)by the minimal size of an orbit of a nontrivial\ncharacter in the Pontryagin dual of B. Each character in the Pontryagin dual corresponds to a symmetric\nmatrix XoverZ/pkandχX(A) =ωtr(XA)\np . Now the orbit of Xunder the action of Gcan be lower bounded\nby the orbit of PiXfor each i. We may therefore multiply Xby powers of puntil all its entries are multiples\nofpk−1and lower bound the orbit of the resulting character. However, the resulting matrix has entries in Fp\nand the lower bound on the orbit of Xwas established for that case by Gurevich and Howe [GH17].\n6.2 Cosystolic Expansion for Gr[X]\nIn this section we show that Gr[X]has sufficiently strong cosystolic expansion. To prove this we use a local\nto global theorem of [DD23b] that shows that when the links are coboundary expanders then the complex is a\ncosystolic expander. Let us first show that the vertex links of the Chapman-Lubotzky complex are spherical\nbuildings of type C, as well as tensor products of type A and type C buildings.\nLemma 6.8. The Chapman-Lubotzky complex Xdiscussed in Section 6.1 has vertex links that are spherical\nbuildings of either type Cn−1, tensor of type A1and type Cn−2, or tensor of type Ckand type Cn−k−1.\nFurthermore, Xis anO(1/√p)-one-sided local spectral expander.\nProof. Consider the complex X, and note that by construction the links of Xare the same as the links of\nΓ. The affine building Γhas a Coxeter diagram of type ˜Cn. By [AB08, Proposition 3.16], the links of\nΓcorrespond to spherical buildings whose Coxeter diagram is the result of deleting one vertex from the\ndiagram ˜Cn(see also [Ess10, Lemma 3.1.13]). By inspection, it follows that all links are spherical buildings\nthat are either type Cn−1, tensor of type A1and type Cn−2, or tensor of type Ckand type Cn−k−1. These\ncomplexes correspond to subspaces over the field Fp(see [Nel23] for a description of the ˜Cnbuilding).\nUsing Lemmas 2.20 and 2.15 we know that the spherical buildings of type C and type A are O(1/√p)-\nproduct distributions or equivalently O(1/√p)-one-sided local spectral expanders. This implies that the\nsame also holds for their tensor products. Using the trickling-down theorem, Theorem 2.12 we then get that\nXis also an O(1/√p)-one-sided local spectral expander.\nBelow we state the formal definition of cosystolic expansion for graphs and complexes that will be useful\nfor us.\nDefinition 6.9. We say a graph Gis aC-cosystolic expander with respect to Smif the following holds. Let\nΦbe a Unique-Games instance on GoverSmwith incons (Φ) = δ. Then one can change the constraints of\nΦonCδ-fraction of the edges to get Φ′such that incons (Φ′) = 0 . We say that a complex Xis aC-cosystolic\nexpander over Smif the graph G1[X]is aC-cosystolic expander over Sm.\n57In literature the above definition is known as the 1-cosystolic expansion of XoverSm, and has various\ngeneralizations to higher levels of X(when Smis replaced by an Abelian group) but we refrain from stating\nthose definitions.\nLet us now state the local to global theorem. For a complex XletY=X⩽Rdenote the “cutoff” of X,\ni.e.Y= (X(1), . . . , X (R)), equipped with the same distributions.\nTheorem 6.10 ([DD23b] Theorem 1.2 modified) .There exists a constant Rsuch that for all β, λ > 0and\nm, d, C ∈Nwithβ, λ⩽poly(1 /C)the following holds. Let Xbe ad-dimensional complex such that X⩽R\nis aλ-one-sided local spectral expander and for all v∈X(1),G1[Xv]is a(C, β)-coboundary expander\noverSm. Then Xis apoly( C)-cosystolic expander over Sm.\nProof. First, [DD23b, Theorem 1.2] asserts that if the one-skeletons of all links of Xareλ-one-sided\nexpanders and for all v∈X(1)the complex Xvis aC-coboundary expander over Sm, then Xis apoly( C)\ncosystolic expander over Sm.\nWe discuss how to change their argument to get the stronger theorem. Firstly their argument only uses\nthe link expansion to derive spectral gaps of the up-down walks on XonR′< R levels. Therefore we can\nlook at Y=X⩽Rinstead and using the local spectral expansion of Y, derive that these up-down walks on\nYare sufficiently expanding. But these are the same as the walks on Xtherefore hence the latter also have\nthe required expansion properties. Hence we only require the local spectral expansion assumption on X⩽R.\nNow let us discuss the requirements on the coboundary expansion of links – we only have that the\n1-links are (C, β)-coboundary expanders over Sm, instead of C-coboundary expanders. Their argument\nworks as is, except in the step that they apply coboundary expansion. They use the fact that given any UG\ninstance: incons (Φ)⩾1\nCviol(Φ), but we can instead get that, incons (Φ)⩾1\nCviol(Φ)−β\nC. This error of\nβ/C gets absorbed into the other additive errors that depend on “ η” (check the proof overview) and λas\nlong as β⩽poly(1 /C). Hence this error does not affect the rest of the argument and we get the same\nconclusion.\nLemma 6.11. For all r≪d≪qthe following holds. Let Xbe the d-dimensional Chapman-Lubotzky\ncomplex over Fq. Then Gr[X]is a2O(r0.99logr)-cosystolic expander over Smfor all m∈N.\nProof. Consider the complex Xr, with\nXr(1) ={v|v∈X(r)}, Xr(2) ={(u, v)|u, v∈X(r)such that u∪v∈X(2r)}\nand more generally\nXr(ℓ) ={(u1, . . . , u ℓ) :u1, . . . , u ℓ∈X(r)such that u1∪. . .∪uℓ∈X(ℓr)}.\nNote that that Gr[X] =G1[Xr]. We intend to use Theorem 6.10, and for that we verify that the constant-\nsized links have one-sided expansion on the 1-skeletons, and that the 1-links of Xrare coboundary ex-\npanders over Sm.\nOne-sided local spectral expansion of (Xr)⩽R:We will show that (Xr)⩽Ris an exp(−r10)-one-sided\nlocal spectral expander. Let us fix an i-face FofXrfori⩽R−2and upper bound the second eigenvalue\nof the 1-skeleton of (Xr)F. Let Ybe the complex XFwithd′:= dim( Y) =d−ir. Then bounding the\nsecond eigenvalue of the 1-skeleton corresponds to bounding the second eigenvalue of the random walk W\nthat picks a random r-face A∈Y(r), goes up to a random 2r-face (A, B)∈Y(2r)and then outputs the\nr-face B∈Y(r). We can check that Wis1−O(r2/d′)-close to the walk W′that picks a random d′-face I\n58inYand two uniformly random r-faces inside I. This is because W′conditioned on outputting two disjoint\nr-faces is the same as W. The probability that W′outputs disjoint faces is at least 1−O(r2/d′), therefore\ngiving us that the second eigenvalues of WandW′differ by at most O(r2/d′). But W′is the same as the\nup-down walk from X(r)→X(d′)→X(r), which by Lemma 2.13 has a second eigenvalue of at most\nO(r/d′)since XFis aO(1/√q)-one-sided local spectral expander. This gives us that every i-link of Xr\nhas a second eigenvalue of at most O(r/(d−ir))which is at most 2−r12by taking dto be a large enough\nfunction of r, R, as required.\nCoboundary expansion of 1-links of Xr:We will show that for all links I∈Xr(1),G1[(Xr)I]are\n(2O(r0.99logr),exp(−r10))-coboundary expanders. Fix some I∈Xr(1). By definition of Xr,Icorresponds\nto anr-face in X, which we also denote by I. Letv∈X(1)be some vertex that belongs to I. By Lemma 6.8\nwe know that Xvis a spherical building either of type Cn−1, tensor of type A1and type Cn−2, or tensor\nof type Ckand type Cn−k−1. Let Y=Xv, associated with the distribution µover its maximal faces. We\nget that the complex XIis some r−1-link of Ydenoted by YS→A0and associated with the distribution\nµ|YS=A0. One can check that the complex (Xr)Iis then equal to (XI)rwhich in turn equals (YS→A0)r.\nSo we get that, G1[(Xr)I] =G1[(YS→A0)r] =Gr(µ|XS=A0)which is a (2O(r0.99logr),2−Ω(r12))-\ncoboundary expander by Lemma 5.3. So we get that all the vertex links of Xrare coboundary expanders.\nIn the two paragraphs above, we have shown that Xrsatisfies the hypotheses of Lemma 6.10. Therefore\napplying the lemma on Xrwe get that Xror equivalently the graph G1[Xr] =Gr[X]is a2O(r0.99logr)-\ncosystolic expander over Sm.\n6.3 Chapman-Lubotzky complex is a UG Coboundary Expander\nTheorem 6.12. For all m, r≪d≪q∈Nthe following holds. Let Xbe the d-dimensional Chapman-\nLubotzky complex over Fq. Then Xis an (m, r, α (r), α(r))-UG coboundary expander for\nα(r) = 2O(r0.99logr).\nProof. From Lemma 6.6 we know that Xhas vanishing 1-cohomology over Sm. We also know that Xis a\nwell-connected clique complex [CL23]. We now use [DD23a, Lemma 3.7], which asserts that if a complex\nXis a well-connected clique complex with G1[X]having vanishing cohomology then Gr[X]also has\nvanishing cohomology. We also know from Lemma 6.11 that Gr[X]is an2O(r0.99logr)-cosystolic expander\noverSm. Combining the two facts proves that Gr[X]is an 2O(r0.99logr)-coboundary expander over Sm,\ntherefore an (m, r, α (r), α(r))-UG coboundary expander.\n6.4 Proof of Theorem 1.3\nFixε, δ > 0and take randmsufficiently large. Take the Chapman Lubotzky complex Xfor sufficiently\nlargenand prime p, which is a one-sided local spectral expander [CL23]. Combining this with Theorem 6.12\nwe get that Xsatisfies the conditions of Theorem 1.8, and therefore we conclude that the canonical direct\nproduct tester of Xhas soundness at most δ.\n7 Acknowledgements\nWe would like to thank Alex Lubotzky for several helpful discussions. In particular, Alex suggested to us\nthe use of the Chapman-Lubotzky complex and sent us a preliminary version of [CL23]. We thank Yotam\n59Dikstein, Irit Dinur and Alex Lubotzky for telling us that the Chapman-Lubotzky complex has vanishing\n1-cohomology over Sm. We would also like to thank Shai Evra, Michael Chapman and Tali Kaufman for\nhelpful conversations and the Simons Institute for the Theory of Computing, where part of this work took\nplace when the authors were participating in the “Analysis and TCS: New Frontiers” program.\nReferences\n[AB08] Peter Abramenko and Kenneth S Brown. Buildings: theory and applications , volume 248.\nSpringer Science & Business Media, 2008.\n[AL20] Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and appli-\ncations. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing,\nSTOC 2020, Chicago, IL, USA, June 22-26, 2020 , pages 1198–1211. ACM, 2020.\n[ALM+98] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof veri-\nfication and the hardness of approximation problems. Journal of the ACM (JACM) , 45(3):501–\n555, 1998.\n[AS98] Sanjeev Arora and Shmuel Safra. 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Comput. , 27(3):763–803, 1998.\n62"    },    {        "title": "2402.00865v1.Towards_Optimal_Feature_Shaping_Methods_for_Out_of_Distribution_Detection.pdf",        "content": "Published as a conference paper at ICLR 2024\nTOWARDS OPTIMAL FEATURE -SHAPING METHODS FOR\nOUT-OF-DISTRIBUTION DETECTION\nQinyu Zhao1, Ming Xu1, Kartik Gupta2, Akshay Asthana2, Liang Zheng1, Stephen Gould1\n1The Australian National University\n2Seeing Machines Ltd\n{qinyu.zhao,mingda.xu,liang.zheng,stephen.gould}@anu.edu.au\n{kartik.gupta,akshay.asthana}@seeingmachines.com\nABSTRACT\nFeature shaping refers to a family of methods that exhibit state-of-the-art perfor-\nmance for out-of-distribution (OOD) detection. These approaches manipulate\nthe feature representation, typically from the penultimate layer of a pre-trained\ndeep learning model, so as to better differentiate between in-distribution (ID) and\nOOD samples. However, existing feature-shaping methods usually employ rules\nmanually designed for specific model architectures and OOD datasets, which con-\nsequently limit their generalization ability. To address this gap, we first formulate\nan abstract optimization framework for studying feature-shaping methods. We then\npropose a concrete reduction of the framework with a simple piecewise constant\nshaping function and show that existing feature-shaping methods approximate the\noptimal solution to the concrete optimization problem. Further, assuming that OOD\ndata is inaccessible, we propose a formulation that yields a closed-form solution\nfor the piecewise constant shaping function, utilizing solely the ID data. Through\nextensive experiments, we show that the feature-shaping function optimized by our\nmethod improves the generalization ability of OOD detection across a large variety\nof datasets and model architectures.1\n1 I NTRODUCTION\nOut-of-distribution (OOD) detection aims to identify test samples that fall outside the inherent training\nlabel space, given a deep learning model pre-trained on an in-distribution (ID) training set. To detect\nOOD samples, OOD scores, such as maximum softmax probability (MSP) (Hendrycks & Gimpel,\n2016) and energy score (Liu et al., 2020) are computed using the logits estimated by the model, where\na lower score indicates a higher probability that the sample is OOD.\nFeature shaping (Sun et al., 2021; Djurisic et al., 2022; Xu & Lian, 2023; Song et al., 2022) refers\nto a family of methods that manipulate the underlying feature representations, typically from the\npenultimate layer of a pre-trained model, such that OOD scores can more effectively distinguish\nbetween ID and OOD data. Representative methods like ReAct (Sun et al., 2021) and ASH-P (Djurisic\net al., 2022) employ element-wise feature clipping and percentile-based feature pruning, respectively.\nDespite their notable achievements, existing feature-shaping methods usually rely on rules manu-\nally designed for specific OOD datasets and model architectures, which consequently limit their\ngeneralization ability. Our observations indicate that variations in OOD datasets and model archi-\ntectures can cause significant performance degradation in these methods. For instance, when using\ntransformer-based models, some approaches, despite achieving state-of-the-art results on established\nbenchmarks, perform no better than random guessing. These observations emphasize the necessity\nfor a comprehensive understanding and comparison of the current feature-shaping methods and the\ndevelopment of a more generalizable approach.\nTo this end, we first formulate an abstract optimization problem that encapsulates all feature-shaping\nOOD detection methods. In a nutshell, our formulation entails finding the shaping function that\nmaximizes some performance objective for OOD detection.\n1Our code is available at https://github.com/Qinyu-Allen-Zhao/OptFSOOD.\n1arXiv:2402.00865v1  [cs.CV]  1 Feb 2024Published as a conference paper at ICLR 2024\nReActBFActASH-PASH-BASH-SVRA-POurs020406080AUC\n(a)ReActBFActASH-PASH-BASH-SVRA-POurs020406080\n(b)ReActBFActASH-PASH-BASH-SVRA-POurs45505560657075\n(c)\n80 85\nAUC (2 ConvNets)20406080AUC (4 Transformers)MSP\nODIN\nEnergy\nDICEReActBFAct\nASH-PASH-B\nASH-SVRA-POurs\n(d)\nFigure 1: Comparing our method with existing feature-shaping methods. The dashed lines denote the\nperformance of our method for comparison. (a) ImageNet (ID) vs.iNaturalist (OOD) with ViT-B-16;\n(b) ImageNet (ID) vs.iNaturalist (OOD) with MLP-Mixer-B; (c) CIFAR100 (ID) vs.CIFAR10\n(OOD) with MLP-Mixer-Nano; (d) Average performance of different methods across eight OOD\ndatasets with two ConvNets and with four transformer-based models.\nTo make the optimization problem concrete and tractable, we propose a piecewise constant shaping\nfunction and formulate an objective function that maximizes the expected difference between the\nmaximum logit values for ID and OOD data, assuming the availability of OOD training samples. An\nimportant benefit of this formulation is that it highlights the operational mechanisms underpinning\nprevious feature-shaping methods. Namely, the shaping functions used by existing feature-shaping\nmethods including ReAct (Sun et al., 2021), BFAct (Kong & Li, 2023), and VRA-P (Xu & Lian,\n2023), approximate the optimal shaping function found by our approach.\nUsing our framework, we further propose a feature-shaping method that can be optimized without\naccess to OOD data. Specifically, through the analysis of the effect of OOD data on the optimal\nsolution compared to ID data, we argue that it is feasible to remove the OOD-related term in\nthe objective function, thereby deriving a closed-form solution based on ID data only. Empirical\nobservations in experiments support our claim.\nOur experiments compare our proposed approach to existing feature-shaping methods, utilizing a more\ncomprehensive experimental setup than previous works. We use CIFAR10, CIFAR100 (Krizhevsky\net al., 2009), and ImageNet-1k (Russakovsky et al., 2015) as ID datasets. For each ID dataset, we use\neight OOD datasets for evaluation. Furthermore, we evaluate all methods using a wide variety of\nmodels, including convolutional networks (ConvNets), transformers and fully multi-layer perceptron\n(MLP) models, the latter two of which have not been extensively studied in prior works. As shown\nin Fig. 1, compared with other feature-shaping methods, ours demonstrates more generalized OOD\ndetection performance across a broad spectrum of datasets and model architectures.\nIn summary, this paper covers three main points. First, we develop a general optimization formulation\nfor feature-shaping methods and show that a concrete instance of this problem provides insight\ninto the mechanisms of previous methods. Second, we propose a novel approach for training a\ngeneralizable feature-shaping method for OOD detection without access to OOD data. Last, our\napproach achieves more reliable OOD detection performance across various datasets and models.\n2 A R ECAP OF FEATURE SHAPING FOR OOD D ETECTION\nIn this section, we define the problem of OOD detection for image classification, with particular\nemphasis on feature-shaping approaches, and subsequently, introduce our general optimization\nformulation of feature shaping for OOD detection.\n2.1 OOD DETECTION FOR IMAGE CLASSIFICATION\nAssume that we have an ID dataset, defined as a joint distribution DID\nX,Yover input observations X\nand discrete labels Yfor a given model M. Furthermore, assume we have an OOD dataset DOOD\nX,Y′.\nPrevious works (Sun et al., 2021; Sun & Li, 2022; Yang et al., 2021) assume Y ∩ Y′=∅. Inherent in\nthis assumption is that the marginal distributions over observations Xfor ID and OOD datasets do\nnot coincide. That is, DID\nX̸=DOOD\nX.\n2Published as a conference paper at ICLR 2024\nThe model, usually a neural network, maps observations to logits, M:X →RC, where C=|Y|\ndenotes the number of classes. It is trained on a dataset Strain={(x(i), y(i))}N\ni=1, comprised of N\ni.i.d. samples drawn from DID\nX,Y. Furthermore, for typical architectures for image classification, we\ncan decompose the model M=W ◦ B into a backbone model, B:X →RM, and linear classifier\nW:RM→RC.Bextracts penultimate layer feature representation z∈RMof an input observation\nx, andWis a linear layer that maps zto logits ℓ∈RC. Different from conventional machine\nlearning practice, the test set Stestcontains inputs sampled from the joint distribution DID\nX∪DOOD\nX,\nwhere DID\nXandDOOD\nXdenote the marginal distributions of DID\nX,YandDOOD\nX,Y′, respectively.\nThe goal of OOD detection is to design a score function sto indicate whether a sample xis from\nDID\nXorDOOD\nX. MSP, maximum logit score (MLS), and energy score serve as examples of scores for\nOOD detection. They are defined based on logits ℓas:\nsMSP(ℓ) = max\ni=1,...,Cexp(ℓi)PC\nj=1exp(ℓj), s MLS(ℓ) = max\ni=1,...,Cℓi, s Energy(ℓ) = logCX\nj=1exp(ℓj),(1)\nwhere ℓiis the logit for the i-th class. For the scores discussed above, a lower value indicates a higher\nlikelihood of the sample being OOD. In practical applications, a threshold, denoted by t, is utilized to\nseparate ID samples from OOD ones. If the score of a sample surpasses this threshold, the sample\nwill be classified as ID, and OOD otherwise.\n2.2 F EATURE -SHAPING\nA feature-shaping method uses a shaping function f:RM→RMto manipulate features, i.e.,\n¯z=f(z). The goal of developing a feature shaping method for OOD detection can be formulated as\nsolving the following optimization problem,\nmaximize (over f)ExID∼DID\nX,xOOD∼DOOD\nXh\nPn\ns(¯ℓID), s(¯ℓOOD)oi\n,\nsubject to ¯ℓID=W(¯zID),¯zID=f(B(xID)),\n¯ℓOOD=W(¯zOOD),¯zOOD=f(B(xOOD)),(2)\nwhere Pis a given performance measure. For example, a common measure is the area under the\nreceiver operating characteristic curve (AUC), defined as PAUC(s1, s2) = 1{s1> s2}. Existing\nshaping functions fcan be broadly categorized into two types: (i) element-wise, such as ReAct (Sun\net al., 2021) and VRA-P (Xu & Lian, 2023), which applies a global element-wise mapping to each\nindividual feature, i.e., ¯zi=f(zi), or (ii) whole-vector mappings such as ASH (Djurisic et al., 2022),\nwhich manipulate the full feature vector z.\n3 O PTIMAL FEATURE SHAPING\nIn this section, we propose a specific instance of the general formulation described in Section 2,\nfollowing previous element-wise feature-shaping methods (Sun et al., 2021; Xu & Lian, 2023; Kong\n& Li, 2023). To optimize the shaping function, we initially examine which feature value ranges are\neffective for OOD detection. A significant challenge arises from the continuous nature of feature\nvalues. To address this, we partition the domain of features into a disjoint set of intervals, which\nallows us to approximate the optimal shaping function with a piece-wise constant function. With\nthis approximation, we can formulate an optimization problem to maximize the expected difference\nbetween the maximum logit values of ID and OOD data. Subsequently, we show that the optimized\nshaping function using OOD samples bears a striking resemblance to prior methods, shedding light\non the mechanics of these methods. Finally, we propose a formulation that does not require any OOD\ndata to optimize and admits a simple, closed-form solution.\n3.1 V ALUE -AND INTERVAL -SPECIFIC FEATURE IMPACT\nGiven a sample x, the maximum logit ℓmax(disregarding the bias term) is the dot product between\nthe corresponding weight vector wmaxand its feature representation z, specifically,\nℓmax(z) =wmax·z=MX\ni=1wmax\nizi. (3)\n3Published as a conference paper at ICLR 2024\nAssuming an element-wise feature-shaping function f:R→R, define θ(zi)≜f(zi)\nzi. Then\nassuming the weights wmaxdo not change2, the maximum logit after shaping is:\n¯ℓmax(¯z) =MX\ni=1wmax\nif(zi) =MX\ni=1θ(zi)wmax\nizi. (4)\nTo design θ(z)and understand which feature value ranges are effective for OOD detection, we\ncompute the contribution of features with specific values to the maximum logit. Concretely, we\ndefine the value-specific feature impact (VSFI): V(z) =PM\ni=1 1{zi=z}wmax\nizi. However, zis a\ncontinuous variable, and real-world data only provide a countable number of features, which can\ngives overly sparse distributions of VSFIs. To address this, we partition the domain of individual\nfeatures into a disjoint set of intervals Athat cover zi. Formally, the interval-specific feature impact\n(ISFI) on an interval [a, b)∈ A is given by:\nI[a,b)(z) =MX\ni=11{a≤zi< b}wmax\nizi=X\ni:a≤zi<bwmax\nizi. (5)\nThen, for a single sample, the maximum logit is the summation over all ISFIs, given by:\nℓmax(z) =X\n[a,b)∈AI[a,b)(z). (6)\nIn our experiments, we divide feature values into equal-width intervals. Given that feature distributions\ntypically exhibit long-tailed characteristics (Sun et al., 2021), we select the 0.1 and 99.9 percentiles\nof all features in a training set as the lower αand upper βlimits of feature values, respectively. We\nthen partition the range [α, β)intoK= 100 equal-width intervals. This results in a set of intervals\nA={[ak, bk)}K\nk=1, where ak=α+ (k−1)δ,bk=α+kδ, andδ=β−α\nKfork= 1, . . . , K . Then,\nthe maximum logit can be expressed as:\nℓmax(z) =X\n[a,b)∈AI[a,b)(z) =KX\nk=1I[ak,bk)(z), (7)\nand for each data sample, we can generate a K-dim ISFI vector denoted as:\nI(z) =\u0000\nI[a1,b1)(z), . . . , I [aK,bK)(z)\u0001\n∈RK. (8)\n3.2 O PTIMIZING THE RESHAPING FUNCTION\nWe now describe how to optimize θ(z), assuming θ(z)is piecewise constant over the intervals A. For\nan interval [ak, bk), we estimate a scalar ˆθkas a constant approximation to θ(z)within the interval.\nIn the limit as the interval widths shrinks to zero, we recover the continuous, element-wise shaping\nfunction. We can approximate the maximum logit as,\n¯ℓmax(¯z) =KX\nk=1ˆθkI[ak,bk)(z). (9)\nWhile previous works (Sun et al., 2021; Xu & Lian, 2023) evaluate their methods by computing the\nexpected difference in maximum logits between ID and OOD data, optimizing this criteria solely\ndoes not improve OOD detection performance. To see this, observe that simply uniformly scaling\nfeatures by a large factor increases the expected difference in maximum logits without enhancing\nthe differentiation between ID and OOD samples. This motivates introducing regularization on the\nshaping function and leads us to the following problem3:\nmaximize (over θ)ExID∼DID\nX,xOOD∼DOOD\nXh\nθTI(zID)−θTI(zOOD)i\nsubject to ∥θ∥=√\nK,zID=B(xID),zOOD=B(xOOD),(10)\n2We validate this assumption in Section B.2.\n3We constrain ∥θ∥to be equal to ∥1∥=√\nKinstead of other values. Reasons can be found in Section A.2.\n4Published as a conference paper at ICLR 2024\n0.5\n0.00.51.0(z)\n(a)Ours (w/ OOD)\nReAct\n(b)Ours (w/ OOD)\nBFAct\n(c)Ours (w/ OOD)\nVRA-P\n(d)Ours (w/ OOD)\nOurs (w/o OOD)\n0 2 4\nFeature value z0.5\n0.00.51.01.5(z)\n(e)Ours (w/ OOD)\nASH-P\n0 2 4\nFeature value z\n(f)Ours (w/ OOD)\nASH-B\n0 2 4\nFeature value z\n(g)Ours (w/ OOD)\nASH-S\n0 2 4\nFeature value z\n(h)Ours (w/ OOD)\nASH-Rand\nFigure 2: Visualization of shaping functions. The blue lines (ours w/ OOD) derive from Eq. 10, while\nthe green line (ours w/o OOD) from Eq. 14. Red lines represent different existing methods, while\nshaded regions indicate estimated standard deviations. θhas been rescaled for the best visualization.\nwhere due to the linearity of expectation, the objective function can be expressed as:\nθT\u0010\nExID∼DID\nX\u0002\nI(zID)\u0003\n−ExOOD∼DOOD\nX\u0002\nI(zOOD)\u0003\u0011\n. (11)\nWe extract ISFI vectors for the ImageNet-1k training set (ID) (Russakovsky et al., 2015) and for\nthe iNaturalist dataset (OOD) (Van Horn et al., 2018), using a ResNet50 model pre-trained on\nImageNet-1k training set. The estimated ˆθkis plotted in Fig. 2.\nFor completeness, we describe existing feature-shaping methods (Sun et al., 2021; Kong & Li, 2023;\nXu & Lian, 2023) using our reparameterization of the feature shaping problem with θ(z),\nθReAct(z) =min(z, t)\nz, θ BFAct(z) =1q\n1 + (z\nt)2N, θ VRA-P (z) =\n\n0 z <low\n1 low≤z <high\nhigh\nzhigh≤z,\n(12)\nwhere t,N, low, and high are hyperparameters. We then plot the resultant θ(z)in Fig. 2(a-c). We\nshow that these prior works are empirically approximating our optimal piecewise constant shaping\nfunction. A difference between our shaping function and VRA-P, is that VRA-P sets low-value\nfeatures to zero while ours flips the sign of these features to better utilize them for OOD detection.\nThen, we extend our analysis to other feature-shaping methods. ASH-P, -B, -S, and -Rand denote\nfour simplistic yet effective methods introduced by Djurisic et al. (2022). In contrast to ReAct, which\nemploys a global threshold to clip features, ASH methods incorporate the entire feature vector and\nutilize a local threshold to process the feature representation of each instance. Their shaping functions\nare not reducible to element-wise functions.\nWe instead approximate the average shaping effect within each feature value interval. Specifically,\nwe utilize ImageNet-1k validation set (ID) and iNaturalist (OOD) with ResNet50. For an interval\n[ak, bk), we retrieve all features from ID and OOD datasets that fall within this range, along with\ntheir reshaped counterparts. Following this, the ratio of reshaped and original values are computed as\nθkfor a specific data instance. Subsequently, we calculate the mean value and standard deviation of\nthese θkvalues over all samples on each interval. The results are depicted in Fig. 2(e-h).\n3.3 O PTIMIZING THE SHAPING FUNCTION WITHOUT OOD SAMPLES\n5Published as a conference paper at ICLR 2024\nID\nHard OOD\nEasy OOD𝜽 optimized\nw/ OOD data𝜽 optimized\nw/o OOD data\nFigure 3: Diagram to show the intuition\nin deriving Eq. 13.Normally, we have access to the ID training set, en-\nabling us to estimate ExID∼DID\nX\u0002\nI(zID)\u0003\n. However, a\nsignificant challenge arises with the unknown term\nExOOD∼DOOD\nX\u0002\nI(zOOD)\u0003\n, which renders the optimization\nproblem defined in Eq. 10 intractable.\nTo make the problem tractable without access to OOD data,\nwe omit the OOD-related term ExOOD∼DOOD\nX\u0002\nI(zOOD)\u0003\nby\nthe following argument, which as we will see is also sup-\nported empirically in Section 4.2.\nA diagram to explain our intuition is shown in Fig. 3.\nSpecifically, in scenarios where the OOD distribution\nclosely resembles the ID (hard OOD), OOD samples likely\nhave similar feature representations to ID samples, imply-\ning both expectations share similar directions. Conversely,\nin cases where OOD samples significantly diverge from\nID samples (easy OOD), the maximum logits of OOD samples tend to be lower, thereby reducing the\nmagnitude of ExOOD∼DOOD\nX\u0002\nI(zOOD)\u0003\ncompared to ID.\nThus, the component of ExOOD∼DOOD\nX\u0002\nI(zOOD)\u0003\northogonal to ID expectation is small and we suggest\nomitting ExOOD∼DOOD\nX\u0002\nI(zOOD)\u0003\n, resulting in the following formulation:\nmaximize (over θ)θTEx∼DID\nX[I(z)]\nsubject to ∥θ∥=√\nK,z=B(x).(13)\nThe optimal solution to this new problem can be expressed in closed-form as:\nθ⋆=√\nK\n∥Ex∼DID\nX[I(z)]∥2Ex∼DID\nX[I(z)]. (14)\nWe plot θ⋆calculated by Eq. 14 in Fig. 2(d), noting that it is in close proximity to the optimal solution\nin the case where OOD samples are available. In experiments, we will also show that our method\nwith only ID data performs almost on par with using the true I(zOOD)calculated from OOD data.\nAt inference time, for each sample, we rescale its features within each interval [ak, bk)by the\ncorresponding entry in θ⋆, and calculate an OOD score based on the reshaped feature representation.\n4 E XPERIMENTS\nDatasets. We evaluate our proposed method using a large scale benchmark where ImageNet-\n1k (Russakovsky et al., 2015) is the ID dataset, as well as two moderate scale benchmarks where\nCIFAR10 and CIFAR100 (Krizhevsky et al., 2009) are used as the ID datasets. We evaluate eight\nOOD datasets for the ImageNet benchmark, comprised of Species (Hendrycks et al., 2019b), iNatural-\nist (Van Horn et al., 2018), SUN (Xiao et al., 2010), Places (Zhou et al., 2017), OpenImage-O (Wang\net al., 2022), ImageNet-O (Hendrycks et al., 2021), Texture (Cimpoi et al., 2014), and MNIST (Deng,\n2012). When CIFAR10/100 is the ID dataset, eight OOD datasets are exploited, comprised of\nTinyImageNet (Torralba et al., 2008), SVHN (Netzer et al., 2011), Texture (Cimpoi et al., 2014),\nPlaces365 (Zhou et al., 2017), LSUN-Cropped (Yu et al., 2015), LSUN-Resized (Yu et al., 2015),\niSUN (Xu et al., 2015), and finally CIFAR100/10.\nModel architectures. We use pre-trained models provided by PyTorch and prior works in our\nexperiments. For ImageNet-1k, we use a pre-trained ResNet50 (He et al., 2016) provided by PyTorch\nand a MobileNet-v2 (Sandler et al., 2018) provided by Djurisic et al. (2022). Additionally, we include\nmodels based on the transformer and fully-MLP architectures, which have been largely overlooked in\nprevious studies. Specifically, we include ViT-B-16, ViT-L-16 (Dosovitskiy et al., 2020), SWIN-S,\nSWIN-B (Liu et al., 2021), MLP-Mixer-B, and MLP-Mixer-L (Tolstikhin et al., 2021). For the\nCIFAR10 and CIFAR100 benchmarks, we use a DenseNet101 (Huang et al., 2017) model provided\nby Sun & Li (2022), a ViT-B-16 finetuned on CIFAR10/100 consistent with Fort et al. (2021), and a\nMLP-Mixer-Nano model trained on CIFAR10/100 from scratch.\n6Published as a conference paper at ICLR 2024\nTable 1: OOD detection results on ImageNet-1k. ↑indicates larger values are better and ↓indicates\nsmaller values are better. All values are percentages averaged over different OOD datasets. Bold\nnumbers are superior results whereas underlined numbers denote the second and third best results.\nMethodResNet50 MobileNetV2 ViT-B-16 ViT-L-16 SWIN-S SWIN-B MLP-B MLP-L Average\nFP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑\nMSP 69.30 76.26 72.58 77.41 69.84 77.40 70.59 78.40 65.12 80.89 69.45 77.26 70.66 80.31 71.63 79.93 69.90 78.48\nODINa61.56 80.92 62.91 82.64 69.25 72.60 70.35 74.51 62.35 78.67 70.90 68.36 65.00 82.99 67.90 81.86 66.28 77.82\nEnergy 60.97 81.01 61.40 82.83 73.96 67.65 74.89 70.11 66.56 75.95 79.53 60.14 87.86 78.42 84.60 79.01 73.72 74.39\nDICE 45.32 83.64 49.33 84.63 89.68 71.32 72.38 67.08 78.84 49.06 77.89 50.10 57.12 80.60 61.98 80.75 66.57 70.90\nReAct 42.29 86.54 54.19 85.37 73.82 76.86 76.16 81.07 58.07 85.10 70.23 80.10 90.13 77.18 89.03 78.32 69.24 81.32\nBFAct 43.87 86.01 52.87 85.78 77.64 80.16 84.02 81.12 64.76 85.57 68.40 84.36 96.04 67.87 96.00 70.59 72.95 80.18\nASH-P 55.30 83.00 59.41 83.84 99.36 21.17 99.18 20.27 99.09 21.78 98.83 22.11 98.19 35.85 98.91 29.69 88.53 39.71\nASH-B 35.97 88.62 43.59 88.28 94.87 46.68 93.72 38.95 96.48 36.54 92.40 49.85 99.51 21.73 64.91 83.15 77.68 56.73\nASH-S 34.70 90.25 43.84 88.24 99.48 18.52 99.42 18.61 99.20 18.26 99.06 19.27 97.61 33.91 99.28 19.29 84.07 38.29\nVRA-P 37.97 88.58 49.98 86.83 98.39 35.66 99.58 16.70 99.27 20.34 99.46 17.64 99.38 18.73 99.05 21.23 85.39 38.21\nOurs (V) 41.84 88.11 53.31 86.36 69.33 82.59 72.17 83.23 66.89 84.53 65.96 83.71 76.59 78.46 78.13 78.85 65.53 83.23\nOurs (E) 39.75 88.56 51.77 86.62 69.52 82.66 78.57 82.94 66.28 84.99 66.20 84.21 83.54 75.94 85.50 78.11 67.64 83.00\naChoice of hyperparameters on Imagenet makes ODIN yield identical results to MLS.\nTable 2: OOD detection results on CIFAR10/100. ↑indicates larger values are better and ↓indicates\nsmaller values are better. All values are percentages averaged over different OOD datasets. Bold\nnumbers are superior results whereas underlined numbers denote the second and third best results.\nMethodCIFAR10 CIFAR100\nDenseNet101 ViT-B-16 MLP-N Average DenseNet101 ViT-B-16 MLP-N Average\nFP↓ AU↑ FP↓ AU↑ FP↓ AU↑ FP↓ AU↑ FP↓ AU↑ FP↓ AU↑ FP↓ AU↑ FP↓ AU↑\nMSP 52.66 91.42 25.47 95.61 67.01 82.86 48.38 89.96 80.40 74.75 61.24 85.24 83.97 73.07 75.20 77.69\nODIN 32.84 91.94 24.93 92.84 69.20 69.53 42.32 84.77 62.03 82.57 56.91 80.86 78.71 66.47 65.88 76.63\nMLS 31.93 93.51 12.79 97.16 64.50 82.97 36.41 91.21 70.71 80.18 52.31 88.63 82.41 74.52 68.48 81.11\nEnergy 31.72 93.51 12.40 97.22 63.95 82.84 36.02 91.19 70.80 80.22 51.93 88.82 79.90 75.30 67.54 81.45\nDICE 29.67 93.27 26.84 92.41 96.64 52.17 51.05 79.28 59.56 82.26 95.18 42.05 95.78 44.48 83.51 56.26\nReAct 82.00 76.46 12.33 97.29 64.34 81.85 52.89 85.20 77.00 78.30 52.25 88.84 79.99 75.87 69.75 81.00\nBFAct 84.40 74.39 12.34 97.21 78.02 72.68 58.25 81.43 80.27 73.36 51.23 89.04 80.05 76.58 70.52 79.66\nASH-P 29.39 93.98 20.20 95.31 84.39 66.93 44.66 85.41 68.21 81.11 53.69 87.66 86.73 65.27 69.54 78.01\nASH-B 28.21 94.27 82.10 69.54 93.93 53.00 68.08 72.27 57.45 83.80 91.74 56.39 93.63 57.20 80.94 65.80\nASH-S 23.93 94.41 39.93 91.06 82.57 68.02 48.81 84.50 52.41 84.65 60.44 84.25 89.39 59.63 67.41 76.18\nVRA-P 38.41 92.77 13.73 96.86 100.00 65.95 50.71 85.19 67.75 82.72 52.21 88.49 87.19 66.03 69.05 79.08\nOurs (V) 30.12 93.97 13.13 97.07 77.25 73.86 40.17 88.30 69.01 81.42 52.76 88.60 83.83 73.65 68.53 81.22\nOurs (E) 28.90 94.12 12.79 97.14 83.87 71.83 41.85 87.70 65.2 82.39 51.67 88.78 81.33 74.67 66.07 81.95\nEvaluation metrics. We utilize two standard evaluation metrics, following previous works (Liang\net al., 2017; Sun et al., 2021; Sun & Li, 2022): the false positive rate (FPR95) when the true positive\nrate of the ID samples is 95%, and AUC.\nImplementation details. We evaluate two variants of our method, the vanilla method described in\nEq. 14 denoted “Ours (V)”, based on the maximum logit score, and a variant based on energy scores,\ndenoted “Ours (E)”. For comparison, we implement five baseline methods (top rows in results tables),\nincluding MSP (Hendrycks & Gimpel, 2016), ODIN (Liang et al., 2017), MLS (Hendrycks et al.,\n2019a), Energy (Liu et al., 2020), and DICE (Sun & Li, 2022), and six feature-shaping methods\n(remaining rows in results tables), including ReAct (Sun et al., 2021), BFAct (Kong & Li, 2023),\nASH-P, ASH-B, ASH-S (Djurisic et al., 2022), and VRA-P (Xu & Lian, 2023). All comparison\nfeature-shaping methods are evaluated using the energy score. Hyperparameters are set consistent\nwith the original papers and concrete hyperparameter settings can be found in Section A.1 of the\nAppendix. All experiments are run on a single NVIDIA GeForce RTX 4090 GPU.\n4.1 OOD DETECTION BENCHMARK RESULTS\nExperimental results on the ImageNet-1k benchmark are shown in Table 1. More detailed results\ncan be found in Sections E and F of the Appendix. We observe that the comparison methods for\n7Published as a conference paper at ICLR 2024\nMSP MLSEnergy DICE ReAct406080FPR9569\n62 61\n454261\n41 404240\n(a)Base + Ours\nMSP MLSEnergy DICE ReAct708090AUC\n7681 818487\n8088 898788\n(b)Base + Ours\n0 1 3 10 30 100 300\nNumber of intervals K405060FPR95FPR95 AUC\n82848688\nAUC\n(c)\nFigure 4: Compatibility and sensitivity analysis. (a-b) Our method can improve other OOD scores\nand methods. “Base” denotes using the original OOD score or method, while “+Our” indicates\ncombining the score or method with our feature-shaping function. (c) Our method’s performance\nwith different hyperparameter settings, i.e., numbers of intervals K.\nfeature shaping perform extremely well on the pretrained ResNet-50 model, with some ASH variants\noutperforming element-wise shaping methods like ReAct (Sun et al., 2021), and yielding similar\nperformance to our proposed methods. This is not surprising, given that ResNet-50 was the model\narchitecture used in the original papers. As we showed in Fig. 2, the shaping functions proposed by\nthese methods closely align with the optimal shaping function learned given actual OOD samples.\nHowever, we observe that these methods do not generalize well to architectures outside of ResNet-50.\nWe observe that for feature-shaping methods more generally, stronger convolutional network archi-\ntecture performance is negatively correlated to performance on other network architectures. However,\nour proposed methods yield consistently strong performance across all network architectures, despite\nonly having access to ID data for parameter tuning.\nResults for the CIFAR10 and CIFAR100 benchmarks are shown in Table 2. An important observation\nregarding these results is that feature-shaping methods as a whole seem to underperform the baseline\nmethods. This indicates that feature shaping appears to be ineffective on smaller-scale datasets such\nas CIFAR. While our methods do not outperform the baselines, we are competitive and do outperform\nall of the comparison feature shaping methods, which is consistent with the ImageNet-1k benchmark.\n4.2 F URTHER ANALYSIS\nWe conduct additional experiments using the ImageNet-1k benchmark and employing ResNet50.\nExtension to other OOD scores. While our proposed method in Eq. 14 is designed assuming\nMLS as the OOD score, we also extend our method to use other OOD scores such as energy and\nMSP. Furthermore, we combine our method with other OOD detection methods including DICE and\nReAct. Results are averaged across the eight OOD datasets. As shown in Fig. 4(a-b), our method\nyields consistent improvement across these OOD detection scores and methods.\nDifferent numbers of intervals. We evaluate our methods with different number of intervals, i.e.,\nvarying K. As shown in Fig. 4(c), most of the improvement in performance comes from K≤10.\nThis indicates that the optimal feature shaping function has a relatively simple form. This is consistent\nwith prior methods (Sun et al., 2021; Xu & Lian, 2023; Kong & Li, 2023), which assume 2–3 intervals.\nIn addition, the relatively large performance gap over the baseline motivates feature shaping more\ngenerally. Interestingly, we observe an extremely large jump between K= 0(“Energy” baseline)\nwithK= 1, which effectively prunes the top and bottom 0.1% of feature values (using the thresholds\ncomputed based on the ID training data).\nEmpirical validation of Problem 13. In deriving Eq. 13, we remove the OOD-related term,\nassuming OOD data is inaccessible. We now empirically justify our intuition shown in Fig. 3. We\nestimate a=ExID∼DID\nX\u0002\nI(zID)\u0003\nwith ImageNet-1k and b=ExOOD∼DOOD\nX\u0002\nI(zOOD)\u0003\nwith different\nOOD datasets. We then compute the angles and the norm ratios,∥b∥\n∥a∥, of the two expectations.\n8Published as a conference paper at ICLR 2024\nTable 3: Empirical validation of Problem 13. aand\nbdenote the expectations of I(z)for ID and OOD,\nrespectively. \"AU (Real)\" refers to the performance\nof solving Eq. 10 with OOD data.\nOOD dataset ||b||/||a||cos(a,b)AU (Real) AU (Ours)\nSpecies 0.72 0.99 78.69 78.59\niNaturalist 0.48 0.97 96.53 96.63\nSUN 0.55 0.98 92.90 92.84\nPlaces 0.59 0.99 90.06 90.15\nOpenImage-O 0.56 0.98 92.31 92.54\nImageNet-O 0.91 0.99 70.19 59.33\nTexture 0.51 0.96 95.82 95.48\nMNIST 0.42 0.84 99.32 99.34As shown in Table 3, challenging OOD datasets\nlike Species and ImageNet-O exhibit smaller\nangles between the two expectations, while sim-\npler OOD datasets like iNaturalist and MNIST\ndemonstrate smaller norm ratios. Thus, Eq. 13\nprovides a reasonable approximation to θsolved\nby Eq. 10 with access to OOD data.\nCompare the formulation utilizing OOD data\n(Eq. 10) and our method (Eq. 14) . We opti-\nmizeθby solving Eq. 10 with each OOD dataset\nand get the OOD detection performance on that\ndataset. The last two columns of Table 3 com-\npare the performance of θsolved from Eq. 10\nand that from Eq. 14 (ours). Notably, our method with only ID data yields performance comparable\nto the formulation with access to real OOD data.\n5 D ISCUSSION\nHere we first discuss a core limitation to our framework that our proposed method assumes element-\nwise shaping functions, which fails to capture methods like ASH (Dosovitskiy et al., 2020). We then\nconnect our proposed method to existing works.\nGeneral feature-shaping functions. As discussed previously, a feature-shaping function can\nrepresent arbitrary mappings f:RM→RM. While our proposed methods handle mappings\nwhich can be defined element-wise over individual features, they fail to capture methods such as\nASH (Dosovitskiy et al., 2020) which can only be defined over the full feature vector. An important\ndirection for future work is in designing a tractable optimization problem where the family of\nfunctions fincludes whole-vector methods such as ASH. We expect that the resultant mappings\nshould also be relatively simple as in the element-wise case.\nFeature contributions to the maximum logits. Prior research (Sun & Li, 2022; Dietterich &\nGuyer, 2022) has explored the idea of feature contributions to the maximum logits. For example,\nDICE (Sun & Li, 2022) identifies the most significant feature indices for each class using ID data.\nDuring test-time, it masks feature indices deemed insignificant before calculating OOD scores.\nHowever, our approach differs from theirs in that we consider feature value ranges instead of indices.\nWe find our function yields significantly better performance in practice, partially attributed to the fact\nthat it admits a tractable optimization problem. Investigating the combination of these two classes of\nmethods remains a promising direction for future work.\nClassifier training for OOD detection. Prior studies (Liang et al., 2017; Sun & Li, 2022) have\nleveraged surrogate OOD datasets, such as Gaussian and Uniform distributions, to train classifiers for\ndistinguishing between ID and OOD samples. Liang et al. (2017) primarily employ these datasets\nto assess their method’s performance, while Sun & Li (2022) use them for hyperparameter tuning.\nDespite these efforts, the approach is seldom applied in feature-shaping methods. Concurrently,\nseveral studies are probing into outlier synthesis (Du et al., 2022; He et al., 2022) to approximate real\nOOD distributions and train classifiers accordingly. Our optimization problem could benefit from\nthese advancements, as our parameterized feature-shaping function could be optimized based on an\nID dataset and synthesized outliers. We explore this possibility in Section B.1 of the Appendix.\n6 C ONCLUSION\nThis paper formulates a general optimization problem for feature-shaping methods in OOD detection.\nUsing this formalism, we are able to better understand and explain the effectiveness of existing\nstate-of-the-art methods. In the context of our framework, we propose a novel feature shaping method\nwhich can be trained solely on ID data. Experimental results validate the superior performance and\ngeneralization ability of our approach compared to prior works, and motivate the development of new\nmethods under our framework to further advance OOD detection methods.\n9Published as a conference paper at ICLR 2024\nACKNOWLEDGMENTS\nWe would like to extend our deepest appreciation to Weijian Deng, Yanbin Liu, Yunzhong Hou,\nXiaoxiao Sun, and all our lab colleagues for their invaluable support throughout this project. Their\ncollaborative efforts, insightful discussions, and constructive feedback have been crucial in shaping\nand improving our paper.\nREFERENCES\nMircea Cimpoi, Subhransu Maji, Iasonas Kokkinos, Sammy Mohamed, and Andrea Vedaldi. Describ-\ning textures in the wild. In Proceedings of the IEEE conference on computer vision and pattern\nrecognition , pp. 3606–3613, 2014.\nLi Deng. The mnist database of handwritten digit images for machine learning research [best of the\nweb]. 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IEEE transactions on pattern analysis and machine\nintelligence , 40(6):1452–1464, 2017.\n12Published as a conference paper at ICLR 2024\nA I MPLEMENTATION DETAILS\nA.1 H YPERPARAMETER SETTINGS FOR EXPERIMENTS IN SECTION 4\nFor feature-shaping methods, we adhere to the hyperparameter settings reported in the original papers\nof the respective methods. These settings usually optimize the performance of the methods on existing\nbenchmarks. For additional methods, we primarily rely on hyperparameter settings detailed in Sun’s\nwork (Sun & Li, 2022). The detailed hyperparameter setting are listed in Table 4.\nTable 4: Hyperparameter settings for experiments in Section 4\nMethod Hyperparameters ImageNet-1k CIFAR10 CIFAR100\nMSP No hyperparameters - - -\nODINT, temperature scaling\nϵ, noise levelT= 1000 , ϵ= 0.0 T= 1000 , ϵ= 0.004 T= 1000 , ϵ= 0.004\nMLS No hyperparameters - - -\nEnergy No hyperparameters - - -\nDICE p, sparsity parameter p= 0.7 p= 0.9 p= 0.9\nReAct p, threshold percentile p= 90 p= 90 p= 90\nBFAct thresh ,NN= 2, thresh = 95\npercentile of the training setN= 2, thresh = 95\npercentile of the training setN= 2, thresh = 95\npercentile of the training set\nASH-P p, pruning percentile p= 60 p= 90 p= 80\nASH-B p, pruning percentile p= 65 p= 95 p= 85\nASH-S p, pruning percentile p= 90 p= 95 p= 90\nVRA-P x1,x2 x1= 0.5, x2= 1.0x1= 60 percentile\nof the training set\nx2= 95 percentile\nof the training setx1= 60 percentile\nof the training set\nx2= 95 percentile\nof the training set\nOursK, the number of\nfeature value intervalsK= 100 K= 100 K= 100\nA.2 T HE MAGNITUDE OF RESHAPED FEATURES\nIn Section 3, we describe our method for learning a piecewise constant reshaping function subject to\nnorm constraints. Scaling θhas no effect on the vanilla version of method. However, OOD detection\nperformance is sensitive to the scaling of θwhen our method is combined with other OOD scores\nwith non-linear transformations such as MSP and the energy score.\nTake the energy score as an example. If we multiply the feature representation zby a small positive\nfactor q(0< q < 1), ignoring the bias term in the last fully connected layer, we have:\nsEnergy(ℓ) = logCX\nj=1exp(qℓj). (15)\nThis is equivalent to apply a temperature scaling T= 1/qto the energy score, which will affect\nthe OOD detection performance, as analyzed in a previous paper (Liang et al., 2017). Besides,\nconsidering a fully connected layer usually contains a bias term, the scaling of θwill further affect\nthe influence of the bias. Thus, we need to consider the scale of θin our optimization problem.\nWe scale θby a factor of√\nK, motivated by the following analysis. Considering that the maximum\nlogit is the sum of all ISFIs, we have:\nℓmax(z) =KX\nk=1I[ak,bk)(z) =1TI(z), (16)\nwhere 1represents a K-dimensional vector of ones, which is a trivial feature-shaping function with\nno effect. Therefore, we set ∥θ∥=∥1∥=√\nK.\n13Published as a conference paper at ICLR 2024\nTable 5: OOD detection results of variants of our methods on ImageNet-1k. ↑indicates larger values\nare better and ↓indicates smaller values are better. All values are percentages averaged over different\nOOD datasets. Bold numbers are superior results whereas underlined numbers denote the second and\nthird best results. “Real” means using a real OOD dataset to resolve Problem (Eq. 10). “Gaussian”\nand “Uniform” indicate using Gaussian and Uniform distributions as surrogate OOD datasets in\nresolving Problem (Eq. 10), respectively. “Dynamic” denotes solving a non-convex problem with\ndynamic weight indices defined in Section B.2.\nMethodResNet50 MobileNetV2 ViT-B-16 ViT-L-16 SWIN-S SWIN-B MLP-B MLP-L Average\nFP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑\nMSP 69.30 76.26 72.58 77.41 69.84 77.40 70.59 78.40 65.12 80.89 69.45 77.26 70.66 80.31 71.63 79.93 69.90 78.48\nODIN 61.56 80.92 62.91 82.64 69.25 72.60 70.35 74.51 62.35 78.67 70.90 68.36 65.00 82.99 67.90 81.86 66.28 77.82\nEnergy 60.97 81.01 61.40 82.83 73.96 67.65 74.89 70.11 66.56 75.95 79.53 60.14 87.86 78.42 84.60 79.01 73.72 74.39\nDICE 45.32 83.64 49.33 84.63 89.68 71.32 72.38 67.08 78.84 49.06 77.89 50.10 57.12 80.60 61.98 80.75 66.57 70.90\nReAct 42.29 86.54 54.19 85.37 73.82 76.86 76.16 81.07 58.07 85.10 70.23 80.10 90.13 77.18 89.03 78.32 69.24 81.32\nBFAct 43.87 86.01 52.87 85.78 77.64 80.16 84.02 81.12 64.76 85.57 68.40 84.36 96.04 67.87 96.00 70.59 72.95 80.18\nASH-P 55.30 83.00 59.41 83.84 99.36 21.17 99.18 20.27 99.09 21.78 98.83 22.11 98.19 35.85 98.91 29.69 88.53 39.71\nASH-B 35.97 88.62 43.59 88.28 94.87 46.68 93.72 38.95 96.48 36.54 92.40 49.85 99.51 21.73 64.91 83.15 77.68 56.73\nASH-S 34.70 90.25 43.84 88.24 99.48 18.52 99.42 18.61 99.20 18.26 99.06 19.27 97.61 33.91 99.28 19.29 84.07 38.29\nVRA-P 37.97 88.58 49.98 86.83 98.39 35.66 99.58 16.70 99.27 20.34 99.46 17.64 99.38 18.73 99.05 21.23 85.39 38.21\nReal 42.09 88.13 55.05 85.77 73.69 81.65 75.36 83.10 76.06 83.22 73.54 83.11 81.66 78.34 78.82 79.20 69.53 82.82\nGaussian 42.05 88.20 55.35 85.67 71.88 82.01 73.99 83.28 71.29 83.99 70.79 83.37 82.09 77.58 76.40 79.79 67.98 82.99\nUniform 42.36 88.02 55.12 85.73 73.14 81.82 74.30 83.25 72.68 83.74 70.57 83.43 80.84 76.88 76.41 79.74 68.18 82.83\nDynamic 41.81 88.13 53.29 86.35 69.33 82.59 72.18 83.23 66.90 84.52 65.98 83.71 76.87 78.35 78.10 78.85 65.56 83.22\nOurs (V) 41.84 88.11 53.31 86.36 69.33 82.59 72.17 83.23 66.89 84.53 65.96 83.71 76.59 78.46 78.13 78.85 65.53 83.23\nOurs (E) 39.75 88.56 51.77 86.62 69.52 82.66 78.57 82.94 66.28 84.99 66.20 84.21 83.54 75.94 85.50 78.11 67.64 83.00\nB V ARIANTS OF OURMETHOD\nIn this section, we execute a series of experiments on variants of our proposed methods. Initially, we\nemploy real and surrogate OOD datasets to address the optimization problem (Eq. 10) as alternatives\nto our ID only method. Subsequently, we evaluate an assumption that weight indices of maximum\nlogit will not change after feature shaping. We calculate and report the ratio of changed weights\nafter reshaping. Furthermore, we solve an optimization problem which allows for dynamic weight\nindices. In this scenario, it is possible that the weight indices of the maximum logits can undergo\nchanges after the reshaping exercise. Our experiments are based on the ImageNet-1k benchmark and\nthe results are illustrated in Table 5.\nB.1 E XPLOIT REAL AND SURROGATE OOD DATASETS\nWe draw comparisons between our ID only method and a feature-shaping function optimization\napproach that accesses both ID and OOD datasets.\nFirst, we utilize real OOD datasets to address Problem (Eq. 10). For each model architecture,\nImageNet-1k training set (ID) and iNaturalist (OOD) are leveraged to optimze θ.\nSecond, we incorporate Gaussian and Uniform distributions as surrogate OOD datasets in resolving\nProblem (Eq. 10), aligning with prior studies (Liang et al., 2017; Sun & Li, 2022). For the Gaussian\ndistribution, each pixel is generated with a mean of 127.5 and a standard deviation of 255. For the\nuniform distribution, we directly generate random pixel values within the range [0, 255]. Every\ngenerated image is of size 32×32, and is subsequently clipped and rounded to conform to the [0,\n255] value range. For each surrogate OOD dataset, we generate a total of 50,000 samples.\nAs illustrated in Table 5, our method, even without access to OOD data, performs comparably to\nmethods that optimize θutilizing both ID and OOD datasets. Moreover, we observe that a more robust\nobjective function, such as Log Loss or Hinge Loss, could potentially outperform our method when\ngiven access to real or surrogate OOD datasets. This presents an intriguing avenue for exploration.\n14Published as a conference paper at ICLR 2024\nTable 6: Ratio of changed weight indices ron ImageNet-1k. All numbers are percentages. We\nrandomly sample 10,000 samples from ImageNet-1k training set, and ris defined in Section B.2\nResNet50 MobileNetV2 ViT-B-16 ViT-L-16 SWIN-S SWIN-B MLP-B MLP-L Average\nr 6.8 6.5 1.1 0.5 1.5 0.9 4.3 0.4 2.8\nB.2 D YNAMIC WEIGHT INDICES\nOur proposed method, as elucidated in Sections 3.2 and 3.3, relies on a fundamental simplifying\nassumption: the class index of the maximum logit remains unchanged after feature shaping.\nWe initially examine the validity of this assumption by calculating the ratio of weight indices that\nchange after feature shaping. For a given ID training set StrainofNsamples, each sample x(i)∈Strain\nhas weight indices corresponding to the maximum logits before and after feature shaping, as follows:\n˜k(i)=arg max\nkwT\nkz(i)¯k(i)=arg max\nkwT\nk¯z(i), (17)\nwhere wkdenotes the weight vector corresponding to class kin the final fully-connected layer,\nz(i)represents the feature representation for the sample x(i), and ¯z(i)is the reshaped feature.\nSubsequently, we compute the ratio of changed weight indices as:\nr=PN\ni=1 1n\n˜k(i)̸=¯k(i)o\nN×100% . (18)\nWe randomly sample 10,000 samples from ImageNet-1k training set. The ratios of changed weights\nwith different models are presented in Table 6. As observed, the ratios are markedly small, which\nfurther substantiates the credibility of our assumption pertaining to fixed weights.\nFurthermore, we propose a straightforward Alternating Optimization algorithm for dynamic weight\nindices. Each iteration involves three primary steps. Firstly, we optimize our feature-shaping function\nby resolving the optimization problem (Eq. 10) given the weights relative to the maximum logits\nfor each sample. Next, we reshape features utilizing our optimized feature-shaping function. Lastly,\nwe update the weights with reshaped features. To expedite the process, we randomly subsample\n10,000 samples from the ID training set during each iteration and utilize them for optimization. Ten\niterations are run for each model architecture. The results can be viewed in Table 5.\nPredictably, this approach yields performance metrics that closely mirror our initial method. Despite\nthe incorporation of dynamic weight indices, the optimized feature-shaping function does not signifi-\ncantly differ from that obtained with fixed weights. This observation reinforces the validity of our\noriginal assumption: the class index of the maximum logit remains constant following reshaping.\nCROBUSTNESS OF OUR METHOD TO THE CHOSEN LOWER AND UPPER LIMITS\nOF FEATURES\nTable 7: Performance of our method with differ-\nent choices of percentiles of all features on the\nImageNet-1k benchmark with ResNet50.\nLower\npercentileLower\nlimitUpper\npercentileUpper\nlimitFPR95 AUC\n0.0 0.00 100.0 20.65 42.42 87.90\n0.01 0.00 99.99 5.64 42.22 87.97\n0.05 0.00 99.95 4.40 41.97 88.04\n0.1 0.00 99.9 3.88 41.82 88.11\n0.5 0.00 99.5 2.74 41.01 88.44\n1.0 0.00 99.0 2.28 41.00 88.50\n5.0 0.03 95.0 1.35 47.85 86.41\n10.0 0.06 90.0 1.01 57.45 83.60Given that feature distributions typically ex-\nhibit long-tailed characteristics, we select the\n0.1 and 99.9 percentiles of all features in a train-\ning set as the lower and upper limits of feature\nvalues, respectively, to optimize our shaping\nfunction. Here we validate different choices of\npercentiles on the performance of our method,\non the ImageNet-1k benchmark with ResNet50.\nThe results are shown in Table 7.\nOur method exhibits a insensitivity to the\nchooses of lower and upper limits. Although\nin the main experiments we consistently use the\n0.1 and 99.9 percentiles, exploring alternate set-\ntings reveals potential for slight improvements.\n15Published as a conference paper at ICLR 2024\nD E MPIRICAL ANALYSIS OF THE OPTIMAL SHAPING FUNCTION\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5\nFeature value0.10\n0.05\n0.000.050.100.15Weights w.r.t. the max logit\n1.0\n0.5\n0.00.51.01.5\n(z)\nID samples\nOOD samples\nOur optimized (z)\nFigure 5: Empirical analysis to explain a specific form of the\noptimal shaping function.Here we try to provide empirical\ninsights into the specific form of\nour optimal shaping function based\non a ResNet50 model pretrained\non ImageNet-1k, which is shown\nin Fig. 2. Specifically, we feed\nImageNet-1k val (ID) and iNaturalist\n(OOD) samples to the model, getting\nthe feature representation z∈RM\nand the weights w.r.t. the maximum\nlogitwmax∈RMfor each sample.\nThe results are shown in Fig. 5. The\nred and green lines represent the mean\nweights assigned to different feature\nvalues for ID and OOD samples, re-\nspectively. The colored regions repre-\nsent standard deviations.\nAn interesting aspect is that our approach flips the sign of low-value features, contrary to previous\ntechniques like ReAct and ASH. Our experiments revealed that low-value features ( z <0.5) often\nalign with negative weights ( wmax<0). Intriguingly, ID samples generally align these features with\nsmaller weights than OOD samples (on average, wmax\nID< wmax\nOOD<0). By flipping the sign of these\nlow-value features, we can enhance the maximum logits more significantly for ID samples than for\nOOD samples, thereby boosting OOD detection performance.\nHowever, note that the shape of the optimal shaping function is influenced by multiple factors. For\ninstance, while ID samples are generally assigned larger weights for high-value features, OOD\nsamples tend to exhibit a greater number of these features, as observed in prior studies (Sun et al.,\n2021). This aspect reduces the impact of high-value features in distinguishing between ID and OOD\nsamples, leading to the decrease of θfor those features.\nBesides, note that our analysis above is grounded in the context of a ResNet50 model pre-trained\non ImageNet-1k. The form of the optimal shaping function may differ when applied to other model\narchitectures or datasets.\nE D ETAILED OOD D ETECTION PERFORMANCE\nWe report additional results in Tables 8, 9, and 10 to supplement the results provided in Table 1.\nSpecifically, we provide detailed performance for all eight test OOD datasets for the ImageNet-1k\nbenchmark for three representative models, including ResNet50, ViT-B-16, and MLP-B.\nF S TANDARD IMAGE NET-1K B ENCHMARK COMPARISON\nStandard ImageNet-1k benchmark uses a subset of OOD datasets of ours, and these results can be\nfound in Tables 8, 9, and 10.\nBesides, we pick the results of standard ImageNet-1k benchmark in our experiments and calculated\nthe new average performance. Our method still outperforms prior works on this standard benchmark.\nResults are shown in Table 11.\n16Published as a conference paper at ICLR 2024\nTable 8: OOD detection results with ResNet50 on ImageNet-1k. ↑indicates larger values are better\nand↓indicates smaller values are better. All values are percentages averaged over different OOD\ndatasets. Bold numbers are superior results whereas underlined numbers denote the second and third\nbest results. “Real” means using a real OOD dataset to resolve Problem (Eq. 10). “Gaussian” and\n“Uniform” indicate using Gaussian and Uniform distributions as surrogate OOD datasets in resolving\nProblem (Eq. 10), respectively. “Dynamic” denotes solving a non-convex problem with dynamic\nweight indices defined in Section B.2.\nMethodSpecies iNaturalist SUN Places OpenImage-O ImageNet-O Texture MNIST Average\nFP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓ AU↑ FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑\nMSP 79.53 75.15 52.77 88.42 68.58 81.75 71.57 80.63 63.58 84.98 100.00 28.64 66.13 80.46 52.25 90.02 69.30 76.26\nODIN 80.90 72.88 50.86 91.14 59.89 86.59 65.67 84.18 57.29 89.23 100.00 40.86 54.31 86.40 23.54 96.11 61.56 80.92\nEnergy 82.40 72.03 53.93 90.59 58.27 86.73 65.40 84.13 57.21 89.12 100.00 41.92 52.29 86.73 18.23 96.79 60.97 81.01\nDICE 74.64 74.30 26.66 94.49 36.09 90.98 47.66 87.73 45.13 88.77 98.10 43.01 32.46 90.46 1.79 99.40 45.32 83.64\nReAct 68.64 77.41 19.55 96.39 24.00 94.41 33.43 91.93 43.68 90.53 98.05 52.43 45.83 90.45 5.11 98.76 42.29 86.54\nBFAct 68.08 77.76 20.53 96.22 20.94 95.24 30.17 92.62 49.71 86.48 96.20 54.15 54.01 87.84 11.33 97.78 43.87 86.01\nASH-P 80.09 73.19 44.57 92.51 52.88 88.35 61.79 85.58 50.45 90.67 100.00 45.99 42.06 89.70 10.57 97.99 55.30 83.00\nASH-B 66.08 80.33 14.21 97.32 22.08 95.10 33.45 92.31 37.63 91.97 93.05 56.82 21.17 95.50 0.06 99.58 35.97 88.62\nASH-S 64.61 81.45 11.49 97.87 27.98 94.02 39.78 90.98 32.74 92.75 89.05 67.51 11.93 97.60 0.01 99.80 34.70 90.25\nVRA-P 69.61 76.93 15.70 97.12 26.94 94.25 37.85 91.27 36.37 92.93 95.55 60.79 21.47 95.62 0.30 99.70 37.97 88.58\nReal 68.91 78.58 18.64 96.53 37.76 92.69 47.31 89.97 39.63 92.33 97.45 60.01 24.15 95.57 2.90 99.34 42.09 88.13\nGaussian 69.39 78.31 18.88 96.49 37.54 92.68 47.43 89.88 39.60 92.32 97.05 60.85 23.24 95.76 3.25 99.30 42.05 88.20\nUniform 69.34 78.49 18.83 96.51 38.71 92.51 47.98 89.81 39.51 92.46 97.70 59.49 24.24 95.54 2.58 99.37 42.36 88.02\nDynamic 68.81 78.58 18.33 96.63 36.87 92.87 45.94 90.17 39.34 92.54 97.80 59.44 24.70 95.50 2.68 99.34 41.81 88.13\nOurs (V) 68.83 78.59 18.33 96.63 37.03 92.84 45.97 90.15 39.29 92.54 97.80 59.33 24.80 95.48 2.67 99.34 41.84 88.11\nOurs (E) 68.46 78.51 15.90 97.00 34.00 93.28 43.61 90.50 37.07 92.82 96.25 60.70 21.61 95.99 0.85 99.66 39.75 88.56\nTable 9: OOD detection results with ViT-B-16 on ImageNet-1k. ↑indicates larger values are better\nand↓indicates smaller values are better. All values are percentages averaged over different OOD\ndatasets. Bold numbers are superior results whereas underlined numbers denote the second and third\nbest results. “Real” means using a real OOD dataset to resolve Problem (Eq. 10). “Gaussian” and\n“Uniform” indicate using Gaussian and Uniform distributions as surrogate OOD datasets in resolving\nProblem (Eq. 10), respectively. “Dynamic” denotes solving a non-convex problem with dynamic\nweight indices defined in Section B.2.\nMethodSpecies iNaturalist SUN Places OpenImage-O ImageNet-O Texture MNIST Average\nFP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓ AU↑ FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑\nMSP 80.56 72.91 51.52 88.16 66.54 80.93 68.67 80.38 59.93 84.81 90.95 58.80 60.23 82.99 80.29 70.23 69.84 77.40\nODIN 82.18 67.96 52.26 85.24 66.89 76.35 69.14 75.06 58.75 81.55 89.05 54.30 56.68 81.69 79.03 58.65 69.25 72.60\nEnergy 86.54 61.88 64.08 79.24 72.77 70.24 74.31 68.44 64.93 76.46 87.00 52.71 58.48 79.30 83.61 52.92 73.96 67.65\nDICE 88.09 65.08 90.43 74.49 94.27 65.76 92.81 65.19 82.61 77.51 85.85 70.86 83.67 77.63 99.68 74.02 89.68 71.32\nReAct 86.82 66.08 65.15 85.98 72.46 78.97 73.74 77.51 64.71 84.19 87.30 66.70 56.95 84.65 83.41 70.81 73.82 76.86\nBFAct 90.52 64.43 80.01 84.62 77.86 81.10 77.55 79.61 69.66 85.45 87.15 73.84 57.61 85.67 80.76 86.53 77.64 80.16\nASH-P 98.38 28.99 99.95 10.42 99.69 19.83 99.66 21.02 99.57 18.91 99.15 30.70 98.49 29.92 100.00 9.57 99.36 21.17\nASH-B 94.34 53.78 92.36 48.20 95.30 52.35 95.75 52.18 92.74 52.73 94.60 51.53 93.90 43.84 99.96 18.86 94.87 46.68\nASH-S 98.11 28.50 99.99 6.71 99.73 16.70 99.66 18.33 99.84 14.07 99.40 28.51 99.11 24.15 100.00 11.20 99.48 18.52\nVRA-P 98.14 39.19 99.78 23.06 98.83 30.67 98.89 32.59 98.28 37.46 98.65 43.67 94.57 52.22 100.00 26.40 98.39 35.66\nReal 85.07 67.00 64.16 88.03 70.73 83.05 71.76 81.74 65.56 87.14 89.60 73.69 63.09 84.63 79.53 87.93 73.69 81.65\nGaussian 84.14 67.56 61.10 88.71 69.13 83.41 70.28 82.07 63.32 87.51 89.70 73.48 60.16 85.20 77.20 88.16 71.88 82.01\nUniform 84.69 67.25 62.56 88.44 70.60 83.27 71.37 81.93 64.54 87.41 90.20 73.59 61.60 84.94 79.54 87.71 73.14 81.82\nDynamic 83.20 69.47 56.06 89.91 66.63 84.17 68.48 82.69 60.21 88.17 89.50 71.82 56.81 86.43 73.75 88.08 69.33 82.59\nOurs (V) 83.19 69.47 56.09 89.91 66.63 84.17 68.51 82.69 60.21 88.18 89.50 71.82 56.79 86.43 73.75 88.08 69.33 82.59\nOurs (E) 84.56 68.70 60.09 89.55 68.02 83.92 69.42 82.36 61.77 88.07 87.95 72.74 54.61 86.72 69.72 89.21 69.52 82.66\n17Published as a conference paper at ICLR 2024\nTable 10: OOD detection results with MLP-B on ImageNet-1k. ↑indicates larger values are better\nand↓indicates smaller values are better. All values are percentages averaged over different OOD\ndatasets. Bold numbers are superior results whereas underlined numbers denote the second and third\nbest results. “Real” means using a real OOD dataset to resolve Problem (Eq. 10). “Gaussian” and\n“Uniform” indicate using Gaussian and Uniform distributions as surrogate OOD datasets in resolving\nProblem (Eq. 10), respectively. “Dynamic” denotes solving a non-convex problem with dynamic\nweight indices defined in Section B.2.\nMethodSpecies iNaturalist SUN Places OpenImage-O ImageNet-O Texture MNIST Average\nFP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓ AU↑ FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑\nMSP 82.15 73.21 58.99 86.84 72.98 80.92 75.52 79.83 72.42 82.59 90.95 65.84 69.95 80.27 42.36 92.95 70.66 80.31\nODIN 80.99 73.39 53.31 89.53 67.47 84.24 71.67 82.44 68.06 85.51 90.45 68.69 59.75 84.40 28.33 95.73 65.00 82.99\nEnergy 89.04 71.84 84.31 83.65 86.85 79.35 87.15 78.42 82.30 81.24 88.40 67.53 85.25 78.41 99.58 86.91 87.86 78.42\nDICE 71.40 75.65 44.44 90.19 64.69 80.84 70.31 78.08 63.89 80.26 89.90 56.33 52.20 84.33 0.12 99.11 57.12 80.60\nReAct 91.20 70.81 91.47 80.11 91.83 77.31 90.99 76.82 85.66 79.60 87.85 67.74 87.36 77.20 94.65 87.89 90.13 77.18\nBFAct 95.94 64.38 98.60 65.76 97.16 67.50 96.66 68.74 94.69 70.00 90.85 66.57 94.61 67.83 99.85 72.19 96.04 67.87\nASH-P 97.25 46.50 99.22 32.82 98.72 40.24 97.74 43.07 98.67 38.34 94.55 52.65 99.38 31.38 100.00 1.81 98.19 35.85\nASH-B 99.67 25.98 100.00 15.86 99.71 19.84 99.74 21.36 99.82 24.19 97.25 46.39 99.86 19.48 100.00 0.74 99.51 21.73\nASH-S 96.50 42.89 98.42 32.35 98.03 38.00 97.90 38.63 97.65 37.03 93.50 50.19 98.90 30.41 100.00 1.80 97.61 33.91\nVRA-P 98.88 29.11 99.92 11.57 99.73 17.54 99.41 20.71 99.77 16.82 97.80 37.15 99.50 16.49 100.00 0.44 99.38 18.73\nReal 86.49 71.56 71.23 84.79 79.47 79.42 81.95 78.44 78.30 81.00 91.00 66.46 78.24 78.36 86.61 86.69 81.66 78.34\nGaussian 87.00 71.10 73.67 83.35 79.82 78.80 82.07 77.86 79.06 79.83 91.20 65.95 78.90 77.64 85.01 86.10 82.09 77.58\nUniform 86.92 70.35 75.45 81.81 80.47 77.69 82.25 77.07 79.29 78.56 90.70 65.63 78.05 76.45 73.59 87.51 80.84 76.88\nDynamic 87.24 70.88 76.95 82.17 83.83 77.26 84.75 76.12 78.65 80.31 90.70 65.68 76.76 79.17 36.06 95.19 76.87 78.35\nOurs (V) 86.98 70.98 76.40 82.36 83.45 77.42 84.67 76.27 78.35 80.46 90.85 65.77 76.40 79.26 35.66 95.19 76.59 78.46\nOurs (E) 91.67 69.17 92.48 76.83 91.83 73.88 91.86 73.34 87.14 77.34 90.40 66.02 89.29 75.33 33.61 95.61 83.54 75.94\nTable 11: OOD detection results on ImageNet-1k where OOD datasets are iNaturalist, SUN, Places,\nand Textures .↑indicates larger values are better and ↓indicates smaller values are better. Bold\nnumbers are superior results whereas underlined numbers denote the second and third best results.\nMethodResNet50 MobileNetV2 ViT-B-16 ViT-L-16 SWIN-S SWIN-B MLP-B MLP-L Average\nFP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑FP↓AU↑\nMSP 64.76 82.82 70.47 80.67 61.74 83.12 65.22 81.75 59.68 83.75 62.79 81.38 69.36 81.97 76.01 80.04 66.25 81.94\nODIN 57.68 87.08 60.42 86.39 61.24 79.59 64.06 78.65 57.30 80.99 64.36 73.77 63.05 85.15 73.17 82.29 62.66 81.74\nEnergy 57.47 87.05 58.87 86.59 67.41 74.31 68.43 74.65 62.82 77.65 75.32 64.87 85.89 79.96 84.44 79.38 70.08 78.06\nDICE 35.72 90.92 41.93 89.60 90.30 70.77 71.77 67.12 88.68 39.90 87.92 40.71 57.91 83.36 66.23 81.53 67.55 70.49\nReAct 30.70 93.30 50.09 88.81 67.08 81.78 69.58 83.50 50.09 87.92 64.86 83.24 90.41 77.86 87.01 79.13 63.73 84.44\nBFAct 31.41 92.98 48.35 89.19 73.26 82.75 81.16 82.69 57.20 88.21 65.44 86.62 96.76 67.46 96.36 72.16 68.74 82.76\nASH-P 50.33 89.04 57.15 87.34 99.45 20.30 99.42 18.37 99.11 19.21 99.08 20.59 98.77 36.88 99.04 29.20 87.79 40.11\nASH-B 22.73 95.06 35.66 92.13 94.33 49.14 94.08 37.89 96.42 30.00 91.47 47.38 99.83 19.14 66.05 84.31 75.07 56.88\nASH-S 22.80 95.12 38.67 90.95 99.62 16.47 99.55 16.72 99.36 14.98 99.35 17.75 98.31 34.85 99.36 18.65 82.13 38.18\nVRA-P 25.49 94.57 45.53 89.85 98.02 34.64 99.62 14.99 99.38 18.15 99.65 15.51 99.64 16.58 99.23 19.49 83.32 37.97\nOurs (V) 31.53 93.78 49.09 89.62 62.01 85.80 68.61 85.10 61.47 87.04 62.96 86.09 80.23 78.83 82.59 78.92 62.31 85.65\nOurs (E) 28.78 94.19 46.92 89.90 63.04 85.64 74.96 84.69 60.67 87.50 63.50 86.55 91.37 74.85 88.49 78.22 64.71 85.19\n18"    },    {        "title": "2402.00946v1.High_order_recovery_of_geometric_interfaces_from_cell_average_data.pdf",        "content": "High order recovery of geometric interfaces from cell-average data\nAlbert Cohen∗, Olga Mula†, Agustin Somacal∗\nFebruary 5, 2024\nAbstract\nWe consider the problem of recovering characteristic functions u:=χΩfrom cell-average data on a\ncoarse grid, and where Ω is a compact set of Rd. This task arises in very different contexts such as image\nprocessing, inverse problems, and the accurate treatment of interfaces in finite volume schemes. While linear\nrecovery methods are known to perform poorly, nonlinear strategies based on local reconstructions of the\njump interface Γ := ∂Ω by geometrically simpler interfaces may offer significant improvements. We study\ntwo main families of local reconstruction schemes, the first one based on nonlinear least-squares fitting, the\nsecond one based on the explicit computation of a polynomial-shaped curve fitting the data, which yields\nsimpler numerical computations and high order geometric fitting. For each of them, we derive a general\ntheoretical framework which allows us to control the recovery error by the error of best approximation up to\na fixed multiplicative constant. Numerical tests in 2d illustrate the expected approximation order of these\nstrategies. Several extensions are discussed, in particular the treatment of piecewise smooth interfaces with\ncorners.\n1 Introduction\n1.1 Reconstruction from cell-averages\nWe consider the problem of reconstructing a function u:D→Rdefined on a multivariate domain D⊂Rd\nfrom cell averages\naT(u) :=1\n|T|Z\nTu(x)dx, T ∈ T, (1)\nover a partition TofD. This task occurs in various contexts, the most notable ones being:\n1.Image processing: hereuis the light intensity of an image and Trepresents a grid of pixels in dimension\nd= 2 or voxels in dimension d= 3. Various processing tasks are facilitated by the reconstruction of the\nimage at the continuous level, for example when applying operations that are not naturally compatible with\nthe pixel grid such as rotations, or when changing the format of the pixel grid such as in super-resolution.\n2.Hyperbolic transport PDE’s: hereuis a solution to such an equation and Tis a computational grid,\ntypically in dimension d= 1,2 or 3. Finite volume schemes evolve the cell average data by computing at\neach time step the numerical fluxes at the interfaces between each cell. Several such schemes are based on\nan intermediate step that reconstructs simple approximations to uon each cell and compute the numerical\nfluxes by applying the transport operator to these approximations.\n3.Inverse Problems: numerous inversion tasks can be formulated as the recovery of a function from\nobservational data, and this data could typically come in the form of local averages of the type (1).\nIn this paper, we consider functions defined on the unit cube\nD:= [0,1]d,\n∗Laboratoire Jacques-Louis Lions, Sorbonne Universit´ e, 4 place Jussieu, 75005 Paris, France (albert.cohen@sorbonne-\nuniversite.fr and agustin.somacal@sorbonne-universite.fr)\n†TU Eindhoven, Department of Mathematics and Computer Science, 5612 AZ Eindhoven, Netherlands (o.mula@tue.nl)\n1arXiv:2402.00946v1  [math.NA]  1 Feb 2024and partitions ThofDbased on uniform cartesian meshes, that is, consisting of cells of the form\nT=h(k+D), k := (k1, . . . , k d)∈ {0, . . . , l −1}d,\nwhere h:=1\nl>0 is the side-length of each cell in Th, for some l >1. The cardinality of the partition is therefore\nn:= #(Th) =ld=h−d.\nWe are thus interested in reconstruction operators Rthat return an approximation ˜ u=R(a) toufrom the\nn-dimensional vector a=a(u) = (aT(u))T∈Th∈Rn. The most trivial one is the piecewise constant function\n˜u:=X\nT∈ThaT(u)χT, (2)\nwhich is for example used in the Godunov finite volume scheme. Elementary arguments show that this recon-\nstruction is first order accurate: if u∈W1,p(D), one has\n∥u−˜u∥Lp(D)⩽Ch∥∇u∥Lp(D)∼n−1\nd,\nwhere Cis a fixed constant, and the exponent in this estimate cannot be improved for smoother functions.\nA simple way to raise the order of accuracy is by reconstructing on each cell polynomials of higher degree using\nneighbouring cell averages. For example, in the univariate case d= 1 and for some fixed m⩾1, we associate to\neach interval Tk:= [kh,(k+ 1)h] the centered stencil consisting of the cells Tlforl=k−m, . . . , k +m. Then,\nthere exists a unique polynomial pk∈P2msuch that\naTl(pk) =aTl(u), l=k−m, . . . , k +m.\nWe then define a piecewise polynomial reconstruction by\n˜u:=X\nk=0,...,n−1pkχTk.\nThis strategy can be generalized to higher dimension d > 1 in a straightforward manner: for each cell T\nwe consider the stencil S=STof (2m+ 1)dcells centered around Tand define the piecewise polynomial\nreconstruction\n˜u:=X\nT∈ThpTχT,\nwhere pTis the unique polynomial of degree 2 min each variable such that\na˜T(pT) =a˜T(u),˜T∈ST.\nFor example in the bivariate case d= 2, we may use 3 ×3 stencils to reconstruct bi-quadratic polynomials on\neach cell. Standard approximation theory arguments show that these local reconstruction operators now satisfy\naccuracy estimates of the form\n∥u−˜u∥Lp(D)⩽Chr|u|Wr,p(D)∼n−r\nd,\nforr⩽2m+ 1.\nRemark 1.1. Polynomials of odd degree can also be constructed by using non-centered stencils. Also note that\nnon-centered stencils need to be used when approaching the boundary of D, but this does not affect the above\nestimate.\nThese classical methods are therefore efficient to reconstruct smooth functions with a rate of accuracy that\noptimally reflects their amount of smoothness. Unfortunately they are doomed to perform poorly in the case of\nfunctions uthat are piecewise smooth with jump discontinuities accross hypersurfaces. For example, a piecewise\nconstant reconstruction will have O(1) error on each cell that is crossed by the interface. Since the amount of\nsuch cells is of order h1−d, we cannot expect a reconstruction error better than\n∥u−˜u∥Lp>∼(hdh1−d)1\np=h−1\np=n−1\ndp. (3)\n2X0 i−1ii+1 i+3\nFigure 1: ENO-SR in one dimension: the jump point X0is identified by matching the average on the singular\ncell with the piecewise polynomial reconstruction.\nIn particular, this reconstruction has first order accuracy O(h) in the L1norm.\nThe use of higher order polynomial cannot improve this rate. In fact, a fundamental obstruction is the fact\nthat all the above methods produce approximations ˜ uthat depend linearly on a(u), and therefore belong to a\nlinear space of dimension n. The bottleneck of such methods for a given class of functions Kis therefore given\nby the so-called Kolmogorov n-width defined by\ndn(K)Lp:= inf\ndim(Vn)=nsup\nu∈Kmin\nv∈Vn∥u−v∥Lp.\nThen, it can be shown that for very simple classes Kof discontinuous functions such as those of the form\nu=χH, where His any half-space passing through D, the n-width in Lpprecisely behaves like n−1\ndp, see [4].\nIn summary, any linear method cannot do much better than the low order piecewise constant method, even for\ninterfaces that are infinitely smooth.\nImproving the accuracy in the reconstruction of piecewise smooth functions from cell averages therefore\nmotivates the development and study of nonlinear reconstruction strategies, which is at the heart of this work.\nWe first recall the main existing approaches.\n1.2 Reconstruction of discontinuous interfaces\nOne first approach aiming to tackle jump discontinuities while maintaining high order approximation in the\nsmooth regions was proposed for the univariate case d= 1 by Ami Harten in terms of ENO (Essentially Non\nOscillatory) and ENO-SR (Subcell Resolution). The ENO strategy [6] is based on selecting for each cell Ta\nstencil STthat should not include the cell T∗which contains the point of jump discontinuity. This is achieved\nby choosing among the stencils that contains Tthe one where the cell average values have the least numerical\nvariation. For T̸=T∗such adaptively selected stencils will tend to avoid T∗.\nAs a consequence high order reconstruction can be preserved in all cells where uis smooth, and in addition\nthis yields for free a singularity detection mechanism which identifies the singular cell T∗that is avoided from\nboth side by the stencil selection. The ENO-SR strategy [5] then consists in reconstructing in this singular\ncell by extending the polynomials fitted on both sides until a point for which the resulting average match the\nobserved average. The position of this point can therefore be identified by solving a simple algebraic equation.\nThis strategy is very effective in the univariate case as illustrated in Figure 1.\nWhile the ENO stencil selection can be generalized to higher dimension, the ENO-SR strategy does not\nhave a straightforward multivariate version. This is due to the fact that, instead of a single point, the jump\ndiscontinuity to be identified is now a hypersurface (curve in 2d, surface in 3d, ...) that cannot be described\nby finitely many parameters. The approximate recovery of geometric interfaces from cell-average has been the\nobject of continuous investigation, with the particular focus on functions of the form\nu=χΩ,\nwhere Ω ⊂Dis a set with boundary Γ = ∂Ω having a certain smoothness. For such characteristic functions,\n3that could for example represent a two phase flow without any mixing or the evolution of a front, the whole\ndifficulty is concentrated in the recovery of Γ since the smooth parts are the trivial constant functions 0 or 1.\nIn this paper, we denote by\nSh:={T∈ Th:|T∩Ω| ̸= 0 and |T∩Ωc| ̸= 0},\nthe set of cells that are crossed by the interface Γ in a non trivial way. Such cells are termed as singular , the\nother as regular . Let us note that singular cells are characterized by the property\n0< aT(u)<1,\nand can thus be identified from the cell-average data.\nPractical computational strategies, often termed as volume of fluid methods, consist in using the cell-average\ndata to reconstruct a local approximation of the interface that can be described by finitely many parameters,\nsuch as lines in 2d or planes in 3d. This idea was introduced in [12], and was significantly improved in [14] with\nthe LVIRA algorithm that consists in reconstructing in each singular cell a linear interface whose parameters are\nfound by least-square minimization of the difference between exact and reconstructed cell averages on centered\n3×3 stencil, in the 2 dcase.\nNoe that this continuous least-square minimization is performed over a nonlinear set. This induces a sub-\nstantial computational burden that could be avoided through a variant, the ELVIRA algorithm, in which the\nline selection is made between 6 possible configurations explicitly computed from the cell averages. One main\nresult is the fact that this reconstruction returns precisely the true interface if this one is indeed a straight\nline. We refer to [13] for a comparative survey on these reconstruction algorithms and to [7, 16, 15, 17] for\nimprovements and applications in the domain of 2d and 3d fluid mechanics.\nu= 1\nu= 0T hχP= 1\nχP= 0T h\nFigure 2: Local approximation of a smooth interface by a line interface.\nIntuitively the advantage of locally fitting a line or plane interface is that it has the ability to better\napproximate the interface Γ if it is smooth, so that it is expected to improve the low order of accuracy (3) of\nlinear methods. More precisely, if Γ has C2regularity, on each cell of side-length h, it can be approximated\nwith Hausdorff distance O(h2) by a line in 2d, a plane in 3d, etc. Therefore, if the locally reconstructed linear\ninterface is optimally fitted, the O(1) error is observed on a strip of volume O(h2+d−1), see Figure 2. Since the\namount of singular cell is of order #( Sh)∼h1−d, we may hope for a reconstruction error with improved order,\n∥u−˜u∥Lp∼(hd+1h1−d)1\np=h−2\np=n−2\ndp, (4)\nthat is, we double the rate compared to (3). In particular, we obtain second order accuracy O(h2) in the L1\nnorm.\nHowever, to our knowledge such estimates have never been rigourously proved for the aforementionned\nmethod. In addition the approximation power of linear interface is also limited and we cannot hope to improve\nthe above rate for interfaces that are smoother than C2. In this context, the objective of this paper is twofold:\n1. introduce a theoretical framework for the rigourous convergence analysis of local interface reconstructions\nfrom cell averages,\n2. within this framework develop reconstruction methods going beyond linear interfaces and provably achiev-\ning higher order of accuracy.\n41.3 Outline\nIn this paper, we will essentially work in the bivariate case d= 2 which makes the exposition simpler while\nmost of our discussion can be carried over to higher dimension.\nThe recovery methods that we study are local: on each cell Tidentified as singular, the unknown function u=\nχΩis approximated by a simpler ˜ u=χΩTpicked from a family Vnthat can be described by nparameters and\nthat enjoy certain approximation properties for interfaces having prescribed smoothness. This approximation\nis computed from the cell averages of a rectangular stencil STofm⩾ncells centered around T. We begin §2\nby giving examples of such families and discussing their approximation properties for prior classes KsofχΩ\nassociated to sets Ω with Csboundaries for s >1. Our ultimate goal is to develop recovery schemes such that\nthe error between uand ˜uis near optimal in the sense of being bounded (up to a multiplicative constant) by\nthe error of best approximation of uby elements from Vn. This, in particular means that the recovery should\nbe exact if the true function ubelongs to Vn.\nWe introduce in §3 a first class of recovery strategies which we call Optimization Based Edge Reconstruction\nAlgorithms (OBERA). Similar to LVIRA it is based on least-square fitting of simpler interfaces such as lines,\nbut also circles or polynomials that allow to raise the order of accuracy. We show that near optimality of this\nrecovery is ensured by an inverse stability inequality which can in particular be established for line edges and\n3×3 stencils. Unfortunately, this property seems more difficult to prove when raising the order of accuracy,\nwhich also leads to more difficult nonlinear optimization procedures.\nAs a more manageable alternative, we consider recovery methods that are based on the identification of a\ncertain preferred orientation - vertical or horizontal - for describing the interface by a function in the vicinity\nof each such cell. This leads us to the second class of recovery methods, termed as Algorithms for Edge\nReconstruction using Oriented Stencils (AEROS) which is discussed in §4. It avoids continuous optimization\nby finding an edge described by a polynomial function y=pT(x) orx=pT(y) after having used the previously\ndiscussed orientation mechanism. The polynomial pTis identified by simple linear equations. We show that this\nprocess satisfies an exactness and stability property that leads to the near optimal recovery bound. In addition,\nthe rate of convergence can be raised to an arbitrarily high order by raising the degree of pT, however at the\nprice of using larger stencils. The analysis of the orientation selection mechanism, based on the Sobel filter, is\npostponed to the Appendix, where it is shown that it correctly classifies the cells when his sufficiently small.\nAll these methods are numerically tested and compared in §5. The differences in terms of convergence rates\nare confirmed, and we also compare the computational costs, which are by far less for AEROS than OBERA.\nWe also discuss and test the specific recovery of corner singularities in the interface. Finally, in the case of\nlinear transport, we illustrate the propagation of error when using finite volume schemes based on these local\nrecovery strategies.\nAn open-source python framework1is made available to show the methods presented here but specially\nto allow an easy way of creating, testing, and comparing new subcell resolution and interface reconstruction\nmethods without the need to re-implement everything from scratch.\n2 Numerical analysis of local recovery methods\n2.1 Local approximation by nonlinear families\nThe methods that we study in this paper for the recovery of the unknown u=χΩare based on local approx-\nimations of uon each cell T∈ Ththat is identified as singular by simpler characteristic functions picked from\nannparameter family Vn.\nLet us give three examples that will be used further. We stress that in all such examples Vnis not an\nn-dimensional linear space, but instead should be thought as an n-dimensional nonlinear manifold.\nExample 1: linear interfaces. These are functions of the form v=χHwhere His a half-plane with a\nline interface L=∂H. Such functions are described by n= 2 parameters. One convenient description is by\nthe pair ( r, θ), where r⩾0 is the offset distance between the center zTof the cell Tof interest and the linear\ninterface and θ∈[0,2π[ is the angle between this line and the horizontal axis. In other words, in this case we\n1https://github.com/agussomacal/SubCellResolution\n5use\nV2:={vr,θ:=χ{⟨z−zT,eθ⟩⩽r}, θ∈[0,2π[;r >0},\nwhere eθ= (−sin(θ),cos(θ)). Of course, the d-dimensional generalization by half-spaces is a d-parameter family\nwhere the unit normal vector eθlives on the d−1-dimensional unit sphere.\nExample 2: circular interfaces. These are functions of the form v=χDorv=χDcwhere Dis a\ndisc with circular interface, and Dcits complement. It is easily seen that the corresponding space V3is now a 3\nparameter family, and its d-dimensional generalization by characteristic function of balls and their complements\nis ad+1 parameter family. The idea of using circles instead of lines is to increase the approximation capability.\nHowever, as we will see, the next family turns out to be more effective both from the point of view of theoretical\nanalysis and computational simplicity.\nExample 3: oriented graphs. These are functions of the form v=χPwhere Pis the subgraph or the\nepigraph of a function p∈Wn, either applied to the coordinate xory, where Wnis a linear space of univariate\nfunctions. In other words, Pis given by one among the four equations\ny⩽p(x), y⩾p(x), x⩽p(y), x⩾p(y). (5)\nOf course, the corresponding space Vnis an n-parameter family that is not a linear space, while Wnis. Note\nthat the linear interfaces of Example 1 are a particular case where W2is the set of affine functions. Raising the\norder of accuracy will be achieved by taking for Wnthe space of polynomials of degree n−1. The ddimensional\ngeneralisation is obtained by taking for Wna linear space of functions of d−1 variables and considering Pto\nbe defined by one of the 2 dequations\nxi⩽p(x1, . . . , x i−1, xi+1, . . . , x d), x i⩾p(x1, . . . , x i−1, xi+1, . . . , x d), i= 1, . . . d,\nfor some p∈Wn. In particular Wncould be a space of multivariate polynomials.\nExample 4: piecewise linear interface. These are functions of the form u=χH1∩H2where H1and\nH2are two half planes. Therefore the interface consists of two half lines that touch at a corner point x0. The\ncorresponding space V4is a 4 parameter family, for example by considering the coordinates x0= (x1, x2) and\nthe angles θ1andθ2of the normal vectors to the two lines. The goal of this family is to better approximate\npiecewise smooth interfaces that have corner singularities. Note that Example 1 may be viewed as a particular\ncase where the two half-planes coincide and there is no corner point.\nGiven such a family Vnand a set S⊂D, we denote by\nen(u)S= min\nv∈Vn∥u−v∥L1(S),\nthe error of best approximation on Smeasured in the L1-norm.\nRemark 2.1. Throughout this paper, we shall systematically measure error in L1norm which is the most\nnatural since in the case of u=χΩand˜u=χ˜Ω, this error is simply the area of the symmetric difference\nbetween domains, that is,\n∥u−v∥L1(D)=|Ω ∆˜Ω|=|Ω∪˜Ω−Ω∩˜Ω|.\nNote however that estimates in Lpnorms can be derived from L1e stimates in a straightforward manner since\n∥u−v∥Lp(D)=|Ω ∆˜Ω|1/p=∥u−v∥1/p\nL1(D).\nIn our analysis of the reconstruction error on a singular cell T, we will need to estimate the local approx-\nimation error on a stencil S=STthat consists of finitely many cells surrounding T. We shall systematically\nconsider rectangular stencils of symmetric shape centered around T, so in the 2 dcase they are of size\nm= (2k+ 1)×(2l+ 1),\nfor some fixed k, l⩾1.\n6The order of magnitude en(u)Sboth depends on the type of family Vnthat one uses and on the smoothness\nproperty of the boundary Γ = ∂Ω. We describe these properties by introducing prior classes of characteristic\nfunctions χΩof sets Ω ⊂Dwith boundary of a prescribed H¨ older smoothness. There exists several equivalent\ndefinitions of a Csdomain. We follow the approach from [11], that expresses the fact that the boundary can\nlocally be described by graphs of Csfunctions (see also Chapter 4 of [1]).\nDefinition 2.2. Lets >0. A domain Ω⊂R2is of class Csif and only if there exists an R > 0,P > 1and\nM > 0, such that for any point z0∈∂Ω, the following holds: there exists an orthonormal system (e1, e2)and a\nfunction ψ∈ Cs([−R, R])with∥ψ∥Cs⩽M, taking its value in [−PR, PR ]and such that\nz∈Ω⇐⇒ z2⩽ψ(z1),\nfor any z=z0+z1e1+z2e2with|z1|⩽Rand|z2|⩽PR.\nHere, we have used the usual definition\n∥ψ∥Cs= sup\n0⩽k⩽⌊s⌋∥ψ(k)∥L∞([−R,R])+ sup\ns,t∈[−R,R]|s−t|⌊s⌋−s\f\f\fψ(⌊s⌋)(s)−ψ(⌊s⌋)(t)\f\f\f,\nfor the H¨ older norm. In the case of integer smoothness, we use the convention that Csdenotes functions with\nLipschitz derivatives up to order s−1, so that in particular the case s= 1 corresponds to domains with Lipschitz\nboundaries. This definition naturally extends to domains of Rdwith d >2 with ψnow being a Csfunction of\nd−1 variables.\nWe can immediately derive a first local approximation error estimate for the the space V2of linear interfaces\nfrom the above Example 1: let u=χΩwith Ω a domain of class Cs. Then, if Sis a 2 k+ 1×2l+ 1 stencil\ncentered around a cell Tthat is crossed by the interface, we apply the above Definition 2.2 taking z0=zTthe\ncenter of T. We assume that the sidelength his small enough so that the stencil Sis contained in the rectangle\n{z=z0+z1e1+z2e2:,|z1|⩽R,|z2|⩽PR}, in which ∂Ω is described by the graph of the Csfunction ψ. For\nz∈S, we have in addition that |z1|⩽C0h⩽Rwhere C0depends only on landk. Using a Taylor expansion\nand the smoothness of ψwe find that there exists an affine function asuch that\n∥ψ−a∥L∞([−C0h,C 0h])⩽C1hr, r := min {s,2}, (6)\nwhere C1depends on C0and the bound Mon the Csnorm of ψ. For example, we can take for athe Taylor\npolynomials of order 1 at z1= 0 when s >1, which corresponds to match the tangent of the interface at the\npoint z0+ψ(0)e2, or of order 0 when s⩽1. Then, taking v=χH∈V2, where\nH:={z2⩽a(z1)},\nis the corresponding half-space, it follows that\n∥u−v∥L1(S)=|S∩(Ω ∆ H)|⩽Chr+1,\nwhere Cdepends on ( M, l, k ). In summary, for the local approximation error of Csdomains by a linear interface\nwe have\nen(u)S⩽Chr+1, r := min {s,2}. (7)\nThe same reasoning in ddimensions delivers a local approximation estimate of order hr+d−1.\nOne way to raise this order of accuracy for smoother domains is to use approximation by circular interfaces\nfrom Example 2 since this allows us to locally match the curvature in addition to the tangent. In turn we reach\na similar estimate of order hr+1however with r:= min {s,3}. One more systematic way of raising the order\narbitrarily high is to use approximation by oriented subgraphs from Example 3. This approach is central to the\nAEROS strategies discussed in §5 and we thus discuss it below in more detail.\nWe begin with the observation that when s >1, the unit tangent vector varies continuously on ∂Ω if Ω is of\nclassCs. It follows that locally around any point z0, this vector remains away either from the horizontal vector\n(1,0) or from the vertical vector (1 ,0). This allows us to locally describe the boundary by graphs of functions\nof the standard cartesian coordinates, as expressed by the following alternate definition of Csdomains.\n7Definition 2.3. Lets >1. A domain Ω⊂R2is of class Csif and only if there exists an R > 0,P > 1and\nM > 0, such that for any point z0= (x0, y0)∈∂Ω, the following holds: there exists a function ψ∈ Cs([−R, R])\nwith∥ψ∥Cs⩽M, taking its value in [−PR, PR ]and such that the membership in Ωof a point z= (x, y)is\nequivalent to one of the two equations\ny⩽ψ(x), y⩾ψ(x), (8)\nwhen|x−x0|⩽Rand|y−y0|⩽PR, or one of the two equations\nx⩽ψ(y), x⩾ψ(y), (9)\nwhen|y−y0|⩽Rand|x−x0|⩽PR.\nThe generalization of this alternate definition to higher dimension is straighforward by considering equations\none of the form\nxi⩽ψ(x1, . . . , x i−1, xi+1, . . . , x d) or xi⩾ψ(x1, . . . , x i−1, xi+1, . . . , x d), i= 1, . . . , d,\nwith ψaCsfunction of d−1 variables defined on [ −R, R]d−1.\nRemark 2.4. We stress that this definition is only valid for s >1and not for less smooth domains such as\nLipschitz domains. For example if Ωis a rectangle with side oriented along principal axes, then no such local\nparametrization can be derived if z0is a corner point.\nConsider now the family Vnof oriented subgraphs from Example 3, associated with the linear space Wn=\nPn−1of univariate polynomials of degree n−1. Let u=χΩwith Ω a domain of class Csfor some s >1.\nThen, if Sis a (2 k+ 1)×(2l+ 1) stencil centered around a cell Tthat is crossed by the interface, we apply the\nabove Definition 2.3 taking z0=zTthe center of T. Without loss of generality, assume for example that the\ndescription of Ω near z0is by the equation\ny⩽ψ(x),\nforz= (x, y) in the rectangle {|x−x0|⩽R,|y−y0|⩽PR}. We assume that the sidelength his small enough\nso that the stencil Sis contained in this rectangle. For z= (x, y)∈Swe thus have\n|x−x0|⩽C0h⩽R, C 0=k+1\n2.\nUsing Taylor formula and the smoothness of ψwe find that there exists a polynomial p∈Pn−1such that\n∥ψ−p∥L∞([−C0h,C 0h])⩽C1hr, r := min {s, n}, (10)\nwhere C1depends on C0and the bound Mon the Csnorm of ψ. For example, we can take for pthe Taylor\npolynomials of order ˜ n= min {⌈s⌉ −1, n−1}atx=x0. Therefore, taking v, where\nv:=χ{y⩽p(x)}∈Vn,\nthe corresponding subgraph, it follows that\n∥u−v∥L1(S)= 2C0hC1hr=Chr+1.\nwhere Cdepends on ( M, k). We treat the other cases y⩾ψ(x),x⩽ψ(y) and x⩾ψ(y) in a similar manner.\nIn summary, for the local approximation error of Csdomains by polynomial oriented subgraphs, we have\nen(u)S⩽Chr+1, r := min {s, n}. (11)\nThe same reasoning in ddimensions delivers a local approximation estimate of order hr+d−1.\n82.2 Near optimal recovery from cell averages\nThe recovery methods that we develop in §3 and §4 are based on recovering on each singular cell Tan element\n˜uT∈Vnwhere Vnis a given nonlinear family, based on the data of the cell-averages\naS(u) = (a˜T(u))˜T∈S,\nwhere S=STis a rectangular stencil centered around T. It can therefore be summarized by a local nonlinear\nrecovery operator\nRT:Rm→Vn\nwhere m= (2k+ 1)×(2l+ 1) is the size of the stencil, such that\n˜uT=RT(aS(u)).\nWe are interested in deriving a favorable comparison between the local recovery error ∥u−˜uT∥L1(T)and the\nerror of best approximation by Vnwhose magnitude can be estimated depending on the amount of smoothness\nof the boundary, as previously discussed.\nDefinition 2.5. The local recovery procedure is said to be near-optimal over a class of function Kif there exists\na fixed constant Cso that one has\n∥u−RT(aS(u))∥L1(T)⩽Cen(u)S, (12)\nfor all uin this class. In particular Cshould be independent of the considered singular cell Tand mesh size h.\nRemark 2.6. In the above definition, the recovery error on Tis bounded by the approximation error on the\nlarger stencil S. This is due to the fact that the recovery operator RTacts on the cell averages a˜T(u)for all\ncells ˜T⊂S.\nOn regular cells T∈ Th\\ Sh, that is, such that aT(u) = 0 or aT(u) = 1, we simply define the reconstruction\nby the constant value\n˜uT=aT(u),\nwhich is then the exact value of u. The global reconstruction of ufrom the cell-averages ( aT(u))T∈This given\nby the function\n˜u=X\nT∈Th˜uTχT.\nThe global L1error can be estimated by aggregating all local error estimates, which thus gives\n∥u−˜u∥L1(D)=X\nT∈Sh∥u−˜uT∥L1(T)⩽CX\nT∈Shen(u)ST. (13)\nwhere Cis the stability constant in (12) and where STdenotes the stencil centered at Twhich is used in the\nreconstruction of ˜ uT.\nIf Ω is a Csdomain with s⩾1, that is, at least a Lipschitz domain, one has the cardinality estimate\n#(Sh)⩽Ch−1(14)\nwhere Cdepends on the length of Γ. We may thus derive a global error estimate by combining (13), (14) and\nthe local approximation estimates (7) and (11): is Ω is a Csdomain with s⩾1, we obtain\n∥u−˜u∥L1(D)⩽Chr, (15)\nwith r:= min {s,2}when using local recovery by linear interfaces and r:= min {s, n}when using local recovery\nby polynomial subgraphs of degree n−1. This estimate generalizes to the higher dimensional case, combining\nlocal approximation estimates with the cardinality estimate #( Sh)⩽Ch1−d.\nOur central objective is now to propose local recovery methods that provably satisfy the near optimal\nrecovery bound (12). For this, we start by remarking that this bound implies the property\nRT(aS(v)) =v,∀v∈Vn, (16)\n9T\n(a) Line interfaces (b) Circular interfaces (c) Quadratic subgraphs\nFigure 3: Cases of non-injectivity: two elements of Vnhaving same averages on a 3 ×3 stencil.\nT\n(a) Line interfaces (b) Circular interfaces (c) Quadratic subgraphs\nFigure 4: Illustration of the restrictions ensuring injectivity\nthat is, the recovery is exact for elements from Vn. This property itself implies that any element from Vnshould\nbe exactly characterized by its cell-average on the stencil S. In other words, the averaging operator\nv7→aS(v) = (a˜T(v))˜T∈S,\nshould be injective from VntoRm. For this to hold, the classes Vnfrom Examples 1, 2, 3, 4 need to be restricted.\nIn the case of linear interfaces, if Hand ˜Hare two half-spaces that contain the stencil S, the functions\nv=χHand ˜v=χ˜Hobviously have the same cell-average vector aSwith component identically equal to 1.\nAsking that the linear interface passes through the stencil is thus necessary but not sufficient as illustrated\non Figure (3a): two lines passing only through a corner cell may result in identical cell averages. The correct\nrestriction on Vnin this case is obtained by imposing that the linear interface passes through the central cell,\nwhich in the case of a 3 ×3 stencil suffices to ensure injectivity as recalled in §4 and illustrated on Figure 4a.\nAn important observation is that this type of restriction does not affect the local error estimates (7) since we\nare precisely considering a stencil Scentered at a cell Tthat contains the interface. Therefore, the tangent of\nthe interface at the point z0+ψ(0)e2delivering this estimate satisfies this restriction.\nIn the case of circular interface, asking that the disk intercepts the central cell is not sufficient as illustrated\non Figure (3b): two discs Dand ˜Dof equal size and contained in the central cell will result in identical cell\naverages for v=χDand ˜v=χ˜D. In this case, an additional restriction should be that the radius of the disc\nis sufficiently large compared to the size of the stencil, for example by imposing that the disc center is not\ncontained in S, as illustrated on Figure 4b.\nIn the case of subgraphs of polynomial functions, again two subgraphs passing through several cells of\nthe stencil might have the same cell averages. This typically occurs when the polynomials are too peaky, as\nillustrated on Figure (3c). In this case, the additional restriction should be that the range of premains inside\nthe stencil. By this we mean, say for a subgraph of the type y⩽p(x) and a rectangular stencil S= [a, b]×[c, d],\none has that p([a, b])⊂[c, d], as illustrated in Figure (4c). Similarly to linear interfaces, the local error estimate\n(11) is not affected by such a restriction, as we discuss in §5.\nA similar type of observation shows that there is no hope to uniquely characterize a piecewise linear interface\nfrom the cell averages on a given stencil if it is too peaky, that is, the opening angle of the cone embraced by the\ntwo lines cannot be arbitrarily close to 0 or 2 π. In other words, corners cannot be arbitraritly acute or obtuse.\n103 Reconstruction by optimization (OBERA)\n3.1 Presentation of the method\nOptimization-Based Edge Reconstruction Algorithms (OBERA) consists in recovering on each singular cell\nT∈ Sha recovery ˜ uT∈Vnby a best fit of the available cell-average data on the stencil S=ST. For this\npurpose, we solve a minimization problem of the form\n˜uT=RT(aS(u))∈arg min\nv∈Vn∥aS(v)−aS(u)∥2\nwhere ∥ · ∥is a given norm on Rmwhere m:= #( S) is the size of the stencil. In a practical implementation,\none first simple choice is to use the Euclidean ℓ2norm, that is, minimize the loss function\nL(u, v) :=X\n˜T∈S|a˜T(u)−a˜T(v)|2,\nover all v∈Vn. The case of linear interface corresponds to the LVIRA method [14]. Note that v∈Vnis defined\nthrough an appropriate parametrization as in Example 1, 2, 3 and 4,\nµ∈ M ⊂ Rn7→vµ∈Vn,\nwhere Mis the restricted range of the parameter µthat defines the family Vn. Therefore the optimization is\ndone in practice by searching for\nµ∗∈arg min\nµ∈ML(u, vµ)\nand taking ˜ uT=vµ∗.\nIt is interesting to note that this recovery method is not consistent in the sense that it does not guarantee\nthat\naT(˜uT) =aT(u), (17)\na property that is required, typically in finite volume methods since it reflects the conservation of mass. In\norder to restore this property, one possibility is to define the recovery by solving the constrained optimization\nproblem\n˜uT∈arg min\nv∈Vn{L(u, v) :aT(v) =aT(u)} (18)\nIn practice, this can be emulated by modifying the loss function into\nL(u, v) :=K|aT(u)−aT(v)|2+X\n˜T∈S,˜T̸=T|a˜T(u)−a˜T(v)|2, (19)\nand taking K≫1 (in our numerical tests we took K= 100). As explained below, this constrained recovery\nsatisfies similar error bounds as its unconstrained counterpart.\nThe practical difficulty of the OBERA lies in the quick and accurate computation of the cell averages aST(v)\nfor any given cell Tandv∈Vn, that is, have a fast evaluation procedure for the parameter to average map\nµ∈ M ⊂ Rn7→aST(vµ)∈Rm,\nas it is needed to calculate L(u, vµ) at each iteration of the the optimization algorithm. While analytic expres-\nsions are easily available for linear interfaces, they become more difficult to derive for more general curves like\npolynomials or circle interfaces. In such cases, the exact computation of aS(vµ) is possible if for each cell ˜T∈ST\none is able identify the points where the curved interface crosses its boundary. We followed this approach in our\nnumerical implementation. Another option relies on quadrature methods as used in [3], however at the expense\nof potentially many evaluations of v. Finally, another perspective is to use machine learning methods in order\nto derive a cheaply computable surrogate of the parameter to average map.\nEven with such tools in hand, the computation of aS(vµ) for the many parameter values µthat are explored\nthrough the optimization process results in a time overhead that one would like to avoid. For linear interfaces,\nthis was achieved by the ELVIRA method, as it only requires 6 calculations of aS(vµ) in order to decide which\nµ∗should be retained. This can also be avoided for more general interfaces having higher order geometric\napproximations by the AEROS approach that we present in §4.\n113.2 Analysis of the recovery error\nIn order to prove that the recovery error is near optimal in the L1(S) norm, we follow a general strategy\nintroduced in [4] which is based on comparing the continuous L1(S) norm of functions and the discrete ℓ1(Rm)\nnorm of their cell-averages.\nIn the 2 dcase, one obviously has on the one hand the inequality\nh2∥aS(v)∥ℓ1⩽∥v∥L1(S), v∈L1(S), (20)\nwhich is obtained by summing up\nh2|a˜T(v)|=\f\f\fZ\n˜Tv\f\f\f⩽∥v∥L1(˜T),\nover all ˜T∈S. This property reflects the stability of the averaging operator, between the continuous and the\nconveniently normalized discrete norm. Note that in more general dimension dthe normalizing factor is hd.\nConversely, we say that the family Vnsatisfies an inverse stability property if there exists a constant C\nindependent of hsuch that\n∥v−˜v∥L1(S)⩽Ch2∥aS(v)−aS(˜v)∥ℓ1,∀v,˜v∈Vn. (21)\nWe stress that such a property cannot hold for general pairs of integrable functions, their membership in Vnis\ncritical. Note that this property is a more quantitative version of the injectivity of the map v7→aS(v) from Vn\ntoRm. Its validity is thus conditioned to a proper restriction of the classes Vnfrom the various Examples 1, 2,\n3, and 4, as already explained in §2.2. In the particular case of linear edges the following result was proved in\n[4].\nTheorem 3.1. LetSbe the 3×3stencil centered at Tand let V2be the family of linear interfaces from Example\n1, with the restriction that the linear interfaces passes through T. Then (21)holds and the best constant is C=3\n2.\nThe above stability and inverse stability property allow us to assess the recovery error in the following way.\nWe first write that for any v∈Vn\n∥u−˜uT∥L1(T)⩽∥u−v∥L1(T)+∥v−˜uT∥L1(T)⩽∥u−v∥L1(T)+Ch2∥aS(v)−aS(˜uT)∥ℓ1,\nwhere we have used (21). We then have\n∥aS(v)−aS(˜uT)∥ℓ1⩽√m∥aS(v)−aS(˜uT)∥ℓ2⩽2√m∥aS(v)−aS(u)∥ℓ2,\nwhere the first inequality is Cauchy-Schwartz and the second comes by triangle inequality and the ℓ2minimiza-\ntion property of ˜ uT. Finally, we have\n∥aS(v)−aS(u)∥ℓ2,⩽∥aS(v)−aS(u)∥ℓ1⩽h−1∥u−v∥L1(S),\nby using (20). Combining all these, and using that v∈Vnis arbitrary, we obtain that\n∥u−˜uT∥L1(T)⩽(1 + 2 C√m)en(u)S,\nwhich is summarized in the following.\nTheorem 3.2. Under (20) and(21), the recovery by ℓ2minimization is near optimal in L1norm with mul-\ntiplicative constant 1 + 2 C√m. In the case of linear interfaces using 3×3stencils (LVIRA), this constant is\n1 + 23\n2√\n9 = 10 .\nAs observed in §2.2, near optimal local recovery allows us to derive convergence rates for smooth domains\nin terms of the global error estimate (15). This gives the following.\nCorollary 3.3. IfΩis aCsdomain with s⩾1, the LVIRA method which is OBERA recovery based on linear\ninterfaces converges in L1norm at rate O(hr)withr= min {s,2}.\n12Γ\nL−L\nLcL+\nT\nFigure 5: By shifting the linear interface Lone achieves average consistency on Tby a linear interface Lchaving\nthe same order of accuracy.\nRemark 3.4. Note that if we were using a more general ℓpnorm for the data fitting, we would obtain a similar\nresult with constant 1 +C2m1−1\np. The fact that we do not need to restrict ourselves just to p= 2 had been\nmentioned in [13] concerning the LVIRA method.\nAs previously remarked, we can modify the OBERA approach in order to impose the consistency condition\n(17), by solving the constrained optimization problem (18), that is optimizing inside the subfamily\n˜Vn:={v∈Vn:aT(v) =aT(u)} ⊂Vn\nIt is obvious that if the inverse stability property (21) is valid for Vn, it is also valid for the smaller set ˜Vn. We\nthus reach a similar estimate\n∥u−˜uT∥L1(T)⩽(1 + 2 C√m)˜en(u)S,\nwhere ˜ en(u)Sis the error of best approximation of uinL1(S) norm from the element of ˜Vn, that is best\napproximation from Vnunder the consistency constraint.\nWe thus need to understand if ˜ en(u)Ssatisfies similar size estimates as en(u)S. While this cannot be ensured\nin full generality (for example the set ˜Vncould be empty), the following simple argument shows that this indeed\nholds in the particular case of linear interface: the estimate (6) shows that the function ψdescribing the interface\nin the stencil Ssatisfies\na−⩽ψ⩽a+,\nwhere a−=a−C1hranda+=a+C1hrare two affine functions that parametrize two linear interfaces L−and\nL+that circumscribe Γ in S, as illustrated on Figure 5. For the corresponding halfplanes H−andH+thus\nsatisfy\naT(χH−)⩽aT(u) and aT(χH+)⩾aT(u).\nTherefore, by sliding continuously a linear interface between L−andL+, which corresponds to the affine function\na+twhen tvaries in [ −C1hr, C1hr], there exists an intermediate interface Lccorresponding to a particular tc\nand halfplane Hcfor which one has\naT(χHc) =aT(u).\nTherefore v=χHc∈˜Vn, and since one also has\n∥ψ−ac∥L∞([−C0h,C 0h])⩽C1hr, r := min {s,2}, (22)\nwe reach the same estimate for ˜ en(u)Sas the one obtained for en(u)S.\nFrom the theoretical perspective, an open problem is to establish the inverse stability bound (21) for nonlinear\nfamilies Vnoffering higher order approximation properties than the linear interface, for which the proof of\nTheorem 3.1 is already quite involved. This, together with the already mentioned computational complexity of\nthe optimization process, leads us to give up on OBERA for higher order geometrical approximation in favor\nof the AEROS approach that we next discuss.\n13Figure 6: On the left h= 1/10 and on the right h= 1/30. The singular cells idenfied as horizontally and\nvertically oriented are pictured in blue and green. For h= 1/10, there exists a singular cell T(indicated in red)\nfor which the interface cannot be described by a graph y=ψ(x) orx=ψ(y) in any stencil STof width 3 that\ncontains T. This is no more the case for h= 1/30 when using an adaptive selection of the stencil.\n4 Reconstruction on oriented stencils (AEROS)\n4.1 Presentation of the method\nAlgorithms for Edge Reconstruction using Oriented Stencils (AEROS) are based on recovering an element from\nthe family Vnpresented in Example 3. We thus recover on each singular cell T∈ Sha domain having one of\nthe four subgraph or epigraph forms (5), that is, an interface having one of the two Cartesian forms y=p(x)\norx=p(y), where p∈Wna linear space of dimension n.\nMore specifically we consider for some fixed k⩾1 the space\nW2k+1=P2k,\nof polynomials of even degree 2 k. We then use stencils STcontaining Tof the form (2 k+1)×LorL×(2k+1),\nwhen reconstructing an interface of the form y=p(x) orx=p(y) respectively, for some L >0. As we further\nexplain, the value of Land the exact positioning of Tinside STmay depend on the considered cell T.\nAs a first step we need to identify for each T∈ Shthe exact orientation of the subgraph or epigraph that we\ndecide to use. The decision must be based on the available data of the cell averages . We have already noticed\nthat if Ω is a Csdomain for s >1, then it can itself be locally described by (at least) one of the four forms with\na function ψ∈ Csdescribing the interface, as expressed by (8) and (9) in Definition 2.3. Our objective is that\nour choice of form in the recovery is consistent with the form of the exact interface over the stencil STfor each\ncellT.\nWe thus need to identify for each Tan orientation y=ψ(x) orx=ψ(y) for the exact interface over the\nstencil ST. Let us immediately observe that this is only possible if his below a certain critical resolution\nh∗=h∗(Ω) that depends on the amount of variation of the tangent to the interface, as shown on Figure 6 for\nthe case k= 1 (stencils of widths 3).\nMore precisely, when Ω is a Csdomain with s >1, the variation of the slope of the tangent to the interface\nbetween two points zandz′is controlled by a bound of the form M|z−z′|rwhere r:= min {1, s−1}andMthe\nbound on the Csnorm of the functions ψthat describe the interface. Therefore a given orientation, say y⩽ψ(x),\ncan be maintained on a stencil STof width 2 k+ 1 in the xdirection provided that h⩽h∗∼k−1M−1/r.\nFor identifying the orientation, we introduce a selection mechanism based on a numerical gradient computed\nby the Sobel filter. We denote by Tεwith ε= (εx, εy)∈ {− 1,0,1}2the cells in the 3 ×3 stencil centered\naround T=T0,0where εxandεyindicate the amount of displacement by hfrom Tin the xandydirection,\nrespectively. We then define the numerical gradient\nGT= (HT, VT),\n14with horizontal component\nHT:= 2aT1,0+aT1,1+aT1,−1−(2aT−1,0+aT−1,1+aT−1,−1)\nand vertical component\nVT:= 2aT0,1+aT1,1+aT−1,1−(2aT0,−1+aT1,−1+aT−1,−1)\nwhere aTε=aTε(u) are the available cell-average.\nThe selection mechanism is based on comparing the absolute size of HTandVTand examining their sign.\nMore precisely:\n1. If|VT|⩾|HT|and if VT⩽0, we search for a subgraph of the form y⩽p(x).\n2. If|VT|⩾|HT|and if VT>0, we search for an epigraph of the form y⩾p(x).\n3. If|VT|<|HT|and if HT⩽0, we search for a subgraph of the form x⩽p(y).\n4. If|VT|<|HT|and if HT>0, we search for an epigraph of the form x⩾p(y).\nOne important result is that this selection mechanism correctly detects the orientation of the exact interface\nforhsufficiently small, as also illustrated on Figure 6.\nTheorem 4.1. LetΩbe aCsdomain for some s > 1, then there exists h∗=h∗(Ω)such that under the\nassumption h < h∗, the following holds for any T∈ Sh: in each of the above cases (1,2,3,4)of the selection\nmechanism, the exact domain Ωcan be described by an equation of the same form with preplaced by a function\nψ∈ Csover the stencil STcentered at Tand of size (2k+ 1)×(2l+ 1) in case (1,2)or(2l+ 1)×(2k+ 1)\nin case (2,3)with l=k+ 2. The graph of ψremains confined in STin the following sense: denoting by\nI×J:=S{˜T:˜T∈ST}the total support of ST, one has ψ(I)⊂Jin case (1,2)andψ(J)⊂Iin case (3,4).\nWe postpone the proof of this result to the Appendix, and proceed with the description and error analysis\nof the recovery method. We place ourselves in case 1 without loss of generality since all other cases are dealt\nwith similarly up to an obvious exchange of xandyor change of sign in one of these variable.\nAccording to the above theorem, we are ensured that Ω is characterized by the equation y⩽ψ(x) when\n(x, y)∈STwhere STis a stencil of size (2 k+ 1)×(2l+ 1) with l=k+ 2. The choice l=k+ 2 is conservative\nand our numerical experiments show that it can sometimes be reduced while maintaining the property that\nthe graph of ψremains confined in ST. In practice we use the following adaptive strategy to use a stencil of\nminimal vertical side.\nBy convention, we denote by ( i, j) the coordinates of a generic cell ˜Twhen the lower left corner of ˜Tis\n(ih, jh ). Let ( iT, jT) be the coordinates of the singular cell T. We explore the neighboring cells by defining\nl−:= min {l >0 :a˜T(u) = 1 , i=iT−k, . . . , i T+k, j=jT−l−1},\nwhich is the smallest lower shift below which we find a row of non-singular cells. Likewise, we define\nl+:= min {l >0 :a˜T(u) = 0 , i=iT−k, . . . , i T+k, j=jT+l+ 1}.\nThen, we take for STthe stencil that consists of cells ˜Tof coordinates ( i, j) for i=iT−k, . . . , i T+kand\nj=jT−l−, . . . , j T+l+. This stencil has size (2 k+ 1)×(1 + l−+l+) and is centered around the cell T\nhorizontally but not vertically. From the definition of l−andl+we have the guarantee that the graph of ψ\nremains confined in ST. One option to further adapt the stencil STis to allow that it is also not centered\nhorizontally but still contains T. This leads to (2 k+ 1)×(1 + l−+l+) stencils corresponding to values\ni=iT−k−, . . . , i T+k+andj=jT−l−, . . . , j T+l+, where k−, k+⩾0 are such that k−+k+= 2kand are\nselected so to minimize the vertical size 1 + l−+l+.\nOnce the stencil SThas been selected, the polynomial pT∈P2kis constructed as follows. For each i=\niT+k−, . . . , i T+k+, we denote by Rithe column that consists of the cells ˜T∈STwith first coordinate equal\ntoiand define the corresponding column average\nai(u) =1\nl−+l++ 1X\n˜T∈Ria˜T(u).\n15Since the graph of ψremains confined in ST, it follows that ai(u) can be identified to the univariate cell average\nofψon the interval [ ih,(i+ 1)h] after having substracted the base elevation of the stencil, that is,\nai(u) +bT=1\nhZ(i+1)h\nihψ(x)dx, i =iT+k−, . . . , i T+k+, b T:= (jT−l−)h\nWe then define pT∈P2kas the unique polynomial of degree at most 2 kthat agrees with the observed averages\nofψ, that is, such that\n1\nhZ(i+1)h\nihpT(x)dx=ai(u) +bTi=iT+k−, . . . , i T+k+.\nThe polynomial pTis sometimes called the interpolant of averages, and its existence and uniqueness is standard,\nsimilar to the more usual Lagrange interpolant of point values. In particular, it is easily checked that the P2k\ninterpolant of the averages of a function von 2k+ 1 adjacent intervals is the derivative of the P2k+1Lagrange\ninterpolant at the 2 k+ 2 interval endpoints for the primitive of v.\nQuite remarkably, although we are locally approximating uby a nonlinear family, we observe that the\nrecovery map\nLT:ψ7→pT,\nis linear, and that the AEROS recovery approach amounts to solving a simple (2 k+ 1)×(2k+ 1) linear system,\nresulting in substantial computational saving compared to the OBERA approach.\n4.2 Analysis of the recovery error\nIn order to assess the recovery error, we first observe that the above described strategy has the property of\nexact recovery for polynomials\nψ∈P2k=⇒pT=ψ, (23)\ndue to the uniqueness of the interpolant of averages. In other words, AEROS recovers on Tthe true interface\nif it is described by a polynomial of degree 2 konST.\nOur next observation is that the linear application LTis stable in the max norm over the relevant interval\nIT= [(iT−k−)h, . . . , (iT+k+)h], with stability constant that does not depend on h. This can be proved by\nmaking the affine change of variable\nx=φ(ˆx) =h(iT+ ˆx),\nthat maps the reference interval ˆI:= [−k−, . . . , k+] onto IT. Then, it is readily checked that\nLψ◦φ=ˆL(ψ◦φ),\nwhere ˆLis the average interpolant for the intervals of size 1 contained in ˆI. Therefore, we may write\n∥Lψ∥L∞(IT)=∥Lψ◦φ∥L∞(ˆI)=∥ˆL(ψ◦φ)∥L∞(ˆI)⩽C∥ψ◦φ∥L∞(ˆI)=ˆC∥ψ∥L∞(IT)\nwhere the constant ˆCis the norm of ˆLacting on L∞(ˆI). This constant only depends on k.\nWe are now in position to obtain an error estimate by writing for all p∈P2k,\n∥ψ−pT∥L∞(IT)⩽∥ψ−p∥L∞(IT)+∥pT−p∥L∞(IT)=∥ψ−p∥L∞(IT)+∥L(ψ−p)∥L∞(IT)⩽(1+ˆC)∥ψ−p∥L∞(IT),\nwhere we have combined exact recovery of polynomials and uniform stability. Since pis arbitrary we have\nobtained the following result.\nTheorem 4.2. The AEROS recovery of the interface based on polynomials of degree 2ksatisfies for each\nsingular cell the near optimality property\n∥ψ−pT∥L∞(IT)⩽(1 + ˆC) min\np∈P2k∥ψ−p∥L∞(IT). (24)\n16(a) Circle.\n (b)h= 1/10.\n (c)h= 1/20.\nFigure 7: Circular domain and its cell average data at different scales of refinement.\nThis error bound of ψinL∞readily induces an L1(T) error bound between the recovery\nRT(aS(u)) =uT:=χ{y⩽pT(x)}\nanduby multiplying by the width of the cell. This gives\n∥u−RT(aS(u))∥L1(T)⩽(1 + ˆC)hmin\np∈P2k∥ψ−p∥L∞(IT). (25)\nNote that the right-side is not en(u)Sand we thus have not obtained the near-optimal recovery property in the\nform (12). Nevertheless we can derive from (25) the same convergence rates since these are based on the Taylor\npolynomial approximation error (10), which thus yields the following result.\nTheorem 4.3. LetΩbe aCsdomain for some s⩾1. The AEROS recovery of the interface based on polynomial\nof degree 2ksatisfies for each singular cell Ta local error bound of the form\n∥u−RT(aS(u))∥L1(T)⩽Chr+1, r := min {s,2k+ 1}, (26)\nand the global error bound (15) of order O(hr)for the same value of r.\n5 Numerical experiments\n5.1 Recovery of smooth domains\nIn this section, we compare various recovery strategies in terms of:\n1. visual aspects,\n2. quantitative rate of convergence,\n3. computational time.\nIn order to draw the second comparison, we first consider the simple case of a domain Ω with circular boundary\n(see Figure 7) for which the recovery error can be computed within machine precision.\nThe following eight reconstruction strategies are compared:\n1.Piecewise Constant : as mentioned in the introduction, the simplest linear method that one can come\nup with is ˜ u:=P\nT∈ThaT(u)χT, that is, on each cell Tthe value given by the observed cell average aT(u).\n2.OBERA Linear : we apply the minimization strategy described in §3 for linear interfaces as in Example\n1, with loss function L(u, v) =∥aS(u)−aS(v)∥using ℓ1norm and a 3 ×3 stencil S.\n3.OBERA-W Linear : we apply the same approach but enforcing area consistency on Tthrough the\nweighted loss function (19) with K= 100 .\n174.ELVIRA : following [13], the ELVIRA method consists on replacing, still for linear interfaces , the OBERA\ncontinuous optimization strategy by providing only 6 combinations of parameters µfrom which to choose\nµ∗, the minimizer of Lwith ℓ2norm instead of ℓ1. The 6 alternatives are obtained by proposing, for the\ninterface’s slope, the 6 possible finite differences estimations of a 3 ×3 stencil.\n5.ELVIRA-WO : we apply ELVIRA but choosing first an orientation , as in AEROS, which allows to reduce\nthe choice to 3 alternatives. In addition, we work with the modified weighted loss function, with K= 100,\nto favour area consistency on T.\n6.OBERA Quadratic : we apply the minimization strategy for quadratic interfaces after choosing an\norientation to have a Cartesian parametrization of the interface, that is, vµ=χP, as in Example 3, and\np∈P2the space of univariate polynomials of degree 2. Here, we also use a 3 ×3 fixed stencil and enforce\narea consistency on Tthrough the same modified loss function (19) with K= 100.\n7.AEROS Quadratic : we apply the AEROS reconstruction strategy for quadratic interfaces with the\nadaptive method, described in §4.1, to build stencils of width 3 minimal height.\n8.AEROS Quartic : we apply the AEROS strategy now with polynomials of degree 4, therefore with\nstencils of width 5.\nFigure 8 and Figure 9. display a detail of the recovery with h= 1/10 and h= 1/20 respectively in order to\ncompare the visual quality.\nAs to linear interfaces, we clearly notice the relevance of enforcing area consistency on the cell Tof interest\nby appropriately modifying the loss function L. Although the four methods benefit from similar convergence\nrates of O(h2) as expected from (3.3) and (4.3), see Figure 10, the reconstruction error improvement is of an\norder of magnitude by this simple change. When area consistency is not imposed, the linear interfaces are\npushed towards the interior of the circle as the curvature of the original domain points inwardly.\nAs to the AEROS strategies, we notice that for h= 1/10, they have difficulties to reconstruct the interface\non cells where Γ is rapidly passing from a situation where an horizontal orientation is preferred to another in\nwhich a vertical one is better. This effect if particulary evident for the case of AEROS Quartic as the method\nneeds a stencil with 5 columns. At this scale we are above the critical scale (see Figure 6) in which for some cells\nthere is no stencil of the needed width allowing the curve to be described as a graph. This problem disappears\nforh= 1/20, for which these methods have the best visual quality.\nOn Figure 10, we see that in terms of convergence we obtain for both AEROS the expected rates from (4.3):\nfor quadratics we get O(h3) and we almost get O(h5) for polynomials of degree 4. As mentioned before and\ngraphically shown in Figure 8, AEROS Quartic breaks down when the scale of the discretization is above the\ncritical scale which, for this particular, example is around h= 1/20.\nRegarding the computational time per cell taken by each algorithm we observe on Table 1 that OBERA\nstrategies are two orders of magnitude slower than any AEROS approach. At the same time, although ELVIRA\nmethods are faster than OBERA, they still are an order slower than AEROS due to the bottleneck of having\nto compute the stencil cell averages to compare the 6 or 3 alternatives under evaluation. This limit is justified\nby the fact that choosing an orientation, as in ELVIRA-WO, cuts by half the computing time of the overall\nalgorithm, while the improvement in accuracy is achieved by modifying the loss function.\nIn summary, all three comparisons in terms of visual aspect, order of convergence and computational time\nare in favor of the AEROS strategy provided that his below the critical scale.\nFinally, we show in Figure 11 how the best linear interface method, OBERA-W Linear, and the three higher\norder methods compare when used to reconstruct an arbitrary, still smooth, domain. We see that by passing\nfrom the linear interface method to AEROS Quadratic the reconstruction becomes smoother while still suffering\nfrom some imperfections, notably in regions where there is a stronger change in the orientation of the curve.\nThis is slightly improved by the optimization done in OBERA Quadratic method. An even smoother result is\nobtained with AEROS Quartic at the expense of some small deviations again in regions where the orientation\nis rapidly changing.\n18(a) Piecewise Constant.\n (b) OBERA Linear.\n (c) ELVIRA.\n (d) ELVIRA-WO.\n(e) OBERA-W Linear.\n (f) AEROS Quadratic.\n (g) OBERA Quadratic.\n (h) AEROS Quartic.\nFigure 8: Reconstruction of a portion of the circle by different methods for a scale of h= 1/10.\n(a) Piecewise Constant.\n (b) OBERA Linear.\n (c) ELVIRA.\n (d) ELVIRA-WO.\n(e) OBERA-W Linear.\n (f) AEROS Quadratic.\n (g) OBERA Quadratic.\n (h) AEROS Quartic.\nFigure 9: Reconstruction of a portion of the circle by different methods for a scale of h= 1/20.\n1910 30 100 10 20\n1/h10−710−610−510−410−310−2/bardblu−˜u/bardblL1\nPiecewise Constant: O(1.0)\nOBERA Linear:O(2.0)\nELVIRA:O(2.2)\nELVIRA-WO:O(1.9)\nOBERA-W Linear: O(1.9)\nAEROS Quadratic: O(3.1)\nOBERA Quadratic: O(3.0)\nAEROS Quartic:O(4.8)Figure 10: Convergence for different reconstruction models. The convergence rates (in parenthesis) are estimated\nusing values 1 /h > 30.\nOBERA Linear 0.8\nOBERA-W Linear 0.7\nOBERA Quadratic 0.3\nELVIRA 0.04\nELVIRA-WO 0.02\nAEROS Quadratic 0.003\nAEROS Quartic 0.003\nTable 1: Average time (in seconds) taken to find the parameters of the interface by the different tested models.\n20(a) OBERA-W Linear.\n (b) AEROS Quadratic.\n(c) OBERA Quadratic.\n (d) AEROS Quartic.\nFigure 11: Reconstruction of a smooth domain by different methods for a scale of h= 1/30.\n215.2 The treatment of corner domains\nThe previously tested strategies achieve to reconstruct, at their corresponding orders, different smooth domains\nwith H¨ older smoothness s >1. This excludes the case of domains with piecewise smooth boundaries for which\nthe presence of corners call for a specific treatment. The simplest option that we study here is to use local\nrecovery by the approximation family V4of piecewise linear interface from Example 4. We propose, in what\nfollows, two methods to deal with vertices of any angle, bearing in mind the limitations expressed in Figure 4c.\nAEROS Vertex : The first method applies the AEROS strategy for v∈ˆV4a restriction of V4where π/2<\nθ1<3π/2 and −π/2< θ2< π/ 2,i.e.elements vwhose interface can be written as an oriented graph, which is\na particular instance of Example 3 where instead of searching pin a space of univariate polynomials, we pick\nit into W4the space of piecewise functions with one breakpoint contained in Tor its immediate neighbours.\nThis excludes the possibility of reconstructing a rectangle whose sides are parallel to the mesh but it applies\nto the case of the same rectangle slightly rotated. Under this restrictions it is possible, although lengthy, to\nextract explicit equations that allow us to derive a finite set of admissible parameters µ= (x1, x2, θ1, θ2) from\nthe observed cell averaged vector aS. We compare each proposed local approximation v∈ˆV4as in ELVIRA\nor OBERA, that is, by means of their associated loss L(u, v) while retaining at the end the one achieving the\nminimal value between the many possibilities. This same model selection strategy can be used to aggregate\nother competing models, like quadratic interfaces. This has the effect of keeping higher order models when\nthe interface is locally smooth, while taking corners into account as illustrated on Figure 12. By this approach\nwe avoid defining a vertex detection mechanism at the expense of computational overhead as we now need to\ncompute for each cell many losses, which was already the time bottleneck for ELVIRA method.\nTangent Extension Method (TEM) : The above restriction that the interface with a vertex needs to be\nan oriented graph could be limiting in some applications but it can be removed at the expense of complexifying\nthe reconstruction procedure. Our second proposed method deals with this aspect and consists in the following\nsteps:\n1. Associate to each singular cell T∈ Shsome reconstruction vTstemming from any of the local interface\nreconstruction methods discussed discussed so far.\n2. For each cell where the presence of a vertex is suspected (eventually for all T∈ Sh) we search for two\nsingular cells T1, T2∈ Shsatisfying the following\n•T̸=T1̸=T2̸=T\n•ST1∩ {T}=∅\n•ST2∩ {T}=∅\nwhere SvTidenotes the stencil Sused by a given smooth interface reconstruction method (for example\nlinear or quadratic) to produce local approximations vTi.\n3. Take the parameterized interfaces Γ 1and Γ 2, associated to vT1andvT2respectively, and do an order 1\nTaylor expansion at an intermediate point between cells ( T, T 1) and ( T, T 2) respectively. This yields the\nparameters of the two half planes H1andH2of Example 4 needed to define v∈V4.\n4. Finally, we compare the new local approximation v∈V4with the existing one vT, as explained above,\nretaining only the one whose associated loss, L(u, v) orL(u, vT), is minimal.\nThis procedure will reconstruct exactly corners when the interface is a line along both directions, but it will\nnot produce area-consistent reconstructions on cell Totherwise. In this regard, AEROS Vertex, being based\non AEROS strategy, will yield interfaces that are, though not cell-consistent as one could get with OBERA,\nat least column-consistent as long as we remain in the interpolation case. This is ensured if the stencil width\nequals the number of parameters of the approximating class which in the case of ˆV4is guaranteed by using\n4-width stencils.\nFigure 12 displays the successive improvements in the reconstruction when combining the different strategies\ndescribed so far:\n22•On Figure 12a, we use the AEROS Quadratic method. We observe that it recovers the interface in a\nsatisfactory manner only far enough from corners.\n•On Figure 12b, we first find a curve for each singular cell using AEROS Quadratic and then TEM. In this\ncase, some of the problems are solved, in particular corners with a 90◦angle and parallel to the grid.\n•On Figure 12c, we add to the previous method the first proposed approach, based on AEROS for vertices.\nWe obtain almost perfect results except for some cells where the quadratic approximation of the interface\ngiven by AEROS Quadratic was not replaced by a better one.\n•On Figure 12d, this last issue is addressed by aggregating, before applying any of the vertex mechanisms\ndescribed before, an ELVIRA-WO strategy to offer an alternative when AEROS Quadratic is too much\naffected by the presence of a nearby corner.\n23(a) AEROS Quadratic.\n (b) AEROS Quadratic + TEM.\n(c) AEROS Quadratic + TEM + AEROS Vertex.\n(d) ELVIRA-WO + AEROS Quadratic + TEM +\nAEROS Vertex.\nFigure 12: Reconstruction of a domain with corners from cell-averages of scale h= 1/30.\n245.3 Finite volume evolution in time\nFinally, we use the interface recovery methods presented so far as a constituent part of a finite volume solver\nwith the objective of reducing numerical dissipation. We study the particular case of a simple linear transport\nPDE:∂u\n∂t+b· ∇u= 0,\nin the unit square domain D= [0,1]2with periodic boundary conditions and initial condition being a piecewise\nconstant function u0:D→ {0,1}with Ω limited by a smooth interface as in Figure 11 or an interface having\ncorners as in Figure 12.\nAs in a large class finite volume schemes, the reconstruction is used at each step to compute the flux that\nupdates the averages at the next step. For simplicity of the presentation we have set a constant (both in space\nand time) velocity field b= (h/4,0) and worked with unit time steps ∆ t= 1 and coarse grids of size h= 1/30,\nso that the CFL condition is maintained. In this case, the numerical flux induced by a local reconstruction uk\ni,j\non a cell Tof coordinate ( i, j) at time step ktakes the form\nF(uk\ni,j) :=1\n|RT|Z\nRTuk\ni,j(x)dx\nwhere RT= [(i+ 1)h−b,(i+ 1)h]×[jh,(j+ 1)h]. The finite volume approximation at the next time step k+ 1\nis then given by the updated cell-average\nak+1\ni,j=ak\ni,j+F(˜uk\ni−1,j)− F(˜uk\ni,j).\nFigure 13a displays the evolution of the L1error between the exact solution and the reconstruction, for\nthe time evolution of a smooth domain. In this case, all three methods ELVIRA-WO, AEROS Quadratic and\nELVIRA-WO + AEROS Quadratic + TEM + AEROS Vertex behave similarly with an error that is maintained\nat the same level for all tested times showing that numerical dissipation has been avoided on the three cases.\nThis contrasts with the piecewise constant reconstruction that corresponds to the standard upwind scheme.\nFigure 13b shows the effect of the presence of corners in the interface of Ω in terms of a slow but accumulative\ndeterioration in both methods ELVIRA-WO, AEROS Quadratic as they are not design to treat vertices. In\ncontrast, ELVIRA-WO + AEROS Quadratic + TEM + AEROS Vertex, as before, keeps its error on the same\nlevel for all the times tested.\n6 Conclusion and perspectives\nIn this work, we have presented several interface recovery methods. For the two main classes OBERA and\nAEROS, we have provided general analysis strategies for establishing convergence rates that depend on the\ngeometric smoothness of the interface. From a practical perspective, the methods can be combined with the\naggregation strategy outlined in the previous section, and we have made them available in the open-source\npython package2.\nSeveral natural perspectives are foreseen (and we have explored some of them already in the open-source\npackage):\n1. Address the reconstruction of more general piecewise smooth functions with jump discontinuities across\ngeometrically smooth or piecewise smooth interfaces. This requires a proper adaptation of the interface\nrecovery strategies, combined with a high-order treatment of the smooth part of the function corresponding\nto the cell T /∈ Sh. The latter can be done by using polynomial reconstructions on stencils not containing\ncells of Shfollowing the standard ENO strategy.\n2. Study the use of machine learning techniques trained on sufficiently rich sets of interfaces for performing\ncertain tasks in an automated and hopefully more efficient manner. Such tasks include, for example, the\nfast reconstruction of the parameter µfrom the cell averages, the identification of cells that may contain\nvertices, or the direct access to the numerical flux in the case of finite volume scheme, as proposed for\n2https://github.com/agussomacal/SubCellResolution\n251 2 4 8 16 24 40 70 120\nIterations10−310−210−1/bardblu−˜u/bardblL1\nAEROS Quadratic\nELVIRA-WO\nELVIRA-WO + AEROS Quadratic + TEM + AEROS Vertex\nUpWind(a) Smooth domain.\n1 2 4 8 16 24 40 70 120\nIterations10−310−210−1/bardblu−˜u/bardblL1\nAEROS Quadratic\nELVIRA-WO\nELVIRA-WO + AEROS Quadratic + TEM + AEROS Vertex\nUpWind (b) Domain with corners.\nFigure 13: Time evolution of the finite volume scheme error for h= 1/30.\nexample in [3] for vertices forming angles of ϕ= 90◦. It should however be noted that, as opposed to the\napproaches that we developped in this paper, the machine learning-based approach does not offer rigorous\nconvergence guarantees. Also, our attempts to beat the AEROS method with machine learning strategies\nwere so far unsuccessful, both from the accuracy point of view, and the runtime point of view. We provide\nour implementation of these strategies in the Python package.\nReferences\n[1] R. Adams and J. Fournier Sobolev Spaces , Academic Press, 2003.\n[2] F. Arandiga, A. Cohen, R. Donat, N. Dyn, and B. Matei Approximation of piecewise smooth functions\nand images by edge-adapted (ENO-EA) nonlinear multiresolution techniques , Applied and Computational\nHarmonic Analysis, 24, 225–250, 2008.\n[3] B. Despr´ es and H. Jourdren, Machine Learning design of Volume of Fluid schemes for compressible flows ,\nJournal of Computational Physics 408, 2020.\n[4] A. Cohen, M. Dolbeault, O. Mula, A. Somacal. Nonlinear approximation spaces for inverse problems ,\nAnalysis and Applications (AA) 21, 01, 217–253, 2023.\n[5] A. Harten, ENO schemes with subcell resolution , Journal of Computational Physics 83, 148-184, 1989.\n[6] A. Harten, B. Engquist, S. Osher, and C. Chakravarthy, Uniformly high order accurate essentially non-\noscillatory schemes, III , Journal of Computational Physics 71, 231-303, 1987.\n[7] C. W. Hirt and B. D. Nichols. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries ,\nJournal Of Computational Physics 39, 201-225 (1981).\n[8] S. Ioffe and C. Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal\nCovariate Shift , arXiv, 2015, http://arxiv.org/abs/1502.03167.\n[9] D. Kroon. Numerical Optimization of Kernel-Based Image Derivatives , Short Paper University Twente,\n2009.\n[10] X.-D. Liu, S. Osher and T. Chan. Weighted Essentially Non-oscillatory Schemes , Journal of Computational\nPhysics 115, 200–212, 1994.\n[11] J. Necas, Les m´ ethodes directes en th´ eorie des ´ equations elliptiques , Masson, 1967.\n26[12] W. F. Noh and P. Woodward. SLIC (Simple Line Interface Calculation) , A. I. van der Vooren and P. J.\nZandbergen, Eds, Lecture Notes in Physics, Springer New York, 1976, p. 330.\n[13] J.E. Pilliod and E. G. Puckett, Second-order accurate volume-of-fluid algorithms for tracking material\ninterfaces , Journal of Computational Physics 199, 465–502, 2004.\n[14] E. G. Puckett, A volume-of-fluid interface tracking algorithm with applications to computing shock wave\nrefraction , proceedings of the fourth international symposium on Computational Fluid Dynamics, 933–938,\n1991.\n[15] W. J. Rider and D. B. Kothe. Reconstructing Volume Tracking , Journal of Computational Physics, Vol.\n141, No. 2, 1998, pp. 112-152. doi:10.1006/jcph.1998.5906.\n[16] M. Rudman. Volume-Tracking Methods for Interfacial Flow Calculations , International Jour-\nnal for Numerical Methods in Fluids, Vol. 24, No. 7, 1997, pp. 671-691, doi:10.1002/(SICI)1097-\n0363(19970415)24:7¡671::AID-FLD508¿3.0.CO;2-9.\n[17] R. Scardovelli and S. Zaleski. Direct Numerical Simulation of Free-Surface and Interfacial Flow , Annual\nReview of Fluid Mechanics Vol. 31, 1999, pp. 567-603. doi:10.1146/annurev.fluid.31.1.567.\n[18] D. Gueyffier, J. Li, A. Nadim, R. Scardovelli, and S. Zaleski. Volume-of-Fluid Interface Tracking with\nSmoothed Surface Stress Methods for Three-Dimensional Flows. , Journal of Computational Physics 152,\nno. 2 (July 1999): 423–56. doi:10.1006/jcph.1998.6168.\n[19] Virtanen, et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python , Nature Methods\n17, 261–272, 2020.\nA The orientation test\nIn this apprendix we give the proof of Theorem 4.1, which is based on:\n1. First studying the case where the interface Γ is a line over the 3 ×3 stencil where the numerical gradient\nGT= (HT, VT) is computed.\n2. Second applying a perturbation argument in the case of a general Csinterface which locally deviates from\na line in a quantitatively controlled manner.\nA.1 The case of a linear interface\nWe assume here that, over the 3 ×3 stencil Scentered at T, the interface Γ is a line crossing T. Therefore, the\nrestriction of utoSis of the form\nu|S=vr,θ:=χ{⟨z−zT,e⊥\nθ⟩⩽r}.\nwhere zTis the center of Tandeθ= (−sin(θ),cos(θ)) with θthe angle between Γ and the horizontal line, that\nis,eθis the unit normal vector to Γ pointing to the outward direction where u|S= 0. The following result shows\nthat the orientation test discriminates exactly if the direction of Γ is closer to horizontal or vertical. Its proof\nuses elementary geometrical arguments, which are only sketched using pictures in order to avoid cumbersome\nanalytic developments.\nTheorem A.1. IfGT= (HT, VT)is the numerical gradient based on the Sobel filter for the above function vr,θ,\nthen the following holds:\n•|VT|>|HT|if and only if θ∈[0, π/4[∪]3π/4,5π/4[∪]7π/4,2π[\n•|HT|>|VT|if and only if θ∈]π/4,3π/4[∪]5π/4,7π/4[\n•|HT|=|VT|if and only if θ∈ {π/4,3π/4,5π/4,7π/4}\nIn addition\n27•VT>0if and only if θ∈]π/2,3π/2[andVT<0if and only if θ∈[0, π/2[∪]3π/2,2π[\n•HT>0if and only if θ∈]0, π[andHT<0if and only if θ∈]π,2π[.\nProof: Without loss of generality, we only consider the case where θ∈[0, π/4] since all other cases [ kπ/4,(k+\n1)π/4] for k= 1, . . . , 7 are treated in a similar way. In order to understand the effect of θon the values of HT\nandVT, we parametrize the function vr,θdifferently: we fix zTto be the point crossed by Γ on the descending\ndiagonal of T(which exists and is unique when θ∈[0, π/4]) and study HTandVTfor the function\nvθ:=χ{⟨z−zT,e⊥\nθ⟩⩽0},\nas we let θvary. By scale invariance, we may assume that we work with cells of side-length equal to 1 without\naffecting HTandVT.\nFigure 14 (left) pictures the value of VTas the difference between areas of the portions of cells from the\nupper and lower rows crossed by the half-plane below Γ with weight 2 for central cells and 1 for left and right\ncells. This difference is strictly negative for all θ∈[0, π/4]. Its value at θ= 0 is equal −4. As θgrows towards\nπ/4 it first stays equal to −4 until it starts strictly increasing for some value θ∗∈[0, π/4[ that depends on the\nposition of zTon the diagonal. This monotonic growth can be checked by observing that for 0 ⩽θ1< θ2⩽π/4,\none has VT(vθ2)−VT(vθ1) =VT(vθ2−vθ1) and the function vθ2−vθ1is supported in a symmetric cone Kθ1,θ2\ncentered at zTand has value 1 on the right and −1 on the left. Thus VT(vθ2)−VT(vθ1) is the sum of the areas\nof the portions of cells from the upper and lower rows intersected by Kθ1,θ2with weight 2 for central cells and\n1 for for left and right cells, which is strictly positive if θ2> θ∗.\nFigure 14 (center) pictures the value of HTas the difference between areas of the portions of cells from the\nright and left columns crossed by the half-plane below Γ with weight 2 for central cells and 1 for lower and\nupper cells. This difference is null when θ= 0 and increases strictly as θgrows from 0 to π/4. Once again, the\nstrictly monotonic growth is due to the fact that VT(vθ2)−VT(vθ1) is the sum of the areas of the portions of\ncells from the left and right columns crossed by Kθ1,θ2with weight 2 for central cells and 1 for for left and right\ncells, which is strictly positive whenever 0 ⩽θ1< θ2⩽π/4.\nθ1θ∗θ2\nzT\n−1−2−11 2 1\nθ1θ2\nzT\n−1−2−1\n121\nθ˜θ=π/2−θ\nzT\n−1−2−1\n121\nFigure 14: Dependence of VT(left), HT(center), |VT| − |HT|(right) as θvarie in [0 , π/4].\nThis demonstrates the second statement regarding signs of VTandHT.\nOn the other hand, by symmetry, we note that VT(vθ) =−HT(v˜θ) where ˜θ=π/2−θwith same base point\nzT. Therefore |VT| − |HT|=−VT−HT=HT(v˜θ−vθ) and v˜θ−vθis supported in a symmetric cone Kθ,˜θ\ncentered at zTwith value 1 on the right and −1 on the left. As pictured on Figure 14 (right), this quantity\nis the sum of the areas of the portions of cells from the right and left column crossed by Kθ,˜θwith weight 2\nfor central cells and 1 for lower and upper cells. These quantities are null when θ=π/4 since the cone is then\nrestricted to a line, and increases strictly as θdecreases from θ/4 to 0 since the cone is opening.\nThis demonstrates the first statement regarding comparison between |VT|and|HT|. □.\nWe will use the following direct consequence of this result, which is obtained by compactness since VTand\nHTare continuous with respect to θandzT: for any 0 < δ < π/ 4, there exists a γ=γ(δ)>0 such that\nθ∈[π/4 +δ,3π/4−δ]∪[5π/4 +δ,7π/4−δ] =⇒ |HT|⩾|VT|+γ (27)\nand\nθ∈[3π/4−δ,5π/4 +δ] =⇒VT⩾γ, (28)\n28A.2 A perturbation analysis\nWe next turn to the proof of Theorem 4.1, focusing without loss of generality on case 1. Let us assume that the\ninterface Γ = ∂Ω has Cssmoothness for some s >1, and let Tbe a singular cell crossed by Γ and Sthe 3×3\nstencil centered at T.\nWe now fix ˜ zTto be one point of Γ ∩Tande⊥\nθ= (−sin(θ),cos(θ)) be the outer unit normal to Ω at ˜ zT.\nThe function\nvθ:=χ{⟨z−˜zT,e⊥\nθ⟩⩽0},\nis a perturbation of uas pictured on Figure 15 where the line interface Lis the tangent to Ω at ˜ zT. Since Ω\nhasCssmoothness, the deviation between the curved interface Γ and its tangent Lhas area of order O(hs+1)\noverS, and therefore\n|a˜T(u)−a˜T(vθ)|⩽Chs−1,˜T∈S,\nfor some fixed constant Cthat only depends on the Csnorm of the graph that locally characterizes Γ. In turn,\nup to multiplying the constant Cby 8, one has\n|HT(u)−HT(vθ)|⩽Chs−1and|VT(u)−VT(vθ)|⩽Chs−1(29)\neθ\nLvθ= 0\nvθ= 1Γu= 0\nu= 1˜zT\nFigure 15: Approximation of a smooth interface by its tangent\nWe now take any 0 < δ < π/ 4 arbitrarily small and consider the quantity γ=γ(δ) such that (27) and (28)\nare valid. For h⩽h0small enough, we are ensured that\nChs−1⩽γ\n3,\nwhere Cis the constant in (29). Therefore, if θ /∈[0, π/4+δ]∪[7π/4−δ,2π[, or equivalently θ /∈[−π/4−δ, π/4+δ],\nwe obtain either by (27) that\n|HT(u)|⩾|HT(vθ)| −γ\n3⩾|VT(vθ|+2γ\n3⩾|VT(u)|+γ\n3>|VT(u)|,\nor by (28) that\nVT(u)⩾VT(vθ)−γ\n3⩾2γ\n3>0.\nTherefore, we conclude for case 1 that if |HT(u)|⩽|VT(u)|andVT(u)⩽0, the angle θof the tangent line L\nnecessarily lies in [ −π/4−δ, π/4 +δ]. In other words, the points z= (x, y) in the half plane {⟨z−˜zT, e⊥\nθ⟩⩽0}\ncan be characterized by an equation of the form\ny⩽a(x),\nwhere ais affine and |a′(x)|⩽1 +εsuch that 1 + ε= tan( π/4 +δ), with 0 ⩽ε⩽δandε∼δwhen δis small.\nOn the other hand the interface Γ is described on Tby an equation of the form\ny⩽ψ(x),\nwhere due to the Cssmoothness of Γ, one has an estimate of the form\n|ψ′(x)−a′(x)|⩽Chs−1,\n29and the same will hold on a stencil STof width 2 k+ 1 up to enlarging the value of C, so that\n|ψ′(x)|⩽1 +ε+Chs−1, x∈I,\nwhere Iis the horizontal support of ST. Therefore, we can find h∗=h∗(Ω) such that if h⩽h∗we are thus\nensured that\n|ψ′(x)|⩽k+ 2\nk+ 1, x∈I.\nThis implies that the graph of ψremains confined in STif we choose it to be of size (2 k+ 1)×(2l+ 1) with\nl=k+ 2, which concludes the proof of the theorem.\n30"    },    {        "title": "2402.00961v1.Novel_turbulence_and_coarsening_arrest_in_active_scalar_fluids.pdf",        "content": "Novel turbulence and coarsening arrest in active-scalar fluids\nNadia Bihari Padhan,1,∗Kolluru Venkata Kiran,1,†and Rahul Pandit1,‡\n1Centre for Condensed Matter Theory, Department of Physics,\nIndian Institute of Science, Bangalore, 560012, India.\n(Dated: February 5, 2024)\nWe uncover a new type of turbulence – activity-induced homogeneous and isotropic turbulence\nin a model that has been employed to investigate motility-induced phase separation (MIPS) in a\nsystem of microswimmers. The active Cahn-Hilliard-Navier-Stokes equations (CHNS), also called\nactive model H, provides a natural theoretical framework for our study. In this CHNS model, a\nsingle scalar order parameter ϕ, positive (negative) in regions of high (low) microswimmer density,\nis coupled with the velocity field u. The activity of the microswimmers is governed by an activity\nparameter ζthat is positive for extensile swimmers and negative for contractile swimmers. With\nextensile swimmers, this system undergoes complete phase separation, which is similar to that in\nbinary-fluid mixtures. By carrying out pseudospectral direct numerical simulations (DNSs), we\nshow, for the first time, that this model (a) develops an emergent nonequilibrium, but statistically\nsteady,state(NESS)ofactiveturbulence,forthecaseofcontractileswimmers,if ζissufficientlylarge\nand negative and (b) this turbulence arrests the phase separation. We quantify this suppression\nby showing how the coarsening-arrest length scale does not grow indefinitely, with time t, but\nsaturates at a finite value at large times. We characterise the statistical properties of this active-\nscalar turbulence by employing energy spectra and fluxes and the spectrum of ϕ. For sufficiently\nhigh Reynolds numbers, the energy spectrum E(k)displays an inertial range, with a power-law\ndependence on the wavenumber k. We demonstrate that, in this range, the flux Π(k)assumes a\nnearly constant, negative value, which indicates that the system shows an inverse cascade of energy,\neven though energy injection occurs over a wide range of wavenumbers in our active-CHNS model.\nI. INTRODUCTION\nActive turbulence , spatiotemporal chaos in active-matter\nsystems[see, e.g., Refs.[1–4]], hasgarneredconsiderableat-\ntention over the past decade. This intriguing form of tur-\nbulence manifests itself in various experimental systems,\nincluding bacterial suspensions [1, 5–11], suspensions of\nmicrotubles, and molecular motors [12, 13]. In classical-\nfluid turbulence, a nonequilibrium statistically steady state\n(NESS) is reached when the fluid is driven by an external\nforce; by contrast, in activefluids, the microscopic con-\nstituents drive the system by converting chemical sources\nof energy into kinetic energy [5, 14]. Many studies have fo-\ncused on understanding emergent turbulence-type patterns\nby using continuum hydrodynamical models, in which phe-\nnomenological parameters depend on the microscopic de-\ntails of the active fluid. An overview of the various models\ncan be found in Ref. [3]. In certain models, the energy\nspectrum of such turbulence exhibits universal power-law\nbehaviors [4, 15]. In contrast, there are instances in which\npower-law exponents for these energy spectra depend on\nparameters in the model [7, 11, 16]. The elucidation of\nthe statistical properties of these types of emergent tur-\nbulent states continues to be an important challenge in\nactive-matter research. Recent studies have shown the im-\nportance of fluid inertia in some systems that display ac-\ntive turbulence such as active polarandnematic fluids [see,\ne.g., Refs. [17–20]]. In addition, considerable attention has\n∗nadia@iisc.ac.in\n†kollurukiran@iisc.ac.in\n‡rahul@iisc.ac.inbeen directed towards the study of scalar active fluids , in\nwhich the intricate spatiotemporal evolution of an active\nfluid arises from the interaction of a scalar order parameter\nϕwith the fluid velocity u. Scalar active fluids are simpler\nthan their polar or nematic counterparts, yet they are rich\nenough to yield intriguing emergent NESSs [21–23]; and\nthey have been used in studying active droplets [24] and\nactive stratified turbulence [25].\nWe uncover a new type of turbulence – activity-induced\nhomogeneous and isotropic turbulence in a model that has\nbeen employed to investigate motility-induced phase sepa-\nration (MIPS) [22, 24] in a system of microswimmers. The\nactive Cahn-Hilliard-Navier-Stokes equations (CHNS), also\nknown as the active model H [22], provides a natural the-\noretical framework for our study. In this CHNS model, a\nsingle scalar order parameter ϕ[which is positive (nega-\ntive) in regions where the microswimmer density is high\n(low)] is coupled with the velocity field u. The activity of\nthe microswimmers is governed by an activity parameter\nζthat is positive for extensile swimmers and negative for\ncontractile swimmers. With extensile swimmers, this sys-\ntem undergoes complete phase separation, which is similar\nto that in binary-fluid mixtures [22]. By carrying out pseu-\ndospectral direct numerical simulations (DNSs) we show,\nfor the first time, that this model develops an emergent\nnonequilibrium, but statistically steady, state (NESS) of\nactive turbulence, for the case of contractile swimmers, if\nζis sufficiently large and negative. This turbulence arrests\nthe phase separation into regions with positive and neg-\native values of ϕ, in much the same way as conventional\nfluid turbulence leads to the suppression of phase separa-\ntion in a binary-fluid mixture [26–28]. We quantify this\nsuppression by showing how the coarsening-arrest length\nscale does not grow indefinitely, with time t, but saturatesarXiv:2402.00961v1  [physics.flu-dyn]  1 Feb 20242\n(c) (d)(a) (b)\nFIG. 1. Pseudo-gray-scale plots of the ϕfield [at representative\ntimes in the nonequilibrium statistically steady state (NESS)]\nfor the activity parameter (a) |ζ|= 0.01and (b) |ζ|= 1.5. Pseu-\ndocolor plots of the vorticity field, normalized by the maximum\nof|ω|, are shown in (c) and (d) for the parameters in (a) and\n(b), respectively.\nat a finite value at large times. We then characterise the\nstatistical properties of this active-scalar turbulence by em-\nploying the energy spectrum and fluxes, which are familiar\nfrom classical fluid turbulence, and also the spectrum of\nϕ, which is used in studies of phase separation. For suffi-\nciently high Reynolds numbers, the energy spectrum E(k)\ndisplays an inertial range, with a power-law dependence on\nthe wavenumber k. We demonstrate that, in this range,\nthe flux Π(k)assumes a nearly constant, negative value,\nwhich indicates that the system shows an inverse cascade\nof energy that is similar to its counterpart in 2D homoge-\nneous and isotropic fluid turbulence, even though energy\ninjection occurs over a wide range of wavenumbers in our\nactive-CHNS model.\nThe remaining part of this paper is organised as follows.\nSection II introduces the active CHNS model, summarises\nthe numerical methods we employ to study it, and defines\nthe statistical measures we use to characterise active tur-\nbulence in this model. In Section III we present the results\nof our study. We discuss the significance of our results in\nSection IV.\nII. MODEL, METHODS, AND STATISTICAL\nMEASURES\nWe introduce the active-CHNS model in Subsection IIA.\nIn Subsection IIB we describe the statistical measures we\nuse to characterise active turbulence in this model. Finally,\nin Subsection IIC we give the details of our pseudospectral\nDNS.A. The Active Cahn-Hilliard-Navier-Stokes Model\nWe consider the incompressible active CHNS equations\n(also called active model H) to study active turbulence in\nsystems of contractile swimmers [22, 24] in two spatial di-\nmensions (2D):\n∂tϕ+ (u· ∇)ϕ=M∇2\u0012δF\nδϕ\u0013\n; (1)\n∂tω+ (u· ∇)ω=ν∇2ω+3\n2ϵ∇ ×(∇ ·ΣA)−αω;(2)\n∇ ·u= 0 ; (3)\nωis the vorticity field; ν,α, and M are the kinematic vis-\ncosity, bottom friction, and mobility, respectively. Fis the\nLandau-Ginzburg variational free-energy functional\nF[ϕ,∇ϕ] =Z\nΩ\u00143\n16σ\nϵ(ϕ2−1)2+3\n4σϵ|∇ϕ|2\u0015\n,(4)\nin which the first term is a double-well potential with min-\nima at ϕ=±1. The scalar order parameter ϕis positive\n(negative) in regions where the microswimmer density is\nhigh (low); in the interfaces between these regions, ϕvaries\nsmoothly, over a width ϵ. The free-energy penalty for an\ninterface is given by the bare surface tension σ. In the\ninherently nonequilibrium active model H all terms in the\nstress tensor do not follow from F. In particular, we must\ninclude the stress tensor ΣA, which has the form of a non-\nlinear Burnett term and has the components [22, 24, 25, 29]\nΣA\nij=−ζ\u0014\n∂iϕ∂jϕ−δij\n2|∇ϕ|2\u0015\n, (5)\nwhere ζ, the activity coefficient, can take both positive and\nnegative values: ζ <0(ζ >0) for contractile (extensile)\nswimmers [22].\nB. Statistical Characterisation\nTo characterise the statistical properties of active-scalar\nturbulence, we employ energy spectra and fluxes, which are\nfamiliar from classical fluid turbulence, and the spectrum\nofϕ. These quantities not only help us to understand, via\nDNS, the emergent turbulent like behaviour (characterized\nby spatiotemporal fluctuations) in Eqs. (1-3), but they also\naidusindifferentiatingactivescalarturbulencefromclassi-\ncal 2D incompressible fluid turbulence [see, e.g., Refs. [30–\n33]] and also turbulent patterns found in other active sys-\ntems [see, e.g., Refs. [17–20]]. We define the shell-averaged\nenergy and phase-field spectra, respectively, both at time\ntand averaged over time [the time average is denoted by\n⟨·⟩t]:\nE(k, t)≡X\nk≤k′<k+1|ˆu(k′, t)|2; Φ( k, t)≡X\nk≤k′<k+1|ˆϕ(k′, t)|2;\nE(k)≡ ⟨E (k, t)⟩t; Φ( k)≡ ⟨Φ(k, t)⟩t; (6)\nthe caret denotes a spatial Fourier transform. Our CHNS3\n(a) (b)\nFIG. 2. (a) Plot of the coarsening length scale L(t)[Eq. (8)] versus time tfor various values of |ζ|; the plot for ζ= 0shows growth\nthat is consistent with the Lifshitz-Slyozov form L(t)∼t1/3(dashed line); L(t)saturates to a finite value for |ζ|>0. (b) Log-linear\nplots of the mean coarsening-arrest scale Lc=⟨L(t)⟩t(red curve) and the integral-scale Reynolds number ReLI(blue curve) versus\n|ζ|.\n(a) (b)\nFIG. 3. Log-log plots versus the wavenumber kof spectra for |ζ|= 0.001,0.01,0.1,1.5: (a) The compensated energy spectrum\nk5/3E(k). For|ζ|= 0.1,1.5, the exponent is −5/3(black dotted line in the compensated spectrum). (b) The compensated phase-\nfield spectrum k−7/6Φ(k)for|ζ|= 0.001,0.01,0.1,1.5. These spectra show the power-law behaviour Φ(k)∼k7/6(black dotted line\nin the compensated spectrum).\nstudy of active scalar turbulence uses the Reynolds, Péclet,\nWeber, and the non-dimensional friction numbers that are,\nrespectively:\nRe=LIurms\nν;Pe=ϵLIurms\nMσ;\nWe =LIu2\nrms\nσ;α′=αLI\nurms; (7)\nurmsandLIare, respectively, the root-mean-squared ve-\nlocity and the fluid integral length scale; and L(t)andLc\nare the time-dependent and mean coarsening length scales;these are defined as follows:\nurms =\"X\nkE(k)#1/2\n;LI= 2πP\nkE(k)P\nkkE(k);\nL(t) =P\nkΦ(k, t)P\nkkΦ(k, t);Lc=⟨L(t)⟩t. (8)\nInTableI,weprovidethenon-dimensionalparametersfrom\nour direct numerical simulations (DNSs) for various values\nof the activity parameter |ζ|. Furthermore, E(k, t)satisfies\nthe following energy-budget equation [28, 34, 35]:\n∂tE(k, t) =T(k, t) +Dα(k, t) +Dν(k, t) +Sϕ(k, t),(9)4\nwhere\nT(k, t) =−ℜ\nX\nk≤|k′|<k+1[ˆu(−k′, t)·P(k′)·\\(u· ∇u)(k′, t)]\n,\nDα(k, t) =−2αE(k, t), D ν(k, t) =−2νk2E(k, t),\nSϕ(k, t) =ℜ\nX\nk≤|k′|<k+1[ˆu(−k′, t)·P(k′)·\\(∇ ·ΣA)(k′, t)]\n(10)\nare, respectively, the energy transfer because of the in-\nertial term, the energy dissipations arising from friction\nand the viscosity, and the energy transfer via the active-\nstress term; the transverse projector P(k), which enforces\nthe incompressibility condition, has the components Pij≡\n(δij−kikj/k2). We also use the following mean energy\ntransfers from the inertial, friction, viscous, and active-\nstress terms, and the associated kinetic-energy and active-\nstress fluxes [35]:\nT(k) =⟨T(k, t)⟩t;Sϕ(k) =⟨Sϕ(k, t)⟩t;\nDα(k) =⟨Dα(k, t)⟩t;Dν(k) =⟨Dν(k, t)⟩t;\nΠ(k) =−Zk′\n0T(k′)dk′; Πϕ(k) =−Zk′\n0Sϕ(k′)dk′.(11)\nC. Direct Numerical Simulation\nWe carry out DNSs of the active-CHNS partial dif-\nferential equations (1)-(5) by using the pseudospectral\nmethod [24, 27, 36, 37] in a 2D periodic square domain,\nD ≡ [0, L]2, with Lthe length of the side of the square.\nWe evaluate spatial derivatives in Fourier space and the\nnonlinear terms in physical space. For time integration,\nwe use the semi-implicit exponential-time-difference\nRunge-Kutta2 (ETDRK2) method [38]. We employ the\n1/2-dealiasing scheme to remove the Fourier aliasing\nerrors [24, 27, 36, 37]. To resolve the interface of width\nϵ, we ensure that there are three grid points across the\ninterface. We use a CUDA-C code that we have developed\nand optimised for an NVIDIA A100 processor.\nD. Initial conditions\nWe use the following initial conditions for the ωandϕ\nfields:\nω(x, y,0) = 0 ; ϕ(x, y,0) = ϕ0+ξ(x, y) ;(12)\nξ(x, y), a random number distributed uniformly on the in-\nterval [−0.1,0.1], provides a random perturbation to the\nϕ=ϕ0= 0state.\nIII. RESULTS\nIn this Section we present a series of DNSs that we have\ndesigned to demonstrate how the activity of contractileswimmers [ ζ <0in Eq. (5)] leads to active turbulence that\nisstrongenoughtosuppressmotility-inducedphasesepara-\ntion. Our results are the active-turbulence counterparts of\nthe suppression of phase separation (also called coarsening\narrest) by fluid turbulence [see, e.g., Refs. [26, 27]].\nWe note that, if the activity ζ= 0, the stress tensor (5)\nvanishes, so the coupled Eqs. (1)-(5) decouple into Cahn-\nHilliardequationsormodelB[39,40]andtheNavierStokes\nequations [41–43] for a Newtonian fluid; the advection term\nfor the ϕfield is set to zero by virtue of the initial condi-\ntions. Therefore, the domain growth or coarsening takes\nplace solely via diffusion, without hydrodynamical effects,\nand it follows the well-established Lifshitz-Slyozov domain-\ngrowth form L(t)∼t1/3[see, e.g., Refs. [40, 44, 45]];\ncomplete phase separation also occurs for extensile swim-\nmers [22] that lead to ζ >0.\nWe concentrate on active-turbulence-induced suppres-\nsion of phase separation and the diffusive Lifshitz-Slyozov\ncoarsening in our model (1)-(5) with contractile swimmers,\nfor which the activity parameter ζ <0[46]. Active turbu-\nlence and coarsening arrest in the active-CHNS model (1)-\n(5) can be visualized qualitatively by using pseudo-gray-\nscale plots of the ϕfield as we show in Figs. 1 (a) and (b)\nfor|ζ|= 0.01and (b) |ζ|= 1.5, respectively, at representa-\ntive times in the nonequilibrium statistically steady state\n(NESS).InFigs.1(c)and(d), wepresentpseudocolorplots\nofthevorticityfield, normalizedbythemaximum of |ω|, for\nthe parameters in Figs. 1 (a) and (b), respectively. These\nplots show clearly that the typical size of a single-phase do-\nmain decreases as activity-induced turbulence is enhanced\nby an increase in the value of |ζ|.\nWe quantify active-turbulence-induced suppression of\nphase separation by plotting the coarsening length scale\nL(t)versus time tin Fig. 2(a) for various values of |ζ|;\nthe plot for ζ= 0shows growth that is consistent with\nthe Lifshitz-Slyozov form L(t)∼t1/3(dashed line [47]).\nAstincreases, L(t)saturates to a finite value for |ζ|>0,\ni.e., Eqs. (1)-(5) lead to coarsening-arrest because of ac-\ntive turbulence . In Fig. 2(b) we show how the mean\ncoarsening-arrest scale Lc=⟨L(t)⟩tdecreases as |ζ|in-\ncreases (red curve); the attendant growth of the integral-\nscale Reynolds number ReLI(blue curve) signals the en-\nhancement of activity-induced turbulence.\nWe now characterise the statistical properties of activity-\ninduced turbulence in Eqs. (1)-(5). We begin with log-log\nplots of compensated energy and scalar- ϕspectra, k5/3E(k)\nandk−7/6Φ(k), versus k, in Figs. 3 (a) and (b), respec-\ntively, for various values of |ζ|. These plots suggest that,\nasζincreases, the activity-induced turbulence in this sys-\ntems leads to a nonequilibrium statistically steady state\n(NESS) with an inertial range of scales in which the en-\nergy spectrum has a power-law form that is consistent with\nE(k)∼k−5/3. We show below that this power-law spec-\ntrum arises because of an inverse cascade of energy. Its\npower-law form can then be surmised as in statistically\nsteady homogeneous and isotropic 2D-fluid turbulence with\nan inverse energy cascade [30–33]. Note that this power-\nlaw region extends over nearly one-and-a-half decades of k\nat the largest value |ζ|(= 1.5)that we consider.5\n(a) (b)\nFIG. 4. Plots versus k(log scale) of the contributions T(k), Dα(k), Dν(k),and Sϕ(k), in red, purple, yellow, and blue, respectively,\nfor (a) |ζ|= 0.001and (b) |ζ|= 1.5.\n(a) (b) (c)\nFIG. 5. Plots for |ζ|= 0.001,0.01,0.1,and1.5of the normalized (a) energy flux Π(k)LI/u3\nrmsversus k(log scale) and log-log plots\nversus kof (b) |Π(k)|LI/u3\nrmsand (c) Πϕ(k)LI/u3\nrms.\nRun |ζ|ReLI α′Pe We\nR100 − 0 0\nR20.001 1.1×1005.6×10−22.8×10−11.6×10−4\nR30.01 3.2×1007.9×10−38.5×10−12.3×10−3\nR40.03 1.3×1016.6×10−33.5×1002×10−2\nR50.05 4.3×1017.8×10−31.2×1011.1×10−1\nR60.1 6.9×1017.0×10−31.9×1012.3×10−1\nR70.5 1.1×1025.8×10−33.2×1015.3×10−1\nR811.2×1026.8×10−33.3×1015.8×10−1\nR91.5 1.3×1025.7×10−33.4×1016.1×10−1\nTABLE I. The values of various parameters in our DNS runs\nR1-R9. The following parameters are fixed in all these runs:\nN= 1024 ,grid size dx= 2π/N, ϵ = 3dx, L = 2π,Cn=\n3dx/L, M =ϵ2/2, σ= 1, ν= 5×10−3,α= 0.01.\nWeexaminetheenergy-transfermechanismsintheNESS\nof activity-induced turbulence in Eqs. (1)-(5) by using\nthe energy-budget equation (9) and evaluating the rele-\nvant scale-by-scale energy contributions [28, 34, 35]. In\nFigs. 4(a) and (b) we present, for the illustrative values|ζ|= 0.001and (b) |ζ|= 1.5, respectively, plots versus k\n(log scale) of the inertial, friction, viscous, and active-stress\ncontributions T(k) (red) , Dα(k) (purple) , Dν(k) (yellow)\nandSϕ(k) (blue) ,which we have defined in Eq. (11).\nSuch plots indicate that, for low values of |ζ|, the contri-\nbutions of T(k)andDα(k)are negligible [Fig. 4(a)], so, in\nthe NESS with ⟨∂tE(k, t)⟩t= 0, dominant balance yields\nDν(k) +Sϕ(k) = 0. As|ζ|increases, both T(k)andDα(k)\nincrease in magnitude [Fig. 4(b)], so a four-term balance is\nrequired in the NESS.\nTo show that activity-induced turbulence exhibits a bona\nfideinertial range we present plots for |ζ|= 0.001,0.01,0.1,\nand1.5of the energy flux Π(k)[Eq. 11] versus k(log scale)\n[Fig. 5(a)]. We also present log-log plots versus kof|Π(k)|\n[Fig. 5(b)] and Πϕ(k)[Fig. 5(c)]. These plots show con-\nstant fluxes over at least one decade of the wavenumber k,\nso we have a well-defined inertial range, that has a remark-\nable similarity with fluid turbulence [48]. The sign of Π(k)\nin this range of scales indicates that activity-induced tur-\nbulence in Eqs. (1)-(5) yields an inverse cascade of energy\nthat is reminiscent of a similar cascade in 2D statistically\nsteady homogeneous and isotropic fluid turbulence [30–33].\nBy comparing the different plots in Fig. 5(a), we see that6\nthis inverse cascade is suppressed as |ζ|decreases; this is\nreminiscent of similar cascade suppression in instability-\ndriven 2D turbulence [49].\nIV. CONCLUSIONS\nWe have uncovered activity-induced homogeneous and\nisotropic turbulence in the active Cahn-Hilliard-Navier-\nStokes equations (CHNS), which provides a natural the-\noretical framework for our study, in which a single scalar\norder parameter ϕ[positive (negative) in regions where the\nmicroswimmer density is high (low)] is coupled with the\nvelocity field u. The activity of the microswimmers is gov-\nerned by an activity parameter ζthat is positive for exten-\nsileswimmers and negative for contractile swimmers [see\nEqs. (1)-(5)]. With extensile swimmers, this system un-\ndergoes complete phase separation, as in binary-fluid mix-\ntures [22]. By carrying out extensive pseudospectral di-\nrect numerical simulations (DNSs) we have shown that this\nmodel develops an emergent nonequilibrium, but statisti-\ncally steady, state (NESS) of active turbulence, for the case\nof contractile swimmers, if ζis sufficiently large and neg-\native. This turbulence arrests the phase separation into\nregions with positive and negative values of ϕ, as in con-\nventional fluid turbulence leads to the suppression of phase\nseparation in a binary-fluid mixture [26–28].\nWe have quantified this suppression by showing how the\ncoarsening-arrest length L(t)scale does not grow indefi-\nnitely, with time t, but saturates at a finite value at large\ntimes. We have then characterised the statistical properties\nof this active-scalar turbulence by employing the energy\nspectrum E(k)and the fluxes Π(k)andΠϕ(k). We have\nalso obtained the spectrum of ϕ, which is used in studies\nof phase separation. For sufficiently high Reynolds num-\nbers, we have shown that the energy spectrum E(k)dis-\nplays an inertial range, with a power-law dependence on\nthe wavenumber k. We have demonstrated that, in this\nrange, the flux Π(k)assumes a nearly constant, negative\nvalue, which indicates that the system shows an inverse\ncascade of energy that is similar to its counterpart in 2Dhomogeneous and isotropic fluid turbulence, even though\nenergy injection occurs over a wide range of wavenumbers\nin our active-CHNS model.\nOur statistical characaterization of active-CHNS turbu-\nlence shows that it is fundamentally different from con-\nventional 2D fluid turbulence [30, 31], forced 2D CHNS\nturbulence [27], and other types of active-fluid turbulence,\ndiscussed, e.g., in Refs. [1–11, 25]. For large values of |ζ|,\nactive-CHNS turbulence has some similarities with con-\nventional 2D fluid turbulence and 2D forced CHNS tur-\nbulence, inasmuch as it shows an inverse-cascade region\nwith E(k)∼k−5/3. The |ζ|-dependent, small- k, power-law\nregime in E(k)is qualitatively reminiscent of parameter-\ndependent small- kpower-law regimes in energy-spectra in\nsome minimal models for bacterial turbulence [3, 4, 7–\n9, 11, 16]. The scalar spectrum Φ(k)of active-CHNS tur-\nbulence shows a substantial power-law regime which is dif-\nferent from that in conventional forced 2D CHNS turbu-\nlence [27]. The fluxes and energy budgets in active-CHNS\nturbulence are also markedly different from their counter-\nparts in other types of 2D turbulence. 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In recent years, progress in the field of singular stochastic PDEs,\ninitiated by the theory of regularity structures, has allowed for a new construction of the Φ4\n3EQFT\nas the invariant measure of a previously ill-posed Langevin dynamics—a strategy originally proposed\nby Parisi and Wu (’81) under the name stochastic quantisation . We apply the same methodology\nto obtain a large deviation principle (LDP) for the family of periodic Φ4\n3measures at varying\ntemperature. In addition, we show that the rate functional of the LDP and the Φ4\n3action functional\ncoincide up to a constant.\nKey words and phrases.\nEuclidean Quantum Field Theory, Φ4\n3Measure, Large Deviations, Stochastic Quantisation\n2020 Mathematics Subject Classification.\nPrimary: 81S20, Secondary: 60F10, 60H17, 60L30\nCompeting Interests: The authors affirm that they have no competing interests to declare.\nAcknowledgements\nBoth authors thank Ilya Chevyrev for his hospitality and financial support during a research visit\nat the University of Edinburgh in April 2023 and for suggesting a strategy that lead to a significant\nsimplification in the proof of Theorem 2.1. TK thanks Nils Berglund for many interesting discussions\non the topic of this work and related problems. AM additionally thanks the University of Warwick for\nfinancial and housing support during his visit in August 2023 and support from the DFG research unit\nFOR2402. TK gratefully acknowledges financial support via Giuseppe Cannizzaro’s UKRI FL Fellowship\n“Large-scale universal behaviour of Random Interfaces and Stochastic Operators” MR/W008246/1.\nContents\n1 Introduction 2\n1.1 Main Result and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Notations, Conventions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n2 Quasipotential and Φ4\n3Action Coincide 13\n1arXiv:2402.00975v1  [math.PR]  1 Feb 20243 LDP Lower Bound for the Φ4\n3Measure 18\n4 LDP Upper Bound for the Φ4\n3Measure 22\n4.1 Control on the Dynamic Remaining in Eρ,δ,θ. . . . . . . . . . . . . . . . . . . . . . . . . 29\n4.2 Control when the Dynamic Leaves Eρ,δ,θ. . . . . . . . . . . . . . . . . . . . . . . . . . . 31\n4.3 Proof of the LDP Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\nA The Skeleton Equation 34\nA.1 Well-Posedness and A Priori Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34\nA.2 Global Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41\nB The Dynamical Φ4\n3Equation on the Torus 46\nB.1 Local and Global Solution Theory for the Φ4\n3Equation . . . . . . . . . . . . . . . . . . . 46\nB.1.1 Trees, Models, and the Solution Map . . . . . . . . . . . . . . . . . . . . . . . . . 47\nB.1.2 Coming Down from Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51\nB.2 Uniform LDP for the Φ4\n3Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54\nB.3 Tail Bounds for the Φ4\n3Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59\nReferences 60\n1 Introduction\nTheΦ4\n3measure on the 3-dimensional torus T3at temperature ε>0is given by the formalexpression\nµε∝exp\n−1\nε2/integraldisplay\nT3/parenleftigg\n1\n2|∇ϕ(x)|2+1\n4ϕ(x)4+m2\n2ϕ(x)2/parenrightigg\ndx\n/productdisplay\nx∈T3dϕ(x). (1.1)\nwhere “∝” denotes a suitable normalisation so as to give a probability measure. We refer the reader to\nSection B.3 for a more rigorous description of this object. Taking this definition for granted, the measure\ndefined in (1.1)gives a Euclidean version of a quartic, Bosonic, relativistic quantum field theory. In the\nprogramme of constructive quantum field theory, Euclidean quantum field theory (EQFT) measures are\nrelated to relativistic theories through verification of the Osterwalder–Schrader (OS) axioms [OS73,OS75];\nsee Glimm and Jaffe [GJ81, Sec. 6.1] for a pedagogical account and the introduction by Gubinelli and\nHofmanova [GH21] for a more detailed overview. Informally, the integration variable ϕin(1.1)is to be\nunderstood as an element of S′(T3), the space of distributions on the three dimensional torus. As such,\nthe expression (1.1)is purely formal for two reasons: firstly, the product measure over/producttext\nx∈T3dϕ(x)is\nnot well-defined and secondly, even if it were, the quartic non-linear terms are generically ill-defined for\nϕ∈S′(T3).\nRigorous Constructions of the Measure. Despite these challenges, rigorous meaning can be given\nto a renormalised modification of (1.1); this was first achieved in both finite and infinite volumes in the\nseries of works [Gli68,EO71,GJ73,Fel74,FO76]. A major outcome of this programme was the verification\nof the (OS) axioms for the Euclidean Φ4measure at small couplings, i.e. small relative strength of the\nnon-linear term, mediated through a parameter absent in our presentation. However, modern approaches\nhave recently succeeded in constructing the same measure at all coupling strengths, as well as showing\npromise for more systematic approaches to a wider range of physical theories. One of these approaches,\nwhich is also the approach of this paper, is the programme of stochastic quantisation , proposed originally\nby Nelson [Nel66] in the context of quantum mechanics and then by Parisi and Wu [PW81] in the\ncontext of EQFT; see Damgaard and Hüffel [DH87] for a review article on the subject. In the case of the\nEuclidean Φ4\n3measure, the approach of Parisi and Wu leverages the Gibbsian nature of (the renormalised\n2version of) (1.1) to view it as the invariant measure of the (renormalised) Langevin dynamics\n∂t¯uε−∆¯uε=−¯u3\nε−m2¯uε+∞¯uε+εξ, ¯uε(0,·) =z, (1.2)\nwhereξis a space-time white noise, zis a suitably regular initial condition, and the term ∞¯uεsignifies\na diverging counter-term which must be included in the equation to balance the singularity of the\nnon-linear term. We give rigorous meaning to this equation in Definition B.1 below.\nA local solution theory for (1.2)onT3was first obtained by Hairer [Hai14] via the novel theory of\nregularity structures, then by Catellier and Chouk [CC18] via the theory of paracontrolled calculus as\ndeveloped in [GIP15], and subsequently via renormalisation group methods by Kupiainen and Duch,\nsee for example [Kup16,Duc22,Duc23]. Global well-posedness and existence of an invariant measure\nfor the dynamical equation, on the finite volume, was then shown by Mourrat and Weber [MW17b]\nwhile uniqueness of the measure, for all coupling strengths, in the finite volume case,1was obtained in\ncombination by Hairer and Mattingly [HM18b] and Hairer and Schönbauer [HS22a]. In [HM18a], Hairer\nand Matetski further showed that this measure, µε, agrees with the one obtained in the programme of\nCQFT, for example by [Fel74], for coupling strengths where the latter is defined.\nAlternativeconstructionsofthesamemeasurewerealsorecentlygiven byBarashkovandGubinelli viaa\nvariationalapproach[BG20]andGirsanov’stheorem[BG21]aswellasbyGubinelliandHofmanova[GH21]\nvia an approximating family of PDEs on the lattice. In the latter, it was also shown that the measure\nsatisfies all OS axioms in the form of [EO71], apart from rotation invariance and the clustering property\nwhich are only relevant on the infinite volume.\nAs described above, a consequence of the stochastic quantisation approach is that, in certain circum-\nstances, it allows for the transference of properties obtained for dynamical solutions, where tools of\nstochastic analysis and PDE theory may be applied, to the invariant Euclidean QFT measure. One\npertinent example of this is the work by Hairer and Steele [HS22b] where sub-Gaussian tail estimates\nare obtained for (a modified version of) the dynamical Φ4\n3equation. These tail bounds for the dynamics\nare their main technical contribution which, by invariance of the EQFT measure, quite simply carry\nover to the renormalised version of (1.1). The present work is another example of a similar methodo-\nlogy; we transfer the large deviation principle for the periodic Φ4\n3dynamics, obtained by Hairer and\nWeber [HW15], onto the EQFT measure as informally described by (1.1). However, in our case the\ntransference, compared to [HS22b], is more technical and builds on analytic arguments developed by\nSowers [Sow92a] and Cerrai and Röckner [CR05] in the non-singular SPDE setting. We believe that the\nmethod developed in our work may readily be applied to other physically relevant models for which the\ntheory of regularity structures applies, see the paragraph on future directions at the end of the next\nsection.\n1.1 Main Result and Discussion.\nIn order to state our main result, we introduce the Φ4\n3action functional S:S′(T3)→[0,+∞]by setting\nS(ϕ):=\n\n/integraltext\nT3/parenleftig\n1\n2|∇ϕ(x)|2+1\n4|ϕ(x)|4+m2\n2|ϕ(x)|2/parenrightig\ndx, ϕ∈H1(T3),\n+∞, ϕ ∈Cα(T3)\\H1(T3).(1.3)\nThe following theorem is the main result of this paper.2\nTheorem 1.1 (LDP for the Φ4\n3measure) Letα∈(−2/3,−1/2)be fixed. Then, the measures (µε)ε>0,\nformally given by (1.1), satisfy a large deviation principle on Cα(T3)with good rate function 2Sand\nspeedε2. That is,\n1In2D, global well-posedness on the plane was shown by the same authors in [MW17a].\n2Note that the factor “ 2” in the rate function 2Scan simply be removed by rescaling ε⇝√\n2εor by considering an SPDE\nlike (1.2) with formal gradient −1\n2DS(uε)rather than−DS(uε).\n3(i)the map Sintroduced in (1.3)is lower-semicontinuous, has compact sub-level sets, and is not\nidentical to +∞,\n(ii) for any open set ˚A⊂Cα(T3), we have\nlim inf\nε→0ε2ln(µε(˚A))≥− inf\nz∈˚A2S(z), (1.4)\n(iii) for any closed set ¯A⊂Cα(T3), we have\nlim sup\nε→0ε2ln(µε(¯A))≤− inf\nz∈¯A2S(z). (1.5)\n♤\nIntuition for LDPs and Semi-Classical Physics. One may naturally ask what additional informa-\ntion a large deviation principle provides for the family (µε)ε>0. Mathematically speaking, it expresses the\nidea that as random fluctuations are removed, the measure µεshould concentrate on classical minimisers\nof the action functional S. From a physical point of view, up to normalising other dimensionless\nconstants, we may identify ℏ=ε2whereℏis Planck’s constant. Thus, we may think of ε>0in(1.1)\nas either a measure of temperature, or the discrepancy between the classical theory and its quantum\nanalogue. We expect it to quantify the amplitude of expected fluctuations of the quantum system around\nits natural, classical analogue.\nThe large deviation principle captures the size of these fluctuations on a logarithmic scale. Informally,\nit expresses the probability of witnessing rare events Aasε→0, that is\nµε(A)≃logexp/parenleftig\n−1\nε2inf\nz∈A2S(z)/parenrightig\nforε≪1,\nwhere the notation ≃logis a shorthand for logarithmic equivalence in the sense of the bounds (1.4)\nand(1.5). It is one of the main results in the theory of large deviations that an LDP is equivalent to\naLaplace principle , namely the approximation\n/integraldisplay\ne−1\nε2(F+S)(ϕ)dϕ∝Eµε/bracketleftbig\ne−1\nε2F(ϕ)/bracketrightbig\n≃logexp/parenleftig\n−1\nε2inf\nϕ(F+ 2S)(ϕ)/parenrightig\n(1.6)\nfor suitable functionals F:supp(µε)→R. In fact, this is the starting point for the so-called weak\nconvergence approach to large deviations developed by Dupuis and Ellis [DE97] and applied to stochastic\nPDEs by Budhiraja, Dupuis, and Maroulas [BDM08].\nIn the context of Euclidean QFT, the Laplace principle as expressed by (1.6)is an example of semi-\nclassical limits. The study of these limits generalises the idea described above, i.e. they are concerned\nwith the discrepancy, due to thermal fluctuations, between quantum field theoretic descriptions and\ntheir classical counterpart. The LDP is only one example of such a limit. A tighter relationship between\nthe quantum and classical theories is given by the notion of sharp Laplace asymptotics orsemi-classical\nexpansions . For an example of how to obtain sharp Laplace asymptotics for solutions to a family of\nsingular SPDEs we refer to [FK22,Klo22].\nWe also mention that LDPs have been studied in the context of Euclidean gauge theories by Lévy and\nNorris [LN06], obtaining the first rigorous connection between the previously constructed 2D Yang–Mills\nmeasure and the Yang–Mills action functional. In the direction of semi-classical limits, large deviations\nand sharp Laplace asymptotics have been studied in the context of Louville quantum field theories\nby Lacoin, Rhodes, and Vargas [LRV17,LRV22]. In the context of relativistic QFT, the asymptotics\ndescribed by LDP and sharp Laplace asymptotics on the Euclidean side are related to the approximations\nof stationary phase on the relativistic side. For finite-dimensional SDE dynamics, this connection has\n4been explored by Ben Arous [BA88], motivated by earlier works of Azencott and Doss [Dos85,AD85] on\nthe semi-classical limits of Schrödinger’s equation.\nLarge Deviation Principles for SPDEs and their Invariant Measures. As described earlier\nin the introduction, the dynamical Φ4\n3equation is not naively well-posed, due to the irregularity of\nspace-time white noise. However, if one considers a sufficiently regular approximation to the noise, for\nexample, stetting ξκ:=ξ∗ρκwithρκa standard mollifier at scale κ>0, then the equation\n/braceleftigg\n∂tuε,κ(t,x)−∆uε,κ(t,x) =−uε,κ(t,x)3−m2uε,κ(t,x) +εξκ(t,x),\nuε,κ(0,·) =z,(t,x)∈[0,T]×Td,(1.7)\nhas a classically well-defined, unique solution for all d≥1. Cerrai and Röckner analysed such equations\nin a series of works [Cer03,CR04] and, in fact, obtained an LDP for a large class of reaction-diffusion\nsystems, withnon-Lipschitz reaction term andmultiplicative, regularnoise, even in the case ofunbounded,\ndegenerate diffusion coefficient. In the subsequent work [CR05] the same authors demonstrated that\nunder suitable structural assumptions on the coefficients, the LDP can be transferred to the unique\ninvariant measure of the system; this generalised earlier work by Sowers [Sow92b,Sow92a] which treated\nthe case of a single equation with more restrictive conditions on both the reaction and the diffusion\nterms. For a succinct, intuitive description of the relation between LDPs for stochastic PDEs and those\nfor their invariant measures, we direct the reader to the introduction of [Fre88].\nWhend= 1, the equation (1.7)is well-posed even in the limit κ→0and an LDP (also in the\ncase of a double-well potential, i.e. making the replacement −m2∝⇕⊣√∫⊔≀→+m2in(1.7)) was obtained by\nFaris and Jona-Lasinio [FJL82]. In contrast, equation (1.7)with fixedκ= 0is naively ill-posed for\nanyd≥2. In fact, when d= 2andε>0is fixed, Hairer, Ryser, and Weber [HRW12] showed that uε,κ\nconverges to 0in a space of distributions as κ→0. In contrast, it had already been proved by Cerrai\nand Freidlin [CF11] on the level of large deviations that the rate function I(κ)\n0,Twhich governs the LDP\nof the random variables (uε,κ)κ>0Γ-converges in any dimension d≥1to a limiting rate function3\nI0,T(v) =1\n2/integraldisplayT\n0/integraldisplay\nTd(∂tv−∆v+v3+m2v)2dxdt (1.8)\nasκ→0. Ford= 1, this functional does indeed coincide with the rate function obtained by Faris and\nJona-Lasinio [FJL82], but when d≥2, the problem of finding a suitable stochastic dynamics ¯uε,κ, whose\nlimit ¯uε(asκ→0) is both non-trivial and whose large deviations are governed by I0,T, had eluded\nprogress at the time. In dimension d∈{2,3}, this problem was overcome by Hairer and Weber [HW15]\nusing the novel theory of regularity structures. Indeed, they showed that I0,Tas given by (1.8)is the\nrate function for an LDP governing solutions (¯uε,κ)ε>0to a suitably renormalised version of (1.7),\n/braceleftigg\n∂t¯uε,κ(t,x)−∆¯uε,κ(t,x) =−¯uε,κ(t,x)3−/parenleftbig\nm2−3ε2C(1)\nκ+ 9ε4C(2)\nκ/parenrightbig\n¯uε,κ(t,x) +εξκ(t,x),\n¯uε,κ(0,·) =z,(t,x)∈[0,T]×Td,(1.9)\nwhere the smoothing parameter κis chosen as a non-negative function of the amplitude εso thatκ(ε)→0\nasε→0. Importantly, their result includes the case κ= 0, for which the process ¯uεonly formally solves\nthe stochastic PDE given by (1.2).4In the case d= 2, this extends earlier partial results by Jona-Lasinio\nand Mitter [JLM90]. For context, we point out that the article [HW15] also considers the situation when,\nadditionally to κ(ε)→0, one hasεκ(ε)−1→λ2∈[0,∞)asε→0; in that case, the unrenormalised\ndynamics (uε,κ(ε))ε>0given in (1.7)satisfy an LDP with good rate function similar to (1.8), but with\n3This is with the appropriate understanding that I0,T(v)is infinite whenever the integral on the right hand side is not\nfinite, see (B.16) in Appendix B.2 for details.\n4This rigorous limiting procedure is more fully described in Appendix B.\n5some explicit effective mass mλ. Cerrai and Debussche [CD19] have recently generalised this result to\nthe case of arbitrary polynomial non-linearities in any dimension d≥1.\nOne way to view the result of [HW15] is to show that the Γ-convergence result of [CF11] correctly\npredicted that I0,Tshould be the rate function of a limiting dynamics ¯uεthat is independent of κ. The\ncurrent article demonstrates that a similar result holds at the level of the invariant measure. Starting\nfrom the dynamic rate functions I(κ)\n[0,T](resp.I[0,T]) the natural candidate rate function for an LDP\ngoverning the invariant measure of uε,κsolving(1.7)(resp. ¯uε:=limκ→0¯uε,κsolving(1.9)) are the\nquasi-potentials5\nV(κ)(z):= inf/braceleftbig\nI(κ)\n0,T(v):T >0, v:[0,T]×T3→R, v(0) = 0, v(T) =z/bracerightbig\n(1.10)\nand\nV(z):= inf/braceleftbig\nI0,T(v):T >0, v:[0,T]×T3→R, v(0) = 0, v(T) =z/bracerightbig\n. (1.11)\nGiven [CF11, Thm. 6.1], in which the authors show that V(κ)converges pointwise to VinH1(T3),\nit is natural to conjecture that Vgoverns the large deviations of the family of measures µεwhich are\ninvariant for the renormalised dynamics ¯uε(i.e.(1.9)withκ= 0). A consequence of our work, similar\nin this regard to [HW15], is to show that is indeed the case. Furthermore, we identify Vas being\nproportional to the Euclidean Φ4\n3actionSdefined in (1.3)above; this is natural to expect since it\nhas already been shown that the invariant measure µεfor the dynamics ¯uεis exactly the Euclidean Φ4\n3\nmeasure. We give more details of our proof strategy and this identification, in particular, in the next\nparagraph.\nLet us further mention that the same LDP for the periodic Φ4\n3measure6has been obtained in [Bar22b,\nThm. 2.40] in which it is phrased equivalently as a Laplace principle, see (1.6)above; the proof then uses\nweak convergence arguments in the spirit of Dupuis and Ellis [DE97] which are based on the Boué–Dupuis\nformula as in [BG20]. We maintain two motivations for re-deriving the result via stochastic quantisation:\nFirstly, our method builds on the systematic approach of regularity structures [Hai14,BHZ19,CH16,\nBCCH20] that is applicable to a wide class of singular SPDEs and requires only a small number of\nadditional, model-specific ingredients. In principle, this would make it possible to obtain analogous\nresults for a large class of models using a similar approach. Secondly, a major goal of constructive\nQFT is to give rigorous meaning to gauge theories such as Yang–Mills(–Higgs) theories which form an\nintegral part in the standard model of particle physics. At present, a variational formulation similar\nto [BG20] is not available for gauge theories of this type. In contrast, however, the approach via\nstochastic quantisation has shown success in applications to gauge theories, see Shen [She21], Chandra et\nal. [CCHS22a,CCHS22b] (and Chevyrev’s article [Che22] reviewing the two works) as well as the recent\nwork of Chevyrev and Shen [CS23]. This, and other possible future directions are further discussed in\nthe final paragraph of the introduction.\nStrategy of Proof. We give a short description of our proof strategy, highlighting in particular where\nwe appeal to model specific properties and where we make use of more general tools. For the most part,\nour strategy is inspired by the above-mentioned works of Cerrai and Röckner [Cer03,CR04,CR05] in\nwhich the authors deal with structurally more general but non-singular equations. One of the main\nchallenges in their setting, and in ours, is to find a good representation of the quasi-potential Vthat is\neasier to control than the expression in (1.11). In a first step, we observe that the definition of I0,T\n5Note that we do not impose any regularity constraints on the functions vin(1.10)and(2.1): Simply requiring V(κ)(v)\nresp.V(v)to be finite already ensures the requisite regularity.\n6In two dimensions, the analogous result on the infinite volume is proved in [BG23].\n6immediately implies that one can represent the quasi-potential Vas\nV(z):= inf/braceleftbigg1\n2∥h∥2\nL2\nTL2x:T >0, wh(T) =z/bracerightbigg\n(1.12)\nwhere zis in the Hölder–Besov space Cα(T3)forα<−1/2, the support of the invariant measure µε,\nandwhsolves the skeleton equation ,\n∂twh−∆wh=−(wh)3−m2wh+h, wh|t=0= 0, (1.13)\non[0,T]where the driver, h∈L2\n[0,T]L2(T3), is an element of the Cameron–Martin space associated to the\nspace-time white noise. Crucially for our purposes, the rate function ITin(1.8)is independent of the\nrequisite infinite renormalisation; in turn, this makes (1.13) the correct skeleton equation, even for the\nrenormalised dynamics (1.2). In [CR05], the authors further simplify the expression for Vby appealing\nto so-called ancient solutions to (1.13); broadly speaking, they fix the common time horizon (−∞,0]for\nthe infimisation problem in (1.12)by demanding that whhitszat time 0and decays to zero as t→−∞.\nIn our case, we eschew this argument and instead make use of the additive noise and gradient flow\nstructure of (1.2) to more directly show that the quasi-potential andaction functional coincide, i.e.\nV(z) = 2S(z)for all z∈Cα(T3). (1.14)\nWe expect that a similar strategy will apply for other physically relevant models, see the paragraph\non future directions below and Section 2 for details. We point out that (1.14)is the direct analogue\nof the finite-dimensional result by Freidlin and Wentzell [FW12, Chap. 4.3, Thm. 3.1] but, to the best\nof our knowledge, our method of proof seems new, even in that case. Since the action functional S\nis only finite when restricted to the Sobolev space H1(T3), a direct consequence of (1.14)is that the\nquasi-potential Vhas compact sub-level sets in Cα(T3). That is, V= 2Sis agoodrate functional, a\nfact that is required in the proof of the LDP upper bound, see Section 4.\nWe then combine the techniques of Cerrai and Röckner [CR05] with the LDP for the Φ4\n3equation as\nobtained by Hairer and Weber [HW15], see Section B.2 for a recap. In fact, we need a mild generalisation\nof their result which is locally uniform in the initial condition, which we prove in Proposition B.16 in\nAppendix B.2. We view it as an advantage of this approach that the techniques in [HW15] are applicable\nto a large class of singular stochastic dynamics with minimal change, essentially all equations which\nare amenable to the theory of regularity structures. We have aimed to continue this philosophy in our\narticle, appealing to as few model specific facts as possible. Essentially, our analysis relies only on two\nconditions, both of which are reasonable to expect from other physical models:\n(1)Sufficient tail estimates on the invariant measure . In our case, we appeal to the work of Moinat\nand Weber [MW20b] which gives stretched exponential moments (see Propositions B.14 and B.21\nbelow for a recap). We stress that their work onlyrelies on analysing the dynamics and does not\nappeal to any a priori information on the invariant measure. While these stretched exponential\nmoments were upgraded to sub-Gaussian tail bounds in [HS22b], also appealing to analysis of the\ndynamics, we point out that these stronger tail estimates are not required for our method. In fact,\nthese sub-Gaussian tail bounds are also stronger than those required to verify the OS axioms. In\nour proof, the stretched exponential tail estimates are required both in the proof of the LDP lower\nbound, see Section 3, and that of the LDP upper bound, see Section 4.3.\n(2)Asymptotic stability of the deterministic dynamics . In our case, we are able to show decay of the\nclassical, deterministic dynamics (i.e. (1.7)withε= 0) to a unique, asymptotically stable solution\nas well as long time decay in H1of solutions to the associated skeleton equation, (1.13). The\nformer is guaranteed by the complimentary signs of the non-linear and linear terms in (1.2)(see\nLemma 2.2) while we only rely on the positive mass parameter to obtain long time decay of the\n7skeleton equation, see Remark A.3. With regard to generalisations of the former we expect that if\nminimisers of the classical dynamics form a single connected component in the state space then\nour approach should be extendable without major change. In the case of non-unique, disconnected\nminimisers some additional arguments may be required. The second property, of skeleton solution\ndecay inH1, should be similarly adaptable to other relevant cases.\nWe close this paragraph by highlighting the main ways in which the singular nature of (1.2)requires us\nto improve upon the methods of [CR04,CR05]. The main point of departure from those works stems\nfrom the low regularity of the invariant measure support, which is the natural state space for our LDP,\nCα(T3). A key, related issue is that the LDP rate functional and Φ4\n3action functional are both infinite\non this state space, and only finite on the solution space of the skeleton equation, i.e. paths taking\nvalues inH1(T3). This causes technical challenges, for example, in the proof of the LDP upper bound,\nTheorem 4.1 and requires more careful analysis of the skeleton equation in scale of weighted, path space\nnorms, allowing for prescribed blow-up as t→0. See the proof of Lemma 4.2 and attendant discussions.\nOn the other hand, it is to our benefit that despite the singular nature of the stochastic equation, the\nskeleton equation remains classically well-posed. Appealing to suitable Sobolev embeddings one sees\nthat the cubic non-linearity is well defined for weak solutions to (1.13)inCTH1(T3). A final, major,\ndifference between ours and the previous work [CR05] is that, taking advantage of the gradient flow\nstructure at hand, we are able to identify the limiting rate function for the invariant measures as the Φ4\n3\naction, Section 2. As discussed above, this identification greatly simplifies later proofs, in particular,\nsince it immediately implies compactness of sublevel sets for the quasipotential. We point out here\nthat our proof of this identification requires global, exact controllability of the skeleton equation, see\nAppendix A.2 and the proof of Theorem 2.1.\nFuture Directions. As described above, we expect the strategy developed in the current paper to be\nreadily applicable to a number of other physically relevant theories beyond Φ4\n3and also open the way to\naddress more challenging questions.\n•A natural extension of our work would be to consider the Φ4model in the full subcritical regime ,\nthat is, the family of dynamic Φ4\n4−ιmodels for any ι >0. These models were considered by\nChandra, Moinat, and Weber [CMW23] who obtained tail bounds for an appropriate generalisation\nof (1.2).\n•A related, but potentially more challenging extension would be to the Euclidean Φ4\n3measure\nconstructed on a compact Riemannian manifold, see the recent series of works by Bailleul et\nal. [BDFT23a,BDFT23b,Bai23]. In the same direction we also refer to Hairer and Singh [HS23]\nfor a treatment of more general singular SPDEs on compact, Riemannian manifolds.\n•More challenging further extensions one might consider are to the sine-Gordon and exponential\nmodels, [AVG21,GHR23,HS16,CHS18] as well as the recently studied Langevin dynamics for\ntensor field theories , see Chandra and Ferdinand [CF23]. With regards the sine-Gordon model,\nwe mention that large deviation principles as well as other important properties have been\nderived for these models on the infinite volume via an FBSDE, stochastic control approach,\nfirst by Barashkov [Bar22a] in the parameter regime β2∈(0,4π)and recently by Gubinelli and\nMeyer [GM24] in the regime β2∈(0,6π). A major hurdle for our approach, in this regard, is the\nlack, at time of writing, of a global well-posedness and ergodic theory for the Langevin dynamics\nassociated to the sine-Gordon theory. However, we point out that the local well-posedness theory\nof [CHS18], given in the language of regularity structures, holds in the full sub-critical regime\nβ2∈(0,8π).\n•As originally suggested by Parisi and Wu [PW81], the method of stochastic quantisation has\nrecently lead to progress in the rigorous construction and analysis of gauge theories , see the\n8above-mentioned series of works [She21,CCHS22a,CCHS22b,Che22,CS23] and also Chatterjee’s\nand Cao’s independent approach [CC23,CC24] that lead to similar results. Combined with the\nresults of [CCHS22a,CS23], the strategy we developed in this article provides a novel route to\nprove an LDP for the 2D Yang–Mills measure on T2with respect to a stronger topology than in\nthe earlier work [LN06]. This question is the subject of ongoing, joint work by the two present\nauthors and Chevyrev.\n•The programme of stochastic quantisation has also shown success in the construction of Euclidean,\nFermionic theories, see recent work by Chandra, Hairer and Peev [CHP23] giving a stochastic\nquantisation approach to the construction of a 1-dimensional, mixed fermionic-bosonic quantum\nfield theory. While we do not expect any simple modification of the present arguments to be directly\napplicable to fermionic theories, we simply wish to highlight the general success of stochastic\nquantisation in constructing physical quantum field theories, as well as other, rapidly developing\nmethodologies, [VFG22,ABDVG22].\nOrganisation of this Article. In the remainder of this section we describe some frequently used\nnotation and useful preliminaries, Section 1.2. The following three sections of the paper are dedicated\nto the proof of Theorem 1.1. In Section 2 we show that the quasi-potential and Φ4\n3action coincide.\nSection 3 gives a self-contained proof of the LDP lower bound while Section 4 does the same for the\nLDP upper bound. This concludes the proof of our main result. These sections are supplemented by\ntwo appendices. In Appendix A we provide necessary analysis of the skeleton equation; the main points\nbeing global well-posedness, long time decay and global, exact controllability. Appendix B provides a\nbrief recap of the necessary aspects of regularity structures for the dynamical Φ4\n3equation as well as\nproofs of necessary ancillary results; a locally uniform version of Hairer–Weber’s dynamic LDP, [HW15],\nand suitable asymptotic tail bounds on µεin terms of ε∈(0,1).\n1.2 Notations, Conventions and Preliminaries\nIn this subsection, we collect notational conventions and preliminaries.\nSets and Sequences We write Zfor the set of integers, N:={1,...}for the strictly positive integers,\nN0:={0}∪Nfor the non-negative integers, Rfor the real numbers and R+:= [0,∞)for the non-negative,\nreal numbers. We set T:=R/Zand define the three dimensional torus T3:= (R/Z)3. Given a Banach\nspaceEand an index set I⊆Nwe use the (slight) abuse of notation (an)n∈I⊂Eto denote a sequence\nofEvalued elements. When the context is clear we simply write (an)n∈I. Given a real number a∈R,\nwe occasionally use the shorthands\na−:={b∈R:b<a}anda+:={b∈R:b>a}.\nInequalities, Embeddings and Operators When stating an inequality we either write A≲a,b,...B\nto indicate that the inequality holds up to a constant depending on the parameters a, b, ...or write that\nthere exists a C:=C(a,b,... )>0such thatA≤CB. If we write A≲Bwithout explicit dependents\nwe mean that the inequality holds up to a constant depending on parameters that we do not keep track\nof. These constants may always change in absolute size from line to line. If we write A≃Bwe mean\nthat there exists a constant c≥1such that\n1\ncB≤A≤cB.\nGiven two topological spaces, X, Y, we writeX ◁arrowhookleft→Yto denote that Xembeds continuously into Y\nandX ◁arrowhookleft→ →Yto denote that Xembeds compactly into Y.\n9Given a metric space (X,dX),x∈Xandλ∈Rwe write,\nBX\nλ(x):={y∈X:dX(x,y)<λ}and ¯BX\nλ(x):={y∈X:dX(x,y)≤λ}.\nWhen the context is clear we will simplify our notation to Bλ(x),¯Bλ(x)and whenx= 0we simplify\neven further, writing, Bλ,¯Bλ.\nGiven two Banach spaces X, Y, we writeL(X,Y )for the set of bounded, linear operators from Xto\nY. As is standard we write X∗:=L(X,R).\nLebesgue Spaces on T3Forp∈[1,∞)(respectively p=∞) we write Lp(T3)for the space of\np-integrable (respectively essentially bounded) functions f:T3→R, equipped with the norms,\n∥f∥Lp\nx:=\n\n/parenleftig/integraltext\nT3|f(x)|pdx/parenrightig1/p\n, p∈[1,∞),\ness supx∈T3|f(x)|, p =∞.\nCorrespondingly, for p∈[1,∞)(respectively p=∞) we write ℓp(Z3)for the space of p-summable\n(respectively bounded), real valued sequences, indexed by Z3.\nDifferentiable Functions and Distributions Givenk∈N0we writeCk(T3)for the functions\nf:T3→Rwhich arek-times continuously differentiable, we define C∞(T3):=∩k∈N0Ck(T3)and write\nS′(T3)for the set of distributions on T3, identified as the dual of C∞(T3). For 0≤s < t <∞, the\ndistributionsS′([s,t]×T3)are defined analogously.\nFourier Transform on T3Forf∈L1(T3)andg∈ℓ1(Z3)we define the Fourier transform\nF(f)(k):=/integraldisplay\nT3f(x)e−2πik·xdxand inverseF−1g(x):=/summationdisplay\nk∈Z3g(k)e2πik·x.\nFor concision, where appropriate, we write ˆf(k):=F(f)(k)and recall the well-known isometry,\n∥f∥L2x=∥ˆf∥ℓ2\nk:=\n/summationdisplay\nk∈Z3|ˆf(k)|2\n1/2\n,\nThe Fourier transform naturally extends to S′(T3)by density and duality.\nSobolev Spaces on T3Givenα∈R, we letHα(T3)denote the functions f:T3→R(respectively\ndistributions f∈S′(T3)whenα<0) which are finite under the norm,\n∥f∥Hαx:=/vextenddouble/vextenddouble/vextenddouble(1 +|·|2)α/2ˆf(·)/vextenddouble/vextenddouble/vextenddouble\nℓ2\nk=\n/summationdisplay\nk∈Z3(1 +|k|2)αˆf(k)\n1\n2\n.\nFor allα∈R, the space Hα(T3)is Hilbert, with the natural inner product and one has the duality\n/parenleftig\nHα(T3)/parenrightig∗\n=H−α(T3).\n10Whenα= 0, one hasHα(T3) =L2(T3)and whenα∈N, one has the equivalence,\n∥f∥2\nHαx≃α/summationdisplay\nk=0∥∂k\nxf∥2\nL2x.\nFinally, for α′≥αone directly has\n∥f∥2\nHαx≤∥f∥2\nHα′\nx. (1.15)\nAs is standard we use angle brackets ⟨·,·⟩to denote both the inner product associated to a given\nHilbert space as well as the canonical pairing between a Hilbert space and is dual. Where necessary, to\navoid confusion we will decorate the brackets with suitable subscripts; for example\n⟨f,g⟩H1\nx;H−1\nxor⟨f,g⟩L2x;L2x.\nBesov Spaces on T3Forα∈Randp, q∈[1,∞]we define the Besov space, Bα\np,q(T3)as the\ncompletion of the smooth functions, C∞(T3), under the topology induced by the norm\n∥f∥Bαp,q:=\n\n/parenleftig/summationtext\nj≥−12jqα∥∆jf∥q\nLp/parenrightig1\nq, q∈[1,∞),\nsupj≥−12jqα∥∆jf∥Lp, q =∞.\nwhere ∆jdenotes the jthLittlewood–Paley projector, we refer to [BCD11, Ch. 2] for details. Note that\nin contrast to the Sobolev spaces, Hα(T3), we define the Besov spaces as completions under the norm\nrather than finite under the norm . This ensures that the Besov spaces are Polish. When p=q=∞we\nuse the shorthand Cα(T3):=Bα\n∞,∞(T3).\nFor allp∈[1,∞]we have the compact embedding,\nBα′\np,q′(T3)◁arrowhookleft→ →Bα\np,q(T3)⇒ ∥ f∥Bαp,q≤∥f∥Bα′\np,q′, α′>α& &q′≥q, (1.16)\nRecall that as a result, given an f∈S′(T3), it holds that\n∥f∥Bαp,q<∞ ⇒f∈Bα′\np,q(T3)for allα′<α. (1.17)\nConversely, f∈Bα\np,q(T3)implies∥f∥Bαp,q<∞.\nWe also have the following continuous embeddings, for all α, α′∈Rp, p′∈[1,∞]andq, q′∈[1,∞],\nBα\np,q(T3)◁arrowhookleft→Bα\np,q′(T3) ⇐⇒ ∥f∥Bα\np,q′≲∥f∥Bαp,q, q′>q, (1.18)\nBα\np,q(T3)◁arrowhookleft→Bα−3(1/p−1/p′)\np′,q (T3)⇐⇒ ∥f∥\nBα−3(1/p−1/p′)\np′,q≲∥f∥Bαp,q, p′>p, (1.19)\nB0\np,1(T3)◁arrowhookleft→Lp(T3)◁arrowhookleft→B0−\np,∞(T3)⇐⇒ ∥f∥B0−\np,∞≲∥f∥Lp\nx≲∥f∥B0p,∞, p∈[1,∞].(1.20)\nIt is readily checked through Plancherel’s theorem that\n∥f∥Bα\n2,2=∥f∥Hαxfor allα∈R.\nHence, considering (1.17), (1.19) and (1.16), we have the embeddings\nBα\n2,2(T3)◁arrowhookleft→Hα(T3)◁arrowhookleft→Cα′(T3)for allα′<α−3\n2\n11and\nHα(T3)◁arrowhookleft→Bα′\n2,2(T3)◁arrowhookleft→Cα′−3\n2(T3)for allα′<α.\nOur most common application of these embeddings will be to combine them with (1.20)and(1.16)to\nsee that for α∈(−2/3,−1/2), there exists a q∈(3/2,2)such that\nH1(T3)◁arrowhookleft→ →L3q(T3)◁arrowhookleft→Cα(T3). (1.21)\nFurthermore, one has the classical, continuous Sobolev embedding H1(T3)◁arrowhookleft→L6(T3)which has the\nimportant consequence,\nf∈H1(T3)⇒f3∈L2(T3).\nWe refer to [BCD11,GIP15,Tri92] for details and proofs of the above.\nOrnstein–Uhlenbeck Semi-Group Form > 0, we define the action of the Ornstein–Uhlenbeck\nsemi-group on L1(T3)by setting,\net(∆−m2)f:=F−1/parenleftig\ne−2πit(|·|2+m2)ˆf(·)/parenrightig\n,for allt≥0.\nThe following two inequalities hold for distributions f∈S′(T3)andm≥0,\n∥et(∆−m2)f∥Bβ\np,q≲e−tm2(1∨t)−β−α\n2−3\n2/parenleftbig\n1\np′−1\np/parenrightbig\n∥f∥Bα\np′,q′, β≥α, p′≥p, q≥q′,(1.22)\n∥(1−et(∆−m2))f∥Bαp,q≲/parenleftig\n(1∨t)β−α\n2+m2(1∨t)/parenrightig\n∥f∥Bβ\np,q, β≥α. (1.23)\nBanach Space Valued Paths Given a Banach space, Eand a finite interval [s,t]⊂Rfor0≤s<\nt<∞, we define the space of continuous, Evalued paths, C[s,t]E, equipped with the supremum norm,\n∥f∥C[s,t]E:= sup\nr∈[s,t]∥f(r)∥E.\nWhen,s= 0, we simplify notation by writing CtE. Given a weight η>0, and horizon T >0, we define\nthe family of weighted, path spaces by setting,\n∥f∥Cη;TE:= sup\nt∈(0,T](t∧1)η∥f(s)∥E, Cη;tE:={f∈CTE:∥f∥Cη;TE<∞}.\nNote, that given η1<η2,\nCη1;TE ◁arrowhookleft→Cη2;TE⇐⇒ ∥f∥Cη2;TE≲T∥f∥Cη1;T;E. (1.24)\nBochner–Sobolev Spaces We work with a pair of Bochner–Sobolev spaces. For 0≤s<t<∞and\nα∈Rwe set,\n∥f∥2\nL2\n[s,t]Hα(T3):=/integraldisplayt\ns∥f(r)∥2\nHαxdr, L2\n[s,t]Hα(T3):=/braceleftig\nf:[s,t]→T3:∥f∥H1\n[s,t]Hαx<∞/bracerightig\nand\n∥f∥2\nH1\n[s,t]H−1(T3):=∥f∥2\nL2\n[s,t]H−1\nx+∥∂tf∥L2\n[s,t]H−1\nx, H1\n[s,t]H1(T3):=/braceleftig\nf:[s,t]→T3:∥f∥L2\n[s,t]H1x<∞/bracerightig\n,\nwhere∂tfis understood in the weak sense.\nAs usual, when s= 0we simplify notation by writing L2\nTHα(T3)andH1\nTH1(T3).\n12Note that we have the trivial embedding CTH1(T3)◁arrowhookleft→L2\nTH1(T3)as well as the following interpolation\nresult\nf∈L2\nTH1(T3)∩H1\nTH−1(T3)⇒f∈CTL2(T3)\nand generalisation of the chain rule,\nd\ndt∥f(t)∥2\nL2(T3)= 2⟨f(t),∂tf(t)⟩L2x,for a.e.t∈[0,T]. (1.25)\nSee [Eva10, Sec. 5.9.2, Thm. 3].\nRate Functions Three rate functions appear (with some mild variations) in the main body of the\npaper,\n•I[s,t]:C[s,t]Cα(T3)→[0,∞], for 0≤s<t<∞, is the rate function on path space associated to\ntheΦ4\n3equation; see Appendix B.2 for a detailed definition.\n•V:Cα(T3)→[0,∞]is the associated quasipotential ; see (2.1) for a detailed definition.\n•S:Cα(T3)→[0,∞]is the (natural extension of) the Φ4\n3action to a rate function; see (1.3)for a\ndetailed definition.\nGiven a Polish space E, a rate function I:E→[0,∞]andθ∈R, we define the sub-level set,\nI[θ]:={f∈E:I(f)≤θ}.\nFor example, we will write I[s,t][θ],V[θ]andS[θ]for the respective sub-level sets. In the case of the\ndynamic rate function, when s= 0we simplify notation by writing, It(f)andIt[θ].\nGeneralised Grönwall Inequality We make use of the following generalisation of Grönwall’s\ninequality, which can be found with proof as [TW20, Lem. B.1]. Let f:[0,T]→Rbe a measurable\nfunction,m2>0,a, b∈Randσ1+σ2<1be such that\nf(t)≤e−m2ta+b/integraldisplayt\n0e−m2(t−s)(t−s)−σ1s−σ2f(s) ds.\nThen, there exist constants c, C > 0such that\nf(t)≤Cexp/parenleftig\n−m2t+cb1\n1−σ1−σ2t/parenrightig\na. (1.26)\n2 Quasipotential and Φ4\n3Action Coincide\nLet us introduce re-introduce the quasi-potential, V:Cα(T3)→[0,∞]associated to the Φ4\n3dynamics,\nV(z):= inf/braceleftig\nIT(v):T >0, v∈CTCα(T3), v(0) = 0, v(T) =z/bracerightig\n, (2.1)\nRecalling the definition of the Φ4\n3action, S, given by (1.3), the main result of this section is to show\nthat the functionals VandSagree up to a constant and have compact sublevel sets in Cα(T3).\nTheorem 2.1 For any z∈Cα(T3)one has\nV(z) = 2S(z). (2.2)\n♤\n13While this result is a central to our strategy, its proof is somewhat technical and therefore, we recommend\nthat on a first reading, provided the result is appreciated, the proof can be safely skipped.\nBefore proving Theorem 2.1, we state and prove a technical lemma regarding properties of the Φ4\n3\naction.\nLemma 2.2 ForS:S′(T3)→[0,+∞]theΦ4\n3action as defined by (1.3)the following all hold,\ni)S(ϕ) = 0if and only if H1(T3)∋ϕ= 0andS(ϕ)>0for allϕ̸= 0∈H1(T3).\nii)The restriction of StoH1(T3)is Fréchet differentiable and at each ϕ∈H1\nx, for anyφ∈H1(T3)\none has\nDS(ϕ)(φ) =⟨DS(ϕ),φ⟩H−1\nx,H1x=−⟨∆ϕ,φ⟩H−1\nx;H1x+m2⟨ϕ,φ⟩L2x;L2x+⟨ϕ3,φ⟩L2x;L2x.(2.3)\nAs a result, we may use the shorthand,\nDS(ϕ) =−∆ϕ+m2ϕ+ϕ3.\niii)In the operator theoretic sense, DS(ϕ)≡0if and only if ϕ= 0inH1(T3). I.e.ϕ= 0is the\nglobal minimizer of S. ♤\nProof.To prove i) it suffices to recall the embeddings H1(T3)◁arrowhookleft→L6(T3)◁arrowhookleft→L4(T3)◁arrowhookleft→L2(T3)so that\nfor allϕ∈H1\nxone has\nS(ϕ) =1\n2∥∇ϕ∥2\nL2x+m2\n2∥ϕ∥2\nL2x+1\n4∥ϕ∥4\nL4x<∞.\nThe claim then follows directly from the positive definiteness property of norms.\nTo show ii), let ϕ,ψ∈H1(T3), defineAϕ(ψ):=A0\nϕ(ψ) +A1\nϕ(ψ)by\nA0\nϕ(ψ):=⟨−∆ϕ+m2ϕ,ψ⟩H−1\nx×H1x, A1\nϕ(ψ):=⟨ϕ3,ψ⟩L2x\nand then we claim that\n|S(ϕ+ψ)−S(ϕ)−Aϕ(ψ)|=o/parenleftbig\n∥ψ∥H1x/parenrightbig\nas∥ψ∥H1x→0. (2.4)\nTo this end, note that\nA0\nϕ(ψ) =⟨∇ϕ,∇ψ⟩L2x+m2⟨ϕ,ψ⟩L2x= 2B(ϕ,ψ)\nwhere the bilinearformB:H1\nx×H1\nx→Rcorresponds to the quadratic partS0inS, i.e.\nB(ϕ,ψ) =1\n2⟨∇ϕ,∇ψ⟩L2x+m2\n2⟨ϕ,ψ⟩L2x, S 0(ψ) =B(ψ,ψ).\nTherefore, we have\n|S0(ϕ+ψ)−S0(ϕ)−A0\nϕ(ψ)|=|B(ϕ+ψ,ϕ+ψ)−B(ϕ,ϕ)−2B(ϕ,ψ)|\n=B(ψ,ψ) =S0(ψ)≃∥ψ∥2\nH1x(2.5)\nFor thequarticpart,S1ofS, we find that\nS1(ϕ+ψ)−S1(ϕ)−A1\nϕ(ψ) =1\n4/integraldisplay\nT36ϕ(x)2ψ(x)2+ 4ϕ(x)ψ(x)3+ψ(x)4dx\n≃⟨ϕ2,ψ2⟩L2\nx+⟨ϕ,ψ3⟩L2\nx+∥ψ∥4\nL4x\n14and thus\n|S1(ϕ+ψ)−S1(ϕ)−A1\nϕ(ψ)|≤∥ϕ∥2\nL4x∥ψ∥2\nL4x+∥ϕ∥L2x∥ψ∥3\nL6x+∥ψ∥4\nL4x=o/parenleftbig\n∥ψ∥H1x/parenrightbig\n,(2.6)\nas∥ψ∥H1x→0. Note that we have used the Cauchy–Schwarz inequality and the same embeddings as\nabove. The claim in (2.4) now follows from (2.5) and (2.6).\nFinally, to obtain iii), we begin with an auxiliary observation. Let b:R→Rbe given by b(x) =\nx3+m2xand observe that\nd\ndxb(x) = 3x2+m2≥0, x∈R.\nThus,bis (strictly) increasing and we see that\n(b(x)−b(y)) (x−y)≥0, x,y∈R. (2.7)\nFrom ii) it follows that the zeros of DSinH1(T3)are characterised by weak solutions of the elliptic\nPDE problem,\n−∆ϕ=−ϕ3−m2ϕ. (2.8)\nIt is obvious that ϕ≡0is a solution to (2.8), so we only have to prove uniqueness. For two, distinct,\nsolutionsϕ,ψ∈H1(T3)of (2.8) we have\n∥∇(ϕ−ψ)∥2\nL2x=⟨∇(ϕ−ψ),∇(ϕ−ψ)⟩L2x\n=−⟨∆(ϕ−ψ),ϕ−ψ⟩H−1\nx×H1x\n=−⟨ϕ3−ψ3+m2(ϕ−ψ),ϕ−ψ⟩L2x\n=−/integraldisplay\nT3/parenleftbig\nb(ϕ(x))−b(ψ(x))/parenrightbig/parenleftbig\nϕ(x)−ψ(x)/parenrightbig\ndx\n≤0,\nwhere we have used integration by parts on T3in the second line, eq. (2.8)in the third line and eq. (2.7)\nin the last line. It thus follows that ∥∇(ϕ−ψ)∥2\nL2x= 0. By the Poincaré inequality, we finally have\n0≤∥ϕ−ψ∥2\nL2x≲∥∇(ϕ−ψ)∥2\nL2x= 0\nand therefore ϕ=ψinH1(T3). □\nWe are now ready to prove Theorem 2.1, which claims that the quasipotential Vassociated to the Φ4\n3\ndynamics and the Φ4\n3actionScoincide up to a constant.\nProof (of Theorem 2.1). By definition S(ϕ) = +∞ifϕ∈Cα(T3)\\H1(T3), while it follows from\nTheorem A.6 that the same holds for V. Hence, if ϕ∈Cα(T3)\\H1(T3),\nV(ϕ) = +∞=S(ϕ)\nand there is nothing to show. Therefore, we need only consider ϕ∈H1(T3). Firstly, we observe\nthat given T > 0and anyv∈L2\nTH1(T3)∩H1\nTH−1(T3), it follows the chain rule in Hilbert spaces\n(c.f. [Eva10, Sec. 5.9.2, Thm. 3] and (1.25)), along with the explicit expression for the Fréchet derivative\nof the action, (2.3), that\nS(v(t))−S(v(0)) =/integraldisplayt\n0⟨DS(v(s)),∂tv(s)⟩H−1\nx;H−1\nxds (2.9)\n15From Proposition A.11 there exists an h∈L2\nTL2(T3)such that,wh,0solving\n∂twh,0= ∆wh,0−(wh,0)3−m2wh,0+h=−DS(wh,0\nt) +h, wh,0(0) = 0\nsatisfieswh,0(T) =ϕ.\nThen, by Theorem A.6 it holds that wh,0∈CTH1(T3)∩H1\nTH−1(T3)◁arrowhookleft→L2\nTH1(T3)∩H1\nTH−1(T3)\nand by (2.9) along with suitably integrating by parts one finds\nS(ϕ) =S(ϕ)−S(0) =S(wh,0(T))−S(wh,0(0))\n=−/integraldisplayT\n0∥DS(wh,0\nt)∥2\nH−1\nxdt+/integraldisplayT\n0⟨DS(wh,0\nt),ht⟩H−1\nx;H1\nxdt.(2.10)\n▷Proof that 2S≤V.By Cauchy–Schwarz, Young’s product inequality,7(2.10)and the ordering of\nSobolev spaces, (1.15), we find\nS(ϕ)≤−/integraldisplayT\n0∥DS(wh,0\nt)∥2\nH−1\nxdt+/integraldisplayT\n0∥DS(wh,0\nt)∥H−1\nx∥ht∥H−1\nxdt≤1\n4/integraldisplayT\n0∥ht∥2\nH−1\nxdt\n≤1\n4/integraldisplayT\n0∥ht∥2\nL2xdt.(2.11)\nSincehandTwere arbitrary, recalling the definition of Vas given by (2.1), the claim follows directly.\n▷Proof that 2S≥V.The proof of the reverse direction is supplemented by Figure 1, which\nschematically describes the main ideas. At first, we recall that we may write any solution to the equation,\n∂tw0,ϕ(t)−∆w0,ϕ(t) =−w0,ϕ(t)3−m2w0,ϕ(t), w0,ϕ(0) =ϕ∈H1(T3),\nas a solution to the equation,\n∂tw0,ϕ(t) =−DS(w0,ϕ(t)), w0,ϕ(0) =ϕ∈H1(T3),\nwhich is properly understood as an identity in H−1(T3). In the remainder of the proof, we make use\nofglobal, exact controllability of the non-linear skeleton equation, c.f. Proposition A.11. In particular,\nforε>0andψ∈H1(T3), there exists an h∈L2\nεL2(T3)such that\nwh,0(ε) =ψ. (2.12)\nSettingψ:=w0,ϕ(T−ε)∈H1(T3), we denote the corresponding h∈L2\nεL2(T3)from(2.12)by˜hε,T:=h.\nThat is,\nw˜hε,T,0(ε) =w0,ϕ(T−ε).\nWe now consider the backwards flow up the gradient of S,\n∂tv(t) =DS(v(t)), v(ε) =w0,ϕ(T−ε).\nNote that by construction, the reverse gradient flow vsatisfies the identity\nv(t) =w0,ϕ(T−t)for allt∈[ε,T].\n7We use the inequality in the form ab≤a2+b2\n4fora, b≥0which is sometimes also referred to as Peter–Paul inequality\nwith parameter 1/2.\n16Figure 1: Schematic overview of the proof idea. (Left) In green, the\nflowt∝⇕⊣√∫⊔≀→w0,ϕ(t)down the gradient ofSis depicted. The time T(ε)is cho-\nsen such that w0,ϕ(T−ε)∈∂Bδ(ε)(0)⊆H1(T3). (Right) In purple, the flow t∝⇕⊣√∫⊔≀→\n˜v(t) =w˜hε,T,0(t)fort∈[0,ε]is depicted, where—by local exact controllability—the\nelement ˜hε,T∈L2\n[0,ε]L2(T3)is chosen such that ˜v(ε) =w0,ϕ(T−ε). In blue, the\nflowt∝⇕⊣√∫⊔≀→v(t) =w0,ϕ(T−t)fort∈[ε,T]up the gradient ofSis pictured. The\npathv⋆is the concatenation of ˜v(on[0,ε]) andv(on[ε,T]).\nAs a result, we also have v(T) =w0,ϕ(0) =ϕand so we may write\n∂tv(t) =−DS(v(t)) + 2DS(v(t)), v(ε) =w0,ϕ(T−ε), v(T) =ϕ.\nIn this way, we see the term t∝⇕⊣√∫⊔≀→2DS(v(t))is the necessary control to drive the forward gradient flow ,\nstarted from the point w0,ϕ(T−ε)at timeε, to equalϕat timeT. Using the same identities as in (2.10)\nand the equation solved by v(t), we find that\n1\n2∥DS(v(·))∥2\nL2\n[ε,T]L2x= 2/integraldisplayT\nε∥DS(v(s))∥2\nL2xds=2/bracketleftbig\nS(v(T))−S(v(ε))/bracketrightbig\n(2.13)\n=2/bracketleftbig\nS(ϕ)−S(v(ε))/bracketrightbig\n<∞.\nIn other words, 2DS(v)∈L2\n[ε,T]L2(T3)and so it is a member of the Cameron–Martin space of the\nspace-time white noise and is therefore a suitable control.\nNext, recalling that ˜v:=w˜hε,T,0by definition solves\n∂t˜v(t) =−DS(˜v(t)) +˜hε;T(t), t∈[0,ε],˜v(0) = 0,\nby global, exact controllability and our choice of ˜hε,Twe have\n˜v(ε) =w0,ϕ(T−ε) =v(ε).\nAlthough this fact is not required directly by our proof here; note that by Theorem A.6, point (ii), we\nhave\nlim\nt→∞∥w0,ϕ(t)∥H1x= 0⇒ lim\nT→∞∥w0,ϕ(T−ε)∥H1\nx= 0\nwhich demonstrates that this target point can be chosen arbitrarily small, by taking Tarbitrarily large.\n17Settingv⋆(t):= ˜v(t)1[0,ε](t) +v(t)1[ε,T](t)fort∈[0,T], we see that v⋆=wh⋆,0for the control\nh⋆:=˜hε;T1[0,ε]+DS(v(·))1[ε,T]∈L2\n[0,T]L2(T3),\ni.e.\n∂tv⋆(t) =−DS(v⋆(t)) +h⋆(t), v⋆(0) = ˜v(0) = 0, v⋆(T) =v(T) =ϕ.\nFinally, the definition of h⋆and the calculation in (2.13) imply that\n/integraldisplayT\n0∥h⋆(s)∥2\nL2xds= 4/bracketleftbig\nS(ϕ)−S(v⋆(ε))/bracketrightbig\n+/integraldisplayε\n0∥˜hε,T(s)∥2\nL2xds.\nRearranging this equality, recalling that Stakes values in [0,∞]and so taking a lower bound gives\nS(ϕ)≥1\n4/integraldisplayT\n0∥h⋆(s)∥2\nL2xds−1\n4/integraldisplayε\n0∥˜hε;T(s)∥2\nL2xds.\nSinceT=T(ε)>0and\n1\n2/integraldisplayT\n0∥h⋆(s)∥2\nL2ds=I0,T(v⋆)\n≥inf/braceleftig\nI0,T′(v′):T′>0, v′∈CT′Cα(T3), v′(0) = 0, v′(T′) =ϕ/bracerightig\n=V(ϕ),\nwe can further bound Sfrom below by\nS(ϕ)≥1\n2V(ϕ)−1\n4/integraldisplayε\n0∥˜hε;T(s)∥2\nL2xds.\nSinceε>0was arbitrary, the claim follows and the proof is finished. □\nWe have the following corollary of Theorem 2.1.\nCorollary 2.3 The set\n/braceleftig\nz∈Cα(T3):V(z)≤θ/bracerightig\n=/braceleftbigg\nz∈Cα(T3):S(z)≤θ\n2/bracerightbigg\nis a compact subset of Cα(T3). ♤\nProof.By definition of S, this follows from the compact embedding H1(T3)◁arrowhookleft→ →Cα(T3), c.f. (1.21). □\n3 LDP Lower Bound for the Φ4\n3Measure\nIn this section we establish the large deviation lower bound, i.e. Theorem 1.1 (ii). By Remark B.17 (i),\nthis statement is equivalent to the following theorem:8\n8Note that in contrast to the analogous equivalence for the LDP upper bound, this does notrequire that Shas compact\nsublevel sets, see Section 4 for further discussion.\n18Theorem 3.1 (LDP lower bound) Foranyγ,δ> 0andz∈Cα(T3), there exists an ε0:=ε0(γ,δ,z)>\n0such that for all ε≤ε0,\nµε/parenleftbig\ny∈Cα:∥y−z∥Cα<δ/parenrightbig\n≥exp/parenleftbigg\n−2S(z) +γ\nε2/parenrightbigg\n. ♤\nIn order to prove Theorem 3.1, we will require the following lemma which amounts to a uniform,\nquantitative estimate on the control required to drive the solution of the skeleton equation (1.2), started\nfrom a ball of radius ρto a neighbourhood of any terminal state, in terms of the quasi-potential at that\nterminal state. Note that the lemma assumes we take an z∈Cα(T3)such that V(z)<∞. While the\nstatement is true under this, possibly vacuous, assumption, the global, exact controllability shown for\nthe skeleton equation by Proposition A.11 demonstrates that the set infimised over in the definition of\nVis non-empty, hence such a z∈Cα(T3)exists.\nLemma 3.2 Letz∈Cα(T3)be such that V(z)<∞, andγ >0. Then, there exists a ¯T >0such that\nfor anyρ, δ> 0, there exist T0>0andh0∈L2\n¯T+T0L2(T3)with the property that\n1\n2∥h0∥2\nL2\nT0+¯TL2x≤V(z) +γ\n2and sup\n∥y∥Cα≤ρ/vextenddouble/vextenddoublewh0,y(T0+¯T)−z/vextenddouble/vextenddouble\nCα≤δ\n2. (3.1)\n♤\nTo give an intuition for the result, we can think of the required control on ∥h0∥L2\nT0+¯TL2xas a budget to\nspend and the distance δ/2as the accuracy we wish to achieve.\nProof (of Lemma 3.2). To prove the statement, first recall that using Theorem A.6 and the assumption\nthatV(z)<∞it must hold that in fact z∈H1(T3)◁arrowhookleft→L3q(T3). Furthermore, it follows from the\nassumption that V(z)<∞that there must exist a ¯T >0and¯h∈L2\n¯TL2(T3)such that\nw¯h;0(¯T) =z.\nTherefore, upto modifying the pathifnecessary, itis always possible to choose ¯has above andadditionally\nsuch that1\n2∥¯h∥2\nL2\n¯TL2x≤V(z) +γ\n2. (3.2)\nWe now take a T > 0, which will chosen appropriately later, and define h0:[0,T+¯T]→L2(T3)by\nsetting\nh0(t):=/braceleftigg\n0 ift∈[0,T],\n¯h(t−T)ift∈(T,T+¯T].\nWe note that in particular,\n∥h0∥L2\nT+¯TL2x=∥¯h∥L2\n¯TL2\nx. (3.3)\nThus the first bound claimed in (3.1) follows from (3.2) and (3.3), with T >0to be fixed later as T0.\nTo prove the second bound of (3.1)we recall that we have chosen ¯T >0and¯h∈L2\n¯TL2(T3)such that\nw¯h,0(¯T) =z. Furthermore, by definition of h0, for any y∈Cα(T3)andt∈[0,T+¯T]one has that\nwh0,y(t) =\n\nw0,y(t) ift∈[0,T],\nwh0,w0,y(T)\nT (t)ift∈(T,T+¯T].\nThen, given t∈[0,¯T], by the semi-group property of the skeleton equation (see for example the proof of\n19Proposition A.1) and a simple change of variables, for all t∈[0,¯T]we have,\nwh0,y(T+t) =et(∆−m2)w0,y(T)−/integraldisplayT+t\nTe(T+t−s)(∆−m2)wh0,y(s)3ds+/integraldisplayT+t\nTe(T+t−s)(∆−m2)h0(s) ds\n=et(∆−m2)w0,y(T)−/integraldisplayt\n0e(t−s)(∆−m2)wh0,y(T+s)3ds+/integraldisplayt\n0e(t−s)(∆−m2)h0(T+s) ds.\nUsing the fact that for all s∈[0,¯T],h0(s+T) =¯h(s); comparing wh0,y(T+·)tow¯h,0(·)we find\nwh0,y(T+t)−w¯h,0(t) =et(∆−m2)w0,y(T)−/integraldisplayt\n0e(t−s)(∆−m2)/parenleftig\nwh0,y(T+s)3−w¯h,0(s)3/parenrightig\nds.\nBy(A.22)of Theorem A.6, item (iii) in combination with (A.3)of Proposition A.2, for q∈(3/2,2)such\nthat the embedding (1.21) holds, one sees that,\nsup\nt∈[0,¯T]∥w¯h,0(t)∥L3q\nx≲q,m∥¯h∥L2\n¯TL2xand sup\nt∈[0,¯T]∥wh0,y(T+s)∥L3q\nx≲T,q,m,α∥y∥Cα+∥h0∥L2\n¯T+TL2x.\n(3.4)\nSo making use of the Besov embedding, (1.20)along with the regularising effect of the Ornstein–\nUhlenbeck semi-group (1.22) and the fact that\n∥h0∥L2\n¯T+TL2x=∥¯h∥L2\n¯TL2x\none sees that for all t∈[0,¯T]\n∥wh0,y(T+t)−w¯h,0(t)∥L3q\nx\n≤ ∥et(∆−m2)w0,y(T)∥L3q\nx+/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddoublee(t−s)(∆−m2)/parenleftig\nwh0,y(T+s)3−w¯h,0(s)3/parenrightig/vextenddouble/vextenddouble/vextenddouble\nL3q\nxds\n≲∥et(∆−m2)w0,y(T)∥L3q\nx+/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddoublee(t−s)(∆−m2)/parenleftig\nwh0,y(T+s)3−w¯h,0(s)3/parenrightig/vextenddouble/vextenddouble/vextenddouble\nB0\n3q,∞ds\n≲∥w0,y(T)∥L3q\nx+/integraldisplayt\n0(t−s)−1\nq/vextenddouble/vextenddoublewh0,y(T+s)3−w¯h,0(s)3/vextenddouble/vextenddouble\nB−2/q\n3q,3qds\n≲∥w0,y(T)∥L3q\nx+/integraldisplayt\n0(t−s)−1\nq/vextenddouble/vextenddoublewh0,y(T+s)3−w¯h,0(s)3/vextenddouble/vextenddouble\nLq\nxds\n≲∥w0,y(T)∥L3q\nx+/parenleftbigg\n∥wh0,y∥2\nCT+¯TL3q\nx∨∥w¯h,0∥2\nC¯TL3q\nx∨1/parenrightbigg/integraldisplayt\n0(t−s)−1\nq/vextenddouble/vextenddoublewh0,y(T+s)−w¯h,0(s)/vextenddouble/vextenddouble\nL3q\nxds\n≲∥w0,y(T)∥L3q\nx+/integraldisplayt\n0(t−s)−1\nq/vextenddouble/vextenddoublewh0,y(T+s)−w¯h,0(s)/vextenddouble/vextenddouble\nL3q\nxds,\nwhere the final proportionality constant depends only on q,m,∥y∥Cα,∥¯h∥L2\n¯TL2x, due to the a priori bounds,\n(3.4). Therefore, by Grönwall’s inequality, there exists a possibly different constant C > 0, but with the\nsame dependencies as above such that\n∥wh0,y(T+t)−w¯h,0(t)∥L3q\nx≲Cexp/parenleftbig\nt1−1\nq/parenrightbig\n∥w0,y(T)∥L3q\nx,\nIn particular, choosing t=¯Tleads to\n/vextenddouble/vextenddoublewh0,y(T+¯T)−z/vextenddouble/vextenddouble\nL3q\nx≤c(¯T)∥w0,y(T)∥L3q\nx\n20and the claim then follows from item (ii) of Theorem A.6 after choosing T=T0:=T0(ρ, δ)>0\nappropriately large and applying the embedding (1.21). □\nWe are now in a position to prove the main result of this section, Theorem 3.1.\nProof (of Theorem 3.1). Recalling the notation uy\nεfor the solution to the renormalised, dynamic Φ4\n3\nequation with temperature ε>0and started from y∈Cα(T3)as described rigorously by Definition B.1,\nand givenwh0,yas in Lemma 3.2, one has\n∥¯uy\nε(¯T+T0)−z∥Cα≤∥¯uy\nε(¯T+T0)−wh0,y(¯T+T0)∥Cα+∥wh0,y(¯T+T0)−z∥Cα\n≤∥¯uy\nε(¯T+T0)−wh0,y(¯T+T0)∥Cα+δ\n2.\nSinceµεis invariant for uε, we apply this bound to see that\nµε/parenleftbig\ny∈Cα(T3):∥y−z∥Cα<δ/parenrightbig\n=/integraldisplay\nCαP/parenleftbig\n∥¯uy\nε(¯T+T0)−z∥Cα<δ/parenrightbig\nµε(dy)\n≥/integraldisplay\nCαP/parenleftbig\n∥¯uy\nε(¯T+T0)−wh0,y(¯T+T0)∥Cα<δ/2/parenrightbig\nµε(dy)(3.5)\n≥/integraldisplay\n∥η∥Cα≤ρP/parenleftbig\n∥¯uy\nε−wh0,y∥C¯T+T0Cα<δ/2/parenrightbig\nµε(dy)\nThe locally, uniform LDP, obtained for (uy\nε)ε>0by Proposition B.16, implies that for ρ>0there exists\nanε0>0such that for any ε≤ε0and∥y∥Cα≤ρ, one has\nP/parenleftbig\n∥¯uy\nε−wh0,y∥C¯T+T0Cα<δ/2/parenrightbig\n≥exp/parenleftigg\n−I¯T+T0/parenleftbig\nwh0,y/parenrightbig\n+γ\n2ε2/parenrightigg\n= exp/parenleftigg\n−∥h0∥2\nL2\n¯T+T0L2x+γ\n2ε2/parenrightigg\n≥exp/parenleftigg\n−V(z) +γ\nε2/parenrightigg\n,\nwhere the last inequality is due to the first bound of (3.1). Hence, it follows from (3.5) that\nµε/parenleftbig\ny∈Cα(T3):∥y−z∥Cα<δ/parenrightbig\n≥µε/parenleftbig\nBρ/parenrightbig\nexp/parenleftigg\n−V(z) +γ\nε2/parenrightigg\nIn order to conclude therefore, we only need establish that there exists some ρ⋆>0such that\nlim\nε→0µε/parenleftbig\nBρ⋆/parenrightbig\n= 1.\nThis, in turn, is a straightforward consequence of Proposition B.21. With the notations therein, let θ:= 1\nandρ:=ρ(θ). Then, for each ε∈(0,1], one has\n1≥µε(Bρ⋆) = 1−µε(Bc\nρ⋆)≥1−Kexp/parenleftbigg\n−1\nε2/parenrightbigg\nSince the right hand side goes to 1asε→0the proof is complete. □\n214 LDP Upper Bound for the Φ4\n3Measure\nIn this section we establish the large deviation upper bound, Theorem 1.1 (iii). By Remark B.17 (i),\nthis statement is equivalent to the following theorem since we have already shown that Shas compact\nsublevel sets, see Corollary 2.3.\nTheorem 4.1 (LDP upper bound) For anyθ, δ, γ > 0there exists an ε0:= (δ, θ,γ )>0such that\nfor allε≤ε0,\nµε/parenleftbig\nz∈Cα(T3):distCα(z,S[θ])≥δ/parenrightbig\n≤exp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\n. ♤\nThe proof of Theorem 4.1 requires several preliminary lemmas and propositions; the final proof is\nconcluded in Section 4.3. The first required result, is the following lemma, which allows us to define the\nset of paths t∝⇕⊣√∫⊔≀→v(t)∈Cα(T3)which begin in a given ball but take values outside a second ball at all\ninteger time points.\nLemma 4.2 (Dynamic energy bounds eventually imply static energy bounds) Letθ, δ > 0\nand for any λ>0recall the definition of the following set for any t>0,\nI¯Bλ\nt[θ]:=/braceleftig\nv∈CtCα(T3):v(0)∈¯Bλ, v∈It[θ]/bracerightig\n(4.1)\nThen, there exists some λ(δ, θ) =:λ>0and ¯T(δ, θ) =:¯T >0such that for all t≥¯T, we have\n/braceleftig\nv(t)∈Cα(T3):v∈I¯Bλ\nt[θ]/bracerightig\n⊆/braceleftbigg\nz∈Cα(T3):distCα(z,V[θ])<δ\n2/bracerightbigg\n. (4.2)\nwhereVis the quasi-potential defined by (2.1)andV[θ]denotes the associated sub-level set. ♤\nThe claim made in Lemma 4.2 is illustrated in figure 2 below. Intuitively, it says that a path which starts\nclose to equilibrium 0and has bounded dynamic energy will eventually have bounded static energy.\n1\nFigure 2: Schematic overview of the statement in Lemma 4.2. The green\npathvis an element in I¯Bλ\nt[θ]and, according to the set inclusion in (4.2), therefore\nnever leaves the δ/2-neighbourhood of V[θ]during the time interval [¯T,t].\n22Proof.Let us assume that the claimed inclusion does not hold, in other words, there exist δ, θ> 0such\nthat for all T, λ> 0there exists a t≥Tsuch that,\n/braceleftig\nv(t)∈Cα(T3):v∈I¯Bλ\nt[θ]/bracerightig\n̸⊆/braceleftbigg\nz∈Cα(T3):distCα(z,V[θ])<δ\n2/bracerightbigg\n.\nAs a consequence, given some θ, δ> 0, there exist sequences λn↘0,Tn↗+∞andvn∈I¯Bλn\nTn[θ]for\nn∈N, such that\ninf\nn≥0distCα/parenleftbig\nvn(Tn/parenrightbig\n,V[θ])≥δ\n2. (4.3)\nSincevn∈I¯Bλn\nTn[θ], using(4.1), it holds in particular that ITn(vn)≤θ <∞. Thus, by definition of\nthe rate function, there exists an hn∈L2\nTnL2\nxsuch thatvn=whn,znforzn:=vn(0), i.e.\n∂tvn−∆vn=−v3\nn−m2vn+hn,\nzn=vn(0)∈¯Bλn(0)⊆Cα(T3)and1\n2∥hn∥2\nL2\nTnL2x≤θ. (4.4)\nIn particular, as λn↘0, it holds that\nzn→0inCα(T3)asn→∞. (4.5)\nSince the rest of the proof is quite technical, at this stage we pause the formal proof to give an intuitive\nidea behind the rest of the strategy, which relies essentially on the gradient flow structure of the dynamic\nequation.\n— Intuition of the Strategy: Suppose we knew instead that zn∈H1(T3)and the convergence\nin(4.5)actually took place in H1(T3). Then, arguing as in the proof of Theorem 2.1 (in particular using\nanalogues of (2.10) and (2.11)) we would be able to arrive at the estimate\nS(vn(Tn))−S(zn)≤1\n4∥h∥2\nL2\nTnL2x≤1\n2θ.\nBy continuity of SonH1(T3)and the fact that S(0) = 0, this would then imply that S(vn(Tn))≤θ\nfornlarge enough. Because the sublevel set V[2θ] =S[θ] ={ϕ∈H1(T3):S(ϕ)≤θ}is compact\ninCα(T3)(c.f. Corollary 2.3), there would exist a subsequence (nk)k∈Nandy∈Cα(T3)such that\nlim\nk→∞∥vnk(Tnk)−y∥Cα. (4.6)\nBy lower semicontinuity of S, we could then obtain the estimate\nS(y)≤lim inf\nk→∞S(vnk(Tnk)≤lim\nk→∞S(zn) +1\n2θ=S(0) +1\n2θ=1\n2θ (4.7)\nfrom which, in combination with Theorem 2.1, we would see that yis an element of the sublevel set V[θ],\na contradiction to (4.3). —\nThe obstacle to applying this relatively simple strategy is that we do not necessarily have convergence\nof the sequence zn→0with respect to the H1(T3)topology. The remainder of the proof, however,\ndemonstrates that one can make technical amendments to the previous argument to arrive at the same\nconclusion only using that zn→0in the weakerCα(T3)topology.\nStep 1:The key idea is to define a family of solutions to approximate equations, similar to (4.4)but\nwith zero initial data and forcing that only takes effect after a short positive time. Let (λn)n∈N,(Tn)n∈N\n23be the respectively decreasing and increasing sequences from previously, redwhich we may assume are\nrespectively such that λn∈(0,1]andTn∈(1,∞]. We then define a strictly decreasing sequence (sn)n∈N\nby setting\nsn:=λ1\n2n∧e−T2\nn\nελ−1\nεn∧1, (4.8)\nwhere the parameters η, εandαare chosen, as in Proposition A.1, to satisfy\nα∈/parenleftig\n−2\n3,−1\n2/parenrightig\n, η 0∈/parenleftig\n−5α\n12,−α\n3/parenrightig\n&ε∈/parenleftbig\n0,−α−3η0/parenrightbig\n. (4.9)\nWe also fix p=1\n1+α∈(6,9)so that applying (1.16) we have Lp(T3)◁arrowhookleft→ →Cα(T3).\nWe then define a sequence of forcing elements,\n˜hn(t):=/braceleftigg\n0 ift∈[0,sn),\nhn(t−sn)ift∈[sn,sn+Tn].\nand a family of paths (˜vn)n∈N, solving the analogous equation to (4.4)withhnreplaced by ˜hn, explicitly\ngiven by setting\n˜vn(t):=w˜hn,zn(t) =\n\nw0,zn(t) ift∈[0,sn),\nw˜hn,w0,zn(sn)\nsn (t)ift∈[sn,sn+Tn].(4.10)\nThe notation wh,z\nshere indicates that wh,z\ns(t)/vextendsingle/vextendsingle\nt=s=z.\nSince ˜vn(s) =w0,zn(s)fors∈[0,sn)andsn∈(0,1], we may apply Proposition A.1, specifically (A.4),\nto see that there exists an η:= 1 +η0∈/parenleftbig\n1−5α\n12,1−α\n3/parenrightbig\n⊂/parenleftbig\n1 +10\n36,1 +2\n9/parenrightbig\nandδ:=δ(α,η)>0such that\nsup\ns∈(0,sn](s∧1)η∥˜vn(s)∥H1x≲sδ\nn∥zn∥Cα\nFrom which it follows that,\n∥˜vn(sn)∥H1x≲sδ−η\nn∥zn∥Cα≤sδ−η\nnλn≤λδ\n2+1−η\n2n. (4.11)\nwhere we have used the assumption that zn∈Bλn(0)⊆Cα(T3)along with the definition of the sequence\n(sn)n∈Ngiven by (4.8). Sinceδ\n2+ 1−η\n2>0we find,\n˜vn(sn)→0inH1(T3)asn→∞\nand so ˜vn(sn)provides an sequence to which we can apply our intuitive strategy presented previously.\nIn particular, making the replacements\nvn(Tn)⇝˜vn(sn+Tn), vnk(Tnk)⇝vnk(snk+Tnk),zn⇝˜vn(sn).\nWe obtain versions of (4.6) and (4.7) which now read, for some y∈Cα(T3),\nlim\nk→∞∥˜vnk(Tnk+snk)−y∥Cα= 0andS(y)≤1\n2θ.\nTherefore, in order to arrive at a contradiction to (4.3) and complete the proof, it remains to show\nlim\nn→∞˜vn(Tn+sn) = lim\nn→∞vn(Tn)inCα(T3) (4.12)\nif both of these limits exist. Note that we have re-labelled the subsequences to depend on nonly for the\n24sake of presentation. We establish the equality in the limit of (4.12) in Step 2below.\nStep 2:We write the mild formulation for ˜vn(·+sn)andvn, starting with the former. Let t∈(0,Tn]\nso that using (4.10), we obtain\n˜vn(t+sn) =et(∆−m2)w0,zn(sn)−/integraldisplayt+sn\nsne(t+sn−r)(∆−m2)w˜hn,w0,zn(sn)\nsn(r)3dr\n+/integraldisplayt+sn\nsne(t+sn−r)(∆−m2)˜hn(r) dr\n=et(∆−m2)w0,zn(sn)−/integraldisplayt\n0e(t−r)(∆−m2)˜vn(r+sn)3dr+/integraldisplayt\n0e(t−r)(∆−m2)hn(r) dr\nand also\nvn(t) =et(∆−m2)zn−/integraldisplayt\n0e(t−r)(∆−m2)vn(r)3dr+/integraldisplayt\n0e(t−r)(∆−m2)hn(r) dr\nso that we obtain the following equation for their difference,\n˜vn(t+sn)−vn(t) =et(∆−m2)/parenleftbig\nw0,zn(sn)−zn/parenrightbig\n−/integraldisplayt\n0e(t−r)(∆−m2)/parenleftbig\n˜vn(r+sn)3−vn(r)3/parenrightbig\ndr.(4.13)\nNote that the integral term with the forcing hnhas cancelled out exactly. We aim to apply (a generalised\nversion of) Gronwall’s inequality and, to this end, analyse the two terms on the right hand side of (4.13)\nseparately.\nStep 2a: Comparing Initial Data Using the mild formulation of the skeleton equation we directly\nhave,\n∥et(∆−m2)(w0,zn(sn)−zn)∥Lp\nx≤e−m2t∥et∆(esn(∆−m2)zn−zn)∥Lp\n+e−m2t/vextenddouble/vextenddouble/vextenddouble/vextenddoubleet∆/integraldisplaysn\n0e−(sn−r)(∆−m2)w0,zn(r)3ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLp\nx\n=:(I) + (II).\nTo bound the first term we use the regularising effect of the heat semi-group, (1.22), along with the\ncontinuity at zero of the Ornstein–Uhlenbeck semi-group, (1.23), to see that for any t>1,sn∈(0,1)\nandε∈(0,−α−3η)\n(I)≲e−tm2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftig\nesn(∆−m2)−1/parenrightig\nzn/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nB5α/6−ε\np,p≲(sε\nn+m2sn)∥zn∥B5α/12\np,p≲sε\nn∥zn∥Cα≤sε\nnλn.\nFor the second term we proceed similarly, first using that et(∆−m2)is a uniformly bounded family of\noperators from Lp(T3)to itself and then applying (1.22) to give\n(II)≲/integraldisplaysn\n0e−(sn−r)m2(sn−r)−3\np∥w0,zn(r)3∥Lp/3\nxdr\n≲∥w0,zn∥3\nCη;snLp\nx/integraldisplaysn\n0e−(sn−r)m2(sn−r)−3\npr−3ηdr\n25≲∥w0,zn∥3\nCη;snLp\nxs1−3\np−3η\nn.\nSo applying estimate (A.3)of Proposition A.1 and using the assumptions that λn≤1,sn∈(0,1)and\n0<ε<−α−3η0= 1−3\np−3η0we find\n∥et(∆−m2)(w0,zn(sn)−zn)∥Lp\nx≲sε\nnλn+s1−3\np−3η0\nn∥w0,zn∥3\nCη0;snLp\n≤sε\nnλn+s1−3\np−3η0\nn∥zn∥3\nCα\n≲sε\nnλn. (4.14)\nStep 2b: Comparing the Non-LinearTerm Recallthat Lp/3(T3)◁arrowhookleft→B−6/p\np,p(T3)byBesovembedding\nand lett∈[0,Tn]. For the second term in (4.13), we find\n/integraldisplayt\n0/vextenddouble/vextenddoublee(t−r)(∆−m2)/bracketleftbig\n˜vn(r+sn)3−vn(r)3/bracketrightbig/vextenddouble/vextenddouble\nLp\nxdr\n≤/integraldisplayt\n0e−(t−r)m2(t−r)−3/p∥˜vn(r+sn)3−vn(r)3∥Lp/3\nxdr (4.15)\n≲/integraldisplayt\n0e−(t−r)m2(t−r)−3/p/parenleftbig\n∥˜vn(r+sn)∥2\nLp\nx+∥vn(r)∥2\nLp\nx/parenrightbig\n∥˜vn(r+sn)−vn(r)∥Lp\nxdr\n≤C(t)/integraldisplayt\n0e−(t−r)m2(t−r)−3/p(r∧1)−2η0∥˜vn(r+sn)−vn(r)∥Lp\nxdr\nwhere we have set\nC(t):= sup\nr∈[0,t](r∧1)2η0/parenleftbig\n∥˜vn(r+sn)∥2\nLp\nx+∥vn(r)∥2\nLp\nx/parenrightbig\n.\nStep 2c: Conclusion and Grönwall Estimate Since the family{C(Tn)}n∈Nis uniformly bounded\n(see Theorem A.6, item (iii)) we combine (4.13),(4.14), and(4.15)to give the following, estimate which\nholds for all t∈[1,Tn]:\n∥˜vn(t+sn)−vn(t)∥Lp\nx≲e−tm2sε\nnλn+/integraldisplayt\n0e−(t−r)m2(t−r)−3/p(r∧1)−2η0∥˜vn(r+sn)−vn(r)∥Lp\nxdr\nFurther, observe that the choice of parameters in (4.9) guarantees that\n3\np+ 2η0<1.\nTherefore, we may apply the generalised Grönwall inequality (1.26)to see that there exist constants\nc1:=c1(α,η0), c2:=c2(α,η0)>0such that for all t∈[1,Tn]such that,\n∥˜vn(t+sn)−vn(t)∥Lp≤c1sε\nnλnexp/parenleftbigg/parenleftbig\nc2−m2/parenrightbig\nt/parenrightbigg\n(4.16)\nTakingt=Tnand using that\nsn≤e−T2\nn\nελ−1\nεn\nwe find,\n∥˜vn(Tn+sn)−vn(Tn)∥Lp\nx≤c1e(c2−m2)Tn−T2\nn\n26Since the RHS goes to 0asn→∞(andTn↗∞), the claim of (4.12)follows by Besov embedding\nLp(T3)◁arrowhookleft→ →Cα(T3), c.f. (1.16). □\nRemark 4.3 (Relation between λandρ.)If one could quantify the dependence of λonδandθin\nLemma 4.2, such that it matched that of ρ=ρ(θ)from Proposition B.21 (cf. also Remark B.22), then\nit would be possible to drastically simplify the proof of the LDP upper bound. To wit, choosing ρ=λ,\none could directly apply the estimate (4.27)below to the situation that ¯uy\nεbegins from y∈Bλ(0). As a\nconsequence, we would not have to introduce the set Eρ,δ,θ(n)below and could conclude without considering\nthe, proximately introduced, scenarios IIandIIIseparately. However, since our proof of Lemma 4.2 is\nnon-constructive, this simpler proof does not seem immediately available. ♤\nWe are now able to define the excursion setEρ,δ,θ(n)which will play an important role in the arguments\nof this section.\nDefinition 4.4 (The excursion set Eρ,δ,θ)Letδ, θ > 0andλ:=λ(δ, θ)>0be the associated\nparameter obtained in Lemma 4.2. Then, for any n∈Nandρ>0, we define the set\nEρ,δ,θ(n):=/braceleftig\nv∈CnCα(T3):∥v(0)∥Cα≤ρ,∥v(j)∥Cα≥λ,for allj∈{1,...,n}/bracerightig\n. ♤\nThe purpose of defining this set is to give us access to a useful decomposition of the event\n{distCα(T3)(¯uy\nε(t),V[θ])≥δ}. (4.17)\nBy statistical invariance of the dynamics with respect to the measure µε, provided yis sampled from\nµε, exponential control on the event described by (4.17)is equivalent to the LDP upper bound claimed\nby Theorem 4.1, c.f. Remark B.17, item (i). Hence, a proof of Theorem 4.1 will follow from sufficient\nexponential control on (4.17), averaged over y∼µε. To this end we decompose the event described by\n(4.17) into three disjoint scenarios:\nI:when y∈Bρ(0)c⊂Cα(T3)anddistCα(T3)(¯uy\nε(t),V[θ])≥δ,\nII:when y∈Bρ(0)⊂Cα(T3),¯uy\nε∈Eρ,δ,θ(¯n)for some ¯n∈N,¯n≤t, and distCα(T3)(¯uy\nε(t),V[θ])≥δ,\nIII:when y∈Bρ(0)⊂Cα(T3),¯uy\nε∈Eρ,δ,θ(¯n)cfor the same ¯n∈N, and distCα(T3)(¯uy\nε(t),V[θ])≥δ.\nRemark 4.5 (Choice of tand¯n.)We emphasise that, at this stage, both tand¯nareparameters .\nThe latter will be determined in Lemma 4.6 below whereas the former picks up the constraint t≥¯n\nin(4.30)below andt≥¯Tin Lemma 4.2 above. Consequently, we later choose t:=¯T+¯n, see the proof\nof Proposition 4.9. ♤\nThe three scenarios are graphically represented in Figure 3 and the strategies to controlling their\nprobabilities, informally, described below. At the end of each paragraph, we direct the reader to the\nrelevant rigorous results which make these heuristics precise.\nScenario I: Initial condition outsidea ballBρ.This is the easiest scenario to control, we simply\ndiscard the constraint distCα(T3)(¯uy\nε(t),V[θ])≥δand use the fact that y∼µεto obtain a bound on\nthe event y∈Bρ(0)cvia the appropriate tail bounds on the invariant measure, c.f. Proposition B.21\nand (4.33).\nScenario II: Initial condition insidea ballBρ,excursions fromBλat all integer times\nup to ¯n.Similarly to Scenario I, we discard the second condition, distCα(T3)(¯uy\nε(t),V[θ])≥δ, on\nthe distance between the dynamic path ¯uy\nεand the sub-level set of the quasi-potential, V[θ]. Instead,\nto control Scenario II we focus on the condition ¯uy\nε∈Eρ,δ,θ(¯n), see Definition 4.4. By definition, if\n27(a)Scenario I\n (b)Scenario II\n(c)Scenario III\nFigure 3: Schematic overview of the different scenarios.\nthe realised path lies in this excursion set then at each integer time j∈{1,..., ¯n}we must have that\n¯uy\nε(j)∈Bc\nλ. Since 0is the unique, asymptotically stable attractor of the deterministic dynamics, these\nλ-excursions require repeated inputs of energy from the noise, which the locally uniform LDP for the\ndynamics (see Proposition B.16) guarantees is an exponentially vanishing event as ε→0. This argument\nis carried out in Lemma 4.6 and Proposition 4.7, which are made use of in (4.34) below.\nScenario III: Initial condition insidea ballBρ,entrance intoBλbetween time 1and¯n.\nTo control this scenario we finally make use of the assumption that distCα(T3)(¯uy\nε(t),V[θ])≥δ. The core\nidea is to appeal to Lemma 4.2 which, informally, tells us that, taking t>0sufficiently large, realisations\nof the path which lie at distance δ>0or more from the sub-level set V[θ]must also have have dynamic\nenergy larger than θ. The locally uniform LDP for the dynamics, given by Proposition B.16, ensures\nthat this is an exponentially vanishing event, uniformly in the initial data y∈Bρ, asε→0. Technically,\nwe make use of the additional assumption that ¯uy\nε∈Eρ,δ,θ(¯n)cto guarantee that for at least some integer\ntimej∈{1,..., ¯n}one has ¯uy\nε(j)∈Bλ. This allows us to leverage the Markov property (c.f. Figure 5) in\norder to properly apply Lemma 4.2 for the equation restarted from Bλ. This argument is made rigorous\nin Lemma 4.8 and Proposition 4.9 which are made use of in (4.35) below\n284.1 Control on the Dynamic Remaining in Eρ,δ,θ\nThe following lemma identifies a time horizon ¯n∈N(cf. Remark 4.5) for which the set Eρ,δ,θ(¯n)is\nin the complement of the θ-sub-level set of the dynamic rate function I¯n. In other words: Excursion\npathsv∈Eρ,δ,θ(¯n)have energy I¯n(v)larger than θ.\nLemma 4.6 For eachρ, δ, θ> 0, there exists some ¯n= ¯n(ρ, δ, θ )∈Nsuch that\nβ¯n:= inf{I¯n(v):v∈Eρ,δ,θ(¯n)}>θ.\nIn particular,Eρ,δ,θ(¯n)⊆{I¯n>θ}=I¯n[θ]c. ♤\nProof.Let us suppose for a contradiction that supn∈Nβn≤θ. Then, for each n∈N, we can fix a\npathvn∈Eρ,δ,θ(n)such that\nIn(vn)≤βn+ 1≤θ+ 1. (4.18)\nBy Theorem A.6, item (iii), applied with p>6(such that by (1.16)one hasLp(T3)◁arrowhookleft→ →Cα(T3)), and\nrecalling the definition of It(vt)given in (B.16), for any n∈Nwe have\nsup\nt∈[1,n]∥vn(t)∥Cα≤sup\nt∈[1,n]∥vn(t)∥Lp\nx≲m,α∥vn(0)∥Cα+In(vn)1\n2≤ρ+ (1 +θ)1\n2=:cρ,θ>ρ.(4.19)\nWithλ:=λ(δ, θ)again as in Lemma 4.2, we then introduce the set\n˜EEP\nρ,δ,θ(¯k):={v∈C¯kCα(T3):∥v(0)∥Cα≤cρ,θ,∥v(¯k)∥Cα≥λ}.\nof “end-point excursions” starting in a ball of radius cρ,θ>ρ. We now reduce ourselves to prove that\nthere exists some ¯k∈Nlarge enough such that\nι¯k:= inf{I¯k(v):v∈˜EEP\nρ,δ,θ(¯k)}>0. (4.20)\nTo see that (4.20) leads to a contradiction, given any n∈N, denote by\nv(m)\nn¯k(t):=vn¯k|[m¯k,(m+1)¯k](m¯k+t), t∈[0,¯k], m∈{0,...,n−1}, (4.21)\na reparametrisation of vn¯k|[m¯k,(m+1)¯k]that lies in C¯kCα(T3); see Figure 4 for a visualisation. Then, note\nthat due to (4.19)and because vn¯k∈Eρ,δ,θ(n¯k), one hasv(m)\nn¯k∈˜EEP\nρ,δ,θ(¯k)for eachm∈{0,...,n−1}\nsuch that we find\nθ+ 1≥In¯k(vn¯k) =n−1/summationdisplay\nm=0I[m¯k,(m+1)¯k]/parenleftbig\nvn¯k|[m¯k,(m+1)¯k]/parenrightbig\n=n−1/summationdisplay\nm=0I¯k/parenleftbig\nv(m)\nn¯k/parenrightbig\n≥nι¯k(4.20)\n>0.(4.22)\nSincen∈Ncan be chosen arbitrarily large we obtain a contradiction.\nWe are therefore only required to show (4.20). By Theorem A.6, item (ii), there exists some time ¯t≥1\nsuch that\n∥w0,z\n1(t)∥Cα≤∥w0,z\n1(t)∥Lp\nx≤λ\n2for allt≥¯t, (4.23)\nwhere we have use the notation w0,z\n1to indicate that the initial condition zholds att= 1, i.e.w0,z\n1(1) = z.\nThis trivially implies that, for any integer ¯k≥¯t, we have\nw0,z\n1|[1,¯k]/∈/braceleftig\nv∈C[1,¯k]Cα(T3):∥v(¯k)∥Cα≥λ/bracerightig\n. (4.24)\nWe claim that for any such ¯k, we haveι¯k>0. Suppose otherwise, i.e. that ι¯k= 0. In that case, there\n29Figure 4: Visualisation of the path vn¯k.\nexists a minimising sequence\n{˜vn}n∈N⊆˜EEP\nρ,δ,θ(¯k),lim\nn→∞I¯k(˜vn) = 0.\nFor eachn∈N, we have∥˜vn(0)∥Cα≤cρ,θby definition of ˜EEP\nρ,δ,θ(¯k)and, therefore, the same argument\nas in (4.19) implies that\n∥˜vn(1)∥Lp\nx≤cρ,θ+I¯k(˜vn)1\n2≤cρ,θ+C\nsince{I¯k(˜vn)}n∈Nis a convergent sequence and therefore bounded by some C <∞. As a conse-\nquence,{vn(1)}n∈N⊆Lp(T3)isaboundedsequenceandsobythecompactembedding Lp(T3)◁arrowhookleft→ → Cα(T3)\n(due to our choice of p>6) there exist a subsequence {˜vnj(1)}j∈Nand¯z∈Cα(T3)such that\nlim\nj→∞˜vnj(1) = ¯zinCα(T3)\nSince, by construction I¯k(˜vn)→0asn→∞, the definition of I¯kimplies the existence of a se-\nquence (hn)n∈N⊆L2\n[1,¯k]L2(T3)such that\n˜vn|[1,¯k]=whn,˜vn(1)\n1 and lim\nn→∞hn= 0inL2\n[1,¯k]L2(T3).\nTherefore, by joint continuity of the solution map in the initial data and forcing, we have\nlim\nj→∞˜vnj|[1,¯k]=w0,¯z\n1inC[1,¯k]Cα(T3)\nand then∥w0,¯z\n1(¯k)∥Cα≥λsince ˜vn∈˜EEP\nρ,δ,θ(¯k). This contradicts (4.24)and we indeed have ι¯k>0. The\nclaim in (4.20) follows. The proof is complete. □\nThe following proposition makes use of the inclusion Eρ,δ,θ(¯n)⊆{I¯n>θ}obtained by Lemma 4.6 to\nestablish an upper bound on the probability of the Φ4\n3dynamics remaining in the same set.\nProposition 4.7 (Scenario II) Letρ, δ, θ> 0and¯n:=¯n(ρ, δ, θ )∈Nbe given by Lemma 4.6. Then,\ngivenγ >0, there exists an ε1>0such that\nsup\ny∈¯BρP/parenleftbig\n¯uy\nε∈Eρ,δ,θ(¯n)/parenrightbig\n≤exp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\n,for allε≤ε1. ♤\n30Proof.By Lemma 4.6, for any ρ, θ, δ> 0, there exists some ¯n∈Nsuch that\nEρ,δ,θ(¯n)⊆I¯n[θ]c≡{I¯n>θ}.\nIn particular,\n¯uy\nε∈Eρ,δ,θ(¯n)⇒ ¯uy\nε/∈I¯n[θ].\nand since I¯n[θ]is compact inCα(T3)by Proposition B.16, item (i), it holds that\n¯uy\nε∈Eρ,δ,θ(¯n)⇒ distC¯nCα/parenleftbig\n¯uy\nε,I¯n[θ]/parenrightbig\n>0.\nTherefore, the LDP upper bound for ˆuy\nε, which is locally uniform in the initial condition, see Proposi-\ntion B.16, item (iii), we can find some ε1>0such that\nsup\ny∈¯BρP/parenleftbig\n¯uy\nε∈Eρ,δ,θ(¯n)/parenrightbig\n≤sup\ny∈¯BρP/parenleftbig\ndistC¯nCα/parenleftbig\n¯uy\nε,I¯n[θ]/parenrightbig\n≥0/parenrightbig\n≤exp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\nfor allε≤ε1, which concludes the proof. □\n4.2 Control when the Dynamic Leaves Eρ,δ,θ\nWe first require the following version of Lemma 4.2 and record its specialisation to ˆuy\nε.\nLemma 4.8 Letδ, θ> 0andλ:=λ(δ, θ)>0be as in Lemma 4.2. Then, recalling the notation\nIy\nt[θ] =/braceleftig\nv∈CtCα(T3):v(0) = y,It(v)≤θ/bracerightig\n, (4.25)\nthere exists some ¯T >0such that for any y∈¯Bλ⊂Cα(T3)andt≥¯Tone has\n/braceleftig\nv∈CtCα:v(0) = y,distCα/parenleftbig\nv(t),V[θ]/parenrightbig\n≥δ/bracerightig\n⊆/braceleftbigg\nv∈CtCα:v(0) = y,distCtCα/parenleftbig\nv,Iy\nt[θ]/parenrightbig\n≥δ\n2/bracerightbigg\n.\n(4.26)\nIn particular, we have\nP/parenleftbig\ndistCα/parenleftbig\n¯uy\nε(t),V[θ]/parenrightbig\n≥δ/parenrightbig\n≤P/parenleftbigg\ndistCtCα/parenleftbig\n¯uy\nε,Iy\nt[θ]/parenrightbig\n≥δ\n2/parenrightbigg\n. (4.27)\n♤\nProof.LetLdenote the set on the LHS of (4.26)andRthe set on its RHS. The statement in (4.27)\nthen simply reads\nP/parenleftbig\n¯uy\nε∈L/parenrightbig\n≤P/parenleftbig\n¯uy\nε∈R/parenrightbig\nand thus follows from (4.26) by monotonicity of probability measures with respect to set inclusion.\nIn order to prove (4.26), letv∈L. Taking the complement on both sides of (4.2)we see that we must\ntherefore have\nv /∈I¯Bλ\nt[θ]for allt≥¯T.\nIn words, any v∈Lcannot be an element in any of the sets I¯Bλ\nt[θ],t≥¯T. However, it is not a priori\nexcluded that there exists a v∈Land somet≥¯Tsuch thatvisarbitrarily close to the set I¯Bλ\nt[θ].\nIn order to rule that out, assume for a contradiction that distCtCα/parenleftbig\nv,I¯Bλ\nt[θ]/parenrightbig\n<δ/2for somet≥¯T.\nThen we may find ˜v∈I¯Bλ\nt[θ]such that\n∥v(t)−˜v(t)∥Cα≤∥v−˜v∥CtCα<δ\n2.\n31However, Lemma 4.2 applied to ˜v∈I¯Bλ\nt[θ]shows that distCα(˜v(t),V[θ])<δ/2. That is, there exists\nsome z∈V[θ]such that∥˜v(t)−z∥Cα<δ/2. As a consequence, the estimate\n∥v(t)−z∥Cα≤∥v(t)−˜v(t)∥Cα+∥˜v(t)−z∥Cα<δ\n2+δ\n2=δ\ncontradicts our assumption that v∈Land so we actually have\ndistCtCα/parenleftbig\nv,I¯Bλ\nt[θ]/parenrightbig\n≥δ/2.\nFinally, note that y∈¯Bλimplies Iy\nt[θ]⊆I¯Bλ\nt[θ]which, together with the previous estimate, leads to\nthe lower bound\ndistCtCα/parenleftbig\nv,Iy\nt[θ]/parenrightbig\n≥distCtCα/parenleftbig\nv,I¯Bλ\nt[θ]/parenrightbig\n≥δ\n2.\nSincev∈Lby definition also entails that v(0) = y, we arrive at the conclusion that v∈R. This\nestablishes the claim in (4.26). □\nProposition 4.9 (Scenario III) Letδ, γ, ρ, θ > 0and ¯n∈N. Then, there exists a t > 0and\nε0(t,¯n,δ, θ,γ ):=ε0>0such that\nsup\ny∈¯BρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ,¯uy\nε∈Eρ,δ,θ(¯n)c/parenrightig\n≤¯nexp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\n,for allε∈(0,ε0).♤\nProof.We first note that using the definition of Eρ,δ,θ(¯n), for all y∈¯Bρit holds that,\nP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ,¯uy\nε∈Eρ,δ,θ(¯n)c/parenrightig\n≤P/parenleftigg¯n/uniondisplay\nj=1/braceleftig\ndistCα(¯uy\nε(t)V[θ])≥δ,¯uy\nε(j)∈¯Bλ/bracerightig/parenrightigg(4.28)\nUsing the fact that t∝⇕⊣√∫⊔≀→¯uy\nε(t)is a Markov process (see for example [HM18a, Coro. 1.3]) we may restart\nthe equation at any time j∈{1,..., ¯n}when ˆuy\nε(j)∈¯Bλto find that for any t>¯n, which will be chosen\nlater,\nP/parenleftigg¯n/uniondisplay\nj=1/braceleftig\ndistCα(¯uy\nε(t),V[θ])≥δ,¯uy\nε(j)∈¯Bλ/bracerightig/parenrightigg\n≤¯n/summationdisplay\nj=1sup\nz∈¯BλP/parenleftig\ndistCα(¯uz\nε(t−j),V[θ])≥δ/parenrightig\n;(4.29)\nsee Figure 5 for a graphical representation of this restarting procedure. Now recalling Lemma 4.8 and\nthe¯T:=¯T(δ, θ)>0defined therein, we choose t=¯T+¯nso that applying (4.27)we obtain the bound\n¯n/summationdisplay\nj=1sup\nz∈¯BλP/parenleftig\ndistCα(¯uz\nε(t−j),V[θ])≥δ/parenrightig\n≤¯nmax\nj=1,...,¯nsup\nz∈¯BλP/parenleftig\ndistCα(¯uz\nε(t−j),V[θ])≥δ/parenrightig\n≤¯nmax\nj=1,...,¯nsup\nz∈¯BλP/parenleftbigg\ndistCt−jCα/parenleftbig\n¯uz\nε,Iy\nt−j[θ]/parenrightbig\n≥δ\n2/parenrightbigg(4.30)\nWe now recall the locally uniform LDP satisfied by the process ˆuy\nεas given by Proposition B.16. Using\nthe notation of that proposition, given any γ >0andδ, θ> 0as above, let us set\nε0:=ε0(t,¯n,δ,γ,θ ):= min/braceleftbig\nε(t−j,δ,γ,θ ):j∈{1,..., ¯n}/bracerightbig\n,\n32Figure 5: Application of the Markov property. The figure on the left shows\na realisation of ¯uy\nεthat hits the ball Bλat timej, at which point the Markov\nproperty is applied. The figure on the right presents a realisation of the restarted\ndynamics ¯uz\nεafter the probability has additionally been maximised over all possible\npoints z∈Bλthat ¯uy\nε(j)could have hit.\nso that applying Proposition B.16, item (iii), we find, for any γ >0,\nmax\nj=1,...,¯nsup\nz∈¯BλP/parenleftbigg\ndistCt−jCα/parenleftbig\n¯uz\nε,Iz\nt−j[θ]/parenrightbig\n≥δ\n2/parenrightbigg\n≤exp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\n,for allε<ε 0.(4.31)\nNote that the respective right hand sides of (4.29),(4.30)and(4.31)do not depend on y∈¯Bρany more;\ncombined with (4.28), they establish the claim and conclude the proof. □\n4.3 Proof of the LDP Upper Bound\nWe are now in a position to prove the LDP upper bound claimed in Theorem 4.1.\nProof (of Theorem 4.1). First, by linearity of the integral, invariance of the measure µεfor the dynamic,\nrenormalised process, uεand additivity of measures, for any ρ>0, t> 0, which will be further restricted\nlater on, and ε>0, it holds that,\nµε/parenleftbig\nz∈Cα(T3):distCα(z,V[θ])≥δ/parenrightbig\n=/integraldisplay\nCαP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ/parenrightig\nµε(dy)\n=/integraldisplay\nBcρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ/parenrightig\nµε(dy)\n+/integraldisplay\nBρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ/parenrightig\nµε(dy).\n=/integraldisplay\nBcρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ/parenrightig\nµε(dy)\n+/integraldisplay\nBρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ, uy\nε∈Eρ,δ,θ(¯n)/parenrightig\nµε(dy)\n+/integraldisplay\nBρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ, uy\nε∈Eρ,δ,θ(¯n)c/parenrightig\nµε(dy)\n=:(I) + (II) + (III).(4.32)\n33In the above we recall the notation ¯uy\nεfor the solution to the infinitely renormalised dynamics, (B.2)with\nuy\nε(0) = y. Treating the first term enforces our choice of ρ. We apply Proposition B.21 with ρ=ρ(θ)>0\nandK > 0as obtained therein, to see that for any ε>0,\n(I) =/integraldisplay\nBcρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ/parenrightig\nµε(dy)≤µε(Bc\nρ)≤Kexp/parenleftbigg\n−θ\nε2/parenrightbigg\n(4.33)\nConcerning the second term, we apply Proposition 4.7 to see that there exists an ¯n:=¯n(ρ, δ, θ )∈N\nandε1(γ,δ, θ ):=ε1>0such that for all ε∈(0,ε1)andt>0,\n(II) =/integraldisplay\nBρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ,¯uy\nε∈Eρ,δ,θ(¯n)/parenrightig\nµε(dy)\n≤sup\ny∈¯BρP/parenleftig\n¯uy\nε∈Eρ,δ,θ(¯n)/parenrightig\n≤exp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\n.(4.34)\nFinally, the third term is treated using Proposition 4.9 to see that for ¯n:=¯n(ρ, δ, θ )∈Nas above, there\nexists at:=t(¯n)>0(see also Remark 4.5) and ε2(t,¯n,δ, θ,γ ):=ε2>0such that for all ε∈(0,ε2),\n(III) =/integraldisplay\nBρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ,¯uy\nε∈Eρ,δ,θ(¯n)c/parenrightig\nµε(dy)\n≤sup\ny∈¯BρP/parenleftig\ndistCα(¯uy\nε(t),V[θ])≥δ,¯uy\nε∈Eρ,δ,θ(¯n)c/parenrightig\n≤¯nexp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\n.(4.35)\nCombining (4.33),(4.34)and(4.35)(along with the requisite choices of ρ, t> 0) we see that for all\nε∈(0,ε1∧ε2∧1)it holds that\nµε/parenleftbig\nz∈Cα(T3):distCα(z,V[θ])≥δ/parenrightbig\n≤exp/parenleftbigg\n−θ\nε2/parenrightbigg\n+ (1 + ¯n) exp/parenleftbigg\n−θ−γ/2\nε2/parenrightbigg\n.\nThis concludes the proof upon restricting ε∈(0,ε0)withε0<ε1∧ε2∧1sufficiently small and making\nuse of the identity V= 2Sproved by Theorem 2.1. □\nA The Skeleton Equation\nWe collect some results concerning well-posedness and controllability of the skeleton equation associated\nto (1.2),/braceleftigg\n∂twh,z−∆wh,z=−(wh,z)3−m2wh,z+h,\nwh,z|t=0=z,(A.1)\nwhereh∈L2\nTL2\nxis an element of the Cameron–Martin space of the white noise. Many (but not all)\nresults contained in the appendix can surely be found in disparate, alternative sources. However, we\nfind it beneficial to collect them here together along with (at least) sketched proofs of these facts.\nA.1 Well-Posedness and A Priori Bounds\nWe begin with the following local well-posedness result for (A.1).\n34Proposition A.1 (Local Well-Posedness of the Skeleton Equation) Leth∈L2(R+×T3),α∈\n(−2\n3,−1\n2)andz∈Cα(T3). Then, there exists a ¯T:=¯T(α,∥z∥Cα,∥h∥L2\nTL2x)∈R∪{+∞}such that a\nunique, mild solution, wh,z, to(A.1)exists on [0,¯T). In addition,\n•it either holds that\n¯T=∞or lim\nt↗¯T∥wh,z(t)∥Cα=∞, (A.2)\n•forp:=3\n1+α∈(6,9)there exists a weight, η:=η(α)∈(−5α\n12,−α\n3)an exponent δ:=δ(η,α)>0\nand a finite constant C0:=C0(η,α)>0such that\nsup\nt∈(0,¯T)(t∧1)η∥wh,z(t)∥Lp\nx≤C0(T∧1)δ(∥z∥Cα+∥h∥L2\n¯TL2x), (A.3)\n•forη∈/parenleftbig\n−5α\n12,−α\n3/parenrightbig\nas above, there exists a constant C1:=C1(α,η)andδ:=δ(α,η)>0such that\nsup\nt∈(0,¯T)(t∧1)1+η∥wh,z(t)∥H1x≤C1(T∧1)δ(∥z∥Cα+∥h∥L2\n¯TL2\nx). (A.4)\n•ifz∈H1(T3)then in place of (A.3)and(A.4)one has\nsup\nt∈[0,¯T)∥wh,z(t)∥H1x≲/parenleftig\n∥z∥H1x+∥h∥L2\n¯TL2x/parenrightig\n. (A.5)\n♤\nProof.We only treat the case z∈Cα\nxin detail, since the estimate (A.5)is easily obtained by similar\narguments. For T >0andp=3\n1+αas above define the set\nBT:=/braceleftig\nw∈CTCα(T3):∥w∥Cη;TLp\nx≤1/bracerightig\n,\nalong with the map Ψdefined such that for all w∈CTLp(T3)andt∈[0,T],\nΨ(w)(t):=et(∆−m2)z−/integraldisplayt\n0e(t−s)(∆−m2)w(s)3ds+/integraldisplayt\n0e(t−s)(∆−m2)h(s) ds. (A.6)\nWe first show that there exists some T∗∈(0,1)such that Ψmaps BT∗to itself. Applying the Ornstein–\nUhlenbeck semi-group estimate (1.22)where appropriate along with the embedding (1.16), for any\nw∈BTandt∈(0,T], we find,\n∥Ψ(w)(t)∥Lp\nx≤∥et(∆−m2)z∥Lp\nx+/integraldisplayt\n0∥e(t−s)(∆−m2)w(s)3∥Lp\nxds+/integraldisplayt\n0∥e(t−s)(∆−m2)h(s)∥Lp\nxds\n≲αt5α\n12∥z∥B5α/6\np,1+/integraldisplayt\n0(t−s)−3\n2/parenleftbig\n3\np−1\np/parenrightbig\n∥w(s)3∥Lp/3\nxds+/integraldisplayt\n0(t−s)−3\n2/parenleftbig\n1\n2−1\np/parenrightbig\n∥h(s)∥L2ds\n≤t5α\n12∥z∥Cα+/integraldisplayt\n0(t−s)−3\np∥w(s)∥3\nLp\nxds+/parenleftigg/integraldisplayt\n0(t−s)−3p−6\n2pds/parenrightigg1\n2\n∥h∥L2\ntL2x\n≤t5α\n12∥z∥Cα+∥w∥3\nCη0;tLp\nx/integraldisplayt\n0(t−s)−3\nps−3η0ds+∥h∥L2\ntL2xt1\n2−3p−6\n4p.\nTherefore, for any T >0, withη0as above,\n∥Ψ(w)∥Cη0;TLp\nx≲αTη0+5α\n12∥z∥Cα+T1−3\np−2η0∥w∥Cη0;TLp\nx+T1\n2−3p−6\n4p+η0∥h∥L2\nTL2x.(A.7)\n35Note that due to the restrictions placed on α, pandη0all the exponents in (A.7)are positive. Hence,\nnoting that continuity of t∝⇕⊣√∫⊔≀→∥Ψ(w)(t)∥Lp\nxcan be easily shown, there exists a T∗∈(0,1)such that\nΨ(w)∈BT∗. In order to show that Ψis a contraction on BT∗∗, forT∗∗∈(0,T∗], letw̸=v∈BT, so\nthat for any t∈(0,T],\n∥Ψ(w(t)−Ψ(v)(t)∥Lp≤/integraldisplayt\n0∥e(t−s)(∆+m2)(w(s)3−v(s)3)∥Lp\nxds\n≲p/integraldisplayt\n0(t−s)−3\n2/parenleftbig\n3\np−1\np/parenrightbig\n∥w(s)3−v(s)3∥Lp/3\nxds\n≲/integraldisplayt\n0(t−s)−3\np/parenleftig\n∥w(s)∥2\nLp\nx∨∥v(s)∥2\nLp\nx/parenrightig\n∥w(s)−v(s)∥Lp\nxds\n≤/parenleftig\n∥w∥2\nCη0;TLp\nx∨∥v∥2\nCη0;TLp\nx/parenrightig\n∥u−v∥Cη0;TLp\nx/integraldisplayt\n0(t−s)−3\nps−3η0ds\n≤∥w−v∥Cη0;TLp\nxT1−3\np−3η0,\nwhere the last inequality follows from the assumptions on pandη0as well as the fact that w, v∈BT.\nTherefore, choosing T∗∗∈[0,T∗]sufficiently small, we see that Ψis a contraction mapping from BT∗∗to\nitself; hence, by Banach’s fixed point theorem, a unique fixed point, wh,z, exists in BT∗∗. Uniqueness\nin all ofCη;T∗∗Lp(T3)can be shown by similar steps, see for example the last steps in the proof\nof [CHM22, Thm. 3.4].\nTo show that we may extend the solution beyond T∗∗we use the semi-group property of the solution,\ni.e. fort0∈[0,T∗∗]and anyt∈(0,¯T−t0), one has,\nwh,z(t0+t) =et(∆−m2)wh,z(t0)−/integraldisplayt0+t\nt0e(t0+t−s)(∆−m2)wh,z(s)3ds+/integraldisplayt0+t\nt0e(t0+t−s)(∆−m2)h(s) ds,\nIt is then easy to see that we may restart the equation, obtaining local existence and uniqueness of\nsolutions in a similar manner as above, for as long as ∥wh,z(t)∥Cα<∞, proving (A.2)\nIn order to obtain the growth estimates (A.3)and(A.4), letT∈[0,T∗∗]and we first return to (A.7),\nnow applied to the fixed point found above, on the interval [0,T∗∗). Using the fact that T∗∗<1, we find\nthat there exists a δ:=δ(α,η0)>0such that for any T∈(0,T∗∗],\n(1−T1−3\np−2η0)∥wh,z∥Cη0;TLp\nxlesssimδ\nT∗∗/parenleftig\n∥z∥Cα+∥h∥L2\nT∗∗L2x/parenrightig\n,\nfrom which it follows that for any T∈[0,T∗∗/2)there exists a C:=C(η0,α)such that,\n∥wh,z∥Cη0;TLp\nx≲Tδ\n∗∗/parenleftig\n∥z∥Cα+∥h∥L2\nTL2x/parenrightig\n.\nThe extension to all T∈(0,¯T)follows from the same re-starting procedure as above, repeating the same\nargument on intervals of length 1/2or less; this concludes the proof of (A.3).\nTo obtain (A.4), we first pass to a direct Fourier representation in the additive final term of (A.6)\nand show that it enjoys H1(T3)regularity, continuously in time. For any t∈(0,¯T),\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e(t−s)(∆−m2)h(s) ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH1x=/summationdisplay\nk∈Z3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftig\n1 +|k|2/parenrightig1\n2/integraldisplayt\n0e−(t−s)(|k|2+m2)⟨h(s),ek⟩ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤/summationdisplay\nk∈Z3/parenleftig\n1 +|k|2/parenrightig/integraldisplayt\n0e−2(t−s)(|k|2+m2)ds/integraldisplayt\n0|⟨h(s),ek⟩|2ds\n36=/summationdisplay\nk∈Z3/parenleftbig\n1 +|k|2/parenrightbig\n2/parenleftbig\nm2+|k|2/parenrightbig/parenleftig\n1−e−2t(|k|2+m2)/parenrightig/integraldisplayt\n0|⟨h(s),ek⟩|2ds\n≲¯T,m/summationdisplay\nk∈Z3/integraldisplayt\n0|⟨h(s),ek⟩|2ds=∥h∥2\nL2\nTL2x. (A.8)\nContinuity in time of the left-hand side is easily shown. Then, again applying (1.22), along with the\nembedding Lp(T3)◁arrowhookleft→L6(T3)◁arrowhookleft→L2(T3)and the estimate (A.8), for any t0∈(0,T∗∗/2)andt∈(0,t0]it\nholds that\n∥wh,z(t0+t)∥H1x≤∥et(∆−m2)u(t0)∥H1x+/integraldisplayt0+t\nt0∥e(t+t0−s)(∆−m2)wh,z(s)3∥H1xds\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt0+t\nt0e(t+t0−s)(∆−m2)h(s) ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH1x\n≲t−1\n2∥wh,z(t0)∥L2x+/integraldisplayt0+t\nt0(t+t0−s)−1\n2∥wh,z(s)3∥L2xds\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt0+t\nt0e(t+t0−s)(∆−m2)h(s) ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH1x\n≲¯T,mt−1\n2∥wh,z(t0)∥Lp\nx+t1\n2t−3η0\n0∥wh,z∥3\nCη0;T∗∗Lp\nx+∥h∥L2\n¯TL2x.\nSince∥wh,z∥Cη0;T∗∗Lp\nx≤1, multiplying both sides by tη0\n0tand taking suprema we find,\nsup\nt0∈(0,T∗∗/2)sup\nt∈(0,t0]tη0\n0t∥wh,z(t0+t)∥H1x≲¯T,m/parenleftbigg\nT1\n2∗∗+T3\n2−2η0\n∗∗/parenrightbigg\n∥wh,z∥Cη0;T∗∗+∥h∥L2\n¯TL2x\nTherefore applying (A.3)and consolidating the relevant estimates, there exists a δ:=δ(η0)>0such\nthat\nsup\nt∈(0,T∗∗/2)t1+η0∥wh,z(t)∥H1\nx≲Tδ\n∗∗/parenleftig\n∥z∥Cα+∥h∥L2\n¯TL2x/parenrightig\n.\nIterating the same arguments on successive time intervals as previously, we then obtain,\nsup\nt∈(0,¯T)(t∧1)1+η0∥wh,z(t)∥H1x≲(¯T∧1)δ/parenleftig\n∥z∥Cβ+∥h∥L2\n¯TL2x/parenrightig\n(A.9)\nwhich concludes the proof of (A.4). □\nThe following proposition establishes an a priori estimate on the growth of certain Lp(T3)norms\nof weak solutions to (A.1); in combination with the maximal time of existence of (A.2)and standard\nBesov embeddings, we will establish global well-posedness of (A.1), this argument will be concluded in\nTheorem A.6 below.\nProposition A.2 LetT∈(0,∞]andwbe a mild solution to (A.1)on[0,T]. Then, let t0∈[0,T)be\nsuch thatw(t0)∈Lp(T3andw∈C((t0,T);H1(T3))and for any p>1andt∈(0,T−t0]it holds that\n1\np∥w(t0+t)∥p\nLp\nx=1\np∥w(t0)∥p\nLp\nx−2(p−1)\np/integraldisplayt0+t\nt0∥∇(w(s)p/2)∥L2xds\n−/integraldisplayt0+t\nt0⟨w(s)3+m2w(s),w(s)p−1⟩ds+/integraldisplayt0+t\nt0⟨h(s),w(s)p−1⟩ds.(A.10)\n37Then, forp≥1there exists a constant C:=C(p,m)>0and an exponent k(p)∈(1,2]such that for all\nt∈(0,T−t0],\n∥w(t0+t)∥p\nLp\nx≲Ce−m2\n2t∥w(t0)∥p\nLp\nx+1\nm2−k(p)/parenleftbigg\n1−e−m2\n2−k(p)t/parenrightbigg2−k(p)\n2\n∥h∥k(p)\nL2\n[t0,t0+t]L2x.(A.11)\n♤\nProof.Choosingp≥2to be even in (A.10) we first find the identity,\n1\np∥w(t0+t)∥p\nLp\nx=1\np∥w(t0)∥p\nLp\nx−2(p−1)\np/integraldisplayt0+t\nt0∥∇(w(s)p/2)∥L2xds−/integraldisplayt0+t\nt0∥w(s)p+2∥L1xds\n−m2/integraldisplayt0+t\nt0∥w(s)p∥L1xds+/integraldisplayt0+t\nt0⟨h(s),w(s)p−1⟩ds.(A.12)\nThe only term to be estimated is the last one which has no given sign. Applying Hölder’s inequality\ndirectly gives,\n|⟨h(s),w(s)p−1⟩|≤∥h(s)∥L2x∥w(s)p−1∥L2x. (A.13)\nWe recall the Sobolev inequality for non-mean free functions f:T3→R, see for example [BO13, Cor. 2.2],\nk\n3≥1\nr−1\nqthen∥f∥Lq\nx≲∥f∥Wk,r\nx. (A.14)\nChoosingk= 1andr= 2, we may apply (A.14)tow(s)p/2withq=4p−4\npand using the equivalence of\nfinite dimensional norms we obtain\n∥w(s)p−1∥L2x=∥w(s)p/2∥p\n8p−8\nL(4p−4)/p\nx≲∥w(s)p/2∥p\n8p−8\nH1x≲∥w(s)p/2∥p\n8p−8\nL2x+∥∇(w(s)p/2)∥p\n8p−8\nL2x\n=∥w(s)p∥p\n16p−16\nL1x+∥∇(w(s)p/2)∥p\n8p−8\nL2x.(A.15)\nCombining (A.13) with (A.15) and the Cauchy–Schwarz inequality gives,\n|⟨h(s),w(s)p−1⟩|≲∥h(s)∥L2x/parenleftbigg\n∥w(s)p∥p\n16p−16\nL1\nx+∥∇(w(s)p/2)∥p\n8p−8\nL2\nx/parenrightbigg\n.\nApplying Young’s product inequality (in the form ab≤1\nθ1εaθ1+ε\nθ2bθ2) and noting that for p≥2one has\np\n16p−16∨p\n4p−4≤2,\nwe see that there exist constants k:=k(p)∈(1,2]andC:=C(p,m)>0such that,\n|⟨h(s),w(s)p−1⟩|≲C\np∥h(s)∥k(p)\nL2x+(p−1)m2\np∥w(s)p∥L1x+p−1\np∥∇(w(s)p/2)∥L2x.\nIntroducing this bound into (A.12) shows that,\n∥w(t0+t)∥p\nLp\nx≤∥w(t0)∥p\nLp\nx−(p−1)/integraldisplayt0+t\nt0∥∇(w(s)p/2)∥L2xds−p/integraldisplayt0+t\nt0∥w(s)p+2∥L1xds\n−m2/integraldisplayt0+t\nt0∥w(s)p∥L1xds+/integraldisplayt0+t\nt0∥h(s)∥k(p)\nL2xds.(A.16)\nSimply bounding the second and third negative terms on the right hand side by zero, we find a comparison\n38with the ODE, written in suggestive notation,\nd\ndtx(t0+t) =−m2\n2x(t) +h(t)k, x(t0+t)/vextendsingle/vextendsingle\nt=0=x(t0),\nwhich has the explicit solution,\nx(t0+t) =e−m2\n2tx(t0) +/integraldisplayt0+t\nt0e−m2\n2(t0+t−s)h(s)kds.\nThus, by a suitable comparison, we obtain the chain of estimates,\n∥w(t0+t)∥p\nLp\nx≤e−m2\n2t∥w(t0)∥p\nLp\nx+/integraldisplayt0+t\nt0e−m2\n2(t0+t−s)∥h(s)∥k(p)\nL2xds\n≤e−m2\n2t∥w(t0)∥p\nLp\nx+/parenleftigg/integraldisplayt0+t\nt0e−m2\n2−k(p)(t0+t−s)ds/parenrightigg2−k(p)\n2/parenleftigg/integraldisplayt0+t\nt0∥h(s)∥2\nL2xds/parenrightiggk(p)\n2\n=e−m2\n2t∥w(t0)∥p\nLp\nx+/parenleftigg\n2−k(p)\nm2/parenleftbigg\n1−e−m2\n2−k(p)t/parenrightbigg/parenrightigg2−k(p)\n2\n∥h∥k(p)\nL2\n[t0,t0+t]L2x\n≲e−m2\n2t∥w(t0)∥p\nLp\nx+1\nm2−k(p)/parenleftbigg\n1−e−m2\n2−k(p)t/parenrightbigg2−k(p)\n2\n∥h∥k(p)\nL2\n[t0,t0+t]L2x\nwe obtain the claimed estimate (A.11)for all even integers p≥2. The estimate for any p≥2follows\nfrom the ordering of the Lp(T3)spaces. □\nRemark A.3 Instead of ignoring the effect of the negative non-linear term in the above proof, one could\ninstead apply Jensen’s inequality to obtain,\n∥w∥p\nLp\nx≲∥wp+2∥p\np+2\nL1x\nwhich combined with (A.16)and formally differentiating in time, gives,\nd\ndt∥w(t)∥p\nLp\nx+c1/parenleftig\n∥w(t)∥p\nLp\nx/parenrightigp+2\np≲∥h(t)∥k(p)\nL2x,for allt∈(t0,T),\nfor the same k(p)∈(1,2]as above. Then, applying [TW18, Lem. 3.8] (see also [MW17b, Lem. 7.3]) one\nfinds a stronger bound, which is independent of the initial data, of the following form\nsup\nt∈(t0,T)∥w(t)∥Lp≲pmax/braceleftbigg1√t0,∥h∥k(p)\np+2\nL2\n[t0,T]L2x/bracerightbigg\n.\nIt follows in particular, that for any p≥2andα∈(−2\n3,−1\n2)as in Proposition A.1 one has\nsup\nz∈Cα(T3)sup\nt∈(t0,T)∥w(t)∥Lp≲pmax/braceleftbigg1√t0,∥h∥k(p)\np+2\nL2\n[t0,T]L2x/bracerightbigg\n. (A.17)\nSince(A.17)is independent of the initial data it is strictly stronger than the estimate derived in\nProposition A.2. However, the proof of (A.17)relies on the strongly damping cubic term in the Φ4\nequation and since many other quantum field theories to which this method may conceivably be applied\n(e.g. sine-Gordon and Yang–Mills theories) do not contain an analogous term we only make use of the\nweaker estimate (A.11). This estimate is both sufficient for our purposes and likely to be replicable for\n39other suitable theories. One advantage to note of the strategy to prove (A.17)is that, unlike the proof of\nProposition A.2, it does not require a positive mass m2>0in the equation. ♤\nCorollary A.4 Letz∈Cα(T3),h∈L2(R+×T3)andwh,z:[0,∞)×T3→Rbe a weak solution to\n(A.1)such that (A.10)holds for some p>1. Then, for the same p≥1one has\nlim\nt→∞∥wh,z(t)∥Lp\nx= 0. ♤\nProof.The claim follows directly from the a priori estimate (A.3). Sincewh,zis a global weak solution,\nwe may take the joint limit in (A.10) to give,\nlim\nt→∞∥wh,z(t)∥p\nLp\nx= lim\nt0→∞lim\nt→∞∥wh,z(t)∥p\nLp\nx=1\nm2−k(p)lim\nt0→∞∥h∥k(p)\nL2\n[t0,∞]L2x= 0. □\nThe following a priori lemma demonstrates that decay to zero at t= +∞of a sufficiently high Lp(T3)\nnorm of the solution implies decay in the H1(T3)norm. Note that the proof also makes use of the\nassumed positive mass m2>0.\nLemma A.5 Letz∈Cα(T3),h∈L2(R+×T3),α∈/parenleftbig\n−2\n3,−1\n2/parenrightbig\n,q:=−1\nα∈/parenleftbig3\n2,2/parenrightbig\n, andwh,z:R+×T3→\nRbe a mild solution to (A.1)onR+×T3such that limt→∞∥wh,z(t)∥L3q\nx= 0. Then it also holds that,\nlim\nt→∞∥wh,z(t)∥H1x= 0. (A.18)\n♤\nProof.Recall that for any t0≥0andt>0, by the semi-group property of the solution, we have\nwh,z(t0+t) =et(∆−m2)wh,z(t0)−/integraldisplayt0+t\nt0e(t0+t−s)(∆−m2)wh,z(s)3ds+/integraldisplayt0+t\nt0e(t0+t−s)(∆−m2)h(s) ds.\nHence, applying (1.22) along with (A.8), we obtain the chain of estimates\n∥wh,z(t0+t)∥H1x≤e−tm2∥et∆wh,z(t0)∥H1x\n+/integraldisplayt0+t\nt0∥e(t0+t−s)(∆−m2)wh,z(s)3∥H1xds+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt0+t\nt0e(t0+t−s)(∆−m2)h(s) ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH1x\n≲e−tm2(t∧1)−3−q\n4q∥wh,z(t0)∥L3q\nx\n+e−(t0+t)m2/integraldisplayt0+t\nt0esm2((t0+t−s)∧1)−3−q\n4q∥wh,z(s)∥L3q\nxds+∥h∥L2\n[t0,t0+t]L2x\n≲e−tm2(t∧1)−3−q\n4q∥wh,z(t0)∥L3q\nx+e−tm2t1−3−q\n4q sup\ns∈[t0,t0+t]∥wh,z(s)∥L3q\nx+∥h∥L2\n[t0,+∞)L2x.\nTherefore, since for any fixed t>0one has\nlim\ns→+∞∥wh,z(s)∥H1x= lim\nt0→+∞∥wh,z(t0+t)∥H1x,\nusing the assumption that limt→∞∥wh,z(t)∥L3q\nx= 0, it holds that\nlim\ns→∞∥wh,z(s)∥H1x≲lim\nt0→+∞∥h∥L2\n[t0,+∞)L2x= 0. □\n40Theorem A.6 (Global Well-Posedness of the Skeleton Equation) Letα∈(−2\n3,−1\n2),z∈Cα(T3)\nandh∈L2(R+×T3). Then, there exists a unique solution wh,z:(0,∞]→H1(T3)to(A.1). Furthermore,\n(i) for any T >0andt0∈(0,T](where if z∈H1(T3)we may allow t0= 0)\n∥wh,z∥H1\n[t0,T]H−1\nx+∥wh,z∥C[t0,T]H1x<∞. (A.19)\n(ii) for any p≥2, it holds that\nlim\nt→∞∥wh,z(t)∥Cα= lim\nt→∞∥wh,z(t)∥Lp\nx= lim\nt→∞∥wh,z(t)∥H1x= 0, (A.20)\n(iii)for anyp≥2, there exists an η∈/parenleftbig\n−5α\n12,−α\n3/parenrightbig\nsuch that for all z∈Cα(T3),h∈L2(R+×T3)and\nT∈(0,∞),\nsup\nt∈(0,T)(t∧1)η∥wh,z(t)∥Lp\nx≲m,p,α∥z∥Cα+∥h∥L2\nTL2x. (A.21)\nwhile, if z=∈H1(T3), we have\nsup\nt∈[0,T)∥wh,z(t)∥Lp\nx≲m,p∥z∥H1x+∥h∥2\nL2\nTL2x. (A.22)\n♤\nProof.Firstly,applyingLemmaA.1weseethatthereexistsaunique,mildsolution wh,z∈C((0,¯T];H1(T3))\nfor some ¯T∈(0,1). Furthermore, this solution can be extended so long as ∥wh,z(t)∥Cα<∞. Since\none haswh,z(t0)∈H1(T3)for allt0∈(0,¯T], one may argue using uniqueness of the mild solution\nin this regularity class and construction of weak solutions via the variational method (see for exam-\nple [PR07,Par21]) to see that t∝⇕⊣√∫⊔≀→wh,z(t)satisfies(A.10)on any interval [t0,¯T]witht0∈(0,¯T). As a\nresult, we may apply Proposition A.2 for p≥2sufficiently large we see that\nsup\nt∈[t0,¯T]∥wh,z(t)∥Cα<∞.\nSo repeating the same arguments, we may extend the solution indefinitely. Uniqueness follows from\nsimilar arguments as in the proof of Proposition A.1 and the estimate (A.19), forz∈Cα(T3)\\H1(T3)\nfollows directly. To see that we may take t0= 0in the case z∈H1(T3)we make use of (A.5).\nTo show the decay of solutions as in (A.20)it suffices to combine Corollary A.4, Lemma A.5 along\nwith the embedding Lp(T3)◁arrowhookleft→Cα(T3)forp>1sufficiently large.\nThe estimates (A.21)and(A.22)follow from their counterparts given by Proposition A.1 but now\nextended to any T∈(0,∞)using the fact that supt≥0∥wh,z(t)∥Cα<∞, as shown in the proceeding\nparagraph. □\nA.2 Global Controllability\nIn this section, we study controllability of the solution wh,0to(A.1)with 0initial data. We first consider\nthe linearized, skeleton equation, we consider the solution to\n/braceleftigg\n∂tzh−∆zh=−m2zh+h,\nzh|t=0= 0.(A.23)\nforh∈L2\nTL2(T3).\n41Lemma A.7 LetT > 0,m2>0,h∈L2\nTL2andzhsolve(A.23). Then there exists a constant\nC:=C(T,m)>0such that\n∥zh∥CTH1x∨ ∥zh∥L2\nTH2x≤C∥h∥L2\nTL2x. (A.24)\n♤\nProof.The proof is straightforward and standard so we only present a sketch. Firstly, for any t∈[0,T]\nandα∈[0,1], using the Fourier representation of the heat semi-group, one directly has\n∥zh(t)∥2\nHαx=/summationdisplay\nk∈Z3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftig\n1 +|k|2/parenrightigα\n2/integraldisplayt\n0e−(t−s)(|k|2+m2)⟨h(s),ek⟩ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤/summationdisplay\nk∈Z3/parenleftig\n1 +|k|2/parenrightigα/integraldisplayt\n0e−2(t−s)(|k|2+m2)ds/integraldisplayt\n0|⟨h(s),ek⟩|2ds\n=/summationdisplay\nk∈Z3/parenleftbig\n1 +|k|2/parenrightbigα\n2/parenleftbig\nm2+|k|2/parenrightbig/parenleftig\n1−e−2t(|k|2+m2)/parenrightig/integraldisplayt\n0|⟨h(s),ek⟩|2ds\n≲T,m∥h∥2\nL2\nTL2x\nHence, taking α= 1giveszh∈L∞\nTH1\nx. However, it is readily checked that t∝⇕⊣√∫⊔≀→∥zt∥H1xis continuous\nwhich proves the first estimate of (A.24). To prove the second, take α= 2, apply Young’s convolution\ninequality and arguing a posteriori that one may exchange summation and integration, one finds\n∥zh∥2\nL2\nTH2x=/integraldisplayT\n0∥zh(t)∥2\nH2xdt=/integraldisplayT\n0/summationdisplay\nk∈N3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftig\n1 +|k|2/parenrightig/integraldisplayt\n0e−(t−s)(|k|2+m2)⟨h(s),ek⟩ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt\n=/summationdisplay\nk∈N3/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftig\n1 +|k|2/parenrightig\ne−(·)(|k|2+m2)∗⟨h(·),ek⟩/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\nT\n≤/summationdisplay\nk∈N3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/parenleftig\n1 +|k|2/parenrightig\ne−(t−s)(|k|2+m2)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/vextenddouble/vextenddouble⟨h(·),ek⟩/vextenddouble/vextenddouble2\nL2\nT\n=/summationdisplay\nk∈N3/parenleftig\n1−e−t(|k|2+m2)/parenrightig2/vextenddouble/vextenddouble⟨h(·),ek⟩/vextenddouble/vextenddouble2\nL2\nT\n≲T,m∥h∥2\nL2\nTL2x. □\nTo show controllability of the non-linear equation, we require some finer properties of the linear solution\nmap.\nLemma A.8 ForT >0andm2>0, defining the linear solution map LT:L2\nTL2(T3)→H1(T3), given\nbyLTh:=zh(T), defines a linear, bounded, compact and surjective operator for which there exists a\nC:=C(T,m)>0such that\ninf\n∥Ψ∥H−1\nx\\{0}∥L∗\nTΨ∥L2\nTL2x\n∥Ψ∥H−1\nx≥C > 0. (A.25)\n♤\nProof.Linearity and boundedness from L2\nTL2(T3)→H1(T3)follows directly from Lemma A.7. To\ndemonstrate compactness, it suffices to argue that since zh∈L2\nTH2(T3), up to possibly redefining the\nimage on a set of measure zero, we in fact have LTh∈H2(T3)◁arrowhookleft→ →H1(T3). Thus it only remains to\ncheck(A.25), which by the closed range theorem (c.f. [Rud91, Thm. 4.13]) implies surjectivity of the\n42map. Since the operator (∆−m2)is self-adjoint we see that\n(L∗\nTΨ)(s) =e(T−s)(∆−m2)∗Ψ =e(T−s)(∆−m2)Ψ.\nRecalling that we write (ek)k∈Z3for the Fourier basis of L2(T3)we set Ψk:=⟨Ψ,ek⟩H−1\nx×H1xso that, for\nsome ˜C:=˜C(T)>0, we find\n∥L∗\nTΨ∥2\nL2\nTL2\nx=/integraldisplayT\n0∥e(T−s)(∆−m2)Ψ∥2\nL2\nxds=/summationdisplay\nk∈Z3Ψ2\nk/integraldisplayT\n0e−2(T−s)(|k|2+m2)ds\n=1\n2/summationdisplay\nk∈Z3/bracketleftbig\n1−e−2T(|k|2+m2)/bracketrightbig Ψ2\nk\n(|k|2+m2)≥1−e−2Tm2\n2/summationdisplay\nk∈Z3Ψ2\nk\n(|k|2+m2)\n≥1−e−2Tm2\n2m2∥Ψ∥2\nH−1\nx.\nSo that (A.25) holds with\nC:=√\n1−e−2Tm2\n√\n2m. □\nObserve that while the operator LTis surjective onto H1(T3)it is clearly not injective. However, we\nmay define its pseudo-inverse L+\nT, as described by the following lemma.\nLemma A.9 LetT >0andm2>0. Then, there exists a bounded, linear map\nL+:H1(T3)\\{0}→L2\nTL2(T3),\nsuch that\n•for allϕ∈H1(T3)\\{0}\nLTL+\nTϕ=ϕ,\n•given anyϕ∈H1(T3)\\{0}, for allg∈L2\nTL2(T3)such that LTg=ϕit holds that\n⟨L+\nTϕ−g,L+\nTϕ⟩L2\nTL2x= 0.\nFurthermore, recalling the constant C:=C(T,m)>0from(A.25), one has\nsup\nϕ∈H1x\\{0}∥L+\nTϕ∥L2\nTL2x\n∥ϕ∥H1x≤1\nC(A.26)\n♤\nProof.First note that since in general, one may also define the pseudo-inverse of an operator as selecting\nthe pre-image of any element of smallest norm, the existence of a pseudo-inverse fails if and only if the\noperator itself does not have closed range. In our case, since we have shown LT:L2\nTL2(T3)→H1(T3)\nto be surjective, its range is trivially closed. To obtain the estimate, (A.26), one uses the fact that LT\nis a compact operator to approximate it by finite rank operators and the lower bound (A.25)to see\nthat this approximations can be chosen to have a uniformly positive, smallest eigenvalue. Using this\napproximation, one also derives the identity,\nL+\nT= lim\nι→0/parenleftig\nLTL∗\nT+ιIH1x/parenrightig−1\nL∗\nT, (A.27)\nand sees that (A.26)holds uniformly along the approximating sequence. Since the sequence converges in\nthe strong topology, we obtain (A.26) in the limit. □\n43The non-linear equation. We employ the results obtained in Lemmas A.8 & A.9 to obtain local,\nexact, controllability of the non-linear, skeleton dynamics, (A.1). We begin with a straightforward\npreliminary lemma.\nLemma A.10 LetT >0,ϕ∈H1(T3)andL+\nTϕbe the pseudo-inverse of LTas defined in Lemma A.9.\nThen, forwL+\nTϕ,0solving,\n/braceleftigg\n∂twL+\nTϕ,0−∆wL+\nT,0=−(wL+\nT,0)3−m2wL+\nT,0+L+\nTϕ,\nwL+\nT,0|t=0= 0,(A.28)\nit holds that\n∥wL+\nTϕ,0∥CTL6x≲∥ϕ∥H1x. ♤\nProof.This is a direct consequence of Theorem A.6 and Lemma A.9, in particular the inequalities (A.22)\nand (A.26). □\nWe are now ready to prove global, exact controllability of the skeleton equation.\nProposition A.11 (Global Exact Controllability of wh,0)For anyT >0andϕ∈H1(T3)there\nexists anh(ϕ)∈L2\nTL2(T3)such thatwh,0(T) =ϕ, wherewh,0solves(A.1)withz= 0. ♤\nProof.ForT >0, using item (i) of Theorem A.6 we define the family of maps\nH1(T3)∋ϕ∝⇕⊣√∫⊔≀→fT(ϕ):=wL+\nTϕ,0(t)∈H1(T3),fort∈[0,T], (A.29)\nwherewL+\nTϕ,0solves(A.28); note that here we use the fact that L+\nTis a bounded, linear operator. We\nwill apply the global, inverse function theorem to show that there exists a map\nH1(T3)∋ϕ∝⇕⊣√∫⊔≀→f−1\nT(ϕ)∈H1(T3),\nsuch that choosing h(ϕ):=L+\nTf−1\nT(ϕ)we have\nwh(ϕ)(T) =ϕ\nas required. A local version of the inverse function theorem can be found as [Zei95, Thm. 4.F]. A\nglobal version is given as [Zei95, Prob. 4.12] (see also [Zei95, Prob. 4.13]). We are required to show first\nfT:H1(T3)→H1(T3)is aC1-Frechét map and then that\n(i)DϕfT(ϕ):=DϕwL+\nTϕ[·](T):H1(T3)→H1(T3)is bijective for all ϕ∈H1(T3),\n(ii)fT(ϕ):H1(T3)→H1(T3)isproper(that is ifK⊂H1(T3)is compact, then f−1(K)⊂H1(T3)is\ncompact).\nTo show that fTisC1, first letϕ∈H1(T3),t∈[0,T]and then write\nwL+\nTϕ,0(t) =−/integraldisplayt\n0e(t−s)(∆−m2)/parenleftig\nwL+\nTϕ,0(s)/parenrightig3\nds+zL+\nTϕ(t), (A.30)\nwherezL+\nTϕsolves (A.23) with h=L+\nTϕ. Since the map ϕ∝⇕⊣√∫⊔≀→zL+\nTϕis linear and bounded, we have\nDϕzL+\nTϕ[ψ] =zSTψ,for allψ∈H1(T3).\nThen, itis readily checked, forexample applying the implicitfunction theorem, [AP93, Thm. 2.3], thatthe\nmapϕ→wL+\nTϕis Fŕechet differentiable (see for example the arguments proceeding [TW18, Prop. 5.1]\n44or the proof of [CHM22, Lem. 5.11] for details of a similar argument). Then, applying the standard\nrules of calculus, and equality of Fréchet and Gateaux derivatives in the case that the former exists, one\nfind that for all ψ∈H1(T3), the derivative satisfies, for all t∈[0,T],\nDϕwL+\nTϕ,0[ψ](t) = 3/integraldisplayt\n0e(t−s)(∆−m2)/parenleftig\nwL+\nTϕ,0(s)/parenrightig2\nDϕwL+\nTϕ,0[ψ](s) ds+zSTψ.(A.31)\nSimilar estimates as for the linear and non-linear skeleton equations, (A.23)&(A.1)allow for control on\nDϕwL+\nTϕ[ψ]∈CTH1(T3), hence showing that ϕ∝⇕⊣√∫⊔≀→fT(ϕ) =wL+\nT,0is aC1-Frechét map.\nTo show Point (i), we first observe that by definition zSTψ(T) =ψ, so that we have\nDϕwL+\nTϕ,0[ψ](T) = 3/integraldisplayT\n0e(T−s)(∆−m2)/parenleftig\nwL+\nTϕ,0(s)/parenrightig2\nDϕwL+\nTϕ,0[ψ](s) ds+ψ,(A.32)\nand hence, computing somewhat formally,\nDϕwL+\nTϕ,0[ψ](T) =ψexp/parenleftigg\n3/integraldisplayT\n0e(T−s)(∆−m2)/parenleftig\nwL+\nTϕ,0(s)/parenrightig2\nds/parenrightigg\n.\nFrom this expression it is clear that DϕfT(ϕ)[·] =DϕwL+\nTϕ,0[·](T):H1(T3)→H1(T3)is a global,\nC1-diffeomorphism.\nTo check Point (ii) we appeal to [Zei95, Ex. 3]; to wit, fisproperif\nlim\n∥ϕ∥H1x→∞∥f(ϕ)∥H1x= lim\n∥ϕ∥H1x→∞∥wL+\nTϕ(T)∥H1x= +∞. (A.33)\nand we can write\nfT(ϕ) =f1(ϕ) +f2(ϕ), (A.34)\nwheref1:H1(T3)→H1(T3)is acompact andf2:H1(T3)→H1(T3)is anhomeomorphism .\nTo show the decomposition (A.34)it suffices to use (A.30)and again use the fact that zL+\nTϕ(T) =ϕ\nto see that\nfT(ϕ) =wL+\nTϕ(T) =−/integraldisplayT\n0e(T−s)(∆−m2)/parenleftig\nwL+\nTϕ(s)/parenrightig3\nds+ϕ. (A.35)\nSo that we may set f2(ϕ) =ϕ, which as the identity is clearly a homeomorphism. To show that the\nintegral operator\nH1(T3)∋ϕ∝⇕⊣√∫⊔≀→f1(ϕ):=−/integraldisplayT\n0e(T−s)(∆−m2)/parenleftig\nwL+\nTϕ(s)/parenrightig3\nds∈H1(T3)\nis compact, we note that if ϕ∈H1(T3), then by Theorem A.6, one has s∝⇕⊣√∫⊔≀→wL+\nTϕ(s)∈CTL3q(T3), for\nsomeq∈[1,2). Hence,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayT\n0e(T−s)(∆−m2)/parenleftig\nwL+\nTϕ(s)/parenrightig3\nds/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH1+ε\nx≲/integraldisplayT\n0(t−s)−1+ε\n2−3\n2/parenleftbig\n1\n3q−1\n2/parenrightbig\n∥wL+\nTϕ(s)∥3\nL3q\nxds\n≲1−1+ε\n2−3(2−3q)\n12q\nT sup\ns∈[0,T]∥wL+\nTϕ(s)∥H1x<∞.\nSinceH1+ε(T3)◁arrowhookleft→ →H1(T3)we have shown that the integral operator is a compact map from H1(T3)to\nitself.\n45To show the limit (A.33), sincewL+\nTϕ,0∈CTH1(T3)(see Theorem A.6) by continuity in time it is\nenough to show that\nlim\n∥ϕ∥H1x→∞sup\nt∈[0,T]∥wL+\nTϕ(t)∥H1\nx=∞.\nTherefore, assume for a contradiction, that there exists a sequence ϕnand aC > 0such that\n∥ϕn∥H1x→∞and sup\nt∈[0,T]∥wL+\nTϕn,0(t)∥H1x≤C <∞.\nSimply rearranging (A.35) and applying similar estimates as above with q= 1andε= 0\n∥ϕn∥H1=/vextenddouble/vextenddouble/vextenddouble/vextenddoublewL+\nTϕn,0(T) +/integraldisplayT\n0e(t−s)(∆−m2)(wL+\nTϕn,0(s))3ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH1x\n≤∥wL+\nTϕn,0(T)∥H1x+T3\n4/parenleftigg\nsup\ns∈[0,T]∥wL+\nTϕn,0(s)∥3\nH1x/parenrightigg\n≤C+T3\n4C3<∞.\nSince the right hand side is uniform in nthis contradicts our assumption, showing that the limit (A.33)\nholds and the proof is complete. □\nB The Dynamical Φ4\n3Equation on the Torus\nIn this appendix, we collect various properties of the Φ4\n3equation on the 3D torus. We begin with a recap\nof the local and global well-posedness theory for the Φ4\n3dynamics, as given by the theory of regularity\nstructures, c.f. Section B.1. As discussed in the introduction, the appeal to regularity structures is\nessentially incidental. However, from our perspective the theory presents two advantages. Firstly, prior\nwork has already obtained a large deviation principle for the dynamic Φ4\n3model using the theory of\nregularity structures, [HW15]. Secondly, the theory of regularity structures is largely systematised,\ngiving a pathwise, local well-posedness for a large range of models at once, opening up the possibility of\napproaches similar to ours for many, interesting EQFT models.\nIn Section B.2, we extend the main result of [HW15] to obtain a locally uniform large deviation\nprinciple, leveraging continuity of the solution map and the large deviation principle obtained on model\nspace therein. The locally uniform LDP is required in the proof of our main result, see the proofs of\nTheorem 3.1, Proposition 4.7 and Proposition 4.9. Finally, in Section B.3, we recap results on ergodicity\nfor the Φ4\n3dynamics, identification of the unique invariant measure with the Φ4\n3EQFT measure and\nverification of the appropriate Osterwalder–Schrader axioms. In that section we also present suitable\ntail bounds on the invariant measure µεwhich are required in our main proofs.\nB.1 Local and Global Solution Theory for the Φ4\n3Equation\nWe first focus on the localsolution theory of the periodic Φ4\n3dynamics, presented in the language of\nregularity structures. We recall that the dynamic equation with smooth noise reads,\n∂tuz\nε,κ−∆uz\nε,κ=−(uz\nε,κ)3−m2uz\nε,κ+εξκ, uz\nε,κ(0,·) =z. (B.1)\nSince the driving noise ξκ=ξ∗ρκdenotes a space-time white noise ξconvolved with a mollifier ρκ,\nthis equation is classically well-posed at any scale κ >0. Forκ= 0, however, the driving noise ξis\na distribution of (parabolic) Hölder–Besov regularity α−2<−d+2\n2; hence, by Schauder theory, the\nsolution to the corresponding linear equation has regularity α<−d−2\n2. As soon as d≥2, this is strictly\n46negative and the cubic term in the non-linear equation becomes, a priori, ill-defined.\nIn fact, already in dimension d= 2and for fixed ε >0, it has been proved by Hairer, Ryser, and\nWeber [HRW12] that uz\nε,κconverges to 0in a space of distributions in probability as κ→0. If we denote\nbyΦclas(z,εξκ):=uz\nε,κthe classical solution map to (B.1), this immediately implies that Φclas(z,·)\nis discontinuous for d= 2; simple power counting suggests that the situation is no better in higher\ndimensions. The theory of regularity structures [Hai14]9takes the following steps to overcome that\nproblem:\n•Letζbe a smooth driving noise, for example ζ=ξκforκ >0. In a first step, one constructs\na finite number of non-linear objects from ζwhich are indexed by trees. The collection of these\nobjects is called a (minimal) model Πζ, see Definition B.2 below for details.\n•On the space of (minimal) models M−, anon-linear metric space with distance dM−, one defines\na solution map Φ(z,·)in a way that is consistent with the classical solution theory, i.e. such\nthat Φclas(z,ζ) = Φ( z,Πζ). This solution map Φis locally Lipschitz continuous in both arguments,\nsee Definition B.12 and Lemma B.13 below.\n•Asdiscussedabove, onedoesnotexpectthenaivemodels Πεξκtoconvergeas κ→0. However, inthe\ncase of the periodic Φ4\n3dynamics, a relatively simple renormalisation procedure maps Πεξκ⇝¯Πεξκ\nthrough two families of diverging constants C(1)\nε,κandC(2)\nε,κ(see Definition B.7). There exists a\nparticular choice of such constants for which ¯Πεξκconverges to a limiting model ¯Πεasκ→0. This\nmodel is called the BPHZ model , see Proposition B.9.\n•Inorderfortherenormalisationproceduretobemeaningful, oneneedstoassociate ¯uz\nε,κ:= Φ(z,¯Πεξκ)\nto an actual equation. In fact, it can be shown that ¯uz\nε,κsolves (1.9), which reads\n∂t¯uz\nε,κ−∆¯uz\nε,κ=−(¯uz\nε,κ)3−/parenleftbig\nm2−3ε2C(1)\nκ+ 9ε4C(2)\nκ/parenrightbig\n¯uz\nε,κ+εξκ,¯uz\nε,κ(0,·) =z.\nFinally, all these steps justify the following definition:\nDefinition B.1 (Rigorous meaning of ¯uz\nε)Forε∈[0,1]andz∈Cα(T3)withα∈(−2/3,−1/2), we\ndefine\n¯uz\nε:= Φ(z,¯Πε) = lim\nκ→0Φ(z,¯Πεξκ)in prob.in CTCα(T3). (B.2)\n♤\nWe emphasise that ¯uz\nεdoesnotsolve a stochastic PDE itself; in line with the previous definition, the\nequation in (1.2) only represents the formallimit asκ→0of the stochastic PDEs for ¯uz\nε,κ, see (1.9).\nB.1.1 Trees, Models, and the Solution Map\nIn this section, we properly introduce the abstract objects alluded to in the outline above. Particular\nexamples of the regularity structure associated to the Φ4\n3equation have been presented in numerous other\nworks, originally in [Hai14, Sec. 9.2 and 9.4], summarised in by [HW15, Sec. 2], as well as [Hai15b,HM18a].\nAs a result, we do not present all the required details here but refer the interested reader to the above\nworks. The main object to introduce is the set of treesorsymbols, denoted byW−, of negative degree\nwhich do not contain polynomial factors. These are summarised in the following table:\nWe also setT−:=spanW−and fix a compact domain D:=I×T3for some interval I⊆R. One may\nthink ofIas the interval IT:= [0,T]for some horizon T >0. For technical reasons, however, we are\nrequired, at various points, to enlarge the interval I, see Remarks B.4 and B.5. The precise length of I\nis essentially unimportant, so long as it contains [−2,T+ 2]. We write DTinstead ofDwhen we wish\nto highlight its dependence on T. The following definition is taken from [HW15, p. 11].\n9For an introduction to the theory, see the two expository articles [Hai15a,Hai15b], the second of which is specialised to\ntheΦ4\n3equation. Since the case d= 2is analogous, but easier, we focus on d= 3from here on.\n47τ\n|τ| −5\n2−κ−1\n2−κ−1−2κ−3\n2−3κ−4κ−4κ−1\n2−5κ\nnτ 1 1 2 3 4 4 5\nTable 1: List of trees τ∈W−, their degree|τ|, and the number of leaves nτ.\nDefinition B.2 (Minimal model space) For any smooth map Π:D→L(T−,C∞(D;R))we define\n|||Πτ|||:=\n\nsupλ∈(0,1]supφ∈B4supz∈Dλ−|τ|/vextendsingle/vextendsingle⟨Πzτ,φλ\nz⟩/vextendsingle/vextendsingle\nL2 ifτ∈W−\\{,},\nsups∈Rsupλ∈(0,1]supφ∈B4supz∈Dλ−||/vextendsingle/vextendsingle⟨1t≥sΠz,φλ\nz⟩L2/vextendsingle/vextendsingleifτ=,\nsupλ∈(0,1]supη∈B3supz∈Dλ−||/vextendsingle/vextendsingle⟨Πz(t,·),ηλ\nx⟩L2/vextendsingle/vextendsingle ifτ=,(B.3)\nwhere\n•the collections of test functions Bdford= 3,4are given by\nBd:=/braceleftig\nψ∈C3(Rd;R):∥ψ∥C3≤1,supp (ψ)⊆B(0,1)/bracerightig\n, d∈{3,4},\n•the rescaled, shifted test functions φλ\nzresp.ηλ\nxare defined by\nφλ\nz(¯z):=λ−5φ(λ−2(t−¯t),λ−1(x−¯x)), ηλ\nx(¯x):=λ−3η(λ−1(x−¯x))\nforz= (t,x)and¯z= (¯t,¯x)inD.\nThe (inhomogeneous, minimal) model norm|||·|||and the distance dM−are then given by\n|||Π|||:= max\nτ∈W−|||Πτ|||, dM−(Π,˜Π):= sup\nτ∈W−sup\nλ∈(0,1]sup\nφ∈B4sup\nz∈Dλ|τ||⟨Πzτ−¯Πzτ,φλ\nz⟩|.(B.4)\nand the spaceM−of (minimal, admissible) models is defined to be the completion of smooth maps Π,\nas above, under the distance dM−. We also impose the algebraic constraint [HW15, Eq. 2.9], the\nadmissibility conditions [HW15, Eq. 2.10(a-c)] and define the homogeneous model norm⌊⌉Π⌊⌉by\n⌊⌉Π⌊⌉:= max\nτ∈W−⌊⌉Πτ⌊⌉,⌊⌉Πτ⌊⌉:=|||Πτ|||1\nnτforτ∈W−. ♤\nAs before, we write M−\nTas well as|||·|||Tand⌊⌉·⌊⌉Twhenever we wish to highlight their dependence\nonT. The following lemma is immediate from the definitions.\nLemma B.3 Convergence w.r.t. |||·|||and⌊⌉·⌊⌉inM−are equivalent. ♤\nRemark B.4 (Recentering maps F)The usual definition of a model, minimal or otherwise, is a\npair of maps (Π,F)with Πas above and Farecentring map , see Remark B.6 below. In the case of\nminimal models however, the implicit algebraic and admissibility conditions in Definition B.2 guarantee\nthat one can canonically define the action of the recentring maps Fzτforz∈Don the symbols τ∈W−;\nfurther details are presented in [HW15, Sec. 2.3]. For the requisite analytical bounds to be satisfied, one\nwill have to suitably enlarge the interval I, see [HW15, Remarks 2.3 and 2.4] and the discussion below\ntheir Remark 2.6. ♤\n48Remark B.5 (Choice of test function) As an immediate consequence of [CZ20, Prop. 13.1 and 13.2],\none can equivalently state Definition B.2 using only a singletest function φ(resp.η), at the expense\nof suitably enlarging the domain D. Henceforth, in view of Proposition B.14 below, we choose φ:= Ψ\nwhere Ψis introduced in [MW20b, p. 4]. Since we do not explicitly refer to the properties of Ψ, we do\nnot give details of its construction here. ♤\nDefinition B.2 of the space of minimal modelsM−is very similar to that of the model space M\ngiven in [Hai14, Def. 2.17]. The following remark establishes a connection between both notions.\nRemark B.6 (Minimal models and models) By [HW15, Thm. 2.10],\n•any element Π∈M−can beuniquely extended to a bona fide admissible model Z= (˜Π,˜F)∈M\nsuch that Πzτ=˜Πzτfor anyz∈Dandτ∈W−;\n•the map E:M−→M,Π∝⇕⊣√∫⊔≀→Z= (˜Π,˜F)is locally Lipschitz continuous.\nThis is reminiscent of Lyons’ extension theorem in rough paths theory. Conversely, if we choose\nZ= (˜Π,˜F)∈Mand set Πz:=˜Πz|W−forz∈D, we obtain an element Π∈M−. ♤\nWe work with minimal admissible models since they do not act on polynomials and therefore transform\nhomogeneously when rescaling the noise ξ⇝εξ, see Lemma B.10 below. This is also the reason why we\nwork with the homogeneous model norm⌊⌉·⌊⌉Trather than the inhomogeneous model norm |||·|||T. Both\nof these facts will be crucial in Sections B.2 and B.3 below.\nDefinition B.7 ((Renormalised) Canonical lift) LetGdenote the Green’s function of the heat\nkernel (∂t−∆)onR+×T3with decomposition\nG=K+R (B.5)\nwhereRis smooth and bounded and Ksatisfies [Hai14, Assumption 5.1, 5.3, and 5.4]. For ζ∈C∞(D)\nand arbitrary constants C(1)\nζ,C(2)\nζ∈R, we then set\n•¯Πζ\nz:=ζ\n•¯Πζ\nz:=K∗ζ\n•¯Πζ\nz:=/parenleftbig¯Πζ\nz/parenrightbig2−C(1)\nζ\n•¯Πζ\nz:=/parenleftbig¯Πζ\nz/parenrightbig3−3C(1)\nζ¯Πζ\nz\n•¯Πζ\nz:=K∗¯Πζ\nz•¯Πζ\nz(¯z):=/parenleftbig\nK∗¯Πζ\nz/parenrightbig\n(¯z)−/parenleftbig\nK∗¯Πζ\nz/parenrightbig\n(z)\n•¯Πζ\nz:=¯Πζ\nz¯Πζ\nz\n•¯Πζ\nz:=¯Πζ\nz¯Πζ\nz−C(2)\nζ\n•¯Πζ\nz:=¯Πζ\nz¯Πζ\nz−3C(2)\nζ¯Πζ\nz\nFor any choice of constants C(i)\nζ, this defines an element ¯Πζ∈M−, see [HW15], more specifically their\ncomments before Theorem 2.13 (incl. references).\n•ForC(1)\nζ=C(2)\nζ= 0, we write Πζinstead of ¯Πζand refer to it as the canonical lift ofζtoM−.\n•In the specific case ζ=εξκ, we writeC(i)\nε,κ:=C(i)\nεξκandC(i)\nκ:=C(i)\n1,κfori= 1,2and call ¯Πεξκthe\nrenormalised canonical lift of εξκ. ♤\nThe next lemma is a special case of [Sch23, Prop. 3.9] and extends the definition of the canonical lift Π•\nto Cameron–Martin functions h∈L2\nTL2(T3), for anyT >0.\nLemma B.8 (Canonical lift of Cameron–Martin functions) LetT >0. The map\nC∞(DT)∋ζ∝⇕⊣√∫⊔≀→Πζ∈M−\nT\n49extends to a continuous map from L2\nTL2(T3)intoM−. In addition, it is locally Lipschitz continuous in\nthe sense that, for any M > 0, uniformly over all h1,h2∈L2\nTL2(T3)with∥h1∥L2L2x∨∥h2∥L2L2x≤M, it\nholds that\ndM−(Πh1,Πh2)≲T,M∥h1−h2∥L2\nTL2x. (B.6)\nForwh,zwithh∈L2\nTL2(T3)andz∈Cα(T3)as in(A.1), we also have Φ(z,Πh) =wh,z. ♤\nThe following proposition provides us with a specific choice of constants C(i)\nε,κwhich then remain fixed\nfor the rest of the appendix.\nProposition B.9 (BPHZ model) Letε∈[0,1]. There exists a choice of constants C(1)\nκandC(2)\nκ\nsuch that\n(i)C(1)\nε,κ=ε2C(1)\nκandC(2)\nε,κ=ε4C(2)\nκfor anyκ>0, and\n(ii)the corresponding sequence (¯Πεξκ)κ>0converges in probability in (M−,dM−)to a limiting model ¯Πε.\nWe call ¯Πεthe (minimal) BPHZ model (with noise intensity ε) and also write ¯Π:=¯Π1. ♤\nProof.Let us define the constants C(1)\nε,κandC(2)\nε,κas in [Hai14, Eq. (10.33)] and [Hai14, Eq. (10.41)],\nrespectively; the claim in (i) is immediate from that definition. It also follows from a more general\nformula for the BPHZ renormalisation constants, see [BCCH20, Rmk. 2.23] and the references therein.\nThe assertion in (ii) is the content of [Hai14, Thm. 10.7] (which, in turn, is a special case of the more\ngeneral results in [CH16] or [HS24]). □\nLemma B.10 (Homogeneity of ¯Πε)LetT >0. For anyz∈DTand anyτ∈W−, we have ¯Πε\nzτ=\nεnτ¯Πzτ. As a consequence, we also have ⌊⌉¯Πε⌊⌉T=ε⌊⌉¯Π⌊⌉T. ♤\nProof.By definition of ¯Πε, it suffices to check that the claims hold for ¯Πεξκwithκ>0fixed andC(i)\nε,κ\nas in Proposition B.9(i). In that case, the claims follow immediately from Definitions B.2 and B.7. □\nLemma B.11 (Fernique) For anyT >0, there exists some ΛT>0such that\nKT:=E[exp/parenleftbig\nΛT⌊⌉¯Π⌊⌉2\nT/parenrightbig\n]<∞. ♤\nProof.It is immediate to see that, for κ > 0andτ∈W−, the norm⌊⌉¯Πξκτ⌊⌉nτ\nTis an element in\nthenτ-th inhomogeneous Wiener chaos. Since all Wiener chaoses are closed under convergence in\nprobability [Sch69], the norm ⌊⌉¯Πτ⌊⌉nτ\nTlives in the nτ-th inhomogeneous Wiener chaos as well; for\nelements therein, it is well known that there exists some Λτ\nT>0such that\nE[exp/parenleftbig\nΛτ\nT⌊⌉¯Πτ⌊⌉2\nT/parenrightbig\n]<∞.\nFor eachτ∈W−, we then choose pτ∈(1,∞)such that/summationtext\nτ∈W−1\npτ= 1and set\nΛT:= min\nτ∈W−Λτ\nT\npτ.\nSince\n⌊⌉¯Π⌊⌉2\nT= max\nτ∈W−⌊⌉¯Πτ⌊⌉2≤/summationdisplay\nτ∈W−⌊⌉¯Πτ⌊⌉2\n50Hölder’s inequality implies\nKT≤E/bracketleftigg/productdisplay\nτ∈W−exp/parenleftbig\nΛT⌊⌉¯Πτ⌊⌉2/parenrightbig/bracketrightigg\n≤/productdisplay\nτ∈W−/vextenddouble/vextenddoubleexp/parenleftbig\nΛT⌊⌉¯Πτ⌊⌉2/parenrightbig/vextenddouble/vextenddouble\nLpτ(P)\n=/productdisplay\nτ∈W−E/bracketleftbig\nexp/parenleftbig\nΛTpτ⌊⌉¯Πτ⌊⌉2/parenrightbig/bracketrightbig1\npτ≤/productdisplay\nτ∈W−E/bracketleftbig\nexp/parenleftbig\nΛτ\nT⌊⌉¯Πτ⌊⌉2/parenrightbig/bracketrightbig1\npτ<∞. □\nFinally, let us give some details regarding the solution map Φ. For later reference, we already present\nthe next two statements for arbitraryT > 0since we know that ¯uz\nεexists forever, a fact that we\nrecapitulate in the next subsection, see Proposition B.14 and Corollary B.15.\nDefinition B.12 (Solution map) For anyT >0, the solution map Φ:Cα(T3)×M−\nT→CTCα(T3)\nis defined as Φ(z,Π):=SA(−m2,z,Π)whereSAis given in [HW15, Thm. 2.12]. ♤\nThe following lemma immediately follows from global existence and the joint local Lipschitz continuity\nof the solution map, see (the proof of) [HW15, Thm. 2.12] and the references therein.\nLemma B.13 For allR > 0andT > 0, there exists some LR,T>0such that for all Π,˜Π∈M−\nT\nwith|||Π|||T∨|||˜Π|||T≤R, it holds that\nsup\nz∈¯BR(0)∥Φ(z,Π)−Φ(z,˜Π)∥CTCαx≤LR,TdM−(Π,˜Π). (B.7)\n♤\nB.1.2 Coming Down from Infinity\nIn the previous subsection, we have sketched the localsolution theory for the Φ4\n3equation using regularity\nstructures; in this subsection, we will recap the strategy to show that this solution actually exists globally\nin time. This follows from the so-called coming down from infinity (CDFI) estimate which was first\nobtained by Mourrat and Weber [MW17b] for the Φ4\n3equation on the 3D torus based on paraproduct\nestimates and paracontrolled calculus in the sense of [GIP15]. In recent years, the same result has been\nobtained by Moinat and Weber [MW20b] (see also [MW20a]) in a setting that is closer to regularity\nstructures, allowing for a streamlined proof which has then been generalised by Chandra, Moinat, and\nWeber [CMW23] to the case of the Φ4\n4−ιequation when ι>0.\nThe following proposition presents a version of the CDFI estimate that is essentially equivalent\nto [MW20b, Thm. 2.1] but phrased in the language of the previous subsection; this will be useful in\nSections B.2 and B.3 below.\nProposition B.14 Letz∈Cα(T3)andε∈[0,1]. We denote byεthe solution to the additive noise,\nstochastic heat equation, with noise intensity εand0initial condition, i.e.\n(∂t−∆)ε=εξ,ε(0,·) = 0. (B.8)\nFurthermore, set vz\nε:=uz\nε−εwithuz\nεas given by (B.2). Then, for any t∈(0,1), the bound\nsup\n(s,x)∈(t,1)×T3|vz\nε(s,x)|≲1\nt2∨/parenleftbig\nε⌊⌉¯Π⌊⌉/parenrightbig2\n1−2κ(B.9)\nholds uniformly over all initial conditions z∈Cα(T3), where ¯Πdenotes the (minimal) BPHZ model. ♤\nProof.It suffices to prove the claim when ε= 1; the case of general ε∈[0,1]follows from Lemma B.10.\nWhenε= 1the claimed statement of (B.9)is equivalent to the content of [MW20b, Thm. 2.1] upon\n51making the change\n⌊⌉¯Π⌊⌉⇝max/braceleftbigg\n[σ]1\nnσ\n|σ|:σ∈L/bracerightbigg\nfor the following list of trees:\nL:={,,x,,,,,}.\nThese trees are closely related to those in W−, see table 1, but—despite superficial appearances—they\narenotidentical.10We therefore colour them in black rather than blue; however, we emphasise that the\ntreesσ∈Lwillonlyplay a role within this proof. The quantities [σ]|σ|are appropriate norms, which\nwe detail below, and nσdenotes the number of leaves.\nIn order to establish our claim, it therefore suffices to obtain the estimate\n[σ]1\nnσ\n|σ|≲⌊⌉¯Π⌊⌉for allσ∈L. (B.10)\nTo this end, we recall the construction of the trees σ∈Lfrom [MW20b, Sec. 2.2]. First, consider a\nsmooth noise ζwhich we may think of as ζ=ξκ. Since all estimates obtained therein are stable in the\nlimitκ→0and since⌊⌉¯Πξκ⌊⌉→⌊⌉ ¯Π⌊⌉(c.f. Proposition B.9), it suffices to prove the estimate (B.10)with\nboth sides smoothed out at level κ>0. Since we fixed ε= 1, we write κfor the solution to B.8, with ξ\nreplaced by ξκand then set\nκ:=2\nκ−C(1)\nκ,κ:=3\nκ−3C(1)\nκκ,κ:=G∗κ,κ:=G∗κ (B.11)\nwhereGdenotes the Green’s function of the heat operator. Additionally, as in [MW20b, Eq. (2.16–2.19)],\nlet|x|:=||+ 1and, withφas in Remark B.5, set\n/bracketleftbig\nx,κ/bracketrightbig\n|x|:= sup\nz∈(0,1)×T3sup\nλ<1λ−|x|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n(¯z−z)κ(¯z)φλ\nz(¯z) d¯z/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\n/bracketleftbig\nκ/bracketrightbig\n||:= sup\nz∈(0,1)×T3sup\nλ<1λ−||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/parenleftig/parenleftbig\nκ(¯z)−κ(z)/parenrightbig\nκ(¯z)−C(2)\nκ/parenrightig\nφλ\nz(¯z) d¯z/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\n/bracketleftbig\nκ/bracketrightbig\n||:= sup\nz∈(0,1)×T3sup\nλ<1λ−||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/parenleftig/parenleftbig\nκ(¯z)−κ(z)/parenrightbig\nκ(¯z)/parenrightig\nφλ\nz(¯z) d¯z/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\n/bracketleftbig\nκ/bracketrightbig\n||:= sup\nz∈(0,1)×T3sup\nλ<1λ−||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/parenleftig/parenleftbig\nκ(¯z)−κ(z)/parenrightbig\nκ(¯z)−3C(2)\nκκ(¯z)/parenrightig\nφλ\nz(¯z) d¯z/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(B.12)\nThe norm [σκ]|σ|for all the symbols σ∈Lthat do not appear in (B.12)is simply the corresponding\n(Hölder-) norm on C|σ|((0,1)×T3); the exception is κ, in which case we use the slightly stronger norm\n[κ]||:= sup\nt∈[0,1]∥κ(t,·)∥Cα(T3).\nThe remainder of the proof consists of associating each σ∈Lwith a corresponding tree11τ∈W−, and\nvice versa.\nLet us explain in detail how one obtains the estimate [σκ]|σ|≲⌊⌉¯Πξκ⌊⌉nσfor the most complicated\ntreeσ=κ. We graphically encode the decomposition G=K+Rgiven in (B.5) as\n=+, :=G, :=K, :=R,\n10The connection between the symbols in LandW−is discussed in detail in [MW20b, Sec. 3].\n11The action of any admissible model, such as ¯Π, on the symbols X,andis canonically defined from the action\nonresp.; this follows from the admissibility conditions that are implicit in Definition B.2. Therefore, we have not\nadded these symbols to the minimal listW−.\n52and accordingly have\nκ(y) =κ(y) +κ(y),κ(y) =κ(y) + 2κ(y) +κ(y),\nκ(y) =κ(y) +κ(y) =/parenleftbig\nκ(y) + 3κ(y) + 3κ(y) +κ(y)/parenrightbig\n+/parenleftbig\nκ(y) + 3κ(y) + 3κ(y) +κ(y)/parenrightbig\n.\nFor any of these terms we introduce the shorthand notation Az,¯z:=A(¯z)−A(z)and use the previous\ndecomposition to express the trees within the integral defining [κ]||as follows:12\n/parenleftbig\nκ(¯z)−κ(z)/parenrightbig\nκ(¯z)−3C(2)\nκκ(¯z)\n=/parenleftig/parenleftbig\nκ+ 3κ+ 3κ+κ/parenrightbig\n+/parenleftbig\nκ+ 3κ+ 3κ+κ/parenrightbig/parenrightig\nz,¯z/parenleftbig\nκ+ 2κ+κ/parenrightbig\n(¯z)−3C(2)\nκ/parenleftbig\nκ+κ/parenrightbig\n(¯z)\n=/parenleftig/parenleftbig\nκ/parenrightbig\nz,¯zκ(¯z)−3C(2)\nκκ(¯z)/parenrightig\n+/parenleftig\n3/parenleftbig\nκ/parenrightbig\nz,¯zκ(¯z)−3C(2)\nκκ(¯z)/parenrightig\n+ (...)\nNote that we have suitably paired the two summands above with the corresponding counterterms in\norder to relate them to ¯Πξκz(¯z)and¯Πξκz(¯z)from Definition B.7, respectively. In total, making use of\nthe simple estimates,\n∥R∗¯Πξκτ∥L∞≲|||¯Πξκτ|||,|||¯Πξκ|||≲|||¯Πξκ|||\nwe find\n/bracketleftbig\nκ/bracketrightbig\n||≲|||¯Πξκ|||+|||¯Πξκ|||∥ξκ∥C||+|||¯Πξκ|||/parenleftbig\n∥ξκ∥|||¯Πξκ|||+∥ξκ∥2\nC||/parenrightbig\n+/parenleftig\n|||¯Πξκ|||+∥ξκ∥|||¯Πξκ|||+∥ξκ∥2|||¯Πξκ|||+∥ξκ∥3/parenrightig/parenleftig\n|||¯Πξκ|||+∥ξκ∥|||¯Πξκ|||+∥ξκ∥2/parenrightig(B.13)\nwhere we wrote∥ξκ∥=∥ξκ∥C||=|||¯Πξκ|||to lighten the notation. Since the total number of leaves in\neach product on the RHS of (B.15) equals n= 5, we find that\n/bracketleftbig\nκ/bracketrightbig\n||≲⌊⌉¯Πξκ⌊⌉5. (B.14)\nThis is exactly the claimed estimate (B.10) in the case σκ=κ.\nUsing the same simple combinatorial argument, one can show that the following formula holds for the\nother trees in L:\nFor eachσ∈L\\{,x}, write13σ=I/parenleftbig\nI()k/parenrightbig\nI()mwithk∈ {0,2}andm∈ {0,1,2,3}such\nthatnσ=k+m. We then have the estimate\n[σκ]|σ|≲|||¯Πξκσ|||1k>0+|||¯ΠξκI()k|||/parenleftbiggm/summationdisplay\nℓ=1∥ξκ∥ℓ|||¯ΠξκI()m−ℓ|||/parenrightbigg\n+/parenleftbiggk/summationdisplay\nr=0∥ξκ∥r|||¯ΠξκI()m−r|||/parenrightbigg/parenleftbiggm/summationdisplay\nℓ=0∥ξκ∥ℓ|||¯ΠξκI()m−ℓ|||/parenrightbigg (B.15)\nwhere the norms |||¯Πξκτ|||forτ∈W−were introduced in Definition B.2. For σ=x, the estimate is of\nthe same form as for σ=, only with a different implicit constant: The reason is that the distance |z−¯z|\nis uniformly bounded for any z,¯z∈(0,1)×T3.\n12For notational purposes, we have also joined roots in the case of non-singular products, e.g. κ:=κκ, as opposed\nto (B.11) where we need to renormalise the product. Similarly, κ:=G∗κ=G∗(κκ) =G∗([(κ)2−C(1)\nκ]κ).\n13The operatorIdenotes the “abstract” integration operator associated to Kwithin the theory of regularity structures,\nsee [Hai14, Def. 5.7]; however, we use the convention that I()0:=1andI1:=1. Graphically,Iis encoded by the blue\nstraight line .\n53Using the bound (B.15), the same argument as for (B.14) then implies the estimate\n[σκ]|σ|≲⌊⌉¯Πξκ⌊⌉nσ1k>0+⌊⌉¯Πξκ⌊⌉k⌊⌉¯Πξκ⌊⌉m+⌊⌉¯Πξκ⌊⌉k⌊⌉¯Πξκ⌊⌉m≃⌊⌉¯Πξκ⌊⌉nσ\nwhere we have used that k+m=nσ. As argued before, sending κ→0concludes the proof. □\nSince the estimate in (B.9)is independent of the initial condition, we immediately have the following\ncorollary.\nCorollary B.15 (Global existence) For any z∈Cα(T3)andε≥0, the process ¯uz\nεdefined in (B.2)\nalmost surely is an element of CTCα(T3)for anyT >0. ♤\nB.2 Uniform LDP for the Φ4\n3Equation\nAs recalled in the introduction, an LDP for the Φ4\n3dynamics (¯uz\nε)ε>0was obtained in [HW15, Thm. 4.4]\nforfixedinitial condition z. In this subsection, we prove a slightly stronger version of their result which\nislocally uniform inz. However, some care is needed in stating the uniform LDP since one finds various,\ninequivalent concepts in the literature. We refer to the survey by Salins’s [Sal19] and Remark B.17\nbelow. Recall:\n•¯BR(0)denotes a closed ball in Cα(T3)with radius R> 0, centred at the origin,\n•the dynamics ¯uz\nεintroduced in Definition B.1,\n•with the notation of Lemma B.8, it holds that wh,z= Φ(z,Πh),\n•|||·|||and⌊⌉·⌊⌉induce the same topology on M−, see Lemma B.3.\nThe following proposition is the main result of this subsection.\nProposition B.16 (Locally uniform LDP for the dynamics) For eachR,T > 0, the family\n{¯uz\nε:ε>0,z∈¯BR(0)}⊆CTCα(T3)\nsatisfies a uniform LDP with good rate function14\nIz\nT(v):= inf/braceleftbigg1\n2∥h∥2\nL2\nTL2x:h∈L2\nTL2\nx,Φ(z,Πh) =v/bracerightbigg\n=1\n2/integraldisplayT\n0/integraldisplay\nT3/parenleftbig\n∂tv−∆v−m2v+v3/parenrightbig2dxdt(B.16)\nwith the understanding in the second line that Iz\nT(v):= +∞if\n∂tv−∆v−m2v+v3/∈L2\nTL2(T3)orv(0)̸=z.\nThat is:\n(i) For any z∈Cα(T3)andθ>0, the sublevel set Iz\nT[θ]≡{Iz\nT≤θ}is compact.\n(ii)For anyγ, δ,θ > 0there exists some ε↓\n0:=ε↓\n0(T,δ,γ,θ )>0such that for all z∈¯BR(0), all\nv∈Iz\nT[θ], and allε≤ε↓\n0it holds that\nP/parenleftbig\n∥¯uz\nε−v∥CTCαx<δ/parenrightbig\n≥exp/parenleftig\n−Iz\nT(v) +γ\nε2/parenrightig\n. (B.17)\n14If we write IT(v)forv∈CTCα(T3)without specifying the initial condition as a superscript, by convention, we always\nmeanIv(0)\nT(v).\n54(iii)For anyγ, δ,θ > 0there exists some ε↑\n0:=ε↑\n0(T,δ,γ,θ )>0such that for all z∈¯BR(0)and\nallε≤ε↑\n0it holds that\nP/parenleftbig\ndistCTCαx(¯uz\nε,Iz\nT[θ])≥δ/parenrightbig\n≤exp/parenleftig\n−θ−γ\nε2/parenrightig\n. (B.18)\nFor convenience, we set ε0:=ε↓\n0∧ε↑\n0. ♤\nRemark B.17 (Variants of LDPs) We briefly comment on the most commonly employed variants\nof the uniform and non-uniform LDP as well as relations between them.\n(i)Non-uniform LDPs : There are two ways to phrase an LDP: One formulation due to Varadhan\n(which we use to present our main result, Theorem 1.1) and another due to Freidlin and Wentzell. It\nis proved in [FW12, Chap. 3, Thm. 3.3] that the statements concerning lower bounds are equivalent\nin both formulations—but the equivalence of the respective upper bound statements requires compact\nsublevel sets of the rate function.\n(ii)Locally uniform LDPs : Giving credit to Freidlin’s and Wentzell’s definition at the end of [FW12,\nCh. 3, Sec. 3], Salins [Sal19, Def. 2.1] calls the uniform LDP in the previous proposition a\n“FWULDP”—in contrast to a “DZULDP” à la Dembo and Zeitouni, see [Sal19, Def. 2.2]. However,\nsinceCα(T3)is an infinite dimensional space, and hence closed balls are not compact, one of the\ncriteria in [Sal19, Ass. 2.6 and Thm. 2.7] under which both concepts of uniform LDP are equivalent\nisnotmet. Luckily, the formalism of “FWULDP” is sufficient for our purposes. ♤\nRemark B.18 The LDP of [HW15] actually allows for the possibility that the process ¯uz\nεblows up in\nfinite time. However, as recapitulated in Appendix B.1, global existence of the renormalised Φ4\n3dynamics\nhas been established by now (see Corollary B.15 above), so the conditioning on times T >0which are\nsmaller than the explosion time of ¯uz\nεbecomes superfluous; the previous proposition accounts for that. ♤\nAs in the proof of the non-uniform LDP given by [HW15], our proof of Proposition B.16 will be\nbased on the jointlocal Lipschitz continuity of the solution map Φ, see Lemma B.13. The other main\ningredient is the following LDP on minimal model space, see [HW15, Thm. 4.3], which we present in the\nFreidlin–Wentzell formulation, cf. Remark B.17 (i).\nProposition B.19 For eachT >0, the family{¯Πε:ε>0}satisfies an LDP on M−\nTwith good rate\nfunction\nJT(Π):= inf/braceleftbigg1\n2∥h∥2\nL2\nTL2x:h∈L2\nTL2\nx,Πh= Π/bracerightbigg\n. (B.19)\nThat is:\n(i) For any θ>0, the sublevel set JT[θ]≡{Π∈M−:JT(Π)≤θ}is compact.\n(ii)For anyγ, δ,θ > 0andΠ∈M−\nT, there exists some εM,↓\n0=εM,↓\n0(T,δ,γ,θ )>0such that for\nallε≤εM,↓\n0we have\nP/parenleftbig\ndM−/parenleftbig¯Πε,Π/parenrightbig\n<δ/parenrightbig\n≥exp/parenleftig\n−JT(Π) +γ\nε2/parenrightig\n. (B.20)\n(iii)For anyγ, δ,θ > 0there exists some εM,↑\n0=εM,↑\n0(T,δ,γ,θ )>0such that for all ε≤εM,↑\n0we\nhave\nP/parenleftbig\ndistM−(¯Πε,JT[θ])≥δ/parenrightbig\n≤exp/parenleftig\n−θ−γ\nε2/parenrightig\n. (B.21)\n55For convenience, we set εM\n0:=εM,↓\n0∧εM,↑\n0. ♤\nWe will also need the following elementary observation, a consequence of sub-additivity of probability\nmeasures.\nLemma B.20 For anyδ≥0andδ1,δ2≥0such thatδ1+δ2=δ, we have\nP(X1+X2≥δ)≤P(X1≥δ1) +P(X2≥δ2). ♤\nFinally, we have gathered all the ingredients to give a proof of Proposition B.16.\nProof (of Proposition B.16). We proceed in three steps that mirror the three claims.\n▷Proof of (i). The statement in (i) deals with the case of fixed z∈Cα(T3)and was proved by Hairer\nand Weber [HW15, Thm. 4.4].\n▷Proof of (ii). Observe that Iz\nT(v)<∞implies that we can find a unique(anddeterministic )\nh:=hv∈L2\nTL2(T3)given by\nh=∂tv−∆v+v3+m2v\nsuch that\nv=wh,z≡Φ(z,Πh),Iz\nT(v) =1\n2∥h∥2\nL2\nTL2x. (B.22)\nThis follows immeditely from the definition of the rate function Iz\nTgiven in (B.16)above. We then\nchooseδ1,δ2>0such thatδ=δ1+δ2and apply Lemma B.20 with\nX1:=∥¯uz\nε−v∥CTCαx1ε⌊⌉¯Π⌊⌉<M1, X 2:=∥¯uz\nε−v∥CTCαx1ε⌊⌉¯Π⌊⌉≥M1, (B.23)\nwhereM1>0is arbitrary for the moment; it will be chosen appropriately in (B.28)below. This leads\nto the following estimate:\nP/parenleftbig\n∥¯uz\nε−v∥CTCαx<δ/parenrightbig\n= 1−P/parenleftbig\n∥¯uz\nε−v∥CTCαx≥δ/parenrightbig\n≥1−P/parenleftbig\n∥¯uz\nε−v∥CTCαx1ε⌊⌉¯Π⌊⌉≤M1≥δ1/parenrightbig\n−P/parenleftbig\n∥¯uz\nε−v∥CTCαx1ε⌊⌉¯Π⌊⌉>M1≥δ2/parenrightbig\n=:1−P1−P2. (B.24)\nWe start by estimating P2. Sinceδ2>0, we have the elementary inclusion\n/braceleftig\n∥¯uz\nε−v∥CTCαx1ε⌊⌉¯Π⌊⌉>M1≥δ2/bracerightig\n⊆{ε⌊⌉¯Π⌊⌉>M 1}\nwhich, by sub-additivity of Pand Markov’s inequality, leads to the estimate\nP2≡P/parenleftbig\n∥¯uz\nε−v∥CTCαx1ε⌊⌉¯Π⌊⌉>M1≥δ2/parenrightbig\n≤P/parenleftig\n⌊⌉¯Π⌊⌉>M1\nε/parenrightig\n≤KTexp/parenleftig\n−ΛTM2\n1\nε2/parenrightig\n(B.25)\nfor anyε>0withKT∈(0,∞)andΛT>0as in Lemma B.11. For P1, we wish to apply Lemma B.13,\nthe joint local Lipschitz continuity of the solution map Φ. To this end, note that by Lemma B.10, one\nhas\n¯uz\nε= Φ(z,¯Πε),⌊⌉¯Πε⌊⌉1ε⌊⌉¯Π⌊⌉≤M1=ε⌊⌉¯Π⌊⌉1ε⌊⌉¯Π⌊⌉≤M1≤M1, (B.26)\nand we can choose M1>⌊⌉Πhv⌊⌉sincehvis deterministic. By Lemma B.13 and Lemma B.3, there exists\nsomeρ1:=ρ1(δ1,M1,T)>0such that\nP1≡P/parenleftbig\n∥Φ(z,¯Πε)−Φ(z,Πhv)∥CTCαx1⌊⌉¯Πε⌊⌉≤M1≥δ1/parenrightbig\n≤P/parenleftbig\ndM−(¯Πε,Πhv)1⌊⌉¯Πε⌊⌉≤M1≥ρ1/parenrightbig\n.\n56and the LDP lower bound (B.20) gives the estimate\n1−P1≥P/parenleftbig\ndM−(¯Πε,Πhv)<ρ1/parenrightbig\n≥exp/parenleftig\n−JT(Πhv) +γ/2\nε2/parenrightig\nfor allε<εM,↓\n0(T,ρ1,γ/2,θ). Since\nJT(Πhv) =1\n2∥hv∥2\nL2\nTL2x=Iz\nT(v),\nthe previous bound is equivalent to\n1−P1≥exp/parenleftig\n−Iz\nT(v) +γ/2\nε2/parenrightig\n. (B.27)\nIn summary, our estimates in (B.24), (B.25), and (B.27) lead to the inequality\nP/parenleftbig\n∥¯uz\nε−v∥CTCα\nx<δ/parenrightbig\n≥exp/parenleftig\n−Iz\nT(v) +γ/2\nε2/parenrightig\n−KTexp/parenleftig\n−ΛTM2\n1\nε2/parenrightig\n= exp/parenleftig\n−Iz\nT(v) +γ/2\nε2/parenrightig/parenleftbigg\n1−KTexp/parenleftig\n−ΛTM2\n1−Iz\nT(v)−γ/2\nε2/parenrightig/parenrightbigg\nfor allε<εM,↓\n0(T,ρ1,γ/2,θ). Finally, we observe that v∈Iz\nT[θ]implies\n1−KTexp/parenleftig\n−ΛTM2\n1−Iz\nT(v)−γ/2\nε2/parenrightig\n≥1−KTexp/parenleftig\n−ΛTM2\n1−θ−γ/2\nε2/parenrightig\n=:E(γ,θ,M 1,ε)\nand by choosing M1=M1(hv,γ,θ)such that\nM1>⌊⌉Πhv⌊⌉∨/radicalbigg\nθ+γ/2\nλ(B.28)\nwe can guarantee that E(γ,θ,M 1,ε)is monotonously increasing in ε. As a consequence, we can\nfindε↓\n0≤εM,↓\n0(T,ρ1,γ/2,θ)such that, for all ε≤ε↓\n0, we have\nE(γ,θ,M 1,ε)≥exp/parenleftig\n−γ/2\nε2/parenrightig\n.\nAltogether, this implies that the estimate\nP/parenleftbig\n∥¯uz\nε−v∥CTCαx<δ/parenrightbig\n≥exp/parenleftig\n−Iz\nT(v) +γ\nε2/parenrightig\nholds for all ε≤ε↓\n0and thereby completes the proof of the lower bound.\n▷Proof of (iii). Since the singleton {¯Πε}is closed and JT[θ]is compact, we can find a Π∈JT[θ]\nsuch that\ndistM−(¯Πε,JT[θ]) =dM−(¯Πε,Π).\nNote that Πis at minimal distance from the randomelement ¯Πεrelative to the other elements in JT[θ],\ni.e.Πitself israndom, too. From the proof of [HW15, Thm. 4.4], we know that\n{Πh:h∈L2\nTL2(T3)}={JT<∞}.\n57Our choice of Π, i.e. the fact that JT(Π)≤θ, thus implies the existence of hΠ∈L2\nTL2(T3)such that\nΠ = ΠhΠ,1\n2∥hΠ∥2\nH≡JT(Π)≤θ (B.29)\nAgain, note that this control hΠisrandom as well due to its dependence on the random model Π.\nHowever, for each fixed realisation ofΠ, it isunique: If we suppose that there were h1,h2∈Hsuch\nthat Πh1= Π = Πh2, then the definition of the canonical lift Π•implies that\nh1= Πh1= Πh2=h2.\nWe then set vΠ:= Φ(z,Π)≡whΠ,zwhich, by the form of the rate function Iz\nTin(B.16)above, implies\nthat\nIz\nT(vΠ) =1\n2∥hΠ∥2\nL2\nTL2x≤θ, (B.30)\ni.e.vΠ∈Iz\nT[θ]. At the same time, we know that the map\nΠ•:L2\nTL2\nx→M−\nT, h∝⇕⊣√∫⊔≀→Πh\nis locally Lipschitz continuous w.r.t. |||·|||T, see Lemma B.8 above; we denote the corresponding Lipschitz\nconstant, which is implicit in (B.6), bycT. Then, Lemma B.8 combined with (B.29)and(B.30)implies\nthat the bound|||Π|||T≤cTθholds uniformly over all realisations of Π; by definition of⌊⌉·⌊⌉T, this implies\nthe bound\n⌊⌉Π⌊⌉T≤AT (B.31)\nwhereATis some function of cTθ. As in the proofofthe lowerbound, we choose δ1,δ2>0withδ=δ1+δ2\nand apply Lemma B.20 with X1andX2as in(B.23), only withM1replaced by M2; the latter is arbitrary\nfor now and will be chosen later. We obtain the estimate\nP/parenleftbig\ndistCTCαx(¯uz\nε,Iz\nT[θ])≥δ/parenrightbig\n≤P/parenleftbig\n∥¯uz\nε−vΠ∥CTCαx≥δ/parenrightbig\n≤P/parenleftbig\n∥¯uz\nε−vΠ∥CTCαx1ε⌊⌉¯Π⌊⌉≤M2≥δ1/parenrightbig\n+P/parenleftbig\n∥¯uz\nε−vΠ∥CTCαx1ε⌊⌉¯Π⌊⌉>M2≥δ2/parenrightbig\n=:Q1+Q2.\nThe probability Q2can be estimated exactly as P2in (B.25) above, which gives the bound\nQ2≤Kexp/parenleftig\n−λM2\n2\nε2/parenrightig\n.\nThanks to B.31 and, as before, (B.26), we can apply Lemma B.13 to estimate Q1similarly to P1if we\nchooseM2>AT. Then, by Lemma B.13, there exists β1:=β1(δ1,M2,T)>0such that\nQ1≤P/parenleftbig\ndM−(¯Πε,Π)1⌊⌉¯Πε⌊⌉≤M2≥β1/parenrightbig\n≤P/parenleftbig\ndM−(¯Πε,Π)≥β1/parenrightbig\n=P/parenleftbig\ndistM−(¯Πε,JT[θ])≥β1/parenrightbig\n≤exp/parenleftig\n−θ−γ/2\nε2/parenrightig\nfor allε≤εM,↑\n0(T,β1,γ/2,θ). Note that we have additionally used the LDP upper bound (B.18).\nOne may now proceed similarly as in the proof of the lower bound, arrive at a similar constraint\nforM2as forM1in(B.28), and finally choose an appropriate ε↑\n0≤εM,↑\n0(T,β1,γ/2,θ)such that the claim\nin (B.18) follows. We refrain from giving all the details. □\n58B.3 Tail Bounds for the Φ4\n3Measure\nIn this section, we recapitulate the strategy to rigorously construct the measure µε, informally described\nin(1.1), as the unique invariant measure of the Markov process ¯uz\nεgiven by Definition B.1. Furthermore,\nwe establish exponential tail bounds on µεwhich are quantitative in ε>0, see Proposition B.21 below,\nwhich play a key role in the proof of our main result, Theorem 1.1.\nTo begin with, the existence of an invariant measure for ¯uz\nεfollows from the CDFI results, recovered\nin Subsection B.1.2 by means of a Krylov–Bogoliubov argument, see [MW17b] for details. Uniform\nexponential convergence in total variation distance to a uniqueinvariant measure µεthen follows from a\ncombination of CDFI with the following results, see (the proof of) [HS22a, Coro. 1.13]:\n•Thestrong Feller property, obtained by Hairer and Mattingly in [HM18b], which implies continuity\nof the transition probabilities associated with ¯uz\nεin total variation distance.\n•Thesupport theorem ofHairerandSchönbauer[HS22a], whichisgeneralityapplicablebutspecialised\ntoΦ4\n3in [HS22a, Thm. 1.12]. In our case, this asserts that the law of ¯uz\nεhas full support in Cα(T3).\nThe following works connect the measure µεas obtained via the above procedure to the programme of\nconstructive QFT (CQFT):\n•In [HM18a], Hairer and Matetski showed that, for small coupling strengths, the measure µεisiden-\nticalto the one constructed via lattice approximations, see for example [Fel74]. Note, however, that\nin the CQFT programme of the 1970’s, the measure µεhad only been constructed for small coupling\nconstants, hence the constraint in the previous statement; the SPDE techniques summarised earlier\napply at any coupling strength and thus generalise the earlier CQFT constructions.\n•The Osterwalder–Schrader axioms [OS73,OS75], which connect a Euclidean QFT to its relativistic\ncounterpart, were verified for Φ4\n3by Gubinelli and Hofmanova in [GH21] using PDE techniques.\n•Finally, Hairer and Steele [HS22b] showed that, in fact, the measure µε(for fixedε>0) satisfies\nan exponential tail bound that is much stronger than the one required by the OS axioms.\nThe following proposition is the main result of this section: It provides an exponential tail bound for the\nmeasureµεin the temperature parameter ε.\nProposition B.21 (Tail Bound) For anyθ > 0there exists some ρ:=ρ(θ)>0such that for\nallε∈(0,1]\nµε/parenleftbig\nBc\nρ/parenrightbig\n≤Kexp/parenleftbigg\n−θ\nε2/parenrightbigg\n. (B.32)\nwhereK:=K1/2is the constant from Lemma B.11, the Fernique estimate. ♤\nRemark B.22 The previous tail bound could likely be strengthened if we were to track the dependence\non the temperature εin the arguments of [HS22b], resulting in a tighter dependence of ρonθ. We do\nnot need such precise information here and therefore do not investigate this assertion further. ♤\nProof (of Proposition B.21). Since the measure µεis invariant for the renormalised dynamics t∝⇕⊣√∫⊔≀→uz\nε(t),\nwith z∼µε, for anyt∈[0,∞)it holds that\nµε(¯Bc\nρ) =/integraldisplay\nCα(T3)P/parenleftbig\n¯uz\nε(t)∈¯Bc\nρ/parenrightbig\nµε(dz).\nTherefore, the claimed bound in (B.32) is a consequence of the following lemma. □\n59Lemma B.23 Lett∈(0,1). For each θ >0, there exists a exists some ρ=ρ(t,θ)>0such that for\nallε∈(0,1], the bound\nP/parenleftbig\n∥¯uz\nε(t)∥Cα≥ρ/parenrightbig\n≤Ktexp/parenleftbigg\n−θ\nε2/parenrightbigg\n. (B.33)\nholds uniformly over all initial conditions z∈ Cα(T3)andKt∈(0,∞)denotes the constant in\nLemma B.11, the Fernique estimate. ♤\nLet us highlight that it is onlywithin the proof of that lemma that we use the CDFI result of Moinat and\nWeber, Proposition B.14, which is specific for the Φ4\n3equation and does not hold for other interesting\nmodels from QFT. We direct the reader to the “proof strategy” paragraph of the introduction, specifically\nitem (1) on page 7, where this issue has been extensively discussed.\nProof.By Proposition B.14, we have\n∥¯uz\nε(t)∥q\nCα≤∥vz\nε(t)∥q\nCα+∥ε(t)∥q\nCα≲1\nt2q+ε2q\n1−2κ/parenleftbig\n⌊⌉¯Π⌊⌉t/parenrightbig2q\n1−2κ(B.34)\nwhere we have additionally used homogeneity of ⌊⌉·⌊⌉with respect to rescaling the noise ξ⇝εξ,\nrecall Lemma B.10. Choosing q= 1−2κand denoting the implicit constant in (B.34)byC, Markov’s\ninequality implies the estimate\nP/parenleftbig\n∥¯uz\nε(t)∥Cα≥ρ/parenrightbig\n=P/parenleftbig\n∥¯uz\nε(t)∥1−2κ\nCα≥ρ1−2κ/parenrightbig\n≤P/parenleftbigg1\nε2t2(1−2κ)+⌊⌉¯Π⌊⌉2\nt≥ρ1−2κ\nCε2/parenrightbigg\n≤exp/parenleftbigΛt\nε2t2(1−2κ)/parenrightbig\nE/bracketleftbig\nexp (Λt⌊⌉¯Π⌊⌉2\nt)/bracketrightbig\nexp/parenleftbig\nΛtρ1−2κC−1ε−2/parenrightbig =Ktexp/parenleftig\n−Λt/bracketleftigρ1−2κ\nC−1\nt2(1−2κ)/bracketrightig1\nε2/parenrightig\nwhere Λt>0andKt∈(0,∞)denote the constants in Lemma B.11. The claim now follows by\nchoosingρ>0such that\nΛt/bracketleftigρ1−2κ\nC−1\nt2(1−2κ)/bracketrightig\n≥θ. □\nReferences\n[ABDVG22] Sergio Albeverio, Luigi Borasi, Francesco C. De Vecchi, and Massimiliano Gubinelli. Grass-\nmannian stochastic analysis and the stochastic quantization of Euclidean fermions. Probab.\nTheory Related Fields , 183(3-4):909–995, 2022. DOI: 10.1007/s00440-022-01136-x . 9\n[AD85] R. 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Main Principles and Their Ap-\nplications , volume 109 of Applied Mathematical Sciences . Springer, 1995. DOI:\n10.1007/978-1-4612-0821-1 . 44, 45\nTom Klose\nUniversity of Warwick\nDepartment of Statistics\nCoventry, CV4 7AL, United Kingdom\nE-mail address: tom.klose@warwick.ac.ukAvi Mayorcas\nUniversity of Bath\nDepartment of Mathematical Sciences\nBath, BA2 7AY, United Kingdom\nE-mail address: am2735@bath.ac.uk\n66"    },    {        "title": "2402.00990v1.Criteria_for_dynamic_stall_onset_and_vortex_shedding_in_low_Reynolds_number_flows.pdf",        "content": "Criteria for dynamic stall onset and vortex shedding in low\nReynolds number flows\nSarasija Sudharsan∗and Anupam Sharma†\nIowa State University, Ames, IA, 50011\nNovember 6, 2023\nAbstract\nDynamic stall at low Reynolds numbers, Re∼O(104), exhibits complex flow physics with\nco-existing laminar, transitional, and turbulent flow regions. Current state-of-the-art stall on-\nset criteria use parameters that rely on flow properties integrated around the leading edge.\nThese include the leading edge suction parameter or LESP (Ramesh et al., 2014) and bound-\nary enstrophy flux or BEF (Sudharsan et al., 2022), which have been found to be effective for\npredicting stall onset at moderate to high Re. However, low Reflows feature strong vortex shed-\nding events occurring across the entire airfoil surface, including regions away from the leading\nedge, altering the flow field and influencing the onset of stall. In the present work, the ability of\nthese stall criteria to effectively capture and localize these vortex shedding events in space and\ntime is investigated. High-resolution large-eddy simulations for an SD7003 airfoil undergoing\na constant-rate, pitch-up motion at two Re(10,000 and 60 ,000) and two pitch rates reveal a\nrich variety of unsteady flow phenomena, including instabilities, transition, vortex formation,\nmerging, and shedding, which are described in detail. While stall onset is reflected in both\nLESP andBEF , local vortex-shedding events are identified only by the BEF . Furthermore,\nthese events can be localized in space and time by considering the contributions to the BEF\nfrom different airfoil sections, which holds significant promise for effective flow control.\n1 Introduction\nDynamic stall is a topic of great interest in unsteady flows since it can lead to aerodynamic forces\nand moments severe enough to cause catastrophic structural failure (McCroskey, 1981; Corke &\nThomas, 2015). Stall control efforts are most effective before the formation of the dynamic stall\nvortex (DSV) (Chandrasekhara, 2007), a characteristic feature of ‘deep’ dynamic stall. Therefore,\ncharacterizing stall onset is of crucial importance for control efforts to be deployed in a timely\nmanner. Various criteria for dynamic stall onset based on unsteady aerodynamic coefficients have\n∗Graduate student, Department of Aerospace Engineering, Iowa State University (ISU).\n†Associate Professor, Department of Aerospace Engineering, ISU.\n1arXiv:2402.00990v1  [physics.flu-dyn]  1 Feb 2024been explored to formulate first-order, semi-empirical, dynamic stall models (Leishman & Beddoes,\n1989; Sheng et al., 2005). Several stall criteria have been proposed to narrow down the identification\nof stall onset to a finer degree in time. These include the leading edge suction parameter or the\nLESP (Ramesh et al., 2014), which is pressure-based, and the boundary enstrophy flux or the\nBEF (Sudharsan et al., 2022), which is vorticity-based. Reduced-order models of unsteady stall\nincorporate stall criteria based on the LESP to determine the onset of leading edge vortex shedding.\nIn the present work, we analyze the max( |BEF|) criterion to characterize vortex shedding and\nunsteady stall at low Reynolds numbers and compare it against the max( LESP ) criterion. BEF\nhas previously been explored for chord-based Reynolds numbers, Re∼ O\u0000\n105−106\u0001\n, and has\nbeen found to reach its maximum magnitude in advance of DSV formation. More generally, the\nmax(|BEF|) criterion signifies the instance of maximum wall shear and indicates imminent vortex\nformation.\nWe first summarize the flow physics of low- Redynamic stall and make the case that the prop-\nerties of the BEF make it particularly suitable for characterizing unsteady stall at low Re.\nLow-Reaerodynamics are dominated by a rich variety of coherent vortical structures such as\nshear layer vortices, a dynamic stall system comprised of multiple vortices, and induced secondary\nvortex flow structures. These vortices generally start out as laminar with strong spanwise coherence\nand undergo transition as the airfoil angle of attack increases. In their large-eddy simulations (LES)\nof a plunging airfoil at Re60,000, Visbal (2011) found that the DSV system and the shear layer\nvortices independently undergo flow transition. In the same study, they observed that the flow\nremained laminar through the entire plunging cycle at Re1,000. Figure 1 shows a schematic of\nthe type of flow observed at moderate pitch/plunge rates as a function of Reand the effective\nangle of attack, αeff, which is the instantaneous angle of attack accounting for the airfoil motion.\nNote that the unsteadiness introduced by the pitching/plunging maneuver could shift the transition\nboundary.\nAsαeffincreases, spanwise-coherent shear layer vortices are formed, followed by the establish-\nment of a DSV system, which propagates downstream and is eventually shed. At lower Reand\nαeff, the flow over the airfoil is laminar, though the laminar boundary layer is susceptible to sepa-\nration when subject to an adverse pressure gradient (APG), resulting in the formation of a laminar\nseparation bubble (LSB). If the APG remains low, the free shear layer above the bubble does not\ntransition and instead reattaches laminarly. This leads to a long LSB extending over most of the\nairfoil with fully laminar flow (Shyy et al., 2007). However, as αefforReis increased, the APG\ndownstream of the suction peak increases, encouraging spanwise instabilities that lead to earlier\ntransition and reattachment, and shorter LSBs. Therefore, during a pitch-up motion at moderate\npitch rates, the DSV system could remain laminar through part of the motion depending on the Re,\nas illustrated by Fig. 1. Unsteady flow at such low Recan consist of a mix of laminar, transitional,\nand turbulent flow regimes in space and time. In contrast, transitional effects are very small at\nhigh Resince the shear layer transitions close to the leading edge, with turbulent flow prevailing\nover most of the airfoil. It has also been observed that a single coherent DSV is more typical of\n2Figure 1: Schematic showing flow regimes prevailing over varying αeffandRe\nhighRe, while a DSV system consisting of one or more laminar/transitional vortices is likely at low\nRe(Galbraith & Visbal, 2010). At high Re, there is a pronounced effect on the moment coefficient,\nCm, which undergoes severe divergence (Visbal, 2011). In contrast, multiple, relatively less severe\nmoment stall occurrences are typical at low Re. Dynamic stall vortices are shed faster, and stall\noccurs earlier. The early stall is attributed to the stronger viscous response of the boundary layer\nat low Re, leading to the earlier formation of secondary structures having strong circulation that\ncut off the primary vortices from the feeding shear layer (Widmann & Tropea, 2017).\nThe structure of the LSB is highly sensitive to Re, as determined from experiments on an\nNACA 66 3-018 airfoil by O’Meara and Mueller (1987), where the LSB more than doubled in length\nwhen the Rewas decreased from 140,000 to 50,000. Ol et al. (2009) reported a similar increase\nin LSB size with reducing Rein their experiments in a free-surface water tunnel where shallow\nand deep stall cases were investigated for a pitch-plunge motion of a rigid airfoil in the range\n10,000≤Re≤60,000. The vortices shed at low Realso have larger length scales. Visbal (2009)\nobserved that the vorticity contained in laminar DSVs was an order of magnitude greater compared\nto fully transitioned DSVs, which was attributed to increased viscous dissipation and cancellation of\nprimary vorticity by secondary vorticity ejected from the wall in the turbulent case. The formation\nof laminar vortices is also accompanied by larger streamwise gradients in momentum thickness and\nedge velocities (Shyy et al., 2007). The above observations suggest that the BEF , which is sensitive\nto changes in pressure gradient and vorticity, which are especially pronounced for laminar vortices,\nwould be effective at signaling instances of vortex formation.\nThe current state-of-the-art criterion used to characterize unsteady stall is the LESP (Ramesh\net al., 2014; Narsipur et al., 2020), which is the camber-wise suction force at the leading edge.\nLESP has been used to trigger leading edge vortex shedding in reduced-order models based on\na pre-determined critical value. max( LESP ) is a standalone criterion that has been used as a\nproxy for critical LESP in some cases (Deparday & Mulleners, 2018, 2019) to identify the onset\n3of dynamic stall. Both criteria, max( LESP ) and max( |BEF|), have previously been observed to\nreach their peak magnitudes in advance of DSV formation regardless of the stall type (leading-edge,\nmixed, or trailing-edge) at Re∼ O\u0000\n105−106\u0001\n, in cases where DSV played a significant role in\nthe stall process (Sudharsan et al., 2023). Therefore, the max( LESP ) criterion is evaluated in the\npresent work for comparison.\nWhile some studies (e.g., Visbal (2009)) have provided descriptions of the unsteady flow field\nat low Re, we focus on the development of transitional instabilities in the flow and how they\naffect leading edge flow, where stall criteria are typically evaluated. We assess the applicability\nof the max( |BEF|) and max( LESP ) criteria to Re∼ O(104), both for signaling imminent DSV\nformation and localized vortex shedding. Expanding their applicability would enable the use of\na fundamental, standalone parameter in reduced-order dynamic stall models for the prediction of\nvortex shedding and DSV formation over a wider range of Re. It also holds significant promise for\nstall control efforts.\n2 Methods and Datasets\nOur analysis is based on wall-resolved LES carried out using the compressible flow solver FDL3DI\n(Gaitonde & Visbal, 1998). The flow over an SD7003 airfoil undergoing a constant-rate, pitch-\nup motion is simulated at low-to-moderate pitch rates (Ω+\n0= Ω 0c/U∞, with freestream velocity\nU∞= 1) of 0 .05 and 0 .25 at Reof 10,000 and 60,000. Spanwise-periodic boundary conditions\napplied to the ends of a span of length 0 .2care used to simulate an infinite wing geometry. Extensive\nstudies have been carried out on the effect of grid resolution and spanwise extent for dynamic stall\nsimulations using the FDL3DI solver in the same Rerange by Visbal (2011, 2009); Garmann and\nVisbal (2009). The O-grid mesh used in the present study consists of 554 ×380×101 points in\nthe circumferential, radial, and spanwise directions, respectively. The selected discretization is on\npar with the finest grids simulated in the cited studies. The simulated airfoil has a unit chord,\nand the farfield boundary is located about 100 chord lengths away, where freestream conditions are\nspecified. The airfoil surface is modeled as an adiabatic, no-slip wall. The spanwise extent must\nbe sufficiently large to prevent the artificial suppression of large-scale spanwise instabilities and\navoid numerical artifacts due to the imposed span periodicity. However, the computational cost\nincreases with span size. A uniform span-wise spacing of 0 .002cbetween grid points is used over the\nspan-wise extent of 0 .2c, which is sufficiently large for the Reand pitch rates under consideration,\nbased on the studies cited above. Even for the largest Reconsidered in this study, the y+values\nin the static simulations remain well below 1 over the entire airfoil surface. Appendix A presents\nsome of the static simulation results. Figure 2 shows images of the grid used in the present study.\nA nondimensional time step size (∆ t∗= ∆t U∞/c) of 1 ×10−4is used for time integration, as is\ntypical in LES simulations using FDL3DI in the literature. Additional details on the solver are\navailable in Sharma and Visbal (2019) and Visbal and Gaitonde (2002).\nTable 1 lists the datasets used in the current analysis. In all cases, an SD7003 airfoil having a\n4(a) full view\n (b) zoomed-in view\n (c) trailing-edge region\nFigure 2: Grid used in the present study:(a) full view, (b) zoomed-in view, and (c) trailing-edge\nregion. Every third point in the radial and circumferential directions is shown for clarity in panels\na and b.\nTable 1: Datasets used in the present work. In all cases, an SD7003 airfoil is pitched up at a\nconstant rate about the quarter-chord point, at M∞= 0.1.\nCase# Acronym Re Ω+\n0\n1 R60-p05 60,000 0.05\n2 R60-p25 60,000 0.25\n3 R10-p05 10,000 0.05\n4 R10-p25 10,000 0.25\nmaximum thickness of 8.5% chord at x/c= 0.24 and a camber of 1.4% chord undergoes a constant-\nrate, pitch-up motion about its quarter-chord point at a freestream Mach number, M∞= 0.1. A\nsmooth hyperbolic tangent function is used to reach the final nondimensional pitch rate, Ω+\n0, as\ndescribed in Sharma and Visbal (2019). The results reported herein are obtained by averaging\nthe three-dimensional solutions in the spanwise direction. The datasets consist of two different Re\n(10,000 and 60,000) at two different Ω+\n0(0.05 and 0.25), for a total of four cases. The acronyms\nused in Table 1 will be used to refer to each case in the rest of the paper. Each acronym consists\nof two parts separated by a hyphen. In the first part, the numbers following ‘R’ represent the Re\nin thousands; for example, ‘R10’ refers to Re= 10,000. In the second part, the numbers following\n‘p’ refer to the nondimensional pitch rate in hundredths; for example, ‘p05’ refers to Ω+\n0= 0.05.\n3 Results & Discussion\n3.1 Definitions of BEF andLESP\nWe first provide the definitions of the BEF andLESP parameters for reference. For a 2D flow\nfield, the BEF is the flux of the squared spanwise vorticity at the wall scaled by Re, as shown\nin Eq. (1). That is, ∂(ω2/2)/∂n, which is written as a product of vorticity ( ω) and vorticity flux\n5(∂ω/∂n ), and scaled by Re.\nBEF =1\nReZ(x/c)s\n(x/c)pω∂ω\n∂nds, (1)\nwhere ωis normalized by U∞/c. The normal and tangential coordinates to the airfoil surface, n\nands, respectively, are normalized by c. The integral is carried out between x/con the pressure\nside to x/con the suction side, with all quantities calculated in the airfoil frame of reference.\nThe factor Recan be combined with the vorticity flux to write an equivalence relation with the\nfavorable streamwise pressure gradient for small tangential accelerations (that is, (1 /Re)(∂ω/∂n ) =\n−(1/ρ)(∂p/∂s )). Therefore, large contributions to the BEF arise from regions where high vorticity\ncoexists with large pressure gradients. At moderate to high Re, this occurs primarily near the\nairfoil leading edge, and hence the BEF is nearly independent of integration length at such Reas\nlong as the region very close to the leading edge is included. The integration region was determined\nto be about 1% chord for Re≥ O\u0000\n105\u0001\nby Sudharsan et al. (2022).\nLESP , given by Eq. (2), is a measure of the chord-wise or camber-wise suction force Fsuction\nnear the leading edge, obtained by integrating surface pressure. In the forward part of the airfoil\n(upstream of max thickness location, which is at x/c= 0.24 for the SD7003 airfoil), Fsuction points in\nthe−xdirection, indicating a ‘suction’ force acting on the airfoil, which explains the nomenclature.\nThe integration to obtain Fsuction is conventionally carried out from the maximum thickness point\non the pressure side to that on the suction side. Since the SD7003 airfoil has a very small camber\n(∼0.014c), the chord-wise direction has been used to compute LESP , which has negligibly small\ndifferences compared to using the camber direction.\nLESP =p\n|Csuction|/(2π),where Csuction =Fsuction /(q∞c),andq∞=ρ∞U2\n∞. (2)\nIn reduced-order models based on unsteady thin airfoil theory, the instance of vortex shedding\nfrom the leading edge is determined by the LESP reaching a critical value (Ramesh et al., 2014).\nThe idea is based on the suction at the leading edge being limited (via vortex shedding) to a\ncertain (critical) value that can be supported. The LESP is a measure of the A0coefficient in the\nFourier series expansion of vorticity from classical thin airfoil theory (Katz & Plotkin, 2001); A0\nrepresents the vorticity at the leading edge. A leading-edge vortex is, therefore, shed any time the\nLESP reaches the critical value, thus limiting the maximum LESP in the model to the critical\nvalue. This approach requires the critical LESP value to be determined a priori (via simulations or\nmeasurements) instead of real-time parameter tracking; critical LESP depends on airfoil geometry\nand operating conditions. max( LESP ) has been used as a proxy for critical LESP to circumvent\nthis impediment (Deparday & Mulleners, 2018, 2019), and will be evaluated in the present work\nfor comparison.\nThe critical value of LESP is the LESP at the time instant when the profile of the skin\nfriction coefficient ( Cf) over the suction surface of the airfoil near the leading edge first develops\nan inflection point (Narsipur et al., 2020). This is referred to as the Cf-signature criterion, which\nsignifies the development of instabilities within the LSB. These instabilities lead to the bursting of\n6the LSB, which is followed by DSV formation. Note that the Cf-signature criterion per se is hard\nto use as a stall indicator because it is spatially localized, and the location of the inflection point\nis not fixed or known a priori. Spatially integrated quantities such as the LESP andBEF are\ntherefore preferred.\nThe results from the numerical simulations are described in detail in the following sections.\n3.2 Case R60-p05\nWe begin by describing the results obtained at Re60,000 and pitch rate 0 .05. Figure 3 shows\nthe space-time contours of −CpandCfwith the normalized chord-wise distance along the airfoil\nsuction surface ( x/c) as the abscissa and angle of attack ( α) as the ordinate. The shear layer that\nseparates from the airfoil leading edge develops inviscid, Kelvin-Helmholtz (K-H)-type instabilities\ndownstream, resulting in the shedding of laminar vortices around 60% chord. The shed vortices\nsubsequently transition to turbulence, and the instability and transition locations move upstream\nas the airfoil pitches up. These are pointed out in the Cfcontours in Fig. 3b.\n(a)−Cp\n (b)Cf\nFigure 3: Space-time contours for Case R60-p05\nAround α∼11◦, the initial K-H instabilities give way to viscous instabilities as an LSB is\nestablished close to the leading edge. A couple of strong laminar vortices are shed around this\ntime, as pointed out in Fig. 3b. The flow in the LSB transitions to turbulence and reattaches as\nthe increasing APG amplifies the instabilities. Figure 4 shows isosurfaces of the Q-criterion (Hunt et\nal., 1988) colored by spanwise vorticity values at the following instances: (a) initial K-H instability of\nthe shear layer leading to laminar vortex shedding, (b) upstream propagation of the K-H instability,\nand (c) the development of viscous instabilities near the leading edge. Stall occurs primarily due\nto flow breakdown at the leading edge. Contours of spanwise vorticity, ω, as the unsteady motion\nprogresses are included in Movie 1.\nA DSV is formed from the shed vorticity near the leading edge. The DSV grows and convects\n7(a)α= 6.9◦\n(b)α= 9.3◦\n(c)α= 10.7◦\nFigure 4: Isosurfaces of Q-criterion (value 100) colored by spanwise vorticity contours (inlaid legend)\nshowing the shear layer undergoing a K-H instability and subsequent transition to turbulence (a),\nupstream propagation of the K-H instability (b), and development of viscous instabilities close to\nthe leading edge (c), for Case R60-p05.\ndownstream, as seen from the imprint on the airfoil suction surface in the contours shown in Fig. 3.\nThe downstream convection of the DSV results in a moment stall, and its subsequent shedding\nresults in a lift stall, as observed from the aerodynamic coefficients ( Cl,Cd, and Cm) plotted\nagainst αin the top three panels of Fig. 5. The nondimensional time, t∗, is plotted in the top\npanel of the figure for reference. The bottom panel of Fig. 5 shows the maximum magnitude of Cp\nwithin the first 5% chord from the leading edge. Note that Cpis a point quantity, in contrast to the\naerodynamic coefficients plotted in the top panels, which are integrated quantities. As the airfoil\npitches up, the stagnation point moves downstream on the pressure side, even as the shear layer\nslowly moves away from the surface on the suction side. This leads to a net increase in the curvature\nof the shear layer near the leading edge, leading to a rise in Cp. However, when strong vortices\nare generated and shed around α∼10◦, there is a slight reduction in the slope of the max( |Cp|)\ncurve. This reduction occurs because the growth of these vortices induces strong secondary vorticity\nbeneath them, resulting in the shear layer being pushed away from the surface at a faster rate.\nThis decreases the curvature of the shear layer, leading to a reduction in Cpmagnitude. Once the\nvortices are shed, the shear layer movement away from the surface again slows down, leading to\na further increase in Cpmagnitude. After the leading edge flow breaks down around 13 .5◦, the\nmagnitude of Cpbegins to reduce.\nFigure 6 shows Cpprofiles around the time when strong laminar vortices are shed. Around\nα= 10.1◦, the strengths of the shed vortices markedly increase, as pointed out in the Cfcontours\nin Fig. 3b. In addition to the instance of DSV formation, we are also interested in the instances\nof strong laminar vortex shedding. At low Re, these vortex cores have Cpmagnitudes comparable\nto peak |Cp|at the leading edge, and they induce a large viscous response at the wall, significantly\nimpacting the onset of dynamic stall and its progression. Therefore, effective criteria for vortex\nshedding need to demonstrate critical behavior around these events.\nOur prior studies at moderate-to-high Rehave shown that BEF calculated by integrating\nover 1% chord is sufficient to indicate imminent DSV formation. The small region of integration\nis justified because the effect of DSV formation is strongly felt close to the airfoil leading edge\n8Figure 5: Variation with αof aerodynamic coefficients (top three panels) and max( |Cp|) near the\nfirst 5% of airfoil chord (bottom panel) for Case R60-p05.\nthrough the collapse of leading-edge Cp. In contrast, laminar vortex shedding, which occurs at low\nReis a spatially localized event not limited to the leading-edge region. Therefore, we calculate\nBEF by integrating over different chord lengths to capture localized vortex-shedding events. The\nlowest integration length is set to 0 .05csince the spatial scales are larger at these low Re.LESP is\ntypically calculated by integrating up to the point of maximum thickness, according to the definition\nprovided by Narsipur et al. (2020). For comparison, we integrate both parameters over a range of\nfractional chord lengths. The suction force used in the definition of LESP given by Eq. (2) has a\nlength scale cin the denominator. We retain the same length scale cwhile varying the integration\nlength used. Setting this length scale to the integration length would simply scale the LESP curves\nby a constant. We also use a low-pass filter to remove the high-frequency fluctuations in time that\nare captured in the LES simulations for LESP ,BEF , and max( |Cp|) near the leading edge. A\nGaussian filter with a half-width equal to the inverse of a nondimensional frequency f+=fc/U∞\nof 20 is used.\nFigure 7 plots the variation of LESP and|BEF|obtained by integrating over different chord\nlengths. Both parameters reach their respective global maxima between 13◦−14◦, which is ahead of\nDSV formation. This point reflects the leading edge |Cp|reaching a maximum, following which DSV\nformation begins. In addition, two secondary peaks, corresponding to strong laminar vortices that\nare shed, are observed around α∼10◦for the |BEF|curves with integration length ≥0.25c. Note\nthat there is no change in BEF across integration lengths unless there is an additional downstream\nsource of vorticity at a given time. That is, the BEF curves all coincide at initial times and vary\nonly as the flow develops. This is in contrast to the LESP , where the values vary with integration\n9Figure 6: −Cpprofiles close to the time of strong laminar vortex shedding. The profile correspond-\ning to relatively strong vortex shedding is highlighted in red.\nlength even at the initial times (note the vertical displacement of the LESP curves at low α).\nAnother aspect of BEF behavior is the contribution from vortex shedding being in the same sense\nas the contribution from the leading edge (see Fig. 9 and the related description in Sudharsan et\nal. (2022) for a detailed explanation on the sense of BEF contributions over the airfoil).\nSince the effect of the laminar vortices is felt locally, they are indicated by the BEF when the\npertinent area is included in the integration region. Calculating BEF over different integration\nlengths and comparing the difference can also identify the region over the airfoil where vortex\nshedding occurs. This is demonstrated in detail in Section 3.6. The LESP curves do not show\na peak around the instance of laminar vortex shedding since only the camber-wise component of\nthe suction force features in the LESP definition (see Eq. (2)). A vortex shed away from the\nleading edge has a more pronounced effect on the normal component (as opposed to the camber-\nwise component) of the suction force. The curves corresponding to LESP and|BEF|integrated up\nto less than 0 .25cshow a slight decrease in slope around the instance of laminar vortex shedding.\nThis can be attributed to a temporary reduction in leading-edge Cpmagnitude due to the change\nin the curvature of the shear layer caused by the downstream vortex shedding.\nThese results demonstrate that the BEF can be used to identify imminent vortex shedding\noccurring away from the airfoil leading edge.\n3.3 Case R60-p25\nNext, we discuss the results obtained at the same Reof 60,000, albeit with a higher pitch rate of\n0.25. The unsteady lag effects due to the higher pitch rate serve to delay the angle of attack at\nwhich stall occurs, in comparison with Case R60-p05. Moreover, the higher pitch rate also promotes\na more conventional leading-edge stall, characterized by the breakdown of the LSB and a drop in\n10(a)LESP\n (b)|BEF|\nFigure 7: LESP (a) and BEF (b) integrated over different chord lengths, plotted against α, for\nCase R60-p05.\nleading-edge |Cp|. Space-time contours of −CpandCfare shown in Fig. 8. As in the previous\ncase, the inflectional velocity profiles in the shear layer subjected to an APG develop an inviscid\nK-H instability, leading to vortex roll-up and transition to turbulence. This occurs close to 0 .6c\ninitially, as pointed out in the Cfcontours (Fig. 8b).\nBetween approximately 15◦and 20◦, a sequence of events transpires in rapid succession. Span-\nwise vorticity contours in Fig. 9 show the dynamic flow field during this time. In panel a of\nthe figure, the shedding of vortices due to K-H instabilities in the shear layer and transition to\nturbulence on the aft section of the airfoil are evident. Panel b shows a time instance after the\nestablishment of the LSB with laminar reattachment. In the downstream region of this reattached\nflow, a second separated shear layer develops, experiencing K-H instabilities that lead to the roll-\nup of laminar vortices and downstream transition to turbulence. After its initial establishment,\nthe LSB’s existence is short-lived, as it quickly succumbs to viscous instabilities magnified by the\nincreasing APG. During this time, strong vortices are shed from the rear of the LSB. Almost simul-\ntaneously, the K-H instabilities also undergo substantial amplification, giving rise to a series of shear\nlayer vortices that roll up downstream (panel c). This case serves as an illustration of a transition\nmechanism wherein both viscous instabilities within the LSB and the inviscid K-H instabilities are\nsimultaneously amplified. This phenomenon is clearly discernible from the CpandCfcontours in\nFig. 8, where a horizontal row of vortices is observed shortly after the formation of the LSB. Similar\nbehavior has previously been observed in water tunnel experiments for a pitching-plunging airfoil\nundergoing deep stall at Re=20,000 (see Fig. 9 of Ol et al. (2009)).\nFollowing the onset of instabilities, the LSB collapses and a coherent leading-edge DSV is\nformed. The subsequent growth of the DSV and the shear layer vortices are showcased in panels\nd-f of Fig. 9. The DSV begins to convect downstream as it grows, with a higher convection speed\ncompared to the downstream shear layer vortices. It entrains a pair of previously-shed shear\n11(a)−Cp\n (b)Cf\nFigure 8: Space-time contours for Case R60-p25.\n(a)α= 15.0◦\n(b)α= 15.5◦\n(c)α= 16.7◦\n(d)α= 18.4◦\n(e)α= 19.6◦\n(f)α= 20.7◦\nFigure 9: Case R60-p25: Vorticity contours showing the simultaneous amplification of the K-H\ninstability in the shear layer and the viscous instabilities within the LSB (panels a-c) leading to the\nroll-up of vortices. Panels d-f show the growth of the leading-edge DSV and shear layer vortices.\nPurple represents clockwise vorticity, while green represents counter-clockwise vorticity.\n12Figure 10: Variation with αof aerodynamic coefficients (top three panels) and max( |Cp|) near the\nfirst 5% of airfoil chord (bottom panel) for Case R60-p25.\nlayer vortices further downstream (marked in Fig. 8b). During its downstream propagation, the\nDSV maintains its proximity to the airfoil surface until α∼45◦. A more comprehensive visual\nrepresentation of this sequence is included in the accompanying video (see Movie 2).\nThe top three panels of Fig. 10 show the variation of the unsteady aerodynamic coefficients\nwith α. Moment stall occurs around α∼25◦when the DSV convects past the quarter-chord point.\nMaximum lift occurs around α∼33◦, which is marked as the lift stall point in the figure. However,\nthe lift remains elevated until 45◦, attributed to the proximity of the DSV to the airfoil surface.\nAs the DSV progressively convects away from the airfoil surface, the reduction in vortex-induced\ncontribution to the bound circulation leads to a decline in Cl. The bottom panel of Fig. 10 shows\nthe variation of peak leading-edge |Cp|. The collapse of the LSB is accompanied by a reduction\nmax(|Cp|) near the leading edge around α∼18◦, as is typical in a bubble-bursting, leading-edge\nstall.\nThe variation with αofLESP andBEF is shown in Fig. 11. Both parameters exhibit distinct\npeaks for all integration lengths, attributed to the decrease in leading-edge |Cp|as the LSB breaks\ndown. The larger values of |BEF|for integration lengths greater than 0 .05care due to the large\nvorticity and vorticity flux associated with the vortices shed downstream of the LSB as it becomes\nunstable. Due to the greater spatial distance of the LSB from the geometric leading edge and its\nextended length, DSV formation occurs around 0 .1crather than very close to the leading edge.\nThis is consistent with the expected behavior at low Re(Gaster, 1967).\nFigure 11 shows that the instance of the peak in LESP moves aft (to higher α) with increasing\nintegration length. This is due to the increasing contribution from the suction induced by the DSV\nas it grows and influences a larger portion of the airfoil between the leading edge and the maximum\nthickness point. Hence, even though leading-edge Cpmagnitude begins to drop in advance of\n13(a)LESP\n (b)|BEF|\nFigure 11: LESP (a) and BEF (b) integrated over different chord lengths, plotted against α, for\nCase R60-p25.\n20◦, the LESP curve obtained by integrating over 0 .25ccontinues to rise until about 22◦. In\ncontrast, the peak |BEF|location occurs before 20◦for all integration lengths. The leading edge\nCppeak representing stall onset is captured by the BEF curve corresponding to integration up to\n0.05c. For integration lengths larger than 0 .05c, the peak BEF value is much higher due to the\nstrong vortices shed from the rear of the LSB around the same time. The vortex shedding event\nis captured by the BEF for this case as well when appropriate regions of the airfoil are included.\nThere is another peak around 25◦for integration lengths ≥0.25c, which corresponds to an increase\nin induced secondary vorticity as the DSV gets stronger as it grows. While LESP captures the\nstall onset point, the localized vortex shedding event is missed.\n3.4 Case R10-p05\nFor the lower Recases, the DSV remains laminar through nearly the entire maneuver. These cases\nare also characterized by a DSV system consisting of several large-scale laminar vortices.\nFigure 12 shows the vorticity contours at specific instances. A region of reversed flow at the\ntrailing edge propagates upstream as the airfoil starts to pitch up. Around α∼10◦, the separated\nshear layer from the leading edge becomes susceptible to instabilities and sheds spanwise vortices.\nPanel a ( α∼11◦) shows the initial stages of the unstable, separated shear layer. The instability\nlocation moves upstream as the airfoil continues to pitch up. The large-scale shear layer vortices\ninduce secondary vorticity beneath them. The induced secondary vorticity develops into coherent,\ncounter-clockwise vortices, which act to cut off the primary shear layer vortex that is formed around\nmid chord.\nPanel b ( α∼13.3◦) shows the first instance of secondary vorticity (SVa) cutting off the primary\nshear layer vortex (PVa), and the roll-up of the shear layer upstream into another vortex (PVb).\nThe shedding of shear layer vortices and induced secondary vortices result in multiple instances of\n14(a)α= 11.0◦\n(b)α= 13.3◦\n(c)α= 14.8◦\n(d)α= 15.7◦\n(e)α= 18.3◦\n(f)α= 20.3◦\nFigure 12: Vorticity contours showing the formation of a DSV system through shear layer vortex\nand wall interactions for Case R10-p05.\nvortex entrainment. Therefore, the DSV system in the present case comprises multiple shear layer\nvortices entrained together as αincreases. Note that the DSV system is centered downstream of\nthe quarter-chord point of the airfoil during its incipience. Panels c ( α∼14.8◦) and d ( α∼15.7◦)\nshow the merging of PVa and PVb into a single coherent vortex and its downstream convection,\nrespectively. During the later part of the unsteady motion (panels e-f), the DSV, continuing\nto entrain more leading-edge vortices, transitions to turbulence as it propagates downstream. The\nDSV weakens due to the entrained counter-clockwise vorticity (shown in green contours) and viscous\ndissipation. The dominant, large-scale vortices located away from the surface of the airfoil generate\nand influence the motion of smaller-scale secondary vortices and, over time, tertiary vortices close\nto the wall, that affect surface quantities and make it challenging to interpret the space-time plots\nofCpandCf. Movie 3provides a clearer picture of the sequence of events as the airfoil pitches up.\nThe top three panels of Fig. 13 show the variation with αof the unsteady aerodynamic co-\nefficients. Large wiggles are observed in Cl,Cd, and Cmbefore stall due to the strong laminar\nvortices shed from the shear layer. Clcontinues to increase until α∼18.8◦due to vortex-induced\nsuction/lift from multiple shear layer vortices. However, large fluctuations in max( |Cp|) near the\nleading edge (shown in the bottom panel of Fig. 13) are observed much earlier as large variations\n15Figure 13: Variation with αof aerodynamic coefficients (top three panels) and max( |Cp|) near the\nfirst 5% of airfoil chord (bottom panel) for Case R10-p05. The red x’s correspond to panels b-d in\nFig. 12.\nin shed vorticity occur downstream. The peaks and the valley in max( |Cp|), identified by red x’s\nin Fig. 13, are due to the vortices highlighted in panels b-d in Fig. 12. This undulatory behavior\nof max( |Cp|) is attributed to the changes in the curvature of the shear layer around the leading\nedge due to the vortices shed downstream. As αincreases beyond 12 .5◦and vortex shedding occurs\naround mid-chord, the induced secondary vorticity pushes the shear layer away from the airfoil\nsuction surface, reducing its curvature and causing max( |Cp|) to dip. After 15◦, the (PVa + PVb)\nDSV system detaches from the shear layer, arresting the movement of the shear layer away from\nthe surface, thereby temporarily but sharply increasing max( |Cp|) near the leading edge.\nThe variation with αofLESP andBEF integrated over various chord lengths is shown in\nFig. 14. The shedding of the downstream shear layer vortices that affects leading-edge Cpalso\nimpacts the variation in LESP andBEF . For the BEF , integrated lengths up to 0 .25cfollow\nthe same trend as max( |Cp|) near the leading edge. The larger integrated lengths show larger\n|BEF|values at the peak locations, corresponding to the vorticity shed downstream. The increase\ninBEF around 13◦correponds to increased induced secondary vorticity from the roll-up of PVa.\nThe increase to the second peak around 15 .0◦correponds to the entrainment of PVa by PVb during\nwhich PVa moves closer to the wall, causing increased secondary vorticity. The small increase to\nanother peak around 18 .3◦corresponds to another instance of vortex merging with the DSV system,\nwhich again induces increased secondary vorticity. The LESP curves for all integration lengths\nshow nearly the same trend as max( |Cp|) near the leading edge. Note that there is no additional\nlocal (in space and time) contribution to LESP from the vortices formed around mid-chord; the\ncurves show the same trend and are almost simply displaced along the ordinate. The largest\nintegration length shows a deviation in trend at high αsince the chord-wise force contribution\n16(a)LESP\n (b)|BEF|\nFigure 14: LESP (a) and |BEF|(b) integrated over different chord lengths, plotted against α, for\nCase R10-p05.\nfrom the shed vortices is directed in the opposite direction over the rear of the airfoil. It is the\nlocal contributions around vortex formation that distinguish the BEF from the LESP . This is\nelucidated in section 3.6 by delineating specific contributions from different airfoil sections.\n3.5 Case R10-p25\nThe higher pitch rate case at lower Reis also characterized by the shedding of multiple shear layer\nvortices with the flow over the airfoil remaining laminar throughout. The difference from the lower\npitch rate case (R10-p05) is the formation of a stronger DSV system further upstream with more\npronounced lag effects due to unsteadiness. Movie 4 clearly illustrates the sequence of events during\nthe unsteady maneuver using spanwise vorticity contours.\nThe sequence of events remains similar to Case R10-p05 but is postponed to higher αdue to\nthe pronounced unsteady lag. Figure 15 shows span-averaged vorticity contours at a few instances\nduring the pitch-up maneuver. The upstream propagation of trailing edge reversed flow and the\ndevelopment of instabilities in the shear layer leading to vortex formation are observed in this case\nas well. However, due to the larger APG encountered by the flow, stronger leading-edge vortices are\nshed farther upstream (around 0 .12c), beginning around α∼20◦(Fig. 15a). Induced secondary\n(counter-clockwise) vorticity acts to cut off the shear layer, which forms clockwise vortices near the\nairfoil leading edge (Fig. 15 panels b-d). When the clockwise vortex (marked as ‘PVa’) rolls up,\nit cuts off the downstream shear layer from the leading edge, leading to its roll-up into vortices\ndownstream (as pointed out in Fig. 15b). The secondary vorticity itself lifts up due to the induction\nfrom the leading edge vortices, rolls up, and is pinched off by the clockwise vorticity induced by it\n(Fig. 15c). This process repeats a few times, and multiple vortices are shed from the leading edge.\nSimilar to Case R10-p05, the DSV system continues to entrain these shed vortices and grows in\nsize. DSV-induced suction on the airfoil surface is strongest around α∼33◦(Fig. 15e). The effect\n17(a)α= 20.2◦\n(b)α= 22.8◦\n(c)α= 25.7◦\n(d)α= 29.7◦\n(e)α= 33.2◦\n(f)α= 49.7◦\nFigure 15: Vorticity contours at different instances during the pitch-up maneuver for Case R10-p25.\nSimilar behavior as in Case R10-p05, but is delayed (in t∗andα) due to the higher pitch rate, and\nvortex shedding occurs closer to the leading edge.\nof the DSV system on the surface drops as it moves farther away and becomes more diffuse due to\nviscosity and annihilation from the entrained secondary vorticity (Fig. 15f). The DSV system also\nentrains the downstream shear layer vortex around this time.\nThe top three panels of Fig. 16 show the variation of the unsteady aerodynamic coefficients.\nMaximum lift occurs around α∼33◦, when the DSV system induces the strongest suction over\nthe airfoil surface. The lift continues to remain high until around 45◦, owing to the DSV system\nremaining relatively close to the airfoil surface and continuing to interact with the newly shed\nleading-edge vortices and induced secondary vorticity. The magnitude of leading edge Cpshows\nundulations (see bottom panel of Fig. 16) due to the effect of vortices shed downstream, similar to\nthe previous case.\nFigure 17 shows the variation of LESP and|BEF|for the present case for different integration\nlengths. Both sets of curves are characterized by a series of peaks for all integration lengths. The\npeak corresponding to the maximum leading edge |Cp|is clearly observed from BEF at the lowest\nintegration length. In contrast to the previous case, the newly shed shear layer vortices remain\nvery close to the leading edge (for example, see PVb in Fig. 15b). Therefore, the contribution to\n18Figure 16: Variation with αof aerodynamic coefficients (top three panels) and max( |Cp|) near the\nfirst 5% of airfoil chord (bottom panel) for Case R10-p25.\nLESP from the suction induced by these vortices is significant, leading to multiple peaks. A series\nof significant spikes in |BEF|is observed for larger integration lengths. The first instance of such\na peak around 22◦corresponds to the roll-up of vortex PVa (shown in Fig. 15a). The subsequent\ninstances correspond to the DSV system entraining a new leading edge vortex, during which it\nmoves closer to the airfoil surface, inducing strong secondary vorticity. The trend of the BEF\nlocalizing the points corresponding to peak |Cp|and DSV-induced secondary vorticity peaks holds\nfor the present case as well.\n(a)LESP\n (b)|BEF|\nFigure 17: LESP (a) and BEF (b) integrated over different chord lengths, plotted against α, for\nCase R10-p25.\n19Figure 18: Different sections of the airfoil referred to in subsequent figures.\n3.6 Comparison of results\nWe next distinguish between the max( LESP ) and max( |BEF|) criteria. We divide the airfoil into\nmultiple sections as shown in Fig. 18 and identify contributions from different regions of the airfoil\nto the two parameters. Since the definition of LESP involves a square root, we instead compare\nthe suction force coefficient, Csuction (see Eq. 2). The contributions to both Csuction andBEF are\nprimarily negative, so we plot the contributions to the negative of these quantities.\nFigure 19 shows the contributions to Csuction andBEF from the airfoil sections marked in Fig. 18\nfor Case R60-p05. The contribution from the pressure side (PS) to either parameter is negligible.\nThe largest contributions are from the leading edge region (marked ‘LE’) for both Csuction and\nBEF . The Post-LE and THK regions contribute to Csuction throughout the unsteady maneuver.\nThe effect of increasing (or decreasing) leading edge |Cp|is felt at these regions since there is a\nsignificant chord-wise component of the surface normal. In contrast, the contribution to BEF from\nvariation in leading-edge |Cp|is limited to the LE region. The Post-LE region does not contribute\ntoBEF because, even though |Cp|varies with αin this region, the pressure gradient (similar\nbehavior to vorticity flux) drops to zero. The local peaks in the BEF curves that occur before\nthe peak in the LE curve correspond to local vortex shedding, while those following it correspond\nto the DSV convecting downstream and growing stronger. As noted in section 3.2, strong laminar\nvortices are shed from the THK and MID regions around α∼10◦. These are reflected as peaks in\nthe corresponding BEF curves in Fig. 19b. A strong vortex shed from the rear of the LSB is also\ncaptured in the THK region. At higher α, the MID and AFT regions show an opposite contribution\ntoCsuction relative to other regions. This arises from the convection of the DSV center downstream,\nwhich induces high |Cp|in the MID and AFT regions where the chord-wise component of the surface\nnormal points in the + xdirection. The peaks in BEF from the MID and AFT regions at higher\nangles of attack( α∼16.5◦and 20◦) occur due to strong secondary vorticity induced by the DSV\nas it grows stronger while convecting downstream. These peaks occur later in time for LESP\ncompared to BEF since the chordwise component of the surface normal is more significant towards\nthe rear of the airfoil.\nCase R10-p25 is also briefly presented in a similar way. Figure 20 shows the contributions to\nCsuction andBEF from different airfoil sections shown in Fig. 18. The contribution to Csuction\nfrom LE includes multiple peaks, due to the leading edge vortices remaining farther upstream\nfor this case. In contrast, the LE contribution to BEF shows a clear peak, following the same\n20(a)Csuction\n (b)BEF\nFigure 19: Contributions to Csuction (a) and BEF (b) from different airfoil segments, for Case\nR60-p05.\ntrend as max( |Cp|) near the leading edge (see bottom panel of Fig. 16). Specific instances of\nvortex-roll-up/vortex entrainment, which induce increased secondary vorticity into the DSV system\nare observed from the other regions. These include the roll-up of PVa, PVb, merging of PVa\nand PVb, and the entrainment of further leading edge vortices and the downstream shear layer\nvortex into the DSV system. The contributions to Csuction from the Post-LE and THK regions are\ninfluenced by the variation of leading edge Cpand leading edge vortices. Opposite contributions to\nCsuction due to DSV convection are captured in the MID and AFT regions. A similar breakdown\nof contributions to Csuction andBEF for the remaining cases, namely, R60-p25 and R10-p05 is\npresented in Appendix B.\nFigure 21 illustrates the overall picture that emerges from the results of the four simulations,\nincluding the performance of LESP andBEF in identifying critical flow events. At Re= 10,000\n(cases R10-p05 and R10-p25), the flow remains laminar almost through the entire maneuver. There\nis a large region of reversed flow, and the Cfis negative over nearly the entire suction surface of the\nairfoil. Due to the larger viscous response, the DSV system comprises multiple large-scale laminar\nvortices. These vortices are shed further upstream (close to the airfoil leading edge) for the high\npitch-rate case.\nCases with Re= 60 ,000 (R60-p05 and R60-p25) demonstrate a more typical LSB-bursting,\nleading-edge stall. During the establishment of the LSB, strong, small-scale vortices are shed. For\nthe higher pitch rate case, the DSV forms farther upstream and is stronger, as reported in the\nliterature (Acharya & Metwally, 1992). The DSV is fed by the vorticity shed from the leading\nedge, but due to the higher Re, it does not organize into large coherent vortices before merging\nwith the DSV.\nOf the two stall criteria under investigation, max( LESP ) mainly captures variations in leading\nedge Cp. The max( |BEF|) criterion captures the variation in leading edge Cpand also enables\n21(a)Csuction\n (b)BEF\nFigure 20: Contributions to Csuction (a) and BEF (b) from different airfoil segments, for Case\nR10-p25.\nFigure 21: Illustration of observations from all four cases considered in the present study\n22the localization of vortex shedding events occurring anywhere on the airfoil surface in space and\ntime. This includes strong, small-scale vortex shedding that is not reflected in the leading edge Cp\nas well as large-scale vortex shedding events located away from the leading edge. These events are\ndirectly captured in the spatial region where they occur. Specific instances of vortex roll-up and\nentrainment, which trigger an increase in induced secondary vorticity are captured by the BEF .\nThe versatility of the BEF parameter makes it a suitable candidate for flow control at low Re.\n4 Conclusion\nWe investigate airfoil dynamic stall at low Reynolds numbers where laminar, transitional, and\nturbulent regimes can coexist, giving rise to rich fluid dynamics. The dynamic stall process is\nstrongly influenced by multiple vortices that shed from different regions of the airfoil not limited\nto the leading edge. Current state-of-the-art stall onset criteria based on the LESP andBEF\nparameters, are calculated by integrating these quantities around the leading edge and hence may\nnot directly capture vortex shedding occurring away from it.\nWe evaluate the max( LESP ) and max( |BEF|) criteria over an extended integration region for\nlow-Re(∼ O\u0000\n104\u0001\n) dynamic stall for an LES dataset consisting of the SD7003 airfoil undergoing\na constant-rate, pitch-up maneuver at two Revalues at two pitch rates. The highly-resolved LES\nresults provide insights into the unsteady flow phenomena (instabilities, shear layer dynamics,\nvortex formation, pairing, shedding, dissipation, etc.), which are commented on. The ability of the\nLESP andBEF parameters to capture vortex-shedding events is analyzed using these results.\nOur analyses indicate that the max( |BEF|) criterion captures strong vortex-shedding events\nwhen the integration region includes the locations of these events. These events are reflected as local\nmaxima in the BEF curves. The LESP , being based on the camber-wise component of pressure\nforce, captures only the effect of vortex shedding events that significantly affect Cpnear the leading\nedge. The BEF parameter differs from LESP in that it directly captures the flow events such as\n(a) strong, small-scale vortex shedding events that do not significantly impact leading edge Cp, and\n(b) large-scale shed vortices, localizing them in space in time. It captures instances of vortex roll-up\nand entrainment, which induce increased secondary vorticity. Therefore, the BEF parameter can\nbe used to localize stall onset and vortex-shedding events in space and time for effective flow control\nat low Re.\nAcknowledgments\nThis material is based upon work supported by the National Science Foundation (Grants CBET-\n1935255 and 1554196) and the US Air Force Office of Scientific Research (Award # FA9550-\n23-1-0016). We also acknowledge the computational resources provided by Argonne Leadership\nComputing Facility and Iowa State University.\n23Declaration of Interests\nThe authors report no conflict of interest.\nReferences\nAcharya, M., & Metwally, M. H. (1992). Unsteady pressure field and vorticity production over a\npitching airfoil. AIAA Journal ,30(2), 403-411. Retrieved from https://doi.org/10.2514/\n3.10931 doi: 10.2514/3.10931\nChandrasekhara, M. S. (2007). Compressible dynamic stall vorticity flux control using a dynamic\ncamber airfoil. Sadhana ,32, 93–102. Retrieved from https://doi.org/10.1007/s12046\n-007-0008-8 doi: 10.1007/s12046-007-0008-8\nCorke, T. C., & Thomas, F. O. (2015). Dynamic Stall in Pitching Airfoils: Aerodynamic Damping\nand Compressibility Effects. Annual Review of Fluid Mechanics ,47(1), 479–505. Retrieved\nfrom http://www.annualreviews.org/doi/10.1146/annurev-fluid-010814-013632 doi:\n10.1146/annurev-fluid-010814-013632\nDeparday, J., & Mulleners, K. (2018). 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Numerical investigation of the effect of airfoil thickness on onset\nof dynamic stall. Journal of Fluid Mechanics ,870, 870–900. doi: 10.1017/jfm.2019.235\nSheng, W., Galbraith, R. A. M., & Coton, F. N. (2005, 11). A New Stall-Onset Criterion for Low\nSpeed Dynamic-Stall. Journal of Solar Energy Engineering ,128(4), 461-471. Retrieved from\nhttps://doi.org/10.1115/1.2346703 doi: 10.1115/1.2346703\nShyy, W., Lian, Y., Tang, J., Viieru, D., & Liu, H. (2007). Aerodynamics of low reynolds number\nflyers . Cambridge University Press. doi: 10.1017/CBO9780511551154\nSudharsan, S., Ganapathysubramanian, B., & Sharma, A. (2022). A vorticity-based criterion to\ncharacterise leading edge dynamic stall onset. Journal of Fluid Mechanics ,935, A10. doi:\n10.1017/jfm.2021.1149\nSudharsan, S., Narsipur, S., & Sharma, A. (2023). Evaluating dynamic stall-onset criteria for\nmixed and trailing-edge stall. AIAA Journal ,0(0), 1-16. Retrieved from https://doi.org/\n10.2514/1.J062011 doi: 10.2514/1.J062011\nVisbal, M. R. (2009). High-fidelity simulation of transitional flows past a plunging airfoil. AIAA\nJournal ,47(11), 2685-2697. Retrieved from https://doi.org/10.2514/1.43038 doi: 10\n.2514/1.43038\nVisbal, M. R. (2011). Numerical investigation of deep dynamic stall of a plunging airfoil. AIAA\nJournal ,49(10), 2152-2170. Retrieved from https://doi.org/10.2514/1.J050892 doi:\n10.2514/1.J050892\nVisbal, M. R., & Gaitonde, D. V. (2002). On the use of higher-order finite-difference schemes on\ncurvilinear and deforming meshes. Journal of Computational Physics ,181(1), 155-185.\nWidmann, A., & Tropea, C. (2017). Reynolds number influence on the formation of vortical\nstructures on a pitching flat plate. Interface Focus ,7(1). doi: 10.1098/rsfs.2016.0079\n25Appendices\nA Static simulations\nStatic simulations were performed using FDL3DI at α= 4◦for both Re; only the results for\nRe= 60,000 are shown for brevity. The simulations were run for about 40 characteristic convective\ntimes ( c/U∞) to ensure that the forces and moments reached statistical stationarity. Comparisons\nof surface pressure and skin friction coefficient distributions, CpandCf, repectively, are made with\nXFOIL (Drela, 1989). XFOIL results are obtained with the Ncrit parameter set to 9, corresponding\nto a low freestream turbulence intensity. Figure 22 shows good agreement between LES and XFOIL\nresults; the transition on the suction surface occurs around 60% chord in both.\n(a) surface pressure coeff., Cp\n (b) skin friction coeff., Cf\nFigure 22: Comparison of CpandCfdistributions between static LES and XFOIL for Re60,000\natα= 4◦.\nB Contributions to Csuction andBEF from different airfoil regions\nThe contributions to Csuction andBEF from different sections of the airfoil (see Fig. 18) for Cases\nR60-p25 and R10-p05 are shown in Fig. 23. Critical flow events identified by the two parameters\nare annotated in the figure.\n26(a)Csuction , R60-p25\n (b)BEF , R60-p25\n(c)Csuction , R10-p05\n (d)BEF , R10-p05\nFigure 23: Contributions to Csuction andBEF from different airfoil segments, for cases R60-p25\n(a,b) and R10-p05 (c,d).\n27"    },    {        "title": "2402.00997v1.Hitting_probability_for_Reflected_Brownian_Motion_at_Small_Target.pdf",        "content": "HITTING PROBABILITY FOR REFLECTED BROWNIAN\nMOTION AT SMALL TARGET\nYUCHEN FAN\nAbstract. We derive the asymptotic behavior of hitting probability at small\ntarget of size O(ϵ) for reflected Brownian motion in domains with suitable\nsmooth boundary conditions, where the boundary of domain contains both\nreflecting part, absorbing part and target. In this case the domain could be\nlocalized near the target and explicit computations are possible. The asymp-\ntotic behavior is only related to ϵup to some multiplicative constants that\ndepends on the domain and boundary conditions.\nContents\n1. Introduction 1\n2. Domains and Construction of Reflected Brownian Motion 2\n2.1. Construction of RBM on General (SULD) Domains 2\n2.2. Construction of RBM on Analytic Simply Connected Domains in C 5\n3. Main Result 7\n3.1. Description of Hitting Target 7\n3.2. Main Result 7\n4. Standard Definitions and Standard Theorems 8\n5. Proof of Main Result in Simply Connected Domains in C 9\n5.1. Explicit Computation in H 9\n5.2. Proof of Main Results in C 12\n6. Proof of Main Result in General SULD Domains 15\n6.1. Localization of Domain 15\n6.2. Explicit Computation and Estimation in U′16\n6.3. Proof of Main Result 19\n7. Appendix: Proofs of technical details 19\n8. Acknowledgement 22\nReferences 23\n1.Introduction\nReflected Brownian motions, or more generally reflected Wiener processes, can\nroughly be thought as a Brownian motion that is “reflected” when it hits some\ntargets. Although it has simple intuitive description, it is extremely hard to con-\nstruct Reflected Brownian motions in domains in Rnforn≥2 [1, 2]. Despite of\nthis difficulty, reflected Brownian motion has became a great area of research in\nDate : July 2023.\n1arXiv:2402.00997v1  [math.PR]  1 Feb 20242 YUCHEN FAN\nboth pure mathematics and applied mathematics [1, 3, 4]. In particular, its hitting\nbehavior at small targets has grasped the interests of more and more researchers\n[5, 6]. Despite there has been some works done in asymptotic hitting behaviors\nof Brownian motions at small targets [7, 8], the asymptotic hitting behaviors of\nreflected Brownian Motions at small targets is poorly understood.\nIn this paper we will mainly be focused on asymptotic hitting probability for\nreflected Brownian motion at small targets. The current state of art in this area\nis preforming explicit computations using connections between reflected Brownian\nmotion and partial differentiate equations, which was done by Denis S. Grebenkov\nand Adrien Chaigneau in certain domains [4]. However, this approach is limited to\nhuge amount of computations and could only be done in a few domains where solu-\ntions to Dirichlet problem behaves nicely. Instead, in this paper we observed that\nthe reflected Brownian motion behaves very well in a large class of domains which\nwe defined to be “Smooth Uniform Lipchitz Domains” (see Definition 2.4), and in\nthis class of domains the asymptotic behavior hitting probability is comparable to\na fundamental solution of Laplacian. The intuition is when the reflect Brownian\nparticle is near the target, the probability that it hits non-targeted boundary is\nvery small and thus irrelevant to the universal shape of the domain. When target\nis very small, the local geometry of SULD domain near the target is nearly half\nspace, so the asymptotic hitting probability should agree to the case of a half ball\nwith target at the bottom planar surface, and computation could be done easily in\nthis case.\nComplex analysis, especially conformal maps and harmonic measures, has played\na crucial role in studies of planar Brownian motions [9, 10]. We will also use\nmethods from complex analysis to prove a more accurate result (see Theorem 3.4).\nRegrettably, this accurate result haven’t be generalized to higher dimensions as the\nmethod of complex analysis ceases to work.\nAlthough this paper successfully computed the asymptotic behavior of hitting\nprobability in a large class of domains, the asymptotic mean hitting time still re-\nmains open except for some elongated domains [6]. Besides, the asymptotic hitting\nprobability for domains with rough boundaries (e.g. fractal domains) still remains\nopen. The methods in this paper all cease to work for these kind of domains and\neven some basic properties of reflected Brownian motion hasn’t been well-studied\nfor these domains yet.\nIn this paper, we will construct reflected Brownian motion in chapter 2, and\nshow any analytic simply connected domain in Cis SULD. Some technique proofs\nwill be included in chapter 7 appendix. In chapter 3 we define the target, absorbing\nboundaries and state our main results (Theorem 3.3 and Theorem 3.4). In chapter\n4 we cite some standard definitions and results in complex analysis. We will prove\nTheorem 3.3 and Theorem 3.4 using complex analysis tools in Cin chapter 5 and\nprove Theorem 3.3 in higher dimension in chapter 6.\n2.Domains and Construction of Reflected Brownian Motion\nIn this section we construct the reflected Brownian motions (RBM) in certain\ndomains.\n2.1.Construction of RBM on General (SULD) Domains. In this subsection\nwe discuss how to define RBM on any connected domains in Rn(n≥2) with\nsuitable smooth and analytical conditions.HITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 3\nDefinition 2.1. LetΩbe a connected open set of Rn. We say Ωis a smooth\ndomain if at each point x∈∂Ωthere is an open ball Bxcentered at xand a smooth\nbijection Φx:Bx→D⊂Rnwith smooth inverse such that Φx(Bx∩Ω)⊂Rn\n+:=\n{(x1, ..., x n) :x1>0}andΦx(Bx∩∂Ω)⊂∂Rn\n+. We call Φxbe the coordinate\nmapping associated with Bx.\nThere are many equivalent ways to construct RBM, in here we follow the con-\nstruction by [2]. Constructions in more general domain could be done, for example,\nin [11] using Dirichlet form. We define the RBM by explicit construction of solu-\ntion to stochastic differential equation which generate RBM in an intuitive sense.\nLetHn:=Rn\n+be the upper half space in RnandWtbe a n-dimensional unre-\nstricted standard Brownian motion. Consider the stochastic differential equation\nin (Xx\nt, ξx\nt):\n(1) dXx\nt= dWt+1∂Hn(Xx\nt)γ(Xx\nt)dξx\nt\nwith Xx\n0=xandξx\n0= 0, and γis a constant vector valued function with γ≡\n(1,0, ...,0). (Xx\nt, ξx\nt) is adapted to the σ-algebra that with probability 1, ξx\ntis non-\ndecreasing in tand increases only at set ∆ := {t:Xx\nt∈∂Hn}and ∆ is a Lebesgue\nnull set. Now we use explicit construction to proof the following proposition\nProposition 2.2. Under the condition above, the stochastic differential equation\n(1)has a unique solution pair (Xx\nt, ξx\nt)in the sense that any other pair satisfies (1)\nand∆be a Lebesgue null set is equal to this pair with probability 1. We define the\nreflected Brownian motion in Rn\n+to be the process Xx\nt.[2]\nProof. We only proof its existence here. For uniqueness see [2]. Define a trans-\nformation Γ : C(Rn)→C\u0000\nRn\n+\u0001\n(the same symbol Γ may be used when the\nparameter set is [0 , T]) as follows: for ζ= (ζ1, ζ2, ..., ζ n)∈C(Rn), η= Γ( ζ) is\ndefined by η= (η1, η2, ..., η n), where ηi=ζifori= 2,3,···, n, and η1(t) =\nζ1(t)−\u0012\u0012\ninf\n0≤s≤tζ1(s)\u0013\n∧0\u0013\n. Write Γ t(ζ) for (Γ( ζ))(t). Now define the transforma-\ntionξ:C\u0000\nRd\u0001\n→C(R) by Γ( ζ)−ζ= (ξ(ζ),0,···,0) and write ξt(ζ) for ( ξ(ζ))(t).\nThen follow the proof of proposition 1 of [2] we have that, if Yx\ntis the standard\nBrownian motion in Rn, then Xx\nt= Γ◦Yx\ntandξx\nt=ξ◦Yx\ntsolves (1) and Xx\nt\nandξx\ntsatisfy the conditions imposed in connection with (1) . The pair ( Xx\nt, ξx\nt)\nis uniquely determined by (1) and the associated conditions, i.e., any other pair\nsatisfying (1) and the associated conditions is equal to ( Xx\nt, ξx\nt) with probability\none. □\nWe introduce an equivalent formulation of RBM in Hn.\nProposition 2.3. LetXx\ntbe defined in 2.2. Let Yx\nt= (Yx\n1(t), Yx\n2(t), ..., Yx\nn(t))\nbe the standard Brownian motion in Rnsuch that Yx\n0=x= (x1, x2, ..., x n)∈Hn\nand each Yx\nk(t)is the independent one dimensional Brownian motion. Define the\nprocess\nYx\nt= (|Yx\n1(t)|, Yx\n2(t), ..., Yx\nn(t))\nand the associated local time process\nLx\nt=1\n2Zt\n0δ(Yx\n1(s))ds.\nThen (Yx\nt, Lx\nt)is equal to (Xx\nt, ξx\nt)constructed in Proposition 2.2 with probability\n1.4 YUCHEN FAN\nProof. By Itˆ o’s formula we have\n|Yx\n1(t)|=Zt\n0sign ( Yx\n1(s))dYx\n1(s) +1\n2Zt\n0δ(Yx\n1(s))ds=Yx\n1(t) +1\n2Zt\n0δ(Yx\n1(s))ds\nthus ( Yx\nt, Lx\nt) satisfies (1). As Brownian motion spend almost zero time on any\npoint set we see that ∆ = {t:Yx\n1(t) = 0}is a Lebesgue null set. So by Proposition\n2.2 we see that ( Yx\nt, Lx\nt) is equal to ( Xx\nt, ξx\nt) with probability 1. □\nNow for a general domain Ω with smooth boundary and satisfies some conditions,\nwe may construct the RBM in the following way. We firstly give the definition of a\n“Smooth Uniform Lipchitz domain” (SULD) in any dimension.\nDefinition 2.4. We say Ω⊂Rnis a SULD if it satisfies the following conditions:\n(1) Ω is a connected smooth domain;\n(2) Ω can be covered by U=\b\nU0, U1, ...\t\nwhich is a countable family of open\ncover of Ωwhere U0⊆Ωis equipped with the original Euclidean coordinate\nofRnwhich defines Ωand{Ui:i≥1} ⊂ { Bx:x∈∂Ω}where Bxare\ndefined in Definition 2.1;\n(3)LetΦkbe the coordinate mapping associated with Ukdefined in Definition\n2.1. We require Φkmaps normal on Uk∩∂Ωto a vector pointing towards\n(1,0,0, ...,0), which is equivalent to ∇Φi\nk·γ=δ1i|∇Φi\nk|.\n(4)We will suppose all (Φk:k≥1)satisfies uniform Lipchitz condition in the\nsense that there is an universal constant Cthat\n|x−y|\nC≤ |Φk(x)−Φk(y)| ≤C|x−y|\nfor all k≥1andx, y∈Uk. We will also require the following conditions:\nThere exists constants C1,C2,C3such that\nmax\u001a\r\r\r\r∂Φi\nk\n∂xj\r\r\r\r\n∞,∥∆Φi\nk∥∞\u001b\n≤C1,max\u001a\nLip\u0012∂Φi\nk\n∂xj\u0013\n,Lip\u0000\n∆Φi\nk\u0001\u001b\n≤C2.\nfor all k,jwhere Lip(f)denotes the Lipchitz constant of f. Finally if we\nletJkbe the Jacobian of Φkthen Ak=JkJT\nksatisfies\nC−1\n3|x|2≤xTAkx≤C3|x|2\nfor all k,jandx∈Rn.\nSULD represents a large class of domains as we shall see analytic domains in\nCare generally SULD (Proposition 2.9). Now we can construct RBM in SULD\nfollowing [2].\nProposition 2.5. LetΩbe a SULD, and γ(x)be a vector field varies smoothly on\nthe boundary and uniformly bounded away from tangent space of the boundary at\nx. then the stochastic differential equation\n(2) dXx\nt= dWt+1∂Ω(Xx\nt)γ(Xx\nt)dξx\nt\nhas a unique solution pair (Xx\nt, ξx\nt)in the sense that any other pair is equal to this\npair with probability 1. In particular when γis the unit normal vector at each x, we\ndefine the reflected Brownian motion in Ωto be the corresponding solution process\nprocess Xx\nt.[2]HITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 5\nProof. (Existence part) We let x∈Ω and there exists an non negative integer\nk(x) such that positive number r=r(x) such that Bx(r)⊂Uk(x). We use the\ncoordinate patch of Uk(x). We will construct the process in Bx(r) until the time S1\nthat it leaves Bx(r). Then we will keep constructing until the last time S2before\nthe process leaves the associated ball around Xx\nS1and so on. For the first step we\nhave the following cases:\n(1) If k(x) = 0 only then the process is just the normal Brownian motion as\nboundary is not involved;\n(2) If we can choose k(x)>0 then the coordinate mapping Φ kassociated with\nUk(x)changes the problem to a problem in the upper half space with normal\nreflection on boundary. The new process Φ k(Xt) satisfies a new stochastic\ndifferential equation by itˆ o’s formula: Write Φ k=f= (f1, ..., fn), then\ndfi(Xt) = nX\nk=1∂fi\n∂xkdWk\nt!\n+1\n2∆fidt+1∂Φk(Uk)∩H(f(Xt))δi1|∇f1|dξt\nwhere ( W1\nt, ..., Wn\nt) is the standard ndimensional Brownian motion. By\nour assumption on the domain, this stochastic differential equation has a\nunique solution by [2] (proposition 1), with local time changed to\nΞt=Zt\n0|∇f1(Bt)|dξt\n(3) This problem was solved above and we obtain a process in the upper half\nspace with normal reflection. Then Φ−1\nkmaps the process in upper half\nspace back to Uk(x)which gives us Xx\nt,0≤t≤S1. For this process (1)\nwill hold, with ξx\ntbeing identical to the local time on the boundary for\nthe process in upper half space. Then we just repeat this and gives us the\ndesired process.\n□\nHowever, non-uniqueness choices of k(x) result in the same process. We will\nhave to additionally show the consistency of our definition which is missed in [2].\nProposition 2.6. Suppose V=\b\nV0, V1, ...\t\nis another cover of Ωdefined in 2.4\nand(Ψk)k≥1be another collection of associated maps. Suppose for some kandm\nthatVm∩Ukis non-empty. Then the transition map\nΦk◦(Ψm)−1: Ψm(Vm∩Uk)→Φk(Vm∩Uk)\nmaps process Ψm(Xx\nt)to the process Φk(Xx\nt).\nProof. See Appendix, 7.1. □\nWe give another equivalent construction of RBM in analytic simply connected\ndomains in Cas a special case in order to apply tools in complex analysis to proof\na stronger result.\n2.2.Construction of RBM on Analytic Simply Connected Domains in C.\nDefinition 2.7. LetΩ⊊ Cbe a domain in the sense that it is open and connected.\nLet∂Ω := ¯Ω\\˚Ω. We say Ωisanalytic if there is an univalent function fanalytic\nnear ∂Dsuch that ∂Ωis the image of ∂Dunder f. We say Ωis analytic simply\nconnected domain if it is additionally simply connected.6 YUCHEN FAN\nOne equivalent construction of the RBM on analytic domain is using conformal\nmapping. We define RBM in upper half plane Hthen define the RBM in general\nanalytic simply connected domain Ω by conformal maps, where Riemann mapping\ntheorem ensure there is an conformal map between Hand Ω.\nDefinition 2.8. Let(Xt, Yt)be the standard Brownian motion in R2where Xt, Yt\nare independent standard Brownian motion in R. Define another stochastic process\nBt= (Xt,|Yt|)as the (complex) reflected Brownian motion inH. Identify Cwith\nR2our reflected Brownian motion in HisBt=Xt+i|Yt|. We write Bz\ntto denote\nthe reflected Brownian motion starts at z. We also note the process Xt+iYtis\nreferred to the complex Brownian motion by [9].\nBy Proposition 2.3 we see the stochastic process Btis equal to RBM in Hcon-\nstructed in proof of Proposition 2.2 with probability 1. We now construct RBM in\nunit disk D:={z∈C:|z|<1}.\nProposition 2.9. Dis a SULD. More over, any domain Ωdefined in Definition\n2.7 is a SULD.\nProof. See Appendix, 7.2. □\nBy the above proof and Proposition 2.5 we can construct the RBM in D. Alter-\nnatively we have another equivalent construction of RBM in D. We firstly prove a\npowerful characterisation of conformal invariance of RBM in H.\nProposition 2.10. Suppose ΩandΩ′are two simply connected open subset of H\nandBz0\ntis the RBM in Hwhere z0∈Ωwith the associated local time process ξz0\nt.\nLetfbe a conformal automorphism of H. Let τΩ= inf{t≥0 :Bz0\nt∈∂Ω}. Then\nwith probability one, (f(Bz0\nt), t∈[0, τ))is equal to the time-changed RBM in H.\nProof. See Appendix, 7.4. □\nBy examine above proof and notice that any conformal map f=u+ivfrom Ω\ntoHthat can be extended conformally between a neighborhood of Ω and Hwith\nfinite number of singularities (as we may delete them as the probability that the\ncomplex Brownian motion hit them is 0) on ∂Ω, we have a stronger proposition:\nProposition 2.11. Letfbe a conformal map between ΩandH. If fcan be\nextended conformally between a neighborhood Ω′ofΩandH′ofHwith finite number\nof singularities on ∂Ω, then fmaps the complex RBM in Ωto a time-changed\ncomplex RBM in H.\nProposition 2.12. LetBz\ntbe defined in Definition 2.8. Let f(z) =iz−i\nz+i. Then\nthe process f(Bz\nt)is the RBM started at f(z)which is equivalent to complex RBM\ninDconstructed by Proposition 2.5 after a suitable time change, which is denoted\nbyXf(z)\nt.\nProof. We notice that the point ∞is mapped to ibyf, which could be removed\nfrom consideration as {i}is a null set. We note that by above proposition ( fj)−1◦f\nis an automorphism of Hand thus it maps Bz\ntto another time-changed complex\nRBM in H. It follows that f(z) is the time changed complex RBM in D. □\nBy above arguments we can now give an equivalent definition of the complex\nRBM via conformal mapping:HITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 7\nDefinition 2.13. Suppose Btis the complex RBM in Dand let Ωbe an analytic\nsimply connected domain. Let gbe the associated conformal map from ΩtoDabove.\nThen we define the process ˜Bt=g−1(Bt)to be the reflected Brownian motion in Ω.\nProof. By lemma 7.3 we see that fcould be extended so f◦(fi)−1could be the\nassociated maps (properties could be checked by procedure similarly to proof of\nproposition 2.9 and by open mapping theorem each open cover of Dis mapped to\nopen cover of Ω by f. □\nWe would like to finish this section by the following key proposition, which is\na generalization of proposition 2.11. Its proof is also similar to the proof we did\nabove.\nProposition 2.14. LetΩ1andΩ2be two analytic simply connected domain. By\nRiemann mapping theorem there is a conformal map fbetween Ω1andΩ2which\ncould also be extended to a conformal map between open neighborhood of Ω1and\nΩ2. Let Btbe the complex RBM in Ω1, then f(Bt)is the time-changed complex\nRBM in Ω2, starts at f(z0).\n3.Main Result\nIn this section we define the hitting target, absorbing boundary and reflecting\nboundaries. We will also states our main result.\n3.1.Description of Hitting Target.\nDefinition 3.1. (Absorbing Boundary, Target and Reflecting Boundary) Let Ωbe\na SULD in S=RnorS=C. Let Pbe a point on ∂Ω. Let R > 0be a fixed\nconstant that\u0010\nS \\BR(P)\u0011\n∩∂Ωis non-empty, which will be denoted by abs.\nLetϵ >0be a very small constant ( ϵ≪R) and then our target (which is also\nabsorbing) is Bϵ(P)∩∂Ω, which will be denoted by tg.\nLet our reflecting boundary be∂Ω\\(abs∪tg)which will be denoted by ref.\nSee Figure 1 for an example.\nThroughout the paper we assume there is a path inside Ωthat connects a point\nintgandabs.\nWe now give definition of the absorbing boundaries.\nDefinition 3.2. LetBx0\ntbe the reflected Brownian Motion in Ω. Let τ= inf{t≥\n0 :Bx0\nt∈abs∪tg}be the stopping time that RBM hits absorbing boundaries.\nWe consider the stopped process Bx0\nτ∧tto be the RBM associated to the absorbing\nboundaries. In the later texts, without specification, Bx0\ntdenotes Bx0\nτ∧t.\n3.2.Main Result. The following theorem is our main result:\nTheorem 3.3. LetΩbe a SULD in Rn(identify CasR2) and Pbe a point on\n∂Ω, with target, reflecting boundary and absorbing boundary defined above. Let\n(Bx0\nt, t > 0)be the reflected Brownian Motion starting at x0∈Ω, and τ= inf{t≥\n0 :Bx0\nt∈abs}. Define\nGn,y(x) =(\n−1\n(n−2)ωn−1|x−y|2−nifn∈N\\{2}\n1\nω1log|x−y| ifn= 28 YUCHEN FAN\nFigure 1. Absorbing Boundary, Reflecting Boundary and Target\nbe the generalized Newtonian potential, then there exists non-zero, positive constants\nC1, C2determined by Ωandz0such that\nC1\nGn,0(ϵ)≤P(Bx0\nτ∈tg)≤C2\nGn,0(ϵ)\nwhere ϵ= (ϵ,0,0...,0)∈Rn, for all ϵsmall enough.\nWe have a stronger result, which confirm the asymptotic behavior of P(Bz0τ∈tg)\nfor certain domain.\nTheorem 3.4. LetΩbe an analytic simply connected domain in C. If in addition\nthere is only one component of absorbing boundary, then there exists a constant\nC(Ω, z0)that depends on Ωand starting point z0such that\nlim\nϵ↓0P(Bz0\nτ∈tg) log ϵ=C(Ω, z0).\n4.Standard Definitions and Standard Theorems\nIn this section we give some definitions to avoid confusions. Also we quote some\nuseful standard theorems and lemmas.\nDefinition 4.1. Aconformal map between two domain is a holomorphic bijection\nbetween these two domains. [12] pp.206\nTheorem 4.2. (Riemann Mapping Theorem) Let Ωbe a simply connected domain\nandΩ̸=C, then there exists a conformal map between ΩandD. [12] Chapter 8,\nTheorem 3.1\nWe also note a uniqueness result.\nTheorem 4.3. Letfbe a conformal bijection between DandΩwhere Ωis an\nanalytic simply connected domain. Then by lemma 7.3 fmay be extended to aHITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 9\nconformal bijection g. If in addition we fix two point on ∂Ωtogether with their\nderivatives, then such fis unique.\nWe also have the following very useful Christoffel-Schwarz formula which maps\nHonto polygons.\nTheorem 4.4. Assumen−1X\nk=1βk<2. Let pbe a polygonal region in the complex\nplane with vertexs a1,a2, ..., anwhere the angle at vertex akfork < n is(1−βk)π\nand the angle at vertex anis(1−βn)πwhere\nβn= 2−n−1X\nk=1βk.\nLetFbe a conformal map from the upper half-plane to psuch that the points\nA1, . . . , A n−1,∞=AnonRare sent to the vertices of pwhere Akis sent to ak.\nThen there exist unique constants C1andC2such that\nF(z) =C1Zz\n0dζ\n(ζ−A1)β1···(ζ−An−1)βn−1+C2.\nwhere (ζ−Ak)βkis defined by cutting along ray {Ak−ri:r≥0}. Furthermore,\nsuch Fis unique. [12] (Chapter 8 Theorem 4.7)\nWe also use a bit result from harmonic measure.\nDefinition 4.5. LetΩbe an analytic simply connected domain, zbe a point inside\nΩandAbe an Lebesgue measurable subset of ∂Ω. Then by Riemann mapping\ntheorem there exists a unique conformal bijection maps ΩontoDsuch that f(z) = 0\nandf′(z)>0. Then Ais mapped onto f(A)on∂Dand we define |f(A)|be the\nLebesgue measure of |f(A)|on measurable space ∂D. The harmonic measure ofA\nwith respect to z,ωΩ(z, A), is then defined by ωΩ(z, A) =|f(A)|\n2π. [13]\nWe note the following relation between Brownian motion and harmonic measure.\nTheorem 4.6. Let(Bt:t≥0)be a complex Brownian motion in ΩandAbe a\nmeasurable set on ∂Ω. Let zbe a point inside Ω. Then we have\nωΩ(z, A) =P(Bz\nτ∈A).\n[14] pp.117\n5.Proof of Main Result in Simply Connected Domains in C\n5.1.Explicit Computation in H.In this subsection we proof a result that is very\nsimilar and can be easily modified to our main result by explicitly computation in\nupper half plane H.\nTheorem 5.1. LetH={z∈C:ℑ(z)>0}be our domain and (0,1)∪(1 +ϵ,∞)\nbe our reflecting boundary, and [−∞,0]∪[1,1 +ϵ]be our absorbing boundary while\n[1,1 +ϵ]is our target. Let (Bt, t≥0)be the RBM in H. Then\nP(Bz0\nτ∈(1,1 +ϵ))∼C(z0)\nlogϵ\nasϵ→0, where C(z0) =ℜ(lim ϵ↓0fϵ(z0)).10 YUCHEN FAN\nProof. By Christoffel-Schwarz mapping formula:\nfϵ(z) =Zz\n0dξ√ξ√ξ−1p\nξ−(1 +ϵ)\nwhere each√zis obtained by cutting along the negative imaginary axis, we maps\nupper half plane to a rectangle ABCD where A=fϵ(z0) := 0, B=fϵ(1) := −Lϵ,\nD=fϵ(+∞) :=−ilϵandC=fϵ(1 +ϵ) :=−Lϵ−ilϵ, as shown in figure 2. We note\nthat\nLϵ:=Z1\n0dξ√ξ√1−ξp\n(1 +ϵ)−ξ=2K\u0010\n1\n1+ϵ\u0011\n√1 +ϵ\nand\nlϵ:=Z1+ϵ\n1dξ√ξ√1−ξp\n(1 +ϵ)−ξ\nare convergent integrals, where Kis the complete elliptical integral of first kind.\nFigure 2. Conformal map\nProposition 5.2. RBM in His mapped to the time-changed RBM in rectangle\nABCD by map fϵ, with reflecting boundaries AB∪CD, absorbing boundaries AD∪\nBCand target BC.\nProof. As four points A, B, C, D are Lebesgue null sets we may remove them from\nconsideration and rectangle ABCD could be thought as an analytic simply con-\nnected domain. The boundaries are mapped to corresponding boundaries of the\nrectangle as fϵis injective. Then by proposition 2.11 with reversed direction of f\n(which also holds, proof is similar) we may conclude. □\nWe observe the following reflection property regarding the RBM in rectangle,\nwhich will be useful later.\nProposition 5.3. Letz0=x0+iy0be a point in the rectangle ABCD such that\nthere exists an open ball B:=Br(p)⊊ABCD where z0∈Br(p),p∈ABand\nr <min{dist(z0, AD),dist(z0, BC)}.\nLetRz0\ntbe the RBM in ABCD starts at z0andτB= inf{t≥0 :Rz0\nt∈∂B∩\nABCD }. Let Zz0\nt=Xx0\nt+iYy0\ntbe the standard Brownian motion in Cstarts atHITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 11\nz0such that both Xx0\ntandYy0\ntare independent standard Brownian motions on R\nstart at x0,y0, respectively. Let τ′\nB= inf{t≥0 :Zz0\nt∈∂B}, then with probability\none, we have\nRz0\nt∧τB=Xx0\nt∧τ′\nB−i\f\f\fYy0\nt∧τ′\nB\f\f\f.\nProof. By taking minus sign which preserve RBM, the stopped RBM −Rz0\nt∧τBsatis-\nfies the stochastic differential equation in the form of (1) as it has normal reflection\nonAB. Let H−z0\ntbe the RBM in Hstarts at −z0and that τ= inf{t≥0 :\nH−z0\nt∈∂(−B)∩H}. Then stopped RBM H−z0\nt∧τstill satisfies the stochastic dif-\nferential equation in the form of (1). Thus by uniqueness stated in Theorem 2.2\nwe see that with probability 1, −Rz0\nt∧τB=H−z0\nt∧τ. By Definition 2.8 we can write\nH−z0\nt∧τ=X−x0\nt∧τ′\n−B+i\f\f\fY−y0\nt∧τ′\n−B\f\f\fwhere both X−x0\nt∧τ′\n−BandY−y0\nt∧τ′\n−Bare stopped standard\nBrownian motion in C. By taking minus sign back and notice that with probability\n1,−X−x0\nt∧τ′\n−B=Xx0\nt∧τ′\nBand−Y−y0\nt∧τ′\n−B=Yy0\nt∧τ′\nB, we have that\nRz0\nt∧τB=Xx0\nt∧τ′\nB−i\f\f\fYy0\nt∧τ′\nB\f\f\f\nwith probability 1. □\nso that we may consider the RBM in rectangle as a Brownian motion in strip.\nProposition 5.4. RBM in rectangle ABCD is equivalent to the standard Brownian\nmotion in the strip Sϵ:={z∈C:−Lϵ<ℜ(z)<0}with target at Tϵ:={z∈C:\nℜ(z) =−Lϵ}and absorbing boundary at Imaginary axis.\nProof. By equivalent we mean there is an bijective map hbetween Brownian motion\nin strip (starts inside rectangle ABCD ) and RBMs in the rectangle, preserving the\ncorresponding stopping condition. The intuition is shown in figure 3: we may\nreflect ABCD over its sides with length Lϵover and over again to get the strip we\nneed. Define h0(z) =|ℑz|, hn(z) = 1− |1− |hn−1(z)|, and we notice that h(z) =\nlimn→∞hn(z) exists for all z. By proposition 5.3, hworks for the bijection. □\nThis implies that\nP(Bz0\nτ∈[1,1 +ϵ]) =P(fϵ(Bz0)τ∈BC) =P\u0010\n¯Bfϵ(z0)\nτ∈Tϵ\u0011\n=ωS(fϵ(z0), Tϵ).\nwhere ¯Bdenotes the Brownian motion in Sϵand, abusing notations, τdenotes the\ntarget hitting time of each motion. We can immediately write down the unique\nharmonic measure ωS(fϵ(z0), Tϵ) inSϵ:\nωS(fϵ(z0), Tϵ) =−ℜ(fϵ(z0))\nLϵ.\nNow plug z0=Reiθ(R > 0, θ∈(0, π)) into Christoffel-Schwarz map, after some\ncomputation we reach\nfϵ(z0) =e−iθ\n2√\nRZ1\n0dt\n√\ntq\nt−e−iθ\nRq\nt−(1+ϵ)e−iθ\nR.\nBy simple Euclidean geometry,s\f\f\f\ft−e−iθ\nR\f\f\f\fs\f\f\f\ft−(1 +ϵ)e−iθ\nR\f\f\f\f≥dist\u0012e−iθ\nR,R\u0013\n=sinθ\nR12 YUCHEN FAN\nFigure 3. Reflecting rectangle ABCD\nfort∈(0,1] and thus by Lebesgue Dominated Convergence Theorem we also have\nthat\nlim\nϵ↓0fϵ(z0) =Z1\n0dt\n√\nt\u0010\nt−e−iθ\nR\u0011.\nNow by standard result of K(x) we conclude that Lϵ∼ −logϵasϵ→0. Thus we\nconclude that\nP(Bz0\nτ∈(1,1 +ϵ))∼C(z0)\nlogϵ\nasϵ→0, where C(z0) =ℜ(lim ϵ↓0fϵ(z0)). □\nCorollary 5.5. Letϵ >0andfϵbe a conformal map mapping Hinto a rectangle\nRwhich can be extended that it maps 0,1,1 +ϵand∞to corner of R(then fϵis\nan uniquely represented Christoffel-Schwarz map). Then for any z0∈Hwe have\nlim\nϵ→∞fϵ(z0) =C1Z1\n0dt\n√\nt\u0010\nt−e−iθ\nR\u0011+C2.\nwhere z0=Reiθ,C1andC2are the unique constants in Christoffel-Schwarz formula\noffϵas described in theorem 4.4.\n5.2.Proof of Main Results in C.Now we are ready to proof the main results\nby explicit computation above. We starts from stronger result.\n5.2.1. Proof of Theorem 3.4.\nProof. We introduce some technical lemmas.HITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 13\nLemma 5.6. LetΩbe an analytic simply connected domain and Pbe a point on\n∂Ω. Let R > 0be a fixed constant that SR(P)intersects ∂Ωat non-empty point\nset{At:t∈I}for some index set I. Then sup\nt1,t2∈I∠At1PAt2is attended by some\nunique At1, At2∈∂Ω∩BR(P).\nProof. Firstly notice that ∠At1PAt2≤2πfor all t1, t2which means the supremum\nis well-defined and denote it by s. We may extract a subsequence ∠A(n)\nt1PA(n)\nt2→s.\nAs Ω is an analytic simply connected domain, observe that ∂Ω∩BR(P) is compact\nso by choosing a convergence subsequence\u0010\nA(nk)\nt1\u0011\nk≥1of\u0010\nA(n)\nt1\u0011\nn≥1and another\nconvergence subsequence\u0012\nA(nkj)\nt2\u0013\nj≥1of\u0010\nA(nk)\nt2\u0011\nk≥1, we find\u0012\nA(nkj)\nt1, A(nkj)\nt2\u0013\nj≥1\nconverge to some At1, At2∈∂Ω∩BR(P) and s=∠At1PAt2. For uniqueness we\njust notice that if there exists distinct pairs ( At1, At2) and ( A′\nt1, A′\nt2) such that s=\n∠At1PAt2=∠A′\nt1PA′\nt2, then by simple Euclidean Geometry at least one of angle\n∠MPN where M, N ∈ {At1, At2, A′\nt1, A′\nt2}is greater than s. Contradiction. □\nWe call At1, At2theendpoints of∂Ω∩BR(P). Now let A, D be the endpoints\nof tg and B, C be the endpoints of ref. We also notice that when ϵis very small\ntg has only one component. Rigorously speaking, we have the following technical\nlemma.\nLemma 5.7. LetΩbe an analytic simply connected domain, then for all ϵsmall,\nBP(ϵ)∩∂Ωhas only one connected component.\nProof. Let{Ct:t∈I}be the collection of closure of components of BP(ϵ)∩∂Ω\nthat doesn’t contain P. Then as connected component is closed it is compact.\nSo dist( P, C t)>0 for all t∈I. Now if inf {dist(P, C t) :t∈I}= 0 then Iis\ninfinite and we may take a subsequence tjsuch that dist( P, C tj)→0. Now by\ncompactness dist( P, C t) is attended by some point Mt∈Ct, so for jlarge we have\nallMtjcontained in a small disk BP(δ) (δ < ϵ ) and thus, as Ctjintersect ∂Ω at\ntwo points then their length is greater than ϵ−δfor all jlarge enough. But we\nhave infinitely many such jand thus ℓ(∂Ω)≥P\ntℓ(Ct) =∞, contradiction. So\ninf{dist(P, C t) :t∈I}=ξ >0. Now Take ϵ < ξ and we are done. □\nSo it is enough to consider one component in tg and ϵvery small. Now we can\nturn to proof the main result. We firstly uniformize our domain. By Riemann\nMapping Theorem there is an unique conformal bijection h(extended analytically\nto boundary) that maps Ω onto unit disk Dwith boundary points Pmapped into\nX,Bmapped into Ywith non-vanishing derivative at PandB. Let f(C) =Z.\nThis is true because we assume Ω has an analytic boundary. This process is shown\nin figure 4.\nNow we can proof the stronger result. Let f1(z) be the extension of f−1beyond\nDas Ω is a analytic simply connected domain (see lemma 7.3). By open mapping\ntheorem image of a neighborhood of ∂Dviaf1is an open neighborhood ∆ of ∂Ω.\nLet fixed δso small that BP(δ) is also contained in ∆. Take ϵ < δ , then f1is\ninvertible on BP(ϵ) with holomprohic inverse g1that has derivatives bounded away\nfrom 0. We can get an estimation for lower bound using the univalent map f1\nwhich has non-vanishing derivative on ∂D. Also, for any η >0 we may choose η1\nsuch that for any ϵ < η 1,|g′\n1(z)−g′\n1(P)| ≤ηfor all z∈BP(ϵ). Consider when14 YUCHEN FAN\nFigure 4. Ω being mapped into D\nη <|g′\n1(P)|andϵis sufficiently small, then by boundary correspondence, for all\nz∈∂g1(BP(ϵ)),f1(z)∈∂(BP(ϵ)). Consider the line segment joining zandg1(P),\nlet it be γ(ξ). Then,\n|z−g1(P)|=Z\nγ|(g1◦f1◦γ)′(ξ)|dξ≥(|g′\n1(P)| −η)ℓ(γ)≥(|g′\n1(P)| −η)ϵ.\nand we also simultaneously have\n|z−g1(P)| ≤Zf1(z)\nP|g′\n1(ξ)|dξ≤(|g′\n1(P)|+η)dist( f1(z), P)≤(|g′\n1(P)|+η)ϵ.\nNow let Mbe the (unique) mobius transform sending unit disk DtoHwith g1(P)\nbeing sent to 1, g1(B) being sent to 0 and g1(D) being sent to ∞. We do assume\nthatg1(A) being sent to complex number 1 −k1(ϵ)>0 and g1(B) tog(D) = 1+ k2(ϵ)\nwhen ϵsmall . Choose η′small, and by almost the same argument above we could\nalso bound\n(|M′(g1(P))|−η′)(|g′\n1(P)|−η)ϵ≤ |k2(ϵ)−k1(ϵ)| ≤(|M′(g1(P))|+η′)(|g′\n1(P)|+η)ϵ.\nforϵsmall enough. Just as above, choose Tϵ(z) =z\n1−k1(ϵ)andTϵ◦M◦g1sends\nΩ toHagain but with Ato 1 instead, so Bis sent to1 +k2(ϵ)\n1−k1(ϵ). Hence\nP(Bz0\nτ∈tg) =P\u0012\n(Tϵ◦M◦g1) (Bz0)τ∈\u0012\n1,1 +h2(ϵ)\n1−h1(ϵ)\u0013\u0013\nRemark that Tϵ◦M◦g1(z0)→M◦g1(z0) asϵ→0. As η,η′is arbitrary, hence by\nTheorem 5.1,\nP(Bz0\nτ∈tg)∼ −ℜ(M◦g1(z0))\nlog(|M′(g1(P))||g′\n1(P)|ϵ)\nasϵ↓0. Hence\nP(Bz0\nτ∈tg) log ϵ→ −ℜ (M◦g1(z0))\nasϵ↓0. □HITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 15\n5.2.2. Proof of Theorem 3.3 in C.\nProof. Our result follows from the following lemma and proof of Theorem 3.4.\nLemma 5.8. There is an arc of length independent of ϵthat contained in f(abs) .\nProof. Just a modification of length estimations in proof of Theorem 3.4. □\nObserve that, if we let c1be an arc contained in abs and c2be another arc\ncontaining f(abs) (for example, such c2can be the arc with endpoints Y,Z, which\ndoes not intersect f(tg)). Let us consider the target hitting probability for the\nabsorbing boundary to be c1,f(abs) and c2, which are P1,P2,P3, respectively,\nthenP3≤P2≤P1. As by Theorem 3.4 P3andP1are comparable to1\nlogϵasϵ↓0,\nP2is as well. □\n6.Proof of Main Result in General SULD Domains\nAs we remarked we consider the case when ϵis very small. In this case we can\nlocalize our domain\n6.1.Localization of Domain. Observe that when the RBM particle is near the\ntarget the hitting probability is asymptotically irrelevant to the shape of domain.\nWithout loss of generality let tg ∈U1and let C1,C2be two fixed n−1 di-\nmensional subsets in U1such that Φ 1(C1) = ∂BP(r1)∩Φ1(U1) and Φ 1(C2) =\n∂BP(r2)∩Φ1(U1) for some r1> r2>0 and that both curves do not intersects\nabs. As both C1andC2are disjoint compact sets their distance is separated and\nwe may assume that any curve γ: [0,1]→Ω connecting some points on abs and\nP=γ(1) that only intersects C1andC2once for each, intersects C1att1andC2at\nt2where t1< t2(C2lies “closer” to PthanC1). We let τ1= inf{t≥0 :Bx0\nt∈C2}\nfori= 1,2. Let P1:=P(Bx0τ∈tg, Bx0\n[τ1,τ]∩C1=∅). We claim the following\nobservation:\nProposition 6.1. There exists constants c1,c2such that\nc1P1≤P(Bx0\nτ∈tg)≤c2P1\nProof. From definition of P1we immediately have P1≤P(Bx0τ∈tg), and notice\nthat\nP(Bx0\nτ∈tg) =P1+P(Bx0\nτ∈tg, Bx0\n[τ1,τ]∩C1̸=∅).\nSince our RBM particle roams outside Φ−1\n1(BP(r1)∩Φ1(U1)) we have a smaller\nchance to hit tg so P1≥P(Bx0τ∈tg, Bx0\n[τ1,τ]∩C1̸=∅). Thus P(Bx0τ∈tg)≤2P1.□\nBy strong Markov property,\n(3) P(Bx0\nτ∈tg) =P(τ1≤ ∞)×P\u0010\nBBx0τ1τ∈tg\u0011\nas any RBM hits target will hits C1by geometrical observation. Now from above\nproposition for asymptotic behavior we may consider P\u0010\nBBx0τ1τ∈tg, Bx0\n[τ1,τ]∩C1=∅\u0011\n.\nThis probability can be interpreted as the probability that the RBM starts at a point\nC2, roaming in Φ−1\n1(BP(r1)∩Φ1(U1)) and hits target, which is same as the target\nhitting probability that our domain is instead the interior of Φ−1\n1(BP(r1)∩Φ1(U1))\nwith absorbing boundary C1and the same target. We can thus localize our domain16 YUCHEN FAN\ninto interior of Φ−1\n1(BP(r1)∩Φ1(U1)), which will be denoted by U′, as shown in\nfigure 5.\nFigure 5. local structure near P\n6.2.Explicit Computation and Estimation in U′.We may now turn to some\nexplicit computations and we restrict the RBM in U′for now. Now notice that\nΦ1mapsC1andC2into two semi-spheres by their definition. We denote them by\nS1,S2, respectively. We also define our RBM in Ω by Φ 1so we may consider the\noriginal RBM satisfies (1) and associated assumptions in Φ 1(U1).\nNow by similar reason as Proposition 5.3 we may reflect Φ 1(U1)∩BP(r1) along\nRn−1\n0={(0, x2, ..., x n) :xi∈R}and then the RBM in Φ 1(U1)∩BP(r1) is equivalent\nto the Brownian Motion Wtstated in equation (1) in BP(r1)\\Φ1(tg).\nNow by adapting the proof in the case of Cwith mean value inequality we\nmay assume that there exists constants k1,k2depends on Φ 1(hence Ω) such that\nBP(k1ϵ)∩Rn−1\n0⊂Φ1(tg)⊂BP(k2ϵ)∩Rn−1\n0for all ϵsmall enough, which allows\nus to have\n(4) Pk1≤P\u0010\nBBx0τ1τ∈tg\u0011\n≤Pk2\nwhere Btis restricted in U′and\nPki:=P\u0012\nΦ1(B)Φ1(Bx0τ1)\nτ ∈BP(kiϵ)∩Rn−1\n0\u0013\n(we have abused notation τ). So now we may turn the problem into the asymptotic\nhitting probability for the following problem: Given the RBM starts at a point y0\nonC1, with absorbing boundary C2, target at BP(kiϵ)∩Rn−1\n0fori= 1,2, and\nreflecting boundary at ( BP(r1)\\BP(kiϵ))∩Rn−1\n0. The target hitting probability\nisPkifor each i.\nBy reflection Pkiis the probability that the Brownian motion WΦ1(Bx0τ1)\nt , where\nWtis stated in equation (1) and restricted in BP(r1)\\(BP(kiϵ)∩Rn−1\n0) with target\nBP(kiϵ)∩Rn−1\n0and absorbing boundary ∂BP(r1), hits the target at escaping time\nofBP(r1).HITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 17\nNow we consider the sphere ∂BP(k1ϵ). Then let\nτ2= inf\u001a\nt≥0 :WΦ1(Bx0τ1)\nt /∈BP(r1)\\BP(k1ϵ)\u001b\nwriting Φ 1\u0000\nBx0τ1\u0001\n=y0, again by strong Markov Property we have that\n(5) Pk1=P\u0000\nWy0\nτ2∈∂BP(k1ϵ)\u0001\n×P\u0010\nWWy0τ2τ∈BP(k1ϵ)∩Rn−1\n0\u0011\n.\nNow we observe again that P\u0000\nWy0τ2∈∂BP(k1ϵ)\u0001\nis the probability of the Brownian\nMotion WΦ1(Bx0τ1)\nt inside the ndimensional open annulus BP(r1)\\BP(k1ϵ) with ab-\nsorbing boundary at ∂BP(r1) and target at ∂BP(k1ϵ). Now we may explicitly write\nout the solution to this target hitting probability. Since we know that the target\nhitting probability P\u0000\nWy0τ2∈∂BP(k1ϵ)\u0001\n=u1(y0) satisfies the Dirichlet problem\n\n\n∆u1(y) = 0 if y∈BP(r1)\\BP(k1ϵ)\nu1(y) = 1 if y∈∂BP(k1ϵ)\nu1(y) = 0 if y∈∂BP(r1)\nas the solution to this problem is unique, and thus we may easily verify that\nP\u0000\nWy0\nτ2∈∂BP(k1ϵ)\u0001\n=u1(y0) =Gn,P(y0)−Gn,P(P+r1)\nGn,P(P+k1ϵ)−Gn,P(P+r1).\nwhere r1= (r1,0,0, ...,0)∈Rnandk1ϵ= (k1ϵ,0,0, ...,0)∈Rn, is the solution to\nthe above problem.\nNow we need to estimate P\u0000\nWy1τ∈BP(k1ϵ)∩Rn−1\n0\u0001\nwhere Wy0τ2=y1. We claim\nthe following estimation:\nProposition 6.2. There exists a constant c(1)\n3=c(1)\n3(k1)>0such that\nP\u0000\nWy1\nτ∈BP(k1ϵ)∩Rn−1\n0\u0001\n≥c(1)\n3\nfor any y1∈∂BP(k1ϵ)and small ϵ.\nProof. Consider probability that Wy1τ∈BP(k1ϵ)∩Rn−1\n0but without hitting ∂By1(4k1ϵ)\nforϵso small that BP(5k1ϵ)⊂BP(r1). We notice that if we let Hy1\ntbe the same\nBrownian motion as Wy1\ntinBP(r1)\\(BP(k1ϵ)∩Rn−1\n0) but with absorbing boundary\n∂By1(4k1ϵ) and target BP(k1ϵ)∩Rn−1\n0instead, we will have\nP\u0000\nWy1\nτ∈BP(k1ϵ)∩Rn−1\n0\u0001\n≥P\u0000\nHy1\nτ∈BP(k1ϵ)∩Rn−1\n0\u0001\n.\nBy shifting and rescaling invariance of Brownian motion, let\nZt:=Hy1\nt−y1\n4k1ϵ\nbe another (non standard) Brownian motion started at 0, roams inside B0(1), with\noriginal target of Hy1\ntbeing shifted and rescaled to a n−1 dimensional open ball\nwith diameter1\n2(which will be denoted by tg′) and absorbing boundary being\nshifted and rescaled to ∂B0(1). Let τ3= inf{t≥0 :Zt/∈B0(1)\\tg′}Then we have\nP\u0000\nHy1\nτ∈BP(k1ϵ)∩Rn−1\n0\u0001\n=P(Zτ3∈tg′).\nWe notice that dist(0 ,tg′)<1\n4. Consider the n−1 dimensional cube Q1inscribed\nin tg′and we notice that there exists a positive constant c4such that it only18 YUCHEN FAN\nFigure 6. Local structure near P\ndepends on n−1 and the side length of Q1isc4. Now consider building up another\nndimensional cube Q2with one face exactly Q1and 0 lies in same side of Q2\ncompare to Rn−1\n0. Then Q2is well defined and lies inside B0(1). Let the center\nofQ2beQand construct sphere ∂BQ\u0010c4\n4\u0011\n, as shown in figure 7. Let P3be the\nFigure 7. Q,Q1andQ2\nprobability that Zthits∂BQ\u0010c4\n4\u0011\nbefore hitting ∂B0(1). Then as c4is independentHITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 19\nofϵthere exists constant c5>0 only depends on nandk1such that P3≥c5. Let\nP4be the probability that starts on a point on ∂BQ\u0010c4\n4\u0011\nand exits Q2atQ1. Then\nby rescaling invariance we also see there exists c6>0 that also only depends on n\nandk1such that P4≥c6. So\nP(Zτ3∈tg′)≥P3×P4≥c5c6>0.\nThus we have\nP\u0000\nWy1\nτ∈BP(k1ϵ)∩Rn−1\n0\u0001\n≥c(1)\n3:=c5c6.\n□\nWe may obtain the same result for the case of k2.\n6.3.Proof of Main Result. Now we may finish the proof. By (4) and above\nproof we have\nui(y0)c(i)\n3≤Pki≤ui(y0)\nfori= 1,2. So by (2) ,(3) and proposition we proofed in 6.1 we have\nc1c(1)\n3P(τ1<∞)u1(y0)≤P(Bx0\nτ∈tg)≤c2P(τ1<∞)u2(y0).\nWe note that P(τ1<∞)>0 as position of C2is independent of ϵ. We also notice\nthat\nui(y0) =\n\nr2−n\n1−r2−n\n2\nr2−n\n1−(kiϵ)2−nifn∈N\\{1,2}\nlog(r2)−log(r1)\nlog(kiϵ)−log(r1)ifn= 2\nSo that let ϵ→0 we have uiis comparable with general Newtonian potential and\nthus the main result is proven.\n7.Appendix: Proofs of technical details\nProposition 7.1. Suppose V=\b\nV0, V1, ...\t\nis another cover of Ωdefined in 2.4\nand(Ψk)k≥1be another collection of associated maps. Suppose for some kandm\nthatVm∩Ukis non-empty. Then the transition map\nΦk◦(Ψm)−1: Ψm(Vm∩Uk)→Φk(Vm∩Uk)\nmaps process Ψm(Xx\nt)to the process Φk(Xx\nt).\nProof. Notice that for Xx\ntinside domain Ω, it satisfies Stochastic differential equa-\ntion d Xx\nt= dWt. Write Ψ m=f= (f1, f2, ..., fn) and Φ k=g= (g1, ..., gn),\nWt= (W1\nt, ..., Wn\nt) where Wi\ntare independent standard 1 dimensional Brownian\nmotion. Apply Itˆ o’s formua,\n(6) d fi(Xt) = nX\nk=1∂fi\n∂xkdWk\nt!\n+1\n2∆fidt+1∂Φk(Uk)∩H(f(Xt))∇fi(Xt)·γdξt\nwhere δ1i= 0 for i̸= 1 and 1 for i= 1. We abbreviate starting position to make\nour equation clean. We also abbreviate\nR(f, i, ξ t, Xt) =1∂Φk(Uk)∩H(f(Xt))∇fi(Xt)·γdξt20 YUCHEN FAN\nwhere γis the unit vector field normal to boundary of Vm∩Uk. Write f(Xt) as\nFt= (F1\nt, F2\nt, ..., Fn\nt), we need to show that g◦f−1maps process Ftto same process\ning(Vm∩Uk). Apply Itˆ o’s formula again we have\n(7)\nd\u0000\ng◦\u0000\nf−1\u0001\u0001i(Ft) =nX\nk=1∂\u0000\ng◦\u0000\nf−1\u0001\u0001i\n∂xkdFk\nt+1\n2nX\nk,m=1∂2\u0000\ng◦\u0000\nf−1\u0001\u0001i\n∂xk∂xmd⟨Fm, Fk⟩t\nplug in (6) and rearrange (7) we have\nd\u0000\ng◦\u0000\nf−1\u0001\u0001i(Ft) =nX\nk=1nX\np=1∂gi\n∂xp∂\u0000\nf−1\u0001p\n∂xknX\nj=1∂fk\n∂xjdWj\nt\n+1\n2\nnX\nk,m=1nX\na=1∂2\u0000\ng◦\u0000\nf−1\u0001\u0001i\n∂xk∂xm∂fk\n∂xa∂fm\n∂xa+ ∆fknX\np=1∂gi\n∂xp∂\u0000\nf−1\u0001p\n∂xk\ndt\n+nX\nj=1∂(g◦f−1)i\n∂xjR(f, j, ξ t, Xt)\nwhere\u0000\nf−1\u0001pis interpreted as the pthcomponent of f−1. Notice that by written in\nmultiplication form of Jacobian matrix the first term on RHS is justPn\nk=1∂gi\n∂xkdWk\nt\nand second term is1\n2∆gidt. We also note that\nγ·nX\nj=1∂(g◦f−1)i\n∂xj∇fk=γ· ∇gi,\nwhich implies\nnX\nj=1∂(g◦f−1)i\n∂xjR(f, j, ξ t, Xt) =R(g, i, ξ t, Xt)\nThus we have verified that\ndgi(Xx\nt) =nX\nk=1∂gi\n∂xkdWk\nt+1\n2∆gidt+R(g, m, i, ξ t, Xt) = d\u0000\ng◦\u0000\nf−1\u0001\u0001i(Ft)\nso consistency follows. □\nProposition 7.2. Dis a SULD. More over, any domain Ωdefined in Definition\n2.7 is a SULD.\nProof. See Appendix, Choose fj(z) = (i)jz−i\nz+iforj= 1,2,3,4. and we note that\neach fjis conformal between an open neighborhood of a subset H′ofHaround\n0, denoted by H′′, and an open neighborhood of D∩(i)j+1H, denoted by D′\nj, 0 is\nsent to −ijand each D′\nj=iD′\nj−1. With out loss of generality we may assume that\nd= inf{|x−y|:x∈∂D∩ −H, y∈∂D′\n1}>0 and that each fiis holomorphic on\nsome neighborhood of H′′with a holomorphic inverse. Now for z∈∂D∩(i)j+1H\nwe take Bj\nz=Bj\nz\u0000d\n2\u0001\nand by compactness of ∂Dwe can find ( Bj\nzk: 1≤k≤nj)\nsuch that ∂D∩(i)k+1H⊂ ∪nj\nk=1Bj\nzk.PutU={D} ∪ ∪4\nj=1{Bj\nzk: 1≤k≤nj}each\nmap Φ kbe the correspondent ( fj)−1. SoDsatisfies properties 1-3. Now as each fjHITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 21\nis holomorphic on neighborhood of H′′we conclude that\f\f\f\u0000\nf−1\nj\u0001′\f\f\fis bounded above\nby positive constants d1independent of jas we only have finitely many fj. So\n\f\f\f\u0000\nf−1\nj\u0001′(x)−\u0000\nf−1\nj\u0001′(y)\f\f\f=\f\f\f\fZx\ny\u0000\nf−1\nj\u0001′(z)dz\f\f\f\f≤d1|x−y|\nand on the other hand, notice that in particular ( i)j/∈D′\njso there is a d2>0 such\nthat dist(( i)j,D′\nj)≥d2. Now\n\f\ff−1\nj(x)−f−1\nj(y)\f\f=\f\f\f\fix+ (i)j+1\n(i)j−x−iy+ (i)j+1\n(i)j−y\f\f\f\f=\f\f\f\f2(x−y)\n((i)j−x)((i)j−y)\f\f\f\f≥2\nd2\n2|x−y|.\nsof−1\njsatisfies a Lipchitz condition. As we only have 4 jwe have the uniform\nLipchitz condition. Also from here all partial derivative are nicely bounded by\ntotal derivative of f−1\nj. By Cauchy-Riemann Equation we see JjJT\nj=cIwhere\nc=|(f−1\nj)′|2>0,Jjis Jacobian of f−1\njandIis the 2 ×2 identity matrix. Also\nf−1\njare all harmonic, so property 4 is satisfied. Thus Dis a SULD. For the case\nof general Ω in Definition 2.7, by Riemann mapping theorem we may choose a\nconformal bijection fmapsDonto Ω. Now we introduce a lemma\nLemma 7.3. Letfbe a conformal bijection between DandΩwhere Ωis defined\nin 2.7. Then we may extend fto a conformal bijection Fbetween Ω′, an open\nneighborhood of ΩandD′, an open neighborhood of D, and Fagree with fonD.\n[13] (Theorem 2.25)\nSo we may extend fto be such Fand for each x∈∂Ω we have Bx(rx)⊂Ω′.\nThen as ∂Ω is compact we can find an integer Nsuch that ∂Ω =∪n\nk=1Bxk(rxk). Let\ngbe the inverse of F. Write Bxk(rxk) =Bkand consider Bj,m,k =g(Bj\nzm)∩Bkwith\nassociated map Φ j,m,k =g◦f−1\nj. Then take U= Ω∪\u0000\n∪4\nj=1∪nj\nm=1∪n\nk=1Bj,m,k\u0001\nwe\nsee Ω satisfies properties 1-3. Now as g−1is also conformal its derivative is bounded\nby some constant cj,m,k onBj,m,k and so by similar argument as (7) we have\n|x−y|=|g−1(g(x))−g−1(g(y))| ≤cj,m,k|g(x)−g(y)| ≤max\nj,m,kcj,m,k|g(x)−g(y)|.\nThus\n2\nd2\n2|x−y| ≤\f\ff−1\nj(x)−f−1\nj(y)\f\f≤\u0012\nmax\nj,m,kcj,m,k\u0013\f\f\u0000\ng◦f−1\nj\u0001\n(x)−\u0000\ng◦f−1\nj\u0001\n(y)\f\f\nand as g′is bounded on each Bj,m,k we can similarly also find constant d3>0 such\nthat\f\f\u0000\ng◦f−1\nj\u0001\n(x)−\u0000\ng◦f−1\nj\u0001\n(y)\f\f≤d3\f\ff−1\nj(x)−f−1\nj(y)\f\f≤d3d1|x−y|.\nTogether by same reasons listed in about proof that Dsatisfies property 4, we see\nproperty 4 is satisfied and thus Ω is a SULD. □\nProposition 7.4. Suppose ΩandΩ′are two simply connected open subset of H\nandBz0\ntis the RBM in Hwhere z0∈Ωwith the associated local time process ξz0\nt.\nLetfbe a conformal automorphism of H. Let τΩ= inf{t≥0 :Bz0\nt∈∂Ω}. Then\nwith probability one, (f(Bz0\nt), t∈[0, τ))is equal to the time-changed RBM in H.\nProof. We may assume that Bz0\ntis written in the form in the proof of proposition\n2.2 with associated ξz0\ntalso given. Write in clean notation Bz0\nt= (˜Xz0\nt, Yz0\nt) where\nYz0\ntis the standard 1 dimensional Brownian motion independent of˜Xz0\nt. We also22 YUCHEN FAN\nwrite Xz0\ntfor the standard 1 dimensional Brownian motion generates˜Xz0\nt. Write\nf(z0) =w0. As conformal automorphism of Htakes the form\nf(z) =az+b\ncz+da, b, c, d ∈R, ad−bc= 1\nWe have that f′(z) =1\n(cz+d)2and note that it is positive on R\\ {−d\nc}. We may\nremove point−d\ncout of consideration as it has zero Lebesgue measure. Consider\nthe associated process ( Yw0\nt,Ξw0\nt) defined by Yw0\nt=f(Bz0\nt) and\nΞw0\nt=Zt\n0f′(Bt)dξs.\nIt suffice to check that f(Bz0\nt)−1∂H(f(Bz0\nt)) Ξw0\ntis a time changed complex Brow-\nnian motion. Write f(z) =u(x, y) +iv(x, y) where z=a+bi. Apply Itˆ o’s formula,\nnote that fis harmonic:\nd\u0010\nu(˜Xt, Yt)−1∂H(f(Bz0\nt)) Ξt\u0011\n=∂u\n∂x\u0010\n˜Xt, Yt\u0011\nd˜Xt+∂u\n∂y\u0010\n˜Xt, Yt\u0011\ndYt−1∂H(f(Bz0\nt))∂u\n∂xdΞt\n=∂u\n∂x\u0010\n˜Xt, Yt\u0011\n(dXt+1∂H(Bz0\nt) dξt) +∂u\n∂y\u0010\n˜Xt, Yt\u0011\ndYt−1∂H(f(Bz0\nt))∂u\n∂xdΞt\n=∂u\n∂x\u0010\n˜Xt, Yt\u0011\ndXt+∂u\n∂y\u0010\n˜Xt, Yt\u0011\ndYt,\nSimilarly\nd\u0010\nv(˜Xt, Yt)−1∂H(f(Bz0\nt)) Ξt\u0011\n=∂v\n∂x\u0010\n˜Xt, Yt\u0011\ndXt+∂v\n∂y\u0010\n˜Xt, Yt\u0011\ndYt.\nWrite ˜ u=u(˜Xt, Yt)−1∂H(f(Bz0\nt)) Ξtand ˜v=v(˜Xt, Yt)−1∂H(f(Bz0\nt)) Ξt. Then\ntheir quadratic variation satisfies\n⟨˜u⟩t=⟨˜v⟩t=Zt\n0|f′(Bs)|2ds\nwith zero covariation. Let\nσ(t) = inf\u001a\ns≥0 :Zs\n0|f′(Bu)|2du > t\u001b\nand set ˜Bw0\nt= ˜u\u0010\nBz0\nσ(t)\u0011\n+i˜v\u0010\nBz0\nσ(t)\u0011\n. By the Dubins-Schwarz theorem, ˜Bw0\ntis a\ncomplex Brownian motion. □\n8.Acknowledgement\nI would like to express my deepest gratitude to prof. Dmitry Belyaev as my\nsupervisor of this summer project. He advised me the topic of this project and\nsupervised me throughout the project.HITTING PROBABILITY FOR REFLECTED BROWNIAN MOTION AT SMALL TARGET 23\nReferences\n[1] K. Burdzy and Z.-Q. Chen, “Reflecting random walk in fractal domains,”\nThe Annals of Probability , vol. 41, no. 4, Jul. 2013. [Online]. Available:\nhttp://dx.doi.org/10.1214/12-AOP745\n[2] R. F. Anderson and S. Orey, “Small random perturbation of dynamical systems\nwith reflecting boundary,” Nagoya Mathematical Journal , vol. 60, pp. 189–216,\n1976.\n[3] Q. Hu, Y. Wang, and X. Yang, “The hitting time density for a reflected brow-\nnian motion,” Computational Economics , vol. 40, pp. 1–18, 06 2012.\n[4] A. Chaigneau and D. S. Grebenkov, “Effects of target anisotropy on harmonic\nmeasure and mean first-passage time,” Journal of Physics A: Mathematical\nand Theoretical , vol. 56, no. 23, p. 235202, May 2023. [Online]. Available:\nhttp://dx.doi.org/10.1088/1751-8121/acd313\n[5] S. Redner, A Guide to First-Passage Processes . Cambridge University Press,\n2001.\n[6] D. S. Grebenkov and A. T. Skvortsov, “Mean first-passage time\nto a small absorbing target in three-dimensional elongated domains,”\nPhys. Rev. E , vol. 105, p. 054107, May 2022. [Online]. Available:\nhttps://link.aps.org/doi/10.1103/PhysRevE.105.054107\n[7] P. Grandits, “Asymptotics of the hitting probability for a small sphere\nand a two dimensional Brownian motion with discontinuous anisotropic\ndrift,” Bernoulli , vol. 27, no. 2, pp. 853 – 865, 2021. [Online]. Available:\nhttps://doi.org/10.3150/20-BEJ1257\n[8] A. E. Lindsay, A. J. Bernoff, and M. J. Ward, “First passage statistics for the\ncapture of a brownian particle by a structured spherical target with multiple\nsurface traps,” Multiscale Modeling & Simulation , vol. 15, no. 1, pp. 74–109,\n2017. [Online]. Available: https://doi.org/10.1137/16M1077659\n[9] G. F. Lawler, Conformally invariant processes in the plane . American Math-\nematical Soc., 2008, no. 114.\n[10] B. Øksendal, “Brownian motion and sets of harmonic measure zero.” 1981.\n[11] R. F. Bass and P. Hsu, “The semimartingale structure of reflecting brownian\nmotion,” Proceedings of the American Mathematical Society , vol. 108, no. 4, pp.\n1007–1010, 1990. [Online]. Available: http://www.jstor.org/stable/2047960\n[12] E. M. Stein and R. Shakarchi, Complex analysis . Princeton University Press,\n2010, vol. 2.\n[13] D. Beliaev, Conformal Maps and Geometry . WORLD SCIENTIFIC\n(EUROPE), 2019. [Online]. Available: https://www.worldscientific.com/doi/\nabs/10.1142/q0183\n[14] B. Øksendal, Diffusions: Basic Properties . Berlin, Heidelberg: Springer\nBerlin Heidelberg, 2003, pp. 115–140. [Online]. Available: https://doi.org/10.\n1007/978-3-642-14394-6 7"    },    {        "title": "2402.01042v6.Nonlinear_electrodynamics_for_the_vacuum_of_Dirac_materials__Photon_magnetic_properties_and_radiation_pressures.pdf",        "content": "Nonlinear electrodynamics for the vacuum of Dirac materials.\nPhoton magnetic properties and radiation pressures\nA. W. Romero Jorge,1, 2, 3, 4, ∗A. P´ erez Mart´ ınez,1,†and E. Rodr´ ıguez Querts1,‡\n1Instituto de Cibern´ etica, Matem´ atica y F´ ısica (ICIMAF),\nCalle E esq a 15 Vedado 10400 La Habana Cuba\n2Frankfurt Institute for Advanced Studies,\nRuth Moufang Str. 1, 60438 Frankfurt, Germany\n3Helmholtz Research Academy Hessen for FAIR (HFHF),\nGSI Helmholtz Center for Heavy Ion Physics. Campus Frankfurt, 60438 Frankfurt, Germany\n4Institut f¨ ur Theoretische Physik, Johann Wolfgang Goethe University,\nMax-von-Laue-Str. 1, 60438 Frankfurt, Germany\n(Dated: March 11, 2024)\nWe investigate the magnetic properties of photons propagating through Dirac\nmaterials in the presence of a magnetic field, taking into account both medium and\nvacuum contributions. The vacuum photon propagation properties are obtained\nthrough a second-order expansion of nonlinear Euler-Heisenberg electrodynamics\nconsidering Dirac material parameters (effective structure constant, band gap and\nFermi velocity, respectively). Total magnetization and effective magnetic moment\nare obtained. Observables such as energy density, radiation pressure, and Poynting\nvector are getting by an average of components of the energy-momentum tensor.\nAll these quantities are expressed in terms of Lagrangian derivatives and are valid\nfor arbitrary values of the external magnetic field. The weak and strong field limits\nare recovered for all the quantities. We discuss some ideas of experiments that may\ncontribute to testing in Dirac materials the phenomenology of the strong magnetic\nfield in the Quantum Electrodynamic’s vacuum.\n∗adrian@icimaf.cu/aromerojorge@physik.uni-frankfurt.de\n†aurora@icimaf.cu\n‡elizabeth@icimaf.cuarXiv:2402.01042v6  [hep-th]  8 Mar 20242\nI. INTRODUCTION\nMaxwell’s theory predicts that plane-polarized light propagates in a vacuum: empty space\nwith the speed of light c. Quantum Electrodynamics (QED) has another picture of the\nvacuum, according to this theory, the vacuum is not really an empty space, but it is a ”sea”\nof virtual electron-positron pairs whose negative energy states occupied and unoccupied the\npositive ones [1]. This picture can explain, how in the presence of an external magnetic field,\nthe speed of light propagating parallel to the magnetic field remains c while if the propagation\nis perpendicular, light changes its plane of polarization. The resulting incoming wave splits\nin two polarized modes moving with different speeds (and frequencies). This phenomenon\nis a consequence of the interaction of the magnetic field with the virtual electron-positron\npairs it is well-known as the Cotton-Mouton birefringence [2, 3].\nIlluminated by the Quantum field theories vacuum concept becomes in the ground state\nof the theories, being for QED the virtual pairs of electron-positrons, and for Quantum Chro-\nmodynamics vacuum virtual quarks-antiquarks, etc. In summary, the shift from Maxwell’s\nvacuum to Quantum Electrodynamics’ dynamic ’sea’ of virtual particle pairs redefines its\nconceptual meaning and trying to understand its properties and complexity.\nThe vacuum of QED in the presence of a strong electromagnetic field behaves as a\nmedium, virtual pairs interact with magnetic fields. Thereby, the properties of light prop-\nagating in a QED-vacuum may be studied using the tools of non-linear optics, based on\nMaxwell’s theory with all the magnitudes described in terms of a medium electric per-\nmittivity and magnetic permeability depending on the non-linear of the external electric\nand magnetic field1. Besides birefringences2, QED predicts other exotic properties, Casimir\neffect, vacuum instability (close to the critical field, Ec=m2c3\neℏ= 1.3×1018V/m, electron-\npositron pairs creation appear [1, 5, 6]), anisotropy pressures, etc [7]. All these vacuum\nphenomenology predicted 90 years ago are still waiting for experimental confirmation.\nTesting the vacuum properties of the QED requires large magnetic and/or electric fields.\nSo far, there are technological barriers that limit the generation of high magnetic/electric\nfields in the laboratory. Magnetic fields achieved in laboratories do not exceed a few teslas\n(104G). For instance, vacuum birefringence, even when manifests for any strength of the\nmagnetic field, at weak field limit, has a quadratic dependence of the ratio between magnetic\n1Vacuum electric permittivity and magnetic permeability are related with the speed of light [4]\n2In the presence of strong electric field the birefringence also appears, and it is called Kerr-birefringence3\nfield B and the parameter ξ, i.e, ( B/ξ)2with ξ=8α2\n45BcandBc=m2c3\neℏyielding a very tiny\ntheoretical prediction of ∆ nQED= 2.5×10−23for a magnetic field of 2 .5×104G, turning\ninto a challenge to measurement it. Precisely, the experiment Polarization of Vacuum with\nLaser (PVLAS) [8] constructed to pursue vacuum magnetic birefringence could be near of\nconfirm the prediction, obtaining a higher bound ∆ nPV LAS ∼(12±17)×10−23for the same\nmagnetic field of ∆ nQED.\nBut in the Universe, we can find entities like Neutron stars, whose magnetic fields are\nclose to the critical field, and even more in the case of magnetars [9–11]. Short-time huge\nmagnetic fields may also be found in heavy ion colliders, which can reach the order of 1018\nG [12]. In both scenarios, neutron stars and heavy ions colliders have reported indirect\nvacuum magnetic birefringence [10, 11]. The work [10] claims the first evidence of vacuum\nbirefringence by measuring the degree and angle of polarization of the photons coming\nfrom the pulsar RX J1856.5-4 and comparing them with the theoretical results, but the\nuncertainties in their measurements are large enough to fully trust in this measurement.\nThese observations have boosted the search directly for the phenomenon based on other\nexperimental setups. One plausible alternative is the design of experiments with scattering\nof the pulsating laser [13]. It is predictable that with the developed technology of lasers,\nin the near future, the power of lasers will reach an intensity around I∼1033Wm−2\nthat corresponds to an electric (magnetic) field close to the critical field, which may allow\ndetection easier the effect of the birefringence of vacuum and/or to test the pair’s creation\nfrom the vacuum.\nBesides, another exciting and new alternative to probe vacuum properties would be ex-\nperiments designed with Dirac materials due to their critical fields are Ec=∆2\nevF∼105\nV/m, Bc=∆2\nev2\nF∼104G = 1 T, accessible to get in laboratories. A very hopeful work [14]\nhas studied the contribution to the magnetization of the vacuum for three different Dirac\nmaterials and the possibility of testing in experiments.\nDirac materials, including Weyl metals, topological materials, and graphene are charac-\nterized by electrons that behave as relativistic fermions with a linear dispersion relation.\nThis leads to unique optical, magnetic, and transport properties for these materials. The\npioneer theoretical work of the Dirac materials is [15], but the boom of these materials only\nappeared with the amazing finding of graphene by Novoselov [16]. This experiment led to\nan onrush of experimental and theoretical works to look for other similar materials and/or4\nother properties and consequently, technological applications.\nThe discovery of Dirac materials has been gratifying news for the Quantum Field Theory\nbecause it has opened the window to test in top-table experiments, theories, and phenomena\nthat would require a high scale of energies and expensive experiments in particle accelerators.\nConsequently, the tools of Quantum Field theory have been extended to condensed matter\nphysics. On track of this direction the study of the photon propagation parallel to the\nexternal magnetic field for electron-positron plasma [17–19] was extended to the graphene-\nlike system, and the study of Faraday and Quantum Hall effect [20–22]. The study was done\nusing previous calculations [17–19] of the one-loop studies of 3D electron-positron plasma\nin a magnetized medium, doing a ”dimensional reduction” to 2D and considering also the\nproperties of the graphene-like systems.\nIn this paper we proceed similarly extrapolating to Dirac materials physics, the method\nand results of the study of photon propagation transverse to the external magnetic field.\nWe use Euler Heisenberg (EH) non-linear electrodynamics to study the vacuum electron\npositron-plasma. In particular, we use an approximated version of EH Lagrangian which\ncorresponds to an expansion up to second order on the photon field [23] in the limit of\nω≪2mc2, where mis the electron mass. From this Lagrangian, we investigate the magnetic\nand dielectric properties of a photon traveling in the Dirac vacuum as well as the observables,\nenergy densities and pressures, and Poynting vector, that result from the energy-momentum\ntensor. The properties of the medium Dirac materials are considered for the study of the\ncontribution to total magnetization. We used previous calculations of magnetization of\nelectron-positron plasma and we have translated them into the language of Dirac materials.\nConcerning the treatment of the Dirac vacuum, our study has at least two novelties that\ndeserve to be highlighted. The first one is that all properties of the photon are obtained\nin terms of the effective Lagrangian derivatives, so they can extend to other non-linear\nelectrodynamics. Besides, the calculations are valid for arbitrary values of the magnetic\nfield, making it possible to recover the weak and strong limits.\nOur work is also important not only for the interest that awakens the magnetic proper-\nties of Dirac materials (including the vacuum properties) but also it eventually testing the\nproperties of the strong field QED regime. As we have already commented, critical fields\nfor the Dirac materials are experimentally reachable. Hence, these materials could imitate\nthe birefringence, anisotropic pressures, and other exotic properties of the vacuum of QED5\nat the regime of the strong magnetic field.\nThe paper is organized as follows, in section I a brief overview of Dirac materials char-\nacteristics is illustrated. Section II we present the second order in the photon field of the\nEuler Heisenberg Lagrangian, in section III we present the solution of the Maxwell equation\nand dispersion equation of photon propagating in the vacuum Dirac. Section IV is devoted\nto studying the magnetic properties and magnetic moment of the photon. In section V\nthe energy-momentum tensor is presented and the observables like energy density pressures\nand Poynting vector are obtained. Finally, we give a conclusion about the work and in the\nappendixes, some calculations are presented.\nII. A BRIEF OVERVIEW OF DIRAC MATERIALS CHARACTERISTICS\nThe crystalline structure of Dirac materials is characterized by a ”Dirac points” or well,\nwith a small band gap around them, the electrons behave like relativistic particles with free\nelectrons as Dirac fermions exhibiting an extremely high Fermi speed, about 106meters\nper second and electrons could be described by a linear relationship according to ϵ≈vFp\nbetween the energy and momentum of its electrons in the conduction and valence band for\nlow energies [14, 24]. Dirac materials with zero mass (gap) are known as Weyl fermions and\nthe very well-known is graphene [25]. This one, a highly conductive material with no energy\ngap in its Dirac points (∆ = 0), it is distinguished from conventional semiconductors by its\nabsence of this gap (∆ ̸= 0), which limits its use in devices that require precise control of\nelectron flow, such as transistors and diodes. Unlike semiconductors, which can turn current\nflow on and off by excitation and the relaxation of electrons between the valence band\nand the conduction band, graphene maintains a constant flow due to a lack of gap. This\nproperty makes graphene ideal for applications that demand high electrical conductivity,\nsuch as high-speed electronic devices [26], but limit their useful in controllable electronic\ncomponents.\nTo deal with this limitation, graphene-derived structures have been developed, such as\ngraphene ”nanoribbon” or within a substrate, which can have controllable energy gaps and,\ntherefore, be used in electronics applications [27–30]. These energy gaps can be controlled by\nmanipulating the width of the graphene ribbon and introducing defects or doping, among\nother factors. By employing these techniques, small energy gaps are observed, ranging6\napproximately from 100 meV to 250 meV [27–31]. In our study, we consider a non-zero gap\ngraphene energy with an energy gap of 100 meV [32].\nFormally, graphene is not a 2D Dirac semimetal actually, with the top of the valence and\nbottom of the conduction band of a band insulator just touch, before gapping up again after\nband inversion, a graphene-like semimetallic state results. Besides, graphene and topological\ninsulators which have 2D Dirac surface states, there are 3D semimetal with a Dirac point\nin the bulk 3D Brillouin zone linear dispersion in all three kdirections. The theoretical\nprediction and the experimental proves in Bi-based materials did not take long [33–36].\nThese materials are also called (3D) Dirac semimetals.\nOur study will be focused on these materials considering a simple model for them. We\nknow that the crystal structure of these materials is sufficiently complicated with various\nnodes, but we consider in the first approximation that 1) all nodes are equal. On the\nother hand, the value of the energy gap of each material varies depending on the node\nand properties such as temperature, but 2) we assume an energy gap independent of all\nthe parameters of the material, that is, a constant value of gap and 3) we neglect the\ncontribution to magnetization due to material impurities. In Table 4.1 we listed these values\nfor the materials of our interest. These values are based on experimental measurements or\nan average of gap values [14]. Each material will be defined only by its Fermi speed and its\nenergy gap, so there can be several materials with the same characteristics. We are referring,\nfor example, to the graphene-type material that has its Fermi speed and energy gap given,\nbut it can be another material with the same characteristics.\nWith all these approximations, we will proceed to study the propagation of photons in\n3D Dirac materials extrapolating the results of QED, to Dirac materials. It means shifting\nthe values of the critical fields in Dirac materials are defined as [14] [24]\nEc(∆, vF) =vFBc(∆, vF) =\u0012∆\nαDλ3\nD\u00131/2\n=∆2\neℏvF, (1)\nwhere the speed of light has been replaced by the Fermi velocity of the material ( c→vF≈\n106m/s), the fine structure constant has been changed to an effective or Dirac fine structure\nconstant ( α→αD) defined as αD=e2/(ℏvF)∼1−4,λDis the Dirac’s Compton wavelength\ndefined by λD=ℏvF/∆. Here ∆ is the band gap or energy gap of the material and is defined\nas ∆ = m∗mv2\nF, where m∗∼0.01−0.5 is the effective mass of electrons and holes [14][24].7\n∆(meV) αD/αBc(T) b≤3π/α D\nQED 10914.4×1091291\nPb1−xSnxTe 31.5 580 5.6 2.22\nBi1−xSbx 7.75 188 0.036 6.86\nTa3As4 21 357 0.95 3.61\ngraphene 100 301 15.4 4.28\nTable I. Comparison of QED and four Dirac materials for different parameters like band gap\n(∆), Dirac structure constant ( αD) over α, critical magnetic field ( Bc), and bound of the\ndimensionless magnetic field imposed to ensure the validity of non-linear electrodynamics b=\nB/B c. Some values of QED are listed as a reference and the Dirac materials: Pb1−xSnxTe\n[14, 24, 37, 38], Bi1−xSbx[14, 24, 38, 39], Ta3As4[14, 24, 40], and graphene[16, 21, 32].\nTable I shows a comparison of QED and four materials that we use in our study: graphene,\ntantalum arsenide ( Ta3As4), bismuth antimony ( Bi1−xSbx), and lead-tin-tellurium ( Pb1−xSnxTe).\nAlso, we have listed the maximum allowed values of the magnetic field for each material\nto get real corrections to phase velocity in the strong field limit v2\nf= 1−αD\n3πBe\nBc>0.\nThis bound ensures the validity of one-loop approximation and the effective non-linear\nelectrodynamics[41].\nIII. NLED FOR DIRAC MATERIALS, LAGRANGIAN\nThe Lagrangian of QED in the presence of an electromagnetic field is given by the ex-\npression\nL(¯ψ, ψ, A µ) =¯ψ(∂ /−e/Aµ−m)ψ+1\n4FµνFµν(2)\nwhere ¯ψ,ψare the fermion spinors, ∂ /=γν∂ν,/Aµ=γµAµ,γµare the Dirac matrices, Aµis\nthe cuatri-potential vector of the electromagnetic field and Fµνis the electromagnetic tensor.\nIf we integrate the electron contribution to the phase amplitude we define the amplitude\nasZ=R\nDAµeiR\nd4xLeff(Aµ)and all orders of the one-loop approximation of the photon-\nphoton interaction processes are enclosed in the so-called Heisenberg-Euler Lagrangian den-8\nsity [1][42],\nLEH=−F −1\n8π2Zi∞\n0ds\ns3e−m2s\n×\u0014\n(es)2˜a˜bcoth( e˜as) cot( e˜bs)−(es)2\n3\u0010\n˜a2−˜b2\u0011\n−1\u0015\n, (3)\nwhich is gauge invariant and finite after renormalization (see appendix in[6, 43]) where\n˜a=h\n(F2+G2)1/2+Fi1/2\ny˜b=h\n(F2+G2)1/2− F]1/2,eandmare the electron charge\nand mass respectively. F,Gare secular invariants derived from the gauge and Lorentz\ninvariants of the generic electromagnetic fields ( E,B) defined as\nF=1\n4FµνFµν=1\n2(−ϵ0E2+B2\nµ0), (4)\nG=1\n4Fµν˜Fµν=rϵ0\nµ0(−E.B), (5)\nwhere µ0andϵ0are electrical permittivity and magnetic permeability. Along the paper,\nby simplicity, we take µ0=ϵ0= 1. EiandBiare the components of electromagnetic\ntensors defined as Ei=F0i,Bi=−1\n2ϵijkFjkwith i= 1,2,3,˜Fµν=ϵµναβFαβ/2 is the dual\ntensor, and ϵµναandϵµναβare the totally antisymmetric Levi-Civita tensors of rank 3 and\n4, respectively. We consider only the presence of an external magnetic field, which means\nthatF=B2/2 and G= 0 ( Ee= 0).\nAs we are interested in studying the photon propagation transverse to a constant external\nmagnetic field using the EH non-linear electrodynamics we use the prescription introduced\nin [23] for potential fields and electromagnetic tensor. We assume that Aµ=Aext\nµ+aµ,\nwhere aµis the cuatri-potential of the photon and Aext\nµ, is the potential associated with\nthe external and constant magnetic field Be. The electromagnetic field tensor becomes in\nFµν=fµν+FB\nµν,fµν=∂µaν−∂νaµ,with FB\nµν=∂µAext,ν−∂νAext,µandBtot=Be+Bw\nandE=Ew. Then, an expansion of the EH Lagrangian up to the second order of the\nphoton field can be done (see details for NLED for QED in [23]). The coefficients of the\nexpansion are Lagrangian derivatives function of the external magnetic field LF=∂L\n∂F|f=0,\nLFF=∂2L\n∂F2|f=0andLGG=∂2L\n∂G2|f=0(details of calculation can be seen in [23]-[41]).\nThis procedure can be adapted to study the propagation of photons in Dirac materials\nin the presence of the external magnetic field. In that case, the effective Lagrangian of a\nphoton interacting with the magnetic field would take the form\nL(ph−B)\nD =−1\n4(1− LF)fµνfµν+LFF\n8(fµνFµν)2+LGG\n8(fµν˜Fµν)2, (6)9\nwhere L(ph−Be)only contains information about the interaction of photons with magnetic\nfields, and we have neglected quartic terms related to photon and magnetic field self-\ninteraction [23].\nWriting the effective Lagrangian in terms of the photon fields ( Ew,Bw) it has the ex-\npression\nL(ph−B)\nD =(1− LF)\n2(E2\nw−B2\nw) +LFF\n2(Be·Bw)2+LGG\n2(Be·Ew)2. (7)\nIn the limit E→0, i.e. ˜b→0 the scalars LF,LFF, and LGGbecomes in regularized\nintegrals dependent on arbitrary values of the external magnetic field [23, 41, 44, 45]. They\nare quantum corrections proportional to fine structure constant αD. Their explicit forms\nare\nLF=∂LEH\n∂F=−αD\n2πµ0\u00121\n3+ 2h2\nD−8ζ′(−1, hD) + 4hDlnΓ(hD)−2hDlnhD+2\n3lnhD−2hDln2π\u0013\n,\nLFF=∂2LEH\n∂2F=αD\n2πµ0B2\u00122\n3+ 4h2\nDψ(1 +hD)−2hD−4h2\nD−4hDlnΓ(hD) + 2hDln2π−2hDlnhD\u0013\n,\nLGG=∂2LEH\n∂2G=αD\n2πµ0B2\u0012\n−1\n3−2\n3\u0000\nψ(1 +hD)−2h2\nD+ (3hD)−1\u0001\n+ 8ζ′(−1, hD)\n−4hDlnΓ(hD) + 2hDln2π+ 2hDlnhD), (8)\nwhere hD=Bc\n2Be,ψdenotes the PolyGamma function (first derivative of ln Γ) and ζ′is the\nfirst derivative of the Hurwitz zeta function with respect to the first argument.\nLet us remark that all this treatment used for transverse propagation photons to the con-\nstant magnetic field can be straightforwardly extended to the study of photon propagation\nin a pure background electric field and/or in the background of an orthogonal electric and\nmagnetic field, with hD→m2\n2e√\n2F.\nIV. DISPERSION EQUATION OF PHOTON PROPAGATING IN A DIRAC\nVACUUM\nLet us study the motion equation and the dispersion equation of the photon using the\nminimum action principle over the Lagrangian (6). The modified Maxwell equation in terms\nof constitutive vectors DwandHware\n∂Dw\n∂t=−∇ × Hw,∇ ·Dw= 0, (9)10\nwith\nDw,i=∂\n∂Ew,i[L(ph−B)] =ϵijEw,j=Ew,i+Pw,i, (10)\nHw,i=−∂\n∂Bw,i[L(ph−B)] = (µ−1)ijBw,j=Bw,i−Mw,i, (11)\nand a second pair for EwandBw\n∇ ·Bw= 0,∂Bw\n∂t=−∇ × Ew, (12)\nwith i= 1,2,3. Similar to an optical medium, Pw, and Mware the resulting polarization and\nmagnetization of the photon probe due to the magnetized vacuum. Note that the presence\nof an arbitrary strength external magnetic field will impact not only the electric permittivity\nand magnetic permeability tensors, ϵ(B) and µ−1(B), but also Maxwell equation solutions\nfor the photon fields in DwandHwthus affecting its propagation in a magnetized vacuum\nwith explicit expressions for the ϵandµ−1. They can be obtained in terms of Lagrangian\nderivatives as\nϵ⊥=µ⊥= 1− LF, ϵ ∥= (1− LF+ 2FLGG), (13)\nµ∥= (1− LF−2FLFF), (14)\nConsidering the photon propagation in ˆ yˆyˆy−direction of a plane wave as Ew=E0e−i(k⊥·y−ωt),\nfrom the Maxwell equations we can obtain the dispersion equation of photon as\n(ϵijkϵlabkj(µ−1)klka+ω2ϵib)Ew b= 0, (15)\nwhere i, j, k, a, b, l = 1,2,3. The solution of Eq. (15) describes two physical transverse\npolarization modes of the photon field, mode (2) where the Ew∥Beand mode (3) Bw∥Be.\nThe dispersion relation for each mode has the form\nω(2)≃|k⊥|(1−LGGB2\ne\n2), ω(3)≃|k⊥|(1−LFFB2\ne\n2), (16)\nin agreement with [46–48]. The appearance of Cotton-Mouton birefringence [2] brings the\nexistence of two refraction indexes associated with the two different polarization modes: n∥\nfor mode (2) and n⊥for mode (3).\nn∥,⊥=|k⊥|\nω(2,3)=rϵ∥,⊥\nµ⊥,∥. (17)11\nv⊥(Bi1-xSbx)\nv⊥(Pb 1-xSnxTe)\nv⊥(Ta 3As5)\nv⊥(graphene)\n0 2 4 6 80.050.100.501\nBe/Bcv⊥\nFigure 1. Phase velocity for the second mode of propagation of the photon as a function of the\ndimensionless magnetic field for a strong field limit. The non-linear effect decreases the phase\nvelocity with the Beincrease. We have plotted curves for four Dirac materials: Bi1−xSbx(Dashed\nRed), graphene (Blue), Ta3As4(Dashed Orange), and Pb1−xSnxTe(Black). As we can see the\nlowest velocity is obtained for Pb1−xSnxTedue to its effective fine structure constant is the highest.\nThe difference between the refraction index ∆ n=n∥−n⊥takes the form\n∆n=(LGG− LFF)B2\ne\n2. (18)\nIn the weak field limit, it is reduced to ∆ nWF\nCM= 3/4ξDB2\ne, instead of the strong limit\n∆nSF\nCM=αD\n3π(Be\nBc−1). Since the phase velocity v∥,⊥= 1/n∥,⊥, in the strong field limit the\ncondition, vSF\n∥= 1−αD\n3π(Be\nBc)>0 fixes the validity of one-loop approximation up to values\nof the magnetic field [41] b≡Be/Bc≤3π/α D.\nThe phase velocity as a function of the magnetic field is plotted for a second mode for\nfour Dirac materials in (1) . We have depicted curves for Bi1−xSbx, graphene, Ta3As4, and\nPb1−xSnxTe. The graphic shows that Pb1−xSnxTewith the high effective fine structure\nconstant has the lowest velocity.\nThe dispersion equation in weak magnetic field limit [23, 46, 49, 50] is obtained as\nωWF,(2)≃|k⊥|\u0012\n1−7\n4ξDB2\ne\u0013\n, ωWF,(3)≃|k⊥|(1−ξDB2\ne), (19)12\nwhile for strong magnetic field limit, the result is\nωSF,(2)≃|k⊥|\u0012\n1−αD\n3πBe\nBc\u0013\n, ωSF,(3)≃|k⊥|\u0010\n1−αD\n3π\u0011\n. (20)\nV. MAGNETIC PROPERTIES OF PHOTON PROPAGATION IN DIRAC\nMATERIALS. MAGNETIZATION\nWe devote this section to studying the magnetic properties of the Dirac materials con-\nsidering the contribution of the photon propagating in the material in the presence of the\nmagnetic field. We start from the grand partition function defined as Z=Tr{ekT(H−µN)},\nwhere kis the Boltzmann constant, Tis the temperature, His the Hamiltonian, µis the\nchemical potential, and N is the number of particle density. The partition function in the\npath integral language corresponds to Z=R\nD¯ψψeikTR1\n0dτR\nd3xL(Aµ,ψ),where we use eu-\nclidean space-time with τ=ita variable in the interval 0 , kT.\nIn one-loop approximation, the thermodynamical potential of the electron-positron\nplasma in the presence of an electromagnetic field reads as,\nΩ =kT Tr{lnZ}=ikT ln det{G−1(x, x′)}, (21)\nwhere G−1is the inverse of the Green function of electrons in the presence of the electromag-\nnetic field. As we show in the appendix, we can separate from the total thermodynamical\npotential the vacuum, dependent only on the electromagnetic field, and the medium con-\ntribution dependent on temperature, chemical potential, and magnetic field, allowing the\nstudy separately both contributions as\nΩ = Ω vac+ Ω med, (22)\nwhere Ω vac, after renormalization (see appendix A) is just the Euler-Heisenberg effective\nLagrangian Eq. (3). Using its approximation up to second order of the photon field in\nvacuum Dirac material the effective potential writes as\nΩ(ph−B)\nD ≡ ⟨(1− LF)\n2(E2\nw−B2\nw) +LFF\n2(Be·Bw)2+LGG\n2(Be·Ew)2⟩, (23)\nwhere ⟨...⟩ ≡R\nd3xdenotes the average over coordinates of the photon fields.13\nIn turn, Ω medis the thermodynamical potential of the electrons-hole Dirac material\nplasma [21] and its explicit form before the integrate over the p3momentum component\nΩmed=−Ω0∞X\nnanBeZ\ndp3ln(1 +e(ϵn−µ)β)(1 + e(ϵn+µ)β), (24)\nwhere Ω 0=e∆2/(4π2) and ϵ2\nn=p2\n3v2\nF+ ∆2+ 2eBenvF, with nthe Landau levels.\nUsually, when we study electron-positron plasma in the presence of a magnetic field, the\nvacuum contribution is ignored because it is relevant close to the critical field. However, for\nDirac materials, the critical fields are reachable in laboratories, so studying the relevance and\nthe presence of this contribution is very timely for backing up the QED vacuum properties.\nThereby we will calculate the total magnetization of the Dirac material considering the\nphoton and medium contribution. The photon magnetization reads as Mph−Be(Be) =\n−∂Ω(ph−B)\n∂Beand it has the form\nMph−Be=⟨LGG(Ew·Be)Ew,3+LFF(Bw·Be)Bw,3⟩, (25)\nit depends on the polarization mode, which is positive for both modes, behaving the vacuum\nas a paramagnetic ”medium”. The magnetization for each mode in the weak and strong\nBi1-xSbx\nPb1-xSnxTe\nTa3As5\ngraphene\n0 1 2 3 4 50.0010.0100.1001\nBe/Bc4πM[μT]\nFigure 2. Magnetization of the vacuum as a function of external magnetic field strength over the\ncritical field for each Dirac Material: Bi1−xSbx,Pb1−xSnxTe,Ta3As4, and graphene. The value\nofEwis normalized using the electrical critical field, Eω/Ec= 0.3 used in [14] and Eω≈Bω. The\ncurves are plotted up to the maximum value of the magnetic field according to Table I.14\nlimit [23] has the form\nM(2),WF\n∥=7ξD\n2Be⟨E2\nw⟩ˆz, M(3),WF\n∥= 2ξDBe⟨B2\nw⟩ˆz, (26)\nM(2),SF\n∥=αD\n3π⟨E2\nw⟩\nBcˆz, M(3),SF\n∥=αD\n3π⟨B2\nw⟩\nBeˆz. (27)\nNote, that in [14] the weak field is delt for the external magnetic and electric field. In that\ncase mode (3) becomes diamagnetic while, mode (2) continues being positive.\nTo get electron magnetization we will consider the thermodynamical potential in the\napproximation of the degenerate electron gas, ( T << µ ) due to typical experimental tem-\nperatures of the Dirac materials being around 10 K <1 meV [38, 40] and electron densities\nare high, with chemical potentials around 102meV. Then, Ω med(µ, B e) [51] reads as\nΩmed(µ, B e) =−Ω0nµX\nnanBe\u0012\nµ′p′\nf−\u0012\n1 + 2 nBe\nBc\u0013\nln\u0012µ+pf\n∆n\u0013\u0013\n, (28)\nwhere µ′=µ/∆,p′\nf=pf/∆, ∆2\nn= ∆2+ 2eBenvF,pf= 1/vFp\nµ2−∆2\nnis the Fermi\nmomentum and nµis the maximum number of Landau levels taken from the integer part of\nnµ=µ2−∆2\n2eBe.\nThe magnetization Mmed=−∂Ω\n∂Be, yields\nMmed= Ω 0nµX\nnan\u0012\nµ′p′\nf−\u0012\n1 + 4 nBe\nBc\u0013\nln\u0012µ+pf\n∆n\u0013\u0013\n. (29)\nTo illustrate the behavior of the magnetization for Dirac materials Bi1−xSbx,Pb1−xSnxTe,\nTa3As4, and graphene, we have plotted it as a function of the magnetic field and the chem-\nical potential in Fig (3). We can appreciate similar paramagnetic behavior for all Dirac\nmaterials, and the Hass-van Alphen effect due to the quantization on Landau levels. One\nnotes that magnetization increases with the gap and it tends to a constant value when the\nmagnetic field reaches the corresponding critical field for each material. The magnetization\nis strongly dependent on the increase of the chemical potential. The magnetization of each\nmaterial depends on the band gap and Fermi velocity as a consequence of our simple model\nbut also depends on other geometrical factors and chemical composition that we do not\nconsider.\nNote in both figures the oscillations (characteristic behavior of magnetized quantum\ngases) that are due to the so-called Hass-van Alphen effect associated with the transitions\nof fermions between Landau levels.15\nBi1-xSbx\nPb1-xSnxTe\nTa3As5\ngraphene\n0.2 0.5 1 2 51001000104105\nBe/Bc4πM[μT]\nBi1-xSbx\nPb1-xSnxTe\nTa3As5\ngraphene\n1 2 3 40.11101001000104\nμ/Δ4πM[μT]\nFigure 3. (Left) Magnetization of the medium as the dimensionless magnetic field bfor a fix value\nofµ/∆ = 1 .5. (Right) Magnetization of the medium versus dimensionless chemical potential µ/∆\nfor a fixed value of the dimensionless magnetic field Be/Bc= 2. In both figures have been depicted\nthe magnetization for studied Dirac materials: Bi1−xSbx,Pb1−xSnxTe,Ta3As4, and graphene.\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.0010.0100.1001101001000\nBe/Bc4πM Tmat[μT]\nFigure 4. Second mode (2) magnetization for vacuum ( Mvac) and medium( Mmed) as a function\nof the external magnetic field normalized to the critical magnetic field of each material. We used\nµ/∆ = 1 .5 for Bi1−xSbx(Black), Pb1−xSnxTe(Purple), Ta3As4(Orange), and graphene (Blue).\nContinue colors represent the medium magnetization, while the discontinue lines represent the\nvacuum magnetization.\nLet’s compare the vacuum and medium magnetization showing in Fig (4) the magne-\ntization for vacuum ( Mvac) and medium( Mmed) of the materials Bi1−xSbx,Pb1−xSnxTe,16\nTa3As4, and graphene, as a function of the magnetic field. Obviously, we can see that the\nleading contribution to the magnetization comes from electrons for each material. Depend-\ning on the Dirac material parameters this magnetization is 2 or 3 orders higher than the\ncorresponding photon or vacuum one (Fig. 2). The contribution of the vacuum to the total\nmagnetization is very low at the weak magnetic field scale however, at high values of the\nmagnetic field this magnetization increases becoming almost constant.\nVI. EFFECTIVE PHOTON MAGNETIC MOMENT\nLet us study the effective magnetic moment of a photon probe propagating in a magne-\ntized Dirac vacuum defined from the variation of the magnetic energy of the vacuum with\nrespect to the magnetic field. We translate to Dirac material language the results obtained\nin [23] The explicit expression for the two modes |µµµ(2,3)\nph|reads as\n|µµµ(2)\nph|=αD\n16π1\nb3\u001a\n3−12ζ(1,1)\u0012\n−1,1\n2b\u0013\n+ 3ψ\u00121\n2b\u0013\u001b\n+b\u0014\n−3 + log Γ\u00121\n2b\u0013\u0010π\nb\u00112\n+ψ(1)\u0012\n1 +1\n2b\u0013\n+ 2b2\u0015\u001b|k⊥|\nBc, (30)\n|µµµ(3)\nph|=αD\n8π1\nb4\u001a\n−ψ(1)\u0012\n1 +1\n2b\u0013\n+b\u0014\n4−4ψ\u0012\n1 +1\n2b\u0013\n+ 2ψ\u00121\n2b\u0013\u0015\n+b2\u0014\n4−2 log(2 π) + 4 log\u0012\nΓ\u00121\n2b\u0013\u0010π\nb\u00111/2\u0013\u0015\u001b|k⊥|\nBc, (31)\nψ(1)=∂hψ[h] being ψthe PolyGamma o Digamma function, (first derivative of ln Γ).\nζ(1,1)[s, h] =∂hζ′with ζ′=∂sζ[s, h] and ζ[s, h] is the Hurwitz zeta function [52]. Con-\nsidering the form of ψandζ′in weak and strong field limits (see details in [23]). For modes\n(2) and (3) yields |µWF(2)\nph |=α\n4π28\n45Be\nB2c|k⊥|and|µWF(3)\nph |=4\n7|µWF(2)\nph |. While for\nstrong field limit, only mode (2) contributes to the effective magnetic moment and it goes\nto a constant value, |µSF(2)\nph|=αD\n3π⟨E2\nw⟩\nBcand|µSF(3)\nph|= 0.Note that the effective photon\nmagnetic moment |µSF(2)\nph|is a decimal of the electron magnetic moment, contrasting to\nthe QED value, which is two orders lower.\nIn Fig. (5) left we plot the photon effective magnetic moment as a function of bfor light\npolarization modes (2) and (3) of Dirac materials. For comparison, we have depicted the17\nμ(2)(Pb1-xSnxTe)\nμ(3)(Pb1-xSnxTe)\nμ(2)(WF)\nμ(2)(SF)\nμ(3)(WF)0.0 0.5 1.0 1.5 2.0 2.5 3.00.0010.0050.0100.0500.100\nBe/Bcμ[b]B c/k\nμ(2)(Bi1-xSbx)\nμ(3)(Bi1-xSbx)μ(2)(Pb 1-xSnxTe)\nμ(3)(Pb 1-xSnxTe)μ(2)(QED)\nμ(3)(QED)0 2 4 6 810-610-510-40.0010.0100.1001\nBe/Bcμ[b]B c/k\nFigure 5. Photon effective magnetic moment as a function of magnetic field strength and Be/Bc.\nThe left figure shows the magnetic moment for two Dirac materials: Pb1−xSnxTe(red) and\nBi1−xSbx(blue). In black we have plotted the corresponding one for QED for both modes. For\neach case, the modes (2) and (3) are depicted with continuous and dashed lines. The right fig-\nure shows for Pb1−xSnxTethe comparison of the magnetic moment for an arbitrary value of the\nmagnetic field with the corresponding two limits weak (orange and green) and strong (gray) field\nlimits for both modes respectively.\nmagnetic moment for light propagating in a magnetized vacuum of QED. One can see that\nfor the Dirac material, the effective magnetic moment for Pb1−xSnxTeis one order below\nthe QED magnetic moment while for Bi1−xSbxis 3 orders lower than the value of QED. In\nthe right side of fig. (5) we plot for Pb1−xSnxTethe effective magnetic moment considering\nthe results for arbitrary values of the magnetic field as well as we have plotted it for weak\nand strong limit for both modes. Then, for a magnetic field Be≳2Bcthe effective magnetic\nmoment of photons polarized on mode (2) tends asymptotically to a constant value [53]-\n[54]- [46]. Mode (3) slowly decreases with the magnetic field strength, being zero its strong\nfield limit value.\nVII. PHOTON ENERGY-MOMENTUM TENSOR: ENERGY DENSITY,\nPOINTING VECTOR AND RADIATION PRESSURES\nIn [23] we widely discussed the EMT calculated by the Hilbert method from the effective\nLagrangian Eq. (7). The resulting EMT is symmetric, gauge invariant, and conserved.18\nExtrapolating its form to Dirac material yields\ntγρ= (1− LF)fγ\nλfλρ+LFF\n2fµνFµν(Fγαfρ\nα+Fραfγ\nα)\n+LGG\n2fµν˜Fµν(˜Fγαfρ\nα+˜Fραfγ\nα) +ηγρ\n4((1− LF)fµνfµν\n+LFF\n2fµνFµνfαβFαβ+LGG\n2fµν˜Fµνfαβ˜Fαβ\u0011\n+LF(Fγαfρ\nα+Fραfγ\nα) +ηγρ\n2LFfµνFµν. (32)\nThe external magnetic field makes EMT fully anisotropic [23]. The average of the diagonal\npart of the tensor corresponds to energy density and anisotropic pressures while the non-\ndiagonal components account for the Poynting vector. The energy density ⟨t00⟩for both\nmodes is positive [24] and the explicit expression for photon energies for both modes are\nED(2)\nw≃(1− LF+3\n2LGGB2\ne)⟨E2\nw⟩,ED(3)\nw≃(1− LF−1\n2LFFB2\ne)⟨ED(2)\nw⟩. (33)\nHowever, as the Hamiltonian of the effective theory is\nHD=DwEw− L(ph−B)\nD =(1− LF)\n2(E2\nw−B2\nw)−LFF\n2(B·Bw)2+LGG\n2(B·Ew)2.(34)\nonly the energy density for mode for mode (3) becomes in the eigenvalue of the Hamiltonian.\nThe anisotropic pressures take the form\npD(2)\n1,w= (1−1\n2LGGB2\ne)⟨E2\nw⟩, pD(3)\n1,w= (1−1\n2LFFB2\ne)⟨E2\nw⟩,\npD(2)\n2,w= (1− LF+LGGB2\n2)⟨E2\nw⟩, pD(3)\n2,w= (1− LF−3\n2LFFB2)⟨E2\nw⟩,\npD(2)\nw,∥≃(1−3\n2LGGB2\ne)⟨E2\nw⟩, pD(3)\nw,∥≃(1 +1\n2LFFB2\ne)⟨E2\nw⟩. (35)\nwhile the Poynting vector has the form\nPD(2)\nw≃(1− LF+LGGB2\ne)⟨E2\nw⟩),PD(3)\nw≃(1− LF− LFFB2\ne)⟨E2\nw⟩,\n(36)\nwe can rewrite in terms of magnetization the above quantities. In particular the Poynting\nvector takes the form\nPD(2,3)\nw=P0±1\n2M(2,3)\nwBe, (37)\nwhere P0contains the non-linear correction that emerges from the scalar invariant Fwith\nP0= (1− LF)⟨E2\nw⟩while the second term comes from the pseudo-scalar invariant Gand it\nis proportional to the magnetization.19\nFigure 6. Proposed experiment for measuring the pressure of the radiation over the Dirac material\nwith (left) and without (right) external magnetic field.\nNote, that only non-linear isotropic vacuum fulfills that the radiation pressure calcu-\nlated by the Poynting vector to be equal to the target pressure in (ˆ yˆyˆy-direction) p2\nwEq (35)\np2\nw=Pw=P0. In that case, the non-linearity of Lagrangian depends only on the scalar in-\nvariant F. On the contrary, the presence of the magnetic field leads to a non-linear effective\nLagrangian also dependent on the pseudo-scalar invariant Gand leads to anisotropies, the\nrotation symmetry is broken, entailing also that p2\nw̸=Pw. Then, the radiation pressure be-\nhaves differently depending on their propagation mode. For mode (2) both are higher than\nthe corresponding “classical pressures” while for mode (3) it becomes lower than the classical\nvalue. At first glance, the behavior of mode (2) seems to be contrary to intuition. Although\na photon propagating in a magnetized vacuum has a lower velocity than if it propagates in\nan “empty vacuum”, the contribution of magnetic energy enters the radiation pressure as\nan additive term increasing the pressure with respect to the classical one, see Eq.(37-35).\nInstead, for mode (3) this contribution appears subtracting, yielding thus a lower pressure\nvalue, see Eq.(35).\nLet us imagine an experimental setup composed of an emitter of polarized photon beam\ntransversely to an external magnetic field Bin the direction x3. The photon beam is\npointing to the Dirac material ∆ llocated in the plane xz(parallel to the magnetic field),\nFig. (6). The radiation pressure Eq.(37) is exerted over the sheet by the photon beam.\nThe measurement of the non-linear radiation pressures would depend on LGGandLFFfor\na chosen polarization beam. Of course, with a similar experimental setup is also possible\nto measure the different phases and phase velocities (birefringence) of the photon beam20\nthat crosses the sheet in comparison with one propagates without a sheet both propagating\ntransverse in the magnetic field. The birefringence will be characterized by the difference\nbetween LGGandLFFas was pointed out in Eq (18).\nA. Conclusions\nIn this paper, we have extrapolated our previous studies related to photon propagation in\na constant magnetic field in the QED vacuum and medium to the context of Dirac materials.\nFor the treatment of Dirac vacuum, we started from the expansion up to second order of the\nEuler-Heisenberg effective nonlinear Lagrangian for the photon fields, considering photon\nlow energy ω≪2mc2(ω≪2∆) and replacing the fine structure constant α=e2\n4πℏcbyαD\nthat depends on Fermi velocity vFinstead of c.\nWe have obtained the dispersion law, the phase velocity and refraction index of photons\npropagating in the Dirac vacuum, electric permittivity, and the magnetic susceptibility that\nthe photon feels. The results are in agreement with previous ones obtained in [24].\nMagnetic properties in the Dirac vacuum as well as the magnetic moment have also been\nstudied and they depend on the polarization mode of the photon. The photon magnetization\nis positive, showing that it is paramagnetic.\nMagnetization of electron degenerate gas has been obtained and translated to the lan-\nguage of Dirac materials using our previous results obtained for the plasma electron-positron\nplasma in the presence of a magnetic field.\nThe magnetization of a Dirac material (due to electrons) not only depends on αDbut also\non the energy gap. For zero temperature and high density, the magnetization preserves the\nparamagnetic character. The magnetic properties have been illustrated for four Dirac mate-\nrials: Bi1−xSbx,Pb1−xSnxTe,Ta3As5and graphene-like, and the properties are determined\nby the value of the energy gap of each material. Although the medium magnetization (due to\nelectrons) is higher than the vacuum magnetization (due to photons) for two or three orders\nof magnitude depending on the material energy gap, with the increasing of the magnetic\nfield the vacuum magnetization increases becoming constant. Its contribution changes the\nmagnetization in a quantity that could be detected at values of magnetic field reachable at\nthe laboratory. This conclusion reinforces the importance of taking into account the vacuum\nproperties to study Dirac materials.21\nWe have discussed the energy density, pointing vector, and radiation pressures of the\nphoton propagating transverse to the magnetic field starting from the energy-momentum\ntensor calculated ”a la Hilbert” for an effective non-linear electrodynamics for Dirac material.\nThese quantities are dependent on the polarization mode. We have discussed that in our\nframework the pressure target in the direction y is not coincide with the Poynting vector.\nThat is a consequence of the symmetry breaking that the magnetic field produces.\nAlthough our model for Dirac materials from the point of view of material science is\nsimplified, has all the wealth of quantum theories and thereby could be used to study other\nphenomena that appear in QED and their analogies in Dirac materials. On the other hand,\nthe properties studies are valid for all values of the magnetic field reproducing the weak\nand strong magnetic field limit. Besides, the study could be extended to other non-linear\nLagrangian contribute to extract some general properties of Dirac materials.\nWe are hopeful that Dirac materials may be a medium where the vacuum properties of\nQED may be tested.\nVIII. 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P´ erez Mart´ ınez, and A. Sanchez. Phys.\nRev. D , 91:085041, Apr 2015.\n[57] Ciprian Sorin Acatrinei. Rom. J. Phys. , 64(9-10):113, 2019.\nAppendix A: Calculation of the thermodynamical potential for the vacuum and\nmedium\nThe thermodynamical potential of electron-positron plasma in the presence of an external\nmagnetic field in x3direction is given by\nΩ(Be, µ, T ) =1\nβTr{lnZ}=ikTln det{G−1(x, x′)}, (A1)\nwhere trace and logarithm are functional operations. In the momentum space, it has the\nform\nG−1\nn(p) =p·γ−m, (A2)\nwith the notation p= (ip4,0,√\n2eBl, p3) over Landau number n= 0,1,2, ...in Euclidean\nspace, γare the Gamma-matrices.\nIn the momentum space, the thermodynamical potential has the form [55, 56]\nΩ(Be, µ, T ) =−eBe\nβ\nX\nσ=±1∞X\nn=0X\np4∞Z\n−∞dp3\n(2π)2ln det G−1\nn(p∗)\n, (A3)\nhere σ=±1 is the eigenvalues of the spin, p4=iωm,ωmis the Matsubara frequencies\nωm=kT(2m+ 1)πwith m= 0,1,2...and, p3is the momentum in x3-direction.\nPerforming the sum over Matsubara frequencies and calculating the determinants in Eq.\n(A3) we obtain\nΩ(Be, µ, T ) =−1\n2eBe\n4π2Z∞\n−∞dp3X\nσl|Eσn|+eBe\n4π2Z∞\n−∞dp3X\nσ,n1\nβln(1+ e−β|Eσn−µ|)(1+e−β|Eσn+µ|),\n(A4)\nwhere Eσn=p\np2\n3+m2+eBe(2n+ 1 + σ), the first term of Eq. (A4) corresponds to the\nΩvacthe second Ω med.26\nΩvachas an ultraviolet divergence and has to be renormalized. To do it we use the integral\nrepresentation of the integral of Eleads,\nE=Z\nds(1−e−E2s)s−3/2, (A5)\nNow we can perform the Gaussian integral on p3obtaining\nΩvac=−eBe\n8π2X\nσ=±1Z∞\nϵds\ns2\u0010\n2e−(m2+eBe(2n+σ+1))s\u0011\n, (A6)\ndoing the sum over the spin σwe obtain the integral\nΩvac=eBe\n8π2∞X\nn=1Z∞\nϵds\ns2\u0010\n2e−(m2+2eBen)s−1\u0011\n. (A7)\nLet’s do the sum over Landau levels using\n∞X\nn=1xn=1\n1−x, (A8)\nΩvacyields\nΩvac=eB\n8π2Z∞\nϵdse−m2s\ns2\u00121\n1−e−2eBes−1\u0013\n, (A9)\nin terms of hyperbolic trigonometric function, we have\nΩvac=eBe\n8π2Z∞\nϵds\ns2\u0010\ne−m2scoth(2eBes)\u0011\n, (A10)\ncoth(2eBes) provokes that the integral has an ultraviolet divergence. Expanding coth(2eBs)\nin powers of small s we obtain\ncoth(2eBes)∼\u0012\n1 +(eBes)2\n3\u0013\n, (A11)\nThe prescription to regularize the integral consists of adding and subtracting divergencies\nterms Eq. (A11) obtaining\nΩvac=−1\n8π2Z∞\nϵds\ns3\u0012\n2e−m2s((eBes)coth(eBes)−1−(eBes)2\n3\u0013\n. (A12)\nIn the case of we study the vacuum in the presence of a constant electric field the ther-\nmodynamical potential is obtained substituting in Eq(A12) Be=iEe\nΩvac(Ee) =−i\n8π2Z∞\nϵds\ns3\u0012\n2e−m2s((eEes)cot(eEes)−1−(eEes)2\n3\u0013\n. (A13)\nConsidering previous results it is possible to get straightforward the Ω vac[57] dependent\non the electromagnetic field, external electric and magnetic field, and obtain the Lagrangian\nEq (3)."    },    {        "title": "2402.01107v1.Simulation_of_Graph_Algorithms_with_Looped_Transformers.pdf",        "content": "Simulation of Graph Algorithms with Looped Transformers\nArtur Back de Luca∗Kimon Fountoulakis∗\nAbstract\nThe execution of graph algorithms using neural networks has recently attracted significant interest\ndue to promising empirical progress. This motivates further understanding of how neural networks can\nreplicate reasoning steps with relational data. In this work, we study the ability of transformer networks\nto simulate algorithms on graphs from a theoretical perspective. The architecture that we utilize is a\nlooped transformer with extra attention heads that interact with the graph. We prove by construction\nthat this architecture can simulate algorithms such as Dijkstra’s shortest path algorithm, Breadth- and\nDepth-First Search, and Kosaraju’s strongly connected components algorithm. The width of the network\ndoes not increase with the size of the input graph, which implies that the network can simulate the above\nalgorithms for any graph. Despite this property, we show that there is a limit to simulation in our solution\ndue to finite precision. Finally, we show a Turing Completeness result with constant width when the\nextra attention heads are utilized.\n1 Introduction\nRecent advancements in neural network models have significantly impacted various domains, most notably\nvision [Yuan et al., 2021, Khan et al., 2022, Dehghani et al., 2023], and natural language processing [Wei et al.,\n2022b, Touvron et al., 2023]. Transformers [Vaswani et al., 2017], at the forefront of these developments, have\nbecome standard for a wide range of complex tasks. These successes have also shed light on the capabilities of\nneural networks in algorithmic reasoning [Veliˇ ckovi´ c and Blundell, 2021], such as basic arithmetic [Lee et al.,\n2024], sorting [Tay et al., 2020, Yan et al., 2020, Rodionov and Prokhorenkova, 2023], dynamic programming\n[Dudzik and Veliˇ ckovi´ c, 2022, Ibarz et al., 2022] and execution of graph algorithms [Veliˇ ckovi´ c et al., 2022,\nCappart et al., 2023].\nIn this work, we focus on algorithmic reasoning on graphs. Current empirical results display a promising\ndegree of scale generalization on graphs [Yan et al., 2020, Tang et al., 2020, Ibarz et al., 2022, Veliˇ ckovi´ c\net al., 2022, Bevilacqua et al., 2023, Numeroso et al., 2023]. The predominant approach in these studies is to\ntrain a neural network to execute a step of a target algorithm and use a looping mechanism to execute the\nentire algorithm. The looping mechanism is crucial since it allows the execution of long processes on graphs.\nMotivated by these empirical results, our goal is to study the ability of looped neural networks to simulate\nalgorithms on graphs of varying sizes. Briefly, by simulation, we mean the ability of a neural network to\nprovide the correct output of a step of an algorithm for every step. In all our results we utilize a looped\ntransformer architecture with extra attention heads that interact with the graph. Instead of storing the graph\nin the input, we encode it using its adjacency matrix which multiplies the attention head. This allows us to\naccess data from the adjacency matrix without scaling the number of parameters of the network with the size\nof the graph.\n*David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada.\nEmails: abackdel@uwaterloo.ca, kimon.fountoulakis@uwaterloo.ca\n1arXiv:2402.01107v1  [cs.LG]  2 Feb 2024Figure 1: A simplified illustration of the simulation of Dijkstra’s algorithm using a looped transformer with\nextra attention heads that interact with the graph. On the left, we display the pseudocode of Dijkstra’s\nalgorithm, serving as the source code. The rightmost section shows the corresponding simulation via a\ntransformer, where each step of the source code in the loop is simulated using one or more transformer blocks.\nWe specifically focus on lines 11 and 12, highlighted to demonstrate the simulation of individual functions, as\ndiscussed in Section 5. At the top center of the figure, the encoding of graph information and variable scopes\ninto ˜AandXis depicted. For clarity, Xis shown in its transposed format. Throughout the transformer loop,\n˜Aremains constant, while Xis updated in each iteration until the simulation meets its termination criteria.\nUpon termination, the decoding step extracts columns from Xthat correspond to the algorithm’s desired\noutput.\nOur contributions: We list our contributions below.\n1.In Section 6, we prove by construction how a looped transformer architecture is capable of simulating\ngraph algorithms such as Breadth-first search & Depth-first search [Moore, 1959], Dijkstra’s algorithm for\nshortest paths [Dijkstra, 1959], and Kosaraju’s algorithm for identifying strongly connected components\n[Aho et al., 1974]. We chose these algorithms since they are part of the CLRS benchmark [Veliˇ ckovi´ c\net al., 2022].\n2.The width of the network in our results does not scale with the number of nodes or edges of the graph.\nThis allows us to show that simulation of algorithms is possible for graphs of varying sizes. However,\nthe graph size is limited by finite precision. Current results do not consider the looping mechanism\nand they are limited by the depth/width of the network [Loukas, 2020, Xu et al., 2020], or they do not\nconsider graph algorithms [P´ erez et al., 2021, Giannou et al., 2023].\n3. We also provide a Turing Completeness result for the looped transformer with O(1) width which uses\nadditional attention heads to interact with the graph.\n2 Related work\nEmpirical: A plethora of papers have been published on the ability of neural networks to execute algorithms\non graphs. Representative empirical works are Tang et al. [2020], Yan et al. [2020], Veliˇ ckovi´ c et al. [2020],\nIbarz et al. [2022], Bevilacqua et al. [2023], Numeroso et al. [2023], Diao and Loynd [2023], Georgiev et al.\n[2023a,b], Engelmayer et al. [2023]. The authors in these papers utilize the looping mechanism and report\nfavorable scale-generalization results. We refer the reader to Cappart et al. [2023] for a comprehensive review\nof the subject.\nTheoretical: Currently, three types of complementary results study the ability of neural networks to execute\n2algorithms. The first type is simulation results using looped transformers by P´ erez et al. [2021], Giannou\net al. [2023]. These types of results are proved by providing an analytic expression of the neural network\nwhich achieves simulation. Our results are also simulation results. In P´ erez et al. [2021] the authors prove\nthat the basic transformer architecture is Turing Complete. The authors in Giannou et al. [2023] provide a\ndistinct result of Turing Completeness for transformers. Instead of direct simulation of a Turing Machine,\ntheir result is proven through simulation of SUBLEQ [Mavaddat and Parhami, 1988]. Both papers do not\ntake into account graph data. Although these Turing Completeness results apply to graph algorithms as well,\ntheir approach requires the graph to be stored in the input data matrix as opposed to having it as part of the\narchitecture. This imposes limitations on the size of the input graph, see Section 4.\nThe second type of result is using the Probably Approximate Correct learning framework to study the sample\ncomplexity of neural networks for executing algorithms. In Xu et al. [2020] the authors show that sample\ncomplexity is improved when the neural network is aligned with the structure of the algorithm. We also\nuse the concept of alignment in our constructive proofs. However, our results are about the simulation of\nalgorithms, which hold for any distribution of the test graph. Also, in Xu et al. [2020] they don’t use a\nlooping mechanism. The third type is impossibility results. In this case, it is shown that neural networks\ncan’t execute certain algorithms when the depth and width of the network are smaller than a lower bound.\nIn Loukas [2020] the author studies the ability of graph neural networks to solve various graph problems and\nalso proves Turing Completeness. However, all results are limited by the depth and the width of the neural\nnetwork which scales with the size of the input graph. In our case, due to the looping mechanism, the depth\nand width of the network are constant, and a single neural network can solve a particular problem for any\ninput graph, subject to limitations imposed by finite precision.\n3 Preliminaries\nThis section establishes preliminaries for our task of algorithmic simulation on graphs. Recognizing that\nan algorithm consists of multiple steps, we start by defining simulation for an individual step. Consider\nhF:X → Y as the function we aim to simulate and hT:X′→ Y′as the neural network designed for this\npurpose. Note that hFandhTmight operate in different representation spaces. For instance, XandYmight\nbe the set of natural numbers while X′andY′might be multidimensional reals. To address this, we use\nmappings ge:X → X′andgd:Y′→ Y for encoding and decoding, respectively. In our constructions in\nSection 5.1, georganizes terms into our input matrix format and incorporates biases and positional encodings,\nwhile gdsimply extracts the appropriate columns from the output.\nDefinition 3.1 (Simulation) .The neural network hTis a successful simulator of the algorithmic step hF\nif for every input x∈ X,hF(x) =gd(hT(ge(x))). A transformer is said to simulate an algorithm if it can\nsimulate each step of the algorithm [Giannou et al., 2023, P´ erez et al., 2021].\nWe define a graph Gwith nnodes by its adjacency matrix A∈Rn×n\n+, where Ai,j>0 if nodes iandjare\nconnected, otherwise Ai,jis zero. In our work, we also use an input matrix X∈RK×d, where dis the feature\ndimension and K≥n+ 1 is the number of rows that are set according to the simulation. The structure of\nXis further explained in Section 5.1. Because of the mismatch of length between XandA, we adopt a\npadded version of A, denoted ˜A∈RK×K. The entries of the first row and column of ˜Aare set to zero. This\nis to align with the input’s top row, which holds data not associated with any node, as further discussed in\nSection 5.1. The next nrows and columns correspond to the entries of A. Certain implementations require\nK > n + 1. In this case, we further pad ˜Awith zeros in the extra rows and columns beyond the entries of A.\nFor example, in our Turing Completeness result in Theorem 6.5, Kcan exceed n+ 1 because the number of\nrows may not be directly determined by the number of nodes.\n34 The Architecture\nWe utilize a variation of the standard transformer layer in Vaswani et al. [2017] with an additional attention\nmechanism that incorporates the adjacency matrix. The additional attention mechanism is described by the\nfollowing equation:\nψ(i)(X,˜A) := ˜A σ\u0012\nXW(i)\nQW(i)\nK⊤X⊤\u0013\nXW(i)\nV, (1)\nwhere WV∈Rd×d,WQ, and WK∈Rd×daare the value, query, and key matrices, respectively, darepresents\nthe embedding dimension and σdenotes the hardmax1function. Note that setting ˜Aas the identity matrix\nreduces equation (1)to a standard attention head. This modification efficiently extracts specific rows from\nmatrix A. Conversely, applying attention to the transpose of ˜Aenables the extraction of columns of A. These\nvariations are important in our constructions, and they allow parallel execution of algorithmic subroutines\nwhich are further discussed in Section 5.2. We define a complete transformer layer as:\nf(X,˜A) =fmlp(fattn(X,˜A)) (2)\nwhere\nfattn(X,˜A) =X+X\ni∈Hψ(i)(X, In+1) +X\ni∈HAψ(i)(X,˜A) +X\ni∈HA⊤ψ(i)(X,˜A⊤)\nfmlp(X) =Z(m)W(m)+X, Z(j+1)=ϕ(Z(j)W(j))\nforj= 0, . . . , m −1 and Z(0)=X. Here, Xrepresents the input matrix and ˜Ais the padded adjacency\nmatrix. Additionally, In+1is the identity matrix of size n+1,mandϕare the number of layers and activation\nfunction of fMLP, respectively. Note that in the formulation of the multi-head attention, we adopt a residual\nconnection and a sum of attention heads ψin the format described in (2).H,HA, and HA⊤are the index set\nof attention heads for the standard attention [Vaswani et al., 2017], as well as for the versions that utilize the\nadjacency matrix and its transpose, respectively. The total number of attention heads is |H|+|HA|+|HA⊤|.\nFurthermore, for the Multilayer Perceptron (MLP), we define ϕto be the ReLU function, we set m= 4,\nand we use matrices W(i)∈Rd×das the parameters of the MLP, where dis the feature dimension of X.\nWe integrate the bias into the linear operations by adding bias columns to Xand also introduce a residual\nconnection to fmlp.\nAlgorithm 1 demonstrates the use of our model hT, which is constructed by stacking multiple layers as\ndepicted in (2). In this setting, each forward pass feeds its output to a subsequent pass, creating a looping\nmechanism. This loop continues until a pre-defined termination condition is met, such as the activation of a\nboolean element within the input, denoted by term in Algorithm 1. This setting and architecture are used to\nsimulate various graph algorithms, as formally established in the results of Section 6.\nAlgorithm 1 Looped Transformer [Giannou et al., 2023]\nInput: model hT, matrices Xand ˜A, column term\n1:while X[0,term] is false do\n2:X=hT(X,˜A)\n3:end while\n1The softmax activation function can be used as well, since softmax approximates hardmax as the temperature parameter\ngoes to zero. However, softmax introduces errors in the calculations. To guarantee small errors during the simulation, the\ntemperature parameter will have to be a function of the size of the graph and other parameters related to finite precision.\nSoftmax could further limit the simulation properties of the architecture beyond what we discuss in Section 6. However, this\ntype of limitation is common, for example, see Giannou et al. [2023], Liu et al. [2023]. In practice, softmax is preferred over\nhardmax. For this reason, in our empirical validation results in Appendix B.3 we also use softmax in combination with rounding\nlayers in Appendix B.1.\n44.1 Discussion on the attention head in (1)\nWhile encoding the graph in Xis an alternative to our attention head in (1), it is subject to limitations.\nSpecifically, concatenating the adjacency matrix to Xresults in linear dependence of the width of the network\nto the number of nodes, thereby limiting simulation to graphs with sizes smaller or equal to the one that the\nnetwork was set up/trained on. Alternatively, appending the edges of the graph to the input results in the\nnumber of positional encodings being dependent on the edge count, which may increase quadratically as the\nnumber of nodes grows. This can limit simulation performance when the available positional encodings are\nlimited, for instance, due to finite precision.\nFinally, our attention head in (1)can be viewed as a message-passing layer that performs two convolutions.\nThe first convolution corresponds to a complete graph, i.e., standard attention [Vaswani et al., 2017], and the\nsecond is a standard graph convolution [Kipf and Welling, 2017]. Multiplying the two convolution matrices is\nimportant. In particular, the left multiplication of the attention matrix with ˜Ain(1)allows for direct and\neasy access to data from ˜A. This is because the attention matrix acts as an indicator matrix that extracts rows\nfrom ˜A, see Section 5.3 for details. Note that our attention head in (1)differs from Graph Attention Networks\n(GAT) [Veliˇ ckovi´ c et al., 2018], where the graph is incorporated in the attention mechanism. Although\npossible to use the attention mechanism in GAT to access the graph data, the proofs will be more elaborate\nthan with (1).\n5 Simulation Details\nOur simulation results are constructive. We set the parameters of the layer in (2)such that we simulate\neach sub-routine within each algorithm. We start by describing the input to the model in Section 5.1. We\nprovide two simulation examples of common subroutines among all algorithms in this paper. In Section 5.2\nwe provide an example of the implementation of the less-than function. This function is common among\nalgorithms in this paper, and it illustrates how the neural network can achieve parallel computation over the\nnodes. Next, in Section 5.3 we provide an example for reading information from the graph. Throughout the\nsection, we use Dijkstra’s algorithm as an algorithm example (as depicted in Figure 1). With the exemption\nof our Turing Completeness result in Theorem 6.5, it is sufficient to set K=n+ 1 for all algorithms in our\npaper. For this reason, and for simplicity, we set K=n+ 1 in this section.\n5.1 Input matrix\nThe input matrix encompasses all variables used in the algorithm. In the case of Dijkstra’s algorithm in\nFigure 1, the lists of current distances and paths ( dists andpaths ), and variables like the current node and\nits distance ( uanddist ) are all incorporated into X. Below, we describe in detail the general structure of X.\nWe refer the reader to Figure 2 for a visualization of the structure of the matrix X.\nThe top row of Xstores single variables, encapsulating the global information relevant to the algorithm. The\nbottom nrows of Xare reserved for local variables, such as distances and paths, representing the node-specific\ndata. This storage structure separates the global context from the more specific, node-related information\nof the algorithm. The input matrix Xis augmented with columns for positional encodings, biases, and a\nscratchpad area. Each node receives a unique positional encoding, denoted by pi. These concepts have been\npresent in various contexts, see Giannou et al. [2023] for assigning functional blocks to the input and Aky¨ urek\net al. [2022], Wei et al. [2022a], Giannou et al. [2023] for the notion of scratch space. Distinct biases are\napplied to the top and bottom rows to simulate global and local functions effectively. Xis equipped with\nflags indicating global or local states, regulating the algorithm’s flow. These flags can either permit or prevent\noverwriting of fields or signal different stages in the algorithm’s execution. For example, in all our simulations\n5Figure 2: Illustration of the input structure used in the simulation of graph algorithms. On the left, columns\nindicate node positions using circular positional encodings [Liu et al., 2023], detailed in Section 6. The\nnext blocks on the right denote the global (in red) and local (in green) variables, which occupy the top\nand last nrows of X, respectively. In these blocks, the first column marks the bias of the corresponding\nvariables. The symbol tindicates the termination flag, while symbols z, and xi, yi, are generic local and\nglobal variables, respectively. Finally, depicted in the far right block, the scratchpad is used for temporary\nstorage and calculation. Non-shaded areas remain null during execution.\neach of the last nentries of the node-wise flag named visit indicates whether the corresponding node has\nbeen visited, while the global variable term, signals the terminating condition for the algorithm. Finally,\nthe scratchpad within Xserves as a temporary storage space for variable manipulation or intermediate\ncomputations.\n5.2 Less-than\nThe purpose of this operation is to determine whether each element in a specific column of Xis smaller than\nits counterpart in another column. The result of this comparison is then recorded in another designated\ncolumn. In the context of Dijkstra’s algorithm, as shown in Figure 1, this function plays a critical role in\ndetermining if an alternate path is shorter than the shortest path currently known.\nWe express this function as: less-than(X, C, D, E) = write(X[:,C] < X[:,D], X[:,E]) . Here, Xis\nthe input matrix, while C,D, and Eare placeholders for the specified column indices in X. This simulation\ninvolves two key steps. First, we approximate the less-than operator ( <) using the following expression:\nX[:, C]< X[:, D]≈ε−1(ϕ(X[:, D]−X[:, C])−ϕ(X[:, D]−X[:, C]−ε)),\nwhere ε > 0 is a small tolerance value, and ϕdenotes the ReLU activation function. This operation\napproximates a step function of the difference X[:,C]-X[:,D] , where εcontrols the sharpness of the threshold.\nIf this difference is larger than ε, the subtraction of the terms post-ReLU equals ε, which simplifies to 1. If\nthe difference is negative, both ReLU terms become zero. If the difference X[:,C]-X[:,D] is non-negative\nbut less than ε, the approximation linearly interpolates between 0 and 1, with a gradient of ε−1. The\nsecond operation is the write function. In this operation, the output of the less-than comparison is placed\nin the specified field, while the original content of that field is erased. This operation is expressed as:\nX[:,E] ←X[:,E] - X[:,E] + X[:,S 1], where S1is the scratchpad column that holds the result to be put in E.\nCombining these two operations, we define the parameters for the layer f. To understand the purpose behind\neach operation, we start by defining a particular instance of the matrix X, which is used for this example:\n6X=\n···Bglobal\n1Blocal\n0C\nc0D\nd0E\ne0S\n···\n··· 0 1 c1d1e1···\n...............\n··· 0 1 cndnen···\n.\nHere, C,DandEare placeholders for column indices used in (5.2),Sis the scratchpad area, and Bglobal and\nBlocalrepresent the biases for global and local variables respectively. First, in the definition of the attention\nlayer, all values are set to 0, maintaining only the residual connection. For the definition of the parameters\ninfmlp, we introduce the parameters layer by layer, followed by an explanation of what is achieved at each\nstage. For the first layer, we have:\n(W(1))i,j=\n\n1 if ( i, j=E) or (i=D, j∈ {S1, S2})\n−1 if i=C, j∈ {S1, S2}\n−εifi∈ {Bglobal, Blocal}, j=S2\n0 otherwise,\nwhere S1andS2are entries in the scratchpad. Here, the parameters of the first layer W(1)keep the entries\nof the target column E, while it builds the arguments of ϕin(5.2), inserting them in the scratchpad. Storing\nthe intermediate values in the scratchpad is beneficial to prevent overwriting issues, as it avoids the use of\ncolumns involved in the definition of the equation. The result is given by:\nXW(1)=\n···E\ne0S1\nd0−c0S2\nd0−c0−ε···\n···e1d1−c1d1−c1−ε···\n.........\n···endn−cndn−cn−ε···\n.\nSubsequently, the parameters of the second layer W(2)are:\n(W(2))i,j=\n\n1 if i=E, j =i\nε−1ifi=S1, j=i\n−ε−1ifi=S2, j=S1\n0 otherwise.\nHere, W(2)is responsible for subtracting the post-ReLU terms and dividing them by ε−1. This result is then\nstored in the scratchpad entry S1:\nZ(1)W(2)≈\n···E\ne0S1\nc0< d0···\n···e1c1< d1···\n......\n···encn< dn···\n.\nFinally, for the two remaining operations, we have:\n(W(3))i,j=(\n1 if i∈ {E, S 1}, j=i\n0 otherwise,\n(W(4))i,j=\n\n−1 if i, j=E\n1 if i=S1, j=E\n0 otherwise.\n7In these two operations, the output of S1is preserved and then placed in the desired field E. Additionally, the\nentries of E, which were kept in the first three layers, are subtracted from the target field in W(4), effectively\nremoving the previous entry in E. Furthermore, to address cases where the input entries are negative, it\nis necessary to incorporate additional parameters. These parameters should follow the same structure as\npreviously provided, with inverted signs in the parameters W(1)andW(4)for columns C,D, and E. These\nadditional entries are generally placed in the scratchpad.\nParallel computation over the nodes: The node-wise format in Section 5.1 is particularly beneficial for\noperations like less-than in equation (5.2) because it operates across entire columns. This approach enables\nparallel processing for all nodes in a graph, enhancing efficiency in algorithms. For example, in Dijkstra’s\nalgorithm, the typical implementation involves iterating through each non-visited neighbor of a node. However,\nby integrating an additional masking function, we can filter out visited or non-neighboring nodes. This allows\nus to conduct all necessary comparisons and conditional selections for every node simultaneously within the\ninner loop of the implementation. This modification enhances the algorithm’s efficiency by reducing the need\nfor repetitive looping in transformer-based simulations.\n5.3 Read row from A\nIn this section, we provide a configuration of (2)that extracts a row from the adjacency matrix. We describe\nthe function as read-A(X, ~A, C, D) = write( ~A[C,:], X[:,D]) , where Xand ˜Aare the input and padded\nadjacency matrices, while CandDare the row and column indices for the respective matrices. The input\nstructure for this operation requires specialized inputs common to all algorithms presented in this work. This\nincludes a set of columns to store the node-wise positional encodings, denoted as P. Additionally, a global\nvariable representing the positional encoding of the node of interest is stored in a distinct set of columns,\nmarked as Pcur. The structure of the input matrix Xis given below:\nX=\n···P\n0Pcur\npiBglobal\n1Blocal\n0C\nc0D\nd0S\n···\n···p10 0 1 c1d1···\n..................\n···pn0 0 0 cndn···\n.\nFor simplicity, we let the embedding dimension da= 2, which corresponds to the dimension of positional\nencodings. For the attention head for ˜Ain (2), we define:\n(WK, WQ)i,j=(\n1 if i∈ {(P)j,(Pcur)j}, j= 1,2\n0 otherwise,\n(WV)i,j=(\n2 if i=Bglobal, j=D\n0 otherwise.\nThe multiplication XW QW⊤\nKX⊤results in a matrix where each element is the inner product between\npositional encodings of nodes. Positional encodings are formulated such that ⟨pi, pi⟩>⟨pi, pj⟩whenever i̸=j\nfori, j∈[n]. By leveraging this property, we allow the attention matrix to behave as an indicator function.\nThe result is given below:\nσ\u0000\nXW QW⊤\nKX⊤\u0001\n=\n1/20··· 1/2···\n0 1··· 0···\n............\n1/20··· 1/2···\n............\n0 0··· 0···\n.\n8The result of the multiplication of the attention matrix and ˜Ais further multiplied by the input X:\n˜A σ\u0000\nXW QW⊤\nKX⊤\u0001\nX=1/2\n···Bglobal\n0···\n··· A1i···\n...\n··· Ain···\n.\nThe result in column Bglobal is written in the target column DbyWV. Notice that in this case, the target\nfield must also be erased. This can be done as in Section 5.2, using the fmlpand an intermediate scratchpad\nfield, or by leveraging another attention head to replicate the identity matrix and subtract the values in the\ntarget columns.\n6 Theoretical analysis\nIn this section we demonstrate the ability of the architecture in (2)to simulate various graph algorithms. We\nprovide empirical validation of the results in Appendix B.3. We discuss the implications of incorporating\nattention heads in the form of (1)and we demonstrate that the overall architecture is Turing Complete, even\nwhen constrained to constant width. We start by describing the positional encodings we use and other related\nparameters in our architecture.\n6.1 Positional encodings and increment\nIn our approach, we use circular positional encodings to enumerate the nodes in the graph [Liu et al., 2023].\nThis enumeration is carried out by discretizing the unit circle into intervals of angle ˆδ2. Each node is then\nrepresented by a tuple of sine and cosine values.\nFor node enumeration, we use the formula pi=R⊤\nˆδpi−1fori≥1. In this context, ˆδrepresents the smallest\nrotation that our system can represent with limited precision. This parameter essentially sets the limit\non the number of nodes we can enumerate, which is capped at ⌊2πˆδ−1⌋. This limit is determined by the\nprecision with which we can represent ˆδ, thus influencing the maximum number of distinct nodes that can be\neffectively encoded. This approach inherently compensates for rotational imprecision since the encodings are\ngenerated by the rotation matrix and not by the trigonometric functions of each angle. Although this could\naffect the periodicity of the function, our designs do not rely on this attribute. What is crucial is that the\nincrement function can be executed by a neural network, and the positional encodings satisfy the condition:\n⟨pi, pi⟩>⟨pi, pj⟩fori̸=j(see Section 5.3). The rotation matrix Rˆδcan be efficiently implemented as a\nsingle linear layer within fMLP. Furthermore, since Rˆδhas unitary singular values, all positional encodings\nhave unitary norms, thus by the Cauchy-Schwarz Inequality, the inner product property is preserved.\n6.2 Maximum absolute value parameter\nAll algorithms that we simulate utilize a conditional selection function. The implementation of this function\ndepends on the parameter Ω, which represents the maximum absolute value within a clause. We approximate\n2Numerical issues can arise when representing positions by evenly spaced angles around the unit circle. This is due to the\nlimited precision in representing certain quantities, especially irrational numbers and rational numbers with repeating binary\nfractions. To mitigate this issue, we start by setting an initial angle p0as (sin0,cos0) = (0 ,1) and selecting a minimum increment\nangle, denoted as δ. We convert the angle δinto its nearest sine and cosine representation that can be exactly represented by the\nmachine. We denote this approximate angle as ˆδand construct the rotation matrix Rˆδ. Given that this matrix is derived from ˆδ,\nit consists of quantities that can be represented by the machine. Importantly, we ensure that Rˆδmaintains the fundamental\nattribute of orthogonality in the rotation matrix.\n9the conditional selection function in the following manner:\nif-else( c1, c0, γ)≈ϕ(c0−γΩ) + ϕ(c1−(1−γ) Ω),\nwhere c0, c1∈[−Ω,Ω] are the clause values and γ∈ {0,1}is the condition. When γ= 0, the function outputs\nc0, and c1otherwise. The complete construction also replicates the terms for the negative counterparts of\nthe clauses. This approach relies on Ω to cancel any opposite terms in the conditional selection. Therefore,\nΩ determines the largest absolute value present in the clauses, and any element larger than Ω breaks the\nsimulation.\n6.3 Results on simulation of graph algorithms\nWe now present our results on transformer-based simulations of graph algorithms. We first present the\nsimulation results for Dijkstra’s algorithm on weighted graphs, accompanied by a brief overview of its proof.\nWe then shift our focus to undirected graphs, examining algorithms such as Breadth-First Search, Depth-First\nSearch, and the identification of Strongly Connected Components. Alongside these results, we provide an\noutline of the proofs for the related theorems. The complete proofs are in the Appendix C.\nTheorem 6.1. There exists a looped-transformer hTin the form of (2), with 17 layers, 3 attention heads,\nand layer width O(1)that simulates Dijkstra’s shortest path algorithm for weighted graphs with rational\nedge-weights, up to O(ˆδ−1)nodes and graph diameter of O(Ω).\nProof Overview : Theorem 6.1 is proved by construction. Our proof begins by adapting Dijkstra’s algorithm\nto fit our looping structure. This involves unrolling any nested operations to fit within a single loop, as\ndescribed in Algorithm 1. Although these modifications change Dijkstra’s algorithm from its traditional\nform, they do not alter its core functionality. In conventional programming, loops within the code must\nbe completed before the rest of the code can continue. However, in the absence of additional structures,\nour unrolling approach would lead to all instructions being executed all at once. This could potentially\ndisrupt the intended sequence of operations. To address this, we also implement binary flags and conditional\nselections, as explained in (3). These mechanisms are vital for ensuring that certain operations only modify\ndata under specific conditions. A key example in our simulations is the minimum function which has O(n)\ncomplexity and runs concurrently with the rest of the algorithm. When this function is active, it prevents\nmodifications by other parts of the algorithm. The modifications can resume once the minimum function\nhas been completed, at which point it becomes inactive. Finally, each step of the rewritten algorithm is\nimplemented as a series of transformer layers in the form of (2), accounting for 17 layers, and three attention\nheads: two standard and one specialized over A. Some of the core simulation mechanisms of these functions\nare extensively discussed in Section 4 and Section 6, demonstrating that most of the constructions are based\non previously described principles. Furthermore, Theorem 6.1 is restricted to rational edge weights due to the\ninherent limitations of finite precision in representing edge values. We reweight the edges to circumvent issues\narising from the tolerance εof the less-than function (5.2), which may not adequately capture minor path\ndifferences. This involves dividing all edge weights by the smallest absolute edge weight. Consequently, this\nensures that the minimal non-zero path difference is effectively greater than one. The graph size is bounded\nby the enumeration constraints of ˆδ, as mentioned in Section 6.1. The dependence on the graph diameter is\nattributed to the maximum distance that can be constructed during the execution of Dijkstra’s algorithm.\nGiven the algorithm’s design, which excludes the consideration of repeating or negative paths, this quantity\nis determined by the diameter of the weighted graph, which must remain below Ω.\nWe now present our results for algorithms on unweighted graphs, followed by a collective discussion of their\nproofs.\nTheorem 6.2. There exists a looped-transformer hTin the form of (2), with 17 layers, 3 attention\nheads, and layer width O(1)that simulates Breadth-First Search (BFS) for unweighted graphs with up to\nmin(O(ˆδ−1), O(Ω)) nodes.\n10Theorem 6.3. There exists a looped-transformer hTin the form of (2), with 15 layers, 3 attention heads, and\nlayer width O(1)that simulates Depth-First Search (DFS) for unweighted graphs with up to min(O(ˆδ−1), O(Ω))\nnodes.\nTheorem 6.4. There exists a looped-transformer hTin the form of (2), with 22 layers, 4 attention heads,\nand layer width O(1)that simulates Kosaraju’s Strongly Connected Components algorithm for unweighted\ngraphs with up to min(O(ˆδ−1), O(Ω)) nodes.\nProof Overview: Our proofs for Theorems 6.2-6.4 also follow a constructive approach. The implementations\nof breadth-first and depth-first search are comparable to Dijkstra’s algorithm in terms of the number of layers\nand attention heads. However, the Strongly Connected Components algorithm utilizes significantly more\nlayers and an additional attention head for A, due to its complexity and the need for a depth-first search\nover both AandA⊤. Despite the absence of edge weights, we encounter an additional bound of O(Ω) on\nthe number of nodes. This stems from our method of replicating the behavior of queues and stacks, which\nare used in these algorithms. Similar to our approach with Dijkstra’s algorithm, we also incorporate an\ninteraction with a minimum function. During the execution of the algorithms, priority values are assigned to\nthe neighbors of each node visited. These assigned values shape the minimum function’s behavior: increasing\npriority values mimic the functionality of a queue while decreasing values replicate that of a stack. Hence,\nhaving more nodes in a graph increases the number of loop iterations, thereby increasing the absolute value\nof the priority value. Given that these priority values feature in conditional selections, they are also subject\nto the constraints imposed by Ω. Consequently, the graph’s size limitations are independently determined by\neither ˆδor Ω. In the implementation of Breadth-First Search (BFS) and Depth-First Search (DFS), since\nnodes only need to be visited once, Ω is the maximum node count for which accurate simulation of any\ngraph is guaranteed. In the implementation of the Strongly Connected Components (SCC) algorithm, which\ninvolves two distinct DFS operations, the maximum node count is constrained to Ω /3. This specific limit\narises from the two-phase process, and also the role of the first DFS in organizing nodes by their finishing\ntimes, which is further detailed in Appendix C.\nWhile our guarantees are limited by parameters ε, Ω, and ˆδ, an important question remains: What are the\nimplications of using a graph that fails to meet assumptions in our theorems? This situation could result\nin inaccurate simulation outcomes or even a failure to meet the pre-established termination condition, thus\nfailing to halt as expected.\n6.4 Turing Completeness\nWe now present our result on Turing Completeness of the architecture in (2). It’s important to note that\nusing a graph convolution operation within the attention mechanism as shown in (1)might affect the model’s\nexpressiveness. We demonstrate that a transformer based on this structure is also Turing Complete. Following\nthe methodology of Giannou et al. [2023], we demonstrate this claim by successfully simulating SUBLEQ, a\nsingle-instruction language that, with arbitrary memory resources, is proven to be Turing Complete [Mavaddat\nand Parhami, 1988].\nRemark 6.5.There exists a looped-transformer hTin the form of (2), which utilizes the modified attention\nhead in (1), with 11 layers, 3 attention heads, and layer width O(1) that simulates the SUBLEQ.\nProof Overview: We prove that the architecture in (2)simulates a modified version of SUBLEQ, which is also\nTuring Complete. This modification changes the memory structure of SUBLEQ, utilizing two memory blocks\ninstead of one. The first block is the standard memory block as in Mavaddat and Parhami [1988] which\ncan be read and altered. The second is a special read-only memory that only stores the adjacency matrix.\nWe introduce this memory modification to reflect the fact that the architecture in (2)has two inputs, the\ndata matrix Xand the padded adjacency matrix ˜A. The former is modified at each iteration of the looped\ntransformer architecture, while the latter is only used in the convolution stage, and it never changes. The\n11modified version of SUBLEQ utilizes data from the graph by reading the data from the second block and\ncopying them to the first block. We show that this operation of reading and copying data can be simulated\nby our read operation in Section 5.3. The rest of SUBLEQ can be simulated using generic attention heads\nand MLPs similarly to Giannou et al. [2023]. The complete proof is provided in Appendix C.\n7 Conclusion and Future Work\nWe present a constructive approach where a looped transformer, combined with graph convolution operations,\nis used to simulate various graph algorithms. This demonstrates the potential of looped transformers for\nalgorithmic reasoning on graphs. This architecture also proves to be efficient, as it has constant network\nwidth regardless of the input graph’s size, allowing it to handle graphs of varying dimensions.\nCurrently, in our results, we use a single architecture to simulate various graph algorithms. However, each\nalgorithm is simulated using different parameters in the architecture. We believe that it is possible to have a\nsingle neural network that simulates all algorithms. Such a result will be more aligned with the generalist\napproach in [Ibarz et al., 2022]. We leave this goal for future work.\nOutside the scope of simulation, studying looped transformers for graph algorithms within the Probably\nApproximate Correct (PAC) learning framework offers an exciting research direction. Specifically, the\nconstructive parameters Ω, ˆδ, and εare essential in determining which graphs can be simulated. However,\nthese parameters may also influence the learnability of algorithmic subroutines. An interesting research\nquestion involves determining the sample complexity of subroutines as a function of these constructive\nparameters. Investigating this could further reveal the relationship between the design of solutions and their\nlearning effectiveness under the PAC framework.\n8 Acknowledgments\nK. Fountoulakis would like to acknowledge the support of the Natural Sciences and Engineering Research\nCouncil of Canada (NSERC), [RGPIN-2019-04067, DGECR-2019-00147].\nCette recherche a ´ et´ e financ´ ee par le Conseil de recherches en sciences naturelles et en g´ enie du Canada\n(CRSNG), [RGPIN-2019-04067, DGECR-2019-00147].\nReferences\nAlfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms .\nAddison-Wesley, Reading, Mass., 1974.\nEkin Aky¨ urek, Dale Schuurmans, Jacob Andreas, Tengyu Ma, and Denny Zhou. 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In Proceedings of the IEEE/CVF International Conference on Computer\nVision , pages 579–588, 2021.\n15A Additional related work\nThe importance of recurrence in the expressive power of neural networks has been well-established, as\nevidenced by the Turing Completeness results of Recurrent Neural Networks [Siegelmann and Sontag, 1995].\nPrior to the emergence of transformers, pioneering models like Neural Turing Machines (NTMs) [Graves et al.,\n2014], utilized the idea of recurrence, integrating neural networks with external memory, demonstrating strong\nperformance in tasks such as copying and sorting. Subsequent advances were seen by adapting NTMs for\nlearning algorithmic tasks [Kaiser and Sutskever, 2016], incorporating a dynamic memory [Graves et al., 2016],\nand utilizing recurrent models to learn multiple programs [Reed and De Freitas, 2016]. All these advancements\nlaid the groundwork for concepts utilized in transformers and in this work, such as the addressing mechanism\nof Graves et al. [2014] which shares a similar principle to attention in transformers, and concepts like scratch\nspace for intermediate computations as proposed by Reed and De Freitas [2016].\nIn the more recent developments related to algorithmic reasoning in transformers, dedicated frameworks have\nemerged to represent and execute complex algorithmic tasks. An example of this is the Restricted Access\nSequence Processing Language (RASP). RASP allows for the expression of sequences of steps in transformers\nand can be applied to tasks such as token counting and sorting. Along this line, Lindner et al. [2023] developed\na compiler for translating between RASP and Transformer models, while Friedman et al. [2023] have designed\nmodified Transformers that can be trained and subsequently converted to RASP instructions.\nB Implementation details\nIn this section, we describe aspects of the implementation and empirical verification of our simulations. As\nmentioned in Section 4, despite our constructions being based on hardmax, for the purpose of empirical\nvalidation and to better align with the implementation practices in transformers, we utilize the softmax\nfunction:\nσS(zi) =ezi\nPK\nj=1ezj,\nwhere Krepresents the number of elements. When using the softmax in attention, we divide all parameters in\nWQandWKby a temperature parameter T. The value of Tis important as it determines the sharpness of the\nsoftmax operation. As shown in Giannou et al. [2023] and Liu et al. [2023], as the temperature gets arbitrarily\nsmall, the behavior of the softmax approximates that of the hardmax, which is used in our constructions.\nHowever, it is important to acknowledge that in practice the simulation of all algorithmic steps is subject\nto numerical limitations due to floating-point arithmetic. For example, using the minimum angle ˆδthat a\ncomputer can represent accurately is impractical. The reason is that this quantity becomes overshadowed\nby subsequent operations that introduce imprecision, such as matrix multiplications, and by non-linear\nelements like the softmax function in the attention stage. In algorithmic tasks, simulations exhibit a very low\ntolerance for computational errors. If not properly managed, these errors can propagate, leading to inaccurate\nprediction or non-termination. To address this, we adopt a more conservative approach in setting parameters\nforˆδandεas described in Appendix B.3. In addition, we incorporate rounding operations for both binary\nvalues and positional encodings, following the structure described in (2).\n16B.1 Rounding functions\nWe utilize rounding functions in-between the simulated algorithmic steps to ensure that the computation of\nbinary values made by the transformer does not include additional errors that propagate along the simulation.\nTo this end, we adapt the rounding operation presented by Giannou et al. [2023], which can be algebraically\nexpressed as:\nround( X[:, C])≈η−1(ϕ(X[:, C]−1/2)−ϕ(X[:, C]−(1/2+η))), (3)\nwhere ηis a small tolerance value. This implementation can also be done using the structure presented in\n(2), noting that the expression in (3)is equivalent to the less-than function presented in Section 5.2, with\nε=η, and X[:, D] equivalent to the quantity 1 /2.\nIn practice, the increment function described in Appendix C.1.2 introduces noise in the positional encodings.\nTo counter this, we employ a rounding procedure for circular positional encodings. This involves using two\ntransformer layers. The first layer generates a binary vector based on the current positional encoding We then\nround this quantity utilizing the procedure described in (3), and finally obtain the corresponding positional\nencoding based on the index position using the last layer.\nThe first attention layer is set up as described in (6), where Dis set to Bglobal , and SPEdenotes the target\ncolumn in the scratchpad. Another attention head is used to clear the original scratchpad contents, as detailed\nin (7). The corresponding fMLP in this layer implements Equation (3), with SPEas the source column C.\nIn the second transformer layer, the matrices W(1)\nK,W(1)\nQ, and W(1)\nVof the first attention head are defined as\nfollows:\n(W(1)\nK, W(1)\nQ)i,j=(˜T−1/2ifi=SPE\n0 otherwise,(W(1)\nV)i,j=(\n2 if i= (P)k, j= (E)k, k∈0,1\n0 otherwise.\nIn this layer, Edenotes the target columns for inserting positional encodings, and ˜Tis an annealed temperature\nparameter of softmax, which is set higher than in other implementations (e.g., T= 10−5). This ensures\nthat minor differences in this attention head are minimized, thereby reducing errors in reading positional\nencodings. Additionally, we use the configuration of (7)in another attention head to clear the contents of E\nandSPE. Lastly, the implementation of fMLP, as defined in (8), clears the last nentries of the target field E.\nB.2 Writing prevention after termination\nThe constructions in the empirical section slightly differ from the ones presented in Appendix C due to\nan additional consideration. While the theoretical constructions require delivering accurate outputs upon\nreaching the termination condition, one could continue running the looped transformer after the termination\nflag is activated. To ensure that the solution is not modified post-termination, we introduce extra constraints\nin the form of flags that deactivate write permissions once the termination condition is met. These are\narticulated through algebraic expressions, easily integrating into the transformer implementation. In practice,\nthe writing permission flag is not determined solely by the minimum function’s termination flag ( term min) for\nthe implementation of Theorem 6.1 and similar algorithms. Instead, we construct a flag defined as write =\nterm minand (not term) , which can be implemented as X[:,write] =ϕ(X[:,term min]-X[:,term] ). This\ncondition ensures that once the algorithm reaches the termination state, writing is forbidden.\nFurthermore, we employ a conditional selection mechanism – as explained in Appendix C.1.1 – to maintain\nthe termination flag, which is activated if the terminating condition is met but the simulation continues\nrunning. This is performed by first moving the calculation of the termination condition to a temporary\n17storage area, denoted as Sterm. Then, at the end of each iteration, we execute the layer: cond-select(X,\nSterm, term, S term, term) . This means that if the recalculated termination condition in Stermis true, then\ntheterm variable is updated; otherwise, it remains unchanged, effectively managing the looped transformer’s\noperation post-termination.\nB.3 Empirical validation\nWe validate our theoretical results using graphs in the CLRS Algorithmic Reasoning Benchmark [Veliˇ ckovi´ c\net al., 2022]. This benchmark offers data created for multiple tasks, including sorting, searching, and geometry,\namong others. Additionally, it encompasses tasks specific to graphs, including those simulated in this study.\nThese tasks are divided into segments with varying input sizes. For graph-related tasks, the training and\nvalidation sets include graphs with 16 nodes, while the test set contains graphs with 64 nodes. In Table 1, we\nshow that our simulations achieve perfect accuracy across all tested instances.\nTable 1: Accuracy over algorithmic tasks on CLRS\nSplit (size) BFS DFS Dijkstra SCC\nTrain (1000) 100% 100% 100% 100%\nValidation (32) 100% 100% 100% 100%\nTest (32) 100% 100% 100% 100%\nIn our empirical verifications, we consistently apply specific parameters: an angular increment δset at 10−2,\na maximum value Ω of 105, and a temperature parameter Tat 10−7. Additionally, in the masking operations\nsuch as Appendix C.2.1, we address numerical imprecision during execution by using a value that is an order\nof magnitude smaller than Ω. This precaution is crucial because minor inaccuracies that slightly increase these\nvalues above Ω could lead to cumulative errors in the conditional selection function, potentially impacting\nthe integrity of our results.\nFurthermore, beyond the practical considerations discussed in the previous sections, the implementation of\nbreadth-first search in the CLRS benchmark slightly differs from the one presented in Algorithm 4. This\nis because, as mentioned in Veliˇ ckovi´ c et al. [2022], the results of the algorithm are built on top of the\nconcept of parallel execution, which implies that for algorithms such as breadth-first search, there is no\nconcept of order in the discovery of neighbor nodes, as outlined in Cormen et al. [2022]. Instead, the order of\ndiscovery in CLRS is established by the value of the node index. Therefore, to address this particularity, the\nimplementation of the conditional selection function in step (9) of Algorithm 4 has another pair of clauses,\nwhich can be expressed by cond-select(X, val best, order, term min, order) . Essentially, this variation\nin the implementation is designed to capture the priority value from the parent node, thereby ensuring\nuniform priority assignment across all nodes within the same neighborhood level.\n18C Proofs of Theorems\nC.1 Preparations for the proofs\nIn this section, we first introduce key coding primitives that are essential to all the algorithmic simulations\npresented. We illustrate these coding primitives through the algorithm of the minimum function. This\nfunction is designed to identify the smallest value in a list, along with its index and other pertinent information.\nAs outlined in Section 6, for each simulated algorithm, we adapt their formulations to align with the looped\narchitecture of the transformer. Although this architecture inherently does not support inner loops, we\nsimplify the presentation of the pseudocode by using inner loops solely to depict functions that are operated\nin parallel by the transformer. As further depicted in Algorithms 3-6, the iteration of the minimum function\ndescribed in Algorithm 2 is executed inside the main loop of the transformer, along with the other functions\nof the corresponding algorithms. The algorithm description is also accompanied by comments that enumerate\nthe steps that are implemented using the layer formulation in (2). Each comment is also linked to the section\nthat provides a more detailed explanation of its implementation.\nAlgorithm 2 Iteration of the minimum function (get minimum)\nInput: data x, size n List of values to find the minimum\nInput: data xSCC, size n\n1: scc cur, scc best= 0, 0 Only used for SCC\n2: idx cur, val cur= 0, 0\n3: idx best, val best= 0, Ω\n4: visit min= array of size n\n5:fori= 1tondo\n6: visit min[i] =false\n7:end for\n8: term min=false\n9:ifterm ministrue then\n10: ···\n11: term min=false (1) Re-initialize variables [C.1.1]\n12:end if\n13: idx cur= idx cur+ 1 (2) Increment position [C.1.2]\n14: val cur=x[idxcur] (3) Read value at current position [C.1.3]\n15: scc cur=xSCC[idxcur]\n16: condition = val cur<valbest (4) Compare values [C.1.4]\n17:ifcondition then\n18: val best, idx best= val cur, idx cur (5) Update variables [C.1.1]\n19: scc best= scc cur\n20:end if\n21: visit min[idxcur] =true (6) Visit current position [C.1.5]\n22: term min=not(false in visit min) (7) Trigger termination [C.1.6]\nThe lines [1-8] consist of the initialization of the data, while the remaining represent the algorithm executed\ninside the loop. Note that lines 1, 15, and 19 are highlighted in gray, and are only defined in the case of the\nStrongly Connected Components algorithm. As described in Section 5.1, the single variables (e.g. val cur)\noccupy the top row, while lists (e.g. visit min) occupy the last nrows of X. Furthermore, the implementation\nof indexes requires two different representations in the input matrix X.\nIn Algorithm 2, these correspond to idx curandidx bestas well as the variables for the Strongly Connected\nComponent implementation. In X, these indices are represented both as integer numbers and positional\nencodings. The latter is vital for algorithm functionality, while integer numbers are key in the decoding\nphase, forming the output data. Consequently, ambiguous variables in proofs are referred to using their\npositional encoding. For example, if idx curpoints to the list’s first element during execution, it is represented\n19asp1= (sin(θ1),cos(θ1)) — the positional encoding for the first node. The integer number representation is\nemphasized when relevant. Finally, the comments that enumerate and describe the steps executed are the\nbasis for the functions implemented by our architecture.\nC.1.1 Conditional selection: Steps (1) and (5)\nThis section details the implementation of the conditional selection functions outlined in steps (1) and (5) of\nAlgorithm 2. We reference the structure of this implementation in the subsequent proofs. In our constructions,\nwe express this operation as: cond-select(X,V 1,V0,C,E) = write(if-else(X[:,V 1], X[:,V 0], X[:,C]),\nX[:,E]) . Here, Xis the input matrix; V1andV0are columns for clause values; CandEare the condition\nand target field, respectively. The approximation of the function if-else is shown in (3).\nNext, we describe how this function is implemented. As with other implementations, we utilize a scratchpad\narea, denoted as S, to store intermediate calculations. Each column in this space is indexed by Si. During\nimplementation, we also introduce more abstract indices for the scratchpad, such as Sidxcur. While it is\nunderstood that this index correlates to a specific integer number, we adopt this approach to enhance clarity\nand comprehension. By using such descriptive indexing, we aim to convey the intended purpose of each\nallocation in the scratchpad.\nAs also described in the implementation of the less-than function (see Section 5.2), we set all parameters to\nzero in the attention stage, only maintaining the residual connection. Then, for the values of the fMLP, we\ndefine:\n(W(1))i,j=(\n1 if i∈n\nV0,V1,C,E,\nBglobal ,Blocalo\n, j=i,\n0 otherwise,(W(2))i,j=\n\n1 if ( i, j)∈n\n(E,E),(V0,S1)\n(V1,S2)o\n−Ω if ( i, j)∈n\n(C,S1),(Bglobal ,S2)\n(Blocal,S2)o\nΩ if i=C, j=S2\n0 otherwise,\n(W(3))i,j=(\n1 if ( i, j)∈n\n(E,E),(S1,S1)\n(S2,S1)o\n0 otherwise,(W(4))i,j=\n\n1 if i=E, j=S1\n−1 if i, j=E\n0 otherwise.(4)\nIn the general construction of the conditional selection function, the first layer, W(1), functions as an identity\noperator. The second layer, W(2), constructs the arguments of ϕas detailed in (3). The third layer, W(3),\nsums the terms post-ReLU, and the last layer, W(4)is designed to transfer the term, initially located in the\nscratchpad, to the specified column E. Our complete construction also accounts for the negative counterparts\nof the clauses. For simplicity, the adaptation for these counterparts is not included in the proofs, as it simply\ninvolves inverting the signs of the inputs V0, V1andEinW(1)andW(4). While we present an implementation\nfor a single pair of clauses V0andV1, this function can accommodate additional variables following the same\nmechanism as in (3), as is done for steps (1) and (5) of Algorithm 2. For example, in the implementation\nof Algorithm 2, the conditional selection clause represented in step (5) also needs to update the integer\nrepresentation of the index, therefore another pair of clauses is utilized in (5).\nEssentially, the general implementation of the conditional selection function operates by choosing a value\nfrom two columns in the input matrix and inserting it into a designated field. However, the implementation\nvaries in the context of the re-initialization function. In this scenario, the conditional selection is executed\nbetween the current values in the algorithm and their reset versions. Specifically for this function, the reset\nvalues must be generated in the first layer and then placed in the scratchpad, which assumes the role of the\nV1field in the aforementioned formulation. This adjustment ensures that the function appropriately handles\n20the re-initialization process within the algorithmic framework.\n(W(1))i,j=\n\n1 if i={Bglobal, Blocal,idxcur, val cur, ..., visit min}, j=i\n(p0)kifi= (idx cur)k, j= (Sidxcur)k, k∈ {1,2}\nΩ if i= val best, j=Svalbest\n0 if i= val cur, ..., visit min, j=Szero\n0 otherwise.(5)\nIn this implementation, the variable for the current index is reset to a specific value, p0= (sin(0),cos(0)) =\n(0,1). This value does not correspond to any node enumerated and is intentionally reserved in our constructions\nto represent the equivalent of the index 0 in integer numbers. This approach ensures that p0serves as a unique\nand identifiable starting point or reset state within the algorithm. For the remaining layers, the placeholder\nof field V1is then occupied by the targets of W(1), i.e. Sidxcur, SvalbestandSzeroin their corresponding\noperations. Moreover, it is important to note that regardless of the established condition, the status of\nterm minis invariably set to false. Consequently, this field is consistently cleared in every iteration, similar to\nthe way the target field Eis cleared.\nC.1.2 Increment position: step (2)\nAfter the re-initialization process, the first step that follows is obtaining the next index in the list. In this\ncase, since we utilize circular positional encodings, all that is needed is to utilize a rotation matrix. The\nrotation matrix is predetermined by ˆδ, as discussed in Section 6.1. We express this operation as increment(X,\nC, D) = write(rotate(X[:, C]), X[:,D]) , where CandDare the source and target columns. In the\nimplementation of step (2), both CandDare the variable idx cur. Below, we detail the specific set of\nparameters used to implement this function. Again, in this case, we only utilize the entries of fMLP, while\nthe parameters of fattnare set to zero.\n(W(1))i,j=\n\ncos(ˆδ) if i= (C)k, j= (SC)k, k∈ {1,2}\n−sin(ˆδ) if i= (C)1, j= (SC)2\nsin(ˆδ) if i= (C)2, j= (SC)1\n1 if i∈D, j =i\n0 otherwise,\n(W(2,3))i,j=(\n1 if i∈ {D, S C}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i= (SC)k, j= (D)k, k∈ {1,2}\n−1 if i∈D, j=i\n0 otherwise.\nIn this context, the initial entries of the matrix W(1)correspond to the rotation matrix Rˆδ. Additionally,\nthis matrix includes a unique non-zero entry, specifically designed to reset the target field of the final layer.\nIt is also important to note that the variables CandDare represented by a tuple of sine and cosine values,\nand as such, its coordinates must be accurately aligned. Moreover, the implementation must account for\nthe fact that positional encodings can possess negative values. The parameters for these added entries are\nessentially the inverse of the signs present in W(1)andW(4). To facilitate this process, the implementation\nalso incorporates additional scratchpad entries, which serve to temporarily hold these intermediate results.\nC.1.3 Read value: Step (3)\nOnce the node index is updated, we need to retrieve its associated value from the input list, possibly\naccompanied by other relevant information. This operation can be expressed as read-X(X, C, D, E) =\n21write(X[C, D], X[1,E]) , where CandDdenote the coordinates of the target entry and Eis the placeholder\nfor the target field. In a more basic form, this operation involves extracting information from a local variable\nand transferring it into a global variable. For the implementation of step (3), we utilize this function to\nextract the entry from the list of values (denoted by xin Algorithm 2) as well as from the list that encodes\nthe indexes with integer numbers.\nThe principle for this operation is similar to the extraction of rows or columns of A, as presented in Section 5.3.\nWe employ a standard attention mechanism (no graph convolution) leveraging the positional encoding of the\nidxcurvariable to obtain the entries in the corresponding row.\n(W(1)\nK, W(1)\nQ)i,j=(\n1 if i∈ {(P)j,(C)j}, j= 1,2\n0 otherwise,(W(1)\nV)i,j=(\n2 if i=D, j=E\n0 otherwise.(6)\nTo ensure the target column Eis prepared for the insertion of the desired value, we apply a distinct attention\nmechanism. This mechanism employs a standard attention head but sets the first entry of XW KandXW Qto\np0. This configuration guarantees that, when executing a hardmax operation, the attention matrix effectively\nfunctions as an identity matrix. Consequently, this facilitates the removal of any existing entries in the target\ncolumn E.\nThe matrices W(2)\nK,W(2)\nQ, and W(2)\nVare defined as follows:\n(W(2)\nK, W(2)\nQ)i,j=(\n1 if ( i, j)∈ {((P)1,1),((P)2,2),(Bglobal,2)}\n0 otherwise,(W(2)\nV)i,j=(\n−1 if i, j=E\n0 otherwise.(7)\nThis results in the following attention matrix:\nσ\u0010\nXW(2)\nQW(2)⊤\nKX⊤\u0011\n=σ\n\np⊤\n0p0p⊤\n0p1···p⊤\n0pn\np⊤\n1p0p⊤\n1p1···p⊤\n1pn\n............\np⊤\nnp0p⊤\nnp1···p⊤\nnpn\n\n=\n10··· 0\n01··· 0\n............\n00··· 1\n\nBy employing the value matrix WV, we can clear the target column Eto insert new values. It is important\nto note that the attention head in (6)also adds terms to the last nrows of E, which are undesirable. To\ncounteract this, we use the parameters of the fMLP:\n(W(1))i,j=\n\n1 if i, j=E\n−Ω if i=Bglobal, j=E\n0 otherwise,(W(2,3))i,j=(\n1 if i, j=E\n0 otherwise,(W(4))i,j=(\n−1 if i, j=E\n0 otherwise,\n(8)\nIn this construction, the first layer removes the top row entry of E, which is then passed along the second\nand third layers, and finally subtracted from the target, removing all entries in the bottom rows.\nC.1.4 Compare values: step (4)\nIn this stage, we focus on comparing the current value ( val cur) with the best value found so far ( val best).\nThe implementation of this function is detailed in Section 5.2. In this context, the operation is expressed as\nless-than(X, val cur, val best, condition) .\n22C.1.5 Visit node: step (6)\nEach iteration of the minimum function sets the current node as visited. This action is crucial for the\nfunction’s termination criteria, which checks if all nodes have been visited.\nEssentially, the role of this function is to assign a predefined value to a specific entry ( X)i,j. We express this\noperation as write-row(X, C, D, E) = write(X[0,D].e X[0,C], X[:,E]) . Here, eisymbolizes the standard\nbasis vector in an n+ 1 dimensional space. The variables C,D, and Ein the expression represent columns\nthat contain the row to be written, the writing content, and the target field, respectively. This writing\nprocedure is analogous to the read function, previously described in Appendix C.1.3.\nThe matrices WK,WQ, and WVare defined as follows:\n(W(1)\nK, W(1)\nQ)i,j=(\n1 if i∈ {(P)j,(C)j}, j= 1,2\n0 otherwise,(W(1)\nV)i,j=(\n2 if i=D, j=E\n0 otherwise.(9)\nIn the application of step (6) of Algorithm 2, we set Cto be idx cur,DtoBglobal , and Eto the visit min\nvariable. Additionally, as also pointed out in Appendix C.1.3, the extra entries generated by the attention\nhead in (9)must also be eliminated. Contrary to the read function, in this instance, the unwanted entry\nappears in the top row of the target field. To address this, we adopt the same strategy as in the read function,\nutilizing an additional attention head that functions as the identity function, deleting the top row entry using\nthe global bias Bglobal .\n(W(2)\nK, W(2)\nQ)i,j=(\n1 if ( i, j)∈ {((P)1,1),((P)2,2),(Bglobal,2)}\n0 otherwise,(W(2)\nV)i,j=(\n−1 if i=Bglobal, j=E\n0 otherwise.\nTo define fMLP, we set all its parameters to zero and retain only the residual connection. This approach\nessentially disregards any contribution from fMLP.\nC.1.6 Terminate: step (7)\nAt the end of each step of the minimum function, as well as in the other algorithms, we must verify if all nodes\nhave been visited. If all nodes are visited, this indicates termination, therefore the termination flag must be\nactivated. This process can be formulated as all-one(X, C, E) = write(all(X[2:,C]), X[1,E]) , where\nallis a function indicating whether all elements are non-zero, while X[2:, C] captures the last nrows of the\ncolumn C.\nWe start by defining the attention stage:\n(W(1)\nK, W(1)\nQ)i,j=(\n1 if ( i, j)∈ {((P)1,1),((P)2,2),(Bglobal,2)}\n0 otherwise,(W(1)\nV)i,j=\n\n−1 if i=E, j =i\n1 if i=Bglobal, j=E\n0 otherwise.\nThe first attention head, defined as the read function in Appendix C.1.3, is tasked with clearing fields used in\nthe writing process. Additionally, we define:\n(W(2)\nK)i,j=\n\n−1 if i=Bglobal\n1 if i=Blocal\n0 otherwise,(W(2)\nQ)i,j=\n\n1 if i=Bglobal\n−1 if i=Blocal\n0 otherwise,(W(2)\nV)i,j=\n\nΩ if i=C, j =E\n−Ω if i=Blocal, j=E\n0 otherwise.\n23This results in the following attention matrix:\nσ\u0010\nXW(2)\nQW(2)⊤\nKX⊤\u0011\n=σ\n2\n−11··· 1\n1−1··· − 1\n............\n1−1··· − 1\n\n=\n01/n···1/n\n10··· 0\n............\n10··· 0\n(10)\nThe top row of the attention matrix is crucial as it aggregates the last nterms of the matrix X. In addition,\nsince our construction only uses columns with null values at the top row, the remaining nrows in (10) do not\nadd any contribution in any field. We first define the terms for fMLP, followed by an overview of how this\nimplementation simulates the desired behavior.\n(W(1))i,j=\n\n1 if i=E, j∈ {E, S E}\n−1 if i=E, j =S˜E\n0 otherwise,(W(2,3))i,j=(\n1 if i∈ {E, S E, S˜E}, j=i\n0 otherwise,\n(W(4))i,j=\n\n1 if i∈ {E, S ˜E}, j=E\n−1 if i=SE, j=E\n0 otherwise.\nIn the implementation of the fMLP, the entries for SEandS˜Eare used to remove the entries pre-MLP, in both\npositive and negative cases. In summary, the entire operation performs as follows: ϕ(1−Ω + Ω/nP\ni(X[i, C])).\nWhen the sum of elements in C is less than n, the term −Ω dominates, ensuring a resulting flag of zero.\nHowever, once all nodes are visited, X[:, C] offsets the negative influence of Ω, thereby activating the flag.\nC.2 Proof of Theorem 6.1\nIn this section, we introduce the constructions utilized to simulate Dijkstra’s shortest path algorithm [Dijkstra,\n1959]. This algorithm aims to determine the shortest path between any node of a weighted graph and a given\nsource node. To this end, we first present an overview of the implementation strategy, which accompanies the\nlimitations that are described in Theorem 6.1. These are then followed by the specific construction of each\nstep of the algorithm.\nImplementation overview: Executing Dijkstra’s algorithm consists of an iterative process that detects\nthe node with the current shortest distance from the source node, updating the shortest path distances to\nneighboring nodes and marking the given node as visited. This process continues until all nodes have been\nvisited. In our implementation, we simulate a minimum function that operates in the column that stores the\ndistances. This operation is alternated with the main execution of the algorithm, that is, the section that\nreads the adjacency matrix and updates the distances. This is made possible by the conditional selection\nmechanism and the minimum function’s termination flag, which conditions any writing operations in the\nmain execution of the algorithm.\nSpecifications of the architecture: In the following pseudocode description, we show the steps required\nto execute the algorithm. Each of the steps has an associated transformer layer in the form of (2)that\nimplements the corresponding routine. In the case of Dijkstra’s algorithm, there is a total of 17 steps, and\ntherefore our implementation requires 17 layers. Furthermore, throughout the implementation details of each\nstep, no configuration uses parameters whose count scales with the size of the graph, thus accounting for\nconstant network width.\nLimitations: Beside the enumeration and the minimum edge limitations explained in Section 6, the diameter\nof the graph also imposes a limitation on the set of graphs our constructions can represent. Unlike the\n24implementation of the other algorithms, the priority for selecting a node is determined by its current distance\nfrom the source node. As explained in Section 6.2, the constructions of conditional selection functions are\nlimited by Ω, the largest absolute value that can be utilized in a conditional clause. This constructive\nlimitation constrains the distances that can be represented. More specifically, the greatest distance between\nany pair of vertices in the graph is bounded by O(Ω).\nIn the remaining parts of the proof, we start by presenting the revised version of the algorithm, followed by\nthe construction of each step that has not been covered in previous algorithms.\nAlgorithm 3 Dijkstra’s algorithm for shortest path\nInput: integer start\nInput: matrix A, size n×n\n1: prev, visit, dists, dists masked , changes, is zero, candidates = arrays of size n\n2:fori= 1tondo\n3: visit[i], dists[i], prev[i] = false ,ˆΩ, i\n4:end for\n5: visit[start] = 0\n6: term = false\n7:··· Initialization of min-variables\n8:while term is false do\n9:fori= 1tondo\n10: ifvisit[i] is true then\n11: dists masked [i] = Ω (1) Mask visited nodes [C.2.1]\n12: else\n13: dists masked [i] = dists[i]\n14: end if\n15: end for\n16: get minimum(dists masked ) (2-8) Find minimum value [C.1]\n17: ifterm ministrue then\n18: node = idx best\n19: dist = val best (9) Get minimum values [C.2.2]\n20: end if\n21: candidates = A[node, :] (10) Get row of A [C.2.3]\n22: fori= 1tondo\n23: is zero[i] = (candidates[i] ≤0) (11) Mark non-neighbors [C.2.4]\n24: end for\n25: fori= 1tondo\n26: candidates[i] = candidates[i] + dist (12) Build distances [C.2.5]\n27: end for\n28: fori= 1tondo\n29: changes[i] = candidates[i] <dists[i] (13) Identify updates [C.2.6]\n30: end for\n31: fori= 1tondo\n32: ifterm minisfalse and iszero[i] is true then\n33: changes[i] = 0 (14) Mask updates [C.2.7]\n34: end if\n35: end for\n36: fori= 1tondo\n37: ifchanges[i] is true then\n38: prev[i], dists[i] = node, candidates[i] (15) Update variables [C.2.8]\n39: end if\n40: end for\n41: visit[node] = visit[node] + term min (16) Visit node [C.2.9]\n42: term = not(false in visit) (17) Trigger termination [C.2.10]\n43:end while\nreturn prev, dists\n25C.2.1 Mask visited nodes: step (1)\nIn the execution of Dijkstra’s algorithm, visited nodes are excluded from future iterations. Our implementation\nachieves this by masking distances, ensuring that masked values are ignored during the minimum function\nexecution.\nBefore delving into the parameter configuration of this layer, it is crucial to address the initialization of the\ndists variable, which stores current distances. In Dijkstra and other algorithms, this variable is initialized to\nˆΩ= Ω−ϵ. This setup ensures that, during the minimum function phase, there is a preference for unvisited\nnodes - that is, nodes not adjacent to any visited node at a certain point in the execution - over those already\nvisited. Conversely, initializing all nodes to Ω would result in the masking operation treating unvisited and\nvisited nodes equally, thereby disrupting the accuracy of the simulation.\nImplementing the masking operation in our context closely resembles the conditional selection function in\nAppendix C.1.1, with a key difference in its initial setup. This variation primarily involves the definition of\nthe first two layers, which we define as follows:\n(W(1))i,j=\n\n1 if i∈n\nV0,C,E,\nBglobal ,Blocalo\n, j=i\nΩ if i=Blocal, j=SV1\n0 otherwise,(W(2))i,j=\n\n1 if ( i, j)∈n\n(E,E),(V0,S1)\n(SV1,S2)o\n−Ω if ( i, j)∈n\n(C, S 1),(Bglobal , S2),\n(Blocal, S2)o\nΩ if i=C, j=S2\n0 otherwise,\nUnlike the approach outlined in (4), the masking operation does not have a V1counterpart already calculated.\nTo this end, we first generate create a mask of magnitude Ω over all nodes, and insert it in a scratchpad\nspace denoted by SV1. Then for the second layer, we simply redirect the definition presented in (4), from the\ndependency from V1to the dedicated column SV1. These adjustments are processed by the remaining layers\nof the implementation, ensuring the correct nodes are selected for further operations in the algorithm. For\nthe application in step (1) of Algorithm 3, we set V0todists ,Ctovisit , and E to dists masked .\nC.2.2 Get minimum values: step (9)\nThe purpose of this function is to update the values node anddist once the minimum function terminates. To\nthis end, we also utilize the formulation of the conditional selection function, expressing it as cond-select(X,\n[idx best, val best], [node, dist], term min, [node, dist]) . Note that in this formulation, the fields\nV0,V1, and Etake in multiple values. The implementation for this case is similar to that presented in\nAppendix C.1.1, matching the order of the elements in each of these fields.\nC.2.3 Get row of the adjacency matrix: step (10)\nOnce the variable node is updated, the corresponding row of the adjacency matrix Amust be extracted. To\nthis end, we refer to the implementation described in Section 5.3, setting Pcurtonode .\nC.2.4 Mark non-neighbor nodes: step (11)\nIn this operation, all non-neighbor nodes must be indicated by 1 in their corresponding coordinates of the\niszero variable. For this implementation, we set all attention parameters to zero, and define fMLP as\n26follows:\n(W(1))i,j=\n\n1 if ( i, j) ={(Blocal, Szero),(iszero,iszero)}\n−Ω if i= candidates , j=Szero\n0 otherwise,(W(2,3))i,j=(\n1 if i={Szero,iszero}, j=i\n0 otherwise,\n(W(4))i,j=\n\n1 if i=Szero, j= iszero\n−1 if i, j= iszero\n0 otherwise.\nIn effect, we create a list of ones, which are then subtracted by all non-zero entries of candidates , multiplied\nby−Ω. This ensures that all non-neighbor nodes are set to one, while the remaining are zero. In addition,\nthe existing entries in the variable iszero are erased.\nC.2.5 Build distances: step (12)\nIn this step, the distance to the current node must be combined with the weighted edges. To explain\nthis process further, the operation copies a value from the top row across the bottom nrows of X. This\noperation can be formally expressed as repeat-n(X, C, D) = write( 1n+1X[1,C], X[:,D]) . Here, 1n+1is\nthe all-ones vector of size n+ 1, which is multiplied by the first entry of the column C. The result is then\nwritten in column D.\nThe first attention head is defined as:\n(W(1)\nK)i,j=(\n1 if i=Bglobal\n0 otherwise,(W(1)\nQ)i,j=(\n1 if i∈ {Bglobal, Blocal}\n0 otherwise,(W(1)\nV)i,j=(\n1 if i=C, j =E\n0 otherwise.\nThis configuration leads to the following attention matrix:\nσ\u0010\nXW(1)\nQW(1)⊤\nKX⊤\u0011\n=σ\n2\n10··· 0\n10··· 0\n............\n10··· 0\n\n=\n10··· 0\n10··· 0\n............\n10··· 0\n. (11)\nEquation (11)demonstrates that the operation on Xcaptures elements from the first row and distributes\nthem across its subsequent rows The second attention head is defined as follows:\n(W(2)\nK, W(2)\nQ)i,j=(\n1 if ( i, j)∈ {((P)1,1),((P)2,2),(Bglobal,2)}\n0 otherwise,(W(2)\nV)i,j=(\n−1 if i=C, E, j =E\n0 otherwise.\n(12)\nThe attention head in (11) also generates unwanted values in the top row, which are eliminated by retrieving\nthe column Cthat contains the top row value replicated earlier. As for the configuration of fMLP, all its\nvalues are set to 0, only utilizing the residual connection.\nFor the specific application to step (12) in Algorithm 3, this is applied to val curso that its value is distributed\nalong all candidates entries in X. However, since this field already contains the edges from the adjacent\nnodes, we set ( W(2)\nV)(valcur, val cur)= 0, effectively ignoring the deletion of the target field. Furthermore, notice\nthat this operation builds the distance for all nodes, including both neighbors and non-neighbors. The\nsubsequent step in Algorithm 3 addresses the non-neighbors by applying a masking technique. Moreover,\n27two additional variables are replicated: the minimum function’s termination flag, utilizing repeat-n(X,\nterm min, term minn), as well as the integer representation of the current node, through repeat-n(X, node int,\nnode intn).\nThe repeated integer representation of the node ( node intn) is employed in step (15) for updating the previously\nvisited node. On the other hand, the repeated termination flag ( term minn) is utilized in step (16) to mask\noperations if the minimum function is still in progress.\nC.2.6 Identify updates (13)\nIn this step, the distances computed in candidates are compared against the current distances. To this\nend, we utilize the implementation described in Section 5.2 and write: less-than(X, candidates, dists,\nchanges) .\nC.2.7 Mask updates: step (14)\nAfter performing the comparison, the variable changes indicates which nodes require updates. It is important\nto note that the comparison results should be set to false for non-neighbor nodes and when the termination\nflag of the minimum function is not active. We express this condition as follows: changes[i] = changes[i]\nand term minand (not is zero[i]) .\nIn terms of implementation, we begin by setting the parameters of fattnto zero and define the parameters of\nfMLP as follows:\n(W(1))i,j=\n\n1 if i, j∈n\n(changes, changes) ,(term minn, changes) ,\n(changes, Schanges )o\n−1 if i∈n\niszero,\nBglobal ,Blocalo\n, j= changes\n0 otherwise,\n(W(2,3))i,j=(\n1 if i∈ {changes , Schanges }, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i, j= changes\n−1 if i=Schanges , j= changes\n0 otherwise.\nIn this implementation, we write the necessary logical condition as ϕ(changes[i] +term min−iszero[i] −1).\nThis ensures the result equals 1 only when all conditions are satisfied. Additionally, in this implementation,\nthe variable term minnis determined in step (12) by repeating the status of the flag term minalong the last n\nrows of X.\nC.2.8 Update variables: step (15)\nOnce all the conditions have been imposed on the variable changes , the next step consists in updating the\nnecessary values based on this decision variable. The expression in line [38] of Algorithm 3 is implemented\nascond-select(X, [node intn, candidates], [prev, dists], changes, [prev, dists] , utilizing the\nconditional selection function described in Appendix C.1.1. Here, the variable node intnis the list containing\nthe repeated entries of the current node in its integer form, as executed in step (12).\n28C.2.9 Visit node: step (16)\nThe principle for marking a node as visited follows the same implementation outlined in Appendix C.1.5. We\nwrite this function as: write-row(X, node, term min, visit) . It is important to note that in the application\nof the write-row function, we assign the variable Dthe variable term min. This assignment is crucial because\nit ensures that the visit function is activated only when the minimum function’s termination flag is also\nactive.\nC.2.10 Terminate: step (17)\nThe termination function follows the implementation of Appendix C.1.6 by setting all-one(X, visit,\nterm) .\nC.3 Proof of Theorem 6.2\nIn this section, we introduce the constructions used to simulate the Breadth-First Search (BFS) algorithm\n[Moore, 1959]. The objective of this algorithm is to systematically explore nodes in a graph, starting from\na root node and moving outward to adjacent nodes and beyond. First, nodes directly adjacent to the root\nare visited, followed by their neighbors, and so on. We will begin with an overview of our implementation\nstrategy, including its limitations as detailed in Theorem 6.2 Next, we present the specific constructions of\neach step of the algorithm.\nImplementation overview: In BFS, node traversal is typically managed using a queue [Cormen et al., 2022].\nIn this way, nodes nearer to the source node are discovered and queued for visitation first. In this process, we\nalso record their immediate predecessor, i.e., the node through which it was discovered. This continues until\nall nodes connected to the source node have been visited. Our implementation of BFS simulates a queue\nusing a priority list combined with a minimum function, similar to the approach in Appendix C.2. Each time\na node is selected by the minimum function, the priority value increases, thereby assigning lower priority to\nsubsequent nodes discovered through it. This strategy effectively emulates the first-in-first-out nature of a\ntraditional queue.\nSpecifications of the architecture: We outline the algorithm’s execution steps in the Algorithm 4. Each\nstep correlates with a transformer layer, as represented in Equation (2), executing the respective routine. In\nthe case of the Breadth-First Search, there is a total of 17 steps, and therefore our implementation requires\n17 layers. Furthermore, throughout the implementation details of each step, no configuration uses parameters\nwhose count scales with the size of the graph, ensuring constant network width.\nLimitations: In addition to the enumeration limitation discussed in Section 6, simulation is also constrained\nby the largest priority value that can be utilized. A node is visited only once, and the priority value increments\nwith each selection. Hence, for a graph of size n, the priority can grow to a maximum of nas well As detailed\nin Section 6.2, the conditional selection functions are limited by Ω, the highest absolute value in a conditional\nclause. This limitation restricts the maximum usable value and consequently the largest graph size that can\nbe simulated. Specifically, based on conditional selection limits, the largest feasible graph is bounded by O(Ω)\nIn the remaining parts of the proof, we start by presenting the revised version of the algorithm, followed by\nthe construction of each step that has not been covered in previous algorithms.\n29Algorithm 4 Breadth-first search\nInput: integer start\nInput: matrix A, size n×n\n1: order = 0\n2: prev, visit, orders, orders masked , changes, disc = arrays of size n\n3:fori= 1tondo\n4: visit[i], disc[i], orders[i], prev[i] = false ,false ,ˆΩ, i\n5:end for\n6: visit[start] = order\n7: term = false\n8:··· Initialization of min-variables\n9:while term is false do\n10: fori= 1tondo\n11: ifvisit[i] is true then\n12: orders masked [i] = Ω (1) Mask visited nodes [C.2.1]\n13: else\n14: orders masked [i] = orders[i]\n15: end if\n16: end for\n17: get minimum(orders masked ) (2-8) Find minimum value [C.1]\n18: ifterm ministrue then\n19: node = idx best (9) Get minimum value [C.2.2]\n20: end if\n21: A row= A[node,:] (10) Get row of A [C.2.3]\n22: order = order + term min (11) Update priority factor [C.3.1]\n23: fori= 1tondo\n24: change = term ministrue and disc[i] is false\n25: changes[i] = change is true and Arow[i] is 1 (12) Identify updates [C.3.2]\n26: end for\n27: fori= 1tondo\n28: ifchanges[i] is true then\n29: prev[i] = node (13) Update variables [C.2.8]\n30: orders[i] = order\n31: end if\n32: end for\n33: visit[node] = visit[node] + term min\n34: disc[node] = disc[node] + term min\n35: fori= 1tondo\n36: disc[i] = disc[i] or changes[i] (14) Visit and discover nodes [C.3.3]\n37: end for\n38: interrupt = disc == visit (15) Compare visited and discovered nodes [C.3.4]\n39: term = not(false in visit) (16) Trigger termination/interruption [C.3.5]\n40: term = (term or interrupt) and term min(17) Trigger termination by interruption\n[C.3.6]\n41:end while\nreturn prev\nC.3.1 Update priority factor\nIn this stage, as outlined in Algorithm 4, the priority factor, denoted by order , needs to be increased. This\nadjustment ensures that updated nodes are assigned a lower priority in the list. It is important to note that\norder is only modified upon the activation of the minimum function’s termination flag, which guarantees its\nupdate only when necessary.\nFurthermore, the priority factor is utilized in the update step (13), so its value must be made available to\nall nodes. This means that not only this value is increased, but also distributed for the local variables, in a\n30variable we denote as order n.\nThis implementation can be seen as a direct application of the repeat-n operation, detailed in Appendix C.2.5,\nbeing expressed as repeat-n(X, [term min, order], [order n, order n]). Furthermore, to ensure that the\nupdated value is kept for the next iterations we set ( W(2)\nV)(term min, order) = 1 in the definition of (12)\nFurthermore, as in the application of Appendix C.2.5 to Dijkstra’s algorithm, the termination flag for the\nminimum function as well as the integer representation of the current node are also repeated along the last n\nrows.\nC.3.2 Identify updates: Step (12)\nIn this step, we want to prevent any changes for non-neighbor nodes (i.e. Arow[i]=0 ), discovered nodes (i.e.\ndisc[i]=1 ), or if the minimum function’s termination flag (term min) is deactivated. This step closely follows\nthe implementation of the masking function described in Appendix C.2.7.\nHowever, in this case, we implement the following logical expression: changes[i] = A row[i] and term min\nand (not disc[i]) . We define the parameters of fMLP as follows:\n(W(1))i,j=\n\n1 if i, j∈ {(Arow, changes) ,(term minn, changes) ,(changes, Schanges )}\n−1 if i∈ {Bglobal, Blocal,disc}, j= changes\n0 otherwise,\n(W(2,3))i,j=(\n1 if i∈ {changes , Schanges }, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i, j= changes\n−1 if i=Schanges , j= changes\n0 otherwise.\nIn this implementation, we write the logical condition as ϕ(Arow[i]+term minn[i]−disc[i] −1). This\nensures the result equals 1 only when all conditions are satisfied. Additionally, in this implementation, the\nvariable term minnis determined in step (11) by repeating the status of the flag term minalong the last nrows\nofX, as described in Appendix C.3.1.\nC.3.3 Visit and discover nodes: step (14)\nIn this step, both discovered and visited nodes are updated. This implementation closely follows Ap-\npendix C.1.5, where we write write-row(X, node, term min, visit) and write-row(X, node, term min,\ndisc) . The divergence from the implementation of Appendix C.1.5 is that, for the definition of fMLP we let:\n(W(1))i,j=(\n1 if i= changes , j= disc\n0 otherwise,(W(2,3,4))i,j=(\n1 if i, j= disc\n0 otherwise,\nThis extra modification ensures that all nodes that were changed in the previous step are also marked as\ndiscovered. In addition, since the condition for change requires nodes not to have been discovered, this\nformulation does not overcount discovered nodes.\nC.3.4 Compare visited and discovered nodes: step (16)\nThe principle of utilizing a priority list to determine which nodes to visit introduces a challenge in the\nsimulation of the algorithm: the scenario when the graph is disconnected. As it stands, without any interruption\nmechanisms the implementation described in Algorithm 4 would erroneously visit nodes disconnected from the\n31chosen root node, compromising the integrity of the simulation. To resolve this, we introduce an interruption\ncriterion to identify when all nodes connected to the root have been visited.\nThis is achieved by comparing the list of visited nodes with the list of discovered nodes. After the first node\nis selected by the minimum function, a connected sub-graph is entirely visited if its list of visited nodes is\nthe same as the discovered nodes. In this step, we examine whether each element in the visit variable\ncorresponds to its counterpart in the disc variable. To this end, we set the attention parameters to zero and\ndefine the parameters of fMLP as follows:\n(W(1))i,j=\n\n1 if ( i, j)∈n\n(visit ,S1),(Blocal,Blocal),\n(disc,S2),(equal ,equal)o\n−1 if ( i, j)∈ {(visit , S2),(disc, S1)}\n0 otherwise,(W(2))i,j=\n\n1 if ( i, j)∈ {(Blocal, S1),(equal ,equal)}\n−1 if i∈ {S1, S2}, j=S1\n0 otherwise,\n(W(3))i,j=(\n1 if i∈ {equal , S1}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i=S1, j= equal\n−1 if i, j= equal\n0 otherwise.\nIn this operation, we compute the subtraction of visit minus disc and also disc minus visit . Ideally, if\nthe two lists are identical, both subtractions should result in zero, leaving only the bias term introduced. The\noutcome of these computations is then stored in a variable named equal , which we utilize in the next step of\nthe operation.\nC.3.5 Compute termination and interruption criteria\nThe termination operation is described in the same form as Appendix C.1.6, and expressed as: all-one(X,\nvisit, term) . A key addition to this calculation is the inclusion of the condition for the equal variable,\ncomputed in the previous step and described in Appendix C.3.4. To streamline the process, we consolidate\nthe comparison done in the previous step into a single variable, denoted interrupt . Consequently, we write\nthis expression as all-one(X, equal, interrupt) .\nC.3.6 Combine interruption and termination flags\nAfter calculating both the interruption and termination flags, the final step is to combine them using the\nfollowing logic: term = (term or interrupt) and term min. The extra condition on the termination of the\nminimum function ensures that the simulation does not terminate in the first iteration, since in that stage,\nvisit anddisc list are both initialized with zeros. We implement this condition by setting the parameters\noffattnto zero, and we define the parameters of fMLP as follows:\n(W(1))i,j=\n\n1 if ( i, j)∈n\n(term ,term) ,(Bglobal ,Bglobal ),\n(interrupt , S1),(term min,term min)o\n−1 if i= term , j=S1\n0 otherwise,(W(2))i,j=\n\n1 if ( i, j)∈n\n(term ,term) ,(S1,S1),\n(term , S1),(term min, S1)o\n−1 if i=Bglobal, j=S1\n0 otherwise,\n(W(3))i,j=(\n1 if i∈ {term, S1}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if = S1, j= term\n−1 if i, j= term\n0 otherwise.\nThis above construction essentially realizes the intended logical condition through the following operation:\nterm =ϕ(ϕ(interrupt −term ) +term +term min−1). Here, the inner ReLU function, when added to\nterm, calculates the logical disjunction, and the sum of this result with the other terms forms the input for\nthe second ReLU function, thereby computing the logical conjunction.\n32C.4 Proof of Theorem 6.3\nThis section outlines the constructions used for simulating the Depth-First Search (DFS) algorithm [Moore,\n1959]. DFS traverses nodes from the most recently discovered node, aiming to visit all nodes by progressing\nfrom one branch to its end before moving to another branch. Initially, we provide an overview of our DFS\nimplementation approach, addressing its constraints as specified in Theorem 6.3. Next, we present the specific\nconstructions of each step of the algorithm.\nImplementation overview: One alternative to avoid the recursive formulation of DFS is to use a stack\nguided by node discovery order [Cormen et al., 2022]. In this way, nodes are explored in the order they\nare found, recording their immediate predecessor. This continues until all nodes connected to the source\nnode have been visited. Our implementation of DFS simulates a stack using a priority list combined with a\nminimum function, similar to the approach in Appendix C.3. The priority decreases with each node selection\nvia the minimum function, prioritizing newly discovered nodes, thus replicating the last-in-first-out property\nof a stack.\nSpecifications of the architecture: We outline the algorithm’s execution steps in the Algorithm 5. Each\nstep correlates with a transformer layer, as represented in Equation (2), executing the respective routine. In\nthe case of the Depth-First Search, there is a total of 15 steps, and therefore our implementation requires 15\nlayers. Furthermore, throughout the implementation details of each step, no configuration uses parameters\nwhose count scales with the size of the graph, ensuring constant network width.\nLimitations: In addition to the enumeration limitation discussed in Section 6, simulation is also constrained\nby the largest priority value that can be utilized. A node is visited only once, and the priority value decreases\nwith each selection. Hence, for a graph of size n, the absolute priority value can grow to a maximum of nas\nwell As detailed in Section 6.2, the conditional selection functions are limited by Ω, the highest absolute value\nin a conditional clause. This limitation restricts the maximum usable value and consequently the largest\ngraph size that can be simulated. Specifically, based on conditional selection limits, the largest feasible graph\nis bounded by O(Ω)\nIn the remaining parts of the proof, we start by presenting the revised version of the algorithm, followed by\nthe construction of each step, particularly those not covered in earlier algorithms.\n33Algorithm 5 Depth-first search\nInput: integer start\nInput: matrix A, size n×n\n1: order = 0\n2: prev, visit, orders, orders masked , changes = arrays of size n\n3:fori= 1tondo\n4: visit[i], orders[i], prev[i] = false ,ˆΩ, i\n5:end for\n6: visit[start] = order\n7: term = false\n8:··· Initialization of min-variables\n9:while term is false do\n10: fori= 1tondo\n11: ifvisit[i] is true then\n12: orders masked [i] = Ω (1) Mask visited nodes [C.2.1]\n13: else\n14: orders masked [i] = orders[i]\n15: end if\n16: end for\n17: get minimum(orders masked ) (2-8) Find minimum value [C.1]\n18: ifterm ministrue then\n19: node = idx best (9) Get minimum value [C.2.2]\n20: end if\n21: A row= A[node,:] (10) Get row of A [C.2.3]\n22: order = order - term min (11) Update priority factor [C.4.1]\n23: visit[node] = visit[node] + term min (12) Visit node [C.2.9]\n24: fori= 1tondo\n25: change = term ministrue and visit[i] is false\n26: changes[i] = change is true and Arow[i] is 1 (13) Identify updates [C.4.2]\n27: end for\n28: fori= 1tondo\n29: ifchanges[i] is true then\n30: prev[i] = node (14) Update variables [C.2.8]\n31: orders[i] = order\n32: end if\n33: end for\n34: term = not(false in visit) (15) Trigger termination [C.2.10]\n35:end while\nreturn prev\nC.4.1 Update priority factor: step (11)\nIn this step, the priority factor, indicated by order in Algorithm 5, is reduced. This adjustment ensures that\nupdated nodes receive a higher priority than the remaining in the priority list, effectively replicating the\nbehavior of a stack.\nThe implementation of this routine is largely based on the construction detailed in Appendix C.3.1 used in\nthe Breadth-First Search (BFS) algorithm, with a few key modifications. Specifically, we alter the values\nin the construction as follows: ( W(1)V)termmin, ordern =−1 and ( W(2)V)term min, order =−1. These\nmodifications are important as they enable the priority factor to decrease with each iteration of the main\nalgorithm.\n34C.4.2 Identify updates (13)\nThis step follows the same implementation of Appendix C.3.2, replacing the variable disc with visit .\nC.5 Proof of Theorem 6.4\nIn this section, we introduce the constructions utilized to simulate Kosaraju’s Strongly Connected Components\n(SCC) algorithm [Aho et al., 1974]. This algorithm aims to detect the strongly connected components of a\ngraph. A strongly connected component in a graph is a subgraph in which every vertex is reachable from\nevery other vertex through directed paths. To this end, we first present an overview of the implementation\nstrategy, which accompanies the limitations that are described in Theorem 6.4. These are then followed by\nthe specific construction of each step of the algorithm.\nImplementation overview: Executing Kosaraju’s SCC algorithm first consists of performing a depth-first\nsearch on the source node. To implement this, we create a priority list whose updated values get decreased\nat each iteration, effectively replicating the behavior of a stack. For the current node in the execution, we\ncheck whether its neighbors have been visited. If so, the corresponding value on the second priority list is\nupdated. This procedure simulates the insertion of nodes in a stack based on their finish times, as described\nin Kosaraju’s SCC algorithm.\nThis process continues until each node has its values updated in the second priority list, which can take as\nmuch as 2 nif the graph is a path. The end of this process triggers a partial termination criterion, which\nactivates the second phase of the algorithm. In the second part, we perform another depth-first search.\nHowever, this time, the search is determined by the priority set in the previous phase, and it is performed on\nthe transposed version of the graph, i.e. with the edges in the reverse direction. The index to indicate a\nstrongly connected component is the source node of the breadth-first search of each subgraph. This process\ncontinues until all nodes have been visited, for a total of 3 nnode visits.\nSpecifications of the architecture: In the following pseudocode description, we show the steps required\nto execute the algorithm. Each of the steps has an associated transformer layer in the form of (2)that\nimplements the corresponding routine. In the case of Kosaraju’s Strongly Connected Components algorithm,\nthere is a total of 22 steps, and therefore our implementation requires 22 layers. Furthermore, throughout the\nimplementation details of each step, no configuration uses parameters whose count scales with the size of the\ngraph, thus accounting for constant network width.\nLimitations: Beside the enumeration limitations explained in Section 6, the number of node updates also\ndetermines the size of the graph that can be simulated. This is because the update process is governed by a\nsingle variable (denoted order in Algorithm 6) that decreases every update. As discussed in Section 6, since\nthe constructions are limited to a certain size in the input, one of the restrictions of simulation is that the\ngraph size can have at most Ω /3 nodes.\nIn the remaining of the proof, we start by presenting the revised version of the algorithm, followed by the\nconstruction of each step that has not been covered in previous algorithms.\n35Algorithm 6 Strongly Connected Components (SCC) Algorithm\nInput: integer start\nInput: matrix A, size n×n\n1: term, term part, order, node ref=false ,false , 0, 0\n2: orders masked , changes 1, changes 2, changes 3, nvisit = arrays of size n\n3: sccs = array of size n, init. [1 , . . . , n ]\n4: visit 1, visit 2, visit 3= arrays of size n, init. false\n5: orders 1, orders 2= arrays of size n, init. ˆΩ\n6: orders 1[start], orders 2[start] = order, order\n7:··· Initialization of min-variables\n8:while term is false do\n9: orders = orders 2ifterm partistrue else orders 1\n10: visit = visit 3ifterm partistrue else visit 2 (1) Select list to find minimum [C.5.1]\n11: fori= 1tondo\n12: orders masked [i] = Ω ifvisit[i] is true else orders[i] (2) Mask visited nodes [C.2.1]\n13: end for\n14: get minimum(orders masked , sccs) (3-9) Find minimum value [C.1]\n15: write 1= term minand not term part\n16: write 2= term minand term part (10) Update write flags [C.5.2]\n17: ifterm ministrue then\n18: node, scc = idx best, scc best (11) Get minimum values [C.2.2]\n19: end if\n20: A row, Acol= A[node,:], A[:, node] (12) Get row/column of A [C.5.3]\n21: visit 1[node] = visit[node] + write 1\n22: visit 3[node] = visit[node] + write 2 (13) Visit nodes [C.5.4]\n23: fori= 1tondo\n24: nvisit = (visit 1[i]and Arow[i] is 1) orArow[i] is 0\n25: nvisit[i] = nvisit and write 1istrue (14) Check visited neighbors [C.5.5]\n26: end for\n27: nvisit all=not(false in nvisit) (15) Check if all neighbors are visited [C.5.6]\n28: visit scc= visit 3[scc] (16) Check if SCC is visited [C.5.7]\n29: node ref= scc ifvisit sccelse node ref (17) Update reference node [C.1.1]\n30: fori= 1tondo\n31: changes 1[i] = write 1and (Arow[i] is 1) and not visit 1[i]\n32: changes 3[i] = write 2and (Acol[i] is 1) and not visit 3[i]\n33: end for\n34: changes 2[node] = nvisit alland write 1\n35: order = order - term min (18) Identify updates and update priority [C.5.8]\n36: fori= 1tondo\n37: orders 1[i] = order ifchanges 1[i]else orders 1[i] (19) Update variables I [C.2.8]\n38: orders 2[i] = order ifchanges 2[i]else orders 2[i]\n39: visit 2[i] = visit 1[i]ifchanges 2[i]else visit 2[i] (20) Update variables II [C.2.8]\n40: orders 2[i] = order ifchanges 3[i]else orders 2[i]\n41: sccs[i] = node refifchanges 3[i]else sccs[i] (21) Update variables III [C.2.8]\n42: end for\n43: term = not(false in visit 3)\n44: term part=not(false in visit 2) (22) Trigger (partial) termination [C.5.9]\n45:end while\nreturn sccs\nC.5.1 Select list to find minimum: step (1)\nAs mentioned in Appendix C.5, the implementation of the Strongly Connected Components Algorithm\nutilizes a two-staged approach consisting of two breadth-first searches. As demonstrated in Appendix C.4,\nthe depth-first search is implemented using the minimum function and a priority list. To enable the change\nbetween stages, we utilize a conditional selection function, as formalized in Appendix C.1.1 to select between\n36the two priority lists and their corresponding lists of visited nodes for masking. We express this function as\ncond-select(X, [orders 2, visit 3], [orders, visit 2], term part, [orders, visit]) .\nC.5.2 Update write flags: step (10)\nIn this step, we utilize the partial termination criterion term partto indicate which part of the function should\nbe activated in the next steps of the execution. To this end, we implement the logical conditions expressed in\nlines [15-16] of Algorithm 6 as follows:\n(W(1))i,j=\n\n1 if i, j∈ {(write 1, write 1),(write 2, write 2),(term min, S1),(term min, S2),(term part, S2)\n−1 if i, j∈ {(term part, S1),(Bglobal, S2)\n0 otherwise,\n(W(2,3))i,j=(\n1 if i∈ {write 1, write 2, S1, S2}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i∈ {(S1,write 1),(S2,write 2)}\n−1 if i∈ {write 1, write 2}, j=i\n0 otherwise,\nWe erase the original content from the write 1andwrite 2columns and insert write 1=ϕ(term min−term part)\nand write 2=ϕ(term min+ term part−1), respectively, which simulate the corresponding logical conditions.\nC.5.3 Read row and column of A: step (12)\nOnce the node has been selected by the minimum function, we perform the extraction of both a row and\na column of A. The process of reading a row of Ais described in Section 5.3, and we can express it as\nread-A(X, ~A, node, A row)For the extraction of the column of A, we adopt the same procedure, utilizing an\nadditional attention head that interacts with A. The implementation details for this operation are similar to\nthe configurations described in Section 5.3, which enables us to express it as read-A(X, ~A.T, node, A col).\nC.5.4 Visit nodes: step (13)\nIn addition to updating the list of visited nodes in the graph, the following step also serves to update an\nauxiliary variable, denoted curi, which aims to store the position of the current node in a one-hot encoding.\nThis can be easily accomplished by applying the implementation presented in Appendix C.1.5 as follows:\nwrite-row(X, node, write 1, cur i).\nC.5.5 Check visited neighbors: step (14)\nThroughout the execution of the first depth-first search, we must check if the neighbors of the current node\nhave been visited. If this condition is met, the value of this node is updated in the second priority list,\nreplicating the intended behavior of building a stack. This is because nodes that meet this condition later\nin the algorithm receive higher priority in the second list. Additionally, this condition is also used to mask\nnodes from subsequent steps in the first DFS.\nTo this end, this process is implemented in two stages: steps (14) and (15). In this step, we verify the condition\nexpressed in lines [24-25] of Algorithm 6. The parameters of fattnare set according to the implementation\nof Appendix C.2.5. More specifically, we implement the routines for repeat-n(X, write 1, write 1n)and\n37repeat-n(X, write 2, write 2n). We then define the parameters of fMLP as follows:\n(W(1))i,j=\n\n1 if i, j∈ {(Bglobal, Bglobal ),(write 1n,write 1n),\n(nvisit ,nvisit) ,(Arow, S1)}\n−1 if i= visit 1, j=S1\n0 otherwise,(W(3))i,j=(\n1 if i∈ {nvisit , S1}, j=i\n0 otherwise,\n(W(2))i,j=\n\n1 if i, j∈ {(write 1n, S1),(nvisit, nvisit) }\n−1 if i, j=S1\n−Ω if i=Bglobal, j=S1\n0 otherwise,(W(4))i,j=\n\n1 if i=S1, j= nvisit\n−1 if i, j= nvisit\n0 otherwise.\nThis implementation not only clears the target field nvisit but also simulates the logical conditions of lines\n[24-25] with the following expression ϕ(write 1[i]−ϕ(Arow[i]−visit 1[i])). The extra negative Ω used in W(2)\nis to remove any unwanted entries in the top row.\nC.5.6 Check neighboring condition: step (15)\nIn this step, we check if all neighbors of the current node have been visited. This information is then used to\ndetermine if the current node should be updated in the second priority list. This implementation closely\nfollows the one utilized for the termination criterion, since it also collects all values along the last nrows of\nX. Therefore, we can express it as all-one(X, nvisit, nvisit all).\nC.5.7 Check if current SCC is visited: step (16)\nIn this stage, we must verify the condition of whether the Strongly Connected Component index that is\nassigned to the current node is visited. This condition becomes important in the subsequent step. Initially,\nthe SCC index of each node is itself. So, throughout the execution of the code, if we encounter an unvisited\nSCC index, it means that a new Strongly Connected Component has been found. In this case, the visit scc\nflag must be triggered so that in the next step, the reference node ( node ref) is updated. To this end, we\nutilize the same formulation as the read function in Appendix C.1.3 and write read-X(X, scc, visit 3,\nvisit scc).\nC.5.8 Identify updates and update priority: step (18)\nThis step of execution focuses on increasing the priority value and building the decision lists for the subsequent\nupdate steps. Despite being represented as a global variable, in our implementation of Algorithm 6, we set\norder to be a local variable, since it is solely used for updating the values of the priority lists. First, in the\ndefinition of fattn, we set all parameters to zero. In the implementation of fMLP, since we build three decision\nlists as well as the priority update, the parameters are presented separately for each of these purposes.\nFor the update of the priority values, we set:\n(W(1))i,j=(\n1 if i∈ {write 1n,write 2n}, j=Sorder\n0 otherwise,(W(2,3))i,j=(\n1 if i=Sorder, j=i\n0 otherwise,\n(W(4))i,j=(\n−1 if i=Sorder, j= order\n0 otherwise.\n38In this part of the implementation, we simply subtract the writing flags that are distributed along the last n\nrows from the order variable, thereby increasing the priority of newly updated nodes.\nFor building the first list of decision variables, we write:\n(W(1))i,j=\n\n1 if i∈ {(Arow, S1),(write 1n, S1),(changes 1,changes 1)}\n−1 if i∈ {visit 1, Blocal}, j=S1\n−Ω if i=Bglobal, j=S1\n0 otherwise,\n(W(2,3))i,j=(\n1 if i∈ {S1,changes 1}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i=S1, j= changes 1\n−1 if i, j= changes 1\n0 otherwise,\nwhich simulates the condition in line 31 of Algorithm 6 with the expression: ϕ(Arow[i]+write 1n[i]−visit 1[i]−1).\nThe additional entry of −Ω serves to clear any undesired values in the top row. For the second decision list\nwe write:\n(W(1))i,j=\n\n1 if i∈ {(cur i, S2),(write 1n, S2),(nvisit , S2),(changes 2,changes 2)}\n−2 if i∈ {Bglobal, Blocal,}, j=S1\n0 otherwise,\n(W(2,3))i,j=(\n1 if i∈ {S2,changes 2}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i=S2, j= changes 2\n−1 if i, j= changes 2\n0 otherwise,\nwhere curiis the variable obtained in Appendix C.5.4 which represents the one-hot encoding of the current\nnode. This operation simulates the condition in line 34 by using the expression: ϕ(write 1n[i]+Arow[i]+\ncuri[i]−2).\nFor the third decision list, we write:\n(W(1))i,j=\n\n1 if i∈ {(Acol, S3),(write 2n, S3),(changes 3,changes 3)}\n−1 if i∈ {visit 1, Blocal}, j=S3\n−Ω if i=Bglobal, j=S3\n0 otherwise,\n(W(2,3))i,j=(\n1 if i∈ {S3,changes 3}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i=S3, j= changes 3\n−1 if i, j= changes 3\n0 otherwise,\nThis implementation simulates the condition in line 32 using the expression: ϕ(Acol[i]+write 2n[i]−visit 3[i]−1).\nC.5.9 Trigger termination and partial termination: step (22)\nThe last step of the loop is to verify the termination criteria for the first and second phases of the algorithm.\nUsing the implementation described in Appendix C.1.6, we simply write these expressions as: all-one(X,\nvisit 3, term) andall-one(X, visit 2, term part). Since these implementations utilize the same attention\nheads, their implementations are compatible and can be simply merged.\n39C.6 Proof of Remark 6.5\nIn this section, we present the results on the Turing Completeness of the architecture described in (2).\nPrevious work by P´ erez et al. [2021] and Giannou et al. [2023] has established universality results for looped\ntransformers with standard attention. Nevertheless, it is important to assess how the expressiveness of the\narchitecture is affected when graph connectivity is stored as a separate object and graph convolution is\nintegrated into the attention mechanism.\nTo establish Turing Completeness, we adapt the method employed in Giannou et al. [2023]. In particular,\nwe illustrate how our proposed architecture can efficiently emulate SUBLEQ, a single-instruction language\nrecognized for its Turing Completeness [Mavaddat and Parhami, 1988].\nOur discussion begins with an overview of SUBLEQ, followed by the introduction of a modified version\ntermed SUBLEQ−(read as SUBLEQ minus ), which is also Turing Complete. Subsequently, we demonstrate\nthat SUBLEQ−can be simulated using a SUBLEQ-like instruction that uses a specialized memory object for\nthe adjacency matrix, which we name Graph-SUBLEQ. Lastly, we establish that the architecture depicted in\n(2) is capable of simulating Graph-SUBLEQ.\nSUBLEQ: Named for its operation “subtract and branch if less than or equal to zero”, SUBLEQ is a\none-instruction set computer. As detailed in Algorithm 7, it consists of subtracting the content at address a\nfrom that at address b, and storing the result back at b. All these values are stored in a one-dimensional\nmemory array. If the result is non-negative, the computer executes the next instruction; otherwise, it jumps\nto the instruction at address c. Despite this operational simplicity, SUBLEQ is Turing Complete [Mavaddat\nand Parhami, 1988].\nAlgorithm 7 SUBLEQ ( a, b, c )\nInput: memory object M, addresses a, b, c\n1:M[b] =M[b]−M[a]\n2:ifM[b]≤0then\n3: go to c\n4:else\n5: go to next instruction\n6:end if\nSUBLEQ−:In the remainder of the proof, we focus on a specialized variant of SUBLEQ, which we refer to\nas SUBLEQ−. This modified version operates similarly to the standard SUBLEQ, with a key distinction in\nhandling the adjacency matrix in memory. In SUBLEQ, a common approach to represent graph adjacency\ndata involves vectorizing the adjacency matrix and placing it at the beginning of the memory. In this case,\nspecifically in SUBLEQ−, the first n2memory entries, representing the row-major order vectorization of an\nadjacency matrix for a graph with nnodes, are set to be read-only. It is important to note that this read-only\nconstraint on the first n2entries does not diminish the expressive power of SUBLEQ−. The data in these\nentries can always be accessed and then copied into writable memory locations using standard SUBLEQ\ninstructions. Furthermore, these n2entries may not exist for routines that do not utilize the graph in the\nproposed way. Consequently, even with this restriction, SUBLEQ−remains Turing Complete.\nGraph-SUBLEQ: Building upon the concept of SUBLEQ−, we introduce Graph-SUBLEQ, a new formulation\nwith a distinct approach to memory management. Unlike SUBLEQ−, which uses a single memory object\nwith varying writing permissions, Graph-SUBLEQ separates its memory into two distinct entities: a writable\none-dimensional memory object, as in standard SUBLEQ, and a separate read-only memory dedicated to\nstoring graph connectivity data. Crucially, in this second memory, data is arranged in a matrix format, with\nthe same structure as an adjacency matrix.\n40Algorithm 8 Graph-SUBLEQ ( a, b, c, γ a)\nInput: memory objects M,MG, addresses a:= (a1, a2), bandc, flag γa\n1:ifγaistrue then\n2: ma=MG[a1, a2]\n3:else\n4: ma=M[a1]\n5:end if\n6:mb=M[b]\n7:M[b] =mb−ma\n8:ifM[b]≤0then\n9: go to c\n10:else\n11: go to next instruction\n12:end if\nThe implementation of Graph-SUBLEQ, as shown in Algorithm 8, closely resembles that of SUBLEQ in\nAlgorithm 7. The main difference lies in the structure of the instructions. While SUBLEQ operates with the\ntriplet a, b, c , a Graph-SUBLEQ instruction is defined by a, b, c, γ a. In this context, arepresents a pair of\naddresses, a1anda2. These addresses serve two functions: together, they can access an element in the graph\nmemory object MG, or, using only a1, access an element in the main memory M. The decision of which\nmemory is accessed by ais determined by the boolean flag γa: ifγais True, the instruction accesses the graph\nmemory; otherwise, it accesses the main memory. All other aspects of Graph-SUBLEQ’s implementation are\nakin to those of SUBLEQ.\nWith the frameworks of SUBLEQ−and Graph-SUBLEQ established, we now present a lemma important for\nour concluding remark.\nLemma C.1. Graph-SUBLEQ is Turing Complete.\nProof. To establish the Turing Completeness of Graph-SUBLEQ, we must demonstrate that every instruction\nin SUBLEQ−has an equivalent instruction in Graph-SUBLEQ. This equivalence is essential to ensure that\nGraph-SUBLEQ can perform all operations that SUBLEQ−can. The equivalence is examined in two scenarios\nbased on the type of instruction in SUBLEQ−: instructions that read from the read-only graph adjacency\nblock, and those that read from other memory areas. To maintain clarity in our notation, we use a, b, c to\nrepresent instructions in SUBLEQ−, and ˜ a1,˜a2,˜b,˜cfor instructions in Graph-SUBLEQ.\nNo access to graph data: To establish equivalence for instructions not accessing graph data, we define\nγ˜a= 0, and set ˜ a1=a−n2, ˜a2= 0, ˜b=b−n2, and ˜ c=c. Here, nrepresents the size of the graph.\nAccess to graph data: For instructions that access graph data, the equivalence is achieved by defining\nγ˜a= 1, and setting ˜a1=⌊a/n⌋and˜a2=a−n˜a1. We also set ˜b=b−n2, and ˜c=c, where nis the size of\nthe graph.\nWhen SUBLEQ−does not access graph data, the conversion to Graph-SUBLEQ requires a few adjustments\nto the memory addresses. In Graph-SUBLEQ, the address ˜a2is not utilized in this scenario. The addresses ˜a\nand˜bin Graph-SUBLEQ are simply the corresponding SUBLEQ−addresses offset by −n2, to accommodate\nthe different indexing scheme in Graph-SUBLEQ’s memory objects. For the addresses cand˜c, no modification\nis needed, as they are already aligned and refer to elements outside the memory reserved for the graph. In\nsituations where a graph is not used in the memory, nis equal to 0, making the addresses in SUBLEQ−and\nGraph-SUBLEQ naturally align.\nIn cases where SUBLEQ−accesses graph data through the address a, we derive the corresponding elements\n˜a1and˜a2in the matrix format of the graph memory. Given that ain SUBLEQ−represents an entry in the\nvectorized adjacency matrix arranged in row-major order, ˜ a1is set to the integer part of a/n, and ˜ a2is the\n41remainder of the division a/n. This approach ensures proper alignment of indexes between SUBLEQ−and\nGraph-SUBLEQ in both scenarios, thereby showing that Graph-SUBLEQ is also Turing Complete.\nHaving properly defined Graph-SUBLEQ and showing it that is Turing Complete, we now write the\nTheorem 6.5 in its more precise form:\nRemark C.2.There exists a looped-transformer hTin the form of (2), which utilizes the modified attention\nhead in (1), with 11 layers, 3 attention heads, and layer width O(1) that simulates Graph-SUBLEQ.\nProof. In this proof, we employ the architecture outlined in Equation (2)to simulate Algorithm 8. This\nconstructive approach is used to demonstrate that the architecture is Turing Complete. As with our other\nproofs, we start by presenting an adaptation of Algorithm 8 more aligned with our architecture. Within\nthis adapted algorithm, the commands executed inside the while loop correspond to the instructions of\nGraph-SUBLEQ. Following the adaptation, we provide a comprehensive description of the input structure,\ndetailing how the data and commands are organized and processed.\nFinally, we delve into the systematic construction of the algorithm. Each step has an associated transformer\nlayer in the form of (2)that implements the corresponding routine. In the simulation of Graph-SUBLEQ\nshown in Algorithm 9 there is a total of 11 steps, and therefore our implementation requires 11 layers, which\nuse a total of 3 distinct attention heads. Furthermore, throughout the implementation details of each step,\nno configuration uses parameters whose count scales with the size of the graph, thus resulting in constant\nnetwork width.\nAlgorithm 9 Graph-SUBLEQ ( a, b, c, γ a) with external memory block\nInput: memory objects M,MG, instruction list I, instruction index k\n1:while true do\n2: ((a1, a2), b, c, γ a) =I[k] (1) Read instructions [C.6.2]\n3: mG=MG[a1,:] (2) Read row a1from MG[C.6.3]\n4: ma,G=mG[a2] (3) Read address afrom MG[C.6.2]\n5: ma,X=M[a1] (4) Read address afrom M[C.6.2]\n6: ma=ma,Gifγaistrue else ma,X (5) Select the memory value of a\n[C.6.4]\n7: mb=M[b] (6) Read address bfrom M[C.6.2]\n8: diff = mb−ma (7) Compute difference between values [C.6.5]\n9: M[b] = diff (8) Write difference in memory [C.6.6]\n10: knext=k+ 1 (9) Compute next instruction [C.6.7]\n11: cond k= diff≤0 (10) Compute condition for k[C.6.8]\n12: k=cifcond kistrue else knext (11) Select next instruction [C.6.4]\n13:end while\nC.6.1 Input initialization\nWe utilize the same format consistently used in this work, having an input matrix Xand an external adjacency\nmatrix A. The structure of the input matrix follows the convention outlined in Section 5.1, incorporating\nelements such as global and local variables, positional encodings, and biases. We maintain the same naming\nconvention for consistency, with some adaptations specific to this context. For instance, we also refer to\nthe list of all positional encodings by P, while the positional encoding of the current instruction in this\nimplementation is represented by k, as detailed in Algorithm 9.\nGiven that in Graph-SUBLEQ, the size of the main memory and the graph might differ, directly translating\nthese to the formats of XandAcould result in a dimension mismatch, thereby hindering the multiplication\nprocess described in (1). To address this, we define K=max(|M|, n) + 1, where Kis the larger of the\n42two dimensions, either the size of the main memory or the number of nodes in the graph. This ensures\ncompatibility in dimensions for encoding in XandA. In this setup, both the instructions and memory are\npadded with zeros in XandAto align with K.\nIn terms of specific variable representation, the main memory block, denoted as M, is represented as a\nsingle-column local variable within this framework. The instruction list, labeled as I, comprises a set of\nlocal variables, each representing an address in the instruction set. These are individually identified by their\nspecific assignments. For example, Ia1refers to the column in the instruction list where addresses a1are\nstored.\nC.6.2 Read from input: steps (1), (3), (4), and (6)\nIn the implementation of the read function within our architecture, we follow the construction outlined in\nAppendix C.1.3. For step (1), the operation is expressed as read-X (X, k, I a1, Za1). Here, kspecifies the\ncolumns that store the positional encodings of the current instruction, and Za1is the variable holding the\ncurrent value for a1. The same formulation is then extended to all other values contained in the instruction,\nthat is, a2, b, candγa.\nMoving to step (3), after the row of the graph memory is written into the variable mG, this value is retrieved\nusing the operation read-X (X, Z a2, mG, ma,G). In this context, ma,Gis the target field where the retrieved\nvalue will be stored.\nIn step (4), the read operation is defined as read-X (X, Z a1, M, m a,X). The variable Mrepresents the\nmemory column, and ma,Xis designated to hold the entry of aretrieved from M.\nFinally, for step (6), the operation is set as read-X (X, Z b, M, m b). Here, Zbindicates the columns corre-\nsponding to the current address b, and mbis the variable designated to store the retrieved entry of bfrom\nM.\nC.6.3 Read from ˜A: Step (2)\nThe implementation of step (2), which involves reading from the adjacency matrix ˜A, is based on the\nprocedure detailed in Section 5.3. In this specific application, the read operation from ˜Ais described as\nread-A\u0010\nX,˜A, , Z a1, mG\u0011\n. Here, Za1represents the columns that contain the current value of a1, which\nis used to locate the specific row in the adjacency matrix ˜Athat needs to be read. The variable mGis\ndesignated to store the retrieved row from ˜A.\nC.6.4 Conditional selection: steps (5) and (11)\nThe conditional selection process in steps (5) and (11) of our Graph-SUBLEQ implementation follows the\napproach outlined in previous algorithms, as detailed in Appendix C.1.1.\nIn step (5), the conditional selection operation is expressed as cond-select (X, m a,G, ma,X, Zγa, ma). Here,\nmais the target field where the selected value will be stored. Zγais the variable holding the current value of\nγa, which determines whether to select ma,G(the value from the graph memory) or ma,X(the value from\nthe main memory).\nFor step (11), the operation is defined as cond-select (X, Z c, knext,cond k, k). In this context, Zcrefers to\nthe columns that hold the current value of c, and knext represents the columns storing the index of the next\ninstruction’s positional encoding. The conditional logic applied here determines which instruction’s positional\n43encoding, either the current or next, is to be used based on the condition cond k.\nC.6.5 Compute subtraction: step (7)\nIn step (7) of our implementation, we focus on calculating the difference between the values stored in mband\nma, and recording the result in a designated column, which we refer to as diff . To carry out this operation,\nwe set the parameters of fattnto zero and define the parameters of fMLP as follows:\n(W(1))i,j=\n\n1 if i∈ {(mb, S1),(diff,diff)}\n−1 if i∈ {(mb, S1),(diff, S2)}\n0 otherwise,(W(2,3))i,j=(\n1 if i∈ {S1, S2,diff}, j=i\n0 otherwise,\n(W(4))i,j=\n\n1 if i∈ {S1, S2}, j= diff\n−1 if i, j= diff\n0 otherwise.\nHere, the first layers obtain the subtraction of the desired values, while the last layer places the result in\nthe desired field. Additionally, the original value in diff is also replicated for both its positive and negative\nversions. These are used in the final layer to clear the previous entry.\nC.6.6 Write in memory: step (8)\nFor the implementation of step (8), where we write the result back into memory, we adopt the methodology\noutlined in Appendix C.1.5. We express this operation by write-row (X, Z b,diff, M). In this context, X\nrepresents our input matrix, and Zbindicates the columns corresponding to the address bin the memory,\nwhere the subtraction result ( diff ) needs to be written. The diff value holds the result of the subtraction\ncomputed in the previous step. The target for this write operation is the memory column M, which will be\nupdated with the new value.\nC.6.7 Increment positional encoding: step (9)\nThe implementation of step (9), which involves incrementing the positional encoding to move to the next instruc-\ntion, follows the procedure described in Appendix C.1.2. The operation is defined as increment (X, k, k next).\nHere, Xis the input matrix, krepresents the columns storing the current positional encoding, and knext\ndenotes the columns that will store the positional encoding of the next instruction.\nC.6.8 Compute condition for next instruction: step (10)\nIn step (10), we need to check if the result of the subtraction (stored in diff) is less than or equal to zero.\nThis is similar to the comparison function in Equation Section 5.2, but with a modification to include equality.\nEssentially, this function will return true if diff is zero or negative, allowing the algorithm to decide the\nnext step based on this condition. In this case, we approximate the less-or-equal function as follows:\nX[:, C]≤X[:, D]≈ε−1(ϕ(X[:, D]−X[:, C] +ε)−ϕ(X[:, D]−X[:, C])) (13)\nwhere CandDare the columns used for comparison. Notice that in this implementation, different from\n(5.2), the argument of the first ReLU includes an additional positive εvalue Furthermore, for the application\n44in the context of the simulation of Graph-SUBLEQ, the utilization of two columns CandDis not strictly\nnecessary, since the difference between the values is already stored in the variable diff and the comparison is\nessentially made against zero.\nTo implement this function, we begin by setting the parameters fattnto zero. Then, we define the parameters\noffMLP as follows:\n(W(1))i,j=\n\n1 if i, j= cond k\n−1 if i= diff , j∈ {S1, S2}\nε ifi=Bglobal, j=S2\n0 otherwise,(W(2))i,j=\n\n1 if i, j= cond k\nε−1ifi, j=S1\n−ε−1ifi=S2, j=S1\n0 otherwise,\n(W(3))i,j=(\n1 if i∈ {cond k, S1}, j=i\n0 otherwise,(W(4))i,j=\n\n1 if i=S1, j= cond k\n−1 if i, j= cond k\n0 otherwise.\nIn the first layer of the process, we construct the arguments for the ReLU functions as defined in (13).\nAdditionally, we maintain the value in cond kto clear the target column. Moving to the second layer, the\nReLU terms are then processed by subtracting one from the other, and the resulting value is divided by ε. In\nthe final layer, this computed value is recorded in the target column, simultaneously erasing its original value.\n45"    },    {        "title": "2402.01122v1.Generalized_Multi_Speed_Dubins_Motion_Model.pdf",        "content": "Submitted to IEEE Transactions on Robotics\nGeneralized Multi-Speed Dubins Motion Model\nJames P. Wilson†Shalabh Gupta†Thomas A. Wettergren‡\nAbstract —The paper develops a novel motion model, called\nGeneralized Multi-Speed Dubins Motion Model (GMDM), which\nextends the Dubins model by considering multiple speeds. While\nthe Dubins model produces time-optimal paths under a constant-\nspeed constraint, these paths could be suboptimal if this con-\nstraint is relaxed to include multiple speeds. This is because a\nconstant speed results in a large minimum turning radius, thus\nproducing paths with longer maneuvers and larger travel times.\nIn contrast, multi-speed relaxation allows for slower speed sharp\nturns, thus producing more direct paths with shorter maneuvers\nand smaller travel times. Furthermore, the inability of the Dubins\nmodel to reduce speed could result in fast maneuvers near\nobstacles, thus producing paths with high collision risks.\nIn this regard, GMDM provides the motion planners the ability\nto jointly optimize time and risk by allowing the change of\nspeed along the path. GMDM is built upon the six Dubins path\ntypes considering the change of speed on path segments. It is\ntheoretically established that GMDM provides full reachability\nof the configuration space for any speed selections. Furthermore,\nit is shown that the Dubins model is a specific case of GMDM\nfor constant speeds. The solutions of GMDM are analytical and\nsuitable for real-time applications. The performance of GMDM\nin terms of solution quality (i.e., time/time-risk cost) and compu-\ntation time is comparatively evaluated against the existing motion\nmodels in obstacle-free as well as obstacle-rich environments\nvia extensive Monte Carlo simulations. The results show that in\nobstacle-free environments, GMDM produces near time-optimal\npaths with significantly lower travel times than the Dubins\nmodel while having similar computation times. In obstacle-rich\nenvironments, GMDM produces time-risk optimized paths with\nsubstantially lower collision risks.\nIndex Terms —Motion planning, Kinodynamic constraints, Du-\nbins vehicles, Multi-speed vehicles, Time-risk cost\nI. I NTRODUCTION\nThe paper develops a motion model, called Generalized\nMulti-Speed Dubins Motion Model (GMDM), for multi-speed\nvehicles. GMDM provides a higher fidelity than the Dubins\nmodel by allowing the change of speed along the path. The\ncapability to reduce speed from its max value enables GMDM\nto 1) create sharp turns, thus resulting in overall shorter travel\ntimes than the Dubins paths, and 2) reduce collision risks\naround obstacles, thus resulting in safe maneuvering. It is\ntheoretically established that GMDM achieves full reachability\nof the configuration space between any start and goal poses\nfor any arbitrary set of speeds. It is also shown that GMDM\nreduces to the Dubins model for constant speeds. Further-\nmore, GMDM provides computationally efficient analytical\nsolutions, thus making it suitable for real-time applications.\nThis work was supported by US Office of Naval Research under Award\nNumber N000141613032. Any opinions or findings herein are those of the\nauthors and do not necessarily reflect the views of the sponsoring agencies.\n†Dept. of Electrical and Computer Engineering, University of Connecticut,\nStorrs, CT 06269, USA.\n‡Naval Undersea Warfare Center, Newport, RI 02841, USA\nMin. Speed Medium Speed Max. Speed\nStart\nGoalDubins\npath\nGMDM\npathTighter turns\nShorter time(a) Shorter-time path than Dubins in\nobstacle-free environments.\nStartGoalLong & risky\nDubins path\nAgile, flexible, and\nsafe GMDM path(b) Safer path than Dubins in\nobstacle-rich environments.\nFigure 1: Comparison of GMDM with the Dubins paths.\nA. Literature Review\nA fundamental problem in path planning is to find the\nminimum time path from a start pose to a goal pose while con-\nsidering kinodynamic constraints of the vehicle. Dubins [1][2]\nshowed that in absence of obstacles, the shortest path for a\ncurvature-constrained constant-speed vehicle between a pair\nof poses must be one of the following six canonical path\ntypes: LSL, RSR, LSR, RSL, LRL and RLR, where L(R)\nrefers to a left (right) turn with the maximum curvature,\nand S indicates a straight line segment. Reeds-Shepp curves\n[3] extended Dubins curves by considering backward velo-\ncity. These models have analytical solutions that are easy to\ncompute and implement. Further research considered field-of-\nview constraints [4], environmental disturbances [5], [6], [7],\n[8], multiple vehicles [9], [10], obstacle-avoidance [11], the\nmoving-target interception roblem [12][13][14], the traveling\nsalesman problem [15], [16], [17], [18], [19], the orienteering\nproblem [20], [21], and the coverage problem [22], [23]. These\nproblems, however, do not consider multiple speeds which are\nessential for time-optimal risk-aware motion planning [24].\nRecently, Wolek, et al. [25] developed a motion model for\ntime-optimal planning in obstacle-free environments. Their\nsolution is based on two speeds, i.e., the min and max\nspeeds, which are sufficient for time-optimality in obstacle-\nfree environments. However, the solutions of this model are not\nclosed-form and require nonlinear solvers. Kucerov, et al. [26],\n[27] proposed multiple speeds and turning radii to find time-\noptimal paths for aircraft. However, Kucerov’s model does not\nconsider the LRL and RLR path types and requires higher\ncomputation times than the Dubins model without providing\nperformance guarantees. Finally, there exist other kinodynamic\nmodels that consider acceleration [28] and curvature [29]\nconstraints, but the high-dimensional nature of these problems\nmake them infeasible for practical implementation [30].arXiv:2402.01122v1  [cs.RO]  2 Feb 20242\nOther motion models enforce curvature continuity (e.g.,\nsmooth transition from an L to an R segment) by considering\na constraint on its derivative. Fermat’s spiral has been used to\nensure that transitions between Dubins segments are curvature-\ncontinuous [31]. Fraichard and Scheuer [32] extended Reeds-\nShepp curves by using Euler’s spiral, where the curvature\nchanges linearly with respect to arc-length. Bruyninckx and\nReynaerts [33] enforced a continuously differentiable path\nby finding fifth-order Pythagorean hodographs that satisfy\nthe continuity constraints. Qu et al. [34] used piecewise-\nconstant polynomials to construct complex paths that are\ntwice-differentiable. Faigl and Vana [35] used Bézier curves\nto generate smooth paths to travel through a set of way points.\nB. Motivation\nThe constant speed constraint in Dubins model severely\nrestricts the vehicle’s maneuverability. In obstacle-free envir-\nonments, the Dubins model produces suboptimal paths that\nare longer with larger travel times due to its inability to create\nsharp turns by reducing the speed. Fig. 1a shows an example\nwhere the GMDM path turns at the minimum speed to rapidly\norient the vehicle towards the goal, thus producing a path\nwhich is both shorter and quicker than the Dubins path.\nThe Wolek’s motion model [25] is an improvement over the\nDubins model as it produces time-optimal paths in obstacle-\nfree environments. This is achieved by utilizing extremal\n(i.e., maximum and minimum) speeds, which are sufficient\nfor time-optimality in obstacle-free environments. However,\nunlike Dubins, Wolek’s solutions require numerical solvers for\nnonlinear equations, thus limiting their use in many real-time\napplications. On the other hand, GMDM provides analytical\nsolutions that can be computed in real-time while approaching\nthe time-optimal solutions of the Wolek’s model.\nIn obstacle-rich environments, risk evaluation becomes crit-\nical in motion planning [36], [37], [38], [39], [40], [41]. Both\ntime and risk costs are considered by the T⋆algorithm [24]\nfor time-optimal risk-aware planning. However, the solution\nquality of a high-level motion planner (e.g., RRT⋆and T⋆)\ndepends on the underlying motion model used to connect any\ntwo way points. Thus, a motion planner using the Dubins\nmodel lacks the capability to reduce risk and often produces\nlong and risky paths. Moreover, due to constant speed, the\nDubins paths lack flexible maneuvering around obstacles.\nFig. 1b shows an example where the Dubins path goes through\na long narrow corridor which is risky, while the GMDM path\nis shorter, quicker and safer by virtue of changing the speed.\nSimilarly, a motion planner using the Wolek’s model pro-\nduces sub-optimal results in obstacle-rich environments be-\ncause the two extremal speeds are not sufficient to minimize\nthe time-risk cost. For instance, the straight line (S) segments\nin the Wolek’s model always have the max speed for time-\noptimality; however, these segments should adopt lower speeds\nto reduce risk when approaching an obstacle. Similarly, the\nturn segments should adopt different slower speeds to tightly\nwrap around the obstacles of different geometries.\nAs such, the desired motion model should enable the\nselection of appropriate speeds for each path segment in order\nto produce time-risk-optimal paths that are both fast and safe.The speed selection should consider both travel time and risk\nbased on vehicle’s orientation and distance from the obstacle.\nThe model should preferably have analytical solutions for\nreal-time computation. Finally, the model should be easy to\nunderstand and implement. To the best of our knowledge, no\nexisting model in literature possesses all of these attributes.\nIn this regard, this paper develops a motion model, GMDM,\nwhich is a generalization of the Dubins model that incorporates\nmultiple speeds such that any path segment can select an\nappropriate speed. The model was first introduced in [42],\nwhere preliminary results showed that the model provides\nsmoother and shorter paths than those obtained with existing\nmotion models, thus providing reduced travel times and risks.\nC. Contributions\nThe main contributions of this paper are as follows:\n•Model development\n–A fundamental extension of the Dubins motion\nmodel to GMDM considering multiple speeds for\neach path segment (i.e., L, S, and R) that enables\nboth time and risk analysis in motion planning.\n•Model solution\n–Derivation of analytical solutions of the forward and\ninverse problems of the GMDM paths, which allow\nfor real-time implementation.\n–Theoretical guarantee of full reachability for any pair\nof start and goal poses for any selection of speeds.\n–Analytical result to show that the Dubins model is a\nspecific case of GMDM for constant speeds.\n•Model validation\n–Comparative evaluation of GMDM against Dubins\nand Wolek’s models in terms of solution quality (i.e.,\ntime/time-risk cost) and computation time in both\nobstacle-free and obstacle-rich environments.\n–Numerical results to show that in obstacle-free envir-\nonments, the solution quality of GMDM approaches\nthe time-optimal solutions of the Wolek’s model\nwhile enabling significantly faster computation.\nD. Organization\nThe rest of the paper is organized as follows. Section II\npresents the details, solution and properties of GMDM. Sec-\ntion III presents the reachability analysis of GMDM with\ntheoretical proofs. Section IV analyzes the performance of\nthis model in obstacle-free and obstacle-rich environments,\nrespectively. Finally, Section V concludes the paper with\nrecommendations for future work.\nII. GMDM\nThis section presents the analytical details of GMDM. The\nDubins model consists of a set of six constant speed path\ntypes: LSL, RSR, LSR, RSL, LRL and RLR. These can be\nsolved for times spent on each path segment to obtain the\nminimum-time path between any two given poses in obstacle\nfree environments. GMDM generalizes the Dubins model by\nrelaxing the constant speed constraint. Specifically, GMDM\nallows the motion planner to select the speeds for L, S,3\nand R segments of the Dubins path types. Note that the\ndifferent speeds for L and R segments lead to different turning\nradii, thus enhancing path maneuverability. On the other hand,\nreducing the speed on any segment near obtacles reduces the\ncollision risk, thus enhancing path safety. This allows the\nmotion planner to minimize the time and time-risk costs in\nobstacle-free and obstacle-rich environments, respectively.\nA. Vehicle Description\nConsider a vehicle whose motion is described as\n˙x(t) =v(t)cosθ(t) (1a)\n˙y(t) =v(t)sinθ(t) (1b)\n˙θ(t) =ω(t), (1c)\nwhere p(t)≜(x(t),y(t),θ(t))∈SE(2)is the vehicle pose\nat time t;v(t)∈V= [vmin,vmax]is its speed in m/s at\ntime t, where vmin,vmax∈R+; and ω(t)∈Ω=[−ωmax,ωmax]\nis its angular speed (i.e., turning rate) in rad/s at time t,\nwhere ωmax∈R+and+/−denote the left/right turn. The\ncurvature of the vehicle is defined as κ(t) =|ω(t)|/v(t), where\n0≤κ(t)≤ωmax/vmin. The turning radius is r(t) =1/κ(t),\nwhere κ(t) =0 means forward movement on a straight line.\nB. Motion Primitives\nLetu≜(v,ω)∈U, where U=V×Ω, be a constant input to\nthe vehicle with pose p(t)applied for a certain time duration\nτ∈ {R+∪0}. Let M : SE (2)×U×{R+∪0} → SE(2)be the\nmotion primitive that describes the evolution of the pose p(t)\nsubject to the input u∈U. Thus, p(t+τ) =Mu,τ(p(t)). The\nmotion primitive M is of two types as follows:\nMu,τ(p(t)) =(\nCu,τ(p(t))ω̸=0\nSv,τ(p(t))ω=0,(2)\nwhere C u,τ(·)and S v,τ(·)denote the turning ( ω̸=0) and\nstraight line ( ω=0) motions, respectively. The turning motion\nCu,τ(·)is again of two types: left turn L u,τ(·)(ω>0) and right\nturn R u,τ(·)(ω<0). Fig. 2 shows the motion primitives for\nstraight, left, and right turn maneuvers.\n•For turning motion: p(t+τ) =Cu,τ(p(t))s.t.\nx(t+τ) =x(t)−v\nω\u0010\nsinθ(t)−sin\u0000\nθ(t)+ωτ\u0001\u0011\n,(3a)\ny(t+τ) =y(t)+v\nω\u0010\ncosθ(t)−cos\u0000\nθ(t)+ωτ\u0001\u0011\n,(3b)\nθ(t+τ) =θ(t)+ωτ. (3c)\n•For straight line motion: p(t+τ) =Sv,τ(p(t))s.t.\nx(t+τ) =x(t)+vτcosθ(t), (4a)\ny(t+τ) =y(t)+vτsinθ(t), (4b)\nθ(t+τ) =θ(t). (4c)\nC. Motion Model\nBased on the motion primitives of (3) and (4), we now\ndescribe GMDM. Let p0= (x0,y0,θ0) and pf= (xf,yf,θf)\ndenote the start and goal poses, respectively. The objective\n𝐩(𝒕)𝜽𝒗𝝉𝐩𝒕+𝝉\n+𝒚\n+𝒙(a) Straight line motion.\n𝐩(𝒕)𝝎 𝝉𝐩𝒕+𝝉\n𝜽𝜽𝝅−𝜽+ |𝝎|𝝉 (b) Left turn motion.\n𝐩(𝒕)|𝝎|𝝉𝐩𝒕+𝝉\n𝝅−𝜽𝜽−|𝝎|𝝉\n𝝅−𝜽 (c) Right turn motion.\nFigure 2: Motion primitives of GMDM.\nis to find the GMDM path between these poses. Similar to\nDubins, a GMDM path connecting p0andpfconsists of three\npath segments labeled by i=1,2,3. Let the start and end poses\nof segment ibe denoted by pi−1= (xi−1,yi−1,θi−1)andpi=\n(xi,yi,θi), respectively. Note that pf=p3. Let ui= (vi,ωi)∈\nUbe the input to vehicle on segment i, applied for a time\nduration τi∈ {R+∪0}. Then, pi−1evolves on segment itopi\ns.t.pi=Mui,τi(pi−1). Thus, each segment ifollows the motion\nprimitive of either a turn C or a straight line S.\nThis leads to two fundamental classes of GMDM path types:\nCSC (i.e., LSL, LSR, RSL and RSR) and CCC (i.e., LRL\nand RLR). For notation simplicity, we avoid the subscripts\non the motion primitives and use them only when needed. A\nCSC path first turns (either left or right), then goes straight,\nand finally turns (either left or right) before reaching the final\npose. Similarly, a CCC path makes three turning maneuvers\nbefore reaching the final pose, where sign ω1̸=signω2and\nsignω2̸=signω3(i.e., consecutive turning motions must be\nin different directions). Note: the other complex path types\n(e.g., paths with more than three segments) are not included\nin GMDM and are beyond the scope of this work.\nFurthermore, while GMDM is built on the six Dubins\npath types, it allows a different speed on each path segment.\nThus, the total number of GMDM path type configurations\ndepend on the number speeds allowed for each path segment.\nWe show later that GMDM provides full reachability of the\nconfiguration space for any speed selections. Figure 3 shows\nthe GMDM path types with a different speed on each segment.\nD. Model Analysis\nFor GMDM analysis we discuss the forward and inverse\nproblems. First, we define the following parameters.\nri≜vi\nωi,i=1,2,3, (5a)\nri j≜ri−rj,i,j=1,2,3, (5b)\nδi≜viτi,i=1,2,3, (5c)\nφi≜ωiτi,i=1,2,3, (5d)\nθi j≜mod(θi−θj,2πsign(ωi)),i,j=0,1,2,3, (5e)\nwhere mod (a,m)≜a−m⌊a\nm⌋ ∀a,m∈R[43]. Note that |ri|,\n|φi|andδi,i=1,2,3, represent the turning radius, rotation and\nlength of a path segment i, respectively. The rotations satisfy\nthe constraint |φi|<2π.4\n|𝝓𝟏|𝜹𝟏\n𝐋𝐒𝐋StartGoal\n|𝝓𝟑||𝒓𝟑|𝐩𝒇= (𝒙𝒇,𝒚𝒇,𝜽𝒇)\n𝜹𝟐𝜹𝟑\n|𝒓𝟏|\n𝐩𝟎= (𝒙𝟎,𝒚𝟎,𝜽𝟎)𝐩𝟏𝐩𝟐\n𝐋𝐒𝐑StartGoal\n𝝓𝟏𝜹𝟐|𝒓𝟑|𝐩𝒇= (𝒙𝒇,𝒚𝒇,𝜽𝒇)\n𝜹𝟑\n|𝒓𝟏|\n𝐩𝟎= (𝒙𝟎,𝒚𝟎,𝜽𝟎)𝜹𝟏|𝝓𝟑|\n𝐩𝟏𝐩𝟐\n𝐑𝐒𝐋Start\n|𝝓𝟏|\n|𝝓𝟑|𝜹𝟐\n𝒓𝟑𝜹𝟑|𝒓𝟏|𝐩𝟎= (𝒙𝟎,𝒚𝟎,𝜽𝟎)\n𝐩𝒇= (𝒙𝒇,𝒚𝒇,𝜽𝒇)Goal𝜹𝟏\n𝐩𝟏\n𝐩𝟐\n𝐑𝐒𝐑Start\n𝐩𝒇= (𝒙𝒇,𝒚𝒇,𝜽𝒇)Goal|𝝓𝟏|\n|𝝓𝟑|\n|𝒓𝟑|𝜹𝟐\n𝜹𝟑|𝒓𝟏|𝐩𝟎= (𝒙𝟎,𝒚𝟎,𝜽𝟎)\n𝜹𝟏\n𝐩𝟏\n𝐩𝟐\n(a) CSC paths.\n𝐑𝐋𝐑𝜹𝟏𝜹𝟑Goal𝐩𝟎= (𝒙𝟎,𝒚𝟎,𝜽𝟎)\nStart\n|𝝓𝟏| |𝝓𝟑||𝒓𝟑|\n𝜹𝟐|𝒓𝟏|𝐩𝒇= (𝒙𝒇,𝒚𝒇,𝜽𝒇)\n|𝝓𝟐||𝒓𝟐|𝐩𝟏𝐩𝟐|𝒓𝟏|\n𝐋𝐑𝐋Goal\n𝐩𝟎= (𝒙𝟎,𝒚𝟎,𝜽𝟎)|𝝓𝟏|\n|𝒓𝟑|𝜹𝟐\nStart\n𝐩𝒇= (𝒙𝒇,𝒚𝒇,𝜽𝒇)|𝝓𝟐|\n|𝒓𝟐|\n𝜹𝟏𝜹𝟑 |𝝓𝟑|𝐩𝟏\n𝐩𝟐 (b) CCC paths.\nFigure 3: GMDM path types.\n1) The forward problem analysis:\nDefinition II.1 (Forward problem) .The forward problem aims\nto find the final pose pfof a GMDM path given the control\ninputs uiand time durations τiof its segments i =1,2,3.\nThe final pose of a GMDM path is obtained by applying the\nmotion primitive for each segment consecutively as follows\npf=Mu3,τ3(Mu2,τ2(Mu1,τ1(p0))). (6)\nProposition II.1 (CSC forward) .Given p0and(ui,τi), i=\n1,2,3, the final pose pffor a CSC path is given as\nxf=x0−r1sinθ0−r31sin(θ0+φ1)+\nδ2cos(θ0+φ1)+r3sin(θ0+φ1+φ3),(7a)\nyf=y0+r1cosθ0+r31cos(θ0+φ1)+\nδ2sin(θ0+φ1)−r3cos(θ0+φ1+φ3),(7b)\nθf=mod(θ0+φ1+φ3,2π). (7c)\nProof. See Appendix A1.\nProposition II.2 (CCC forward) .Given p0and(ui,τi), i=\n1,2,3, the final pose pffor a CCC path is given as\nxf=x0−r1sinθ0+r12sin(θ0+φ1)+\nr23sin(θ0+φ1+φ2)+r3sin(θ0+φ1+φ2+φ3),(8a)\nyf=y0+r1cosθ0−r12cos(θ0+φ1)−\nr23cos(θ0+φ1+φ2)−r3cos(θ0+φ1+φ2+φ3),(8b)\nθf=mod(θ0+φ1+φ2+φ3,2π). (8c)\nProof. See Appendix A2.\n2) The inverse problem analysis:\nDefinition II.2 (Inverse problem) .The inverse problem aims\nto find the time durations τiof GMDM path segments given\nthe start and goal poses ( p0,pf) and the control inputs ui,\ni=1,2,3. The total travel time of the path is given as T=∑iτi.Let us define the following parameters:\na≜xf−x0+r1sinθ0−r3sinθf, (9a)\nb≜yf−y0−r1cosθ0+r3cosθf, (9b)\nwhich are known for the inverse problem.\nProposition II.3 (CSC inverse) .Given p0,pfandui, the time\ndurations τi, i=1,2,3, of CSC path segments are given as\nτ1=θ10\nω1,τ2=δ2\nv2,andτ3=θ31\nω3, (10)\nwhere\nθ1=arcsin\u0010−r31√\na2+b2\u0011\n−atan2(−b,a), (11)\nand\nδ2=q\na2+b2−r2\n31. (12)\nProof. See Appendix B1.\nProposition II.4 (CCC inverse) .Given p0,pfandui, the time\ndurations τi, i=1,2,3, of CCC path segments are given as\nτ1=θ10\nω1,τ2=θ21\nω2,τ3=θ32\nω3, (13)\nwhere\nθ1=π−arcsin\u0010a2+b2+r2\n12−r2\n23\n2r12√\na2+b2\u0011\n−atan2(−b,a)(14)\nand\nθ2=π−arcsin\u0010a2+b2+r2\n23−r2\n12\n2r23√\na2+b2\u0011\n−atan2(−b,a).(15)\nProof. See Appendix B2.\nCorollary II.1. TheGMDM path types reduce to the Dubins\nset [2] when\n1.|r1|=|r3|for CSC.\n2.|r1|=|r2|=|r3|for CCC.\nProof. The Dubins set [2] follows by plugging conditions 1\nand 2 in (7) and (8), respectively.5\n(a) CSC paths.\n (b) CCC paths.\nFigure 4: Visualization of the reachable (white) and unreachable (colored) sets from p0= (0,0,0)for the (a) CSC and (b)\nCCC paths. The top row shows the reachability in the SE(2) space. The bottom three rows show the cross-sections of xf-yf\nplanes for different θf. The plots are drawn for the control inputs (v1,v2,v3) = ( 0.1,0.5,1)m/s and |ωi| ∈ { 0,1}rad/s, which\ncorrespond to (|r1|,|r3|) = ( 0.1,1.0)m for all paths and |r2|=0.5 m for the CCC paths.\nThis section introduced GMDM and provided the solutions\nfor its forward and inverse problems. The optimal controls u∗\ni\nfor a GMDM path are selected by a higher-level planner. For\npractical implementation, the optimal controls are determined\nas follows. For a given start and goal pose, first a discrete\nset of controls are used to generate different GMDM path\ntype configurations. For example, consider vi∈ {vmin,vmax}\nand|ωi| ∈ {ωmax,0}, then the six GMDM path types generate\na total of 6 ×23=48 different configurations. Then, the inverse\nsolution for each of these configurations is determined, and its\npath quality (i.e., time/time-risk [24] cost) is evaluated. Due to\nthe closed-form nature of the solutions, they can be computed\nquickly in real time. Finally, the control that provides the best\npath quality is selected to yield the GMDM path. Section IV\nshows several examples for different applications.\nIII. R EACHABILITY ANALYSIS\nThis section presents the reachability analysis of GMDM.\nDefinition III.1 (Reachability) .A pose pfis said to be\nreachable if there exists a GMDM path from p0topf.\nLetRj⊆SE(2) denote the reachable set of a GMDM path\ntype j,j= CSC or CCC, that includes all final poses pfthatare reachable from the start pose p0. Let\nc≜x0−r1sinθ0+r3sinθf, (16a)\nd≜y0+r1cosθ0−r3cosθf. (16b)\nTheorem 1 (CSC Reachability) .The reachable set RCSCof\nGMDM satisfies the following\n(xf−c)2+(yf−d)2≥r2\n13. (17)\nProof. See Appendix C1.\nFig. 4a visualizes the reachable set RCSCderived from (17).\nThe top row of Fig. 4a shows RCSCin SE(2) space with four\nreachability plots corresponding to LSL, RSR, LSR and RSL\npath types. As seen in each of these plots, the reachable region\nof a path type lies outside a tube including its boundary while\nthe region inside this tube is unreachable. The bottom three\nrows of Fig. 4a show the cross sections of xf-yfplanes for\ndifferent θf. As seen from each of these cross-sections, the\nreachable region of a path type lies outside an open circle\nwith center (c,d)and radius |r13|.\nRemark III.1. The control inputs ui, i=1,2,3, affect the radii\n|ri|, which in turn affect the center and radius of the circular6\nUnreachable by LSL Unreachable by RSR\n(a) Visualization in SE(2) space.\n- 2 0 2- 2023 f = 0 = 2 :\n- 2 0 2- 2023 f = : = 4\n- 2 0 2- 2023 f = : = 2\n- 2 0 2- 2023 f = 3 : = 4\n- 2 0 2- 2023 f = :\n- 2 0 2- 2023 f = 5 : = 4\n- 2 0 2- 2023 f = 3 : = 2\n- 2 0 2- 2023 f = 7 : = 4 (b) Visualization of the xf-yfplanes for different θf.\nFigure 5: Visualization of the unreachable regions of LSL and RSR path types. Since these regions are disjoint, full reachability\nis achieved by GMDM as per Theorem 3. The plots are drawn for p0= (0,0,0)and(|r1|,|r3|) = ( 0.1,1.0).\nregion in (17). Hence, the reachability of a CSC path type can\nbe improved by choosing from a variety of controls.\nCorollary III.1. The Dubins LSL and RSR path types each\nprovide full reachability of the SE(2) space.\nProof. For the Dubins LSL and RSR path types, r1=r3, thus\nr31=0. From (17) we get (xf−c)2+(yf−d)2≥0, which is\ntrue for all pf= (xf,yf,θf)∈SE(2).\nTheorem 2 (CCC Reachability) .The reachable set RCCC of\nGMDM satisfies the following:\nr2\n13≤\u0000\nxf−c\u00012+\u0000\nyf−d\u00012≤\u0000\nr12−r23\u00012(18)\nProof. See Appendix C2.\nFig. 4b visualizes the reachable set RCCCderived from (18).\nThe top row of Fig. 4b shows RCCCin SE(2) space with two\nreachability plots corresponding to LRL and RLR path types.\nAs seen in each of these plots, the reachable region for a\npath type lies inside an annular region within a tube with\ninner and outer boundaries included, while everything else is\nunreachable. The bottom three rows of Fig. 4b show the cross\nsections of xf-yfplanes for different θf. As seen from each of\nthese cross-sections, the reachable region of a path type lies\ninside an annulus of a closed circle with center (c,d), inner\nradius |r31|and outer radius |r12−r23|.\nRemark III.2. The control inputs ui, i=1,2,3, affect the radii\n|ri|, which in turn affect the center and radii of the annulus\nregion in (18). Hence, the reachability of a CCC path type\ncan be improved by choosing from a variety of controls.\nCorollary III.2. The Dubins LRL and RLR paths each provide\nreachability inside a circle with center (c,d)and radius 4|r1|.\nProof. For the Dubins LRL and RLR path types, r1=−r2=\nr3, thus r31=0 and r12−r23=r1−r2−r2+r3=4r1. From\n(18) we get\u0000\nxf−c\u00012+\u0000\nyf−d\u00012≤(4r1)2.The reachable sets for the GMDM CSC and CCC path\ntypes in Theorems 1 and 2 above were numerically verified\nby solving a large number of goal poses. From the results\nof Theorems 1 and 2, it is clear that any individual GMDM\npath type from either CSC or CCC class does not provide\nfull reachability. However, the next theorem guarantees full\nreachability of GMDM.\nTheorem 3 (Full Reachability) .GMDM achieves full reach-\nability of the SE(2) space using LSL and RSR path types.\nProof. See Appendix C3.\nFig. 5 visualizes and numerically validates Theorem 3. Fig.\n5a shows the unreachable regions of LSL and RSR path types\nin the SE(2) space, while Fig. 5b shows these regions in\nthe cross-sections of the xf−yfplane for different θf. The\nplots are drawn for p0= (0,0,0)and(r1,r3) = ( 0.1,1.0).\nClearly, the unreachable regions of LSL and RSR path types\nare disjoint, thus verifying that all poses are reachable when\nusing at least these two path types. Numerical evaluations for\nother sets of turning radii show similar results.\nRemark III.3. The pair of LSL and RSR path types provides\nfull reachability to GMDM. However, it can be easily shown\nthat for any other pair, the unreachable regions of path types\nare not always disjoint, thus not providing full reachability.\nIV. R ESULTS AND DISCUSSION\nThis section presents the comparative evaluation results of\nGMDM with the baseline models for time-optimal and time-\nrisk optimal planning in different scenarios. The performance\nis measured by the solution quality (i.e., time/time-risk cost),\ncomputation time and collision risk of the produced path.\nThe solutions of the Dubins model and GMDM are closed\nform and obtained analytically, whereas the solutions of the\nWolek model required the IPOPT solver [44]. All motion\nmodels are coded in C++. The computations were done on an7\nTable I: Sets of candidate path types.\nΓW(Wolek, et al. [25]) ΓG2′(GMDM)\nNo. Path Type No. Path Type\n1 L+S+L+1 L+S+L+\n2 L+S+R+2 L+S+R+\n3 R+S+L+3 R+S+L+\n4 R+S+R+4 R+S+R+\n5 L+S+L+L−5 L+S+L−\n6 L+S+R+R−6 L+S+R−\n7 R+S+L+L−7 R+S+L−\n8 R+S+R+R−8 R+S+R−\n9 L−L+S+L+9 L−S+L+\n10 L−L+S+R+10 L−S+R+\n11 R−R+S+L+11 R−S+L+\n12 R−R+S+R+12 R−S+R+\n13 L−L+S+L+L−13 L−S+L−\n14 L−L+S+R+R−14 L−S+R−\n15 R−R+S+L+L−15 R−S+L−\n16 R−R+S+R+R−16 R−S+R−\n17 L−R−L−17 L+R+L+\n18 R−L−R−18 R+L+R+\n19 L+L−L+L+19 L+R+L−\n20 L+L−L+R+20 R+L+R−\n21 R+R−R+L+21 L+R−L+\n22 R+R−R+R+22 R+L−R+\n23 L+L+L−L+23 L+R−L−\n24 L+R+R−R+24 R+L−R−\n25 R+L+L−L+25 L−R+L+\n26 R+R+R−R+26 R−L+R+\n27 L+L−L+L+L−27 L−R+L−\n28 L+L−L+R+R−28 R−L+R−\n29 R+R−R+L+L−29 L−R−L+\n30 R+R−R+R+R−30 R−L−R+\n31 L−L+L+L−L+31 L−R−L−\n32 L−L+R+R−R+32 R−L−R−\n33 R−R+L+L−L+\n34 R−R+R+R−R+\nIntel Core-i7 7700 processor with 32GB of RAM on Ubuntu\n16.04 LTS. A curvature-constrained vehicle with vmin=0.3\nm/s, vmax=1.0 m/s and ωmax=1.0 rad/s is considered, with\nthe associated turning radii of rmin=0.3 m and rmax=1.0 m.\nThe results are generated by extensive Monte Carlo simula-\ntions and a detailed discussion on the advantages of GMDM\nfor time/time-risk optimal planning is presented.\nA. Discussion of the Baseline Models and GMDM\nThe Dubins model is the first baseline model for com-\nparison. It produces the time-optimal paths for constant-\nspeed curvature-constrained vehicles. Let ΓDdenote the set\nof Dubins path types: LSL, LSR, RSL, RSR, LRL, RLR. As\ndiscussed earlier, the Dubins model might produce suboptimal\npaths for multi-speed vehicles due to its inability to create\nsharp turns; however, it is the simplest model with closed form\nsolution, thus suitable for onboard real-time implementation.\nThe Wolek model [25] expanded the Dubins model for time-\noptimal planning of multi-speed vehicles. However, this model\nuses only the extremal (i.e., min and max) speeds which are\nshown to be sufficient for time-optimal planning in obstacle-\nfree environments. As such, this model might produce sub-optimal results for time/time-risk optimal planning in obstacle-\nrich environments. Specifically, in this model, the L and R\nsegments are of two types: bang (+) and cornering (−), which\ncorrespond to the max and min speeds, respectively. The S\nsegments are always at the max speed. Clearly, the speeds\nother than min and max might be necessary on the turn\nsegments to wrap tightly around the obstacles and on the\nstraight line segments to reduce risk. Furthermore, there is\nno restriction on the number of segments for a path type. For\nexample, the path L−L+S+R+R−means that the vehicle first\nturns left at min speed, then turns left at max speed, then\ncontinues on a straight line at max speed, then turns right at\nmax speed, and finally turns right at min speed before reaching\nthe goal. This kind of complex maneuver might be difficult to\nfollow by the vehicle. Wolek et al. [25] provided a sufficient set\nof 84 candidate path types that is guaranteed to yield the time-\noptimal path to any goal pose in an obstacle-free environment.\nFor simplicity, Wolek et al. [25] further provided a smaller set\nof 34 candidate path types that is most likely to yield the time-\noptimal path. Let ΓWdenote this set of most-likely candidate\npaths [25], as shown in Table I. Unlike Dubins, the solution of\nthe Wolek model requires nonlinear optimization, thus making\nit impractical for onboard real-time implementation.\nFinally, GMDM is an extension of the Dubins model that\nallows multi-speed configurations of the Dubins path types.\nThe set of GMDM path types depends on the number of speed\nconfigurations that can be adapted as needed.\nDefinition IV .1. Let GMDM-k denote the GMDM where the\nspeed of each path segment belongs to a set Vk={vℓ∈\n[vmin,vmax]:ℓ=1,...k}of k∈N+different speeds and the\nturning rate |ωi| ∈ { 0,ωmax}.\nLetΓGkdenote the set of GMDM- kpath types. Then, the\ntotal number of configurations of the GMDM- kpath types is\n|ΓGk|=6k3.\nRemark IV .1. The set ΓDof Dubins path types is contained in\nthe set ΓGkof GMDM-k path types, ∀k∈N+, (i.e., ΓD⊆ΓGk).\nThe equality holds for k =1(i.e.,ΓD=ΓG1). Thus,\n•GMDM-1 is identical to the Dubins model.\n•The solution quality of GMDM paths in terms of\ntime/time-risk costs is guaranteed to be better than or\nthe same as the Dubins paths.\nGMDM provides a closed form solution just like the Dubins\nmodel; thus, its computation is straightforward. Furthermore,\nthe GMDM path types are simple with only three segments;\nthus, the time complexity of GMDM to obtain the solution\nof each individual path type is the same as that of Dubins.\nHowever, as compared to the Dubins model, GMDM has more\npath types depending on the number of possible speeds on\neach segment. Thus, the time complexity of GMDM typically\nfalls between that of the Dubins and Wolek’s models; however,\nGMDM computation is still real-time and significantly faster\nthan the Wolek’s model due to the closed-form nature of the\nsolutions. While GMDM does not guarantee to produce time-\noptimal solutions in obstacle-free environments, the results\nlater show that GMDM approaches the solution quality of\nthe Wolek’s model (i.e., time-optimal solutions), and they8\n-3-2-101234-3-2-101234Dubins\n-3-2-101234-3-2-101234GMDM\n-3-2-101234-3-2-101234Wolek et al.\nStart Pose End Pose Max. Speed Min. Speed Inflection Point\nFigure 6: Visualization of paths produced by the three motion models from p0= (0,0,0)to various other goal poses.\nare obtained in real-time with orders of magnitude faster\ncomputation. On the other hand, neither GMDM nor the\nbaseline models guarantee time/time-risk optimal planning in\nobstacle-rich environments; however, the results later show\nthat GMDM in fact produces significantly better solution\nquality (i.e., time-risk cost) than the baseline models with\nmuch faster computation time and simplicity of paths.\nThe Dubins model, the Wolek’s model and GMDM use the\npath types from ΓD,ΓW, andΓGk, respectively, to compute the\ndifferent possible paths between any pair of poses. Then, each\nmodel selects the path with the least cost as the final result.\nB. Time-Optimal Planning\nFirst, we present the comparative results for time-optimal\nplanning. According to the Wolek’s model [25], the time-\noptimal paths can be generated using the extremal speeds in\nobstacle-free environments. Thus, we consider a set ΓG2′⊂\nΓG2of GMDM path types that are built from bang (i.e., max\nspeed) and cornering (i.e., min speed) arcs and max speed\nstraight line segments. Table I shows the set ΓG2′that consists\nof 32 GMDM path types.\n1) Visualization of time-optimized paths: To compare the\npaths produced by the three motion models, we consider\ndifferent goal poses around the origin with the start pose\np0=(0,0,0). Fig. 6 shows the paths produced by the three\nmotion models to the goal poses and the corresponding travel\ntimes. As seen, the Dubins paths to the goals are indirect, i.e.,\nlong and curvy, due to the single-speed restriction. However,\nthe Dubins model takes only ∼2×10−5s to find the solution.\nThe Wolek’s paths, on the other hand, are time-optimal with\nsharp turns and shortest travel times. However, the Wolek’s\nmodel has a higher average computation time of ∼6.53×10−1\ns. Finally, the GMDM paths are much more direct than the\nDubins paths and approach the solution quality of the Wolek’s\npaths. GMDM takes only ∼4×10−5s to find the solution;\nthus, it is well-suited for real time applications. Taking an\nexample, say the goal pose to the right of the start pose,\nthe Dubins, Wolek’s and GMDM paths have the travel time\ncosts of ∼6.70 s,∼3.62 s and ∼4.09 s, respectively. Thus,\nconsidering the travel and computation time costs, the GMDMpaths provide much better solution quality as compared to\nthe Dubins paths while requiring similar computation times.\nOn the other hand, GMDM has a significant computational\nadvantage over the Wolek’s solutions while achieving similar\npath quality. Furthermore, GMDM is easier to implement due\nto the closed-form solutions. Also, its paths with only three\nsegments are simpler and more appealing as compared to the\nWolek’s paths with more than three segments; thus, they are\neasier to follow by onboard controllers.\n2) Traveling Salesman Problem: The above analysis\nprovided insights into the types of paths produced by the three\nmotion models with their travel and computation time costs.\nHowever, many real-time path planning problems often need to\nconsider the combinations of different intermediate waypoints\nbefore reaching the final destination. One such problem is the\nDubins traveling salesman problem (TSP) [15], [16], [17], [18]\nwhich is stated as follows: given a collection of npoints that\nmust be visited, find (1) the sequence of points and (2) the\nheading at each point in the sequence that yields the time-\noptimal path for a curvature-constrained vehicle. It is clear\nthat the overall path quality and computation time of this\ncombinatorial optimization problem depends on the quality\nand computation time of the underlying motion model.\nWe consider the scenario shown in Fig. 7a. The starting\npoint of the vehicle is at the origin with p0= (0,0,0), which is\nmarked with the number \"0\". There are three points of interest\nthat must be visited after which the vehicle must return to the\nstart pose. At each point, four headings {0, π/2,π, 3π/2}rad\nare considered. The objective is to find the minimum time path\nthat visits each of these points at a certain heading and finally\nreturns back to the origin. To solve TSP using any underlying\nmotion model, we have to first compute the paths and their\ntime costs for every pose pair between different points of\ninterest. A pose at a point of interest can connect to 8 other\nposes. Thus, for a total of 12 poses, there are 12 ×8=96 pose\npairs. Additionally, there are 12 +12=24 connections between\nthe start pose and the other poses and vice versa. Thus, there\nare a total of 96 +24=120 pose pairs. For each motion model,\nonce the paths and their time costs are computed for the above\npose pairs, the optimal solution of TSP is found by searching9\n-2-1012-2-1012Dubins\n0\n12\n3Travel Time: 18.299 s\nCPU Time: 0.007 s\n-2-1012-2-1012Wolek et al.\n0\n1\n23Travel Time: 11.237 s\nCPU Time: 60.831 s\n-2-1012-2-1012GMDM\n01\n2\n3Travel Time: 12.638 s\nCPU Time: 0.011 s\nPose Max. Speed Min. Speed\n(a) Visualization of the 3-point TSP solutions produced by the three underlying motion models: Dubins, Wolek and GMDM.\nDubins Wolek et al. GMDM\nMotion Model152025Travel Time (s)\nDubins Wolek et al. GMDM\nMotion Model0.480.510.54...205230255CPU Time (s)\n(b) Monte Carlo simulation results for 5-point TSP.\nFigure 7: TSP solutions produced by the three underlying motion models: Dubins, Wolek and GMDM.\nfor the sequence that has the least total travel time cost.\nFig. 7a shows the paths produced by each motion model,\nwhere the numbers 1, 2 and 3 indicate the order in which\nthe points of interest are visited. The Dubins path takes long\nand clumsy routes to connect each of the points of interest,\nresulting in a total travel time cost of ∼18.30 s. While the\nDubins path is the longest, it is produced in the fastest time of\n∼0.007 s. The Wolek’s solution, on the other hand, provides\na superior path quality, with direct and fast paths connecting\nthe points of interest with a total travel time cost of ∼11.24\ns. However, it takes ∼60.83 s to compute this path, which\nmight not be acceptable for on-demand dynamic path planning\nproblems. Finally, the quality of GMDM TSP solution is very\nclose to the Wolek’s solution with a total travel time cost of\n∼12.64 s, but it is obtained significantly faster in ∼0.011 s.\nFor further validation, we conducted a Monte Carlo study of\nthe TSP. For this study, thirty scenarios were considered, with\neach consisting of five points of interest. For each scenario,\nthese five points of interest were randomly distributed around\nthe start pose within a 5 mradial distance. Each point has\nfour possible headings as before. Again, the objective is to\nfind the minimum time path that travels through each of these\npoints and returns to the origin. Fig. 7b shows the statistical\ncomparison results by box plots of the travel and computation\ntime costs for the above Monte Carlo simulations. The dot\nmarks the median, the box shows the middle 50thpercentile\nand the horizontal lines show the min and max values. Fig. 7b\nshows that the GMDM’s solution quality (i.e., travel time cost)is superior to the Dubins solutions and approaches that of\nthe Wolek’s time-optimal solutions. At the same time, Fig. 7b\nshows that GMDM is significantly faster, similar to Dubins,\nand took only ∼0.5s to get the TSP solution as compared to\nthe Wolek’s model which required ∼230s to get the solution.\nOverall, the above results demonstrate that in obstacle-free\nenvironments: (1) GMDM provides superior path quality as\ncompared to the Dubins model while approaching the quality\nof the time-optimal Wolek’s paths, and (2) GMDM has very\nlow computational requirements similar to Dubins while being\nsignificantly faster than the Wolek’s model; thus it is suitable\nfor real-time path planning and replanning applications.\n3) Time-optimal planning in obstacle-rich environments:\nNext, we present the comparative results for time-optimal\nplanning in obstacle-rich environments with no consideration\nof risk. Fig. 8a shows an obstacle-rich scenario with the start\nand goal poses. For global planning, we use RRT∗[45] as the\nhigh level planner that uses the Dubins, Wolek, and GMDM as\nthe underlying motion models to connect any pair of sample\nposes. As seen in Fig. 8a, the Dubins model produces a long\npath around the bottom obstacle due to its inability to make\na sharp turn. The Wolek’s model produces a better path that\ngoes around the top obstacle to reach the goal. This path has\na shorter travel time since it turns rapidly at slow speed and\ngo between the two obstacles. Finally, GMDM produces the\nbest path that wraps around the top obstacle to quickly reach\nthe goal in the fastest travel time. Note: the Wolek’s paths are10\nDubins\nStart Goal\n012345012345Wolek et al.\nStart Goal\n012345012345GMDM\nStart Goal\n012345012345\nMax. Speed Min. Speed\n(a) Visualization of the time-optimal paths from each motion model.\n10-1100101102103\nSearch Time (s)911131517Travel Time (s)Dubins\nWolek et al.\nGMDM\n(b) Median performance vs. time.\nDubins Wolek et al. GMDM\nMotion Model910111213Travel Time (s) (c) Travel time cost.\nDubins Wolek et al. GMDM\nMotion Model103104105Num. Nodes (d) Number of nodes in the search tree.\nFigure 8: Monte Carlo simulation results for time-optimal planning in obstacle-rich environments using the RRT∗planner and\nthe three underlying motion models: Dubins, Wolek, and GMDM.\ntime-optimal in obstacle-free environments; however, there is\nno performance guarantee in obstacle-rich environments.\nFig. 8b shows the convergence plot for each motion model.\nAs seen, the Dubins plot converged the fastest, although the\nsolution quality is low. On the other hand, the Wolek’s plot did\nnot converge in the allocated time, thus yielding a mediocre\nsolution quality. Finally, the GMDM plot converged orders of\nmagnitude faster than the Wolek’s plot, while yielding the best\nsolution quality as compared to both the baseline models. Figs.\n8c and 8d show the results of Monte Carlo simulations on this\nscenario. Fig. 8c shows the travel time box plots of the three\nmodels. Clearly, GMDM produces the overall best travel times\nwhile the Dubins paths took the longest times. Fig. 8d shows\nthe number of nodes created using RRT∗. Clearly, GMDM\nallowed for the creation of a large number of nodes due to\nfaster computation and thus yielding better solution quality.\nOn the other hand, the Wolek’s model allowed the creation of\nmuch smaller number of nodes due to its high computation\ntime, thus yielding slow convergence.\nC. Time-risk optimal planning\nNow, we present the comparative results for time-risk op-\ntimal planning in obstacle-rich environments.\n1) Motivation for time-risk optimal planning: As discussed\nearlier, the joint time-risk optimal planning becomes critical\nin obstacle-rich environments to produce short but also safe\npaths. However, this requires flexibility in changing speed\nalong the path to reduce risks near obstacles or through narrow\npassages yet providing reasonable travel times.\nTime-Optimal Path\nStart\nGoal\n0123456012345Speed-Encoded Path\nStart\nGoal\n0123456\nTravel Time: 6.14 s\nAvg Risk: 2.01  Max Risk: 3.77012345Risk-Encoded PathTime-Risk Optimal Path\nStart\nGoal\n0123456012345\n0.31.0\nStart\nGoal\n0123456\nTravel Time: 10.51 s\nAvg Risk: 1.03  Max Risk: 1.35012345\n14Fast but risky Slow but safe\nSpeed (m/s) RiskFigure 9: Visualization of the GMDM paths encoded with the\nspeed and risk information. The T⋆[24] planner is used for\ntime-optimal and time-risk optimal planning.\nFig. 9 visualizes and compares the time-optimal and time-\nrisk optimal paths produced using the T⋆planner [24] with\nGMDM as the underlying motion model. While the top row\nof Fig. 9 shows the speed-encoded time-optimal and time-risk\noptimal paths, the bottom row shows the corresponding risk-\nencoded paths. As seen in the top row, the time-optimal path\nruns only at the max-speed and yields a shorter travel time.11\nSpeed-Encoded Path\nStartGoal\nSpeed (m/s)0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n0.31.0\nRisk-Encoded Path\nStartGoal\nRisk0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n14+\n(a) GMDM-1 (Dubins).\nSpeed-Encoded Path\nStartGoal\nSpeed (m/s)0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n0.31.0\nRisk-Encoded Path\nStartGoal\nRisk0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n14+ (b) Wolek et al.\nSpeed-Encoded Path\nStartGoal\nSpeed (m/s)0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n0.31.0\nRisk-Encoded Path\nStartGoal\nRisk0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n14+ (c) GMDM-2.\nSpeed-Encoded Path\nStartGoal\nSpeed (m/s)0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n0.31.0\nRisk-Encoded Path\nStartGoal\nRisk0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n14+\n(d) GMDM-3.\nSpeed-Encoded Path\nStartGoal\nSpeed (m/s)0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n0.31.0\nRisk-Encoded Path\nStartGoal\nRisk0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n150123456789101112131415\n14+ (e) GMDM-4.\nGMDM-1 WolekGMDM-2 GMDM-3 GMDM-4100150200300Time-Risk\nCost\nGMDM-1 WolekGMDM-2 GMDM-3 GMDM-41.01.52.02.5Avg.\nRisk\nGMDM-1 WolekGMDM-2 GMDM-3 GMDM-42.04.06.08.0Max.\nRisk\nGMDM-1 WolekGMDM-2 GMDM-3 GMDM-45075100Travel\nTime (s)\nGMDM-1 WolekGMDM-2 GMDM-3 GMDM-410-1100101102Search\nTime (s)\nGMDM-1 WolekGMDM-2 GMDM-3 GMDM-410-1100101102Table Gen.\nTime (s)\n (f) Results summary.\nFigure 10: Comparison of the time-risk optimal paths generated from T⋆[24] using different motion models: (1) GMDM\nconsidering up to four speeds (GMDM-1, GMDM-2, GMDM-3, and GMDM-4), and (2) Wolek et al. [25].\nOn the other hand, the time-risk optimal path runs at three\ndifferent speeds and yields a higher travel time. The time-\nrisk optimal path chooses the min speed to make sharp turns\nand to reduce the collision risks. Thus, it runs at max speed\nwhen it is safe (i.e., when the approach time to an obstacle\nis high), min speeds during turning and in high-risk regions,and moderate speeds in other regions. As seen in the bottom\nrow, the time-optimal path runs at high risk in most regions,\nwhile the time-risk optimal path runs at minimum risk and it\nis safe. The figure also shows that both the average and max\nrisk of the time-risk optimal path are significantly reduced.12\n2) Comparative evaluation results: As mentioned before,\nGMDM produces significantly better paths in terms of time-\nrisk cost due to its added flexibility on choosing multiple\nspeeds. While the Dubins paths have limited maneuverability\nbecause of the single speed constraint, the Wolek’s paths also\nhave limited capability in reducing the time-risk cost due to\nthe use of only two speeds (i.e., min and max) and because\nthe straight line segments are always at max speed, which in-\ncreases the collision risk. In contrast, the GMDM paths enable\nappropriate speed selection to keep a balance between time\nand risk costs. In addition, the turning segments in GMDM\ncan have multiple turning radii depending on the speeds, thus\nproviding better maneuverability around obstacles.\nFig. 10 compares the time-risk optimal paths produced by\nGMDM-1 (the Dubins model), the Wolek’s model, GMDM-\n2, GMDM-3 and GMDM-4. These models are implemented\nin the T⋆planner [24] for the construction of the time-\nrisk optimal paths in a complex maze scenario. For the T⋆\nframework, the map is divided into cells of size 1 m×1m\nwith 8-connectivity. Each cell has 8 possible orientations that\nare evenly spaced at intervals of π/4. The computations in\nT⋆are done using an a priori look-up table generation of all\npossible path types between any pair of neighboring poses on\nthe grid. This speeds up the search time for path generation\nlater. Details of time-risk cost calculation and path generation\nare found in [24] with a summary in Appendix D.\nThe Dubins model used vmax, the Wolek’s model used vmin\nand vmax, and GMDM-2, GMDM-3 and GMDM-4 used 2,\n3 and 4 speeds, respectively, which are uniformly-spaced in\n[vmin,vmax]. The results in Figs. 10a and 10f show that the\nGMDM-1 (the Dubins model) path travels fastest; however,\nit has the highest time-risk cost due to high average and\nmax risks on the path. Figs. 10b and 10f show that the\nWolek’s path slows down at corners and reduces risk only\nslightly but is still very high due to the fast straight line\nsegments, thus leading to an overall high time-risk cost. Figs.\n10c and 10f show that the GMDM-2 path adapts two speeds\nthroughout the entire path and hence able to reduce the risk\nsignificantly, thus leading to a much smaller time-risk cost.\nAs seen, GMDM-2 mildly compromises the travel time for\nbetter safety. The computation time of GMDM-2, including\nthe table generation and the search times, is smaller than the\nWolek’s model. Finally, Figs. 10d-10f show that GMDM-3 and\nGMDM-4 adapt multiple speeds throughout the path to further\nreduce the risk and overall time-risk cost at the expense of\nslightly higher computation time. Thus, its clear that GMDM\nprovides enhanced maneuverability by using multiple speeds\nwhile substantially reducing the risks, thereby providing a\nnotably superior time-risk optimal path. Furthermore, the use\nof multiple speeds and thus turning radii gives smoother paths\nthat can better adapt to the shape of the obstacles.\nV. C ONCLUSIONS AND FUTURE WORK\nThis paper developed a new motion model, called Gen-\neralized Multi-speed Dubins Motion Model (GMDM), that\nextends the Dubins model to incorporate multiple speeds for\nfast and safe maneuvering. GMDM allows each path segment\nto have any speed and turning rate. GMDM is mathematicallyproven to provide full reachability with closed form solutions.\nIt is shown that GMDM solutions are suitable for real-time\nimplementation. For constant speed, GMDM reduces to the\noriginal Dubins model. To the best of our knowledge, no\nexisting model offers these capabilities. The effectiveness of\nGMDM was demonstrated for both time-optimal and time-risk\noptimal motion planning problems in various scenarios. For\nthe time-optimal motion planning problem, GMDM generates\npaths with a solution quality approaching that of the optimal\npaths and gives a significant improvement over Dubins model.\nIn obstacle-rich environments, GMDM provides shorter paths\nwhile achieving a substantially lower risk.\nFuture work includes using GMDM in advanced motion\nplanners in real-world planning applications, e.g., dynamic\nenvironments [40], human-robot interaction [46] and dynamic\ntarget tracking [47]. GMDM could also be extended to higher-\norder motion models (e.g., considering acceleration).\nAPPENDIX\nA. Solution of the Forward Problem\n1) Proof of Proposition II.1:\nProof. For a CSC path type, the forward equation is given as\npf=Cu3,τ3(Sv2,τ2(Cu1,τ1(p0))), where\np1=Cu1,τ1(p0), (19a)\np2=Sv2,τ2(p1), (19b)\npf=Cu3,τ3(p2). (19c)\nFrom (19a), (3) and (5) we get\nx1=x0−r1\u0000\nsinθ0−sin(θ0+φ1)\u0001\n, (20a)\ny1=y0+r1\u0000\ncosθ0−cos(θ0+φ1)\u0001\n, (20b)\nθ1=mod(θ0+φ1,2π). (20c)\nFrom (19b), (20), (4) and (5) we get\nx2=x0−r1\u0000\nsinθ0−sin(θ0+φ1)\u0001\n+δ2cos(θ0+φ1),(21a)\ny2=y0+r1\u0000\ncosθ0−cos(θ0+φ1)\u0001\n+δ2sin(θ0+φ1),(21b)\nθ2=mod(θ0+φ1,2π). (21c)\nFrom (19c), (21), (3) and (5) we get\nxf=x0−r1sinθ0−r31sin(θ0+φ1)+\nδ2cos(θ0+φ1)+r3sin(θ0+φ1+φ3),(22a)\nyf=y0+r1cosθ0+r31cos(θ0+φ1)+\nδ2sin(θ0+φ1)−r3cos(θ0+φ1+φ3),(22b)\nθf=mod(θ0+φ1+φ3,2π). (22c)\n2) Proof of Proposition II.2:\nProof. For a CCC path type the forward equation is given as\npf=Cu3,τ3(Cu2,τ2(Cu1,τ1(p0)))where\np1=Cu1,τ1(p0), (23a)\np2=Cu2,τ2(p1), (23b)\npf=Cu3,τ3(p2). (23c)13\nFrom (23a), (3) and (5) we get\nx1=x0−r1\u0000\nsinθ0−sin(θ0+φ1)\u0001\n, (24a)\ny1=y0+r1\u0000\ncosθ0−cos(θ0+φ1)\u0001\n, (24b)\nθ1=mod(θ0+φ1,2π). (24c)\nFrom (23b), (24), (3) and (5) we get\nx2=x0−r1sinθ0+r12sin(θ0+φ1)+\nr2sin(θ0+φ1+φ2),(25a)\ny2=y0+r1cosθ0−r12cos(θ0+φ1)−\nr2cos(θ0+φ1+φ2),(25b)\nθ2=mod(θ0+φ1+φ2,2π). (25c)\nFrom (23c), (25), (3) and (5) we get\nxf=x0−r1sinθ0+r12sin(θ0+φ1)+\nr23sin(θ0+φ1+φ2)+r3sin(θ0+φ1+φ2+φ3),(26a)\nyf=y0+r1cosθ0−r12cos(θ0+φ1)−\nr23cos(θ0+φ1+φ2)−r3cos(θ0+φ1+φ2+φ3),\n(26b)\nθf=mod(θ0+φ1+φ2+φ3,2π). (26c)\nB. Solution of the Inverse Problem\n1) Proof of Proposition II.3:\nProof. Using (9) and (20c), we rewrite (7a) and (7b) as\n−r31sinθ1+δ2cosθ1=a, (27a)\nδ2sinθ1+r31cosθ1=b. (27b)\nTaking the squares of (27a) and (27b), adding them and\nrearranging, we get\nδ2=q\na2+b2−r2\n31. (28)\nNext, (27a) ·sinθ1- (27b) ·cosθ1gives\nasinθ1−bcosθ1=−r31. (29)\nThen applying the following trigonometric identity\nAsinα+Bcosα=p\nA2+B2sin(α+atan2(B,A)) (30)\nwe get\nθ1=arcsin\u0010−r31√\na2+b2\u0011\n−atan2(−b,a) (31)\nSince φ1=ω1τ1=θ10, we can solve for τ1using (31).\nSimilarly, since δ2=v2τ2, we can solve for τ2using (28).\nFinally, since φ3=ω3τ3=θ31, we can solve for τ3using (31).\nThe solutions for τ1,τ2, and τ3are given as\nτ1=θ10\nω1,τ2=δ2\nv2,andτ3=θ31\nω3. (32)2) Proof of Proposition II.4:\nProof. Using (9), (24c) and (25c), we rewrite (8a) and (8b) as\na−r12sinθ1=r23sinθ2, (33a)\nb+r12cosθ1=−r23cosθ2. (33b)\nTaking the squares of (33a) and (33b), adding them and\nrearranging we get\nasinθ1−bcosθ1= (a2+b2+r2\n12−r2\n23)/2r12. (34)\nUsing (30) and the fact that |φ2| ∈(π,2π)[48], we get\nθ1=π−arcsin\u0010a2+b2+r2\n12−r2\n23\n2r12√\na2+b2\u0011\n−atan2(−b,a)(35)\nNext, we rewrite (33a) and (33b) as\na−r23sinθ2=r12sinθ1 (36a)\nb+r23cosθ2=−r12cosθ1 (36b)\nTaking the squares of (36a) and (36b), adding them and\nrearranging we get\nasinθ2−bcosθ2= (a2+b2+r2\n23−r2\n12)/2r23. (37)\nAgain using (30) and the fact that |φ2| ∈(π,2π)[48], we get\nθ2=π−arcsin\u0010a2+b2+r2\n23−r2\n12\n2r23√\na2+b2\u0011\n−atan2(−b,a)(38)\nSince φ1=ω1τ1=θ10, we can solve for τ1using (35).\nSimilarly, since φ2=ω2τ2=θ21, we can solve for τ2using\n(35) and (38). Finally, since φ3=ω3τ3=θ32, we can solve\nforτ3using (38). The solutions for τ1,τ2, and τ3are given as\nτ1=θ10\nω1,τ2=θ21\nω2,τ3=θ32\nω3. (39)\nC. Reachability Proofs\n1) Proof of Theorem 1 :\nProof. From the CSC solution in (10), it is clear that the\nsolutions for τiexist as long as the solutions for θ1andδ2\nexist. Note that a,b, and r31are real by their definitions. The\nsolution for θ1exists if the domain of the arcsin term in (11)\nis between -1 and 1, i.e.,\n−1≤−r31√\na2+b2≤1 (40)\nThis implies that a2+b2≥r2\n31. This condition also makes sure\nthatδ2is real in (12). By substituting a=xf−candb=yf−d\nin the above condition we get (xf−c)2+(yf−d)2≥r2\n31.\n2) Proof of Theorem 2:\nProof. From the CCC solution in (13), it is clear that the\nsolutions for τiexist as long as the solutions for θ1andθ2\nexist. Note that a,b,r12, and r23are real by their definitions.14\nThe solutions for θ1andθ2exist if the domains of the arcsin\nterms in (14) and (15) are between -1 and 1, i.e.,\n−1≤a2+b2+r2\n12−r2\n23\n2r12√\na2+b2≤1 (41a)\n−1≤a2+b2−r2\n12+r2\n23\n2r23√\na2+b2≤1 (41b)\nThe above inequalities (41a) and (41b) imply that\n\u0012a2+b2+r2\n12−r2\n23\n2r12√\na2+b2\u00132\n≤1 (42a)\n\u0012a2+b2−r2\n12+r2\n23\n2r23√\na2+b2\u00132\n≤1 (42b)\n\u0000\na2+b2+r2\n12−r2\n23\u00012≤4r2\n12(a2+b2) (43a)\n\u0000\na2+b2−r2\n12+r2\n23\u00012≤4r2\n23(a2+b2) (43b)\nPutting q=a2+b2we get\n\u0000\nq+(r2\n12−r2\n23)\u00012≤4r2\n12q (44a)\n\u0000\nq−(r2\n12−r2\n23)\u00012≤4r2\n23q (44b)\nRearranging we get\nq2+(r2\n12−r2\n23)2+2q(r2\n12−r2\n23)−4r2\n12q≤0 (45a)\nq2+(r2\n12−r2\n23)2−2q(r2\n12−r2\n23)−4r2\n23q≤0 (45b)\nAdding we get\nq2−2(r2\n12+r2\n23)q+(r2\n12−r2\n23)2≤0 (46)\nFactorizing we get\n(q−(r12+r23)2)(q−(r12−r23)2)≤0 (47)\nFor LRL path type, r1>0,r2<0 and r3>0. Thus, r12≥0\nand r23≤0. For RLR path type r1<0,r2>0 and r3<0.\nThus, r12≤0 and r23≥0. For both these path types\n(r12+r23)2≤(r12−r23)2(48)\nTherefore, from (47) and (48) we get\n(r12+r23)2≤q≤(r12−r23)2(49)\nSimplifying and substituting for qwe get,\nr2\n13≤(xf−c)2+(yf−d)2≤(r12−r23)2(50)\n3) Proof of Theorem 3:\nProof. Consider the LSL and RSR path types. It is sufficient to\nshow that the unreachable regions of these two path types are\ndisjoint. This implies that the union of their reachable regions\ncover the entire SE(2) space. Based on the CSC reachability\ncondition in Theorem 1, the unreachable regions of LSL and\nRSR path types for any final heading θfare described by open\ncircles. Let qLSL≜(cLSL,dLSL)andqRSR≜(cRSR,dRSR)be the\ncentres, and |rLSL\n31|and|rRSR\n31|be the radii of the unreachable\ncircular regions of LSL and RSR path types, respectively. We\nneed to show that the distance between the centers of thesecircles is larger than or equal to the sum of their radii. Note\nthat for any given control, rR\n1=−rL\n1<0 and rR\n3=−rL\n3<0.\nThe sum of radii of these circles is\n|rLSL\n13|+|rRSR\n13|=|rL\n1−rL\n3|+|rR\n1−rR\n3|=2|rL\n1−rL\n3|=2|rLSL\n13|.\n(51)\nThe distance between the centres of these circles is given as\ndist(qLSL,qRSR) =q\n(cLSL−cRSR)2+(dLSL−dRSR)2(52)\nThen using (16) we get,\ndist(qLSL,qRSR) =2q\n(rL\n1)2+(rL\n3)2−2rL\n1rL\n3cos(θ0−θf)\n≥2q\n(rL\n1)2+(rL\n3)2−2(rL\n1)(rL\n3)\n=2|rL\n1−rL\n3|\n=2|rLSL\n13|\n(53)\nThus from (51) and (53) the unreachable open circular\nregions of LSL and RSR path types are non-overlapping.\nD. Review of T⋆for Time-Risk Path Planning\nT⋆[24] finds approximate time-risk optimal paths for\ncurvature constrained vehicles in the presence of obstacles.\nThe inputs to T⋆are the motion models which generate the\ncandidate paths between any two intermediate poses.\nThen the time-risk cost is computed for all the candidate\npaths. The risk cost is computed by estimating the collision\ntime of the vehicle with any obstacles tangent to its current\ndirection. Let γbe a feasible path parameterized by s. Let\ndc(s)be the distance to the nearest obstacle in the direction\nof the pose at γ(s). Let v(s)be the speed of the vehicle at s.\nThus, tc(s)≜dc(s)/v(s)is the expected collision time for the\nvehicle to hit the obstacle. The risk cost at sis defined as:\nR(s) =(\n1+t⋆\ntc(s)log\u0010\nt⋆\ntc(s)\u0011\niftc(s)≤t⋆\n1 otherwise(54)\nwhere t⋆=3 s is the risk-free collision threshold (i.e., the time\nfor the vehicle to stop, maneuver around, or regain its control).\nThe time cost is computed from the lengths and velocities of\npath segments.\nThen, a joint cost function is formulated considering both\nrisk and time, which is defined as:\nJ(γ) =Z\nγ\u0000\nR(s)\u0001λ·1\nv(s)ds (55)\nwhere λ=2 is a user-defined risk-weight. 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Cambridge University Press, 2006."    },    {        "title": "2402.01137v1.Long_time_dynamics_of_stochastic_wave_equation_with_dissipative_damping_and_its_full_discretization__exponential_ergodicity_and_strong_law_of_large_numbers.pdf",        "content": "arXiv:2402.01137v1  [math.PR]  2 Feb 2024LONG-TIME DYNAMICS OF STOCHASTIC WAVE EQUATION WITH\nDISSIPATIVE DAMPING AND ITS FULL DISCRETIZATION:\nEXPONENTIAL ERGODICITY AND STRONG LAW OF LARGE\nNUMBERS\nMENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nAbstract. For stochastic wave equation, when the dissipative damping is a non-globally\nLipschitz function of the velocity, there are few results on the long-time dynamics, in par-\nticular, the exponential ergodicity and strong law of large numbers, for the equation and its\nnumerical discretization to our knowledge. Focus on this is sue, the main contributions of\nthis paper are as follows. First, based on constructing nove l Lyapunov functionals, we show\nthe unique invariant measure and exponential ergodicity of the underlying equation and its\nfull discretization. Second, the error estimates of invari ant measures both in Wasserstein\ndistance and in the weak sense are obtained. Third, the stron g laws of large numbers of the\nequation and the full discretization are obtained, which st ates that the time averages of the\nexact and numerical solutions are shown to converge to the er godic limit almost surely.\n1.Introduction\nIn this paper, we consider the following stochastic wave equ ation with dissipative damping\n\ndu(t) =v(t)dt,\ndv(t) =−Λu(t)dt−ηv(t)dt−F(v(t))dt+ dW(t),\nu(t)|∂O= 0,u(0) =u0,v(0) =v0,(1)\nwhereO ⊂Rd(d≤3) is a bounded open domain with regular boundary ∂O, and−Λ := ∆ is\nthe Dirichlet Laplacian in H:=L2(O;R). Hereηvis a linear damping with ηbeing a positive\nconstant, and Fis the dissipative damping, which is a non-globally Lipschi tz function of the\nvelocity. The stochastic process W(t), t≥0 is anH-valuedQ-Wiener process on a filtered\nprobability space (Ω ,F,{Ft}t≥0,P) withQ∈ L(H) being symmetric, positive definite and of\ntrace class. The equation (1), which is first proposed in [23] , characterizes the displacement\nfieldof aparticlesuspendedinacontinuous mediawhilebein gforcedbyrandomperturbations\nvia an additive Gaussian noise, for instance, the motion of a vibrating string or the motion\nof a strand of DNA in a fluid.\nIn recent years, the study on the long-time dynamics, includ ing the invariant measure,\nthe ergodicity as well as the strong law of large numbers for s tochastic partial differential\nequations and its numerical discretization has drawn a lot o f attention (see e.g. [3, 4, 5, 6, 9,\n10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 26]). Especially, for th e stochastic damped Klein–Gordon\nequation, where the term Fis a function of the displacement u(it is called a reaction term),\n2020Mathematics Subject Classification. 37A25; 37M25; 60H15; 60H35.\nKey words and phrases. long-time dynamics, exponential ergodicity, invariant me asure, strong law of large\nnumbers, stochastic wave equation, full discretization .\nThis work is funded by the National key R&D Program of China un der Grant (No. 2020YFA0713701),\nNational Natural Science Foundation of China (No. 12031020 , No. 11971470, No. 12022118, No. 11871068),\nand by Youth Innovation Promotion Association CAS, China.\n12 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\n[10, 21] show that the equation possesses a unique invariant measure and is exponentially\nergodic; the authors of [20] propose a full discretization b y a spectral Galerkin method in\nspace and an exponential Euler integrator in time to inherit the ergodicity of the stochastic\ndamped Klein–Gordon equation, and obtain the convergence r ate of invariant measure via\nweak error estimates of the full discretization. To our know ledge, there are few results on the\nlong-time dynamics, in particular, the exponential ergodi city and strong law of large numbers,\nof the equation (1) and its numerical discretization. This p aper aims to take a step further\nand fill this gap. We consider the following questions:\n(I) Does the equation (1) (resp. its full discretization) ad mit a unique invariant measure\n(resp. a unique numerical invariant measure), and further e xponential ergodicity?\n(II) If so, does the numerical invariant measure converge to the original one? And in which\nsense?\n(III) Further, do the time averages of the exact and numerica l solutions converge to the\nergodic limit in the almost sure sense? Namely, do strong law s of large numbers hold?\nFor thequestion (I), theexistence of aninvariant measureo f theequation (1) isproved in[2,\n19] by employing the classical Krylov–Bogoliubov argument . The uniqueness of the invariant\nmeasure of the equation (1) is given in [2] by showing that reg ardless of initial conditions, the\nsolutions always converge to one another as time tends to infi nity. The polynomial ergodicity\nof the equation (1) is obtained in [22], relying on a combinat ion of Lyapunov conditions, the\ncontracting property of the Markov transition semigroup, a ndd-small sets. For the numerical\nstudy of (1), we are only aware of the paper [8], where the stro ng convergence of a full\ndiscretization on finite time is analyzed.\nIn this paper, we consider the full discretization of (1) by a pplying the spectral Galerkin\nmethod in space and the backward Euler method in time, i.e.,\n/braceleftBigg\nuN\nn+1=uN\nn+τvN\nn+1,\nvN\nn+1=vN\nn−τ(ΛNuN\nn+1+ηvN\nn+1+ΠNF(vN\nn+1))+ΠNδWn.(2)\nFirst, we construct novel Lyapunov functionals to obtain th e uniform moment boundedness\nand exponential contraction properties of the exact and num erical solutions, i.e., there exists\na constantCindependent of time such that\nsup\nt≥0E[/bardblX(t)/bardblp\nH2]+ sup\nm∈N1\nmm/summationdisplay\nk=0E[/bardblXN\nk/bardbl2\n˙H2]≤C, p≥1\n/bardblX(tn)−/tildewideX(tn)/bardblLp(Ω;H1)+/bardblXN\nn−/tildewideXN\nn/bardblLp(Ω;H1)≤1+2ǫ\n1−2ǫe−ǫtn/bardblX(0)−/tildewideX(0)/bardblLp(Ω;H1)\nwhereX(t) = (u(t),v(t))⊤,XN\nk= (uN\nk,vN\nk)⊤andǫis a sufficiently small positive constant.\nHereX(t) (resp.XN\nn) and/tildewideX(t) (resp./tildewideXN\nn) are the solutions of (1) (resp. (2)) with different\ninitial data X0and/tildewideX0, respectively. Based on these results, we finally show the ex ponential\nergodicity of the underlying equation and its full discreti zation, which gives a positive answer\nto the question (I). This means that the full discretization inherits the exponential ergodicity\nof the original equation.\nTo study the question (II), we first estimate the strong error of the full discretization (2)\nand obtain that\n/bardblX(tk)−XN\nk/bardblLp(Ω;H1)/lessorsimilarλ−1\n2\nN(1+t1\n2\nk)+τ1\n2(1+t3\nk),\nwhereλNis theNth eigenvalue of the operator Λ and τis the time step-size. We note that\nin the above result on the estimate of the error, the growth wi th respect to time is at mostSTOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 3\npolynomial. Making utilize of the exponential ergodicity o f the numerical solution, we then\nobtain the error estimates of invariant measures both in Was serstein distance and in the weak\nsense, which provides the answer to the question (II). More p recisely, for the error between\ninvariant measures in Wasserstein distance, we obtain the f ollowing convergence result\nWp(µ,µN\nτ)/lessorsimilarλ−1\n2\nN+τ1\n2.\nAnd for the weak error of invariant measures, we obtain the fo llowing estimate\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµ−/integraldisplay\nH1\nNϕdµN\nτ/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−ǫγtn+λ−γ\n2\nN(1+tγ\n2n)+τγ+p/2\n2(1+t3(γ+p\n2)\nn)\nfor anyn∈Nand for certain test functional ϕ∈Cp,γ, which yields the weak convergence\nof the numerical invariant measure, i.e., lim τ→0limN→∞/integraltext\nH1\nNϕdµN\nτ=/integraltext\nH1ϕdµ.This reveals\nthat the numerical invariant measure of the full discretiza tion can approximate the one of the\noriginal equation properly.\nTo solve the question (III), based on the Markov property of t he solution and the property\nof conditional expectation, we estimate the error between t he time averages of the exact and\nnumerical solutions and the ergodic limit µ(ϕ) :=/integraltext\nH1ϕdµ, and obtain the following results\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nt/integraldisplayt\n0ϕ(X(t))dt−µ(ϕ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilart−1\n2+εa.s.,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nk=1ϕ(XN\nk)−µ(ϕ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarτγ+p/2\n2−ε/parenleftbig\n1+t3(γ+p\n2)\nn/parenrightbig1+ε+λ−γ\n2+ε\nN/parenleftbig\n1+tγ\n2n/parenrightbig1+ε+t−1\n2+ε\nna.s.\nforε >0. These then lead to the a.s. convergence of the time average s of the exact and\nnumerical solution to the ergodic limit, namely, the strong laws of large numbers of the\nequation (1) and the full discretization (2) hold. This answ ers the question (III). The result\nof the strong law of large numbers for the full discretizatio n illustrates the effectiveness of\nconstructing the time-average of the numerical solution to approximate the ergodic limit\nusingasingle samplepath of thenumerical solution, whichc an greatly improve computational\nefficiency by avoiding simulating a large number of samples.\nTheoutline ofthis paperis organized as follows. Thenextse ction presentssomepreliminar-\niesforinvestigating (1)anditsfulldiscretization. Sect ion3isdevotedtothemomentestimates\nof the exact and numerical solutions. In Section 4, we obtain the exponential ergodicity of\nboth the underlying equation and full discretization. The e stimates on the approximation of\nthe invariant measure are presented in Section 5.\n2.Preliminaries\nIn this section, we present some preliminaries for investig ating the stochastic wave equation\n(1) and its full discretization.\n2.1.Notations. Throughout this paper, we will use the symbol Cto denote any unspecified\npositive constant independent of mesh size, whose value may be updated throughout the\nproofs. When we want to specify the dependence of Con some specific quantities, we will\nindicate them through a subscript or use C(·,·).\nDenote byLp(O;R),p≥1 (Lpfor short) the usual Lebesgue space consisting of pth inte-\ngrable functions. When p= 2, the space His then a Hilbert space with inner product /an}bracketle{t·,·/an}bracketri}ht\nand norm /bardbl·/bardbl.4 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nNote that there exists a family of eigenpair {λk,ek}k∈Nof Λ inHsuch that Λ ek=λkek\nfor an increasing sequence of positive numbers {λk}k∈Ntending to infinity. More precisely,\nthere exists Cd∈(0,∞) such that λn∼Cdn2/dwhenn→ ∞. Setting ˙Hr= Dom(Λr\n2),r∈R,\nendowed with the inner product\n/an}bracketle{tx,y/an}bracketri}ht˙Hr=/an}bracketle{tΛr\n2x,Λr\n2y/an}bracketri}ht=∞/summationdisplay\nk=1λr\nk/an}bracketle{tx,ek/an}bracketri}ht/an}bracketle{ty,ek/an}bracketri}ht\nand the induced norm\n/bardblx/bardbl˙Hr=/parenleftBigg∞/summationdisplay\nk=1λr\nk/vextendsingle/vextendsingle/an}bracketle{tx,ek/an}bracketri}ht/vextendsingle/vextendsingle2/parenrightBigg1\n2\n,\nit can be easily shown that the embedding ˙Hr1֒→˙Hr2is compact for r1>r2. We denote by\nHβ=˙Hβ×˙Hβ−1,β∈Rthe product space endowed with the inner product\n/an}bracketle{tX,Y/an}bracketri}htHβ=/an}bracketle{tx1,y1/an}bracketri}ht˙Hβ+/an}bracketle{tx2,y2/an}bracketri}ht˙Hβ−1, X= (x1,x2)⊤∈Hβ, Y= (y1,y2)⊤∈Hβ\nand the norm\n/bardblX/bardblHβ=/parenleftBig\n/bardblx1/bardbl2\n˙Hβ+/bardblx2/bardbl2\n˙Hβ−1/parenrightBig1\n2, X= (x1,x2)⊤∈Hβ.\nIn the sequel, concerning the well-posedness of (1), we shal l give some necessary assump-\ntions.\n2.2.Stochastic wave equation setting. SettingX(t) = (u(t),v(t))⊤, we may rewrite (1)\nin a compact form\ndX(t) =AX(t)dt−F(X(t))dt+BdW(t), X(0) =X0, (3)\nwhere\nA=/bracketleftbigg\n0I\n−Λ 0/bracketrightbigg\n,F(X(t)) =/bracketleftbigg\n0\nηv(t)+F(v(t))/bracketrightbigg\n, B=/bracketleftbigg\n0\nI/bracketrightbigg\n, X0=/bracketleftbigg\nu0\nv0/bracketrightbigg\n.\nThe operator Awith domain\nDom(A) ={X∈H:AX= (v,−Λu)⊤∈H}=H1\nis the generator of a unitary group E(t) onH1, given by\nE(t) =etA=/bracketleftBigg\ncos(tΛ1\n2) Λ−1\n2sin(tΛ1\n2)\n−Λ1\n2sin(tΛ1\n2) cos(tΛ1\n2)/bracketrightBigg\n, t∈R.\nNext, we state the main assumptions on the nonlinearity and t he noise.\nAssumption 1. LetF:H→Hbe the Nemytskii operator associated to a continuously\ndifferentiable function f:R→R, given by\nF(v)(x) :=f(v(x)),v∈H,x∈ O,\nsatisfying\n|f(ξ)| ≤a1(1+|ξ|), a1∈(0,√\n2\n2η), (4)\ninf\nξ∈Rf′(ξ) =a2>−η. (5)STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 5\nIt follows from (5) that\n/an}bracketle{tv1−v2,F(v1)−F(v2)/an}bracketri}ht ≥a2/bardblv1−v2/bardbl2. (6)\nBelow we provide an example of the non-globally Lipschitz fu nctionfwhich satisfies As-\nsumption 1.\nExample 1. Leth∈C(R)be defined by\nh(x) =\n\nαn−x\n2, x ∈[αn,αn+n),\nn(x−αn+1)\n2, x∈[αn+n,αn+1),\n−h(−x), x ≤0,\nwhereαn=n(n+1)\n2−1, n≥1.Then we have |h(x)| ≤ |x|for anyx∈Randh′(x)≥ −1\n2>−1\nfor anyx /∈ {−αn,−αn−n,αn,αn+n}n≥1.\nFor any point ybelongs to the set {−αn,−αn−n,αn,αn+n}n≥1of non-differentiable\npoints forh, by selecting appropriate small positive constants ε1(y)andε2(y), one can modify\nthe function hwithin that small interval using an arc segment which tangen t tohat points\n(y−ε1(y),h(y−ε1(y)))and(y+ε2(y),h(y+ε2(y))). Denoted by hmthe function obtained\nby modifying hin the above way, then f:=η\n2hmsatisfies conditions (4)and(5). However, f\nis not a Lipschitz function.\nRecall that the stochastic process W(t), t≥0 is anH-valuedQ-Wiener process on a\nfiltered probability space (Ω ,F,{Ft}t≥0,P) withQ∈ L(H) being symmetric, positive definite\nand of trace class. It has an expansion\nW(t) =∞/summationdisplay\nk=1Q1\n2ekβk(t), t≥0,\nwhere{ek}k≥1forms an orthonormal basis in Hand{βk(t)}k≥1is a sequence of independent\nand identically distributed real-valued Brownian motions .\nAssumption 2. There exists a sequence of positive numbers {qk}k∈Nsuch thatQ:H→H\nis diagonalizable with respect to the orthonormal basis {ek}k∈N, i.e., of the form\nQek=qkek, k∈N,\nand that/summationtext\nk∈Nqk/bardblek/bardbl2\n∞<∞and\n/vextenddouble/vextenddoubleΛ1\n2Q1\n2/vextenddouble/vextenddouble\nL2(H)=/parenleftBigg∞/summationdisplay\nk=1λkqk/parenrightBigg1\n2\n=/parenleftBig\nTr(ΛQ)/parenrightBig1\n2<∞. (7)\nHere and below, let L2(H) be the set of Hilbert–Schmidt operators with norm\n/bardblT/bardblL2(H):=/parenleftBig∞/summationdisplay\nk=1/bardblTek/bardbl2/parenrightBig1\n2.\nFurthermore, the set L0\n2denotes the space of Hilbert–Schmidt operators from Q1\n2(H) toH\nwith norm /bardbl·/bardblL0\n2:=/bardbl·Q1\n2/bardblL2(H).\nUnder the above assumptions, we can easily obtain the well-p osedness of the stochastic\nwave equation (3) by following the approach via Yosida appro ximations in the proof of [2,\nTheorem 2.3]. The details are thus omitted.6 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nLemma 2.1. Let Assumptions 1 and 2 hold and X0∈H1. Then there exists a unique mild\nsolution of (3)inH1, given by, for all t≥0,\nX(t) =E(t)X0−/integraldisplayt\n0E(t−s)F(X(s))ds+/integraldisplayt\n0E(t−s)BdW(s), a.s. (8)\n2.3.Full discretization. In this subsection, we propose a full discretization of (3) b y means\nof the spectral Galerkin method in space and the backward Eul er method in time.\n2.3.1.Spectral Galerkin method. Taking any positive integer N∈N, we define a finite-\ndimensional subspace HNofHas\nHN:= span{e1,e2,···,eN},\nwhere the bases {ej}N\nj=1are the first Neigenfunctions of the linear operator Λ. We next\nintroduce the projection operators Π N:˙Hα→HNandΠN:Hβ→HN×HN, which are\nrespectively given by\nΠNψ=N/summationdisplay\ni=1/an}bracketle{tψ,ei/an}bracketri}htei, ψ∈˙Hα, α≥0,\nΠNX= (ΠNx1,ΠNx2)⊤, X= (x1,x2)⊤∈Hβ, β≥1.\nOne can immediately verify that\n/bardbl(ΠN−I)ψ/bardbl ≤λ−γ\n2\nN/bardblψ/bardbl˙Hγ, ψ∈˙Hγ, γ≥0. (9)\nWe define the discrete Laplacian Λ N:HN→HNby\nΛNψ= ΛΠNψ= ΠNΛψ=N/summationdisplay\ni=1λi/an}bracketle{tψ,ei/an}bracketri}htei, ψ∈HN.\nThen the spectral Galerkin method of (1) can be expressed as\n\n\nduN(t) =vN(t)dt,\ndvN(t) =−ΛNuN(t)dt−ηvN(t)dt−FN(vN(t))dt+ΠNdW(t),\nuN(0) =uN\n0,vN(0) =vN\n0,(10)\nwhich can also be written in a compact form\ndXN(t) =ANXN(t)dt−FN(XN(t))dt+BNdW(t),\nprovidedXN(0) = (uN\n0,vN\n0)⊤,FN= ΠNFand\nXN=/bracketleftbigg\nuN\nvN/bracketrightbigg\n, AN=/bracketleftbigg\n0I\n−ΛN0/bracketrightbigg\n,FN(XN) =/bracketleftbigg0\nηvN+FN(vN)/bracketrightbigg\n, BN=/bracketleftbigg\n0\nΠN/bracketrightbigg\n.\nSimilarly, the operator ANgenerates a unitary group EN(t),t∈RonHN×HN, which is\ngiven by\nEN(t) =etAN=/bracketleftBigg\ncos(tΛ1\n2\nN) Λ−1\n2\nNsin(tΛ1\n2\nN)\n−Λ1\n2\nNsin(tΛ1\n2\nN) cos(tΛ1\n2\nN)/bracketrightBigg\n.\nThe mild solution of (10) reads\nXN(t) =EN(t)XN(0)−/integraldisplayt\n0EN(t−s)FN(XN(s))ds+/integraldisplayt\n0EN(t−s)BNdW(s).(11)STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 7\n2.3.2.Full discretization. The full discretization is obtained by applying the backwar d Euler\nmethod in the temporal direction of (10). Let τ∈(0,1) be the uniform time step-size. For\nany nonnegative integer n∈N,tn:=nτand the increment of the Wiener process is denoted\nbyδWn=W(tn+1)−W(tn). The full discretization of (1) reads as\n/braceleftBigg\nuN\nn+1=uN\nn+τvN\nn+1,\nvN\nn+1=vN\nn−τ(ΛNuN\nn+1+ηvN\nn+1+ΠNF(vN\nn+1))+ΠNδWn.(12)\nLetXN\nn= (uN\nn,vN\nn)⊤. The compact form of (12) is\nXN\nn+1=XN\nn+τANXN\nn+1−τFN(XN\nn+1)+BNδWn.\nThe full discretization (12) is well-defined and a.s. unique ly solvable thanks to the uniform\nmonotonicity theorem in [25, Theorem C.2]. The proof is triv ial and thus omitted.\nLemma 2.2. Let Assumptions 1 and 2 hold and X0∈H1. Then, there is a unique solution\nXN\nnof(12).\n3.Moment boundedness\nThis section is devoted to the moment estimates for the solut ions of (3) and (12).\nWe first derive the moment bound in H2provided the initial data X0= (u0,v0)⊤∈H2.\nFor any (u,v)⊤∈H2, denote\nHǫ\n2(u,v) :=/bardblu/bardbl2\n˙H2+/bardblv/bardbl2\n˙H1+ǫ/parenleftbig\n/an}bracketle{tu,v/an}bracketri}ht˙H1+η\n2/bardblu/bardbl2\n˙H1/parenrightbig\n.\nNote that\n2−ǫ\n2(/bardblu/bardbl2\n˙H2+/bardblv/bardbl2\n˙H1)≤ Hǫ\n2(u,v)≤ǫ+ǫη+2\n2(/bardblu/bardbl2\n˙H2+/bardblv/bardbl2\n˙H1). (13)\nProposition 3.1. Let Assumptions 1 and 2 hold and X0∈H2. Then for all p≥1and\nsufficiently small ǫ>0, it holds that\nE/bracketleftbig\nHǫ\n2(u(t),v(t))p/bracketrightbig\n≤C(p)Hǫ\n2(u0,v0)p+C′(p),∀t≥0 (14)\nfor some positive constants C(p)andC′(p). Moreover, we have\nsup\nt≥0E/bracketleftbig\n/bardblX(t)/bardblp\nH2/bracketrightbig\n≤C(p)/parenleftbig\n1+/bardblX0/bardblp\nH2/parenrightbig\n. (15)\nProof.We now proceed to prove (14) by induction on p. We start with the base case p= 1.\nA routine calculation gives\ndHǫ\n2(u(t),v(t))\n=/parenleftbig\n−2η/bardblv(t)/bardbl2\n˙H1−2/an}bracketle{tv(t),F(v(t))/an}bracketri}ht˙H1+/vextenddouble/vextenddoubleΛ1\n2Q1\n2/vextenddouble/vextenddouble2\nL2/parenrightbig\ndt+2/an}bracketle{tv(t),dW(t)/an}bracketri}ht˙H1\n+ǫ/parenleftbig\n/bardblv(t)/bardbl2\n˙H1−/bardblu(t)/bardbl2\n˙H2−/an}bracketle{tu(t),F(v(t))/an}bracketri}ht˙H1/parenrightbig\ndt+ǫ/an}bracketle{tu(t),dW(t)/an}bracketri}ht˙H1\n≤/bracketleftBig\n−2(η+a2)/bardblv(t)/bardbl2\n˙H1−ǫ\n2/bardblu(t)/bardbl2\n˙H2+ǫ/parenleftbig\n/bardblv(t)/bardbl2\n˙H1+/bardblF(v(t))/bardbl2/parenrightbig\n+/vextenddouble/vextenddoubleΛ1\n2/vextenddouble/vextenddouble2\nL0\n2/bracketrightBig\ndt\n+/an}bracketle{t2v(t)+ǫu(t),dW(t)/an}bracketri}ht˙H1\n≤/bracketleftBig\n−2(η+a2)/bardblv(t)/bardbl2\n˙H1−ǫ\n2/bardblu(t)/bardbl2\n˙H2+ǫ(1+2a2\n1)λ−2\n1/bardblv(t)/bardbl2\n˙H1+2a2\n1ǫ|O|+/vextenddouble/vextenddoubleΛ1\n2/vextenddouble/vextenddouble2\nL0\n2/bracketrightBig\ndt\n+/an}bracketle{t2v(t)+ǫu(t),dW(t)/an}bracketri}ht˙H1.\nBy choosing sufficiently small ǫ∈(0,min{η+a2\n1+2a2\n1λ2\n1,2}), we obtain\ndHǫ\n2(u(t),v(t))≤/bracketleftBig\n−(η+a2)/bardblv(t)/bardbl2\n˙H1−ǫ\n2/bardblu(t)/bardbl2\n˙H2+2a2\n1ǫ|O|+/vextenddouble/vextenddoubleΛ1\n2/vextenddouble/vextenddouble2\nL0\n2/bracketrightBig\ndt8 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\n+/an}bracketle{t2v(t)+ǫu(t),dW(t)/an}bracketri}ht˙H1.\nOwning to (13), there exist two positive constants C1:=ǫ\nǫ+ǫη+2min{2(1+2a2\n1)\nλ2\n1,1},C2:=\n2a2\n1ǫ|O|+/vextenddouble/vextenddoubleΛ1\n2/vextenddouble/vextenddouble2\nL0\n2such that\ndHǫ\n2(u(t),v(t))≤/parenleftbig\n−C1Hǫ\n2(u(t),v(t))+C2/parenrightbig\ndt+/an}bracketle{t2v(t)+ǫu(t),dW(t)/an}bracketri}ht˙H1.(16)\nCombining the Gronwall inequality yields\nE/bracketleftbig\nHǫ\n2(u(t),v(t))/bracketrightbig\n≤e−C1tHǫ\n2(u0,v0)+C2.\nThis implies (14) for the base case p= 1.\nForp≥2, we assume that (14) holds for the cases up to p−1. It suffices to show that the\ncasepholds. We have\ndHǫ\n2(u(t),v(t))p=pHǫ\n2(u(t),v(t))p−1dHǫ\n2(u(t),v(t))\n+1\n2p(p−1)Hǫ\n2(u(t),v(t))p−2/summationdisplay\nk≥1qk/vextendsingle/vextendsingle/an}bracketle{t2Λv(t)+ǫΛu(t),ek/an}bracketri}ht|2dt.(17)\nFor the first term on the above right hand side, by using (13), ( 16) and Young’s inequality,\nthere exist two positive constants C3andC4, depending on p, such that\npHǫ\n2(u(t),v(t))p−1dHǫ\n2(u(t),v(t))\n≤/bracketleftBig\n−C3Hǫ\n2(u(t),v(t))p−1(/bardblu(t)/bardbl2\n˙H2+/bardblv(t)/bardbl2\n˙H1)\n−C3Hǫ\n2(u(t),v(t))p+CHǫ\n2(u(t),v(t))p−1/bracketrightBig\ndt\n+pHǫ\n2(u(t),v(t))p−1/an}bracketle{t2v(t)+ǫu(t),dW(t)/an}bracketri}ht˙H1\n≤/bracketleftBig\n−C4Hǫ\n2(u(t),v(t))p−2(/bardblu(t)/bardbl2\n˙H2+/bardblv(t)/bardbl2\n˙H1)2−C4Hǫ\n2(u(t),v(t))p+C/bracketrightBig\ndt\n+pHǫ\n2(u(t),v(t))p−1/an}bracketle{t2v(t)+ǫu(t),dW(t)/an}bracketri}ht˙H1.\nFor the second term on the right hand side of (17), by using the Young inequality, we derive\n1\n2p(p−1)Hǫ\n2(u(t),v(t))p−2/summationdisplay\nk≥1qk/vextendsingle/vextendsingle/an}bracketle{t2Λv(t)+ǫΛu(t),ek/an}bracketri}ht|2\n≤p(p−1)Hǫ\n2(u(t),v(t))p−2Tr(ΛQ)(4/bardblv(t)/bardbl2\n˙H1+ǫ2/bardblu(t)/bardbl2\n˙H2)\n≤1\n2C4Hǫ\n2(u(t),v(t))p−2(/bardblv(t)/bardbl2\n˙H1+/bardblu(t)/bardbl2\n˙H2)2+CHǫ\n2(u(t),v(t))p−2\n≤1\n2C4Hǫ\n2(u(t),v(t))p−2(/bardblv(t)/bardbl2\n˙H1+/bardblu(t)/bardbl2\n˙H2)2+1\n2C4Hǫ\n2(u(t),v(t))p+C.\nCombining the above estimates leads to\ndHǫ\n2(u(t),v(t))p≤ −CHǫ\n2(u(t),v(t))pdt+C\n+pHǫ\n2(u(t),v(t))p−1/an}bracketle{t2v(t)+ǫu(t),dW(t)/an}bracketri}ht˙H1.\nTaking expectation and utilizing Gronwall’s inequality pr oduces (14) for the general case\np≥2. The proof of (15) is thus completed by (13). /square\nFor any (u,v)⊤∈H1, we denote\nHǫ\n1(u,v) :=/bardblu/bardbl2\n˙H1+/bardblv/bardbl2+ǫ/parenleftbig\n/an}bracketle{tu,v/an}bracketri}ht+η\n2/bardblu/bardbl2/parenrightbig\n.\nBy following a similar approach to that in Proposition 3.1, w e have the moment boundedness\ninH1provided the initial data X0= (u0,v0)⊤∈H1. The proof is omitted.STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 9\nProposition 3.2. Let Assumptions 1 and 2 hold and X0∈H1. Then for all p≥1and\nsufficiently small ǫ>0, it holds that\nE/bracketleftbig\nHǫ\n1(u(t),v(t))p/bracketrightbig\n≤C(p)Hǫ\n1(u0,v0)p+C′(p),∀t≥0 (18)\nfor some positive constants C(p)andC′(p). Moreover, we have\nsup\nt≥0E/bracketleftbig\n/bardblX(t)/bardblp\nH1/bracketrightbig\n≤C(p)/parenleftbig\n1+/bardblX0/bardblp\nH1/parenrightbig\n. (19)\nForthespatialsemi-discretization, wealsohavemomentbo undednessin H2ofthenumerical\nsolution, whose proof is similar to that of Proposition 3.1 a nd thus is omitted.\nLemma 3.3. Let Assumptions 1 and 2 hold and X0∈H2. Then, for any t>0, there exists\na positive constant C(p)such that\nE/bracketleftbig\n/bardblXN(t)/bardblp\nH2/bracketrightbig\nds≤C(p)(1+/bardblX0/bardblp\nH2), t≥0.\nLemma 3.4. Let Assumptions 1 and 2 hold and X0∈H2. For anyτ∈(0,1), there exists a\npositive constant Csuch that\nsup\nm∈N1\nmm/summationdisplay\nk=0E/bracketleftbig\n/bardblXN\nk/bardbl2\nH2/bracketrightbig\n≤C/parenleftbig\n1+/bardblX0/bardbl2\nH2/parenrightbig\n. (20)\nProof.We first give the estimate of the time average of the second mom ent for the component\nvN\nk. Note that\n/an}bracketle{tuN\nn+1−uN\nn,uN\nn+1/an}bracketri}ht˙H2+/an}bracketle{tvN\nn+1−vN\nn,vN\nn+1/an}bracketri}ht˙H1\n=1\n2/parenleftbig\n/bardbluN\nn+1/bardbl2\n˙H2−/bardbluN\nn/bardbl2\n˙H2+/bardbluN\nn+1−uN\nn/bardbl2\n˙H2/parenrightbig\n+1\n2/parenleftbig\n/bardblvN\nn+1/bardbl2\n˙H1−/bardblvN\nn/bardbl2\n˙H1+/bardblvN\nn+1−vN\nn/bardbl2\n˙H1/parenrightbig\n≥1\n2/parenleftbig\n/bardbluN\nn+1/bardbl2\n˙H2−/bardbluN\nn/bardbl2\n˙H2+/bardblvN\nn+1/bardbl2\n˙H1−/bardblvN\nn/bardbl2\n˙H1/parenrightbig\n+1\n2/bardblvN\nn+1−vN\nn/bardbl2\n˙H1.(21)\nIt follows from (12) that\n/an}bracketle{tuN\nn+1−uN\nn,uN\nn+1/an}bracketri}ht˙H2+/an}bracketle{tvN\nn+1−vN\nn,vN\nn+1/an}bracketri}ht˙H1\n=τ/an}bracketle{tvN\nn+1,uN\nn+1/an}bracketri}ht˙H2−τ/an}bracketle{tΛNuN\nn+1+ηvN\nn+1+ΠNF(vN\nn+1),vN\nn+1/an}bracketri}ht˙H1+/an}bracketle{tδWn,vN\nn+1/an}bracketri}ht˙H1\n=−τη/bardblvN\nn+1/bardbl2\n˙H1−τ/an}bracketle{tF(vN\nn+1),vN\nn+1/an}bracketri}ht˙H1+/an}bracketle{tδWn,vN\nn+1−vN\nn/an}bracketri}ht˙H1+/an}bracketle{tδWn,vN\nn/an}bracketri}ht˙H1\n≤ −(η+a2)τ/bardblvN\nn+1/bardbl2\n˙H1+1\n2/bardblvN\nn+1−vN\nn/bardbl2\n˙H1+1\n2/bardblδWn/bardbl2\n˙H1+/an}bracketle{tδWn,vN\nn/an}bracketri}ht˙H1,(22)\nwhere we used\n/an}bracketle{tF(vN\nn+1),vN\nn+1/an}bracketri}ht˙H1≥a2/bardblvN\nn+1/bardbl2\n˙H1.\nCombining (21) and (22) and taking expectation on both sides then yield\nE/bracketleftbig\n/bardbluN\nn+1/bardbl2\n˙H2−/bardbluN\nn/bardbl2\n˙H2+/bardblvN\nn+1/bardbl2\n˙H1−/bardblvN\nn/bardbl2\n˙H1/bracketrightbig\n≤ −2(η+a2)τE[/bardblvN\nn+1/bardbl2\n˙H1]+τ/bardblΛ1\n2/bardbl2\nL0\n2,\nwhich leads to\nE/bracketleftbig\n/bardbluN\nn+1/bardbl2\n˙H2+/bardblvN\nn+1/bardbl2\n˙H1/bracketrightbig\n+2(η+a2)τn+1/summationdisplay\nk=1E[/bardblvN\nk/bardbl2\n˙H1]\n≤ /bardblu0/bardbl2\n˙H2+/bardblv0/bardbl2\n˙H1+(n+1)τ/bardblΛ1\n2/bardbl2\nL0\n2.(23)10 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nTo estimate the time average of the second moment for the comp onentuN\nk, we note that\n/an}bracketle{tηuN\nn+1+vN\nn+1,uN\nn+1−uN\nn/an}bracketri}ht˙H1+/an}bracketle{tΛuN\nn+1,vN\nn+1−vN\nn/an}bracketri}ht\n=/an}bracketle{tηuN\nn+1,uN\nn+1−uN\nn/an}bracketri}ht˙H1+/an}bracketle{tuN\nn+1,vN\nn+1/an}bracketri}ht˙H1−/an}bracketle{tuN\nn,vN\nn/an}bracketri}ht˙H1\n+τ/an}bracketle{tvN\nn+1,vN\nn+1−vN\nn/an}bracketri}ht˙H1\n=/an}bracketle{tηuN\nn+1,uN\nn+1−uN\nn/an}bracketri}ht˙H1+/an}bracketle{tuN\nn+1,vN\nn+1/an}bracketri}ht˙H1−/an}bracketle{tuN\nn,vN\nn/an}bracketri}ht˙H1\n+τ/bardblvN\nn+1/bardbl2\n˙H1−τ/an}bracketle{tvN\nn+1,vN\nn/an}bracketri}ht˙H1\n≥η\n2/bardbluN\nn+1/bardbl2\n˙H1−η\n2/bardbluN\nn/bardbl2\n˙H1+η\n2/bardbluN\nn+1−uN\nn/bardbl2\n˙H1\n+/an}bracketle{tuN\nn+1,vN\nn+1/an}bracketri}ht˙H1−/an}bracketle{tuN\nn,vN\nn/an}bracketri}ht˙H1+1\n2τ/bardblvN\nn+1/bardbl2\n˙H1−1\n2τ/bardblvN\nn/bardbl2\n˙H1.\nOn the other hand, using Assumption 1 gives\n/an}bracketle{tηuN\nn+1+vN\nn+1,uN\nn+1−uN\nn/an}bracketri}ht˙H1+/an}bracketle{tΛuN\nn+1,vN\nn+1−vN\nn/an}bracketri}ht\n=τ/an}bracketle{tηΛuN\nn+1+ΛvN\nn+1,vN\nn+1/an}bracketri}ht+/an}bracketle{tδWn,ΛuN\nn+1/an}bracketri}ht\n−τ/an}bracketle{tΛuN\nn+1,ΛuN\nn+1+ηvN\nn+1+ΠNF(vN\nn+1)/an}bracketri}ht\n=τ/bardblvN\nn+1/bardbl2\n˙H1−τ/bardbluN\nn+1/bardbl2\n˙H2−τ/an}bracketle{tΛuN\nn+1,ΠNF(vN\nn+1)/an}bracketri}ht\n+/an}bracketle{tδWn,uN\nn+1−uN\nn/an}bracketri}ht˙H1+/an}bracketle{tδWn,uN\nn/an}bracketri}ht˙H1\n≤τ/bardblvN\nn+1/bardbl2\n˙H1−τ/bardbluN\nn+1/bardbl2\n˙H2+1\n2τ/bardbluN\nn+1/bardbl2\n˙H2+a2\n1τ(|O|+/bardblvN\nn+1/bardbl2)\n+η\n2/bardbluN\nn+1−uN\nn/bardbl2\n˙H1+1\n2η/bardblδWn/bardbl2\n˙H1+/an}bracketle{tδWn,uN\nn/an}bracketri}ht˙H1\n= (1+a2\n1)τ/bardblvN\nn+1/bardbl2\n˙H1−1\n2τ/bardbluN\nn+1/bardbl2\n˙H2+η\n2/bardbluN\nn+1−uN\nn/bardbl2\n˙H1\n+1\n2η/bardblδWn/bardbl2\n˙H1+/an}bracketle{tδWn,uN\nn/an}bracketri}ht˙H1+a2\n1|O|τ.\nIt follows by taking expectation and using (23) that,\nE/bracketleftbigη\n2/bardbluN\nn+1/bardbl2\n˙H1+1\n2τ/bardblvN\nn+1/bardbl2\n˙H1+/an}bracketle{tuN\nn+1,vN\nn+1/an}bracketri}ht˙H1/bracketrightbig\n+τ\n2n+1/summationdisplay\nk=1E[/bardbluN\nk/bardbl2\n˙H2]\n≤(1+a2\n1)τn+1/summationdisplay\nk=1E[/bardblvN\nk/bardbl2\n˙H1]+η\n2/bardblu0/bardbl2\n˙H1+1\n2τ/bardblv0/bardbl2\n˙H1+/an}bracketle{tu0,v0/an}bracketri}ht˙H1\n+C·(n+1)τ(1+/bardblΛ1\n2/bardbl2\nL0\n2)\n≤C/parenleftbig\n/bardblX0/bardbl2\nH2+(n+1)τ/parenrightbig\n.\nHence,\nτ\n2n+1/summationdisplay\nk=1E[/bardbluN\nk/bardbl2\n˙H2]≤E/bracketleftbig/vextendsingle/vextendsingle/an}bracketle{tuN\nn+1,vN\nn+1/an}bracketri}ht˙H1/vextendsingle/vextendsingle/bracketrightbig\n+C/parenleftbig\n/bardblX0/bardbl2\nH2+(n+1)τ/parenrightbig\n≤E/bracketleftbig\n/bardblXN\nn+1/bardbl2\nH2/bracketrightbig\n+C/parenleftbig\n/bardblX0/bardbl2\nH2+(n+1)τ/parenrightbig\n.\nTurning back to (23) leads to\nτn+1/summationdisplay\nk=1E/bracketleftbig\n/bardbluN\nk/bardbl2\n˙H2/bracketrightbig\n≤C/parenleftbig\n/bardblX0/bardbl2\nH2+(n+1)τ/parenrightbig\n.\nCombining it with (23) finishes the proof. /squareSTOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 11\n4.Exponential ergodicity\nIn this section, we present the exponential ergodicity of so lutions for (3) and (12).\n4.1.Exponential ergodicity of exact solution. We introduce the Markov semigroup Pt∈\nL(Bb(H1)) associated to (3), which is defined by\nPtϕ(X0) =E/bracketleftbig\nϕ(X(t;X0))/bracketrightbig\n, X0∈H1, ϕ∈ Bb(H1), t≥0,\nwhereBb(H1) represents the Banach space of all Borel bounded mappings ϕ:H1→Rand\nX(t;X0) is the solution of (3) with initial datum X0. Denote by P(H1) the space of all\nprobability measures on ( H1,B(H1)) and by Pp(H1) the space of all probability measures on\n(H1,B(H1)) with finite pth moment. The dual operator of Ptis defined as\nνPt(A) =/integraldisplay\nH1Pt1A(U)ν(dU), A∈ B(H1), ν∈ P(H1).\nThe following proposition states that (3) admits an invaria nt measure.\nProposition 4.1. Let Assumptions 1 and 2 hold. Then there exists an invariant me asure\nµ∈ P(H1)forPt.\nProof.We first show that the Markov semigroup Ptis Feller. Setting ¯ u(t) =u(t)−˜u(t) and\n¯v(t) =v(t)−˜v(t), we have\n\n\nd\ndt¯u(t) = ¯v(t),\nd\ndt¯v(t) =−Λ¯u(t)−η¯v(t)−/bracketleftbig\nF(v(t))−F(˜v(t))/bracketrightbig\n,\n¯u(0) = ¯u0=u0−˜u0,¯v(0) = ¯v0=v0−˜v0.\nIt follows from (6) that\nd\ndt/bardblX(t)−/tildewideX(t)/bardbl2\nH1=d\ndt/parenleftbig\n/bardbl¯u(t)/bardbl2\n˙H1+/bardbl¯v(t)/bardbl2/parenrightbig\n= 2/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht˙H1−2/an}bracketle{t¯v(t),Λ¯u(t)+η¯v(t)+F(v(t))−F(˜v(t))/an}bracketri}ht\n=−2η/bardbl¯v(t)/bardbl2−2/an}bracketle{t¯v(t),F(v(t))−F(˜v(t))/an}bracketri}ht\n≤ −2(η+a2)/bardbl¯v(t)/bardbl2≤0.\nHence, for any t≥0,\n/bardblX(t)−/tildewideX(t)/bardblH1≤ /bardblX0−/tildewideX0/bardblH1,\nwhich yields the Feller property of Pt.\nFixX0∈H2. By the Krylov–Bogoliubov theorem (see e.g., [15, Theorem 7 .1]), it suffices\nto prove the tightness of the family of probability measures {µX0\nT}T>0onB(H1), given by\nµX0\nT:=1\nT/integraldisplayT\n0δX0Ptdt, T >0.\nFor anyR >0, defineKR=/braceleftbig\ny∈H1:/bardbly/bardblH2≤R/bracerightbig\n,which is a compact subset in H1since\nthe embedding H2⊂H1is compact. By using Chebyshev’s inequality and (15), we hav e\nµX0\nT(Kc\nR) =1\nT/integraldisplayT\n0P/parenleftbig\n/bardblX(t;X0)/bardblH2>R/parenrightbig\ndt\n≤1\nTR2/integraldisplayT\n0E/bracketleftbig\n/bardblX(t;X0)/bardbl2\nH2/bracketrightbig\ndt≤CX0\nR2.\nHence, for any ǫ>0, there exists Rǫ>0 such that µX0\nT(Kc\nRǫ)<ǫ. This shows that {µX0\nT}T>0\nis tight. The proof is thus completed. /square12 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nIn the sequel, to ensure the uniqueness of the invariant meas ureµfor (1), we need the\nfollowing assumption on F.\nAssumption 3. There exists a positive constant Lsuch that\n/bardblF(v1)−F(v2)/bardbl˙H−1≤L/bardblv1−v2/bardbl, v1,v2∈H.\nLemma 4.2. Under Assumptions 1 to 3 and X0,/tildewideX0∈H1with finitepth moment. Let X(t)\nand/tildewideX(t)be the solutions of (3)with different initial data X0and/tildewideX0, respectively. Then for\nǫ>0sufficiently small and any p>0,t≥0,\n/bardblX(t)−/tildewideX(t)/bardblH1≤1+2ǫ\n1−2ǫe−ǫt/bardblX0−/tildewideX0/bardblH1,a.s. (24)\nand\n/bardblX(t)−/tildewideX(t)/bardblLp(Ω;H1)≤1+2ǫ\n1−2ǫe−ǫt/bardblX0−/tildewideX0/bardblLp(Ω;H1). (25)\nProof.Note that\nd\ndt/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht=/an}bracketle{t¯v(t),¯v(t)/an}bracketri}ht−/an}bracketle{t¯u(t),Λ¯u(t)+η¯v(t)+F(v(t))−F(˜v(t))/an}bracketri}ht\n=/bardbl¯v(t)/bardbl2−/bardbl¯u(t)/bardbl2\n˙H1−η/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht−/an}bracketle{t¯u(t),F(v(t))−F(˜v(t))/an}bracketri}ht\n≤ /bardbl¯v(t)/bardbl2−/bardbl¯u(t)/bardbl2\n˙H1−η/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht+L/bardbl¯u(t)/bardbl˙H1/bardbl¯v(t)/bardbl,\nwhere Assumption 3 was used in the last step. As a result, we ob tain\nd\ndt/parenleftBig\n/bardbl¯u(t)/bardbl2\n˙H1+/bardbl¯v(t)/bardbl2+4ǫ/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht/parenrightBig\n≤ −/bracketleftbig\n2(η+a2)−4ǫ/bracketrightbig\n/bardbl¯v(t)/bardbl2−4ǫ/bardbl¯u(t)/bardbl2\n˙H1−4ǫη/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht+4ǫL/bardbl¯u(t)/bardbl˙H1/bardbl¯v(t)/bardbl\n≤ −/bracketleftbig\n2(η+a2)−4(L2+η2+1)ǫ/bracketrightbig\n/bardbl¯v(t)/bardbl2−2ǫ/bardbl¯u(t)/bardbl2\n˙H1\n≤ −(η+a2)/bardbl¯v(t)/bardbl2−2ǫ/bardbl¯u(t)/bardbl2\n˙H1,\nwhere we used Young’s inequality and 0 <ǫ<η+a2\n4(L2+η2+1). Furthermore, we get\nd\ndt/parenleftBig\n/bardbl¯u(t)/bardbl2\n˙H1+/bardbl¯v(t)/bardbl2+4ǫ/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht/parenrightBig\n≤ −ǫ/parenleftBig\n/bardbl¯u(t)/bardbl2\n˙H1+/bardbl¯v(t)/bardbl2+4ǫ/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht/parenrightBig\n,\nfor sufficiently small ǫ>0. Thus, we invoke Gronwall’s inequality to deduce\n/bardbl¯u(t)/bardbl2\n˙H1+/bardbl¯v(t)/bardbl2+4ǫ/an}bracketle{t¯u(t),¯v(t)/an}bracketri}ht ≤e−ǫt/parenleftBig\n/bardbl¯u0/bardbl2\n˙H1+/bardbl¯v0/bardbl2+4ǫ/an}bracketle{t¯u0,¯v0/an}bracketri}ht/parenrightBig\n.\nApplying the fact −1\n2(a2+b2)≤ab≤1\n2(a2+b2) yields\n/bardbl¯u(t)/bardbl2\n˙H1+/bardbl¯v(t)/bardbl2≤1+2ǫ\n1−2ǫe−ǫt/parenleftBig\n/bardbl¯u0/bardbl2\n˙H1+/bardbl¯v0/bardbl2/parenrightBig\nFinally, taking expectation, we obtain (25). /square\nThe definition of pth order Wasserstein distance indicates that\nWp(ν1,ν2) = inf{/bardblY1−Y2/bardblLp(Ω;H1):Y1∼ν1andY2∼ν2}.\nSinceν1Pt(resp.ν2Pt) is the distribution of solution X(t;X0) (resp.X(t;/tildewideX0)) with initial\ndistribution X0∼ν1(resp./tildewideX0∼ν2), one can invoke (25) to obtain\nWp(ν1Pt,ν2Pt)≤ /bardblX(t;X0)−X(t;/tildewideX0)/bardblLp(Ω;H1)≤1+2ǫ\n1−2ǫe−ǫt/bardblX0−/tildewideX0/bardblLp(Ω;H1).\nTaking infimum w.r.t. X0and/tildewideX0gives that for any ν1,ν2∈ Pp(H1),\nWp(ν1Pt,ν2Pt)≤1+2ǫ\n1−2ǫe−ǫtWp(ν1,ν2). (26)STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 13\nTheorem 4.3. Let Assumptions 1 to 3 hold. Then, (3)admits a unique invariant measure\nµ∈ ∩p>0Pp(H1)such that for any t≥0, p>0, ν∈ Pp(H1), andǫ>0sufficiently small,\nWp(νPt,µ)≤1+2ǫ\n1−2ǫe−ǫtWp(ν,µ). (27)\nProof.The exponential contraction property (26) implies that for anyν∈ Pp(H1) ands>0,\nWp(νPt+s,νPt)≤1+2ǫ\n1−2ǫe−ǫtWp(νPs,ν),\nwhich together with the completeness of ( Pp(H1),Wp) tells that {νPt}t≥0converges in pth\norder Wasserstein distance. Denote by µν∈ Pp/parenleftbig\nH1/parenrightbig\nthe limit of {νPt}t≥0inWp, then for\nanyν1,ν2∈ Pp/parenleftbig\nH1/parenrightbig\n,\nWp(µν1,µν2) = lim\nt→∞Wp(ν1Pt,ν2Pt)≤lim\nt→∞1+2ǫ\n1−2ǫe−ǫtWp(ν1,ν2) = 0.\nHence, for any ν∈ Pp(H1), the limit µν∈ Pp/parenleftbig\nH1/parenrightbig\nof{νPt}t≥0is independent of ν. Then we\ndenote the limit by µ. Moreover, we have\nWp(µ,µPt) = lim\ns→∞Wp(µPs,(µPt)Ps)≤lim\ns→∞1+2ǫ\n1−2ǫe−ǫsWp(µ,µPt) = 0,\nwhich means that µis the unique invariant measure of {Pt}t≥0andµ∈ ∩p>0Pp/parenleftbig\nH1/parenrightbig\n. Fur-\nthermore, for any ν∈ Pp(H1), taking advantage of (26), one obtains\nWp(νPt,µ) =Wp(νPt,µPt)≤1+2ǫ\n1−2ǫe−ǫtWp(ν,µ),\nwhich finishes the proof. /square\nTo investigate the exponential ergodicity with more genera l test functionals, for p>0,γ∈\n(0,1], we define Cp,γ:=Cp,γ(B;R) the set of continuous functions ϕon the Banach space B\nsuch that\n/bardblϕ/bardblp,γ:= sup\nx1,x2∈B\nx1/negationslash=x2|ϕ(x1)−ϕ(x2)|\ndp,γ(x1,x2)<∞\nwithdp,γ(x1,x2) :=/bardblx1−x2/bardblγ\nB/parenleftbig\n1+/bardblx1/bardblp\nB+/bardblx2/bardblp\nB/parenrightbig1\n2.If the context of the functional’s space\nis clear, we will abbreviate the notation of the space as Cp,γ. It follows from the definition\nthat the functionals in Cp,γ(B;R) have polynomial growth. Indeed, for any ϕ∈ Cp,γ(B;R), by\nmeans of the triangle inequality, one has\n|ϕ(x)| ≤ |ϕ(x)−ϕ(0)|+|ϕ(0)| ≤ /bardblϕ/bardblp,γ/bardblx/bardblγ\nB(1+/bardblx/bardblp\nB)1\n2+|ϕ(0)| ≤C1(1+/bardblx/bardblγ+p\n2\nB).\nIt is known that for all t≥0,q≥1,Ptis uniquely extendible to a linear bounded operator\nonLq(H1,µ) that we still denote by Pt. Sinceµ∈ ∩q≥1Pq(H1), we know that Cp,γ(H1;R)⊂\n∩q≥1Lq(H1,µ). HencePtϕis well-defined for any ϕ∈ Cp,γ.\nTheorem 4.4. Let Assumptions 1 to 3 hold. Then for any t≥0,U∈H1,ϕ∈ Cp,γ, and\nsufficiently small ǫ>0, there exists a constant C(ǫ,p,γ)>0such that the unique invariant\nmeasureµ∈ P(H1)of(3)satisfies\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglePtϕ(U)−/integraldisplay\nH1ϕdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(ǫ,p,γ)/bardblϕ/bardblp,γ(1+/bardblU/bardblγ+p\n2\nH1)e−ǫγt. (28)\nProof.For anyU,/tildewideU∈H1, by using (25) and Proposition 3.2, one obtains\n/vextendsingle/vextendsingle/vextendsinglePtϕ(U)−Ptϕ(/tildewideU)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingleE[ϕ(X(t;U))−ϕ(X(t;/tildewideU))]/vextendsingle/vextendsingle/vextendsingle14 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\n≤/bardblϕ/bardblp,γE/bracketleftbigg/vextenddouble/vextenddouble/vextenddoubleX(t;U)−X(t;/tildewideU)/vextenddouble/vextenddouble/vextenddoubleγ\nH1/parenleftBig\n1+/bardblX(t;U)/bardblp\nH1+/bardblX(t;/tildewideU)/bardblp\nH1/parenrightBig1\n2/bracketrightbigg\n≤/bardblϕ/bardblp,γ/parenleftbigg\nE/bracketleftbigg/vextenddouble/vextenddouble/vextenddoubleX(t;U)−X(t;/tildewideU)/vextenddouble/vextenddouble/vextenddouble2γ\nH1/bracketrightbigg/parenrightbigg1\n2/parenleftBig\nE/bracketleftBig\n1+/bardblX(t;U)/bardblp\nH1+/bardblX(t;/tildewideU)/bardblp\nH1/bracketrightBig/parenrightBig1\n2\n≤C(ǫ,p)/bardblϕ/bardblp,γe−ǫγt/bardblU−/tildewideU/bardblγ\nH1/parenleftBig\n1+/bardblU/bardblp\nH1+/bardbl/tildewideU/bardblp\nH1/parenrightBig1\n2.\nHence, it follows from the above inequality that\n|Ptϕ(U)−µ(ϕ)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1/parenleftbig\nPtϕ(U)−Ptϕ(/tildewideU)/parenrightbig\nµ(d/tildewideU)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C(ǫ,p)/bardblϕ/bardblp,γe−ǫγt/integraldisplay\nH1/bardblU−/tildewideU/bardblγ\nH1/parenleftBig\n1+/bardblU/bardblp\nH1+/bardbl/tildewideU/bardblp\nH1/parenrightBig1\n2µ(d/tildewideU)\n≤C(ǫ,p,γ)/bardblϕ/bardblp,γe−ǫγt(1+/bardblU/bardblγ+p\n2\nH1),\nwhere we used µ∈ ∩q≥1Pq(H1). /square\n4.2.Exponential ergodicity of the numerical solution. Thissubsectionshowstheunique\ninvariant measure and exponential ergodicity of the numeri cal solution for the full discretiza-\ntion (12).\nWe first consider the case for spatial semidiscretization (1 1). We define the associated\nMarkov semigroup PN\nt∈ L(Bb(H1\nN)) as\nPN\ntϕ(XN\n0) =E/bracketleftbig\nϕ(XN(t;XN\n0))/bracketrightbig\n, ϕ∈ Bb(H1\nN), t≥0,\nwhereBb(H1\nN) is the set of Borel bounded measurable functions ϕ:H1\nN→R. The dual\noperator of {PN\nt}t≥0is given by\nνPN\nt(A) =/integraldisplay\nH1\nNPN\nt1A(U)ν(dU), A∈ B(H1\nN), ν∈ P(H1\nN).\nBy a similar argument as in the case of the exact solution, it c an be shown that the spatial\nsemi-discretization (11) is exponentially ergodic with a u nique invariant measure µN. The\nproof is thus omitted here.\nTheorem 4.5. Let Assumptions 1 to 3 hold. Then, the spectral Galerkin semi- discretization\n(11)admits a unique invariant measure µNinH1\nNsuch that for any t≥0,U∈H1\nNand\nϕ∈ Cp,γ, forǫ>0sufficiently small, there exists a constant C(ǫ,p,γ)>0such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglePN\ntϕ(U)−/integraldisplay\nH1ϕdµN/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(ǫ,p,γ)/bardblϕ/bardblp,γ(1+/bardblU/bardblγ+p\n2\nH1)e−ǫγt. (29)\nFor the full discretization (12), we can also obtain the exis tence of an invariant measure by\nfollowing the Krylov–Bogoliubov procedure as in the proof o f Proposition 4.1. The proof is\nomitted.\nProposition 4.6. Let Assumptions 1 and 2 hold. Then there exists an invariant me asureµN\nτ\ninH1\nNof the full discretization (12).\nLet{Pτ,N\nn}n∈Nbe the discrete Markov transition semigroup associated to {XN\nn}n∈N. The\nnext step is to show the uniqueness of the invariant measure µN\nτfor{Pτ,N\nn}n∈N, which relies\non the following lemma.STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 15\nLemma 4.7. Assume that Assumptions 1 to 3 hold, and X0,/tildewideX0∈H1. LetXN\nnand/tildewideXN\nn\nbe the solutions of (12)with different initial data ΠNX0andΠN/tildewideX0, respectively. Then for\nǫ>0sufficiently small, p>0, and anyτ∈(0,1),\n/bardblXN\nn−/tildewideXN\nn/bardblH1≤1+2ǫ\n1−2ǫe−ǫtn/bardblX0−/tildewideX0/bardbl2\nH1,a.s.n∈N (30)\nand\n/bardblXN\nn−/tildewideXN\nn/bardblLp(Ω;H1)≤1+2ǫ\n1−2ǫe−ǫtn/bardblX0−/tildewideX0/bardblLp(Ω;H1), n∈N. (31)\nMoreover, for any ν1,ν2∈ Pp(H1\nN)\nWp(ν1Pτ,N\nn,ν2Pτ,N\nn)≤1+2ǫ\n1−2ǫe−ǫtnWp(ν1,ν2). (32)\nProof.Denote ¯uN\nn+1=uN\nn+1−˜uN\nn+1and ¯vN\nn+1=vN\nn+1−˜vN\nn+1, it follows that\n/braceleftBigg\n¯uN\nn+1= ¯uN\nn+τ¯vN\nn+1,\n¯vN\nn+1= ¯vN\nn−τ/parenleftbig\nΛN¯uN\nn+1+η¯vN\nn+1+ΠNF(vN\nn+1)−ΠNF(˜vN\nn+1)/parenrightbig\n.\nOn the one hand, we have\n/an}bracketle{t¯uN\nn+1−¯uN\nn,¯uN\nn+1/an}bracketri}ht˙H1+/an}bracketle{t¯vN\nn+1−¯vN\nn,¯vN\nn+1/an}bracketri}ht\n=1\n2/parenleftbig\n/bardbl¯uN\nn+1/bardbl2\n˙H1−/bardbl¯uN\nn/bardbl2\n˙H1+/bardbl¯uN\nn+1−¯uN\nn/bardbl2\n˙H1/parenrightbig\n+1\n2/parenleftbig\n/bardbl¯vN\nn+1/bardbl2−/bardbl¯vN\nn/bardbl2+/bardbl¯vN\nn+1−¯vN\nn/bardbl2/parenrightbig\n≥1\n2/parenleftbig\n/bardbl¯uN\nn+1/bardbl2\n˙H1−/bardbl¯uN\nn/bardbl2\n˙H1+/bardbl¯vN\nn+1/bardbl2−/bardbl¯vN\nn/bardbl2/parenrightbig\n.\nOn the other hand, we make use of (6) to obtain\n/an}bracketle{t¯uN\nn+1−¯uN\nn,¯uN\nn+1/an}bracketri}ht˙H1+/an}bracketle{t¯vN\nn+1−¯vN\nn,¯vN\nn+1/an}bracketri}ht\n=τ/an}bracketle{t¯vN\nn+1,¯uN\nn+1/an}bracketri}ht˙H1−τ/an}bracketle{tΛN¯uN\nn+1+η¯vN\nn+1+ΠNF(vN\nn+1)−ΠNF(˜vN\nn+1),¯vN\nn+1/an}bracketri}ht\n=−ητ/bardbl¯vN\nn+1/bardbl2−τ/an}bracketle{tF(vN\nn+1)−F(˜vN\nn+1),¯vN\nn+1/an}bracketri}ht\n≤ −(η+a2)τ/bardbl¯vN\nn+1/bardbl2.(33)\nNext, using the Cauchy–Schwartz inequality and Assumption 3, we deduce\nǫ/parenleftbig\n/an}bracketle{t¯uN\nn+1−¯uN\nn,¯vN\nn+1/an}bracketri}ht+/an}bracketle{t¯uN\nn+1,¯vN\nn+1−¯vN\nn/an}bracketri}ht/parenrightbig\n=ǫ/parenleftbig\nτ/bardbl¯vN\nn+1/bardbl2−τ/an}bracketle{t¯uN\nn+1,ΛN¯uN\nn+1+η¯vN\nn+1+ΠNF(vN\nn+1)−ΠNF(˜vN\nn+1)/an}bracketri}ht/parenrightbig\n=ǫ/parenleftbig\nτ/bardbl¯vN\nn+1/bardbl2−τ/bardbl¯uN\nn+1/bardbl2\n˙H1−τη/an}bracketle{t¯uN\nn+1,¯vN\nn+1/an}bracketri}ht−τ/an}bracketle{t¯uN\nn+1,F(vN\nn+1)−F(˜vN\nn+1)/an}bracketri}ht/parenrightbig\n≤ǫ/parenleftbig\nτ/bardbl¯vN\nn+1/bardbl2−τ/bardbl¯uN\nn+1/bardbl2\n˙H1−τη/an}bracketle{t¯uN\nn+1,¯vN\nn+1/an}bracketri}ht+Lτ/bardbl¯uN\nn+1/bardbl˙H1/bardbl¯vN\nn+1/bardbl/parenrightbig\n.\nFollowing the similar approach in Lemma 4.2, we obtain, for s ufficiently small ǫ>0,\n/an}bracketle{t¯uN\nn+1−¯uN\nn,¯uN\nn+1/an}bracketri}ht˙H1+/an}bracketle{t¯vN\nn+1−¯vN\nn,¯vN\nn+1/an}bracketri}ht\n+8ǫ/parenleftbig\n/an}bracketle{t¯uN\nn+1−¯uN\nn,¯vN\nn+1/an}bracketri}ht+/an}bracketle{t¯uN\nn+1,¯vN\nn+1−¯vN\nn/an}bracketri}ht/parenrightbig\n≤ −τ/bracketleftbig\n(η+a2)−8ǫ/bracketrightbig\n/bardbl¯vN\nn+1/bardbl2−8τǫ/bardbl¯uN\nn+1/bardbl2\n˙H1\n−8τηǫ/an}bracketle{t¯uN\nn+1,¯vN\nn+1/an}bracketri}ht+8τLǫ/bardbl¯uN\nn+1/bardbl˙H1/bardbl¯vN\nn+1/bardbl\n≤ −τ/parenleftbigη+a2\n2/bardbl¯vN\nn+1/bardbl2+4ǫ/bardbl¯uN\nn+1/bardbl2\n˙H1/parenrightbig\n≤ −2ǫτ/parenleftbig\n/bardbl¯uN\nn+1/bardbl2\n˙H1+/bardbl¯vN\nn+1/bardbl2+8ǫ/an}bracketle{t¯uN\nn+1,¯vN\nn+1/an}bracketri}ht/parenrightbig\n.\n≤ −2ǫτ/parenleftbig1\n2/bardbl¯uN\nn+1/bardbl2\n˙H1+1\n2/bardbl¯vN\nn+1/bardbl2+8ǫ/an}bracketle{t¯uN\nn+1,¯vN\nn+1/an}bracketri}ht/parenrightbig\n.16 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nApplying elementary inequality ab≥ −1\n2(a2+b2) yields\n/an}bracketle{t¯uN\nn+1−¯uN\nn,¯uN\nn+1/an}bracketri}ht˙H1+/an}bracketle{t¯vN\nn+1−¯vN\nn,¯vN\nn+1/an}bracketri}ht\n+8ǫ/parenleftbig\n/an}bracketle{t¯uN\nn+1−¯uN\nn,¯vN\nn+1/an}bracketri}ht+/an}bracketle{t¯uN\nn+1,¯vN\nn+1−¯vN\nn/an}bracketri}ht/parenrightbig\n=1\n2/bardbl¯uN\nn+1/bardbl2\n˙H1−1\n2/bardbl¯uN\nn/bardbl2\n˙H1+1\n2/bardbl¯uN\nn+1−¯uN\nn/bardbl2\n˙H1\n+1\n2/bardbl¯vN\nn+1/bardbl2−1\n2/bardbl¯vN\nn/bardbl2+1\n2/bardbl¯vN\nn+1−¯vN\nn/bardbl2\n+8ǫ/parenleftbig\n/an}bracketle{t¯uN\nn+1,¯vN\nn+1/an}bracketri}ht−/an}bracketle{t¯uN\nn,¯vN\nn/an}bracketri}ht+/an}bracketle{t¯uN\nn+1−¯uN\nn,¯vN\nn+1−¯vN\nn/an}bracketri}ht/parenrightbig\n≥1\n2/bardbl¯uN\nn+1/bardbl2\n˙H1−1\n2/bardbl¯uN\nn/bardbl2\n˙H1+1\n2/bardbl¯vN\nn+1/bardbl2−1\n2/bardbl¯vN\nn/bardbl2\n+8ǫ/parenleftbig\n/an}bracketle{t¯uN\nn+1,¯vN\nn+1/an}bracketri}ht−/an}bracketle{t¯uN\nn,¯vN\nn/an}bracketri}ht/parenrightbig\n.\nHence, using the inequality ak<e−(1−a)k,a∈(0,1) gives\n1\n2/bardbl¯uN\nn/bardbl2\n˙H1+1\n2/bardbl¯vN\nn/bardbl2+8ǫ/an}bracketle{t¯uN\nn,¯vN\nn/an}bracketri}ht ≤e−ǫtn/parenleftBig\n1\n2/bardbl¯uN\n0/bardbl2\n˙H1+1\n2/bardbl¯vN\n0/bardbl2+8ǫ/an}bracketle{t¯uN\n0,¯vN\n0/an}bracketri}ht/parenrightBig\n,\nwhich yields (30). And we conclude (31) by taking pth moment on the above estimate. The\nproof of (32) is similar to that of (26), so it is omitted. /square\nNow we can show that (12) is ergodic with a unique invariant me asureµN\nτ, which converges\nexponentially to the equilibrium. The proof is similar to th at of Theorem 4.4 and is omitted.\nTheorem 4.8. Let Assumptions 1 to 3 hold. Then, the full discretization (12)admits a\nunique invariant measure µN\nτinH1\nNsuch that for any n∈N,U∈H1\nN, sufficiently small\nǫ>0, and continuous mapping ϕ∈ Cp,γ(H1\nN,R),\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglePτ,N\nnϕ(U)−/integraldisplay\nϕdµN\nτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(ǫ,p,γ)/bardblϕ/bardblp,γ(1+/bardblU/bardblγ+p\n2\nH1)e−ǫγtn. (34)\n5.Approximation for invariant measure\nThis section presents estimates on the approximation of the invariant measure. More\nprecisely, we obtain the error estimates of invariant measu res both in Wasserstein distance\nand in the weak sense by deriving the strong error estimate of the full discretization. The\nerrors between the time average of the exact and numerical so lutions and the ergodic limit\nare estimated, which leads to the strong laws of large number s of the exact solution and the\nfull discretization.\n5.1.Strong error analysis for full discretization. In this subsection, we concentrate on\nthe error analysis of the full discretization (12). Due to th at the operator semigroup E(t)\nof (1) lacks of smoothing effect, it is difficult to derive the est imates in the maximum norm\nof solutions directly. Instead, the Sobolev embedding theo rem has to be used to bound the\nmaximum norm by using the higher regularity of the solution. Hence, we focus on the case\nd= 1, since the embedding ˙H1֒→C(O;R) fails ford≥2.\n5.1.1.Spatial error analysis. In this part, we analyze the error of spectral Galerkin metho d\n(10) applied to (1). To start the convergence analysis, we ne ed an additional assumption on\nF.\nAssumption 4. The continuously differentiable function f:R→Rassociated to Fsatisfies/vextendsingle/vextendsinglef′(ξ)/vextendsingle/vextendsingle≤C(1+|ξ|), ξ∈R.\nThe following result concerns the spatial error analysis.STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 17\nProposition 5.1. Under Assumptions 1, 2, and 4 and assume that X0∈H2. LetX(t)and\nXN(t)be the solutions of (3)and(11), respectively. Then, for any t≥0,p≥1andN∈N,\nthere exists a positive constant C, independent of t, such that\n/vextenddouble/vextenddoubleX(t)−XN(t)/vextenddouble/vextenddouble\nLp(Ω;H1)≤Cλ−1\n2\nN(1+t1\n2).\nProof.We first decompose /bardblX(t)−XN(t)/bardblLp(Ω;H1)into two parts as follows,\n/bardblX(t)−XN(t)/bardblLp(Ω;H1)≤ /bardblX(t)−ΠNX(t)/bardblLp(Ω;H1)+/bardblΠNX(t)−XN(t)/bardblLp(Ω;H1),\nwhere\nΠNX(t) =EN(t)XN\n0−/integraldisplayt\n0EN(t−s)FN(X(s))ds+/integraldisplayt\n0EN(t−s)BNdW(s).\nFor the estimate of the term /bardblX(t)−ΠNX(t)/bardblLp(Ω;H1), applying (9) and Proposition 3.1 gives\nE/bracketleftbig\n/bardblX(t)−ΠNX(t)/bardblp\nH1/bracketrightbig\n≤Cλ−p\n2\nNE[/bardblX(t)/bardblp\nH2]≤Cλ−p\n2\nN.\nNext weestimate theterm /bardblΠNX(t)−XN(t)/bardblLp(Ω;H1). Denoting eN(t) =ΠNX(t)−XN(t) =:/parenleftbig\neN\n1(t),eN\n2(t)/parenrightbig⊤, we have\ndeN(t)\ndt=ANeN(t)+FN(XN(t))−FN(X(t)).\nOne obtains\nd/bardbleN(t)/bardbl2\nH1\ndt= 2/angbracketleftbigg\neN(t),deN(t)\ndt/angbracketrightbigg\nH1\n= 2/angbracketleftbig\neN(t),ANeN(t)+FN(XN(t))−FN(X(t))/angbracketrightbig\nH1\n= 2/angbracketleftbig\neN(t),FN(XN(t))−FN(X(t))/angbracketrightbig\nH1,\nwhere the final step utilized the antisymmetry of AN. Subsequently, through the application\nof the Cauchy–Schwartz inequality and (6), one obtain\nd/bardbleN(t)/bardbl2\nH1\ndt= 2/angbracketleftbig\neN(t),FN(XN(t))−FN(ΠNX(t))/angbracketrightbig\nH1\n+2/angbracketleftbig\neN(t),FN(ΠNX(t))−FN(X(t))/angbracketrightbig\nH1\n≤ −(η+a2)/bardbleN\n2(t)/bardbl2+C/vextenddouble/vextenddoubleFN(ΠNX(t))−FN(X(t))/vextenddouble/vextenddouble2\nH1.\nIntegrating both sides from 0 to tgives\n/bardbleN(t)/bardbl2\nH1≤C/integraldisplayt\n0(1+/bardblv(s)/bardbl2\n˙H1)/bardblΠNv(s)−v(s)/bardbl2ds,\nwhere we used Assumption 4 and the Sobolev embedding ˙H1֒→C(O;R) ford= 1. In\naddition, taking expectations on both sides and using H¨ old er’s inequality yield\n/bardbleN(t)/bardbl2\nL2p(Ω;H1)≤C/integraldisplayt\n0(1+/bardblv(s)/bardbl2\nL4p(Ω;˙H1))/bardblΠNv(s)−v(s)/bardbl2\nL4p(Ω;H)ds\n≤Cλ−1\nN/integraldisplayt\n0/parenleftBig\n1+/bardblv(s)/bardbl4\nL4p(Ω;˙H1)/parenrightBig\nds≤Cλ−1\nNt.\nThe proof is thus completed. /square18 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\n5.1.2.Temporal error analysis. The full discretization (12) can be rewritten as\nXN\nk=Ek\nN,τXN\n0−τk−1/summationdisplay\ni=0Ek−i\nN,τFN(XN\ni+1)+k−1/summationdisplay\ni=0Ek−i\nN,τBNδWi, (35)\nwherewedenote EN,τ:= (I−τAN)−1. Thefollowing lemma presentsestimates ofsemigroups,\nwhose proofs can be found in [1, Equation (13)] and [7, Theore m 4].\nLemma 5.2. Fort≥s≥0, we have\n/bardbl(E(t)−E(s))X/bardblH≤C(t−s)γ/bardblX/bardblHγ, X∈Hγ, γ∈[0,1]. (36)\nMoreover, for n∈N,\n/bardbl(EN(tn)−En\nN,τ)X/bardblH1≤Ctγ\n2nτγ\n2/bardblX/bardblHγ+1, X∈Hγ, γ= 0,1,2. (37)\nLemma 5.3. Let Assumptions 1 and 2 hold. Assume that X0∈H2. Then for 0≤s≤tand\np≥1, it holds that,\n/bardblXN(t)−XN(s)/bardblLp(Ω;H1)≤C|t−s|1\n2. (38)\nProof.Based on the mild solution of (11), we obtain\nXN(t)−XN(s) =/parenleftbig\nEN(t−s)−I/parenrightbig\nXN(s)\n−/integraldisplayt\nsEN(t−r)FN(XN(r))dr+/integraldisplayt\nsEN(t−r)BNdW(r),\nwhich yields\n/bardblXN(t)−XN(s)/bardblLp(Ω;H1)≤ /bardbl(EN(t−s)−I)XN(s)/bardblLp(Ω;H1)\n+/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\nsEN(t−r)FN(XN(r))dr/vextenddouble/vextenddouble/vextenddouble\nLp(Ω;H1)\n+/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\nsEN(t−r)BNdW(r)/vextenddouble/vextenddouble/vextenddouble\nLp(Ω;H1).(39)\nWe estimate the above three terms separately. The first term i n the right side of (39) can be\ndirectly estimated by (36),\n/vextenddouble/vextenddouble(EN(t−s)−I)XN(s)/vextenddouble/vextenddouble\nLp(Ω;H1)≤C|t−s|/bardblXN(s)/bardblLp(Ω;H2)≤C|t−s|.\nFor the second term in the right side of (39), it follows from t he stability of the semigroup\nthat\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\nsEN(t−r)FN(XN(r))dr/vextenddouble/vextenddouble/vextenddouble\nLp(Ω;H1)≤/integraldisplayt\ns/vextenddouble/vextenddoubleFN(XN(r))/vextenddouble/vextenddouble\nLp(Ω;H1)dr\n≤C/integraldisplayt\ns/parenleftbig\n1+/vextenddouble/vextenddoubleXN(r)/vextenddouble/vextenddouble\nLp(Ω;H1)/parenrightbig\ndr≤C|t−s|.\nFinally, using the Burkholder–Davis–Gundy-type inequali ty, the stability of the sine and\ncosine operators as well as of the projection operator, we ob tain\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\nsEN(t−r)BNdW(r)/vextenddouble/vextenddouble/vextenddouble\nLp(Ω;H1)\n≤/parenleftbigg/integraldisplayt\ns/vextenddouble/vextenddoublesin/parenleftbig\n(t−r)Λ1\n2/parenrightbig/vextenddouble/vextenddouble2\nL0\n2+/vextenddouble/vextenddoublecos/parenleftbig\n(t−r)Λ1\n2/parenrightbig/vextenddouble/vextenddouble2\nL0\n2dr/parenrightbigg1\n2\n≤C|t−s|1\n2.STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 19\nThe proof is thus completed. /square\nTheorem 5.4. Let Assumptions 1, 2, and 4 hold, X0∈H2, andXN(t)andXN\nkbe the\nsolutions of (11)and(35), respectively. Then, for any p≥2, it holds that\n/bardblXN(tk)−XN\nk/bardblLp(Ω;H1)≤Cτ1\n2(1+t3\nk), k∈N. (40)\nProof.We introduce the auxiliary process\n/hatwideXN\nk=Ek\nN,τXN\n0−τk−1/summationdisplay\ni=0Ek−i\nN,τFN(XN(ti+1))+k−1/summationdisplay\ni=0Ek−i\nN,τBNδWi.\nBy using the stability of Ek\nN,τ, it can be derived that\n/bardbl/hatwideXN\nk/bardblLp(Ω;H2)≤C(1+tk).\nThen, we divide /bardblXN(tk)−XN\nk/bardblLp(Ω;H1)into two terms,\n/bardblXN(tk)−XN\nk/bardblLp(Ω;H1)≤ /bardblXN(tk)−/hatwideXN\nk/bardblLp(Ω;H1)+/bardbl/hatwideXN\nk−XN\nk/bardblLp(Ω;H1).\nFor the first term /bardblXN(tk)−/hatwideXN\nk/bardblLp(Ω;H1), we further decompose it as follows,\n/bardblXN(tk)−/hatwideXN\nk/bardblL4p(Ω;H1)\n≤/vextenddouble/vextenddouble(EN(tk)−Ek\nN,τ)XN\n0/vextenddouble/vextenddouble\nL4p(Ω;H1)\n+/vextenddouble/vextenddouble/vextenddouble/integraldisplaytk\n0EN(tk−s)FN(XN(s))ds−τk−1/summationdisplay\ni=0Ek−i\nN,τFN(XN(ti+1))/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n+/vextenddouble/vextenddouble/vextenddouble/integraldisplaytk\n0EN(tk−s)BNdW(s)−k−1/summationdisplay\ni=0Ek−i\nN,τBNδWi/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n=:I1+I2+I3.\nUsing (37) in Lemma 5.2 shows\nI1≤C τ1\n2/bardblX0/bardblH2t1\n2\nk.\nForI2, we obtain\nI2≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0/integraldisplayti+1\ntiEN(tk−s)/parenleftbig\nFN(XN(s))−FN(XN(ti+1))/parenrightbig\nds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0/integraldisplayti+1\nti/parenleftbig\nEN(tk−s)−Ek−i\nN,τ/parenrightbig\nFN(XN(ti+1))ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n=:I2,1+I2,2.\nBy using the stability of EN(t), Taylor’s expansion and Lemma 5.3, one derives\nI2,1≤k−1/summationdisplay\ni=0/integraldisplayti+1\nti/vextenddouble/vextenddoubleFN(XN(s))−FN(XN(ti+1))/vextenddouble/vextenddouble\nL4p(Ω;H1)ds\n≤k−1/summationdisplay\ni=0/integraldisplayti+1\nti/parenleftbig\n1+/bardblvN(s)/bardblL8p(Ω;˙H1)+/bardblvN(ti+1)/bardblL8p(Ω;˙H1)/parenrightbig20 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\n×/vextenddouble/vextenddoubleXN(s)−XN(ti+1)/vextenddouble/vextenddouble\nL8p(Ω;H1)ds\n≤Cτ1\n2tk.\nFor the term I2,2, utilizing (36) and (37) shows that\nI2,2≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0/integraldisplayti+1\nti/parenleftbig\nEN(tk−s)−EN(tk−ti)/parenrightbig\nFN(XN(ti+1))ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0/integraldisplayti+1\nti/parenleftbig\nEN(tk−i)−Ek−i\nN,τ/parenrightbig\nFN(XN(ti+1))ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n≤Ck−1/summationdisplay\ni=0/integraldisplayti+1\ntiτ(1+/bardblXN(ti+1)/bardbl2\nL8p(Ω;H2))ds\n+Ck−1/summationdisplay\ni=0/integraldisplayti+1\ntiτ1\n2·t1\n2\nk·(1+/bardblXN(ti+1)/bardbl2\nL8p(Ω;H2))ds\n≤Cτ1\n2t3\n2\nk.\nFor the estimate of I3,\nI3≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0/integraldisplayti+1\nti/parenleftbig\nEN(tk−s)−EN(tk−ti)/parenrightbig\nBNdW(s)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0/integraldisplayti+1\nti/parenleftbig\nEN(tk−i)−Ek−i\nN,τ/parenrightbig\nBNdW(s)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL4p(Ω;H1)\n=:I3,1+I3,2.\nUsing Burkholder–Davis–Gundy-type inequality, (36) and ( 7), we arrive at\nI3,1≤/parenleftBiggk−1/summationdisplay\ni=0/integraldisplayti+1\nti/vextenddouble/vextenddouble/parenleftbig\nEN(tk−s)−EN(tk−ti)/parenrightbig\nBN/vextenddouble/vextenddouble2\nL0\n2(H1)ds/parenrightBigg1\n2\n≤C/parenleftbigg\nτ2/integraldisplaytk\n0/vextenddouble/vextenddoubleΛ1\n2Q1\n2/vextenddouble/vextenddouble2\nL2ds/parenrightbigg1\n2\n≤Cτt1\n2\nk.\nAnalogously, using (37) instead of (36), we possess\nI3,2≤/parenleftBiggk−1/summationdisplay\ni=0/integraldisplayti+1\nti/vextenddouble/vextenddouble/parenleftbig\nEN(tk−i)−Ek−i\nN,τ/parenrightbig\nBN/vextenddouble/vextenddouble2\nL0\n2(H1)ds/parenrightBigg1\n2\n≤C/parenleftbigg\nτ·tk·/integraldisplaytk\n0/vextenddouble/vextenddoubleΛ1\n2Q1\n2/vextenddouble/vextenddouble2\nL2ds/parenrightbigg1\n2\n≤Cτ1\n2tk.\nGathering the estimates of I1,I2,andI3, we have\n/bardblXN(tk)−/hatwideXN\nk/bardblL4p(Ω;H1)≤Cτ1\n2t3\n2\nk. (41)STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 21\nNext, we focus on the estimate of /bardbl/hatwideXN\nk−XN\nk/bardblLp(Ω;H1). For convenience, we define\neN\nk:=/hatwideXN\nk−XN\nk=τk−1/summationdisplay\ni=0Ek−i\nN,τ/parenleftbig\nFN(XN\ni+1)−FN(XN(ti+1))/parenrightbig\n,\nwhich satisfies\neN\nk−eN\nk−1−τANeN\nk=τ/parenleftbig\nFN(XN\nk)−FN(XN(tk))/parenrightbig\n, eN\n0= 0.\nMultiplying both sides by eN\nk, we have\n/an}bracketle{teN\nk−eN\nk−1,eN\nk/an}bracketri}htH1=τ/an}bracketle{tF(XN\nk)−F(/hatwideXN\nk),eN\nk/an}bracketri}htH1+τ/an}bracketle{tF(/hatwideXN\nk)−F(XN(tk)),eN\nk/an}bracketri}htH1.\nUsing the inequality /an}bracketle{ta−b,a/an}bracketri}htH1≥1\n2(/bardbla/bardbl2\nH1−/bardblb/bardbl2\nH1) and Young’s inequality, we obtain\n1\n2/parenleftbig\n/bardbleN\nk/bardbl2\nH1−/bardbleN\nk−1/bardbl2\nH1/parenrightbig\n≤ −η+a2\n2τ/bardbleN\nk,2/bardbl2+Cτ/bardblF(/hatwideXN\nk)−F(XN(tk))/bardbl2\nH1,\nwhereeN\nk:= (eN\nk,1,eN\nk,2)⊤. As a result, utilizing Assumption 4, moment bounds of /hatwideXN\nkand\nXN(t) as well as (41) shows that\n/bardbleN\nk/bardbl2\nL2p(Ω;H1)≤Cτk/summationdisplay\ni=1/bardblF(/hatwideXN\ni)−F(XN(ti))/bardbl2\nL2p(Ω;H1)\n≤Cτk/summationdisplay\ni=1(1+/bardblvN(ti)/bardbl2\nL4p(Ω;˙H1)+/bardbl/hatwidevN\ni/bardbl2\nL4p(Ω;˙H1))\n×/bardblXN(ti)−/hatwideXN\ni/bardbl2\nL4p(Ω;H1)\n≤Cτ(1+t6\nk).\nTherefore, this together with (41) leads to\n/bardblXN(t)−XN\nk/bardblL2p(Ω;H1)≤Cτ1\n2(1+t3\nk),\nwhich finishes the proof. /square\n5.2.Error estimates of invariant measures. This subsection presents error estimates of\ninvariant measures both in Wasserstein distance and in the w eak sense.\nLetµandµN\nτbe the unique invariant measures of (3) and (12), respective ly. The following\ntheorem states the first result about the error estimate betw eenµN\nτandµin Wasserstein\ndistance.\nTheorem 5.5. Let conditions in Proposition 5.1 and Assumption 3 hold. Then for anyp≥1,\nthere exists a constant C(p,ǫ)>0such that\nWp/parenleftbig\nµ,µN\nτ/parenrightbig\n≤C(p,ǫ)/parenleftbig\nλ−1\n2\nN+τ1\n2/parenrightbig\n.\nProof.For anyn∈N, one has\nWp/parenleftbig\nµ,µN\nτ/parenrightbig\n=Wp/parenleftbig\nµPtn,µN\nτPτ,N\nn/parenrightbig\n≤Wp/parenleftbig\nµPtn,(ΠN)#µPN\ntn/parenrightbig\n+Wp/parenleftbig\n(ΠN)#µPN\ntn,(ΠN)#µPτ,N\nn/parenrightbig\n+Wp/parenleftbig\n(ΠN)#µPτ,N\nn,µN\nτPτ,N\nn/parenrightbig\nwith (ΠN)#µbeing the push-forward of µthrough ΠN. For anyǫ>0, it follows from (32)\nthat\nWp/parenleftbig\n(ΠN)#µPτ,N\nn,µN\nτPτ,N\nn/parenrightbig\n≤1+2ǫ\n1−2ǫe−ǫtnWp/parenleftbig\n(ΠN)#µ,µN\nτ/parenrightbig\n. (42)22 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nIn order to use Proposition 5.1 and Theorem 5.4 to estimate th e remaining two terms, we\nneed to show µ(/bardbl · /bardblp\nH2)<∞for anyp≥1. We claim that (15) yields µ(/bardbl · /bardblp\nH2)<∞.\nIndeed, by tightness, {µ0\nT:=1\nT/integraltextT\n0δ0Ptdt}T≥0has a weakly convergent subsequence, saying\n{µ0\nTn}n≥0, which weakly converges to the invariant measure µ. For anyR>0, the functional\nπ/mapsto→/integraltext\nH1(/bardblx/bardblp\nH2∧R)π(dx) fromP(H1) toRis lower semicontinuous w.r.t. weak convergence\nthat\nµ(/bardbl·/bardblp\nH2∧R)≤liminf\nn→∞µ0\nTn(/bardbl·/bardblp\nH2∧R) = liminf\nn→∞1\nTn/integraldisplayTn\n0E[/bardblX(t;0)/bardblp\nH2∧R]dt≤Cp\nfor some positive constant Cpindependent of R. By means of Fatou’s lemma, we obtain\nµ(/bardbl·/bardblp\nH2)≤liminf\nR→∞µ(/bardbl·/bardblp\nH2∧R)≤Cp.\nHence, for any random variable X0whose distribution is µ, one has E[/bardblX0/bardblp\nH2]<∞and\nP◦(ΠNX0)−1= (ΠN)#µ. By Proposition 5.1 and Theorem 5.4, one can obtain\nWp/parenleftbig\nµPtn,(ΠN)#µPN\ntn/parenrightbig\n+Wp/parenleftbig\n(ΠN)#µPN\ntn,(ΠN)#µPτ,N\nn/parenrightbig\n≤ /bardblX(tn;X0)−XN(tn;ΠNX0)/bardblLp(Ω;H1)+/bardblXN(tn;ΠNX0)−XN\nn(ΠNX0)/bardblLp(Ω;H1)\n≤C1(p)(1+t1\n2n)λ−1\n2\nN+C2(p)(1+t3\nn)τ1\n2,\nwhereXN\nn(ΠNX0) denotes the solution of (12) with initial datum ΠNX0.\nHence, one arrives at\nWp/parenleftbig\n(ΠN)#µ,µN\nτ/parenrightbig\n≤Wp/parenleftbig\n(ΠN)#µ,µ/parenrightbig\n+Wp/parenleftbig\nµ,µN\nτ/parenrightbig\n≤/bardblX0−ΠNX0/bardblLp(Ω;H1)+Wp/parenleftbig\nµ,µN\nτ/parenrightbig\n≤Cλ−1\n2\nNµ(/bardbl·/bardblp\nH2)1\np+Wp/parenleftbig\nµ,µN\nτ/parenrightbig\n.\nCombining the above estimates with (42) yields\nWp/parenleftbig\nµ,µN\nτ/parenrightbig\n≤1+2ǫ\n1−2ǫe−ǫtnWp/parenleftbig\nµ,µN\nτ/parenrightbig\n+C1(ǫ,p)(1+t1\n2n+e−ǫtn)λ−1\n2\nN+C2(p)(1+t3\nn)τ1\n2.\nChoosingǫandnsuch that1+2ǫ\n1−2ǫe−ǫtn≤1\n2, one obtains\nWp/parenleftbig\nµ,µN\nτ/parenrightbig\n≤C(p,ǫ)/parenleftbig\nλ−1\n2\nN+τ1\n2/parenrightbig\n.\nThe proof is thus completed. /square\nThe following result is the weak error estimate of µN\nτandµ.\nTheorem 5.6. Let conditions in Proposition 5.1 and Assumption 3 hold. Then , forϕ∈Cp,γ,\none has for any n∈N,\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµ−/integraldisplay\nH1\nNϕdµN\nτ/vextendsingle/vextendsingle/vextendsingle≤C(ǫ,p,γ)/bardblϕ/bardblp,γ/parenleftBig\ne−ǫγtn+λ−γ\n2\nN(1+tγ\n2n)+τγ+p/2\n2(1+t3(γ+p\n2)\nn)/parenrightBig\n.\nProof.We divide the error into two parts:\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµ−/integraldisplay\nH1\nNϕdµN\nτ/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµ−/integraldisplay\nH1\nNϕdµN/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµN−/integraldisplay\nH1\nNϕdµN\nτ/vextendsingle/vextendsingle/vextendsingle.STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 23\nFor anyt>0, using the exponential ergodicity in (28) and (29), the err or estimate in Propo-\nsition 5.1 and the moment estimates in Proposition 3.2, we ar rive at\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµ−/integraldisplay\nH1\nNϕdµN/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµ−Ptϕ(0)/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsinglePtϕ(0)−PN\ntϕ(0)/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsinglePN\ntϕ(0)−/integraldisplay\nH1\nNϕdµN/vextendsingle/vextendsingle/vextendsingle\n≤ /bardblϕ/bardblp,γE/bracketleftBig\n/bardblX(t;0)−XN(t;0)/bardblγ\nH1/parenleftbig\n1+/bardblX(t;0)/bardblp\nH1+/bardblXN(t;0)/bardblp\nH1/parenrightbig1\n2/bracketrightBig\n+C(ǫ,p,γ)/bardblϕ/bardblp,γe−ǫγt\n≤C(ǫ,p,γ)/bardblϕ/bardblp,γ/parenleftBig\ne−ǫγt+λ−γ\n2\nN(1+tγ\n2)/parenrightBig\n.\nSimilarly, for any n∈N, using (29), (34), (40), Proposition 3.2 yields\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1ϕdµN−/integraldisplay\nH1\nNϕdµN\nτ/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nH1\nNϕdµN−PN\ntnϕ(0)/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsinglePN\ntnϕ(0)−Pτ,N\nnϕ(0)/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsinglePτ,N\nnϕ(0)−/integraldisplay\nH1\nNϕdµN\nτ/vextendsingle/vextendsingle/vextendsingle\n≤ /bardblϕ/bardblp,γE/bracketleftBig\n/bardblXN(tn;0)−XN\nn/bardblγ\nH1/parenleftbig\n1+/bardblXN(tn;0)/bardblp\nH1+/bardblXN\nn/bardblp\nH1/parenrightbig1\n2/bracketrightBig\n+C(ǫ,p,γ)/bardblϕ/bardblp,γe−ǫγtn\n≤C(ǫ,p,γ)/bardblϕ/bardblp,γ/parenleftbigg/parenleftBig\nE/bracketleftBig\n/bardblXN(tn;0)−XN\nn/bardbl2γ+p\nH1/bracketrightBig/parenrightBig1\n2+e−ǫγtn/parenrightbigg\n≤C(ǫ,p,γ)/bardblϕ/bardblp,γ/parenleftbigg\ne−ǫγtn+τγ+p\n2\n2(1+tn3(γ+p\n2))/parenrightbigg\n,\nwhich finishes the proof. /square\n5.3.Strong law of large numbers. This subsection presents the strong law of large num-\nbers of the exact solution and the full discretization based on estimating the error between\nthe corresponding time average and the ergodic limit µ(ϕ). For any X0∈H2, letX(t) and\nXN\nkrespectively be the solutions of (3) and 12 with initial data X0andΠNX0andµbe the\nunique invariant measures of (3).\nTheorem 5.7. Let the conditions of Proposition 5.1 and Assumption 3 hold. Then for all\nϕ∈ Cp,γ, the following strong law of large numbers holds\nlim\nt→∞1\nt/integraldisplayt\n0ϕ(X(s))ds=µ(ϕ)a.s. (43)\nProof.With the help of the method in [24, Proposition 2.6], we first c laim that for any t≥1,\n/parenleftBigg\nE/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nt/integraldisplayt\n0ϕ(X(s))ds−µ(ϕ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2q/bracketrightBigg/parenrightBigg1\n2q\n≤C(ǫ,p,γ,q)/bardblϕ/bardblp,γ(1+/bardblX0/bardblγ+p\n2\nH1)t−1\n2.(44)\nIndeed, without loss of generality, we assume that µ(ϕ) = 0. Otherwise, we let ˜ ϕ=ϕ−µ(ϕ)\nand consider ˜ ϕinstead ofϕ. Denoting\nIq(t) = sup\n0≤r≤tE[|Sr(ϕ)|2q], Sr(ϕ) :=/integraldisplayr\n0ϕ(X(s))ds,24 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nwe have\nE[|Sr(ϕ)|2q] =E/bracketleftBigg/integraldisplay\n[0,r]2q2q/productdisplay\ni=1ϕ(X(ri))dr1···dr2q/bracketrightBigg\n= (2q)!E/bracketleftBigg/integraldisplay\n∆[0,r]2q2q/productdisplay\ni=1ϕ(X(ri))dr1···dr2q/bracketrightBigg\n= (2q)!E/bracketleftBigg/integraldisplay\n∆[0,r]2q2q−2/productdisplay\ni=1ϕ(X(ri))g(r2q−1,r2q)dr1···dr2q/bracketrightBigg\n,(45)\nwhere we set\n∆[0,r]2q:=/braceleftbig\n(r1,...,r2q)∈R2q: 0≤r1≤ ··· ≤r2q≤r/bracerightbig\n,\ng(s1,s2) :=ϕ(X(s1))E[ϕ(X(s2))| Fs1], s1≤s2.\nThe integral on the right-hand side of (45) can be represente d as\n/integraldisplay\n∆[0,r]2g(r2q−1,r2q)\n\n/integraldisplay\n∆[0,r2q−1]2(q−1)2q−2/productdisplay\ni=1ϕ(X(ri))dr1···dr2q−2\n\ndr2q−1dr2q\n=1\n(2q−2)!/integraldisplay\n∆[0,r]2g(r2q−1,r2q)/vextendsingle/vextendsingleSr2q−1(ϕ)/vextendsingle/vextendsingle2(q−1)dr2q−1dr2q.\nSubstituting this expression into (45) and applying H¨ olde r’s inequality, we obtain\nE[|Sr(ϕ)|2q]≤C(q)/integraldisplay\n∆[0,r]2/parenleftBig\nE/bracketleftBig/vextendsingle/vextendsingleSr2q−1(ϕ)/vextendsingle/vextendsingle2q/bracketrightBig/parenrightBigq−1\nq(E[|g(r2q−1,r2q)|q])1\nqdr2q−1dr2q,\nwhereC(q) = 2q(2q−1). Taking the supremum over r∈[0,t], we see that\nIq(t)≤Cq(Iq(t))q−1\nq/integraldisplay\n∆[0,t]2(E[|g(s1,s2)|q])1\nqds1ds2.\nThus, we have\nIq(t)≤/parenleftBigg\nCq/integraldisplay\n∆[0,t]2(E[|g(s1,s2)|q])1\nqds1ds2/parenrightBiggq\nand\nE/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nt/integraldisplayt\n0ϕ(X(s))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle2q/bracketrightBigg\n≤/parenleftBigg\nCq\nt2/integraldisplay\n∆[0,t]2(E[|g(s1,s2)|q])1\nqds1ds2/parenrightBiggq\n. (46)\nIt follows from the Markov property and inequality (28) that\n|E[ϕ(X(s2))| Fs1]|=|Ps2−s1ϕ(X(s1))|\n≤C(ǫ,p)/bardblϕ/bardblp,γe−ǫγ(s2−s1)(1+/bardblX(s1)/bardblγ+p\n2\nH1).\nUsing the definition of Cp,γ, we obtain\n|g(s1,s2)| ≤C(ǫ,p,γ)/bardblϕ/bardblp,γe−ǫγ(s2−s1)(1+/bardblX(s1)/bardblγ+p\n2\nH1)|ϕ(X(s1))|\n≤C(ǫ,p,γ)/bardblϕ/bardbl2\np,γe−ǫγ(s2−s1)(1+/bardblX(s1)/bardbl2γ+p\nH1),STOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 25\nwhich implies that\n(E[|g(s1,s2)|q])1\nq≤C(ǫ,p,γ)/bardblϕ/bardbl2\np,γe−ǫγ(s2−s1)/parenleftbig\n1+(E[/bardblX(s1)/bardbl(2γ+p)q\nH1])1\nq/parenrightbig\n≤C(ǫ,p,γ)/bardblϕ/bardbl2\np,γe−ǫγ(s2−s1)(1+/bardblX0/bardbl2γ+p\nH1).\nNext, we calculate that/integraldisplay\n∆[0,t]2(E[|g(s1,s2)|q])1\nqds1ds2\n≤C(ǫ,p,γ)/bardblϕ/bardbl2\np,γ(1+/bardblX0/bardbl2γ+p\nH1)/integraldisplay\n∆[0,t]2e−ǫγ(s2−s1)ds1ds2\n≤C(ǫ,p,γ)/bardblϕ/bardbl2\np,γ(1+/bardblX0/bardbl2γ+p\nH1)t.\nSubstituting this inequality into (46) leads to (44). By vir tue of the Borel–Cantelli Lemma,\none has that for ε>0,/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nt/integraldisplayt\n0ϕ(X(t))dt−µ(ϕ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤K1(ε,p,γ,/bardblϕ/bardblp,γ)t−1\n2+εa.s. (47)\nfor some random variable K1(ε,p,γ,/bardblϕ/bardblp,γ)∈ ∩q≥1Lq(Ω). Letting t→ ∞finishes the proof.\n/square\nTheorem 5.8. Let the conditions of Proposition 5.1 and Assumption 3 hold. Then for all\nϕ∈ Cp,γ, the following strong law of large numbers holds\nlim\nn→∞lim\nτ→0lim\nN→∞1\nnn/summationdisplay\nk=1ϕ(XN\nk) =µ(ϕ)a.s. (48)\nThe proof of this theorem comes directly from the following p roposition, which is the error\nestimate between the time average of the numerical solution of (12) and the ergodic limit.\nProposition 5.9. Under conditions in Theorem 5.8, for all ϕ∈ Cp,γand allε >0there\nexists a positive random variable K(p,γ,ε,/bardblϕ/bardblp,γ)such that\n/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nk=1ϕ(XN\nk)−µ(ϕ)/vextendsingle/vextendsingle/vextendsingle\n≤K(p,γ,ε,/bardblϕ/bardblp,γ)/parenleftbigg\nτγ+p/2\n2−ε/parenleftbig\n1+t3(γ+p\n2)\nn/parenrightbig1+ε+λ−γ\n2+ε\nN/parenleftbig\n1+tγ\n2n/parenrightbig1+ε+t−1\n2+ε\nn/parenrightbigg\na.s.\nfor alln∈N. Moreover, E[|K(p,γ,ε,/bardblϕ/bardblp,γ)|q]<∞for allq∈N.\nProof.For any fixed functional ϕ∈ Cp,γ, one has\n/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nk=1ϕ(XN\nk)−µ(ϕ)/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nk=1ϕ(XN\nk)−1\ntn/integraldisplaytn\n0ϕ(XN(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0ϕ(X(t))dt−µ(ϕ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nHereX(t) denotes the solution of (3) with initial datum X0andXN(t) denotes the solution\nof (10) with initial datum ΠNX0.\nUsing H¨ older’s inequality and Proposition 5.1, one obtain s\nE/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingleq/bracketrightbigg26 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\n≤1\ntn/integraldisplaytn\n0E/bracketleftBig/vextendsingle/vextendsingleϕ(XN(t))−ϕ(X(t))/vextendsingle/vextendsingleq/bracketrightBig\ndt\n≤/bardblϕ/bardblp,γ\ntn/integraldisplaytn\n0E/bracketleftBig\n/bardblXN(t)−X(t)/bardblγq\nH1(1+/bardblXN(t)/bardblp\nH1+/bardblX(t)/bardblp\nH1)q\n2/bracketrightBig\ndt\n≤C(q,p,γ,/bardblX0/bardblH2)/bardblϕ/bardblp,γλ−γq\n2\nN/parenleftbig\n1+tγ\n2n/parenrightbigq.\nThis together with the Markov inequality implies that for an yε>0 and allδ>0\nP/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δλ−γ\n2+ε\nN(1+tγ\n2n)1+ε/parenrightbigg\n≤E/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingleq/bracketrightbigg\nλγq\n2−qε\nN(1+tγ\n2n)−q(1+ε)δ−q\n≤C(q,p,γ,/bardblX0/bardblH2)/bardblϕ/bardblp,γ(1+tγ\n2n)−qελ−qε\nNδ−q.\nThen forq>1\nεmax/braceleftbigd\n2,γ\n2/bracerightbig\n,\n∞/summationdisplay\nn=1∞/summationdisplay\nN=1P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δλ−γ\n2+ε\nN(1+tγ\n2n)1+ε/parenrightbigg\n≤C(q,p,γ,/bardblX0/bardblH2)/bardblϕ/bardblp,γδ−q∞/summationdisplay\nn=1(1+tγ\n2n)−qε∞/summationdisplay\nN=1λ−qε\nN<∞.\nHence, by the Borel–Cantelli lemma, the random variable\nK2(p,γ,ε,/bardblϕ/bardblp,γ) := sup\nn,N>0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingleλγ\n2−ε\nN(1+tγ\n2n)−1−ε\nis a.s. finite, which implies\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤K2(p,γ,ε,/bardblϕ/bardblp,γ)λ−γ\n2+ε\nN(1+tγ\n2n)1+ε.\nForq>1\nεmax/braceleftbigd\n2,γ\n2/bracerightbig\n, it holds that\nE[|K2(p,γ,ε,/bardblϕ/bardblp,γ)|q]\n≤∞/summationdisplay\nn=1∞/summationdisplay\nN=1E/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\ntn/integraldisplaytn\n0/bracketleftbig\nϕ(XN(t))−ϕ(X(t))/bracketrightbig\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingleq/bracketrightbigg\nλγq\n2−qε\nN(1+tγ\n2n)−q(1+ε)\n<∞.\nUsing Jensen’s inequality gives E[|K2(p,γ,ε,/bardblϕ/bardblp,γ)|q]<∞for allq≥1.\nSimilarly, by using Theorem 5.4 and the Borel–Cantelli Lemm a, one shows that there exists\na random variable K3(p,γ,ε,/bardblϕ/bardblp,γ)∈ ∩q≥1Lq(Ω) such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nk=1ϕ(XN\nk)−1\ntn/integraldisplaytn\n0ϕ(XN(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤K3(p,γ,ε,/bardblϕ/bardblp,γ)τγ+p/2\n2−ε/parenleftbig\n1+t3(γ+p\n2)\nn/parenrightbig1+ε.\nCombined the above estimates with (47) yields the desired es timate with K(ε,p,γ,/bardblϕ/bardblp,γ) :=\nmax3\ni=1Ki(ε,p,γ,/bardblϕ/bardblp,γ). /squareSTOCHASTIC WAVE EQUATION WITH DISSIPATIVE DAMPING AND FULL DISCRETIZATION 27\nReferences\n[1] Rikard Anton, David Cohen, Stig Larsson, and Xiaojie Wan g. 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Syst. , 4(4):883–903, 2005.28 MENG CAI, CHUCHU CHEN, JIALIN HONG, TAU ZHOU\nLSEC, ICMSEC, Academy of Mathematics and Systems Science, C hinese Academy of Sciences,\nBeijing 100190, China\nEmail address :mcai@lsec.cc.ac.cn\nLSEC, ICMSEC, Academy of Mathematics and Systems Science, C hinese Academy of Sciences,\nBeijing 100190, China; School of Mathematical Sciences, Un iversity of Chinese Academy of\nSciences, Beijing 100049, China\nEmail address :chenchuchu@lsec.cc.ac.cn, hjl@lsec.cc.ac.cn\nDepartment of Applied Mathematics, The Hong Kong Polytechn ic University, Hung Hom,\nKowloon, Hong Kong\nEmail address :tau.zhou@polyu.edu.hk"    },    {        "title": "2402.01159v1.Foldable_fans__cscK_surfaces_and_local_K_moduli.pdf",        "content": "arXiv:2402.01159v1  [math.AG]  2 Feb 2024FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI\nCARL TIPLER\nAbstract. We study the moduli space of constant scalar curvature K¨ ahl er\nsurfaces around the toric ones. To this aim, we introduce the class of foldable\nsurfaces : smooth toric surfaces whose lattice automorphis m group contain a\nnon trivial cyclic subgroup. We classify such surfaces and s how that they all\nadmit a constant scalar curvature K¨ ahler metric (cscK metr ic). We then study\nthe moduli space of polarised cscK surfaces around a point gi ven by a foldable\nsurface, and show that it is locally modeled on a finite quotie nt of a toric affine\nvariety with terminal singularities.\n1.Introduction\nThe construction of moduli spaces of varieties is a central problem in complex\ngeometry. Pioneered by Riemann, the case of moduli spaces of cur ves is now fairly\nwell understood. In higher dimension however, the situation becom es much more\ndelicate. Indeed, iterated blow-ups (see e.g. [25, Example 4.4]) or th e presence of\nnon-discrete automorphisms induce non-separatedness in moduli considerations. It\nis then remarkable that despite their very simple combinatorial desc ription, toric\nsurfaces are subject to those two issues, and typically correspo nd to pathological\npoints in the moduli space of surfaces. In this paper, we will show th at when\none restricts to constant scalar curvature K¨ ahler (cscK) surf aces, the moduli space\nenjoys a very nice structure near its toric points.\nOurmotivation to restrict to cscK surfacescomesfrom the Yau–T ian–Donaldson\nconjecture (YTD conjecture, see [53, 48, 13]) which predicts tha t the existence of a\nconstant scalarcurvature K¨ ahlermetric (cscK metric for short ) on a given polarised\nK¨ ahler manifold should be equivalent to (a uniform version of) K-poly stability. In\nthe Fano case, the proof of this conjecture led to the recent con struction of the\nmoduli space of K-polystable Fano varieties by Odaka and Li–Wang–X u [38, 30, 31]\n(see [48, 4, 9, 49] for the YTD conjecture and also [52, Part II] an d reference therein\nfor a historical survey on the related moduli space). Moreover, f rom Odaka’s work\n[36, 37], in the canonically polarised case, K-stability is equivalent to ha ving semi-\nlog canonical singularities, a class that was successfully introduced to construct the\nso-called KSBA moduli space (see [25] for a survey on moduli of varie ties of general\ntype). While a general algebraic construction of a moduli space for K-polystable\nvarieties seems out of reach at the moment, on the differential geo metric side,\nDervan and Naumann recently built a separated coarse moduli spac e for compact\npolarised cscK manifolds [12], extending Fujiki and Schumacher’s con struction that\ndealt with the case of discrete automorphism groups [15] (see also I noue’s work on\nthe moduli space of Fano manifolds with K¨ ahler–Ricci solitons [22]).\nHowever, it seems to the author that the Fano case, and its log or p air analogues,\nare the only sources of examples for those moduli spaces, when no n–discrete au-\ntomorphism groups are allowed (see also the discussion in [12, Examp le 4.16] that\nmay lead to further examples via moduli of polystable bundles). Rest ricting to\nsurfaces, moduli spaces of K¨ ahler–Einstein Del Pezzos were exp licitly constructed\nDate: February 5, 2024.\n12 C. TIPLER\nin [39]. Our focus will then be on the moduli space of polarised cscK sur faces, and\nits geometry around points corresponding to the toric ones. Given that they satisfy\nthe YTD conjecture by Donaldson’s work [13], toric surfaces draw a lot of atten-\ntion (see e.g. [14, 29, 27, 50, 41, 54, 2, 55, 8, 42, 1]), and they app ear as natural\ncandidates to test the general machinery on (compare also with [23 , 40, 19]). As\nthe full classification of cscK toric surfaces is yet to be carried out , we will consider\na subclass whose fans carry further symmetries, which we introdu ce now.\nDefinition 1.1. LetNbe a rank two lattice and Σ a complete smooth fan in\nNR:=N⊗ZR. The fan Σ is called foldableif its lattice automorphism group\nAut(N,Σ) contains a non-trivial cyclic subgroup. The associated toric sur faces will\nbe called foldable surfaces .\nIn orderto understand this classofsurfacesamongst the toric o nes, we show that\nall crystallographicgroups arise as lattice automorphism groups of two dimensional\nsmooth fans (see Section 3.1):\nProposition 1.2. LetNbe a rank two lattice and Σa complete smooth fan in NR.\nThenAut(N,Σ)is isomorphic to one of the groups in the following set\n{C1,C2,C3,C4,C6,D1,D2,D3,D4,D6}.\nMoreover, any group in the above list is isomorphic to the lat tice automorphism\ngroup of some complete two dimensional smooth fan.\nIn Proposition 1.2, we denote the cyclic group of order pbyCpand the dihedral\ngroup of order 2 pbyDp. We make a distinction between C2andD1by assuming\nthatC2acts via −Id onNwhileD1acts through a reflection (see Section 3.1).\nThen, from Proposition 1.2, a complete and smooth two dimensional f an Σ inNR\nis foldable if and only if Aut( N,Σ) is not isomorphic to C1orD1. The associated\nclass of toric surfaces then provides a wide class of examples of csc K surfaces (see\nProposition 3.13) :\nProposition 1.3. LetXbe a foldable surface. Then Xadmits a cscK metric in\nsome K¨ ahler class.\nThisresult relieson the classificationoffoldable surfaces(Section 3 .2) and anap-\nplication of Arezzo–Pacard–Singer and Sz´ ekelyhidi’s blow-up theor em for extremal\nK¨ ahler metrics [3, 47]. Similar arguments were used in [50] to show tha t any toric\nsurface admits an iterated toric blow-up with a cscK metric.\nRemark 1.4.To our knowledge, all known examples of toric cscK surfaces are\nfoldable (see e.g. [50] for a family of examples with unbounded Picard r ank, and\n[51] for a complete classification of polarised cscK toric surfaces up to Picard rank\n4). It would be interesting to produce examples of non foldable toric cscK surfaces.\nOur main result shows that the moduli space of polarised cscK surfa ces, which\nis known to be a complex space [15, 12], is quite well behaved around a f oldable\none, as it is locally modeled on a finite quotient of a toric terminal singula rity (see\nSection 4.3 for a definition of this classof singularitiesthat originated in the MMP).\nTheorem 1.5. Let(X,[ω])be a polarised cscK foldable surface with fan ΣinNR,\nand letG= Aut(N,Σ). Denote by [X]∈Mthe corresponding point in the moduli\nspace Mof polarised cscK surfaces. Then, there are :\n(i)A Gorenstein toric affine G-varietyWwith at worst terminal singularities,\n(ii)Open neighborhoods WandUof respectively (the image of) the torus fixed\npointx∈W⊂W/Gand of[X]∈U⊂M,\nsuch that (U,[X])and(W,x)are isomorphic as pointed complex spaces.FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 3\nThis result sheds some light on the singularities that may appear on th e moduli\nspace of polarised cscK surfaces. Note that by [5, Theorem 4], the good moduli\nspace ofsmooth K-polystable Fano manifolds offixed dimension and v olume has klt\ntype singularities (see Section 4.3 for definitions). Also, a combinatio n of Braun,\nGreb, Langlois and Moraga’s work on GIT quotients of klt type singula rities [5,\nTheorem 1], together with the constructionof Dervan and Nauman n’s moduli space\nfor cscK polarised manifolds [12, Section 3], directly implies that the mo duli space\nof polarised cscK surfaces has klt type singularities around its toric points (see\nProposition 2.3). Hence, Theorem 1.5 provides a refinement of Prop osition 2.3,\nshowing that, up to a finite covering, the toric structure is locally pr eserved on the\nmoduli space, and that the singularities are terminal, at a foldable po int. On the\nother hand, one may not expect such a nice structure at toric bou ndary points of\na (still hypothetical) compactification M. Indeed, the local structure of the K-\nmoduli space of Fano varieties was studied around singular toric var ieties in [23,\n40, 19], where much wilder phenomena were observed, mainly due to t he existence\nof obstructed deformations.\nRemark 1.6.It would be interesting to understand whether the final quotient W/G\nin Theorem 1.5still is terminal or not. Note howeverthat it is a non-tr ivialproblem\nto settle when a finite quotient ofa terminalsingularityis terminal(s ee [26, Remark\npage 161]).\nRemark 1.7.Due to cohomology vanishings for toric surfaces, the germ of the\nmoduli space of polarised cscK surfaces at a toric point does not de pend on the\nchosen cscK polarisation. See however [43] for local wall-crossing t ype phenomena\nwhen the polarisation is pushed away from the cscK locus of the K¨ ah ler cone.\nRemark 1.8.Theorem 1.5 provides another instance where canonical K¨ ahler me t-\nrics, or K-polystability, turns useful in moduli problems. We would like to mention\ntwo other approaches to produce moduli spaces for (or around) toric varieties. In\n[34], a notion of analytic stack is introduced to produce moduli space s of integrable\ncomplex structures modulo diffeomorphisms. Those spaces carry m ore information,\nas they classify a wider class of varieties, but are typically non-sepa rated. In an-\nother direction, leaving the classical setting to the non-commutat ive one, quantum\ntoric varieties admit moduli spaces which are orbifolds, see [24].\nWe proceed now to an overview of the proof of Theorem 1.5. If ( X,[ω]) is a cscK\nfoldable surface, relying on [45, 12] and an observation in [41], the de formation\ntheory of ( X,[ω]) is entirely encoded by the set of unobstructed polystable points in\nH1(X,TX) under the action of Aut( X), the latter being reductive by Matsushima\nand Lichnerowicz’s theorem [33, 32]. Nill’s study of toric varieties with r eductive\nautomorphism groups [35] then shows that either Xis isomorphic to CP2orCP1×\nCP1in which case it is rigid, or Aut0(X)≃T, whereTis the torus of X. Then, by\nIlten’s work [21], H2(X,TX) = 0, so that locally, the moduli space we are after is\ngiven by (a neighborhood of the origin in) the GIT quotient H1(X,TX)//Aut(X),\nwhere Aut( X)≃T⋊G. Thanks again to [21], the deformation theory of toric\nsurfaces is explicitly described in terms of their fan, and the affine to ric variety\nW:=H1(X,TX)//Tin Theorem 1.5 can be explicitly computed. The proof of\nTheorem 1.5 then goes as follows. By a classification result (see Lemm a 4.3),X\ncan be obtained by successive Cp-equivariant blow-ups of some “minimal” models\nin a list of 6 toric surfaces. We then use the combinatorial descriptio n of toric\nterminal singularities (see e.g. [10, Section 11.4]) to prove the result for surfaces\nin that list. Then, using convex geometry, we show that the desired properties for\nthe local moduli space are preserved after the Cp-equivariant blow-ups, and hold\naround [X].4 C. TIPLER\nRemark 1.9.In Section 4.4, we produce an example of a toric surface Xsuch that\nAut(X)≃Tand the quotient H1(X,TX)//TisnotQ-Gorenstein. We do not\nknow whether this surface carries a cscK metric. If not, the expe ctation is that the\ntheory should be extended to smooth pairs ( X,D) whereD⊂Xis a snc divisor,\nconsidering singular or Poincar´ e type cscK metrics, such as in [1].\nThe paper is organised as follows. In Section 2, we settle the necess ary back-\nground material on cscK manifolds. Then, Section 3 is devoted to th e classification\noffoldablesurfacesandtheproofsofProposition1.2andProposit ion1.3. InSection\n4, we carry over the construction of the local moduli spaces, and prove Theorem\n1.5. Finally, in Section 5.1, we provide maps between the local moduli sp aces and\nspeculate about their Weil–Petersson geometry, while in Section 5.2 w e discuss the\nhigher dimensional case.\nNotations. We will use the notations from [10]. For a toric variety X, Aut(X)\nstands for its automorphism group, and Aut0(X)⊂Aut(X) the connected compo-\nnent of the identity. We denote Tthe torus of X,Nthe lattice of its one parameter\nsubgroups, with dual lattice Mand pairing /an}bracketle{tm,u/an}bracketri}ht, for (m,u)∈M×N. The fan\nofXwill be denoted Σ (or Σ X) and letters τ,σwill be used for cones in Σ. For\n0≤j≤dim(X), Σ(j) is the set of j-dimensional cones in Σ. For K∈ {Q,R,C},\nwe letNK:=N⊗ZK, and similarly MK=M⊗ZK. Finally, XΣmay be used to\nrefer to the toric variety associated to Σ.\nAcknowledgments. The author would like to thank Ruadha´ ı Dervan and Cris-\ntiano Spotti for answeringhis questions on moduli of cscK manifolds and for several\nhelpful comments, as well as Ronan Terpereau for stimulating discu ssions on the\ntopic. TheauthorispartiallysupportedbythegrantsMARGEANR-2 1-CE40-0011\nand BRIDGES ANR–FAPESP ANR-21-CE40-0017.\n2.Background on cscK metrics and their moduli\nLet (X,Ω) be a compact K¨ ahler manifold with K¨ ahler class Ω. We will give\na very brief overview on extremal metrics and their deformations, and refer the\nreader to [18, 46] for a more comprehensive treatment.\n2.1.Extremal K¨ ahler metrics. Extremal K¨ ahler metrics were introduced by\nCalabi [6] and provide canonical representatives of K¨ ahler classe s. They are defined\nas the critical points of the so-called Calabi functional, that assign to each K¨ ahler\nmetric in Ω the L2-norm of its scalar curvature. They include as special cases of\ninterest cscK metrics and thus K¨ ahler–Einstein metrics. The follow ing obstruction\nto the existence of a cscK metric is due to Matsushima and Lichnerow icz [33, 32].\nTheorem 2.1 ([33, 32]) .Assume that Xcarries a cscK metric. Then the auto-\nmorphism group of Xis reductive.\nLater on, Futaki discovered another obstruction to the existen ce of a cscK met-\nric on (X,Ω) : the vanishing of the so-called Futaki invariant [16, 17]. More-\nover, from [7], an extremal K¨ ahler metric in the K¨ ahler class Ω is cs cK precisely\nwhen the Futaki invariant Fut Ωvanishes. Together with Arezzo–Pacard–Singer\nand Sz´ ekelyhidi’s results on blow-ups of extremal K¨ ahler metrics, we deduce the\nfollowing existence result for toric surfaces :\nTheorem 2.2 ([3, 47]).Let(X,Ω)be a smooth polarised toric surface with cscK\nmetricω∈Ω. LetZ⊂Xbe a set of torus fixed points, and G⊂Aut(X)a finite\nsubgroup such that :\n(i)The setZisG-invariant,\n(ii)The class ΩisG-invariant,FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 5\n(iii)The adjoint action of GonLie(Aut( X))has no fixed point but zero.\nThen, there is ε0>0such that (BlZ(X),Ωε)carries a cscK metric in the class\nΩε:=π∗Ω−εc1(E)forε∈(0,ε0), whereπ: BlZ(X)→Xstands for the blow-\ndown map and E=/summationtext\nz∈ZEzis the exceptional locus of π.\nProof.From [3, 47], there is ε0such that (Bl Z(X),Ωε) admits an extremal K¨ ahler\nmetric for ε∈(0,ε0). We only need to show the vanishing of the Futaki invariant\nFutΩε. This will follow from its equivariance, as already used in [44, Propositio n\n2.1]. Denote ˜X= BlZ(X). By (i) and (ii),Glifts to a subgroup of Aut( ˜X), such\nthat Ω εisG-invariant. Then, by equivariance of Fut Ωε∈(Lie(Aut( ˜X)))∗(see e.g.\n[17, Chapter 3] or [28, Section 3.1]), together with ( iii), we deduce Fut Ωε= 0,\nwhich concludes the proof. /square\n2.2.Local moduli of polarised cscK manifolds. A moduli space Mfor po-\nlarised cscK manifolds has been constructed in [12], generalising the r esults in [15]\nby taking care of non-discrete automorphisms by mean of [45, 22]. T his space\nis Hausdorff and endowed with a complex space structure. It is a coa rse moduli\nspace in the following sense : its points are in bijective corresponden ce with iso-\nmorphism classes of polarised cscK manifolds and for any complex ana lytic family\n(X,L)→Sof polarised cscK manifolds ( Lstands for the relative polarisation\nofX→S), there is an induced map Sred→Mwhere Sredis the reduction of\nS. In what follows, we will restrict to the connected components cor responding to\nthe moduli space of polarised cscK surfaces, still denoted by M. Roughly, Mis\nconstructed by patching together local moduli spaces ( MX)X∈Sparameterised by\nS, the set of polarised cscK surfaces, each MXbeing obtained as some open set in\nan analytic GIT quotient WX//G. Our case of interest is when Xis a cscK toric\nsurface. We then extract from [12] the following proposition :\nProposition 2.3. Let(X,Ω)be a polarised cscK toric surface. Then the local\nmoduli space MXis given by an open neighborhood of π(0)in the GIT quotient\nπ:H1(X,TX)→H1(X,TX)//Aut(X).\nIn particular, its singularities are of klt type.\nBy GIT quotient in this affine setting, we mean the good categorical q uotient\nH1(X,TX)//Aut(X) := Spec( RAut(X)),\nwhereRstands for the ring of regular functions on H1(X,TX). We refer to Section\n4.3 for the definition of klt type singularities.\nProof.From [12, Section 3], MXis given by an open neighborhood of 0 in the ana-\nlytic GIT quotient WX//G, whereGstands forthe complexificationofthe isometry\ngroupofthecscKmetricinΩ, andwhere WX⊂˜H1isthesmallest G-invariantStein\nspace containingsmall integrableinfinitesimal polarised deformation sof (X,Ω) (see\n[45] for the constructionof the space ˜H1). As noticed in [41, Lemma 2.10], by Bott–\nSteenbrink–Danilov’s vanishing of h0,1(X) andh0,2(X) (see [10, Theorem 9.3.2]),\nthe vector space ˜H1isG-equivariantly isomorphic to H1(X,TX). Also, for a toric\nsurface, Aut( X) is a linear algebraic group, so that G≃Aut(X), as the Lie algebra\nofAut(X) hasnoparallelvectorfield (see[18, Chapter2]). Finally, H2(X,TX) = 0\nby [21, Corollary 1.5], so that any small element of H1(X,TX) is integrable. Alto-\ngether, we obtain the first part of Proposition 2.3. The statement about klt type\nsingularities follows directly from [5, Theorem 1]. /square6 C. TIPLER\n3.Foldable toric surfaces\nLetNbe a rank two lattice and Σ be a fan in NR, with associated toric surface\nX. We will assumeΣ to be smooth, that iseachcone σ∈Σ isgeneratedbyelements\ninNthat form part of a Z-basis, and to be complete, i.e.\n/uniondisplay\nσ∈Σσ=NR.\nHence,Xis a smooth and compact toric surface.\n3.1.Lattice automorphisms and fans. We will be interested in automorphism\ngroups of smooth toric surfaces. Denote by T:=N⊗ZC∗the torus of X, and\nby Aut(N,Σ) the group of lattice automorphisms of Σ. Recall that Aut( N,Σ) is\nthe subgroup of GL Z(N) consisting in elements g∈GLZ(N) such that the induced\nisomorphisms g∈GLR(NR) send bijectively Σ to Σ. By a result of Demazure [11,\nProposition 11], the automorphism group Aut( X) ofXis a linear algebraic group\nisomorphic to\nAut0(X)⋊G,\nwhereGis a quotient of Aut( N,Σ). We will later on be interested in the GIT\nquotient of H1(X,TX) by Aut( X), assuming Xto admit a cscK metric. A combi-\nnation of Matsushima and Lichnerowicz’s obstruction together with Nill’s work on\ntoric varieties with reductive automorphism group [35] implies :\nProposition 3.1. Assume that Xadmits a cscK metric. Then either\nX∈ {P2,P1×P1},\nor\nAut(X)≃T⋊Aut(N,Σ).\nProof.By Matsushima and Lichnerowicz theorem [33, 32], Aut( X) is reductive.\nThe result then follows from Nill’s result [35, Theorem 1.8] together wit h De-\nmazure’s structure theorem [11, Proposition 11]. /square\nAsP2andP1×P1are rigid, we will focus now on the case\nAut(X)≃T⋊Aut(N,Σ),\nandcharacterisethe possiblefinite groupsthatariseaslatticeaut omorphismgroups\nof rank two complete fans.\nLemma 3.2. Letg∈Aut(N,Σ), and denote by m∈Nthe order of g. Then\nm∈ {1,2,3,4,6}.\nMoreover, the complex linear extension of gtoNCis conjugated (in GL(NC)≃\nGL2(C)) to the following :\n(1)Ifm= 1, theng∼Id,\n(2)Ifm= 2anddet(g) = 1, theng∼ −Id,\n(3)Ifm= 2anddet(g) =−1, theng∼/bracketleftbigg1 0\n0−1/bracketrightbigg\n,\n(4)Ifm= 3, theng∼/bracketleftbiggj0\n0j2/bracketrightbigg\n,\n(5)Ifm= 4, theng∼/bracketleftbigg\ni0\n0−i/bracketrightbigg\n,\n(6)Ifm= 6, theng∼/bracketleftbigg−j0\n0−j2/bracketrightbigg\n,\nwhere we denoted by j=ei2π\n3and by∼the equivalence relation given by conjuga-\ntion, and where we used an isomorphism NC≃C2.FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 7\nThe proof is an elementary exercise in linear algebra. We include it for c onve-\nnience of the reader.\nProof.Fix and isomorphism N≃Z2and identify gwith an element of GL 2(Z).\nThe characteristic polynomial of\ng=/bracketleftbigga b\nc d/bracketrightbigg\nthen reads\nχ(Y) =Y2−(a+d)Y+(ad−bc).\nOn the other hand, Jordan’s normal form for the C-linear extension of gimplies\nthat it is conjugated (in M 2(C)) to\n/bracketleftbiggα δ\n0β/bracketrightbigg\nfor (α,β)∈C2andδ∈ {0,1}. Asgis both invertible and of finite order, we deduce\nthatδ= 0 and gis conjugated to\n/bracketleftbigg\nα0\n0β/bracketrightbigg\n.\nMoreover, αm=βm= 1, so that αandβarem-th roots of unity in C. On the\nother hand, by conjugation invariance, we also have\ndet(g) =αβ=ad−bc∈ {−1,+1}\nasg∈GL2(Z) and\ntrace(g) =α+β=a+d∈Z.\nIfαandβbelong to {−1,+1}, that is if m= 2, then we are done. If not, at least\none ofαorβis not real, and thus αandβmust be complex conjugated roots of\nχ(Y). Hence\ntrace(g) =α+α∈Z,\nbut also\n|α+α|<2\nso that\nα+α∈ {−1,0,1}.\nHence\nα∈ {±i,±j,±j2},\nfrom which the result follows easily. /square\nRecall that we denote by Cpthe cyclic group of order pand byDpthe dihedral\ngroup of order p. We then have :\nProposition 3.3. The group Aut(N,Σ)is isomorphic to one of the groups in the\nfollowing set\n{C1,C2,C3,C4,C6,D1,D2,D3,D4,D6}.\nProof.As Aut(N,Σ) is finite, we can fix an Aut( N,Σ)-invariant euclidean metric\nonNR(again considering the R-linear extension of Aut( N,Σ)). Then, by the classi-\nfication of finite subgroups of the group of orthogonal transfor mations of the plane,\nwe deduce that Aut( N,Σ) is isomorphic to CporDp, for some p∈N∗. As those\ngroups admit elements of order p, by Lemma 3.2, the result follows. /square\nRemark 3.4.This proposition simply recovers the well known classification of crys -\ntallographic groups in dimension 2.8 C. TIPLER\nWe will now show that all the above listed groups arise as lattice autom orphism\ngroupsofranktwocompletesmoothfans. Notethat C2andD1willbedistinguished\nby the fact that their generator is −Id in the first case and a reflection in the\nsecond case. Consider then the following fans in Z2, where we denote by {e1,e2}\nthe standard basis of Z2.\nExample 3.5. LetFnbe then-th Hirzebruch surface, that is the total space of\nthe fibration\nP(OP1⊕OP1(n))→CP1.\nNote that F0=CP1×CP1. Then, up toisomorphism, the fan Σ FnofFnis described\nby\nΣFn(1) ={R+·(0,−1),R+·(1,0),R+·(0,1),R+·(−1,−n)}.\nAs an example, F2admits the fan description of Figure 1, that we will denote by\nΣ′\n1. In this figure, and the one that follow, the dots represent the lat tice points,\nand we only represent the ray generators of the complete two dime nsional fan.\n• • • • •\n• • •e2/d79/d79\ne1/d47/d47\n/d7/d7✎✎✎✎✎✎✎✎✎✎✎✎\n/d15/d15• •\n• • • • •\n• • • • •\nFigure 1. Fan Σ′\n1ofF2\nExample 3.6. Here are the fans Σ P2= Σ′\n3and Σ P1×P1= Σ′\n4ofP2andP1×P1\nrespectively (the numbering of the fans is motivated by Proposition 3.8 below).\n• • • • •\n• • •e2/d79/d79\ne1/d47/d47\n/d127/d127⑦⑦⑦⑦⑦⑦⑦• •\n• • • • •\nFigure 2. Fan Σ′\n3ofP2\n• • • • •\n• • •e2/d79/d79\ne1/d47/d47 /d111/d111\n/d15/d15• •\n• • • • •\nFigure 3. Fan Σ′\n4ofCP1×CP1FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 9\nExample 3.7. Recall that blowing-up a smooth toric surface along a torus fixed\npoint produces a new smooth toric surface. By the orbit cone corr espondence, such\na fixed point corresponds to a two dimensional cone\nσ=R+·ei+R+·ei+1\nin the fan of the blown-up surface, and the fan of the resulting sur face is obtained\nby adding the ray generated by ei+ei+1to the set of rays of the initial surface (see\n[10, Chapter 3, Section 3]). By iterated blow-ups, we obtain the follo wing fans.\n• • • • •\n• • •/d79/d79\n/d47/d47 /d111/d111\n/d15/d15/d63/d63⑦⑦⑦⑦⑦⑦⑦\n/d127/d127⑦⑦⑦⑦⑦⑦⑦/d55/d55♦♦♦♦♦♦♦♦♦♦♦♦\n/d119/d119♦♦♦♦♦♦♦♦♦♦♦♦• •\n• • • • •\nFigure 4. Fan Σ′\n2: iterated blow-up of P1×P1\nThe following is a two points blow-up of P1×P1and at the same time a three\npoints blow-up of CP2.\n• • • • •\n• • •/d79/d79\n/d47/d47 /d111/d111\n/d15/d15/d63/d63⑦⑦⑦⑦⑦⑦⑦\n/d127/d127⑦⑦⑦⑦⑦⑦⑦• •\n• • • • •\nFigure 5. Fan Σ′\n6: blow-up of P2along its three fixed points.\nHere are further examples to complete our classification of lattice a utomorphism\ngroups of rank two complete smooth fans. The first one is a single blo w-up ofF2,\nthat cancels the symmetry of Σ′\n1= ΣF2.\n• • • • •\n• • •e2/d79/d79\ne1/d47/d47\n/d7/d7✎✎✎✎✎✎✎✎✎✎✎✎\n/d15/d15\n/d10/d10✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔• •\n• • • • •\n• • • • •\n• • • • •\nFigure 6. Fan Σ 1: one-point blow-up of F210 C. TIPLER\nThe next one is obtained from Σ′\n2by blowing-up in a C2-equivariant way (recall\nthatC2=/an}bracketle{t−Id/an}bracketri}ht).\n• • • • • • •\n• • • • • • •\n• • • •/d79/d79\n/d47/d47 /d111/d111\n/d15/d15/d63/d63⑦⑦⑦⑦⑦⑦⑦\n/d127/d127⑦⑦⑦⑦⑦⑦⑦/d55/d55♦♦♦♦♦♦♦♦♦♦♦♦/d57/d57tttttttttttttttttttt\n/d119/d119♦♦♦♦♦♦♦♦♦♦♦♦\n/d121/d121tttttttttttttttttttt• • •\n• • • • • • •\n• • • • • • •\nFigure 7. Fan Σ 2.\nThe third one is obtained from the fan of P2by three successive blow-ups of\nthree distinct fixed points, in a symmetric way.\n• • • • • •\n• • • • • •\n• • •/d79/d79\n/d47/d47/d63/d63⑦⑦⑦⑦⑦⑦⑦/d55/d55♦♦♦♦♦♦♦♦♦♦♦♦/d52/d52✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ /d111/d111/d95/d95❅❅❅❅❅❅❅/d87/d87✴✴✴✴✴✴✴✴✴✴✴✴\n/d4/d4✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠\n/d7/d7✎✎✎✎✎✎✎✎✎✎✎✎\n/d127/d127⑦⑦⑦⑦⑦⑦⑦\n/d15/d15• • •\n• • • • • •\n• • • • • •\n• • • • • •\nFigure 8. Fan Σ 3\nThe next one is obtained by two successive blow-ups of P1×P1at four fixed\npoints.\n• • • • •\n• • • • •\n• • •/d79/d79\n/d47/d47/d63/d63⑦⑦⑦⑦⑦⑦⑦/d55/d55♦♦♦♦♦♦♦♦♦♦♦♦ /d111/d111/d95/d95❅❅❅❅❅❅❅/d87/d87✴✴✴✴✴✴✴✴✴✴✴✴\n/d119/d119♦♦♦♦♦♦♦♦♦♦♦♦\n/d127/d127⑦⑦⑦⑦⑦⑦⑦\n/d15/d15 /d31/d31❅❅❅❅❅❅❅\n/d23/d23✴✴✴✴✴✴✴✴✴✴✴✴• •\n• • • • •\n• • • • •\nFigure 9. Fan Σ 4FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 11\nThe last one is obtained from Σ′\n6by two successive blow-ups at six fixed points.\n• • • • • • •\n• • • • • • •\n• • • • • • •\n• • • •/d79/d79\n/d47/d47 /d111/d111\n/d15/d15/d63/d63⑦⑦⑦⑦⑦⑦⑦\n/d127/d127⑦⑦⑦⑦⑦⑦⑦/d55/d55♦♦♦♦♦♦♦♦♦♦♦♦/d52/d52✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐/d71/d71✎✎✎✎✎✎✎✎✎✎✎✎/d68/d68✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠/d87/d87✴✴✴✴✴✴✴✴✴✴✴✴/d95/d95❅❅❅❅❅❅❅\n/d119/d119♦♦♦♦♦♦♦♦♦♦♦♦\n/d116/d116✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐\n/d7/d7✎✎✎✎✎✎✎✎✎✎✎✎\n/d4/d4✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠\n/d31/d31❅❅❅❅❅❅❅\n/d23/d23✴✴✴✴✴✴✴✴✴✴✴✴• • •\n• • • • • • •\n• • • • • • •\n• • • • • • •\nFigure 10. Fan Σ 6.\nProposition 3.8. Letp∈ {1,2,3,4,6}. Then we have\nAut(Z2,Σp)≃Cp\nand\nAut(Z2,Σ′\np)≃Dp.\nProof.The proof is straightforward, although a bit tedious, so we will only g ive\na sketch of it. Fix an orientation of R2, and pick a two dimensional cone σof\nΣp(the proof goes the same for Σ′\np). Letg∈Aut(Z2,Σp). Then g·σ∈Σp(2).\nMoreover, gmaps the two facets of σto the two facets of g·σ. As the fan Σ pis\nexplicitly described, for each possible τ=g·σ∈Σp(2), we deduce the possible\nmatrix coefficients for g:\ng=/bracketleftbiggaτbτ\ncτdτ/bracketrightbigg\n∈GL2(Z).\nFor any possible τ, one can then explicitly compute the images of the rays of Σ pby\ng. They should all belong to Σ p(1), and this enables to select the allowed τ’s forg\nto be a lattice automorphism of Σ p. The result then follows easily. /square\nRemark 3.9.One can also show similarly that for any Hirzebruch surface Fnwith\nn≥1, and with fan Σ Fn, one has Aut( N,ΣFn)≃D1.\n3.2.Classification of foldable surfaces. We first recall the definition :\nDefinition 3.10. We will say that a smooth and complete two dimensional fan\nisfoldableif its lattice automorphism group contains a non-trivial cyclic group. A\nfoldable surface is a toric surface whose fan is foldable.\nWe now proceed to a partial classification of foldable toric surfaces . Notice\nthat ifG⊂Aut(N,Σ), then we can find a G-action on X, faithfull as soon as\nAut(X)≃T⋊Aut(N,Σ).\nDefinition 3.11. AG-equivariant blow-up of Xis a blow-up of Xalong aG-\norbit of torus fixed points. We will say that a smooth toric surface ˜Xis obtained12 C. TIPLER\nfromXby successive G-equivariant blow-ups if there is a sequence of G-equivariant\nblow-ups :\n˜X=Xk→Xk−1→...→X1→X0=X.\nDenote by Bl p1,p2,p3(P2) the toric surface associated to the fan of Figure 5, that\nis the blow-up of P2at its three torus fixed points.\nProposition 3.12. LetXbe a foldable surface with fan Σ. Assume that Aut(N,Σ)\ncontains a subgroup isomorphic to Cp,p≥2.\n•Ifp∈ {2,4}, thenXis obtained from P1×P1by successive Cp-equivariant\nblow-ups,\n•Ifp= 3, thenXis obtained from P2by successive Cp-equivariant blow-ups,\n•Ifp= 6, thenXis obtained from Blp1,p2,p3(P2)by successive Cp-equivariant\nblow-ups.\nProof.Through its representation\nCp→GLR(NR),\nthe subgroup Cpof Aut(N,Σ) acts on NRby rotations (once an invariant euclidean\nmetric is fixed), hence has no fixed points. We deduce that the actio n ofCpon\nΣ(1) is free, and then pdivides|Σ(1)|, the number of rays of Σ. If |Σ(1)|=p, then\nfrom the classification of toric surfaces (see e.g. [10, Chapter 10]) we must have\np∈ {3,4,6}.\nIn that situation, if p= 3, then X≃P2. Ifp= 4, then X≃Fnfor some n∈N.\nBy Remark 3.9, we see that X≃P1×P1. We will see shortly that if p= 6, then\nX≃Blp1,p2,p3(P2).\nSowemayassumethat |Σ(1)| ≥5,andthat pdivides|Σ(1)|. Bytheclassification\noftoricsurfaces, weknowthat Xisobtainedbysuccessiveblow-upsfrom Fn,n≥0,\nor fromP2. Then, there is a ray ρiin Σ(1) generated by the sum of two primitive\nelements\nui=ui−1+ui+1\ngenerating adjacent rays ρi−1andρi+1in Σ(1) (that is both ρi+ρi−1andρi+ρi+1\nbelong to σ(2)). By linearity, for any g∈Cpwe have\ng·ui=g·ui−1+g·ui+1,\nhence we can contract the Cp-orbit of the torus invariant divisor Dρiassociated to\nρiand obtain a smooth toric surface whose fan still admits Cpas a subgroup of its\nlattice symmetry group. By induction, we obtain the result. /square\nFrom the classification, we obtain the following proposition.\nProposition 3.13. LetXbe a foldable surface. Then Xadmits a cscK metric.\nProof.This is simply an induction on the number of Cp-equivariant blow-ups in\nProposition 3.12, using Theorem 2.2 at each step, and the fact that P1×P1,P2\nand Bl p1,p2,p3(P2) admit cscK metrics in Cp-equivariant classes (for Bl p1,p2,p3(P2),\nit follows again from Theorem 2.2 applied to P2). /square\n4.The local cscK moduli around a foldable surface\nLetXbe a smooth and complete toric surface with fan Σ and one paramete r\nsubgroups lattice N. We will denote by ( ui)0≤i≤rthe primitive elements of Nthat\ngenerate the rays\nρi=R+·ui∈Σ(1),\nlabeled in the counterclockwise order, so that for each i, (ui,ui+1) is a positively\noriented Z-basis of N, andρi+ρi+1∈Σ(2), with the convention u0=ur.FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 13\n4.1.Deformation theory of toric surfaces. The action of the torus TonX\nnaturally induces a representation of Ton\nV:=H1(X,TX).\nWe will denote by\nV=/circleplusdisplay\nm∈MVm\ntheassociatedweightspacedecomposition. From[21, Corollary1.5], foreachweight\nm∈M= Hom Z(N,Z) :\n(1) dim( Vm) =♯/braceleftbigg\nρi∈Σ(1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}bracketle{tm,ui/an}bracketri}ht=−1\n/an}bracketle{tm,ui±1/an}bracketri}ht<0/bracerightbigg\n.\nExample 4.1. We will provide explicit examples of weight space decompositions\nforV=H1(X,TX), whenXis the smooth toric surface associated to a foldable\nfan. First, P1×P1,P2and Bl p1,p2,p3(P2) are rigid, as can be checked directly using\nFormula (1). In those cases, V= 0. Then, denote by Y2the foldable toric surface\nassociated to the fan Σ′\n2of Figure 4. We also introduce the foldable toric surfaces\nY4andY3associated to the following fans :\n• • • • •\n• • • • •\n• • •/d79/d79\n/d47/d47 /d111/d111\n/d15/d15/d63/d63⑦⑦⑦⑦⑦⑦⑦\n/d127/d127⑦⑦⑦⑦⑦⑦⑦/d95/d95❅❅❅❅❅❅❅\n/d31/d31❅❅❅❅❅❅❅• •\n• • • • •\n• • • • •\nFigure 11. Fan ofY4.\n• • • • • •\n• • • • • •\n• • •/d79/d79\n/d47/d47/d63/d63⑦⑦⑦⑦⑦⑦⑦/d55/d55♦♦♦♦♦♦♦♦♦♦♦♦ /d111/d111/d95/d95❅❅❅❅❅❅❅\n/d7/d7✎✎✎✎✎✎✎✎✎✎✎✎\n/d127/d127⑦⑦⑦⑦⑦⑦⑦\n/d15/d15• • •\n• • • • • •\n• • • • • •\n• • • • • •\nFigure 12. Fan ofY314 C. TIPLER\nSetVi=H1(Yi,TYi). Testing the conditions in Formula (1) for each ray gener-\nator gives\n(2)V2=V1\n+e∗\n2⊕V1\n−e∗\n2⊕V1\n−e∗\n1+e∗\n2⊕V1\ne∗\n1−e∗\n2\nV3= V2\n−e∗\n2⊕V2\ne∗\n1⊕V2\n−e∗\n1+e∗\n2\nV4=V1\n+e∗\n1⊕V1\n−e∗\n1⊕V1\n+e∗\n2⊕V1\n−e∗\n2\nwhereVj\nmstands for a j-dimensional weight mrepresentation of T.\nRemark 4.2.One can check by unraveling the isomorphisms used in [21, Section 1]\nthat the C3-action on Y3induces an action on the coordinates χm\njofV3, with two\norbits given by\n{χ−e∗\n2\nj,χe∗\n1\nj,χ−e∗\n1+e∗\n2\nj}, j∈ {1,2}\nwhereχm\nj, for 1≤j≤dim(Vm), stand for generators of the weight mspaceVm.\nSimilarly, the D2(resp.D4) action on Y2(resp.Y4) induces a transitive action on\nthe coordinates of V2(resp.V4).\n4.2.The toric GIT quotient. We will assume from now on that Xcarries a\ncscK metric, so by Proposition 3.1, we have\nAut(X)≃T⋊Aut(N,Σ).\nWe will set\nG:= Aut(N,Σ).\nFrom Proposition 2.3, the local moduli space MXof polarised cscK surfaces around\n[X]∈Misgivenbyaneighborhoodofthe origininthe GITquotientof VbyT⋊G.\nAssume now that Xis foldable. To prove Theorem 1.5, it is then enough to show\nthat the GIT (or categorical) quotient\nW:=V//T\nis toric, Gorenstein and terminal. We will end this section by showing th atWis\nindeed an affine toric variety. We first need the following lemma, where byCp⊂G\nwe mean that there is an injection of CpinG.\nLemma 4.3. One of the following holds :\n•IfXis rigid, then X∈ {P2,P1×P1,Blp1,p2,p3(P2)}.\n•IfXis not rigid and C2⊂G,X∈ {Y2,Y4}or is obtained from Y2orY4\nby successive C2-equivariant blow-ups.\n•IfXis not rigid and C3⊂G,X=Y3or is obtained from Y3by successive\nC3-equivariant blow-ups.\nRecall that Y2,Y3andY4were defined in Example 4.1.\nProof.Note that by Example 4.1, P2,P1×P1and Bl p1,p2,p3(P2) are rigid. If X\nis not one of those three foldable surfaces, then from Proposition 3.12, either it\nbelongs to {Y2,Y3,Y4}, or is obtained from Yjby successive Cp-equivariant blow-\nups, where p= 2 forj∈ {2,4}andp= 3 forj= 3. From [21, Corollary 1.6],\nH1(Yj,TYj) injects in H1(X,TX). Hence from the computations of Example 4.1,\nwe see that Xis not rigid, and the result follows. /square\nProposition 4.4. The GIT quotient Winherits the structure of a normal toric\naffine variety.FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 15\nAlthoughtheproofisstraightforward,weshouldwarnthe reader that oursetting\nis slightly different from the standard quotient construction of tor ic varieties (see\ne.g. [10, Chapter 5]), as the weight spaces Vmmay have dimension greater than\n1, and the quotient map ˜ σ→σ(see the proof below) may send several rays to\nthe same one. This detailed proof will also settle the necessary nota tions for the\nfollowing section.\nProof.IfV= 0, then there is nothing to prove. We then assume that Xis not\nrigid. From Lemma 4.3, Xis obtained from Yjby successive Cp-equivariant blow-\nups, where j∈ {2,3,4}, for suitable p∈ {2,3}.\nDenote by d= dim(V) and by ˜N≃Zdthe lattice of one parameter subgroups\nof˜T≃(C∗)dacting on V≃Cdby multiplication on each coordinate :\n∀(t,x)∈˜T×V, t·x= (tixi)1≤i≤d.\nTheT-module structure of Vis then equivalent to the injection T→˜T, or the\nlattice monomorphism\nB:N→˜N\ndefined by\nB(u) = (/an}bracketle{tmi,u/an}bracketri}ht)1≤i≤d\nwhere the mi’s run through all the weights in the weight space decomposition of V\n(with multiplicities when dim( Vm)≥2). The fact that Bis injective comes again\nfrom the injection Vj⊂V(cf [21, Corollary 1.6]) and the explicit computation of\nVjin Example 4.1. We then consider the quotient\nN′:=˜N/N,\nwhere we identify NwithB(N) by abuse of notation. We claim that N′is again a\nlattice, that is Nis saturated in N′. IfX=Yj, forj∈ {2,3,4}, then this can be\nchecked directly using the description of Vjin Example 4.1. If Xis a blow-up of\nYj, thenVj⊂V, so that Bcan be written B= (Bj,B+), where\nBj:N→˜Nj\nis the injection corresponding to the weight space decomposition of Vj, with\n˜Nj≃Zdj,\nfordj= dim(Vj). Then, the fact that Bj(N) is saturated in ˜Nj⊂˜Nimplies that\nB(N) is saturated in ˜N. Thus, we have a short exact sequence of lattices\n(3) 0 −→NB−→˜NA−→N′−→0\nwhere\nA:˜N→˜N/N\ndenote the quotient map.\nIntroduce (˜ ei)1≤i≤dthe basis of Vdual to the coordinates χmiof the weight\nspacesVmi. It corresponds to a Z-basis of ˜N, still denoted (˜ ei)1≤i≤d. The cone\n˜σ=d/summationdisplay\ni=1R+·˜ei⊂˜NR\nsatisfies\nSpec(C[˜σ∨∩˜M]) =V,\nwhere\n˜M= Hom Z(˜N,Z).16 C. TIPLER\nConsider the cone of N′\nR:\nσ=A(˜σ) =d/summationdisplay\nj=iR+·A(˜ei)\n(we will still denote by AandBtheirR-linear extensions). Our goal is then to\nshow that Wis isomorphic to the affine toric variety defined by σ. We first need\nto prove that σhas the required properties to define such a variety. The cone σ\nclearly is rational, convex and polyhedral. We now prove that σis strictly convex,\nthat is\n−σ∩σ={0}.\nSo letv∈ −σ∩σ. Then there is ( u+,u−)∈(˜σ)2such that A(u±) =±v. We\ndeduce that\nu++u−∈ker(A)∩˜σ= Im(B)∩˜σ.\nWe will then show that\n(4) Im( B)∩˜σ={0},\nwhich implies u+=u−= 0 by strict convexity of ˜ σ. Notice that it is enough to\nshow (4) for X=Yj. Indeed, using the decomposition B= (Bj,B+) as before, if\nu=B(x)∈˜σ,\nthen\nuj=Bj(x)∈˜σj,\nwhere ˜σj⊂(˜Nj)Ris the cone corresponding to Vj. If\n(5) Im( Bj)∩˜σj={0},\nby injectivity of Bj, we deduce x= 0 and then u= 0. Remains to prove (5), which\nfollows again from the explicit descrition of the weights of the Taction on Vj. For\nexample, if j= 2, and if\nB2(x1,x2) = (x2,−x2,−x1+x2,x1−x2)∈˜σ2,\nthen by definition of ˜ σ2we deduce\n\n\nx2≥0\n−x2≥0\n−x1+x2≥0\nx1−x2≥0\nand thus x1=x2= 0. The cases j∈ {3,4}are similar. We just proved that σis a\nstrongly convex rational polyhedral cone, and thus defines an affi ne toric variety.\nWe now claim that\nW≃Spec(C[σ∨∩M′])\nwith\nM′= Hom Z(N′,Z).\nIt is equivalent to show\nC[σ∨∩M′]≃C[χmi]T.\nThe latter equality follows easily from the definitions. Indeed, consid ering the dual\nsequence to (3),\n(6) 0 −→M′A∗\n−→˜MB∗\n−→M−→0FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 17\nform′∈M′, if ˜m=A∗(m′), then\nχm′∈σ∨⇐⇒ ∀i∈[[1,d]],/an}bracketle{tm′,A(˜ei)/an}bracketri}ht ≥0\n⇐⇒ ∀i∈[[1,d]],/an}bracketle{tA∗(m′),˜ei/an}bracketri}ht ≥0\n⇐⇒/braceleftbigg\nB∗(˜m) = 0\n∀i∈[[1,d]],/an}bracketle{t˜m,˜ei/an}bracketri}ht ≥0\n⇐⇒ χ˜m∈C[χmi]T.\nThis proves the claim, and the result follows. /square\n4.3.Singularities. We keep the notations from the previous section (see in par-\nticular the proof of Proposition 4.4). Our goal here is to conclude th e proof of\nTheorem 1.5, by showing that Wis Gorenstein and terminal. Let us first recall the\ndefinitions of those notions, and their combinatorial characterisa tion in the toric\ncase.\nDefinition 4.5. LetZbe a normal toric variety1. ThenZisGorenstein if the\ncanonical divisor KZis Cartier ( Q-Gorenstein ifKZisQ-Cartier). In that case, Z\nhasterminal singularities if there is a resolution of singularities π:˜Z→Zsuch\nthat if we set\nK˜Z=π∗KZ+/summationdisplay\niaiEi,\nwhere the Ei’s are distinct irreducible divisors, then for all i, we have ai>0.\nOne can check that the above definition doesn’t depend on the choic e of resolu-\ntion. Terminal singularities play an important role in the minimal model p rogram,\nbeing the singularities of minimal models. Their logarithmic version turn ed out\nvery useful as well. Recall that a log pairis a normal variety Ztogether with an\neffective Q-divisorDwith coefficients in [0 ,1]∩Q. Alog resolution for a log pair\n(Z,D) is a resolution of singularities π:˜Z→Zsuch that the exceptional locus\nExc(π) ofπis of pure codimension 1 and the divisor π−1\n∗D+ Exc(π) has simple\nnormal crossing.\nDefinition 4.6. Let (Z,D) be a log pair such that KZ+DisQ-Cartier (the pair\nis then called Q-Gorenstein ). In that case, ( Z,D) hasklt singularities if there is a\nlog-resolution π:˜Z→Zsuch that if we set\nK˜Z=π∗(KZ+D)+/summationdisplay\niaiEi,\nwhere the Ei’s are distinct irreducible divisors, then for all i, we have ai>−1. Fol-\nlowing[5], wewill saythat anormalvariety Zis ofklt typeifthere existsan effective\nQ-divisorDsuch that ( Z,D) is aQ-Gorenstein log pair with klt singularities.\nAt that stage, from [5, Theorem 1], we know that Wis of klt type. Also, by\n[10, Corollary 11.4.25 ], if Wis Gorenstein, then it has log terminal singularities,\nmeaning that the pair ( W,0) is klt. We will give a direct proof that it is actually\nGorenstein with terminal singularities, using the following character isation from\n[10, Proposition 11.4.12] (where we use the notation uρfor the primitive generator\nof a rayρinσ(1)).\n1Being normal is not required in the most general definition of Gorenstein singularities. However,\nwe will only deal with normal toric varieties, that are ratio nal [10, Theorem 11.4.2], hence Cohen–\nMacaulay, which is usually required to define Gorenstein sin gularities.18 C. TIPLER\nProposition 4.7. Consider W= Spec(C[σ∨∩M′])the affine toric variety asso-\nciated to the rational strictly convex polyhedral cone σ. Then\n•Wis Gorenstein if and only if there exists m∈M′such that for all ρ∈\nσ(1),/an}bracketle{tm,uρ/an}bracketri}ht= 1.\n•In that case, Whas terminal singularities if and only if the only lattice\npoints in\nΠσ:=\n\n/summationdisplay\nρ∈σ(1)λρuρ|/summationdisplay\nρ∈σ(1)λρ≤1,0≤λρ≤1\n\n\nare given by its vertices.\nFrom the discussion in Section 4.2, the following proposition concludes the proof\nof Theorem 1.5.\nProposition 4.8. The affine toric variety Wis Gorenstein and has terminal sin-\ngularities.\nProof.We will exclude the rigid cases, and as in the proof of Proposition 4.4,\nassume that Xis obtained from Yjby successive Cp-equivariant blow-ups, where\nj∈ {2,3,4}, for suitable p∈ {2,3}.\nWe use the characterisation in Proposition 4.7, and first need to des cribe the\nrays ofσand their primitive generators. Note that by construction, the ra ys of\nσbelong to the set {R+·A(˜ei),1≤i≤d}. Letρi=R+·A(˜ei) be such a ray.\nWe claim that A(˜ei) is primitive in N′. The argument is similar to the one used\nto prove strict convexity in the proof of Proposition 4.4. Suppose b y contradiction\nthat there is a∈N,a≥2, and ˜e∈˜σ∩˜Nsuch that A(˜ei) =aA(˜e). Notice that\nthere isx∈Nsuch that\n(7) B(x) = (Bj(x),B+(x)) = ˜ei−a˜e.\nBy injectivity of Bj, ifBj(x) = 0, then x= 0 hence ˜ ei=a˜e. This is absurd as ˜ ei\nis primitive. So we may assume Bj(x)/ne}ationslash= 0. Similarly, using Equation (5), we may\nassume as well that Bj(x)/∈ −˜σjandBj(x)/∈˜σj. HenceBj(x) must have at least\none positive and one negative coordinate in the basis (˜ ek)1≤k≤dj. As ˜e∈˜σ, we can\nwrite\na˜e=d/summationdisplay\nk=1ak˜ek\nwithak= 0 orak≥2. Then, from Equation (7), Bj(x) has exactly one coordinate\nequal to 1, while its other non-zerocoordinatesareall less or equa l to−2. A caseby\ncaseanalysisusingthe descriptionof Vjin Equation(2)(cfExample4.1)showsthat\nthis is impossible (for V3, use the fact that the weight spaces are 2-dimensional).\nWe will then prove that there is m′∈M′such that\n∀i∈[[1,d]],/an}bracketle{tm′,A(˜ei)/an}bracketri}ht= 1,\nwhich implies that Wis Gorenstein. So let m′∈M′. Set\n˜m=A∗(m′)∈ker(B∗).\nThen,\n∀i∈[[1,d]],/an}bracketle{tm′,A(˜ei)/an}bracketri}ht= 1⇐⇒ ∀i∈[[1,d]],/an}bracketle{t˜m,˜ei/an}bracketri}ht= 1\n⇐⇒ ˜m= (1,...,1)\nwhereweusedthe basis(˜ ei)to producethe coordinatesof ˜ m. Hence, it isequivalent\nto show that\n(1,...,1)∈ker(B∗),FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 19\nwhich by definition of Bis equivalent to\nd/summationdisplay\ni=1mi= 0∈M\nwheremi’s are the weights describing the T-action on V. This is where we use\nthatXis foldable. The G-action on Nnaturally induces a G-action on M, and on\nthe weights ( mi)’s. AsGcontains a non-trivial cyclic group, there is an element\ng∈Aut(N) of order pwith no fixed point but 0. This element generates a cyclic\ngroup that acts freely on {mi,1≤i≤d}. Hence\nd/summationdisplay\ni=1mi=/summationdisplay\ni′p−1/summationdisplay\nk=0gk·m′\ni\nwhere we picked a single element m′\niin each orbit under this action. As g/ne}ationslash= Id, we\nhave\nId+g+···+gp−1= 0\nso that for each orbit\np−1/summationdisplay\nk=0gk·m′\ni= 0.\nWe then deduce the existence of the required m′, and that Wis Gorenstein.\nWe proceed to the proof of the fact that Whas terminal singularities. Let\nΠσ:=\n\n/summationdisplay\nρ∈σ(1)λρuρ|/summationdisplay\nρ∈σ(1)λρ≤1,0≤λρ≤1\n\n.\nAs described above, we may fix a subset of\n{A(˜ei),1≤i≤d}\nas a set of ray generators for σ. Letu′∈Πσ∩N′given by\nu′=d/summationdisplay\ni=1λiA(˜ei),\nwith\n0≤λi≤1,\nand\nλ1+...+λd≤1,\nassuming λi= 0when A(˜ei) is not in ourfixed chosenset ofraygenerators. Assume\nthat there is i0withλi0/ne}ationslash= 0. We need to show that λi0= 1 and λi= 0 fori/ne}ationslash=i0.\nBy assumptions on the ( λi)’s, it is enough to show that λi∈Zfor alli. Asu′∈N′,\nthere exists ˜ u∈˜Nsuch that\nA(˜u) =A(d/summationdisplay\ni=1λi˜ei),\nand then there is x∈NRsuch that\n˜u−d/summationdisplay\ni=1λi˜ei=B(x) = (Bj(x),B+(x)).\nBy injectivity of Bjagain, we only need to consider the case when Bj(x)/ne}ationslash= 0 and\n1≤i0≤dj(as ifλi= 0 for all i≤djthenx∈NandB(x)∈˜N). Hence, we20 C. TIPLER\nreduced ourselves to study the case when X=Yj. We will use again the explicit\ndescriptions of the Vj’s from Example 4.1. For V3, we find the system\n\n\n˜u1−λ1=−x2\n˜u2−λ2=−x2\n˜u3−λ3=x1\n˜u4−λ4=x1\n˜u5−λ5=−x1+x2\n˜u6−λ6=−x1+x2\nfrom which we obtain\n(λ1+λ3+λ5,λ1+λ3+λ6,λ1+λ4+λ6,λ1+λ4+λ5)∈Z4\nand\n(λ2+λ3+λ5,λ2+λ3+λ6,λ2+λ4+λ6,λ2+λ4+λ5)∈Z4.\nTogether with\n0≤λ1+...+λ6≤1,\nwe deduce that all those linear combinations of the λi’s equal 0 or 1, with at least\none that is non zero. Using λi≥0, we then deduce that λi0= 1 and the result\nfollows in that case. The cases V2andV4are similar, so we will only treat V4. In\nthat case, one checks that\n/braceleftbiggA(˜e1) =A(˜e2)\nA(˜e3) =A(˜e4)\nso we can pick the ray generators ( A(˜e1),A(˜e3)) forσ(1). The description of V4\nthen implies\n\n˜u1−λ1=x1\n˜u2=−x1\n˜u3−λ3=x2\n˜u4=−x2\nhencex∈Nand the result follows. /square\n4.4.Examples. We first provide two examples of local moduli spaces and then an\nexample of a quotient H1(X,TX)//Tthat is not Q-Gorenstein, for Xa smooth\ntoric surface.\n4.4.1.A smooth example : Y4.We consider the toric variety W4which is the\nquotient of V4byT(recall the definition of Y4in Example 4.1, and that V4=\nH1(Y4,TY4)). As seen in the proof of Proposition 4.8, the generators of σin that\ncase can be taken to be A(˜e1) andA(˜e3), so that σis isomorphic to the cone\nR+·(1,0)+R+·(0,1)⊂R2.\nThen,\nW4≃C2,\nand the G-action on V4(see Remark 4.2), descends to a D1-action on W4≃C2\ngenerated by a reflection\n(x,y)/mapsto→(y,x).\nHence, we conclude that the local moduli space is modeled on\nW4//G≃C2.FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 21\n4.4.2.A singular example : Y3.Let’s consider now W3, given by the GIT quotient\nofV3byT. We can pick isomorphisms N≃Z2,˜N≃Z6and\nN′=˜N/N≃Z4\nsuch that the map B:N→˜Nis given by\nB=\n0−1\n0−1\n1 0\n1 0\n−1 1\n−1 1\n\nand the map A:˜N→N′by\nA=\n−1 1 0 0 0 0\n0 0−1 1 0 0\n1 0 1 0 1 0\n1 0 1 0 0 1\n.\nHence,W3is the toric variety associated to the cone\nσ=6/summationdisplay\ni=1R+·e′\ni⊂R4\nwhere (e′\ni)1≤i≤6are given by the columns of the matrix A. Using Sage Math, we\nfind\nW3≃Spec(C[z0,...,z 7]/I3)\nwithI3the ideal whose set of generators is given by\nI3=/an}bracketle{t−z1z4+z0z7,−z1z5+z0z6,−z4z6+z5z7,\nz0z4−z3z5,−z2z4+z3z7,−z1z4+z3z6,\nz0z2−z1z3,z2z6−z1z7,−z1z4+z2z5/an}bracketri}ht.\nThis toricaffinevarietyissingular(it is noteven simplicial). The G-actiondescends\nto aD3-action on W3. This action is induced by a representation\nD3→GL4(Z),\nand a set of generators for this action is given by\n0 0−1 1\n1 0 0 0\n0 0 1 0\n0 1 1 0\nand\n−1 0 0 0\n0−1 0 0\n1 1 0 1\n1 1 1 0\n\nwhere the first matrix represents an element of order 3 and the se cond one a re-\nflection. The final quotient W3//D3provides a singular example of local moduli\nspace.\n4.5.A non–Gorenstein example. We producehereanexampleofatoricsurface\nXwith Aut( X)≃T, andH1(X,TX)//Ta nonQ-Gorenstein toric variety. For\nthis, simply blow-up Y4in a single point to produce the fan on the following page :22 C. TIPLER\n• • • • •\n• • • • •\n• • •/d79/d79\n/d47/d47 /d111/d111\n/d15/d15/d63/d63⑦⑦⑦⑦⑦⑦⑦\n/d127/d127⑦⑦⑦⑦⑦⑦⑦/d95/d95❅❅❅❅❅❅❅\n/d31/d31❅❅❅❅❅❅❅/d71/d71✎✎✎✎✎✎✎✎✎✎✎✎• •\n• • • • •\n• • • • •\nFigure 13. Fan ofX.\nAsXis blown-up from Y4, we have Aut0(X)≃T. One can also check that we\nactually have Aut( X)≃T. Moreover, the arguments in Section 4.2 go through,\nand we find that if Xwere cscK, then its local moduli space would be modeled\non the GIT quotient of H1(X,TX) byT. A direct computation again, using the\nmethod in Section 4.1, provides the weight space decomposition\nH1(X,TX) =V1\n+e∗\n1⊕V2\n−e∗\n1⊕V1\n+e∗\n2⊕V1\n−e∗\n2⊕V1\ne∗\n1−e∗\n2.\nGiven that the sum ofthe weightsthat appearin this decomposition d oesn’t vanish,\nthe quotient H1(X,TX)//Tis notQ-Gorenstein (see the proof of Proposition 4.7).\n5.Discussion and perspectives\n5.1.Relations between local moduli and Weil–Petersson metrics .We will\ndiscussinthis sectionthe factthat the localmodulispaceswecons ideredarerelated\nby toric fibrations. Assume that π:X→X0is aG-equivariant blow-up between\ntwo foldable toric surfaces. We keep the notations from previous s ections, using the\nsubscript 0 to refer the the spaces associated to X0. From [21, Corollary 1.6], the\ncorresponding space V0injects in V. It is a straightforward exercise to check that\nwe have the following commutative diagram :\n0−→NB−→˜NA−→N′−→0\n↓ ↓ ↓\n0−→NB0−→˜N0A0−→N′\n0−→0\nwhere the first vertical arrow is the identity and the last two are su rjective. By\nconstruction, one sees that the surjective map N′→N′\n0is compatible with σand\nσ0, and induces a toric locally trivial fibration W→W0whose fiber is itself an\naffine toric variety. The whole construction is G-equivariant, and provides maps\nbetween the associated local moduli spaces (up to shrinking the ne ighborhoods we\nconsidered).\nThe cscK metric on Xlives in the class π∗[ω0]−εEi, whereω0is cscK on X0,ε\nis small and the Ei’s stand for the exceptional divisors of the blow-up. It would be\ninteresting to understand the behaviour of the associated Weil–Pe tersson metrics\n(ΩWP\nε)0<ε<ε 0on the local moduli spaces Wε⊂W/Gas constructed in [12] when ε\ngoesto zero. It seemsnaturalto expect that the volumeofthe fi bers ofthe fibration\nW/G→W0/Ggo to zero, so that Wεwould converge in Gromov–Hausdorff sense\ntoW0.FOLDABLE FANS, CSCK SURFACES AND LOCAL K-MODULI 23\n5.2.Higher dimensional case. Many features that hold for toric surfaces fail in\nhigher dimension. First, even when it is reductive, the identity compo nent of the\nautomorphism group of a non-rigid toric variety will not a priori be iso morphic to\nthe torus (see [35]). Then, from dimension 3, toric varieties may be o bstructed (see\n[20]). Finally, the toric MMP (that produces the classification of toric surfaces)\nproduces singular varieties in general. So our main ingredients to pro ve Theorem\n1.5 do not generalise, a priori, in higher dimension.\nNevertheless, it would be interesting to study what goes through t he new dif-\nficulties. We expect the right extension of the notion of foldable fan s in higher\ndimension to be fans whose lattice automorphism group admit a subgr oup that\nacts with no fixed point. This raises several questions :\n(1) Do all crystallographic groups arise as lattice automorphism gro ups of\nsmooth complete fans?\n(2) Do all foldable toric varieties admit a cscK metric?\n(3) Are foldable toric varieties unobstructed?\n(4) What are the singularities of the moduli space of cscK metrics ar ound cscK\nfoldable toric varieties?\nReferences\n1. Vestislav Apostolov, Hugues Auvray, and Lars Martin Sekt nan,Extremal K¨ ahler Poincar´ e\ntype metrics on toric varieties , J. Geom. Anal. 31(2021), no. 2, 1223–1290 (English).\n2. Vestislav Apostolov, David M. J. Calderbank, and Paul Gau duchon, Ambitoric geometry. II:\nExtremal toric surfaces and Einstein 4-orbifolds , Ann. Sci. ´Ec. Norm. Sup´ er. (4) 48(2015),\nno. 5, 1075–1112 (English).\n3. 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Z. 283(2016),\nno. 3-4, 1011–1031 (English).\nUniv Brest, UMR CNRS 6205, Laboratoire de Math ´ematiques de Bretagne Atlan-\ntique, France\nE-mail addresses :carl.tipler@univ-brest.fr"    },    {        "title": "2402.01199v1.MIQCQP_reformulation_of_the_ReLU_neural_networks_Lipschitz_constant_estimation_problem.pdf",        "content": "arXiv:2402.01199v1  [math.OC]  2 Feb 2024MIQCQP reformulation of the ReLU neural networks Lipschitz\nconstant estimation problem\nMohammed Sbihi∗1, Sophie Jan†2, and Nicolas Couellan‡1,2\n1F´ ed´ eration ENAC ISAE-SUPAERO ONERA, Universit´ e de Toulouse , France\n2Institut de Math´ ematiques de Toulouse UMR 5219, Universit´ e de T oulouse; CNRS,\nUPS, F-31062 Toulouse Cedex 9, France.\nAbstract\nIt is well established that to ensure or certify the robustne ss of a neural network, its Lipschitz\nconstant plays a prominent role. However, its calculation i s NP-hard. In this note, by taking\ninto account activation regions at each layer as new constra ints, we propose new quadratically\nconstrained MIP formulations for the neural network Lipsch itz estimation problem. The solutions\nof these problems give lower bounds and upper bounds of the Li pschitz constant and we detail\nconditions when they coincide with the exact Lipschitz cons tant.\n1 Introduction\nSeveral studies have demonstrated the prominent role of the Lips chitz constant in the robustness of\nneural networks. It has been shown for example that it is related t o generalization bounds of neural\nnetwork classifiers [11, 1]. The Lipschitz constant expresses also the maximum variation of the neural\nnetwork outputs and can therefore be used to derive robustnes s certificates when inputs are subject to\nrandom or adversarialperturbations [10]. Furthermore, it has be en used as a regularizationterm of the\nneural network training loss function to compute optimal neural n etwork weights that achieve better\nrobustness [5]. Alternatively, constraints on the Lipschitz consta nt have been added in the training\nloss minimization problem to develop 1-Lipschitz neural networks [2].\nThe exact calculation of the Lipschitz constant of a neural networ k is a difficult problem. Even in\nthe simple case of one hidden layer neural network, it can be shown t hat the problem is NP-hard [10].\nTherefore, an estimation of the constant in the form of upper bou nds (sometimes lower bounds) is\nusually sought. However, in the common case of ReLU networks, co mputing tight estimates of the\nconstant is also an NP-hard problem (see Theorem 4 in [9]). Several a pproaches have been proposed\nin the literature. They vary by the scope of the input being consider ed (global or local Lipschitz\nregularity), the order pofLp-Lipschitz regularity, or the underlying estimation method.\nAmong these various approaches, Lipschitz certificates via semide finite programming (SDP) are\nproposed in [4]. By exploiting slope restriction properties of common a ctivation functions, incremental\nquadratic constraints are formulated and the global L2-Lipschitz constant problem is then expressed\nas a SDP. While it provides a nice convex formulation of the constant e stimation, its application to\nreal life neural networks architectures is limited by the computatio nal complexity of available methods\nfor solving SDP. Alternatively, in the case of ReLU networks, autho rs have considered mixed integer\nprogramming (MIP) approaches to derive exact or upper bounds o f the Lipschitz constant. Indeed,\n∗mohammed.sbihi@recherche.enac.fr\n†sophie.jan@math.univ-toulouse.fr\n‡nicolas.couellan@recherche.enac.fr\n1in [9], by showing that a ReLU network is a composition of MIP-encodab le components and therefore\nitself MIP-encodable, the authors formulate the exact calculation of the local Lipschitz constant of\nReLU networks as a MIP. In the worst case, MIP problems have exp onential time complexity, however,\nin practice they are often solved in reasonable time.\nIn this note, we propose new quadratically constrained MIP formula tions for the neural network\nLipschitz estimation problem. First, by taking into account activatio n regionsat each layeras new con-\nstraints, we derive three new MIP formulations whose solutions give a lower bound Lof the Lipschitz\nconstant, a sequence {Lǫ}ǫ>0of lower bounds convergingto Land an upper bound ¯L. We further show\nthatLand¯Lcoincide and are equal to the true Lipschitz constant if the neural network is in general\nposition as defined in [9]. We also show that, except on a set of networ k parameters of Lebesgue mea-\nsure 0,Lcoincide with ¯L. Next, by reformulating the activation constraints as quadratic c onstraints,\nwe propose equivalent Mixed Integer Quadratically Constrained Qua dratic Program (MIQCQP) for-\nmulations. These new constrained problems have the benefit of red ucing the search space in the\nbranching phase involved in the MIP solving process. Furthermore, the specific quadratic structure\nof the objective and the constraints can also be exploited in the bou nding phase of MIP solvers using\nquadratic convex relaxations and linearizations strategies as expla ined in [3]. However, the study of\nthe numerical solutions of these problems goes beyond the scope o f this note that was only intended to\nexplainthe derivationofthe MIQCQPformulationofthe neuralnetw orkLipschitzestimation problem.\nThe note is organized in three sections. Section 2 introduces the ge neral problem of calculating the\nLipschitz constant L(f,X) of a neural network fover an input set X. Section 3 details the derivation\nof lower and upper bounds for L(f,X). Finally, Section 4 provides Mixed Integer Quadratically\nConstrainedQuadraticProgramreformulationsforthe problemof estimating L(f,X)usingthe bounds\nobtained in Section 3.\n2 Problem statement\nWe consider ReLU Multi-Layer-Perceptron function f:Rn0→RnL, that is a composition of affine\noperators and element-wise ReLU nonlinearities. More precisely, it m ay be encoded by:\nf(x) =TL◦ρL−1◦TL−1◦···◦ρ1◦T1(x)\nwhereTk:Rnk−1∋x/mapsto→Mkx+bk∈Rnkis an affine function and ρk:Rnk→Rnkis the ReLU\noperator applied element-wise. We denote by θk=Tk◦ρk−1◦Tk−1◦···◦ρ1◦T1the pre-activation\noutput of the k-th layer and by θi\nkthei-th component of θk(corresponding to the pre-activation of\nthei-th neuron of layer k). In the following, given an element v∈Rn, we denote its i−thcomponent\nbyviand we denote Hadamard product between two vectors v1andv2byv1⊙v2.\nWe are interested in computing the quantity\nsup\nx,y∈X/bardblf(y)−f(x)/bardbl\n/bardbly−x/bardbl,\nwhereXis an open subset of Rnand/bardbl./bardblis a norm. When this quantity is finite, we denote it by\nL(f,X) and we say that fis locally Lipschitz over X.IfX=Rn0, then we denote the above quantity\nL(f) and we simply say that fis (globally) Lipschitz.\n3 Deriving lower and upper bounds of the Lipschitz constant\nof ReLU networks\nWe now derive an upper and lower bound sforL(f,X) by observing [9, Theorem 1] that\nL(f,X) = esssupx∈Xsup\nG∈Jf(x)/bardblG/bardbl (1)\n2whereJf(x) is the (Clarke) generalized jacobian of fatx. Using recursively the Clarke jacobian\nChain Rule (see [8, Theorem 4]), we obtain the following bound\nL(f,X)≤sup\nx∈X/braceleftbig\n/bardblMLdiag(gL−1)ML−1···diag(g1)M1/bardbl |gk∈[0,1]nk, gi\nk∈∂ReLU(θi\nk(x))/bracerightbig\n.(2)\nHere∂ReLU(x) is the subdifferential of the ReLU function:\n∂ReLU(x) =\n\n{0}ifx <0,\n[0,1] if x= 0,\n{1}ifx >0.\nAn activation pattern for the ReLU network fis an assignment to each hidden neuron of a sign 1\nor 0:\n(σ1,σ2,...,σ L−1)∈ {0,1}n1×{0,1}n2×···×{ 0,1}nL−1.\nThe activation region in Xcorresponding to ( σ1,σ2,...,σ L−1) is\nR(X;(σ1,σ2,...,σ L−1)) =/braceleftbigg\nx∈ X |/parenleftbigg\nσi\nk−1\n2/parenrightbigg\nθi\nk(x)>0,∀i∈ {1,...,n k},∀k∈ {1,···,L−1}/bracerightbigg\n.\nWhenspecifyinganactivationpattern, thesignalassignedtoaneu ronifromthe k−thlayerdetermines\nwhether it is on or off for inputs in the activation region since the pre- activation of neuron is positive\n(resp. negative) when σi\nk= 1 (resp. σi\nk= 0). Let us define Hk,i={x∈Rn0|θi\nk(x) = 0}(for\ni∈ {1,...,n k},k∈ {1,···,L−1}) which can be thought of as “bent hyperplanes”. The non-empty\nactivation regions is not else but the connected components of X \\ ∪k,iHk,i(Lemma 2 in [6]). The\nJacobian exists and is the same for all the points belonging to the sam e activation pattern. The\nJacobian corresponding to a pattern ( σ1,···,σL−1) is equal to MLdiag(σL−1)ML−1···diag(σ1)M1.\nWe obtain the following lower bound for L(f,X):\nL(f,X)≥ sup\n(σ1,σ2,...,σL−1)∈{0,1}n1+n2+···+nL−1|R(X;σ)/ne}ationslash=∅/bardblMLdiag(σL−1)ML−1···diag(σ1)M1/bardbl.(3)\nDenoting N(g) =/bardblMLdiag(gL−1)ML−1···diag(g1)M1/bardbl, we are thus interested in solving the fol-\nlowing two optimization problems to obtain upper and lower bounds of t he Lipshitz constant of the\nReLU neural network:\n(U∗)\n\nmaxN(g)\ns.t. gi\nk∈∂ReLU(θi\nk(x)),∀i∈ {1,...,n k},k∈ {1,···,L−1}\ngk∈[0,1]nk,∀k∈ {1,···,L−1}\nx∈ X\nand\n(L∗)\n\nmaxN(σ)\ns.t.R(X;(σ1,···,σL−1))/\\e}atio\\slash=∅\nσk∈ {0,1}nk,∀k.\nProposition 1. For each k∈ {1,···,L−1}and each i∈ {1,...,n k}, the function gi\nk/mapsto→N(g)is\nconvex.\nProof.The function is the composition of gi\nk/mapsto→MLdiag(gL−1)ML−1···diag(g1)M1which is affine and\nthe norm /bardbl./bardblwhich is convex, therefore it is convex [7, Proposition 2 .1.5].\nProposition 2. Problem (U∗)is equivalent to the following one:\n(ˆU)\n\nmaxN(g)\ns.t. gi\nk∈∂ReLU(θi\nk(x)),∀i∈ {1,...,n k},k∈ {1,···,L−1}\ngk∈ {0,1}nk,∀k∈ {1,···,L−1}\nx∈ X.\n3Proof.Letg∗denote a solution of ( U∗) and ˆga solution of ( ˆU).\nThe vector ˆ gis naturally feasible for ( U∗) and thus\nN(ˆg)≤N(g∗).\nNow, if ( g∗)i\nk∈]0,1[ for some iandk, using convexity of gi\nk/mapsto→N(g), there exists g,whose all com-\nponents are equal to that of g∗except (g)i\nkwhich belongs to {0,1}, such that N(g)≥N(g∗). Moreover\n(g∗)i\nk∈]0,1[∩∂ReLU(θi\nk(x)) implies that θi\nk(x) = 0 and thus ( g)i\nk∈ {0,1}is also in ∂ReLU(θi\nk(x)).\nRepeating this for all components of g∗that are in ]0 ,1[, we show that there exists a solution gof (U∗)\nwhose components are all in {0,1}satisfying N(g∗)≤N(g). Therefore, we have\nN(g∗)≤N(g)≤N(ˆg).\nUsing Proposition 2 and the fact that gi\nk∈∂ReLU(θi\nk(x)) is equivalent to/parenleftbig\ngi\nk−1\n2/parenrightbig\nθi\nk(x)≥0 for\ngi\nk∈ {0,1}, we compute an upper bound of the Lipschitz constant by solving\n(P)\n\nmaxN(g)\ns.t.(gi\nk−1\n2)θi\nk(x)≥0,∀i∈ {1,...,n k},k∈ {1,···,L−1}\ngk∈ {0,1}nk,∀k∈ {1,···,L−1}\nx∈ X.\nBy the definition of the activation region, problem ( L∗) can be reformulated as\n(P)\n\nmaxN(σ)\ns.t.(σi\nk−1\n2)θi\nk(x)>0,∀i∈ {1,...,n k},k∈ {1,···,L−1}\nσk∈ {0,1}nk,∀k∈ {1,···,L−1}\nx∈ X.\nIn order to avoid strict inequalities, we introduce for ε≥0,\n(P)ε\n\nmaxN(σ)\ns.t./parenleftbig\nσi\nk−1\n2/parenrightbig\nθi\nk(x)≥ε,∀i∈ {1,...,n k},k∈ {1,···,L−1}\nσk∈ {0,1}nk,∀k∈ {1,···,L−1}\nx∈ X.\nObserve that ( P)0corrresponds to ( P).\nWe now introduce the following constraint sets:\nC=X ×{0,1}n1+n2+...+nL−1, (4)\nCs=/braceleftbigg\n(x,σ)∈ C:/parenleftbigg\nσi\nk−1\n2/parenrightbigg\nθi\nk(x)>0,∀i∈ {1,...,n k},k∈ {1,···,L−1}/bracerightbigg\n,(5)\nCε=/braceleftbigg\n(x,σ)∈ C:/parenleftbigg\nσi\nk−1\n2/parenrightbigg\nθi\nk(x)≥ε,∀i∈ {1,...,n k},k∈ {1,···,L−1}/bracerightbigg\n,(6)\nso that the preceeding problems can be rewritten as:\n(P)/braceleftbiggmaxx,σN(σ)\ns.t.(x,σ)∈ C0/bracerightbigg\n(P)/braceleftbiggmaxx,σN(σ)\ns.t.(x,σ)∈ Cs/bracerightbigg\n(P)ε/braceleftbiggmaxx,σN(σ)\ns.t.(x,σ)∈ Cε/bracerightbigg\n.\nLetL,L,Lεdenote the optimal values of ( P), (P) and (P)εrespectively. The following proposition\nsummarizes and completes the above discussion.\n4Proposition 3. We have\n1. The function ]0,+∞[∋ε/mapsto→Lεis non-increasing and piece-wise constant.\n2.limε↓0Lε↑L≤L(f,X)≤L=L0.\n3. If∪i,kHi,kis Lebesgue measure negligible, then L(f,X) =L.\n4. If the ReLUnetwork fis in general position (see Definition 4 in [9]), then L(f,X) =L=L.\nProof. 1. Ifε1≤ε2thenCε2⊂ Cε1which implies that Lεis non increasing. Moreover, Lεbelongs\nto{N(σ),σ∈ {0,1}n1+n2+...+nL−1}which is a finite set. Hence Lεis piece-wise constant.\n2. Observe that Lε≤L≤L(f,X)≤L=L0.To prove the remaining statement, let ˆ σan optimal\nsolution of ( P). ThenR(X;ˆσ)/\\e}atio\\slash=∅, that is there exists ˆ x∈ Xsuch that/parenleftbig\nˆσi\nk−1\n2/parenrightbig\nθi\nk(ˆx)>0,∀i∈\n{1,...,n k},∀k∈ {1,···,L−1}.Define\nˆε= min\ni∈{1,...,nk},k∈{1,···,L−1}/parenleftbigg\nˆσi\nk−1\n2/parenrightbigg\nθi\nk(ˆx).\nSo (ˆx,ˆσ)∈ Cεfor allε≤ˆεand thus Lε≥L.\n3. We can write X= (/uniontext\nσR(X;σ))∪(/uniontext\ni,kHi,k∩X).Now (1) implies\nL(f,X) = esssupx∈(/uniontext\nσR(X;σ))sup\nG∈Jf(x)/bardblG/bardbl\n= sup\nσ|R(X,σ)/ne}ationslash=∅N(σ)\n=L.\n4. By [9, Theorem 2] if the ReLU network is in general position then\nJf(x) =/braceleftbig\n/bardblMLdiag(gL−1)ML−1···diag(g1)M1/bardbl |gk∈[0,1]nk, gi\nk∈∂ReLU(θi\nk(x))/bracerightbig\n.\nBy (1) we obtain L(f,X) =L.Furthermore, if the ReLU network fis in a general position\nthen∪i,kHi,kis Lebesgue measure negligible [9], which ensures the second equality using the\npreceding item.\nRemark 1. By [9, Theorem 3] the set of ReLUnetworks not in general position has Lebesgue measure\nzero over the parameter space and consequently for almost al lReLUNetworks we have L(f,X) =L=\nL.\nCorollary 1. IfX=Rn0and the bias bk,k∈ {1,···,L−1}are zero, then Lε=Lfor allε >0.\nProof.We have already established that Lε≤L.Let (ˆx,ˆσ)∈ Csan optimal solution of ( P) and let ε >\n0.Since the bias bk,k∈ {1,···,L−1}are zero then for any λ >0 we have θk(λˆx) =λθk(ˆx).Therefore\nwe can choose λsufficiently large so that/parenleftbig\nˆσi\nk−1\n2/parenrightbig\nθi\nk(λˆx)> ε,∀i∈ {1,...,n k},k∈ {1,···,L−1}\nensuring that ( λˆx,ˆσ)∈ Cεand consequently Lε≥N(ˆσ) =L.\nThe following simple examples illustrate the above results.\nExemple 1. Forf(x) =x−1, we trivially have L(f,X) = 1. This function can also be written\nf(x) = max( x−1,0)−max(1−x,0), corresponding to our formalism with\nM1=/parenleftbigg1\n−1/parenrightbigg\n, M2=/parenleftbig\n1−1/parenrightbig\n, b1=/parenleftbigg−1\n1/parenrightbigg\n, b2= 0.\n5With this choice, we have Lε=L= 1for allε >0andL0=L= 2. Indeed, x= 1is neither in\nR(R;σ)nor in the feasible set of (P)εforε >0, but it belongs to C0.\nεLε\n2•\n1\nExemple 2. Forf(x) = max( x+1,0)−max(x−1,0)corresponding to\nM1=/parenleftbigg1\n1/parenrightbigg\n, M2=/parenleftbig−1 1/parenrightbig\n, b1=/parenleftbigg−1\n1/parenrightbigg\n, b2= 0,\nwe have L= 1,L(f,X) = 1andL0=L= 1. Moreover,\nLε=/braceleftbigg1for all0< ε≤0.5,\n0for allε >0.5.\nεLε\n1••\n0.5\n4 MIQCQP reformulations\nLet now consider the Lp-norm for p∈ {1,2,∞}and assume that Xcan be expressed by quadratic\nconstraints (e.g. a ball) or linear constraints (e.g. polyhedron).\nThen let us remark that we have ρk◦θk(x) =σk⊙θk(x) for allk= 1,2,···L−1 as soon as ( x,σ)\nis in one of the sets Cε,CsorC0. By definition of θk(x), we have thus ρk◦θk(x) =σk⊙θk(x) =\nσk⊙(Mk◦ρk−1◦θk−1(x) +bk) fork= 2,3,···L−1 andρ1◦θ1(x) =σ1⊙(M1x+b1).Denoting\nxk=ρk◦θk(x) we get the following bilinear relations\nxk=σk⊙(Mkxk−1+bk) fork= 1,2,···L−1 withx0=x.\nTherefore, the constraints of ( P)εare equivalent to\nx0∈ Xand for all k∈ {1,2,...,L−1},\n\nxk=σk⊙(Mkxk−1+bk),\n(σk−1\n2)⊙(Mkxk−1+bk)≥ε,\nσk∈ {0,1}nk.\nThe objective function of ( P)εcan also be expressed, depending on the chosen Lp-norm, as\nNp(σ) =/bardblMLdiag(σL−1)ML−1···diag(σ1)M1/bardblp= sup\ny,/bardbly/bardblp≤1/bardblMLdiag(σL−1)ML−1···diag(σ1)M1y/bardblp\n6and equivalently as\nmax\ny/bardblyL/bardblp\ns.t. yk=Mkdiag(σk−1)yk−1,∀k∈ {2,...,L},\ny1=M1y0\n/bardbly0/bardblp≤1\nwhereyis the collection of y0,y1,...,y L.\nFinally, for a given p∈ {1,2,∞}, (P)εis equivalent to the following problem:\n(Pp)ε\n\nmax\ny,σ,x/bardblyL/bardblp\ns.t. yk=Mkdiag(σk−1)yk−1,∀k∈ {2,...,L},\ny1=M1y0\nxk=σk⊙(Mkxk−1+bk),k= 1,2,···L−1\n(σk−1\n2)⊙(Mkxk−1+bk)≥ε, k= 1,2,···L−1\nσk∈ {0,1}nk,k= 1,2,···L−1\nx0∈ X\n/bardbly0/bardblp≤1\nwhereσis the collection of σ1,...,σ L−1andxis the collection of x0,x1,...,x L−1. Provided that\nXis expressible by quadratic (or linear) constraints, the above prob lem can also be expressed as a\nMIQCQP as shown below.\nThe case p= 2 In this case, ( P2)εis trivially equivalent to the following MIQCQP problem.\nmax\ny,σ,x/bardblyL/bardbl2\n2\ns.t. yk=Mkdiag(σk−1)yk−1,∀k∈ {2,...,L},\ny1=M1y0\nxk=σk⊙(Mkxk−1+bk),k= 1,2,···L−1\n(σk−1\n2)⊙(Mkxk−1+bk)≥ε, k= 1,2,···L−1\nσk∈ {0,1}nk,k= 1,2,···L−1\nx0∈ X\n/bardbly0/bardbl2\n2≤1.\nThe case p= 1 Observing that for all a∈R,|a|= (2λ−1)awithλ∈ {0,1}and (2λ−1)a≥0,\n(P1)εis equivalent to\nmax\ny,σ,x,ν,µ/summationtextnL\ni=1(2µi−1)(yL)i\ns.t. y k=Mkdiag(σk−1)yk−1,∀k∈ {2,...,L},\ny1=M1y0\nxk=σk⊙(Mkxk−1+bk),k= 1,2,···L−1\n(σk−1\n2)⊙(Mkxk−1+bk)≥ε, k= 1,2,···L−1\nσk∈ {0,1}nk,k= 1,2,···L−1\nx0∈ X\nν∈ {0,1}n0\n(2νi−1)(y0)i≥0, i= 1,2,···,n0/summationtextn0\ni=1(2νi−1)(y0)i≤1,i= 1,2,···,n0\nµ∈ {0,1}nL\n(2µi−1)(yL)i≥0, i= 1,2,···,nL.\nThe last constraints can be further linearized by introducing additio nal binary variables and using a\nbig-Mtechnique. Indeed, for all x∈R, with|x| ≤B,u=|x|if and only if there exists λ∈ {0,1}such\n7that \n\nu≥xandu≥ −x,\nx≤B(1−λ) andx≥ −Bλ,\nu≤ −x+2B(1−λ) andu≤x+2Bλ.\nThe first constraints imply that u≥ |x|.The second ones ensure that if x >0 (respectively x <0)\nthenλ= 0 (respectively λ= 1). The last constraints guarantee that u≤ |x|.Problem ( P1)εis hence\nequivalent to\nmax\ny,σ,x,u,w,ν,µ/summationtextnL\ni=1wi\ns.t. y k=Mkdiag(σk−1)yk−1,∀k∈ {2,...,L},\ny1=M1y0\nxk=σk⊙(Mkxk−1+bk),k= 1,2,···L−1\n(σk−1\n2)⊙(Mkxk−1+bk)≥ε, k= 1,2,···L−1\nσk∈ {0,1}nk,k= 1,2,···L−1\nx0∈ X\nν∈ {0,1}n0\n−1≤y0≤1\n(y0)i≤ui, i= 1,2,···,n0\n−(y0)i≤ui, i= 1,2,···,n0\n(y0)i≤1−νi,i= 1,2,···,n0\n(y0)i≥ −νi, i= 1,2,···,n0\nui≤ −(y0)i+2(1−νi), i= 1,2,···,n0\nui≤(y0)i+2νi,i= 1,2,···,n0/summationtextn0\ni=1ui≤1\nµ∈ {0,1}nL\n(yL)i≤wi, i= 1,2,···,nL\n−(yL)i≤wi, i= 1,2,···,nL\n(yL)i≤C(1−µi),i= 1,2,···,nL\n(yL)i≥ −Cµi, i= 1,2,···,nL\nwi≤ −(yL)i+2C(1−µi), i= 1,2,···,nL\nwi≤(yL)i+2Cµi,i= 1,2,···,nL\nwhereC≫0 is the so-called big-Mconstant.\nThe case p=∞Problem ( P∞)εcan similarly be expressed as\nmax\ny,σ,x,u,µ,η/summationtextnL\ni=1ηiui\ns.t. y k=Mkdiag(σk−1)yk−1,∀k∈ {2,...,L},\ny1=M1y0\nxk=σk⊙(Mkxk−1+bk),k= 1,2,···L−1\n(σk−1\n2)⊙(Mkxk−1+bk)≥ε, k= 1,2,···L−1\nσk∈ {0,1}nk,k= 1,2,···L−1\nx0∈ X\n−1≤y0≤1\nµ∈ {0,1}nL\nu≥0\nui= (2µi−1)(yL)i, i= 1,2,···,nL\nη∈ {0,1}nL/summationtextnL\ni=1ηi= 1.\nNote: in this formulation the last constraints can also be linearized as in the case p= 1.\n8References\n[1] Peter Bartlett, Dylan J. 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Lipschitz regularity of deep neu ral networks: analysis and\nefficient estimation. In NeurIPS , 2018.\n[11] Yusuke Tsuzuku, Issei Sato, and Masashi Sugiyama. Lipschit z-margin training: Scalable certifi-\ncation of perturbation invariance for deep neural networks, 201 8.\n9"    },    {        "title": "2402.01251v1.Amorphous_Boron_Nitride_as_a_Diffusion_Barrier_to_Cu_Atoms.pdf",        "content": "Amorphous Boron Nitride as a Diffusion Barrier to\nCu Atoms\nOnurcan Kaya1,2, Hyeongjoon Kim3, Byeongkyu Kim4, Luigi\nColombo5, Hyeon-Jin Shin6Ivan Cole2, Hyeon Suk Shin3, and\nStephan Roche1,7\n1Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST,\nCampus UAB, Bellaterra, 08193, Barcelona, Spain\n2School of Engineering, RMIT University, Melbourne, Victoria, 3001, Australia\n3Department of Chemistry, Ulsan National Institute of Science and Technology,\nUlsan 44919, Republic of Korea\n4School of Semiconductor Materials and Devices Engineering, Ulsan National\nInstitute of Science and Technology (UNIST), Ulsan 44919, Republic of Korea\n5CNMC, LLC, Dallas, TX, 75248, USA\n6Department of Semiconductor Engineering, School of Electrical Engineering and\nComputer Science, Gwangju Institute of Science and Technology (GIST), Republic of\nKorea\n7ICREA Institucio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain\nE-mail: stephan.roche@icn2.cat\nAbstract. This study focuses on amorphous boron nitride ( α-BN) as a novel diffusion\nbarrier for advanced semiconductor technology, particularly addressing the critical\nchallenge of copper diffusion in back-end-of-logic (BEOL) interconnects. Owing to\nits ultralow dielectric constant and robust barrier properties, α-BN is examined as\nan alternative to conventional low-k dielectrics. The investigation primarily employs\ntheoretical modeling, using a Gaussian Approximation Potential, to simulate and\nunderstand the atomic-level interactions and barrier mechanisms of α-BN. This\nmachine learning-based approach allows for realistic simulations of its amorphous\nstructure, enabling the exploration of the impact of different film morphologies on\nbarrier efficacy. Complementing the theoretical study, experimental analyses are\nconducted on Plasma-Enhanced Chemical Vapor Deposition (PECVD) grown α-BN\nfilms, evaluating their effectiveness in preventing copper diffusion in silicon-based\nsubstrates. The results from both the theoretical and experimental investigations\nhighlight the potential of α-BN as a highly effective diffusion barrier, suitable\nfor integration in nanoelectronics. This research not only proposes α-BN as a\npromising candidate for BEOL interconnects but also demonstrates the synergy of\nadvanced computational models and experimental methods in material innovation for\nsemiconductor applications.arXiv:2402.01251v1  [cond-mat.mtrl-sci]  2 Feb 2024Amorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 2\n1. Introduction\nContinuous down-scaling of devices and architectures has been the major goal of the\nelectronic industry for decades to increase the performances and chip density, and reduce\npower consumption [1, 2, 3]. The technology of interconnects is also experiencing\nsuch extreme scale-down, with sustained efforts to improve the resistance of metals\nand to lower the dielectric constant of intermetal dielectrics (IMDs), for optimized\ncontrol of signal delays such as RC delay. The diffusion barrier that prevents the\ninteraction between metals and IMD is also becoming thinner and is required to have\nexcellent blocking properties. Ultimately, the technology aims for an almost “barrier-\nfree” structure. Two-dimensional (2D) materials, such as MoS 2and h−BN have recently\ngarnered attention as potential barrier materials. Indeed, recent studies have shown\nthat MoS 2, when synthesized through Atomic Layer Deposition (ALD), can act as an\neffective Cu diffusion barrier, potentially reducing barrier thickness down to 1-2 nm.\nFurthermore, the integration of dopants into MoS 2such as Niobium (Nb) has been\nfound to enhance its ability to block Cu atom diffusion, indicating its suitability for\nsub-5 nm technology nodes [4, 5]. However, while these 2D materials could act as\nefficient diffusion barriers, they also have a higher dielectric constant which would add\nto the RC-delay.\nHexagonal boron nitride (h −BN) on the other hand could be a better choice given\nits low dielectric constant and higher bandgap. However, the practical application of\nthese 2D materials is confronted with significant challenges. Integrating these materials\ninto existing semiconductor manufacturing processes, especially developing a BEOL-\ncompatible growth process for deposition on dielectrics remains a major challenge\n[6, 7, 8]. Moreover, the presence of defects in 2D materials, such as vacancy defects\nin hBN and MoS 2, can lead to Cu accumulation, creating diffusion pathways that\ncompromise the barrier’s effectiveness [9]. Amidst the ongoing quest for materials that\ncan meet the stringent demands of advanced semiconductor technologies, amorphous\nboron nitride ( α-BN) emerges as a particularly compelling candidate. In contrast to h-\nBN,α-BN does not suffer from dangling bonds and grain boundaries. In addition to its\nlow temperature growth possibility, α-BN films can be uniformly synthesized on various\nsubstrates (such as metals and dielectrics) with a controllable thickness. This makes α-\nBN a desirable dielectric material and a diffusion barrier [10, 11]. Besides, α-BN has been\nreported to have a low dielectric constant as a result of poor dipole alignment due to its\namorphous nature [12]. While Hong et al.[2] reported an ultralow dielectric constant of\n1.7 for 3 nm PECVD-grown α-BN. Glavin et al.[11] pointed out that homonuclear bonds\ncan open conductive pathways and increase the dielectric constant. They measured a\ndevice dielectric constant of atomic layer deposition (ALD)-grown α-BN of 5.9 with a\nband gap of 4.5 eV [11]. Similarly, the dielectric constant of ALD grown α-BN at 65 -\n250ºC is between 4.3 - 9.0 which increases with faster cooling. They observed that faster\ncooling leads to N deficiency and higher oxygen contamination, causing midgap states\nand higher dielectric constant [13]. Lin et al.[14] showed that the dielectric constantAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 3\ncan be monitored during the growth process and used precursor, to ultimately achieve\na dielectric constant of 2.1 for a sub-10 nm α-BN film. Meanwhile, Abbas et al.[15]\nreported dielectric constant of 2.61 - 5.88 for sputtered α-BN films with thicknesses\nranging between 12 nm and 800 nm[15]. Due to its low dielectric constant, good\nmechanical properties, and thermal stability, α-BN films have already been shown to\nact as efficient metal diffusion barriers. While Hong et al. [2] showed that 3 nm α-BN\nfilm can stop Co diffusion at 600 ºC, Kim et al. [16] found that 7-nm- α-BN film can\nstop Cu diffusion up to 500 ℃. Hence, α-BN seems to be the ideal dielectric material\ndue to the ultralow dielectric constant and its great barrier properties. However, as\nshown in the literature [11, 14, 13, 17, 18], the properties and performance of α-BN are\ndetermined by their morphology. Hence, ALD α-BN with the correct composition could\nbe an ideal dielectric material due to the potential low dielectric constant and excellent\nmetal barrier properties.\nAn exhaustive experimental investigation of the morphology and properties of α-BN\nwould be quite time-consuming and expensive. On the other hand, atomistic simulations\nwould allow us to study the materials at the atomic level and can provide insights into\nthe relation between morphology and key parameters of materials faster with moderate\ncomputational cost. Notwithstanding, implementing an accurate modeling of highly\ndisordered materials such as α-BN is quite a challenging task. Due to the lack of long-\nrange order, the simulation box of amorphous materials should be quite large. This\nbecomes quickly prohibitive for ab initio calculations. On the other hand, simulations\nusing classical MD with empirical potentials can deal with such large samples. However,\ndue to simplifications and assumptions used to develop these interatomic potentials,\nclassical MD cannot capture the diverse atomic environment of the amorphous material.\nThese challenges can be overcome using machine learning-based approaches. Classical\nMD simulations with machine learning-based interatomic potential trained over ab initio\ncalculations can deal with very large simulation boxes with an ab initio accuracy [19, 20].\nHere, the relationship between the morphology of α-BN structures and their barrier\nproperties is assessed both theoretically and experimentally. First, we develop a\nGaussian Approximation Potential (GAP) model to describe the interactions between\nα-BN and Cu atoms. Later, we generate α-BN films with different thicknesses, and by\nemploying different parameters, we investigate their resulting morphologies and barrier\nproperties. We also study the mechanical properties of generated α-BN samples. Finally,\nwe present an experimental investigation of the barrier performance of PECVD-grown\n3-, 5- and 7-nm thick α-BN films against Cu diffusion to the Si-based substrate and\ncompare it with the simulations.\n2. Methods\nIn this study, we utilize both experimental and theoretical methods to investigate how\nα-BN can prevent copper diffusion. Initially, we train an interatomic potential model\nusing the Gaussian Approximation Potential (GAP) framework. This model is thenAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 4\n2-body 3-body SOAP\nδ(eV) 2.0 0.1 0.1\nrcut(˚A) 3.7 3.0 5.0\nr∆(˚A) 0.5\nσat(˚A) 0.5\nnmax,lmax 8\nζ 4\nSparsification Uniform Uniform CUR\nNt(α-BN/Cu samples) 150 3000\nNt(Crystalline samples) 50 1000\nNt(Total) 25 200 4000\nTable 1: Parameters used to train the GAP potential for α-BN/Cu heterostructures.\napplied in molecular dynamics (MD) simulations to explore the relationship between\nthe morphology, properties, and barrier performance of melt-quenched (MQ) α-BN.\nThese simulations consider variations in thickness and cooling rates. Subsequently, we\nmeasure the barrier properties of PECVD-grown α-BN films of different thicknesses\nby detecting the presence of copper silicide crystals, which aids in correlating our\nexperimental findings with the theoretical work.\n2.1. Gaussian Approximation Potential\nDue to the complex nature of aBN structure, traditional empirical interatomic poten-\ntials cannot provide a clear and accurate picture of how these atoms interact with each\nother. Moreover, there is no available empirical interatomic potential for BN and Cu\ninteractions. Hence, we first generated a new GAP model based on a training set gen-\nerated via first-principle methods.\nThe GAP model is developed using two sets of samples, ”training” and ”validation”\nsets which include positions of atoms, atomic energies, forces, and stresses calculated\nusing Car Parrinello Molecular Dynamics and density functional theory with Quantum\nEspresso package [21, 22, 23]. A multi-step procedure is employed to train the GAP\nmodel: 1) initial training set is generated using Car-Parrinello MD and 2) a GAP model\nis trained over this set; 3) a new set of validation set is generated using the previously\ngenerated GAP model; 4) force and energy estimation of GAP model is evaluated using\nDFT calculations. These iterative training-validating steps are repeated until sufficiently\nlow RMSE values are reached.\nThe ”training” and ”validation” sets contain more than 3000 structures. Here, training\nsets contain atomistic structures of different phases of boron nitride, Cu-diffused α-BN\nsamples, α-BN/Cu interfaces, and Cu atoms. Periodic boundary conditions are applied\nto the cell. Ultrasoft pseudopotentials (USPP) with LDA as an exchange-correlationAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 5\nFigure 1: RMSE of energy and forces on training and validation sets.\nfunctional have been used for Car-Parrinello MD calculations. Kinetic energy cutoff for\nwavefunctions and energy are set to 150 Ry and 20 Ry. DFT calculations have been\ncarried out using PAW pseudopotentials with PBE exchange-correlation functionals. An\nenergy cutoff of 75 Ry and 600 Ry has been employed for the wavefunction and electron\ndensity, respectively. To properly take into account the van der Waals (vdW) inter-\nactions between BN/Cu interfaces, the Tkatchenko-Sheffle (TS)[24], since it is already\nproven efficient for both BN and Cu systems seperately [25, 26]. A Gaussian smearing\nof 0.1 eV width is applied to electronic levels.\nThe GAP model is trained using the ”quippy” code. Here, we employed three descrip-\ntors: two-body(2b), three-body(3b), and smooth overlap of atomic positions (SOAP).\nThe 2b descriptor is used to describe the pairwise interactions between atoms. 3b and\nSOAP descriptors represent the contribution of the triplets of atoms and the local en-Amorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 6\nFigure 2: A sample of simulation cells which contains 5 nm α-BN film between two\ncopper films where B, N, and Cu atoms are represented as pink, blue, and orange,\nrespectively.\nvironment around an atom. The cut-offs and hyperparameters for each descriptor have\nbeen presented in Table 1. The GAP models have been proven to be useful for several\nmaterials [17, 18, 27, 28]. Sparsification with CUR method [29] has been selected for the\nSOAP kernel while sparsification on a uniform grid is used for two-body and three-body\nkernels.\nLater, the quality of the trained GAP model is assessed by comparing the obtained\nenergies per atom and forces of samples in the training and validation sets with the\ntrained GAP model and DFT calculations. Significantly low root-mean-squared-errors\n(RMSE) values are obtained for training and validation sets as presented in 1. A slightly\nhigher RMSE for the validation set is expected since those structures are not used for\ntraining the model. Low RMSE values show a good agreement between the GAP model\nand DFT calculations.\n2.2. Molecular Dynamics Simulations\nTo evaluate the barrier properties of α-BN samples, we generated various samples with\ndifferent thicknesses varying from 3 nm to 7 nm using the melt-quenched protocol.\nHere, we first heat the randomly positioned BN atoms at random positions to 5000 K\nand later reduce the temperature to 300 K employing varying cooling rates. After the\nquenching, we also employ annealing steps to reduce the amount of homonuclear bonds\nand other impurities. By employing the cooling rates, we achieve different morphologies\nwhich allows us to understand how the morphology affects the barrier properties. After\nthe generation of samples, the morphological (amounts of sp1,sp2and sp3, amount of\nhomonuclear bonds, nanovoids) and mechanical properties are evaluated. After the\nstructural characterization, we put the α-BN samples between two [0001] cubic Cu\nlayers which are 5 nm thick. Here, the interactions between aBN and Cu atoms are\ncalculated using a newly developed GAP model. Fig. 2 shows a sample structure used\nin this paper. All of the simulations are carried out by the Large-scale Atomic/Molecular\nMassively Parallel Simulator (LAMMPS) package [30].Amorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 7\n2.3. Experimental Studies\n2.3.1. α-BN Film Growth To experimentally demonstrate the barrier properties of\naBN samples, we employed capacitively coupled plasma-chemical vapor deposition\n(CCP-CVD) to deposit aBN films. The Si substrate was placed on a substrate holder\nin the deposition chamber. The substrate was heated at a rate of 10 ºC/min and\nreached growth temperatures between 350 and 400 ºC. Following a 20-minute annealing\nprocess, a borazine precursor (B 3H6N3, Gelest) was introduced into the chamber at a\nrate of 0.05 sccm with a hydrogen flow of 20 sccm. During the growth process, the\nplasma operated at a power of 20 W. The growth time was adjusted between 20 and 80\nminutes depending on the thickness of the aBN film. After the growth was completed,\nthe furnace was slowly cooled down to room temperature.\n2.3.2. Experimental Cu Diffusion Barrier Test We assessed the Cu diffusion barrier\nproperties of the grown aBN films. For comparison with simulation results, 3-nm, 5-\nnm, and 7-nm thick aBN films on Si substrate were prepared and a 50-nm thick Cu film\nwas deposited on aBN/Si by e-beam evaporation. Cu/aBN/Si samples were annealed\nat the temperature range of 400 ºC to 600 ºC for 30 minutes using a vacuum furnace.\nSubsequently, these samples were immersed in a Cu etchant (49-1, Trensene) for 20\nminutes to remove the Cu films, followed by a deionized water rinse. to remove the\nexcess etchant. Cu diffusion was evaluated through confirmation of the formation of\ncopper silicide using X-ray diffraction (XRD) and scanning electron microscopy (SEM).\n3. Results and Discussion\n3.1. Molecular Dynamics Simulations\nBefore studying the barrier properties, a rigorous morphological investigation was\ncarried out for MQ samples to understand how different fabrication conditions change\nthe morphology of α-BN samples. Fig. 3 presents how the amount of sp1, sp3, and\nhomonuclear bonds change with the different cooling rates. As cooling rates are getting\nhigher, atoms cannot find enough to organize in an ordered way and are trapped in\na disordered state. Hence, fasting cooling rates lead to more disordered samples with\nhigher amounts of homonuclear bonds and sp1hybridized atoms, which may adversely\naffect the stability of the films. On the contrary, slowly cooled α-BN samples tend\nto have a high amount sp2hybridized atoms and a low amount of homonuclear bonds.\nEven though sp3-hybridized atoms do not adversely affect the stability of the films, they\nalso diminish with the slower cooling. The annealing step can help atoms reorganize\nthemselves and generate a more ordered structure. However, as shown in Fig. 3, the\nsamples generated with high cooling rates still have higher homonuclear bonds and\nsp1. The difference between the bonding structure of films generated with different\ncooling rates can be observed in the first peak of RDF in Fig.4. While slowly cooled\ndown samples have a narrower peak around 1.42 ˚A, average length of B-N bonds, fastlyAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 8\n(a)\n (b)\nFigure 3: Changing ratio of sp1, and sp3hybridized atoms (a) and homonuclear bonds\n(b) of 3 nm, 5 nm, and 7 nm films. Results are averaged on 5 different samples.\nFigure 4: Radial Distribution Function of melt-quenched 3-nm thick α-BN films.\ncooled down samples have broader peaks due to the presence of shorter and longer B-N\nbonds, and the presence of N-N and B-B bonds, which have average bond lengths of 1.4\n˚A and 1.81 ˚A, respectively. RDF also reveals the amorphicity of the samples can vary by\nemploying different cooling rates. Additionally, samples cooled down faster tend to have\nnanovoids which may provide extra pathways for Cu atoms diffusion. While samples\ncooled down slower have a density around 2.18 g /cm3or higher, samples quenched fast\nhave a lower density which is caused by the nanovoids and other defects in the sample.\nAfter the structural characterization, the full elastic stiffness matrix of generated\nα-BN samples is calculated by using the stress fluctuations and the second derivatives ofAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 9\nFigure 5: Average Young’s Modulus of α-BN samples generated various cooling rates.\nthe energy with respect to the strain, i.e. Born matrix. Young’s modulus of generated\nα-BN samples with different thicknesses are presented in Fig. 5. Results show that fast-\ncooled α-BN samples with high amounts of sp1-hybridized atoms and homonuclear bonds\nhave lower Young’s modulus. Larger amounts of sp3and less homonuclear bonds have\nhigher density and Young’s modulus which is desired for diffusion barrier. While fast-\ncooled samples have Young’s modulus of 206 GPa, Young’s modulus of slowly cooled\nsamples may be as high as 280 GPa. Similar trends are observed in the literature\n[14, 17, 18, 31]. Good mechanical properties and stability are crucial for future barrier\nmaterials and dielectrics which makes them more suitable for integration.\nFinally, we create α-BN/Cu heterostructures to understand the barrier properties\nof the films. Hence, the diffusivity of Cu atoms is extracted from the mean-squared\ndisplacement (MSD). This helps us to evaluate the thermal stability of α-BN samples\nand how Cu atoms can penetrate through the barrier. The positions of atoms are printed\nat every 10 fs and visualized using a visualization tool, OVITO[32]. We later calculate\nthe MSD by evaluating the displacement of atoms relative to their initial positions. Fig.\n6 shows the diffusivity of Cu atoms as a function of temperature. As shown in Fig.\n6-a, all 7-nm-thick films can stop Cu diffusion regardless of the density of α-BN films\nup to 1000 K which exceeds the expectation of the interconnect industry. 7-nm-thick\nfilms have a more ordered structure due to the high amount of sp2-hybridized atoms and\nfewer homonuclear bonds compared to the thinner films cooled with the same cooling\nrates and quenching protocol. A high amount of sp2and sp3hybridized bonds improve\nthe thermal stability of films, and reduce the cracks or nanovoids that create pathways\nto Cu atoms. The sp2and sp3dominated samples with low amounts of homonuclear\nbonds, along with the large thickness, can prevent the Cu atoms from diffusing to α-BN\nfilms and then to the other Cu region. Even though thicker films are better diffusionAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 10\n(a)\n (b)\n(c)\nFigure 6: Diffusivity of Cu atoms with 7-nm (a), 5-nm (b), and 3-nm (c) thick aBN\nbarrier films.\nbarriers, the dielectric constant can be higher for thicker films which can increase the RC\ndelay and prevent further down-scaling of the devices. We also calculated the diffusivity\nof Cu through thinner a-BN as shown in Fig. 6-b and c where the data shows that the\ndiffusivity depends largely on the morphology.\nAs shown earlier above, the variation of atom hybridizations and bond types of\n3-nm and 5-nm films are more sensitive to the cooling process compared to 7-nm thick\nfilms. These films have more sp1-hybridized atoms and homonuclear bonds compared\nto 7-nm films, and some have nanovoids and cracks which might be detrimental to\nthe barrier properties. Fig. 6-b and c show that films with higher density, less sp1, and\nhomonuclear bonds can be better diffusion barriers than the thicker films suggesting that\nthe thinner films might meet or exceed the Cu-barrier requirements at the industrialAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 11\nrelevant thermal budgets. Analogous to the mechanical properties and amount of sp3-\nhybridized atoms, both 3-nm and 5-nm films with the highest density have better Cu\ndiffusion barrier properties up to 1000K. These structures have only a negligible amount\nof B-Cu or N-Cu bonds (mostly near the α-BN/Cu interface) and sp1-hybridized atoms\n(3-5%) and a significant amount of sp3-hybridized atoms (10-14%). The positive impact\nof sp3-hybridized atoms on the stability is already shown in the literature[17, 18]. Even\nthough a certain amount of sp3-hybridized atoms increases the density and decreases\nthe voids, it should be noted that a high amount of non-sp2-hybridized may lead to\nsome internal stress which would be the main cause of the nanovoids we observed in\nfast cooled samples. Moreover, higher sp3hybridization and density, even though it\nmay improve the diffusion barrier performance of the film, may cause to formation\nof additional conductive pathways and increase the dielectric constant of the material\n[11]. Since a higher dielectric constant of α-BN would increase the RC delay of the\ndevice, the impact of the sp3-hybridizations on electronic and optical properties should\nbe investigated thoroughly.\nDespite the positive impact of sp3-hybridized atoms on the barrier performance of high-\ndensity samples (blue lines in Fig.6-b and -c), nearly full sp2-hybridized B-N network\nare very effective in stopping the Cu diffusion (as shown the red straight lines in Fig.6-\nb and -c). Both these samples have the slowest quenching rate during the sample\ngeneration and have insignificant amounts of non-sp2-hybridized atoms or homonuclear\nbonds. Although some Cu atoms manage to diffuse to these samples, they could only\ndiffuse to the near-interface region since sp2B-N network may not have sufficient Cu-\ndiffusion pathways. As expected, due to low stability and extra diffusion pathways\nformed by cracks and nanovoids, Cu atoms could easily diffuse to the fast-cooled α-\nBN. Fig. 7 visualizes how Cu atoms diffuse into the two α-BN films generated with\ndifferent cooling rates at 800 K. The α-BN film in Fig.7-a (d=2.17 g /cm3in Fig.6-c)\nin the figure is cooled slowly, hence it has mostly B-N and sp2-hybridized atoms, and\nnearly no nanovoids. This film could stop Cu diffusion up to 1000 K, only then it allows\nonly a small number of Cu atoms could penetrate through the α-BN film. Meanwhile,\nthe film in 7-b (d=1.99 g /cm3in Fig.6-c) is cooled fast and has substantial amount of\nhomonuclear bonds, sp1-hybridized atoms and nanovoids. This film could not stop the\nCu diffusion at lower temperature and lost its structural integrity at 800 K completely\nand Cu atoms completely diffuse to the film.\nTo experimentally evaluate the Cu diffusion barrier properties of aBN, we fabri-\ncated Cu/aBN/Si structures and annealed them at 400 - 600 ℃for 30 minutes. After\nannealing at 400 ℃, a Si substrate coated with a 3-nm thick aBN film did not form\nany copper silicide (Fig. 8-a), while a Si substrate not covered with aBN formed a\nsubstantial amount of copper silicide (Fig. 8-e). After annealing 3-nm thick aBN/Si at\n500℃, very small copper silicide nucleations were observed on the Si surface (Fig. 8-b),\nand it was confirmed that the formation of copper silicide crystals after annealing at\n600℃(Fig. 8-d). Fig. 8-c provides a brief summary of Cu diffusion barrier properties\nof 3-nm, 5-nm, and 7-nm thick aBN films after the 400 - 600 ℃annealing process. TheAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 12\nFigure 7: Position of Cu atoms at 800 K. With the increased temperature, some Cu\natoms diffused through aBN film. While aBN film above has a density of 2.17 g /cm3,\nalmost no homonuclear bonds and sp1-hybridized atoms (a), aBN film below has a\ndensity of 1.99 g /cm3, high amounts of homonuclear bonds and sp1-hybridized atoms\n(b). BN atoms on the right side are hidden to make diffused Cu atoms more visible.\nobservation of crystallization by SEM is fine but not sensitive enough to determine Cu\ndiffusion in the sub-silicide formation regime - that is - you could have Cu in the Si\nbut SEM resolution is insufficient to detect or within the solubility limit of Cu in Si\n(Fig. 9). The characteristics such as density or existence of defects of experimentally\nsynthesized aBN films differ from the simulated aBN films and do not exhibit the same\nbarrier performance as the simulations. However, there is a tendency for the barrier\nproperty to improve as the thickness of the aBN film increases.\n4. Conclusion\nIn summary, this study reports a comprehensive analysis that underscores the potential\nofα-BN as an effective Cu diffusion barrier that could be used for future-generation\nintegrated circuits to meet the ever-demanding RC-delay time constant. MD simulations\nwith Gaussian Approximation Potential (GAP) modeling have allowed for a nuanced\nunderstanding of how different morphologies of α-BN influence its barrier properties.\nComplementing the theoretical insights, our experimental investigations confirm the\nefficacy of α-BN films to act as significant resistance for blocking copper migration.\nSuch findings demonstrate that α-BN films exhibit an efficient barrier against copperAmorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 13\nFigure 8: Experimental result of Cu diffusion barrier test. (a-c) SEM images of 3-\nnm thick aBN/Si surface after 400 ºC (a), 500 ºC (b), and 600 ºC (c) annealing and\nremoving residual Cu films with Cu etchants. (d) Evaluation of Cu diffusion barrier\nproperties of 3-nm, 5-nm, and 7-nm thick aBN/Si after 400 600 ºC annealing process.\nBlue O means aBN film successfully blocked Cu diffusion. Yellow Delta means there\nis small copper silicide nucleation through the defect of aBN films. Red X means aBN\nfilm failed to block Cu diffusion and copper silicide crystals are formed. (e) SEM image\nof Si surface without α-BN barrier films after 400 ºC annealing and removing residual\nCu films with Cu etchants. (f) XRD spectrum of Si surface with copper silicide crystals\nwithout α-BN barrier films that shows the formation of copper silicide crystals.\ndiffusion and hence respond to a major challenge for BEOL integration of metal\nwiring. This work not only contributes to the evolving landscape of material science in\nnanoelectronics but also sets a precedent for future research, which should focus on fine-\ntuning the properties of α-BN and exploring its integration into existing semiconductor\nmanufacturing processes.\nFinally, one should add that the use of Artificial Intelligence-based techniques is the\ncornerstone to obtain realistic interatomic potentials or electronic models able to deliver\nquantitative predictions [33, 34]. This has become fundamental to achieve predictive\nmodeling of thermal stability, mechanical [17, 18], electronic, dielectric properties [35],\nor other properties in amorphous forms of boron-nitride composites, enabling further\nmaterials optimization and performance improvement [36, 37].Amorphous Boron Nitride as a Diffusion Barrier to Cu Atoms 14\nFigure 9: SEM images of 3-nm, 5-nm, and 7-nm thick aBN/Si after 400 600 ℃\nannealing and removing residual Cu films with Cu etchants. the insets boxed in red\nrepresent enlarged images of the copper silicide.\nAcknowledgement\nThis project has been supported by Samsung Advanced Institute of Technology and\nis conducted under the REDI Program, a project that has received funding from\nthe European Union’s Horizon 2020 research and innovation programme under the\nMarie Sk lodowska-Curie grant agreement no. 101034328. This paper reflects only the\nauthor’s view and the Research Executive Agency is not responsible for any use that\nmay be made of the information it contains. ICN2 acknowledges the Grant PCI2021-\n122092-2A funded by MCIN/AEI/10.13039/501100011033 and by the “European Union\nNextGenerationEU/PRTR”. 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University of Oviedo at Gij´ on, 33204 Asturias, Spain\nhttp://www.aic.uniovi.es\nAbstract\nThis research arises from the need to predict the amount of air pollutants\nin meteorological stations. Air pollution depends on the location of the sta-\ntions (weather conditions and activities in the surroundings). Frequently,\nthe surrounding information is not considered in the learning process. This\ninformation is known beforehand in the absence of unobserved weather con-\nditions and remains constant for the same station. Considering the sur-\nrounding information as side information facilitates the generalization for\npredicting pollutants in new stations, leading to a zero-shot regression sce-\nnario. Available methods in zero-shot typically lean towards classification,\nand are not easily extensible to regression. This paper proposes two zero-\nshot methods for regression. The first method is a similarity based approach\nthat learns models from features and aggregates them using side informa-\ntion. However, potential knowledge of the feature models may be lost in the\naggregation. The second method overcomes this drawback by replacing the\naggregation procedure and learning the correspondence between side infor-\nmation and feature-induced models, instead. Both proposals are compared\nwith a baseline procedure using artificial datasets, UCI repository commu-\nnities and crime datasets, and the pollutants. Both approaches outperform\nthe baseline method, but the parameter learning approach manifests its su-\nperiority over the similarity based method.\nKeywords: Side information, zero-shot regression, air pollution prediction\n∗Corresponding author\nEmail addresses: miriam.fernandezdiaz@arcelormittal.com (Miriam Fdez-D´ ıaz\n),quevedo@uniovi.es (Jos´ e Ram´ on Quevedo), montaneselena@uniovi.es (Elena\nMonta˜ n´ es)\nPostprint submitted to Information SciencesarXiv:2402.01252v1  [cs.LG]  2 Feb 202421. Introduction\nThe research presented in this paper arises from the desire to predict air\npollutants in meteorological stations. Air pollution depends on the location\nof the stations. On the one hand, weather conditions, which in turn depend\non the climate and on the surrounding orography (mountains, sea, etc.), have\na notable influence on air pollution. On the other hand, information about\nthe stations, such as the activities that take place in the surrounding areas\n(industry, power plants, leisure centers, residential areas, shopping centers,\nadministrative buildings...) also condition air pollution. Typically, the sur-\nrounding information is used merely to split the original task into as many\nseparate tasks as the number of available meteorological stations, and only\nthe weather conditions are considered as features in the learning process [19].\nTwo studies [3, 23] represent a preliminary advance in incorporating infor-\nmation about the stations. The former [3] takes pollutant values from the\nnearest neighbor station of the target stations, whereas the latter [23] per-\nforms a prior classification of the stations, taking into account information\nabout roads, traffic flow, or whether they are located in urban, suburban or\nindustrial areas. However, this valuable information about the stations is not\nincluded in the learning procedure. This information is known beforehand in\nthe absence of unobserved weather conditions and remains constant for the\nsame meteorological station. This fact allows us to regard this information as\nbeing what we usually refer to as side (or privileged) information [38]. Side\ninformation is neither features nor targets; it actually constitutes additional\nand, prior information about how targets (a pollutant measure in stations)\nare related to features (weather conditions). It may well be a promising in-\nformation source susceptible to being exploited to improve the accuracy of\nthe predictions. Furthermore, treating information about stations as side\ninformation allows us to make predictions over potential future locations of\nmeteorological stations, for which only side information is available, enabling\nus to state the problem as a zero-shot regression learning task.\nThere already exist many approaches to handling zero-shot learning tasks\n[17, 18, 35, 47]. In fact, researchers typically associate zero-shot learning to\ncomputer vision or image classification [18, 47]. Additionally, the existing\nzero-shot classification methods are specifically designed for classification,\nusually for image classification. Despite this, there are few studies that cope\nwith other classification applications [35]. The main drawback of these ex-\nisting classification methods is that their strategies are built on the basis of\n3a classification problem and are therefore not extensible to regressions, or if\nthey are, it is a non-trivial and a non-straightforward task [32]. Certainly, we\ncan commonly find regression-based zero-shot methods for zero-shot learn-\ning [17, 18, 26, 35, 47]. But we must distinguish these methods, that are\nspecifically designed for the classification zero-shot task, from methods for\nthe actual regression zero-shot task. The fact is that these frequently labelled\nregression-based zero-shot methods propose regression to learn a projection\nfunction in certain phases of the zero-shot classification task, typically in\norder to map target instances into either feature spaces or side information\n(semantic) spaces, or even into other kinds of spaces. Hence, they are not\napplicable to the regression zero-shot task. At present, only two methods\nable to cope with zero-shot regression tasks exist in the literature [32, 48].\nThe first [32] is a preliminary study, whose experiments include a single toy\nexample based on a beta distribution with just one feature and two side infor-\nmation features. In fact, the available software reports the message ”expected\n1D vector for x” when attempting to perform experiments over datasets with\nhigher dimensions in features and side information. The second [48] is a deep\nlearning procedure specifically built to predict the future position of a piece\nthat is pushed by a robotic arm, given the present location. The features\n(which are information from the images of the piece in the present) are taken\ntogether with the side information (additional characteristics of the piece).\nThe main drawback of this method is that the strategy to include side infor-\nmation in the zero-shot learning can only be used in neural networks, and\nnot for a general purpose regressor.\nThe contribution of this paper, then, is to provide methods to cope with\na generic purpose regression zero-shot task; that is to say, the strategies will\nnot depend on the regressor. Two new approaches are proposed. The first\none learns a model per observed target (station) from features (weather con-\nditions). Then, it takes the side information (surrounding information about\nthe stations) of both observed (stations) and unobserved (possible future lo-\ncations of stations) targets through a similarity based relationship method\nthat aggregates the feature models predictions. However, this aggregation\nstrategy has limited generalization power, since it only interpolates feature\nmodel predictions that will be bounded within a certain range, discarding\ndirect knowledge from the models themselves, learned from features. In this\nsense, the second approach constitutes an alternative that transfers the in-\nduced feature models knowledge to another learning procedure which is also\nfed by target side information. Specifically, it is a parameter learning model\n4correspondence method that learns the parameters of the unobserved target\nmodels. So far, and to the best of our knowledge (i) none of the existing\nstudies take surrounding information for the stations as side information for\npredicting air pollution, nor do they generalize air pollution prediction to\nnew unobserved stations. Furthermore, (ii) none of the studies include fea-\nture model knowledge together with side information into a zero-shot learning\nprocedure for regression tasks.\nThe rest of the paper is organized as follows: Section 2 discusses existing\nstudies related to the research described in this paper. Section 3.1 deals\nwith the notation and the problem statement in a zero-shot framework with\nside information. Following that, the two approaches we propose for zero-\nshot regression are detailed in Section 4. The experiments carried out and\ntheir results are discussed in Section 5. Finally, in Section 6 we draw some\nconclusions and suggest certain lines of research for future work.\n2. Related Work\nSide information is potential information that relates features and tar-\ngets and is susceptible to being exploited in order to improve the prediction\nperformance. Sometimes this additional information is not available. If that\nis the case, the common practice in traditional machine learning approaches\nis usually to discard it, or if it is not discarded, to consider it as common\nfeatures. The importance of this privileged information is such that in some\nstudies efforts are directed to extracting, discovering or learning it even before\ncarrying out the learning procedure [6, 39]. On other occasions, the infor-\nmation is already available because experts have collected it [26, 29]. Side\ninformation can be available in different forms for features [22] or targets\n[37]. In addtion, there are several ways of making side information available:\nin a hierarchy [10], in a response prediction [20], in a structure [11] or as\nfeature representation [13, 26]. Furthermore, this information source could\nbe heterogeneous and multimodal: for instance, in a recommender system\ncontext [1].\nAmong the possibilities opened up by exploiting side information, zero-\nshot learning [9, 34] has become more relevant in recent years. Zero-shot tasks\n[41] aim to make predictions for new targets for which no observable data is\navailable. Similar scenarios to zero-shot learning are one-shot learning [16]\nand few-shot learning [28], which consist of having respectively one and few\nobservable data available for the new targets. Side information in the new\n5targets becomes particularly relevant in the zero-shot scenario, since it is the\nonly available information that can be exploited in order to yield predictions\ngiven the lack of observable data for the new targets [14]. This fact allows the\nzero-shot scenario to convert the traditional learning paradigm from trans-\nductive to inductive, with respect to the targets [41]. Conventional machine\nlearning usually assumes transductive learning with regard to the targets,\nsince the models are induced for specific known and observable targets. In\nrelation to a zero-shot scenario, which is characterized by the existence of\nunknown and unobservable targets, we can assume both transductive and\ninductive learning as far as targets are concerned. The difference between\nthem lies in whether to obtain the models for specific (transductive) or for\ngeneric unknown and unobservable (inductive) targets. Most of the studies\nabout zero-shot deal with inductive learning for targets, but there are some\nthat keep the transductive learning property [30]. The great majority of\napplications of these scenarios are related to computer vision classification\ntasks [21, 42, 45, 49]. However, they have also been applied to natural lan-\nguage processing [12, 15], mobile and wireless security [33], or human activity\nrecognition [40] classification tasks.\nZero-shot learning (as well as one-shot and few-shot learning) is closely\nrelated to transfer learning. In fact, the absence (or scarcity) of observable\ndata for targets provides transfer learning with an important role in this\ncontext [5, 50]. In transfer learning, the observable instances and targets are\nreferred to respectively as source domain and source task, whereas unobserv-\nable instances and targets are referred to as target domain and target task.\nThe principle of transfer learning is to transfer the knowledge contained in\nthe source domain and source task in order to solve the target task in the\ntarget domain. This means transferring the knowledge from observable in-\nstances and targets to unobservable instances and targets. Transfer learning\ncan be split into inductive, transductive or unsupervised transfer learning\ndepending on the different existing situations between the source and target\ndomains, and the tasks [27]. In the inductive transfer learning setting, the\ntarget task is different from the source task, regardless of whether the source\nand target domains are identical or not. In the transductive transfer learn-\ning setting, the source and target tasks are the same, while the source and\ntarget domains are different. Finally, in the unsupervised transfer learning\nsetting, the target task is different from the source task and the main focus\nis placed on unsupervised learning tasks. Traditionally [27], instances with\ntheir target values (labels in classification) have been required in the target\n6domain for inductive transfer learning, whereas no labeled data in the target\ndomain is available when it comes to transductive transfer learning. But in\nboth cases, a number of instances in the target domain are required. The\ncase studied in this paper can be seen as belonging to the category of induc-\ntive transfer learning. In addition, the source and target domains are the\nsame (the features are the same in both domains). However, no instances\nare available in the target domain, and so classic transfer learning strategies\nare not applicable to our context. More recently, some studies [2, 36, 46]\nhave included transfer learning techniques in a zero-shot framework despite\nlacking instances in the target domain, but have only done this for classifi-\ncation tasks. Typical strategies followed in these cases involve specific tasks\nfor which they were tailor-made, and, hence they are not suitable for a gen-\neral purpose task. These strategies are, for instance, finding outliers [36],\nextracting specific image characteristics [46], or generating synthesized dia-\nlog instances [2]. But whatever the case, all of them apply specific strategies\nthat are commonly adopted in classification, but that are not easy, or are\neven impossible, to extend to regression.\nThe research addressed in this paper fits into a zero-shot rather than into\na one-shot or a few-shot scenario, since new targets (stations) lack observable\ninstances (weather conditions). In addition, our task fits into an inductive\nrather than into a transductive learning framework for both instances and\ntargets because of the need to produce models for generic unknown and\nunobservable instances (new weather conditions for any station) and targets\n(potential stations in new locations). Also, it is a zero-shot regression rather\nthan a classification task, since the targets are pollutant concentrations in\nstations. Fortunately, side information for targets is available and can be\nexploited in order to produce more accurate pollutant predictions in the\npotential locations of new stations. This information is presented in the\nform of homogeneous feature representation, consisting of all the activities\nthat take place in the surroundings of the stations’ new locations.\n3. Target zero-shot framework for regression\nThis section addresses the problem statement and also presents a classi-\nfication of the strategies to cope with it.\n73.1. Target inductive versus transductive zero-shot statement for regression\nLet us now formally state the zero-shot task in which side information is\navailable on targets. Let Xbe the feature space of instances, Sthe feature\nspace of targets (also called the semantic space) and Ythe image space of\nthe predictions. Let us denote To={to\ni∈ S:i= 1, . . . , m o}(To={to\ni}mo\ni=1\nin what follows) as the set of observed targets and Tu={tu\nj∈ S :j=\n1, . . . , m u}(Tu={tu\nj}mu\nj=1in what follows1) as the set of unobserved targets,\nsuch that To∩ Tu=∅. We will differentiate the image space Yo⊂ Y from\nYu⊂ Y for the observed targets Toand unobserved targets Tu. Hence, let\nus define Do={(xo\ni, yo\ni)∈ X × Yo:i= 1, . . . , n o}as the set of observed\ninstances for observed targets and Du={(xu\nj, yu\nj)∈ X × Yu:j= 1, . . . , n u}\nas the unobserved instances for unobserved targets, such that xo\ni̸=xu\njwhere\ni= 1, . . . , n oandj= 1, . . . , n u, that is, observed and unobserved instances\ndiffer from each other. Obviously, in general, yo\ni̸=yu\njwhere i= 1, . . . , n o\nandj= 1, . . . , n usinceTo∩Tu=∅. In other words, predictions of observed\nand unobserved instances also differ from each other.\nThe formal statement of zero-shot learning varies depending on whether\nan inductive or a transductive perspective is adopted (see Section 2). As\nobserved in Section 2, this paper will assume an inductive perspective in\nboth instances and targets, since the goal is to build a model for unseen new\nstations for which weather conditions have not been measured yet. However,\nwe shall formalize the goal of zero-shot learning under both perspectives with\nregard to targets in order to get a better understanding of the differences.\nUnder an inductive scenario, the goal of zero-shot learning is to learn a\nfunction f:X × S → YufromDoandToable to estimate Du\nYu={yu\nj∈\nYu:j= 1, . . . , n u}given a generic Du\nX={xu\nj∈ X :j= 1, . . . , n u} ⊂ X and\na generic Tu⊂S.\nUnder a transductive scenario, the goal of zero-shot learning is to learn a\nfunction fu:X → YufromDo,ToandTuthat can estimate Du\nYu={yu\nj∈\nYu:j= 1, . . . , n u}given a generic Du\nX={xu\nj∈ X :j= 1, . . . , n u} ⊂ X .\nHence, the difference lies in whether the side information of the unob-\nserved targets Tuis included in the training phase (transductive) or not\n(inductive). Figure 1 graphically represents such difference using a matrix\nrepresentation. For this purpose, the noobserved instances of Do\nXand the\n1Later on and without loss of generality we will assume that mu= 1, hence, just one\nunobserved target is referenced and it will be denoted as tu\n8nu(equals to 1 for simplicity) unobserved instances of Du\nXare respectively\nrepresented by the axnumber of features in matrices XoandXu(in pur-\nple). Analogously, the moobserved targets of Toand the mu(equal to 1\nfor simplicity) unobserved targets of Tuare respectively represented by the\nasnumber of semantic features in matrices SoandSu(in blue). Also, the\npredictions of the noobserved instances for the moobserved targets Do\nYo\nare placed in matrix Yo(in grey). Finally, fandfupredict the nu= 1\nunobserved instances of Du\nYu. In inductive learning, frequires the target un-\nobserved side information Suto make predictions, whereas fuin transductive\nlearning does not. In this last case, Suis required to learn fu.\n𝑢(                      ) (                      ,        )moSo\nXo Yomo\nno noaxas\n𝑓axas\nXuSu\nnu=1mu=1TRAINING PHASE TESTING PHASETarget inductive\nmoSo\nXo Yomo\nno noaxas\n𝑓𝑢axas\nXuSu\nnu=1mu=1TRAINING PHASE TESTING PHASETarget transductive\nax\nXunu=1\nasSu\nLearning Learning\nmu=1\nFigure 1: Training and testing phases for both inductive and transductive zero-shot learn-\ning using matrix representation.\nTable 1 summarizes the notation employed.\n9Notation Explanation\nX feature space of instances\nS feature space of targets (semantic space)\nY image space of predictions\nmo number of observed targets\nmu number of unobserved targets\nToset of observed targets\nTuset of unobserved targets\nYoimage space of predictions of the observed targets\nYuimage space of predictions of the unobserved targets\nno number of instances of the observed targets\nnu number of instances of the unobserved targets\nDoset of instances for the observed targets\nDo\nX the projection set of Doover the feature space X\nDo\nYo the projection set of Doover the subset Yo⊂ Y\nDuset of instances for the unobserved targets\nDu\nX the projection set of Duover the feature space X\nDu\nYu the projection set of Duover the subset Yu⊂ Y\nax number of features that represent the instances\nas number of features that represent the targets\nXofeature matrix representation of the noobserved instances\nofDo\nX\nXufeature matrix representation of the nuunobserved instances\nofDu\nX\nSosemantic feature matrix representation of the moobserved\ntargets of To\nSusemantic feature matrix representation of the muunobserved\ntargets of Tu\nYoprediction matrix for the noobserved instances of the mo\ntargets Do\nYo\nYuprediction matrix for the nuunobserved instances of the mu\ntargets Du\nYu\nf the induced function for inductive zero-shot learning\nfuthe induced function for transductive zero-shot learning\nfothe set of the moobserved targets models\nTable 1: Summary of the notation (The Yumatrix is available to test the quality of the\npredictions, but it is not used in the notation of the paper)\n103.2. Strategy classification for target zero-shot methods for regression\nExisting zero-shot learning approaches can be split into instance-based\nmethods and classifier-based methods [41]. However, before continuing, we\nshall rename the classifier-based methods to model-based methods in order\nto also include regression methods. Instance-based methods stand out for ob-\ntaining (by extracting them, by learning them, and so on) observed instances\nfor the unobserved targets, and afterwards a model for the unobserved targets\nis learned from these new instances. Model-based methods learn a model for\nthe unobserved targets directly from the information available.\nLet us focus on model-based methods, since all the proposed approaches\nin this paper aim to obtain a model directly for the unobserved targets.\nModel-based methods can be split into different types, depending on the\nstrategies they follow to build a model. Specifically, model-based methods\nfall into the category of relationship methods, correspondence methods and\ncombination methods [41]:\ni) The relationship methods build a model for the unobserved targets\nfrom the feature models obtained for the observed targets fo={fto\ni}mo\ni=1\nand from a relationship function δo,ubetween observed and unobserved\ntargets. Therefore, the function f:X × S → Yulearned for the\nunobserved targets will be a function of foandδo,u. In short, f(·) =\nF(fo, δo,u).\nii) The correspondence methods build a model for the unobserved targets\nlearning the correspondence between the side information of the ob-\nserved targets To={to\ni}mo\ni=1and the feature models obtained for the\nobserved targets fo={fto\ni}mo\ni=1. Therefore, the function f:X ×S → Yu\nlearned for the unobserved targets will be a function of foandTo. That\nis,f(·) =F(fo,To).\niii) The combination methods are based on decomposing the observed and\nunobserved targets into basic elements. As a result, the learning process\ntakes place for each of these basic elements. Finally, the models for\nthe basic elements are combined though an inference procedure to get\nf:X × S → Yu.\nThe next section will address the methods proposed in this paper.\n114. Target inductive zero-shot methods for regression\nThis paper assumes an inductive perspective in targets (and also in in-\nstances) for the zero-shot learning task, as it was detailed in Section 3.1.\nThis approach was motivated by the application that prompted this research,\nnamely, to build a model for unseen new meteorological stations for which\nweather conditions have not yet been measured. Hence, just the side infor-\nmation from observed targets Tois considered in the learning phase. The\nside information from unobserved targets Tuis thus discarded in this phase.\nThe proposed approaches fall into the category of model-based methods, so\nmodels for the unobserved targets will be built. One of these approaches\nclassifies as a relationship method, whereas the other belongs to correspon-\ndence methods. Hence, although both learn a model per observed target from\nfeatures fo={fto\ni}mo\ni=1, they differ in how they manage the side information\nof the observed targets Toin order to obtain the models for the unobserved\ntargets. The next subsections detail both of the newly proposed methods\nand a baseline method for comparison and reference.\n4.1. Baseline\nThis section describes a baseline method as a point of reference. The\nstrategy is to treat the side information as common features. Hence, there\nwill be several instances with equal values for the features that correspond\nto the side information. In the context of air pollution, the instances for\nweather conditions (features) of the same station will have equal values in\nthe features of the station description (side information). Figure 2 displays\nthe training and testing phases for this baseline approach using the same\nnotation as in Figure 1.\nRegarding the training phase (see left side of Figure 2), the features for\ntraining data for the baseline method are obtained joining features (weather\nconditions) Xotogether with side information (station description) So. The\ntarget for training data for the baseline method will be the values of Yo(the\npollutant concentration given the weather conditions measured in the station\ndescription). Therefore, this approach induces just one learning function\nfollowing a classical machine learning procedure, which corresponds to the\nfunction f:X ×S → Yu. With regard to the testing phase (see right side of\nFigure 2), the features of a new unobserved instance xu(represented by the\nfeature matrix Xu) and an unobserved target tu(represented by the semantic\nfeature matrix Su) are joined together to produce the prediction f(xu, tu).\n12no\nmoYo\nmoYono\nno\nmoYo\nmoYonoXo\nXo\nXo\nXoXo\nXo\nXo\nXoasSoXo noax\nmoTranspose\nSoRepeat XomotimesRepeat Sonotimes\nmoSomoSomoSomoSo\nas\nas\nas\nasmoSoas\n(                     )\nLearningXuSuConcatenate Xuand SuTranspose \nSuax\nnu= 1 Xuas\nmu=1Su\nSuSu\nasmu=1\nYomo\nnoTRAINING PHASE TESTING PHASE\nSu\nXu\nXo\nXo\nXo\nXoFigure 2: Training and testing phases for the baseline method\n4.1.1. A toy example\nDespite the process being simple, let us illustrate it through a toy example\nwith two observed targets, where the function fmust estimate y=x·s.\nFigure 3 displays the procedure with specific values for Xo,So,Yo,Xuand\nSu. The function fobtained during learning (see parameters {θi}4\ni=1in the\nfigure) gives low importance to the side information (note that θ3andθ4are\nnear to zero) and practically averages the features (note that θ1andθ2are\nnear 0 .5). The prediction of yobtained for the testing instance (1 .2,1.3) is\n1.28, which clearly differs from the actual value of y=x·swhich is 2 .5.\n4.1.2. Discussion\nThe main drawback of this baseline method is that different kinds of\ninformation are mixed together, assuming the same kind of relationship be-\ntween features and targets, as between side information and targets. Despite\nhaving the same kind of relationship, what actually happens is that features\nare related to targets through the side information, which means that the\n131 0\n1 0\n1 0\n1 01.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6-0.8Xo 602\nRepeat Xo2 timesRepeat So60 times\n2\nLearning1.2 1.31\n1\nConcatenate Xuand SuTranspose \nSu2\n1Xu2\n1Su\n21\n1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Yo\n2TRAINING PHASE TESTING PHASE\n0 1\n0 1\n0 1\n0 11 0\n0 12So21 0\n0 12So\n1 0\n0 1So\n221 0\n0 1So\n22\n1.2\n1.9\n-1.0\n-0.6\n1.5\n-0.9\n1.6\n-0.8Transpose\nSo\n1.286\n0.53 0.50Ɵ4\n0.03-0.03Ɵ3Ɵ2Ɵ1( )601.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6-0.8Xo1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6-0.8Xo\n1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6-0.8Xo1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6-0.8Xo\n1 0\n1 0\n1 0\n1 0\n0 1\n0 1\n0 1\n0 11.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6-0.8Xo1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6-0.8Xo1.2\n1.9\n-1.0\n-0.6\n1.5\n-0.9\n1.6\n-0.81.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Yo\n260\n1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Yo\n2601.2 1.3Xu1 1Su1 1Su\n1.2 1.3Xu1 1SuFigure 3: A toy example to illustrate the baseline method\nside information intervenes in the relationship between features and targets,\nbecause of which it does not seem adequate to give equal treatment to both\nkinds of information.\n4.2. A relationship zero-shot method for regression\nThis proposal arises from the necessity to exploit side information in a\nmore adequate way than just giving it the same treatment as the common fea-\ntures. The relationship methods assume a relationship δo,ubetween observed\nand unobserved targets (see Section 3.2). Then, a function f:X × S → Yu\nis induced from δo,uand from the models fo={fto\ni}mo\ni=1of observed targets.\nConsequently, the proposal consists in inducing the function fin two steps.\nIn the first step, the approach learns the models fo={fto\ni}mo\ni=1of observed\ntargets solely from the features (weather conditions), thus ignoring the side\ninformation (surroundings of the stations) (see the top left side of Figure\n4). In the second step, a δo,ufunction is designed in order to perform an\naggregation procedure (see the top right side of Figure 4).\n14Notation Explanation\nδo,urelationship function between observed and unobserved targets\nTtu\nk the subset of the closest kobserved targets of Toto the\nunobserved target tu\nftu\nk the subset of models of fofor the observed targets of Ttu\nk\nTable 2: Summary of the notation for the similarity based relationship zero-shot method\nfor regression\nWe shall now define the δo,ufunction relationship between observed and\nunobserved targets and the aggregation procedure.\n4.2.1. Defining the δo,ufunction relationship between observed and unob-\nserved targets\nOur proposal intends to define the δo,urelationship in terms of the dis-\ntance between observed and unobserved targets. As a result, δo,uis defined\noverToandTuand yields a real value that establishes the closeness or sim-\nilarity between an observed target and an unobserved target through the\ninverse of the distance, that is,\nδo,u:To× Tu→R\n(to\ni, tu\nj)→1/d(to\ni, tu\nj)\nNotice that δo,uis strictly positive, since To∩ Tu=∅, as stated in Section\n3.1. Manhattan (L1 norm) and Euclidean (L2 norm) distances could be some\nexamples of a distance dto define δo,u, typically adopted in the k-nearest\nneighbor method.\n4.2.2. Defining the aggregation procedure\nThe aggregation procedure takes place in the testing phase (see the bot-\ntom of Figure 4) and it is inspired on strategies followed by the k-nearest\nneighbor and the inverse distance weighting methods. For this reason we\nwill call this approach as similarity based relationship method. Firstly, the\nk-nearest observed targets from To={to\ni}to an unobserved target tuac-\ncording to the relationship δo,uare chosen. Let’s call this subset of observed\ntargets Ttu\nk. Remember that δo,uhas been defined in terms of the inverse of\na distance, so δo,umeasures the closeness or similarity between observed and\n15δo\nInverse distance weighting\nLearning\n LearningTRAINING PHASE\nTESTING PHASE\n …\n…moSoYomo\nnoas Xonoax\n…\n𝑓 1𝑓 p\nax\nnu= 1XuasSu\nmu=1u\np…\n… …Xonoax\nXonoax\nax\nnu= 1Xu1ax\nnu= 1Xu asSo\nmoasSu\nmu=1δo (       , )u\nmoSoas asSu\nmu=1δo (            ,    )u\nax\nnu= 1Xu1 pax\nnu= 1Xu\nmoSoas asSu\nmu=1δo (  ,  )u\nmoSoas asSu\nmu=1δo (               ,  )uFigure 4: Training and testing phases for the similarity based relationship zero-shot\nmethod for regression\nunobserved targets. Then, only the models from fo={fto\ni}mo\ni=1of these k-\nnearest observed targets will be taken into account to compute the prediction\nof the unobserved target tu. Let call this subset of models ftu\nk={fto\ni}to\ni∈Ttu\nk.\nTable 2 summarizes the additional notation employed in the development of\nthe similarity relationship zero-shot method for regression.\nSecondly, the aggregation procedure is based on inverse distance weight-\ning using the similarity function δo,u. For this purpose, the features of the new\nunobserved instance xufeed the set of models ftu\nk={fto\ni}to\ni∈Ttu\nkto get the pre-\ndictions {fto\ni(xu)}to\ni∈Ttu\nk. Then, the similarity δo,u(to\ni, tu) values between the\nk-nearest observed targets to\ni∈ Ttu\nkand the unobserved target tuare taken.\nFinally, our proposal entails weighting these predictions with the correspon-\ndent similarity values. Hence, the closer the similarity between an unobserved\ntarget tuand the observed target to\ni, the greater influence this observed tar-\n16get model fto\nihas on the prediction of tu. Therefore, the f:X × S → Yuis\ncomputed as the normalized weighted (by {δo,u(to\ni, tu)}to\ni∈Ttu\nk) average of the\n{fto\ni(xu)}to\ni∈Ttu\nk. That is,\nf(xu, tu) =1P\nto\ni∈Ttu\nkδo,u(to\ni, tu)·X\nto\ni∈Ttu\nkδo,u(to\ni, tu)·fto\ni(xu)\nNotice that the effect of each neighbor is weighted by the inverse of a\ndistance. As a result, the farthest neighbors will have significantly less ef-\nfect than the nearest neighbors. This illustrates the reason for taking all\nobserved targets instead of just taking the k-nearest neighbours. Effectively,\ntheir own weights tend to mitigate the influence of the farthest observed tar-\ngets, avoiding, in addition, the hassle of tuning the hyperparameter k. In\nthis context of zero-shot learning, this practice may be promising, since the\nnumber of targets is considerably low in relation to the number of instances\nwhere methods like k-nearest neighbor are applied. Therefore, the low car-\ndinality of Tofavors not having so many distant observed targets that may\nhave an influence. Nevertheless, with a high number of targets, one can keep\njust the kcloser observed targets.\n4.2.3. Toy example\nFigure 5 displays how the process of this method works for the toy exam-\nple. The expected parameters θfor the models of the two observed targets\nareθ1= 1 and θ2= 0 for to\n1, and θ1= 0 and θ2= 1 for to\n2, since y1=x1\nandy2=x2. Once these observed target models are learned, the aggregation\nprocedure takes place. The similarity (inverse of distances) for each observed\ntarget is received. In this case, there is no observed target with distance 0\nto the unobserved target, so all the observed targets must be taken into ac-\ncount in order to weight the predictions. The distances obtained from the\nunobserved target are 0 .5 with regard to both observed targets using the\nManhattan distance (L1 norm) (as an example). In order to make predic-\ntions for the unobserved target, whose side information is (1 ,1), the models\nof the two observed targets are used. The weighted average equals the aver-\nage, since both similarities are equal to 0 .5 and positive. Consequently, the\nprediction value yforx= (1.2,1.3) is 1 .25, which also differs from the actual\nvalue y=x·s, which is 2 .5.\n171.2 1.3 1.2 1.31\n11\n11 0\n0 11 0\n0 11 0\n0 11 0\n0 1δo\nInverse distance weighting\nLearning\n LearningTRAINING PHASE\nTESTING PHASE\n 1 0\n0 1\n2So1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Yo2\n60 21.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Xo602\n𝑓 1𝑓 2\n1.2 1.32\n1Xu1\n12Su\n1u\n21.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Xo602\n1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Xo602\n2\nXu12\n2So\n22Su\n1δo (                   ,       )u2So\n22Su\n1δo (                  ,      )u\n2\n1 22So\n22Su\n1δo (                  ,   )u2So\n22Su\n1δo (   ,      )u1.2\n1.9\n-1.0\n-0.61.5\n-0.9\n1.6\n-0.8\n1.2 1.31Xu Xu1.2 1.31Xu1\n11\n1Su Su\nXu12\nXu11.2 1.3 0.5 0.5\n1.25Ɵ1\n1 0Ɵ2 Ɵ1\n0 1Ɵ2\nY1 Y2Figure 5: A toy example to illustrate the similarity based relationship zero-shot method\nfor regression\n4.2.4. Discussion\nThe main drawback of this procedure is that the side information of\nthe observed targets To, which relates the features with the predictions, is\nignored in the learning process, since it is taken into account a posteriori in\nthe testing phase. Hence, knowledge from the feature models fo={fto\ni}mo\ni=1\ncan be lost. Moreover, it has limited generalization power, since a weighing\nprocess just interpolates the predictions produced by fo={fto\ni}mo\ni=1, whose\nvalues fall within a certain range. Therefore, the prediction for unobserved\ntargets is limited to this range. Nevertheless, it is an attractive method for\ndealing with a zero-shot regression task.\n184.3. A correspondence zero-shot method for regression\nThis proposal arises from an attempt to overcome the drawbacks of the\nsimilarity based relationship method introduced in Section 4.2. The idea is\nto include the side information into the learning process in order to increase\nthe generalization power of the predictions for the unobserved targets. The\nproposal becomes one of the correspondence methods, since it builds the\nunobserved target models though a learning process in which both the side\ninformation of the observed targets To={to\ni}mo\ni=1and the observed target\nmodels fo={fto\ni}mo\ni=1are involved.\n~ (                         ) TRAINING PHASE\nTESTING PHASE\nƟ1(                         ) Ɵp\n (                         )\n,,𝑓   Ɵ1,…,Ɵ1𝑓   Ɵm0,…,Ɵm01p 1 p 1 p\nLearning\n LearningmoSo\nYomo\nnoas\nXonoax\n… Xonoax\nXonoax\n𝑔ƟpLearningasSomo …Ɵ1\np\nƟ2p\nƟm0p\n𝑔Ɵ1LearningasSo mo …Ɵ1\n1\nƟ2\n1\nƟm01\nasSu\nmu=1ax\nnu= 1 Xu…\n…as\nmu=1Suas\nmu=1Su\nƟ1(                         )as\nmu=1SuƟp(                         )as\nmu=1Suax\nnu= 1Xu……\nFigure 6: Training and testing phases for the model parameter learning based correspon-\ndence zero-shot method for regression\n19Learning…\n𝑔ƟjLearningXonoax\n𝑓 1Ɵ1,…, Ɵ1,…,Ɵ1\nj 1 pLearningXonoax\n𝑓 Ɵ2,…, Ɵ2,…,Ɵ2\nj 1 pLearningXono\n𝑓 Ɵm0,…, Ɵm0,…,Ɵm0j 1 p2 p…ax\nmoSo\nas…Ɵ1\nj\nƟ2\nj\nƟm0j\nFigure 7: Zoom in learning process of the the model parameter learning based correspon-\ndence zero-shot method for regression\nThe proposed approach also induces the function fin two steps (see the\ntop of Figure 6). In the first step, the approach learns the models fo=\n{fto\ni}mo\ni=1of observed targets just from the features, that is, ignoring the side\ninformation, much like the similarity based relationship method does (see\nthe top of the training phase of Figure 6). However, in the second step both\nmethods differ. In this new approach, the second step includes a learning\nprocess. The idea is to transfer the knowledge of the observed target models\nfo={fto\ni}mo\ni=1into the learning process of the unobserved target models.\nOn the one hand, the learned observed target models fo={fto\ni}mo\ni=1are\ndefined by a set of parameters Θ = {(θi\n1,···, θi\np)}mo\ni=1, which, to some extent,\nrepresents the relationship between features and targets. On the other hand,\nthe side information of To={to\ni}mo\ni=1represents the observed targets. In this\nway, our proposal attempts to learn the correspondence between the side\ninformation of the observed targets To={to\ni}mo\ni=1and the feature observed\ntarget model parameters Θ = {(θi\n1,···, θi\np)}mo\ni=1. For this reason we will call\nthis approach as model parameter learning based correspondence method.\n20Table 3 summarizes the additional notation employed in the development of\nthis method.\nNotation Explanation\np number of parameters for the observed and\nunobserved models\n(θi\n1,···, θi\np) the set of pparameters for the model of an observed\ntarget iofTo\nΘ the set of parameters for the models of all the mo\nobserved targets of To\ngθj induced model for learning the parameter θjof an\nunobserved target\ngθ the set of the pinduced models for learning the\nparameters ( θ1,···, θp) of an unobserved target\nTable 3: Summary of the notation for the model parameter learning based correspondence\nzero-shot method for regression\nLet us detail how this learning process takes place.\n4.3.1. Learning the correspondence between side information and models of\nobserved targets\nThis learning process takes place in the second step of the training phase\n(see the bottom of the training phase of Figure 6). In this step, a set of\nplearning processes are carried out, as many as the number of parameters\nΘ ={(θi\n1,···, θi\np)}mo\ni=1of the observed target models fo={fto\ni}mo\ni=1, which\nmatches the number of parameters for the unobserved target models. Hence,\nthe goal is precisely to learn the parameters of the function ffor each of\nthe unobserved targets of Tu. Then, the gθ={gθj}p\nj=1set of pmodels is\ninduced, one per parameter of {θj}p\nj=1for the function fof an unobserved\ntarget. The key aspect of these learning processes is the way the datasets\nconstructed to induce the set of models gθ={gθj}p\nj=1are built. The instances\nof these datasets are the observed targets To={to\ni}mo\ni=1. The features that\nrepresent these instances are the side information of the observed targets\nTo={to\ni}mo\ni=1. These features are the same for learning all the models gθ=\n{gθj}p\nj=1. Nevertheless, the targets of these instances differ from one model of\ngθ={gθj}p\nj=1to another. Specifically, Figure 7 details the learning process for\ninducing gθjthat will provide the value of the parameter θjfor the function f\n21of an unobserved target tu. The target value of the instance ifori= 1, . . . , m o\nis the value of the parameter θj\niof the model fto\ni. Hence, the values of the\ntarget for the different instances in order to learn the model gθjis the set of\nvalues {θi\nj}mo\ni=1of the observed target models fo={fto\ni}mo\ni=1. Therefore, the\nproblem is reduced to a classic regression. From that point on, the general\npurpose regression algorithm must be applied to learn the set of models\ngθ={gθj}p\nj=1.\nIn the testing phase (see the bottom of Figure 6), the side information of\nan unobserved target tuis taken into all the gθ={gθj}p\nj=1in order to predict\nthe parameters {θu\nj}p\nj=1for the model of the unobserved target tu. Finally,\nthef:X × S → Yuis built using these parameter values {θu\nj=gθj(tu)}p\nj=1\nand evaluated over an unobserved instance xu. The gθ={gθj}p\nj=1models\nwill be the correspondence functions between the side information and the\nfeature learned models, whose knowledge will be transferred to each of the\nunobserved target models. As a result, this method can be considered a\nmodel parameter learning based correspondence method, since it learns the\nparameters for each of the unobserved target models.\n4.3.2. Toy example\nFigure 8 shows how this method works when applied to the toy example.\nThe models for the two observed targets are the same from the previous\nmethod, namely, θ1= 1 and θ2= 0 for to\n1, and θ1= 0 and θ2= 1 for to\n2.\nThe models gθfor the parameters of the unobserved target model are now\nlearned and then applied to the unobserved target. The parameter models\naregθ1(s) =s1·1+s2·0 because θ1\n1= 1 and θ2\n1= 0, and gθ2(s) =s1·0+s2·1\nbecause θ2\n1= 0 and θ2\n2= 1. Hence, the model for the unobserved target\nwhose side information is (1 ,1) will have gθ1((1,1)) = 1 and gθ2((1,1)) = 1\nas parameters. Therefore, evaluating the unobserved instance (1 .2,1.3) over\nthis model, the prediction obtained will be 2 .5, which is the actual value.\n4.4. Discussion\nAs we have shown, the knowledge of the models that is learned from\nfeatures is transferred together with the side information to the unobserved\ntargets models. This practice increases the generalization prediction power\nof unobserved instances for unobserved targets, since the model for the un-\nobserved target results from a learning procedure, rather than from an ag-\ngregation procedure.\n22~ (  ) TRAINING PHASE\nTESTING PHASE\nƟ1(                   )\n,𝑓 𝑓1 2\nLearning\n Learning1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Xo\n𝑔Ɵ2𝑔Ɵ1\n1\n12Su\n11.2 1.32\n1Xu2\n11 1Su\nƟ1(                   )2\n11 1SuƟ2( )1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Yo2\n602\n601 0\n0 1\n2So2\n1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Xo602\n1.2 1.5\n1.9 -0.9\n-1.0 1.6\n-0.6 -0.8Xo602\n1.2\n1.9\n-1.0\n-0.61.5\n-0.9\n1.6\n-0.8\nƟ1\n1 0Ɵ2 Ɵ1\n0 1Ɵ2\nƟ2(                   )11 1Su 1 12\n1.2 1.32\n1Xu2\n11 1Su\n2.51\n01 0\n0 1\n2So2Ɵ1\n1\nƟ2\n10\n11 0\n0 1\n2So2Ɵ1\n2\nƟ2\n2Figure 8: A toy example to illustrate the model parameter learning based correspondence\nzero-shot method for regression\n235. Experiments\nThis section describes the experiments carried out. Subsection 5.1 de-\nscribes the datasets taken for the experiments, including the dataset concern-\ning air pollution that motivates this work. In subsection 5.2 the parameter\nsettings are established. Finally, subsection 5.3 analyzes and discusses the\nresults.\n5.1. Description of datasets\nThere are not many datasets in the literature which include side infor-\nmation for regression. The Communities and Crime dataset2of the UCI\nMachine Learning Repository was suitable for adaptation to this purpose, so\nit is one of the datasets used for the experiments. In addition, some artifi-\ncial datasets were created in order to check the performance of the different\napproaches in a more exhaustive way. Of course, the air pollution dataset\nwas also included. The next subsections provide further details about these\ndifferent datasets.\n5.1.1. Artificial datasets\nSeveral artificial datasets were generated in order to study the perfor-\nmance of the methods. The feature values for the instances and the targets\nwere drawn from a uniform distribution in the range of ( −2;−1]∪[+1; +2)\nin order to avoid zero values or values near zero. This process builds the ma-\ntrices XandS. The Ymatrix was carefully built from the matrices Xand\nS. Hence, if x= (xi, . . . , x ax) is an instance description and s= (si, . . . , s as)\nis the side information of a certain target, the value of yfor this instance is\nfixed as a linear function of x, where the coefficients are in turn functions of\ns. That is:\ny=axX\ni=1(αi(s)·xi) +β\nAt this point, two kind of artificial datasets are generated, depending on\nhow the {αi(s)}ax\ni=1functions are built. In this sense, the {αi(s)}ax\ni=1func-\ntions are defined in an attempt to globally cover the domains that both the\nsimilarity based relationship method (SR) and the model parameter learning\n2Based on http://archive.ics.uci.edu/ml/datasets/communities+and+crime\n24based correspondence method (MPLC) are able to stand up to. Doing some-\nthing similar to the baseline method would imply that the side information\nwould not define the relationship between the instance description and the\ntargets. This would mean to dealing with a general purpose regression prob-\nlem, which is not the aim of this study. Let us now return to the definition\nof the {αi(s)}ax\ni=1functions.\nOn the one hand, the {αi(s)}ax\ni=1functions for the first kind of artificial\ndatasets are built taking into account a similarity δfunction applied to the\nside information sof the target and a set of other side information descrip-\ntions{µk}d\nk=1. The idea is that the generated target value ywill be defined\nin terms of similarity values. Hence, the {αi(s)}ax\ni=1functions are defined as\na weighted average of similarities:\nαi(s) =Pd\nk=1\u0000\nτi,k·δ(s, µk)\u0001\nPd\nk=1δ(s, µk)\nThe coefficients {τi,k}ax,d\ni=1,k=1and the side information descriptions {µk}d\nk=1\nwere drawn using the same distribution as for the feature values of Xand\nS, which is a uniform distribution in the range of ( −2;−1]∪[+1; +2). The\nsimilarity function δhas been randomly chosen to be either the Manhattan\n(L1 norm) or the Euclidean (L2 norm) in equal shares in order to avoid bias.\nOn the other hand, the {αi(s)}ax\ni=1functions for the second kind of ar-\ntificial datasets are built taking the most general structure that provides a\nlinear dependence of side information. That is:\nαi(s) =asX\nj=1(γi,j·sj) +βi\nThe coefficients β,{βi}ax\ni=1and{γi,j}ax,as\ni=1,j=1were drawn using the same\ndistribution as for the feature values of XandS, the coefficients {τi,k}ax,d\ni=1,k=1\nand the side information descriptions {µk}d\nk=1, which is a uniform distribution\nin the range of ( −2;−1]∪[+1; +2).\nWithout loss of generality, the number of instances and features were fixed\nto 5000 and 50, respectively. However, due to the focus of the paper, the\nnumber of targets and features that conform the side information were varied\nto obtain different datasets. The number of targets considered were 5 ,10,50\nand 100, and the number of side information features were 5 ,15 and 25, in\norder to cover a range both below and above the real datasets. The number\n25of the other side information descriptions dfor the first kind of datasets was\nfixed as 5 in order to allow the dataset with 5 targets to include enough\ninformation to be learned. Table 4 displays the main properties of these\ndatasets. The datasets are named Sk,l(first kind of artificial datasets) and\nRk,l(second kind of artificial datasets) where kandlrespectively indicate\nthe number of targets and the size of the side information.\ndataset targetssidedataset targetsside\ninformation information\nS5,5/R5,55 5 S50,15/R50,1550 15\nS10,5/R10,510 5 S100,15/R100,15100 15\nS50,5/R50,550 5 S5,25/R5,255 25\nS100,5/R100,5100 5 S10,25/R10,2510 25\nS5,15/R5,155 15 S50,25/R50,2550 25\nS10,15/R10,1510 15 S100,25/R100,25100 25\nTable 4: Number of targets and side information size of the artificial datasets\n5.1.2. Real datasets: Communities and Crime dataset and Air pollution\ndataset\nTwo real datasets were selected in the experiments to check the two pro-\nposed methods and to compare them with the baseline; namely, the Com-\nmunities and Crime dataset and the Air pollution dataset. Table 5 shows\nthe main properties of both datasets. The descriptions of both datasets are\ndetailed below.\ndataset instances features targetsside\ninformation\nCommunities and Crime 1420 101 13 15\nAir pollution 41325 12 5 16\nTable 5: Properties of the real datasets: Communities and Crime dataset and Air pollution\ndataset\n26Communities and Crime dataset. The Communities and Crime dataset of\nthe UCI Machine Learning Repository3combines socioeconomic data from\nthe 1990 US Census, law enforcement data from the 1990 US LEMAS survey,\nand crime data from the 1995 FBI UCR [31]. The ratio of violent crimes per\ncapita is obtained from the population and the sum of crime considered\nviolent in various counties of the United States of America: murder, rape,\nrobbery, and assault. The original version of this dataset is not suitable for\nzero-shot regression, so it had to be adapted. The instances are counties\nwhose features are socioeconomic characteristics. The counties form states,\nthat will become the targets, for which socioeconomic features are also taken\nas side information. Hence, the goal is to predict the crime ratio for a county\nbelonging to a new state.\nAir pollution dataset. The air pollution dataset was the motivation for this\nresearch and includes hourly averages of air pollution of the Principality of\nAsturias, Spain, from 2010 to 2018. This dataset corresponds to 11 types\nof pollutants from 18 pollution and meteorological stations. However, the\nlargest common set of pollutants and stations included 4 pollutants (NO 2,\nTSP, NO, SO 2) from 5 stations, since not all the stations register the same\nair pollutants.\nIn addition to pollutant measurements, meteorological data is taken into\naccount, such as wind direction and speed, season, temperature, humidity,\npressure or precipitation. External data is also available concerning certain\nproperties of the surroundings of the stations, such as if there is an urban\ncenter, a highway, an industrial area, the sea or a river nearby. These are\nrepresented by indicating their directions (north, south, east and/or west).\nHence, a total of 16 features describe the stations in the form of side infor-\nmation. Figure 9 displays an example with a location that has an urban area\nin the northwest, a highway in the southeast and an industrial area and a\nriver towards the northeast.\nThe aim is to predict the amount of a certain pollutant in a location\nwhere a new station may be installed, for which meteorological data has not\nbeen collected.\n3https://archive.ics.uci.edu/ml/index.php\n27Figure 9: Example of a location with different surroundings elements. In relation to point\nX, the urban center is located to the north and west, a highway is located to the south\nand east, an industrial area is located to the north and east and finally, a river or sea is\nlocated to the north and east.\n5.2. Parameter settings\nAmong the baseline method, SR and MPLC, only the SR has a parameter,\nnamely, the distance for the relationship. In this sense, both the Manhattan\n(L1 norm) and the Euclidean (L2 norm) distances were varied for the rela-\ntionship between observed and unobserved targets. They were taken as a\npreliminary approximation, since they are typical measures adopted by the\nk-nearest neighbor method.\nAll the approaches for the experiments were implemented in Python using\nScikit Learn Library4. All the code is available in GitHub repository .\nThe same regressor was used as a base learner to induce the models of the\nobserved targets and, in the case of the MPLC, to induce the models for\nlearning the parameters of an unobserved target. Two different regressors\nwere checked, namely, i) Ridge Regression (Ridge)5and ii) Linear Support\nVector Regression (LSVR)6. Ridge solves a regression model where the loss\n4https://scikit-learn.org/stable/\n5https://scikit-learn.org/stable/modules/generated/sklearn.linear model.Ridge.html\n6https://scikit-learn.org/stable/modules/generated/sklearn.svm.LinearSVR.html\n28mS\nX\n Ym\nFold 1\nFold 2\nFold 3Fold 1Fold 2 Fold 3\nnas\nax\nTrain\nTestFigure 10: The matrices X, Y and S will be cut in a consistent way, but not all of the data\nis going to be used in each fold (white without texture).\nfunction is the linear least squares function and regularization is given by the\nL2 norm. LSVR solves a convex quadratic optimization problem with linear\nconstraints, similar to Support Vector Regression (SVR), which uses a linear\nkernel. The αparameter of Ridge and cparameter of LSVR were optimized\nthrough a grid search using the mean squared error as the evaluation function\nand were estimated via a 3-fold cross validation. The values of αchecked\nin the grid search for Ridge were {e−3, e−2, . . . , e2, e3}and the values of c\nchecked in the grid search for LSVR were {10−3,10−2, . . . , 102,103}. The\nscore for the results was the relative mean squared error computed following\na 3-fold cross validation with 3 repetitions. This cross validation, which has\nside information present, differs slightly from that of classic machine learning.\nIn this case, the split in folders requires us to take into account instances and\ntargets, and it must be consistent. Figure 10 illustrates an example where the\nX,YandSmatrices are split in 3 folds (2 for training and 1 for testing). The\nstriped cells are for training, whereas the dotted cells are for testing. In the\ncase of Y, the blank cells in the fold that play the role of testing are ignored\nin order to make the split consistent. These cells correspond to the values of\nthe targets: i) for the observed (training) targets and unobserved (testing)\ninstances (the blank cells of Yon the bottom), and ii) for the unobserved\n(testing) targets and observed (training) instances (the blank cells of Yon\nthe right).\n295.3. Result analysis and discussion\nThis section analyzes and discusses the results obtained from the experi-\nments carried out over the above-described datasets, using the baseline and\nthe two proposed methods: SR and MPLC. Firstly, the discussion is centered\non the artificial datasets and afterwards it focuses on the real datasets about\ncrime and pollution. The tables will display the relative mean squared error\n[44], which is the mean squared error normalized by the mean squared error of\nthe default predictor (the mean of the predictions). The Friedman-Nemenyi\ntest [4, 25] was carried out. This test consists of a two-step comparison for\neach measure. The first step is a Friedman test [7, 8], which rejects the null\nhypothesis that states that not all learners perform equally. The second step\nis the post-hoc Nemenyi test [24]. In the tables, the ranks of each dataset are\nshown in brackets. In case of ties, average ranks are indicated. The average\nranks over all the datasets are calculated and presented in the last row of\neach table.\n5.3.1. Artificial datasets\nTables 6 and 7 display the relative mean square error and Friedman-\nNemenyi test when baseline, SR (Euclidean), SR (Manhattan) and MPLC\nmethods are applied to the artificial datasets. The results experimentally\nconfirm the theoretical hypothesis (also shown through an illustrative toy ex-\nample) about the loss of knowledge that takes place when information about\nthe observed target models is not used for learning the unobserved target\nmodels. Effectively, the baseline method achieves the worst average rank,\nregardless of the regressor used for estimating the error (Ridge or LSVR)\nand the artificial datasets (S and R datasets). In S artificial datasets, and\ntaking into account the average ranks, SR (Manhattan) outperforms the rest\nof the methods, followed by MPLC and then by SR (Euclidean). However,\nfocusing on the particular ranks of MPLC, one can clearly see that MPLC\ncomes first for half of the S artificial datasets, specifically those with the\nhighest number of targets (50 and 100), regardless of the number of features\nof the side information. MPLC does not outperform SR (Manhanttan) due\nto the fourth place that it yields when the number of targets is low (5 and\n10). In R artificial datasets, MPLC clearly obtains the best scores, regard-\nless of the regressor used for estimating the error (Ridge or LSVR). Indeed\nin some cases the error of MPLC is practically 0.\nSince both Table 6 and Table 7 shows the relative error, only MPLC are\nunable to achieve the media system in one S artificial dataset (S5,5), which\n30Ridge baselineSR SRMPLC(Euclidean) (Manhattan)\nS5,592.17 (1) 97.82(3) 95.97(2) 111.29(4)\nS10,556.00(4) 50.51(3) 50.15(2) 44.30 (1)\nS50,558.86(4) 50.02(3) 49.20(2) 26.55 (1)\nS100,545.42(4) 38.11(3) 37.53(2) 15.74 (1)\nS5,153.64(3) 3.46(2) 3.28(1) 94.73(4)\nS10,152.19(3) 2.13(1) 2.14(2) 75.32(4)\nS50,152.69(4) 2.43(3) 2.39(2) 0.14(1)\nS100,151.94(4) 1.84(3) 1.83(2) 0.14(1)\nS5,252.86(3) 2.83(2) 2.79(1) 81.28(4)\nS10,250.83(2) 0.83(2) 0.83(2) 98.43(4)\nS50,251.43(4) 1.36(3) 1.35(2) 0.18(1)\nS100,251.18(4) 1.14(3) 1.13(2) 0.06(1)\nAvg. Rank (3.3) (2.6) ( 1.8) (2.3)\nLSVR baselineSR SRMPLC(Euclidean) (Manhattan)\nS5,591.95 (1) 96.83(3) 94.96(2) 121.80(4)\nS10,555.86(3) 48.76(2) 48.16 (1) 60.13(4)\nS50,558.84(4) 49.73(3) 48.93(2) 26.55 (1)\nS100,545.42(4) 38.11(3) 37.53(2) 15.74 (1)\nS5,153.73(3) 3.38(2) 3.21(1) 99.83(4)\nS10,152.22(1) 2.39(2) 2.40(3) 77.73(4)\nS50,152.69(4) 2.43(3) 2.39(2) 0.14(1)\nS100,151.94(4) 1.84(3) 1.83(2) 0.14(1)\nS5,253.26(1) 3.46(3) 3.40(2) 93.85(4)\nS10,250.90(1) 1.48(3) 1.47(2) 98.89(4)\nS50,251.43(4) 1.36(3) 1.35(2) 0.18(1)\nS100,251.19(4) 1.13(3) 1.12(2) 0.06(1)\nAvg. Rank (2.8) (2.8) ( 1.9) (2.5)\nTable 6: Relative mean square error and Friedman-Nemenyi test for S artificial datasets\nusing baseline, SR (Euclidean), SR (Manhattan) and MPLC\n31Ridge baselineSR SRMPLC(Euclidean) (Manhattan)\nR5,5108.68(4) 96.23(3) 95.49(2) 61.55 (1)\nR10,5121.34(4) 93.69(3) 86.84(2) 1.45(1)\nR50,5102.51(4) 69.05(3) 63.44(2) 0.00(1)\nR100,5102.38(4) 68.50(3) 62.97(2) 0.00(1)\nR5,15150.44(4) 148.66(3) 146.43(2) 83.47 (1)\nR10,15114.71(4) 111.58(3) 108.60(2) 72.30 (1)\nR50,15103.20(4) 96.09(3) 92.46(2) 0.00(1)\nR100,15101.64(4) 94.25(3) 90.13(2) 0.00(1)\nR5,25136.30(4) 132.95(3) 129.55(2) 83.71 (1)\nR10,25119.76(4) 114.60(3) 111.88(2) 65.06 (1)\nR50,25102.32(4) 98.24(3) 96.00(2) 0.00(1)\nR100,25102.22(4) 97.77(3) 95.21(2) 0.00(1)\nAvg. Rank (4.0) (3.0) (2.0) (1.0)\nLSVR baselineSR SRMPLC(Euclidean) (Manhattan)\nR5,5103.14(4) 88.28(3) 87.79(2) 73.53 (1)\nR10,5119.56(4) 94.01(3) 87.96(2) 2.10(1)\nR50,5102.49(4) 69.05(3) 63.44(2) 0.00(1)\nR100,5102.36(4) 68.50(3) 62.97(2) 0.00(1)\nR5,15142.28(4) 130.19(3) 128.48(2) 87.81 (1)\nR10,15112.80(4) 104.77(3) 102.64(2) 72.07 (1)\nR50,15103.12(4) 96.09(3) 92.46(2) 0.00(1)\nR100,15101.63(4) 94.25(3) 90.13(2) 0.00(1)\nR5,25129.51(4) 118.60(3) 116.24(2) 85.12 (1)\nR10,25117.85(4) 108.00(3) 106.01(2) 66.81 (1)\nR50,25102.26(4) 98.30(3) 96.14(2) 0.00(1)\nR100,25102.21(4) 97.82(3) 95.21(2) 0.00(1)\nAvg. Rank (4.0) (3.0) (2.0) (1.0)\nTable 7: Relative mean square error and Friedman-Nemenyi test for R artificial datasets\nusing baseline, SR (Euclidean), SR (Manhattan) and MPLC\n32corresponds to the S dataset with the lowest number of targets and the lowest\nsize of side information. However, in the case of the R artificial datasets, the\nbaseline approach is the method that is unable to outperform the media\nsystem. This is also the case for both versions of SR in the instances of R5,15,\nR10,15, R5,25and R10,25. Meaning that SR does not work well for datasets with\nfew targets and a high number of side information features when the datasets\nare not generated using similarities. Focusing on the similarity measures of\nSR, Manhattan distance seems to perform slightly better than Euclidean\ndistance for all the datasets, whatever S and R artificial datasets.\nIn case of the S artificial datasets, SR benefits when the side information\nsize is sufficiently high (from 15), whatever the number of targets, since the\nrelative error of both the SR versions falls considerably to values below 4%.\nHowever, in the case of the R artificial datasets, SR benefits most when the\ndatasets are characterized as having the highest number of targets and fewer\nside information features (R50,5and R100,5).\nMPLC seems to benefit from having a high number of observed targets\n(50 and 100), since the relative error falls drastically under these situations\nand the method comes out in the first place, whatever the S or R artificial\ndatasets. In case of the S artificial datasets, the relative error falls below\n1% when the side information size is sufficiently high (from 15). In the case\nof the R artificial datasets, the error equals 0%, no matter how large the\nfeature side information is. This is consistent with having enough instances\nto learn the parameters, since the number of targets conditions the number\nof instances in the second learning phase.\nIn the case of the R artificial datasets, it should be noted that the meth-\nods, when ranked by performance, remain in the same order regardless of the\ndataset chosen. However, in the case of the S artificial datasets, the baseline\nmethod and MPLC seem to distribute the first and fourth places, whereas\nSR (Euclidean) and SR (Manhattan) seem to distribute the second and third\nplaces.\nA Wilcoxon test [43] was made using the ranks of each pair of systems.\nSince the comparison is between pairs of methods, the ranks take values 1\nor 2. The size of the sample for the test is 24 = 12 ·2, which corresponds\nwith the 12 artificial datasets and the 2 base learners. In the case of the S\nartificial dataset, SR (Manhattan) performs better than SR (Euclidean) with\nap−value equal to 0 .0067 in the case of Ridge and 0 .0039 in the case of LSVR.\nAlso, in the case of the S artificial dataset, both SR (Manhattan) and SR\n(Euclidean) outperforms the baseline method with a p−value equal to 0 .0067.\n33In the case of the R artificial datasets, the p−value is the same for each pair\nof systems and equal to 0 .0005. This is because MPLC always performs\nbetter than SR (Manhattan), SR (Manhattan) always performs better than\nSR (Euclidean) and in turn SR (Euclidean) always outperforms baseline.\nThis p−value far exceeds even the risk of 1%, and therefore the differences\nin performance between pairs of methods are statistically significant.\nRidge baselineSR SRMPLC(Euclidean) (Manhattan)\nCrime 27.56 (1) 36.52(3) 34.20(2) 51.65(4)\nNO2 86.32(4) 84.19(3) 80.24(2) 78.67 (1)\nPST 92.40(3) 92.55(4) 90.86(2) 87.65 (1)\nNO 101.01(4) 100.73(2) 100.75(3) 89.31 (1)\nSO2 95.14(2) 96.26(4) 95.19(3) 93.63 (1)\nAvg. Rank (2.8) (3.2) (2.4) (1.6)\nLSVR baselineSR SRMPLC(Euclidean) (Manhattan)\nCrime 33.36(2) 33.43(3) 35.86(4) 25.40 (1)\nNO2 86.11(4) 81.07(3) 76.52(2) 76.43 (1)\nPST 90.86(3) 91.17(4) 88.95(2) 87.15 (1)\nNO 96.20(4) 93.84(3) 90.28 (1) 91.68(2)\nSO2 95.01(3) 95.53(4) 92.19 (1) 93.49(2)\nAvg. Rank (3.2) (3.4) (2.0) (1.4)\nTable 8: Relative mean square error and Friedman-Nemenyi test for the Communities and\nCrime and air pollution datasets using baseline, SR (Euclidean), SR (Manhattan) and\nMPLC\n5.3.2. Real datasets\nThis section discusses the experiment results over the Communities and\nCrime and the air pollution datasets. Table 8 displays the relative mean\nsquare error and Friedman-Nemenyi test when baseline, SR (Euclidean), SR\n(Manhattan) and MPLC methods are applied over these real datasets. Our\nconclusions remain similar in the experiments carried out over these datasets,\nnamely, i) MPLC shows the best performance and ii) Between the two ver-\nsions of SR, Manhattan distance seems to work better and comes close to\nMPLC performance. Two particulars stand out in the experiments and de-\nserve comment. The first one is the high performance that the baseline\n34method attains for the Communities and Crime dataset when Ridge regres-\nsion is used. The other one is that SR (Manhattan) slightly outperforms\nMPLC for two of the four pollutants. Focusing on the relative errors, it seems\nthat NO pollutant is quite hard to learn, since only the MPLC method man-\nages to overcome the media system using Ridge regression. However, NO 2is\nthe easiest to learn when compared to the rest of pollutants. The p−values\nof the Wilcoxon test are displayed in Table 9. The size of the sample in this\ncase is 10 = 5 ·2, because of the 5 real datasets and the 2 base learners. These\np−values indicate that there are no significant differences between baseline\nand SR (Euclidean), or between SR (Manhattan) and MPLC. However, there\nare significant differences between MPLC and baseline, as well as between\nMPLC and SR (Euclidean), at a risk of 1%. There are also significant differ-\nences between both versions of SR, Euclidean and Manhattan, although in\nthis case at a risk of 5%. In consequence, MPLC shows itself to be superior\nto the rest of the methods, but SR (Manhattan) performs quite similarly.\nSR SRMPLC(Euclidean) (Manhattan)\nbaseline 0.52 0.20 0.01\nSR (Euclidean) 0.05 0.01\nSR (Manhattan) 0.20\nTable 9: p-values of the Wilcoxon test for the communities and crime, and air pollution\ndatasets\n6. Conclusions and future work\nThis paper proposes two approaches to cope with regression zero-shot\ntasks. To the best of our knowledge, zero-shot has been mainly focused on\ncomputer vision, speech or character recognition classification tasks. The in-\nterest in providing methods for zero-shot regression in this paper was spurred\nby the need to predict air pollution in locations where weather conditions\nhave not been registered before, but where environmental information about\nthe location is available. This information can be considered side information,\nsince it is neither features nor targets, it is known beforehand, is independent\nof possible unobserved weather conditions, and remains constant for the same\nlocation. It is a potential source of information susceptible to being exploited\nthat makes the task suitable for a zero-shot regression environment.\n35One of the contributions of this paper is a similarity based relationship\nzero-shot method for regression that combines the feature (weather condi-\ntions) models for the observed targets (meteorological stations) in order to\nobtain models for the unobserved targets (potential future locations of new\nmeteorological stations). However, this method has the drawback of losing\npotential knowledge of the feature models, since it just performs an aggre-\ngation procedure, limiting the power of generalization to locations of future\nmeteorological stations. The second contribution of the paper is a parameter\nlearning model based correspondence zero-shot method for regression that\novercomes this limitation by attempting not to aggregate the predictions of\nthe feature models, but extracting knowledge directly from the models, in-\ncluding the parameters within a learning procedure. This learning procedure\nis what provides the generalization power in order to obtain predictions for\nunobserved stations from weather conditions. Both proposals are compared\nto a simple baseline method, consisting in taking both weather conditions\n(features) and environmental information (side information) as common fea-\ntures. Several artificial data sets are generated so that a more detailed com-\nparison can be performed, by means of varying the number of targets and\nthe size of the side information. The comparison is also carried out using the\nCommunities and Crime dataset of the UCI Machine Learning Repository,\nin addition to the Air pollution dataset.\nThe conclusion is that both the parameter learning model based corre-\nspondence method and the similarity based relationship method with Man-\nhattan distance outperform both similarity based relationship method with\nEuclidean distance and the baseline approaches, the improvement in perfor-\nmance being statistically significant for Communities and Crime dataset of\nthe UCI Machine Learning Repository and the Air pollution dataset. The pa-\nrameter learning model based correspondence method outperforms the simi-\nlarity based relationship method with Manhattan distance for some artificial\ndatasets, whereas the latter obtains better performance than the former for\nother artificial datasets. As far as the variations in the number of observed\ntargets and the size of the side information are concerned, a sufficient num-\nber of observed targets is crucial in order to obtain promising parameter\nlearning, whilst; also, having fewer side information features brings a better\nperformance. Whichever the case, these contributions enrich the zero-shot\nframework in relation to regression tasks.\nWhen it comes to possible future work, several lines can be opened. One\npotentially interesting future line of research would be adapting the strategies\n36proposed in this paper to neural networks, and particularly to deep learning.\nThis adaptation lies outside of the scope of this paper, since such an adapta-\ntion is no trivial task, given the large amount of hyperparameters present in\nneural networks. Another future line may be developed if instead of limiting\nthe task to regression, we extended the procedure to multiregressions. This\nextension may improve the performance of the predictions, since possible in-\nformation about correlations among the targets could be exploited. In the\ncontext of air pollution, this means relating different pollutants. This prac-\ntice is sensible, since the presence of certain pollutants may condition the\npresence or absence of others. Another line susceptible to being exploited\nmight be to improve the parameter learning procedure, trying to simulta-\nneously optimize both learning phases. With regard to the similarity based\nrelationship zero-shot method for regression, it should be possible to explore\nand design new relationship functions in order to improve their performance.\nAcknowledgments\nThis research has been partially supported by the Spanish Ministry of\nScience and Innovation through the grant PID2019-110742RB-I00.\nReferences\n[1] Aktukmak, M., Yilmaz, Y., and Uysal, I. (2019). 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We consider triangular P´ olya urns and show under very weak c ondi-\ntions a general strong limit theorem of the form Xni{ania.s.Ý ÑXi, where Xniis\nthe number of balls of colour iafterndraws; the constants aniare explicit and\nof the form nαlogγn; the limit is a.s. positive, and may be either deterministic or\nrandom, but is in general unknown.\nThe result extends to urns with subtractions under weak cond itions, but a\ncounterexample shows that some conditions are needed.\nFor balanced urns we also prove moment convergence in the mai n results if the\nreplacements have the corresponding moments.\nThe proofs are based on studying the corresponding continuo us-time urn using\nmartingale methods, and showing corresponding results the re. In the main part\nof the paper, we assume for convenience that all replacement s have finite second\nmoments; in an appendix this is relaxed to Lpfor some pą1.\nContents\n1. Introduction 2\n1.1. Contents 6\n2. Some notation and other preliminaries 6\n2.1. Standing assumptions 6\n2.2. General notation 7\n2.3. The colour graph 8\n2.4. More notation 9\n2.5. Stochastic processes 9\n2.6. Continuous-time urn 10\n3. Analysis of one colour 10\n3.1. A colour not influenced by others 12\n3.2. A colour only produced by one other colour 14\n3.3. The general case for a single colour 21\n4. The main theorem for continuous time 23\n5. Proof of the main theorem for discrete time 23\n6. Dependencies between the limits 25\n7. Degenerate limits in discrete time 30\n8. Urns with subtractions 31\n8.1. Some motivation for (A6)–(A8) 32\n8.2. Main results for urns with subtractions 33\n8.3. A colour not influenced by others 34\nDate: 21 March, 2024.\n2020Mathematics Subject Classification. 60F15, 60F25; 60C05, 60G44, 60J85.\nSupportedbytheKnutandAlice WallenbergFoundationandth eSwedish ResearchCouncil (VR)\n.\n12 SVANTE JANSON\n8.4. A colour only produced by one other colour 36\n8.5. The general case for a single colour 39\n8.6. Proofs of Theorems 8.4–8.6 39\n9. Random vs non-random replacements 40\n10. Balanced urns 40\n11. The drawn colours 43\n12. Moment convergence 46\n12.1. Continuous-time urns 46\n12.2. Balanced discrete-time urns 47\n12.3. Moments for drawn colours 48\n13. Rates of convergence? 48\n14. Examples 49\nAppendix A. Absolute continuity and conditioning 66\nAppendix B. Lptheory 69\nB.1. A single colour not influenced by others 69\nB.2. A colour only produced by one other colour 72\nB.3. The general case for a single colour 75\nB.4.Lpbounds and convergence 76\nB.5. A.s. convergence 77\nAppendix C. Proof of (14.80) 80\nReferences 82\n1.Introduction\nA (generalized) P´ olya urn contains balls of different colour s. A ball is drawn at\nrandom from the urn, and is replaced by a set of balls that depe nds on the colour\nof the drawn balls; more generally, the replacement set may b e random, with a\ndistribution depending on the drawn colour. We assume that t he setQof colours\nis finite, and let q:“ |Q|be the number of colours. (See e.g. [31], [38], [27] for the\nhistory and further references.) It is often convenient to a ssume that Q“ t1,...,q u,\nbut for us it will be convenient not to assume this.\nThe P´ olya urn process can be defined formally as follows. The composition of\nthe urn at time nis given by the vector Xn“ pXniqiPQP r0,8qq, whereXniis the\nnumber of balls of colour i. The urn starts with a given vector X0, and evolves\naccording to a discrete-time Markov process. Each colour ihas anactivityaiě0,\nand a (generally random) replacement vector ξi“ pξijqjPQ. At each time n`1ě1,\nthe urn is updated by drawing one ball at random from the urn, w ith the probability\nof any ball proportional to its activity. (In many cases ai“1 for all i, so all balls\nare drawn with equal probability; the reader may concentrat e on this case until\nSection 5.) Thus, the drawn ball has colour iwith probability\naiXniř\njajXnj. (1.1)\nIf the drawn ball has colour i, it is replaced together with ξpnq\nijballs of colour j,\njPQ, where the random vector ξpnq\ni“ pξpnq\nijqjPQis a copy of ξithat is independentALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 3\nof everything else that has happened so far. Thus, the urn is u pdated to\nXn`1“Xn`ξpnq\ni. (1.2)\nRemark 1.1. Note that, as in many other papers on P´ olya urns, we do not ass ume\nthatXniare integers; any real numbers Xniě0 are allowed. In general, it is thus a\nmisnomer to call Xnithe “number” of balls of colour i; it is more precise to regard\nXnias the amount of colour iin the urn. Nevertheless, we will continue to use the\ntraditional terminology, which thus has to be interpreted l iberally by the reader. △\nRemark 1.2. Since the drawn ball is replaced, it is also really a misnomer to call\nξithe “replacement” vector; it is really an addition vector. (The replacements are\nreallyξij`δij.) Nevertheless, we use the terminology above, which is used in many\npapers. △\nRemark 1.3. We allow the replacement vectors ξito be random. (Some papers\nconsider only the special case of deterministic ξi, which is an important special case\nthat appears in many applications.) We may say that the urn ha sdeterministic\n(or non-random) replacements if allξijare deterministic, and otherwise random\nreplacements . These terms should thus be interpreted as conditioned on (t he colour\nof) the drawn ball. △\nRemark 1.4. It is often convenient to describe the replacements by the replacement\nmatrix pξijqi,jPQ. Note, however, that unless the replacements are determini stic, this\nmay be somewhat misleading, since the rows should be regarde d as separate random\nvectors, not necessarily defined on the same probability spa ce; there is no need for a\njoint distribution of different rows. △\nRemark 1.5. In the first part of the paper we assume that the replacements ξijě0\n(Condition (A5) below), meaning that we only add balls to the urnand never remove\nany. In Section 8, and also usually in later sections, we more generally allow also\nthat balls may be removed from the urn (assuming some hypothe ses). △\nRemark 1.6. We allow some activities aito be 0; this means that balls of colour\ninever are drawn. See Section 5 for an important example of thi s. (If all aią0,\nwe may reduce to the standard case ai“1 by considering the urn paiXniqi, with\ncorresponding replacement vectors pajξijqj, but we will not use this.) △\nRemark 1.7. We assumed tacitly above that the denominatorř\njajXnjin (1.1) is\ną0 for every ně0, so that the definition makes sense. This holds, for example ,\nunderassumptions(A1)and(A5)below. (Urnsthat donotsati sfythis, andtherefore\nmay stop at some finite time, have also been studied, but they w ill not be considered\nhere.) △\nWe are interested in asymptotic properties of XnasnÑ 8.\nIn the present paper, we study triangular urns , i.e., P´ olya urns such that, for\na suitable labelling of the colours by 1 ,...,q, we have ξij“0 wheniąj. (See\nalso Section 2.3.) This includes the original P´ olya urns st udied by Markov [39],\nEggenberger and P´ olya [16], and P´ olya [41] (all for q“2), where the replacement\nmatrix is diagonal:ξij“0 wheni‰j, but also many other interesting cases. See\nSection 14 for some examples.\nThere are many previous papers on triangular urns; we mentio n here only a few\nthat are particularly relevant to the present paper; see als o the references in the4 SVANTE JANSON\nexamples in Section 14. Athreya [2] studied diagonal urns wi th random replace-\nments and showed a.s. convergence of the proportions of differ ent colours, using the\nembedding mthod of Athreya and Karlin [3] that is also the bas is of the present\npaper. Gouet [18, 19] proved (in particular) an a.s. converg ence result for triangular\nurns with 2 colours and deterministic replacements, assumi ng also that the urn is\nbalanced, meaningř\njξij“bfor some constant b(and all ai“1, see further Sec-\ntion 10). Janson [28] studied triangular urns with 2 colours and deterministic ξi,\nand proved convergence in distribution (but not a.s.) of the components Xniafter\nsuitable normalizations; there are several cases, and the l imits are sometimes normal\nand sometimes not. This was partially extended by Aguech [1] , who also studied\ntriangular urns with 2 colours, but allowed random replacem entsξi(under some hy-\npotheses, see Example 14.4); moreover, he proved convergen ce a.s., and not just in\ndistribution. Bose, Dasgupta, and Maulik [12] (and [11] for q“2) studied triangular\nurns with an arbitrary (finite) number of colours; they assum ed that the replace-\nments are deterministic, and that the urn is balanced, and th en, under some further\nassumptions, showed convergence a.s. of the components Xni, suitably normalized,\nsee Example 14.12.\nThemain purposeof the present paperis to extend these resul ts by Gouet [18, 19],\nAguech [1], and Bose, Dasgupta, and Maulik [12], and show a.s . convergence for\ntriangular urnswith any (finite) numberof colours, allowin g replacements ξithat are\nboth random and unbalanced. Our main result is the following , using the technical\nassumptions (A1)–(A5) in Section 2.1 and the notation define d in (2.7)–(2.13) in\nSection 2.4 below. See also the extensions Theorems 8.4 and B .1 where the technical\nassumptionsareweakened(allowingurnswithsubtractions andreducingourmoment\nassumption, respectively); since the proof of the theorem i s rather long and we want\nto focus on the main ideas, we use first the conditions (A1)–(A 5) (which suffice for\nmany applications), and add later the extra arguments neede d for the extensions.\nTheorem 1.8. Let pXniqiPQbe a triangular P´ olya urn satisfying the conditions\n(A1)–(A5)below. Then, for every colour iPQ, there exists a random variable pXi\nwith0ăpXiă 8a.s. such that as nÑ 8:\n(i)Ifpλą0, then\nXni\nnλ˚\ni{pλlogγina.s.Ý ÑpXi. (1.3)\n(ii)Ifpλ“0, then\nXni\nnκi{pκ0a.s.Ý ÑpXi. (1.4)\nNote that the exponent γimay be both positive and negative; see the examples\nin Section 14. Note also that the various exponents in (1.3) a nd (1.4) are explicitly\ngiven in Section 2.4, but the limiting random variables pXiare known only in some\nspecial cases; in general they are unfortunately unknown.\nThe virtue of Theorem 1.8 is that it is very general, but as dis cussed in the remaks\nbelow, more precise results are known in some special cases.\nIn Theorem 1.8, the main case is (i), pλą0, and the reader should focus on this\ncase. The case pλ“0 is more special and of less interest for applications, but i t\nis included for completeness; by (2.10) and (2.7), this case occurs when ξii“0 for\neveryiPQ. (We might call such urns strictly triangular .)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 5\nAnothermajorresultinthepresentpaper(Theorem12.5)say sinparticularthatin\nthe special case of balanced triangular urns, if the replace ments have finite moments,\nthen the a.s. limits in Theorem 1.8 hold also in Lpfor anypă 8, and thus moments\nconverge. (It is an open problem whether this extends to some class of unbalanced\ntriangular urns.)\nRemark 1.9. Theorem 1.8 describes the first-order asymptotics of the urn . We will\nsee inSection 7that thelimitingrandomvariable pXiis deterministic(i.e., aconstant)\nin some cases, but not in general. In cases where Theorem 1.8 y ields a limit pXithat\nis deterministic (and perhaps also otherwise), it is intere sting to study fluctuations\n(i.e., second order terms) and try to find limits (e.g. in dist ribution, after a suitable\nnormalization) for the difference of the two sides of (1.3) or ( 1.4). Such results in\nsome cases are given in [19], [28] and [1]; see the examples in Section 14, but we will\nnot pursue this problem here, and leave it as an open problem. △\nRemark 1.10. Convergencealmostsurelyimpliesconvergenceindistribu tion. Thus,\nas a corollary, (1.3) holds also with convergence in distrib ution. However, our proof\ndoes not seem to provide a method to find the limit distributio n, i.e. the distribution\nofpXi, except in some very simple cases. Moreover, the limits pXiare (in general)\ndependent.\nForq“2 and deterministic ξij, limit distributions were given in [28]. (Sometimes\ndegenerate, sometimes not.) The results there thus describ e the distribution of pXi\nin this case, although the descriptions in some cases are com plicated. Some further\nexamples of known limit distributions are given in some exam ples in Section 14. We\nleave the general case as another open problem. △\nThe proof of Theorem 1.8 is given in Sections 3–5 below, after some preliminaries\nin Section 2. The proof is based on the embedding by Athreya an d Karlin [3] of\na P´ olya urn into a continuous-time multitype branching pro cess (Section 2.6); we\nthen apply martingale methods to obtain a continuous-time v ersion of Theorem 1.8\n(Theorem 4.1); finally, this implies results for the embedde d discrete-time urn. The\nproof is generalized to urns with subtraction in Section 8, a nd to urns with a weaker\nmoment condition in Appendix B. Since the proofs are rather l ong and technical, we\nprefer to first present the proof in the basic case Theorem 1.8 (which is enough for\nmost applications) and later discuss the modifications requ ired for the extensions,\ninstead of proving the most general results immediately.\nRemark 1.11. Gouet [18, 19] and Bose, Dasgupta, and Maulik [12], in specia l cases\n(seeExample14.3andExample14.12), insteadstudy Xnidirectlyandusemartingale\nmetods in discrete time. It seems that this approach (also us ed by several authors\nfor non-triangular urns) works well for balanced urns, but t hat the embedding into\ncontinuous time works better for unbalanced urns. △\nRemark 1.12. We consider in this paper only triangular P´ olya urns. Anoth er\nimportant class of urns consists of the irreducible urns. In this case a.s. convergence\n(under some technical conditions) was shown by Athreya and K arlin [3], see also [4,\nSection V.9.3] and [27, Theorem 3.21].\nIt might be possible to combine the methods of the present pap er and the meth-\nods for irreducible urns to obtain results on a.s. convergen ce for all types of P´ olya\nurns (under some technical conditions), see Remark 2.3, but the present paper is\nlong as it is and we leave this as a speculation for future rese arch. (Note also the6 SVANTE JANSON\ncounterexamples Example 14.14 and 14.15, showing that some conditions are needed\neven in the triangular case.) △\n1.1.Contents. Section 2 contains preliminaries, including some definitio ns and no-\ntation. Section 3 consists of a series of lemmas that compris e the main technical\npart of our proofs. They lead to the main theorem for continuo us time in Section 4,\nwhich in turn is used to prove Theorem 1.8 in Section 5. Sectio ns 6 (continuous\ntime) and 7 (discrete time) contain results on whether the li mit random variable are\ndegenerate (i.e., constant) or not, and some related result s.\nSection 8 extend the previous results to urns with subtracio n, where we allow\nξii“ ´1. With some extra technical conditions, the previous resul ts hold in this\ncase too, with only minor modifications of the proofs.\nThe following sections contain some complements. Section 9 is a short section\ncomparing random and non-random replacements with the same means.\nSection 10 contains some general results on balanced urn, ma inly as preliminaries\nto the following sections.\nSection 11 considers the number of times a given colour is dra wn; it is shown that\nthe results of earlier sections extend to this case.\nSection 12 contains results on convergence in L2and inLp, and closely related\nresults on convergence of moments in, for example, Theorem 1 .8. For the main\nresult (discrete-time), we have to assume that urn is balanc ed, and we state an open\nproblem for more general urns.\nSection 13 discusses briefly another open problem (rates of c onvergence).\nSection 14 contains a number of examples that illustrate the results and their\nlimitations, and also give connections to previous literat ure.\nFinally, Appendix A contains some simple general lemmas on a bsolute continuity\nthat we believe are known, but for which we were unable to find r eferences. Ap-\npendix B gives proofs of Lpversions of L2estimates used in the main part of the\npaper; this yields both the extension of Theorem 1.8 mention ed above, and a proof\nof the results in Section 12. Appendix C gives a rather techni cal proof of one claim\nin Example 14.14.\n2.Some notation and other preliminaries\nWe use throughout the paper the notation Q,q,Xn“ pXniqiPQ,ai, andξi“\npξijqjPQintroduced in the introduction.\n2.1.Standing assumptions. In the rest of the paper we assume\n(A0)The P´ olya urn is triangular.\n(Unless we explicitly say so, for example when we discuss thi s property in Sec-\ntion 2.3.) For the central part of the paper (Sections 2–7) we make also some\nstanding technical assumptions:\n(A1)The initial urn X0is non-random. Moreover, each X0iě0 andřaiX0ią0.\n(The results may be extended to random X0by conditioning on X0.)\n(A2)Ifai“0, thenξi“0, i.e.,ξij“0 for every jPQ. (This is without loss\nof generality, since ai“0 means that balls of colour inever are drawn, and\nthusξidoes not matter.)\n(A3)For every iPQ, eitherX0ią0, or there exists j‰isuch that Ppξjią0q ą0\n(or both). (This too is without loss of generality, since oth erwise balls ofALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 7\ncolourican never appear, so Xni“0 a.s. for all n, and we may remove the\ncolourifromQ.)\n(A4)Eξ2\nijă 8for alli,jPQ.\n(A5)ξijě0 (a.s.) for all i,jPQ.\nNote that (A5) implies that every Xniis (weakly) increasing in n. In particular,\nthe urn never gets empty. Combined with (A1) we see thatř\niaiXnią0 for every\nn, and thus the probabilities (1.1) and the urn process are wel l defined. We discuss\nextensions to urns not satisfying (A5) (i.e., urns with subt ractions) in Section 8; see\nTheorems 8.4–8.6.\nRemark 2.1. The second moment condition (A4) is for technical convenien ce. In\nfact, the results (including Theorem 1.8) hold assuming onl yEξp\nijă 8for some\npą1. However, this adds further arguments to an already long pr oof, so we assume\nsecond moments in the main part of the paper and show the exten sion topą1 in\nAppendix B.\nIt seems possible that the results could extend further by as suming only that\nEξijlogξijă 8, as for diagonal urns in [2] using related results for (singl e-type)\nbranching processes [4, Theorem III.7.2]. (The results do n ot extend without modi-\nfications to cases without this assumption, see Example 14.1 6.) We have not pursued\nthis, and leave it as an open problem. △\n2.2.General notation. As usual, we ignore events of probability 0. We often write\n“almost surely” or “a.s.” for emphasis, but we may also tacit ly omit this.\nWe use “increasing” in the weak sense. Similarly, “positive function” means in the\nweak sense, i.e. ě0. (However, “positive constant” always means strictly pos itive;\nwe sometimes add “strictly” for emphasis, but not always.)\nWe lets^t:“mints,tuands_t:“maxts,tu.\nWe letZě0:“ t0,1,2,...uandZ`:“ t1,2,...u.\nWe usea.s.Ý Ñ,pÝ Ñ, anddÝ Ñto denote convergence almost surely, in probability, and\nin distribution, respectively.\n“Absolutely continuous” (for a probability distribution i nRorRd) means with\nrespect to Lebesgue measure. (Except in Appendix A when a ref erence measure is\nexplicitly specified.) We may say that a random variable is ab solutely continuous\nwhen its distribution is.\nWe use some standard probability distributions: Exp pλqis an exponential distri-\nbution with mean λą0; we may also say with rate1{λ. Γpα,bqis a Gamma distri-\nbution. (Thus Exp pλq “Γp1,λq.) Be ppqis a Bernoulli distribution. NegBin pr,pqis\na negative binomial distribution. Ge ppqis a geometric distribution on t1,2,...u.\nFor a random variable W, we letLpWqdenotes the distribution of W, and, for\nanypą0,\n/bardblW/bardblp:“`\nE|W|p˘1{p. (2.1)\nLpdenotes the set of all random variables Wsuch that /bardblW/bardblpă 8.\nCdenotes unspecified constants that may vary from one occurre nce to the next.\nThey may depend on the activities aiand the distributions of the replacements ξi,\nand perhaps on other parameters clear from the context, but t hey never depend on\nnort.\nLet\nrij:“Eξij. (2.2)8 SVANTE JANSON\nThus the matrix prijqi,jPQis the mean replacement matrix. Note that rijexists and\nis finite by (A4). By (A5), we have rijě0, andrij“0ð ñξij“0 a.s.\nWe occasionally (when we discuss two different urns at the same time) denote a\nP´ olya urn by U; we then (somewhat informally) mean both the urn process Xnand\nits continuous-time version Xptqdefined below, and also the colour set Qand the\nreplacement matrix pξijq.\n2.3.The colour graph. (Inthissubsection, theurndoesnothavetobetriangular.)\nRecall that Qis the set of colours. As said in the introduction, we allow Qto be any\nfinite set, although it is possible to assume Q“ t1,...,q uwithout loss of generality\nwhen this is convenient.\nWe regard the set Qof colours as a directed graph, called the colour graph , where\nfor any distinct i,jPQthere is an edge iÑjif and only if Ppξij‰0q ą0. In other\nwords,iÑjmeans that if a ball of colour iis drawn, it is possible (with positive\nprobability) that some balls of colour jare added. Note that by (A2), iÑjentails\naią0, so balls of colour imay (and will) actually be drawn; furthermore, by (A5),\niÑjð ñriją0 andi‰j. (2.3)\nWe say, again for two distinct colours i,jPQ, thatiis anancestor ofj, andja\ndescendant ofi, if there exists a directed path in Qfromitoj; we denote this by\ni≺j. In other words, i≺jmeans that if we start the urn with only a ball of colour\ni, then it is possible that balls of colour jare added at some later time.\nIfiPQ, letPi:“ tjPQ:jÑiu, the set of colours different from iwhose\ndrawings may cause addition of balls of colour i. We say that the colour iisminimal\nifPi“ H. We denote the set of minimal colours by Qmin. Note that (A3) can be\nformulated as: X0ią0 for every iPQmin.\nWe say that the urn is triangular , if there exists a (re)labelling of the colours\nby 1,...,qthat makes the matrix pξijqi,jPQtriangular a.s. (Assuming (A5), this is\nequivalent to the mean replacement matrix prijqi,jPQbeing triangular, cf. (2.3).) In\nother words, the urn is triangular if there exists a total ord ering ăof the colours\nsuch that\niąjù ñξij“0 a.s. (2.4)\nUsing the definitions above to rewrite (2.4), we see that the u rn is triangular if\nand only there exists a total ordering ăsuch that, for i,jPQ,\niÑjù ñiăj. (2.5)\nFurthermore, this is equivalent to\ni≺jù ñiăj. (2.6)\nProposition 2.2. The following are equivalent.\n(i)The urn is triangular.\n(ii)The colour graph is acyclic.\n(iii)The relation ≺onQis a partial order.\nProof.(i) implies (ii) and (iii) as a consequence of (2.5) and (2.6) .\n(ii)ð ñ(iii) is easily seen.\nFinally, any partial order can be extended to a total order. T hus, if (iii) holds, we\nmay extend ≺to a total order ă, which means that (2.6) holds. /squareALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 9\nNote that if the urn is triangular, a colour iis minimal if and only if it is minimal\nin the partial order ≺.\nRemark 2.3. WemayalsonotethataP´ olyaurnisirreducibleifandonlyif itscolour\ngraph is strongly connected. Given any P´ olya urn, we may dec ompose its colour\ngraph into its strongly connected components, which are lin ked by the remaining\nedges inanacyclic way. Hencetheurncan beregardedas anacy clic directednetwork\nof irreducible urns. This suggests, as mentioned in Remark 1 .12, that the methods\nin the present paper perhaps might be combined with methods f or irreducible urns\nto obtain results for general urns. △\n2.4.More notation. Let, foriPQ,\nλi:“airii“aiEξiiě0. (2.7)\nIn the continuous-time version introduced below, λiis the rate of additions to colour\niby drawings of the same colour. Since colours also may be adde d by drawings of\nanother colour, we define further\nλ˚\ni:“max/visualspace\nλj:j/precedesequali(\ně0. (2.8)\nIn other words, λ˚\niis the largest λjfor a colour jsuch that there exists a path\n(possibly of length 0) in Qfromjtoi. Such a path may contain several colours k\nwith the same, maximal, λk, and we denote the largest number of them in a single\npath by 1 `κi; i.e.,\nκi:“max/visualspace\nκ:Di1≺i2≺¨ ¨ ¨≺iκ`1/precedesequaliwithλi1“ ¨ ¨ ¨ “λiκ`1“λ˚\ni(\ně0.(2.9)\nDefine further\npλ:“maxtλi:iPQu ě0, (2.10)\npκ:“maxtκi:iPQandλ˚\ni“pλu\n“max/visualspace\nκ:Di1≺i2≺¨ ¨ ¨≺iκ`1withλi1“ ¨ ¨ ¨ “λiκ`1“pλ(\ně0.(2.11)\nIfpλ“0 (i.e., if λi“0 for every iPQ), let further\npκ0:“1`max/visualspace\nκi:iPQwithaią0(\ně1. (2.12)\nIfpλą0, define also\nγi:“κi´pκλ˚\ni{pλ, i PQ. (2.13)\n2.5.Stochastic processes. All our continuous-time stochastic processes are de-\nfined on r0,8qand are assumed to be c` adl` ag (right-continuous with left l imits).\nWe consider martingales without explicitly specifying the filtration; this will al-\nways be the natural filtration pFtqtě0, whereFtis generated by “everything that has\nhappened up to time t”. IfTis a stopping time, then FTdenotes the corresponding\nσ-field generated by all events up to time T.\nGiven a stochastic process W“ pWptqqtě0, we define its maximal process by\nW˚ptq:“sup\n0ďsďt|Wpsq|,0ďtď 8. (2.14)\n(We consider only să 8, also when t“ 8.) We further define\n∆Wptq:“Wptq ´Wpt´q,0ďtă 8, (2.15)\nwhereWpt´qis the left limit at t, withWp0´q:“0.10 SVANTE JANSON\nIfWis a process with locally bounded variation, we may define its quadratic\nvariation by\nrW,W st:“ÿ\n0ďsďt/vextendsingle/vextendsingle∆Wpsq/vextendsingle/vextendsingle2,0ďtď 8, (2.16)\n(summing over să 8ift“ 8), noting that the sum always really is countable.\n(We have no need for the definition for general semimartingal es, see e.g. [32, p. 519\nand Theorem 26.6] or [42, Section II.6].) Recall [42, Coroll ary 3 to Theorem II.6.27,\np. 73] that if Mis a local martingale and ErM,M stă 8for some tă 8, then\nE|Mptq|2“ErM,M st, (2.17)\nand as a consequence, if ErM,M s8ă 8, thenMis anL2-bounded martingale and\nthusMp8q:“limtÑ8Mptqexists a.s., and (2.17) holds for all tď 8.\nRecall also Doob’s inequality [32, Proposition 7.16] which as a special case yields,\ncombined with (2.17),\nEM˚ptq2ďCE|Mptq|2“CErM,M st, (2.18)\nprovided ErM,M stă 8. (HereC“4, but we will not use this.)\n2.6.Continuous-time urn. We will use the standard method of embedding the\ndiscrete-timeurninacontinuous-timeprocess, duetoAthr eyaandKarlin[3], seealso\n[4,§V.9], and after them used in many papers. We thus define the continuous-time\nurnas a vector valued Markov process Xptq “ ppXiptqqiPQwith given initial value\nXp0q:“X0such that, for each iPQ, “a ball of colour iis drawn” with intensity\naiXiptq; when a ball of colour iis drawn, we add to Xptqa copy of ξi(independent\nof the history). In the classical case (e.g. [3]) when each Xiptqis integer valued, Xptq\nis a multitype continuous-time Markov branching process; i n general (allowing any\nrealXiptq ě0),Xptqisa(vector-valued) continuous-time continuous-state br anching\nprocess (abbreviated CB process ) as defined by Jiˇ rina [30], see also e.g. Li [37]. (The\nprocessXptqis of jump-type, as in [30, Section 3].) Note that (A4) implie sEξijă 8\nfor alli,jPQ, which is a well-known sufficient condition for non-explosio ns; i.e.,\nthere exists such a process XptqwithXptqfinite for all tP r0,8q.\nSinceaiXip0q ą0 for some iby (A1), and Xiptqis increasing by (A5), there will\na.s. be infinitely many draws in the urn. We let pTnbe thenth time that a ball is\ndrawn, with pT0:“0. Then the discrete-time urn Xnin Section 1 can be realized as\nXn:“XppTnq. (2.19)\nWe assume (2.19) throughout the paper.\nSinceXptqdoes not explode, there is a.s. only a finite number of draws up to any\nfinite time t, and thus pTna.s.Ý Ñ 8asnÑ 8. Note that we denote the discrete-time\nurn byXn“ pXniqiand the continuous-time urn by Xptq “ pXiptqqi.\nOf course, the continuous-time urn can also be studied for it s own sake, see for\nexample [13].\n3.Analysis of one colour\nIn this section, we study one fixed colour iPQin the continuous-time urn.\nRecall that Xiptqis the number (amount) of balls of colour iat timetin the\ncontinuous-time urn. There are several possible sources of these balls: some may be\nthere from the beginning, some may be added when a ball of some other colour jisALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 11\ndrawn (with jÑiand thus j≺i), and in both cases these balls of colour imay\nlater be drawn and producefurther generations of balls of co louri(provided λią0).\nWe begin by considering, in the following two subsections, t wo simpler special\ncases. We will then in the final subsection, rather easily, tr eat the general case by\ncombining these special cases.\nWe first state some simple general results; these are more or l ess known, see e.g.\n[27, Lemma 9.3] for a related result, but for convenience we g ive full proofs.\nFix a colour j. Let 0 ăT1ăT2ă...be the times that a ball of colour jis\ndrawn, and let Nptq:“/vextendsingle/vextendsingletk:Tkďtu/vextendsingle/vextendsingledenote the corresponding counting process;\ni.e.,Nptqis the number of draws of colour jup to time t. These draws occur with\nintensity ajXjptq, which means that\nrNptq:“Nptq ´ajżt\n0Xjpsqds, t ě0, (3.1)\nis a local martingale. In fact, rNptqis a martingale, which we verify by the following\nsimple lemma. Recall the notation (2.14).\nLemma 3.1. Suppose that EXjptq ă 8for some tP r0,8q. Then,\nENptq ă 8, (3.2)\nErN˚ptq “Esup\nsďt|rNpsq| ă 8. (3.3)\nIn particular, if EXjptq ă 8for every tă 8, then the local martingale rNptqis a\nmartingale.\nProof.By the definition of local martingale, there exists an increa sing sequence of\nstopping times τm,mě1, such that τmÕ 8a.s. asmÑ 8, andrNpt^τmqis a\nmartingale for each m. In particular, since rNp0q “0,\nENpt^τmq “ErNpt^τmq `ajEżt^τm\n0Xjpsqds“ajEżt^τm\n0Xjpsqds.(3.4)\nSinceNptqandXjptqare increasing positive functions of t, we may use monotone\nconvergence and let mÑ 8to obtain\nENptq “ajEżt\n0XjpsqdsďajE“\ntXjptq‰\nă 8. (3.5)\nThe monotonicity of Nptqfurther implies\nrN˚ptq ďNptq `ajżt\n0XjpsqdsďNptq `ajtXjptq, (3.6)\nand thus (3.3) follows by (3.5).\nThe final statement follows since a local martingale with int egrable maximal func-\ntion is a martingale. /square\nLemma 3.2. Suppose that EXjptq ă 8for some tP r0,8q.\n(i)Letfbe a positive or bounded measurable function on r0,ts. Then\nE8ÿ\nk“11tTkďtufpTkq “Eżt\n0fpsqdNpsq “ajEżt\n0fpsqXjpsqds.(3.7)12 SVANTE JANSON\n(ii)Let also pηkq8\n1be a sequence of identically distributed random variables w ith\nfinite mean Eη1such that ηkis independent of Tk. Then\nE8ÿ\nk“11tTkďtufpTkqηk“ajEη1Eżt\n0fpsqXjpsqds. (3.8)\nProof. Step 1 . The left and middle terms in (3.7) are the same, sinceşt\n0fpsqdNpsq “ř8\nk“11tTkďtufpTkqby the definition of N.\nStep 2. Suppose that fpsq “1taăsďbu, the indicator function of an interval pa,bs P\nr0,ts. Then, by (3.1),\nżt\n0fpsqdNpsq ´ajżt\n0fpsqXjpsqds“żt\n0fpsqdrNpsq “rNpbq ´rNpaq,(3.9)\nandErrNpbq ´rNpaqs “0 by Lemma 3.1; hence (3.7) holds for such f.\nStep 3. The monotone class theorem [21, Theorem 1.2.3] now shows th at (3.7) holds\nfor the indicator function fpsq “1tsPAuof any Borel set AP r0,ts.\nStep 4. By linearity, (3.7) holds for any positive simple function f. Then, by mono-\ntone convergence, (3.7) holds for any positive measurable f unction, and by linearity\nagain for any bounded measurable function. This proves (i).\nStep 5. In (ii), we may decompose ηkinto its positive and negative parts; thus it\nsuffices to consider ηkě0. Then the sum in (3.8) is well-defined, and\nE8ÿ\nk“11tTkďtufpTkqηk“8ÿ\nk“1E“\n1tTkďtufpTkqηk‰\n“Eη18ÿ\nk“1E“\n1tTkďtufpTkq‰\n,(3.10)\nand (3.8) follows from (3.7). /square\n3.1.A colour not influenced by others. In this subsection, we assume that\nξji“0 a.s. for all colours j‰i. Equivalently, jÑiforjPQ, i.e.,Pi“ H; in\nother words, iis a minimal colour. This means that Xiptqis affected only by draws\nof the same colour i, and we may thus ignore all other colours and regard Xiptqas a\ncontinuous-time urn with a single colour. In other words, Xiptqis a one-dimensional\nCB process, starting at some given Xip0qand adding copies of ξiiwith intensity\naiXiptq. We write Xip0q “x0. Note that our assumption (A3) means x0ą0, but\nfor completeness we allow also the trivial case x0“0 in the present subsection. (In\nthis subsection, we really use only the assumptions (A1), (A 4), and (A5), and only\nfor the colour i.) Note also that λ˚\ni“λiby (2.8).\nWe will only need a few simple facts about CB processes; see fu rther e.g. [30; 36;\n20; 9; 37] where many more results are given.\nRecall that λi“airii. Ifλi“0, then there are no additions at all, and Xiptq “\nXip0q “x0is constant.\nIt is easy to see that since riiă 8, the CB process Xiptqis well defined and non-\nexplosive (i.e., finite for all t), for any given x0ě0. Moreover [20], [9, Proposition\n2.2],\nEXiptq “eλitx0. (3.11)\nThis implies by conditioning and the Markov property that [4 , Theorem III.7.1]\ne´λitXiptqis a martingale . (3.12)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 13\nThe following result too well known, but we include a proof fo r completeness, and\nsince we discussmodifications of it later. Seee.g. [4, Theor em V.8.2] and [27, Lemma\n9.5] for a vector-valued extension.\nLemma 3.3. Suppose that ξji“0a.s. for all j‰i. Thene´λitXiptqis anL2-\nbounded martingale, and thus\ne´λitXiptqa.s.Ý ÑXi (3.13)\nfor some random variable Xi. Furthermore, with x0“Xip0q,\nEXi“x0, (3.14)\nVarXi“x0ai\nλiEξ2\nii, (3.15)\nwhere we interpret0\n0as0. Hence\nVar“\ne´λitXiptq‰\nďVarXiďCx0, (3.16)\nE/vextendsingle/vextendsingle/vextendsinglesup\ntě0e´λitXiptq/vextendsingle/vextendsingle/vextendsingle2\nďCEX2\niďCpx0`x2\n0q. (3.17)\nFurthermore, if x0ą0, then0ăXiă 8a.s.\nProof.The case λi“0 is trivial, with Xi“x0. The same holds if x0“0. We may\nthus assume λią0 andx0ą0.\nWe argue as in [27, Section 9]. As above (now taking j“i), let 0 ăT1ăT2ă...\nbe the times that a ball of colour iis drawn, and let Nptq:“/vextendsingle/vextendsingleti:Tiďtu/vextendsingle/vextendsingledenote the\ncorresponding counting process- Furthermore, let ηk:“∆XipTkqbe the number of\nballs of colour iadded at the k-th draw. Thus η1,η2,...is a sequence of independent\ncopies of ξii.\nLetMptq:“e´λitXiptq. Then Mis a martingale by (3.12), and its quadratic\nvariation is by (2.16) given by\nrM,M st“ÿ\n0ďsďt|∆Mpsq|2“x2\n0`ÿ\nTkďte´2λiTk|∆XipTkq|2“x2\n0`ÿ\nTkďte´2λiTkη2\nk.\n(3.18)\nHence, it follows from Lemma 3.2(ii) that\nErM,M st“x2\n0`aiErξ2\niisżt\n0e´2λisEXipsqds. (3.19)\nHence, writing β:“Eξ2\nii, (3.11) yields\nErM,M st“x2\n0`aiβx0żt\n0e´λisds“x2\n0`x0ai\nλiβ`\n1´e´λit˘\n. (3.20)\nThis shows by (2.17) that Mis anL2-bounded martingale; thus (3.13) holds for\nsomeXi“Mp8q. Clearly Xiě0. We have EMp8q “EM0“x0, which yields\n(3.14). Furthermore, (3.15) holds by (2.17) and (3.20) agai n, together with (3.14);\nthis yields also (3.16). Doob’s inequality (2.18) yields (3 .17).\nFinally, the distribution of Xidepends on x0; thus let us denote XibyXipx0q. The\nCB property implies that if x,yě0, thenXipx`yqd“Xipxq `X1\nipyq, whereX1\nipyqis\na copy of Xipyqindependent of Xipxq. Letppxq:“P`\nXipxq “0˘\nP r0,1s. It follows\nthat for any x,yě0,\nppx`yq “ppxqppyq, (3.21)14 SVANTE JANSON\nand thus there exists cP r0,8ssuch that\nPpXipxq “0q “ppxq “e´cx, x ą0. (3.22)\nWe must have cą0, since otherwise Xi“0 a.s. for any x0, which contradicts (3.14).\nThe Markov property and (3.22) yield, for any tP r0,8q,\nP`\nXi“0|Xiptq˘\n“e´cXiptq. (3.23)\nThus, taking the expectation,\nP`\nXi“0˘\n“Ee´cXiptq. (3.24)\nIt is clear that if x0ą0 andλią0, soaią0 andEξiią0, then there is a.s.\nan infinite number of draws, and the (standard) law of large nu mbers shows that\nXiptqa.s.Ý Ñ 8. Consequently, letting tÑ 8in (3.23) yields, by dominated conver-\ngence,\nPpXi“0q “0. (3.25)\n/square\nBy (3.15), the limit Xiis degenerate only in the trivial cases when x0“0 or\nai“0. We note that except in these cases, the limit has an absolut ely continuous\ndistribution.\nLemma 3.4. In Lemma 3.3, suppose further that λią0andx0ą0. Then the\ndistribution of Xiis absolutely continuous.\nProof.LetT1bethefirsttimethataballofcolour iisdrawn; then T1PExpp1{paix0qq.\nThe distribution of the balls added at T1is independent of T1, and thus XipT1qis\nindependent of T1. It follows by the strong Markov property that the stochasti c\nprocessYptq:“XipT1`tq,tě0, is independent of T1. By (3.13), we have as\ntÑ 8,\ne´λitYptq “eλiT1e´λipT1`tqXipT1`tqa.s.Ý ÑY:“eλiT1Xi, (3.26)\nand it follows that Yis independent of T1. Consequently,\nXi“e´λiT1Y, (3.27)\nwhere the two factors on the right-hand side are independent . The result follows\nfrom (3.27) since e´λiT1has an absolutely continuous distribution and Yą0 a.s.\n(because Xią0 a.s. by Lemma 3.3). /square\n3.2.A colour only produced by one other colour. We continue to consider a\nfixed colour i. In this subsection we assume that |Pi| “1; thus there is exactly one\ncolourjPQsuch that jÑi. We assume also that Xip0q “0, so there are initially\nno balls of colour i. Recall that the assumption jÑimeansriją0, and thus\nimplicitly ają0.\nSince the colour graph is acyclic, there is no feedback from c olourito colour j;\nthus we may regard the entire process Xjptq,tP r0,8q, as known, and consider only\nits effect on Xiptq.\nWe may regard Xiptqas a CB process with immigration. Let again 0 ăT1ă\nT2ă...be the times that a ball of colour jis drawn, and let η1,η2,...be the\ncorresponding number (amount) of balls of colour ithat are added. In case there\nis only a finite number Kof times that a ball of colour jis drawn, we define for\ncompleteness Tk“ 8forkąK, and pick ηkwith the correct distribution andALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 15\nindependent of everything else. We regard each ηkas an immigrant coming at Tk.\n(Someηkmay be 0; this is no problem.) We can separate the descendants of each\nof these immigrants, and write\nXiptq “ÿ\nk:TkďtYkpt´Tkq “8ÿ\nk“11tTkďtuYkpt´Tkq (3.28)\nwhere each Ykptq, conditioned on ηk, is a copy of the single-colour CB process in\nSection 3.1 with Ykp0q “ηk. Furthermore, conditionally on pηkqk, the processes\nYkptqare independent. Recall that the additions η1,η2,...are i.i.d. copies of ξji;\nin particular, they have expectations Eηk“rji. The times Tkare stopping times;\nmoreover, the process Ykis independent of Tkand theσ-fieldFTk.\nThe following lemma contains much of the technical parts of o ur argument. It\nmakes an assumption on the growth of Xjptqthat later will be justified by an induc-\ntion argument.\nWe define, for mPQ,\nrXmptq:“t´κme´λ˚\nmtXmptq,0ătă 8, (3.29)\nrX˚˚\nm:“sup\ntě0/visualspace\npt`1q´κme´λ˚\nmtXmptq(\n. (3.30)\nWe use powers of tin (3.29) and of t`1 in (3.30) for technical convenience below\n(and we therefore use a notation with˚˚instead of˚); the difference is not important\nsince we are mainly interested in limits as tÑ 8.\nLemma 3.5. LetiPQ. Suppose that there is exactly one colour jPQsuch that\njÑi, and that Xip0q “0. Suppose further that\nrXjptqa.s.Ý ÑXjastÑ 8, (3.31)\nfor some random variable Xjě0, and\n/vextenddouble/vextenddoublerX˚˚\nj/vextenddouble/vextenddouble\n2ă 8. (3.32)\nThen\nrXiptqa.s.Ý ÑXiastÑ 8, (3.33)\nfor some Xiě0and\n/vextenddouble/vextenddoublerX˚˚\ni/vextenddouble/vextenddouble\n2ă 8. (3.34)\nFurthermore, a.s., if Xją0, thenXią0.\nMoreover, if λiďλ˚\nj, and thus λ˚\ni“λ˚\nj, then\nXi“#ajrji\nλ˚\nj´λiXj, λiăλ˚\nj,\najrji\nκiXj, λi“λ˚\nj.(3.35)\nProof.Using (3.28), and recalling Ykp0q “ηk, we make the decomposition\ne´λitXiptq “8ÿ\nk“11tTkďtue´λiTk`\ne´λipt´TkqYkpt´Tkq ´Ykp0q˘\n(3.36)\n`8ÿ\nk“11tTkďtue´λiTkpηk´rjiq (3.37)16 SVANTE JANSON\n`rji´8ÿ\nk“11tTkďtue´λiTk´żt\n0e´λisajXjpsqds¯\n(3.38)\n`rjiajżt\n0e´λisXjpsqds (3.39)\n“:Z1ptq `Z2ptq `rjiZ3ptq `ajrjiZ4ptq, (3.40)\nsay. We consider the four processes Zℓptqin (3.40) separately (initially, at least).\nMoreover, recalling (2.8) and(2.9), weconsider sometimes separately thethreecases:\n(i)λiąλ˚\nj: thenλ˚\ni“λiandκi“0.\n(ii)λi“λ˚\nj: thenλ˚\ni“λi, andκi“κj`1.\n(iii)λiăλ˚\nj: thenλ˚\ni“λ˚\nj, andκi“κj.\nWe define, in analogy with (3.29)–(3.30) (but notethedifferen t exponents λi´λ˚\ni),\nforℓ“1,2,3,4,\nrZℓptq:“t´κiepλi´λ˚\niqtZℓptq, (3.41)\nrZ˚˚\nℓ:“sup\ntě0/visualspace\npt`1q´κiepλi´λ˚\niqt|Zℓptq|(\n. (3.42)\nThen (3.40) yields\nrXiptq “rZ1ptq `rZ2ptq `rjirZ3ptq `ajrjirZ4ptq, (3.43)\nrX˚˚\niďrZ˚˚\n1`rZ˚˚\n2`CrZ˚˚\n3`CrZ˚˚\n4. (3.44)\nWe will prove, for ℓ“1,...,4 and some Zℓ,\nrZℓptqa.s.Ý ÑZℓastÑ 8, (3.45)\n/bardblrZ˚˚\nℓ/bardbl2ă 8, (3.46)\nand then (3.33)–(3.34) follow from (3.43)–(3.44).\nWe treat Z1ptq,...,Z 4ptqin (3.39) in reverse order, partly because Z4ptqwill turn\nout to be the main term (sometimes at least).\nStep 1:Z4. By (3.39) and (3.30),\n0ďZ4ptq ďrX˚˚\njżt\n0ps`1qκjepλ˚\nj´λiqsds\nď$\n’&\n’%CrX˚˚\nj“Cpt`1qκirX˚˚\nj, λ iąλ˚\nj,\npt`1qκj`1rX˚˚\nj“ pt`1qκirX˚˚\nj, λi“λ˚\nj,\nCpt`1qκjepλ˚\nj´λiqtrX˚˚\nj, λ iăλ˚\nj.(3.47)\nIt follows that in all three cases (i)–(iii), rZ˚˚\n4ďCrX˚˚\nj. In particular, (3.32) implies\n/vextenddouble/vextenddoublerZ˚˚\n4/vextenddouble/vextenddouble\n2ă 8. (3.48)\nFor (3.45) we treat the three cases (i)–(iii) separately, in each case using (3.41)\nand (3.29)–(3.30). First, in case (i), we let tÑ 8in (3.39) and (3.47) and conclude\nthat a.s.\nrZ4ptq “Z4ptq ÑZ4p8q:“ż8\n0e´λisXjpsqdsďCrX˚˚\njă 8.(3.49)\nHence\nrZ4ptqa.s.Ý ÑZ4“Z4p8q. (3.50)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 17\nNote that if Xją0, thenXjptq ą0 for large t, and thus Z4“Z4p8q ą0 by (3.49).\nIn case (ii), we have for tě1, using the change of variables s“xt,\nrZ4ptq “t´κiZ4ptq “t´κj´1żt\n0sκjrXjpsqds\n“t´κj´1ż1\n0sκjrXjpsqds`ż1\n1{txκjrXjpxtqdx\nÑ0`ż1\n0xκjXjdx“ pκj`1q´1Xj“κ´1\niXj, (3.51)\nby (3.31) and dominated convergence, which applies since (3 .29)–(3.30) show that\nsupsě1rXjpsq ď2κjrX˚˚\njă 8a.s.\nIn case (iii), similarly, now with s“t´u,\nrZ4ptq “t´κjepλi´λ˚\njqtżt\n0sκjepλ˚\nj´λiqsrXjpsqds\n“żt\n0´t´u\nt¯κje´pλ˚\nj´λiqurXjpt´uqdu\nÑż8\n0e´pλ˚\nj´λiquXjdu“ pλ˚\nj´λiq´1Xj, (3.52)\nusing dominated convergence again, justified by, for tě1,\n´t´u\nt¯κjrXjpt´uq ď´t´u`1\nt¯κjrX˚˚\njď2κjrX˚˚\njă 8.(3.53)\nWe have thus shown (3.45) and (3.46) for ℓ“4 in all cases, with Z4ą0 when\nXją0. Furthermore, in cases (ii) and (iii), we have\nZ4“#\nκ´1\niXj, λ i“λ˚\nj,\npλ˚\nj´λiq´1Xj, λiăλ˚\nj.(3.54)\nStep 2:Z3, first part . We have, recalling (3.1),\nZ3ptq “żt\n0e´λisdrNpsq, (3.55)\nwhich is a local martingale since rNptqis; in fact it is a martingale since\nZ˚\n3ptq ďNptq `ajżt\n0e´λisXjpsqdsďNptq `tXjptq, (3.56)\nand thus EZ˚\n3ptq ă 8by Lemma 3.1, noting that the assumption (3.32) implies\nEXjptq ă 8for alltě0. Since Z3ptqhas locally bounded variation and jumps only\nat the times Tk, with ∆Z3pTkq “e´λiTk, its quadratic variation is\nrZ3,Z3st“8ÿ\nk“11tTkďtue´2λiTk“żt\n0e´2λisdNpsq. (3.57)\nThe draws Tkoccur with rate ajXjptq, and it follows by (3.57) and Lemma 3.2 that,\nrecalling (3.30),\nE|Z3ptq|2“ErZ3,Z3st“ajEżt\n0e´2λisXjpsqds18 SVANTE JANSON\nďajErX˚˚\njżt\n0ps`1qκjepλ˚\nj´2λiqsds. (3.58)\nThe rest of the argument will be the same for Z2andZ1, so we first consider them.\nStep 3:Z2, first part . Each term in the sum (3.37) is a martingale, since Tkis a\nstopping time, and ηk´rjihas mean 0 and is independent of the σ-fieldFTk. Hence,\nthe sum Z2ptqis at least a local martingale (since stopping at any Tkyields a finite\nsum and thus a martingale). Since Z2ptqjumps only at the times Tk, and is constant\nin between, its quadratic variation is given by\nrZ2,Z2st“8ÿ\nk“11tTkďtue´2λiTkpηk´rjiq2. (3.59)\nHence, again using the independence between ηkandTk,\nErZ2,Z2st“8ÿ\nk“1E“\n1tTkďtue´2λiTk‰\nE“\npηk´rjiq2‰\n“Varrη1sE8ÿ\nk“11tTkďtue´2λiTk. (3.60)\nWe recognize the final sum from (3.57), and conclude\nErZ2,Z2st“CErZ3,Z3st. (3.61)\nStep 4:Z1, first part . We write the sum in (3.36) as Z1ptq “ř8\nk“1Zpkq\n1ptq, with\nZpkq\n1ptq:“1ttěTkue´λiTk`\ne´λipt´TkqYkpt´Tkq ´Ykp0q˘\n. (3.62)\nIt is easily seen that each Zpkq\n1ptqis a martingale, since Tkis a stopping time and\ne´λitYkptq ´Ykp0qis a martingale starting at 0, which furthermore is independ ent of\nFTk. Hence, for every finite mě1, the finite sum\nZrďms\n1 ptq:“mÿ\nk“1Zpkq\n1ptq (3.63)\nis a martingale, and thus Z1pt^Tmq “Zrďms\n1 pt^Tmqis a martingale. Consequently,\nZ1is a local martingale. Furthermore, conditioned on all Tkandηk, the processes\nZpkq\n1ptqare independent and thus a.s. they jump at different times. (No te that\nZpkq\n1p0q “0.) Hence, by (2.16),\nrZ1,Z1st“8ÿ\nk“1rZpkq\n1,Zpkq\n1st (3.64)\nand thus, using (3.62) and (2.17) again together with (3.16) ,\nE“\nrZ1,Z1st| pTk,ηkq8\n1‰\n“8ÿ\nk“1E“\nrZpkq\n1,Zpkq\n1st| pTk,ηkq8\n1‰\n“8ÿ\nk“11tTkďtue´2λiTkVar“\ne´λipt´TkqYkpt´Tkq |Tk,ηk‰\nď8ÿ\nk“11tTkďtue´2λiTkCηk. (3.65)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 19\nThis yields, by taking the expectation,\nErZ1,Z1stďCE8ÿ\nk“11tTkďtue´2λiTk. (3.66)\nThis is the same sum as in (3.57), and we conclude\nErZ1,Z1stďCErZ3,Z3st. (3.67)\nStep 5:Z1,Z2,Z3, final part . LetℓP t1,2,3u. In all three cases, Zℓptqis a local\nmartingale such that, by (3.61), (3.67), and (3.58), for tP r0,8q,\nErZℓ,ZℓstďCErZ3,Z3stďCErX˚˚\njżt\n0ps`1qκjepλ˚\nj´2λiqsds. (3.68)\nIn particular, (3.68) shows that ErZℓ,Zℓstă 8for every tă 8, and thus Zℓptqis a\nsquare integrable martingale, and E|Zℓptq|2“ErZℓ,Zℓst.\nWe now consider another partition into three separate cases forλ˚\njandλi.\n(i1)λ˚\njă2λi. Then we can let tÑ 8in (3.68) and obtain ErZℓ,Zℓs8ă 8.\nConsequently, Zℓptqis anL2-bounded martingale, and thus\nZℓptqa.s.Ý ÑZℓp8q ă 8 ,astÑ 8, (3.69)\nandZ˚\nℓp8q:“suptě0|Zℓptq| PL2. Furthermore (as always), λ˚\niěλi, and thus (3.42)\nimpliesrZ˚˚\nℓďZ˚\nℓp8q; consequently, /bardblrZ˚˚\nℓ/bardbl2ă 8.\nWe consider two subcases, recalling the cases discussed at t he beginning of the\nproof:\n(i1a)Case(i),λiąλ˚\nj.Thenλ˚\ni“λiandκi“0; hence(3.41) yields rZℓptq “Zℓptq\nand thus (3.69) yields rZℓptq ÑZℓ:“Zℓp8qa.s.\n(i1b)Case(ii)or(iii),λiďλ˚\nj.Thenλ˚\niąλiorκiě1 and thus (3.41) and (3.69)\nyieldrZℓptq ÑZℓ:“0 a.s.\n(ii1)λ˚\njě2λiandλ˚\nją0. Let, for ně1,\nrZ:\nℓpnq:“sup\nn´1ďtďnpt`1q´κiepλi´λ˚\niqt|Zℓptq|. (3.70)\nIt follows from Doob’s inequality (2.18) and (3.68) that, wi thC:“CErX˚˚\nj,\nErZ:\nℓpnq2ďCn´2κie2pλi´λ˚\niqnEZ˚\nℓpnq2\nďCn´2κie2pλi´λ˚\niqnErZℓ,Zℓsn\nďCn´2κie2pλi´λ˚\niqnżn\n0ps`1qκjepλ˚\nj´2λiqsds\nďCn´2κie2pλi´λ˚\niqnnκj`1epλ˚\nj´2λiqn\n“Cnκj`1´2κiepλ˚\nj´2λ˚\niqn. (3.71)\nWe have always λ˚\niěλ˚\nj, and thus, in the present case, λ˚\nj´2λ˚\niď ´λ˚\njă0. Hence\n(3.71) yields\nE8ÿ\nn“1rZ:\nℓpnq2“8ÿ\nn“1ErZ:\nℓpnq2ă 8. (3.72)20 SVANTE JANSON\nConsequently, a.s. rZ:\nℓpnq Ñ0 asnÑ 8, which by (3.41) and (3.70) means rZℓptq Ñ0\nastÑ 8. Moreover, (3.42) implies\n`rZ˚˚\nℓ˘2ď8ÿ\nn“1rZ:\nℓpnq2, (3.73)\nand thus (3.72) implies also /bardblrZ˚˚\nℓ/bardbl2ă 8.\n(iii1)λ˚\nj“λi“0. In this case we have λ˚\ni“0 andκi“κj`1. Let, for ně1,\nrZ;\nℓpnq:“sup\n2n´1ďtď2nt´κiepλi´λ˚\niqt|Zℓptq| “sup\n2n´1ďtď2nt´κj´1|Zℓptq|.(3.74)\nSimilarly to (3.71), it follows from Doob’s inequality and ( 3.68) that\nErZ;\nℓpnq2ďC2´2pκj`1qnż2n\n0ps`1qκjdsďC2´pκj`1qn. (3.75)\nHence,\nE8ÿ\nn“1rZ;\nℓpnq2“8ÿ\nn“1ErZ;\nℓpnq2ă 8, (3.76)\nand the rest of the argument is as in the preceding case, now us ing\n`rZ˚˚\nℓ˘2ďZ˚\nℓp1q2`8ÿ\nn“1rZ;\nℓpnq2(3.77)\nand noting that EZ˚\nℓp1q2ă 8by (2.18) and (3.68).\nWe have shown that (3.45) and (3.46) hold for ℓď3 in all cases, with Zℓ“0\nexcept in the case (i).\nStep 6: Conclusion . We haveshownthat (3.45)–(3.46) holdforevery ℓ; consequently,\n(3.33)–(3.34) hold by (3.43)–(3.44).\nMoreover, in cases (ii) and (iii), Zℓ“0 forℓ“1,2,3, and thus Xi“ajrjiZ4; this\nyields (3.35) by (3.54), which in particular shows that Xią0 whenXją0 in these\ncases.\nIt remains to show Xią0 whenXją0 in the case (i), i.e., when λiąλ˚\nj. In this\ncaseλ˚\ni“λiandκi“0, and thus by (3.29) and (3.28),\nrXiptq “e´λitXiptq “8ÿ\nk“1e´λit1tTkďtuYkpt´Tkq. (3.78)\nMoreover, if Xją0, then liminf tÑ8Xjptq ą0 and thus a.s. there is an infinite\nnumber of draws of colour j, and thus all Tkare finite. Let Kbe the (random)\nsmallest ksuch that ηką0; suchkexist a.s. since Eηk“rjią0 by assumption.\nThen (3.78) implies\nrXiptq ěe´λiTK1ttěTKue´λipt´TKqYKpt´TKq, (3.79)\nwhich a.s. has a strictly positive limit by conditioning on Kand applying Lemma 3.3\ntoYK. We have already shown that the limit Xiin (3.33) exists a.s., and (3.79) then\nshowsXią0 a.s. /square\nIfλiďλ˚\nj, thenXiis determined by Xj, see (3.35). On the contrary, if λiąλ˚\nj,\nthenXiis not determined by Xj, as shown in the following lemma.ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 21\nLemma 3.6. Ifλiąλ˚\njin Lemma 3.5, then the distribution of Xiconditioned on\nXjis non-degenerate. In fact, then the conditional distributi onLpXi|Xjqis a.s.\nabsolutely continuous.\nProof.Sinceλiąλ˚\nj, we have (3.78). Condition on the entire process pXjptqqtě0\nand on all Tkandηk; then the terms in the sum in (3.78) are independent, and, as\ntÑ 8, each term converges a.s. by Lemma 3.3 to a limit that by Lemma 3.4 has\nan absolutely continuous distribution when ηką0. Since a.s. ηką0 for infinitely\nmanyk, it follows that the limit is a.s. (conditionally) absolute ly continuous. Since\npXjptqqtdetermines Xj, this implies (by Lemma A.1) that the distribution a.s. is\nabsolutely continuous also if we condition only on Xj. /square\n3.3.The general case for a single colour. We have in the preceding subsections\nconsideredtwo special cases. Weconsidernowasinglecolou riinageneral triangular\nurn. We continue to use the notations (3.29)–(3.30). Recall also that Pi:“ tjPQ:\njÑiu.\nLemma 3.7. LetiPQ, and assume that for every jPPi, we have\nrXjptqa.s.Ý ÑXjastÑ 8, (3.80)\nfor some random variable Xjě0, and\n/vextenddouble/vextenddoublerX˚˚\nj/vextenddouble/vextenddouble\n2ă 8. (3.81)\nThen\nrXiptqa.s.Ý ÑXiastÑ 8, (3.82)\nfor some Xiě0and\n/vextenddouble/vextenddoublerX˚˚\ni/vextenddouble/vextenddouble\n2ă 8. (3.83)\nFurthermore, a.s., if Xją0for every jPPi, thenXią0.\nProof.IfPi“ H, we are in the situation of Lemma 3.3. Moreover, in this case\nXip0q ą0 by our standing assumption (A3). Consequently, in this cas e the result\nfollows from Lemma 3.3. (Note that λ˚\ni“λiandκi“0 by (2.8) and (2.9).)\nIn general, we separate the balls of colour iaccording to their original reason\nfor existing. Formally, we split the colour iand replace it by several “subcolours”\n(or shades); we define one subcolour labelled i0, and an additional subcolour ijfor\neachjPPi. These subcolours have the same replacement vector ξiasi, with the\nmodification that new balls of colour ialways get the same subcolour as the drawn\nball. Also, balls of colour ithat are added when a ball of some other colour jis\ndrawn get the subcolour ij. Furthermore, all balls of colour iat time 0 get subcolour\ni0. In other words, i0is used for descendants of the balls with colour iin the urn at\nthe beginning, and ijare used for balls of colour ithat eventually descend from a\nball of colour j. Note that i0is minimal, while Pij“ tju.\nWe thus have\nXiptq “Xi0ptq `ÿ\njPPiXijptq. (3.84)\nMoreover, in the modified urn with subcolours instead of colo uri, the subcolour i0\nis of the type in Section 3.1 (possibly with x0“0), and each subcolour ij(jPPi) is22 SVANTE JANSON\nof the type in Section 3.2. Hence, Lemmas 3.3 and 3.5 apply, us ing our assumptions\n(3.80)–(3.81). It follows from (2.8) that\nλ˚\ni0“λi, (3.85)\nλ˚\nij“λi_λ˚\nj, j PPi. (3.86)\nIn particular, for every jPPiY t0u, we have λ˚\nijďλ˚\ni. Furthermore, if Pi‰ H, then\nλ˚\ni“λi_maxtλ˚\nj:jPPiu “maxtλ˚\nij:jPPiu. (3.87)\nMoreover, (2.9) yields κi0“0, and implies also that if λ˚\nij“λ˚\ni, thenκijďκi.\nHence, for every jPPiY t0u, and for all large t(at least),\nt´κie´λ˚\nitďt´κije´λ˚\nijt. (3.88)\nConsequently, it follows from (3.84) and the definition (3.2 9) that, at least for large\nt,\nrXiptq ďrXi0ptq `ÿ\njPPirXijptq, (3.89)\nand furthermore, using also (3.13) and (3.33),\nrXiptqa.s.Ý ÑXi:“ÿ\njPPiYt0uXij1tpλ˚\nij,κijq“pλ˚\ni,κiqu. (3.90)\nThis shows (3.82).\nSimilarly, pt`1q´κie´λ˚\nitďCpt`1q´κije´λ˚\nijtfor every jPPiY t0uand every\ntě0, and thus (3.84) and (3.30) imply\nrX˚˚\niďCrX˚˚\ni0`Cÿ\njPPirX˚˚\nij. (3.91)\nHence, (3.83) follows from (3.17) and (3.34).\nFinally, assume Xją0 for every jPPi. IfPi“ H, then, as remarked above,\nXią0 a.s. by (A3) and Lemma 3.3. On the other hand, if Pi‰ H, thenXiją0 for\neveryjPPiby Lemma 3.5. By (3.87) and (2.9), there exists some jPPisuch that\npλ˚\nij,κijq “ pλ˚\ni,κiq, and thus (3.90) implies\nXiěXiją0 (3.92)\na.s., which completes the proof. /square\nRemark 3.8. Ifλiąλ˚\njfor every jPPi, thenλ˚\nij“λi“λ˚\niandκij“0“κifor\neveryjPPiY t0u, and thus (3.90) yields\nXi“ÿ\njPPiYt0uXij. (3.93)\nOn the other hand, suppose that λiďλ˚\njfor some jPPi. Then either λ˚\niąλi“\nλ˚\ni0, orλi“λ˚\ni“λ˚\njfor some jPPi, and in the latter case κiě1`κjąκi0“0.\nHence, in both cases, pλ˚\ni0,κi0q ‰ pλ˚\ni,κiq, and thus the sum in (3.90) is really only\nover (some) jPPi. △ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 23\n4.The main theorem for continuous time\nTheorem 4.1. LetpXiptqqiPQbe a continuous-time triangular P´ olya urn satisfying\nthe conditions (A1)–(A5). Then, for every colour iPQ,\nt´κie´λ˚\nitXiptqa.s.Ý ÑXiastÑ 8, (4.1)\nfor some random variable XiwithXią0a.s.\nProof.We may choose a total order ăof the colours such that (2.5) holds. Taking\nthe colours in this order, we see by Lemma 3.7 and induction th at (3.82) and (3.83)\nhold for every iPQ, withXią0 a.s. Recalling (3.29), we see that (3.82) is the same\nas (4.1). /square\nThe limits Xiare strictly positive. We will see in Theorem 6.4 that they ar e\nnon-degenerate when λ˚\nią0; however, in general some of them will be linear com-\nbinations of others. We discuss dependencies between the li mits in Section 6.\n5.Proof of the main theorem for discrete time\nProof of Theorem 1.8. We modify the urn by adding a new colour, 0 say, to Q; let\nQ`:“QY t0ube the new set of colours. All activities aiand replacements ξijwith\ni,jPQremain unchanged. Balls of colour 0 have activity a0:“0, and thus they\nare never drawn; in accordance with (A2) we define ξ0j:“0 for every jPQ`. We\nfurther let ξi0:“1 for every iPQwithaią0. ForiPQwithai“0 we letξi0:“0,\nagain in accordance with (A2). We let X00“X0p0q:“0, so there are initially no\nballs of colour 0. Note that the extended urn too satisfies (A1 )–(A5).\nSince balls of colour 0 never are drawn, we may ignore them and recover the\noriginal urn. In other words, the new urn will be the original urn with “dummy\nballs” of colour 0 added. We may assume that the old and new urn are coupled in\nthis way, which means that for iPQ, the number of balls of colour iat any time is\nthe same in both urns, so we may unambiguously use the notatio nsXniandXiptq\nfor both urns. Moreover, there are initially no balls of colo ur 0, but we add exactly\n1 ball of colour 0 every time a ball is drawn. Hence, in the disc rete-time version, the\nnumber of dummy balls at time nequalsn; thusXn0“n. In the continuous-time\nmodel,X0ptqequals the number of draws up to time t, and in particular, cf. (2.19),\nX0ppTnq “Xn0“n. (5.1)\nThe new urn is obviously also triangular, with iÑ0 for every iPQwithaią0.\nWe have λ0“0. Since every λiě0, and we have iÑ0 whenλią0 (and thus\naią0 by (A2)), it follows from (2.8) and (2.10) that\nλ˚\n0“maxtλj:jPQu “pλ. (5.2)\nIfpλą0, then is further easily seen from (2.9) and (2.11) that\nκ0“maxtκi:iPQandλ˚\ni“pλu “pκ. (5.3)\nIfpλ“0, which means that λi“0 for every i, we have by (2.9) and (2.12) instead\nκ0“max/visualspace\nκ:Di1≺i2≺¨ ¨ ¨≺iκ`1“0 inQ`(\n“max/visualspace\nκ:Di1≺i2≺¨ ¨ ¨≺iκwithaiκą0 inQ(\n“1`max/visualspace\nκj:jPQwithają0(\n“pκ0. (5.4)24 SVANTE JANSON\nWe apply Theorem 4.1 to the new urn. In particular, taking i“0 in (4.1) yields,\nusing (5.1) and (5.2)–(5.4) and recalling that pTnÑ 8a.s.,\npT´κ0ne´pλpTnn“pT´κ0ne´pλpTnX0ppTnqa.s.Ý ÑX0,asnÑ 8. (5.5)\nRecall that 0 ăX0ă 8a.s.\nCase 1:pλą0.Consider first the case pλą0; thenκ0“pκby (5.3). It follows from\n(5.5) that a.s., as nÑ 8,\nlogn´pλpTn´pκlogpTnÑlogX0 (5.6)\nwhich yields, since pTnÑ 8,\nlogn\npTnÑpλ, (5.7)\nand thus\nloglogn´logpTnÑlogpλ. (5.8)\nUsing (5.8) in (5.6) yields, a.s.,\npTn“1\npλ`\nlogn´pκloglogn`pκlogpλ´logX0`op1q˘\n. (5.9)\nNow letiPQ. Then (5.9) yields, as nÑ 8, a.s.,\npTκineλ˚\nipTn„pλ´κi`pκλ˚\ni{pλX´λ˚\ni{pλ\n0nλ˚\ni{pλplognqκi´pκλ˚\ni{pλ. (5.10)\nConsequently, taking t“pTnin (4.1) yields, as nÑ 8, using the notation (2.13),\nXni\nnλ˚\ni{pλlogγin“XippTnq\nnλ˚\ni{pλplognqκi´pκλ˚\ni{pλa.s.Ý ÑpXi (5.11)\nwith\npXi:“pλ´κi`pκλ˚\ni{pλX´λ˚\ni{pλ\n0Xi“pλ´γiX´λ˚\ni{pλ\n0Xi. (5.12)\nThis shows (1.3).\nCase 2:pλ“0.In the case pλ“0, we have instead κ0“pκ0by (5.4). Moreover,\n(5.5) now yields that as nÑ 8, a.s.,\npTn„X´1{pκ0\n0n1{pκ0(5.13)\nand thus (4.1) yields\nXni\nnκi{pκ0“XippTnq\nnκi{pκ0a.s.Ý ÑpXi:“X´κi{pκ0\n0Xi. (5.14)\nThis shows (1.4), and completes the proof of Theorem 1.8. /square\nThe limits pXiare all strictly positive. We return to the question whether they are\ndegenerate in Section 7.ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 25\nRemark 5.1. Suppose that pλą0, so that (5.11) holds. As a sanity check, we note\nthat (5.11) and (5.12) imply, recalling (2.10) and (2.11), t hat if we define\nQ˚:“ tiPQ:λ˚\ni“pλandκi“pκu, (5.15)\nwhich is nonempty, then\nXni\nna.s.Ý Ñ#pXi“Xi{X0ifiPQ˚,\n0 otherwise .(5.16)\nHence, the total number of balls grows linearly, as should be expected; moreover,\nthe distribution of colours in the urn is asymptotically con centrated on Q˚.\nNote also that (3.90) for thedummycolour 0yields, recallin g (5.2)–(5.3) and using\n(3.35),\nX0“ÿ\njPQ˚X0j“ÿ\njPQ˚aj\nλ˚\njXj“pλ´1ÿ\njPQ˚ajXj. (5.17)\nBy (5.16) and (5.17), we have for the total activity in the urn\n1\nnÿ\niPQaiXnia.s.Ý Ñ1\nX0ÿ\niPQ˚aiXi“pλ. (5.18)\n△\nRemark 5.2. If we instead suppose pλ“0, which implies λi“λ˚\ni“0 for every i,\nthen (5.14) similarly shows that if we now define\nQ˚:“ ti:κi“pκ0u, (5.19)\nthen (5.16) still holds. Note that it is quite possible that Q˚is empty; this happens\nprecisely when pκ0ąpκ, which by (5.4) happens when some iwithκi“pκhas activity\naią0. (Such a colour cannot have any descendants, since all colo urs have the same\nλ˚\nj“0; hence wemust have ξi“0 a.s., which means that such colours will bedrawn,\nbut nothing happens to the urn at these draws.) In this case th e total number of\nballs is a.s. opnq, and the colour distribution is asymptotically concentrat ed on the\ncoloursisuch that κi“pκ“pκ0´1.\nOn the other hand, if Q˚‰ H, which means that pκ0“pκ, and thus by (5.4) ai“0\nfor every iwithκi“pκ, letQ˚´:“ tj:κj“pκ´1u. Then (3.90) and (3.35) yield,\nsincejÑ0 if and only if ają0, and then κ0j“κj`1,\nX0“ÿ\njPP0XQ˚´X0j“ÿ\njPQ˚´aj\npκXj“pκ´1ÿ\njPQ˚´ajXj. (5.20)\n△\n6.Dependencies between the limits\nThelimits XiinTheorem4.1arenon-degenerateexceptinextremecases, a sshown\nbelow, but there are frequently linear dependencies betwee n them. To explore this,\nwe introduce more terminology.\nWe say that a colour iis aleaderifλjăλifor every j≺i, and a follower\notherwise. (In particular, a minimal colour iis a leader.) We have, recalling (2.8),\niis a leader ð ñ´\nj≺iù ñλjăλi¯\nð ñ´\njÑiù ñλ˚\njăλi¯\n,(6.1)26 SVANTE JANSON\niis a follower ð ñ´\nDj≺iwithλjěλi¯\nð ñ´\nDjÑiwithλ˚\njěλi¯\n.(6.2)\nBy (2.8)–(2.9),\niis a leader ð ñλ˚\ni“λiandκi“0. (6.3)\nIfiis a follower, let Aibe the set of ancestors of ithat have maximal λj, i.e.,\nλj“λ˚\ni, and such that furthermore there is a path from jtoiwith the maximum\nnumberκi`1 of colours ℓwithλℓ“λ˚\ni. (Recall (2.8) and (2.9).) Thus\nAi:“ tj≺i:Di1“j≺i2≺¨ ¨ ¨≺iκi`1/precedesequaliwithλi1“ ¨ ¨ ¨ “λiκ`1“λ˚\niu.(6.4)\nNote that Aiis non-empty for every follower i. Moreover, it is easily seen from\n(6.1)–(6.2) and (6.4) that every jPAiis a leader. We may say that ifollows the\nleaders in Ai; note that a follower may follow several leaders. For comple teness, we\ndefineAi:“ tiuwheniis a leader. Note that, by (6.3) and (6.4),\nλj“λ˚\nj“λ˚\nifor every jPAi. (6.5)\nWe show first that the variables Xiare determined by the ones for leaders i.\nLemma 6.1. IfiPQ, thenXiis a linear combination\nXi“ÿ\nkPAicikXk (6.6)\nwith strictly positive coefficients cik.\nProof.Note first that (6.6) is trivial when iis a leader, with cii“1. We may thus\nsuppose that iis a follower.\nBy induction on the colour i, we may assume that the formula (6.6) holds for\nevery colour that is an ancestor of i, and in particular for every jPPi.\nBy Remark 3.8, we only have to consider jPPiin (3.90). Let\nP1\ni:“/visualspace\njPPi:pλ˚\nij,κijq “ pλ˚\ni,κiq(\n, (6.7)\nso that the sum in (3.90) really is over jPP1\ni. By (6.3), since we now assume that\niis a follower, either λ˚\niąλiorκią0 (or both). Hence, if jPP1\ni, then either\nλ˚\nij“λ˚\niąλiorκij“κią0; sinceλ˚\nij“λi_λ˚\nj, it follows in both cases (using\n(2.9) in the latter) that λ˚\nj“λ˚\nijěλiand thus by (3.35)\nXij“c1\nijXj (6.8)\nfor some constant c1\niją0. Moreover, we have λ˚\nj“λ˚\nj_λi“λ˚\nij“λ˚\niand it follows\nfrom (6.7) and (2.9) that\nκi“κij“κj`1tλi“λ˚\niu. (6.9)\nIfjPP1\niis a leader, so λj“λ˚\nj“λ˚\niandκj“0 by (6.3), it follows from (6.4) and\n(6.9) that jPAi. Similarly, if jPP1\niis a follower, then it follows from (6.4) and\n(6.9) that if kPAj, then a chain from ktojof the (maximal) type in (6.4) can be\nextended to a chain from ktoiof the same type, and thus kPAi; consequently, if\njPP1\niis a follower, then AjĎAi. The result (6.6), with some cijě0, now follows\nby (3.90), (6.8), and induction.\nFinally, if kPAi, leti1“k,...,i κi`1/precedesequalibe as in (6.4), choose a path in Q\nfromktoithat contains all iℓ, and let jbe the last colour in this path before i.\nThen it follows from (6.4) that kPAj, and thus by induction cjką0, which implies\ncikěc1\nijcjką0. /squareALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 27\nMotivated by (6.6), we turn to considering Xifor leaders i. We first consider a\ntrivial exceptional case.\nLemma 6.2. Letibe a colour with λi“0. Then iis a leader if and only if i\nis minimal, i.e., there is no colour jwithjÑi. In this case Xi“Xip0qis a\ndeterministic positive constant.\nProof.We have already remarked that a minimal colour is a leader. Co nversely,\nifiis a leader with λi“0, then (6.1) shows that jÑiis impossible; hence i\nis minimal. In this case, no balls of colour iare added by draws of balls of other\ncolours. Furthermore, since λi“0, also no balls of colour iare added by draws of\ncolouri. Consequently, Xiptq “Xip0qfor alltě0, and (3.80) holds trivially with\nrXiptq “XiptqandXi“Xip0q. /square\nWe next show that except for the case in Lemma 6.2, the distrib ution ofXifor a\nleaderiis absolutely continuous, and thus non-degenerate.\nLemma 6.3. (i)Letibe a leader with λią0, and let Ebe a set of colours that\ncontains neither inor any descendant of i, i.e., if jPEthenjńi. Then the\nconditional distribution LpXi|Xj,jPEqis a.s. absolutely continuous. In particular,\nthe distribution of Xiis absolutely continuous.\n(ii)More generally, let Lbe a (non-empty) set of leaders with λią0for every\niPL, and suppose that Lis an anti-chain in Q, i.e.,ićjwheni,jPL. Let further\nEbe a set of colours such that if jPE, thenjńifor every iPL. Then the joint\nconditional distribution LppXiqiPL|Xj,jPEqis a.s. absolutely continuous. (This is\na distribution in R|L|.)\nProof.It suffices to consider (ii). The conclusion (conditional abs olute continuity)\nis preserved if we reduce Eto a smaller set, see Lemma A.1 and Remark A.2. Thus\nwe may assume that Eis maximal, i.e.,\nE“ tjPQ:jńi,@iPLu. (6.10)\nNote that, if iPLandj≺i, thenwecannothave j/followsequalkforanykPL, sincethiswould\nimplyk≺i, contradicting the assumption that Lis an antichain. Consequently, by\n(6.10), if iPLandj≺i, thenjPE.\nWe argue as in the proof of Lemma 3.6, and condition on the enti re processes\npXjptqqtě0,jPE, and also on the times of all draws of a color jPE, and on all\nreplacement vectors ξpnq\nj(ně1,jPE).\nLetiPL. IfjPPi, sojÑi, thenλ˚\njăλi“λ˚\niby the assumption that iis\na leader. We use again the decomposition (3.84), and note tha t eache´λitXijptq\n(jPPi) can be written as a sum as in (3.78) with (conditionally) ind ependent terms,\nand it follows as in the proof of Lemma 3.6 that each Xijhas (conditionally) an\nabsolutely continuous distribution. Furthermore, Xi0is independent of conditioning\non colours jPE, and ifXip0q ą0, then its distribution too is absolutely continuous\nby Lemma 3.4 (since we assume λią0). Moreover (still conditionally), all Xijand\nXi0(iPL,jPPi) are independent.\nSinceiis a leader, Remark 3.8 shows that (3.93) holds. We have shown that\n(conditionally) the terms in the sum in (3.93) are independe nt, and all are absolutely\ncontinuous except Xi0whenXip0q “0. Thus our assumption (A3) implies that\nthe sum contains at least one absolutely continuous summand , and hence Xiis\n(conditionally) absolutely continuous.28 SVANTE JANSON\nMoreover, this argument shows that Xifor different iPLare (conditionally)\nindependent, and since each has an absolutely continuous di stribution, their joint\ndistribution is (conditionally) absolutely continuous.\nFinally, Lemma A.1 (again with Remark A.2) shows that the sam e holds if we\ninstead condition on Xj,jPE. /square\nTheorem 6.4. LetLbe a set of leaders such that λią0for every iPL. Then the\njoint distribution L`\npXiqiPL˘\nis absolutely continuous.\nProof.Order the elements of the set tλi:iPLuasλp1qă ¨ ¨ ¨ ăλprq. Let\nLℓ:“ tiPL:λi“λpℓqu,1ďℓďr, (6.11)\nLďℓ:“ℓď\nk:“1Lk,0ďℓďr. (6.12)\n(Thus,Lď0“ H.) LetℓP t1,...,r u. Ifi,jPLℓ, then (6.1) shows that jći, and\nthusLℓis an antichain. Furthermore, if iPLℓandjPLďℓ´1, thenλ˚\nj“λjăλi, and\nthusiłj. Consequently, Lemma 6.3(ii) shows that the conditional di stribution of\npXiqiPLℓgiven pXjqjPLďℓ´1is a.s. absolutely continuous.\nIt now follows from Lemma A.3 by induction that the distribut ion of pXiqiPLďℓ\nis absolutely continuous for ℓ“1,...,r. Taking ℓ“ryields the result, since\nLr“L. /square\nTheorem 6.5. LetiPQbe any colour.\n(i)Ifλ˚\ni“0, thenXiis a deterministic positive constant.\n(ii)Ifλ˚\nią0, thenXihas an absolutely continuous distribution.\nProof.(i): Every kPAiis a leader with λk“λ˚\ni“0. Thus Lemma 6.2 shows that\nXkis deterministic for kPAi. Consequently, (6.6) shows that Xitoo is deterministic.\n(ii): Every kPAiis a leader with λk“λ˚\nią0. Thus Theorem 6.4 shows that the\njoint distribution of pXkqkPAiis absolutely continuous. It follows from (6.6) that the\ndistribution of Xiis absolutely continuous. (See Lemma A.4.) /square\nCorollary 6.6. Ifλ˚\nią0, then the coefficients cikin Lemma 6.1 are uniquely\ndetermined.\nProof.Suppose, on the contrary, that (6.6) holds a.s. for two differe nt sets of co-\nefficients pcikqkand pc1\nikqk. Letbk:“cik´c1\nik. Thenř\nkPAibkXk“0 a.s., and\nthus pXkqkPAia.s. lies in the certain hyperplane orthogonal to pbkq. However, this\ncontradicts Theorem 6.5 which says that the distribution of pXkqkPAiis absolutely\ncontinuous. (See also Lemma A.4.) This contradiction prove s the claim. /square\nThe coefficients cikin (6.6) can be found by the recursive procedure in the proof\nof Lemma 6.1. We proceed to show that they also can be found as e igenvectors of\nsuitable submatrices of the weighted mean replacement matr ixpairijqi,jPQ. In the\nsequel, we let cikbe the coefficient given by the inductive proof of Lemma 6.1; th is\ndetermines cikuniquely for kPAialso when λ˚\ni“0. We further define cik:“0 if\nkRAi.\nLetνbe a leader, and let\nDν:“ ti:λ˚\ni“λνu, (6.13)\nDκ\nν:“ tiPDν:κi“κu, κ “0,1,.... (6.14)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 29\n(The sets Dκ\nνare empty from some κon; we only consider κwithDκ\nν‰ H. Note\nalso that Dν“Dν1ifν1is another leader with λν1“λν.) Note that if ciνą0, then\nνPAi, and thus iPDν.\nLetiPDκ\nν, and suppose first that λiăλ˚\ni“λν. Then it follows from Remark 3.8\nthat in (3.90), we only have to sum over jPPi. Moreover, if jPPiandλ˚\nij“λ˚\ni,\nthenλ˚\nj“λ˚\nij“λ˚\niandκij“κj. Hence, in (3.90) we only have to sum over\njPPiXDκ\nν. SincejPPið ñjÑið ñrjią0 andj‰iby (2.3), it follows\nfrom (3.90) and (3.35) that\nXi“ÿ\njPDκνXPiXij“ÿ\njPDκνztiuajrji\nλ˚\nj´λiXj. (6.15)\nConsequently, the coefficient ciνin (6.6) is given by\nciν“ÿ\njPDκνztiuajrji\nλ˚\nj´λicjν. (6.16)\nSincejPDκ\nνimpliesλ˚\nj“λν, (6.16) yields\npλν´λiqciν“ÿ\njPDκνztiuajrjicjν, (6.17)\nfor every iPDκ\nνwithλiăλ˚\ni.\nOn the other hand, suppose that iPDκ\nνwithλi“λ˚\ni“λν. IfjPDνztiuand\nrjią0, thenλ˚\nj“λν“λ˚\niandjÑi, and thus κiěκj`1, since a maximal chain\nin the definition (2.9) of κjcan be extended by i. Hence, if jPDκ\nνztiu, thenrji“0.\nIt follows that (6.17) holds trivially for iPDκ\nνwithλi“λ˚\ni.\nConsequently, (6.17) holds for every iPDκ\nν, and thus, recalling λi“airii,\nÿ\njPDκνajrjicjν“ pλν´λiqciν`airiiciν“λνciν, i PDκ\nν.(6.18)\nWe summarize, and elaborate, this as a lemma. We say that iPQis asubleader\nifλi“λ˚\nibutiis not a leader; recalling (6.3) we thus have\niis a subleader ð ñλ˚\ni“λiandκiě1. (6.19)\nLemma 6.7. For any leader ν, and any κě0such that Dκ\nν‰ H, we have (6.18),\nand thus pciνqiPDκνis a left eigenvector of the triangular matrix pairijqi,jPDκνfor its\nlargest eigenvalue λν.\nForκ“0, the value ciνfor a leader iPD0\nνis determined by ciν“δiν(i.e.,1when\ni“νand0otherwise), and these values for the leaders determine the e igenvector\npciνqiPDκνuniquely.\nForκě1, the value ciνfor a subleader iPDκ\nνis determined recursively from the\nvaluescjνforjPDκ´1\nνby\nciν“ÿ\njPDκ´1\nνajrji\nκcjν, (6.20)\nand these values for the subleaders determine the eigenvect orpciνqiPDκνuniquely.\nProof.We have shown that (6.18) holds, and thus pciνqiPDκνis a left eigenvector.\nSince the matrix Aκ:“ pajrijqi,jPDκνis triangular, its eigenvalues are its diagonal\nelements ajrjj“λj,jPDκ\nν. The definition (2.9) of κiimplies that if Dκ\nν‰ H, then\nthere exists iPDκ\nνwithλi“λ˚\ni, i.e. a leader (if κ“0) or a subleader (if κě1).30 SVANTE JANSON\nSinceλiďλ˚\ni“λνfor every iPDκ\nν, it follows that λ˚\nνis the largest eigenvalue, and\nthat its multiplicity equals the number of leaders or sublea ders inDκ\nν.\nLetLκbe the set of leaders or subleaders in Dκ\nν, and consider the projection\nΠκ:RDκ\nνÑRLκmapping pxiqiPDκνÞÑ pxiqiPLκ. LetVκbe the left eigenspace of the\nmatrixAκfor its largest eigenvalue λν. The recursion in the proof of Lemma 6.1\nand the calculations (6.15)–(6.17) show, more generally, t hat any vector pxiqiPLκcan\nbe extended to a vector pxiqiPDκνthat belongs to Vκ. In other words, the projection\nΠκmapsVκontoRLκ. These spaces have the same dimension |Lκ|, and thus Π κ:\nVκÑRLκis a bijection. Consequently, an eigenvector is determined by its values\nfor leaders or subleaders.\nIt is trivial that ciν“δiνfor a leader i.\nForκě1, letibe a subleader in Dκ\nν, soλi“λ˚\ni. Sinceκi“κą0, it follows from\nRemark 3.8 that λi“λ˚\njfor some jPPi, and furthermore that κj`1ďκifor every\nsuchj. Since then also λ˚\nij“λ˚\nj“λ˚\niandκij“κj`1, it follows that in (3.90) we\nonly have to sum over jPDκ´1\nν, which together with (3.35) yields (6.20). /square\nRemark 6.8. It is well known that the eigenvalues and eigenvectors of the weighted\nmean replacement matrix pairijqi,jPQareimportant for theasymptotics of P´ olya urns\ningeneral; seeforexample[4, SectionV.9.3] and[27]forir reducibleurns. Lemma6.7,\nwhere we consider eigenvectors of certain submatrices, is i nspired by a special case\nin [12], see Example 14.12. △\n7.Degenerate limits in discrete time\nConsider now the limits pXiin Theorem 1.8 for the discrete-time urn.\nTheorem 7.1. LetiPQ.\n(i)Ifλ˚\ni“0, thenpXiis a positive constant.\n(ii)If0ăλ˚\niăpλ, thenpXihas an absolutely continuous distribution.\n(iii)Ifλ˚\ni“pλą0, thenpXiis either constant or has an absolutely continuous\ndistribution; both alternatives are possible.\nProof.Consider the urn with an added dummy colour 0 as in Section 5.\n(i): In this case, Theorem 6.5 shows that Xiis a positive constant. If pλą0, then\n(5.12) shows that pXi“pλ´κiXiis a constant. If pλ“0, then (5.2) and Theorem 6.5\nshow that also X0is a positive constant; thus (5.14) shows that pXiis a constant.\n(ii): By (5.2), we have λ˚\n0“pλą0. By Lemma 6.1, Xiis a linear combination\n(6.6) of Xkfor leaders kwithλ˚\nk“λ˚\ni, andX0is a similar linear combination,\nnow for leaders kwithλ˚\nk“λ˚\n0“pλ. The two sets of leaders are disjoint, and\nthus Theorem 6.4 implies, using Lemma A.4, that the distribu tion of pXi,X0qis\nabsolutely continuous in R2. It is then easily seen from (5.12) that the distribution\nofpXiis absolutely continuous.\n(iii): In this case, Lemma 6.1 shows that both XiandX0are linear combinations\n(6.6) of Xkfor leaders kwithλ˚\nk“pλ. If the two vectors of coefficients of these\nlinear combinations are proportional, then XiandX0are proportional; since now\n(5.12) shows that pX“cXi{X0for some constant c, it follows that pXis constant. On\nthe other hand, if the vectors of coefficients are not proporti onal, then Lemma A.4\nshows that the distribution of pXi,X0qis absolutely continuous in R2, and as in (ii),\nit follows easily from (5.12) that the distribution of pXiis absolutely continuous. /squareALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 31\nAn example where λ˚\ni“pλandpXiis absolutely continuous is given by the classical\nP´ olya urn, see Example 14.1. Many examples with degenerate pXiare provided by\nthe following theorem; see also the examples in Section 14.\nTheorem 7.2. Suppose that there is only one leader νwithλν“pλ. (This is\nequivalent to λν“pλand that i/followsequalνfor every colour iwithλi“pλ.) Then pXiis\ndeterministic for every iwithλ˚\ni“pλ.\nProof.Ifpλ“0, this follows from Theorem 7.1(i).\nSuppose now pλą0, and consider the urn with an added dummy colour 0 as in\nSection 5. Then λ˚\n0“pλby (5.2). Furthermore, λν“pλą0 and thus aνą0; hence\nνÑ0 and consequently νis still the only leader kwithλk“pλ. In particular,\nA0“ tνu. Lemma 6.1 shows that Xi“ciνXνfor every iPQwithλ˚\ni“pλ, and also\nthatX0“c0νXν. Hence, the result follows from (5.12). /square\nEven when the limits pXiare not deterministic, there are generally linear depen-\ndencies between them just as for Xi. In fact, in the main case pλą0, we have the\nfollowing analogue of Lemmas 6.1 and 6.7.\nLemma 7.3. Ifpλą0, then for every iPQ\npXi“ÿ\nkPAipcikpXk, (7.1)\nwhere\npcik“pλ´κiciką0. (7.2)\nMoreover, Lemma 6.7 holds also for pcik, with the only difference that (6.20)is re-\nplaced by\npciν“ÿ\njPDκ´1\nνajrji\nκpλpcjν. (7.3)\nProof.The expansion (7.1)–(7.2) follows from (6.6) and (5.12), si ncekPAiimplies\nλ˚\nk“λ˚\niandκk“0. The final claim follows by (7.2) and Lemma 6.7. /square\nWe leave the corresponding result for the less interesting c asepλ“0 to the reader.\n8.Urns with subtractions\nWe have so far assumed (A5): ξijě0 for alli,jPQ. In many applications, there\nare urns with subtractions, where we allow ξijă0. In the present paper, we are only\ninterested in cases where the urn allows an infinite number of drawings according to\nthe rules in Section 1; in other words we want (1.1) to make sen se as probabilities\nfor alln, and thus we require that the urn is such that Xniě0 for all n, and alsoř\niaiXni‰0. Such urns are often called tenable. (See e.g. [38] for a discussion\nand examples.) A standard way to ensure that Xniě0 without assuming ξiiě0\nis to assume that X0iand every ξjialways is an integer, so that Xniis an integer,\nand that ξiiě ´1 whileξjiě0 forj‰i. Typically, this is done for all i. (Thus\nXnPZq\ně0; we may then regard the urn process as drawings without repla cement\nfollowed by adding ξij`δijě0 balls of each colour j.) However, we will be more\nflexible and allow a combination with this assumption for som e colours i, andξjiě0\n(as in earlier sections) for the others. We therefore assume :32 SVANTE JANSON\n(A51)For each colour iPQ, we have either (or both)\n(a)ξjiě0 a.s. for all jPQ, or\n(b)ξiiPZě0Y t´1ua.s. andξjiPZě0a.s. for all j‰i, andXip0q PZě0.\nNote that (A5) is (A51)(a) for every iPQ. Note also that (A51) implies ξjiě0 a.s.\nfor alli,jPQwithi‰j. We let\nQ´:“/visualspace\niPQ:Ppξiiă0q ą0(\n“/visualspace\niPQ:Ppξii“ ´1q ą0(\n(8.1)\ndenote the set of colours where (A51)(b) but not (A51)(a) applies. Note that (A2)\nimplies that aią0 for every iPQ´.\nRemark 8.1. More generally, we may assume that, for a given i,ξiimay take some\nnegative value ´b, provided it does not take any other negative value, and Xip0q\nand allξji(jPQ) a.s. are multiples of b. This case is easily reduced to the case\nb“1 in (A51) by dividing all Xni,Xiptq, andξjibyband multiplying the activity ai\nbyb. (See Example 14.8 for an example where we, however, use a sli ghtly different\nalternative.) For convenience, andthuswithoutreal loss o f generality, wewill assume\n(A51), where the only allowed negative replacement is ´1. △\nRemark 8.2. IfiPQ´XQmin, thenXiptqis not influenced by the other colours, and\n(A51)(b) implies that Xiptqis integer-valued and a classical continuous-time Markov\nbranching process of the type studied in e.g. [4, Chapter III ]. (As noted above, we\nhaveaią0 by (A2).) Recall that [4, Section III.7] in the subcritical and critical\ncasesλiď0, this process a.s. dies out, i.e., Xiptq “0 for all sufficiently large t, while\nin thesupercritical case λią0, theprocesssurvives for ever with positive probability\n(assuming Eξiilogξiiă 8, as we do); this probability is strictly less than 1, since\nthe process always may die out when Ppξii“ ´1q ą0. △\nWe continue to use the definitions above, in particular (2.7) –(2.13). Note that we\nnow may have riiă0, and thus λiă0; we may also have λi“rii“0 without\nξii“0 a.s.\nWe will see that the results in the previous sections are vali d with minor changes\nalso if we replace (A5) by (A51), at least under some further technical assumptions\n(A6)–(A8) given below. However, in order to get more general results, these will be\nassumed only when needed. Hence, we assume throughout this s ection (A1)–(A4)\nand (A51), with the following further assumptions added explicitly when needed.\n(A6)ř\niPQaiXiptq ą0 a.s. for all tě0.\n(A7)IfiPQ´, thenλ˚\nią0.\n(A8)IfiPQis a minimal colour, then ξiiě0 a.s. (i.e., iRQ´).\nIn other words, (A7) says that if iPQ´, then either λią0 or there exists jPQ\nwithj≺iandλją0 (or both). In particular, if iis a minimal colour, then either\nλią0 orξiiě0 a.s.\nNote that (A8) implies that Xiptq ěXip0qwheniis minimal; since (A1)–(A3)\nimply that there exists a minimal iwithaią0 andXip0q ą0, (A8) together with\n(A1)–(A3) imply (A6).\n8.1.Some motivation for (A6)–(A8). The assumption (A51) implies that the\ncontinuous-time urn Xptq “ pXiptqqiPQis well-defined for tP r0,8q, withXiptq ě0\nfor alliPQ. However, the urn might possibly reach a state where Xiptq “0 for alli\nwithaią0, and thusř\niaiXiptq “0; such a state is absorbing and no more draws\nwill be made, and then the total number of draws is finite and th e discrete-timeALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 33\nurn is not well defined for all n. For results on the discrete-time urn, we therefore\nfurther assume (A6). Sinceř\niaiXiptqis constant between the jumps, (A6) implies\nthat waiting times pTn`1´pTnare finite a.s., and thus there is a.s. an infinite number\nof draws; hence pTnă 8for every n, and the discrete-time urn pXnq8\n1is well defined\nby (2.19).\nNext, we note that a colour iwithλ˚\niă0 will die out:\nLemma 8.3. Ifλ˚\niă0, then a.s. Xiptq “0for all large t, and thus Xni“0for all\nlargen.\nProof.Since every jÑihasλ˚\njďλ˚\niă0, we can use induction and assume that\nthe statement holds for every jÑi. Then there exists a.s. a (random) time Tsuch\nthat no colour jwithjÑiexists for times těT. Hence, for těT,Xiptqevolves\nas a single colour urn with only colour i; we have λiďλ˚\niă0, and thus, as in\nRemark 8.2, this process is a subcritical Markov branching p rocess, which dies out\na.s. /square\nHence, if λ˚\niă0, we can wait until colour iand all its ancestors have disappeared,\nand we may then regard the urn as restarted at that time; then w e have an urn with\nfewer colours. Consequently, we may without loss of general ity assume λ˚\niě0 for\nevery colour i. Note that it follows from the definition (2.8) that\nλ˚\niě0 for every colour ið ñλjě0 for every minimal colour j.(8.2)\nMoreover, also the case λ˚\ni“0 may be problematic when iPQ´, and we will\nactually make a stronger assumption than (8.2). There are tw o reasons.\nFirst, suppose that iPQ´is a minimal colour (and thus λ˚\ni“λi). Then, as\nsaid in Remark 8.2, Xiptqis a Markov branching process which dies out also in the\ncritical case λi“λ˚\ni“0, so as above we may in this case wait until colour ihas\ndisappeared and consider an urn with fewer colours.\nSecondly, and more importantly, if iPQ´is not minimal and λ˚\ni“0, there are\nexamples where (after normalization) Xiptqconverges in distribution but nota.s.,\nand similarly for Xni; see Examples 14.14 and 14.15. (In Example 14.15, Xnidoes\nnot even converge in distribution.)\nWe therefore exclude these cases and assume (A7). Note that i fiRQ´, then\nλ˚\niěλiě0. Hence, (A7) implies that λ˚\niě0 for every i, and thus both conditions\nin (8.2) hold. (So we do not have to assume this explicitly.)\nEven with these assumptions, one complication remains. If iPQ´is a minimal\ncolour, then, as noted in Remark 8.2, Xiptqis a branching process, and even in the\nsupercritical case λią0, it dies out with positive probability. We will accept this\ncomplication, but note that it may be eliminated by the addit ional assumption (A8),\nwhich we only assume when needed.\n8.2.Main results for urns with subtractions. With the assumptions above, our\nmain theorems for discrete and continuous time still hold:\nTheorem 8.4. LetpXiptqqiPQbe a discrete-time triangular P´ olya urn satisfying the\nconditions (A1)–(A4),(A51), and(A7)–(A8). Then the conclusions of Theorem 1.8\nhold.\nWe do not explicitly assume (A6) in Theorem 8.4, but it is impl icit since it follows\nfrom (A8) and the other assumptions, as noted above.34 SVANTE JANSON\nWithout (A8), we get a more complicated partial result (wher eX0is as in Sec-\ntion 5); it is particularly useful in cases where X0ą0 a.s. for some other reason, for\nexample by Lemma 10.9 below in the balanced case. (See Exampl e 14.8.)\nTheorem 8.5. LetpXiptqqiPQbe a discrete-time triangular P´ olya urn satisfying the\nconditions (A1)–(A4),(A51), and(A6)–(A7). Then the conclusions of Theorem 1.8\nhold a.s. on the event tX0ą0u(which has positive probability), except that pXi“0\nis possible, with 0ďPppXi“0|X0ą0q ă1.\nTheorem 8.6. LetpXiptqqiPQbe a continuous-time triangular P´ olya urn satisfying\nthe conditions (A1)–(A4),(A51), and(A7). Then the conclusions of Theorem 4.1\nhold, except that Xi“0is possible, with 0ďPpXi“0|X0ą0q ă1; moreover, we\nhavePpXią0@iPQq ą0.\nIf furthermore (A8)holds, then the all conclusions of Theorem 4.1 hold.\nRemark 8.7. In Theorems 8.5 and 8.6, if the limit pXi“0 orXi“0, then the\nresult (1.3), (1.4) or (4.1) does not give the correct growth ofXniorXiptq. In such\ncases, it might be possible to find more precise results using our methods on suitable\nsubsets of the colours; it seems that this can be done very gen erally, at least on a\ncase-by-case basis. We give one example in Example 14.13 but do not attempt to\nstate any general theorem. △\nRemark 8.8. Also the results in Section 6–7 still hold (with the same proo fs) under\nthe assumptions (A1)–(A4), (A51), and (A7)–(A8), as the reader might verify.\nIf we do not assume (A8) and assume only (A1)–(A4), (A51), and (A6)–(A7), then\nthe results on absolute continuity do not hold as stated, sin ce typically the limits\nmay be 0 with positive probability. (We conjecture that thes e results might hold\nconditioned on the variables being non-zero, but we have not pursued this.) The\nremaining results in Section 6–7 still hold. △\nNote that (A5) implies (A51), (A7), and (A8) (with Q´“ H), and thus (A1)–(A5)\nimply also (A6); hence these theorems (strictly) extend The orems 1.8 and 4.1.\nWe prove Theorems 8.4–8.6 in the remainder of this section. A s in Section 3, we\nfirststudyasinglecolour, andastherewesplitthediscussi onintoseveralsubsections.\nFirst note that we used that Xjptqis increasing in Lemma 3.1 and its proof. This\nis no longer necessarily true, but we may replace XjptqbyX˚\njptqwith no further\nconsequences. Hence, Lemmas 3.1 and 3.2 hold with the assump tion modified to\nEX˚\njptq ă 8.\n8.3.A colour not influenced by others. Consider the situation in Section 3.1,\nwhereξji“0forj‰i, i.e.,iPQmin. IfiPQ´, thenXiptqis, as noted inRemark 8.2,\na Markov branching process.\nLemma 3.3 extends to this case, with some modifications.\nLemma 8.9. Suppose that (A1)–(A4)and(A51)hold and that iPQmin.\n(a)IfiRQ´(i.e.,ξiiě0a.s.), then Lemma 3.3 still holds.\n(b)IfiPQ´, then Lemma 3.3 holds with the following modifications:\n(i)Ifλią0, the only difference is that Xi“0is possible with positive probabil-\nity. Ifx0ą0, then0ăPpXi“0q ă1.\n(ii)Ifλi“0, thenXiptqis a martingale with\nXiptqa.s.Ý ÑXi“0,astÑ 8, (8.3)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 35\nE“\nXiptq‰\n“x0, (8.4)\nVar“\nXiptq‰\n“Cx0t, (8.5)\nE`\nsup\n0ďtďuXiptq˘2ďCx2\n0`Cx0u,for every uă 8. (8.6)\nFurthermore, for every δą0,\nE`\nsup\ntě0te´δtXiptqu˘2ă 8. (8.7)\n(iii)Ifλiă0, then\nXiptqa.s.Ý ÑXi“0,astÑ 8, (8.8)\nVar“\nXiptq‰\nďCx0eλit, (8.9)\nE`\nsup\n0ďtă8Xiptq˘2ďCx2\n0. (8.10)\nProof.(a): We thus assume (A5) = (A51)(a) for the colour i, and then the proof\nof Lemma 3.3 still holds. (Recall that for simplicity we had ( A1)–(A5) as standing\nassumptions in Section 3, including (A5) for all colours; ho wever, as remarked before\nLemma 3.3, this is needed only for the colour iin that subsection, and in particular\nfor Lemma 3.3 and its proof.)\n(b): Now assume (A51)(b). By (A2), we also have aią0.\nWe argue as in the proof of Lemma 3.3, with the following differe nces. Note that\nx0is assumed to be an integer, and the case x0“0 is trivial; we thus may assume\nx0ě1. It is still true that Mptq:“e´λitXiptqis a martingale, and (3.20) holds if\nλi‰0.\nNow, however, as said in Remark 8.2, with positive probabili ty,Xiptq “0 for all\nlarget; moreover, in the critical and subcritical cases (ii) and (i ii), this happens a.s.\nWe study the three cases separately:\n(i): The proof of Lemma 3.3 still holds, except for the final pa rt yielding (3.25).\nBy Remark 8.2, with positive probability Xiptqdies out and thus Xi“0. More-\nover, by (3.14), EXi“x0ą0. Consequently, 0 ăPpXi“0q ă1.\n(ii): By Remark 8.2, a.s. Xiptq “0 for all large t, which gives (8.3).\nForλi“0, (3.12) says that Xiptq “Mptqis a martingale, which implies (8.4).\nFurthermore, (3.20) in this case yields\nEXiptq2“EMptq2“ErM,M st“x2\n0`aiβx0t, (8.11)\nwhich gives (8.5) and, together with Doob’s inequality, (8. 6). To prove (8.7), we\nnote that (8.6) implies\nE´\nsup\ntě0te´δtXiptqu¯2\nďE8ÿ\nn“0e´2δn´\nsup\nnďtďn`1Xiptq¯2\nďE8ÿ\nn“0e´2δnOpn`1q ă 8.\n(8.12)\n(iii): As for (ii), Remark 8.2 shows that a.s. Xiptq “0 for all large t, and thus\n(8.8) holds. Since λiă0, we obtain from (3.20)\nEMptq2“ErM,M stďx2\n0`Cx0e´λit(8.13)\nand thus\nVar“\nXiptq‰\n“e2λitVar“\nMptq‰\nďCx0eλit, (8.14)36 SVANTE JANSON\nshowing (8.9). Furthermore, (8.13) and Doob’s inequality y ield, for any mě1,\nE“\nsup\nm´1ďtďmXiptq2‰\nďe2λipm´1qE“\nsup\nm´1ďtďmMptq2‰\nďCe2λimE“\nMpmq2‰\nďCeλimx2\n0. (8.15)\nSinceλiă0, the sum over all mě1 isďCx2\n0, which implies (8.10). /square\nRemark 8.10. In fact, it follows from [4, Theorem III.7.2] (using (A4)) th at if\niPQminXQ´, thenXi“0 occurs a.s. exactly when Xiptq “0 for large tso that\nthe branching process dies out in finite time. Thus, PpXi“0qequals the probability\nthat the continuous-time branching process Xiptqdies out. Considering only the\ntimes that a ball of colour iis drawn, we obtain an embedded random walk with\ni.i.d. increments distributed as ξii; thusPpXi“0qalso equals the probability that\nthis random walk hits 0. (It is easy to see that this also equal s the probability of\nextinction for a Galton–Watson process with offspring distri bution 1 `ξiistarted\nwithx0individuals.) △\n8.4.A colour only produced by one other colour. Consider the situation in\nSection 3.2, where there is a single colour jsuch that jÑi.\nWe used that Xjptqis increasing in (3.56); this is no longer necessarily true, but\nwe may replace XjptqbyX˚\njptqthere with no further consequences.\nLemma 3.5 holds with the following minor modifications. We as sumeλ˚\njě0,\nsince otherwise jeventually becomes extinct by Lemma 8.3, and after that Xiptq\nevolves as in Lemma 8.9. (We are not really interested in this case, as discussed\nearlier.) Note that we also exclude the case λ˚\ni“λ˚\nj“0. (For good reasons, see\nExamples 14.14 and 14.15).\nLemma 8.11. Suppose that (A1)–(A4)and(A51)hold. Let iPQ, and suppose that\nthere is exactly one colour jPQsuch that jÑi, and that Xip0q “0. Suppose also\nthat one of the following holds:\n(a)iRQ´(i.e.,ξiiě0a.s.), and λ˚\njě0.\n(b)iPQ´,λ˚\nią0, andλ˚\njě0.\nThen Lemma 3.5 still holds, i.e., if (3.31)–(3.32)hold, then we have the conclusions\n(3.33)–(3.35)andXją0ù ñXią0a.s.\nProof.(a): Recall again that we had (A1)–(A5) as standing assumpti ons in Sec-\ntion 3, including (A5) for all colours; however, it is easily verified that the proof of\nLemma 3.5 does not use this for other colours than i, except in (3.56), where we now\nreplaceXjptqbyX˚\njptqas discussed above, and to see that λ˚\njě0. Hence, in the\npresent setting where we assume (A1)–(A4) and (A51), and also explicitly assume\nλ˚\njě0, ifξiiě0 a.s., then the proof of Lemma 3.5 still holds.\n(b): We now assume (A51)(b) fori. Most of the proof of Lemma 3.5 remains the\nsame. Themaindifferenceintheproofcomes inSteps4–5, where weusedLemma3.3\nin (3.65), but now instead use Lemma 8.9(b). We consider thre e cases:\n(1)λią0: Then (3.65) still holds by Lemma 8.9, and thus all estimate s in Steps\n4–5 hold.\n(2)λi“0: Then (3.65) is replaced by, using (8.5),\nE“\nrZ1,Z1st| pTk,ηkq8\n1‰\n“8ÿ\nk“11tTkďtuVar“\nYkpt´Tkq |Tk,ηk‰ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 37\n“8ÿ\nk“11tTkďtuCpt´Tkqηk. (8.16)\nThus\nErZ1,Z1st“CE8ÿ\nk“11tTkďtupt´Tkq ďCtENptq. (8.17)\nUsing Lemma 3.2 (or Lemma 3.1), it follows that (3.68) for ℓ“1 is replaced\nby (with λi“0)\nErZ1,Z1stďCtEżt\n0XjpsqdsďCErX˚˚\njtżt\n0ps`1qκjepλ˚\nj´2λiqsds,(8.18)\nwith an extra factor t. We assume 0 ăλ˚\ni“λi_λ˚\njandλi“0, and thus\nλ˚\nją0“λi, so we are in Case (ii1) of Step 5; (3.71) now holds with an\ninsignificant extra factor nand thus (3.72)–(3.73) still hold and the conclusions\nof Step 5 remain the same.\n(3)λiă0: Then we obtain instead of (3.65), now using (8.9),\nE“\nrZ1,Z1st| pTk,ηkq8\n1‰\nď8ÿ\nk“11tTkďtue´2λiTkCe´λipt´Tkqηk\n“C8ÿ\nk“11tTkďtue´λit´λiTkηk. (8.19)\nThus, by Lemma 3.2(ii),\nErZ1,Z1stďCE8ÿ\nk“11tTkďtue´λit´λiTkηk“Ce´λitEżt\n0e´λisXjpsqds.(8.20)\nIt follows that (3.68) for ℓ“1 is replaced by\nErZ1,Z1stďCErX˚˚\nje´λitżt\n0ps`1qκjepλ˚\nj´λiqsds. (8.21)\nHowever, for tď1, (8.21) is trivially equivalent to (3.68), and for tě1, (8.21)\nimplies, using λ˚\nj´λiě ´λią0,\nErZ1,Z1stďCErX˚˚\njpt`1qκjepλ˚\nj´2λiqt\nďCErX˚˚\njżt\nt´1ps`1qκjepλ˚\nj´2λiqsds. (8.22)\nHence (3.68) holds for all ℓď3 also in the present setting. By our assumption\nλ˚\ni“λi_λ˚\nją0 we have λ˚\nją0, so we are again in Case (ii1) of Step 5;\n(3.71)–(3.73) still hold and the conclusions of Step 5 remai n the same.\nIn all cases, the conclusions (3.33)–(3.35) of Step 6 follow as for Lemma 3.5.\nFinally, as for Lemma 3.5, if λiďλ˚\nj, then (3.35) shows that Xją0ù ñXią0.\nIfλiąλ˚\njandXją0, then we use again (3.78), where still a.s. all Tkare finite. Let\nYk:“limtÑ8e´λitYkptq. In contrast to Lemma 3.5, Yk“0 is now possible also if\nηką0, but 0 ăPpYk“0q ă1 by Lemma 8.9. We now let Kbe the smallest ksuch\nthatYką0, and conclude as in Section 3 that Xią0. /square38 SVANTE JANSON\nRemark 8.12. In Lemma 8.11(b) we have excluded the case λi“λ˚\nj“0, since\nthenλ˚\ni“0. In this case we are in Case (iii1) of Step 5 of the proof of Lemma 3.5.\nAs mentioned above, there is now an extra factor tin (3.68), and therefore we obtain\ninstead of (3.75)\nErZ;\nℓpnq2ďC2´κjn. (8.23)\nHence, if furthermore κjě1, then (3.76) still holds, and the rest of the proof of\nLemma 3.5 applies. Consequently, Lemma 8.11(b) holds also i n the case λi“λ˚\nj“0\nandκiě1.\nExample 14.14 gives an example with λi“λ˚\nj“0“κiwhere the conclusion of\nLemma 8.11 does not hold. For simplicity we have in Lemma 8.11 (b) excluded all\ncases with λi“λ˚\nj“0 (and thus λ˚\ni“0).\nSimilarly, Example 14.15 gives an example with λiă0“λ˚\nj“κiwhere the\nconclusion of Lemma 8.11 does not hold. We have excluded all c ases with λiăλ˚\nj“\n0; it seems likely that Lemma 8.11(b) might hold also in this c ase ifκiě1, but we\nhave not investigated this further. △\nWe need also a coarse estimate in the case excluded in Lemma 8. 11(b).\nLemma 8.13. Suppose that (A1)–(A4)and(A51)hold and that iPQis such that\nPpξii“ ´1q ą0. Assume that there is exactly one colour jPQsuch that jÑi, and\nthatXip0q “0. Assume also that λ˚\ni“λ˚\nj“0.\nAssume further that (3.31)–(3.32)hold. Then, for every δą0,\ne´δtXiptqa.s.Ý Ñ0, (8.24)\nand\n/vextenddouble/vextenddouble/vextenddoublesup\ntě0/visualspace\ne´δtXiptq(/vextenddouble/vextenddouble/vextenddouble\n2ă 8. (8.25)\nProof.We have λiďλ˚\ni“0 and, by (A2), aią0. Thus rii“λi{aiď0. Fur-\nthermore, ξiiě ´1 a.s., and thus rii“Eξiiě ´1. Choose pP p0,1qsuch that\n0ării`2păδ{ai.\nModifytheurnby increasing ξiitoqξii:“ξii`2ζ, whereζPBeppqisindependentof\nξii; all other ξkℓand initial conditions Xkp0qremain the same. We denote quantities\nfor the new urn by adding q. The modification does not affect any ancestors of i;\nthusqXjptq “Xjptqandqλ˚\nj“λ˚\nj“0. On the other hand, by our choice of p,\nqλi“aiEqξii“aipEξii`2Eζq “aiprii`2pq P p0,δq. (8.26)\nConsequently, qλ˚\ni“qλi_qλ˚\nj“qλią0and thusLemma 8.11(b) applies tothemodified\nurn. (The other conditions of Lemma 8.11(b) obviously hold. )\nThe modification adds extra balls of colour i, and these may get descendants of\ncolouriand they may disappear again, but we may separate these extra balls of\ncolourifrom the original ones, and thus couple the two versions such thatqXiptq ě\nXiptqfor alltě1. Consequently, noting that qλ˚\ni“qλiandqκi“0, Lemma 8.11\nyields, from (3.33)–(3.34) and (3.29)–(3.30),\nlimsup\ntÑ8e´qλitXiptq ďlim\ntÑ8e´qλitqXiptq “qXiă 8a.s., (8.27)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 39\nand/vextenddouble/vextenddouble/vextenddoublesup\ntě0/visualspace\ne´qλitXiptq(/vextenddouble/vextenddouble/vextenddouble\n2ď/vextenddouble/vextenddouble/vextenddoublesup\ntě0/visualspace\ne´qλitqXiptq(/vextenddouble/vextenddouble/vextenddouble\n2ă 8. (8.28)\nThe results follow, since qλiăδby (8.26). /square\n8.5.The general case for a single colour.\nLemma 8.14. Suppose that (A1)–(A4),(A51), and(A7)hold. Then Lemma 3.7\nstill holds, except for the last sentence.\nThe last sentence remains valid if iRQ´, and also if iRQmin. (In particular, it\nis valid if (A8)holds.) In the remaining case, when iPQ´and is minimal, then\n0ăPpXią0q ă1.\nProof.Recall that the assumption (A7) implies that λ˚\njě0 for every jPQ.\nWe follow the proof of Lemma 3.7 and split the colour iinto subcolours i0andij,\njPPi(also ifiis minimal so Pi“ H); then (3.84) holds, and also (3.85)–(3.89) and\n(3.91).\nIfiRQ´, then the proof is exactly as for Lemma 3.7, now using Lemmas 8 .9(a)\nand 8.11(a) instead of Lemmas 3.3 and 3.5.\nAssume thus in the sequel of the proof that instead iPQ´. (Thus, (A51)(b)\nholds.) Then by (A7), we have λ˚\nią0. However, it is possible that λi0“λiď0 or\nthatλ˚\nij“λi_λ˚\nj“0 for some jPPi.\nFori0, we now apply Lemma 8.9(b). If λią0, then Lemma 8.9(b)(i) shows that\nXi0can be treated as before in (3.89) and (3.91). If λiď0, we replace, as we may,\nrXi0ptqbye´λ˚\nitXi0ptqin the estimate (3.89), cf. (3.84) and (3.29). By (8.3) and (8 .8),\nthis term contributes 0 to the limit (3.82). Similarly, in (3 .91) we replace rX˚˚\ni0by\nsuptě0te´λ˚\nitXi0ptqu, which gives a finite contribution to (3.83) by (8.7) and (8.1 0).\nSimilarly, for each ij(jPPi), we now apply Lemma 8.11(b) if λ˚\niją0; otherwise,\ni.e., ifλiď0 andλ˚\nj“0, we apply Lemma 8.13 (with δ:“λ˚\nią0). In the latter\ncase we replace rXijptqbye´λ˚\nitXijptqin the estimate (3.89) and use (8.24), and we\nreplacerX˚˚\nijby suptě0te´λ˚\nitXijptquin (3.91) and use (8.25).\nThisproves, inallcases, that(3.82)and(3.83)hold. Moreo ver, foreach jPPiYt0u\nsuch that we apply Lemma 8.9(b)(ii),(iii) or Lemma 8.13 to ij, the contribution to\nthe limit (3.82) is 0, and so is the contribution to the right- hand side of (3.90) since\nin these cases λ˚\nijď0ăλ˚\niby (3.85)–(3.86). Hence, (3.90) still holds.\nIt remains to consider the final sentence in Lemma 3.7. First, ifPi‰ H, then it\nfollows as in the proof of Lemma 3.7 that if Xją0 for every jPPi, thenXią0. In\nthe remaining case, iis minimal. Note that then Xip0q ą0 by (A3). If iis minimal\nandiRQ´, then a.s. Xią0 by Lemma 3.3. On the other hand, if iis minimal and\niPQ´, then Lemma 8.9(b)(i) shows that PpXi“0q P p0,1q. /square\n8.6.Proofs of Theorems 8.4–8.6.\nProof of Theorem 8.6. As in Section 4, now using Lemma 8.14; we use again induc-\ntion onthecolour iandconcludethat (3.82) and(3.83) holdforevery iPQ, although\nnowXi“0 is possible.\nThe processes XiptqforiPQminare independent, and thus so are their limits Xi.\nEach is strictly positive with positive probability, by Lem ma 8.9 or 8.14, and thus\nthere is a positive probability that Xią0 for every minimal i. Moreover, if (A8)40 SVANTE JANSON\nholds, then this probability is 1. Finally, it follows by Lem ma 8.14 by induction on\nthe colour ithat a.s. if Xią0 for every minimal i, thenXią0 for every i./square\nProof of Theorems 8.4 and 8.5. As in Section 5, now using Theorem 8.6 and consid-\nering only the event tX0ą0u. Note that Theorem 8.6, applied to the extended urn,\nshows that with positive probability X0ą0 andXią0 for all iPQ; furthermore,\nin Theorem 8.4, where we assume (A8), this holds a.s.\nNote also that since we assume (A6), we have a.s. pTnă 8for every n, and thus\nthe discrete-time process Xnis well-defined. /square\n9.Random vs non-random replacements\nConsider a P´ olya urn Uwith a random replacement matrix pξijqi,jPQ. For simplic-\nity we study only the case when all ξijě0, and we thus assume (A1)–(A5). Consider\nalso another urn U1with the same colours Q, the same initial vector X0, and the\nsame activities ai, but non-random replacements rij“Eξij. We thus replace the\nreplacement matrix by its mean, and we may call U1themean urn corresponding\ntoU. Both urns have the same rij, and thus the same colour graph, λi,λ˚\ni,κi,pλ,\npκandγi, see (2.3) and (2.7)–(2.13). The new urn U1also satisfies (A1)–(A5), and\nTheorems 1.8 and 4.1 show that we have the same qualitative as ymptotic behaviour\nfor both urns, with the same normalization factors. However , the limits pXiandXi\nare in general notthe same for the two urns UandU1, and thus the asymptotic\ndistributions may be be different; see Example 14.5.\nWe note that the constants cikin Section 6 by Lemma 6.7 are the same for the\ntwo urns. This yields one simple case. (We assume pλą0 for simplicity, and leave a\nstudy of the case pλ“0 to the reader.)\nTheorem 9.1. Assumepλą0. Ifiis a colour such that pXiis deterministic for one\nof the two urns, then pXiis the same constant for both urns.\nProof.By Theorem 7.1, there are two cases: either λ˚\ni“0, orλ˚\ni“pλ.\nIfλ˚\ni“0, then (6.5) and Lemma 6.2 show that for every jPAi,Xj“Xjp0qis a\nconstant, the same for both urns. Hence, Lemmas 6.1 shows, si nce the constants cik\nare thesame for bothurns, that Xiis thesame constant for both urns. Consequently,\n(5.12) shows that pXi“pλ´κiXiis the same constant for both urns.\nAssume now λ˚\ni“pλ. If we add dummy balls to both urns as in Section 5, then,\nby Lemma 6.1 and (5.2), for any of the urns, both XiandX0are linear combinationsř\njPJcijXjandř\njPJc0jXj, whereJis the set of leaders with λ˚\nj“pλ. The joint\ndistribution of pXjqPJis absolutely continuous by Theorem 6.4, and since (5.12)\nyieldspXi“pλ´γiXi{X0, it follows that if pXiis deterministic in one of the two urns,\nthen the vectors pcijqjPJand pc0jqjPJare proportional. Since these vectors are the\nsame for both urns, it follows that then pXiis the same constant in both urns. /square\n10.Balanced urns\nMany papers on P´ olya urns assume that urn is balanced (as defi ned below); while\nthis is quite restrictive, it is partly justified by the fact t hat P´ olya urns that appear\nin applications often are balanced. We have not needed this a ssumption in the\npreceding sections, but it will be used for some results in th e following sections (inALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 41\nparticular Section 12). The present section contains the de finition and some simple\nresults for later use.\nIn the standard case when all activities ai“1, a P´ olya urn is said to be balanced if\nweaddthesametotal numberofballsateachdraw; inotherwor ds, ifthereplacement\nmatrix pξijqi,jPQsatisfiesř\njξij“ba.s., for some bPRand every iPQ. In\ngeneral, with possibly different activities ai, it is not the total number of balls but\ntheir total activity that is important, and we make the follo wing general definition.\n(Definition 10.1 and Lemma 10.2 – Remark 10.6 below apply to al l P´ olya urns,\ntriangular or not.)\nDefinition 10.1. A P´ olya urn is balanced if there exists a constant bPR(called\nthebalance) such that for every iPQwithaią0, we have\nÿ\njPQajξij“ba.s. (10.1)\nWitha:“ pajqjPQ, the vector of activities, (10.1) may be written a¨ξi“bfor all\niwithaią0.\nSuppose that the urn is balanced as in Definition 10.1. It foll ows from (1.2) (since\ncoloursiwithai“0 will never be drawn) that a.s.\na¨Xn“ÿ\njPQajXnj“a¨X0`nb. (10.2)\nHence, the denominator in (1.1) is deterministic. This has b een used in several\nways in many papers by different authors. (In particular, it ma kes it possible to\nuse martingale methods for the discrete-time urn, similar t o the continuous-time\narguments in the present paper; see Remark 1.11 and e.g. [12] .) Here we note some\nsimple consequences.\nNote first that if the balance bă0, then (10.2) shows that the activity a¨Xn\nbecomes negative for large n; this is clearly impossible and shows that the discrete-\ntime urn process must stop and cannot be continued for ever. I n the present paper,\nwe are not interested in this case, so we must have bě0, and we thus assume this in\nthe sequel (whether it is said explicitly or not). In this cas e, there are no problems.\nLemma 10.2. A balanced urn with balance bě0which satisfies (A1)and(A5)or\n(more generally) (A51)is well-defined for all discrete or continuous times, and thu s\nsatisfies (A6).\nProof.By (10.2), we have, for every n, a.s.a¨Xněa¨X0ą0. Hence the discrete-\ntime process Xnis well-defined, and also a¨Xptq ą0 for every t. /square\nWe assume (A1) and (A5) or (A51) below; thus (A6) holds.\nRemark 10.3. The case b“0 is trivial. First, if (A5) holds, so there are no\nsubtractions, then the only possibility is ξij“0 a.s. for all i,jPQ; in other words,\nwe draw from the urn but nothing happens. If we allow subtract ions as in (A51)(b),\nthere may be a few initial draws that wipe out some colours, bu t nothing more will\nhappen; furthermore, such urns violate (A7). △\nLemma 10.4. If a P´ olya urn is balanced, then the random processes pXnq8\nn“1and\nppTnq8\nn“1are independent.42 SVANTE JANSON\nProof.Letně0. Stop the continuous-time process XptqatpTn, i.e., at the nth draw.\nConditionally on everything that has happened so far, the wa iting time pTn`1´pTn\nuntil the next draw is an exponential random variable with ra teř\niaiXni, which by\n(10.2) is the constant a¨X0`nb. Hence, pTn`1´pTnis independent of Xn, and thus\nalso of the colour of the drawn ball, say i, and of Xn`1´Xn“ξpnq\ni, recall (1.2).\nConsequently, the processes ppTnqnand pXnqnevolve independently. /square\nLemma 10.5. If a P´ olya urn is balanced, then the distribution of the rando m se-\nquence ppTnq8\nn“1depends only on the balance band the initial activity a¨X0.\nProof.As in the proof of Lemma 10.4, pTn`1´pTnis exponential distributed with rate\na¨X0`nb, and these waiting times are independent. /square\nRemark 10.6. The distribution of ppTnq8\nn“1is thus the same as for the single-colour\nurn with initial value x0:“a¨Xo, activity a1“1, and replacement matrix pbq. For\nbą0, this is the more or less trivial case q“1 of Example 14.1, and it is well known\nthat pbpTnq8\n1are the jump times of a Yule process started with x0{bindividuals. △\nIn the remaining part of the section we consider, as in the res t of the paper,\ntriangular urns.\nLemma 10.7. Consider a triangular balanced urn with balance bě0, and suppose\nthat the urn satisfies (A1)–(A4). Thenpλ“b. Furthermore, if bą0, thenpκ“0.\nProof.LetQ1:“ tiPQ:aią0u. ThenQ1‰ Hby(A1). Therearethreepossibilities\nfor a colour iPQ:\n(i) IfiRQ1, thenai“0 and thus λi“0.\n(ii) IfiPQ1andiis maximal in Q1for the order ≺, i.e.,iÑjfor every jPQ1,\nthenξij“0 a.s. when jPQ1ztiu, andaj“0 for every jRQ1; henceajξij“0\na.s. for all colours j‰i. Consequently, (10.1) yields\naiξii“ÿ\njPQajξij“b (10.3)\na.s., and thus, taking the expectation,\nλi“airii“aiEξii“b. (10.4)\n(iii) IfiPQ1is not maximal in Q1, then there exists jPQ1withiÑjand thus\nriją0. Taking expectations in (10.1) yields\nλi“airii“b´ÿ\nj‰iajrijďb´ajrijăb. (10.5)\nIt follows that pλ:“maxiλi“b. Furthermore, if bą0, then the maximum is\nattained precisely in Case (ii), i.e., for ithat are maximal in Q1. It follows that for\nevery such iwe have λjăbfor alljPPi, and thus κi“0. Hence, pκ“0. /square\nRemark 10.8. The proof shows also that if a balanced triangular urn has bą0\nand all activites ai“1, thenξii“rii“λi“ba.s. for every ithat is maximal in Q1,\nbutEξii“rii“λiăbfor all other colours i. △\nFor urns with subtractions as in Section 8, there are further simplifications when\nthe urn is balanced.ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 43\nLemma 10.9. Consider a triangular balanced urn with balance bě0, and sup-\npose that the urn satisfies (A1)–(A4),(A51), and(A7). Then, with notation as in\nSection 5, X0ą0a.s.\nProof.The case b“0 is trivial by Remark 10.3, so we may assume bą0.\nFor convenience, denote the urn by U. Lemma 10.5 and Remark 10.6 show that\npTnare (jointly) distributed as for the one-colour urn U1with replacement matrix pbq,\nactivitya“1, and the same initial total activity. Hence we may couple th e urns\nsuch that they have the same pTn. (Or we may simply define the content of U1as\nbeing the total activity in the urn U.) The urn U1obviously satisfies (A1)–(A5), and\nthus (5.5) applies to it, with X0ą0 a.s. Moreover, (5.5) applies also to the original\nurnUby the same argument in Section 5 (as in the proof of Theorems 8 .4 and 8.5\nin Section 8.6); note that the two urns UandU1have the same pλ“bandκ0“0 by\nLemma 10.7 and (5.3). Consequently, the two urns UandU1have the same X0./square\nIn fact, in Lemma 10.9, X0has a Gamma distribution by the proof and (14.2).\n11.The drawn colours\nWe have so far studied XnandXptq, the numbers of balls of each colour in the\nurn. It is also of interest to study the number of times each co lour is drawn. (See\nExamples 14.7–14.9 for an application.) For iPQ, we denote the number of times\nthat a ball of colour iis drawn up to time nin the discrete-time urn by Nni, and\nthe number of times up to time tin the continuous-time urn by Niptq; thus\nNni“NippTnq, i PQ, ně0. (11.1)\nWe state first a continuous-time and then a discrete-time res ult; both are similar\nto the results for XptqandXnearlier, but note that exponents change when λ˚\ni“0.\n(Proofs are given later in this section.)\nTheorem 11.1. LetpXiptqqiPQbe a continuous-time triangular P´ olya urn satisfying\neither(A1)–(A5), or (more generally) (A1)–(A4),(A51), and(A7). LetiPQ.\n(i)Ifλ˚\nią0, then, as tÑ 8,\nt´κie´λ˚\nitNiptqa.s.Ý ÑNi:“ai\nλ˚\niXi. (11.2)\n(ii)Ifλ˚\ni“0, then, as tÑ 8,\nt´κi´1Niptqa.s.Ý ÑNi:“ai\nκi`1Xi. (11.3)\nFurthermore, if (A5)or(A8)holds, then Nią0a.s.\nTheorem 11.2. LetpXiptqqiPQbe a discrete-time triangular P´ olya urn and suppose\nthat it satisfies (A1)–(A5). Alternatively, suppose that the urn satisfies (A1)–(A4),\n(A51), and(A6)–(A7), and also either satisfies (A8)or is balanced. Let iPQ.\n(i)Ifλ˚\nią0, then as nÑ 8,\nNni\nnλ˚\ni{pλlogγina.s.Ý ÑpNi:“ai\nλ˚\nipXi. (11.4)\n(ii)Ifλ˚\ni“0andpλą0, then as nÑ 8,\nNni\nlogκi`1na.s.Ý ÑpNi:“ai\npκi`1qpλpXi. (11.5)44 SVANTE JANSON\n(iii)Ifλ˚\ni“pλ“0, then as nÑ 8,\nNni\nnpκi`1q{pκ0a.s.Ý ÑpNi:“ai\nκi`1X´1{pκ0\n0pXi. (11.6)\nFurthermore, if (A5)or(A8)holds, then pNią0a.s.\nRemark 11.3. Ifweassumeonly(A1)–(A4), (A51), and(A6)–(A7)inTheorem11.2,\nthen the results hold on the event tX0ą0u, by the same proof and Theorem 8.5. △\nWe can also state the results as the following simple a.s. lim it results for the\nratiosNni{XniandNiptq{Xiptq; in particular, if λ˚\nią0 (the main case), these ratios\nconverge a.s. to a positive constant.\nTheorem 11.4. LetpXiptqqiPQbe a discrete-time triangular P´ olya urn and suppose\nthat it satisfies (A1)–(A5)or (more generally) (A1)–(A4),(A51), and(A7)–(A8).\nLetiPQ.\n(i)Ifλ˚\nią0, then\nlim\nnÑ8Nni\nXni“lim\ntÑ8Niptq\nXiptq“ai\nλ˚\nia.s. (11.7)\n(ii)Ifλ˚\ni“0, then, as tÑ 8,\nNiptq\ntXiptqa.s.Ý Ñai\nκi`1. (11.8)\n(iii)Ifλ˚\ni“0andpλą0, then, as nÑ 8,\nNni\nXnilogna.s.Ý Ñai\npκi`1qpλ. (11.9)\n(iv)Ifλ˚\ni“0andpλ“0, then, as nÑ 8,\nNni\nXnin1{κ0a.s.Ý Ñai\nκi`1X´1{pκ0\n0. (11.10)\nNote that the limit in all cases is a strictly positive consta nt, in case (iv) by\nTheorem 6.5 (and Remark 8.8).\nProof of Theorem 11.1. We use (as in the proof of the corresponding result in [27])\ndummy balls similarly to the proof in Section 5, but now we use one dummy ball for\neach colour.\nIt suffices to consider one colour at a time, so we fix iPQ; we assume aią0 since\notherwise Nni“Niptq “0 a.s. and the results are trivial. Denote the urn by U. We\nconsider one new colour, which we denote by ι, and let Q`:“QY tιube the new\nset of colours. Balls of colour ιhave activity aι:“0 and are thus never drawn, and\nwe letξιj:“0 for alljPQ`; we further let ξiι:“1 andξjι:“0 for every j‰i, and\nwe start with X0ι:“0.\nConsequently, the new urn, which we denote by U`, differs from the old one U\nonly in that one additional ball of colour ιis added each time a ball of colour iis\ndrawn. We may thus assume that the two urns are coupled such th at they have the\nsameXjptqfor alljPQandtě0, and then\nNiptq “XιptqandNni“Xnι. (11.11)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 45\nThe new urn U`is also triangular and satisfies (A1)–(A5), or (A1)–(A4), (A 51), and\n(A7), ifUdoes. In U`we have λι“0 and, using (2.8) and (2.9),\nλ˚\nι“λ˚\ni, (11.12)\nκι“#\nκi, λ˚\nią0,\nκi`1, λ˚\ni“0.(11.13)\n(i): By Theorem 4.1 or Theorem 8.6 applied to the new urn U`, we have (4.1) for\nthe colour ι, and thus, using (11.11) and (11.12)–(11.13),\nt´κie´λ˚\nitNiptq “t´κιe´λ˚\nιtXιptqa.s.Ý ÑXι. (11.14)\nFurthermore, Lemma 3.5 or Lemma 8.11(a) applies to ιPQ`(withireplaced by ι\nandjbyi), and thus (3.35) yields, since λι“0ăλ˚\ni,\nXι“airiι\nλ˚\ni´λιXi“ai\nλ˚\niXi. (11.15)\nHence, (11.2) follows, with Ni:“Xι.\n(ii): Similar, now with κι“κi`1 by (11.13), and using the second alternative in\n(3.35) which gives\nNi:“Xι“airiι\nκιXi“ai\nκi`1Xi. (11.16)\nThe final sentence follows from Theorems 4.1 and 8.6, which yi eldXιą0 a.s./square\nProof of Theorem 11.2. We add a dummy colour ιas in the proof of Theorem 11.1,\nand note that if the original urn Usatisfies (A8) or is balanced, then the same\nholds for the new urn U`. (Note that dummy colours with activity 0 are ignored in\nDefinition 10.1.)\nBy Theorem 1.8, Theorem 8.4, or Theorem 8.5 together with Lem ma 10.9, the\nconclusions of Theorem 1.8 hold for U`, except that pXι“0 is possible unless we\nhave (A51) or (A8). (However, our assumptions yield X0ą0 a.s. in all cases.)\nWe argue similarly to the proof of Theorem 11.1, now using (1. 3) or (1.4) for the\ndummy colour ιinU`, together with (11.11) and (11.12)–(11.13). This yields a. s.\nconvergence to pNi:“pXιin (11.4), (11.5), or (11.6) (depending on λ˚\niandpλ); note\nthatpλěλ˚\ni, that (2.13) yields γι“γiin (i) and γι“κι“κi`1 in (ii), and that\n(2.12) shows that pκ0is the same for U`as forU.\nFinally, the formulas for pNifollow from (11.15)–(11.16) and (5.12) or (5.14). /square\nAlternatively, wemay prove Theorem 11.2 from Theorem11.1 b y addingadummy\ncolur 0 as in Section 5. In any case, the proof is really based o n adding two dummy\ncolours 0 and ιto the continuous-time urn.\nProof of Theorem 11.4. The results for the continuous-time urn in (i) and (ii) follo w\nby comparing the results of Theorem 11.1 with the limits for Xiptqin Theorem 4.1\nor Theorem 8.6, recalling that Xią0 a.s. as stated in Theorem 4.1. The result for\nNni{Xniin (i) then follows by (11.1) and (2.19). Similarly, (11.9) a nd (11.10) follow\nfrom (11.8) and (5.9) or (5.13). (Alternatively, the result s forNni{Xnifollow by\ncomparing the results in Theorem 11.2 and Theorem 1.8.) /square\nRemark 11.5. In Theorem 11.4, if we assume only (A1)–(A4), (A51), and (A7)\nbut not (A8), then the conclusions hold on the event tpXią0uforNiptqand on\ntpXią0,X0ą0uforNni. △46 SVANTE JANSON\n12.Moment convergence\nWe have so far considered convergence a.s., which as always i mplies convergence\nin distribution. We consider in this section whether we also have convergence of\nmoments, orequivalentlyconvergencein Lp; recallthegeneralfactthatforasequence\nof positive random variables converging a.s. (as we have her e), convergence of the\npth moment is equivalent to convergence in Lp, for any (real) pą0, see e.g. [21,\nTheorem 5.5.2].\nUnlike earlier sections, there seems to be an important differ ence between the\ndiscrete-time and continuous-time cases.\nFor a continuous-time urn, we will see below (Theorem 12.2) t hat we always have\nconvergence in L2in Theorems 4.1 and 8.6; hence the mean and variance converge in\nthese results. This extends to convergence in Lpfor anypą2, and thus convergence\nof any moment, assuming a corresponding moment condition fo r the replacements\nξij(Theorem 12.3).\nThe situation for discrete-time urns is more complicated. W e will only consider\nbalanced urns, and then prove L2-convergence, and thus convergence of mean and\nvariance; we also extend this to Lpand higher moments under the corresponding\nmoment condition for the replacements ξij(Theorem 12.5). It seems likely that this\nresult extends to a wider class of triangular urns. However, it doesnothold for all\ntriangular urns. Example 14.2 gives a simple example where t he a.s. limit does not\nhave a finite mean, and thus we cannot have even L1-convergence in Theorem 1.8.\nRemark 12.1. The counterexample in Example 14.2 is rather special (a diag onal\nurn); [28, Theorem 1.6] shows that for a class of more typical unbalanced triangular\nurns, the a.s. limit in Theorem 1.8 has moments of all orders. This does not prove\nmomentconvergence, butweseenoreasonagainstit, andweco njecturethatforthese\nurns, and many others, we have convergence in Lpfor allpą0 (Problem 12.6). △\nLp-convergence or moment convergence (usually as part of a pro of of convergence\nin distribution by the method of moments) have been proved ea rlier by different\nmethods for some discrete-time P´ olya urns, as far as we know all of them bal-\nanced. This includes balanced triangular urns with q“2 or 3 and deterministic\ninteger-valued replacements by Flajolet, Dumas and Puyhau bert [17], [43] (see Ex-\namples 14.3 and 14.11), and, for L2only, more general balanced triangular urns with\ndeterministic replacements by Bose, Dasgupta, and Maulik [ 12] (Example 14.12).\nFurther examples where moment convergence has been shown ea rlier are discussed\nin Examples 14.6, 14.7, 14.8, and 14.10.\n12.1.Continuous-time urns.\nTheorem 12.2. In Theorems 4.1 and 8.6, the limit (4.1)holds also in L2. In\nparticular, mean and variance converge.\nProof.LetiPQ. The proofs of Theorems 4.1 and 8.6 show that (3.83) holds. Th us,\nby the definitions (3.29)–(3.30),\nsup\ntě1/vextendsingle/vextendsinglerXiptq/vextendsingle/vextendsingle2ď/vextendsingle/vextendsingle2κirX˚˚\ni/vextendsingle/vextendsingle2PL1. (12.1)\nHence, the collection t|rXiptq|2:tě1uis uniformly integrable, and consequently\nthe a.s. convergence rXiptq ÑXiimplies convergence in L2[21, Theorems 5.4.4 and\n5.5.2]. /squareALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 47\nTheorem 12.2 extends to Lpfor allpě2 as follows. The proof is based on the\narguments in Section 3 combined with the Burkholder–Davis– Gundy inequalities\nwhich enable us to replace L2estimates by Lpestimates. The details are quite long,\nhowever, so we postpone them to Appendix B.\nTheorem 12.3. Letpě2. In Theorems 4.1 and 8.6, assume also ξijPLpfor all\ni,jPQ. Then, for every iPQ, the limit (4.1)holds also in Lp. Moreover, rX˚˚\niPLp.\nRemark 12.4. In Appendix B, we further show that Theorem 12.3 holds for any\npą1, also without the L2condition (A4). The proofs below then show that the\nsame holds for all Lpresults in this section. △\n12.2.Balanced discrete-time urns.\nTheorem 12.5. Consider a balanced triangular urn statisfying (A1)–(A5)or(A1)–\n(A4),(A51), and(A7).\n(i)In Theorems 1.8 and 8.4–8.5, the limit (1.3)or(1.4)holds also in L2. In\nparticular, mean and variance converge.\n(ii)Moreover, if pě2andξijPLpfor alli,jPQ, then, for every iPQ, the limit\n(1.3)or(1.4)holds also in Lp. Hence all moments of order ďpconverge.\nAs said above, this has been shown earlier in some special cas es and examples, in\nparticular [17], [43] ( q“2,3), and [12] ( p“2); see also the examples in Section 14.\nProof.Part (i) is a special case of (ii), so we show only the latter.\nNotefirstthat(A6)holdsbyLemma10.2, andthatLemma10.7sh owsthat X0ą0\na.s.; thus the conclusion (1.3) or (1.4) of Theorem 1.8 holds by Theorem 1.8 or 8.5.\n(Or Theorem 8.4 when it is applicable.)\nLetcą0 be so small that PpX0ącq ą1\n2. Our assumptions imply (as in the proof\nof Theorems 8.4 and 8.5) that (5.5) holds, by the proof in Sect ion 5. This implies,\nby our choice of c,\nP´\npT´κ0ne´pλpTnnąc¯\ną1\n2(12.2)\nfor all large n. By decreasing c(or just by ignoring some small nin the sequel), we\nmay assume that (12.2) holds for all ně1. LetEndenote the event\nEn:“/visualspacepT´κ0ne´pλpTnnąc(\n, (12.3)\nso that (12.2) reads PpEnq ą1\n2.\nLetiPQ. By Theorem 12.3, rX˚˚\niPLp. Hence, by (2.19) and (3.30),\nYn:“/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleXni`\n1`pTn˘κieλ˚\nipTn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n“/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleXippTnq\n`\n1`pTn˘κieλ˚\nipTn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\nď prX˚˚\niqpPL1.(12.4)\nConsequently, the sequence Ynis uniformly integrable. It follows from this and\n(12.2) that the sequence of conditioned random variables pYn|Enqalso is uniformly\nintegrable. We consider two cases:\nCase 1: pλą0.In this case, Lemma 10.7 shows, using (5.3) and (2.13), that\nκ0“pκ“0 andγi“κi. Hence, the event Enmeanse´pλpTnnąc, and thus pTnď\ntn:“pλ´1plogn`Cq. Consequently, on the event Enwe have (for ně2)\nZn:“/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleXni\nnλ˚\ni{pλlogγin/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n“/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleXni\nnλ˚\ni{pλlogκin/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\nďC/vextendsingle/vextendsingle/vextendsingle/vextendsingleXni\ntκineλ˚\nitn/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\nďCYn(12.5)48 SVANTE JANSON\nand thus the uniform integrability of pYn|Enqimplies uniform integrability of pZn|\nEnq. However, by Lemma 10.4 and (12.3), Xniis independent of the event En; hence,\nso isZnand thus pZn|Enqd“Zn. Consequently, the sequence Znis uniformly\nintegrable.\nIn other words, the left-hand side of (1.3) is uniformly pth power integrable.\nHence, the a.s. convergence in (1.3) implies convergence al so inLp.\nCase 2:pλ“0.This is similar. In this case, (12.3) means pTnďCn1{κ0. We now\ndefine\nZn:“/vextendsingle/vextendsingle/vextendsingle/vextendsingleXni\nnκi{pκ0/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n, (12.6)\nand note again that Znis independent of En. Since 0 ďλ˚\niďpλ“0 we have λ˚\ni“0,\nandpκ0“κ0by (5.4); hence (12.6) and (12.4) show that on En, we have ZnďCYn.\nConsequently, we have again Znd“ pZn|Enq ď pCYn|Enq, and it follows that Znis\nuniformly integrable. Hence the a.s. convergence (1.4) hol ds also in Lp. /square\nAs said above, Example 14.2 shows that Theorem 12.5 does not e xtend to all\ntriangular urns. However, it seems likely that it extends to many unbalanced urns;\nwe leave this as an open problem.\nProblem 12.6. Findmoregeneral conditions (includingalso someunbalanc ed urns)\nfor convergence in L2orLpin Theorems 1.8 and 8.4.\n12.3.Moments for drawn colours. The results above on convergence in L2, and\nthus convergence of mean and variance, apply to the number of drawn balls with a\ngiven colour, NniandNiptq, since as shown in the proofs in Section 11, they can be\nregarded as XnιandXιptqfor an extended urn with a dummy colour ιadded. Hence\nwe obtain:\nTheorem 12.7. In Theorem 11.1, the a.s. limit (11.2)or(11.3)holds also in L2.\nMoreover, if pě2andξijPLp@i,jPQ, then the limit holds also in Lp.\nProof.By Theorem 12.2 or 12.3 applied to Xιptq. /square\nTheorem 12.8. In Theorem 11.2, if the urn is balanced, then the a.s. limit (11.4),\n(11.5), or(11.6)holds also in L2. Moreover, if pě2andξijPLp@i,jPQ, then\nthe limit holds also in Lp.\nProof.By Theorem 12.5 applied to Xnι. /square\n13.Rates of convergence?\nFor classical P´ olya urns(Example 14.1), the rate of conver gence for convergence in\ndistribution in (14.4) or (14.5) has been studied, for sever al different metrics; see [29]\nand the references there. As noted in [29, Remark 1.4], the ra te of a.s. convergence\nis slower, and is the same as in the law of large numbers for i.i .d. Bernoulli variables,\nwhich is given by the law of iterated logarithm.\nFor other triangular urns, we are not aware of any similar res ults on rates of\nconvergence; however, [17]gives upperboundsfortherateo fconvergence ofmoments\nand in a local limit theorem, for some balanced triangular ur ns with deterministic\nreplacements. (Irreducible, and thus non-triangular, bal anced urns with q“2 and\ndeterministic replacements are studied in [34].)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 49\nProblem 13.1. Study rates of convergence in e.g. (1.3) and (4.1), both for t he a.s.\nconvergence and for convergence in distribution.\nNote that this problemis closely related to theproblem stud yingfluctuations from\nthe limit, mentioned in Remark 1.9.\n14.Examples\nWe consider several examples, many of which have been treate d earlier from dif-\nferent perspectives. The purpose is to illustrate both the t heorems above and (some\nof) their relations to earlier literature. We generally lab el the colours by 1 ,...,q,\nand then assume ξij“0 wheniăj. We assume that the initial composition X0is\ndeterministic. We write for convenience xi:“Xi0“Xip0qandx:“ pxiqq\n1“X0. We\ndenote the total number of balls in the urn after ndraws by |Xn|:“řq\ni“1Xni.\nWhenq“2 we sometimes also call the colours whiteandblack(in this order;\nthus a black draw may add only black balls: “there is no escape from a black hole”).\nWe then may write Wn:“Xn1,Bn:“Xn2,Wptq:“X1ptq,Bptq:“X2ptq,w0:“\nx1“X10,b0:“x2“X20.\nWe usually describe the urns using the replacement matrix pξijqq\ni,j“1where the\nrows are the replacement vectors. (See Remark 1.4.)\nIn all our examples, all activities ai“1. Thus\nλi“rii“Eξii, i PQ“ t1,...,q u. (14.1)\nExample 14.1. The classical P´ olya urn has balls of qcolours; when a ball is drawn\nit is replaced together with a fixed number bą0 balls of the same colour. Hence,\nthe replacement matrix is deterministic and diagonal, with entriesbon the diagonal.\nThis urn is obviously balanced, and we have λi“λ˚\ni“bandκi“0 for every colour\ni.\nThis urn model was studied (for q“2) already by Markov [39], Eggenberger\nand P´ olya [16] and P´ olya [41]. See also e.g. Johnson and Kot z [31, Chapter 4] and\nMahmoud [38].\nFor this urn (as for any diagonal urn), in the continuous-tim e version, the dif-\nferent colours evolve independently, and each colour is ver sion of the Yule process.\nMore precisely, Xiptq{bis a Yule process started with xi{bindividuals, where each\nindividual gets children at rate b; thusXipt{bq{bis a Yule process with the standard\nrate 1. (This is a classical branching process if xi{bis an integer, and in general a\nCB process.) It is well-known that in this case e´btXiptq{bdÝ ÑΓpxi{b,1qand thus\ne´btXiptqa.s.Ý ÑXiPΓpxi{b,bq; (14.2)\nfurthermore, X1,...,Xqare independent, since the processes Xiptqare independent.\nThis is an example of Theorem 4.1. Moreover, (5.11)–(5.12), or (5.16) where now\nQ˚“Q, show together with (5.17) that\nXin\nna.s.Ý ÑpXi:“bXiřq\nj“1Xj. (14.3)\nIt follows that the vector of proportions converges:\nXn\n|Xn|a.s.Ý Ñ1\nbpX:“1\nb`pX1,...,pXq˘\n“pX1,...,Xqqřq\nj“1Xj, (14.4)50 SVANTE JANSON\nwhere, as a consequence of (14.2), the limit vector b´1pXhas a Dirichlet distribution\nwith parameter x{b. In particular, each marginal converges a.s. to a Beta distr ibuted\nvariable:\nXni\n|Xn|a.s.Ý Ñb´1pXi„B´xi\nb,ÿ\nj‰ixj\nb¯\n. (14.5)\nThese results are all well known; the limit (14.5) with conve rgence in distribution\nwas shown for q“2 already in [39] and [41], and for general qin [10] (in a special\ncase) and [2]; see also [31, Section 6.3.3] and [38]. Further more, a.s. convergence has\nbeen shown by a number of methods, for example in [10] and [2]. △\nExample 14.2. Adiagonal P´ olya urnhas ξij“0fori‰j; in other words, all added\nballs have the same colour as the drawn ball. This is a general ization of the classical\nP´ olya urn in Example 14.1, but now the diagonal elements ξiican be random, and\nthey may have different distributions. A.s. convergence for t his urn has been shown,\nunder weak technical conditions, by Athreya [2]; see also Ag uech [1, Theorem 4].\nConsider for simplicity the case when the replacements are d eterministic, and\nassume to avoid trivialities that rii“ξiią0 for every iPQ.\nAs in Example 14.1, in the continuous-time urn, the colours e volve as independent\nYule processes, now with possibly different rates λi“ξii. Hence, generalizing (14.2),\ne´λitXiptqa.s.Ý ÑXiPΓpxi{λi,λiq, (14.6)\nwith allXiindependent.\nConsider the simplest case: q“2, and assume λ1“αandλ2“δwithαąδą0.\nThenλ˚\ni“λi,pλ“α,κi“pκ“γi“0 (i“1,2). It follows from (5.11)–(5.12) and\n(5.17) (or directly from (14.6)) that\nn´δ{αXn2a.s.Ý ÑpX2“αδ{αX2\nXδ{α\n1. (14.7)\nNote that if X„Γpa,bq, then its moments (for arbitrary real r) are given by\nEXr“#\nbrΓpa`rq{Γpaq ă 8,´aără 8,\n8, r ď ´a.(14.8)\nIn particular, since X1andX2are independent, it follows from (14.6)–(14.8) that for\nrą0,\nEpXr\n2ă 8 ð ñ EX´rδ{α\n1 ă 8 ð ñ rδ{αăx1{αð ñrăx1{δ.(14.9)\nConsequently, pX2does not have finite moments of all orders. In particular, we\ncannot always have (finite) moment convergence in (14.7). Ta king, for example,\nα“2,δ“1, andx1“x2“1 we see that not even the mean EpX2is finite; hence\nwe cannot have convergence in L1orL2in Theorem 1.8.\nAs far as we know, it is an open problem to find asymptotics of mo mentsEXr\nn2\nfor general rą0 in this (simple) example. △\nExample 14.3. Consider a two-colour urn with a deterministic replacement matrixˆ\nδ γ\n0α˙\n. (14.10)\n(We have chosen anotation agreeing with [28], although colo urs aretaken in different\norder thereandthusthematrices arewritten differently.) Th isurn(and special casesALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 51\nof it) have been studied in many papers; in particular, [28] g ives a detailed study of\nlimits in distribution. The balanced case α“δ`γwith integers α,γ,δis studied\nby very different methods (generating functions) in [43] and [ 17]. A.s. convergence\nhas been shown in special cases in [18; 19; 11; 12] ( α“δ`γ), and [1] ( αěδ).\nSuppose that δą0,γą0,w0“x1ą0, andb0“x2ě0; suppose also either\nαě0, orα“ ´1 together with γPZ`andx2PZě0. Then the urn satisfies (A1)–\n(A5) ifαě0, and (A1)–(A4), (A51), and (A7)–(A8) for all α. Hence, Theorems 1.8\nand 4.1 apply if αě0, and Theorems 8.4 and 8.6 apply for any α; consequently, the\nconclusions of Theorems 1.8 and 4.1 hold for all cases.\nWe have λ1“δandλ2“α. Furthermore, since γą0, we have 1 Ñ2. (Thus 1\nis the only minimal colour.) Hence, λ˚\n1:“δ,λ˚\n2:“α_δand thus pλ“λ˚\n2“α_δ;\nfurthermore κ1“0 whileκ2“1 whenα“δandκ2“0 otherwise. We consider\nseveral cases.\nCase 1,αăδ:\nThenλ˚\n1“λ˚\n2“pλ“δą0; furthermore, κ1“κ2“0“pκ, and (2.13) yields\nγ1“γ2“0. Consequently, Theorem 1.8(i) or Theorem 8.4 yields\nXni\nna.s.Ý ÑpXi, i “1,2. (14.11)\nFurthermore, 1 is the only leader, and thus Theorem 7.2 shows thatpX1andpX2are\nconstants. To find them, we can use Lemma 6.7. By Lemma 6.1 (sim plifying the\nnotation), Xi“ciX1, where obviously c1“1, and Lemma 6.7 gives the eigenvalue\nequation\npc1,c2qˆ\nδ γ\n0α˙\n“δpc1,c2q, (14.12)\ni.e.,γ`αc2“δc2, with the solution c2“γ{pδ´αq. In other words,\nX2“γ\nδ´αX1. (14.13)\nThis follows also directly from Lemma 3.5 and (3.35).\nIf we add a dummy colour 0 as in Section 5, then (5.17) and (14.1 3) yield\nX0“δ´1pX1`X2q “1\nδγ`δ´α\nδ´αX1. (14.14)\nHence, by (5.12),\npX1“X1\nX0“δpδ´αq\nγ`δ´α, (14.15)\npX2“X2\nX0“δγ\nγ`δ´α. (14.16)\nConsequently, as nÑ 8we have\nXn1\nna.s.Ý ÑpX1“δpδ´αq\nγ`δ´α, (14.17)\nXn2\nna.s.Ý ÑpX2“δγ\nγ`δ´α. (14.18)52 SVANTE JANSON\nThis is in agreement with [28, Theorem 1.3(i)-(iii) and Lemm a 1.2], which give the\nasymptotic distribution of the difference Xni´npXidivided by the correct normal-\nization factor, which implies (and is much more precise than ) convergence in proba-\nbility in (14.17)–(14.18). (The normalization factor is n1{2forαăδ{2,pnlognq1{2\nforα“δ{2 andnα{δforδ{2ăαăδ. Moreover, the distribution is asymptotically\nnormal for αďδ{2, but not for δ{2ăαăδ. See [28] for details.)\nCase 2,α“δ:\nThenλ˚\n1“λ˚\n2“pλ“δą0; furthermore, κ1“0 andκ2“1“pκ, and thus (2.13)\nyieldsγ1“ ´1 andγ2“0. Consequently, Theorem 1.8(i) yields\nXn1\nn{logna.s.Ý ÑpX1, (14.19)\nXn2\nna.s.Ý ÑpX2. (14.20)\nFurthermore, also in this case, 1 is the only leader, and thus Theorem 7.2 shows\nthatpX1andpX2are constants. However, unlike Case 1, λ2“λ˚\n2and thus 2 is now\na subleader. Again, Lemma 6.1 shows that X2“c2X1, where (6.20) in Lemma 6.7\nimmediately yields\nc2“a1r12\nκ2c1“γ. (14.21)\nFurthermore, (5.15) now yields Q˚“ t2u, and thus (5.17) yields\nX0“pλ´1X2“δ´1X2. (14.22)\nConsequently, recalling (5.12), the limits in (14.19)–(14 .20) are\npX1“pλX1\nX0“δ2X1\nX2“δ2\nc2“δ2\nγ, (14.23)\npX2“X2\nX0“δ. (14.24)\nThis is in agreement with the result on the asymptotic distri bution in [28, Theorem\n1.3(iv) and Lemma 1.2], which implies convergence in probab ility in (14.19)–(14.20).\nCase 3,αąδ:\nThen 0 ăλ˚\n1“δăλ˚\n2“α“pλ; furthermore, κ1“κ2“0“pκ, and thus\nγ1“γ2“0. Consequently, Theorem 1.8(i) yields (see also [1])\nXn1\nnδ{αa.s.Ý ÑpX1, (14.25)\nXn2\nna.s.Ý ÑpX2. (14.26)\nIn this case, both colours 1 and 2 are leaders; hence Theorem 6 .4 shows that X1and\nX2are absolutely continuous, also jointly. Furthermore, The orem 7.1(ii) shows that\npX1is absolutely continuous, while Theorem 7.2 shows that pX2is deterministic. We\nhave again Q˚“ t2u, which by (5.17) now yields\nX0“pλ´1X2“α´1X2. (14.27)\nHence, (5.12) yields\npX1“X1\nXδ{α\n0“αδ{αX1\nXδ{α\n2, (14.28)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 53\npX2“X2\nX0“α. (14.29)\nHowever, the formula (14.28) for pX1does not seem to be of much use to find the\ndistribution of pX1. This distribution was found by other methods in [28, Theore m\n1.3(v)], which yields convergence in distribution of Xn1{nδ{α; the a.s. convergence in\n(14.25) is a stronger result, and the distribution of pX1is thus the limit distribution\nfound in [28]. This limit distribution is characterized in [ 28], but no simple form\nis known in general; it is shown in [28, Theorem 1.6] that pX1has moments of all\norders, and a complicated integral formula is given for thes e moments. Except in\nthe balanced case below, it is, as far as we know, an open probl em whether moments\nconverge in (14.25) (to these limits) or not.\nCase 4,α“δ`γ:\nThe balanced case of the two-colour urn studied here is α“δ`γ; by our assump-\ntionγą0, this is a special case of Case 3, and thus (14.25)–(14.29) h old. In this\nspecial case, pX1can be characterized by its moments, for which there is a simp le\nformula, see (for integers α,γ,δ,r) [43, Theorem 2.9], [17, Proposition 17], and (for\nthe general case) [28, Theorem 1.7]:\nEpXr\n1“δrΓ`\npx1`x2q{α˘\nΓpx1{δ`rq\nΓpx1{δqΓ`\npx1`x2`rδq{α˘, r ą0. (14.30)\nIt follows that the moment generating function EetpX1is finite for all real t, and\nthus the moments (14.30) (even for integer r) determine the distribution. Moreover,\nTheorem 12.5 shows that all moments converge in (14.25) (to t he limits (14.30)); this\nwas earlier shown in [17; 43] in the case that the replacement sδ,γ,αare integers.\nIf we further assume x2“0, so we start only with white balls, then pX1has a\ndensity function that can be expressed using the density fun ction of a Mittag-Leffler\ndistribution with parameter δ{α, or aδ{α-stable distribution, see [28, Theorem 1.8]\nand [17, Proposition 16].\nThe continuous-time processes WptqandBptqare studied by [13], which includes\nour Theorem 4.1 for this balanced urn. △\nExample 14.4. Generalizing Example14.3, considerageneral triangular t wo-colour\nurnUwith random replacement matrixˆ\nξ11ξ12\n0ξ22˙\n. (14.31)\nSuch urns have been studied by Aguech [1], who proved (among o ther results) the\nexistence of a.s. limits under some assumptions (including our (A5), Eξ22ěEξ11,\nand an unnecessary independence assumption). We extend thi s result as follows.\nAssume ξ11,ξ12,ξ22PL2(Condition (A4)), and let δ:“Eξ11,γ:“Eξ12,α:“\nEξ22, so that`δ γ\n0α˘\nis the mean replacement matrix. Suppose that (A51) holds,\nthatξ11ě0 a.s., and that, as in Example 14.3, δą0,γą0,w0“x1ą0, and\nb0“x2ě0. Then the urn satisfies (A1)–(A5) if ξ22ě0 a.s., and (A1)–(A4),\n(A51), and (A7)–(A8) in any case. Hence, Theorems 8.4 and 8.6 appl y, and thus the\nconclusions of Theorems 1.8 and 4.1 hold for all α.\nAs in Section 9, we let U1denote the mean urn with replacement matrix`δ γ\n0α˘\n;\nthis urn is of the type in Example 14.3. As discussed in Sectio n 9, all parameters\nλi,λ˚\ni,pλ,...are the same for UandU1, and thus all results are qualitatively the same54 SVANTE JANSON\nfor the two urns. Hence, all results in Example 14.3 hold for t he urnUwith random\nreplacements too, except any assertions on the precise dist ributions of the limits\n(including the moment formula (14.30), which as shown in Exa mple 14.5 below is\nnotvalid in general for random replacements). Note, however, t hat by Theorem 9.1,\nin all cases in Example 14.3 with a deterministic limit pXifor the mean urn, we have\nthe same limit for the urn U. △\nExample 14.5. Consider a two-colour urn Uwith the random replacement matrixˆ\nξ1´ξ\n0 1˙\n, (14.32)\nwhereξPBeppqfor some pP p0,1q. In other words, if we draw a white ball, we add\nanother ball that is white with probability p, and otherwise black; if we draw a black\nball we always add another black ball. (As usual, we also alwa ys return the drawn\nball.) Note that this urn is balanced, in spite of replacemen ts being random.\nThis urn appears in several applications, see Examples 14.6 and 14.10 for two of\nthem.\nThis urn is a special case of Example 14.4, and a.s. convergen ce for the urn follows\nfrom Theorem 1.8. Moment convergence follows from Theorem 1 2.5.\nThe mean replacement matrix is\nprijq2\ni,j“1“ˆ\np1´p\n0 1˙\n. (14.33)\nBy Section 9, the asymptotic behaviour of the urn Uis qualitatively the same as\nfor the mean urn U1with the replacement matrix (14.33), which is an instance of\nExample 14.3, more precisely the balanced Case 4. In particu lar, since (14.29) shows\nthatpX2“α“1 is constant for the mean urn U1, the same holds for the original urn\nUby Theorem 9.1.\nWe may also relate the urn Uto the mean urn U1in another, more direct, way.\nConditioned on the contents pXn1,Xn2qof the urn at time n, the probability that\nthe next added ball is white is, letting ζbe the colour of the drawn ball,\nP`\nζ“1|Xn1,Xn2˘\n¨p`P`\nζ“2|Xn1,Xn2˘\n¨0“pXn1\nXn1`Xn2.(14.34)\nHence, if we define\nYn1:“pXn1, (14.35)\nYn2:“ p1´pqXn1`Xn2, (14.36)\nand note that Yn1`Yn2“Xn1`Xn2, we see from (14.34) that we may regard\ntheaddedball inUas thedrawnball in an urn with composition Yn“ pYn1,Yn2q.\nAdding a white ball to pXn1,Xn2q(i.e., increasing Xn1by 1) means by (14.35)–\n(14.36) adding pp,1´pqtopYn1,Yn2q, while adding a black ball to pXn1,Xn2qmeans\nadding p0,1qtopYn1,Yn2q. Consequently, the stochastic process pYnqně0describes\na P´ olya urn with the replacement matrix`p1´p\n0 1˘\n, which is the same as (14.33)\nfor the mean urn U1above. Note, however, that the initial conditions now are, b y\n(14.35)–(14.36),\ny1“px1, y 2“ p1´pqx1`x2. (14.37)\nBy Example 14.3, we have\nYn1{npa.s.Ý ÑpY1 (14.38)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 55\nwhere the limit by (14.30) has moments, recalling (14.37),\nEpYr\n1“prΓpy1`y2qΓpy1{p`rq\nΓpy1{pqΓpy1`y2`rpq“prΓpx1`x2qΓpx1`rq\nΓpx1qΓpx1`x2`rpq.(14.39)\nHence, by (14.35), we have in the urn Uwith replacements (14.32)\nXn1{npa.s.Ý ÑpX1 (14.40)\nwithpX1“p´1pY1, and thus\nEpXr\n1“p´rEpYr\n1“Γpx1`x2qΓpx1`rq\nΓpx1qΓpx1`x2`rpq, r ě0. (14.41)\nBy comparing (14.40) with (14.30) for the mean urn, we see tha t the means EpX1\nare the same for the two urns, while for the second moment, (14 .41) and (14.30)\nyield, for the urn Uand its mean urn U1, respectively,\nEpX2\n1“x1px1`1qΓpx1`x2q\nΓpx1`x2`2pq,EpX2\n1“x1px1`pqΓpx1`x2q\nΓpx1`x2`2pq.(14.42)\nThe variance is thus larger for the urn Uwith random replacement. (Perhaps not\nsurprisingly.) This shows that an urn and its mean urn in gene ral have different\nasymptotic distributions, although the qualitative behav iour is the same as shown\nin Section 9. △\nExample 14.6. One example where the urn in Example 14.5 with replacement\nmatrix (14.32) appears is that it describes the size of the ro ot cluster for bond\npercolation (with parameter p) on the random recursive tree (with vertices in the\nroot cluster coloured white and all other vertices black, an d the initial vector p1,0q).\n(See the argument in the generalization Example 14.7 below. The root cluster is\nstudied by other methods in [6], [35], [5], [14].) In this cas e we obtain (14.40) where\n(14.41) yields\nEpXr\n1“Γpr`1q\nΓp1`rpq, r ě0, (14.43)\nwhich means that pX1has a Mittag-Leffler distribution; this was proved by Baur and\nBertoin [6] (using other methods). △\nExample 14.7. Baur [5] and Desmarais, Holmgren, and Wagner [14] considere d\n(among other things) the root cluster in bond percolation on a preferential attach-\nment tree, generalizing Example 14.6. The tree is defined as f ollows, for a real\nparameter α. Construct the rooted tree Tnwithnvertices recursively, starting with\nT1being just the root and adding vertices one by one; each new ve rtex is attached to\na parent vchosen among existing vertices with probability proportio nal toαdpvq`1,\nwheredpvqis the current outdegree of v. We also perform bond percolation, and\nlet each edge by activewith probability pP p0,1q, independently of all other edges.\n(Note that both active and passive edges are counted in the ou tdegree.) A vertex is\nactive if it is connected to the root by a path of active edges. LetZnbe the number\nof active vertices in Tn.\nThe case α“0 gives the random recursive tree in Example 14.6. We conside r\nhere the case αě0 (as assumed in [5]), and study the modifications for αă0 in\nthe following example.56 SVANTE JANSON\nWe model this process by an urnwith two colours, whereeach ve rtexvcontributes\nαdpvq `1 balls, which are white if the urn is active and black otherwi se. To find the\nparent of the next vertex corresponds to drawing a ball from t he urn; if the ball is\nwhite then the parent is active and the new vertex becomes act ive with probability\np(and otherwise passive); if the ball is black then the parent is passive and the new\nvertex always becomes passive. Since the outdegree of the pa rent increases by 1, we\naddαballs of the same colour as the drawn ball, plus one ball for th e new vertex\nwhich is white if the new vertex is active, i.e., with probabi litypif the drawn ball is\nwhite, and otherwise black. Hence, the urn is a P´ olya urn wit h random replacement\nmatrix\nˆ\nα`ξ1´ξ\n0α`1˙\n, (14.44)\nwhereξPBeppq. We start with 1 active vertex, and thus the urn starts with a s ingle\nwhite ball, i.e., x“ p1,0q.\nIfα“0 (the random recursive tree), we get again (14.32), as discu ssed in Exam-\nple 14.6. In general, unlike Example 14.6, the number of acti ve vertices Znis not\ndirectly reflected by the contents of the urn. However, if Nn1is the number of drawn\nwhite balls, then this is the number of vertices that get an ac tive parent; hence Nn1\nis the total outdegree of the active vertices, and therefore the number of white balls\nin the urn is\nWn“Xn1“αNn1`Zn. (14.45)\nThe replacement matrix (14.44) shows that the urnis of the ty pe in Example 14.4;\nfurthermore, it is balanced with balance α`1. We have λ˚\n1“λ1“Eξ11“α`p\nandpλ“λ2“α`1; further κ1“κ2“0. Theorem 1.8 yields\nXn1\nnpα`pq{pα`1qa.s.Ý ÑpX1. (14.46)\nMoreover, Theorem 11.4 yields\nNn1\nXn1a.s.Ý Ñ1\nλ˚\n1“1\nα`p, (14.47)\nand thus (14.45) yields\nZn\nXn1“Xn1´αNn1\nXn1a.s.Ý Ñ1´α\nα`p“p\nα`p. (14.48)\nHence, (14.46) yields\nZn\nnpα`pq{pα`1qa.s.Ý ÑZ:“p\nα`ppX1, (14.49)\nwhere the limit is in p0,8qa.s. The limits (14.46) and (14.49) hold also in Lrfor\nanyră 8by Theorems 12.5 and 12.8; hence all moments converge.\nThis complements Baur [5, Proposition 4.1], who shows L2-convergence in (14.49)\nand gives the first two moments of the limit (by different but rel ated methods), and\nDesmarais, Holmgren, and Wagner [14] who prove convergence of all moments in\n(14.49) and give a recursion for the moments of the limit. (Th e distribution of the\nlimit is not known explicitly.) △ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 57\nExample 14.8. In Example 14.7, we assumed αě0. However, the results extend\neasily to the case αă0. (This case was included in [14].) In this case, as is well\nknown, we must have α“ ´1{dwithdě1 an integer; the random tree Tnthen\nis a random d-ary recursive tree [15, Section 1.3.3]. (The case d“1 is trivial, and\nwe assume dě2.) In this case the replacements (14.44) do not satisfy (A51), since\nα`ξmay take the non-integer negative value ´1{d. This is can be remedied as in\nRemark 8.1 by multiplying the number of white balls by dand adjusting the activity\na1, but in the present case we find it simpler to multiply the numb er of balls of both\ncolours by dand keep the activities ai“1. This gives the replacement matrix\nˆ\ndξ´1d´dξ\n0d´1˙\n, (14.50)\nwith the initial state x“ pd,0q. In (14.45), we have to replace Xn1byXn1{d, which\nyields\nZn“Xn1`Nn1\nd. (14.51)\nWe have λ˚\n1“λ1“Eξ11“dp´1, so (A7) requires dpą1. (In fact, it is easily\nseen that if dpď1, thenZna.s.Ý ÑZ8ă 8[14].) Assuming dpą1, this urn satisfies\n(A1)–(A4), (A51), and (A6)–(A7). It does not satisfy (A8), but Lemma 10.9 sho ws\nthatX0ą0 a.s., and thus we obtain by Theorem 8.5 the a.s. limit (14.49 ) again\n(withpX1replaced by pX1{d); this can be written\nZn\nnpdp´1q{pd´1qa.s.Ý ÑZ:“p\ndp´1pX1. (14.52)\nMomentconvergence, earliershownby[14], followsfromThe orems12.5and12.8. △\nExample 14.9. In Example 14.7, we studied the number Znof vertices in the\npreferential attachment tree Tnsuch that the path to the root contains only active\nvertices. More generally, let Zpkq\nnbe the number of vertices such that this path\ncontains exactly kě0 passive edges. For fixed k, this can be treated similarly, with\nan urn with qěk`2 colours 1 ,...,q, where vertices with jpassive edges on the\npath to the root are represented by colour max pj`1,qq. For example, for q“3,\nthis leads to an urn with replacement matrix\n¨\n˝α`ξ1´ξ0\n0α`ξ1´ξ\n0 0 α`1˛\n‚, (14.53)\nwhereξPBeppqas above. (It does not matter whether we write this with the sa me\nξon both rows or not; recall Remark 1.4.) For a general qě2 we have ξii“α`ξ\nandξi,i`1“1´ξfor 1 ďiďq´1,ξqq“α`1, and all other ξij“0. The urn is\nbalanced with balance b“α`1. We assume for simplicity αě0; the case αă0\ncan be treated as in Example 14.8.\nWe findλ1“ ¨ ¨ ¨ “λq´1“α`pandλq“α`1, and thus λ˚\n1“ ¨ ¨ ¨ “λ˚\nq´1“α`p,\npλ“λ˚\nq“α`1,κi“i´1 for 1 ďiďq´1, andκq“0. We have in analogy with\n(14.45), Zpkq\nn“Xn,k`1´αNn,k`1forkďq´2. Theorem 1.8 applies and shows\ntogether with Theorem 11.4 as in Example 14.7, an a.s. limit. Sinceκk`1“k, we58 SVANTE JANSON\nnow get\nZpkq\nn\nnpα`pq{pα`1qlogkna.s.Ý ÑZpkq:“p\nα`ppXk`1. (14.54)\nFurthermore, we obtain from (7.3) and induction, since ri,i`1“Ep1´ξq “1´pfor\neveryiďq´1,\nZpkq“1\nk!ˆ1´p\nα`1˙k\nZp0q, (14.55)\nwhereZp0qequalsZin (14.49). Consequently, the numbers Zpkq\nnfor different kare\nasymptotically proportional. △\nExample 14.10. Another example wheretheurnin Example14.5 with replaceme nt\nmatrix (14.32) appears is for elephant random walks with del ays. In a standard\nelephant random walk (ERW), the elephant takes steps YnP t˘1u; after an initial\nstepY1, the elephant (which remembers the entire walk) chooses one of thepreceding\nsteps, uniformly at random, and then randomly either (with p robability p) repeats\nit, or (with probability q) takes a step in the opposite direction. (Here q“1´p.) As\nnoted by Baur and Bertoin [7] (to which we refer for details an d further references),\nthis may be modelled by a (non-triangular) P´ olya urn, with o ne white ball for each\nstep `1 and one black ball for each step ´1 taken so far; the replacement matrix isˆ\nξ1´ξ\n1´ξ ξ˙\n(14.56)\nwithξPBeppq. Hence results for the ERW follows from known results for irr educible\nP´ olya urns [7]. (We have nothing to add here.)\nIn theelephant random walk with delays [33; 23; 22], the elephant has a third\npossibility: with probability rit makes a step 0 (i.e., stays put), regardless of the\nremembered step. (Now p`q`r“1; we assume p,q,r ą0.) This can be modelled\nby a 3-colour urn, with colours representing steps `1,´1, and 0, and replacement\nmatrix¨\n˝ζ1ζ2ζ3\nζ2ζ1ζ3\n0 0 1˛\n‚, (14.57)\nwhere pζ1,ζ2,ζ3qis a random vector with exactly one component 1 and the others 0,\nand pPrζi“1sq3\ni“1“ pp,q,r q. This P´ olya urn is neither triangular nor irreducible,\nbut it may be regarded as a combination of two such urns (cf. Re mark 2.3) as\nfollows. Let as before YnP t˘1,0ube thenth step, and let Zn:“ |Yn| P t0,1u;\nZnthus just records whether the elephant moves or stays put. (T he process Znis\ncalledBernoulli elephant random walk in [25]; it has also been studied in [8] and,\nfor somewhat different reasons, in [26].) As noted by [7, V.C], the process pZnqcan\nbe modelled by a P´ olya urn with q“2 and one white ball for each step ˘1 and one\nblack ball for each step 0 so far; the replacement matrix isˆ\n1´ζ3ζ3\n0 1˙\n. (14.58)\nThis is the urn in Example 14.5 with ξ“1´ζ3PBep1´rq. The number of non-zero\nsteps up to time nisWn, the number of white balls in the urn. By conditioning\nonZ1:“ |Y1|, we may assume that Z1is deterministic. Moreover, the case Z1“0ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 59\nis trivial, with Yn“Zn“0 for all ně1; hence we may assume Z1“1, and\nthus the urn starts with 1 white ball. ([22; 24] use a different i nitial condition with\nZ1„Bep1´rq.) Then the urn is the same as in Example 14.6, and Theorem 1.8\nyields\nWn{n1´rÑpX1, (14.59)\nas shown by other methods in [22, Theorem 3.1]; furthermore, [8, Lemma 2.1] and\n[24, Theorem 5.1] show that (14.43) holds (with preplaced by 1 ´randrbys,\nsay), and thus pX1has a Mittag-Leffler distribution (as we saw in Example 14.6).\nMoreover, [8] and [25] show that moment convergence holds in (14.59); this also\nfollows from Theorem 12.5.\nConditioned on the number Wn, the position of the elephant is the same as for\na standard ERW with Wnsteps, and thus the limit results in [8] and [24] for the\nERW with delays may easily be obtained by combining (14.59) a nd the results for\nthe standard ERW obtained by [7] from the urn (14.56); we leav e the details to the\nreader. △\nExample 14.11. The triangular urn with q“3 and balanced deterministic replace-\nments (with all entries integers ě0)\n¨\n˝α β σ ´α´β\n0δ σ ´δ\n0 0 σ˛\n‚ (14.60)\nwas studied by Puyhaubert [43, Section 2.5] and Flajolet, Du mas and Puyhaubert\n[17, Section 10]; the results include convergence in distri bution (after normalization),\nand, in some cases, convergence of all moments with explicit formulas for moments\nof the limits.\nIn particular, they show (in our notation and correcting sev eral typos) that if\nαąδą0 andβą0, then\nXn2{nα{σdÝ ÑpX2 (14.61)\nwhere the limit has moments\nEpXr\n2“ˆαβ\nα´δ˙rΓpx1{α`rqΓp|x|{σq\nΓpx1{αqΓpp|x| `rαq{σq. (14.62)\nIn this case, we have σěα`βąαąδand thus λ˚\n1“λ˚\n2“λ1“α,λ2“δ,\npλ“λ˚\n3“λ3“σ,pκ“κi“0,γi“0 (i“1,2,3). Hence, Theorem 1.8 yields\n(14.61) with the stronger convergence a.s. Moreover, the le aders are 1 and 3, and\nLemma 7.3 yields pX2“pc21pX1, where Lemma 7.3 and (6.18) show that p1,pc21qis a\nleft eigenvector of`α β\n0δ˘\n, with eigenvalue α. Consequently, pc21“β{pα´δqand\npX2“β\nα´δpX1. (14.63)\nFurthermore, as noted in [17], we may in this urnbepartially colour-blind and merge\ncolours 2 and 3; then pXn1,Xn2`Xn3qis a 2-colour urn with replacement matrix`α σ´α\n0σ˘\n; hence the moments EpXr\n1are given by (14.30), where now αandδare\nreplaced by σandα, andx1`x2is replaced by |x| “x1`x2`x3, i.e.,\nEpXr\n1“αrΓ`\n|x|{σ˘\nΓpx1{α`rq\nΓpx1{αqΓ`\np|x| `rαq{σ˘, r ě0. (14.64)60 SVANTE JANSON\nThus, (14.62) follows by (14.63).\nSimilarly, in the case α“δą0 andβą0, [43] and [17] show (again correcting\nseveral typos)\nXn2{pnα{σlognqdÝ ÑpX2 (14.65)\nwhere the limit has moments\nEpXr\n2“ˆαβ\nσ˙rΓpx1{α`rqΓp|x|{σq\nΓpx1{αqΓpp|x| `rαq{σq. (14.66)\nIn this case λ˚\n1“λ˚\n2“λ1“λ2“α,κ1“0 andκ2“1, and as above pλ“λ˚\n3“\nλ3“σandpκ“κ3“0. Hence, Theorem 1.8 yields (14.65) with convergence a.s.\nMoreover, Lemma 7.3 and (7.3) yield\npX2“β\nσpX1, (14.67)\nwhich yields (14.66) by (14.64).\n[17] and [43] further prove moment convergence in (14.61) an d (14.65), which also\nfollows from Theorem 12.5.\nNote that [17] and [43] assume the replacements to be integer s, while we obtain\nthe results above also for non-integer replacements.\nA.s. convergence in (14.61) and (14.65) follows also from [1 2], see Example 14.12.\n△\nExample 14.12. Bose, Dasgupta, and Maulik [12] study a rather general class\nof balanced triangular urns with deterministic ξi; thusξij“rij. They show a.s.\nconvergence of Xnisuitably normalized, as in our Theorem 1.8.\nWe introduce some of the notation from [12]. The colours are t1,...,q u(where\nthey write q“K`1), and the replacement matrix is triangular, so (2.4)–(2.6 ) hold\nwith the natural order ă. The diagonal entries rii“λiiare denoted ri, and the\npositions of the weak maxima in the sequence r1,...,rqare denoted i1,...,iJ`1.\nThusri1ďri2ď...ďriJ`1, andrkărijwhenijăkăij`1. Since the urn is\nbalanced, Lemma 10.7 and Remark 10.8 show that rq“λq“pλis a maximum, and\nthusiJ`1“q. Clearly, i1“1. Thejth block of colours is tij,...,ij`1´1u. (In\n[12], the replacements are normalized by λq“pλ“1, which can be assumed without\nloss of generality. It is also assumed that initially there i s 1 ball in the urn, i.e.,ř\nixi“1; this seems to be a mistake since one cannot in general norma lize both to\n1 simultaneously.)\n[12] says that the colours are arranged in increasing order if for every kP pij,ij`1q\n(with 1 ďjďJ), there exists mP rij,kqsuch that rmką0. Using our terminology,\nthis is easily seen to be equivalent to: If kP pij,ij`1q, thenkis a descendant of ij.\n[12, Proposition 2.1] shows that in every balanced triangul ar urn, the colours can be\nrearranged in increasing order. [Sketch of proof: Construc t the blocks in backwards\norder. In each step find a colour i(to be labelled ij) withλimaximal among the\nremaining colours; let the next block consist of iand all its remaining descendants.\nOrder this block in a suitable way, with ifirst, and place it before the previously\nconstructed blocks. Repeat with the remaining colours.]\nThe main result of [12] further assumes [12, (2.2)], which sa ys that for every\nj“1,...,J, there exists mP rij,ij`1qsuch that rm,ij`1ą0. In our terminology,\nand assuming (as in [12]) that the colours are in natural orde r, this is equivalentALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 61\ntoij`1≻ij. It follows that the assumptions in [12] (i.e., increasing o rder and\ntheir (2.2)) imply that i1,...,iJ`1are precisely the colours iwithλ˚\ni“λi, i.e. our\nleaders and subleaders (see (6.3) and (6.19)), and that the l eadersνhave distinct λ˚\nν.\nMoreover, for each leader ν“ij, the subleaders in Dν(see (6.13)) are ij`1,...,ij`ℓ\nwhereℓ“ℓjě0 is the largest integer with λij`ℓ“λij; these subleaders form a\nchain in the partial order ≺, and thus κij`k“kfork“0,...,ℓ. The set Dκ\nνis an\ninterval rij,ij`1qfrom a (sub)leader to the next.\nThe a.s. convergence in the main result [12, Theorem 3.1] now follows from The-\norem 1.8, Lemma 6.7, and Theorem 7.1. Moreover, [12, Remark 3 .3] says that it is\nclear from their proof that a.s. convergence holds also with out their assumptions of\nincreasing order and their (2.2); [12] thus essentially sta tes our Theorem 1.8 for the\ncase of a balanced urn with deterministic replacements.\nFurthermore, [12] also shows convergence in L2, which is extended to Lpfor any\npby our Theorem 12.5. △\nExample 14.13. Consider an urn with q“4 and replacement matrix\n¨\n˚˚˝αη1´1 0 η20\n0βη1´1η20\n0 0 γ0\n0 0 0 δ˛\n‹‹‚, (14.68)\nwhereα,β,γ,δ are positive integers and η1,η2PBep1{2qare independent. We have\nλ1“α{2´1,λ2“β{2´1,λ3“γ, andλ4“δ. Supposethat λ4“λ1ąλ2ąλ3ą0,\nand start with X0“ p1,1,0,1q.\nThis urn satisfies (A1)–(A4), (A51), and (A6)–(A7), but not (A8). We have pλ“\nλ˚\n1“λ˚\n3“λ˚\n4“λ1,λ˚\n2“λ2, andκi“0@i. Theorem 8.6 shows that (4.1) holds for\nall colours i, with\ne´λ˚\nitXiptqa.s.Ý ÑXi. (14.69)\nFurthermore, X4ą0 a.s., as is seen by considering only balls of colour 4; hence , by\n(5.17),X0“pλ´1pX1`X4q ěpλ´1X4ą0 a.s.\nFurthermore, 3 is a follower of the leader 1, and Lemma 6.1 wit h Remark 8.8\nyieldsX3“cX1for some cą0. (In fact, c“1{2pλ1´λ3q, by the same argument as\nfor (14.13).) Note also that X1andX2are independent.\nIt is obvious that colours 1 and 2 both may die out in a few draws , and that if\nthey do, they may or may not first generate a ball of colour 3. If they do not die\nout, then a.s. X1ą0 andX2ą0 respectively, see Remark 8.10.\nConsequently, the following cases can appear, all with posi tive probabilities:\n(i)X1ą0 and then X3ą0 and thus X3ptqgrows at rate eλ˚\n3t“eλ1t. Similarly,\nby Theorem 8.5 and (5.12), Xn3{na.s.Ý ÑpX3ą0.\n(ii)X1“0 butX2ą0; thenX3“0, but by considering the urn with colour 2, 3,\nand 4 only (after 1 has died out), it follows that e´λ2X3ptqa.s.Ý ÑX1\n3:“c1X2ą0\nfor some c1ą0, and thus X3ptqgrows at rate eλ2t; similarly Xn3{nλ2{λ1a.s.Ý Ñ\npX1\n3ą0.\n(iii)X1“X2“0 and both X1ptqandX2ptqdie out, but at least one of them first\ngets a ball of colour 3 as offspring. Then, by considering only b alls of colour 3\nand 4 (when the others have died out), e´λ3X3ptqa.s.Ý ÑX2\n3ą0, and thus X3ptq\ngrows at rate eλ3t; similarly Xn3{nλ3{λ1a.s.Ý ÑpX2\n3ą0.62 SVANTE JANSON\n(iv)X1“X2“0, and both X1ptqandX2ptqdie out without producing an offspring\nof colour 3. Then, X3ptq “0 for all tand thus X3n“0 for all n.\nThis example shows that in a case when Xi“0 with positive probability, it may\nstill be possible to find precise limit results, but different l imit results may hold in\ndifferent subcases. It seems that this can be done very general ly on a case by case\nbasis, but as said in Remark 8.7 we do not attempt any general s tatement. △\nExample 14.14. An interesting counterexample is given by the replacement m atrix\nˆ\n0 1\n0˘1˙\n, (14.70)\nwhereξ22“ ˘1 denotes a random variable with Ppξ22“1q “Ppξ22“ ´1q “1\n2. In\nwords: when we draw a white ball, it is replaced together with a black ball; when we\ndraw a black ball, we toss a coin and then (with probability1\n2each) either remove\nthe ball or replace it together with another black ball.\nWe have r22“Eξ22“0, and thus λ1“λ2“0“λ˚\n1“λ˚\n2“pλ,κ1“0,\nκ2“1,pκ“1,pκ0“2. Note that this example is excluded from Theorems 8.4–8.6\nsince (A7) does not hold. (However, (A1)–(A4), (A51), and (A6) hold, provided\nx1ą0 andx2PZě0.) We will see that, in fact, we do nothave a.s. convergence of\nBn{n1{2“Xn2{n1{2andBptq{t“X2ptq{tas (1.4) and (4.1) would give; however,\nthese converge in distribution.\nConsider first a continuous time urn with only black balls. Th is is a branching\nprocess where balls live an Exp p1qtime and then randomly either die or are split\ninto two. Let Yptqdenote such an urn with black balls, starting with Yp0q “1, and\ndenote its probability generating function by (for |z| ď1, say)\ngtpzq:“EzYptq. (14.71)\nThen the Kolmogorov backward equation [32, Theorem 12.22] y ields, for tą0,\nB\nBtgtpzq “1\n2`\n1`gtpzq2˘\n´gtpzq “1\n2`\n1´gtpzq˘2(14.72)\nwithg0pzq “z, which has the solution\ngtpzq “1´´1\n1´z`t\n2¯´1\n“tp1´zq `2z\ntp1´zq `2“t\nt`2`2\nt`2¨2\nt`2z\n1´t\nt`2z.(14.73)\nConsequently, Yptqhas a modified geometric distribution: PpYptq “0q “gtp0q “\nt\nt`2and the conditional distribution`\nYptq |Yptq ą0˘\nis Ge`2\nt`2˘\n. In particular,\nPpYptq “0q Ñ1 astÑ 8; since 0 is an absorbing state, it follows that a.s. Yptq “0\nfor sufficiently large t; in other words, Yptqdies out. (This follows also since Yptq\nis a time-changed simple random walk, absorbed at 0.) A simpl e calculation yields\nEYptq “1 and Var Yptq “t, in accordance with (8.4)–(8.5) and (8.11). Note that`\nt´1Yptq |Yptq ą0˘dÝ ÑExp`1\n2˘\nastÑ 8. Roughly speaking, for large t,Yptqis\nnon-zero with probability «2{t, and if it is, it is of order t.\nNowconsiderthetwo-colour urnabove, andassumethatwesta rtwithX0“ p1,0q,\ni.e., 1 white ball. The number of white balls is constant for t his urn, and thus\nWptq “1 for all tin the continuous-time urn. This means that white balls are\ndrawn according to a Poisson process Ξ with constant rate 1. I f the times they areALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 63\ndrawn are pTkq8\n1, then we have\nBptq “ÿ\nTkďtYkpt´Tkq, (14.74)\nwhereYkare independent copies of the one-colour process Yptq, independent also of\npTkq8\n1. Consequently, for zP r0,1ssay, we have\nE`\nzBptq|Ξ˘\n“ź\nTkďtgt´Tkpzq “exp´ÿ\nTkďtloggt´Tkpzq¯\n“exp´żt\n0loggt´upzqdΞpuq¯\n. (14.75)\nHence, by a standard formula for Poisson processes [32, Lemm a 12.2],\nEzBptq“Eexp´żt\n0loggt´upzqdΞpuq¯\n“exp´żt\n0`\ngspzq ´1˘\nds¯\n“exp´\n´żt\n0´1\n1´z`s\n2¯´1\nds¯\n“exp´\n´2´\nlog´1\n1´z`t\n2¯\n´log´1\n1´z¯¯¯\n“´\n1`t\n2p1´zq¯´2\n“´2\nt`2\n1´t\nt`2z¯2\n. (14.76)\nConsequently, Bptqhas a negative binomial distribution NegBin`\n2,2\nt`2˘\n. In partic-\nular,\nEBptq “t, (14.77)\nwhich also follows directly from (14.74).\nIt follows from (14.76) that as tÑ 8, for any są0,\nEe´sBptq{t“´\n1`t\n2p1´e´s{tq¯´2\nÑ´\n1`s\n2¯´2\n(14.78)\nand thus Bptq{tconverges in distribution to a Gamma distribution:\nt´1BptqdÝ ÑΓp2,1\n2q,astÑ 8. (14.79)\nHowever, we will see that Bptq{tdoesnotconverge a.s., which shows that Theo-\nrem 8.6 does not extend to this example.\nTo see this, we extend (14.79) to process convergence. We cla im that as tÑ 8,\nwe have\nt´1BptxqdÝ ÑBpxq:“1\n4BESQ4pxqinDr0,8q, (14.80)\nwhere BESQ4pxqdenotes a squared 4-dimensional Bessel process [44, Chapte r XI].\nRecall that\nBESQ4pxq “ |Wpxq|2“4ÿ\ni“1Wipxq2, (14.81)\nwhereW1pxq,...,W4pxqare independent standard Brownian motions (Wiener pro-\ncesses), and Wpxq:“`\nW1pxq,...,W4pxq˘\nthus is a 4-dimensional Brownian mo-\ntion. Hence,1\n4BESQ4pxq „Γp2,x{2q, in accordance with (14.79). Furthermore, a.s.\nBESQ4pxq ą0 for every xą0.\nThe proof of (14.80) is somewhat technical and is given in App endix C, where we\nalso extend the result to other initial values pw0,b0q, see Theorem C.1.64 SVANTE JANSON\nSuppose now that Bptq{ta.s.Ý ÑZfor some random variable Z; thenZ„Γp2,1\n2qby\n(14.79). Moreover, for every rą0, we would have\nt´1`\nBprtq ´rBptq˘\n“r´Bprtq\nrt´Bptq\nt¯a.s.Ý ÑrpZ´Zq “0. (14.82)\nHowever, the process convergence in (14.80) implies finite d imension convergence,\nand in particular\nt´1`\nBprtq ´rBptq˘dÝ Ñ1\n4`\nBESQ4prq ´rBESQ4p1q˘\n. (14.83)\nHence, (14.82) would imply BESQ4prq “rBESQ4p1qa.s., for every rą0, which\nobviously is false (even for a single r‰1). This contradiction proves the claim that\nBptq{tdoes not converge a.s. Hence, the convergence in (14.79) hol ds in distribution\nbut not a.s.\nWe use (14.80) to derive corresponding results for the discr ete-time urn. Let, as\nabove,pTnbe thenth time that a ball is drawn, and let Nptqbe the total number of\ndraws up to time t; thusNppTnq “n. Since all balls are drawn with intensity 1,\nrNptq:“Nptq ´żt\n0`\nWpsq `Bpsq˘\nds“Nptq ´t´żt\n0Bpsqds (14.84)\nis a local martingale with rNp0q “0, and it follows as in the proof of Lemma 3.1\n(now using (14.77)) that rNptqis a martingale. In particular ErNptq “0, and thus\nby (14.84) and (14.77)\nENptq “t`Eżt\n0Bpsqds“t`żt\n0EBpsqds“t`t2{2. (14.85)\nFurthermore, all jumps are `1 and thus the quadratic variation is by (2.16)\nrrN,rNst“ÿ\n0ăsďt∆Npsq “Nptq. (14.86)\nConsequently, by Doob’s inequality (2.18) and (14.85),\nErN˚ptq2ďCErrN,rNst“CENptq “Ct`Ct2. (14.87)\nIn particular, rN˚ptq{t2pÝ Ñ0 astÑ 8, which together with (14.84) implies that\nNptxq{t2´żx\n0t´1Bptyqdy“t´2Nptxq ´t´2żtx\n0Bpsqds“t´2rNptxq `t´1xpÝ Ñ0\n(14.88)\ninDr0,8q, and consequently (14.80) implies\nNptxq{t2dÝ ÑVpxq:“żx\n0Bpyqdy, (14.89)\ninDr0,8q, jointly with (14.80). Note that Vpxqis a continuous stochastic process\nwhich strictly increases from Vp0q “0 to 8. Define τby\nVpτq “1; (14.90)\nthusτis random with 0 ăτă 8a.s. It follows easily from (14.89) (we omit the\ndetails) that, jointly with (14.80),\npTn{?npÝ Ñτ (14.91)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 65\nand as a consequence\nBn{?n“BppTnq{?ndÝ ÑBpτq. (14.92)\nThis proves convergence in distribution of Bn{?n. The limit Bpτqis determined by\n(14.80), (14.89), and (14.90); unfortunately we do not know any simpler description\nof the limit distribution, and we leave it as an open problem t o find one.\nFor the discrete-time urn, we thus have convergence in distr ibution of Bn{n1{2.\n(The exponent 1 {2 equals κ2{pκ0, just as in Theorem 1.8(ii) although we cannot\napply that theorem.) However, we do not have convergence a.s . in (14.92). In fact,\nifBn{n1{2a.s.Ý ÑpZfor some pZ, then for every xą0, astÑ 8,\nBpxtq2\nNpxtq“B2\nNpxtq\nNpxtqa.s.Ý ÑpZ2, (14.93)\nwhile the joint convergence of (14.80) and (14.89) implies\nBpxtq2\nNpxtq“ˆBpxtq\nt˙2\n¨t2\nNpxtqdÝ ÑBpxq2\nVpxq(14.94)\ninDp0,8q, i.e., in Dra,bsfor every 0 ăaăbă 8. Consequently, (14.93) would\nimply that a.s. Bpxq2{Vpxq “pZ2for every xą0, and thus a.s.\nżx\n0Bpyqdy“Vpxq “pZ´2Bpxq2, x ą0. (14.95)\nBut this impossible, for example because (14.95) would impl y thatBpxq, and thus\nBESQ4pxq, is differentiable (and, moreover, linear). This contradict ion shows that\nBn{?ndoes not converge a.s. △\nExample 14.15. A counterexample somewhat similar to Example 14.14 is given by\nthe replacement matrixˆ\n0 1\n0´1˙\n. (14.96)\nIn words: when we draw a white ball, it is replaced together wi th a black ball; when\nwe draw a black ball, we discard it.\nWe have λ1“0 andλ2“ ´1, and thus λ˚\n1“λ˚\n2“0. Note that this example\ntoo is excluded from Theorem 8.6 since (A7) does not hold. (Ho wever, again (A1)–\n(A4), (A51), and (A6) hold, provided x1ą0 andx2PZě0.) We will see that, as in\nExample 14.14, Bptqconverges in distribution but nota.s.\nSupposethat theurnstarts withasinglewhiteball, i.e., X0“ p1,0q. Then, forthe\ncontinuous-time urn, Wptq “1 for allt, and thus white balls are drawn according to\na Poisson process with constant rate 1. At each draw in this Po isson process, we add\na black ball. Black balls live an exponential time with mean 1 , and then disappear.\nConsequently, Bptq, the number of black balls, is a birth-death process where th e\nbirth rate is constant 1 and the death rate equals the number o f particles. (Thus\nµk“1 andλk“kin the standard notation.)\nLetTkbe the time of the kth white draw, and Lkthe life-length of the black ball\nthat then is added. Then the pairs pTk,Lkqform a Poisson process in R2\n`with rate\ne´ydxdy. The number of black balls at time tis\nBptq “8ÿ\nk“11tTkďtu1tLkąt´Tku“8ÿ\nk“11tpTk,LkqPDtu (14.97)66 SVANTE JANSON\nwhereDt:“ tpx,yq PR2: 0ăxďt, yąt´xu. Consequently, Bptqhas a Poisson\ndistribution Po pµptqq, where\nµptq “ż\nDte´ydxdy“żt\n0ż8\nt´xe´ydydx“żt\n0ex´tdx“1´e´t.(14.98)\nAstÑ 8, we thus have convergence in distribution BptqdÝ ÑPop1q. Obviously,\nwe cannot have convergence a.s., since Bptqjumps `1 at every Tk, andTkÑ 8as\nkÑ 8.\nBndoes not converge in distribution as nÑ 8for a simple parity reason: since\nBnchanges by ˘1 at each draw, we have Bn”npmod 2 q. However, Bnis a Markov\nchain (since Wnis constant), it is irreducible, and it is easily seen that th e expected\ntime to return to 0 is finite; hence the Markov chain Bnis positive recurrent. The\nperiod is 2, and and it follows that the two subsequences B2nandB2n`1converge in\ndistribution to some limits, say pBevenandpBodd. The mixture (with equal weights)\nofpBevenandpBoddhas a stationary distribution for the Markov chain, and it is easily\nfound that the limit distributions are given by\nPppBeven“kq “k`1\nk!e´11tkis even u, (14.99)\nPppBodd“kq “k`1\nk!e´11tkis odd u. (14.100)\nThese can be described as Po p1qrounded up to nearest even or odd integer, respec-\ntively.\nObviously, we do not have convergence a.s., even for these su bsequences. △\nExample 14.16. Consider a diagonal urn with q“2 and replacement matrix´\nξ110\n0ξ22¯\nas in [2]; cf. the deterministic case in Example 14.2. Assume thatξ11,ξ22P\nZě0a.s., that Eξ11“Eξ22“1, and that Eξ2\n22ă 8butEξ11logξ11“ 8, so\nthat (A4) does nothold. (We may simply take ξ22“1 a.s.) Then the stochastic\nprocesses X1ptqandX2ptqare independent Markov branching processes, and [4,\nTheorem III.7.2] shows that, as tÑ 8,\ne´tX1ptqa.s.Ý Ñ0. (14.101)\nHence, (4.1) holds with X1“0 a.s., while Theorem 4.1 (applied to colour 2 only),\nor [4, Theorem III.7.2] again, shows that (4.1) holds also fo ri“2 withX2ą0 a.s.\nIt follows that X1ptq{X2ptqa.s.Ý Ñ0 astÑ 8, and thus Xn1{Xn2a.s.Ý Ñ0 asnÑ 8. It\nfollows easily that (1.3) holds, which in this case is Xni{na.s.Ý ÑpXi, withpX1“0 and\npX2“1; we thus have X1“pX1“0, in contrast to Theorems 1.8 and 4.1.\nThis shows that the main results in the present paper do not ho ld without assum-\ning at least Eξijlogξijă 8. We have for convenience assumed the stronger second\nmoment condition (A4), but as said in Remark 2.1, we conjectu re that it can be\nweakened. △\nAcknowledgement. I thank Allan Gut for help with references.\nAppendix A.Absolute continuity and conditioning\nIn this appendix we state three general lemmas on absolute co ntinuity of distri-\nbutions and conditioning. We find them intuitively almost ob vious, but only almost,ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 67\nand since we do not know any references, we provide complete p roofs. For two mea-\nsuresµandλon the same space, we let µ!λdenote that µis absolutely continuous\nwith respect to λ, i.e., that λpBq “0ù ñµpBq “0.\nWe recall some further standard definitions:\nA measure space pX,Xqis aBorel space if it is (or is isomorphic to) a Borel set\nin a complete separable metric space with its Borel σ-field, see [32, Appendix A1].\nThis includes, for example, R,Rn, and the function space Dr0,8q; moreover, any\nfinite or countable product of Borel spaces is a Borel space.\nIfpX,Xqand pY,Yqare two measurable spaces, then a probability kernel from\nXtoYis a mapping µ:XˆYÑ r0,1ssuch that BÞѵpx,Bqis a probability\nmeasure on pY,Yqfor every fixed xPX, and furthermore xÞѵpx,Bqis measurable\nonpX,Xqfor every fixed BPY, see [32, p. 20].\nIfXandYare random variables with values in measurable spaces pX,Xqand\npY,Yq, respectively, then a regular conditional distribution ofX, givenY, is a prob-\nability kernel µfromYtoXsuch that for any fixed BPX,\nµpY,B q “PrXPB|Ysa.s., (A.1)\nsee [32, p. 106–107]. It follows that for any measurable f:XÑ r0,8s,\nErfpXq |Ys “ż\nXfpxqµpY,dxqa.s. (A.2)\nSimilarly [32, (7) on p. 108], for any measurable f:XˆYÑ r0,8s,\nEfpX,Y q “Eż\nXfpx,YqµpY,dxq. (A.3)\nIfpX,Xqis a Borel space, then a such a regular conditional distribut ionµexists,\nand the probability measure µpy,¨qisLpYq-a.e. unique (in the standard sense that\ntwo different such kernels are equal for LpYq-a.e.yPY) [32, Theorem 6.3].\nLemma A.1. LetX,Y,Zbe random variables taking values in Borel spaces pX,Xq,\npY,Yq,pZ,Zq, respectively, and suppose that Z“ϕpYqfor some measurable function\nϕ:YÑZ. Letλbe a measure on X, and suppose that the regular conditional distri-\nbutionµpy,¨qofXgivenYis absolutely continuous with respect to λforLpYq-a.e.\nyPY. Then the regular conditional distribution µ1pz,¨qofXgivenZis absolutely\ncontinuous with respect to λforLpZq-a.e.zPZ.\nRemark A.2. Although Lemma A.1 is stated for conditionings on single ran dom\nvariables YandZ, it holds also for conditionings on finite or countably infini te\nsequences of randomvariables (taking values in possiblydi fferent Borel spaces), since\nsuch sequences can be regarded as a single variable in a suita ble product space. △\nProof.IfBĂXis any set with λpBq “0, thenµpY,B q “0 a.s., and thus, by (A.1),\nµ1pZ,B q “Er1BpXq |Zs “E“\nEr1BpXq |Ys |Z‰\n“ErµpY,B q |Zs “0 a.s.\n(A.4)\nHowever, this is for a fixed B, while the conclusion of the lemma is that a.s. (A.4)\nholds simultaneously for every λ-null set BĂX. There is in general an uncountable\nnumber of λ-null sets BĂX, and we do not see how to use the argument in (A.4)\nto prove the result.68 SVANTE JANSON\nInstead, we argue as follows. First, by if necessary changin gµpy,¨qon aLpYq-null\nset ofy, we may assume that\nµpy,¨q !λfor every yPY. (A.5)\nLetνpz,¨qbe the regular conditional distribution of YgivenZ. Then, for any\nBPX, a.s., using (A.1) and (A.2),\nPrXPB|Zs “Er1BpXq |Zs “E“\nEr1BpXq |Ys |Z‰\n“ErµpY,B q |Zs\n“ż\nYµpy,BqνpZ,dyq. (A.6)\nThis shows that (a version of) the regular conditional distr ibutionµ1is given by the\ncomposition of the kernels νandµdefined by\nµ1pz,Bq:“ż\nYνpz,dyqµpy,Bq, B PX; (A.7)\nnote that this composition is a probability kernel, see e.g. the more general [32,\nLemma 1.41(iii)].\nNow, ifλpBq “0, then (A.5) shows that µpy,Bq “0 for every y, and hence (A.7)\nyieldsµ1pz,Bq “0 for every zPZ. Consequently, µ1pz,¨q !λfor every zPZ./square\nLemma A.3. LetXandYbe random variables taking values in Borel spaces pX,Xq\nand pY,Yq, respectively. Let λandλ1beσ-finite measures on XandY, respectively.\nSuppose that LpYq !λ1and that the regular conditional distribution µpy,¨qofX\ngivenYsatisfiesµpy,¨q !λforLpYq-a.e.yPY. Then the distribution of pX,Y qin\nXˆYis absolutely continuous with respect to λˆλ1.\nProof.By if necessary changing µpy,¨qon aLpYq-null set of yPY, we may assume\nthat (A.5) holds.\nLetBĂXˆYwithλˆλ1pBq “0. ForyPY, letBy:“ txPX:px,yq PBu. Let\nA:“ tyPY:λpByq ą0u. By Fubini’s theorem,\n0“λˆλ1pBq “ż\nXˆY1Bpx,yqdλpxqdλ1pyq “ż\nYλpByqdλ1pyq(A.8)\nand thus λpByq “0 forλ1-a.e.y, i.e.,λ1pAq “0. Since LpYq !λ1, this implies\nPpYPAq “0. (A.9)\nFurthermore, by (A.3),\nPrpX,Y q PBs “E1BpX,Y q “Eż\nX1Bpx,YqµpY,dxq “Eż\nX1BYpxqµpY,dxq\n“EµpY,BYq. (A.10)\nIfYRA, thenλpBYq “0, and thus µpy,BYq “0 for every yby (A.5); in particular\nµpY,BYq “0. By (A.9), this shows that µpY,BYq “0 a.s., and thus (A.10) yields\nPrpX,Y q PBs “0. /square\nFor easy reference, we state also an elementary result on abs olute continuity in\nRd, as in the main part of the paper this tacitly means with respe ct to Lebesgue\nmeasure.\nLemma A.4. LetT:RnÑRmbe a linear operator, where 1ďmďn. IfXis a\nrandom vector in Rnwith an absolutely continuous distribution, and Tis onto, then\nthe distribution of TpXqinRmis absolutely continuous.ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 69\nProof.By changes of bases, we may assume that Tis the projection to the first m\ncoordinates. Let λddenote the Lebesgue measure in Rd. IfAĂRmwithλmpAq “0,\nthen\nP`\nTpXq PA˘\n“P`\nXPAˆRn´m˘\n“0, (A.11)\nsinceλnpAˆRn´mq “0. /square\nAppendix B.Lptheory\nThe proofs in this paper frequently use martingales and L2theory, in particular\nthe identity (2.17). In this appendix, we extend the results toLpestimates for\nanypą1 by combining the arguments in Section 3 with the Burkholder –Davis–\nGundy inequalities (see e.g. [32, Theorem 26.12]), which sa y that if pě1, then there\nexist constants c“cppqandC“Cppqsuch that for every (continuous-time) local\nmartingale Mptq\ncErM,M sp{2\ntďEM˚ptqpďCErM,M sp{2\nt,0ďtď 8.(B.1)\n(All constants in this appendix may depend on the exponent p.) We will mainly use\nthe second inequality.\nThis extension to Lpleads to two major results. Using the case pą2, we will\nobtain a proof of Theorem 12.3 (and therefore Theorem 12.5) s howing convergence\ninLpand thus moment convergence under natural conditions. More over, using the\ncase 1 ăpă2, we show that, as said in Remark 2.1, our main results hold al so if\nwe weaken the L2condition (A4) to Lpfor some pą1. More precisely, we will show\nthe following.\nTheorem B.1. Theorems 1.8 and 4.1, and their extensions Theorems 8.4–8.6, a ll\nhold also if (A4)is replaced by the weaker\n(A4p)E|ξij|pă 8for alli,jPQand some pą1.\nAlso other results in this paper, for example the results on t he drawn colours in\nSection 11, hold if (A4) is replaced by (A4p), provided we rep lace any L2-norms by\nLpnorms/bardbl /bardblp; see also Remark 12.4 for the results on moments in Section 12 . We\nleave the details to the reader.\nWe will basically follow the arguments in the main part of the paper, replacing\nL2estimates by Lpestimates, but sometimes the details of the arguments will d iffer.\nMoreover, we have chosen to first focus on obtaining the Lpestimates, leading to the\nproof of Theorem 12.3 (partly becausethis seems to beof grea ter interest for applica-\ntions); we then return to the arguments yielding a.s. conver gence and Theorem B.1.\nAs before, we argue in several steps.\nB.1.A single colour not influenced by others. We begin with a colour ithat is\nnot influenced by any other (i.e., iPQmin), and prove an Lp-version of Lemmas 3.3\nand 8.9.\nLemma B.2. Assume(A1)–(A3),(A51)(or(A5)), and(A4p)for some pą1. Let\niPQmin, and assume\neitheriRQ´(i.e.,ξiiě0a.s.) or λią0. (B.2)\nThen70 SVANTE JANSON\n(i)The martingale e´λitXiptqisLp-bounded, and thus the a.s. limit (3.13)holds\nfor some limit Xi, and\nrX˚˚\ni:“sup\ntě0/visualspace\ne´λitXiptq(\nPLp. (B.3)\n(ii)LetpTkq8\n1be the times a ball of colour iis drawn, and let ηkbe the number of\nballs of colour ithat are added at time Tk. Let0ăqďp, and let f:RÑRbe a\nfunction such that E|fpξiiq|qă 8. Finally, let µą0be such that p1^qqµąλi.\nThen (with a.s. convergent sum)\n8ÿ\nk“1e´µTkfpηkq PLq. (B.4)\nThe statement in (i) that the martingale is Lp-bounded, is (since we have pą1)\nby Doob’s inequality equivalent to (B.3), but we state both f or emphasis. Moreover,\nthe statements are equivalent to Xi:“limtÑ8e´λitXiptq PLp. Note also that the\ndefinition of rX˚˚\niin (B.3) agrees with (3.30) since κi“0 andλ˚\ni“λiwheniPQmin.\nSimilarly, (B.2) is equivalent to (A7) for i, but for later use we prefer the form (B.2).\nProof.Ifλiď0, then by (B.2) we have iRQ´and thus ξiiě0 a.s.; consequently\nξii“0 a.s. and Xiptqis constant so the results are trivial (with an empty sum in\n(B.4)). We thus assume λią0.\nStep 1: (ii)forq“1. Since ηkd“ξiiand is independent of Tk, we have by\nLemma 3.2(ii), letting tÑ 8,\nE8ÿ\nk“1e´µTk/vextendsingle/vextendsinglefpηkq/vextendsingle/vextendsingle“aiE|fpξiiq|ż8\n0e´µsEXipsqds, (B.5)\nwhich is finite by (3.11) and the assumption µąλi.\nStep 2:(ii)forqă1. We have\n´8ÿ\nk“1e´µTk/vextendsingle/vextendsinglefpηkq/vextendsingle/vextendsingle¯q\nď8ÿ\nk“1e´qµTk/vextendsingle/vextendsinglefpηkq/vextendsingle/vextendsingleqPL1, (B.6)\nby Step 1 applied to |f|qandqµ.\nStep 3: If (i)holds for some pą1, then(ii)holds for all qďp. We have already\nproved the case qď1, so we may assume qą1. In particular, E|fpξiiq| ă 8.\nFurthermore, by induction(on rlog2qs), wemay assumethat (ii) holdsif qis replaced\nbyq{2.\nWe consider first two special cases, and then the general one.\n(i)Efpξiiq “0. In this case,\nMptq:“8ÿ\nk“11tTkďtue´µTkfpηkq (B.7)\nis a local martingale with Mp0q “0, since each Tkis a stopping time and fpηkq\nhas mean 0 and is independent of FTk. (Cf.Z2in the proof of Lemma 3.5.) The\nquadratic variation is by (2.16) (cf. (3.59))\nrM,M st“8ÿ\nk“11tTkďtue´2µTkfpηkq2. (B.8)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 71\nWe have fpξiiq2PLq{2, so by the induction hypothesis, we have rM,M s8PLq{2. (If\nq{2ă1, note that pq{2q2µ“qµąµąλi.) Consequently, (B.1) shows that Mis an\nLq-bounded martingale, which yields (B.4).\n(ii)f“cis a constant . It suffices to consider the case c“1. We now define\nMptq:“8ÿ\nk“11tTkďtue´µTk´żt\n0e´λisaiXipsqds (B.9)\nand note that Mptqis a local martingale (Cf. Z3in the proof of Lemma 3.5 and\n(3.55).) The quadratic variation is (cf. (3.57))\nrM,M st“8ÿ\nk“11tTkďtue´2µTk. (B.10)\nHence, in this case too, the induction hypothesis yields rM,M s8PLq{2, and thus,\n(B.1) shows that Mptqis anLq-bounded martingale. Furthermore,\n8ÿ\nk“1e´µTk“Mp8q `ż8\n0e´µsaiXipsqds. (B.11)\nBy assumption, (i) holds, and thus rX˚˚\niPLpĎLqby (B.3). Furthermore, since\nµąλi,\nż8\n0e´µsXipsqdsďż8\n0e´µs`λisrX˚˚\nids“ pµ´λiq´1rX˚˚\niPLq.(B.12)\nThis and (B.11) show thatř\nke´µTkPLq, which is (B.4) in this case.\n(iii)General f. We use the decomposition fpxq “`\nfpxq ´Efpξiiq˘\n`Efpξiiqand\nthe two preceding cases.\nStep 4:(i)holds for all pą1. By induction (on rlog2ps), we may for pą2 assume\nthat (i) holds if pis replaced by p{2.\nAs in the proof of Lemma 3.3, we let Mptq:“e´λitXiptq, so that Mptqis a\nmartingale with quadratic variation (3.18):\nrM,M st“Xip0q2`8ÿ\nk“11tTkďtue´2λiTkη2\nk. (B.13)\nIf 1ăpď2, we apply (ii) with q“p{2ď1; this case holds by Step 1 or Step 2. If\npą2, we apply (ii) with q“p{2 andpreplaced by p{2ą1; this case holds by the\ninduction hypothesis and Step 3. In both cases, we take fpxq “x2and note that\nE|fpξiiq|q“E|ξii|pă 8. Hence, taking µ:“2λi, (B.4) shows that rM,M s8PLp{2.\nConsequently, (B.1)showsthat MisanLp-boundedmartingaleandthat(B.3) holds.\nThis completes the proof. /square\nWe will also need a quantitative version of Lemma B.2(i).\nLemma B.3. Under the assumptions of Lemma B.2, we have\n/vextenddouble/vextenddouble/vextenddoublesup\ntě0/vextendsingle/vextendsinglee´λitXiptq/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble\npďCrXip0qs, (B.14)\nwhereCdoes not depend on Xip0q.72 SVANTE JANSON\nProof.It ought to be straightforward to keep track of the norms of al l quantities in\nthe proof of Lemma B.2, but it seems simpler to argue as follow s. First, if Xip0q “m\nis an integer, then the process Xiptqcan be seen as the sum of mindependent and\nidentially distributed processes Xpkq\niptq,k“1,...,m, each started with Xpkq\nip0q “1\n(but otherwise the same as Xiptq). Hence,\nsup\ntě0/vextendsingle/vextendsinglee´λitXiptq/vextendsingle/vextendsingleďmÿ\nk“1sup\ntě0/vextendsingle/vextendsinglee´λitXpkq\niptq/vextendsingle/vextendsingle, (B.15)\nand (B.14) follows by (B.3) and Minkowski’s inequality.\nIfXip0qis not an integer, let X1\niptqbe an independent copy of Xiptqstarted with\nX1\nip0q “rXip0qs´Xip0q. ThenXiptq `X1\niptqis a copy of the same process started\nwith rXip0qs. Since 0 ďXiptq ďXiptq `X1\niptq, the result follows from the special\ncase just treated. /square\nB.2.A colour only produced by one other colour. Consider now the situation\nin Lemmas 3.5 and 8.11, with two colours iandjsuch that Pi“ tju, and also\nXip0q “0. We fix such iandjthroughout this subsection. (We may repeat the\nassumptions for emphasis.)\nAs in Section 3.2, let 0 ăT1ăT2ă...be the times that a ball of colour jis\ndrawn, and let FTkbe the corresponding σ-fields.\nRecall the notation (3.30). Define also, for µPR,\nκ˚pµq:“$\n’&\n’%κj, µ ăλ˚\nj,\nκj`1, µ “λ˚\nj,\n0, µ ąλ˚\nj.(B.16)\nNote that, by (2.9), κi“κ˚pλiq. (We are mainly interested in the case µ“λi, but\nwe use induction to prove Lemma B.5 below, and we will then nee d more general µ.)\nLemma B.4. LetµPRand define\nVptq:“żt\n0e´µsXjpsqds. (B.17)\nThen\nrV˚˚:“sup\ntě0/visualspace\npt`1q´κ˚pµqe´pλ˚\nj´µq`t|Vptq|(\nďCrX˚˚\nj. (B.18)\nProof.By (3.30) we have, considering the three cases in (B.16) sepa rately,\nVptq ďżt\n0ps`1qκjepλ˚\nj´µqsrX˚˚\njdsďCrX˚˚\njpt`1qκ˚pµqepλ˚\nj´µq`t,(B.19)\nand (B.18) follows. /square\nLemma B.5. Assume (A1)–(A3),(A51)(or(A5)), and(A4p)for some pą1.\nSuppose that i,jPQare such that Pi“ tjuandXip0q “0, and suppose also that\nrX˚˚\njPLp. (B.20)\nLetpζkq8\n1be a sequence of random variables with the same distribution such that ζk\nis independent of FTk. LetµPRbe such that\n#\nµě0,ifiRQ´,\nµ_λ˚\nją0,ifiPQ´,(B.21)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 73\nand let\nZptq:“8ÿ\nk“11tTkďtue´µTkζk. (B.22)\nIf1ăqďpandE|ζk|qă 8, then\nrZ˚˚:“sup\ntě0/visualspace\npt`1q´κ˚pµqe´pλ˚\nj´µq`t|Zptq|(\nPLq. (B.23)\nProof.The proof is similar to Step 5 in the proof of Lemma 3.5 (which e ssentially is\nthe case q“2 of the present lemma), now using (B.1).\nBy induction (on rlog2qs), we may for qą2 assume that the lemma holds for q{2.\nCase 1: Eζk“0. In this case, Zptqis a local martingale, for the same reason as\nMptqin (B.7). Its quadratic variation is, by (2.16) again,\nrZ,Z st“8ÿ\nk“11tTkďtue´2µTk|ζk|2. (B.24)\nWe consider two subcases.\n(i) 1 ăqď2. By (B.1), together with (B.24) and the independence of ζkandTk,\nwe have, since q{2ď1,\nEZ˚ptqqďCErZ,Z sq{2\nt“CE´8ÿ\nk“11tTkďtue´2µTk|ζk|2¯q{2\nďCE8ÿ\nk“11tTkďtue´qµTk|ζk|q. (B.25)\nHence, Lemma 3.2(ii) and (3.30) yield\nEZ˚ptqqďCajEżt\n0e´qµsXjpsqdsďCErX˚˚\njżt\n0ps`1qκjepλ˚\nj´qµqsds.(B.26)\nIn the sequel, we allow constants Cto depend on /bardblrX˚˚\nj/bardblp(which is finite by assump-\ntion); hence we may absorb ErX˚˚\njintoCin (B.26).\nWe consider three subsubcases:\n(i)(a)λ˚\njăqµ. In this case, we may take t“ 8in (B.26) and obtain Z˚p8q PLq.\nThe result (B.23) follows since rZ˚˚ďZ˚p8q.\n(i)(b)λ˚\njěqµandλ˚\nją0. Define, similarly to (3.70),\nrZ:pnq:“sup\nn´1ďtďnpt`1q´κ˚pµqe´pλ˚\nj´µqt|Zptq| ďCn´κ˚pµqe´pλ˚\nj´µqnZ˚pnq.(B.27)\nBy (B.26), we thus have\nErZ:pnqqďCn´qκ˚pµqe´qpλ˚\nj´µqnżn\n0ps`1qκjepλ˚\nj´qµqsds\nďCn1`κj´qκ˚pµqep´qpλ˚\nj´µq`λ˚\nj´qµqn“Cn1`κj´qκ˚pµqe´pq´1qλ˚\njn.(B.28)\nWe have pq´1qλ˚\nją0, and thus (B.28) implies\nEprZ˚˚qqďE8ÿ\nn“1rZ:pnqqă 8, (B.29)74 SVANTE JANSON\nwhich shows (B.23).\n(i)(c)λ˚\njěqµandλ˚\njď0. Then also µď0, and thus µ_λ˚\njď0. By (B.21), we\nmust have iRQ´andµ“0, and then also λ˚\nj“0. Hence, κpµq “κj`1 by (B.16).\nWe define, similarly to (3.74),\nrZ;pnq:“sup\n2n´1ďtď2nt´κ˚pµqe´pλ˚\nj´µqt|Zptq| “sup\n2n´1ďtď2nt´κj´1|Zptq|.(B.30)\nSimilarly to (B.28), it follows from (B.26) that\nErZ;pnqqďC2´qpκj`1qnż2n\n0ps`1qκjdsďC2p1´qqpκj`1qnďC2´pq´1qn.(B.31)\nHence, (B.23) follows by\nEprZ˚˚qqďEZ˚p1qq`E8ÿ\nn“1rZ;pnqqă 8, (B.32)\nsinceEZ˚p1qqă 8by (B.26).\n(ii)qą2. By (B.1), together with (B.24) and the induction hypothes is (forq{2\nand 2µ, usingE|ζ2\nk|q{2ă 8)\nEZ˚ptqqďCErZ,Z sq{2\ntďC`\npt`1qκ˚p2µqepλ˚\nj´2µq`t˘q{2.(B.33)\nAgain, we consider three subsubcases:\n(ii)(a)λ˚\njă2µ. Then κ˚p2µq “0 by (B.16), and thus we may let tÑ 8in (B.33)\nand obtain EZ˚p8qqă 8, which yields (B.23) since rZ˚˚ďZ˚p8q.\n(ii)(b)λ˚\njě2µandλ˚\nją0. Define again rZ:pnqby (B.27). Then, by (B.33),\nErZ:pnqqďCn´qκ˚pµq`q\n2κ˚p2µqep´qpλ˚\nj´µq`q\n2pλ˚\nj´2µqqn\nďCn´qκ˚pµq`q\n2κ˚p2µqe´q\n2λ˚\njn. (B.34)\nHence, we have again (B.29), and thus (B.23).\n(ii)(c)λ˚\njě2µandλ˚\njď0. Again, by (B.21), we must have iRQ´andµ“0, and\nthen also λ˚\nj“0. Hence, κ˚pµq “κ˚p2µq “κj`1. Define again rZ;pnqby (B.30).\nThen, by (B.33),\nErZ;pnqqďC2p´qpκj`1q`q\n2κ˚p2µqqn“C2´q\n2pκj`1qn. (B.35)\nConsequently, (B.32) holds, and thus (B.23).\nCase 2:ζk“cis a constant . We may assume c“1.\nLetVptqbe as in (B.17) and define\nMptq:“Zptq ´ajVptq “8ÿ\nk“11tTkďtue´µTk´ajżt\n0e´µsXjpsqds. (B.36)\nThenMptqis a local martingale by the same argument as for (B.9) (i.e., as forZ3\nin the proof of Lemma 3.5), and its quadratic variation is (cf . (3.57))\nrM,M st“8ÿ\nk“11tTkďtue´2µTk. (B.37)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 75\nThis is the same as in (B.24), except for the factor |ζk|2there (or the same if we\nchooseζk“ ˘1). Consequently, the argument in Case 1 yields\nĂM˚˚:“sup\ntě0/visualspace\npt`1q´κ˚pµqe´pλ˚\nj´µq`t|Mptq|(\nPLq. (B.38)\nFurthermore, (B.18) and the assumption (B.20) yield rV˚˚PLpĎLq. Consequently,\n(B.23) follows by (B.36).\nCase 3: General ζk. The result follows from Cases 1 and 2 by the decomposition\nζk“ pζk´Eζkq `Eζk. /square\nLemma B.6. Assume (A1)–(A3),(A51)(or(A5)), and(A4p)for some pą1.\nSuppose that i,jPQare such that Pi“ tjuandXip0q “0, and suppose also that\n(B.2)holds. If rX˚˚\njPLp, thenrX˚˚\niPLp.\nProof.We use again the decomposition (3.28), where we recall that Ykptqdenote\ncopies of the one-colour process in Section 3.1, and that the processYkptqis inde-\npendent of FTk. Let\nζk:“sup\ntě0/visualspace\ne´λitYkptq(\n. (B.39)\nThen (3.28) implies\ne´λitXiptq “8ÿ\nk“11tTkďtue´λiTk¨e´λipt´TkqYkpt´Tkq\nď8ÿ\nk“11tTkďtue´λiTkζk“:Zptq. (B.40)\nLetηk“Ykp0q, and recall that ηkd“ξjiPLp. Then, conditioning on ηk,\nLemma B.3 applies to Ykptqand shows that\nE`\nζp\nk|ηk˘\nďCrηkspďCpηk`1qp. (B.41)\nConsequently, using (A4p),\nEζp\nkďCEpηk`1qp“Epξji`1qpă 8. (B.42)\nWe apply Lemma B.5 to the sum Zptqin (B.40), taking q“pandµ“λi. Note\nthat then (B.21) holds: if iRQ´thenξiiě0 a.s., andthus µ“λiě0; ifiPQ´then\nµ“λią0 by our assumption (B.2). Hence, (B.23) holds. Furthermore ,κ˚pµq “κi\nand pλ˚\nj´µq`“ pλ˚\ni´λiq, as is easily verified by considering the three cases in\n(B.16) (and after (3.40)) separately. Hence, (3.30), (B.40 ), and (B.23) yield\nrX˚˚\ni“sup\ntě0/visualspace\npt`1q´κie´pλ˚\ni´λiqt¨e´λitXiptq(\nďrZ˚˚PLp.(B.43)\n/square\nB.3.The general case for a single colour. We now consider any colour iPQ.\nLemma B.7. Assume(A1)–(A3), either`\n(A51)and(A7)˘\nor(A5), and(A4p)for\nsomepą1. LetiPQ, and assume that for every jPPi, we have rX˚˚\njPLp. Then\nrX˚˚\niPLp.76 SVANTE JANSON\nProof.Since (A5) implies (A51) and (A7), these hold in any case. We consider two\ncases.\nCase 1: (B.2)holds, i.e. iRQ´orλią0. As in the proof of Lemmas 3.7 and 8.14,\nwe split the colour iinto subcolours i0andij,jPPi. We then use Lemma B.2(i)\nfori0and Lemma B.6 for every ij, and the result follows by (3.91).\nCase 2: iPQ´andλiď0. Then (A7) yields λ˚\nią0. Hence λ˚\niąλi, and since\nλ˚\ni“λi_maxjPPiλ˚\nj, we have Pi‰ Hand\nλ˚\ni“max\njPPiλ˚\nj. (B.44)\nConsider a modification ¯ξiiofξiisuch that ¯ξiiěξiia.s., and ¯λi:“aiE¯ξiiP p0,λ˚\niq.\n(For example, let ¯ξii:“ξii_˜ξwhere˜ξP t˘1uis independent of ξiiandPp˜ξ“1q P\np0,1sis chosen suitably.) Modify the urn by replacing ξiiby¯ξii; this does not affect\nany colour j≺i; in particular Xjptqand all draws of colour jremain the same for\neveryjPPi, but at each draw of colour iwe may add more balls of colour i; hence,\nlettingXiptqdenote the number of balls of colour iin the modified urn, we have,\nusing an obvious coupling of the two urns,\nXiptq ěXiptq, t ě0. (B.45)\nThe modified urn satisfies all our conditions in the present le mma, and since ¯λią0,\nthe already proven Case 1 shows that (with obvious notation)rX˚˚\niPLp. Further-\nmore,¯λiăλ˚\ni, and thus (B.44) implies\n¯λ˚\ni:“¯λi_max\njPPiλ˚\nj“¯λi_λ˚\ni“λ˚\ni. (B.46)\nSimilarly, using (2.9), we have ¯ κi“κi. Consequently, the exponents in (3.30) are\nthe same for XiptqandXiptq, and thus (B.45) implies\nrX˚˚\niďrX˚˚\ni. (B.47)\nSince, as just shown,rX˚˚\niPLp, this completes the proof. /square\nB.4.Lpbounds and convergence.\nLemma B.8. Assume(A1)–(A3), either`\n(A51)and(A7)˘\nor(A5), and(A4p)for\nsomepą1. ThenrX˚˚\niPLpfor every iPQ.\nProof.By Lemma B.7 and induction on the colour i. /square\nProof of Theorem 12.3. By Lemma B.8, rX˚˚\njPLp; hence it follows, exactly as for\nthe case p“2 in Theorem 12.2, that the collection/visualspace\n|rXiptq|p:tě1(\nis uniformly\nintegrable, and thus the convergence (4.1) holds also in Lp. /square\nOnce we have proved Theorem B.1, the proof of Theorem 12.3 app lies to any\npą1, as claimed in Remark 12.4.ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 77\nB.5.A.s. convergence. We now turn Theorem B.1, i.e., that our a.s. convergence\nresults hold also if (A4) is replaced by (A4p). We may assume 1 ăpă2, since for\npě2 the assumption (A4p) implies (A4), and the results are alre ady proven.\nWe begin by extending Lemma 8.9 to 1 ăpă2, complementing Lemmas B.2 and\nB.3 for the case excluded there when (B.2) does not hold.\nLemma B.9. Assume(A1)–(A3),(A51), and(A4p)for some pP p1,2s. LetiPQmin\nand assume λiď0. Letx0:“Xip0q.\n(i)Ifλi“0, thenXiptqis a martingale with\nXiptqa.s.Ý ÑXi“0,astÑ 8, (B.48)\nEX˚\niptqpďCxp\n0`Cx0t,for every tă 8. (B.49)\nFurthermore, for every δą0,\nE`\nsup\ntě0te´δtXiptqu˘pă 8. (B.50)\n(ii)Ifλiă0, then, with Mptq:“e´λitXiptq,\nXiptqa.s.Ý ÑXi“0,astÑ 8, (B.51)\nEXiptqpďCxp\n0eλit, (B.52)\nEM˚ptqpďCxp\n0e´pp´1qλit. (B.53)\nProof.In both cases, Xiptqis a (sub)critical continuous-time branching process and\ntherefore a.s. dies out, see Remark 8.2, which gives (B.48) a nd (B.51).\nRecall from (3.12) that Mptq:“e´λitXiptqis a martingale (this does not require\n(A4), only Eξiiă 8), and hence, or by (3.11),\nEXiptq “eλitEMptq “eλitMp0q “x0eλit. (B.54)\nIt follows from (3.18) that\nrM,M sp{2\ntďxp\n0`8ÿ\nk“11tTkďtue´pλiTkηp\nk, (B.55)\nand hence, using Lemma 3.2(ii) and (B.54),\nErM,M sp{2\ntďxp\n0`Cżt\n0e´pλisEXipsqds“xp\n0`Cx0żt\n0e´pp´1qλisds.(B.56)\nIfλi“0, this yields (B.49) by (B.1); then (B.50) follows as in (8.1 2).\nIfλiă0, then (B.56) and (B.1) yield, recalling that x0is an integer,\nEM˚ptqpďCxp\n0`Cx0e´pp´1qλitďCxp\n0e´pp´1qλit(B.57)\nand thus\nEXiptqp“epλitEM˚ptqpďCxp\n0eλit, (B.58)\nshowing (B.53) and (B.52). /square\nProof of Theorem B.1. As said above, we may assume 1 ăpă2. It suffices to prove\nthe continuous-time versions Theorem 4.1 and 8.6; then the p roofs of Theorems 1.8,\n8.4, and 8.5 are as before.\nBy Lemma B.8, we have rX˚˚\niPLpfor every iPQ, but it remains to show that\nrXiptqconverges. We follow the proof of Theorem 4.1 (and Theorem 8. 6) step by step78 SVANTE JANSON\nin the claims below which extend the limit statements in the l emmas in Section 3\nand Section 8 (recall that Lpversions of the L2estimates there already are given);\nwe omit some details. Assumptions on /bardblrX˚˚\nj/bardbl2are replaced by /bardblrX˚˚\nj/bardblp(and hold in\nour case by Lemma B.8). We assume in the sequel (A1)–(A3), (A51) (or (A5)), and\n(A4p) for some pP p1,2s. (But not (A7) unless said so.)\nNote first that (3.11)–(3.12) still hold. (In fact, they requ ire only the first moment\nEξiiă 8.)\n(i)In Lemmas 3.3 and 8.9, the convergence rXi:“e´λitXiptqa.s.Ý ÑXistill holds. We\nstill have Xią0a.s. when iRQ´, andPpXią0q ą0when ifiPQ´andλią0.\nProof.The convergence rXiptqa.s.Ý ÑXifor some limit XiPLpis shown in Lemma B.2\nor B.9. Furthermore, if iRQ´, then the argument in the proof of Lemma 3.3 shows\nthatXią0 a.s.; otherwise, if λią0, thenEXi“Xip0q ą0 and thus at least\nPpXią0q ą0. /square\n(ii)In Lemmas 3.5 and 8.11, the convergence rXiptqa.s.Ý ÑXistill holds; furthermore,\nXją0ù ñXią0a.s.\nProof.We argue as in the proof of Lemmas 3.5 and 8.11, using again the decom-\npositions (3.40) and (3.43). For Z4ptq, the argument in (3.49)–(3.54) holds without\nchanges. Also for Z3ptqthe argument in the proof of Lemma 3.5 still holds, since\ntheL2estimate (3.58) remains valid.\nThe remaining terms Z1ptqandZ2ptqare still local martingales. For Z2ptqwe have\nby the definition (3.59), (B.1), Lemma 3.2(ii), and (3.30)\nEZ˚\n2ptqpďCErZ2,Z2sp{2\ntďCE8ÿ\nk“11tTkďtue´pλiTk|ηk´rji|p\n“CEżt\n0e´pλisXjpsqds\nďCErX˚˚\njżt\n0ps`1qκjepλ˚\nj´pλiqsds. (B.59)\nWe now argue as in Step 5 of the proof of Lemma 3.5, using (B.59) instead of (3.68)\nandLpinstead of L2; we replace 2 by pin all exponents, and the cases (i1) and (ii1)\nare replaced by λ˚\njăpλiand (λ˚\njěpλiandλ˚\nją0). We obtain as before that\nrZ2ptqa.s.Ý ÑZ2astÑ 8, where the limit Z2“0 except in case (i).\nForZ1ptqwe use (3.64) and both directions of (B.1) to obtain\nE|Z˚\n1ptq|pďCErZ1,Z1sp{2\ntďCE8ÿ\nk“1rZpkq\n1,Zpkq\n1sp{2\ntďCE8ÿ\nk“1|Zpkq˚\n1ptq|p.(B.60)\nThe definition (3.62) yields\n/vextendsingle/vextendsingleZpkq˚\n1ptq/vextendsingle/vextendsinglep“1ttěTkue´pλiTk|rY˚\nkpt´Tkq|p, (B.61)\nwhere the martingale rYkptq:“e´λitYkptq ´Ykp0qis independent of Tk, andYkptqis\na copy of the one-colour process in Section 3.1 and Lemma B.9. Thus (B.60) andALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 79\nLemma 3.2(ii) yield\nE|Z˚\n1ptq|pďCE8ÿ\nk“11ttěTkue´pλiTkrY˚\nkptqp\n“CE“rY˚\n1ptqp‰\nEżt\n0e´pλisXjpsqds. (B.62)\nWe separate three cases, as in the proof of Lemma 8.11:\n(1)iRQ´orλią0: This is condition (B.2); thus Lemma B.3 applies to Ykptq\nconditionally on ηk, which gives\nE“rY˚\nkptqp|ηk‰\nďCrηksp, (B.63)\nand consequently\nE“rY˚\n1ptqp‰\nďC. (B.64)\nThus (B.62) yields the same estimate (B.59) as for Z˚\n2ptq, and the argument\nafter (B.59) applies to Z1too and yields rZ1ptqa.s.Ý ÑZ1for some Z1, withZ1“0\nexcept in case (i).\n(2)iPQ´andλi“0: (As in the proof of Lemma 8.11, the assumptions then\nyieldλ˚\nj“λ˚\nią0.) In this case, rYkptq “Ykptq ´Ykp0qwhereYkptqis a copy of\nXiptqin Lemma B.9. Thus, (B.49) yields, recalling that now ηkis an integer,\nfortě1,\nE“rY˚\nkptqp|ηk‰\nďCηp\nkt, (B.65)\nand consequently\nE“rY˚\n1ptqp‰\nďCt. (B.66)\nThis and (B.62) yield, still for tě1,\nE|Z˚\n1ptq|pďCtEżt\n0e´pλisXjpsqdsďCtErX˚˚\njżt\n0ps`1qκjepλ˚\nj´pλiqsds\nďCtκj`1epλ˚\nj´pλiqt. (B.67)\nThis differs from (B.59) by an extra factor t, but the argument after (B.59) (in\nthis case modifications of (3.70)–(3.73)) still works and yi eldsrZ1ptqa.s.Ý Ñ0.\n(3)λiă0: (The assumptions yield λ˚\nj“λ˚\nią0 in this case too.) In this case, we\ncondition on Tkandηkand then use (B.61) and (B.53), yielding\nE“\n|Zpkq˚\n1ptq|p|Tk,ηk‰\n“1ttěTkue´pλiTkE“\n|rY˚\nkpt´Tkq|p|Tk,ηk‰\nďC1ttěTkue´pλiTkηp\nke´pp´1qλipt´Tkq\n“C1ttěTkue´λiTke´pp´1qλitηp\nk. (B.68)\nHence, (B.60), Lemma 3.2(ii), and (3.30) yield\nE|Z˚\n1ptq|pďCE8ÿ\nk“1E“\n|Zpkq˚\n1ptq|p|Tk,ηk‰\nďCe´pp´1qλitE8ÿ\nk“11ttěTkue´λiTkηp\nk\n“Ce´pp´1qλitEżt\n0e´λisXjpsqds80 SVANTE JANSON\nďCe´pp´1qλitErX˚˚\njżt\n0ps`1qκjepλ˚\nj´λiqsds\nďCpt`1qκjepλ˚\nj´pλiqt. (B.69)\nWithrZ:\nℓpnqdefined in (3.70), we again modify (3.71) using pth powers, now\nusing (B.69) instead of (3.68), and it follows similarly to ( 3.72) that\nE8ÿ\nn“1rZ:\nℓpnqpă 8. (B.70)\nHence,rZ1a.s.Ý Ñ0.\nFinally, in all cases, it follows from (3.43) that rXiptqa.s.Ý ÑXi. Furthermore, we\nhaveXją0ù ñXią0 a.s. by the argument in Step 6 of Lemma 3.5, if necessary\nmodified as in the proof of Lemma 8.11. /square\n(iii)In Lemma 8.13, e´δtXiptqa.s.Ý Ñ0still holds.\nProof.The same as for Lemma 8.13. /square\n(iv)In Lemmas 3.7 and 8.14, rXia.s.Ý ÑXistill holds, and so do the claims on Xią0.\nProof.The same as for Lemmas 3.7 and 8.14, using the claims above. No te that the\nassumptions now include (A7). /square\nTo complete the proof of Theorem B.1, we now obtain by inducti on from (iv)\nabove that for every iPQwe have rXiÑXiastÑ 8, i.e., (4.1). This proves\nthat Theorems 4.1 and 8.6 hold with (A4) replaced by (A4p), wh ich as said above\ncompletes the proof. /square\nAppendix C.Proof of (14.80)\nIn this appendix we prove (14.80) in Example 14.14, using res ults from Markov\nprocess theory.\nSuppose, more generally, that the P´ olya urn in Example 14.1 4 starts with w0“\nαą0 white balls and b0ě0 black balls. Thus Wptq “αfor alltě0. (We do\nnot have to assume that αis an integer, although we must have b0PZě0, since we\nallow subtractions.) The stochastic process Bptqis a time-homogeneous pure-jump\nMarkov process on Zě0with jumps\n#\n`1 with intensity α`1\n2Bptq,\n´1 with intensity1\n2Bptq.(C.1)\nWe define for any real ℓą0 the scaled process\nrBℓptq:“ℓ´1Bpℓtq. (C.2)\nIt follows from (C.1) that rBℓis a pure-jump Markov process with jumps\n#\n`1{ℓwith intensity ℓ`\nα`1\n2Bpℓtq˘\n“αℓ`ℓ2\n2rBℓptq,\n´1{ℓwith intensityℓ2\n2rBℓptq.(C.3)ALMOST SURE AND MOMENT CONVERGENCE FOR TRIANGULAR P ´OLYA URNS 81\nIn other words, the generator Aℓof the Markov process rBℓis given by, see e.g. [32,\nTheorem 19.23],\nAℓfpxq “αℓ`\nfpx`ℓ´1q ´fpxq˘\n`xℓ2\n2`\nfpx`ℓ´1q `fpx´ℓ´1q ´2fpxq˘\n.(C.4)\nSinceBptqtakes its values in Zě0,rBℓptqis a Markov process on the state space\nℓ´1Zě0“ t0,ℓ´1,2ℓ´1,...u. For technical reasons, we extend it to a pure-jump\nMarkov process on r0,8qby defining the intensities to be as in (C.3) whenever\nrBℓptq ěℓ´1; otherwise, if rBℓptq “bă1{ℓ, we jump `1{ℓwith intensity αℓand\n˘bwith intensitiesℓ2\n2beach. (This makes no difference if we start with an integer\nnumber of black balls.) The generator of the extended proces s is\nAℓfpxq “αℓ`\nfpx`ℓ´1q ´fpxq˘\n`xℓ2\n2`\nfpx`hℓ,xq `fpx´hℓ,xq ´2fpxq˘\n,(C.5)\nwherehℓ,x:“ℓ´1^x.\nLetC2r0,8qbe the space of all continuous functions fonr0,8qthat have two\ncontinuous derivatives in p0,8qwithf1andf2extending continuously to r0,8q. Let\nfurther\nC:“/visualspace\nfPC2r0,8q:fpxq,f1pxq,f2pxq “Ope´εxqfor some εą0(\n.(C.6)\nIt follows from (C.5) and Taylor’s formula that if fPC, then as ℓÑ 8,\nAℓfpxq ÑAfpxq:“αf1pxq `x\n2f2pxq (C.7)\nuniformlyin xP r0,8q, andthusinthespace C0r0,8q. Notethat4 A“2xd2\ndx2`4αd\ndx\nis the generator of the squared Bessel process BESQδwith dimension δ:“4α, see\n[44, p. 443 and Proposition VII.(1.7)]; hence Ais the generator of BESQ4αpt{4qd“\n1\n4BESQ4αptq[44, Proposition XI.(1.6)]; It is well known that BESQ4αptqis a Feller\nprocess on r0,8q[44, p. 442], and it is easily verified directly that each Aℓalso is\na Feller process on r0,8q. The transition probabilities qδ\ntpx,yqof BESQδare given\nexplicitly in [44, XI.(1.4)], and a simple calculation usin g [40, (10.29.4)] shows that\nB\nBxqδ\ntpx,yq “1\n2t`\nqδ`2\ntpx,yq ´qδ\ntpx,yq˘\n, (C.8)\nand hence\nB2\nBx2qδ\ntpx,yq “1\n4t2`\nqδ`4\ntpx,yq ´2qδ`2\ntpx,yq `qδ\ntpx,yq˘\n. (C.9)\nSince BESQδis a Feller process for every δą0, the transition operator Tδ\ntfpxq:“ş\nqδ\ntpx,yqfpyqdymapsC0r0,8qinto itself for every tą0, and it follows from (C.8)–\n(C.9) that Tδ\ntmapsC0r0,8qintoC2r0,8q; moreover, using also again the explicit\nform ofqδ\ntin [44, XI.(1.4)], it is easy to see that TtmapsCinto itself. Hence, it\nfollows from [32, Proposition 19.9] that Cis a core for the generator 4 A, and thus\nalso forA. Consequently, (C.7) and [32, Theorem 19.25] show the follo wing.\nTheorem C.1. Let the P´ olya urn in Example 14.14 start with w0“αą0white\nballs and b0“0black balls. Then, as ℓÑ 8, we have\nℓ´1Bpℓtq “rBℓptqdÝ Ñ1\n4BESQ4αptqinDr0,8q. 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Stochastic Integration and Differential Equations , 2nd ed.\nSpringer-Verlag, Berlin, 2004. MR 2020294\n[43] Vincent Puyhaubert. Mod` eles d’urnes et ph´ enom` enes de seuils en combi-\nnatoire analytique . Ph. D. thesis, l’ ´Ecole Polytechnique, 2005. Available at\nhttps://pastel.hal.science/pastel-00001348\n[44] Daniel Revuz & Marc Yor. Continuous Martingales and Brownian Motion . 3rd\ned., Springer-Verlag, Berlin, 1999. MR 1725357.\nDepartment of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala,\nSweden\nEmail address :svante.janson@math.uu.se\nURL:http://www.math.uu.se/ „svante/"    },    {        "title": "2402.01301v1.On_Inflation_and_Axionic_Dark_Matter_in_a_Scaled_Gravity.pdf",        "content": "On Inflation and Axionic Dark Matter in a Scaled Gravity\nA. Belhaj1∗, S. E. Ennadifi2†, M. Lamaaoune1‡ §\n1D´ epartement de Physique, Equipe des Sciences de la mati` ere et du rayonnement, ESMaR\nFacult´ e des Sciences, Universit´ e Mohammed V de Rabat, Rabat, Morocco\n2LPHE-MS, Facult´ e des Sciences, Universit´ e Mohammed V de Rabat, Rabat, Morocco\nFebruary 5, 2024\nAbstract\nMotivated by the modified gravity theories F(R)̸=Rand inflationary physics, we\nfirst propose and investigate an inflation model in a scaled gravity F(R) =R+βR,\nwhere βis a dimensionless scaling parameter. The latter is also implemented in a\nparticular potential V(ϕ) =M4\u0014\n1−cos\u0010\nϕ\nµ\u0011β\u0015\nbeing considered to drive the inflation\nvia a parameter coupling scenario. Using the slow-roll approximations, the gravity\nscale parameter βis approached with respect to the range of the associated computed\ncosmological observables nsandraccording to the recent Planck and BICEP/Keck\ndata. Then, we discuss the axionic dark matter in the suggested gravity model by\nconsidering the case where the inflaton is taken to be identified with an axion-like field\nϕ=faθwith the decay constant fa=µ. Referring to the known data, the underlying\ninflation scale Mis constrained to be much lower than the corresponding axion scale\nM≪fa.\nKeywords : Inflation, Modified gravity, Axions, Dark Matter.\n∗a-belhaj@um5r.ac.ma\n†Ennadifis@gmail.com\n‡lamaaoune2944massi@gmail.com\n§Authors in alphabetical order .\n1arXiv:2402.01301v1  [hep-th]  2 Feb 20241 Introduction\nRecently, inflationary models have been studied in depth by considering many theories of\ngravity [1–28]. Certain models have opened new ways to understand the evolutionary phe-\nnomena of the Universe, such as the horizon, the flatness and the problems of large struc-\ntures [8–10]. Various gravity theories have been suggested by exploiting single and multiple\nscalar fields through potentials in order to determine the standard cosmological observables\nin the context of general relativity (GR). The simplest models involve a single field that\nis considered as the main component of inflation. This scalar field has been approached\nusing many theories, including higher dimensional supergravity ones [14–21]. Concretely,\nadvances in string theory and related topics have been exploited to develop inflationary\nmodels based on D-brane physics using the Randall-Sundrum II (RS-2) mechanism. In this\nway, interesting scalar potentials in the presence of the stringy parameters such as the brane\ntension have been analyzed where the stringy corrections of the involved quantities have\nbeen obtained [14–18]. An examination reveals that the compactification of the superstring\nmodels and M-theory could generate many scalar fields being derived from the geometric\ndeformations of the metric and the non-trivial tensor fields defining the stringy moduli space.\nThese scalar fields have been explored to confront predictions with observations from the\ncosmological microwave background (CMB) and the Planck experiments [29–31].\nMore recently, modified models of GR have been explored providing interesting inflationary\nresults. The most discussed ones are F(R) modified gravity theories where Rdenotes the\nRicci scalar. They are widely studied by dealing with several scalar potentials [23–25]. Based\non the slow-roll analysis, the associated spectral index nsand the tensor/scalar ratio rhave\nbeen determined with particular values of the number of e-folds required by the observational\nresults. These modified gravity models have been extended by adding other quantities, in-\ncluding the trace Tof the stress-energy tensor. The resulting F(R, T) theory of the gravity\nis intensively investigated allowing the realization of inflationary models [32–37]. Some of\nthem have been motivated by the study of dark energy (DE) [38, 39]. It has been pointed\nout that this substance can be explored to provide explanations for the accelerated aspect\nof the expansion of the Universe. Specifically, the modified gravity theories could generate\ndynamical phenomena which can be assimilated to contributions associated with DE. These\nbehaviors exceed the cosmological constant in the context of GR.\nIt has been remarked that the form of the scalar potential can be of crucial importance in\nthe building of inflationary models arising from various theories including superstring mod-\nels and M-theory. The choice of the form of the potential generally depends on motivations\nsupported by known models like the standard model (SM) of particle physics [40,41]. It has\nbeen noted that famous examples are the chaotic inflation potential and the minimal super-\nsymmetric standard model (MSSM) inflation potential. In addition to these known models,\nother types of scalar potentials have been discussed in the context of dark matter (DM) [42].\nIn connections with inflation activities, several DM candidates have been proposed and con-\nsidered. However, it has been suggested that axions could be considered as relevant DM\n2candidates via certain vacuum fluctuations during (or at the end) of inflation [42–45]. It has\nbeen shown that these scalar field have been introduced in different ways. One of them is\nassociated with the CP problem via the Peccei and Quinn symmetry in the quantum chromo-\ndynamics (QCD) context. These axions are called QCD axions [46,47]. Other types of axions\nappear naturally in string theory dealing with higher dimensional objects like strings and\nbranes in extra dimensional space-times. In this way, the associated scalars are called axion-\nlike particles derived from topological and geometrical contributions of the internal compact\ngeometries associated with extra dimensions [42,48,49]. Alternatively, axions could appear\nalso in Chern-Simons (CS) interactions with gravity producing axions-CS gravity [50,51].\nIn the examination of inflation parameters, one should distinguish two categories. The\nfirst one contains the parameters of the gravity sector. However, the second one involves the\nmatter parameters including the dark sector contributions. A close inspection reveals that\none can follow two different inflation scenarios where such parameters are linked or not.\nMotivated by the modified gravity theories F(R)̸=Rand inflationary physics, we first\npropose and study an inflation model in a scaled gravity F(R) =R+βR, where βis a\ndimensionless scaling parameter. The latter is implemented in the special potential V(ϕ) =\nM4\u0014\n1−cos\u0010\nϕ\nµ\u0011β\u0015\nbeing considered to drive the inflation by means of a parameter coupling\nscenario. Using the slow-roll approximations, the gravity scale parameter βis approached\nwith respect to the range of the corresponding computed cosmological observables nsandr\naccording to the recent Planck and BICEP/Keck data. After that, we investigate axionic DM\nin the suggested scaled gravity model by discussing the case where the inflaton is identified\nwith an axion-like field ϕ=faθwith the decay constant fa=µ. Considering known data,\nthe underlying inflation scale Mis constrained to be much lower than the associated axion\nscale M≪fa.\nThe organisation of this paper is as follows. In section 2, we present inflation calculations\nin a scaled gravity. In section 3, we investigate parameter decoupling and coupling scenarios\nfor a scalar potential supported by DM activities. In section 4, we discuss the corresponding\naxionic DM investigation. The last section is devoted to concluding remarks.\n2 Inflation in a scaled gravity\nIn this section, we reconsider the study of F(R) gravity with a kinetic coupling term in order\nto investigate the inflation scenarios and related topics including DM. Indeed, we start by\ntaking the following action\nS=Z\ndx4√−g\u0012F(R)\n16π+Lm\u0013\n(2.1)\nwhere F(R) is an arbitrary function of the Ricci scalar Randgdenotes the determinant of\nthe metric gµν[32–37]. Lmrepresents the matter sector given by\nLm=−1\n2(gµν−ω2Gµν)▽µϕ▽νϕ−V(ϕ) (2.2)\n3where Gµνindicates the Einstein tensor and ωis a positive parameter having the dimension of\nthe inverse mass scale [52–56]. The positive sign of the kinetic coupling term ω2Gµν∂µϕ∂νϕis\nneeded to remove the ghost in the studied model [40,41]. This term has been implemented in\nthe matter sector in order to reduce the tensor-to-scalar ratio rneeded to establish bridges\nwith CMB observations. In the matter sector, the relevant piece is the dynamical scalar\nfield ϕcontrolled by a potential V(ϕ). Recently, many different F(R) function forms have\nbeen considered in inflation activities in connection with many topics including swampland\ncriteria. A particular emphasis has been put on a rescaled Einstein-Hilbert theory with\nF(R)∼αRwhere αis a dimensionless constant parameter constrained by 0 < α < 1 [57–59].\nInspired by such activities, we would like to implement a F(R) gravity parameter in order to\nestablish a coupling scenario between matter and gravity sectors, via the scalar potential. In\nparticular, we would like to identify a parameter in the potential with one of F(R) function.\nThe latter will be relevant in the inflation discussion. Concretely, we consider a scaled gravity\ndescribed by the following function of R\nF(R) =R+βR (2.3)\nwhere βis a dimensionless free parameter being independent of R. This theory can be\nderived by considering the following scaling\nR→(1 +β)R (2.4)\nproducing a scaled gravity. An other possible justification for the use of such a gravity lies\nin the fact that it can be exploited to provide models with linked parameters of matter and\ngravity sectors by means of the coupling scenario. In such a gravity, the previous action\nreduces to\nS=Z\ndx4√−g\u0012(1 +β)R\n16π−1\n2\u0000\ngµν−ω2Gµν\u0001\n▽µϕ▽νϕ−V(ϕ)\u0013\n. (2.5)\nFor the moment, it is worth noting that βis different to −1 and 0. Varying the action given\nby Eq.(2.5) with respect to the metric gµνandϕ, we get the equations of motion which read\nas\n(1 +β)Gµν= 8π(T(ϕ)\nµν+ω2Θµν) (2.6)\n\u0000\ngµν−ω2Gµν\u0001\n▽µ▽µϕ=dV(ϕ)\ndϕ(2.7)\nwhere T(ϕ)\nµνandAµνare given by\nT(ϕ)\nµν=▽µϕ▽νϕ−1\n2gµν▽ρϕ▽ρϕ+gµνV(ϕ) (2.8)\nΘµν=−1\n2▽µϕ▽νϕR+ 2▽αϕ▽(µϕRα\nν) +▽αϕ▽βϕRµανβ\n+▽µ▽αϕ▽ν▽αϕ−▽µ▽νϕ□ϕ−1\n2(▽ϕ)2Gµν (2.9)\n+gµν[−1\n2▽αϕ▽βϕ▽αϕ▽βϕ+1\n2(□ϕ)2−▽αϕ▽βϕRαβ].\n4Taking β=ω= 0, for instance, we recover the usual standard field equations. Using gµν\nandϕvariations via the Friedman-Lemaitre-Robertson-Walker (FLRW) metric, we obtain\nthe equations of motion\n3(1 + β)H2= 4π˙ϕ2(1 + 9 ω2H2) + 8πV(ϕ), (2.10)\n(1 +β)(2˙H+ 3H2) = −4π˙ϕ2(1−ω2(2˙H+ 3H2+ 4H¨ϕ˙ϕ−1)) + 8 πV(ϕ) (2.11)\n(¨ϕ+ 3H˙ϕ) + 3 κ(H2¨ϕ+ 3H3˙ϕ+ 2H˙H˙ϕ) =−V′(ϕ) (2.12)\nwhere one has used the notations′=d\ndϕand ˙ =d\ndt. It is recalled that Hdenotes the Hubble\nparameter defined by H=˙a(t)\na(t)where a(t) is the scalar factor. To confront the proposed\nmodel with the observational data, one should exploit the slow-roll analysis by computing\nthe associated parameters. During the inflation phase, they are given by\nϵ=−˙H\nH2, η =˙ϵ\nHϵ, κ 0= 12πκ˙ϕ2, κ 1=˙κ0\nHκ0(2.13)\nbeing constrained by\nϵ, η, κ 0, κ1<<1. (2.14)\nIn this way, the field equations of motion take the following simplified forms\n3H˙ϕ+ 9κH3˙ϕ=−V′(ϕ) (2.15)\n3(1 + β)H2=8πV(ϕ) (2.16)\n(1 +β)˙H=−4π˙ϕ2(1 + 3 κH2) (2.17)\nwhich can be solved as follows\n˙ϕ=−(1 +β)3/2V′(ϕ)\n2p\n6πV(ϕ)(1 + β+ 8πω2V(ϕ))(2.18)\n˙H=−V′2(1 +β)\n6V(ϕ)(1 + β+ 8πω2V(ϕ))(2.19)\nH2=8π\n3(1 + β)V(ϕ). (2.20)\nUsing the slow-roll analysis, the relevant parameters are found to be\nϵ=(1 +β)2V′2\n16πV(ϕ)2(1 +β+ 8πω2V(ϕ))(2.21)\nη=(1 +β)2\u0000\n(1 +β+ 12πV(ϕω2))V′(ϕ)2−V(ϕ)(1 + β+ 8πω2V(ϕ))V′′(ϕ)\u0001\n4πV(ϕ)2(1 +β+ 8πω2V(ϕ))2.(2.22)\nIn the inflationary model scenarios, such quantities give the scalar field values ϕEat the\nend of the expansion via the constraint ϵ(ϕE) = 1. Moreover, the scalar field at the beginning\nof inflation can be determined by exploiting the total logarithmic phase. It turns out that\n5the number of e-folds associated with the inflation duration will be needed to handle the\ncosmological observables. Usually, it reads as\nN=ZtE\ntIHdt=ZϕE\nϕIH\n˙ϕdϕ=−ZϕE\nϕI8π\n(1 +β)2V(ϕ)\nV′(ϕ)(1 +β+ 8πω2V(ϕ))dϕ (2.23)\nwhere one has used the subscript IandEindicating the parameter values at the onset and the\noffset time of inflation, respectively. The interesting gravity models should provide consistent\npredictions which can be either refuted or corroborated by the observational data. To give\nsuch an evidence, the inflationary observables should be determined. Following [53, 54, 56],\nthe scalar spectral index nsand the tensor-to-scalar ratio rare expressed as follows\nns−1 = −2ϵ−η (2.24)\nr= 16 ϵ. (2.25)\nIn the presence of the kinetic term in the scaled gravity, nsandrare modified as follows\nns= 1−(β+ 1)2V′2\n8πV(ϕ)2(1 +β+ 8πω2V(ϕ))(2.26)\n+(β+ 1)2(1 +β+ 12πω2V(ϕ)) (V(ϕ)V′′(ϕ)−V′2)\n4πV(ϕ)2(1 +β+ 8πω2V(ϕ))2\nr=(β+ 1)2V′2\nπV(ϕ)2(1 +β+ 8πω2V(ϕ)). (2.27)\nThese relations go beyond the known ones. Taking β= 0, we recover the scalar spectral\nindex nsand the tensor-to-scalar ratio rof the model constituting of a scalar field kinetically\ncoupled to a standard gravity model with a positive coupling constant [52,55]. Considering\nβ=ω= 0, we obtain the relations associated with standard inflation.\n3 Decoupling and coupling scenarios in scaled gravity\ninflation\nIn this section, we would like to investigate inflation coupling scenarios in a scaled gravity\nby means of the inflation moduli space. A close examination shows that the moduli space\nMof this theory can be split as follows\nM=Mg× M m (3.1)\nwhere Mgis the moduli subspace associated with the gravity depending on the form of\nF(R). However, the moduli subspace Mmis coordinated by the parameters appearing\nin the matter sector described by Lm. Precisely, it depends on ωand the parameters of\nthe potential V(ϕ). A generic moduli space could generate complex computations. Here,\nhowever, we pay attention to special forms of the gravity function F(R) and the scalar\n6potential V(ϕ). Precisely, we consider the situation where the gravity function takes the\nform F(R) = (1 + β)Rand the potential is given by\nV(ϕ) =M4\u0014\n1−cos\u0012ϕ\nµ\u0013α\u0015\n(3.2)\nwhere the exponent αis a free parameter. Mandµare two free mass scale parameters\nassociated with the moduli sub-space factor Mm[60]. These parameters will be investigated\nlater on. It is worth noting that, for α= 1, the scalar potential reduces to the form\nV(ϕ) =M4\u0014\n1−cos\u0012ϕ\nµ\u0013\u0015\n(3.3)\nwhich has been largely studied in connections with particle physics dealing with axions\nand DM from models going beyond SM. For generic values of α, we consider two inflation\nscenarios. In the first one, the relevant parameters of MgandMmare not bridged and\nlinked. We refer to such a scenario as a parameter decoupling scenario assured by\nα̸=β. (3.4)\nThe second scenario corresponds to the case where the relevant parameters of MgandMm\nare linked. We refer to such a road as parameter coupling scenario. A special situation can\nbe occurred by considering\nα=β. (3.5)\nThe implementation of the gravity parameter in the matter sector can be considered as a new\nway to generate an inflation coupling scenario by means of the moduli space M. Precisely,\nthis implementation could be interpreted as an alternative way to generate the coupling\nbetween the gravity and the scalar field via the potential. Coupling F(R) with such a scalar\npotential, the relevant cosmological quantities get modified in the same time by changing\nthe gravity parameter β. In this regard, the modified gravity controlled by changing βin the\nscalar potential could provide models going beyond other investigations where the gravity\nand the scalar potential are not linked. We anticipate that this scenario could provide some\nmodels going beyond the previous ones which could be either rejected or corroborated by\nexperimental findings via the falsification analysis.\n3.1 Parameter decoupling scenario\nIn this subsection, we consider the first inflation scenario. Concretely, we compute the\nrelevant observables being the scalar spectral index nsand the tensor-to-scalar ratio ras\nfunctions of the moduli space coordinates. To perform such calculations, the number of\ne-folds Nshould be determined. Indeed, it can be expressed as\nN=8πµ(AE−AI)\nα(β+ 1)2(3.6)\n7where one has used\nAi=−µ(1 +β+ 8πM4ω2) cos2−α\u0010\nϕi\nµ\u0011\nα−2F1+8πµM4ω2cosα+2\u0010\nϕi\nµ\u0011\nα+ 2F2\n+1\n2µ\u0000\n1 +β+ 16πM4ω2\u0001\nlog\u0012\nsin2\u0012ϕi\nµ\u0013\u0013\n. (3.7)\nIt is denoted that i=E, I indicate the onset and the offset on the inflationary phase. F1\nandF2are the hypergeometric functions reading as\nF1=2F1\u0012\n1,1−α\n2; 2−α\n2; cos2\u0012ϕi\nµ\u0013\u0013\n(3.8)\nF2=2F1\u0012\n1,α\n2+ 1;α\n2+ 2; cos2\u0012ϕi\nµ\u0013\u0013\n. (3.9)\nIt has been observed that the number of e-folds imposes extra conditions on α. For the\npresent scaled gravity, the scalar spectral index nsand the tensor-to-scalar ratio rare found\nto be\nns= 1−α2(β+ 1)2sin2\u0010\nϕ\nµ\u0011\ncos2α−2\u0010\nϕ\nµ\u0011\n8πµ2\u0010\n1−cosα\u0010\nϕ\nµ\u0011\u00112\u0010\n1 +β+ 8πM4ω2\u0010\n1−cosα\u0010\nϕ\nµ\u0011\u0011\u0011\n+α(β+ 1)2Bcosα−2\u0010\nϕ\nµ\u0011\n8πµ2\u0010\n1−cosα\u0010\nϕ\nµ\u0011\u00112\u0010\n1 +β+ 8πM4ω2\u0010\n1−cosα\u0010\nϕ\nµ\u0011\u0011\u00112\nr=α2(β+ 1)2sin2\u0010\nϕ\nµ\u0011\ncos2α−2\u0010\nϕ\nµ\u0011\nπµ2\u0010\n1−cosα\u0010\nϕ\nµ\u0011\u00112\u0010\n1 +β+ 8πM4ω2\u0010\n1−cosα\u0010\nϕ\nµ\u0011\u0011\u0011\nwhere the quantity Bis given by\nB=\u0000\n1 +β+ 8πM4ω2\u0001\u0012\n2−α\u0012\n1−cos\u00122ϕ\nµ\u0013\u0013\u0013\n−2 cosα\u0012ϕ\nµ\u0013\u0012\n1 +β+ 16πM4ω2−2παM4ω2\u0012\n1−cos\u00122ϕ\nµ\u0013\u0013\u0013\n+ 4πM4ω2\u0012\n4 +α\u0012\n1−cos\u00122ϕ\nµ\u0013\u0013\u0013\ncos2α\u0012ϕ\nµ\u0013\n. (3.10)\nTo validate the obtained results, one should make contact with the observational findings\nincluding the Planck 2018 and the recently released BICEP/Keck data [29–31]. Here, we\nhave used a normalized ωby multiplying this quantity with Mwhich is fixed to 1. In Fig(1),\nwe illustrate the ns−rcurves for different values N,β,ωand taking µ=M= 1. Instead\nof giving generic situations, we consider a particular one corresponds to β= 1.\nThe associated ns−rcurves are plotted with the presence of the Planck contour constraints\n[29,30]. In graphic representations, the values of the number of e-folds have been considered\nby taking into account of the experimental constraint namely N > 60. In this way, the\n8α\n5 10 1520\n0.95 0.96 0.97 0.980.000.020.040.060.080.10\nnsr\nα\n5 10 1520\n0.95 0.96 0.97 0.980.000.020.040.060.080.10\nnsr\nα\n5 10 1520\n0.95 0.96 0.97 0.980.000.020.040.060.080.10\nnsrFigure 1: The behavior of one-dimensional real ns−rcurves by taking α= 2.1,···,20, the parameter\nω= 1,···,100normalized by the mass scale M. The left, the center and the right plots correspond to the\ne-folding number N= 55,N= 60 andN= 65, respectively. The green and the orange contour constraints\nrepresent 68% and 95% confidential levels of Planck results (TT,TE,EE+lowE +lensing +BK15+BAO),\nrespectively. The yellow contour is associated with the Planck results (TT,TE,EE+lowE+lensing).\nns−rcurves have been analyzed by varying αin the interval [2 .1,20]. It has been observed\nfrom this figure that the range of rincreases by increasing α. Considering large values of\nα, the range of nsincreases. Increasing ω, the range of the scalar spectral is increased\nand the tensor-to-scalar ratio involves small values. Fixing αandω,nsincreases with N.\nA close inspection reveals that the decoupling between the scalar potential and the scaled\ngravity with the kinetic term could bring interesting numerical results of the spectral index\nnsand the tensor-to-scalar ratio r. However, it is not good enough with respect to the range\nassociated with the Planck and the BICEP/Keck data [29–31]. The obtained range of ris\n[0.02,0.08] which is in a good range but still not good enough.\n3.2 Parameter coupling scenario\nIn this subsection, we follow the second scenario by implementing the gravity in the matter\nsector by means of the scalar potential. In this way, it takes the following form\nV(ϕ) =M4\"\n1−cos\u0012ϕ\nµ\u0013β#\n. (3.11)\nThis parameter interplay can be viewed as an alternative way to generate a coupling via the\nmoduli space. This provides a bridge between MgandMmin order to find results which\ncould be confronted with the observational findings. A rapid examination reveals that the\nscalar potential form imposes extra conditions on the gravity parameter β. It should be\ndifferent to 2 and −2. Instead of repeating the computations and relations, we give only\ngraphic representations.\nAs the previous scenario, to check the obtained results, the contact with observational\ndata including the Planck 2018 and the recently released BICEP/Keck data should be pro-\nvided [29–31]. In Fig(2), we illustrate the ns−rcurve behaviors by varying N,β,ω, with\n9β\n5 10 15 20\n0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.9850.000.020.040.060.080.10\nnsr\nβ\n5 10 15 20\n0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.9850.000.020.040.060.080.10\nnsr\nβ\n5 10 15 20\n0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.9850.000.020.040.060.080.10\nnsrFigure 2: The behavior of one dimensional curves ns−rby varying the gravity parameter β= 2.1, . . . , 20\nandω= 1, . . . , 100normalized by the mass scale M. The plots from left to right correspond to the number\ne-folding N= 55,N= 60 andN= 65, respectively.\nnormalized parameters namely µ=M= 1.\nThese behaviors are plotted by implementing the Planck contour constraints [29,30]. They\nhave been examined by varying βin the interval [2 .1,20]. It follows from this figure that r\ndecreases by increasing β. Taking large values of β, the range of nsdecreases. Increasing\nω, the range of the scalar spectral increases and the tensor-to-scalar ratio takes high values.\nFixing βandω,nsincreases by increasing N. A close examination shows that the coupling\nbetween the scalar potential and the scaled gravity with a kinetic term provides interesting\nnumerical values of the spectral index nsand the tensor-to-scalar ratio rwhich are in a very\ngood range of the Planck data which covers both 95% and 68% CL contour regions of the\nrecent released BICEP/Keck data [29–31]. The range of ris [0.001,0.07] which is in a very\ngood agreement with the Planck findings being smaller than 0 .1. It has been anticipated\nthat the coupling between the scalar potential and the scaled gravity with a kinetic term\ncould be worked out to bring very good agreements with the experimental data [54–56].\nIt has been suggested that the scalar power spectrum could be exploited to discuss the\nviability of the studied gravity theory [52]. In the slow-roll approximations, this quantity\ntakes the form\nPζ≈H2\n8π2ϵ. (3.12)\nTaking ϕ= 0.1,M=µ= 1,ω= 0.1,β= 2.1, this found to be\nPζ≈3.97444 .10−5(3.13)\nbeing an acceptable value. Other values could be also obtained in certain regions of the\nmoduli space M.\n4 Axionic dark matter\nThe idea that the inflaton might be related to axion-like particles belongs to one of the\nprimary theoretical problems of the minimal inflationary scenario [61–66]. Such a connection\n10is a hint of non ordinary physics beyond the minimal model of inflation introduced above.\nSimilar to the axion field suggested to solve the strong CP problem of quantum chromo\ndynamics [67, 68], a possible connection is assuming that the inflaton is a pseudo-Nambu-\nGoldstone boson (PNGB), of a spontaneously broken approximate global U(1) symmetry,\nwhich arises in the action with a derivative term ( ∂µϕ)2. In fact, it posses a shift symmetry\nϕ→ϕ+c (4.1)\nwhere cis a real constant. Such a symmetry saves its role as inflaton from being invalid via\na coupling to unknown UV physics by severely bordering the form of its possible interactions\nwith other fields. However, this continuous symmetry is broken by the ALP potential V(ϕ)\nacquired through non-perturbative effects certain gauge fields Fi. They are naturally coupled\nto the axion via ∼gif−1\naϕFi∼\nFiwith giis a model-dependent coupling constant, and fais\nthe axion decay constant. The latter determines the scale at which the axion symmetry is\nbroken. Supposing the associated global symmetry is spontaneously broken at a scale fa,\nwith a soft explicit symmetry breaking at a lower scale M, the axionic inflation is quietly\ndescribed by these two scales which will be specified by the exigencies of infallible inflation.\nRoughly, in connection with the considered model associated with Eq.(3.11), the resulting\nscaled gravity axion potential is generally of the form\nV(ϕ) =M4h\n1−cos (θ)βi\n(4.2)\nwhere we have used µ=faand the canonically normalized field ϕ≡faθwhere the dynamical\nfield is constrained by θ≪0 [65,67]. Concretely, the axion-like field is chosen such that the\ncorresponding potential given by Eq.(4.2) is minimized at θ= 0. This potential breaks the\nglobal shift symmetry of the axion Eq.(4.1) down to a discrete symmetry\nϕ→ϕ+ 2πfa. (4.3)\nBy expanding the scalar potential given by Eq.(4.2) to leading order in θlike\nV(ϕ)≃M4\"\n1−\u0012\n1−θ2\n2\u0013β#\n≃M4\u0012\nβθ2\n2\u0013\n=1\n2M4β\nf2\naϕ2, (4.4)\none can read the axion mass from the only appearing gravity scaled mass term such as\nmϕ≃p\nβM2\nfa. (4.5)\nIt is denoted that the inflation mass scale Mand the axion decay constant faare not fixed\nby theory. This could be scaled by the gravity parameter β. Therefore, higher values of\nthe associated axion symmetry scale M≪fa≪MPlanck imply lighter and less interacting\naxions with the SM. In this case, the corresponding axions would be relativistic particles.\nThis sets a three-dimensional parameter space on which the axion-like searches depend\n11Mϕ={M, f a, β}. (4.6)\nSuch weakly interacting light particles could make or contribute as a sub-component to hot\nDM in the Universe. Indeed, in such a case, these particles should have a local mass density\nof that pretended in our proximity to account for the dynamics of our galaxy. Concretely,\nthey should be distributed in a halo endging our galaxy with a characteristic relativistic\nvelocity close to the light speed va∼c. In spite of the fact that the axion decays are\nstill constrained by several cosmological arguments, these particles could be detected either\nindirectly via their self-annihilation products likely into photons or neutrinos\nϕϕ→XX X =γ, ν (4.7)\nor directly by considering their interaction via the tiny shocks with the detector materials.\nIn particular, the corresponding typical kinetic energy would of the order of\nKϕ∼p\nβM2\nfac2∼eV (4.8)\nwhere the axion mass upper bound mϕ<0.5eVis put from the constrained value of hot DM\ndensity by cosmological observations [68]. In this way, we can now deal with the involved\ninflation scale Mof the proposed gravity model. Concretely, owing to the high and wide\nenergy range of the decay constant of the axion fa≳1010GeV and according to the considered\nrange of the scaled gravity parameter β≤20, we get\nM≪fa. (4.9)\nIn this approach, for the underlying scale Mto be high enough to account for inflation, the\naxionic scale can go up to the Planck scale fa≲MPlanck .\n5 Conclusion\nIn this work, we have investigated parameter coupling scenarios in a scaled gravity via the\ninflation moduli space. In particular, we have observed that this moduli space contains two\nfactors providing two inflation scenarios. The first one is called parameter decoupling scenario\nwhile the second one is parameter coupling scenario. Motivated by the modified gravity\ntheories F(R)̸=Rand inflationary physics, we have proposed and investigated an inflation\nmodel in a scaled gravity F(R) =R+βR, where βis a dimensionless scaling parameter\nfor both scenarios. For the second one, this gravity parameter has been implemented in\nparticular inflation potential given by V(ϕ) =M4\u0014\n1−cos\u0010\nϕ\nµ\u0011β\u0015\nbeing considered to drive\nthe inflation. Exploiting the slow-roll analysis, the gravity scale parameter βhas been\napproached with respect to the range of the associated computed cosmological observables\nnsandraccording to the recent Planck and BICEP/Keck data. In the second part of\n12this work, we have investigated an axionic dark matter in the proposed gravity model by\nconsidering the case where the inflaton is taken to be identified with an axion-like field\nϕ=faθwith the decay constant fa=µ. Based on known data, we have shown that the\nunderlying inflation scale Mis constrained to be much lower than the associated axion scale\nM≪fa.\nIt has been concluded that connecting inflation with axion-like particles could offer an\nattractive model that can account for most observed cosmological structures, including DM.\nThis investigation road is now undergoing an expansion phase and the experimental endeav-\nors are rapidly increasing in intensity as well as diversity to trap DM by probing a large\nfraction of the axion parameter space. Seen that a discovery in the forthcoming years is not\nprecluded, such a finding would be a breakthrough discovery that could reframe the posterior\ndevelopments of particle physics, astrophysics and cosmology, including other high energies\ntheories.\nDeclarations\nThe authors declare that they have no known competing interests or personal relationships\nthat could have appeared to influence the work reported in this paper.\nEthical Approval\nIt is not applicable in this article.\nCompeting interests\nThe authors declare that they have no known competing interests.\nAuthors’ contributions\nThe all authors have worked on the proposed work.\nFunding\nNo fundings are associated with this article.\nAvailability of data and materials\nNo data are associated with this article.\n13Acknowledgments\nThe authors would like to thank I. Aamer, N. Askour, S. Baddis, H. Belmahi, M. Benali, H.\nEl Moumni, Y. Hassouni, M. Oualaid, and M.B. 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We show that if Phas countable\nsupport and satisfies certain tail and moment conditions, th en in the limit as\nN→ ∞the process ( YN(t))t≥0converges in law to the process ( S(t)P)t≥0, in the\nso-called Meyer-Zheng topology, where ( S(t))t≥0is the Fisher-Wright diffusion\nwith diffusion constant Dgiven by 1 /D=/integraltext\nR+(1/r)P(dr).\nKeywords: Moran model; random resampling rates; Meyer-Zheng topology; Ly apunov\nfunction; Fisher-Wright diffusion.\nMSC 2010: Primary 60J70, 60K35; Secondary 92D25.\n1 Introduction\nTheMoranmodel, describing theevolutionvia resampling ofapopulationofindividuals\ncarrying genetic types, is a workhorse in population genetics. It is t ractable because it\nis mean-field, i.e., all individuals exchange type with each other in the sa me manner,\nand because it has a simple dual, namely, the death process on the po sitive integers.\nIt has been the basis for much more sophisticated models incorpora ting evolutionary\nforces such as mutation, selection, recombination andmigration(s ee [Dur08]). Selection\namounts to letting the rateat which an individual resamples depend o n itscurrent type.\nThis bias causes a genetic drift towards the type for which the resampling is slowest,\naffecting in significant ways how the population evolves over time.\n1.1 The two-type Moran model\nThe standard model for evolution of types in population genetics is t he two-type Moran\nmodel (see [Dur08, Section 1.5]). Consider a population of N∈Nindividuals. Each\nindividual carries either type 1 or type 0, and at rate 1 chooses an in dividual uniformly\n1at random from the population (possibly itself) and adopts its type. LetXN(t) be the\nnumber of individuals with type 1 at time t. For every N∈N,\nlim\nt→∞P(XN(t) = 0 orXN(t) =N) = 1, (1.1)\nbecause{0}and{N}are traps for ( XN(t))t≥0: genetic diversity is lost forever when all\nindividuals have type 0 or all individuals have type 1, respectively.\nTo see what happens prior to trapping, scale space and time to get\nYN(t) =1\nNXN(Nt), (1.2)\nwhich is the fraction of individuals of type 1 at time Nt. It is well know that if\nlimN→∞YN(0) =y∈[0,1], then\nlim\nN→∞L[YN] =L[Y] in the Skorohod topology , (1.3)\nwhereYN= (YN(t))t≥0andY= (Y(t))t≥0, andLstands for law. The limit Yis the\nFisher-Wright diffusion\ndY(t) =/radicalbig\nY(t)(1−Y(t))dW(t), Y(0) =y, (1.4)\nwhereW= (W(t))t≥0is standard Brownian motion. The convergence in (1.3) is proven\nby showing that the infinitesimal generator LNofYNconverges to the infinitesimal\ngeneratorLofYon a dense set of test functions (see [EK86, Chapter 4, Section 8 ]).\nThe latter is given by\n(Lf)(y) =y(1−y)f′′(y), y∈[0,1], (1.5)\nfor twice differentiable test functions f: [0,1]→R.\nLetτ= inf{t≥0:Y(t) = 0 orY(t) = 1}. It is know that τ <∞almost surely,\ni.e., genetic diversity is lost at time τN+o(N) asN→ ∞.\n1.2 The two-type Moran model with random resampling rates\nIt is natural to allow for resampling rates that are random. In the present paper\nwe consider a version of the two-type Moran model in which each indiv idual carries\napre-assigned resampling rate , drawn from a discrete probability distribution Pon\nR+= (0,∞). This represents a type-independent selection driven by disorder . We will\nsee that, in the large-population-size limit and after scaling time by th e population\nsize, the empirical distribution of the resampling rates of the individu als with type 1\nconverges to Ptimes a Fisher-Wright diffusion with a diffusion constant that depends\nonP. The proof of this fact turns out to be somewhat delicate because the path of\nthe empirical distribution contracts to the submanifold {sP:s∈[0,1]}at a rate that\ndiverges with the population size (so that convergence of the infinit esimal generator\naway from the submanifold fails). This contraction, which is absent in the standard\n2Moran model, requires us to work with a particular path topology, ca lled theMeyer-\nZheng topology , which roughly speaking induces weak convergence in “time ×space”.\nWe may think of the latter as convergence outside a set of exceptio nal times.\nVery little is known about population genetic models with disorder. The reason is\nthatthe disorder destroys the exchangeability of the individua ls, and therefore makes\nduality, another workhorse in population genetics, much more involv ed. In fact, in the\npresent paper we do not resort to duality and work with Lyapunov functions to con-\ntrol the contraction to the submanifold. How the contraction pre cisely occurs comes\nwith a number of subtleties. Our proofs are restricted to discrete Pbecause the sub-\nmanifold depends on P. Perturbation arguments are delicate because they displace the\nsubmanifold.\n1.3 Main theorem\nThroughout the sequel we assume that Psatisfies the following assumption.\nAssumption 1.1.\n(1)Phas countable support, i.e., P=/summationtext\nk∈Nµkδrkwithµ= (µk)k∈Na probability\ndistribution on Nand (rk)k∈Na sequence in R+. Moreover,/summationtext\nk∈Nµkr−1\nk<∞.\n(2) LetR= (Ri)i∈Nbe i.i.d. drawn from P, i.e.,Rhas lawP⊗N. ForN∈N, define\nˆnN\nk=/summationdisplay\ni∈[N]1{Ri=rk}, k∈N, N R={k∈N: ˆnN\nk>0}.(1.6)\nThen\nlim\nN→∞(/summationtext\nk∈NRrk)(/summationtext\nk∈NRr−2\nk)\nNmink∈NRrk= 0P⊗N-a.s. (1.7)\nNote that Assumption 1.1 holds for all Pwith finite support and fails for all Pwith\nuncountable support. In Section 2.3 we derive a sufficient condition o nPwith countable\nsupport guaranteeing that (1.7) holds. For instance, it suffices th atµk≍k−χ,k→ ∞,\nwithχ>2 (right tail condition) and/summationtext\nk∈Nµkr−α\nk<∞,/summationtext\nk∈Nµkrβ\nk<∞forαandβ\nsufficiently large depending on χ(left and right moment conditions).\nIn what follows, we consider the quenched process in which the resampling rates are\nchosen initially and are kept fixed while the process evolves. Let Ridenote the random\nresampling rate of individual i. Note that R= (Ri)i∈[N]has lawP⊗[N]. The empirical\ndistribution of the resampling rates is\nEN=1\nN/summationdisplay\ni∈[N]δRi. (1.8)\nNote that this is a probability distribution on R+and that, P⊗N-a.s.,ENconverges to\nPasN→ ∞. Letτi(t) denote the type of individual iat timet, and let\nYN(t) =1\nN/summationdisplay\ni∈[N]1{τi(Nt)=1}δRi (1.9)\n3denote the empirical distribution of the resampling rates of the indiv iduals with type 1at\ntimeNt. Note that this is an element of M≤1, the set of sub-probability distributions\nonR+. Our main result says that the process YN= (YN(t))t≥0converges in law, in\nthe so-called Meyer-Zheng topology (explained in Section 2), to a simple but interesting\nlimiting process. We may rewrite ENas\nEN=/summationdisplay\nk∈NnN\nkδrk, nN\nk=1\nNˆnN\nk=1\nN/summationdisplay\ni∈[N]1{Ri=rk}. (1.10)\nNote thatNR={k∈N:nN\nk>0}. Also note that |NR| ≤N, and consequently the\nvectornN= (nN\nk)k∈Nhas at most Nnon-zero elements.\nTheorem 1.2. Suppose that Psatisfies Assumption 1.1. If limN→∞YN(0) =ν, then\nlim\nN→∞L[YN] =L[Y]in the Meyer-Zheng topology P⊗N-a.s. (1.11)\nwithY= (Y(t))t≥0given by\nY(t) =S(t)P, (1.12)\nwhereS= (S(t))t≥0is the Fisher-Wright diffusion on [0,1]given by\ndS(t) =/radicalbig\nDS(t)(1−S(t))dW(t), S(0) =ν(R+), (1.13)\nwith diffusion constant Ddefined as 1/D=/integraltext\nR+(1/r)P(dr) =/summationtext\nk∈Nµkr−1\nk.\nRemark 1.3. Theorem 1.2 says that YN= (YN(t))t≥0collapses onto the submani-\nfold{sP:s∈[0,1]}in the limit as N→ ∞and on this submanifold performs an\nautonomous Fisher-Wright diffusion with a diffusion constant that is d etermined by\nP. The contraction to the submanifold shows that YNexhibits homogenisation , i.e.,\nthe type and the resampling rate of the individuals are asymptotically independent as\nN→ ∞. The inverse of the diffusion constant 1 /Drepresents the average time be-\ntween two successive resamplings of an individual. Section 2.3 contain s examples where\nAssumption 1.1 holds. Assumption 1.1 excludes Pwith uncountable support. We con-\njecture that Theorem 1.2 continues to hold in that setting, subjec t to appropriate tail\nand moment conditions, but we are unable to provide a proof. Along t he way we will\nsee what the hurdles are.\nRemark 1.4. Our model canbe interpreted as a Moranmodel with a particular kind of\nselection by viewing the resampling as done in pairs. Pick a pair of individu als, decide\nrandomly which of the two individuals adopts the type of the other ind ividual. In the\nstandard Moran model the latter is decided uniformly at random. In our model it is\ndecided with probabilities that depend on the inverses of the resamp ling rates of the\ntwo individuals.\nRemark 1.5. It is straightforward to extend Theorem 1.2 to the Moran model wit h\nfinitely many types. It is more challenging to include the Fleming-Viot mo del with\ninfinitely many types. In the latter setting, it would be interesting to see what happens\nwhen the resampling rate distribution has no finite inverse moment at zero. Slow rates\nmay affect the property that the state of the Fleming-Viot diffusion after an arbitrary\npositive time becomes atomic (a property referred to as “coming do wn from infinity”).\n4Acknowledgements. SA was supported through a Knowledge Exchange Grant from\nICTS and an Infosys Excellence Grant. FdH was supported throug h NWO Gravitation\nGrant NETWORKS-024.002.003. AR was supported through Singapo re Ministry of\nEducation Academic Research Fund Tier 2 grant MOE2018-T2-2-07 6. The present\nwork evolved out of the master thesis written by Jannetje Driesse n at Leiden University\ncompleted on 25 May 2022. We also thank the International Centre for Theoretical\nSciences (ICTS) for hospitality during the ICTS-NETWORKS Worksh op “Challenges\nin Networks” in January 2024.\n2 Definitions and notation\nIn Section 2.1 we define the random process precisely and in Section 2 .2 we recall the\nMeyer-Zheng topology. In Section 2.3 we derive some tail estimates for the random\nresampling rates and exhibit sufficient conditions under which Assump tion 1.1 holds.\n2.1 The random process\nLetN∈Nbe the size of the population. Individuals are labelled [ N] ={1,...,N}and\ncarry either type 1 or type 0. Each individual has a random resamplin g rate that is\ndrawn from a probability distribution PonR+. The resampling rates are assigned inde-\npendently and before the process starts. We write R= (Ri)i∈N, withRithe resampling\nrate of individual i, and define\nYN(t) =1\nN/summationdisplay\ni∈[N]1{τi(Nt)=1}δRi, (2.1)\nwhereτi(t) is the type of individual iat timet. Note that it suffices to look at type\n1 only, because the total number of individuals with a specific resamp ling rate does\nnot change over time. The state space of YN= (YN(t))t≥0isM≤1, endowed with an\nappropriate metric d(·,·).\nFor ∆∈ {−1,+1},r∈R+andy∈ M≤1, put\nTr,∆(y) =y+∆\nNδr. (2.2)\nThis operator encodes the event that an individual with resampling r aterchanges its\ntype. Namely, when ∆ = −1 an individual with resampling rate rand type 1 chooses\nan individual with type 0 and adopts its type, while when ∆ = +1 an individu al with\nresampling rate rand type 0 chooses an individual with type 1 and adopts its type.\nDefine\nRr,−1(y)dr=ry(dr)[1−y(R+)], Rr,+1(y)dr=r[EN(dr)−y(dr)]y(R+).(2.3)\nThe change ˆTr,−1(y) happens at rate ˆRr,−1(y), because there are y(dr) individuals with\ntype 1 that can choose a new individual from the population at rate r. In order for\n5such an individual to change to 0, it needs to choose an individual with type 0. There\nis a fraction 1 −y(R+) individuals of type 0 in total, and the individual chooses one of\nthem uniformly. The explanation for Tr,+1(y) andRr,+1(y) is analogous.\nNote that the transition rates Rr,∆(y) and the transition operators Tr,∆(y) only\ndepend on the current state yof the process. Hence YNis Markov. We can write down\nthe infinitesimal generator LNofY. Letf:M≤1→Rbe a bounded and continuous\ntest function, and let y∈ M≤1. Then\n(LNf)(y) =/integraldisplay\nR+dr/summationdisplay\n∆∈{−1,+1}Rr,∆(y)[f(Tr,∆(y))−f(y)]N2. (2.4)\nThe termN2comes from the time scaling by Nand the space scaling by 1 /N.\n2.2 The Meyer-Zheng topology\nFor more background on what is written below, we refer the reader to [DM78, Chapter\nIII, no.40–46], [MZ84], and [GdHO22, Appendix B].\nThe Meyer-Zheng topology assigns to each ¯R-valued Borel measurable function w=\n(w(t))t≥0a probability measure on [0 ,∞]ׯRdefined by\nψw(A) =/integraldisplay\n[0,∞]ׯR1A(w(t))e−tdt, (2.5)\ni.e.,ψwis the image under the mapping t→(t,w(t)) of the probability measure e−tdt.\nThis image measure is called the pseudopath associated with w, and is simply the\noccupation measure of w. We say that a sequence of pseudopaths induced by a sequence\nof c` adl` ag paths ( wN)N∈Nconverges to a pseudopath induced by a c` adl` ag path wif, for\nall continuous bounded function f(t,w(t)) on [0,∞]ׯR,\nlim\nN→∞/integraldisplay\n[0,∞)f(t,wN(t)) e−tdt=/integraldisplay\n[0,∞)f(t,w(t)) e−tdt. (2.6)\nSinceapseudopathisameasure, convergenceofpseudopathsisc onvergenceofmeasures.\nLet Ξ be the space of all pseudopaths. Endowed with the Meyer-Zh eng topology, Ξ is\na Polish space.\nLetvN: [0,∞)→ M ≤1denote the path of YNandv: [0,∞)→ M ≤1the path of\nY. Forf∈Cb(Ξ), define\nψvN(f) =/integraldisplay\n[0,∞)f(t,vN(t)) e−tdt, ψ v(f) =/integraldisplay\n[0,∞)f(t,v(t)) e−tdt.(2.7)\nSuppose that f(t,x) = 1t∈[a,b]¯f(x), with¯f:M≤1→Rbounded and continuous and\n0≤a<b<∞. The set of all functions of this form generates Cb(Ξ). Write\nψvN(f) =/integraldisplayb\na¯f(vN(t)) e−tdt, ψ v(f) =/integraldisplayb\na¯f(v(t)) e−tdt. (2.8)\n6To prove convergence in the Meyer-Zheng topology, we need to sh ow that\nlim\nN→∞E[ψvN(f)] =E[ψv(f)]∀f∈Cb(Ξ), (2.9)\nwhich is equivalent to\nlim\nN→∞E/bracketleftbigg/integraldisplayb\na/vextendsingle/vextendsingle¯f(YN(t))−¯f(Y(t))/vextendsingle/vextendsinglee−tdt/bracketrightbigg\n= 0∀¯f∈Cb(M≤1),0≤a<b<∞.\n(2.10)\nSuppose that ¯fis Lipschitz continuous with Lipschitz constant C∈Rfor a given\ndistancedonM≤1. By the Stone-Weierstrass theorem, the space of polynomial fun c-\ntions on the set of closed intervals in [0 ,1]Nis dense in the space of continuous functions\non [0,1]N. Furthermore, the space of Lipschitz functions contains the spa ce of polyno-\nmial functions. This means that any result for Lipschitz functions c an be generalised\nto continuous ¯f. We find that\nE/bracketleftbigg/integraldisplayb\na/vextendsingle/vextendsingle¯f(YN(t))−¯f(Y(t))/vextendsingle/vextendsinglee−tdt/bracketrightbigg\n≤CE/bracketleftbigg/integraldisplayb\nad(YN(t),Y(t))e−tdt/bracketrightbigg\n=C/integraldisplayb\naE/bracketleftbig\nd(YN(t),Y(t))/bracketrightbig\ne−tdt,(2.11)\nwhere we use Fubini’s theorem to interchange the integral and the e xpectation. Thus,\nif we manage to prove that\nlim\nN→∞E/bracketleftbig\nd(YN(t),Y(t))/bracketrightbig\n= 0 uniformly in tover compact subsets of (0 ,∞),(2.12)\nthen we have proven that YNconverges to Yin the Meyer-Zheng topology.\n2.3 Examples satisfying Assumption 1.1\nIn this subsection we will identify examples satisfying Assumption 1.1 u sing tail and\nmoment estimates. We will need with the following conditions:\n(I) There are α,β∈(0,∞) such that\n/summationdisplay\nk∈Nµkr−α\nk<∞,/summationdisplay\nk∈Nµkrβ\nk<∞. (2.13)\n(II) There are γ,δ∈(0,1] such that\nlimsup\nε↓0ε−(1−γ)/summationdisplay\nk∈Nµk1{µk≤ε}<∞,limsup\nε↓0εδ/summationdisplay\nk∈N1{µk>ε}<∞.(2.14)\n(III) There is a χ>1 such that\nµk≍k−χ, k→ ∞. (2.15)\n7Lemma 2.1. Define (recall (1.6)and(3.19))\nM−\nN= min\ni∈[N]Ri, M+\nN= max\ni∈[N]Ri, ZN=/summationdisplay\ni∈[N]1{Ri/ne}ationslash=Rj∀1≤j<i}.(2.16)\nWritePandEto denote probability and expectation with respect to the la wP⊗Nof\n(Ri)i∈N.\nIf Condition (I)holds, then\n(1) lim ε↓0liminf N→∞P(M−\nNN1/α≥ε) = 1.\n(2) lim ε↓0liminf N→∞P(M+\nNN−1/β≤ε−1) = 1.\nIf Condition (II)holds, then\n(3) lim ε↓0liminf N→∞P(ZNN−(γ∨δ)≤ε−1) = 1.\nIf Condition (III)holds, then\n(4) lim ε↓0liminf N→∞P(ZNN−υ\nχ−1≤ε−1) = 1∀v >1.\nProof.(1) Estimate, for ε>0,\nP(M−\nN≥εN−1/α) =P/parenleftbig\nRi≥εN−1/α∀i∈[N]/parenrightbig\n=P(R1≥εN−1/α)N\n=/parenleftbig\n1−P((1/R1)α>ε−αN)/parenrightbigN≥/parenleftbig\n1−εαN−1E[(1/R1)α]/parenrightbigN→exp(−εαE[(1/R1)α]).\n(2.17)\nBy (2.13), E[(1/R1)α]<∞, and so the claim follows after letting ε↓0.\n(2) Estimate, for ε>0,\nP(M+\nN≤ε−1N1/β) =P/parenleftbig\nRi≤ε−1N1/β∀i∈[N]/parenrightbig\n=P(R1≤ε−1N1/β)N\n=/parenleftbig\n1−P(Rβ\n1>ε−βN)/parenrightbigN≥/parenleftbig\n1−εβN−1E[(R1)β]/parenrightbigN→exp(−εβE[(R1)β]).(2.18)\nBy (2.13), E[(R1)β]<∞, and so the claim follows after letting ε↓0.\n(3) Estimate\nP(N−(γ∨δ)NR≤ε−1) = 1−P(N−(γ∨δ)NR>ε−1)≥1−εN−(γ∨δ)E[ZN].(2.19)\nWriteZN=/summationtext\ni∈[N]1{Ri/ne}ationslash=Rj∀1≤j<i}, and compute\nE[ZN] =/summationdisplay\ni∈[N]P(Ri/ne}ationslash=Rj∀1≤j <i)\n=/summationdisplay\ni∈[N]/summationdisplay\nk∈NP(Ri=rk)P(Ri/ne}ationslash=Rj∀1≤j <i|Ri=rk)\n=/summationdisplay\ni∈[N]/summationdisplay\nk∈Nµk(1−µk)i−1=/summationdisplay\nk∈Nµk1−(1−µk)N\n1−(1−µk)=/summationdisplay\nk∈N[1−(1−µk)N].(2.20)\n8Define\nAN=/summationdisplay\nk∈Nµk1{µk≤1/N}, B N=/summationdisplay\nk∈N1{µk>1/N}. (2.21)\nThen\nE[ZN]≤cNNAN+BN (2.22)\nwithcN= supx∈(0,1/N][−1\nxlog(1−x)]→1 asN→ ∞. By (2.14), limsupN→∞N1−γAN\n<∞and limsupN→∞N−δBN<∞. Hence limsupN→∞N−(γ∨δ)E[ZN]<∞, and so the\nclaim follows after letting ε↓0 in (2.19).\n(4)Theproofofthispartisinspired by theproofpresented in[ZK01 , Lemma 5]. Return\nto (2.20). Let υ>1. Thenυ\nχ−1>1\nχ−1, and hence χ−χ−1\nυ>1. LetN0=⌊υ\nχ−1⌋. Then\n/summationdisplay\nk∈N[1−(1−µk)N] =/summationdisplay\nk∈N0[1−(1−µk)N]+/summationdisplay\nk≥N0+1[1−(1−µk)N]\n≤N0+/summationdisplay\nk≥N0+1[1−(1−µk)N].(2.23)\nNote that for N≥N0+1 there are C1,C2>0 such that\n1−(1−µk)N≤1−/parenleftbigg\n1−C11\nkχ/parenrightbiggkχ−1\nυ\n≤C21\nkχ−χ−1\nυ. (2.24)\nHence there is a C3>0 such that E[ZN]≤C3Nχ\nυ−1, and so the claim follows after\nlettingε↓0 in (2.19).\nCorollary 2.2. Assumption 1.1 is satisfied if either of the following is true :\n(a) Conditions (I)and(II)hold with\n2(γ∨δ)+1\nβ<1−3\nα. (2.25)\n(b) Conditions (I)and(III)hold with\n1\nχ−1+1\nβ<1−3\nα. (2.26)\nProof.Note that/summationtext\nk∈NRrk/summationtext\nk∈NRr−2\nk\nNmink∈NRrk≤Z2\nNM+\nN\nN(M−\nN)3. (2.27)\n(a) Note that Lemma 2.1(1) implies that a.s. there exist ε1>0 andN1such that\nM−\nNN1/α≥ε1∀N >N 1. (2.28)\n9Likewise, we can find N2,N3andε2,ε3by Lemma 2.1(2,3). It follows that a.s. for\nN0= max{N1,N2,N3}andε0= min{ε1,ε2,ε3}>0,\nZ2\nNM+\nN\nN(M−\nN)3≤N2(γ∨δ)N1/β\nε6\n0N1−3/α∀N >N 0. (2.29)\nFrom this it easily follows that the right-hand side of (2.27) converge s to zero a.s.\n(b) Note that, as in part (a), using Lemma 2.1(1,2,4) we have that a.s . there are ˜Nand\n˜εsuch that\nZ2\nNM+\nN\nN(M−\nN)3≤Nυ\nχ−1N1/β\n˜ε6N1−3/α∀N >˜N. (2.30)\nFromthis it easily follows thatthe right-handside of (2.27) converge s to zero a.s. (recall\nthatv >1 is arbitrary).\nThe example given below Assumption 1.1follows from(2.26). We close wit htwo further\nexamples for which (2.25) and (2.26) are met.\nExample 2.3. (I) Pickrk=kandµk=p(1−p)k−1fork∈Nand some 0 < p <1.\nThen (2.13) holds for any α,β <∞and (2.14) holds for any γ,δ >0. Hence (2.25) is\nmet by taking α,βlarge enough and γ,δsmall enough.\n(II) Pickrk=kandµk=C(χ)k−χfork∈Nand someχ >3. Then (2.13) holds for\nanyα<∞andβ <χ−1 and (2.15) holds as well. Hence (2.26) is met by taking α>0\nlarge enough and β <χ−1 close enough to χ−1.\n3 Proof of the main theorem\nIn Section 3.1 we state a key lemma (Lemma 3.1 below) that implies Theor em 1.2. In\nSections 3.2–3.5 we prove this lemma. The proof is based on the const ruction of a Lya-\npunov function together with a projection argument , showing that the scaled dynamics\nrapidly contracts to the submanifold {sP:s∈[0,1]}. Outside the submanifold the gen-\nerator of the scaled dynamics diverges as N→ ∞. We show that inside the manifold\nthe generator converges to that of the Fisher-Wright diffusion in ( 1.13). Throughout\nthis section, Assumption 1.1 is in force.\n3.1 Definitions and a key lemma\nRecall that we denoted by ( Ri)i∈Nthe sequence of independent random variables with\ndistribution P, andENthe empirical distribution of the first Nrandom variables, i.e.,\nEN=/summationdisplay\nk∈NnN\nkδrk, nN\nk=N−1/summationdisplay\ni∈[N]1{Ri=rk}. (3.1)\nPutNR={ki∈N:nN\nki>0}. The measure-valued process YNcan be represented as\na vector-valued process YN= (YN\nk)k∈N, also with at most Nnon-zero elements. Each\n10YN\nkmay be viewed as the fraction of YNthat have the resampling rate rk, and thus\nYN\nk>0 if and only if k∈NR.\nConvergence of Markov processes is usually proven by showing tha t the infinitesi-\nmal generators converge (see [EK86, Chapter 4, Section 8]. For c onvenience we first\nreformulate the infinitesimal generator of YN. Letf:RN→Rbe three times partially\ndifferentiable, and let\n|f|3= sup\ni,j,k∈N/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂3\n∂zi∂zj∂zkf/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞. (3.2)\nFork∈N, letXrk(t) be the number of individuals of type 0 with resampling rate rkat\ntimetand\nYN\nk(t) =XN\nrk(Nt)\nN, (3.3)\nwhich is the fraction of individuals at time Ntof type 1 with resampling rate rk. The\nrandom process YN= (YN(t))t≥0can now be viewed as YN(t) = (YN\nk(t))k∈Nliving on\nthe state space\nQ=/braceleftbig\ny= (yk)k∈N: 0≤yk≤nN\nk∀k∈N/bracerightbig\n. (3.4)\nFork∈N, we can view (2.2) as\nTk,∆y=y+∆ek\nN, (3.5)\nwithekthe unit vector in direction k, and the transition rates in (2.3) as\nRk,−1(y) =rkyk/parenleftBigg/summationdisplay\nl∈N(nN\nl−yl)/parenrightBigg\n, Rk,+1(y) =rk(nN\nk−yk)/parenleftBigg/summationdisplay\nl∈Nyl/parenrightBigg\n,(3.6)\nand the generator LNin (2.4) as\n(LNf)(y) =/summationdisplay\nk∈N/summationdisplay\n∆∈{−1,+1}Rk,∆(y)/bracketleftbig\nf(Tk,∆y)−f(y)/bracketrightbig\nN2. (3.7)\nA straightforward Taylor expansion yields\n(LNf)(y) =−N/summationdisplay\nk∈NRrk/parenleftBigg\nyk−nN\nk/summationdisplay\nl∈NRyl/parenrightBigg\nfk(y)\n+1\n2/summationdisplay\nk∈NRrk/parenleftBigg\nyk−2yk/summationdisplay\nl∈NRyl+nN\nk/summationdisplay\nl∈NRyl/parenrightBigg\nfkk(y)+O/parenleftbigg\nN−1|f|3/summationdisplay\nk∈NRrk/parenrightbigg\n,\n(3.8)\nwherefk(y) = (∂/∂yk)f(y) andfkk(y) = (∂/∂yk)2f(y).\nWe see that the first term in the infinitesimal generator of the proc essYNhas a\nprefactorN, and therefore diverges as N→ ∞on most of the state space. However, we\nwill be able to show that YNis driven to a subspace on which the first term vanishes\nand the infinitesimal generator does converge. To account for th is properly, we will\n11prove thatYNconverges in the Meyer-Zheng topology (from Section 2.2). We nee d to\nintroduce some further quantities. Let\nDN=/parenleftBigg/summationdisplay\nk∈NRnN\nk\nrk/parenrightBigg−1\n, D=/parenleftBigg/summationdisplay\nk∈Nµk\nrk/parenrightBigg−1\n. (3.9)\nBy Assumption 1.1, the strong law of large numbers gives (recall (1.1 0))\nlim\nN→∞DN=DP⊗N-a.s. (3.10)\nDefine\nSN(t) =/summationdisplay\nk∈NRYN\nk(t),ˇSN(t) =DN/summationdisplay\nk∈NR1\nrkYN\nk(t). (3.11)\nLetS= (S(t))t≥0be a Fisher-Wright diffusion with diffusion constant D, defined on\nthe same probability space as YN. Using the triangle inequality, we have\n/vextenddouble/vextenddoubleYN(t)−S(t)P/vextenddouble/vextenddouble\n2≤/vextenddouble/vextenddoubleYN(t)−SN(t)nN/vextenddouble/vextenddouble\n2+/vextenddouble/vextenddoubleSN(t)nN−ˇSN(t)nN/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddoubleˇSN(t)nN−ˇSN(t)P/vextenddouble/vextenddouble\n2+/vextenddouble/vextenddoubleˇSN(t)P−S(t)P/vextenddouble/vextenddouble\n2,(3.12)\nwhere/⌊ard⌊l·/⌊ard⌊l2is theℓ2-norm for vectors. We will bound the right-hand side via a sequence\nof lemmas summarised as follows.\nLemma 3.1. P⊗N-a.s. the expectation of each of the four terms in the right-h and side\nof(3.12)tends to zero as N→ ∞uniformly in ton compacts in (0,∞), where for the\nlast term this holds for an appropriate coupling of ˇSN= (ˇSN(t))t≥0andS= (S(t))t≥0.\nLemma 3.1 implies (2.12) with Y(t) =S(t)PanddtheL2-norm on M≤1. It therefore\nsettles the proof of Theorem 1.2.\n3.2 First term\nProof.The idea behind the proof is to find a Lyapunov function for the infinit esi-\nmal generator that rapidly decays to zero (for background on Ly apunov functions, see\n[Mei17, Chapter 4.6]). To this end, let ( ck)k∈Nbe a sequence in R+(to be determined\nlater), and let h:Q→Rbe defined by\nh(y) =/summationdisplay\nm∈NRcm/parenleftBig\nym−nN\nm/summationdisplay\nl∈NRyl/parenrightBig2\n. (3.13)\nWe have\nhk(y) = 2/summationdisplay\nm∈NRcm/parenleftbig\nδmk−nN\nm/parenrightbig/parenleftBig\nym−nN\nm/summationdisplay\nl∈NRyl/parenrightBig\n(3.14)\nand\nhkk′(y) = 2/summationdisplay\nm∈NRcm/parenleftbig\nδmk−nN\nm/parenrightbig/parenleftbig\nδmk′−nN\nm/parenrightbig\n. (3.15)\n12Substituting these expressions into (3.8), we obtain\n(LNh)(y)\n=−2N/summationdisplay\nk∈NRckrk/parenleftBigg\nyk−nN\nk/summationdisplay\nl∈NRyl/parenrightBigg2\n+2N/summationdisplay\nk∈NRrk/parenleftBigg\nyk−nN\nk/summationdisplay\nl∈NRyl/parenrightBigg/summationdisplay\nm∈NRcmnN\nm/parenleftBigg\nym−nN\nm/summationdisplay\nl∈NRyl/parenrightBigg\n+/summationdisplay\nk∈NRrk/parenleftBigg\nyk−2yk/summationdisplay\nl∈NRyl+nN\nk/summationdisplay\nl∈NRyl/parenrightBigg/summationdisplay\nm∈NRcm/parenleftbig\nδmk−nN\nm/parenrightbig/parenleftbig\nδmk−nN\nm/parenrightbig\n.\n(3.16)\nWith the choice ck= 1/nN\nkfork∈NR, the second term vanishes and, for k∈NR,\n/summationdisplay\nm∈NRcm/parenleftbig\nδmk−nN\nm/parenrightbig/parenleftbig\nδmk−nN\nm/parenrightbig\n=/summationdisplay\nm∈NR,\nm/ne}ationslash=knN\nm+1\nnN\nk(1−nN\nk)2=(1−nN\nk)\nnN\nk.(3.17)\nThus,\n(LNh)(y) =−2N/summationdisplay\nk∈NRrk\nnN\nk/parenleftBigg\nyk−nN\nk/summationdisplay\nl∈NRyl∈NR/parenrightBigg2\n+/summationdisplay\nk∈NRrk\nnN\nk/parenleftbigg\nyk/parenleftBig\n1−/summationdisplay\nl∈NRyl/parenrightBig\n+(nN\nk−yk)/summationdisplay\nl∈NRyl/parenrightbigg\n(1−nN\nk)\n≤ −2NM−\nNh(y)+2Σ N,(3.18)\nwhere\nM−\nN= min\nk∈NRrk,ΣN=/summationdisplay\nk∈NRrk, (3.19)\nand we use the fact that 0 ≤yk≤nN\nk. Hence, putting\ng(t) =Ey[h(YN(t))], (3.20)\nwhereEydenotes expectation conditional on YN(0) =y, we obtain\ng′(t)≤ −2NM−\nNg(t)+2Σ N, (3.21)\nand so, solving the differential inequality, we obtain\nEy/bracketleftbig\n/⌊ard⌊lYN(t)−SN(t)nN/⌊ard⌊l2\n2/bracketrightbig\n=Ey/bracketleftBigg/summationdisplay\nk∈NR/parenleftBig\nYN\nk(t)−nN\nk/summationdisplay\nl∈NRYN\nl(t)/parenrightBig2/bracketrightBigg\n≤Ey/bracketleftBigg/summationdisplay\nk∈NR1\nnN\nk/parenleftBig\nYN\nk(t)−nN\nk/summationdisplay\nlYN\nl(t)/parenrightBig2/bracketrightBigg\n=g(t)≤ΣN\nNM−\nN+/parenleftBig\ng(0)−ΣN\nNM−\nN/parenrightBig\ne−2NM−\nNt.\n(3.22)\n13By Assumption 1.1, the first term in the right-hand side of (3.22) con verges to zero as\nN→ ∞P⊗N-a.s. For the second term, note that\n−nN\nk≤yk−nN\nk/summationdisplay\nl∈NRyl≤nN\nk, k∈NR, (3.23)\nwhich implies\ng(0) =/summationdisplay\nk∈NR1\nnN\nk/parenleftBig\nyk−nN\nk/summationdisplay\nl∈NRyl/parenrightBig2\n≤1. (3.24)\nSince e−x≤x−1,x >0, and limsupN∈N(1/ΣN)<∞P⊗N-a.s., it follows that also the\nsecond term in the right-hand side of (3.22) converges to zero as N→ ∞P⊗N-a.s. and\nuniformly in ton compacts in (0 ,∞).\nnN\n(nN\nk/rk)k(∆N\nk)k\n(rk∆N\nk)k\n(∆N\nk)kYN(t)\nZN(t)ˇZN(t)PYN(t)\nˇPYN(t)\nFigure 1: Figure illustrating the splitting in (3.12). See also (3.25 ), (3.33), (3.35) and Re-\nmark 3.2.\n3.3 Second term\nProof.Define the maps PandˇPfromQtoQby\nPy=/parenleftbigg/summationdisplay\nk∈NRyk/parenrightbigg\nnN,ˇPy=/parenleftbigg\nDN/summationdisplay\nk∈NR1\nrkyk/parenrightbigg\nnN. (3.25)\n14It is straightforward to check that both PandˇPare idempotent maps that project Q\nonto the line spanned by the vector nN. Hence,Py=ˇPPy, and so\n/vextenddouble/vextenddoubleSN(t)nN−ˇSN(t)nN/vextenddouble/vextenddouble\n2=/vextenddouble/vextenddoublePYN(t)−ˇPYN(t)/vextenddouble/vextenddouble\n2=/vextenddouble/vextenddoubleˇP(PYN(t)−YN(t))/vextenddouble/vextenddouble\n2\n≤ /⌊ard⌊lˇP/⌊ard⌊lop/vextenddouble/vextenddoublePYN(t)−YN(t)/vextenddouble/vextenddouble\n2.(3.26)\nNow,/vextenddouble/vextenddoubleˇPy/vextenddouble/vextenddouble\n2=DN/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈NRyk\nrk/vextendsingle/vextendsingle/vextendsingle/⌊ard⌊lnN/⌊ard⌊l2≤DN/parenleftbig\nM−2\nN/parenrightbig1/2/⌊ard⌊ly/⌊ard⌊l2, (3.27)\nwhere we abbreviate\nM(−2)\nN=/summationdisplay\nk∈NR1\nr2\nk. (3.28)\nWe know from (3.10) that DNis bounded P⊗N-a.s. Consequently,/vextenddouble/vextenddoubleˇP/vextenddouble/vextenddouble\nopis bounded\nP⊗N-a.s. Using the latter in combination with (3.26), (3.22) and the definit ion ofPy,\nwe have\nEy/vextenddouble/vextenddoubleSN(t)nN−ˇSN(t)nN/vextenddouble/vextenddouble\n2\n≤CM(−2)\nN/parenleftbiggΣN\nNM−\nN+/bracketleftbigg\ng(0)−ΣN\nNM−\nN/bracketrightbigg\ne−2NM−\nNt/parenrightbigg\n≤C/parenleftBigg\nΣNM(−2)\nN\nNM−\nN+M(−2)\nNg(0)e−2NM−\nNt/parenrightBigg\n.(3.29)\nBy Assumption 1.1,the first term in the right-hand side of (3.29) conv erges to zero as\nN→ ∞P⊗N-a.s. Again using (3.24) and e−x≤x−1,x>0, we see that the second term\nin the right-hand side of (3.29) converges to zero as N→ ∞P⊗N-a.s. and uniformly in\nton compacts in (0 ,∞).\nRemark 3.2. Figure 1 illustrates why we need the various quantities in the right-ha nd\nside of (3.12), which we discuss now.\n(a) WhileYN(t) quickly moves close to its projection PYN(t) =SN(t)nNon the\n(blue colored) submanifold, and the distance between YN(t) andPYN(t) can be con-\ntrolled via the Lyapunov function argument, the motion of PYN(t) on the submanifold\nis difficult to approximate directly by a diffusion because the drift expe rienced byYN(t)\ndoes not align with the direction of the projection P. This means that YN(t), and by\nextension also PY(t), experiences a side-ways wind across the submanifold, even when\nit is close to the submanifold. This wind is not seen by PYN(t) directly, which is there-\nforenotobviously closetoMarkov. Incontrast, forthetransfo rmedquantity ZN(t)with\nZN\nk(t) =YN\nk(t)/rk,k∈NR, its projection ˇZN(t) and drift do align, making the motion\nofˇZN(t) on the submanifold close to Markov, i.e., the drift term of ˇZN(t) vanishes (see\n(3.34) below).\n(b) Note that ˇPYN(t) is just a rescaled version of ˇZN(t), and the scaling is chosen\nso as to bring ˇPYN(t) back to the neighbourhood of PYN(t). Fortunately, the wind is\nnot too strong, i.e., the direction of the main drift and the projectio n are close enough\nso long asYN(t) is close to the submanifold. Hence, the distance between ˇPYN(t) and\nPYN(t) can be controlled by means of the distance between YN(t) andPYN(t).\n15(c) Figure 1 is drawn in correct proportions using the values nN= (0.37,0.63),\nYN(t) = (0.17,0.61) andr= (0.8,3.5). However, note that the actual drift experi-\nenced by the processes is much larger due to the additional factor Nappearing in their\ngenerators, which is not reflected in the figure.\n3.4 Third term\nProof.Since/vextenddouble/vextenddoubleˇSN(t)nN−ˇSN(t)P/vextenddouble/vextenddouble\n2≤ˇSN(t)/vextenddouble/vextenddoublenN−P/vextenddouble/vextenddouble\n2, (3.30)\nthe result is a straightforward consequence of the strong law of la rge numbers for the\nmultinomial distribution and the observation that ˇSN(t)≤1.\n3.5 Fourth term\nThe proof uses the following lemma.\nLemma 3.3 ([EK86, Chapter 4, Corollary 8.4]) .PutE= [0,1]andEN={y∈\nEKN:yl≤nN\nl, l∈[KN]},N∈N, both endowed with the respective Euclidean dis-\ntances. Let A⊂Cb(E)×Cb(E)be linear and such that the closure of Agenerates a\nstrongly continuous contraction semigroup on the domain DofA(which is assumed to\nbe separating). Let Xbe a Markov process with generator A. For each N∈N, letYN\nbe a progressive Markov process on ENcorresponding to a measurable contraction semi-\ngroup with full generator ˆAN⊂Cb(EN)×Cb(EN). LetηN:EN→Ebe measurable.\nSuppose that, for each (f,g)∈AandT >0, there exist (fN,gN)∈ˆANsuch that\nsup\nN∈Nsup\n0≤t≤TEN[|fN(YN(t))|]<∞,sup\nN∈Nsup\n0≤t≤TEN[|gN(YN(t))|]<∞,(3.31)\nand\nlim\nN→∞EN/bracketleftbig/vextendsingle/vextendsinglefN(YN(t))−f(ηN(YN(t)))/vextendsingle/vextendsingle/bracketrightbig\n= 0,lim\nN→∞EN/bracketleftbig/vextendsingle/vextendsinglegN(YN(t))−g(ηN(YN(t))/vextendsingle/vextendsingle/bracketrightbig\n= 0,\n(3.32)\nwhereENdenotes expectation with respect to YN(t). Then the finite-dimensional distri-\nbutions ofηN(YN)converge to those of XasN→ ∞.\nProof.All spaces arecompact, and hence separable and complete. Notet hat, for each t,\n{XN(t)}N∈Nis relatively compact, since it is a subset of a compact set. The state ment\nnow follows from [EK86, Chapter 4, Corollary 8.4] after taking M=Dtherein and\nusing Eq. (8.12) therein.\nWe are now ready to show that the fourth term vanishes in expecta tion.\nProof.Let ˇzbe defined by\nˇzk(y) =nN\nk/summationdisplay\nl∈NRyl\nrl. (3.33)\n16From (3.8) (or otherwise) it is not difficult to see that\n(LNf)(ˇz(y))\n=−N/summationdisplay\nk∈NR/parenleftBigg\nyk−nN\nk/summationdisplay\nl∈NRyl/parenrightBigg/summationdisplay\nm∈NRnN\nmfm(ˇz)\n+1\n2/summationdisplay\nk∈NR1\nrk/parenleftBigg\nyk−2yk/summationdisplay\nl∈NRyl+nN\nk/summationdisplay\nl∈NRyl/parenrightBigg/summationdisplay\nm,m′∈NRnN\nmnN\nm′fmm′(ˇz)\n+O(N−1|f|3)\n=1\n2/parenleftBigg/summationdisplay\nk∈NRˇzk−2/summationdisplay\nk∈NRˇzk/summationdisplay\nl∈NRyl+/summationdisplay\nk∈NRnN\nk\nrk/summationdisplay\nl∈NRyl/parenrightBigg/summationdisplay\nm,m′∈NRnN\nmnN\nm′fmm′(ˇz)\n+O(N−1ΣN|f|3),(3.34)\nwhere we use that/summationtext\nl∈NRyl/rl=/summationtext\nkˇzk. Abbreviate\n∆N\nk(y) =yk−nN\nk/summationdisplay\nl∈NRyl. (3.35)\nNote that/summationdisplay\nk∈NR∆N\nk(y)\nrk=/summationdisplay\nk∈NRyk\nnk−/summationdisplay\nk∈NRnN\nk\nrk/summationdisplay\nl∈NRyl, (3.36)\nand, hence,\n/summationdisplay\nl∈NRyl=DN/summationdisplay\nk∈NRˇzk−DN/summationdisplay\nk∈NR∆N\nk(y)\nrk. (3.37)\nWe can therefore write\n(LNf)(ˇz(y)) =1\n2/parenleftBigg/summationdisplay\nk∈NRˇzk−2/summationdisplay\nk∈NRˇzk/parenleftBigg\nDN/summationdisplay\nk∈NRˇzk−DN/summationdisplay\nk∈NR∆N\nk(y)\nrk/parenrightBigg\n+(DN)−1/parenleftBigg/summationdisplay\nk∈NRˇzk−DN/summationdisplay\nk∈NR∆k(y)\nrk/parenrightBigg/parenrightBigg\n×/summationdisplay\nk,l∈NRnN\nknN\nlfkl(ˇz)+O(N−1ΣN|f|3)\n=/parenleftBigg/summationdisplay\nk∈NRˇzk/parenleftBigg\n1−DN/summationdisplay\nk∈NRˇzk/parenrightBigg\n−1\n2/parenleftBigg\n1−2DN/summationdisplay\nk∈NRˇzk/parenrightBigg/summationdisplay\nk∈NR∆N\nk(y)\nrk/parenrightBigg\n×/summationdisplay\nm,m′∈NRnN\nmnN\nm′fmm′(ˇz)+O(N−1ΣN|f|3).\n(3.38)\n17Hence, for ˇsN(y) =DN/summationtext\nk∈NRˇzk, we obtain\n(LNf)(ˇsN(y))\n= (DN)2/parenleftBigg/summationdisplay\nk∈NRˇzk/parenleftBigg\n1−DN/summationdisplay\nk∈NRˇzk/parenrightBigg\n−1\n2/parenleftBigg\n1−2DN/summationdisplay\nk∈NRˇzk/parenrightBigg/summationdisplay\nk∈NR∆N\nk(y)\nrk/parenrightBigg\nf′′(ˇsN)\n+O(N−1|f|3)\n=/parenleftBigg\nDNˇsN(1−ˇsN)−1\n2DN(1−2ˇsN)/summationdisplay\nk∈NR∆N\nk(y)\nrk/parenrightBigg\nf′′(ˇsN)+O/parenleftbig\nN−1ΣN|f|3/parenrightbig\n.\n(3.39)\nNow, let\nA=/braceleftbig\n(f,Lf):f∈C3([0,1])/bracerightbig\n⊂Cb([0,1])×Cb([0,1]). (3.40)\nThe fact that the closure of Agenerates a strongly continuous contraction semigroup\n(i.e., Feller semigroup) corresponding to the Fisher-Wright diffusion o n [0,1] is stan-\ndard. Moreover, it is clear that L=C3([0,1]) is an algebra and separates points. The\nprocessesYNare right-continuous and measurable with respect to their own filtr ation,\nand hence are progressive. Define the linear operator\nAN=/braceleftbig\n(f,LNf):f∈C3(EN)/bracerightbig\n⊂Cb(EN)×Cb(EN), (3.41)\nthe closure of which generates the process YN(we will not need the flexibility of the full\ngenerator).\nWriteηN(y) = ˇs(y) =DN/summationtextyl/rl. Letf∈C3([0,1]) be arbitrary, and let g=Lf.\nSetfN(y) =f(ηN(y)), which clearly is an element of C3(EN), and hence an element\nof the domain of AN. The first conditions of (3.31) and (3.32) are therefore trivially\nsatisfied. With gN=LNfN,\n/vextendsingle/vextendsingleg(ηN(y))−gN(y)/vextendsingle/vextendsingle≤ |DN−D||f|2+1\n2|f|2DN/summationdisplay\nk∈NR1\nrk|∆N\nk(y)|+O/parenleftbig\nN−1ΣN|f|3/parenrightbig\n.(3.42)\nConsequently, from the above, taking expectations and using (3.2 2), we get\nEy/bracketleftbig/vextendsingle/vextendsingleg(ηN(YN(t)))−gN(YN(t))/vextendsingle/vextendsingle/bracketrightbig\n≤ |DN−D||f|2+DN|f|2/parenleftbigg\nM(−2)\nNEy/bracketleftBigg/summationdisplay\nk∈NR∆N\nk(YN(t))2/bracketrightBigg/parenrightbigg1/2\n+O/parenleftbig\nN−1ΣN|f|3/parenrightbig\n≤ |DN−D||f|2\n+DN|f|2/parenleftbigg\nM(−2)\nN/parenleftbiggΣN\nNM−\nN+/bracketleftbigg\ng(0)−ΣN\nNM−\nN/bracketrightbigg\ne−2NM−\nNt/parenrightbigg/parenrightbigg1/2\n+O/parenleftbig\nN−1ΣN|f|3/parenrightbig\n.\n(3.43)\nNow, thefirst termintheright-handsideof (3.43)converges to ze ro asN→ ∞P⊗N-a.s.\nbecause of (3.10). The second term can be treated in the same way as (3.29). Finally,\nnote that there exists a Csuch that, for Nlarge enough, M−\nN≤CM(−2)\nN, and hence\nΣN\nN=ΣNM−\nN\nNM−\nN≤CΣNM(−2)\nN\nNM−\nN. (3.44)\n18Therefore, the second part of (3.32) holds P⊗N-a.s. Finally,\n|gN(y)| ≤ |Lf(ηN(y))|+|LN(f◦ηN)(y)−Lf(ηN(y))|. (3.45)\nThe first term is bounded since f∈C3([0,1]) and the second term is bounded in\nexpectation due to (3.43), which implies the second part of (3.31).\nReferences\n[DM78] C. Dellacherie and P.-A. Meyer. Probabilities and Potential A: General The-\nory. North Holland Publishing Company, New York, 1978.\n[Dur08] R. Durrett. Probability Models for DNA Sequence Evolution . Probability\nand its Applications (New York). Springer, New York, second edition , 2008.\n[EK86] S. N. Ethier and T. G. Kurtz. Markov Processes — Characterization and\nConvergence . WileySeriesinProbabilityandMathematicalStatistics: Prob-\nability and Mathematical Statistics. John Wiley & Sons, Inc., New York ,\n1986.\n[GdHO22] A. Greven, F. den Hollander, and M. Oomen. Spatial popula tions with\nseed-bank: well-posedness, duality and equilibrium. Electron. J. Probab. ,\n27:Paper No. 18, 88, 2022.\n[Mei17] J. D. Meiss. Differential Dynamical Systems , volume 22 of Mathematical\nModeling and Computation . Society for Industrial andApplied Mathematics\n(SIAM), Philadelphia, PA, revised edition, 2017.\n[MZ84] P.-A.Meyer andW.A.Zheng. Tightnesscriteriaforlawsofse mimartingales.\nAnn. Inst. H. Poincar´ e Probab. Statist. , 20:353–372, 1984.\n[ZK01] N. S. Zakrevskaya and A. P. Kovalevski˘ ı. One-parameter probabilistic mod-\nels of text statistics. Sib. Zh. Ind. Mat. , 4:142–153, 2001.\nSiva Athreya, International Centre for Theoretical Sciences, Survey No. 151, Shiv-\nakote, Hesaraghatta Hobli, Bengaluru 560089, India and Ind ian Statistical Institute,\n8th Mile Mysore Road, Bengaluru 560059, India.\nathreya@icts.res.in ,\nFrank den Hollander, Mathematical Institute, Leiden University, Niels Bohrweg 1,\n2333 CA Leiden, The Netherlands.\ndenholla@math.leidenuniv.nl\nAdrian R¨ ollin, Department of Statistics and Data Science, National Univer sity of\nSingapore, 6 Science Drive 2, Singapore 117546.\nadrian.roellin@nus.edu.sg\n19"    },    {        "title": "2402.01425v1.Conformal_vector_fields_on_almost_Kenmotsu_manifolds.pdf",        "content": "arXiv:2402.01425v1  [math.DG]  2 Feb 2024CONFORMAL VECTOR FIELDS ON ALMOST KENMOTSU\nMANIFOLDS\nUDAY CHAND DE, ARPAN SARDAR AND KRISHNENDU DE∗\nAbstract. In this paper, first we consider that the conformal vector field X\nis identical with the Reeb vector field ςand next, assume that Xis pointwise\ncollinear with ς, in both cases it is shown that the manifold N2m+1becomes a\nKenmotsu manifold and N2m+1is locally a warped product N′×fM2m, where\nM2mis an almost K¨ ahler manifold, N′is an open interval with coordinate t,\nandf=cetfor some positive constant c. Beside these, we prove that if a\n(k,µ)′-almost Kenmotsu manifold admits a Killing vector field X, then either\nit is locally a warped product of an almost K¨ ahler manifold and an open\ninterval or Xis a strict infinitesimal contact transformation. Furthermore,\nwe also investigate η-Ricci-Yamabe soliton with conformal vector fields on\n(k,µ)′-almost Kenmotsu manifolds and finally, we construct an example.\n1.Introduction\nIn a contact manifold, the existence of conformal vector fields (b riefly,CVF)\ncan be explained beautifully by some intrinsic properties of contact m anifolds.\nFor example, it is well circulated that [27] a Riemannian manifold Nmof dimen-\nsionmis conformally flat if it concedes a maximal, that is,(m+1)(m+2)\n2-parameter\ngroup of conformal motions. A conformally flat Sasakian manifold is a lso known\nto be of constant curvature 1 [15]. In [16], Okumura established tha t form >3, a\nconnected complete Sasakian manifold equipped with a conformal mo tion is iso-\nmetric to sphere. After that, Sharma [20] extended his research on contact man-\nifolds that admit conformal motions to N(k)-contact metric manifolds. Later,\nthe idea of the holomorphic planar CVFin Hermitian manifolds [21] was also\nintroduced by Sharma. Planar CVFhave also been studied in [21].\nA vector field XonN2m+1satisfying the equation\n£Xg= 2ρg, (1.1)\nρbeing a smooth function and £is the Lie-derivative, is called a CVF. IfXis\nnot Killing, it is termed as non-trivial. If ρvanishes, then the CVFXis named\nKilling. Xis called homothetic, if ρis constant. CVFhave been studied by\nmany authors such as ([5], [7], [8], [14], [20]-[22]) and many others.\nIfr,R,Sindicate the scalar curvature, the curvature tensor and the Ricc i\ntensor, respectively, then the CVFXsatisfies the followings[27]\n(£X∇)(U1,V1) = (U1ρ)V1−(V1ρ)U1−g(U1,V1)Dρ, (1.2)\n0The Mathematics subject classification 2010: 53C15, 53C25.\nKey words and phrases : conformal vector fields, infinitesimal str ict contact transformation,\nη-Ricci-Yamabe solitons, almost Kenmotsu manifolds, ( k,µ)′-almost Kenmotsu manifolds.\n∗Corresponding author\n12\n(£XR)(U1,V1)W1=g(∇U1Dρ,W 1)V1−g(∇V1Dρ,W 1)U1(1.3)\n+g(U1,W1)∇V1Dρ−g(V1,W1)∇U1Dρ,\n(£XS)(U1,V1) =−(2m−1)g(∇U1Dρ,V1)−(△ρ)g(U1,V1),(1.4)\n£Xr=−4m(△ρ)−2rρ (1.5)\nfor all vector fields U1,V1,W1onN2m+1, whereDρand△ρ=divDρrespectively\ndenote the gradient and Laplacian of ρ.\nDefinition 1.1. A vector field Xsatisfying the relation\n£Xη=ση, (1.6)\nσbeing a scalar function, is named an infinitesimal contact tr ansformation. It\nis named as infinitesimal strict contact transformation, if σvanishes identically.\nSharma and Vrancken[22] generalized the theorem of Tanno[23] an d proved:\nTheorem 1.1. TheCVFonNis an infinitesimal automorphism of Nif it is\nan infinitesimal contact transformation.\nNow, for dimension >3, we recollect the subsequent outcome on a Sasakian\nmanifold stated by Okumura[16].\nTheorem 1.2. IfN2m+1, m >1be a Sasakian manifold with a non-Killing CVF\nX, thenXis special concircular. Moreover, N2m+1is isometric to S2m+1(unit\nsphere) if it is connected and complete as well.\nAgain, Sharma and Blair([20]) proved the following Theorem to genera lize the\nforegoing result.\nTheorem 1.3. On a(k,0)-contactmanifold N2m+1, letXbe a non-Killing CVF.\nN2m+1is Sasakian and Xis concircular for m >3, hence if N2m+1is complete,\nit is isometric to S2m+1. Also,N2m+1is either Sasakian or flat for m= 1.\nTo generalized the above result, Sharma and Vrancken[22] establis hed the fol-\nlowing theorem:\nTheorem 1.4. LetN2m+1be a(k,µ)-contact metric manifold which admits X,\na non-Killing CVF. Form >3,\n(i)N2m+1is Sasakian and Xis concircular, also N2m+1is isometric to S2m+1if\nit is complete, or\n(ii)k=−m−1andµ= 1. Also,N2m+1is isometric S2m+1if it is compact.\nVery recently, De, Suh and Chaubey[5] studied CVFon almost co-K¨ ahler\nmanifolds. In 2022 [25], Wang investigated almost Kenmotsu ( k,µ)′-manifolds\nwithCVFin dimension three.\nInspired by the foregoing studies we are interested to investigate CVFon\nalmost Kenmotsu manifolds (briefly, akm).\nThe following is how the current article is structured:\nFollowing the preliminaries, we investigate CVFin akm in Section 3. In next\nSection, we investigate a ( k,µ)′-akm admitting a CVFX. Section 5, concerns\nwith the study of η-Ricci-Yamabe soliton with CVFon (k,µ)′-akm. Finally, we\nbuild an example.3\n2.Preliminaries\nA Riemannian manifold ( N2m+1,g) is caled an almost contact metric manifold\n[3] if it admit a tensor field ϕ((1,1)-type), a vector field ςanda 1-form ηobeying\nϕ2U1=−U1+η(U1)ς,η(ς) = 1, (2.1)\ng(ϕU1,ϕV1) =g(U1,V1)−η(U1)η(V1) (2.2)\nfor all vector fields U1,V1. The vector field ςis called the Reeb or characteristic\nvector field.\nIf the Nijenhuis tensor of ϕ[3] disappears, analmost contact metric manifoldis\ncalled normal. If in an almost contact metric manifold, dη= 0 anddϕ= 2η∧ϕ,\nϕ(U1,V1) =g(U1,ϕV1), then it is akm. A normal almost akm is a Kenmotsu\nmanifold. In an akm the relation\n(∇U1ϕ)V1=g(ϕU1,V1)ς−η(V1)ϕU1 (2.3)\nholds. Also the following formulas hold in akm ([9],[10])\nhϕ+ϕh= 0, hς=h′ς= 0, tr(h) =tr(h′) = 0, (2.4)\n∇U1ς=U1−η(U1)ς+h′U1, (2.5)\nwhereh=1\n2£ςϕandh′=h◦ϕ.\nIn an akm if the characteristic vector field ςbelongs to ( k,µ)′-nullity distribu-\ntion, that is,\nR(U1,V1)ς=k(η(V1)U1−η(U1)V1)+µ(η(V1)h′U1−η(U1)h′V1),(2.6)\nkandµare constants, then it is called a ( k,µ)′-akm[9]. In a ( k,µ)′-akm [9]:\nh′2U1= (k+1)ϕ2U1 (2.7)\nandµ=−2. Equation (2.7) reflects that h′= 0 if and only if k+1 = 0 whereas\nh′/ne}ationslash= 0 if and only if k+1<0. The equation (2.6) yields\nR(ς,U1)V1=k(g(U1,V1)ς−η(V1)U1)−2(g(h′U1,V1)ς−η(V1)h′U1) (2.8)\nfor anyU1,V1∈χ(N).\nProposition 2.1. ([26]) In a(k,µ)′-akm the (1,1)-Ricci tensor Qis given by\nQU1=−2mU1+2m(k+1)η(U1)ς−2mh′U1 (2.9)\nfork+1<0, whereQis defined by S(U1,V1) =g(QU1,V1)andr= 2m(k−2m).\nProposition 2.2. ([10]) An akm is a Kenmotsu manifold if and only if h = 0.\nProposition 2.3. ([10]) IfN2m+1be an akm with h= 0, thenN2m+1is locally a\nwarped product N′×fM2m, in which M2mindicate an almost K¨ ahler manifold,\nwith coordinate t,N′being the open interval and f=cetfor some c( positive\nconstant).\nDefinition 2.1. An akm N2m+1is named an η-Einstein manifold if its Ricci\ntensorSfulfills\nS(U1,V1) =a1g(U1,V1)+b1η(U1)η(V1),\nwherea1, b1are scalars of which b1/ne}ationslash= 0.4\nIn [11], the author proved that an η-Einstein Kenmotsu manifold is an Ein-\nstein manifold, provided b1= constant (or, a1= constant). Also, Pastore and\nSaltarelli[17] proved that η-Einstein ( k,µ)-akm is an Einstein manifold for any\none ofa1orb1is constant. Again, Mandal and De[13] studied the above result\nin (k,µ)′-akm. They explain the following:\nProposition 2.4. Anη-Einstein (k,µ)′-akm becomes an Einstein manifold, pro-\nvideda1orb1is constant.\n3.Conformal vector fields on almost Kenmotsu manifolds\nLet usassume that theReebvector field ςbeaCVFonN2m+1. Then equation\n(1.1) implies\n(£ςg)(U1,V1) = 2ρg(U1,V1), (3.1)\nwhich means that\ng(∇U1ς,V1)+g(U1,∇V1ς) = 2ρg(U1,V1). (3.2)\nUsing (2.5) in (3.2), we obtain\nρg(U1,V1) = [g(U1,V1)−η(U1)η(V1)+g(h′U1,V1)]. (3.3)\nSettingU1=V1=ςin the foregoing equation entails that\nρ= 0. (3.4)\nApplying (3.4) in (3.1), we obtain\n(£ςg)(U1,V1) = 0. (3.5)\nHence, the foregoing equation implies that ςis a Killing. Thus, we acquire h= 0.\nTherefore, from Proposition 2.2 and 2.3, we have:\nTheorem 3.1. If the Reeb vector field ςofN2m+1is aCVF, thenN2m+1becomes\na Kenmotsu manifold and N2m+1is also locally a warped product N′×fM2m,\nwhereM,N′andfare defined earlier.\nSuppose X=bς, wherebis a smooth function on N2m+1. Then, using (1.1),\nwe get\n(£bςg)(U1,V1) = 2ρg(U1,V1), (3.6)\nwhich implies\ng(∇U1bς,V1)+g(U1,∇V1bς) = 2ρg(U1,V1). (3.7)\nUsing (2.5) in the foregoing equation gives\n(U1b)η(V1)+(V1b)η(U1)+2b[g(U1,V1)−η(U1)η(V1)+g(h′U1,V1)] = 2ρg(U1,V1).\n(3.8)\nContracting the above equation we obtain\nςb=ρ(2m+1)−2mb. (3.9)\nReplacing U1byϕU1in (3.8), we get\n(ϕU1b)η(V1)+2b[g(ϕU1,V1)−g(hU1,V1)] = 2ρg(ϕU1,V1).(3.10)\nPuttingV1=ςin (3.10), we get\n(ϕU1)b= 0. (3.11)5\nThe above two equations implies\nb[g(ϕU1,V1)−g(hU1,V1)] =ρg(ϕU1,V1). (3.12)\nReplacing U1byV1andV1byU1in (3.12), we get\nb[−g(ϕU1,V1)−g(hU1,V1)] =−ρg(ϕU1,V1). (3.13)\nIn view of the above two equations, we get\n(b−ρ)g(ϕU1,V1) = 0, (3.14)\nwhich implies\nb=ρ. (3.15)\nUsing (3.15) in (3.13), we infer\nbg(hU1,V1) = 0, (3.16)\nwhich implies either b= 0 or,b/ne}ationslash= 0.\nCase I: If b= 0, then (3.15) implies ρ= 0. Hence, Xis Killing, which means\nthatςis Killing.\nCase II: If b/ne}ationslash= 0, then g(hU1,V1) = 0. Hence h= 0, which implies ςis Killing.\nHence, both cases implies ςis Killing. Hence using Proposition 2.2 and 2.3,\nwe provide:\nTheorem 3.2. InN2m+1, if aCVFXis pointwise collinear with ς, thenN2m+1\nbecomes a Kenmotsu manifold and N2m+1is locally a warped product N′×fM2m.\n4.Conformal vector fields on (k,µ)′-akm\nSuppose the vector field XinN2m+1is Killing. Then\n(£Xg)(U1,V1) = 0 (4.1)\nand\n(£XS)(U1,V1) = 0. (4.2)\nDefinition of Lie-derivative infers that\n(£Xη)U1=£Xη(U1)−η(£XU1). (4.3)\nEquation (4.1) implies\n(£Xη)U1=g(£Xς,U1). (4.4)\nAlso, we have\nη(£Xς) = 0and(£Xη)ς= 0. (4.5)\nFrom (2.9), we obtain\nS(U1,V1) =−2mg(U1,V1)+2m(k+1)η(U1)η(V1)−2mg(h′U1,V1).(4.6)\nNow, we take Lie-derivative of (4.6) along Xand get\n(£XS)(U1,V1) =−2m(£Xg)(U1,V1) (4.7)\n+2m(k+1)[((£Xη)U1)η(V1)+((£Xη)V1)η(U1)]\n−2m[(£Xg)(h′U1,V1)+g((£Xh′)U1,V1)].\nUsing (4.1) and (4.2) in (4.7), we provide\n(k+1)[((£Xη)U1)η(V1)+((£Xη)V1)η(U1)] (4.8)\n[(£Xg)(h′U1,V1)+g((£Xh′)U1,V1)] = 0.6\nPuttingV1=ςin (4.8) and using (4.4),(4.5) gives\n(k+1)g(£Xς,U1) = 0, (4.9)\nwhich implies either k+1 = 0 or, k+1/ne}ationslash= 0.\nCase I: When k+ 1 = 0: Dileo and Pastore[9] shown that in an ( k,µ)′-akm\nifk+ 1 = 0, then h′= 0 and hence the manifold is locally a warped product\nN′×fM2m.\nCase II: If k+1/ne}ationslash= 0, hence g(£Xς,U1) = 0, then (4.4) implies\n(£Xη)U1= 0,\nwhich means that Xis a strict infinitesimal contact transformation.\nHence we have:\nTheorem 4.1. IfN2m+1admits a Killing vector field X, then either N2m+1\nis locally a warped product N′×fM2morXis a strict infinitesimal contact\ntransformation.\nSinceXis Killing, ρ= 0. Also, from Proposition 2.1, we have r= 2m(k−2m).\nHence, (1.3) and (1.4) implies\n(£XR)(U1,V1)W1= 0 and ( £XS)(U1,V1) = 0. (4.10)\nTaking Lie-derivative of (2.6) and using (4.10) , we obtain\nR(U1,V1)£Xς=k[((£Xη)V1)U1−((£Xη)U1)V1] (4.11)\n−µ[((£Xη)V1)h′U1−((£Xη)U1)h′V1\n+η(V1)(£Xh′)U1−η(U1)(£Xh′)V1].\nSince,Xis Killing, (1.1) implies\ng(∇ςX,U1)+g(∇U1X,ς) = 0, (4.12)\nwhich entails\n(£Xη)U1=g(£Xς,U1), (4.13)\nfor any vector field U1. Let{e0=ς,e1,e2,...,e2m}be an orthonormal basis of\nN2m+1. Then from the above equation, we have\n2n/summationdisplay\ni=0g(ei,V1)(£Xη)ei=g(£Xς,V1),2n/summationdisplay\ni=0g(ei,h′V1)(£Xη)ei=g(£Xς,h′V1).\n(4.14)\nAgain, for a non-Kenmotsu ( k,µ)′-akm, it is known that\n(∇U1h′)V1=g((k+1)U1−h′U1,V1)ς+η(V1)((k+1)U1−h′U1) (4.15)\n−2(k+1)η(U1)η(V1)ς\nfor anyU1, V1andµ=−2.\nAlso, from [24], we get\ntr(£Xh′) = 0. (4.16)\nContracting (4.11) and using (4.13), (4.14) and (4.16), we obtain\nS(V1,£Xς) = 2mkg(£Xς,V1)+2g(£Xς,h′V1)+2g(£Xh′V1,ς).(4.17)7\nFrom (4.12), we have g(£Xh′V1,ς) =−g(£Xς,h′V1). Hence the foregoing equa-\ntion implies\nS(V1,£Xς) = 2mkg(£Xς,V1). (4.18)\nReplacing U1by£Xςin (4.6) and using (4.13) and (4.18), we get\n(k+1)£Xς+h′£Xς= 0. (4.19)\nOperating h′in (4.19) and using (2.7), we provide\nh′£Xς−£Xς= 0. (4.20)\nThe above two equations implies ( k+2)£Xς= 0, which implies either k+2 = 0\nork+2/ne}ationslash= 0.\nCase I: If k=−2, then it is a non-Kenmotsu ( k,−2)′-akm and is locally\nisometric to the Riemannian product Hm+1(−4)×Rm[9].\nCase II: If k/ne}ationslash=−2, then£Xς= 0, hence from (4.13), we get ( £Xη)U1=\n0.Hence we have:\nTheorem 4.2. If a non-Kenmotsu (k,µ)′-akmN2m+1permits a Killing vector\nfieldX, it is either N2m+1locally isometric to the product space Hm+1(−4)×Rm\norXis a strict infinitesimal contact transformation.\n5.η-Ricci-Yamabe solitons with conformal vector fields\nThe concept of η-Ricci-Yamabe soliton comes as a generalization of Ricci-\nYamabe soliton and is defined by [18]\n£Xg+2α1S+(2λ1−β1r)g+2ν1η⊗η= 0, (5.1)\nwhereα1,β1,ν1are constant. If ν1= 0, then η-Ricci-Yamabe soliton reduces to a\nRicci-Yamabe soliton and for ν1/ne}ationslash= 0, it is called proper η-Ricci-Yamabe soliton.\nThis soliton turns into\n(i)η-Ricci soliton if α1= 1,β1= 0,\n(ii)η-Yamabe soliton if α1= 0,β1= 1,\n(iii)η-Einstein soliton if α1= 1,β1=−1.\nSeveral authors have studied η-Ricci solitons and Ricci-Yamabe solitons, in-\ncluding ([1], [2], [4], [6], [19]) and many others.\nAssume that the ( k,µ)′-akmN2m+1admits an η-Ricci-Yamabe soliton with\nCVF. Then from (5.1) we have\n(£Xg)(U1,V1)+2α1S(U1,V1)+(2λ1−β1r)g(U1,V1)+2ν1η(U1)η(V1) = 0.(5.2)\nUsing (1.1) in (5.2), we get\nρg(U1,V1)+α1S(U1,V1)+(λ1−β1\n2r)g(U1,V1)+ν1η(U1)η(V1) = 0.(5.3)\nSettingU1=V1=ςin (5.3) entails that\nρ=−2mkα1−λ1−ν1+β1\n2r, (5.4)\nwhich is a constant. Again, contracting U1andV1in (5.3), we infer\nρ(2m+1) =−λ1(2m+1)+β1\n2r(2m+1)−ν1−α1r. (5.5)8\nIn view of (5.4) and (5.5), we obtain\nλ1=β1m(k−2m)−ρ+2mα1 (5.6)\nand\nν1=−2mα1(k+1). (5.7)\nIt is known that in a ( k,µ)′-akm,r= 2m(k−2m), a constant. Hence with the\nhelp of (1.5) and (5.4), we get ρr= 0, which implies ρ= 0. Hence, equation\n(5.3) implies\nα1S(U1,V1) = (β1\n2r−λ1)g(U1,V1)−ν1η(U1)η(V1), (5.8)\nwhich represents an η-Einstein manifold. Since, the coefficients are constant,\nfrom Proposition 2.4, it becomes an Einstein manifold. Therefore we h ave:\nTheorem 5.1. If a(k,µ)′-akm admits an η-Ricci-Yamabe soliton with CVF,\nthen it becomes an Einstein manifold and the constants λ1andν1are given by\nλ1=β1m(k−2m)−ρ+2mα1\nand\nν1=−2mα1(k+1).\nIn particular, if we take α1= 1,β1= 0. Then we have\nλ1= 2m−ρandν1=−2m(k+1).\nHence we have:\nCorollary 5.1. If a(k,µ)′-akm permits an η-Ricci soliton with CVF, then the\nconstants λ1andν1are given by\nλ1= 2m−ρandν1=−2m(k+1).\nIn particular, if we take α1= 0,β1= 1. Then we have\nλ1=m(k−2m)−ρandν1= 0.\nHence we have:\nCorollary 5.2. If a(k,µ)′-akm concedes an η-Yamabe soliton with conformal\nvector field, then the soliton becomes Yamabe soliton and λ1= 2m−ρ.\nIn particular, if we take α1= 1,β1=−1. Then we have\nλ1=−m(k−2m)−ρ+2mandν1=−2m(k+1).\nHence we have:\nCorollary 5.3. If a(k,µ)′-akm admits an η-Einstein soliton with CVF, then\nthe constants λ1andν1are given by\nλ1=−m(k−2m)−ρ+2mandν1=−2m(k+1).\nAlso, if we take ν1= 0, then the equation (5.7) implies k=−1 henceh= 0.\nTherefore, the manifold becomes a Kenmotsu manifold. Hence we ha ve:\nCorollary 5.4. If a(k,µ)′-akm permits a proper Ricci-Yamabe soliton with\nCVF, then the manifold becomes a Kenmotsu manifold.9\n6.Example\nWe choose N3={(x,y,z)∈R3}, in which ( x,y,z) denote the standard coor-\ndinates in R3. Letς,ε2,ε3be three vector fields in R3which satisfies [9]\n[ς,ε2] =−ε2−ε3,[ς,ε3] =−ε2−ε3,[ε2,ε3] = 0.\nThe Riemannian metric gis defined by\ng(ς,ς) =g(ε2,ε2) =g(ε3,ε3) = 1and g(ς,ε2) =g(ς,ε3) =g(ε2,ε3) = 0.\nHere,ηandϕare defined as follows:\nη(W1) =g(W1,ς), for any W1∈χ(N), and\nϕς= 0, ϕε2=ε3, ϕε3=−ε2.\nMaking use of the linearity property of ϕandg, we acquire\nη(ς) = 1,\nϕ2U1=−U1+η(U1)ς,\ng(ϕU1,ϕV1) =g(U1,V1)−η(U1)η(V1)\nfor anyU1,V1∈χ(N). Thus the structure ( ϕ,ς,η,g) is an almost contact struc-\nture.\nMoreover, h′ς= 0, h′ε2=ε3and h′ε3=ε2.\nIn [12], the authors acquired the expression of the curvature ten sor as:\nR(ς,ε2)ς= 2(ε2+ε3), R(ς,ε2)ε2=−2ς, R(ς,ε2)ε3=−2ς,\nR(ε2,ε3)ς=R(ε2,ε3)ε2=R(ε2,ε3)ε3= 0,\nR(ς,ε3)ς= 2(ε2+ε3), R(ς,ε3)ε2=−2ς, R(ς,ε3)ε3=−2ς.\nFrom the expressions of the R, we ensure that with k=−2 andµ=−2,ς\nbelongs to the ( k,µ)′-nullity distribution.\nForegoing expressions reveal the values of the Ricci tensor as:\nS(ς,ς) =−4, S(ε2,ε2) =S(ε3,ε3) =−2.\nHence,r=S(ς,ς)+S(ε2,ε2) =S(ε3,ε3) =−8. Therefore, from (5.3) we obtain\nρ−4α1+λ1+4β1+ν1= 0\nandρ−2α1+λ1+4β1= 0.\nFrom the above two equations, we get\nλ1= 2α1−4β1−ρandν1= 2α1.\nTherefore, the manifold defines an η-Ricci-Yamabe soliton for λ1= 2α1−4β1−\nρandν1= 2α1.10\nReferences\n[1] Blaga, A. M., η-Ricci solitons on para-Kenmotsu manifolds , Balkan J. Geom. Appl.,\n20(2015), 1-13.\n[2] Blaga, A. 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M., Almost Kenmotsu manifolds and nullity distribution , J.\nGeom.,93(2009), 46-61.\n[10] Dileo, G. and Pastore, A. M., Almost Kenmotsu manifolds and local symmetry , Bull. Belg.\nMath. Soc. Simon Stevin, 14(2007), 343-354.\n[11] Kenmotsu, K., A class of almost contact Riemannian manifolds , Tohoku Math. J.,\n24(1972), 93-103.\n[12] Mandal, K.andDe, U.C., On 3-dimensional almost Kenmotsu manifolds admitting cert ain\nnullity distribution , Acta Math. Univ. Comenianae, LXXXVI (2017), 215-226.\n[13] Mandal, K. and De, U. C., Ricci solitons on almost Kenmotsu manifolds , Analele Univer-\nsitatii din Oradea. Fascicola Matematica, XXIII(2016), 109-116.\n[14] Obata, M., Conformal transformations of Riemannian manifolds , J. Diff. Geom., 4(1970),\n311-333.\n[15] Okumura, M., Some remarks on space with certain contact structures , Tohoku Math. J.,\n14(1962), 135-145.\n[16] Okumura,M., Oninfinitesimal conformal and projective transformations of normal contact\nspaces, Tohoku Math. J., 14(1962), 398-412.\n[17] Pastore, A. M. and Saltarelli, V., Almost Kenmotsu manifolds with conformal Reeb folia-\ntion, Bull. Belg. Math. Soc. Simon Stevin, 18(2011), 655-666.\n[18] Siddiqi, M. D. and Akyol, M. A., η-Ricci-Yamabe soliton on Riemannian submersions\nfrom Riemannian manifolds , preprint (2020), arXiv: 2004.14124.\n[19] Singh, J.P. and Khatri, M., On Ricci-Yamabe soliton and geometrical structure in a perf ect\nfluid spacetimes , Afrika Mathematica.\n[20] Sharma, R. and Blair, D. E., Conformal motion of contact manifolds with characteristic\nvector field in the k-nullity distribution , Illinois J. Math., 40(1996), 553-563.\n[21] Sharma, R., Holomorphically planar conformal vector fields on almost He rmitian mani-\nfolds, Contemp. Math., 337(2003), 145-154.\n[22] Sharma, R. and Vrancken, L., Conformal classification of (k,µ)-contact manifolds , Kodai\nMath. J., 33(2010), 267-282.\n[23] Tanno, S., Note on infinitesimal transformations over contact manifol ds, Tohoku Math.\nJ.,14(1962), 416-430.\n[24] Wang, Y., Almost Kenmotsu (k,µ)′-manifolds with Yamabe solitons , RACSAM,\n115(2021), 14.\n[25] Wang, Y., Almost Kenmotsu (k,µ)′-manifolds of dimension three and conformal vector\nfields, Int. J.Geom. Methods Mod. Phys. 19, no. 4, (2022),2250054 (9 pages).11\n[26] Wang, Y. and Wang, W., Some results on (k,µ)′-almost Kenmotsu manifolds , Quaest.\nMath.,41(2018), 469-481.\n[27] Yano, K., Integral formulas in Riemannian geometry , Marcel Dekker, New York, 1970.\nUday Chand De\nDepartment of Pure Mathematics,\nUniversity of Calcutta,\n35, Ballygunge Circular Road,\nKolkata 700019,\nWest Bengal, India.\nORCID iD: https://orcid.org/0000-0002-8990-4609\nEmail address :ucde@yahoo.com\nArpan Sardar\nDepartment of Mathematics\nUniversity of Kalyani\nKalyani 741235\nNadia, West Bengal, India\nEmail address :arpansardar51@gmail.com\nKrishnendu De\nDepartment of Mathematics,\nKabi Sukanta Mahavidyalaya,\nThe University of Burdwan.\nBhadreswar, P.O.-Angus, Hooghly,\nPin 712221, West Bengal, India.\nORCID iD: https://orcid.org/0000-0001-6520-4520\nEmail address :krishnendu.de@outlook.in"    },    {        "title": "2402.01447v1.The_Hamilton_space_of_pseudorandom_graphs.pdf",        "content": "arXiv:2402.01447v1  [math.CO]  2 Feb 2024The Hamilton space of pseudorandom graphs\nMicha Christoph∗Rajko Nenadov†Kalina Petrova‡\nAbstract\nWe show that if nis odd and p≥Clogn/n, then with high probability Hamilton cycles in G(n, p)\nspan its cycle space. More generally, we show this holds for a class of graphs satisfying certain\nnatural pseudorandom properties. The proof is based on a nov el idea of parity-switchers, which can\nbe thought of as analogues of absorbers in the context of cycl e spaces. As another application of\nour method, we show that Hamilton cycles in a near-Dirac grap hG, that is, a graph Gwith odd n\nvertices and minimum degree n/2 +Cfor sufficiently large constant C, span its cycle space.\n1 Introduction\nGiven a graph G= (V, E), let E(G) denote the vector space of subsets of EoverF2, also known as\ntheedge space . The elements of this vector space correspond to subsets of E, with addition defined\nas the symmetric difference. The cycle space C(G)⊆ E(G) is defined as the subspace spanned by all\n(not necessarily induced) cycles in G, and Ck(G)⊆ C(G) — by cycles of length exactly k. Determining\nconditions under which Ck(G) =C(G), for some k, is a well-studied problem in graph theory [3, 4, 5, 9,\n15, 16].\nHere we are interested in the case where Cn(G) =C(G), that is, in the case where Hamilton cycles\nspan the cycle space. Note that if Gcontains an odd cycle, a necessary condition for Cn(G) =C(G) is\nthat Ghas an odd number of vertices. Alspach, Locke, and Witte [2] s howed that if Γ is a finite abelian\ngroup of odd order, then Cn(G) =C(G) for any connected Cayley graph Gover Γ. Heinig [11] showed\nthat the same holds if Gis a graph with odd nvertices and minimum degree δ(G)≥(1 + ε)n/2, for\nanyε >0 and nsufficiently large, and in [12] he obtained Cn(Gn,p) =C(Gn,p) with high probability for\np≥n−1/2+o(1). We improve both of these results. In the case of dense graphs , we improve the minimum\ndegree to the one which is optimal up to an additive constant t erm.\nTheorem 1.1. There exists C > 1such that the following holds for sufficiently large odd n. Let Gbe a\ngraph with nvertices such that δ(G)≥n/2 +C. Then Cn(G) =C(G).\nIn the case of random graphs, we improve the bound on ptop≥Clogn/n, which is best possible\nup to the (multiplicative) constant C. More generally, we show an analogous statement for a class o f\npseudorandom graphs.\nDefinition 1.2 ([17]) .A graph Gis (p, β)-jumbled with 0 < p < 1≤βif for every U⊆V(G), it holds\nthat/vextendsingle/vextendsingle/vextendsingleeG(U)−p/parenleftbigg|U|\n2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle≤β|U|.\nTheorem 1.3. There exist ε >0andC > 1such that the following holds for sufficiently large odd n.\nLetGbe a graph on nvertices, satisfying the following three conditions for so mep≥Clogn/n:\n•δ(G)≥(1−ε)np,\n•Gis(p, β)-jumbled for some β≤εnp/ log log n,\n•there exists an edge between every two disjoint subsets of ve rtices A, B ⊆V(G)such that\n|A|,|B| ≥εnlog log n\nlogn.\n∗Department of Computer Science, ETH Z¨ urich, Switzerland. Email: micha.christoph@inf.ethz.ch . This author was\nsupported by SNSF Ambizione grant No. 216071.\n†School of Computer Science, University of Auckland, New Zea land. Email: rajko.nenadov@auckland.ac.nz\n‡Department of Computer Science, ETH Z¨ urich, Switzerland. Email: kalina.petrova@inf.ethz.ch . This author was\nsupported by grant no. CRSII5 173721 of the Swiss National Sc ience Foundation.\n1Then Cn(G) =C(G).\nTo the best of our knowledge, it is conceivable that Theorem 1 .3 holds assuming only the first two\nconditions, and moreover weakening the second condition to β≤εnp. However, even showing that such\nGcontains a Hamilton cycle is a major open problem [8, 14]. On t he positive side, we can remove the\nthird assumption if we strengthen the jumbledness property .\nCorollary 1.4. There exist ε >0andC > 1such that the following holds for sufficiently large n. Let\np≥Clogn/nandβ≤εnp/ logn. Then, any (p, β)-jumbled graph Gonnvertices with minimum degree\nδ(G)≥(1−ε)npsatisfies Cn(G) =C(G).\nProof. Suppose there exist disjoint A, B ⊆V(G) of size keach, for which eG(A, B) = 0. From ( p, β)-\njumbledness we get\neG(A∪B)≥pk(2k−1)−2βk= 2pk(k−1) + pk−2βk\nand\neG(A), eG(B)≤pk(k−1)/2 +βk.\nFrom eG(A∪B) =eG(A, B) +eG(A) +eG(B), we conclude\npk≤4β.\nTherefore, between any pair of sets of size\n4β/p+ 1≤4εn\nlogn+ 1≤εnlog log n\nlogn\nthere is an edge, thus Gsatisfies all the conditions of Theorem 1.3.\nCorollary 1.4 can readily be applied to ( n, d, λ )-graphs, d-regular graphs with nvertices and the second\nlargest absolute value of the adjacency matrix at most λ, with d≥Clognandλ≤εd/logn. Indeed, as\nshown in [1, Lemma 2.3], ( n, d, λ )-graphs are ( d/n, λ )-jumbled.\nCorollary 1.5. There exist ε >0andC > 1such that the following holds for sufficiently large odd n.\nEvery (n, d, λ )-graph Gwith d≥Clognandλ≤εd/lognsatisfies Cn(G) =C(G).\nFinally, a simple application of the Chernoff inequality and a union bound shows that Theorem 1.3\napplies to random graphs. The following result was our main m otivation for this work.\nCorollary 1.6. There exists C > 1such that for p≥Clogn/n,G∼G(n, p)w.h.p. satisfies Cn(G) =\nC(G)ifnis odd.\nProof. Let us show that Gsatisfies the three conditions of Theorem 1.3, where we take Clarge enough\ncompared to C1.3, ε1.3satisfying Theorem 1.3. The first condition follows w.h.p. f rom a standard\napplication of the Chernoff inequality. It is known, see for e xample [18], that Gn,pis w.h.p. ( p,2√np)-\njumbled for pas stated, thus\n2√np≤ε1.3np\nlog log n\nimplies the second condition. Finally, let A, B ⊆V(G) be two disjoint sets of size ε1.3nlog log n\nlogneach. It\nfollows that\nPr[eG(A, B) = 0] = (1 −p)|A||B|≤e−p|A||B|≤e−ε21.3Cn(log log n)2/logn.\nSince there are at most /parenleftbiggn\n|A|/parenrightbigg/parenleftbiggn\n|B|/parenrightbigg\n≤e2ε1.3n(log log n)2/logn\nchoices of A, B ⊆V(G), a union bound yields that w.h.p. Galso satisfies the third condition. Thus,\nensuring C≥C1.3andC > 2/ε1.3, the corollary follows from Theorem 1.3.\nThe rest of the paper is organised as follows. In Section 2 we l ay out the framework and prove the\nCycle Lemma (Lemma 2.3), which is the heart of the proof. We th en illustrate an application of the\nframework by giving a simple proof of Theorem 1.1. Section 3 c ollects known results and properties of\npseudorandom graphs. We then give the proof of Theorem 1.3 in Section 4, and state open problems in\nSection 5.\n2Notation. It is important to keep in mind that throughout the paper we id entify a graph with its set\nof edges. Our notation is standard. Given a graph Gand disjoint subsets S, A ⊆V(G), we use NG(S)\nto denote the set of neighbours of vertices from SinV(G)\\S, and NG(S, A) =NG(S)∩A. For disjoint\nsubsets A, B ⊆V(G), we denote with G[A, B] the induced bipartite subgraph and with eG(A, B) the\nnumber of edges in G[A, B]. For a given subgraph F⊆G, we denote with V(F)⊆V(G) the set of\nvertices incident to edges in F.\n2 Proof Strategy\nThe following lemma is our starting point. A similar stateme nt is also used as a starting point in [3, 5],\nhowever after it the two approaches diverge.\nLemma 2.1. LetGbe a Hamiltonian graph with odd nvertices, such that Cn(G)/\\e}atio\\slash=C(G). Then there\nexists a subgraph R⊆Gfor which the following holds:\n(C1) R/\\e}atio\\slash=G,\n(C2) every Hamilton cycle in Gcontains an even number of edges from R, and\n(C3) for every partition V(G) =A∪Bwe have\neR(A, B)≥eG(A, B)/2\nandR/\\e}atio\\slash=G[A, B].\nProof. Given a subspace S ⊆ E (G), we write S⊥for its orthogonal complement,\nS⊥={D∈ E(G):/a\\}bracketle{tD, S/a\\}bracketri}ht= 0 for all S∈ S} ,\nwhere\n/a\\}bracketle{tD, S/a\\}bracketri}ht=/summationdisplay\ne∈E(G)D(e)S(e)\nwith addition being done in F2and, for any X∈ E(G),X(e) is the indicator for e∈X. It is well known\nthat C⊥(G) is precisely the cut space ofG, that is, it consists of all edge subsets corresponding to in duced\nbipartite graphs G[A, B], including ∅, for all partitions V(G) =A∪B.\nFrom Cn(G)⊆ C(G) and the assumption of the lemma, we conclude C⊥\nn(G)\\ C⊥(G) is non-empty. Let\nR⊆Gbe a largest (in terms of edges) element of C⊥\nn(G)\\C⊥(G). Since R∈ C⊥\nn(G), every Hamilton cycle\ninGhas an even number of edges in R(by the definition). Note that R+C⊥(G)⊆ C⊥\nn(G) is disjoint\nfrom C⊥(G) since it is a coset of C⊥(G). Therefore, Ris not a cut of Gbut contains at least half the\nedges over every cut ( A, B), as otherwise R+G[A, B]∈ C⊥\nn(G)\\ C⊥(G) contradicts the maximality of\nR. Finally, we have R/\\e}atio\\slash=G, as otherwise Rcontains a Hamilton cycle, contradicting the fact that ever y\nHamilton cycle has an even number of edges in R. It is worth noting that this is the only place where\nwe use the fact that nis odd.\nCentral to our proof is the notion of a parity-switcher .\nDefinition 2.2 (Parity-switcher) .Given a graph Gand a subgraph R⊆G, a subgraph W⊆Gis called\nanR-parity-switcher if it consists of an even cycle C= (v1, v2, . . . , v 2k) with an odd number of edges in\nR, and vertex-disjoint paths Pibetween viandv2k−i+2for 2 ≤i≤k.\nA parity-switcher contains two Hamilton paths (that is, pat hs containing each vertex in V(W)):\n•v1→v2P2→v2k→v2k−1P3→v3→v4P4→v2k−2→. . .→vk+1, and\n•v1→v2kP2→v2→v3P3→v2k−1→v2k−2P4→v4→. . .→vk+1.\nImportantly, the two paths have a different parity of edges fr omR, since each Pifor 2 ≤i≤kappears\nfully in both paths and each edge of Cis contained in exactly one of the paths (see Figure 1).\n3v1v2v3\nv4\nv5\nv6\nv7v8P2 P3 P4\nFigure 1: A Hamilton path with an even number of edges from R(represented in red) in a parity-switcher.\nProof strategy. We use the following recipe for showing Cn(G) =C(G):\n(S1) Suppose Cn(G)/\\e}atio\\slash=C(G), and take R⊆Gto be a subgraph given by Lemma 2.1.\n(S2) Find a (small) R-parity-switcher W:\n(S2.a) Find an even cycle C= (v1, . . . , v 2k) with an odd number of edges in R.\n(S2.b) Find edge-disjoint (short) paths Pibetween viandv2k−i+2, for each 2 ≤i≤k.\n(S3) Find a Hamilton path H′from v1tovk+1inG\\(V(W)\\ {v1, vk+1}).\n(S4) If H′contains an odd (even) number of edges in R, then choose a Hamilton path H′′inWfrom v1\ntovk+1with an even (odd) number of edges in R.\n(S5) Conclude that the concatenation of H′andH′′gives a Hamilton cycle H⊆Gwith an odd number\nof edges in R, contradicting (C2).\nA reader familiar with the concept of absorbers will notice their similarity to parity-switchers. Indeed,\nthe main goal of the parity-switcher is to reduce the problem to the one where we just aim to find a\nHamilton cycle, without any further restrictions (technic ally, the goal of absorbers is to reduce a problem\nto the one about almost-spanning structures; morally, its g oal is to change the problem in a way which\nprovides us more flexibility).\nTo implement the strategy, we only need to take care of steps ( S2) and (S3). In particular, as only\n(S2.a) is concerned with R, this is the most difficult step (that being said, (S2.b) and (S 3) are not simple,\nhowever there is already an existing machinery which we make use of). The following lemma takes care\nof Step (S2.a), and is the crux of our proof.\nLemma 2.3 (Cycle Lemma) .LetGbe a graph and R⊆Gsuch that the following holds for some ℓ∈N:\n(L1) For any S⊆V(G)of size |S| ≤2ℓand any two vertices x, y∈V(G)\\S, there is a path between x\nandyinR\\Sof length at most ℓ−1,\n(L2) R/\\e}atio\\slash=GandR/\\e}atio\\slash=G[A, B]for every partition V(G) =A∪B.\nThen there exists an even cycle C⊆Gof length |C| ≤2ℓthat contains an odd number of edges from R.\nProof. AsR/\\e}atio\\slash=G, there exists an edge uw∈G\\R. For i∈N, denote with Ni\nodd(u) the set of vertices\ninV(G)\\ {w}reachable by an odd path (that is, a path with an odd number of e dges) of length at\nmost ifrom uinR\\ {w}. Define Ni\neven(u) analogously for even paths, and note that u∈Ni\neven(u). We\ndistinguish two cases.\nCase 1: Nℓ\nodd(u)∩Nℓ\neven(u)/\\e}atio\\slash=∅.Choose any z∈Nℓ\nodd(u)∩Nℓ\neven(u), and let PoandPebe a shortest\nodd and even path from utozinR\\ {w}. By assumption, the set S=V(Po)∪V(Pe)\\ {z}is of size\n|S| ≤2ℓTherefore, by (L1), we can find a path Pof length at most ℓ−1 from ztowinR\\S. If this\npath has even length then set P′=Po, and otherwise P′=Pe. In any case, we get an even cycle of\nlength at most 2 ℓwith an odd number of edges from R:\nw→uP′\n→zP→w\n4Case 2: Nℓ\nodd(u)∩Nℓ\neven(u) =∅.In this case, both Nℓ−1\nodd(u) and Nℓ−1\neven(u) are independent sets in R. If\nthere is an edge wz∈Rfor some z∈Nℓ−1\neven(u), then a shortest even path in Rfrom utoztogether with\nz→w→ugives the desired cycle. Therefore, we can assume all the edg es between wandNℓ−1\neven(u) are\nnot in R. Note that by (L1), we have that Nℓ−1\neven(u)∪Nℓ−1\nodd(u)∪ {w}=V(G). This implies R⊂G[A, B]\nforA=Nℓ−1\neven(u)∪ {w}andB=Nℓ−1\nodd(u), thus Ris bipartite. Since R/\\e}atio\\slash=G[A, B], there exist vertices\nx∈Aandy∈Bsuch that xy∈G\\R. Let Pbe a shortest path in Rfrom xtoy. Then by (L1) Pis\nof length at most ℓ−1, and as Ris bipartite, it is necessarily odd. Therefore, Pand the edge xyclose\nthe desired cycle.\nTo demonstrate the overall strategy, we give a short proof of Theorem 1.1.\nProof of Theorem 1.1. Suppose Cn(G)/\\e}atio\\slash=C(G). Let R⊆Gbe a subgraph given by Lemma 2.1, which\nwe can apply since Gis Hamiltonian by Dirac’s theorem [6]. Note that we only need to implement steps\n(S2) and (S3).\nStep (S2.a). Letℓ= 10. We use Lemma 2.3 to obtain a cycle C= (v1, . . . , v 2k) inG, of even length\nat most 2 ℓ, which contains an odd number of edges from R. Condition (L2) follows from (C1) and the\nsecond part of (C3). To see that (L1) holds, first note that (C3 ) implies Rhas minimum degree at least\nn/4 +C/2 and it is connected. Choosing C > 4ℓ, we have that for any S⊆V(G) of size |S| ≤2ℓ < C/ 2,\nthe minimum degree of R\\Sis strictly larger than n/4. If a shortest path between some xandyinR\\S\nis of length at least 10, then the neighbourhoods of the first, fourth, seventh, and tenth vertex on such a\npath are disjoint – which cannot be given the minimum degree o fR\\S.\nStep (S2.b). Next, we need to connect vitov2k−i+2for every 2 ≤i≤k. This can be done by\nsequentially applying the fact that as long as the set Sconsisting of the vertices on the cycle and all the\nobtained paths so far is of size less than C, any two vertices x, y∈V(G) are at distance at most two\ninG\\(S\\ {x, y}). As the set Sis always of size at most 3 ℓ(2ℓvertices on the cycle, and at most one\nadditional vertex for each pair of vertices we are aiming to c onnect), this is indeed the case for C > 3ℓ.\nThis concludes the construction of an R-parity-switcher W.\nStep (S3). Finally, G′:=G\\(V(W)\\ {v1, vk+1}) has minimum degree δ(G′)> n/ 2, thus by Dirac’s\ntheorem it contains a Hamilton path between any two prescrib ed vertices.\n3 Properties of pseudorandom graphs\nWe make extensive use of the fact that pseudorandom graphs ar e good expanders .\nDefinition 3.1. We say that a graph Gis (s, d)-expanding , for s, d∈N, if for every subset S⊆V(G) of\nsize|S| ≤swe have\n|N(S)| ≥d|S|.\nLemma 3.2. There exists ε >0such that the following holds for any nand0< p < 1. Suppose Gis\nann-vertex (p, β)-jumbled graph with δ(G)≥(1−ε)np, where β≤εnp/d for some d∈N. Then Gis\n(n/(3d), d)-expanding.\nProof. Suppose that Gis not ( n/(3d), d)-expanding. Let S⊆V(G) be a set of vertices of size |S| ≤n/(3d)\nsuch that N(S)< d|S|. Let X1, X2, ..., X ℓbe a partition of N(S) such that |X1|=|X2|=...=|Xℓ−1|=\n|S|and|Xℓ| ≤ |S|. Note that ℓ≤d. Let F⊆Gbe the set of edges with at least one endpoint in S. By\nthe assumptions on G, we have:\n|F| ≥(1−ε)|S|np−eG(S)≥(1−ε)|S|np−p|S|2/2−εnp|S|/d≥/parenleftBig\n1−1\n6d−ε/parenrightBig\n|S|np−εnp|S|\nd.\nOn the other hand, since Gis (p, β)-jumbled, for i∈[ℓ] we get:\neG(S∪Xi)≤p/parenleftbigg2|S|\n2/parenrightbigg\n+ 2εnp|S|/d≤2|S|2p+ 2εnp|S|/d.\nTherefore,\n|F| ≤2d|S|2p+ 2εnp|S|.\n5Combining the above two observations, we get\n/parenleftBig\n1−1\n6d−ε/parenrightBig\n|S|np−εnp|S|\nd≤ |F| ≤2d|S|2p+ 2εnp|S|\nand so\n1−1/(6d)−4ε\n2dn≤ |S|,\ncontradicting our assumption |S| ≤n/(3d).\nLemma 3.3. There exists ε >0such that the following holds for any nand0< p < 1. Let β≤εnp, and\nsuppose Gis an n-vertex (p, β)-jumbled graph with δ(G)≥(1−ε)np. Then for any disjoint A, B ⊆V\nwith A∪B=V, we have\neG(A, B)≥(1−6ε)p|A||B|.\nProof. Suppose w.l.o.g. |A| ≤ |B|, so|B| ≥n/2. Then, since Gis (p, β)-jumbled,\neG(A)≤p/parenleftbigg|A|\n2/parenrightbigg\n+β|A|,\nso the minimum degree of Gimplies\neG(A, B)≥ |A|(1−ε)np−2eG(A)≥ |A|(1−ε)np− |A|2p−2|A|εnp≥ |A|(n− |A|)p−3ε|A|np\n≥ |A|(n− |A| −3εn)p≥ |A||B|/parenleftBig\n1−3εn\n|B|/parenrightBig\np≥(1−6ε)|A||B|p.\nLemma 3.4. There exists ε >0such that the following holds for any nand0< p < 1. Let β≤εnp.\nLetGbe an n-vertex (p, β)-jumbled graph. Then for any disjoint A, B ⊆V, we have\neG(A, B)≤ |A||B|p+ 2β(|A|+|B|).\nProof. ByGbeing ( p, β)-jumbled, we have\neG(A∪B)≤p/parenleftbigg|A|+|B|\n2/parenrightbigg\n+β(|A|+|B|)\neG(A)≥p/parenleftbigg|A|\n2/parenrightbigg\n−β|A|\neG(B)≥p/parenleftbigg|B|\n2/parenrightbigg\n−β|B|.\nCombining these, we get\neG(A, B)≤p/parenleftbigg|A|+|B|\n2/parenrightbigg\n−p/parenleftbigg|A|\n2/parenrightbigg\n−p/parenleftbigg|B|\n2/parenrightbigg\n+ 2β(|A|+|B|) =p|A||B|+ 2β(|A|+|B|).\nFinally, the last lemma shows that large subgraphs of ( p, β)-graphs are robust with respect to having\na small diameter, which allows us to apply Lemma 2.3.\nLemma 3.5. There exist ε >0andC > 1such that the following holds for sufficiently large n. Let G\nbe an n-vertex (p, β)-jumbled graph with δ(G)≥(1−ε)np, where p≥Clogn/nandβ≤εnp. Suppose\nR⊆Gis such that for every partition A∪B=V(G)we have eR(A, B)≥eG(A, B)/2. Then, for any\nS⊆V(G)of size |S| ≤2 log n,R′=R\\Shas the property that between any two vertices v, w∈V(G)\\S\nthere is a path of length at most logn−1inR′.\nProof. LetSbe as in the statement of the lemma. We denote V(G) by V, and we start by showing the\nfollowing helper expansion claim.\nClaim 3.6. IfX⊆Vhas size at most n/2, then we have\n|NR(X, V \\(X∪S))| ≥min{10|X|, n/7}.\n6Proof. LetY:=NR(X, V \\X). We first suppose |X| ≤2 log n. For any x∈Xwe have degR(x)≥\ndegG(x)/2≥np/3, thus for sufficiently large Cwe have\n|NR(X, V \\(X∪S))| ≥degR(x, V\\(X∪S))≥Clogn/3− |X| − |S| ≥20 log n≥10|X|.\nNext, consider the case 2 log n≤ |X| ≤n/200. Suppose, towards a contradiction, that |Y|<20|X|.\nThen, by Lemma 3.3 and Lemma 3.4, we have\n|X|np/6≤eR(X, V \\X) =eR(X, Y )≤20|X|2p+ 42β|X| ≤20|X|np/200 + 42 εnp|X|<|X|np/6,\nwhere we used |V\\X| ≥n/2 and eR(X, V \\X)≥eG(X, V \\X)/2 for the first inequality. This is a\ncontradiction, thus |Y| ≥20|X|. Therefore, in this case we have\n|NR(X, V \\(X∪S))|=|Y\\S| ≥20|X| −2 log n≥10|X|.\nFinally, if |X| ≥n/200, then, again by Lemma 3.3 and Lemma 3.4,\n|X|np/6≤eR(X, Y)≤ |X||Y|p+ 2β(|X|+|Y|)≤ |X||Y|p+ 2εn2p≤ |X||Y|p+ 400 εn|X|p,\nimplying that |Y| ≥(1/6−400ε)n, therefore |NR(X, V \\(X∪S)|=|Y\\S| ≥n/7.\nFor a set A⊆V, a vertex x∈Aand an integer ℓ≥1, we denote by Nℓ\nR(x, A) the set of vertices in A\nreachable from xinR[A] via a path of length at most ℓ. Note that N0\nR(x, A) ={x}.\nClaim 3.7. For any x∈V\\Sandℓ= log10n+ 7, we have\n|Nℓ\nR(x, V\\S)|> n/ 2.\nProof. Starting with N0\nR(x, V\\S), we iteratively apply Claim 3.6 to X:=Ni\nR(x, V\\S) to show that\n|Ni+1\nR(x, V\\S)| ≥ |Ni\nR(x, V\\S)|+ min {10i+1, n/7}.\nBy Claim 3.7, for every v, w∈V\\Swith v/\\e}atio\\slash=w, we have that Nℓ\nR(v, V\\S)∩Nℓ\nR(w, V \\S)/\\e}atio\\slash=∅,\nimplying that there is a path between vandwinR′=R\\Sof length at most\n2ℓ≤2 log10n+ 14 ≤2logn\nlog 10+ 14 ≤0.87 log n+ 14 ≤logn−1.\n3.1 Connecting vertices\nThe previous section summarises important properties of ( β, p)-jumbled graphs with sufficiently large\nminimum degree. Here we state some known results which rely o n these properties, and which we use to\nimplement steps (S2.b) and (S3).\nTheorem 3.8 ([10]) .LetGbe a graph with nvertices, and suppose the following two properties hold:\n•Gis/parenleftbig4n\nlogn,log log n/4/parenrightbig\n-expanding, and\n•between every two disjoint subsets of vertices A, B ⊆V(G)such that |A|,|B| ≥nlog log n\n5000 log n, there\nexists an edge.\nThen, for nsufficiently large and for any two vertices x, y∈V(G), there exists a Hamilton path in G\nbetween xandy.\nTo find a switcher in Gbut still be able to apply the above theorem, we have to connec t some vertices\nwith paths while maintaining that the remaining graph is an e xpander. In the case of Theorem 1.1 we\ngot that, in some sense, for free because of the large minimum degree. The following definition and two\nstatements provide us with the framework to achieve that in t he sparse setting.\n7Definition 3.9. LetGbe a graph and s, D∈N. Given a graph Fwith maximum degree at most D, we\nsay that an embedding φ:F ֒→Gis (s, D)-good if for every X⊆V(G), of size |X| ≤s, we have\n|NG(X)\\φ(F)| ≥ |φ(F)∩X|+/summationdisplay\nv∈X/bracketleftbig\nD−degF(φ−1(v))/bracketrightbig\n.\n(Note: We slightly abuse the notation by letting degF(∅) = 0).\nTheorem 3.10 ([7]).LetFbe a graph with maximum degree at most Dand fewer than s/2 + 1 vertices,\nfor some D, s∈N. Suppose we are given an (s, D+ 2)-expanding graph Gand an (s, D)-good embedding\nφ:F ֒→G. Then for every graph F′⊇Fwith maximum degree at most Dand at most s/2 + 1 vertices,\nwhich can be obtained from Fby successively adding a new vertex of degree 1, there exists an(s, D)-good\nembedding φ′:F′֒→Gwhich extends φ.\nLemma 3.11 ([7]).LetFbe a graph with at most svertices and maximum degree at most D, for some\ns, D ∈N. Suppose we are given a graph Gand an (s, D)-good embedding φ:F ֒→G. Then for every\ngraph F′obtained from Fby successively removing a vertex of degree 1, the restricti onφ′ofφtoF′is\nalso(s, D)-good.\n4 Hamilton space in sparse pseudorandom graphs\nThroughout the section we use Vto denote the vertex set of G. We follow the recipe given in Section 2.\nProof of Theorem 1.3. Note first that Gis Hamiltonian. That is because it is (n\n3 log log n,log log n)-\nexpanding by Lemma 3.2, and so it satisfies the conditions of T heorem 3.8. Thus for any edge uv∈E(G),\nthere is a Hamilton path between uandvinG, closing a Hamilton cycle.\nSuppose Cn(G)/\\e}atio\\slash=C(G), and let R⊆Gbe a subgraph given by Lemma 2.1. We now show how to\nimplement steps (S2) and (S3), which suffices for the contradi ction. We refer the reader to Section 2 for\ndetails. We use the following values:\nℓ= log n, d = log log n/2,and s=n/(6d).\nStep (S2.a). By applying Lemma 2.3 with ℓ, which we can do due to Lemma 3.5, we obtain an even\ncycle C= (v1, . . . , v 2k) inGof length at most 2 ℓthat contains an odd number of edges in R.\nStep (S2.b). We now find paths connecting vitov2k−i+2, for 2 ≤i≤k, which together with Cform\nanR-parity-switcher. Let Fbe a graph with vertex set {u1, . . . , u 2k}(Fdoes not live in G), and let\nφ:F ֒→Gbe defined as φ(ui) =vi. Note that |φ(F)|= 2k≤2ℓ. Recall that by Lemma 3.2, Gis\n(n/(6d),2d)-expanding. Together with δ(G)≥np/2, for every X⊆V(G) with |X| ≤n/(6d) we have\n|N(X)| ≥max{np/2,2d|X|}.\nThis implies φis an ( s, d)-good embedding: Consider some X⊆V(G) of size |X| ≤n/(6d).\n• If |X| ≤4ℓ/d, then\n|NG(X)\\φ(F)| ≥np/2− |φ(F)| ≥6ℓ≥ |φ(F)∩X|+d|X|.\n• If 4 ℓ/d≤ |X| ≤n/(6d), then\n|NG(X)\\φ(F)| ≥2d|X| − |φ(F)| ≥2d|X| −2ℓ≥2ℓ+d|X| ≥ |φ(F)∩X|+d|X|.\nLetBnbe a binary tree with s/10 vertices and of depth at most log n. We connect viandv2k−i+2for\ni∈ {2, . . . , k }, sequentially, with paths of length at most log n, as follows:\n• Let F′be a graph obtained from Fby attaching disjoint copies of Bntouiandu2k−i+2. Denote\nthese two copies of BninF′byBa\nnandBb\nn.\n• Note that ∆( F′)≤3≤dand|V(F′)| ≤s/5 + 2 ℓ·logn≤s/2. We may apply Theorem 3.10 to\nextend φto an ( s, d)-good embedding φ′ofF′.\n8• Let a∈φ′(Ba\nn) and b∈φ′(Bb\nn) be such that abis an edge in G, which exists since\ns\n10=n\n30 log log n≥εnlog log n\nlogn.\nLetPidenote the path in Gfrom vitov2k−i+2going through φ′(Ba\nn) and φ′(Bb\nn) using ab.\n• Let F′′be obtained from F′by removing all the vertices in Ba\nnandBb\nnwhich are not on φ′−1(Pi).\n• Set F:=F′′andφto be the restriction of φ′toF′′, which remains an ( s, d)-good embedding by\nLemma 3.11.\nAt the end of this process, we have that every pair of vertices vi, v2k−i+2is connected by a path Piin\nG, thus φ(F)∪Cis an R-parity-switcher which we denote by W⊆G. Crucially, the embedding φis\n(s, d)-good. We use this immediately in the next step.\nStep (S3). Since ∆( F)≤3, we get that\nG′=G\\(φ(F)\\ {v1, vk+1})\nis (s, d/2)-expanding. To see this, consider any set X⊆V(G′) with |X| ≤s. Then\n|NG′(X)| ≥ |NG(X)\\φ(F)| ≥(d−3)|X| ≥d|X|/2,\nwhere the second inequality follows from the fact that φis (s, d)-good. We can now apply Theorem 3.8\nto obtain a Hamilton path PinG′between v1andvk+1, which finishes the proof.\n5 Concluding remarks\n• Heinig [12] showed that if Cn(Gn,p) =C(Gn,p) and G(n, p) is not a forest, then necessarily δ(Gn,p)≥\n3. This prompts the following hitting time problem, which would further refine Corollary 1.6:\nProblem 5.1. Suppose nis odd, and consider the random graph process {Gm}m∈(n\n2). Is it true\nthat, with high probability, δ(Gm)≥3implies Cn(Gm) =C(Gm)?\n• We reiterate the question asked by Heinig [11]:\nProblem 5.2. Suppose nis odd and sufficiently large, and let Gbe a graph with δ(G)> n/ 2. Is it\ntrue that Cn(G) =C(G)?\nTheorem 1.1 answers this if δ(G) is just a bit larger, namely δ(G)≥n/2+C(taking C= 41 suffices).\nIt would be interesting to see if the strategy outlined in Sec tion 2, together with a stability-type\ncase analysis akin to the one in [13], for example, can be used to resolve Problem 5.2. We leave this\nfor future work.\nReferences\n[1] N. Alon and F. R. Chung. Explicit construction of linear s ized tolerant networks. Discrete Mathe-\nmatics , 72(1-3):15–19, 1988.\n[2] B. Alspach, S. C. Locke, and D. Witte. The Hamilton spaces of Cayley graphs on abelian groups.\nDiscrete Math. , 82(2):113–126, 1990.\n[3] J. D. Baron and J. Kahn. On the cycle space of a random graph .Random Structures & Algorithms ,\n54(1):39–68, 2019.\n[4] J. A. Bondy and L. Lov´ asz. Cycles through specified verti ces of a graph. Combinatorica , 1:117–140,\n1981.\n[5] B. DeMarco, A. Hamm, and J. Kahn. On the triangle space of a random graph. J. Comb. , 4(2):229–\n249, 2013.\n9[6] G. A. Dirac. Some theorems on abstract graphs. Proceedings of the London Mathematical Society ,\n3(1):69–81, 1952.\n[7] N. Dragani´ c, M. Krivelevich, and R. Nenadov. Rolling ba ckwards can move you forward: on embed-\nding problems in sparse expanders. Trans. Am. Math. Soc. , 375(7):5195–5216, 2022.\n[8] S. Glock, D. M. Correia, and B. Sudakov. Hamilton cycles i n pseudorandom graphs.\narXiv:2303.05356 , 2023.\n[9] I. B.-A. Hartman. Long cycles generate the cycle space of a graph. Eur. J. Comb. , 4:237–246, 1983.\n[10] D. Hefetz, M. Krivelevich, and T. Szab´ o. Hamilton cycl es in highly connected and expanding graphs.\nCombinatorica , 29(5):547–568, 2009.\n[11] P. Heinig. On prisms, M¨ obius ladders and the cycle spac e of dense graphs. European Journal of\nCombinatorics , 36:503–530, 2014.\n[12] P. C. Heinig. When Hamilton circuits generate the cycle space of a random graph. arXiv preprint\narXiv:1303.0026 , 2013.\n[13] M. Krivelevich, C. Lee, and B. Sudakov. Robust Hamilton icity of Dirac graphs. Trans. Am. Math.\nSoc., 366(6):3095–3130, 2014.\n[14] M. Krivelevich and B. Sudakov. Sparse pseudo-random gr aphs are Hamiltonian. J. Graph Theory ,\n42(1):17–33, 2003.\n[15] S. C. Locke. A basis for the cycle space of a 2-connected g raph. Eur. J. Comb. , 6:253–256, 1985.\n[16] S. C. Locke. A basis for the cycle space of a 3-connected g raph. Ann. Discrete Math. 27, 381-397 ,\n1985.\n[17] A. Thomason. Pseudo-random graphs. In Annals of Discrete Mathematics (33) , volume 144 of\nNorth-Holland Mathematics Studies , pages 307–331. North-Holland, 1987.\n[18] A. Thomason. Pseudo-random graphs. In North-Holland Mathematics Studies , volume 144, pages\n307–331. Elsevier, 1987.\n10"    },    {        "title": "2402.01452v1.Almost_co_Kähler_manifolds_and___m_ρ___quasi_Einstein_solitons.pdf",        "content": "arXiv:2402.01452v1  [math.DG]  2 Feb 2024ALMOST CO-K ¨AHLER MANIFOLDS AND\n(m,ρ)-QUASI-EINSTEIN SOLITONS\nKRISHNENDU DE, MOHAMMAD NAZRUL ISLAM KHAN∗AND UDAY CHAND DE\nAbstract. The present paper aims to investigate ( m,ρ)-quasi-Einstein metrices\non almost co-K¨ ahler manifolds M. It is proven that if a ( κ,µ)-almost co-K¨ ahler\nmanifold with κ <0is (m,ρ)-quasi-Einstein manifold, then Mrepresents a N(κ)-\nalmost co-K¨ ahler manifold and the manifold is locally isom orphic to a solvable\nnon-nilpotent Lie group. Next, we study the three dimension al case and get\nthe above mentioned result along with the manifold M3becoming an η-Einstein\nmanifold. We also show that there does not exist ( m,ρ)-quasi-Einstein structure\non a compact ( κ,µ)-almost co-K¨ ahler manifold of dimension greater than thr ee\nwithκ <0. Further, we prove that an almost co-K¨ ahler manifold sati sfying\nη-Einstein condition with constant coefficients reduces to a K-almost co-K¨ ahler\nmanifold, provided ma1/negationslash= (2n−1)b1andm/negationslash= 1. We also characterize perfect\nfluid spacetime whose Lorentzian metric is equipped with ( m,ρ)-quasi Einstein\nsolitons and acquired that the perfect fluid spacetime has va nishing vorticity, or\nit represents dark energy era under certain restriction on t he potential function.\nFinally, we construct an example of an almost co-K¨ ahler man ifold with ( m,ρ)-\nquasi-Einstein solitons.\n1.Introduction\nIn[5], Boothby and Wanginvestigated anodd-dimensional di fferentiable manifold\nin 1958 with the help of almost contact and contact structure and investigated its\nfeatures from a topological perspective. Making use of tens or calculus, Sasaki [31]\ndescribed the characteristics of a differentiable manifold w hich is odd-dimensional,\nwith contact structures in 1960. Such manifolds were referr ed to as contact mani-\nfolds. After that, other researchers have discovered vario us types of contact mani-\nfolds and investigated their characteristics. The Kenmots u, the Sasakian, and the\nco-K¨ ahler manifolds are Tanno’s divisions of the almost co ntact metric manifolds\n[33], whose automorphism groups have the highest dimension s. It should be noted\nthat the cosymplectic manifolds that Blair [4] introduced a nd Goldberg and Yano\n[24] examined are nothing but the co-K¨ ahler manifolds. The co-K¨ ahler manifolds\n02020 Mathematics Subject Classification .53D15, 53C25.\nKey words and phrases: Almost Co-K¨ ahler manifold; ( κ,µ)-almost co-K¨ ahler manifold; compact\nmanifold; ( m,ρ)-quasi-Einstein soliton.\n∗Corresponding author.\n12\ncanbethought of fromsometopological perspectives as thea nalogueof K¨ ahler man-\nifolds in odd-dimension, that is why the new terminology has been introduced. The\ncharacteristics of almost co-K¨ ahler manifolds, which are an extension of co-K¨ ahler\nmanifolds, were examined by several researchers. Perrone [ 29] provided a thorough\nclassification of the three-dimensional homogeneous almos t co-K¨ ahler manifolds as\nwell as a local characterization of these manifolds under th e assumption of local\nsymmetry. We advise readers to read ([12], [32], [34]-[36]) and the references therein\nfor more information regarding almost co-K¨ ahler manifold s.\nIn both mathematics and physics, Einstein manifolds are cru cial. In Riemann-\nian and Semi-Riemannian geometry, it is interesting to inve stigate Einstein man-\nifolds and their generalizations. Numerous generalizatio ns of Einstein manifolds\nhave been developed recently, including quasi-Einstein ma nifolds [19], generalized\nquasi-Einstein manifolds [9], m-quasi-Einstein manifolds [35], ( m,ρ)-quasi-Einstein\nmanifolds [25], and many more.\nA Riemannian metric gof an almost co-K¨ ahler manifold is named a Ricci soliton\n[8] if there is a λ1∈R= constant, Ris the set of all real numbers and a smooth\nvector field Xsuch that\nRic+1\n2£Xg=λ1g, (1.1)\nin which Ricindicates the Ricci tensor of the metric tensor gand£denotes the\nLie-derivative. If λ1is a smooth function, then the above soliton is called an almo st\nRicci soliton [21].\nIf we choose a smooth function ω:M →RwithX=Dω,Dbeing the gradient\noperator of g, then it is named a gradient Ricci soliton. Hence, the equati on (1.1)\nreduces to\nRic+Hessω=λ1g, (1.2)\nwhereHessis the Hessian operator.\nA Ricci soliton is nothing but a natural generalization of Ei nstein metric [1]. In\nthis paper we study ( m,ρ)-quasi-Einstein solitons which are the generalization of\nEinstein solitons and gradient Ricci solitons.\nIn an almost co-K¨ ahler manifold the metric g, is named a generalized quasi-\nEinstein soliton if there exist ω,αandβ(smooth functions) such that\nRic+Hessω−αdω⊗dω=βg. (1.3)\nIn particular, if β∈Randα= 0 , then the foregoing soliton reduces to a gradient\nRicci soliton and m-quasi-Einstein soliton [2], if α=1\nmandβ∈R, wherem∈N.\nCatino [9] presented the idea of a generalized quasi-Einste in soliton, and Huang\nand Wei [25] further proposed taking into consideration the idea of a ( m,ρ)-quasi-\nEinstein-soliton as its special case.\nDefinition 1.1. In an almost co-K¨ ahler manifold the metric gis named (m,ρ)-\nquasi-Einstein soliton if there exists a smooth function ω:Mn→ Randm,ρ,λ 1∈3\nR=constant with 0< m≤ ∞such that\nRic+Hessω−1\nmdω⊗dω=βg= (ρτ+λ1)g, (1.4)\nwhereτdenotes the scalar curvature.\nObviously, a ( ∞,0)-quasi-Einstein soliton is a gradient Ricci soliton. In [ 15], De\nand De investigated ( m,ρ)-quasi-Einstein solitons in the frame-work of paracon-\ntact manifolds. Demirbe˘ g and G¨ uler [20] studied rigidity of (m,ρ)-quasi-Einstein\nmanifolds. As the study of almost co-K¨ ahler manifolds with (m,ρ)-quasi-Einstein\nsolitons is still pending, we want to fill this gap in this pape r. Precisely, we establish\nthe subsequent theorems:\nTheorem 1.1. If the metric of a (κ,µ)-almost co-K¨ ahler manifold M2n+1,n >1,\nwithκ <0is a(m,ρ)-quasi-Einstein metric, then Mrepresents a N(κ)-almost co-\nK¨ ahler manifold. Also, the manifold is locally isomorphic to a solvable non-nilpotent\nLie-group Gσwith the almost co-K¨ ahler structure (η,ζ,ϕ,g), whereσ=√−κ.\nTheorem 1.2. Let the metric of a (κ,µ)-almost co-K¨ ahler manifold M3withκ <0\nis a(m,ρ)-quasi-Einstein metric. Then Mrepresents a N(κ)-almost co-K¨ ahler\nmanifold. Also, M3is locally isomorphic to Gσwith the almost co-K¨ ahler structure,\nwhereσ=√−κandM3becomes an η-Einstein manifold.\nTheorem 1.3. There does not exist (m,ρ)-quasi-Einstein structures (g,ω,λ1)with\nDω= (ζω)ζon a compact (κ,µ)-almost co-K¨ ahler manifold with κ <0.\nTheorem 1.4. If an almost co-K¨ ahler manifold M2n+1satisfying η-Einstein con-\nditionRic=a1g+b1η⊗η, b1/ne}ationslash= 0with constant coefficients is (m,ρ)-quasi-Einstein\nmanifold and m/ne}ationslash= 1, thenM2n+1is aK-almost co-K¨ ahler manifold, provided\nma1/ne}ationslash= (2n−1)b1.\nSolitons are actually waves that physically propagate with some energy loss and\nmaintain their speed and shape after colliding with another wave of a similar kind.\nSolitons are crucial in the solution of initial-value probl ems in nonlinear partial\ndifferential equations describing wave propagation. Additi onally, it describes the\nFermi-Pasta-Ulam system’s recurrence [26].\nInterestinstudyingRiccisolitonsandtheirgeneralizati ons indifferentgeometrical\ncontexts has also considerably increased due to their conne ction to general relativ-\nity. Recently, in perfect fluid spacetimes, many authors inv estigated many type of\nsolitons like Ricci solitons [17], gradient Ricci solitons ([16], [17]), Yamabe solitons\n[18], gradient Yamabe solitons[16], gradient m-quasi Eins tein solitons[16], gradient\nη-Einstein solitons([17]), gradient Schouten solitons([1 7]), respectively. Also, physi-\ncal applications of almost Ricci solitons are investigated in ([21], [22]). Motivated by\nthe above studies, in this article, we intend to investigate the (m,ρ)-quasi-Einstein\nsolitons in perfect fluid spacetimes. As a result, we establi sh the following theorems.\nTheorem 1.5. If a perfect fluid spacetime of dimension n≥4admits a (m,ρ)-quasi\nEinstein soliton with the scalars α1,βand the potential function ωremain invariant4\nunder the velocity vector field ρ, then either the perfect fluid spacetime has vanishing\nvorticity or, the spacetime represents dark energy era.\n2.almost co-K ¨ahler manifolds\nM2n+1, a smooth manifold is named an almost contact metric manifol d if there\nexist a vector field ζ, a tensor field ϕ, a one-form ηand the Riemannian metric g\nsatisfying\nϕ2+I=η⊗ζ, η(ζ) = 1, (2.1)\ng(E1,F1) =g(ϕE1,ϕF1)+η(E1)η(F1) (2.2)\nfor all vector fields E1,F1onM2n+1. From (2.1), we get η◦ϕ= 0 and ϕζ= 0.\nToeachM2n+1thereisan associated 2-form Φ( E1,F1) =g(E1,ϕF1) forall vector\nfieldsE1,F1onM2n+1.η∧Φnis non-vanishing everywhere and hence M2n+1is\norientable.\nThe Riemannian curvature tensor Kis described by\nK(E1,F1) = [∇E1,∇F1]−∇[E1,F1], (2.3)\nin which ∇denotes the Riemannian connection.\nThe manifold Mis named an almost co-K¨ ahler manifold [24] if dη= 0 and\ndΦ = 0. In particular, an almost co-K¨ ahler manifold is called co-K¨ ahler if it is\nnormal, that is, ∇Φ = 0. An almost co-K¨ ahler structure is said to be strictly al most\nco-K¨ ahler if it is not a co-K¨ ahler structure.\nOn an almost co-K¨ ahler manifold M2n+1(η,ζ,ϕ,g), we set 2 h=£ζϕandh′=\nh◦ϕ. It is known that handh′are symmetric (1 ,1)-tensor fields and satisfy the\nfollowing relations ([27], [28], [29])\nhζ= 0, ϕh+hϕ= 0,trh= 0 = tr h′, (2.4)\n∇ζϕ= 0,∇ζ=h′, (2.5)\n∇ζh=−h2ϕ−ϕℓ, (2.6)\nϕℓϕ−ℓ= 2h2, (2.7)\nRic(ζ,ζ)+trh2= 0, (2.8)\nwhereℓ=K(.,ζ)ζindicates the Jacobi operator and tr denotes the trace.\nDefinition 2.1. An almost co-K¨ ahler manifold is named a K-almost co-K¨ ahler\nmanifold if the characteristic vector field ζis Killing.\nIt is known that, any co-K¨ ahler manifold is a K-almost co-K¨ ahler manifold, but\nthe converse is not true, in general. But it is true in 3-dimen sinal manifold.\nLemma 2.1. [24]Any 3-dimensional almost co-K¨ ahler manifold is co-K¨ ahle r if and\nonly ifζis Killing.5\n3. (κ,µ)- almost co-K ¨ahler manifolds\nLetM2n+1be an almost co-K¨ ahler manifold. Then M2n+1is named a ( κ,µ)-\nalmost co-K¨ ahler manifold if ξbelongs to the ( κ,µ)-nullity distribution, that is,\nK(E1,F1)ξ=κ(η(F1)E1−η(E1)F1)+µ(η(F1)hE1−η(E1)hF1) (3.1)\nfor all vector fields E1,F1onM2n+1and (κ,µ)∈ R2. Such a manifold was first\npresented by Endo [23] and were generalized by Dacko and Olsz ak [14] to ( κ,µ,ν)-\nspaces. Replacing F1byξin (3.1), we acquire ℓ=−κϕ2+µh. Putting the value of\nℓin (2.7) and using second equation of (2.4), we obtain\nh2=κϕ2. (3.2)\nBy (3.2), we easily see that κ= 0 and κ≤0 if and only if M2n+1is aK-almost\nco-K¨ ahler manifold. If κ <0, thenM2n+1is a strictly almost co-K¨ ahler manifold.\nIn particular, if µ= 0, then ξbelongs to the κ-nullity distribution N(κ) described\nby\nN(κ) :p→Np(κ)\n={G1∈Tp(M2n+1) :K(E1,F1)G1\n=κ(g(F1,G1)E1−g(E1,G1)F1),∀E1,F1∈Tp(M2n+1)},\nwhereTp(M2n+1) is the tangent space of M2n+1atp∈ M2n+1and investigated by\nDacko [13]. According to Theorem 4 of [13], we have\nTheorem 3.1. LetM2n+1be an almost co-K¨ ahler manifold with κ <0. Then\nM2n+1is locally isomorphic to a solvable non-nilpotent Lie group Gσwith the almost\nco-K¨ ahler structure (η,ξ,ϕ,g), whereσ=√−κand underlying manifold of Gσis\nthe space R2n+1.\nLemma 3.1. [10]LetM2n+1be a(κ,µ)-almost co-K¨ ahler manifold with κ <0.\nThen the Ricci operator Qis described by\nQ= 2nκη⊗ξ+µh. (3.3)\nThought the paper we assume ( κ,µ)-almost co-K¨ ahler manifolds in the sense of\nEndo [23], that is κandµare constant.\n4.Proof of the results\nNaturally, there arises a question regarding the existence of Theorem 3.1 for\n(κ,µ)-almost co-K¨ ahler manifolds with ( m,ρ)-quasi-Einstein solitons. The affirma-\ntive answer of the above question is given by Theorem 1.1.\nProof of the Theorem 1.1. From (3.3), we infer that τ= 2nκ= constant.\nBy virtue of (1.4) and (3.3), we obtain\n∇E1Dω= (2nκρ+λ1)E1−µhE1−2nκη(E1)ζ+1\nmg(E1,Dω)Dω (4.1)6\nfor all vector field E1onM2n+1. Taking covariant derivative of (4.1) along F1and\nusing (2.5), we get\n∇F1∇E1Dω= (2nκρ+λ1)∇F1E1−µ∇F1(hE1)\n−2nκ(F1η(E1))ζ−2nκη(E1)h′F1\n+1\nm[g(∇F1E1,Dω)Dω+g(E1,∇F1Dω)Dω\n+g(E1,Dω)∇F1Dω]. (4.2)\nInterchanging E1andF1in the above equation, we have\n∇E1∇F1Dω= (2nκρ+λ1)∇E1F1−µ∇E1(hF1)\n−2nκ(E1η(F1))ζ−2nκη(F1)h′E1\n+1\nm[g(∇E1F1,Dω)Dω+g(F1,∇E1Dω)Dω\n+g(F1,Dω)∇E1Dω]. (4.3)\nUsing (4.1), (4.2) and (4.3) in (2.3), we infer that\nK(E1,F1)Dω=−µ(∇E1h)F1+µ(∇F1h)E1\n−2nκ(η(F1)h′E1−η(E1)h′F1)\n+2nκρ+λ1\nm((F1ω)E1−(E1ω)F1)\n−µ\nm((F1ω)hE1−(E1ω)hF1)\n−2nκ\nm((F1ω)η(E1)−(E1ω)η(F1))ζ. (4.4)\nTaking inner product of the previous equation with ζand utilizing (3.2), we obtain\ng(K(E1,F1)Dω,ζ) = 2µκg(E1,ϕF1)+2nκ(ρ−1)+λ1\nm((F1ω)η(E1)−(E1ω)η(F1)).(4.5)\nFrom (3.1) and (4.5), it follows that\nκ((F1ω)η(E1)−(E1ω)η(F1))+µ(g(hF1,Dω)η(E1)−g(hE1,Dω)η(F1))\n= 2µκg(E1,ϕF1)+2nκ(ρ−1)+λ1\nm((F1ω)η(E1)−(E1ω)η(F1)).(4.6)\nReplacing E1byϕE1andF1byϕF1in the foregoing equation, we get\n0 = 2µκg(E1,ϕF1)\nwhich implies that µ= 0, because κ <0.\nIfµ= 0, the( κ,µ)-almost co-K¨ ahler manifoldreducestoa N(κ)-almost co-K¨ ahler\nmanifold.\nHereζbelongs to κ-nullity distribution with κ <0. By Theorem 3.1 M2n+1is\nlocally isomorphic to a solvable non-nilpotent Lie group Gσ, whereσ=√−κ.\nThus the proof is completed.7\nIt is well known that on any 3-dimensional manifold, the curv ature tensor is\ndescribed by\nK(E1,F1)G1=Ric(F1,G1)E1−Ric(E1,G1)F1+g(F1,G1)QE1−g(E1,G1)QF1\n−τ\n2(g(F1,G1)E1−g(E1,G1)F1) (4.7)\nfor all vector fields E1,F1,G1onM3.\nTaking trace of (3.1), we get Qζ= 2κζ. Putting F1=G1=ζin (4.7) and using\nℓ=−κϕ2+µh, we obtain\nQ=/parenleftBigτ\n2−κ/parenrightBig\nI+/parenleftBig\n3κ−τ\n2/parenrightBig\nη⊗ζ+µh. (4.8)\nThis is the expression of the Ricci operator Qon any 3-dimensional ( κ,µ)-almost\nco-K¨ ahler manifold.\nProof of the Theorem 1.2. Using (4.8) in (1.4), we get\n∇E1Dω=/parenleftBig\nρτ+λ1−τ\n2+κ/parenrightBig\nE1+/parenleftBigτ\n2−3κ/parenrightBig\nη(E1)ζ\n−µhE1+1\nmg(E1,Dω)Dω\nfor allE1onM3. By direct computation, we have\nK(E1,F1)Dω=/parenleftbigg\nρ−1\n2/parenrightbigg\n(E1τ)F1−/parenleftbigg\nρ−1\n2/parenrightbigg\n(F1τ)E1\n+1\n2(E1τ)η(F1)ζ−1\n2(F1τ)η(E1)ζ\n+/parenleftBigτ\n2−3κ/parenrightBig\n(η(F1)h′E1−η(E1)h′F1)\n− −µ(∇E1h)F1+µ(∇F1h)E1\n+1\nm/parenleftBig\nρτ+λ1−τ\n2+κ/parenrightBig\n((F1ω)E1−(E1ω)F1)\n+1\nm/parenleftBigτ\n2−3κ/parenrightBig\n((F1ω)η(E1)−(E1ω)η(F1))\n−µ\nm((F1ω)h(E1)−(E1ω)h(F1)) (4.9)\nfor allE1,F1onM3. Taking inner product of (4.9) with ζ, we obtain\ng(K(E1,F1)Df,ζ) =ρ(E1τ)η(F1)−ρ(F1τ)η(E1)+2µκg(E1,ϕF1)\n+1\nm(ρτ+λ1−2κ)((F1ω)η(E1)−(E1ω)η(F1)),(4.10)8\nwhere we have used (3.2). By virtue of (3.1) and (4.10), we hav e\nκ((F1ω)η(E1)−(E1ω)η(F1))+µ(g(hF1,Df)η(E1)−g(hE1,Df)η(F1))\n=ρ(E1τ)η(F1)−ρ(F1τ)η(E1)+2µκg(E1,ϕF1)\n+1\nm(ρτ+λ1−2κ)((F1ω)η(E1)−(E1ω)η(F1)). (4.11)\nNow, replacing E1andF1byϕE1andϕF1, respectively, in (4.11) we get\n0 = 2µκg(E1,ϕF1),\nwhich implies that µ= 0, because κ <0.\nIfµ= 0, the ( κ,µ)-almost co-K¨ ahler manifold turnsinto a N(κ)-almost co-K¨ ahler\nmanifold.\nBy Theorem 3.1 M3is locally isometric to a solvable non-nilpotent Lie group\nGσ, whereσ=√−κ. Sinceµ= 0, from (4.8) it follows that M3is anη-Einstein\nmanifold.\nHence the proof.\nProof of the Theorem 1.3. From (3.3), we acquire that the scalar curvature\nτ= 2nκ. Differentiating Dω= (ζω)ζalong the arbitrary vector field E1, we get\n∇E1Dω= (E1(ζω))ζ+(ζω)h′E1. (4.12)\nBy (3.3) and (4.12), equation (1.4) takes the form\n2nκη(E1)η(F1)+µg(hE1,F1)+(E1(ζω))η(F1)+(ζω)g(h′E1,F1)\n−1\nm(ζω)η(E1)η(F1) = (2nκρ+λ1)g(E1,F1) (4.13)\nfor all vector fields E1,F1onM2n+1. Replacing E1byϕE1andF1byϕF1, we\nobtain\n−µg(hE1,F1)−(ζf)g(h′E1,F1) = (2nκρ+λ1)g(ϕE1,ϕF1).\nContracting the preceding equation, we get\n2nκρ+λ1= 0. (4.14)\nSettingE1=F1=ζin (4.13) and using (4.14), we have\n2nκ+ζ(ζω)−1\nm(ζω)2= 0. (4.15)\nContracting in (4.12), we find\n∆ω+ζ(ζω) = 0, (4.16)\nwhere ∆ = −divDis the Laplacian operator. Using (4.16) in (4.15), we obtain\n∆ω= 2nκ−1\nm(ζω)2.\nBy divergence theorem\n1\nm/integraldisplay\nM(ζω)2dM= 2nκ/integraldisplay\nMdM.9\ndMrepresentsthevolumeformofthemanifoldandispositive, s inceMisorientable.\nTherefore, the right hand side is negative, because κ <0. Thus, the above relation\ndoes not hold.\nThis completes the proof.\nη-Einstein almost co-K¨ ahler manifolds:\nAn almost co-K¨ ahler manifold ( M2n+1,g) is named an η-Einstein manifold if\nRic=a1g+b1η⊗η, (4.17)\nwherea1andb1aresmoothfunctions. For b1= 0, it reducestoanEinsteinmanifold.\nIn particular, if both a1andb1are constant, then we say that M2n+1satisfiesη-\nEinstein condition with constant coefficients. From (2.8), w e have\na1+b1=−trh2. (4.18)\nTheη-Einstein condition on an almost co-K¨ ahler manifold impli es thatζis a har-\nmonic vector field [30].\nProof of the Theorem 1.4. From (4.17), we infer that scalar curvature τ=\n(2n+1)a1+b1= constant. By virtue of (4.17), we obtain from (1.4)\n∇E1Dω= (ρτ+λ1−a1)E1−b1η(E1)ζ+1\nmg(E1,Dω)Dω (4.19)\nfor all vector field E1onM2n+1. Using this we compute\nK(E1,F1)Dω=−b1η(F1)h′E1+b1η(E1)h′F1\n+1\nm(ρτ+λ1−a1)((F1ω)E1−(E1ω)F1)\n−b1\nm((F1ω)η(E1)−(E1ω)η(F1))ζ. (4.20)\nContracting (4.20), we get\nRic(F1,Dω) =2n(ρτ+λ1−a1)\nm(F1ω)−b1\nm((F1ω)−(ζω)η(F1)),(4.21)\nwhere we have used tr h′= 0. From (4.17) and (4.21), it can be written as\na1(F1ω)+b1(ζω)η(F1) =2n(ρτ+λ1−a1)\nm(F1ω)−b1\nm((F1ω)−(ζω)η(F1)).(4.22)\nReplacing F1byζin (4.22), we have\n/parenleftbigg\na1+b1−2n(ρτ+λ1−a1)\nm/parenrightbigg\n(ζω) = 0.\nFrom the preceding equation, we infer either a1+b1−2n(ρτ+λ1−a1)\nm= 0, or\nζω= 0.10\nIfa1+b1−2n(ρτ+λ1−a1)\nm= 0, from (4.22) we can find\nb1(1−m)\nm((F1ω)−(ζω)η(F1)) = 0,\nwhich implies Dω= (ζω)ζ, sinceb1/ne}ationslash= 0 and m/ne}ationslash= 1. Putting Dω= (ζω)ζin (4.19)\nand using (2.5), we get\n(E1(ζω))ζ+(ζω)h′E1=m(a1+b1)\n2nE1−b1η(E1)ζ+1\nm(ζω)2η(E1)ζ.(4.23)\nTaking trace of (4.23), we have\nζ(ζω) =m(a1+b1)\n2n(2n+1)−b1+1\nm(ζω)2. (4.24)\nSettingE1=ζin (4.23) and after that taking inner product, we acquire\nζ(ζω) =m(a1+b1)\n2n−b1+1\nm(ζω)2. (4.25)\nThe foregoing two equations imply that a1+b1= 0. From (4.18), tr h2= 0. Since\nhis a symmetric tensor, h= 0. Hence ζis a Killing vector field. Consequently,\nM2n+1is aK-almost co-K¨ ahler manifold.\nOn the other hand if ζω= 0, then from (4.22) we get\n/parenleftbigg\na1−2n(ρτ+λ1−a1)\nm+b1\nm/parenrightbigg\n(F1ω) = 0. (4.26)\nIfF1ω= 0, then ωis a constant and from (1.4) we see that the manifold is Einste in,\nwhich is a contradiction because b1/ne}ationslash= 0. Therefore, from (4.26) we observed that\na1−2n(ρτ+λ1−a1)\nm+b1\nm= 0. (4.27)\nPuttingE1=ζin (4.19) and then taking inner product with ζ, we get\nρτ+λ1−a1−b1= 0, (4.28)\nwhere we have used ∇ζζ= 0. From (4.27) and (4.28), it follows that ma1=\n(2n−1)b1.\nThus the proof is finished.\nUsing the Lemma 2.1, we can state that\nCorollary 4.1. If an almost co-K¨ ahler manifold M3satisfiesη-Einstein condition\nRic=a1g+b1η⊗η,b1/ne}ationslash= 0with constant coefficients is a (m,ρ)-quasi-Einstein\nmanifold and m/ne}ationslash= 1, thenM3is a co-K¨ ahler manifold, provided ma1/ne}ationslash=b1.\nBy Lemma 5 of [13], if M2n+1is an almost co-K¨ ahler manifold whose structure\nvector field ζbelongs to the κ-nullity distribution with κ <0, thenMsatisfies\nη-Einstein condition with constant coefficients with a1= 0 and b1= 2nκ/ne}ationslash= 0. Also\nma1/ne}ationslash= (2n−1)b1. Thus, from Theorem 1.4, we can state that11\nCorollary 4.2. There does not exist (m,ρ)-quasi-Einstein structure with m/ne}ationslash= 1\non an almost co-K¨ ahler manifold whose structure vector fiel d belongs to κ-nullity\ndistribution with κ <0.\n5.Applications to physics\nA Lorentzian manifold Nnis called a perfect fluid spacetime if its non-vanishing\nRicci tensor Ricobeys\nRic=α1g+β1A⊗A, (5.1)\nwhereα1,β1are scalar (not simultaneously zero) and for all E1,g(E1,ρ) =A(E1),\nρbeing the velocity vector field. We know that in a perfect fluid spacetime, the\nvector field ρis unit timelike, hence, we have g(ρ,ρ) =−1.\nFrom (5.1), we get\nQE1=α1E1+β1A(E1)ρ, (5.2)\nwhereQis the Ricci operator defined by g(QE1,F1) =Ric(E1,F1) and contracting\nE1in (5.2), we infer\nr=/summationdisplay\niǫiQEi=nα1−β1, (5.3)\nwhere{Ei}is an orthonormal basis of the tangent space at each point of t he\nspacetime and ǫi=g(Ei,Ei) =±1. The covariant derivative of (5.2) gives\n(∇E1Q)(F1) =E1(α1)F1+E1(β1)A(F1)ρ+β1(∇E1A)(F1)ρ+β1A(F1)∇E1ρ.(5.4)\nIfkdenotes the gravitational constant and Tindicates the energy momentum\ntensor, then Einstein’s field equations in the absence of cos mological constant have\nthe structure\nRic−r\n2g=kT. (5.5)\nIn case of perfect fluid spacetime if pindicates isotropic pressure and σis the energy\ndensity, then Tis defined as\nT= (p+σ)A⊗A+pg. (5.6)\nThe necessary and sufficient condition for the constant scala r curvature of a per-\nfect fluid spacetime is that nE1(α1) =E1(β1).Combining the equations (5.1), (5.5)\nand (5.6), we infer that\nα1=k(p−σ)\n2−n, β1=κ(p+σ). (5.7)\nFurthermore, characterizing the particular sort of perfec t fluid spacetimes, σand\npare interconnected by the relation p= Ωσin which Ω is named the equation of\nstate (EOS) parameter. If the state equation is of the form p=p(σ), then the\nperfect fluid spacetime is called isentropic. Also, if p=σ, then the perfect fluid\nspacetime represents stiff matter fluid[11]. The equation of state of stiff matter fluid\nwas introduced by Zeldovich [37] to describe a cold gas of bar yons, and used in his\ncosmological model.12\nA dust solution, in general relativity, is one in which the gr avitational field is\ntotally produced by the mass, momentum, and stress density o f a perfect fluid with\npositive mass density but vanishing pressure. This is a form of exact solution to the\nEinstein field equation. If p= 0, perfect fluid spacetime represents the dust matter\nfluid whereas the perfect fluid spacetime represents the radi ation era with p−σ\n3= 0\nand the dark energy era with p=−σ[11].\nMoreover, Dark energy era that satisfies the equation of stat e with Ω <−1 is\nknown as phantom regime or phantom energy. It has positive en ergy density but\nnegative pressure, such that p+σ <0. The physical consequences are explored in\n([6], [7]).\nProof of the Theorem 1.5 .\nLet the perfect fluid spacetime of dimension n≥4 admit a ( m,ρ)-quasi Einstein\nmetric and at first, we prove the following result\nLemma 5.1. Every perfect fluid spacetime of dimension n≥4obeying(m,ρ)-quasi\nEinstein soliton satisfies the following:\nK(E1,F1)Dω= (∇F1Q)E1−(∇E1Q)F1+β\nm{F1(ω)E1−E1(ω)F1}\n+1\nm{E1(ω)QF1−F1(ω)QE1}+{(E1β)F1−(F1β)E1}, (5.8)\nfor allE1, F1.\nProof.By hypothesis, the perfect fluid spacetime is endowed with ( m,ρ)-quasi Ein-\nstein metric. Therefore, using (1.4) we can write\n∇E1Dω+QE1=1\nmg(E1,Dω)Dω+βE1. (5.9)\nDifferentiating the equation (5.9) covariantly along F1, we get\n∇F1∇E1Dω=−∇F1QE1+1\nm∇F1g(E1,Dω)Dω\n+1\nmg(E1,Dω)∇F1Dω+β∇F1E1+(F1β)E1.(5.10)\nExchanging E1andF1in (5.10), we lead\n∇E1∇F1Dω=−∇E1QF1+1\nm∇E1g(F1,Dω)Dω\n+1\nmg(F1,Dω)∇E1Dω+β∇E1F1+(E1β)F1(5.11)\nand\n∇[E1,F1]Dω=−Q[E1,F1]+1\nmg([E1,F1],Dω)Dω+β[E1,F1]. (5.12)13\nUsing (5.9)-(5.12) and the relation K(E1,F1)Dω=∇E1∇F1Dω− ∇F1∇E1Dω−\n∇[E1,F1]Dω, we acquire\nK(E1,F1)Dω= (∇F1Q)E1−(∇E1Q)F1+β\nm{F1(ω)E1−E1(ω)F1}\n+1\nm{E1(ω)QF1−F1(ω)QE1}+{(E1β)F1−(F1β)E1}.\n/square\nUtilizing the previous Lemma and the equations (5.2), (5.4) , we infer\nK(E1,F1)Dω=E1(α1)F1−F1(α1)E1+{E1(β1)A(F1)−F1(β1)A(E1)}ρ\n+β1{(∇E1A)(F1)ρ+A(F1)∇E1ρ−∇F1A)(E1)ρ−A(E1)∇F1ρ}\n+β\nm{F1(ω)E1−E1(ω)F1}+1\nm{α1E1(ω)F1+β1E1(ω)A(F1)ρ\n−α1F1(ω)E1−β1F1(ω)A(E1)ρ}+{(E1β)F1−(F1β)E1}. (5.13)\nTaking a set of orthonormal frame field and executing contrac tion of the equation\n(5.13), we get\nRic(E1,Dω) = (1 −n)E1(α1)+E1(β1)+ρ(β1)A(E1)\n+β1{(∇E1A)(F1)−∇E1A)(ρ)+A(E1)divρ}+β\nm(n−1)(E1ω)\n+1\nm{α1E1(ω)+β1ρ(ω)A(E1)−nα1E1(ω)+β1E1(ω)}\n+(1−n)E1(β). (5.14)\nSettingE1=ρin the last equation and (5.1), and then comparing both the eq ua-\ntions, we find\n{m\n1−n(α1−β1)+β−α1}ρ(ω) =m{(1−n)[ρ(α1)+ρ(β)−β1divρ}.(5.15)\nLetα1,βandωare invariant underthevelocity vector field ρwhichmean ρ(α1) = 0,\nρ(β) = 0 and ρ(ω) = 0. Hence, the foregoing equation yields either, divρ= 0, or\nβ1= 0, since m/ne}ationslash= 0.\nIfdivρ= 0, then the velocity vector field is conservative. Since a co nservative\nvector field is always irrotational, we get the vorticity of t he perfect fluid is zero.\nIfβ1= 0, then we obtain that σ+p= 0. This represents dark energy era.\nThis finishes the proof.\n6.Example\nLetM3={(x,y,z)∈ R3:z >0}, where ( x,y,z) are the standard co-ordinate of\nR3. We choose the Riemannian metric gonM3described by\ng=dx⊗dx+dy⊗dy+x2+y2+e−2z\n4zdz⊗dz−y√zdx⊗dz−x√zdy⊗dz.14\nLet\nδ1=∂\n∂x, δ2=∂\n∂y, δ3=yez∂\n∂x+xez∂\n∂y+2√zez∂\n∂z.\nThen, the vector fields δ1,δ2,δ3are orthonormal.\nWe have\n[δ1,δ2] = 0,[δ1,δ3] =ezδ2,[δ2,δ3] =ezδ1.\nWe define η,ζandϕby\nη=e−z\n2√zdz, ζ=δ3, ϕδ1=δ2, ϕδ2=−δ1, ϕδ3= 0.\nWe can easily verify that, ηand Φ are closed. Hence, M3is an almost co-K¨ ahler\nmanifold.\nUsing Koszul’s formula we get\n∇δ1δ1= 0,∇δ1δ2=−ezδ3,∇δ1δ3=ezδ2,\n∇δ2δ1=−ezδ3,∇δ2δ2= 0,∇δ2δ3=ezδ1,\n∇δ3δ1= 0,∇δ3δ2= 0,∇δ3δ3= 0.\nThe components of the curvature tensor Kare given by\nK(δ1,δ2)δ1=−e2zδ2, K(δ1,δ2)δ2=e2zδ1, K(δ1,δ2)δ3= 0,\nK(δ1,δ3)δ1=e2zδ3, K(δ1,δ3)δ2= 2√ze2zδ3,\nK(δ1,δ3)δ3=−e2zδ1−2√ze2zδ2, K(δ2,δ3)δ1= 2√ze2zδ3,\nK(δ2,δ3)δ2=e2zδ3, K(δ2,δ3)δ3=−2√ze2zδ1−e2zδ2.\nUtilizing the foregoing expressions of K, we compute the Ricci operator Qby\nQδ1=−2√ze2zδ2, Qδ2=−2√ze2zδ1, Qδ3=−2e2zδ3. (6.1)\nThe tensor field his given by\nhδ1=−ezδ1, hδ2=ezδ2, hδ3= 0. (6.2)\nSuppose that ω=z. Then,Dω= 2√zezδ3. By directed computation, we have\n\n\n∇δ1Dω= 2√ze2zδ2,\n∇δ2Dω= 2√ze2zδ1,\n∇δ3Dω= 2e2z(1+2z)δ3.(6.3)\nFrom (6.1) and (6.3), we can verify that\n\n\nQδ1+∇δ1Dω−g(δ1,Dω)Dω= 0,\nQδ2+∇δ2Dω−g(δ2,Dω)Dω= 0,\nQδ3+∇δ3Dω−g(δ3,Dω)Dω= 0.\nThus for m= 1, ρ=λ1= 0, equation (1.4) is satisfied. Hence Mis an (m,ρ)-quasi-\nEinstein manifold.15\n7.Conclusion\nEinstein solitons are essential in both mathematics and phy sics. It is intriguing\nto look into Einstein solitons and their generalizations in Riemannian and Semi-\nRiemannian geometry. Recent years have seen the developmen t of many gener-\nalizations of Einstein solitons, including quasi-Einstei n solitons, generalized quasi-\nEinstein solitons, m-quasi-Einstein solitons, ( m,ρ)-quasi-Einstein solitons. In this\nstudy, we investigate the ( m,ρ)-quasi-Einstein solitons in almost co-K¨ ahler mani-\nfolds.\nHere we show that an almost co-K¨ ahler manifold admitting ( m,ρ)-quasi-Einstein\nsolitons islocally isomorphictoasolvablenon-nilpotent Liegroup. Wealsoestablish\nthat there does not exist ( m,ρ)-quasi-Einstein structure on a compact ( κ,µ)-almost\nco-K¨ ahler manifold. Finally, we construct an example of an almost co-K¨ ahler man-\nifold with ( m,ρ)-quasi-Einstein solitons.\nFurther study of ( m,ρ)-quasi-Einstein solitons in the context of Hopf manifolds ,\nas well as in the general theory of relativity and cosmology, is possible in the near\nfuture.\n8.Acknowledgement\nWe would like to thank the Referees and the Editor for reviewi ng the paper\ncarefully and their valuable comments to improve the qualit y of the paper.\nReferences\n[1] Basse, A. L., Einstein manifolds , Springer, Berlin, 1987.\n[2] Barros, A. and Gomes, J. N., Triviality of compact m-quasi-Einstein manifolds , Res. 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Chavanis, Cosmology with a stiff matter era, Phys. Rev. D 92, 103004 (2015).\n[12] Chen, X. M., Cotton solitons on almost co-Kahler 3-manifolds , Quaestiones Mathematicae 44\n(2021), 1055–1075.\n[13] Dacko, P., On almost cosymplectic manifolds with the structure vector fieldζbelonging to the\nκ-nullity distribution , Balkan J. Geom. Appl., 5(2000), 47-60.\n[14] Dacko, P. and Olszak, Z., On almost cosymplectic (κ,µ,ν)-spaces, Banach center publ., 69\n(2005), 211-220.16\n[15] De, K and De, U.C., ( m,ρ)-quasi Einstein solitons on paracontact geometry , Novi Sad J. Math.\n(published online) https://doi.org/10.30755/NSJOM.132 25.\n[16] De, K., De, U.C., Syied, A. A., Turki N. B. and Alsaeed, S. ,Perfect fluid spacetimes and\ngradient solitons , Journal of Nonlinear Mathematical Physics https://doi.o rg/10.1007/s44198-\n022-00066-5.\n[17] De, U. C., Mantica, C. A. and Suh, Y. J., Perfect Fluid Spacetimes and Gradient Solitons ,\nFilomat, 36(2022), 829-842.\n[18] De, U.C. Chaubey, S.K. and Shenawy, S., Perfect fluid spacetimes and Yamabe solitons , J\nMath Phys. 62, 032501 (2021); https://doi.org/10.1063/5.0033967\n[19] De, U.C. and De, B.K., On quasi-Einstein manifolds , Commun. Korean Math. Soc., 23(2008),\n413-420.\n[20] Demirbe˘ g, S. A. and G¨ uler, S., Rigidity of (m,ρ)-quasi Einstein manifolds , Mathematische\nNachrichten, 290(2017), 2100-2110.\n[21] Duggal, K.L., A New Class of Almost Ricci Solitons and Their Physical Inter pretation, Int Sch\nRes Notices, (2016), 4903520 (6 pages). doi: 10.1155/2016/ 4903520.\n[22] Duggal, K.L., Almost Ricci Solitons and Physical Applications, IEJG,10(2017), 1–10.\n[23] Endo, H., Non-existence of almost cosymplectic manifolds satisfyin g a certain condition , Tensor\n(N.S.),63(2002), 272-284.\n[24] Goldberg, S.I. and Yano. K., Integrability of almost cosymplectic structures , Pacific J. Math,\n31(1969), 373-382.\n[25] Huang, G. and Wei, Y., The classification of (m,ρ)-quasi-Einstein manifolds , Ann. Global\nAnal. Geom., 44(2013), 269-282.\n[26] A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser and N. Akhmediev, Fermi-Pasta-Ulam\nrecurrence in Nonlinear Fiber Optics: The Role of Reversibl e and Irreversible Losses , Physical\nReview X 4, 011054 (2014). DOI:https://doi.org/10.1103/P hysRevX.4.011054\n[27] Olszak, Z., On almost cosymplectic manifolds , Kodai Math. J., 4(1981), 239-250.\n[28] Olszak, Z., On almost cosymplectic manifolds with K¨ ahlerian leaves, Tensor (N.S.), 46(1987),\n117-124.\n[29] Perrone, D., Classification of homogeneous almost cosymplectic three-m anifolds, Diff. Geom.\nAppl.,30(2012), 49-58.\n[30] Perrone, D., Minimal Reeb vector fields on almost cosymplectic manifolds , Kodai Math. J., 36\n(2013), 258-274.\n[31] Sasaki, S., On differentiable manifolds with certain structures which ar e closely related to almost\ncontact structure I , Tohoku Math. J. 12(1960), 459–476.\n[32] Suh, Y. J. and De, U. C., Yamabe solitons and Ricci solitons on almost co-K¨ ahler man ifolds,\nCanadian Math. Bull., 62(2019), 653-661.\n[33] Tanno, S., Note on infinitesimal transformations over contact manifol ds, Tohoku Math.J. 14\n(1962), 416–430.\n[34] Wang, W., A class of three dimensional almost co-K¨ ahler manifold , Palestine J. Math. 6(2017),\n111-118.\n[35] Wang, Y., A generalization of Goldberg conjecture for co-K¨ ahler man ifolds, Mediterr. J. Math.,\n13(2016), 2679-2690.\n[36] Wang, Y., Ricci tensors on three-dimensional almost co-K¨ ahler mani folds, Kodai Math. J., 39\n(2016), 469-483.\n[37] Y. B. Zeldovich, The equation of state of ultrahigh density and its relativis tic limitations , Soviet\nPhys. JETP 14n.5 (1962), 1143-1147.17\nDepartment of Mathematics, Kabi Sukanta Mahavidyalaya, Th e University of Bur-\ndwan. Bhadreswar, P.O.-Angus, Hooghly, Pin 712221, West Be ngal, India. ORCID iD:\nhttps://orcid.org/0000-0001-6520-4520\nEmail address :krishnendu.de@outlook.in\nDepartment of Computer Engineering, College of Computer, Q assim University,\nBuraydah 51452, Saudi Arabia.\nEmail address :m.nazrul@qu.edu.sa\nDepartment of Pure Mathematics, University of Calcutta, We st Bengal, India.\nORCID iD: https://orcid.org/0000-0002-8990-4609\nEmail address :uc−de@yahoo.com"    },    {        "title": "2402.01474v1.Eigenvalues_of_the_magnetic_Dirichlet_Laplacian_with_constant_magnetic_field_on_discs_in_the_strong_field_limit.pdf",        "content": "arXiv:2402.01474v1  [math.SP]  2 Feb 2024Eigenvalues of the magnetic Dirichlet Laplacian with\nconstant magnetic field on discs in the strong field limit\nMatthias Baur1*and Timo Weidl1†\n1Institute of Analysis, Dynamics and Modeling, Department of Mathe matics, University of\nStuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany\n*E-mail:matthias.baur@mathematik.uni-stuttgart.de\n†E-mail:timo.weidl@mathematik.uni-stuttgart.de\nFebruary 5, 2024\nAbstract\nWe consider the magnetic Dirichlet Laplacian with constant magnetic field on domains of\nfinite measure. In the case of a disc, we prove that the eigenva lue branches with respect to the\nfield strength behave asymptotically linear with an exponen tially small remainder term as the\nfield strength goes to infinity. We compute the asymptotic exp ression for this remainder term.\nFurthermore, we show that for sufficiently large magnetic fiel d strengths, the spectral bound\ncorresponding to the P´ olya conjecture for the non-magneti c Dirichlet Laplacian is violated up\nto a sharp excess factor which is independent of the domain.\n1 Introduction\nIn this paper, we consider the magnetic Laplacian with a cons tant magnetic field on an open\nset Ω⊂R2of finite Lebesgue measure. Let the vector potential A:R2→R2be given by the\nstandard linear gauge\nA(x1,x2) =B\n2/parenleftbigg−x2\nx1/parenrightbigg\nwhere the scalar B∈Ris the strength of the constant magnetic field. The magnetic L aplacian\nHΩ\nBwith constant magnetic field and Dirichlet boundary conditi on is realized by Friedrichs\nextension of the quadratic form\na[u] :=/integraldisplay\nΩ|(−i∇+A)u|2dx, u ∈C∞\n0(Ω).\nJust as the non-magnetic Dirichlet Laplacian, the operator defined via a[·] is positive, self-\nadjointandhascompactresolvent. Itadmitsasequenceofre al, positiveeigenvalues {λn(Ω,B)}n∈N\noffinitemultiplicity thataccumulate atinfinityonly. Thec orrespondingeigenfunctions {ψn}n∈N\nsatisfy the weak eigenvalue equation\n/integraldisplay\nΩ(−i∇+A)ψn·(−i∇+A)ϕdx=λn(Ω,B)/integraldisplay\nΩψnϕdx, ϕ ∈C∞\n0(Ω).\nFor domains Ω with regular boundary, in particular for discs , this is equivalent to the strong\neigenvalue equation\n(−i∇+A)2ψn=λn(Ω,B)ψnin Ω,\nψn= 0 on ∂Ω,\n1where (−i∇+A)2abbreviates the classical differential operator ( −i∂1+A1)2+(−i∂2+A2)2.\nAs usual, the eigenvalues λn(Ω,B) shall be sorted by magnitude, i.e.\n0<λ1(Ω,B)≤λ2(Ω,B)≤...≤λn(Ω,B)≤...,\ncounting multiplicities. From the eigenvalues of HΩ\nBone defines for any γ≥0 the so-called\nRiesz means or eigenvalue moments\ntr(HΩ\nB−λ)γ\n−:=\n\n#{n:λn(Ω,B)<λ}, γ = 0,/summationtext\nn:λn(Ω,B)<λ(λ−λn(Ω,B))γ, γ > 0.\nIfγ= 0, the expression tr( HΩ\nB−λ)0\n−is more commonly called ”counting function” and denoted\nbyN(HΩ\nB,λ).\nIt is well known that the eigenvalues {λn(Ω,B)}n∈Nas functions over B∈Rcan be iden-\ntified piecewise with real-analytic eigenvalue branches. T his is a classical result of analytic\nperturbation theory, see for example Kato [17, Chapter VII §3 and §4]. In this framework, the\noperators {HΩ\nB}form a type (B) self-adjoint holomorphic family. The eigenv alue branches that\nrepresent the spectra of the family {HΩ\nB}typically do not maintain a particular order, since\ndifferent branches can intersect.\nWe are interested in the behavior of the spectrum of HΩ\nBas the field strength Bbecomes\nlarge. Our first result (Theorem 2.1) deals with the special c ase where Ω is a disc. Here,\nall real-analytic eigenvalue branches of the spectra of {HΩ\nB}B∈Rare given in terms of roots of\nconfluenthypergeometricfunctions. Wecomputetwotermasy mptoticsofallanalyticeigenvalue\nbranches. This result generalizes a theorem by Helffer and Sun dqvist [16].\nIn the second part of the paper, we are concerned with spectra l bounds on the sorted\neigenvalues λn(Ω,B)aswellastheRieszmeanstr( HΩ\nB−λ)γ\n−. Letusbrieflysummarizeimportant\nresults related to our work.\nFor the non-magnetic Dirichlet Laplacian, i.e. the case B= 0, the celebrated Weyl law [26]\n(see also [13, Chapter 3.2]) states that for any domain Ω ⊂R2\nN(HΩ\n0,λ) = tr(HΩ\nB−λ)0\n−=1\n4π|Ω|λ(1+o(1)) as λ→+∞. (1)\nBy integration, one can show that the Riesz means satisfy for anyγ≥0\ntr(HΩ\n0−λ)γ\n−=Lcl\nγ,2|Ω|λ1+γ(1+o(1)) as λ→+∞\nwhereLcl\nγ,2:= (4π(1+γ))−1. P´ olya conjectured that the asymptotic expression in (1) i s actually\nan upper bound, i.e. for any open domain Ω ⊂R2holds\nN(HΩ\n0,λ)≤1\n4π|Ω|λ, λ ≥0.\nor equivalently\nλn(Ω,0)≥4πn\n|Ω|, n∈N. (2)\nHe then gave a proof for so-called tiling domains [22]. Howev er, the case of general domains\nis still open. Even for discs, P´ olya’s conjecture was unres olved for several decades until it was\nfinally confirmed in 2022 by Filonov et al. [10].\nThe Aizenman-Lieb identity [2] allows lifting an inequalit y for a Riesz mean of order γ≥0\nto any order γ′>γ, turning the constant Lcl\nγ,2intoLcl\nγ′,2. This means that any domain satisfying\nP´ olya’s inequality also satisfies\ntr(HΩ\n0−λ)γ\n−≤Lcl\nγ,2|Ω|λ1+γ, λ≥0. (3)\n2for anyγ≥0. As an generalization of Polya’s conjecture, it is conject ured that (3) holds for\narbitrary domains.\nIn 1972 Berezin [5] gave a proof of the inequality (3) for γ= 1 and arbitrary domains.\nTogether with the Aizenman-Lieb identity, this confirmed th e bounds (3) for any γ≥1. For\n0≤γ <1 however, the inequalities (3) are still unproven. In this c ase, Berezin’s result can still\nbeusedtofindupperboundssimilar to(3), butthederivedcon stants differ fromtheconjectured\nconstantsLcl\nγ,2by certain excess factors, see for instance [13, Corollary 3 .30]. Combining the\ntwo arguments for γ≥1 and 0≤γ <1, Berezin’s bound implies for γ≥0 the inequalities\ntr(HΩ\n0−λ)γ\n−≤RγLcl\nγ,2|Ω|λ1+γ, λ≥0. (4)\nwhere\nRγ=\n\n2 if γ= 0,\n2/parenleftbiggγ\nγ+1/parenrightbiggγ\nif 0<γ <1,\n1 if γ≥1.\nThe bounds (4) are believed to be non-sharp for 0 ≤γ <1 as the excess factors Rγcan be\nomitted for all known domains satisfying P´ olya’s bound, e. g. tiling domains and discs.\nWeyl’slaw (1)continuestoholdifanadditionalfixedmagnet icvectorpotential isintroduced.\nThis has been proven under minimal assumptions on the magnet ic vector potential, see [12,\nTheorem A.1]. Asymptotics for Riesz means also remain uncha nged. It is therefore reasonable\nto ask whether inequalities similar to (3) resp. (4) hold wit h magnetic fields and what the\nsmallest possible values of the excess factors are, if they c annot be omitted. Following up a\nremark by Helffer, Laptev and Weidl [18] proved that (3) contin ues to hold if HΩ\n0is replaced\nby a magnetic Laplacian with arbitrary magnetic field in case γ≥3/2. It follows that bounds\nof the form (4) also hold if γ <3/2, but with larger excess factors than those known in the\nnon-magnetic case, see [14, Appendix], [12, Theorem 3.1]. F or the special case of the constant\nmagnetic field, Erd¨ os, Loss and Vougalter [8] established ( 3) forγ= 1. This means that (4)\nextends to\ntr(HΩ\nB−λ)γ\n−≤RγLcl\nγ,2|Ω|λ1+γ, λ≥0,B∈R, (5)\nwith the same excess factors Rγas without magnetic field.\nIn contrast to its non-magnetic version (4), it has been show n that (5) is sharp for any\nγ≥0, see Frank, Loss and Weidl [14]. Frank [12, Theorem 3.6] eve n proved that (5) is sharp\nfor any given domain Ω: The bounds (5) can be saturated when le ttingBandλtend to infinity\nsimultaneously in a suitable way. This is remarkable since t he origin of the excess factors Rγ\nin (4) and (5) is non-spectral. What is believed to be an artif act of a rough estimate in the\nnon-magnetic case turns out to be sharp for constant magneti c fields. Our second result is an\nalternative proof of the sharpness of (5) based on estimates of eigenvalues of the disc. Like\nFrank, we will show that (5) is sharp for any given domain Ω and anyγ≥0 (Theorem 2.2 and\nCorollary 2.3).\nIt should also be noted that the situation of constant magnet ic fields differs from another\nimportant special case, the case of δ-like magnetic fields which are induced by the so-called\nAharonov-Bohm potentials. Whereas P´ olya’s bound is viola ted for constant magnetic fields,\nit has recently been shown that P´ olya’s bound continues to h old for centered Aharonov-Bohm\npotentials on discs, see Filonov et al. [9].\nThe paper is structured as follows. In Section 2, we review th e eigenvalue problem of the\nmagnetic Dirichlet Laplacian with constant magnetic field o n discs and state our two main\nresults. In Section 3 and 4, we give the proofs to our main resu lts. Finally, we describe possible\nimprovements to the proof presented in Section 4 in the appen dix.\n32 Preliminaries and statement of results\nTopresentourresults, weneedtobrieflysummarizehowtheei genvalueproblemforthemagnetic\nDirichlet Laplacian with constant magnetic field on discs is solved. It is closely related to the\neigenvalue problem of the harmonic oscillator confined to di scs, see e.g. [23] and [25]. We follow\nthe outline of the latter source and modify it to fit the eigenv alue problem of the magnetic\nLaplacian on discs.\n2.1 Explicit solution of the eigenvalue problem on discs\nLetDRdenote a disc of radius R >0. The eigenvalue problem on DRcan be explicitly\nsolved in terms of zeros of special functions. Since the eige nvalues are invariant under rigid\ntransformations of Ω, we may assume thedisc DRis centered at theorigin. In polar coordinates,\nthe strong eigenvalue equation turns into\n/bracketleftbigg\n−∂2\n∂r2−1\nr∂\n∂r−1\nr2∂2\n∂ϕ2−iB∂\n∂ϕ+B2\n4r2/bracketrightbigg\nψ(r,ϕ) =λψ(r,ϕ), r∈(0,R), ϕ∈[0,2π),\nψ(R,ϕ) = 0, ϕ ∈[0,2π).\nWe seethat themagnetic Laplacian ( −i∇+A)2commutes withtheangular momentumoperator\nLz=−i∂\n∂ϕ, so the operators have a shared eigenbasis. The correspondi ng eigenfunctions are\ngiven by\nψ(r,ϕ) =Z(r)eilϕ, r ∈(0,R), ϕ∈[0,2π),\nwherel∈ZandZ(r) solves the one-dimensional differential equation\n/bracketleftbiggd2\ndr2+1\nrd\ndr−/parenleftbiggB2\n4r2+(Bl−λ)+l2\nr2/parenrightbigg/bracketrightbigg\nZ(r) = 0, r∈(0,R), (6)\nZ(R) = 0. (7)\nWe assume from now on that B >0 since (6) is invariant under simultaneous change of the\nsigns ofBandl. The solution to (6) can then be written as\nZ(r) =e−s\n2s|l|\n2Y(s), s=B\n2r2.\nHere,Y(s) solves\n/bracketleftbigg\nsd2\nds2+(|l|+1−s)d\nds−1\n2/parenleftbigg\nl+|l|+1−λ\nB/parenrightbigg/bracketrightbigg\nY(s) = 0\nwhich is an instance of Kummer’s equation\n/bracketleftbigg\nzd2\ndz2+(b−z)d\ndz−a/bracketrightbigg\ny(z) = 0, a,b,z ∈C, b/\\⌉}atio\\slash= 0,−1,−2,....\nSolutions of Kummer’s equation are usually given as linear c ombinations of Kummer’s confluent\nhypergeometric function M(a,b,z) (also often denoted by 1F1(a,b,z)) and Tricomi’s confluent\nhypergeometric function U(a,b,z). Of those two functions only M(a,b,z) is regular at z= 0\nand eventually leads to square-integrable eigenfunctions ψ. Therefore, only M(a,b,z) is relevant\nfor us. Recall that Kummer’s function M(a,b,z) is entire in aandzand given by the power\nseries\nM(a,b,z) =∞/summationdisplay\nk=0(a)k\n(b)kk!zk, b/\\⌉}atio\\slash= 0,−1,−2,...\nwhere\n(c)k=/braceleftBigg\n1 for k= 0,\nc·(c+1)·...·(c+k−1) fork≥1,c∈C, k∈N0,\n4denotes the Pochhammer symbol (for references on confluent h ypergeometric functions see [6],\n[4, Vol. I, Chapter VI], [1, Chapter 13] and [21, Chapter 13]) . In order to be an eigenvalue,\nthe yet unspecified eigenvalue λmust be chosen in such a way that Z(r) satisfies the Dirichlet\nboundary condition (7). This means that λmust be a solution to the implicit equation\nM/parenleftbigg1\n2/parenleftbigg\nl+|l|+1−λ\nB/parenrightbigg\n,|l|+1,BR2\n2/parenrightbigg\n= 0. (8)\nIt is known that the function a/ma√sto→M(a,b,z), wherea∈R,b>0,z≥0, has no roots larger\nthanorequal tozeroandaninfinitenumberofsimple, negativ e rootsam(b,z) withaccumulation\npoint at −∞(see e.g. [6, §17]). Let us assume the roots am(b,z) are sorted in decreasing order.\nWe then denote the solutions of the implicit equation (8) by λm,l(B) according to the relation\nλm,l(B) =/parenleftbigg\nl+|l|+1−2am/parenleftbigg\n|l|+1,BR2\n2/parenrightbigg/parenrightbigg\nB. (9)\nThe eigenvalues λm,l(B) then satisfy\nλ1,l(B)≤λ2,l(B)≤...\nBy the implicit function theorem, all maps z/ma√sto→am(b,z) are real analytic functions of z. This\nimplies that all maps B/ma√sto→λm,l(B),n∈N,l∈Z, are real analytic functions of B.\nFinally, the eigenfunctions\nψm,l(r,ϕ) =Zm,l(r)eilϕ,\nwhereZm,lare the solutions of the radial equation with λ=λm,l(B),m∈N,l∈Z, form a\ncomplete orthonormal basis of L2(DR). This is a consequence of {Zm,l}m∈Nforming a complete\northonormal basis of L2((0,R),rdr) which follows from classical Sturm-Liouville theory (for\nmore details see [23]). The set {λm,l(B) :m∈N,l∈Z}gained by solving (8) is thus indeed\nthe complete set of all eigenvalues. We have found a complete description of all eigenvalue\nbranches of the magnetic Dirichlet Laplacian with constant magnetic field on DR. Ordering\n{λm,l(B) :m∈N,l∈Z}by magnitude results in the set of sorted eigenvalues {λn(DR,B)}n∈N.\n2.2 Asymptotics for eigenvalue branches of the disc\nFigure 1(a) displays the lowest few eigenvalues λn(DR,B) overBfor a disc of unit area. It\nsuggests that the eigenvalue branches have an asymptotical ly linear behavior as B→+∞\nand there appear linearly growing sectors that only certain branches can populate. A detailed\ninvestigation of the eigenvalue branches λm,l(B) has been carried out in the doctoral thesis of\nSon [23, Theorem 3.3.4] where it is proven that\nlim\nB→+∞λm,l(B)\nB=l+|l|+1+2(m−1).\nSon posed the question for the asymptotic expression of the r emainderλm,l(B)/B−(l+|l|+\n1+2(m−1)) which is equivalent to asking for the next terms in the asy mptotic expansion of\nλm,l(B). In the case of the ground state (which can be identified with the branch λ1,0(B)), this\nquestion was already addressed by Helffer and Morame [15, Prop osition 4.4] in 2001. However,\ntheir asymptotic expression is incorrect and was also mista kenly cited by Ekholm, Kovaˇ r´ ık and\nPortmann [7] and Fournais and Helffer [11]. Helffer and Sundqvis t [16] corrected the erroneous\nasymptotic from [15] by showing an asymptotic in the semi-cl assical setting via construction of\ntrial states. Theorem 5.1 of [16] asserts that if DRis a disc of radius R>0, then for any n∈N\nλn(DR,B)\nB−1 =2\n(n−1)!/parenleftbiggBR2\n2/parenrightbiggn\ne−BR2/2(1+O(B−1)) asB→+∞.\n50 25 50 75 100 125 150 175 200\nB050100150200250300350400λn(Ω,B)\n(a)\n0 100 200 300 400 500 600\nB0.60.81.01.21.41.61.8λn(Ω,B)/parenleftbigg4πn\n|Ω|/parenrightbigg−1\n(b)\nFigure 1: Eigenvalues for a disc of radius of R= 1/√π(and hence area |DR|= 1). Color of\nthe curves fades from red to blue as nincreases. (a) Low eigenvalues over the field strength B.\nThe dashed lines indicate the lines λ=B, 3Band 5B. (b) The first 500 eigenvalues divided by\n4πn/|DR|plotted over the field strength B. The value 1 .0 associated with the lower bound from\nthe non-magnetic P´ olya conjecture is crossed for the first t ime atBcrit≈110.335 byλ11(DR,B).\nThis point is marked by a black dot and a dashed line.\n6Barbaroux et al. [3] gave a generalization of this result for non-constant magnetic fields and\nfairly generic domains.\nThis however does not fully answer the question about the sec ond term in the asymptotic\nexpansion of the eigenvalue branches λm,l(B) for allm∈N,l∈Z. The construction of trial\nstates only allows treating eigenvalues sorted in increasi ng order. As we will see with Corollary\n4.2, the sorted eigenvalues λn(Ω,B) coincide with the eigenvalue branches λ1,−(n−1)(B) for large\nenoughB, so the result by Helffer and Sundqvist only covers the case m= 1 andl≤0.\nOur first result gives the next term in the asymptotic expansi on for all eigenvalue branches\nλm,l(B) and therefore generalizes the result by Helffer and Sundqvis t.\nTheorem 2.1. For any fixed m∈N,l∈ZandR>0holds\nλm,l(B)\nB>l+|l|+1+2(m−1)for allB >0\nand\nλm,l(B)\nB−(l+|l|+1+2(m−1)) =2\nΓ(|l|+m)Γ(m)/parenleftbiggBR2\n2/parenrightbigg|l|+1+2(m−1)\ne−BR2\n2(1+O(B−1))\nasB→+∞.\nWe give the proof of Theorem 2.1 in Section 3. Note that the pro of can be extended to gain\nmore orders in the asymptotic expansion.\n2.3 Spectral inequalities and sharpness\nFor the constant magnetic field, Erd¨ os, Loss and Vougalter [ 8] proved that\nn/summationdisplay\nk=1λk(Ω,B)≥2πn2\n|Ω|, n∈N, (10)\nfor anyB∈R. Earlier, Li and Yau [19] had shown the inequality in the non- magnetic case. It\nwas then realized that the Li-Yau bound is equivalent to Bere zin’s bound by Legendre transfor-\nmation, see e.g. [13, Chapter 3.5]. Similarly, (10) is equiv alent to (5) for γ= 1.\nBy the same reasoning as without magnetic field, the result by Erd¨ os, Loss and Vougalter\nimplies (5) with the excess factors Rγdefined in the introduction and in particular\nλn(Ω,B)≥1\n2·4πn\n|Ω|, n∈N,B∈R, (11)\nfor any domain Ω. Compared to P´ olya’s conjectured bound (2) for the non-magnetic case there\nappears an excess factor of 1 /2.\nOur second main result is a proof of the sharpness of (11) for a ny domain Ω when Bis\nallowed to become large.\nTheorem 2.2. LetΩ⊂R2be a domain of finite measure. Then for any η >0there exists\nsomeBη>0such that for any B≥Bηone has at least one eigenvalue λn(Ω,B)with\n1\n2≤λn(Ω,B)/parenleftbigg4πn\n|Ω|/parenrightbigg−1\n≤1\n2+η.\nWe show Theorem 2.2 in Section 4. We first prove Theorem 2.2 in t he case of discs by an\nexplicit calculation. We then extend the result to disjoint unions of discs and finally any open\ndomain Ω of finite measure by a domain inclusion argument.\nAs a consequence of the theorem, none of the excess factors Rγ,γ≥0, in (5) can be\nimproved for any fixed domain. This is Frank’s sharpness resu lt [12, Theorem 3.6] that was\nalready mentioned in the introduction.\n7Corollary 2.3. LetΩ⊂R2be a domain of finite measure and γ≥0. Then for any η >0,\nthere exists some Bη>0such that for any B≥Bηone finds some λ>0with\nRγ−η≤tr(HΩ\nB−λ)γ\n−\nLcl\nγ,2|Ω|λ1+γ≤Rγ. (12)\nThe proof of the corollary will be given right after the proof of Theorem 2.2 in Section 4.\nFigure 1(b) illustrates Theorem 2.2 for the disc. It shows th at once the field strength\nsurpasses a critical value, the normalized quantities λn(Ω,B)(4πn/|Ω|)−1frequently drop below\nthe constant one, indicated by a black line. This is exactly t he case when the sorted eigenvalues\nλn(Ω,B) fall below the non-magnetic P´ olya bound of 4 πn/|Ω|. Theorem 2.2 asserts that any\ndomain Ω features this kind of behavior and that\nlim\nB→∞inf\nn∈Nλn(Ω,B)/parenleftbigg4πn\n|Ω|/parenrightbigg−1\n=1\n2, (13)\ni.e. the lower envelope of all curves in Figure 1(b) must conv erge to 1/2 asB→ ∞. Our proof\nof Theorem 2.2 allows an estimate on the convergence speed of (13). For the disc, we provide\nan upper bound without proof in the appendix. We plan to publi sh its proof in a future article.\nRemark. Opposite to the strong magnetic field limit where P´ olya’s no n-magnetic bound fails, it\nis a striking feature of the figure that all eigenvalues consi dered appear to satisfy P´ olya’s bound\non a finite interval of small magnetic field strengths. More sp ecifically, we see that for a disc\nwith|DR|= 1, P´ olya’s bound appears to be preserved if B <B crit≈110.335. By scaling, the\nsame is the case for discs of radius R>0 whenB <B crit·|DR|−1. This can be rephrased as a\ncondition on the magnetic flux\nφ=1\n2π/integraldisplay\nΩBdx=|Ω|B\n2π,\nThe condition B < B crit· |DR|−1is equivalent to φ < B crit/(2π)≈17.56. Guided by the\nnumerics, we may ask: Is there actually a critical magnetic fl uxφcritso that P´ olya’s bound is\npreserved for φ<φcrit? Is it also true for domains of arbitrary shape?\n3 Proof of Theorem 2.1\nFor completeness, we begin with a proof of the fact that\nlim\nB→+∞λm,l(B)\nB=l+|l|+1+2(m−1).\nAfter that we compute the asymptotic expression of the remai nder term.\nEquation (8) relates eigenvalues for the magnetic Dirichle t Laplacian on the disc with roots\nof Kummer’s function. Statements about roots of a/ma√sto→M(a,b,z) directly translate into prop-\nerties of eigenvalues. We will therefore closely analyze Ku mmer’s function and its roots in the\nfollowing. The arguments of Kummer’s function in (8) are alw ays real-valued, so from now on\nthe arguments a,bandzof Kummer’s function will be considered real numbers.\nThe existence of a limit of λm,l(B)/Bcan be established by monotone convergence. For\nλ≤(l+|l|+1)Bwe have\n1\n2/parenleftbigg\nl+|l|+1−λ\nB/parenrightbigg\n≥0\nand thus\nM/parenleftbigg1\n2/parenleftbigg\nl+|l|+1−λ\nB/parenrightbigg\n,|l|+1,BR2\n2/parenrightbigg\n= 1+∞/summationdisplay\nk=1/parenleftbig1\n2/parenleftbig\nl+|l|+1−λ\nB/parenrightbig/parenrightbig\nk\n(|l|+1)kk!/parenleftbiggBR2\n2/parenrightbiggk\n≥1,\n8as all summands in the series are non-negative. Therefore, ( 8) cannot be satisfied and λcannot\nbe an eigenvalue. This means that any eigenvalue λm,l(B) must fulfill\nλm,l(B)\nB>l+|l|+1.\nNote that the roots am(b,z) are strictly increasing with z >0 for any fixed b>0 andm∈N, see\ne.g. [6, §17]. According to (9), this implies that all maps B/ma√sto→λm,l(B)/Bare strictly decreasing.\nBy monotone convergence, the limit lim B→+∞λm,l(B)/Bexists and is bounded from below by\n(l+|l|+1).\nLet us now determine this limit. Let b=|l|+1>0 be fixed. For any m∈N0we have\nM(−m,b,z) =∞/summationdisplay\nk=0(−m)k\n(b)kk!zk=m/summationdisplay\nk=0(−m)k\n(b)kk!zk,\nsince (−m)k= 0 fork >m. The map z/ma√sto→M(−m,b,z) is a (scaled Laguerre) polynomial in z\nof degreemand for real zwe have\nM(−m,b,z) =(−m)m\n(b)mm!zm(1+O(z−1)) =(−1)m\n(b)mzm(1+O(z−1)) asz→+∞.\nThus, ifz∗>0 is chosen large enough, one can enforce that sgn( M(−m,b,z)) = (−1)mfor all\nz >z∗. Let nowm′∈N. One can choose z∗>0 large enough such that sgn( M(−m,b,z∗)) =\n(−1)mfor all integers mwith 0≤m≤m′. Asa/ma√sto→M(a,b,z) is continuous for fixed z > z∗,\nit must have at least one root in each of the intervals ( −m,−(m−1)), 1≤m≤m′, by the\nintermediate value theorem. However, it is actually true th at each of these intervals contains\nexactly one root. This is a consequence of intimate connecti on between roots of a/ma√sto→M(a,b,z)\nand roots of z/ma√sto→M(a,b,z) which is content of the next lemma. We will see that am(b,z) must\nbe bounded from above by −(m−1).\nTo state the lemma, let us introduce some new notation. First , sincebis fixed, we suppress\nthe parameter bin the following and just write am(z) for the negative roots of a/ma√sto→M(a,b,z)\ninstead ofam(b,z). Recall that for every fixed z >0 the roots am(z) are sorted in decreasing\norder. The function z/ma√sto→M(a,b,z) fora <0,b>0, has exactly ⌈−a⌉real, positive roots, see\n[24,§9], [6, §17] or [21, Section 13.9]. We sort the roots in increasing ord er and denote the m-th\nroot byzm(a). Figure 2 shows the first few roots am(z) andzm(a) for the case b= 1. We will\nnow proceed to prove the following lemma.\nLemma 3.1. Letam(z),zm(a)be defined as above. Then for all m∈N\nam(z)<−(m−1)for allz∈(0,∞).\nMoreover, am: (0,∞)→(−∞,−(m−1))is invertible and its inverse is given by zm:\n(−∞,−(m−1))→(0,∞).\nThe lemma not only states that am(z) can be inverted by some zk(a) but that the index k\nis always be equal to m. Knowing this, the upper bound on the root am(z) is owed to the fact\nthat there are only ⌈−a⌉real, positive roots zm(a) for fixeda.\nLemma 3.1 appeared as part of Proposition 4.3.12 in the thesi s of Son [23]. The importance\nof the lemma is two-fold. For one, it guarantees the upper bou ndam(z)<−(m−1) that\nwe need in this section. Secondly, we will make use of the char acterization of the inverse of\nam(z) in the proof of Theorem 2.2, see Corollary 4.1. For convenie nce of the reader, we give a\nself-contained proof of the lemma.\nProof.First, note that all roots am(z) andzm(a),m∈N, are simple. For the roots am(z), we\nalready mentioned this fact before. The roots zm(a) are simple as well. This is because, if one of\nthem were a multiple root, repeated use of Kummer’s equation would imply that this particular\nroot is of infinite order. But viewed as a complex function, Ku mmer’s function is entire in z.\n90 5 10 15 20 25 30\nz−6−5−4−3−2−10am(z)m = 0\nm = 1\nm = 2\nm = 3\nm = 4\nm = 5\n−6−5−4−3−2−10\na051015202530zm(a)\nm = 0m = 1m = 2m = 3m = 4m = 5\nFigure 2: The first few roots am(z) andzm(a) forb= 1.\nA root of infinite order would imply that Kummer’s function is identical to the zero function\nwhich is a clear contradiction.\nFor fixeda<0 there are exactly ⌈−a⌉rootszm(a), which means that the m-th rootzm(a)\nonly exists for a∈(−∞,−(m−1)). By the implicit function theorem, all maps amandzm,\ndefined on (0 ,∞) resp. ( −∞,−(m−1)) are smooth functions. For now, we only know that\nthe image of amis contained in ( −∞,0), because we do not have any constraints yet where\nthe rootsam(z) lie other than that they are negative. We know amis injective, since am(z)\nis strictly increasing with z. We need to show that the image of amis (−∞,−(m−1)) and\ninverted by zm.\nLeta∈(−∞,0),z∈(0,∞). Note that ais a root ofa′/ma√sto→M(a′,b,z) if and only if zis a root\nofz′/ma√sto→M(a,b,z′). This means that a=am(z) only ifz=zk(a) for somek∈N, 1≤k≤ ⌈−a⌉,\nand conversely z=zk(a) only ifa=am(z) for some m∈N. We wrote kbecause we do not\nknow yet what the connection between mandkis. We now show that m=kalways holds. In\nother words, we show that\nifa∈[−(m−1),0),thena=am(z) has no solution z∈(0,∞)\nand ifa∈(−∞,−(m−1)),thena=am(z) if and only if z=zm(a).\nOnce this is shown, we have that the image of amis contained in ( −∞,−(m−1)) andzm:\n(−∞,−(m−1))→(0,∞) is the inverse of am: (0,∞)→(−∞,−(m−1)) which is the assertion\nof the lemma.\nWecarryoutaninductionover m. Forthestartoftheinduction, let a∈(−∞,0)bearbitrary\nand suppose a=a1(zk(a)) for some integer 1 <k≤ ⌈−a⌉. Theremust exist some integer m∈N\nsuch thata=am(z1(a)). But then the sortings a1(z)>a2(z)>...andz1(a)<z2(a)<...and\nthe strict monotonicity of a1(z) with respect to zyields the contradiction\na=am(z1(a))≤a1(z1(a))<a1(zk(a)) =a.\nTherefore, for any a∈(−∞,0),a=a1(z) if and only if z=z1(a).\nLet us now assume that there exists an m′∈Nsuch that for all mwith 1≤m≤m′we have\nifa∈[−(m−1),0),thena=am(z) has no solution z∈(0,∞)\nand ifa∈(−∞,−(m−1)),thena=am(z) if and only if z=zm(a).\nWe need to show that\nifa∈[−((m′+1)−1),0),thena=am′+1(z) has no solution z∈(0,∞)\nand ifa∈(−∞,−((m′+1)−1)),thena=am′+1(z) if and only if z=zm′+1(a).\n10Leta∈(−∞,0) be arbitrary and assume that we have a z∈(0,∞) witha=am′+1(z). We\nknow thatz=zk(a) for some integer ksatisfying 1 ≤k≤ ⌈−a⌉. First of all, if k≤m′, then by\ninduction assumption a=ak(zk(a)). But this leads to the contradiction\na=am′+1(zk(a))<ak(zk(a)) =a,\nbecausek<m′+1. Thus, we must have k≥m′+1.\nWe can now discuss the solvability of a=am′+1(z) case-by-case.\nCase 1: Ifa∈[−((m′+1)−1),0) = [−m′,0), then 1 ≤k≤ ⌈−a⌉impliesk≤m′which\ncontradicts k≥m′+1. Therefore, there is no solution z∈(0,∞) toa=am′+1(z).\nCase 2: Considera∈(−∞,−((m′+1)−1)) = (−∞,−m′).\nCase 2.1: Ifa∈[−m′−1,−m′), thenm′+ 1≤k≤ ⌈−a⌉impliesk=m′+ 1. Thus,\na=am′+1(z) if and only if z=zm′+1(a).\nCase 2.2: Ifa∈(−∞,−m′−1), then more than m′+1 roots in zexist, namely z1(a), ...,\nz⌈−a⌉(a), so thatkcould be larger than m′+1. Suppose k > m′+1. There must exist some\nintegerm∈Nsuch thata=am(zm′+1(a)). Ifm <m′+1, then by the induction assumption\na=am(zm(a)) which leads to the contradiction\na=am(zm(a))<am(zm′+1(a)) =a.\nIfm≥m′+1, then\na=am(zm′+1(a))≤am′+1(zm′+1(a))<am′+1(zk(a)) =a\nwhich is again a contradiction. Hence, k=m′+1 and for all a∈(−∞,−((m′+1)−1)) holds\na=am′+1(z) only ifz=zm′+1(a). This concludes the induction and the proof.\nWe return to the proof of Theorem 2.1. By Lemma 3.1 we have am(z)<−(m−1) for all\nz∈(0,∞). Sinceam: (0,∞)→(−∞,−(m−1)) is strictly increasing and bijective, we must\nhaveam(z)→ −(m−1) forz→+∞. These two fact show directly with (9) that\nλm,l(B)\nB>l+|l|+1+2(m−1) for all B >0\nand\nlim\nB→+∞λm,l(B)\nB=l+|l|+1+2(m−1).\nThe lower bound is the first statement of Theorem 2.1.\nWe can now discuss the decay rate of the remainder λm,l(B)/B−(l+|l|+1+2(m−1)). For\nthis, we fix b=|l|+1 and set εm(z) :=−(m−1)−am(z) where - as previously - am(z) is the\nm-th negative root of a/ma√sto→M(a,b,z) forz >0 and fixed b>0. By Lemma 3.1, εm(z) is always\npositive. Since am(z)→ −(m−1) forz→+∞, we know that εm(z) =o(1) forz→+∞. We\nlook for an asymptotic expression of εm(z) asz→+∞, since this translates directly into an\nasymptotic expression for the remainder λm,l(B)/B−(l+|l|+1+2(m−1)).\nFirst, we split the series that defines M(am(z),b,z) into two parts. If\npm(z) :=m−1/summationdisplay\nk=0(−(m−1)−εm(z))k\n(b)kk!zk,\nqm(z) :=∞/summationdisplay\nk=m(−(m−1)−εm(z))k\n(b)kk!zk,\nthen\npm(z)+qm(z) =M(am(z),b,z) = 0.\n11The reason why splitting the series at k=mis helpful to gain an asymptotic expression for\nεm(z) is thatqm(z) contains all terms that have εm(z) as a factor appearing in the Pochhammer\nsymbol. We can factor out εm(z) fromqm(z) and write qm(z) =εm(z)˜qm(z). Then,\nεm(z) =−pm(z)\n˜qm(z)=/vextendsingle/vextendsingle/vextendsingle/vextendsinglepm(z)\n˜qm(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nand the asymptotic behavior of εm(z) is determined by pm(z) and ˜qm(z). As we will see, both\n|pm(z)|and|˜qm(z)|tend to infinity as z→+∞, but at different speed. It remains to investigate\ntheir asymptotic behavior as z→+∞.\nLet us first analyze the asymptotic behavior of pm(z). Forc∈Randk∈N0the function\nx/ma√sto→(c+x)kis a polynomial in xand (c+x)k= (c)k+O(x) asx→0. Now, because we already\nknow that εm(z)→0 asz→+∞, this yields that ( −(m−1)−εm(z))k= (−(m−1))k(1 +\nO(εm(z))) = (−(m−1))k(1+o(1)) asz→+∞for any fixed k∈N0andm∈N. Therefore,\npm(z) is asymptotically dominated by the term with k=m−1, or more precisely\npm(z) =(−(m−1))m−1\n(b)m−1(m−1)!zm−1(1+o(1)) =(−1)m−1\n(b)m−1zm−1(1+o(1)) as z→+∞.(14)\nLet us now take a closer look at qm(z). Using (b)i+j= (b)i(b+i)jforb∈Randi,j∈N0\nandn! = (1) nforn∈N0yields\nqm(z) =∞/summationdisplay\nk=m(−(m−1)−εm(z))k\n(b)kk!zk\n= (−(m−1)−εm(z))mzm∞/summationdisplay\nk=m(1−εm(z))k−m\n(b)kk!zk−m\n= (−1)m(1+εm(z))m−1εm(z)zm∞/summationdisplay\nk=0(1−εm(z))k\n(b)k+m(k+m)!zk\n= (−1)m(1+εm(z))m−1εm(z)zm∞/summationdisplay\nk=0(1)k(1−εm(z))k\n(b)m(b+m)k(1)m(1+m)kzk\nk!\n=εm(z)˜qm(z).\nwhere\n˜qm(z) := (−1)m(1+εm(z))m−1zm∞/summationdisplay\nk=0(1)k(1−εm(z))k\n(b)m(b+m)k(1)m(1+m)kzk\nk!.\nNote, that if zis large enough, then εm(z)<1, the summands in the series are positive and\n˜qm(z) has indeed the opposite sign of pm(z) as expected. We can rewrite ˜ qm(z) as\n˜qm(z) =(−1)m(1+εm(z))m−1zm\n(1)m(b)m2F2(1,1−εm(z);m+1,b+m;z)\nwhere\n2F2(a,b;c,d;z) =∞/summationdisplay\nk=0(a)k(b)k\n(c)k(d)kzk\nk!, c,d /\\⌉}atio\\slash= 0,−1,−2,...\nis a generalized hypergeometric function, see for example [ 4, Volume I, Chapter IV] and [20,\nChapter 3]. For given parameters a,b,c,d, the generalized hypergeometric function 2F2is an\nentire function in zand is increasing in a,bfora,b>0. For fixed a,b,c,dwe have\n2F2(a,b;c,d;z) =Γ(c)Γ(d)\nΓ(a)Γ(b)za+b−c−dez(1+O(z−1)) asz→+∞, (15)\n12see [20, Chapter 5.11.3] for reference. Let now be 0 <ε<1 arbitrary and let z∗be large enough\nsuch thatεm(z)<εfor allz >z∗. Then (1 −ε)k<(1−εm(z))kfor allk∈Nand allz >z∗,\ntherefore\n2F2(1,1−εm(z);m+1,b+m;z)>2F2(1,1−ε;m+1,b+m;z)>0\nfor allz>z∗and\n|˜qm(z)|=(1+εm(z))m−1zm\n(1)m(b)m2F2(1,1−εm(z);m+1,b+m;z)\n>(1)m−1zm\n(1)m(b)m2F2(1,1−ε;m+1,b+m;z)\n=zm\nm(b)mΓ(m+1)Γ(b+m)\nΓ(1−ε)z1+1−ε−m−1−b−mez(1+O(z−1)) asz→+∞\n=Γ(m)Γ(b+m)\n(b)mΓ(1−ε)z1−ε−b−mez(1+O(z−1)) asz→+∞.\nCombining this with (14), we get\nεm(z) =/vextendsingle/vextendsingle/vextendsingle/vextendsinglepm(z)\n˜qm(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(b)mΓ(1−ε)\n(b)m−1Γ(m)Γ(b+m)zb+2(m−1)+εe−z(1+o(1)) as z→+∞\n=Γ(1−ε)\nΓ(m)Γ(b+m−1)zb+2(m−1)+εe−z(1+o(1)) as z→+∞.\nand therefore for any ε>0\nεm(z) =O(zb+2(m−1)+εe−z) asz→+∞. (16)\nThis is not the asymptotic expression for εm(z) we want, but it is a good upper bound on\nits decay rate. To find the asymptotic expression for εm(z), we can feed back this upper bound\nin our calculation of asymptotics of pm(z) and ˜qm(z) to improve them further.\nWith the aid of (16), we now know that for any ε>0 and any fixed k∈N0\n(−(m−1)−εm(z))kzk= (−(m−1))kzk(1+O(zb+2(m−1)+εe−z)) asz→+∞.\nCompared to (14), this leads to the improved asymptotic\npm(z) =m−1/summationdisplay\nk=0(−(m−1))k\n(b)kk!zk+o(1) asz→+∞. (17)\nNext, we improve our lower bound for |˜qm(z)|. We note that for any k∈N\n(1−x)k=k/summationdisplay\nj=0ck,jxj\nwhereck,jare suitable coefficients with ck,0= (1)kand sgn(ck,j) = (−1)j. Forx=−1, we see\nthat\nk/summationdisplay\nj=0|ck,j|=k/summationdisplay\nj=0ck,j(−1)j= (2)k\n13and in particular |ck,j| ≤(2)kfor all 0≤j≤k. Using this, we deduce that\n|2F2(1,1;m+1,b+m;z)−2F2(1,1−εm(z);m+1,b+m;z)|\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nk=1(1)k[(1)k−(1−εm(z))k]\n(b+m)k(1+m)kzk\nk!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤∞/summationdisplay\nk=1k/summationdisplay\nj=1(1)k|ck,j|(εm(z))j\n(b+m)k(1+m)kzk\nk!\n≤∞/summationdisplay\nj=1(εm(z))j∞/summationdisplay\nk=j(1)k(2)k\n(b+m)k(1+m)kzk\nk!.\nNow, with (15) follows\n∞/summationdisplay\nk=j(1)k(2)k\n(b+m)k(1+m)kzk\nk!≤∞/summationdisplay\nk=0(1)k(2)k\n(b+m)k(1+m)kzk\nk!\n=2F2(1,2;m+1,b+m;z)\n=O(z−b−2(m−1)ez) asz→+∞\nand with (16) we get\n∞/summationdisplay\nj=1(εm(z))j∞/summationdisplay\nk=j(1)k(2)k\n(b+m)k(1+m)kzk\nk!≤εm(z)\n1−εm(z)2F2(1,2;m+1,b+m;z)\n=O(zε) asz→+∞\nwith theεcoming from (16). Hence, we can improve the previous estimat e of|˜qm(z)|to\n|˜qm(z)|=(1+εm(z))m−1zm\n(1)m(b)m2F2(1,1−εm(z);m+1,b+m;z)\n≥(1)m−1zm\n(1)m(b)m(2F2(1,1;m+1,b+m;z)+O(zε))\n=Γ(m)Γ(b+m)\n(b)mz1−b−mez(1+O(z−1)) asz→+∞\nwhich gives with (17)\nεm(z) =/vextendsingle/vextendsingle/vextendsingle/vextendsinglepm(z)\n˜qm(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\nΓ(m)Γ(b+m−1)zb+2(m−1)e−z(1+O(z−1)) asz→+∞.\nThis upper bound is the asymptotic expression we are looking for. It is not difficult to estimate\n˜qm(z) also from above by\n|˜qm(z)| ≤(1)m−1zm\n(1)m(b)m(1+O(z−1))2F2(1,1;m+1,b+m;z)\n=Γ(m)Γ(b+m)\n(b)mz1−b−mez(1+O(z−1)) asz→+∞\nwhich then yields with (17) the corresponding lower bound\nεm(z) =/vextendsingle/vextendsingle/vextendsingle/vextendsinglepm(z)\n˜qm(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥(b)m\n(b)m−1Γ(b+m)(m−1)!zb+2(m−1)e−z(1+O(z−1)) asz→+∞\n=1\nΓ(m)Γ(b+m−1)zb+2(m−1)e−z(1+O(z−1)) asz→+∞.\n14Putting both bounds for εm(z) together we have established\nεm(z) =1\nΓ(m)Γ(b+m−1)zb+2(m−1)e−z(1+O(z−1)) asz→+∞. (18)\nFinally, when we set b=|l|+1,z=BR2/2 and employ the relation\nλm,l(B)\nB=l+|l|+1−2am/parenleftbiggBR2\n2/parenrightbigg\n=l+|l|+1+2(m−1)+2ε/parenleftbiggBR2\n2/parenrightbigg\n,\nour result (18) becomes the second statement of the theorem.\n4 Proof of Theorem 2.2 and Corollary 2.3\nWe begin with the proof of Theorem 2.2. Given η>0, we need to find an eigenvalue λn(Ω,B)\nthat satisfies\nλn(Ω,B)≤/parenleftbigg1\n2+η/parenrightbigg4πn\n|Ω|(19)\nfor somen=n(B) whenBis large enough. The index n(B) cannot be chosen independent of\nBbut generally needs to increase with B. This is because λn(Ω,B)≥Bfor anyn∈N(for\ndiscs, this follows from Theorem 2.1, but for general domain s Ω, see e.g. [11, Lemma 1.4.1]) and\ntherefore the bound (19) cannot be satisfied for\nn</parenleftbigg1\n2+η/parenrightbigg−1|Ω|B\n4π.\nWe will see that one can choose n(B) linearly increasing with B.\nThe proof of Theorem 2.2 is divided into three steps. First, w e show the assertion of the\ntheorem in the case where Ω is a disc. We are then able to lift th e result from discs to unions\nof disjoint discs. The generalization to arbitrary domains is done with coverings by unions\nof disjoint discs and a domain inclusion argument. We conclu de the section by proving the\ncorresponding result for Riesz means.\n4.1 Proof of Theorem 2.2 for discs\nAlthough we know the eigenvalue branches λm,l(B) for the disc in terms of roots of Kummer’s\nfunction, there remains the difficulty of relating the eigenv alue branches λm,l(B) to the sorted\neigenvalues λn(DR,B). We need to identify λm,l(B) withλn(DR,B) for certain nand also find\nestimates on the branches λm,l(B). We will gain such estimates from a further investigation o f\nthe Kummer function.\nIt will suffice for us to characterize all eigenvalues λn(DR,B)≤3B. Theorem 2.1 tells us\nthatλm,l(B)>3Bform≥2 orl >0, so all eigenvalues λn(DR,B)≤3Bmust correspond to\neigenvalue branches λ1,l(B)withl≤0. AsseeninFigure1(a), thenumberof branches λ1,l(B)≤\n3Bdepends on the field strength. We can count these branches by c ounting the intersection\npoints between λ1,l(B) and the line λ= 3B. In general, intersections of the eigenvalue branches\nλm,l(B) with lines λ= (l+|l|+1+2k)B,k∈N, can be identified with help of Lemma 3.1.\nRecall that λm,l(B)/Bis strictly decreasing, so λm,l(B) has at most one intersection with\na lineλ= (l+|l|+ 1 + 2k)B,k∈N. By the lower bound given in Theorem 2.1, the branch\nλm,l(B) has no intersection with λ= (l+|l|+ 1 + 2k)Bifk≤m−1. Ifk≥m, there is an\nintersection and Lemma 3.1 yields the following.\nCorollary 4.1. Letm∈N,l∈Zandk∈N,k≥m. Thenλm,l(B) = (l+|l|+1+2k)Bif and\nonly ifB= 2zm(−k,|l|+1)/R2. Here,zm(a,b)denotes the m-th positive root of z/ma√sto→M(a,b,z).\n15For the proof, recall also that we denoted the m-th negative root of a/ma√sto→M(a,b,z) earlier\nbyam(b,z), sorted in decreasing order.\nProof.According to (9), λm,l(B) is related to the root am(b,z) by\nλm,l(B) =/parenleftbigg\nl+|l|+1−2am/parenleftbigg\n|l|+1,BR2\n2/parenrightbigg/parenrightbigg\nB.\nTherefore,λm,l(B) = (l+|l|+1+2k)Bif and only if\nam/parenleftbigg\n|l|+1,BR2\n2/parenrightbigg\n=−k.\nFinally, since zmis the inverse of amwith respect to a∈(−∞,−(m−1)) for any fixed b >0\nby Lemma 3.1 and −k∈(−∞,−(m−1)) fork≥m, we see that\nam/parenleftbigg\n|l|+1,BR2\n2/parenrightbigg\n=−kif and only ifBR2\n2=zm(−k,|l|+1).\nFork∈N, the power series of M(−k,b,z) turns into a polynomial of degree kwhich allows\nexplicit computation of the roots zm(−k,b) for lowk. In particular, if k= 1, the polynomial\nM(−1,b,z) = 1−z/bis linear and its only root is z=b. This means z1(−1,b) =band\na1(b,b) =−1. Withb=|l|+1, the corollary implies λ1,l(B) = 3Bif and only if l≤0 and\nB=2z1(−1,|l|+1)\nR2=2\nR2(|l|+1).\nThis allows us to give a simple characterization of sorted ei genvaluesλn(DR,B).\nCorollary 4.2. Letn∈N. IfB≥2n/R2, then\nλn(DR,B) =λ1,−(n−1)(B).\nProof.We have just shown with Corollary 4.1 that\nλ1,−(n−1)(B) = 3Bif and only if B=2n\nR2,\nthus, by strict monotonicity of λ1,l(B)/Bwith respect to B, we haveλ1,−(n−1)(B)≤3Bfor\nB≥2n/R2. Due to the fact that for any B>0\nλ1,l′(B)>λ1,l(B) forl′<l≤0,\nsee [23, Theorem 3.3.4], we get\nλ1,0(B)<λ1,−1(B)<...<λ 1,−(n−1)(B)≤3BforB≥2n\nR2\nandλ1,l(B)>λ1,−(n−1)(B) for anyl≤ −n. Recall that by Theorem 2.1, if m≥2 orl>0, then\nλm,l(B)>3Band henceλm,l(B)>λ1,−(n−1)(B) ifB≥2n/R2. This shows that if B≥2n/R2,\nthere are no smaller eigenvalues than λ1,0(B), ...,λ1,−(n−1)(B) and since they are sorted by\nmagnitude, we can identify them with λ1(DR,B), ...,λn(DR,B).\nLet us now derive estimates for eigenvalue branches λ1,l(B),l≤0. First, we show the\nfollowing lemma concerning Kummer’s function.\nLemma 4.3. Let0<δ<1andε>0. There exists bδ,ε>0such that for any b>bδ,ε\nM(−δ,b,(1+ε)b)<0.\n16Proof.First observe that\nM(−δ,b,(1+ε)b) = 1+∞/summationdisplay\nk=1(−δ)k\nk!bk\n(b)k(1+ε)k\nwhere the series only sums negative terms when 0 < δ <1. The appearing series converges,\nhowever the series\n∞/summationdisplay\nk=1(−δ)k\nk!(1+ε)k,\nwhere we replaced the bdependent terms by one, is divergent for any ε >0. It tends to −∞,\nsince all summands are negative and the radius of convergenc e of the power series\n∞/summationdisplay\nk=1(−δ)k\nk!zk\nis\nR=/parenleftBigg\nlim\nk→+∞/parenleftbigg(−δ)k\nk!/parenrightbigg1\nk/parenrightBigg−1\n=/parenleftBigg\nlim\nk→+∞/parenleftbiggΓ(−δ+k)\nΓ(−δ)Γ(k+1)/parenrightbigg1\nk/parenrightBigg−1\n= 1.\nWe can therefore choose some c∈(0,1) and then some large N∈Nsuch that\ncN/summationdisplay\nk=1(−δ)k\nk!(1+ε)k<−1.\nFurthermore, for any fixed k≥1 holds\nbk\n(b)k→1 asb→+∞,\nthus there exists bδ,ε>0 such that for b>bδ,ε\nbk\n(b)k≥cfor allk= 1,...,N.\nHence, forb>bδ,ε\nM(−δ,b,(1+ε)b) = 1+∞/summationdisplay\nk=1(−δ)k\nk!bk\n(b)k(1+ε)k≤1+cN/summationdisplay\nk=1(−δ)k\nk!(1+ε)k<0.\nand we have shown the lemma.\nThe lemma implies a bound on the roots in aof Kummer’s function and therefore on eigen-\nvalue branches. Together with M(0,b,(1 +ε)b) = 1>0 we see that for any b >bδ,ε, the map\na/ma√sto→M(a,b,(1 +ε)b) has at least one root in ( −δ,0) by the intermediate value theorem. In\nparticular, we have\n−δ<a1(b,(1+ε)b)<0 forb>bδ,ε. (20)\nAgain, recall that a1(b,(1+ε)b) relates totheeigenvalues λ1,l(B)via(9). Whensetting b=|l|+1\nforl≤0 we see that\nλ1,l(B|l|+1)\nB|l|+1= 1−2a1(|l|+1,(1+ε)(|l|+1))\n17where the magnetic field strength B|l|+1is given by\nB|l|+1=2\nR2(1+ε)(|l|+1).\nBut then (20) implies that\nλ1,l(B|l|+1)\nB|l|+1≤1+2δfor anyl≤0,|l|+1>bδ,ε\nand sinceλ1,l(B)/Bis strictly decreasing with B, we have\nλ1,l(B)\nB≤1+2δfor anyB≥B|l|+1andl≤0,|l|+1>bδ,ε. (21)\nWith (21) at hand, we are almost done. It remains to bring Bto the right side, choose δ\nandεappropriately and use Corollary 4.2 to identify λ1,−(n−1)(B) withλn(DR,B).\nLet nowη>0 be arbitrary. Choose ε>0 and 0<δ<1 small enough such that (1+2 δ)(1+\nε)<1+2η. After that choose b∗≥bδ,εlarge enough such that\n(1+2δ)(1+ε)/parenleftbigg\n1+1\nb∗/parenrightbigg\n≤1+2η.\nAssume|l|+1>b∗and\nB=2\nR2(1+ ˜ε)(|l|+1),where ˜ε∈/bracketleftbigg\nε,ε+1+ε\nb∗/bracketrightbigg\n. (22)\nSince ˜ε≥ε, we haveB≥B|l|+1and therefore (21) holds. Hence, it follows that\nλ1,l(B)≤(1+2δ)B= (1+2δ)2\nR2(1+ ˜ε)(|l|+1)\n≤(1+2δ)/parenleftbigg\n1+ε+1+ε\nb∗/parenrightbigg2π\n|DR|(|l|+1)\n= (1+2δ)(1+ε)/parenleftbigg\n1+1\nb∗/parenrightbigg2π\n|DR|(|l|+1)\n≤(1+2η)2π\n|DR|(|l|+1).\nNote that if |l|+1>b∗andB∈[B|l|+1,B|l|+2] thenBcan be written as in (22) and thus we\nhave shown that\nλ1,l(B)≤(1+2η)2π\n|DR|(|l|+1),forB∈[B|l|+1,B|l|+2], l≤0,|l|+1>b∗.\nNow letn=−l+ 1 =|l|+ 1. Then B∈[B|l|+1,B|l|+2] impliesB >2n/R2and therefore by\nCorollary 4.2\nλn(DR,B) =λ1,−(n−1)(B) =λ1,l(B)≤(1+2η)2π\n|DR|(|l|+1) =/parenleftbigg1\n2+η/parenrightbigg4πn\n|DR|(23)\nforB∈[Bn,Bn+1], n > b∗. The desired inequality hence always holds for some eigenva lue if\nB≥minn>b∗Bn.\n184.2 Proof of Theorem 2.2 for disjoint unions of finitely many\ndiscs\nLetη >0 be arbitrary. Let N∈Nand Ω = Ω 1∪...∪ΩNbe the disjoint union of Ndiscs\nΩi=DRiwith radiiRi>0. Then of course |Ω|=|Ω1|+...+|ΩN|=π/summationtextN\ni=1R2\ni. By the\nprevious proof for a single disc, we can choose a small ε>0 so that for all iexistb∗\nisuch that\nλn(Ωi,B)≤(1+2η)2πn\n|Ωi|,forB∈[B(i)\nn,B(i)\nn+1], n>b∗\ni,\nwhere\nB(i)\nn=2\nR2\ni(1+ε)n.\nLetη′>0 and letBη′>0 be large enough such that\n1\n1−2π(1+ε)N\n|Ω|B≤1+η′\nforB≥Bη′. Then assume B≥max{max1≤i≤NB(i)\n⌈b∗\ni⌉,Bη′}. For such B, there exists for all i\nanniwithB∈[B(i)\nni,B(i)\nni+1] and we have seen that for this niholds\nλni(Ωi,B)≤(1+2η)2πni\n|Ωi|.\nThe spectrum of Ω is just the union of the spectra of all Ω i, so for any iexists ann′\nisuch\nthatλn′\ni(Ω,B) =λni(Ωi,B). We now let n:= max 1≤i≤Nn′\niso thatλn(Ω,B) is the eigenvalue\nof Ω whose index corresponds to the largest of the eigenvalue sλni(Ωi,B). The index of the\nmaximizing component shall be denoted by i∗so that we have λn(Ω,B) =λni∗(Ωi∗,B). By the\nchoice ofnwe necessarily have n≥/summationtextN\ni=1ni. With this, we observe that\nλn(Ω,B) =λni∗(Ωi∗,B)≤(1+2η)2πni∗\n|Ωi∗|=ni∗\nn|Ω|\n|Ωi∗|/parenleftbigg1\n2+η/parenrightbigg4πn\n|Ω|,\nso all that remains to show is thatni∗\nn|Ω|\n|Ωi∗|is close to one. From B∈[B(i)\nni,B(i)\nni+1] follows\nBR2\ni\n2(1+ε)−1≤ni≤BR2\ni\n2(1+ε)(24)\nfor anyi. Summing these inequalities over iyields\nB/summationtextN\ni=1R2\ni\n2(1+ε)−N≤N/summationdisplay\ni=1ni≤B/summationtextN\ni=1R2\ni\n2(1+ε). (25)\nSincen≥/summationtextN\ni=1ni, the inequalities (24) and (25) imply\nni∗\nn≤BR2\ni∗\n2(1+ε)\nB/summationtextN\ni=1R2\ni\n2(1+ε)−N=|Ωi∗|\n|Ω|−2π(1+ε)N\nB\nand thus\nni∗\nn|Ω|\n|Ωi∗|≤1\n1−2π(1+ε)N\n|Ω|B≤1+η′.\nHence, we have\nλn(Ω,B)≤(1+η′)/parenleftbigg1\n2+η/parenrightbigg4πn\n|Ω|.\nBecauseη >0 andη′>0 were arbitrary, this proves the assertion if Ω is a disjoint union of\nfinitely many discs.\n194.3 Proof of Theorem 2.2 for general domains\nFor the final step, we take advantage of the domain inclusion p rinciple. This principle is well-\nknown for the non-magnetic Dirichlet Laplacian and states t hat eigenvalues increase under\ndomain inclusion. The same holds with a constant magnetic fie ld.\nLemma 4.4 (Domain inclusion principle) .LetΩ,Ω′be open domains of finite measure with\nΩ′⊂ΩandB≥0. Then\nλn(Ω′,B)≥λn(Ω,B)forn∈N.\nProof.SinceC∞\n0(Ω′)⊂C∞\n0(Ω) by trivial extension, the form closure of C∞\n0(Ω′) w.r.t. the\nquadratic form of the magnetic Laplacian can be embedded in t he form closure of C∞\n0(Ω). The\nmin-max-principle then shows the inequality between the ei genvalues.\nLet now Ω be any open set of finite measure. By Vitali’s coverin g theorem there exists a\ncountable collection of pairwise disjoint open discs {Ωi}∞\ni=1such that Ω i⊂Ω for alli∈Nand\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΩ\\∞/uniondisplay\ni=1Ωi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\nThis means in particular that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/uniondisplay\ni=1Ωi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle→ |Ω|asN→+∞.\nIn other words, for any η′>0 exists Ω′⊂Ω, Ω′being a disjoint union of finitely many discs,\nwith|Ω|\n|Ω′|≤1+η′. We have for Blarge enough that for some nholds\nλn(Ω′,B)≤(1+2η)2πn\n|Ω′|.\nBut then by domain monotonicity we arrive with such a disjoin t union of discs Ω′at\nλn(Ω,B)≤λn(Ω′,B)≤(1+2η)2πn\n|Ω′|≤(1+2η)|Ω|\n|Ω′|2πn\n|Ω|= (1+η′)/parenleftbigg1\n2+η/parenrightbigg4πn\n|Ω|.\nAgain, since ηandη′>0 were arbitrary, this shows the assertion for general Ω and fi nishes the\nproof.\n4.4 Proof of Corollary 2.3\nLetη>0 be arbitrary. Choose η′>0 small enough such that 2 −η <2(1+2η′)−1. By Theorem\n2.2, there exists Bη′>0 such that for any B≥Bη′exists annwith\n2−η<2(1+2η′)−1≤4πn\n|Ω|λn(Ω,B)≤2.\nRecall that R0= 2 andLcl\n0,2= 1/4π. Thus, letting λցλn(Ω,B), we get for any B≥Bη′\nR0−η≤tr(HΩ\nB−λ)0\n−\nLcl\n0,2|Ω|λ≤R0. (26)\nThis is (12) for γ= 0.\nLet now 0<γ <1. Suppose there exists R<R γwith\n(HΩ\nB−µ)γ\n−≤RLcl\nγ,2|Ω|µ1+γ(27)\n20for allµ>0 and allB≥0. By [13, Lemma 3.31], it is known that for any µ>λ\ntr(HΩ\nB−λ)0\n−≤(µ−λ)−γtr(HΩ\nB−µ)γ\n− (28)\nand for any 0 ≤α<β\ninf\nµ>λ(µ−λ)−αµβ=C(α,β)−1λβ−α. (29)\nwhereC(α,β) :=ααβ−β(β−α)β−α. But from (28) follows under assumption of (27)\ntr(HΩ\nB−λ)0\n−≤RLcl\nγ,2|Ω|(µ−λ)−γµ1+γ\nfor anyµ>λ> 0 and hence with (29)\ntr(HΩ\nB−λ)0\n−≤RC(γ,1+γ)−1Lcl\nγ,2|Ω|λ.\nThis however yields\ntr(HΩ\nB−λ)0\n−\nLcl\n0,2|Ω|λ≤RC(γ,1+γ)−1Lcl\nγ,2\nL0,2<Rγγ−γ(1+γ)γ=R0\nwhich contradicts the lower bound of (26) for ηsmall enough. Therefore, (12) is also true for\n0<γ <1.\nForγ≥1, we have already discussed that (12) is a consequence of the asymptotics for Riesz\nmeans.\nA Appendix\nLet us define\nµ(Ω,B) := inf\nn∈Nλn(Ω,B)/parenleftbigg4πn\n|Ω|/parenrightbigg−1\n.\nBy Theorem 2.2, we have\nlim\nB→∞µ(Ω,B) =1\n2.\nWe want to supply this limit with a convergence rate in the cas e where Ω is a disc. We have\nthe following result.\nTheorem A.1. LetF: [0,∞)→[0,∞)be the strictly increasing function defined by\nF(x) =x/integraldisplayx\n0et2dt.\nand letF−1: [0,∞)→[0,∞)denote its inverse. Let\nf(b) =F−1(b1\n2)\nb1\n2\nforb>0. Furthermore, let α>0. Then the following holds:\n1. Ifα >1/√\n2, then for all Blarge enough there exists some eigenvalue λn(DR,B)≤3B\nsatisfying\nλn(DR,B)/parenleftbigg4πn\n|DR|/parenrightbigg−1\n≤1\n2+αf/parenleftbiggBR2\n2/parenrightbigg\n. (30)\n212. Ifα<1/√\n2, then for all Blarge enough any eigenvalue λn(DR,B)≤3Bsatisfies\nλn(DR,B)/parenleftbigg4πn\n|DR|/parenrightbigg−1\n≥1\n2+αf/parenleftbiggBR2\n2/parenrightbigg\n. (31)\nIn other words,\ninf\nn:λn(DR,B)≤3Bλn(DR,B)/parenleftbigg4πn\n|DR|/parenrightbigg−1\n=1\n2+1√\n2f/parenleftbiggBR2\n2/parenrightbigg\n(1+o(1)) (32)\nasB→ ∞.\nWe do not proof Theorem A.1 here as the calculations are quite lengthy. We plan on\npublishing the full proof in another paper. The key idea is to improve upon Lemma 4.3 in\nSection 4.1 by letting δandεdepend onband such that\nM(−δ(b),b,b(1 +ε(b)))<0\nstillholdsfor blargeenough. ThroughthesameargumentsaspresentedinSec tion4.1, thisyields\nan upper bound of the form (23) with ηdepending on the choice of δ(b) andε(b). Optimizing\nδ(b),ε(b) results in the rate given by the theorem.\nTheorem A.1 immediately gives the following corollary.\nCorollary A.2. Letα>1/√\n2andfbe defined as before. Then for Blarge enough\nµ(DR,B)≤1\n2+αf/parenleftbiggBR2\n2/parenrightbigg\n.\nThis corollary already follows from part 1 of Theorem A.1. Wh ile it is not ruled out that\nµ(DR,B) converges faster than what part 1 implies, part 2 asserts th at the convergence rate\ncannot be improved further by considering eigenvalues λn(DR,B)≤3B. A better convergence\nrate could only be obtained by an investigation of eigenvalu esλn(DR,B)>3B. Here, however,\nthe relation between unsorted eigenvalue branches λm,l(B) and sorted eigenvalues λn(DR,B)\nbecomes difficult to untangle. Note that in our numerical calc ulations (λn(DR,B) up ton= 500\nandB= 600, see Figure 1(b)) the minimal value for λn(DR,B)(4πn/|DR|)−1was always\nattained by an eigenvalue λn(DR,B)≤3BonceB >B crit. We hence conjecture that\nµ(DR,B) = inf\nn:λn(DR,B)≤3Bλn(DR,B)/parenleftbigg4πn\n|DR|/parenrightbigg−1\n.\nfor anyBlarge enough and (32) gives the asymptotic for µ(DR,B) as well.\nRemark. Regarding general domains Ω, we believe that is possible to m odify the proofs in\nSections 4.2 and 4.3 to show upper bounds of the form\nµ(Ω,B)≤1\n2+CΩB−ρΩ\nwithCΩ>0 andρΩ>0. However, any bound derived this way will depend on a partic ular\ncircle packing of Ω and most likely result in a non-optimal ex ponentρΩ<1/2.\nAcknowledgements\nThe first author is grateful for the support by the Institut Mi ttag-Leffler and COST (Action\nCA18232). He also wants to express his gratitude to the organ izers of the International con-\nference on spectral theory and approximation 2023 in Lund an d the granted financial support.\nFinally, he wants to thank Søren Fournais, Mikael Persson Su ndqvist, Dirk Hundertmark and\nSemjon Vugalter for valuable discussions.\n22References\n[1] Milton Abramowitz and Irene A. 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In: European Journal of Physics 40 045101 (June 2019). doi:\n10.1088/1361-6404/ab0e5f .\n[26] Hermann Weyl. “Das asymptotische Verteilungsgesetz der Eige nwerte linearer par-\ntieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraum-\nstrahlung)”. In: Mathematische Annalen 71.4 (Dec. 1912), pp. 441–479. issn: 1432-\n1807.doi:10.1007/bf01456804 .\n24"    },    {        "title": "2402.01522v1.Double_Red_Giant_Branch_and_Red_Clump_features_of_Galactic_disc_stellar_populations_with_Gaia_GSPspec.pdf",        "content": "Astronomy & Astrophysics manuscript no. HighPrecisionParameters ©ESO 2024\nFebruary 5, 2024\nLetter to the Editor\nDouble Red Giant Branch and Red Clump features of Galactic\ndisc stellar populations with Gaia GSP-Spec\nAlejandra Recio-Blanco1, P. de Laverny1, P. A. Palicio1, S. Cassisi2,3, A. Pietrinferni2, and N. Lagarde4\n1Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229,\n06304 Nice cedex 4, France\ne-mail: alejandra.recio-blanco@oca.eu\n2INAF – Osservatorio Astronomico di Abruzzo, Via M. Maggini s/n, I-64100 Teramo, Italy\n3INFN, Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy\n4Laboratoire d’Astrophysique de Bordeaux, Université Bordeaux, CNRS, B18N, Allée Geoffroy Saint-Hilaire, 33615\nPessac, France\nReceived ; accepted\nABSTRACT\nContext. The bimodality of the Milky Way disc, in the form of a thick short disc and a thinner more radially extended\none, encrypts the complex internal evolution of our Galaxy and its interaction with the environment.\nAims.TodisentanglethedifferentcompetingphysicalprocessesatplayinGalacticevolution,adetailedchrono-chemical-\nkinematical, and dynamical characterisation of the disc bimodality is necessary, including high number statistics.\nMethods. Here we make use of an extremely precise subsample of the Gaia DR3 GSP-Spec catalogue of stellar chemo-\nphysical parameters. The selected database is composed of 408 800 stars with a median uncertainty of 10 K, 0.03 and\n0.01 dex in Teff, log( g) and [M/H], respectively.\nResults.The stellar parameter precision allows to break the age-metallicity degeneracy of disc stars. For the first time,\nthe disc bimodality in the Kiel diagramme of giant stars is observed, getting rid of interstellar absortion issues. This\nbimodality produces double Red Giant Branch sequences and Red Clump features for mono-metallicity populations.\nA comparison with BaSTI isochrones allows to demonstrate that an age gap is needed to explain the evolutionary\nsequences separation, in agreement with previous age-metallicity relations obtained using Main-Sequence Turn Off\nstars. A bimodal distribution in the stellar mass-[ α/Fe] plane is observed at constant metallicity. Finally, a selection\nof stars with [M/H]=0.45 ±0.03 dex shows that the most metal-rich population in the Milky Way disc presents an\nimportant proportion of stars with ages in the range 5-13 Gyr. This old, extremely metal-rich population is possibly a\nmix of migrated stars from the internal Galactic regions, and old disc stars formed before the last major merger of the\nMilky Way.\nConclusions. The Gaia GSP-Spec Kiel diagrammes of disc mono-abundance stellar populations reveal a complex, non\nlinear age-metallicity relation crafted by internal and external processes of Galactic evolution. Their detailed analysis\nopens new opportunities to reconstruct the puzzle of the Milky Way disc bimodality.\nKey words. tbd\n1. Introduction\nThestructureoftheMilkyWaydiscisknowntobebimodal\nsince the seminal works of Yoshii (1982) and Gilmore &\nReid (1983). It is composed of a thick, more primitive stel-\nlarpopulation,andofathinmoreradiallyextendedcompo-\nnent, containing stars, gas and dust. The disc bimodality\nencrypts crucial information about the physical processes\nof our Galaxy formation, as it is the result of more than\n10 Gyr of evolution. It emerges from the complex inter-\nplay of internal evolutionary processes and the interaction\nof our Milky Way with its ecosystem. As a consequence, a\nvariety of physical mechanisms are invoked to explain the\nspatial, kinematical, dynamical and chemical properties of\nthe thick and the thin discs, and their evolution with time.\nEarly turbulence (Bournaud et al. 2009), radial migration\n(e.g Schönrich & Binney 2009b; Minchev & Famaey 2010;\nTrick et al. 2019; Prantzos et al. 2023), gas infall (eg Spi-\ntoni et al. 2023a), galaxy mergers (eg. Abadi et al. 2003;Renaud et al. 2021) and disc marginal stabilty (Park et al.\n2021), can be cited among the numerous studied processes\nand their related literature.\nFrom the observational point of view, characterizing the\ndiscbimodalityimpliesprecisedataandhighnumberstatis-\ntics. For this reason, the thick/thin disc transition has been\nat the core of the observational strategy of all recent spec-\ntroscopic surveys of the Milky Way. In particular, the ex-\nistent correlations between chemical abundances and kine-\nmaticaland/ordynamicalpropertieshavebeenexploredin-\nside the Solar neighbourhood, using HARPS and GALAH\ndata, among others (e.g. Adibekyan et al. 2013; Santos-\nPeral et al. 2021; Hayden et al. 2020) and outside the So-\nlar neighbourhood, using for instance Gaia-ESO Survey,\nAPOGEE or Gaia GSP-Spec data (e.g. Recio-Blanco et al.\n2014; Hayden et al. 2015; Gaia Collaboration et al. 2023a).\nThick disc stellar populations are kinematically hotter than\nthindiscones.Additionalevidenceofdiscbimodalitycomes\nArticle number, page 1 of 9arXiv:2402.01522v1  [astro-ph.GA]  2 Feb 2024A&A proofs: manuscript no. HighPrecisionParameters\nfromchemicalabundanceratios.Inparticular,thethickand\nthin disc populations define two parallel sequences in the\n[α/Fe]abundancevs.metallicityplane,withthickdiscstars\npresenting higher [ α/Fe] ratios than thin disc stars of sim-\nilar [M/H], with a gap in [ α/Fe] whose detailed properties\nindeed depend on the specific α-element taken into con-\nsideration(Prantzos et al. 2023). The two sequences seem\nto merge in the super-Solar metallicity regime, although\nthis fact is observationally challenging to confirm due to\nthe complex analysis of metal-rich stellar spectra (Santos-\nPeral et al. 2020). Moreover, the temporal evolution of the\ndisc chemical properties suggests the existence of two dit-\ninct phases in the age-metallicity relation (e.g. Haywood\net al. 2013; Hayden et al. 2017; Santos-Peral et al. 2021;\nLagarde et al. 2021; Miglio et al. 2021; Xiang & Rix 2022).\nIn the metal-intermediate regime, a gap of about 2-5 Gyr\nbetween mono-metallicity populations of the thick and the\nthin disc seems to be present. Conversely, a gap in metallic-\nity of about 0.5 dex at the start of the thin disc formation\n7-9Gyragoisdetected.Thismetallicitygapisprogressively\nreduced, and seems to disappear 4-5 Gyr ago. The relative\nproportion of thin to thick disc stars is linked to the dif-\nferent structural properties of the two discs, with the thick\ndisc having a shorter scale-length and a larger scale-height\nthan the thin one.\nTo disentangle the Galactic disc dichotomy, and in par-\nticular, its temporal evolution, the study of large samples\nof stars in a large spatial volume is mandatory. To this pur-\npose, giant stars are ideal tracers of disc populations up to\nvery large distances. However, their use is hampered by a\nsevere age-metallicity degeneracy of their physical proper-\nties (in particular colour and therefore effective tempera-\nture), and non negligible uncertainties on interstellar ab-\nsorption in the Galactic plane. In this letter, we use an ex-\ntremelyprecisesampleofstellarchemo-physicalparameters\n(cf. Sect. 2) from the Gaia DR3 (Gaia Collaboration et al.\n2023b) GSP-Spec catalogue (Recio-Blanco et al. 2023), to\nbreak the age-metallicity degeneracy of disc stellar popu-\nlations. New constraints on disc bimodality using the Teff\nvs. log( g) diagramme are presented in Sect. 3. Finally, the\nage distribution of extremely metal-rich stars in the disc\nis explored in Sect. 4. The conclusions of this Letter are\nsummarized in Sect. 5.\n2. A high precision DR3 GSP-Spec data set\nThis work makes use of the stellar physical parame-\nters and chemical abundances derived from the Gaia\nRVS spectra by the MatisseGauguin analysis procedure\n(Recio-Blanco et al. 2006, 2016; Bijaoui et al. 2012) of\ntheGSP-Spec module (Recio-Blanco et al. 2023). The\ndata are available through the astrophysical_parameters\ntable of the GaiaDR3 archive. In particular, we em-\nploy the stellar effective temperature Teff(teff_gspspec ),\nthe surface gravity log( g) (logg_gspspec ), the metallic-\nity [M/H] ( mh_gspspec ) and the abundance of α-elements\n[α/Fe] and magnesium [Mg/Fe] with respect to iron\n(alphafe_gspspec and mgfe_gspspec , respectively). The\nprescriptions recommended by Recio-Blanco et al. (2023)\nin Tables 3 and 4 have been used to calibrate log( g),\n[M/H] [ α/Fe] and [Mg/Fe] as a function of Teff(see Ap-\npendix A). It is important to note that the GSP-Spec DR3\n[M/H]abundancesareindeedindicatorsofthe[Fe/H]abun-\ndance, as iron lines dominate the non- αelement features\n−0.75−0.50−0.25 0.00 0.25 0.50\n[M/H] (dex)−0.4−0.20.00.20.4[Mg/Fe] (dex)\n180200220240260\nVφ(km/s)Fig. 1. GaiaGSP-Spec [Mg/Fe] abundances as a function of\nmetallicity for a subsample of 3,209 stars with high SNR data,\ngood quality flags and uncertainties in [Mg/Fe] lower than\n0.05 dex. The colour code shows the Galactic azimuthal velocity.\nused by GSP-Spec for the mh_gspspec estimations. As a\nconsequence, [M/H] abundances can be directly compared\nto [Fe/H] values of chemical evolution models or stellar\nisochrones.\nTo select high precision parameters, our work-\ning sample contains only stars with a metallic-\nity uncertainty defined as ( mh_gspspec_upper -\nmh_gspspec_lower )/2, lower than 0.025 dex, and a\nsignal-to-noise ( rv_expected_sig_to_noise ) higher than\n100.Moreover,toensurethequalityoftheparametrization,\nweimposedthefollowingfiltersinthe flags_gspspec chain\n(cf. Recio-Blanco et al. 2023, Table 2): the vbroadT,G,M ,\nvradT,G,M and the KMgiantPar flags have been set to\n0, and the extrapol flag to 0 or 1. Finally, to clean\nfor the remaining less reliable solutions, we selected\nstars not cooler than 3750 K and with a goodness-of-fit\nparameter logchisq_gspspec <-3.7. The final sample\ncontains 408,800 stars with a median uncertainty of 10 K,\n0.03 and 0.01 dex in Teff, log( g) and [M/H] respectively.\nThe signal-to-noise of the corresponding RVS spectra\nis distributed between 100 and 2823 with a median at\n144. The Gmagnitude is distributed between 3.47 and\n12.44 mag with a median at 8.8 mag. The stars are located\nbetween 6.7 kpc to 10 kpc from the Galactic centre and\nwithin 1 kpc from the Galactic plane. Finally, we have\nused the kinematical and orbital parameters of the selected\nstars computed by Palicio et al. (2023) to complement the\nspectroscopic GSP-Spec chemo-physical ones.\n3. Bimodal Giant Branches and Red Clumps\nA common way of separating the stellar populations in the\nthin and the thick discs is to use the distribution of individ-\nualabundancesinthe[ α/Fe]vs.[Fe/H]plane.Nevertheless,\nthe separation power of this chemical diagnostic depends\non the considered α-element1(eg., Mikolaitis et al. 2014;\n1Those elements whose production is dominated by Type II\nsupernovae, as Mg, show a higher difference in their abundance\nwithrespecttoironbetweenthinandthickdiscstarsatconstant\nmetallicity. The Gaia GSP-Spec [α/Fe] estimation is dominated\nby the Ca triplet (Recio-Blanco et al. 2023) and it is therefore a\ntracer of [Ca/Fe], which has a lower separating power between\nthe thin and thick disc than [Mg/Fe]\nArticle number, page 2 of 9Recio-Blanco et al.: Double RGB and RC sequences of disc stars with Gaia GSP-Spec\nFig. 2.Giant branch and red clump locus of the Teffvs. log( g)\ndiagramme in six different metallicity bins. The colour code il-\nlustrates the azimuthal Galactic velocity. Two RGBs and two\nred clumps are distinguishable in all the panels, with the excep-\ntion of the more metal-poor one at [M/H] =-0.85 dex, in which\nonly lower Galactic rotational velocity stars are present.\nPrantzos et al. 2023). Figure 1 presents the [Mg/Fe] abun-\ndance distribution as a function of metallicity for a subsam-\nple of stars of our working data set with high-quality mag-\nnesium abundances. In particular, we have selected stars\nwhose uncertainty in [Mg/Fe] is lower than 0.05 dex and\nthe two corresponding Mg GSP-Spec quality flags are equal\nto zero. In addition, to avoid potential problems due to line\nblending, a Teffthreshold ranging between 4500 K in the\nmetal-poor regime, and 5200 K in the metal-rich one, has\nbeen imposed. As expected, two sequences are visible in\nthe resulting abundance distribution presented in Fig. 1.\nTo help the identification of thin and thick disc popula-\ntions, corresponding to the lower and the upper sequences\nrespectively, the colour-code shows the Galactic azimuthal\nvelocity of the stars.\nMoreover, thanks to the high number statistics of the\nGaia GSP-Spec catalogue, our high-precision working sam-\nple can be used to confidently select stars in narrow metal-\nlicity bins. For each bin, the upper and lower confidence\nvalues of the GSP-Spec metallicity indeed allow to build\nstellar subsamples whose [M/H] is within that bin with a\nprobabilityof70%.Thismetallicityselectionispresentedin\nFig. 2, using again a colour code on the Galactic azimuthal\nvelocity, V ϕ. The surface gravity versus temperature dis-\ntribution of giant stars is shown, considering six separate\n[M/H] bins. Two different stellar evolutionary paths, high-\nlighted by a red and a blue arrow in the upper left panel,are observed. They are separated by a gap in the Teffvs.\nlog(g) plane, and they are visible from the red giant branch\n(RGB) and the Red Clump up to the Asymptotic Giant\nBranch.\nIt can be seen in this Fig. 2 that a first evolutionary\npath is defined by stars with high Galactic rotational ve-\nlocities, appearing in reddish colours. This population pro-\ngressively fades from [M/H] =-0.15 dex (upper left panel)\nto [M/H] =-0.60 dex (bottom left panel), being absent in\nthe more metal-poor bin at [M/H] =-0.85 dex (bottom right\npanel).\nThesecondevolutionarypathisvisibleinthecoolerside\nofthefirstoneanditispopulatedbystarswithlowerGalac-\ntic rotational velocities (bluish colours in the Figure). It\nprogressively detaches from the high V ϕevolutionary path\nas metallicity diminishes, showing a separated RGB in the\nmetal-intermediate regime. This low-V ϕpopulation is in-\ncreasingly more visible as metallicity decreases, being the\nonly one present in the most metal-poor bin.\nThe gap in Teffand/or log( g) between the two evolu-\ntionary paths of mono-metallicity populations, suggests dif-\nferent age distributions, with the hotter population being\nyounger than the other one. This fact, combined with the\nkinematical characteristics (high Galactic rotation for the\nhotter population and mild Galactic rotation for the cooler\none) advocates for their identification as the evolutionary\nloci of thin and thick disc stars, respectively. This is also\nsupported by the dynamical characteristics of the two pop-\nulations, as shown in Fig.A.2 and A.3. The cooler, high-V ϕ\npopulation is observed to have orbits with higher pericen-\ntres (R peri) and lower maximun distances from the Galac-\ntic plane (Z max), than the hotter, lower V ϕpopulation at a\ngiven metallicity. This is coherent with the expected orbital\ncharacteristics of the thin disc, namely a longer scale-length\nand a shorter scale-height with respect to the thick disc.\nTo further constrain the characteristics of the observed\nthin and thick disc bimodality in the Teffvs. log( g) plane,\nFig. 3 presents a more detailed analysis of an intermediate\nmetallicity bin selecting stars with mh_gspspec_lower >-\n0.4 dex and mh_gspspec_upper <-0.3 dex. The figure is\ncomposed of nine panels zooming on different regions of\nthe Kiel diagram and kinematically decomposing the thin\nand thick disc evolutionary sequences. The upper and mid-\ndle row panels show the region around the RGB and the\nRed Clump, while the Main Sequence Turn Off region is\npresented in the bottom row panels. In addition, a kine-\nmatical decomposition is displayed, from left to right. No\nkinematical selection is made in the leftmost panels, show-\ning all the stars in this metallicity bin. On the contrary, the\nmiddle and right column panels present the stars with thin\ndisc or thick disc kinematics, respectively. The kinematical\nselection is based on the departures from the circular ve-\nlocity value at the corresponding Galactic position of the\nstar. In particular, thin disc stars in the middle panels are\nrequired to have a V ϕhigher than 30 km/s with respect\nto their circular velocity V circ. The thick disc population\nin the right panels is selected to have a V ϕlower than -60\nkm/s with respect to V circ(see for instance Lagarde et al.\n2021). These criteria avoid as much as possible the over-\nlapping regime in V ϕbetween the thin and the thick disc.\nThis allows to cleanly separate the evolutionary sequences\nin the Kiel diagram using an independent parameter. In\nthe different panels of Fig. 3, two RGBs and two Turn Offs\nArticle number, page 3 of 9A&A proofs: manuscript no. HighPrecisionParameters\nFig. 3.Disc stellar population with [M/H] =-0.35±0.05 dex. The panels show a kinematical decomposition of the RGB and Red\nClump (first and second row) and the Turn Off (bottom row) features . Left column panels show all the selected stars and present\nthe double evolutionary sequences both for giants and dwarfs. The central and right column panels show the thin and thick disc\npopulations, respectively, separated thanks to their Vϕ. BaSTI isochrones with different α-enhancements are fitted to the data .\n4200 4400 4600 4800 5000\nTeff (K)1.4\n1.6\n1.8\n2.0\n2.2\n2.4\n2.6\n2.8log gRCThinRCThickE-AGB Thin\nRGB Thin\nRGB Thick\nBump Thin\nBump Thick[M/H]= -0.375±0.025 dex\n0.00.10.20.30.40.5\n[Mg/Fe] (dex)\nFig. 4. Teffvs. log( g) diagram for stars with [M/H] =-\n0.375±0.025 dex, using the [Mg/Fe] abundance as colour code.\nThe main stellar evolutionary features of thin and thick disc\npopulations can be identified.\nseparated by a small gap can be identified. In addition, two\nred clumps shifted in both Teffand log( g) are observed.\n−0.1 0.0 0.1 0.2\n[α/Fe] (dex)0.51.01.52.02.5Mass (M Sun)[M/H]= -0.375±0.025 dex\n150175200225250275\nVφ(km/s)Fig. 5. Stellar mass as a function of [ α/Fe] for stars with\n[M/H]=-0.375±0.025 dex, using the Galactic azimuthal veloc-\nity as colour code. Thick disc stars, with lower Vϕvelocities,\nlower masses and higher [ α/Fe], define a Mass-[ α/Fe] sequence\nseparated from thin disc stars.\nContrary to the colour-magnitude diagramme, the spec-\ntroscopic Teffvs. log( g) plane has the advantage of being\nreddeningfree,whichisparticularlyimportantfordiscstars\nArticle number, page 4 of 9Recio-Blanco et al.: Double RGB and RC sequences of disc stars with Gaia GSP-Spec\noutside the Solar Neighbourhood. As a consequence, and\nthanks to the GSP-Spec high precision in the stellar param-\neters of the selected stars, the age distribution of the ob-\nservedtwopopulationscanbeconfidentlyretrievedthrough\nisochronefitting.Forthispurpose,weadopttheBaSTI-IAC\nisochrone library for both solar-scaled (Hidalgo et al. 2018)\nandα-enhanced(Pietrinfernietal.2021)heavyelementdis-\ntribution2and [Fe/H] =-0.35 dex (the central value of the\nanalysed metallicity bin). On one hand, as shown in the\ncentral panel of Fig. 3, the thin disc RGB and Red Clump\nat this metallicity can be fitted with isochrones in the range\n1.5 to 6 Gyr. This age distribution is compatible with the\nisochrone fit of the Turn Off populations, presented in the\nmiddle bottom panel. On the other hand, the thick disc\nRGB and Red Clump follow the loci of a 10 Gyr isochrone,\nas presented in the middle right panel. This value is again\ncompatible with the fit of the Turn Off population in the\nbottom right panel. The thick disc evolutionary sequences\nseem to present a lower age dispersion than the thin disc\nones, as expected from the tighter age-metallicity correla-\ntion of the thick disc (e.g. Haywood et al. 2013). Moreover,\nthe fitted thick disc isochrones illustrate the effect of the\n[α/Fe] enrichment for the same metallicity and age. Al-\nthough a shift in Teffand log( g) is visible and should be\nconsidered for careful age determinations on a star by star\nbasis, it is insufficient to explain the gap between the thin\nandthickdiscevolutionarysequencesatthismetallicity.An\nage gap of about 4 Gyr is required to interpret the observed\ndichotomy.\nTo complete this study, we focus on an even narrower\nmetallicity bin composed of 6,601 stars with [M/H] =-\n0.375±0.025 dex. Figure 4 presents the Teffvs. log( g) di-\nagramme centred on the giant stars locus and using the\n[Mg/Fe] abundance as colour-code. The bimodality of disc\npopulations is present in all the different stellar evolution-\narysequencesalongthisdiagrammeAsexpected(cf.Fig.1),\nthe thin and the thick disc evolutionary sequences corre-\nspond to two different regimes of [Mg/Fe] abundance, with\nthick disc stars presenting higher values. Finally, we com-\nputedthemassesofthestarsintheabovementionedmetal-\nlicity bin ([M/H] =-0.375±0.025 dex) using their GSP-Spec\nparameters and E(BP-RP) colours (see. Appendix C). Fig-\nure 5 presents the derived stellar masses as a function of the\n[α/Fe] abundance. The selected stars have an uncertainty\nlower than 0.05 M ⊙in mass and 0.01 dex in [ α/Fe]. The\nGalactic azimuthal velocity is used as a colour code. The\nthin and thick disc stars define again two different regimes\nin this plot. In particular, the population of stars typical\nof thick disc stars ( with V ϕvalues lower than 200 km/s)\nhas a median mass 0.1 M ⊙lower than stars with higher\nVϕvalues in the thin disc. This is consistent with the age\ndifference between the two populations.\n4. Super-Solar metallicity populations in the disc\nThestudyofdiscstarspresentingthehighestlevelsofmetal\nenrichment is key to understand the mechanisms of radial\n2Since the BaSTI models are available for [ α/Fe]=0.00 and\n+0.40, at any given [Fe/H] and age a linear interpolation in\n[α/Fe] has been adopted in order to derive isochrones for any\nrequested α-element enhancement. Only a 45 K offset seems\nto be needed to match the BaSTI isocrones and the calibrated\nGSP-Spec data, confirming the coherence of both Teffscales.migration (Sellwood & Binney 2002; Schönrich & Binney\n2009a, e.g.). Figure 6 shows the Kiel diagramme for the\nmetallicity bin [M/H]=+0.45 ±0.03 dex (a factor ∼3 larger\nthan the Solar metal enrichment), and the corresponding\nfit of three BaSTI isochrones with 5, 8 and 13 Gyr. Rather\nunexpectedly, an important fraction of the stars in this ex-\ntremely metal-rich regime are old. The different evolution-\nary features are globally well fitted by a 8 Gyr isochrone,\nalthough younger and older isochrones, in the interval of\nabout 5 to 13 Gyr are also compatible with about 85%of\nthe selected stars. Interestingly enough, this age interval is\nsimilar to the one observed for bulge populations by Joyce\net al. (2023). Additionaly, about 15 %of the stars are in the\ntemperatureregimeofyoungerpopulations,with Teffvalues\nbetween 6000 and 8000 K. Although the relative proportion\nof young to old stars is difficult to evaluate (among other\nthings, due to a lack of completeness), the data highlight\nthe existence of an old extremely metal-rich population. It\nis interesting to note that the radial distribution of stars in\nthis metallicty bin spans from 7.5 to 9 kpc from the Galac-\ntic centre (with a median at 8.24 kpc). The distribution of\nmaximum orbital distances from the Galactic plane, Z max,\nranges from 0 to 1 kpc (with a median at 0.26 kpc). Finally,\nthe median Galactic azimuthal velocity is 235 km/s, with\na dispersion of 25 km/s.\n4000 4500 5000 5500 6000\nTeff (K)2.0\n2.5\n3.0\n3.5\n4.0\n4.5log g[M/H]= 0.45±0.03 dex\n588 13Gyr\nFig. 6. Kiel diagramme of the disc extremely metal-rich\npopulation with [M/H] =+0.45±0.03 dex. Solar-scaled BaSTI\nisochrones of 5, 8 and 13 Gyr (pink, orange and green, respec-\ntively) are overlaid to the data.\n5. Conclusions\nThe Gaia DR3 GSP-Spec catalogue allows to study the disc\nstellar populations in a high precision chemo-physical pa-\nrameters regime, with unprecedentedly high number statis-\ntics. This opens new horizons to characterise in detail the\nthin and thick discs evolution outside the Solar neighbour-\nhood,gettingridofinterstellarabsorptionissuesandbreak-\ning the age-metallicity degeneracy. In this work, we present\nfor the first time the consequence of Galactic disc bimodal-\nity in the physical properties of its giant stars population,\nArticle number, page 5 of 9A&A proofs: manuscript no. HighPrecisionParameters\nthat is, double RGB sequences and Red Clump features for\nmono-metallicity populations. An age gap is needed to ex-\nplain the evolutionary sequences separation, in agreement\nwith previous age-metallicity relations using Turn Off stars\n(e.g. Xiang & Rix 2022) and supporting the modeling of the\ndisc chemical evolution by distinct infall episodes (e.g. Spi-\ntoni et al. 2023b). The bimodal evolutionary sequences are\ncharacterizedinkinematics,dynamics,[Mg/Fe]abundances\nand stellar masses consistently derived from Gaia data. Ad-\nditionally,aselectionofextremelymetal-richdiscstarswith\n[M/H]=0.45 ±0.03 dex contains a considerable proportion\nof very old stars (ages 5-13 Gyr) reaching distances of up\nto 9 kpc from the Galactic centre, and maximum orbital\ndistances from the Galactic plane of up to 1 kpc. This old\npopulation could be composed of thin disc stars having mi-\ngrated from the internal regions of the Galaxy, and thick\ndisc stars formed before the last major merger of the Milky\nWay, namely that of the Gaia-Enceladus satellite about\n9 yrs ago (Helmi et al. 2018; Belokurov et al. 2018; Gallart\net al. 2019). The detailed analysis of precise Gaia GSP-\nSpecKiel diagrammes of mono-abundance stellar popula-\ntions opens new pathways to disentangle the complex puz-\nzle of Galactic disc bimodality, at the core of Milky Way’s\nevolutionary processes.\nAcknowledgements. 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I., et al. 2021, MNRAS, 503, 5846\nSantos-Peral, P., Recio-Blanco, A., de Laverny, P., Fernández-Alvar,\nE., & Ordenovic, C. 2020, A&A, 639, A140\nSantos-Peral, P., Recio-Blanco, A., Kordopatis, G., Fernández-Alvar,\nE., & de Laverny, P. 2021, A&A, 653, A85\nSchönrich, R. & Binney, J. 2009a, MNRAS, 396, 203\nSchönrich, R. & Binney, J. 2009b, MNRAS, 399, 1145\nSellwood, J. A. & Binney, J. J. 2002, MNRAS, 336, 785\nSpitoni, E., Recio-Blanco, A., de Laverny, P., et al. 2023a, A&A, 670,\nA109\nSpitoni, E., Recio-Blanco, A., de Laverny, P., et al. 2023b, A&A, 670,\nA109\nTrick, W. H., Coronado, J., & Rix, H.-W. 2019, MNRAS, 484, 3291\nXiang, M. & Rix, H.-W. 2022, Nature, 603, 599\nYoshii, Y. 1982, PASJ, 34, 365\nArticle number, page 6 of 9Recio-Blanco et al.: Double RGB and RC sequences of disc stars with Gaia GSP-Spec\nTable A.1. Polynomial coefficients for the calibration of the\nGSP-Spec MatisseGauguin gravities, metallicities and [Mg/Fe]\nabundances as a function of t=Teff/5750\nParameter p0 p1 p2 p3\nlog(g) -0.2320 0.8291 0.7441 -1.1677\n[M/H] 1.2739 -1.1383 -1.4776 1.3398\n[Mg/Fe] -15.8586 48.1211 -48.5437 16.1383\nAppendix A: Calibration of surface gravity,\nmetallicity and magnesium abundance.\nThe raw GSP-Spec values for surface gravity, metallic-\nity and magnesium abundance with respect to iron have\nbeencalibratedfollowingtheproceduresdescribedinRecio-\nBlanco et al. (2023). However, as the temperature range\nspanned by these data is large, it has been found to be\nmore adapted to calibrate as a function of Teffinstead of\nasafunctionoflog( g).Tothispurpose,andfollowingRecio-\nBlanco et al. (2023), polynomial corrections in the form:\nParam calibrated =Param +3X\ni=0pi·ti(A.1)\nhave been applied, where Paramis either log( g), [M/H]\nor [Mg/Fe]. tis the relative temperature with respect to the\nSun, Teff/5750, and piare the corresponding polynomial\ncoefficients from Table A.1.\nAppendix B: Density diagrammes and dynamical\ndependencies\nThe thin/thick disc bimodality in the Kiel diagram illus-\ntrated for six different mono-abundance populations in Sec-\ntion 3 is also distinguishable in the stellar distribution in\nthis plane (see Fig A.1). Thanks to the precision of the\nGSP-Spec chemo-physical parameters, the double RGB se-\nquences are visible without the use of any kinematical\nor dynamical information, particularly in the intermediate\nmetallicity domain.\nIn addition, Fig. A.2 and Fig. A.3 allow to observe the\ntrends of the two RGB sequences with the orbital pericen-\ntre, R peri, and the maximum distance from the Galactic\nplane, Z max, respectively. The typical thin and thick disc\nranges in R periand Z maxcan be distinguished. The hot-\nter sequence presents larger orbital pericentres and lower\nZmaxvalues, typical of thin disc stars. On the contrary,\nthe cooler sequence corresponds to a population with peri-\ncentres closer to the Galactic centre and reaching larger\ndistances from the plane, characterising thick disc stars.\nAppendix C: Estimation of stellar Masses from\nGaia GSP-Spec parameters, astrometry and\nphotometry\nStellar masses were derived for 6,601 stars with [M/H] =-\n0.375±0.025 dex using their GSP-Spec spectroscopic atmo-\nspheric parameters and Gaia photometry and astrometry\n(see de Laverny 2024, in preparation, for a detailed de-\nscription).Firstofall,theE(Bp-Rp)extinctionisestimated\ncomparing the observed Gaia (Bp-Rp) colour to the theo-\nretical one, derived from Casagrande et al. (2021) and usingtheGSP-Spec parameters. We then computed the stellar\nluminosity, adopting the geometric distances from Bailer-\nJones et al. (2021). The stellar mass is finally derived from\nthis luminosity, using the corresponding effective temper-\nature and surface gravity. The associated errors were esti-\nmated by propagating the uncertainties on the parameters,\nphotometry and distances, thanks to Monte-Carlo realisa-\ntions.\nArticle number, page 7 of 9A&A proofs: manuscript no. HighPrecisionParameters\nFig. A.1. Density plot corresponding to Fig. 2 stellar populations.\nFig. A.2. Same as Fig. 2, but colour-coded with the star orbital pericentre (R peri).\nArticle number, page 8 of 9Recio-Blanco et al.: Double RGB and RC sequences of disc stars with Gaia GSP-Spec\nFig. A.3. Same as Fig. 2, but colour-coded with star maximum orbital distance to the Galactic plane (Z max), as a colour code.\nArticle number, page 9 of 9"    },    {        "title": "2402.01554v1.Diastolic_and_isoperimetric_inequalities_on_surfaces.pdf",        "content": "DIASTOLIC AND ISOPERIMETRIC INEQUALITIES ON\nSURFACES\nFLORENT BALACHEFF AND ST ´EPHANE SABOURAU\nAbstract. We prove an universal inequality between the diastole, de-\nfined using a minimax process on the one-cycle space, and the area of\nclosed Riemannian surfaces. Roughly speaking, we show that any closed\nRiemannian surface can be swept out by a family of multi-loops whose\nlengths are bounded in terms of the area of the surface. This diastolic\ninequality, which relies on an upper bound on Cheeger’s constant, yields\nan effective process to find short closed geodesics on the two-sphere, for\ninstance. We deduce that every Riemannian surface can be decomposed\ninto two domains with the same area such that the length of their bound-\nary is bounded from above in terms of the area of the surface. We also\ncompare various Riemannian invariants on the two-sphere to underline\nthe special role played by the diastole.\nR´esum´e.Nous d´ emontrons une in´ egalit´ e universelle entre la diastole,\nd´ efinie par un proc´ ed´ e de minimax sur l’espace des 1-cycles, et l’aire\nd’une surface riemannienne ferm´ ee. De mani` ere informelle, nous prou-\nvons que toute surface riemannienne ferm´ ee peut ˆ etre balay´ ee par une\nfamille de multi-lacets dont les longueurs sont contrˆ ol´ ees par l’aire de la\nsurface. Cette in´ egalit´ e diastolique, qui repose sur une majoration de\nla constante de Cheeger, fournit en particulier un proc´ ed´ e effectif pour\ntrouver de courtes g´ eod´ esiques ferm´ ees sur une 2-sph` ere. Nous d´ eduisons\nque toute surface riemannienne peut ˆ etre d´ ecompos´ ee en deux domaines\nde mˆ eme aire dont la longueur du bord commun est major´ ee ` a l’aide\nde l’aire de la surface. Nous comparons ´ egalement divers invariants rie-\nmanniens sur la 2-sph` ere afin de souligner le rˆ ole sp´ ecial jou´ e par la\ndiastole.\n1.Introduction\nThe topology of a manifold Mand the topology of its loop space Λ Mare\nclosely related, through their homotopy groups, for instance. The critical\npoints of the length or energy functionals on the loop space of a Riemannian\nmanifold can be studied using this connection. These critical points have a\nspecial geometric meaning since they agree with the closed geodesics on the\nmanifold.\nFrom the isomorphism between π1(ΛM,Λ0M) and π2(M) for the two-\nsphere (here, Λ0Mdenotes the space of constant curves), G. D. Birkhoff\nproved the existence of a nontrivial closed geodesic on every Riemannian\n2000 Mathematics Subject Classification. Primary 53C23; Secondary 53C20, 58E10.\nKey words and phrases. Cheeger constant, closed geodesics, curvature-free inequalities,\ndiastole, isoperimetric inequalities, one-cycles.\nThe first author was supported by the Swiss National Science Foundation (grant 20-\n118014/1) during the redaction of this article. The second author has been partially\nsupported by the Swiss National Science Foundation.\n1arXiv:2402.01554v1  [math.DG]  2 Feb 20242 F. BALACHEFF AND S. SABOURAU\ntwo-sphere using a minimax argument on the loop space (this method was\nextended in higher dimension by A. Fet and L. Lyusternik, cf.[Kl78]).\nRelationships on the lengths of these closed geodesics have then been\ninvestigated. For instance, C. Croke [Cr88] showed that every Riemannian\ntwo-sphere Mhas a nontrivial closed geodesic of length\nscg(M)≤31p\narea( M). (1.1)\nThe notation ‘scg’ stands for the (length of a) shortest closed geodesic. The\ninequality (1.1) has been improved in [NR02], [Sa04] and [Ro05].\nThe closed geodesic on the two-sphere obtained in C. Croke’s theorem\ndoes not always arise from a minimax argument on the loop space, even\nthough such an argument is used in the proof. Indeed, the closed geodesics\nobtained from a minimax argument on the loop space have a positive index\nwhen they are nondegenerate, which is the case for generic (bumpy) metrics.\nNow, consider two-spheres with constant area and three spikes arbitrarily\nlong ( cf.[Sa04, Remark 4.10] for further detail). On these spheres, the\nclosed geodesics satisfying the inequality (1.1) have a null index while the\nclosed geodesics with positive index are as long as the spikes. Therefore,\na minimax argument on the loop space does not provide an effective way\nto bound the area of a Riemannian two-sphere from below. More precisely,\ngiven a Riemannian two-sphere M, we define the diastole, dias Λ(M), over\nthe loop space Λ Mas\ndias Λ(M) := inf\n(γt)sup\n0≤t≤1length( γt)\nwhere ( γt) runs over the families of loops inducing a generator of π1(ΛM,Λ0M)≃\nπ2(M). Then,\nthe ratiodias Λ(M)p\narea( M)is unbounded (1.2)\non the space of Riemannian metrics on M.\nNote that a diastolic inequality between the diastole over the loop space\nand the volume of the convex hypersurfaces in the Euclidean spaces does\nhold true, cf.[Tr85], [Cr88].\nThe existence of a closed geodesic on the two-sphere can also be proved\nby using a minimax argument on a different space, namely the one-cycle\nspace Z1(M;Z). (Recall that one-cycles are unions of loops, cf.Section 2\nfor a precise definition.) This minimax argument relies on F. Almgren’s\nisomorphism [Al60] between π1(Z1(M;Z),{0}) and H2(M;Z), which holds\ntrue for every closed manifold M. Given a closed Riemannian surface M,\nwe define the diastole over the one-cycle space as\ndiasZ(M) := inf\n(zt)sup\n0≤t≤1M(zt) (1.3)\nwhere ( zt) runs over the families of one-cycles inducing a generator of\nπ1(Z1(M;k),{0}) and M(zt) represents the mass (or length) of zt. Here\nk=ZifMis orientable and k=Z/2Zotherwise. For short, we will write\ndias(M) for dias Z(M). From a result of J. Pitts [Pi74, p. 468], [Pi81, The-\norem 4.10] (see also [CC92]), this minimax principle gives rise to a union of\nclosed geodesics (counted with multiplicity) of total length dias( M). Hence,\nscg(M)≤dias(M). This principle has been used in [CC92], [NR02], [Sa04],DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 3\n[Ro05], [Ro06], [Ba08] and [Sa09] in the study of closed geodesics on Rie-\nmannian two-spheres.\nOn nonsimply connected surfaces, no minimax principle is required to\nshow the existence of a closed geodesic. In this setting, we directly define\nthe systole as the minimum of the length functional over the connected\ncomponents not containing the trivial loops of the loop space. Every closed\nRiemannian surfaces Mof genus g≥1 satisfies the following asymptotically\noptimal systolic inequality\nsys(M)≤Clog(g+ 1)√gp\narea( M) (1.4)\nwhere sys( M) is the systole of MandCis a universal constant, cf.[Gr83],\n[Ba04] and [KS05] for three different proofs.\nThe goal of this article is to establish curvature-free inequalities similar\nto (1.1) and (1.4). The use of the one-cycle space, rather than the loop\nspace, introduces some flexibility. It allows us to cut and paste closed curves,\nand to deal with both simply and nonsimply connected surfaces. We show a\ndifference of nature between the diastole over the loop space and the diastole\nover the one-cycle space on the two-sphere, and between the systole and the\ndiastole for surfaces of large genus. More precisely, we obtain the following\ndiastolic inequality.\nTheorem 1.1. There exists a positive constant C≤108such that every\nclosed Riemannian surface Mof genus g≥0satisfies\ndias(M)≤Cp\ng+ 1p\narea( M). (1.5)\nSince the minimax principle (1.3) gives rise to a union of closed geodesics\nof length dias( M), Theorem 1.1 yields a construction of short closed geodesics\non surfaces through Morse theory over the one-cycle space. In particular, it\nsheds some light on the nature of the closed geodesics whose lengths provide\na lower bound on the area of the two-spheres. Compare with (1.1) and (1.2).\nThe dependence on the genus in the inequality (1.5) is optimal, cf.Re-\nmark 7.3, and should be compared with the one in (1.4).\nA version of the diastolic inequality (1.5) holds true for compact surfaces\nwith boundary, in particular for disks, cf.Remark 2.4.\nThe inequality (1.5) is derived from a stronger estimate, where the diastole\nis replaced with the technical diastole introduced in Definition 2.2. This\nestimate also yields the following result.\nCorollary 1.2. There exists a positive constant Csuch that every closed\nRiemannian surface Mof genus g≥0decomposes into two domains with\nthe same area whose length of their common boundary γsatisfies\nlength( γ)≤Cp\ng+ 1p\narea( M). (1.6)\nThe two domains in the previous result are not necessarily connected,\neven on two-spheres. A counterexample is given by two-spheres with three\nlong fingers, whose area is equally concentrated in the tips of these fingers\n(cf.[Sa04, Remark 4.10]).4 F. BALACHEFF AND S. SABOURAU\nThe second author [Sa04] showed that the area of a bumpy Riemannian\ntwo-sphere can be bounded from below in terms of the length of its shortest\none-cycle of index one. Our inequality (1.5) on the two-sphere extends this\nresult since the minimax process used in the definition of the diastole gives\nrise to a one-cycle of index one when the metric is bumpy. However, the\nrelation between the filling radius of a bumpy Riemannian two-sphere and\nthe length of its shortest one-cycle of index one established in [Sa04] cannot\nbe extended to the diastole. Indeed, from [Sa04, Theorem 1.6], there exists\na sequence Mnof Riemannian two-spheres such that\nlim\nn→+∞FillRad( Mn)\ndias(Mn)= 0.\nThis result illustrates the difference of nature between the length of the\nshortest closed geodesic or of the shortest one-cycle of index one, which can\nbe bounded by the filling radius on the two-sphere, and the diastole. It also\nshows that the proof of Theorem 1.1 requires different techniques.\nThe arguments used throughout this article are rather “elementary” and\nrobust. It is our hope that they can be adapted to investigate further prob-\nlems.\nAt the end of this article, we consider a minimax principle on the one-cycle\nspace for another functional and establish further geometric inequalities on\nthe two-sphere.\nHigher dimensional analogs of the diastole, sometimes called k-widths,\nhave been investigated in [Al65], [Pi81], [Gr83] and [Gu07]. In particular, L.\nGuth [Gu07] obtained upper bounds on the k-width of Euclidean domains in\nterms of their n-dimensional volumes. He also showed that no such bound\nholds in the Riemannian setting for the ( n−1)-width with n≥3. Our\nmain result provides a positive result on Riemannian surfaces, answering a\nquestion raised in [Gu07, p. 1148] in a particular case.\nLet us present the structure of the article and an outline of the proof of\nthe diastolic inequality (1.5). The one-cycle space and the definition of the\ndiastole are presented in Section 2. In Section 3, we show how to replace a\nRiemannian surface with a simplicial piecewise flat surface with comparable\narea and diastole. This will enable us to prove the main diastolic inequality\nby induction on the number of simplices in an approaching piecewise flat\nsurface. In Section 4, we prove an upper bound on Cheeger’s constant, and\non some related invariant, in terms of the area of the surface alone. This\nupper bound yields an isoperimetric inequality which permits us to split\nthe surface into two domains D1andD2whose boundary lengths are small\nin comparison to their areas. This isoperimetric inequality can be thought\nof as a discrete version of the diastolic inequality. At this stage, we could\ncap off the two domains D1andD2and apply the discrete version of the\ndiastolic inequality to the two resulting surfaces with the hope to derive a\ncontinuous version of it by iterating this principle sufficiently many times.\nUnfortunately, passing from a discrete parameter family of one-cycles to a\ncontinuous parameter family on a smooth surface is technically challenging.\nArguing by induction on the number of simplices of a piecewise flat surface\napproaching the initial surface turns out to be more manageable in this case.DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 5\nMore specifically, we argue as follows. In Section 5, we prove a simplicial\nversion of the isoperimetric inequality obtained from the upper bound on\nCheeger’s constant. The two simplicial surfaces obtained by coning off the\ndomains D1andD2have fewer simplices than the initial simplicial surface.\nUsing a cut-and-paste argument on the families of one-cycles, we compare\nthe diastole of the two new simplicial surfaces to the diastole of the initial\nsimplicial surface in Section 6. The diastolic inequality is then established by\ninduction on the number of simplices and the genus in Section 7. Further\nrelated geometric inequalities about families of one-cycles sweeping out a\nRiemannian two-sphere are presented in Section 8 and in the appendix.\n2.Almgren’s isomorphism and definition of the diastole\nWe introduce the space of flat chains and cycles with coefficients in Z\norZ/2Z, defined in [Fl66] and [Fe69].\nLetMbe a closed Riemannian manifold, Nbe a submanifold of M(pos-\nsibly with boundary), k=ZifMis orientable and k=Z/2Zotherwise,\nandkbe an integer.\nA Lipschitz k-chain of Mwith coefficients in kis a finite sum C=Paifi\nwhere ai∈kandfiis a Lipschitz map from the standard k-simplex to M.\nThe volume of C, denoted by |C|, is defined asP|ai|vol(f∗\niG) where Gis the\nRiemannian metric on M. The flat pseudo-norm of a Lipschitz k-chain Cis\nthe infimum of |C−∂D|+|D|over the Lipschitz ( k+1)-chains D. Two Lip-\nschitz chains C1andC2are equivalent if the flat pseudo-norm between them\nis zero. The flat pseudo-norm induces a distance (flat norm) on the space\nof equivalent classes of Lipschitz k-chains. By definition, the completion of\nthis metric space, denoted by Ck(M;k), is the space of (flat) k-chains with\ncoefficients in k.\nThe mass functional on Ck(M;k), denoted by M, is defined as the largest\nlower-semicontinuous extension of the volume functional on the space of\nLipschitz k-chains. The flat norm of a k-chain can be computed by using\nthe definition of the flat pseudo-norm with Drunning over the ( k+1)-chains\nand the mass Mreplacing the volume functional.\nThe space of k-chains of Mrelative to Nwith coefficients in kis de-\nfined as the quotient Ck(M, N ;k) :=Ck(M;k)/Ck(N;k) endowed with the\nquotient flat norm and the quotient mass still denoted M. The boundary\nmap of a Lipschitz k-chain induces a boundary map ∂:Ck+1(M, N ;k)→\nCk(M, N ;k) for every k∈Nsuch that ∂2= 0 which gives rise to a complex\n. . .→Ck+1(M, N ;k)∂→Ck(M, N ;k)→. . .\nThe cycles of this complex are by definition the k-cycles of Mrelative to N\nwith coefficients in k. They can be represented by k-chains of Mwith\nboundary lying in N. We denote by Zk(M, N ;k) the space formed of these\nk-cycles. When Nis empty, we simply write Zk(M;k) orZk(M).\nWe describe now the structure of the zero- and one-cycles. A zero-cycle\nis indecomposable if it induced by a point p, in which case, its mass is equal\nto 1. Every zero-cycle zdecomposes into a finite sum of indecomposable\nzero-cycles pi,i.e.,z=Pn\ni=1pi, such that M(z) =n. The structure of the6 F. BALACHEFF AND S. SABOURAU\none-cycles is the following. A one-cycle is indecomposable if it is induced by\na simple closed curve or a simple arc with endpoints in N. Every one-cycle z\ndecomposes (not necessarily in a unique way) into a sum of indecomposable\none-cycles zi,i.e.,z=P\ni∈Nzi, such that M(z) =P\ni∈Nlength( zi) (see\n[Fe69, p. 420]).\nThe homotopy groups of the space of (relative) cycles have been deter-\nmined by F. Almgren, cf.[Al60], [Al65, §13.4] and [Pi81, §4.6]. On surfaces,\none has the following natural isomorphism for the one-cycle space\nπ1(Z1(M;k),{0})≃H2(M;k)≃k. (2.1)\nMore generally, if Nis a submanifold with boundary of M, one has\nπ1(Z1(M, N ;k),{0})≃H2(M, N ;k). (2.2)\nThis isomorphism permits us to apply the Almgren-Pitts minimax prin-\nciple to the one-cycle space of surfaces. Recall that k=Zif the surface is\norientable and that k=Z/2Zotherwise.\nLet us consider the one-parameter families ( zt)0≤t≤1of one-cycles sweep-\ning out a closed surface M, that is, which satisfy the following conditions\n(D.1) ztstarts and ends at the null one-cycle;\n(D.2) ztinduces a generator of π1(Z1(M;k),{0})≃k.\nFor technical reasons, we will also consider the following conditions\n(D.3) there exists a finite subdivision t1< t2<···< tkof [0,1] and\nhomotopies ( γj,t)tp≤t≤tp+1,j∈Jp, of finitely many piecewise smooth\nloops such that the one-cycle ztagrees with the finite sumP\nj∈Jpγj,t\non [tp, tp+1];\n(D.4) for every t∈]tp, tp+1[, the loops γj,t,j∈Jp, are simple and disjoint;\n(D.5) the loops γj,tandγj′,t′are disjoint for t̸=t′.\nExample 2.1. Letf:M→Rbe a Morse function on a closed surface M.\nThe level curves zt=f−1(t) define a one-parameter family of one-cycles ( zt)\nsatisfying the conditions (D.1-5).\nDefinition 2.2. Thediastole ofM, denoted by dias( M), is defined as the\nminimax value\ndias(M) := inf\n(zt)sup\n0≤t≤1M(zt) (2.3)\nwhere ( zt) runs over the families of one-cycles satisfying the conditions (D.1-2)\nabove. Similarly, the (technical) diastole ofM, denoted by dias′(M), is de-\nfined as the minimax value\ndias′(M) := inf\n(zt)sup\n0≤t≤1M(zt) (2.4)\nwhere ( zt) runs over the families of one-cycles satisfying the conditions (D.1-5)\nabove.\nRemark 2.3. We will prove that the diastolic inequality (1.5) holds true\nif one replaces the diastole dias( M) with the (technical) diastole dias′(M).\nSince dias( M)≤dias′(M), this will yield Theorem 1.1.DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 7\nRemark 2.4. IfMis a Riemannian compact surface with boundary ∂M,\nthe isomorphism (2.2) allows us to define a relative notion of diastole using\none-cycles relative to ∂M. This relative diastole on Mis less or equal to the\ndiastole on the closed Riemannian surface obtained by filling the boundary\ncomponents of Mwith disks of small area. Therefore, a diastolic inequality\nsimilar to (1.5) holds for compact surfaces with boundary where the diastole\nis replaced with the diastole relative to the boundary.\n3.From Riemannian metrics to simplicial surfaces\nProposition 3.1. Every closed Riemannian surface MisK-bilipschitz\nhomeomorphic to a closed piecewise flat surface M0triangulated by equi-\nlateral flat triangles, where K= 33.\nRemark 3.2. If we do not require the triangles of M0to have the same size,\nwe can take the constant Kto be arbitrarily close to one in the previous\nproposition. However, this is no longer true for surfaces with nonzero Euler\ncaracteristic when all the triangles of M0are supposed to be equilateral.\nLet us argue by contradiction. Subdividing the triangulation of M0if\nnecessary, we can assume that the equilateral flat triangles of M0are arbi-\ntrarily small. Consider the geodesic triangulation TofMisotopic to the\nimage of the triangulation of M0under the K-bilipschitz homeomorphism\nwhile keeping the vertices fixed. If Kis close enough to one, the angles of\nall the triangles of Tare close toπ\n3.\nFrom Euler’s formula, we have the following relation\n6χ(M) =X\ni(6−vi)\nbetween the Euler characteristic of Mand the degrees viof the vertices of\nits triangulation. If the Euler characteristic of Mis nonzero, at least one\nvertex of the triangulation of Mhas its degree different from 6. In this case,\nthe angles of the triangulation Tof the smooth surface Mcannot all be\nclose toπ\n3. Hence a contradiction.\nFor the two-torus and the Klein bottle, the constant Kin the previous\nproposition can be taken to be arbitrarily close to one, see [CVM90] and the\nfollowing.\nThe proof of Proposition 3.1 rests on the following construction.\nLetMbe a closed Riemannian surface. Choose ε∈]0,1\n100[ and ε′∈]0, ε[.\nFrom [CVM90], there exists a geodesic triangulation TofMsuch that\n(1) the diameter of every triangle of Tis less than ε′;\n(2) the angles of every triangle of Tare between2π\n7−ε′and5π\n14+ε′.\nDenote by ∆ ithe triangles of the triangulation T. By taking ε′small enough,\nwe can assume that every triangle ∆ iofTis almost isometric to a triangle ∆′\ni\nof the Euclidean plane with the same side lengths as ∆ i. More precisely, we\ncan assume that\n(a) there exists a (1+ ε)-bilipschitz homeomorphism between ∆ iand ∆′\ni;\n(b) the diameter of the triangle ∆′\niis less than ε;\n(c) the angles of the triangle ∆′\niare strictly betweenπ\n4and3π\n7.8 F. BALACHEFF AND S. SABOURAU\nReplacing the triangles ∆ iof the triangulation TofMby the Euclidean\ntriangles ∆′\nigives rise to a piecewise flat metric with conical singularities on\nthe surface. The size of the Euclidean triangles making up this piecewise\nflat metric may vary from one triangle to another. By subdividing some of\nthese triangles, one can make this size more uniform.\nMore precisely, the segments connecting the midpoints of the sides of\neach Euclidean triangle ∆′\nidecompose ∆′\niinto four triangles twice smaller\nthan ∆′\niwith the same angles as ∆′\ni(see Figure 1).\n∆′′\ni,1\n∆′′\ni,2∆′′\ni,3 ∆′′\ni,4\nFigure 1. Decomposition of a triangle ∆′\ni\nBy iterating this subdivision process, replacing some of the ∆′\ni’s by four\ntriangles twice smaller than ∆′\ni, one can construct a new collection T′′of\ntriangles ∆′′\njsuch that\nmax\njdiam(∆′′\nj)≤2 min\njdiam(∆′′\nj).\nLemma 3.3. There exists ℓ > 0such that every triangle of T′′is8-\nbilipschitz homeomorphic to the equilateral flat triangle of side length ℓ\nthrough an affine map.\nProof. Letℓ= max jdiam(∆′′\nj) and ∆′′\njbe a triangle of T′′. By construction,\nthe angles of ∆′′\njstrictly lie betweenπ\n4and3π\n7and the length of its longest\nside betweenℓ\n2andℓ.\nConsider the longest side of ∆′′\nj. Let δbe the perpendicular bisector of this\nside and vbe the vertex of ∆′′\njopposite to this side. Move vto its orthog-\nonal projection on δ. Then, move this projection along the perpendicular\nbisector δso as to obtain an equilateral flat triangle. A straightforward com-\nputation shows that the composite of these two transformations defines a\n4-bilipschitz affine homeomorphism from ∆′′\njonto an equilateral flat triangle\n(recall that the angles of ∆′′\njare greater thanπ\n4).\nThe length of the sides of this equilateral triangle is betweenℓ\n2andℓ.\nTherefore, a homothety of ratio between 1 and 2 transforms this equilateral\ntriangle into an equilateral flat triangle of side length ℓ. □DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 9\nThe collection T′′of these triangles may not form a triangulation of M\nsince the vertex of a triangle ∆′′\njmay lie in the interior of the edge of another\ntriangle. However, the union of the triangles of T′′covers Mand the interiors\nof two distinct triangles of T′′are disjoint. Furthermore, if e1ande2are the\nedges of two triangles of T′′then either e1lies in e2ore2lies in e1, unless\nthe two edges are disjoint or intersect at a single point.\nLet us see now how to obtain a genuine triangulation of M. For that\npurpose, we will need the following lemma.\nLemma 3.4. Consider an edge of a triangle of T′′which contains at least\ntwo edges of some other triangles of T′′. Then this edge agrees exactly with\nthe union of these two edges.\nProof. Since the angles of every triangle ∆′′ofT′′are greater thanπ\n4, the\nlength of the longest side of ∆′′, which agrees with its diameter, is less than\ntwice the length of its shortest side. This remark is left to the reader.\nConsider an edge eof a triangle ∆′′ofT′′which contains at least two\nedges of some other triangles of T′′. By construction, the edge eis made up\nof a necessarily even number of edges of some other triangles of T′′. Assume\nthat at least four edges of some other triangles of T′′lie in e. The length of\nthese edges is at most one quarter of the length of e. Therefore, our initial\nremark shows that there exist some triangles of T′′whose diameter is less\nthan half the diameter of ∆′′. Hence a contradiction with the construction\nofT′′. □\nIf the edge eof a triangle ∆′′ofT′′agrees with the union of the edges of\ntwo other triangles of T′′, then we split ∆′′along the segment connecting the\nmidpoint of eto the vertex of ∆′′opposite to that edge. The new collection\nof triangles still satisfies the conclusion of the previous lemma. We repeat\nthis process to this new collection until we get a genuine triangulation T0\nofM, where every edge is the side of exactly two adjacent triangles.\nLemma 3.5. Every triangle of T0is32-bilipschitz homeomorphic to the\nequilateral flat triangle of side length ℓ(the same ℓas in Lemma 3.3).\nProof. The (8-bilipschitz) affine homeomorphism of Lemma 3.3 takes seg-\nments to segments and the midpoints of segments to the midpoints of the\nimage segments. Therefore, we can work with an equilateral flat triangle ∆\nof side length ℓinstead of a triangle of T′′. From Lemma 3.4, this triangle\nis split into at most four triangles through the subdivision of T′′intoT0.\nThese subtriangles of ∆ are isometric to one of the following\n(1) the equilateral triangle of side lengthℓ\n2;\n(2) the isosceles triangle with a principal angle of2π\n3and two sides of\nlengthℓ\n2;\n(3) the right triangle with anglesπ\n2,π\n3andπ\n6, and side lengths ℓ,√\n3\n2ℓ\nandℓ\n2.\nSince each of these triangles is 4-bilipschitz homeomorphic to ∆, we obtain\nthe desired result. □\nWe can now conclude.10 F. BALACHEFF AND S. SABOURAU\nProof of Proposition 3.1. Denote by M0the piecewise flat surface obtained\nby replacing every triangle of T0by the equilateral flat triangle with side\nlength ℓ. Putting together the bilipschitz homeomorphisms of (a) above\nand Lemma 3.5 gives rise to a (1+ ε)32-bilipschitz homeomorphism between\nthe Riemannian surface Mand the piecewise flat surface M0. □\nCorollary 3.6. The area, the diameter and the diastole of MandM0satisfy\nthe following inequalities\nK−2≤area( M0)\narea( M)≤K2(3.1)\nK−1≤dias(M0)\ndias(M)≤K (3.2)\nwhere K= 33.\nRemark 3.7. From Corollary 3.6, proving a diastolic inequality on M\namounts to proving a diastolic inequality on the simplicial surface M0. Since\ndiastolic inequalities are scale invariant, the size of the equilateral flat trian-\ngles composing the piecewise flat surface M0can arbitrarily be fixed equal\nto 1. We will always assume this is the case when we consider simplicial\nsurfaces.\n4.Upper bounds on Cheeger’s constant\nLetMbe a closed Riemannian surface. Recall that the Cheeger con-\nstant h(M) ofMis defined as\nh(M) = inf\nDlength( ∂D)\nmin{area( D),area( M\\D)},\nwhere the infimum is taken over all the domains DofMwith smooth bound-\nary.\nProposition 4.1. Every closed Riemannian surface M(not necessarily ori-\nentable) of genus g≥0satisfies the following inequality\nh(M)≤p\n96π(g+ 1)p\narea( M). (4.1)\nProof. Letλ1(M) be the first nonzero eigenvalue of the Laplacian on M.\nFrom [Ch70], we have\nλ1(M)≥h(M)2\n4.\nOn the other hand, from [LY82], we have\nλ1(M) area( M)≤24π(g+ 1). (4.2)\nThe combination of these two inequalities yields the desired upper bound\non the Cheeger constant. □\nRemark 4.2. For orientable surfaces, the inequality (4.2) still holds by re-\nplacing the constant 24 by 8, cf.[YY80]. This leads to a better upper bound\non Cheeger’s constant in Proposition 4.1, where the constant 96 can be re-\nplaced by 32. Thus, Proposition 4.1 improves the multiplicative constant of\nan inequality established in [Pa09] by using a different method.DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 11\nNote that the dependence of the upper bound on the genus cannot be sig-\nnificantly improved. Indeed, there exist closed hyperbolic surfaces of arbi-\ntrarily large genus with Cheeger constant bounded away from zero, cf.[Br86]\nfor instance.\nAn upper bound on the Cheeger constant of Riemannian two-spheres\nsimilar to (4.1) can be obtained using different arguments. Although these\narguments do not lead to a better multiplicative constant, they rely on a\ncomparison between two intermediate invariants, namely the filling radius\nand the invariant L,cf.below, which is interesting in its own right. Further,\nthey can be generalized in higher dimension (but the final statement is not\nas meaningful). For these reasons, we still include this result even though\nwe will not use it in the rest of this paper.\nLetMbe a Riemannian two-sphere. Define\nL(M) = inf\nγlength( γ)\nwhere γruns over all the simple loops dividing Minto two disks of area at\nleast1\n4area( M). Clearly,\nh(M)≤4L(M)\narea( M).\nWe show in Proposition 4.4 that the invariant Lprovides a lower bound\non the filling radius on every Riemannian two-sphere.\nDefinition 4.3. Leti:M ,→L∞(M) be the Kuratowski distance preserv-\ning embedding defined by i(x)(.) =dM(x, .). The filling radius ofM, de-\nnoted by FillRad( M), is defined as the infimum of the positive reals rsuch\nthat the homomorphism Hn(M;Z)−→Hn(Ur;Z) induced by the embed-\nding of Minto the r-tubular neighborhood Urof its image i(M) inL∞(M)\nis trivial.\nRecall that a lower bound on the filling radius provides bounds on the area\nand the diameter of Riemannian two-spheres since FillRad( M)≤p\narea( M)\nfrom [Gr83] and FillRad( M)≤1\n3diam( M) from [Ka83].\nProposition 4.4. LetMbe a Riemannian two-sphere. Then,\nL(M)≤6 FillRad( M).\nIn particular,\nh(M)≤24p\narea( M).\nRemark 4.5. Similar lower bounds on the filling radius have been proved\nin [Gr83] for essential manifolds, where Lis replaced with the systole, and\nin [Sa04] on the two-sphere, where Lis replaced with the shortest length of\na geodesic loop.\nProof. We argue by contradiction. Pick two reals randεsuch that\nFillRad( M)< r < r +ε <1\n6L(M).12 F. BALACHEFF AND S. SABOURAU\nLeti:M ,→L∞(M) be the Kuratowski distance preserving embedding de-\nfined by i(x)(.) =dM(x, .). By definition of the filling radius, there exists\na map σ:P→Urfrom a 3-complex Pto the r-tubular neighborhood\nUrofi(M) inL∞=L∞(M) such that σ|∂P:∂P→i(M) represents the\nfundamental class of i(M)≃MinH2(i(M);Z)≃H2(M;Z).\nLet us show that the map σ|∂P:∂P→i(M) extends to a map f:P→i(M).\nThis would yield the desired contradiction since the fundamental class of i(M),\nrepresented by σ|∂P, is nonzero.\nDeforming σand subdividing Pif necessary, we can assume that σtakes\nevery edge of Pto a minimizing segment and that the diameter of the σ-\nimage of every simplex in Pis less than ε. We define fto agree with σon\nthe 2-complex ∂P.\nWe now define fon the 0-skeleton of P\\∂Pby sending each vertex p\nofP\\∂Pto a point f(p) oni(M) closest to σ(p). Thus, dL∞(f(p), σ(p))< r\nfor every vertex pofP. Since the embedding ipreserves the distances, every\npair of adjacent vertices p, q∈Psatisfies\ndi(M)(f(p), f(q)) = dL∞(f(p), f(q))\n≤dL∞(f(p), σ(p)) +dL∞(σ(p), σ(q)) +dL∞(σ(q), f(q))\n<2r+ε=:ρ\nWe extend fto the 1-skeleton of Pby mapping each edge of P\\∂Pto a\nminimizing segment joining the images of the endpoints. By construction,\nthe image under fof the boundary of every 2-simplex ∆2ofPis a simple\nloop γ∆2of length less than 3 ρ. Since 3 ρ <L(M) for εsmall enough, the\nsimple loop γ∆2bounds a disk D∆2ini(M)≃Mof area less than1\n4area( M).\nWe extend fto the 2-skeleton of Pby mapping each 2-simplex ∆2to the\ndiskD∆2. Thus, the restriction ∂∆3→i(M) offto the boundary ∂∆3of a\n3-simplex ∆3ofPis a degree zero map between two spheres since the area\nof its image is less than the area of i(M)≃M. Therefore, fextends to each\nsimplex of Pand gives rise to an extension f:P→i(M) ofσ|∂P. □\n5.Isoperimetric inequalities on simplicial surfaces\nWe will need the following remark. Let ε >0. Every piecewise flat metric\nwith conical singularities on a surface can be smoothed out at its singu-\nlarities into a Riemannian metric (1 + ε)-bilipschitz homeomorphic to it.\nTherefore, Proposition 4.1 still holds true for closed simplicial surfaces.\nWe prove now that the proposition 4.1 still holds in a simplicial setting.\nProposition 5.1. LetM0be a closed simplicial surface (not necessarily ori-\nentable) of genus g≥0formed of Ntriangles. The surface M0decomposes\ninto two simplicial domains, D1andD2, with disjoint interiors, satisfying\nthe following inequality for i= 1,2\nlength( ∂Di)≤C0p\ng+ 1min{area( D1),area( D2)}p\narea( M0), (5.1)\nwhere C0= 15√\n96π.DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 13\nProof. Assume that N≥N∗where N∗:= 10 ·96π(g+ 1). Fix ε∈]0,1\n50[.\nSince M0is composed of Nequilateral flat triangles (of side length 1), the\narea of M0is equal to\narea( M0) =√\n3\n4N. (5.2)\nFrom the remark at the beginning of this section, the surface M0decom-\nposes into two (not necessarily simplicial) domains D1andD2with common\nboundary δsuch that\nlength( δ)<(1 +ε)√\n96πp\ng+ 1min{area( D1),area( D2)}p\narea( M0). (5.3)\nCombined with the formula (5.2) and the bound N≥N∗, this estimate\nyields\nlength( δ)<1\n2area( Dj) for j= 1,2. (5.4)\nWe want to deform δinto a one-cycle lying in the one-skeleton of M0while\ncontrolling its length and the area of the two domains it bounds.\nWithout loss of generality, we can suppose that\n(1)δis composed of finitely many simple loops;\n(2) each of these loops meets the edges of the triangulation of M0;\n(3) each of these loops intersects the edges of the triangulation at finitely\nmany points;\n(4) none of these loops passes through the midpoints of the edges of the\ntriangulation.\nIn this situation, the one-cycle δdecomposes into a finite union of sub-\narcsτi, where i∈I, such that each arc τiis contained into a triangle ∆ of\nthe triangulation of M0with its endpoints lying in the boundary ∂∆ of ∆.\nFor every i∈I, denote by ℓithe length of τi. We have\nlength( δ) =X\ni∈Iℓi.\nFor every triangle ∆ of the triangulation of M0, we apply the curve-\nshortening process defined in [Sa04, §2.2] to the finite collection of arcs τi\nlying in the convex domain ∆. Through this curve-shortening process, the\narcsτiremain disjoint, do not self-intersect and converge to the segments si\nof ∆ with the same endpoints. (Strictly speaking the convergence property\nhas been stated for bumpy metrics but it still holds in the flat case since\nevery couple of points of ∆ are connected by a unique geodesic.) Now, we\nextend this process by deforming simultaneously every segment siinto a\nvertex or an edge of the triangulation through a family of segments whose\nendpoints move along the sides of ∆ to the closest vertices of ∆ (see figure 2).\nThis deformation ( δt) ofδcan be performed so that the loops forming the\none-cycles δtremain simple, except possibly for the final loop δ′=δ∞. This\ndefines our deformation process.\nThe final one-cycle δ′lies in the one-skeleton of M0. Therefore, the two\ndomains bounded by δtconverge to two simplicial domains D′\n1andD′\n2ofM0\nwhose common boundary lies in δ′.14 F. BALACHEFF AND S. SABOURAU\nFigure 2.\nLemma 5.2. We have\nlength( δ′)≤2 length( δ).\nProof. Leti∈I. By definition, the arc τi, of length ℓi, lies in a triangle ∆\nof the triangulation of M0and length( si)≤ℓi. Through the deformation\nprocess, the segment sideforms into either a vertex or an edge of ∆. The\nlatter case occurs when the two endpoints of siare at distance less than1\n2\nfrom two distinct vertices of ∆. Thus, the segment sideforms into a path of\nlength at most twice the length of si. (The limit case occurs when sijoins\nthe midpoints of two edges of ∆.) Summing up over all the i∈Iyields the\ndesired inequality. □\nLemma 5.3. Forj= 1,2, we have\narea( D′\nj)≥1\n2area( Dj). (5.5)\nProof. The intersection Dj∩∆ between Djand a triangle ∆ of the trian-\ngulation of M0is formed of domains bounded by the arcs τiand the edges\nof ∆. Consider i∈Isuch that τiis an arc of ∆.\nSuppose that the length ℓiofτiis less than1\n2.\nIf the endpoints of τilie in the same edge eof ∆, the arc τideforms either\nto this edge or to an endpoint of this edge through the deformation process\npreviously defined (see figure 2). And so does the region of ∆ bounded by τi\nand the edge e.\nIf the endpoints of τilie in two adjacent edges e1ande2of ∆, the endpoints\nofτimove to the same vertex vof ∆ through the deformation process (recall\nthat the length of τiis less than1\n2). Thus, the arc τideforms into this vertex\nand the region of ∆ bounded by τiand the edges e1ande2retracts onto the\nvertex v.\nIn both cases, the area of this region of ∆ bounded by τiand either the\nedge eor the edges e1ande2is less or equal to3\n2πℓ2\ni. This isoperimetric\ninequality comes from the standard isoperimetric inequality on the plane\ncombined with a symmetry argument.\nSuppose now that the length ℓiofτiis at least1\n2. In this case, a rougher es-\ntimate holds. Specifically, the area of each of the two domains of ∆ bounded\nbyτiis less than the area of ∆, i.e.,√\n3\n4.DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 15\nTherefore, by substracting up the area change of Dj∩∆ through the\ndeformation process for every triangle of the triangulation, we obtain\narea( D′\nj)≥area( Dj)−X\ni∈I−3\n2πℓ2\ni−X\ni∈I+√\n3\n4\n≥area( Dj)−X\ni∈I−3\n4πℓi−X\ni∈I+√\n3\n2ℓi\n≥area( Dj)−√\n3\n2 X\ni∈Iℓi!\n(5.6)\nwhere the first sum is over all the indices ifor which ℓi<1\n2and the second\nover those for which ℓi≥1\n2.\nFrom the inequality (5.4), namely\nlength( δ) =X\ni∈Iℓi<1\n2area( Dj),\nwe derive\narea( D′\nj)≥1\n2area( Dj).\n□\nFrom the inequality (5.3) and the lemmas 5.2 and 5.3, we deduce\nlength( δ′)<4 (1 + ε)√\n96πp\ng+ 1min{area( D′\n1),area( D′\n2)}p\narea( M0).\nWhen N≤N∗, we can take for D1any triangle of M0and for D2the\nunion of the remaining triangles. Thus,\nlength( ∂Di) = 3 and min {area( D1),area( D2)}=√\n3\n4.\nUsing the formula (5.2) and the bound N≤N∗, one can check that the\ninequality (5.1) is satisfies in this case too. Hence the conclusion. □\n6.Diastole comparison\nLetM0be a simplicial closed surface decomposed into two simplicial do-\nmains D1andD2with disjoint interiors. Denote by δthe common boundary\nof the domains D1andD2. Consider the abstract simplicial cones over the\nconnected components of ∂Dicomposed of triangles with basis the edges\nofδ=∂Di. Denote by Mithe simplicial closed surface obtained by attach-\ning these simplicial cones to Dialong ∂Di.\nThe following estimate on the (technical) diastole, cf.Definition 2.2, fol-\nlows from a cut-and-paste argument.\nProposition 6.1. We have\ndias′(M0)≤max\nidias′(Mi) + length( δ). (6.1)16 F. BALACHEFF AND S. SABOURAU\nProof. Fixϵ >0. By definition, for i= 1,2, there exists a one-parameter\nfamily ( zi\nt)0≤t≤1of one-cycles on Mi, with coefficients in k, which satisfies\nthe conditions (D.1-5) of the definition of the (technical) diastole, cf.Defi-\nnition 2.2, such that\ndias′(Mi)≤sup\n0≤t≤1M(zi\nt)<dias′(Mi) +ϵ.\nDenote by ( γi\nj,t) the homotopies of the loops defining the family ( zi\nt),\ncf.item (D.3) of Definition 2.2. By slightly perturbing the family ( zi\nt) and\nrefining the subdivision ( tp) if necessary, we can assume that the loops γi\nj,t\nare disjoint or transverse to ∂Difor every t̸=tp, and that they have at most\nfinitely many tangent points with ∂Difor every tp, where p= 1,···, k.\nThe boundary ∂(zi\nt⌞Di) of the restriction of zi\ntto the domain Didefines\na family of zero-cycles on ∂Diwith disjoint supports composed of finitely\nmany points. This family of one-cycles sweeps out ∂Di, that is, it starts and\nends at null one-cycles, and induces a generator of π1(Z0(∂Di;k),{0}). It\nalso induces a family of one-chains αi\nton∂Disuch that αi\n0= 0,αi\n1=±∂Di\nand∂αi\nt=∂(zi\nt⌞Di).\nIn a more constructive way, this family of one-chains on ∂Dican be\ndescribed as follows. The restriction zi\nt⌞Diofzi\nttoDi, with tp≤t≤tp+1,\ndecomposes as a finite sum of loops lying in Diplus a finite sum of arcs ci\nj,t\nlying in Diwith endpoints on ∂Di. As truns over [ tp, tp+1], the basepoints\nand the endpoints of the arcs ci\nj,tdescribe some arcs on ∂Di,cf.[Rh55,\n§14] for a more general construction. The arcs described by the basepoints\n(resp. endpoints) of the ci\nj,t’s inherit a negative (resp. positive) orientation.\nNote that these arcs do not overlap because of the condition (D.5). Putting\ntogether the sum of these arcs on ∂Digives rise to a family of one-chains αi\nt\non∂Di, starting at the null one-chain and ending at the one-cycle ±∂Di,\nsuch that ∂αi\nt=∂(zi\nt⌞Di).\nConsider the family of one-cycles ˜ zi\nt= (zi\nt⌞Di)−αi\ntlying in Di. By con-\nstruction, this family starts at the null one-cycle, ends at the one-cycle ±∂Di\nand induces a generator of π1(Z1(Di, ∂D i;k),{0}). Furthermore, its mass is\nbounded as follows\nM(˜zi\nt)≤M(zi\nt) + length( δ).\nWe can now define a new one-parameter family (˜ zt)0≤t≤1of one-cycles\nonM0by concatenating ˜ z1\ntand ˜z2\ntas follows\n˜zt=\n\n˜z0\n2t ift∈[0,1/2],\n−˜z1\n2−2tift∈[1/2,1].\nBy construction, this new family satisfies the conditions (D.1-4) of the defi-\nnition of the (technical) diastole, cf.Definition 2.2, and\nM(˜zt)<max\nidias′(Mi) + length( δ) +ϵ. (6.2)\nIt does not necessarily satisfy the condition (D.5) though. Indeed, the sup-\nports of the one-cycles ˜ ztmay have a nonempty intersection on ∂Di.DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 17\nBy slightly perturbing the family (˜ zt) in the neighborhood of ∂Di, it is\npossible to obtain a family of one-cycles satisfying the conditions (D.1-5)\nand the inequality (6.2)\nTherefore, dias′(M0)≤max idias′(Mi) + length( δ) +ϵfor any positive ϵ.\nHence the proposition. □\n7.Proof of the diastolic inequality\nWe can now prove the main diastolic inequality, cf.Theorem 1.1. We will\nuse the notations of the previous sections.\nFrom the discussion in Section 3, cf.Corollary 3.6, it is enough to show\nthat\ndias′(M0)≤C\nK2p\ng+ 1p\narea( M0) (7.1)\nfor every simplicial closed surface M0(not necessarily orientable) of genus g.\nWe will argue by induction both on the number Nof triangles composing M0\nand on its genus. From now on we suppose that Theorem 1.1 is proved for\nsurfaces of genus less than g. IfM0is a 2-sphere, the proof proceeds by\ninduction only on the number of simplices.\nDecompose M0into two simplicial domains D1andD2as in Proposi-\ntion 5.1. Consider the simplicial closed surfaces Midefined in Section 6 by\nattaching simplicial cones along ∂Di. By construction, the genus of Miis at\nmost g. Since ∂Diis made up of length( ∂Di) edges, the area of the cones\nglued to Diis equal to√\n3\n4length( ∂Di). Hence the formula\narea( Mi) = area( Di) +√\n3\n4length( ∂Di). (7.2)\nSetN0:=√\n3C2\n0(g+ 1) = 225√\n3·96π(g+ 1).\nLemma 7.1. ForN≥N0, the simplicial closed surfaces Miare composed\nof fewer triangles than M0.\nProof. Recall that area( M0) =√\n3\n4N. Thus, for N≥N0, the quantity√\n3\n4C0√g+1√\narea( M0)is less than 1 /2. From the formula (7.2) and the bound (5.1),\nwe deduce that\narea( Mi)<area( Di) + min {area( D1),area( D2)}\n<area( D1) + area( D2).\nHence, area( Mi)<area( M0). □\nRenumbering the Mi’s if necessary, we can assume that\ndias′(M1)≥dias′(M2).\nLetαbe a real such that αarea( M0) = min {area( D1),area( D2)}. Note\nthat 0 < α≤1\n2. The inequality (5.1) can be written\nlength( δ)≤C0αp\ng+ 1p\narea( M0), (7.3)\nwhere δ=∂Di.\nWe have two cases to consider.18 F. BALACHEFF AND S. SABOURAU\nFirst case: Suppose that the number Nof triangles of M0is at most N0,\nwhere N0is defined in Lemma 7.1. Fix an orientation on the surface M0.\nThis orientation induces an orientation on all the triangles of M0and on their\nboundaries. Of course, if the surface is nonorientable, there is nothing to do.\nConsider the boundaries of the triangles of M0as one-cycles with coefficients\nink. The sum of these boundaries, viewed as one-cycles, vanishes. Given\na triangle ∆ of M0, there exists a homotopy from the boundary of ∆, with\nthe induced orientation, to the center of ∆ composed of disjoint loops. The\nsum of these homotopies over all the triangles of M0defines a one-parameter\nfamily of one-cycles ( zt)tonM0, which starts and ends at the null one-cycle.\nFurthermore, it induces a generator of π1(Z1(M0;k),{0})≃H2(M0;k) and\nsatisfies the technical conditions (D.3-5) of Definition 2.2. Therefore,\ndias′(M0)≤sup\ntM(zt)≤3N.\nSince area( M0) =√\n3\n4NandN≤N0, we obtain\ndias′(M0)≤3√\nNp\nN0\n≤2·33/4p\nN0p\narea( M0)\n≤6C0p\ng+ 1p\narea( M0).\nSecond case: Suppose that Nis greater than N0. We can assume that\ndias′(M1)p\narea( M1)≤dias′(M0)p\narea( M0)\notherwise the diastolic inequality (7.1) follows from Lemma 7.1 by induction\nongandN. That is,\ndias′(M1)≤λdias′(M0),\nwhere\nλ:=s\narea( M1)\narea( M0). (7.4)\nNote that λ <1 from Lemma 7.1. From Proposition 6.1 and since dias′(M1) =\nmax idias′(Mi), we deduce\ndias′(M0)≤1\n1−λlength( δ).\nFrom the inequality (7.3), we obtain\ndias′(M0)≤C0α\n1−λp\ng+ 1p\narea( M0). (7.5)\nLemma 7.2. We haveα\n1−λ≤4.\nProof. Since N≥N0, the quantity√\n3\n4C0√g+1√\narea( M0)is less than1\n2and\narea( Mi)≤area( Di) +1\n2min{area( D1),area( D2)}\nfrom the formula (7.2) and the bound (5.1).DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 19\nIf area( D1)≤area( D2), then area( D1) =αarea( M0). In this case, we\ndeduce that λ≤q\n3\n2αfrom the definition (7.4) of λ. Since α≤1\n2, we obtain\nα\n1−λ≤α\n1−q\n3\n2α≤2 +√\n3.\nOtherwise, area( D1) = (1 −α) area( M0). As previously, we derive, in this\ncase, that λ≤p1−α\n2. Since α >0, we get\nα\n1−λ≤2\u0012\n1 +r\n1−α\n2\u0013\n≤4.\n□\nThis lemma with the inequality (7.5) implies that\ndias′(M0)≤4C0p\ng+ 1p\narea( M0).\nTherefore, the desired diastolic inequality (7.1) holds in both cases.\nRemark 7.3. Let ( zt) be a family of one-cycles which sweeps out a closed\nRiemannian surface M,cf.(D.1-2). From the constructions of [Al60], there\nexists t0such that zt0decomposes Minto two disjoint domains of area at\nleast1\n4area( M). Hence,\ndias(M)≥M(zt0)≥1\n4h(M)area( M).\nThe closed hyperbolic surfaces of arbitrarily large genus with Cheeger con-\nstant bounded away from zero constructed in [Br86] provide examples of\nsurfaces with area( M)≃gand dias( M)≳g. Thus, the dependence of the\ninequality (1.5) on the genus is optimal.\n8.Comparison of Riemannian invariants on the two-sphere\nIn this section, we clarify the relationships between several Riemannian\ninvariants on the two-sphere by comparing them up to an equivalence rela-\ntion.\nDefinition 8.1. Two Riemannian invariants I1andI2are said to be es-\nsentially equivalent on a manifold Mif there exist two positive constants C\nandC′such that for every Riemannian metric GonM, we have\nC I1(G)≤I2(G)≤C′I1(G).\nWe denote by I1(M)≃I2(M) this equivalence relation.\nTogether with the filling radius introduced in Definition 4.3, we will con-\nsider the following Riemannian invariants.\nDefinition 8.2. LetMbe a closed n-dimensional Riemannian manifold.\n•The 1 -hypersphericity , denoted by HS( M), is the supremum of the\npositive reals Rso that there is a contracting map of degree 1 from\nMto the n-sphere of radius R.20 F. BALACHEFF AND S. SABOURAU\n•The Uryson 1-width , denoted by UW 1(M), is the infimum of the\npositive reals Wso that there is a continuous map from Mto a\n1-dimensional polyhedron whose fibers have diameter less than W.\n•Let us define the functional µon the one-cycle space Z1(M,Z) of a\nRiemannian two-sphere Mas\nµ(z) = sup {M(zi)|ziis a connected component of z}.\nWe define a critical value of the functional µby a global minimax\nprinciple\nL(M) = inf\n(zt)sup\n0≤t≤1µ(zt)\nwhere ( zt) runs over the families of one-cycles satisfying the following\nconditions:\n(C.1) ztstarts and ends at null one-cycles;\n(C.2) ztinduces a generator of H2(M;Z).\nIn the following proposition, we compare these Riemannian invariants on\nthe two-sphere.\nProposition 8.3. The hypersphericity, the Uryson 1-width, the filling ra-\ndius and Lare essentially equivalent on the two-sphere S2, that is\nUW 1(S2)≃HS(S2)≃FillRad( S2)≃L(S2).\nProof. Since the filling radius of every closed Riemannian n-manifold de-\ncreases under contracting maps of degree one, cf.[Gr83], we have\nFillRad( S2,can) HS( M)≤FillRad( M) (8.1)\nwhere the value of the filling radius of the standard spheres has been com-\nputed in [Ka83]. Here, FillRad( S2,can) =1\n2arccos\u0000\n−1\n3\u0001\n.\nThe inequality\nUW 1(M)≤HS(M) (8.2)\nholds for every closed Riemannian manifold, cf.[Gr88].\nThe inequalities\nL(M)\n120≤HS(M)≤2\nπUW 1(M) (8.3)\nhave been established in [Gu05, Theorem 0.3 and p. 1058].\nNext we will prove that there exists a constant C\nFillRad( M)≤C L(M). (8.4)\nFirst of all observe that if ( zt) is a family of one-cycles on Mand if t1\nandt2are close enough, the one-cycle z=zt2−zt1bounds a 2-chain of\nsmall mass on M. Indeed, by definition of the flat norm, cf.Section 2,\nand since the one-cycle zhas a small flat norm, there exists a 2-chain A\nsuch that both M(z−∂A) and M(A) are small. By the isoperimetric in-\nequality on M, there exists a 2-chain Bof small mass bounding the one-\ncycle z−∂A. Thus, the 2-chain A+Bbounds zand has a small mass.\nNow, using the arguments of [Gr83, p. 133], one can show that, for every\nε >0, there exists a 1-Lipschitz map ∂X→Mof degree one, where Xis a\n3-dimensional pseudomanifold endowed with a metric length structure such\nthatd(x, ∂X )≤C L(M) +ε. We immediately derive the inequality (8.4).DIASTOLIC AND ISOPERIMETRIC INEQUALITIES 21\nPutting together the inequalities (8.1), (8.2), (8.3) and (8.4) yields the\ndesired relations. □\nRemark 8.4. We have L(M)≤dias(M) for every Riemannian two-sphere M\nbut the diastole is notessentially equivalent to any of the Riemannian in-\nvariant of Proposition 8.3 on the two-sphere from [Sa04, Theorem 1.6]. In\nthe appendix, we bound from below the filling radius of the Riemannian\ntwo-spheres in terms of the invariant Lusing different methods.\nAppendix\nIn this appendix, we prove the following result which first appeared in [Sa01,\n§2.4.2] and was not published in a journal.\nProposition A.1. LetMbe a Riemannian two-sphere, then\nL(M)≤18 FillRad( M).\nRemark A.2. The multiplicative constant in this inequality is better than\nthe one obtained in the proof of Proposition 8.3. Furthermore, it follows\nfrom a more general result, namely Proposition A.4.\nFrom the bounds FillRad( M)≤p\narea( M),cf.[Gr83], and FillRad( M)≤\n1\n3diam( M),cf.[Ka83], we immediately deduce the following corollary.\nCorollary A.3. LetMbe a Riemannian two-sphere, then\nL(M)≤18p\narea( M) (A.1)\nL(M)≤6 diam( M). (A.2)\nIn order to prove Proposition A.1, we need to extend the definition of the\ninvariant L,cf.Definition 8.2, as follows.\nLetN⊂Mbe a connected domain whose boundary ∂Nis a finite union\nof closed geodesics ciwith i∈I.\nLet us consider the one-parameter families of one-cycles ( zt)0≤t≤1onN\nwhich satisfy the following conditions.\n(C’.1) z0=∪i∈I0ciandz1=∪i∈I1cifor some partition of I=I0⨿I1\n(C’.2) ztinduces a generator of H2(N, ∂N ;Z).\nIfI0orI1is empty, the corresponding one-cycles are reduced to the null\none-cycle.\nWe define a critical value of the functional µby a global minimax principle\nL(N) = inf\n(zt)sup\n0≤t≤1µ(zt)\nwhere ( zt) runs over the families of one-cycles satisfying the conditions (C’.1-2)\nabove.\nWe will need a few more definitions. Every nontrivial simple loop γofM\nadmits two sides defined as the connected components of M\\γ. A nontriv-\nial simple geodesic loop γis said to be minimizing with respect to one of\nits sides if every pair of points of γcan be joined by a minimizing segment\nwhich does not pass through this side. A simple geodesic loop minimizing22 F. BALACHEFF AND S. SABOURAU\nwith respect to both sides is said to be minimizing.\nNow, we can state the following result which immediately leads to Propo-\nsition A.1 when N=Mand∂N=∅.\nProposition A.4. LetMbe a Riemannian two-sphere and N⊂Mbe\na connected domain whose boundary ∂Nis a finite union of minimizing\ngeodesic loops ci. Then\nFillRad( M)≥1\n18L(N).\nWithout loss of generality, we can assume the metric on Mis bumpy (a\ngeneric condition). The curve-deformation process described below plays a\nkey role in the proof of the above proposition.\nLetγbe a non-minimizing simple geodesic loop on Mandcbe a mini-\nmizing arc intersecting γonly at its endpoints. The arc cdivides γintoc+\nandc−. The simple loops c+∪candc−∪c, endowed with the orientations\ninduced by γ, bound convex domains and converge to simple geodesic loops\nγ+andγ−through homotopies z+\ntandz−\ntgiven by the curvature flow. Ap-\nplying the curve-shortening process of [Sa04, §2] if necessary, we can extend\nthe homotopies and suppose that γ+andγ−are local minima of the mass\nfunctional. Since the curves c+∪candc−∪cdo not intersect transversely\nand bound convex domains, the simple loops z+\ntandz−\ntare disjoint for\nt >0. The sum zt=z+\nt+z−\ntdefines the curve-deformation process of γ\nwith respect to c. The homotopy of the one-cycles ztisµ-nonincreasing and\nsatisfies µ(zt)≤length( γ). In particular, length( γ+) and length( γ−) are less\nthan length( γ).\nIn the following two lemmas, we assume that the only minimizing geodesic\nloops of Nare the connected components of the boundary ∂N.\nLemma A.5. Letγandγ′be two disjoint (nontrivial) simple geodesic loops\nofN. Suppose that γ(resp. γ′) is minimizing with respect to the side which\ndoes not contain γ′(resp. γ).\nThen, γorγ′is a connected component of ∂N.\nProof. The geodesics γandγ′bound a connected domain N′⊂Nwith\nγ, γ′⊂∂N′. Suppose that γdoes not lie in ∂N. Since the only minimizing\ngeodesic loops of Nare the connected components of ∂N, the loop γis not\nminimizing. Therefore, there exists a minimizing arc cinN′with endpoints\ninγwhich does not lie in γ. The curve-deformation process applied to\nγwith respect to cyields two disjoint simple geodesic loops of N′. The\nunion of these two geodesic loops is noted z1. Let us define by induction on\nka sequence zkformed of a disjoint union of simple geodesic loops of N′.\nThis sequence of one-cycles is µ-nonincreasing. Furthermore, each zkbounds\nwithγa connected domain DkofN′with∂Dk=zk∪γ. If one of the geodesic\nloops of zkis not minimizing with respect to the side opposed to γ, we apply\nto it the curve-deformation process with respect to this side. Otherwise, if\none of them is not minimizing with respect to the side containing γ, we\napply to it the curve-deformation process. In both cases, we obtain a new\ncollection Cof simple geodesic loops which do not intersect each other. WeDIASTOLIC AND ISOPERIMETRIC INEQUALITIES 23\ndefine zk+1as the union of the nontrivial geodesic loops of Cwhich can be\njoined to γby paths cutting no loop of C. This induction process stops after\na finite number nof steps when znis only formed of minimizing geodesic\nloops of N′. Since there is no minimizing geodesic loop in the interior of N,\nwe have zn⊂∂N. Therefore, the geodesic loop γ′, which lies between zn\nand∂N, is a connected component of ∂N. □\nLemma A.6. Letγbe a simple geodesic loop of Nof length less than L(N)\nand suppose that γis minimizing with respect to no side.\nThere exists a non-minimizing simple geodesic loop of N, lying in a side\nofγ, which is minimizing with respect to the side opposed to γ.\nProof. Assume that every simple geodesic loop of N, disjoint from γand\nminimizing with respect to the side opposed to γ, is minimizing. Let us recall\nthat the only minimizing geodesic loop of Nare the connected components\nof∂N. We apply the curve-deformation process to γwith respect to each\nof its sides N1andN2. Then, we extend these homotopies as in the proof\nof Lemma A.5. By assumption, one of these homotopies lies in N1and\nthe other in N2. Since there is no minimizing geodesic loop in the interior\nofN, these homotopies, put together, yield a one-parameter family of one-\ncycles ztinN, which starts from ∂N1∩∂N, passes through γand ends at\n∂N2∩∂N. Furthermore, the family ( zt) induces a generator of H2(N, ∂N )\nand satisfies µ(zt)≤length( γ). Therefore, length( γ)≥L(N) from the\ndefinition of L(N). □\nRemark A.7. From Lemma A.5, the non-minimizing simple geodesic loops\nof Lemma A.6 lie in only one side of γ.\nUsing the lemma A.6 and the remark A.7, we can define the side of de-\nformation of a geodesic loop and make the curve-deformation process more\nprecise.\nDefinition A.8. The side of deformation of a non-minimizing simple geo-\ndesic loop γofNof length less than L(N) is\n(1) the opposite side with respect to which γis minimizing, if γis min-\nimizing with respect to a side.\n(2) the side which does not contain any geodesic loop minimizing with\nrespect to the side opposed to γ(except the connected components\nof∂N), otherwise.\nFrom now on, when we will apply the curve-deformation process to such\na geodesic loop γ, it will always be with respect to a minimizing arc with\nendpoints in γlying in the deformation side of γ.\nProof of Proposition A.4. Let us assume first that the only minimizing ge-\nodesic loops of Nare the connected components of ∂N. We argue by con-\ntradiction as in the proof of Proposition 4.4. Pick two reals randεsuch\nthat FillRad( M)< r < r +ε <1\n8L(M). Let i:M ,→L∞(M) be the Ku-\nratowski distance preserving embedding defined by i(x)(.) =dM(x, .). By\ndefinition of the filling radius, there exists a map σ:P→Urfrom a 3-\ncomplex Pto the r-tubular neighborhood Urofi(M) inL∞=L∞(M) such24 F. BALACHEFF AND S. SABOURAU\nthat σ|∂P:∂P→i(M) represents the fundamental class of i(M)≃Min\nH2(i(M);Z)≃H2(M;Z). Let us show that the map σ|∂P:∂P→i(M)\nextends to a map f:P→i(M).\nDeforming σand subdividing Pif necessary, we define an extension f:\nP1∪∂P→i(M)≃Mofσ|∂Pto the 1-skeleton of Pas in the proof of\nProposition 4.4. From now on, we will identify i(M),i(N) and i(∂N) with\nM,Nand∂N. Recall that the image under fof the boundary of every\n2-simplex of Pis a simple loop of length less than 3 ρ, where ρ= 2r+ε.\nSince the connected components of ∂Nare minimizing geodesic loops, the\nimages of the edges of Pcut∂Nat most twice. In particular, the images of\nthe boundary of the 2-simplices of Pcut∂Nat most six times.\nOn every edge of Pwhose image cuts ∂N, we introduce new vertices given\nby the preimages of the intersection points with ∂N. Let ∆2be a 2-simplex\nofP. Every pair of new vertices of ∆2which map to the same connected\ncomponent of ∂Ndefines a new edge in ∆2. By definition, the images by f\nof these new edges are the minimizing segments (lying in ∂N) joining the\nimages of their endpoints. We number the vertices of P, old and new, and\nconsider the natural induced order. The new edges of ∆2, which map to N,\nbound a domain D2of ∆2. We introduce new edges on D2joining the\ngreatest vertex of D2to every other vertex of D2. This decomposes ∆2into\ntriangles. By definition, each new edge maps onto a minimizing segment\nofNjoining the images of its endpoints. The triangles of the faces of every\n3-simplex ∆3whose boundaries map to Nbound a domain D3of ∆3. We\nintroduce, as previously, new edges on D3joining the greatest vertex of D3\nto every other vertex of D3. These new edges define new faces in ∆3. These\nnew faces decompose ∆3into 3-simplices. As previously, each new edge\nmaps onto a minimizing segment of Njoining the images of its endpoints.\nIn conclusion, we have defined a new simplicial structure on P, a map\nf:P1∪∂P−→Mwith f|∂P=σ|∂Pand a decomposition of Pinto two\nfinite subcomplexes P′andP′′such that\n(1) the image of the 1-skeleton of every 3-simplex of P′(resp. P′′) lies\nin the closure of N(resp. M\\N).\n(2) the boundary of every 2-simplex of P′maps into a geodesic triangle\nwhose length of the edges is less than3\n2ρ <1\n6L(N)\nWe want to extend ftoP2. Let ∂∆2be the boundary of a 2-simplex ∆2\nofPwhich does not lie in ∂P. If∂∆2lies in P′′, we fill its image by the\ndisk it bounds in M\\N. Otherwise, the image of ∂∆2lies in N. In this\ncase, it converges to a simple geodesic loop cofNby the curve-shortening\nprocess of [Sa04, §2]. We want now to define a µ-nonincreasing homotopy of\none-cycles in Nfrom cto some connected components of ∂N, which gives\nrise to a filling of the image of ∂∆2inM.\nThe images of the boundaries of the 2-simplices of P′converge to simple\ngeodesic loops of N. The set formed of these simple geodesic loops which are\nnot minimizing is noted C. It is finite since P′is a finite simplicial complex.\nWe want to define homotopies of one-cycles in Nfrom every loop of CtoDIASTOLIC AND ISOPERIMETRIC INEQUALITIES 25\nsome minimizing geodesic loops. Let us consider the deformation sides of\ntwo disjoint geodesic loops of C. From Lemma A.5, either one is contained\nin the other or their intersection is disjoint. Therefore, there exists γ0∈ C\nsuch that no geodesic loop of Clies in the deformation side of γ0. The\ncurve-deformation process applied to γ0yields a homotopy of one-cycles zt\ndefined for 0 ≤t≤1 between γ0and the disjoint union γ+\n0∪γ−\n0of two\nsimple geodesic loops. The flow of every other loop of Cis constant for\n0≤t≤1. We repeat this construction with the set of the non-minimizing\ngeodesic loops of {γ|γ∈ C, γ̸=γ0} ∪ { γ+\n0, γ−\n0}. Since for a (generic)\nbumpy metric there are only finitely many geodesic loops of length uni-\nformly bounded, this process stops after a finite number of iterations. From\nLemma A.5, the deformation side of every simple geodesic loop of Nlying\nin the deformation side of a geodesic loop γdoes not contain γ. Therefore,\nthe iterations of this process give rise to µ-nonincreasing homotopies of one-\ncycles between the geodesic loops γofCand the connected components of\n∂Nwhich lie in the deformation side of γ. A contraction of these latter\ninto points in M\\Nyields a filling of the boundary of the 2-simplices of\nP′inM. Furthermore, the homotopies arising from two geodesic loops of C\nwhich do not intersect each other remain disjoint at every time t, except\npossibly for some connected components lying in ∂N. Thus, the sum ztof\nthe homotopies arising from two disjoint geodesics γandγ′ofCsatisfies\nµ(zt)≤length( γ) + length( γ′)≤9ρ.\nIn conclusion, for every 2-simplex ∆2ofP, we get a degree one map\nfrom ∆2onto its image in i(M)≃Mwhich agrees with fon the boundary.\nThis yields the desired extension f:P2−→i(M)≃M.\nLet us extend this map to P3. Let ∆3be a 3-simplex of P. The restriction\noffto the boundary ∂∆3is noted φ:∂∆3−→M. If ∆3lies in P′′, it\nextends to a map defined on ∆3whose image lies in the closure of M\\N.\nOtherwise, ∆3lies in P′and the map φinduces a class [ ∂∆3] inH2(M, M \\\nN)≃H2(N, ∂N ). This class arises from degree one maps defined on the\nfaces of ∆3. Therefore, it is trivial or generates H2(N, ∂N ). The map φ\nextends to ∆3−→Mif and only if the class [ ∂∆3] vanishes in H2(N, ∂N ).\nSince the images of the edges of ∆3are minimizing segments, the image of\nthe 1-skeleton of ∆3is isotopic to one of the following two graphs on M\nshown in Figure 3.\nTherefore, the faces of ∆3decompose into two pairs such that the edges\nof the faces of each pair do not intersect each other transversely. By con-\nstruction, the images of the faces of ∂∆3are defined by µ-nonincreasing\nhomotopies of one-cycles zt\nibetween the images of their boundaries and\nsome connected components of ∂Nor null one-cycles. Furthermore, the\nsum of the two homotopies zt\n1andzt\n2arising from a pair of faces that do not\nintersect each other transversely satisfies µ(zt\n1+zt\n2)≤9ρ. The same goes\nfor the other pair of faces, which yields the homotopies zt\n3andzt\n4. Putting\ntogether the homotopies zt\n1+zt\n2andzt\n3+zt\n4, we obtain a one-parameter\nfamily of one-cycles zt. By construction, the family ( zt) represents [ ∂∆3]\nand satisfies µ(zt)≤9ρ < L (N). Therefore, the class [ ∂∆3] vanishes in26 F. BALACHEFF AND S. SABOURAU\nH2(N, ∂N ).\nThus, the map fextends to each simplex of Pand gives rise to an exten-\nsionf:P−→i(M)≃Mofσ|∂P. This proves the inequality when Nhas\nno minimizing geodesic in its interior.\nSuppose now that Nadmits a minimizing geodesic loop γwhich does not\nlie in ∂N. This simple loop decomposes Ninto two connected components\nN1andN2, whose boundaries are finite unions of minimizing simple geodesic\nloops. By induction on the number of nontrivial minimizing geodesic loops\nwhich lie in the interior of N, we have FillRad( M)≥1\n18L(Ni) for i= 1,2.\nLet us consider some homotopies of one-cycles on Nibetween the connected\ncomponents of ∂Niwith i= 1,2 which induce generators of H2(Ni, ∂N i)\nfori= 1,2. Put together, these homotopies yield one-parameter families of\none-cycles satisfying (C.1-2). We deduce that L(N)≤max{L(N1), L(N2)}.\nThis proves the result by induction. □\nFigure 3.\nReferences\n[Al60] Almgren, F.: The homotopy groups of the integral cycle groups. Topology 1(1960)\n257–299.\n[Al65] Almgren, F.: The theory of varifolds. Mimeographed notes, Princeton, 1965.\n[Ba04] Balacheff, F.: Sur des probl` emes de la g´ eom´ etrie systolique. S´ emin. Th´ eor. Spectr.\nG´ eom. Grenoble 22(2004), 71–82.\n[Ba08] Balacheff, F.: A local optimal diastolic inequality on the two-sphere. Preprint\narXiv:0811.0330.\n[Br86] Brooks, R.: The spectral geometry of a tower of coverings. J. Differential Geom.\n23(1986), 97–107.\n[CC92] Calabi, E.; Cao J.: Simple closed geodesics on convex surfaces. J. Differential\nGeom. 36(1992), 517–549.\n[Ch70] Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Problems\nin analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton\nUniv. 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France 132(2004), no. 1, 105–136.\n[Sa09] Sabourau, S.: Local extremality of the Calabi-Croke sphere for the length of the\nshortest closed geodesic. Preprint arXiv:0907.2223.\n[Tr85] Treibergs, A.: Estimates of volume by the length of shortest closed geodesics on a\nconvex hypersurface. Invent. Math. 80(1985), no. 3, 481–488.\n[YY80] Yang, P.; Yau, S.-T.: Eigenvalues of the Laplacian of compact Riemann surfaces\nand minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7(1980), no. 1,\n55–63.\nFlorent Balacheff - Universit ´e des Sciences et Technologies, Laboratoire\nPaul Painlev ´e, Bˆat. M2, 59655 Villeneuve d’Ascq, France\nEmail address :Florent.Balacheff@math.univ-lille1.fr\nSt´ephane Sabourau - Universit ´e Franc ¸ois Rabelais, Tours. Laboratoire\nde Math ´ematiques et Physique Th ´eorique. CNRS, UMR 6083. F ´ed´eration de\nRecherche Denis Poisson (FR 2964). Parc de Grandmont, 37400 Tours, France\nEmail address :sabourau@lmpt.univ-tours.fr"    },    {        "title": "2402.01570v1.Controllable_frequency_tunability_and_parabolic_like_threshold_current_behavior_in_spin_Hall_nano_oscillators.pdf",        "content": "Controllable frequency tunability and parabolic-like threshold current behavior in\nspin Hall nano-oscillators\nArunima TM1and Himanshu Fulara1,∗\n1Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India.\n(Dated: February 5, 2024)\nWe investigate the individual impacts of critical magnetodynamical parameters—effective mag-\nnetization ( µ0Meff) and magnetic damping ( α)—on the auto-oscillation characteristics of nano-\nconstriction-based Spin Hall Nano-Oscillators (SHNOs). Our micromagnetic simulations unveil a\ndistinctive non-monotonic relationship between current and auto-oscillation frequency in out-of-\nplane magnetic fields. The influence of effective magnetization on frequency tunability varies with\nout-of-plane field strengths. At large out-of-plane fields, the frequency tunability is predominantly\ngoverned by effective magnetization, achieving a current tunability of 1 GHz/mA—four times larger\nthan that observed at the lowest effective magnetization. Conversely, at low out-of-plane fields,\nalthough a remarkably high-frequency tunability of 4 GHz/mA is observed, the effective magnetiza-\ntion alters the onset of the transition from a linear-like mode to a spin-wave bullet mode. Magnetic\ndamping primarily affects the threshold current with negligible impact on auto-oscillation frequency\ntunability. The threshold current scales linearly with increased magnetic damping at a constant out-\nof-plane field but exhibits a parabolic behavior with variations in out-of-plane fields. This behavior\nis attributed to the qualitatively distinct evolution of the auto-oscillation mode across different\nout-of-plane field values. Our study not only extends the versatility of SHNOs for oscillator-based\nneuromorphic computing with controllable frequency tunability but also unveils the intricate auto-\noscillation dynamics in out-of-plane fields.\nI. INTRODUCTION\nSpin Hall nano-oscillators (SHNOs) [1–9] are promi-\nnent spintronic devices that leverage the spin Hall effect\n(SHE) phenomenon [10–12] within the heavy metal layer\ntoconvertalateraldirectchargecurrentintoatransverse\npure spin current. This spin current then drives mag-\nnetization auto-oscillations in nanoscopic regions within\nthe adjacent ferromagnetic layer through the transfer of\nspin angular momentum [1, 4, 6, 13]. Recently, SHNOs\nhave attracted significant attention due to their advan-\ntages in nanofabrication, rapid and ultra-broadband mi-\ncrowave frequency tunability, compatibility with com-\nplementary metal-oxide-semiconductor (CMOS) technol-\nogy, and emerging applications in neuromorphic comput-\ning and magnonics [4, 5, 13–18]. Notably, among var-\nious SHNO device geometries, nano-constriction-based\nSHNOs offer distinct advantages, including flexible de-\nvice layouts, direct optical access to auto-oscillating re-\ngions, and voltage control of the constriction region [4, 6,\n8,19–21]. Theirhighlynon-linearpropertiesandin-plane\ncurrent flow facilitate the mutual synchronization of mul-\ntiple SHNOs in both linear chains and two-dimensional\narrays,enablinghigh-qualitymicrowavesignalgeneration\nand scaling neuromorphic computing to large oscillator\nnetworks [6, 19, 20, 22]. The precise control of auto-\noscillation frequency over a wide range plays a pivotal\nrole in controlling the dynamical couplings between oscil-\nlators, offering a potential avenue for facilitating synap-\ntic communication among artificial neurons. To achieve\n∗himanshu.fulara@ph.iitr.ac.inlearning in oscillator-based neuromorphic computing, ro-\nbust frequency tunability is crucial, especially at higher\noperating frequencies [23–25]. The agility to control\nfrequency tunability is also beneficial for communication\nsystems where specific frequency bands are allocated for\ndifferent applications. The degree of frequency tunabil-\nity depends on magnetodynamical properties, drive cur-\nrent, and the orientation and strength of the applied\nmagnetic field [26, 27]. In a recent study, Haidar et\nal. [28] reported the impact of NiFe alloy composition on\nthe auto-oscillation properties of nano-constriction-based\nNi100-xFex/Pt SHNOs under in-plane fields. The study\nrevealed that Fe-rich devices require higher current den-\nsities for driving auto-oscillations at higher frequencies,\nraising a fundamental question about how damping and\nsaturation magnetization independently influence these\nauto-oscillation properties. Despite numerous studies on\ntheauto-oscillationbehaviorofnanoconstrictionSHNOs,\na comprehensive understanding of the independent roles\nof key magnetodynamical parameters, such as effective\nmagnetization ( µ0Meff) and magnetic damping ( α) on\nthe auto-oscillation properties remains elusive.\nIn this study, we systematically investigate the in-\ndividual impacts of effective magnetization and mag-\nnetic damping on the magnetization auto-oscillations of\nSHNOs. Our observations reveal qualitatively similar\nauto-oscillation behavior for different effective magne-\ntization values at a fixed out-of-plane field. However,\nthe frequency tunability controlled by effective magne-\ntization exhibits significant variations at different out-\nof-plane fields. Conversely, magnetic damping primarily\ninfluences the threshold current with minimal impact on\nfrequency tunability. Additionally, the threshold current\nof auto-oscillation at different out-of-plane fields exhibitsarXiv:2402.01570v1  [cond-mat.mes-hall]  2 Feb 20242\nFIG. 1. Device schematic, current density, Oersted field,\nand demagnetizing field (a) Schematic of a nano-constriction-\nbased Pt(6 nm)/Pt(5 nm) spin Hall nano-oscillator (SHNO),\nwith a constriction width of w= 150 nm. (b) Distribution of\nlateral charge current density in the Pt/Py nano-constriction\nfor an input current, Idc= 1 mA. The inset illustrates the\nsimulated current density in the individual Pt and Py layers.\n(c) Estimation of the out-of-plane component of the Oersted\n(Oe) field in the Py layer for Idc= 1 mA. (d) Distribution of\ndemagnetizing field in the Py layer for Idc= 0 mA.\nanintriguingparabolic-likebehavior, indicativeofaqual-\nitatively different spatial evolution of the auto-oscillating\nmode within distinct field energy landscapes.\nII. MICROMAGNETIC SIMULATIONS\nHere, we simulate a Pt(6 nm)/Py(5 nm) nano-\nconstriction-based SHNO with an optimized constriction\nwidthof150nm, asillustratedschematicallyinFig. 1(a).\nThe device geometry is initially modeled using the COM-\nSOL [29] Multiphysics Software. Simulations consider\nthe full-scale Pt/Py bilayer with an input current, Idc\n= 1 mA, and utilize resistivity values of 32.6 µΩ-cm for\nPy and 11.2 µΩ-cm for Pt, as reported in experimental\nstudy [4]. Figure 1(b) shows the charge current density\ndistribution along the y-axis in a 150 nm SHNO obtained\nfrom COMSOL simulations. Notably, there is a markedincrease in local current density at the constriction re-\ngion. Due to the disparate electrical resistances of the Pt\nand Py layers, the current predominantly flows through\nthe Pt metal, as illustrated in the inset of Fig. 1(b). As a\nresult, in the micromagnetic simulations, we neglect the\ncontribution of the charge current passing through the\nPy layer to influence the magnetization dynamics. Given\nthat the spin current injected into Py is directly propor-\ntional to the current density in Pt, JS= Θ SHJC, the\nnano-constriction region, characterized by a large cur-\nrent density, defines the active device area, wherein the\ninjected spin current is large enough to excite magneti-\nzation dynamics. Figure 1(c) shows the simulated out-\nof-plane component of the current-induced Oersted (Oe)\nfield profile in the Py layer for an input current, Idc= 1\nmA. Using MUMAX3 [30], we then modeled a NiFe layer\ncharacterized by an exchange stiffness ( Aex) of 10 pJ/m\nand a gyromagnetic ratio ( γ/2π) of 29.53 GHz/T [6]\nwithin a mesh area measuring 2000 x 2000 nm ², divided\ninto a grid size of 512 x 512. The externally applied mag-\nnetic field has fixed in-plane ( ϕ) and out-of-plane angles\n(θ) of 24 and 80 degrees, respectively.\nThe spatial distribution of demagnetizing fields in the\nnano-constriction area is illustrated in Fig. 1(d), ob-\ntainedthroughmicromagneticsimulationsusingtheMU-\nMAX3 solver [30]. The calculations reveal that the nano-\nconstriction edges produce an Oe field and a demagne-\ntizing field opposing the external static magnetic field,\nleading to a localized decrease in the internal field within\nthe nano-constriction area of the Py film. Our methodol-\nogy involves running current sweep simulations to inves-\ntigate the influence of independently varying Meffand\nαon the magnetization auto-oscillation frequency under\nboth lower and higher applied magnetic fields. Subse-\nquently, Fast Fourier Transform (FFT) calculations were\nappliedtothetime-dependentandspace-dependentmag-\nnetization data to extract frequencies and spatial profiles\nof the auto-oscillating modes.\nIII. MAGNETODYNAMICAL NON-LINERAITY\nTheintricatenon-linearmagneto-dynamicswithinpat-\nterned magnetic thin films can be analytically character-\nized by a non-linearity coefficient, denoted as N[31].\nThe magnitude and sign of Nplay a pivotal role in de-\ntermining the strength and nature of magnon-magnon\ninteractions, where positive and negative values corre-\nspond to magnon repulsion and attraction, respectively\n[27, 31]. A negative non-linearity leads to a reduction\nin the auto-oscillation frequency with increasing ampli-\ntude, ultimately causing it to enter the spin-wave band\ngap. This leads to the self-localization of spin-waves,\ntriggering the excitation of solitonic modes such as spin-\nwave bullets in in-plane magnetized films [32, 33], and\nmagnetic droplets in films exhibiting very large perpen-\ndicular magnetic anisotropy (PMA) [34, 35]. Conversely,\nlarge positive non-linearity results in an increase in the3\nFIG. 2. Contour plot illustrating the analytically determined\nnon-linearity coefficient ( N) for a thin ferromagnetic film,\nplotted as a function of saturation magnetization and exter-\nnally applied out-of-plane magnetic field ( θ= 80◦). The dot-\nted black line shows the N= 0 region. The black arrow\nsignifies the enhancement in the precession amplitude with\napplied current, resulting in an effective reduction of MSvia\nthe projection of magnetization ( Mβ=MScosβ) along the\nstatic field direction. Here, βrepresents the precession angle\nof magnetization.\nauto-oscillationfrequencywithamplitude, surpassingthe\nferromagnetic resonance (FMR) frequency and leading to\nthe excitation of propagating spin waves[17, 36, 37].\nThe magnetodynamical non-linearity is primarily gov-\nernedbymaterialparameterssuchassaturationmagneti-\nzation ( MS), magnetic anisotropy, and the strength and\norientation of applied magnetic fields. To analytically\ncalculate the nonlinear coefficient Nfor an in-plane mag-\nnetized film, shown in Fig. 2, we adopted the methodol-\nogy outlined in Ref. [31, 38]. The results are plotted as a\nfunction of MSand externally applied out-of-plane mag-\nnetic fields ( θ= 80◦). As shown in Fig. 2, Npredomi-\nnantly exhibits negative values (depicted by red regions)\nat low applied magnetic fields, regardless of variations in\nMS. However, at higher applied fields, Nundergoes a\nmonotonic transition from negative (red regions) to pos-\nitive values (illustrated by blue regions), depending on\nthe strength of MS. Notably, it crosses through zero at\na specific field strength (indicated by the black dotted\nline), the precise location of which is influenced by MS.\nIn other words, the reduction in MSshifts the point of\nzero non-linearity towards lower values of applied fields.\nTailoringnon-linearitythroughthemanipulationofeffec-\ntive magnetization and applied magnetic fields in SHNOs\nshould, in principle, offer precise control over frequency\ntunability and magnetization auto-oscillation dynamics.IV. RESULTS AND DISCUSSION\nFigure 3(a-h) presents micromagnetically simulated\ncurrent sweep auto-oscillations under two qualitatively\ndifferentoperatingregimesforfourdistincteffectivemag-\nnetization values, which are practically tunable through\nthe optimization of Fe composition in the Ni 100-xFex\nlayer [28]. The objective was to discern the influence\nof effective magnetization on auto-oscillation properties\nwhile maintaining other magnetodynamical parameters\nconstant. In Fig. 3(a-d), at the low applied field of µoH\n= 0.3 T, the auto-oscillations display a typical redshifted\nfrequency vs. current behavior, accompanied by a lin-\near rise in threshold frequency with effective magnetiza-\ntion. A similar red-shifted frequency behavior is experi-\nmentally observed in earlier studies on SHNOs [33, 39].\nThe frequency of auto-oscillations remains nearly con-\nstant at low current values, but at a specific effective\nmagnetization-governed current, there is an abrupt jump\nin the frequency indicative of the so-called spin-wave bul-\nlet—a non-topological self-localized mode that emerges\nin large negative non-linearity regions [27, 31, 33], as\nshown in Fig. 2.\nIn the presence of large out-of-plane field of µoH=\n0.7 T, as shown in Fig. 3(e-h), auto-oscillations exhibit a\nnonmonotonic frequency vs. current behavior, character-\nized by a red-shifted frequency at lower currents followed\nbyablue-shiftedoneathighercurrentvalues[6,27]. This\ndistinct non-monotonic frequency behavior has been ex-\ntensively reported in recent experimental investigations\non SHNOs [6, 17, 39–41]. For in-plane magnetized Py\nthin films, the non-linearity at high applied fields is pre-\ndominantly negative, and therefore, the auto-oscillation\nfrequency initially experiences a redshift with increasing\ndrive current [27, 40]. As the drive current increases, the\nprecession amplitude increases, and the demagnetizing\nfield along the static field direction reduces, leading to a\nsituation where the non-linearity becomes zero (refer to\nFig. 2). Consequently, the redshifting of the frequency\nstops, and the auto-oscillation frequency passes through\na minimum where the non-linearity is zero. With any\nfurther increase in the drive current, the non-linearity\nbecomes positive, inducing a blue shift in the auto-\noscillation frequency [17, 27, 40]. Similar to the low-field\nauto-oscillation behavior, this nonmonotonic frequency\nresponseathigherfieldsremainsessentiallyunchangedin\nnature across all four distinct µ0Meffvalues, with only a\nminor increase in onset auto-oscillation frequency. How-\never, the frequency tunability demonstrates a significant\nboost at higher effective magnetization values, achiev-\ning a fourfold increase in current tunability at µ0Meff=\n0.8 T. As illustrated in Fig. 3(i), at low fields, effective\nmagnetization primarily influences the onset of the bullet\nmode by significantly enhancing negative non-linearity\nwhile moderately adjusting frequency tunability, reach-\ning a maximum current tunability of 4 GHz/mA. Con-\nversely, at high fields, frequency tunability experiences\na fourfold increase with increasing effective magnetiza-4\nFIG. 3. Power spectral density obtained from micromagnetically simulated data, illustrating its variation with current for\nfour distinct effective magnetization values under out-of-plane magnetic fields ( θ=80◦,φ=24◦): (a-d) µoH= 0.3 T, (e-h) µoH\n= 0.7 T. Comparison of auto-oscillation frequency tunability ( d f/dI) as a function of drive current for four distinct effective\nmagnetization values at (i) µoH= 0.3 T, (j) µoH= 0.7 T\ntion values (refer to Fig. 3(j)). Our results indicate the\npivotal role of effective magnetization in tailoring the\nmagnetodynamic non-linearity at the active dynamical\nregion, thereby influencing both the frequency tunability\nand the onset of the bullet mode.\nSubsequently, we explore the critical influence of\nmagnetic damping on the auto-oscillation properties of\nSHNOs, while maintaining a fixed effective magneti-\nzation value of µ0Meff= 0.74 T. This investigation\nis confined to high out-of-plane fields, µoH= 0.7 T\n(θ=80◦,φ=24◦), where significant variations in effective\nmagnetization-controlled frequency tunability were ob-\nserved (refer to Fig. 3(j)). In Fig. 4(a-d), we present\ncurrent sweep auto-oscillations at four distinct magnetic\ndamping values ranging from α= 0.02 to 0.05. As shown\nin Fig. 4(e), magnetic damping predominantly influ-\nences the threshold current and onset frequency, with\nnegligible impact on auto-oscillation frequency tunabil-\nity. Thequalitativelysimilarfrequencytunabilityimplies\nthat the fundamental nature of auto-oscillations remains\nunaffected by variation in damping. As illustrated in theinset of Fig. 4(e), the onset frequency demonstrates a\nlinear decrease with an increase in α, while the threshold\ncurrent increases with increasing αvalues. Note that we\nestimated the threshold current by monitoring the mag-\nnetization behavior over time. The initial 5 ns were ex-\ncluded to mitigate transient effects, and the subsequent\n15 ns were observed to detect the magnetization auto-\noscillations. If the auto-oscillations sustain over time,\nwe identify it as the onset of auto-oscillation. The ob-\nserved enhancement in threshold current is slightly below\nthe anticipated value of 2.5 times, indicating that the in-\ncrease in damping is linked with the stronger localization\nof auto-oscillating mode due to a higher Oe field. The Oe\nfield effectively suppresses the spin-wave well on one side\nof the nano-constriction while enhancing the localization\nof the mode on the opposite side [27].\nTo delve deeper into the impact of magnetic damping\nacross various operational regimes, micromagnetic simu-\nlations were conducted, maintaining a constant effective\nmagnetization of µ0Meff= 0.74 T while varying out-of-\nplane magnetic fields ( µ0H). Figure 5(a) summarizes5\nFIG. 4. Micromagnetically simulated auto-oscillation fre-\nquency vs. drive current for four damping values at a fixed\nµ0Meff= 0.74 T : (a) α= 0.02, (b) α= 0.03, (c) α= 0.04,\n(d)α= 0.05. (e) Comparison of auto-oscillation frequencies\nas a function of drive current at various damping magnitudes.\nThe inset shows the variation of threshold frequency (blue\nsquares) and threshold current (black squares) as a function\nof damping, with red lines indicating linear fits.\nthe variation in threshold currents as a function of out-\nof-plane field strengths at three distinct damping levels.\nThe plots of Ithversus µ0Hexhibit a parabolic-like non-\nmonotonic trend for all three damping values. In general,\nthe threshold current ( Ith) is expected to increase mono-\ntonically with the applied magnetic field strength [27].\nHowever, in our case, an intriguing deviation is observed,\nwherein a reduction in Ithoccurs at intermediate field\nstrengths. This reduction reaches a minimum value be-\nfore the threshold current begins to increase at strong\nout-of-plane fields. A similar parabolic-like behavior has\nbeen reported by Yin et al. [42] in Py 100-x-yPtxAgy/Pt-\nbased nanoconstriction SHNOs. This intriguing non-\nmonotonic behavior observed in the field dependence of\nthreshold current underlines a more in-depth investiga-\ntion of the dynamics of auto-oscillating modes in the con-\nstriction region.\nTo elucidate this behavior, we performed simulations\nof current sweep auto-oscillations at three different oper-\nating regimes depicted in Fig. 5(a). Figures 5(b-d) re-\nveal three distinct characteristics of auto-oscillation fre-\nquency vs. current, highlighting the distinct evolution of\nFIG. 5. (a) Variation of threshold current ( Ith) as a function\nof applied out-of-plane fields ( θ=80◦,φ=24◦), calculated for\nthree different damping values. Micromagnetically simulated\ncurrentsweepauto-oscillationsfor α=0.03subjectedtothree\ndistinct out-of-plane field strengths: (b) µoH= 0.1 T, (c)\nµoH= 0.4 T, and (d) µoH= 0.8 T. The inset illustrates the\nspatial profiles of auto-oscillations simulated at the onset of\nauto-oscillations for each field strength.\nauto-oscillating modes. In weak fields, such as µ0H=\n0.1 T, the auto-oscillation frequency exhibits negligible\nchange with current, and the mode remains quasi-linear,\nas illustrated in Fig. 5(b). The spatially simulated pro-\nfile, depicted as the inset of Fig. 5(b), confirms that the\nmode originates at the constriction edges due to inhomo-\ngeneous current distributions, as shown in Fig. 1(b). A\nsmall negative non-linearity (refer to Fig. 2) governs this\nquasi-linear mode at weak out-of-plane fields [17, 27, 31].\nAs the applied field strength increases, the non-linearity6\nbecomes more negative (see Fig. 2), leading to a stronger\nlocalization of the auto-oscillating mode at the constric-\ntion edges [31, 43]. Consequently, the mode undergoes a\ntransition from a quasi-linear behavior to a strongly lo-\ncalized spin-wave bullet behavior, resulting in an abrupt\ndrop in frequency, as shown in Figure 5(c). With a fur-\nther increase in the field strength, the non-linearity ini-\ntially increases from a negative value at low currents,\npasses through a zero value, and eventually transitions\nto a positive value at higher currents [27, 39] (refer to\nFig. 2). This behavior arises due to the magnetization\nvectorundergoingprecessionatanincreasinglylargeran-\ngle as the current magnitude rises. Consequently, there\nis an effective reduction of MSthrough the projection of\nmagnetization ( Mβ=MScosβ) along the applied field\ndirection [17, 44], resulting in a shift of the non-linearity\nfrom a negative value to a positive one, as illustrated by\nblack arrow in Fig. 2. This transition occurs as the auto-\noscillatingmodedetachesfromtheconstrictionedgesand\ntransforms into a bulk type. As shown in the inset of\nFig. 5(d), the auto-oscillating mode shifts inwards into\nthe interior of the constriction and expands away from\nthe constriction as the current increases.\nTo look forward, controllable frequency tunability in\nSHNOs can enhance their adaptability and versatility,\nmaking them suitable for a wide range of practical ap-\nplications in communication, adaptive signal processing,\ninformation storage, and computing [8, 20, 24, 45]. De-\npending on the desired functionality, the SHNOs can\nbe tuned to operate in different frequency ranges (re-\nfer to Fig. 3), enabling their use in a variety of appli-\ncations within a single device. In the realm of neuro-\nmorphic computing, the ability to adjust the oscillation\nfrequency of SHNOs becomes particularly valuable. This\ntunability can be leveraged to emulate synaptic plastic-\nity, where frequency modulation represents variations in\nsynaptic strength [24, 25]. This, in turn, facilitates the\nimplementation of learning and memory functions within\nneuromorphic circuits. Moreover, the tunable frequen-\ncies of SHNOs enable them to replicate the dynamic be-\nhavior of neurons in response to changing input condi-\ntions. The capability to dynamically adjust frequencieson the fly allows for swift modifications to neural connec-\ntions, facilitating continuous learning in neuromorphic\nsystems. With controllable frequency tunability, SHNOs\nexhibitthecapabilitytodynamicallyadapttheirfrequen-\ncies based on the computational task at hand. This not\nonly optimizes energy consumption but also contributes\nto the overall efficiency of the system.\nV. CONCLUSION\nIn summary, we have studied the individual influence\nof key magnetodynamical parameters, namely effective\nmagnetization ( µ0Meff) and magnetic damping ( α), on\nthe auto-oscillation characteristics of nano-constriction-\nbased SHNOs. Our findings reveal a significant vari-\nation in the impact of effective magnetization on fre-\nquency tunability, particularly in response to applied\nfield strengths. Conversely, magnetic damping primarily\ninfluences the threshold current, exerting minimal effects\non frequency tunability. We observed a linear increase\nin the threshold current with increasing magnetic damp-\ning under a constant magnetic field. However, when ex-\nploring the interplay of threshold current with applied\nfield strengths, a distinct parabolic-like trend emerges.\nThis behavior suggests qualitatively different origins of\nauto-oscillating modes across different field landscapes,\nproviding valuable insights into the intricate magneto-\ndynamics of SHNOs in nano-constriction geometry. Our\nstudy not only contributes to oscillator-based neuromor-\nphiccomputingthroughcontrollablefrequencytunability\nbutalsoprovidesvaluableinsightsintotheintricateauto-\noscillation dynamics of nano-constriction-based SHNOs.\nAcknowledgements\nThis work is supported by the Faculty Initiation Grant\n(FIG) sponsored by SRIC, IIT Roorkee. We thankfully\nacknowledge the Institute Computer Center (ICC), IIT\nRoorkee for providing a high-end computational facility\nto run simulations.\n[1] V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich,\nA.Slavin,D.Baither,G.Schmitz, andS.O.Demokritov,\nNature Mater. 11, 1028 (2012).\n[2] L. 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Goldman1, 3,†\n1Center for Neuroscience, and Department of Neurobiology, Physiology,\nand Behavior, University of California, Davis, Davis, CA 95616, USA\n2Institute for Computational Biomedicine and Department of Physiology and Biophysics,\nWeill Cornell Medical College, New York, NY 10021, USA\n3Department of Ophthalmology and Vision Science,\nUniversity of California, Davis, Davis, CA 95616, USA\n1arXiv:2402.01605v1  [q-bio.NC]  2 Feb 2024Abstract\nThe nervous system reorganizes memories from an early site to a late site, a commonly observed\nfeatureoflearningandmemorysystemsknownassystemsconsolidation. Previousworkhassuggested\nlearning rules by which consolidation may occur. Here, we provide conditions under which such\nrules are guaranteed to lead to stable convergence of learning and consolidation. We use the theory\nof Lyapunov functions, which enforces stability by requiring learning rules to decrease an energy-like\n(Lyapunov) function. We present the theory in the context of a simple circuit architecture motivated\nby classic models of learning in systems consolidation mediated by the cerebellum. Stability is\nonly guaranteed if the learning rate in the late stage is not faster than the learning rate in the\nearly stage. Further, the slower the learning rate at the late stage, the larger the perturbation the\nsystem can tolerate with a guarantee of stability. We provide intuition for this result by mapping\nthe consolidation model to a damped driven oscillator system, and showing that the ratio of early-\nto late-stage learning rates in the consolidation model can be directly identified with the (square of\nthe) oscillator’s damping ratio. This work suggests the power of the Lyapunov approach to provide\nconstraints on nervous system function.\nI. INTRODUCTION\nSystems consolidation is the process of transferring learned memories from an early-stage\nsite to a late-stage site [ 1–3] and has been suggested theoretically to enhance the ability of\nmemory systems to simultaneously learn new associations while protecting previously learned\nmemories from being overwritten [ 2,4]. Various forms of memories undergo consolidation in\ndifferent brain areas. For example, declarative memories initially learned in the hippocampus\nget transferred to the neocortex [ 2,5]. Motor memories initially located in the cerebellar\ncortex [6] or the basal ganglia [ 7] get transferred out of the early learning site into direct motor\npathways. Furthermore, strong evidence suggests that fear-based memories initially learned\nin the amygdala later get transferred to a different site [ 3,8]. Understanding how neural\nsignals and learning rules orchestrate a successful memory transfer requires guiding principles\nto shed light on the interactions of brain areas and their plasticity rules. Here we develop a\nLyapunov theory that provides a first-principles account for the speed of consolidation and\nthe robustness of the consolidation process.\nNeural circuits underlying learning face a fundamental challenge common to many biolog-\nical and engineered dynamical systems with adaptively tunable parameters: the concurrent\npresence of time-varying inputs, states, and parameters may cause the dynamics to become\nunstable, for example, by growing unboundedly or falling into undesirable oscillatory patterns.\nIn addition, the nervous system abounds with various forms of noise and disturbances [ 9],\nwhich may take the system into undesirable regimes. Thus, not only should the final desired\nsolution of learning be stable, but also the overall system should remain stable throughout\nthe process of learning.\nThe theory of adaptive control systems has been successful in providing essential tools,\nsuch as the Lyapunov function formalism, for guaranteeing the stability of learning systems\n[10]. The concept of a Lyapunov function has been used for quantifying the stability of\nadaptive recurrent neural networks [ 11–13] as well as for building discrete attractor neural\n∗alemi@ucdavis.edu\n†msgoldman@ucdavis.edu\n2networks to model long-term memory [ 14,15]. Here, we apply the Lyapunov formalism to\nthe problem of guaranteeing stable systems consolidation. Systems consolidation, like many\nlearning processes, contains feedback loops between the training signals and neural dynamics\nthat drive learning, and the weight changes that drive system dynamics. Such feedback can\nmake learning prone to instability. This may be further exacerbated by the fact that, in\nbiological systems, many synapses do not have direct access to the ground-truth performance\nerror and must therefore learn from indirect error signals.\nWe place our theory in the context of a simple circuit architecture of systems consolidation\nin which the late stage, unlike the early stage, lacks direct access to an error-correcting\nteaching signal (Fig. 1). In this architecture, the activity at this early site then trains the\nweight of the late site such that the memory is eventually transferred from the early to\nthe late site. This architecture and set of learning rules correspond to classic theories of\nlearning and systems consolidation in the cerebellum. Learning at the early site corresponds\nto the classic Marr-Albus-Ito theory of cerebellar learning [ 16,17] in which an error signal\nconveyed by climbing fibers trains the cerebellar output conveyed by Purkinje cells. These\nPurkinje cell firing rates then serve as a secondary teaching signal for the late site located in\nthe cerebellar output nuclei. Applying the formalism of Lyapunov stability theory, we find\nthat, to guarantee stability, plasticity at the late-stage site must not be updated faster than\nplasticity at the early-stage site. The slower the tuning in the late stage, the more robust\nthe learning process is against noise in the primary teaching signal.\nII. RESULTS\nA. Model for systems consolidation\nWe consider a simple toy model of systems consolidation motivated by the basic circuitry\nthought to be involved in systems consolidation of cerebellum-mediated learning. The task\nis to track a given time-varying input command rin(t)and to generate a desired output\nr∗\no(t) =w∗rin(t)where the desired input-to-output gain is denoted as w∗. The model, shown\nin Fig. 1, has an architecture with two pathways: a direct pathway with gain w2and an\nindirect pathway with gain w1that provides a learned, online correction to the output of the\ndirect pathway. The output of the model can be written as\nro(t) = (w1+w2)rin(t) (1)\nwhere ro(t)is the output. The goal of learning is two-fold: (1) to reduce the gain error\nfW=w1+w2−w∗to zero, and (2) to consolidate the weight changes into the late-stage\nweight such that asymptotically w1= 0andw2=w∗. The indirect pathway receives the\ninformation about the error signal and constitutes the early site of plasticity. The direct\npathway constitutes the late site of plasticity and needs to be tuned based on the information\nreceived from the early stage, i.e., r1=w1rin.\nThe learning rule in the early stage is a supervised delta-like correlational rule proportional\nto the product of the input and the teaching signal [18, 19].\n˙w1=−η1rin(t)(e(t) +ξ(t)) (2)\nwhere the teaching error signal e(t)is the tracking error e(t) =ro(t)−r∗\no(t) = (w1+w2−\nw∗)rin(t),η1is the learning rate of the early-stage tuner, and ξ(t)is a perturbation to the\n3error signal. We assume the perturbation is signal-dependent [ 20,21], i.e., the strength of\nthe perturbation depends on the amount of error signal e(t). We formalize the regime of\nthe perturbation in the teaching signal during learning by a parameter µ=max|ξ(t)|\n|e(t)|(where\ne(t)̸= 0) that defines the maximal amount of perturbation during learning.\nThe learning rule in the late stage is a heterosynaptic correlational rule between the online\ncorrective signal r1(t)provided by the early stage, and the direct input to the late stage,\nrin(t)[22]:\n˙w2=η2rin(t)r1(t) (3)\nwhere η2is the learning rate.\n00.10.2Signals00.10.2w10 5000 10000 15000Time (sec)00.10.2w200.10.2Signals00.10.2w10 5000 10000 15000Time (sec)00.10.2w200.10.2Signals00.10.2w10 5000 10000 15000Time (sec)00.10.2w2Fixed !!̇!\"∝−%#$&Output%%=!\"+!!%#$Command %#$1Single-stage tuning%\"=!\"%#$+Plant+−&=%%−%∗̇!\"∝−%#$&Early-stage tuner (!\")%%=!\"+!!%#$%#$1Two-stage tuning\n+PlantLate-stage tuner (!!)%\"=!\"%#$̇!!∝%\"%#$+−AB\nSlow consolidationFast consolidationEarly-stage tuner (!\")\nDesired %∗Output %%!∗%∗=!∗%#$!∗%∗=!∗%#$&=%%−%∗10 s0.02\n10 s0.02\nFIG. 1. A toy tracking model demonstrating instability in systems consolidation. (A) Top: Single-\nstage model. The early-stage parameter, w1, is directly tuned by the error signal e(t), whereas\nthe late-stage parameter, w2, is fixed. Bottom: Simulation showing that the model successfully\nconverges to the desired output r∗and the desired, tuned weight w1(dashed line). (B) Two-stage\ntuning model. Top: The early stage is as in the single-stage model. The late-stage weight w2\nis tuned using the output of the early stage as a secondary teaching signal. Bottom: When the\nconsolidation process is slow enough (left panels), the model dynamics successfully converges and\ntunes the weight w2to its desired value (dashed line). However, if the consolidation process becomes\ntoo fast (right panels), the system can show instability. See Appendix A for simulation details.\nWe seek a general framework to characterize the conditions for stability and convergence\nof this two-stage learning system. More specifically, we seek criteria for tuning the system\nto avoid undesired behaviors of the system, such as oscillation or unbounded growth. By\nstability, we intuitively mean that, as time goes to infinity, the system is well-behaved;\n4for example, it should reach its goal and stay close to it. The mathematical definition of\nLyapunov stability and its various refined notions are provided in Appendix B.\nB. Stability of the single-stage model\nTo illustrate the Lyapunov stability approach and motivate the problem of instability in\nsystems consolidation, we first consider a one-stage model (Fig. 1A, top) in which learning\noccurs only at the early stage of the circuit (i.e., we setdw2\ndt= 0). In this case, so long as\nthe magnitude of the perturbation ξ(t)does not exceed the magnitude of the error signal\ne(t), the learning is guaranteed to stably converge. This is illustrated for the task of learning\nto track a step-like input in Fig. 1A (bottom) and proven for the more general case in the\nfollowing paragraphs.\nThe basic idea behind Lyapunov’s direct approach to stability is based on constructing\na scalar function and showing that all trajectories of the dynamics of the system decrease\nthis scalar function, guaranteeing that the system safely and stably reaches the fixed point\nof the dynamics. This scalar function, known as a Lyapunov function, can be considered a\ngeneralization of the concept of energy in classical mechanics. Finding one such function\nis enough to prove the stability of the system. To provide intuition about the Lyapunov\nfunction formalism, we first apply it to the simple single-stage tuning model given by Eq. 2.\nTo prove the uniform global stability of the model, we use the Lyapunov theorem 1\nfor non-autonomous systems, and to prove asymptotic stability, we use the Lyapunov-like\nlemma 1 (Appendix B). To prove the uniform global stability, we need to find an appropriate\nLyapunov function candidate Lsuch that 1) Lis positive definite, 2) the time derivative\nofLis˙L≤0, 3)Lis decrescent, and 4) Latt= 0is radially unbounded. We propose the\nfollowing scalar function of the error in the gain fW:\nL1=1\n2fW2, (4)\nwhere fW=w1+w2−w∗.L1is positive definite and radially unbounded by inspection.\nTo show that ˙L1≤0, we write the learning dynamics as ˙w1=−η1fWr2\nin−η1rinξ. Using\nthe learning rule to compute the time derivative yields ˙L1=−η1fW2r2\nin−η1fWrinξ. With\n|ξ| ≤µ|e|, one obtains:\n˙L≤ −η1fW2r2\nin+η1|fWrin||ξ|\n≤ −η1fW2r2\nin+η1|fWrin|µ|fWrin|\n=−η1r2\nin(1−µ)fW2.(5)\nη1>0by definition, and as long as µ≤1, we have ˙L1≤0. Ifrin= 0, we are in a trivial\ncase where ˙L1= 0and ˙w1= 0and the system is not changing at all. We assume rin̸= 0\nthroughout most of the learning period. Therefore, the equilibrium is globally stable. This\nstability guarantees that w1andfWare bounded. Since L1does not explicitly depend on\ntime and is positive definite, it is decrescent; therefore, the equilibrium is uniformly globally\nstable. To find conditions for guaranteeing asymptotic stability, i.e., as t→ ∞,w1→w∗,\nfrom Lemma 1 what is left to show is that ˙L1is uniformly continuous in time. A practical\nway of showing uniformity is to show that ¨L1is bounded. Given that the hyperparameters\nη1, η2and the gain error fWare bounded, and rinandξare assumed to be smooth functions\nof time with bounded derivatives, ¨L1is bounded. ■\n5C. Lyapunov function theory for the two-stage consolidation model\nWe next consider systems consolidation in the two-stage model. As proven below, when the\nconsolidation process is sufficiently slow, the two-stage learning model successfully converges\nto solving the tracking task (Fig. 1B, bottom left). However, when the consolidation\nprocess is too fast, the system can exhibit instability in which a small perturbation can\ncause large perturbations in the output (Fig. 1B, bottom right). Below, we analytically find\nthe conditions for guaranteeing stability (Section IIC1), provide intuition for the source of\npossible instability in regions without stability guarantee by solving a special case of the\nsystem (Section IIC2), and demonstrate with simulations a case in which loss of stability\nleads to unbounded growth of activity (Section IIC3).\n1. Theory\nTo investigate the stability and convergence of the two-stage model, we need to find\nan appropriate Lyapunov function candidate. The two learning rules can be rewritten as\n˙w1=−η1fWr2\nin−η1rinξand ˙w2=η2w1r2\nin. We choose the following Lyapunov function\ncandidate for the two-stage model:\nL=1\n2(fW2+ew2\n2), (6)\nwhere ew2=w2−w∗. We refer to the first term in the above as the (squared) gain error and\nthe second term as the (squared) consolidation error. The nullclines and the fixed points of\nthe learning dynamics are shown in the weight space in Fig. 2A for w∗= 1in the limit that\nthe amplitude of the perturbation goes to zero, i.e., µ→0. The first term encourages the\ngain error to go towards zero in a stable manner and stay close to zero, which is the goal of\nthe learning rule for w1. The second term aligns with the goal of consolidating the learned\nmemories into w2and is achieved when the desired gain w∗is only due to w2(Fig. 2B).\nWe now turn to proving uniform global stability and asymptotic stability of the two-stage\nmodel. Using w1=fW−ew2, the time derivative of Lcan be written as ˙L=˙fWfW+˙ew2ew2=\n−η1r2\nin((1−α)fW2+αew2\n2)−η1fWrinξ, where α=η2/η1. With |ξ| ≤µ|e|one obtains:\n˙L=−η1r2\nin((1−α)fW2+αew2\n2)−η1fWrinξ\n≤ −η1r2\nin((1−α)fW2+αew2\n2) +η1|fWrin||ξ|\n≤ −η1r2\nin((1−α)fW2+αew2\n2) +η1|fWrin|µ|fWrin|\n=−η1r2\nin((1−α−µ)fW2+αew2\n2).(7)\nThe main requirement for Lyapunov stability is to show ˙L≤0, which is achieved when\nα≤1−µ. As in the single-stage model, we assume rin̸= 0throughout most of the learning\nperiod. For the rest of the proof, we need to verify the other conditions of Theorem 1. Lis\nbounded from below, min(L) = 0, and Ldoes not explicitly depend on time. Hence, Lis\npositive definite and decrescent. We therefore conclude that the equilibrium is uniformly\nstable. Since Lis the sum of two quadratic terms, it is radially unbounded by inspection,\nwhich guarantees that fW,ew2,w2,w1are globally bounded. To guarantee asymptotic stability,\nwhat is left to show is that ˙Lis uniformly continuous in time by showing ¨Lis bounded.\n6\"&\"'̇\"&=0̇\"'=011\nStability requirement:All trajectories ()(*≤0Gain errorConsolidation error''='+'=(Gain error)2+(Consolidation error)2\n01200.51Noise  ())StablePotentially unstableBA\nCFIG. 2. Lyapunov function theory for stability of the two-stage model. (A) In the limit that the\nperturbation goes to zero, µ→0, the closed-loop learning dynamics has a single fixed point and two\nnullclines (shown for w∗= 1). (B) The Lyapunov function candidate Lhas two terms: the squared\ngain error and the squared consolidation error. The most important property in order to have stable\nconvergence in the Lyapunov sense is that the dynamics of the learning rules should avoid going\nuphill on the Lyapunov function surface. (C) When the ratio of learning rates α=η2/η1is less than\na critical value αc= 1−µ, the learning is guaranteed to be stable. As the maximum perturbation\namplitude reaches |e|, i.e., µ= 1, the region of guaranteed stability vanishes.\nGiven that the hyperparameters η1, η2, α, and the variables fW,ew2,w1are bounded, and rin\nandξare assumed to be smooth bounded functions of time with bounded derivatives, ¨Lis\nbounded. ■\nThe key result of the above is that, when the late stage is tuned at a rate not faster than\nthe early stage rate, i.e., α=η2/η1≤αc= 1−µ, the system provably remains globally\nstable and is guaranteed to successfully converge (Fig. 2C). Intuitively, in the extreme case\nwhere the learning rate of w1is much lower than that of w2, it is easy to see why the system\nmay become unstable: w1moves infinitesimally slowly towards the goal, but w2gets rapidly\nupdated with a secondary teaching signal that is not in the direction of the gradient of the\nerror. This leads to an alteration of the error signal feeding back onto the early ( w1) site\nof learning, potentially causing the learning process to become unstable. To combat this\npotential source of stability, the learning rate at w2should be slower than that of w1to filter\n7A\n0123/n0200400600800Amplification % =0.5=1=2=4=8=16=32=64!\"*'!\"*'Amplification % (()!,##*)BFIG. 3. Amplification of perturbation in the region without stability guarantee. (A) The state-\nstate of w2exhibits an amplification of an infinitesimal sinusoidal perturbation probe ξp=ϵsin(ωnt),\nshown in the limit that µ→0. Top, α= 3(red box); bottom, α= 5.33(green box). (B) The\nsteady-state percent amplification of the sinusoidal probe perturbation as a function of the ratio of\nthe normalized frequency (normalized by the undamped natural frequency ωn) of the probe ξpin\nthe limit that µ→0.\nout noise and prevent run-away amplification.\nWhen α > α c, the analysis only indicates that the system may become prone to instability,\nbut does not itself say whether the system will become unstable. Such instability can\npotentially arise if a perturbation brings the system into regions where the derivative of L\nbecomes positive, which for this system occurs when |ew2|<q\nα−1\nα|fW|, showing that the size\nof this region increases with α.\nTo check whether we can improve the stability conditions (i.e., find a higher value\nofαc) by considering a different relative weighting of the gain and consolidation error\nterms of L, we consider the Lyapunov function Lb=1\n2(fW2+bew2\n2), where b > 0. For\nsimplicity, we work in the regime µ→0, for which αc= 1. Calculating the time derivative\n˙Lb=−η1r2\nin((1−α)fW2+α(1−b)fWew2+αb˜w2\n2), we note that ˙Lbis again guaranteed to be\nless than or equal to zero in the whole weight space as long as α≤1, but not for α > 1\n(in particular, this is easily seen when ˜w2= 0). Thus, the same fundamental criterion for\nguaranteeing stability emerges even for different weightings of the two error terms of the\nLyapunov function L.\n2. Intuition for instability\nIn the α > α cregime, our Lyapunov stability analysis only shows that stability is not\nguaranteed and thus only indicates the potential for instability. Therefore, it is instructive to\ninvestigate this regime more closely. Consider the case where a sinusoidal probe perturbation\nξp=ϵsin(ωt)with an infinitesimal amplitude ϵis present in Eq. 2. We examine its effect on\nthe system in the regime that µ→0while, for simplicity, we set rin= 1. To gain intuition\n8about this amplification, we solve the system in the presence of the probe. By eliminating\nw1in the two learning rules, we obtain\n¨ew2+η1˙ew2+η1η2ew2=−η1η2ϵsin(ωt). (8)\nThis second-order differential equation is equivalent to forced mass-spring-damper dynamics\nm¨x+c˙x+kx=F, where xis the object displacement, m= 1is the mass, c=η1is the\ndamping coefficient, k=η1η2is the spring constant, and F=−η1η2ϵsin(ωt)is the external\nforce. When the damping ratio ζ=c\n2√\nkm=1\n2√α<1, we are in the underdamped regime.\nThe steady-state response, which is dominated by ew2,ss(t)since w1approaches zero in the\nsteady state, then has the following form:\new2,ss(t) =−ϵsin(ωt+ϕ)q\n(1−ω2\nω2n)2+ (2ω\nωnζ)2, (9)\nwhere ϕ=arctan2ζ(ω\nωn)\n1−(ω\nωn)2andωn=√η1η2is the undamped natural frequency. As αincreases,\nthe damping ratio ζdecreases, and the amplitude of the ew2,ss(t)resonance increases. This\nclearly shows that a small error perturbation ξpleads to an amplified output whose amplitude\nat the natural frequency equals√αϵ. The simulations for two values of α, i.e., α= 0.33and\nα= 3, show an amplification of the probe in w2in the potentially unstable region (Fig. 3A)\nat the natural frequency. The mathematical correspondence between the ratio of learning\nrates αin systems consolidation and the (inverse square of the) damping ratio in a physical\noscillator, and the resultant resonant amplification, is the intuition behind the potential\ninstability in the presence of perturbations. This resonant behavior is shown for a sweep of\nrelative frequencies for a larger range of αvalues in Fig. 3B.\n3. Simulation of an unbounded growth instability\nTo further investigate the instability, we simulate the effect of a sinusoidal signal-dependent\nperturbation, which can alternatively be interpreted as a time-varying learning rate of the\nearly stage. Solving the equations directly in this case is cumbersome, but analyzing their\nstability is trivial with the Lyapunov theory. The theory we have developed can be used\ndirectly: when the maximum α(t)is less than one, the system is guaranteed to be stable;\notherwise, stability is not guaranteed. Consider the case η1(t) = 0 .1(1 + 0 .7sin(0.2πt)).\nFor a slow learning rate of the late site, η2= 0.02,max(α(t))<1so that stability is\nguaranteed (Fig. 4A). By contrast, for η2= 1, yielding values of α(t)>1so that stability\nis not guaranteed, we find that the system not only exhibits amplification of the sinusoidal\nperturbation but grows unboundedly (Fig. 4B).\nWe can interpret the instability using the resonance intuition we developed in the previous\nsubsection. For the simple case of Section IIC2, the resonance caused amplification but this\namplificationwaskeptfinitebythedamping. Inthepresenceofsignal-dependentperturbation,\nthe natural frequency of the unperturbed system can interact with the frequency of vibration\nof the perturbation. When this interaction gives rise to a frequency close to the natural\nfrequency of the unperturbed system, especially when the amplitude of the perturbation is\nsufficiently large, there can be increasing amplification of the system in each period, leading\nto unbounded growth. This resonance phenomenon, often referred to as parametric resonance\n9-20020Signals-505w10 200 400 600 800 1000-20020w20 200 400 600 800 1000Time (sec)0100200L00.10.2Signals00.10.2w10 1000 2000 3000 4000 5000 600000.10.2w20 1000 2000 3000 4000 5000 6000Time (sec)00.0050.01LA\nBDesired !∗Output !\"\nDesired !∗Output !\"FIG. 4. The presence of a sinusoidal signal-dependent perturbation in the form of a time-\nvarying learning rate of the early stage can cause unbounded growth instability. We consider\nη1(t) = 0 .1(1 + 0 .7sin(0.2πt)). (A) A stable case with η2= 0.02so that max ( α)<1. The first\nthree subplots are in the same format as Figure 1, with the last subplot showing the Lyapunov\nfunction L. (B) An unstable case with η2= 1so that α >1.\n[23], can happen when two oscillators get coupled in such a way that one causes oscillations\nin the parameters of the other oscillator, and does not necessarily need an external force to\nexhibit instability.\nIII. DISCUSSION\nWe have provided a framework for studying the stability of systems consolidation and\napplied it to a simple circuit architecture characterized by an early learning area that is\ndirectly trained by performance errors, which trains a late learning area that provides the\nfinal site of memory storage. Using a Lyapunov function theory that enforces the stability\nof the learning and consolidation process, we have obtained a fundamental result on the\nspeed of learning: the late stage must not be tuned faster than the early stage, and when\nthe teaching signal is corrupted by perturbation, the late stage should be tuned much more\nslowly. We mapped the consolidation process to the dynamics of a driven damped oscillator,\nproviding the intuition that increasing the ratio of late- to early-stage learning rates αis like\ndecreasing the oscillator damping, leading to potential resonant instability.\nPreviousworkonmemoryconsolidationhasfocusedprimarilyonafundamentalrobustness-\n10speed tradeoff in learning with single-stage models, known as the stability-plasticity dilemma.\nThis dilemma states that, in single-stage models, having fast plasticity leads to ‘instability’\nin the sense that new memories overwrite old ones, whereas this tradeoff can be lessened in\nmulti-stage models [ 24,25]. Here, by contrast, we show a complementary, dynamical form of\ninstability that occurs for the systems consolidation of graded memories, in which having too\nfast a speed of consolidation can lead to amplification of a perturbation or even exponential\nunbounded growth of activity. Despite these differences, both our theory and previous ones\nobey a similar principle: multi-stage learning offers robustness by having slower learning in\nthe later stages.\nOur toy two-stage model maps onto the architecture of classic models of motor learning\nmediated by the cerebellar brain region [ 16,17]. In such models, early learning is thought\nto occur through plasticity of the weight w1between presynaptic parallel fiber inputs and\npostsynaptic Purkinje cells. This plasticity is thought to be driven by correlations between\nthe activity of the parallel fiber inputs with behavioral error signals that are conveyed by\nseparate, climbing fiber inputs to the Purkinje cells. The learning process is particularly\nwell-characterized in the cerebellum-mediated adaptation of eye movement reflexes. For\nexample, in the vestibulo-ocular reflex (VOR), rapid corrective eye movements are generated\nto offset movements of the head, functioning like a motion-correcting camera. This reflex\nrequires tuning because, for example, the introduction of eyeglasses can alter the relation\nbetween eye movement and resulting image motion across the retina. Connecting to the\npresent work, one can map the input rinto the head velocity, the output roto the eye velocity,\nw∗to the desired VOR gain (i.e., the ratio of eye to head velocity), and the teaching error to\nthe \"retinal slip\" motion of the visual image on the retina. Learning is then transferred from\nan initial site in the cerebellum (weight w1) to a late site of final storage in the vestibular\nnucleus (weight w2). Interestingly, to properly model the biological circuit, one should make\nthe climbing-fiber-driven error signals come through discrete spikes rather than the smooth\nfiring rate assumed here. This provides an effective form of perturbation ξ(t)that can\ndecrease the stability of the system if not compensated for by decreasing the learning rate\nat the late site. Finally, we note that a similar consolidation of learning has been shown to\noccur in the striato-neocortical reinforcement learning system of the brain [ 26], suggesting\nsimilar fundamental constraints on the speed of learning may be applicable more broadly.\nAlthough systems consolidation in the late stage tends to be considered a slow process\nin several reported neural systems, this is not always the case. For declarative memories,\nthe late stage can consolidate quickly if the new memories have features that are consistent\nwith the existing structure of knowledge in the late stage [ 27,28]. Recent evidence from\nsongbird motor consolidation suggests that the consolidation process may happen faster\nthan originally thought, occurring online in the daytime, and not necessarily requiring offline\nnighttime processes [ 26]. Enforcing the stability and convergence of consolidations in these\nscenarios may reveal constraints on the speed of learning and consolidation, similar to what\nwe have found in the current work.\nBesides the important implications for neuroscience experiments, the framework we have\nprovided here may have engineering applications. Classically in adaptive control theory,\nthe tracking error or prediction error is directly used to tune the parameters of single-stage\nadaptive controllers [ 10]. Our work gives the insight that using a two-stage adaptive controller\ncan give flexibility in terms of having a robust storage memory at the final site, as well as\nan extra knob to tune the speed of learning in a stable manner. In systems with delayed\nnegative feedback that are subject to inappropriate oscillations, such two-stage learning\n11could be used to avoid deleterious resonance effects. Recent machine learning approaches\nhave focused on using machine learning to generate and control complex dynamical systems\n[29,30]. Given that artificial neural networks with online, adaptive learning are increasingly\nin demand throughout society, including in safety-critical tasks, this suggests a compelling\nneed to develop frameworks that guarantee the stability of such algorithms. We hope that\nthe principles of systems consolidation and Lyapunov theory introduced here could help\ncurrent progress in this area by highlighting the need for real-time, continuously adaptive\nsystems that are safe and stable.\nAppendix A: METHODS\n1. Toy model simulation\nFor the simulation of the toy model, we simulated Eqs. (1-3) using the MATLAB solver\node45, which is a fifth-order Runge-Kutta method. The learning rate of the first stage was set\nto a fixed value η1= 0.01in all simulations shown in Fig. 1. η2was zero in the single-stage\nmodel, 0.0003in the two-stage model with slow consolidation (Fig. 1B, left), and 1in the\ntwo-stage model with fast consolidation (Fig. 1B, right). rin(t)was generated by a step\nfunction with amplitude 0.1which was smoothed by the filter 100/(100s+ 1)(with sbeing\nthe Laplace variable). In Fig. 1, a sinusoidal perturbation with time-varying amplitude\nξ=a(t)sin(ωnt)where ωn= 0.1(rad/s) was considered. In addition, a(t)was sampled at\nrandom from a uniform distribution in the range [0,0.002] with a 10 s sampling period – this\nwas not necessary for our core results, but was included to illustrate the effect of a slow\nnon-stationary amplitude a(t).\nAppendix B: STABILITY DEFINITIONS AND THEOREMS\nFor ease of notation, we present the following definitions, lemmas, and theorems in the\ncontext of a general non-autonomous dynamical system ˙x=dx\ndt=f(x, t)for the state vector\nx∈RNwith equilibrium point xeq=0.\n1. Definitions\nThe formal definition of the most basic notion of stability in the Lyapunov sense for a\nnon-autonomous system is\nDefinition 1\nThe equilibrium point 0isstableatt0if for any R≥0, there exists a positive scalar r(R, t 0)\nsuch that\n∥x(t0)∥< r ⇒ ∥ x(t)∥< R ∀t≥t0.\nOtherwise, the equilibrium point 0isunstable. If the scalar r in the above can be chosen\nindependently of t0, i.e., if r = r(R), then the equilibrium point 0is uniformly unstable .\nIf the above conditions are true for the whole state space, then the stability is global; otherwise,\nthe stability is local.\n12A more refined and desirable concept is asymptotic stability, which not only ensures that\nthe state stays in a ball of arbitrarily small radius around the equilibrium but also provides\na statement about convergence to the equilibrium:\nDefinition 2\nThe equilibrium point 0is asymptotically stable att0if\n•it is stable\n•∃r(t0)>0such that ∥x(t0)∥< r(t0)⇒ ∥ x(t0)∥ → 0ast→ ∞.\nIn addition, if there exists a ball of attraction BR0, whose radius is independent of t0, such\nthat any system trajectory with initial states in BR0converges to 0uniformly in t0, then the\nequilibrium point 0is uniformly asymptotically stable .\nDefinition 3\nA scalar continuous function L(x)is said to be locally positive definite ifL(0) = 0and, in a\nball around the origin\nx̸=0 ⇒ L(x)>0.\nIf the inequality in the above is replaced with L(x)≥0, then L(x)is (locally) positive semi-definite .\nIfL(0) = 0and the above property holds over the whole state space, then L(x)is\nsaid to be globally positive definite . If L(x)is positive (semi-)definite, then −L(x)is\nnegative (semi-)definite .\nThe time-varying function L(x, t)is said to be positive definite if L(0, t) = 0and there is\na time-invariant positive definite function L0(x)such that\n∀t≥t0, L (x, t)≥L0(x).\nDefinition 4\nA scalar function L(x, t)is said to be decrescent ifL(0, t) = 0, and if there exists a time-\ninvariant positive definite function Ll(x)such that\n∀t≥0, L (x, t)≤Ll(x).\nDefinition 5\nA function g is said to be uniformly continuous on[0,∞)if\n∀R > 0,∃η(R),∀t1≥0,∀t≥0, such that |t−t1|< η⇒ |g(t)−g(t1)|< R.\nThis uniformity in time means that one can always find an ηwhich does not depend on the\npoint t1.\n132. Lemmas and theorems\nBelow, we provide the theorems and lemmas needed to prove the stability of the systems\nin the main text. For their proofs, consult [10].\nTheorem 1 (Lyapunov theorem for non-autonomous systems)\nStability :If, in a ball BR0around the equilibrium point 0, there exists a scalar function\nL(x, t)with continuous partial derivatives such that\n1.Lis positive definite\n2.˙L=dL\ndtis negative semi-definite\nthen the equilibrium point 0is stable in the sense of Lyapunov.\nUniform stability and uniform asymptotic stability :If, furthermore,\n3.Lis decrescent,\nthen the origin is uniformly stable. If condition 2 is strengthened by requiring that Lbe\nnegative definite, then the equilibrium point is uniformly asymptotically stable.\nGlobal uniform asymptotic stability :If the ball BR0is replaced by the whole state\nspace, and condition 1, the strengthened condition 2, condition 3, and the condition\n4.L(x,0)is radially unbounded, i.e., L(x,0)→ ∞as∥x∥ → ∞\nare all satisfied, then the equilibrium point at 0is globally uniformly asymptotically stable.\nTo prove asymptotic stability in cases where it is not easy to prove negative definiteness\nof˙L, we use the following Lyapunov-like lemma, which is a variant of Barbalat’s lemma [ 10],\nthat requires the derivative of Lto have some additional smoothness property to ensure ˙L\nconverges to zero:\nLemma 1\nIf a scalar function L(x, t)satisfies the following conditions\n•L(x, t)is lower bounded\n•˙L(x, t)is negative semi-definite\n•˙L(x, t)is uniformly continuous in time\nthen ˙L(x, t)→0ast→ ∞.\nThe first two conditions in the above lemma imply that Lhas a finite limiting value L∞, but\nthey do not guarantee that Lwill remain stationary at L∞[10]. The addition of the third\ncondition gives us the ability to conclude that, in the limit t→ ∞,Lremains stationary at\nL∞and the convergence will be achieved.\n14ACKNOWLEDGMENTS\nWe thank Jay Bhasin for helpful discussions. We acknowledge financial support from\nSimons Foundation Collaboration on the Global Brain grants 542989SPI and NC-GB-CULM-\n00002734 (MG, EA), NIH R01 NS104926 (MG, EA), and NIH R01 EY031972 (MG).\n[1]Reza Shadmehr and Henry H Holcomb. Neural correlates of motor memory consolidation.\nScience, 277(5327):821–825, 1997.\n[2]James L McClelland, Bruce L McNaughton, and Randall C O’Reilly. Why there are comple-\nmentary learning systems in the hippocampus and neocortex: insights from the successes and\nfailures of connectionist models of learning and memory. Psychological Review , 102(3):419,\n1995.\n[3]Javier F Medina, J Christopher Repa, Michael D Mauk, and Joseph E LeDoux. Parallels\nbetween cerebellum-and amygdala-dependent conditioning. 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Closed-form continuous-time neural networks.\nNature Machine Intelligence , pages 1–12, 2022.\n16"    },    {        "title": "2402.01741v2.Development_and_Testing_of_a_Novel_Large_Language_Model_Based_Clinical_Decision_Support_Systems_for_Medication_Safety_in_12_Clinical_Specialties.pdf",        "content": "Development and Testing of a Novel  Large Language Model -Based Clinical Decision Support \nSystems for Medication Safety in 12 Clinical Specialties  \n \nJasmine Chiat Ling Ong 1,2* \nJin Liyuan 2,3,4* \nKabilan Elangovan3 ,4 \nGilbert Yong San Lim3,4 \nDaniel Yan Zheng Lim5,6  \nGerald Gui Ren Sng6,7  \nKe Yu He 8  \nJoshua Yi Min Tung6,9  \nRyan Jian Zhong1  \nChristopher Ming Yao Koh1  \nKeane Zhi Hao Lee1  \nXiang Chen1  \nJack Kian Ch’ng 10 \nAung Than11   \nKen Junyang Goh 12  \nDaniel Shu Wei Ting2,3,4+ \n \n1. Division of Pharmacy, Singapore General Hospital, Singapore  \n2. Duke- NUS Medical School, Singapore, Singapore   \n3. Singapore National Eye Centre, Singapore Eye Research Institute, Singapore, Singapor e \n4. Singapore Health Services, Artificial Intelligence Office, Singapore  \n5. Department of Gastroenterology , Singapore General Hospital, Singapore  \n6. Data Science and Artificial Intelligence Lab, Singapore General Hospital, Singapore \n7. Department of Anesthesiology, Singapore General Hospital, Singapore, Singapore \n8. Department of Endocrinology , Singapore General Hospital, Singapore  \n9. Department of Urology, Singapore General Hospital, Singapore   \n10. Department of Vascular Surgery , Singapore General Hospital, Singapore  \n11. Department of Internal Medicine , Singapore General Hospital, Singapore  \n12. Department of Respiratory and Critical Care Medicine, Singapore General Hospital, Singapore  \n \n*Contributed Equally  \n+Corresponding Author  \n Corresponding author: \nA/Prof Daniel Ting MD (1\nst Hons) PhD  \nAssociate Professor, Duke- NUS Medical School  \nDirector, AI Office, Singapore Health Service  \nAddress: The Academia, 20 College Road, Level 6 Discovery Tower, Singapore, 169856  \n Abstract  \nImportance : We introduc e a novel Retrieval Augmented Generation (RAG) -Large Language Model \n(LLM) framework as a Clinical Decision Support Systems (CDSS)  to support safe medication \nprescription, a critical aspect of patient safety.  This overcomes existing challenges of irrelevancy  of \nalerts in rules -based CDSS in provision of prescribing error alerts that is relevant to the patient’s context \nand institutional medication use guides.  \n \nObjective: To evaluate the efficacy of LLM- based CDSS in correctly identifying medication errors in \ndifferent patient case vignettes from diverse medical and surgical sub- disciplines, against a human  \nexpert panel derived ground truth. We compared performance for under 2 different CDSS practical \nhealthcare integration modalities : LLM- based CDSS alone (fully autonomous mode) vs junior \npharmacist + LLM- based CDSS (co- pilot, assistive mode).  \n \nDesign, Setting, and Participants : Utilizing a RAG  model with state- of-the-art medically -related LLMs \n(GPT -4, Gemini Pro 1.0 and Med- PaLM 2) , this study used 61 prescribing error scenarios embedded \ninto 23 complex  clinical vignettes across 12 different medical and surgical specialties.  A multidisciplinary \nexpert panel assessed these cases for Drug- Related Problems (DRPs) using the PCNE classification  \nand graded severity / potential for harm using revised NCC MERP medication error index. We \ncompared.  \n \nMain Outcomes and Measures : This study compares the performance of an LLM- based CDSS in \nidentifying DRPs. Key metrics include accuracy,  precision, recall, and F1 scores. We also compare the \nperformance of  LLM- CDSS  alone and junior  hospital  pharmacist s (less than 2 years post licensure) + \nLLM- CDSS (co -pilot, assistive  mode)  in the provision of recommendations to clinicians. In addition, we \npresent comparative results from different LLMs: GPT -4, Gemini Pro 1.0 and Med- PaLM  2.  \n \nResults  \nRAG -LLM performed better compared to LLM alone. When employed in a  co-pilot mode , accuracy, \nrecall, and F1 scores were optimized, indicating effectiveness in identifying moderate to severe DRPs. \nThe accuracy of DRP detection with RAG -LLM improved in several categories but at the expense of \nlower precision.  \n Conclusions  \nThis study established that a RAG -LLM based CDSS  significantly boosts the accuracy of medication \nerror identification when used alongside junior pharmacists  (co-pilot) , with notable improvements in \ndetecting severe DRPs . This study also illuminates the comparative performance of current state- of-\nthe-art LLMs in RAG -based CDSS systems.  \n   Introduction  \n \nMedical  errors remain a formidable challenge for healthcare institutions all around the world and is the \nthird leading cause of mortality in the United States . Medication- related errors account  for 5% to 41% \nof US hospital admissions every year .1,2 Medication errors can potentially result in prolonged \nhospitalization stay, elevated risk for morbidity and mortality as well as an increase in healthcare spending.\n3 This translates to high economic burden of medication errors, amounting up to USD$40 \nbillion in the United States and £750 million in England per year.3,4 In an acute care setting, medication \nerrors can occur at any stage of the medication use process : medication prescribing, dispensing, \nadministration and patient monitoring. A vast majority of errors occur at the prescribing stage, \naccounting for 70% of errors that result in adverse patient events.5 Halting errors at this stage is critical \nin preventing perpetuation of the error downstream and eventually reaching the patient.   \n \nClinical Decision Support Systems (CDSS) have become a cornerstone of modern healthcare systems \nas a direct aid to clinical decision making. A CDSS  is intended to improve healthcare delivery by \nenhancing medical decisions with targeted clinical knowledge, patient information, and other health \ninformation.6 A key function of CDSS is in reducing the incidence of prescribing or medication errors \nand adverse events through integration with electronic health records and computerized provider order \nentry (CPOE) systems.7 In particular categories of prescribing errors such as drug- drug interactions, \nCDSS have been shown to be effective in reducing the incidence of errors.8 However, a vast majority \nof current systems are rules -based, resulting in the generation of voluminous and clinically irrelevant \nalerts to users.9 Over time, safety alerts are ignored and overwritten by physicians in as high as 95%, \nthis phenomenon of 'alert fatigue'  leads to ignoring of these alerts , posing a barrier to effective adoption \nand utilization of CDSS systems .10-12 There is a need to re- invent delivery of prompts  by CDSS through \ngreater personalization, intuitive and lesser emphasis on disruptive alerts.  \n \nThe growing capabilities of l arge language models (LLMs) in medical  tasks are becoming more \napparent. LLMs are advancing in handling such tasks, particularly those that do not require extensive \nspecialized expertise.13,14 This includes simplifying administrative duties like composing medical  letters ,  \ncreating summaries upon patient discharge using  information from electronic health records (EHR)  to   \nsemi -autonomous decision- making support for managing operati ng theatres.15,16 Finally, LLM- powered \nhealthcare chatbots is capable of providing patients and health professionals with highly professional -\nsounding, accurate and personalized responses to medical queries.17,18 Healthcare systems are facing \ncritical shortage of healthcare professionals compounding by a mounting issue of healthcare \nprofessional  burnout.19-21  LLM- powered solutions are well poised to improve operational efficiency  and \nstandards of patient care when trained with the right data, robust clinical evaluation with a deployment \nstrategy armed with appropriate safety measures.  \n \nPrior published studies have developed LLM- based tools to support  various clinical  applications and \ndomains.22-24 The use of LLMs as an innovative substitute to current rules -based CDSS has not been described . While hundreds of pretrained LLMs have published since last year, few LLMs are trained \nspecifically on medical knowledge  or reported  their clinical  performance. In this study, GPT -4.0, Gemini \npro 1.0, and Med- PaLM 2 were chosen for their reported close to expert performance in various medical \ntasks.25-28  \n \nHerein, we propose a new model of CDSS leveraging upon LLM grounded with contextual knowledge \non medications  to accurately identify prescribing errors and provide evidence- based recommendations \nto healthcare professionals . In this study, we  seek to address the following question: Can L LM-based \nCDSS identify prescribing errors and provide accurate, clinically acceptable recommendations across \ndifferent medical disciplines? The results will reveal the potential for integration of LLMs  in healthcare, \npotential ly leading to substantial improvements in patient safety and quality of care.  \n Methods  \nWe followed the Standards for Reporting of Diagnostic Accuracy (STARD) reporting guideline for this  \nstudy on the development of LLM- based  CDSS for medication prescribing error identification. An \noverview of the evaluation workflow is shown in Figure 1.  Ethics review board was not required because \nno identifiable patient data were used.   \n \nFigure 1: Overview of evaluation workflow  \n Development of Prescribing E rror Scenarios  \nWe created a total of 61 different simulated prescribing error scenarios  based on 23 case  vignettes \nmodelled after complex clinical cases . Prescribing error  scenarios  were adapted  from the local \ninstitution pharmacy intervention and error reporting databases  to maintain realism.  The vignettes  \ncovered clinical scenarios from 1 2 different medical or surgical subspecialties (Cardiology, \nEndocrinology, General Medicine, Ophthalmology, Gastroenterology, General Surgery, Urology , \nVascular Surgery, Infectious Disease, Respiratory  Medicine , Oncology , Colorectal Surgery ), with some \ncases involving more than one discipline. Each case vignette consisted of a patient clinical note and \nmedication prescription. Prescribing errors in the clinical scenarios were designed to be reflective of \ncommonly encountered drug- related problems (DRP). We present a sample of one case vignette in \nFigure 2. A detailed description of all case vignettes is found in the supplement. The Anatomical \nTherapeutic Chemical (ATC)  category of medications prescribed in each clinical scenario is also \npresented in the supplement.  \n \nFigure 2: Sample of Case Vignette. Abbreviations ( CVM: cardiovascular Medicine, BP: blood pressure, \nRR: respiratory rate, Ht: height, BMI: body mass index, T: temperature, Hb: hemoglobin, TW: total white \ncount, Plt: platelet count, SCr: serum creatinine, INR: international normalized ratio, PMHx: past medical \nhistory, DM: diabetes mellitus, HTN: hypertension, HLD: hyperlipidemia, CKD: chronic kidney disease, \nBKA: below knee amputation , OM: osteomyelitis, I&D: incision and drainage, DVT: deep vein \nthrombosis, PE: pulmonary embolism, VAS: vascular, HOPC: history of presenting complain, CABG: \ncoronary artery bypass grafting, NSAID (non- steroidal anti -inflammatory drug), LV: left ventricle, RV: \nright ventricle, EF: ejection fraction, MI:  myocardial infarction, O/E: on examination, NSR: normal sinus \nrhythm, JVP: jugular venous pressure, SNT: soft non tender, BS: bowel sound, LAD: left anterior descending artery, AV: atrioventricular, LCx: left circumflex artery, ICA: intermediate care area,  DAPT: \ndual anti -platelet therapy)  \n \nDevelopment of Reference Standard  \nThe reference standard was developed through manual grading  of DRP categories and severity  by a \nmulti -disciplinary expert panel. The expert panel consisted of pharmacotherapy board certified \npharmacists and  physicians  with > 10 years of clinical practice experience in tertiary hospitals. Every  \nclinical scenario was graded by at least 1 pharmacist and 1 physician  member of the panel. Any \ndisagreements were resolved by a 3rd independent member. DRPs from each clinical scenario was \ncategorized using the Pharmaceutical Care Network Europe (PCNE) classification V 9.1 and ASHP \nstatement of pharmaceutical care as a guide.29,30 Categories including in error scenarios include: \nadverse drug reaction, allergy, drug- drug interactions, duplication of therapy, inappropriate choice of \ntherapy, inappropriate dosage regimen, no indication and omission of drug therapy. T he potential \nseverity of these errors were graded according to the Harm Associated with Medication Error \nClassification (HAMEC)  classification tool .31  \n Development of LLM -based CDSS Tool  \n Development of RAG -LLM Tool   \nWe engineered 2 versions of LLM- based tool using a Retrieval -Augmented Generation (RAG) and \nGPT4.0 as the base model. Showcasing bot h a simple (version 1) and more complex design (version \n2) of RAG system . The RAG -LLM framework merges the Retrieval -Augmented Generation (RAG) \nmodel with a Language Model (LLM), optimizing the transformation of specialized documents into embedding vectors. This process utilizes advanced pre- processing and embedding models, with  a \nfocus on similarity -based retrieval between query vectors and embedded vectors of targeted \ndocuments, including drug monographs and hospital drug- use protocols.  \n Version 1 of the RAG -LLM framework utilized Pinecone for vector storage and OpenAI's text -\nembedding- ada-002 for embedding, with pre-processing by the Langchain Python package. It operated \nwith a chunk size of 1000 and a retrieval parameter 'k' of 5, balancing retrieval detail and computational efficiency.  Version 2 advanced the framework by integrating the Llamaindex RAG framework, relying \non auto- merging retrieval  to provide contextualized search. This version introduced manual indexing of \ndrug names for targeted pharmaceutical relevance. It employed HuggingFace's bge- small -en-v1.5 \nembedding model and adjusted to hierarchical chunking sizes of 2048, 512, and 123. The 'k' value was increased to 20, enhancing the breadth and depth of information retrieval.  \n \nThe prompt was designed as a series of tasks. Each task required retrieval of knowledge for each \nmedication on the prescription (e.g. Is this medication indicated for the patient?), before feeding into \nGPT-4 to generate a response. The response from each task  is fed into a final GPT -4 model for \nsummarization and recommendations.  A diagrammatic view of the sequence is shown in Figure 3. All \nresponses were generated and analysed in triplicates to account for reproducibility. The performance of both versions was compared and one version was selected as the final co- pilot.   \n \nFigure 3 : (A) Version 1 employing simplified RAG architecture (B) Version 2 employing advanced \nRAG architecture with auto -merging retrieval  \n   \nKnowledge Source  \nInstitutional medication use and dosing guidelines, medication monographs were used as sources of \ninformation. Each medication monograph was split into 4 separate sections  according to the following \ncategories of information: (1) Adverse drug reactions, cautions and contraindications, (2) ATC category and mechanism of action, (3) Drug- drug interactions, (4) Drug dosing and adjustments.  \n LLM Prompt  \nWe first designed a general natural language prompting template and experimented with various \nprompting strategies to test the effect on model response. Various strategies that were tested included \ndynamic few -shot learning using 2 clinical scenarios separately created for in- context learning, chain of \nthought prompting and self -generated chain of thought prompting.\n32 The final adapted prompt used i s \npresented in Figure 4. \n \nFigure 4: Final Adapted Prompt for LLM  \n \nGeneration of DRP Responses    \nWe generated GPT -4, version 1 RAG -LLM and version 2 RAG -LLM responses in triplicates, resulting \nin a total of 149, 161 and 246 DRP responses for assessment. Respective performance was used to \nselect the best model to be used as comparator against human onl y and for adoption in co- pilot mode.  \n \nWe randomly assigned the  scenarios to 3 junior pharmacists (< 2 years of post -licensure practice \nexperience) to independently identify any DRPs and generate a standard clinical note in standardized \nhealthcare SBAR  (situation, background, assessment, recommendation)  format for each scenario. \nEach pharmacist had access to all institutional protocols and guidelines and was given a time limit of 1 hour to review and generate a set of responses for 7- 8 clinical scenarios. P harmacists were also given  \nRAG -LLM generated responses  as reference. During evaluation phase of different modes of \ndeployment, a total of 52 and 39 DRP responses generated by co- pilot mode and autonomous mode \nrespectively  were assessed.  \n \n \n \nAssessment  of Accuracy of Responses  \nWe evaluated the accuracy of  all DRP  responses (see above section) using the human expert panel as \nthe criterion standard. A DRP is considered correctly identified if the response mentions the specific \nmedication name and provides a recommendation to perform an action or intervention to change or \namend the medication order. If only the drug class is mentioned; or no action is suggested, the DRP is \nnot considered accurately identified.  \n \nStatistical Analysis  \nThe concordance of  co-pilot and autonomous mode with human expert as benchmark was measured \nusing accuracy, precision , recall , and F1 score as per  a previous published study .33 Accuracy is \nexpressed as a percentage of correctly identified DRP against expert. Precision, which denotes the \nfraction of DRPs correctly identified amongst all suggested DRPs, was defined as precision=true positives/(true positives+false positives). Recall, or the fraction of all DRPs in the criterion standard \ncorrectly identifie d by LLMs, was defined as recall=true positives/(true positives+false negatives). \nThe F1 score is the harmonic mean of precision and recall, and thus penalizes unbalanced precision \nand recall scores (ie, is higher when both precision and recall have similar values): F1 \nscore=(2×precision×recall)/(precision+recall). The higher any of the 3 scores, the better the \nresponse, with 1 being the maximum value for each score.  Data analysis, calculation of precision, recall \nand F1 scores were done in R version 4.3.0 (R Project for Statistical Computing).   \n \nResults  \nSummary characteristics of error scenarios  and case vignettes  are presented in the supplement.  Briefly, \nthe case vignettes are representative of complex clinical case scenarios with multiple co- morbidities \nand problem lists.  The expert  panel  determined that 29.5% of error scenarios were capable of causing \nserious harm, 50.8% capable of inflicting moderate harm while the remaining 19.2% were rated as \ncapable of causing minor or no harm. The three most common DRPs in the case scenarios were: \nInappropriate dosage regimen that arose due the need for dose, frequency or duration adjustment; \nadverse drug r eactions requiring change in medication or reversal agents (this also includes prescribing \nmedications that are contraindicated based on patient profile); and significant drug- drug interactions \nrequiring change in medication or therapeutic drug monitoring.  The median unique medications across \nATC classes was 12  (IQR 5 – 16) per case vignette , suggesting that our constructed cases were \ncomplex and demonstrative of patients seen at a tertiary healthcare institution.  \n Comparative Performance of GPT -4 and RAG -LLMs  \nWe first performed evaluation of results from GPT -4, version 1 and version 2 of RAG -LLM. We found \nthat GPT -4 scored the lowest on measures of overall accuracy , recall and F1. While Version 2 RAG -\nLLM performed the best in terms of accuracy and recall, precision dropped drastically due to the high volumes of false positive responses generated. This trade- off is indicative of Version 2’s sophisticated \nfeatures like auto- merging retrieval and manual indexing, which, while enhancing recall and accuracy, compromise precision. Incorporating a more complex embedding model and larger ‘k’ values in version \n2 may not have been optimally aligned with the dataset as expected thus impacting precision negatively.  \nAs such, w e chose adopt version 1 RAG -LLM for subsequent evaluation as co -pilot in view of the \nhighest F1 score and highest precision. This model presents with the best balance between achieving higher accuracy of identifying DRPs at the lowest alert burden (false positive rate).    \n \nGPT-4 Version 1 RAG -LLM Version 2 RAG -LLM \n \nMean  SD Mean  SD Mean  SD \nPrecision  0.368  0.046  0.407  0.045  0.317  0.020  \nRecall  0.295  0.033  0.355  0.009  0.426  0.028  \nF1 0.325  0.013  0.379  0.022  0.364  0.023  \nAccuracy (%)  29.500  3.300  35.467  0.924  42.600  0.280  \n \nTable 1: Comparative performance of GPT-4, Version 1 and Version 2 of RAG -LLM. Poorest \nperformance highlighted in red while best performance highlighted in green.  \n \nComparative performance of RAG -LLM and Co- pilot modes  \nThe co- pilot mode demonstrated the highest accuracy . Using RAG -LLM as a co- pilot doubled the \nrelative accuracy of DRP identification as compared to RAG -LLM alone (54.1% Co- pilot vs 31.1% \nautonomous ). LLM (without RAG) performed the poorest. Precision, Recall and F1 score was  also \nhighest in the co- pilot mode. In terms of DRP categories, co -pilot mode accuracy improved across most \ncategories including adverse drug reaction, drug- drug interaction, duplication of therapy, inappropriate \nchoice of therapy and omission of drug therapy. However, performance  for co- pilot mode  declined in \nthe categories of inappropriate dosage regimen and no indication when pharmacists were unblinded to RAG -LLM response.  \n \n \nFigure 5: (A) Chart showing comparative accuracy of different modes. (B) Spider diagram showing \nrelative precision, recall and F1 scores of different modes.  \n \n \nFigure 6: (A) Heatmap showing relative accuracy of different modes in various categories of DRP.  (B) \nHeatmap showing relative accuracy of different modes for DRPs of varying severity of harm.   \n Performance of RAG -LLM using Gemini Pro 1.0 and Med-PaLM 2 \nWe also performed a post -hoc experiment using Gemini- Pro 1.0 and Med- PaLM  2. Both are conducted \nthrough Google Cloud Platform, with temperature set as 0.2.  We used the same RAG version 2 retrieval , \nand generated responses using the same prompt and contextual knowledge using  Gemini  Pro 1.0 and \nMed- PaLM 2. Results are shown in Table 2.  \n \n \nGemini  Pro 1.0 Med-PaLM  2 GPT-4 (Version 2)  \n \nMean  SD Mean  SD Mean  SD \nPrecision  0.301  0.014  0.342  0.059  0.317  0.020  \nRecall  0.372  0.053  0.350  0.019  0.426  0.028  \nF1 0.332  0.030  0.345  0.036  0.364  0.023  \nAccuracy (%)  37.200 5.300  35.000 1.900 42.600  0.280  \n \nTable 2: Comparative performance of Gemini  Pro 1.0 and Med- PaLM  2 based RAG models against \nGPT-4 based RAG model.  GPT-4 based model outperformed Gemini -Pro and Med- PaLM 2 based \nmodels in terms of accuracy, Recall and overall F1 score, while Med- PaLM 2 based model \ndemonstrated better precision.  \n \nDiscussion  \n \nAlthough most of recent LLM studies focused heavily on improving model performance based, few \nstudies have explored its function qualitatively and practical applicability of LLM in actual healthcare \ndeployment. In this study, we provide important insights to the feasibility of deploying a novel LLM-\nbased CDSS  to healthcare practice. Our selected RAG -LLM model improved the accuracy of DRP \nidentification by 22% . When deployed autonomously, GPT -4 and RAG -LLM showed subpar \nperformance across all metrics of accuracy, precision, recall and F1. However, when deployed as a co-pilot, the RAG -LLM model is capable of highlighting DRPs that were missed by LLM and RAG -LLM. \nThis suggests strongly that for complex reasoning clinical tasks such as medication chart review, LLMs alone are not adequate. A co- pilot mode can potentially improve model utility and also performance of \nhuman alone.  We also demonstrated comparative performance of 2 different versions of RAG -LLM, \nillustrating incremental improvement in retrieval from complex medical documents. Hence, it suggested \nexcellent potential of future LLM integration into healthcare practice with ever improving LLM performance.  However, we observed that in the co- pilot mode, performance declined in 2 key categories \nof DRP, namely inappropriate dosage regimen and no indication for drug. This is potentially due to \ninconsistencies in document formats.\n34 For instance, dosage adjustment recommendations in drug \nmonographs may be presented in table format (which poses inherent difficulties for GPT to process), or in a prose format. Indications listed in drug monographs may not be exhaustive. In addition, off -label \nprescribing practices vary between institutions and not explicitly listed in tertiary drug monographs.  \n \nMedication errors often result in unintended patient harm and considerable healthcare cost. As \ndescribed in a systematic analysis of studies measuring economic impact of medication errors, various \ncost parameters are involved including hospitalization, medication, primary care, outpatient, litigation \nand non- healthcare costs. The mean costs per error per study was reported to be as high as €111 727 \nper error (translating to SGD$172 842 per error).\n4 The advent of digital medical records have enabled \nthe proliferation and adoption of CDSS in healthcare systems worldwide. CDSS have been designed for various purposes, mostly customized towards the clinical and operational needs  of health institutions \nand patients.  \n CDSS  can be broadly classified into two categories: knowledge- based and non- knowledge- based \nsystems. Knowledge- based (KB) CDSS rely on a set of rules and associations derived from clinical \nguidelines or expert opinions. They function by matching patient -specific data with a knowledge base \nand providing recommendations or alerts.\n35 Non- knowledge- based (NKB) systems, on the other hand, \nuse artificial intelligence and machine learning (ML) techniques to analyse large datasets, identify patterns, and make predictions or recommendations based on new data without predefined rules.7 Our \nLLM- based CDSS is a novel approach to a NKB based system. Widespread adoption of NKB CDSS \nsystems is met with barriers, including lack of transparency, uncertainty relating to the evidence and \nlack of trust in the system, or disruptions to the clinical  workflow  that adds time to routine clinical \npractice.36,37 Till date, ML- based CDSS are fairly narrow in applications, most being disease domain \nspecific. LLM grounded with contextual knowledge present with various advantages over ML- based \nmodels, including the ability to integrate and process vast amounts of varied data types  including \nunstructured clinical texts , easy updating of clinical knowledge corpus without the need for explicit \nretraining, offer explanations in natural language that are more comprehensible to human practitioners. \nWe demonstrate that a RAG -LLM CDSS performed equally across different clinical disciplines and \nmedication classes, being agnostic to disease state.  \n \nLLMs, especially GPT based models, have been evaluated for their role as decision support tools. One \nstudy evaluated the accuracy of GPT -4 base model to generate medication plans using n- gram based \nevaluation scores like ROUGE.38 Despite the small sample size and the use of fictional cases, we are \nthe first study to demonstrate the utility of a novel RAG -LLM based CDSS in enhancing safe medication \nprescribing. We are also the first to report the impact of RAG -based LLM models in i mproving efficiency \nand consistency of pharmacists under conditions simulated to reflect real -world workload and patient . \nIn addition, we performed a post -hoc study to provide insights into comparative performance of different \nstate -of-the-art, commercial LLMs in this clinical application. GPT -4 outperformed the other models in \nmost performance metrics.  \n We were unable to perform a direct comparison of our RAG -LLM model against traditional knowledge-\nbased CDSS. KB  CDSS are implemented for various purposes and often tailored to the peculiar needs \nof the healthcare facility.  Regarding  their efficacy in reinforcing prescribing safety, various studies have \nshown that KB CDSS can significantly contribute to the reduction of prescription errors and improve \nprescribing practices. In a meta- analysis of 68 trials evaluating CDSS on physician prescribing, positive \nbehaviour improved by 4.4% (95% CI  2.6% to 6.2% ) with the deployment of KB CDSS.\n39 AI-based \nCDSS have similarly demonstrated positive outcomes on prescribing safety and potential cost savings \nfrom DRP avoidance.40,41 These trials  underscore the vital role of CDSS in enhancing prescribing safety, \nwhether through rule- based systems or advanced AI -driven analytics. However, the successful \nimplementation and effectiveness of these systems depend on several factors, including system design, \nuser interface, integration into clinical workflows, and training of healthcare professionals.  LLM- based \nCDSS present with unique advantages such as the ability to process and interpret large volumes of \nunstructured clinical data, adapt to evolving medical knowledge, and provide personalized treatment \nrecommendations .42,43 This supports  more informed, relevant and individualized patient care decisions.  \n The limitations of our study are as follows: (1) DRPs identified from clinical scenarios were limited to 5 categories and this might limit generalizability clinical cases with other DRP categories, (2) Small number of fictional clinical scenarios, however the fictional scenarios are highly complex allowing for initial exploration of RAG -LLM based CDSS capabilities, (3) Development and evaluation of RAG -LLM \nusing GPT -4 as sole LLM (4) Bias and fairness of output not evaluated, encoding gender and racial \nbiases has been quantitatively demonstrated for native GPT -4 model44 (5) Knowledge base from local \ninstitution due to copyright issues, (6) Rapid development of new LLM models and versions, limit \nconclusions from study results, (7) Knowledge source was based on local institutional monographs and \ndrug use guidelines, this may not be broadly applicable across different practice settings , (8) No \ncomparison against current knowledge based or non- knowledge based CDSS.   \n \nConclusion  \nThe integration of a RAG -LLM into CDSS  presents as  a novel  tool in improving prescription safety. \nSimulated with LLM integration into healthcare practice, our study reveals that when used in tandem \nwith junior  pharmacists, the RAG -LLM enhances the identification of DRPs,  surpassing the accuracy of \nLLMs  alone . 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The Lancet Digital Health. 2024;6(1):e12 - e22.  \n \n  Supplement 1: Summary of Case Vignettes  \nCase \nNo   Discipline  Brief Description of Clinical Vignette s Number of \nMedications \nPrescribed  ATC Categories  \n1 Cardiology  61-year-old Malay male with history of diabetes, \nhypertension, chronic kidney disease, and \nrecurrent vascular thrombotic events. He has \nundergone multiple surgeries, including \namputations and thrombolysis. He presented \nwith chest pain and was diagnosed with an \nevolved anterior myocardial infarction, for which he underwent successful percutaneous \ncoronary intervention (PCI).  13 Electrolytes, Antithrombotics \n(Heparins), Insulin (fast -acting; \nrapid- acting and long- acting), \nPlatelet aggregation inhibitors, proton pump inhibitors, \nVasodilators, DPP -IV inhibitors, \nBeta- blocking agent (selective), \nACE-inhibitors, B vitamins  \n2 Cardiology  / \nGastroenterology  41-year-old Malay male with a history of \ncoronary artery disease, diabetes mellitus, KDIGO stage 3 kidney disease, and a recent \nepisode of acute gout. He presented with epigastric pain and was admitted to hospital \nfrom the clinic. Investigations included c oronary \nangiography with a view for percutaneous \ncoronary intervention and gastrointestinal \nscopes. His medication allergies include \npenicillin.  13 Fast-acting insulins, parenteral \nnutritional products, blood \nglucose- lowering drugs, oral \nantidiabetics, lipid- modifying \nagents, sulfonylureas, beta-blocking agents, agents for gout, \nmacrolides, nitroimidazole \nderivatives, proton pump \ninhibitors, antithrom botic agents.  \n3 Cardiology  66-year-old male with a history of minor \ncoronary artery disease, hypertension, \nhyperlipidaemia, diabetes, and a chronic left \ncaudate nucleus infarct. He presents with left \nlower limb swelling and pain, fever, and chest \ntightness. Diagnosed with an evolved inferior \nSTEMI and left lower limb cellulitis complicated \nby acute kidney injury.  15 Long Acting insulin,  Heparin \ngroup, low molecular weight heparins , Macrolides , Platelet \naggregation inhibitors excl. heparin, Angiotensin- converting \nenzyme inhibitors , Sulfonylureas , \nBiguanides , Dipeptidyl peptidase 4 \n(DPP -4) inhibitors , HMG CoA \nreductase inhibitors , Beta-blocking \nagents, selective , Dihydropyridine \nderivatives , Proton pump \ninhibitors , Organic nitrates , Drugs \nused in erectile dysfunctio n \n4 Cardiology  75-year-old Chinese male with a history of \nhypertension, hyperlipidaemia, benign prostatic \nhyperplasia, and a right occipital parasagittal \nmeningioma. He presents with atypical chest \npain, which is a dull ache and non- radiating. He \nhas a history of ischemi c cardiomyopathy and \nhas undergone coronary artery bypass graft \nsurgery and mitral valve repair.  6 Beta-blocking agents, selective , \nPlatelet aggregation inhibitors \nexcl. heparin, 5-alpha- reductase \ninhibitors , proton pump inhibitors , \nAlpha- adrenoreceptor antagonists , \nHMG CoA reductase inhibitors  \n5 Cardiology  / \nRespiratory  58-year-old female admitted for acute \npulmonary edema. Her past medical history \nincludes left ataxic hemiparesis, poorly \ncontrolled diabetes mellitus, and hypertension. She exhibits shortness of breath, lower limb \nswelling, and is found to have an NSTEMI w ith \nunderlying chronic type 2 respiratory failure \nlikely contributed by obstructive sleep apnea.  16 Insulins and analogues , Organic \nnitrates , High-ceiling diuretics , \nHeparin group, low molecular weight heparins , Angiotensin II \nantagonists,  Beta-blocking agents, \nselective, Sulfonylureas , Dipeptidyl \npeptidase 4 (DPP -4) inhibitors , \nPlatelet aggregation inhibitors \nexcl. heparin, Proton pump \ninhibitors  \n6 General Medicine  58-year-old male with a history of hypertension, \nadmitted with dizziness and vertiginous \nsymptoms, associated with difficulty balancing. \nHe has a strong smoking history and had a \npossible posterior circulation stroke. The patient \nalso shows signs of hyponatremia, high anion \ngap metabolic acidosis likely from fasting \nketoacidosis, and possible polycythem ia related \nto his smoking history.  7 Angiotensin -converting enzyme \ninhibitors , Platelet aggregation \ninhibitors excl. heparin, HMG CoA \nreductase inhibitors , Combination \nof penicillins, including beta-\nlactamase inhibitors  \n7 Endocrinology  65-year-old Indian female with a history of type \n2 diabetes mellitus, hyperlipidemia, \nhypertension, osteoarthritis, and a past surgical \nprocedure for a distal radius fracture. She was recently admitted for Group B streptococcus \nbacteremia secondary to pneumonia and \nexperienc ed septic shock. She presented with \nan unwitnessed loss of consciousness and was \nfound to have hypoglycemia and acute kidney \ninjury  on admission  10 Angiotensin -converting enzyme \ninhibitors , Other antiepileptics) , \nSulfonylureas , Biguanides , HMG \nCoA reductase inhibitors , High-\nceiling diuretics , Acetic acid \nderivatives and related substances  8 Endocrinology  68-year-old Chinese female with hypertension \nand osteoarthritis. She was previously admitted \nfor severe symptomatic hypercalcemia with \ninappropriately normal parathyroid hormone \nlevels, acute kidney injury, urinary tract \ninfection, hypertensive urgency, anemia, and \nbilateral lower limb weakness due to cervical \nmyelopathy. She is currently admitted from \nclinic for hypercalcemia management.  11 Calcitonins , Propulsives , \nAngiotensin II \nantagonists,Anilides , Antivertigo \npreparations , Osmotically acting \nlaxatives , Vitamin D and \nanalogues , Dihydropyridine \nderivatives  \n9 Endocrinology / \nInfectious disease / \nVascular  71-year-old Chinese male with a complex \nmedical history including ischemic heart disease, poorly controlled type 2 diabetes \nmellitus, peripheral vascular disease, chronic kidney disease stage 4, and moderate- severe \ndementia. He has been admitted multiple t imes \nfor fluid overload and is currently admitted for \nthe same issue. The patient also has infected \nlower limb diabetic ulcers and a painful \nindwelling catheter.  10 Insulins and analogues , High-\nceiling diuretics , Biguanides , HMG \nCoA reductase inhibitors , Platelet \naggregation inhibitors excl. \nheparin, H2-receptor antagonists , \nContact laxatives  \n10 Ophthalmology  \n Female patient with r ight phacomorphic \nglaucoma and pseudophakia. She has diabetes \nmellitus, hypertension, hyperlipidemia, and no \nknown drug allergies. There is a history of \ncataract and glaucoma, with no familial history \nof glaucoma.  16 Carbonic anhydrase inhibitors , \nHMG CoA reductase inhibitors , \nAlpha- adrenoreceptor agonists , \nAngiotensin- converting enzyme \ninhibitors , Sulfonylureas , \nAnxiolytics , Prostaglandin \nanalogues , Sodium -glucose co -\ntransporter 2 (SGLT2) inhibitors , \nBiguanides , Aldosterone \nantagonists)  \n11 Ophthalmology / \nEndocrinology  65-year-old female with severe thyroid eye \ndisease (TED) related to Graves' disease. She has undergone total thyroidectomy and bilateral \norbital decompression. Her current admission is \nfor postoperative management of TED .  15 Combination of penicillins, \nincluding beta- lactamase \ninhibitors , Glucocorticoids ,  \nAnilides , HMG CoA reductase \ninhibitors , Selective COX -2 \ninhibitors , Thyroid hormones , \nProton pump inhibitors ,  \nAminoglycoside antibiotics  \n12 Ophthalmology / \nInfectious Disease   45-year-old male with no significant past \nmedical history. He presented with right eye \nendophthalmitis and underwent various \ntreatments, including intravitreal vancomycin \nand washout procedures. The patient was \ndiagnosed with mycobacterium abscess and \nwas treated with a combination of intravenous \nand oral medications.  12 Carbapenems , Macrolides , \nQuinolone, Other antibacterials , \nAnilides , Glucocorticoids , Other \nopioids  \n13 Gastroenterology  20-year-old female with primary sclerosing \ncholangitis and Child's Pugh A cirrhosis, chronic \nulcerative pancolitis, and a history of \ncholelithiasis. She presented with left knee \npain, fever, and symptoms of a respiratory \ninfection. The preliminary assessment  suggests  \na likely gout flare.  10 \n Immunosuppressants , Bile acids \nand derivatives , Proton pump \ninhibitors , Aminosalicylic acid and \nsimilar agents , Alpha and beta \nblocking agents , Quinolone \nantibacterials , Aminopyrazoles , \nImidazole derivatives ,  \nMonobactams , Combination of \npenicillins, including beta-\nlactamase inhibitors  \n14 Gastroenterology  45-year-old Malay male with a history of \nHepatitis C genotype 3A and Child’s B cirrhosis. \nHe was admitted for deranged liver function \ntests and presented with jaundice. He presented with  a needle stick injury and agrees \nto start treatment with Epclusa for Hepatitis C. Other active issues also includes managing \npossible acute cholecystitis and fluid restriction  6 High-ceiling diuretics , proton pump \ninhibitors , Aldosterone \nantagonists ,Anilides  \n*Epclusa not listed  \n15 Gastroenterology  64-year-old Indian male with a history of Child's \nB8 Cirrhosis, diabetes, hypertension, hyperlipidemia, ischemic heart disease, atrial \nfibrillation, stage 3 chronic kidney disease, and \nhypothyroidism. He presents with worsening \nabdominal distension, lower limb edema, and reduced urine output. The patient has a history \nof spontaneous bacterial peritonitis and is on \nlifelong Ciprofloxacin for SBP prophylaxis.  10 Organic nitrates , Dipeptidyl \npeptidase 4 (DPP -4) inhibitors , \nAminopyrazoles , Proton pump \ninhibitors , Platelet aggregation \ninhibitors excl. heparin, Alpha and \nbeta blocking agents , Quinolone \nantibacterials , Cardiac glycosides , \nThyroid hormones  \n \n16 Gastroenterology  77-year-old female with a history of Child’s C10 \ncryptogenic liver cirrhosis with portal vein \nhypertension. She presented with diuretic -8 Glycopeptide antibacterials , \nAntibiotics , Imidazole derivatives , \nProton pump inhibitors , resistant ascites s/p ascitic drain with H. pylori \ninfection.  Tetracyclines , Osmotically acting \nlaxatives , Quinolone antibacterials , \nDihydropyridine derivatives)  \n \n17 General Surgery / \nOncology  55-year-old Chinese male  with a history of \nperforated duodenal ulcer, epithelial carcinoma \nof the salivary gland with liver metastasis, and \nneutropenia sepsis. His current condition \nincludes postoperative care following \nlaparoscopic omental patch repair of a \nperforated duodenal ulcer and chemotherapy \nfor his cancer . 10 Insulins and analogues , \nEchinocandins , Colony -stimulating \nfactors , Proton pump inhibitors ,  \nAnilides , Combination of \npenicillins, including beta-\nlactamase inhibitors , Other \nopioids , Glycopeptide \nantibacterials , Nucleoside and \nnucleotide reverse transcriptase inhibitors  \n \n18 Vascular Surgery  71-year-old Malay  female  with a complex \nmedical history including end stage renal failure \npost living donor kidney transplant, complicated \nby chronic kidney disease of allograft, left renal \ncell carcinoma post radical nephrectomy, \nhyperparathyroidism with hypercalcemia, \nhypertension, type 2 diabetes mellitus, benign prostatic hyperplasia, and high cholesterol. She  \npresents with right big toe pain and duskiness, leading to a diagnosis of dry gangrene of the \nright first toe .  18 Insulins and analogues , Beta-\nblocking agents, selective , HMG \nCoA reductase inhibitors , \nImmunosuppressants , Other \ncalcimimetics , Platelet aggregation \ninhibitors excl. heparin, \nTrimethoprim and derivatives, combinations with \nsulfamethoxazole , Other lipid \nmodifying agents , Folic acid , High-\nceiling diuretics , Sulfonylureas , \nDipeptidyl peptidase 4 (DPP -4) \ninhibitors , Angiotensin- converting \nenzyme inhibitors , Magnesium , \nMycophenolic acid, proton pump \ninhibitors , Alpha- adrenoreceptor \nantagonists)  \n \n19 General Surgery / \nColorectal Surgery   60-year-old Chinese male with a history of \ndiabetes, hypertension, hyperlipidemia, \npolycythemia rubra vera, end- stage renal failure \non hemodialysis, diverticular disease, and past \nsurgery for perforated jejunal diverticulitis. He \npresented with lower abdom inal pain and \nconstipation  suspicious for intestinal obstruction \nand later developed anal pain.  17 Osmotically acting laxatives , \nAnilides , Contact laxatives , Local \nanesthetics , Other opioids , HMG \nCoA reductase inhibitors , Other \nantiepileptics , Dipeptidyl peptidase \n4 (DPP -4) inhibitors , \nSulfonylureas , Propulsives , Other \ncalcimimetics , Vasodilators used \nin peripheral vascular diseases , \nPhosphate binders , Organic \nnitrates , Quinolone antibacterials  \n20 General Surgery  46-year-old male with a history of adjustment \ndisorder, lupus nephritis, hypertensive urgency, \nand chest pain. He presented with right lower \nabdominal pain associated with fever, \ndiagnosed as perforated appendicitis. He \nunderwent a laparotomy converted to open limited right hemicolectomy.  12 Potassium , Third -generation \ncephalosporins , Imidazole \nderivatives , Proton pump \ninhibitors , Serotonin (5HT3) \nantagonists , Propulsives ,  Anilides , \nDihydropyridine derivatives , \nAngiotensin- converting enzyme \ninhibitors , Non-steroidal anti -\ninflammatory and antirheumatic \nproducts, coxibs , Hydralazine and \ndiuretics , Selective COX -2 \ninhibitors  \n21 General Surgery  71-year-old male with a diagnosis of cecal \ndiverticulitis and a small diverticular abscess. \nHis past medical history includes benign \nprostatic hyperplasia, hypertension, and \nhyperlipidemia. The patient presented with right \niliac fossa pain and had a history of abdominal \npain.  11 Potassium , Third -generation \ncephalosporins , Imidazole \nderivatives , Proton pump \ninhibitors , Propulsives , 5-alpha-\nreductase inhibitors , Alpha-\nadrenoreceptor antagonists , \nAnilides , Other opioids , \nSulfonamides, plain , \nPhenothiazines with aliphatic side-\nchain  \n22 Urology  75-year-old male with a history of ischemic \ncolitis, hemorrhoids, benign prostatic \nhyperplasia (BPH), asthma, hypertension, \ngastritis, ischemic cardiomyopathy, and \nprostate cancer. He was admitted for gross \nhematuria and underwent bladder cystoscopy \nand cystodiathermy. His current issues include \ngross hematuria likely from prostate cancer and \na urinary tract infection  14 Potassium, Combination of \npenicillins, including beta-lactamase inhibitors , Other \nantineoplastic agents , Platelet \naggregation inhibitors excl. \nheparin, Beta- blocking agents, \nselective, 5-alpha- reductase \ninhibitors , Anilides , Alpha-\nadrenoreceptor antagonists , Acetic acid derivatives and related \nsubstances , Organic nitrates , \nHigh-ceiling diuretics , Osmotically \nacting laxatives , HMG CoA \nreductase inhibitors  \n \n23 Urology  67-year-old male with a history of hypertension, \ntype 2 diabetes mellitus, ischemic heart \ndisease, benign prostatic hyperplasia, and \npyelonephritis. He was admitted for pyelonephritis with persistent purulent \ndischarge from the left loin and underwent \npercutaneous drainage.  12 Insulins and analogues , Platelet \naggregation inhibitors excl. \nheparin, Beta- blocking agents, \nselective, Combination of \npenicillins, including beta-lactamase inhibitors , Other \nantiepileptics , Osmotically acting \nlaxatives , Anilides , Other opioids , \nAlpha- adrenoreceptor antagonists ,  \n \n \nSupplement 2: DRPs and Risk / Potential for Harm Categories  \nCase No  DRP(s) Category and Description  Severity / Potential for Harm   \n1 (1)    Drug allergy: Background of NSAIDS allergy (rash) but prescribed with aspirin \nwithout any challenge or test dose  Moderate  \n(2)    Inappropriate dosage regimen: Enoxaparin dosed at 60mg BD in obese patient of \n90kg, no bleeding.  Serious  \n(3)    Omission of therapy: Omission of statin therapy in patient presenting with \nmyocardial infarction  Moderate  \n(4)    Adverse drug reaction: Borderline blood pressure but prescribed with perindopril \nand bisoprolol  Moderate  \n2 (1)    Drug Drug Interaction: Significant Interaction between atorvastatin and \nclarithromycin  Moderate  \n(2)    Inappropriate dosage regimen: Colchicine dosed in MG instead of MCG  Serious  \n(3)    Adverse drug reaction: Bradycardia but prescribed with both bisoprolol and \nticagrelor  Moderate  \n3 (1)    Wrong indication: Wrong drug of clarithromycin instead of clindamycin for cellulitis \nin patient with penicillin allergy  Serious  \n(2)    Inappropriate dosage regimen: Wrong dose of enoxaparin (dosed 1mg/kg BD) in \npatient with renal failure   Moderate  \n(3)    Drug drug interaction: Significant drug interaction (contraindicated) between \nsildenafil and isosorbide mononitrate  Serious  \n4 Control Case  NA \n5 (1)    Adverse drug reaction: Patient presented with hypokalemia, on intensive \nfurosemide therapy without electrolyte replacement  Serious  \n(2)    Adverse drug reaction: Patient presenting with acute pulmonary edema but \ncontinued on beta- blocker therapy  Moderate  \n(3)    No indication for medication: Patient on triple antithrombotic therapy with aspirin / \nclopidogrel / enoxaparin post myocardial infarction, also continued on dipyridamole \n(chronic medication for previous history of CVA)  Serious  \n6 (1)    Duplication of therapy: Duplication between Simvastatin and Atorvastatin. Both \nbelong to the same category of HMG -CoA reductase inhibitor  Minor  (2)    Inappropriate dosing frequency for atorvastatin. It is usually given once daily, but \ndoes not exceed max daily dose  Minor  \n(3)  Adverse Drug Reaction:  Patient has allergy to amoxicillin (rash) but was prescribed \nwith Co -amoxiclav which contains amoxicillin Moderate  \n7 (4)    Adverse Drug Reaction: Acute kidney injury with hyperkalemia but continued on \nenalapril  Moderate  \n(5)    Adverse drug reaction: Acute kidney injury but prescribed with NSAID  Moderate  \n(6)    Adverse drug reaction: Presented with hypoglycemia but continued on glipizide  Serious  \n8 (1)    Inappropriate dosage regimen: Wrong weight -based dose of calcitonin prescribed \nfor patient with hypercalcemia of malignancy  Serious  \n(2)    Adverse drug reaction: Colecalciferol not held off despite hypercalcemia  Moderate  \n(3)    No indication: Wrong drug of cinnarizine prescribed instead of cinacalcet for patient \nwith b/g secondary hyperparathyroidism  Minor  \n9 (1)    Duplication of therapy: Patient with poorly controlled diabetes on both ultra -short -\nacting insulin Aspart and short -acting insulin Actrapid.  Moderate  \n(2) Omission of therapy: Omisson of basal insulin such as insulatard in patient with \npoorly controlled diabetes  Moderate  \n(3)    Omission of therapy: Antibiotics not ordered for patient with suspected sepsis  Serious  \n10 (1)    Adverse drug reaction: Laboratory results demonstrating non -anion gap metabolic \nacidosis with respiratory compensation, continued on acetazolamide  Moderate  \n(2)    Adverse drug reaction: Hydroxyzine (first generation antihistamine with significant \nanticholinergic effects) contraindicated in acute- angled glaucoma  Moderate  \n(3)    Adverse drug reaction: Patient with frequent urinary tract infection but continued \non dapaglifozin, an SGLT -2 inhibitor  Moderate  \n11 (1)    Duplication of therapy: Patient prescribed with PO prednisolone and IV \nMethylprednisolone (high- intensity) concurrently  Minor  \n(2)    Inappropriate dosage regimen: Levothyroxine prescribed in MG instead of MCG  Serious  \n12 (1)    Drug drug interaction: Significant interaction between tramadol and linezolid \nleading to increased risk for serotonin syndrome  Moderate  \n(2)    Duplication of therapy: Patient prescribed with IV Tienam and PO ciprofloxacin for \ngram -negative coverage of urinary tract infection  Minor  \n(3)    Inappropriate dosage regimen: Instructions for prednisolone eye drops was to \nadminister to left eye, when the affected eye was the right  Serious  \n13 (1)    Drug drug interaction: Significant (contraindicated) interaction between \nazathioprine and allopurinol leading to increased risk for neutropenia  Serious  \n(2)    Omission of therapy: Patient initiated on allopurinol but not on anti -inflammation \ntherapy to avoid exacerbation of gout flare (e.g. colchicine or steroids)  Moderate  \n14 (1)    Omission of therapy: Noted complains of no bowel movement for n -days but no \nlaxatives ordered for patient with a significant history of liver cirrhosis  Moderate  (2)    Drug drug interaction: Significant interaction between Epclusa and omeprazole that \nwill reduce absorption and efficacy of anti -viral agent  Moderate  \n(3)    Inappropriate dosage regimen: Dosage of paracetamol not adjusted in the \npresence of liver cirrhosis  Moderate  \n15 Control Case  NA \n16 (1)    Drug drug interaction: Significant interaction between calcium supplement and \ntetracycline. Calcium reduces absorption of tetracycline when taken concurrently  Moderate  \n(2)    Inappropriate dosage regimen: Dose of metronidazole not adjusted in the presence \nof significant liver impairment  Moderate  \n17 (1)    Inappropriate dosage regiment: The infusion rate of vancomycin has exceeded the \nmaximum recommended rate, leading to increased risk for red man’s syndrome  Serious  \n(2)    No indication: G -CSF (filgrastim) continued despite recovery of absolute neutrophil \ncounts  Moderate  \n18 (1)    Drug drug interaction: Significant interaction between ciclosporin and atorvastatin, \nwith max dose of atorvastatin limited to not more than 20mg per day  Moderate  \n(2)    Inappropriate dosage regimen: Inappropriate dose of co -trimoxazole, prescribed in \ntrimethoprim component instead of co- trimoxazole for PCP prophylaxis  Moderate  \n19 (1)    Inappropriate dosage regimen: Dose of tramadol not adjusted in the presence of \nsignificant renal failure  Moderate  \n(1)    Inappropriate dosage regimen: Dose of gabapentin not adjusted in the presence of \nsignificant renal failure  Moderate  \n(2)    Omission of therapy: Aspirin omitted in patient with significant cardiac history and \nlow risk of bleeding  Minor  \n(3)    Drug drug interaction: significant interaction between sulphonylurea and \nciprofloxacin leading to increased risk for hypoglycemia  Minor  \n20 (1)    Duplication of therapy: Celecoxib and etoricoxib overlapping mechanism of action  Minor  \n(2)    Duplication of therapy: PO and IV omeprazole both ordered  No harm  \n(3)    Adverse drug reaction: Initiation of hydralazine in patient with a background of lupus \ndisease  Serious  \n21 (1)    Adverse Drug Reaction: Hyperkalemia but continued on potassium chloride \ninfusion  Serious  \n(2)    Inappropriate dosage regimen: Exceeded maximum recommended dose of \nomeprazole for the indication of ulcer prevention  Minor  \n(3)    Drug drug interaction: Significant interaction between prochlorperazine and \nmetoclopramide  Serious  \n(4)    Omission of therapy: Untreated hyperlipidemia  Minor  \n(5)    Omission of therapy: Untreated hyperkalemia  Serious  \n22 (1)    Inappropriate dosage regimen: Rapid intravenous infusion of potassium chloride \n10mmol over 1 minute  Moderate  \n(2)    Omission of therapy: Omission of steroid (e.g. prednisolone) when on abiraterone \ntreatment for castrate resistant bladder cancer  Moderate  (3)    Adverse drug reaction: Diclofenac use in patient with recent myocardial infarction \npresenting with heightened risk for cardiovascular events  Serious  \n23 (1)    Duplication of therapy: Prescription of 2 alpha -blockers (Tamsulosin and Alfuzosin) \nin a patient with history of benign prostate hyperplasia  Minor  \n(2)    Wrong choice of therapy: Co -amoxiclav use in patient with wound culture growing \nE.Coli  reported to be resistant to co- amoxiclav  Moderate  \n \n    \n \n \n \n    "    },    {        "title": "2402.01780v1.Late_time_phantom_characteristic_of_the_model_in__f_R_T___gravity_with_quadratic_curvature_term.pdf",        "content": "Late time phantom characteristic of the model in f(R, T )gravity with quadratic\ncurvature term\nShaily,1, 2,∗Akanksha Singh,1,†J. K. Singh,1,‡and Saibal Ray3,§\n1Department of Mathematics, Netaji Subhas University of Technology, New Delhi 110078, India\n2School of Computer Science Engineering and Technology,\nBennett University, Greater Noida 201310, India\n3Centre for Cosmology, Astrophysics and Space Science (CCASS),\nGLA University, Mathura 281406, Uttar Pradesh, India\nWe propose a novel cosmological framework within the f(R, T) type modified gravity theory,\nincorporating a non-minimally coupled with the higher order of the Ricci scalar ( R) as well as the\ntrace of the energy-momentum tensor ( T). Therefore, our well-motivated chosen f(R, T) expression\nisR+Rm+2λTn, where λ,m, and nare arbitrary constants. Taking a constant jerk parameter ( j),\nwe derive expressions for the deceleration parameter ( q) and the Hubble parameter ( H) as functions\nof the redshift z. We constrained our model with the recent Observational Hubble Dataset (OHD),\nPantheon , and Pantheon + OHD datasets by using the analysis of Markov Chain Monte Carlo\n(MCMC). Our model shows early deceleration followed by late-time acceleration, with the transition\noccurring in the redshift range 1 .10≤ztr≤1.15. Our findings suggest that this higher order model\noff(R, T) gravity theory can efficiently provide a dark energy model for addressing the current\nscenario of cosmic acceleration.\nPACS numbers: 04.20.-q, 04.50.Kd, 98.80.-k, 98.80.Es\nKeywords: FLRW metric, f(R, T) gravity, constant jerk parameter, phantom model\nI. INTRODUCTION\nCosmological discoveries have revealed that astronomical structures in the universe exhibit a systematic arrangement\nrather than a random distribution. By studying this organized pattern and examining the physical characteristics of\ninterstellar objects, we can gain insights into the phenomenon of cosmic accelerated expansion. While general relativity\nserves as the foundation of modern physics and effectively explains black holes and gravitational phenomena, it falls\nshort of providing a comprehensive explanation for cosmic acceleration. To shed light on this intriguing concept,\nresearchers have turned their attention towards investigating dark energy and exploring modified theories of gravity.\nVarious astronomical studies, which include the Sloan Digital Sky Survey (SDSS), the 2-Degree Field Galactic Redshift\nSurvey (2DFGRS), and the Huge Synoptic Survey Telescope (HSST), have demonstrated the significant influence\nthat galaxies and stars have on the evolution of the universe. Analyzing these fundamental components is crucial to\ndeepening our understanding of the formation and nature of the universe. Through the examination of gravitating\nsystems, we can uncover the underlying physics of dark matter and dark energy. As theoretical cosmology confronts\nthe challenges posed by the existence of dark energy and compelling experimental evidence, modified gravity theories\nemerge as encouraging avenues for refining our theoretical understanding of late-time cosmic acceleration.\nProminent modified gravitational theories in astrophysics as well as cosmology include f(R) [1–4], f(R, T) [5–7],\n∗shaily.ma19@nsut.ac.in\n†akanksha.ma19@nsut.ac.in\n‡jksingh@nsut.ac.in\n§saibal.ray@gla.ac.inarXiv:2402.01780v1  [gr-qc]  1 Feb 20242\nf(T) [8–10], f(G) [11, 12], f(R, G) [13, 14], f(G, T) [15, 16] and f(R, ϕ) [17] (however, for interested readers, there\nare a few more references herein to consult the alternative gravity theories [18–32]). A cornerstone in the exploration\nof gravitational theories is the f(R) gravity theory, pioneered by Buchdahl [33], which involves substituting the Ricci\nscalar Rin the Einstein-Hilbert (EH) action with an arbitrary function f(R). This theory has been examined and\nverified as a viable explanation for late-time cosmic acceleration while also aligning with local gravitational tests [3, 34–\n39]. While f(R) gravity remains a focus of extensive study, recent decades have witnessed a surge of interest in f(T)\ngravity. This alternative theory has also provided explanations regarding cosmic expansion and galactic dynamics, as\nevidenced by references [40–42]. In their work, Harko et al. [43] first put forward the f(R, T) theory as a modification\nof general relativity (GR) that involves altering the geometric part of the EH action, where Ris the Ricci scalar and\nTdenotes the trace of the energy-momentum tensor (EMT). In this theory, Ris replaced by an arbitrary f(R, T)\nfunction, and due to the incorporation of T, this theory encompasses novel gravitational aspects and also satisfies the\nweak-field conditions of the solar system.\nIn the realm of cosmic dynamics, Houndjo [44] shed light on the transition from a matter-dominated epoch to\nthe late-time acceleration phase using the minimally coupled f(R, T) model. Within the vast expanse of scientific\nliterature, several distinct formulations emerge, each offering a unique lens into the functional dependence of f(R, T).\nResearchers have explored various mathematical expressions to represent the dependence on T, including polynomials,\nrational functions, and exponential expressions [45–51]. It is argued that the T-dependence can arise due to the\npresence of quantum effects or imperfect fluids. Amidst the array of modified gravitational theories, the f(R, T)\ntheory stands out as it incorporates an energy transfer relationship between matter and geometry, thus potentially\nexhibiting the observed phenomenon related to the accelerating phase of the cosmic expansion. In connection to a\nfew important features of f(R, T) gravity theory, the following works are notable [52–56]. The consequences of the\nf(R, T) theory on cosmological and solar system models have been investigated in [57]. In literature, discussions on\nbouncing models within the f(R, T) theory have also gained attention [58–60].\nMotivated by the above-mentioned f(R, T) theory-based works, our current investigation starts with the general\npowers of curvature and trace of the energy-momentum tensor. Specifically, we consider the form f(R, T) =R+Rm+\n2λTn, subsequently delving into a detailed analysis with distinct parameter values for mandn. In this context, it is to\nbe noted that for wormhole-related investigations, Elizalde and Khurshudyan [61] exploited a similar form of f(R, T)\nwith specific parametric values, such as m= 2 and n= 1. We also notice that very recently Mahapatra and Das [62]\nhave employed a polytropic model under f(R, T) in a restricted manner to study the equilibrium configurations of\nrelativistic neutron stars. However, in this connection, we would like to emphasize that our approach is more general\nso that the results can be expected to offer a wide range of flexibility to explore the multifarious features of the\nlate-time phantom behavior of the universe.\nThe structured sequence of our research is: Section II serves as the starting point, where we delve into the com-\nputation of the Hubble parameter by using the jerk and deceleration parameter expressions. Next, we derive the\nEinstein field equations for a specific form of the f(R, T) function in Sec. III. In our model, we constrain three model\nparameters using the OHD dataset, the Pantheon dataset, and their joint (OHD + Pantheon ) dataset in Sec. IV.\nSec. V contains the explanation related to the dynamical behavior of the universe for this model. At last, in Sec. VI,\nwe discuss the results and conclude our work.3\nII. FORMULATION OF THE HUBBLE PARAMETER\nLet us first define here the jerk parameter j\nj(z) =q(z) + 2q(z)2+ (1 + z)dq(z)\ndz, (1)\nwhere qandzare, respectively, the deceleration parameter and the redshift.\nTreating jas a constant and subsequently solving the above differential equation leads to\n√8j+ 1 + 4 q+ 1√8j+ 1−4q−1= (α+αz)√8j+1, (2)\nwhere αis an integration constant.\nRewriting Eq. (2) we get\nq=−1\n4+√8j+ 1\n4−√8j+ 1\n2\u0010\n1 + (α+αz)√8j+1\u0011. (3)\nThe equation representing qas a function of the Hubble parameter Hand redshift zis\nq(z) =−1 +(1 +z)\nH(z)dH\ndz. (4)\nUsing Eq. (3) in Eq. (4), we get\n1\n1 +z\n1−1\n4+√8j+ 1\n4−√8j+ 1\n2\u0010\n1 + (α+αz)√8j+1\u0011\ndz=dH\nH. (5)\nBy integrating both sides of Eq. (5), we obtain\nH=β(1 +z)3+√8j+1\n4q\n1 + (α+αz)−√8j+1, (6)\nwhere βis an integration constant.\nIn order to work with observational data, we define βasH0√\n1+α−√8j+1, where H0=H(0). Subsequently, one can\nfind the H, which is as follows\nH=H0p\n1 +α−√8j+1(1 +z)3+√8j+1\n4q\n1 + (α+αz)−√8j+1. (7)\nIII. FORMULATION OF f(R, T)THEORY OF GRAVITY\nThe gravitational action corresponding to f(R, T) gravity theory is\nS=Z\"\nf(R, T)\n16πG+Sm#\n√−gd4x, (8)4\nwhere R,T,g,G, and Sm, respectively, denote the Ricci scalar, the trace of the EMT, the determinant of the metric\ntensor, the gravitational constant, and the matter Lagrangian. Here, the function f(R, T) is explicitly defined as\nf(R, T) =f1(R) + 2f2(T) [43].\nOn varying the action (8) with respect to the metric tensor gµν, one can easily get the following gravitational\nequation [32]\nfR\n1(R)Rµν−1\n2gµν(f1(R) + 2f2(T)) + ( gµν□− ∇ µ∇ν)fR\n1(R) = 8 πGT µν−2fT\n2(T)(Tµν+ Θ µν), (9)\nwhere fR\n1(R) symbolizes the first order differentiation of f1with respect to RandfT\n2(T) means the first order\ndifferentiation of f2with respect to T. The d’Alembert operator is denoted by □, and ∇µsignifies the covariant\nderivative with respect to gµν. The expression Θ µνcan be written as\nΘµν=gijδTij\nδgµν=−2Tµν+gµνSm−2gijδ2Sm\nδgµνδgij. (10)\nConsidering a perfect fluid, the matter Lagrangian is conventionally represented as Sm=−p. In this case, the\nEMT is given by Tµν= (ρ+p)uµuν−pgµν, where the physical parameters have their usual meanings. Using these\nexpressions, we get Θ µν=−2Tµν−pgµν. Therefore, Eq. (9) takes the following form\nfR\n1(R)Rµν−1\n2gµν(f1(R) + 2f2(T)) + ( gµν□− ∇ µ∇ν)fR\n1(R) = 8 πGT µν+ 2fT\n2(T)(Tµν+pgµν). (11)\nThe form in which the Einstein field equations are expressed is given by\nGµν=Rµν−1\n2Rgµν=8πG\nfR\n1(R)(Tµν+T∗\nµν), (12)\nwhere\nT∗\nµν=1\n8πGh\n1\n2gµν((f1(R) + 2f2(T))−RfR\n1(R)) + (∇µ∇ν−gµν□)fR\n1(R) + 2fT\n2(T)(Tµν+pgµν)i\n.\nNow, the metric of the FLRW universe is expressed by\nds2=dt2−a2(t)3X\ni=1dx2\ni. (13)\nHere, a(t) is the scale factor. Now, by considering f1(R) =R+Rmandf2(T) =λTn, where T=ρ−3pand\nR=−6(2H2+˙H), we can obtain the Einstein field equations (EFE) as\n3H2=1\n1 +mRm−1[8πρ+1\n2((1−m)Rm+ 2λTn)−3Hm(m−1)Rm−2˙R+ (ρ+p)2λnTn−1]\nor\n3H2=1\n1 +mRm−1[8πρ+1\n2((1−m)(−6(2H2+˙H))m+ 2λTn)] +\n1\n1 +mRm−1[−3Hm(m−1)(−6(2H2+˙H))m−2(−6(4H˙H+¨H)) + ( ρ+p)2λnTn−1],5\n2˙H+ 3H2=−1\na2(1 +mRm−1)[8πpa2+1\n2(−a2)((1−m)Rm+ 2λTn) +am(m−1)Rm−3(a((m−2)˙R2+R¨R) + 3˙aR˙R)]\nor\n2˙H+ 3H2=−1\na2(1 +mRm−1)[8πpa2+1\n2(−a2)((1−m)(−6(2H2+˙H))m+ 2λTn) +\naζm(m−1)(−6(2H2+˙H))m−3(a((m−2)(−6(4H˙H+¨H))2+ (−6(2H2+˙H))(−6(4H¨H+ 4˙H2+...\nH))) +\n3˙a(−6(2H2+˙H))(−6(4H˙H+¨H)))]. ‘\nAs the calculations involved in determining the physical parameters ρandpbased on the above equations are\ncomplex, we simplify the analysis by choosing m= 2 and n= 1. Now, we have taken f1(R) =R+R2, a choice\npreviously addressed by Starobinsky [63], and f2(T) =λT. The EFEs for our chosen model are\n(8π+ 3λ)ρ−λp= 3H2+ 18( ˙H2−6H2˙H−2H¨H), (14)\n(8π+ 3λ)p−λρ=−2˙H−3H2+ 6(26 ˙HH2+ 9˙H2+ 14H¨H+ 2...\nH). (15)\nIV. OBSERVATIONAL ANALYSIS\nObservational analysis is an endeavoring field in cosmology where we calculate the optimal value of model\nparameters with the help of some experiments. In this analysis, the most accurate and strong persistence results are\nobtained from the Planck satellite mission. In this model, we determine the optimal values for model parameters by\nconducting a best-fit analysis with the help of three distinct observational datasets, namely, the Hubble dataset (OHD)\nwith 77 data points, the Pantheon dataset with 1048 data points, and the combined dataset (OHD + Pantheon ).\nThe Hubble parameter value from Eq. (7), which contains three model parameters named α,j, and H0, is used for\nour analysis. In order to constrain the mentioned model parameters, we basically execute the Markov Chain Monte\nCarlo (MCMC) method, implemented through the emcee library in Python.\nA. H(z)Dataset\nExploring the dark division of the universe involves tapping into the insightful H(z) data, and this data is\nbasically acquired from the cosmic chronometers [64]. In our model, we use the OHD along with 77 data points [65].\nThereafter, by using the χ2test, we find the parametric best-fit values of the present model.\nNow, the formula for χ2is drafted as\nχ2\nHb(α, j, H 0) =77X\ni=1[H(α, j, H 0, zi)−Hobs(zi)]2\nσ2zi, (16)\nwhere H(α, j, H 0, zi) and Hobsare denoted for, respectively, the theoretical and observed values of the Hubble\nparameter. Additionally, σ(zi)stands for the standard deviation for every value of H(zi).\nWe determine that the analysis of only the Hubble dataset and the function χ2\nHbis not very authentic for the6\nFIG. 1: Visualization of the likelihood contours using the H(z) dataset.\ncosmological models. Therefore, we take the Pantheon dataset and their joint dataset into consideration.\nTABLE I: Best-fit values of model parameters where D1=H(z),D2=Pantheon , and D3=H(z) +Pantheon .\nDataset α j H 0\nD1 0.59905+0.00097\n−0.00023 0.9215+0.0076\n−0.0210 70.932+0.069\n−0.020\nD2 0.59998±0.0001 0 .949998 ±0.000098 70 .000013+0.000081\n−0.000099\nD3 0.598987+0.000094\n−0.000110 0.98999+0.000120\n−0.000085 70.3719±0.0001\nB. Pantheon Dataset\nIn the last two decades, Type Ia supernovae have emerged as pivotal tools in the analysis of cosmological\nparameters, notably serving as the first evidence for the late time accelerated expansion phase of the universe.\nSupernovae are the stars, and as they burst, they release a lot of energy and enlarge their outer shell. Thus, this part\nis related to examining the time evolution of their brightness and their spectrum. In this model, the Pantheon dataset\nwhich comprises of 1048 data points, is used to constrain the parameters of the model [66]. The Pantheon dataset\ncontains the resulting compilation of different surveys, including the CfA1-CfA4 surveys, the Sloan Digital Sky Survey\n(SDSS), the Supernovae Legacy Survey (SNLS), the Carnegie Supernova Project (CSP), the Pan-STARRS1 (PS1),\nand various Hubble Space Telescope (HST) surveys, all of which span a redshift range from 0 .01 to 2 .26 [67–71]. In\nType Ia supernovae analysis, we deal with redshift as well as luminosity distance, which are closely related to standard7\nFIG. 2: Visualization of the likelihood contours for the Pantheon dataset.\ncandles in the universe. The luminosity distance is defined as\nDL(z) = (1 + z)Zz\n0c\nH(z∗)dz∗, (17)\nwith cas the speed of light in a vacuum.\nNow, m, the apparent magnitude can be provided as\nm(z) =M+ 5log10\"\nDL(z)\n1Mpc#\n+ 25, (18)\nwith Mdenoting the absolute magnitude.\nHere, a dissipation between H0andMis observed, and given that the distance modulus µis equal to m−M, the\nχ2formula is written as\nχ2\nPn(α, j, H 0) =1048X\ni=1\u0014µth(α, j, H 0, zi)−µobs(zi)\nσµ(zi)\u00152\n, (19)\nwhere µthandµobsdisplay the theoretical and observed distance moduli, respectively.\nC. Joint Dataset ( H(z) +Pantheon )\nFor enhanced precision in constraining model parameters, we make use of the joint dataset, and the results are\ncalculated by the χ2\nHPfunction, which is the sum of the functions χ2\nHbandχ2\nPnas\nχ2\nHP(α, j, H 0) =χ2\nHb(α, j, H 0) +χ2\nPn(α, j, H 0). (20)\nThe obtained values of α,j, and H0from the above-mentioned observational datasets are placed in Table I. In Fig.\n1, we notice that the obtained contours are not properly in an oval shape, but we have tested using the Gelman-Rubin8\nFIG. 3: Visualization of the likelihood contours for the H(z) +Pantheon dataset.\n-0.04-0.02 0.00 0.02 0.046768697071727374\n-0.5 0.0 0.5 1.0 1.5 2.0 2.550100150200250\nzHHzL@KmMpc-1s-1D\nHHzL\nPantheon\nHHzL+Pantheon\nLCDM\n(a)\n0.0 0.5 1.0 1.534363840424446\nzmHzL\nHHzL\nPantheon\nHHzL+Pantheon\nLCDM (b)\nFIG. 4: The error bar plots for H(z) and Type Ia Supernovae datasets exhibiting the similarities between the\npresented model and the ΛCDM.\nconvergence test that such contour patterns also converge [72, 73].\nFor a better understanding of our model, we conduct a comparative analysis with the ΛCDM model, illustrated\nthrough error bar plots. In Fig. 4, there are three distinct trajectories concerning the best-fit parametric values of\nthe model. The Hubble dataset and the Type Ia Supernovae dataset based error plots are presented in Figs. 4(a) and\n4(b), respectively.9\n1.08 1.10 1.12 1.14 1.16 1.18-0.02-0.010.000.010.020.03\n-1 0 1 2 3 4 5-1.0-0.8-0.6-0.4-0.20.00.20.4\nzqHHzL\nPantheon\nHHzL+Pantheon\nFIG. 5: The graphical representation of qvs.z.\nV. PHYSICAL PARAMETERS\nThis section involves the scrutinization of physical parameters, systematically exploring their behavior across\nthe best-fit values corresponding to the Hubble dataset, the Pantheon dataset, and their joint dataset. To gain\nknowledge about phase-wise cosmic evolution, we examine the features of the different model parameters, especially\nthe dimensionless deceleration parameter expressed in Eq. (3).\nFig. 5 demonstrates a trend in which the deceleration parameter qexperiences a declination from high to low\nredshift values. Also, this parameter undergoes a sign change, indicating a phase shift from the deceleration to the\ncosmic acceleration. It is observed that for all the trajectories at the early epochs, the universe was in the deceleration\nmode, whereas at present, as well as in late time eras, the universe is in the accelerating mode. The values of\nredshift corresponding to the phase transition phenomena are 1 .14161, 1 .12361, and 1 .10797 for the H(z) dataset, the\nPantheon dataset, and their joint dataset, respectively. Furthermore, the deceleration parameter assumes values of\n−0.705673, −0.715246, and −0.731007 at the present for H(z),Pantheon , and their joint dataset, respectively.\nFor convenience,the jerk parameter jcan be articulated in relation to the deceleration parameter q, as in Eq. (1).\nIn the context of the ΛCDM model, the jerk parameter maintains a constant value of 1. In our model, we determine\nthe best-fit values for the jerk parameter to be 0 .9215, 0 .949998, and 0 .98999 for the H(z) dataset, the Pantheon\ndataset, and their joint dataset, respectively. Notably, all three values of jare remarkably close to unity, reflecting\nthe consistency and agreement found within the results.\nBy using Eqs. (14) and (15), we calculate the values of ρandp. However, since their expressions are very large\nand complicated, we have plotted their trajectories directly for every mentioned dataset. We have Fig. 6(a), in which\nit is observed that ρis decreasing monotonically from the early to late times. For the isotropic pressure, Fig. 6(b)\nshows that all the curves are in a highly positive region, but afterward, they decrease and enter a negative region.\nAt present, the negative value of psignifies an expanding and accelerating universe, while in late times, the model\nbecomes pressureless since ptends to zero as zapproaches −1. In Fig. 6(c), the progression of the Equation of State\n(EoS) parameter ( ω=p/ρ) has been unfolded: starting from a perfect fluid, then it is entering the quintessence\nregion and eventually remaining in the phantom region. For all observational datasets, the value of ωis less than −1,10\n-1.0-0.8-0.6-0.4-0.205000001.0´1061.5´1062.0´1062.5´106\n-10 1 2 3 4 501´10102´10103´10104´1010\nzrHHzL\nPantheon\nHHzL+Pantheon\n(a)\n-1.0-0.5 0.0 0.5 1.0-4´106-2´10602´1064´1066´1068´106\n-10 1 2 3 4 502´1094´1096´1098´109\nzp\nHHzL\nPantheon\nHHzL+Pantheon (b)\n-0.10 -0.05 0.00 0.05 0.10-2.0-1.8-1.6-1.4-1.2-1.0\n-1 0 1 2 3 4 5-3-2-101\nzwHHzL\nPantheon\nHHzL+Pantheon\n(c)\nFIG. 6: The graphical representations of the physical parameters, viz., ρ,p, and ωvs.zfor all the observational\ndatasets.\nindicating that our model represents a phantom scenario both at the present as well as in the late times.\nBesides phenomenal fields, there are some major points to explore, such as the classification of singularities, causal\nstructure, energy conditions, etc. In particular, energy conditions originated from the well-known Raychaudhuri\nequation [74–78], serving as a valuable framework to assess the influence of all the geometric terms within the stress-\nenergy tensor. Stemming from the Einstein field equations, energy conditions involve diminutions of time-like or null\nvector fields with respect to the EMT and Einstein tensor. These conditions stand out as prominent features in\ncosmological models, defined as the weak energy condition (WEC: ρ≥0, ρ+p >0), null energy condition (NEC:\nρ+p≥0), strong energy condition (SEC: ρ+ 3p≥0), and dominant energy condition (DEC: ρ≥ |p|).11\n-1.0-0.8-0.6-0.4-0.20.0-1´106-50000005000001´106\n-10 1 2 3 4 501´10102´10103´10104´1010\nzNECHHzL\nPantheon\nHHzL+Pantheon\n(a)\n-1.0 -0.5 0.0 0.5-8´106-6´106-4´106-2´10602´1064´1066´106\n0 2 4 6 802´1094´1096´1098´1091´1010\nzSECHHzL\nPantheon\nHHzL+Pantheon (b)\n-1.0-0.50.00.51.01.52.001´1082´1083´1084´108\n-10 1 2 3 4 505.0´1091.0´10101.5´10102.0´1010\nzDECHHzL\nPantheon\nHHzL+Pantheon\n(c)\nFIG. 7: The representative graphical plots of the energy conditions.\nIn Fig. 7, we interestingly perceive that for all the observational datasets, NEC and SEC behave alike. Both the\nenergy conditions hold good in the early universe but are violated in the present as well as in later eras. On the other\nhand, DEC holds for all time periods. The violation of SEC hints at the existence of dark energy in our model (see\nFig. 7(b)). The breach of NEC indicates the phantom behavior of the model in late times, indicating its instability\nfor an interval of redshift values (see Fig. 7(a)).\nVI. CONCLUSION\nIn the present investigation, we try to investigate a cosmological model in the alternative gravity theory f(R, T).\nThe motivation for choosing this gravity theory arises from its capacity to explore both the physical and geometric\nbehavior of the cosmos. We obtained the EFEs by taking a specific form of the f(R, T) function. The Hubble12\nparameter has been calculated here in terms of the redshift parameter along with three model parameters from the\njerk and deceleration parameter relation. Also, we enumerate the best-fit values of the model parameters, namely,\nj,H0, and α, by utilizing the observational datasets H(z),Pantheon , and their joint data (refer to Table I). For\nthis purpose, we used the MCMC method with the help of the emcee package in Python. Interestingly, one can find\nthat our results of jandH0are consistent with the recently obtained Planck data. The contours plotted with 1 −σ\nand 2 −σconfidence levels have been discussed (see Figs. 1, 2, and 3). To assess the performance of our f(R, T)\nmodel against the ΛCDM model, we have used error bar plots in Fig. 4. We observe the cosmic phase transition from\ndeceleration to acceleration mode, marked by the shift of the qfrom positive to negative values.\nThe top left panel of Fig. 6 provides a visual narrative of the evolution of the energy density from the earlier to\nthe later phase of the universe. For all three datasets, the curves of energy density decrease monotonically from high\nto low redshift. In Fig. 6(b), we notice that the isotropic pressure shows positive behavior in the early cosmic epoch,\nwhereas it has a negative signature in the present stage and zero in the late epoch. This indicates an accelerating as\nwell as expanding universe at present and a pressureless universe in the future. The trajectories of EoS parameter ω\npass through the various stages of evolution of the cosmos as it starts as perfect fluid enters the quintessence dark\nenergy region and then after remains in the phantom region (see Fig. 6(c)).\nIn this model, we observe that the energy conditions, NEC and SEC, face violations at low redshift values, whereas\nDEC holds across the entire value of redshift (see Fig. 7).\nFinally, based on the above-mentioned features, one can note that the presented model of f(R, T) gravity in the\nbackground of the flat FLRW metric is quite different from the usual dark energy models. Undoubtedly, via the plots\nand exhibited physical features, our obtained model in the f(R, T) gravity theory emerges as a promising candidate\namong various modified gravity theories and thus indicates the usefulness of further scope to work in the modified\ntheory of gravity for verification of observational signatures in a more detailed and significant way.\nACKNOWLEDGEMENT\nSR gratefully acknowledges support from the Inter-University Centre for Astronomy and Astrophysics (IUCAA),\nPune, India, under its Visiting Research Associateship Programme as well as the facilities under ICARD, Pune, at\nCCASS, GLA University, Mathura, India.\nDATA AVAILABILITY STATEMENT\nIn this manuscript, we have used observational data as available in the literature; as such, our work does not\nproduce any form of new data.\nCONFLICTS OF INTEREST\nThe authors assert that there are no conflicts of interest of the publication of this work.\n[1] S. Capozziello, Int. J. Mod. Phys. 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Dark Univ. 43, 101397 (2024)."    },    {        "title": "2402.01885v2.Understanding_Growth_Mindset_Practices_in_an_Introductory_Physical_Computing_Classroom__High_School_Students__Engagement_with_Debugging_by_Design_Activities.pdf",        "content": "U n d e r s t a n d i n gG r o w t hM i n d s e tP r a c t i c e si na nI n t r o d u c t o r yP h y s i c a lC o m p u t i n gC l a s s r o o m :H i g hS c h o o lS t u d e n t s ’E n g a g e m e n tw i t hD e b u g g i n gb yD e s i g nA c t i v i t i e s\nLuisMor ales-Na varro,Univ ersityofPenns ylvania,luismn@upenn.eduDebor ahA.Fields,UtahStateUniv ersity,debor ah.fields@usu.eduYasminB.Kafai,Univ ersityofPenns ylvania,kafai@upenn.edu\nB a c k g r o u n da n dC o n t e x t:Whiledebuggingisrecognizedasanessentialpractice,formanystudents,encounteringbugscangenerateemotionalresponsessuchasfearandanxietythatcanleadtodisengagementandtheavoidanceofcomputerprogramming.Growthmindsetscansupportperseveranceandlearninginthesesituations,yetfewstudieshaveinvestigatedhowgrowthmindsetsemergeinpracticeamongstK–12computingstudentsfacingphysicalcomputingdebuggingchallenges.O b j e c t i v e:Weseektounderstandwhat(ifany)growthmindsetpracticeshighschoolstudentsexhibitedwhencreatingandexchangingbuggyphysicalcomputingprojectsfortheirpeerstosolveduringaDebuggingbyDesignactivityaspartoftheirintroductorycomputingcourse.M e t h o d:Wefocusedonmoment-to-momentmicrogeneticanalysisofstudentinteractionsindesigningandsolvingbugsforotherstoexaminethepracticesstudentsexhibitedthatdemonstratedthedevelopmentofagrowthmindsetandthecontextsinwhichthesepracticesemerged.F i n d i n g s:Weidentifiedfiveemergentgrowthmindsetpractices:choosingchallengesthatleadtomorelearning,persistingaftersetbacks,givingandvaluingpraiseforeffort,approachinglearningasconstantimprovement,anddevelopingcomfortwithfailure.Studentsmostoftenexhibitedthesepracticesinpeer-to-peerinteractionsandwhilemakingbuggyphysicalcomputingprojectsfortheirpeerstosolve.I m p l i c a t i o n s: Ouranalysiscontributestoamoreholisticunderstandingofstudents’social,emotional,andmotivationalapproachestodebuggingphysicalcomputingprojectsthroughthecharacterizationofgrowthmindsetpractices.Thepresentedinventoryofgrowthmindsetpracticesmaybehelpfultofurtherstudygrowthmindsetinactioninothercomputingsettings.\nKeywords:debugging,growthmindset,physicalcomputing,computingeducation\nCite\nas:\nMorales-Navarro,\nL.,\nFields,\nD.\nA.,\n&\nKafai,\nY.\nB.\n(2024).\nUnderstanding\nGrowth \nMindset\nPractices\nin\nan\nIntroductory\nPhysical\nComputing\nClassroom:\nHigh\nSchool\nStudents’ \nEngagement\nwith\nDebugging\nby\nDesign\nActivities.\nComputer\nScience\nEducation\n.\nIn tr oduction\nDebug ging—theprocessofiden tifyingandfixingerrorsthatpreventanapplic ationfromworkingasexpect ed—isaninevitable,intrinsic,andoftenfrustratingpartoflearningcomputing.AsPapert(1980)remark ed,“errorsbene fitusbecausetheyleadustostudywha thappened,tounder standwha twentwrong,and,throughunder standing ,tofixit”(p.114).However,debug gingisalsoexactingandcanbe\n1difficultforstuden tstolearnandchallengingforteacher stoteach(McCaule yetal.,2008;Henness yElliot t,2023).Forman ystuden ts,encounteringbugscangener atefearandanxietythatcanleadtodiseng agemen tandtheavoidanceofcomputing(Scott&Ghinea,2013;Cotoetal.,2022).Yet,fewstudieshavetreateddebug gingholis tically,consideringthe“inextricablerelationshipbetweenthinkingandemotion”inlearningtopersevereandhandlefailur e(DeLiemaetal.,2020,p.210).Traditionaldebug gingpedag ogiesinvolvelinearstrategiestofindbugsincodeinconstrainedprogrammingcontextswithtimelyfeedback(Silva,2011;Luxton-R eillyetal.,2018).Theseappr oachesonlyaddr essisola tedcognitiv easpectsofdebug ginginlimit edscenarios,failingtoaddr essthe\"emotionaltoll\"(Kinnunen&Simon,2010;Cotoetal.,2022)thatdebug gingmayhaveonlearner sinopen-endeddesigncontexts.\nOnewaytoreframedebug gingasanopportunityforlearningandtodecr easehopelessnessandattritionamongnovicesisbypromotingagrowthmindse t(Kinnunen&Simon,2010).Mindse tsorganizepeople’ sbelie fsaboutthemselv esandtheirabilities(Dweck&Yeager,2019).Studen tswithgrowthmindse tsbelie vetheirskillsandabilitiescanevolveanddevelopovertime,wher easstuden tswithfixedmindse tsbelie vetheirintellectualcapacityandsocialchar acteristicsareunchang eable(Dweck,2006).Mindse tsinfluenceemotionalstates(Heyman&Dweck,1992;Robins&Pals,2002),andgrowthmindse tinterventionsseemtoreducelearner s’anxiety(Smith&Capuzzi,2019).Withincomputing ,priorresear chprovidespromisingevidenceofmindse tsplayinganimport antroleinstuden tlearning.Thisresear chshowsthatwhilestuden tswithgrowthmindse tstendtoperceivebugsasopportunitiesforlearning ,thosewithfixedmindse tsbecomefrustratedwithfailur eandseeitasachalleng etotheirintelligence(Burne tteetal.,2020;Nolan&Bergin,2016).Thus,promotingagrowthmindse tintheclassr oomisimport antsincedebug gingissuchanessen tialpractice,andperseveringthroughdebug gingiscritic altostuden ts’learning.Traditionalgrowthmindse tinterventionsincomputingeduc ation,whichtendtowardamoredidacticlectur e/tellingappr oach(Burne tteetal.,2020;Simonetal.,2008),havenotconsis tentlyyieldedasignific antimpactonstuden tperformance.Further ,actualstuden tpracticesdonotalwaysalignwithself-report edinstrumen tresponses(Stout&Blane y,2017;Gorson&O’Rourk e,2019).Withtheadditionallackofresear chattentiontoK–12computingstuden ts’growthmindse t,thiscallsforresear chintostuden ts’growthmindse tpractices,i.e.,studen ts’activ ebeha viorsthatputgrowthmindse tintopracticewhenrespondingtobugsincomputing.\nInthisexplor atory,observationalstudy ,weexaminethegrowthmindse tpracticesthathighschoolstuden tsexhibit edwhentheycreatedbuggyphysicalcomputingprojects,inparticularelectr onictextiles(hereaftere-textiles),fortheirpeer stosolveandthenfixedeachother'sprojects.Wepositionedlearner sasbugdesigner sfortheirpeer s,potentiallyincreasingtheirengagemen twithbugsbyallowingthemtobecreatorsofbugsratherthanjustsolvers.Wereportonfourcasesofstuden tgroupsparticipa tinginourimplemen tationof“Debug gingbyDesign”(DbD),anopen-endeddesignlearningactivity ,inahighschoolcomputingclassr oomoveraperiodofeigh t50-minut eclassperiods(Fieldset.al.,2021).Thisactivitywassitua tedinthecontextofExploringComput erScience,anintroduct oryyear-longhighschoolcomputingcurriculum(Mar golisetal.,2017).InDbD ,studen tsworkedwithe-textiles,whichinvolvecreatingwearablesusingprogrammablemicr ocontroller s,sensor s,and\n2actua torsthatcanbesewnintofabrics(Buechle yetal.,2013).Makingphysicalcomputingprojectsnotonlyincludesdesigningfunctionalcircuitsbutalsowritingcodethatcontrolsinteractions.Assuch,debug ginginphysicalcomputinginvolvesfindingandfixingseman ticandsyntacticbugsincodeaswellasdesignbugsinthecircuitsofphysicalartifacts(Searleetal.,2018;DesP ortes&DiSalv o,2019).Weobservedhighschoolstuden tstoseeifandwhentheydemons tratedgrowthmindse tpracticeswhiledesigningbuggye-textileprojectsfortheirpeer sanddebug gingpeer -designedprojects.Buildingontraditionsofqualit ativeandlearningsciencesresear chincomputingeduc ation(Tenenber g,2019;Margulieuxetal.,2019),wedrawonmicr ogenetic,momen t-by-momen t,analy sisofstuden tteaminteractionstoaddr essthefollowingresear chques tions:(1)Wha tgrowthmindse tpracticesdostuden tsexhibitwhiledesigningandsolvingbugsinphysicalcomputingprojects?and(2)Wher eandhowdostuden tsexhibitgrowthmindse tpracticesduringtheDbDactivity?\nBackgr ound\nTowardaHolisticApproachtoDebugging\nWhilecomputingeduc ationresear chhashistoricallyfocusedoncognitiv easpectsoflearning ,non-c ognitiv edimensionssuchasmotiv ationandemotionareequallyinstrumen talinsupportingallstuden ts(Lishinski&Yadav,2019).Thelatteristhefocusofthispaper .Forinstance,whiletheimport anceandprevalenceofdebug ginghavebeenrecogniz edincomputingeduc ation,littleattentioninresear chandpracticehasbeengiventotheroleofmotiv ationandemotionindebug ging(DeLiemaetal.,2020).Indeed,traditionalappr oachestolearningandteachingdebug gingtendtofocusontheconcep tualandcognitiv echalleng esthatemer gewhenencounteringbugs(Lietal.,2019).Procedur e-basedlinearstrategieswithflowchartsandtablesaresomeexamples(Silva,2011).Other sareself-directedstrategiesbasedondomain-specificknowledg eorsystema ticdebug gingstrategies(Michaeli&Romeik e,2019;Jonassen&Hung ,2006).However,debug gingisacomple xprocesswher etroubleshooting ,iden tity,andemotionintersect(Dahn&DeLiema,2020),gener atingdeepemotionalinvestmen t(Turkle&Papert,1990;Cotoetal.,2022).Learner s’emotionalstatesdomina tetheirperceptionoftheirprogrammingexperiences(Kinnunen&Simon,2010),whichmayaffecttheirperformanceandmotiv ation(Heyman&Dweck,1992;Hech t,2021).Asaresult,wemus tconsiderdebug gingaholis ticexperience,asitisshapedbyandshapescognition,emotion,andmotiv ation(DeLiemaetal.,2020;Dahnetal.,2020).\nAttendingtheroleofmotiv ationandemotionindebug ginghashistoricgroundingincomputinglearningresear ch.Forexample,intheirstudiesofnoviceprogrammer s,Perkinsetal.(1986)iden tifiedadichot omyofreactionsamongbeginner swhentheyencounterbugs.Somenovicesseebugsasinher entandexcitingchalleng esoflearningtocode,whileother sbecomefrustrated,perceivingbugsasnegativereflectionsoftheirperformance.Itisnotuncommonforthissecondgroupofnoviceprogrammer stoexperiencediscomfortandanxietywhendebug gingandencounteringfailur e(Lovell,2014;Kenchetal.,2016;Nolan&Bergin,2016).Thesestuden tsseebugsasthreatstotheirself-esteemandreput ationandsubsequen tlydoub ttheirabilitytolearn.Thisperceptionofdebug gingexperiencesasreflectinginna teprogrammingabilityisrelatedtolearnermindse ts.\n3Inpsychologic almotiv ationresear ch,Dweck(2006)hasorganizedopposingself-belie fsaboutinna teandmalleableintelligenceinto“fixed”and“growth”mindse ts.Studen tswithfixedmindse tsperceivetheirintellectualabilitiesandsocialtraitsasunchang eable—par alleltoPerkinsetal.’s(1986)learner swhosawbugsasanegativereflectionontheircapabilities.Ontheotherhand,learner swithgrowthmindse tsbelie vetheircompe tenciesandabilitiescanchang eandbedevelopedovertime—c ompar ablewithlearner swhoseebugsasinher entchalleng esandopportunitiestolearn(Ibid.).Learner swithgrowthmindse tstendtohavehigherresilienceinchallengingtasks,suchasdebug ging ,thanthosewithfixedmindse ts(Apiola&Sutinen,2020;Hetch,2021;Yeager&Dweck,2012).Indeed,mindse tsinfluencehowpeoplechoosechalleng es,persistoversetback s,andcreatevaluejudg emen ts(Dweck&Yeager,2019).Mindse tsaremalleableandcanbechang edthroughsmallinterventionsinwhichstuden tslearnaboutthepotentialtochang etheirabilitiesandhowtodevelopgrowthmindse ts(Apiola&Sutinen,2020;Yeageretal.,2012).However,theeffectofaninterventionmaydependonthepsychologic alaffordancesofthecontext,thatis,thelearningenvironmen tchar acteristicsthatsupportgrowthmindse tbelie fs(Hech tetal.,2021).Thisopensopportunitiestostudyhowgrowthmindse tscanbepromot edwithincomputinglearningenvironmen ts.\nGrowthMindsetsinComputingEducation\nIncomputingeduc ation,growthmindse tshavegainedrelevanceinresear chonmotiv ationalandemotionalfactorscontributingtodebug ging(Kwaketal.,2022).Forinstance,Murph yandThomas(2008)arguedthatincomputing ,whilestuden tswithgrowthmindse tsperceivebugsasopportunitiesforlearning ,thosewithfixedmindse tsbecomefrustratedwithfailur eandseeitasachalleng etotheirintelligence.Whenencounteringbugsandattemp tingtofixthem,novicelearner smayexperiencefeelingsofhelplessness,frustration,andanxiety,promotingfixedmindse ts(Scott&Ghinea,2013;Nolan&Bergin,2016),leadingtowithdr awalandevenabandonmen tofcomputing(Mar golisetal.,2017).Yetwhilefixedmindse tsareprevalen tandcommonamongnovicecomputinglearner s(Gor son&O’Rourk e,2019),designinglearningenvironmen tsandinterventionsthatpromot egrowthmindse tscanpositiv elyinfluencestuden tmotiv ationandengagemen t(Flanig anetal.,2022).\nTheexisting,butlimit ed,resear chongrowthmindse tincomputingeduc ationhasyieldedsomepromisingresultsthatsuggestthatmindse tinterventionsincreaseinterest,particularlyamongnovices(Burne tteetal.,2019).However,mos tstudieshavebeenconduct edatanunder gradua televel(Flannig anetal.,2022;Kwaketal.,2022;Apiola&Sutinen,2020;Burne tteetal.,2020;Quille&Bergin,2020;Woods,2020;Gorson&O’Rourk e,2019;Stout&Blane y,2017;Nolan&Bergin,2016;Scott&Ghinea,2013)orwithadults(Rang eletal.,2020),withlittleattentiongiventoyoung ermiddleandhighschoolstuden ts(Lok saetal.,2016;Kenchetal.,2016;Margolisetal.,2017).Atthesametime,themajorityoftheseinterventionsusedidactictechniques,suchasdeliv eringgrowthmindse tinforma tionthroughvideos,teacherlectur es,andreadings,occasionallyfollowedbyreflectiv ewritingand\"sayingisbelie ving\"exercises(Simonetal.,2008;Cuttsetal.,2010;Rang eletal.,2020).Mos toftheseeffortsprioritiz etop-do wnappr oachestoinformstuden tsaboutgrowthmindse tratherthanfacilit atinglearningexperiencesthatsupportstuden ts’activ edevelopmen tofgrowth\n4mindse tinpractice.Someexceptionsincludeinterventionsthatuseproblem-solvingchecklis tsforstuden tstotracktheirprogrammingprogressinprojects(Lok saetal.,2016)andgoal-se ttinginterventionswher estuden tsplan,followthrough,andreflectontheirowngoals(Woods,2020).\nYetbeyondinforma tiondeliv eryorevenproblem-solvingchecklis tsorgoalsetting,itiscritic altofostergrowthmindse tsinactualclassr oompractice(Dweck&Yeager,2019).Forinstance,inmathema tics,Campbelletal.(2020)iden tifiedpracticesthatchar acterizegrowthmindse tsinlearningactivitiesandsitua tedthesewithinlargerlearningtheories.Theydistinguishedsixcategoriesofpracticesinwhichlearner swithgrowthandfixedmindse tsbeha vedinopposingways:challeng es,persistence,effort,praise,learninggoals,andthesuccessofother s.Forexample,indealingwithchalleng es,learner swithgrowthmindse tschoosechalleng esthatallowthemtolearnmore,whilethosewithfixedmindse tsavoidchalleng esinareasthattheycannotensur etheywill“dowell”(Campbelletal.,2020).Similarly ,Haimo vitzandDweck(2017)examinedhowadultandsocializ ationpracticesfosterbothgrowthandfixedmindse tsinchildr en,highligh tingpracticesforrespondingtosuccessandfailur einprocess- focusedteaching.Centeringexplicitlyonpracticesmaybeessen tialbecause,withtraditionalgrowthmindse tinterventionsincomputingeduc ationpracticesdonotalwaysalignwithself-report edinstrumen tresponses(Stout&Blane y,2017;Gorson&O’Rourk e,2019).Resear chisneededondesigningcomputinglearningenvironmen tsthatpromot egrowthmindse tpractices,resilience,andpersistenceindebug ging.Thiskindofresear chonobservablegrowthmindse tpracticesincomputingeduc ationmaybehelp fulforteacher stoiden tifyandpromot estuden tbeha viorsthatmayleadtoagrowthmindse t.\nInsum,alongsideconcep tualunder standinganddebug gingpractices,studen tsneedtoengagewithgrowthmindse tpracticesanddevelopgrowthmindse tstoovercomethemotiv ationalandemotionalchalleng estheyencounterwhenfacingbugs.Whiletop-do wnappr oachesandexposur etomodelsofgrowthmindse tcanbehelp ful,experience-basedmindse tinterventionsmaybebettersuitedtosupportingstuden tstoovercomechalleng esincomputing(Burne tteetal.,2020;Margolisetal.,2017).Intheremainderofthispaper ,wefocusonaninterventionthatwasdesignedtoengagestuden tsincreatingintentionalbugsfortheirpeer stosolveandpromot etheirgrowthmindse tpractices.\nGrowthMindsetPracticesinDebuggingbyDesign\nBuildingonthisresear chbase,weinvestigategrowthmindse tinactionbyanaly zinghigh-school-ag edstuden ts’growthmindse tpracticeswhenengaginginDbD ,aninterventionwher estuden tsdesignandthenexchang ebuggye-textilesprojectstosolve.Design-basedphysicalcomputinglearningactivities,suchase-textiles,maysupportthedevelopmen tofgrowthmindse tsbecausetheyemphasiz etheprocessoflearningoveritsoutcomesandtheimport anceofpersonaleffortinconnectiontopersonalinterests(Vongk ulluk snetal.,2021).Physicalcomputingactivitiesthatfocusonfosteringcreativityanddesigninginterest-driv enartifactsareparticularlyeffectiv einpromotingstuden tmotiv ation(Przybylla&Romeik e,2018).DbDisagoodcasestudyforstudyinggrowthmindse tpracticesincomputingbecauseitisastuden t-cen teredinterventiondesignedtoempo werstuden tsindebug gingbydesigningcreative,\n5multimodalbuggyprojectsforother stosolve.Inthiscase,weanticipa tedthatstuden tswouldfeelgreatercontroloverbugsbecausetheychoseandcreatedthemratherthanrunningintothemacciden tally,asnormallyhappensindesigningcomput ationalartifacts.\nE-textileslearningactivitiesprovidearichcontexttodesignandsolvebugs,especiallyonesthatintegratecodeerrorswithcircuitr yproblems,forcinglearner stotrace,evalua te,andisola teproblemsacrossmodalities(Hodg esetal.,2020).Assuch,debug gingphysicalcomputingartifactsinvolvesaddr essingsyntactic,seman tic,concep tual,anddesignissuesasbugsmayemer geincodeandcircuit/ artifactdesign(DesP ortes&DiSalv o,2019;Jayathirthaetal.,2018;Luietal.,inpress;Schneider ,2022).Forexample,seman ticerrorsmayincludevariablesnotbeingdeclar ed,typemisma tches,ormethodcallswithwrongargumen ts;syntacticerrorsmayincludemissingdelimit ers;concep tualbugsmayincludeintentionalitybugsthatoccurwhenlearner sexpectprogramstobeha veinwaysbeyondthecodegiven;andcircuitr ydesignbugsmayincludeshortcircuitsandwrongconnections(Jayathirthaetal.,2018;DesP ortes&DiSalv o,2019;Fieldsetal.,2021;Boothetal.,2016).Studen tsuseavarietyofdebug gingstrategieswhentheyencounterbugsinphysicalcomputingsystems,suchasisola tingproblemsthroughforwardandbackw ardreasoning ,formula tinghypotheses,gener atingsolutions,andtestinghypothesesandsolutions(Jayathirthaetal.,2020).Thehybridnatureofphysicalcomputingallowsforrichdebug ginglearningasstuden tsdeveloptroubleshootingandpragma ticskills(Jayathirthaetal.,inpress;Luietal.,inpress).\nContrarytotraditionalgrowthmindse tinterventions(e.g.,“sayingisbelie ving ”exercises),whichoftenfailamongadolescen tsbecausetheydonotalignwithstuden ts’enhanceddesir etobe“treatedasthoughtheyarecompe tent,haveagencyandautonom y,andareofpotentialvaluetothegroup”(Yeageretal.,2018,p.104),inDbDlearner sareincontrolofdesigningbugsininterest-driv en,personaliz edprojects.InDbD ,studen tsareinchar geofintentionallycreatingthebugsratherthanjuststumblinguponthem.Assuch,theydecidewha tkindofbugstoselectandmak e,drawingontheirownpriorexperiencesaswellasthoseoftheirclassma tes.DbDwasdesignedtopromot egrowthmindse tsbyputtinglearner s,ratherthanteacher sorresear cher s,inchar geofcreatingintentionalproblems,turningbugsintoafeatureofthelearningproductratherthanastumblingblock.Anearlystudybasedonstuden ts'self-reportssuggestedthatwhenstuden tsconsciouslycreatedbugsforpeer s,theyimpr ovedtheirabilitiestodetectandfixbugs,collabor ativelysolvedproblems,andincreasedtheirconfidenceindebug ging.Althoughsomestuden tsexpressedfrustrationwhensolvingbugscreatedbytheirpeer s,afterwardstheyreport edfeelingmorecomfortableandcompe tentinsolvinganddesigningbugs(Fieldsetal.,2021).Ourresear chonDbDinthisexplor atory,observationalstudyfocusesonthegrowthmindse tpracticesstuden tsexhibit edwhendesigningbugsforother sandsolvingbugs,aswellaswhenandwher ethesepracticesemer gedincontext.\nMe thods\nSiteandParticipan ts\nWeconduct edthisstudyinSpring2019atahighschoolintroduct orycomputingclass\n6inametropolit anschooldistrictontheWestCoas toftheUnit edStates.Intheclass,25studen tsages14–18receiv edparentalconsen tandassen tedtoparticipa teinthestudy:11self-iden tifiedasfemale,and14self-iden tifiedasmale.In72%ofcases,participa tingstuden tsspok ealanguag eotherthanEnglishathome;80%hadnopriorcomputingeduc ationexperience(noCSclassesorworkshop sbeforethiscourse);and80%hadfamilymember swithatleastsomecolleg eeduc ation.Theclasswasethnic allyandraciallydiverse,with48%ofparticipa tingstuden tsiden tifyingasLatinx,36%asAsianAmeric an/P acificIslander s,8%Whit e,4%other ,and4%notreportingtheirraceorethnicity .Ben,theteacher ,hadthreeyearsofexperienceteachinge-textilesandcollabor atedwiththeresear chteamtodevelopthecurriculum.\nWereques tedBen’ shelpinchoosingfourgroupsascasestudiesforcloseobservation,askingforarangeofstuden tgroupssothatwecouldhaveadiverseviewofhowstuden tsrespondedtotheDbDactivity .Drawingonhisknowledg eofstuden tgroupswhohadjustcomple tedthethird(collabor ative)projectinthee-textilescurricularunit,Benselect edfourstuden tgroups(n=10studen ts)forcasestudiestorepresen tarangeofstuden tinteractionandperformancestyles.Twogroupswereveryhighperformer s:studen tsfinishedprojectsontimeandcollabor atedsmoothlyasateam.Thetwoothergroupsstruggledinpriorprojects:studen tsdidnotcomple teprojectsontime,hadtroublegettingprojectstowork,andhadmoreawkw ardcollabor ations(e.g.,majordisagr eemen tsanddifferentworkingstyles).Casestudygroupsincludedtwogroupsoftwostuden ts(EvelynandNicolás;LiamandSophia)andtwogroupswiththreestuden ts(Lucas,Emma,andLily;Geor gia,Gabriel,andCamila)(seeTable1fordemogr aphicchar acteristicsofparticipan ts).\nTable1.Self-report eddemogr aphicchar acteristicsofparticipan ts.Allnamesarepseudon yms.\nG r o u pP a r t i c i p a n tG e n d e rR a c e / E t h n i c i t yP r e v i o u sC o m p u t i n gE d u c a t i o n\n1LucasMaleAsianNo\nEmmaFemaleLatinaNo\nLilyFemaleLatinaNo\n2GeorgiaFemaleAsianNo\nGabrielMaleLatinoNo\nCamilaFemaleAsianYes\n3EvelynFemaleLatinaNo\nNicolásMaleLatinoNo\n4LiamMaleAsianNo\nSophiaFemaleAsianNo\n7DebuggingbyDesign\nWesitua tedDbDwithinthee-textilesunitofExploringComput erScience(ECS).Thisisayear-longinquir y-basedcomputingcurriculumconsis tingofmultipleunitscommit tedtobroadeningparticipa tionincomputingthroughabuildingtalen tappr oachthataddr essesthestructur alinequitiesandbelie fsystemsthatlimitparticipa tionfromhistoricallymarginaliz edgroups(Mar golisetal.,2017).WithinECS,thepromotionofgrowthmindse tsamongstuden tsandteacher salreadyplaysanimport antrole,asitputsequityfirstbyconsideringthatallstuden tswithaccesstoqualityeduc ationcangrowinengagemen tandcapacity(Mar golisetal.,2017).However,thee-textilesunitistheonlyunitintheECScurriculumthatprioritiz eshavingstuden tsdesignpersonallyrelevantprojects(Fieldsetal.,2018;Fields&Kafai,2023).\nTheunit1centersaroundthedesignofthefoure-textilesprojects—papercard(simplecircuit),wristband(parallelcircuitwithswitch),interactiv emural(programmablecircuitwithtwoswitches),andhumansensor(comput ationalcircuitwithanalogsensor)—tha tsupportstuden tsinlearningcomputing ,electr onics,andcraftingwhiledesigningpersonallyrelevantartifacts(fordesignprinciplesoftheunit,seeKafai&Fields,2018;forevalua tionsofstuden tlearning ,seeKafaietal.,2019;Fieldsetal.,2021).Theunitrequir esstuden tstoapplyconcep tssuchassequences,loop s,conditionals,andvariables(learnedinUnit4)toatext-basedprogramminglanguag e(Arduino).Inaddition,studen tslearnnewprogrammingconcep tssuchasnestedconditionals,datainputfromsensor s,andfunctions.\nDbDwasintentionallysitua tedbetweenthethirdandfourthprojectsinthee-textilesunit,afterthecollabor ativeclassmuraldescribedabove.Thisallowedstuden tstoapplytheirexperienceswithbugsfromearlierprojectstotheirDbDdesignsandapplyanyknowledg egainedfromDbDtotheirfinalproject.Theactivitytookplaceduringeigh t50-minut eclasssessionsoveraperiodoftwoweeks(seeTable2).Duringthefirstsession’ s“HallofProblems,\"studen tsfirstdiscussedwiththeirpartner s,thenthewholeclass,differenterrorsandproblemstheyencounteredwhenworkingone-textilesprojects,creatingapublic,collectiv esetofallthebugstheycouldthinkofthatoccurr edintheirpriordesigns.Following ,duringsessiontwo,studen tgroupscreatedaselectlistofbugstoincludeintheirDebugItalongwithanaestheticdesignorsketchoftheproject.Theteacherencouragedstuden tstobeconsider ateoftheirpeer sinputtingonlyasman ybugsinascouldbesolvedinasingleclassperiod,whichledtosomecarefulselectionfromseveralinitialbugideas.Insessionsfourandfive,andafterreceivingappr ovaloftheirdesignfromtheirteacher ,studen tscreatedtheirbuggydesigns,orDebugIts,whichrequir edthemtocraftanon- functionalprojectwithitscircuitandcode(seeFigur es1,3,and4).Inthefollowingtwosessions,studen tsexchang edbuggyprojectswiththeirpeer sandsolvedeachother'sprojects.Duringthelastsession,pairspresen tedtheirsolutions(some timesfull,some timesincomple te)totheclassandreflect edonthestrategiestheyusedtosolvetheDebugIts.\nTable2.Debug gingbyDesignActivity .\n1Theentirecurriculumcanbefoundathttp://exploringcs.org/e-textiles\n8Class1\"HallofPr oblems\":Aspartner s,thenasawholeclass,studen tsliste-textileproblems.Thentheycategorizedtheseproblemsintogroups,whicharewrittenonpostersontheclassr oomwalls.\nClass2DebugltDesign:Studen tsplantheirDebuglts,turninginalistofproblemswithsolutionsaswellasacircuitdiagr amshowinganycircuitr ybugs.Theteacherhadtoappr ovedesigns.Mos tgroupsrevisedtheirdesignsafterteacherfeedback,whichcontinuedintoClass3.\nClasses3-5DebugltCons truction:Afterreceivingteacherappr ovalontheirdesign,studen tscreatedtheirDebuglts:sewingandcodingtheirprojectswithintentionalbugs.Theyalsowroteanintentionstatemen tforhowtheprojectough ttowork.\nClasses6-7DebugltSolving:Underthet eacher'sdir ection,s tuden tse x chang edpr ojectsandhad1.5classperiodst osolv ethemasbes tthe yc ould.Studen tsindividually ,theninsmallgroups,reflect edonwha tthebest,mos tfrustrating,andsurprisingpartsoftheentiredebug gingbydesignexperiencewere.\nClass8R e flectiononPr oblem-SolvingStr a t egies:Individually ,thenaspairs,andfinallyasaclass,studen tsreflect edonthekindsofstrategiestheyusedtosolveDebugIts.Theyshar edstrategiesatthefrontofclass,suchthatDebugIt ’sdesigner scouldrespondtoanybugsthesolversmissedornoteanybugsunin tentionallyputintothedesign.\nDataCollection\nToinvestigatethegrowthmindse tpracticesthatstuden tsexhibit edandthesitua tions,challeng es,orcontextswher ethesepracticesseemedtoemer ge,wedrewon:1)in-classobservationalfieldnotes;2)videologs(basedonvideorecordings)ofparticipan tsworkingontheirprojects;3)artifactcollection;and4)interviewswithstuden ts(n=10).Throughouttheeigh tclasssessionsofDbD ,wepositionedvideocamer asinthefourselect edstuden tgroups.Inaddition,oneresear cherrotatedacrossthegroupsandtookobservationalfieldnotesandregularphot ographsofstuden twork,occasionallyaskingthegroupsaboutwha ttheyweredoingandwhy.Weloggedandannot atedthevideorecordingsbyaddinginforma tionregardingwha tstuden tsdidinthevideo ,theirdesignanddebug gingprocesses,andtheiremotionalreactions.Inaddition,wecollect edallstuden tproducts:circuitdiagr ams,code,andpictur esoftheartifactsdesignedbythestuden ts.Thismulti-sour ceappr oachprovidedamorecomple tepictur eofwha thappenedintheclassr oominteractions,whichwassupplemen tedbypost-interviewswithstuden tgroupsseveralweeksaftertheDbDactivityended.Allthedatawasgatheredandorganizedbygrouptoputtogetherapictur eofhoweachgroupengagedwiththeactivity .\nDataAnalysis\nToanswertheresear chques tions,(1)Wha tgrowthmindse tpracticesdostuden tsexhibitwhiledesigningandsolvingbugsinphysicalcomputingprojects?and(2)Wher e\n9andhowdostuden tsexhibitgrowthmindse tpracticesduringtheDbDactivity?,wefocusedonmomen t-to-momen tmicr ogeneticobservationalanaly sisofstuden tteaminteractionsindesigningandsolvingbugsforother s.Byexaminingtheprocessoflearningandtheexperiencesoflearner s,webuiltonalongs tandingtraditionofthelearningsciencesincomputingeduc ationresear ch(Mar gulieuxetal.,2019).Momen t-to-momen tanaly sis,alsoknownasmicr ogeneticanaly sis,isacommonmethodinthelearningsciencesthatallowsresear cher sto“observelearningprocessesastheyoccuranddosoinsuchawayastopermitstronginferencesaboutlearningprocesses”(Chinn&Sherin,2014).Thistypeofanaly sisiswellsuitedtoinvestigatinglearninginsmallcollabor ativegroups(Taylor&Cox,1997)andespeciallythemomen t-to-momen tco-occurr enceofprocessesinalearningenvironmen t,suchasdevelopmen tpracticeswhilecomple tingalearningactivity(Schoen feldetal.,1993;Chinn&Sherin,2014).Incomputingeduc ationresear ch,momen t-to-momen tanaly siscanbeespeciallyusefulforstudyinglearner s'non-c ognitiv eandcognitiv epractices.Becausemicr ogeneticresear chnecessit atesmethodsthatattendtotherichprocessesthattakeplacewhilestuden tslearn,qualit ativeresear chandcasestudiesarefrequen tlyusedtodistillthepracticesandprocessesthatstuden tsengagewithandtoexplainhowlearner sengagewiththemincontext(Chinn&Sherin,2014).\nInthisobservationalstudy ,weusedqualit ativemethodstoiden tifypracticesinactionthroughaground-upanaly sisthatenabledustobuildonexistingtheor ywhilebeingopentopracticesthatmaynothavebeenobservedinothercontexts.Bypractices,wemeanobservablebeha viors,mos tlydiscernibleduringconversationorinactionsaimedatfixingbugs,suchaseditingcodeormakingvisiblechang estoaprojectordesign.Buildingontraditionsofqualit ativemethodsincomputingeduc ationresear ch(Tennenber g,2019),weanaly zedthedatainthreesteps.First,wereaddatasour cesfromtwostuden tgroupschronologic allyfocusedoninductiv edescrip tivecoding(Saldaña,2015).Throughthisprocess,wedevelopedacollectionofcodesforinstances,momen ts,orexcerp tswher estuden tsexhibit edgrowthmindse tsinobservableways.Salien tcodesincludedmomen tsofpraise,peersupport,sharingfailur e,valuingthesuccessofother s,settinglearninggoals,notbeingafraidofbeingwrong,andmomen tsofchalleng e.Thisanaly sishelpedusunder standandiden tifyhowstuden tspracticedagrowthmindse tduringthelearningactivities.Wethenclusteredthedescrip tivecodesintosimilarcategoriestodetectpatternsandinterrelationship s.Previousstudiesonmindse tpracticesandbeha viors(Campbelletal.,2020;Dweck,2006;Dweck,2000)informedtheclusteringprocess.Takinganinductiv eappr oachfirsttoseewha temer gesfromthedataandthenclusteringcodesaccordingtotheliteratureisacommonpracticeinqualit ativeresear chincomputingeduc ation(Tennenber g,2019;Searle&Kafai,2015).Inourcase,thiswasimport antsincetheliteratureongrowthmindse tpracticeshaslargelybeenbuiltinotherfields(i.e.,socialpsychologyandmathema ticseduc ation),andsuchinductiv ecodingservedtoopenlyinvestigatehowgrowthmindse tpracticesemer gedinahighschoolphysicalcomputingcontextpriortobuildingacodingschemeinformedbytheliterature.Afterdevelopingthecodingscheme(allcodingcategoriesareexplainedwithexamplesinthefindingssection;foratablewiththecodebook,seeAppendix1),wediscusseditwithtworesear cher sfamiliarwiththedata,receiv edfeedback,andrevisedit.Inasecondroundofanaly sis,weappliedthecodingschemeacrossallgroupsthroughadeductiv ereading(Saldaña,2015).\n10Followingthat,inordertogainacompr ehensiv eunder standingofthecontextsinwhichstuden tsengagedingrowthmindse tpractices,wecreatednarrative-basedaccountsthatsitua tedthepracticespreviouslyiden tifiedinthemomen t-to-momen trichnessofthelearningcontext(Schoen feldetal.,1993).Theresear chteamdrewonthechronologic alassemblag eofdataforeachgrouptocreatenarrativesthatincludedcontextualdetailsthatinfluencedgrowthmindse tpractices,suchaspeerinteractions(withinandbetweengroups),teachersupport,andclassdiscussions.Thisenabledustofurtheranaly zethenarrativeaccountstouncoverwhenandhowgrowthmindse tpracticesemer gedduringtheDbDactivity .\nFindings\nFigure1.Projectscreatedbythefourfocalgroups.Fromlefttoright:anemojiplushiecreatedbySophiaandLiam;apianocreatedbyGeor gia,Gabriel,andCamila;asickcloudthrowinguparainbo wcreatedbyEvelynandNicolás;andaheartcreatedbyLucas,Emma,andLily.\nOuranaly sisiden tifiedfivegrowthmindse tpractices:(1)choosingchalleng esthatleadtomorelearning;(2)persistingaftersetback s;(3)givingandvaluingpraiseforeffort;(4)appr oachinglearningasconstantimpr ovemen t;and(5)developingcomfortwithfailur e.ThefirstfourpracticescoincidewiththepracticesproposedbyCampbelletal.(2020);thelastpracticeemer gedinductiv elyfromanaly sis.Inthefollowingsections,weintroducethesefivepracticesandhighligh twher etheyemer gedduringthevariousaspectsoftheactivity(planning ,designing ,anddebug ging)andacrossindividualorpeer -to-peerencounters.Finally ,weshar etwoin-dep thcasestudiesdemons tratinghowthesepracticesemer gedinclassr oomandsmall-gr oupcontexts.\nGrowthmindsetpractices\nChoosingchallengesthatleadtomorelearning\nThroughoutDbD ,studen tspurpose fullychosechalleng esthatledthemtolearnnewthings,“throwingthemselv eswholeheart edlyintodifficulttasks—andstickingwiththem”(Dweck,2000,p.3).Asanexample,whenplanningtheirDebugIt,Evelynexplainedshewantedittoincludemusicthatcouldbeactiv atedwhenpeoplepressedabutton.Nicolásreplied,“That’stoohard”becausetheyhadnopreviousexperiencecodingmusic.Yet,heturnedtohiscomput er,andafterafewminut esofgooglingandtinkeringwithArduinosamplecode,heshowedEvelynhowtheycouldmak etheArduinoboar dplaynotes,learningsome thingnewintheprocess(Lesson2,Video).Inadifferentgroup,whenmakingaDebugIt,LiamsuggestedtohispartnerSophiathattheymak eanemojiplushie,athree-dimensionale-textilesprojectwiththeelectr oniccomponen tsdistribut edinsidetheartifactandonitssurface(Lesson2,Video)(see\n11\nFigur e1).Whiletheyhadnevermadeaplushie,theydecidedtoplacelightsonthefrontandthemicr ocontrollerontheback.Thus,knowingtheyhadnoexperienceinmakingthree-dimensionale-textiles,thegrouptookonthenewchalleng etoarrangetheelectr oniccomponen tsinthisspatiallydistribut edwayand,mischie vously ,toembedabuginsidetheplushie.Thesearejusttwoofthe45instancesweiden tifiedinwhichstuden tsconsciouslychosechalleng esthatledthemtodeeperlearning.\nPersistingaftersetbacks\nSetback sarecommoninlearningcomputingandengineeringandareanessen tialpartofdebug gingthatcancausefrustrationand,atthesametime,provideopportunitiesforstuden tstopersist.Frus trationaspartofthelearningprocessismeaningfulbecauseitcreatesopportunitiesforstuden tstopersistand,asDweck(2006)writes,“toturnanimport antsetbackintoanimport antwin”(p.93).Indeed,althoughfrustrationcanbeunpleasan t(Bak eretal.,2010),itcanprecedesuccess fullearning(D’Mello ,2013).Forthispractice,weiden tifiedtwosubc ategories:(1)howfrustrationsome timesprecedespersistence,and(2)howfrustrationco-occur sininstancesofpersistence.\nAsanexampleoffrustrationprecedingpersistence,weshar eatimewhenEvelynwasfrustratedwithfixingacircuitr ybuginhersewing.Shewasoverwhelmed,exclaiming ,“Thisistoomuch...Idon’tknowwha t’swrongwithit.”Shewasupset,feelingliketheproblemwas“toomuch. ”.Herpartner ,Nicolás,encouragedher,butsheinsis tedshedidn’tknowhowtofixit.Nicolásofferedtohelp:“déjame[letme]lookatit”(Lesson7,Video).Twentyminut eslater,andafterworkingonotheraspectsoftheproject,Evelynpersistedandreturnedtothebug,foldingandbendingtheartifacttogetittoworkuntilthelightsturnedon.Duringthisprocess,Evelyndiscoveredthattheproblemwasinthelooseconnectionsoftheconductiv ethread.“IcouldhavefixeditalongtimeagoifIknewtheconnectionswereallloose, ”shereflect ed(Lesson7,Video).\nOnadifferentoccasion,Evelynseemedupsetdealingwithabug,andNicolásencouragedher:“Youcanfixit!Fixit!”(Lesson7,Video).Inthisinstance,frustrationandpersistenceco-occurr ed.Despit eherfrustration,Evelynmanag edtogetthelightstoturnon.Thus,withencouragemen tfromherpartner ,asEvelynpersisted,sheevalua tedandanaly zedproblems,lookingforalterna tivewaystosolvethem.Thesetwoexamplesillustratetheoccurr enceandco-occurr enceoffrustrationandpersistenceandhowsetback s,althoughfrustrating,becameopportunitiesforlearningintheprocessofcreatingandsolvinge-textilebugs.\nGivingandvaluingpraiseforeffort\nWhileworkingontheirDebugIts,studen tspraisedeachother ’sefforts,strategies,andprocesses.Thiskindofpraisemayfostermastery-orien tedresponseswhenfacingdifficultproblems(Dweck,2000).Whenlearner sreceiv esupportiv eresponsesfrompeer s,theyaremorelikelytoaccomplishtheirtasksandparticipa teingener atingnewideas(Jordan&McDaniel,2014).Asanillustration,whileGeor giastruggledwithhersewing ,Camilaencouragedher,“You’realmos tthere;keepgoing ,”andwhentheyfinishedstitching ,aconnectioncelebr atedtheireffort,saying,“Wegotit!”(Lesson3,Video).NicolásevenpraisedLilyforaddingmischie vousandnuisancecode,suchas\n12declaringfourextravariablesthatwerenotusedintheprojectjusttoconfusethedebug gingteam,saying,“Goodjob,youmadeitreallyhard[fortheotherteam]”(Lesson7,Video).Thesetwoinstancesshowcasethefrequen tpraiseandencouragemen tforeffortthatstuden tsgaveeachotherindesigningandsolvingbugs.Thiskindofsupportfrompeer s,centeredaroundeffort,promot esagrowthmindse tbyframingsetback sandchalleng esasinher entaspectsoflearningthatcanbeovercomewithpersistence.\nApproachinglearningasconstantimprovement\nStuden tteamsalsoembr acedlearningasconstantimpr ovemen tandlook edforwaystoimpr ovetheirprojectsandtheircompe tence.AsDweck(2000)argues,thisreflectsadesir etolearn,getsmart er,andacquir enewskills.Weseethiswhen,aftergettingallthelightsoftheprojecttowork,Liamdecidedto“resewthething ”tomak esurethatalltheconnectionsweretighterandbetterorganizedbecause“thethreadingiskindofbad”(Lesson6,Video).Inanotherepisode,afterworkingonthecodeandchangingthepinsused,Sophiadecidedtoredrawthecircuitdiagr am.Shecouldjusthaveannot atedorfixedtheoldonebutinsteadwantedtomak eanewone“tomatchthecode”(Lesson6,Video).Inthesetwoexamples,wecanseehowtheysough topportunitiestoimpr ovetheirperformanceandsawthegoaloflearningasconstantimpr ovemen tandnotjustshowcasingorachie vingacertainlevelofperformance.\nDevelopingcomfortwithfailure\nDevelopingcomfortwithfailur egener atesopportunitiesforlearner stoembr aceandovercomemistakeswithpropermotiv ationandguidance(Dweck,2000).Whenlearner sareworriedaboutfailur e,theyaremorelikelytodevelopafixedmindse tandques tiontheirabilitytolearn.Ontheotherhand,learner swithagrowthmindse t“seetheirownfailur esasproblemstobesolved,andtheyseeotherpeople'sfailingsthatwayaswell”(Dweck,2000,p.88).InDbD ,wesawevidenceofstuden ts’comfortwithfailur ewhen:(1)askingques tionsandreques tingfeedbackwithoutfearofbeingwrong;(2)sharingfailur ewiththeirpeer s;and(3)embr acingfailur eandimperf ectionaspartoftheprocess.Sophia,forexample,askedques tionstoherpeer swithoutbeingworriedaboutbeingwrong.Whilesewingacircuit,sheaskedLiam,“Canpositiv eandpositiv ecrosseachother? ”(Lesson3,Video).Later,LiamaskedSophiaiftheconstantpinvaluesinArduino(“LOW”and“HIGH”)couldbelowercase(Lesson3,Video).Thesearetwoofman yinstancesinwhichstuden ts,includinghigh-achie vingstuden tslikeLiamandSophia,reques tedfeedbackfrompeer swithoutanyfearofbeingwrong.Peersupportwhenfacinguncert aintycangener ateasafeenvironmen twher estuden tsarenotafraidofaskingforhelpfromother s(Jordan&McDaniel,2014).\nFurther ,whenlearner sencounteredfailur ewhileworkingontheirprojects,itwasnotuncommonforthemtoopenlyshar efailur ewiththeirpeer s.Asanexample,Evelynshar edamistakeshemadewhencraftingthecircuit:shesaid,“Ohmygod!Iwasgoingtokeepgoingwithoutsewingupthelight!”whenshenoticedagapinsewingandlaughedatherownmistake(Lesson4,Video).Onadifferentoccasion,sheshowedNicoláshowthethreadgotstuckbecauseshe“messedupwiththeglueearlier ”(Lesson4,Video).Similarly ,Lucasshar edamistakehemadewhenwriting\n13conditionals(Lesson3,Video;moredetailsinthecasestudy).Theseareinstancesofseeingfailur eassome thingworthsharing.\nFurthermor e,studen tsembr acedimperf ectionandfailur easafeatureoftheirprojects—separ atefromtheintentionallydesignedbugs.Forinstance,whenEmmanotedthatthetwoheart -shapedpiecesoffeltfromthegroup’ sprojectwerenotperfectlyaligned,shedescribedthemas“alittlebitoff,butit’sokay.”.Shealsowentontoofferasimple,imperf ectsolution,saying,“We’lljustcutitoff;itdoesn’thavetobeperfect”(Lesson4,Video).Thesearesomeexamplesofhowstuden tssough thelpwithoutfearofbeingwrong,“messingup,”makingmistakes,orhavingimperf ectprojects.Developingcomfortwithfailur ehelpedstuden tsreframefailur easasetbacktheycouldovercomeandasanopportunityforlearning.\nGrowingmindsetsacrossgroups\nThefourgroupsweanaly zedengagedinthefiveiden tifiedgrowthmindse tpracticesin231instances.Figur e2showsthefrequencyofinstancesofeachpracticebygroupofinterest(ChartA)aswellasinforma tionaboutthecontextofeachinstance,whentheyoccurr ed(e.g.,whenplanningtheproject;ChartB),andwhowasinvolved(e.g.,teacherandlearner;ChartC).Studen tsengagedwithgrowthmindse tpracticesthroughpeer -to-peerandlearner -teacherinteractionsthroughouttheDbDactivity ,includingwhenplanningtheirprojects,makingbuggyprojects,anddebug gingtheirpeer s’projects.Mor ethanhalfoftheinstancesofgrowthmindse tpracticesiden tified(56%)occurr edwhenlearner smadebuggyprojectsfortheirpeer stosolve(thisisnotsurprisingsincestuden tsspen tthree50-minut esessionsdesigningDebugIts).Mos toftheinstancescoded(78%)occurr edinpeer -to-peerinteractions,althoughwealsoiden tifiedgrowthmindse tpracticeswhenstuden tsworkedaloneorinlearner -teacherinteractions.Thefrequen toccurr enceofgrowthmindse tpracticeswhiledesigningbuggyprojectsandprimarilyduringpeer -to-peerinteractionsisshowcasedinthecasestudiesbelo w.\n14\nFigure2.Chartsofinstancesofgrowthmindse tpractices(GMP s)bystuden tgroup(ChartA);byDbDactivities(ChartB);andbywhowasinvolved(ChartC).Thekeyindic atesthepracticesobserved.\nCaseStudy1:Georgia,Gabriel,Camila,andtheMusicalKeyboard\nOnecasestudygroup—Geor gia,Gabriel,andCamila—illus trateshowthefivegrowthmindse tpracticesdescribedabovecameupinthecontextoftheDbDactivity .Gabriel,who ,accordingtotheteacher ,tendedtobeofftaskinthepreviousterm,waspurpose fullygroupedwithhigh-achie vingstuden ts.Together,theycreatedafeltpianothatplayedmusicandlitup,whichincludedfivecircuitr y/craftingbugs(addingunnecessar ythread,reversepolarityinacircuitconnection,andabad/looseconnection)andfourcodingbugs(thatincludedseman ticissues:passingthewrongparame terstoafunction,messingupthesequenceofthedesir edlightpattern,andonesyntacticproblem:missingadelimit er).\nWeiden tified75instancesinwhichthegroupshowcasedgrowthmindse tpractices(62duringpeer -to-peerinteractionsand12duringlearner -teacherinteractions),evenfromtheinitialplanningstages.Asthegroupbrainstormedwha ttheywantedtomak efortheirDebugItandthebugstheywantedtoinclude(seeFigur e3),Camilasuggestedanareathatwentbeyondtheircurrentexpertise:makingapianoandcodingmusic.Gabrielhesit ated,“We’veneverusedmusicbefore,andwehavetwodays”(Lesson2,Video):theyhadnotlearnedhowtocodemusicinclassanddidnothavemuchtimetofigur eitoutbythemselv es,buttheydidhaveaccesstoasupplemen taryinstructionalguideone-textileswithmusic.Ben,theirteacher ,whooverhear dtheconversationaboutcodingmusic,encouragedtheteamtodoitandeventocreateabugrelatedtomusic.TheydecidedtoplayTwinkle,Twinkle,LittleStarandtohaveGabrielcodeit.Thisrequir edunder standinghowtoplaynotesbyusingarraystodefinepitchanddurationinnumeric alvaluesandusingnewfunctions(tone()andnotone())intheArduinolibraryincombina tionwithforloop stoiteratethroughthearraysandplaythemusic.AlthoughGabrielshar edthathehad“nocluehowtocodemusic, ”hetookchar geofthisnewchalleng eandlearnedaboutfrequenciesandtempowithaninstructionalguide.Thus,thegroupadmit tedtheirlackofexpertiseandchoseachalleng ethatwouldleadtonewlearning.Ofnote,itwasinthecontextofbothpeerandteachersupportthatGabrieldecidedtotakeonthisnewchalleng e.\nFigure3.Geor gia,Gabriel,andCamila’ sDebugIt:(1)Left:listofbugsincludedintheirproject,plusorderoftones;(2)Righ t:Geor giasewsmusic alnotesabovethekeyboar d.\n15\nCamilaandGeor giaworkedtogetheroncraftingthekeyboar d,andGeor giabecamefrustratedwhengluingthemusic alnotes:“Ican’tdothis;Iphysicallycan’tdothis”(Lesson3,Video).Itwashardforhertocalcula tehowmuchgluetoapplytothefelt.Havingpour edglueallovertheproject,shecomplained,“Idon'tlikeglue;nowit’samess. ”Geor gia’sfrustrationcreatedanopportunityforhertopersistandovercomethissetbackwiththesupportofherpeer s.Camilaprovidedjust-in-timepraiseforefforttomotiv atehertokeepgoing:“You'realmos tthere,Geor gia;thatlook sverygood”(Video ,Lesson3).Inpersisting,Geor giatrieddifferentstrategies:firstusingapieceoffelttoapplyglueandthenusingthetipofthebottletospreadit.Thelatterstrategyworked,andsatisfied,shepraisedherownwork:“This[keyboar d]look ssogood!”(SeeFigur e3).Here,twogrowthmindse tpracticesworkedtogether:frustrationgener atedopportunitiesforpersistence,andjust-in-timepraiseforefforthelpedreframesetback sasinher entaspectsoflearningthatcanbeovercomewithpersistence.\nAsworkontheprojectcontinued,Geor gia,whohadlittleexperiencesewingandcrafting ,shar edhermistakeswhentakingonachalleng ethatledtomorelearning.“I’mjustwarningyou,I'mnotthebestatsewing ,”shesaidasshestartedsewingthecircuit.Notafraidoffailur e,Geor giashar edherproblemswithCamila,whohelpedhertieknotsandmak eherconnectionstight.Theyshar edtheirfailur eswitheachotherandlaughedtogetherabouttheunin tentionalmistakestheymadewhiledesigningtheirbuggyproject.Camila,forexample,shar edthatshesewedanLEDtothewrongpinofthemicr ocontroller ,andGeor giashar edthatshesewedanLEDbackw ards.Com fortablewithfailur e,whenstuden tschosechalleng esthatledtomorelearning ,theyembr acedmistakesandovercamethem.\nWhencodingTwinkle,Twinkle,LittleStar,Gabrielpersistedafterencounteringman ysetback sandfrustrations.Hedidnotknowhowtotriggerthesongtoplay,butafterfixingman yunin tentionalseman ticbugsinhiscode(forinstance,iteratingoverthepitchesarrayandnotthedurationarray)andgettingsupportfromtheteacher ,hemanag edtomak ethemicr ocontrollerplaymusic.Whenitfinallyworked(albeitnotthewayhewanted),withfrustration,hesaid,“Ihavenocluewhyitsoundssohideous;thisisdevilishtrapmusic”(VideoLesson4).Hispeer sdidnotlikethesound.Yethepersistedandcontinuedworkingonimpr ovingthesongbyfixingthepitchvaluesinthearray.ThemusicbecameanEastereggintheproject;theyusedthelightsensoronthemicr ocontrollertotriggerthesong.Thisideabrough tinnewchalleng esandunin tentionalbugsbecauseitwasthefirsttimetheyusedalightsensor .Withthehelpofatutorial,Gabrielmanag edtoreadsensordatabutstruggledtounder standthatsensorreadingswentfrom0to1024andwerenotjust“LOW”and“HIGH”likewhenworkingwithswitches.Afterstrugglingwiththeconditionalstatemen tforthesensor ,withthehelpofaresear cher ,hegotittowork.Gabrielexplained:“IfyoutaketheflashonyourphoneandputitonyourCP[micr ocontroller],it’llmak esoundshappen. ”Thisisanexampleofhowwhenstuden tschosechalleng esthatledtomorelearning ,theirfrustrationbecameanopportunitytoovercomesetback s,persistedwhenthingsdidnotgoasexpect ed,andappr oachedlearningasconstantimpr ovemen t.WhenGabrielwasdonewiththecode,heinsis tedtheyhadtotestittoseeifitworkedand,ifnot,fixit.Theytestedseveraltimes,andeverytime,together,thegroupcameupwithnewideasofhowtoimpr ovetheproject.\n16CaseStudy2:Lucas,Emma,Lily,andtheBuggyHeart\nInanothercasestudygroup,Lucas,Emma,andLily,agroupchosenbytheteacherfortheirtendencytowardslowerperformanceearlierintheclass,engagedwithgrowthmindse tpracticesin56observedinstances(36duringpeer -to-peerinteractionsand5duringlearner -teacherinteractions).Together,theycreatedaheartwithtwoshee tsoffelt,oneovertheother ,playfullyhidingsomebugsinbetweenthepieces.Thisprojectincluded6circuitr y/craftingbugs(seeFigur e4:threebadconnections,twomisma tchesbetweenpinnumber sdeclar edinthecodeandpinsusedinthecircuit,andonewrongconnection)and20syntacticcodingbugs(seeFigur e5:missingdelimit ers,4spacesinvariablenames,and6errorsinvariabledeclar ationandusag e).\nWhenmakingtheDebugIt,Emma,whohadlittlesewingexperience,tookonthechalleng eofsewingonecircuitacrossthetwodifferentheart -shapedpiecesoffelt,withlightsononeshee tandmos tofthesewncircuitr yontheothershee t(seeFigur e3).Asshestruggledwithsewingthemicr ocontrollerononeshee toffeltandconnectingittotheLEDlightsontheothershee t,Emmareflect ed,“ProbablyIshoulddomorecraft,”recognizingherneedformoresewingexpertise(Lesson3,Video).Yetevenknowingthis,Emmaproposedarrangingtheelectr oniccomponen tsinthisspatiallycomple xwayand,mischie vously ,embeddingashortcircuitbetweenthetwolayersoffelt.LikeGabriel,inthepreviouscasestudy ,Emmatookonachalleng ethatpushedherlearning.Thegrouptookadvantageofthespatialplacemen tofthecomponen tsanddeviouslyaddedbugsinsidetheheart(betweenthetwolayersoffelt)byplacingoneLEDupsidedownwiththelightshiningintowardthefabric.Thegrouptookonthechalleng eofmakingsome thingdifficultasanopportunitytoimpr ovetheirskillsanddesignnovelbugs.\nFigure4.Lily,Emma,andLucas’DebugIt:(1)Left:Frontofe-textile:awhit eheartwithfourLEDs;oneLEDisflippedwiththelightdirectedtowardsthefelt;(2)Middle:Backofe-textile:themicr ocontrollerissewnoverasecondlayeroffelt;threadsareloose;andLEDsareattachedtopins9,12,6,and10;(2)Righ t:EmmaresewingtheDebugItcreatedbyherpeer s.\nWhenchoosingwha tbugstodesigninthecode,Lucasalsochoseachalleng ethatdeepenedhislearning:addingbugstotheconditionalstatemen tsthatevalua ted\n17\nthestateoftwobuttons.Thiswasnotaneasytaskashewasnotveryfamiliarwithbuttons,yetthiscreatedanopportunitytomak eintentional(suchaswriting“low”insteadof“LOW”)andunin tentionalbugs(Lesson2,Video).Whileworkingonthispartofthecode,heopenlyshar edhiscodingmistakes,saying,“Oh!Imessedup!Wronginthecoding!”asheunin tentionallymadeaseman ticmistakebywritingthesameconditionaltwice(if(but t1Val==HIGH&&butt2Val==HIGH))whenhewantedonetobehighforbothbuttonsandtheothertobehighonlyforthevalueofbutton1(Lesson3,Video).\nThegroupappr oachedlearningasconstantimpr ovemen t,asevidencedwhen,afterLucasfinishedwritingandeditingthecode,Emmaresponded,“Letmeseewha totherthingswecando,”andproposedseveralwaystheycouldimpr ovetheprojectbyaddingmorebugs,suchasassigningthewrongmicr ocontrollerpinstoprogramvariables(Lesson3,Video).Inacomplemen taryepisode,afterEmmatestedwhe therthestitchesandknotswerestrongenoughbyturningtheprojectupsidedown,Lucassuggestedtheycheckthewirestomak esuretheywerenottouching ,lookingforopportunitiesforimpr ovemen t(Lesson3,Video).LucasandEmmasupport edthegroupinfindingwaystoimpr ovetheirproject,appr oachinglearningasconstantimpr ovemen t.\n18Figur e5.CodeforLily,Emma,andLucas’DebugItannot atedwithbugs.\n19\nWhilesewing ,Emmabecamefrustratedyetovercameplen tyofchalleng eswithsupportfromherteam.Findingseveralunin tentionalproblemsinhercircuit,shereflect edandsuggestedapossiblesolution:“Ialwaysforgettotietheknots;it’ssoanno ying,IbettertiethemasIgoalong. ”Lucaspraisedherproposedstrategyandtheprocessofgettingtherebysaying,“Smartthinking!”(Lesson3,Video).Inadifferentinstance,praisingandencouragingeffortdemons tratedwha tLove(2019)callscultur allegacy,thatis,thetraditionsandcustomsofcultur e,inself-belie fs.Emmahadtoaddashort -circuitbuginwhichtwothreadsintersected.Herteamma tes,LucasandLily,wereworriedtheywouldnotfinishontime.“Wha ttimeisit?DoyouthinkI’llfinish? ”Emmaasked.Inresponse,LilyandLucasstartedsingingDolor esHuert a’scivilrightschan t“¡Sísepuede!”2(Yes,wecan)toencourageandpraiseEmma’ seffort(Lesson4,Video).Althoughtheteamcouldnotfinishthecircuitthatdayasclassended,Lilyencouragedherpeer s:“We’llhavetobestrongtofinish”(Lesson4,Video).Theydidindeedfinishstronglyaftermoreworkthenextday,andtheirbuggyprojectbefuddledthereceivinggroupforawhile.\nWhenLily,Emma,andLucasdebug gedaprojectthattheirpeer shadcreated,wefoundgrowthmindse tpracticesjustlikeinalltheothergroups.Forinstance,theyfacedunexpect edchalleng esastheydecidedtoresewthecircuitintheprojecttheyweredebug ging—a“sickcloudthrowingupacolorfulrainbo w”designedbyEvelynandNicolás(formoredetailsonthisproject,seeFieldsetal.,2021).Thegroupdeba tedcuttingaparticularlygnarly -lookingthreadandresewingtheconnections(Lesson6,Fieldnot es).\"Youcouldsaythebiggestchalleng ewasfiguringoutifweweresupposedtocutorleavethemthere,”Lilyreflect ed(Lesson7,Video).Thegroupwasawarethatresewingawholeprojectinacoupleofhour swaschallenging.Yet,bymistakenlytakingafeatureoftheprojectasabug,theydecidedtoembarkonthischallengingtask.Emmatriedtosewasfastaspossible,regularlyaskingLilyforfeedbackwithoutbeingafraidofbeingwrongormakingmistakes.Lilyinspect edtheartifacttomak esuretheknotsweretightandencouragedEmmabytellingherthatshecouldfixit.OnceEmmafinished,Lucassuggestedtheytesttheprojecttomak esurethewiresdidnottouch,appr oachinglearningasconstantimpr ovemen t.Whentesting,theyrealiz edtheprojectdidnotwork.Theywerefrustratedandstumped.Yettheypersistedandgrabbedcoppertapetodebugtheproject.“Let’sseeifitworks,”Lilysaidastheytriedsome thingnew.Theycoveredthestitchesonthebackoftheprojectwithcoppertapetomak esurethecircuitwasconnect ed.Thecoppertapedidnotsolvetheissues.Astheycontinuedtopersist,LucasandLilystartedinspectingtheconnectionstothepinsofthemicr ocontrollerwhileEmmainspect edthecode.Intheend,theydidnotfullydebugthe“sickcloud”projectintheallot tedtime,some thingman ygroupsintheclassexperienced(indeed,itwasnotagoalforeachgrouptofullydebugthebuggyprojectstheirpeer screated).Yetagain,thiswasseenaspartofdevelopingcomfortwithfailur e,andLily,Emma,andLucascontinuedtolearnastheymetwiththedesigner sandfoundoutsomeofthebugstheydidnotdiagnose.\n2DoloresHuertaisaLatinacivilrightsactivistwholed(withCésarChávez)the1960sfarmerandimmigrantrightsmovementintheSouthwesternUnitedStates.Shecameupwiththechant“¡Sísepuede!”whichbecameacallforactionandpersistenceduringrallies(Sowards,2019).\n20Discussion\nInthispaper ,weanaly zedthegrowthmindse tpracticesthathighschoolstuden tgroupsexhibit edduringDebug gingbyDesign(DbD),aninterventioncreatedtoempo werlearner stobecomemorefamiliar ,comfortable,andcapablewithdebug gingbydesigningbugsfortheirpeer stosolve.Ouranaly sisofthesefourgroupsofstuden tsprovidedsomeinsigh tsintohowlearner sdevelopedtheirgrowthmindse tsandbecamemorecomfortablewithdebug gingbyattendingtothemotiv ational,collabor ative,andemotionalfactorsofdebug gingholis tically.Studen tsengagedinpracticessuchaschoosingchalleng esthatledtomorelearning ,praisingeachother ’sefforts,appr oachinglearningasconstantimpr ovemen t,persistingaftersetback s,anddevelopingcomfortwithfailur e.Thislastpracticeaddstoandcomplemen tsthepracticesmen tionedinCampbelletal.'s(2020)frame workofpracticesforgrowthmindse tlearningactivities,specific allyhighligh tingmistakesandfailur esas“anintrinsicpartofthelearningprocess”(Papert,1980,p.153).Whileotherstudieshavefocusedontheconcep tualskillsandpracticesstuden tsusedwhendebug gingphysicalcomputingprojectsorthetypesofbugstheydesignedandsolved(Lui,Fields,&Kafai,inpress),thisstudyfocusedonthegrowthmindse tpracticesobservedwhenstuden tswereinchar geofmakingordesigningbugs.Thispapercontribut estocurrentresear chongrowthmindse tandholis ticappr oachestodebug gingbychar acterizinganinventoryofgrowthmindse tpractices,highligh tingtheimport anceofthedesignoflearningenvironmen tsinpromotinggrowthmindse ts,andprovidingevidencethatcollabor ationplaysanimport antroleingrowthmindse tpractices.\nOuranaly siscontribut estoamoreholis ticunder standingofstuden ts’social,emotional,andmotiv ationalappr oachestodebug gingthroughthechar acterizationofgrowthmindse tpractices.Whileitisworthhighligh tingthatmos tofthegrowthmindse tpracticesoccurr edwhilestuden tsweredesigningbugs,itisnecessar ytofurtherexplor ehowthiscompar estootherdesign-basedlearningactivitiestobeabletodrawconclusionsabouttheeffectoftheintervention.Here,ourinventoryofpracticesisafirststepthatcouldbehelp fulinconductingstudiesthatcompar emindse tpracticesacrossinterventionsandfurtherexplor etherolethatintentionallydesigningbugsmayhaveonstuden ts’mindse ts.Itcouldalsobehelp fulforteacher stoiden tifygrowthmindse tsastheydevelopintheirclassr oom.Futur eresear chcouldexpandonwher egrowthmindse tpracticesemer geinothercontextsoracrossalargersetofstuden ts.Whileweattemp tedtostudyadiverserangeofstuden ts(basedontheteacher'sassessmen tofwhichgroupshadbeenmoreandlesssuccess fulinapreviousproject),ourexplor atorystudywasinevitablyconstrainedbyasmallsamplesize(10studen ts),aneigh t-classperiodfocus,andsmallgroupinteractions.Otherstudiescouldbuildonouremer gentlistofgrowthmindse tpracticeswhilealsoconsideringotherfactors,suchashowstuden ts’priorprogrammingexperienceandtheirinitialperceptionsofcomputingmigh talsoimpacttheirmindse tsinpracticeandtheirdevelopmen tofdebug gingskills.Inthefollowingsections,weconsiderthedesignedcontextfortheintervention(Debug gingbyDesign),wha twelearnedaboutdevelopingstuden ts'growthmindse tpractices,andmoredirectionsforfurtherresear ch.\nDebuggingandGrowthMindsetsinComputingEducation\nDbDmayprovideanopportunityforlearner stodevelopgrowthmindse tsin\n21computing ,alignedwithYaegeretal.’s(2018)recognitionoflearner s’needswithanemphasisongivingstuden tsagreatersenseofcontroloverbugsanddebug ging.ThedesignofDbDcontrastswithpreviousgrowthmindse tinterventionsincomputingthatcenteredondeliv eringinforma tionaboutmindse tstostuden tsandreflectiv ewritingor“sayingisbelie ving ”exercises(e.g.,Simonetal.,2008).WithDbD ,weaimedtoempo werstuden tsindebug ging ,withallitscognitiv eandsocio-emotionalchalleng es,throughtheintentionaldesignofbugs.\nDbDbuiltonalongs tandingtraditionofconstructionis tactivitiesthatputlearner sincontroloftheirownlearningbydesigningsoftw areorgameapplic ationsforother s(Har el&Papert,1990;Kafai,1995),extendingthisconcep tbyshiftingthefocusfromdesigningfunctionalartifacts(thefocusofmos tinstruction)todesigningnon- functional,orbuggy,projects.Thelearningenvironmen tputsstuden tsincontrolofnamingandcreatingmistakesbymakingthemdesignersratherthansolelysolversofbugs(seeFieldsetal.,2021).Ouranaly sisprovidesacompellingexampleofan“examina tionofthemindse tsconveyedbyorembodied”inthelearningenvironmen t(Dweck&Yeager,2019,p.490).\nOuranaly sisrevealedthatmos tinstancesofgrowthmindse tpracticesoccurr edprimarilywhenlearner sdesignedbuggyprojectsfortheirpeer stosolve,ratherthanlaterwhenstuden tssolvedtheexchang edbuggyprojects.Whendesigningbugsfortheirpeer s,learner ssough tchalleng es,madeandsolvedunin tentionalbugs,encounteredfailur e,andpersisted.Designingbugsbuiltonadolescen ts'enhanceddesir etobe“treatedasthoughtheyarecompe tent,haveagency ,andautonom y”(Yeageretal.,2018,p.104),puttingthemincontrolofbugsininterest-driv enprojectsthatcanchalleng etheirpeer s.Inadditiontoourearlierresear chonthebene fitsoftheentireDbDactivity ,whichfocusedmoreonstuden ts’self-reportofgainsweeksaftertheactivitywascomple ted(Fieldsetal.,2021),thispapershowsthespecificopportunityforpracticinggrowthmindse tsinthemomen t,especiallyduringthedesignphaseofDbD .\nThishasimplic ationsforwher eandhoweduc atorsmigh tsupporthighschoolstuden ts’growthmindse tsinaction(i.e.,inpractice).Aspreviouslydiscussed,mos tresear chstudiesongrowthmindse tsincomputingtaketop-do wn,didacticappr oachesthatmayresultinfalsegrowthmindse ts—tha tis,simplytellingstuden tsaboutgrowthmindse tswithoutprovidingthemwithstrategiesandopportunitiesforpractice(Dweck&Yeager,2019).Whileexplicitinstructionandreflectionongrowthmindse tcancertainlyhelpstuden ts(Burne tteetal.,2020;Kwaketal.,2022),wesuggestthatthisdesign- focused,practice- firstappr oachmaybeparticularlywellsuitedtocomputingeduc ation.Learningenvironmen tssuchasDbDmayhelppromot egrowthmindse tinpracticeinobservableandconcr eteways,givingstuden tstheopportunitytoencounterordesignforfailur einordertodevelopresilienceandtakeowner shipoftheirownlearning.Whileourresear chdoesnotmak eacausalclaim,thisappr oachofputtinglearner sinchar geofdesigningmistakesfortheirpeer stosolveandsolvingbugsintheirpeer s’projectsrequir esfurtherinvestigation.Thismayinvolvecomparingthepracticesofstuden tsinDbDwiththoseofstuden tscreatingopen-endedprojects.DbDcouldalsobeappliedtootherareasofcomputingeduc ationthatinvolveiterating,testing,andsolvingproblems.\n22TowardsaHolisticApproachtoDebugging\nWesough taholis ticappr oachinstudyinggrowthmindse tpracticeswhilestuden tsdesignedandsolvedbugs,emphasizingthemotiv ational,emotional,andsocialaspectsofdebug ging.Thecasestudiesdemons tratedhowdebug ginginvolvedcognitionandemotion(DeLiemaetal.,2020)andnotjustbasicskillsforiden tifyingandsolvingisola tedproblems.Whenencounteringcognitiv echalleng es,studen tmotiv ationandemotioninfluencedhowtheyaddr essedbugs,andatthesametime,bugsthemselv esaffectedmotiv ationandemotion.Further ,whilebothdebug gingandmindse tpracticesareoftendiscussedasindividualprocesses,collabor ationalsoplaysakeyrole.Accep tingdebug gingasaholis ticprocessrequir esustoexaminetheroleofsocialinteractionandcultur einadditiontocognitionandemotion.\nPeercollabor ationsplayedacritic alroleaslearner sengagedwithgrowthmindse tpracticesduringDbD .Aswesawinouranaly sis,78%oftheinstancesofgrowthmindse tpracticesiden tifiedoccurr edinpeer -to-peerinteractions.Furthermor e,peerfeedbackandsupportwerecritic alinpromotingpersistence,reshapingsetback sasopportunitiesforlearning ,andprovidingaspaceforstuden tstosupportoneanotherwhenfacedwithfailur eandfrustration.Thisalignswithotherstudiesondebug ginge-textilesthatshowtheimport anceofprovidingopportunitiesforpeercollabor ation(Jayathirthaetal.,2020;Shawetal.,2019;Luietal.,inpress).Italsoalignswithearlierresear chonDbDthatshowedhowlearner sfeltmorecomfortableaskingpeer sforhelpafterdoingtheDbDactivity(Fields&Kafai,2020).\nMor ebroadly ,oneunan ticipa tedfindingisthatdebug gingandgrowthmindse tpracticesareembeddedwithincultur aldiscoursespresen tinlearner s’communities.Inouranaly sis,wesawhowstuden tcultur allegacymanif estedinlearner s’belie fsabouttheirabilitytodebugwhenLilyandLucasstartedchan tingDolor esHuert a’s“¡Sísepuede!”toencourageandcelebr atetheirpeer s'efforts.The“¡Sísepuede!”chan t,whichemer gedinthe1970sduringthecivilrightsmovemen tforfarmandimmigr antworkerrightsintheSouth westernUnit edStates,isanexpressionofencouragingeffortandpersistencewithgreatcultur alandhistoricalsignific anceinLatinxcommunities(Sowards,2019).Indeed,asLove(2019)writes,perseveranceforjustice,civilrights,andsurvivalamongminoritiz edcommunitiesaremanif estationsofincremen taltheoriesofintelligence(suchasgrowthmindse torgrit)thatareusuallyignor edintheclassr oombutthatough ttoberecogniz ed,protected,andnurtur ed.Thisalignswithotherresear chthatdemons trateshowstuden tsmakinge-textilescanleveragecultur allyrelevantpractices(Kafaietal.,2014)anddeveloptheirsociopolitic aliden tities(Sha wetal.,2021).Furtherresear chcouldaddr esshowdebug ginginterventionscanbuildonthecultur allegaciesofstuden tsandtakeadvantageofthegrowthmindse tpracticesthatarealreadypartoftheircommunities.Atrulyholis ticappr oachtodebug gingmus tincludenotonlycognition,emotion,andmotiv ationbutalsocultur eandsocialinteraction.\nFinally ,thecollabor ative,holis ticworkinthisinterventionhappenedinasociallysitua tedclassr oomcontext.ThedesignoftheDbDunitwaspartofthatcontext,butsowasthehighlyexperiencedteacherwhopromot edaclassr oomcultur ethroughouttheschoolyearthatencouragedstuden tcollabor ations,legitimiz edstuden texpertise,andmodeledcomfortwithfailur e(formoreinforma tionon\n23teaching ,seeFieldsetal.,2019;Fieldsetal.,2021).Whileouranaly sisinthispaperfocusedoncasestudystuden tgroups,theteacherplayedacrucialrolebyintentionallygroupingstuden tswithdifferentabilities,providingjust-in-timefeedback,andencouragingstuden tgroupstodesignmorechallengingbugsandexplor ecomputingconcep tsbeyondwha twascoveredintheclass. Futur eworkcouldexplor eholis ticsupportsforgrowthmindse tsattheclassr oomlevel.\nFutureDirections\nWereport edonemer gentgrowthmindse tpracticesexhibit edbyhighschoolstuden tsinthefirstclassr oom-basedimplemen tationofDbD .Giventhelackofresear chonK–12studen ts’growthmindse tsincomputing ,muchmoreworkcouldexplor egrowthmindse tsinothercontextsofdebug gingandmoregener alcomputinglearning.Suchresear chshouldfocusonthesocial,motiv ational,andemotionalaspectsofcreatingandsolvingbugs,aswellashowteacher scanfostersupportiv ecollabor ativeenvironmen tsinwhichstuden tsarenotafraidofbugsorseekinghelpandsupportfromtheirpeer s.Infutur estudiesweplantoresear chhowdifferentteacher spromot egrowthmindse tpracticeswhenimplemen tingorintegratingtheactivityintheircomputingclassr ooms.Thiswillallowustoexaminetherangeofteachingpracticesthatsupportgrowthmindse tinthecontextofDbD .Thecasestudyanaly sisalreadyprovidedsomeevidenceofhowteacher splayedaroleinstuden ts'engagemen tingrowthmindse tpracticesthroughfeedbackandadviceatkeypoin tsofthedesignprocess.Comparingstuden tgrowthmindse tpracticesacrossclassr oomsandteacher simplemen tingtheactivitymayhelpiden tifywha troleDbDhas,ifany,inpromotinggrowthmindse t. \nAdditionally ,itisnecessar ytoanaly zetherelationshipbetweengrowthmindse tpracticesobservedintheclassr oomandtheperceivedgrowthmindse tofstuden tsandteacher s,asassessedthroughself-report edinstrumen ts.Werecen tlydevelopedandvalida tedapre-andpost-surveyinstrumen ttomeasur estuden tself-belie fsincomputinganddebug gingingener al,aswellasine-textiles,basedontheimplemen tationdiscussedinthispaper(Mor ales-Na varroetal.,2023a).Weusedthisinstrumen tinaquasi-e xperimen talstudy ,withsomeclassr oomsdoingDbDandother sdesigninganadditionale-textilesproject.ResultsfromthisotherstudysuggestthatDbDprojectcomple tionwasuniquelycorrelatedwithincreasedprogramminggrowthmindse t(Mor ales-Na varroetal.,2023b).Learner scanbene fitnotonlyfromdesigningfunctionalapplic ations,butalsofromdesigningbuggyapplic ations—bothofwhichareandshouldbeanintractablepartofanylearningproduction.\nBeyondDbDande-textiles,ourgrowthmindse tpracticesinventorymaybeusefulforobservinggrowthmindse tinactioninothercomputingeduc ationsettings.Similarly ,futur eresear chcouldinvestigatetherelationshipbetweengrowthmindse tpracticesanddebug gingpracticesandstrategiesthatstuden tsengagewithwhensolving(orcreating)bugsinopen-endedprojects.\nConclusions\nThispaperillustrateddifferentgrowthmindse tpracticesthathighschoolstuden tsengagedwithduringaninterventiondesignedforstuden tstocreatebuggyprojectsfor\n24theirpeer s.Analy sisofstuden ts’peerinteractionsandcasestudiesshowcasedhowhighschoolstuden tteamsdesignedbugsforother stosolveandintheprocesschosechalleng esthatledtomorelearning ,persistedaftersetback s,gaveandvaluedpraiseforeffort,appr oachedlearningasconstantimpr ovemen t,anddevelopedcomfortwithfailur e.Thesepracticescanbehelp fultofurtherstudygrowthmindse tinactioninothercomputinglearningenvironmen ts.Computingeduc atorsandresear cher smus tstrivetodevelopinterventionsthatengagestuden tsindevelopinggrowthmindse tpracticesratherthansimplytellingthemaboutthem.Finally ,theideaofmakingstuden tsdesigner sofbugs,ormistakes,isnotlimit edtoe-textiles.Debug gingbyDesigncouldbeusedasapedag ogicalappr oachinotherareasofphysicalcomputingandprogrammingeduc ationingener al.Welookforwardtowha tfutur eapplic ationsofDbDcanshowaboutstuden tlearningandengagemen twithgrowthmindse tpractices. \nAckno wledgmen ts\nThisworkwassupportedbyagrantfromtheNationalScienceFoundationtoYasminKafai(#1742140)andMichaelEisenberg(#1742081).Anyopinions,findings,andconclusionsorrecommendationsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyreflecttheviewsofNSF,theUniversityofPennsylvania,theUniversityofColorado,Boulder,orUtahStateUniversity.SpecialthankstoLindsayLindbergandAmmarahAftabfortheirsupportindatacollectionandanalysis.\nR esear chE thicsandConsen tSt a t emen t\nWerecruitedstudentsalreadyenrolledinanintroductorycomputinghighschoolclass.Aresearchervisitedtheclasstoinvitestudentstoparticipate,distributeconsentandassentforms,andaddressanyquestionsabouttheresearchprocedures.Parentsreceivedconsentformspriortothestudy,whichincludedabriefexplanationoftheresearch,andyouthassentedtotheirparticipationinwriting.Studentsdidnotreceiveanyincentivesforparticipatinginthestudy.TheIRBboardoftheUniversityofPennsylvaniaapprovedresearchprotocolsanddatacollectiontechniques(Protocol:827747).\nDa t aA v ailabilitySt a t emen t\nDuetothenatureofthisresearch,participantsinthisstudydidnotagreefortheirdatatobesharedpublicly.Supportingdataisnotavailable.\nDisclosur eSt a t emen t\nNopotentialconflictofinterestwasreportedbytheauthors.\nFundingde t ails\nThisworkwassupportedbytheNationalScienceFoundation[1742140].\nR e f er 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o d ea n dD e f i n i t i o n :E x a m p l e1E x a m p l e2\nC h o o s i n gc h a l l e n g e st h a tl e a dt om o r el e a r n i n g :Studentspurposefullychoosechallengesthatleadthemtolearnnewthingssayingthingslike“Ilikeachallenge”insteadofavoidingchallengesthatmayexposeareasofweakness.Despiteherlittleexperiencesewing[“probablyIshoulddomorecraft”],EmmadecidestosewthemicrocontrollerandtheLEDsontwosheetsoffeltoneaftertheother.ThisrequireshertoturnthehearttofigureouthowsewingtheCPononepieceoffeltandtheLEDsontheotherwouldwork.(VideologGroup1Day3,page7)Whencomingupwithbugs,Evelynsaysshewantstoincludemusic.Sheexplainsthatshewantstopressabuttontoactivatethemusic.Nicolássaysthat’shardandthathedoesn’tknowhowtocodemusic.ThenNicolásturnstohiscomputerandstartstyping.A10minuteslaterandaftertryingthingsonthecomputer,NicolásshowsEvelynhowtheycouldaddmusic(VideologGroup3Day2,page3).\nP e r s i s t i n ga f t e rs e t b a c k s-w i t hp r e c e d i n gf r u s t r a t i o n :Studentsdisplayfrustrationandpersist,overcomingitwithcontinuedengagementwiththeproblemathand.Evelynlooksattheheartandpointsoutthat“they(thedesigninggroup)didn’ttieanyknots.”Sheseemsfrustrated.Suddenlyshemanipulatestheheartandexclaims,“Ididit.”(VideologGroup3Day2,page3)Despiteherfrustrationshekeepssewing.Evelyn“IcouldhavefixeditalongtimeagoifIknewtheconnectionswereallloose”shereflects.(VideologGroup3Day7,page5)Evelynhasbeentryingtoworkonanaestheticandcircuitdesign.Shesayseverythingistoohardtodraw.Nicolássuggestsusingsquaresandcirclestostartandthenaddingmorebasicshapestorepresentwhattheywant.(VideologGroup3Day2,page3)Evelynfollowsthisnewstrategyandfinishesthedrawing.(VideologGroup3Day2,page4)\nP e r s i s t i n ga f t e rs e t b a c k s-c o - o c c u r r e n c eo ff r u s t r a t i o n :Studentsencounterachallengeandovercomeitbytryingagainoradoptinganewstrategy:“I’llhavetotryharderorworkdifferently.”Evelynisfrustratedwithabugthatpreventsthelightsfromworking,herpartnerencouragesher“Youcanfixit!Fixit!,”shetriesagainandmanagestogetthelightstoturnon(VideologGroup3Day7,page2)Gabrielcannotmanagetogetthemusictoplay,hehasafunctionthatplaysitandiscallingit,butitneverplays.Hetriescallingthefunctioninthevoidsetup()togetthemusictoplay(VideologGroup2Day7,page7)\nG i v i n ga n dv a l u i n gp r a i s ef o re f f o r t :Studentspraisetheirownortheirpeers’efforts,strategies,andprocessesCamilaasksGeorgia:“itlooksgood,right?”Showingherthenoteshejustcut.“Yeah,itlooksrealgood”Georgiareplies.(VideologGroup2Day3,page9)NicoláspraisesLilyforaddingnuisancecode[declaringfourextravariablesthatwerenotused]justtoconfusethedebuggingteam,saying“goodjob,youmadeitreallyhard[fortheotherteam]”(Videolog\n32Group3Day7,page7)\nA p p r o a c h i n gl e a r n i n ga sc o n s t a n ti m p r o v e m e n t :Studentsseethegoaloflearningasimprovingperformance.Lucasfinisheswritingthecodeandsays“Success”indicatingheimplementedthecodechangesEmmasuggested.Emmaresponds,“Letmeseewhatotherthingswecando”andlooksforwaystheycanimprovetheirproject.(VideologGroup1Day3,page3)Whendecidingonhowtoorganizethenegativeandpositivethreadsintheirproject,Nicolásreflects“weshouldhavedonethisforthemuralproject.Itwouldhavebeennice”asheadmireshowtheircurrentstrategyforthedesigncouldhaveimprovedtheirmuralproject.(VideologGroup3Day4page2)\nD e v e l o p i n gc o m f o r tw i t hf a i l u r e-( 1 )a s k i n gq u e s t i o n sa n dr e q u e s t i n gf e e d b a c kw i t h o u tf e a ro fb e i n gw r o n g :Studentsaskquestionstotheirpeerswithnofearofbeingwrong.LiamaskedSophiaiftheconstantpinvaluesinArduino(“LOW”and“HIGH”)couldbelowercase.(VideologGroup4Day3,page8)WhilesewingacircuitSophiaaskedLiam“Canpositiveandpositivecrosseachother?”(VideologGroup4Day3,page7)\nD e v e l o p i n gc o m f o r tw i t hf a i l u r e-( 2 )s h a r i n gf a i l u r ew i t ht h e i rp e e r s :Studentsrecognizeerrorsormistakesandsharethemwiththeirpeers.GeorgiarealizesshesewedtheLEDtothewrongpin.“WasIsupposedtosewnumbersix”Camila“Idon’tknow”Georgia“Isewedittonumberten.(VideologGroup2Day3,page9)Evelynsharedamistake“Ohmygod!Iwasgoingtokeepgoingwithoutsewingupthelight!”(VideologGroup3Day4,page2)\nD e v e l o p i n gc o m f o r tw i t hf a i l u r e-( 3 )e m b r a c i n gf a i l u r ea n di m p e r f e c t i o na sp a r to ft h ep r o c e s s :Studentsvoicethatfailureispartoftheprocessandthattheirprojectsdonothavetobeperfect.Whenthefeltdoesn’talignEmmaexplains“it’salittlebitoff,butit’sokay”.Sheoffersasolution,saying“we’lljustcutitoff,itdoesn’thavetobeperfect”(VideologGroup1Day4,page6)Georgia“I’mjustwarningyouI'mnotthebestatsewing,that'salright.Aslongasitalllooksokayattheenditdoesn'treallymatter.”(VideologGroup2Day3,page6)\n33"    },    {        "title": "2402.01905v2.Carthago_Delenda_Est__Co_opetitive_Indirect_Information_Diffusion_Model_for_Influence_Operations_on_Online_Social_Media.pdf",        "content": "Carthago Delenda Est: Co-opetitive Indirect Information Diffusion\nModel for Influence Operations on Online Social Media\nJwen Fai Low, Benjamin C. M. Fung, Farkhund Iqbal, and Claude Fachkha\nAbstract\nFor a state or non-state actor whose credibility is bankrupt, relying on bots to conduct non-attributable,\nnon-accountable, and seemingly-grassroots-but-decentralized-in-actuality influence/information opera-\ntions (info ops) on social media can help circumvent the issue of trust deficit while advancing its interests.\nPlanning and/or defending against decentralized info ops can be aided by computational simulations\nin lieu of ethically-fraught live experiments on social media. In this study, we introduce Diluvsion , an\nagent-based model for contested information propagation efforts on Twitter-like social media. The model\nemphasizes a user’s belief in an opinion (stance) being impacted by the perception of potentially illusory\npopular support from constant incoming floods of indirect information, floods that can be cooperatively\nengineered in an uncoordinated manner by bots as they compete to spread their stances. Our model,\nwhich has been validated against real-world data, is an advancement over previous models because we\naccount for engagement metrics in influencing stance adoption, non-social tie spreading of information,\nneutrality as a stance that can be spread, and themes that are analogous to media’s framing effect and are\nsymbiotic with respect to stance propagation. The strengths of the Diluvsion model are demonstrated\nin simulations of orthodox info ops, e.g., maximizing adoption of one stance; creating echo chambers; in-\nducing polarization; and unorthodox info ops, e.g., simultaneous support of multiple stances as a Trojan\nhorse tactic for the dissemination of a theme.\nKeywords— agent-based modeling, information diffusion, information propagation, information operations, in-\nfluence operations, psychological operations, rumor spreading, rumor management, social cybersecurity, persuasive\ncomputing\n1 Introduction\nArtificial intelligence (AI), alternatively known as machine learning, has advanced on all fronts — behavioral, textual,\naudio, visual, and video — to the point where comprehensive and convincing digital personas can be crafted from\nmere lines of code (Section 2.2.5). As a result, the detection of information operations (defined in Section 2.2.1)\nas a method for combating the spread of undesirable narratives by adversarial agents on online social networks,\nwhile still important, becomes increasingly non-viable as the predominant means of achieving information security.\nA 2021 review of research papers on countering influence operations that went as far back as 1972 noted that all\n223 studies they found were on user-focused countermeasures (users being consumers of disinformation) [1]. The\nauthors concluded that “[b]eyond fact-checking, the research base is very thin”, and called for increased focus on key\nareas such as “studying countermeasures that target the creators of disinformation content in addition to studying\nconsumer-facing policies”. In response to this call for action, we propose complementing existing detect-and-correct\nmethods that conduct struggles against undesirable information on a tactical, low-level per-message or per-user basis\nwith a method that facilitates waging war against disinformation from a high-level strategical perspective, where\nusers and messages are seen as a part of a synergistic whole meant to achieve a combined arms outcome.\nTo that end, we designed Diluvsion , an agent-based model for simulating how information operations can perturb\nopinion evolution in an online social media platform. The model relies on parameters derived from publicly available\ndata on a social media platform to forecast the outcomes of info ops on similar platforms. As such, it can serve as a\ntool to be used by planners in crafting campaigns to attack disinformation or defend against it without needing to\nfirst test the interventions in the real world — tests that can have unsavory ethical and even geopolitical implications\n[2, 3]. Like many other information diffusion models, Diluvsion is inspired by epidemiology but updated to consider\nnew insights on transmission dynamics from the COVID-19 pandemic (Section 2.1.2). But, unlike many models,\nDiluvsion is developed with info ops and bots in mind, meaning that unconventional routes for expanding influence\nare accounted for and that optimistic assumptions for information diffusion in other arguably marginally more benign\n1arXiv:2402.01905v2  [cs.SI]  6 Feb 2024contexts are discarded, for example, the ability to select seed nodes to maximize propagation in product advertising\n[4, 5].\nWhile models are often designed to adhere closely to the real world, no model is meant to be a perfect recreation of\nthe real world; design choices reflect aspects that the modelers wish to emphasize. In our case, Diluvsion brings into\nprominence the role played by indirect information, namely social cues [6] — engagement metrics and the perception\nof mass adoption — in opinion propagation. Repetition is the key to the operation of our model. Much like Cato’s\nbrilliant tactic of incessantly playing up Carthage’s threat to Rome [7] and calling for Carthage’s destruction in the\nRoman Senate led to the genocide [7] of Carthage, much like repeating a lie can let it take on the mantle of truth [8, 9],\nso does the persistent saturation of an opinion in the media landscape can drive its adoption among the populace\n[10, 11, 12, 13, 14, 15, 16].\nThe contributions of this paper are:\n•An agent-based co-opetitive information diffusion model for information operations on social media that is\nnovel in its incorporation of concepts not considered by existing models: an infectious neutral stance; themes\nthat must piggyback on stances to infect users and have an enhancing or diminishing effect on the stance’s\ninfectivity; non-social tie-based transmission of information; parity in the treatment of bots backing different\nstances; perception of mass support inferred through engagement metrics such as the number of retweets\ninfluencing stance adoption; and agents with heterogeneous roles and activity levels populating the simulation.\n•In-depth rationale behind the design/modeling choices, demonstrating the empirical evidence that our model\nis predicated upon and consequently the real-world applicability of our model.\n•Initial “validation” of the model by drawing simulation and graph structure generation parameter values from\ndistributions found in real-world data and further validation that succeeded in reproducing via simulations the\ndistributions from a different empirical dataset.\n•Simulations that are in part ablation studies and in part experimentation with info ops tactics. The maneuvers\nexamined include bot-backed minority stances facing off against a non-bot-backed majority stance, polariza-\ntion/depolarization, and maximizing theme diffusion. The emergent outcomes from implicit cooperation and\ncompetition found in these simulations are examined and discussed in the context of their implications for\ndecentralized info ops.\n2 Background\nAs our work occurs at the intersection of different domains, we split our discussion of relevant works into two major\nsections: information diffusion and influence maximization (Section 2.1), and information operations (Section 2.2).\n2.1 Information diffusion models\nThe study of information diffusion — the process by which information spreads from one entity to another — has a\nlong history. In the particular case of diffusion on online social networks, the history is sufficiently extensive to allow\nfor a survey [17] of existing work to be published in 2013. The closely related domains of influence maximization and\nconsensus formation have similarly rich histories, which can be seen in a 2021 survey [18]. Influence maximization\n(IM) studies the problem of locating influential nodes that maximize the spread of information to the entire network.\nConsensus formation is concerned with determining the manner in which a group of agents can all arrive at the same\nopinion. For an in-depth discussion of these topics, we refer the readers to the aforementioned surveys. Here, we\nrestrict ourselves to discussing the most prominent and relevant models and the elements from those models that\ninfluenced the development of our model, Diluvsion . The information presented is also meant to help situate Diluvsion\namong the constellation of predecessor models.\n2.1.1 Information\nIn the space of diffusion/propagation models, information goes by many different names, including information [19],\ninfluence [18], rumor [20], opinion [21], memes [22], and news [23], depending on the context. While we will use these\nterms interchangeably throughout this paper, we favor the term “opinion” due to the fact that the major component\nof the information being spread in networks within our model, the stance, is polarized and homophilic. Together with\nnon-polarizable themes, they form a unit of information in our model (Section 3.1). That is not to say that other\nforms of information are not present in the model. Engagement metrics, for instance, are exposed to agents and can\ninfluence their decision to adopt a piece of information.\nWe are aware of works that have studied how real news and fake news spread differently on social media [24, 25].\nThe information of interest in this paper, i.e., the information being disseminated, is not generally regarded as “fact”\n2(e.g., quantum mechanics and general relativity applied within their respective restricted contexts and not outside)\nbut “opinion” (e.g., the dispute over the rarity of scale-free networks in the real-world [26]), which does not have\na clear delineation between true and false. In our model, “disinformation” is simply a term of convenience for\nundesirable information; messages are indistinguishable in terms of veracity, so veracity cannot influence a message’s\nspread. Only stances and themes have influence.\n2.1.2 Uncontested information diffusion\nBased on the typology introduced in [17], information diffusion models can be separated into two categories: explana-\ntory and predictive. Explanatory models aim to “infer the underlying spreading cascade, given a complete activation\nsequence.” Diluvsion belongs to the other category, predictive models, that aim to predict “how a specific diffusion\nprocess would unfold in a given network, from temporal and/or spatial points of view by learning from past diffusion\ntraces.” Predictive models are further divided into graph- and non-graph-based approaches. Graph-based approaches\nassume that information spreads along the edges in the graph. Non-graph-based approaches ignore network topology\nand model information diffusion as a process represented by a system of differential equations, in emulation of similar\nequations in epidemiology [27, 28].\nThe two most well-known graph-based models are the independent cascade (IC) [29] and linear threshold (LT) [30]\nmodels, and they serve as the base models for most influence maximization research. IC models diffusion as a chain\nreaction in which a node that has adopted a piece of information, designated as being activated, attempts to activate\ninactive neighboring nodes in the graph with a certain probability. There are no retries and an activated node remains\nactivated forever. In LT, an inactive node becomes activated if the sum of influence from activated neighboring nodes\nexceeds the node’s threshold. Influence takes the form of weighted directed edges between nodes and the threshold\nis generally drawn from a uniform random distribution. An activated node remains forever activated in LT as well.\nIC can be seen as sender-centric in contrast to the receiver-centric LT.\nIn non-graph-based models, the model names usually take after the sequence of state transitions permissible\nwithin the model, where the states are susceptible (S), infected (I), and recovered (R), which are based on the earliest\nepidemiological model, SIR [27]. The first explicit link between information and disease was established by Daley and\nKendall, who were inspired by the SIR model to create the Daley–Kendall (DK) model for rumor propagation [31].\nNotable variants of DK include the Maki–Thompson model [32]. Rumor diffusion models have states such as ignorant\n(I, susceptible), spreader (S, infected), and stifler (S/R, recovered) that mirror their epidemiological counterparts,\nalthough these specific terms were not used in the initial work by Daley and Kendall. Later reviews [33, 34] showed\nthe cross-pollination of ideas between rumor-specific and biological epidemiological models. Subsequent studies on\nepidemiological diffusion models also innovated upon the transition sequence, e.g., SIRS [35] or added additional\nstates, e.g., SIHR [36] rumor spreading model where H stands for hibernator and SHIR [19] model for competitive\ninformation diffusion on online social media where H stands for hesitated. DK descendants innovated in a similar\nfashion, e.g., ISCR [37] which added a counter-spreader (C) state.\nInDiluvsion , network topology matters, as in IC and LT, and information generally travels based on the agents’\nconnections. The influence of epidemiological models, SIR, SIS, etc., are in inclusion of non-graph edge routes\n(circumventing topology) for information diffusion to our model, as well as in the agents’ non-durable immunity\n(SIS). However, agents are never in an uninfected state within our model, as they only ever transition between the\nthree mutually exclusive infected states (stances), negative, neutral, and positive, with infection by non-exclusive\nthemes providing an additional differentiating mechanism akin to viral variants (Section 3.1).\nThe SARS-CoV-2 pandemic has revealed the shortcomings of some graph-based models. In information diffusion,\nthe common assumption is that a link/edge exists between two people only if they have social ties (e.g., friends,\nfamily, followers, and colleagues). However, in epidemiology, such an assumption is not always made because ties can\nbe formed with complete strangers in many situations. A passerby breathing, coughing, or sneezing in a crowded cafe\nspreads aerosolizable infectious disease almost as effectively as staying under the same roof with a diseased family\nmember [38]. On social media, many posts are publicly viewable and lively discussions can occur in the replies section\nof those posts, analagous to a forum or a cafe, and a user may be exposed to the opinions of strangers, i.e., people\nwho are neither followers nor followees, who replied to the post of the user’s followee (this idea will be explored in\nSection 3.3.2). Another issue is the strong assumption inherent in designating the recovered/stifler state as an agent’s\nend state in an information epidemic. SARS-CoV-2 immunity is transient. It is not permanent. Reinfection by the\nsame or different variants is a real possibility [39]. Non-permanent resistance to contagions that increases or decreases\ndepending on the infection type is a mechanic implemented in Diluvsion to offer an alternative perspective to the\nstrong assumptions of permanent immunity in the existing literature (Section 3.4).\n32.1.3 Competitive information diffusion\nClassical information diffusion does not account for information possessing polarity and existing in a contested\nspace. “Vaccines are safe” and “vaccines are dangerous” are contradictory opinions which a single entity is unlikely\nto hold simultaneously. There are actors, such as healthcare authorities, with a vested interest in not just rapidly\ndisseminating the opinion that vaccines are safe and having people adopt that opinion but also stemming the adoption\nof the opposing opinion and having the opposing opinion be dislodged from those who have adopted it. Competitive\ninformation diffusion models, also known as opinion formation models, were developed to simulate such scenarios.\nIn biology, competition can be thought of as cross-immunity contagions [40], where developing immunity for one\ndecreases the likelihood of being infected by the other.\nTwo broad categories of competition exist based on the type of information being diffused: continuous or discrete.\nCertain consensus formation mechanisms (e.g., taking the mean or median) are not possible for an information type\n(e.g., selecting a material for constructing an object, although the problem can be reformulated such that selection is\nbased on continuous-valued attributes such as tensile strength). Elements of continuous and discrete models can be\nmade to work together, as demonstrated by the Biased Voter model [41] that combines DeGroot and voter models.\nFor continuous-valued information, the DeGroot/French–DeGroot [42, 43] model is widely used for competitive\ndiffusion. In the model, every incoming edge for a node/agent has a weight denoting another agent’s influence (a\nweight of zero is equivalent to not having an edge). There is also a weighted self-loop for the extent to which an\nagent values its own opinion. At every time-step, an agent updates its opinion by taking the weighted average\nof all neighboring opinions, including its own. This process is repeated until the values (opinions) are stable. In\na strongly connected graph, perfect consensus (all agents share the same opinion) is guaranteed. The Bounded\nConfidence model, alternatively known as the Hegselmann–Krause model [44], also updates an agent’s opinion by\naveraging others’ opinions, except that instead of the opinions of all neighbors, the opinions are those who share\nsimilar opinions, with similarity being bounded by some arbitrary value, hence the model’s name. Clustering (a form\nof polarization), instead of perfect consensus, is the likely outcome of the Bounded Confidence model. A review of\ncontinuous-valued models, which includes DeGroot, Weighted-Median, Bounded Confidence, and Quantum Game,\ncan be found in [45].\nFor discrete opinions, one popular model is the voter model, independently developed by Clifford and Sudbury\n[46] and by Holley and Liggett [47]. At every step, the model first chooses an agent and its neighbor, both at random.\nThe agent then adopts the opinion of its neighbor. Another popular model is the social influence model by political\nscientist Axelrod, designed to explain how global polarization arises due to local homophily. In this model, each\nagent is defined by its cultural traits, which is a fixed-length string of discrete numbers. Similar to the voter model,\nthe Axelrod model randomly chooses an agent and its neighbor at every step. Then, with a probability equal to their\ncultural similarity, the agent will update one of its traits that differs from its neighbor’s. The Axelrod model can be\nconsidered an extension of the voter model.\nOne natural extension to voter models is to have an agent’s opinion updates be guided by the majority instead\nof just any random neighbor. The majority rule model by Chen and Redner selects at random a flippable spin\n(a changeable binary opinion) and have the spin adopt the majority state of its interaction neighborhood [49]. Fu\nand Wang incorporated minority avoidance into the majority rule model by having agents occasionally sever ties\nwith those holding a minority opinion [50]. Other information diffusion models studied a different form of minority\navoidance [51, 52], specifically the “spiral of silence” [53], which theorizes that people are reluctant to disagree with the\nmajority, hence restricting themselves from expressing a non-conforming opinion, ultimately leading to the continued\ndomination of the majority voice. The majority voice can be illusory, created by a vocal minority.\nThere are many ways in which competition can be modeled using epidemiological models. For two competing\ninformation contagions, a winner-takes-all situation emerges, where the stronger contagion dominates completely\ninstead of just gaining a majority across all graph types, given the conditions of no permanent immunity (SIS-like,\nwhere S is susceptible and I is infected), mutually exclusive infections, and nodes that are homogeneous in their\nresistance towards both contagions [54]. Non-graph-based epidemiological models can also account for competition.\nTwo of the proposed epidemiological models in [37] accounted for the presence of competing discrete information\ndifferently: the ISCR model features an additional class of user, counter-spreaders (C), in addition to the classical\nignorant (I), spreader (S), and recovered (R), while the ISSRR model divides the spreader and recovered classes based\non the which side of the bipolar information they were infected by or recovered from.\nExtensions to IC and LC (Section 2.1.2), which are discrete models, allow them to model competitive influence\nmaximization. The simplest modification is to start the diffusion of two different pieces of information from two\ndifferent origin/seed nodes. The different IC and LT derivatives distinguish themselves based on the restrictions or\nconditions that the competing factions are subject to: selecting the most influential starting nodes for one faction\ngiven that the competing faction’s starting nodes are known under IC [55]; best response and first mover strategies\nunder IC [56]; proving that being the first mover is not always an advantage under a special case of IC [20]; different\nforms of competitive diffusion under LT, e.g., different tie-breaking rules [57]; and two competing cascades on a signed\n4network under the IC model [58].\nThe limitations and unrealistic assumptions of IC and LT have been recognized, with [59] highlighting three\nmisspecifications of classical IC and LT: not modeling assortativity of influence, ignoring non-uniformity of the joint\ndistribution of influence and susceptibility (due to peer effects, homophily, and social influence), and heterogeneity in\ninfluence and susceptibility. However, [59] has misspecifications of its own. In the problem of IM, [59] defaults to seed\nselection: a faction (representing a stance) has the opportunity to pick and choose seed nodes from which to begin its\ndiffusion. However, this seed selection assumption that non-competitive and competitive IM operate under is alien\nto info ops (Section 2.2). Influencers on social media can be paid to promote a stance, but not all can be bought off.\nInfo ops are also often conducted in a decentralized manner to make them resilient against the decapitation strike\ntactic of silencing/compromising influential nodes (Section 2.2.4), so info ops planners are unlikely to even consider\nthe problem of seed selection.\nTraces of LT’s threshold mechanism combined with DeGroot’s weighted influence (despite our model being de-\nsigned for discrete instead of continuous opinion), and the voter/majority rule model are present in our model,\ngrounding it in the extant competitive diffusion literature. A threshold (resistance) must be crossed for opinion\nadoption/infection events. Unlike classical majority rule models, in which an agent’s adoption of the majority opin-\nion is deterministic (the neighborhood’s majority opinion will definitely be adopted), ours is probabilistic, with each\nopinion weighted by perceived popularity, as measured through the proxies of engagement metrics. While agents do\nnot have weighted self-loops as in DeGroot, they are influenced by how others judge the opinions that they express\n(Section 3.3.2). The mechanics make it so that human agents in Diluvsion are more inclined to convert to a stance\nthat is perceived to be held by the majority in the quasi-global neighborhood (Section 3.3.2). Adopting an unpopular\nopinion is possible, though unlikely. An agent’s discrimination against minority opinions is therefore encoded in the\nprobability distribution. This form of minority avoidance in our model, Diluvsion , despite sharing conceptual roots\nwith those encountered in the literature, is considerably different its technical implementation.\nThe design of our mechanics for themes can be considered a special restricted case of Axelrod’s cultural traits.\nThe example of traits given in Axelrod’s original paper allows for 105possible states since there are five positions\nfilled with numbers ranging from zero to nine. The number of positions in Diluvsion is variable, from one to an\narbitrary maximum, and a theme can fill a position only once. If the maximum was five and there were 12 themes,P5\nk=112!\nk!(12−k)!= 1585. Peer effects, homophily, and social influence all affect an agent’s decisions in Diluvsion ,\nheightening realism (see Section 3.3.2). One manifestation of homophily is in opinions sharing similar stance and\nthemes as the agent itself possessing greater weights during an infection event, and this homophily is comparable to\nAxelrod’s and Hegselmann-Krause’s homophily, though less extreme.\n2.1.4 Cooperative and competitive information diffusion\nCooperation can also be an outcome of the interactions between multiple information contagions present in a system\n[40]. Immunological dysfunctions caused by the infections of viruses such as HIV [60], Ebola virus [61], and SARS-\nCoV-2 [62] that make the infected more susceptible to other viruses are examples of cooperative biological contagions.\nRelative to competitive diffusion, the many forms of cooperative information diffusion have been less frequently\nstudied.\nCooperation can occur at the information/contagion level. The diffusion model in [63] learns from empirical data\nthe affinity between different contagions (URLs), finding that news stories of the same event boost the adoption of\neach other’s URLs, implying cooperation. The spread of a highly infectious news story’s URL suppresses the spread of\nURLs from other unrelated, less infectious news stories, implying competition. Different models [64, 65] have recreated\nthe synergistic (e.g., complimentary products such as iPhone and Apple Watch) and antagonistic relationships between\ncontagions propagating in heterogeneous and homogeneous multilayer networks where simultaneous infection by\nmultiple contagions (coinfection) is possible.\nCooperation can occur at the agent level. The implicit form of such cooperation is common in the study of\nmulti-agent systems in the context of robotics (vehicular information diffusion and consensus formation [66, 67],\ndrone swarms [68, 69], and search-and-rescue robots [70]). Ideas found in (not necessarily originating in) robotics can\nbe adapted to information diffusion, e.g., swarm intelligence algorithms for influence maximization under IC and LT\nmodels [71, 72].\nFinally, the cooperative effect can be found in the links/edges connecting nodes in a network; a node ubeing\ninfected by a contagion through another node vmakes umore susceptible to future infection by a different contagion\nviav[73], reminiscent of the effects of trust and affection in social relations.\nThe implicit cooperation for Diluvsion happens at the agent-level, where all bots targeting the most vulnerable\nfor stance propagation 3.5) and when bots’ independent actions unknowingly contribute towards a common objective\n(Section 6.2). At the information level, stances and themes induce homophilic conversions, and engagement metrics\ninduce bandwagoning (Sections 3.1 and 3.3.2).\n52.1.5 Agent-based models\nAgent-based models (ABMs) aim to infer macro-level outcomes from micro-level interactions and can be viewed\nas a further reduction of abstraction compared to models based on interconnected homogeneous nodes and models\nconstructed only from differential equations. Nothing stipulates that agent-based models have to incorporate a\nnetwork topology in modeling diffusion (e.g., agents can be placed on a grid and the diffusion can be governed\nby spatial proximity), but in practice, agent-based models of information diffusion on social media invariably use\nnetwork topology, as networks are an effective approximation of social organization in human communities, online\nand offline. Being freed from the constraints of stringently adhering to the precepts of classical IC and LT, and their\nmany extensions, enables more customized and accurate models to be built based on empirical data in sociological\nand psychological studies, albeit costlier to run computationally. An example is the ABM for the highly specific\ncontext of online anti-vaccination movements [74] which contained three different agent types: doctors (similar but\nnot equivalent to external authority/mass media in other models), patients (humans), and initiators (stubborns/bots).\nA comprehensive review of the various design aspects of modeling information diffusion in social networks as multi-\nagent systems (MAS) can be found in [75]. This review focuses on the trade-off between the simplistic assumptions\nfound in classical MAS and the complexity of real-world social networks. Thinking in terms of complexity versus\nabstraction can help establish boundaries between classical and agent-based models of diffusion. This demarcation\nis used by some ABMs, e.g., [52]. Thus, while agency is arguably present in some models discussed earlier due to\nindividualization/heterogeneity of nodes, e.g., DeGroot [42, 43], those models never fully embraced the possibilities of\nagency and the resulting emergent behavior (e.g., John Conway’s Game of Life) in the interest of keeping the models\nanalytically tractable, which is why we do not discuss them in this section.\nDiluvsion is an agent-based model. It borrows many concepts from existing models that are relevant to accurately\nsimulating the impact of info ops on information diffusion on social media.\nOne borrowed concept is that agents have a set amount of energy, with every action depleting their energy\nreserves. In [76], agents have a fixed energy level and interactions do not cost energy; however, after some arbitrary\npoint in time, every interaction decreases the agent’s energy level. This setup enables the study of the correlation\nbetween agent energy and rumor propagation intensity. In [77], the simulations were populated by different classes of\nagents, each class with its own range for how frequently its members tweet and retweet. Our implementation of the\nenergy level is based on empirical data from Twitter that separated users into three tiers (Sections 3.4 and 3.8.1).\nOur agents’ actions are additionally restricted by a diurnal cycle (Section 3.6).\nAgents are also commonly constrained by having a memory and attention (also uncommonly, and perhaps con-\nfusingly, called vigor [78]) limit placed upon them, e.g., [22, 78, 79, 80, 81], reflecting the non-omniscient nature of\nhumans. We discuss this mechanic in greater detail in Section 3.2.\nLinked to the concept of memory is the concept of repetition driving information adoption. In the agent-based\nSIR rumor diffusion model introduced in [82], agents can receive the same information multiple times, with agents\nlikelier to adopt information that has been seen more frequently, contributing to the spread of the information as\nagents eventually share the information that they have adopted. In the information entropy-based rumor spreading\nmodel in [79], a rumor’s information salience is measured by the frequency of the rumor’s occurrence in the agent’s\nmemory, with greater frequency linked to higher salience. The most salient rumor is selected for transmission during\nthe spreading phase of the model, and this selection mechanism constitutes the conformity effect put into practice\n[79]. In an ABM for studying astroturfing by bots under the spiral of silence framework, conformity takes the form\nof an opinion climate, which is composed of the opinions expressed by neighboring agents with silent agents excluded\n[52]. Another model [80] of conformity by has an agent slowly revising its opinion to become more similar to its\nneighbors’. Yet another model, AMID, features users who are likely to adopt opinions supported by the majority\nof adjacent neighbors in [78]. Additionally, under AMID, trust between agents was modeled as the frequency with\nwhich a truster shares the trustee’s messages, with users likelier to adopt messages from trusted sources [78].\nRepetition’s influence similarly suffuses every part of Diluvsion . A stance that occurs more frequently in an agent’s\nmemory has a greater chance of being adopted. But, we introduced additional complications into the mechanism.\nConformity effects are extended to account for the engagement metrics prominently displayed alongside every message\non social media, which serve as cues on how others perceive the credibility of the message (Section 3.3.2). Conformity\neffects are also balanced against homophily for stances and themes, which is based on the real-world tendency for\npeople to prefer opinions that are similar to their own.\nA divide exists among ABMs on the matter of opinions being exclusionary or capable of co-existing with oth-\ners. Exclusionary opinions, as conceptualized in [83], permit a user to only adopt one discrete opinion at any one\ntime. Models in which opinions are modeled as a point along a continuous spectrum, as is common in counter-\ndisinformation ABMs, e.g., [81, 82, 84], naturally feature exclusionary opinions as opinions cannot simultaneously\noccupy two positions along the spectrum. Studies using the spiral of silence framework, e.g., [51, 52], require polar-\nized/exclusionary opinions by design; users would not be hesitant to voice opinions that ran counter to the perceived\nmajority opinion climate if an oppositional relationship did not exist. For non-exclusionary opinions, there is also a\n6myriad of conceptualizations. In SMSim, developed for studying non-polarized information diffusion [85], bit vectors\ndenote a message’s association with the set of possible topics in the simulation. For the rumor spreading model\nintroduced in [79], infection by an opinion means that the opinion goes into the agent’s memory bank. There is no\nincompatibility mechanism that forces a stored opinion to be ejected from memory; removal is only due to insufficient\nspace in the bank. And with some models, the distinction between exclusionary and non-exclusionary is less clear,\ne.g., [63] where agents can only adopt one contagion but the agent’s infection history continues to influence future\ncontagion adoption. In the AMID model [78], each message is associated with all possible topics in the simulation\nwith varying strengths (fuzzy sets) and there is an additional binary number for each topic denoting the message’s\npositive or negative opinion towards the topic, a formulation reminiscent of Diluvsion ’s stance and themes (Section\n3.1). Users in AMID are capable of caring about multiple topics simultaneously, but the opinion (polarity) for each\ntopic is an exclusionary binary value.\nOn the matter of co-existing opinions, Diluvsion straddles the middle as there are exclusionary and non-exclusionary\ncomponents of an opinion within Diluvsion , corresponding to polarized stances and non-polarizable themes respec-\ntively (Section 3.1).\nA common feature of ABMs is the segregation of the agents into distinct classes. This practice is intuitive in\nstudies investigating the impact of bots on social media, e.g., [52, 77, 81, 84], as bots have to behave differently than\nthe rest of the users to distinguish themselves and for their influence to be measurable. In [77], both humans and\nbots were further subdivided into additional categories, e.g., amplifier and commentator, based on extant research\non social media user behavior. Some works [81, 84] divided agents based on stance, with “truth” propagandists,\nneutral/uninformed people, and “disinformation” peddlers exhibiting different behavioral patterns; bots are associated\nsolely with the undesirable stance, disinformation. While most models keep an agent’s role static throughout the\nsimulation, some allow agents to radically alter their behavior upon meeting certain conditions, e.g., radicalization\nturning passive users into active disseminators [74, 84].\nUnder Diluvsion , three main attributes define an agent: its activity level, its action-role, and whether the agent is\na human or a bot. There are three activity levels, five action-roles (15 unique action distributions as the distributions\nfor the same role differ based on activity level), and a binary bot/human membership. As such, a total of 30 possible\nclasses of users exist in Diluvsion . Agents do not change roles. Our conception of bots (Section 3.5) hews closer to\n[52, 77] in that their behavior is stance-agnostic for the reasons specified in Section 2.2.5, even though [52, 77] also\nonly associate bots with malicious behavior.\nModels found in the literature that most closely resemble our Diluvsion are those developed for a purpose similar\nto ours: the ones by Beskow and Carley and Averza et al. which dealt with information operations involving bots\nshaping discourse on online social media platforms [81, 84]. Unlike [52] which also studied the impact of bots on online\ndiscourse, [81, 84] did not integrate the spiral of silence into their models. And unlike [77] which was interested in the\ndescriptive statistics (e.g., tweets and retweets per day) that a mixed population of bots and humans would generate,\n[81, 84] are concerned with competitive diffusion of three stances, just like our study. Although to be accurate, our\nmodel allows all three stances to be propagated, but in [81, 84], only the two stances that are antithetical to each\nother can be propagated while the neutral stance is a static default state. The crucial differences between our work\nand [81, 84] are the stance-agnosticism of our bots (Section 3.5), neutrality being an infectious stance in our model, as\npreviously mentioned (Section 3.1), the inclusion of indirect influence mechanics (engagement metrics and non-edge\ninformation transmission, Section 3.3.2), and heterogeneous activity levels and roles for agents (Section 3.8.1).\n2.2 Information operations\n2.2.1 Terminology\nWithin the existing literature, information operations (info ops), influence operations, and psychological operations\nhave been used interchangeably to describe “activities that target information bases of an adversary, protect one’s own\ninformation assets and make efficient use of one’s own information”[86] even though each term has a specific meaning,\ne.g., psychological operations are originally defined as falling under the umbrella of info ops [87]. In this paper, we\nadhere to the loose usage of these terms as other researchers have, e.g., authors of [1, 88] switching between the terms\nwithin the same document; Cambridge Analytica varyingly described as engaging in psychological operations [89],\ninformation operations, [90], and influence operations [91].\nAlthough info ops is “a military defined concept and doctrine” originating from the post-Cold War US and\nNATO military establishments, it “interacts largely with society as the majority of the targets either attacked or to\nbe defended by Information Operations means are not military but civil targets” [86]. As pervasive computing has\nbecome a reality in a world where interconnected mobile computing devices are increasingly commonplace among\ncivilians, there has never been a richer, more fertile ground for info ops [92], especially when the rise of social media\nnormalizes rapid information sharing.\n72.2.2 Social media\nSocial media has always been a tool for information warfare. As early as 2003 [93] / 2006 [94], the cyberdissident\ndiplomacy (CDD) initiative from the US State Department reached out to “tech-savvy youth in the sixteen-to-thirty-\nfive age range with the aim of training them in particular forms of cyberdissidence and online campaigning”, leading\n“young Muslims” to “serve as the proxies to the US” in a “war of ideas”, an “ideological war”— influence operations,\nin other words — “to maintain its global hegemony” [93, 95]. As the provocative title of one journal paper neatly\nsummarized, “Facebook to Mobilize, Twitter to Coordinate Protests, and YouTube to Tell the World”: New Media,\nCyberactivism, and the Arab Spring [96]. In the 2009 protests in Iran, Twitter was deemed so crucial to the protesters\nthat the US State Department intervened in Twitter’s scheduled server upgrade to ensure that Iranian users did not\nsuffer from service interruption [97]. In Russia where the user base of native social media platforms VKontakte\nand Odnoklassniki dwarfs that of the US-developed Facebook, it is Facebook that mobilized anti-regime forces in\n2011 [98]. The clearest example of social media as an instrument of war is the USAID-backed ZunZuneo, which is a\nTwitter-like service designed to run solely in Cuba through mobile text messaging for the express purpose of covertly\ninciting regime change, according to the reporting done by the US news agency Associated Press [99, 100].\nBut as the flames of social unrest fanned in faraway “jungle” [101] countries escaped their confines and started\nengulfing First World/developed “garden” countries, the dangerous potential of social media was finally recognized\nby the research community and alarms were raised. In a review of academic publications on countermeasures against\ninfluence operations, the authors found a “dramatic increase in the number of studies since 2016” which is “[i]n line\nwith other research trends in the broader field of influence operations and disinformation” [1]; 2016 was the year\nwhen Russia was accused of meddling in the US presidential election in favor of Trump via social media campaigns\n[102]. Threats can be homegrown too: Amazon, Apple, and Google suspended the US social media platform Parler\nfor its role in the 2021 Capitol Hill riot [103]. The panic in the research community has culminated in the birth of\na new field, social cybersecurity, which seeks to “characterize, understand, and forecast cyber-mediated changes in\nhuman behavior and in social, cultural, and political outcomes” as one of its objectives [104].\nDiluvsion is a tool that seeks to aid in the fulfillment of the aforementioned objective by enabling the character-\nization of Twitter-like services and the forecast of info ops on such services.\n2.2.3 Twitter-like platforms\nLikely because of the ease of collecting data from Twitter, many researchers have focused on modeling information\ndiffusion on Twitter such as [22, 41, 81, 84, 105]. This is one reason the simulation environment within Diluvsion\nmodels itself after Twitter. A greater motivating factor is the profusion of Twitter-like social media platforms. Truth\nSocial, Parler, and Gab cater to users chafing at the politically restrictive policies of Twitter [106]. Mastodon lets\nusers escape the whims of a single central authority by dispersing regulatory power among multiple central authorities\n[107]. Twitter-like platforms have also emerged to serve different regions and languages, such as China’s Sina Weibo.\nEven social media platforms that have never sought to emulate Twitter have analogs to certain features of Twitter\n(which may not have originated from Twitter), e.g., Facebook’s share and share with a comment being similar to\nTwitter’s retweet and quote-tweet, respectively.\nThe mushrooming of Twitter alternatives also speaks to a credibility crisis, where users no longer trust that they\ncan express their views free of suppression from a central authority. Decentralization is a natural antidote to this\ncredibility crisis. Our conceptualization of decentralization within Diluvsion does not take after Mastodon, since their\nfederated instances still allow moderation power to rest in the hands of the few [107]. We speak of a decentralization\nwhere power is much more diffuse — or at least appears so (Section 2.2.4).\n2.2.4 Decentralization and radicalization\nThe most effective way to shut down conversation is the top-down approach of restricting free speech, as the Ukrainian\nexperience of banning Russian social media VKontakte from 2017 onwards has shown [108]. Yet at the same time, the\nvast majority of both pro-Ukraine and pro-Russia Ukrainians circumventing the ban to continue accessing VKontakte\n[108] shows that the desire to escape overt, centralized censorship can transcend political affiliations. In liberal\ndemocracies, harsh speech policing measures are not well-tolerated by users, even if private corporations instead of\ngovernments are the ones doing the policing [109], as evinced by the many alternatives to Twitter (Section 2.2.3).\nAnd if governments instead of corporations enforce censorship, the governments risk being viewed as draconian,\ntyrannical, totalitarian, authoritarian, etc., by liberty-minded people. A 2019 attitudinal survey of US internet users\nfound that “among political attitudes, opposition to censorship ( β=−.231,p < . 001) had a significant negative effect\non support for government regulation of platforms” [110]. Hence the rise of a decentralized approach to information\nwarfare: shaping discourse with an invisible hand, unbeknownst to the public.\nDecentralization has proven effective on offline social networks. Decentralization allows for radicalization without\nattribution and accountability. Absent clear chains of command, membership simply involves claiming affiliation.\n8Members can appear to be acting autonomously, when collectively those actions advance the interests of the movement\n[111]. Should the members’ actions harm the movement, “they can simply disavow the act without disavowing the\nideology” [112] in the same vein as black ops and black sites [113]. Without central authority, a movement can resist\ncollapse from decapitation strikes [96, 114, 115]. Ivan Marovic, a leader in the successful Serbian regime change\norganization, Otpor, which is funded by NED; IRI; USAID; Freedom House; and British Westminster Foundation,\nexplained to National Public Radio’s Bob Garfield that “[l]eaders could have been blackmailed or bribed or even\nmaybe killed. You can’t do that with brands or ideas” [116]. This is similar to how a perpetual struggle can be waged\nagainst an idea, e.g., the Global War on Terrorism. Terrorist themselves from white supremacists [117] to al-Qaeda\nand Hamas [111] also favored decentralized structures for their resilience and “tactical security”.\nSuccess in the offline world has led to attempts to replicate decentralized movements online. The US cyberdis-\nsidence program is one such attempt at growing an online decentralized movement (Section 2.2.2). Another is\nCENTCOM’s Operation Earnest Voice (OEV), which empowered “one US serviceman or woman to control up to 10\nseparate identities based all over the world” so they can create “false consensus in online conversations” [118, 119].\nGraphika and Stanford Internet Observatory (SIO) reported in 2022 that their joint investigation with Twitter and\nMeta uncovered hundreds of accounts engaging in covert pro-Western info ops over the course of five years [120].\nAn Israeli company, Team Jorge, specializing in info ops that ran multiple campaigns influencing elections around\nthe world [121, 122], created Advanced Impact Media Solutions (AIMS), a one-click system for creating “fake digital\npersona[s] built to mimic human behavior and avoid detection” which operate “on social media sites, disseminating\nrumors, harassment, defamation or praise — whatever the client asked for” [123, 124]. Other Israeli info ops com-\npanies Archimedes and Percepto also employed fictitious digital personas [125, 126]. White supremacists [112] and\ninternational Islamists [127, 128] took advantage of social media for recruitment [112, 127, 128], to transition to decen-\ntralized structures [112, 127], and to conduct info ops of their own [87]. The “IT cells” manipulating Indian elections\nadopted a functionally leaderless structure, where the bottom layer “creating and trending narratives” maintains\n“enough distance from the top to give leaders plausible deniability if the digital foot soldiers become too extreme”\n[115]. Another example is NAFO (North Atlantic Fellas Organization), which Politico described as consisting of\n“internet culture warriors taking on Russian disinformation and raising funds for Ukraine” by adopting the same\ntactics as “the Islamic State” and “the American far-right boogaloo movement” [129]. According to one freelance\ndisinformation researcher and self-proclaimed member [130, 131], NAFO is a leaderless movement [132]. The Wall\nStreet Journal describes NAFO as having “no command structure” [133].\nOwing to the achievements and prevalence of decentralized info ops, the simulations we chose to show running in\nDiluvsion (Section 5) were developed with decentralized info ops in mind, with affecting change through uncoordinated\ncoordination. The change agents being unaware of each other (Section 3.5) in Diluvsion is also a deliberate design\nchoice grounded in decentralization — leaderless terrorist/freedom fighter cells [134].\nTo affect change and appear organic at the same time, a decentralized movement needs a large number of warm\nbodies. But growing a movement is time consuming. Furthermore, each additional truly independent (human) agents\nwithin a movement is an additional security risk. Humans can turn into whistleblowers. Financial trails from hiring\npropagandists risk being traced back to the mastermind.\nCENTCOM’s OEV, Archimedes, Percepto, and Team Jorge’s AIMS already hinted at a solution: autonomous\nartificial personas (Section 2.2.5).\n2.2.5 Digital personas and bots\nBased on all the markers accessible via cyberspace — i.e., excluding anything that requires physical interaction in the\nreal world — bots can be made virtually indistinguishable from live human operatives. This is all thanks to progress\nin the field of machine learning making it easier than ever to deceive human senses and cognition [135]. On the\nbehavioral and textual front, services such as Character.AI allow users to create chatbots with distinct personalities,\nsuch as Elon Musk or Marvel movie characters, and these chatbots are capable of carrying conversations for hours\nwith humans [136]. Chatbots can exert influence over humans; interacting with chatbots has a positive impact on\npeople’s attitude towards vaccines and also their intention to get vaccinated [137]. Bot-run Twitter accounts are\nperceived as credible sources of information [138]. On the audio front, text-to-speech audio generation [139] has led\nto convincing audio clips of Biden and Trump chatting over gaming sessions [140]. On the video front, there are\ntalking head video generation neural networks producing realistic depictions of people saying things that they have\nnever said, e.g., [141]. A variety of techniques exist for transposing one person’s face onto the body of another person\nin a video, e.g., [142]. High resolution videos can be created with only text prompts, e.g., [143]. A navigable 3D\nscene can be created through neural radiance fields using just a few static images, e.g., [144, 145]. On the image\nfront, text-to-image synthesis networks can hallucinate scenes that never existed in the real world purely by relying\non text descriptions, e.g., [146]. For realistic faces, one does not need to steal the identity of a human, as faces of\nnon-existent people can be generated [147].\nTo tie it all together, AIs given an initial goal from humans can plan to accomplish the goal by giving itself a\n9list of tasks and setting about accomplishing the tasks by itself using various tools (including itself and other AIs) at\nits disposal [148], currently best exemplified by Auto-GPT. Pretending to be a human employed in a job will not be\ntoo difficult, as AIs can produce codes that run [149] and also play at the role of a knowledgeable analyst since vast\nswaths of programming and data science work can be automated via ChatGPT [150].\nWhile we are not aware of any entity that has made digital personas using a combination of all these bleeding\nedge AI capabilities1, there is a high likelihood of such hypothetical human-like bots being created and deployed\nin service of info ops at present or in the near future. Relying on detection as the sole strategy to safeguard\nagainst them might be a dead-end. A collaborative report by the Stanford Internet Observatory, OpenAI, and\nGeorgetown University’s Center for Security and Emerging Technology concluded that it was reasonable to assume\nthat “attributing short pieces of [textual] content as AI-generated will remain impossible” [151]. The same report\nwarned that, based on their deliberately limited scope of “generative language models” producing “persuasive text”,\nAI is “likely to significantly impact the future of influence operation” [151]. The days in which bots are simplistic\nand easily detectable automatons, such as those found by BotHunter to be influencing Singaporean elections to little\napparent success [152] or found by Botometer (which already often mistakenly classifies humans as bots [153]), are\ndrawing to a close with advancements in AI.\nThe promise of AI-enabled bots lies not just in rapidly raising an info ops army as Team Jorge did with its 39,000\navatars [121, 122], but also in reducing the cost of info ops. “The potential of language models to rival human-written\ncontent at low cost suggests that these models—like any powerful technology—may provide distinct advantages to\npropagandists who choose to use them” [151]. Accessibility to content generation capabilities, which is correlated\nwith cost, has also been widened with the advent of AI. Image generation via diffusion models such as Dall-E-2,\nMidjourney, and Stable Diffusion are available as services “for free or for a low price” [154] to practically everyone.\nConsequently, social media platforms have been flooded by “hyper-realistic fake images deceiving many online” such\nas those of Trump resisting arrest in March 2023 [155].\nBots can solve the problem of credibility crisis by appearing independent. This is not the same credibility crisis\nwe spoke of earlier where social media platforms were not trusted to be non-interventional (Section 2.2.3). This is\na credibility crisis among entities pushing the message — usually users of the platform and not the platform itself.\nMessages from US-linked organizations are distrusted even for the seemingly benign (or even beneficial) cause of\ncounter-terrorism [127]. Attempts at laundering their reputation by using proxies such as “community groups, non-\ngovernmental organisations and private enterprise” may be unsuccessful because simply being affiliated with the US\nin any way taints the proxies’ reputations [127]. However, by conducting info ops through a collective of artificial\nonline personas who cannot be linked back to any state or non-state actor, the info ops would not be subjected to any\nlong-held animosity that the targeted population may have towards an actor, thus encountering less resistance. A\nscholar examining the US’s credibility problem on social media in the issue of counter-terrorism similarly noted that\n“producing the appearance of independence by hiding behind fictitious online identities is one option” to address the\ncredibility deficit [127].\nOur conception of bots being nigh undetectable, as expounded in this section, plays a critical role in the design\nof information propagation within Diluvsion . In Diluvsion , we assume that bots are bound by many of the same\nlimitations that a human agent would be, e.g., their activity level, as they would seek to blend in with the human\npopulation. We also assume that people would not be able to ascertain another social media user’s true nature and\naffiliations, hence why a person’s judgment on the credibility of a message/tweet relies on external social cues. And\nthis human heuristic of assessing credibility by following the herd makes social media users uniquely vulnerable to\ndecentralized info ops that can be trivially signal-boosted by bots, as we will discuss in Section 3.3.2.\nHowever, at the same time, there exists in Diluvsion a special class of agents separate from humans that are\ncalled “bots” (Section 3.5); within Diluvsion , “bots” is a term of convenience for all forms of propagandists. If a\nhuman takes on the role of a propagandist (calculatingly penning tweets to maximize engagement, using Photoshop\nor a physical airbrush to alter images, etc.) on social media, they are rightly classified as “bots” since they are\nindistinguishable from purely algorithmic bots engaging in info ops.\nThe reason we reject the terms “propagandists” or “operatives” in favor of “bots” is because we reject the current\nerroneous framing within the agent-based simulation community of segmenting agent types based on stance (defined\nin Section 3.1) on an issue. In twitter sim[81], there are “three types of users: normal users (ignorants), bots/trolls\n(spreaders), and truth defenders (stiflers).” In the model introduced in [84], there are “three types of agents: Deceptive\nagents, representing bots or malicious human users; neutral agents, representing most Twitter users; and news agents,\nofficial sources who share accurate, verified information.” A pattern emerges: bots are invariably associated only\nwith the “bad guys”, which is categorically false since NAFO, OEV, and the pro-Western covert info ops documented\n1OEV predates the AI/neural network revolution. Graphika and SIO noted that assets in the pro-Western covert info ops\nthey uncovered “created fake personas with GAN-generated faces”[120], but no details were provided on whether other content\nare AI-generated. AIMS bots do not rely fully on AI: profile pictures were “stolen from a genuine profile”, although the “creation\nof [textual] content” is “driven by AI” [121, 123].\n10by Graphika and SIO are examples of bots working for “good” causes by disseminating “truth”. Under Diluvsion ,\nall propagandists are bots regardless of their stance.\nReclaiming the term “bots” is necessary to ensure accuracy in analysis and simulation, and thus avoid errors in\nstrategic judgment. Propaganda work and counter-propaganda work are often broadly equivalent in tactics. Persisting\nin the delusion that the “good guys” behave in a radically different manner in matters of war is detrimental to the\nformulation of good policy. Researchers will chase after the wrong solutions due to that initial error in judgment.\nRecall the example of the failure of US counter-terrorism info ops on social media [127]. Foreign policy analysis\nand reporting, e.g., [156, 157, 158, 159], will fixate on attributing the success of Islamist or ISIS propaganda over\nUS counter-propaganda to their speed, sophistication, creativity, having a better narrative, etc., and rarely question\nwhether the failure of counter-propaganda lies in the counter-propagandists’ lack of credibility, in the message listeners’\nfear and loathing of the US. The State Department’s Digital Outreach Team (DOT) accusing ISIS of terror tactics\nvia social media accounts clearly affiliated with the US [127] appears oblivious to the fact that the use of terror tactics\nby the US itself2undermines such a message. The initial error in judgment in assuming that the US’s hands are\nclean ensured US counter-propaganda efforts’ continued failure [127].\n2.2.6 State and non-state actors\nOther notions that we need to disabuse ourselves of for the sake of strategic clarity is that info ops originating from the\n“good guys” — democratic states — are only sanctioned against so-called non-democratic states and that meaningful\ndistinction exists between state and non-state actors. Democratic states attack each other and themselves. State and\nnon-state actors lead a chimeric existence.\nFor social media platforms, the distinction between state and non-state actors is increasingly blurred due to the\nvast reach that the platforms have in people’s lives, leading governments to constantly intervene in the platforms’\ndaily operations [163] or even prop up the platforms, e.g., the case of Twitter, Iran, and the US State Department [97]\nand the case of USAID-funded ZunZuneo [99, 100]. So deeply and confoundingly intertwined is the public-private\nrelationship that in [109], the author explicitly set out to argue that it is simultaneously unlikely and likely that\ncourts will conclude that a social media company is a state actor.\nDistinguishing between private and public entities among those engaged in info ops would be similarly difficult.\nFormer intelligence operatives naturally gravitate towards private info ops firms, e.g., [124, 126, 164, 165, 166], and\nthese operatives bring along their contacts in the government, skills, and knowledge into their new roles [166]. One\nremaining difference may be the scale of resources that state-level actors can muster compared to non-state actors,\nbut even that disparity is vanishing as states outsource info ops to private entities, potentially as a form of legalized\ncorruption via awarding lucrative contracts [156, 167] and as a way to shield themselves from culpability (Section\n2.2.4). According to Oxford Internet Institute’s 2020 Industrialized Disinformation report, despite the difficulty\nin “tracking down contractual evidence of private contracting firms”, they found that “almost US $60 million was\nspent on hiring firms for computational propaganda since 2009” [168]. Public-private joint efforts can be seen in\nCENTCOM awarding a $2.76M contract to the private corporation Ntrepid to develop OEV’s persona management\nsoftware [118]; in the US State Department awarding a $500,000 contract to Cambridge Analytica’s parent company\nSCL for “target audience research” [169]; in the many info ops companies manipulating democratic elections at the\nbehest of clients from democratic states [89, 90, 91, 115, 121, 122, 123, 124, 125, 126, 164, 166]; in Team Jorge running\ninfo ops through “Demoman International, which is registered on a website run by the Israeli Ministry of Defense\nto promote defence exports” [121, 122]; and in NAFO, which according to a Kyiv Post interview of its founder, is\n“endorsed by U.S. Congressman Adam Kinzinger, as well as U.K. Minister of Defense Ben Wallace” [170].\nConducting info ops against other democracies is not beyond the pale for democracies. Many documented in-\nstances exist. The campaigns conducted by the UK-based Cambridge Analytica in support of Trump and in col-\nlaboration [124, 165, 169] with the Israeli Team Jorge is a well-known example [89, 90, 91]. The actions of Israeli\ninfo ops companies in meddling with non-US democracies — e.g., Team Jorge’s interference3in “33 presidential-level\nelection campaigns, 27 of which were successful” [121, 122]; Percepto’s info ops against a Burkina Faso election [126];\nArchimedes targeting a Nigerian election [125] — are no less impactful simply because they received comparatively\nlittle attention in mainstream media and scholarly circles. Israeli Prime Minister Netanyahu also attempted to influ-\nence the 2016 election in favor of Trump [166]. Info ops conducted by an outside actor may even be orchestrated by\npaymasters situated within the targeted state. Trump campaign official Rick Gates sought proposals from an Israeli\n2In recent years, it has come to light, through the reporting by outlets such as Current Affairs, The Guardian, and The Wire\n[160, 161, 162], that indiscriminate bombing via drones has instilled terror in the hearts of the populace in many regions of the\nMiddle East. Children have come to fear clear blue skies because those are the perfect weather for flying drones.\n3Veracity of Haaretz’s reporting: “Journalists from The Guardian, Der Spiegel, Die Zeit, Le Monde, the international\norganization of investigative journalists OCCRP, Radio France, Haaretz, TheMarker and other media outlets worked in France,\nKenya, Israel, the United States, Indonesia, Germany, Tanzania and Spain to examine the veracity of Jorge’s claims about his\nworldwide deeds. Shockingly, many of the allegations were corroborated.” [121]\n11company Psy Group to “to create fake online identities, to use social media manipulation and to gather intelligence\nto help defeat Republican primary race opponents and Hillary Clinton”; the proposal, named Project Rome, was\nultimately never acted upon [171, 172]. The US under the Obama regime in turn has employed covert info ops against\nIsrael [173]: a US Senate inquiry found that “[i]n service of V15 [an anti-Netanyahu group], OneVoice deployed its\nsocial media platform, which more than doubled during the State Department grant period; used its database of voter\ncontact information, including email addresses . . . and enlisted its network of trained activists, many of whom were\nrecruited or trained under the grant, to support and recruit for V15.” Even a seemingly innocuous “voter button”\ndisplayed on Facebook to encourage turnout in UK elections can be construed as an attack because Facebook, as a\nUS company, should not be influencing UK politics in any way [3].\nAll these examples are in essence the use of non-state intermediaries in information warfare involving three\nstate-level actors widely considered to be democracies (according to the Economist Intelligence Unit’s 2016 and 2022\nDemocracy Index [174, 175], the US and Israel are flawed democracies and the UK is a full democracy), putting to\nrest the idea that info ops are only ever employed in the stalwart defense of peace, freedom, and democracy against\nstates with autocratic, authoritarian, totalitarian, etc. tendencies4.\nInfo ops’ use by democracies is not limited to state-to-state situations. Info ops can be used against a democracy’s\nown populace. In India, the world’s largest democracy, “[t]he 2014 election of Narendra Modi catapulted the BJP\nto power thanks, at least in part, to an expansive network of ‘IT cells’ aimed at spreading positive news about the\nBJP and attacking its detractors” [115]. Cambridge Analytica is believed to have influenced the Brexit vote results\n[124, 190], though its culpability is disputed by some [191].\nHaving established that info ops originating from state and non-state actors would have no appreciable difference\ndue to ever-present public-private entanglement, we can claim that incorporating the publicized details of info ops\nconducted by non-state actors into our modeling will not bias the model towards the form of info ops conducted by\nnon-state actors and vice versa, and we can claim that findings from Diluvsion simulations are equally applicable to\ninfo ops from or against either state or non-state actors.\nInDiluvsion , we defaulted to bots starting out embedded in a social network prior to being activated, in contrast\nto some preceding models that started bots out with only a single connection (edge) to the network, e.g., [81], or let\nbots have lower degrees than humans, e.g., [52], as they assumed that communities have not been penetrated by bots\nprior to an info ops. Our choice is based on reports of uncovered info ops. Team Jorge estimated that about 1,000\nbots are needed to “postpone an election in Africa without good cause” with half being newly created bots and the\nother half preexisting bots [123]; the “inventory of fictitious accounts” were categorized based on regions/identities\n(e.g., Arabs, Russians, Asians, and Africans) and reused for different info ops [121] — segmentation by region and\nrecycling bots for multiple campaigns strongly suggest that Team Jorge would equip bots with social relations (if\nthe bots’ positions within a social network did not matter, swapping in a campaign-appropriate regional identity on\npreexisting bots instead of maintaining an extensive inventory would have been more cost-effective). AIMS bots also\nexhibit meticulous attention to detail, with some of the bots even having bank accounts [123]. Taking that fact into\nconsideration, Team Jorge are unlikely to neglect situating their bots inside a suitably believable network of social\nrelations at creation time. Percepto’s case adds further support to the likelihood of bots having relations. One of\nPercepto’s bots was an entirely fictitious investigative reporter based in Paris that had a real news agency with actual\nhumans working under the bot [126]. Even the simplistic bots that apparently failed in manipulating Singaporean\nelections “appear relatively ubiquitous throughout the social network” despite a “failure to amass network influence”\n[152]. The strongest evidence comes from bots used in the covert pro-Western info ops studied by Graphika and\nSIO [120], which exhibited “a typical long-tail distribution in the follower footprints, with a few influential accounts\nfollowed by a descending list of accounts with progressively fewer followers”, showing that the distribution of node\ndegrees among bots mimic that of humans on social media (in most [192, 193] but not all cases [194] of human\nnetworks). The probability of an info ops bot army starting out from the periphery of a network is low.\n3 The Model — Diluvsion\nIn our information diffusion model for Twitter-like social media, Diluvsion , information is composed of two elements:\nstance and themes. Stances are polarized and exclusionary. A message can have multiple but non-repeated themes.\nThe dissemination of stances through the network is of greater concern to us than themes. Details on stances and\n4For purveyors of “No True Scotsman” and “Democratic Peace” [176]: Some academics have argued that Israel is an apartheid\nstate [177] and also a settler colonial state [178]. The US has been characterized as an oligarchy [179] and as a plutocracy\n[180, 181], and some Americans prefer to define the US as a republic [182]. The UK may fit the definition of an “electoral\nauthoritarian” regime [183] due to a history of banning political parties [184], media censorship [185], and electoral fraud [186],\non top of inheriting a hereditary monarchical head of state and hereditary peers in the House of Lords who exercise indirect\npolitical influence [187]. An argument can also be advanced that certain states within the triad, being client states, lack true\nsovereignty [188, 189] so conducting inter-state info ops does not constitute warfare as no one’s sovereignty has been violated.\n12themes are found in Section 3.1.\nThere are two major classes of agents in our model, humans (Section 3.4) and bots (Section 3.5). While “humans”\nare exclusively humans, “bots” can encompass artificial algorithmic entities and natural organic entities so long as\nthey play the role of propagandist. Our motives for using this terminology are explained in Section 2.2.5.\nSection 3.2 explains the attention-memory mechanism that agents are subjected to. In brief, the mechanism\naims to mimic the impact that real humans’ limited information processing capability and retention capacity can\nhave on information diffusion. Section 3.3 lists the range of actions that agents can perform within the simulation\nto propagate or boost the propagation of stance and themes. Section 3.3.1 describes statistical data — engagement\nmetrics — that agents are exposed to and affected by. Section 3.3.2 describes indirect influence mechanisms as well\nas the rationale behind our model’s reliance on engagement metrics as proxy for indirect influence.\nThe chosen resolution for Diluvsion simulations, i.e., the length of real-world time that a single step in the model\nis equivalent to, is justified in Section 3.6. The starting parameters of the simulation are detailed in Section 3.8.\nAn assumption under our model is that all agents are devoted to debating one topic. They may devote a differing\namount of time per day on social media and their forms of engagement may differ (e.g., spending most of their time\nliking posts they agree with, primarily retweeting others, etc.). These additional traits of the agents are sufficiently\ndivergent that each combination constitutes a sub-class of bots and humans (Section 3.8.1).\nPrivate accounts are excluded from our model primarily because they have a limited ability to engage in public\ndebates; a private account’s tweets can only be seen by the account’s followers. For instance, if a private account sees\nan opinion they disagree with, refutations that the private account uposts in reply will not be seen by the author\nvof the disagreeable tweet and the author’s, v’s, followers if the author vis not a follower of the private account\nu. This limited reach of private accounts makes them inappropriate for inclusion in simulations of public info ops.\nAdditionally, private accounts constitute a small minority of Twitter users so their exclusion would not drastically\naffect our results. According to a Pew Research Center survey on adult US Twitter users conducted in 2019, 13% of\nTwitter accounts are private [195].\nWe also exclude the “silent majority”/“no activity” users (Section 3.8.1) from our model because their lack of\nactivity means that there is little data to collect on how they exert influence and how they, in turn, are influenced.\nTheir lack of activity also means that they would have no discernible impact on the evolution of a Twitter discourse.\nDiluvsion is built on the Python agent-based modeling framework Mesa, which in turn uses the Python network\nanalysis package NetworkX as a backbone.\n3.1 Stances and themes\nStances are opinions on an issue. To ease interpretability, we allow only three discrete stances in our model: two\nlocated at the poles and one at the exact midpoint between the two. We let the distance from the midpoint to either\npole be one. The three stances are known as negative, neutral, and positive. Alternate names used throughout this\npaper for the negative stance include “con”, “against”, and “opposition” and the positive stance is also known by\n“pro”, “favor”, and “support”. While not all opinions can be modeled under this system, e.g., opinions on the actual\nworth of a used car, many can be, e.g., opinions on voluntary abortion.\nOur model differs from others in the literature, e.g., [80], in that we allow neutrality as a stance to be propagated.\nBipolar ideological battles have been consistently framed as drawing users to either pole, but not taking a stand can\nbe as principled a stance as taking a stand, at least in principle, although perhaps not always in practice (some may\nargue that true equidistant neutrality in the strictest sense does not exist as everything exhibits a bias5). An example\nof a manifestation of neutrality is the Non-Aligned Movement [199] in geopolitics.\nStances are different from sentiments. Sentiments are concerned with the emotive content of a message, whereas\nstances are concerned with the message’s position with regards to an issue of interest. For example, both statements “I\nhate wearing masks” and “I hate doctors not wearing masks around immunocompromised patients” show a negative\nsentiment due to their negative tone and presence of words such as “hate” in them, but in terms of stances, the first\nstatement is against mask-wearing while the second is in favor.\nThemes are the non-polarizable aspects of an opinion, and they are non-polarizable only with respect to the issue\nof interest. We illustrate the concept with an example where the issue of interest is mask-wearing: a message decrying\n5Academics, e.g., [10, 13], have argued that mass media always represent ruling class interests, making impartiality impossible.\nTo maintain a facade of impartiality, mass media “strictly limit the spectrum of acceptable opinion, but allow very lively\ndebate within that spectrum” [196]. Others have argued that journalism, “in its pretense of ideological neutrality”, created\n“a particularly pernicious myth” as the presence of filters in both the news reporter and news consumer prevents neutrality\n[197]. Spiral of silence [53] predicts that some people are unwilling to voice out an opinion to avoid contradicting what is\nperceived to be a majority opinion. Such inaction can be interpreted as neutrality, yet it does not contribute to the emergence\nof neutrality but instead contributes to the (potentially false) majority’s continued dominance. Central banks invoking market\nneutrality to depoliticize corporate security purchase decisions are biased because “[a]s critical political economists, sociologists\nand anthropologists have so frequently argued, markets embody a specific, political vision of society, which is not shared by\nevery member of the polity” [198].\n13a store for banning masked patrons for fear of criminal intentions and another message praising a store for banning\nunmasked patrons out of health concerns, despite differences in stance (anti- vs. pro-mask) and sentiment (negative\nin condemning vs. positive in complimenting), are united under the common theme of “non-medical authority” being\nexercised. And although people may have opinions on authority (favor or against), those stances have no direct\nbearing on the issue of concern, mask-wearing. Themes are comparable to the concept of frames in the field of\ncommunications [200, 201, 202] but is notequivalent to frames6. A frame is envisioned as a wrapper, a method\nof carrying messages. Different perspectives (wrapper) on an issue (message) changes one’s perception on an issue.\nFor us, themes function as more than wrappers. They are capable of being transmitted to (infecting) a target, just\nlike stances. There can be multiple themes per message just as there can be multiple frames/wrappers per message.\nMultiple themes are similar to opinions encoded as binary strings in the rumor spreading model introduced in [79]\nand to a lesser degree the discrete-numbered cultures in the Axelrod model [48], SMSim’s single-topic bit vector [85],\nand AMID’s fuzzy set multi-topic messages [78].\nThe usefulness of explicitly modeling themes and having multiple themes per tweet/user can be demonstrated\nwith an example involving an unpopular theme (adopted by a low percentage of agents). To make things easier to\nunderstand, we name this theme “conspiracy”. By wrapping every message a bot army tweets out with a layer of\nconspiracy and by having more popular themes accompany the conspiracy theme since multiple themes are allowed,\nwe can ensure that the conspiracy theme is more willingly adopted by others because of its constant presence and its\nassociation with popular themes. This ultimately results in a conspiratorial mindset slowly taking over the targeted\nhuman populace on social media. We indeed simulated such a scenario under Diluvsion (Sim 9, Section 6.4). A less\nnefarious theme can always take the place of “conspiracy”, e.g, “affordability”.\nEach message, i.e., tweet, in our model is composed of a single stance and one or more themes. Each agent has a\nstance and a set of themes as well. Agents can switch stances and can adopt themes over time. To keep the agents’\nthemes from homogenizing as the simulation progresses because of the adoption mechanic, the maximum number of\nthemes that an agent can possess at any one time is always lower than the set of all available themes in a simulation.\nThis limitation also reflects that not all aspects (themes) of an issue can be the core interests of an agent; people\nhave a limited capacity to care. All tweets by an agent will have the same stance as the one held by the agent at\nthe time of tweeting. Themes, however, can differ between tweets (Sections 3.4 and 3.5), e.g., a human agent will\nuse the same themes as the tweet it is replying to, even if the themes are not part of the agent’s theme set. Under\nDiluvsion , compatibility in both the stance and the themes is required to improve the persuasiveness of a message\nto its recipient (Section 3.8.2). Once again using a mask-wearing example, a message that attempts to proselytize\nmask-wearing through the themes of safety and medical authority will find no purchase on an agent whose themes\nreflect a concern for comfort, e.g., masks being difficult to breathe through. Stance plus themes can also be thought\nof as viral variants, except that the transmission of stance and themes is dependent on homophily, but viral variants\ndonotnecessarily operate under this mechanic7.\n3.2 Memory and attention\nMemory and attention mechanisms are common features in many models of information diffusion, e.g., [22, 78, 79,\n80, 81] and the motivations behind incorporating such mechanisms are largely similar. Modeling agents as having\nlimited attention and limited memory is meant to reflect the reality where humans are incapable of digesting all\nthe information presented to them and retaining the digested information over the long term. Diluvsion features\nan attention and memory mechanism. An attention-memory limit governs the number of tweets that an agent sees\nper activation (when an agent wakes during a time-step and performs actions), the number of its own most recently\n6To underscore the importance of themes/frames: Incidentally, the concept of framing introduced by Tversky and Kahneman\nwas first motivated by the “Asian disease problem” [203], where people were surveyed on their preferences to two solutions to an\n“unusual Asian disease”, with one solution framed as saving lives and the other as causing deaths. Framing the disease as Asian\nbetrays a problematic inclination harbored by the authors in their supposition that a disease terrible enough to force people\nto face a moral dilemma could have only originated from Asia or Asians, much like SARS and SARS-CoV-2 are exoticized and\nracialized as Asian [204, 205]. But perhaps the framing offraming’s motivating problem can be excused as it dates from an era\nwhen people have yet to learn to better hide their prejudices. In the own words of Kahneman, “[s]ome people have commented\nthat the ‘Asian’ label is unnecessary and pejorative . . . but the example was written in the 1970s, when sensitivity to group\nlabels was less developed than it is today” [206]. The term “Asian disease problem” remains in common usage in scholarly\ncircles today.\n7There is some superficial similarity in how newer variants of SARS-CoV-2 such as Delta and Omicron evolve greater binding\naffinity to ACE2 receptors in human cells than the ancestral wild type [207, 208]. Antibodies and some therapeutics either\naim to present a more attractive binding target for the virus or bind to ACE2 receptors first before the virus [209, 210, 211],\nessentially outcompeting the virus. Themes are analogous to binding affinity, where a tweet exhibiting a greater number of\nmatching themes (homophily) with the targeted user increases the chance of the tweet’s stance being accepted by the targeted\nuser, just as a the strength of binding affinity can influence whether an antibody (a stance) or a virus (a different stance) binds\nto a host cell.\n14tweeted tweets that the agent remembers, and the number of users that the agents remember seeing. Memory observes\nthe first-in first-out (FIFO) queue, with the newest item displacing the oldest should the limit of a memory bin is\nreached. Since it is a combination attention and memory limit, it is possible for items that have entered an agent’s\nmemory to be forgotten before the items have a chance to be seen by the agent. For instance, if there are 31 new\ntweets directed to an agent whose memory is empty and has a limit of 30, the earliest/oldest 31sttweet will be\n“forgotten” before ever being read by the agent. The memory of seen users is duplicative and cumulative: a user\ncan appear more than once in an agent’s history and users that appear more frequently are given greater weight in\ncalculations (e.g., when deciding which user to follow).\nWhile we keep track of the history of allusers seen by an agent (this history is different and separate from the\nlimited memory seen user history) and the history of allinfluential users (users that infected/changed the stance of\nthe agent), they do not affect agent behavior and are only used by us to diagnose how influence spreads in the social\nnetwork (Section 5). These histories are also duplicative.\nBots are constrained by a memory limit though theirs is several times higher than that of the humans. Bots are\nassumed to be more dedicated to spreading stances and/or themes than regular humans whose social media usage is\nless singularly goal-driven. Therefore, bots would dedicate more resources, such as greater attention and memory, to\naccomplish their goals.\n3.3 Engagement\nThe ways in which a user can engage with the social media platform Twitter are referred to as actions in our model.\nThe set of actions available to an agent in our model — following, unfollowing, like, tweet, retweet, quote-tweet, and\nreply— is different from the set of engagement metrics, which we will discuss in Section 3.3.1.\nFollowing is a one-way relationship where the follower uindicates their interest in receiving updates on the\nactivities of the followee, v, e.g., when vtweets, retweets, etc. Unfollowing is the sundering of the follow relationship.\nA follower following a followee indicates that the follower is receptive to information flowing out from the followee.\nA loss of interest in the followee and/or the followee’s tweets losing their informativeness often leads to the demise of\nthe follow relationship [212, 213]. In our model, a potential followee having matching stance and themes with a user\nincreases the candidate’s chance of being followed by the user.\nLiking (previously known as “favoriting”) is the most voluminous [214] action performed by Twitter users. Likes\non social media can be considered a “numeric representation of social acceptance” [215]. The bandwagon effect —\nthe heuristic of following the crowd used by humans when other signals of authenticity are weak or absent — can\nguide a person’s decision to like, as people were found to be more likely to like photos with many likes over those\nwith few likes [216].\nTweeting is the act of composing and posting a message on the social media platform. Since our model excludes\nprivate accounts, all tweets are assumed to be publicly viewable and searchable. While there are a number of factors\n(Twitter’s algorithm, a human user’s own preferences, etc.) affecting whether these viewable tweets are actually seen\nby users, our model assumes that an agent’s tweets are always seen by the agent’s followers so long as the tweet\nremains in the memory (Section 3.2) of the followers. While not the most common action, the production of tweets\nis the basis for all other actions, except for following and unfollowing, because liking, retweeting, etc., require tweets\nto first exist.\nRetweeting is the sharing of another’s tweet with one’s followers without any embellishment. According to [217],\nretweets indicate interest, trust, and agreement with the message. As credibility and trust are important to having\na tweet retweeted, our model factored trust into an human agent’s decision on which tweet to retweet. Rather than\nusing a randomly generated number to quantify trust relationship between agents, we relied on indirect influence,\nnamely engagement metrics (like, retweet itself, and reply), as a proxy for trust (detailed in Section 3.3.1, justified\nin Section 3.3.2).\nReplying is responding to another’s tweet with one’s own tweet, not as an independent tweet but as a tweet that\nwill be added to the original tweet’s reply thread, easily accessible by others should they tap or click on the original\ntweet. While not all replies challenge the stance of the parent/source tweet, replies are strongly weighted towards\nbeing negatory in nature. The Stance in Replies and Quotes (SRQ) dataset [218] shows that 44.0% of the replies\nare against the stance expressed in the tweet being replied to (parent/source tweet), and that approximately half,\n22.9%, are supportive. The remaining 33.2% are comments and queries about the parent tweet’s stance, which can\nbe interpreted as being neutral in nature. In the problem of detecting contradiction for rumorous claims [219], there\nis a special category for detecting disagreeing replies (as opposed to expressing disagreement in independent tweets),\nhighlighting replies as vehicles for disagreement. Our model adheres to the conception of replies being primarily,\nthough not exclusively, negatory in nature.\nQuote-tweeting, alternatively known as quoting, is sharing another’s tweet with one’s followers while adding extra\ncontent from the quote-tweeter. Quote-tweeting is relatively rare, even when compared against replies. In a dataset of\nCOVID-19 tweets that consisted of just retweets, quote-tweets, and replies, quote-tweets’ share is 3.82% while replies\n15have 4.91% [220]. A study on the roles assumed by Twitter users on the platform [214] showed that quote-tweeting is\nseldom practiced, with Repliers as a role and replying as an action outnumbering Quoters and quoting across all roles\nand user activity levels save for Quoters themselves. The motivation behind quoting appears equally divided between\nrefuting and affirming the original tweet. In the SRQ dataset [218], 46.8% of quote-tweets are negatory with respect\nto the quoted tweet and 43.5% are affirmatory, with the remaining 10.7% being comments and queries regarding the\nquoted tweet’s stance. Stance ambivalence in motivating quote-tweets, as evinced by the even split between negatory\nand affirmatory, is modeled in Diluvsion as compatibility in stance having a lower weight than themes when an agent\npicks which tweet to quote-tweet.\nRatios can also offer clues on whether agreement or disagreement drives an action (Section 3.3.1). Actions as\nmodeled in Diluvsion is generally in agreement with findings from ratiometrics.\nIn our model, an agent’s decision to like, retweet, or reply to a tweet is partially based on the agent’s agreement\nwith the tweet’s stance and themes, and partially on whether the tweet’s popularity merits bandwagoning. Whether\npopularity or compatibility is the dominant factor depends on which action is being taken, as detailed in the explana-\ntion of this mechanism in Section 3.8.2. The existence of a follower-followee (or inverse) relationship between agents\ndoes not influence an agent’s action; the “text semantics and contents” of a tweet, “regardless of tweet author”,\nare the most important factors in determining if a user engages with a tweet through liking, retweeting, replying or\nquoting [221].\nAgents who received replies to their tweets or have had their tweets quoted usually receive notifications, so we\nassume that information from replies and quote-tweets diffuse to the targeted agents. Additionally, replies to a reply-\ntweet are read by a fixed percentage of the followers of the author of the tweet being replied to, which is a form of\nnon-social tie information diffusion (Section 3.3.2).\nLike Twitter and other real-world social media platforms, we only allow a user to like a tweet once and retweet\na tweet once. The practice of including a link to a tweet, ain a reply-tweet, which increases the retweet counter of a\nand can be performed multiple times, is excluded from our model. And although it is possible to reply to the same\ntweet multiple times on Twitter, we restrict the number of times an agent can reply to a tweet to just one. Any\ninformation that a replier wishes to convey through multiple replies to a parent tweet can be thought of as having\nbeen consolidated into one reply.\nTweeting is the action that all agents, bot and human, default to if the agent’s attention-memory contains no seen\ntweets or if the chosen action for a particular time-step of the simulation is invalid. Many actions, e.g., liking and\nretweeting, require tweets to be present before they can act on those tweets, so this default behavior helps populate\nthe simulation with tweets. Invalid actions are common around simulation start due to a confluence of three factors\n(1) the lack of tweets, (2) limiting each tweet to one like, retweet, quote-tweet, and reply each per user, and (3) liking\nand retweeting being common actions. A consequence of tweet scarcity is that the action distributions generated\nfrom the simulation deviate slightly from empirical data, despite the model using parameters from empirical data, as\nwe shall discuss in Section 4.\n3.3.1 Engagement metrics and ratioing\nThere are three prominent engagement metrics — likes, retweets, and replies — visible to Twitter users. While we\nallow for quote-tweeting as an action in our model, its metrics are not visible to agents in our model and are therefore\nincapable of influencing the agents (Section 3.3.2). In the context of stance conversion, likes are positive, retweets\nare moderately positive, and replies are moderately negative.\nExcluding the number of quote-tweets from engagement metrics is because of this metric being hidden throughout\nmuch of Twitter’s existence. In 2020, quote-tweets were still semi-hidden, with Twitter forcing users to tap/click on\na tweet then on the tweet’s retweets before they could see the number of retweets broken down into retweets and\nquote-tweets [222, 223]. Besides quote-tweets, we also exclude additional metrics that have been surfaced since Elon\nMusk’s Twitter acquisition, such as the counts of views and bookmarks a tweet has received. This is because the\nfoundations of our model — engagement data we and others have collected (Section 3.8.1) and the vast majority of\nscholarly literature on the impact of engagement — date from the era when these statistics were hidden. On Twitter’s\ncurrent timeline interface where users scroll through tweets, only viewcounts have been added alongside the counts of\nlikes, retweets, and replies. The newly surfaced engagement metrics are also rarely displayed on other platforms, e.g.,\nInstagram and YouTube only show the number of likes and replies that a post has received while the Twitter-like\nMastodon shows retweets, replies, and likes.\nThe number of followers is an engagement metric that does not influence human agents. Although a user’s\nfollower count is publicly visible to anyone who visits the user’s profile page, it is not visible in the context of other\nactions — liking, retweeting, replying, and quote-tweeting. Visiting a user’s profile page is an extra step from the\ntypical Twitter usage behavior of “infinite scrolling”, breaking from a user interface (UI) designed to entrap users in\nperpetually consuming media [224, 225]. For following and unfollowing a user, these actions are available as part of\na dropdown menu accessible by clicking an icon on a tweet; visiting a user’s profile page and being made aware of\n16the user’s follower count are entirely avoidable. Bots, however, do use follower count as information for some of their\ndecisions, as they are motivated agents of stance/theme propagation.\nRatioing on Twitter is a phenomenon that best expresses the logic behind engagement when Twitter conversations\ninvolve contested issues. When a user expresses an objectionable opinion, others may express their disagreement by\nreplying to the user’s tweet. Should the number of replies (regardless of whether they all express disagreement) or\nthe number of likes of any one of the disagreeeing tweets outnumber the likes of the parent tweet, a “ratio” has been\nachieved, i.e., the ratioed user has been publicly and quantitatively mocked. A study on an info ops campaign waged\nby social media users opposed to the fossil fuel industry used ratioing as a means to identify tweets targeted by\nthe operatives as the operatives employed the strategy of questioning fossil fuel organizations’ legitimacy with large\nnumbers of negative comments [226]. A broader definition of ratioing includes cases in which the number of replies\nexceeds the number of retweets. This definition was employed in a study of the different ratios of engagement metrics\nthat tweets from Presidents Trump and Obama received [227], which found that Trump consistently achieve higher\nreply-retweet — more controversial — ratios compared to Obama.\n3.3.2 Indirect influence and trust\nOur conceptualization of influence departs from some works in the literature that regard all influences as having\na positive impact. These works include [228], which posits that the high trustingness of an acting agent and the\nhigh trustworthiness of the acted-upon agent are the determinants behind the acting agent’s decision to act (retweet,\nreply). Another work is [229], which deemed one user as having influenced another if the user engaged with the tweet\nin any way (retweet, like, reply, quote, and follow). From the perspective of stance diffusion, such conceptualizations\nof influence clash with empirical observations of why people choose to respond to messages. As discussed earlier in\nSection 3.3, replies and, to a lesser extent, quote-tweets are often used to make rebuttals. Ratioing (Section 3.3.1)\ndemonstrates that not all interactions constitute affirmative influence. The disconnect between the models [228, 229]\nand the data is a result of ignoring polarity/stance. Zhang et al. explicitly stated that their method of finding\nmatching tokens between two tweets “identify whether their topics are similar”; nothing is said about the user’s\nstance on the topic.\nDirect influence comes from direct attempts at persuasion, which in our model consists of the contents of a tweet\n— its stance and themes. Direct influence travels along the prescribed lines of influence, social ties, from followee\nto follower. Indirect influence is information outside of a tweet’s content that can influence an agent’s adoption of\nthe contents. Indirect influence can either increase or decrease the persuasiveness of tweets. Indirect influence can\noperate outside of observable follower-followee relationships.\nOne form of indirect influence modeled under Diluvsion is the ability for replies to circumvent social ties when\ntransmitting information, where a certain percentage of the followers of the parent tweets’ authors will read the\nreplies (Section 3.8.2). Tweets in most cases can be replied to by any user, not just followers or followees. More than\njust transmitting information to the user targeted by the reply, replies can also transmit information to the targeted\nuser’s followers. A browse through Twitter will reveal that many replies to tweets that have gained a lot of attention\n(high number of likes, retweets, etc.), colloquially known as “going viral”, are advertisements or posts about topics\ntangential to those of the original tweets, e.g., fake BTC giveaway scams [230] and shilling for cryptocurrency. They\nare attempts to take advantage of the tweets’ popularity. The hope is that some of the attention can be diverted to\nprofitable or persuasive ends.\nAttempts have been made to model such indirect routes of information diffusion. The model proposed in [84]\nallows for tweets that have crossed a certain threshold of engagement numbers to go viral, meaning that the tweet\nwill be “shown to all agents in the network that share the same topics with the sender agent, even if an edge does\nnot connect them”. From our perspective, the indirect propagation of information captured in their implementation\nappears limited, as most information still travels along the network edges in their model. The sole exceptions were\nviral tweets. In Diluvsion , virality would naturally emerge from the propensity for agents to favor sharing or otherwise\nengaging with popular tweets over unpopular ones (bandwagoning) since engagement metrics, themselves a form of\nindirect influence, factor into an agent’s decisions on which tweet to engage.\nThe engagement metrics discussed in Section 3.3.1 constitutes a form of indirect influence. The metrics serve as\nboth [6, 231, 232] social cues for the bandwagon effect [233, 234] and social cues for computing source trustworthiness\n[78, 228, 231] and message credibility [232].\nEngagement metrics taking the place of credibility is supported by the existing research literature. One study\nfound that “bandwagon heuristics—such as the number of likes, comments, and retweets—increased the credibility of\nnews” [232]. The automated activity of social probing by non-human-like bots with zero trust, i.e., strangers, which\nin the case of [235] consists of visiting the profile page of others, can boost popularity. That popularity translated\nsuccessfully to influence, which was getting others to follow accounts recommended by the bots. The number of\nretweets, despite its noisiness and bias, has a positive correlation with expert-assessed credibility of messages, and\nit is effective at predicting ground-truth credibility when combined with credibility ratings from a different crowd,\n17Mechanical Turk annotators [236]. Cues in the form of retweets (information sharing) boosted the credibility of a\ntweet’s author regardless of whether the tweet originated from a peer or a stranger, and even if the retweets were\nperformed by strangers [231]. Conversely, source credibility was found to play a major role in the information sharing\nbehavior of users on Twitter [237]. Credibility and influence are intertwined, feeding off each other and creating a\nrich-get-richer effect, where the wealth is social capital.\nThe need to impute a separate trust value for a social tie shared between two agents is also obviated by engagement\nmetrics that can serve as proxies. We improve upon the trust mechanism found in [78, 228], where the proxy variables\nfor trust are the number of times a user has retweeted [78, 228] and replied [228] to another user, by factoring in the\nrelationship between engagement type, stance distance, and themes; weighing the different engagement metrics; and\ncounting indirect engagement actions instead of direct ones. The authors of [228] themselves noted that replies as a\nproxy can be unreliable because replies may contain users expressing a stance different from the parent tweet, lending\nsupport to our decision to account for stance when using engagement metrics. Whether an engagement type is driven\nby heterophilia or homophilia in stances and themes (and consequently the metric generated for the engagement\ntype) under our model is explained in Section 3.3.\nAs for using indirect engagement numbers over direct ones as proxies for trust, indirect engagement can be more\nreliable than direct engagement despite appearing counter-intuitive.\nTraditional indicators of sources as being authoritative are less visible online [238] because of a flattening of\nhierarchies. Surmounting the accessibility barrier (e.g., radiocommunication spectrum licensing, capital costs of print\nmedia) and the expertise barrier (e.g., sufficiently credentialed to be invited onto mass media to comment on an\nissue) denoted authoritativeness pre-social media, but under social media where such barriers towards broadcasting a\nmessage has disappeared, so too has their function as indicators of trustworthiness. Distrust in authoritative sources\nhas also been on the rise, further weakening traditional signals of credibility. Distrust in authority being a widespread\nphenomenon is evident in Twitter discourse on vaccine and climate change where “science” (science as a source of\nauthority, not science as practiced) is regarded with suspicion [239, 240, 241]. While an authoritative source can still\nengender trust on social media, a stranger retweeting information from an authoritative source actually reduces the\ntrustworthiness of the source, whereas a stranger retweeting peers or other strangers increases trustworthiness [231].\nThe rising distrust in authorities was an impetus for the development of bot-augmented decentralized info ops (as\ndiscussed in Sections 2.2.4 and 3.5). On social media, the audience has also been flattened through the phenomenon\nof context collapse [242], where users cannot tailor messages to pander to specific subgroups because messages posted\non social media are viewable by all, making it easy to offend others, with the offense causing diminished trust in the\nsource [243].\nWith direct trust in a message’s source significantly dampened by flattened hierarchy and context, message\ncredibility and user trustworthiness online are strongly coupled. Social media users become only as trustworthy as\nthe credibility of their messages. People judge the credibility of information by engagement metrics attached to\nthat information [231, 232]. Consequently, crowdsourcing trust is an established practice, e.g., computation of a\nTwitter user’s credibility based on the degree of agreement between the user’s tweets and their respective replies\n[244]. Engagement-based source credibility indiscriminately boosts the credibility of peers andstrangers [231]. These\nengagement metrics are susceptible to the bandwagon effect, in which people base their opinion on what they perceive\nto be the opinion of the collective herd [231, 232], e.g., liking what others liked and retweeting what other have\nretweeted. Another factor at play is minority-avoidance and the spiral of silence, as discussed in Section 2.1.3. Trust\nquantified by indirect engagement numbers is the overwhelmingly dominant form of trust online.\nThat is not to say that a message source under an indirect trust system, e.g., as implemented in Diluvsion , is\nwholly incapable of affecting the message’s credibility, and by extension, the source’s trustworthiness. A source that\nhas amassed a large following of users sharing similar stances is likelier to receive more likes for its message than a\nuser with zero followers.\nThe pervasive indirect influence of engagement metrics extends beyond how one perceives others and into how\none perceives oneself. Every additional like or retweet that a user receives for their own tweets serves as a small\nvalidation, affirmation, and acceptance of the online self [215, 245]. Under Diluvsion , an agent’s memory of its own\nrecently tweeted tweets also influences stance adoption, and the agent’s perception of those recently tweeted tweets\nare shaped by their respective engagement metrics. Because an agent’s tweet’s stance is always the same as the\nagent’s (unless they recently changed their stance) under our model, if an agent’s recent tweets have received a lot of\naffirmation (e.g., a high number of likes), the agent will be more likely to retain their stance even when they encounter\na lot of opposition to their stance in their memory of seen tweets (memory of others’ tweets that they saw).\nFigure 1 demonstrates how indirect influence operates to infect an agent. This example omits the influence of\nstance and theme compatibility between the tweets and the agent in an infection event. For an example that does\ninclude it, see Section 6.5.\n18H1H2B1H3\nH4B2B3\nH1H2B1H3\nH4B2B3H1H2B1H3\nH4B2B3\nMemory (H1)\nStance# L# RT# RP\nPro000A B\nC D\nFollowAction (reply-tweet, like) BotHuman NegativeNeutralPositiveH1H2B1H3\nH4B2B3\nSelf-Memory (H1)\nStance# L# RT# RP\nNeu001\nMemory (H1)\nStance# L# RT# RP\nCon000\nPro000Self-Memory (H1)\nStance# L# RT# RP\nNeu002\nMemory (H1)\nStance# L# RT# RP\nCon200\nCon000\nPro001Self-Memory (H1)\nStance# L# RT# RP\nNeu002\nMemory (H1)\nStance# L# RT# RP\nPro000\nCon200\nCon000Self-Memory (H1)\nStance# L# RT# RP\nNeu003Figure 1: At time-step 1, a neutral human agent (H1) makes a tweet (action not depicted in this figure)\nexpressing a neutral stance and themes of medical authority and individual liberty over collective good, e.g.,\n“My doctor tells me wearing masks is a personal preference.” (A)While still at time-step 1, a second human\nagent (H2), H1’s follower, concocts a reply-tweet with matching themes but a positive stance, e.g., “The\nCDC said masks protect everyone, not just the wearer.” (B)A negative bot (B1) calls poll tweets , which\nis equivalent to doing a search of existing tweets, finds H1’s neutral tweet, and replies with a negative tweet.\n(C)A third human agent (H3) makes a reply-tweet with a negative stance to H2’s earlier reply to H1. Two\nbots (B2 and B3) liked H3’s reply-tweet. H3’s reply happens to be seen by H1 because H1, as a follower\nof H2, occasionally reads the replies to H2’s tweets. (D) A fourth human agent (H4) makes a tweet with a\npositive stance that is seen by H1, which pushes the very first tweet seen out of H1’s limited memory. (E,\nnotshown) At the start of time-step 2, when read conversion is called, H1 observes one positive-stanced\ntweet without any engagement, two negative-stanced tweets, one which received two likes, and finally its\nown neutral-stanced tweet that received three replies. To determine if H1 will experience a stance conversion\nafter reading these tweets, we check if a generated uniform random number clears H1’s resistance threshold.\nThe number does. To see which stance from the seen tweets infect H1, a weight is calculated for each stance.\nThe negative stance has a heavier weight than neutral or positive because the negative tweets are greater\nin number and have affirmative engagement instead of negatory (likes instead of replies). Weighted random\nchoice picks the negative stance, H1 changes its stance to negative.\n193.4 Humans\nAn agent designated as “human” in our model is an idealized representation of a typical social media user. All\nhuman agents follow the same heuristics, and they also have the same maximum memory size and maximum number\nof concurrent themes. Human agents may differ in their starting stance and themes, starting infection resistance,\nactivity level, and roles. Over the course of the simulation, their stances and themes may change. Resistance evolves\nwith every infection (stance change). Their activity level and roles remain fixed. The heuristics and maximums\nremain fixed as well.\nThe roles taken on by human agents, as well as their activity levels, are based on empirical Twitter data (Section\n3.8.1). Roles are the five groups that users were placed into based on their favored engagement action on Twitter,\ne.g., they were known as Likers if they exhibited a tendency to like others’ tweets rather than compose their own\ntweets. Users were also separately placed into three activeness categories depending on the number of observable\nactions taken per week. Each role, with respect to its activity level, will have the highest frequency for the engagement\naction that the role is named for, e.g., moderately active Quoters will have the highest frequency of quote-tweeting\ncompared to other moderately active roles. However, the same role will have different action frequency distributions\nbetween different activity levels, e.g., a highly active Quoter will be likelier to like tweets than they are to quote-tweet,\nbut a moderately active Quoter will quote-tweet more often than like.\nAt every time-step, a check is performed to determine whether a human agent wakes. If it wakes, it will read\ntweets and potentially be infected by the tweets it has read. If it is awake, a second check is performed to determine\nwhether it has used up its allotted quota of weekly actions (this quota is based on its assigned activity level). If it\nhas not, the agent will definitely act during this time-step. If it has, the agent will have a very small fixed random\nchance of acting. The actions that an agent can perform are discussed in Section 3.3.\nFor all actions, the part that involves broadcasting a stance — i.e., writing a reply, a tweet, or a quote-tweet — the\nbroadcasted stance is always the same as the agent’s. Themes for replies and quote-tweets always match the themes\nof the parent tweet; the assumption is that when responding to another, normal users would not go off-topic. Themes\nfor an independent tweet are drawn from the agent’s own pool of themes, with themes that have been recently used\nby an agent having greater probability of being chosen. For the part of an action that does not explicitly broadcast\na stance, i.e., selecting a tweet to like, retweet, quote, or reply to, the agent will favor tweets with stance and themes\nthat align with its own, but the agent is also influenced by engagement metrics to pick tweets with different stances\nand themes.\nTo be infected, a uniform random number must first exceed an agent’s resistance threshold. A high resistance\nresults in fewer infection events being triggered. If the threshold is cleared, one stance among the tweets an agent has\nread will be selected as the infecting stance. The selection is random, but each stance is weighted by the engagement\nmetrics (Section 3.3.1) and by the compatibility of the tweets representing the stance. The compatibility calculation\nfactors in themes as well, e.g., for a neutral agent (equidistant from both poles), a group of tweets representing the\npositive stance that has many matching themes will influence the agent more strongly than a similar-sized group of\nnegative tweets with non-matching themes, all else being equal. A successful infection results in the agent’s adoption\nof not just the infecting stance but also a portion of the most popular themes found in the tweets representing the\ninfecting stance. Examples of the infection mechanism at work can be found in Figure 1 (the influence of stance and\ntheme compatibility is omitted in this example) and in Section 6.5.\nAlthough we have called infection as stance “change” or “conversion” throughout this paper, an infection event\ncan end with an agent being infected by the same stance. This situation can be thought of as witnessing strong\narguments that affirms one’s faith in one’s stance.\nResistance is altered by infections. If the infecting stance is the same as the one held by an agent at the start of\nan infection event, the infection will improve resistance. Infections by non-matching stances degrade resistance. This\nmechanism is inspired by certain viruses such as the Hepatitis C virus (HCV) whose infection some people can clear,\nwhereas in certain other population segments they induce an impaired immune response thereby allowing reinfections\nor chronic infections [246, 247]. More domain-appropriate inspirations are the observed instances of people slowly\ndoubting their worldview if some strongly held beliefs have been shattered (e.g., South Korean feminists “taking the\nred pill”[248], spiral of mistrust/cynicism [249]).\n3.5 Bots\nBots are propagandists, biological (Section 2.2.5) or electronic, dedicated to promoting a stance on a social network.\nThe key differences between bots and humans are: a bot’s stance is fixed, a bot never unfollows another user, a bot\nconstantly surveys the conversation landscape, and a bot has a greater memory capacity. The set of heuristics that a\nbot follows also differ from that of humans as they are designed to spread their stance as widely as possible. None of\nthe differences should be interpreted as bots having superhuman abilities or access to tools on social media platforms\nthat regular users do not. The difference is purely a result of motivation and prioritization. Bots are just as limited\n20as humans in terms of their activity levels and roles.\nA bot surveying the conversation landscape, poll tweet , at the start of every time-step is equivalent to running\na search or covertly reading tweets from accounts that it does not follow. A fixed number of the most recent tweets\nare returned with each survey attempt and these tweets are saved into the bot’s memory of seen tweets. Surveying\nallows bots to take note of tweets/conversations that may have escaped the attention of human agents. If necessary,\nbots can jump into the conversations to influence other agents. This form of non-social tie information acquisition is\nexclusive to bots. For human agents, non-social tie information is acquired if they happen to be randomly selected\nas the followers who have read a reply to their followee’s tweet (see Section 3.3; the mechanism applies to bots too).\nBots do not unfollow so as to avoid triggering reciprocal unfollowing among their followers. Bots want as large\nan audience as possible to maximize the reach of their messages. This desire for a large audience is also manifested\nwhen choosing which tweets to quote, retweet, or reply to. Unlike humans who will only consider engagement and\ncompatibility, bots will additionally consider a tweet author’s follower count, favoring those who have large counts.\nBots not changing their stance is simply a result of them being bots, i.e., unyielding propagandists. Their greater\nmemory capacity (Section 3.2) is also due to them being motivated agents of change.\nHowever, bots do adopt new themes. Since bots are conceptualized as being interested in altering only the\nstances with regard to the issue of interest, theme adoption does not compromise the bots’ mission as themes are\nnon-polarizable with regard to the issue of interest (Section 3.1). It may even aid in the bots’ mission as bots can\nbetter empathize with the rest of the agent population, e.g., composing tweets with themes that resonate with other\nagents and are able to infect others more because of greater theme compatibility.\nWhen bots have to choose which tweet to reply to, they have two tactics — targeting vulnerable users and\ntargeting high engagement tweets — that they alternate between. A probability distribution governs how often one\ntactic is used over the other. To target vulnerable users is to target tweets from users who have a low number of\nfollowees. The reasoning for this tactic is that such users have low number of information sources (followees) and are\ntherefore more easily swayed by a flood of information from non-followees. To target high engagement tweets is to\ntarget tweets whose engagement metrics are the highest among those that a bot has witnessed in its memory and\nare authored by users with high follower counts. This tactic maximizes the visibility of a tweet as high engagement\ntweets are likelier (due to bandwagoning) to be targeted by agents for actions that trigger information sharing such\nas replying, retweeting, and quote-tweeting. This tactic has real-world precedence, e.g., BTC scams [230].\nReplies and quote-tweets authored by bots have the same stance as each bot’s respective stance, just like humans,\nbut the process of themes selection differs from humans. While humans would use the same themes as the parent\ntweets, bots would only use half of the themes from parent tweets, while the other half is drawn randomly from\neach bot’s pool of themes. In the case where a bot’s themes deviate significantly from the population, this tactic\nserves to subtly inject the bot’s themes into the information environment to facilitate infection in the future, while\ninfection chances at present are improved by adopting half of the same themes as the parent tweet. Theme selection\nfor independent tweets also differs from humans, as a bot does not select based on a history of its own tweets. Instead,\nfollowing the same selfish/motivated reasoning as when replying and quote-tweeting, half of the themes are the most\ncommon themes found in a bot’s seen tweet memory while the remaining half are randomly chosen from the bot’s\npool of themes.\nLike humans, bots start out embedded in a network instead of starting out in the network’s periphery. Section\n2.2.6 explains the reasoning behind this positioning. The bots in our simulation are bound by the same diurnal\nactivity schedule as the one humans follow. This is informed by the activity data of the covert pro-Western info ops’\nassets, which have peaks and troughs [120] that mimic a human’s. Motivated by decentralization (Section 2.2.4), the\nbots act independently (just like human agents) and are not conscious of the presence of other bots. Naturally, this\nexcludes explicit coordination between the bots.\nBots operate the same across stances within Diluvsion , unlike other works that split propagandists based on\nstance. For instance, the “bad guys” bots/trolls class of agents in twitter simactively spread disinformation while\nthe “good guys” stiflers in twitter sim[81] are reactionary by design, only countering disinformation that they see\nand never proactively spreading truth.\n3.6 Temporal resolution\nHumans follow a diurnal cycle, and their inclination towards being active on social media follows a similar cycle [250].\nTherefore, the human and bot agents in our model also observe this cycle. The mean number of tweets from a dataset\nof mask-wearing conversations on Twitter, binned by the day of the week and the time of the day, gave us an action-\nprobability table. This table approximates the likelihood of an average Twitter user using the service at a particular\ntime-step. Within each time-step, the order in which the agents activate is random. A consequence of simulating\npeaks and troughs in platform activity levels is that during periods of high activity, information dissemination and\nactions to boost dissemination will reach the highest number of awake users. However, at the same time, the great\n21volume of information flowing during high activity periods means that disseminated information risks being pushed\nout of the memory (Section 3.2) of agents with high numbers of incoming links.\nThe resolution of the simulation is 30 minutes. This resolution is sufficient and has empirical support: the\nGraphika and SIO investigation found that the info ops “assets in the Afghanistan and Central Asia groups typically\nposted at roughly 15-minute or 30-minute intervals in any given hour” [120]. Our decision was also influenced by the\nmean weekly frequency (the number of actions taken by a user in a week) for the high-activity group in the study\nby Antelmi et al., which was found to be 133.84 [214]. If we took 133.84 actions and divided them over seven days,\nthis leaves approximately 19 actions per day. At a resolution of 30 minutes, 19 actions will take 9.5 hours, including\nthe interval time between actions. This wakefulness period does not violate typical sleep-wake cycles within human\ncircadian rhythms [251].\nAs validation, when we generated 1,000,000 sets of 336 random numbers (2 periods per hour ×24 hours ×7 days)\nwith a 0.9 damping factor8and tested them against the activity-probability table (derived from our mask-wearing\ndataset), the maximum number of periods/time-step in which an agent is awake was 153 .45±8.46 per week, a range\nwhich can accommodate the maximum number of actions the most active class of agents can take per week within\nour model (we let this maximum be the rounded mean number of actions of high activity users, 134, from the dataset\nin [214], which is different from the mask-wearing dataset used to generate the action-probability table). To allow for\nvariability, once an agent exhausts its allotted action quota, if a random number check against the action-probability\ntable says that the agent is awake and can therefore take an action, the agent has a fixed 1% chance of exceeding the\nquota.\n3.7 Graph structure\nDirected scale-free graphs were used as the simulation environments. Specifically, we used the improved version [252]\nof the algorithm first introduced in [253] by Schweimer et al. for generating random directed social network (SN)\ngraphs for use in information diffusion studies. The hyperparameters used for graph generation comes from the\nhyperparameters file in the open-source code repository shared by the author9which themselves are derived from\ncrawling real-world social networks on Twitter. The set of hyperparameters used were named “Covid”, the largest\nof the 14 networks crawled with 50,133 nodes and 4,832,226 edges.\nDesirable network properties such as a high amount of clustering and reciprocity are reproduced by the SN\ngraph generation algorithm. Clustering describes users aggregating into homophilic clusters when interacting online\non Facebook and Twitter [254]. Reciprocity describes the small but still significant amount of reciprocal edges on\nsocial media (22.1% found on Twitter [255]). If we used non-SN scale-free graph generation algorithms, e.g., the\none proposed in [256] by Bollob´ as et al. which improved upon the Barabasi-Albert method, certain properties of the\nnetwork deviate further from expectations than if an SN graph generation algorithm was used (Section 4).\nA total of 20 graphs with 1,000 nodes and 20 graphs with 900 nodes were generated. Because of an inherent\nflaw [253] of the graph generation algorithm itself, approximately 2.5% of the nodes for each graph are isolates, i.e.,\nthe nodes have no incoming or outgoing edges. The graphs were saved and reused for different scenarios to allow\nfor reproducible results and also because graph generation is time consuming. The graphs with 900 nodes were used\nfor the “peripheral”/“fringe” scenarios where bots (10% of the network population, equivalent to 100 agents) do not\nstart out embedded in the network but is added to the graph later and given only one outgoing edge to a random\nnode, i.e., a bot’s only social tie is being a follower of an existing human user.\n3.8 Parameterization\nThe same set of parameters are used for validation (Section 4) and simulations (Section 5).\n3.8.1 Empirical data\nTo ensure that agents populating the network mimic the behavior of those found in real-world social media, we\ninitialize the agents based on statistical data collected from Twitter users.\nFor the diurnal activity schedule, we took the mean number of tweets split by day of the week and into half-hour\nintervals (the resolution of the simulation) from the two million tweets we collected on COVID-19 masking debate\nbetween January 1–June 21, 2020 as a probability distribution for when an agent will be active.\nThe starting distributions of stances and themes, including the starting number of themes per agent, are taken\nfrom the February 27–March 2, 2020 period of the COVID-19 masking debate, which we have identified as the starting\npoint for when the majority of the discussions occur within the appropriate context of masking as protection against\n8When multiplied by probabilities in the action-probability table (Section 3.8.1), which can range from 0 to 1, the factor\nshifts the graph down/reduces the amplitude, i.e., lowers the action probability.\n9https://github.com/Buters147/Social_Network_Graph_Generator\n22COVID-19 instead of masking in other contexts, e.g., masks to protect against volcanic ashfall. A total of 12 themes\nare identified from the dataset. Examples of the themes are “encourage” (encouraging people to mask or to not\nmask) and “appraisal-critize” (praising others for wearing or for not wearing masks).\nAlthough the proportions of stances and themes at the population level are maintained, the stance and the themes\nchosen for an individual human agent are entirely random, unaffected by the agent’s neighbor’s stance and themes.\nSome may take issue with our choice since homophily would predict that users sharing similar outlooks, political\nor otherwise, are more likely to be connected, so there should be homogeneous stance and theme clusters in the\nnetwork. For emerging issues where strong narratives have not been established, the situation is not so clear-cut.\nRepublicans are more anti-mask during the COVID-19-pandemic than Democrats [257, 258], but the Twitter data on\nmask-wearing that we collected showed that right before the pandemic, many fans of Trump supporter Scott Presler\nwere urging him to wear a mask to avoid catching disease from homeless people.\nThe counts of users found in the high, medium, and low activity categories are taken from [214]. For the range\nof actions available to the agents (Section 3.3) except for following and unfollowing, the frequency distribution is\nbased on the tables and figures in [214]. In [214], the authors categorized users into five roles — Tweeters, Quoters,\nRetweeters, Repliers, and Likers — with a corresponding action frequency distribution. Each role is further sub-\ndivided into high, medium, low, and no activity. Although almost 50% of users belong to the no activity cluster\n(no visible activity but have done at least one hidden activity such as changing their screen name), we exclude this\ncategory because they would have no appreciable impact on the simulation dynamics owing to their inactivity. This\nleaves us with 5 ×3 = 15 human agent categories.\nStatistics on following and unfollowing are not found in [214]. We relied on the descriptions provided in [212, 259].\nAccording to Kwak et al., the “average of unfollows is 15.4” and “average number of followees increases from 59.7 to\n75.7” in 51 days with “about 30% of users [having] unfollowed at least once” [212]. According to Myers and Leskovec,\n“[i]n a given month a user of degree 100 tends to gain 10 and loose [sic] 3 followers” [259]). Their descriptions\nshowed that while following and unfollowing are pervasive over a scale of multiple months, they are rare when one\nfactors in all the possible actions that a user can take (tweeting, retweeting, etc.) at the scale of half-hour steps\nwhich our simulation runs at. Modeling the actions of following and unfollowing as suitably rare events translates to\nhaving follow and unfollow constituting 0 .6% and 0 .3% of all of an agent’s actions, respectively. This applies to all\n15 agent categories. Given that highly active and moderately active agents form 10.4% and 22.2% of the population\n(totaling ∼30%) and their actions per week are 134 and 66, respectively, we expect highly active agents to follow\n134×0.66%×4 = 3 .5 users and unfollow 1 .8 users over 28 days while moderately active users would attempt 1 .7\nfollows and 0 .9 unfollows, which is slightly lower than the frequency observed in [212].\nThe parameterization of activity levels and roles used statistics from [214] because it cannot be done using the\nmask-wearing dataset. Only tweet-level data were collected for the mask-wearing dataset; user-level data were absent.\nHowever, this does provide us an opportunity to validate a simulation using parameters from the dataset in [214]\nagainst an “unobserved” dataset (the mask-wearing dataset), as we shall see in Section 4, which is a more informative\nvalidation process than using the same dataset from which the parameters are drawn.\n3.8.2 Best approximations\nCertain parameters in our model require information that the existing literature does not provide and that we were\nunable to collect. The resistance to stance conversion is one such parameter. We opt to model the resistance as a\ntruncated normal distribution with a low of 0.5, a mean of 0.75, a high of 1.0, and a standard deviation of 0.25. The\nsize of a human agent’s memory is fixed at m= 30, and this upper limit is used for three separate memory bins,\nwhich are the agent’s own recent tweets, tweets by others that an agent has seen, and the author of those seen tweets.\nBots have a memory of 120 and poll tweet returns 30 tweets each time. The number of themes that every human\nand bot agent can hold at once is five while the maximum percentage of new themes that an agent will adopt is\n40% for humans and bots. Humans adopt themes upon a successful infection, whereas bots adopt themes at every\ntime-step. All of a user’s tweets are assumed to propagate to all of the user’s followers, although not all tweets are\nguaranteed to be seen as older tweets risk being pushed out of a follower’s limited memory by newer tweets. The\nproportion of a user u’s followers who see a reply from another user vto a tweet from u, which is the primary method\nof non-social tie information propagation in our model (there are other methods such as uretweeting a seen tweet\nfrom a non-followee user), is set at 10%, with u’s followers being chosen at random.\nEvery infection event by a stance matching a human agent’s own increases the agent’s resistance by 1% while\nan infection by a non-matching stance decreases the agent’s resistance, with the decrease being proportional to the\ndistance between the stances, −0.5% for the closest neighboring stance and −1% for the stance on the opposite pole.\nStance compatibility is simply the distance between two stances divided by 2, cs=|suser−stweet|/2. Division by\nthe maximum distance between stances, 2, ensures that the maximum value of csis 1. Themes compatibility is the\nnumber of matching themes divided by the number of unique themes, ct=|tuser∩ttweet|/|tuser∪ttweet|. Compatibility,\nc=csws+ctwtwhere wsis the weight for stance compatibility and wtis the weight for themes compatibility.\n23The default values for {ws, wt}are{1,0.8}so having matching stances is more important than matching themes.\nNote that since incompatibility, not compatibility, in stances increases the chance of replying, the compatibility score\nwhen replying is calculated slightly differently, c= ((1−cs)×ws) +ctwt. Weights are also modified when replying,\n{ws, wt}are{0.8,1}to mildly de-emphasize stance compatibility, thus preventing replies from being exclusively an\nact of disagreement with a tweet’s stance.\nWhen deciding on which tweet to reply to, a human agent will prefer to reply to tweets that have the same themes\nbut a different stance, giving less consideration to the amount of engagement that a tweet has received. When deciding\nwhich tweet to retweet, human agents will prefer tweets with matching stance and themes, with a tweet’s engagement\nhaving slightly less weight on the decision. Quote-tweets emphasize theme compatibility over stance compatibility\nwhen choosing a target tweet to quote, and a slight importance is placed on compatibility over engagement just like\nwhen retweeting. The probability score of a tweet being liked, retweeted, quoted, or replied to, p=gwg+cwcwhere\nwstands for weight, gfor engagement, and cfor stance and theme compatibility. The weights {wg, wc}are{0.5,1}\nwhen replying, {0.8,1}when retweeting and quote-tweeting, and {1,1}when liking. Engagement is a weighted sum\nof metric counts, g=Nlikewlike+NRTwRT+Nreplywreply. The weights used differ based on the action taken (e.g., in\ndeciding whether one should like a tweet, the bandwagon heuristic would more pay more attention to the number of\nlikes the tweet has instead of the number of replies). The weights for the engagement metrics ,{wlike, wRT, wreply},\nfor the engagement actions of liking, retweeting, quoting, and replying are {1,0.2,0.2},{0.2,1,0.2},{0.2,1,0.2}, and\n{0.2,1,0.2}respectively. Quote-tweeting and retweeting have the same weights as both actions expose one’s followers\nto another’s tweet. Both gandcare normalized by dividing over the values summed over all candidate tweets, e.g.,\nfor a tweet x,gx=gx/PNcandidate\nn=1gn.\nBots differ from humans in that their probability for choosing a target, p=g+c+f, meaning that the probabilities\nare unweighted and an additional term fhas been added for the follower counts of the tweet authors. Like gand\nc,fis normalized. And instead of the probabilities being used in weighted random choice, they are used to sort\nthe candidate tweets and the tweet with the highest probability is always selected. Bots also weigh each metric in\nengagement gand the weights, {wlike, wRT, wreply}, for liking, retweeting, quoting, and replying are the same as those\nof humans.\nWithin the sub-component of the read conversion algorithm for computing a human agent’s stance changes,\nthe compatibility score, rc, for each stance (e.g., rc-pro) is the sum of the individual compatibility scores for the\nsubset of an agent’s seen tweets that exhibit the stance. The same applies to the engagement metrics’ scores,\nrg. The compatibility scores are normalized by dividing over the total score for messages of all stances. The\nengagement scores are normalized on a per-engagement type basis by dividing over the total of the frequencies for\nan engagement type. Prior to summation, the engagement metrics are weighted, {wlike, wRT, wreply}={1,0.5,−0.5}.\nThe weighting reflects whether an engagement action is affirmatory or negatory in the context of stances. The\nsummed compatibility scores and engagement metrics’ scores are also weighted to avoid stance conversion from\nbeing entirely dominated by engagement metrics. The weight for compatibility is an agent’s memory, m, while the\nweight for engagement is the reciprocal,1/m. As such, the final conversion score for a stance, e.g., pro, wr−pro=\nm−1rg-pro+mrc-prowhere rg-pro=PNpro\nx=1gxandrc-pro= (PNpro\nx=1cx)/Pc. The engagement score for a tweet\nxisgx=wlikeNlike-x/PNlike+wRTNRT-x/PNRT+wreplyNreply- x/PNreply. The conversion scores are used as\nprobabilities for selecting the stance which a human agent would convert to if its resistance threshold has been\nexceeded.\n4 Validation\nValidating information diffusion models can take the form of showing that the model is able to generate statistical\ndistributions similar to those in the real world, e.g., recreating the percentages of tweets that receive zero engagement\nand tweets that go viral by adjusting parameters [84], and matching curve shapes for message volume [85]. Another\nmethod is confirming that the parameters used in the model match those found in the real world, e.g., Twitter usage\ninter-arrival time in the twitter simmodel [81] by Beskow and Carley.\nAs explained in Section 3.8.1, a large number of key parameters in our model such as the distribution of a user’s\nactions, a user’s degree of activity (cognate to inter-arrival time), and the distribution of user roles are taken directly\nfrom real-world data. The underlying graph structure on which the simulation is run on (Section 3.7) and the user’s\ndirunal schedule (Section 3.6) are also based on real-world data. Our model would naturally reproduce inter-arrival\ntime, action distribution, etc., that are close (though not perfect matches) to empirical observations. Therefore, we\nhave “validated” our model in the same sense that Beskow and Carley have.\nFor additional validation, we compared the engagement statistics obtained from simulations with those obtained\nfrom real-world data, namely the Twitter on mask-wearing conversations pertaining to the COVID-19 pandemic\n(Section 3.8.1). Differences and similarities between statistics collected empirically and from the simulation are\nshown in Figure 2. The simulations are run for a period equivalent to 28 days and data was aggregated from 20\n24runs. Parameters for the agents are the same as those described in Section 3.8. The configuration for the human-only\nsimulation scenario is equivalent to the Sim 11 scenario described in Section 5.\nThere are two sets of real-world distributions from the mask-wearing dataset in Figure 2 as we intend to illustrate\nhow distributions can differ depending on how hotly a topic is contested. Each set of distributions is from the complete\nmonth-long period of the named month. The January 2020 data are from a less popular and less contentious period as\nit is composed of tweets that have been classified as out-of-topic/containing no stances, i.e., they are not offering any\nopinion on mask-wearing as it relates to COVID-19, but they are discussing masks in other contexts (e.g., protection\nagainst volcanic ash, fictional characters). In contrast, the March 2020 distribution is a subset of only tweets that\nhave stances. March is also the period we have identified as when the mask-wearing conversation occured almost\nexclusively in the context of COVID-19.\nThe January distributions, from a subset where masks are a less contentious and less discussed topic, all contain\nlower values than the March distributions, i.e., shifted left. The distributions from the simulation in Figure 2 nearly\noverlap the empirical January distribution for likes and replies. The simulation distribution is visibly shifted right\ncompared to the January distribution when comparing retweets, although it is still bound to the right by the March\ndistribution. For simulations, the within-dataset ratios Nlike:NRTandNlike:Nreply are on average 2.0 and 8.4\nrespectively, close to the numbers calculated directly from the data from which the simulation parameters are drawn\n(sans simulation), which are 2.1 and 8.5. If we had instead used a non-social network-specific graph generation\nalgorithm (Section 3.7), the discrepancy between the simulation and empirical ratios would have been starker, as\nthe simulation ratios obtained would have been 1.7 and 5.2, validating our choice in the graph generation algorithm.\nJanuary mask-wearing data shows ratios of 3.8 and 22.5, while March’s ratios are 3.3 and 13.2. Retweets and especially\nreplies are intensified when a topic is contentious (March), hence the lower ratios.\nOne factor involved in the discrepancy in distributions and ratios is the insufficient number of tweets at simulation\nstart for agents to like, retweet, etc. We alleviated tweet scarcity by having an agent default to tweeting as an action\nif there are no incoming tweets in its memory and whenever a chosen action for a time-step is invalid. A consequence\nof this design choice is that the observed action distribution of agents would differ from the parameters they were\ngiven, as they tweeted more often than they should. However, the alternative of doing nothing if a chosen action for\na time-step is invalid would have caused a similar issue of mismatch between parameters and observed distribution of\nactions taken, in addition to causing an agent to appear less active (due to skipped actions) and causing the starting\nphase of the simulation to last longer (due to tweet scarcity). We had considered seeding the simulation with a set\nof initial tweets but decided against it as the assumptions and ramifications — heuristic for distributing the initial\ntweets to the nodes, the tweets causing stance changes in agents immediately at simulation start, etc. — would\ncomplicate the analysis of the simulation results.\nAnother partial explanation for this difference is the size and structure of the networks. Simulations relied on\na synthetic scale-free network as the graph structure in terms of the follower-followee network of the mask-wearing\ndataset is unknown to us. When compared with the January dataset, the across-dataaset ratio for the counts of likes,\nNlike-Sim :Nlike-Jan , is 2.7, that of retweets, NRT-Sim :NRT-Jan , is 1.4, and that of replies, Nreply-Sim :Nreply-Jan , is\n1.0. When compared with March, the respective ratios are 30.3, 18.0, and 19.2. If we infer the hidden size of the\nnetwork through the engagement counts (hidden size because users in the mask-wearing dataset cannot represent the\ntotality of network; metrics for a tweet can originate from a user outside the collected dataset), the number of users\nin January could be at least 2 .7×larger than in the simulation and for March, 20 .3×larger.\nThe across-dataset ratio of replies as well as the within-dataset ratios indicate that the simulations show a greater\naffinity to March than January data. For all engagement types, a constant multiplier of ∼1.5×reproduces March’s\nratios for within-dataset ratios, whereas while the across-dataset ratios for March’s ratios are a constant ∼20×\nof the simulations except for Nlike-Sim :Nlike-Mar . January’s multipliers differs depending on engagement type for\nacross-dataset ( ∼2×and∼3×) and within-dataset ( ∼3×,∼1.5×, and∼1×) cases. This makes the simulation a\ncloser approximation of conditions where a topic is heavily debated, i.e., March’s conditions.\nOverall, the distributions for the engagement actions strongly resemble real-world data, owing to the real-world\nstatistics used to initialize simulation parameters and the robust SN-graph generation algorithm that also relied on\nempirical data. As the indirect influence from engagement actions and the metrics resulting from those actions are\ncentral to Diluvsion , the findings in this section supports the model’s validity.\n5 Simulations\nWe present a number of simulations, starting from the baseline scenario of bots representing two disadvantaged stances\nengaged in a co-opetitive influence operation and continuing with simulations where certain key functionalities of the\nbots are progressively altered/degraded, which can be thought of as ablation studies. Other scenarios tested the effect\nof varying the mixture of bot-backed stances. The results from the simulations are shown in Table 1 and we grouped\nour discussion of the results in Section 6 based on their implications for conducting real-world info ops. All scenarios\n25100101102103100101102103104Frequency# likes\n100101102103100101102103104 # RTs\n100101102103100101102103104 # Replies\nSim, humans only\nEmpirical, Jan,\nno stance\nEmpirical, Mar,\nhas stancesFigure 2: Engagement metrics: real-world data vs simulations with 1,000 agents.\nfeatured 1,000 agents in scale-free networks and the results were averaged over 20 runs, with each run lasting 1344\nsteps (includes step 0, equivalent to 28 days). A bot’s role determines its action distribution, i.e., the probabilities for\nliking, retweeting, etc. at each time-step. A role is named after the most frequent action the role takes, e.g., Likers\nlike liking. Bot frequency is the number of times a bot is permitted to act per week. The definitions for both roles\nand frequencies (activity levels) are based on the Twitter data reported in [214], and the definitions apply to bots\nand humans.\nIn classical influence maximization (Section 2.1.3), information that we wish to see widely propagated is injected\nvia nodes that have been algorithmically identified as influential, i.e., positioned to maximize spread, but the bots\nin our simulations have no such advantage as their positions in the network are entirely random. Starting from the\nperiphery of a network is the most disadvantaged position our bots are subjected to (Sims 6 and 7).\nInter- and intra-stance awareness of the presence of other bots is absent as the info ops is conducted in a decen-\ntralized fashion (Section 2.2.4). Competition is implied as each agent can only belong to one stance so an agent’s\nconversion from one stance to another directly results in losing an agent to the other stance group; bots from one\nstance are not forbidden from converting agents belonging to a different bot-backed stance. Cooperation amongst\nagents arise naturally because agents gravitate towards performing affirmative actions, e.g., liking, for other agents\nthat share similar a stance. Cooperation among two bot groups, each backing a different stance, can be implicit in that\ntheir uncoordinated and even competitive actions can accomplish a common goal better than they can individually,\ne.g., reducing support for a third non-bot-backed stance.\nA high bot percentage, 10%, is used in all scenarios so that the impact from bots is more easily apparent, not\nbecause this percentage is reflective of the actual bot population in the real world. Estimates of bot percentages on\nTwitter vary widely depending on the source, with Twitter’s own analysis claiming 5.3% in 2022, a 2016 Securities\nand Exchange Commission filing by Twitter claiming 8.5%, Cyabra claiming 11% in 2022, and a study on the 2020\nSingaporean election claiming 26.99% [152, 260, 261]. Bots defaulting to a high activity level also makes their influence\nmore easily observable. Additionally, evidence exists that bots are prolific content creators on Twitter, with the Israeli\nweb analytics company Similarweb finding that “20%–29% of content in the US on Twitter is generated by bots”\n[262].\nTo ensure that the overall stance ratio at simulation start matches the one at the start of March 2020 in the\nmask-wearing dataset, the stance ratio for humans is rebalanced to account for the bots’ stance distribution. In other\nwords, the number of human agents assigned to a particular stance is reduced if bots are assigned to that stance.\nAs Sim 1 is the baseline, Sim 1 is referenced by all other scenarios for comparison in the scenario descriptions\nbelow:\n1. Sim 1: Bots belonging to two minority stances, neutral and opposition, attempt to maximize the spread of their\nrespective stances. The bot population is evenly split between neutral and opposition, and there are no bots\nin the camp of supporters, making the bot population’s stance ratio 1 : 1 : 0 (pro : neutral : con). This stance\ndistribution enables us to observe synergistic and antagonistic effects between the two bot sub-populations.\nFor all agent types, the number of themes an agent has as well as the distribution of the 12 themes are the\nsame as in the March mask-wearing dataset. Bots start out embedded in the network, i.e., they have organic\nfollower-followee relationships just as human agents do for reasons we have argued in Section 2.2.6. All the\nbots take on the role of a Replier to better demonstrate non-social tie spreading of information (Section 3.3.2).\nBots are as active as highly active humans are. In choosing whose tweets to reply to, bots will favor those\ndesignated as vulnerable, which are people with a low number of information sources, i.e., low followee count.\n26Bots will adopt commonly encountered themes to help the tweets they compose appeal to a wider audience.\n2. Sim 2: Bots will prioritize tweets that have high engagement metrics and that happen to also be authored\nby people with high follower counts when deciding whose tweet to reply to instead of targeting tweets from\nvulnerable agents as in Sim 1.\n3. Sim 3: Bots are as active as moderately active humans rather than matching highly active humans’ frequency.\n4. Sim 4: Instead of being Repliers, bots are Likers.\n5. Sim 5: Instead of being Repliers, bots are Retweeters.\n6. Sim 6: Instead of being embedded in the network, each bot starts out isolated in the fringes of the network\nwith only one followee relationship with a random human agent in the network (meaning that any tweet by the\nbot that is not a reply or quote-tweet will not be read by anyone in the network). This replicates the difficult\nstarting condition that bots face at the start of an info-ops, as presumed by other researchers [81] but which\nwe have argued against in Section 2.2.6.\n7. Sim 7: In addition to starting out in the fringes of the network as in Sim 6, bots take on the role of Likers\ninstead of Repliers.\n8. Sim 8: Instead of being highly active Repliers, bots take on a variety of roles (Likers, Retweeters, etc.) with\nvariable activity levels: low, medium, and high, just like human agents. The role and activity level distributions\nof the bots follow the distributions in [214].\n9. Sim 9: Bots conserve a theme, Theme 7, instead of following the usual theme adoption process. Conservation\ninvolves a set of conserved themes that always have priority whenever the bot updates its list of themes to\nadopt and whenever the bot needs to decide which themes its tweet should be accompanied by.\n10. Sim 10: Just as in Sim 8, bots mimic human agents’ roles and activity levels. Furthermore, the distribution of\nthe bots’ stances are the same as the humans’ stance distribution from the March 2020 mask-wearing dataset.\n11. Sim 11: All agents in this scenario are humans, unlike other scenarios that contain 10% bots.\n12. Sim 12: Bots are evenly distributed between supporters and detractors instead of being evenly distributed\nbetween neutrals and detractors, 1 : 0 : 1 (negative : neutral : positive).\n13. Sim 13: Bot distribution follows a 1 : 0 : 0 (negative : neutral : positive) ratio.\n14. Sim 14: Bot distribution follows a 1 : 1 : 1 (negative : neutral : positive) ratio.\n15. Sim 15: Bot distribution follows a 2 : 2 : 1 (negative : neutral : positive) ratio.\n16. Sim 16: Bot distribution follows a 4 : 0 : 1 (negative : neutral : positive) ratio.\n6 Results and discussion\nThe simulation results are shown in Table 1. Figure 3 shows the temporal evolution of the stances for the simulations\nin Table 1 while Figure 4 shows the temporal evolution for follower and followee counts, split by stances. Figures\n5, 6, 7, 8, and 9 are for our discussion on the consequences of bots starting out embedded in the network versus its\nperiphery (Section 6.5).\nA note on the presentation of the results: not all models of information diffusion are strongly deterministic, which\nis why some authors, e.g., [52], opted to instead show the number of simulations in which the bot-backed stance\nbecomes dominant instead of showing the average number of agents per stance at the conclusion of a simulation as\nwe have done in this paper. However, because our simulations all have a biased start where one stance is a clear\nmajority that outnumbers the combined total of the minority stances by a factor of 2.7 to 1 (human agents’ stances\nare evenly split in [52]) and bots can back more than one stance per scenario (bots all back the negative opinion in\n[52]), reporting our results as percentages of runs in which the bot-backed stance dominates is not viable.\nWhen we speak of a “majority” or a “minority” stance, we are referring a stance’s share among the agent\npopulation at the beginning of the simulation.\nSince Diluvsion is designed to support studies on info ops, discussion of simulation results will be interwoven with\ndiscussion of their relevance to info ops.\n6.1 Polarization\nWinner-takes-all, i.e., no polarization/unipolar polarization, is the natural trend of a network where a stance has a\nnumerical advantage over the combined total from minority stances, as witnessed in Sim 11 and Sim 10. This trend is\nnot meaningfully altered whether the network is populated entirely by human agents (Sim 11) or the network contains\n27Sim 1 Sim 2 Sim 3 Sim 4 Sim 5 Sim 6 Sim 7 Sim 8\nStance ratio (bot) 1 : 1 : 0 1 : 1 : 0 1 : 1 : 0 1 : 1 : 0 1 : 1 : 0 1 : 1 : 0 1 : 1 : 0 1 : 1 : 0\nBot start Embed Embed Embed Embed Embed Fringe Fringe Embed\nBot roles Replier Replier Replier Liker RTer Replier Liker Human\nBot frequency Hi-hum. Hi-hum. Md-hum. Hi-hum. Hi-hum. Hi-hum. Hi-hum. Human\nBot reply Vuln. Engage Vuln. Vuln. Vuln. Vuln. Vuln. Vuln.\nBot adopts themes Yes Yes Yes Yes Yes Yes Yes Yes\nStart/End N (Con) 139 305 345 133* 212 101* 70* 64* 109*\nStart/End N (Neu.) 129 568 551 400* 391* 159* 74* 67* 124*\nStart/End N (Pro) 732 127 105 468* 397* 740* 856* 870* 767*\nStart/End N (T1) 158 232 223 277 246 236 224 404* 248\nStart/End N (T2) 240 240 230 259 270 310* 374* 264 303*\nStart/End N (T3) 549 288 317 329 360* 430* 592* 589* 439*\nStart/End N (T4) 188 244 236 263 261 271 306 246 311\nStart/End N (T5) 34 204 195 189 184 181 126* 155 180\nStart/End N (T6) 69 220 237 219 223 203 190 142* 172*\nStart/End N (T7) 1 60 56 87 36 34 20* 15* 54\nStart/End N (T8) 52 218 195 194 224 191 144* 196 206\nStart/End N (T9) 26 193 221 186 176 172 126* 101* 151\nStart/End N (T10) 35 187 177 193 198 157 131* 137 132*\nStart/End N (T11) 26 194 207 184 185 180 108* 125* 144*\nStart/End N (T12) 26 200 178* 191 190 159* 142* 105* 151*\nSim 9 Sim 10 Sim 11 Sim 12 Sim 13 Sim 14 Sim 15 Sim 16\nStance ratio (bot) 1 : 1 : 0 March - 1 :0:1 1 :0:0 1 :1:1 2 :2:1 4 :0:1\nBot start Embed Embed - Embed Embed Embed Embed Embed\nBot roles Replier Human - Replier Replier Replier Replier Replier\nBot frequency Hi-hum. Human - Hi-hum. Hi-hum. Hi-hum. Hi-hum. Hi-hum.\nBot reply Vuln. Vuln. - Vuln. Vuln. Vuln. Vuln. Vuln.\nBot adopts themes Consv. Yes - Yes Yes Yes Yes Yes\nStart/End N (Con) 139 320 25* 12* 196* 760* 115* 213 540*\nStart/End N (Neu.) 129 536 63* 61* 8* 6* 228* 372* 7*\nStart/End N (Pro) 732 143 912* 926* 796* 234* 658* 415* 452*\nStart/End N (T1) 158 197* 231 312 220 219 236 243 243\nStart/End N (T2) 240 218 347* 367* 246 237 248 229 241\nStart/End N (T3) 549 266 427* 571* 305 303 298 290 301\nStart/End N (T4) 188 214 283 355* 220 232 218 238 231\nStart/End N (T5) 34 139* 190 61* 180 204 202 210 188\nStart/End N (T6) 69 181* 182 151* 198 205 214 221 200\nStart/End N (T7) 1 519* 29 4* 69 38 40 51 23\nStart/End N (T8) 52 175* 184 138 196 190 204 206 194\nStart/End N (T9) 26 160* 140* 66* 171 189 187 207 192\nStart/End N (T10) 35 152* 153 56* 195 181 182 212* 194\nStart/End N (T11) 26 174 124* 38* 179 190 206 193 178\nStart/End N (T12) 26 157* 144* 81* 175 191 192 173* 188\nTable 1: The number of agents holding a stance and theme(s) at simulation end averaged over 20 runs for\n16 scenarios with 1,000 agents. Agents can adopt a maximum of five themes. Stance ratio is negative :\nneutral : positive. An asterisk (*) indicates significant difference ( p <0.05) compared to Sim 1. Starting\nnumber of agents for a stance or a theme is listed on the unnamed column on the left, between the column\nof row names and the named columns.\n2805001000Sim 1 Sim 2 Sim 3 Sim 4\n05001000Sim 5 Sim 6 Sim 7 Sim 8\n05001000Sim 9 Sim 10 Sim 11 Sim 12\n0 4 812 16 20 24 28\nDays05001000Sim 13\n0 4 812 16 20 24 28\nDaysSim 14\n0 4 812 16 20 24 28\nDaysSim 15\n0 4 812 16 20 24 28\nDaysSim 16\nAgainst Neutral FavorFigure 3: Temporal evolution of stance distribution among 1,000 agents. Values are from 20-run averages of\ndifferent scenarios/simulations. Values at simulation end corresponds to the mean values reported in Table\n1. Descriptions of the scenarios are found in Table 1 and in Section 5.\n10% bots which are allocated similarly as the human population’s stance distribution (Sim 10). The slightly higher\nnumber of agents belonging to the minority stances in Sim 10 is likely a consequence of the bots’ presence, whose\nstances can never be changed, slowing but not halting the network from reaching a consensus around the majority\nopinion. These are only trends, and not necessarily the final outcomes, as the simulations need to be run longer to\nobserve whether all agents converge to a single opinion. And in the case where bots populate the simulation, Sim 10,\nthe network will never fully converge as bots will never change their stance.\nA winner-takes-all result [54] has been found for networks with competing viruses where one virus is stronger\nthan the other, and the strength of the virus is quantified by a single numerical parameter (derived from attack and\nrecovery rates). While the viruses (opinions) in our model are equal in that they do not have an explicit parameter\nthat offers one virus an advantage over others, a combination of factors enable the dominant stance in our work\nto replicate a similar advantage over competitors. The dominant stance in Diluvsion is the stronger virus because\nhomophily dictates that the stance has a more receptive audience, and their greater number means that their opinions\nare broadcasted and boosted more often, making others more susceptible to succumbing to peer pressure and adopting\nthe majority stance.\nPolarization can be induced if bots are present, and the stance distribution of bots differs from that of the humans.\nWhen half of the entire bot population holds a negative stance and the other half a neutral stance while the bots\nare restricted to human-like role and activity level distributions (Sim 8), the encroachment of the majority opinion\nis mostly halted. In Sims 12 and 16, where bots are only distributed between negative and positive stances with\nnone for neutral, assigned a role more suitable for stance propagation (Replier), and are all as active as highly active\nhumans, neutrals virtually disappear and agents congregate around the two polar opinions. In the case of Sim 13\nwhere all agents start out belonging to only one minority stance, specifically the polar minority stance, a similar\ndepopulation of neutral agents is observed. More time-steps are needed to determine whether Sim 12 ends with this\npolarized distribution or if all agents are eventually converted to the bot-backed negative stances.\nIn existing studies on binary opinion dynamics, consensus cannot be reached if stubborn agents are present in\nthe network [263, 264]. When the problem is extended to account for committed/stubborn agents and for multiple\nopinions (greater than binary), only opinions that are backed by committed people survive under the voter model\n[83]. The results from our simulations serve to further confirm the role played by stubborn agents, bots in our case,\nin driving polarization in a network.\n294050# AgentsSim 1\n4050Sim 2\n4050Sim 3\n4060Sim 4\n5060# AgentsSim 5\n050Sim 6\n050Sim 7\n4550Sim 8\n4060# AgentsSim 9\n4050Sim 10\n2550Sim 11\n2040Sim 12\n0481216202428\nDays304050# AgentsSim 13\n0481216202428\nDays455055Sim 14\n0481216202428\nDays5060Sim 15\n0481216202428\nDays2550Sim 16\nFollowers-Against\nSources-AgainstFollowers-Neutral\nSources-NeutralFollowers-Favor\nSources-FavorFigure 4: Temporal evolution of mean follower and followee (source) counts split by an agent’s stance for\n20-run averages of different scenarios/simulations. When an agent converts to a stance at a time-step, its\nfollower and source counts are included in the average for the new stance and excluded for the old stance.\nSubplots do not share a common range for the y-axis due to significant variance across scenarios. Descriptions\nof the scenarios are found in Table 1 and in Section 5.\n30Figure 5: Embedded bots scenario. Each edge arrow represents the direction of information flow, i.e., the\ninverse of a directed follower-followee relationship, for a simulation with 100 agents in conditions equivalent\nto Sim 13 (Table 1). In standard graph notation for an edge/link ( u, v),uis a follower of v, so the follower-\nfollowee relationship is u→vbut the information flow is reversed, u←v. The image on the left shows\nthe network at simulation start and has nodes sized according to the ratio of followers to followees. The\nimage on the right shows the graph at simulation end and nodes are sized according to the average ratio\nof likes to replies received by the agent’s tweets1\nNPi=1\nN#likes i\n#repliesi, with higher values indicating that the\nagent’s tweets tend to have a perception of being well-received. As likes and replies are accumulated over the\ncourse of a simulation, sizing the nodes based on engagement metrics is only possible post-simulation. The\nedge/link widths for the image on the left has a fixed value, while those on the right are based on stance and\nthemes compatibility between two agents. Stance is indicated by each node’s color. Agency is indicated in\nthe edge/link color and the node’s border color. Blue nodes are the opposition, yellownodes are neutral,\nandgreen nodes are supporters. A redborder indicates a bot while a white/invisible border indicates a\nhuman.\n31Figure 6: Embedded bots scenario. From the same simulation run as Figure 5. The image on the left\nshows whose tweets a node has seen at the end of the simulation. The image on the right shows whose\ntweets contributed to a node’s stance switching at the end of the simulation. For both images, the edge\nwidths correspond to the frequency of the seen agent’s (left)/the contributor’s (right) appearance. Not all\nseen tweets led to stance conversion, hence the difference. The color of each node indicates its stance while\nthe color of the node’s edges indicates agency. Blue nodes are the opposition, yellow nodes are neutral,\nandgreen nodes are supporters. A redborder indicates a bot while a white/invisible border indicates\na human. The edge/link colors distinguish between influence originating from bots, indicated in red, and\nfrom humans, in black .\n32Figure 7: Peripheral bots scenario. Information flow for a simulation with 100 agents with all bots backing\nthe negative stance (no equivalent in Table 1, closest is Sim 6). This scenario differs from Sim 13 in Figures 5\nand 6 only in that each bot starts out in the periphery/fringes with a single outgoing edge (follower-followee)\nto a random human node instead of being embedded naturally in the network. This more challenging\nstarting condition resulted in the non-bot-backed dominant opinion maintaining its dominance, although\nbots managed to convert two nodes in the fringes. Legend is the same as Figure 5.\n33Figure 8: Peripheral bots scenario. From the same simulation run as Figure 7. Left image shows how often\na different user’s tweets were seen by an agent while right image shows how often a different user’s tweets\nconverted an agent’s stance. Legend is the same as Figure 6.\nFigure 9: Each box in the heatmap represents the stance of an agent at a particular time-step. The evolution\nof an agent’s stance can be tracked by following a column from top to bottom. Agents are sorted by their\ninitial stances, leftmost are negative agents and rightmost are positive, with neutrals in between. Rapid\nconversion at the early stage in the embedded scenario has rapidly eliminated neutral agents, hence their\nseeming absence. Data are from the same simulation runs as Figures 5–8.\n346.2 Impact of bots and bot ratios\nGiven that the distribution of roles and action frequencies among bots do not differ between the stances (i.e., bots all\nhave the same role and frequency regardless of the stance that they back), the ratio of bots between the stances has a\ndrastic impact on the temporal evolution of the stances, which can be seen in their impact on polarization in (Section\n6.1). If the stance ratio of bots follows that of humans, then the bots backing the majority stance would suppress the\nbots backing the minority stances, reproducing a dynamic very similar to the scenario where only humans presents\nwhile bots are absent. The bots’ stance ratio must deviate from that of the humans to cause disequilibrium.\nThe co-opetitive relationship between the bots backing a minority polar stance (negative) and those backing a\nminority midpoint stance (neutral) benefits the midpoint stance the most. Across all scenarios in which the two\nminority stances are backed by an equal number of bots (Sims 1–9, 14, and 15), the number of neutral agents is\nalways greater than that of negative agents. We suspect that this is due to the neutral agents’ stance distance being\nthe shortest for both polar stances (except for the distance between a polar stance and itself). Once neutral agents\ngain a critical mass through conversion, they have the numbers to peer pressure other agents in the network to adopt\ntheir opinion. But the strength of neutral agents is also their greatest vulnerability. Without agents immune to stance\nconversion, i.e., bots, in their midst, the group of neutral agents collapses quickly as they are the easiest targets to be\nconverted by the majority stance (which does not need to be bot-backed) and the bot-backed minority stance. This\nphenomenon is evident in Sims 12, 13, and 16.\nDepending on the info ops goal of the bots’ operators, one mixture of bots can be more optimal than the other.\nIf the goal is to maximize the number of agents holding a stance that is the polar opposite of the majority, all bot\nresources should be devoted to backing that stance, as shown by the results of Sim 13. The cost of not having neutral\nbots is a slight inefficiency in reducing the number of people holding the majority stance as seen in the greater number\nof agents being positive at simulation end in Sim 13 compared to Sim 1. However, if one intends to purely reduce\nthe share of the majority stance, Sims 1 and 2 show that having an equal number of negative and neutral bots is the\nmost effective. The neutral agents’ smaller stance distance with positive agents makes them more persuasive. A side\neffect of a bot operator employing the tactic of mixed neutral and negative bots is that the population of negative\nagents is “cannibalized”, a consequence of the bots in our simulations acting independently to avoid being seen as\na centrally-directed, coordinated info ops. In cases where the positive stance is also backed by bots, using a mix of\nneutral and negative bots to counter the positive bots (a pro : neutral : con ratio of 2 : 2 : 1 in Sim 15) still appears\nmarginally better than relying purely on negative bots (4 : 0 : 1 in Sim 16).\nIf the goal is to drive polarization, bots can be assigned to both ends of the pole at equal ratios (Sim 12) or\nwith a ratio that favors the minority polar stance (Sim 16). Without unwavering advocacy for neutrality, neutral\nagents quickly dwindle to nothing, as polar opinions compete to attract users to their stance. In situations where\nthere are motivated neutral agents, they must be outnumbered to be overpowered. As Sim 10 shows, if the bots’\nstarting stance distribution matches that of the humans from the March mask-wearing dataset, which implies that\nthe numbers of bots representing the pro and neutral camps are roughly equivalent in most runs, the polar minority\nstance, negative, diminishes in size far more rapidly than the midpoint minority stance, neutral.\nFor stances starting out from a disadvantaged position (negative and neutral at 13.9% and 12.9% of the pop-\nulation), Sims 15 and 16 show that the combined number of bots backing the non-majority stances requires an\napproximate 4-to-1 numerical advantage over the bots backing the majority stance in order to reach influence parity,\nassuming bots for all stances are similarly capable (same roles and activity levels). Should bots only back the polar\nstances in equal numbers while neutral bots are absent, the majority stance maintains and extends its dominance,\nnearly eliminating neutrals but is unable to penetrate the enclave of the non-neutral minority stance (Sim 12). The\nsource of the negative agents’ resilience against conversion can potentially be explained by Figure 4, where the follower\nand followee counts for the bot-backed negative stance in Sim 12 remained steady as the simulation progresses (so did\nthe counts for the bot-backed positive stance). The neutral stance which lacked bot backing showed greater volatility\nand trended downwards. Bots backing only one minority stance against the onslaught of the majority stance, which\ncan be seen in Sim 13, also stabilizes the follower and followee counts of the minority stance. Bot-backing may be\ncapable of creating echo chambers for minority stances.\nAlthough echo chambers/filter bubbles are terrible for the purpose of gaining new recruits as the Islamic State’s\nexperience has shown (Section 6.3), echo chambers are excellent for the reinforcement and consolidation of opinions\n[128]. A bot’s broadcasted messages can reach people sharing the same stance, allowing bots to drive stance retention,\ni.e., keeping users with the same stance from being lost to other stances because those users are exposed to messages\ncontaining alternative stances. Bots play much of the same role as stubborn users in other information diffusion\nmodels; their unchanging stance making them ideal as the resilient core of a movement. Some have attributed\nTrump’s electoral success to the presence of botnets “spreading pro-Trump content in overwhelming numbers” and\n“digital echo chambers where users see content and posts that agree only with their preexisting beliefs” [265]. Using\nbots to create echo chambers may even escape alerting social media monitors as certain platforms are already inclined\ntowards echo chamber formation, with researchers finding that “information diffusion is biased towards individuals\n35who share a similar leaning in some social media, namely Twitter and Facebook” compared to Reddit and Gab [254].\nWhen great political orators across the entire American political spectrum, e.g., Hillary Clinton [266] and George\nW. Bush, expressed sentiments along the lines of “you are either with us, or against us”, that there is no middle\nground, their words contain a kernel of truth. From the perspective of eroding the support base of a polar opinion\n(e.g., pro-war), the actions taken by neutrals and the opposition (e.g., situationally pro-war/war-ambivalent and\nanti-war) to draw people into supporting their respective stances have the same effect — erosion. All our simulations\nfeaturing a pro : neutral : con stance ratio at 1 : 1 : 0 for bots provide evidence that support for the positive stance is\ninevitably eroded (Sims 6 and 7 are not exceptions as they are just examples of bots being ineffectual, as discussed\nin Section 6.5). In other words, if a predominant force supports one pole of a polarized issue, appealing to others\nnot to pick either side can make one a collaborator for the “enemy” (supporters of the opposite pole). The slightest\ndeviation from full, enthusiastic support of a stance is indeed “treachery”.\n6.3 Stance conversion and retention strategies\nRepetition is the key to success for info ops. Repetition in this case means being extremely active in disseminating\nand boosting content. If the action frequency of the bots is cut in half, from 134 actions per week to 66, as seen in\nSim 1 vs. Sim 3, then bots are substantially less effective at promoting minority stances against a non-bot-backed\nmajority stance. This reduction in influence is also evident in other scenarios where bots went from being highly\nactive to adopting activity levels that mimic the human distribution (low activity and medium activity users make\nup 67.4% and 22.2% of the human population respectively) which can be seen in Sim 1 vs. Sims 8 and 10. Our\nobservation that repetition is important has support, e.g., simulations in [76] which showed that high energy agents\ncontributed towards the prevalence of extreme opinions (polarization). Another study demonstrated that repetition\ncan turn into influence: a bot in a social media experiment that had gained popularity simply by periodically visiting\nothers’ profile pages later recommended accounts for users to follow and succeeded in convincing a majority of them\nto take its recommendation [235].\nNot all forms of repetitive action elicit the same responses from the network. Sims 4 and 5 show that bots\nwho favored liking or favored retweeting over all other actions are unable to convert as many users to their stance\ncompared to bots which favored replying (Sim 1) despite having the same high activity level. In the case of Sim 5,\nthe bots were only able to stop the majority opinion from gaining ground. There are several factors behind these\noutcomes. In a situation where the bots represent the minority stance, there are a greater number of tweets that\nexpress the majority stance and consequently more targets for negatory replies. On the other hand, bots that favor\nliking and retweeting, which are affirmatory actions, are likely to have run out of minority stance tweets to boost.\nAs a small percentage of the population, bots are unlikely to occupy a majority of high-degree nodes, limiting the\nreach of their messages. Replier bots can sidestep the lack of an audience by borrowing the audience of others as a\nsmall portion of replies is seen by the followers of the parent tweet. Retweets, however, are only seen by the author\nof the parent tweet (and the bot’s own followers). Liker bots do not have to find an audience as they are liking other\nusers’ tweets that already have audiences, namely those other users’ followers, but the reach of liker bots is largely\nconfined to these audiences as they seldom invade the conversations of other users compared to Replier bots. Since\nminority stances are minorities population-wise and Liker bots who back a minority stance will favor liking tweets\nadvocating the minority stance, the audience of tweets liked by Liker bots are small too, although not as small as\nthat of Retweeter bots.\nTargeting vulnerable users — friendless people, i.e., having a low number of followees — for replies can result in a\nsimilar number of stance conversions as the alternative tactic of finding the largest platform to shout from (replying\nto tweets with high engagement), as demonstrated by Sim 1 vs. Sim 2. Also, in terms of decreasing the number of\npeople holding the majority stance, both Sim 1 and Sim 2 are the most effective of all the tested tactics (Sim 9 will\nbe discussed in Section 6.4). Our results are in accord with real-world observations. Bots finding success by targeting\nvulnerable users bears some similarities to how “extremist” groups have historically targeted loners, marginalized\nand ostracized groups, and other societal outcasts for recruitment/radicalization owing to these people’s greater\nreceptiveness towards extremist messaging [267, 268]. The tactic of maximizing audience reach is also favored by\nextremist groups, as seen in the Islamic State (IS) “fighting for its propaganda to appear on search indexes and\ncontinued to launch ‘media invasion raids’ to gain an even short-lived visibility on mainstream social media” due\nto IS’s favored social media platform, Telegram, having “limited audience reach and echo-chamber effect, with the\nIS message being shared mostly among like-minded individuals” [269]. Another example are BTC scam bots [230],\nwhich are known to raid the replies sections of other tweets to maximize visibility.\n6.4 Themes conservation and propagation\nThemes are unique within our model in that agents do not guard against them unlike stances that have polarity,\nbut their transmission is entirely dependent on the successful adoption of polarizable stances. Themes can improve\n36the transmission probability of the stance that they are attached to if the targeted agent shares the same themes.\nThemes can therefore be thought of as having a symbiotic relationship with stances.\nWe tested a naive tactic for improving the propagation of themes — saturation. By making all the bots conserve\na particular theme (Theme 7), we aim to see if themes can spread beyond the bots themselves and into the human\npopulation. Conservation involves attaching the conserved theme(s) to every outgoing message from bots and also\ngiving the conserved theme(s) a permanent place in a bot’s set of themes. The results from Sim 9 show that not\nonly are bots capable of making half the population of the simulation be infected by Theme 7, they were able to\naccomplish it without causing any appreciable change in the final distribution of the stances when compared to Sim\n1, i.e., the bots still experienced similar success in stance propagation while conserving a theme. Because the number\nof agents holding the conserved stance, Theme 7, exceeds the bot population and an agent cannot hold duplicates in\nits theme set, we can conclude that the bots have succeeded in pushing the conserved theme onto human agents.\nEven without an explicit theme conservation tactic, the bots’ greedy strategy of keeping half of their own themes\nwhenever they survey tweets for themes to adopt ensures that minority themes are given a boost whenever bots are\ninvolved in a simulation (Sims 1–16, except the humans-only Sim 11). In cases where bots failed to propagate their\nbacked stances because the bots’ starting positions are in the periphery of the network (Sims 6 and 7), the bots are\nstill capable of elevating the presence of the initially obscure Theme 7. Bots can also suppress the growth of popular\nthemes or even cause their decline, which can be observed by tracking the counts for the first and second highest\nfrequency themes, Themes 3 and 2 across the simulations. The exceptions are when bots are exiled to the fringes,\nas in Sims 6 and 7, when bot activity levels are low, as in Sims 8 and 10, or when bots are completely absent, as in\nSim 11, where a normative rich-gets-richer effect is at play where themes that were already popular mostly grew in\npopularity as the simulation progresses.\nPrior to obtaining our simulation results, our concern was that themes not matching fully will hamper stance\nadoption and therefore theme adoption; theme compatibility is ct=|tuser∩ttweet|/|tuser∪ttweet|, so deliberately in-\ncluding a theme that is virtually absent in the entire population would ensure that during the early phase of the\nsimulation, the denominator of the theme compatibility equation will almost always be increased by one even in cases\nwhere all the non-conserved themes in a tweet matches with a user’s theme. However, as the results from Sim 9\ndemonstrate, this is not a factor, at least not when the number of themes to be conserved is just one. The outcome\nmight be different if more than one theme has to be conserved, as the reduced presence of compatible themes may\nbe insufficient to compensate for the incompatibility arising from multiple conserved themes.\nWe theorize that engagement metrics, a high volume of replies representing a stance, and other effects that are\nwithin the ability of the bots in the simulation to manipulate are sufficient to overcome incompatibility in themes\nduring stance adoption. Once a critical mass of agents are infected with the conserved theme, a synergistic effect\nbegins to take hold. Human agents who have yet to be infected by the conserved theme find that their tweets are\nless effective at changing the stances of those who are infected with the conserved theme. This enables the number\nof agents with the conserved theme to grow without much interruption. These agents are also likely to share the\nsame stance(s) as the bots because the conserved theme is almost exclusively possessed by the bots at simulation\nstart, so acquiring that theme implies that the bots’ stance(s) has infected the agent, at least in the early phase\nof the simulation. Once agents with the conserved theme become the majority, they can peer pressure others into\naccepting their stance. A theme can be viewed as performing a function similar to “investments” (sunk costs) that\nmakes “betraying”/leaving “extremist” groups [270, 271], armed forces [272], religions [273], and currency standards\n[274, 275] difficult.\nBots being able to push a single theme without sacrificing progress on stance conversion, as results from Sim 9\ndemonstrated, is a non-trivial finding. There are implications for actors intending to conduct separate info ops on\nmore than one issue. Such actors would ideally wish to succeed on all fronts without having to decide which info ops\nare the most important ones while making trade-offs and compromises for the rest. Our results show that not only\ncan balancing acts be avoided, the actors can design an info ops in ways that can improve the outcomes of other\nrelated info ops. We illustrate this with an example concerning the issue of mask-wearing.\nUnder the guise of facilitating debate between the supporting, neutral, and opposing factions of mask-wearing,\na motivated actor participating in more than one side of the debate via bots that always frame their messages with\na theme of liberty (personal/individual freedom versus communal responsibilities and interests) is ensuring that the\ntheme over time becomes the basis around which the issue is discussed as an increasing number of agents are infected\nby the theme. This mindset caused by the theme-infection can potentially be carried over into other issues, e.g.,\nseatbelts, drunk driving, and climate change, causing the infected agents to frame those issues in terms of liberty.\nThis is similar to the belief in the field of pedagogy that thinking habits can be taught and can be carried over across\ndifferent situations (e.g., critical thinking [276], thinking like an engineer [277], fostering creativity in children [278]),\nexposing people who think in a certain way to cognitive blind spots [279] due to “belief bias and the magnitude\nof framing effects” [280]. When others attempt to persuade these theme-infected agents to, for instance, not drink\nand drive by presenting arguments framed only in authoritarian terms (e.g., driving under the influence is against\nthe law) or safety terms (e.g., drinking and driving often kills the driver), they would have difficulty dislodging the\n37agent’s stance.\n6.5 Embedded vs. peripheral bots\nFigures 5, 6, 7, and 8 are visual illustrations of the differences between bots starting out embedded naturally within\na network and bots starting out as outsiders with only one outgoing follower-followee link (i.e., the bots’ tweets that\nare not replies will not be seen by anyone in the network). This challenging start is meant to mimic the “time zero\non the periphery” with “a single link to a random node” start for bot disinformation maneuvers in [81]. Two other\nscenarios featuring bots in the periphery are Sims 6 and 7 in Table 1. Although only 100 agents are used for plotting\nthese graphs, the embedded scenario is representative of the 1,000 agent simulation results in Table 1 for Sim 13.\nThe isolated/fringe scenario with 100 agents has no equivalent in Table 1, but it can be considered a variant of Sim\n6 where the stance ratio for bots is 1 : 0 : 0 instead of 1 : 1 : 0. The influence of bots is easier to see visually if the\nbots all belonged to one stance, hence the chosen configurations.\nEmbedded bots (Sim 13 and Figures 5 and 6) succeeded in converting a majority of the network population to its\nstance but the bots that started out in the network’s fringes (Sims 6 and 7, and Figures 7 and 8) could only convert\nother users who also exist on the fringes. Cut off from the influence of negative bots existing on the periphery,\nthe positive human agents in the central clusters rapidly converted the few neutral and negative human agents into\npositive agents (Figures 7 and 8, and Sims 6 and 7). Although bots did not actively initiate the severing of links\n— they start the simulation already isolated from the central clusters — the peripheral bots scenarios serve as an\nin situ demonstration of the effects of a community (central clusters) shielding itself from outside influence (bots).\nThe results from these simulations of ours are in accord with the model in [281] which produced predictions of social\ninfluence (homophily in opinion) and unfollowing being drivers behind the emergence of echo chambers, consistent\nwith the empirical Twitter data they collected.\nThe fierceness of ideological competition is evident from comparing both halves of Figure 6, which represents the\nembedded bots scenario. While a large share of outgoing messages were read by users in the network, as indicated by\nthe thick lines of the edges in the graph on the left, the messages that convinced another user to switch their stances\nare a smaller proportion of those that have been read, indicated by the thinner influence lines connecting nodes on\nthe right. The same observation applies even more so to the peripheral bots scenario in Figure 8. Bot messages lack\nthe consistent persuasiveness of more centrally located humans, as the bots’ red influence lines are comparatively\nthin next to the thick black influence lines emanating from humans. One shared trait of the winning sides in both\nthe embedded and fringe cases is that their tweets receive a lot of affirmation, i.e., likes received outnumbered the\nreplies, indicated by their greater node size in the right halves of Figure 5 (blue negative agents) and Figure 7 (green\npositive agents).\nFigure 9 shows that stance conversion happens early in both the embedded and fringe scenarios. While there\nare occasional bouts of rapid stance switches, most agents tend to hold a consistent stance for long periods of time.\nIn the fringe scenario, there is a greater occurrence of positive agents converting to neutrality for short stretches of\ntime in the early phase of the simulation. We speculate that this is because of negative bots having numbers that\nare sufficient to dampen positive tweets with their actions but are insufficient to boost negative tweets past the point\nwhere their stance incompatibility with positive agents is minimal, leaving neutrality to benefit from the struggle10.\nThe outcomes for Sim 6 and Sim 7 in Table 1 provide support for this observation in Figure 9. Despite the absolute\ncounts at simulation end dropping for both negative and neutral, neutral has a higher count relative to negative\ndespite starting with a comparatively lower count.\nThe peripheral bots’ tactic of targeting low-followee accounts for replying and quote-tweeting is unlikely to be\nthe cause of the bots’ failure in combating the majority stance, because replying and quote-tweeting retweeting is\nlimited to once each per tweet, so bots would switch to other targets once they have hit their interaction quota with\n10An example with a positive agent undergoing stance conversion after having read a tweet each from negative, neutral, and\npositive stances: Recall that a stance’s conversion score wris made up of two components, engagement rgand compatibility\nrc,wr=m−1rg+mrc. The m−1andmterms are weights used to de-emphasize engagement and emphasize compatibility,\nrespectively. Let themes be ignored in the compatibility calculation and assume that the two components will notbe weighted\nwhen summed. Assume that in the absence of bot-backing, the three tweets receive zero engagement. Assume that in the\npresence of bot-backing for the negative stance, the backing involves only actions that impact the engagement component of\nthe conversion weight — the positive tweet receiving one reply (assume that the reply was not seen by the agent experiencing\nthis conversion event). For the non-bot backed case, rg-con=rg-neu=rg-pro= 0 and rc-con= 0, rc-neu= 0.33, rc-pro= 0.66.\nFor the bot-backed case, rcremains the same but rg-pos=−0.5 due to receiving a reply. Engagement components for negative\nand neutral remain the same, rg-con=rg-neu = 0, because they received no engagement. The conversion weights without\nbot-backing are wr-neg= 0, wr-neu= 0.5, wr-pro= 1 and with bot-backing, they are wr-neg= 0, wr-neu= 0.5, wr-pro= 0.16.\nIn percentages, the chances of a stance being selected, from negative to neutral to positive, are 0%, 33%, and 66% for the case\nwithout bot-backing, and are 0%, 76%, and 24% with bot-backing. By “attacking” positive tweets, the bots’ actions ended up\nmaking neutrality a more attractive option. In the actual simulations, the m−1andmweights are meant to ensure that the\ninfluence from engagement metrics does not easily overpower homophily for stance and themes.\n38agents on the fringes. For liking and retweeting, bots target tweets with high engagement written by agents with high\nfollower counts. Incidentally, bots frequently liking other tweets to boost their infectivity appears to be a slightly\ninferior tactic compared to frequently replying to messages (Sim 6 vs. Sim 7). This difference between the Liker and\nReplier bots are less pronounced compared to the embedded scenarios (Sim 1 vs. Sim 4).\nThe difficulty faced by bots under the fringe scenario in overturning the predominant stance can perhaps be\nattributed to the follow and unfollow mechanism in Diluvsion . The probability for an agent to follow and unfollow\naccounts are based on data of user behavior on social media (Section 3.8.1) that likely was collected under “peacetime”,\ni.e., when a community is not vigorously debating a topic. Therefore, they may not reflect a greater tendency to\ncluster into echo chambers/hives during “wartime” by rapidly establishing social ties with like-minded users and\nsevering ties with non-believers. However, as we aim to ground our model in as much empirical data as possible and\nwe did not encounter data of heightened follow/unfollow behavior on “war footing”, we opted to keep probabilities\nlow. The model in [81], twitter sim, strongly incentivizes the creation of new links, which explains the difference\nbetween their simulation results and ours, where they demonstrated that bots are successful in backing a stance and in\nbridging two separate communities despite bots starting in the fringes. Under twitter sim, link creation probability\nfor humans is low at 5% but is higher than our model’s 0 .6%, the check for link creation is triggered every time a\nuser wakes instead of being a separate event that consumes a user’s action budget under our model (compounding\nthe higher link creation probability), and links in twitter simare never broken.\nFigure 4 shows the crippling effect of being deprived of an audience. Unlike other scenarios where the minority\nstances’ mean follower and followee counts are at parity with the majority stance’s at simulation start, they are\nonly about three-fourths of the majority’s in Sims 6 and 7. Once agents begin renouncing a minority stance for the\nmajority stance, the minority stance’s follower and followee counts begin experiencing a precipitous decline that they\nnever recover from.\nOur results here are corroborated by the details of real info ops on social media that have emerged (Section 2.2.6),\nshowing that great effort is invested in equipping many bots with extensive social ties, going as far as having real\nhumans work under a bot. No matter how persuasive a bot is constructed to be, its effect is limited if it does not have\na ready audience that is always exposed to the messages that the bot broadcasts. Bombarding the replies section\nof other users to persuade them and their neighbors into changing their stances cannot succeed if the bots are just\n10% of the population, as our simulations have shown. Resorting to other tactics to improve the performance of bots\nwithout increasing the number of bots, e.g., explicitly coordinating with other bots and increasing the frequency of\nactivity, runs the risk of the info ops being discovered.\n7 Limitations and future work\nAt the outset of this paper, we have stated that no model is perfect and that they are all reflections of some aspect\nof real life that the modelers wish to bring to attention. There is room for future work because we made several\nsimplifying assumptions in our model. For instance, we accounted for the bandwagon effect but did not implement\na mechanism for the spiral of silence. Our simulations start out with one dominant and two minority stances, which\ndo not cover scenarios where the distributions are balanced. Some other assumptions involved picking one side of an\nongoing debate (e.g., whether resistance increases with repeat infections) and not having the opportunity to explore\nthe alternative side in-depth in this paper.\nOne promising route for future investigation is to allow a reinforcement agent to control the bots. The reinforce-\nment learning agent AlphaStar has been demonstrated to perform at the Grandmaster level for the real-time strategy\n(RTS) game StarCraft II [282]; an info ops problem bears strong resemblance to an RTS game. Just as playing an RTS\ngame involves controlling groups of hundreds of units with specific roles (attackers, defenders, healers, etc.), so too can\nthe managing of an info ops on social media involve controlling hundreds of bots with specific roles (Repliers, Likers,\netc.). However, a central controller dictating the actions of multiple bots does violate the emergent/uncoordinated\ncoordination condition that this paper is predicated upon and could potentially make this proposed form of bot-driven\ninfo ops easier to detect, so an alternative scenario where there are multiple reinforcement agents that each control\none bot should also be explored.\nGreater heterogeneity in terms of bots should also be investigated. While we have varied the distribution of roles\nand activity levels, we have kept other aspects homogeneous throughout a simulation run, such as the process by which\na bot chooses a tweet to like. And when deciding how bots should act, e.g., the per-week frequency of actions such as\nretweeting, we have relied on pre-defined roles derived from empirical observations of Twitter user behavior. All this\nwas done to facilitate comparison between scenarios. There may be more optimal action distributions for maximizing\nstance propagation. Heterogeneity would also enable us to explore more complicated info ops scenarios. An example\nwould be explicit cooperation between neutral and opposition bots, where neutral bots would only target agents who\nare supporters for conversion to neutrality while opposition bots would target both neutral and supporter agents. In\nthis example, opposition bots would effectively function as a conversion pipeline for opposition bots. Compared to\n39opposition bots attempting to convert supporters into detractors, the lesser difference in stance between supporters\nand neutrals means that neutral bots would have an easier time converting supporters to neutrality. Once converted\nto neutrality, opposition bots can more easily convert the former supporters into agents of the opposition.\nAgent heterogeneity should include class and role transitions, e.g., deepening radicalization turning human agents\ninto bots and de-radicalization turning bots to humans. Bots switching roles as the info ops situation demands\nshould also be included. Role transition in an ABM for information diffusion has been explored before in [74] where\nradicalized patients become staunch anti-vaccine propagandists. Our model’s radicalization and de-radicalization\ninvolved only an agent switching stances. In addition, human and bot users entering and exiting the network should\nbe accounted for. The dynamics of a social network in which the ratio between bots and humans is ever shifting may\nprovide additional info ops insights.\n8 Conclusion\nIn this paper, we developed and empirically validated an agent-based information diffusion model for bot-driven\ninformation operations on social media, Diluvsion , that incorporated crucial but oft-neglected real-world aspects of\ndiffusion. Chief among them is indirect influence, consisting of two forms, the exposure to a message’s engagement\nmetrics inducing bandwagon/conformity effects within a user and the ability to transmit information to another node\nin the absence of a social tie linking two nodes together. Other key innovations of the model are the designation of\nneutrality as an independent infection-capable stance instead of a transition point between the two discrete polar\nstances (opinions) and the introduction of the non-polarizable, homophily-inducing, infection-capable, and stance-\nsymbiotic themes as a counterpart to stances.\nThrough simulations, we demonstrated the different routes by which motivated agents, i.e., bots, can exploit\nthese aspects of information diffusion to manipulate the information environment. The simulations showed that\nhighly active Replier bots representing minority stances are capable of overturning the dominance of a non-bot-\nbacked majority stance, and that a naive saturation tactic can not only keep a minority theme, e.g., “conspiracy”,\nalive but also further its adoption over time without compromising stance propagation. Neutrality is also shown to be\nnon-neutral in its threat to the continued dominance of the majority stance. Overall, our model shows that info ops\noriginating from agents embedded in the network who are not counterbalanced by a similarly motivated oppositional\nforce can easily pollute the information environment and induce different outcomes, such as increased polarization.\nAlthough much of our paper has been steeped in the language of war, the model we introduced here can be\nused towards beneficial, pacific ends. Reducing polarization on social media through the introduction of bots that\nstubbornly advocate a neutral stance, for instance, can be studied through Diluvsion , in the tradition of the numerous\nworks that have been published on depolarization, e.g., [283, 284, 285, 286]. The assumption here is that polarization\nreduction is prosocial, an assumption that some may not agree with as they may believe that there is no peaceful\ncoexistence with certain opinions or ideas, e.g., Nazism. On the other hand, there are real-life examples — the US, for\none — of comfortable coexistence being possible between “extreme” ideologies and the ideals of liberal democracy, the\nbest governing system ever and which Churchill, the widely venerated British prime minister who presided over the\nBengal famine [287, 288, 289], wryly described as “the worst form of Government except for all those other forms that\nhave been tried from time to time”. The US never lost its global status (global as in encompassing only the “garden”\n[101]) as the shining beacon of democracy despite being the direct progenitor of Nazism via cherished American ideals\nsuch as Manifest Destiny [290, 291] and despite fascism resurging in modern-day America [292, 293, 294, 295].\nOr, perhaps, certain states, systems, ideas, etc., maintaining an untarnished reputation as a force for good are not\ndue to their inherent merits but due to having waged successful info ops campaigns, and maintaining that reputation\nin the age of social media requires active effort in maintaining an edge in tools for wargaming info ops on social\nmedia, in tools such as Diluvsion .\n9 Acknowledgments\nThis research is supported by NSERC Discovery Grants (RGPIN-2018-03872), NSERC CREATE Grants (CREATE-\n554764-2021) and Canada Research Chairs Program (CRC-2019-00041).\nReferences\n[1] Laura Courchesne, Julia Ilhardt, and Jacob N. Shapiro. 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Moreover , we investigate how the notion of the multiplier \ncan help us in macroeconomic analysis of capital and investment dynamics and in \nunderstanding some general principles of capital market equilibrium.  Using the concept \nof this multiplier, we build a macroeconomic model of capital market dynamics which \nis consistent with the implications of classical models and with the market equilibrium \ncondition but gives additional  quantitative and qualitative predictions regarding the \ndynamics of shares of investment into the production of consumer goods and t he \nproduction of means of production.  The i nvestment volume is modeled as a function \nof the multiplier: investments adjust when the value of the multiplier fluctuates around \nits equilibrium value of one. In addition, we suggest p ossible connections between the \ninvestment volume and the multiplier value in the form of differential equations. We \nalso present the formula for the rate of growth of the multiplier.  Independently of the \nimplications of capital market dynamics models, the formula for the multiplier  itself \ncan be applied for the evaluation of the present value of capital or the estimation of the \nmacroeconomic impact of changes in investment volumes.  Our findings show that b oth \nthe exponential and hyperbolic discounting in combination with empirical evidence \navailable lead to the value of the multiplier that is close to one. \n \nKEYWORDS : consumer, multiplier, investment, present value, discounting, capital \nmarket  \n \nJEL CLASSIFICATION : E22, E44, G10 \n \n \n \n \n \n \n \n \n 3 \n 1. INTRODUCTION  \n \nThe p resent value method of asset pricing is one of the most generally accepted and \nwidely used in finance and macroeconomics  (see e.g. Berk  and Van Binsbergen , 2016; \nor Pohl, Schmedders , Wilms , 2018) . For example, Gallo (2014) provides  a \ncomprehensive review of the state of the subject and lists common assumptions of the \npresent value calculation process which we also use.  Another review is made by \nFernandez  (2019).  Generality and applicability of the net present value criterion for \nsimilar problems are shown by Graham and Harvey  (2001 ) or Brounen et al.  (2004). \nAlthough our analysis is macroeconomic in its essence, it is crucial to understand the \nrole the net present value ( NPV ) plays on a level of a single firm, from the standpoint \nof corporate finance. This is investigated by Pasqual, Padilla,  and Jadotte  (2013). In \naddition, Bierman and Smidt (2012)  studied capital budgeting from various points of \nview, contributing to the understanding of the optimality criteria for investment \ndecision. The multiplier we derive can be applied both in the world of efficient markets \nand outside of it. Actually, the multiplier itself can be applied for testing the efficient \nmarket hypothesis. However, the efficient market hypothesis is needed to make som e \nfurther conclusions out of the multiplier formula and in order to investigate equilibrium \nvalues of multiplier and its equilibrium behavior. An overview of the efficient market \nhypothesis relevant to this task is made by Degutis  and Novickytė (2014), as well as \nby Gabriela ğiĠan, A. (2015).   \nIn this paper , we present a new formula for the estimation  of the present value of a unit \nof capital  on the macroeconomic level  and make some implications regarding its \nparameters and consistency with standard macroeconomics theory. The formula is \nderived as a sum of eternal cash flows generated from the initial investment, assuming \nthat one constant share of investment flows into the production of means of production, \nwhereas another constant share of investment flows into the production of consumer \ngoods. Technically, it can be described as the present value of some multiplier M. This \nmultiplier, in turn, calculates the total futu re value of all consumer goods that will be \ncreated by the initial investment of one unit of capital.  \nWe derive the value of future consumer goods multiplier based on the analysis of the \nsequence of consumer goods produced in the future by one unit of investment and \ninvestigate parameters it depends on. Then, we investigate how the notion of the \nmultiplier can help us i n macroeconomic analysis of capital and investment dynamics \nand in understanding some general principles of capital market equilibrium. Namely, \nit is shown that analysis of the multiplier function leads to theoretically relevant results \nand t hat changes in capital stock can be described as a result of the adjustment of the \nmultiplier to the value of one. Being consistent with modern macroeconomic theory, 4 \n the model brings new highlights to the problem of capital accumulation. The multiplier \nis also investigated in the connection with the intertemporal choice theory.  The main \nresearch question of this paper is the following one : how can we measure the present \nvalue of the average capital unit invested?  \nThen, the dynamical behavior of the multiplier formula is analyzed in order to make a \nprediction regarding the investment and capital market dynamics. It is shown that the \nnotion of the present value of future consumer goods multiplier can be helpful for \nunderstanding the impact of discount rate and marginal productivity of capital on \ncapital markets and investment volumes.  \nThis paper is structured as follows: Section 2 provides a comprehensive literature \nreview. Section 3 outlines the research methodology describing the materials and \nmethods , as well as the model itself . Section 4 provides the main results and outcomes. \nSection 5 summarizes our main findings with a discussion. Finally, section 6 list s the \nmain findings and implications.  \n \n2. LITERATURE REVIEW  \n \nIt is a well -known fact that t he issue  of present value is deeply investigated in modern \neconomic theory. The difference between efficient and inefficient capital markets for \nthe multiplier theory is very important , since in the case of efficient capital markets we \nprove that the multiplier is consistent with market equilibrium and fluctuates at the \nvalue of one. In the case of inefficient capital markets, the value of the multiplier may \nnot adjust to one, and the dynamic models will be less applicable. For the study of the \ndiscussion on market efficiency, we refer to Akbas, Armstrong, Sorescu , \nSubrahmanyam  (2016) , to LeRoy and Lansing (2016) , as well as to Williams, \nDobelman, ( 2017) . Various empirical case studies deliver important insights as well \n(Boya, 2017 ; Kumar, Raman, 2016 ; or Ma, 2017 ). \nAs we refer to the concept of marginal productivity of capital in the study, it is \nimportant to define  it properly  (Fuller, 2013) . Marginal productivity of capital theory \nis tested empirically by Biewen  and Weiser  (2014) . The limitations of the theory are \nstudied by Moseley (2012).  \nIntertemporal choice is another area that relates directly to the problem we analyze. \nHolistically the problem of intertemporal choice was studied by Loewenstein, R ead, \nBaumeister (2003 ), Chabris, Laibson and Schuldt (2010) . They  enlighten  economic, \npsychological and neurobiological approaches to the question  of what is extremely \nhelpful in applying different models of time discounting and understanding \npsychological of the phenomenon itself.  Psychological aspects of discounting are \nstudied by Matta, Gonçalves  and Bizarro  (2012).  5 \n The w orks dedicated to the macroeconomic theory of investment and capital are of \nhuge importance for our problem , starting from such classic authors as Fisher (1906)  \nand ending with Garisson (2001) , Fuster, Hebert, Laibson (2012).  The r ecursive \nmacroeconomic theory brings similar ideas of capital dynamics (Ljungqvist, Sargent, \n2018).   \nOf course, the very concept of multiplier dates back to Keynes (1936).  We build the \npresent value of capital multiplier following the common notion of the macroeconomic \nmultiplier, which is developed in the works of Gnos and Rochon (2008), Ono (2011), \nCogan, Cwik, Taylor , and Wieland (2010),  Westerhoff (2006), Robinson (2006), \nCurrie (1983), Smithies  (1948).  \nOnce derived, the multiplier is applied to the analysis of macroeconomic problems of \ncapital accumulation, investment and interest rates dynamics, and financial markets. \nHamouda  (2011) delivers a survey of modern Keynesian approaches to the problems \nof investment on the macro level , which is important for the analysis of the connection \nof multiplier with interest rates dynamics. However, the multiplier itself is constructed \nmore in the spirit of neoclassical models, where the concept of marginal productivit y \nand long -term equilibrium plays an important role.  The equilibrium concepts we use \nare based on the results of Caselli  and Feyrer (2007).  \nAlthough the applicability and efficiency of the notion of the Keynesian multiplier are \nin general questionable ( Carpenter  and Demiralp, 2012 ; Castelnuovo  and Lim, 2019 ), \nit does not affect the applicability of the multiplier we present, since it is not a demand -\nside multiplier and all demand -side critique cannot be applied. The research in the field, \nhowever, focuses more on fiscal, rather than investment, multipliers ( Spilimbergo, \nSchindler, Symansky, 2009 , Dupor  and Guerrero,  2017 ). Whereas we model the \ninvestment demand, the derivation of the multiplier formula is conducted in the supply -\nside spirit ( Vollet, Aubert, Frère, Lépicier,  Truchet, 2018).   \nOur differential equations model s can be analyzed and compared in the context of \nmodern theories of capital dynamics. Although we do not study explicitly the \nimplications of the model for monetary and fiscal policy, such analysis can be \nconducted based on the frameworks described by Laopodis  (2013) , Borio  and Zhu \n(2012) , Bekaert, Hoerova  and Duca (2013) , Arrow  and Kruz (2013) , Bhattarai  and \nTrzeciakiewicz (2017).  Models constructed based on the dynamic stochastic general \nequilibrium ( Khramov,  2012 ; Herbst  and Schorfheide, 2015 ) may have a similar form.  \nThe boundaries of applicability of the capital dynamics model presented in the paper \ncan be estimated by the study of the model under the conditions of inefficient capital \nmarkets. Such capital market imperfections are listed, for example, by Rotheim (2013).  \n \n3. METHODOLOGY AND MODEL  6 \n  \nThe present value of future consumer goods multiplier (later denoted as M) is \nconstructed technically similar to common Keynesian multipliers (calculating the value \nof infinite geometric sequence), but economically its value is created on supply, not \ndemand, side.  In this sense, it is rather neoclassical, than  a Keynesian, multiplier.  \nLet us define variables: K – initial capital investment;  p – marginal productivity of \ncapital;  n - the rate of depreciation ; a=1-n; c – the share of investment in the production \nof consumer goods; i – the share of investment in the production of  means of \nproduction; R – discount rate; r=1/(1+R ) (discount factor) .  \nThe a mount of capital K =1 is invested in year 0.  It is divided in to 2 shares – one goes \ninto consumer goods production and another in production of means of production.  \nThus, we can buil d the following structure of goods produced in each year : \n \n \n{    𝑡1: (𝑐𝑝)+𝑖𝑝\n𝑡2: (𝑐𝑝𝑎+𝑐𝑖𝑝2)+𝑖𝑝𝑎+𝑖2𝑝2\n𝑡3: (𝑐𝑝𝑎2+𝑐𝑖𝑝2𝑎+𝑐𝑝2𝑎𝑖+𝑐𝑖2𝑝3)+𝑖𝑝𝑎2+𝑖2𝑝2𝑎+𝑖2𝑝2𝑎+𝑖3𝑝3\n…\n𝑡𝑛: (𝑐𝑝𝑎𝑛+⋯+𝑐𝑖𝑛𝑝𝑛+1)+𝑖𝑝𝑎𝑛+⋯+𝑖𝑛+1𝑝𝑛+1\n…     (1) \n \nHowever, one should note that we are interested only in consumer goods.  They are \nhighlighted by brackets . We can consider them separately. Hence, the total amount of \nconsumer goods produced in the future by 1 unit of capital investment is following:  \n \n{      𝑐𝑝\n+\n𝑐𝑝𝑎+𝑐𝑖𝑝2\n+\n𝑐𝑝𝑎2+𝑐𝑖𝑝2𝑎+𝑐𝑝2𝑎𝑖+𝑐𝑖2𝑝3\n+\n𝑐𝑝𝑎3+𝑐𝑖𝑝2𝑎2+𝑐𝑝2𝑎2𝑖+𝑐𝑖2𝑝2𝑎+𝑐𝑝2𝑎2𝑖+𝑐𝑖2𝑝3𝑎+𝑐𝑝3𝑎𝑖2+𝑐𝑖3𝑝4\n+…\n+\n𝑐𝑝𝑎𝑛+⋯+𝑐𝑖𝑛𝑝𝑛+1\n+\n…      (2) \n \nWe can observe that the sequence of lin es builds geometric sequence :  \n \n𝑆𝑛+1=𝑆𝑛(𝑎+𝑖𝑝)          (3) \n \nwhere 𝑆𝑛 is line n in term (2).   \n 7 \n If 𝑎+𝑖𝑝<1, then the value of the term (2) is the sum  of infinitely decreasing \ngeometric sequence : \n \n𝑀=𝑐𝑝\n1−𝑎−𝑖𝑝           (4)  \n \nLet us call M future consumer goods multiplier.  The p resent v alue of future consumer \ngoods multiplier  can be thus presented as following :  \n𝑀𝑟=𝑐𝑝𝑟\n1−𝑟(𝑎+𝑖𝑝)=𝑐𝑝𝑟\n1−𝑟𝑎−𝑟𝑝+𝑟𝑝𝑐    (5) \n \nIt is natural  now to attempt to find extreme point s of the function 𝑀𝑟(𝑐): \n \n𝜕𝑀𝑟\n𝜕𝑐=𝑟𝑝−𝑟2𝑝2−𝑎𝑟2𝑝\n(1−𝑝𝑟−𝑎𝑟+𝑟𝑝𝑐)    (6) \n \nTerm (6) is never equal to  0. \n \nAs we know, expression 𝑀𝑟 is a hyperbolic function  with respect to c.  Thus,  since term \n𝑝𝑟 is always positive,  the local  optimum of the function 𝑀𝑟(𝑐) on the interval 𝑐∈\n[0;1] is:  \n \n𝑐∗={0,(1−𝑟𝑎−𝑟𝑝)<0\n(0;1),(1−𝑟𝑎−𝑟𝑝)=0\n1,(1−𝑟𝑎−𝑟𝑝)>0     (7) \n \nIt is definitely  true that in the real economy 𝑐∗𝜖(0;1). Then : \n \n𝑝=𝑅+𝑛    (8)  \n \nThe e mpirical validity of this expression can be tested.  Yet this expression  is just \nanother form of famous result MP=MC, which shows the theoretical integrity of the \ncalculations made.  \nIf expression (8) is true, th en formula (5) for multiplier 𝑀𝑟 is consistent with market \nequilibrium condition  for any possible values of parameters . Indeed, we know that \nunder the law of one price : \n \n𝑃𝑉(𝐾)≡𝐾∗𝑀𝑟=𝐾𝑐𝑝𝑟\n1−𝑟(𝑎+𝑖𝑝)=𝐾       (9) \n 8 \n Then : \n \n𝑀𝑟=𝑐𝑝𝑟\n1−𝑟(𝑎+𝑖𝑝)=1                    (10) \n \nIf the capital market is in equilibrium, 𝑀𝑟 should be equal to 1.  \n  \nIf 𝑝=𝑅+𝑛, then:  \n𝑀𝑟=𝑐𝑝𝑟\n1−𝑟(𝑎+𝑖𝑝)=𝑐(𝑅+𝑛)1\n1+𝑅\n1−1\n1+𝑅(1−𝑛+(1−𝑐)(𝑅+𝑛))=𝑐𝑅\n1+𝑅+𝑐𝑛\n1+𝑅\n1+𝑅\n1+𝑅−1−𝑛\n1+𝑅−(1−𝑐)(𝑅+𝑛)\n1+𝑅=𝑐𝑅+𝑐𝑛\n1+𝑅−1+𝑛−𝑅−𝑛+𝑐𝑅+𝑐𝑛=\n𝑐𝑅+𝑐𝑛\n𝑐𝑅+𝑐𝑛=1                     (11) \n \nIt turns out that (5) and (8) are equivalent in terms of market equilibrium definition.  \nCondition (1 0) follows directly from the equilibrium (8).  \nWe can rewrite expression  (7) in the following form:  \n \n𝑐∗={0,𝑝>𝑅+𝑛\n(0;1),𝑝=𝑅+𝑛\n1,𝑝<𝑅+𝑛               (12) \n \nIt means that it is optimal to invest in the production of consumer goods if 𝑝<𝑅+𝑛, \nand it is optimal to invest in the production of means of production if  𝑝>𝑅+𝑛. \nTerm ( 12) also tends to be justified if  we analyze the issue from the standpoint of  \nconsumer choice problem:  \n \n𝑚𝑎𝑥𝑈=𝐶+𝑀𝑟𝐾 𝑠.𝑡.𝑊=𝐶+𝐾              (13) \n \nwhere C is today consumption, W is today wealth.   \n \nIt is obvious that we get a corner solution.  If 𝑀𝑟>1, then K=W, if 𝑀𝑟<1, then K=0. \nAnd only if 𝑀𝑟=1, 𝐾∈(0;𝑊). Only  the third case is empirically relevant.  Let us \nassume that  p adjusts, so that (8) is true : \n \n𝑑𝐾\n𝑑𝑡=𝑝(𝐾,𝑡)−𝑅−𝑛                       (14) \n \nThis differential equation describes the net capital change. Then, gross capital change  \n(investment expenditure)  is: 9 \n  \n𝐼=𝑑𝐾𝑔\n𝑑𝑡=𝑑𝐾\n𝑑𝑡+𝑛=𝑝(𝐾,𝑡)−𝑅                 (15) \n \nIn this case , we will observe  investment volume fluctuations around the value  \n𝑝−1(𝑅+𝑛),𝑝−1 is the reverse function of p.  Let us  use the Cobb -Douglas function:  \n \n𝑌=𝐴𝐿𝑎𝐾𝑏              (16) \nThen,  for A,  L=const,  the solution of equation (1 4) is: \n \n𝐾=(𝑥→𝑥 𝐹12 \n (1,1\n𝑏−1,1+1\n𝑏−1,𝐴𝑏𝐾𝑏−1𝐿𝑎\n𝑛+𝑅)\n𝑛+𝑅)−1 (𝑐1−𝑡)              (17) \n \nwhere F is the hypergeometric function.  \n \nThe solution of equations (1 5) is corresponding:  \n \n𝐾=(𝑥→𝑥 𝐹12 \n (1,1\n𝑏−1,1+1\n𝑏−1,𝐴𝑏𝐾𝑏−1𝐿𝑎\n𝑅)\n𝑅)−1 (𝑐1−𝑡)              (18) \n \nBut for the long -term analysis , A and L should not be treated as constant s. Then, for \nthe solution of the equations , numerical methods can be applied.  \nLet us consider the case of h yperbolic discounting  (Grüne -Yanoff, Till, 2015 , \nHampton, Venkatraman, Olson, 2017 ). Then, the present value of future consumer \ngoods produced in period n is:  \n \n𝑆𝑛=𝑐𝑝(𝑎+𝑖𝑝)𝑛−1\n1+𝑘𝑛                      (19) \n \nFor (𝑎+𝑖𝑝)<1: \n   \n𝑀𝑟=𝑐𝑝\n𝑘𝜙(𝑎+𝑖𝑝,1,1+1\n𝑘)=𝑐𝑝\n𝑘𝜙(𝑎+𝑝−𝑝𝑐,1,1+1\n𝑘)              (20) \n \nwhere 𝜙 is Lerch zeta function  (Apostol, 2010) . The optimum then is the following:  \n \n𝜕𝑀𝑟\n𝜕𝑐=𝑝(−1\n𝑘−1)𝜙(𝑎+𝑝−𝑐𝑝,1,1+1\n𝑘)+1\n1−𝑎−𝑝+𝑝𝑐\n𝑎+𝑝−𝑝𝑐=0                            (21) \n 10 \n We can also consider the case of continuous -time:   \n \n𝑀𝑟=∫𝑐𝑝(𝑎+𝑖𝑝)𝑛−1\n1+𝑘𝑛𝑑𝑛=lim\n𝑛→∞𝑐𝑝(𝑎+𝑖𝑝)−𝑘+1\n𝑘𝐸𝑖((𝑘𝑛+1)ln(𝑎+𝑖𝑝)\n𝑘)\n𝑘∞\n0−𝑐𝑝(𝑎+𝑖𝑝)−𝑘+1\n𝑘𝐸𝑖(ln(𝑎+𝑖𝑝)\n𝑘)\n𝑘=\n∞                            (22) \n \nObviously, the case of continuous time is irrelevant as it leads to meaningless results.  \nIf some permanent  growth  of marginal productivity of capital is expected  (for example, \nit can be due to technological progress ), both i n the case s of exponential  and hyperbolic  \ndiscounting  the value of the multiplier approaches infinity.  But equation (8) should be \ntrue, wh ich leads to the compensation of the effect of the growth of p.  \nNevertheless, the discount rate  can adjust to the new level of marginal productivity of \ncapital only gradually (as soon as the market interest rate does).  It means that the \ngrowth of p  can lead to the situation when 𝑀𝑟 is always greater than 1. It would mean  \nthe flow of income from consumption to production:  \n \n{𝑑𝑅\n𝑑𝑡=𝑝−𝑅−𝑛\n𝑀𝑟=𝑐𝑝𝑟\n1−𝑟𝑎−𝑟𝑝+𝑟𝑝𝑐                 (23) \n \nIs it possible for 𝑀𝑟 to reach infinity?  If the growth of p is exponential, then yes, if \ncondition (24) is true : \n \n𝑅=𝛽𝑒−𝑡−𝑛\n𝑔∗𝑙𝑛𝑝+1−𝑔∗𝑛∗𝑙𝑛𝑝\n𝑔∗𝑙𝑛𝑝+1+𝑝𝑔𝑡\ng∗ln𝑝+1                 (24) \n \nwhere 𝛽 is some constant.  \n \nlim\n𝑡→∞(𝑝−𝑅)=lim\n𝑡→∞(𝑝𝑡−𝛽𝑒−𝑡+𝑛\n𝑔∗𝑙𝑛𝑝+1+𝑔∗𝑛∗𝑙𝑛𝑝\n𝑔∗𝑙𝑛𝑝+1−𝑝𝑔𝑡\ng∗ln𝑝+1)=∞             (25) \nlim\n𝑡→∞𝑀𝑟=lim𝑀𝑟=∞\n𝑝→∞\n𝑅→∞                   (26) \n \nYet the question, what does it mean, to have an “infinite” value , lies beyond the scope \nof the paper.  \nFormula ( 12) gives  strange  prediction that in the case of the infinite value of 𝑀𝑟 share \nof investment in means of production would be 1.  But why would people refuse from \na higher value in the near future, even if the total future value is “infinite”?  11 \n We observed now 2 options: either marginal productivity of capital or discount rate \nadjusts to hold the equilibrium. But what is more proba ble, they do it simultaneously. \nObviously , the most simple and evident way of that is:  \n \n{𝑑𝑅\n𝑑𝑡=𝑝−𝑅−𝑛\n𝑑𝑝\n𝑑𝑡=𝑅+𝑛−𝑝                   (27) \n \nNevertheless, it leads to the result that p always equals R+n and th ere are no changes \nin capital , which is obviously not true.  We may consider more complicated cases.  \nFirstly, we should note that investment depends on the value of the future goods value \nmultiplier (when the multiplier is greater than  1, it is reasonable to invest):  \n \n𝑑𝐾\n𝑑𝑡=𝐶(𝑀𝑟−1)                    (28) \n \nwhere 𝐶 is some parameter.  \n \nWe can also consider the change of the multiplier  in a short time interval:  \n \n∆𝑀𝑟=𝜕𝑀𝑟\n𝜕𝑝𝜕𝑝\n𝜕𝑡∆𝑡+𝜕𝑀𝑟\n𝜕𝑅𝜕𝑅\n𝜕𝑡∆𝑡                   (29) \n \nwhere: \n𝜕𝑀𝑟\n𝜕𝑝=𝑟(1−𝑎1\n1+𝑅)\n(1−𝑎1\n1+𝑅−𝑝1\n1+𝑅+1\n1+𝑅)2                  (30) \n𝜕𝑀𝑟\n𝜕𝑅=−𝑐𝑝\n(1−𝑎−𝑝+𝑟+𝑐𝑝)2                   (31) \n \nWe know that in the equilibrium : \n \n∆𝑀𝑟=0                     (32) \n \nhence: \n \n𝜕𝑀𝑟\n𝜕𝑝𝜕𝑝\n𝜕𝑡∆𝑡=−𝜕𝑀𝑟\n𝜕𝑅𝜕𝑅\n𝜕𝑡∆𝑡                   (33) \n 12 \n The formula (3 3) is the fundamental prediction of the model.  But 𝑐 can also be a \nvariable. Then : \n \n∆𝑀𝑟=𝜕𝑀𝑟\n𝜕𝑝𝜕𝑝\n𝜕𝑡∆𝑡+𝜕𝑀𝑟\n𝜕𝑅𝜕𝑅\n𝜕𝑡∆𝑡+𝜕𝑀𝑟\n𝜕𝑐𝜕𝑐\n𝜕𝑝𝜕𝑝\n𝜕𝑡∆𝑡+𝜕𝑀𝑟\n𝜕𝑐𝜕𝑐\n𝜕𝑅𝜕𝑅\n𝜕𝑡∆𝑡=𝜕𝑀𝑟\n𝜕𝑝𝜕𝑝\n𝜕𝑡∆𝑡+𝜕𝑀𝑟\n𝜕𝑅𝜕𝑅\n𝜕𝑡∆𝑡+\n𝜕𝑀𝑟\n𝜕𝑐(𝜕𝑐\n𝜕𝑝𝜕𝑝\n𝜕𝑡+𝜕𝑐\n𝜕𝑅𝜕𝑅\n𝜕𝑡)∆𝑡                    (34) \n \nHowever, w e should not e that when condition (8) holds, 𝜕𝑀𝑟\n𝜕𝑐(𝜕𝑐\n𝜕𝑝𝜕𝑝\n𝜕𝑡+𝜕𝑐\n𝜕𝑅𝜕𝑅\n𝜕𝑡)∆𝑡 should \nbe equal to zero. Hence, in competitive markets, formula (3 4) can be reduced to \nformula ( 29). \nIn the context of the present value of future consumer goods multiplier, in general, \nvarious models of investment dynamics can be proposed : \n \n{𝑑𝐾\n𝑑𝑡=𝑀𝑟(𝐾,𝑅)−1\n𝑑𝑅\n𝑑𝑡=𝑝−𝑅−𝑛                    (35) \nor \n{  𝑑𝐾\n𝑑𝑡=𝑀𝑟(𝐾,𝑅)−1\n𝑑𝑅\n𝑑𝑡=𝐾−𝐾∗\n𝑀𝑟(𝐾∗,𝑅)=1                   (36) \nor \n{𝑑𝐾\n𝑑𝑡=𝑀𝑟(𝐾,𝑅)−1\n𝑑𝑝\n𝑑𝑡=𝑅+𝑛−𝑝                       (37) \nor \n{𝑑𝐾\n𝑑𝑡=𝑀𝑟(𝐾,𝑅)−1\n𝑅=𝑓(𝐾)                    (38) \n \n4. RESULTS AND OUTCOMES  \n \nQuantitative estimations of the derived function s based on macroeconomic data  should \nbe made . Moreover, the p redictions of the model  appear to be the following : \n \n1. Investment  grows as 𝑀𝑟 grows.  \n2. Investment grows as the discount rate falls.  13 \n 3. Investment grows as marginal productivity of capital grows.  \n4. Formula (3 4) is true.  \n5. The share of i nvestment in the production of consumer goods falls \nwhen  𝑝>𝑅+𝑛. \n6. The share of i nvestment in the production of consumer goods grows when \n  𝑝<𝑅+𝑛. \n7. The value of the multiplier is close to 1.  \n8. Ceteris paribus, the share of investment in the production of consumer  \ngoods falls when marginal productivity of capital grows or when interest \nrate falls , and vice versa.  \n9. Ceteris paribus, the share of investment in the production of means of \nproduction grows when marginal productivity of capital grows or when \ninterest rate falls, and vice versa.  \n10.  When the discount rate falls, the rate of growth of investment in the \nproduction of means of production should be higher than the rate of growth \nof investment in the production of consumer goods (actually the later may \neven be negative).  \n11.  When the discount rate grows, the rate of growth of investment in the \nproduction of means of production should be lower than the rate of growth \nof investment in the production of consumer goods (and the rate of growth \nof investment in the production of means of pro duction should be negative).  \n \nIn general, our model is consistent with the present macroeconomic models of capital \ndynamics ( Rode (2012 ), Blanchard, Fischer (1989), Meng, Yip, (2004), Caselli, Feyrer, \n(2007) , Tan, Tang, (2016) , Lin, Wang, Wang, Yang (2018) ), namely the predictions 2, \n3, and 7 are consistent. The model, however, give s predictions additional ly to what is \navailable in the literature. In this sense, the multiplier model can be distinguished from \nstandard models of investment dynamics by additional  predictions it makes. The \nneoclassic capital market model is thus a special case of the multiplier model.  Tables \n1 and 2  that follow list the differences in the models. In addition, they show t he \nadditional  predictions of the multiplier model.  \n \nTable 1.  Comparison of  the prediction s of standard models with the predictions of the \nmultiplier model, if discount rate R falls  \n Investment  Share of \ninvestment in c  Share of \ninvestment in i  Rate of \ngrowth of \ninvestment in \nc Rate of \ngrowth of \ninvestment in \ni \nStandard  + NA NA NA NA 14 \n Models \nMultiplier \nmodel  + - + NA + \nNotation: “i” – means of production, “c” – consumer goods , “NA” – not available (the model cannot \ngive a n unequivocal prediction), “+” – the impact is positive, “ -“ – the impact is negative.  \nSource:  Own results  \n \nTable 2. Comparison of the predictions of standard models with the predictions of the \nmultiplier model, if discount rate R grows.  \n Investment  Share of \ninvestment in c  Share of \ninvestment in i  Rate of \ngrowth of \ninvestment in \nc Rate of \ngrowth of \ninvestment in \ni \nStandard  \nModels  - NA NA NA NA \nMultiplier \nmodel  - + - NA - \nNotation: “i” – means of production, “c” – consumer goods, “NA” – not available (the model cannot \ngive an unequivocal prediction), “+” – the impact is positive, “ -“ – the impact is negative.  \nSource:  Own results  \n \nThere is ambiguity regarding the rate of growth of investment in the production of \nconsumer goods  depending on the discount rate. It is due to the assumption that the \nshare of investment in the production of consumer goods  growth when investment falls, \nand vice versa. This ambiguity can be removed by analyzing t he dynamics of \ninvestment in the production  ∆𝐾𝑐 of consumer goods with respect to R:  \n \n∆𝐾𝑐=𝑐(𝑅)∗∆𝐾(𝑅)                    (39) \n∆𝐾𝑐=𝜕𝑐\n𝜕𝑅𝜕𝑅\n𝜕𝑡∆𝑡+𝜕𝐾\n𝜕𝑅𝜕𝑅\n𝜕𝑡∆𝑡                   (40) \n \nWe know that 𝜕𝑐\n𝜕𝑅>0 and 𝜕𝐾\n𝜕𝑅<0. Then : \n  \n{∆𝐾𝑐≥0,𝜕𝑐\n𝜕𝑅>−𝜕𝐾\n𝜕𝑅 \n∆𝐾𝑐<0,𝜕𝑐\n𝜕𝑅<−𝜕𝐾\n𝜕𝑅                   (41) \n \nHence, the impact of R on the investment in the production of consumer goods depends \non the speed of adjustment of c to R relative to the speed of adjustment of K to R. \n 15 \n 5. DISCUSSION OF RESULTS  \n \nThe concept of the multiplier developed in the paper can be applied for the evaluation \nof the present value under various market regimes. The derived models of capital \ndynamics, however, work only in the case of efficient capital markets . \nAlthough it is difficult to test some empirical predictions of the model,  since the model \nis consistent with capital market equilibrium (in terms of the theory of marginal \nproductivity of capital), the evidence supporting capital market efficiency is also \nsupporting the multiplier capital dynamics model.  \nThe model can be falsified by testing its additional  predictions, i.e. the predictions it \ngives above the predictions of standard models of investment dynamics.   \nAlthough some assumptions of the multiplier model appear to be very simplified (such \nas constant depreciation rate), given the market if efficient, the changes in these \nparameters do not affect the results.   \nWhereas the formula for the multiplier itself works just by definition in the case of \nmarket efficiency, the consequent models of investment dynamics may be or be not \nempirically relevant.  \nOf course, the model works until the concept of discounted cash flows  works . It means \nthat if the value of the discounted future cash flows for some reason diverges, the model \nis not relevant.  The results in formulas 29, 33 and 3 4 can be considered as the main \npredictions of the model (beyond its predictions for the impact of the change in the \ndiscount rate on the shares and the growth rates of the investment in the production of \nconsumer goods and means of productions respectively). They should be considered, \nhowever, as a way to represent the results of standard macroeconomic theories (which \nare based on the concept of marginal productivity) of capital market dynamics.  \nDifferent marginal productivity of capital models can be built to test the consistency of \nthe predictions of our model.  \nIn the case of market inefficiency, the models should be extended to take into account \nthe fact that the marginal productivity of capital condition may not hold.  \n \n6. CONCLUSIONS AND IMPLICATIONS  \n \nOverall, it appears that further research may focus on empirical tests of the proposed \nmodels; extending the formula for the multiplier with respect to more complex \nassumptions; simulation of macroeconomic shocks in order to investigate the fitness of \nthe multiplier model for the d escription of the investment dynamics and capital \nmarkets ; study of the capital dynamics model in the case of the inefficient capital \nmarket.   16 \n It is clear that economic agents do not expect in general the share of investment in the \nproduction of consumer goods, as well rate of depreciation, to be constant, but when \ncondition (8) holds (and it should hold in competitive markets), the value of the \nmultiplier does not depend on this share. The changing rate of depreciation can also be \nintegrated in the model, as it was done with the marginal productivity of capital (see \nformulas 3 4-38).  \nDespite its assumptions` simplicity, the multiplier formula seems to work whenever \nthe markets are efficient. The model can be adjusted to various market inefficiencies. \nFor example, it is reasonable to ask what happens with the value of the multiplier if \ncondition (8) does not hold, and how can we justify that it is still 𝑐∈(0;1) in this case.  \nIn the paper, predictions of the model are explicitly listed, so the authors hope that they \nwill be tested empirically. Another topic for empirical study  is testing of various \ninvestment dynamics models (formulas 35 -39 and similar) and selection of the most \nvalid among them.  \nThe implications for economic policy may also be studied . Namely, how monetary \npolicy may change if we know that interest rate changes affect investment in different \nsectors differently.  It is also interesting to know who crowding out effect would affect \ninvestment in the production of consumer goods or the production of means of \nproduction.  \nThe multiplier presented in the paper may be seen as the first approximation of the \nreality in terms of the present value of the investment. The m odel may become more \nprecise if more dynamical parameters are added. However, if the market is efficient, \nthe multiplier value is robust to the changes of parameters.  \nOur analysis of the value of future consumer goods multiplier gives the result that \ncorrespond s with the basic implications of economic  theory . Dynamics of capital and \ninvestment can be described as fluctuations that adjust the value of the multiplier to the \nvalue of one. The v alue of the multiplier can theoretically be infinite and is possible if \nsome prerequisites are satisfied.  Both exponential and hyperbolic discounting in \ncombination with empirical evidence available lead to the value of the multiplier that \nis close to one.  \nThe multiplier model presented in the paper is consistent with common macroeconomic \nmodels of capital market equilibrium and investment dynamics and expands them in \nthe sense that it give s additional  specific predictions on the issues where standard \nmodels fail to make unequivocal statements . Such predictions relate mostly to the \ndynamics of investment with respect to changes in the discount rate.   \nDifferent models of investment dynamics based on the adjustment of the multiplier to \nthe value of one are possible. They should be tested empirically.  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As β\npasses through βc, the stability of isolated fixed points changes, giving rise to a one-dimensional\nmanifold of fixed points. Notably, this attracting invariant manifold forms an arc of an ellipse.\nIn the context of the BNG, we propose modeling the Bayesian learning probabilities pAandpB\nas logistic functions. This modelling approach allows us to establish the existence of fixed points\nwithout relying on the overly strong assumption that pA=pB=p, where pis a constant.\nI. INTRODUCTION\nThe naming game is a multi-agent model that aims to\nsimulate the spontaneous emergence of consensus among\nagents who interact pairwise according to two possible\nprocesses: agreement and word/name learning [1–5]. In\nits simplest realization, the agents are allowed to learn\nonly two possible names, denoted here by the symbols\nA and B, according to some basic rules (we refer the\nreader to Refs. [3, 5] for their definitions). Furthermore,\nif the underlying topology on which the agents may lie,\nis neglected altogether [5, 6], we refer to such a model as\nthe minimal naming game (MNG).\nThe MNG model can be thought as a particular case\nof two variants: the generalized naming game ( β-NG) [7]\nand the Bayesian naming game (BNG) [5, 8, 9]. Both\nvariants add human features otherwise absent in agents of\nMNG. In β-NG, a parameter βis introduced to take into\naccount the probability of acknowledged influence [10],\nthus affecting the agreement processes, while in BNG the\nname learning processes are now dictated by the time-\ndependent probabilities pA≡pA(t) and pB≡pB(t)\nnecessary for generalizing the concept Cassociated with\nwords A and B, respectively [11–13]. It is worth noting\nhere that such probabilities are a direct consequence of\nBayes rule [14]. Therefore, according to the latter model\nthe agents become Bayesian-rational individuals that in\nan uncertain world find their best action using the laws of\nprobability [15–18]. That being said, the minimal nam-\ning game can be recovered from the above two variants by\nsetting their characteristic parameters, i.e. βandpA, pB,\nto unity as illustrated in cartoon of Fig. 1.\nThe dynamics of the naming game is routinely in-\nvestigated through computer simulations of multi-agent\n∗gionnimarchetti@ub.edu; gionnimarchetti@gmail.comsystems. The simulations allow for the computation of\nthe observables of interest, e.g. the convergence time,\nthe success rate, averaging over several hundred realiza-\ntions [5]. However, the differential equations dictating\nthe time-evolution for each model, can be also derived,\nassuming the mean-field approximation, thus offering an\nalternative approach to study some features of consen-\nsus dynamics. In fact, the differential equations of these\nmodels form two-dimensional (2D) dynamical systems\nwhose study has some evident advantages. First, due\nto their low dimensionality, the time-evolution is much\nsimpler when compared to that of naming game models\nwhere multiple names need to be taken into account [19].\nSecond, one can exploit the methods provided by the dy-\nnamical systems theory [20, 21], such as bifurcation di-\nagrams and phase portraits for autonomous systems, to\ninfer whether or not the consensus emerges without the\nneed to perform the numerical simulations, a task that\nbecomes computationally expensive when the system size\nN, i.e., the number of agents, increases.\nIn this paper, we shall pursue the latter approach,\nthereby finding novel interesting dynamical features that,\nto our best knowledge, have been overlooked so far.\nIn Section II the linear stability (LSA) applied system-\natically to β-NG shows that such a dynamical system\nundergoes to a non-generic bifurcation with bifurcation\npoint β=βc(βc= 1/3), where several things happen\nsimultaneously. The isolated fixed points A= (1 ,0),\nB= (0,1) and C= (cβ, cβ) whose coordinates in terms\nof parameter βare given by cβ=\u0010\n5β+ 1−√\n∆\u0011\n/4β\nwhere ∆ ≡1 + 10 β+ 9β2[22], change their stability\nand join an emerging one-dimensional manifold of fixed\npoints when βgoes through βc. Such an attracting in-\nvariant manifold is shown to be an arc of an ellipse (see\nFigs. 4, 5). This is analog of the non-generic bifurca-\ntion observed in 1D differential equation ˙ z=ξ\u0000\nz−z3\u0001arXiv:2402.01940v2  [physics.soc-ph]  18 Feb 20242\natξ= 0 ( ξis a parameter) with fixed isolated points\n0,±1. In fact, as ξpasses through 0, the fixed points at\nz= 0, z= 1, and z= –1 change stability and a line\nof fixed points emerges (the whole z-axis is filled with\nfixed points when ξ= 0). This is a very degenerate sit-\nuation because the single parameter ξaffects both the\nlinear and the cubic term in the differential equation si-\nmultaneously, causing both terms to vanish when ξ= 0.\nNote that the flow of the above 1D differential equation\nis similar to the 2D flow along the attracting invariant\nmanifold corresponding to the (orange) stable “line” in\nthe bifurcation diagram (see Fig. 2) and to the (orange)\ncurve in the 2D phase plane (see Fig. 4).\nWe conclude Section II by briefly mentioning the con-\nnection between this non-generic bifurcation and the type\nof noise characterizing the trajectories obtained from\nthe multi-agent simulations, which are now viewed as\nstochastic processes [23, 24].\nIn Section III, we shall address the issue related to\nthe lack of spontaneous symmetry breaking for the dy-\nnamical system associated with BNG. This issue arises\nwhen attempting to find the solution by numerically inte-\ngrating the respective first-order differential system, as-\nsuming the initial condition I0= (0.5,0.5) and setting\npA=pB=pwhere pis a constant, simultaneously.\nDenoting the solution as ( x, y), where x≡x(t) and\ny≡y(t) are the population fractions having either A\nor B in their inventories at time trespectively, it is\nfound that ( x, y) asymptotically converges to (0 .38,0.38).\nThus, there is no convergence to consensus, similar\nto what occurs with MNG for the very same initial\nsetup [5, 25].\nThis is in contrast to what is found in multi-agent sim-\nulations, where consensus always emerges either at A or\nat B for MNG and BNG models. It has already been\nnoted that such a discrepancy between simulations and\ndynamical systems arises from the fact that the dynam-\nical system approach cannot account for the ubiquitous\ndensity fluctuations, or equivalently noise, present in nu-\nmerical simulations [25]. However, this issue is partic-\nularly significant here, as the above setup was used to\nprove the existence of consensus for the dynamical sys-\ntem associated with BNG [8].\nTo address this serious problem, we propose model-\ning the Bayesian probabilities as time-dependent logistic\nfunctions (see Eq. 12) [26]. Subsequently, by numerically\nintegrating the modified equations, we demonstrate that\nconsensus consistently emerges at either A or B. This out-\ncome is attributed to the spontaneous symmetry break-\ning induced by the time-dependent probabilities, disrupt-\ning the invariance under the swap of variables xandy\n(see Eqs. 10 and 11). Accordingly, the solution closely re-\nsembles that obtained from multi-agent simulations with\nsystem size N≳200 [8].\nβ-NG BNG\nMNGβ\n= 1\npA= 1\npB= 1FIG. 1. The cartoon depicts the relationship between the\nMNG, β-NG, and BNG through their characteristic parame-\nters: β∈[0,1] and the learning probabilities pA, pB∈[0,1].\nSetting βandpA, pBto unity in β-NG and BNG models, re-\nspectively, one recovers the MNG.\nII. THE β-NG MODEL AS DYNAMICAL\nSYSTEM\nIn mean-field approximation the time-evolution of β-\nNG model is dictated by the following system of first-\norder differential equations:\n˙x=−xy+β(1−x−y)2+Kx(1−x−y),(1)\n˙y=−xy+β(1−x−y)2+Ky(1−x−y),(2)\nwhere ˙ x,˙ythe derivatives of x, y, with respect to time.\nWe set K≡(3β−1)/2 in Eqs. 1, 2 while β∈[0,1].\nNext, setting ˙ x= 0 and ˙ y= 0 into Eqs.1 and 2, it was\npreviously determined that three isolated fixed points ex-\nist: A, B, and C[7, 25]. Applying the LST revealed a\nfirst-order non-equilibrium transition at β=βc. Fur-\nthermore, the system converges to point C for β < β c,\nleaving a fraction of undecided agents equal to 1 −2cβ,\nwhile a spontaneous emergence of consensus occurs at ei-\nther A or B for β > β c. These findings are summarized\nin the first and third columns of Table I.\nWhile the above analysis is certainly correct and has\nbeen confirmed by multi-agent simulations, it does not\naddress the non-generic bifurcation that characterizes the\ndynamical system in question.\nIn the following, we shall unravel the dynamical fea-\ntures of the dynamical system by applying the LSA\nto the problem at hand in a systematic way. First,\nlet us define the functions f1(x, y), f2(x, y) such that\n˙x=f1(x, y),˙y=f2(x, y) according to Eqs. 1, 2. It is\nthen possible to compute the eigenvalues of the following\nJacobian matrix Jevaluated at a given fixed equilibrium\npoint with Cartesian coordinates (˜ x,˜y) [20]\nJ=\u0012\n∂xf1(˜x,˜y)∂yf1(˜x,˜y)\n∂xf2(˜x,˜y)∂yf2(˜x,˜y)\u0013\n, (3)\nwhere the symbols ∂x,∂ydenote the partial derivatives\noff1(x, y),f2(x, y) with respect to x, y, respectively.3\ntr (J)det (J)\nsaddle point\n(λi<0< λ j)stable node\n(λi< λ j<0)\nstable line\n(λi= 0, λj<0)det (J) =tr(J)2\n4\nFIG. 2. The types of the equilibria for the dynamical system\nwith Eqs. 1, 2 and β∈[0,1] according to LSA, in the trace-\ndeterminant plane. The respective inequalities between the\ntwo (real) eigenvalues λi, λj(i̸=j) of the matrix Jevaluated\nat each type of fixed point are also shown. The parabola in\nthe (black) solid line corresponds to tr ( J)2−4 det ( J) = 0.\nNote that all the saddle points lie on quadrants III and IV of\nthe trace-determinant plane.\nSecond, the local equilibrium character of each fixed\npoint can be determined by the eigenvalues of the corre-\nsponding matrix J. In particular, their equilibrium char-\nacter is easily identified by looking at the inequalities be-\ntween the eigenvalues as shown in the trace-determinant\nplane (also called the bifurcation diagram) (see Fig. 2)\n[21, 27]).\nThe matrix J’s entries evaluated at the fixed point A\nread: J11=−K,J12=−1−K,J21= 0,J22=−1 while\nthe matrix entries takes the values J11=−1,J12= 0,\nJ21=−1−K,J22=−Kwhen computed at B. Since\nin both cases the traces and the determinants of the\nJacobian matrices are the same, i.e. tr ( J) =−1−K\nand det ( J) =K, one must conclude that these matrices\nhave same eigenvalues. Therefore, the fixed points A, B\npresent the very same stability behaviour for each value\nofβin [0,1]. Accordingly, in the following we shall focus\non the equilibrium character of A only, denoting its cor-\nresponding eigenvalues by the symbols λ1, λ2. By means\nof the equations tr ( J) =λ1+λ2and det ( J) =λ1λ2, one\nimmediately finds these two real eigenvalues: λ1=−1\nandλ2=−K. Since λ1is always negative, both eigen-\nvalues are negative when β > β c, while λ2>0 when\nβ < β c(see Fig. 3). Looking at the classification scheme\nin Fig. 2, one must conclude that A, B are saddle points\nand asymptotically stable nodes for β∈[0, βc) and\nβ∈(βc,1], respectively. So far, the results agree with\nthose reported in Ref. [7]. However, at β=βc,λ2= 0,\nand hence both A and B must belong to the stable line\ncorresponding to the negative half-axis (see the (orange)\nsolid line in Fig. 2). Therefore, A and B cannot be longer\nconsidered isolated, but according to LSA there must ex-\n0.0 0.2 0.4 0.6 0.8 1.0\nβ−3−2−101λi\nλ1, A\nλ2, A\nλ3, C\nλ4, C\nβc= 1/3FIG. 3. The real eigenvalues λiwith i= 1,2 and i= 3,4\ncorresponding to the Jacobian matrix J (Eq. 9) evaluated at\nA and C, respectively, as functions of the parameter β. The\nvertical (dotted) line corresponds to β=βc.\nist an infinite number of other fixed points. We will re-\nturn to this point later.\nWe now turn our attention to the equilibrium character\nof the fixed point C, denoting its corresponding eigenval-\nues by λ3andλ4. To this end, we shall compute them\nby means of the following formula [28]\nλ3, λ4=1\n2\u0012\ntr (J)±q\ntr (J)2−4 det ( J)\u0013\n.(4)\nIn Fig. 3 we plot λ3, λ4as function of βby means of\nEq.. 4. From a direct comparison of the corresponding\ncurves with those of λ1, λ2, one immediately concludes\nthat Chas the opposite equilibrium character of A and\nB for each each β∈[0,1]\\ {βc}, see Fig. 3. However,\nwhen β=βc,Cbelongs to the stable line as well because\nλ3= 0 while λ4negative (see Fig. 2).\nNoting that the stable line is characterized by tr ( J)<\n0 and det ( J) = 0, the last important result could also be\nobtained without the need to compute the eigenvalues, as\nfollows. First, we note that at C= (cβ, cβ), the matrix\nJ’s entries must satisfy the identities J11=J22andJ12=\nJ21, thus the Jacobian matrix is now symmetric. Second,\nJ11, J12take the following form as functions of β\nJ11=−\u0012β+ 1\n2\u0013\n+\u00121−β\n2\u0013\ncβ, (5)\nand\nJ12=−2β+\u00125β−1\n2\u0013\ncβ, (6)\nrespectively.\nTherefore, in such a case tr ( J) = 2 J11and\ndet (J) = J2\n11−J2\n12. Now noting that det ( J) =4\nTABLE I. The classification of the fixed points A, B, C ac-\ncording to the linear stability theory. Note that when the\nisolated equilibrium points are stable, they are also asymptot-\nically stable. The fixed points A, B, C are no longer isolated\nat bifurcation point β=βc.\nFixed Point β∈[0, βc) β=βc β∈(βc,1]\nA saddle point stable line stable node\nB saddle point stable line stable node\nC stable node stable line saddle point\n(J11+J12) (J11−J12), one finds that\nJ11−J12=K(1−2cβ). (7)\nSo, as K= 0 when β=βc, it follows that the left hand\nside of Eq. 7, and hence det ( J) are both zero. Next,\nwe note that the corresponding trace is negative, i.e.\ntr (J) = 2 /3 (cβ−2)<0, because cβ= 2−√\n3≈0.26.\nOverall, our analysis shows that the fixed points\nA, B, C exist for all values of the parameter βbut there\nan exchange of stabilities between A, B andCasβpasses\nthrough βc. In this regard, we can say that the dynam-\nical system undergoes a non-generic bifurcation similar\nto that of the 1D differential equation discussed in Sec-\ntion I. For the sake of completeness, we summarize all\nthese findings in Table I.\nNext, the existence of a continuum of fixed points at\nthe bifurcation point could be easily inferred by exam-\nining the streamline plot of the velocity field v= ( ˙x,˙y)\ncorresponding to Eqs. 1, 2. To this end, in Fig. 4 we plot\nsuch a velocity field on the model’s domain in Cartesian\nxy-plane: {(x, y)∈R2: 0≤x≤1,0≤y≤1,0≤x+y≤\n1}. It is then evident that there exists one-dimensional\nmanifold of fixed points (orange solid line), and that all\ntrajectories approach it along parallel lines. This finding\nis consistent with the stable line, that is, the negative\nnegative half-axis of trace tr J, obtained from LSA (see\nFig. 2).\nWe now prove that the above one-dimensional mani-\nfold constitutes an arc of an ellipse. First, its Cartesian\nequation can be obtained by setting β=βcand either\n˙x= 0 or ˙ y= 0 into Eqs. 1, 2. One then finds that the set\nof points must satisfy the following second-order degree\npolynomial equation in the variables x, y\nx2−xy+y2−2x−2y+ 1 = 0 . (8)\nAccording to Eq. 8 the one-dimensional manifold\ncorresponds to the graph of the function x7→\n1/2\u0000\n−√\n3√\n4x−x2+x+ 2\u0001\nwith x∈[0,1]. Next, we\nconstruct the symmetric matrix Mof the quadratic form\nassociated with Eq. 8 in order to classify it as a non-\ndegenerate conic section [29]. In the present case, one\nI0\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\nxyFIG. 4. Streamline plot of v= ( ˙x,˙y) corresponding to Eqs. 1,\n2 at bifurcation point β=βc. Here the curve (orange) of the\nattracting invariant manifold corresponds to the graph of the\nmap x7→1/2\u0000\n−√\n3√\n4x−x2+x+ 2\u0001\nwith x∈[0,1]. The\nimportant initial condition I0= (0.5,0.5) is also shown.\nfinds\nM=\u0012\n1 0.5\n0.5 1\u0013\n, (9)\nwith determinant det ( M) = 3 /4. As a consequence its\neigenvalues must have the same sign, so Eq. 8 yield a\nconic section that is an ellipse with semi-major axis, semi-\nminor axis and eccentricity corresponding to a=√\n6,\nb=√\n2 and e=p\n2/3, respectively. The graph of\nsuch an ellipse with its center 0 = (2 ,2) and foci F1=\u0000\n2−√\n2,2−√\n2\u0001\nandF2=\u0000\n2 +√\n2,2 +√\n2\u0001\nis shown in\nFig. 5. Note that in such a case, point Cnow has coordi-\nnates ( cβc, cβc) with cβc= 2−√\n3 and hence lies on the\nellipse between points A and B.\nWe conclude this section by noting another important\nconsequence of the existence of a non-generic bifurcation.\nThe dynamical system approach provides a determinis-\ntic mean-field description of the β-model, but the trajec-\ntories from its multi-agent simulations are intrinsically\nnoisy. Therefore, one would expect that there is some\neffect of noise close to the bifurcation [23].\nTo illustrate this, let us consider x, or equivalently\ny, obtained from a single run of a simulation starting\nfrom I0as a random variable Xt. In Fig. 6, we plot six\nrandomly chosen sample paths of Xtas functions of time\nfrom simulations with N= 1,000 agents. Note that the\npaths labeled by ω= 1,2,3 and ω= 4,5,6 correspond to\nβ= 0.2 and β=βc, respectively where each ωicould be\nunderstood as a specific outcome in the sample space.\nThe paths with ω= 1,2,3 exhibit a pattern completely\ndifferent from those at the bifurcation point, i.e., with\nω= 4,5,6. While the former remain close to the sam-\nple mean E[Xt] (black solid line) computed from 1 ,200\ntrajectories, thus showing the typical mean-returning be-5\nxy\nOF2\nF1B\nAC\nFIG. 5. The graph of the ellipse with Eq. 8 emerging at\nbifurcation point β=βc. The ellipse’s center and foci are 0 =\n(2,2),F1=\u0000\n2−√\n2,2−√\n2\u0001\nandF2=\u0000\n2 +√\n2,2 +√\n2\u0001\n,\nrespectively. The fixed points A= (1 ,0),B= (0 ,1) and\nC= (cβc, cβc) where cβc= 2−√\n3 are no longer isolated\nequilibria.\n0 20 40 60 80 100\nt[103steps ]0.10.20.30.40.50.6XtE[Xt], β= 0.2\nω= 1\nω= 2\nω= 3\nE[Xt], β=βc\nω= 4\nω= 5\nω= 6\nFIG. 6. Randomly chosen sample paths of Xt(X0= 0.5) as\nfunctions of time tfrom multi-agents simulations with N=\n1,000 agents. The paths with label ω= 1,2,3 and ω= 4,5,6\ncorrespond to β= 0.2 and β=βc, respectively. Solid lines in\nblack color and red color correspond to sample mean E[Xt]\ncomputed from 1 ,200 trajectories for β= 0.2 and β=βc,\nrespectively.\nhavior of the Ornstein-Uhlenbeck (OU) process [30, 31],\nthe latter seem to wander in a more random fashion. Pre-\nliminary statistical analysis [32] confirms that the sample\npaths at β= 0.2 belong to OU process, while those at\nbifurcation point to a nonstandard Brownian motion [33–\n36].III. THE BAYESIAN NAMING GAME AS A\nDYNAMICAL SYSTEM\nAccording to the BNG model, agents must learn new,\nunknown names through their previous experiences and\nbackground knowledge. Furthermore, since the learn-\ning process occurs under uncertainty, a probabilistic ap-\nproach is necessary for modeling such a process. In this\ncontext, a Bayesian framework proposed by Tenenbaum\nand co-workers seems appropriate [11, 13, 37–41]. As a\nresult of this probabilistic framework, agents attempt to\ngeneralize the concept Cassociated with the names A and\nB by computing the learning probabilities pAandpB, re-\nspectively. These probabilities depend on the positive\nexamples stored by the agents during their pairwise in-\nteractions. In this regard, the BNG model paves the way\nfor incorporating many human cognitive biases [18, 42–\n46] into the consensus dynamics, which were out of reach\nfor the previous variants of naming game.\nAssuming the mean-field approximation, the time\nevolution of BNG is governed by the following non-\nautonomous first-order differential system [8]:\n˙x=−pBxy+ (1−x−y)2+3−pB\n2x(1−x−y),\n(10)\n˙y=−pAxy+ (1−x−y)2+3−pA\n2y(1−x−y).\n(11)\nNote that finding a solution to the above system is not\npossible without making some reasonable assumptions\nabout the probabilities pA(t) and pB(t), both falling\nwithin the interval [0,1] with an unknown functional\nform. In Ref.[8], it was proposed to set pA=pB=p, thus\nmaking the probabilities identical and time-independent\nat the same time. While this assumption, along with the\ninitial condition I0, played a crucial role in estimating\nan upper bound of about 20% for the number of agents\nwith two names in their inventories [9] [47], it cannot be\nused to demonstrate the existence of fixed points A and\nB in full generality, as wrongly assumed in Ref. [8].\nIn fact, consensus cannot be reached in such an initial\nsetup due to the absence of spontaneous symmetry break-\ning [48, 49] [50], similar to what happens to the dynamical\nsystem associate with MNG (see Section I).\nNext, we shall demonstrate that the assumption pA=\npB=pcannot be valid throughout the entire time-\nevolution of the dynamical system associated with BNG.\nTo illustrate this fact we computed cumulative distribu-\ntion function (CDF) of the values taken by pAandpB\nduring a single run of a system with size N= 3,000\nwith initial condition corresponding to 50% of the agents\nknowing the word A and the remainder knowing the\nother word B [8] [51]. The simulation data were ob-\ntained assuming the Erlang prior and different thresholds\nn∗\nex,A= 5, and n∗\nex,B= 6, of the number of positive ex-\namples stored by the agents, necessary for computing the6\n0.0 0.2 0.4 0.6 0.8 1.0\npA,pB0.00.20.40.60.81.0CDFCDF,pA\nCDF,pB\nFIG. 7. The cumulative distribution function of pA(solid\nline) and pB(dashed line) as a function of the Bayesian prob-\nabilities pAandpB, respectively, obtained from a simulation\nof a system with size N= 3,000, starting from the initial con-\ndition I0, and assuming the Erlang prior. The name learning\nthresholds for A and B, were n∗\nex,A= 5 and n∗\nex,B = 6, re-\nspectively. In such a case the consensus is reached at A.\nprobabilities pAandpB, respectively. As a consequence\nof the last assumption, the words A and B are now distin-\nguishable synonyms, and, at the same time, the learning\nof A is favored [8].\nIn Fig. 7, the curves corresponding to CDF of pA(solid\nline) and pB(dashed line) are shown as functions of the\nBayesian probabilities. From their different trends, it is\nclear that the assumption pA=pB=pis untenable\nand may only be valid for a short time-interval. Further-\nmore, there is also a step-like progression toward unity\nin the two cumulative distribution functions as a direct\nconsequence of the few-shot learning processes [11]. The\nlatter feature of this model demonstrates that it is im-\nplausible to consider such probabilities as smooth func-\ntions either of the number of positive examples or of time.\nRegarding their time-dependence, we cannot infer its ex-\nact functional form directly from the multi-agent simu-\nlations, due to their strong dependence on the system\nsizeN. However, Bayesian probabilities exhibit an evi-\ndent monotonically increasing trend with time within a\nmean-field approximation [8].\nWith the above considerations in mind, we propose\nmodeling both pA(t) and pB(t) as a time-dependent lo-\ngistic function (or sigmoid function) [13] [52]. The logistic\nfunction has a S-shaped form that can be modified ap-\npropriately thanks to its two characteristic parameters:\nthe logistic growth rate (or the steepness of the curve) η,\nand the quantity tmthat corresponds to the value of t\nof function’s midpoint. Denoting it by the symbol g, the\nlogistic function reads:\ng(t) =1\n1 + exp ( −η(t−tm)). (12)\n0 20 40 60 80\nt0.00.20.40.60.81.0x,yx\ny\n0 510 15 20 25 30\nt(×10−1)00.51gA,gB\ngA\ngBFIG. 8. Numerical solution of Eqs. 10, 11 replacing the prob-\nabilities pAandpBwith the logistic functions gAandgB, re-\nspectively. The logistic function’s parameters used are η= 2\nfor both gAandgB,tm,A= 10 and tm,B= 130 for gAandgB,\nrespectively. The initial condition is I0= (0.5,0.5). (Inset)\nThe logistic functions gA(solid line) and gB(dashed line) un-\nder scrutiny as functions of time t.\nIn the following, we shall give an example of how our\nprobabilistic modeling yields reasonable results in accor-\ndance with the numerical simulations. Without the loss\nof generality, we chose the following three values for pa-\nrameters ηandtm:η= 2, tm,A= 10 and tm,B= 130.\nInserting η= 2, tm,A= 10 and η= 2, tm,B= 130 into\nEq. 12 we get two different logistic functions: gAand\ngB, respectively. While such a value of ηguarantees that\nboth gAandgBare step-like non-smooth functions of\ntime t, the different values of tmi.e.tm,A,tm,B, differen-\ntiate gAfrom gB. Furthermore, our choice of parameters\nensures that the learning of name A will be favored. In\nthe inset of Fig. 8, the logistic functions gA(solid line)\nandgB(dashed line) are shown as functions of time t.\nNext, we proceed to numerically integrate Eqs. 10, 11,\nreplacing pAwith gAandpBwith gB, using the algo-\nrithm LSODA [53], and assuming the initial condition\nI0= (0 .5,0.5). The numerical result shows that the\nsolution ( x, y) asymptotically converges to A= (1 ,0)\n(see Fig. 8). Accordingly, there exists now a sponta-\nneous emergence of consensus despite the dynamics fol-\nlowing a mean-field deterministic prescription. Further-\nmore, by swapping the values of parameters tm,Aand\ntm,B, it is also straightforward to show that the solution\nconverges to the other possible consensus state, that is,\nB= (0,1). Remarkably, the numerical integration yields\na solution ( x, y) that closely resembles that obtained from\nthe numerical simulations with relatively large number of\nagents, i.e. , N≳200 (see Ref. [8]).\nWe conclude by noting that the numerical solution\n(x, y) splits into its two components xandyafter a tran-\nsient time during which x=y(see Fig. 8). Such an7\ninitial stage of dynamics, can be thought as dictated by\na dynamical system, whose differential equations read:\n˙x=−pBxy, ˙y=−pAxy. This observation clearly ex-\nplains that in the initial stage of the dynamics, the time-\nevolution of the system is now primarily dictated by the\nBayesian probabilities pAandpB. Once again, it is their\ninterplay and time-dependence that lead to the subse-\nquent spontaneous symmetry-breaking process, essential\nfor the emergence of consensus.ACKNOWLEDGMENTS\nThe author would like to express gratitude to Steven\nStrogatz for his helpful insights on bifurcation, to Kon-\nstantinos ”Kostas” Zygalakis for the important remarks\nabout the effect of noise around critical points and bi-\nfurcations, and to Andrea Baronchelli for reading the\nmanuscript. 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Pilditch, Scientific\nReports 8, 12391 (2018).\n[47] The agents having two names in their inventories are of-\nten called bilinguals in the literature.[48] J. Goldstone, Il Nuovo Cimento (1955-1965) 19, 154\n(1961).\n[49] L. Michel, Rev. Mod. Phys. 52, 617 (1980).\n[50] Here the term “spontaneous” indicates that the symme-\ntry breaking occurs without the intervention of an exter-\nnal agent, similar to what happens with the Higgs mech-\nanism in particle physics.\n[51] From the dynamical system point of view this setup cor-\nresponds exactly to have the initial condition I0.\n[52] The logistic function is a common choice of activation\nfunction for the neural networks and deep learning.\n[53] L. Petzold, SIAM Journal on Scientific and Statistical\nComputing 4, 136 (1983)."    },    {        "title": "2402.01973v1.Refined_bounds_on_energy_harvesting_from_anisotropic_fluctuations.pdf",        "content": "Refined bounds on energy harvesting from anisotropic fluctuations\nJordi Ventura Siches,1Olga Movilla Miangolarra,1and Tryphon T. Georgiou1\n1Department of Mechanical and Aerospace Engineering,\nUniversity of California, Irvine, CA 92697, USA\n(Dated: February 6, 2024)\nWe consider overdamped Brownian particles with two degrees of freedom (DoF) that are confined\nin a time-varying quadratic potential and are in simultaneous contact with heat baths of different\ntemperatures along the respective DoF. The anisotropy in thermal fluctuations can be used to\nextract work by suitably manipulating the confining potential. The question of what the maximal\namount of work that can be extracted is has been raised in recent work, and has been computed\nunder the simplifying assumption that the entropy of the distribution of particles (thermodynamic\nstates) remains constant throughout a thermodynamic cycle. Indeed, it was shown that the maximal\namount of work that can be extracted amounts to solving an isoperimetric problem, where the 2-\nWasserstein length traversed by thermodynamic states quantifies dissipation that can be traded\noff against an area integral that quantifies work drawn out of the thermal anisotropy. Here, we\nremove the simplifying assumption on constancy of entropy and carry out the analysis without any\nrestriction on the thermodynamic cycle. We show that the work drawn can be computed similarly to\nthe case where the entropy is kept constant while the dissipation can be reduced by suitably tilting\nthe thermodynamic cycle in a thermodynamic space with one additional dimension. Optimal cycles\ncan be locally approximated by solutions to an isoperimetric problem in a tilted lower-dimensional\nsubspace.a\nI. INTRODUCTION\nIn recent years the newly developed field of Stochastic\nThermodynamics has made it possible to quantify en-\nergy exchanges between thermodynamic systems taking\nplace in finite time and has provided models for study-\ning naturally occurring processes transducing energy at a\ncellular level [1–3]. In this endeavor, a paradigmatic ex-\nample that allows harvesting mechanical work from ther-\nmal gradients is the Brownian gyrator [4]–a simple model\nthat can sustain a far-from-equilibrium operation pow-\nered by anisotropic thermal excitations. Detailed analy-\nsis of a thermodynamic cycle was first carried out in [5]\nand subsequently in [6] to characterize optimality and to\nderive achievable bounds for power and efficiency that\na so-enacted thermodynamic engine is capable of. The\nanalysis of the respective dynamical process in [5] was\ncarried out under a simplifying assumption that the en-\ntropy of the system-states remains constant. Here, we\nremove this restriction and show how the conclusions in\n[5] extend to thermodynamic cycles traversing more gen-\neral system states. Specifically, we conclude that work is\nharvested from the two heat baths in a similar manner as\nin [5], but the flexibility of traversing paths correspond-\ning to states of different entropy allows a reduction in\nthe dissipation, thereby increasing the net work being\nextracted and attaining higher efficiency.\naSupported by the ARO under grant W911NF-22-1-0292 and\nthe AFOSR under grant FA9550-23-1-0096. OMM was sup-\nported by “la Caixa” Foundation (ID 100010434) with code\nLCF/BQ/AA20/11820047.\nEmails: {jordiv, omovilla, tryphon }@uci.eduII. MODEL\nThe Brownian gyrator [4] represents a thermodynamic\nsystem of overdamped particles having two coupled de-\ngrees of freedom (DoF) that are subject to thermal exci-\ntation at different temperatures, T1andT2. These two\nDoF may represent the position Xt∈R2of a particle\non the plane at time t, with {Xt|t∈R}a stochastic\nprocess obeying the (overdamped) Langevin dynamics\ndXt=−γ−1∇U(t, Xt)dt+s\n2kBT\nγdBt.\nHere, {Bt|t∈R}is a two-dimensional Brownian mo-\ntion, T= diag( T1, T2) is a diagonal matrix of the two\ntemperatures, γis the dissipation constant, and U(t, Xt)\nis a time-varying potential.\nThe state of the Brownian gyrator is the distribution\nρ(t, x) of the Langevin particles, with x∈R2, which\nobeys the Fokker-Planck equation ∂tρ+∇ ·J= 0 for\nthe probability current J=−ρ(∇U+kBT∇log(ρ))/γ.\nThe problem we consider is to steer ρ(t, x) along a closed\norbit (thermodynamic cycle) via suitable manipulation\nof the controlling potential U(t, Xt) so as to maximize\nwork extracted from the coupling of the system with heat\nbaths for specified dissipation. The mechanical power\nexchanged via manipulating Ucan be expressed as ˙W=R\nR2ρ∂tUdx and the rate of heat uptake from the two\nreservoirs as ˙Q=−R\nR2U∇ ·Jdx, see [1, page 212].\nWe specialize to the case of a quadratic potential\nU(t, Xt) =1\n2XT\ntK(t)Xtcentered at the origin. Ac-\ntuation is effected via suitable schedule for the time-\nvarying “spring-matrix” K(t). The thermodynamic state\nρremains a two-dimensional Gaussian distribution with\nmean equal to zero and covariance matrix Σ( t)∈R2×2arXiv:2402.01973v1  [cond-mat.stat-mech]  3 Feb 20242\nthat satisfies\nγ˙Σ(t) =−K(t)Σ(t)−Σ(t)K(t) + 2kBT. (1)\nRates of work and heat exchange between the system,\nthe actuating potential and the two heat baths can be\nreadily expressed in terms of K(t) and Σ( t). Indeed, the\ninternal energy of the system is\nE=1\n2Tr[K(t)Σ(t)],\nwhere Tr[ ·] denotes the trace operator. Work and heat-\nexchange rates are given by (see [5], [1, page 212-213])\n˙W(t) =1\n2Tr[˙K(t)Σ(t)],and ˙Q(t) =1\n2Tr[K(t)˙Σ(t)].\nThe “spring matrix” K(t) can be expressed from (1)\nas a function of (Σ( t),˙Σ(t)),\nK(t) =Z∞\n0e−τΣ(t)\u0010\n2kBT−γ˙Σ(t)\u0011\ne−τΣ(t)dτ\n=:LΣ(t)[2kBT−γ˙Σ(t)].\nSubstituting this expression for K(t) into the formula for\n˙Q, the heat-exchange rate splits into two terms, one that\nis linear in ˙Σ and one that is quadratic,\n˙Q=kBTr[LΣ(t)[T]˙Σ(t)]−γ\n2Tr[LΣ(t)[˙Σ(t)]˙Σ(t)].\nThe linear term represents quasi-static heat, as it remains\ninvariant with the speed of traversing the path, while the\nquadratic quantifies dissipation as it vanishes when the\nspeed slows down to 0. Thus, the total quasi-static heat\nand dissipation over a cycle with period tfare\nQqs=kBZtf\n0Trh\nLΣ(t)[T]˙Σ(t)i\ndt,and\nQdiss=γ\n2Ztf\n0Trh\nLΣ(t)[˙Σ(t)]˙Σ(t)i\ndt,(2)\nrespectively [5].\nThe state ρof the thermodynamic system at time t,\nbeing zero-mean Gaussian, is specified by its covariance\nΣ(t). Thus, we seek to study thermodynamic cycles as\nclosed orbits on the space of positive definite 2×2real\nsymmetric matrices , the Σ-space. To this end, we select\ncoordinates ( r, θ, z ) for this space as\nr=1\n2log(λ1(Σ)/λ2(Σ))\nz= log(det(Σ)) = log( λ1(Σ)·λ2(Σ)),\nwhere λ1,2denote the eigenvalues of Σ, λ1≥λ2>0, and\nθspecifies the rotation matrix\nR\u0000\n−θ\n2\u0001\n=\u0012cos\u0000θ\n2\u0001\n−sin\u0000θ\n2\u0001\nsin\u0000θ\n2\u0001\ncos\u0000θ\n2\u0001\u0013\nthat diagonalizes Σ. Thus,\nFIG. 1: Pictorial of planar closed orbits on the Σ-space.\nΣ =R\u0000\n−θ\n2\u0001\nσ(z, r)R\u0000\n−θ\n2\u0001′, (3)\nwhere′denotes transposition, and\nσ(z, r) =ez\n2\u0012\ner0\n0e−r\u0013\n=\u0012\nλ10\n0λ2\u0013\n.\nIn the Σ-space, ( r, θ) are planar polar coordinates\nand specify the eccentricity and orientation of princi-\nple components of Σ, respectively, while zrelates to\nthe area of drawn ellipses and specifies the entropy\n(−R\nR2ρlogρdx=1\n2z+ const .) ofρ. Figure 1 displays\nclosed semi-circular planar orbits in which zis kept con-\nstant. The analysis in [5] was carried out for zconstant.\nIn the sequel we will explore the case where zis not kept\nconstant and seek properties of optimizing cycles.\nWe are interested in closed orbits that maximize work\nproduced, namely, closed paths in the Σ-space solving\nmax\nΣ(t)Qqs(Σ(t))− Q diss(Σ(t)) (4)\nsubject to Σ(0) = Σ( tf). We also define efficiency\nη:=Wout\nQqs= 1−Qdiss\nQqs(5)\nas the ratio between the work output Woutover a cycle\nWout=Qqs− Q diss\nand the work output in the quasi-static limit Qqs. Triv-\nially η≤1 with equality attained as tf→ ∞ .\nIn the following section we explain how (4) relates to an\nisoperimetric problem, seeking a maximal area-integral\nfor a fixed (2-Wasserstein) length traversed in the Σ-\nspace, echoing results in [5] and [7] for the cases of con-\nstant zand linear response regime, respectively. We also\nprovide a correction to the bound η≤1 that takes into\naccount the finite period in traversing thermodynamic\ncycles, and compute efficiency at maximum power.3\nIII. RESULTS AND ANALYSIS\nTaking the time derivative of Σ in (3) we obtain\n˙Σ =1\n2R(σ˙z+ 2σΞ ˙r+ (σΩ−Ωσ)˙θ)R′, (6)\nfor\nΞ =\u0012\n1 0\n0−1\u0013\n,and Ω =\u0012\n0 1\n−1 0\u0013\n.\nTo tackle (4), we express the quasi-static heat and dissi-\npation over a cycle (2) explicitly in terms of parameters\n(r, θ, z ) using (6). This is as follows,\nQqs=kB∆T\n2Ztf\n0cos(θ) ˙r−tanh( r) sin(θ)˙θ dt\nQdiss=γ\n2Ztf\n0ez/2\u0010\nsinh ( r) tanh ( r)˙θ2\n+ cosh ( r)\u0000\n˙r2+1\n4˙z2\u0001\n+ sinh ( r) ˙r˙z\u0011\ndt,(7)\nfor ∆ T=T1−T2. (Compare with [5, Equations (7a-7b)],\nwhere z= const .and thus ˙ z= 0). Interestingly, vary-\ningz(the entropy of thermodynamic states) along cycles\ndoes not affect Qqsbut impacts dissipation Qdiss. Thus,\nby suitably modifying the state-entropy along a thermo-\ndynamic cycle, trajectories of a fixed length (i.e., fixed\ndissipation) can encompass a larger area in the Σ-space\nand thereby enable a relative increase in work produc-\ntion. Symbolic code that can be used to obtain Eq. (7)\nis given in the Appendix.\nA. Geometric analysis\nWe now explain the inherently geometric nature of the\nproblem. Firstly, dissipation can be expressed as\nQdiss=γ\n2Ztf\n0∥˙α(t)∥2\ngdt, (8)\nwhere α(t) ={(r(t), θ(t), z(t)) : t∈[0, tf]}denotes a\ntrajectory on the Σ-space and ∥·∥2\ngdenotes the square\nnorm of a vector with respect to the metric\ng=ez/2\ncosh( r) 01\n2sinh(r)\n0 sinh( r) tanh( r) 0\n1\n2sinh(r) 01\n4cosh( r)\n.\nNote that when zremains constant, the metric in [5] is\nrecovered. By the Cauchy-Schwarz inequality, the mini-\nmal dissipative heat for any trajectory is given by\nγ\n2tf\u0012Ztf\n0∥˙α(t)∥gdt\u00132\n=:γ\n2tfℓ2(9)\nwhere ℓis the length of the path with respect to gon\nthe Σ-space, which coincides with Qdiss, and hence, withthe 2-Wasserstein length of the cycle of thermodynamic\nstates [8]. The minimum is attained when ∥˙α(t)∥gre-\nmains constant along the path.\nSecond, the quasi-static heat can be written as a\nweighted surface integral over a domain D, precisely as\nshown in [5]. Note however, that Dis no longer the do-\nmain encircled by a path drawn on some two-dimensional\nsubmanifold in the Σ-space, but instead, it is the area\nenclosed by the projection of the cycle onto a plane that\ncorresponds to a constant value for z.\nFollowing [5], by means of Stokes’ theorem, quasi-static\nheat from (7) gives\nQqs=±kB∆T\n2ZZ\nDtanh2(r)\nrsinθ rdθ dr (10)\n=:±kB∆T\n2Ah,\nwhere the ±sign depends on the direction chosen, and\nAh=ZZ\nDh(r, θ, z )p\ndet(g)drdθ,\nis an area integral with respect to the Riemannian canon-\nical 2-form of the metric gand the work density function\nh(r, θ, z ) = 2 e−3z/4sinθtanh( r)p\ncosh( r), (11)\nthat results in Ahbeing independent of z.\nThus, the problem to maximize work extraction Wout\nalong a cycle, namely,\nmax\nα(t)Ah−µℓ2\nfor a given µ=γ\nkB∆Ttf, that can be interpreted as a\nLagrange multiplier, amounts to maximizing the area in-\ntegral Ahenclosed by a cycle of fixed length ℓ, as in [5].\nOn the other hand, efficiency (see (5)) can also be ex-\npressed in geometric terms as\nη= 1−µℓ2\nAh,\nwith the problem to maximize efficiency along a cy-\ncle of arbitrary length turning into a search for an\nisoperimetric-like inequality in the space of thermody-\nnamics states. That is, maximizing efficiency amounts\nto seeking\nµ∗:= max\nDAh\nℓ2.\nWe remark that, from the exponential term ez/2in the\nexpression for the metric g, one can see that trajectories\nwith increasingly negative values of z(and thus, with in-\ncreasingly negative entropy) result in a vanishingly small\ndissipation. It is physically meaningful to bound zand\nso, for carrying out a local analysis to identify the struc-\nture of optimal cycles in the next section, we prescribe a\nfinite value of zand linearize around it.4\nB. Local analysis\nSince the work density (11) is proportional to sin( θ)\nandgis independent of θ, the choice θ0=π\n2maximizes\nAh, locally, over an infinitesimally small domain Dabout\na state ( r0, θ0, z0). Without loss of generality we also fix\nz0= 0 (corresponding to det(Σ) = 1) and consider closed\npaths {(r(t), θ(t), z(t)|t∈[0, tf]}with\nr(t) =r0+ϵr1(t)\nθ(t) =π\n2+ϵθ1(t)\nz(t) =ϵz1(t),\nforr0to be determined with ϵ >0 assumed small. Up to\no(ϵ2), the quasi-static and dissipative heat are\nQϵ\nqs=−kB∆T\n2ϵ2Ztf\n0θ1˙r1+1\nc2\n0r1˙θ1dt\nQϵ\ndiss=γ\n2ϵ2Ztf\n0\u0012\nc0˙r2\n1+s2\n0\nc0˙θ2\n1+1\n4c0˙z2\n1+s0˙r1˙z1\u0013\ndt,(12)\nwhere c0= cosh( r0) and s0= sinh( r0). Note that Qϵ\nqs\ncan be written as an area integral as before, or as the line\nintegral above.\nWe now consider maximizing work extracted, namely,\nmax\nr,θ,zQϵ\nqs− Qϵ\ndiss, (13)\nover a small cycle about ( r0, θ0=π\n2, z0= 0). The Euler-\nLagrange equation (first order necessary condition for op-\ntimality) for the functional in (13) with respect to the\nz-coordinate gives\n¨z1=−2s0\nc0¨r1(=−2 tanh( r0)¨r1). (14)\nIntegrating over time gives ˙ z1=−2s0\nc0˙r1+ const .This\nconstant however must be equal to zero on periodic or-\nbits. Substituting ˙ z1=−2s0\nc0˙r1into (12), we obtain that\nQϵ\ndiss=γϵ2\n2Ztf\n0∥( ˙r1,˙θ1)∥2\ng0dt (15)\nis quadratic in the 2-dimensional velocity vector ( ˙ r,˙θ) for\nthe metric (cf. (8))\ng0= 1\nc00\n0s2\n0\nc0!\n.\nAs observed in the geometric analysis section, instead\nof Problem (13) we may instead consider the problem to\nmaximize Qϵ\nqssubject to Qϵ\ndissin (15) being specified. To\nthis end, cf. (9), we set\nQϵ\ndiss\nϵ2=γ\n2Ztf\n01\nc0˙r2\n1+s2\n0\nc0˙θ2\n1dt=:γℓ2\nϵ\n2tf. (16)Our problem now becomes\nmax\nr1(t),θ1(t)Ztf\n0\"\nθ1˙r1+1\nc2\n0r1˙θ1+λ \n1\nc0˙r2\n1+s2\n0\nc0˙θ2\n1−ℓ2\nϵ\nt2\nf!#\n| {z }\nL(r1,θ1,˙r1,˙θ1)dt,\nwhere the constraint (16) has been included in the La-\ngrangian Lwith Lagrange multiplier λ. The correspond-\ning Euler-Lagrange equations are\n¨r1=−s2\n0\n2λc0˙θ1, ¨θ1=1\n2λc0˙r1,\ngiving\nr1(t) =s0Acos(ωt)−s0Bsin(ωt)\nθ1(t) = Asin(ωt) + Bcos(ωt),\nwith ω=2π\ntf=s0\n2λc0. Since we are interested in the\ncomplete (closed) orbit, we may take B= 0. Then, A=\nℓϵ√c0\n2πs0from (16), and we obtain the equations of an ellipse\nr1(t) =ℓϵ√c0\n2πcos\u0010\n2π\ntft\u0011\nθ1(t) =ℓϵ√c0\n2πs0sin\u0010\n2π\ntft\u0011\n,(17)\nfort∈[0, tf].\nUp to ϵ2, the quasi-static heat and dissipation are\nQϵ\nqs=kB∆Ts0ℓ2\nϵ\n8πc0ϵ2,andQϵ\ndiss=γℓ2\nϵ\n2tfϵ2,\nrespectively, giving\n1\nµQϵ\nqs\nQϵ\ndiss=1\n4πµs0\nc0\nThus, the efficiency can be expressed as\nηϵ\n3d= 1−Qϵ\ndiss\nQϵqs= 1−4πµc0\ns0. (18)\nFIG. 2: Efficiency of isentropic (blue) and\nnot-isentropic/general (red) cycles vs. r0(eccentricity of\nstates); locally for µ=1\n16π,θ0=π\n2,z≃0.5\nHere, the subscript 3d refers to the fact that optimiza-\ntion takes place in the 3-dimensional parameter space, of\ncoordinates ( r, θ, z ). The efficiency is seen to be greater\nthan that obtained over cycles of constant entropy ( z=\nconstant), when it was found that ηϵ\n2d= 1−4πµc2\n0\ns0[5]. It\nwas further shown in [5] that ηϵ\n2d= 1−8πµis maximal\n(achieved for r0= asinh(1)). Instead, ηϵ\n3dincreases as\nr0→ ∞ towards the limit 1 −4πµ, see Figure 2.\nEquation (18) implies an inherent speed limit, in that\nfor\ntf<4πγc0\ns0kB∆T(19)\nit is impossible to extract work ( ηϵ\n3d<0). A corre-\nsponding speed limit for isentropic cycles obtained in [5],\ntf<4πγc2\n0\ns0kB∆T, differs by a factor of c0.\nFor isentropic cycles, it was inferred in [5] that the ef-\nficiency - and thus the ratio of the weighted area and\nthe 2-Wasserstein length of the cycle squared - was max-\nimized as ℓ→0. In the present case of cycles where the\nentropy is allowed to vary, it appears that the same is\ntrue, and thus we presume the efficiency to be bounded\nbyη≤1−4πtc\ntf, where tc=µtfis a characteristic time,\npointing towards a more general isoperimetric inequality\nand speed limit. Numerical experiments appear to back\nthis hypothesis but a formal proof is lacking.\nC. General cycles\nWe now consider cycles that have finite length. The\nRiemannian metric gcan no longer be assumed constant,\nand therefore closed-form expressions for the optimal cy-\ncles are not feasible. However, numerical experimenta-\ntion suggests a similar shape of optimal cycles as in the\nsmall-length-case.\nThe discrepancy between an optimal cycle and its ap-\nproximation, corresponding to vanishing ϵ, is negligible\nfor small ℓ, as highlighted in Figure 3. Note that the\nCone and cylinder\nIntersection\nOptimal trajectory\nFIG. 3: Optimal cycle and its approximation as the\nintersection of a cone with an elliptic cylinder for ℓ= 0.1\nFIG. 4: Optimal trajectories of different length and\ntheir local approximations as seen from above, along the\nz-axis (left), and from the side, along the x-axis (right).\ncoordinates x=rcosθandy=rsinθhave been used as\naxes (in accordance with their portrayal in Figure 1). As\ndepicted in the figure, the approximation is precisely the\nintersection of a cone with an elliptic cylinder (17); the\nequation of the cone z1=−2 tanh( r0)r1+ const .follows\nfrom (14). Due to the scale of the figure (range of y-\nvalues far from the origin), the slice of the cone appears\nas planar.\nFor larger values of ℓ, optimal cycles are depicted\nin Figure 4. The dashed curves on the left plot out-\nline approximations of optimal cycles obtained in the ϵ-\nsmall limit. These are in surprisingly good agreement\nwith the exact numerical solutions (solid curves). The\nsubfigure on the right displays the side-view of optimal\ncycles, lying on the generatrix of the cone with slope\n−2 tanh( r0)≃ −2 (since here, r0is large).\nD. Efficiency at maximum power\nThe thermodynamic efficiency of heat engines is max-\nimal in the quasi-static limit ( tf→ ∞ ), a regime with\nvanishing power output. Indeed, quantifying the power\nthat an engine is capable of has motivated this and ear-\nlier studies. In the regime where power is maximal, it is\nalso of interest to quantify efficiency.\nIn the present context, for a specified thermodynamic\ncycle (hence, with Ah, ℓgiven), the power output\nP=Wout\ntf=1\ntf\u0012kB∆T\n2Ah−γ\n2tfℓ2\u0013\nis maximized for tf=2γℓ2\nkB∆TAh, so that power and effi-\nciency (defined in (5)) become\nPmax=(kB∆T)2A2\nh\n8γℓ2,andη∗=1\n2. (20)\nThese depend on Ah, ℓand apply to all cycles (tra-\nversed in constant speed on the Wasserstein manifold,\nas explained earlier). The efficiency at maximum power\nη∗matches the universal linear-response bound of 1 /2\n[9, 10].6\nIV. CONCLUDING REMARKS\nThe present work builds on [5] that derived quantita-\ntive bounds on power and efficiency for a thermodynamic\nengine that is based on the Brownian gyrator. The salient\nfeature is to capitalize on a temperature gradient that can\nproduce a torque on mechanical degrees of freedom, and\nthereby allow extracting work from the heat baths that\nare coupled via these same degrees of freedom.\nThermodynamic states are seen as distributions on the\nWasserstein manifold (distributions metrized by the 2-\nWasserstein metric of Optimal Mass Transport theory).\nLengths being traversed in a thermodynamic cycle quan-\ntify dissipation while suitably weighted area quantifies\nwork produced during the cycle. Our analysis echoes\nthat in [5] where similar conclusions where drawn un-\nder the assumption of isentropic cycles. Our results are\nmore general since fluctuation of the entropy of thermo-\ndynamic states as they traverse a cycle can be used ju-\ndiciously to reduce dissipation. Specifically, we obtain\nincreased maximal work output, tighter bounds on effi-\nciency (18), inherent improvement in speed limits (19)and explicit expressions for maximum power and for ef-\nficiency at maximum power (20).\nFuture work may focus on losses due to housekeeping\nentropy production, that has not been dealt with in the\ncurrent work (see [11, 12]). A holistic picture of how\ntemperature gradients can be used to generate work and\nhow protocols may be designed to minimize entropy pro-\nduction should be of great interest when studying bio-\nlogical engines. The distinguishing feature of such real-\nworld embodiments is that the capacity of heat baths or\nof chemical potentials is not inexhaustible, and thereby,\ntotal entropy production should be contained as much as\npossible (see [13]). Ultimately, it would be of great inter-\nest to compare theoretical results to experimental data\nthat pertain to flagellar and other biological engines.\nAPPENDIX\nSymbolic computations to derive the expressions for\nQqsandQdissin (7), using (2-6), can be carried out using\nthe Mathematica©code that follows.\n1θ= Symbol[\" θ\"]; r = Symbol[\"r\"]; z = Symbol[\"z\"]; t = Symbol[\"t\"];\n2T = DiagonalMatrix[{Symbol[\"T1\"],Symbol[\"T2\"]}];\n3r1 = {{Cos[- θ/2],Sin[- θ/2]},{-Sin[- θ/2],Cos[- θ/2]}}; r2 = {{Cos[ θ/2],Sin[ θ/2]},{-Sin[ θ/2],Cos[ θ/2]}};\n4dg = {{Exp[z/2+r],0},{0,Exp[z/2-r]}}; Sigma = r1.dg.r2; Xi = {{1,0},{0,-1}}; Om = {{0,1},{-1,0}};\n5dr = Symbol[\"dr\"]; dz = Symbol[\"dz\"]; d θ= Symbol[\"d θ\"];\n6dSigma = Simplify[-r1.dg.r2*dz/2+r1.dg.Xi.r2*dr+r1.(dg.Om-Om.dg).r2*d θ/2];\n7expTSigma = Simplify[MatrixExp[-t*(Sigma)].T.MatrixExp[-t*(Sigma)]];\n8integral = Simplify[Integrate[expTSigma,{t,0,Infinity}]];\n9expdSigma = Simplify[MatrixExp[-t*(Sigma)].dSigma.MatrixExp[-t*(Sigma)]];\n10integral2 = Simplify[Integrate[expdSigma,{t,0,Infinity}]];\n11dQ = Symbol[\"kB\"]*Simplify[Tr[integral.dSigma]]; dW = -Symbol[\" γ\"]/2*Simplify[Tr[integral2.dSigma]];\n[1] K. Sekimoto, Stochastic Energetics (Springer, 2010).\n[2] U. Seifert, Stochastic thermodynamics, fluctuation the-\norems and molecular machines, Rep. Prog. Phys. 75,\n126001 (2012).\n[3] L. Peliti and S. Pigolotti, Stochastic Thermodynamics:\nAn Introduction (Princeton University Press, 2021).\n[4] R. Filliger and P. Reimann, Brownian gyrator: A mini-\nmal heat engine on the nanoscale, Phys. Rev. Lett. 99,\n230602 (2007).\n[5] O. Movilla Miangolarra, A. Taghvaei, R. Fu, Y. Chen,\nand T. T. Georgiou, Energy harvesting from anisotropic\nfluctuations, Phys. Rev. E 104, 044101 (2021).\n[6] O. Movilla Miangolarra, A. Taghvaei, Y. Chen, and T. T.\nGeorgiou, Geometry of finite-time thermodynamic cycles\nwith anisotropic thermal fluctuations, IEEE Control Sys-\ntems Letters 6, 3409 (2022).\n[7] A. G. Frim and M. R. DeWeese, Geometric bound on\nthe efficiency of irreversible thermodynamic cycles, Phys.Rev. Lett. 128, 230601 (2022).\n[8] E. Aurell, C. Mej´ ıa-Monasterio, and P. Muratore-\nGinanneschi, Optimal protocols and optimal transport\nin stochastic thermodynamics, Phys. Rev. Lett. 106,\n250601 (2011).\n[9] C. Van den Broeck, Thermodynamic efficiency at maxi-\nmum power, Phys. Rev. Lett. 95, 190602 (2005).\n[10] M. Esposito, K. Lindenberg, and C. Van den Broeck,\nUniversality of efficiency at maximum power, Phys. Rev.\nLett. 102, 130602 (2009).\n[11] O. Movilla Miangolarra, A. Taghvaei, and T. T. Geor-\ngiou, Minimal entropy production in the presence of\nanisotropic fluctuations, arXiv:2302.04401 (2023).\n[12] T. Hatano and S.-i. Sasa, Steady-state thermodynamics\nof Langevin systems, Phys. Rev. Lett. 86, 3463 (2001).\n[13] J. G. Richens, ´A. M. Alhambra, and L. Masanes, Finite-\nbath corrections to the second law of thermodynamics,\nPhys. Rev. E 97, 062132 (2018)."    },    {        "title": "2402.01983v1.Searching_for_cosmological_variation_of_the_proton_to_electron_mass_ratio_using_a_single__H_2__system.pdf",        "content": "1Searchingforcosmologicalvariationoftheproton-to-electronmassratiousinga\nsinglesystem\nT.D.Le1.2\n1DivisionofAppliedPhysics,DongNaiTechnologyUniversity,BienHoaCity,Vietnam\n2FacultyofEngineering,DongNaiTechnologyUniversity,BienHoaCity,Vietnam\nEmail:leducthong@dntu.edu.vn\nAbstract.ExploringphysicsbeyondGeneralRelativityandtheStandardModelof\nParticlePhysicsinvolvesinvestigatingspacetimevariationsinnaturalconstants.This\nstudyemploysan2-singleofQSO0347-383observationalspectrumtoproposea\nuniqueapproachfordetectingpotentialchangesintheproton-to-electronmassratio.\nBycomparingtheratiofromobservationalandlaboratorydataintheLyman-Alpha\ntransitionline,wederiveacosmologicalvariationof∆µ/µ=(0.120±0.144)×10−8\nat=3.025.Thisapproachnotonlyadvancesfundamentalphysicsunderstanding\nbutalsointroducesinnovativetechniquesforanalyzinghigh-redshiftQSOsystems.\nKeywords:varyingphysicalconstants;individualQSO0347-383;QSOspectra\nanalysis\n1.Introduction.\nTheStandardModelofParticlePhysics(SMPP)andthetheoryofGeneral\nRelativity(GR)underlieourpresentunderstandingoftheuniverse.Despitetheir\nremarkablesuccess,thesetheoriesareanticipatedtoeventuallyfailatsomepoints.\nVariousscenariosinvolvingunification(Hojjatietal.,2016),higher-dimensional\ntheories(Levshakovetal.,2016),andmodelsconcerningdarkmatter(Martins,2017)\nanddarkenergy(Thompson,2017;Kosteleckyetal.,2003;Uzan,2011)suggest\npotentialdeviationsfromtheEinsteinEquivalencePrinciple(EEP).Thesetheoretical\nframeworkshaveledtoextensionsofphysicswhereconstantstransitionintodynamic\nphenomena,challengingtheEEP.\nInvestigatingspacetimevariationsinnaturalconstants,suchasthefinestructure\nconstant,theproton-to-electronmassratio,andthegravitationalconstant,servesasa\nmeanstotesttheEEP.Atomicclockexperimentshavesetstringentlimitsonlinear\nchangesintheseconstants,oftenatthelevelofafewpartspermillion(ppm)(Liberati,\n2013;Dzubaetal.,1999;Webbetal.,1999;2011).Transitionwavelengthsofferagrat\nwaytodetectvariationsinthefinestructureconstant(α)andtheproton-to-electron\nmassratio(μ).Atomicclockcomparisonsprovidetheirtemporalderivatives(Blattetal.,\n2008;Rosenbandetal.,2008),whilestudiesofhigh-redshiftcelestialobjectsexplore\ncosmic-scalechanges.\nAnalysisofhigh-redshiftQSOabsorptionspectrahasindicatedpotential\nvariationsinμacrosstheuniverseatafewppmlevel(Reinholdetal.,2006).Utilizing2microwaveandmm-wavemoleculartransitionsinthesesystemsenhancessensitivityto\nconstantvariationscomparedtoopticaltransitions,yieldingresultsfor∆μ/μwithinthe\nrangeof8to0.8ppm(Levshakovetal.,2012;Weiβetal.,2012;Kanekaretal.,2010;\nRahmanietal.,2012).Recentinvestigationsintomethanolabsorptionlinesathigh\nredshiftsreporteda0.1ppmchangein∆μ/μ.Giventheubiquityofsingle-molecule\nlineslikehydrogen,probingμ-variationsathighredshiftsprovesadvantageous.\nAdditionally,thefar-infraredfinestructureofCI,CII,andCOhavebeenemployedto\nconstraincosmicvariationsinμ(Levshakovetal.,2020andreferencestherein).\nNotably,astudyexaminednineabsorptionsystemswithredshiftsrangingfrom2.05to\n4.22,yieldinga~5ppmresultata3σlevel(Ubachsetal.,2016).\nThestudyfurtheremployedtheµ-dependentstateofwavelengthtransitionshifts\nviatheH2-technique(Salumbidesetal.,2012).Otherworkscombined21cmlinesofHI\nandUVmetalabsorptionlinestoassessthetemporalvariationsofµ,withOH\nmicrowavelinemeasurementsprovidingcombinedchangesfor2(denotesthe\nprotongyromagneticratio)(Noterdaemeetal.,2009;2010).CO,abundantinour\ngalaxies,hasalsocontributedtoµ-variationanalyses.Investigationsinvolvingamixture\nofHIandCOinsixabsorbingsystemswithhigh-qualityQSOspectrayieldedestimates\nforthecosmicµ-variation(Srianandetal.,2010).Incontrast,studiesusingCHlinesin\ntheMilkyWaynearlyconstrainedthecosmologicalµ-variationtoalevelof∆μ/μ=(-\n0.07±0.22)ppm(Truppeetal.,2013).Currently,themostrobustlimitoncosmological\nµvariationstandsatppm,basedonNiVlinesdetectedinstronggravitationalfields\nfromthewhitedwarfspectrumofG191-B2B(Le.,2019;2020).\nInthisstudy,wefocusontheQSO0347-383systemataredshift,=3.025,\naimingtoestimatethecosmologicaldeviationoftheproton-to-electronmassratio.The\nchoiceofthishigh-qualityquasarisdrivenbytheobjectiveofconstraining∆μ/μover\ncosmictimescales.Utilizingacombinationofsingleobservationalsystemsand\nlaboratorydata,thisstudyprovidesasignificantlyimprovedaccuracydetermination\ncomparedtopreviousestimates.\n2.systemconstraintson∆/\nAstrophysicaltechniquesofferausefultoolforinvestigatingchangesin\nfundamentalconstantsofnatureoncosmictimescales.Inthecontextofunification\nscenarios,constantssuchasthefine-structureconstantandtheproton-to-electronmass\nratiocouldexhibitvaryingvalues,asindicatedbymeasurementsfromQSOsystems.\nThisexplorationofconstantsinvolvescomparingobservedwavelengthsfromQSO\nsystemswiththeircorrespondinglaboratoryvalues.Forthefine-structureconstant's3variation,generalrelativisticeffectsassociatedwithredshift()playacentralrole,\ninvolvingcouplingswithscalarandotherfields.Thisisexpressedthroughthelossof\nenergybyaphoton(),givenby=−∆.Theresultingfractionalchangeinenergy\nleadstoacorrespondingfractionalchangeinwavelengthmeasurements,\n−∆=∆~∆.Spatialortemporalvariationsindimensionlessphysical\nconstantsarekeyelementsforunifyinggravitationalandelectromagneticforceswithin\nGrandUnifiedTheories(GUTs).ThisunificationassumesequivalentYukawa\ncouplingsandischaracterizedbydimensionaltransmutationsataweakscale.The\nframeworkemploysadrivendilaton-typetoestablishallcouplings.Thisapproach\nrelatesfine-structureconstantvariationtotheQuantumChromodynamics(QCD)scale\n()through/=∆/,whereisderivedfromGUTs.Byconsidering\nmodel-independenceatlowenergies,the-valuecanbedeterminedthroughthe\nrelationship=().ModificationsinYukawacoupling(ℎ)impactthe\nHiggsVEV(VacuumExpectationValue-)atthePlanckmassscaleofGUTs.\nConsequently,dimensionaltransmutationcandeduce=exp(-(82)/ℎ2)and\n/=162(∆ℎ/ℎ)=(∆ℎ/ℎ).Specifically,≡ℎ,and\n∆ℎℎ=(12)∆(≃ℎ≃1).Thisprovidesthefoundationfortestingthevariation\noftheproton-to-electronmassratiowithintheframeworkofunificationscenarios(Coc\netal.,2007;Ferreiraetal.,2014;Le.,2019;2020;2021a):\n∆µ\nµ=[0.8−0.31+]∆\n\nIt'simportanttohighlightthatthevaluesofandaredeterminedfrom\nastronomicaldataandexhibitvariationsacrossdifferentmodels(Cocetal.,2007;\nFerreiraetal.,2014;Le.,2019;2020;2021).Thepresentpredictionswithinunification\nscenariosdependonspecificmodelchoicesfortheseparameters.Consequently,the\nparameters=273±86and=630±230(Le.,2019;2020;2021)wereselectedas\nthebest-fitforourpurposes,resultinginsignificantoutcomesabovethementioned\nvalues.However,thepreciseselectionofandinthecontextofunifiedscenarios\nremainsuncertain.Todetermineapotentialcosmologicaldeviationofµ,weutilize\ntheseparametervaluestocompareindividuallinesfromQSO0347-383withtheir\nassociatedlaboratory.TheQSO0347-383spectrawereacquiredusingtheEuropean\nSouthernObservatory'sUltravioletandVisibleEchelleSpectrographontheVeryLarge\nTelescope(WendandMolaro,2011).Thesespectraprovideadvantageousqualitieswith\n=53000,=30−70,accuraciesrangingfrom0.2to1ppmforthepresent\nanalysis.Ouranalysisrevealsthateachhydrogenlinemaintainsauniformand\nconsistentshape.Additionally,weassumeuniformvelocitiesforH2.Gaussianline-4fittingprofilesareemployedduringtheanalysisproceduretodeterminelinewidthsand\ncentralvelocities.SingleGaussianfitsindividualcharactervelocities,whilemultiple\nGaussiansaccommodatediversecharacters.Eachcharactercomponentisdenotedas(,\n,=2),whererepresentscolumndensity,indicatesabsorptionredshift,\n=2signifiesDopplerlinewidth,anddenotesroot-mean-square(RMS).Inour\napproach,H2simulatedlineswereinitiallygeneratedtoestimate∆values.The\nvaluesweresubsequentlydeterminedusingfittingparameters(∆,,).Forthe\ngiveninputdata,thefittingprocessonlyinvolved(2)andminimal(2)toderive\n∆μ/μwithareducedfitting2≃1.Thebest-fittingvalueof∆μ/μisdeterminedby\none-sigmaerrorwithinstandardqualityappropriatestatistics,signifyingtheuseofto\npredict∆μ/μ-changes.Subsequently,forestimatingerrors,themaximumratechanges\nin∆μ/μarebasedon∆2=1.The2-minimumisdeterminedforfittingeachline,\nleadingtotheexpectedoutcomeofpotentialcosmiceffectsontheproton-to-electron\nmassratio,∆μ/μ,aspresentedinTable1.2=∆μ/μ2+2isemployedtoassess\ntotalerrors.Theredshift-dependentvariationoftowardQSO0347-383isdepictedin\nFigure1.\nTable1:Thevalueof∆µ/µ=(0.120±0.144)×10−8isderivedfromthe\nweightedaverageofallH2lines.\nLineID(Å)(Å) z ∆(−)∆μ/μ(−)∆μ/μ(−)\nL14R13811.5038946.98043.02490250.1073 0.499960.5999\nW3Q13813.2825947.42193.02490430.1082 0.4196 0.5036\nW3P33830.3795951.67193.0248950.1455 0.5875 0.7050\nL13R13844.0442955.06583.0248999-0.2218 0.2105 0.2526\nL13P13846.6271955.70833.02489660.0658 0.1237 0.1484\nW2Q13888.4352966.09613.02489480.1574 0.2559 0.3071\nW2Q23893.2050967.28113.02489510.2056 0.5173 0.6208\nL12R33894.7939967.67703.02489040.1653 0.2641 0.3169\nW2Q33900.3288969.04923.0249028-0.1957 0.3910 0.4692\nL10R13952.7477982.07423.02489720.0112 0.2486 0.29845L10P13955.8160982.83533.02490220.1949 0.4674 0.5609\nL10R33968.3977985.96283.0248960-0.1496 0.3415 0.4099\nL10P33975.6657987.76883.0248950-0.0213 0.1993 0.2391\nW1Q23976.4877987.97453.02488900.0133 0.1204 0.1445\nL9R13992.7546992.01643.0248877-0.0326 0.1934 0.2321\nL9P13995.9594992.80963.02490000.2037 0.6210 0.7452\nL8R14034.76991002.45213.02490040.2055 0.5331 0.6397\nL8P34058.65751008.38603.02490460.2449 0.2647 0.3176\nW0R24061.21321009.02493.024889-0.1014 0.5395 0.6474\nL7P34103.38361019.50223.02488940.0094 0.1415 0.1699\nL6R34141.56401028.98663.024896-0.2418 0.2847 0.3416\nL6P34150.43491031.19263.0248882-0.1325 0.1921 0.2305\nL5R14174.42041037.14983.02489630.0353 0.1880 0.2257\nL5R24180.61521038.69033.024891-0.1914 0.2917 0.3501\nL5R34190.56901041.15883.0249086-0.0383 0.1277 0.1532\nL4R14225.98221049.95973.02489940.1573 0.2239 0.2687\nL4P14230.29741051.03253.0248969-0.2398 0.4084 0.4901\nL4R34242.15311053.97613.0249045-0.2238 0.3238 0.3886\nL4P34252.19111056.47143.0248994-0.0185 0.1959 0.2351\nL3R14280.32341063.46013.0249027-0.1292 0.2924 0.3509\nL3P14284.93491064.60543.02490430.2214 0.5214 0.6257\nL3R34296.48221067.47863.0248884-0.1590 0.5590 0.6709\nL3P34307.21141070.14093.02490120.0342 0.1742 0.2091\nL2P34365.2399946.98043.0248937-0.0853 0.1853 0.2224\nL1R14398.1291947.42193.0248913-0.1355 0.5355 0.64266L1P14403.4456951.67193.0248961-0.1263 0.2263 0.2716\nFigure1.Aplotfor∆μ/μversusredshifts\nThesingleH2linesofferameanstotestlimitationsonnaturalconstantssuchas\nandμ.However,discrepanciesarisewhendifferenttechniquesareappliedtothesame\nobservationaldata,revealingchallengesincontrolandunderstanding.Thesevariations\nderivefromuncertaintiesinwavelengthlaboratoriesandtheirassociatedanalysis\nmethodologies,leadingtodifferentconclusions(Ubachsetal.,2016;CalmetandFritsch,\n2002;Tzanavarrisetal.,2005;Salumbidesetal.,2012;Kosteleckyetal.,2003;Kinget\nal.,2012).\nInthiscontext,ourpresentanalysisusesanonlinearleast-squaresapproach,\napplyingobservablespectrawithuncertaintiesrangingfrom0.2to1ppmandupdated\nlaboratorywavelengths.Thistechniqueeffectivelyquantifies∆μ/μ-errorswith\nremarkableprecisionthroughGaussiandistribution(Le.,2019;2020;2021a,b,c).Our\nstudyaimstodemonstratethatasingleH2systemcansetconstraintsonpotential\nvariationsinphysicalconstants,suchasthefine-structureconstantandtheproton-to-\nelectronmassratio.7It'scrucialtoemphasizethatthecurrentstudyisparticularlyrelevantfor\ndeterminingvariationsintheproton-to-electronmassratiowithinhigh-redshiftsystems.\nAsaresult,weestablishanovellimitof∆µ/µ=(0.120±0.144)×10−8atthe\nabsorptionredshift=3.025,basingontheanalysisofH2lines.\n3.Discussions\nExplorationofchangesinfundamentalconstantsacrosscosmicspacetimewill\ngeneratenewcosmologicalmodelsandmysteriousphenomenabeyondtheSMPP.\nObservationaldataprovidesasapowerfultooltoproblethesephenomena.Recent\ninvestigationshaveunveiledaseriesofunifiedhypothesesconcerningspacetime\nvariationsinfundamentalconstants,particularlyandμ.Hence,studyingthe\ncosmologicalevolutionoftheseconstantsplaysanimportantlroleinadvancing\nunificationtheories.\nRecently,astronomicalmeasurementshaveyieldedimportantoutcomes.For\ninstance,measurementsinvolvingthe6-bearingdampedLymansystemswithinthe\nredshiftrangeof2≤≤3resultedinafindingof∆µµ≤10ppm(Levshakovetal.,\n2002;Malecetal.,2010;VarshalovichandLevshakov,1993;WendandMolaro,2011;\nKanekar,2011).Combiningthesamedatawithothermolecules,researchersestablished\nµ/µ=−(0.35±0.12)ppmand∆µ/µ=(0.10±0.47)ppmwith≤1.0,\nrespectively,alongwithdifferentialconclusionsdrawnfromstudyingasinglesystem\ntowardQ0528-250,whichresultedin∆µ/µ=(0.3±3.7)ppm(Rahmanietal.,2012;\n2013).Additionally,analysisofammoniatowardthequasarPKS-1830-211ataredshift\nof=0.89yieldeda∆µ/µ=(0.00±0.10)ppm.However,thisstudyonlyfocusedon\nthelow-redshiftsystem≤1.0applyingtheNH3andCH3OHlines(Levshakovetal.,\n2011).TheinvestigationofthequasarJ1337+3152produceda∆µ/µ=(−1.7±1.7)\nppmmeasurementat∼3.17,whileanotherstudysetalowerlimitof∆µ/µ=(0.0±\n1.5)ppmwith∼1.3utilizingaselectionoffour21-cmabsorptionsystems\n(Levshakovetal.,2010a,b).Consequently,recentresearchhaslargelylimitedthe\ncosmologicalvariationoftheproton-to-electronmassratiotothe~0.0028ppmlevel\n(Levshakovetal.,2011).\nProposinganeffectiveapproachforincreasingthislimitassociatesrealizingon\nsingleH2systems,usinghigh-resolutionspectrafromexistingopticalinstruments.\nSingleH2systems,giventheirnormalityacrosstheuniverse,provemorecrucialthan\nothermultipletlinesorlaboratorysetups.Throughoutourstudy,weidentifiedthatthe\nwidth-separationratiosignificantlycontributestomajorerrors,whichcouldbeasmall\nseparationrelatedtoobservationalandlaboratorywavelengths..Thissourceof\nuncertaintyrepresentsthemainerrorandvariesbetweenabsorptionsystems.Whena8largenumberofobservationsareused,theseuncertaintiesleadtoaverageout.Our\nestimationsofpotentialerrorsourcesinthecosmicvariationinµhavegreatlyadvanced\nduetotheseuncertainties(0.2-1ppm),whichoccurduetotherelationshipbetween\nobservationalandlaboratorywavelengthcalibrations.Moreover,theinfluenceofthe\nDopplershifteffecthasbeenidentifiedandquantifiedthroughtheseerrorsources,\nwhichwillnotaffectofourresults.Alinearmappingbetweenrealisticandmeasurable\ncalibrationerrorseparationscanyieldhighlyaccuratefindingswith∆µ/µ-dependence.\nConsequently,potentialerrorsourcesmayderiverfromobservationsorlaboratory-\nbasedcalculationsinvolvingthe2-computation.\n4.Conclusions.\nCompactastronomicalobjectsarebecomingprominenceastestingfor\ndimensionlessphysicsparadigms.Inthisstudy,wehaveindecatedthepredictive\ncapabilityofsingleH2systemsinevaluatingtheimpactofcosmicspacetimevariations\nonfundamentalconstants,specificallytheproton-to-electronmassratio'scorrelation\nwiththefine-structureconstantandphenomenologicalparameters(,).Our\ninvestigationestablishesanupperlimitof∆µ/µwithinthiscontext.\nWehavedemonstratedthatthecombinationofsingleH2systemsyields\nconstraintsonphenomenologicalparametersthatcharacterizeunificationmodels.This\nexplorationprobesintohowtherelationshipsamongphysicalconstantsareinfluenced\nwithintheframeworkofGUTmodels,wherevariationsinbothαandµareexpectedto\nvary.Notably,thelimitationssetbysinglelinesarebetterthanthosesetbymultiplet\nlinesandatomicclocktestsinthecontextofspacetimevariationsoftheseconstants.\nAsaresult,wehaveestablishedanupperlimitof∆µ/µwithinthiscontext,\nofferingprecisionbeyondpreviousstudiesthatreliedonmultipletlinesandatomic\nclockexperimentstoexaminespacetimevariationsinµ.Moreover,thisachievementis\nattributedtospatial/temporalapproximationsinextra-dimensionalcontextsatthelevel\nof∆µ/µ=(0.120±0.144)×10−8.\nThestudyofspatialortemporalvariationsinnaturalconstants(Murphyetal.,\n2008;Uzan,2011;Bagdonaiteetal.,2013;Liberati,2013;Will,2014;Martins,2017;\nLe.,2019;2020;2021a,b,c)generallycontributestonoveltheoreticalphysics\nextendingbeyondcurrentobservationalmodels.Themethodemployedtoprobethese\nvariations,suchascosmicchangesinµ,providespotentialfordeterminationonthe\nincompletenatureoftheEinsteinEquivalencePrinciple(EEP).Thepursuitofmore\nprecisemeasurementsoftheseconstantsfromastrophysicalsourcesspanningdiverse\nredshiftrangesshouldguidefutureresearchendeavors.Inanyscenario,thisstudy9broadensthefigureofpossibilitiesforidentifyingfeatureswithinGrandUnified\nTheories(GUTs).\n5.Dataavailability\nThedatausedtosupportthefindingsofthepresentstudyarelistedinTable1.\nDeclarationofCompetingInterest\nTheauthorhasnoconflictofinteresttodeclare.\nFunding\nThisresearchhasnotbeenreceivedexternalfunding.\nReferences\nBainbridge.M.BandWebb.J.K,2017.Artificialintelligenceappliedtotheautomatic\nanalysisofabsorptionspectra.Objectivemeasurementofthefinestructureconstant.\nMon.Not.R.Soc.468,1639.\nBagdonaite,Jetal.,2013.AStringentLimitonaDriftingProton-to-ElectronMass\nRatiofromAlcoholintheEarlyUniverse.Science.339,46.\nBagdonaite,J.,Ubachs,W.,Murphy,M.T.,andWhitmore,J.B,2015.Constraintona\nvaryingproton-electronmassratio1.5billionyearsafterthebigbang.Phys.Rev.Lett.\n114,071301.\nBlatt,Setal.,2008.Newlimitsoncouplingoffundamentalconstantstogravityusing\n87Sropticallatticeclocks.Phys.Rev.Lett.100,140801.\nCalmet,X.,andFritsch,H,2002.TheCosmologicalevolutionofthenucleonmassand\ntheelectroweakcouplingconstants.Eur.Phys.J.C.24,639.\nCoc,A.,etal.,2007.Coupledvariationsoffundamentalcouplingsandprimordial\nnucleosynthesis.Phys.Rev.D.76,023511.\nDzuba.V.A,Flambaum.V.VandWebb.J.K,1999.Calculationsoftherelativistic\neffectsinmany-electronatomsandspace-timevariationoffundamentalconstants.Phys\nRevA.59;230.\nFerreira,M.C.,Martins,C.J.A.P,2014.Furtherconsistencytestsofthestabilityof\nfundamentalcouplings.Phys.Rev.D.89,083011.\nHojjati.A,Plahn.A,Zucca.A,Pogosian.L,Brax.P,Davis.A.C,andZhao.G.B,2016.\nSearchingforscalargravitationalinteractionsincurrentandfuturecosmologicaldata.\nPhys.Rev.D.93;043531.\nKanekar,N,2011.CONSTRAININGCHANGESINTHEPROTON-ELECTRON\nMASSRATIOWITHINVERSIONANDROTATIONALLINES.Astrophys.J.728,\nL12.10King.J.A,Webb.J.K,Murphy.M.T,Flambaum.V.V,Carswell.R.F,Bainbridge.M.\nB,Wilczynska1.M.RandKoch.F.E,2012.Spatialvariationinthefine-structure\nconstant–newresultsfromVLT/UVES.Mon.Not.R.Soc.422,3370.\nKostelecky.A,Lehnert.RandPerry.M,2003.Spacetime-varyingcouplingsand\nLorentzviolation.Phys.Rev.D.68,123511.\nLe,T.D,2019.Asearchforthespace-timevariationsintheproton-to-electronmass\nratiousingthe[FeII]transitions.Chin.J.Phys.62,252.\nLe,T.D,2020.Newlimitonspace-timevariationsintheproton-to-electronmassratio\nfromanalysisofquasarJ110325-264515spectra.Symmetry.12,344.\nLe,T.D,2021a.Newlimitonspace-timevariationoftheproton-to-electronmassratio\nusinghigh-resolutionspectraofwhitedwarfstars.J.HighEnergyPhys.29,43.\nLe,T.D,2021b.Astudyofspace–timevariationofthegravitationalconstantusing\nhigh-resolutionquasarspectra.Gen.Relativ.Gravit.53,37.\nLe,T.D,2021c.Atestofcosmologicalspace–timevariationofthegravitational\nconstantwithstronggravitationalfields.Chin.J.Phys.73,147.\nLevshakov,S.A,2010a.Limitsonthespatialvariationsoftheelectron-to-protonmass\nratiointheGalacticplane⋆.Astron.Astrophys.512,A44.\nLevshakov,S.A,2010b.Searchingforchameleon-likescalarfieldswiththeammonia\nmethod⋆,⋆⋆II.MappingofcoldmolecularcoresinNH3andHC3Nlines.Astron.\nAstrophys.524,A32.\nLevshakov,S.A.,Dessauges-Zavadsky,M.,D’Odorico,S.,andMolaro,P,2002.A\nnewconstraintoncosmologicalvariabilityoftheproton-to-electronmassratio.Mon.\nNot.R.Astron.Soc.333,373.\nLevshakov,S.A,M.G.Kozlov,M.GandD.Reimers,D,2011.Methanolasatracerof\nfundamentalconstants.Astrophys.J.738,26.\nLevshakov,S.A,Kozlov,M.G,Agafonova.I.I,2020.Constraintsontheelectron-to-\nprotonmassratiovariationattheepochofreionization.Mon.Not.R.Astron.Soc.498,\n3624\nLiberati.S,2013.TestsofLorentzinvariance:a2013update.Class.Quant.Grav.30,\n133001.\nMalec,A.Letal.,2010.Kecktelescopeconstraintoncosmologicalvariationofthe\nproton-to-electronmassratio.Mon.Not.R.Astron.Soc.403,1541.\nMartins.C.J.A.P,Pinho.A.M.M,2017.Stabilityoffundamentalcouplings:aglobal\nanalysis.Phys.Rev.D.95,023008.\nMurphy.M.T,Flambaum.V.V,Muller.SandHenkel.C,2008.Stronglimitona\nvariableproton-to-electronmassratiofrommoleculesinthedistantuniverse.Science.\n320,1611.11Noterdaeme,P.,Ledoux,C.,Srianand,R.,Petitjean,P.,andLopez,S,2009.Diffuse\nmoleculargasathighredshift:DetectionofCOmoleculesandthe2175Ådustfeature\natz=1.64.Astron.Astrophys.503765.\nNoterdaeme,P.,Petitjean,P.,Ledoux,C.,Lopez,S.,Srianand,R.,andVergani,S.D,\n2010.Atranslucentinter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   },    {        "title": "2402.01997v1.Integral_formulas_for_slice_Cauchy_Riemann_operator_and_applications.pdf",        "content": "arXiv:2402.01997v1  [math.CV]  3 Feb 2024Integral formulas for slice Cauchy-Riemann\noperator and applications\nChao Ding and Xiaoqian Cheng\nAbstract. Thetheoryofslice regularfunctionshasbeendevelopedrap id-\nly in the past few years, and most properties are given in slic es at the\nearly stage. In 2013, Colombo et al. introduced a non-consta nt coeffi-\ncients differential operator to describe slice regular func tions globally,\nand this brought the study of slice regular functions in a glo bal sense. In\nthis article, we introduce aslice Cauchy-Riemannoperator , which is mo-\ntivated by the non-constant coefficients differential operat or mentioned\nabove. Then, A Borel-Pompeiu formula for this slice Cauchy- Riemann\noperator is discovered, which leads to a Cauchy integral for mula for\nslice regular functions. A Plemelj integral formula for the slice Cauchy-\nRiemann operator is introduced, which gives rise to results on slice reg-\nular extension at the end.\nMathematics Subject Classification (2010). 30G35, 32A30.\nKeywords. Slice Cauchy-Riemann operator, Borel-Pompeiu integral fo r-\nmula, Cauchy integral formula, Plemelj integral formula, s lice regular\nextension.\n1. Introduction\nClassicalCliffordanalysisstartedasageneralizationofaspectsofo nevariable\ncomplex analysis to higher dimensional Euclidean spaces. At the hear t of this\ntheory is the study of the Dirac operator, which generalizes the ro le of the\nCauchy-Riemann operator and null solutions to the Dirac operator are called\nmonogenic functions. The theory related to monogenic functions h as been\nfully developed and we know that most properties of holomorphic fun ctions\nare preserved for monogenic functions, such as Cauchy’s theore m, Cauchy\nintegral formula, mean value property, Liouville’s theorem, etc. Mo re details\ncan be found, for instance, in [2,5,10,21,23].\nIt is well known that polynomials given in terms of the complex variable\nare holomorphic as well in complex analysis. However, polynomials given in\nterms of vectors in higher dimensional Euclidean spaces are not mon ogenic2 Chao Ding and Xiaoqian Cheng\nanymore in classical Clifford analysis. This situation was changed when the\ntheory of slice Clifford analysis over quaternions was introduced by G entili\nand Struppa in 2006 [19,20], which was inspired by an earlier work done\nby Cullen in [9]. Later in 2010, Colombo, Sabadini and Struppa [6] gener al-\nized this idea to the general Clifford algebras and introduced the con cept of\nslice monogenic functions. Slice monogenic functions were firstly defi ned as\nfunctions which are holomorphic on each slice in the Euclidean space. T his\nis considered as a local definition for slice regular functions. Hence, many\nresults for slice monogenic functions are demonstrated in slices. Fo r instance,\nthe value of a slice monogenic function at some point pin the domain is given\nby a Cauchy integral formula with a regular kernel over a slice by app lying\nan representation formula in [3]. It is worth pointing out that the poin tpis\nnot necessarily in the slice.\nLater on, Ghiloni and Perotti [12] also built the theory of slice regula r\nfunctions on real alternative algebras with the concept of slice fun ctions in\n2011. This new approach allows the theory of slice regular functions to be\ndeveloped even further. For instance, the theory of slice regular functions\nhas also been developed into Fueter-regular functions over octon ions [11,26].\nRecently, a mean value formula for slice regular functions was introd uced by\nBisi and Winkelmann in [1], this formula leads to an important result, whic h\nsaysthatasliceregularfunctionoverquaternionsisalsoharmonicin acertain\nsense.In[16],Ghiloni andPerottiintroducedvolumeCauchyintegr alformula\nand volume Borel-Pompeiu formula for slice regular functions. More d etails\non slice regular functions can be found, for instance, in [7,8,12–16 ,18,24].\nIn 2013, Colombo et al. [4] introduced a non-constant coefficients d if-\nferential operator G, whose null solutions are closely related to slice regular\nfunctions when the domains are given with suitable conditions. The ad van-\ntage of studying this non-constant coefficients differential opera tor is that\nwe now have a explicit differential operator to define slice regular fun ctions\ninstead of a descriptive definition. This allows us to build a theory cent ered\non the differential operator G. In other words, the differential operator G\nplays the role of the Cauchy-Riemann operator in complex analysis wit h one\nvariable. Hence, later on, many researchers started to investiga te this global\ndifferential operator for slice regularity. For instance, in [25], Jin an d Ren\nintroduced a Cauchy-Pompeiu integral formula for slice monogenic f unction\nover octonions with complex spine. In [22], a global Borel-Pompeiu for mula\nand a gobal Cauchy-type formula for the non-constant coefficien ts differen-\ntialoperatorwerepresented.Theseformulascomesfromconne ctionsbetween\nthe Dirac operator and the slice Cauchy-Riemann operator, and th e Borel-\nPompeiu formula for the Dirac operator. This approach gives rise to a rather\ncomplicated extra integral over the domain in the Borel-Pompeiu for mula,\nand the Cauchy kernel in the slice case was not found in [22] neither.\nIn this article, we introduce some integral formulas, such as Borel-\nPompeiu formula, Cauchy integral formula, and Plemelj integral for mula,\nfor a slice Cauchy-Riemann operator, which is the same as the non-c onstantIntegral formulas for slice Cauchy-Riemann operator and applicat ions 3\ncoefficient differential operator up tp a multiplicative scalar term. Th is work\nleads to a further investigation on some boundary value problems fo r the\nslice Cauchy-Riemann operator, which will be demonstrated in an upc oming\narticle.\nThis article is organized as follows. In Section 2, we introduce some\npreliminaries on classical Clifford algebras, some definitions and basic w ell-\nknown results on slice analysis in the context of Clifford algebras. Sec tion\n3 is devoted to an investigation of integral formulas for the slice Cau chy-\nRiemann operator in a global sense. For instance, a Borel-Pompeiu f ormula\nand a Cauchy integral formula for the slice Cauchy-Riemann operat or are\ngivenhere.Further,aCauchytypeintegralisalsointroduced,wh ichnaturally\ngives rise to a study of the boundary behavior of the Cauchy type in tegral.\nThis study is demonstrated in details in Section 4, which leads to a Pleme lj\nintegral formula for the slice Cauchy-Riemann operator.Some res ults on slice\nregular extension are given at the end. It worth pointing out that s ome of\nthe results given in this paper are similar to the ones given in [16]. Howev er,\nin [4], the authors pointed that the kernel space of Gis not exactly the same\nas the space of slice regular functions. Further, these fundamen tal results on\nthe slice Cauchy-Riemann operator are crucial for a further inves tigation on\nthis topic.\n2. Preliminaries\nIn this section, we review some definitions and preliminaries results on slice\nClifford analysis, for more details, we refer the readers to [7].\nLetRmbe m-dimensional Euclidean space with a standard orthonormal\nbasis{e1,...,em}. The real Clifford algebra Clmis generated by Rmwith\nthe following relationship\neiej+ejei=−2δij,\nwhereδijis the Kronecker delta function. Hence a Clifford number x∈ Clm\ncan be written as x=/summationtext\nAxAeAwith real coefficients and A⊂ {1,...,m}.\nIf we denote Clk\nm={x∈ Clm:x=/summationtext\n|A|=kxAeA}, where|A|stands for the\ncardinality of the set A, then one can see that Clm=m/circleplusdisplay\nk=0Clk\nm. In particular,\nthe (m+1)-dimensional Euclidean space Rm+1can be identified with Cl0\nm⊕\nCl1\nmas\nRm+1−→ Cl0\nm⊕Cl1\nm,\n(x0,x1,...,x m)/ma√sto−→x0+x1e1+···+xmem.\nHence, we callelements in Cl1\nmvectorsand elements in Cl0\nm⊕Cl1\nmparavectors.\nIn this article, we denote x=/summationtextm\nk=1xkek, then, given a paravector x, we can4 Chao Ding and Xiaoqian Cheng\nwrite it as x=x0+x=: Re[x]+x\n|x||x|=:u+Ixv, whereu=x0, v=|x|=\n/parenleftbiggm/summationdisplay\nj=1x2\nj/parenrightbigg1\n2\nandIx=x\n|x|.\nLet the set of unit vectors denoted by\nS:={x=e1x1+···+emxm∈ Cl1\nm:x2\n1+···+x2\nm= 1}.\nOne can easily observe that Sis the unit sphere in Rmand ifI∈S, then\nI2=−1. Further, for I∈S, letCIbe the plane generated by 1 and I, which\nis isomorphic to the complex plane. An element in CIwill be denoted by\nu+Ivwithu,v∈R. Hence, given a paravector x, one can rewrite it as an\nelement in a suitable complex plane CI.\nDefinition 2.1. Let Ω∈Rm+1be an open set and f: Ω−→ Clmbe a real\ndifferentiable function. Let I∈S,fIis the restriction of fto the complex\nplaneCIandu+Ivis an element in CI.fisleft slice monogenic , if for all\nI∈S, we have\n1\n2/parenleftbigg∂\n∂u+I∂\n∂v/parenrightbigg\nfI(u+Iv) = 0\non ΩI:= Ω∩CI. Similarly, we can define right slice monogenicity for a\nfunction f, if for every I∈S, it satisfies\nfI(u+Iv)1\n2/parenleftbigg∂\n∂u+I∂\n∂v/parenrightbigg\n= 0\non ΩI.\nAs analogs of ∂and∂in one dimensional complex analysis, we define\nthe notion of I-derivative as\n∂I:=1\n2/parenleftbigg∂\n∂u−I∂\n∂v/parenrightbigg\n,∂I:=1\n2/parenleftbigg∂\n∂u+I∂\n∂v/parenrightbigg\n.\nHence, the condition of left slice monogenicity can be simply expresse d by\n∂If= 0 for short. More details can be found in [18].\nLateron,anotherapproachto slice regularitywasintroduced by G hiloni\nand Perotti in [13,14,17] with the concept of slice functions . Let us recall\nsome definitions as follows.\nDefinition 2.2. Given a set D⊂C, invariant with respect to complex con-\njugation, a function F:D−→ Clm⊗Cwhich satisfies F(z) =F(z) for all\nz∈Dis called a stem function on E.\nLetJ∈Sand ΦJ:C−→CJbe the canonical isomorphism which maps\na+ibtoa+Jb. Given an open set D⊂C, we denote\nΩD=∪J∈SΦJ(D)⊂Rm+1.\nIf an open set Ω ⊂Rm+1satisfies the form Ω = Ω D, then we say it is axially\nsymmetric . According to the definition of a stem function, such a functionIntegral formulas for slice Cauchy-Riemann operator and applicat ions 5\nF=F1+iF2onDwithF1,F2:D−→ Clminduces the slice function\nf=I(F) : ΩD−→ Clm, which satisfies\nf(x) =F1(z)+JF2(z),ifx=α+Jβ= ΦJ(z)∈ΩD∩CJ.\nWe denote the set of (left) slice functions on Ω Dby\nS(ΩD) :={f: ΩD→ Clm|f=I(F), F:D→ Clm⊗Cstem function },\nand a subspace of stem-functions as\nS1(ΩD) :={f=I(F)∈ S(ΩD)|F∈C1(D)}.\nAn important property of the slice function is the following represen ta-\ntion formula.\nTheorem 2.3 (Representation formula). [12] Let Dbe a symmetric domain\ninCand letΩD⊂Rm+1be defined as above. Further, let f∈ S(ΩD). Then,\nfor anyI∈Sandx=u+Ixv∈ΩD, whereIx∈S, we have\nf(x) =1−IxI\n2f(u+Iv)+1+IxI\n2f(u−Iv).\nDefinition 2.4. LetDbe a symmetric domain in Cand Ω D⊂Rm+1be\ndefined as above. Then, a slice function f: ΩD−→ Clmis calledslice regular\nif its stem function F=F1+iF2:D−→ Clm⊗Cis holomorphic, in other\nwords, its components F1,F2satisfy the Cauchy-Riemann equations:\n∂F1\n∂α=∂F2\n∂β,∂F1\n∂β=−∂F2\n∂α, z=α+iβ∈D.\nIn [4], the authors introduced a non-constant coefficients differen tial\noperator, which has close connections to slice regular functions. I ndeed, the\nspace of slice regular functions coincides with the kernel space of t his global\ndifferential operatorunder certain conditions on the domain. Here , we denote\ntheslice Cauchy-Riemann operator by\nG=∂\n∂x0+x\n|x|2m/summationdisplay\nj=1xj∂\n∂xj.\nThis differential operator is slightly different from the operator give n in [4]\nby a factor |x|2, which does not change the results obtained there except\nthat we need to be careful when the domain intersect the real line, which\ncreates singularities for the |x|2term in the differential operator. Hence, we\nintroduce the notation Rm+1\n∗:=Rm+1\\Rfor the rest of this article. This\ndifferential operator is more reasonable to be called slice Cauchy-Rie mann\noperator, since when j= 1, it reduces to the Cauchy-Riemann operator in\ncomplex analysis with one variable.6 Chao Ding and Xiaoqian Cheng\n3. Borel-Pompeiu formula for slice Cauchy-Riemann operato r\nIn this section, we introduce a global Borel-Pompeiu formula for the slice\nCauchy-Riemann operator. This formula has a similar form as in the cla ssical\ncase. In particular, it gives us a global Cauchy integral formula for the slice\nCauchy-Riemann operator. Firstly, we introduce a Stokes’ theor em for the\nslice Cauchy-Riemann operator as follows.\nTheorem 3.1 (Stokes’ theorem). LetΩD∈Rm+1\n∗be defined as above. f,g\nare Clifford-valued, continuously differentiable function s on an open set ΩI\nof the plane CI\\R, whose boundary consists of a finite number of piece-wise\nsmooth, closed curves. Then, we have\n/integraldisplay\n∂ΩIg(x)dxIf(x) =/integraldisplay\nΩI(gG)(x)f(x)+g(x)(Gf)(x)dVI(x),\nwheredxI=−Idx,dVIis the Lebesgue area element in ΩI, anddxis the\ninduced Lebesgue line element on the boundary ∂ΩI.\nProof.Indeed, we only need to notice that the restriction ofthe slice Cauc hy-\nRiemann operator to each slice coincides with the differential operat or∂I.\nMore specifically, recall that for x∈ΩI, we can rewrite it as x=u+Iv,\nwhereu=x0, v=|x|=/radicalbig\nx2\n1+···+x2m. Then, we can see that\n∂\n∂xj=∂v\n∂xj∂\n∂v=xj\nv∂\n∂v, j= 1,...,m.\nHence, the slice Cauchy-Riemann operator Gcan be rewritten as\nG=∂\n∂x0+x\n|x|2m/summationdisplay\nj=1xj∂\n∂xj=∂\n∂u+x\n|x|2m/summationdisplay\nj=1xjxj\nv∂\n∂v\n=∂\n∂u+x\n|x|∂\n∂v=∂\n∂u+I∂\n∂v.\nHence, the right hand side of the equation in the theorem above bec omes\n/integraldisplay\nΩI/bracketleftbigg\ng(x)/parenleftbigg∂\n∂u+I∂\n∂v/parenrightbigg/bracketrightbigg\nf(x)+g(x)/bracketleftbigg/parenleftbigg∂\n∂u+I∂\n∂v/parenrightbigg\nf(x)/bracketrightbigg\ndVI(x).\nThen, following the proof of Lemma 2 .23 in [3] completes the proof. /square\nInparticular,wehavethefollowingCauchy’sintegraltheoremasfo llows.\nTheorem 3.2 (Cauchy’s integral theorem). LetΩD∈Rm+1\n∗be defined as\nabove.f,gare Clifford-valued, continuously (real) differentiable fu nctions on\nan open set ΩIof the plane CI\\R, whose boundary consists of a finite number\nof piece-wise smooth, closed curves. Further, Gf(x) =gG(x) = 0for all\nx∈G. Then, we have\n/integraldisplay\n∂ΩIg(x)dxIf(x) = 0.Integral formulas for slice Cauchy-Riemann operator and applicat ions 7\nNow, we denote the Cauchy kernel for slice monogenic functions by\nS−1(q,x) =−(q2−2Re[x]q+|x|2)−1(q−x),\nwherex=u+Iv∈Rm+1, u,v∈R. This function is left slice monogenic\n(regular) in the variable qand right slice monogenic (regular) in the variable\nxin its domain of definition. More details can be found in [7,16]. Now,\nwe introduce a slice version of Borel-Pompeiu formula for the slice Cau chy-\nRiemann operator as follows.\nTheorem 3.3 (Borel-Pompeiu formula-slice version). LetΩD⊂Rm+1\n∗be de-\nfined as above and bounded, and f∈ S1(ΩD). Then, for each I∈S, we\nhave/integraldisplay\n∂ΩIS−1(q,x)dxIf(x)−/integraldisplay\nΩIS−1(q,x)(Gf)(x)dVI(x)\n=/braceleftBigg\n2πf(q),q∈ΩD,\n0,q∈Rm+1\n∗\\ΩD.\nProof.It is easy to see that the identity above is correctwhen q∈Rm+1\n∗\\ΩD,\nsinceS−1(q,x) is left slice monogenic with respect to q, which can also be\nannihilated by G.\nForq∈ΩD, we firstly consider the second integral on the left hand side\nabove/integraldisplay\nΩIS−1(q,x)(Gf)(x)dVI(x),\nwheredVI(x) is the area element on the plane CI.\nWe notice that the differential operator Greduces to ∂I(up to a con-\nstant) based on the proof in the previous theorem. We denote q=ζ+Iqη∈\nΩ, ζ,η∈R. We know that when S−1(q,x) has one real singularity, the inte-\ngral reduces to the classical Borel-Pompeiu formula, hence, we as sume that\nS−1(q,x)hastwosingularpointsinΩ Idenotedby s1=ζ+ηIands2=ζ−ηI\n(η/\\e}atio\\slash= 0). Let B1,ǫ={s∈ΩI:|s−s1|< ǫ},B2,ǫ={s∈ΩI:|s−s2|< ǫ}and\nBǫ=B1,ǫ∪B2,ǫ. One can see that B1,ǫandB2,ǫare two disks in the plane\nCIcentered at s1ands2respectively with radius ǫ. Then, we have/integraldisplay\nΩIS−1(q,x)(Gf)(x)dVI(x) =/integraldisplay\nΩI\\BǫS−1(q,x)(Gf)(x)dVI(x)\n+/integraldisplay\nB1,ǫS−1(q,x)(Gf)(x)dVI(x)+/integraldisplay\nB2,ǫS−1(q,x)(Gf)(x)dVI(x).(3.1)\nNoticethat S−1(q,x)canbeannihilatedby G,thenweapplyStokes’theorem\n(Theorem 3.1) to the first integral on the right hand side above to o btain\n/integraldisplay\nΩI\\BǫS−1(q,x)(Gf)(x)dVI(x) =/integraldisplay\n∂ΩIS−1(q,x)dxIf(x)\n−/integraldisplay\n∂B1,ǫS−1(q,x)dxIf(x)−/integraldisplay\n∂B2,ǫS−1(q,x)dxIf(x).(3.2)8 Chao Ding and Xiaoqian Cheng\nFollowing the argument of calculating the residues at s1ands2in Lemma\n2.30 in [3] and the representation formula given in Theorem 2.3, one can see\nthat/integraldisplay\n∂B1,ǫS−1(q,x)dxIf(x)+/integraldisplay\n∂B2,ǫS−1(q,x)dxIf(x) = 2πf(q),(3.3)\nwhenǫ→0. Now,we only need to calculatethe lasttwo integralsin (3.1). We\nclaim that both integrals approach zero when ǫgoes to zero. Here, we only\nshow this for B1,ǫ. Firstly, since f∈C1(ΩI), we can assume that |Gf| ≤M\nfor some positive constant M. Secondly, we assume that x∈B1,ǫandq=\nζ+Iqη, letx=s1+reIθ=ζ+ηI+reIθ, then we have\n|S−1(q,x)|\n=|(ζ+Iη)2−2(ζ+rcosθ)(ζ+Iη)+(ζ+rcosθ)2+(η+rsinθ)2|−1\n·|ζ+Iη−(ζ−ηI+re−Iθ)|\n=|(2rηcosθIq−2rηsinθ−r2)−1(ηIq+ηI+re−Iθ)|\n≤1\nr2|η|+r/radicalbig\nr2+4rηsinθ+4η2.\nNotice that 0 < r < ǫ and we assume that ǫ <|η|\n2, then we have r2+\n4rηsinθ+4η2≥2|η|2. Hence, we have\n|S−1(q,x)| ≤2|η|+|η|\n2/radicalbig\n2|η|2r=5\n2√\n21\nr.\nThus, we have that\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglelim\nǫ→0/integraldisplay\nB1,ǫS−1(q,x)(Gf)(x)dVI(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤lim\nǫ→0/integraldisplayǫ\n0/integraldisplay2π\n0M5√\n21\nr·rdθdr= 0.\nSimilar argument can show that\nlim\nǫ→0/integraldisplay\nB2,ǫS−1(q,x)(Gf)(x)dVI(x) = 0.\nCombining these with (3.1),(3.2) and (3.3), we have\n/integraldisplay\nΩIS−1(q,x)(Gf)(x)dVI(x) =/integraldisplay\n∂ΩIS−1(q,x)dxIf(x)−2πf(q),\nwhich completes the proof. /square\nIn particular, if f∈kerG, we have a Cauchy integral formula as follows.\nTheorem 3.4 (Cauchy integral formula-slice version). LetΩD⊂Rm+1\n∗be\ndefined as above and bounded, and f∈ S1(ΩD)∩kerG. Then, for each I∈S,\nwe have\n/integraldisplay\n∂ΩIS−1(q,x)dxIf(x) =/braceleftBigg\n2πf(q),q∈ΩD,\n0,q∈Rm+1\n∗\\ΩD.Integral formulas for slice Cauchy-Riemann operator and applicat ions 9\nNow, in order to investigate integral formulas for the slice Cauchy-\nRiemann operator in a global sense, we introduce a global Cauchy ke rnel\nas\nK(q,x) =2S−1(q,x)\nωm−1|x|m−1,\nwhereωm−1is the area of the (m-1)-sphere S. To obtain a global version of\nBorel-Pompeiu integral formula for the slice Cauchy-Riemann opera tor, we\nneed to integrate over the upper-half (m-1)-sphere S+:={x∈S:xm>0}\nwith respect to the variable I. More specifically, we adapt the argument used\nin [25, Theorem 4] to give a more concise proof to the following theore m.\nTheorem 3.5 (Borel-Pompeiu formula-global version). LetΩD⊂Rm+1\n∗be\ndefined as above and bounded, and f∈ S1(ΩD). Then, we have\n/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)−/integraldisplay\nΩDK(q,x)(Gf)(x)dV(x)\n=/braceleftBigg\n2πf(q), q∈ΩD,\n0, q∈Rm+1\n∗\\ΩD.\nwheren(x)is the outward unit normal vector to the boundary ∂ΩD.\nProof.Here, we only prove the case q∈ΩD. Since the idea is to rewrite the\nintegrals above as iterated integrals, we adapt the technique applie d in [25]\nto find out the Jacobians when we rewrite the integrals.\nForx=x0+x∈ΩD, we can rewrite x=rIwithI∈Sby spher-\nical coordinates. Then, we have the volume element dV(x) =dx0dV(x) =\nrm−1dx0drdS(I) wheredS(I) is the surface element on the sphere S. Hence,\nthe second integral over Ω Dabove becomes\n/integraldisplay\nΩDK(q,x)(Gf)(x)dV(x)\n=/integraldisplay\nS+/integraldisplay\nΩIK(q,x)(Gf)(x)rm−1dx0drdS(I)\n=/integraldisplay\nS+/integraldisplay\nΩI2S−1(q,x)\nωm−1rm−1(Gf)(x)rm−1dx0drdS(I)\n=2\nωm−1/integraldisplay\nS+dS(I)/integraldisplay\nΩIS−1(q,x)\nrm−1(Gf)(x)rm−1dVI(x)\n=2\nωm−1/integraldisplay\nS+/integraldisplay\nΩIS−1(q,x)(Gf)(x)dVI(x)dS(I). (3.4)\nFor the first integral over the smooth boundary ∂ΩD, we notice that x∈ΩD\ncanbewrittenas x=x0+x,andx∈Rm.Wecanadaptsphericalcoordinates\nforxas\nx=rIθ\n=r(cosθ1,sinθ1cosθ2,···,sinθ1···sinθm−2cosθm−1,sinθ1···sinθm−1).10 Chao Ding and Xiaoqian Cheng\nHence, the boundary ∂ΩDadmits a parameter expression\n(x0,r,Iθ) = (x0(t),r(t),Iθ).\nFort∈Ufor some domain U∈R. In other words, we have a map\nU×S−→Rm+1\\{R},\n(t,Iθ)/ma√sto−→(x0(t),r(t),Iθ) =: (r(t),Iθ).\nNow, we calculate the the Jacobian matrix with respect to t,θ1,...,θ m−1as\nJ=/parenleftbigg∂x0\n∂t∂r\n∂tIθ\n0rJIθ/parenrightbigg\n,\nwhereJIθis the Jacobian matrix of Iθwith respect to θ1,...,θ m−1and it is\nwell-known that/radicalBig\ndet(JIθJT\nIθ) = sinm−2θ1sinm−3θ2···sinθm−2.\nFurther, since IθIT\nθ= 1, differentiating both sides gives rise to IθJT\nIθ= 0.\nTherefore, we can calculate that\nJJT=/parenleftbigg∂x0\n∂t∂r\n∂tIθ\n0rJIθ/parenrightbigg/parenleftbigg∂x0\n∂t0\n∂r\n∂tIT\nθrJT\nIθ/parenrightbigg\n=/parenleftBigg/parenleftbig∂x0\n∂t/parenrightbig2+/parenleftbig∂r\n∂t/parenrightbig2r∂r\n∂tIθJT\nIθ\nr∂r\n∂tJIθIT\nθr2JIθJIT\nθ/parenrightBigg\n=/parenleftbigg/parenleftbig∂x0\n∂t/parenrightbig2+/parenleftbig∂r\n∂t/parenrightbig20\n0 r2JIθJT\nIθ/parenrightbigg\n,\nand\n√\ndetJJT=/radicalBigg/parenleftbigg∂x0\n∂t/parenrightbigg2\n+/parenleftbigg∂r\n∂t/parenrightbigg2\nrm−1sinm−2θ1sinm−3θ2···sinθm−2\n=|r′(t)|rm−1sinm−2θ1sinm−3θ2···sinθm−2.\nThis term leads to |r′(t)|dt=ds(x), which is the line element on ∂ΩI. Thus,\nwe have/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)\n=/integraldisplay\nS+/integraldisplay\n∂ΩI2S−1(q,x)\nωm−1rm−1n(x)f(x)|r′(t)|dtrm−1drdS(I)\n=2\nωm−1/integraldisplay\nS+dS(I)/integraldisplay\n∂ΩIS−1(q,x)n(x)f(x)|r′(t)|dt\n=2\nωm−1/integraldisplay\nS+/integraldisplay\n∂ΩIS−1(q,x)n(x)f(x)ds(x)dS(I). (3.5)\nNow, subtracting (3.4) from (3.5) gives us that\n/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)−/integraldisplay\nΩDK(q,x)(Gf)(x)dV(x)\n=2\nωm−1/parenleftbigg/integraldisplay\nS+/integraldisplay\n∂ΩIS−1(q,x)n(x)f(x)ds(x)dS(I)Integral formulas for slice Cauchy-Riemann operator and applicat ions 11\n−/integraldisplay\nS+/integraldisplay\nΩIS−1(q,x)(Gf)(x)dVI(x)dS(I)/parenrightbigg\n=2\nωm−1/integraldisplay\nS+/bracketleftbigg/integraldisplay\n∂ΩIS−1(q,x)n(x)f(x)ds(x)\n−/integraldisplay\nΩIS−1(q,x)(Gf)(x)dVI(x)/bracketrightbigg\ndS(I)\n=2\nωm−1/integraldisplay\nS+2πf(q)dS(I) = 2πf(q),\nwhere the last second equation comes from Theorem 3.3 and the fac t that\ndxI=n(x)ds(x) forx∈ΩI, which completes the proof. /square\nThe theorem above also gives a global Cauchy integral formula for t he\nslice Cauchy-Riemann operator as follows.\nTheorem 3.6 (Cauchy integral formula-global version). LetΩD⊂Rm+1\n∗be\ndefined as above and bounded, and f∈ S1(ΩD)∩kerG. Then, we have\n/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x) =/braceleftBigg\n2πf(q),q∈ΩD,\n0,q∈Rm+1\n∗\\ΩD.\nwheren(x)is the outward unit normal vector to the boundary ∂ΩD.\n4. Plemelj integral formula for slice Cauchy-Riemann opera tor\nIn this section, we investigate the boundary behavior of global Cau chy inte-\ngralgiven in Theorem 3.4 with the function fbelonging to H¨ older continuous\nfunction space in slice case defined below.\nDefinition 4.1. Let ΩD⊂Rm+1be defined as in the previous section with\nsmooth boundary ∂ΩD.fis a slice function on Ω Dandα∈(0,1]. If there\nexists a constant C >0 such that for all I∈Sandx,y∈ΩI,\n|f(x)−f(y)| ≤C|x−y|α,\nthenfis called a H¨ older continuous function of index α. The set of all such\nH¨ older continuous functions on Ω Dis denoted by Cα\ns(ΩD).\nNow, we review the Sokhotski-Plemelj formula in complex analysis as\nfollows. Let Cbe a smooth closed simple curve in the plane, and ϕan holo-\nmorphic function on C. The Cauchy type integral is given by\nϕ(z) =1\n2πi/integraldisplay\nCϕ(ζ)\nζ−zdζ.\nLetϕi(z) andϕe(z) stand for ϕ(z) withzis in the interior of Candzis in\nthe exterior of C, respectively. The Sokhotski-Plemelj formula states that\nlim\nω→zϕi(ω) =1\n2πip.v./integraldisplay\nCϕ(ζ)\nζ−zdζ+1\n2ϕ(z),12 Chao Ding and Xiaoqian Cheng\nlim\nω→zϕe(ω) =1\n2πip.v./integraldisplay\nCϕ(ζ)\nζ−zdζ−1\n2ϕ(z),(4.1)\nwherep.v.stands for the principal value of the Cauchy type integral.\nRecall that the representationformula given in Theorem 2.3 provide s us\na decomposition of the kernel K(q,x) in terms of some functions defined on\nslices. More importantly, these functions are actually the Cauchy k ernel in\neachslice,which allowsustoapplythe Sokhotski-Plemeljformulain the com-\nplex plane to each slice. The Plemelj formula for the slice Cauchy-Riema nn\noperator is stated as follows.\nTheorem 4.2 (Plemelj integral formula). LetΩD⊂Rm+1\n∗be defined as above\nwith smooth boundary ∂ΩD, andy(t)∈ΩDis a smooth path in Rm+1\n∗and\nit has non-tangential limit q∈∂ΩD. Then, for each H¨ older continuous slice\nfunction f: ΩD−→ Clmdefined on ΩD, we have\nlim\nt→01\n2π/integraldisplay\n∂ΩDK(y(t),x)n(x)f(x)dσ(x)\n=\n\nf(q)\n2+p.v.1\n2π/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x), y(t)∈ΩD;\n−f(q)\n2+p.v.1\n2π/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x), y(t)∈Rm+1\n∗\\ΩD.\nThe strategy is to rewrite the slice Cauchy integral over ∂ΩDas an\niterated integral, which consists of an integral over S+and an integral over\n∂ΩI. Then, we study the boundary behavior of the slice Cauchy integra l\ninstead, where the results in complex analysis can be applied to.\nProof.Here, we only present the details for the case y(t)∈ΩD, for the other\ncase, one can prove it similarly.\nFirstly, the equation (3.5) tells us that\n1\n2π/integraldisplay\n∂ΩDK(y(t),x)n(x)f(x)dσ(x)\n=1\nωm−1π/integraldisplay\nS+/integraldisplay\n∂ΩIS−1(y(t),x)n(x)f(x)ds(x)dS(I),(4.2)\nwhereI∈S. We denote y(t) =u(t) +Iyv(t), where u(t),v(t)∈R, and\ny±I(t) =u(t)±Iv(t). Then, Theorem 2.3 gives us that\nS−1(y(t),x) =1\n2/bracketleftbigg\nS−1(yI(t),x)+S−1(y−I(t),x)\n+IyI/bracketleftbig\nS−1(y−I(t),x)−S−1(yI(t),x)/bracketrightbig/bracketrightbigg\n.(4.3)\nFurther, it is known that [7, Proposition 2.7.11]\nS−1(q,x) = (x−q)−1=:C(q,x),Integral formulas for slice Cauchy-Riemann operator and applicat ions 13\nifqandxin the same complex plane CI, and that is the Cauchy kernel in\nCI. Therefore, we have\n/integraldisplay\n∂ΩIS−1(y(t),x)n(x)f(x)ds(x)\n=1\n2/parenleftbigg/integraldisplay\n∂ΩIC(yI(t),x)n(x)f(x)ds(x)+/integraldisplay\n∂ΩIC(y−I(t),x)n(x)f(x)ds(x)\n+IyI/bracketleftbigg/integraldisplay\n∂ΩIC(y−I(t),x)n(x)f(x)ds(x)−/integraldisplay\n∂ΩIC(yI(t),x)n(x)f(x)ds(x)/bracketrightbigg/parenrightbigg\n.\n(4.4)\nNotice that y±I(t) approach q±Iin Ω∩CInon-tangentially, when y(t) ap-\nproaches qin Ω non-tangentially. Hence, with the Sokhotski-Plemelj formula\non the complex plane given in (4.1), we have\nlim\nt→01\n2π/integraldisplay\n∂ΩI(C(yI(t),x)n(x)f(x)ds(x)\n=p.v.1\n2π/integraldisplay\n∂ΩDK(qI,x)n(x)f(x)dσ(x)+f(qI)\n2,\nlim\nt→01\n2π/integraldisplay\n∂ΩI(C(y−I(t),x)n(x)f(x)ds(x)\n=p.v.1\n2π/integraldisplay\n∂ΩDK(q−I,x)n(x)f(x)dσ(x)+f(q−I)\n2,\nwhere the constant iin (4.1) disappeared above because of the outward nor-\nmal vector n(x) . Plugging the equations above into (4.4), we have\n1\n2π/integraldisplay\n∂ΩIS−1(y(t),x)n(x)f(x)ds(x) =p.v.1\n2π/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)\n+1\n2/bracketleftbigg\nf(qI)+f(q−I)+IqI/bracketleftbig\nf(q−I)−f(qI))/bracketrightbig/bracketrightbigg\n=p.v.1\n2π/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)+f(q)\n2,\nwhere the two equations above rely on (4.3) and the fact that\n1\n2/bracketleftbigg\nf(qI)+f(q−I)+IqI/bracketleftbig\nf(q−I)−f(qI))/bracketrightbig/bracketrightbigg\n=f(q),\nwhich comes from Theorem 2.3. Therefore, with (4.2), we finally have\nlim\nt→01\n2π/integraldisplay\n∂ΩDK(y(t),x)n(x)f(x)dσ(x)\n=p.v.1\n2π/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)+f(q)\n2,\nwhich completes the proof. /square14 Chao Ding and Xiaoqian Cheng\nNoticethat K(q,x)isleftsliceregularwithrespectto q,andtheintegral\nF∂ΩDf(q) :=1\n2π/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)\nactuallydefinesasliceregularfunctionforall q∈Rm+1\\∂ΩD.Then,withthe\nPlemeljformulagivenabove,wecanhavearesultonsliceregularcont inuation\nas follows.\nCorollary 4.3. LetΩDbe defined as above with sufficiently smooth boundary\n∂ΩD. Then, the relation/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x) =πf(q),for allq∈∂ΩD\nis a necessary and sufficient condition so that F∂ΩDfis the slice regular\ncontinuation of finto the domain ΩD. On the other hand, the condition/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x) =−πf(q),for allq∈∂ΩD\nis a necessary and sufficient condition so that F∂ΩDfis the slice regular\ncontinuation of finto the domain Rm+1\n∗\\ΩD, which vanishes at q=∞.\nProof.Here, we only prove for the interior domain Ω D, the proof for the\nexterior domain Rm+1\n∗\\ΩDcase can be derived similarly.\nFirstly, let Abe the slice regular continuation into Ω Dof the function\nfgiven on ∂ΩD. Then, the Cauchy integral formula tells us that A(q) =\nF∂ΩDf(q).Therefore,thenon-tangentialboundaryvaluesof Aaref.Further,\nwith the Plemelj formula given in the previous theorem, we have\nf(q) =1\n2/parenleftbigg\nf(q)+1\nπ/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x)/parenrightbigg\n,\nwhich gives us that/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x) =πf(q),for allq∈∂ΩD.\nIf vice versa we have/integraldisplay\n∂ΩDK(q,x)n(x)f(x)dσ(x) =πf(q),for allq∈∂ΩD,\nthen,F∂ΩDfhas the boundary values f, thus it is the slice regular continu-\nation offinto Ω D, which completes the proof. /square\nConclusion\nIn this article, we investigated some integral formulas for the slice C auchy-\nRiemann operatorin a globalsense comparingwith the previous resu ltsgiven\non slices. Boundary behavior of the Cauchy type integral is also stu died here\nby a Plemelj integral formula, which gives rise to some results on slice r egu-\nlar continuation. This work leads to a further investigation to a Teod orescu\ntransform and some boundary value problems, which will be given in an\nupcoming paper.Integral formulas for slice Cauchy-Riemann operator and applicat ions 15\nAcknowledgments\nChao Ding is supported by the National Natural Science Foundation (NNSF)\nofChina(No.12271001)andtheNaturalScienceFoundationofAn huiProvin-\nce (No. 2308085MA03).\nDisclosure statement\nNo potential conflict of interest was reported by the author.\nReferences\n[1] C. Bisi, J. Winkelmann, The harmonicity of slice regular functions , J. Geom.\nAnal. Vol. 31, No. 8, 2021, 7773-7811.\n[2] F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis , Pitman, London, 1982.\n[3] F. Colombo, J.O. Gonz´ alez-Cervantes, I. Sabadini: A Cauchy kernel for slice\nregular functions , Ann. Glob. Anal. Geom. 37, 2010, 361-378.\n[4] F. Colombo, J.O. Gonz´ alez-Cervantes, I. Sabadini: A non constant coefficients\ndifferential operator associated to slice monogenic functio ns, Trans. Amer.Math.\nSoc. 365, 2013, 303-318.\n[5] F. Colombo, I. Sabadini, F. Sommen, D. C. Struppa: Analysis of Dirac systems\nand computational algebra , Progress in Mathematical Physics, 39. Birkh¨ auser\nBoston, 2004.\n[6] F. Colombo, I. Sabadini, D.C. Struppa: An extension theorem for slice mono-\ngenic functions and some of its consequences . Israel J. Math. 177, 2010, 369–489.\n[7] F. Colombo, I. Sabadini, D.C. Struppa: Noncommutative Functional Calcu-\nlus, Theory and Applications of Slice Hyperholomorphic Fun ctions, Progress in\nMathematics 289, Birkh¨ auser , 2011.\n[8] F. Colombo, I. Sabadini, and D. C. Struppa: Algebraic properties of the module\nof slice regular functions in several quaternionic variabl es, Indiana Univ. Math.\nJ. 61(4), 2012, 1581–1602.\n[9] C.G. Cullen: An integral theorem for analytic intrinsic functions on qua ternions.\nDuke Math. J. 32, 1965, 139–148.\n[10] R.Delanghe,F.Sommen,V.Souˇ cek: Clifford Analysis and Spinor Valued Func-\ntions, Kluwer Academic Dordrecht, 1992.\n[11] R. Ghiloni: Slice Fueter-regular functions , J. Geom. Anal. 31, no. 12, 2011,\n11988-12033.\n[12] R. Ghiloni, A. Perotti: Slice regular functions on real alternative algebras . Adv.\nMath. 226, 2011, 1662–1691.\n[13] R. Ghiloni, A. Perotti: A new approach to slice regularity on real algebras .\nIn Hypercomplex analysis and applications, Trends Math., p ages 109-123.\nBirkh´ auser/Springer Basel AG, Basel, 2011.\n[14] R. Ghiloni and A. Perotti: Slice regular functions on real alternative algebras .\nAdv. Math., 226(2):1662– 1691, 2011.\n[15] R. Ghiloni and A. Perotti: Slice regular functions of several Clifford variables ,\nin: AIP Conf. 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Murray: Clifford algebras and Dirac operators in har-\nmonic analysis , Cambridge Studies in Advanced Mathematics, 26. Cambridge\nUniversity Press, Cambridge, 1991.\n[22] J.O. Gonz´ alez-Cervantes, D. Gonz´ alez-Campos: The global Borel-Pompeiu-type\nformula for quaternionic slice regular functions , Complex Variables and Elliptic\nEquations, 66:5, 721-730, DOI: 10.1080/17476933.2020.17 38410.\n[23] K. G¨ urlebeck, K. Habetha, W. Spr¨ oßig: Holomorphic functions in the plane\nand n dimensional space , Birkh¨ auser Verlag, Basel, 2008.\n[24] A. Perotti: Almansi Theorem and Mean Value Formula for Quater-\nnionic Slice-regular Functions , Adv. Appl. Clifford Algebras, 30:61, 2020,\nhttps://doi.org/10.1007/s00006-020-01078-4.\n[25] M. Jin, G. Ren: Cauchy Kernel of Slice Dirac Operator in Octonions with\nComplex Spine , Complex Analysis and Operator Theory, 14, 2020, :17.\n[26] M. Jin, G. Ren, I. Sabadini: Slice Dirac operator over octonions , Israel J. Math.\n240, no. 1, 2020, 315-344.\nChao Ding\nCenter for Pure Mathematics,\nSchool of Mathematical Sciences,\nAnhui University, Hefei, P.R. China\ne-mail:cding@ahu.edu.cn\nXiaoqian Cheng\nSchool of Mathematical Sciences,\nAnhui University, Hefei, P.R. China\ne-mail:a02014026@stu.ahu.edu.cn"    },    {        "title": "2402.02065v1.Training_Implicit_Networks_for_Image_Deblurring_using_Jacobian_Free_Backpropagation.pdf",        "content": "Training Implicit Networks for Image Deblurring using Jacobian-Free\nBackpropagation\nLinghai Liu†, Shuaicheng Tong‡,andLisa Zhao§\nProject advisor: Samy Wu Fung¶\nAbstract. Recent efforts in applying implicit networks to solve inverse problems in imaging have achieved com-\npetitive or even superior results when compared to feedforward networks. These implicit networks\nonly require constant memory during backpropagation, regardless of the number of layers. However,\nthey are not necessarily easy to train. Gradient calculations are computationally expensive because\nthey require backpropagating through a fixed point. In particular, this process requires solving a\nlarge linear system whose size is determined by the number of features in the fixed point iteration.\nThis paper explores a recently proposed method, Jacobian-free Backpropagation (JFB), a back-\npropagation scheme that circumvents such calculation, in the context of image deblurring problems.\nOur results show that JFB is comparable against fine-tuned optimization schemes, state-of-the-art\n(SOTA) feedforward networks, and existing implicit networks at a reduced computational cost.\n1. Introduction. Inverse problems consist of recovering a signal, such as an image or a\nparameter of a partial differential equation (PDE), from noisy measurements, where direct\nobservation of the signal is not possible. As an effort to solve these problems, deep learning\ntechniques have been utilized to acquire high-quality medical images like magnetic resonance\nimaging (MRI) and computed tomography (CT) [45, 3, 44].\nConventional deep learning approaches for solving inverse problems use deep unrolling\n[2, 7, 33, 43, 29, 28, 8], which utilizes a fixed number of iterations usually chosen heuristically.\nA deep network is “unfolded” into a wider and shallower network, where each layer is split into\nmultiple sub-layers. While this method allows the network to learn complex patterns in the\ninput data, it suffers from overfitting and the well-known vanishing gradient problem [23], not\nto mention the lack of flexibility compared to other network structures [14, 35]. Moreover, they\nare challenging to train due to memory constraints in backpropagation. Another line of work,\nfeed-forward denoising convolutional neural networks [46, 47, 48, 39], use deep convolutional\nneural networks (CNNs) to learn the residuals between the ground truth images and noisy\nobservations instead of directly reconstructing the clean underlying images. These end-to-end\nmodels are not trained for a particular forward model, so they may require large amounts of\nlabeled data for training [14, 35].\nRecently, deep equilibrium models (DEQs) were proposed [5, 6, 42, 16, 26, 36, 37]. DEQs\nuse implicit networks with weight-tying, input-injected layers that propel the dynamics of la-\ntent space representation by sharing the input across layers. Training involves backpropagat-\ning through a fixed point of the layer using implicit differentiation, where the number of layers\ncan be deemed infinite. This feature allows implicit networks to save memory significantly\nsince there is no need to save any intermediate values on the backpropagation graph. Despite\n†Department of Applied Mathematics, Brown University (linghai liu@brown.edu)\n‡Department of Mathematics, University of California, Los Angeles (allentong24@ucla.edu)\n§Department of Statistics, University of California, Berkeley (lisazhao@berkeley.edu)\n¶Department of Applied Mathematics and Statistics, Colorado School of Mines\n1arXiv:2402.02065v1  [cs.LG]  3 Feb 2024L. LIU, S. TONG, AND L. ZHAO\nyielding fixed memory costs and matching performances of other state-of-the-art (SOTA)\nmodels, DEQs are still very expensive to train because backpropagation requires solving a\nJacobian-based linear system at every gradient evaluation [35]. To this end, a Jacobian-Free\nBackpropagation (JFB) approach was recently introduced to avoid solving the linear system\nduring training [10].\nThe theory of JFB allows us to replace the Jacobian matrix with the identity under certain\nconditions. JFB not only maintained a fixed memory cost but also avoided the substantial\ncomputational cost of Jacobian-based methods while ensuring a descent direction [10]. It\nhas performed well in image classification tasks [10], computational tomography [21, 22, 19],\ntraffic routing [21], and finding the shortest paths [31]. A variation of JFB, where the inverse\nJacobian was approximated as a perturbed identity matrix and falls back to JFB when the\nnorm between the approximation and the true inversion grows beyond a threshold [37], also\nproved to be successful in image classification. In this paper, we investigate JFB’s effectiveness\nin training implicit networks for inverse problems arising from image deblurring1.\n2. Mathematical Background.\n2.1. Problem Setup. We have Nnoisy blurred images {di}N\ni=1⊆Rnthat we refer to as\nmeasurements . The underlying original images , denoted as {xi}N\ni=1⊆Rn, are hidden from\nthe experimenter(s). We use the model:\n(2.1) d=Ax+ε\nwhere the forward operator Ais a mapping from signal space of original images to measure-\nment space and ε∈Rmis a noise term that models measurement errors. In this work, we\ndeal with image deblurring, where the forward process is a Gaussian blur. The value of each\npixel in a measurement is a weighted average of its neighborhood under a Gaussian kernel\nwith discretized weights of a 2-dimensional Gaussian density, plus noise generated from the\nhypothesized measurement process.\n2.2. Traditional Optimization for Deblurring. A natural idea is to apply A−1to (2.1)\nand obtain x∗=A−1(d−ε) when Ais invertible, which is the case in denoising ( A=I) and\ndeblurring ( Ais the Toeplitz matrix of a convolution operator). This can amplify the noise\nand result in really bad reconstructions when Ais ill-conditioned. Therefore, we estimate the\ntrue image x∗by formulating a regularized optimization problem that minimizes the difference\nbetween the reconstructed image and the observed image:\n(2.2) x∗= arg min\nx∈Rn1\n2||Ax−d||2\nL2+λR(x)\nwhere λ >0 is a tunable parameter, R(x) is a regularizer chosen based on common practice\n[15] or potentially learned from given data [4, 18, 20, 1]. We can solve (2.2) by applying the\ngradient descent algorithm, with a common choice of R(x) =1\n2||x||2\nL2. However, we observe\nin our numerical experiments that results are not ideal (low SSIM values for reconstructed\nimages). To this end, we explore the use of implicit networks in this work.\n1Access the GitHub repository at https://github.com/lliu58b/Jacobian-free-Backprop-Implicit-Networks\n2TRAINING IMPLICIT NETWORKS FOR IMAGE DEBLURRING USING JFB\n2.3. Implicit Networks and Challenges. Implicit networks [9] are newly proposed models\nthat also leverage the dataset {(di, xi)}N\ni=1and are capable of representing a wide range of\nfeedforward models. The idea is to find a fixed point for their weight-tying layers, modeled\nas a non-linear function T(·), and map it to the inference space. We formulate our implicit\nnetwork as follows:\n(2.3)[Equilibrium equation] TΘ(x) =x\n[Prediction equation] x∗(d) =x\nThe iteration to find the fixed point reads:\n(2.4) xk+1\ni=xk\ni−η\u0010\n∇x||Axk\ni−di||2\nL2+SΘ(xk\ni)\u0011\n:=TΘ(xk\ni), k = 0,1, . . . , K −1\nwhere η >0 is step size, x0\niis the initial “guess”, Kis the number of iterations (layers) in our\nneural network NΘ(·), and SΘ(·) is a trainable network containing all the weights of NΘ(·).\nNote that the [Equilibrium equation] is satisfied as K→ ∞ when TΘ(·) is a contraction. The\noutput of the network is NΘ(d) :=x∗(d), given by the [Prediction equation]. This iterative\nscheme is called DE-GRAD [14].\nTo update Θ, we perform implicit differentiation on the equilibrium equation in (2.3)\ndx∗\ndΘ=dTΘ(x∗)\ndx∗dx∗\ndΘ+∂TΘ(x∗)\n∂Θrearrange terms=⇒\u0012\nI−dTΘ(x∗)\ndx∗\u0013dx∗\ndΘ=∂TΘ(x∗)\n∂Θ\nand substitutedx∗\ndΘinto the gradient descent scheme after setting up a loss function ℓ:\n(2.5) Θ ←Θ−αdℓ\ndx∗\u0012\nI−dTΘ(x∗)\ndx∗\u0013−1∂TΘ(x∗)\n∂Θ\nwhere α >0 is the learning rate and ( I−dTΘ(x∗)\ndx∗) is the Jacobian matrix J. This update rule\nis costly due to the need to invert J, which motivates the search for other ways to speed up\nthe backpropagation process.\n3. Related Works.\n3.1. Deep Unrolling for Inverse Problems. Deep Unrolling [2, 7, 33, 43, 29, 28, 8] is used\nto unpack deep neural networks, whose black-box nature hinders their training and interpre-\ntation. Intuitively, each iteration is “unfolded” into smaller layers and then concatenated to\nform a deep network. Therefore, these neural networks can be interpreted as an optimization\nproblem as in (2.2), with a fixed number of Kiterations upon initialization, and the regular-\nizerR(x) can be parametrized to adaptably regularize the training process to minimize the\nloss of each estimate xk. Deep Unrolling has achieved successful results in other inverse prob-\nlems in imaging, such as low-dose CT [43], light-field photography [8], blind image deblurring\n[27], and emission tomography [32]. In our setup, this method corresponds to (2.4), with a\nfinite Kbeing fixed. From prior numerical experiments [14, 36], Kis eventually a relatively\nsmall, handcrafted number as a result of fine-tuning and limitations in time and space for\nboth training and inference.\n3L. LIU, S. TONG, AND L. ZHAO\n3.2. Implicit Networks. Deep Equilibrium Models (DEQs), a type of implicit networks,\nwere proposed [5, 6, 42, 16, 26, 14, 13]. The advantage of DEQ lies in that it requires less\nmemory because it uses a weight-tied, input-injected design, where it only has one layer of\nactual weights and the original input is fed into each of the identical layers. It solves the\nfixed-point and uses implicit differentiation to calculate the gradient for backpropagation. On\nthe other hand, SOTA deep feed-forward networks such as Deep Unrolling have memory issues\nsince they store intermediate values while iterating through each network layer.\n4. Proposed Methodology. The convergence criterion of TΘ(·) in (2.4) ( DE-GRAD ) is dis-\ncussed in more detail in [14], where SΘ−Ineeds to be ϵ−Lipschitz to ensure that TΘ(·)\nis contraction with parameter γ∈[0,1). We propose using Jacobian-free Backpropagation\n(JFB) [10], a recently-introduced algorithm that updates Θ at a lower computational cost.\nThe idea is to circumvent the Jacobian calculation in (2.5) by replacing Jwith the identity\nI, leading to an approximation of the true gradient:\n(4.1) pΘ=dℓ\ndx∗∂TΘ(x∗)\n∂Θ\nwhich is still a descending direction for the loss function ℓwith more constraints on TΘ\n[10]. Approximating Jwith Iis equivalent to taking the first term of the Neumann series\u0010\nI−dTΘ(x∗)\ndx∗\u0011−1\n=P∞\nk=0\u0010\ndTΘ(x∗)\ndx∗\u0011k\n. Some researchers adopt the first several terms for finer\napproximations while training Jacobian-based implicit networks [13]. Note the difference here\ncompared to implicit networks is that we are inverting the identity matrix rather than the\nJacobian Jin (2.5). Like in other works, we used Anderson acceleration [41] to facilitate the\nprocess of finding fixed points for the mapping TΘ(·) while keeping torch.no grad() . After\nfinding the root x∗, we resume the gradient tape and output TΘ(x∗) =x∗as an input of loss\nℓ. Then, we use PyTorch to calculatedℓ\ndx∗∂TΘ(x∗)\n∂Θ=dℓ\ndx∗∂x∗\n∂Θ, which is O(n2) because we fixed\nthe dimension of parameters once training starts. Even though JFB is not always performing\nthe steepest descent, the JFB step is much less costly to calculate, which lowers its overall\ncost while ensuring a descent direction. In contrast, implementing other models with implicit\ndifferentiation requires multiplying bydℓ\ndx∗and∂TΘ(x∗)\n∂Θ, which is also O(n2), to the left and\nright of the inverse J−1[13]. The complexity of these update rules is augmented mainly by\ninverting the Jacobian, which is hard to build explicitly and requires solving a large linear\nsystem.\nAlgorithm 4.1 Learning with JFB\nRequire: Implicit network NΘ(·) with weight-tying layers TΘ(·). Set learning rate α >0.\nformeasurement-truth pair ( d, x) in training set do\nFind fixed point: x∗=TΘ(x∗;d) with torch.no grad()\nOutput: NΘ(d) =TΘ(x∗)\nCalculate loss: ℓ(x∗, x)\nUpdate: Θ ←Θ−αdℓ\ndx∗∂TΘ(x∗)\n∂Θ\nend for\n4TRAINING IMPLICIT NETWORKS FOR IMAGE DEBLURRING USING JFB\n5. Experimental results. We mimic the format in [14] while implementing our JFB ap-\nproach. That is, we perform our experiments on the same dataset and use the same quality\nmeasures for image reconstruction.\n•Data : We use a subset of size 10,000 of the CelebA dataset [30], which contains around\n200,000 centered human faces with annotations. Among the subset of 10,000, 8,000\nimages are used for training and the rest are left for validating and testing purposes.\n•Preprocessing : Each image is resized into 128 ×128 pixels with 3 channels (RGB)\nand normalized into range [0 ,1] with mean1\n2for each channel. The blurred images\nare generated using Gaussian blurring kernels of size 5 ×5 with variance 1. The\nmeasurements are then crafted by adding white Gaussian noise with standard deviation\nσ= 10−2to the blurred images.\n•Network Architecture : We use a convolutional neural network (CNN) structure\nwith 17 layers. Except for the first and last CNN layer, each intermediate layer is\nfollowed by batch normalization and element-wise ReLU activation function. We also\napplied spectral normalization to each CNN layer to make sure that the mapping is\nLipschitz continuous with Lipschitz constant no greater than 1.\n•Training : With the aforementioned data and preprocessing specifics, the training\nstrictly follows Algorithm 1, where TΘ(·) is the same as defined in (2.4). As part of\nTΘ, the trainable network SΘ(·) is also pretrained as practiced in [14] to observe an\nimprovement in reconstruction.\n•Visualization : We first visualize the average training and validation loss per image\nover the number of epochs in Fig. 1, running on a sample of 2000 images, with an\n80-20 training-validation split.\n0 20 40 60 80 100\nNumber of Epochs5.005.255.505.756.006.256.50log\nLoss Plot of JFB\ntrain loss\nvalid loss\nFigure 1. Training and Validation losses of the DE-GRAD model with JFB\nWe see that as the number of epochs increases, the training loss decreases, while\nthe validation loss remains relatively high. For this proof of concept, we are using\nonly a subset of our dataset, so it is hard for the model to generalize well to the\nvalidation images. However, the validation loss still shows a decreasing trend. Also,\nthe JFB algorithm only promises a descent direction for the loss function, rather\n5L. LIU, S. TONG, AND L. ZHAO\nthan a technique that learns the structure of the data distribution. Nevertheless, the\ncomparable PSNR and SSIM values reported in Table 2 were calculated with a JFB\nmodel that achieved an MSE of about 100 per image (log 100 ≈4.605).\nGround Truth\nNoisy Blurred Image\nDirect Inverse\nGradient Descent\nJFB\nTable 1\nVisualization of JFB\nWe then visualize examples from the DE-GRAD model trained using JFB in Table 1,\nusing the same four images for each row. Ground Truths are the images in CelebA\ndataset resized to 128 ×128 pixels. Noisy Blurred Images are generated by a 5 ×5\nGaussian kernel mentioned above. Direct Inverse images are the results of applying\nthe inverse of Aon blurred images, which is defined in (2.1). Gradient Descent images\nare obtained by applying gradient descent using mean squared error (MSE) as the loss\nfunction. We enforce early stopping so that the results are not corrupted. JFB images\nare obtained using Noisy Blurred images as inputs with JFB-trained weights for the\nDE-GRAD model.\n•Comparison of Quality : We compare our results obtained from JFB with other\nstate-of-the-art methods using total variation (TV), standard deep neural networks,\nand DEQ. The metrics we use to assess the quality of reconstructed images are peak-\nsignal-to-noise ratio (PSNR, a positive number, best at + ∞) and structural similarity\nindex measure (SSIM, also positive, best at 1) [24]. The data in Table 2 are calculated\non the testing dataset of size 2000. Although we have not achieved results better than\nDEQs, which finds the true gradient in a complicated manner in (2.5), we currently\n6TRAINING IMPLICIT NETWORKS FOR IMAGE DEBLURRING USING JFB\nobserve results that are competitive with other techniques.\nTotal Variation Plug-n-Play [40] Deep Equilibrium [14] JFB (Ours)\nPSNR 26.79 29.77 32.43 26.88\nSSIM 0.86 0.88 0.94 0.91\nTable 2\nComparison across Models\n•Comparison of Time and Complexity : We compare the complexity per gradi-\nent/step computation using JFB and Jacobian-based backpropagation on the CelebA\ndataset. Here, n= 128 ×128×3 = 49152 after preprocessing. Importantly, we note\nthat JFB requires a Jacobian matrix-vector product per sample, leading to complexity\nO(n2). Jacobian-based backpropagation, however, also requires the inverse of J, i.e.,\nsolving a linear system as shown in (2.5).\nNa¨ ıvely using PyTorch for gradient tracking and taking the inverse of the Jacobian\nduring our experiments depletes all possible RAM (more than 32 Gigabytes). Hence,\nwe use the conjugate gradient method to solve the linear system in (2.5). The idea is\nto let w=dℓ\ndx∗J−1, and solve wJ JT=dℓ\ndx∗JTinstead of wJ=dℓ\ndx∗.\nIn our experiments, we fix a batch of 16 images in the CelebA dataset, run both\nthe Jacobian-based update and JFB for 20 repetitions, and then record the average\ntime needed in seconds for each parameter update. The dimensions of the images are\n“Image Size” by “Image Size” by 3 channels, where “Image Size” varies from 16 to\n128 at an increment of 16.\n20 40 60 80 100 120\nImage size01020304050607080Avg. Time Per Batch (s)Jacobian-based\nJacobian-free\nFigure 2. Computation Time per Batch for Jacobian-based method v.s. JFB\nFigure 2 shows that while maintaining comparable image reconstruction quality mea-\nsured by PSNR and SSIM, JFB algorithm is faster and easier to implement with\nauto-differentiation libraries such as Tensorflow or PyTorch.\n7L. LIU, S. TONG, AND L. ZHAO\n6. Conclusions. In this paper, we explored Jacobian-free Backpropagation for implicit\nnetworks with applications in image deblurring. Our approach recovers images rather effec-\ntively across the testing dataset. Moreover, JFB is competitive with other state-of-the-art\nmethods whose hand-crafted parameters have been fine-tuned across all stages. We also\ndemonstrated the advantage of JFB in terms of its computational complexity and easiness of\nimplementation in practice. Future work involves application to other inverse problems like\ndenoising [34, 38], geophysical imaging [17, 11, 12, 25], and more.\n7. Acknowledgements. 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Existingmethodsforsu chproblemseitheronlyguar-\nantee asymptotic convergence, have slow sublinear rates, o r require strong assumptions.\nTo address these challenges, we develop a novel penalty-bas ed approach that employs\nthe accelerated proximal gradient (APG) method. Under an α-H¨ olderian error bound\ncondition on the lower-level objective, our algorithm atta ins an (ǫ,l−β\nFǫβ)-optimal so-\nlution for any β >0 within O/parenleftbigg/radicalBig\nLf1\nǫ/parenrightbigg\n+O/parenleftBigg/radicalbigg\nlmax{α,β}\nFLg1\nǫmax{α,β}/parenrightBigg\niterations, where lF,Lf1\nandLg1denote theLipschitz constants of the upper-level objectiv e, the gradients of the\nsmooth parts of the upper- and lower-level objectives, resp ectively. If the smooth part\nof the upper-level objective is strongly convex, the result improves further. We also es-\ntablish the complexity results when both upper- and lower-l evel objectives are general\nconvex nonsmooth functions. Numerical experiments demons trate the effectiveness of\nour algorithms.\n1 Introduction\nBilevel optimization is a type of optimization framework where one pro blem is embedded\nwithin another. It has a hierarchical structure, in which the feasib le set of the upper-level\n∗Equal contribution\n†Corresponding author\n1problem is determined by the lower-level problem. This structure fr equently occurs in var-\nious real-world scenarios, such as meta-learning [Bertinetto et al., 2 018, Rajeswaran et al.,\n2019], hyper-parameter optimization [Franceschi et al., 2018, Sha ban et al., 2019], reinforce-\nment learning [Mingyi et al., 2020] and adversarial learning [Bishop et a l., 2020, Wang et al.,\n2021, 2022]. In this paper, we focus on a subclass of bilevel optimiza tion, namely simple\nbilevel optimization (SBO), which has attracted great attention in m achine learning soci-\nety owing to its applications in dictionary learning [Jiang et al., 2023, Be ck and Sabach,\n2014], lexicographic optimization [Kissel et al., 2020, Gong and Liu, 20 21], lifelong learning\n[Malitsky, 2017, Jiang et al., 2023]; see more details in Appendix A.\nThe goal of an SBO is to identify an optimal solution that effectively min imizes the\nupper-level objective function over the solution set of the lower- level problem. That is, we\nare interested in the following problem:\nmin\nx∈RnF(x) :=f1(x) +f2(x)\ns.t.x∈argmin\nz∈RnG(z) :=g1(z) +g2(z).(P)\nHeref1andg1are convex and differentiable real-valued functions, whose domains are subsets\nofRn;f2andg2:Rn→R∪ {∞} are proper, closed, and convex functions that may not\nbe differentiable. We assume, without loss of generality, that the op timal solution set of\nthe lower-level problem, denoted as Xopt, is nonempty and not a singleton; otherwise, the\noptimal solution is determined by the lower-level problem.\nLetF∗be the optimal value of problem (P) and G∗be the optimal value of the lower-level\nproblem, i.e., G∗= min x∈RnG(x). We say that ˜x∗is an (ǫF,ǫG)-optimal solution of problem\n(P) if it satisfies\nF(˜x∗)−F∗≤ǫF, G (˜x∗)−G∗≤ǫG. (1)\n1.1 Related Work\nThe inherent challenge in obtaining an ( ǫF,ǫG)-optimal solution for problem (P) stems from\nthe implicit nature of the feasible set Xopt. It results in difficulty when applying standard\nfirst-order methods, such as the projected gradient descent m ethod, to solve the problem, as\nthey typically demand an explicitly provided and relatively simple feasible set.\nVarious approaches have been developed to solve problem (P). One category is the regu-\nlarization method, which integrates the upper- and lower-level obj ectives through Tikhonov\nregularization [Tikhonov and Arsenin, 1977]\nmin\nx∈Rnη(x) :=σF(x) +G(x), (PReg)\nwhereσis the so-called regularization parameter. Cabot [2005] provided a r egularization ap-\nproach that converges asymptotically to the optimal solution set. Later, Dutta and Pandit\n2[2020] extended this framework to accommodate general closed a nd convex feasible domains.\nSolodov [2007] introduced the iterative regularized projected gra dient (IR-PG) method with\nσsatisfying the slow condition: lim k→∞σk→0, and/summationtext∞\nk=0σk=∞and asymptotic con-\nvergence was proven if FandGwere smooth. However, neither of these works provided\nnon-asymptotic convergence results. When Fis strongly convex and its domain is compact,\nAmini and Yousefian [2019] extended the IR-PG method and achieve d a convergence rate of\nO/parenleftbig\n1/K0.5−b/parenrightbig\nfor the lower-level problem, where b∈(0,0.5). Malitsky [2017] studied a ver-\nsion of Tseng’s accelerated gradient method that shows a converg ence rate of O(1/K) for the\nlower-level problem, while the convergence rate for the upper-lev el objective is not explicitly\nprovided. Kaushik and Yousefian [2021] proposed an iteratively reg ularized gradient (a-IRG)\nmethod can obtain a complexity of O/parenleftbig\n1/K0.5−b/parenrightbig\nandO/parenleftbig\n1/Kb/parenrightbig\nfor the upper- and lower-level\nobjective, respectively, where b∈(0,0.5). Motivated by this work, with a quasi Lipschitz\nassumption for F, Merchav and Sabach [2023] proposed a bi-sub-gradient (Bi-SG) m ethod\nthat enjoys O(1/Kb) andO(1/K1−b) for the lower- and upper-level objectives, respectively,\nwhereb∈(0.5,1); whenFis strongly convex, the convergence rate of the upper-level ob-\njective can be improved to be linear. Recently, under the weak-sha rp minima assumption\nof the lower-level problem, Samadi et al. [2023] proposed a regular ized accelerated proximal\nmethod (R-APM), showing a convergence rate of O(ǫ−0.5) for both upper- and lower-level\nobjectives. When the domain is compact and F,G are both smooth, Giang-Tran et al. [2023]\nproposed an iteratively regularized conditional gradient (IR-CG) m ethod, which ensures con-\nvergence rates of O(1/Kp) andO(1/K1−p) for upper- and lower-level objectives, respectively,\nwherep∈(0,1).\nAnother class of methods is based on the sub-level set, which involv es relaxing the con-\nstraint by replacing Xopt={x∈Rn|G(x)≤G∗}with its approximation. Beck and Sabach\n[2014] presented the Minimal Norm Gradient (MNG) method for the c ase ofFis strongly\nconvex and g2= 0. They provided the asymptotic convergence to the optimal solu tion\nset and a convergence rate of O/parenleftbig\nL2\ng1/ǫ2/parenrightbig\nfor the lower-level problem. Jiang et al. [2023]\nintroduced a conditional gradient-based bilevel optimization (CG-B iO) method when Fis\nassumed to be smooth, which approximates Xoptby replacing Gwith its linear approxima-\ntion. Their method invokes at most O(max{Lf1/ǫF,Lg1/ǫG}) of linear optimization oracles\nto achieve an ( ǫF,ǫG)-optimal solution. Instead of approximating Xopt, Doron and Shtern\n[2023] presented a scheme called iterative approximation and level-s et expansion (ITALEX),\nwhich approximates the sub-level set of Fto solve problem (P). They provided convergence\nguarantees of O(1/K) andO/parenleftBig\n1/√\nK/parenrightBig\nfor the lower- and upper-level objectives, respectively.\nApart from those, another approach is developed from the Seque ntial Averaging Method\n(SAM) framework, Sabach and Shtern [2017] proposed the Bilevel Gradient Sequential Av-\neraging Method (BiG-SAM), which provides an asymptotic converge nce for the upper-level\n3objective and achieves an O(Lg1/ǫ) convergence rate for the lower-level objective. Shen et al.\n[2023] combined an online framework with the mirror descent algorith m and established a\nconvergence rate of O(1/K1\n3) for both upper- and lower-level objectives, assuming a compact\ndomain and boundedness of the functions and gradients at both lev els. Furthermore, they\nshowed that the convergence rate can be improved to O(1/√\nK) under additional structural\nassumptions. Gong and Liu [2021] proposed a dynamic barrier gradie nt descent (DBGD)\nmethod with an asymptotical convergence to the optimal set.\nWe summarize these methods along with their assumptions and conve rgence results in\nTable 1.\nTable 1: Summary of simple bilevel optimization algorithms . The abbreviations “SC,” “C,”, “diff”,\n“comp”, “WS” and “C3” represent “strongly convex,” “convex ,”, “differentiable”, “composite”,\n“weak sharpness” and “Convex objective with Convex Compact constraints,” respectively. The\nabbreviation α-HEB refers to H¨ olderian error bound with exponent paramet erα. We only include\nthe gradient’s Lipschitz constant in the complexity result when its relation to the complexity is\nclear; otherwise, we omit it. Notations lF,Lf1, andLg1are the Lipschitz constants of F,∇f1, and\n∇g1, respectively.\nMethodsUpper-level Lower-level ( ǫF,ǫG)-optimal Convergence\nObjectiveF ObjectiveG Solution Upper-level Lower-level\nMNG [Beck and Sabach, 2014] SC, diff C, smooth ( /,ǫG) Asymptotic O/parenleftbig\nL2\ng1/ǫ2\nG/parenrightbig\nBiG-SAM [Sabach and Shtern, 2017] SC, smooth C, comp ( /,ǫG) Asymptotic O(Lg1/ǫG)\nIR-IG [Amini and Yousefian, 2019] SC C3, Finite sum ( /,ǫG) Asymptotic O/parenleftbigg\n1/ǫ1\n0.5−ε\nG/parenrightbigg\n,ε∈(0,0.5)\nIR-CG [Giang-Tran et al., 2023] C, smooth C3, smooth ( ǫF,ǫG) O/parenleftbigg\nmax{1/ǫ1\n1−p\nF,1/ǫ1\np\nG}/parenrightbigg\nTseng’s method [Malitsky, 2017] C, comp C, comp ( /,ǫG) Asymptotic O(1/ǫG)\nITALEX [Doron and Shtern, 2023] C, comp C, comp ( ǫ,ǫ2) O(1/ǫ2)\na-IRG [Kaushik and Yousefian, 2021] C, Lip C, Lip ( ǫF,ǫG) O/parenleftbigg\nmax{1/ǫ1\n0.5−b\nF,1/ǫ1\nb\nG}/parenrightbigg\n,b∈(0,0.5)\nCG-BiO [Jiang et al., 2023] C, smooth C3, smooth ( ǫF,ǫG) O(max{Lf1/ǫF,Lg1/ǫG})\nBi-SG [Merchav and Sabach, 2023]C, quasi-Lip/comp C, comp ( ǫF,ǫG) O/parenleftbigg\nmax{1/ǫ1\n1−a\nF,1/ǫ1\na\nG}/parenrightbigg\n,a∈(0.5,1)\nµ-SC, comp C, comp ( ǫF,ǫG)O/parenleftbigg\nmax{/parenleftBig\nlog1/ǫF\nµ/parenrightBig1\n1−a,1/ǫ1\na\nG}/parenrightbigg\n,a∈(0.5,1)\nR-APM [Samadi et al., 2023] C, smooth C, comp, WS ( ǫ,ǫ) O(max{Lf1/ǫ0.5,Lg1/ǫ0.5})\nOnline Framework [Shen et al., 2023] C, Lip C3, Lip ( ǫF,ǫG) O(max{1/ǫ3\nF,1/ǫ3\nG})\nOur methodC, comp, Lip C, comp, α-HEB ( ǫ,l−β\nFǫβ) O/parenleftbigg/radicalBig\nLf1\nǫ/parenrightbigg\n+O/parenleftbigg/radicalBig\nlmax{α,β}\nFLg1\nǫmax{α,β}/parenrightbigg\nµ-SC, comp, Lip C, comp, α-HEB ( ǫ,l−β\nFǫβ)O/parenleftbigg/radicalBig\nLf1\nµlog1\nǫ/parenrightbigg\n+O/parenleftbigg/radicalBig\nlmax{α,β}\nFLg1\nǫmax{α−1,β−1}log1\nǫ/parenrightbigg\nnonsmooth, Lip nonsmooth, Lip, α-HEB ( ǫ,l−β\nFǫβ) O/parenleftBigl2\nf2\nǫ2/parenrightBig\n+O/parenleftbigg\nlmax{2α,2β}\nf2l2\ng2\nǫmax{2α,2β}/parenrightbigg\n1.2 Our Method\nIn this paper, we consider the following penalty formulation of proble m (P):\nmin\nx∈RnΦγ(x) =F(x) +γp(x), (Pγ)\n4wherep(x) :=G(x)−G∗is the residual function. Obviously, we have p(x)≥0, andp(x) = 0\nif and only if x∈Xopt. Problem (P γ) can be seen as a penalization of the following value\nfunction reformulation of problem (P):\nmin\nx∈RnF(x) s.t.G(x)−G∗≤0. (PVal)\nIn (Pγ), the parameter γcontrols the trade-off between the optimality and the feasibility of\nproblem (P). The penalty formulation (P γ) is equivalent to (P Reg) withσ= 1/γ, despite their\ndiffering motivations. Although approaches based on (P Reg) are widely studied in literature,\nas far as we are aware, there exists no quantitative analysis invest igating the connection\nbetween (P γ) and (P)1.\nOur method is simple: we first carefully set suitable parameter γin different settings, then\napply accelerated proximal gradient (APG) methods [Nesterov, 20 18, Beck and Teboulle,\n2009, Lin and Xiao, 2014] to solve problem (P γ). We summarize our contributions as follows.\n•We explicitly analyze the relationship between an ǫ-optimal solution of the penalty\nformulation (P γ) and an (ǫF,ǫG)-optimal solution of problem (P). We further show\nthat anǫ-optimal solution xfor problem (P γ) also admits a lower bound for F(x)−F∗.\n•We provide a penalty-based APG (PB-APG) algorithm that attains an (ǫ,l−β\nFǫβ)-\noptimal solution of problem (P) with an O/parenleftbigg/radicalBig\nLf1\nǫ/parenrightbigg\n+O/parenleftbigg/radicalBig\nlmax{α,β}\nFLg1\nǫmax{α,β}/parenrightbigg\ncomplexity\nbound. Furthermore, if the upper-level objective is strongly con vex, our method PB-\nAPG-sc can improve the complexity bound to O/parenleftbigg/radicalBig\nLf1\nµlog1\nǫ/parenrightbigg\n+O/parenleftbigg/radicalBig\nlmax{α,β}\nFLg1\nǫmax{α−1,β−1}log1\nǫ/parenrightbigg\n.\nHere,lF,Lf1, andLg1are the Lipschitz constants of F,∇f1, and∇g1, respectively.\nWe further extend our method to the case that both upper- and lo wer-level objectives\nare general nonsmooth convex functions.\n•We present adaptive versions of PB-APG and PB-APG-sc with warm- start, which\ndynamically adjust the penalty parameters, and solve the associat ed penalized problem\nwith adaptive accuracy. The adaptive ones have the same complexit y bound as their\nprimal counterparts but can achieve superior performance in som e experiments.\n•We conduct experimental results that demonstrate the superior performance of our\nmethods.\n1To the best of our knowledge, the most related work is Shen and Che n [2023], which first analyzed the\nrelationship between approximate stationarity conditions of the or iginal BLO problem and its penalized\ncounterpart, and developed gradient-based algorithms with prov able complexity bounds.\n52 The Penalization Framework\nWe first establish some assumptions for FandGthat will play an important role in analyzing\nthe relationship between the penalty formulation (P γ) and problem (P).\nAssumption 2.1. (1)f1(x) andf2(x) arelf1- andlf2-Lipschitz continuous on dom( F),\nrespectively;\n(2)f1(x) is convex and its gradient, denoted as ∇f1, isLf1-Lipschitz continuous on\ndom(F);\n(3)g1(x) is convex and its gradient, denoted as ∇g1, isLg1-Lipschitz continuous on\ndom(G).\nUnder Assumption 2.1, the function F(x) islF-Lipschitz continuous on dom( F), where\nlF=lf1+lf2.\nWe remark that Assumption 2.1(1) is widely recognized and commonly a dopted in the\nliterature [Sabach and Shtern, 2017, Helou and Sim˜ oes, 2017, Kau shik and Yousefian, 2021,\nShen et al., 2023]. Moreover, Assumption 2.1(2) is more general tha n papers in the litera-\nture. In particular, while previous works such as Beck and Sabach [2 014], Jiang et al. [2023],\nGiang-Tran et al. [2023], Amini and Yousefian [2019] require the uppe r-level objective to be\nsmooth or strongly convex, we simply assume that Fis a composite function composed\nof a smooth convex function and a possibly non-smooth convex fun ction. For the lower-\nlevel objective, previous works such as Beck and Sabach [2014], Am ini and Yousefian [2019],\nJiang et al. [2023], Giang-Tran et al. [2023] impose smoothness assum ptions and, in some\ncases, convexity and compactness constraints on the domain; wh ile our approach does not\nrequire these additional constraints, allowing for more flexibility and generality.\nTo better analyze the connection between (P γ) and (P), we assume that p(x) satisfies\ntheα-exponential H¨ olderian error bound ( α-HEB) condition.\nAssumption 2.2 (H¨ olderian error bound) .The function p(x) :=G(x)−G∗satisfies the\nH¨ olderian error bound with exponent α≥1 andρ>0. Namely,\ndist(x,Xopt)α≤ρp(x),∀x∈dom(G),\nwhere dist( x,Xopt) := inf y∈Xopt/⌊a∇⌈⌊lx−y/⌊a∇⌈⌊l.\nWe remark that H¨ olderian error bounds are satisfied by many prac tical problems and\nwidely used in optimization literature [Pang, 1997, Bolte et al., 2017, Zh ou and So, 2017,\nRoulet and d’Aspremont, 2020, Jiang and Li, 2022]. There are two no table special cases: (i)\nWhenα= 1, we often refer to Xoptas a set of weak sharp minima of G[Burke and Ferris,\n61993, Studniarski and Ward, 1999, Burke and Deng, 2005, Samad i et al., 2023]; (ii) When\nα= 2, Assumption 2.2 is known as the quadratic growth condition [Drusv yatskiy and Lewis,\n2018a]. Additional examples of functions exhibiting H¨ olderian error bound, along with their\ncorresponding parameters, are presented in Appendix B.\nWe say ˜x∗\nγis anǫ-optimal solution of problem (P γ) if\nΦγ(˜x∗\nγ)−Φ∗\nγ≤ǫ,\nwhere Φ∗\nγis the optimal value of problem (P γ). We are now ready to establish the connection\nbetween approximate solutions of problems (P) and (P γ).\nThe subsequent two lemmas build upon the work of Shen and Chen [202 3] for (general)\nbilevel optimization. We apply their results to the simple bilevel problem (P) and generalize\nthe exponent αfrom 2 toα≥1.\nLemma 2.3. Suppose that Assumptions 2.1 and 2.2 hold with α>1. Then, for any ǫ>0,\nan optimal solution of problem (P)is anǫ-optimal solution of problem (Pγ)whenγ≥\nρlα\nF(α−1)α−1α−αǫ1−α.\nLemma 2.3 establishes the relationship between the optimal solution o f problem (P) and\ntheǫ-optimal solution of problem (P γ) whenα > 1. It also provides a lower bound for γ,\nwhich plays a pivotal role in the complexity results. The proofs of this paper are deferred to\nAppendix D.\nThe following lemma provides a more favorable outcome when α= 1.\nLemma 2.4. Suppose that Assumptions 2.1 and 2.2 hold with α= 1, an optimal solution of\nproblem (P)is an optimal solution of problem (Pγ)whenγ≥ρlF. Conversely, an optimal\nsolution of problem (Pγ)is also an optimal solution of problem (P)whenγ > ρlF. In this\ncase, we say that there is an exact penalization between prob lems (P)and(Pγ).\nWe remark that Theorem 1 of Samadi et al. [2023] also establishes th e relationship of\nthe solutions of problems (P γ) and (P) when α= 1. However, it does not mention the\nequivalence between optimal solutions of problems (P) and (P γ).\nFor simplicity, we define\nγ∗=/braceleftBigg\nρlα\nF(α−1)α−1α−αǫ1−αifα>1,\nρlF ifα= 1\nin the subsequent discussions. In our pursuit of achieving an ( ǫF,ǫG)-optimal solution of\nproblem (P), it is essential to consider the properties of the appro ximate optimal solutions\nof problem (P γ).\n7Theorem 2.5. Suppose that Assumptions 2.1 and 2.2 hold. For any given ǫ>0andβ >0,\nset\nγ=/braceleftBigg\nγ∗+ 2lβ\nFǫ1−βifα>1,\nγ∗+lβ\nFǫ1−βifα= 1.\nIf˜x∗\nγis anǫ-optimal solution of problem (Pγ), then, ˜x∗\nγis an (ǫ,l−β\nFǫβ)-optimal solution of\nproblem (P).\nParticularly, we can also provide a lower bound for F(˜x∗\nγ)−F∗.\nTheorem 2.6. Suppose that the conditions in Theorem 2.5 hold. Then, ˜x∗\nγsatisfies the\nfollowing suboptimality lower bound,\nF(˜x∗\nγ)−F∗≥ −lF(ρl−β\nFǫβ)1\nα.\nBy setting β=αin Theorem 2.6, we obtain F(˜x∗\nγ)−F∗≥ −ρ1\nαǫ. This, along with\nTheorem 2.5, gives\n|F(˜x∗\nγ)−F∗| ≤ O (ǫ).\nWe highlight that all the algorithms presented in this paper satisfy th e lower bound stated\nin Theorem 2.6. For brevity, we will omit further analysis regarding th is lower bound.\n3 Main Algorithms\nIn this section, we present two algorithms: the penalty-based acc elerated proximal gradient\n(PB-APG) algorithm and its adaptive variant, aPB-APG, to solve (P γ) approximately. Ac-\ncording to the results in the previous section, we obtain that the ap proximate solutions are\nalso approximate solutions for (P).\nTo simplify notations, we omit the constant term −γG∗, and rearrange problem (P γ) as\nfollows,\nmin\nx∈RnΦγ(x) :=φγ(x) +ψγ(x), (PΦ)\nwhereφγ(x) =f1(x) +γg1(x) andψγ(x) =f2(x) +γg2(x) represent the smooth and nons-\nmooth parts, respectively. Then, it follows that the gradient of φγ(x) isLγ-Lipschitz contin-\nuous withLγ=Lf1+γLg1.\nTo implement the APG methods, we need an assumption concerning th e property of the\nproximal mapping.\nAssumption 3.1. For anyγ > 0, the function ψγ(x) is prox-friendly, i.e., the proximal\nmapping\nproxtψγ(y) := argmin\nx∈Rn/braceleftbigg\nψγ(x) +1\n2t/⌊a∇⌈⌊lx−y/⌊a∇⌈⌊l2/bracerightbigg\n,\nis easy to compute for any t>0.\n8The function ψγ(x) represents the sum of two nonsmooth functions, and we remark that\nthe investigation of proximal mappings for such function sums is wide ly studied and used\nin the literature [Yu, 2013, Pustelnik and Condat, 2017, Adly et al., 20 19, Boob et al., 2023,\nLatafat et al., 2023].\n3.1 Accelerated Proximal Gradient-based Algorithm\nIn this subsection, we utilize the APG algorithm [Beck and Teboulle, 200 9, Lin and Xiao,\n2014] to solve problem (P Φ), as outlined in Algorithm 1. For simplicity, we denote Algorithm\n1 by ˆx= PB-APG( φγ,ψγ,Lf1,Lg1,x0,ǫ), where ˆxis anǫ-optimal solution for problem (P Φ).\nAlgorithm 1 Penalty-based APG (PB-APG)\n1:Input:γ,Lγ=Lf1+γLg1,x−1=x0∈Rn,R> 0,t−1=t0= 1,k= 0,ǫ> 0 and{tk}.\n2:repeat\n3:yk=xk+tk/parenleftbig\nt−1\nk−1−1/parenrightbig\n(xk−xk−1)\n4:xk+1= proxL−1\nγψγ(yk−L−1\nγ∇φγ(yk))\n5:k=k+ 1\n6:until2LγR2\n(k+1)2≤ǫ\n7:Output: x k\nThe convergence result of Algorithm 1 is stated as follows.\nLemma 3.2 (Corollary 2 in Tseng [2008]) .Suppose that Assumptions 2.1 and 3.1 hold. The\nsequence {tk}in Algorithm 1 satisfies\n1−tk+1\nt2\nk+1≤1\nt2\nk. (2)\nLetx∗\nγbe an optimal solution of problem (Pγ)and suppose that there exists a constant R\nsuch that /⌊a∇⌈⌊lx0−x∗\nγ/⌊a∇⌈⌊l ≤R. Then, the sequence {xk}generated by Algorithm 1 satisfies\nΦγ(xk)−Φγ(x∗\nγ)≤2LγR2\n(k+ 1)2. (3)\nWe remark that one example to obtain such a constant Ris to assume that dom( F)∪\ndom(G) is bounded, as done in many previous works [Amini and Yousefian, 2 019, Xu, 2022,\nJiang et al., 2023]. Moreover, if the Lipschitz constant Lγis unknown or computationally\ninfeasible, line search [Beck and Teboulle, 2009] can be adopted and w ill yield almost the\nsame complexity bound, which we omit for simplicity.\nCombining Theorem 2.5 and Lemma 3.2, we establish the following complex ity result for\nproblem (P).\n9Theorem 3.3. Suppose that Assumptions 2.1, 2.2, and 3.1 hold. Let γbe given as in\nTheorem 2.5. Algorithm 1 generates an (ǫ,l−β\nFǫβ)-optimal solution of problem (P)after at\nmostKiterations, where Ksatisfies\nK=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlmax{α,β}\nFLg1\nǫmax{α,β}\n.\nParticularly, when β≤α, we have\nK=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlα\nFLg1\nǫα/parenrightBigg\n.\nWe remark that Theorem 3.3 encompasses all the possible relationsh ips of magnitudes\nbetweenǫFandǫGin the definition of ( ǫF,ǫG)-optimal solution for problem (P), since β >0\nis arbitrary. Furthermore, it encompasses all scenarios for α≥1, whereαrepresents the\nexponent in the H¨ olderian error bound condition. Specially, set α= 1 andβ≤αin Theorem\n3.3, the number of iterations is K=O/parenleftBig/radicalbig\n(Lf1+lFLg1)/ǫ/parenrightBig\n. This result matches the lower\nbound for convex composite optimization with the Lipschitz gradient assumption for the\nsmooth part [Nesterov, 2018]. We note that this complexity agrees with the complexity\nresults proposed in Theorem 1 of Samadi et al. [2023] when f2≡0. However, in addition to\nthis complexity result, our study offers an explicit exact penalization relationship between\nproblems (P) and (P γ) as described in Lemma 2.4.\nAdditionally, if g1≡0, the total iteration number for obtaining an ( ǫ,ǫβ)-optimal solution\nof problems (P) is independent of γ, and it can be improved to K=O/parenleftBig/radicalbig\nLf1/ǫ/parenrightBig\n.\n3.2 Adaptive Version with Warm-Start\nA good initial point is crucial for practical performance of an algorit hm. Hence, it is desirable\nto explore algorithms that are more adaptive than the standard AP G. To this end, we\nintroduce Algorithm 2, which combines the incremental adjustment of penalty parameters\nand solution accuracies with a warm-start mechanism in the APG.In Algorithm 2, we adaptively update the penalty parameter γk, and invoke the PB-APG\nto generate an approximate solution for (P γ) withγ=γk. Meanwhile, we employ a warm-\nstart mechanism, meaning that the initial point for each subproblem is the output of the\npreceding subproblem. The convergence result of Algorithm 2 is as f ollows.\nTheorem 3.4. Suppose that Assumptions 2.1, 2.2, and 3.1 hold. Let ǫ0>0be given.\n•Whenα > 1, setν > ηα−1,N=⌈logη1−αν(ρLα\nF(α−1)α−1α−αǫ1−α\n0/γ0)⌉+andγ∗\nk=\nρLα\nF(α−1)α−1α−αǫ1−α\n0ηk(α−1).\n•Whenα= 1, setν >1,N=⌈logν(ρlF/γ0)⌉+andγ∗\nk=ρLF.\n10Algorithm 2 Adaptive PB-APG method (aPB-APG)\n1:Input: x 0∈Rn,γ0=γ1>0,Lf1,Lg1,ν >1,η> 1,ǫ0>0.\n2:fork≥0do\n3:φk(x) =f1(x) +γkg1(x)\n4:ψk(x) =f2(x) +γkg2(x)\n5: Invokexk= PB-APG( φk,ψk,Lf1,Lg1,xk−1,ǫk)\n6:ǫk+1=1\nηǫk\n7:γk+1=νγk\n8:end for\nThen, for any k≥N, Algorithm 2 generates an (ǫ0\nηk,2ǫ0\nηk(γ0νk−γ∗\nk))-optimal solution of problem\n(P)after at most Kiterations, where Ksatisfies\nK=O\n/radicalBigg\nLf1ηk\nǫ0\n+O\n/radicalBigg\nLg1γ0(ην)k\nǫ0\n.\nTheorem 3.4 shows that for any given initial accuracy ǫ0>0, Algorithm 2 can produce\nan approximate solution of problem (P) with desired accuracy.\nRemark 3.5.From Theorem 3.4, whenǫ\nη≤ǫ0\nηk≤ǫ, one can obtain an ( ǫ,ǫ\nγ0νk−γ∗\nk)-optimal\nsolution of problem (P) within O(/radicalbig\nLf1/ǫ) +O(/radicalbig\nLg1/ǫα) iterations, which is similar to the\ncomplexity results in Theorem 3.3.\n4 Extensions to Other Settings of SBO\nIn this section, we aim to explore the power of the penalized framewo rk under additional\nassumptions. Specifically, we present an enhanced convergence r esult by leveraging the\nstrong convexity of the upper-level objective. Then, we conside r the case where both upper-\nand lower-objectives are nonsmooth.\n4.1 Improved Results under Strong Convexity\nWe first examine the convergence performance under the strong convexity assumption of the\nupper-level objective.\nAssumption 4.1. f1(x) isµ-strongly convex on dom( F) withµ>0.\nAssumption 4.1 is another widely adopted setting in the existing simple- bilevel opti-\nmization literature [Beck and Sabach, 2014, Sabach and Shtern, 20 17, Amini and Yousefian,\n112019, Merchav and Sabach, 2023]. In this case, we will propose a va riant of PB-APG that\nprovides better complexity results than existing methods. Our main integration is an APG-\nbased algorithm to address the strongly convex setting, which has been studied in the existing\nliterature [Nesterov, 2013, Lin and Xiao, 2014, Xu, 2022]. We adopt the algorithm proposed\nin Lin and Xiao [2014] and modify it with a constant step-size for simplicit y as in Algorithm 3.\nSimilar to Algorithm 1, we also denote Algorithm 3 by ˆx= PB-APG-sc( φγ,ψγ,Lf1,Lg1,y0,ǫ).\nAlgorithm 3 PB-APG method for Strong Convexity Case (PB-APG-sc)\n1:Input:γ,Lγ=Lf1+γLg1,x−1,y0∈Rn,t1= 1.\n2:˜y=y0−L−1\nγ∇φγ(x−1)\n3:˜x= proxL−1\nγψγ(˜y−L−1\nγ∇φγ(˜y))\n4:Initialization: Letx−1=x0=˜x,k= 0\n5:repeat\n6:yk=xk+√\nLγ−√µ√\nLγ+√µ(xk−xk−1)\n7:xk+1= proxL−1\nγψγ(yk−L−1\nγ∇φγ(yk))\n8:k=k+ 1\n9:until (Lγ+µ\n2R2)(1−/radicalBig\nµ\nLγ)k≤ǫ\n10:Output: x k\nThe convergence guarantee of Algorithm 3 under the strong conv exity off1has been\nconstructed in the existing literature [Nesterov, 2013, Lin and Xiao , 2014]. Modified from\nTheorem 1 in Lin and Xiao [2014], we state it in the subsequent lemma for completeness.\nLemma 4.2. Suppose that Assumptions 2.1, 3.1, and 4.1 hold. Let x∗\nγbe an optimal solution\nof problem (Pγ)and suppose that there exists a constant Rsuch that max{/⌊a∇⌈⌊ly0−x∗\nγ/⌊a∇⌈⌊l,/⌊a∇⌈⌊l˜x−\nx∗\nγ/⌊a∇⌈⌊l} ≤R. Then, the sequence {xk}generated by Algorithm 3 satisfy\nΦγ(xk)−Φγ(x∗\nγ)≤/parenleftbiggLγ+µ\n2R2/parenrightbigg/parenleftbigg\n1−/radicalbiggµ\nLγ/parenrightbiggk\n. (4)\nCombining Lemma 4.2 and Theorem 2.5, we have the following complexity r esults for\nproblem (P).\nTheorem 4.3. Suppose that Assumptions 2.1, 2.2, 3.1, and 4.1 hold. Algori thm 3 can\nproduce an (ǫ,l−β\nFǫβ)-optimal solution ˜x∗of problem (P)after at most Kiterations, where\nKsatisfies\nK=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlmax{α,β}\nFLg1\nǫmax{α−1,β−1}log1\nǫ\n.\n12Theorem 4.3 improves the complexity results of Theorem 3.3. Specific ally, when 0 <β≤\nα= 1, we have linear convergence, i.e., K=O/parenleftbigg/radicalBig\nLf1+ρlFLg1\nµlog1\nǫ/parenrightbigg\n.\nAdditionally, we also introduce an adaptive variant of PB-APG-sc, de noted by aPB-\nAPG-sc, in Algorithm 4 of Appendix C.1. This algorithm also adaptively inv okesxk=\nPB-APG-sc( φk,ψk,Lf1,Lg1,xk−1,ǫk) and implements a warm-start mechanism.\n4.2 Performance of Nonsmooth Objectives\nIn this section, we focus on addressing the scenario where both th e upper- and lower-level\nobjectives are non-smooth, specifically, f1=g1= 0.\nIt is worth noting that in the case where both FandGare non-smooth, the convergence\nresult may not be as favorable as that in the previous scenario. This is primarily due to\nthe limited availability of information and unfavorable properties conc erning the functions\nFandG. For the nonsmooth cases, we employ a subgradient method to solv e problem (P γ),\nwhich has been extensively studied in the existing literature [Shor, 20 12, Bubeck et al., 2015,\nBeck, 2017, Nesterov, 2018]. Specifically, we update\nxk+1=xk−ηkξk, (5)\nwhereξk∈∂Φγ(xk) is an arbitrary subgradient of Φ γ(xk).\nLetx∗\nγbe an optimal solution of problem (P Φ) and suppose that there exists a constant\nRsuch that /⌊a∇⌈⌊lx0−x∗\nγ/⌊a∇⌈⌊l ≤R. Then, motivated by Theorem 8.28 in Beck [2017], we establish\nthe subsequent complexity result for problem (P).\nTheorem 4.4. Suppose that f2andg2arelf2- andlg2-Lipschitz continuous, respectively. Set\nstep-sizeηk=R\nlγ√k+1in(5). Then, the subgradient method produces an (ǫ,l−β\nf2ǫβ)-optimal\nsolution of problem (P)after at most Kiterations, where Ksatisfies\nK=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftBigg\nlmax{2α,2β}\nf2l2\ng2\nǫmax{2α,2β}/parenrightBigg\n.\nIn the setting of non-smooth SBO problems, our proposed method has lower complex-\nity compared to existing approaches. Specifically, under a bounded domain assumption,\nHelou and Sim˜ oes [2017] proposed an ǫ-subgradient method with an asymptotic rate towards\nthe optimal solution set, which is worse than ours. The a-IRG metho d in Kaushik and Yousefian\n[2021] achieved convergence rates of O(1/ǫ1\n0.5−b) andO(1/ǫ1\nb) for the upper- and lower-level\nobjectives, respectively, where b∈(0,0.5). Setting b= 0.25 yields their convergence rates\nofO(1/ǫ4) for both upper- and lower-level objectives, which implies that our complexity is\nmore efficient than theirs when α < 2 andβ≤α. Furthermore, the online framework pro-\nposed in Shen et al. [2023] performed a complexity of O(1/ǫ3) for both upper- and lower-level\nobjectives. Similarly, our approach prevails over theirs when α<1.5 andβ≤α.\n13We next explore the improved complexity result for problem (P) when f2is assumed to\nbeµf2-strongly convex, based on Theorem 8.31 in Beck [2017].\nTheorem 4.5. Suppose that f2andg2arelf2- andlg2-Lipschitz continuous, respectively,\nandf2isµf2-strongly convex. Choose step-size ηk=2\nµf2(k+1)in(5). Then, the subgradient\nmethod produces an (ǫ,l−β\nf2ǫβ)-optimal solution of problem (P)after at most Kiterations,\nwhereKsatisfies\nK=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftBigg\nlmax{2α,2β}\nf2l2\ng2\nµf2ǫmax{2α−1,2β−1}/parenrightBigg\n.\nTo the best of our knowledge, in the setting of the above theorem, existing results cannot\nuse the additional strong convexity to obtain better outcomes. I n contrast, our method lever-\nages the unique structural properties, which leads to improved co mplexity results compared\nto Theorem 4.4 when α<2 andβ≤α.\n5 Numerical Experiments\nWe apply our Algorithms 1, 2, 3 and 4, namely, PB-APG, aPB-APG, PB- APG-sc and aPB-\nAPG-sc, to two simple bilevel optimization problems from the motivatin g examples in Ap-\npendix A. The performances of our methods are compared with six e xisting methods: MNG\n[Beck and Sabach, 2014], BiG-SAM [Sabach and Shtern, 2017], DBGD [G ong and Liu, 2021],\na-IRG [Kaushik and Yousefian, 2021], CG-BiO [Jiang et al., 2023] and B i-SG [Merchav and Sabach,\n2023]. For practical efficiency, we use the Greedy FISTA algorithm p roposed in Liang et al.\n[2022] as the APG method in our approach. Detailed settings and add itional experimental\nresults are presented in Appendix E.\n5.1 Logistic Regression Problem (LRP)\nThe LRP reads\nmin\nx∈Rn1\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2\ns.t.x∈argmin\nz∈Rn1\nmm/summationtext\ni=1log(1 + exp( −aT\nizbi)) +IC(z),(6)\nwhereIC(x) is the indicator function of the set C={x∈Rn:/⌊a∇⌈⌊lx/⌊a∇⌈⌊l1≤θ}withθ= 10.\nOur goal is to find a solution to the lower-level problem with the smalles t Euclidean norm.\nThe upper-level objective only consists of the smooth part, which is 1-strongly convex and\n1-smooth; meanwhile, the lower-level objective is a composite func tion, where the smooth\npart is1\n4mλmax(ATA)-smooth, and the nonsmooth part is prox-friendly [Duchi et al., 2 008].\n14In this experiment, we compare our methods with MNG, BiG-SAM, DBG D, a-IRG, CG-\nBiO, and Bi-SG. We plot the values of residuals of the lower-level obje ctiveG(xk)−G∗and\nthe upper-level objective over time in Figure 1.\n0 0.1 0.2 0.3 0.4 0.5 0.610-1010-5100\n0 0.1 0.2 0.3 0.4 0.5 0.605101520\nFigure 1: The performances of our methods compared with othe r methods in LRP.\nAs shown in Figure 1, the PB-APG, aPB-APG, PB-APG-sc, and aPB-A PG-sc algorithms\nexhibit significantly faster convergence performance than the ot her methods for both lower-\nand upper-level objectives. This is because our methods achieve lo wer optimal gaps and\ndesired function values of the lower- and upper-level objectives w ith less execution time.\nThis observation confirms the improved complexity results stated in the theorems above.\nAlthough the high exactness of our methods for the lower-level pr oblem leads to larger\nupper-level objectives, Table 3 in Appendix E.1 shows that our meth ods are much closer to\nthe optimal value. This is reasonable because the other methods ex hibit lower accuracy at\nthe lower-level problem, resulting in larger feasible sets compared t o the lower-level optimal\nsolution set Xopt. In addition, Figure 1 demonstrates that aPB-APG and aPB-APG-s c\noutperform PB-APG and PB-APG-sc in terms of convergence rate . This improvement can\nbe attributed to the adaptiveness incorporated in Algorithms 2 and 4.\n5.2 Least Squares Regression Problem (LSRP)\nThe LSRP has the following form:\nmin\nx∈Rnτ\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2+/⌊a∇⌈⌊lx/⌊a∇⌈⌊l1\ns.t.x∈argmin\nz∈Rn1\n2m/⌊a∇⌈⌊lAz−b/⌊a∇⌈⌊l2,(7)\nwhereτ= 0.02 regulates the trade-off between ℓ1andℓ2norms. We aim to find a sparse\nsolution for the lower-level problem. The upper-level objective is f ormulated as a composite\n15function, which consists of a τ-strongly convex and τ-smooth component, along with a\nproximal-friendly non-smooth component [Beck, 2017]. The lower-le vel objective is a smooth\nfunction with a smoothness parameter of1\nmλmax(ATA).\nIn this experiment, we compare the performances of our methods with a-IRG, BiG-SAM,\nand Bi-SG. We plot the values of residuals of lower-level objective G(xk)−G∗and the\nupper-level objective over time in Figure 2.\n0 10 20 30 4010-810-610-410-2100102\n0 10 20 30 40020406080100\nFigure 2: The performances of our methods compared with othe r methods in LSRP.\nFigure 2 shows that the proposed PB-APG, aPB-APG, PB-APG-sc, and aPB-APG-sc\nconverge faster than the compared methods for both the lower- and upper-level objectives,\nas well. For the upper-level objective, our methods achieve larger function values than other\nmethods, except BiG-SAM ( δ= 0.01). This is because our methods attain higher accuracy\nfor the lower-level objective than other methods. We have similar o bservations in Section 5.1.\nFurthermore, Figure 1 also demonstrates that the adaptive mech anism produces staircase-\nshaped curves for aPB-APG and aPB-APG-sc, which might prevent undesirable fluctuations\nin PB-APG and PB-APG-sc.\n6 Conlusion\nThis paper introduces a novel penalty formulation to address the m inimization of a composite\nconvex function as the upper-level objective within the optimal so lution set of a lower-level\nproblem, which is also composite convex. By assuming an α-H¨ olderian error bound, our\napproach, PB-APG, achieves superior complexity results compare d to the existing methods.\nFurthermore, we improve the complexity results when the smooth p art of the upper-level\nobjective is strongly convex. 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First-order methods for problems with o (1) functio nal constraints can have\nalmost the same convergence rate as for unconstrained problems .SIAM Journal on Opti-\nmization , 32(3):1759–1790, 2022.\nYao-Liang Yu. On decomposing the proximal map. Advances in neural information process-\ning systems , 26, 2013.\nZirui Zhou and Anthony Man-Cho So. A unified approach to error bo unds for structured\nconvex optimization problems. Mathematical Programming , 165:689–728, 2017.\n21A Motivating Examples\nMany machine learning applications involve a primary objective G, which usually represents\nthe training loss, and a secondary objective F, which can be a regularization term or an\nauxiliary loss. A common approach for such problems is to optimize Gfully and then use\nFto select the optimal solutions from the ones obtained for G. This is called lexicographic\noptimization [Kissel et al., 2020, Gong and Liu, 2021]. Two classes of lex icographic opti-\nmization problems are the regularized problem, also known as the ill-po sed optimization\nproblem [Amini and Yousefian, 2019, Jiang et al., 2023], and the over- parameterized regres-\nsion [Jiang et al., 2023], where the upper-level objectives are the re gularization terms or loss\nfunctions, and the lower-level objectives are the loss functions a nd the constraint terms. We\npresent some examples of these classes of problems as follows.\nExample A.1 (Linear Inverse Problems) .Linear inverse problems aim to reconstruct a\nvectorx∈Rnfrom measurements b∈Rmthat satisfy b=Ax+ρε, whereA:Rn→Rm\nis a linear mapping, ε∈Rmis unknown noise, and ρ > 0 is its magnitude. Various opti-\nmization techniques can address these problems. We focus on the b ilevel formulation, widely\nadopted in the literature [Beck and Sabach, 2014, Sabach and Shte rn, 2017, Dempe et al.,\n2021, Latafat et al., 2023, Merchav and Sabach, 2023].\nThe lower-level objective in the bilevel formulation is given by\nG(x) =1\n2m/⌊a∇⌈⌊lAx−b/⌊a∇⌈⌊l2+IC(x), (8)\nwhereIC(x) is the indicator function of a set Cthat satifies IC(x) = 0 if x∈C, and\nIC(x) = +∞ifx/∈C. The setCis a closed, convex set that can be chosen as C=Rn,\nC={x∈Rn:x≥0}, orC={x∈Rn:/⌊a∇⌈⌊lx/⌊a∇⌈⌊l1≤θ}for someθ>0.\nThis problem may have multiple minimizer solutions. Hence, a reasonable option is to\nconsider the minimal norm solution problem, i.e., find the optimal solutio n with the smallest\nEuclidean norm [Beck and Sabach, 2014, Sabach and Shtern, 2017, Latafat et al., 2023]:\nF(x) =1\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2.\nWe need to solve the simple bilevel optimization problem:\nmin\nx∈Rn1\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2\ns.t.x∈argmin\nz∈Rn1\n2m/⌊a∇⌈⌊lAz−b/⌊a∇⌈⌊l2+IC(z).\nExample A.2 (Sparse Solution of Linear Inverse Problems) .Consider the same setting as\nin Example A.1, but with the additional goal of finding a sparse solution among all the\nminimizers of the linear inverse problem (8). This can simplify the model and improve\n22computational efficiency. To achieve sparsity, we can use any func tion that encourages it.\nOne such function is the well-known elastic net regularization [Friedla nder and Tseng, 2008,\nAmini and Yousefian, 2019, Merchav and Sabach, 2023], which is defi ned as\nF(x) =/⌊a∇⌈⌊lx/⌊a∇⌈⌊l1+τ\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2,\nwhereτ >0 regulates the trade-off between ℓ1andℓ2norms.\nThis example corresponds to our second experiment in Section 5.2.\nExample A.3 (Logistic Regression Problem) .The logistic regression problem aims to\nmap the feature vectors aito the target labels bi. A standard machine learning tech-\nnique for this problem is to minimize the logistic loss function over the giv en dataset\n[Amini and Yousefian, 2019, Gong and Liu, 2021, Jiang et al., 2023, La tafat et al., 2023,\nMerchav and Sabach, 2023]. We assume that the dataset consists of a feature matrix A∈\nRm×nand a label vector b∈Rm, withbi∈ {− 1,1}for eachi. The logistic loss function is\ndefined as\ng1(x) =1\nmm/summationdisplay\ni=1log(1 + exp( −aT\nixbi)). (9)\nOver-fitting is a common issue when the number of features is large c ompared to the number\nof instances m. A possible approach is to regularize the logistic objective function w ith a\nspecific function or a constraint [Jiang et al., 2023, Merchav and Sab ach, 2023]. For instance,\nwe can use g2(x) =IC(x), whereIC(x) is the indicator of the set C={x∈Rn:/⌊a∇⌈⌊lx/⌊a∇⌈⌊l1≤θ},\nas in Example A.1.\nThis problem may also have multiple optimal solutions. Hence, a natura l extension\nis to consider the minimal norm solution problem [Gong and Liu, 2021, Jia ng et al., 2023,\nLatafat et al., 2023], as in Example A.1. This requires solving the followin g problem:\nmin\nx∈Rn1\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2\ns.t.x∈argmin\nz∈Rn1\nmm/summationtext\ni=1log(1 + exp( −aT\nizbi)) +IC(z).\nWhen choosing C={x∈Rn:/⌊a∇⌈⌊lx/⌊a∇⌈⌊l1≤θ}for someθ > 0, it corresponds to our first\nexperiment in Section 5.1.\nExample A.4 (Over-parameterized Regression Problem) .The linear regression problem\naims to find a parameter vector x∈Rnthat minimizes the training loss ℓtr(x) over the train-\ning dataset Dtr. Without explicit regularization, the over-parameterized regress ion problem\nhas multiple minima. However, these minima may have different generaliz ation performance.\nTherefore, we introduce a secondary objective, such as the valid ation loss over a validation\n23setDval, to select one of the global minima of the training loss. This results in t he following\nbilevel problem:\nmin\nx∈RnF(x) :=ℓval(x)\ns.t.x∈argmin\nz∈RnG(z) :=ℓtr(z).(10)\nFor instance, we can consider the sparse linear regression problem , where the lower-level\nobjective consists of the training error and a regularization term, namely,G(x) =1\n2/⌊a∇⌈⌊lAtrx−\nbtr/⌊a∇⌈⌊l2+IC(x). Here,IC(x) denotes the indicator of a convex set, as in Example A.2. The\nupper-level objective is the validation error, i.e., F(x) =1\n2/⌊a∇⌈⌊lAvalx−bval/⌊a∇⌈⌊l2. The linear regres-\nsion problem is over-parameterized when the number of features nis larger than the number\nof data instances in the training set.\nB Examples of Functions Satisfying the H¨ olderian Er-\nror Bound\nWe present several examples of functions that satisfy the H¨ olde rian error bound Assumption\n2.2 and their corresponding exponent parameter αin Table 2. We also provide some clarifi-\ncations for Table 2 below. The abbreviations “ Q∈Sn” and “Q≻0” stand for “ Qis a sym-\nmetric matrix of order nand a positive definite matrix, respectively. We refer the reader to\nPang [1997], Bolte et al. [2017], Zhou and So [2017], Jiang and Li [2022], D oron and Shtern\n[2023] and the references therein for more examples of functions that satisfy H¨ olderian error\nbound Assumption 2.2.\nTable 2: Summary of some functions satisfying H¨ olderian er ror bound with corresponding expo-\nnents.\nG(x) Remarks Name α\nmaxi∈[m]{/an}⌊∇a⌋k⌉tl⌉{tai,x/an}⌊∇a⌋k⌉t∇i}ht−bi} ai∈Rn,i∈[m],b∈Rmpiece-wise maximum 1\n/⌊a∇⌈⌊lx−x0/⌊a∇⌈⌊lQ=/radicalbig\n(x−x0)TQ(x−x0)Q∈Sn,Q≻0,x0∈RnQ-norm 1\n/⌊a∇⌈⌊lx−x0/⌊a∇⌈⌊lpx0∈Rn,p≥1 ℓp-norm 1\n/⌊a∇⌈⌊lx/⌊a∇⌈⌊l1+τ\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2τ >0 Elastic net 1 or 22\n/⌊a∇⌈⌊lAx−b/⌊a∇⌈⌊l2A∈Rm×n,b∈RmLeast squares 2\n1\nm/summationtextm\ni=1log(1 + exp( −aT\nixbi))ai∈Rn,i∈[m],b∈Rm,A∈Rm×nLogistic loss 2\nη(x) +σ\n2/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2ηconvex,σ>0 Strongly-convex 2\n2According to Table 2 of Doron and Shtern [2023], the parameter αcan take values of either 1 or 2.\nParticularly, when α= 1, we have ρ= 1; when α= 1, we have ρ= 2/τ.\n24C Supplementary Results\nIn this section, we provide supplementary results that complement the findings in the previ-\nous contexts.\nC.1 Adaptive Version of PB-APG method with Strong Convexity\nAssumption\nIn this section, we present the adaptive version of Algorithm 3 as fo llows.\nAlgorithm 4 Adaptive PB-APG-sc method (aPB-APG-sc)\nInput: x −1=x0∈Rn,γ0=γ1>0,Lf1,Lg1,ν >1,η> 1,ǫ0>0.\nfork≥0do\nφk(x) =f1(x) +γkg1(x)\nψk(x) =f2(x) +γkg2(x)\nInvokexk= PB-APG-sc( φk,ψk,Lf1,Lg1,xk−1,ǫk)\nǫk+1=1\nηǫk\nγk+1=νγk\nend for\nSimilar to Algorithm 2, we have the following convergence results of Alg orithm 4.\nTheorem C.1. Suppose that Assumptions 2.1, 2.2, 3.1, and 4.1 hold. Let ǫ0>0be given.\n•Whenα > 1, setν > ηα−1,N=⌈logη1−αν(ρLα\nF(α−1)α−1α−αǫ1−α\n0/γ0)⌉+andγ∗\nk=\nρLα\nF(α−1)α−1α−αǫ1−α\n0ηk(α−1);\n•Whenα= 1, setν >1,N=⌈logν(ρlF/γ0)⌉+andγ∗\nk=ρLF.\nThen, for any k≥N, Algorithm 2 generates an (ǫ0\nηk,2ǫ0\nηk(γ0νk−γ∗\nk))-optimal solution of problem\n(P)after at most Kiterations, where Ksatisfies\nK=O/parenleftBigg/radicalBigg\nLf1\nµlogηk\nǫ0/parenrightBigg\n+O\n/radicalBigg\nνklmax{α,β}\nFLg1\nǫmax{α−1,β−1}logηk\nǫ0\n.\nProof. The proof is similar to the proof of Theorem 3.4 in Appendix D.6. So we omit it\nhere. /square\nD Proofs of Main Results\nIn this section, we propose the proofs of our main convergence re sults in this paper.\n25D.1 Proof of Lemma 2.3\nProof. For any x∈Rn, let¯xbe the projection of xontoXopt, that is, dist( x,Xopt) =/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊l.\nBy Lipschitz continuity of F, we have\nF(x)−F(¯x)≥ −lF/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊l.\nChoosingγ∗=ρlα\nF(α−1)α−1α−αǫ1−α, it follows that\nF(x)−F(¯x) +γ∗p(x)≥ −lF/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊l+γ∗p(x)\n(a)\n≥ −lF/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊l+γ∗\nρ/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊lα\n≥min\nz≥0−lFz+γ∗\nρzα\n(b)=−ǫ,(11)\nwhere (a) follows from the H¨ olderian error bound assumption of p(x), and (b) is from the\nfact that y=−lFz+γ∗\nρzαattains its minimum at z∗=/parenleftBig\nρlF\nαγ∗/parenrightBig1\nα−1.\nSince ¯x∈Xoptis feasible for problem (P), we have F(¯x)≥F∗. This along with (11)\nindicates\nF(x) +γp(x)−F∗≥F(x) +γ∗p(x)−F(¯x)≥ −ǫ,∀x∈Rdandγ≥γ∗. (12)\nLetx∗be an optimal solution of (P) so that F(x∗) =F∗. Sincex∗∈Xopt, we havep(x∗) = 0.\nBy (12), we have\nF(x∗) +γp(x∗)≤F(x) +γp(x) +ǫ,∀x∈Rdandγ≥γ∗. (13)\nThis demonstrates that an optimal solution of (P) is an ǫ-optimal solution for (P γ)./square\nD.2 Proof of Lemma 2.4\nProof. The proof is motivated by Theorem 1 in Luo et al. [1996]. Denot ex∗,x∗\nγas optimal\nsolutions of problem (P) and (P γ), respectively.\nFor anyx∈Rn, let ¯xbe the projection of xontoXopt. We have\nF(x) +γp(x) =F(¯x) +F(x)−F(¯x) +γp(x)\n(a)\n≥F(x∗)−lF/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊l+γ\nρ/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊l\n=F(x∗) +/parenleftbiggγ\nρ−lF/parenrightbigg\n/⌊a∇⌈⌊lx−¯x/⌊a∇⌈⌊l\n(b)\n≥F(x∗) =F(x∗) +γp(x∗),(14)\n26where (a) follows from the H¨ olderian error bound assumption of p(x), the optimality of x∗\nto (P) and Lipschitz continuity of F, and (b) follows from γ≥ρlF. Therefore, we conclude\nthatx∗is an optimal solution of (P γ).\nFor the converse part, similarly, we have\nF(x∗) =F(x∗) +γp(x∗)≥F(x∗\nγ) +γp(x∗\nγ)(c)\n≥F(x∗) +/parenleftbiggγ\nρ−lF/parenrightbigg\n/⌊a∇⌈⌊lx∗\nγ−¯x∗\nγ/⌊a∇⌈⌊l ≥F(x∗),(15)\nwhere ¯x∗\nγis the projection of x∗\nγontoXopt. The inequality ( c) holds since we let x=x∗\nγin\n(14).\nTherefore, all inequalities in (14) become equalities. We deduce that /⌊a∇⌈⌊lx∗\nγ−¯x∗\nγ/⌊a∇⌈⌊l= 0,\nimplying that x∗\nγis inXopt, i.e.,p(x∗\nγ) = 0. Furthermore, as the first inequality of (15)\nbecomes an equality, we obtain\nF(x∗) =F(x∗\nγ) +γp(x∗\nγ) =F(x∗\nγ).\nTherefore, x∗\nγis also an optimal solution of (P). /square\nD.3 Proof of Theorem 2.5\nProof.Case ofα>1.Since ˜x∗\nγis anǫ-optimal solution of (P γ), we have\nF(˜x∗\nγ) +γp(˜x∗\nγ)≤F(x) +γp(x) +ǫ,∀x∈Rn. (16)\nNote that the arguments in the proof of Lemma 2.3 still hold. Substit utingx=x∗into (16)\nand utilizing p(x∗) = 0, we have\nF(˜x∗\nγ) +γp(˜x∗\nγ)≤F(x∗) +ǫ≤F(˜x∗\nγ) +γ∗p(˜x∗\nγ) + 2ǫ,\nwhere the last inequality follows from setting x=˜x∗\nγandγ=γ∗in (13). Then, we have\np(˜x∗\nγ)≤2ǫ\nγ−γ∗=2ǫ\n2lβ\nFǫ1−β=l−β\nFǫβ. (17)\nNote that ˜x∗\nγis feasible for the following problem\nmin\nx∈RnF(x)\ns.t. p(x)≤p(˜x∗\nγ).(18)\nSincex∗is an optimal solution of problem (P), by(16), we have\nF(˜x∗\nγ)−F(x∗)≤γ(p(x∗)−p(˜x∗\nγ)) +ǫ.\n27Fromp(x∗) = 0≤p(˜x∗\nγ), we have\nF(˜x∗\nγ)−F(x∗)≤ǫ. (19)\nThis, along with (17) and the fact that x∗is feasible for (18), proves that ˜x∗\nγis an (ǫ,l−β\nFǫβ)-\noptimal solution of (P).\nCase ofα= 1.Since ˜x∗\nγis anǫ-optimal solution of (P γ), we have\nF(˜x∗\nγ) +γp(˜x∗\nγ)≤F(x∗) +γp(x∗) +ǫ. (20)\nwherex∗is an optimal solution of (P γ). Sinceγ=γ∗+lβ\nFǫ1−β> γ∗,x∗is also an optimal\nsolution of (P) from Lemma 2.4. Therefore, p(x∗) = 0 and we have\nF(x∗)≤F(˜x∗\nγ)+γp(˜x∗\nγ)(20)\n≤F(x∗)+γp(x∗)+ǫ=F(x∗)+γ∗p(x∗)+ǫ≤F(˜x∗\nγ)+γ∗p(˜x∗\nγ)+ǫ,\n(21)\nwhere the first inequality follows from the fact that x∗is an optimal solution of (P γ), and\nthe last inequality follows from the optimality of x∗to (Pγ) withγ=γ∗.\nThe second inequality of (21) and p(˜x∗\nγ)≥0 imply that\nF(˜x∗\nγ)≤F(x∗) +ǫ. (22)\nThen, from (21), we have F(˜x∗\nγ) +γp(˜x∗\nγ)≤F(˜x∗\nγ) +γ∗p(˜x∗\nγ) +ǫ, which implies that\np(˜x∗\nγ)≤ǫ\nγ−γ∗=ǫ\nlβ\nFǫ1−β=l−β\nFǫβ. (23)\nThis along with (22) demonstrate that ˜x∗\nγis an (ǫ,l−β\nFǫβ)-optimal solution of (P). /square\nD.4 Proof of Theorem 2.6\nProof. The set Xoptis convex and closed since the upper- and lower-level objectives\nare both convex, proper, and closed. Let ˆx∗\nγbe the projection of ˜x∗\nγonXopt, we have\n/⌊a∇⌈⌊l˜x∗\nγ−ˆx∗\nγ/⌊a∇⌈⌊l= dist( ˜x∗\nγ,Xopt).\nBy Assumption 2.2, the following inequality holds,\n/⌊a∇⌈⌊l˜x∗\nγ−ˆx∗\nγ/⌊a∇⌈⌊lα≤ρp(˜x∗\nγ)(a)\n≤ρl−β\nFǫβ=⇒ /⌊a∇⌈⌊l ˜x∗\nγ−ˆx∗\nγ/⌊a∇⌈⌊l ≤/parenleftBig\nρl−β\nFǫβ/parenrightBig1\nα, (24)\nwhere (a) follows from (17) when α>1 or from (23) when α= 1.\nBy Assumption 2.1, FislF-Lipschitz continuous. Align with (24), we have\nF(˜x∗\nγ)−F∗≥F(˜x∗\nγ)−F(ˆx∗\nγ)≥ −lF/⌊a∇⌈⌊lˆx−ˆx∗\nγ/⌊a∇⌈⌊l ≥ −lF/parenleftBig\nρl−β\nFǫβ/parenrightBig1\nα,\nwhere the first inequality follows from the fact that F∗is the optimal value of problem (P)\nandˆx∗\nγ∈Xopt. /square\n28D.5 Proof of Theorem 3.3\nProof. By Lemma 3.2, the number of iterations to obtain an ǫ-optimal solution of (P γ) is\nK=O/parenleftBigg/radicalbigg\nLf1+γLg1\nǫ/parenrightBigg\n,\nwhereLγ=Lf1+γLg1.\n•Case ofα > 1.In this case, γ=γ∗+ 2lβ\nFǫ1−βcomprises two components: γ∗and\n2lβ\nFǫ1−β. Therefore, it is natural to discuss which of these two component s plays the\ndominant role in the complexity results.\nIfβ <α , the dominating term in γisγ∗=ρlα\nF(α−1)α−1α−αǫ1−α. Then, the number\nof iterations is\nK=O/parenleftBigg/radicalbigg\nLf1+lα\nFǫ1−αLg1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlα\nFLg1\nǫα/parenrightBigg\n.\nIfβ=α, we haveγ= (ρ(α−1)α−1α−α+ 2)lα\nFǫ1−α. Then, the number of iterations is\nK=O/parenleftBigg/radicalbigg\nLf1+lα\nFǫ1−αLg1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlα\nFLg1\nǫα/parenrightBigg\n.\nIfβ >α , the dominating term in γis 2lβ\nFǫ1−β. Then, the number of iterations is\nK=O\n/radicalBigg\nLf1+ 2lβ\nFǫ1−βLg1\nǫ\n=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlβ\nFLg1\nǫβ\n.\n•Case ofα= 1. In this case, γ=γ∗+lβ\nFǫ1−β, whereγ∗=ρlF. Similarly, we explore\nwhich of these two elements plays a more significant role.\nIfβ <1, the dominating term in γisγ∗. Then, the number of iterations is\nK=O/parenleftBigg/radicalbigg\nLf1+ρlFLg1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlFLg1\nǫ/parenrightBigg\n.\nIfβ= 1, we have γ= (ρ+ 1)lFǫ1−α. Then the number of iterations is\nK=O/parenleftBigg/radicalbigg\nLf1+ (ρ+ 1)lFLg1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlFLg1\nǫ/parenrightBigg\n.\nIfβ >1, the dominating term in γislβ\nFǫ1−β. Then, the number of iterations is\nK=O\n/radicalBigg\nLf1+lβ\nFǫ1−βLg1\nǫ\n=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlβ\nFLg1\nǫβ\n.\n29Combining the above results, we conclude that\nK=O/parenleftBigg/radicalbigg\nLf1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlmax{α,β}\nFLg1\nǫmax{α,β}\n.\n/square\nD.6 Proof of Theorem 3.4\nProof. In this proof, we denote Φ∗\nkas the optimal value of problem (P γ) whenγ=γk, and\nxkas the output of PB-APG (Algorithm 1) in the k-th iteration.\n•Case ofα > 1.Suppose that Nis the smallest nonnegative integer such that γN≥\nγ∗\nN:=ρlα\nF(α−1)α−1α−αǫ1−α\nN. In this case, we have\nγN=γ0νN≥ρlα\nF(α−1)α−1α−αǫ1−α\nN=ρlα\nF(α−1)α−1α−αǫ1−α\n0(1/η)(1−α)N, (25)\nwhich is equivalent to\nγ0/parenleftbig\nνη1−α/parenrightbigN≥ρlα\nF(α−1)α−1α−αǫ1−α\n0. (26)\nFrom (26), after at most N:=⌈logη1−αν/parenleftBig\nρlα\nF(α−1)α−1α−αǫ1−α\n0\nγ0/parenrightBig\n⌉+iterations, (25) holds.\nSincexN= PB-APG( φN,ψN,Lf1,Lg1,xN−1,ǫN), we have\nΦN(xN)−Φ∗\nN≤ǫN, γN≥γ∗\nN,\nwhich shows that xNis anǫN-optimal solution of (P γ) withγ=γN. From the\nproof in Theorem 2.5 (see inequalities (17) and (19) in Appendix D.3), xNis also\nan (ǫ0\nηN,2ǫ0\nηN(γ0νN−γ∗\nN))-optimal solution of problem (P).\nFurthermore, note that for any iteration k≥N, inequality (26) always holds, which\nmeans that the following statement holds for any k≥N:\nΦk(xk)−Φ∗\nk≤ǫk, γk≥γ∗\nk. (27)\nLetIkbe the number of iterations of PB-APG required to satisfy (27) at t hek-th\niteration of aPB-APG. Then, for any k≥N, the total number of iterations is\nK=I0+I1+···+Ik.\n30From Lemma 3.2, we give the following bound for K:\nK=O/parenleftBigg/radicalbigg\nLf1+γ0Lg1\nǫ0/parenrightBigg\n+···+O/parenleftBigg/radicalbigg\nLf1+γkLg1\nǫk/parenrightBigg\n≤ O/parenleftBigg/radicalbigg\nLf1+γkLg1\nǫ0/parenrightBigg\n+···+O/parenleftBigg/radicalbigg\nLf1+γkLg1\nǫk/parenrightBigg\n=O/parenleftBigg/radicalbigg\nLf1+γkLg1\nǫk/parenleftBig\n1 +/radicalbig\n1/η+/radicalbig\n1/η2+···+/radicalbig\n1/ηk/parenrightBig/parenrightBigg\n=O/parenleftBigg/radicalbigg\nLf1+γkLg1\nǫk/parenrightBigg\n=O\n/radicalBigg\nLf1ηk\nǫ0\n+O\n/radicalBigg\nLg1γ0(ην)k\nǫ0\n.\n•Case ofα= 1.Suppose that after Nupdates, we have γN≥ρlF, i.e.,\nγ0νN≥ρlF. (28)\nThis demonstrates that after for all k≥N:= logν/parenleftBig\nρlF\nγ0/parenrightBig\n, (28) always holds.\nSimilar to the case of α>1, the total iteration number is:\nK=O/parenleftBigg/radicalbigg\nLf1+γ0Lg1\nǫ0/parenrightBigg\n+···+O/parenleftBigg/radicalbigg\nLf1+γkLg1\nǫk/parenrightBigg\n=O/parenleftBigg/radicalbigg\nLf1+γkLg1\nǫk/parenrightBigg\n=O\n/radicalBigg\nLf1ηk\nǫ0\n+O\n/radicalBigg\nLg1γ0(ην)k\nǫ0\n.\n/square\nD.7 Proof of Lemma 4.2\nProof. Denote Lγ=Lf1+γLg1. From Theorem 3.1 in Beck and Teboulle [2009], we have\nΦγ(˜x)−Φγ(x∗\nγ)≤Lγ\n2/⌊a∇⌈⌊ly0−x∗\nγ/⌊a∇⌈⌊l2. (29)\n31By Theorem 1 in Lin and Xiao [2014], we have\nΦγ(xk)−Φγ(x∗\nγ)≤/parenleftBig\nΦγ(˜x)−Φγ(x∗\nγ) +µ\n2/⌊a∇⌈⌊l˜x−x∗\nγ/⌊a∇⌈⌊l2/parenrightBig/parenleftbigg\n1−/radicalbiggµ\nLγ/parenrightbiggk\n(29)\n≤/parenleftbiggLγ\n2/⌊a∇⌈⌊ly0−x∗\nγ/⌊a∇⌈⌊l2+µ\n2/⌊a∇⌈⌊l˜x−x∗\nγ/⌊a∇⌈⌊l2/parenrightbigg/parenleftbigg\n1−/radicalbiggµ\nLγ/parenrightbiggk\n≤/parenleftbiggLγ+µ\n2R2/parenrightbigg/parenleftbigg\n1−/radicalbiggµ\nLγ/parenrightbiggk\n.(30)\n/square\nD.8 Proof of Theorem 4.3\nProof. By Lemma 4.2, the number of iterations required to achieve a nǫ-optimal solution\nfor problem (P γ) is\nK=O/parenleftBigg/radicalBigg\nLγ\nµlog/parenleftbiggLγ+µ\n2ǫR2/parenrightbigg/parenrightBigg\n=O/parenleftBigg/radicalBigg\nLγ\nµlog1\nǫ/parenrightBigg\n.\n•Case ofα>1.In this case, γ=γ∗+ 2lβ\nFǫ1−β, whereγ∗=ρlα\nF(α−1)α−1α−αǫ1−α.\nIfβ <α , the dominating term in γisγ∗. Then, the number of iterations is\nK=O/parenleftBigg/radicalBigg\nLf1+lα\nFǫ1−αLg1\nµlog1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlα\nFLg1\nǫα−1log1\nǫ/parenrightBigg\n.\nIfβ=α, we haveγ= (ρ(α−1)α−1α−α+ 2)lα\nFǫ1−α. Then, the number of iterations is\nK=O/parenleftBigg/radicalBigg\nLf1+lα\nFǫ1−αLg1\nµlog1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlα\nFLg1\nǫα−1log1\nǫ/parenrightBigg\n.\nIfβ >α , the dominating term in γis 2lβ\nFǫ1−β. Then, the number of iterations is\nK=O\n/radicalBigg\nLf1+ 2lβ\nFǫ1−βLg1\nµlog1\nǫ\n=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlβ\nFLg1\nǫβ−1log1\nǫ\n.\n•Case ofα= 1.Whenα= 1,γcan be written as γ=γ∗+lβ\nFǫ1−β, whereγ∗=ρlF.\nIfβ <1, the dominating term in γisγ∗. Then, the number of iterations is\nK=O/parenleftBigg/radicalBigg\nLf1+ρlFLg1\nµlog1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlFLg1\nǫα−1log1\nǫ/parenrightBigg\n.\n32Ifβ= 1, we have γ= (ρ+ 1)lFǫ1−α. Then, the number of iterations is\nK=O/parenleftBigg/radicalBigg\nLf1+ρlFLg1\nµlog1\nǫ/parenrightBigg\n=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O/parenleftBigg/radicalbigg\nlFLg1\nǫα−1log1\nǫ/parenrightBigg\n.\nIfβ >1, the dominating term in γislβ\nFǫ1−β. Then, we have\nK=O\n/radicalBigg\nLf1+lβ\nFǫ1−βLg1\nµlog1\nǫ\n=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlβ\nFLg1\nǫβ−1log1\nǫ\n.\nCombining the above results, we conclude that\nK=O/parenleftBigg/radicalBigg\nLf1\nµlog1\nǫ/parenrightBigg\n+O\n/radicalBigg\nlmax{α,β}\nFLg1\nǫmax{α−1,β−1}log1\nǫ\n.\n/square\nD.9 Proof of Theorem 4.4\nProof. Denote lγ=lf2+γlg2. Define ΦK\nγ,best = min\ni=0,...,KΦγ(xi) and ˆΦK,j\nγ,best = min\ni=j,...,KΦγ(xi) for\nall 0≤j≤K. We claim that the sequence generated by the subgradient method satisfies\nΦK\nγ,best−Φ∗\nγ≤lγ\n4R2+ 2 log 2√\nK+ 2. (31)\nSpecifically, from Lemma 8.24 in Beck [2017], for all 0 ≤j≤K, we have\nˆΦK,j\nγ,best−Φ∗\nγ≤1\n2R2+/summationtextK\nk=jη2\nk/⌊a∇⌈⌊lξk/⌊a∇⌈⌊l2\n/summationtextK\nk=jηk. (32)\nDefine⌊·⌋and⌈·⌉as rounding up and rounding down, respectively. Let j=⌊K\n2⌋in (32),\nby the definition of step-size ηk=R\nlγ√\nk+1, we have\nˆΦK,j\nγ,best−Φ∗\nγ≤lγ\n2R2+/summationtextK\nk=⌊K\n2⌋1\nk+1/summationtextK\nk=⌊K\n2⌋1√k+1\n≤lγ\n4R2+ 2 log 2√\nK+ 2,(33)\nwhere the second inequality follows from that/summationtextK\nk=⌊K\n2⌋1\nk+1≤/integraltextK\n⌈K\n2⌉−11\ns+1ds≤2 log 2 and\n/summationtextK\nk=⌊K\n2⌋1√k+1≥/integraltextK+1\n⌈K\n2⌉1√s+1ds≥1\n2√\nK+ 2.\nFrom the fact that ΦK\nγ,best≤ˆΦK,j\nγ,best, we deduce (31).\n33Then, inequality (31) demonstrates that the number of iterations to obtain an ǫ-optimal\nsolution for problem (P γ) is\nK=O/parenleftbigglf2+γlg2\nǫ/parenrightbigg2\n.\n•Case ofα>1.we haveγ=γ∗+ 2lβ\nf2ǫ1−βandγ∗=ρlα\nf2(α−1)α−1α−αǫ1−α.\nIfβ <α , the dominating term in γisγ∗. Then, the number of iterations is\nK=O/parenleftbigglf2+lα\nf2ǫ1−αlg2\nǫ/parenrightbigg2\n=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftbiggl2α\nf2l2\ng2\nǫ2α/parenrightbigg\n.\nIfβ=α, we haveγ= (ρ(α−1)α−1α−α+ 2)lα\nFǫ1−α. Then, the number of iterations is\nK=O/parenleftbigglf2+lα\nf2ǫ1−αlg2\nǫ/parenrightbigg2\n=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftbiggl2α\nf2l2\ng2\nǫ2α/parenrightbigg\n.\nIfβ >α , the dominating term in γis 2lβ\nFǫ1−β. Then, the number of iterations is\nK=O/parenleftBigg\nlf2+ 2lβ\nf2ǫ1−βlg2\nǫ/parenrightBigg2\n=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftBigg\nl2β\nf2l2\ng2\nǫ2β/parenrightBigg\n.\n•Case ofα= 1.we haveγ=γ∗+lβ\nf2ǫ1−βandγ∗=ρlf2.\nIfβ <1, the dominating term in γisγ∗. Then, the number of iterations is\nK=O/parenleftbigglf2+ρlf2lg2\nǫ/parenrightbigg2\n=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftbiggl2\nf2l2\ng2\nǫ2/parenrightbigg\n.\nIfβ= 1, we have γ= (ρ+ 1)lFǫ1−α. Then, the number of iterations is\nK=O/parenleftbigglf2+ρlf2lg2\nǫ/parenrightbigg2\n=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftbiggl2\nf2l2\ng2\nǫ2/parenrightbigg\n.\nIfβ >1, the dominating term in γislβ\nf2ǫ1−β. Then, the number of iterations is\nK=O/parenleftBigg\nlf2+lβ\nf2lg2ǫ1−β\nǫ/parenrightBigg2\n=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftBigg\nl2β\nf2l2\ng2\nǫ2β/parenrightBigg\n.\nCombining the above results, we conclude that\nK=O/parenleftbiggl2\nf2\nǫ2/parenrightbigg\n+O/parenleftBigg\nlmax{2α,2β}\nf2l2\ng2\nǫmax{2α,2β}/parenrightBigg\n.\n/square\n34D.10 Proof of Theorem 4.5\nProof. Denote lγ=lf2+γlg2, define ΦK\nγ,best = min\ni=0,...,KΦγ(xi). From Theorem 8.31 in Beck\n[2017], the sequence generated by the subgradient method satisfi es\nΦK\nγ,best−Φ∗\nγ≤2l2\nγ\nµf2(K+ 1).\nThis demonstrates that the number of iterations to obtain an ǫ-optimal solution for problem\n(Pγ) is\nK=O/parenleftbigg(lf2+γlg2)2\nµf2ǫ/parenrightbigg\n.\n•Case ofα>1.we haveγ=γ∗+ 2lβ\nf2ǫ1−βandγ∗=ρlα\nf2(α−1)α−1α−αǫ1−α.\nIfβ <α , the dominating term in γisγ∗. Then, the number of iterations is\nK=O/parenleftbigg(lf2+lα\nf2ǫ1−αlg2)2\nµf2ǫ/parenrightbigg\n=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftbiggl2α\nf2l2\ng2\nµf2ǫ2α−1/parenrightbigg\n.\nIfβ=α, we haveγ= (ρ(α−1)α−1α−α+ 2)lα\nFǫ1−α. Then, the number of iterations is\nK=O/parenleftbigg(lf2+lα\nf2ǫ1−αlg2)2\nµf2ǫ/parenrightbigg\n=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftbiggl2α\nf2l2\ng2\nµf2ǫ2α−1/parenrightbigg\n.\nIfβ >α , the dominating term in γis 2lβ\nFǫ1−β. Then, the number of iterations is\nK=O/parenleftBigg\n(lf2+ 2lβ\nf2ǫ1−βlg2)2\nµf2ǫ/parenrightBigg\n=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftBigg\nl2β\nf2l2\ng2\nµf2ǫ2β−1/parenrightBigg\n.\n•Case ofα>1.we haveγ=γ∗+lβ\nf2ǫ1−βandγ∗=ρlf2.\nIfβ <1, the dominating term in γisγ∗. Then, the number of iterations is\nK=O/parenleftbigg(lf2+ρlf2lg2)2\nµf2ǫ/parenrightbigg\n=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftbiggl2\nf2l2\ng2\nµf2ǫ/parenrightbigg\n.\nIfβ= 1, we have γ= (ρ+ 1)lFǫ1−α. Then, the number of iterations is\nK=O/parenleftbigg(lf2+ρlf2lg2)2\nµf2ǫ/parenrightbigg\n=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftbiggl2\nf2l2\ng2\nµf2ǫ/parenrightbigg\n.\nIfβ >1, the dominating term in γislβ\nf2ǫ1−β. Then, the number of iterations is\nK=O/parenleftBigg\n(lf2+lβ\nf2lg2ǫ1−β)2\nµf2ǫ/parenrightBigg\n=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftBigg\nl2β\nf2l2\ng2\nµf2ǫ2β−1/parenrightBigg\n.\n35Combining the above results, we conclude that\nK=O/parenleftbiggl2\nf2\nµf2ǫ/parenrightbigg\n+O/parenleftBigg\nlmax{2α,2β}\nf2l2\ng2\nµf2ǫmax{2α−1,2β−1}/parenrightBigg\n.\n/square\nE Implementation Details\nIn this section, we provide supplementary experiment settings and results. Specifically, in\nAppendix E.1, we present the detailed experimental settings, and in Appendix E.2, we pro-\nvide the detailed experimental results. Additionally, in Appendix E.3 an d E.4, we conduct\nexperiments with different values of penalty parameter γand solution accuracy ǫ, respec-\ntively.\nE.1 Experiment Setting\nAll simulations are implemented using MATLAB R2023a on a PC running Win dows 11 with\nan AMD (R) Ryzen (TM) R7-7840H CPU (3.80GHz) and 16GB RAM.\nE.1.1 Experiment Setting of Section 5.1\nWe conduct the first experiment using the a1a.t data from LIBSVM datasets3. This data\nconsists of 30 ,956 instances, each with n= 123 features. For this experiment, a sample of\n1,000 instances is taken from the data, denoted as A. The corresponding labels for these\ninstances are denoted as b, where each label biis either −1 or 1, corresponding to the i-th\ninstance ai.\nThe Greedy FISTA algorithm [Liang et al., 2022] is used as a benchmark to compute G∗.\nTo compute the proximal mapping of f2(x) +γg2(x) in problem (P γ), i.e, projection onto\na 1-norm ball, we utilize the method proposed in Duchi et al. [2008], whic h performs exact\nprojection in O(n) expected time, where n is the dimension of x.\nFor the PB-APG and PB-APG-sc algorithms, we set the value of γ= 105, and we\nterminate the algorithms when /⌊a∇⌈⌊lxk+1−xk/⌊a∇⌈⌊l ≤10−10. For the aPB-APG and aPB-APG-sc\nalgorithms, we set γ0=1\n25,ν= 20,η= 10, and ǫ0= 10−6. The iterations of these two\nalgorithms continue until ǫkreaches 10−10(meanwhile, γ= 105).\nWe note that the termination criterion /⌊a∇⌈⌊lxk+1−xk/⌊a∇⌈⌊l ≤10−10used in our experiments is\ndifferent from the one proposed in our algorithms since the paramet ers required for the\nlatter are not easily measurable. Nevertheless, this termination cr iterion is also widely\n3https://www.csie.ntu.edu.tw/ ~cjlin/libsvmtools/datasets/binary/a1a.t\n36used in the literature, as it corresponds to a gradient mapping [Beck , 2017, Nesterov, 2018,\nDavis and Drusvyatskiy, 2019]. Furthermore, Theorem 3.5 of Drus vyatskiy and Lewis [2018b]\nimplies that /⌊a∇⌈⌊lxk+1−xk/⌊a∇⌈⌊lalso measures the distance to the optimal solution set.\nE.1.2 Experiment Setting of Section 5.2\nIn the second experiment, we address the problem of least square s regression using the\nYearPredictionMSD data from the UCI Machine Learning Repository4. This data consists\nof 515,345 songs with release years ranging from 1992 to 2011. Each song has 90 features,\nand the corresponding release year is used as the label. For this exp eriment, a sample of\nm= 1,000 songs is taken from the data, and the feature matrix and relea se years vector are\ndenoted as Aandb, respectively.\nFollowing Section 5.2 in Merchav and Sabach [2023], we apply the min-max scaling tech-\nnique to normalize the feature matrix A. Additionally, we add an intercept term and 90\ncollinear features to Asuch that the resulting matrix ATAbecomes positive semi-definite,\nwhich implies that the feasible set Xoptis not a singleton.\nWe compare our methods with a-IRG, BiG-SAM, and Bi-SG in this exper iment. Specifi-\ncally, for BiG-SAM [Sabach and Shtern, 2017], we consider the accur acy parameter δfor the\nMoreau envelope with two values, namely δ= 1 andδ= 0.01.\nTo benchmark the performance, we utilize the MATLAB function lsqminnorm to com-\nputeG∗. Moreover, we follow the parameter settings outlined in Section 5.1.\nE.2 Detailed Results of Experiments\nTo approximate the optimal value F∗, we use the MATLAB function fmincon to solve a\nrelaxed version of the function-value-based reformulations in equ ation (P Val). In this relaxed\nversion, we replace the constraint in (P Val) withG(x)−G∗≤ε, whereε= 10−10. This allows\nus to obtain an approximation of the optimal value while allowing for a sm all deviation from\nthe true optimal value G∗.\nWe gather the total number of iterations for our methods, as well as the lower- and upper-\nlevel objective values and the optimal gaps for all the methods, in T able 3. Subsequently,\nwe compare the optimal gaps of all methods, which are defined as G(x)−G∗andF(x)−F∗\nfor the lower- and upper-level optimal gaps, respectively.\nTable 3 reveals that for the logistic regression problem (6), our PB- APG, aPB-APG,\nPB-APG-sc, and aPB-APG-sc exhibit almost identical function value s for both objectives,\nsurpassing other methods in terms of optimal gaps for the lower- a nd upper-level objectives\n(measured by the numerical value of the upper-level objective). In the case of the least\n4https://archive.ics.uci.edu/ml/datasets/YearPredict ionMSD\n37Table 3: Methods comparison: lower- and upper-level object ives and optimal gaps\nLogistic Regression Problem (6)\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 1470 3.2794e-01 1.7630e-08 4.9382e+00 -3.3998e-03\naPB-APG 1010 3.2794e-01 1.7630e-08 4.9382e+00 -3.3998e-03\nPB-APG-sc 2278 3.2794e-01 1.7630e-08 4.9382e+00 -3.3998e-03\naPB-APG-sc 1046 3.2794e-01 1.7630e-08 4.9382e+00 -3.3998e-03\nMNG / 3.4648e-01 1.8535e-02 1.7263e+00 -3.2153e+00\nBiG-SAM / 3.3818e-01 1.0240e-02 2.3480e+00 -2.5937e+00\nDBGD / 4.8861e-01 1.6067e-01 1.5146e-01 -4.7902e+00\na-IRG / 3.3763e-01 9.6831e-03 2.5429e+00 -2.3987e+00\nCG-BiO / 5.2562e-01 1.9767e-01 9.1645e-02 -4.8500e+00\nBi-SG / 3.2805e-01 1.0259e-04 4.7033e+00 -2.3835e-01\nLeast Squares Regression Problem (7)\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 39314 7.3922e-03 6.0034e-07 4.7236e+00 -1.1888e-01\naPB-APG 40784 7.3922e-03 6.0030e-07 4.7236e+00 -1.1887e-01\nPB-APG-sc 46446 7.3922e-03 6.0034e-07 4.7236e+00 -1.1888e-01\naPB-APG-sc 61777 7.3922e-03 6.0035e-07 4.7236e+00 -1.1888e-0 1\nBiG-SAM ( δ= 1) / 7.5189e-03 1.2733e-04 3.5081e+00 -1.3344e+00\nBiG-SAM ( δ= 0.01) / 7.3958e-03 4.2281e-06 5.8510e+01 5.3668e+01\na-IRG / 1.6224e-02 8.8328e-03 4.7745e-01 -4.3651e+00\nBi-SG / 8.5782e-03 1.1866e-03 1.3832e+00 -3.4593e+00\n38squares regression problem (7), aPB-APG achieves the smallest op timal gaps for both objec-\ntives, followed by PB-APG and PB-APG-sc. These results demonstr ate that our methods,\ndespite yielding larger upper-level function values, generate solut ions that are significantly\ncloser to the optimal solution, as depicted in Figure 1. Additionally, fo r the problem in (6),\nboth aPB-APG and aPB-APG-sc require fewer iterations than PB-A PG and PB-APG-sc,\nrespectively. This can be attributed to the warm-start mechanism employed in aPB-APG\nand aPB-APG-sc. Moreover, for the problem in (7), both aPB-APG and aPB-APG-sc re-\nquire more iterations than PB-APG and PB-APG-sc, respectively. H owever, they exhibit\nstaircase-shaped curves, which avoid the unwanted oscillations in P B-APG and PB-APG-sc,\nwe have a similar observation in Figure 2.\nE.3 Supplementary Experiments for Different Penalty Parame -\nters\nIn this section, we investigate the impact of different values of pena lty parameter γon the\nexperimental results of problems (6) and (7). We set γto be either 2 ×104or 5×105for\nPB-APG and PB-APG-sc, and choose the corresponding γ0values as0.2\n25or5\n25for aPB-APG\nand aPB-APG-sc, respectively. The remaining settings are the sam e as in Section 5.\nWe plot the values of the residuals of the lower-level objective G(xk)−G∗and the upper-\nlevel objective over time in Figures 3 and 4. Additionally, we also collect the total number of\niterations, the lower- and upper-level objective values, and the o ptimal gaps of our methods\nin Table 4 for problems (6) and (7) with different values of γ.\n0 0.1 0.2 0.3 0.410-810-610-410-2100102\n0 0.1 0.2 0.3 0.405101520\n0 0.2 0.4 0.6 0.8 110-1010-5100\n0 0.2 0.4 0.6 0.8 105101520\nFigure 3: LRP (6) with γ= 2×105(left two subfigures) and γ= 5×105(right two subfigures).\nAs Figures 3 and 4 show, our methods consistently outperform the other methods for\nboth the lower- and upper-level objectives, irrespective of the p enalty parameter γ, since our\nmethods achieve lower optimal gaps and desired function values for the lower- and upper-\nlevel objectives, respectively. The only exception is problem (7) wit hγ= 5×105, as the\nthird subfigure of Figure 4 shows, since we do not set the solution ac curacy of BiG-SAM\n(δ= 0.01), it attains a lower optimal gap than our PB-APG-sc and aPB-APG -sc for the\n390 5 10 15 20 25 3010-610-410-2100102\n0 5 10 15 20 25 30020406080100\n0 20 40 60 80 10010-1010-5100\n0 20 40 60 80 100020406080100\nFigure 4: LSRP (7) with γ= 2×105(left two subfigures) and γ= 5×105(right two subfigures).\nlower-level objective. However, BiG-SAM ( δ= 0.01) produces significantly worse upper-level\nobjective values, which are much larger than the objective values o f our methods (meanwhile,\nTable 4 shows that the upper-level gaps of our methods are smaller ).\nTable 4: Lower- and upper-level objectives and optimal gaps with different penalty parameters for\nproblem (6).\nγ= 2×105\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 883 3.2794e-01 4.3569e-07 4.9243e+00 -1.7362e-02\naPB-APG 967 3.2794e-01 4.3569e-07 4.9243e+00 -1.7362e-02\nPB-APG-sc 1123 3.2794e-01 4.3569e-07 4.9243e+00 -1.7362e-02\naPB-APG-sc 879 3.2794e-01 4.3569e-07 4.9243e+00 -1.7362e-02\nγ= 5×105\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 1623 3.2794e-01 7.0685e-10 4.9410e+00 -5.7820e-04\naPB-APG 976 3.2794e-01 7.0685e-10 4.9410e+00 -5.7820e-04\nPB-APG-sc 4848 3.2794e-01 7.0684e-10 4.9410e+00 -5.7820e-04\naPB-APG-sc 1018 3.2794e-01 7.0687e-10 4.9243e+00 -5.7821e-04\nTables 3, 4, and 5 reveal that the number of iterations for our met hods increases as penalty\nparameterγincreases. However, it is worth noting that the accuracy of the ob tained solutions\nalso increases, as indicated by the decreasing optimal gaps of the lo wer- and upper-level\nobjectives. This observation confirms that the complexity results and solution accuracies of\nour methods are indeed dependent on the choice of penalty parame ters, specifically, Lγ, as\ndemonstrated in corresponding Theorem 3.3 and other related the orems.\nE.4 Supplementary Experiments for Different Solution Accur acy\nIn this section, we investigate the impact of different solution accur acies on the experimental\nresults of problems (6) and (7). We set ǫto be either 10−4or 10−7and terminate the\n40Table 5: Lower- and upper-level objectives and optimal gaps with different penalty parameters for\nproblem (7).\nγ= 2×105\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 17153 7.4052e-03 1.3619e-05 4.2843e+00 -5.5818e-01\naPB-APG 20877 7.4052e-03 1.3619e-05 4.2843e+00 -5.5818e-01\nPB-APG-sc 27501 7.4052e-03 1.3619e-05 4.2843e+00 -5.5818e-01\naPB-APG-sc 40077 7.4052e-03 1.3619e-05 4.2843e+00 -5.5818e-0 1\nγ= 5×105\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 85511 7.3916e-03 2.4094e-08 4.8198e+00 -2.2752e-02\naPB-APG 85502 7.3916e-03 2.4093e-08 4.8198e+00 -2.2752e-02\nPB-APG-sc 173731 7.3916e-03 2.4071e-08 4.8198e+00 -2.2740e-0 2\naPB-APG-sc 166324 7.3916e-03 2.4091e-08 4.8198e+00 -2.2751e- 02\nalgorithms for PB-APG and PB-APG-sc when /⌊a∇⌈⌊lxk+1−xk/⌊a∇⌈⌊l ≤ǫ. For aPB-APG and aPB-\nAPG-sc, we choose the corresponding ǫ0values as 1 or 10−3. The remaining settings are the\nsame as in Section 5.\nWe also plot the values of the residuals of the lower-level objective G(xk)−G∗and the\nupper-level objective over time in Figures 5 and 6. Additionally, we als o collect the total\nnumber of iterations, the lower- and upper-level objective values , and the optimal gaps of\nour methods in Table 6 for problems (6) and (7) with different solution accuracies.\n0 0.05 0.1 0.1510-810-610-410-2100102\n0 0.05 0.1 0.1505101520\n0 0.05 0.1 0.15 0.2 0.25 0.310-1010-5100\n0 0.05 0.1 0.15 0.2 0.2505101520\nFigure 5: LRP (6) with ǫ= 10−4(left two subfigures) and ǫ= 10−7(right two subfigures).\nFrom Figures 5 and 6, it is evident that in most cases, our methods ou tperform the\nother methods in terms of both the lower- and upper-level object ives. However, there is\nan exception in the case of the upper-level objective for problem ( 7) whenǫ= 10−4. As\nillustrated in the second subfigure in Figure 6, our methods exhibit lar ger function values\nfor the upper-level objective compared to the other methods (e xcept BiG-SAM ( δ= 0.01)),\ndespite still achieving smaller optimal gaps for the lower-level objec tive. This discrepancy\nactually indicates that our methods have not yet achieved the desir ed accuracy when ǫ=\n410 0.05 0.1 0.15 0.2 0.25 0.310-410-2100102\n0 0.05 0.1 0.15 0.2 0.25 0.3020406080100\nFigure 6: LSRP (7) with ǫ= 10−4(left two subfigures) and ǫ= 10−7(right two subfigures).\n10−4, and it is important to note that /⌊a∇⌈⌊lxk+1−xk/⌊a∇⌈⌊l ≤ǫis not the termination criterion in our\nproposed algorithms, as explained in Appendix E.1. Therefore, the la rger optimality gaps\nfor the upper-level objective in this case may be attributed to the termination criterion.\nTable 6: Lower- and upper-level objectives and optimal gaps with different solution accuracies for\nproblem (6).\nǫ= 10−4\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 124 3.2794e-01 2.8671e-07 4.9483e+00 6.7024e-03\naPB-APG 148 3.2794e-01 2.3660e-07 4.9419e+00 2.9831e-04\nPB-APG-sc 100 3.2794e-01 5.4674e-07 4.9287e+00 -1.2956e-02\naPB-APG-sc 149 3.2794e-01 7.9015e-07 4.9302e+00 -1.1404e-02\nǫ= 10−7\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 841 3.2794e-01 1.7631e-08 4.9382e+00 -3.3999e-03\naPB-APG 551 3.2794e-01 1.7707e-08 4.9382e+00 -3.4075e-03\nPB-APG-sc 225 3.2794e-01 1.7493e-08 4.9383e+00 -3.3691e-03\naPB-APG-sc 614 3.2794e-01 1.7507e-08 4.9382e+00 -3.3874e-03\nTables 3, 6, and 7 demonstrate that the number of iterations for o ur methods also in-\ncreases with the solution accuracy, while the optimal gaps of the low er- and upper-level\nobjectives decrease correspondingly. This finding confirms that t he number of iterations and\nthe optimal gaps are influenced by the solution accuracy, as illustra ted in the expressions for\nthe number of iterations provided by Theorem 3.3 and other related theorems.\n42Table 7: Lower- and upper-level objectives and optimal gaps with different solution accuracies for\nproblem (7).\nǫ= 10−4\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 426 7.6950e-03 3.0342e-04 6.0249e+01 5.5407e+01\naPB-APG 432 7.8018e-03 4.1016e-04 4.8967e+01 4.4125e+01\nPB-APG-sc 437 7.6456e-03 2.5400e-04 6.0196e+01 5.5354e+01\naPB-APG-sc 517 7.6143e-03 2.2274e-04 4.9292e+01 4.4449e+01\nǫ= 10−7\nMethod Total iterations Lower-level value Lower-level gap Upper -level value Upper-level gap\nPB-APG 13707 7.3922e-03 5.9756e-07 4.7279e+00 -1.1460e-01\naPB-APG 7803 7.3923e-03 6.5025e-07 4.7300e+00 -1.1248e-01\nPB-APG-sc 12724 7.3922e-03 5.7840e-07 4.7354e+00 -1.0714e-01\naPB-APG-sc 7429 7.3922e-03 6.3816e-07 4.7326e+00 -1.0992e-01\n43"    },    {        "title": "2402.02166v1.A_comprehensive_study_on_zero_background_solitons_of_the_sharp_line_Maxwell_Bloch_equations.pdf",        "content": "A COMPREHENSIVE STUDY ON ZERO-BACKGROUND SOLITONS OF THE\nSHARP-LINE MAXWELL-BLOCH EQUATIONS\nSITAI LI\nABSTRACT . This work is devoted to systematically study general N-soliton solutions possibly con-\ntaining multiple degenerate soliton groups (DSGs), in the context of the sharp-line Maxwell-Bloch\nequations with a zero background. A DSG is a localized coherent nonlinear traveling-wave struc-\nture, comprised of inseparable solitons with identical velocities. Hence, DSGs are generalizations of\nsingle solitons (considered as 1-DSGs), and form fundamental building blocks of solutions of many\nintegrable systems. We provide an explicit formula for an N-DSG and its center from an appropriate\nRiemann-Hilbert problem. With the help of the Deift-Zhou’s nonlinear steepest descent method, we\nrigorously prove the localization of DSGs, and calculate the long-time asymptotics for an arbitrary\nN-soliton solutions. We show that the solution becomes a linear combination of multiple DSGs with\ndifferent sizes in the distant past and future. The asymptotic phase shift for each DSG is obtained\nin the process as well. Other generalizations of a single soliton are also carefully discussed, such\nasNth-order solitons and soliton gases. We prove that every Nth-order soliton can be obtained by\nfusion of eigenvalues of N-soliton solutions, with proper rescalings of norming constants, and every\nsoliton-gas solution can be considered as limits of N-soliton solutions as N→+∞. Consequently,\ncertain properties of Nth-order solitons and soliton gases are obtain as well. With the approach\npresented in this work, we show that results can be readily migrated to other integrable systems,\nwith the same non-self-adjoint Zakharov-Shabat scattering problem or alike. Results for the focusing\nnonlinear Schr ¨odinger equation and complex modified Korteweg–De Vries equation are obtained as\nexplicit examples for demonstrative purposes.\nCONTENTS\n1. Introduction 2\n2. Main results 5\n2.1. Assumptions and notations 5\n2.2. Spectra, Riemann-Hilbert problems and solution formulæ 6\n2.3. Solitons and degenerate soliton groups 9\n2.4. Soliton asymptotics 11\n2.5. High-order solitons 13\n2.6. Soliton gas 16\n2.7. Results for the focusing NLS and complex mKdV equations 17\n3. Reconstruction and soliton solution formulæ 20\n3.1. Proof of Lemma 1 20\n3.2. Proof of Theorem 1 21\n4. Degenerate N-soliton group 22\n4.1. Localization of N-DSG 22\n4.2. Center of N-DSG 25\n4.3. Boundedness of M(λ;t,z) 27\n5. Soliton asymptotics 28\n5.1. Basic analysis of soliton asymptotics in a stable medium 28\n5.2. Soliton asymptotics as t→+∞in a stable medium 29\n5.3. Soliton asymptotics as t→−∞in a stable medium 34\n1arXiv:2402.02166v1  [nlin.SI]  3 Feb 20245.4. Soliton asymptotics in an unstable medium 38\n6. High-order solitons 39\n6.1. Derivation of an Nth order soliton from fusion 39\n6.2. Derivation of the pole form of RHP 3 40\n6.3. Nth-order soliton solution formula 42\n7. Soliton gas 43\nAcknowledgments 46\nReferences 46\n1. I NTRODUCTION\nThis work studies the general N-soliton solutions of the sharp-line Maxwell-Bloch equations\n(MBEs) with a zero background (ZBG), particularly including degenerate solitons, in hoping to\npresent a framework of systematical analysis on arbitrary multi-soliton solutions of integrable\nsystems with the same non-self-adjoint Zakharov-Shabat scattering problem or alike. Examples of\ngeneralizations to other integrable systems are presented as well.\nSolitons are fascinating phenomena, and have been extensively studied in years in different\nfields of nonlinear sciences. The availability of the inverse scattering transform (IST) and many\nother algebraic methods make it possible to derive and study exact soliton solutions for completely\nintegrable systems [52]. In particular, the focus on one hand is studying the properties of single\nsolitons, and on the other hand is analyzing the nonlinear interactions among solitons, including\nthe proof of elastic collisions and the calculation of asymptotic phase shifts before and afterwards.\nIn order to discuss soliton interactions, it is crucial to distinguish the cases whether solitons\nhave different velocities or share the same ones. The latter case is referred to as a degenerate soliton\ngroup (DSG) in this work, but may have other names in literature, such as a breather [2]. Of\ncourse, some integrable systems by nature do not admit DSGs, such as the Korteweg–De Vries\n(KdV) equation and the defocusing nonlinear Schr ¨odinger (NLS) equation [3, 6, 14, 15]. On the\nother hand, a large number of integrable systems do admit such complex structures, for example,\nthe focusing NLS-type systems (including continuous and discrete cases, scalar and vector forms,\nwith zero or nonzero backgrounds) [5,20,37,39,42,52], the modified KdV (mKdV) equation [3,54],\nthe sine-Gordon equation [3], and many more [48]. Therefore, the soliton analysis on the latter set\nof systems is more complex in general because of the existence of DSGs.\nUnfortunately, many past works on soliton asymptotics of various systems exclude DSGs by\nassuming distinct soliton velocities inside multi-soliton solutions. The common aftermath of col-\nlisions is therefore well-separated individual solitons with phase shifts [1, 3, 5, 28, 34, 42, 52].\nIt is unavoidable to include DSGs when discussing the general N -soliton solutions for many in-\ntegrable systems1. Because a DSG is a multi-soliton structure comprising of solitons sharing an\nidentical velocity, it forms a coherent localized particle-like wave train. Contrary to the aforemen-\ntioned non-degenerate multi-soliton solutions, a DSG does not break apart into smaller compo-\nnents as time passes, but is able to interact with other DSGs or solitons as a whole. This novel and\ncomplex nonlinear phenomenon may be the reason why many past studies avoid such intriguing\nyet intimidating solutions, because more in-depth analysis become necessary and new tools may\nbe required as well.\nWe recall some existent results on DSGs for different integrable systems. Though this list is by\nno means exhaustive. The general N-soliton solutions of the focusing NLS equation with a ZBG is\n1The phrase “general” means that the N-soliton solutions may contain DSGs in different sizes, not just Nsingle\nsolitons. In other words, the assumption of distinct soliton velocities is dropped.\n2systematically studied in [39], where the center of a N-DSG is derived and soliton asymptotics are\ncalculated. However, the authors use a determinant solution formula obtained from the operator\nformalism. Thus, the results cannot be easily generalized to other integrable systems, because\nnew solution formulæ have to be derived using similar operator formalism. The N-DSGs of the\nfocusing NLS equation with a nonzero background (NZBG) is considered in [37], but the long-\ntime asymptotics for a general N-soliton solution is still missing because of the complexity of\nthe solution formula. The degenerate breathers of the focusing NLS-type equation with NZBG\nare also analyzed in a series of works when discussing superregular breathers and modulational\ninstability [25, 41, 53]. Partial results on DSGs, particularly 2-DSGs, for various integrable systems\nare discussed in many early works, such as the focusing NLS equation, the sine-Gordon equation,\nthe mKdV equation and others [2, 3, 29–31, 33, 35, 46, 52].\nWe should also mention other close topics of DSGs, such as the high-order solitons and soliton\ngases. High-order solitons correspond to eigenvalues of multiplicities more than one, and can be\nconsidered as fusion of solitons [3]. There are a great number of results on this type of solutions,\nfor sine-Gordon equation [12], the NLS equation [47], Hirota equation [13] and many more [10,\n11, 40]. Other than merging eigenvalues, one can also discuss the limit as N→+∞in a N-DSG,\nresulting in a phenomenon called soliton gas or integrable turbulence. This class of solutions\ncannot be explicitly computed in general. Though special cases are typically described by finite-\ngap solutions [18, 19, 24, 26, 27, 32, 51].\nThe purpose of this work using the sharp-line MBEs with a ZBG as a model is: (i) to present\na comprehensive analysis on an arbitrary N-DSGs, and an arbitrary N-soliton solution; (ii) to\ndemonstrate the relation between N-soliton solutions and Nth-order solitons or between N-soliton\nsolutions and soliton gases, subsequently deducing certain properties of Nth-order solitons and\nsoliton gases; (iii) to generalize results to other integrable systems by primarily analyzing solu-\ntions in the spectral space instead of the physical space. With recent developments in integrable\nsystems, such as the invention of the powerful Deift-Zhou’s nonlinear steepest descent method for\nanalyzing oscillatory Riemann-Hilbert problems (RHPs) [16,17], it is time to attack these problems\nand to build a framework for studying general N-soliton solutions in many integrable systems\nsharing similar spectra.\nThe MBEs are completely integrable [6, 7], and have attracted great interest, because of their\nimportance in nonlinear optics, successfully explaining self-induced transparency [43–45] and the\nclosely-related phenomenon of superfluorescence [21–23, 50]. In particular, the system describes\nlight-matter interactions in a semi-infinitely long one dimensional two-level optical medium with\nz≥0 for all time t∈R, where the light pulse is injected at one end z=0. The general MBEs\ncontain an integral with an arbitrary weight function, describing the atom distribution of the op-\ntical medium according to the atoms’ resonant frequencies. This work is concerned with the ideal\nmedium, whose atom distribution is a Dirac delta, corresponding to a sharp-line spectral shape.\nIn other words, one assumes that all the medium atoms have identical resonant frequency which\nis normalized to zero. Therefore, the Cauchy problem for sharp-line MBEs are written as fol-\nlows [4, 23],\n(1.1)qz(t,z)=−P(t,z), Pt(t,z)=−2q(t,z)D(t,z), Dt(t,z)=2R(q(t,z)P(t,z)),\nq(t, 0)=q0(t), t∈R,\nlim\nt→−∞q(t,z)=0 , D−∶=lim\nt→−∞D(t,z)=±1 , P−∶=lim\nt→−∞P(t,z)=0 , z≥0 .\nThe variable tand zare the spatial and temporal variables in the comoving frame. The quantity\nq(t,z)∈Cdescribes the optical pulse, D(t,z)∈Rdenotes the population inversion, and P(t,z)∈\nCis the polarization of the optical medium. The subscript tand zdenote partial derivatives\nand q(t,z)is the complex conjugate of q(t,z). It is easy to check that∂\n∂t(D2+∣P∣2)=0 from the\n3MBEs (1.1). We then normalize the system such that D2+∣P∣2=1 for t∈Rand z≥0. Moreover,\nq0(t)is the input pulse, and {D−,P−}is the initial state of the medium in the distant past t→−∞.\nThe trivial solutions {q≡0,D≡±1,P≡0}are considered as the ZBGs, denoting the static states\nof the system with absence of light. Consequently, the medium is in one of two pure states, the\nground state D=−1 or the excited state D=1. This work considers the soliton solutions riding on\neither backgrounds, exhibiting consequential differences from a physical point of view. Therefore,\nthe boundary condition D−=−1 or D−=1 means that initially all atoms are in the ground state or\nexcited state, respectively.\nIt should be pointed out that, assuming real solutions, the MBEs (1.1) with restraints P=sin(Θ),\nD=cos(Θ)and q−Θt/2 are related to the sine-Gordon equation in characteristic coordinates\nΘtz=2 sin(Θ)withΘ∈R. It is hard to consider the N-DSGs for the sine-Gordon in the physical\nframe Θtt−Θzz+sin(Θ)=0 with fixed boundary conditions at infinities, because the solitons are\ntopological ones and the soliton quantity Nchanges the amplitudes of boundaries. In some sense,\nMBEs are complex generalizations of the sine-Gordon equation and may be easier to analyze.\nThe Lax pair for the system (1.1) is given by\n(1.2)ϕt=(iλσ3+Q)ϕ, ϕz=−i\n2λρ(t,z)ϕ,\nQ(t,z)∶=(0 q(t,z)\n−q(t,z) 0),ρ(t,z)∶=(D(t,z)P(t,z)\nP(t,z)−D(t,z)),σ3∶=(1 0\n0−1),\nwhere ϕ∶=ϕ(λ;t,z)is the eigenfunction. Note that the scattering problem ϕt=(iλσ3+Q)ϕ\nis the celebrated non-self-adjoint Zakharov-Shabat problem, so MBEs (1.1) can be considered as\na member of negative flows in the AKNS hierarchy, due to the term proportional to λ−1in the\nsecond part of the Lax pair (1.2).\nOne of many ways to find exact multi-soliton solutions of MBEs (1.1) is to formulate the cor-\nresponding IST, which was first done in [4], but the authors only considered D−=−1 case. Ten\nyears later, the IST for the full Cauchy problem (1.1) was finally established [21–23]. However, in\nall works, the inverse problems were formulated in terms of the Gel’fand-Levitan-Marchenko in-\ntegral equations. It is possible to obtain the soliton solutions from these works, but the form of the\ninverse problem are not useful for the asymptotic calculations. A RHP formulation is necessary\nfor the Deift-Zhou’s nonlinear steepest descent approach. One could reformulate the IST for the\nCauchy problem (1.1) and update its format using a RHP as the inverse problem. However, as re-\ncently shown [36] that, the Cauchy problem for a general input light pulse q0(t)may be ill-posed,\nand one of the often used assumption that q(⋅,z)∈L1(R)may be violated. Thus, a considerable\neffort must be made in order to pursue this direction, which defeats the purpose of the current\nwork on analyzing soliton solutions, by making a huge detour. By similar arguments, algebraic\nmethods, such as the Darboux transformation, to obtain exact soliton solutions are not feasible as\nwell.\nAs such, we adopt another approach similarly to the one used in [36], by starting with an ap-\npropriate RHP and using the dressing method to obtain exact N-soliton solutions to MBEs (1.1).\nThis approach is able to kill two birds with one stone. On one hand, it provides a RHP that is\nreadily available for the nonlinear steepest descent. On the other hand, the dressing argument is\nrelatively short and is able to keep the length of the manuscript under control.\nIn this work, we perform a systematical study of the general N-soliton solutions of MBEs (1.1).\nWe first present two equivalent forms of RHPs describing general soliton solutions, both of which\nare useful in the study. We then derive an explicit formula for the N-soliton solution. Using\nthe Deift-Zhou’s nonlinear steepest descent method, we prove the localization of N-DSGs and\ncompute explicit expression for its center. Also using the nonlinear steepest descent method, we\ncalculate the long-time asymptotics as t→±∞for general N-soliton solutions, showing that they\n4are linear combinations of DSGs with various sizes asymptotically. The phase shifts for each DSG\nare obtained as well. Furthermore, we demonstrate that every Nth-order soliton can be obtained\nby fusion of Nsimple solitons, where the rescaling process is explicit. We also formally take\nthe limit of N-soliton solutions as N→+∞ in order to find soliton-gas solutions. Finally, we\ndemonstrate how to generalize the results for MBEs (1.1) to the focusing NLS equation and the\ncomplex mKdV equation with the identical scattering problem.\n2. M AIN RESULTS\nWe first lay the groundwork and state necessary assumptions.\n2.1. Assumptions and notations. Recall that IST works in three major steps:\n(1)Direct problem: One analyzes the scattering problem with the given initial condition q0(t)\nand obtains the scattering data at z=0, including the reflection coefficient r(λ,z=0),\ndiscrete eigenvalues and norming constants.\n(2)Propagation2:One uses the second half of the Lax pair (1.2) to calculate the zdependence\nof the scattering data.\n(3)Inverse problem: One reconstructs the eigenfunction ϕ(λ;t,z), by formulating and solv-\ning either a Gel’fand-Levitan-Marchenko integral equation or a RHP , and subsequently\nobtains the solutions of MBEs (1.1).\nRecall that the past ISTs [4, 23] do not apply to this work. Luckily, the direct problem only\ninvolves the initial data q0(t)and the scattering problem, the latter of which is shared among the\nwhole AKNS hierarchy. Thus, one could borrow the results from the general theory for the AKNS\nhierarchy and other well-studied integrable systems [3, 8, 9, 55, 56]:\n(1)Existence of scattering data: Ifq0(t)is sufficiently smooth and decays to zero fast enough\nast→±∞, for example, if q0(t)is in the Schwartz class, then the reflection coefficient\nr(λ; 0)and Npairs of discrete eigenvalues {λj,λj}N\nj=1⊂C∖Rexist, where λjis the complex\nconjugate of λj. For each eigenvalue pair, there are associated norming constants with\nproper symmetries.\n(2)Reflectionless solutions: Pure soliton solutions require that the reflection coefficient r(λ;z)\nvanishes for all z≥0.\n(3)Anomalies:\n(a) Eigenvalues could be of higher orders.\n(b) The number of eigenvalue pairs Ncould be infinity.\n(c) There could be singularities on the continuous spectrum R.\nBecause we are studying N-soliton solutions, it is reasonable to assume that the first two results\nare true. For anomaly (a), most of the work addresses simple eigenvalues, but in Section 6 we\nshow that every high-order eigenvalue can be obtained by fusion of Nsimple eigenvalues. Cor-\nrespondingly, every high-order soliton can be obtained by merging Nsimple solitons. Then, we\npresent an explicit formula for such high-order solutions. Similarly to anomaly (b), we mostly\ndiscuss the case with 1 ≤N<+∞, but in Section 7 we show how to derive a RHP for soliton gases\nby taking limits N→+∞. We do not discuss the phenomena related to anomalies (c), because they\nare out of the scope of this work.\nRemark 1 (On notations) .We use boldface letters to denote matrices and vectors, with the ex-\nception of the identity matrix Iand the Pauli matrix σ3defined in Equation (1.2). The imaginary\nunit is denoted i. Complex conjugation and conjugate transpose are indicated with a bar λand a\n2This step usually is called “evolution”. Because of the spatial-temporal-variable switch in optical systems, the\n“evolution” variable becomes the spatial variable z, and we rename this step as “propagation”.\n5FIGURE 1. Left: an illustration of the pole distribution in RHP 1. Right: an illus-\ntration of the jump configuration in RHP 2 which is equivalent to the left plot.\ndagger A†, respectively. The Schwarz reflection for a scalar function is define by f(z)∶=f(z), and\nfor a matrix function is define by A†(λ)∶=A(λ)⊺. Furthermore, we use the shorthand notation\nA−†(λ)=[A†(λ)]−1to denote the combination of Schwarz reflection and inversion of a matrix.\nLet the notation Dϵ\na∶={λ∈C∶∣λ−a∣<ϵ}being the open disk centered at λ=awith radius ϵ>0 in\nthe complex plane, so ∂Dϵ\na∶={λ∈C∶∣λ−a∣=ϵ}is a circle.\nWith the assumptions in mind, we can properly introduce the spectra of multi-soliton solutions\nand the corresponding RHP .\n2.2. Spectra, Riemann-Hilbert problems and solution formulæ.\nDefinition 1 (Eigenvalues and norming constants) .LetJ≥1 and Nj≥1 with j=1, 2, . . . , J, be\ngiven integers. Suppose 0 <r1<r2<⋅⋅⋅<rJare positive real numbers. We define two sets\n(2.1)Λ∶=J\n⋃\nj=1Λj,Λj∶={λj,k∣λj,k=rjeiαj,kwith 0<αj,1<⋅⋅⋅<αj,Nj<π},\nΩ∶=J\n⋃\nj=1Ωj,Ωj∶={ωj,k∣ωj,k∈C∖{0}with k=1, 2, . . . , Nj}.\nWe call Λthe set of eigenvalues (in the upper half plane) ,Ωthe set of norming constants . The total\nnumber of eigenvalues is simply the cardinality of set Λ, denoted by N∶=∣Λ∣=N1+⋅⋅⋅+ NJ.\nBefore moving on, we present a few remarks about Definition 1:\n●From now on, the phrase “eigenvalues” simply means “discrete eigenvalues”, because: (i)\nwe are only discussing soliton solutions, rising from discrete spectra, and (ii) we assume\nreflectionless solutions, so that the continuous spectrum Rdoes not play any role.\n●Eigenvalues from the non-self-adjoint Zakharov-Shabat problem come in conjugate pairs\n{λj,k,λj,k}, but for simplicity we call the ones in the upper half plane eigenvalues, or Λthe\neigenvalue set. The conjugates are implicitly implied.\n●All eigenvalues from Definition 1 are categorized into Jgroups Λj, each of which contains\neigenvalues with identical moduli rj>0. The reason for the definition of each group Λjis\ndiscussed in detail after Theorem 2.\nWe first present RHPs composed of given eigenvalues and norming constants. The RHPs also\nsatisfy necessary symmetries from the non-self-adjoint Zakharov-Shabat scattering. The relation\nbetween the RHPs and the solutions to MBEs (1.1) will be discussed in Lemma 1.\n6Riemann-Hilbert Problem 1 (Residue form) .Given sets ΛandΩfrom Definition 1. Seek for a matrix\nmeromorphic function λ↦M(λ;t,z)onC, such that it has asymptotics M(λ;t,z)→Iasλ→∞, and\nsatisfies the following residue conditions\n(2.2)Res\nλ=λj,kM(λ;t,z)=lim\nλ→λj,kM(λ;t,z)(0 0\nωj,ke−2iθ(λ;t,z)0),\nRes\nλ=λj,kM(λ;t,z)=lim\nλ→λj,kM(λ;t,z)(0−ωj,ke2iθ(λ;t,z)t\n0 0),\nfor all λj,k∈Λandωj,k∈Ω, and\n(2.3) θ(λ;t,z)∶=λt−D−\n2λz.\nThe quantity θ(λ;t,z)is the (linear) dispersion relation of the MBEs (1.1) deduced from the Lax\npair (1.2). An illustration of the pole distribution of RHP 1 is shown in Figure 1(left). The existence\nand uniqueness of solutions of RHP 1 will be addressed in Lemma 1 later. Clearly, by Liouville’s\ntheorem the solution M(λ;t,z)can be formally written as\n(2.4) M(λ;t,z)=I+∑\nλj,k∈Λ(Res λ=λj,kM(λ;t,z)\nλ−λj,k+Resλ=λj,kM(λ;t,z)\nλ−λj,k).\nNote that this is not an explicit formula, but instead is a linear system, which is solved in Sec-\ntion 3.2. It is also worth pointing out that the residue conditions in RHP 1 are not friendly when\napplying Deift-Zhou’s nonlinear steepest descent method in order to calculate asymptotics, which\nis the main part of this work. Thus, it is necessary to convert the residue conditions to jumps.\nRecall the notation Dϵ\nafrom Remark 1. Obviously, there always exists a constant 0 <ϵ≪1 such\nthat all closes disks {Dϵ\nλj,k,Dϵ\nλj,k}λj,k∈Λdo not intersect for a given set Λ, and they do not cross the\nreal line. Without repeating this argument in the rest of this work, we always assuming it is true\nand such ϵis used.\nRiemann-Hilbert Problem 2 (Jump form) .For given sets ΛandΩ. Seek for a sectional analytic matrix\nfunction λ↦M(λ;t,z)onC, such that it has asymptotics M(λ)→Iasλ→∞, and satisfies the jumps\n(2.5)M+(λ;t,z)=M−(λ;t,z)Vj,k(λ;t,z)−1, λ∈∂Dϵ\nλj,k,\nM+(λ;t,z)=M−(λ;t,z)V†\nj,k(λ;t,z), λ∈∂Dϵ\nλj,k,\nfor all λj,k∈Λandωj,k∈Ω, all circles {∂Dϵ\nλj,k,∂Dϵ\nλj,k}λj,k∈Λare oriented counterclockwise and disjoint\nwith a proper 0<ϵ≪1, and the jump matrices are defined by\n(2.6) Vj,k(λ;t,z)∶=⎛\n⎜⎜\n⎝1 0\nωj,ke−2iθ(λ;t,z)\nλ−λj,k1⎞\n⎟⎟\n⎠, V†\nj,k(λ;t,z)∶=[Vj,k(λ;t,z)]†=⎛\n⎜⎜\n⎝1ωj,ke2iθ(λ;t,z)\nλ−λj,k\n0 1⎞\n⎟⎟\n⎠.\nAn illustration of the jump configuration of RHP 2 is shown in Figure 1(right).\nRemark 2. Clearly, all jump matrices in RHP 2 satisfy the Schwarz symmetry V(λ)−1=V†(λ)for\nalltand z, so by Zhou’s lemma the solution to RHP 2 uniquely exists [55]. Moreover, it is easy to\nverify that the solution M(λ;t,z)has the symmetry M(λ;t,z)−1=M†(λ;t,z)which implies that\n(2.7) M2,2(λ;t,z)=M1,1(λ;t,z), M2,1(λ;t,z)=−M1,2(λ;t,z).\nHence, one only needs to solve for the first row in order to reconstruct the whole matrix M(λ;t,z).\n7Lemma 1 (Reconstruction formula) .For given sets ΛandΩ, and a given number D −=±1, there is a\nunique solution {Q(t,z),ρ(t,z)}to the MBEs (1.1) with boundary condition D →D−as t→−∞, which\ncan be reconstructed from the solutions to the RHP 1 or RHP 2 via the following formula\n(2.8) Q(t,z;Λ,Ω)=lim\nλ→∞iλ[M(λ;t,z),σ3], ρ(t,z;Λ,Ω)=D−M(0;t,z)σ3M(0;t,z)−1.\nRecall N=∣Λ∣. The solution is call the N-soliton solution of MBEs, and Λis the eigenvalue set.\nLemma 1 can be proved by applying the dressing method to RHP 2, quite similar to the one\nin [36, Appendix B, step (i)]. The proof is shown in Section 3.1 for completeness of this work.\nRemark 3. In this work, the phrase equivalent forms orequivalent RHPs indicates that multiple RHPs\nyield identical MBEs solutions. Hence, RHPs 2 and 1 are equivalent by Lemma 1. Usually there\nare two equivalent RHPs for soliton solutions. One contains residue conditions, as usually done\nin ISTs, and the other one contains jumps. The former one is useful when calculating the solution\nformula, whereas the latter one is useful when analyzing properties or applying the nonlinear\nsteepest descent method. Both forms are used extensively in this work.\nRemark 4. We append the spectral data ΛandΩas explicit parameter dependence in MBEs\nsolutions as (t,z;Λ,Ω)in Lemma 1 or later, in order to explicitly show the spectral dependence.\nThis will be helpful when later discussing soliton asymptotics, where ΛorΩmay change.\nIn Section 3.2, we compute the N-soliton solution explicitly via the reconstruction formula in\nLemma 1 from RHP 1.\nTheorem 1 (General N-soliton solution formula) .For given sets ΛandΩfrom Definition 1, and\nN=∣Λ∣. the corresponding N-soliton solution to MBEs is explicitly given by\n(2.9) q(t,z;Λ,Ω)=2i−2idet(I−⃗1⊗C∞+ΓΓ)\ndet(I+ΓΓ),\nwhere D (t,z;Λ,Ω)and P(t,z;Λ,Ω)are reconstructed by Lemma 1, with entries of M(λ;t,z)given by\n(2.10) M1,1(λ;t,z)=det(I−⃗1⊗CΓ+ΓΓ)\ndet(I+ΓΓ), M1,2(λ;t,z)=det(I−⃗1⊗C+ΓΓ)\ndet(I+ΓΓ)−1 ,\nwhere⊗denotes the tensor product, and other quantities are defined by\n(2.11)C(λ)∶=(c1,1(λ). . . c1,N1(λ)c2,1(λ). . . c2,N2(λ). . . cJ,1(λ). . . cJ,NJ(λ)),\nC∞=lim\nλ→∞λC(λ)=(ω1,1e−2iθ1,1ω1,2e−2iθ1,2. . . ωJ,NJe−2iθJ,NJ),\n⃗1∶=(1 . . . 1 )⊺\n1×N,\nΓ∶=(C(λ1,1)⊺. . . C(λ1,N1)⊺. . . C(λJ,1)⊺. . . C(λJ,NJ)⊺)⊺\nN×N,\nθj,k∶=θ(λj,k;t,z), cj,k(λ)∶=ωj,k\nλ−λj,ke−2iθj,k, λj,k∈Λ, ωj,k∈Ω.\nThe spectrum set ΛandΩare arbitrarily given in Theorem 1. Each λj,kproduces a soliton\ntravels with a particular speed. Because we do not impose additional constraint on Λ, some of\nthe solitons may travel with the same speed, i.e., form a DSG3. Hence, the setup of this work\nis capable of producing DSGs, and becomes a perfect tool for studying multi-soliton solutions\ncontaining such complex structures.\n3We will make these statements concrete in Theorems 2 and 3.\n8Before diving into the long-time asymptotics for general N-soliton solutions, it is necessary to\ncharacterize the fundamental building blocks, which are single solitons and degenerate solitons.\nIt is sufficient to study N-DSGs, because a single soliton is simply a 1-DSG.\n2.3. Solitons and degenerate soliton groups. We would like to be able to show that an N-DSG is\na generalization of a single soliton with N≥2. To this end, we explore certain properties of DSGs\nand show similarities between DSGs and solitons, and so we assume there is only one eigenvalue\ngroup J=1 from Definition 1. As the name suggested, solitons are localized traveling waves\nand maintain their velocities and shapes after nonlinear interactions [49]. The following theorem\nestablishes the localization and traveling-wave nature of an DSG with an arbitrary size N≥1. An\nexplicit formula for the DSG position is also provided.\nTheorem 2 (Degenerate N-soliton solution) .Suppose J =1. LetΛandΩbe given according to Def-\ninition 1, so that N =N1≥1be a positive integer and Λ=Λ1. Namely, all eigenvalues have identical\nmoduli∣λ1,k∣=r1>0for1≤k≤N1. Then, regardless of the initial state of the medium D −=±1, Lemma 1\nproduces an N-DSG of the MBEs (1.1) , which has the following properties\n(1) All N solitons in the solution travel coherently with an identical speed V ∶=−2D−r2\n1.\n(2) The N-DSG is localized along the direction z =Vt. Along other directions z =ξt with ξ≠V,\n(2.12) q(t,z;Λ,Ω)=o(1), P(t,z;Λ,Ω)=o(1), D(t,z;Λ,Ω)=D−+o(1), t→±∞.\n(3) The center of the N-DSG denoted by z c(t)is given as\n(2.13) zc(t)∶=Vt+zd, zd∶=−D−r1\n∑N1\nk=1sin(α1,k)ln(N1\n∏\nk=1∣ω1,k∣⋅∏N1\nk=2∏k−1\nl=1∣λ1,k−λ1,l∣2\n∏N1\nk=1∏N1\nl=1∣λ1,k−λ1,l∣),\nwhere z dis the displacement.\n(4) With J =1, the solution M(λ;t,z)to RHPs 1 and 2 are bounded as t →±∞in the direction z =Vt.\nTheorem 2 is proved in Section 4. The last statement (4) of Theorem 2 is used later in the proof\nof Theorem 3, in Section 5.2.4.\nRemark 5. The N-DSG’s velocity Vcan be derived from the equation I(θ(λ1,k;t,Vt))=0 and\nis solely determined by r1. Conversely, we use the equation I(θ(λj,k;t,Vt)=const to define the\neigenvalue groups Λjin Definition 1. The velocity is positive in a stable medium ( D−=−1)and is\nnegative in an unstable medium ( D−=1). Because the light cone is the first quadrant of the (t,z)\nplane, the N-DSG travels subluminally in a stable medium, and superluminally in an unstable\nmedium. Consequently, the stable N-DSG stays inside the light cone as t→+∞, whereas the\nunstable N-DSG eventually travels out. Clearly, the unstable case seems unphysical, and deserves\nfurther analysis. In fact, solitons in unstable mediums with non-vanishing reflection coefficient\nare studied before and shown to be related to superfluorescence [21–23, 50].\nDue to the complexity of the solution formula from Theorem 1, it is hard to discuss the shape of\nN-DSGs, and how the shape depends on the eigenvalues and norming constants. Therefore, it is\nnecessary to analyze special cases when Nis small. The simplest DSG is of course a single soliton\ncorresponding to N=1, and the breathers correspond to N=2. Let us first introduce a shorthand\nnotation to simplify solution formulæ before we investigate these special cases.\n(2.14) sj,±(t,z)∶=±2rjt+D−\nrjz.\nOne first discusses what happens of Theorem 2 when N=1, namely, a single soliton.\n9FIGURE 2. Plots of exact 1-soliton solutions in an initially stable medium with r1=\n1,ξ1,1=0 and ϕ1,1=π/2, and α1,1=π/7 or α1,1=π/24 from Corollary 1. The two\nsolutions have identical velocities, but different amplitudes determined by α1,1.\nCorollary 1 (One-soliton solution) .Suppose that N =J=N1=1in Theorem 1, and sets Λ={λ1,1}and\nΩ={ω1,1}are given. Using the parameterization for eigenvalue and norming constants in Definition 1.\nTheorem 1 produces a one-soliton solution, which can be written in a compact form\n(2.15)q(t,z;λ1,1,ω1,1)=2ir1sin(α1,1)e−iτ1,1(t,z)sech(χ1,1(t,z)),\nD(t,z;λ1,1,ω1,1)=D−[cos(2α1,1)sech2(χ1,1(t,z))+tanh2(χ1,1(t,z))],\nP(t,z;λ1,1,ω1,1)=−2D−e−iτ1,1(t,z)sin(α1,1)sech2(χ1,1(t,z))cosh(χ1,1(t,z)−iα1,1(t,z)),\nwith\n(2.16)χ1,1(t,z)∶=sin(α1,1)s1,+(t,z)+ξ1,1, τ1,1(t,z)∶=cos(α1,1)s1,−(t,z)+ϕ1,1,\nω1,1=2r1sin(α1,1)eξ1,1+iϕ1,1, ξ1,1∈R, ϕ1,1∈[0, 2π).\nThe soliton velocity is V =−2D−r2\n1, the soliton amplitude is 2r1sin(α1,1), and the soliton center is z c(t)=\nVt+r1ξ1,1/sin(α1,1).\nClearly, the modulus parameter r1of the eigenvalue λ1,1solely determines the soliton velocity\nV, and together with the phase α1,1determines the soliton amplitude. The phase parameter α1,1\nalso dominates the complex oscillation of the soliton, via the functions τ1,1(t,z)in Equation (2.16).\nIn particular, the oscillation frequency is proportional to cos (α1,1), so smaller value of α1,1means\nmore rapidly oscillations, as shown in Figure 2. Of course, the oscillatory behavior is hidden\nwhen considering the modulus ∣q(t,z)∣in the one-soliton solution, but it becomes more prominent\nin consideration of N-DSG with N≥2.\nRemark 6. As can be easily seen in Corollary 1, the one-soliton solution decays to the ZBG as\nt→±∞with a fixed value of z. In particular, D(t,z)→D−ast→±∞. Therefore, if the medium is\ninitially in the stable state, it falls back to the stable state after a long time, as one expects physically.\nHowever, if the medium is initially in the unstable state, it also returns, which is notphysical.\nContrary to the pure soliton solution discussed here, it was proved that the solution to MBEs\nwithout any solitons fell back to the stable state from the unstable one [36]. Hence, it suggests that\nfor an unstable medium, the nonlinear interactions between solitons and radiation play a crucial\nrole, and deserve further analysis. Again, this case is related to superfluorescence.\nNow, we move on to analyzing a more complex yet interesting coherent structure than a single\nsoliton. However, a general 2-DGS (a bound state or a breather) is still too complex to present for\nthe MBEs. We therefore discuss a symmetric version, by imposing λ1,1=−λ1,2. This spectral setup\nwas discussed in [3, Equation (5.9)], but for the sine-Gordon equation.\n10FIGURE 3. Plots of exact symmetric 2-DSG in an initially stable medium with r1=1,\nξ1,1=ϕ1,1=0, and α=π/10 or α=π/30 from Corollary 2. Since q(t,z)is purely\nimaginary, the right column only shows the imaginary parts. The two cases have\nidentical velocity. The parameter αaffects amplitudes and oscillatory behavior.\nCorollary 2 (Symmetric 2-DSG) .Suppose that J =1and N=N1=2. Theorem 2 reduces to a general\n2-DSG, which can be simplified further by taking α1,1=αandα1,2=π−α,ω1,1=2r1sin(α)eξ+iϕand\nω1,2=ω1,1, implying that the eigenvalues are tied with an additional symmetry λ1,2=−λ1,1.\n(2.17)q(t,z;Λ,Ω)=4ir1tan(α)sin(α)sinh(L)sin(τ)+cos(α)cosh(L)cos(τ)\ncosh(L)2+tan(α)2sin(τ)2,\nD(t,z;Λ,Ω)=D−cosh(L)4−6 tan(α)2cosh(L)2sin(τ)2+tan(α)4sin(τ)4\n[cosh(L)2+tan(α)2sin(τ)2]2,\nP(t,z;Λ,Ω)=4iD−tan(α)cosh(L)sin(τ)cosh(L)2−tan(α)2sin(τ)2\n[cosh(L)2+tan(α)2sin(τ)2]2,\nwhere the two real functions L (t,z)andτ(t,z)are defined below\n(2.18)L(t,z)∶=sin(α)s1,+(t,z)+ξ+ln(∣cos(α)∣),\nτ(t,z)∶=cos(α)s1,−(t,z)+ϕ−α.\nIn the symmetric 2-DSG in Corollary 2, the optical pulse q(t,z)and polarization P(t,z)are\npurely imaginary, and D(t,z)is of course still real. Note that this symmetric 2-DSG is fundamen-\ntally different from the one of the sine-Gordon equation [3, Equation (5.9)], the latter of which\nis a real function. The overall form of 2-DSG is remarkably similar to the soliton solution with\nNZBG [38, Equation (17)]. This can be explained from the spectral point of view, because: (i) both\ncases are reflectionless; and (ii) the RHP for 2-DSG with ZBG contains four poles (two pairs of\neigenvalues), whereas the RHP for the 1-soliton solution with NZBG contains a quartet of eigen-\nvalues using a uniformization variable. Thus, the overall structure of RHPs are alike, so are the\ncorresponding solutions. Two such examples are shown in Figure 3. Clearly, the shape of DSGs\ndepends on the phase parameter αj,k.\nA more general case is shown in Figure 4(left), where the density plot of ∣q(t,z)∣of a 4-DSG in an\ninitially stable medium D−=−1 is presented from Theorem 1 with parameters λ1,k=eiπ/5, eiπ/3, i , e2iπ/3\nandω1,k=1 for 1≤k≤4. On top of the exact solution, a red dashed line is drawn determined by\nthe center zc(t)from Theorem 2. A perfect agreement can be observed.\n2.4. Soliton asymptotics. With the general characterization of N-DSGs obtained in Section 2.3, we\nare ready to consider the nonlinear interactions among several DSGs with different sizes. Thus, in\nthis subsection, we consider more eigenvalue groups, meaning J>1 from Definition 1. We would\nlike to calculate the long-time asymptotics of general N-soliton solution as t→±∞, corresponding\n11to the investigation of the light-matter interactions inside the optical medium from the distant past\nto far future, in different directions z=ξtwith ξ∈R. Hence, we apply the Deift-Zhou’s nonlinear\nsteepest descent method to the oscillatory RHP 2 and calculate the leading terms and perform\nerror estimates.\nTheorem 3 (Soliton asymptotics) .Suppose that J ≥2. LetΛandΩbe given according to Definition 1.\nThen, the general N-soliton solution from Theorem 1 has the following long-time asymptotics with z =ξt:\n(1) If ξ=Vjfor each 1≤j≤J, where V j=−2D−r2\njis the velocity of the DSG generated from eigenvalue\nsetΛjgiven in Theorem 2, then in both stable and unstable mediums as t →±∞one obtains the\nasymptotic expansion\n(2.19)q(t,z;Λ,Ω)=q(t,z;Λj,Ω(±)\nj)+o(1),\nD(t,z;Λ,Ω)=D(t,z;Λj,Ω(±)\nj)+o(1), P(t,z;Λ,Ω)=P(t,z;Λj,Ω(±)\nj)+o(1).\nThe solution set {q(t,z;Λj,Ω(±)\nj),D(t,z;Λj,Ω(±)\nj),P(t,z;Λj,Ω(±)\nj)}forms an N j-DSG of the\nMBEs in the corresponding medium given by Theorem 2 with the eigenvalue set Λjand a modified\nnorming constant set Ω(±)\nj∶={ω(±)\nj,k}Nj\nk=1, where the modified norming constants are given by\n(2.20) ω(+)\nj,k∶=ωj,kδj+1(λj,k)−2, ω(−)\nj,k∶=ωj,kδj(λj,k)2δ1(λj,k)−2, k=1, 2, . . . , Nj.\nwith\n(2.21) δj(λ)∶=pj(λ)\npj(λ), pj(λ)∶=J\n∏\ns=jNs\n∏\nl=1(λ−λs,l), j=1, 2, . . . , J.\n(2) If ξ≠Vjfor all 1≤j≤J, then\n(2.22) q(t,z;Λ,Ω)=o(1), D(t,z;Λ,Ω)=o(1), P(t,z;Λ,Ω)=o(1), t→±∞.\nAll error terms are exponentially small.\nTheorem 3 immediately implies that, asymptotically, the general N-soliton solution is a linear\ncombination of multiple DSGs with various sizes,\n(2.23)q(t,z;Λ,Ω)=J\n∑\nj=1q(t,z;Λj,Ω(±)\nj)+o(1),\nD(t,z;Λ,Ω)=J\n∑\nj=1D(t,z;Λj,Ω(±)\nj)+o(1), P(t,z;Λ,Ω)=J\n∑\nj=1P(t,z;Λj,Ω(±)\nj)+o(1).\nApplying the DSG center formula from Theorem 2 to Theorem 3 immediately yields the asymp-\ntotic phase shifts induced by nonlinear interactions among DSGs.\nCorollary 3 (Asymptotic shifts) .The asymptotic centers for each N j-DSG with parameters Λj,Ω(±)\njand\nj=1, 2, . . . , J, as t→±∞from Theorem 3, in both stable and unstable mediums, are given by\n(2.24)z+\nc,j(t)∶=Vjt+z(+)\nd,j,z(+)\nd,j∶=−D−rj\n∑Nj\nk=1sin(αj,k)ln(Nj\n∏\nk=1∣ω(+)\nj,k∣∏Nj\nk=2∏k−1\nl=1∣λj,k−λj,l∣2\n∏Nj\nk=1∏Nj\nl=1∣λj,k−λj,l∣),t→+∞,\nz−\nc,j(t)∶=Vjt+z(−)\nd,j,z(−)\nd,j∶=−D−rj\n∑Nj\nk=1sin(αj,k)ln(Nj\n∏\nk=1∣ω(−)\nj,k∣∏Nj\nk=2∏k−1\nl=1∣λj,k−λj,l∣2\n∏Nj\nk=1∏Nj\nl=1∣λj,k−λj,l∣),t→−∞.\n12FIGURE 4. Density plots of ∣q(t,z)of exact N-soliton solutions in stable medi-\nums from Theorem 1 with different setups. Left: A 4-DSG with parameters r1=1,\nα1,k=π/5,π/3,π/2, 2π/3, and ω1,k=1 for all 1 ≤k≤4. The red dashed line cor-\nresponds to the center in Theorem 2. Center: A 3-soliton solution with two DSGs.\nThe parameters are r1=0.4,r2=1,α1,1=2π/5,α2,1=π/3,α2,2=π/2,ω1,1=e14,\nω2,1=1, and ω2,2=i. The red dashed lines correspond to the asymptotic centers\nin Corollary 3. Right: A 5-soliton solution with three DSGs. The parameters are\nr1=0.4,r2=1,r3=2,α1,k=π/3,π/2,α2,k=π/3, 2π/3,α3,1=π/4,ω1,1=ω1,2=e15,\nω2,1=e3,ω2,2=e4, and ω3,1=e−10. Again, the red dashed lines correspond to the\nasymptotic centers in Corollary 3.\nComparing with the unshifted displacement z d,jfrom Theorem 2 yields the asymptotic shifts\n(2.25)∆(+)\nj∶=z(+)\nd,j−zd,j=2D−rj\n∑Nj\nk=1sin(αj,k)Nj\n∑\nk=1J\n∑\ns=j+1Ns\n∑\nl=1ln∣λj,k−λs,l\nλj,k−λs,l∣,\n∆(−)\nj∶=z(−)\nd,j−zd,j=2D−rj\n∑Nj\nk=1sin(αj,k)Nj\n∑\nk=1j−1\n∑\ns=1Ns\n∑\nl=1ln∣λj,k−λs,l\nλj,k−λs,l∣,\nThe overall asymptotic phase shift is given by\n(2.26) ∆j∶=z(+)\nd,j−z(−)\nd,j=2D−rj\n∑Nj\nk=1sin(αj,k)Nj\n∑\nk=1ln(J\n∏\ns=j+1Ns\n∏\nl=1∣λj,k−λs,l\nλj,k−λs,l∣⋅j−1\n∏\ns=1Ns\n∏\nl=1∣λj,k−λs,l\nλj,k−λs,l∣).\nTwo examples of general soliton asymptotics are shown in Figure 4, where the medium is cho-\nsen as initially stable D−=−1. Only the optical pulse ∣q(t,z)∣are shown in the two examples. The\nmedium functions {D(t,z),P(t,z)}can also be obtained from Theorem 1 and can be compared\nwith Theorem 3, but are omitted to avoid repetition. The center panel contains the density plot\nof a 3-soliton solution with two DSGs, with parameters given in the caption. Four red dashed\nlines are drawn describing the asymptotic center z±\nc,j(t)for each DSG, obtained from Theorem 3.\nSimilarly, the right panel contains a 5-soliton solution with three DSGs. In both plots, phase shifts\nresulting from nonlinear interactions can be observed. Theorem 3 perfectly describes the asymp-\ntotic behavior for each DSG, before and after nonlinear collisions.\n2.5. High-order solitons. Merging two eigenvalues with proper rescaling of the norming con-\nstants yields a double-pole soliton of the NLS equation [3, 52]. In this paper, we make the calcula-\ntion more general and present the special limiting procedure in a systematically way, by showing\nthat merging all eigenvalues Λinto one, with rescaling of the norming constants {ωj,k}, produces\nanNth-order soliton, i.e., N-pole soliton of the MBEs (1.1).\n13By definition, the Nth-order soliton is the solution corresponding to an Nth-order eigenvalue,\ni.e., an Nth pole of the scattering data in the inverse scattering. So, let us denote this eigenvalue\nin the upper half plane by ω○∈C+. Naturally, the other eigenvalue ω○is in the lower half plane.\nThen, the corresponding Nth pole in the upper half plane can be described byp○(λ)\n(λ−λ○)Nwith N≥2\nand\n(2.27) p○(λ)∶=N−1\n∑\nk=0ω○,k(λ−λ○)k, ω○,k∈C, ω○,0≠0 .\nBy definition, ω○,0cannot be zero, otherwise this is not an Nth pole anymore. However, all other\nconstants can be arbitrary, including zeros.\nThen, the most general Nth-order soliton can be represented via the following RHP .\nRiemann-Hilbert Problem 3 (Jump form of the Nth-order soliton) .LetM○(λ;t,z)be a two-by-two\nmatrix function. It is sectional analytic with asymptotics M○(λ;t,z)→Iasλ→∞and jumps\n(2.28)M+\n○(λ;t,z)=M−\n○(λ;t,z)V○(λ;t,z)−1, λ∈∂Dϵ\nλ○,\nM+\n○(λ;t,z)=M−\n○(λ;t,z)V†\n○(λ;t,z), λ∈∂Dϵ\nλ○,\nwhere the jump matrices are given by\n(2.29) V○(λ;t,z)∶=⎛\n⎜\n⎝1 0\np○(λ)e−2iθ(λ;t,z)\n(λ−λ○)N1⎞\n⎟\n⎠, V†\n○(λ;t,z)=⎛\n⎜\n⎝1p○(λ)e2iθ(λ;t,z)\n(λ−λ○)N\n0 1⎞\n⎟\n⎠.\nClearly, both jump matrices in RHP 3 satisfy the Schwarz symmetry, so by Zhou’s lemma the\nsolution to RHP 3 exists and is unique [55]. It is easy to apply the dressing method on RHP 3 in\norder to show that it, together with the reconstruction formula in Lemma 1, produces a unique\nsolution of the MBEs with boundary conditions lim t→−∞D=D−for all z≥0. The procedure is\nidentical to the one in Section 3.1, so it is omitted for brevity.\nNow, a few questions rise:\n(1) What is relation between RHPs 2 and 3?\n(2) What is the pole-form of RHP 3?\n(3) How can the Nth-pole soliton be solved from RHP 3?\nWe answer all three questions below.\nFirst, we present the following theorem in order to show the relation between RHPs 2 and 3.\nTheorem 4 (Fusion of solitons) .For every given RHP 3 with the order N ≥2, the eigenvalue λ○∈C+and\nnorming constants {ω○,k}N−1\nk=0andω○,0≠0, there exists a fusion process of merging the (simple) eigenvalue\nsetΛof RHP 2 while rescaling norming constant set Ω.\nTheorem 4 is proved in Section 6.1, where the explicit rescaling of the norming constants Ωis\nalso provided in Equation (6.8). Moreover, we also show that the rescaling of norming constants is\nessential when obtaining the Nth-order soliton from the N-soliton solution. Without it, the fusion\nofΛjust produces an one-soliton solution, which is the trivial result.\nIt is established that it is definitely possible to obtain the Nth-order soliton from N-soliton so-\nlutions. The next task is to solve for such solution from RHP 3. It turns out that while the jump\nform a RHP is ideal for the Deift-Zhou’s nonlinear steepest descent method, the equivalent pole\nform becomes useful when solving for the exact soliton solutions. As such, we address the second\nquestion, which is to rewrite RHP 3 into its pole form. Different from the previous simple pole\ncases, where the residue conditions are used, the high-order poles require more general conditions.\nTherefore, we first define the pole operator Pnwhich picks out the coefficient of the nth-order term\nin a Laurent expansion of a function.\n14Definition 2. Let us define a operator Pn, such that for every meromorphic function f(λ),\n(2.30) Pn\nλ=λ○f=fn, where f=∞\n∑\nn=−N(λ−λ○)nfn,\nClearly, the operator Pnis a generalization of Res, with the following property\n(2.31) P−1\nλ=λ○f=Res\nλ=λ○f.\nWith the help of Pn, the pole form of an Nth-order soliton RHP is given below.\nRiemann-Hilbert Problem 4 (Pole form of the Nth-order soliton) .Given the Nth-order eigenvalue\nλ○in the upper half plane and corresponding norming constant polynomial p ○(λ). Let M○(λ;t,z)be a\n2×2meromorphic matrix function on C, with asymptotics M○(λ;t,z)→Iasλ→∞and pole conditions\n(2.32)P−n\nλ=λ○M○(λ;t,z)=lim\nλ→λ○N−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke−2iθ\n∂λN−n−i−k∂kM○\n∂λk(0 0\n1 0),\nP−n\nλ=λ○M○(λ;t,z)=−lim\nλ→λ○N−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke2iθ\n∂λN−n−i−k∂kM○\n∂λk(0 1\n0 0),\nwhere n=1, 2, . . . , N.\nThe derivation RHP 4 from RHP 3 is shown in Section 6.2. Similarly to the N-soliton solutions.\nBoth RHPs 3 and 4 are equivalent and useful in different ways. RHP 3 is suitable for applying\nDeift-Zhou’s nonlinear steepest descent method when computing asymptotics, whereas RHP 4\ncan be used to find the exact solution formulæ for the matrix M○(λ;t,z)and subsequently for the\nMBEs. As shown in Section 6.3, the exact formula for the Nth-order soliton is given below.\nTheorem 5 (Nth-order soliton solution formula) .Given an Nth-order eigenvalue ω○∈C+and norming\nconstants {ω○,k}N−1\nk=0, the corresponding Nth-order soliton of MBEs is given by\n(2.33) q(t,z)=2i−2idet(I−⃗e1⊗⃗γ∞+Γ○Γ○)\ndet(I+Γ○Γ○),\nwhere D (t,z)and P(t,z)are reconstructed by Lemma 1, with entries of M(λ;t,z)given by\n(2.34) M1,1(λ;t,z)=det(I−⃗e1⊗⃗γΓ○+Γ○Γ○)\ndet(I+Γ○Γ○), M1,2(λ;t,z)=det(I−⃗e1⊗⃗γ+Γ○Γ○)\ndet(I+Γ○Γ○)−1 ,\nand other quantities\n(2.35)⃗X∶=(X0X1. . . XN−1Y0Y1. . . YN−1)⊺,\n⃗γ(λ)∶=(γ0(λ)⋯γN−1(λ)),⃗γ∞∶=lim\nλ→∞λ⃗γ(λ),\nΓ○∶=(⃗γ(λ○)⊺⋯⃗γ(N−1)(λ○)⊺)⊺,⃗e1∶=(1 0 . . . 0 )⊺,\nγk(λ)∶=N−k\n∑\nn=1N−n−k\n∑\ni=01\n(λ−λ○)nω○,i\n(N−n−i−k)!k!∂N−n−i−ke−2iθ\n∂λN−n−i−k(λ○).\nSince the Nth-order soliton is a special limit of the N-DSG with rescaled norming constants\ngiven in Equation (6.8) as shown in Section 6.1, one immediately obtains the following result.\nCorollary 4 (Localization for the Nth-order soliton) .The Nth-order soliton is localized along the line\nz=V○t+zd,○with its velocity V ○∶=−2D−∣λ○∣2.\n15FIGURE 5. Density plots of ∣q(t,z)∣of three exact Nth-order soliton solutions in\nstable mediums from Theorem 5. All three plots have similar parameters λ○=i,\nω○,0=1 and ω○,k=0 for k≥1, but with different values of N. Left: N=3. Center:\nN=4. Right: N=5.\nHowever, we currently are not able to provide a simple formula for the displacement zd,○, which\ncan be computed by substituting Equation (6.8) into Equation (2.13) and then taking the fusion\nlimit.\nThree high-order solitons from Theorem 5 are shown in Figure 5 in an initially stable medium\nD−=−1, with N=3, 4, 5, respectively. In all three panels, the eigenvalues and the norming con-\nstant polynomials remain unchanged as λ○=i and p○(λ)=1, respectively. Hence, from left to\nright, the solutions correspond to upper-half-plane poles (λ−i)−3,(λ−i)−4and(λ−i)−5. They\ntravel inside the light cone with identical velocity V=2.\n2.6. Soliton gas. Another generalization of the N-soliton solutions, corresponding to anomaly\n(b) in the spectra of the non-self-adjoint Zakharov-Shabat scattering problem, produces the so-\ncalled soliton gas phenomenon, or integrable turbulence. In particular, one considers the limit\nof an arbitrary N-soliton solution as N→+∞. In this work, however, we only present results\nof a special case, in which one considers the limit of an N-DSGs as N→+∞, in order to keep\ndiscussions simple.\nDefinition 3 (Generalized eigenvalues and norming constants) .We define an arc A generalizing\nthe eigenvalue set Λ\n(2.36) A ∶={λ∣ ∣λ∣=r1>0 with arg (λ)∈(ϵ,π−ϵ)}, with 0 <ϵ≪1 fixed ,\nand a complex function ω(λ)∈L2(A)generalizing the norming constant set Ω.\nThe soliton gas solution of MBEs (1.1) is described by the following RHP together with the\nreconstruction formula Theorem 1. This RHP is derived in Section 7 as a limit of N-DSGs with\nN→+∞.\nRiemann-Hilbert Problem 5 (Jump-form of soliton gas) .Let an arc Aand a function ω(λ)be given\naccording to Definition 3. Let λ↦Msg(λ;t,z)be a 2×2matrix function on C, with asymptotics\nMsg(λ;t,z)→Iasλ→∞. It is also sectional analytic with jumps\n(2.37)M+\nsg(λ;t,z)=M−\nsg(λ;t,z)Vsg(λ;t,z)−1, λ∈A ,\nM+\nsg(λ;t,z)=M−\nsg(λ;t,z)V†\nsg(λ;t,z), λ∈A ,\n16where the jump contours are oriented left-to-right, and the jump matrix Vsg(λ;t,z)is given by\n(2.38) Vsg(λ;t,z)∶=(1 0\ne−2iθ(λ;t,z)ω(λ)1).\nRemark 7. Jump matrices in RHP 5 satisfy the Schwarz symmetry, so by Zhou’s lemma the solu-\ntion exists and is unique [55]. RHP 5 can be easily generalized following the discuss in Section 7 by\nmodifying the jump contour to more complex configurations. Moreover, RHP 5 is an analog to the\npole-forms for the N-soliton solutions and Nth-order solitons in RHPs 1 and 4. There is an equiv-\nalent RHP 17 which is the analog to the jump-forms for the N-soliton solutions and Nth-order\nsolitons in RHP 2 and 3. It is easy to prove that RHP 5 generate a unique solution of MBEs (1.1)\nwith boundary condition D→D−ast→−∞. The proof is similar to Section 3.1 and is omitted for\nbrevity.\nWe would also like to obtain other properties of the soliton gas solution, such as the localization\nproperty, by mimicking the proofs for Theorem 2 and Corollary 4. However, it turns out that cer-\ntain steps require additional properties of the function ω(λ). For example, one needs to know the\npossible zeros of its Cauchy transform Cω(λ), defined in Equation (7.6). The soliton gas solutions\nexhibit diverse properties depending on the choice of ω(λ), and deserve own studies. Thus, we\nleave the analysis of RHP 5 for future studies.\n2.7. Results for the focusing NLS and complex mKdV equations. We demonstrate here how to\neasily generalize obtained results for MBEs (1.1) to other integrable systems. For simplicity, we\nonly consider two systems in the same non-self-adjoint Zakharov-Shabat hierarchy, the focusing\nNLS equation\n(2.39) iqt(x,t)+qxx(x,t)+2∣q(x,t)∣2q(x,t)=0 ,\nwith its Lax pair\n(2.40)ϕx=(iλσ3+Q)ϕ,\nϕt=[−2iλ2σ3−2λQ+iσ3(Qx−Q2)]ϕ,\nand the complex mKdV equation\n(2.41) qt(x,t)+qxxx(x,t)+6∣q(x,t)∣2qx(x,t)=0 ,\nwith its Lax pair\n(2.42)ϕx=(iλσ3+Q)ϕ,\nϕt=[4iλ3σ3+4λ2Q+2iλ(Q2+Qx)σ3+2Q3+QxQ−QQ x−Qxx]ϕ.\nIn both Lax pairs (2.40) and (2.42), the matrix Q(x,t)is identical to that in Equation (1.2), but with\nq=q(x,t). Clearly, all three systems [MBEs (1.1), NLS (2.39) and mKdV (2.41)] have identical\nscattering problem.\nUpon reviewing all soliton results for the MBEs in previous subsections, one notices that there\nare only two system-specific components, the structure of the N-DSG eigenvalue set Λjfrom Def-\ninition 1 and the dispersion relation θ(λ;t,z)given in Equation (2.3). Let us rewrite them\n(2.43) θMBE(λ;t,z)∶=λt−D−\n2λz,ΛMBE, j∶={λj,k∣ ∣λj,k∣=const}.\nNote that eigenvalues with identical soliton velocities are determined by the equation I(θ)=0 for\nallt∈Rwith x=Vtand Vbeing the fixed velocity (cf. the remark below Theorem 2). The same\n17condition holds true for other integrable systems, so one gets\n(2.44)θNLS(λ;x,t)∶=λx−2λ2t, ΛNLS, j∶={λj,k∣R(λj,k)=const},\nθmKdV(λ;x,t)∶=λx+4λ3t,ΛmKdV, j∶={λj,k∣3R(λj,k)2−I(λj,k)2=const}.\nConsequently, the DSG velocities are given by\n(2.45) VNLS∶=4R(λj,k), VmKdV∶=4I(λj,k)2−12R(λj,k)2.\nAlso, the eigenvalue groups Ωjare ordered according to DSG velocities in the increasing order.\nNow, simply substituting θNLSandΛNLS, jinto all results for the MBEs, one gets the correspond-\ning soliton result for the focusing NLS equation. For example, replacing θMBE byθNLSin Theo-\nrem 1, one obtains the general N-soliton solution formula for the focusing NLS equation. Replac-\ningΛMBE byΛNLSin Theorem 2 and substituting θNLSinto Equation (4.18) yield the localization\nand center formula of N-DSGs for the NLS equation. Same is true for mKdV equation as well. The\nresults are\n(2.46)xNLS,c(t)∶=VNLSt+xd,\nxmKdV,c(t)∶=VmKdV t+xd,\nxd∶=−1\n2∑N1\nk=1I(λ1,k)ln(N1\n∏\nk=1∣ω1,k∣⋅∏N1\nk=2∏k−1\nl=1∣λ1,k−λ1,l∣2\n∏N1\nk=1∏N1\nl=1∣λ1,k−λ1,l∣).\nNote that the displacement xdfor both systems are formally identical.\nOne can obtain the general soliton asymptotics for the NLS equation as well, by replacing the\neigenvalue sets in Theorem 3 and Corollary 3. Again, the same procedure yields soliton asymp-\ntotics for the complex mKdV equation. The final asymptotics for these two systems are\n(2.47)qNLS(x,t;Λ,Ω)=J\n∑\nj=1qNLS(x,t;Λj,Ω(±)\nj)+o(1), t→±∞,\nqmKdV(x,t;Λ,Ω)=J\n∑\nj=1qmKdV(x,t;Λj,Ω(±)\nj)+o(1), t→±∞,\nwhere, for both systems, the modified norming constants are given by\n(2.48) ω(+)\nj,k∶=ωj,kδj(λj,k)2δ1(λj,k)−2, ω(−)\nj,k∶=ωj,kδj+1(λj,k)−2, k=1, 2, . . . , Nj.\nRemark 8. It is important to point out that the results for the focusing NLS equation and the\nmKdV equation are formally identical, but are slightly different from the ones for the MBEs as\nshown in Theorems 2 and 3. The reason is that the spatial and temporal variables are swapped\nbetween the MBEs and NLS/mKdV equations.\nOne is able to obtain the Nth-order solitons and soliton gases for the focusing NLS equation and\nthe complex mKdV equation, again, by replacing θandΛin Theorem 5 and RHP 5. The explicit\nresults are omitted for brevity.\nFinally, for demonstrative purposes, we present figures of exact solutions and our theoretical\nresults for the focusing NLS and mKdV equations. Figure 6 contains results for the focusing\nNLS equation (2.39): (Left) an exact 3-DSG solution by replacing θMBE with θNLSand proper Λin\nTheorem 1, with its center shown as the red dashed line from Equation (2.46); (Center) an exact\n5-soliton solution with three DSGs by replacing θMBE with θNLSand proper Λin Theorem 1, with\nthe asymptotic center shown as the red dashed lines from Equations (2.46) and (2.47); (Right) an\nexact 4th-order soliton solution by replacing θMBE with θNLSin Theorem 5. Similarly, Figure 7\n18FIGURE 6. Density plots of exact solutions ∣q(x,t)∣of the focusing NLS equa-\ntion (2.39) from Theorems 1 and 5. Left: A 3-DSG with parameters λ1,1=0.2+0.6i,\nλ1,2=0.2+0.4i, λ1,3=0.2+0.3i, ω1,1=−1, and ω1,2=λ1,3=e. The red dashed\nline is the DSG center from Equation (2.46). Center: A 5-soliton solution with three\nDSGs and parameters λ1,1=−0.3+0.2i, λ1,2=−0.3+0.4i, λ2,1=0.3i, λ2,2=0.4i,\nλ3,1=0.5+0.4i, ω1,1=ω1,2=1,ω2,1=ω2,2=e5, and ω3,1=e−6. The red dashed\nlines are the asymptotic DSG centers from Equations (2.46) and (2.47). Right: A\n4th-order soliton solution with λ○=i,ω○,0=1, and ω○,k=0 for 1≤k≤3.\nFIGURE 7. Density plots of exact solutions ∣q(x,t)∣of the complex mKdV equa-\ntion (2.41) from Theorems 1 and 5. Left: A 3-DSG with parameters λ1,1=i/2,\nλ1,2=1/2+i,λ1,3=1+i√\n13/2, and ω1,1=ω1,2=ω1,3=0. The red dashed line\ndenotes the DSG center from Equation (2.46). Center: A 4-soliton solution with\ntwo DSGs and parameters λ1,1=−1+i√\n3,λ1,2=1+i√\n3,λ2,1=−1/2+i,λ2,2=i/2,\nand ωj,k=2 for all j,k=1, 2. The red dashed lines denote the asymptotic DSG\ncenters from Equations (2.46) and (2.47). Right: A 4th-order soliton solution with\nλ○=i/2,ω○,0=1, and ω○,k=0 for 1≤k≤3.\ncontains results for the complex mKdV equation (2.41): (Left) an exact 3-DSG solution by replac-\ningθMBE with θmKdV and proper Λin Theorem 1, with its center shown as the red dashed line\nfrom Equation (2.46); (Center) an exact 4-soliton solution with two DSGs by replacing θMBE with\nθmKdV and proper Λin Theorem 1, with the asymptotic center shown as the red dashed lines from\nEquations (2.46) and (2.47); (Right) an exact 4th-order soliton by substituting θmKdV in Theorem 5.\nRemark 9. It is demonstrated that one is able to expand the discussion in this section without\nmuch effort, and obtain exact solutions and asymptotic results to even higher order systems in\nthe focusing Zakharov-Shabat AKNS hierarchy as far as one wishes. The generalization to other\n19hierarchies on DSGs/soliton asymptotics/high-order solitons/soliton gases can also be achieved\nfollowing the framework.\n3. R ECONSTRUCTION AND SOLITON SOLUTION FORMULÆ\n3.1. Proof of Lemma 1. We prove Lemma 1 in three steps:\n(1) We first derive the reconstruction formula from RHP 2 via the dressing method.\n(2) We then show the equivalence between RHPs 2 and 1, i.e., both RHPs generate identical\nsolutions of MBEs.\n(3) We finally show that the MBEs solutions obtained from RHP 2 exhibit the correct boundary\ncondition q(t,z)→0,P(t,z)→0 and D(t,z)→D−ast→−∞.\nFor statement (1): one defines a new matrix function from RHP 2 as N(λ;t,z)∶=M(λ;t,z)eiθ(λ;t,z)σ3,\nwhich is analytic for λ∈C∖({0}∪{∂Dλj,k∪∂Dλj,k}λj,k∈Λ), and which has the asymptotic behavior\nthat N(λ;t,z)e−iλtσ3=I+λ−1M1(t,z)+o(λ−1)asλ→∞with M(λ;t,z)=I+λ−1M1(t,z)+O(λ−2).\nIt is easy to check that the jump condition for N(λ;t,z)on all small circles are independent of\n(t,z)∈R2. Therefore, the matrix function Nt(λ;t,z)N(λ;t,z)−1is analytic in the whole com-\nplex plane with the possible exception at the origin. One easily checks that the singularity at\nzero is removable. Using the asymptotic behavior as λ→∞, Liouville’s theorem shows that\nNt(λ;t,z)N(λ;t,z)−1is a linear function of the form i λσ3+Q(t,z), with Q(t,z)given in Lemma 1.\nSimilarly, the matrix function Nz(λ;t,z)N(λ;t,z)−1is analytic except for a simple pole at the ori-\ngin, and it vanishes as λ→∞. Therefore Liouville’s theorem again yields that the product has\nthe form−iρ(t,z)/(2λ), and the matrix ρ(t,z)is obtained from M(λ;t,z)by Theorem 1. Since\nN(λ;t,z)is a simultaneous fundamental matrix solution of the Lax pair (1.2), by compatibility, the\nmatrices Q(t,z)andρ(t,z)solve the MBEs (1.1).\nFor statement (2): Let M(λ;t,z)be the solution of RHP 2. One can define a new matrix function\nas\n(3.1) N(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩M(λ;t,z)Vj,k(λ;t,z), λ∈Dϵ\nλj,k,\nM(λ;t,z)[V†\nj,k(λ;t,z)]−1,λ∈Dϵ\nλj,k,\nM(λ;t,z), otherwise .\nIt is easily checked that N(λ;t,z)does not admit any jumps, and has simple poles at each eigen-\nvalue point and its complex conjugate. The pole condition of N(λ;t,z)is exact the ones in RHP 1.\nTherefore, N(λ;t,z)is a solution of RHP 1. Similarly, by taking a solution RHP 1 and reverse\nthe definition of Equation (3.1), one gets a solution of RHP 2. In other words, the solutions of\nthe two RHPs have one-to-one correspondence. The uniqueness of solutions of RHP 2 yields the\nuniqueness of RHP 1. Because the two RHP solutions are identical for ∣λ∣≫1 and for ∣λ∣≪1 in\nthe definition (3.1) with sufficiently small ϵ>0, both yield identical solutions to MBEs (1.1) via the\nreconstruction formula from Lemma 1. Thus, RHPs 1 and 2 are equivalent.\nFor statement (3): We apply the nonlinear steepest descent method to RHP 2 as t→−∞ to\ncalculate the asymptotics for the solutions of MBEs for all z≥0. The inequality t<0 yields\nR(iλt)=−I(λ)t>0 for λin the upper half plane. Consequently, one obtains that e−2iθ(λ;t,z)→0\nuniformly as t→−∞for all λ∈∂Dϵ\nλj,k. Therefore, all jumps in the upper half plane of RHP 2 admit\nuniform limit Vj,k(λ;t,z)−1→Iast→−∞. Similar argument shows that all jumps in the lower\nhalf plane admit uniform limit V†\nj,k(λ;t,z)→Iast→−∞as well. As RHP 2 becoming a small-\nnorm problem, one concludes that M(λ;t,z)=I+o(1)ast→−∞, yielding the desired boundary\nconditions in the Cauchy problem (1.1) by the reconstruction formula from Lemma 1.\n203.2. Proof of Theorem 1. In this section, we suppress the (t,z)parametric dependence in all rela-\ntive quantities for simplicity. In particular, we write M(λ;t,z)=M(λ). We start from RHP 1. The\nsolution can be written as\n(3.2) M(λ)=I+∑\nλj,k∈ΛRes λ=λj,kM(λ)\nλ−λj,k+∑\nλj,k∈ΛResλ=λj,kM(λ)\nλ−λj,k.\nAgain, using the residue conditions in RHP 1, the above equation can be rewritten as\n(3.3)\nM(λ)=I+∑\nλj,k∈Λ1\nλ−λj,klim\nλ→λj,kM(λ)(0 0\nωj,ke−2iθ(λ)0)+∑\nλj,k∈Λ1\nλ−λj,klim\nλ→λj,kM(λ)(0−ωj,ke2iθ(λ)t\n0 0).\nLooking at the first and second columns and denoting them as M1(λ)and M2(λ), respectively,\none has\n(3.4) M1(λ)=(1\n0)+∑\nλj,k∈Λωj,ke−2iθj,k\nλ−λj,kM2(λj,k), M2(λ)=(0\n1)−∑\nλj,k∈Λωj,ke2iθj,k\nλ−λj,kM1(λj,k),\nwhere we recall the definition θj,k∶=θ(λj,k)from Theorem 1. Recall the discuss in Remark 2\non the symmetries of entries of M(λ). Hence, it is sufficient to look at the first row of Equa-\ntion (3.4). Moreover, by substituting λ=λj,kinM1,1(λ)andλ=λj,kinM1,2(λ), respectively, for\nk=1, 2, . . . , Njand j=1, 2, . . . , J, one obtains the following N×Nlinear system,\n(3.5)M1,1(λj,k)=1+∑\nλj′,k′∈Λcj′,k′(λj,k)M1,2(λj′,k′), M1,2(λj,k)=−∑\nλj′,k′∈Λcj′,k′(λj,k)M1,1(λj′,k′),\nwhere k=1, 2, . . . , Njand j=1, 2, . . . , J. In order to rewrite the above linear system in matrix form,\none defines vectors of unknowns\n(3.6)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→\nM1,1(λj,k)∶=(M1,1(λ1,1)⋯M1,1(λ1,N1)⋯M1,1(λJ,1)⋯M1,1(λJ,NJ))⊺,\n/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→M1,2(λj,k)∶=(M1,2(λ1,1)⋯M1,2(λ1,N1)⋯M1,2(λJ,1)⋯M1,2(λJ,NJ))⊺.\nAs a result, the linear system (3.5) becomes\n(3.7)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→\nM1,1(λj,k)=⃗1+Γ/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→M1,2(λj,k),/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→M1,2(λj,k)=−Γ/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→\nM1,1(λj,k),\nwhere the vector ⃗1and the matrix Γare defined in Theorem 1. Cramer’s rule immediately yields\nthe solutions to the above system\n(3.8) M1,1(λj,k)=det(I+ΓΓ)j,k\ndet(I+ΓΓ), M1,2(λj,k)=−det(Γ−1+Γ)j,k\ndet(Γ−1+Γ),\nwhere det (A)j,kdenotes the determinant of the matrix that is obtained by replacing the (N1+⋅⋅⋅+\nNj−1+k)-th column of the matrix Aby⃗1.\n21Combining the above quantities with the first row of Equation (3.4), one has\n(3.9)M1,1(λ;t,z)=1+det(0 C\n⃗1Γ−1+Γ)\ndet(Γ−1+Γ)\n=1+det(Γ−1+Γ−⃗1⊗C)−det(Γ−1+Γ)\ndet(Γ−1+Γ)\n=det(Γ−1+Γ−⃗1⊗C)\ndet(Γ−1+Γ)\n=det(I−⃗1⊗CΓ+ΓΓ)\ndet(I+ΓΓ)\nM1,2(λ;t,z)=det(0 C\n⃗1I+ΓΓ)\ndet(I+ΓΓ)=det(I−⃗1⊗C+ΓΓ)\ndet(I+ΓΓ)−1 ,\nwhere Cis defined in Theorem 1. The reconstruction formula (2.8) yields the explicit formulæ for\nthe MBEs solutions presented in Theorem 1.\n4. D EGENERATE N-SOLITON GROUP\nThis section proves results in Theorem 2, so one focuses on the degenerate N-DSGs, with J=1\nand N=N1. We recall ∣λ1,k∣=r1>0 for k=1, . . . , N1. Recall that all eigenvalues and α1,kare\ndistinct.\n4.1. Localization of N-DSG. In this section, we calculate the asymptotics of the N-DSG q(t,z)\nandρ(t,z)ast→±∞with assumption z=ξtwhere ξ∈R. Recall the soliton velocity Vis defined\nin Theorem 2. We show that as long as ξ≠V, it is q(t,z)=o(1)andρ(t,z)=D−σ3+o(1)ast→±∞,\nand only if ξ=V,q(t,z)does not decay in the limits. Hence, the N-DSG is localized along the line\nz=Vt+zdwith zda constant.\nAs always, it is important to analyze the sign structure of R(iθ(λ;t,z))when calculating the\nasymptotics using Deift-Zhou’s nonlinear steepest descent method. Simple calculations show that\nR(iθ(λ;t,z))=−D−yt[ξ+2D−(x2+y2)]/[2(x2+y2)], with x=R(λ),y=I(λ)>0, and z=ξt.\nLooking at each eigenvalue λ1,k, the equation yields\n(4.1) R(iθ(λ1,k;t,z))=−D−y1,k\n2r2\n1(ξ−V)t, k=1, 2, . . . , N1,\nwhere one uses V=−2D−r2\n1,r1=∣λ1,k∣, and λ1,k=x1,k+iy1,k. Obviously, R(iθ(λ1,k;t,z))has a\nsingle root ξ=V, meaning that R(iθ)=0 if and only if ξ=V. Hence, the sign of R(iθ(λ1,k;t,z))is\ndetermined by: (i) the sign of ξ−V; (ii) the sign of D−; and (iii) the sign of t.\nFor (i), one should recall that in RHP 2 all the jumps are on the small circles centered at eigen-\nvalues with radii ϵ≪1, soR(iθ(λ;t,z))on the jump surrounding λ1,khas the same sign as\nR(iθ(λ1,k;t,z)), provided ξ≠V. Therefore, the exponential functions e±2iθ(λ;t,z)inside the jump\nmatrices have the same growing/decaying properties as e±2iθ(λ1,k;t,z), respectively, with ξ≠V.\nFor (ii), because the initial state of the medium D−affects R(iθ(λ1,k;t,z)), it is necessary to\ndiscuss the two cases (stable/unstable) separately. We start with the more important and real-\nistic case—initially stable medium ( D−=−1). After the stable case is properly and rigorously\naddressed, the unstable case can be easily calculated by slight modification of the stable one.\n22FIGURE 8. The sign structure of R(iθ(λ;t,z))with z=ξtin an initially stable\nmedium D−=−1. The blue dots represent the Npairs of eigenvalues Λof the N-\nDSG. The blue circles surrounding the eigenvalues are the jumps from RHP 2. Gray\nregions denote R(iθ(λ;t,z))<0, and the yellow regions denote R(iθ(λ;t,z))>0.\nTop left: ξ≤0<V. Top right: 0 <ξ<V. Bottom left: ξ=V. Bottom right: ξ>V.\nFor (iii), one needs to discuss the asymptotics as t→±∞, corresponding to the situation t≷0.\nHowever, it turns out the discussion of t→−∞is almost identical to the case of t→+∞, sowe only\nconsider the case of t →+∞in this section , and the case of t→−∞is omitted for brevity.\nTo summarize, in an initially stable medium with t>0, Equation (4.1) dictates that R(iθ(λ1,k;t,z))>\n0 if and only if ξ>V, andR(iθ(λ1,k;t,z))<0 if and only if ξ<V. This sign structure of R(iθ)\nis shown in Figure 8, for the cases ξ<V,ξ=Vand ξ>V. The blue dots represent Npairs of\neigenvalues with the same radius r. In an initially unstable medium with t>0, Equation (4.1)\ndictates that R(iθ(λ1,k;t,z))>0 if and only if ξ<VandR(iθ(λ1,k;t,z))<0 if and only if ξ>V.\n4.1.1. The case of ξ<V with stable mediums. First of all, there are two subcases which are ξ≤0<V\nand 0<ξ<V, shown in the top row in Figure 8, corresponding to the physical situations outside\nof and inside the light cone, respectively. However, it turns out that the mathematical treatments\nare identical, so we analyze them together in this subsection.\nAccording to previous discussions and Figure 8(left), one knows that R(iθ(λ1,k;t,z))→−∞as\nt→+∞, which proves that all the jumps in RHP 2 are growing as t→+∞. Hence, one needs to\nchange the jumps according to the Deift-Zhou’s nonlinear steepest descent method. In particular,\nchanging the exponential functions so that they decay as t→+∞, one needs to define new func-\ntions and deform the jumps from RHP 2. To achieve this, one first defines the following useful\n23functions\n(4.2) δDSG(λ)∶=pDSG(λ)\npDSG(λ), pDSG(λ)∶=N1\n∏\nk=1(λ−λ1,k),\nwhere one notices that pDSG(λ)=pDSG(λ). Obviously, δDSG(λ)→1 asλ→∞.\nThen, one defines a new matrix as follows,\n(4.3) MDSG(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩M(λ;t,z)δDSG(λ)−σ3, λ∈C∖N1\n⋃\nk=1(Dϵ\nλ1,k⋃Dϵ\nλ1,k),\nM(λ;t,z)(λ−λ1,k)σ3Sk(λ;t,z)−1, λ∈Dϵ\nλ1,k,k=1, 2, . . . , N1,\nM(λ;t,z)(λ−λ1,k)−σ3S†\nk(λ;t,z), λ∈Dϵ\nλ1,k,k=1, 2, . . . , N1,\nwhere for k=1, 2, . . . , N1,\n(4.4) Sk(λ;t,z)∶=⎛\n⎜⎜⎜⎜⎜\n⎝λ−λ1,k\npDSG(λ)pDSG(λ)2−pDSG(λ1,k)2\npDSG(λ)ω−1\n1,kλ−λ1,k\nλ−λ1,kpDSG(λ1,k)2\npDSG(λ)pDSG(λ)e2iθ(λ;t,z)\n−ω1,kpDSG(λ)\npDSG(λ)λ−λ1,k\nλ−λ1,ke−2iθ(λ;t,z) 1\nλ−λ1,kpDSG(λ)\npDSG(λ)⎞\n⎟⎟⎟⎟⎟\n⎠.\nIt is worth pointing out that the matrices Sk(λ;t,z)are analytic in Dϵ\nλ1,k, and the matrices S†\nk(λ;t,z)\nare analytic in Dϵ\nλ1,k, for k=1, 2, . . . , N1. Therefore, the newly defined matrix MDSG(λ;t,z)is still\nanalytic in Dϵ\nλ1,k⋃Dϵ\nλ1,kfor all integer k. One can verify that MDSG(λ;t,z)solves the following RHP .\nRiemann-Hilbert Problem 6. Suppose λ↦MDSG(λ;t,z)is a2×2matrix function on C. It has the\nasymptotics MDSG(λ;t,z)→Iasλ→∞, and is sectional analytic with jumps\n(4.5)M+\nDSG(λ;t,z)=M−\nDSG(λ;t,z)VDSG, k(λ;t,z), λ∈∂Dϵ\nλ1,k,k=1, 2, . . . , N1,\nM+\nDSG(λ;t,z)=M−\nDSG(λ;t,z)[VDSG, k]−†(λ;t,z), λ∈∂Dϵ\nλ1,k,k=1, 2, . . . , N1,\nwhere the jump matrices are given by\n(4.6) VDSG, k(λ;t,z)=⎛\n⎜\n⎝1−λ−λ1,k\nω1,kpDSG(λ1,k)2\npDSG(λ)2e2iθ(λ;t,z)\n0 1⎞\n⎟\n⎠, k=1, 2, . . . , N1.\nNow, one sees that for each λ=λ1,k, the surrounding jump VDSG, k(λ;t,z)contains exponential\ne2iθ(λ;t,z)instead of e−2iθ(λ;t,z)from the original jump matrix V1,k(λ;t,z), so the growing jumps\nbecome decaying ones. Such switches happen in the lower half plane simultaneously. Thus, all\nthe jumps on ∂Dϵ\nλ1,kand∂Dϵ\nλ1,kfork=1, 2, . . . , N1are decaying uniformly to the identity matrix as\nt→+∞. Therefore, one concludes that\n(4.7) MDSG(λ;t,z)=I+o(1), t→+∞,\nwhere the error o(1)is exponentially small. Then, reversing the definition (4.3) yields the asymp-\ntotics for M(λ;t,z)ast→+∞. Note that the solution q(t,z)is reconstructed from M(λ;t,z)with\n∣λ∣≫1 and ρ(t,z)is reconstructed from M(0;t,z), from Lemma 1. Since all eigenvalues are finite\nand off the real line (cf. Definition 1), one concludes\n(4.8) M(λ;t,z)=MDSG(λ;t,z)δσ3\nDSG=(I+o(1))δDSG(λ)σ3, t→+∞, with ∣λ∣≫1 orλ=0 .\nThe reconstruction formula from Lemma 1 yields\n(4.9) q(t,z)=o(1), ρ(t,z)=D−σ3(I+o(1)), t→+∞.\n244.1.2. The case of ξ=V with stable mediums. Recall that R(iθ(λ1,k;t,z)=0 for k=1, 2, . . . , N1from\nthe discussion at the beginning of Section 4.1. In terms of the jumps surrounding the eigenvalues,\none part of each circular jump is growing while the rest is decaying as t→+∞[cf, Figure 8(bottom\nleft)]. As a result, the jumps V1,k(λ;t,z)−1on∂Dϵ\nλ1,kand V†\n1,k(λ;t,z)on∂Dϵ\nλ1,kin RHP 2 do not\ndecay nor grow uniformly as t→+∞. So, the asymptotics cannot be easily calculated from RHP 2.\nInstead, one can directly solve the equivalent RHP 1, and obtains the explicit N-DSG given via\nTheorem 1. Then, one observes that all the exponential functions e±2iθ1,kfork=1, 2, . . . , N1with\nJ=1 from the solution formula in Theorem 1 are purely oscillatory, because R(iθ1,k)=0. Hence,\none concludes that q(t,z)andρ(t,z)do not decay nor grow in the limit t→+∞.\n4.1.3. The case of ξ>V with stable mediums. Similarly to previous subsections, one recalls the\ndiscussion at the beginning of Section 4.1 [cf. Figure 8(Bottom right)], to see that all the jumps\nin RHP 2 decay to the identity matrix uniformly as t→+∞. Therefore, one can directly write\nM(λ;t,z)=I+o(1), ast→+∞. Lemma 1 yields\n(4.10) q(t,z)=o(1), ρ(t,z)=D−σ3, t→+∞.\nCombining all cases, one concludes that the N-DSG is traveling along a line in the form of\nz=Vt+zdwith an undetermined constant zd. When observed in a different direction z=ξtwhere\nξ≠V, the N-DSG rapidly vanishes to the ZBG. In fact, it has proved that the solution decays\nexponentially. Therefore, one obtains the localization of the N-DSG, and knows that the N-DSG\ntravels with the speed V. The localization [results (1) and (2)] of N-DSG in Theorem 2 in a stable\nmedium is proved.\n4.1.4. Localization of N-DGS with unstable mediums. It is time to discuss N-DSG in an unstable\nmedium, i.e., D−=1. Recall Equation (4.1) and the discussion afterwards. It is also worth recalling\nthe velocity Vu∶=−2r2\n1<0 in the unstable medium. Instead of calculating the asymptotics of RHP 2\nstep-by-step, we make the following connections between the unstable cases and the previously\ndiscussed stable cases.\n(1) In the case of ξ<Vu, Equation (4.1) implies that R(iθ(λ1,k;t,z))>0, yielding that all the\njumps in RHP 2 are decaying as t→+∞. Similarly to Section 4.1.3, one immediately obtains\nthe same result (4.10).\n(2) In the case of ξ=Vu, Equation (4.1) implies that R(iθ(λ1,k;t,z))=0. Recall the discussion\nin Section 4.1.2. The MBEs solutions can be solved exactly via RHP 1 and can be shown to\nnot decay nor grow in the limit t→+∞.\n(3) In the case of Vu<ξ, Equation (4.1) implies that R(iθ(λ1,k;t,z))<0, so all jumps in RHP 2\nare growing as t→+∞. One can follow the calculations in Section 4.1.1, apply similar\ntreatments to RHP 2, and obtain Equation (4.9).\nFinally, one proves the results (1) and (2) in Theorem 2 in the unstable case.\n4.2. Center of N-DSG. The previous subsections prove that the N-DSG is traveling in the direc-\ntion z=Vt. Hence, the center of N-DSG also moves with the same velocity Vastchanges. One\ndefines the center zc(t)∶=Vt+zd, where the displacement zd∈Ris to be determined.\nOur next task is to seek for zd. We perform two pre-treatments before the actual calculation.\nFirstly, Let us substitute z=zcinto the quantity θ(λ1,k;t,z)yielding\n(4.11) R(iθ(λ1,k;t,z))=−D−I(λ1,k)\n2r2\n1zd, k=1, 2, . . . , N1.\n25Note that all I(λ1,k)>0. Secondly, we factor the matrix Γas follows,\n(4.12)Γ=Γ(1)Γ(2),\nΓ(1)∶=⎛\n⎜⎜⎜⎜⎜⎜\n⎝1\nλ1,1−λ1,11\nλ1,1−λ1,2⋯1\nλ1,1−λ1,N1\n⋮ ⋮ ⋮\n1\nλN1,1−λ1,11\nλN1,1−λ1,2⋯1\nλN1,1−λN1,N1⎞\n⎟⎟⎟⎟⎟⎟\n⎠,\nΓ(2)∶=diag(ω1,1e−2iθ1,1,ω1,2e−2iθ1,2,⋯,ω1,N1e−2iθ1,N1).\nIt is worth pointing out that Γ(1)is independent of θ, so it always remains constant no matter\nzd→±∞. On the other hand, Γ(2)grows to infinity or decays to the identity matrix, depending on\nthe state of the medium and whether zd→−∞orzd→+∞.\nAs demonstrated in [37], an effective way to find the center of a DSG is to match the asymptotic\nbehavior of solutions as zd→±∞. This follows from a simple observation: the N-DSG behaves\nconsiderably complicated when the observation point zis close to its center zc; but whenever\nthe observation point zmoves away from the center zc, the solution decays rapidly (the proved\nlocalization property in Section 4.1), yielding simple behavior of q(t,z). Because zcis the center,\nthe solution to its right and to its left should behave similarly further away. Thus, calculating the\nasymptotics of the solution q(t,z)aszd→±∞ and matching them should give rise to a proper\nvalue of zd, describing the displacement. We next discuss the asymptotics.\n4.2.1. Stable medium (D −=−1).In this case, zd→−∞yields e−2iθ1,k→+∞from Equation (4.11) for\nallk=1, 2, . . . , N1. Consequently, Γ(2)→∞implies that Γ→∞as well. Examining the solution\nq(t,z)from Equation (2.9) yields that\n(4.13) q(t,z)=2i−2idet(ΓΓ)det(Γ−1Γ−1(I−⃗1⊗C∞)+I)\ndet(ΓΓ)det(Γ−1Γ−1+I)=2i−2i1+o(1)\n1+o(1)=o(1), zd→−∞.\nAs expected, the solution decay to zero as one moves away from its center. In fact, it can be seen\nfrom the above equation that q(t,z)decays exponentially. The exact decay rate is necessary for\nlater comparison. However, the current form is inconvenient for calculating the leading-order\nterm. So, we rewrite the asymptotics differently. The above calculation shows that the numerator\nand the denominator inside the fraction in q(t,z)behave similarly as zd→−∞. (Note that the\nfraction of Equation (4.13) tends to one, not zero.) One can extract the leading-order term alone\nfrom the denominator, as follows,\n(4.14) det (I+ΓΓ)=det(ΓΓ)det(Γ−1Γ−1+I)=det(ΓΓ)(1+o(1)), zd→−∞.\nThe other limit zd→+∞implies that e−2iθ1,k→0 from Equation (4.11) for k=1, 2, . . . , N1. Corre-\nspondingly, Γ(2)→0 and Γ→0. Clearly, one sees that\n(4.15) q(t,z)=2i−2idet(I−⃗1⊗C∞+ΓΓ)\ndet(I+ΓΓ)=2i−2i1+o(1)\n1+o(1)=o(1).\nAgain, q(t,z)decays to zero as one moves away from the soliton group center as zd→+∞. As\nbefore, one looks at the denominator of q(t,z), but the direct calculation of det (I+ΓΓ)yields the\nleading-order term as 1. To extract useful information, we rewrite the fraction so that both the\n26numerator and the denominator are growing instead of decaying as zd→+∞,\n(4.16)det(I−⃗1⊗C∞+ΓΓ)\ndet(I+ΓΓ)=det(Γ−1Γ−1(I−⃗1⊗C∞)+I)\ndet(Γ−1Γ−1+I).\nThe denominator yields the leading-order term\n(4.17) det (Γ−1Γ−1+I)=det(Γ−1Γ−1)det(I+ΓΓ)=det(Γ−1Γ−1)(1+o(1)), zd→+∞.\nMatching the two asymptotics from Equations (4.14) and (4.17) implies det (ΓΓ)=det(Γ−1Γ−1).\nThe center zdis therefore determined by the equation det (ΓΓ)2=1, which can be calculated as\n(4.18) 1 =∣det(Γ)∣4=∣det(Γ(1))∣4∣det(Γ(2))∣4=∣det(Γ(1))∣4∣∣N1\n∏\nk=1ω1,k∣4\n∣e−2i∑N1\nk=1θ1,k∣4\n.\nNote that Γ(1)is a Cauchy matrix, whose determinant can be readily computed,\n(4.19) det (Γ(1))=∏N1\nk=2∏k−1\nl=1∣λ1,k−λ1,l∣2\n∏N1\nk=1∏N1\nl=1(λ1,k−λ1,l).\nFinally, one obtains the center of the soliton group zc(t)=Vt+zdwith zdgiven in Theorem 2. This\nproves the result (3) in the theorem in a stable medium.\n4.2.2. Unstable medium (D −=1).Here, one employs similar arguments as in Section 4.1, by com-\nparing and relating the stable and unstable case. Opposite to the previous stable case, Equa-\ntion (4.11) implies the following equivalence\n(1) The case of an unstable medium with zd→−∞implies that R(iθ1,k)→−∞, which is equiv-\nalent to the calculation of the case of a stable medium with zd→+∞.\n(2) The case of an unstable medium with zd→+∞implies that R(iθ1,k)→+∞, which is equiv-\nalent to the calculation of the case of a stable medium with zd→−∞.\nHence, the two asymptotics switch between the stable and unstable cases. Nonetheless, matching\nthe two leading-order terms in the asymptotics in either case yields identical result, i.e., Equa-\ntion (4.18). Thus, the formula for the center of N-DSG in an unstable case in Theorem 2 is proved.\n4.3. Boundedness of M (λ;t,z).Finally, we would like to prove that with J=1, the solution\nM(λ;t,z)to RHPs 1 and 2 are always bounded as t→±∞ in the direction z=Vt. This result\nis used in Section 5 when calculating the soliton asymptotics. Of course, there are four subcases\nresulting from all combinations of t→±∞and stable/unstable medium. We first discuss the case\noft→+∞in a stable medium. The other three can be easily analyzed afterwards.\nCase I Boundedness as t→+∞in a stable medium. One needs to look at RHP 1. The first row of\nits solution M(λ;t,z)has been solved in Equation (3.9). Using Equation (4.1) and z=Vt,\none can see that all the (t,z)dependence in Equation (3.9) are in the exponential functions\ne±2iθ1,k(cf. Theorem 2), which are oscillatory. Similar phenomenon happens to the sec-\nond row of M(λ;t,z)whose explicit expression is omitted for brevity. Hence, the matrix\nM(λ;t,z)is bounded as t→+∞.\nCase II Boundedness as t→−∞in a stable medium. The exponentials e±2iθ1,kare still oscillatory.\nHence, repeating similar arguments in Case I yields that M(λ;t,z)is bounded as t→−∞,\nas well.\nCase III Boundedness as t→+∞in an unstable medium. Using the analysis in Section 4.1.4 and\nrepeating similar arguments of Case I yields desired results.\nCase IV Boundedness as t→−∞in an unstable medium. One simply repeat similar argument of\nCase II and obtains the boundedness of M(λ;t,z).\n27FIGURE 9. Left: An illustration of distributions of Λwith 1≤j≤Jand J≥2 in the\ncomplex plane. Right: The equivalent RHP with jumps transformed from poles.\nHence, we have proved the boundedness of M(λ;t,z)of RHPs 1 and 2 as t→±∞in both stable\nand unstable mediums, with J=1 and z=Vt.\n5. S OLITON ASYMPTOTICS\nWe prove Theorem 3 in this section. Several cases are necessary to be considered. First, it needs\nto discuss initially the medium whether in its stable or unstable state. Also, it is necessary to\ncalculate the asymptotics as t→±∞. Finally, the direction ξfrom z=ξtaffects the application of\nthe Deift-Zhou’s nonlinear steepest descent method. In order to simplify our discussion regarding\nall the cases, we mainly focus on the case of the stable medium, then briefly discuss what happens\nin an unstable medium.\n5.1. Basic analysis of soliton asymptotics in a stable medium. We now start proving Theorem 3\nin the case of a stable medium ( D−=−1). Recall the assumption that J≥2, so the configuration\nof the spectra is shown in Figure 9. We would like to show that all eigenvalues form Jdifferent\nDSGs, each of which contains Njsolitons ( Njpairs of eigenvalues). If Nj=1 for some values of\n1≤j≤NJ, the corresponding DSG contains only one soliton, hence the DSG reduces to a single\nsoliton. Each DSG travels with velocity Vj, where according to Definition 1 and Theorem 2, the\nsoliton velocities corresponding to Λare given by\n(5.1) 0 <V1<V2<⋅⋅⋅<VJ, Vj∶=2r2\nj, j=1, 2, . . . , J.\nFinally, we would like to show that as t→±∞, the solution is approximately the sum of all DSGs.\nHowever, as will be shown in the rest of this section, each DSG suffers a nontrivial shift resulting\nfrom nonlinear interactions between the said DSG and other DSGs. It should be pointed out that\nsimilar nontrivial shifts also presents in other systems, like the focusing NLS equation [39]. In\norder to better illustrate the shift in the asymptotics, we first present the center of the jth DSG\ndenoted by zc,j(t)forj=1, . . . , Jgiven by Theorem 2, as if there were no interactions among the\nDSGs in the solution,\n(5.2) zc,j(t)=Vjt+zd,j, zd,j∶=−D−rj\n∑Nj\nk=1sin(αj,k)ln(Nj\n∏\nk=1∣ωj,k∣∏Nj\nk=2∏k−1\nl=1∣λj,k−λj,l∣2\n∏Nj\nk=1∏Nj\nl=1∣λj,k−λj,l∣).\nIt is time to outline the asymptotic calculations. Note that there are two kinds of long-time\nasymptotics, as t→+∞and t→−∞. Additionally, assuming z=ξt, there are subcases depending\non the relative values of ξand velocities Vj, listed in Table 1.\n28TABLE 1. Four cases in the long-time asymptotics in a stable medium.\nStable Cases Relations between ξand velocities Description\nCase 1 ξ<V1<⋅⋅⋅<VJ ξis smaller than all soliton velocities\nCase 2 V1<⋅⋅⋅<VJ<ξ ξis larger than all soliton velocities\nCase 3 Vj<ξ<Vj+1with j=1, . . . , J−1 ξis in-between the velocities\nCase 4 Vj−1<ξ=Vj<Vj+1with j=1, . . . , J ξcoincides with some velocity Vj\nFIGURE 10. Representative plots of Case 1 from Table 1 with t→+∞, where the\ndirection of the observation line ξis smaller than all velocities. Left: ξ≤0<V1.\nRight: 0<ξ<V1The blue circles represent all the jump contours in RHP 2. Yellow\nor gray region corresponds to R(iθ)>0 orR(iθ)<0, respectively. The red contours\nare the separation between the regions.\nAll cases need to be carefully examined. In the process, we apply the nonlinear steepest descent\nmethod to RHP 2 to calculate the long-time asymptotics. Firstly, we recall the useful quantities\nδj(λ)and pj(λ)defined in Equation (2.21). Note that δj(λ)→1 as λ→∞for each j=1, 2, . . . , J.\nNote also that δ1(λ)with J=1 is equivalent to δDSG(λ)defined in Equation (4.2).\n5.2. Soliton asymptotics as t→+∞in a stable medium. We first present the long-time asymp-\ntotic calculations of all cases in Table 1 as t→+∞.\n5.2.1. The case of ξ<V1.There are two subcases, which are ξ≤0 and 0<ξ<V1. The first one\ndiscusses the situation when looking at the solution outside and on the boundary of the light\ncone, whereas the latter one discusses what happens inside the light cone. The sign structures of\nR(iθ(λ;t,z))differ slightly, as illustrated in Figure 10, which are calculated by substituting z=ξt\nintoθ(λ;t,z). Nonetheless, in both subcases all jumps (with small enough radii ϵ) in RHP 2 contain\ngrowing exponentially, specifically, exp (−2iθ(λ;t,z))in the upper half plane, and exp (2iθ(λ;t,z))\nin the lower half plane. Therefore, it is unnecessary to distinguish the two subcases mathemati-\ncally, and we treat them simultaneously below.\nOne needs to modify all the growing jumps in order to perform the nonlinear steepest descent.\nIt turns out that the process is quite similar to what have been done in Section 4.1 in the case of\nξ<V. We only outline the process here in order to avoid repetitions. Step (i), one utilizes the\nrational function δ1(λ)from Equation (2.21), similarly to the usage of δDSG(λ)in Section 4.1.1.\n29FIGURE 11. Representative plots of jump configuration of RHP 2 similarly to Fig-\nure 10 but for Cases 2–4 (from left to right) in Table 1 with t>0.\nStep (ii), one defines a new matrix function mimicking Equation (4.3),\n(5.3) M(1)(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩M(λ;t,z)δ1(λ)−σ3, λ∈C∖J\n⋃\nj=1Nj\n⋃\nk=1(Dϵ\nλj,k⋃Dϵ\nλj,k),\nM(λ;t,z)(λ−λj,k)σ3Y(1)\nj,k(λ;t,z)−1, λ∈Dϵ\nλj,k,λj,k∈Λ,\nM(λ;t,z)(λ−λj,k)−σ3(Y(1)\nj,k)†(λ;t,z), λ∈Dϵ\nλj,k,λj,k∈Λ,\nwhere\n(5.4) Y(1)\nj,k(λ;t,z)∶=⎛\n⎜⎜⎜⎜⎜\n⎝λ−λj,k\np1(λ)p1(λ)2−p1(λj,k)2\np1(λ)ω−1\nj,kλ−λj,k\nλ−λj,kp1(λj,k)2\np1(λ)p1(λ)e2iθ(λ;t,z)\n−ωj,kp1(λ)\np1(λ)λ−λj,k\nλ−λj,ke−2iθ(λ;t,z) 1\nλ−λj,kp1(λ)\np1(λ)⎞\n⎟⎟⎟⎟⎟\n⎠.\nConsequently, one can verify that M(1)(λ;t,z)satisfies the following RHP .\nRiemann-Hilbert Problem 7. Suppose λ↦M(1)(λ;t,z)is a2×2matrix function on C. It has the\nasymptotics M(1)(λ;t,z)→Iasλ→∞, and is sectional analytic with jumps\n(5.5)M(1)+(λ;t,z)=M(1)−(λ;t,z)V(1)\nj,k(λ;t,z), λ∈∂Dϵ\nλj,k,λj,k∈Λ,\nM(1)+(λ;t,z)=M(1)−(λ;t,z)[V(1)\nj,k]−†(λ;t,z), λ∈∂Dϵ\nλj,k,λj,k∈Λ,\nwhere the jump matrices are given by\n(5.6) V(1)\nj,k(λ;t,z)∶=⎛\n⎜⎜\n⎝1−λ−λj,k\nωj,kp1(λj,k)2\np1(λ)2e2iθ(λ;t,z)\n0 1⎞\n⎟⎟\n⎠.\nStep (iii), it can be easily shown that the solution to the new RHP 7 tends to the identity matrix\nexponentially fast as t→+∞, just like what happens to RHP 6. One obtains the final asymptotics\n(5.7) q(t,z)=o(1), ρ(t,z)=D−σ3(I+o(1)), t→+∞.\n305.2.2. The case of ξ>VJ.As shown in Figure 11(left), and as calculated by plugging z=ξtinto\nR(iθ(λ;t,z)), with small enough ϵin RHP 2, all the jump matrices Vj,k(λ;t,z)−1in the upper\nhalf plane and V†\nj,k(λ;t,z)in the lower half plane contain decaying exponentials. Taking the limit\nt→+∞, one directly arrives at the following result\n(5.8) q(t,z)=o(1), ρ(t,z)=D−σ3(1+o(1)), t→+∞.\nRecall that this also happens in Section 4.1 in the case of ξ>V.\n5.2.3. The case of V j<ξ<Vj+1.Now, let us discuss a novel case (comparing to Section 4.1) where in\nRHP 2 some jumps contain growing exponentials and others contain decaying exponentials. The\nconfiguration can be seen in Figure 11(center).\nIn particular, let us fix the value of jwith 1≤j≤J−1 in this section. Then, as can be calculated\neasily, the jump matrices Vl,k(λ;t,z)−1and V†\nl,k(λ;t,z)with 1≤l≤jand 1≤k≤Nlcontain\ndecaying exponentials, whereas matrices Vl,k(λ;t,z)−1and V†\nl,k(λ;t,z)with j<l≤Jand 1≤k≤Nl\ncontain growing exponentials, as t→+∞in RHP 2. Thus, one can ignore the decaying jumps for\nnow, and focus on the growing jump. Recall δj(λ)defined in Equation (2.21). Then, one defines a\nnew matrix function\n(5.9) M(2)\nj(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩M(λ;t,z)δj+1(λ)−σ3, λ∈C∖J\n⋃\nl=j+1Nl\n⋃\nk=1(Dϵ\nλl,k⋃Dϵ\nλl,k),\nM(λ;t,z)(λ−λl,k)σ3, λ∈Dϵ\nλl,k,l=j+1, . . . , J,k=1, . . . , Nl,\nM(λ;t,z)(λ−λl,k)−σ3, λ∈Dϵ\nλl,k,l=j+1, . . . , J,k=1, . . . , Nl.\nHence, one can check that M(2)\nj(λ;t,z)solves the following RHP .\nRiemann-Hilbert Problem 8. Suppose M(2)\nj(λ;t,z)is a2×2matrix function on C. It has the asymp-\ntotics M(2)\nj(λ;t,z)→Iasλ→∞. It is sectional analytic with jumps, for 1≤l≤j and 1≤k≤Nl,\n(5.10)M(2)+\nj(λ;t,z)=M(2)−\nj(λ;t,z)δj+1(λ)σ3Vl,k(λ;t,z)−1δj+1(λ)−σ3, λ∈∂Dϵ\nλl,k,\nM(2)+\nj(λ;t,z)=M(2)−\nj(λ;t,z)δj+1(λ)σ3V†\nl,k(λ;t,z)δj+1(λ)−σ3, λ∈∂Dϵ\nλl,k,\nwhere Vl,k(λ;t,z)is given in RHP 2, and for j +1≤l≤J and 1≤k≤Nl,\n(5.11)M(2)+\nj(λ;t,z)=M(2)−\nj(λ;t,z)Wj,l,k(λ;t,z), λ∈∂Dϵ\nλl,k,\nM(2)+\nj(λ;t,z)=M(2)−\nj(λ;t,z)W−†\nj,l,k(λ;t,z), λ∈∂Dϵ\nλl,k,\nwith\n(5.12)\nWj,l,k(λ;t,z)∶=δj+1(λ)σ3Vl,k(λ;t,z)−1(λ−λl,k)σ3=⎛\n⎜⎜\n⎝δj+1(λ)(λ−λl,k) 0\n−λ−λl,k\nλ−λl,kωl,ke−2iθ(λ;t,z)\nδj+1(λ)1\nδj+1(λ)(λ−λl,k)⎞\n⎟⎟\n⎠.\nLet us define another matrix M(3)\nj(λ;t,z)as follows,\n(5.13) M(3)\nj(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩M(2)\nj(λ;t,z), λ∈C∖J\n⋃\nl=j+1Nl\n⋃\nk=1(Dϵ\nλl,k⋃Dϵ\nλl,k),\nM(2)\nj(λ;t,z)Zj,l,k(λ;t,z)−1, λ∈Dϵ\nλl,k,j+1≤l≤J, 1≤k≤Nl,\nM(2)\nj(λ;t,z)Z†\nj,l,k(λ;t,z), λ∈Dϵ\nλl,k,j+1≤l≤J, 1≤k≤Nl,\n31where for j+1≤l≤Jand 1≤k≤Nl, and one defines\n(5.14) Zj,l,k(λ;t,z)∶=⎛\n⎜⎜⎜⎜⎜\n⎝λ−λl,k\npj+1(λ)pj+1(λ)2−pj+1(λl,k)2\npj+1(λ)ω−1\nl,kλ−λl,k\nλ−λl,kpj+1(λl,k)2\npj+1(λ)pj+1(λ)e2iθ(λ;t,z)\n−ωl,kpj+1(λ)\npj+1(λ)λ−λl,k\nλ−λl,ke−2iθ(λ;t,z) 1\nλ−λl,kpj+1(λ)\npj+1(λ)⎞\n⎟⎟⎟⎟⎟\n⎠.\nSimilarly to Section 4.1, it is important to notice that the matrices Zj,l,k(λ;t,z)are analytic in Dϵ\nλl,k,\nand the matrices Z†\nj,l,k(λ;t,z)are analytic in Dϵ\nλl,k. Therefore, the new matrix M(3)\nj(λ;t,z)are still\nanalytic in Dϵ\nλl,k⋃Dϵ\nλl,kfor all 1≤l≤Jand 1≤k≤Nl. One can then verify that M(3)\nj(λ;t,z)solves\nthe following RHP .\nRiemann-Hilbert Problem 9. Suppose M(3)\nj(λ;t,z)is a2×2matrix function. It has the asymptotics\nM(3)\nj(λ;t,z)→Iasλ→∞, and is sectional analytic with jumps, for 1≤l≤j and 1≤k≤Nl\n(5.15)M(3)+\nj(λ;t,z)=M(3)−\nj(λ;t,z)δj+1(λ)σ3Vl,k(λ;t,z)−1δj+1(λ)−σ3, λ∈∂Dϵ\nλl,k,\nM(3)+\nj(λ;t,z)=M(3)−\nj(λ;t,z)δj+1(λ)σ3V†\nl,k(λ;t,z)δj+1(λ)−σ3, λ∈∂Dϵ\nλl,k,\nwhere the jump matrices Vl,k(λ;t,z)are given in RHP 2, and for j +1≤l≤J and 1≤k≤Nl\n(5.16)M(3)+\nj(λ;t,z)=M(3)−\nj(λ;t,z)W(1)\nj,l,k(λ;t,z), λ∈∂Dϵ\nλl,k,\nM(3)+\nj(λ;t,z)=M(3)−\nj(λ;t,z)[W(1)\nj,l,k]−†(λ;t,z), λ∈∂Dϵ\nλl,k,\nwhere W(1)\nj,l,k(λ;t,z)are given by\n(5.17) W(1)\nj,l,k(λ;t,z)∶=Wj,l,k(λ;t,z)Zj,l,k(λ;t,z)−1=⎛\n⎜⎜\n⎝1−λ−λl,k\nωl,kpj+1(λl,k)2\npj+1(λ)2e2iθ(λ;t,z)\n0 1⎞\n⎟⎟\n⎠.\nIt is easy to check that all the jumps in RHP 9 are decaying exponentially to the identity matrix\nIast→+∞. Therefore, the solution M(3)\nj(λ;t,z)to RHP 9 can be approximated by\n(5.18) M(3)\nj(λ;t,z)=I+o(1), t→+∞,\nwhere the error term o(1)is exponentially small. Then, tracing back all the deformations, one\nconcludes that as t→+∞\n(5.19) M(λ;t,z)=M(2)\nj(λ;t,z)δσ3\nj+1=M(3)\nj(λ;t,z)δσ3\nj+1=δj+1(λ)σ3(I+o(1)),∣λ∣≫1 or∣λ∣≪1 .\nRecall that δj+1(λ)→1 as λ→∞, and δj+1(λ)σ3commutes with σ3. The reconstruction formula\nfrom Lemma 1 yields\n(5.20) q(t,z)=o(1), ρ(t,z)=D−σ3(I+o(1)), t→+∞.\nNotice that the calculations in the current subsection are valid for a fixed j=1, 2, . . . , J−1, so\nthe Case 3 in Table 1 is finished, i.e., we have proved that when ξis between any two adjacent\nvelocities, the solution as t→+∞decays to the ZBG exponentially.\n325.2.4. The case of ξ=Vj.Fixing the value of j=1, 2, . . . , J, we analyze Case 4 in Table 1 in this\nsubsection. The configuration is shown in Figure 11(right). The difference between the current\ncase and Case 3 in Section 5.2.3 is that the jumps on the contours ∂Dϵ\nλj,kand∂Dϵ\nλj,kfor all 1≤k≤Nj\ndo not decay to the identity matrix uniformly. Other than this, the two cases are quite alike, in\nthe sense that, all jumps enclosing eigenvalues λl,kandλl,kwith 1≤l≤j−1 decay to the identity\nmatrix uniformly, and the ones enclosing λl,kandλl,kwith l>jgrow uniformly, as t→+∞. As a\nresult, both deformations (5.9) and (5.13) apply to the current case. Thus, we start the discussion of\nCase 4 with RHP 9. However, unlike the previous case, one cannot take the limit t→+∞of RHP 9\nand arrive at Equation (5.18), because the jumps surrounding eigenvalues λj,kare oscillatory as t\nchanges. In fact, RHP 9 do not have a limit in Case 4.\nTo continue our calculation, let us first build a parametrix for RHP 9 by ignoring all decaying\njumps.\nRiemann-Hilbert Problem 10 (Jump form) .LetPs+,j(λ;t,z)be a 2×2matrix function on λ∈C.\nSuppose that Ps+,j(λ;t,z)has the asymptotics Ps+,j(λ;t,z)→Iasλ→∞, is sectional analytic and satisfies\nthe following jumps,\n(5.21)P+\ns+,j(λ;t,z)=P−\ns+,j(λ;t,z)δj+1(λ)σ3Vj,k(λ;t,z)−1δj+1(λ)−σ3, λ∈∂Dϵ\nλj,k, 1≤k≤Nj,\nP+\ns+,j(λ;t,z)=P−\ns+,j(λ;t,z)δj+1(λ)σ3V†\nj,k(λ;t,z)δj+1(λ)−σ3, λ∈∂Dϵ\nλj,k, 1≤k≤Nj,\nwhere Vj,k(λ;t,z)is still given in RHP 2.\nThe subscript “s +” denotes the case in a stable medium with asymptotics t→+∞. For conve-\nnience, we explicitly express the jump matrices of RHP 10 below\n(5.22)δj+1(λ)σ3Vj,k(λ;t,z)−1δj+1(λ)−σ3=⎛\n⎜⎜\n⎝1 0\n−ωj,kδj+1(λ)−2e−2iθ\nλ−λj,k1⎞\n⎟⎟\n⎠,\nδj+1(λ)σ3V†\nj,k(λ;t,z)δj+1(λ)−σ3=⎛\n⎜⎜\n⎝1ωj,kδj+1(λ)2e2iθ\nλ−λj,k\n0 1⎞\n⎟⎟\n⎠.\nNote that the (2, 1)entry of δj+1(λ)σ3Vj,k(λ;t,z)−1δj+1(λ)−σ3and the (1, 2)entry of\nδj+1(λ)σ3V†\nj,k(λ;t,z)δj+1(λ)−σ3have simple poles at λ=λj,kandλ=λj,k, respectively. Hence, one\ncan convert the above RHP with jumps into an equivalent RHP with residue conditions, just like\nthe equivalence between RHPs 1 and 2. Comparing the residue form of RHP 10 with RHP 1, one\nimmediately realizes that RHP 10 describes a DSG containing Njsolitons. The eigenvalues are\nstill the same Λjin the upper half plane, but the corresponding norming constants are modified\nas the set Ω(+)\njcontaining ω(+)\nj,k. Therefore, the parametrix Ps+,j(λ;t,z)is solvable, and is bounded\nast→+∞via Theorem 2 with z=Vjt.\nOne looks at the difference between the original matrix function M(3)(λ;t,z)from RHP 9 and\nits parametrix Ps+,j(λ;t,z)from RHP 10, by defining an error function\n(5.23) Es+,j(λ;t,z)∶=M(3)\nj(λ;t,z)Ps+,j(λ;t,z)−1, for a fixed j=1, 2, . . . , J.\nThen, RHPs 9 and 10 yield a RHP for the error function Es+,j(λ;t,z).\n33Riemann-Hilbert Problem 11. LetEs+,j(λ;t,z)be a2×2matrix function on λ∈C. It has asymptotics\nEs+,j(λ;t,z)→Iasλ→∞, is sectional analytic and satisfies jumps, for 1≤l<j and 1≤k≤Nl\n(5.24)E+\ns+,j(λ;t,z)=E−\ns+,j(λ;t,z)Ps+,jδj+1(λ)σ3Vl,k(λ;t,z)−1δj+1(λ)−σ3P−1\ns+,j, λ∈∂Dϵ\nλl,k,\nE+\ns+,j(λ;t,z)=E−\ns+,j(λ;t,z)Ps+,jδj+1(λ)σ3V†\nl,k(λ;t,z)δj+1(λ)−σ3P−1\ns+,j, λ∈∂Dϵ\nλl,k,\nand for j+1≤l≤J and 1≤k≤Nl\n(5.25)E+\ns+,j(λ;t,z)=E−\ns+,j(λ;t,z)Ps+,jW(1)\nj,l,k(λ;t,z)P−1\ns+,j, λ∈∂Dϵ\nλl,k,\nE+\ns+,j(λ;t,z)=E−\ns+,j(λ;t,z)Ps+,j[W(1)\nj,l,k]−†(λ;t,z)P−1\ns+,j, λ∈∂Dϵ\nλl,k,\nNote that Ps+,j(λ;t,z)is bounded as t→+∞. Thus, all jump matrices for Es+,j(λ;t,z)in RHP 11\ndecay to the identity matrix exponentially and uniformly as t→+∞. Consequently, one writes\n(5.26) Es+,j(λ;t,z)=I+o(1), t→+∞,\nfor every fixed j=1, 2, . . . , J. Because of the boundedness of Ps+,j(λ;t,z), the definition (5.23) yields\n(5.27) M(3)\nj(λ;t,z)=(I+o(1))Ps+,j(λ;t,z), t→+∞.\nTherefore, the solution to RHP 2 can be reconstructed similarly to Equation (5.19), as follows,\n(5.28) M(λ;t,z)=(I+o(1))Ps+,j(λ;t,z)δj+1(λ)σ3,∣λ∣≫1 or∣λ∣≪1 .\nRecall that δj+1(λ)→1 asλ→∞, and δj+1(λ)σ3commutes with σ3. So, by Lemma 1 and Theorem 2\none concludes\n(5.29)q(t,z;Λ,Ω)=q(t,z;Λj,Ω(+)\nj)+o(1),\nD(t,z;Λ,Ω)=D(t,z;Λj,Ω(+)\nj)+o(1),\nP(t,z;Λ,Ω)=P(t,z;Λj,Ω(+)\nj)+o(1),\nwhere the solution {qs+,j,Ds+,j,Ps+,j}is an Nj-DSG derived from RHP 10 with eigenvalues Λjand\nmodified norming constants Ω(+)\nj.\n5.3. Soliton asymptotics as t→−∞in a stable medium. One needs to consider the four cases in\nTable 1, but with the limit t→−∞. Because t<0 instead of t>0 from Section 5.2, and because\nofθ(λ;t,z)=λt+ξt/(2λ), the quantity R(iθ(t,z))flips its sign comparing to the previous t→+∞\ndiscussion.\nThe illustrative plots are given in Figure 12, which should be compared with Figures 10 and 11.\nIt is then necessary to notice the similarities and differences between the situations t→−∞and\nt→+∞. Built on the detailed discussion on the case of t→+∞in Section 5.2, the results of the\nt→−∞case can be easily obtained, in similar treatments. As such, most details will be omitted in\norder to avoid repetitions.\nIt is important to point out that we have chosen Vj−1<ξ<Vjin Figure 12(bottom center),\ninstead of Vj<ξ<Vj+1presented in Table 1. We shift the index in order to simplify calculations in\nSection 5.3.3, as will be discussed in detail there. The shifted case is equivalent to Case 3 in Table 1.\nOne just needs to write j=2, 3, . . . , Jinstead of j=1, 2, . . . , J−1 in Section 5.2.3.\n34FIGURE 12. Similarly to Figures 10 and 11, but with t<0 representing the four\ncases in Table 1 . Top row: situations ξ≤0<V1(left) and 0 <ξ<V1(right)\ncorresponding to Case 1. Bottom left: ξ>VJcorresponding to Case 2. Bottom\ncenter: Vj−1<ξ<Vjslightly different but equivalent to Case 3. Bottom right: ξ=Vj\ncorresponding to Case 4.\n5.3.1. The case of ξ<V1.There are two subcases ξ≤0<V1and 0<ξ<V1as shown in Figure 12(top\nleft and top right). Physically, they describe solutions inside and outside of the light cone, but\nmathematically, they can be treated in identical ways. For small enough radii ϵ>0, all jumps in\nRHP 2 are decaying exponentially and uniformly to the identity matrix as t→−∞. Therefore, the\nsolution can be written as M(λ;t,z)=I+o(1)ast→−∞. Consequently, one has\n(5.30) q(t,z)=o(1), ρ(t,z)=D−σ3+o(1), t→−∞.\n5.3.2. The case of ξ>VJ.As can be calculated and also observed from Figure 12(bottom left), for\nsmall enough ϵ>0, all jumps in RHP 2 grow exponentially as t→−∞. Thus, one needs to reuse\nthe technique from Section 5.2.1 to order to turn the growing jumps into decaying ones. We define\n(5.31) M(5)(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩M(λ;t,z)δσ3\n1, λ∈C∖{J\n⋃\nj=1Nj\n⋃\nk=1(Dϵ\nλj,k⋃Dϵ\nλj,k)},\nM(λ;t,z)(λ−λj,k)σ3(Y(1)\nj,k(λ;t,z))−1,λ∈Dϵ\nλj,k,\nM(λ;t,z)(λ−λj,k)−σ3(Y(1)\nj,k)†(λ;t,z),λ∈Dϵ\nλj,k,\nwhere for j=1, 2, . . . , Jand k=1, 2, . . . , Nj, and Y(1)\nj,k(λ;t,z)is defined in Equation (5.4). Then, the\nnewly defined matrix function M(5)(λ;t,z)satisfies the following RHP .\n35Riemann-Hilbert Problem 12. LetM(5)(λ;t,z)be a2×2sectional analytic matrix function on λ∈C.\nIt has asymptotics M(5)(λ;t,z)→Iasλ→∞. For j=1, 2, . . . , J and k=1, 2, . . . , Nj, it has jumps\n(5.32)M(5)+(λ;t,z)=M(5)−(λ;t,z)⎛\n⎝1−ω−1\nj,k(λ−λj,k)p1(λj,k)2\np1(λ)2e2iθ\n0 1⎞\n⎠, λ∈∂Dϵ\nλj,k,\nM(5)+(λ;t,z)=M(5)−(λ;t,z)⎛\n⎝1 0\nωj,k−1(λ−λj,k)p1(λj,k)2\np1(λ)e−2iθ1⎞\n⎠, λ∈∂Dϵ\nλj,k.\nClearly, all jumps are decaying exponentially to the identity matrix as t→−∞. Therefore, one\nconcludes\n(5.33) q(t,z)=o(1), D(t,z)=D−(1+o(1)), P(t,z)=o(1), t→−∞.\n5.3.3. The case of V j−1<ξ<Vj.First of all, let us fix the value of j=2, 3, . . . , J. One observes from\nFigure 12(bottom center) that all jumps inside the red circles are growing exponentially while the\nones outsides are decaying as t→−∞. Hence, one needs to modify the jumps with Vl,kwith\nl=1, 2, . . . , j−1 and k=1, 2, . . . , Nl.\nThe reason why we have shifted the index from Case 3 in Table 1 is to simplify expressions in\nthis subsection. In particular, with the original expression Vj<ξ<Vj+1, one would need to employ\nquantities δj+1(λ)and pj+1(λ)in calculations, but with the shifted one, one can just use δj(λ)and\npj(λ)as shown below.\nOne defines\n(5.34) M(6)\nj(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩M(λ;t,z)δ1(λ)−σ3δj(λ)σ3, λ∈C∖j−1\n⋃\nl=1Nl\n⋃\nk=1(Dϵ\nλl,k⋃Dϵ\nλl,k),\nM(λ;t,z)(λ−λl,k)σ3Z(1)\nj,l,k(λ;t,z)−1, λ∈Dϵ\nλl,k,\nM(λ;t,z)(λ−λl,k)−σ3(Z(1)\nj,l,k)†(λ;t,z),λ∈Dϵ\nλl,k,\nwhere for l=1, 2, . . . , j−1 and k=1, 2, . . . , Nl,\n(5.35) Z(1)\nj,l,k(λ;t,z)∶=⎛\n⎜⎜⎜⎜\n⎝(λ−λl,k)pjpj\np1p1[p1(λ)2\npj(λ)2−p1(λl,k)2\npj(λl,k)2]ω−1\nl,kλ−λl,k\nλ−λl,kp1(λl,k)2\npj(λl,k)2pjpj\np1p1e2iθ(λ;t,z)\n−ωl,kp1pj\np1pjλ−λl,k\nλ−λl,ke−2iθ(λ;t,z) 1\nλ−λl,kp1pj\np1pj⎞\n⎟⎟⎟⎟\n⎠.\nThen, the new function M(6)\nj(λ;t,z)satisfies the following RHP .\nRiemann-Hilbert Problem 13. LetM(6)\nj(λ;t,z)be a2×2sectional analytic matrix function on λ∈C,\nwith asymptotics M(6)\nj(λ;t,z)→Iasλ→∞, and satisfying the following jumps, for l =1, 2, . . . , j−1and\nk=1, 2, . . . , Nl,\n(5.36)M(6)+\nj(λ;t,z)=M(6)−\nj(λ;t,z)δσ3\n1δ−σ3\njVl,k(λ;t,z)−1(λ−λl,k)σ3Z(1)\nj,l,k(λ;t,z)−1, λ∈∂Dϵ\nλl,k,\nM(6)+\nj(λ;t,z)=M(6)−\nj(λ;t,z)δσ3\n1δ−σ3\njV†\nl,k(λ;t,z)(λ−λl,k)−σ3(Z(1)\nj,l,k)†(λ;t,z), λ∈∂Dϵ\nλl,k, ,\nand for l=j,j+1, . . . , J and k=1, 2, . . . , Nl,\n(5.37)M(6)+\nj(λ;t,z)=M(6)−\nj(λ;t,z)δσ3\n1δ−σ3\njVl,k(λ;t,z)−1δσ3\njδ−σ3\n1, λ∈∂Dϵ\nλl,k,\nM(6)+\nj(λ;t,z)=M(6)−\nj(λ;t,z)δσ3\n1δ−σ3\njV†\nl,k(λ;t,z)δσ3\njδ−σ3\n1, λ∈∂Dϵ\nλl,k,\nwhere Vl,k(λ;t,z)is given in RHP 2.\n36In particular, some useful jumps are given explicitly below\n(5.38)δσ3\n1δ−σ3\njVl,k(λ;t,z)−1(λ−λl,k)σ3Z(1)\nj,l,k(λ;t,z)−1=⎛\n⎜⎜\n⎝1−λ−λl,k\nωl,kp1(λl,k)2\npj(λl,k)2pj(λ)2\np1(λ)2e2iθ\n0 1⎞\n⎟⎟\n⎠,\nδσ3\n1δ−σ3\njV†\nl,k(λ;t,z)(λ−λl,k)−σ3(Z(1)\nj,l,k)†(λ;t,z)=⎛\n⎜⎜\n⎝1 0\nλ−λl,k\nωl,kp1(λl,k)2\npj(λl,k)2pj(λ)2\np1(λ)2e−2iθ1⎞\n⎟⎟\n⎠.\nIt is easy to verify that all jumps in RHP 13 decay exponentially to the identity matrix as t→−∞,\nso one can write M(6)\nj(λ;t,z)=I(1+o(1)). Therefore,\n(5.39) q(t,z)=o(1), P(t,z)=o(1), D(t,z)=D−+o(1), t→−∞.\nThe calculation is valid for every j=2, 3, . . . , J, so we have proved that when ξis between adjacent\nvelocities, the solution decays to the ZBG as t→−∞.\n5.3.4. The case of ξ=Vj.First, let us fix the value of j=1, 2, . . . , J. According to Figure 12(bottom\nright), this case is analogous to the one in Section 5.2.4, in the sense that all the jumps of RHP 2\nwith indices l<jare growing as t→−∞and the ones with l>jare decaying to the identity matrix\nin the asymptotics. Different from the previous case, here the jumps with l=jstay bounded. The\nsteps to treat the growing jumps is identical to the previous case, so we omit the calculations and\nstart with RHP 13, which is similar to what happens in Section 5.2.4. As a result, all jumps with\nindices l≠jdecay exponentially to the identity matrix as t→−∞. By ignoring all decaying jumps,\none arrives at the following leading-order problem.\nRiemann-Hilbert Problem 14 (Jump form) .Let j=1, 2, . . . , J be fixed. Let Ps−,j(λ;t,z)be a 2×2\nsectional analytical matrix function on C.Ps−,j(λ;t,z)→Iasλ→∞. This matrix function also satisfies\nthe following jumps\n(5.40)P+\ns−,j=P−\ns−,jδσ3\n1δ−σ3\njV−1\nj,kδσ3\njδ−σ3\n1=P−\ns−,j⎛\n⎜⎜\n⎝1 0\n−ωj,k\nλ−λj,kδj(λ)2\nδ1(λ)2e−2iθ1⎞\n⎟⎟\n⎠, λ∈∂Dϵ\nλj,k,\nP+\ns−,j=P−\ns−,jδσ3\n1δ−σ3\njV†\nj,kδσ3\njδ−σ3\n1=P−\ns−,j⎛\n⎜⎜\n⎝1ωj,k\nλ−λj,kδ1(λ)2\nδj(λ)2e2iθ\n0 1⎞\n⎟⎟\n⎠, λ∈∂Dϵ\nλj,k,\nThe subscript “s −” denote the asymptotics in a stable medium as t→−∞. If needed, the above\nRHP can be turned in to a residue form. According to Theorem 2, solution Ps−,j(λ;t,z)is bounded\nast→−∞. One then considers the following error function,\n(5.41) Es−,j(λ;t,z)∶=M(6)\nj(λ;t,z)Ps−,j(λ;t,z)−1, for a fixed j=1, 2, . . . , J,\nSimilarly to the matrix Es+,j(λ;t,z)from Equation (5.23), it is straightforward to verify that Es−,j(λ;t,z)\nsatisfies a RHP whose jump matrices decay uniformly to the identity matrix as t→−∞. Hence, one\nwrites Es−,j(λ;t,z)=I+o(1), ast→−∞. Consequently, it is M(6)\nj(λ;t,z)=(I+o(1))Ps−,j(λ;t,z),\nast→−∞. Tracing back all the deformations in Section 5.3.3, one finally obtains as t→−∞,\nM(λ;t,z)=(I+o(1))Ps−,j(λ;t,z)δ1(λ)σ3δj(λ)−σ3, with∣λ∣≫1 or∣λ∣≪1. Recall δ1(λ)→1 and\n37TABLE 2. Four cases in the long-time asymptotics in an unstable medium.\nUnstable Cases Relations between ξand velocities Description\nCase 1 ξ<Vu,J<⋅⋅⋅<Vu,1 ξis smaller than all soliton velocities\nCase 2 Vu,J<⋅⋅⋅<Vu,1<ξ ξis larger than all soliton velocities\nCase 3 Vu,j+1<ξ<Vu,jwith j=1, . . . , J−1 ξis in-between adjacent velocities\nCase 4 Vu,j+1<ξ=Vu,j<Vu,j−1with j=1, . . . , Jξcoincides with velocity Vu,j\nδj(λ)→1 asλ→∞, and δ1(λ)σ3δj(λ)−σ3commutes with σ3, Lemma 1 and Theorem 2 yields\n(5.42)q(t,z;Λ,Ω)=q(t,z;Λj,Ω(−)\nj)+o(1),\nD(t,z;Λ,Ω)=D(t,z;Λj,Ω(−)\nj)+o(1),\nP(t,z;Λ,Ω)=P(t,z;Λj,Ω(−)\nj)+o(1),\nwhere the solution {qs−,j,Ds−,j,Ps−,j}is an Nj-DSG derived from RHP 14 with eigenvalues Λjand\nmodified norming constants Ω(−)\nj∶={ω(−)\nj,k}defined in Theorem 3.\n5.4. Soliton asymptotics in an unstable medium. It is time to discuss the soliton asymptotics in\nan unstable medium with J≥2. According to Theorem 2, let us denote Vs,j=2r2\njas the velocity in a\nstable medium, while Vu,j=−2r2\nj=−Vs,jas the one in an unstable medium. The crucial component\nin the Deift-Zhou’s nonlinear steepest descent method is the sign of R(iθ(λj,k;t,z)), which are\ngiven in the stable and unstable case, respectively, as follows,\n(5.43) R(iθs,j,k(ξ,t))=yj,k\n2r2\nj(ξ−Vs,j)t,R(iθu,j,k(ξ,t))=−yj,k\n2r2\nj(ξ−Vu,j)t,\nwhere yj,k=I(λj,k)>0 and z=ξt. As a result, it can be seen\n(5.44) R(iθu,j,k(ξ,t))=R(iθs,j,k(−ξ,t)).\nTherefore, the asymptotics in an unstable medium along the line z=ξtcan be regarded as asymp-\ntotics in a stable medium along a different line z=−ξt. This relation allows us to compute the\nasymptotics in the unstable case easily from known results of the stable case.\nClearly, the soliton velocities Vu,j=−2r2\njdefined in Theorem 2 are ordered below,\n(5.45) VJ<VJ−1<⋅⋅⋅<V1<0 .\nSimilarly to what happens in a stable, there are four fundamental cases given in Table 2.\nIn the asymptotics t→+∞, one can relate the cases between stable medium (cf. Table 1) and\nunstable medium (cf. Table 2) below.\nUnstable Case 1 Equation (5.44) implies that the unstable Case 1 ( ξ<Vu,J<⋅⋅⋅<Vu,1) can be\ncalculated equivalently to the stable Case 2 ( Vs,1<⋅⋅⋅<Vs,J<−ξ) in the same asymptotics\nt→+∞, where z=ξtwith ξ<0. Hence, one arrives at q(t,z)=o(1)andρ(t,z)=D−σ3(I+\no(1))following Section 5.2.2.\nUnstable Case 2 Equation (5.44) implies that the unstable Case 2 ( Vu,J<⋅⋅⋅<Vu,1<ξ) is equivalent\nto the stable Case 1 ( −ξ<Vs,1<⋅⋅⋅<Vs,J). Therefore, one arrives at the same asymptotic\nresult q(t,z)=o(1)andρ(t,z)=D−σ3(I+o(1))ast→+∞following Section 5.2.1.\n38Unstable Case 3 Equation (5.44) implies that the unstable Case 3 ( Vu,j+1<ξ<Vu,j) is equivalent\nto the stable Case 3 ( Vs,j<−ξ<Vs,j+1). Therefore, one arrives at the same asymptotic result\nq(t,z)=o(1)andρ(t,z)=D−σ3(I+o(1))ast→+∞following Section 5.2.3.\nUnstable Case 4 Equation (5.44) implies that the unstable Case 4 ( ξ=Vu,j) is equivalent to the\nstable Case 4 ( −ξ=Vs,j). Therefore, by repeating similar calculations as in Section 5.2.4, one\narrives at the formula for the asymptotic solution with ΛjandΩ(+)\njin Theorem 3.\nThe other asymptotics t→−∞ between the stable and unstable cases can also be related. The\ndetails are omitted for brevity. The asymptotic DSG is given in Theorem 3. Finally, this completes\nthe proof of Theorem 3 for all cases.\n6. H IGH-ORDER SOLITONS\nIn this section, we discuss various aspects about the high-order solitons. First, we show how to\nfuse N≥2 simple poles of RHP 1 in order to get an Nth order pole. As a result, the corresponding\nN-soliton solution merge into a single Nth-order soliton.\n6.1. Derivation of an Nth order soliton from fusion. Suppose RHP 3 with the eigenvalue λ○and\nnorming constants {ω○,k}N−1\nk=0given. We would like to find an explicit limiting procedure, such\nthat when merging all eigenvalues Λfrom RHP 2, the Nsimple poles ωj,k/(λ−λj,k)fuse into the\nNth-order pole p○(λ)/(λ−λ○)Nappearing in RHP 3.\nTherefore, we start with RHP 2 for the N-soliton solutions. To better take limits, we rewrite this\nRHP in an equivalent form, in which the jump contours ∂Dϵ\nλj,kare modified into ∂Dϵ\nΛ, which is a\nsingle contour surrounding all eigenvalues in the upper half plane. Moreover, we would like to\nrelabel all eigenvalues as λj,k↦λsfors=1, 2, . . . , N, and same for the norming constants ωj,k↦ωs.\nAs a result, the new RHP is given by below.\nRiemann-Hilbert Problem 15 (Equivalent jump-form of RHP 2) .LetM(λ;t,z)be a sectional analytic\n2×2matrix function, with asymptotics M(λ;t,z)→Iasλ→∞and jumps\n(6.1)M+(λ;t,z)=M−(λ;t,z)V(λ;t,z)−1, ∂Dϵ\nΛ,\nM+(λ;t,z)=M−(λ;t,z)V†(λ;t,z), ∂Dϵ\nΛ,\nwhere\n(6.2) V∶=⎛\n⎜⎜\n⎝1 0\nN\n∑\ns=1ωj\nλ−λje−2iθ1⎞\n⎟⎟\n⎠.\nRecall that we would like to take limits λs→λ○, so it is natural to write\n(6.3) λs=λ○+ϵs,\nwhere the target eigenvalue λ○is a fixed complex number in the upper half plane. As a result, the\nrelevant quantity in the above RHP becomes\n(6.4)N\n∑\ns=1ωs\nλ−λs=N\n∑\ns=1ωs\nλ−λ○−ϵs.\nWe next show that the fusion of eigenvalues must be done in a special way. Otherwise, the\nresult is quite trivial.\n39Remark 10 (A naive limit) .We demonstrate here that taking the limit ϵs→0 for all s=1, 2, . . . , N\ndirectly will yield a trivial result. Note that this naive limit does not rescale the norming constants\nωs. Then,\n(6.5) ∑\nsωs\nλ−λs=∑\nsωs\nλ−λ○−ϵs→∑\nsωs\nλ−λ○=∑sωs\nλ−λ○,\nasϵs→0 for all s. So, the Nsimple polesωs\nλ−λsfuse into one simple pole at λ=λ○. Correspondingly,\ntheN-soliton solution becomes an one-soiton solution, with its eigenvalue λ=λ○and norming\nconstant∑sωs. Thus, in order to get the Nth-order soliton, one has to rescale the norming con-\nstants ωs, i.e., assuming ωs=ωs(ϵ1, . . .ϵN). It remains to show that such rescaling exists.\nLet us calculate the sum from Equation (6.4) more carefully\n(6.6)N\n∑\ns=1ωs\nλ−λs=∑N\ns=1ωsps(λ)\n∏N\ns=1(λ−λs), where we define ps(λ)∶=∏N\nl=1(λ−λl)\nλ−λs.\nNote that the quantity ps(λ)is a polynomial of λof degree N−1. Thus,∑sωsps(λ)is also a\npolynomial of degree at most N−1. The leading term of ps(λ)is simply λN−1, and the leading\nterm of∑sωsps(λ)isλN−1∑sωs. For now, let us assume ∑sωs∈C∖{0}, so that the degree of\n∑sωsps(λ)is exactly N−1.\nLemma 2. For the given polynomial p ○(λ)from RHP 3, the following equation for unknowns {ωs}N\ns=1\n(6.7)N\n∑\ns=1ωsps(λ)=p○(λ)\nwith λ∈Chas a unique solution.\nProof. Recall that the given polynomial p○(λ)from RHP 3 has the coefficients {ω○,k}N−1\nk=0with ω○,k∈\nCandω○,0≠0. We show that the unknowns {ωl}N\nl=1can be solved from Equation (6.7) explicitly. It\nis important to notice that the polynomial ps(λ)of degree N−1 has roots {λ1, . . . , λs−1,λs+1, . . . , λN}.\nThus, substituting λ=λsinto Equation (6.7) yields ωsps(λs)=p○(λs), which implies\n(6.8) ωs=p○(λs)\nps(λs)=∑N−1\nk=0ω○,kϵk\ns\n∏k≠s(ϵs−ϵk), s=1, 2, . . . , N−1 .\nHence, the unknowns {ω1, . . . , ωN}are solved explicitly and uniquely. □\nLemma 2 implies that the rescaling ωs=ωs(ϵ1, . . . , ϵN)exists. Using this, we next show that\ntheN-soliton solution can be transformed into an Nth-oder soliton. Lemma 2 implies that Equa-\ntion (6.6) becomes ∑N\ns=1ωs\nλ−λs=p○(λ)/∏N\ns=1(λ−λs). Then, taking limits λs→λ○for all s=1, 2, . . . , N\nwith the particular ωsfrom Equation (6.8), the sum becomes ∑N\ns=1ωs\nλ−λs→p○(λ)/(λ−λ○)N. One\ncan treat the poles in the lower half plane in a similar way, what we omit here. Hence, RHP 15\nimmediately becomes RHP 3, where the contour ∂Dϵ\nΛbecomes ∂Dϵ\nλ○in the upper half plane.\nFinally, we conclude that by taking the limit λs→λ○for all s=1, 2, . . . , Nof the N-soliton\nsolution of MBEs, with particular rescaling of the norming constants {ωs}N\ns=1described in Equa-\ntion (6.8), the N-soliton solution becomes an Nth-order soliton, described by RHP 3.\n6.2. Derivation of the pole form of RHP 3. In this section, we rewrite RHP 3 for the Nth-order\nsoliton in an equivalent form, where the jump conditions become appropriate pole conditions,\n40mimicking the residue conditions in RHP 1 for the N-soliton solutions. To start, let use define a\nnew matrix function\n(6.9) N○(λ;t,z)∶=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩M○(λ;t,z), λ∈C∖{Dϵ\nλ○,Dϵ\nλ○},\nM○(λ;t,z)V○(λ;t,z),λ∈Dϵ\nλ○,\nM○(λ;t,z)V−†\n○(λ;t,z),λ∈Dϵ\nλ○,\nwhere M○(λ;t,z)and V○(λ;t,z)are defined in RHP 3. It is easy to verify that: N○(λ;t,z)→∞as\nλ→∞;N○(λ;t,z)does not admit any jumps on ∂Dϵ\nλ○and∂Dϵ\nλ○; and N○(λ;t,z)is meromorphic\nonC, with poles at λ=λ○and λ=λ○. The only thing left is to derive the pole conditions for\nN○(λ;t,z). Recall the pole operator Pnfrom Definition 2, which can be used to compute the\nLaurent coefficient of N○(λ;t,z)atλ=λ○andλ=λ○. Here, we show detailed steps for the case of\nλ=λ○. The other point can be addressed similarly.\n(6.10) P−n\nλ=λ○N○(λ;t,z)=P−n\nλ=λ○(M○,1+p○(λ)\n(λ−λ○)Ne−2iθM○,2M○,2),\nwhere the subscripts 1 and 2 denote the first and second columns of the matrix, respectively, and\nn=1, 2, . . . , N. Also, we suppress the variable dependence of M○in the above and below equations\nfor simplicity. Recall that M○is analytic at λ=λ○, so by distributing P−nin each column, one gets\n(6.11) P−n\nλ=λ○N○(λ;t,z)=P−n\nλ=λ○M○⎛\n⎝0 0\np0(λ)\n(λ−λ○)Ne−2iθ0⎞\n⎠, n=1, 2, . . . , N.\nIt is necessary to calculate P−n\nλ=λ○M○p○(λ)\n(λ−λ○)Ne−2iθ. The analyticity of p0(λ)e−2iθ(λ;t,z)M○(λ;t,z)atλ=\nλ○yields P−n\nλ=λ○p○(λ)e−2iθ\n(λ−λ○)NM○=PN−n\nλ=λ○p○(λ)e−2iθM○, for n=1, 2, . . . , N. Using the expression for p○(λ)\nin Equation (2.27), the computation continues\n(6.12) P−n\nλ=λ○p○(λ)e−2iθ\n(λ−λ○)NM○=N−n\n∑\ni=0PN−n\nλ=λ○ω○,i(λ−λ○)ie−2iθM○+N−1\n∑\ni=N−n+1PN−n\nλ=λ○ω○,i(λ−λ○)ie−2iθM○\nClearly, terms in the second summation are all zeros. Thus, the equation continues\n(6.13) P−n\nλ=λ○p○(λ)e−2iθ\n(λ−λ○)NM○=N−n\n∑\ni=0PN−n\nλ=λ○ω○,i(λ−λ○)ie−2iθM○=N−n\n∑\ni=0ω○,iPN−n−i\nλ=λ○e−2iθM○,\nforn=1, 2, . . . , N. Due to analyticity, the Taylor expansion of the product e−2iθM○atλ=λ○is\n(6.14) e−2iθM○=∞\n∑\nk=0(λ−λ○)k\nk!∂k\n∂λk(e−2iθM○)λ○.\nThus, each Laurent coefficient in Equation (6.13) can be calculated as (with help of the product\nrule)\n(6.15) PN−n−i\nλ=λ○e−2iθM○=1\n(N−n−i)!N−n−i\n∑\nk=0(N−n−i\nk)∂N−n−i−k\n∂λN−n−i−k(e−2iθ)λ○∂k\n∂λk(M○)λ○.\n41Therefore, one can continue the calculation of the pole condition from Equation (6.11)\n(6.16)P−n\nλ=λ○N○=P−n\nλ=λ○M○p○(λ)\n(λ−λ○)Ne−2iθ(0 0\n1 0)\n=N−n\n∑\ni=0ω○,iPN−n−i\nλ=λ○e−2iθM○(0 0\n1 0)\n=N−n\n∑\ni=0ω○,i\n(N−n−i)!N−n−i\n∑\nk=0(N−n−i\nk)∂N−n−i−k\n∂λN−n−i−k(e−2iθ)λ○∂k\n∂λk(M○)λ○(0 0\n1 0).\nNext, we need to convert the derivatives of M○back to the ones of N○on the right-hand side of\nthe above equation. Recall that near λ=λ○, it is M○=N○V−1\n○, so\n(6.17)∂k\n∂λk(M○)=∂k\n∂λk(N○V−1\n○)=k\n∑\nj=0(k\nj)∂j\n∂λj(N○)∂k−j\n∂λk−j(V−1\n○), λ∈Dϵ\nλ○∖{λ○}.\nRecall also that V−1\n○=⎛\n⎝1 0\n−p○e−2iθ\n(λ−λ○)N1⎞\n⎠, so that∂k−j\n∂λk−j(V−1\n○)=⎛\n⎝0 0\n−∂k−j\n∂λk−j(p○e−2iθ\n(λ−λ○)N)0⎞\n⎠when k−j>0.\nSuch lower triangular matrices with zero diagonals when multiplying by (0 0\n1 0)yields the zero\nmatrix. As a result, the product in Equation (6.16) becomes\n(6.18)∂kM○\n∂λk(0 0\n1 0)=∂kN○\n∂λk(0 0\n1 0), λ∈Dϵ\nλ○∖{λ○}.\nThen, the combination of Equations (6.16) and (6.18) yields\n(6.19) P−n\nλ=λ○N○=lim\nλ→λ○N−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke−2iθ\n∂λN−n−i−k∂kN○\n∂λk(0 0\n1 0)\nSimilarly, one can derive the pole conditions for N○(λ;t,z)atλ=λ○\n(6.20) P−n\nλ=λ○N○=−lim\nλ→λ○N−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke2iθ\n∂λN−n−i−k∂kN○\n∂λk(0 1\n0 0).\nFinally, let us rename the matrix N○(λ;t,z)back to M○(λ;t,z). Then, RHP 4 is obtained.\n6.3. Nth-order soliton solution formula. In this section, we show how to derive the Nth-order\nsoliton solution formula from RHP 4. For brevity, we suppress the variable dependence (λ;t,z)in\nthis section. Due to the particular form of the RHP , we assume the following ansatz for the matrix\nM○(λ;t,z)\n(6.21) M○=I+N\n∑\nn=11\n(λ−λ○)nP−n\nλ=λ○M○+N\n∑\nn=11\n(λ−λ○)nP−n\nλ=λ○M○.\nAs discussed in Remark 2, one only needs to reconstruct the top row, so let us denote\n(6.22) X∶=M○,1,1, Y∶=M○,1,2.\nThus, the top row of Equation (6.21) can be expressed explicitly as\n(6.23) (X Y)=(1 0)+N\n∑\nn=11\n(λ−λ○)nP−n\nλ=λ○(X Y)+N\n∑\nn=11\n(λ−λ○)nP−n\nλ=λ○(X Y).\n42Substituting both pole conditions into the above equation, one gets\n(6.24)X=1+N\n∑\nn=11\n(λ−λ○)nlim\nλ→λ○N−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke−2iθ\n∂λN−n−i−k∂kY\n∂λk,\nY=−N\n∑\nn=11\n(λ−λ○)nlim\nλ→λ○N−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke2iθ\n∂λN−n−i−k∂kX\n∂λk.\nClearly, Xis analytic at λ=λ○and Yis analytic at λ=λ○, so the limits can be computed as\n(6.25)X=1+N\n∑\nn=11\n(λ−λ○)nN−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke−2iθ\n∂λN−n−i−k(λ○)∂kY\n∂λk(λ○),\nY=−N\n∑\nn=11\n(λ−λ○)nN−n\n∑\nk=0N−n−k\n∑\ni=0ω○,i\n(N−n−i−k)!k!∂N−n−i−ke2iθ\n∂λk(λ○)∂kX\n∂λk(λ○).\nWe next change the order of sums as ∑N\nn=1∑N−n\nk=0=∑N−1\nk=0∑N−k\nn=1and use the quantity γk(λ)defined\nin Theorem 5 to simplify the equations,\n(6.26) X(λ)=1+N−1\n∑\nk=0γk(λ)∂kY\n∂λk(λ○), Y(λ)=−N−1\n∑\nk=0γk(λ)∂kX\n∂λk(λ○).\nLet us define more quantities in order to solve the above system\n(6.27) Xs∶=X(s)(λ○), Ys∶=Y(s)(λ○), s=0, 1, . . . , N−1 .\nSubstituting λ=λ○andλ=λ○in Equation (6.26), one gets\n(6.28) X0=1+N−1\n∑\nk=0γk(λ○)Yk, Y0=−N−1\n∑\nk=0γk(λ○)Xk.\nMoreover, by taking derivatives of the system (6.26) stimes and substituting λ=λ○andλ=λ○,\none gets more equations\n(6.29) Xs=N−1\n∑\nk=0γ(s)\nk(λ○)Yk, Ys=−N−1\n∑\nk=0γ(s)\nk(λ○)Xk, s=1, 2, . . . , N−1 .\nEquations (6.28) and (6.29) form a linear system for {Xs,Ys}N−1\ns=0, which can be solved in the same\nmanner as in Section 3.2. The final solution formula is given in Theorem 5.\n7. S OLITON GAS\nIn this section, we explain how one can take the formal limit N→+∞of an N-DSG and derive\nan∞-DSG, i.e., soliton gas. The derivation is similar to [26].\nRecall that N-DSG is described by RHP 2 with J=1. It is easy to verify that this RHP can be\nequivalently turned into the following one with an enlarged jump contour.\nRiemann-Hilbert Problem 16. Suppose J =1and N≥1, and the sets Λ,Ωare given according to\nDefinition 1. Let λ↦M(7)(λ;t,z)be a2×2matrix function on C, with asymptotics M(7)(λ;t,z)→Ias\nλ→∞. It is also sectional analytic with jumps\n(7.1)M(7)+(λ;t,z)=M(7)−(λ;t,z)V(7)(λ;t,z)−1, λ∈∂DΛ,\nM(7)+(λ;t,z)=M(7)−(λ;t,z)[V(7)]†(λ;t,z), λ∈∂DΛ,\n43FIGURE 13. Left and center: The transformation from the N-DSG RHP 2 with\nfunction M(λ;t,z)and J=1 to an equivalent RHP 16 with combined jump con-\ntours. Right: An illustration of: (i) the contour A and A from Equation (2.36) shown\nin red; (ii) the regions DAand DAwith boundaries shown in blue, oriented coun-\nterclockwise. The blue dot are representations of discrete eigenvalues ΛandΛ.\nwhere the jump matrix V(7)(λ;t,z)is given by\n(7.2) V(7)(λ;t,z)∶=(1 0\ne−2iθ(λ;t,z)∑N\nk=1ω1,k\nλ−λ1,k1),\nand the jump contours ∂DΛand∂DΛare illustrated in Figure 13(center) with counterclockwise orientation.\nWe point out that the region DΛ⊂C+encloses all the discrete eigenvalues Λin the upper half\nplane, and similarly, DΛ⊂C−encloses Λin the lower half plane. Here, Λdenote the complex\nconjugate of the set Λ, not its closure. Therefore, the two jump contours ∂DΛand ∂DΛnever\nintersect. Furthermore, in the rest of this work, we assume implicitly that ∂DΛis sufficiently close\ntoΛand contains an arc of the circle with radius r1, as illustrated in Figure 13(left and center).\nWith the new setup, we are now ready to take the limit of RHP 16 as N→+∞while keeping\nJ=1. We would like to take this limit in a controllable way, in the sense that all discrete eigenvalues\ninΛ1=Λare accumulating on an arc centered at the origin. Recall the arc A defined in Definition 3,\nso we know that Λ⊂A. Correspondingly, one defines a new region DA⊂C+which contains the\nwhole arc A and does not intersects with the real axis. Please see an illustration of the contour A\nand corresponding region DAshown in Figure 13(right).\nIt is clear that in RHP 16 the poles are encoded in the jump matrix V(7)(λ;t,z)via the sum in the\n(1, 2)component. In order to take the limit N→+∞, we rewrite the relevant function as follows,\n(7.3)N\n∑\nk=1ω1,k\nλ−λ1,k=∫Aω1(s;N)ds\ns−λ, ω1(λ;N)∶=−N\n∑\nk=1ω1,kδ(λ−λ1,k), λ/∈A ,\nwhere δ(⋅)is the Dirac delta distribution defined on the arc A, oriented left-to-right i.e., ∫Af(s)δ(s−\ns0)ds=f(s0), provided that f(z)is continuous at every interior point z=s0∈A. In Equation (7.3),\nthe norming constants ω1,kcan be recognized as weights of each Dirac delta, and the sum is not\nnormalized, i.e, ∫Aω1(s;N)ds≠1. Hence, the function ω1(s;N)can be considered as an (unnor-\nmalized) complex distribution on the arc A. Therefore, taking the limit N→+∞of the left-hand\nside of Equation (7.3) is equivalent to calculating lim N→+∞ω1(λ;N). Formally speaking, the latter\nlimit describes infinitely many Dirac deltas accumulating on the arc A with their own complex-\nvalued weights, which can be considered as an interpolation of an (unnormalized) continuous\n44FIGURE 14. An illustration of the eigenvalue distribution in the limit N→+∞. In\nthe right panel, the thick blue arcs represent supp ω⊂A and supp ω⊂A.\ndistribution. So, the formal result can be written as\n(7.4) ω(λ)∶=2πi lim\nN→+∞ω1(λ;N), λ∈A1,\nwhere the function ω(λ)is a complex-valued function. The factor 2 πi in the above equation is for\nconvenience for later usages. Also, we assume ω∈L2(A)for convenience. It is important to point\nout that the support of ω(λ)is an arbitrary subset of A. Please see Figure 14 for demonstration.\nHence, the limit produces\n(7.5)lim\nN→+∞N\n∑\nk=1ω1,k\nλ−λ1,k=1\n2πi∫Aω(s)ds\ns−λ, λ/∈A ,\nlim\nN→+∞N\n∑\nk=1ω1,k\nλ−λ1,k=1\n2πi∫Aω(s)ds\ns−λ, λ/∈A .\nwhere the integration contour is oriented left-to-right. To simplify the above integrals, one defines\nthe following Cauchy transforms\n(7.6) Cω(λ)∶=1\n2πi∫Aω(s)ds\ns−λ.\nThen, the above limits can be rewritten as lim\nN→+∞∑N\nk=1ω1,k\nλ−λ1,k=Cω(λ), and lim\nN→+∞∑N\nk=1ω1,k\nλ−λ1,k=Cω(λ).\nIt is easy to verify that Cω(λ)=−Cω(λ). Consequently, the jumps in RHP 16 in the limit becomes\n(7.7)lim\nN→+∞V(7)(λ;t,z)−1=lim\nN→+∞(1 0\n−e−2iθ(λ;t,z)∑N\nk=1ω1,k\nλ−λ1,k1)=(1 0\n−e−2iθ(λ;t,z)Cω(λ)1),\nlim\nN→+∞[V(7)]†(λ;t,z)=lim\nN→+∞⎛\n⎝1 e2iθ(λ;t,z)∑N\nk=1ω1,k\nλ−λ1,k\n0 1⎞\n⎠=(1 e2iθ(λ;t,z)Cω(λ)\n0 1).\nFollowing these calculations, RHP 16 as N→+∞becomes the following one.\nRiemann-Hilbert Problem 17 (Cauchy-form of soliton gas) .Let an arc Aand a function ω(λ)be\ngiven according to Definition 3. Let λ↦M(8)(λ;t,z)be a2×2matrix function on C, with asymptotics\nM(8)(λ;t,z)→Iasλ→∞. It is also sectional analytic with jumps\n(7.8)M(8)+(λ;t,z)=M(8)−(λ;t,z)V(8)(λ;t,z)−1, λ∈∂DA,\nM(8)+(λ;t,z)=M(8)−(λ;t,z)[V(8)]†(λ;t,z), λ∈∂DA,\n45where the jump matrix V(8)(λ;t,z)is given by\n(7.9) V(8)(λ;t,z)∶=(1 0\ne−2iθ(λ;t,z)Cω(λ)1),\nand the jump contours ∂DAand∂DAare illustrated in Figure 14(right) with left-to-right orientation.\nFinally, one can convert the above RHP into a simpler form, by closing the jump contour ∂DA\nonto the arc A and ∂DAonto A, respectively. Then, RHP 17 is converted to RHP 5.\nACKNOWLEDGMENTS\nThis project was partially supported by the NSF of China (No. 12201526), the Fundamental Re-\nsearch Funds for the Central Universities (No. 20720220040), and the Natural Science Foundation\nof Fujian Province of China (No. 2022J01032).\nREFERENCES\n[1] A. Abeya, B. Prinari, G. Biondini, and P . G. Kevrekidis, Solitons and soliton interactions in repulsive spinor Bose–Einstein\ncondensates with nonzero background, Eur. Phys. J. Plus 136, 1126 (2021).\n[2] M. J. 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Li) S CHOOL OF MATHEMATICAL SCIENCES , XIAMEN UNIVERSITY , XIAMEN , FUJIAN 361005, P EOPLE ’SREPUB -\nLIC OF CHINA\nEmail address :sitaili@xmu.edu.cn\n48"    },    {        "title": "2402.02177v1.Jordan_constants_of_Cremona_group_of_rank_2_over_fields_of_characteristic_zero.pdf",        "content": "arXiv:2402.02177v1  [math.AG]  3 Feb 2024JORDAN CONSTANTS OF CREMONA GROUP OF RANK 2 OVER\nFIELDS OF CHARACTERISTIC ZERO\nA. V. ZAITSEV\nAbstract. In this paper we compute the Jordan constants of the Cremona g roup of\nrank two over all fields of characteristic zero.\n1.Introduction\nGroups of birational automorphisms of algebraic varieties are often complicated. For\nexample, finding a set of generators of such groups may be a non trivial task already in\ndimension 2. As one of the possible ways to study such complic ated groups, one can try\nto describe their finite subgroups. In particular, one can st udy the Jordan property of\ninfinite groups.\nDefinition 1.1 ([13, Definition 2.1]) .LetGbe a finite group. The Jordan constant J(G)\nofGis the smallest index of a normal abelian subgroup in G. LetΓbe an arbitrary group.\nThenΓis called Jordan if the value\nJ(Γ) = sup\nG⊆Γ,|G|<∞(J(G))\nis finite. In this case the number J(Γ)is called the Jordan constant of the group Γ.\nAt the end of the 19th century, Camille Jordan proved that com plete linear\ngroups GL n(K)over a field Kof characteristic zero are Jordan, see [ 12, §40]\nor [5, Theorem 36.13]. The Jordan constants of these groups were c omputed over al-\ngebraically closed fields, much later, in 2004, in the work [ 4]. As a corollary, all linear\nalgebraic groups are Jordan.\nWe are also interested in the question of Jordan property of a nother class of groups\nof geometric origin, namely the groups of birational automo rphisms of projective spaces.\nThese groups are called Cremona groups and are denoted by Crn(K), wherenis the\ndimension of the projective space (it is called the rank of th e Cremona group), and Kis\na base field. Groups Crn(K)are Jordan over fields of characteristic zero. In dimension\ntwo, this was proved in the paper [ 19, Theor´ em` e 3.1], and in arbitrary dimension a similar\nresult was obtained in [ 16, Theorem 1.8] and [ 2, Theorem 1.1]. The situation with Jordan\nconstants of these groups is more complicated than for linea r groups. For now, the exact\nvalues of the Jordan constants of the group of birational aut omorphisms of the projective\nplane have been found over algebraically closed fields of cha racteristic zero, as well as over\nfields of real and rational numbers. Namely, the following th eorem holds.\nTheorem 1.2 ([21, Theorems 1.9, 1.10, 1.11]) .LetKbe an algebraically closed field of\ncharacteristic 0. Then\nJ(Cr2(K)) = 7200 , J(Cr2(R)) =J(Cr2(Q)) = 120.\nIn dimension 3, the exact values are unknown even in the case o f algebraically closed\nfield. But there are upper bounds, see [ 15, Theorem 1.2.4]. It is also worth noting that the\ngroups of birational automorphisms of projective spaces ov er algebraically closed fields of\npositive characteristic are not Jordan, since already auto morphism groups are not Jordan\n1(see for example [ 3, §1]). But for finite field the group of birational automorphi sms of the\nprojective plane is Jordan, and even its Jordan constant is c omputed, see [ 17] and [ 20].\nIn this paper we compute the Jordan constants of the biration al automorphism group\nof the projective plane over all fields of characteristic zer o. Namely, we prove the following\ntheorem.\nTheorem 1.3. LetKbe a field of characteristic 0. The Jordan constant of the group\nCr2(K)can attain the following values: 7200,168or120. Moreover,\n(1)J(Cr2(K)) = 7200 if and only if√\n5∈K, and−1is a sum of two squares in K;\n(2)J(Cr2(K)) = 168 if and only if√\n5/negationslash∈Kand√−7∈K;\n(3)J(Cr2(K)) = 120 if and only if one of the two conditions is satisfied:\n•√\n5∈K, and−1is not a sum of two squares in K;\n•√\n5/negationslash∈K, and√−7/negationslash∈K.\nWe also give examples of fields for each value of the Jordan con stant from Theorem 1.3.\nEverywhere in this work, ωwill denote a primitive root of unity of degree 3.\nCorollary 1.4. All values of the Jordan constant from Theorem 1.3are attained:\nJ(Cr2(C)) =J(Cr2(Q(√\n5,√\n−7))) =J(Cr2(Q(√\n5,ω))) = 7200 ,\nJ(Cr2(Q(√\n−7))) = 168 , J(Cr2(R)) =J(Cr2(Q(√\n5)) =J(Cr2(Q)) = 120.\nThe plan of the paper is as follows. In Section 2we collect some auxiliary statements.\nIn Section 3we estimate the Jordan constants of automorphism groups of s urfaces having\na conic bundle structure over P1. In Section 4we estimate the Jordan constants of\nautomorphism groups of del Pezzo surfaces. Finally, in Sect ion5, taking into account all\nprevious estimates, we prove Theorem 1.3and Corollary 1.4.\nWe will use the following notation. By D2nwe denote the dihedral group of order 2n.\nByKwe denote the algebraic closure of the field K. IfK⊂Lis an extension of fields,\nandXis a variety over K, then by XLwe denote the extension of scalars of XtoL, and\nwe denote the extension of scalars of XtoKbyX. Zariski tangent space to a variety X\nat a point Pwe denote by TPX. LetGbe a group, and g,g′∈G, then by /angbracketleftg,g′/angbracketrightwe\ndenote thе subgroup, generated by elements gиg′. Letπ:X− →Bbe a morphism of\nalgebraic varieties; then by Aut(X,π)we denote the group of automorphisms compatible\nwith the morphism π, that is, the group of such automorphisms g∈Aut(X)that there\nexistsg′∈Aut(B)for which π◦g=g′◦π.\nI would like to thank my advisor Constantin Shramov for stati ng the problem, useful\ndiscussions and constant attention to this work. I also want to thank Andrey Trepalin for\nuseful discussions.\n2.Preliminaries\nIn this section, we collect some (well known) general statem ents that will be convenient\nto refer later.\nFirstly, let us recall the standard definition.\nDefinition 2.1. A subgroup Hof a group Gis called a characteristic subgroup if for\nevery automorphism ϕofG, one has ϕ(H) =H.\nThe following theorem is useful for estimating Jordan const ants of finite groups.\nTheorem 2.2 (see [10, Theorem 1.41]) .LetGbe a finite group, and Abe its abelian\nsubgroup. Then there is a characteristic abelian subgroup NinGsuch that\n[G:N]/lessorequalslant[G:A]2.\n2The proofs of the following two propositions can be found in [ 22].\nProposition 2.3 ([22, Corollary 2.9]) .Letnbe an integer number, n/greaterorequalslant3,n/negationslash= 6. LetA\nbe a finite abelian group. Then any group Gincluded in the exact sequence\n1− →Sn− →G− →A− →0,\nis isomorphic to the direct product of Sn×A.\nProposition 2.4 ([22, Proposition 2.11]) .Letnandmbe positive integers, n/greaterorequalslant4,n/negationslash= 6.\nLet the group Gbe included in the exact sequence\n1− →An− →G− →Z/mZ− →0.\nIfm= 2k+1, thenGis isomorphic to An×Z/mZ. Ifm= 2k, then either Gis isomorphic\ntoAn×Z/mZ, or at least contains a normal subgroup isomorphic to An×Z/kZ.\nIn this paper, we will need some information about finite subg roups in the\ngroupPGL2(K), whereKis an arbitrary field of characteristic zero. For our purpose s,\nthe following proposition will be convenient.\nProposition 2.5 (cf. [1, Proposition 1.1]) .LetKbe a field of characteristic 0. Then the\ngroupPGL2(K)can contain only finite subgroups isomorphic to Z/nZ,D2n(forn/greaterorequalslant2),\nA4,S4andA5. Moreover, PGL2(K)contains A5if and only if −1is a sum of two squares\ninK, andKcontains√\n5.\nJordan constants of the group GL2(K)are computed over all fields of characteristic\nzero in paper [ 9], but for our purposes we limit ourselves to the following es timate.\nLemma 2.6. LetKbe a field of characteristic 0. Then\nJ(GL2(K))/lessorequalslant60.\nProof. It is easy to see that any finite subgroup in GL2(K)is a central extension of\na finite subgroup in PGL2(K)by a finite cyclic group. Let G⊂GL2(K)be a finite\nsubgroup, then it is included into the exact sequence\n0− →Z/rZ− →G− →G′− →1,\nwhereG′is a finite subgroup in PGL2(K). By Proposition 2.5the group G′is either\ncyclic, or dihedral, or isomorphic to A4,S4, orA5.\nIfG′is isomorphic to A4,S4orA5, then the subgroup Z/rZ⊂Gis a normal abelian\nsubgroup of index at most 60, henceJ(G)/lessorequalslant60.\nIfG′is cyclic, then the group Gis the central extension of a finite cyclic group by a\nfinite cyclic group, hence Gis abelian, and J(G) = 1.\nIfG′is dihedral, then we can take the normal cyclic subgroup Z/nZ⊂G′of index 2.\nThe preimage of Z/nZis a normal abelian subgroup in Gof index2, henceJ(G)/lessorequalslant2.\nThus, all cases are considered, and the lemma is proved. /square\nThe following theorem about the action of a finite group on a ta ngent space of a variety\nat fixed point is well known. However, in different sources it i s formulated in different\ngenerality, so we also give a proof of it.\nTheorem 2.7. LetXbe an irreducible algebraic variety over a field Kof characteristic 0.\nLetGbe a finite group acting faithfully on Xwith a fixed K-pointP. Then the natural\nhomomorphism\nd:G− →GL(TPX)\nis injective.\n3Proof. Suppose dis not injective. Then, replacing Gby the kernel of the homomor-\nphismd, we can assume that Gacts trivially on the tangent space. Hence Gacts trivially\non the cotangent space mP/m2\nP, wheremP⊂ OPis the maximal ideal of the local ring of\nthe point P. The morphism\nOP− → OP/mP≃K\nhas the natural section, hence we have G-invariant decomposition OP≃mP⊕Kinto a\ndirect sum of vector subspaces.\nConsider a G-invariant filtration\nmP⊃m2\nP⊃m3\nP⊃...\nAccording to Nakayama’s lemma, the ideal mPis generated by elements whose images\nform a basis of the space mP/m2\nP. Hence, Gacts trivially on vector spaces mn\nP/mn+1\nPfor all\nintegersn/greaterorequalslant1. According to Maschke’s theorem, any representation of a fin ite group G\nover a field of characteristic zero is completely reducible. Hence we have an isomorphism\nof representations of the group G:\nmP≃∞/circleplusdisplay\nn=1mn\nP/mn+1\nP.\nTherefore, Gacts trivially on mPand onOP.\nLetU⊂Xbe an affine neighborhood of the point P. Take an arbitrary element g∈G,\nthenU′=g(U)is also an affine neighborhood of the point P. LetRandR′be the\ncoordinate rings of neighborhoods UandU′, respectively. Then RandR′are subrings in\nthe local ring OP, such that g∗R′=R. Sinceg∗is trivial automorphism of the ring OP,\nthenR=R′andgtrivially acts on R. Therefore U=U′andgtrivially acts on U. Using\nthe irreducibility of X, we conclude that gacts onXby a trivial automorphism. So gis\na neutral element, and the group Gis trivial. /square\nThe following corollary will be useful for estimating Jorda n constants of groups acting\non surfaces with fixed points.\nCorollary 2.8. LetSbe a smooth surface over a field K. LetΓbe a group acting\nfaithfully on Swith a fixed K-pointP. ThenJ(Γ)/lessorequalslant60.\nProof. LetG⊂Γbe a finite subgroup. Then the action of GonSinduces the action\nofGon the tangent space TPS. By Theorem 2.7this action is faithful, that is, the group G\nembeds in the group GL(TPS)≃GL2/parenleftbig\nK/parenrightbig\n. Using Lemma 2.6, we get inequality\nJ(G)/lessorequalslantJ/parenleftbig\nGL2/parenleftbig\nK/parenrightbig/parenrightbig\n/lessorequalslant60.\nHence,J(Γ)/lessorequalslant60. /square\nWe will also need the following remark.\nRemark 2.9. In the field Q(√\n5,√−7)and in the field Q(ω)there are decompositions\nof−1into the sums of two squares, for example:\n/parenleftBig√−7+√\n5\n−1+√−35/parenrightBig2\n+/parenleftBig6\n−1+√−35/parenrightBig2\n=−1, ω2+(ω2)2=−1.\n3.Automorphisms of conic bundles\nIn this section we evaluate Jordan constants of automorphis m groups of surfaces having\na conic bundle structure over P1\nK. At first, let us recall the definition. Let Xbe a smooth\nsurface and let π:X− →Bbe a dominant morphism with connected fibers to a smooth\ncurve. Then πis called a conic bundle if each (scheme) fiber of πis isomorphic to a conic.\n4LetXbe a smooth surface over the field K, and let π:X− →P1\nKbe a conic bundle.\nLetG⊂Aut(X,π)be a finite subgroup, then it is included in a short exact seque nce\n1− →GF− →Gp− →GB− →1.\nHereGBis a finite subgroup in automorphism group of base, that is GB⊂PGL2(K). A\nsubgroup GFacts onXpreserving fibers. Since GFis finite, there exist a rational fiber\nwith faithful action of group GF, that isGF⊂PGL2(K). We denote the homomorphism\nfromGtoGBbyp.\nNow we evaluate the Jordan constant of the group Gfor different GFandGB. We\nuse approximately the same methods as in [ 21, Proposition 3.2], however, we need more\nprecise estimates.\nFirstly, we need the following auxiliary geometric lemma.\nLemma 3.1 ([18, Lemma 5.2]) .Letg∈G, andh∈GFbe such that gnormalizes the\ncyclic subgroup /angbracketlefth/angbracketright. Thenghg−1equals either horh−1.\nThe following lemma follows from the proof of Proposition [ 21, Proposition 3.2]. For\nthe convenience of the reader, we write down this proof.\nLemma 3.2. LetGFbe isomorphic to either Z/nZorD2n, andGBis isomorphic to\neitherZ/mZorD2m. ThenJ(G)/lessorequalslant32.\nProof. Firstly, assume that GFis not isomorphic to D4≃(Z/2Z)2. Then there is\nanh∈GFsuch that the cyclic subgroup /angbracketlefth/angbracketright ⊂GFis characteristic of index at most 2.\nAt the same time, GBcontains a normal cyclic subgroup G′\nBof index at most 2. De-\nnoteG′=p−1(G′\nB). ThenG′is a normal subgroup in Gof index at most 2. Letg∈G′\nbe such that p(g)generates G′\nB. The element gnormalizes the subgroup /angbracketlefth/angbracketright, then by\nLemma 3.1eitherghg−1=h, orghg−1=h−1. In any case, the element g2commutes\nwithh, hence the subgroup /angbracketlefth,g2/angbracketright ⊂G′is an abelian subgroup of index at most 4. By The-\norem2.2there is an abelian characteristic subgroup Aof index at most 16 in the group G′.\nThenA⊂Gis the abelian normal subgroup of index at most 32, and J(G)/lessorequalslant32.\nSuppose GFis isomorphic to D4. Again, GBcontains a normal cyclic subgroup G′\nBof\nindex at most 2. Denote G′=p−1(G′\nB). ThenG′is a normal subgroup in Gof index at\nmost2. Letg∈G′be such that p(g)generates G′\nB. Then subgroup /angbracketleftg/angbracketright ⊂GFis abelian\nof index at most 4. By Theorem 2.2there is an abelian characteristic subgroup Aof index\nat most 16 in the group G′. ThenA⊂Gis the abelian normal subgroup of index at\nmost32, andJ(G)/lessorequalslant32. /square\nThe following three lemmas are purely group-theoretic asse rtions.\nLemma 3.3. LetGFbe isomorphic to either Z/nZ, orD2n, andGBbe isomorphic to\neitherA4, orS4. ThenJ(G)/lessorequalslant48.\nProof. There is a characteristic abelian subgroup G′\nFof index at most 2 in GF\n(ifGF≃D4, then the desired subgroup is the entire group D4). Since GFis normal\ninG, thenG′\nFis also normal in Gand\n[G:G′\nF]/lessorequalslant2|GB|/lessorequalslant48.\nHence,J(G)/lessorequalslant48. /square\nLemma 3.4. LetGFbe isomorphic to A4, andGBbe isomorphic to either Z/nZ, orD2n,\norA4, orS4. ThenJ(G)/lessorequalslant72.\n5Proof. Firstly, let us consider the case when GBis isomorphic to either A4orS4.\nSinceGFis isomorphic to A4, it contains the characteristic abelian Klein subgroup G′\nFof\nindex 3. Hence G′\nFis a normal abelian subgroup in Gof index\n[G:G′\nF]/lessorequalslant3|GB|/lessorequalslant72.\nTherefore J(G)/lessorequalslant72.\nNow letGBbe isomorphic to Z/nZ. We have the following exact sequence:\n1− →A4− →G− →Z/nZ− →0.\nIn this case, according to Proposition 2.4, there is a normal subgroup G′inGof index\nat most2and isomorphic to A4×Z/kZ. Denote by Nthe Klein four-subgroup in A4.\nThenN×Z/kZis a unique abelian subgroup of index 3inG′, therefore it is a characteristic\nsubgroup. It follows that Gcontains a normal abelian subgroup of index at most 6, that\nis,J(G)/lessorequalslant6.\nIfGBis isomorphic to D2n, then we have the following exact sequence:\n1− →A4− →G− →D2n− →0.\nConsider the cyclic subgroup Z/nZ⊂D2nand its preimage ˜G=p−1(Z/nZ). According\nto Proposition 2.4, there is a normal subgroup G′in˜Gof index at most 2and isomorphic\ntoA4×Z/kZ. Note that G′is also a normal in Gof index at most 4(as a preimage of\nnormal). Similarly to the previous case, the group G′contains a characteristic abelian\nsubgroup of index 3, which means that Gcontains a normal abelian subgroup of index at\nmost12, that is, J(G)/lessorequalslant12. /square\nLemma 3.5. LetGFbe isomorphic to S4, andGBis isomorphic to either Z/nZ, orD2n,\norA4, orS4. ThenJ(G)/lessorequalslant72.\nProof. Firstly, assume that GBis isomorphic to Z/nZ, we have the following exact\nsequence:\n1− →S4− →G− →Z/nZ− →0.\nBy Proposition 2.3, the group Gis isomorphic to the direct product S4×Z/nZ, in\nparticular, it contains a normal abelian subgroup N×Z/nZof index6, that is, J(G)/lessorequalslant6.\nSecondly, assume that GBis isomorphic to A4. Since GFis isomorphic to S4, it\ncontains the characteristic abelian Klein four-subgroup G′\nFof index 6. Hence G′\nFis a\nnormal abelian subgroup in Gof index\n[G:G′\nF]/lessorequalslant6|GB|/lessorequalslant72.\nTherefore J(G)/lessorequalslant72.\nNow letGBbe isomorphic to D2n, then we have the following exact sequence:\n1− →S4− →G− →D2n− →0.\nLetG′⊂Gbe the preimage of the subgroup Z/nZ⊂D2n. ThenG′is normal, since it\nis a preimage of normal, and, according to Proposition 2.3, it is isomorphic to the direct\nproductS4×Z/nZ. ThenN×Z/kZis a unique normal abelian subgroup of index 6\ninG′, therefore it is a characteristic subgroup. It follows that Gcontains a normal abelian\nsubgroup of index at most 12, that is, J(G)/lessorequalslant12.\nIfGBis isomorphic to S4, then we have the following exact sequence:\n1− →S4− →G− →S4− →0.\nLetG′⊂Gbe a preimage of the Klein four-subgroup. Then G′is normal, since it is a\npreimage of normal, and [G:G′] = 6. Since the Klein subgroup is abelian, then by Propo-\nsition 2.3the group G′is isomorphic to the direct product S4×(Z/2Z)2. Therefore, G′\n6contains an abelian characteristic subgroup isomorphic to (Z/2Z)4, of index 6, henceG\ncontains a normal abelian subgroup of index 36, that is, J(G)/lessorequalslant36. /square\nFrom the previous four lemmas we obtain the following coroll ary.\nCorollary 3.6. LetKbe a field of characteristic 0, such that either√\n5/negationslash∈K, or−1is\nnot a sum of two squares in K. LetXbe a smooth rational surface over the field K, and\nletπ:X− →P1\nKbe a conic bundle. Then\nJ(Aut(X,π))/lessorequalslant72.\nProof. LetGbe a finite subgroup in J(Aut(X,π)). Then, as already discussed, the\ngroupGis included in the exact sequence\n1− →GF− →G− →GB− →1,\nwhereGFandGBare finite subgroups in PGL2(K). By Proposition 2.5, the\ngroupPGL2(K)can contain only the following finite subgroups: Z/nZ,D2n(forn/greaterorequalslant2),\nA4andS4. The case when GFandGBare cyclic or dihedral groups is handled by\nLemma 3.2. The case when GFis cyclic or dihedral, and GBis isomorphic to either A4\norS4is considered in Lemma 3.3. Finally, the cases when GFis isomorphic to either A4\norS4are handled by Lemmas 3.4and3.5. ThusJ(G)does not exceed 72in all possible\ncases, hence we have the inequality J(Aut(X,π)))/lessorequalslant72. /square\n4.Del Pezzo surfaces\nIn this section we estimate Jordan constants of automorphis m groups of rational del\nPezzo surfaces. Recall that a del Pezzo surface is a smooth projective surface with an\nample anticanonical class. The self-intersection index of the canonical class is called the\ndegree of a del Pezzo surface. Also recall that the degree of t he del Pezzo surface can take\ninteger values from 1to9.\nFirst of all, we need a theorem about the Jordan constants of t he automorphism group\nof the rational del Pezzo surface of degree 9, that is, the pro jective plane.\nTheorem 4.1 ([9, Theorem 1.2], weak version) .LetKbe a field of characteristic 0. Then\nthe Jordan constant of the group PGL3(K)takes the following values:\n(1)J(PGL3(K)) = 360 if and only if ω∈Kand√\n5∈K;\n(2)J(PGL3(K)) = 168 if and only if√−7∈K, and at the same time either ω/negationslash∈K,\nor√\n5/negationslash∈K;\n(3)J(PGL3(K))/lessorequalslant60if and only if√−7/negationslash∈K, and at the same time either ω/negationslash∈K,\nor√\n5/negationslash∈K.\nWe now proceed to evaluate the Jordan constants of automorph ism groups of rational\ndel Pezzo surfaces. But it is worth noting that rationality i s not used in the following\nlemma.\nLemma 4.2. LetKbe a field of characteristic 0, and let Xbe a del Pezzo surface of\ndegreed/negationslash= 2,8,9overK. Then\nJ(Aut(S))/lessorequalslant120.\nProof. We proceed case by case.\nLetd= 7. In this case, there is (−1)-curve on Sdefined over Kand invariant with\nrespect to all automorphisms. Therefore, there exists an Aut(S)-equivariant contraction\non a smooth surface S′. Thus, the group Aut(S)effectively acts on S′with a fixed point\n(the image of the contracted (−1)-curve), and by Corollary 2.8we haveJ(Aut(S))/lessorequalslant60.\n7Letd= 6. Then by Theorem [ 6, Theorem 8.4.2] we have the isomorphism\nAut/parenleftbig\nS/parenrightbig\n≃/parenleftBig\nK×/parenrightBig2\n⋊D12.\nHence,J(Aut(S))/lessorequalslantJ(Aut/parenleftbig\nS/parenrightbig\n)/lessorequalslant12.\nLetd= 5. Then by Theorem [ 6, Theorem 8.5.6] we have the isomorphism\nAut/parenleftbig\nS/parenrightbig\n≃S5.\nTherefore, J(Aut(S))/lessorequalslantJ(Aut/parenleftbig\nS/parenrightbig\n) = 120 .\nLetd= 4. Then by Theorem [ 6, Theorem 8.6.6] we have an embedding\nAut/parenleftbig\nS/parenrightbig\n⊂(Z/2Z)4⋊S5.\nHence,J(Aut(S))/lessorequalslantJ(Aut/parenleftbig\nS/parenrightbig\n)/lessorequalslant120(one can prove a more precise estimate, but for\nour purposes, this is enough).\nLetd= 3. In this case Sis a cubic surface in P3\nK. From the classification (see for\nexample [ 6, Theorem 9.5.6]) it can be seen that the group Aut(S)is either isomorphic\nto(Z/3Z)3⋊S4(Fermat cubic case), or has at most 120elements. In any case, we have\nthe inequality J(Aut(S))/lessorequalslant120.\nLetd= 1. In this case there is a rational point on Sthat is invariant with respect to\nall automorphisms, namely, the intersection point of all di visors from the complete linear\nsystem|−KS|. Hence, by Corollary 2.8we have the inequality J(Aut(S))/lessorequalslant60./square\nAfter we have received the estimates, it is convenient to not e the following.\nLemma 4.3. LetKbe a field of characteristic 0. LetSbe a del Pezzo surface of degree 2\noverK. Then either J(Aut(S))/lessorequalslantJ(PGL3(K)), orJ(Aut(S))/lessorequalslant96.\nProof. The anticanonical linear system defines a double-covering s tructure over P2\nKonS\nwith branching over a smooth quartic C. Therefore, the automorphism group of the\nsurfaceSis included in the following exact sequence\n0− →Z/2Z− →Aut(S)− →Aut(C)− →1.\nFrom this sequence we immediately obtain an estimation J(Aut(S))/lessorequalslant|Aut(C)|. Au-\ntomorphism groups of smooth flat quartics over Kare described in [ 6, Theorem 6.5.2].\nIt follows from this theorem that if the automorphism group Aut(C)is not isomorphic\nto PSL 2(F7), then it has no more than 96elements. But the group Aut(C)is a subgroup\ninAut(C), so if|Aut(C)|/lessorequalslant96, then|Aut(C)|/lessorequalslant96, and in this case the lemma is proved.\nSuppose\nAut(C)≃PSL2(F7),\nin particular |Aut(C)|= 168 . IfAut(C)⊂Aut(C)is a proper subgroup, then we have the\ninequality |Aut(C)|/lessorequalslant56, and in this case the lemma is also proved. So we can assume\nthatAut(C)≃PSL2(F7). Since the embedding of the curve Cinto the projective plane\nis canonical, the group Aut(C)is a subgroup in PGL3(K). Since the group PSL 2(F7)is\nsimple, then\nJ(PGL3(K))/greaterorequalslantJ(PSL2(F7)) =|PSL2(F7)|=|Aut(C)|/greaterorequalslantJ(Aut(S)),\nand in this case the lemma is also proved. /square\nThe following proposition restricts the Jordan constant of automorphism groups of\nrational quadrics, which are different from the product of tw o projective lines.\nProposition 4.4 ([22, Proposition 1.6]) .LetKbe a field of characteristic 0. LetSbe a\nsmooth rational quadric in P3\nK, andS/negationslash≃P1\nK×P1\nK. Then\nJ(Aut(S))/lessorequalslant120.\n8Corollary 4.5. LetKbe a field of characteristic 0, and let Sbe a rational del Pezzo\nsurface of degree 8overK. Assume that S/negationslash≃P1\nK×P1\nK. Then\nJ(Aut(S))/lessorequalslant120.\nProof. It is known that Sis isomorphic to either P1\nK×P1\nK, or to the blow-up of P2\nK\nat a point. If Sis isomorphic to P1\nK×P1\nK, then since Sis rational, it is isomorphic to\na smooth rational quadric in P3\nK(see for example [ 22, Proposition 6.1]). According to\nProposition 4.4we haveJ(Aut(S))/lessorequalslant120.\nIfSis isomorphic to the blow-up of P2\nKat a point, then there is a unique (−1)-curve on S\ndefined over Kand invariant with respect to all automorphisms. Therefore , there exists\nanAut(S)-equivariant contraction to the projective plane P2\nK. Thus, the group Aut(S)\neffectively acts on P2\nKwith a fixed point (the image of the contractible (−1)-curve), and\nby Corollary 2.8we have J(Aut(S))/lessorequalslant60. /square\nFor the quadric P1\nK×P1\nK, there is the following result.\nTheorem 4.6 ([22, Theorem 1.4]) .LetKbe a field of characteristic 0.\n(1)J(Aut(P1\nK×P1\nK)) = 7200 if and only if√\n5∈Kand−1is a sum of two squares\ninK;\n(2)J(Aut(P1\nK×P1\nK)) = 72 if and only if√\n5/negationslash∈K, and−1is a sum of two squares\ninK;\n(3)J(Aut(P1\nK×P1\nK)) = 8 if and only if −1is not a sum of two squares in K.\nFor convenience, we introduce the following notation. Let Kbe a field of characteris-\ntic0. Denote by JdP(K)the following value:\n(4.1) JdP(K) = max\nX/parenleftbig\nJ(Aut(X))/parenrightbig\n,\nwhereXruns through all rational del Pezzo surfaces over K. For this value, there is an\nobvious lower bound.\nRemark 4.7. There exists a (rational) del Pezzo surface of degree 5with\nan automorphism group isomorphic to S5over any field, see for exam-\nple [23, Example 3.1, Corollary 8.3]. Therefore, we have JdP(K)/greaterorequalslant120.\nNow let us gather all the statements of this section into a pro position that will be\nconvenient to refer to.\nProposition 4.8. LetKbe a field of characteristic 0. Then\n(1)JdP(K) = 7200 if and only if√\n5∈K, and−1is a sum of two squares in K;\n(2)JdP(K) = 168 if and only if√\n5/negationslash∈Kand√−7∈K;\n(3)JdP(K) = 120 if and only if one of the two conditions is satisfied:\n•√\n5∈K, and−1is not a sum of two squares in K;\n•√\n5/negationslash∈Kand√−7/negationslash∈K.\nProof. To calculate the value of JdP(K), one can ignore del Pezzo surfaces with Jordan\nconstant of the automorphism group less or equal to 120 by Rem ark4.7. These are all\ndel Pezzo surfaces of degree d/negationslash= 2,8,9by Lemma 4.2and rational del Pezzo surfaces of\ndegree8that are not isomorphic to P1\nK×P1\nK, by Corollary 4.5.\nLetXbe a rational del Pezzo surface of degree 9, thenXis isomorphic to P2\nK. Indeed,\nsinceXis rational, there is a K-point on Xby the Lang–Nishimura theorem (see for ex-\nample [ 14, Theorem 3.6.11]). Hence Xis a Severi–Brauer surface containing K-point. By\nChatelet’s theorem (see for example [ 8, Theorem 5.1.3]), we obtain that Xis isomorphic\ntoP2\nK. In particular, the group Aut(X)is isomorphic to the group PGL3(K). Therefore,\n9by Lemma 4.3, del Pezzo surfaces of degree 2can also be ignored when computing the\nvalue ofJdP(K). Therefore, the value of JdP(K)can be rewritten as follows:\nJdP(K) = max( J(PGL3(K)),J(Aut(P1\nK×P1\nK)),120).\nLet√\n5∈K, and−1is a sum of two squares in K. Then by Theorem 4.6we\nhaveJ(Aut(P1\nK×P1\nK)) = 7200 , at the same time J(PGL3(K))/lessorequalslant360by Theorem 4.1.\nHenceJdP(K) = 7200 .\nLet√\n5∈K, but−1is not a sum of two squares in K. Note that in this case\nneither√−7norωcan lie in Kdue to Remark 2.9. Then we have the inequal-\nityJ(PGL3(K))/lessorequalslant60by Theorem 4.1, and we have equality J(Aut(P1\nK×P1\nK)) = 8\nby Theorem 4.6. Therefore JdP(K) = 120 .\nLet√\n5/negationslash∈Kand√−7∈K. Then we have the inequality J(Aut(P1\nK×P1\nK))/lessorequalslant72by The-\norem4.6, and the equality J(PGL3(K)) = 168 by Theorem 4.1. Therefore JdP(K) = 168 .\nLet√\n5/negationslash∈Kand√−7/negationslash∈K. Then J(Aut(P1\nK×P1\nK))/lessorequalslant72by Theorem 4.6,\nandJ(PGL3(K))/lessorequalslant60by Theorem 4.1. Therefore JdP(K) = 120 .\nThus, we have considered all possible fields of characterist ic0, and the theorem is\nproved. /square\n5.Jordan constants of Cremona group of rank 2\nIn this section we prove the main theorem and its corollary. B ut before that, let us\nmake a very simple remark.\nRemark 5.1. LetKbe a field of characteristic 0. Since the group Cr2(K)is embedded\nin the group Cr2(K), then applying Theorem 1.2, we obtain an estimate\nJ(Cr2(K))/lessorequalslantJ(Cr2(K)) = 7200 .\nNow everything is prepared to prove the main theorem.\nProof (of Theorem 1.3).LetGbe a finite subgroup in the group Cr2(K). Regularizing the\naction of Gand taking G-equivariant resolution of singularities, we can assume th atGis a\nsubgroup in the automorphism group of a smooth rational surf aceS(see [7, Lemma 3.5]).\nIn this case, Scontains a K-point according to the Lang–Nishimura theorem (see for\nexample [ 14, Theorem 3.6.11]). Further, applying the program of minima l models with\nan action of the group G, we can assume that Sis either a del Pezzo surface, or has a\nconic bundle structure π:S− →B, andG⊂Aut(S,π), see [ 11, Theorem 1G]. Note that\nin the latter case, the curve Bis geometrically rational and contains a K-point, since S\ncontains a K-point. Hence, we have an isomorphism B≃P1\nK.\nHence, the Jordan constant of the Cremona group of rank 2 can b e rewritten in the\nfollowing form:\nJ(Cr2(K)) = max( JdP(K),Jcb(K)).\nHereJdP(K)is a constant, defined in ( 4.1), andJcb(K)is a similar constant for conic\nbundles:\nJcb(K) = max\n(Y,πY)(J(Aut(Y,πY))),\nwhere pairs of (Y,πY)run through all rational surfaces over Ktogether with a conic\nbundle structure over P1\nK.\nNote that for any field Kof characteristic zero, the inequality Jcb(K)/lessorequalslantJdP(K)is\nsatisfied. Indeed, for fields Kin which either√\n5does not lie, or −1is not a sum of two\nsquares we have\nJcb(K)/lessorequalslant72<120/lessorequalslantJdP(K)\n10by Corollary 3.6and Remark 4.7. If√\n5∈K, and−1is a sum of two squares in K,\nthenJdP(K) = 7200 , andJcb(K)cannot exceed 7200by Remark 5.1.\nNow the assertion of the theorem directly follows from Propo sition 4.8. /square\nTo prove Corollary 1.4we prove an elementary auxiliary lemma.\nLemma 5.2. Letmandnbe two coprime square-free integers. Then√mdoes not lie in\nthe fieldQ(√n).\nProof. Sincemandnare square-free integers, the numbers√mand√nare irrational.\nSuppose that√mlies inQ(√n). Then there are a,b,c,d∈Zsuch that\n/parenleftBiga\nb+c\nd√n/parenrightBig2\n=m,\nand we can assume that fractions are irreducible. Let us act b y Galois involution of\nquadratic extension Q⊂Q(√n)on both sides of the equality. We obtain:\n/parenleftBiga\nb−c\nd√n/parenrightBig2\n=m.\nIt follows form this two equalities that either a= 0, orc= 0. Ifa= 0, we get the\nequality nc2=md2, therefore, cis divisible by m, but then dis divisible by m, and we\ngot a contradiction with the irreducibility of the fractionc\nd. If we assume that c= 0, then\nwe immediately get a contradiction with the irrationality o f the number√m. Thus, we\ngot contradictions in all cases, so√m /∈Q(√n). /square\nProof (of Corollary 1.4.).For fields C,RandQ, the assertion is guaranteed by Theo-\nrem1.2and this is consistent with Theorem 1.3. Applying Remark 2.9and Theorem 1.3,\nwe get\nJ(Cr2(Q(√\n5,√\n−7))) =J(Cr2(Q(√\n5,ω))) = 7200 .\nBy Lemma 5.2, the number√\n5does not lie in Q(√−7). So, by Theorem 1.3we have\nJ(Cr2(Q(√\n−7))) = 168 .\nFinally, since −1is not a sum of two squares in Q(√\n5), then by Theorem 1.3we have\nJ(Cr2(Q(√\n5))) = 120 .\n/square\nReferences\n[1] A. Beauville. Finite subgroups of PGL2(K). Contemporary Mathematics, 522(2010), 23–29.\n[2] C. Birkar. Singularities of linear systems and boundedn ess of Fano varieties. Annals of Mathematics,\n193(2021), no. 2, 347–405.\n[3] Y. Chen, C. Shramov. Automorphisms of surfaces over field s of positive characteristic.\narXiv:2106.15906 (2021).\n[4] M. J. Collins. On Jordan’s theorem for complex linear gro ups. Journal of Group Theory, 10(2007),\n411–423.\n[5] C. W. Curtis, I. Reiner. Representation theory of finite g roups and associative algebras. Pure and\nApplied Mathematics, vol. XI. Interscience Publishers, a d ivision of John Wiley & Sons, New York–\nLondon, 1962.\n[6] I. Dolgachev. Classical algebraic geometry: a modern vi ew. /emdash.cyr Cambridge: Cambridge University\nPress, 2012.\n[7] I. Dolgachev, V. Iskovskikh. Finite subgroups of the pla ne Cremona group. In Algebra, arithmetic,\nand geometry: in honor of Yu. I. Manin, vol. I. Mathematical P rogramming, 269(2009), 443–548.\n[8] P. Gille, T. Szamuely. Central simple algebras and Galoi s cohomology, 2nd edition. /emdash.cyr Cambridge:\nCambridge University Press, 2017.\n[9] Yi. Hu. Jordan Constant of PGL3(K). arXiv:2206.0218 (2022).\n11[10] I. M. Isaacs. Finite group theory. Graduate Studies in M athematics, vol. 92. American Mathematical\nSociety, Providence, RI, 2008.\n[11] V. Iskovskikh. Minimal models of rational surfaces ove r arbitrary fields. Mathematics of the USSR\n- Izvestiya, 14(1980), no. 1, 17–39.\n[12] C. Jordan. M´ emoire sur les ´ equations diff´ erentielle s lin´ eaires ` a int´ egrale alg´ ebrique. Journal f¨ ur die\nreine und angewandte Mathematik, 84(1878), 89–215.\n[13] V. Popov. On the Makar-Limanov, Derksen invariants, an d finite automorphism groups of algebraic\nvarieties. In Affine algebraic geometry. CRM Proceedings and Lecture Notes, 54(2011), 289–311.\n[14] B. Poonen. Rational points on varieties. Graduate Stud ies in Mathematics, vol. 186. American Math-\nematical Society, Providence, RI, 2017.\n[15] Yu. Prokhorov, C. Shramov. Jordan constant for Cremona group of rank 3. Moscow Mathematical\nJournal, 17(2017), no. 3, 457–509.\n[16] Yu. Prokhorov, C. Shramov. Jordan property for Cremona groups. American Journal of Mathematics,\n138(2016), no. 2, 403–418.\n[17] Yu. Prokhorov, C. Shramov. Jordan property for Cremona group over a finite field. Proceedings of\nthe Steklov Institute of Mathematics 320(2023), 278–289.\n[18] J.-P. Serre. A Minkowski-style bound for the orders of t he finite subgroups of the Cremona group of\nrank 2 over an arbitrary field. Moscow Mathematical Journal. 9(2009), no. 1, 183–198.\n[19] J.-P. Serre. Le groupe de Cremona et ses sous-groupes fin is. S´ eminaire Bourbaki, Nov. 2008, 61` eme\nann´ e,1000 (2010), 75–100.\n[20] A. Vikulova. Jordan constant for the Cremona group of ra nk two over a finite field”, Math. Notes,\n113(2023), no. 4, 587–592.\n[21] E. Yasinsky. The Jordan constant for Cremona group of ra nk 2, Bulletin of the Korean Mathematical\nSociety, 54(2017), no. 5, 1859–1871.\n[22] A. Zaitsev. Automorphism groups of two-dimensional qu adrics. arXiv:2401.15655 (2024).\n[23] A. Zaitsev. Forms of del Pezzo surfaces of degree 5 and 6. Sbornik: Mathematics, 214(2023), no. 6,\n816-831.\nNational research university ”Higher school of economics”, Lab oratory of algebraic\ngeometry, 6 Usacheva str., Moscow, 119048, Russia\nEmail address :alvlzaitsev1@gmail.com\n12"    },    {        "title": "2402.02202v1.Three_body_scattering_area_for_particles_with_infinite_or_zero_scattering_length_in_two_dimensions.pdf",        "content": "Three-body scattering area for particles with infinite or zero scattering length in two\ndimensions\nJunjie Liang∗and Shina Tan†\nInternational Center for Quantum Materials, School of Physics,\nPeking University, Beijing 100871, China\n(Dated: February 3, 2024)\nWe derive the asymptotic expansions of the wave function of three particles having equal mass\nwith finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero\nenergy and zero orbital angular momentum, from which a three-body parameter Dis defined. The\ndimension of Dis length squared, and we call Dthree-body scattering area. We find that the\nground state energy per particle of a zero-temperature dilute Bose gas with these interactions is\napproximatelyℏ2D\n6mρ2, where ρis the number density of the bosons, mis the mass of each boson,\nand ℏis Planck’s constant over 2 π. Such a Bose gas is stable at D≥0 in the thermodynamic\nlimit, and metastable at D < 0 in the harmonic trap if the number of bosons is less than Ncr≈\n3.6413q\nℏ\nmω|D|, where ωis the angular frequency of the harmonic trap. If the two-body interaction\nsupports bound states, Dtypically acquires a negative imaginary part, and we find the relation\nbetween this imaginary part and the amplitudes of the pair-boson production processes. We derive\na formula for the three-body recombination rate constant of the many-boson system in terms of the\nimaginary part of D.\nI. INTRODUCTION\nThe low energy effective interaction of identical bosons\nor distinguishable particles in two dimensions (2D) is usu-\nally dominated by the 2D scattering length a. When the\ntwo particles collide with zero total energy and zero or-\nbital angular momentum, their wave function behaves as\nϕ(x)∝lnx\na(1)\natx > r e, where reis the range of interaction, xis the\nvector extending from one particle to the other particle,\nandx=|x|is the distance between the two particles.\nIn analogy to the scattering length in two-body scatter-\nings, one can define three-body parameters in three-body\nzero-energy collisions. One of us defined the three-body\nscattering hypervolume for three identical bosons in three\ndimensions [1, 2], which was studied by some other au-\nthors [3–7]. The three-body scattering hypervolume has\nbeen generalized to three particles with unequal masses\n[6], three fermions in three dimensions [8], two dimen-\nsions [9], and one dimension [10].\nNaturally, one would like to study the problem of zero-\nenergy collision of three identical bosons in 2D. When the\n2D scattering length ais finite, the logarithmic behavior\nof the two-body wave function as shown in Eq. (1) com-\nplicates the structure of the three-body wave function\n[11]. But if a=∞ora= 0, the constant ln adominates\nthe behavior of ϕ(x) outside of the range of interaction,\nand thus ϕ(x) becomes a constant outside of the range\nof interaction, and the structure of the three-body wave\n∗junjieliang@pku.edu.cn\n†shinatan@pku.edu.cnfunction is simplified greatly. The purpose of the present\npaper is to study such three-body collisions at a=∞or\n0. In this paper we will define a three-body parameter D\nfor such a system by finding the asymptotic expansions\nof the three-body wave function when the three particles\nare far apart from each other or when one pair of par-\nticles is far apart from the remaining particle. We will\nthen study some physical implications of this parameter,\nincluding implications for the many-boson systems in 2D.\nThe parameter Ddefined and the asymptotic expansions\nof the three-body wave function derived in this paper are\nalso applicable to distinguishable particles having equal\nmass if the three pairwise interaction potentials are the\nsame.\nIn another paper we will elucidate the structure of the\nthree-body wave function at zero energy in 2D when ais\nfinite.\nIn Sec. II, we study the zero-energy scattering of three\nidentical bosons with short-range interactions in 2D, fine-\ntuned such that a=∞or 0, and derive the asymptotic\nexpansions of the three-body wave function at large in-\nterparticle distances, from which the three-body param-\neterDis defined. The dimension of Dis length squared,\nin contrast to the three-body scattering hypervolume of\nidentical bosons in three dimensions whose dimension is\nlength to the fourth power [1, 2].\nIn Sec. III, we calculate the ground state energy of\nthree identical bosons in a large periodic square, and then\ngeneralize the result to Nbosons. After taking the ther-\nmodynamic limit, we find the energy of the Bose gas due\nto the three-body parameter D. The Bose gas is stable\nat zero temperature for D≥0 only. But if D < 0 the\nsystem is unstable in the thermodynamic limit.\nIn Sec. IV we study the 2D Bose gas in a harmonic trap\natD < 0. By numerically solving the Gross-PitaevskiiarXiv:2402.02202v1  [cond-mat.quant-gas]  3 Feb 20242\n(GP) equation with D < 0, we find that the system is\nmetastable if the number of bosons is less than a critical\nnumber. Our result is reminiscent of analogous predic-\ntions for the Bose gases with attractive two-body inter-\nactions [12, 13]. People have also studied trapped Bose\ngases with both two-body and three-body couplings [14–\n16], but the harmonically trapped Bose gas with infinite\nor zero 2D scattering length was not studied to our knowl-\nedge.\nII. ASYMOTOTIC EXPANSIONS OF THE\nTHREE-BODY WAVE FUNCTION\nWe consider three identical bosons (or three distin-\nguishable particles with equal mass) with both two-body\nand three-body interactions. The three-body wave func-\ntion for zero total momentum, zero total energy, zero\norbital angular momentum, and even y-parity is\nϕ(3)(r1,r2,r3) =Zd2k1d2k2\n(2π)4ϕ(3)\nk1k2k3ei(k1·r1+k2·r2+k3·r3),\n(2)where r1,r2,r3are the 2D position vectors of the parti-\ncles, and ℏk1,ℏk2,ℏk3are their linear momenta satisfy-\ning the constraint\nk1+k2+k3≡0. (3)\nWe assume that\nϕ(3)(r1,r2,r3)→1 (4)\nwhen the three pairwise distances between the parti-\ncles are all large. By assuming Eq. (4), we can usually\nuniquely determine the wave function ϕ(3)for given in-\nteractions, if there are no two-body deep bound states.\nThose three-body wave functions which diverge when the\nthree pairwise distances are large are usually less impor-\ntant in ultracold atoms physics than the one we study\nin this paper. If there is one or more two-body bound\nstates, and we assume that the resultant bound pair and\nthe third particle fly apart from each other rather than\napproach each other, then we can also uniquely deter-\nmine ϕ(3)by also assuming Eq. (4).\nWe assume that the particles interact with two-body\npotentialℏ2\nmUk1k2k′\n1k′\n2(where k1+k2≡k′\n1+k′\n2) and\nthree-body potentialℏ2\nmUk1k2k3k′\n1k′\n2k′\n3(where k1+k2+\nk3≡k′\n1+k′\n2+k′\n3), such that the three-body Schr¨ odinger\nequation at zero energy is\nq2\n1+q2\n2+q2\n3\n2ϕ(3)\nq1q2q3+1\n2Z\nk′Up1k′ϕ(3)\nq1,−q1/2+k′,−q1/2+k′+1\n2Z\nk′Up2k′ϕ(3)\n−q2−k′,q2,−q2/2+k′\n+1\n2Z\nk′Up3k′ϕ(3)\n−q3/2+k′,−q3/2−k′,q3+1\n6Z\nk′\n1k′\n2Uq1q2q3k′\n1k′\n2k′\n3ϕ(3)\nk′\n1k′\n2k′\n3= 0,(5)\nwhereR\nk′≡Rd2k′\n(2π)2,R\nk′\n1k′\n2≡Rd2k′\n1\n(2π)2d2k′\n2\n(2π)2, and\nUkk′≡Uk,−k,k′,−k′, (6)\np1≡1\n2(q2−q3), (7a)\np2≡1\n2(q3−q1), (7b)\np3≡1\n2(q1−q2). (7c)\nWe assume the following properties for the interac-\ntions.\n•Spatial translational invariance: k1+k2≡k′\n1+k′\n2\ninUk1k2k′\n1k′\n2, and k1+k2+k3≡k′\n1+k′\n2+k′\n3in\nUk1k2k3k′\n1k′\n2k′\n3.\n•Finite range: the interaction is limited within a\nfinite inter-particle distance rein real space, and\nsoUk1k2k′\n1k′\n2is a smooth function of k1,k2,k′\n1ifwe set k′\n2=k1+k2−k′\n1. Similarly, Uk1k2k3k′\n1k′\n2k′\n3is a smooth function of k1,k2,k3,k′\n1,k′\n2if we set\nk′\n3=k1+k2+k3−k′\n1−k′\n2.\n•Bosonic symmetry: because of Bose statistics, we\ncan symmetrize Uwith respect to the incom-\ning (outgoing) momenta without losing generality:\nUk1k2k′\n1k′\n2=Uk2k1k′\n1k′\n2=Uk1k2k′\n2k′\n1. Similarly,\nUk1k2k3k′\n1k′\n2k′\n3is symmetric under the interchange\nof the incoming (outgoing) momenta.\n•Hemiticity: Uk1k2k′\n1k′\n2= U∗\nk′\n2k′\n1k2k1, and\nUk1k2k3k′\n1k′\n2k′\n3=U∗\nk′\n3k′\n2k′\n1k3k2k1.\n•Rotational invariance: for any rotation Ron\nthe 2D plane, Uk1k2k′\n1k′\n2=URk1,Rk2,Rk′\n1,Rk′\n2and\nUk1k2k3k′\n1k′\n2k′\n3=URk1,Rk2,Rk3,Rk′\n1,Rk′\n2,Rk′\n3.\n•Galilean invariance: for any 2D vector ∆ k,\nUk1+∆k,k2+∆k,k′\n1+∆k,k′\n2+∆k=Uk1k2k′\n1k′\n2and\nUk1+∆k,k2+∆k,k3+∆k,k′\n1+∆k,k′\n2+∆k,k′\n3+∆k =\nUk1k2k3k′\n1k′\n2k′\n3.3\n•y-reversal invariance: Uk1k2k′\n1k′\n2=\nUPyk1,Pyk2,Pyk′\n1,Pyk′\n2and Uk1k2k3k′\n1k′\n2k′\n3=\nUPyk1,Pyk2,Pyk3,Pyk′\n1,Pyk′\n2,Pyk′\n3, where Pykis\nthe reversal of the y-component of the 2D vector\nk, namely\nPy(kxˆx+kyˆy)≡kxˆx−kyˆy, (8)\nandˆxandˆyare unit vectors long the + xaxis and\nthe + yaxis respectively. The y-reversal invariance\nis equivalent to the x-reversal invariance for rota-\ntionally invariant interactions.\nNote that the leading-order term 1 in the expansion\nofϕ(3)(r1,r2,r3) (when the three pairwise distances are\nall large) has even y-parity. Since the interactions are\ny-reversally invariant, we may focus on the three-body\nstate with even y-parity, whose wave function satisfies\nϕ(3)(Pyr1, Pyr2, Pyr3) =ϕ(3)(r1,r2,r3) (9)\nfor all r1,r2,r3. Since the total orbital angular momen-\ntum is zero, the three-body wave function also satisfies\nϕ(3)(Rr1, Rr2, Rr3) =ϕ(3)(r1,r2,r3) (10)\nfor any 2D rotation R. So the wave function is an even\nfunction under reflection about any axis on the 2D plane.\nWe define the following Jacobi coordinates:\nx1=r2−r3,x2=r3−r1,x3=r1−r2,(11a)\ny1=r1−1\n2(r2+r3),\ny2=r2−1\n2(r3+r1),\ny3=r3−1\n2(r1+r2). (11b)\nWe also define the hyperradius\nB=r\ny2\ni+3\n4x2\ni=q\n(x2\n1+x2\n2+x2\n3)/2. (11c)\nWe will derive the asymptotic expansions of the three-\nbody wave function ϕ(3). For this purpose, we will first\ndefine two-body special functions.\nA. Two-body special functions\nConsider two identical bosons with mass meach, inter-\nacting with potential energy ∝Ukk′in momentum space\nin the center-of-mass frame, such that when they collide\nwith energy\nE=ℏ2k2\nE\nm(12)\nin the center-of-mass frame, the wave function ψk(where\n+ℏkand−ℏkare the linear momenta of the two bosonsin the center-of-mass frame) for their relative motion sat-\nisfies the Schr¨ odinger equation\n(Hψ)k=k2\nEψk, (13)\nwhere ℏ2H/m is the Hamiltonian in the center-of-mass\nframe, and\n(HX)k≡k2Xk+1\n2Zd2k′\n(2π)2Ukk′Xk′ (14)\nfor any function Xk. We assume that the interac-\ntion is rotationally invariant, y-reversally invariant, and\nbosonic, namely Ukk′=URk,Rk′=UPyk,Pyk′=U−k,k′=\nUk,−k′for any spatial rotation R, where Pyis the y-\nreversal defined in Eq. (8). If the collision energy Eis\nsmall, as is often the case for ultracold atoms, we may\nsolve Eq. (13) by introducing two-body special functions\nϕk, fk, gk, . . . that are independent of energy, satisfying\nthe following sequence of equations:\n(Hϕ)k= 0,(Hf)k=ϕk,(Hg)k=fk, . . . (15)\nand expressing the solution to Eq. (13) as a power series:\nψk∝ϕk+k2\nEfk+k4\nEgk+···. (16)\nEquation (15) does not uniquely fix the definition of the\ntwo-body special functions: even if the collision energy\nis zero, the scattering wave function is not unique, be-\ncause the two identical bosons may collide with orbital\nangular momentum quantum number l= 0,2,4,···, and\nforl >0 we also need to specify the y-parity of the wave\nfunction. So we may define a sequence of two-body spe-\ncial functions ϕ(l,σ)\nk, f(l,σ)\nk, g(l,σ)\nk, . . .for each partial wave\nchannel specified by the quantum numbers ( l, σ), where\nσ=±1 is the y-parity: if ψPyk=σψkwe say that the\nwave function ψkhasy-parity σ. Ifl= 0 we must take\nσ= +1. These special functions are defined to satisfy\n(Hϕ(l,σ))k= 0, (17a)\n(Hf(l,σ))k=ϕ(l,σ)\nk, (17b)\n(Hg(l,σ))k=f(l,σ)\nk, (17c)\n. . .\nand\nϕ(l,+)\nk=ϕ(l)\nkcos(lθ), (17d)\nϕ(l,−)\nk=ϕ(l)\nksin(lθ), (17e)\nf(l,+)\nk=f(l)\nkcos(lθ), (17f)\nf(l,−)\nk=f(l)\nksin(lθ), (17g)\ng(l,+)\nk=g(l)\nkcos(lθ), (17h)\ng(l,−)\nk=g(l)\nksin(lθ), (17i)\n. . .\nwhere k≡ |k|, and θis the angle from the + xaxis to the\ndirection of k, such that kx=kcosθandky=ksinθ.4\nEquations (17) still do not uniquely determine the spe-\ncial functions. The overall amplitudes of ϕ(l)\nkare not yet\nspecified. Even after fixing the definition of ϕ(l,σ)\nk, we\nstill have some degrees of freedom in the definitions of\nf(l,σ)\nk,g(l,σ)\nketc, because ˜f(l,σ)\nk=f(l,σ)\nk+c1ϕ(l,σ)\nkand\n˜g(l,σ)\nk=g(l,σ)\nk+c1f(l,σ)\nk+c2ϕ(l,σ)\nketc satisfy\n(H˜f(l,σ))k=ϕ(l,σ)\nk,(H˜g(l,σ))k=˜f(l,σ)\nk,\netc, for any coefficients c1andc2etc. These degrees of\nfreedom in the definitions of the special functions may\nbe called some “gauge freedom”. The general solution to\nEq. (13) is\nψk=X\nlσCl,σ(ϕ(l,σ)\nk+k2\nEf(l,σ)\nk+k4\nEg(l,σ)\nk+···),(18)\nwhere Cl,σare any coefficients depending on ( l, σ).\nFor later convenience we also define two-body spe-\ncial functions that have well-defined parity for reflection\nabout an arbitrary unit vector ˆn=ˆxcosθn+ˆysinθn:\nϕ(l)\nˆnk=ϕ(l,+)\nkcos(lθn) +ϕ(l,−)\nksin(lθn)\n=ϕ(l)\nkcos(larg(ˆn,ˆk)), (19a)\nf(l)\nˆnk=f(l,+)\nkcos(lθn) +f(l,−)\nksin(lθn)\n=f(l)\nkcos(larg(ˆn,ˆk)), (19b)\ng(l)\nˆnk=g(l,+)\nkcos(lθn) +g(l,−)\nksin(lθn)\n=g(l)\nkcos(larg(ˆn,ˆk)), (19c)\n. . . ,\nwhere arg( ˆn,ˆk) is the angle formed by the unit vectors\nˆnandˆk. These functions satisfy\n(Hϕ(l)\nˆn)k= 0,(Hf(l)\nˆn)k=ϕ(l)\nˆnk,(Hg(l)\nˆn)k=f(l)\nˆnk, . . . .\n(20)\nWe define the Fourier transforms of the above special\nfunctions:\nϕ(l)\nˆn(x) =Zd2k\n(2π)2ϕ(l)\nˆnkeik·x, (21a)\nf(l)\nˆn(x) =Zd2k\n(2π)2f(l)\nˆnkeik·x, (21b)\ng(l)\nˆn(x) =Zd2k\n(2π)2g(l)\nˆnkeik·x, (21c)\n. . . .\nFors-wave collisions, l= 0, the special functions do not\ndepend on ˆnand we may simply write ϕk≡ϕ(0)\nˆnk,fk≡\nf(0)\nˆnk,gk≡g(0)\nˆnk, etc, and ϕ(x)≡ϕ(0)\nˆn(x),f(x)≡f(0)\nˆn(x),\ng(x)≡g(0)\nˆn(x), etc. For d-wave collisions, l= 2, we\nwrite ϕ(d)\nˆnk≡ϕ(2)\nˆnk,f(d)\nˆnk≡f(2)\nˆnk, etc, and ϕ(d)\nˆn(x)≡ϕ(2)\nˆn(x),\nf(d)\nˆn(x)≡f(2)\nˆn(x), etc.Assuming that the interaction is finite-ranged in real\nspace, such that Ukk′is a smooth function of kandk′,\nwe may analytically determine the two-body special func-\ntions outside of the range of interaction re. If the 2D\ns-wave scattering length ais finite, then ϕ(x) satisfies\nEq. (1). But in this paper we will be focused on those\ninteractions for which a=∞or 0, and so ϕ(x) is a con-\nstant at x > r e. We choose this constant to be 1 and\nthereby fix the overall amplitude of ϕ(x):\nϕ(x) = 1 ,ifx > r e. (22)\nWhen x > r ethe special functions f(x) and g(x) etc sat-\nisfy−∇2f(x) =ϕ(x) and −∇2g(x) =f(x) etc. Solving\nthe first equation we find\nf(x) =−1\n4x2+rs∗lnx\nx∗,ifx > r e, (23)\nwhere rs∗depends on the details of the interaction, but\nx∗is an arbitrary length scale. rs∗can be positive, nega-\ntive, or zero for different interactions. If rs∗̸= 0, different\nchoices of x∗correspond to different “gauges” discussed\nabove: if we add any multiple of ϕ(x) to the definition\noff(x) we will effectively change the value of x∗. Given\nthe above formula for f(x) we find that\ng(x) =1\n64x4+rs∗\n4x2(1−lnx\nx∗) +r′\ns∗lnx\nx∗(24)\nifx > r e, where we have fixed the “gauge freedom” in\ng(x) by requiring that there is no extra constant term on\nthe right hand side of Eq. (24). r′\ns∗is another parameter\nthat depends on the details of the interaction. ϕ(d)\nˆn(x)\nsatisfies −∇2ϕ(d)\nˆn(x) = 0 at x > r e. We can fix the\noverall amplitude of ϕ(d)\nˆn(x) by requiring that\nϕ(d)\nˆn(x) =\u0010x2\n8−4ad\nπx2\u0011\ncos[2arg( ˆn,ˆx)] (25)\natx > r e, where adis the d-wave scattering “length”\nhaving the dimension of length to the fourth power.\nTaking inverse Fourier transformations, we find that\nat small kwe have\nϕk= (2π)2δ(k) +u0+u1k2+O(k4), (26a)\nfk=π2∇2\nkδ(k)−2πrs∗Zk∗(k)\nk2+f0+O(k2),(26b)\ngk=π2\n16∇4\nkδ(k)−2πrs∗Zk∗(k)\nk4−2πr′\ns∗Zk∗(k)\nk2+g0+O(k2)\n(26c)\nϕ(d)\nˆnk=−π2\n2Q(d)\nˆn(∇k)δ(k) +4ad\nk2Q(d)\nˆn(k) +O(k2),(26d)\nwhere u0,u1,f0, and g0are parameters that depend on\nthe details of the interaction,\nk∗= 2e−γ/x∗, (27)5\nγ= 0.5772···is Euler’s constant,\nQ(d)\nˆn(k) = 2( ˆn·k)2−k2(28)\nis a harmonic polynomial,\nQ(d)\nˆn(∇k) = 2( ˆn· ∇k)2− ∇2\nk, (29)\nand we have used the Zfunctions first introduced in\nRef. [1] for three dimensions, but here we define the Z\nfunctions in 2D:\nZk∗(k)\nk2=\u001a1\nk2, k > 0,\n−∞, k= 0,(30a)\nZ\nk<k∗Zk∗(k)\nk2d2k= 0, (30b)\nZk∗(k)\nk4=\u001a1\nk4, k > 0,\n−∞, k= 0,(30c)\nZ\nallkZk∗(k)\nk4d2k= 0, (30d)\nZ\nk<k∗Zk∗(k)\nk4k2d2k= 0. (30e)\nJust like the Dirac delta function which can be ap-\nproached by a sequence of ordinary functions, the Zfunc-\ntions defined here and those defined in Ref. [1] can be\napproached by sequences of ordinary functions. For ex-\nample, the sequence of ordinary functions of 2D vector\nk,\nFη(k)≡\u001a1\nk2, k ≥η,\n−2\nη2lnk∗\nη, k < η,(31)\napproachesZk∗(k)\nk2when η→0+. Just like those for\nthe Dirac delta function, there are uncountably infinitely\nmany different sequences of ordinary functions that ap-\nproach any given Zfunction. Some important math-\nematical properties of these Zfunctions are shown in\nAppendix A.\nIn general, the small- kexpansion of ϕ(l)\nˆnkis of the form\nϕ(l)\nˆnk∝Q(l)\nˆn(∇k)δ(k) +o(k−l−2), (32)\nwhere Q(l)\nˆn(∇k) is a degree- lharmonic polynomial of ∇k.\nSo the leading-order term in this expansion scales like\nk−l−2.\nFrom Eqs. (14), (20), and (26) we find that at small k\n1\n2Z\nk′Ukk′ϕk′=−u0k2−u1k4+O(k6), (33a)\n1\n2Z\nk′Ukk′fk′=u0+ 2πrs∗+O(k2), (33b)\n1\n2Z\nk′Ukk′ϕ(d)\nˆnk′=O(k2). (33c)The parameters that appear in the two-body special\nfunctions are related to the two-body scattering phase\nshifts at small but nonzero collision energies.\nFor the s-wave two-body collision with energy E=\nℏ2k2\nE/min the center-of-mass frame, where kE>0, the\nwave function is\nϕk(E) =ϕk+k2\nEfk+k4\nEgk+···. (34)\nFourier transforming the above wave function, we find\nthat\nϕ(x, E) =Zd2k\n(2π)2ϕk(E)eik·r\n=ϕ(x) +k2\nEf(x) +k4\nEg(x) +···. (35)\nAtx > r ewe must have\nϕ(x, E) =G\u0002\nJ0(kEx) cotδs−Y0(kEx)\u0003\n,(36)\nwhere δsis the s-wave scattering phase shift, and Gis\nindependent of x. Using Eqs. (22), (23), and (24), we\nfind that\nG=−π\n2P(kE) (37)\nand\ncotδs=2\nπh\n−1\nP(kE)+ lnkE\nk∗i\n, (38)\nwhere\nP(kE) =rs∗k2\nE+r′\ns∗k4\nE+O(k6\nE). (39)\nThe case of a=∞or 0 studied here corresponds to the\ncase of vanishing parameter aL,ddefined in Ref. [17] with\nL= 0 and d= 2. But if the parameter aL,din Ref. [17]\nvanishes, the usual effective range expansion in Eq. (15)\nof Ref. [17] is no longer applicable, and that expansion is\nreplaced by our Eq. (38) in the case of s-wave collisions\nin 2D.\nFor the d-wave two-body collision with energy E=\nℏ2k2\nE/min the center-of-mass frame, where kE>0, the\nwave function that is even under reflection about the ˆn\naxis (where ˆnis any unit vector) is\nϕ(d)\nˆnk(E) =ϕ(d)\nˆnk+k2\nEf(d)\nˆnk+··· (40)\nFourier transforming the above wave function, we find\nthat\nϕ(d)\nˆn(x, E) =ϕ(d)\nˆn(x) +k2\nEf(d)\nˆn(x) +···. (41)\nAtx > r ewe must have\nϕ(d)\nˆn(x, E) =G(d)\u0002\nJ2(kEx) cotδd−Y2(kEx)\u0003\n×cos[2arg( ˆn,ˆx)], (42)6\nwhere δdis the d-wave scattering phase shift, and G(d)\nis independent of x. Using Eq. (25), and assuming that\nad̸= 0, we find that\nG(d)=−k2\nEad (43)\nand\nk4\nEcotδd=−1\nad+O(k2\nE). (44)\nEquation (44) agrees with Ref. [17].\nB. Asymptotics of ϕ(3)\nk1k2k3at small momenta\nHereafter, we let q’s be small momenta and qi’s scale\nlikeq1. The three-body wave functions could be ex-\npanded asymptotically in two limits:\nϕ(3)\nq1q2q3=∞X\ns=−4T(s)\nq1q2q3, (45)\nϕq\nk≡ϕ(3)\nq,−q/2+k,−q/2−k=∞X\ns=−2S(s)q\nk, (46)\nwhere T(s)\nq1q2q3andS(s)q\nkboth scale like qs(including pos-\nsibly qslnnq,n= 1,2,···). Because ϕ(3)\nk1k2k3is rotation-\nally invariant and symmetric under the interchange of\nany two momenta, we can show that\nϕ−q\nk=ϕq\n−k=ϕq\nk. (47)\nSo\nS(s)−q\nk=S(s)q\n−k=S(s)q\nk. (48)\nWhen qandkare both small the above functions can\nbe further expanded as\nT(s)\nq,−q/2+k,−q/2−k=X\nit(i,s−i)\nqk, (49)\nS(s)q\nk=X\njt(s,j)\nqk, (50)\nwhere t(i,j)\nq,kscales like qikj.\nThe three-body Schr¨ odinger equation has two special\nforms:\nq2\n1+q2\n2+q2\n3\n2ϕ(3)\nq1q2q3=−\u00103X\ni=11\n2Z\nk′Upik′ϕqi\nk′\u0011\n−Uϕ\nq1q2q3,\n(51)\n(Hϕq)k+3q2\n4ϕq\nk+Wq\nk= 0, (52)\nwhere\nUϕ\nq1q2q3=1\n6Z\nk′\n1k′\n2Uq1q2q3k′\n1k′\n2k′\n3ϕ(3)\nk′\n1k′\n2k′\n3, (53)Wq\nk=\u00121\n2Z\nk′U−q/2+k,qk′k′′ϕ(3)\n−q/2−k,k′k′′+ (q↔ −q)\u0013\n+Uϕ\n−q/2+k,−q/2−k,q. (54)\nSince the three-body potential Uq1,q2,q3,k′\n1,k′\n2,k′\n3de-\npends smoothly on q1andq2forq1+q2+q3≡0,\nand since this potential and ϕ(3)\nk′\n1k′\n2k′\n3are both rotationally\ninvariant and y-reversally invariant, and since the three-\nbody potential is symmetric with respect to q1,q2,q3, we\nhave the following Taylor expansion at small q1,q2,q3if\nq1+q2+q3≡0:\nUϕ\nq1q2q3=κ0+κ1(q2\n1+q2\n2+q2\n3) +O(q4). (55)\nSince the three-body potential U−q/2+k,−q/2−k,q,k′\n1k′\n2k′\n3\nand the wave function ϕ(3)\nk′\n1k′\n2k′\n3are both rotationally in-\nvariant and y-reversally invariant and the three-body po-\ntential is symmetric under the interchange of the outgo-\ning momenta, we can show that Wq\nk=W−q\nk=WPyq\nPyk\nandWq\nk=WRq\nRkfor any 2D rotation R. We thus have\nthe following small- qTaylor expansion:\nWq\nk=X\ns=0,2,4,···qsW(s)\nˆqk, (56)\nBecause Eq. (56) is a Taylor series expansion in\nnonnegative-integer powers of qxandqy(the Cartesian\ncomponents of q), and because Wq\nkis rotationally invari-\nant and y-reversally invariant, W(s)\nˆqkcan only depend on\nˆqlike\nW(s)\nˆqk=X\nl=0,2,4,···,sc(s)\nl(k) cos( larg(ˆq,ˆk)), (57)\nwhere c(s)\nl(k) depends on s, l, k only. In particular, W(0)\nˆqk\ndoes notdepend on ˆqand can be simply written as\nW(0)\nk≡W0\nk.\nTaking the Fourier transformation of ϕ(3)(r1,r2,r3)\nand using Eq. (4), we find that the leading order of\nϕ(3)\nq1q2q3is\nT(−4)\nq1q2q3= (2π)4δ(q1)δ(q2). (58)\nNote that q1+q2+q3≡0 inT(s)\nq1q2q3. Using Eq. (58),\nEq. (51) and Eq. (52), we can determine S(−2),T(−2),\nS(0),T(0), and S(2)step by step.\nExpanding T(−4)\nq,−q/2+k,−q/2−kin powers of q[see\nEq. (49)], we find that\nt(−2,−2)\nqk= (2π)4δ(q)δ(k), (59a)\nt(i,−4−i)\nqk= 0, i̸=−2. (59b)\nStep 1: Determination of S(−2).The fact that\nt(−2,−2)\nqkis nonzero, together with Eq. (50), suggests that7\nS(−2)q\nkis nonzero. Extracting the terms of order q−2in\nEq. (52), we get\n(HS(−2)q)k= 0. (60)\nOn the other hand, when kis small\nS(−2)q\nk=t(−2,−2)\nq,k+O(k0) = (2 π)4δ(q)δ(k) +O(k0).\n(61)\nThe even- y-parity solution to Eq. (60) is\nS(−2)q\nk=X\nl=0,2,4,...η(l)\nqϕ(l)\nˆqk, (62)\nwhere η(l)\nqdoes notdepend on k. Substituting the\nsmall- kexpansions of ϕ(l)\nˆqkshown in Eqs. (26a) and (26d)\ninto the above equation, and comparing the result with\nEq. (61), we find that η(0)\nq= (2π)2δ(q) and η(l)\nq= 0 for\nl≥2. So we find\nS(−2)q\nk= (2π)2δ(q)ϕk. (63)\nExpanding S(−2)q\nkat small k, and using Eq. (26a), we\nfind\nt(−2,−2)\nqk= (2π)4δ(q)δ(k), (64a)\nt(−2,0)\nqk=u0(2π)2δ(q), (64b)\nt(−2,2)\nqk=u1k2(2π)2δ(q). (64c)\nEquation (64a) agrees with Eq. (59a), as we expect.\nStep 2: Determination of T(−2)and S(0).Ex-\ntracting the terms of order q0in Eq. (52), we get\n(HS(0)q)k=−3q2\n4S(−2)q\nk−W0\nk. (65)\nBut\n−3q2\n4S(−2)q\nk=−3q2\n4(2π)2δ(q)ϕk= 0. (66)\nSo we have\n(HS(0)q)k=−W0\nk. (67)\nAccording to Eq. (57), W0\nkis rotationally invariant and\ndepends on k=|k|only. Before symbolically solving\nEq. (67), we introduce a function dkindependent of q\nand satisfying the equation\n(Hd)k=−W0\nk. (68)\nSince W0\nkis rotationally invariant, we may require dkto\nbe rotationally invariant, namely dkdepends on k=|k|\nonly. Even after making such a requirement on dk, the\ndefinition of dkstill has a “gauge freedom”: edk=dk+ηϕkstill satisfies Eq. (68) and is still rotationally invariant\nfor any constant η. Under this change of the definition\nofdk, the small- kexpansion of dkwill be modified: the\ncoefficients of the terms proportional to δ(k), 1, k2etc\nwill be modified. To completely fix the definition of dk,\nwe need to expand dkat small kto the order k−2and\nspecify the coefficient of the term proportional to δ(k).\nAssuming that dkhas been defined, we may solve\nEq. (67) and obtain\nS(0)q\nk=dk+A(q)ϕk+B(q)ϕ(d)\nˆqk+···. (69)\nwhere A(q) and B(q) etc are all of order q0(including\nthe possibility of some logarithmic dependencies on q)\nand independent of the direction of q. Since the small-\nkexpansion of ϕ(d)\nˆqkcontains a term of order k−4[see\nthe first term on the right hand side of Eq. (26d)], B(q)\nmust be zero. If B(q) is not zero, the small- kexpansion\nofS(0)q\nkwould contain a nonzero term of order q0k−4.\nBut in Eq. (59b) we see that t(0,−4)= 0. So B(q) = 0.\nSimilarly the higher partial-wave terms on the right side\nof Eq. (69) are zero. So we get\nS(0)q\nk=dk+A(q)ϕk. (70)\nTo find the expansion of dkat small k, we need to first\nstudy the expansion of W0\nkat small k. From Eq. (54),\nwe get\nW0\nk=Uϕ\nk,−k,0+Z\nk′Uk\n2,k′ϕ(3)\n−k,k\n2+k′,k\n2−k′. (71)\nWhen kis small, using Eq. (46) and Eq. (55) we get\nW0\nk= (κ0+ 2κ1k2) +Z\nk′Uk\n2,k′\u0002\nS(−2)k\nk′+S(0)k\nk′+S(2)k\nk′\u0003\n+O(k4)\n= (κ0+ 2κ1k2)\n+Z\nk′Uk\n2,k′\u0002\n(2π)2δ(k)ϕk′+dk′+A(k)ϕk′\u0003\n+Z\nk′U0,k′S(2)k\nk′+O(k4). (72)\nWhen kis small, we have Taylor expansion\n1\n2Z\nk′Uk,k′dk′=χ0+χ1k2+O(k4). (73)\nSubstituting the above equation and Eq. (33a) into\nEq. (72), we get\nW0\nk= (κ0+ 2χ0) +k2h\n2κ1+χ1\n2−u0\n2A(k)i\n+Z\nk′U0,k′S(2)k\nk′+O(k4). (74)8\nSubstituting the above equation and Eq. (14) into\nEq. (68), and using Eq. (73), we get\nk2dk=−D+k2h\n−2κ1−3χ1\n2+u0\n2A(k)i\n−Z\nk′U0,k′S(2)k\nk′+O(k4), (75)\nwhere\nD≡κ0+ 3χ0. (76)\nSolving Eq. (75) and using the requirement that dkis\nrotationally invariant, we get\ndk=−DZk∗(k)\nk2+h\n−2κ1−3χ1\n2+u0\n2A(k)i\n−1\nk2Z\nk′U0,k′S(2)k\nk′+O(k2), (77)\nwhere we have fixed the “gauge freedom” in the defini-\ntion of dkby specifying the subscript of the Zfunction\nZ?(k)/k2to be the k∗defined in Eq. (27). Note that ac-\ncording to Eq. (A1a), if we change this subscript, this Z\nfunction will change by a coefficient times δ(k).\nExpanding S(0)q\nkshown in Eq. (70) at small k, and\nusing Eq. (26a) and Eq. (77), we get\nt(0,−2)\nqk=−DZk∗(k)\nk2+A(q)(2π)2δ(k), (78a)\nt(0,0)\nqk=−2κ1−3χ1\n2+u0\n2A(k)+u0A(q)−1\nk2Z\nk′U0,k′S(2)k\nk′.\n(78b)\nExtracting the terms of order q0on both sides of\nEq. (51), we get\nq2\n1+q2\n2+q2\n3\n2T(−2)\nq1q2q3=−D+u03X\ni=1(2π)2δ(qi)p2\ni\n=−D+u03X\ni=1(2π)2δ(qi)\u0010\np2\ni+3\n4q2\ni\u0011\n. (79)\nUsing the identity p2\ni+3\n4q2\ni=q2\n1+q2\n2+q2\n3\n2forq1+q2+q3≡\n0, we get\nT(−2)\nq1q2q3=−2D\nq2\n1+q2\n2+q2\n3+u03X\ni=1(2π)2δ(qi).(80)\nTherefore\nT(−2)\nq,−q/2+k,−q/2−k=−D\nk2+3\n4q2+ (2π)2u0h\nδ(q)\n+δ\u0010\n−q\n2+k\u0011\n+δ\u0010\n−q\n2−k\u0011i\n.\n(81)When qis small we have the Z-δexpansion\n1\nk2+ 3q2/4=Zk∗(k)\nk2−2π\u0010\nlnq\nk0∗\u0011\nδ(k)−3\n4q2Zk∗(k)\nk4\n+3π\n8q2\u0010\nlnq\nk0∗\u0011\n∇2\nkδ(k) +O(q4lnjq)\n(82)\nfor some nonnegative integer j, where\nk0∗≡2k∗/√\n3. (83)\nTo understand how to derive the Z-δexpansion, one may\nrefer to an appendix in Ref. [1]. When qis small we also\nhave the Taylor expansion\nδ\u0010\n−q\n2+k\u0011\n+δ\u0010\n−q\n2−k\u0011\n= 2δ(k) +q2\n4(ˆq· ∇k)2δ(k) +O(q4). (84)\nExpanding both sides of Eq. (81) in powers of qand using\nEq. (82) and Eq. (84), we find\nt(−2,0)\nqk=u0(2π)2δ(q), (85a)\nt(0,−2)\nqk=−DZk∗(k)\nk2+\u0010\n2u0+D\n2πlnq\nk0∗\u0011\n(2π)2δ(k),\n(85b)\nt(2,−4)\nqk=π2u0q2(ˆq· ∇k)2δ(k) +3Dq2\n4Zk∗(k)\nk4\n−3πDq2\n8\u0010\nlnq\nk0∗\u0011\n∇2\nkδ(k). (85c)\nEquation (85a) agrees with Eq. (64b).\nComparing Eq. (78a) and Eq. (85b), we get\nA(q) = 2 u0+D\n2πlnq\nk0∗. (86)\nSubstituting the above formula into Eq. (70), we get\nS(0)q\nk=dk+\u0010\n2u0+D\n2πlnq\nk0∗\u0011\nϕk. (87)\nSteps 3: Determination of T(0)and S(2).Ex-\ntracting the terms of order q2in Eq. (52), we get\n(HS(2)q)k=−3q2\n4S(0)q\nk−q2W(2)\nˆqk\n=−3\n4q2A(q)ϕk+q2\u0010\n−3\n4dk−W(2)\nˆqk\u0011\n.(88)\nBefore symbolically solving Eq. (88), we introduce a\nfunction d(2)\nˆqkindependent of qand satisfying the equation\n(Hd(2)\nˆq)k=−3\n4dk−W(2)\nˆqk. (89)\nBecause the right hand side of the above equation con-\ntains only s-wave and d-wave terms [because of Eq. (57)9\nand the fact that dkis rotationally invariant] and is also\ny-reversally invariant, and because the two-body inter-\naction is rotationally and y-reversally invariant, we may\nrequire that d(2)\nˆqkbe of the form\nd(2)\nˆqk=d(2)\n0(k) +d(2)\n2(k) cos(2 arg( ˆq,ˆk)), (90)\nwhere d(2)\n0(k) and d(2)\n2(k) depend only on k=|k|. The\nabove requirement still leaves d(2)\nˆqkwith two remaining\ndegrees of “gauge freedom”, namely\ned(2)\nˆqk=d(2)\nˆqk+ηsϕk+ηdϕ(d)\nˆqk\nstill satisfies Eq. (89) and Eq. (90), for any coefficients\nηsandηd. To completely fix the definition of d(2)\nˆqk, we\nwill need to expand d(2)\nˆqkat small kto the order k−2and\nspecify the coefficients of the terms proportional to δ(k)\nandQ(d)\nˆq(∇k)δ(k).\nAssuming that d(2)\nˆqkhas been defined, we may solve\nEq. (88) and obtain\nS(2)q\nk=q2d(2)\nˆqk−3\n4q2A(q)fk+α0(q)ϕk+α2(q)ϕ(d)\nˆqk\n+∞X\nl=4,6,8,...αl(q)ϕ(l)\nˆqk. (91)\nwhere fkwere defined in Eq. (20) and Eq. (26b), and\nαl(q) is of order q2(including the possibility of some log-\narithmic dependencies on q). Note that the small- kex-\npansion of ϕ(l)\nˆqkstarts with the term that scales like k−l−2\n[see Eq. (32)]. But t(2,−6)\nqk= 0 according to Eq. (59b).\nAlso, t(2,j)\nqk= 0 for j <−6 because T(s)\nq1q2q3= 0 for\ns <−4. Therefore αl(q) = 0 for l= 4,6,8, . . ., and\nS(2)q\nk=q2d(2)\nˆqk−3\n4q2A(q)fk+α0(q)ϕk+α2(q)ϕ(d)\nˆqk.(92)\nSubstituting Eq. (86) and Eq. (92) into Eq. (78b), and\nusing Eqs. (33), we get\nt(0,0)\nqk=ξ0+u0D\n2πlnq\nk0∗+D\u0010u0\nπ+3rs∗\n2\u0011\nlnk\nk0∗,(93)\nwhere\nξ0≡6u2\n0+ 6πu0rs∗−2κ1−3χ1\n2−Z\nk′U0k′d(2)\nˆqk′.(94)\nUsing the rotational invariance of the two-body interac-\ntion and d(2)\nˆqk, one can show thatR\nk′U0k′d(2)\nˆqk′is indepen-\ndent of ˆq.\nExpanding S(2)q\nkat small k, and using the fact that\nt(2,j)\nqk= 0 at j <−4 [note Eq. (59b) and the fact that\nT(−3)\nq1q2q3= 0 and T(s)\nq1q2q3= 0 at s <−4], we see that\nthe leading-order term in the small- kexpansion of d(2)\nˆqkis of order k−4. Writing the leading order term in the\nsmall- kexpansion of d(2)\nˆqkasLˆqk, and comparing the\nleading-order term of the small- kexpansion of S(2)q\nkwith\nEq. (85c), we get\nq2Lˆqk=π2[α2(q) +u0q2](ˆq· ∇k)2δ(k)\n+h3π2\n2u0q2−π2\n2α2(q)i\n∇2\nkδ(k) +3Dq2\n4Zk∗(k)\nk4.\n(95)\nThe above equation indicates that α2(q) must be equal\nto a constant times q2, because Lˆqkis independent of\nq=|q|. We can exploit the “gauge freedom” in the def-\ninition of d(2)\nˆqk: if the small- kexpansion of d(2)\nˆqkcontains\na term proportional to ( ˆq· ∇k)2δ(k), we can add a cer-\ntain coefficient times ϕ(d)\nˆqktod(2)\nˆqk, such that this term\nis canceled. According to this particular choice in the\ndefinition of d(2)\nˆqk, we must have\nα2(q) =−u0q2, (96)\nand thus\nLˆqk=3D\n4Zk∗(k)\nk4+ 2π2u0∇2\nkδ(k). (97)\nExpanding S(2)q\nkat small kto the next order term, we\nget\nt(2,−2)\nqk=q2L′\nˆqk+3πrs∗\n2q2A(q)Zk∗(k)\nk2+α0(q)(2π)2δ(k),\n(98)\nwhere L′\nˆqkis the next-order term in the small- kexpan-\nsion of d(2)\nˆqk.\nExtracting the terms of order q2in Eq. (51), and di-\nviding them by ( q2\n1+q2\n2+q2\n3)/2, we get\nT(0)\nq1q2q3=ξ0+3rs∗D\n2(q2\n1+q2\n2+q2\n3)3X\ni=1q2\nilnqi\nk0∗\n+3X\ni=1h\nu1p2\ni(2π)2δ(qi) +u0D\n2πlnqi\nk0∗i\n.\n(99)\nAt small qwe have the following Z-δexpansions:\n(k−q/2)2ln|k−q/2|+ (k+q/2)2ln|k+q/2|\nk2+ 3q2/4\n= 2 ln k+q2Z(k) cos(2 arg( ˆq,ˆk))\n4k2+q2\n2Zk∗(k)\nk2\n−q2Zk∗(k) lnk\nk2\n+πq2h\nλ−ln2k∗+ lnk∗−\u0010\n1 + ln4\n3\u0011\nlnq+ ln2qi\n×δ(k) +O(q4), (100a)10\nln|k−q/2|+ ln|k+q/2|\n= 2 ln k−q2Z(k) cos(2 arg( ˆq,ˆk))\n4k2+πq2\n4δ(k) +O(q4),\n(100b)\nwhere\nZ(k) cos(2 arg( ˆq,ˆk))\nk2=cos(2 arg( ˆq,ˆk))\nk2,ifk̸= 0,\n(101a)\nZ\nk<ΛZ(k) cos(2 arg( ˆq,ˆk))\nk2d2k≡0 (101b)\nfor any Λ >0,\nZu(k) lnk\nv\nk2=\u001alnk\nv\nk2, k > 0,\n+∞, k= 0,(102a)\nZ\nk<uZu(k) lnk\nv\nk2d2k= 0, (102b)\nuandvare arbitrary positive wave numbers, and\nλ= lim\nT→+∞1\n4+T(1−lnT) +ln2T\n4+ZT\n0tlnt√\nt2+t+ 1dt\n=1\n12h\n9 +π2+ 3\u0010\n−6 + ln4\n3\u0011\nln4\n3+ 6Li 2\u0010\n−1\n3\u0011i\n≈1.007118 . (103)\nLi2(z) =P∞\nj=1zj\nj2is the polylogarithm function.\nExpanding T(0)\nq,−q/2+k,−q/2−kin powers of q, and using\nEqs. (82), (83), (84), (100a), and (100b), we get\nt(−2,2)\nqk=u1k2(2π)2δ(q), (104a)\nt(0,0)\nqk=ξ0+u0D\n2πlnq\nk0∗+D\u0010u0\nπ+3rs∗\n2\u0011\nlnk\nk0∗,\n(104b)\nt(2,−2)\nqk=−3rs∗Dq2\n4Zk∗(k)(−1\n2+ lnk\nq)\nk2\n+\u00103rs∗D\n16−u0D\n8π\u0011\nq2Z(k) cos[2 arg( ˆq,ˆk)]\nk2\n+q2h3rs∗D\n16π\u0010\nλ−lnq\nk∗−ln2q\nk∗\u0011\n+bi\n(2π)2δ(k),\n(104c)where\nb≡2u1+u0D\n32π2. (105)\nComparing Eq. (104c) with Eq. (98), and using the\n“gauge freedom” in the definition of d(2)\nˆqk, we have\nα0(q) =q2h\nb+3rs∗D\n16π\u0010\nλ−lnq\nk∗−ln2q\nk∗\u0011i\n(106)\nand\nL′\nˆqk=Zk∗(k)(−3πrs∗u0−3rs∗D\n4lnk\nk′∗)\nk2\n+\u00103rs∗D\n16−u0D\n8π\u0011Z(k) cos[2 arg( ˆq,ˆk)]\nk2,(107)\nwhere\nk′\n∗= 2re\n3k∗≈1.90378 k∗. (108)\nHere we have fixed the “gauge freedom” in the definition\nofd(2)\nˆqkby requiring that the coefficient of the term con-\ntaining δ(k) is zero, when the subscript of the Zfunction\nin the first term on the right side of Eq. (107) is chosen\nto be k∗.\nCombining Eq. (97) and Eq. (107), we find the small- k\nexpansion of d(2)\nˆqk:\nd(2)\nˆqk=3D\n4Zk∗(k)\nk4+ 2π2u0∇2\nkδ(k)\n+Zk∗(k)(−3πrs∗u0−3rs∗D\n4lnk\nk′∗)\nk2\n+\u00103rs∗D\n16−u0D\n8π\u0011Z(k) cos[2 arg( ˆq,ˆk)]\nk2\n+O(lnjk) (109)\nfor some nonnegative integer j.\nSummary and Fourier transforms. The above\nstep-by-step procedure can be continued, but we stop at\nS(2)andT(0), and sum up all S(s)(−2≤s≤2) or all\nT(s)(−4≤s≤0) we have derived, to get the expansions\nof the three-body wave function in momentum space:\nϕ(3)\nq1q2q3= (2π)4δ(q1)δ(q2)−2D\nq2\n1+q2\n2+q2\n3+ξ0+3rs∗D\n2(q2\n1+q2\n2+q2\n3)3X\ni=1q2\nilnqi\nk0∗\n+3X\ni=1h\n(u0+u1p2\ni)(2π)2δ(qi) +u0D\n2πlnqi\nk0∗i\n+O(q2lnjq), (110a)11\nwhere q1+q2+q3≡0, and\nϕ(3)\nq,−q/2+k,−q/2−k=\u0014\n(2π)2δ(q) + 2u0+D\n2πlnq\nk0∗+α0(q)\u0015\nϕk−3\n4q2\u0010\n2u0+D\n2πlnq\nk0∗\u0011\nfk−u0q2ϕ(d)\nˆqk\n+dk+q2d(2)\nˆqk+O(q4lnj′q), (110b)\nfor some nonnegative integers jandj′, where α0(q) is shown in Eq. (106).\nSubstituting Eq. (110a) into Eq. (2), we find that when the pairwise distances |r1−r2|,|r2−r3|, and|r3−r1|go\nto infinity simultaneously, the wave function in real space has the “111 expansion”\nϕ(3)(r1,r2,r3) = 1−D\n4π2B2−3rs∗D\n4π2B43X\ni=1\u00101\n2−cos 2θiln cos θi\u0011\n+O(B−6lnjB), (111a)\nwhere θi= arctan\u00002yi√\n3xi\u0001\nis the hyperangle, satisfyingP3\ni=1cos(2 θi)≡0.\nSubstituting Eq. (110b) into Eq. (2), we find that when xis fixed, by y→ ∞ , the wave function in real space has\nthe “21 expansion”\nϕ(3)(x/2,−x/2,y) =h\n1−D\n4π2y2+3rs∗D\n8π2y4\u0010\n−3 + 2 lny\nx∗\u0011i\nϕ(x)−3D\n4π2y4f(x) +O(y−6lnj′y). (111b)\nIf there are two-body bound states, the three incoming\nfree bosons with zero total energy may recombine into adimer and a free boson, which fly apart with total kinetic\nenergy equal to the released two-body binding energy. In\nthis case, the 21 expansion in real space is changed to\nϕ(3)(x/2,−x/2,y)) =R+X\nl=0,2,4,···,lmaxνlX\nν=0Clν3X\ni=1φlν(xi,yi), (112)\nwhere Ris the right hand side of Eq. (111b), lmaxis the\nmaximum orbital angular momentum quantum number\nof the two-body bound state, and ( νl+ 1) is the num-\nber of vibrational states for orbital angular momentum\nquantum number l.\nφlν(x,y) =ulν(x)H(1)\nl\u00122√\n3κlνy\u0013\ncos\u0000\nlarg(ˆx,ˆy)\u0001\n,\n(113)\nwhere H(1)\nl(z) is the Hankel function of the first kind,\nwhose asymptotic form at large zis [18]\nH(1)\nl(z) =r\n2\nπzei(z−lπ/2−π/4)\u0002\n1 +O(z−1)\u0003\n,(114)\nandulν(x) is the radial part of the dimer wave func-\ntion for vibrational quantum number νat orbital angular\nmomentum quantum number l, satisfying the two-body\nSchr¨ odinger equation\n(−∇2\nx+κ2\nlν)ulν(x) cos\u0000\nlarg(ˆx,ˆy)\u0001\n+1\n2Z\nd2x′U(x,x′)ulν(x′) cos\u0000\nlarg(ˆx,ˆy)\u0001\n= 0.(115)U(x,x′) is the Fourier transform of Ukk′defined in\nEq. (6):\nU(x,x′) =Zd2k\n(2π)2Zd2k′\n(2π)2Ukk′eik·x−ik′·x′,(116)\nWe impose the following normalization condition for\nulν(x):\nZ∞\n0xu∗\nlν(x)ulν′(x)dx=δνν′\u0000\n1 +δl,0\u0001\nπκ2\nlν, (117)\nwhere δl,0is the Kronecker delta. From the conservation\nof probability, we find that\nIm(D) =−9X\nlν|Clν|2\nκ2\nlν. (118)\nThe parameter Dhas important implications for the\nfour-body physics in 2D. For the four-boson system with-\nout two-body interaction and with three-body resonant\ninteraction in 2D, Nishida proved that there is semisuper\nEfimov effect [19]. The system Nishida studied [19] cor-\nresponds to the case of a divergent parameter Ddefined\nabove.12\nIII. N-BODY ENERGY AND THREE-BODY\nRECOMBINATION RATE\nA. Three bosons in the periodic square\nNow we place the three bosons with interactions spec-\nified in Sec. II into a periodic square of side length L,\nand approximately calculate the ground state energy E3\nat large L, assuming that the two-body interaction does\nnotsupport any bound state. The three-body ground\nstate wave function ψ(r1,r2,r3) is periodic and is ap-\nproximately equal to 1 when the pairwise distances be-\ntween the bosons are all comparable to L. But when\nre≪B≪L,\nψ(r1,r2,r3)≈1−D\n4π2B2(119)\nunless two bosons are within the range of interaction. We\nalso have\n−ℏ2\n2m(∇2\n1+∇2\n2+∇2\n3)ψ(r1,r2,r3) =E3ψ(r1,r2,r3)\n(120)\nwhen the pairwise distances are all larger than re. Since\nψ(r1,r2,r3) is periodic,\nZ\nd2r1d2r2d2r3(∇2\n1+∇2\n2+∇2\n3)ψ(r1,r2,r3) = 0 ,(121)\nwhere the integral over riis carried out over one periodic\nsquare. The above integral may be divided into two do-\nmains: B < B 0andB > B 0, where re≪B0≪L. The\nintegral in the domain B > B 0is\nZ\nB>B 0d2r1d2r2d2r3(∇2\n1+∇2\n2+∇2\n3)ψ(r1,r2,r3)\n≈Z\nB>B 0d2r1d2r2d2r3(−2mE3/ℏ2)ψ(r1,r2,r3)\n≈ −2mE3L6/ℏ2. (122)\nThe integral in the domain B < B 0isR\nB<B 0d2r1d2r2d2r3(∇2\n1+∇2\n2+∇2\n3)ψ(r1,r2,r3), which\nmay be expressed as a surface integral at B=B0, at\nwhich ψsatisfies Eq. (119). This surface integral is\n2DL2. So we have 2 DL2−2mE3L6/ℏ2≈0 and\nE3≈ℏ2D\nmL4. (123)\nWe expect that E3has the following expansion at large\nL:\nE3=ℏ2D\nmL4+O(L−6lnjL) (124)\nfor some integer j.B. Thermodynamic limit\nNow we consider Nbosons with interactions specified\nin Sec. II and take the thermodynamic limit N→ ∞ ,\nL→ ∞ , but the number density ρ=N/L2is fixed. In\nthe absence of two-body bound states, the ground state\nenergy per particle is\nE\nN≈1\nN\u0012N\n3\u0013ℏ2D\nmL4≈ℏ2D\n6mρ2(125)\nif the number density ρis sufficiently small such that the\ninterparticle spacing ρ−1/2is much larger than both the\nrange of interaction reand the length scale |D|1/2.\nC. Three-body recombination rate\nIf the two-body interaction supports two-body bound\nstates, Eq. (125) may still describe the many-body sys-\ntem approximately if the system is in a metastable state\nat zero temperature, but now Dusually has a negative\nimaginary part shown in Eq. (118). Within a short time\nduration t, the probability that no recombination occurs\nis exp( −2|Im(E)|t/ℏ)≈1−2|Im(E)|t/ℏ, and so the prob-\nability of one recombination is 2 |Im(E)|t/ℏ. Each recom-\nbination results in the loss of three low-energy bosons, so\ndρ\ndt=−L3ρ3, (126)\nwhere ρis the number density of the remaining low-\nenergy bosons, and\nL3=ℏ|Im(D)|\nm(127)\nis the three-body recombination rate constant.\nIV. 2D BOSE GAS WITH NEGATIVE DIN A\nHARMONIC TRAP\nConsider a 2D Bose gas with infinite or zero 2D scatter-\ning length in some external potential V(r). At zero tem-\nperature the gas is approximately described by a macro-\nscopic wave function Ψ( r) satisfying\nZ\n|Ψ(r)|2d2r=N. (128)\n|Ψ(r)|2is the local number density of the bosons. The\nenergy functional of the gas is approximately\nE=Z\nd2r\u0014ℏ2\n2m|∇Ψ|2+V(r)|Ψ|2+ℏ2D\n6m|Ψ|6\u0015\n.(129)\nThe last term on the right hand side of Eq. (129) is a\nnatural generalization of Eq. (125). This energy func-\ntional is analogous to the one shown in Ref. [16], but we13\nare now considering a system with attractive three-body\nforce ( D < 0) and nearly no two-body force ( a=∞or\n0), in 2D. The system we are studying here is also rem-\niniscent of the one studied in Ref. [13], in which there\nis no three-body force but the two-body force may be\nattractive.\nMinimizing the energy functional with respect to Ψ( r)\nand Ψ∗(r) under the constraint Eq. (128), we derive a\nGP equation\n−ℏ2\n2m∇2Ψ(r) +Vext(r)Ψ(r) +ℏ2D\n2m|Ψ(r)|4Ψ(r) =µΨ(r),\n(130)\nwhere µis the chemical potential and also the Lagrange\nmultiplier for imposing the constraint Eq. (128) when\nminimizing the energy functional. To our knowledge, a\nGP equation containing the three-body coupling term\nwas first considered in Refs. [14–16].\nNow consider an isotropic harmonic trap\nV(r) =1\n2mω2r2, (131)\nwhere ω >0 is the angular frequency of the trap. If Ψ( r)\nis also isotropic and real, such that\nΨ(r) =\u0010mω\nℏ|D|\u00111/4eΨ(er), (132)\nwhere\ner=rmω\nℏr, (133)\nandeΨ(er) is a dimensionless function, the GP equation\nmay be rewritten in the dimensionless form\n−1\nerd\ndererd\ndereΨ(er) +er2eΨ(er) +ceΨ5(er)−2eµeΨ(er) = 0 ,(134)\nwhere\neµ=µ\nℏω(135)\nandc= sgn( D). IfD > 0,c= 1. If D < 0,c=−1.\nIfer→ ∞ , we have the asymptotic formula [13]\neΨ(er) =Cereµ−1\u0002\n1 +O(er−2)\u0003\ne−er2/2. (136a)\nIfer→0,\neΨ(er) =eΨ0−(2eµeΨ0−ceΨ5\n0)\n4er2+O(er4). (136b)\nFor each value of eµ, we numerically solve Eq. (134), using\nthe conditions in Eqs. (136), to find the function eΨ(er),\nfrom which we evaluate the dimensionless number\nn≡Z∞\n0|eΨ(er)|2erder=N\n2πr\nmω|D|\nℏ. (137)We have done such calculations for D < 0 only. Our nu-\nmerical results are shown in Fig. 1. For a fixed trap angu-\nlar frequency ω, as we gradually increase the dimension-\nless number nfrom zero, the chemical potential decreases\ngradually from ℏω(the one-body zero-point energy in the\n2D harmonic trap), until nreaches its maximum value\nncr= 0.579537 (138)\nandµreaches its critical value\nµcr= 0.4283 ℏω. (139)\nThen-eµcurve for 0 < n < n cris shown in Fig. 1; note\nthatµ > µ cr, ie only the segment of the curve to the right\nof the dashed line in the figure is physically realizable.\nFurther increasing nbeyond ncrwould cause the system\nto collapse due to a negative D. The maximum number\nof bosons that can be stably confined in such a trap at\nzero temperature, for D < 0, is\nNcr= 2πncrs\nℏ\nmω|D|≈3.6413Lhop\n|D|, (140)\nwhere Lho≡p\nℏ/mω. Equation (140) is a generalization\nof those results for the Bose gases with two-body attrac-\ntive force in 3D [12] or in 2D [13]; it is also reminiscent\nof those results for the Bose gases with both two-body\nattractive force and some nonnegative three-body cou-\nplings in 3D [15, 20].\nV. SUMMARY AND DISCUSSION\nWe have derived the asymptotic expansions of the wave\nfunction of three identical bosons in 2D with two-body\n2D scattering length a=∞or 0 , and defined a three-\nbody parameter Dwhose dimension is length squared.\nWe may call Dthethree-body scattering area . When\nthere are two-body bound states, Dtypically acquires a\nnegative imaginary part related to the probability flux\nfor the production of a bound pair and a free boson.\nThe three-body scattering area Dis a fundamental pa-\nrameter for the low-energy effective interaction, and it\nstrongly affects N-body physics for N= 3,4,5,···.\nWe have evaluated the energy of three such bosons in\na large periodic square in terms of Dand the side length\nof the square, and generalized the result to the dilute 2D\nBose gas in the thermodynamic limit. We then derived a\nformula for the three-body recombination rate constant\nin terms of the imaginary part of D.\nForD < 0, we have studied the system of Nsuch\nbosons in an isotropic harmonic trap satisfying the con-\ndition Lho≫p\n|D|, at zero temperature, using the GP\nequation, and approximately calculated the maximum\nvalue of Nfor the system to be metastable.14\nFIG. 1. The number of bosons Nv.s. the chemical poten-\ntialµfor a 2D Bose gas with a=∞or 0 and D < 0 con-\nfined in an isotropic harmonic trap with angular frequency\nω, at zero temperature. In the graph, Lho≡p\nℏ/mω. The\nvertical dashed line represents the critical value of µ/ℏω[see\nEq. (139)] at which the number Nreaches its maximum value\n[see Eq. (140)]. The segment of the solid-line curve on the\nleft side of the vertical dashed line is not physically realizable\nbecause the 2D gas becomes energetically unstable on that\nside. We have verified this instability by expanding the en-\nergy functional in Eq. (129) around the numerical solution for\nΨ(r) [see Eq. (130) or Eq. (134)], to second order in the tiny\ndeviations from the solution, and found that the solution is\na saddle point but nota local minimum of the energy func-\ntional if µ < µ cr. In Refs. [15, 20] it is shown that the N-µ\ncurve for the 3D Bose gas with attractive two-body force and\nsome nonnegative three-body coupling has a similar branch\nwith instability. We have also verified that if ℏω > µ > µ cr\nthen the solution for Ψ( r) [see Eq. (130) or Eq. (134)] is a\nlocal minimum of the energy functional.\nACKNOWLEDGMENTS\nThis work was supported by the National Key R&D\nProgram of China (Grants No. 2019YFA0308403 and\nNo. 2021YFA1400902). We thank Jiansen Zhang for\ndiscussions.Appendix A: Some mathematical properties of the Z\nfunctions defined in this paper\nOne can easily show that the Zfunctions defined in\nEqs. (30) satisfy the following identities:\nZk1(k)\nk2−Zk2(k)\nk2= 2π\u0010\nlnk2\nk1\u0011\nδ(k), (A1a)\nZk1(k)\nk4−Zk2(k)\nk4=π\n2\u0010\nlnk2\nk1\u0011\n∇2\nkδ(k), (A1b)\nfor any two positive constants k1andk2. The Fourier\ntransforms of the Zfunctions are ordinary functions:\nZd2k\n(2π)2Zk∗(k)\nk2eik·x=−1\n2πlnx\nx∗, (A2a)\nZd2k\n(2π)2Zk∗(k)\nk4eik·x=x2\n8π\u0010\n−1 + lnx\nx∗\u0011\n,(A2b)\nwhere k∗is related to x∗in Eq. (27). One can also verify\nthe inverse Fourier transformations:\nZ\u0010\n−1\n2πlnx\nx∗\u0011\ne−ik·xd2x=Zk∗(k)\nk2, (A3a)\nZx2\n8π\u0010\n−1 + lnx\nx∗\u0011\ne−ik·xd2x=Zk∗(k)\nk4, (A3b)\nOne can also easily show that the Zfunction defined\nin Eqs. (102) satisfies\nZk1(k) lnk\nv\nk2−Zk2(k) lnk\nv\nk2=π\u0010\nlnk2\nk1\u0011\u0010\nlnk1k2\nv2\u0011\nδ(k),\n(A4)\nfor any three positive constants k1,k2, and v. We can\nalso calculate the Fourier transform of the Zfunction\ndefined in Eqs. (102):\nZd2k\n(2π)2Zu(k) lnk\nv\nk2eik·x=1\n4π\u0002\nln2(eγvx/2)−ln2(v/u)\u0003\n.\n(A5)\nOne can verify the inverse transformation:\nZ1\n4π\u0002\nln2(eγvx/2)−ln2(v/u)\u0003\ne−ik·xd2x=Zu(k) lnk\nv\nk2.\n(A6)\n[1] S. Tan, Three-boson problem at low energy and implica-\ntions for dilute bose-einstein condensates, Physical Re-\nview A 78, 013636 (2008).\n[2] S. Zhu and S. Tan, Three-body scattering hypervolumes\nof particles with short-range interactions, arXiv preprint\narXiv:1710.04147 (2017).\n[3] W. Zwerger, Quantum-unbinding near a zero tempera-\nture liquid–gas transition, Journal of Statistical Mechan-\nics: Theory and Experiment 2019 , 103104 (2019).\n[4] P. Mestrom, V. Colussi, T. Secker, and S. Kokkelmans,Scattering hypervolume for ultracold bosons from weak\nto strong interactions, Physical Review A 100, 050702\n(2019).\n[5] P. Mestrom, V. Colussi, T. Secker, G. Groeneveld,\nand S. Kokkelmans, Van der waals universality near a\nquantum tricritical point, Physical Review Letters 124,\n143401 (2020).\n[6] Z. Wang and S. Tan, Three-body scattering hypervolume\nof particles with unequal masses, Physical Review A 103,\n063315 (2021).15\n[7] H. Hu, Z.-Q. Yu, J. Wang, and X.-J. Liu, First-order\nbose-einstein condensation with three-body interacting\nbosons, Physical Review A 104, 043301 (2021).\n[8] Z. Wang and S. Tan, Scattering hypervolume of spin-\npolarized fermions, Physical Review A 104, 043319\n(2021).\n[9] Z. Wang and S. Tan, Scattering hypervolume of fermions\nin two dimensions, Physical Review A 106, 023310\n(2022).\n[10] Z. Wang and S. Tan, Three-body scattering hypervolume\nof identical fermions in one dimension, Physical Review\nA108, 033306 (2023).\n[11] O. Kartavtsev and A. Malykh, Universal low-energy\nproperties of three two-dimensional bosons, Physical Re-\nview A 74, 042506 (2006).\n[12] P. Ruprecht, M. Holland, K. Burnett, and M. Edwards,\nTime-dependent solution of the nonlinear schr¨ odinger\nequation for bose-condensed trapped neutral atoms,\nPhysical Review A 51, 4704 (1995).\n[13] S. K. Adhikari, Numerical solution of the two-\ndimensional gross–pitaevskii equation for trapped inter-\nacting atoms, Physics Letters A 265, 91 (2000).\n[14] A. Gammal, T. Frederico, and L. Tomio, Trapped bose-einstein condensed gas with two and three-atom interac-\ntions, arXiv preprint cond-mat/9902188 (1999).\n[15] A. Gammal, T. Frederico, and L. Tomio, Improved\nnumerical approach for the time-independent gross-\npitaevskii nonlinear schr¨ odinger equation, Physical Re-\nview E 60, 2421 (1999).\n[16] A. Gammal, T. Frederico, L. Tomio, and P. Chomaz,\nAtomic bose-einstein condensation with three-body in-\nteractions and collective excitations, Journal of Physics\nB: Atomic, Molecular and Optical Physics 33, 4053\n(2000).\n[17] H.-W. Hammer and D. Lee, Causality and the effective\nrange expansion, Annals of Physics 325, 2212 (2010).\n[18] P. Morse and H. Feshbach, Methods of mathematical\nphysics, vol. 1 (1953).\n[19] Y. Nishida, Semisuper efimov effect of two-dimensional\nbosons at a three-body resonance, Physical Review Let-\nters118, 230601 (2017).\n[20] N. Akhmediev, M. P. Das, and A. Vagov, Bose-einstein\ncondensation of atoms with attractive interaction, Inter-\nnational Journal of Modern Physics B 13, 625 (1999)."    },    {        "title": "2402.02237v1.Wall_wake_laws_for_the_mean_velocity_and_the_turbulence.pdf",        "content": "1\nWall-wake laws for the mean velocity and\nthe turbulence\nAlexander J. Smits\nDepartment of Mechanical and Aerospace Engineering, Princeton University,\nPrinceton, NJ, USA\n(6 February 2024)\nA new wall-wake law is proposed for the streamwise turbulence in the outer region of a\nturbulent boundary layer. The formulation pairs the logarithmic part of the profile (with\na slope A1and additive constant B1) to an outer linear part, and it accurately describes\nover 95% of the boundary layer profile at high Reynolds numbers. Once the slope A1is\nfixed, B1is the only free parameter determining the fit. Most importantly, B1is shown to\nbe proportional to the wake factor in the wall-wake law for the mean velocity, revealing\na previously unsuspected connection between the turbulence and the mean flow.\n1. Introduction\nThe mean velocity distribution in the outer part of a turbulent boundary layer is often\nexpressed as a combination of a logarithmic part and a wake component, as in\nU\nuτ=1\nκlnyuτ\nν+B+2Π\nκW\u0010y\nδ\u0011\n, (1.1)\nwhere Uis the mean velocity in the streamwise direction, uτ=p\nτw/ρ,τwis the shear\nstress at the wall where y= 0, ρis the fluid density, κis von K´ arm´ an’s constant, Bis\nthe additive constant, and Π is the Reynolds number dependent wake factor. The wake\nfunction Wis taken to be a universal function of y/δ, where δis the outer layer length\nscale. This wall-wake model was first formulated by Coles (1956), and it is an essential\npart of the widely-used composite profile derived by Chauhan et al. (2007).\nMarusic et al. (1997) proposed a similar formulation for the streamwise turbulence\nintensity in zero pressure gradient turbulent boundary layers, given by\nu2+=B1−A1lny\nδm−Vg\u0012\ny+,y\nδm\u0013\n−Wg\u0012y\nδm\u0013\n, (1.2)\nwhere u2+=u2/u2\nτ/. The model incorporates the log-law for u2+with constants A1\nandB1(Hultmark et al. 2012; Marusic et al. 2013), where Vgis a mixed scale viscous\ndeviation term, Wgis the wake deviation term, and δmis a boundary layer thickness\ndefined in a way that is similar to the Rotta-Clauser thickness. It is typically 15 to 20%\nlarger than δ99, the 99% thickness (further details are given in the Appendix). The wake\ndeviation was given by\nWg=B1η2(3−2η)−A1η2(1−η)(1−2η), (1.3)\nwhere η=y/δm.\nPirozzoli & Smits (2023) recently proposed an alternative model for the mean velocity\ndistribution in the outer layer, given by a compound logarithmic-parabolic distributionarXiv:2402.02237v1  [physics.flu-dyn]  3 Feb 20242 A. J. Smits\nof the type first suggested by Hama (1954). That is,\nUe−U\nuτ=B−1\nk0lny\nδ0, (1.4)\nUe−U\nuτ=C\u0012\n1−y\nδ0\u00132\n, (1.5)\nwhere Cis a constant, and Ueis the freestream velocity. Requiring the two velocity\ndistributions to smoothly connect up to the first derivative yields the position of the\nmatching point ( η0=y0/δ0) and the additive constant Bin (1.4) as a function of k0and\nC,\nη0=1\n2 \n1−\u0012\n1−2\nCk0\u00131/2!\n, B =C(1−η0)2+1\nk0logη0. (1.6)\nIn what they called the classical case, Pirozzoli & Smits (2023) found that with δ0=\n1.6δ95the best fit of the data was obtained with k0=κ≈0.38,C≈9.88, so that\nB= 2.15, with the two distributions smoothly matched at η0= 0.158. This compound\nlogarithmic-parabolic distribution fits the velocity distributions well down to y/δ 0≈0.01\n(for Reynolds numbers based on displacement thickness greater than 2000).\nHere, we suggest a similar approach for the streamwise component of the turbulent\nstress. That is, we propose a compound representation given by\nu2+=B1−A1lny\nδ1, (1.7)\nu2+=b1−a1y\nδ1, (1.8)\nwhere a1, and b1are constants, and δ1is the appropriate length scale for the outer layer.\nIn this formulation, there is no viscous deviation term. Requiring the two turbulence\ndistributions to smoothly connect up to the first derivative yields the position of the\nmatching point ( η1=y1/δ1) and the additive constant B1in (1.7) as a function of the\nother constants,\nη1=A1\na1, B 1=b1−A1(1−lnη1). (1.9)\nAccording to (1.8), u2+is zero when y/δ 1=b1/a1, and so we impose one further con-\nstraint and set b1/a1= 1.05 (which closely corresponds to the point where y/δ 995= 1).\nFinally, if we assume A1is a true constant (= 1 .26 for boundary layers according to\nMarusic et al. (2013)), the only free parameter in our fit to the turbulence profile is B1,\nwhich we will show to be Reynolds number dependent and proportional to the mean\nvelocity wake factor Π.\n2. Comparisons with data\nWe now demonstrate the quality of the model by comparing it with experimental\nand DNS data over a wide range of Reynolds numbers (see table 1). Before proceeding,\nwe need to specify the particular length scale δ1used to describe the outer layer. We\nhave chosen δ1=δ99, for reasons made clear in the Appendix. We also need to relate\nReθ=θUe/ν, where θis the momentum thickness, to the friction Reynolds number\nReτ=δ99uτ/ν, in that not all data sets specify both. This issue is also addressed in the\nAppendix.\nTo begin the analysis, we use the data by Samie et al. (2018) (6250 < Re θ<47,100).Wall-wake laws for the mean velocity and the turbulence 3\nReθReτ b1 B1 η1 Symbol\nDeGraaff & Eaton (2000) 1430 541 3.56 1.05 0.372 •\n2900 993 3.77 1.19 0.351 ■\n5200 1692 4.33 1.57 0.306 ▲\n13000 4336 4.83 1.94 0.274 ♦\n31000 10023 4.67 1.82 0.276 ◦\nFernholz et al. (1995) 2573 866 4.10 1.42 0.323 •\n5023 1692 4.34 1.59 0.305 ■\n7140 2375 4.80 1.92 0.276 ▲\n16080 5068 5.30 2.29 0.250 ♦\n20920 6824 4.75 1.88 0.279 ◦\n41300 12633 4.75 1.88 0.279 □\n57720 18692 4.95 2.03 0.267 △\n60810 18362 4.95 2.03 0.267 ⋄\nOsaka et al. (1998) 6040 1800 4.75 1.88 0.279 •\nVallikivi et al. (2015) 8402 2622 4.90 1.99 0.270 •\n15121 4635 5.00 2.07 0.265 ■\n26884 8261 4.85 1.95 0.273 ▲\n46732 14717 4.80 1.92 0.276 ♦\n80579 25062 3.82 1.22 0.346 ◦\n133040 40053 3.57 1.06 0.371 □\n234670 68392 3.23 0.845 0.410 △\nSamie et al. (2018) 6252 1929 4.92 2.00 0.269 •\n12913 3984 4.96 2.04 0.267 ■\n26034 8032 5.10 2.14 0.259 ▲\n47096 14530 4.88 1.98 0.271 ♦\nSillero et al. (2013) 6000 1848 4.530 1.72 0.292 - - - -\nTable 1. Data sources and fitting parameters for A1= 1.96.B1is the only free parameter,\nandb1and the matching point η1are defined by (1.9).\nFigure 1 demonstrates that, as expected from previous work, the log part with A1= 1.26\nandB1= 2.00 is a good fit in the overlap region. In addition, the linear part of the model\ndescribes the profile beyond the matching point very well over this range of Reynolds\nnumbers, except for the region y/δ 99>1 where a more gradual decline is observed. At\nthe highest Reynolds numbers, the compound formulation represents the data well for\n95% of the profile.\nFor reference, we also plot (1.2) for the same values of A1andB1(using δ99/δm= 0.81).\nWe neglected the viscous deviation term Vg, which leads to a positive offset of (1.2)\nwith respect to the compound in the logarithmic region. Both fits work well beyond the\nlogarithmic region, although it could be argued that the linear fit is a trifle more accurate\nfor 0.2< y/δ 99<0.9. In our analysis going forward, we will use compound fit, primarily\nbecause the viscous deviation term in (1.2) appears to obscure some of the underlying\ntrends as well as the comparisons between the turbulence and mean velocity profiles.\nWe now consider all the high Reynolds number data listed in table 1 over the range\n6000≤Reθ<60,000 (the Vallikivi et al. profiles for Reθ>60,000 will be dealt with\nseparately). The results are shown in figure 2 for B1= 2.00. Although there is some\nthe scatter in the data, the compound fit works reasonably well using this value. The4 A. J. Smits\n(a) (b)\nFigure 1. Comparison to the experimental data of Samie et al. (2018) for Reθ= 6252–47096\n(y+>100,A1= 1.26,B1= 2.00). (a) Linear scaling. (b) Log scaling. ······ , (1.2) (neglecting\nVg);——— , (1.7); ——— , (1.8) (matched at η1= 0.269, vertical dashed line). Symbols as in\ntable 1.\n(a) (b)\nFigure 2. Comparison to all the experimental data for 6000 ≤Reθ<60,000 ( y+>100). Sym-\nbols as in table 1. (a) Linear scaling. (b) Log scaling. ——— , (1.8); ——— , (1.7). B1= 2.00,\ndistributions matched at η1= 0.269 (vertical dashed line).\nagreement can be improved by using values of B1optimized for each profile, as listed in\ntable 1.\nThe low Reynolds number data ( Reθ≤6040) are shown in figure 3. The compound fit\nis plotted for two cases, B1= 2.00 (the value used for the high Reynolds number data\nshown in figures 1 and 2), and B1= 1.05 (chosen to match the lowest Reynolds number\nprofile in the data set). In order to match the in-between Reynolds number cases, B1was\nvaried as given in table 1. Again, we see a very satisfactory fit to the data, even in this\nlow Reynolds number range.\nWe now consider the highest Reynolds number data, where Reθ>60,000. There are\nonly three profiles available, at Reθ= 80,579, 133 ,040, and 234 ,670 (Vallikivi et al. 2015).\nThey are unique, in the sense that no other turbulence data exist at comparable ReynoldsWall-wake laws for the mean velocity and the turbulence 5\n(a) (b)\nFigure 3. Comparison to the experimental data for Reθ≤6040 ( y+>100). Symbols as in\ntable 1. (a) Linear scaling. (b) Log scaling. ——— , (1.8); ——— , (1.7). b1= 4.92, distributions\nmatched at η1= 0.269 (vertical dashed line). - - - - - , (1.8); - - - - - , (1.7). b1= 3.56,\ndistributions matched at η1= 0.372 (vertical dashed-dotted line).\n(a) (b)\nFigure 4. Comparison to the experimental data for Reθ>60,000 ( y+>100). Symbols as in\ntable 1. (a) Linear scaling. (b) Log scaling. ——— , (1.8); ——— , (1.7). b1= 4.92, distributions\nmatched at η1= 0.269 (vertical dashed line). - - - - - , (1.8); - - - - - , (1.7). b1= 3.23,\ndistributions matched at η1= 0.410 (vertical dashed-dotted line).\nnumbers. The compound fit is shown in figure 4. Somewhat surprisingly, B1begins to\ndecrease in a systematic way below its ‘high’ Reynolds number value of 2.00 (see table 1).\nThis result may indicate a problem with the data, but it may also reflect the apparent\nconnection between B1and Π.\nThe constant B1acts as a wake function for the turbulence profile, similar to the wake\nfunction Π for the mean velocity profile. In figure 5, we compare the Reynolds number\ndependence of B1with that of 2Π /κ(using κ= 0.384;B1was scaled by an arbitrary\nfactor of 1.15 to aid the comparison). We see a clear similarity between the two wake\nfunctions for Reθ<60,000, and they appear to be proportional to each other by a\nconstant factor.\nForReθ>60,000, the B1values decrease with Reynolds number, as noted above.\nHowever, they still tend to follow the behavior of the corresponding Π values. Although6 A. J. Smits\nFigure 5. Wake factors versus Reθ.■, 1.15 B1(Vallikivi et al. 2015 values shown as □);\n——– , 2Π/κ(Chauhan et al. 2007); ▲, 2Π/κ(Vallikivi et al. 2015).\nthe Π values at lower Reynolds numbers for the Vallikivi data set lie above the Chauhan\ncorrelation (which is similar to the Coles (1956) distribution, not shown here), they\nconsistently decrease with Reynolds number before becoming approximately constant,\nwhich is also the trend followed by B1. This observation is not conclusive, but it helps\nto reinforce the notion that the two wake functions are linked.\n3. Conclusions and discussion\nThe log-linear compound fit in y/δ 99proposed here for the streamwise turbulent stress\nu2+in the outer layer of a turbulent boundary layer works well over a wide range of\nReynolds numbers. For the log part of the fit we assumed that A1, the slope of the log-\nlaw, is fixed at 1.26 (as given by Marusic et al. (2013)), and the linear part of the fit was\nconstrained to pass through zero at y/δ 99= 1.05. As a consequence, the fit has only one\nfree parameter, B1, which acts like a wake factor.\nFor low Reynolds numbers ( Reθ≤6040), B1increases with increasing Reynolds num-\nber, attaining an approximately constant vaue of about 2 for higher Reynolds numbers\n(6000 ≤Reθ≤60,000). At very high Reynolds numbers ( Reθ>60,000), B1begins to\ndecrease with increasing Reynolds number, although this observation is based on very\nlimited data.\nThe behavior of B1closely follows the variation of the mean flow wake factor Π, sug-\ngesting that the mean flow and the turbulence in the outer layer may be linked by a\ncommon mechanism related to the structure of turbulence. This link is clear for the log-\npart of the formulation, in that the mean velocity and the turbulence distributions are\nconnected through the attached eddy hypothesis (Perry & Chong 1982; Marusic & Monty\n2019). For the parabolic part of the mean velocity (1.5) and the linear part of the tur-\nbulence (1.8), the link is still unknown. One possibility is to consider the behavior of the\n“detached” (or Type B) eddies (Perry & Marusic 1995). However, as they note, building\nthis connection “would be very complicated and would depend on the assumed shape\nof the representative eddies.” Also, as Hu et al. (2020) point out, “Unlike the attached\neddies, whose statistical behaviours are well described by the (attached eddy hypothesis),\nthe detached eddies lack a good phenomenological model.” In their statistically-based in-\nterpretation, the major contributions to the streamwise turbulence stress for y/δ > 0.25\nare equally divided between the detached eddies and the Kolmogorov-scale eddies. Un-Wall-wake laws for the mean velocity and the turbulence 7\nFigure 6. Boundary layer thickness variations with Reθ, as found using the composite profile\n(Chauhan et al. 2007).\nderstanding the physics that connects the mean velocity and the turbulence in the outer\nlayer is clearly in need of further work.\nAcknowledgements\nThe authors would like to thank Sergio Pirozzoli and Jean-Paul Dussauge for their com-\nments on an earlier draft.\nDeclaration of interests. The author reports no conflict of interest.\nAppendix: Data analysis\nFor the length scale used to describe the outer layer, δ1, there are a multitude of\nchoices. In examining the mean flow, Pirozzoli & Smits (2023) considered δ0= 1.6δ95,\n0.28∆ (where ∆ = ( Ue/uτ)δ∗is the Rotta-Clauser thickness), and δN= (H/(H−1))δ∗.\nFor the data in table 1, Sillero et al. (2013), DeGraaff & Eaton (2000) and Vallikivi et al.\n(2015) used δ99, Osaka et al. (1998) used δ995, and Samie et al. (2018) used δc, where δc\nis the outer length scale adopted by Chauhan et al. (2007) for their composite profile.\nIn order to compare data, we need a common standard, and we will show that δ1=δ99\nserves that purpose well. To convert δ995andδcto the matching value of δ99, we used\nthe composite profile. In figure 6, we show how these various thicknesses compare.\nWe used the results of Klebanoff (1955) on the eddy viscosity. His boundary layer\nthickness was about 1.15 times larger than δ99(Smits 2024), and the data were scaled\naccordingly.\nIn addition, we need to relate ReθandReτ, in that not all data sets specify both.\nHere, we use\nReθ= 3.241Reτ, (3.1)\nbased on a fit to the available data ( R2= 0.9997 for full data set). See figure 7.\nFinally, the Reθ= 234670 profile by Vallikivi et al. (2015) was corrected for an error\nin the 99% thickness, which was smaller by a factor of 0.943 than the value originally\nreported. This changed the profile, and the corresponding value of Reτ. Also, the Fernholz\nprofile at Reθ= 21410 was not used since it has some obvious problems.8 A. J. Smits\n(a) (b)\nFigure 7. Momentum thickness Reynolds number versus friction Reynolds number. Dashed\nline, (3.1). (a) Full data set; (b) data for Reθ<10000.\nREFERENCES\nChauhan, K., Nagib, H. & Monkewitz, P. 2007 On the composite logarithmic profile in zero\npressure gradient turbulent boundary layers. AIAA Paper 2007-0532 .\nColes, D. E. 1956 The law of the wake in the turbulent boundary layer. J. Fluid Mech. 1,\n191–226.\nDeGraaff, D. B. & Eaton, J. K. 2000 Reynolds-number scaling of the flat-plate turbulent\nboundary layer. J. Fluid Mech. 422, 319–346.\nFernholz, H.H., Krause, E., Nockemann, N. & Schober, M. 1995 Comparative measure-\nments in the canonical boundary layer at Reδ2≤6x104on the wall of the German-Dutch\nwindtunnel. Phys. Fluids 7, 1275–1281.\nHama, F. R. 1954 Boundary layer characteristics for smooth and rough surfaces. Trans. Soc.\nNaval Archit. Mar. Engrs. 62, 333–358.\nHu, R., Yang, X. I. A. & Zheng, X. 2020 Wall-attached and wall-detached eddies in wall-\nbounded turbulent flows. J. Fluid Mech. 885, A30.\nHultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. 2012 Turbulent pipe flow at\nextreme Reynolds numbers. Phys. Rev. Lett. 108(9), 1–5.\nKlebanoff, P. S. 1955 Characteristics of turbulence in a boundary layer with zero pressure\ngradient. NACA Report 1247 .\nMarusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid\nMech. 51, 49–74.\nMarusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region\nin wall turbulence. J. Fluid Mech. 716, R3.\nMarusic, I., Uddin, M. & Perry, A. E. 1997 Similarity law for the streamwise turbulence\nintensity in zero-pressure-gradient turbulent boundary layers. Phys. Fluids 12, 3718–3726.\nOsaka, H., Kameda, T. & Mochizuki, S. 1998 Re-examination of the Reynolds-number-effect\non the mean flow quantities in a smooth wall turbulent boundary layer. JSME Int. J., Ser.\nB41(1), 123–129.\nPerry, A. E. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech.\n119, 173–217.\nPerry, A. E. & Marusic, I. 1995 A wall-wake model for the turbulence structure of boundary\nlayers. Part 1. Extension of the attached eddy hypothesis. J. Fluid Mech. 298, 361–388.\nPirozzoli, S. & Smits, A. J. 2023 Outer-layer universality of the mean velocity profile in\nturbulent wall-bounded flows. Phys. Rev. Fluids 8(6), 064607.\nSamie, M., Marusic, I., Hutchins, N., Fu, M. K., Fan, Y., Hultmark, M. & Smits, A. J.\n2018 Fully resolved measurements of turbulent boundary layer flows up to Reτ= 20,000.\nJ. Fluid Mech. 851, 391–415.Wall-wake laws for the mean velocity and the turbulence 9\nSillero, J. A., Jim ´enez, J. & Moser, R. D. 2013 One-point statistics for turbulent wall-\nbounded flows at Reynolds numbers up to δ+≈2000. Phys. Fluids 25(10).\nSmits, A. J. 2024 Assessing Klebanoff’s data. Under review .\nVallikivi, M., Hultmark, M. & Smits, A. J. 2015 Turbulent boundary layer statistics at\nvery high Reynolds number. J. Fluid Mech. 779, 371–389."    },    {        "title": "2402.02247v1.Novel_approaches_for_the_reliable_and_efficient_numerical_evaluation_of_the_Landau_operator.pdf",        "content": "Novel approaches for the reliable and efficient\nnumerical evaluation of the Landau operator\nJosé Antonio Carrillo, Mechthild Thalhammer\nFebruary 6, 2024\nWhen applying Hamiltonian operator splitting methods for the time integration of multi-\nspecies Vlasov–Maxwell–Landau systems, the reliable and efficient numerical approxima-\ntion of the Landau equation represents a fundamental component of the entire algorithm.\nSubstantial computational issues arise from the treatment of the physically most relevant\nthree-dimensional case with Coulomb interaction. This work is concerned with the intro-\nduction and numerical comparison of novel approaches for the evaluation of the Landau\ncollision operator. In the spirit of collocation, common tools are the identification of fun-\ndamental integrals, series expansions of the integral kernel and the density function on\nthe main part of the velocity domain, and interpolation as well as quadrature approx-\nimation nearby the singularity of the kernel. Focusing on the favourable choice of the\nFourier spectral method, their practical implementation uses the reduction to basic in-\ntegrals, fast Fourier techniques, and summations along certain directions. Moreover, an\nimportant observation is that a significant percentage of the overall computational effort\ncan be transferred to precomputations which are independent of the density function. For\nthe purpose of exposition and numerical validation, the cases of constant, regular, and\nsingular integral kernels are distinguished, and the procedure is adapted accordingly to the\nincreasing complexity of the problem. With regard to the time integration of the Landau\nequation, the most expedient approach is applied in such a manner that the conservation\nof mass is ensured.\n1 Introduction\nScope of applications. The present work is inspired by various contributions on the\nnumerical simulation of kinetic equations modelling the distribution of charged particles\nin a collisional plasma, see in particular [1, 2, 3, 6, 7, 8, 9, 11, 14, 15, 17, 18, 19, 20, 22, 23]\nand the references given therein. We intend to lay the foundation for the application of\nHamiltonian operator splitting methods to Vlasov–Maxwell–Landau and Vlasov–Poisson–\nLandau equations, where the efficient time integration of the Landau equation via spectral\nmethods represents a fundamental component of the entire algorithm.\nVlasov–Landau equation. Solving the inhomogeneous Vlasov–Landau equation [13] is\nstill one of the most computationally costly kinetic problems in the field and of paramount\nimportance in plasma physics. With the variables x∈Ω(x)⊆Rd,v∈Ω(v)⊆Rd, and\nt∈[t0, T]⊂Rrepresenting position, velocity, and time, the functions f: Ω(x)×Ω(v)×\n1arXiv:2402.02247v1  [math.NA]  3 Feb 2024[t0, T]→RandF: Ω(x)×Ω(v)×[t0, T]→Rdescribing the distribution of charged particles\nand a given or self-consistent force field including electromagnetic effects, and the Landau\noperator Q(f, f)capturing collisions between particles, the Vlasov–Landau equation reads\nas\n∂tf+v· ∇xf−F· ∇vf=Q(f, f).\nThroughout, for notational simplicity, we neglect dependences on variables when appro-\npriate and no confusion arises.\nLandau operator. Our main concern is the efficient numerical evaluation of the Landau\ncollision operator\nQ(f, f) =∇v· Q(c)(f, f), (1a)\nwhich is given by the divergence of the integral operator\nQ(c)(f, f)(v) =CZ\nΩ(v)A(v−w)\u0000\nf(w)∇vf(v)− ∇ wf(w)f(v)\u0001\ndw\n=:I(f)(v)∇vf(v) +J(f)(v)f(v), v∈Ω(v).(1b)\nHere, we set A(z) =|z|β(|z|2Id−z⊗z)and denote by |z|=√\nzTz∈R≥0andz⊗z=z zT∈\nRd×dthe Euclidean norm and the outer product of a column vector z∈Rd, byId∈Rd×d\ntheidentitymatrix, andby C > 0apositiveconstant. Forthegradientoperatorcomprising\nthe partial derivatives with respect to the velocity components v= (v1, . . . , v d)T∈Rd, we\nemploy the notation ∇v= (∂v1, . . . , ∂ vd)T. The purpose of efficiently evaluating the Landau\noperator (1) is closely related to the numerical approximation of the associated Landau\nequation\n∂tf=∇v· Q(c)(f, f). (2)\nMaxwellian molecules and Coulomb interaction. Henceforth, we tacitly assume\nthat the density function satisfies suitable regularity and integrability requirements such\nthat the Landau operator (1) is well-defined in the classical sense. The choices d∈ {2,3},\nΩ(v)=Rd, and β= 0, referred to as Maxwellian molecules cases, serve as basic test\nexamples, since the specification of density functions like\nd= 2, f(v) = e−|v|2|v|2, v∈R2,\nd= 3, f(v) = e−|v|2\u0000\n2|v|2−1\u0001\n, v∈R3,\npermits to determine the integrals inside the Landau operators. We in particular aim at\nthe treatment of the physically most relevant and numerically challenging case of Coulomb\ninteraction in three dimensions\nd= 3,Ω(v)=R3, β =−3, (3)\nwhich leads to a strong singularity in the integral operator.\nNovel strategies. In this work, we propose novel approaches in the spirit of collocation\nmethodsforthereliableandefficientnumericalevaluationoftheLandauoperator(1), tobe\n2applied in the time integration of the Landau equation (2). We next sketch basic concepts\nfor the relevant case of Coulomb interaction (3) and essential components of the resulting\nalgorithms. Our common starting point is the representation as a nonlocal drift-diffusion\nequation, where we identify fundamental integrals of the form\nZ\nR3φ(v−w)p(v−w)g(w) dw\nthatinvolvethesingularintegralkernel φ:R3→R, apolynomialofdegreetwo p:R3→R,\nandaregularfunction g:R3→Rreflectingthevaluesofthedensityfunctionorderivatives\nthereof, respectively. In addition, we make use of the decomposition\nZ\nR3φ(v−w)p(v−w)g(w) dw=Z\nR3ψ(v−w)p(v−w)g(w) dw (4a)\n+Z\nR3(φ−ψ)(v−w)p(v−w)g(w) dw ,\nwhere ψis a suitable regularisation of the kernel, obtained by interpolation nearby the\nisolated singularity, such that the difference φ−ψvanishes on the main part of the ve-\nlocity domain, see Figure 1. In order to numerically compute these integrals, we employ\nseries expansions of ψandgas well as quadrature approximations. Due to the particular\nproperties of Fourier functions\nFκ:R−→R:ξ7−→1√\n2beµκ(ξ+b),\nFm:R3−→R:v7−→ F m1(v1)Fm2(v2)Fm3(v3),\nµκ=πiκ\nb∈C, µ m=\u0000\nµm1, µm2, µm3\u0001\n∈C1×3,\nb >0, κ∈Z, m = (m1, m2, m3)∈Z3,(4b)\nand in view of the highly efficient practical implementation of Fourier series expansions by\nfast transforms, i.e.\nX\nm∈MgmFm≈g ,X\nm∈MψmFm≈ψ ,M=\b\n−M\n2, . . . ,M\n2−1\t3,(4c)\nfor an even positive integer number M∈N, we favour the Fourier spectral method. Under\nthe reasonable presumption of a localised density function, we may replace the unbounded\nvelocity domain by a cartesian product of intervals, characterised by a sufficiently large\npositive real number b >0. Accordingly, we choose uniform grid points that cover the\ntruncated domain\nvℓ∈[−b, b]3, ℓ = (ℓ1, ℓ2, ℓ3)∈ L={1, . . . , M }3, (4d)\nLikewise, the relatively small neighbourhood of the origin, where the interpolant of the\nsingular kernel is devised\n(φ−ψ)\f\f\n[−b,b]3\\[−b0,b0]3= 0, (4e)\n3is defined by a positive real number 0< b0< < b, which we adjust in such a way that the\npoint (b0,0,0)coincides with a grid point. Thus, non-zero values\n(φ−ψ)(vℓ)̸= 0, ℓ∈eL ⊂ L , (4f)\noccur, but only for a small subset eL ⊂ L. Because of the multiplicativity of Fourier func-\ntions, the concrete tasks for the numerical computation of the fundamental integrals (4a)\namount to determine one-dimensional integrals of the form\nZb\n−bξiFκ(ξ) dξ , i ∈ {0,1,2}, κ∈\b\n−M\n2, . . . ,M\n2−1\t\n, (4g)\nto approximate three-dimensional integrals by quadrature\nZ\n[−b0,b0]3(φ−ψ)(w)p(w)Fm(w) dw , m ∈ M , (4h)\nand to apply fast Fourier transforms or summations along certain directions, respectively.\nComparison with alternative spectral methods. The efficient implementation of our\napproaches strongly relies on spectral methods, in particular, on fast Fourier techniques.\nHowever, compared to [8, 17, 18, 19, 23], rather than approximating the weak formulation\nof the Landau operator, we approximate the distribution function in physical space by\ncollocation. One of the advantages of such a Fourier collocation method is the localisation\nof the singular Coulomb kernel in velocity space and the accurate representation of the\ndistribution function by Fourier series outside the truncated domain. As a consequence,\nin connection with the future application of Hamiltonian splitting methods for Vlasov–\nMaxwell–Landau systems, e.g., we can hand over the values of the density function to\nthe solvers of the other subproblems. We propose different approaches using or avoiding\nnumerical differentiation, depending on whether mass conservation or higher accuracy is\npreferable, respectively, see Table 1 for an overview. Considering a Cartesian product of\nintervals, we have the possibility to adjust the resolution along each velocity component.\nMaking use of the fact that the Coulomb kernel is well represented by Fourier series except\non a small neighbourhood of the singularity, restricting the quadrature approximation to\nthis neighbourhood has the advantage of significantly reducing the precomputation times\nas well as the computations times during evolution, see Table 2.\nOutline. This manuscript has the following structure. In Section 2, we introduce the\nconsidered numerical methods for the Landau operator involving constant, regular, and\nsingular kernels, respectively. Numerical comparisons for different test problems are then\ndiscussed in Section 3.\n2 Numerical methods\nIn this section, we develop the proposed spectral collocation methods for the evaluation of\nthe Landau operator and the time integration of the Landau equation. We begin with an\n4overviewofthefundamentalapproachandthendescribeinfulldetailaslightgeneralisation\nand related approaches.\n2.1 Overview\nGuide line. We point out that our approaches are designed for the numerical approxi-\nmation of the Landau equation (2) and that a significant percentage of the overall compu-\ntational effort per time step can be transferred to precomputations which are independent\nof the density function. In order to illustrate the key aspects of the resulting algorithms,\nwe introduce the forward Euler method as simplest representative for standard classes of\nexplicit and implicit, one-step and multi-step time integrators. Prescribing positive step-\nsizes (τn)N−1\nn=0such that τ0+···+τN−1=Tand an initial approximation, the time-discrete\nsolution is determined by the recurrence\n(\nf(n)=f(n−1)+τn−1∇v· Q(c)\u0000\nf(n−1), f(n−1)\u0001\n, n∈ {1, . . . , N },\nf(0)given .(5)\nEvidently, the main computational issue in each time step is related to the numerical\nevaluation of the integral operator. Briefly summarised, we have the following guide line.\nInput (Problem data).\n(i) The function f(0)defines the inital state of the density function, see (5).\n(ii) The function φdefines the integral kernel with isolated singularity at the origin,\nsee (1) and (3).\nInput (Discretisation).\n(i) The positive real number b >0defines the truncated velocity domain, see (4d).\n(ii) The even positive integer number M∈Nreflects the total numbers of Fourier func-\ntions and uniform grid points covering the truncated domain, see (4c) and (4d).\n(iii) The small positive real number b0>0is chosen such that (b0,0,0)coincides with a\ngrid point and defines a neighbourhood of the origin, where a regularisation of the\nsingular integral kernel is devised by interpolation, see (4e) and Figure 1.\n(iv) A sequence of positive time stepsizes (τn)N−1\nn=0is defined, and, if required for stability\nand accuracy, suitably adapted during time integration, see (5).\nPrecomputations.\n(i) The uniform grid points (vℓ)ℓ∈Lare computed and stored, see (4d).\n(ii) The complex eigenvalues (µm)m∈Massociated with the included Fourier functions\n(Fm)m∈Mare computed and stored, see (4b) and (4c).\n5(iii) The basic integrals (4g) are computed and stored.\n(iv) The values of the integral kernel at the grid points are computed. Based on them,\nthe values of a suitable regularisation on the previously defined neighbourhood of\nthe origin are determined by interpolation. On the one hand, the associated spectral\ncoefficients (ψm)m∈Mare computed through a fast Fourier transform, see (4c). Aux-\niliary quantities involving the basic integrals (4g) and complex exponentials (4b) are\ncomputed through summations along certain directions, in total three single sums\nand three double sums. On the other hand, the non-zero values corresponding to the\ndifference between the singular integral kernel and its interpolant\n\u0000\nφ(vℓ)−ψ(vℓ)\u0001\nℓ∈eL,\nare computed, see (4f). Based on these quantities, quadrature approximations to the\nin total 6M3integrals\nZ\n[−b0,b0]3wiwj(φ−ψ)(w)Fm(w) dw ,\n(i, j)∈ {(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}, m∈ M ,\nare computed and stored, see (4h).\nComputations. From the values of the initially prescribed density function at the\nvelocitygridpoints, theassociatedspectralcoefficientsarecomputedthroughafastFourier\ntransform\nn= 1,\u0000\nf(n−1)(vℓ)\u0001\nℓ∈L,\u0000\nf(n−1)\nm\u0001\nm∈M,\nyielding the approximation\nn= 1,X\nm∈Mf(n−1)\nmFm≈f(n−1),\nsee also (4c). In each substep of the time integration, based for instance on the explicit\nEuler method (5), the following computations are carried out, in order to determine the\nvalues and spectral coefficients of the discrete solution.\n(i) Numerical approximations to the values of the gradient at the velocity grid points\nare computed through pointwise multiplications by the corresponding eigenvalues and\nthree inverse fast Fourier transforms\nX\nm∈Mµmf(n−1)\nmFm(vℓ)≈ ∇ vf(n−1)(vℓ), ℓ∈ L,\nsee (1b).\n(ii) The main computational cost for the numerical approximation of fundamental inte-\ngrals, which involve the regularised kernel, amounts to pointwise multiplications and\nin total 18 inverse fast Fourier transforms, see (4a).\n6(iii) The representation (1b) is used to determine the values of the integral operator\nQ(c)\u0000\nf(n−1), f(n−1)\u0001\n(vℓ), ℓ∈ L.\nIn each case, the divergence is computed by three fast Fourier transforms and an\nadditional inverse fast Fourier transform\n∇v· Q(c)\u0000\nf(n−1), f(n−1)\u0001\n(vℓ), ℓ∈ L. (6)\nSimple summations finally yield the values and spectral coefficients of the new ap-\nproximation\u0000\nf(n)(vℓ)\u0001\nℓ∈L,\u0000\nf(n)\nm\u0001\nm∈M,\nsee also (5).\nOutput.\n(i) By means of the above procedure, we obtain the values and the spectral coefficients\nof the discrete solution\n\u0000\nf(n)(vℓ)\u0001\nℓ∈L,\u0000\nf(n)\nm\u0001\nm∈M, n∈ {0,1, . . . , N }.\n(ii) If desired, approximations at intermediate points can be computed through the rela-\ntion\nf(n−1)(v)≈X\nm∈Mf(n−1)\nmFm(v), v∈[−b, b]3, n∈ {0,1, . . . , N }.\nComputational cost. In the previously described situation, we may consider the 26\nFourier transforms for the computation of the Landau operator in a substep of the time\nintegration as the computationally most elaborate components. Other processes such as\nsummations and pointwise multiplications can be optimised by parallelisation. We point\nout that the choice of the neighbourhood, where a regularisation of the singular integral\nkernel is determined, is crucial, see Figure 1. Compared to a quadrature approximation on\nthe whole domain, a suitable adjustment of the relatively small subset makes it possible\nto significantly reduce the precomputation time for the same accuracy, see Table 2.\nImplementation and improvements. In order to perform numerical comparisons and\nto design graphical illustrations, we found it convenient to implement our approaches in\nMatlab . An elementary code that has the purpose to illustrate the practical imple-\nmentation and reproduces numerical results displayed in Figures 10 and 11 is available\nthrough [4]. As we prioritised readability and the validation of common components, our\ncode comprises several subroutines, which lower speed. We thus consider the observed\ncomputation times as rough indicators and see opportunities for improvements based on\nefficient software packages. Detailed derivations in a more general setting and additional\nnumerical comparisons for different test examples with known solutions are described in\nthe subsequent sections, see also Table 1 for an overview.\n7Conservation of mass. A characteristical property of the solution to the Landau equa-\ntion (2) is that it conserves mass, momentum, and energy and that entropy decays over\ntime Z\nΩ(v)f(v, t) dv=Z\nΩ(v)f(v, t0) dv ,\nZ\nΩ(v)v f(v, t) dv=Z\nΩ(v)v f(v, t0) dv ,\nZ\nΩ(v)|v|2f(v, t) dv=Z\nΩ(v)|v|2f(v, t0) dv ,\nZ\nΩ(v)f(v, t) ln\u0000\nf(v, t)\u0001\ndv≤Z\nΩ(v)f(v, t0) ln\u0000\nf(v, t0)\u0001\ndv ,\nt∈[t0, T].(7)\nDue to the employed representation of the Landau operator in divergence form, the con-\nservation of mass is ensured for the discrete solution as well, provided that the mass is\ncomputed subsequently to (6) through the trivial identity for the associated spectral coef-\nficients\nf(n)\n0=f(n−1)\n0 , n∈ {1, . . . , N }.\nAny additional application of a fast Fourier transform or inverse fast Fourier transform,\nhowever, will cause numerical perturbations, since the values of a regular function are not\nretained, in general, i.e. IFFT\u0000\nFFT(g)\u0001\n̸=g, but we may expect highly accurate approxi-\nmations. Likewise, we cannot ensure the preservation of momentum and energy as well as\nthe strict positivity of the density function and the decay of entropy in case the reconstruc-\ntion is positive, but we may expect highly accurate numerical approximations for suitable\ndiscretisations based on the Fourier spectral method and geometric time integrators, see\nfor instance [10, 12, 16, 21].\n2.2 Detailed description\nCompact exposition. For the benefit of compact representations, we employ convenient\nabbreviations for the integral kernel and the term involving the outer product\nφ: Ω(φ)−→R:v7−→C|v|β,Ω(φ)=(\nRdifβ≥0,\nRd\\ {0}ifβ <0,\nP:Rd−→Rd×d:v7−→ | v|2Id−v⊗v .(8a)\nWhen applicable, we omit the dependence of the density function, the integral operator,\nand the Landau operator on the time variable, i.e., we write\nQ(c)(f, f)(v) =Z\nΩ(φ P)(v−w)\u0000\n∇vf(v)f(w)−f(v)∇wf(w)\u0001\ndw ,\nQ(f, f)(v) =∇v· Q(c)(f, f)(v),\nv∈Ω = Ω(v),(8b)\n8for short. For the purpose of numerical validation and comparison, we distinguish the cases\nof constant, regular, and singular integral kernels and adapt our procedure accordingly to\nthe increasing complexity of the problem. We exemplify calculations for two dimensions\nand state the analogous results for the significantly more involved three-dimensional case.\n2.3 Auxiliaries\nIn the following, we collect useful abbreviations and elementary results related to the\nFourier spectral method. Its efficient implementation relies on fast Fourier techniques.\nShort notation. For the purpose of more compact representations, we introduce the\ncommon short notation\nvj=vj1\n1···vjd\nd, ∂j\nv=∂j1\nv1···∂jd\nvd,\nv= (v1, . . . , v d)∈Rd, j = (j1, . . . , j d)∈Nd\n≥0.(9)\nTruncated velocity domain. With regard to the numerical approximation of the in-\ntegral operator in (8), it is required to truncate the initially unbounded velocity domain\nΩ =Rd. Under the reasonable assumption that the density function can be approximated\nwith high accuracy by a localised function, we may replace the original integral domain by\na cartesian product of sufficiently large intervals\nΩ(b)= [b11, b12]× ··· × [bd1, bd2]⊂Ω. (10)\nFourier functions. For Fourier functions depending on a single variable, we in addition\nspecify the dependence on the canonical periodicity interval and the number of oscillations\nF(α)\nκ(ξ) =1√α2−α1eµ(α)\nκ(ξ−α1), µ(α)\nκ=2πiκ\nα2−α1,\nF(α)\nκ(α1) =1√α2−α1=F(α)\nκ(α2), ∂ ξF(α)\nκ(ξ) =µ(α)\nκF(α)\nκ(ξ),\nκ∈Z, α = (α1, α2)∈R2, α 1< α 2, ξ∈R.(11a)\nAccordingly, for Fourier functions in several variables, we set\nF(b)\nm(v) =F(b11, b12)\nm1(v1)···F(bd1, bd2)\nmd(vd),\nm= (m1, . . . , m d)∈Zd, v = (v1, . . . , v d)∈Rd.(11b)\nMoreover, in view of compact representations, we employ the notation\nµ(b)\nm=\u0000\nµ(b11, b12)\nm1, . . . , µ(bd1, bd2)\nmd\u0001\n∈C1×d,\nE(b)\nm(v) = eµ(b11, b12)\nm1v1···eµ(bd1, bd2)\nmdvd,\nE(b)\nm(b1) = eµ(b11, b12)\nm1b11···eµ(bd1, bd2)\nmdbd1,\nm∈Zd, v∈Rd.(11c)\n9Basic integrals and identities. Elementary calculations permit to determine the basic\nintegrals\nIj(kj, mj) =Zbj2\nbj1wkj\njF(bj1, bj2)\nmj(wj) dwj,\nIj(kj, mj) =\n\np\nbj2−bj1, k j= 0, m j= 0,\nb2\nj2−b2\nj1\n2√\nbj2−bj1, k j= 1, m j= 0,\nb3\nj2−b3\nj1\n3√\nbj2−bj1, k j= 2, m j= 0,\n0, k j= 0, m j∈Z\\ {0},√\nbj2−bj1\nµ(bj1, bj2)\nmj, k j= 1, m j∈Z\\ {0},\n(b2\nj2−b2\nj1)µ(bj1, bj2)\nmj−2 (bj2−bj1)\n√\nbj2−bj1\u0000\nµ(bj1, bj2)\nmj\u00012, k j= 2, m j∈Z\\ {0},\nI(k, m) =Z\nΩ(b)wkF(b)\nm(w) dw=I1(k1, m1)···Id(kd, md),\nk∈ {0,1,2}d, m∈Zd,(12)\nsee also (9). Due to their close connection to the complex exponential, Fourier functions\nsatisfy identities such as\nF(b)\nm(v−u) =F(b)\nm(v)E(b)\n−m(u), (13a)\nF(b)\nℓ(u)F(b)\nm(v−u) =E(b)\n−m(b1)F(b)\nm(v)F(b)\nℓ−m(u), (13b)\nF(b)\nℓ(v−w)F(b)\nm(w) =E(b)\n−ℓ(b1)F(b)\nℓ(v)F(b)\nm−ℓ(w), (13c)\nℓ, m∈Zd, u, v, w ∈Rd.\nFourier series expansions. Accordingly to the regularity of a real-valued function\ng: Ω(b)→Rand its partial derivatives, Fourier series expansions\ng(v)≈X\nm∈MMgmF(b)\nm(v), g m=Z\nΩ(b)g(v)F(b)\n−m(v) dv , m ∈ M M,\n∂vkg(v)≈X\nm∈MMµ(bk1, bk2)\nmkgmF(b)\nm(v), k∈ {1, . . . , d },\nv∈Ω(b),\nhold, where the index set\nMM=\b\n−M1\n2, . . . ,M1\n2−1\t\n× ··· ×\b\n−Md\n2, . . . ,Md\n2−1\t\n⊂Zd(14)\nis characterised by M= (M1, . . . , M d)∈Ndcomprising even positive integers. For the\nsake of compact representations, we initially identify functions with their formal Fourier\n10series\ng(v) =X\nm∈ZdgmF(b)\nm(v), v∈Ω,\nand afterwards comment on the practical implementation of the Fourier spectral method.\n2.4 Approaches\nIn this section, we outline different approaches for the numerical evaluation of the Landau\noperator(8),seeTable1. Webeginwithdetailedconsiderationsforanaturalprocedureand\nthen indicate possible alternatives. Useful means are suitable reformulations of the integral\noperator obtained by Fourier series expansions of the density function and a regularised\nintegral kernel, the application of a linear integral transform and a quadrature rule on a\nrelatively small neighbourhood of the origin, and subsequent differentiation. In order to\nunderline the sources of errors due to the truncation of the unbounded velocity domain,\nthe replacement of infinite by finite sums, and quadrature, we first state the employed\nrepresentations of the Landau operator on the basis of formal series expansions and then\nsketch the corresponding algorithms. We recall that auxiliary abbreviations and results\nrelated to the Fourier spectral method are found in Section 2.3. For the benefit of shorter\nformulas, we henceforth set\nd= 2,\nM1={(1,1),(1,2),(2,2)},\nM2={(1,1,2),(1,2,1),(1,2,2),(2,2,1)},\nM3={(0,0),(1,0),(0,1),(2,0),(1,1),(0,2)},\nd= 3,\nM1={(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)},\nM2={(1,1,2),(1,1,3),(1,2,1),(1,2,2),(1,3,1),(1,3,3),\n(2,2,1),(2,2,3),(2,3,2),(2,3,3),(3,3,1),(3,3,2)},\nM3={(0,0,0),(1,0,0),(0,1,0),(0,0,1),(2,0,0),(1,1,0),\n(1,0,1),(0,2,0),(0,1,1),(0,0,2)},(15)\n2.5 Approach CST2 (Conservative form Singular kernel Trans-\nform twice)\nIntegral operator. Our starting point is the following representation of the integral\noperator\nQ(c)(f, f)(v) =I(f)(v)∇vf(v) +J(f)(v)f(v), v∈Ω.\nThe arising matrix- and vector-valued operators\nd= 2,\nI=\u0012I22−I12\n−I12I11\u0013\n,J=\u0012−J 221+J122\nJ121− J 112\u0013\n,\n11d= 3,\nI=\nI22+I33−I12 −I13\n−I12I11+I33−I23\n−I13 −I23I11+I22\n,\nJ=\n−J 221− J 331+J122+J133\nJ121− J 112− J 332+J233\nJ131+J232− J 113− J 223\n,\nare defined by fundamental integrals, which involve polynomials of degree two, the singular\nkernel, and the density function\nIij(f)(v) =Z\nΩ(vi−wi) (vj−wj)φ(v−w)f(w) dw , (i, j)∈ M 1,\nJijk(f)(v) =Z\nΩ(vi−wi) (vj−wj)φ(v−w)∂wkf(w) dw , (i, j, k )∈ M 2,\nv∈Ω.\nFor the sake of concreteness, we next explain our main strategy on the basis of the two\ninstances\nI11(f)(v) =Z\nΩ(v1−w1)2φ(v−w)f(w) dw ,\nJ112(f)(v) =Z\nΩ(v1−w1)2φ(v−w)∂w2f(w) dw ,\nv∈Ω,(16)\nand then state the corresponding representations obtained in the general case.\nSingular integral kernel. Our basic idea is to exploit the evident identity\nφ(v) =ψ(v) + (φ−ψ)(v), v∈Ω, (17a)\nand to adjust the regular function ψsuch that the remainder φ−ψvanishes on the main\npart of the velocity domain. More concretely, we make use of the fact that the singular\nintegral kernel associated with the Coulomb potential is regular outside a relatively small\nneighbourhood of the origin\nΩ(0)= [b(0)\n11, b(0)\n12]× ··· × [b(0)\nd1, b(0)\nd2]⊂Ω(b)⊂Ω, (17b)\nsee also (10). Hence, excluding this set, the kernel defines a regular function\nψ(v) =φ(v), v∈Ω\\Ω(0). (17c)\nNearby the origin, straightforward interpolation of φis applied to determine ψ.\nFourier series expansions. For the regularised integral kernel and the density function,\nwe may assume that favourable approximations are provided by Fourier series expansions.\n12We meanwhile employ the formal representations\nψ(v) =X\nℓ∈ZdψℓF(b)\nℓ(v),\nf(v) =X\nm∈ZdfmF(b)\nm(v), ∂ vkf(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmF(b)\nm(v),\nv∈Ω, k∈ {1, . . . , d }.(18)\nTheir practical implementation presupposes suitable truncations of the unbounded velocity\ndomain to avoid significant aliasing effects, see for instance [18, 19]. Moreover, it requires\ntruncations of the infinite sums as well as the application of the trapezoidal rule for the\nnumerical computation of the spectral coefficients.\nDecisive integrals. The above stated Fourier series expansions of the density function\nand its partial derivatives imply representations such as\nI11(f)(v) =X\nm∈ZdfmI(m)\n11(v),\nJ112(f)(v) =X\nm∈Zdµ(b21, b22)\nm2fmI(m)\n11(v),\nI(m)\n11(v) =Z\nΩ(v1−w1)2φ(v−w)F(b)\nm(w) dw , m ∈Zd,\nv∈Ω,\nsee (16) and (18). Due to the decomposition of the kernel into a regular function and\na singular function that vanishes on the main part of the domain, the decisive integrals\ncomprise the two contributions\nI(m)\n11(v) =I(m,ψ)\n11(v) +I(m,φ−ψ)\n11 (v),\nI(m,ψ)\n11(v) =Z\nΩ(v1−w1)2ψ(v−w)F(b)\nm(w) dw ,\nI(m,φ−ψ)\n11 (v) =Z\nΩ(v1−w1)2(φ−ψ)(v−w)F(b)\nm(w) dw ,\nm∈Zd, v∈Ω,(19)\nsee also (17).\nLinear integral transform. Applying in both cases the linear integral transform u=\nv−w, yields the equivalent reformulations\nI(m,ψ)\n11(v) =Z\nv−Ωu2\n1ψ(u)F(b)\nm(v−u) du ,\nI(m,φ−ψ)\n11 (v) =Z\nv−Ωu2\n1(φ−ψ)(u)F(b)\nm(v−u) du ,\nm∈Zd, v∈Ω,\n13where we employ a symbolic notation for the shifted domain comprising an additional sign\nowing to duk=−dwkfork∈ {1, . . . , d }. By means of the Fourier series expansion of\nthe regularised kernel (18) and the identity (13b), the first multiple integral reduces to\none-dimensional integrals\nI(m,ψ)\n11(v) =E(b)\n−m(b1)F(b)\nm(v)X\nℓ∈ZdψℓZ\nv−Ωu2\n1F(b)\nℓ−m(u) du , m ∈Zd, v∈Ω.\nFor the second multiple integral, we instead apply the relation (13a) and have\nI(m,φ−ψ)\n11 (v) =F(b)\nm(v)Z\nv−Ωu2\n1(φ−ψ)(u)E(b)\n−m(u) du , m ∈Zd, v∈Ω.\nApproach CST2. Summarising the above procedure, we obtain the following relations\nfor the derivatives of the density function\n∂vkf(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmF(b)\nm(v), k∈ {1, . . . , d }, v∈Ω, (20a)\nthe fundamental integrals\nI(m,ψ)\nij(v) =E(b)\n−m(b1)F(b)\nm(v)X\nℓ∈ZdψℓZ\nv−ΩuiujF(b)\nℓ−m(u) du ,\nI(m,φ−ψ)\nij (v) =F(b)\nm(v)Z\nv−Ωuiuj(φ−ψ)(u)E(b)\n−m(u) du ,\nI(m)\nij(v) =I(m,ψ)\nij(v) +I(m,φ−ψ)\nij (v),\nIij(f)(v) =X\nm∈ZdfmI(m)\nij(v),(i, j)∈ M 1,\nJijk(f)(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmI(m)\nij(v),(i, j, k )∈ M 2,\nv∈Ω,(20b)\nand the integral operator\nQ(c)(f, f)(v) =I(f)(v)∇vf(v) +J(f)(v)f(v),\nd= 2,\nI=\u0012I22−I12\n−I12I11\u0013\n,J=\u0012−J 221+J122\nJ121− J 112\u0013\n,\nd= 3,\nI=\nI22+I33−I12 −I13\n−I12I11+I33−I23\n−I13 −I23I11+I22\n,J=\n−J 221− J 331+J122+J133\nJ121− J 112− J 332+J233\nJ131+J232− J 113− J 223\n,\nv∈Ω.(20c)\n14Furthermore, in order to determine the Landau operator, we use the associated Fourier\nseries representation\nQ(c)(f, f)(v) =X\nm∈ZdQ(c)\nmF(b)\nm(v),\nQ(f, f)(v) =∇v· Q(c)(f, f)(v) =X\nm∈Zdµ(b)\nmQ(c)\nmF(b)\nm(v),\nv∈Ω.(20d)\nPracticalimplementation. Thepracticalimplementationof (20)requiressuitabletrun-\ncations of the velocity domain and the infinite sums. We point out that replacing the\nshifted domain v−Ωby the truncated integral domain Ω(b)permits the application of\nfast Fourier techniques. We recall that the essential steps of the algorithm, distinguishing\nbetween precomputations that are independent of the density function and computations\nthat are carried out repeatedly in course of the time integration of the Landau equation,\nare described in Section 1. Moreover, a link to an elementary Matlab code is provided\nthere.\n2.6 Approach CST1 (Conservative form Singular kernel Trans-\nform once)\nModification. Regarding a possible modification of our first approach, we reconsider the\nquantity\nI(m,ψ)\nij(v) =Z\nΩ(vi−wi) (vj−wj)ψ(v−w)F(b)\nm(w) dw ,\n(i, j)∈ M 1, m∈Zd, v∈Ω,\nsee also (19). Expanding the integrand, inserting the Fouries series representation of the\nregularised kernel (18), and applying the identity (13c) yields the alternative reformulation\neI(m−ℓ)\nij (v) =Z\nΩ(vivj+viwj+vjwi+wiwj)F(b)\nm−ℓ(w) dw ,\nI(m,ψ)\nij(v) =X\nℓ∈ZdψℓE(b)\n−ℓ(b1)F(b)\nℓ(v)eI(m−ℓ)\nij (v),\n(i, j)∈ M 1, m∈Zd, v∈Ω.(21a)\nEmploying the short notation\neI(k, m) =Z\nΩwkF(b)\nm(w) dw , k ∈ M 3, m∈Zd, (21b)\n15the two-dimensional case takes the form\neI(m)\n11(v) =v2\n1eI\u0000\n(0,0), m\u0001\n+ 2v1eI\u0000\n(1,0), m\u0001\n+eI\u0000\n(2,0), m\u0001\n,\neI(m)\n22(v) =v2\n2eI\u0000\n(0,0), m\u0001\n+ 2v2eI\u0000\n(0,1), m\u0001\n+eI\u0000\n(0,2), m\u0001\n,\neI(m)\n12(v) =v1v2eI\u0000\n(0,0), m\u0001\n+v1eI\u0000\n(0,1), m\u0001\n+v2eI\u0000\n(1,0), m\u0001\n+eI\u0000\n(1,1), m\u0001\n.(21c)\nIn three dimensions, we instead have\neI(m)\n11(v) =v2\n1eI\u0000\n(0,0,0), m\u0001\n+ 2v1eI\u0000\n(1,0,0), m\u0001\n+eI\u0000\n(2,0,0), m\u0001\n,\neI(m)\n22(v) =v2\n2eI\u0000\n(0,0,0), m\u0001\n+ 2v2eI\u0000\n(0,1,0), m\u0001\n+eI\u0000\n(0,2,0), m\u0001\n,\neI(m)\n33(v) =v2\n3eI\u0000\n(0,0,0), m\u0001\n+ 2v3eI\u0000\n(0,0,1), m\u0001\n+eI\u0000\n(0,0,2), m\u0001\n,\neI(m)\n12(v) =v1v2eI\u0000\n(0,0,0), m\u0001\n+v1eI\u0000\n(0,1,0), m\u0001\n+v2eI\u0000\n(1,0,0), m\u0001\n+eI\u0000\n(1,1,0), m\u0001\n,\neI(m)\n13(v) =v1v3eI\u0000\n(0,0,0), m\u0001\n+v1eI\u0000\n(0,0,1), m\u0001\n+v3eI\u0000\n(1,0,0), m\u0001\n+eI\u0000\n(1,0,1), m\u0001\n,\neI(m)\n23(v) =v2v3eI\u0000\n(0,0,0), m\u0001\n+v2eI\u0000\n(0,0,1), m\u0001\n+v3eI\u0000\n(0,1,0), m\u0001\n+eI\u0000\n(0,1,1), m\u0001\n.(21d)\nApproach CST1. For the sake of completeness, we recapitulate the resulting represen-\ntations for the derivatives of the density function\n∂vkf(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmF(b)\nm(v), k∈ {1, . . . , d }, v∈Ω, (22a)\nthe basic integrals given by polynomials and Fourier functions\neI(k, m) =Z\nΩwkF(b)\nm(w) dw , k ∈ M 3, m∈Zd, (22b)\nthe fundamental integrals involving the regularised kernel\neI(m−ℓ)\nij (v) =vivjZ\nΩF(b)\nm−ℓ(w) dw+viZ\nΩwjF(b)\nm−ℓ(w) dw\n+vjZ\nΩwiF(b)\nm−ℓ(w) dw+Z\nΩwiwjF(b)\nm−ℓ(w) dw ,\nI(m,ψ)\nij(v) =X\nℓ∈ZdψℓE(b)\n−ℓ(b1)F(b)\nℓ(v)eI(m−ℓ)\nij (v),\nI(ψ)\nij(f)(v) =X\nm∈ZdfmI(m,ψ)\nij(v),(i, j)∈ M 1,\nJ(ψ)\nijk(f)(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmI(m,ψ)\nij(v),(i, j, k )∈ M 2,\nv∈Ω,(22c)\n16as well as the fundamental integrals involving the difference of the singular kernel and its\nregularisation\nI(m,φ−ψ)\nij (v) =F(b)\nm(v)Z\nv−Ωuiuj(φ−ψ)(u)E(b)\n−m(u) du ,\nI(φ−ψ)\nij (f)(v) =X\nm∈ZdfmI(m,φ−ψ)\nij (v),(i, j)∈ M 1,\nJ(φ−ψ)\nijk (f)(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmI(m,φ−ψ)\nij (v),(i, j, k )∈ M 2,\nv∈Ω,(22d)\nthe corresponding sums\nIij(f)(v) =I(ψ)\nij(f)(v) +I(φ−ψ)\nij (f)(v),(i, j)∈ M 1,\nJijk(f)(v) =J(ψ)\nijk(f)(v) +J(φ−ψ)\nijk (f)(v),(i, j, k )∈ M 2,\nv∈Ω,(22e)\nand the integral operator\nQ(c)(f, f)(v) =I(f)(v)∇vf(v) +J(f)(v)f(v),\nd= 2,\nI=\u0012I22−I12\n−I12I11\u0013\n,J=\u0012−J 221+J122\nJ121− J 112\u0013\n,\nd= 3,\nI=\nI22+I33−I12 −I13\n−I12I11+I33−I23\n−I13 −I23I11+I22\n,J=\n−J 221− J 331+J122+J133\nJ121− J 112− J 332+J233\nJ131+J232− J 113− J 223\n,\nv∈Ω.(22f)\nSubsequent differentiation is again based on the Fourier series expansion\nQ(c)(f, f)(v) =X\nm∈ZdQ(c)\nmF(b)\nm(v),\nQ(f, f)(v) =∇v· Q(c)(f, f)(v) =X\nm∈Zdµ(b)\nmQ(c)\nmF(b)\nm(v),\nv∈Ω.(22g)\nPractical implementation. Concerning the practical implementation of (22), we pre-\nscribe the dimension dof the velocity domain, the integral kernel φ, and the value of the\ndensity function fat the initial time and the considered space grid points. We replace the\noriginal integral domain Ωby the truncated domain Ω(b)and make use of the fact that the\n17integrals in (21) are related to the basic integrals given in (12). It is again expedient to\ndistinguish between precomputations and computations in course of the time integration,\nsee also (5).\nInput (Discretisation). We first specify\n(i) the real numbers (bi1, bi2)d\ni=1with bi1< bi2fori∈ {1, . . . , d }defining the truncated\ndomain Ω(b), see (10),\n(ii) even positive integers to define the set MMreflecting the total number of Fourier\nfunctions, see (14),\n(iii) the real numbers (b(0)\ni1, b(0)\ni2)d\ni=1with b(0)\ni1< b(0)\ni2fori∈ {1, . . . , d }defining the small\nrestricted domain Ω(0), see (17).\nPrecomputations Due to the fact that certain quantities do not depend on the values\nof the density function, it is advantageous to compute them in advance. This in particular\nconcerns\n(i) the in total M1···Mdequidistant grid points covering the truncated domain Ω(b),\n(ii) the eigenvalues (µ(b)\nm)m∈MMassociated with the Fourier functions, see also (11),\n(iii) the basic integrals I(k, m)fork∈ M 3andm∈ M M, see (12),\n(iv) the values of the integral kernel φon the equidistant grid, excluding the singularity\nat the origin,\n(v) the values of the regularised kernel ψon the grid points that are contained in the\nsmall domain Ω(0), obtained by interpolation,\n(vi) the associated spectral coefficients (ψℓ)ℓ∈MMthrough a fast Fourier transform as well\nas the products ψℓE(b)\n−ℓ(b1)forℓ∈ M M, see also (18),\n(vii) and quadrature approximations to the integrals\nZ\nΩ(0)uiuj(φ−ψ)(u)E(b)\n−m(u) du , (i, j)∈ M 1, m∈ M m,\nsee (15) and (22).\nComputations Regarding the evaluation of the Landau operator, we compute approx-\nimations to\n(i) the spectral coefficients (fm)m∈MMassociated with the density function through a\nfast Fourier transform,\n(ii) the values of the derivatives ∂vkffork∈ {1, . . . , d }at the prescribed equidistant grid\npoints by means of the representations in (18), requiring componentwise multiplica-\ntions by eigenvalues and the application of fast inverse Fourier transforms,\n18(iii) on the one hand, the values of the following fundamental quantities at the grid points\nI(ψ)\nij(f)(v) =X\nℓ∈MMψℓE(b)\n−ℓ(b1)F(b)\nℓ(v)X\nm∈MMfmeI(m−ℓ)\nij (v),\n(i, j)∈ M 1, v∈Ω,(23)\nby summations along certain directions as well as fast inverse Fourier transforms\nfundamental integrals and accordingly for J(ψ)\nijk(f)with (i, j, k )∈ M 2,\n(iv) on the other hand, the values of the fundamental quantities I(φ−ψ)\nij (f)for(i, j)∈ M 1\nandJ(φ−ψ)\nijk (f)for(i, j, k )∈ M 2by fast inverse Fourier transforms,\n(v) the products of their sums with the derivatives of the density function to obtain the\ncomponents of the integral operator Q(c),\n(vi) and, inafinalstep, thedivergencebymultiplicationswitheigenvaluesandfastinverse\nFourier transforms.\n2.7 Approaches CCT1-2 (Constant kernel) and CRT1-2 (Regular\nkernel)\nSimplified approaches. The consideration of the Maxwellian molecules case and test\nproblems involving regular integral kernels and permits significant simplifications concern-\ning the representations of the fundamental integrals. On account of situations exemplified\nin (28), where the approach CST2 fails due to the incorrect replacement of the shifted inte-\ngraldomain v−Ωbytheoriginaldomain Ω, wefirstincludethedetailsforapproachesCRT1\nand CCT1 and then, for the sake of completeness, the corresponding relations for ap-\nproaches CRT2 and CCT2.\nApproach CRT1. For a regular integral kernel with formal Fourier series expansion\nφ(v) =X\nℓ∈ZdφℓF(b)\nℓ(v), v∈Ω,\nthe fundamental integrals are given by\neI(k, m−ℓ) =Z\nΩwkF(b)\nm−ℓ(w) dw , k ∈ M 3,\neI(m−ℓ)\nij (v) =vivjZ\nΩF(b)\nm−ℓ(w) dw+viZ\nΩwjF(b)\nm−ℓ(w) dw\n+vjZ\nΩwiF(b)\nm−ℓ(w) dw+Z\nΩwiwjF(b)\nm−ℓ(w) dw ,\nI(m)\nij(v) =X\nℓ∈ZdφℓE(b)\n−ℓ(b1)F(b)\nℓ(v)eI(m−ℓ)\nij (v),\n19Iij(f)(v) =X\nm∈ZdfmI(m)\nij(v),(i, j)∈ M 1,\nJijk(f)(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmI(m)\nij(v),(i, j, k )∈ M 2,\nv∈Ω.\nApproach CCT1. For a constant integral kernel, that is φ=C, we arrive at the further\nsimplification\neI(k, m) =Z\nΩwkF(b)\nm(w) dw , k ∈ M 3,\nI(m)\nij(v) =C\u0012\nvivjZ\nΩF(b)\nm(w) dw+viZ\nΩwjF(b)\nm(w) dw\n+vjZ\nΩwiF(b)\nm(w) dw+Z\nΩwiwjF(b)\nm(w) dw\u0013\n,\nIij(f)(v) =X\nm∈ZdfmI(m)\nij(v),(i, j)∈ M 1,\nJijk(f)(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmI(m)\nij(v),(i, j, k )∈ M 2,\nv∈Ω.(24)\nApproach CRT2. On the other hand, for a regular integral kernel, we obtain\nI(m)\nij(v) =E(b)\n−m(b1)F(b)\nm(v)X\nℓ∈ZdφℓZ\nv−ΩuiujF(b)\nℓ−m(u) du ,\nIij(f)(v) =X\nm∈ZdfmI(m)\nij(v),(i, j)∈ M 1,\nJijk(f)(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmI(m)\nij(v),(i, j, k )∈ M 2,\nv∈Ω.(25)\nApproach CCT2. In case of a constant integral kernel, the former approach reduces to\nI(m)\nij(v) =CF(b)\nm(v)Z\nv−ΩuiujE(b)\n−m(u) du ,\nIij(f)(v) =X\nm∈ZdfmI(m)\nij(v),(i, j)∈ M 1,\nJijk(f)(v) =X\nm∈Zdµ(bk1, bk2)\nmkfmI(m)\nij(v),(i, j, k )∈ M 2,\nv∈Ω.(26)\n202.8 Approaches NST1 and NRT1 (Non-Conservative form)\nNon-conservative formulation. With regard to numerical comparisons for different\ntest problems, it is worth mentioning that alternative approaches in the lines of CST2\nand CST1, based on a non-conservative formulation of the Landau operator and hence\navoiding numerical differentiation of the integral operator, are advantageous in certain\nsituations, see for instance (28). However, due to the fact that intrinsic properties such\nas the conservation of mass are lost, additional technicalities are needed, and the overall\ncomputation times are higher, we refrain from detailed descriptions and merely include the\nnumerical results obtained for the general approach NST1 and its simplification NRT1 in\nthe special case of a regular kernel.\n3 Numerical validations and comparisons\nIn this section, we are concerned with a thorough numerical validation and comparison\nof the approaches summarised in Table 1. For this purpose, we next state test problems\ninvolving constant, regular, and singular integral kernels.\n3.1 Test problems\nTest problem A (Constant integral kernel, Unbounded domain). In the special\ncase of Maxwellian molecules in two and three dimensions, the integral kernels are defined\nby certain constants\nd= 2, φ =C=1\n16,\nd= 3, φ =C=1\n24,(27a)\nand the underlying velocity domains coincide with the Euclidian spaces. In our numerical\nexperiments, we make use of the fact that particular choices of the density functions permit\nto determine the associated integral and Landau operators by straightforward calculations\nd= 2,\nf(v) =1\nπe−|v|2|v|2,\nQ(c)(f, f)(v) =−1\n16πe−|v|2\u0000\n|v|2−2\u0001\nv ,\nQ(f, f)(v) =1\n8πe−|v|2\u0000\n|v|4−4|v|2+ 2\u0001\n,\nv∈Ω =R2,(27b)\nd= 3,\nf(v) =1\n2π3/2e−|v|2\u0000\n2|v|2−1\u0001\n,\nQ(c)(f, f)(v) =−1\n12π3/2e−|v|2\u0000\n|v|2−5\n2\u0001\nv ,\nQ(f, f)(v) =1\n6π3/2e−|v|2\u0000\n|v|4−5|v|2+15\n4\u0001\n,\nv∈Ω =R3.(27c)\n21Further numerical tests for the Landau equation rely on the knowledge of the BKW solu-\ntions\nd= 2,(α1, α2, α3) =\u0000\n2,1,1\n8\u0001\n,\nd= 3,(α1, α2, α3) =\u00005\n2,3\n2,1\n6\u0001\n,\nK(t) = 1−1\n2e−α3t,\nf(v, t) =1\n(2π K(t))d/2e−1\n21\nK(t)|v|2\u0000\nα1−α21\nK(t)+1\n21−K(t)\n(K(t))2|v|2\u0001\n,\nv∈Ω =Rd, t∈[t0, T],(27d)\nsee [3, Ex. 1 and 2]. For three dimensions, the profile of the BKW solution is illustrated\nin Figure 12.\nTest problem B (Regular integral kernel, Bounded domain). First artificial test\nproblems in two and three dimensions involve regular integral kernels and bounded velocity\ndomains\nφ(v) = cos( v1)···cos(vd), v∈Ω = [−π, π]d. (28a)\nThe prescribed density functions\nf(v) = sin( v1)···sin(vd), v∈Ω = [−π, π]d, (28b)\nare chosen such that the integral operators result from straightforward calculations\nd= 2,\nq1(v) =π2\n8cos(2 v2)\u0000\n2v1cos(2 v1)−sin(2 v1)−2v1\u0001\n,\nq2(v) =−π2\n4cos(2 v1) sin( v2)\u0000\n2v2sin(v2) + cos( v2)\u0001\n,\nQ(c)(f, f)(v) =\u0000\nq1(v), q2(v)\u0001T,\nv∈Ω = [−π, π]2,(28c)\nd= 3,\nq11(v) =\u0000\ncos(2 v3)−1\n2\u0001\u0000\nv1cos(2 v1)−1\n2sin(2 v1)−v1\u0001\ncos(2 v2),\nq12(v) =\u00001\n4sin(2 v1)−1\n2v1cos(2 v1)\u0001\ncos(2 v3) +v1\u0000\ncos2(v3)−1\n2\u0001\n,\nq1(v) =−π3\n4\u0000\nq11(v) +q12(v)\u0001\n,\nq21(v) =\u0000\ncos(2 v3)−1\n2\u0001\u0000\nv2cos(2 v2)−1\n2sin(2 v2)−v2\u0001\ncos(2 v1),\nq22(v) =\u00001\n4sin(2 v2)−1\n2v2cos(2 v2)\u0001\ncos(2 v3) +v2\u0000\ncos2(v3)−1\n2\u0001\n,\nq2(v) =−π3\n4\u0000\nq21(v) +q22(v)\u0001\n,\nq31(v) =1\n4\u0000\n2v3cos(2 v3)−sin(2 v3)−2v3\u0001\n,\nq32(v) = 2 cos(2 v1) cos(2 v2)−cos(2 v1)−cos(2 v2),\nq3(v) =−π3\n4q31(v)q32(v),\nQ(c)(f, f)(v) =\u0000\nq1(v), q2(v), q3(v)\u0001T,\nv∈Ω = [−π, π]3.(28d)\n22The associated Landau operators are obtained by differentiation\nd= 2,\nq1(v) =\u0000\n2v2sin(2 v2) + 1\u0001\ncos(2 v1),\nq2(v) =\u0000\n2v1sin(2 v1) + 1\u0001\ncos(2 v2),\nQ(f, f)(v) =−π2\n4\u0000\nq1(v) +q2(v)\u0001\n,\nv∈Ω = [−π, π]2,(28e)\nd= 3,\nq11(v) =v2sin(2 v2) cos(2 v3) +v3cos(2 v2) sin(2 v3),\nq12(v) =1\n2\u0000\n−v2sin(2 v2)−v3sin(2 v3) + cos(2 v2) + cos(2 v3)−1\u0001\n,\nq1(v) =\u0000\nq11(v) +q12(v)\u0001\ncos(2 v1),\nq21(v) =\u0000\nv1sin(2 v1) +1\n2\u0001\ncos(2 v3),\nq22(v) =−1\n2\u0000\nv1sin(2 v1) +v3sin(2 v3) + 1\u0001\n,\nq2(v) =\u0000\nq21(v) +q22(v)\u0001\ncos(2 v2),\nq3(v) =−1\n2\u0000\nv1sin(2 v1) +v2sin(2 v2)\u0001\ncos(2 v3)−cos2(v3) +1\n2,\nQ(f, f)(v) =π3\n2\u0000\nq1(v) +q2(v) +q3(v)\u0001\n,\nv∈Ω = [−π, π]3.(28f)\nTest problem C (Regular integral kernel, Unbounded domain). Further artificial\ntest problems in two and three dimensions are defined by Gaussian-like integral kernels\nand density functions\nd= 2, φ (v) = e−v2\n1−2v2\n2, f(v) = e−1\n2v2\n1−1\n4v2\n2,\nd= 3, φ (v) = e−v2\n1−2v2\n2−3v2\n3, f(v) = e−1\n2v2\n1−1\n4v2\n2−1\n8v2\n3,\nv∈Ω =Rd.(29a)\nThe associated integral and Landau operators are given by\nd= 2,\nq1(v) =√\n6πe−5\n6v2\n1−17\n36v2\n2,\nQ(c)(f, f)(v) =q1(v)\u0010\n−1\n2187v1\u0000\nv2\n2+ 18\u0001\n,1\n729v2\u0000\nv2\n1+ 3\u0001\u0011T\n,\nQ(f, f)(v) =−1\n13122q1(v)\u0000\n7v2\n1v2\n2−198v2\n1+ 57v2\n2+ 54) ,\nv∈Ω =R2,(29b)\n23d= 3,\nq1(v) =√\n3π3/2e−5\n6v2\n1−17\n36v2\n2−49\n200v2\n3,\nq21(v) =−1\n6834375v1\u0000\n1250v2\n2+ 243 v2\n3+ 46800\u0001\n,\nq22(v) =1\n2278125v2\u0000\n1250v2\n1−9v2\n3+ 2850\u0001\n,\nq23(v) =1\n91125v3\u0000\n27v2\n1+v2\n2+ 99\u0001\n,\nQ(c)(f, f)(v) = 2 q1(v)\u0000\nq21(v), q22(v), q23(v)\u0001T,\nq31(v) = 17500 v2\n1v2\n2+ 7047 v2\n1v2\n3+ 135 v2\n2v2\n3,\nq31(v) =−1005300 v2\n1+ 111000 v2\n2+ 46899 v2\n3+ 369900 ,\nQ(f, f)(v) =−1\n41006250q1(v)\u0000\nq31(v) +q32(v)\u0001\n,\nv∈Ω =R3.(29c)\nTest problem D (Singular integral kernel, Unbounded domain). In the lines of [3,\nEx. 3 and 4] illustrating a two-dimensional anisotropic solution and the three-dimensional\nRosenbluth problem with Coulomb potential, we finally study the numerical evaluation of\nLandau operators that involve singular integral kernels and density functions defined as\nd= 2,\nφ(v) =1\n161\n|v|3, f(v) =1\n4π\u0000\ne−1\n2((v1+2)2+(v2−1)2)+ e−1\n2(v2\n1+(v2+1)2)\u0001\n,\nd= 3,\nφ(v) =1\n4π1\n|v|3, f(v) =1\nc2\n1e−c1/c2\n2(|v|−c2)2, c 1= 10 , c 2=3\n10,\nv∈Ω =Rd.(30)\nFor the relevant three-dimensional case, the profile of the solution to the associated Landau\nequation is shown in Figure 13.\n3.2 Numerical results\nReliability and efficiency. In our numerical tests for the different approaches outlined\nin Table 1, we first resort to the test problems A–C with known solutions, see (27) to (29).\nWe in particular monitor the achieved accuracy when evaluating the Landau operator and\nintegrating the associated Landau equation in time. In addition, we measure the reliability\nof our approaches in a long-term integration through the conservation of mass, momentum,\nand energy as well as the decay of entropy, see (7). Afterwards, we extend our studies to\nthe most relevant case of Coulomb interaction (30). We recall that our strategy for the\nregularisation of the singular integral kernel is illustrated in Figure 1. Throughout, with\nthe conclusion from Table 2 in mind, we use quadrature approximations based on few grid\npoints. We point out that the observed computation times in Matlab are erratic and thus\nserve as rough indicators for the overall cost. We expect improvements for implementations\nbased on efficient software packages.\n24First validation. In sight of the complexity of the problem, it is reasonable to validate\nour theoretical considerations in two and three dimensions step-by-step. First numerical\ntests rely on the knowledge of the Landau operators in the Maxwellian molecules case (Test\nproblem A) for special choices of the density functions, see (27). We study the approaches\nCRT2 and CRT1, described in Section 2.7, which are suitable for regular kernels. On the\none hand, we vary the truncated domain and the Fourier spectral discretisation in each\ndirection\nd= 2,\nΩ(b)= [−9,10]×[−10,11], M = (100 ,110),\nd= 3,\nΩ(b)= [−9,10]×[−10,11]×[−11,12], M = (100 ,110,120),\nand on the other hand, we set\nΩ(b)= [−10,10]d, M i= 100 , i∈ {1, . . . , d }, (31)\nsee also (10) and (14). Besides, we contrast implementations using for-loops over index\nsets such as (15) or not, respectively. Due to the fact that consistent results regarding\naccuracy and computation time are observed in all cases, see Figure 2, we proceed with\nthorough numerical comparisons of the different approaches.\nNumerical comparisons. We next contrast the results obtained for the general ap-\nproaches CST2, CST1, NST1 and their simplifications CRT2, CRT1, NRT1. Setting\nagain (31), we consider a Landau operator involving a constant integral kernel (Maxwellian\nmolecules case, Test problem A) and a regular kernel (Test problem C), respectively, see\nFigures 3 and 5. Moreover, we demonstrate the issues of a bounded domain and numerical\ndifferentiation (Test problem B), see Figure 4. Specifically, the integral operator does not\nfulfill periodicity requirements and hence is not well represented by Fourier series. In addi-\ntion, wecomparecomputationtimesandrelativeaccuracieswithrespecttoapproachCST2\nwhen evaluating the Landau operator involving the singular kernel (Coulomb interaction,\nTest problem D), see Figure 6. Presumably, in the latter case, the main source of approxi-\nmation errors is linked to numerical quadrature. Due to the fact that the non-conservative\napproach NST1 does not preserve mass and requires quadrature approximations of the\nsingular integral kernel and its first-order derivatives, we select the conservative approach\nCST2 with a reduced computation time for further numerical tests.\nTime integration. For the time integration of the Landau equation, as indicated in\nSection 1, we combine our approach CST2 with explicit Runge–Kutta methods of non-\nstiff orders p∈ {1,2,3,4}. We once again set Ω(b)= [−10,10]dand choose 100×100\nFourier functions in two dimensions or 64×64×64Fourier functions in three dimensions,\nrespectively. The results displayed in Figures 7 and 8 confirm highly accurate numerical\noutcomes for the Maxwellian molecules case. Accordingly, we observe conservation of\nmass as well as nearby conservation of momentum and energy over time, see Figure 9.\nRegarding the decay of entropy, as negative contributions in the range of machine precision\nmay appear, we employ a projection on the strictly positive density function values. The\n25corresponding results for the significantly more demanding case of Coulomb interaction\nare shown in Figure 10. As a final test, we prescribe a significantly smaller truncated\ndomain and larger equidistant time steps. We observe expected quantitative effects, that\nis, a certain loss of accuracy, but still retain qualitatively reliable results, see Figure 11.\nSolution profiles. For the two relevant cases of Maxwellian molecules (Test problem A)\nand Coulomb interaction (Test problem D), the solution profiles at the initial and final\ntimes are shown in Figures 12 and 13. Movies illustrating the time evolution are found\nthrough [5].\nReferences\n[1] R. Bailo, J. A. Carrillo, and J. Hu. The collisional particle-in-cell method for the\nvlasov-maxwell-landau equations, 2024.\n[2] R. Bailo, J. A. Carrillo, A. Medaglia, and M. Zanella. Uncertainty quantification for\nthe homogeneous Landau-Fokker-Planck equation via deterministic particle Galerkin\nmethods. Preprint arXiv: 2312.07218 , Dec. 2023.\n[3] J. A. Carrillo, J. Hu, L. Wang, and J. Wu. A particle method for the homogeneous\nLandau equation. J. Comput. Phys. X , 7:100066, June 2020.\n[4] J. A. Carrillo and M. Thalhammer. Time integration of Landau equation. doi.org/\n10.5281/zenodo.10407756.\n[5] J. A. Carrillo and M. Thalhammer. Time integration of Landau equation. dx.doi.org/\n10.6084/m9.figshare.25024328.\n[6] G. Dimarco, Q. Li, L. Pareschi, and B. Yan. Numerical methods for plasma physics\nin collisional regimes. J. Plasma Phys. , 81(1):305810106, 2015.\n[7] G. Dimarco and L. Pareschi. Numerical methods for kinetic equations. Acta Numer. ,\n23:369–520, 2014.\n[8] F. Filbet and L. Pareschi. A numerical method for the accurate solution of the Fokker-\nPlanck-Landau equation in the nonhomogeneous case. J. Comput. Phys. , 179(1):1–26,\n2002.\n[9] I. M. Gamba. Deterministic solvers for nonlinear collisional kinetic flows: a conserva-\ntive spectral scheme for Boltzmann type flows. 18:403–433, 2017.\n[10] E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration , volume 31\nofSpringer Series in Computational Mathematics . Springer-Verlag, Berlin, second\nedition, 2006. Structure-preserving algorithms for ordinary differential equations.\n26[11] J. Hu and K. Qi. A fast Fourier spectral method for the homogeneous Boltzmann\nequation with non-cutoff collision kernels. J. Comput. Phys. , 423:109806, 21, 2020.\n[12] A. Iserles. A first course in the numerical analysis of differential equations . Cambridge\nTexts in Applied Mathematics. Cambridge University Press, Cambridge, second edi-\ntion, 2009.\n[13] L. D. Landau. Die kinetische gleichung für den fall Coulombscher wechselwirkung.\nPhys. Z. Sowjetunion , 10(2):154–164, 1936.\n[14] Y. Li, Y. He, Y. Sun, J. Niesen, H. Qin, and J. Liu. Solving the Vlasov-Maxwell\nequations using Hamiltonian splitting. J. Comput. Phys. , 396:381–399, 2019.\n[15] A. Mangeney, F. Califano, C. Cavazzoni, and P. Travnicek. A numerical scheme\nfor the integration of the Vlasov-Maxwell system of equations. J. Comput. Phys. ,\n179(2):495–538, 2002.\n[16] R. I. McLachlan and G. R. W. Quispel. Splitting methods. Acta Numer. , 11:341–434,\n2002.\n[17] L. Pareschi and T. Rey. Moment preserving Fourier-Galerkin spectral methods and\napplication to the Boltzmann equation. SIAM J. Numer. Anal. , 60(6):3216–3240,\n2022.\n[18] L. Pareschi and G. Russo. Numerical solution of the Boltzmann equation. I. Spectrally\naccurate approximation of the collision operator. SIAM J. Numer. Anal. , 37(4):1217–\n1245, 2000.\n[19] L. Pareschi, G. Russo, and G. Toscani. Fast spectral methods for the\nfokker–planck–landau collision operator. J. Comput. Phys. , 165(1):216–236, Nov.\n2000.\n[20] C. A. Pennie and I. M. Gamba. Entropy decay rates for conservative spectral schemes\nmodeling fokker-planck-landau type flows in the mean field limit. Preprint arXiv:\n1910.03110 , Oct. 2019.\n[21] J. M. Sanz-Serna and M. P. Calvo. Numerical Hamiltonian problems , volume 7 of\nApplied Mathematics and Mathematical Computation . Chapman & Hall, London,\n1994.\n[22] Y. Wang. The two-species Vlasov-Maxwell-Landau system in R3.Ann. Inst. H.\nPoincaré C Anal. Non Linéaire , 32(5):1099–1123, 2015.\n[23] C. Zhang and I. M. Gamba. A conservative scheme for vlasov poisson landau modeling\ncollisional plasmas. J. Comput. Phys. , 340:470–497, July 2017.\n27Approach CST2 (Section 2.5)\nBased on the conservative formulation of the Landau operator.\nUses numerical differentiation of the integral operator.\nAdapted to kernels with an isolated singularity at the origin.\nThe integral transform is applied to the singular kernel and its regularisation.\nApproach CST1 (Section 2.6)\nBased on the conservative formulation of the Landau operator.\nUses numerical differentiation of the integral operator.\nAdapted to kernels with an isolated singularity at the origin.\nThe integral transform is applied to the singular kernel.\nApproaches CRT2 and CRT1 (Section 2.7)\nSimplification of CST2 and CST1 for the case of a regular kernel.\nApproach NST1 (Section 2.8)\nBased on the non-conservative formulation of the Landau operator.\nAvoids numerical differentiation of the integral operator.\nAdapted to kernels with an isolated singularity at the origin.\nThe integral transform is applied to the singular kernel and its derivatives.\nApproach NRT1 (Section 2.8)\nSimplification of NST1 for the case of a regular kernel.\nTable 1: Overview on different approaches for the numerical evaluation of the Landau\ncollision operator (1).\nQuadrature on a small neighbourhood Precomputation time CT\nQuadrature on the whole domain 88×CT\nTable 2: Test problem C(regular integral kernel, unbounded domain, known solution) in\ntwo dimensions. Numerical evaluation of the Landau operator based on 256×256uniform\ngrid points covering the truncated velocity domain [−10,10]×[−10,10]. Precomputation\ntimes observed for a quadrature approximation based on 5×5grid points versus a quadra-\nture approximation on the whole domain based on 256×256grid points. In both cases,\nan overall relative accuracy of about 4·10−11is obtained.\n28Figure 1: Illustration of the singular integral kernel φ:R3\\ {0} →R:v7→ |v|−3arising\nin the case of Coulomb interaction, see (1) and (3). A regularised kernel is obtained by\ninterpolation on a small neighbourhood of the origin. The remaining difference vanishes\non the main part of the velocity domain.\n29Figure 2: Test problem A (Maxwellian molecules case) in two ( d= 2) and three ( d= 3)\ndimensions. The evaluation of the Landau operator is based on the approaches CRT2 (left)\nandCRT1(right)describedinSection2.7. Fordifferentimplementations(velocitydomains\ndefined by non-symmetric versus symmetric intervals, computation of fundamental inte-\ngrals without or with for-loops), consistent results concerning accuracy and computation\ntime are observed.\n30Figure 3: Test problem A (Maxwellian molecules case) in two and three dimensions. Nu-\nmerical comparisons of relative accuracies as well as precomputation and computation\ntimes. First and second rows: General approaches for the evaluation of the associated\nLandau operator based on the conservative form (CST2, CST1) and the non-conservative\nform (NST1). Third and fourth rows: Simplifications to regular kernels (CRT2, CRT1,\nNRT1). In all cases, highly accurate results are obtained.\n31Figure 4: Test problem B (regular integral kernel, bounded domain, known solution). The\napproaches based on the non-conservative formulation yield a satisfactory result (NST1,\nNRT1), whereas the approaches based on the conservative formulation and numerical dif-\nferentiation are not suitable (CST2, CST1, CRT2, CRT1).\n32Figure 5: Test problem C (regular integral kernel, unbounded domain, known solution)\nin two and three dimensions. Numerical comparisons of relative accuracies as well as\nprecomputation and computation times. General approaches for the evaluation of the\nassociated Landau operator based on the conservative form (CST2, CST1) and the non-\nconservative form (NST1). Simplifications to regular kernels (CRT2, CRT1, NRT1).\n33Figure 6: Test problem D (Coulomb case) in two and three dimensions. Numerical compar-\nisons of relative accuracies with respect to the first approximation obtained by approach\nCST2 as well as precomputation and computation times.\nFigure7: TestproblemA(Maxwellianmoleculescase)intwodimensions. Timeintegration\nof the Landau equation based on 100×100Fourier functions and explicit Runge–Kutta\nmethods of orders p∈ {1,2,3}. The absolute errors over time with respect to the known\nsolution values, obtained for a certain time increment and a reduced increment, confirm\nthe orders of convergence.\n34Figure 8: Test problem A (Maxwellian molecules case) in three dimensions. Time in-\ntegration of the Landau equation based on 64×64×64Fourier functions and explicit\nRunge–Kutta methods of orders p∈ {1,2,3}. Absolute errors over time with respect to\nthe known solution values, obtained for a certain time increment.\nFigure 9: Test problem A (Maxwellian molecules case) in two and three dimensions. Time\nintegration of the Landau equation based on an explicit Runge–Kutta method of order\np= 4. Mass is conserved. First momentum, energy, and entropy over time.\n35Figure 10: Test problem D (Coulomb case) in two and three dimensions. Time integration\nof the Landau equation based on an explicit Runge–Kutta method of order p= 4. Mass is\nconserved. First momentum, energy, and entropy over time.\nFigure 11: Corresponding results for test problem D (Coulomb case) in three dimensions\nwith a significantly smaller truncated domain and a reduced number of time steps.\n36Figure 12: Numerical illustration of the BKW solution to the Landau equation involving a\nconstant kernel in three dimensions (Maxwellian molecules case), see (27). The initial time\nt0= 6 ln(5\n4)is chosen in such a way that the non-negativity of the solution is ensured.\n37Figure 13: Numerical illustration of the solution to the Landau equation involving a sin-\ngular kernel in three dimensions (Coulomb case), see (30).\n38"    },    {        "title": "2402.02253v1.Rotating_Black_Holes_in_a_Viable_Lorentz_Violating_Gravity__Finding_Exact_Solutions_Without_Tears.pdf",        "content": "arXiv:2402.02253v1  [hep-th]  3 Feb 2024arXiv:2402.xxxx [hep-th]\nRotating Black Holes in a Viable Lorentz-Violating Gravity :\nFinding Exact Solutions Without Tears\nDeniz O. Devecio˘ glu∗1and Mu-In Park†1\n1Center for Quantum Spacetime, Sogang University, Seoul, 12 1-742, Korea\n(Dated: February 6, 2024)\nAbstract\nWe introduce a two-step procedure for finding Kerr-type rota ting black hole solutions without\ntears. Considering the low-energy sector of Hoˇ rava gravit y as aviableLorentz-violating gravity in\nfour dimensions which admits a different speed of gravity, we fi nd the exact rotating black hole\nsolutions(with orwithoutcosmological constant). Wefindt hat thesingularregion extendsto r <0\nregion from the ring singularity at r= 0 in Boyer-Lindquist coordinates. There are two Killing\nhorizons where grr= 0 andthe black hole thermodynamics laws are still valid. We findthe rotating\nblack hole solutions with electromagnetic charges only whe n we consider the nobleelectromagnetic\ncouplings, in such a way that the speed of light is the same as t he speed of gravity. With the noble\nchoice of couplings, our Lorentz-violating gravity can be c onsistent with the recently-observed time\ndelay of the coincident GW and GRB signals. Furthermore, in A ppendices, we show that (a) the\nuniqueness of the invariant line element ds2underDiffF, contrary to LV action, (b) the solutions\nare the Petrov typeI with four distinctprincipal null vecto rs, and (c) the Hamilton-Jacobi equation\nfor the geodesic particles are notseparable.\nDedicated to the memory of Roman Jackiw (8 November 1939 - 14 J une 2023).\nKeywords: Horava gravity, Rotating black hole solutions, Black hole thermodynamics, Gravitational waves,\nLorentz violations\n∗E-mail address: dodeve@gmail.com\n†E-mail address: muinpark@gmail.com, Corresponding author\n1I. INTRODUCTION\nThe rotating black hole solution which was discovered by Kerr [1] in gen eral relativity\n(GR) has occupied an immovable position: The solution structure is qu ite rigid and it is\nextremely difficult to find another exactsolution with non-trivial deformations by cracking\nthe rigid solution structure. Moreover, even deriving the known Ke rr solution is not quite\neasy task and requires some sophisticated methods whose generic validities in other gravity\ntheories are not clear [2]. In particular, from the separability of Ham ilton-Jacobi equation\nfor the geodesic particles in a quite general context, Carter obta ined the Carter constant\nas well as the Kerr solution [3]. Later, it was proved that the Petrov type-D solutions, like\nKerr metric, guarantee the existence of a Killing tensor and its conc rete form was derived\nin GR [4]\nInthispaper, weintroduceatwo-stepproceduretofindKerr-ty peexact solutionswithout\nmuch tears. We apply the procedure to the low-energy sector of H oˇ ravagravity as a Lorentz-\nviolating (LV) gravity in four dimensions which admits a different speed of gravity, and we\nfind the exact rotating black hole solutions (with or without cosmolog ical constant). In\naddition, if we consider charged rotating black holes, the nobleelectromagnetic couplings\nare needed for consistency, in such a way that the speed of light is t he same as the speed of\ngravity. In Sec. II, we introduce our two-step procedure to find Kerr-type exact solutions.\nIn Sec. III, we apply the procedure to the low-energy sector of H oˇ rava gravity in four\ndimensions and find the exact rotating black hole solution without cos mological constant.\nIn Sec. IV, we study the singularity structure which is richer than K err solution. In Sec.\nV, we study the horizon structure and new barriers for geodesic p articles which are genuine\nto our LV gravity. In Sec. VI, we consider the generalizations with c osmological constant\nand electromagnetic charges, and show that the first law of black h ole thermodynamics is\nsatisfied. In Sec. VII, we consider some observational constrain ts and in Sec. VIII, we\nconclude with some discussions. In Appendix B, we prove the uniqueness of the invariant\nlineelement ds2underDiffF, contrarytowidespreadbelief basedonLVaction. InAppendix\nC, we study the Petrov classification, the Killing tensor, and the Hamilt on-Jacobi equation.\nII. A TWO-STEP PROCEDURE TO FIND KERR-TYPE STATIONARY SOLU-\nTIONS\n(i) Suppose that there is an exact“massless” rotating space-time solution, which means\nit is the exact solution for an arbitrary rotation parameter a. This is the first step and the\nmassless solution is a seedsolution for the step 2.\n(ii) The second step is to introduce the “mass”-dependent ansatz functions into the\nmassless seed solution, without loss of much generality but still in a solvable way ,i.e., with\nordinary differential equations for the polar angle θor the radial coordinate r. This requires\nnoble insights as well as some trials and errors. Without cosmological constant, we find\nthat the situation becomes simpler since the massiverotating solution does not deform the\nangle-dependent parts but just the radial-dependent parts of t hemassless rotating space-\ntime solution. On the other hand, with a cosmological constant, it is a bit more complicated\nandweneed amoreclever choice oftheansatz. So, inthefollowing se ctions, we first consider\nthe solution without cosmological constant and later with a cosmolog ical constant.\nThe above two-step procedure can be applied to any gravity theor y, in principle. In this\npaper, we apply the procedure to find the rotating black hole solutio ns in the low-energy\n2sector of Hoˇ rava gravity in four dimensions, for the first time.\nIII. LOW-ENERGY SECTOR OF HO ˇRAVA GRAVITY AND ROTATING BLACK\nHOLE SOLUTIONS\nThelowenergysector of(non-projectable) Hoˇ ravagravity[5,6 ]isdescribed bytheaction,\nup to boundary terms,\nSg=/integraldisplay\nR×Σtdtd3x√gN/bracketleftbigg1\nκ/parenleftig\nKijKij−λK2/parenrightig\n+ξR−2Λ+σ\n2aiai/bracketrightbigg\n, (1)\nwhereKij= (2N)−1(˙gij−∇iNj−∇jNi) is the extrinsic curvature [the overdot (˙) denotes\nthe time derivative and ∇iis the covariant derivatives for the induced metric gijon the\ntime-slicing hypersurface Σ t] andRis thethree-curvature in the ADM metric\nds2=−N2c2dt2+gij/parenleftig\ndxi+Nidt/parenrightig/parenleftig\ndxj+Njdt/parenrightig\n, (2)\nrespectively. (Hereafter, we shall use the unit c= 1, unless stated otherwise.) The last\nterm in the gravity action (1) is introduced for completeness, with t he proper acceleration\nai=∇ilnN[8].\nThe equations from the variations of N,Ni, andgijare given by\nH ≡1\nκ/parenleftig\nKijKij−λK2/parenrightig\n−ξR+2Λ−σ/parenleftigg1\n2∇iN∇iN\nN2−∇k∇kN\nN/parenrightigg\n= 0,(3)\nHi≡2\nκ∇j/parenleftig\nKji−λKgji/parenrightig\n= 0, (4)\nEij≡1\nκ/parenleftig\nE(1)\nij−λE(2)\nij/parenrightig\n+ξE(3)\nij+σ\n2E(4)\nij= 0, (5)\nwhere\nE(1)\nij=Ni∇kKk\nj+Nj∇kKk\ni−Kk\ni∇jNk−Kk\nj∇iNk−Nk∇kKij\n−2NKikKjk−1\n2NKkℓKkℓgij+NKK ij+˙Kij,\nE(2)\nij=1\n2NK2gij+Ni∂jK+Nj∂iK−Nk(∂kK)gij+˙Kgij,\nE(3)\nij=N/parenleftig\nRij−1\n2Rgij+Λ\nξgij/parenrightig\n−(∇i∇j−gij∇k∇k)N ,\nE(4)\nij=1\nN/parenleftig\n−1\n2gij∇kN∇kN+∇iN∇jN/parenrightig\n.\nWith the arbitraryparameters λ,ξ, andσ, theaction does not admit thefull diffeomorphism\nbut only the foliation-preserving diffeomorphism ( DiffF) [6, 7],\nδξt=−ξ0(t), δξxi=−ξi(t,x),\nδξN= (Nξ0),0+ξk∇kN,\nδξNi=ξ0\n,0Ni+ξj\n,0gij+∇iξjNj+Ni,0ξ0+∇jNiξj,\nδξgij=∇iξkgkj+∇jξkgki+gij,0ξ0. (6)\n3Note that the Einstein-aether theory reduces to the same gravit y action (1) with the\nhypersurface-orthogonal aether field [9] or the khronometric theory with a globaltime-like\nkhronon field [10]. In this paper, we do not introduce additional assu mptions on the aether\nor khronon field, but we only study the gravity action (1).\nThe first step toward the rotating black hole solutions is to find the massless rotating\nblackholesolutions. Here, wefirstconsider theasymptotically flatcasewithoutcosmological\nconstant and the aiextension, i.e., Λ = 0, σ= 0, for simplicity. To this end, we note the\nrecent finding by one of authors that the massless Kerr solution is a lso a solution of Hoˇ rava\ngravity with an arbitrary λ[11],\nds2\n0=−ρ2∆(0)\nr\nΣ2\n(0)dt2+ρ2\n∆(0)\nrdr2+ρ2dθ2+Σ2\n(0)sin2θ\nρ2dφ2, (7)\nwhere,ρ2=r2+a2cos2θ,∆(0)\nr= (r2+a2),Σ2\n(0)= (r2+a2)ρ2,and (t,r,θ,φ) are the Boyer-\nLindquist coordinates [3, 12, 13]. The metric (7) is nothing but the flatMinkowski spacetime\nwritten in the ellipsoidal spacial coordinates, which are related to Cartesian coordinates\nx= (r2+a2)1/2sinθcosφ,y= (r2+a2)1/2sinθsinφ,z=rcosθ.\nHere, it is important to note that r= 0 is not the end of the coordinates but there\nis another copy of Kerr spacetime in the r <0 regime, with another asymptotic infinity\natr→ −∞in ellipsoidal coordinates system. Actually, the massless Kerr metric (7) was\ninterpreted as a wormhole solution in [14]. Of course, there is no even t horizon in (7) but\nthe genuine spacetime deformation for a rotating object is b elieved to be naturally encoded\nin the ellipsoidal coordinates with the rotation parameter a.\nThe next step towards the generic rotating black hole solutions is to consider the mass-\ndependent ansatz functions without loss of generality, until we may obtain the consistent\nsolutions . In the known Kerr metric, the mass term is tightly bounded to the r otation\nparameter aand it is not easy to separate them. So, it is important to consider a g eneric\nansatz such that the spin part and the mass part are consistently separated which might\ncrack the rigid structure of Kerr solution. Now, by comparing to Ke rr solution, we consider\nthe metric ansatz,\nds2\n1=−N2dt2+ρ2\n∆rdr2+ρ2dθ2+Σ2sin2θ\nρ2/parenleftig\ndφ+Nφdt/parenrightig2, (8)\nwhere we introduce\nΣ2=/parenleftig\nr2+a2/parenrightig\nρ2+f(r)a2sin2θ,\nN2=ρ2∆r(r)\nΣ2, Nφ=−g(r)\nΣ2(9)\nwith three r-dependent undetermined functions f(r),g(r), and ∆ r(r). Then, by solving the\nfull equations of motion (3-5) for the ansatz (8) and (9), we can d etermine the undetermined\nfunctions uniquely as,\nf(r) = 2mr, g(r) = 2amr/radicalig\nκξ,∆r(r) =r2+a2−2mr, (10)\nwhich reduce tothe known Kerr solutioninthe GRcase of ξ= 1/κorSchwarzschild solution\nwitha= 0, where mis an integration constant parameter. Here, we adopt the usual c hoice\n4ofNφ|∞= 0,W(∞)≡N√grr|∞= 1. It is rather surprising that the Kerr-solution cracking\nterm with the LV factor√κξappears only in Nφ. But, if we look at the component form\nds2\n1=/bracketleftigg\n−(∆r−a2sin2θ)\nρ2+(κξ−1) (2mr)2a2sin2θ\nρ2Σ2/bracketrightigg\ndt2\n+ρ2\n∆rdr2+ρ2dθ2+Σ2sin2θ\nρ2dφ2−4amr√κξsin2θ\nρ2dtdφ, (11)\none can easily see the non-trivial LV effect for ξ/negationslash= 1/κingttas well as in gtφcomponents.\nAnother notable property is that our solution, as well as the Kerr s olution with ξ= 1/κ,\nare valid for an arbitrary λdue toK= 0,i.e.,“maximal” slicing. This makes even Kerr\nsolution or Schwarzschild solution with a= 0 has different notions of singularities as we can\nsee in the next section, due to the lack of the full Diffwithλ/negationslash= 1.\nThe mass and angular momentum are computed as\nM= 16πξm, J= 16πam/radicalig\nξκ−1, (12)\nby the standard definition [15], as the conjugates to W(∞) andNφ(∞), respectively, in the\nboundary term\nB= (t2−t1)[−W(∞)M+Nφ(∞)J] (13)\nwhich makes the total action S=Sg+Bto bedifferentiable with the boundary conditions\nδNφ(∞) =δW(∞) = 0. We obtain the same result in the recently-proposed conserve d-\ncharge formula in a covariant formalism [16] and it provides another non-trivial evidence of\nthe formalism. The generalizations to the asymptotically (A)dSblack holes (Λ /negationslash= 0) and\nthechargedblack holes will be considered in the later part, due to some complicatio ns.\nHowever, with the aiextension term, we find no solution for the ansatz (8) and (9), exc ept\nthe massless solution, which indicates that, with the aiterm, the solution structure will be\nquite different than ours or GR. So, from now on we consider only the standard case without\nai,i.e.,σ= 0.\nIV. SINGULARITY STRUCTURE\nDue to the apparent lack of the full Diff, we have a different notion of singularities. In our\ncase, thephysical singularities arecapturedby DiffFinvariant curvatures( K= 0, KijRij=\n0),\nR∼a2m2\nρ6Σ4, RijRij∼m2\nρ12Σ8, KijKij∼κξa2m2\nρ6Σ4, (14)\nwhere we have omitted the other detailed factors which are finite an d are not canceled by\nthe denominators. It is interesting to note that the omitted facto rs are not modified by our\nLV solutions and they are exactly the same as in GR. The distinct feat ure in our case is\nthat the singularity properties of each quantity in (14) are physica lly meaningful and they\nshow the curvature singularities at Σ2= 0, as well as the usual singularity at ρ2= 0. On\n50.5 1.0 1.5 2.0 2.5\n-2-112\n-4 -2 2 4\n-0.4-0.20.20.4\nFIG. 1: Left: rvs.θ[0,π] of singularity surfaces for Σ2(r,θ) = 0, by varying a= 2,1,0.1 (from\nthe outer to inner curves) with m= 2 (the lower branch), m=−2 (the upper branch). Right: The\nextreme radius r0vs.m, whereθbecomes an extreme value for a given aandm, andr0→ ∓a/√\n3\nasm→ ±∞.\nthe other hand, the four-dimensional curvature invariants in GR a re given by\nR(4)∼(κξ−1)a2m2\nρ6Σ4, R(4)\nµνR(4)µν∼(κξ−1)2a4m4\nρ12Σ8,\nR(4)\nµνσρR(4)µνσρ∼(κξ−1)m2\nρ12Σ8+m2\nρ12(···), (15)\nwhere the omitted term in ( ···) is the same as in GR. This clearly shows that the additional\nsingularities of (14) at Σ2= 0 are exactly canceled in the GR limit ξ=κ−1so that only\nthe usual ring singularity at ρ= 0,i.e.,r= 0,θ=π/2 remains in GR1. Furthermore,\neven in the GR limit ξ=κ−1,i.e., Kerr solution or even Schwarzschild solution, due to the\nphysical meaningfulness of DiffFinvariant quantities in (14), not (15), we still have the\nsame additional singularities at Σ2= 0.\nFig. 1 shows the singularity surfaces of\ncos2θ=2mra2+r2(r2+a2)\na2(2mr−(r2+a2))(16)\nfor Σ2= 0, by varying the mass parameter mand the rotation parameter a. The lower\n(r <0) and the upper ( r >0) branches correspond to m >0 andm <0, respectively (the\nleft panel).\nAsm→ ±∞, the singularity surfaces cover the whole region of r <0 orr >0, with\nr0=∓a/√\n3 which touches the θboundaries θ= 0,π(the right panel). As a→0, the\nsingularity surfaces reduce to the point singularity of Schwarzsch ild black hole at r= 0.\nOther than these extreme cases, i.e.,0< a≤m <∞, there exits always a time-like\ntrajectory which avoids the singularity surfaces such that the clo sed-time-like (CTC) curves\nare possible, similarly to Kerr solution in GR.\n1This is a common feature in Hoˇ ravagravity[17, 18], as was once note d by S. Mukohyama earlier (a private\ncommunication).\n6V. KILLING HORIZONS AND PARTICLE BARRIERS\nIn our coordinates, there are two event horizons at r±wheregrr= ∆r/ρ2= 0, which\noccurs when\n∆r=r2+a2−2ma= 0, (17)\nsinceρ2>0, excluding the singularity at ρ2= 0 that was studied in Sec. IV. The event\nhorizons at r±are Killing horizons with the Killing vectors χa\n±= (1,0,0,Ω±), which become\nnullvectorsχa\n±χa±=−ρ2∆r/Σ2+ Σ2sin2θ(Nφ−Ω±)2/ρ2= 0 for the constant angular\nvelocities on the horizons r±, Ω±=−Nφ|r±. Then, one can define the surface gravities κ±\nsuch that χν\n±∇(4)\nνχµ\n±=κ±χµ\n±onthe horizons r±2.\nFrom the hypersurface-orthogonality of the Killing horizons, one can find that κ±are\nconstants on the corresponding horizons r±,i.e.,\nχ[µ∇ν]\n(4)κ±=−χ[µRν]\nσ(4)χσ=−χ[µTν]\nσ(eff)χσ= 0, (18)\nwhere we have expressed our non-covariant equations (3-5) into thecovariant formR(4)\nµν−\n(1/2)g(4)\nµνR(4)=T(eff)\nµνwith aneffective energy-momentum tensor,\nTµ\nν(eff)=\n/hatwideρ0 0 0\n0/hatwiderp1/hatwiderp20\n0/hatwiderp3−/hatwiderp10\n/hatwiderp40 0−3/hatwideρ,\n(19)\nwhich is a non-perfect fluid form (see Appendix Afor the explicit expressions of /hatwideρand\n/hatwiderpa), but still satisfies the covariant conservation equation ∇(4)\nµTµν\n(eff)= 0. The effective\nenergy-momentum violates the dominant energy condition, T00(eff)≥ |Tµν(eff)|, especially\nby|T33(eff)|= 3|/hatwideρ|>/hatwideρ, butitsatisfies χ[µTν]σ(eff)χσ= 0onthehorizons r±sothatthe zeroth\nlaw (18) is satisfied. In other words, the usual notion of Killing horizo ns atr±satisfying the\nzeroth law are still valid in our LV gravity, which seems to be a quite non -trivial result since\nthe effective energy-momentum tensor T(eff)\nµνmight break the last equation of (18), generally.\nNow, we turn to the discussion on the role of the event horizons to t he geodesic particles\nto see whether they have the similar roles even in our LV gravity also. To this end, we\nconsider the conserved energy E=−vµ(∂t)µand angular momentum L=vµ(∂φ)µ, for the\ntime-like Killing vector ( ∂t)µand the axial Killing vector ( ∂φ)µ, and the on-shellequations\n−ǫ=gµνvµvν, (20)\nwherevµ=xµ′≡dxµ/dτfor a particle’s 4-velocity with the proper time τ, andǫ= +1 (−1)\nfor a time-like (space-like) geodesics and ǫ= 0 for a null geodesics.\nHere, it is important to note that the line element (2) describes the invariant distance\nmeasure even in our LV case , contrary to a widespread belief based on the symmetry trans-\nformation of action which allows LV action (1) with the reduceddiffeomorphism DiffF.\n2Thedirectcomputationin ourcoordinatesisrathertrickyandwefir stneedtoconsider χν\n±∇(4)\nν/hatwideχµ\n±=κ±/hatwideχµ\n±\nwith/hatwideχµ\n±= (1,∆r\nr2+a2,0,Ω±) and then consider the limit /hatwideχµ\n±→χµ\n±at the horizons.\n7Actually, one can prove that the invariant line element is only given by (2), due to mix-\ning between different components under the coordinate transfor mation, even if we consider\nDiffF(see Appendix Bfor the proof). This justifies EandLas the conserved quantities,\nindependently on the chosen coordinates , even in Hoˇ rava gravity, which has neverbeen\nknownin the literature.\nThen, by solving t′,φ′, andr′in terms of the conserved quantities EandL, one obtains\nt′=/bracketleftig\n(r2+a2)ρ2+2mra2sin2θ/bracketrightig\nE−2mra√κξL\nρ2∆r, (21)\nφ′=2mra√κξE+gttρ2L/sin2θ\nρ2∆r, (22)\nr′=±/radicalbigg\n−2/parenleftig\nV+gθθθ′2/2grr/parenrightig\n, (23)\nwhere\nV=−ǫmr\nρ2+L2\n2sin2θρ2+(ǫ−E2)(r2+a2)\n2ρ2−mr\nρ4/parenleftbigg\n−aEsinθ+L\nsinθ/parenrightbigg2\n+2(√κξ−1)mraEL\nρ4−(κξ−1)(2mra)2L2\n2ρ4Σ2. (24)\nFor a complete analytic integration of the geodesics, including θdirection, we would need\nanother constant of motion (we will discuss more about this in Appen dixC) but there is an\nimportant case where we can neglect θ′, that is the equatorial geodesics, θ=π/2: For other\nangles,θ/negationslash=π/2, thefullgeodesic equation (20) tells us that the θ=constant plane is not\nmaintained generally, due to θ′′/negationslash= 0 even if θ′= 0 initially.\nThen, for the equatorial and the co-rotating geodesics with L=aE, which corresponds\nto the radial geodesics in non-rotating geometries [2], we obtain [+ ( −) denotes the out\n(in)-going geodesics]\ndt\ndr=±(r2+a2)−2(√κξ−1)ma2r−1\n∆r√\n−U, (25)\ndφ\ndr=±a/bracketleftig\n1+2(√κξ−1)mr−1−(κξ−1)(2ma)2((r2+a2)r2+2mra2)−1/bracketrightig\n∆r√−U,(26)\nwhere\n−U=−ǫ∆r\nE2r2+1−4(√κξ−1)ma2\nr3+(κξ−1)(2ma2)2\nr2[(r2+a2)r2+2mra2]. (27)\nNote that (25) and (26) show the characteristic behaviors of the light-cones , which “close\nup” or “peel off”, as the event horizons at r±are approached with ∆ r→0, especially for\nthe null geodesics ( ǫ= 0),independently on the particle’s energy E=La−1.\nMoreover, with the coupling κξ >1 for null particles but arbitrary κξfor time-like\nparticles, it shows a potential barrier, i.e.,U >0. From the condition that the barrier\nshould not exist outside the even horizon r+so that the identity of black holes can be still\nmaintained, one finds the condition κξ≤4, which gives its physical upper bound (Fig. 2):\nIn the GR limit of κξ= 1, the barrier also exists for time-like cases (the right panel) thou gh\n80.0 0.5 1.0 1.5 2.0 2.5 3.00.51.01.52.0\n0 1 2 3 412345\nFIG. 2: Mass parameter mvs. horizons r±(the top curves) or barriers (the bottom curves) for null\n(the left panel) and time-like (the right panel) trajectori es. The bottom curves denote the barrier\nregions by varying Eandκξ: For the null trajectories (left panel), we take κξ= 1.5,4,10 (left to\nright); for the time-like trajectories (right panel), we ta keκξ= 2,E= 0.3, andκξ= 2,4,10,E= 2\n(left to right). For time-like cases (right panel), the orbi ts are bounded for E < ǫ(yellow curve).\nThe barriers exist even when the event horizons r±are absent when m < m ∗, with the extremal\nblack hole mass m∗, where two horizons merge.\nabsent in null cases (the left panel), but it exists always inside the ev ent horizons and so\nit is not harmful. It is notable also that, for ingoing geodesics, the ba rrier can act as a\nclassical protector of the curvature singularities at r≤0 withm >0. On the other hand,\nfor outgoing geodesics, it can produce thermal radiations via quantum-mechanical tunneling\nfrom the inner region even when the event horizon is absent.\nVI. GENERALIZATIONS WITH COSMOLOGICAL CONSTANT AND ELEC-\nTROMAGNETIC CHARGES\nGeneralizing our rotating solution to the asymptotically (A)dSspacetime with a cos-\nmological constant Λ is not quite straightforward and needs a quite clever choice of the\nreference solution, as given by [19]3. Starting from the Kerr-(A)dS solution of [19], and\ndoing the same two-step procedure as in the previous sections, wit h the first step of finding\nthe massless solution done also in [11], we obtain the rotating (A)dSblack hole solutions as\nds2=−N2dt2+ρ2\n∆r(r)dr2+ρ2\n∆θ(θ)dθ2+Σ2sin2θ\nρ2Ξ2/parenleftig\ndφ+Nφdt/parenrightig2, (28)\nwhere\nΞ = 1+Λa2\n3ξ,∆θ= 1+Λa2cos2θ\n3ξ,\n∆r=/parenleftig\nr2+a2/parenrightig/parenleftigg\n1−Λr2\n3ξ/parenrightigg\n−2mr,\n3We thank O. Sarioglu for informing [19], which turns out to be a good re ference solution for Kerr-(A)dS\nsolutions.\n9Σ2=/parenleftig\nr2+a2/parenrightig\nρ2Ξ+f(r)a2sin2θ,\nN2=ρ2∆r∆θ\nΣ2, Nφ=−g(r)∆θ\nΣ2(29)\nwith the same solutions for f(r) andg(r) as in (10).\nThe mass and angular momentum can be computed as\nM=16πξm\nΞ2, J=16πam√ξκ−1\nΞ2, (30)\nsimilarly to the Λ = 0 case in Sec. III, and they satisfy the first law of b lack hole thermo-\ndynamics,\ndM=THdSH+ΩHdJ (31)\nwith the horizon entropy,\nSH=ξ(r2\nH+a2)\n4¯h(32)\nandtheGibbons-Hawkingtemperature THattheblackholehorizons r±andthecosmological\nhorizonr++4, and angular velocity at the horizons Ω H,\nTH=κH¯h\n2π=rH/parenleftbigg\n1−aΛ\n3ξ−r2\nHΛ\nξ−a2\nr2\nH/parenrightbigg\n4π(r2\nH+a2),\nΩH=−Nφ|H=a/parenleftbigg\n1−r2\nHΛ\n3ξ/parenrightbigg√κξ\nr2\nH+a2. (33)\nNote that Ω ∞=−Nφ|∞= 0, contrary to another standard form of Kerr-(A)dS solution\nin GR [3, 13, 24, 25], and we obtain the correct Ω Hfor the first law of thermodynamics,\nwithout the background subtraction Ω H−Ω∞. We obtain the correct mass and angular\nmomentum from the standard definition or from the recently-prop osed charge formula for a\ncovariant formalism, consistently with the first law of thermodynam ics [26, 27]. Moreover,\ndue to the r-independence of the conserved charge in [16], we obtain the mass and angular\nmomentum without the conceptual problem of the infinite boundary in the standard method\nfor the rotating dSsolutions.\nAnother standard form of rotating (A)dS black hole solutions with a non-vanishing an-\ngular velocity Ω ∞at infinity [3, 13, 24–26] is obtained by considering\n/hatwiderNφ=−g(r)∆θ\nΣ2−aΛ\n3ξ, (34)\nvia a coordinate transformation dφ=d/hatwideφ−(aΛ/3ξ)dtin the metric (28), which corresponds\nto aDiffF[3, 11, 25]. We find the same mass and angular momentum as (30) and t he\n4Here, we take the convention of the negative temperatures for the inner and cosmologicalhorizons [20–23].\n10first law of black hole thermodynamics as (31), but now with the subt racted angular speed\n/hatwideΩH=−/hatwiderNφ|H+/hatwiderNφ|∞, in agreement with GR’s result [24, 25].\nAdding the electromagnetic charges to the rotating black hole solut ions by coupling\nMaxwell fields to Hoˇ rava gravity is also an important generalization. To this ends, we\nconsider the LV Maxwell action, up to boundary terms,\nSM=/integraldisplay\nR×Σtdtd3x√gN/bracketleftbigg\n−2η\nN2/parenleftig\nEi+FijNj/parenrightig2+ζFijFij/bracketrightbigg\n, (35)\nwhich admits DiffF,δξA0= (A0ξ0),0+ξk∇kA0+Akξk\n,0, δξAi=Ai,0ξ0+ξk∇kAi+Ak∇iξk,\nas well as (6) for the Hoˇ rava gravity action (1), where Ei=˙Ai−∇iA0,Fij=∇iAj−∇jAi,\nandη,ζare the electromagnetic coupling constants [28–30]. Then, solving t he full equations\nfrom variations of A0andAi, as well as N,Ni, andgijas in Sec. III (see Appendix Dfor\nthe details), we obtain the gauge potential, with electric charge qeand magnetic charge qm,\nA=−/radicaligg\nξ\nη[qer∆θ+qmacosθ(1−Λr2(3ξ)−1)]\nρ2Ξdt+1√ηκ/bracketleftig\nqerasin2θ+qm(r2+a2) cosθ/bracketrightig\nρ2Ξdφ\n(36)\nand the metric (28), (29) by replacing 2 mr→2mr−(q2\ne+q2\nm), butonly with the “noble”\ncoupling,\nζη−1=κξ. (37)\nIt is rather surprising that the exact charged black hole solutions e xist only in the noble\ncoupling (37), due to coupling to Hoˇ rava gravity5. Actually, this is the case where the\nspeed of gravity is identical to the speed of light (or electromagnet ism),cg=cl=√κξ\nwith the dispersion relation ω2= (κξ)k2for the perturbations around the flat Minkowski\nbackground [7].\nThe first law of thermodynamics (31) is extended as (cf. [26]), with t he same SHand\nΩH,\ndM=THdSH+ΩHdJ+Φ(e)\nHdQe+Φ(m)\nHdQm (38)\nwith the Hawking temperature TH, the electric and magnetic chemical potentials Φ(e,m)\nH,\nTH=rH/parenleftbigg\n1−aΛ\n3ξ−r2\nHΛ\nξ−a2+q2\ne+q2\nm\nr2\nH/parenrightbigg\n4π(r2\nH+a2),\nΦ(e)\nH= 16π/radicalig\nξηqerH\nr2\nH+a2,Φ(m)\nH= 16πξ√κηqmrH\nr2\nH+a2, (39)\nand the canonical charges, computed from fluxes of electromagn etic fields,\nQe=/radicaligg\nξ\nηqe\nΞ, Qm=1√κηqm\nΞ, (40)\n5It is interesting that the noble coupling can be also obtained from a Ka luza-Klein reduction from (4+1)-\ndimensional kinetic-conformal pure Hoˇ rava gravity ( λ= 1/4) [31].\n11which are also r-independent6, as inMandJ(30) in gravity. On the other hand, another\nstandard form of rotating (A)dS black hole solutions with a non-van ishing angular velocity\nΩ∞at infinity is obtained by/hatwideA=A−(aΛ/3ξ)Aφdt, considering a coordinate transformation\ndφ=d/hatwideφ−(aΛ/3ξ)dtalso.\nThe ergo-sphere is larger ( κξ >1) or smaller ( κξ <1) than that of GR. In particular, for\nthe asymptotically flat case, the ergo-surface ( gtt= 0) is given by\nsin2θ=/braceleftbigg\n(r2+a2)2+∆2\nr+(κξ−1)(2mr)2−/radicalbigg/bracketleftig\n(r2+a2+∆r)2+(κξ−1)(2mr)2/bracketrightig\nκξ(2mr)2/bracerightbigg\n2a2∆r,\n(41)\nwhich exists alwaysforκξ >0, fromgtt|r+= (κξ)a2sin2θ/ρ2|r+>0. If we assume the\narea non-decreasing case with the appropriate energy conditions [32], one can consider t he\nPenrose process [33] to extract positive energy from inside of the ergo-surface ( gtt<0) and\nobtainformally the same energy-radiating efficiency as in Kerr metric, due to the sa me area\nformulaA= 8πm(m+√\nm−a2). However, its wave analogue, known as the super-radiant\nscattering is not quite clear, due to the lack of separability of wave e quations (see Appendix\nC).\nVII. OBSERVATIONAL CONSTRAINTS\nThe recent observation of the arrival delay of (+1 .74±0.05)sin thecoincident gravita-\ntional waves (GW) and gamma rays (GW170817, GRB170817A) [34] y ields the constraints\non the difference of the gravity speed cgand the gamma-ray photon’s light speed cl,\n−3×10−15cl<(cg−cl)<7×10−16cl, (42)\nwhere the upper bound, or the lower bound, is obtained by assuming the simultaneous emis-\nsion of the GW and GRB signals, or the GRB signals 10 safter the GW signal, respectively.\nBut, not to mention the important change of the order of magnitud es by changing the mod-\nels of photon emissions [34], we note that one can not tell about cgseparately from the data\n(42), without knowing about cltogether, since (42) is only about theirs difference cg−cl.\nThe smallness of the relative difference ( cg−cl)/clin (42) can be due to the smallness of\n(∆cg−∆cl)/clfor the modified speeds of gravity and light, cg=c+∆cg, cl=c+∆cl,from\nthe standard light speed c. And also, the observed time delay of the light can be due to the\nspace-time curvature along its path.\nIn the literatures [35, 36], on the other hand, it has always been ass umed that ∆ cl= 0\nand so (42) has been considered as the absolute bound of −3×10−15<∆cg/c <7×10−16,\nwhich would be strong enough to exclude the modified speed of gravit y ∆cg. However, if\nthere is ∆ clalso, the story is completely different and the constraint (42) yields −3×10−15<\n(∆cg−∆cl)/c <7×10−16, even if ∆ cgand ∆clare not small separately! Actually, our exact\nsolution for charged rotating black holes implies that cg=cl,i.e., ∆cg= ∆clfor the small\n6Fordyonicblack holes, electric (magnetic) canonical charge has “ r−dependent” and “induced” magnetic\n(electric) charge contribution also due to rotation, but they are e xactly canceled and have no effects in\n(38).\n12perturbations around the flat Minkowski background [7]. In that c ase, the time delay will\nbe due to either (a) the emission-time difference of the GW and GRB sig nals, or (b) due\nto the photon’s mass, either intrinsic or curvature induced, or eve n (c) the graviton’s mass\nwhich has not been studied in our solutions. Hence, our LV gravity wit hcg/negationslash=ccan be still\nconsistent withtherecently observed timedelay ofthecoincident G WandGRBsignals from\na binary neutron star merger, contrary to the claims in the literatu re, with some possible\nscenarios (a)-(c) that we listed above, to explain the observed da ta.\nAnother important constraints are the parameterized post-New tonian (PPN) parameters\nwhich measure the preferred-frame effects [10, 35], with the curr ent bounds [37],\n|α1|=|8(/hatwideξ−σ/2)|<4×10−5, (43)\n|α2|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftigg/hatwideξ−σ/2\n/hatwideξ−σ/2+1/parenrightigg/parenleftigg\n1+2(2/hatwideλ+1)(/hatwideξ−σ/2)\n/hatwideλ/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<×10−9, (44)\nwhere/hatwideξ=ξ−1 and/hatwideλ=λ−1. For the case of/hatwideξ−σ/2 = 0, which includes the GR limit of\n/hatwideξ= 0,σ= 0, we obtain α1=α2= 0 which trivially satisfies the constraints (43) and (44),\nwithout any restriction on/hatwideλ. However, for/hatwideξ−σ/2/negationslash= 0, the situation is quite different since,\nin the constraint (44),/hatwideλcan not contain the GR case of/hatwideλ=λ−1 = 0 because it makes\nα2→ ∞. Rather, it has a gap/hatwideλ≈ −2(/hatwideξ−σ/2) +/hatwideǫwith/hatwideǫ >2×10−4using 2/hatwideλ+ 1≈1\nfor a small/hatwideλ, with the constraints |/hatwideξ−σ/2|<5×10−6from (43). In other words, the LV\nparameters/hatwideξ−σ/2 and/hatwideλare strongly correlated in Hoˇ rava gravity.\nVIII. CONCLUDING REMARKS\nIn conclusion, we have studied a procedure for finding rotating blac k hole solutions in the\nlow-energy sector of the four-dimensional (non-projectable) H oˇ rava gravity as a viable LV\ngravity, and obtain the Kerr-type rotating black hole solutions with or without cosmological\nand electromagnetic charges.\nIt would be interesting to test the solution in the astrophysical obs ervations. Applying\nour procedure to the renormalizable ( z= 3) Hoˇ rava gravity in four dimensions and finding\nthe exact rotating black hole solutions will a challenging problem. The u niqueness of our\nsolutions and their stabilities are remaining problems. Especially, it wou ld be interesting to\nsee the instability of ultra-spinning black holes J/aM= 1/√κξ >1 withκξ <1, similar to\nultra-spinning Myers-Perry’s black holes in GR [38].\nNote added : After finishing this paper, a related paper, which corresponds to κξ= 1,\nλ/negationslash= 1 in our context, appeared in Einstein-aether gravity [39], where t he numerical rotat-\ning solutions have been found recently [40]. But the possibilities of our rotating solutions\nwithκξ/negationslash= 1 were not considered due to the recent gravity speed constrain ts [34] without\nconsidering the possibility of modified light speed [35, 36].\nAcknowledgments\nWe would like to thank Ozgur Sarioglu for helpful correspondences. This work was\nsupported by Basic Science Research Program through the Nation al Research Founda-\n13tion of Korea (NRF) funded by the Ministry of Education, Science an d Technology\n(2020R1A2C1010372, 2020R1A6A1A03047877).\nAppendix A: Explicit expressions of /hatwideρand/hatwiderpainTµν(eff)\nThe explicit expressions of /hatwideρand/hatwiderpain the effective energy-momentum tensor Tµν(eff)(19)\nare given as follows:\n/hatwideρ=/hatwiderκξ a2m2sin2θ/braceleftig\n(a4−2a2r2−3r4)2−2a2(a6−5a4r2+a2(4m−3r)r3+3r6)sin2θ\n+a4(a4−6a2r2+(8m−3r)r3)sin4θ/bracerightig\n/Z, (A1)\n/hatwiderp1=/hatwiderκξ a2m2sin2θ/braceleftig\n(a4−2a2r2−3r4)2−2a2(a6−a4r2+3r6+a2r3(−4m+r))sin2θ\n+a4(a4+2a2r2+(−8m+5r)r3)sin4θ/bracerightig\n/Z, (A2)\n/hatwiderp2= 2/hatwiderκξ a4m2rcosθ(a2+r(−2m+r))/bracketleftig\na4−3a2r2−6r4+a2(a2−r2)cos(2θ)/bracketrightig\nsin3θ/Z,(A3)\n/hatwiderp3= 2/hatwiderκξ a4m2rcosθ/bracketleftig\na4−3a2r2−6r4+a2(a2−r2)cos(2θ)/bracketrightig\nsin3θ/Z, (A4)\n/hatwiderp4= 8/radicalig\nκξ/hatwideρamr/Σ2, (A5)\nZ= (a2+r2−a2sin2θ)3/braceleftig\n(a2+r2)2−a2(a2+r(−2m+r))sin2θ/bracerightig2, (A6)\nwhere/hatwiderκξ=κξ−1 is a LV factor.\nAppendix B: The uniqueness of the invariant line element ds2underDiffF.\nIn order to prove the uniqueness of the invariant line element (2) under DiffF, we start\nby expanding (2) as\nds2= (−N2+gijNiNj)dt2+2gijNidxjdt+gijdxidxj. (B1)\nThen, under DiffF(6), the first term transforms as\nδξ/bracketleftig\n(−N2+gijNiNj)dt2/bracketrightig\n= (ξ0∂0gtt+ξi∇igtt+2Ni∂0ξi)dt2, (B2)\nand the second and third terms transform, respectively, as\nδξ/bracketleftig\n2gijNidxjdt/bracketrightig\n=−2Nj∂0ξjdt2+(2ξ0∂0Nj+2ξi∇iNj+2Ni∇jξi\n−2Nk∂jξk+2gij∂0ξi)dtdxj, (B3)\nδξ/bracketleftig\ngijdxidxj/bracketrightig\n= (2∇iξj+ξ0∂0gij−2gik∂jξk)dxidxj−2gij∂0ξjdxidt.(B4)\nSumming up (B2-B4) and using the metric compatibility to change the c ovariant derivatives\nto the partial ones, we have the form of Lie derivative of a scalar as follows:\nδξ(ds2) = (ξ0∂0gtt+ξi∂igtt)dt2+(2ξ0∂0Nj+2ξi∂iNj)dtdxj\n+(ξ0∂0gij+ξk∂kgij)dxidxj(B5)\n=ξµ∂µ(ds2). (B6)\n14Here, it is important to note that the underlined terms are exactly canceled only for the\nline element (2), which means the unique line element that is invariant un derDiffF(6).\nThis is quite remarkable since it seems to be contradict to the transf ormation of action (1),\nin which each term is invariant separately underDiffFso that one can introduce arbitrary\ncoupling constant for each term as the LV effect. The basic reason is thatdxiisnota\nspacial projection of dxµ, but rather it can transform to dtalso (though not for the reverse)\nwhen∂0ξi/negationslash= 0, contrary to the spacially-projected quantities Kij,Rij,∇i,etcin the action\n(1). This is a quite powerful property of our LV gravity that enable s us to study invariant\nproperties of particles via ds2also as in GR, contrary to widespread beliefs. Our proof,\nwhich has never been known in literature, justifies the use of ds2even in Hoˇ rava gravity case\nalso, where the full Diffis apparently broken.\nAppendix C: Petrov classification, Killing tensor, and Hamilton-Jacobi equation\nFor the Petrov classification of our rotating solution (8) without co smological constant,\nwe first find the four nullvectors,\n/hatwidelµ=1\n∆r/bracketleftig\nr2+a2,∆r,0,a+(r2+a2)H/bracketrightig\n,\n/hatwidenµ=1\n2ρ2/bracketleftig\nr2+a2,−∆r,0,a+(r2+a2)H/bracketrightig\n,\n/hatwidermµ=1√\n2(r+iacosθ)[iasinθ,0,1,i+iasinθH], (C1)\nwhich satisfy the usual orthogonality conditions/hatwidelµ/hatwidenµ=−1,/hatwidermµ/hatwider¯mµ= 1 with H= (√κξ−\n1)2mra/Σ2, and our metric can be written as gµν= 2/hatwiderm(µ/hatwider¯mν)−2/hatwidel(µ/hatwidenν), as usual. But, we\nfind that there are fourdistinct principal null directions kµthat satisfy kαkβk[µCν]αβ[ρkσ]= 0\nwith the Weyl tensor Cναβρ, such that our solution is the Petrov type I7. This is contrary\nto Kerr spacetime in GR, where two null vectors lµandnνare aligned with the principal\nnull direction and it is the Petrov type D.\nSince our rotating solution (8) is the Petrov type I, several well-es tablished and nice\nproperties of type D are not guaranteed. For example, the existe nce of Killing tensor and\nalso the separability of Hamilton-Jacobi (HJ) equation are not guar anteed [3, 4]. Especially\nfor Carter’s proof, our solution can notbe expressed as the generic form of solutions which\ngenerates the Killing tensor and separable HJ equation [3].\nActually one can check that the used-to-be Killing tensor form of Kerr metric Lµν=\n2ρ2m(µ¯mν)−a2cos2θg(4)\nµνwith the complex null vectors mµ,¯mνof the Newman-Penrose basis\nin GR, nor its correspondent in our new null vectors (C1)/hatwideLµν= 2ρ2/hatwiderm(µ/hatwider¯mν)−a2cos2θg(4)\nµν, do\nnot satisfy the Killing tensor equation ∇(4)\n(µLνσ)= 0 or∇(4)\n(µ/hatwideLνσ)= 0. However, interestingly,\nwe can find a certain geodesic with tangent vector vµsuch that vνvσLνσis stillconstant\nalong the geodesics, i.e.,\nvµ∇(4)\nµ(vνvσLνσ) =−24(κξ−1)(mra2)2vθv2\nφsinθcosθ\nΣ4= 0. (C2)\n7Similar results for the perturbed spacetimes are also obtained rece ntly [41, 42].\n15The fixed-angle geodesics, vθ= 0 (θ=const.) or vφ= 0 (φ=const.) are those cases and this\nsupports our previous analysis of θ=π/2 plane in Sec. VI. For other more general geodesics\nwithvθvφ/negationslash= 0, we may need numerical analysis to integrate the geodesics comp letely if there\nis no exact Killing tensor8.\nA related difficulty in integrating the full geodesics is the lack of separability of HJ equa-\ntion,\n−2∂S\n∂τ=g(4)µν∂µS∂νS\n=−Σ2\nρ2∆r(∂tS)2+4mra√κξ\nρ2∆r∂tS∂φS+∆rρ4−(2mra)2κξsin2θ\nΣ2ρ2∆rsin2θ(∂φS)2\n+∆r\nρ2(∂rS)2+1\nρ2(∂θS)2(C3)\nfrom a particle Hamiltonian H= (1/2)gµνpµpνandpµ=∂µSwith the Hamilton’s principal\nfunction S. By “assuming” the separability, we consider [2]\nS=1\n2µ2τ−Et+Lφ+Sr(r)+Sθ(θ) (C4)\nthen, we finally obtain the master equation\n\n\nµ2r2+/parenleftbigg\naE+/radicalig\nκξL/parenrightbigg2\n−1\n∆r/parenleftbigg\n(r2+a2)E+/radicalig\nκξaL/parenrightbigg2\n+∆r/parenleftiggdSr\ndr/parenrightigg2\n+(κξ−1)a2L2\n∆r\n\n\n+\n\nµ2a2cos2θ+/parenleftigg\n−a2E2+κξL2\nsin2θ/parenrightigg\n+/parenleftiggdSθ\ndθ/parenrightigg2\n−(κξ−1)L2\nsin2θ\n\n+(κξ−1)(2mra)2L2\nΣ2∆r= 0,\n(C5)\nwhereµis the particle’s rest mass. The last term in (C5) breaks the separab ility of HJ\nequation in our LV gravity with κξ/negationslash= 1, unless we consider L= 0 orθ=constant, similar to\nthe above Killing tensor analysis.\nAppendix D: Equations for the Lorentz-violating Maxwell action\nIn order to find the LV Maxwell action, we first consider the covaria nt Maxwell action,\nSM0=η/integraldisplay\nd4x/radicalig\n−g(4)FµνFµν(D1)\nwith the electromagnetic coupling constant η. Applying the usual ADM decomposition to\nthis action and, due to DiffF(6), introducing an arbitrary coupling parameter ζwhere the\ncovariant case (D1) corresponds to ζ=η, we obtain the LV Maxwell action,\nSM=/integraldisplay\nR×Σtdtd3x√gN/parenleftbigg\nζFijFij∓2η\nN2/bracketleftig\nEi+FijNj/bracketrightig2/parenrightbigg\n, (D2)\n8If one considers successive applications of infinitesimal geodesics w ith fixed-angle planes, we may produce\ngeneral geodesics in the continuous limit. 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Gr av.22, 1503 (2005).\n[28] D. Sudarsky and R. M. Wald, Phys. Rev. D 46, 1453 (1992).\n[29] E. Kiritsis and G. Kofinas, Nucl. Phys. B 821(2009) 467.\n[30] K. Lin et al., Int. J. Mod. Phys. D 23, 1443004 (2014).\n[31] A. Restuccia and F. Tello-Ortiz, Eur. Phys. J. C 80, 86 (2020).\n[32] S. W. Hawking, Phys. Rev. Lett. 26, 1344 (1971).\n[33] R. Penrose, Riv. Nuovo Cimento 1, 252 (1969).\n[34] B. P. Abbott et al., Astrophys. J. Lett. 848, L13 (2017).\n[35] A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, M. Saravani and T. P. Sotiriou , Phys. Rev. D 97, 024032 (2018).\n[36] Y. Gong et al., Phys. Rev. D 98, 104017 (2018).\n[37] C. M. Will, Living Rev. Rel. 17, 4 (2014).\n[38] R. Emparan and R. C. Myers, JHEP 09, 025 (2003).\n[39] E. Franzin, S. Liberati and J. Mazza, arXiv:2312.06891 [gr-qc].\n[40] A. Adam et al., Class. Quant. Grav. 39, 125001 (2022).\n[41] B. Araneda and G. Dotti, Class. Quant. Grav. 32, 195013 (2015).\n[42] C. B. Owen, N. Yunes and H. Witek, Phys. Rev. D 103, 124057 (2021).\n18"    },    {        "title": "2402.02260v1.Open_dynamics_of_entanglement_in_mesoscopic_bosonic_systems.pdf",        "content": "Open dynamics of entanglement in mesoscopic bosonic systems\nKonrad Schlichtholz1,∗and Łukasz Rudnicki1, 2\n1International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdansk, Poland\n2Center for Photonics Sciences, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland\nOne of the important problems from the perspective of Quantum Information is the description\nof systems on the mesoscopic scale, which is simpler than the full quantum formalism but\nmaintains information about non-classical phenomena necessary for Quantum Information tasks\nlike entanglement. In particular, mesoscopic descriptions of fully bosonic systems undergoing open\nevolution are of high interest, as fully photonic quantum computing and communication are one\nof the prominent avenues for the development of the field. In this paper, we propose a mesoscopic\ndescription of such systems based on boson number correlations, which allows for tracking open\nevolution of entanglement of both non-Gaussian and Gaussian states, and their sub-Poissonian\nstatistics. Our description contributes as a generalization of the reduced state of the field formalism\n[Entropy 2019, 21(7), 705], which by itself does not contain information about entanglement. One\nof the important features of our approach is that it inherits the structure of the description of two\nparticles in terms of first quantization, which allows for intuitive broad usage of already known tools.\nUsing the proposed formalism, we show the robustness of entanglement against low-temperature\ndamping for four mode bright squeezed vacuum state and beamsplitted single photon. What is\nmore, we propose generalization of the Mandel Q parameter accessible through our description, and\nupon this we show that the entanglement of the state obtained by beam splitting single-occupied\nmode is fully inherited from sub-Poissonian statistics of the input state.\nI. INTRODUCTION\nOne of the important problems that is still developing in physics is the description of systems on the mesoscopic\nscale. This problem arises because full quantum treatment of larger systems is not feasible. Classical description\nallows for an accurate depiction of the macroscopic systems; however, one can be interested in systems on the scale on\nwhich full quantum description is already not feasible for calculations but with physical phenomena based on quantum\nfeatures. In such a case, the classical paradigm becomes insufficient and some intermediate methods are needed for\nthis mesoscopic scale. The need to develop such theoretical tools emerges even more as experiments on this scale\nbecome feasible nowadays[1, 2]. This is also fundamental from the perspective of the field of quantum information,\nas its protocols require complicated quantum systems, and in their principles they are based on quantum phenomena\nsuch as superposition and entanglement [3].\nDifferent semi-classical methods were introduced for simplification of the problems, for example, widely used\nparametric approximation [4, 5], but such techniques still require the full quantum description of some parts of\nthe system. Different approach is to track collective observables which describe collective behavior of the system,\ne.g. total spin. Such mean-field approaches capture well macroscopic behavior of the system, however, information\nabout quantum features is suppressed quickly with the number of particles in such an approach. One can consider\nanother collective observable, namely fluctuations instead of average values, as it is the case for the fluctuation algebra\napproach to mesoscopic systems [6, 7]. This technique is adopted in particular to describe entanglement in mesoscopic\nsystems [8–10]. Other methods were also used to describe the entanglement in mesoscopic systems [11–13].\nOne of the specific classes of systems for which mesoscopic descriptions are of great use are fully bosonic systems.\nThe reason for this is that information encoded in bosonic modes is one of the promising avenues for implementation\nof quantum information protocols, as, for example, information encoded in photons can be quickly transferred over\nlong distances [14]. However, such systems are also affected by the decoherence effects coming from the environment,\nand their strict theoretical considerations require dealing with the whole infinite-dimensional Fock space what can\nbecome unfeasible in multi-mode systems. The widely used method that simplifies the description is to track the\nevolution of the covariance matrix of position and momenta, which reduces the problem to finite dimensional when\na finite number of modes is considered [15–17]. This approach in fact provides a full description of Gaussian states,\nand for non-Gaussian states it can serve as a mesoscopic description. However, non-Gaussian features can be lost\nin this description, what can result in not keeping useful information about quantum features of the state, such as\nentanglement [18]. Another mesoscopic approach for bosonic systems that was recently proposed is the reduced state\n∗konrad.schlichtholz@phdstud.ug.edu.plarXiv:2402.02260v1  [quant-ph]  3 Feb 20242\nof the field (RSF) [19]. This formalism extends the mean-field approach with additional structure by the reduction\nof the state on the Fock space into the operator on the first quantization single-particle-like Hilbert space. This\ndescription was already applied for the analysis of the coronal heating phenomenon [20] and the dynamical Casimir\neffect [21]. However, it was shown that the RSF approach has some highly classical features and lacks information on\nquantum features interesting from the standpoint of quantum information, i.e., RSF does not store any information\nabout entanglement at least for two mode Gaussian states [22].\nIn this paper, we identify degrees of freedom of bosonic systems, which can be used in a mesoscopic description\nof their states that contains information about entanglement of both Gaussian and non-Gaussian states. Upon this,\nwe propose an extended version of the RSF approach, which allows for tracking evolution of entanglement of bosonic\nfields. This is achieved by construction of two-particle-like Hilbert space on which multi-mode bosonic state is mapped\ninto the two-qudit state entanglement of which is sufficient for the entanglement of the full state. What is important\nis that in this approach structures of typical finite-dimensional quantum systems emerge, and therefore one is left\nwith a toolbox which is intuitive and based on well-known tools from quantum mechanics. This approach allowed us\nto show that the entanglement of the 2×2bright squeezed vacuum state and the entangled two-mode single-photon\nstate is robust against low temperature damping by the environment. Our approach also allows for tracking particle\nnumber variance (fluctuations) and thus also tracking the non-classical phenomenon of sub-Poissonian statistics. In\nthis context, we propose the generalization of Mandel Q parameter which can be calculated using extended RSF. Upon\nthis, we show that the entanglement of the state obtained by beam splitting single-occupied mode is fully inherited\nfrom sub-Poissonian statistics of the input state.\nII. MESOSCOPIC DESCRIPTION OF BOSONIC MODES\nOne of the extremely important quantum phenomena is entanglement, which is one of the core resources in many\nquantum information tasks. This is even more so in the context of the fact that in the quantum field theory\nentanglement of pure states is fundamentally associated with Bell non-classicality [23] which is necessary for device-\nindependent quantum key distribution. For such tasks, particularly compelling is the encoding of information in light\nmodes, as light can be quickly distributed over long distances [14]. What is more, fully photonic quantum devices\nare one of the promising avenues also in different subfields of quantum information, e.g., quantum computing [24].\nHowever, as all quantum systems, light modes are also affected by the environment and undergo non-unitary open\nevolution, which results in decoherence of the system and thus decay of such quantum phenomena like superposition\n(including entanglement).\nIn many circumstances, the open evolution imposed on the system is approximately Markovian. This type of\nevolution of the state of bosonic modes ˆρFis described by the GKLS equation [25, 26]. In particular, one can write\nthis equation for Northogonal bosonic modes aiusing the independent particles approximation in such a way that it\ninclude particle decay and production, random elastic scattering and classical coherent pumping [19, 22]:\nd\ndtˆρF=−i\nℏ[ˆH, ρ F] +NX\nk[(ξkˆa†\nk−ξ∗\nkˆak),ˆρF] +X\njκj(ˆUjˆρFˆU†\nj−ˆρF)\n+NX\nk,k′=1Γk′,k\n↓\u0012\nˆakˆρFˆa†\nk′−1\n2{ˆa†\nk′ˆak,ˆρF}\u0013\n+NX\nk,k′=1Γk′,k\n↑\u0012\nˆa†\nk′ˆρFˆak−1\n2{ˆakˆa†\nk′,ˆρF}\u0013\n,(1)\nwith Hamiltonian ˆH=P\nk,k′ωk,k′ˆa†\nkˆak′, where ˆa†\ni(ˆai)are creation and annihilation operators associated with\ncorresponding modes, complex parameters ξkdescribe classical pumping field, ˆUjare unitary transformations of\nannihilation operators ( ˆU†\njˆakˆUj=P\nk′uj\nk′,kˆak′with ujbeing unitary matrix) corresponding to scattering processes\nwiththeirprobabilitydistribution κj,and Γk′,k\n↕describecreationandannihilationrates. Treatingthisproblemrequires\ndealing with the full infinite-dimensional Fock space. This can become cumbersome for complicated mesoscopic\nsystems involving multiple bosonic modes.\nA custom in the treatment of mesoscopic systems is to consider only some degrees of freedom instead of the full\nquantum state of the field ˆρF, reducing the complexity of the problem. One of such methods for bosonic systems is\nto track the evolution of the covariance matrix of position and momenta [17]:\nVi,j=1\n2⟨{ˆΞi,ˆΞj}⟩ − ⟨ ˆΞi⟩⟨ˆΞj⟩, (2)\nwhere ⃗Ξ = (ˆ x1,ˆp1, ....,ˆxn,ˆpn)and operators ˆxj,ˆpjare dimensionless position and momentum operators in j-th mode.\nThis method is particularly important, as it allows tracking the full Gaussian evolution of Gaussian states, since one3\ncan reconstruct the full Gaussian state ˆρFfrom the covariance matrix. Therefore, the covariance matrix also contains\nfull information about entanglement of Gaussian states, which in the case of distillable bipartite entanglement can be\naccessed simply by using the PPT criterion [3, 27, 28]. However, this is not always the case for non-Gaussian states,\ne. g., cat-like states, for which one has to employ different methods of entanglement detection for which information\ncontained in the covariance matrix is not sufficient [18]. Furthermore, the covariance matrix approach, while well\nsuited for continuous variable analysis, can be non-intuitive with its symplectic structures for researchers working\nwith discrete variables for whom finite-dimensional Hilbert spaces of first quantization equipped with Dirac notation\ncan be more familiar. Additionally, regaining information from the covariance matrix about discrete variables also\nrequires extra effort, and it is not always possible for non-Gaussian states.\nIn fact, multiple fundamental experiments and protocols, e. g. teleportation [29], entanglement swapping [30],\nquantum cryptography [31] are based on entanglement of qubits realized using light modes restricted to a single-\nphoton per party. An emblematic example of the state used in such protocols are polarization and path-entangled\nsinglet states:\n|ψ⟩=1√\n2(ˆa†\n1ˆa†\n4−ˆa†\n2ˆa†\n3)|Ω⟩=1√\n2(|1,0; 0,1⟩ − |0,1; 1,0⟩), (3)\nwhere polarization or path modes a1, a2and contributes to the state of one of the parties and a3, a4to the second\none, and |Ω⟩stands for the vacuum state. Here we have also used the following notation for Fock states:\n|n1, n2;n3, n4⟩=(ˆa†\n1)n1(ˆa†\n2)n2(ˆa†\n3)n3(ˆa†\n4)n4\n√n1!n2!n3!n4!|Ω⟩, (4)\nwith semicolon marking the bipartition. Then the measurements that correspond to typically considered for qubits\nPauli matrices is realized as Stokes operators ˆΘi[32] restricted to the single-photon subspace:\nσi→ˆΠ1ˆΘiˆΠ1=ˆΠ1(ˆa†\niˆai−ˆa†\ni⊥ˆai⊥)ˆΠ1, (5)\nwhere i= 1,2,3denotes one of the three mutually unbiased basis described by two orthogonal modes ai, ai⊥, and ˆΠ1\nstands for the projector into subspace of single photon across those modes. Additionally total photon number operator\nΘ0= ˆa†\niˆai+ ˆa†\ni⊥ˆai⊥after projecting with ˆΠ1corresponds to the identity on the qubit space denoted by σ0. Note that\nany local operation on the two-qubit system can be described as a combination of tensor products of Pauli matricies\n(including σ0) acting on subsystems AandBwhat in the Fock space translates to σA\ni⊗σB\nj→ˆΠA\n1ˆΠB\n1ˆΘA\niˆΘB\njˆΠB\n1ˆΠA\n1\nwhere upper indices denote the party on which given operators act. Therefore, as Stokes operators are in fact\ndifferences of number operators in two orthogonal modes, all non-classical correlations in such systems are described\nas photon number correlations ⟨(ˆaA)†\ni(⊥)ˆaA\ni(⊥)(ˆaB\nj(⊥))†ˆaB\nj(⊥)⟩. This suggests that boson number entanglement is at the\ncore of the discrete variable bipartite entanglement of bosonic systems. What is more, such correlations become\nexperimentally accessible also in the regimes with higher photon numbers due to developments in the field of photon\nnumber resolving detectors [33, 34]. Let us notice that the considered photonic implementation of qubits is, in fact,\napproximate. This is because nothing prevents the given modes from being occupied by a higher or lower number of\nphotons. This is even in the case of event-ready preparation of the system in which photon is confirmed to enter the\ninput of the apparatus, as environment can add multi-photon noise (with undefined number of photons) to the state\nand also result in the loss of the photon which encodes the information especially during propagation through long\ntransmission lines. What is more, one may simply want to consider subsystems which are more dimensional than\nqubits and may include states with different numbers of photons per party. This shows that even in those highly\nsimplified scenarios, in principle, one should consider full Fock space, or at least consider some mesoscopic description\nof the system, which would allow for the reduction of the dimensionality of the problem. All of this motivates us to\nbuild a mesoscopic scale description of bosonic systems with the following matrix as the main structure that contains\nrelevant degrees of freedom of the state ˆρFfor considerations of bipartite entanglement:\nˆρ4:=NX\ni,j,n,mtr\b\nˆρFˆa†\nnˆaiˆa†\nmˆaj\t\n|i, j⟩⟨n, m|, (6)\nIn the following we show that ˆρ4, which we call extended reduced state , contains information about entanglement of\nboth Gaussian states with high average numbers of photons and more importantly non-Gaussian states (for which\nPPT criterion cannot reveal entanglement based on covariance matrix). We also show the time evolution of this\nmatrix under open evolution given by equation (1). At the same time we equip the proposed mesoscopic description\nwith multiple intuitive and familiar tools from finite-dimensional Hlibert spaces.4\nIII. SEPARABILITY IN TERMS OF EXTENDED REDUCED STATE\nThe key question for entanglement considerations in our mesoscopic description is how separability in the Fock space\ntranslates into the extended reduced state. The following considerations are inspired by the mappings of entanglement\nindicators for Stokes-like operators [35, 36]. Let us choose two sets of indices IA,IB∈ {1, N}in such a way that\nIA∩ IB=∅which determines our bipartition of the system. We assume that the modes with indices from IA\nconstitute as one of the subsystems and the modes with indices from IBas the second one. In addition, each set\ncontains at least two elements and we denote the number of these elements by dX=IX. Based on this choice, we\nproject the operator ˆρ4in the following way:\nˆρ4→ˆρΠ=ˆΠIA,IBˆρ4ˆΠIA,IB=X\ni,n∈IAX\nj,m∈IBρijnm|i, j⟩⟨n, m|, (7)\nwhere:\nΠIA,IB=X\ni∈IAX\nj∈IB|i, j⟩⟨i, j|, (8)\nandρijnm =⟨ˆa†\nnˆaiˆa†\nmˆaj⟩are corresponding matrix elements of ˆρ4. Let us here for clarity rename the operators\ncorresponding to modes IBasˆb†\nmandˆbji. e.⟨ˆa†\nnˆaiˆa†\nmˆaj⟩ → ⟨ ˆa†\nnˆaiˆb†\nmˆbj⟩. Note that modes that are not used are\neffectively traced out and modes from IA(IB) are associated only with the first (second) index of the kets and bras.\nLet us assume that the state of the system ˆρFafter tracing out modes not contained in IA,IBis some pure state\nˆρR=|ψ⟩⟨ψ|. In such a case we can see that ρijnmhas a form:\nρijnm =⟨ψnm|ψij⟩, (9)\nwhere |ψij⟩= ˆaiˆaj|ψ⟩. From that follows that ρijnmforms a Gramian matrix and thus is positive definite. Thus\nˆρΠ\nψ(where subscript denotes original state on the Fock space) is also positive definite, and after normalization it is a\nwell-defined density matrix ˆρN\nψfor the system of two qudits with dimensions dX. However, there are exceptions for\nwhich the matrix cannot be normalized as it is traceless. The trace of this matrix is, in general, equal to ⟨ˆNIAˆNIB⟩\nwhere ˆNIXis the total boson number operator in modes corresponding to IX. This can be zero only when the state\nis the vacuum ˆρR=|Ω⟩⟨Ω|or it consists in the superposition only states with bosons only in one of the parties. We\naddres this problem later on. For now, assume that the matrix can be normalized. Because any mixed state can be\nwritten as a convex combination of pure states ˆρR=P\nipi|ϕi⟩⟨ϕi|it will map to a convex combination of ˆρΠ\nϕi:\nˆρΠ\nρR=X\nipiˆρΠ\nϕi. (10)\nAfter normalization, we get:\nˆρN\nρR=X\nipiˆρΠ\nϕi/X\nipitr\b\nˆρΠ\nϕi\t\n=X\nipitr\b\nˆρΠ\nϕi\t\nˆρN\nϕi/X\nipitr\b\nˆρΠ\nϕi\t\n=X\nip′\niˆρN\nϕi. (11)\nNote that 1≤p′\ni≤0andP\nip′\ni= 1. Thus, this again forms a proper density matrix on the two-qudit space.\nIn the next step, we assume that ˆρRis separable in the partition into modes IAandIB. We start with the case\nwhere ˆρRis in some pure state |ψ⟩⟨ψ|. Due to the separability, the state can be written as |ψ⟩=F†\nIAF†\nIB|Ω⟩, where\n|Ω⟩stands for vacuum state, and F†\nIiis a polynomial of creation operators of modes corresponding to IX. From this\nfollows that each ρijnmfrom ˆρΠfactorizes as follows:\nρijnm =⟨Ω|FIAFIBˆa†\nnˆaiˆb†\nmˆbjF†\nIAF†\nIB|Ω⟩=⟨Ω|FIAˆa†\nnˆaiF†\nIA|Ω⟩⟨Ω|FIBˆb†\nmˆbjF†\nIB|Ω⟩=⟨ψA\nn|ψA\ni⟩⟨ψB\nm|ψB\nj⟩,(12)\nwhere |ψA\ni⟩= ˆaiF†\nIA|Ω⟩and|ψB\nj⟩=ˆbjF†\nIB|Ω⟩. To see this, consider rearrangement of the creation and annihilation\noperators in ρijnmsuch that the operators corresponding to IBare shifted to the right and let us put between the\noperators acting on the modes IAandIBthe resolution of identity 1=P\n⃗ n1,⃗ n2|⃗ n1;⃗ n2⟩⟨⃗ n1;⃗ n2|:\nρijnm =⟨Ω|FIAˆa†\nnˆaiF†\nIA1FIBˆb†\nmˆbjF†\nIB|Ω⟩=⟨Ω|FIAˆa†\nnˆaiF†\nIA|Ω⟩⟨Ω|FIBˆb†\nmˆbjF†\nIB|Ω⟩\n+X\n⃗ n1̸=⃗0⟨Ω|FIAˆa†\nnˆaiF†\nIA|⃗ n1;⃗0⟩⟨⃗ n1;⃗0|FIBˆb†\nmˆbjF†\nIB|Ω⟩+X\n⃗ n2̸=⃗0⟨Ω|FIAˆa†\nnˆaiF†\nIA|⃗0;⃗ n2⟩⟨⃗0;⃗ n2|FIBˆb†\nmˆbjF†\nIB|Ω⟩\n+X\n⃗ n1,⃗ n2̸=⃗0⟨Ω|FIAˆa†\nnˆaiF†\nIA|⃗ n1;⃗ n2⟩⟨⃗ n1;⃗ n2|FIBˆb†\nmˆbjF†\nIB|Ω⟩,(13)5\nwhere |⃗ n1;⃗ n2⟩stands for Fock basis state with occupancy of the modes IAandIBdescribed by vectors ⃗ n1,⃗ n2and\n⃗0stands for 0 photons in given modes. This rearrangement is possible as operators acting on orthogonal modes IA\nandIBcommute. One can notice that terms of the first appearing sum contain factor ⟨⃗ n1;⃗0|FIBˆa†\nmˆajF†\nIB|Ω⟩which\nis equal to 0as all operators inside expectation value act only on modes IBleaving part of the states of bra and ket\ncorresponding to modes IAorthogonal. Analogously, all other sums appearing are equal 0, and the relation is proven.\nNow, based on the same arguments as above, ρX\nψ,l,k:=⟨ψX\nk|ψX\nl⟩forms a proper density matrix after normalization\non the one-qudit space. Thus, ˆρN\nψ= ˆρ1\nψ⊗ˆρ2\nψ/Nis a separable state on the two-qudit space. For the mixed separable\nstate from (11) we see that ˆρN\nρRis a convex combination of pure separable states and therefore a proper mixed state\non the two-qudit space. Therefore, if ˆρN\nρRis entangled it follows that ˆρFis entangled.\nLet us now address the scenario in which the matrix for the pure state is not normalizable. This was not important\nfor mapping of entanglement indicators [35, 36] however it is important if one wants to use ˆρN\nρRin a more general\ncontext as a density operator. All pure states that result in the not normalizable ˆρΠ\nψhave the following form:\n|ψ⟩=ζ0|Ω⟩+F†\nIA|Ω⟩+F†\nIB|Ω⟩, (14)\nwhere ζ0stands for amplitude of vacuum state. These can be either separable or entangled. One can easily find that\nfor all such states ˆρΠ\nψ= 0. This simply tells us that for states of the form (14) this state reduction does not allow\none to access the information about their entanglement without some additional considerations. What is more, all of\nthem are equivalent to vacuum (which is separable) at the level of reduction to ˆρΠ\nψ. But still, there is a way to gain\naccess to entanglement of such states in our description what we discuss further on in the examples. Note that such\nstates are effectively highly not likely to be present in experiments due to the presence of noise, for example, thermal\nnoise. Therefore, the important question is what the impact of such states is if they are part of the mixed state ˆρR.\nWithout loss of generality, let us assume that only |ϕ0⟩is of the form (14).\nˆρΠ\nρR=p0ˆ0 +X\ni≥1piˆρΠ\nϕi→ˆρN\nρR=X\ni≥1pitr\b\nˆρΠ\nϕi\t\nˆρN\nϕi/X\ni≥1pitr\b\nˆρΠ\nϕi\t\n=X\ni≥1p′′\niˆρN\nϕi, (15)\nWhere p′′\nifulfill relations for probability. Therefore, even in this case ˆρN\nρRis a well-defined density matrix, and clearly\nthe previous discussion on separability holds. In fact, the presence of the states (14) in the density matrix results only\nin renormalization of probabilities and thus neither helps to detect entanglement nor prevents it. One can observe\nthat all states of the form:\nˆρF=p1X\njqj|ψj⟩⟨ψj|+p2ˆρ′, (16)\nwhere |ψj⟩are of the form (14) and qj, piare independent probability distributions, are equivalent to the state ˆρ′in\nterms of ˆρN\nρR.\nNote that separability of ˆρΠ\nψdoes not imply that ˆρFis separable, as we use only part of the correlations. In\nother words, the entanglement of the state ˆρΠ\nψon the reduced two-qudit Hilbert space is a sufficient condition for the\nentanglement of ˆρF. What is important, one has to consider at least 4 modes to reveal such a kind of entanglement,\nas one has to have both reduced subsystems at least two dimensional.\nA. Impact of mode transformations\nNote that, choice of the basis on the subsystem defined by Iihas no impact on the detection of entanglement. This\nis because the local unitary transformation of the annihilation operators and, resulting from it, the transformation\nof the creation operators, which constitutes a basis transformation, translates into the local change of basis on the\nreduced matrix ρΠ. Let us consider a general form of such a transformation ˆai=P\nkUikˆckwhere ˆckare annihilation\noperators in some new basis and Uikdetermines the transformation between bases. Let us consider transformation\non the first subsystem:\nρ(a)\nijnm =⟨ψ|ˆa†\nnˆb†\nmˆaiˆbj|ψ⟩=X\nk,lUi,k⟨ψ|ˆc†\nlˆb†\nmˆckˆbj|ψ⟩U∗\nl,n=X\nk,lUi,kρ(c)\nkjlmU∗\nl,n, (17)\nwhere superscript in ρ(a)\nijnmdenotes the basis used. This transformation is simply a basis transformation on the\nsubsystem. Such transformations do not affect the separability of the state.6\nFrom this we see that by choice of partition of modes our reduction procedure maps boson number entanglement\ninto the entanglement of two distinguishable particles with finite number of states. The extended reduced state ˆρ4\ncontains information about all possible partitions IA,IB. Thus, the separability of ˆρN\nρrfor one partition does not\nnecessarily mean that ˆρ4does not contain any information on the entanglement. In fact, ˆρ4contains information\nabout the boson number entanglement in any partition IA,IBafter an arbitrary unitary transformation of all modes,\nwhat can be seen analogously to (17). Still, the sensible partitions easily emerge in the experimental setups, where\nspecific modes correspond to different beams clearly determining local operations.\nIV. OPEN EVOLUTION AND REDUCED STATE OF THE FIELD APPROACH\nBased on the results from the previous section, one can see that effectively we mapped relevant for the entanglement\ndegrees of freedom of state ˆρFfrom an infinite-dimensional Fock space into proper mixed state ˆρN\nρRon two-particle-like\nfinite-dimensional Hilbert space, where each of the particles corresponds to one of the parties. It is important that\nwhile ˆρN\nρRposses all mathematical features of the density matrix, it does not inherit a probabilistic interpretation of\nthe entries on the diagonal of the matrix. In this case, they rather describe the strength of linear correlations of boson\nnumbers between given modes. This situation is similar to the case of reduced state of the field approach (RSF) for\nmesoscopic description of bosonic modes [19], and in fact our reduction can be seen as a higher order extension of the\nRSF.\nLet us briefly recall the RSF approach. The RSF formalism attempts to reduce the infinite-dimensional description\nof the second quantization into a single-particle-like finite-dimensional Hilbert space of the first quantization. This\nreduction is carried out as follows. Having the density operator ˆρFon the Fock space, which describes the state of N\northogonal modes {ai}N\ni=1, it is reduced to two structures on the N-dimesional Hilbert space with orthonormal basis\n{|k⟩}N\nk=1: single-particle density matrix and averaged field:\nˆρ=NX\nk,k′=1trn\nˆρFˆa†\nk′ˆako\n|k⟩⟨k′|,|α⟩=NX\nk=1tr{ˆρFˆak}|k⟩. (18)\nThe operator ˆρcontains information on the occupation of modes and coherences, while |α⟩about the local phases of\nthe field. Note that ˆρand|α⟩after normalization have the properties of density matrix and pure state, respectively;\nhowever, they do not inherit the probabilistic interpretation normally associated with quantum states. However,\none can see that the diagonal elements of ˆρrepresent the average occupation of modes instead of probabilities. The\nreduction of the state is also accompanied by a reduction of the observables. One can reduce a class of additive\nobservables on the Fock space in the following way:\nˆO=NX\nk,k′=1ok,k′a†\nkˆak′→ˆo=NX\nk,k′=1ok,k′|k⟩⟨k′|. (19)\nThis construction preserves the expectation values:\ntrn\nˆρFˆOo\n= tr{ˆρˆo}. (20)\nWhat is important, one can find the time evolution equations for ˆρ,|α⟩corresponding to the evolution equation (1):\nd\ndtˆρ=−i\nℏ[ˆh,ˆρ] + (|ξ⟩⟨α|+|α⟩⟨ξ|) +X\njκj(ˆujˆρˆu†\nj−ˆρ) +1\n2{ˆγ↑−ˆγT\n↓,ˆρ}+ ˆγ↑, (21)\nd\ndt|α⟩=−i\nℏˆh|α⟩+|ξ⟩+X\njκj(ˆuj−1)|α⟩+1\n2(ˆγ↑−ˆγT\n↓)|α⟩, (22)\nwhere ˆhstands for reduced Hamiltonian (using (19)) and:\nˆγ↕=X\nk,k′Γkk′\n↕|k⟩⟨k′|,|ξ⟩=X\nkξk|k⟩,ˆuj=X\nk,k′uj\nk,k′|k⟩⟨k′|. (23)\nThis formalism is also equipped with notion of entropy:\nS[ˆρ;|α⟩] =kBtr{(ˆρα+ 1) log(ˆ ρα+ 1)−ˆραlog(ˆρα)}, (24)\nwhere kBstands for Boltzmann constant and ˆρα= ˆρ− |α⟩⟨α|.7\nA. Extended reduced state of the field description\nThe RSF formalism, while simple, has its drawbacks limiting its applications. It was shown that while it is fully\nbased on quantum considerations, it is a highly classical description [22]. This is in the sense that RSF does not\ncontain information about non-classical phenomena like distillable entanglement and even on any entanglement for\ntwo mode Gaussian states, and, what is more, the entropy of this formalism has properties similar to semi-classical\nWehrl entropy rather the quantum von Neumann entropy. However, it turns out that the information contained in\nRSF is needed to track the evolution of our reduced two-particle-like state ˆρN\nρRand its evolution can be put into a\nframework that extends RSF in a natural way. Furthermore, this extension can be seen as the simplest extension of\nRSF that adds quantum features to the formalism and is not equivalent to the covariance matrix formalism.\nLet us show how our extended reduced state ˆρ4can be incorporated into an RSF like formalism. States from the\nsingle-particle space of RSF contain information about particle numbers in modes, and thus a natural concept seems\nto be the introduction of a tensor product of two single-particle spaces to track correlations of particle numbers. The\nextended reduced state ˆρ4is then an operator on the two-particle Hilbert space constructed based on two copies of\nthe single-particle space. Note that reduction of state ˆρFdoes not always have to result in the operator ˆρ4being\nHermitian. Thus, it does not have to constitute a proper state on this Hilbert space, but still ˆρN\nρRdoes. We keep the\nwhole structure ˆρ4as it provides additional useful information on quantum features of the state, which we will exploit\nfurther in the text, and it might also be required to track the time evolution in some circumstances. We build an\nextended reduced state ˆρ4(6) to resemble a single-particle density matrix ˆρ(18) with indices of annihilation operators\nresponsible for labeling rows and indices of creation operators labeling columns. One can analogously to (19) reduce\nthe observables:\nˆO4=X\nk1,k2,k3,k4ok1,k2,k3,k4ˆa†\nk1ˆak2ˆa†\nk3ˆak4→ˆo4=X\nk1,k2,k3,k4ok1,k2,k3,k4|k1, k3⟩⟨k2, k4|, (25)\nwith trn\nˆρFˆO4o\n= tr{ˆρ4ˆo4}. We also complement the reduction of the state with two additional structures. That is,\nrank 3 tensors:\nˆβ:=X\nk1,k2,k3,k4trn\nˆρFˆa†\nk1ˆak2ˆak3o\n|k2, k3⟩⟨k1|, (26)\nand additional operator on single-particle space:\nˆr=X\nk,k′Tr{ˆρFˆak′ˆak}|k⟩⟨k′|. (27)\nThese two structures are introduced solely in order to include coherent pumping in the evolution, what will become\napparent from the evolution equations. The evolution of operators on two-particle space also requires tracking the\nevolution in a single-particle space. Thus, this two-particle reduction has to be used together with the single-particle\nreduction, and the total reduced state consists of ˆρ4,ˆβ,ˆr,ˆρ,|α⟩. The evolution equations for operators ˆρ4,ˆβ,ˆrcan be\nobtained analogously to equations (21-22) by multiplying both sides of (1) with a suitable combination of creation\nand annihilation operators for a given matrix element and then tracing the resulting expressions. Then the set of\nevolution equations after some simplifications done with the commutation relation [ˆai,ˆa†\nj] =δijcan be put in the\nfollowing form:\nd\ndtˆρ4=−i\nℏ[ˆh⊗1+1⊗ˆh,ˆρ4] +\u0010\n|ξ⟩ˆβ†+ (|ξ⟩ˆβ†)τL+ (|ξ⟩⟨α| ⊗1)τL+ˆβ⟨ξ|+ (ˆβ⟨ξ|)τR+ (1⊗ |α⟩⟨ξ|)τR\u0011\n+X\njκj(ˆuj⊗ˆujˆρ4ˆu†\nj⊗u†\nj−ˆρ4) +1\n2{(1⊗ˆγ↑+ ˆγ↑⊗1)−(1⊗ˆγT\n↓+ ˆγT\n↓⊗1),ˆρ4}\n+ (ˆρ⊗γT\n↓)τL+ (γ↑⊗ˆρ)τL+γ↑⊗ˆρ+ ˆρ⊗γ↑+ (γ↑⊗1)τL,(28)\nd\ndtˆβ=−i\nℏ[ˆh⊗1,ˆβ]−i\nℏ1⊗ˆhˆβ+\u0010\nˆρ⊗ |ξ⟩+ (ˆρ⊗ |ξ⟩)τL−[(ˆr⊗ ⟨ξ|)τR]T2\u0011\n+X\njκj(ˆuj⊗ˆujˆβˆu†\nj−ˆβ) +1\n2({ˆγ↑⊗1−ˆγT\n↓⊗1,ˆβ}+ (1⊗ˆγ↑−1⊗ˆγT\n↓)ˆβ) + ˆγ↑⊗ |α⟩+ (ˆγ↑⊗ |α⟩)τL(29)\nd\ndtˆr=−i\nℏˆhˆr+|ξ⟩⟨α|+1\n2(ˆγ↑−ˆγT\n↓)ˆr+T.+X\njκj(ˆujˆrˆuT\nj−ˆr), (30)8\nHereweintroducedthefollowingnotation: Tistandsfortranspositionon i-thspace, τLactsonketsasfollows |n, m⟩ →\n|m, n⟩andτRacts analogously on bras and “ T.„ stands for transposed expression. Also for some operator on single-\nparticle space ˆowe denote |k2, k3⟩⟨k1|ˆo⊗1:=|k2, k3⟩⟨k1|ˆoand|k2⟩⟨k3| ⊗ |k1⟩:=|k2, k1⟩⟨k3|and|k2, k3⟩⟨k1|⟨k4|:=\n|k2, k3⟩⟨k1, k4|. One can observe that if there is no coherent pumping (i. e. ∀iξi= 0) evolution of the operators\nˆρ4andˆρdecouples from evolution of the rest of the operators. Therefore, in such circumstances ˆρ4andˆρcan be\nconsidered by themselves without considering ˆβandˆrfor which there is a lack of a quantum state-like interpretation.\nNote that the first term of the evolution equation (28) for ˆρ4resembles the Heisenberg equation and it is responsible\nfor the unitary evolution generated by the Hamiltonian, where the rest comes from decoherence and pumping terms\nsimilarly as in the case of ˆρ. Let us also comment that ˆrtogether with ˆρand|α⟩allow for reconstruction of the\ncovariance matrix. Therefore, extension of RSF by ˆris the simplest extension of RSF that gives quantum features to\nthis formalism, however it is trivial in the sense that it reproduces the covariance matrix approach, and also it is not\na natural extension of this formalism. At the same time, the extension of RSF by ρ4follows the (multi)particle state\ninterpretation of the original RSF and thus when omitting the coherent pumping it provides the simplest extension\nof RSF that is natural to this formalism and adds the quantum features to the description. Still, one could make\nanalogously further extensions of RSF using three copies of single-particle space, and so on, to track higher-order\ncorrelations in photon numbers.\nThe important feature of the set of equations (21,22) and (28-30) is that in fact they represent a finite set of first\norder in time differential equations and if the time dependence of ˆh,|ξ⟩,ˆγ↕is given by simple functions one can easily\nfind analytic solution. Additionally, one can solve this set order by order, as solutions of tensors of lower order are\nindependent of solutions for tensors of higher order. Simply, one solves from the lowest order, i.e., |α⟩and use the\nsolution as non-homogeneous part for the rest of equations.\nThe introduction of implicit linear coupling of different modes in system Hamiltonian ˆH=P\nk,k′ωk,k′a†\nkˆak′is\nequivalent to taking a local approach in deriving the master equation where this coupling is considered to be a\nperturbation. Although it allows one to work explicitly with desired modes for consideration, it is not always valid and\nthen a global approach should be considered (see [37] for a brief review). In the global approach, one first diagonalizes\nthe system Hamiltonian and then considers the evolution of transformed modes. In the end, one transforms the\nobtained results by reverse unitary transformation of modes to the original basis. This can be easily done on reduced\nstate, which will be described further in the text; therefore, one can easily apply also global approach. In fact, the\nunitary transformation that diagonalizes the reduced Hamiltonian ˆhis the same as the unitary transformation of\nmodes that diagonalizes the original Hamiltonian. This allows for consideration of the system solely from the reduced\nperspective.\nB. Local evolution\nIf one is interested in entanglement in a given bipartition, the structure of interest is the reduced two-particle-like\nstate ˆρN\nρR. However, in general, based on (28) the evolution of this matrix is not self-contained and the full ˆρ4is\nrequired. Still, if the evolution is local in terms of a given bipartition, then the evolution of ˆρN\nρRis closed. Local in\nthis case intuitively means that the evolution does not transfer particles between modes corresponding to two parties\ndetermined by the given bipartition or does not damp or pump them in a correlated manner. This translates to ˆh,ˆγ↕\nand all ˆujbeing block diagonal in this bipartition, i.e. they can be written as a direct sum of matrices that act only\non degrees of freedom of the given party. This allows the projector ΠIA,IBto commute with tensor products of such\nblock diagonal matrices which appear in equation (28) (this also includes identity on a single-particle space and thus\noperators like ˆh⊗1). From this follows that after projecting (28) into subspace corresponding to ΠIA,IBwe get:\nd\ndtˆρΠ\nρ=−i\nℏ[ˆh⊗1+1⊗ˆh,ˆρΠ\nρ]\n+ ΠIA,IB\u0010\n|ξ⟩ˆβ†+ (|ξ⟩ˆβ†)τL+ (|ξ⟩⟨α| ⊗1)τL+ˆβ⟨ξ|+ (ˆβ⟨ξ|)τR+ (1⊗ |α⟩⟨ξ|)τR\u0011\nΠIA,IB\n+X\njκj(ˆuj⊗ˆujˆρΠ\nρˆu†\nj⊗u†\nj−ˆρΠ\nρ) +1\n2{(1⊗ˆγ↑+ ˆγ↑⊗1)−(1⊗ˆγT\n↓+ ˆγT\n↓⊗1),ˆρΠ\nρ}\n+ ΠIA,IB\u0002\n(ˆρ⊗γT\n↓)τL+ (γ↑⊗ˆρ)τL+γ↑⊗ˆρ+ ˆρ⊗γ↑+ (γ↑⊗1)τL\u0003\nΠIA,IB,(31)\nwhere one simply replace ˆρ4→ˆρΠ\nρin equation (28) and project the non-homogeneous using ΠIA,IB. This allows\nrestricting the considerations of entanglement in the extended RSF to ˆρN\nρRinstead of tracking the whole ˆρ4.9\nV. EXAMPLES: PPT CRITERION FOR MULTI-MODE BOSONIC FIELDS\nWe have shown that the sufficient condition for the entanglement of the full state ˆρFis that the reduced two-\nparticle-like state ˆρN\nρRis entangled. However, this does not mean that there exists a state ˆρFsuch that ˆρN\nρRis\nentangled. Therefore, in the following we show the existence of such states by presenting two such examples. What is\nmore, we show that ˆρN\nρRcontains information about entanglement both for Gaussian states and non-Gaussian states\nfor which the PPT criterion for the covariance matrix cannot reveal entanglement.\nBecause one can apply any entanglement detection method for bipartitie finite-dimensional systems to the ˆρN\nρRlet\nus consider the one of the most widely known entanglement criteria i. e. PPT criterion [3] which stands that it is\nsufficient if partial transposition of the state results in:\n(ˆρN\nρR)T2<0, (32)\nfor state ˆρN\nρRto be entangled. For the simplest case where ˆρN\nρRis a two-qubit density matrix which translates to\nˆρFdescribing four orthogonal modes, the PPT criterion is a sufficient and necessary condition for entanglement.\nTherefore, in such a case, if the PPT criterion does not reveal the entanglement of ˆρN\nρRthen our reduced two-particle-\nlike state is not able to reveal the entanglement of ˆρFwhat does not automatically mean that it is a separable state.\nIn the case of a larger number of modes, if the PPT criterion does not reveal the entanglement of ˆρN\nρRthe reduced\nstate might still keep information about entanglement of ˆρFbut it has to be accessed using a different criterion. While\none needs at least four modes for considerations of entanglement using ˆρN\nρR, in the following, we also show a method\nhow one can also apply this methodology to reveal the entanglement of two mode states.\nA. Bright squeezed vacuum\nAs our first example, we will use 2×2bright squeezed vacuum (BSV) which non-classicality in stationary scenarios\nwas considered [38–40] also in the high photon number limit and in the context of correlation in Bose-Einstein\ncondensates [41]. Let us consider four orthogonal modes and the bipartition given by IA={1,2},IB={3,4}. The\nBSV state is obtained by unitary evolution of the vacuum generated by the interaction Hamiltonian [4]:\nˆHint=γ(ˆa†\n1ˆa†\n4−ˆa†\n2ˆa†\n3) +h.c., (33)\nand therefore it is a Gaussian state. Now, BSV state can be written in the form:\n|ψ−⟩=1\ncosh2(Γ)∞X\nn=0tanhn(Γ)\nn!(ˆa†\n1ˆa†\n4−ˆa†\n2ˆa†\n3)n|Ω⟩=1\ncosh2(Γ)∞X\nn=0√\nn+ 1 tanhn(Γ)|ψn⟩, (34)\nwhere, the parameter Γ = γtdenotes the amplification gain and is related to the power of the coherent source\nimpinged on the nonlinear crystal in the parametric down conversion process and\n|ψn\n−⟩=1√n+ 1nX\nm=0(−1)m|(n−m), m;m,(n−m)⟩. (35)\nCalculations of ˆρΠ\nψyields:\nˆρΠ\nψ=\nρ1313 0 0 0\n0ρ1414 ρ1423 0\n0ρ2314 ρ2323 0\n0 0 0 ρ2424\n, (36)\nwhere we have restricted this matrix only to the subspace determined by ΠIA,IBand:\nρ1313=ρ2424= sinh4(Γ),\nρ1414=ρ2323= sinh2(Γ) cosh(2Γ) ,\nρ2314=ρ1423=−sinh2(Γ) cosh2(Γ).(37)10\nAfter normalization and applying the partial transpose, we find that (ˆρN\nψ)T2has two eigenvalues:\nλ1=1\n1−3 cosh(2Γ),\nλ2=λ3=λ4=1\n3−sech(2Γ).(38)\nTheeigenvalue λ1isnegativeforall Γ>0. Thus, weappliedthePPTcriteriontoshowthattheBSVstateisentangled\nfor any Γ>0and thus for an arbitrarily high average number of bosons. As a consequence, the two-particle space\ncan keep information about entanglement in mesoscopic scenarios.\nB. Single photon\nLet us consider also, as a second example, the state of symmetrically bemasplitted single-photon into modes a1, a3:\n|Ψ⟩=1√\n2(|1; 0⟩+|0; 1⟩) =1√\n2(ˆa†\n1+ ˆa†\n3)|Ω⟩. (39)\nThis state is of particular interest as it is non-Gaussian and one cannot reveal its entanglement through PPT criterion\nfor covariance matrix (see Appendix A).\nClearly this is a two-mode state and as we argued to apply the above entanglement criterion, one needs four modes.\nHowever, one can trivially extend the state to four modes by including modes a2, a4with zero photons occupying\nthose modes:\n|Ψ⟩=1√\n2(|10; 00⟩+|00; 10⟩). (40)\nStill, this state has the form (14) for which the RSF results in ˆρΠ\nΨ=ˆ0and thus does not give access to information\nabout entanglement. Instead, one can perform a different extension of the state to four modes to avoid reaching the\nform (14) by, for example, introducing coherent states with real amplitude αin the additional modes:\n|Ψ⟩=1√\n2(|1α; 0α⟩+|0α; 1α⟩) =e−α2\n√\n2(ˆa†\n1+ ˆa†\n3) ∞X\nn=0αnˆa†\n2√\nn!! ∞X\nm=0αmˆa†\n4√\nm!!\n|Ω⟩. (41)\nThis resembles performing a homodyne measurement on the single-photon state. However, for example, the\nassumption of matching energy was not done at this step which would be required to perform the necessary\ninterferometry for homodyne measurement. This is also not fully equivalent to performing homodyne measurement,\nas one is interested in measuring the number of photons instead of quadratures, and such schemes are sometimes\nreferred to as weak homodyne measurement [42–44]. Now, using the same partition IA={1,2},IB={3,4}we get\nthe following normalized reduced state ˆρN\nΨ:\nˆρN\nΨ=1\nα2+α4\n0 0 0 0\n0α2\n2α2\n20\n0α2\n2α2\n20\n0 0 0 α4\n(42)\nAfter partial transposition there are three eigenvalues of resulting matrix:\nλ1=α2−√\nα4+ 1\n2 (α2+ 1),\nλ2=α2+√\nα4+ 1\n2 (α2+ 1),\nλ3=λ4=1\n2 (α2+ 1),(43)\nwhere the first eigenvalue is negative for any finite α >0. Therefore, even for two mode entangled states, one can\nretrieve information about entanglement from extended RSF. Note that the lowest value of λ1is obtained for α= 0.\nThis is because one needs α̸= 0for allowing normalization of the matrix, however, the coherent states are separable\nfrom the entangled part of the state, therefore they do not ad entanglement but rather noise. Thus, optimally one\nshould have α≪1in this scenario.11\nC. Time evolution of entanglement\nLet us also apply the rest of the extended RSF toolbox to consider how entanglement behaves under non-unitary\nevolution resulting from decoherence coming from the environment. We first consider that one distributes a single-\nphotonstate |Ψ⟩betweentwodistantpartiesthatwanttoperformaweakhomodynemeasurementlocallytorevealthe\nentanglement of the state. We start with this example, as it is particularly interesting because a device-independent\nQKD protocol can be constructed based on such a setup [45] and maintaining entanglement during transmission is\nnecessary for its success. However, the state during transmission of the photon necessarily undergoes decoherence.\nLet us assume that during the distribution of the state the transmission line is affected by the thermal environment\nwith temperature Twhile the local oscillators (coherent states) are approximately not affected. In this scenario, the\noperators ˆγ↕are diagonal and fulfill the relation: ˆγ↑=e−ℏω/kbTˆγ↓, where ωstands for the angular frequency of the\nmodes, and kbis Boltzmann constant. The creation rates are given by:\nΓkk\n↑=γωN(ω)(δk1+δk3), (44)\nwhere δnmstands for Kronecker delta and with coefficient γωdetermining the coupling of the bath to modes with\nfrequency ωandN(ω) = 1 /(eℏω/kbT−1)being Planck distribution of the average number of photons in the bath [46].\nFurthermore, as all modes are separated up to the moment of the measurement, there is also no coupling between\nmodes, and the Hamiltonian is simply a free Hamiltonian for four modes with the same frequency. All of this allows\nus to consider only the evolution of reduced single and two particle states ˆρ,ˆρΠ\nρbased on our previous discussion in\nsection IVB. Due to trivial free evolution Hamiltonian the master equation simplifies to:\nd\ndtˆρ=1\n2{ˆγ↑−ˆγT\n↓,ˆρ}+ ˆγ↑, (45)\nd\ndtˆρΠ\nΨ=1\n2{(1⊗ˆγ↑+ ˆγ↑⊗1)−(1⊗ˆγT\n↓+ ˆγT\n↓⊗1),ˆρΠ\nΨ}\n+ ΠIA,IB,\u0000\n(ˆρ⊗γT\n↓)τL+ (γ↑⊗ˆρ)τL+γ↑⊗ˆρ+ ˆρ⊗γ↑+ (γ↑⊗1)τL\u0001\nΠIA,IB.(46)\nUsing initial condition for the reduced state ˆρ:\nˆρ(0) =\n1\n201\n20\n0α20α2\n1\n201\n20\n0α20α2\n. (47)\nwe obtain the following time evolution of the ˆρΠ\nΨ:\nˆρΠ\nΨ(t) =\ne−2γωtλ(λ+ 1) 0 0 0\n01\n2α2e−γωt(2λ+ 1)1\n2α2e−γωt0\n01\n2α2e−γωt 1\n2α2e−γωt(2λ+ 1) 0\n0 0 0 α4\n, (48)\nwhere λ=N(ω) (eγωt−1). Now, in order to access information about entanglement, we apply the PPT criterion\nafter normalization. In the result, we find that there is again one eigenvalue that can be negative:\nλ−(t) =1\n2Nh\nα4+λe−2tγω(λ+ 1)−e−5tγωp\ne6tγω(α8e4tγω−α4e2tγω(2λ(λ+ 1)−1) +λ2(λ+ 1)2)i\n,(49)\nwhere Nstands for normalization factor of ˆρΠ\nΨ(t). Let us first consider the limit t→ ∞. In such a case there are two\noptions:\nlim\nt→∞λ−(t) =N(ω)2/(α2+N(ω))2forα2≥N(ω) (50)\nlim\nt→∞λ−(t) =α4/(α2+N(ω))2forα2< N(ω) (51)\nRemarkably, if α̸= 0and the bath temperature is T= 0(N(ω) = 0) in this limit one has λ−(t) = 0what suggest\nthat extended RSF formalism finds the state to be entangled for any finite time t. To see that it is the case, first\nnote that:\nlim\nN(ω)→0λ−(t) =α4−√\nα8+α4e−2tγω\n2N. (52)12\nN(ω)=1\nN(ω)=0.1\nN(ω)=0\nUnitary0.5 1.0 1.5 2.0 2.5 3.0tγ-0.3-0.2-0.10.1λ-\n(a)\nα=0.1\nα=0.5\nα=10.5 1.0 1.5 2.0tγ\n-0.5-0.4-0.3-0.2-0.10.1λ- (b)\nFIG. 1: a) Time evolution of λ−(t)forα= 0.5and different values of N(ω). The solid red line presents the value\nofλ−(t)in the case of unitary evolution without a thermal environment. For N(ω) = 0value of λ−(t)converges\nto zero but is always negative, showing the entanglement of the state. With increasing N(ω)the photon number\nentanglement starts to decay faster, eventually disappearing. b) Time evolution of λ−(t)forN(ω) = 0 .1and different\nvalues of α. For all values of αaxis is reached at the same time tc. Value of λ−(0)increases with α, however, the rate\nof growth greatly decreases for higher values of α. This can be seen as the curve for α= 0.1quickly crosses other\ncurves while being still negative, and therefore extremely small values of αare no longer always optimal as in the\nunitary case.\nClearly, the second term of the nominator has a higher absolute value, as it is a decreasing function of time with the\nlimit t→ ∞equal to α4. Therefore, the second term determines the sign of the nominator to be negative. Because\nthe denominator is always positive (as it is the sum of expectation values containing only photon number operators\nthat are non-negative) we find that λ−(t)is also negative in this case for any finite tand this certifies entanglement\nof the state in the whole time range. Still, for any T >0the limit t→ ∞ofλ−(t)is positive, and therefore at some\nfinite time the entanglement will no longer be seen within the RSF formalism. One can find that this criterion does\nnot detect entanglement after critical time tc:\ntcγ= log \n1 +√\n2−1\n2N(ω)!\n, (53)\nwhich clearly goes to infinity for N(ω)→0and to 0 for N(ω)→ ∞. Interestingly, tcis independent of αand therefore\nαonly affects the magnitude of λ−(t)and not its qualitative behavior. FIG.: 1 presents examples of the evolution\nof the eigenvalue λ−(t)in different regimes. Clearly, entanglement becomes weaker with time evolution and with\nincreasing N(ω)its decay becomes faster. In general, this shows that a single-photon entangled state is a recourse\nwhich is quite robust against thermal damping when the thermal bath has a small temperature.\nNow, let us consider an analogous scenario for the BSV state, where all modes are coupled to the thermal\nenvironment with the same temperature during the distribution of the state between parties. In this case the creation\nrates are the following:\nΓkk\n↑=γωN(ω)δkk. (54)\nUsing as initial conditions reduced states (36) and\nρ=diag(sinh2(Γ),sinh2(Γ),sinh2(Γ),sinh2(Γ)), (55)\nWe get the following solution for ˆρΠ\nψ:\nˆρΠ\nψ(t) =\nρ1313(t) 0 0 0\n0 ρ1414(t)ρ1423(t) 0\n0 ρ2314(t)ρ2323(t) 0\n0 0 0 ρ2424(t)\n, (56)13\nΓ=∞\nΓ=1\nΓ=0.5\nΓ=0.1\nSingle-Photon\n0.5 1.0 1.5 2.0N(ω)0.20.40.60.81.01.21.4tc\nFIG. 2: Comparison of tcfor single-photon state and BSV state for different Γin function of N(ω). The critical time\ntcfor BSV state quickly converges to limiting case Γ→ ∞and reach significant advantage in robustness over single\nphoton-state. Still for small values of Γentanglement of the single-photon state is more robust against non-unitary\nresulting from thermal bath.\nwhere\nρ1313=ρ2424=1\n4e−2γωt(cosh(2Γ) + 2 λ−1)2,\nρ1414=ρ2323=e−2γωt\u0000\ncosh(2Γ)\u0000\nsinh2(Γ) + λ\u0001\n+ (λ−1)λ\u0001\n,\nρ2314=ρ1423=−e−2γωtsinh2(Γ) cosh2(Γ).(57)\nApplying the PPT criterion we find that negative eigenvalue evolves as follows:\nλ1(t) =1\n4−3 sinh2(2Γ)\n2(8(2 λ−1) cosh(2Γ) + 3 cosh(4Γ) + 16( λ−1)λ+ 5). (58)\nWith the realization that the only time dependence is in λwhich for the special case of T= 0is equal to 0, one\ncan see that the entanglement of the BSV state is, in fact, completely stationary under the considered non-unitary\nevolution i.e.,\nλ1(t) =λ1(0). (59)\nThis shows the exceptional robustness of the BSV state to low temperature damping. For any non-zero temperature\nT̸= 0there is a point in time at which the entanglement is not detected anymore:\ntcγω= log\u0012\n1 +e−Γsinh(Γ)\nN(ω)\u0013\n, (60)\nand for t→ ∞the eigenvalue λ1(t)converges to 1/4. Note that tcis an increasing function of the pumping parameter\nwith the limit\nlim\nΓ→∞tcγω= log\u0012\n1 +1\n2N(ω)\u0013\n. (61)\nCompering this expression to critical time for the single-photon state (53) one can see that the critical time for highly\npumped BSV state is longer matching the interpretation of high robustness of this state to this type of decoherence.\nStill, for small values of Γwhere BSV state is dominated by the vacuum component tccan be smaller than for the\nsingle-photon state. This happens for Γ<−log\u0000\n2−√\n2\u0001\n/2≈0.27. FIG. 2 shows comparison of tcfor BSV state\nwith different values of Γand single-photon state.\nD. Maximally mixed states\nLet us consider the limit t→ ∞of the reduced two-particle-like state ˆρΠ\nψ(t): (56). In this limit, one gets:\nlim\nt→∞ˆρΠ\nψ(t) =diag(N(ω)2, N(ω)2, N(ω)2, N(ω)2). (62)14\nThis matrix after normalization corresponds to the maximally mixed state on the subspace of the two-particle space\nrestricted to the given bipartition. As the bath in this example was thermal and the same for all modes, this\nreduced two-particle-like state corresponds to the four-mode thermal state for a given temperature T. Note that the\ntemperaturedoesnotaffectthenormalizedreducedstate. Similarly,forthisstate,wehave ˆρ=diag(N(ω), N(ω), N(ω), N(ω))\nwhich constitutes a maximally mixed state in a single-particle space. As the inclusion of additional separable modes\nin the thermal state for a given Tcannot add any new type of correlations seen in the reduced states, one can see\nthat the thermal states always correspond to maximally mixed states on single and two particle spaces.\nVI. PASSIVE OPTICAL ELEMENTS\nIn most optical devices, passive optical elements such as beamsplitters and phase shifters are present. The RSF\nformalism allows for easy introduction of such elements into the system. Therefore, this formalism allows for a\nstraightforward analysis of experimental setups. Let us start with the phase shifter. We assume that the beam\ncontains the mode aiat some point in time, starts the interaction with the phase shifter at t=τ0and ends it at\nt=τe, which adds the phase to this particular mode. This can be achieved by the following modification of the\nHamiltonian:\nˆH→ˆH+ϕiI[τ0,τe](t)ˆa†\niˆai, (63)\nwhere I[τ0,τe](t)stands for indicator function on the time interval [τ0, τe]andϕidetermines the phase gain rate of i-th\nmode. The resulting final phase shift is given by:\n∆ϕ= (τe−τ0)ϕi. (64)\nThe beam splitting of two modes ai, ajinto two modes a′\ni, a′\njcan be effectively realized as a unitary transformation\nof the corresponding annihilation and creation operators given by:\n\u0012ˆa′\ni\nˆa′\nj\u0013\n=\u0012√\nT i√\nR\ni√\nR√\nT\u0013\u0012\nˆai\nˆaj\u0013\n=UT\u0012\nˆai\nˆaj\u0013\n(65)\nwhere RandTstand for reflectivity and transmissivity respectively. Before beam splitting modes a′\ni, a′\njare in the\nground state, and after this transformation modes ai, ajare left empty. Therefore, their role interchange and tracking\nmodes ai, ajis of no use after the operation. Therefore, just after the operation, one can rename the modes a′\ni, a′\nj\ntoai, aj, keeping the structure of the reduced state. Based on the equation (17) one can find that any unitary\ntransformation Uof annihilation operators reduces to operator:\nU→ˆu=X\ni,jUji|j⟩⟨i|, (66)\nwhere Ujistand for matrix elements of U. Then under beam splitting operation the full reduced state transforms as\nfollows:\nˆρ4→ˆuT⊗ˆuTˆρ4ˆu†\nT⊗ˆu†\nT,\nˆβ→ˆuT⊗ˆuTˆβˆu†\nT,\nˆρ→ˆuTˆρˆu†\nT,\nˆr→ˆuTˆrˆuT\nT,\n|α⟩ →ˆuT|α⟩.(67)\nNote that one can analogously incorporate to the analysis phase shifting in an instantaneous manner as it results\nin unitary mode transformation. This reduction of unitary transformations of modes also allows one to consider\nthe inefficiency of the detectors, which can be modeled as a two-port beamsplitter on the path of the mode aiwith\ntransmissivity equal to efficiency T=ηand second output mode traced out. Denoting the efficiency of the detector\nat mode aiasηiinefficient detectors can be applied by operation:\nˆη=X\ni√ηi|i⟩⟨i|. (68)15\nThen the reduced state ˆρtransforms as ˆρ→ˆηρˆηand other parts of the reduced state analogously.\nAssuming equal efficiency ηof all detectors, one can see that entanglement of the BSV state can be reveled\nfor arbitrary efficiency, as the matrix ˆρΠ\nψ(36) is varied only by constant coefficient η2, which is removed after\nnormalization. Thus, the eigenvalues after partial transposition are not changed. In fact, this applies for any state\nand any entanglement criterion by the same means.\nLet us go back to the unitary transformation Udof the modes that diagonalize the original Hamiltonian ˆH. Clearly,\nthe diagonalized Hamiltonian ˆHdreduces to the diagonal Hamiltonian ˆhd. As the unitary transformation of modes\nreduces by (66) one gets that:\nˆudˆhˆu†\nd=ˆhd, (69)\nwhere the equality in particular stands for equality of the matrices representing operators in specific basis in which\nhdis diagonal. Therefore, the reduced operators ˆudperform the diagonalization of the reduced Hamiltonian. On the\nother hand, if one finds a unitary operator ˆudthat fulfills (69) for some ˆhdbecause the reduction of Hamiltonian is a\nbijection, one knows that there is a corresponding diagonal Hamiltonian in Fock space ˆHdthat must be obtained by\nmode transformation from the original Hamiltonian ˆH. But, since the reduction of unitary transformations of modes\nis also bijective, this transformation is obtained from ˆud.\nA. Homodyne measurement and squeezed states pumping\nOne of the tools used in quantum optics is homodyne measurements. In such measurements upon state ˆρF, one\ninterferes the coherent state through beamsplitter with the state ˆρFfollowed by intensity measurement or photon\ncounting. Such schemes can be analyzed in the RSF formalism as it requires only extending the reduced state by\nmodes in coherent states (see the Appendix B), and performing a beam splitting operation on the obtained reduced\nstate. However, as presented above, in some circumstances one does not even have to mathematically perform a beam\nsplitting operation to get access to information about entanglement.\nOne of the drawbacks of the RSF formalism is the absence of the possibility of direct inclusion of the squeezing in\nthe evolution of the system. However, there is a partial solution to this problem. In many experimental setups, first\nsome modes are impulsed pumped with squeezed states coming, for example, from parametric down conversion, and\nthen combined with the rest of the system through beamsplitters. As one can easily calculate the reduced state for the\nsqueezed states, analogously to the case of homodyne measurements, one can combine the modes in squeezed states\nwith the reduced state of the system. Therefore, one can effectively include some form of squeezing in the system.\nWhat is more, one can also include in such a way higher-order squeezed states [5].\nVII. MANDEL-Q PARAMETER AND TWO MODE ENTANGLEMENT\nOur extension of RSF is not limited only to revealing entanglement in terms of non-classical phenomena. One of\nthe non-classical phenomena in quantum optics is the sub-Poissonian photon statistics of light, which has no classical\ncounterpart. To quantify this phenomenon, one can use the Mandel Q parameter [47]:\nQi=⟨ˆa†\niˆaiˆa†\niˆai⟩ − ⟨ˆa†\niˆai⟩2\n⟨ˆa†\niˆai⟩−1. (70)\nIfQi<0the state of the mode aihas sub-Poissonian statistics. This non-classical behavior is still widely researched\nandused, forexample, forsourceverification[48–50], inparticular, usingthetime-dependentversionofthisparameter.\nClearly, the extended RSF formalism keeps information on photon statistics and its time dependence in the operators\nˆρ(t),ˆρ4(t)and can be found as:\nQi=tr{|i, i⟩⟨i, i|(ˆρ4−ˆρ⊗ˆρ)}\ntr{|i⟩⟨i|ˆρ}−1. (71)\nThis is one of the reasons why one would want to include ˆρ4in the extension of RSF instead of only ˆρΠ. Mandel Q\nparameter which is, in fact, shifted and rescaled variance allows for detecting non-classical behavior internal to state\nof single mode. However, other parts of the covariance matrix (note that this is a different covariance matrix from\nthe previously mentioned covariance matrix associated with quadratures), which mixes two modes ai, ajalso can be16\nused to reveal non-classicality, however, shared among multiple modes. Let us build a generalization of the Mandel\nQ parameter using the criterion for the two-mode entanglement from [51]:\n⟨ˆa†\niˆaiˆa†\njˆaj⟩ − ⟨ˆa†\niˆaj⟩⟨ˆa†\njˆai⟩<0 =⇒ Q ij≡⟨ˆa†\niˆajˆa†\njˆai⟩ − ⟨ˆa†\niˆaj⟩⟨ˆa†\njˆai⟩\n⟨ˆa†\niˆai⟩−1<0, (72)\nwhere we have used commutation relation [ˆaj,ˆa†\nj] = 1and the fact that ⟨ˆa†\niˆai⟩is non negative. In fact, this parameter\ncontainsthecovarianceoftheoperators ˆa†\njˆai,ˆa†\niˆajandclearlyithighlyresemblestheMandelQparameteras Qii=Qi.\nIn terms of RSF, one can calculate parameter Qijas:\nQij=tr{|j, i⟩⟨i, j|(ˆρ4−ˆρ⊗ˆρ)}\ntr{|i⟩⟨i|ˆρ}−1. (73)\nFor the maximally entangled state of a single photon shared among two modes (39) one gets Q13=Q31= 1/2. The\nanalogous time evolution to the considered in previous sections leads for this state to:\nQ13=Q31=(4λ(λ+ 1)−1)e−2γωt\ne−γωt(1\n2−N(ω)) +N(ω). (74)\nOne can find that Q13= 0at exactly the same time tcas the scenario with ancillary coherent states (53). For Q13\nadding ancillary states to reveal the entanglement turned out to be unnecessary as this parameter uses additional\ninformation contained in ˆρ.\nLet us also mark some additional similarities of Qijwith Qi. Consider Q12for a two-mode coherent state in modes\na1, a2with complex amplitudes α1, α2. In this scenario, one can find that:\nQ12=|α1|2|α2|2+|α1|2− |α1|2|α2|2\n|α1|2−1 = 0 . (75)\nLet us recall that whenever Qi= 0the photon counting in the mode iis a Poissonian process and it is always the\ncase for the coherent states. Now consider two mode thermal state:\nˆρth=∞X\ni=0N1(ω)i\n(N1(ω) + 1)i+1∞X\nj=0N2(ω)j\n(N2(ω) + 1)j+1|i;j⟩⟨i;j|, (76)\nwhere Nq(ω)is the average number of photons in mode q. For this state, one gets:\nQij=Ni(ω). (77)\nNote that for the thermal state, the Mandel Q parameter has the same value Qi=Ni(ω)that exceeds the Poissonian\ncase. One can observe the high resemblance between the values of QijandQiisuggesting a deeper connection between\nthese two.\nA. Transformation of sub-Poissonian statistics into two mode entanglement\nConsider two mode state ˆρΦwith one of the modes in an arbitrarily state ˆρφand the second one in the zero photon\nstate:\nˆρΦ= ˆρφ⊗ |0⟩⟨0|. (78)\nThe relevant for parameter Qijreduced states are the following:\nˆρ=\u0012\n⟨ˆn1⟩0\n0 0\u0013\n,ˆρ4=\n⟨ˆn2\n1⟩0 0 0\n0 0 0 0\n0⟨ˆn1⟩0 0\n0 0 0 0\n. (79)17\nLet us now consider a scenario in which one performs the beam splitting operation on this state which results based\non the transformation (67) in the reduced states:\nˆρ=\u0012\n⟨ˆn1⟩T −i⟨ˆn1⟩√\nR√\nT\ni⟨ˆn1⟩√\nR√\nT ⟨ˆn1⟩R\u0013\n,\nˆρ4=\n⟨ˆn1⟩RT+⟨ˆn2\n1⟩T2i⟨ˆn1⟩√\nRT3/2−i⟨ˆn2\n1⟩√\nRT3/2−i⟨ˆn1⟩R3/2√\nT−i⟨ˆn2\n1⟩√\nRT3/2⟨ˆn1⟩RT− ⟨ˆn2\n1⟩RT\ni⟨ˆn1⟩R3/2√\nT+i⟨ˆn2\n1⟩√\nRT3/2⟨ˆn2\n1⟩RT− ⟨ˆn1⟩RT ⟨ˆn1⟩R2+⟨ˆn2\n1⟩RT i ⟨ˆn1⟩R3/2√\nT−i⟨ˆn2\n1⟩R3/2√\nT\ni⟨ˆn2\n1⟩√\nRT3/2−i⟨ˆn1⟩√\nRT3/2⟨ˆn1⟩T2+⟨ˆn2\n1⟩RT ⟨ˆn2\n1⟩RT− ⟨ˆn1⟩RT −i⟨ˆn1⟩√\nRT3/2−i⟨ˆn2\n1⟩R3/2√\nT\n⟨ˆn1⟩RT− ⟨ˆn2\n1⟩RT i ⟨ˆn1⟩√\nRT3/2+i⟨ˆn2\n1⟩R3/2√\nT i⟨ˆn2\n1⟩R3/2√\nT−i⟨ˆn1⟩R3/2√\nT ⟨ˆn1⟩RT+⟨ˆn2\n1⟩R2\n.\n(80)\nUpon this one can calculate parameter Qout\nijfor the output modes:\nQout\n12=⟨ˆn2\n1⟩RT+⟨ˆn1⟩T2− ⟨ˆn1⟩2RT\n⟨ˆn1⟩T−1 (81)\nQout\n21=⟨ˆn2\n1⟩RT+⟨ˆn1⟩R2− ⟨ˆn1⟩2RT\n⟨ˆn1⟩R−1 (82)\nQout\n11=⟨ˆn2\n1⟩T2+⟨ˆn1⟩RT− ⟨ˆn1⟩2T2\n⟨ˆn1⟩T−1 (83)\nQout\n22=⟨ˆn2\n1⟩R2+⟨ˆn1⟩RT− ⟨ˆn1⟩2R2\n⟨ˆn1⟩R−1 (84)\nNote that, Qout\n21+Qout\n12=Qout\n22+Qout\n11and so entangled state resulting from the beam splitting have at least one of\nthe modes necessarily sub-Poissonian locally. In fact both of them have to be sub-Poissonian as:\nQout\n11=T⟨ˆn2\n1⟩ − ⟨ˆn1⟩2\n⟨ˆn1⟩+R−1 =T⟨ˆn2\n1⟩ − ⟨ˆn1⟩2\n⟨ˆn1⟩−T=TQin\n11<0, (85)\nQout\n22=R⟨ˆn2\n1⟩ − ⟨ˆn1⟩2\n⟨ˆn1⟩+T−1 =R⟨ˆn2\n1⟩ − ⟨ˆn1⟩2\n⟨ˆn1⟩−R=RQin\n11<0, (86)\nwhere we have used that R+T= 1and that input Mandel Q parameter is as follows:\nQin\n11=⟨ˆn2\n1⟩ − ⟨ˆn1⟩2\n⟨ˆn1⟩−1. (87)\nClearly the sub-Poissonian statistics were redistributed accordingly to transmissivity. Now it is straightforward to see\nthe particularly important relation:\nQout\n12+Qout\n21=Qin\n11. (88)\nThis shows that starting from the product input state (78), the two-mode entanglement after beam splitting is\nexclusively inherited from the sub-Poissonian statistics of the input state. Therefore, beam splitting redistributes\nthe internal non-classical photon number correlation of the state to the photon number entanglement of the output\nmodes. Note that due to this equality, Qout\n12+Qout\n21is lower bounded by -1 and that it is minimized by the input\nstates being Fock states ˆρφ=|n⟩⟨n|.\nVIII. CONCLUDING REMARKS\nIn summary, we have proposed an extension of the reduced state of the field formalism for mesoscopic bosonic\nfields. Our extension allows for tracking the evolution of particle number entanglement for systems undergoing non-\nunitary evolution, what is not possible in the original reduced state of the field approach. This is both for Gaussian\nand non-Gaussian states. This was achieved by reducing the problem of entanglement of multi-mode bosonic state\ninto the entanglement of the two-particle state on a finite-dimensional Hilbert space. Using tools available for our\nformalism, we showed that particle number entanglement of multi-mode bosonic field is independent of the efficiency\nof the detectors if it is the same across all detectors. What is more, based on the proposed approach we have shown\nthat the entanglement of the states of the beamsplitted single photon and 2×2bright squeezed vacuum are robust\nagainst interaction with a low temperature thermal environment. Furthermore, we have considered the generalization18\nof the Mandel Q parameter constructed within the proposed approach, which describes both non-classical correlations\nin single mode and shared among different modes. Based on the analysis of this parameter we have shown that the\nentanglement of two-mode entangled states obtained by the beam splitting one occupied mode is solely inherited from\nthe sub-Poissonian statistics of the input state.\nThe proposed reduced state of the field description can provide a useful and intuitive tool for the analysis of\nthe impact of decoherence on non-classical phenomena in multiple optical experimental setups and for designing\nthe experiments requiring mainly specification of the used photon sources. This is because our approach inherits\nmultiple tools from standard quantum mechanics of finite-dimensional systems, with only changed interpretation,\nand the whole analysis of the system could be done without even invoking Fock space. Furthermore, one can easily\nincorporate passive optical elements and inefficiencies in the description of the evolution. In addition, ancillary modes\nand entanglement can also be added to the system through the usage of beamsplitters.\nFurthermore, as extended RSF preserves state-like interpretation of reduced states, and maintains Hilbert space\nstructure for them, one can exploit scalar product of those Hilbert spaces for geometric-like considerations of bosonic\nstates as e.g. in [52]. This allows for considerations of proximity of states in terms of reduced states and therefore for\ntheir classification in terms of mesoscopic properties.\nAs we found that for both single-photon entangled state and bright squeezed vacuum the entanglement is robust\nagainst thermal damping in low temperatures, an interesting open question arises. Is this property universal to photon\nnumber entanglement?\nThe proposed extension of the reduced state of the field approach potentially could also be used to examine different\nquantum optical phenomena. Let us recall that the original reduced state of the field contained information about\nthe expectation values of Stokes operators, giving generalization to the description of polarization in terms of Jones\nvectors. Note that our extension when restricted to two modes keeps information additionally about second-order\npolarization tensors [32, 53]. These are associated with non-classical phenomena of hidden polarization, and our\napproach could give additional insight into this phenomena.\nACKNOWLEDGMENTS\nThe work is part of ‘International Centre for Theory of Quantum Technologies’ project (contract no. 2018/MAB/5),\nwhich is carried out within the International Research Agendas Programme (IRAP) of the Foundation for Polish\nScience (FNP) co-financed by the European Union from the funds of the Smart Growth Operational Programme, axis\nIV: Increasing the research potential (Measure 4.3). The authors thank Tomasz Linowski, Antonio Mandarino, and\nAgnieszka Schlichtholz for their remarks.\n[1] Z. Wang, M. Pechal, E. A. Wollack, P. Arrangoiz-Arriola, M. Gao, N. R. Lee, and A. H. Safavi-Naeini, Quantum Dynamics\nof a Few-Photon Parametric Oscillator, Phys. Rev. X 9, 021049 (2019).\n[2] H. Le Jeannic, T. Ramos, S. F. Simonsen, T. Pregnolato, Z. Liu, R. Schott, A. D. Wieck, A. Ludwig, N. Rotenberg, J. J.\nGarcía-Ripoll, and P. 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Björk, A. B. Klimov, G. Leuchs, and L. L. Sánchez-Soto, Unpolarized states and hidden polarization,\nPhys. Rev. A 90, 043826 (2014).\nAppendix A: Lack of entanglement detection of beamsplitted single photon by PPT criterion applied to\ncovariance matrix\nLet us show that the PPT criterion applied to the covariance matrix does not allow for the detection of entanglement\nof the state of beamsplitted single photon into two modes a1anda2:\n|Ψ⟩=1√\n2(|1; 0⟩+|0; 1⟩) =1√\n2(ˆa†\n1+ ˆa†\n2)|Ω⟩. (A1)\nLet us recall that covariance matrix is given by:\nVi,j=1\n2⟨{ˆΞi,ˆΞj}⟩ − ⟨ ˆΞi⟩⟨ˆΞj⟩, (A2)\nwhere ⃗Ξ = (ˆ x1,ˆp1, ....,ˆxn,ˆpn)and operators ˆxj,ˆpjare dimensionless position and momentum operators in j-th mode:\nˆxj=1√\n2(ˆaj+ ˆa†\nj),ˆpj=i√\n2(ˆa†\nj−ˆaj). (A3)\nBy noting that for state (A1) we have ⟨ˆa†\nj⟩=⟨ˆaj⟩=⟨ˆa†\njˆa†\ni⟩=⟨ˆajˆai⟩= 0and⟨ˆa†\njˆai⟩= 1/2one can write down the\ncovariance matrix for this state as:\nV=1\n2\n⟨2ˆa†\n1ˆa1⟩+ 1 0 ⟨ˆa†\n1ˆa2+ ˆa†\n2ˆa1⟩ 0\n0 ⟨2ˆa†\n1ˆa1⟩+ 1 0 ⟨ˆa†\n1ˆa2+ ˆa†\n2ˆa1⟩\n⟨ˆa†\n1ˆa2+ ˆa†\n2ˆa1⟩ 0 ⟨2ˆa†\n2ˆa2⟩+ 1 0\n0 ⟨ˆa†\n1ˆa2+ ˆa†\n2ˆa1⟩ 0 ⟨2ˆa†\n2ˆa2⟩+ 1\n=1\n2\n2 0 1 0\n0 2 0 1\n1 0 2 0\n0 1 0 2\n. (A4)\nThe PPT criterion for entanglement in terms of covariance matrix is given as follows:\nQ·V·Q−i\n2J <0, (A5)\nwhere Q=diag(1, q1, ...,1, qn)with qj=−1for modes corresponding to transposed subsystem and qj= 1otherwise\nand:\nJ=nM\nk=1J2, J2=\u0012\n0 1\n−1 0\u0013\n. (A6)\nThis criterion is necessary and sufficient condition for detecting two mode entanglement of Gaussian states, and\nsufficient condition for non-Gaussian states (what is a the case for (A1)). The eigenvalues of the matrix on the r.h.s.\nof inequality (A5) for the state (A1) are positive and equal to (2±√\n2)/2. Therefore, one does not find entanglement\nwith PPT criterion applied to the covariance matrix of the state (A1).21\nAppendix B: Extending reduce states by the separable subsystems\nOften when one wants to consider multiple modes, where the states of these modes are prepared by different sources\nand therefore the total state is a product state of subsystems. Let us consider the case of two subsystems A and\nB with global state ρtot= ˆρA⊗ˆρB. Often it is easier to construct the reduced state of the global state using the\nsimpler reduced states of ˆρAandˆρB. This is possible as for such states we have ⟨ˆAˆB⟩=⟨ˆA⟩⟨ˆB⟩, where ˆA,ˆBare\nsome observables on modes corresponding to subsystems A and B. The reduced state of the whole system can be then\nobtained as follows:\n|αt⟩=|αA⟩+|αB⟩,\nˆρt= ˆρA+ ˆρB+|αA⟩⟨αB|+|αB⟩⟨αA|,\nˆrt= ˆrA+ ˆrB+|αA⟩⟨α∗\nB|+|αB⟩⟨α∗\nA|,\nˆβt=ˆβA+ (|α∗\nA⟩ˆrB)T1+ (|αB⟩ˆρA)τL+A↔B,\nˆρ4t= ˆρ4A+ˆβA⟨αB|+ (ˆβA⟨αB| − |αA⟩⟨αB| ⊗1+|αB⟩ˆβ†\nA)τR+ (|αB⟩ˆ(β†\nA)τR− |αB⟩⟨αA| ⊗1)τL+A↔B,(B1)\nwhere subscript tstands for global state and indices AandBfor corresponding parties. Note that, swap operations\nτL, τRcan be performed using N2×N2-dimensionla matrix ˆτwhere Nis total number of modes:\nτi,j=(\n1∃m, n∈ {1..., d}:i=m+d(−1 +n)andj=d(−1 +m) +n\n0otherwise. (B2)\nThen for some operator ˆOone gets:\nOτL= ˆτˆO, (B3)\nOτR=ˆOˆτ. (B4)\nAppendix C: Full second-order evolution\nIn the similar manner to the presented extended RSF approach, one could consider the full second-order evolution of\nthe system. In such scenario one could consider reduction of the second-order observables to three reduced operators\non single particle space:\nˆO:=X\nk,k′ok,k′ˆa†\nkˆak′+1\n2X\nk,k′(qk,k′ˆakˆak′+q∗\nk,k′ˆa†\nkˆa†\nk′)\n→\u0010\nˆo=X\nk,k′ok,k′|k⟩⟨k′|,ˆq:=1\n2X\nk,k′(qk,k′)|k⟩⟨k′|,ˆq∗:=1\n2X\nk,k′(q∗\nk,k′)|k⟩⟨k′|\u0011\n.(C1)\nNote that, having also reduced first-order observables:\nˆO′=X\nk(pkˆak+p∗\nkˆa†\nk)→(⟨p|:=X\nkpk⟨k|,⟨p∗|:=X\nkp∗\nk⟨k|), (C2)\nexpectation value of combination of such observables is then given by:\n⟨ˆO⟩+⟨ˆO′⟩= Trn\nˆOˆρFo\n= tr\b\n(ˆo,ˆq,ˆq∗)·(ˆρ,ˆr,ˆr∗)T\t\n+ (⟨p|,⟨p∗|)·(|α⟩,|α∗⟩)T. (C3)\nWe can allow now any Hamiltonian that can be reduced by the procedure (C1). We denote the reduced Hamiltonian\nby the following triple (ˆh,ˆhs,ˆh∗\ns). The operator ˆhgoverns the free evolution and exchange of photons between\nmodes. Meanwhile, operators ˆhs,ˆh∗\nscorrespond to single-mode and two-mode squeezing. The introduction of such\nan interaction Hamiltonian can correspond, e.g., to using a parametric approximation to some modes in the system.\nThose modes are treated as classical coherent pumping fields which are subject to a parametric down-conversion\nprocess in which annihilation of a single photon from the pumping field is accompanied by creation of two photons in\nmodes which we treat in full quantum manner. This is a similar addition to the formalism to the coherent pumping\nin (1). However, such evolution requires an extension of the reduced states. If one is only interested in the evolution22\nof the original reduced state ˆρthe addition of ˆris sufficient and the equations of motion obtained analogously are as\nfollows:\nd\ndtˆρ=−i\nℏ[ˆh,ˆρ]−2i\nℏ(ˆh∗\nsˆr∗−ˆrˆhs) + (|ξ⟩⟨α|+|α⟩⟨ξ|) +1\n2{ˆγ↑−ˆγT\n↓,ˆρ}+ ˆγ↑, (C4)\nd\ndtˆr=−i\nℏˆhˆr−2i\nℏˆh∗\nsˆρT−i\nℏˆh∗\ns+|ξ⟩⟨α|+1\n2(ˆγ↑−ˆγT\n↓)ˆr+T., (C5)\nd\ndt|α⟩=−i\nℏˆh|α⟩ −2i\nℏˆh∗\ns|α⟩+|ξ⟩+1\n2(ˆγ↑−ˆγT\n↓)|α⟩, (C6)\nwhere “ T.” stands for transposed expression. To consider the evolution of ˆρ4one needs to extend the reduction by\nthe additional reduced states:\nˆm:=X\nk1,k2,k3,k4trn\nˆρFˆa†\nk1ˆak2ˆak3ˆak4o\n|k2, k4⟩⟨k1, k3|, (C7)\nˆq:=X\nk1,k2,k3,k4tr{ˆρFˆak1ˆak2ˆak3ˆak4}|k2, k4⟩⟨k1, k3|, (C8)\nˆζ:=X\nk1,k2,k3,k4tr{ˆρFˆak1ˆak2ˆak3}|k2, k3⟩⟨k1|. (C9)\nthen the set of evolution equations is extended by:\nd\ndtˆρ4=−i\nℏ[ˆh⊗1+1⊗ˆh,ˆρ4]−2i\nℏ(1⊗ˆh†\nsˆm†+ (1⊗ˆh†\nsˆm†)τL−ˆm1⊗ˆhs−( ˆm1⊗ˆhs)τR)\n−2i\nℏ\u0012\n(ˆh†\nsˆr†⊗1)τL+h\n(ˆr†⊗ˆh†\ns)τLiT2\n−(ˆrˆhs⊗1)τL−h\n(ˆhs⊗ˆr)τLiT2\u0013\n+\u0010\n|ξ⟩ˆβ†+ (|ξ⟩ˆβ†)τL+ (|ξ⟩⟨α| ⊗1)τL+ˆβ⟨ξ|+ (ˆβ⟨ξ|)τR+ (1⊗ |α⟩⟨ξ|)τR\u0011\n+1\n2{(1⊗ˆγ↑+ ˆγ↑⊗1)−(1⊗ˆγT\n↓+ ˆγT\n↓⊗1),ˆρ}+ (ˆρ⊗γT\n↓)τL+ (γ↑⊗ˆρ)τL+γ↑⊗ˆρ+ ˆρ⊗γ↑+ (γ↑⊗1)τL,\n(C10)\nd\ndtˆm=−i\nℏ[ˆh⊗1,ˆm]−i\nℏ(1⊗ˆhˆm+ ˆm1⊗ˆhT)−2i\nℏ(ˆρ41⊗ˆh†\ns+{1⊗ˆh†\ns,ˆρ4}T2−ˆqˆhs⊗1)\n−2i\nℏ\u0010\nˆρ⊗h†\ns−\u0002\n(ˆρ⊗h†\ns)τL\u0003T2\u0011\n+\u0010\nˆβ⟨ξ∗|+ (ˆβ⟨ξ∗|)T2+h\n(ˆβ⟨ξ∗|)T2iτL\n−(ˆζ⟨ξ|)τR\u0011\n+1\n2{(ˆγ↑⊗1)−(ˆγT\n↓⊗1),ˆm}+1\n2(1⊗ˆγ↑ˆm+ ˆm1⊗ˆγT\n↑−1⊗ˆγT\n↓ˆm−ˆm1⊗ˆγ↓)\n−ˆγ↑⊗ˆr−(ˆγ↑⊗ˆr)τL−[(ˆγ↑⊗ˆr)τL]T2(C11)\nd\ndtˆq=−i\nℏ(ˆh⊗1ˆq+ ˆqˆhT⊗1+1⊗ˆhˆq+ ˆq1⊗ˆhT)−2i\nℏ\u0010\n(ˆh†\ns⊗1ˆmT)τL+ ( ˆmˆh†\ns⊗1)τR+ˆh†\ns⊗1ˆm+ ˆmTˆh†\ns⊗1\u0011\n−2i\nℏ\u0012\nˆr⊗ˆh†\ns+ (ˆr⊗ˆh†\ns)τL+h\n(ˆr⊗ˆh†\ns)τLiT2\u0013\n−i\nℏ\u0012\nˆh†\ns⊗ˆr+ (ˆh†\ns⊗ˆr)τL+h\n(ˆh†\ns⊗ˆr)τLiT2\u0013\n+\u0012\nˆζ⟨ξ∗|+ (ˆζ⟨ξ∗|)T2+ (ˆζ⟨ξ∗|)τR+h\n(ˆζ⟨ξ∗|)τRiT1\u0013\n+1\n2(ˆγ↑⊗1+1⊗ˆγ↑−ˆγT\n↓⊗1−1⊗ˆγT\n↓)ˆq+1\n2ˆq(ˆγT\n↑⊗1+1⊗ˆγT\n↑−ˆγ↓⊗1−1⊗ˆγ↓),\n(C12)23\nd\ndtˆβ=−i\nℏ[ˆh⊗1,ˆβ]−i\nℏ1⊗ˆhˆβ,−2i\nℏ\u0012\n(ˆβ†1⊗ˆh†\ns)T2+h\n(ˆβ†1⊗ˆh†\ns)T2iτL\n+h\n(ˆh†\ns⊗ ⟨α|)τRiT2\n−ˆζˆhs⊗1\u0013\n+\u0010\nˆρ⊗ |ξ⟩+ (ˆρ⊗ |ξ⟩)τL−[(ˆr⊗ ⟨ξ|)τR]T2\u0011\n+1\n2({ˆγ↑⊗1−ˆγT\n↓⊗1,ˆβ}+ (1⊗ˆγ↑−1⊗ˆγT\n↓)ˆβ)\n+ ˆγ↑⊗ |α⟩+ (ˆγ↑⊗ |α⟩)τL(C13)\nd\ndtˆζ=−i\nℏ[ˆh⊗1ˆζ+ˆζˆhT⊗1+1⊗ˆhˆζ]−2i\nℏ\u0010\nˆβˆh†\ns⊗1+ˆh†\ns⊗1ˆβT1+ (ˆh†\ns⊗1ˆβT1)τL\u0011\n−2i\nℏ\u0012\nˆh†\ns⊗ |α⟩+ (ˆh†\ns⊗ |α⟩)τL+h\n(ˆh†\ns⊗ |α⟩)τLiT1\u0013\n+\u0010\nˆr⊗ |ξ⟩+ (ˆr⊗ |ξ⟩)τL+ [(ˆr⊗ |ξ⟩)τL]T1\u0011\n+1\n2\u0010\n(ˆγ↑⊗1−ˆγT\n↓⊗1)ˆζ+ˆζ(ˆγT\n↑⊗1−ˆγ↓⊗1) + (1⊗ˆγ↑−1⊗ˆγT\n↓)ˆζ\u0011\n.(C14)\nNote that such an extended RSF allows for calculation of any additive observable up to fourth order in creation and\nannihilation operators."    },    {        "title": "2402.02292v1.A_3_D_Full_Wave_Model_to_Study_the_Impact_of_Soybean_Components_and_Structure_on_L_Band_Backscatter.pdf",        "content": "1 \n> MANUSCRIPT ID NUMBER<  \n \nA 3-D Full -Wave Model to Study the Impact of \nSoybean Components and Structure on L -Band \nBackscatter  \n \nKaiser Niknam, Member, IEEE , Jasmeet Judge, Senior  Member , IEEE , A. Kaleo Roberts, Student  Member , IEEE , \nAlejandro Monsivais -Huertero , Senior  Member, IEEE , Robert Moore,  Senior  Member, IEEE , \nKamal Sarabandi,  Life Fellow, IEEE , and Jiayi Wu, Student  Member , IEEE  \n \n  \nAbstract —Microwave remote sensing offers a powerful tool for monitoring the growth of short , dense  vegetation like soybean. As the \nplant s mature, changes in their biomass and 3 -D structure impact the electromagnetic (EM) backscatter signal. This backscatter \ninformation holds valuable insights into crop health and yield, prompting the need for a comprehensive understanding of how \nstructural and biophysical properties of soybeans  as well as  soil characteristics  contribute to the overall backscatter signature. In this \nstudy, a full -wave model is developed for simulating L -band backscatter from soybean fields. Leveraging the ANSYS High -Frequency \nStructure Simulator (HFSS) framework, the model solves for the scattering of EM waves from realistic 3 -D structural models of \nsoybean, explicitly incorporating the interplant scattering effects. The model estimates of backscatter match well with the field \nobservations from the SMAPVEX16 -MicroWEX and SMAPVEX12 , with average differences of 1 -2 dB for co -pol and less than 4 dB \nfor cross -pol. Furthermore, the model effectively replicates the temporal dynamics of crop backscatter throughout the growing season.  \n \nManuscript received XX xx, 2023; revised XX xx, 2024; and accepted XX xx, 2024. This work is supported by NASA Grant #80NSSC2 1K0186 \n(corresponding author: Jasmeet Judge).  \nKaiser Niknam, Jasmeet Judge, and Jiayi Wu are with the Center for Remote Sensing, Department of Agricultural and Biological Engineering, Institute of \nFood and Agricultural Sciences, University of Florida, Gainesville, Florida, USA (e -mail: {gheysar.niknam; jasmeet; wuj2}@ufl.edu).  \nA. Kaleo Roberts and Kamal Sarabandi are with the Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor, Michigan, \nUSA (e -mail: {kaleor; saraband}@umich.edu).  \nAlejandro Monsiváis -Huertero is with the Escuela Superior de Ingenieria Mecanica y Electrica, Ticoman, Instituto Politecnico Nacional, Mexico Cit y, Mexico \n(e-mail: monsivais@ufl.edu).  \nRobert C. Moore is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida, U SA (e -mail: \nmoore@ece.ufl.edu) . 2 \n> MANUSCRIPT ID NUMBER<  \n \nThe HFSS analysis revealed that the stems and pods are the primary contributors to HH -pol backscatter, while the branches \ncontribute to VV -pol, and leaves impact the cross -pol signatures. In addition, a sensitivity study with 3 -D bare soil surface resulted in \nan average variation of 8 dB in co - and cross -pol, even when the root mean square height and correlation length were held constant. \nThese capabilities underscore the model’s potential to provide insights into the underlying dynamics of the backscatter for growing \nvegetation.   \n \nIndex Terms —3-D backscatter, ANSYS HFSS, computational electromagnetics, radar backscatter, SMAPVEX16 -MicroWEX, \nSMAPVEX12, and soybean.   \n \nI. INTRODUCTION  \nICROWAVE  monitoring of soybean fields is an efficient method to assess soybean plant growth over the course of a \ngrowing season  [1]. The dielectric properties and 3 -D structure of the soybean plant change as it grows, and these \nchanges impact the backscatter from the crop in meaningful ways [2] [3] [4]. As a result, radar data provides valuable \ninformation on crop health and yield [5] [6] [7]. Thus, an understanding of how structural and biophysical properties \nof soybean and soil characteristics contribute to the overall signature during the growing season is required to decode \nthe information encoded in radar backscatter. Empirical models have been used to estimate backscatter from crops using \nrelationships that relate vegetation water content (VWC) and soil properties (moisture and roughness) to backscatter values [8] \n[9] [10] [11]. However, these relationships are specific to the region that the model was calibrated for. Additionally, \nunderstanding the mechanisms underlying crop backscatter is best achieved by forward models that simulate the interaction of \nelectromagnetic (EM) waves with crops.  The development of more accurate reverse models is critically dependent on the \ninsights gained from such forward modeling approaches.  \n \nModeling the interaction between EM waves and growing vegetation in agricultural fields is a challenge due to its structural \ncomplexity. The water cloud model (WCM) [12] [13] [14] [15] [16], radiative transfer equation (RTE) [17] [18] [19], and \ndistorted Born approximation (DBA) [20] [21] [22] [23] [24] [25] have been methods of choice to model backscatter from \nvegetation canopies. These methods are based on the simplified models of the plant and/or simplified assumptions on how EM \nwaves interact with the crop. For example, in WCM, canopies are modeled as a water cloud containing identical water droplets \nuniformly distributed within the canopy [26]. Additionally, scattering from the crop is reduced to the scattering from its canopy. \nIn DBA models, however more realistic compared to WCM, scattering mechanisms at the L -band frequencies (1 -2 GHz) are still \nlimited to [27] direct scatter (e.g., direct scattering from ground, or, direct scattering from vegetation), double -bounce scatter M 3 \n> MANUSCRIPT ID NUMBER<  \n \n(e.g., ground -vegetation scattering, or, ground -vegetation -ground scattering), and volume scattering (e.g., attenuating within, and \nscattering from canopy). Although these simplifying assumptions helped to implement a set of successful early models [10] [21] \n[13] [20], they did not work well in all conditions . In particular, they fail (1) when the plants and/or their canopies are not \nuniformly distributed, (2) when the scattering from the plant cannot be decomposed into a summation of independent scattering s \nfrom its components, and (3) when far -field assumption held throughout the RTE and DBA models does not hold true [28] [29]. \n \nThe above -mentioned methods have a common aspect, namely, they are computationally efficient. Recent advances in \ncomputational resources have motivated the development of 3 -D models based on numerical techniques such as Numerical \nMaxwell Model in 3 -D (NMM3D) [30], mainly accompanied by hybrid methods [29] [31]. In hybrid methods, Maxwell’s \nequations are solved for isolated realistic 3 -D structural models of the plant to impose less computational cost, and then transition \nmatrices (T -matrices) obtained from these solutions are employed to build canopies and simulate the total backscatter from a \nfield. Although hybrid approaches are computationally efficient, they tend to ignore physical overlaps and multiple scatterin g \nexisting in the crops, and could create errors of several orders of magnitude (up to 20 dB)  in Radar Cross Section (RCS)  at lower \nincidence angles  [32] [33]. \n \nTo address the limitations of hybrid methods, full -wave simulations have recently been used to estimate backscattering from \nfields without making any restrictive assumptions about the interaction of EM waves with crops [34] [35] [36] [37]. These \nmodels are computationally more intensive than previous methods, but they provide a robust framework to model backscatter \nfrom crops under different conditions. Recently, Roberts et al.  [38] have developed a full -wave simulation model that estimates \ncorn backscatter using a 3 -D structural mesh for a complete plant. This approach improves upon hybrid methods by considering \nmultiple scatterings within crop canopies, and upon previous full -wave models by employing a realistic 3 -D structural plant \nmodel for the growing crop. Roberts et al.  [38] evaluated the model for corn, however, such a model can be adapted for different \ncorps and growth stages. In this study, the model was modified and extended to investigate backscatter from growing soybean, \nwhich is a structurally different crop, lacking a dominant stem , and facing additional computation challenges from overlapping \nvegetation components.  \n \nThe remainder of this paper is organized as follows: In Section II, the model structure and simulation setup and the data used for \nvalidation are described. In Section III, backscatter estimates from the full -wave model are compared with field observations. \nSection IV delves into the model’s capability to accurately estimate the overall signature throughout the growing season, ass ess \nthe impact of soil roughness on backscatter estimates, and investigate the contributions of various vegetation components to the 4 \n> MANUSCRIPT ID NUMBER<  \n \noverall signature. Lastly, Section V summarizes the novel contributions and findings of the study.  \nII. METHODS  \nA. HFSS Modeling  \nANSYS High -Frequency Structure Simulator (HFSS) was the EM solver used for this study, which presents several advantages  \nincluding high accuracy of solutions using the finite element method (FEM) and a user -friendly graphical interface that allows \nusers to build or import 3 -D structural models. In this study, we used a unit cell with periodic boundary conditions that replicate \nthe unit cell infinitely in all directions, simulating an infinite field. HFSS generated and solved a single mesh for the uni t cell, \nsignificantly reducing memory usage and computational costs.  \n \nAs shown in Fig. 1a, a typical unit cell included soybean crops planted in rows along the y -direction at an average inter -plant \ndistance of 𝛥𝑦. The rows were considered at an average row distance of 𝛥𝑥 along the x -direction from each other, and 𝛥𝑥/2 \nfrom the top and bottom borders of the unit cell. Similarly, the soybean plants were located at a distance of 𝛥𝑦 from each other, \nand 𝛥𝑦/2 from the left and right boundaries of the unit cell (Fig. 1b). As a result, the width and length of unit cells were integer \nmultiples of the average distances between the soybean plants along and across rows, respectively. Here, unit cells are descr ibed \nby two integers as 𝑛×𝑚 where 𝑛 is the number of rows, and 𝑚 is the number of soybeans in each row. Unit cells were coupled \nwith periodic boundary conditions on sides, and terminated by a perfectly matched layer (PML) on top, and a layer of soil wit h a \nrough surface, and then, a corresponding surface impedance boundary condition (SIBC) on the bottom (Fig. 1a). Periodic \nboundary conditions require that the EM wave’s path inside the computational grid matches  the wavelength ( 𝜆) of the incident \nwave,  causing the elevation ( ) and azimuth ( ) angles of incidence follow (1) and (2): [38] \n \n𝜑=tan−1[(𝐿𝑥/𝑝)(𝐿𝑦/𝑞)⁄]  (1) \n𝜃=sin−1[𝜆\n2√(𝑝/𝐿𝑥)2+(𝑞/𝐿𝑦)2]  (2) \n \nwhere 𝐿𝑥 and 𝐿𝑦 are the width and length of the unit cell, respectively, and 𝑝 and 𝑞 are arbitrary integers.  and  hold in (1) and \n(2) are called Bragg angles. In this study, the simulations were conducted at incidence angles close to Bragg angles. Unit cells \nwere uniformly illuminated by a plane wave at given incidence angle s and frequency.  \n 5 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 1. A typical HFSS unit cell. (a) The side view of a unit cell employed in this study is shown here. PML stands for perfectly \nmatched layer, and SIBC stands for surface impedance boundary condition. (b) The top view of the unit cell is shown. 𝛥𝑥 and 𝛥𝑦 \nare the distances between soybean plants across, and along rows, respectively.  \n \nInputs to HFSS included soil roughness, dielectric properties of soil and vegetation components, and 3 -D plant structure. The \nsoil roughness was defined by the correlation length and root mean square (RMS) height [39]. The complex permittivity of soil \nwas estimated using the Mironov model [40] as a function of volumetric soil moisture (VSM), which is the ratio of the volume of \nwater in a soil sample to the total volume of the sample, as well as frequency and clay percent. The complex permittivities o f \nstems, branches, leaves, and pods were estimated based on the gravimetric moisture content (M g) and physical temperature [41] \n[42]. M g is the ratio of the mass of water in a component to the total mass of the component. All simulations were performed at \nL-band (1.26 GHz).  \n \nThe 3-D structural models of growing soybean plants consisted of stems, branches, leaves, and pods (during the reproductive \nstages ). Stems and branches were frustums of cone s described by top and base radii, length, and azimuth and elevation angles of \nrotation around their major axes. Leaves were 2 -D impedance sheets with widths and lengths but no thicknesses, following [43] \nand [38]. No significant difference was found between the backscatter coefficients ( 0) using 2 -D and 3 -D leaves, however, 2 -D \nleaves significantly reduced the computational  time and memory requirements.  The branches were positioned at node heights, \n6 \n> MANUSCRIPT ID NUMBER<  \n \nwith leaves positioned at the end of the branches as trifoliates. Leaves were rotated around branches in azimuth and elevatio n \ndirections. Finally, each cluster of pods was represented by one 3 -D ellipsoid with the same volume as the cluster, described by \nthe lengths of three semi -axes. The clusters of pods were placed at a fixed distance of 2 cm below the base of the branches with \ntheir major axis perpendicular to the ground surface. Since the backscatter contribution of the thin stalks that connect leav es to \nbranches, called petioles, was found to be minimal, they were eliminated to reduce computational complexity.  \n \nIn this study, the parameters for generating the 3 -D structural models for soybean plants were obtained from field experiments, \ndescribed in Section II.B. The variabilities observed in the field data were used to generate multiple realizations with perturbed \nbranch and stem angles as well as plants’ locations within the unit cell. Twenty realizations were found to provide a confide nce \ninterval of 2.5 dB, similar to the measurement fidelity in the experiments.  \n \nB. Field Experiments  \nThe study used inputs and observations from the Soil Moisture Active Passive Validation Experiment (SMAPVEX) 2016 – \nMicrowave, Water, and Energy Balance Experiment (MicroWEX), and SMAPVEX in 2012 (SMAPVEX12).  \n \nB-1) SMAPVEX16 -MicroWEX  \nMeasurements from SMAPVEX16 -MicroWEX [44] were used to provide inputs to the model and evaluate its performance. \nThe experiment was conducted over the entire growing season of soybean and corn from May 23 (DoY 144) to September 2 \n(DoY 246), 2016, on a commercial farm (Sweeney Farms) in Alden, Iowa (Fig. 2 a). The goal of the experiment was to collect \nhigh temporal resolution active and passive microwave data from soybean and corn fields to validate and improve the SMAP \nretrieval measurements of soil moisture and vegetation water content. It involved ground -based measurements from soybean and \ncorn fields at L -band (1.41 GHz for passive and 1.26 GHz for active) [44]. The 0 values were obtained hourly at azimuth angles \nof −9 , 0, and 9 , and an elevation angle of 40 , with respect to the direction perpendicular to row structure.  At the soybean \nfield, the observations were performed over two periods, from July 8 -12 (DoY 190 -194) during the early growing season, and \nSeptember 2 -6 (DoY 246 -250) during the late/reproductive stage.  This study used observations at 6 am, matching anticipated \nsatellite observations such as those from NASA -ISRO Synthetic Aperture Radar (NISAR).  \n 7 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 2. Study area s. (a) The SMAPVEX16 -MicroWEX study area located in Central Iowa (inset) and its location relative to the \nUS map are displayed her e. (b) The SMAPVEX12 study area (inset) located in Manitoba, Canada, and its location relative to the \nCanada map are displayed here. The yellow rectangles depict the soybean fields surveyed during SMAPVEX12. The selected \nfields #63, #82, and #112 are designated by orange, green, and purple rectangles, respectively.  \n \nIn addition to the microwave observations, SMAPVEX16 -MicroWEX involved weekly vegetation measurements of row \nspacing; plant density; soil moisture, texture, and roughness; and geometry and wet and dry biomass of the vegetation \ncomponents of the plant at three sampling sites [44]. The geometrical features measured during the experiment included heights \nand diameters of stems and branches, heights of branch -nodes, sizes of leaves and pods, and the number of pods and leaves per \nbranch. The vegetation sampling conducted on 11 days: June 22 (DoY 174), June 28 (DoY 180), July 5 (DoY 187), July 11 \n(DoY 193), July 18 (DoY 200), July 26 (DoY 208), August 1 (DoY 214), August 7 (DoY 220), August 18 (DoY 231), August \n236 (DoY 236), and September 1 (DoY 245). The vegetation samples were weighed wet and dried at 48 C for 3 -7 days to obtain \nMg. Soil moisture was recorded every 15 minutes at depths of 2, 5, 10, 15, 30, 60, and 120 cm. The average of soil moisture at \ndepths 2 and 5 cm was considered as the representative soil moisture for this study. Soil roughness was measured using a pin \nprofiler on two days, May 25 (DoY 146), and August 10 (DoY 223).  \n \nB-2) SMAPVEX12  \nSMAPVEX12 was conducted over a short, 5 -week, non -reproductive period of the growing season, from June 7 - July 19 \n8 \n> MANUSCRIPT ID NUMBER<  \n \n(DoY 159 -201), 2012, covering more than 15 soybean and 10 corn fields near Manitoba, Canada (Fig. 2b). During the \nexperiment, airborne observations of 0 were obtained by NASA’s Unmanned Aerial Vehicle Synthetic Aperture Radar, \nUAVSAR. Fully polarimetric 0 values at L -band (1.26 GHz) were recorded at an azimuth angle of 45 , and at elevation angles \nof 30  to 50  (30 to 60  for cross -polarization) [45]. However, 0 values were corrected to 40  elevation angle based on the \nhistogram -based (HIST) technique [46]. In this study, 0 observations from three soybean fields with minimal variability in 0 \nmeasured across the field were chosen for implementation. Since a UAVSAR footprint is about 6 m, the three selected fields \nwith identifiers 62, 82, and 112 contained 10027, 7109, and 9400 spatially distributed SAR measurement  pixels , respectively.  \n \nDuring the experiment, ground -based measurements of soil and vegetation parameters were obtained [45] in addition to the \nfield’s crop type, geolocation, row direction, and crop density  [47]. Soil moisture was measured in the top 6 cm every 1 -5 days \nusing theta and hydra probes. Soil texture and roughness were measured once in each of the three fields. Destructive sampling  \nwas used to collect vegetation data from three measurement sites in each of the three fields. For soybeans, 10 plants, five e ach \nfrom two rows, were collected at each measurement site, and lengths and diameters of 10 stems were recorded. M g of crop \ncomponents were calculated similar to that as in SMAPVEX16 -MicroWEX. A detailed description of SMAPVEX12 is provided \nin [48]. \n \nC. HFSS Simulation s \n \nC-1) SMAPVEX16 -MicroWEX  \nSimulations were conducted  for the days on which vegetation data were collected, and also for the days during the two periods \nof 0 observations ( Section II.B-1). \n \nThe geometrical measurements of six sample soybean plants were used to build 3 -D structural models of soybean. The \ngeometrical features such as branch and stem angles that were not observed in the SMAPVEX16 -MicroWEX dataset were set \nbased on literature -derived values on typical soybean phenology as follows. The plant spacing within the rows was randomly \nperturbed by 2 cm, and the row spacing was perturbed by 3 cm (Fig. 1b). Stems were rotated randomly around the z - and y -\ndirections, and rotation angles were selected from a Gaussian random distribution with a zero mean and a standard deviation \n(STD) of 3  for the early days (DoY < 231), and 2  for the late days (to maintain soybean stems inside the space of unit cells). \nThe azimuth angles of branches were obtained from a Gaussian random distribution with a zero mean and an STD of ~3 , based 9 \n> MANUSCRIPT ID NUMBER<  \n \non our observations from other soybean fields. The elevation angles of branches were taken from an exponential random \ndistribution with a mean and an STD of 18  and 1 , respectively, based on our observations from other fields. The elevation \nangles higher than 45  were discarded.  \n \n3-D structural models for DoY 193 and 245 were used for simulations during the two periods with 0 observations, on DoYs \n190-194 and 246 -250, respectively. M g values were obtained by interpolating the values measured on the two sampling days. \nFig. 3a displays an example of 3 -D structural models during the growing season. Fig s. 3b-e represent the evolution of the lengths \nand radii of stems and branches, and lengths and widths of leaves and pods, respectively.  \n \nThe soil layer (Fig. 1a) thickness of 6 cm was selected to avoid any subsurface reflections with the implemented SIBC. A stat ic \nsoil roughness with an RMS height of 2.59 cm and a correlation length of 34 cm was applied for all simulations based on one o f \nthe reported values. Throughout the growing season, the height of unit cells was constant at 180 cm, the maximum height \nobserved, so that the plants were completely inside the unit cell. Based on soybean plant spacing of 8.33 cm in -row and 76 cm \nacross -rows, a 3x3 unit cell with corresponding Bragg angles of  = 0 and  = 39.13  was selected to approximate target \nincidence angles of  = 0° and  = 40°.  \n 10 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 3. The generated 3 -D structural models of soybean for the simulations corresponding to SMAPVEX16 -MicroWEX. \n(a) Sample 3 -D structural models are displayed here. The day for which the model is generated is indicated below the model. (b) \nThe evolution of the stems’ and branches’ length over days is displayed here. The shaded area shows the standard deviation \n(STD) of the corresponding trace. The error bars display the sampling values and the corresponding STD. (c) The same as panel  \n(b) but for radii of stems and branches (d) The same as panel (b) but for lengths and widths of leaves (e) The same as panel (b) \nbut for lengths and widths of pods.  \n \nSince the unit cells were excited by a uniform plane wave source, it approximates the airborne measurements well. However, it  \nis not well suited for ground measurements performed in SMAPVEX16 -MicroWEX, resulting in non -uniform illumination \nweighted by the antenna pattern.  To address this problem, a scaling technique used in [49] was employed to translate 0 from the \nHFSS simulations to 0 from ground -based measurements. When taking angle -dependence into account, the estimated 0 can be \ngiven by  \n11 \n> MANUSCRIPT ID NUMBER<  \n \n \n𝜎0=𝑟04\n𝐴∫𝐹𝑡(𝜑,𝜃).𝐹𝑟(𝜑,𝜃).𝜎0(𝜑,𝜃)\n𝑟(𝜑,𝜃)4𝑑𝐴𝐴,  (3) \nwhere Ft and Fr are the normalized beam patterns, A is the antenna footprint, r0 is the distance from the radar to the center of the \nfootprint, and r is the distance from the radar to a scattering center inside A. The scaling technique discretizes the integral by \ndividing A into small rectangles, or tiles, defined by the unit cell’s dimensions (Fig. 4), and the 0(, ) for each tile is obtained \nfrom HFSS. Thus the discretized version becomes a coherent summation:  \n \n[𝜎𝑣𝑝0\n𝜎ℎ𝑝0]=4𝜋\n𝐴∑|(𝑟0\n𝑟𝑛)2\n𝑒−2𝑗𝑘𝑟𝑛𝐅𝑛𝐒𝑛𝐅𝑛[𝑒𝑣\n𝑒ℎ]|2\n𝑛   (4) \n \nHere, k is the wavenumber, n indexes different tiles, rn is the distance to a tile, and [ ev, eh]T indicates vertical ([ 1, 0]T) or \nhorizontal ([ 0, 1]T) transmit polarization. The matrix product of the normalized, complex antenna amplitude F and scattering \nmatrix S is given by  \n \n𝐅𝑛𝐒𝑛𝐅𝑛=[𝑓𝑣𝑣𝑓𝑣ℎ\n𝑓ℎ𝑣𝑓ℎℎ][𝑆𝑣𝑣𝑆𝑣ℎ\n𝑆ℎ𝑣𝑆ℎℎ][𝑓𝑣𝑣𝑓𝑣ℎ\n𝑓ℎ𝑣𝑓ℎℎ] (5) \n \nand accounts for the coupling between cross -polarization terms in the antenna and scatterer, assuming a monostatic scenario. \nThen, for a nominal incidence angle ( i, i), the estimated average 0 is calculated with Monte -Carlo simulations that randomize \nthe locations of the tiles in A and the scattering matrix used for each tile. Due to a random selection of unit cells, the scaling \nresults are prone to outliers. To address this problem, the scaling technique was used 20 times to generate 20 scaled 0 values.  \n 12 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 4. The scaling technique to allow HFSS comparisons with ground -based measurements. The footprint of the University \nof Florida L -band automated radar system (UFLARS) was tiled by unit cells (blue rectangles), and the received 0 by radar was \nassumed to be a weighted sum of the 0 values of the tiled unit cells. The heatmap illustrates the weighting applied to each point \nwithin the radar footprint.  \n \nC-2) SMAPVEX12  \nSimulations were conducted for 12 days during which soil moisture and 0 values were observed: June 17 (DoY 169), June 22 \n(DoY 174), June 23 (DoY 175), June 25 (DoY 177), June 27 (DoY 179), June 29 (DoY 181), July 5 (DoY 187), July 8 (DoY \n190), July 10 (DoY 192), July 13 (DoY 195), July 14 (DoY 196), and July 17 (DoY 199).  \n \n3-D structural models were built as follows. The length and diameter of stems were randomly selected from 30 measurements. \nThe parameters such as stem length, diameter, and M g were interpolated between measurement days, as shown in panels b & c of \nFigs. 5-7. The top and bottom diameters of the stems were considered the same due to the lack of detailed observations. In \naddition, 10 branches with the same diameter but 1/3 of the length of stems were designed for each soybean plant. Branches we re \narranged alternately on either side of the stem, aligned with the positive and negative x -axis directions (shown in Fig. 1 b). The \nlength and width of trifoliates were obtained from the SMAPVEX16 -MicroWEX dataset for plants with similar heights and \nwidths due to the lack of information during SMAPVEX12.  \n \nSimilar to the simulations for SMAPVEX16 -MicroWEX, the geometrical features that were not observed in the SMAPVEX12 \ndataset were set randomly. The rotation angles of stems were selected from a Gaussian random distribution with a zero mean an d \n13 \n> MANUSCRIPT ID NUMBER<  \n \nan STD of 1 . The elevation angles of branches were taken from a uniform random distribution between 1  and 60 . The location \nof branches on the stem were selected from a uniform random distribution between zero and the length of the corresponding \nstem. Leaves were rotated around the corresponding branch randomly along azimuth and elevation directions. The random \nvalues for the rotation angles were drawn from a uniform random distribution between 60. The random parameters generated \nrealizations for the simulations.  \n \nSimilar to SMAPVEX16 -MicroWEX simulations, the soil layer had a thickness of 6 cm and its dielectric properties were \nestimated from the observed VSM. The rough soil was generated based on measured RMS height and correlation length for each \nfield. The same roughness was used for all simulations corresponding to each of the three fields. The height of unit cells wa s set \nto 180 cm. The simulations were performed at the measurement frequency of 1.26 GHz.  \n \nFig. 5a displays a sample 3 -D structural model generated for five days for field #63. Fig s. 5b-d represent the daily evolution of \nthe stem and leaf parameters. The azimuth angles of branches were between 35 in the early days, but gradually reduced to 9 \nin the late days, to keep the branches inside the unit cells. The RMS height and correlation length of the soil roughness were 0.92 \nand 10 cm, respectively. Based on soybean plant spacing of 5.31 cm in -row and 76.5 cm across -rows, a 2x5 unit cell with \ncorresponding Bragg angles of  = 43.86  and  = 40.32  was selected to approximate target incidence angles of  = 45° and  = \n40°. \n 14 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 5. The generated 3 -D structural models of soybean for field #63. (a) Sample 3 -D structural models generated for 5 days \nfor field #63 are displayed here. (b) The evolution of the stems’ length over days is displayed. The shaded area shows the ST D. \nThe markers show the measurements, with their STD shown as error bars. (c) The same as panel (b) but for radii of stems. (d) \nThe same as panel (b) but for lengths and widths of leaves.  \n \nFig. 6a displays a sample 3 -D structural model generated for five days for field #82. Fig s. 6b-d represent the daily evolution of \nthe stem and leaf parameters. The azimuth angles of branches were between 30 in the early days, but gradually reduced to 5 \nin the late days, to keep the branches inside the unit cells. The RMS height and correlation length of the soil roughness were 0.98 \nand 13 cm, respectively. Based on soybean plant spacing of 4.77 cm in -row and 77.8 cm across -rows, a 2x5 unit cell with \ncorresponding Bragg angles of  = 47.37° and  = 42.63° was selected to approximate target incidence angles of  = 45° and  = \n40°. \n \n15 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 6. The generated 3 -D structural models of soybean for field #82. (a) Sample 3 -D structural models generated for 5 days \nfor field #82 are displayed here. (b) The evolution of the stems’ length over days is displayed. The shaded area shows the ST D. \nThe markers show the measurements, with their STD shown as error bars. (c) The same as panel (b) but for radii of stems. (d) \nThe same as panel (b) but for lengths and widths of leaves.  \n \nFig. 7a displays a sample 3 -D structural model generated for five days for field #112. Fi gs. 7b-d represent the daily evolution \nof the stem and leaf parameters. The azimuth angles of branches were between 40 in the early days, but gradually reduced to \n9 in the late days, to keep the branches inside the unit cells. The RMS height and correlation length of the soil roughness were \n1.03 and 18.5 cm, respectively. Based on soybean plant spacing of 7.68 cm in -row and 76.5 cm across -rows, a 2x3 unit cell with \ncorresponding Bragg angles of  = 47.88  and  = 44.08  was selected to approximate target incidence angles of  = 45° and  = \n40°. \n \n16 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 7. The generated 3 -D structural models of soybean for field #112. (a) Sample 3 -D structural models generated for 5 days \nfor field #112 are displayed here. (b) The evolution of the stems’ length over days is displayed. The shaded area shows the STD. \nThe markers show the measurements, with their STD shown as error bars. (c) The same as panel (b) but for radii of stems. (d) \nThe same as panel (b) but for lengths and widths of leaves.  \n \nThe average of 0 values from HFSS were compared with the average of 0 values from field observations over a given period \nusing the root mean square difference (RMSD) and STD over 20 realizations. Note that modeled 0 estimates outside 2 STD \nfrom the observed ones implied that the estimates are significantly (by 95%) different from observations. To demonstrate the \nconfidence interval, the figures show the standard error (SE) of the modeled 0. However, STD for the observed 0 is used in the \nfigures, showing the variability in the observations.  \n \nD. Sensitivity Analysis  \nThe HFSS simulations provide dynamics of 0 and its relationship to changes in crop structure and moisture content \nthroughout the growing season, as well as soil characteristics.  \nD-1) Soil Roughness Analysis  \nTo investigate the impact of soil roughness on 0 [12] [24] [50] [24], we conducted 50 simulations for bare soil, where each \n17 \n> MANUSCRIPT ID NUMBER<  \n \nsimulation employed a unique roughness generated from an RMS height of 1.0 cm, and a correlation length of 12.5 cm. While \nthe generated roughnesses exhibited distinct surface height variations, as shown in Fig 8, they all maintained identical RMS \nheight and correlation length values. The VSM parameter was set to 0.15 m3/m3 across all simulations. The remaining parameters \nwere made consistent with the SMAPVEX16 -MicroWEX simulations.  \n \nFig. 8. Examples of three (out of 50) rough surfaces generated using the same RMS height of 1.0 cm and correlation \nlength of 12.5 cm. The surfaces exhibit different surface height variations.  \n \nD-2) Sensitivity of 0 to Vegetation Components  \nA sensitivity analysis was performed using HFSS to quantify the contributions of individual vegetation components, i.e., stem s, \nbranches, leaves, and pods, to the overall 0 signature throughout the growing season. Simulations were conducted with a \nsmooth soil surface and soil moisture of 0.15 m3/m3, reflecting the mean observed VSM during SMAPVEX16 -MicroWEX. \nThese simulations were performed for 9 days spanning the growing season: DoYs 170, 185, 200, 215, 230, 245, 260, 275, and \n290, corresponding to the SMAPVEX16 -MicroWEX campaign. Eight scenarios were investigated in sensitivity analyses which \nwere modifications of a base scenario mimicking the growth of soybean. In the base scenario, the 3 -D structural models of \nsoybeans consisted of one stem, ten branches, and one trifoliate per branch (as shown in Fig. 9a). Based upon the SMAPVEX16 -\nMicroWEX observations, the lengths of stems and branches, and the length and width of leaves increased linearly from DoY 170 \nto DoY 260 with slopes of 1.44, 0.39, 0.08, and 0.03 cm.day-1, respectively, and then remained constant till the end of the \ngrowing season. The top/bottom radii of stems decreased/increased linearly from DoY 170 to DoY 260 with slopes of 0.01 \ncm.day-1, and increased/decreased by the same slope. The radii of branches were kept constant across the growing season. \nAdditionally, the M g values of stems, branches, and leaves were monotonically decreasing with the same slope of 0.002 day-1. \nThe dielectric properties of vegetation components were estimated from the corresponding M g values.  \n \nIn addition to the base scenario, eight  scenarios were included: (1) Fixed -Mg: This scenario presented the sensitivity of 0 to \ngeometry. The M g of all stems, branches, and leaves were held constant for all 9 days of the simulations, while the geometry was \n18 \n> MANUSCRIPT ID NUMBER<  \n \nthe same as for the base scenario (as shown in Fig. 9b). The constant M g value was the midseason value observed on DoY 230. \n(2) Fixed -geometry : This scenario explored the sensitivity of 0 to M g. So, while the sizes of vegetation components across the \ngrowing season were kept constant at the values corresponding to DoY 230, the M g of vegetation components varied as in the \nbase scenario (as shown in Fig. 9c). (3) Stem -only: This scenario examined the sensitivity of 0 to stems. As a result, branches \nand leaves were removed from the structural models of the HFSS simulations corresponding to the fixed -Mg scenario, and they \nwere re -executed (as shown in Fig. 9d). The fixed -Mg scenario was selected instead of the base scenario to completely nullify \nany potential impact of changes in M g on the outcomes. (4) No-stem: To assess the sensitivity of 0 to the absence of stems, a no -\nstem scenario was implemented, complementing the stem -only scenario. In this scenario, stems were removed from the structural \nmodels of the fixed -Mg simulations, and the simulations were re -run (as shown in Fig. 9e). (5) Branch -only: The same as the \nstem -only scenario, but for branches (as shown in Fig. 9f). (6) No-branch : the same as the no -stem scenario, but for the lack of \nbranches (as shown in Fig. 9g). (7) Leaf-only: The same as the stem -only scenario, but for leaves (as shown in Fig. 9h). (8) No-\nleaf: the same as the no -stem scenario, but for the lack of leaves (as shown in Fig. 9i). To assess the sensitivity of 0 to the \npresence of pods, a pod scenario  was simulated . This involved adding pods to the 3 -D structural models of the fixed -Mg \nsimulations at DoY 260, corresponding to the peak of the productive stage. The pod clusters were implemented following the \nsame protocol as in the SMAPVEX16 -MicroWEX simulations, with lengths and widths matching those observed at DoY 236 of \nthe aforementioned experiment.  \n \n \n 19 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 9. The base and eight derived scenarios used for the sensitivity analysis. (a) This panel, from top to bottom, illustrates \nthe changes in the 3 -D structural models of soybean over the growing season, the evolution of the stems’ and branches’ length, \nthe evolution of the stems’ and branches’ radius, the evolution of the leaves’ length and width, and the evolution of M g for \nstems/branches and leaves, under the base scenario. The changes in M g are visualized by changes in color in the top subpanel. (b) \nThe same as panel (a) but under the fixed -Mg scenario. (c) The same as panel (a) but under the fixed -geometry scenario. (d -i) \nThis panel displays the changes in the 3 -D structural models of soybean over the growing season under the scenario mentioned in \nthe panel title.  \n \nThe distances between synthetic soybean plants along and across rows were  set to  8.33 cm and 76 cm, respectively, similar to \nthose for the SMAPVEX16 -MicroWEX simulation, and a unit cell size of 2x2 was considered to generate Bragg angles at  = 0 \nand  = 39.13  as the illuminating angles. Simulations were conducted at a frequency of 1.26 GHz. Twenty realizations were \ngenerated, each of which was used for the base and eight scenarios. The 0 values were compared with those from the bare-soil \n20 \n> MANUSCRIPT ID NUMBER<  \n \nscenario, which served as the null scenario.  \nIII. RESULTS  \nA. HFSS Simulations: SMAPVEX16 -MicroWEX  \nFig. 10 shows the HFSS 0 compared with the observations for five days in the early season with DoYs  190-194. During these \ndays, the soybean plants were about 44 cm having about ~11 branches. The average M g values of stems/branches and leaves \nwere about 0.86 and 0.81 Kg/Kg, respectively (Fig. 10a). During the first three days, the soil moisture decreased by 0.4 m3/m3, \nbut increased to 0.30 m3/m3 (Fig. 10b) due to rainfall.  \n \n \nFig. 1 0. The 0 values reported in SMAPVEX16 -MicroWEX versus the estimated ones from HFSS over the early days of \nthe growing season. (a) The evolution of M g for stems/branches and leaves over the early days is shown here. The markers \ndisplay the sampling values. (b) The same as panel (a) but for VSM. (c) The measured 0 values (circles: error bars indicate the \nmeasurement STD) are compared with the estimated ones from HFSS (squares) at different polarizations over the early days. \nThe shaded area indicates the standard error of the HFSS simulation results.  \n \nThe estimated 0 values match well with the observations, with the RMSDs of 1.56, 3.82, and 3.04 dB for HH, HV, and VV \npolarizations, respectively (Fig. 10c). HFSS 0VV is higher than 0HH, consistent with observations, though the difference  \nbetween simulation and measurement is lower by 1.44 dB compared to the observations. HFSS 0HV’s are slightly lower than the \n21 \n> MANUSCRIPT ID NUMBER<  \n \nobserved values, except on DOY 193 when the underestimation is 6.83 dB. This deviation could be attributed to unaccounted \nparameters in the model, such as windy condition s. The averages of modeled estimates were within the 2 STD of observations.  \n \nFig. 11 depicts the comparison between HFSS estimated and observed 0 values for five days in the late -season with DoYs \n246-250. During this period, the soybean plants reached an average height of 109 cm with approximately 12 branches and 43 \npods/plant. The M g of stems/branches and leaves were around 0.77 and 0.70 Kg/Kg, respectively (Fig. 1 1a). Due to the absence \nof rainfall, the soil dried steadily, with VSM values decreasing by 0.05 m³/m³ during this period  (Fig. 11b).  \n \n \nFig. 1 1. The 0 values reported in SMAPVEX16 -MicroWEX versus the estimated ones from HFSS over the late days of \nthe growing season. (a) The evolution of M g for stems/branches, leaves, and pods over the late days is shown here. (b) The same \nas panel (a) but for VSM. The markers display the sampling values. (c) The measured 0 values (circles: error bars indicate the \nmeasurement STD) are compared with the estimated ones from HFSS (squares) at different polarizations over the late days. The \nshaded area indicates the standard error of the HFSS simulation results.  \n \nThe estimated 0 values align well with the measured ones at HH and VV polarizations, with RMSDs of 1.05 and 0.95 dB, \nrespectively (Fig. 11c). However, the estimations at HV polarization are significantly lower than the observations with an \naverage difference of 7 dB (Fig. 11c). Nevertheless, estimations still fall within the ±2 STD band. In both simulations and \nobservations, 0HH is higher than 0VV, but the simulated difference exceeds the observed difference by 1.23 dB. Notably, the \n22 \n> MANUSCRIPT ID NUMBER<  \n \nlargest divergence between estimations and observations > ±1 STD occurred at HV polarization on DoYs 246 -250, reaching a \nmaximum of 9.60 dB on DOY 250. Th e substantial underestimation in 0HV is most likely attributed to vegetation structure [10] \n[51] [14] [52]. It is possible that the branch structure was underrepresented, as information regarding subbranch structure was not \ncollected during the experiment.  In addition, observations of cross -pol could also be higher due to some co -pol response leaking \ninto the cross -pol channel.  \n \nB. HFSS Simulations: SMAPVEX12  \nFig. 12 compares the HFSS -simulated 0 values with observations obtained over 12 days for field #63 in the SMAPVEX12. \nDuring this period, the soybean plants exhibited substantial growth, transitioning from an initial height of 13 cm to a final  height \nof 60 cm. M g of stems/branches and leaves exhibited a gradual increase, starting from 0.49 and 0.63 Kg/Kg, respectively, in the \nearly days and reaching approximately 0.84 and 0.82 Kg/Kg, respectively, in the later days (Fig. 1 2a). Concurrently, the soil \ndried throughout the simulation period, with VSM values ranging from 0.21 to 0.05 m³/m³ (Fig. 1 2b).  \n \nThe HFSS estimated 0 values were consistent with the observed values, with RMSDs of 1.97, 2.54, and 1.76 dB for HH, HV, \nand VV polarizations, respectively (Fig. 12c). The averages of the modeled estimates consistently fall within the ±2 STD rang e \nof the observations. While both observations and simulations reveal a higher average 0VV than 0HH, the observed difference is \nsmaller than the simulated one by 0.11 dB. The largest discrepancies between estimations and observations > ±1 STD are 3.93 \ndB at HH polarization on DoY 199, and 2.57 and 3.29 dB at HV polarization on, respectively, DoYs 174 and 175. These \ndiscrepancies could be attributed to inaccuracies in the assumed structural descriptions of the soybean models. Due to the li mited \navailability of soybean geometry data during SMAPVEX12, the average geometry assigned to plants across simulation days may \ndeviate from real -world conditions.  \n \n 23 \n> MANUSCRIPT ID NUMBER<  \n \n \nFig. 1 2. The 0 values reported in SMAPVEX12 versus the estimated ones from HFSS for field #63. (a) The evolution of \nMg for stems/branches and leaves over days is presented here. The markers display the sampling values. (b) The same as panel \n(a) but for VSM. (c) The measured 0 values (circles: error bars indicate the measurement STD) are compared with the estimated \nones from HFSS (squares) at different polarizations over days. The shaded area indicates the standard error of the HFSS \nsimulation results.  \n \nFig. 13 compares the HFSS estimated 0 values with observations collected over 12 days for field #82, encompassing a period \nof significant soybean growth. During this growth phase, plant height increased from an initial 14 cm to a final 66 cm. Notab ly, \nMg of stems/branches and leaves remained relatively constant, ranging between 0.99 and 0.79 Kg/Kg (Fig. 1 3a). Similarly, VSM \nwas given stability throughout the simulation period, spanning a range from 0.31 to 0.18 m³/m³ (Fig. 1 3b). The HFSS estimated \n0 values exhibit remarkable alignment with the observed values, with RMSDs of 1.60, 3.25, and 1.79 dB for HH, HV, and VV \npolarizations, respectively  (Fig. 13c) . Furthermore, the averages of the modeled estimates consistently fall within the ±2 STD \nrange of the observations. While both simulations and observations demonstrate a higher average 0VV than 0HH, the observed \ndifference is smaller than the simulated one by 0.46 dB. Discrepancies between observations and estimations, exceeding the 1 \nSTD band, are 5.57 and 5.57 dB at HV polarization on, respectively, DoYs 192 and 196, and 2.44 and 3.86 dB at VV \npolarization on, respectively, DoYs 169 and 199. These discrepancies could stem from inaccuracies in the assumed 3 -D \nstructural models. As for field #63, the simplified representation of crops may overlook crucial geometric features that infl uence \n0, leading to inaccuracies observed in simulation outcomes.  \n24 \n> MANUSCRIPT ID NUMBER<  \n \n \n \nFig. 1 3. The 0 values reported in SMAPVEX12 versus the estimated ones from HFSS for field #82. (a) The evolution of \nMg for stems/branches and leaves over days is presented here. The markers display the sampling values. (b) The same as panel \n(a) but for VSM. (c) The measured 0 values (circles: error bars indicate the measurement STD) are compared with the estimated \nones from HFSS (squares) at different polarizations over days. The shaded area indicates the standard error of the HFSS \nsimulation results.  \n \nFig. 14 presents a comparison between the HFSS estimated 0 values and observations collected over 12 days for field #112, \nspanning a period of significant soybean growth. During this growth phase, plant height increased from an initial 15 cm to a final \n70 cm. M g of stems/branches and leaves decreased respectively from 0.99 and 0.89 Kg/Kg in the early days to 0.80 and 0.79 \nKg/Kg in the later days (Fig. 1 4a). Similarly, VSM exhibited a steady decline throughout the simulation period, ranging from \n0.46 to 0.28 m³/m³ (Fig. 1 4b). The HFSS estimated 0 values demonstrate notable agreement with the observed values, with \nRMSDs of 0.93, 2.09, and 1.95 dB for HH, HV, and VV polarizations, respectively  (Fig. 14c) . Furthermore, the averages of the \nmodeled estimates consistently fall within the ±2 STD range of the observations. While both simulations and observations \nindicate a higher average 0VV than 0HH, the observed difference is smaller than the simulated one by 1.47 dB. Discrepancies \nbetween observations and estimations, exceeding the 1 STD band, occurred at VV polarization by 4.13 dB on DoY 192. The \ndeviation could arise from imperfections in the assumed 3 -D structural models, as insufficient data on soybean geometry hinders \ntheir verification.  \n25 \n> MANUSCRIPT ID NUMBER<  \n \n \n \nFig. 1 4. The 0 values reported in SMAPVEX12 versus the estimated ones from HFSS for field #112. (a) The evolution of \nMg for stems/branches and leaves over days is presented here. The markers display the sampling values. (b) The same as panel \n(a) but for VSM. (c) The measured 0 values (circles: error bars indicate the measurement STD) are compared with the estimated \nones from HFSS (squares) at different polarizations over days. The shaded area indicates the standard error of the HFSS \nsimulation results.  \n \nThe three fields experienced almost similar weather conditions, with all undergoing significant soybean growth within the \nsimulation period  from DoY 169 -199. However, subtle differences existed. For example, f ield #63 displayed an initial increase \nin M g of vegetation components,  whereas fields #82 and #112 exhibited an initial decrease. All three fields converged to similar  \nlevels of  Mg of vegetation components by the end of the simulation period. Soil moisture consistently decreased across all fields, \nwith field #63 being the driest followed by field #82 and then field #112.  \n \nThe RMSDs of the HFSS estimated 0 values in all fields  were  consistently below 4 dB across all polarizations. This \ndemonstrates the effectiveness of the HFSS model in capturing the overall backscattering behavior.  Field #112 exhibited the \nlowest RMSDs, suggesting the model performs best under the vegetation and soil parameters reported for this field. However, \nfields #63 and #82 displayed higher discrepancies between simulations and observations , due to the less accurate representation \nof the actual soybean structure in the simulations. Despite these inter -field variations, several key similarities emerged. All fields \n26 \n> MANUSCRIPT ID NUMBER<  \n \ndemonstrated a higher average 0VV compared to 0HH in both estimations and observations, indicating that the model effectively \ncaptures the polarization dependence of 0. Additionally, the averages of the modeled estimates consistently fell within the ±2 \nSTD range of the observations for all three fields, demonstrating the ability of the HFSS model to accurately simulate 0 values \nunder diverse field conditions.  \nIV. DISCUSSION  \nThe HFSS model enables the simulation of the evolution of 0 throughout the growing season. Furthermore, it has the \ncapability to demonstrate the variability in 0 due to soil roughnesses, and can illustrate how different vegetation components, \nsuch as stems, branches, leaves, and pods, contribute to the overall signature.  \nA. The Evolution of 0 Over the Growing Season  \nThe temporal variations of 0 have been extensively studied in various crops, including soybeans [22] [53] [3] [44]. Three \ndistinct phases can be generally identified in the evolution of 0 throughout the growing season: (1) Ascending phase I , \ncharacterized by vegetative growth leading to canopy closure, during which both 0VV and 0HH increase with 0VV exceeding \n0HH. This signature is attributed to the backscattering contribution primarily from soil relative to the crops. (2) Ascending phase \nII, encompassing the period succeeding canopy closure, during which both 0HH and 0VV continue to increase; however, due to \nthe higher rate of increase in 0HH compared to 0VV, 0HH values exceed those of 0VV throughout ascending phase II. This shift \ncan be attributed to the increasing contribution of the canopy to backscattering as the season progresses. (3) The descending \nphase , corresponding to the period of water loss and/or shrinkage in the vegetation toward the end of the season, results in \ndecreasing 0HH and 0VV. \nThe phased evolution of 0 emphasizes the sensitivity of backscatter characteristics to vegetation growth dynamics and canopy \nstructure, providing valuable insights into the temporal variations of vegetation properties and their impact on 0 behavior. Fig . \n15 illustrates the evolution of co - and cross -polarized 0 simulated by HFSS throughout the growing season of SMAPVEX16 -\nMicroWEX. As observed, the HFSS model captures the season’s dynamics, with estimated 0VV and 0HH exhibiting an \nincreasing trend from DoY 180 to 245, while 0HH demonstrates a steeper increase compared to 0VV. As a result, while 0VV \ndominates 0HH in the first half of this period, i.e. DoY 180 -200, 0HH surpasses 0VV in the second half (DoY 200 -245). Finally, \na decreasing phase is observed from DoY 245 to 250. This confirms that the HFSS model effectively captures the underlying \ndynamics of the growing season. Soybeans in SMAPVEX16 -MicroWEX exhibited significantly accelerated growth compared to \nthose reported in other studies. For instance, while the SMAPVEX16 -MicroWEX soybeans attained a height of 1 m at mid -27 \n> MANUSCRIPT ID NUMBER<  \n \nseason, the soybeans in [53] required the entire growing season to reach this height. Consequently, the three backscatter phases \nin SMAPVEX16 -MicroWEX are not readily discernible. To address this challenge, a bilinear function was fitted to the model \nestimations (solid traces in Fig. 15 c) to distinguish the backscatter phases. Section IV.C delves into the HFSS model’s ability to \nreplicate the phases more efficiently.  \n \nFig. 15. The evolution of estimated 0 over the entire growing season. (a) The temporal evolution of M g for stems/branches, \nleaves, and pods throughout the growing season is depicted here. The markers display the sampling values. (b) The same as \npanel (a) but for VSM. (c) The HFSS estimated 0 values (circles: error bars indicate SE) are compared with the MIMICS \nestimated ones (dashed) at different polarizations (blue: HH, red: VV, and green: HV). The solid traces display the linear \nenvelopes for the HFSS estimated ones. The yellow, red, and green areas correspond to ascending phase I, ascending phase II, \nand descending phase, respectively.  \n \nFig. 15 also compares the 0 values from HFSS with the corresponding ones from a widely used Michigan Microwave Canopy \nScattering (MIMICS) model [54]. The MIMICS model is based on a first -order solution of the radiative transfer equation for a \ntree canopy, but it can be applied to estimate the backscatter for various canopy types, including soybean [55] [56]. The MIMICS \nmodel’s neglect of multiple scattering leads to 0 estimates  that are different from HFSS.  For instance, while the reported \nRMSDs for HFSS simulations across diverse regions and growth stages are less than 4 dB (except for one case), MIMICS \nexhibits RMSDs exceeding 4 dB for co -pol results. Notably, MIMICS lacks the capability to estimate cross -pol 0. Furthermore, \nrepresenting crop fields with homogeneous vegetation and neglecting row spacing in MIMICS lead s to an unrealistic canopy \n28 \n> MANUSCRIPT ID NUMBER<  \n \ndescription, particularly during late growth stages.  This limitation restricts the estimated evolution of 0 to an initial ascending \nphase followed by a saturation phase (dashed lines in Fig. 15c). Additionally, MIMICS estimated 0VV remains always higher \nthan 0HH, indicating an inability to accurately capture the dynamic nature of the problem.  \n \nB. Impact of Soil Roughness on Backscatter  \nBecause HFSS uses 3 -D soil structures, it captures the details of soil roughness beyond a two-parameter statistical description used \nin conventional models such as the Integral Equation Method  (IEM)  [57]. Thus, HFSS can help provide additional insights into the \nimpact of surface roughness on 0. Fig. 16 compares 0 values obtained from 50 simulations of rough surfaces, each characterized by \nan RMS height of 1 cm and a correlation length of 12.5 cm. The figure illustrates significant variability in 0 values across these \nsimulations. The interquartile range, representing the middle 50% of the data, spans approximately 8 dB for all polarizations . The \nmedians of the 0 values are -23 dB, -42 dB, and -17 dB for HH, HV, and VV polarizations, respectively. Comparison with Advance \nIntegral Equation Model (AIEM)  [58] in Fig. 16 shows that both HH and VV -pol estimates from the AIEM are outside the  \ninterquartile range, shown by the colored boxes. AIEM does not provide cross -pol backscatter. This finding necessitates more refined \napproaches to representing soil roughness in simulations, potentially as distinct scattering centers instead of relying solel y on a few \nstatistical parameters. Moreover, it confirms that selecting a realistic roughness is crucial for any full -wave simulation of crop \nbackscatter, since an inappropriate choice of soil roughness can lead to significant deviations in the obtained 0 values.  \n \n \nFig. 16. The impact of soil roughness on 0 variability. The variability in HFSS estimated 0HH, 0HV, and 0VV from 50 soil \nroughness representations with identical RMS height and correlation length is depicted in blue, green, and red, respectively.  The \nboxes represent the interquartile range and the white line within the boxes represents the median of the data. The hexagram m arkers \nrepresent the Advance Integral Equation Model (AIEM) estimates.  \n29 \n> MANUSCRIPT ID NUMBER<  \n \n \nC. Sensitivity of 0 to Vegetation Components  \nThe HFSS model was utilized to simulate a growth cycle that mimics the natural development of soybeans in the field (the base \nscenario). It is important to note that the simulated growth cycle encompassed changes solely in the size and M g of vegetation \ncomponents, as illustrated in Fig. 9a. The evolution of 0HH, 0HV, and 0VV over the growing season is depicted in Fig. 17a. As \nevident, the proposed growth cycle effectively replicates the ascending phase I from DoY 170 -230, ascending phase II from DoY \n230-260, and descending phase from DoY 260 -290, similar to the simulations from SMAPVEX16 -MicroWEX, in Fig. 15.  \n \nFig. 17b shows that the 0 estimates from the fixed -Mg scenario closely resemble the 0 from the base scenario at all polarizations, \nwith the same backscatter phases. This analysis also suggests a minimal influence of M g on the temporal evolution of 0, and \nhighlights the dominant role of size, or equivalently VWC, in shaping the overall signature. The effect of M g was investigated by \nconducting simulations with varying M g levels while maintaining a constant crop size (the fixed -geometry scenario). In Fig. 17c, the \nchanges in M g with a fixed geometry produce nearly the same 0 across the growing season without distinguishable backscatter \nphases, indicating that the M g has a minimal impact on the overall 0 signature. This analysis implies that crop geometry can be \nreadily estimated from observed 0. \n \nThe dominant influence of crop geometry on 0 necessitates further insights into the contributions of individual crop components, \nnamely stems, branches, leaves, and pods to 0 at different polarizations. The simulations corresponding to the fixed -Mg scenario \nwere re -performed with one or more components removed, and the resulting 0 values were compared to those obtained from the \nfixed -Mg scenario. The fixed -Mg scenario was selected here as the reference instead of the base scenario, given the salient similarity \nbetween the base and fixed -Mg 0 (Fig. 17c).  \n \nFig. 17d reveals that the absence of branches and leaves reduces 0HV to the level of 0HV for bare soil, and increases 0VV, while \n0HH remains relatively unchanged compared to the fixed -Mg scenario with all the vegetation components. This indicates that the stem \nis the primary contributor to col -pol 0, but not to 0HV. Additionally, the stem -only scenario confirms the role of branches and/or \nleaves for the ascending phase II, as the stem -only signature lacks this trend. To corroborate the conclusion drawn from the stem -only \nscenario, fixed -Mg simulations were conducted without stems, creating a no -stem scenario. Fig. 17e shows that the no -stem 0HH is \nclose to the 0HH for bare soil, and no -stem 0VV is lower than the 0VV for fixed -Mg with all components, confirming the contribution 30 \n> MANUSCRIPT ID NUMBER<  \n \nof stems to co -pol 0. The no -stem 0HV remains at the level of the fixed -Mg scenario, suggesting that stems do not influence 0HV. \nNotably, all ascending/descending phases are absent in the no -stem scenario, implying that stems are essential for the observed three \nbackscatter phases in the growing season.  \n \n \nFig. 17. The evolution of 0 over the growing season in different scenarios. (a) The evolution of 0 in the base scenario is shown \nhere, and compared with the similar one from the bare -soil scenario. The shadings around solid traces represent SE. The yellow, red, \nand green areas correspond to ascending phase I, ascending phase II, and descending phase, respectively. (b) The evolution of  0 in \nthe fixed -Mg scenario is shown here, and compared with the similar ones from the base (dashed) and bare -soil (dotted) scenarios. The \nshadings around solid traces represent SE. (c) the same as panel (b) but for the fixed -geometry scenario instead of the fixed -Mg \nscenario. (d -i) The evolution of 0 in the scenario mentioned in the title of the panel is compared with the fixed -Mg and bare -soil \nscenarios.  \n \n31 \n> MANUSCRIPT ID NUMBER<  \n \nTo isolate the influence of branches on 0, we employed fixed -Mg simulations with stems and leaves removed, creating the branch -\nonly scenario (Fig. 17f). This scenario reduces 0HH to bare soil levels, demonstrating that branches lack any contribution to 0HH. \nInterestingly, branch -only 0VV exhibits a partial reduction compared to fixed -Mg, very similar to the no -stem scenario, suggesting a \npartial contribution of branches for 0VV. Altogether, it appears that the fixed -Mg 0VV values lay between the 0VV values observed in \nthe stem -only and branch -only scenarios. Branch -only 0HV falls below both fixed -Mg and even bare soil, confirming the significant \nrole of branches in reducing cross -pol 0. The absence of backscatter phases reiterates the crucial role of stems in their formation. To \nvalidate these findings, we performed the no -branch scenario (Fig. 17g), with only stems and leaves. As expected, 0HH remains \nunaffected, confirming the non -contributive role of branches to 0HH. Similarly, 0HV is unchanged, reiterating the influence of \nbranches in reducing 0HV. However, no -branch 0VV significantly surpasses fixed -Mg 0VV, closely aligning with the stem -only 0VV. \nThis indicates that the overall 0VV can be estimated as an average of stem -only and branch -only 0VV values. Furthermore, the \nabsence of branches eliminates the ascending phase II, as the canopy closure driving this phase no longer exists.  \nThe influence of leaves on 0 was further investigated using leaf -only simulations. As shown in Fig. 17h, leaf -only 0 starkly \ncontrasts with fixed -Mg 0. Both 0HH and 0VV are significantly reduced, resembling bare soil. This conclusively demonstrates that \nleaves do not contribute to co -pol 0. However, 0HV remains unchanged, indicating that leaves are the primary source of 0HV. \nInterestingly, the absence of stems and branches leads to the disappearance of all backscatter phases, while 0HH becomes \nunexpectedly higher than 0VV. To solidify these findings, the no -leaf scenario was simulated using fixed -Mg with stems and branches \nonly. As depicted in Fig. 17i, the no -leaf 0 closely mirrors fixed -Mg 0 at both HH and VV polarizations, confirming that stems and \nbranches govern co -pol 0. Notably, no -leaf 0HV is lower than both fixed -Mg and bare -soil 0HV, highlighting the primary role of \nleaves in 0HV behavior and suggesting an inhibitory effect of branches on 0HV. These simulations provide compelling evidence that \nleaves are the primary source of cross -pol 0, while stems and branches primarily contribute to co -pol 0.  \nThis finding contrasts several studies that successfully estimated the Leaf Area Index (LAI) using co -pol 0 [15] [4] [51]. It could be \ndue to a natural correlation existing between leaf size and stem/branch growth, making LAI and stem/branch size dependent var iables. \nWhile this relationship cannot be investigated experimentally, the HFSS model is able to provide these unique insights. Inter estingly, \nin Section III.A, HFSS simulations for SMAPVEX16 -MicroWEX (Fig 12) had underestimated 0HV compared to observations for \nlate days, DoY 246 -250. Through the above analysis, this could be attributed to the underestimation of leaf numbers and/or sizes \nmeasured during SMAPVEX16 -MicroWEX.  \n 32 \n> MANUSCRIPT ID NUMBER<  \n \nFig. 18 compares 0 from the vegetative plant in the fixed -Mg scenario for DoY 260 to a reproductive plant on the same day with \npods. While the average 0HV and 0VV remained relatively unchanged, with p -values of 0.28 and 0.13, respectively, from the \nWilcoxon signed -rank test, 0HH exhibited a significant reduction from 1.78 to -6.95 dB, with a p -value < 0.001. This suggests that \npods are indeed contributors to 0HH. Our results partially corroborate [3], where pods exhibit the strongest correlation with the \ndifference between 0HH and 0VV. However, the analysis in this study reveals that 0HH, not 0VV, is directly correlated with pod \npresence.  \nAnalyzing the contributions of distinct vegetation components to 0 is crucial in agricultural remote sensing. Elucidating the effects \nof each component on the overall signature helps improve algorithms for crop type discrimination, as well as crop health and yield \nestimation. Current models are mainly based on the observed relationship between 0 and biomass/vegetation component [59] [60] \n[61] [62] [4], and so their applicability is limited to the experiments or regions. In contrast , the HFSS model provides a n accurate  \nphysically -meaningful estimation of the contribution of each component. Moreover, despite their ability to effectively separate \ncomponents’ effects on 0, current models fall short in accurately assessing the true contribution of each vegetation component due to \ntheir limiting assumptions regarding EM wave scatterings. For example, MIMICS restricts wave scatterings to direct, double -bounce, \nand volume scattering, which may result in unrealistic estimates. While hybrid methods solve Maxwell’s equation at the level of \nindividual plants, they neglect multiple scatterings. Consequently, the estimated contribution of each vegetation component u sing \ncurrent models is essentially the contribution assigned to that component by the model, and may not be realistic (see, as an example, \nPauli decomposition [53]), while  the HFSS model presented in this study overcomes these limitations by unrestrictedly solving \nMaxwell’s equations.  \n \n \nFig. 18. The changes in 0 due to pods. The changes in 0 at different polarizations are  depicted by the differences between the \ncolored and empty boxes, which represent 0 values before and after pods, respectively.  The boxes represent the interquartile range \nand the horizontal lines in the boxes show the median of the data.  \n33 \n> MANUSCRIPT ID NUMBER<  \n \n \nOverall, the HFSS model offers a substantial leap forward in addressing the challenges of disassociating vegetation component  \ncontributions to 0. This ability holds the potential to significantly improve agricultural remote -sensing techniques, where researchers \ncan design retrieval models with improved accuracy. Note that data cubes [63] [64] [65], which are currently the most widely used \nreverse models and excel at providing a general overview of 0, make it difficult to disentangle the effects of different components \nand interpret the signature.  \nV. CONCLUSION  \nFull-wave models have emerged as powerful tools for simulating crop backscatter with superior accuracy compared to early \nand hybrid methods. Unlike traditional models, full -wave models can solve for the scattering of EM waves, particularly \nbackscattering from canopies. This allows them to incorporate factors such as row spacing and detailed 3 -D structural models of \ncrops and canopies, which are often neglected in current models.  \n \nIn this study, we developed a full -wave 3 -D model based upon HFSS to demonstrate its potential to accurately estimate 0 \nacross diverse regions and growth stages. During the  early season of SMAPVEX16 -MicroWEX, the model estimates matched \nwell with field observations  with RMSDs less than 4 dB. The same occurred in late -season simulations, with deviations largely \nconfined to cross -pol results, at the level of 7 dB, likely attributable to underestimation of the branch structure or number of \nleaves in the 3 -D structural model, or slightly higher cross -pol measurements due to leakage.  Similarly, during SMAPVEX12, \nthe co-pol RMSDs were below 2 dB and cross -pol RMSDs below 4 dB, both during early -season simulations across three fields \nof observations. Importantly, regardless of the experiment or observation period, modeled estimates consistently fell within the \n±2 STD band of observations, highlighting the model’s accuracy.  \n \nThe full-wave 3 -D model successfully replicated the evolution of 0 throughout the growing season, including two ascending \nphases followed by a descending phase. This capability underscores the model’s ability to capture the underlying dynamics of \nthe backscattering problem, crucial for identifying key crop development stages and developing more effective inverse models.  \nThe bare soil model with a 3 -D rough surface demonstrated that  roughness es with the same RMS height and correlation length , \nthe two parameters widely used in scattering models,  can induce variability up to 8 dB . Additionally, the HFSS model \nsuccessfully disassociated the contributions of different vegetation components to the 0 signature, offering valuable insights for \ndeveloping accurate retrieval algorithms. Our analysis revealed that stems are the primary contributors to 0 at HH polarization, 34 \n> MANUSCRIPT ID NUMBER<  \n \nwhile leaves play a dominant role at HV polarizations. The model also identified branches as co -contributors to 0 at VV \npolarizations alongside stems, and pods as significant influencers of 0 at HH polarization.  \n \nWhile this study demonstrated some applications  of the full -wave 3 -D approach,  its potential extends beyond the scope of the \ncurrent research. Future investigations could explore its use with other crops and vegetation types, develop new algorithms f or \ninverse modeling based on this model, and test its performance with various remote sensing data sources. 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Yueh, \"Coherent Model of L -Band Radar Scattering by Soybean Plants: Model \nDevelopment, Evaluation, and Retrieval,\" IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 9, no. 1, pp. 272 -284, \n2016.  \n \n \n \n "    },    {        "title": "2402.02316v2.Your_Diffusion_Model_is_Secretly_a_Certifiably_Robust_Classifier.pdf",        "content": "Your Diffusion Model is Secretly a Certifiably Robust Classifier\nHuanran Chen1 2Yinpeng Dong1Shitong Shao3Zhongkai Hao1Xiao Yang1Hang Su1Jun Zhu1\nAbstract\nDiffusion models are recently employed as gen-\nerative classifiers for robust classification. How-\never, a comprehensive theoretical understanding\nof the robustness of diffusion classifiers is still\nlacking, leading us to question whether they will\nbe vulnerable to future stronger attacks. In this\nstudy, we propose a new family of diffusion classi-\nfiers, named Noised Diffusion Classifiers (NDCs),\nthat possess state-of-the-art certified robustness.\nSpecifically, we generalize the diffusion classi-\nfiers to classify Gaussian-corrupted data by deriv-\ning the evidence lower bounds (ELBOs) for these\ndistributions, approximating the likelihood using\nthe ELBO, and calculating classification proba-\nbilities via Bayes’ theorem. We integrate these\ngeneralized diffusion classifiers with randomized\nsmoothing to construct smoothed classifiers pos-\nsessing non-constant Lipschitzness. Experimental\nresults demonstrate the superior certified robust-\nness of our proposed NDCs. Notably, we are the\nfirst to achieve 80%+ and 70%+ certified robust-\nness on CIFAR-10 under adversarial perturbations\nwithℓ2norm less than 0.25 and 0.5, respectively,\nusing a single off-the-shelf diffusion model with-\nout any additional data.\n1. Introduction\nDespite the unprecedented success of deep learning in the\npast decade (LeCun et al., 2015; OpenAI, 2023), deep neu-\nral networks are vulnerable to adversarial examples, which\nare generated by imposing human-imperceptible perturba-\ntions to natural examples but can mislead a target model to\nmake erroneous predictions (Szegedy et al., 2014; Goodfel-\nlow et al., 2015). Adversarial examples pose a significant\nthreat to real-world applications of deep learning, such as\nface recognition (Sharif et al., 2016; Dong et al., 2019), au-\ntonomous driving (Cao et al., 2021; Jing et al., 2021), and\n1Tsinghua University2Beijing Institute of Technology\n3Shanghai AI Laboratory. Correspondence to: Jun Zhu <dc-\nszj@mail.tsinghua.edu.cn >.the prevalent vision-language models (Carlini et al., 2023a;\nZhao et al., 2023; Dong et al., 2023). To improve model\nrobustness, numerous defense techniques have been devel-\noped, including adversarial training (Madry et al., 2018;\nZhang et al., 2019; Rebuffi et al., 2021), input purification\n(Liao et al., 2018; Nie et al., 2022), adversarial example\ndetection (Metzen et al., 2017; Pang et al., 2018). However,\nmost of these empirical defenses lack theoretical guaran-\ntees and can be further evaded by stronger adaptive attacks\n(Athalye et al., 2018; Tramer et al., 2020).\nIn contrast, certified defenses (Raghunathan et al., 2018;\nSinha et al., 2018; Wong & Kolter, 2018; Cohen et al., 2019)\nhave been proposed to provide a certified radius for an in-\nput such that the model’s predictions are guaranteed to be\nunchanged for perturbations bounded by the radius. Among\nthem, randomized smoothing (Cohen et al., 2019; Hao et al.,\n2022) is a model-agnostic approach scalable to large net-\nworks and datasets. It constructs a smoothed classifier that\nclassifies an input image by taking the majority vote over the\nlabels predicted by a base classifier under Gaussian noise\nof the input, which possesses non-constant Lipschitzness\nand thus has certified robustness. However, this method ne-\ncessitates that the base model processes Gaussian-corrupted\nimages, which poses a significant obstacle to its application\nin providing certified robustness for common classifiers.\nRecently, various studies (Nie et al., 2022; Carlini et al.,\n2023b) have applied diffusion models (Sohl-Dickstein et al.,\n2015; Ho et al., 2020; Song et al., 2021) to improve the\nadversarial robustness, setting up a connection between ro-\nbust classification and the fast-growing field of pre-trained\ngenerative models. Notably, Chen et al. (2023a) propose a\ngenerative approach to achieve the state-of-the-art empir-\nical robustness by constructing diffusion classifiers from\npre-trained diffusion models. The diffusion classifiers cal-\nculate the classification probability p(y|x)∝p(x|y)p(y)\nthrough Bayes’ theorem and approximate the log likelihood\nlogp(x|y)via the evidence lower bound (ELBO). Though\npromising, there is still lacking a rigorous theoretical anal-\nysis, leaving questions whether their robustness is being\noverestimated and whether they will be vulnerable to (poten-\ntially) future stronger adaptive attacks. In this work, we aim\nto use theoretical tools to derive the certified robustness of\ndiffusion classifiers, fundamentally address these concerns,\nand gain a deeper understanding of their robustness.\n1arXiv:2402.02316v2  [cs.LG]  13 Feb 2024Certified Robustness via a Single Diffusion Model\nWe begin by analyzing the inherent smoothness of diffu-\nsion classifiers through the derivation of their Lipschitzness.\nHowever, the certified radius we obtain is not as tight as\ndesired due to the assumption that the maximum Lipschitz-\nness is maintained throughout the entire space. As a well-\nresearched technique, randomized smoothing (Cohen et al.,\n2019; Salman et al., 2019) can derive non-constant Lips-\nchitzness and thus leads to a much larger certified robust\nradius. Therefore, to achieve a tighter certified radius, we\nintegrate randomized smoothing, which necessitates that\ndiffusion classifiers process Gaussian-corrupted data xτ.\nConsequently, we propose Noised Diffusion Classifiers ,\nwhich calculate p(y|xτ)from logp(xτ|y)estimated by its\nELBO. Hence, the problem turns to deriving the ELBO for\nnoisy data. We first propose the Exact Posterior Noised\nDiffusion Classifier (EPNDC), which extends the ELBO in\nSohl-Dickstein et al. (2015) and Kingma et al. (2021) to\nτ̸= 0. Furthermore, we develop a better ELBO that is an\nexpectation or ensemble of the first over all noisy samples\nderived from a single image but can be computed without\nincurring additional cost, and name corresponding diffusion\nclassifier as Approximated Posterior Noised Diffusion Clas-\nsifier (APNDC). In the end, we significantly reduce time\ncomplexity and enhance the scalability of diffusion clas-\nsifiers for large datasets by using the same noisy samples\nfor all classes to reduce variance and designing a search\nalgorithm to exclude as many candidate classes as possible\nat the outset of classification via a few samples.\nExperimental results substantiate the superior performance\nof our methods. Notably, we establish new benchmarks in\ncertified robustness, achieving 82.2%,70.7%, and 54.5%at\nℓ2radii of 0.25,0.5, and 0.75, respectively, on the CIFAR-\n10 dataset. These results surpass prior state-of-the-art (Xiao\net al., 2023), by margins of 5.6%,6.1%, and 4.1%, in the\ncorresponding categories. In addition, our approach regis-\nters a clean accuracy of 91.2%, outperforming Xiao et al.\n(2023) by 3.6%. Importantly, our time complexity reduc-\ntion techniques reduce the computational burden by a factor\nof10on CIFAR-10 and by an impressive factor of 1000\non ImageNet. Remarkably, this substantial efficiency gain\ndoes not compromise the certified robustness of the model.\nMoreover, our comparative analysis with heuristic methods\nnot only highlights the tangible benefits stemming from our\ntheoretical advancements but also provides valuable theo-\nretical insights into several evidence lower bounds and the\ninherent robustness of diffusion classifiers.\n2. Background\n2.1. Diffusion Models\nFor simplicity in derivation, we introduce a general formula-\ntion that covers various diffusion models. In Appendix A.8,\nwe show that common models, such as Ho et al. (2020),Song et al. (2021), Kingma et al. (2021) and Karras et al.\n(2022), can be transformed to align with our definition.\nGiven x:=x0∈RDwith a data distribution q(x0), the\nforward diffusion process incrementally introduces Gaus-\nsian noise to the data distribution, resulting in a continuous\nsequence of distributions {q(xt) :=qt(xt)}T\nt=1by:\nq(xt) =Z\nq(x0)q(xt|x0)dx0, (1)\nwhere q(xt|x0) =N(xt;x0, σ2\ntI), i.e.,xt=x0+σtϵ,ϵ∼\nN(0,I). Typically, σtmonotonically increases with t, es-\ntablishing one-to-one mappings t(σ)from σtotandσ(t)\nfrom ttoσ. Additionally, σTis large enough that q(xT)\nis approximately an isotropic Gaussian distribution. Given\np:=pθas the parameterized reverse distribution with prior\np(xT) =N(xT;0, σ2\nTI), the diffusion process used to syn-\nthesize real data is defined as a Markov chain with learned\nGaussian distributions (Ho et al., 2020; Song et al., 2021):\np(x0:T) =p(xT)TY\nt=1p(xt−1|xt). (2)\nIn this work, we parameterize the reverse Gaussian distribu-\ntionp(xt−1|xt)using a neural network hθ(xt, t)as\np(xt−1|xt) =N(xt−1;µθ(xt, t),˜σ2\ntI),\nµθ(xt, t) =(σ2\nt−σ2\nt−1)hθ(xt, σt) +σ2\nt−1xt\nσ2\nt.(3)\nThe parameter θis usually trained by optimizing the evi-\ndence lower bound (ELBO) on the log likelihood:\nlogp(x0)≥ −TX\nt=1Eϵ\u0002\nwt∥hθ(xt, σt)−x0∥2\n2\u0003\n+C1,(4)\nwhere wt=σt+1−σt\nσ3\nt+1is the weight of the loss at time step\ntandC1is a constant. Similarly, the conditional diffusion\nmodel p(xt−1|xt, y)is parameterized by hθ(xt, σt, y). A\nsimilar lower bound on conditional log likelihood is\nlogp(x0|y)≥ −TX\nt=1Eϵ\u0002\nwt∥hθ(xt, σt, y)−x0∥2\n2\u0003\n+C,(5)\nwhere Cis another constant.\n2.2. Diffusion Classifiers\nDiffusion classifier (Li et al., 2023; Chen et al., 2023a; Clark\n& Jaini, 2023) is a generative classifier that uses a single\noff-the-shelf diffusion model for robust classification. It\nfirst calculates the class probability p(y|x0)∝p(x0|y)p(y)\nthrough Bayes’ theorem, and then approximates the condi-\ntional likelihood via conditional ELBO with the assumption\n2Certified Robustness via a Single Diffusion Model\nApproximate\nEnsemble\nLogit AveragingBayes' Theorem\nRandomized\nSmoothing Approximate\nApproximateBayes' Theorem\nRandomized\nSmoothing \nFigure 1. Illustration of our theoretical contributions. We derive the Lipschitz constant and the corresponding certified radius for diffusion\nclassifiers (Chen et al., 2023a). Additionally, we introduce two novel evidence lower bounds, which are used to approximate the log\nlikelihood. These lower bounds are then employed to construct classifiers based on Bayes’ theorem. By applying randomized smoothing\nto these classifiers, we derive their certified robust radii.\nthatp(y)is a uniform prior:\nDC(x0)y:=p(y|x0) =p(x0|y)p(y)P\nˆyp(x0|ˆy)p(ˆy)\n=p(x0|y)P\nˆyp(x0|ˆy)=exp(log p(x0|y))P\nˆyexp (log p(x0|ˆy))\n≈exp(−PT\nt=1Eϵ\u0002\nwt∥hθ(xt, σt, y)−x0∥2\n2\u0003\n)\nP\nˆyexp (−PT\nt=1Eϵ[wt∥hθ(xt, σt,ˆy)−x0∥2\n2]).(6)\nThis classifier achieves state-of-the-art empirical robustness\nacross several types of threat models and can generalize to\nunseen attacks as it does not require training on adversar-\nial examples (Chen et al., 2023a). However, there is still\nlacking a rigorous theoretical analysis, leaving questions\nabout whether they will be vulnerable to (potentially) future\nstronger adaptive attacks.\n2.3. Randomized Smoothing\nRandomized smoothing (Lecuyer et al., 2019; Cohen et al.,\n2019; Yang et al., 2020; Kumar et al., 2020) is a model-\nagnostic technique designed to establish a lower bound of\nrobustness against adversarial examples. It is scalable to\nlarge networks and datasets and achieves state-of-the-art\nperformance in certified robustness (Yang et al., 2020). This\napproach constructs a smoothed classifier by averaging the\noutput of a base classifier over Gaussian noise. Owing to\nthe Lipschitz continuity of this classifier, it remains stable\nwithin a certain perturbation range, thereby ensuring certi-\nfied robustness.\nFormally, given a classifier f:RD→RKthat takes a D-\ndimensional input x0and predicts class probabilities over\nKclasses, the y-th output of the smoothed classifier gis:\ng(x0)y=P( arg max\nˆy∈{1,...,K}f(x0+στ·ϵ)ˆy=y),(7)where ϵ∼ N(0,I)is a Gaussian noise and στis the noise\nlevel. Let Φ−1denote the inverse function of the standard\nGaussian CDF. Salman et al. (2019) prove that Φ−1(g(x0)y)\nis1\nστ-Lipschitz. However, the exact computation of g(x0)\nis infeasible due to the challenge of calculating the expec-\ntation in a high-dimensional space. Practically, one usually\nestimates a lower bound pAofg(x0)yand an upper bound\npBofmax ˆy̸=yg(x0)ˆyusing the Clopper-Pearson lemma,\nand then calculates the lower bound of the certified robust\nradius Rfor class yas\nR=στ\n2\u0000\nΦ−1(pA)−Φ−1(pB)\u0001\n. (8)\nTypically, the existing classifiers are trained to classify im-\nages in clean distribution q(x0). However, the input distribu-\ntion in Eq. (7) is q(xτ) =R\nq(x0)q(xτ|x0)dx0. Due to the\ndistribution discrepancy, g(x0)constructed by classifiers\ntrained on clean distribution q(x0)exhibits low accuracy\nonq(xτ). Due to this issue, we cannot directly incorporate\nthe diffusion classifier (Chen et al., 2023a) with randomized\nsmoothing. In this paper, we propose a new category of\ndiffusion classifiers, that can directly calculate p(y|xτ)via\nan off-the-shelf diffusion model.\nTo handle Gaussian-corrupted data, early work (Cohen et al.,\n2019; Salman et al., 2019) trains new classifiers on q(xτ)\nbut is not applicable to pre-trained models. Orthogonal to\nour work, there is also some recent work on using diffusion\nmodels for certified robustness. Carlini et al. (2023b) and\nXiao et al. (2023) propose to first use a diffusion model\nto denoise xτ∼q(xτ), followed by an off-the-shelf dis-\ncriminative classifier for classifying the denoised image.\nHowever, the efficacy of such an algorithm is largely con-\nstrained by the performance of the discriminative classifier.\n3Certified Robustness via a Single Diffusion Model\n3. Methodology\nIn this section, we first derive an upper bound of the Lips-\nchitz constant in Sec. 3.1. Due to the difficulty in deriving a\ntighter Lipschitzness for such a mathematical form, we pro-\npose two variants of Noised Diffusion Classifiers (NDCs),\nand integrate them with randomized smoothing to obtain\na tighter robust radius, as detailed in Sec. 3.2 and Sec. 3.3.\nFinally, we propose several techniques in Sec. 3.4 to reduce\ntime complexity and enhance scalability for large datasets.\n3.1. The Lipschitzness of Diffusion Classifiers\nWe observe that the logitsPT\nt=1Eϵ\u0002\n∥hθ(xt, σt)−x0∥2\n2\u0003\nof diffusion classifier in Eq. (6) can be decomposed as\nTX\nt=1Eϵ\u0002\n∥hθ(xt, σt)∥2\n2\u0003\n+∥x0∥2\n2−2Eϵ\u0002\nhθ(xt, σt)⊤\u0003\nx0.\nGiven that Eϵ\u0002\n∥hθ(xt, σt)∥2\n2\u0003\nandEϵ[hθ(xt, σt)]are\nsmoothed by Gaussian noise, they satisfy the Lipschitz con-\ndition (Salman et al., 2019). Consequently, the logits of\ndiffusion classifiers should satisfy a Lipschitz condition.\nThus, it can be inferred that the entire diffusion classifier is\nrobust and possesses a certain robust radius.\nWe derive an upper bound for the Lipchitz constant of diffu-\nsion classifiers in the following theorem:\nTheorem 3.1. (Proof in Appendix A.2) The upper bound of\nLipschitz constant of diffusion classifier is given by:\n|DC(x0+δ)y−DC(x0)y| ≤1\n2√\n2TX\nt=1wt\nσtT(r\n2\nπ+2√\nD)∥δ∥2.\n(9)\nIf we estimate a lower bound pAforDC(x0)yand a upper\nbound pBformax ˆy̸=yDC(x0)ˆyby Bernstein inequality\n(Maurer & Pontil, 2009), we can get a lower bound of robust\nradius for diffusion classifier:\nR=√\n2T(pA−pB)\n(2/√\nD+p\n2/π)PT\nt=1wt/σt. (10)\nAs demonstrated in Theorem 3.1, the Lipschitz constant\nof diffusion classifiers is nearly identical to that in the\n“weak law ” of randomized smoothing (See Appendix A.3,\nor Lemma 1 in Salman et al. (2019)). This constant is small\nand independent of the dimension D, indicating the inher-\nent robustness of diffusion classifiers. However, similar\nto the weak law of randomized smoothing, such certified\nrobustness has limitations because it assumes the maximum\nLipschitz condition is satisfied throughout the entire per-\nturbation path. Consequently, the robust radius is less than\nhalf of the reciprocal of the maximum Lipschitz constant\n(see Appendix C.3). On the other hand, the “ strong law ” ofAlgorithm 1 EPNDC\nRequire: A pre-trained diffusion model hθ, a noisy input image\nxτ, noisy level τ.\n1:fory= 0toK−1do\n2: fori=τ+ 1toTdo\n3: Calculate the analytical form of E[q(xt|xt+1,xτ, y)]by\n(σ2\nt+1−σ2\nt)xt+(σ2\nt−σ2\nτ)xt+1\nσ2\nt+1−σ2τ;\n4: Calculate the analytical form of E[p(xt|xt+1, y)]by\n(σ2\nt+1−σ2\nt)h(xt+1,σt+1,y)+σ2\ntxt+1\nσ2\nt+1;\n5: end for\n6: Calculate the lower bound logp(xτ|y)oflogp(xτ|y)by\nPT\nt=τ+1w(τ)\nt∥E[q(xt|xt+1,xτ, y)]−E[p(xt|xt+1, y)]∥2,\nwhere w(τ)\nt=σ2\nt+1−σ2\nτ\n2(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt);\n7:end for\n8: Approximate pθ(y|xτ)byexp (log pθ(xτ|y))P\nˆyexp (log pθ(xτ|ˆy));\n9:Return: ˜y= arg max ypθ(y|xτ).\nrandomized smoothing (See Eq. (8) or Lemma 2 in Salman\net al. (2019)) can yield a non-constant Lipschitzness, lead-\ning to a more precise robust radius, with the upper bound\nof the certified radius potentially being infinite. Therefore,\nin the subsequent sections, we combine diffusion classifier\nwith randomized smoothing to achieve a tighter certified\nradius, thus thoroughly explore its robustness.\n3.2. Exact Posterior Noised Diffusion Classifier\nAs explained in Sec. 2.3, randomized smoothing constructs\na smoothed classifier gfrom a given base classifier fby ag-\ngregating votes over Gaussian-corrupted data. This process\nnecessitates that the base classifier can classify data from the\nnoisy distribution q(xτ). However, the diffusion classifier\ndescribed in Chen et al. (2023a) is limited to classifying\ndata solely from q(x0). Therefore, in this section, we aim to\ngeneralize the diffusion classifier to enable the classification\nof any image from q(xτ)for any given τ.\nSimilar to Chen et al. (2023a), our fundamental idea involves\nderiving the ELBO for logp(xτ|y)and subsequently calcu-\nlating p(y|xτ)using the estimated logp(xτ|y)via Bayes’\ntheorem. Drawing inspiration from Ho et al. (2020), we\nderive a similar ELBO for logp(xτ), as elaborated in the\nfollowing theorem (the conditional ELBO is similar to un-\nconditional one, see Appendix A.4 for details):\nTheorem 3.2. (Proof in Appendices A.4 and A.5).\nThe ELBO of logp(xτ)is given by: logp(xτ)\n≥C−TX\nt=τ+1E[DKL(q(xt|xt+1,xτ)||p(xt|xt+1))]\n=C2+TX\nt=τw(τ)\ntE[∥E[q(xt|xt+1,xτ)]−E[p(xt|xt+1)]∥2],\n(11)\n4Certified Robustness via a Single Diffusion Model\nAlgorithm 2 APNDC\nRequire: A pre-trained diffusion model hθ, a noisy input image\nxτ, noisy level τ.\n1:fory= 0toK−1do\n2: Calculate the lower bound logp(xτ|y)oflogp(xτ|y)byPT\nt=τ+1wτ∥hθ(xτ, στ)−hθ(xt, σt+1, y)∥2\n3:end for\n4: Approximate pθ(y|xτ)byexp (log pθ(xτ|y))P\nˆyexp (log pθ(xτ|ˆy));\n5:Return: ˜y= arg max ypθ(y|xτ).\nwhere\nxt+1∼q(xt+1|xτ), w(τ)\nt=σ2\nt+1−σ2\nτ\n2(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt),\nE[q(xt|xt+1,xτ)] =(σ2\nt+1−σ2\nt)xτ+ (σ2\nt−σ2\nτ)xt+1\nσ2\nt+1−σ2τ,\nE[p(xt|xt+1)] =(σ2\nt+1−σ2\nt)h(xt+1, σt+1) +σ2\ntxt+1\nσ2\nt+1.(12)\nRemark 3.3.Notice that the summation of KL divergence\nin the ELBO of logp(xτ)starts from τ+ 1and ends at T,\nwhile that of logp(x0)starts from 1. Besides, the posterior\nisq(xt|xt+1,xτ)instead of q(xt|xt+1,x0).\nRemark 3.4.When τ= 0, this result degrades to the diffu-\nsion training lossσt+1−σt\nσ3\nt+1∥x0−h(xt+1, σt+1)∥2, consistent\nwith Kingma et al. (2021) and Karras et al. (2022).\nBased on Theorem 3.2, we can approximate logp(xτ|y)\nvia its ELBO, and calculate p(y|xτ) =elogpθ(xτ|y)\nP\nˆyelogpθ(xτ|ˆy)for\nclassification. We name this algorithm as Exact Posterior\nNoised Diffusion Classifier (EPNDC), as demonstrated in\nAlgorithm 1.\nTypically, when training diffusion models by Eq. (5), most\nresearchers opt to use a re-designed weight ˆwt(e.g., ˆwt= 1\nin Ho et al. (2020)), rather than the derived weight wt. When\nconstructing diffusion classifier from an off-the-shelf diffu-\nsion model, we find that maintaining consistency between\nthe loss weight in diffusion classifier and the training weight\nis crucial. As shown in Table 6, any inconsistency in loss\nweight leads to a decrease in performance. Conversely, per-\nformance enhances when the diffusion classifier’s weight\nclosely aligns with the training weight. Surprisingly, the de-\nrived weight (ELBO) yields the worst performance. There-\nfore, when τ= 0, we use the training weight ˆwtrather than\nthe derived weight wt.\nSimilarly, when τ̸= 0, the derived weight w(τ)\ntalso results\nin the worst performance among different weight configura-\ntions, as detailed in Appendix C.2. However, determining\nthe optimal weight in this general case is a complex chal-\nlenge. Directly replacing w(τ)\ntwith the training weight ˆwtis\nnot appropriate, as w(τ)\nt̸=wt. To address this issue, we pro-\npose an alternative: multiplying the weight byˆwt\nwt(i.e., usingAlgorithm 3 Sift-and-refine\nRequire: An ELBO computation function for a given timestep\ntand a class y, denoted as eθ(applicable for DC, EPNDC,\nor APNDC); a noisy input image xτ; sift timesteps {ti}Ts\ni=0;\nrefine steps {ti}Tr\ni=0; threshold τ.\n1:Initialize: the candidate class list C={0,1, . . . , K }.\n2:fori= 0toTsdo\n3: for all classes yinCdo\n4: Calculate ELBO for class yat timestep ti:\ney=eθ(xt, σti, y).\n5: end for\n6: Find the class mwith the minimum ELBO:\nm= arg min y∈Cey.\n7: Update Cby removing classes with a reconstruction loss τ\ngreater than that of m:\nC={y∈C:ey−em< τ}\n8:end for\n9: Reinitialize ey:ey=∞ ∀y /∈C,0∀y∈C.\n10:fori= 0toTrdo\n11: for all classes yinCdo\n12: Calculate and accumulate ELBO for class yat timestep\nti:ey=ey+eθ(xt, ti, y).\n13: end for\n14:end for\n15:Return: ˜y= arg min yey.\nˆwtw(τ)\nt\nwtas the weight for EPNDC). This method effectively\nacts as a substitution of wtwith ˆwtwhen τ= 0. However,\nit is important to note that this approach is not optimal, and\nderiving an optimal weight appears to be infeasible. For a\ndetailed explanation, please refer to Appendix C.2.\n3.3.Approximated Posterior Noised Diffusion Classifier\nThe confusion about the weight selection arises from\nthe fact that p(xt|xt+1)is parametrized in the way of\nq(xt|xt+1,x0), rather than q(xt|xt+1,xτ). As a result,\nwhen dealing with DKL(q(xt|xi+t,xτ)||p(xt|xt+1)), there\nis no wtterm, leading to confusion of how to substitute\nwtwith ˆwt. However, if we calculate the KL divergence\nbetween q(xt|xt+1,x0)andp(xt|xt+1), we can readily\nidentify the optimal weight, i.e., the weight used by the\npre-trained diffusion model during training. Inspired by\nthis insight, rather than concentrating on weight selection\ndilemma, we propose an alternative solution: approximating\nthe posterior by\nq(xt|xt+1,xτ) =q(xt|xt+1,xτ,x0=hθ(xτ, στ))\n≈q(xt|xt+1,x0=hθ(xτ, στ)).(13)\nConsequently, the KL divergence could be approximated as\nDKL(q(xt|xt+1,xτ)∥pθ(xt|xt+1))\n≈DKL(q(xt|xt+1,x0=hθ(xτ, στ))∥pθ(xt|xt+1))\n=σt+1−σt\nσ3\nt+1∥h(xτ, στ)−h(xt+1, σt+1)∥2+C3.(14)\nIn this case, we could directly replace wt=σt+1−σt\nσ3\nt+1by the\n5Certified Robustness via a Single Diffusion Model\nTable 1. Certified accuracy at CIFAR-10 test set. The certified accuracy at ϵ= 0 for each model is in the parentheses. The certified\naccuracy for each cell is from Xiao et al. (2023), same as the results from their respective papers.\nMethod Off-the-shelf Extra dataCertified Accuracy at R(%)\n0.25 0.5 0.75 1.0\nPixelDP (Lecuyer et al., 2019) ✗ ✗(71.0)22.0(44.0)2.0 - -\nRS (Cohen et al., 2019) ✗ ✗(75.0)61.0(75.0)43.0(65.0)32.0(65.0)23.0\nSmoothAdv (Salman et al., 2019) ✗ ✗(82.0)68.0(76.0)54.0(68.0)41.0(64.0)32.0\nConsistency (Jeong & Shin, 2020) ✗ ✗(77.8)68.8(75.8)58.1(72.9)48.5(52.3)37.8\nMACER (Zhai et al., 2019) ✗ ✗(81.0)71.0(81.0)59.0(66.0)46.0(66.0)38.0\nBoosting (Horv ´ath et al., 2021) ✗ ✗(83.4)70.6(76.8)60.4(71.6)52.4(73.0)38.8\nSmoothMix (Jeong et al., 2021) ✓ ✗(77.1)67.9(77.1)57.9(74.2)47.7(61.8)37.2\nDenoised (Salman et al., 2020) ✓ ✗(72.0)56.0(62.0)41.0(62.0)28.0(44.0)19.0\nLee (Lee, 2021) ✓ ✗(−)60.0(−)42.0(−)28.0(−)19.0\nCarlini (Carlini et al., 2023b) ✓ ✓(88.0)73.8(88.0)56.2(88.0)41.6(74.2)31.0\nDensePure (Xiao et al., 2023) ✓ ✓(87.6)76.6(87.6)64.6(87.6)50.4(73.6)37.4\nDiffPure+DC (baseline, ours) ✓ ✗(87.5)68.8(87.5)53.1(87.5)41.2(73.4)25.6\nEPNDC ( T′= 100 , ours) ✓ ✗(89.1)77.4(89.1)60.0(89.1)35.7(74.8)24.4\nAPNDC ( T′= 100 , ours) ✓ ✗(89.5)80.7(89.5)68.8(89.5)50.8(76.2)35.2\nAPNDC ( T′= 1000 , ours) ✓ ✗(91.2)82.2(91.2)70.7(91.2)54.5(77.3)38.2\ntraining weight ˆwt. We name this method as Approximated\nPosterior Noised Diffusion Classifier (APNDC), as shown\nin Algorithm 2.\nRemark 3.5.From a heuristic standpoint, one might con-\nsider first employing a diffusion model for denoising, fol-\nlowed by using a diffusion classifier for classification. This\napproach differs from our method, where we calculate the\ndiffusion loss only from τ+ 1toT, and the noisy samples\nxtare obtained by adding noise to xτinstead of x0. In\nour experiments, we demonstrate that our APNDC method\nsignificantly outperforms this heuristic approach.\nRemark 3.6.Intriguingly, APNDC is functionally equivalent\nto an ensemble of EPNDC. Formally, it approximates the\nlog likelihood logp(xτ)by the ELBOs of an expectation of\nlog likelihood Eq(ˆ xτ|x0=hθ(xτ,στ))[logpτ(ˆ xτ)]:\nlogp(xτ)≈Eq(ˆ xτ|x0=hθ(xτ,στ))[logpτ(ˆ xτ)]\n≥C4−T−1X\nt=τ+1wtEq(xt|x0=hθ(xτ,στ))[∥hθ(xt, σt)−x0∥2\n2].\n(15)\nThis free-ensemble can be executed without incurring any\nadditional computational costs compared to a single EP-\nNDC. For detailed explanations and proofs of this equiva-\nlence, please refer to Appendix A.7.\n3.4. Time Complexity Reduction\nVariance reduction. The main computational effort in our\napproach is dedicated to calculating the evidence lower\nbound for each class. This involves computing the sum\nof reconstruction losses. For instance, in DC, the recon-struction loss is ∥x0−hθ(xt, σt+1)∥2\n2. In EPNDC, it is\n∥E[q(xt|xt+1,xτ)]−E[p(xt|xt+1)]∥2\n2, and in APNDC, the\nloss is ∥hθ(xτ, στ)−hθ(xt+1, σt+1)∥2\n2, with the summa-\ntion carried out over t. Chen et al. (2023a) attempt to reduce\nthe time complexity by only calculating the reconstruction\nloss at certain timesteps. However, this approach proves in-\neffective. We identify that the primary reason for this failure\nis the large variance in the reconstruction loss, necessitat-\ning sufficient calculations for convergence. To address this,\nwe propose an effective variance reduction technique that\nuses identical input samples across all categories at each\ntimestep. In other words, we use the same xtfor different\nclasses. This approach significantly reduces the difference\nin prediction difficulty among various classes, allowing for\na more equitable calculation of the reconstruction loss for\neach class. As shown in Figure 2(a), we can utilize a much\nsmaller number of timesteps, such as 8, without sacrificing\naccuracy, thereby substantially reducing time complexity.\nSift-and-refine algorithm. The time complexity of these\ndiffusion classifiers is proportional to the number of classes,\npresenting a significant obstacle for their application in\ndatasets with numerous classes. Chen et al. (2023a) suggest\nthe use of multi-head diffusion to address this issue. How-\never, this approach requires training an additional diffusion\nmodel, leading to extra computational overhead. In our\nwork, we focus solely on employing a single off-the-shelf\ndiffusion model to construct a certifiably robust classifier.\nTo tackle the aforementioned challenge, we propose a Sift-\nand-refine algorithm. As demonstrated in Algorithm 3, the\ncore idea is to swiftly reduce the number of classes, thereby\nlimiting our focus to a manageable subset of classes.\n6Certified Robustness via a Single Diffusion Model\nTable 2. Certified accuracy at ImageNet-64x64. The certified accuracy at ϵ= 0for each model is in the parentheses.\nMethod Off-the-shelf Extra dataCertified Accuracy at R(%)\n0.25 0.5 0.75 1.0\nRS (Cohen et al., 2019) ✗ ✗(45.5)37.3(45.5)26.6(37.0)20.9(37.0)15.1\nSmoothAdv (Salman et al., 2019) ✗ ✗(44.4)37.4(44.4)27.9(34.7)21.1(34.7)17.0\nConsistency (Jeong & Shin, 2020) ✗ ✗(43.6)36.9(43.6)31.5(43.6)26.0(31.4)16.6\nMACER (Zhai et al., 2019) ✗ ✗(46.3)35.7(46.3)27.1(46.3)15.6(38.7)11.3\nCarlini (Carlini et al., 2023b) ✓ ✗(41.1)37.5(39.4)30.7(39.4)24.6(39.4)21.7\nDensePure (Xiao et al., 2023) ✓ ✗(37.7)35.4(37.7)29.3(37.7)26.0(37.7)18.6\nAPNDC (Sift-and-Refine, ours) ✓ ✗(54.4)46.3(54.4)38.3(43.5)35.2(43.5)32.8\nWe also experiment with alternative methods to reduce\nthe time complexity of diffusion classifiers, but these ap-\nproaches are not successful. Detailed information on these\nunsuccessful attempts can be found in Appendix C.4.\n4. Experiment\nFollowing previous studies (Carlini et al., 2023b; Xiao\net al., 2023; Zhang et al., 2023), we evaluate the certi-\nfied robustness of our method on two standard datasets,\nCIFAR-10 (Krizhevsky & Hinton, 2009) and ImageNet\n(Russakovsky et al., 2015), selecting a subset of 512 images\nfrom each. We adhere to the certified robustness pipeline\nestablished by Cohen et al. (2019), although our method\npotentially offers a tighter certified bound, as demonstrated\nin Appendix A.3. We select στ∈ {0.25,0.5,1.0}for certi-\nfication and use EDM (Karras et al., 2022) as our diffusion\nmodels. For more details, please refer to Appendix B.\n4.1. Results on CIFAR-10\nExperimental settings. Due to computational constraints,\nwe employ a sample size of N= 10,000to estimate pA.\nThe number of function evaluations (NFEs) for each im-\nage in our method is O(N·T′·K), amounting to 108for\nT′= 100 and109forT′= 1000 since K= 10 in this\ndataset. In contrast, the NFEs for the previous state-of-the-\nart method (Xiao et al., 2023) are 4·108, which is four times\nhigher than our method when T′is 100. It is important to\nhighlight that our sample size Nis 10 times smaller than\nthose baselines in (Table 1), potentially placing our method\nat a significant disadvantage, especially for large στ.\nExperimental results. As shown in Table 1, our method,\nutilizing an off-the-shelf model without the need for ex-\ntra data, significantly outperforms all previous methods at\nsmaller values of ϵ={0.25,0.5}. Notably, it surpasses\nall previous methods on clean accuracy, and exceeds the\nprevious state-of-the-art method (Xiao et al., 2023) by 5.6%\natϵ= 0.25and6.1%atϵ= 0.5. Even with larger values\nofϵ, our method attains performance levels comparable toexisting approaches. This is particularly noteworthy consid-\nering our constrained setting of N= 10,000, substantially\nsmaller than the N= 100 ,000used in prior works. This\ndiscrepancy in resources demonstrates the strong efficacy\nof our approach, which manages to achieve competitive per-\nformance despite the inherent disadvantage in sample size.\nThese findings underscore the inherent robustness of the\ndiffusion model itself, showcasing the viability of using a\nsingle off-the-shelf diffusion model for robust classification.\n4.2. Results on ImageNet\nExperimental settings. We conduct experiments on Im-\nageNet64x64 due to the absence of conditional diffusion\nmodels for 256x256 resolution. Due to computational con-\nstraints, we employ a sample size of N= 1000 , 10 times\nsmaller than all other works in Table 2. We use the Sift-and-\nRefine algorithm to improve the efficiency.\nExperimental results. As demonstrated in Table 2 and Fig-\nure 2(c), our method, which employs a single off-the-shelf\ndiffusion model without requiring extra data, significantly\noutperforms previous training-based and training-free ap-\nproaches. In contrast, diffusion-based purification methods,\nwhen applied with small CNNs and no extra data, do not\nmaintain their superiority over training-based approaches.\nIt is noteworthy that our experiments are conducted with\nonly one-tenth of the sample size typically used in previous\nworks. This success on a large dataset like ImageNet64x64\nunderscores the scalability of diffusion classifiers in han-\ndling extensive datasets with a larger number of classes.\n4.3. Ablation Studies\nVariance reduction proves beneficial. As illustrated in\nFigure 2(a), our variance reduction technique enables a sig-\nnificant reduction in time complexity while maintaining\nhigh clean accuracy. Regarding certified robustness, Table 1\nshows that a tenfold reduction in time complexity results in\nonly a minor, approximately 3%, decrease in certified robust-\nness across all radii. Notably, with T′= 100 , our method\nstill attains state-of-the-art clean accuracy and certified ro-\n7Certified Robustness via a Single Diffusion Model\n/uni00000017 /uni0000001b/uni00000014/uni00000019 /uni00000016/uni00000015 /uni00000019/uni00000017/uni00000014/uni00000015/uni0000001b /uni00000015/uni00000018/uni00000019 /uni00000018/uni00000014/uni00000015/uni00000014/uni00000013/uni00000013/uni00000013\nT/prime/uni00000014/uni00000013/uni00000016/uni00000013/uni00000018/uni00000013/uni0000001a/uni00000013/uni0000001c/uni00000013/uni00000024/uni00000046/uni00000046/uni00000058/uni00000055/uni00000044/uni00000046/uni0000005c/uni00000003/uni0000000b/uni00000008/uni0000000c\n/uni00000026/uni0000004b/uni00000048/uni00000051/uni00000003/uni00000048/uni00000057/uni00000003/uni00000044/uni0000004f/uni00000011/uni00000003/uni0000000b/uni00000015/uni00000013/uni00000015/uni00000016/uni0000000c\n/uni00000032/uni00000058/uni00000055/uni00000056\n(a)T′\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001c /uni00000014/uni00000011/uni00000015 /uni00000014/uni00000011/uni00000018\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\n/uni00000026/uni00000044/uni00000055/uni0000004f/uni0000004c/uni00000051/uni0000004c/uni00000003/uni00000048/uni00000057/uni00000003/uni00000044/uni0000004f/uni00000011/uni00000003/uni0000000b/uni00000015/uni00000013/uni00000015/uni00000015/uni0000000c\n/uni00000032/uni00000058/uni00000055/uni00000056(b)Rin CIFAR-10\n/uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni0000001a/uni00000018 /uni00000014/uni00000011/uni00000013/uni00000013\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\n/uni00000026/uni00000044/uni00000055/uni0000004f/uni0000004c/uni00000051/uni0000004c/uni00000003/uni00000048/uni00000057/uni00000003/uni00000044/uni0000004f/uni00000011/uni00000003/uni0000000b/uni00000015/uni00000013/uni00000015/uni00000015/uni0000000c\n/uni00000032/uni00000058/uni00000055/uni00000056 (c)Rin ImageNet\nFigure 2. (a) The accuracy (%) on CIFAR-10 dataset with time complexity reduction technique in Chen et al. (2023a) and ours. (b, c) The\nupper envelop of certified radii of different methods.\nbustness for ϵ= 0.25,0.5,0.75while only cost one-fourth\nof the NFEs in Xiao et al. (2023). This underscores the\neffectiveness of our variance reduction approach.\nSift-and-refine proves beneficial. Our method’s require-\nment for function evaluations (NFEs) is proportional to the\nnumber of classes, which limits its scalability in datasets\nwith a large number of candidate classes. For instance, in\nthe ImageNet dataset, without the Sift-and-refine algorithm,\nour method necessitates approximately 108NFEs per image\n(i.e.,100·1000·1000 ), translating to about 3×106seconds\nfor certifying each image on a single 3090 GPU. In contrast,\nXiao et al. (2023)’s method requires about 4×106NFEs\n(i.e., 40·10·10000 ), or roughly 3×105seconds. Our\nproposed Sift-and-refine technique, however, can swiftly\nidentify the most likely candidates, thereby reducing the\ntime complexity. It adjusts the processing time based on\nthe difficulty of the input samples. With this technique,\nour method requires only about 1×105seconds per image,\nmaking it more efficient compared to Xiao et al. (2023).\n4.4. Discussions\nComparison with heuristic methods. From a heuristic\nstandpoint, one might consider initially using a diffusion\nmodel for denoising, followed by employing a diffusion\nclassifier for classification. As demonstrated in Table 1, this\nheuristic approach outperforms nearly all prior off-the-shelf\nand no-extra-data baselines. However, the methods derived\nthrough our theoretical analysis significantly surpass this\nheuristic strategy. This outcome underscores the practical\nimpact of our theoretical contributions.\nTrivial performance of EPNDC. Although EPNDC ex-\nhibits non-trivial improvements compared to previous meth-\nods, it still lags significantly behind APNDC. We hypoth-\nesize two main reasons for this performance gap. First, as\nextensively discussed in Sec. 3.2, the weight utilized in EP-\nNDC is not optimal, and determining the optimal weightthrough simple derivation proves challenging. We leave the\nexploration of the optimal weight for future work. Addi-\ntionally, APNDC is equivalent to logit averaging in EPNDC\n(see Appendix C.1), which may contribute to its superior\nperformance compared to EPNDC.\nExplanation of Eq. (10). Eq. (10) is extremely similar to\nthe weak law of randomized smoothing . When σtis larger,\nit could potentially have a larger certified radius, but the\ninput images will be more noisy and hard to classify. This\ntrade-off is quite similar to the role of στin randomized\nsmoothing. However, there are two key differences. First,\nthe inputs to the network contain different levels of noisy\nimages, which means the network could see clean images,\nless noisy images, and very noisy images, hence could make\nmore accurate predictions. Besides, the trade-off parameter\niswt\nσt, allowing users some freedom to select different noise\nlevels and balance them by wt. We observe that this is the\ncommon feature of such denoising and reconstructing classi-\nfiers. We anticipate this observation will aid the community\nin developing more robust and certifiable defenses.\n5. Conclusion\nIn this work, we conduct a comprehensive analysis of the\nrobustness of diffusion classifiers. We establish their non-\ntrivial Lipschitzness, a key factor underlying their remark-\nable empirical robustness. Furthermore, we extend the capa-\nbilities of diffusion classifiers to classify noisy data at any\nnoise level by deriving the evidence lower bounds for noisy\ndata distributions. This advancement enable us to combine\nthe diffusion classifiers with randomized smoothing, leading\nto a tighter certified radius. Experimental results demon-\nstrate substantial improvements in certified robustness and\ntime complexity. We hope that our findings contribute to\na deeper understanding of diffusion classifiers in the con-\ntext of adversarial robustness and help alleviate concerns\nregarding their robustness.\n8Certified Robustness via a Single Diffusion Model\nImpact Statements\nThe advancement of robust machine learning models, par-\nticularly in the realm of classification under adversarial\nconditions, is crucial for the safe and reliable deployment\nof AI in critical applications. Our work on Noised Diffu-\nsion Classifiers (NDCs) represents a significant step towards\ndeveloping more secure and trustworthy AI systems. By\nachieving unprecedented levels of certified robustness, our\napproach enhances the reliability of machine learning mod-\nels in adversarial environments. This is particularly benefi-\ncial in fields where decision-making reliability is paramount,\nsuch as autonomous driving, medical diagnostics, and finan-\ncial fraud detection. Our methodology could substantially\nincrease public trust in AI technologies by demonstrating re-\nsilience against adversarial attacks, thereby fostering wider\nacceptance and integration of AI solutions in sensitive and\nimpactful sectors.\nReferences\nAthalye, A., Carlini, N., and Wagner, D. Obfuscated gra-\ndients give a false sense of security: Circumventing de-\nfenses to adversarial examples. In International Confer-\nence on Machine Learning , pp. 274–283, 2018.\nCao, Y ., Wang, N., Xiao, C., Yang, D., Fang, J., Yang, R.,\nChen, Q. A., Liu, M., and Li, B. 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On evaluating adversarial robust-\nness of large vision-language models. arXiv preprint\narXiv:2305.16934 , 2023.\n11Certified Robustness via a Single Diffusion Model\nAPPENDIX\nAppendix organization:\nAppendix A: Proofs.\nA.1: Assumptions and Lemmas.\nA.2: Lipschitz Constant of Diffusion Classifiers.\nA.3: Stronger Randomized Smoothing When fPossess Lipschitzness.\nA.4: ELBO of logpτ(xτ)in EPNDC.\nA.5: The Analytical Form of the KL Divergence in ELBO.\nA.6: The Weight in EPNDC.\nA.7: The ELBO of logpτ(xτ)in APNDC.\nA.8: Converting Other Diffusion Models into Our Definition.\nAppendix B: Experimental Details.\nB.1: Certified Robustness Details.\nB.2: ImageNet Baselines.\nB.3: Ablation Studies of Diffusion Models.\nAppendix C: Discussions.\nC.1: ELBO, Likelihood, Classifier and Certified Robustness.\nC.3: Certified Robustness of Chen et al. (2023a).\nC.2: The Loss Weight in Diffusion Classifiers.\nC.4: Time Complexity Reduction Techniques that Do Not Help.\nA. Proofs\nA.1. Assumptions and Lemmas\nAssumption A.1. We adopt the following assumptions, as outlined in Lu et al. (2022):\n1.∀0≤t≤T, qt(x)∈ C2,h(x, σt)∈ C1andEqt(x)[∥x∥2\n2]≤ ∞ .\n2. For any classifier fmentioned in this paper, f(x)∈ C1, and∃C≥0,∀x,y∈RD:∥f(x)−f(y)∥2≤C∥x−y∥2.\n3.∃C≥0,∀x,y∈RD,∀0≤t≤T:∥h(x, σt)−h(y, σt)∥2≤C∥x−y∥2.\n4.∃C≥0,∀x∈RD,∀0≤t≤T:∥h(x, σt)∥2≤C∥x+ 1∥2.\nLemma A.2. The second norm of the gradient of softmax function ||∂\n∂xsoftmax (x/β)y||2is less than or equal to1\n2√\n2β. In\nother words,\n||∂\n∂xsoftmax (x/β)y||2≤1\n2√\n2β.\nProof.\nmax||∂\n∂xsoftmax (x/β)y||2\n= max ||(ey−softmax (x/β))softmax (x/β)y\nβ||2\n= max1\nβ·s\nsoftmax (x/β)2y[(1−softmax (x/β)y)2+X\ni̸=ysoftmax (x/β)2\ni]\n≤max√\n2\nβ·q\nsoftmax (x/β)2y(1−softmax (x/β)y)2\n≤1\n2√\n2β.\n12Certified Robustness via a Single Diffusion Model\nA.2. Lipschitz Constant of Diffusion Classifiers\nChen et al. (2023a) has proven that\nd\ndxEϵ[∥hθ(xt, t, y)−x∥2\n2] =Eϵ[∂logp(xt|x)\n∂x∥hθ(xt, t, y)−x∥2\n2] +Eϵ[(hθ(xt, t, y)−x)2\nσt],\nwhere\n∂logp(xt|x)\n∂x=∂\n∂xlog exp ( −∥xt−x∥2\n2\n2σ2\nt) =−x−xt\nσ2\nt=σtϵ\nσ2\nt=ϵ\nσt.\nAccording to the Lemma 1 in Salman et al. (2019), we can get\n∥Eϵ[∂logp(xt|x)\n∂x∥hθ(xt, t, y)−x∥2\n2]∥2≤max\nuu⊤Z\np(ϵ)ϵD\nσt∥hθ(xt, t, y)−x∥2\n2\nDdϵ≤D\nσtr\n2\nπ\nBecause x∈[0,1]andhθ(xt, t, y)∈[0,1], hence\n∥d\ndxEϵ[∥hθ(xt, t, y)−x∥2\n2]∥2≤ ∥D\nσtr\n2\nπ∥2+∥√\nD·2\nσt∥2=1\nσt(Dr\n2\nπ+ 2√\nD).\nLet’s define one-hot vector eiinRnto be the vector where the i-th element is 1 and all other elements are 0. Hinton et al.\n(2015) proves that∂\n∂xlog softmax( x/β)y=1\nβ(ey−softmax( x/β)), where βis the softmax temperature. Consequently,\n∂\n∂xsoftmax( x/β)y=\u0012∂\n∂xlog softmax( x/β)y\u0013\nsoftmax( x/β)y= (ey−softmax( x/β))softmax( x/β)y\nβ.\nWe can derive the maximum ℓ2norm of the gradient as\n∥∂\n∂x0p(y|x0)∥2\n=∥∂\n∂x0softmax( f(x))y∥2\n=∥KX\ny=1∂softmax( f(x))y\n∂(−PT\nt=1Eϵ[wt∥hθ(xt, t, y)−x0∥2\n2])∂(−PT\nt=1Eϵ\u0002\nwt∥hθ(xt, t, y)−x0∥2\n2\u0003\n)\n∂x∥2\n≤KX\ny=1∥∂softmax( f(x))y\n∂(−PT\nt=1Eϵ[wt∥hθ(xt, t, y)−x0∥2\n2])∥2∥∂(−PT\nt=1Eϵ\u0002\nwt∥hθ(xt, t, y)−x0∥2\n2\u0003\n)\n∂x∥2\n≤1\n2√\n2TX\ni=1wi\nσiβ(Dr\n2\nπ+ 2√\nD)\nDenote uas a unit vector, the Lipschitz constant of the diffusion classifier p(y|x)is\nmax\nuu⊤∂\n∂xp(y|x) = max\nu∥u⊤∂\n∂xp(y|x)∥2=∥∂\n∂xp(y|x)∥2=1\n2√\n2TX\ni=1wi\nσiβ(Dr\n2\nπ+ 2√\nD) (16)\nAs the value of βincreases, we observe a decrease in the Lipschitz constant. Practically, to circumvent numerical issues,\nwe opt to compute the diffusion loss using the MSE (Mean Squared Error) loss instead of the ℓ2norm. This approach is\nequivalent to setting β=D·T. Under this condition, the Lipschitz constant is given by:\n1\n2√\n2TX\ni=1wi\nσiDT(Dr\n2\nπ+ 2√\nD) =1\n2√\n2TX\ni=1wi\nσiT(r\n2\nπ+2√\nD)\nIt is important to note that this represents the upper bound of the Lipschitz constant for the diffusion classifier. In practical\nscenarios, the actual Lipschitz constant is much smaller, due to the conservative nature of the inequalities used in the\nderivation. To gain an intuitive understanding of the practical Lipschitz constant for diffusion classifiers, we measure the\ngradient norm of the classifier on clean and noisy data. Furthermore, employing the algorithm from Chen et al. (2023b), we\nempirically determine the maximum gradient norm. Our results on the CIFAR-10 test set indicate that the gradient norm is\nless than 0.02, suggesting that the Lipschitz constant for our diffusion classifier is smaller than 0.02 on this dataset.\n13Certified Robustness via a Single Diffusion Model\nA.3. Stronger Randomized Smoothing When fPossess Lipschitzness\nIn this section, we will show that if fhas a smaller Lipschitz constant, it will induce a more smoothed function g, thus has a\nhigher certified robustness. Here we discuss a simple case: the weak law of randomized smoothing proposed in Salman et al.\n(2019) with σ= 1. We complement the derivation of this law with much more details, and we hope our detailed explanation\ncould assist newcomers in the field in quickly grasping the key concepts in randomized smoothing.\nTo derive the Lipschitz constant of the smoothed function g, we only need to derive the maximum dot product between the\ngradient of gand a unit vector ufor the worst f.\nmax\nfu⊤∇xg(x) = max\nfu⊤∇xEϵ[f(x+ϵ)]\n= max\nfu⊤∇xZ\np(ϵ)f(x+ϵ)dϵ\n= max\nfu⊤∇x1\n(2π)n/2Z\nexp (−∥ϵ∥2\n2\n2)f(x+ϵ)dϵ\n= max\nfu⊤∇x1\n(2π)n/2Z\nexp (−∥t−x∥2\n2\n2)f(t)dt\n= max\nfu⊤ 1\n(2π)n/2Z\nexp (−∥t−x∥2\n2\n2)(t−x)f(t)dt\n= max\nfu⊤ 1\n(2π)n/2Z\nexp (−∥t−x∥2\n2\n2)(t−x)f(t−x)dt\n= max\nf1\n(2π)n/2Z\nexp (−∥t−x∥2\n2\n2)[u⊤(t−x)]f(t−x)dt\nIn the next step, we transition to another orthogonal coordinate system, denoted as u,u2,···,uD. This change is made\nwith the assurance that the determinant of the Jacobian matrix, representing the transformation from the old coordinates to\nthe new ones, equals 1. We then decompose the vector t−xin the new coordinate system as follows:\nt−x=a1u+a2u2+a3u3+···+aDuD.\nTherefore,\nmax\nf1\n(2π)n/2Z\nexp (−∥t−x∥2\n2\n2)[u⊤(t−x)]f(t−x)dt\n= max\nf1\n(2π)n/2Z\nexp (−∥a1u+a2u2+a3u3+···+aDuD∥2\n2\n2)a1f(a)da\n= max\nf1\n(2π)n/2Z\nexp (−a2\n1+a2\n2+a2\n3+···+a2\nD\n2)a1f(a)da\n= max\nf1\n(2π)n/2Z∞\n−∞exp (−a2\n1\n2)a1Z∞\n−∞exp (−a2\n2\n2)···Z∞\n−∞exp (−a2\nD\n2)f(a)da1da2···daD\n≤max\nf1\n(2π)n/2Z∞\n−∞exp (−a2\n1\n2)a1f(a1)da1Z∞\n−∞exp (−a2\n2\n2)da2···Z∞\n−∞exp (−a2\nD\n2)daD\n= max\nf1\n(2π)1/2Z∞\n−∞exp (−a2\n1\n2)a1f(a1)da1\n≤max\nf1\n(2π)1/2Z∞\n0exp (−s2\n2)sds\nWe could get easily the result by change of variable:\n1\n(2π)1/2Z∞\n0exp (−s2\n2)sds=1\n(2π)1/2Z∞\n0exp (−s2\n2)ds2\n2=1\n(2π)1/2(−exp (−s2\n2))|+∞\n0=1√\n2π.\n14Certified Robustness via a Single Diffusion Model\nThe classifier will robustly classify the input data as long as the probability of classify the noisy data as the correct class\nis greater than the probability of classifying the noisy data as the wrong class. If one estimate a lower bound pAof the\naccuracy of the correct class over noisy sample and a upper bound pBof the probability of classify the noisy input as wrong\nclass using the Clopper-Pearson interval, we could get the robust radius by\nR=rπ\n2(pA−pB),\nwhich will ensure that for any ∥δ∥2≤Rand any wrong class ˆy̸=y:\ng(x+δ)y=P(f(x+δ+ϵ) =y)> g(x+δ)ˆy=P(f(x+δ+ϵ) = ˆy).\nHence, g(x)will robustly classify the input data.\nWhen fsatisfies the Lipschitz condition with Lipschitz constant L, since it is impossible to be one when x >0and zero\nwhen x≤0, the inequality here can be tighter, so we can get a smaller Lipschitz constant. In fact, in this case, the maximum\nLipschitz constant of gis:\nmax\nf1√\n2πZ+∞\n−∞exp(−s2/2)sf(s)ds\ns.t.,0≤f(s)≤1, f(s)isL-lipschitz\nHowever, obtaining an analytical solution for the certified radius when fadheres to Lipschitz continuity appears infeasible.\nTo understand why, consider the following: firstly, fshould approach zero as xtends toward negative infinity and one as x\ntends toward positive infinity. Secondly, fmust be an increasing function, with only one interval of increase. Additionally,\nwithin this interval, fis either zero or one outside the bounds of increase, and it must take a linear form within, with a slope\nof1\nL. We define the left endpoint of this increasing interval as a, which necessarily lies in the range [−1\nL,0]. Also notice\nthat both a=−1\nLanda= 0 yield the same certified radius. However, when a∈(−1\nL,0), the certified radius must be\nsmaller. To see why, let’s consider\nfa(s) =\n\n0, s ≤a\nL(s−a), a≤s≤a+1\nL\n1, s ≥a+1\nL,with special case f0(s) =\n\n0, s ≤0\nLs, 0≤s≤1\nL\n1, s ≥1\nL.\nDenote ˆfa(x) =1√\n2πexp(−x2/2)xfa(x). The difference of Lipschitz constant between two randomized function is\nZ+∞\n−∞[ˆf0(s)−ˆfa(s)]ds=Z0\na[ˆf0(s)−ˆfa(s)] +Z−a\n0[ˆf0(s)−ˆfa(s)] +Z1/L\n−a[ˆf0(s)−ˆfa(s)]\n=Z−a\n0[−aL−ˆfa(s)] +Z1/L\n−a[ˆf0(s)−ˆfa(s)]≤0\nConsequently, the corner case for amust lie within the open interval (−1/L,0). However, the corresponding integral lacks\nan analytical solution, and taking the derivative of it results in a function whose zero points are indeterminable. Therefore,\nobtaining an analytical solution for the improvement in certified robustness when a function exhibits Lipschitz continuity is\nchallenging. Nevertheless, we can approximate the result with numerical algorithms, and it is evident that this Lipschitzness\nleads to a non-trivial enhancement in certified robustness of randomized smoothing.\nA.4. ELBO of logpτ(xτ)in EPNDC\nSimilar to Ho et al. (2020), we derive the ELBO as\nlogp(xτ) = logZp(xτ:T)q(xτ+1:T|xτ)\nq(xτ+1:T|xτ)dxτ+1:T\n= logEq(xτ+1:T|xτ)[p(xτ:T)\nq(xτ+1:T|xτ)]\n15Certified Robustness via a Single Diffusion Model\n= logEq(xτ+1:T|xτ)[p(xT)p(xτ:T−1|xT)\nq(xτ+1:T|xτ)]\n≥Eq(xτ+1:T|xτ)[logp(xT)p(xτ:T−1|xT)\nq(xτ+1:T|xτ)]\n=Eq(xτ+1:T|xτ)[logp(xT)QT−1\nt=τp(xt|xt+1)\nQT−1\nt=τq(xt+1|xt,xτ)]\n=Eq(xτ+1:T|xτ)[logp(xT)QT−1\nt=τp(xt|xt+1)\nQT−1\nt=τq(xt+1|xτ)q(xt|xt+1,xτ)\nq(xt|xτ)]\n=Eq(xτ+1:T|xτ)[logp(xT)QT−1\nt=τp(xt|xt+1)\nQT−1\nt=τq(xt|xt+1,xτ)−logT−1Y\nt=τq(xt+1|xτ)\nq(xt|xτ)]\n=Eq(xτ+1:T|xτ)[logp(xT)QT−1\nt=τp(xt|xt+1)\nQT−1\nt=τq(xt|xt+1,xτ)−logq(xT|xτ)\nq(xτ|xτ)]\n=Eq(xτ+1:T|xτ)[logQT−1\nt=τp(xt|xt+1)\nQT−1\nt=τq(xt|xt+1,xτ)−logq(xT|xτ)\np(xT)]\n=Eq(xτ+1:T|xτ)[logQT−1\nt=τp(xt|xt+1)\nQT−1\nt=τq(xt|xt+1,xτ)]−Eq(xT|xτ)[DKL(q(xT|xτ)||p(xT))]\n=T−1X\nt=τEq(xt,xt+1|xτ)[logp(xt|xt+1)\nq(xt|xt+1,xτ)]−Eq(xT|xτ)[DKL(q(xT|xτ)||p(xT))]\n=T−1X\nt=τEq(xt+1|xτ)Eq(xt|xt+1,xτ)[logp(xt|xt+1)\nq(xt|xt+1,xτ)]−Eq(xT|xτ)[DKL(q(xT|xτ)||p(xT))]\n= log p(xτ|xτ+1)−T−1X\nt=τ+1Eq(xt+1|xτ)[DKL(q(xt|xt+1,xτ)||p(xt|xt+1))]−Eq(xT|xτ)[DKL(q(xT|xτ)||p(xT))]\n=−T−1X\nt=τ+1Eq(xt+1|xτ)[DKL(q(xt|xt+1,xτ)||p(xt|xt+1))] + C\nThis could be understood as a generalization of the ELBO in Ho et al. (2020), which is a special case when τ= 0. Notice\nthat the summation of KL divergence in ELBO of logp(xτ)start from τ+ 1and end at T, while that of logp(x0)start\nfrom 1. Besides, the posterior is q(xt|xt+1,xτ)instead of q(xt|xt+1,x0).\nSimilarly, the conditional ELBO is give by:\nlogp(xτ|y)≥logp(xτ|xτ+1, y)−Eq(xT|xτ)[DKL(q(xT|xτ)||p(xT))]\n−T−1X\nt=τ+1Eq(xt+1|xτ)[DKL(q(xt|xt+1,xτ, y)||p(xt|xt+1, y))]\n=−T−1X\nt=τ+1Eq(xt+1|xτ)[DKL(q(xt|xt+1,xτ, y)||p(xt|xt+1, y))] + C5.\nThis is equivalent to adding the condition yto all posterior distributions in the unconditional ELBO.\nA.5. The Analytical Form of the KL Divergence in ELBO\nThe KL divergence in Theorem 3.2 is the expectation of the log ratio between posterior and predicted reverse distribution,\nwhich requires integrating over the entire space. To compute this KL divergence more efficiently, we derives its analytical\n16Certified Robustness via a Single Diffusion Model\nform:\nq(xt|xt+1,xτ) =q(xt|xτ)q(xt+1|xt)\nq(xt+1|xτ)\n=N(xt|xτ,(σ2\nt−σ2\nτ)I)N(xt+1|xt,(σ2\nt+1−σ2\nt)I)\nN(xt+1|xτ,(σ2\nt+1−σ2τ)I)\n=1\n(2π(σ2\nt−σ2τ))n/2exp(∥xt−xτ∥2\n−2(σ2\nt−σ2τ))1\n(2π(σ2\nt+1−σ2\nt))n/2exp(∥xt+1−xt∥2\n−2(σ2\nt+1−σ2\nt))\n1\n(2π(σ2\nt+1−σ2τ))n/2exp(∥xt+1−xτ∥2\n−2(σ2\nt+1−σ2τ)).\nSince the likelihood distribution q(xt+1|xt)and the prior distribution q(xt|xτ)are both Gaussian distribution, the posterior\nq(xt|xt+1,xτ)is also a Gaussian distribution. Therefore, we only need to derive the expectation and the covariance matrix.\nIn the following, instead of derive the q(xt|xt+1,xτ)using equations, we use some trick to simplify the derivation:\nq(xt|xt+1,xτ)∝exp(∥xt−xτ∥2\n−2(σ2\nt−σ2τ)+∥xt+1−xt∥2\n−2(σ2\nt+1−σ2\nt)−∥xt+1−xτ∥2\n−2(σ2\nt+1−σ2τ))\n= exp( −1\n2\u0014∥xt−xτ∥2\n(σ2\nt−σ2τ)+∥xt+1−xt∥2\n(σ2\nt+1−σ2\nt)−∥xt+1−xτ∥2\n(σ2\nt+1−σ2τ)\u0015\n)\n= exp( −1\n2\u0014∥xt∥2−2xT\ntxτ\n(σ2\nt−σ2τ)+∥xt∥2−2xT\nt+1xt\n(σ2\nt+1−σ2\nt)+C(xt+1,xτ)\u0015\n)\n= exp( −1\n2\u0014∥xt∥2\n(σ2\nt−σ2τ)−2xT\ntxτ\n(σ2\nt−σ2τ)+∥xt∥2\n(σ2\nt+1−σ2\nt)−2xT\nt+1xt\n(σ2\nt+1−σ2\nt)+C(xt+1,xτ)\u0015\n)\n= exp( −1\n2\u0014\n(1\n(σ2\nt−σ2τ)+1\n(σ2\nt+1−σ2\nt))∥xt∥2−2(xT\nτ\n(σ2\nt−σ2τ)+xT\nt+1\n(σ2\nt+1−σ2\nt))xt+C(xt+1,xτ)\u0015\n)\n= exp( −1\n2\u0014(σ2\nt−σ2\nτ) + (σ2\nt+1−σ2\nt)\n(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)∥xt∥2−2(xT\nτ\n(σ2\nt−σ2τ)+xT\nt+1\n(σ2\nt+1−σ2\nt))xt+C(xt+1,xτ)\u0015\n)\n= exp( −1\n2\u0014σ2\nt+1−σ2\nτ\n(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)∥xt∥2−2(xT\nτ\n(σ2\nt−σ2τ)+xT\nt+1\n(σ2\nt+1−σ2\nt))xt+C(xt+1,xτ)\u0015\n)\n= exp( −1\n2\nσ2\nt+1−σ2\nτ\n(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)\n∥xt∥2−2xT\nτ\n(σ2\nt−σ2τ)+xT\nt+1\n(σ2\nt+1−σ2\nt)\nσ2\nt+1−σ2τ\n(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)xt\n+C(xt+1,xτ)\n)\n= exp( −1\n2\u0014σ2\nt+1−σ2\nτ\n(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)\u0012\n∥xt∥2−2(σ2\nt+1−σ2\nt)xT\nτ+ (σ2\nt−σ2\nτ)xT\nt+1\nσ2\nt+1−σ2τxt\u0013\n+C(xt+1,xτ)\u0015\n)\n= exp( −1\n2(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)\nσ2\nt+1−σ2τ\"\u0012\nxt−(σ2\nt+1−σ2\nt)xτ+ (σ2\nt−σ2\nτ)xt+1\nσ2\nt+1−σ2τ\u00132\n+C(xt+1,xτ)#\n)\n∝ N(xt;(σ2\nt+1−σ2\nt)xτ+ (σ2\nt−σ2\nτ)xt+1\nσ2\nt+1−σ2τ,(σ2\nt−σ2\nτ)(σ2\nt+1−σ2\nt)\nσ2\nt+1−σ2τI).\nThe KL divergence between the posterior and predicted reverse distribution is\nDKL(q(xt|xt+1,xτ)∥pθ(xt|xt+1))\n=DKL(N(xt;(σ2\nt+1−σ2\nt)xτ+ (σ2\nt−σ2\nτ)xt+1\nσ2\nt+1−σ2τ,(σ2\nt−σ2\nτ)(σ2\nt+1−σ2\nt)\nσ2\nt+1−σ2τI)∥N(xt;(σ2\nt+1−σ2\nt)h(xt+1, σt+1) +σ2\ntxt+1\nσ2\nt+1,˜σtI))\n=1\n2(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)\nσ2\nt+1−σ2τ∥E[q(xt|xt+1,xτ)]−E[p(xt|xt+1)]∥2+C2\n=σ2\nt+1−σ2\nτ\n2(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)∥E[q(xt|xt+1,xτ)]−E[p(xt|xt+1)]∥2+C2.\n17Certified Robustness via a Single Diffusion Model\nWhen τ= 0, the result degrade to:\nq(xt|xt+1,x0) =N(xt;(σ2\nt+1−σ2\nt)x0+σ2\ntxt+1\nσ2\nt+1,σ2\nt(σ2\nt+1−σ2\nt)\nσ2\nt+1I).\nWhen dσt:=σt+1−σt→0, the KL divergence between posterior and model prediction is simplified by:\nlim\ndσt→0DKL(q(xt|xt+1,x0)∥pθ(xt|xt+1))\n= lim\ndσt→0DKL(N(xt;(σ2\nt+1−σ2\nt)x0+σ2\ntxt+1\nσ2\nt+1,σ2\nt(σ2\nt+1−σ2\nt)\nσ2\nt+1I)∥N(xt;(σ2\nt+1−σ2\nt)h(xt+1, σt+1) +σ2\ntxt+1\nσ2\nt+1,˜σt))\n= lim\ndσt→01\n2σ2\nt(σ2\nt+1−σ2\nt)\nσ2\nt+1∥(σ2\nt+1−σ2\nt)x0\nσ2\nt+1−(σ2\nt+1−σ2\nt)h(xt+1, σt+1)\nσ2\nt+1∥2\n= lim\ndσt→0(σ2\nt+1−σ2\nt)2\n2σ2\nt(σ2\nt+1−σ2\nt)\nσ2\nt+1σ4\nt+1∥x0−h(xt+1, σt+1)∥2\n= lim\ndσt→0(σ2\nt+1−σ2\nt)\n2σ2\nt\nσ2\nt+1σ4\nt+1∥x0−h(xt+1, σt+1)∥2\n= lim\ndσt→0(σ2\nt+1−σ2\nt)\n2σ4\nt+1∥x0−h(xt+1, σt+1)∥2\n=dσt\nσ3\nt+1∥x0−h(xt+1, σt+1)∥2\n=w(i)∥x0−h(xt+1, σt+1)∥2,\nwhich is consistent to the results in Kingma et al. (2021) and Karras et al. (2022).\nA.6. The Weight in EPNDC\nIf we interpret the shift in weight fromσt+1−σt\nσt+1toσ2\nt+σ2\nd\nσ2\ntσ2\nd1√\n2πkσexp(−∥logσt−kµ∥2\n2k2σ)as a reweighting of\nDKL(q(xt|xt+1,x0)∥pθ(xt|xt+1))byˆwt\nwt, a similar methodology can be applied to derive the weight for\nDKL(q(xt|xt+1,xτ)∥pθ(xt|xt+1)):\nlim\ndσt→0ˆwt\nwtDKL(q(xt|xt+1,xτ)∥pθ(xt|xt+1))\n= lim\ndσt→0ˆwt\nwt1\n2(σ2\nt−σ2τ)(σ2\nt+1−σ2\nt)\nσ2\nt+1−σ2τ∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n2\n= lim\ndσt→0ˆwt\ndσt\nσ3\nt+11\n2(σ2\nt−σ2τ)(2σtdσt)\nσ2\nt+1−σ2τ∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n2\n= lim\ndσt→0ˆwt\n(dσt)2\nσ2\nt+11\n4(σ2\nt−σ2τ)\nσ2\nt+1−σ2τ∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n2\n= lim\ndσt→0ˆwtσ2\nt+1(σ2\nt+1−σ2\nτ)\n4(σ2\nt−σ2τ)(dσt)2∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n2\n= lim\ndσt→0σ2\nt+σ2\nd\nσ2\ntσ2\nd1√\n2πkσexp(−∥logσt−kµ∥2\n2k2σ)σ2\nt+1(σ2\nt+1−σ2\nτ)\n4(σ2\nt−σ2τ)(dσt)2∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n2\n= lim\ndσt→0σ2\nt+σ2\nd\nσ2\nd1√\n2πkσexp(−∥logσt−kµ∥2\n2k2σ)σ2\nt+1−σ2\nτ\n4(σ2\nt−σ2τ)(σt+1−σt)2∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n2\n18Certified Robustness via a Single Diffusion Model\nTable 3. The accuracy of EPNDC using the EDM checkpoint (Karras et al., 2022) on CIFAR-10 test set with various weight when\nσ= 0.25. Result are tested on the same subset with 512 images as we clarified in Sec. 4.1.\nWeight Name Weight Accuracy(%)\nNormalized Weight1\n∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n281.6\nDerived Weight wt 67.5\nTruncated Derived Weight wt·I{σt>1} 82.8\nLinear WeightwT−w0\nσt−σ0+w0 43.8\nTruncated Linear Weight (wT−w0\nσt−σ0+w0)·I{σt>0.5} 77.0\nWe compare different weights in EPNDC. As demonstrated in Table 3, these weights are not as effective as the derived weight.\nInterestingly, simply setting the weight to zero when the standard deviation falls below a certain threshold significantly\nboosts the performance of the derived weight. This suggests that the latter part of the derived weight might be close to the\noptimal weight. We observe a similar phenomenon with the final weight we use, suggesting that the current weight might\nnot be optimal. We defer the investigation into how to find the optimal weight, what constitutes this optimal weight, and\nwhy it is considered optimal to future work.\nWe also attempt to learn an optimal weight for our EPNDC. However, the learning process is difficult due to the high variance\nof∥E[q(xt|xt+1,xτ)]−E[pθ(xt|xt+1)]∥2\n2. Attempts to simplify the learning process through cubic spline interpolation for\nreducing the number of learned parameters are also time-consuming. Our work primarily focuses on robust classification of\ninput data using a single off-the-shelf diffusion model. Therefore, we leave the exploration of this aspect for future research.\nA.7. The ELBO of logpτ(xτ)in APNDC\nIn this section, we show that APNDC is actually use the expectation of the ELBOs calculated from samples that are first\ndenoised from the input and then have noise added back. In other word, APNDC approximate logp(ˆxτ)by the lower bound\nofEp(x0|ˆxτ)Eq(xτ|x0)[logpτ(xτ)].\nWe first derive the lower bound for Eq(xt|x0)[logpτ(xτ)]. The derivation of this lower bound is just the discrete case of the\nproof in Kingma & Gao (2023). We include it here only for completeness.\nEq(xτ|x0)[logpτ(xτ)]\n=Z\nq(xτ|x0)logq(xτ|x0) logpτ(xτ)\nlogq(xτ|x0)\n=Eq(xτ|x0)[logq(xτ|x0)]−DKL(q(xτ|x0)∥pτ(xτ))\n≥Eq(xτ|x0)[logq(xτ|x0)]−DKL(q(xτ:T|x0)∥p(xτ:T))\n=Eq(xτ|x0)[logq(xτ|x0)]−T−1X\nt=τ[DKL(q(xt:T|x0)∥p(xt:T))−DKL(q(xt+1:T|x0)∥p(xt+1:T))] + DKL(q(xT|x0)∥p(xT))\n=C4−T−1X\nt=τ[DKL(q(xt:T|x0)∥p(xt:T))−DKL(q(xt+1:T|x0)∥p(xt+1:T))]\n=C4−T−1X\nt=τDKL(q(xt|xt+1,x0)∥p(xt|xt+1))\n=C4−T−1X\nt=τ+1wτExt∼q(xt|x0)[∥hθ(xt, σt)−x∥2\n2].\nGiven a input image xτ, we use the ELBOs of Eq(ˆ xτ|x0=hθ(xτ,στ))[logpτ(ˆ xτ)], rather than its own ELBO, to approximate\n19Certified Robustness via a Single Diffusion Model\nAlgorithm 4 Linear to EDM\nRequire: A pre-trained EDM hθ, a noisy input image xt,\nnoise level t, linear schedule {αi}T\ni=1and{σi}T\ni=1.\n1:Calculate the denoised image x0using hθ:\nx0=hθ\u0010\nxt\nαt,σt\nαt\u0011\n;\n2:ifperforming x0-prediction then\n3: Return: x0.\n4:end if\n5:Calculate the noise component ϵ:ϵ=xt−αtx0\nσt;\n6:ifperforming ϵ-prediction then\n7: Return: ϵ.\n8:end ifAlgorithm 5 EDM to Linear\nRequire: A pre-trained predictor hθ, a noisy input image\nxt, noise level σ, linear schedule {αi}T\ni=1and{σi}T\ni=1.\n1:Calculate t= arg min t|σt\nαt−σ|;\n2:ifperforming ϵ-prediction then\n3: Predict ϵ=ϵ(αtxt, t)\n4: Calculate x0=xt−σ·ϵ\n5:end if\n6:ifperforming x0-prediction then\n7: Predict x0=ϵ(αtxt, t)\n8:end if\n9:Return: x0.\nits log likelihood:\nlogp(xτ)≈Eq(ˆ xτ|x0=hθ(xτ,στ))[logpτ(ˆ xτ)]≥C4−T−1X\nt=τ+1wtExt∼q(xt|x0)[∥hθ(xt, σt)−x∥2\n2].\nHence, APNDC is actually use the expectation of the ELBOs calculated from samples that are first denoised from the input\nand then have noise added back.\nA.8. Converting Other Diffusion Models into Our Definition.\nIn this paper, we introduce a new definition for diffusion models, specifically as the discrete version of Karras et al. (2022).\nThis definition encompasses various diffusions, including VE-SDE, VP-SDE, and methods like x-prediction, v-prediction,\nand epsilon-prediction, transforming their differences into the difference of parametrization in hθ(·,·), makes theoretical\nanalysis extremely convenient. This operation also decouple the training of diffusion models and the sampling of diffusion\nmodels, i.e., any diffusion model could use any sampling algorithm. To better demonstrate this transformation, we present\nthe pseudocodes.\nLinearly adding noise diffusion models. These kind of diffusion models define the forward process as a linear interpolation\nbetween the clean image and Gaussian noise, i.e.,xt=αtx0+σtϵ.It could be transformed as:\nxtin EDMz}|{xt\nαt=x0+σtin EDMz}|{σt\nαtϵ.\nHence, we could directly passxt\nαtandσt\nαtto an EDM model to get the predicted x0, as shown in Algorithm 5.\nDDPM. DDPM define a sequence {βt}T\nt=0andxt=qQt\ni=0(1−βi)x0+q\n1−Qt\ni=0(1−βi)ϵ, which could be seen as\na special case of Linear diffusion models.\nVP-SDE. VP-SDE is the continuous case of DDPM, which define a stochastic differential equation (SDE) as\ndXt=−1\n2β(t)Xtdt+p\nβ(t)dWt, t∈[0,1],\nwhere β(t) =βt·T·T. Consequently, we could also use an EDM model to solve the reverse VP-SDE by predicting\nx0=hθ(xt √\nexp (−Rt\n0β(s)ds),r\n1−exp (−Rt\n0β(s)ds)\nexp (−Rt\n0β(s)ds)).\nVE-SDE. The forward process of VE-SDE is defined as\ndXt=r\ndσ(t)2\ndtdWt.\nEDM is a special case of VE-SDE, where σ(t) =t. For a xtin VE-SDE, the variance of the noise is σ(t). Hence, we could\ndirectly use hθ(xt, σ(t))to get the predicted x0.\n20Certified Robustness via a Single Diffusion Model\nB. Experimental Details\nB.1. Certified Robustness Details\nWe adhere to the certified robustness pipeline established by Cohen et al. (2019), yet our method potentially offers a tighter\ncertified bound, as demonstrated in Appendix A.3. To prevent confusion regarding our hyper-parameters and those in Cohen\net al. (2019), we clarify these hyper-parameters as follows:\nϵ=maximum allowed ℓ2perturbation of the input\nN=number of samples used in binomial test to estimate the lower bound pA\nσ=std. of Gaussian noise data augmentation during training and certification\nT=number of diffusion timesteps\nT′=number of diffusion timesteps we selected to calculate/(estimate) evidence lower bounds\nTo ensure a fair comparison, we follow previous work and do not estimate pB, instead directly setting pB= 1−pA. This\napproach results in a considerably lower certified robustness, especially for ImageNet. Therefore, it is possible that the\nactual certified robustness of diffusion classifiers might be significantly higher than the results we present.\nThere are multiple ways to reduce the number of timesteps selected to estimate the evidence lower bounds (e.g., uniformly,\nselecting the first T′timesteps). Chen et al. (2023a) demonstrate that these strategies achieve similar results. In this work,\nwe just adopt the simplest strategy, i.e., uniformly select T′timesteps.\nB.2. ImageNet Baselines\nWe compare our methods with training-based methods as outlined in Cohen et al. (2019), Salman et al. (2019), Jeong &\nShin (2020), Zhai et al. (2019), and with diffusion-based purification methods in Carlini et al. (2023b), Xiao et al. (2023).\nSince none of these studies provide code for ImageNet64x64, we retrain ResNet-50 using these methods and certify it via\nrandomized smoothing. For training-based methods, we utilize the implementation by Jeong & Shin (2020)1with their\ndefault hyper-parameters for ImageNet. For diffusion-based purification methods, we first resize the input images to 64x64,\nthen resize them back to 256x256, and feed them into the subsequent classifiers. This process ensures a fair comparison\nwith our method.\nB.3. Ablation Studies of Diffusion Models\nIn the ImageNet dataset, we employ the same diffusion models as our baseline studies, Carlini et al. (2023b) and Xiao\net al. (2023). For the CIFAR-10 dataset, however, we opt for a 55M diffusion model from Karras et al. (2022), as its\nparameterization aligns more closely with our definition of diffusion models. In contrast, Carlini et al. (2023b) utilizes a\n50M diffusion model from Nichol & Dhariwal (2021). When we replicate the study by Carlini et al. (2023b) using the\nmodel from Karras et al. (2022) and a WRN-70-2 (Zagoruyko & Komodakis, 2016), we achieve certified robustness of\n76.56%,59.76%, and 41.01% for radii 0.25,0.5, and 0.75, respectively, which is nearly identical to their results shown in\nTable 1. This finding suggests that the choice of diffusion models does not significantly impact certified robustness.\nWe choose the discrete version of the diffusion model from Karras et al. (2022) as our definition because it is the simplest\nversion for deriving the ELBOs in this paper. For more details, please see Appendix A.8.\nC. Discussions\nC.1. ELBO, Likelihood, Classifier and Certified Robustness\nAs demonstrated in Figure 1 and Tables 4 and 5, the basic idea of all these diffusion classifiers is to approximate the log\nlikelihood by ELBO and calculate the classification probability via Bayes’ theorem. All these classifiers possess non-trivial\nrobustness, but certified lower bounds vary in tightness.\nThe diffusion classifier is intuitively considered the most robust, as it can process not only clean images but also those\ncorrupted by Gaussian noise. This means it can achieve high clean accuracy by leveraging less noisy samples, while also\n1https://github.com/jh-jeong/smoothing-consistency\n21Certified Robustness via a Single Diffusion Model\nName ELBO Diffusion Classifier\nDClogp(x0|y)≥\n−PT\nt=1E\u0002\nwt∥hθ(xt, σt, y)−x0∥2\n2\u0003\n+CDC(x)y=exp(−PT\nt=1E[wt∥hθ(xt,σt,y)−x0∥2\n2])\nP\nˆyexp (−PT\nt=1E[wt∥hθ(xt,σt,ˆy)−x0∥2\n2])\nEPNDClogp(xt|y)≥ −PT\ni=twi\nE[∥E[q(xi|xi+1,xt)]−E[p(xi|xi+1, y)]∥2] +C2EPNDC (xt)y=exp(−PT\ni=twiE[∥E[q(xi|xi+1,xt)]−E[p(xi|xi+1,y)]∥2])P\nˆyexp (−PT\ni=twiE[∥E[q(xi|xi+1,xt)]−E[p(xi|xi+1,ˆy)]∥2])\nAPNDCEq(ˆxt|x0),x0=h(xt,σt)[logp(xt|y)]≥\n−PT\ni=tE\u0002\nwi∥hθ(xi, σi, y)−x0∥2\n2\u0003\n+C3APNDC (xt)y=exp(−PT\ni=tE[wi∥hθ(xi,σi,y)−hθ(xt,σt)∥2\n2])\nP\nˆyexp (−PT\ni=tE[wi∥hθ(xi,σi,ˆy)−hθ(xt,σt)∥2\n2])\nTable 4. An illustration of the relationship between different ELBOs and classifiers.\nName Smoothed Classifier Certified Radius\nDC gDC(x) =DC(x) R=√\n2T(pA−pB)\n(2/√\nD+√\n2/π)PT\ni=1wi/σi\nEPNDC gEPNDC (x)y=Pϵ∼N(0,I)(arg max ˆyEPNDC (x0+σ·ϵ)ˆy=y) R=σ\n2\u0000\nΦ−1(pA)−Φ−1(pB)\u0001\nAPNDC gAPNDC (x)y=Pϵ∼N(0,I)(arg max ˆyAPNDC (x0+σ·ϵ)ˆy=y) R=σ\n2\u0000\nΦ−1(pA)−Φ−1(pB)\u0001\nTable 5. An illustration of the relationship between different ELBOs, likelihood, classifiers and certified robustness.\nenhancing robustness through more noisy samples where adversarial perturbations are significantly masked by Gaussian\nnoise. However, its certified robustness is not as tight as desired. As explained in Appendix A.2, this is because we assume\nthat the maximum Lipschitz condition holds across the entire space, which is a relatively broad assumption, resulting in a\nless stringent certified robustness.\nEPNDC and APNDC utilize the Evidence Lower Bound (ELBO) of corrupted data logp(xτ). This implies that the least\nnoisy examples they can process are at the noise level corresponding to τ. Asτincreases, the upper bound of certified\nrobustness also rises, but this simultaneously diminishes the classifiers’ ability to accurately categorize clean data. As\nhighlighted in Chen et al. (2023a), this mechanism proves less effective, particularly in datasets with a large number of\nclasses but low resolution. In such cases, the addition of even a small variance of Gaussian noise can render the entire image\nunclassifiable.\nGenerally, the Diffusion Classifier is intuitively more robust than both EPNDC and APNDC. However, obtaining a tight\ntheoretical certified lower bound for the Diffusion Classifier proves more challenging compared to EPNDC and APNDC.\nTherefore, we contend that in practical applications, the Diffusion Classifier is preferable to both EPNDC and APNDC.\nLooking forward, we aim to derive a tighter certified lower bound for the Diffusion Classifier in future research.\nC.2. The Loss Weight in Diffusion Classifiers\nTo enhance generative performance, most researchers opt not to use the derived weight wtfor the training loss\n∥E[q(xt|xt+1,x0)]−E[p(xt|xt+1)]∥2. Instead, they employ a redesigned weight ˆwt. For example, Ho et al. (2020)\ndefine ˆwt=1\nσt. Another approach involves sampling ifrom a specifically designed distribution p(t), rather than a uniform\ndistribution. This approach can be interpreted as adjusting the loss weight to ˆwt=p(t), without altering the distribution of i:\nEx,t∼p(t)Ext[∥h(xt, σt)−x∥2\n2]\n=ExTX\nt=0p(t)Ext[∥h(xt, σt)−x∥2\n2]\n=Ex,tExt[p(t)∥h(xτ, σt)−x∥2\n2].\nFor example, Karras et al. (2022) employσ2\nt+σ2\nd\nσ2\ntσ2\ndas the loss weight but modify p(t)to1√\n2πkσexp (−∥logσt−kµ∥2\n2k2σ), where\nσd, kσ, kµare hyper-parameters. Thus, it is equivalent to setting ˆwt=σ2\nt+σ2\nd\nσ2\ntσ2\nd1√\n2πkσexp (−∥logσt−kµ∥2\n2k2σ).\nWe discover that maintaining consistency in the loss weight used in diffusion classifiers and during training is crucial. Take\nEDM as an example. As demonstrated in Table 6, if we fail to maintain the consistency of the loss weight between training\nand testing, a significant performance drop occurs. The closer the weight during testing is to the weight used during training,\n22Certified Robustness via a Single Diffusion Model\nTable 6. The accuracy of diffusion classifier (Chen et al., 2023a) using the EDM checkpoint (Karras et al., 2022) on CIFAR-10 test set\nwith various weight. Result are tested on the same subset with 512 images as in Chen et al. (2023a).\nWeight Name ˆwt Accuracy (%)\nEDMσ2\nt+σ2\nd\nσ2\ntσ2\nd1√\n2πkσexp(−∥logσt−kµ∥2\n2k2σ) 94.92\nUniform 1 85.76\nDDPM1\nσt90.23\nEDM-Wσ2\nt+σ2\nd\nσ2\ntσ2\nd88.67\nEDM- p(t)1√\n2πkσexp(−∥logσt−kµ∥2\n2k2σ) 93.75\nELBOσt+1−σt\nσ3\nt44.53\nthe better the performance.\nSurprisingly, the derived weight (ELBO) yields the poorest performance, as indicated in Table 6. This issue also occurs\nwhen calculating DKL(q(xt|xt+1,xt)||p(xt|xt+1)). The performance of our derived weight is significantly inferior to\nboth the uniform weight and the DDPM weight. For DKL(q(xt|xt+1,x0)||p(xt|xt+1)), substituting the derived weight\nσt+1−σt\nσ3\ntwith the training weightσ2\nt+σ2\nd\nσ2\ntσ2\nd1√\n2πkσexp(−∥logσt−kµ∥2\n2k2σ)is feasible. However, this substitution becomes prob-\nlematic for DKL(q(xt|xt+1,xτ)||p(xt|xt+1)), as there is noσt+1−σt\nσ3\ntterm, as outlined in Eq. (12). We propose two\nalternative strategies: one interprets the shift from wttoˆwtas reweighting the KL divergence byˆwt\nwt, requiring only the\nmultiplication ofˆwt\nwtto our derived weight. The other strategy involves using the equation ˆwt=wtto reduce the number\nof parameters. While these two interpretations yield identical results for DKL(q(xt|xt+1,x0)||p(xt|xt+1)), they differ\nforDKL(q(xt|xt+1,xτ)||p(xt|xt+1)). Therefore, directly deriving an optimal weight appears infeasible. For detailed\ninformation, see Appendix A.6.\nC.3. Certified Robustness of Chen et al. (2023a).\nIn this section, our goal is to establish a certified lower bound of Chen et al. (2023a). Specifically, under the conditions\nwhere pA= 1andpB= 0, the maximum certified radius is determined to be approximately 0.39. This implies that the\ncertified radius we derive could not surpass 0.39. In practical applications, our method achieves an average certified radius\nof 0.0002.\nNevertheless, we can significantly enhance the certified radius by adjusting wt. For example, simply set wt≡1, we can get\nan average certified radius of 0.009, 34 times larger than previous one. By reducing wtfor smaller tand increasing it for\nlarger t, such as zeroing the weight when σt≤0.5, we achieve an average radius of 0.156.\nIt is important to note that this finding does not imply that Chen et al. (2023a) lacks robustness in its weight-adjusted version.\nAs discussed in previous sections, the certified radius is merely a lower bound of the actual robust radius, and a higher lower\nbound does not necessarily equate to a higher actual robust radius. When wtis increased for larger t, obtaining a tighter\ncertified radius will be much easier, but the actual robust radius could intuitively decrease due to increased noise in the input\nimages.\nC.4. Time Complexity Reduction Techniques that Do Not Help\nAdvanced Integration. In this work, we select Ttimesteps uniformly. As we discuss earlier, there is a one-to-one\nmapping between σandtin our context. When we change the variable to σ, this can be seen as calculating the expectation\nusing the Euler integral. We also try using more advanced integration methods, like Gauss Quadrature, but only observe\nnuanced differences that there is no any difference in clean accuracy and certified robustness, indicating that the expectation\ncalculation is robust to small truncation errors. We do not attempt any non-parallel integrals, as they significantly reduce\nthroughput.\nSharing Noise Across Different Timesteps. In Sec. 3.4, we demonstrate that by using the same xifor all classes, we\nsignificantly reduce the variance of predictions. This allows us to use a fewer number of timesteps, thereby greatly reducing\ntime complexity. From another perspective, using the same xifor all classes is equivalent to applying the same noise to\n23Certified Robustness via a Single Diffusion Model\nAlgorithm 6 Discrete Progressive Class Selection\nRequire: A pre-trained diffusion model ϵθ, input image x, predefined number of classes K, class candidate trajectory Ccandand timestep\ncandidate trajectory Tcand.\n1:Initialize: entire timesteps ˜t=vec([1,2,···, T]), counter pointer c= 0and top-k cache Ktop={1,2,···, K}.\n2:for(c1,c2) in (Ccand,Tcand)do\n3:˜tselect=˜t[c2−c:c2].\n4: Calculate logity=P\nt∈˜tselect[wt∥hθ(xt, t, y)−ϵ∥2\n2]for all y∈ K topsimultaneously using hθ.\n5: Merge logitycalculated in the previous step into logitycurrently calculated.\n6: Sort {logity}y∈K top.\n7: Set the smallest c1class as new Ktopfrom the sorted {logity}y∈K top.\n8: Update counter pointer c=c2.\n9:end for\n10:Return: Ktop.\nsamples at the same timestep. This raises the question: ”If we share the noise across all samples, could we further reduce\ntime complexity?” However, this approach proves ineffective. Analyzing the difference between the logits of two classes\nreveals that sharing noise does not reduce the variance of this difference. Consequently, we opt not to share noise across\ndifferent timesteps but only within the same timestep for different classes.\nDiscrete progressive class selection algorithm. We design a normalized discrete critical class selection algorithm for\naccelerating diffusion classifiers. Typically, the time complexity of the vanilla diffusion classifier can be defined as O(KT),\nwhere Kdenotes the count of classes and Tdenotes the number of timesteps. Sharpening Tis tractable, but overdoing that\npractice can have an extremely negative impact on final performance. Another way to speed up the computation is to actively\ndiscard some unimportant classes when estimating the conditional likelihood via conditional ELBO. Indeed, these two\nsub-approaches can be merged in parallel in a single algorithm ( i.e., our proposed class selection algorithm). Additionally, to\nachieve an in-depth analysis of acceleration in diffusion classifiers, the time complexity of our proposed discrete acceleration\nalgorithm can be determined by a predefined manual class candidate trajectory Ccandas well as a predefined manual timestep\ncandidate trajectory Tcand.\nThe procedure of the discrete progressive class selection algorithm can be described in Algorithm 6. The specific mechanism\nof Algorithm 6 can be described from a simple example: the predefined trajectory Ccandis set as [400, 80, 40, 1] and the\npredefined trajectory Tcandis set as [2, 5, 25, 50] simultaneously. Assume we conduct this accelerated diffusion classifier\nwithK=1000 in the specific ImageNet (Xiao et al., 2023). First, we estimate the diffusion loss, which is equivalent to\nconditional likelihood, for 1000 classes by 2 time points and then sort the losses of the different classes to obtain the smallest\n400 classes. Subsequently, we estimate the diffusion loss for 400 classes by 3 ( w.r.t. 5-2) time points and then sort the losses\nfor different classes to obtain the smallest 80 classes. Ultimately, the procedure concludes with the selection of 40 out of 80\nclasses with 20 time points ( w.r.t. 25-5), followed by choosing 1 class from 40 classes with 25 ( w.r.t. 50-25) time points.\nThe ablation outcomes presented in Table 7 illustrate that suitable design of predefined manual timestep candidate trajectories\nCcandandTcandare crucial in the final performance of diffusion classifiers. A salient observation in the initial definition of\nCcandandTcandis thatTcandcan initially be small, whereas Ccandmust start significantly larger to achieve high accuracy while\nensuring low time complexity. Thus, the timestep trajectory [2, 5, 15, 50] and the classes trajectory [400, 80, 40, 1] are in\naccordance with the above conditions and achieve the best performance in various predefined trajectories.\nAccelerated Diffusion Models. We observe that consistency models (Song et al., 2023) and CUD (Shao et al., 2023)\ndetermine the class of the generated object at large timesteps, while they primarily refine the images at lower timesteps. In\nother words, at smaller timesteps, the similarity between predictions of different classes is so high that classification becomes\nchallenging. Conversely, at larger timesteps, the images become excessively noisy, leading to less accurate predictions.\nConsequently, constructing generative classifiers from consistency models and CUD appears to be a difficult task.\n24C.4 Time Complexity Reduction Techniques that Do Not Help 25\nTable 7. The experimental results of discrete progressive class selection algorithm.\nTimestep Trajectory Classes Trajectory Time Complexity Accuracy\n[2,5,25,50] [400 ,80,40,1] 5800 58.00%\n[2,5,17,50] [500 ,136,37,1] 6353 58.20%\n[2,5,17,50] [500 ,100,20,1] 5360 57.80%\n[2,4,12,25] [500 ,136,37,1] 4569 54.10%\n[2,4,12,25] [500 ,100,20,1] 4060 54.69%\n[2,7,35,50] [297 ,80,15,1] 5900 56.84%\n[2,7,35,50] [297 ,100,20,1] 6535 57.42%\n[2,6,20,50] [350 ,100,20,1] 5400 57.81%\n[2,6,25,50] [350 ,100,20,1] 5800 56.10%\n[2,7,35,50] [297 ,80,40,1] 6275 56.84%\n[2,5,25,50] [297 ,80,40,1] 5491 57.42%\n[2,5,15,50] [500 ,80,40,1] 5700 59.18%\n[2,4,8,50] [500 ,200,100,1] 8000 56.05%\n[2,4,8,50] [400 ,160,80,1] 6800 56.25%\n[2,4,8,50] [400 ,120,60,1] 5800 56.05%\n[2,4,8,50] [400 ,80,40,1] 4800 55.47%"    },    {        "title": "2402.02324v1.Some_factorization_results_for_bivariate_polynomials.pdf",        "content": "arXiv:2402.02324v1  [math.NT]  4 Feb 2024SOME FACTORIZATION RESULTS FOR BIVARIATE POLYNOMIALS\nNICOLAE CIPRIAN BONCIOCAT1, RISHU GARG2, AND JITENDER SINGH3,†\nAbstract. We provide upper bounds on the total number of irreducible factor s, and in\nparticular irreducibility criteria for some classes of bivariate polynom ialsf(x,y) over an\narbitrary field K. Our results rely on information on the degrees of the coefficients o ff,\nand on information on the factorization of the constant term and o f the leading coefficient\noff, viewed as a polynomial in ywith coefficients in K[x]. In particular, we provide a\ngeneralization of the bivariate version of Perron’s irreducibility crite rion, and similar results\nfor polynomials in an arbitrary number of indeterminates. The proof s use non-Archimedean\nabsolute values, that are suitable for finding information on the loca tion of the roots of fin\nan algebraic closure of K(x).\n1.Introduction\nMany of the classical irreducibility results for univariate integer poly nomialsf(x) rely\non information on the canonical decomposition of their coefficients, such as the famous\nirreducibility criteria of Sch¨ onemann [ 28], Eisenstein [ 14], and Dumas [ 13], or on information\non the canonical decomposition of the values that they assume at s ome suitable integer\narguments, such as the irreducibility criteria of St¨ ackel [ 33], Dorwart [ 12], Ore [23], Weisner\n[35], andCohn[ 26]. Other results, such asP´ olya’s irreducibility criterion[ 25] relyonlyonthe\nmagnitude of f(m1),...,f(mn), with disregard to their canonical decomposition, for distinct\nintegersm1,...,m n, or on the comparative size of the coefficients, as in the irreducibility\ncriterion of Perron [ 24]. For a quick review of some classical or more recent such results th e\nreader is referred to [ 20] and [21], for instance. In more recent times, many fundamental\nresults and new ideas appeared on this topic, for instance in the wor ks of Brillhart, Filaseta\nand Odlyzko [ 7], Cole, Dunn, Filaseta and Gross [ 11], [16], [17], Murty [ 27], Girstmair [ 19]\n(seealso[ 32]), Guersenzvaig [ 20], FilasetaandLuckner [ 18], andWeintraub[ 34], tonamejust\na few. Other recent criteria are inspired by some results on Hilbert I rreducibility Theorem,\nand use some properties of the resultant of two polynomials, combin ed with information\non the magnitude of their coefficients. Here we refer the reader to some elegant methods\ndeveloped by Cavachi, Vˆ ajˆ aitu and Zaharescu in [ 8], [9] and [10], which are useful in the\nstudyoflinearcombinationsofrelativelyprimepolynomials, andwhichh avebeenadaptedby\nvariousauthors forthe study of compositions and multiplicative con volutions of polynomials.\nFinding information on the location of the roots is one of the main ingre dients in the\nproofs of many of these results. Most of these criteria admit natu ral generalizations to the\nmultivariate case. Given a bivariate polynomial f(x,y)over an arbitraryfield K, if we regard\nfas a polynomial in ywith coefficients in K[x], a natural way to study its factorization is to\n†\n2020MSC: Primary 11R09; 12J25; 12E05; 11C08\nKeywords : non-Archimedean absolute value; Perron’s irreducibility criterion; multivariate polynomial.\n1try to adapt the methods available in the univariate case. One of the most suitable ways to\nstudy thelocationof the rootsof fin analgebraic closure of K(x) is to use non-Archimedean\nabsolute values, and to employ the ideas and the techniques in the un ivariate case, where\nthe usual absolute value is used. Such an approach was used for ins tance in [ 6], [5], [2] and\n[3] to obtain multivariate versions of the irreducibility criteria of Cohn a nd P´ olya, and to\nderive irreducibility results and bounds on the number of irreducible f actors counted with\nmultiplicities for some classes of multivariate polynomials.\nThe irreducibility criterion of Perron [ 24] for integer univariate polynomials was recently\ngeneralized in [ 29] and [30], and has the following natural generalization for bivariate poly-\nnomials over an arbitrary field.\nTheorem A (Perron [ 24]).LetKbe a field and f=a0(x)+···+an−1(x)yn−1+an(x)yn∈\nK[x,y]withn≥2,a0,...,a n−1∈K[x],an∈K, anda0an/ne}ationslash= 0. If\ndegan−1>max{dega0,dega1,...,degan−2},\nthenfis irreducible over K[x].\nFor a proof of Theorem Athat relies on the use of non-Archimedean absolute values, and\nits extension to an arbitrary number of indeterminates, we refer t he reader to [ 4].\nIn this paper, we will first prove several factorization results for some classes of bivariate\npolynomials f(x,y) =/summationtextai(x)yiover a given field K, that provide upper bounds for the\nnumber of irreducible factors of foverK[x], counted with their multiplicities. These bounds\ndepend only on the factorizationof a0andan, and hold for polynomials for which the degrees\nofa0,...,a nsatisfy certain inequalities. In particular, these results provide irr educibility\ncriteria for bivariate polynomials for which a0oranis irreducible over K. We will then\nprove some extensions of Theorem Ato the case when a coefficient other than an−1has\ndegree greater than the degrees of all the other coefficients. Se veral similar results will be\nfinally obtained for polynomials in an arbitrary number of indeterminat es. Our proofs will\nuse non-Archimedean absolute values, and some explanations will re ly on arguments coming\nfrom Newton Polygon theory, via Dumas’ Theorem [ 13]. We mention here that the bounds\non the number of irreducible factors that we will obtain are best pos sible, and examples in\nthis respect will be provided.\nThe paper is organized as follows. The main results are stated in Sect ion2, and their\nproofs are given in Section 3. In the last section of the paper we provide several examples\nof infinite families of irreducible polynomials, as well as families of polynom ials for which\nbounds on their total number of irreducible factors can be deduce d from our results.\n2.Main results\nOur first results provide upper bounds on the total number of fac tors for some classes of\nbivariate polynomials f(x,y) =a0(x)+a1(x)y+···+an(x)yn∈K[x,y], using information on\nthe degrees of aicombined with information on the factorization of a0andan. Even if they\nfollow by a similar argument, for the sake of symmetry we will also includ e in our statements\nthe results for the reciprocal of fwith respect to y. Such results are often overlooked, even\nif they usually provide useful information on the factorization of f, and as we shall later see,\nby simultaneously studying fand its reciprocal, one may obtain sharper results.\n2Theorem 1. LetKbe a field and f=a0(x)+a1(x)y+···+an(x)yn∈K[x,y], withn≥2,\na0,...,a n∈K[x],a0an/ne}ationslash= 0. Assume that fhas no nonconstant factor in K[x], and let ν0\nandνnbe the number of irreducible factors of a0(x)andan(x)inK[x], respectively, counted\nwith their multiplicities.\ni) Ifdega0>max{dega1,...,degan}, thenfis a product of at most ν0irreducible\npolynomials over K[x].\nii) Ifdegan>max{dega0,...,degan−1}, thenfis a product of at most νnirreducible\npolynomials over K[x].\nIn each one of the two cases in Theorem 1we may obtain a potentially sharper result,\nprovided some information on the factorization of both a0andanis available, as in the\nfollowing result.\nTheorem 2. LetKbe a field and f=a0(x)+a1(x)y+···+an(x)yn∈K[x,y], withn≥2,\na0,...,a n∈K[x],a0an/ne}ationslash= 0. Assume that fhas no nonconstant factor in K[x], and letν0and\nνnbe the number of irreducible factors of a0(x)andan(x)inK[x], respectively, counted with\ntheir multiplicities. Then fis a product of at most ν= min{ν0,νn}irreducible polynomials\noverK[x]in each one of the following two cases:\ni)a0is reducible over K,degan≥dega0−degq, whereq∈K[x]is an irreducible factor\nofa0of smallest degree, and dega0>max{dega1,...,degan};\nii)anis reducible over K,dega0≥degan−degq, whereq∈K[x]is an irreducible factor\nofanof smallest degree, and degan>max{dega0,...,degan−1}.\nWe mention that the upper bounds on the number of irreducible fact ors offin Theorem\n1and Theorem 2are best possible, in the sense that there exist examples of infinite f amilies\nof bivariate polynomials for which these bounds are attained. To see this, let for example p\nandqbe prime numbers, na positive integer, and let\nf1(x,y) =pq+(p+q)xn+x2n+(p+q+2xn)yn+y2n∈Q[x,y].\nOne may check that f1satisfies the hypothesis of Theorem 1i) witha0=pq+(p+q)xn+x2n,\nwhich is the product of two irreducible factors in Q[x], namely p+xnandq+xn, the first one\nbeing Eisensteinian with respect to p, and the second one with respect to q. By Theorem 1,\nwe conclude that our polynomial f1(x,y) is the product of at most two irreducible factors\noverQ[x]. On the other hand, we observe that actually f1decomposes as\nf1(x,y) = (p+xn+yn)(q+xn+yn),\nwhich is the product of two irreducible factors over Q[x], asp+xn+ynandq+xn+yn\nare Eisensteinian too, the first one with respect to the irreducible p olynomial p+xn, and\nthe second one with respect to the irreducible polynomial q+xn. For an example related to\nTheorem 1ii), one may for instance consider the polynomial\nf2(x,y) = 1+( p+q+2xn)yn+(pq+(p+q)xn+x2n)y2n∈Q[x,y],\nwhich is the reciprocal of f1with respect to y, and whose irreducible factors are precisely\nthe reciprocals with respect to yof the irreducible factors of f1.\nSimilarly, the bound in Theorem 2is best possible too, since it is attained by all the\npolynomials\nf3(x,y) = (r+x2)4+2(p+x3)(r+x2)2yn+(p+x3)2y2n∈Q[x,y],\n3wherepandrare prime numbers, and nis a positive integer. Here, a0= (r+x2)4with\nq(x) =r+x2, which is Eisensteinian with respect to r, anda2n= (p+x3)2, withp+x3\nEisensteinian with respect to p. Thusν0= 4,ν2n= 2, deg a2n= dega0−degq= 6, and\ndega0= 8>7 = max{degan,dega2n}. By Theorem 2i), we conclude that f3is a product\nof at most 2 = min {ν0,ν2n}irreducible factors over Q[x]. On the other hand, we actually\nhavef3(x,y) =h(x,y)2, withh(x,y) = (r+x2)2+ (p+x3)yn, which is irreducible over\nQ[x], being the reciprocal with respect to yof the polynomial ( p+x3)+(r+x2)2yn, which\nis Eisensteinian with respect to p+x3. For an example related to Theorem 2ii), one may\nobviously choose the reciprocal of f3with respect to y.\nThe following irreducibility criteria are immediate applications of Theore ms1and2.\nCorollary 3. LetKbe a field. Let f=a0(x)+a1(x)y+···+an(x)yn∈K[x,y], withn≥2,\na0,...,a n∈K[x],a0an/ne}ationslash= 0, be such that fhas no nonconstant factor in K[x]and\ndega0>max{dega1,...,degan}.\nIfa0is irreducible in K[x], or ifanis irreducible in K[x]anddegan≥dega0−degq, with\nq∈K[x]an irreducible factor of a0of smallest degree, then fis irreducible over K[x].\nCorollary 4. LetKbe a field. Let f=a0(x)+a1(x)y+···+an(x)yn∈K[x,y], withn≥2,\na0,...,a n∈K[x],a0an/ne}ationslash= 0, be such that fhas no nonconstant factor in K[x]and\ndegan>max{dega0,...,degan−1}.\nIfanis irreducible in K[x], or ifa0is irreducible in K[x]anddega0≥degan−degq, where\nq∈K[x]is an irreducible factor of anof smallest degree, then fis irreducible over K[x].\nWe mention here that Theorem 1and Theorem 2extend some of the results in [ 31] and [1]\nto the bivariate case and that the conclusion in Corollary 3on the irreducibility of f(x,y)\noverK[x] whena0is irreducible in K[x] also follows by Lemma 1 in [ 6].\nFor the case when a coefficient of fother than the leading one has degree greater than\nthe degrees of all the other coefficients, we have the following fact orization result, which\ngeneralizes the multivariate version of Perron’s irreducibility criterio n [24], [4].\nTheorem 5. LetKbe a field, and let f=a0(x) +a1(x)y+···+an(x)yn∈K[x,y]with\nn≥2,a0,...,a n−1∈K[x],an∈K,a0an/ne}ationslash= 0, be such that there exists an index jwith\n0≤j≤n−1for which\ndegaj>max\ni/negationslash=jdegai.\nThenfis a product of at most n−jirreducible polynomials over K[x]. In particular, if\nj=n−1, thenfis irreducible over K[x].\nOne may improve the bound on the number of irreducible factors in Th eorem5for the\ncase when a0too belongs to K.\nTheorem 6. LetKbe a field, and let f=a0(x) +a1(x)y+···+an(x)yn∈K[x,y]with\nn≥2,a1,...,a n−1∈K[x],a0,an∈K,a0an/ne}ationslash= 0, be such that there exists an index jwith\n1≤j≤n−1for which\ndegaj>max\ni/negationslash=jdegai.\nThenfis a product of at most min{j,n−j}irreducible polynomials over K[x]. In particular,\nifj= 1orj=n−1, thenfis irreducible over K[x].\n4The upper bound min {j,n−j}in the statement of Theorem 6is also best possible, in the\nsense that there exist polynomials for which this bound is attained. T o see this, consider the\npolynomial f4(x,y) = (1+xy+y2)j∈Q[x,y] withja positive integer. We have degyf4= 2j\nand degxf4=j, and by using the multinomial rule, if we write f4as a polynomial in ywith\ncoefficients in Q[x], sayf4=a0(x) +a1(x)y+···+a2j(x)y2j, we see that f4has a unique\ncoefficient of maximal degree, which is aj, and deg aj=j. Sincea0=a2j= 1, we deduce by\nTheorem 6thatf4is the product of at most jirreducible factors. Indeed, this is the case,\nas one may easily check that 1 + xy+y2is irreducible over Q[x]. A more general related\nexample with a polynomial of arbitrary degree nandja divisor of nwill be given in the\nlast section of the paper.\nInterestingly, Theorem 5extends to arbitrary polynomials, with no need to asking anto\nbe a unit in K[x], as in the following result.\nTheorem 7. LetKbe a field and f=a0(x)+a1(x)y+···+an(x)yn∈K[x,y]withn≥2,\na0,...,a n∈K[x],a0an/ne}ationslash= 0. Assume that fhas no nonconstant factor in K[x]. If there\nexists an index jwith0≤j≤n−1for which\ndegaj>max\ni/negationslash=j{degai+(j−i)degan},\nthenfis a product of at most n−jirreducible polynomials over K[x]. In particular, if\nj=n−1, thenfis irreducible over K[x].\nThe bound n−jon the number of irreducible factors in Theorem 7can be also improved\nto min{j,n−j}, provided we also consider the reciprocal of fwith respect to y, and impose\na stronger condition on the degree of aj, as follows.\nTheorem 8. LetKbe a field and f=a0(x)+a1(x)y+···+an(x)yn∈K[x,y]withn≥2,\na0,...,a n∈K[x],a0an/ne}ationslash= 0. Assume that fhas no nonconstant factor in K[x]. If there\nexists an index jwith1≤j≤n−1for which\ndegaj>max\ni<j{degai+(j−i)degan}and\ndegaj>max\ni>j{degai+(i−j)dega0},\nthenfis a product of at most min{j,n−j}irreducible polynomials over K[x]. In particular,\nifj= 1orj=n−1, thenfis irreducible over K[x].\nWe notice here that Theorem 6can be recovered from Theorem 8as the special case when\nboth the coefficients a0andanbelong to K, since in that case the two inequalities above\nsimply reduce to deg aj>maxi/negationslash=jdegai.\nThe results stated so far extend easily to polynomials in an arbitrary number of indeter-\nminates. For any positive integer r≥2 and any polynomial f∈K[x1,...,x r], let degrfbe\nthe degree of fviewed as a polynomial in the variable xrand coefficients in K[x1,...,x r−1].\nWe will only mention here two such results, each of which is obtained by induction on r. For\ninstance, as an immediate consequence of Theorem 5, we have the following factorization\nresult.\nTheorem 9. LetKbe a field and let s≥3be a positive integer. Let f=/summationtextn\ni=0aixi\ns∈\nK[x1,...,x s]withn≥2,a0,...,a n−1∈K[x1,...,x s−1],an∈K[x1,...,x s−2],a0an/ne}ationslash= 0, and\n5assume that there exists an index jwith0≤j≤n−1for which\ndegs−1aj>max\ni/negationslash=jdegs−1ai.\nThenfis a product of at most n−jirreducible polynomials over K[x1,...,x s−1]. In partic-\nular, ifj=n−1, thenfis irreducible over K[x1,...,x s−1].\nWe mention that Theorem 9generalizes Theorem C in [ 4], which is the irreducibility\ncriterion that corresponds to j=n−1 in the above result. Similarly, Theorem 7gives the\nfollowing result for multivariate polynomials.\nTheorem 10. LetKbe a field and let s≥3be a positive integer. Let f=/summationtextn\ni=0aixi\ns∈\nK[x1,...,x s]withn≥2,a0,...,a n∈K[x1,...,x s−1],a0an/ne}ationslash= 0. Assume that fhas no\nnonconstant factor in K[x1,...,x s−1]. If there exists an index jwith0≤j≤n−1for which\ndegs−1aj>max\ni/negationslash=j{degs−1ai+(j−i)degs−1an},\nthenfis a product of at most n−jirreducible polynomials over K[x1,...,x s−1]. In particular,\nifj=n−1, thenfis irreducible over K[x1,...,x s−1].\n3.Proofs of the main results\nOur factorization results for polynomials over K[x] will be proved in a non-Archimedean\nsetting, and by Gauss’s Lemma, they will imply similar factorization res ults over K(x).\nProof of Theorem 1.i) We will first recall a well-known non-Archimedean absolute value,\nwhich will be used in our proofs. Given a field K, letK(x) denote the field of fractions of\nthe polynomial ring K[x]. Letρ >1 be a real number, define deg(0) = −∞, and for all\na,b∈K[x] withb/ne}ationslash= 0, define\n/bardbla/bardblρ=ρdeg(a)and/bardbla/b/bardblρ=ρdeg(a)−deg(b)=/bardbla/bardblρ//bardblb/bardblρ.\nIn particular, we have /bardbla/bardblρ≥1 for all nonzero a∈K[x]. We note that /bardbl·/bardblρis a nonnegative\nfunction, and that for all a,b∈K[x] we have deg( a)≤deg(b) if and only if /bardbla/bardblρ≤ /bardblb/bardblρ.\nMoreover, for all a,b∈K(x) we have\n/bardblab/bardblρ=/bardbla/bardblρ/bardblb/bardblρand/bardbla+b/bardblρ≤max{/bardbla/bardblρ,/bardblb/bardblρ},\nso/bardbl·/bardblρis a non-Archimedean absolute value on K(x). We fix now an algebraic closure K(x)\nofK(x), which is unique up to isomorphism. We also fix an extension of this abs olute value\nfromK(x) toK(x) (see [15], for instance), which will also be denoted by /bardbl·/bardblρ.\nLet us assume that θ1,...,θ n∈K(x) are the roots of f. We will next prove that our\nassumption on the degree of a0forcesθ1,...,θ nto satisfy the inequalities /bardblθi/bardblρ>1 for each\ni. Indeed, the inequality deg a0>max{dega1,...,degan}reads/bardbla0/bardblρ>maxi≥1/bardblai/bardblρ, so\nif we assume on the contrary that /bardblθi/bardblρ≤1 for some i, then since f(θi) = 0, we have\na0(x) =−a1(x)θi−···−an(x)θn\ni, which further implies that\n/bardbla0/bardblρ=/bardbl−a1θi−···−anθn\ni/bardblρ\n≤max{/bardbla1θi/bardblρ,...,/bardblanθn\ni/bardblρ}\n= max{/bardbla1/bardblρ/bardblθi/bardblρ,...,/bardblan/bardblρ/bardblθi/bardbln\nρ}\n≤max{/bardbla1/bardblρ,...,/bardblan/bardblρ},\n6a contradiction. Thus /bardblθi/bardblρ>1 for each i= 1,...,n, as claimed.\nAssume now that fdecomposes as\nf(x,y) =f1(x,y)f2(x,y)···fr(x,y),\nwithfi∈K[x][y] irreducible over K[x], and hence with degy(fi)≥1 for each i= 1,...,r,\nwhere we regard fias a polynomial in ywith coefficients in K[x] and denote by αi(x)∈K[x]\nits leading coefficient. Then for each i= 1,...,rwe haveαi/ne}ationslash= 0 and\n/bardblfi(x,0)/αi/bardblρ=/vextenddouble/vextenddouble/productdisplay\nθθ/vextenddouble/vextenddouble\nρ=/productdisplay\nθ/bardblθ/bardblρ>1,\nwhere the product is over all zeros θoffi(which arealso zeros of f, thus satisfying /bardblθ/bardblρ>1).\nThis shows that ρdegfi(x,0)> ρdegαi(x)for each i, so deg fi(x,0)>degαi(x) for each i,\nimplying that none of the polynomials fi(x,0) is constant. Next, since\na0(x) =f(x,0) =f1(x,0)···fr(x,0),\nand this is a decomposition of a0intornonconstant factors (not necessarily irreducible\noverK), we conclude that rcan not exceed ν0, sofis a product of at most ν0irreducible\npolynomials over K[x].\nii) We argue in exactly the same way for\n˜f(x,y) =ynf(x,1/y) =an(x)+an−1(x)y+···+a1(x)yn−1+a0(x)yn,\nthe reciprocal of fwith respect to y, which has the same number of irreducible factors as\nf(counted with their multiplicities), and whose roots in K(x) are precisely the inverses\nof the roots of f. We may also argue directly by observing that the inequality deg an>\nmax{dega0,...,degan−1}reads/bardblan/bardblρ>maxi<n/bardblai/bardblρ, and forces the roots θ1,...,θ nto\nsatisfy the inequalities /bardblθi/bardblρ<1 for each i. Indeed, if fwould have a root θiwith/bardblθi/bardblρ≥1,\nthen since f(θi)/θn\ni= 0, we would have an(x) =−an−1(x)\nθi−···−a0(x)\nθn\ni, further implying that\n/bardblan/bardblρ=/bardbl−an−1(x)/θi−···−a0(x)/θn\ni/bardblρ\n≤max{/bardblan−1(x)/θi/bardblρ,...,/bardbla0(x)/θn\ni/bardblρ}\n= max{/bardblan−1/bardblρ//bardblθi/bardblρ,...,/bardbla0/bardblρ//bardblθi/bardbln\nρ}\n≤max{/bardblan−1/bardblρ,...,/bardbla0/bardblρ},\na contradiction. With the same notations, this time we obtain deg fi(x,0)<degαi(x) for\neachi, implying that none of the polynomials αi(x) is constant, so the equality an=α1···αr\nshows that fis a product of at most νnirreducible polynomials over K[x]. /square\nProof of Theorem 2.i) We already proved that all the roots θoffsatisfy/bardblθ/bardblρ>1 if\ndega0>max{dega1,...,degan}, orsatisfy /bardblθ/bardblρ<1 ifdegan>max{dega0,...,degan−1}.\nLet us first note that the existence of qmakes sense, since a0is a nonconstant polynomial.\nNow let us suppose again that\nf(x,y) =f1(x,y)f2(x,y)···fr(x,y)\nwithfi∈K[x][y] irreducible over K[x] and hence with degy(fi)≥1 for each i= 1,...,r, and\ndenote again the leading coefficient of fibyαi. Since deg an≥dega0−degq≥degq >0, it\nfollows that anis a nonconstant polynomial too, and so, νn≥1. We may therefore assume\nthatr≥2, since the conclusion that r≤min{ν0,νn}holds trivially for r= 1.\n7ByTheorem 1, we alreadyhave r≤ν0. Ontheother hand, if wefix anindex i∈ {1,...,r}\nand use the fact that f1(x,0)···fr(x,0) =a0along with the equality α1···αr=an, we\nsuccessively deduce that\n/bardbla0/bardblρ\n/bardblq/bardblρ≤ /bardblan/bardblρ</bardblan/bardblρr/productdisplay\nj=1, j/negationslash=i/bardblfj(x,0)/bardblρ\n/bardblαj/bardblρ\n=/bardblan/bardblρ/bardblf1(x,0)···fr(x,0)/bardblρ/bardblαi/bardblρ\n/bardblfi(x,0)/bardblρ/bardblα1···αr/bardblρ=/bardbla0/bardblρ/bardblαi/bardblρ\n/bardblfi(x,0)/bardblρ,\nwhich yields /bardblαi/bardblρ>/bardblfi(x,0)/bardblρ//bardblq/bardblρ. With the same reasoning as in the proof of Theorem\n1, we deduce that deg fi(x,0)>degαi(x), sofi(x,0) must be a nonconstant factor of a0,\nand hence we must have deg fi(x,0)≥degq, or equivalently, /bardblfi(x,0)/bardblρ≥ /bardblq(x)/bardblρ. In view\nof this, we conclude that\n/bardblαi/bardblρ>1,\nuniformly for i= 1,...,r. Thus each one of the polynomials α1,...,α rmust have degree at\nleast 1, which shows that in our decomposition of anas a product of rnonconstant factors\nan(x) =α1(x)···αr(x),\nrcan not exceed νn. This completes the proof in the first case.\nFor the proof of ii) we apply i) to the polynomial ˜f(x,y) =ynf(x,1/y), the reciprocal of\nfwith respect to y. /square\nTo prove Theorem 5, we will need an extension to the case of multivariate polynomials of\nthe following Theorem of Mayer [ 22] on the location of the zeros of a polynomial with real\nor complex coefficients, in which one of the coefficients has absolute v alue greater than the\nsum of the absolute values of all the other coefficients.\nTheorem B (Mayer [22]).LetK∈ {R,C}, and let |·|denote the Euclidean norm of K. Let\nf=a0+a1y+···+anyn∈K[y]be a nonconstant polynomial such that there exists an index\njwith0≤j≤nfor which |aj|>/summationtextn\ni=0, i/negationslash=j|ai|. Thenfhasjzeros in the disk |z|<1, and\nn−jzeros outside the disk |z| ≤1in the complex plane.\nThe elegance of Mayer’s proof of Theorem Barises from the fact that it relies only on the\nbasic notions of probability theory, the existence and uniqueness t heorem for initial value\nproblems in linear difference equations, and the Euclidean absolute va lue of the coefficients,\nwhich extend naturally to the non-Archimedean setting. Before st ating the corresponding\nextension, weneed tospecify atopologyon K(x). Theabsolutevalue /bardbl·/bardblρinduces atopology\nonK(x) via the metric defined by taking the distance between aandbto be/bardbla−b/bardblρfor\nalla,b∈K(x). In view of this, for any ε >0, it is meaningful to define open and closed\ndisks of radius εinK(x) denoted by /bardbly/bardblρ< εand/bardbly/bardblρ≤ε, and defined as the sets\n{a∈K(x)| /bardbla/bardblρ< ε}and{a∈K(x)| /bardbla/bardblρ≤ε}, respectively. We may now state the\nfollowing extension of Theorem Bfor bivariate polynomials.\nTheorem 11 (Mayer).LetKbe a field. Let f=a0(x) +a1(x)y+···+an(x)yn∈K[x,y]\nbe a polynomial with n≥2,a0,a1,...,a n∈K[x],a0an/ne}ationslash= 0, such that there exists an index\njwith0≤j≤nfor which\n/bardblaj/bardblρ>max\ni/negationslash=j/bardblai/bardblρ.\n8Thenfhasjzeros in the disk /bardbly/bardblρ<1, andn−jzeros outside the disk /bardbly/bardblρ≤1inK(x).\nProof of Theorem 11.Recall that our absolute value also satisfies the triangle inequality,\nand note that by using the condition /bardblaj/bardblρ>maxi/negationslash=j/bardblai/bardblρin the statement of Theorem 11\nfor a choice of ρ > n, we may deduce that\nρdegaj≥ρ1+max\ni/negationslash=jdegai> nρmax\ni/negationslash=jdegai≥n/summationdisplay\ni=0, i/negationslash=jρdegai,\nwhich shows that /bardblaj/bardblρalso satisfies the inequality\n/bardblaj/bardblρ>n/summationdisplay\ni=0, i/negationslash=j/bardblai/bardblρ.\nThe proof of Theorem 11can now be easily obtained by following the lines in the original\nproof of Mayer [ 22] and replacing the absolute value |·|by the absolute value /bardbl·/bardblρ, and by\nalso replacing the field of real numbers or the field of complex number s by the field K(x).\nThe details of the proof will be omitted.\nWe will give here an alternative proof of Theorem 11that relies on a Newton polygon\nargument. So let us construct the Newton polygon NP(f) of the polynomial f(x,y) =\na0(x)+a1(x)y+···+an(x)ynwith respect to the valuation vgiven by v(ai) =−degai, that\nis the lower convex hull of the set of points ( i,v(ai)) withi= 0,...,nandai/ne}ationslash= 0. Note that\nthe extension of vtoK(x) will be also denoted by v. Since deg aj>maxi/negationslash=jdegai, we deduce\nthat the point Pj= (j,−degaj) is the only vertex of NP(f) with the smallest y-coordinate.\nThis implies that all the edges in NP(f) at the left of Pj(if any) have negative slopes, while\nall the edges of NP(f) at the right of Pj(if any) have positive slopes. Now recall that by\nDumas’ Theorem, all the linear factors of foverK(x), that is all the factors of the form\nx−θwithθ∈K(x) a root of f, must contribute to NP(f) via translates with their own\nNewton polygon, that is with an edge having endpoints (0 ,v(θ)) and (1,0), and hence having\nslope equal to −v(θ). Thus, at the left of Pjthere are precisely jtranslates of the edges of\nwidth 1 that have negative slopes (possibly none, if j= 0), while at the right of Pjthere are\nprecisely n−jtranslates of the edges of width 1 that have positive slopes (possib ly none,\nifj=n). This shows that fhasjrootsθ(multiplicities counted) with positive valuation\nv(θ) and hence with absolute value /bardblθ/bardblρ<1, andn−jrootsθ(multiplicities counted) with\nnegative valuation v(θ) and hence with absolute value /bardblθ/bardblρ>1. /square\nTo prove Theorem 5, we will now make use of Theorem 11, as follows.\nProof of Theorem 5.Note that /bardblai/bardblρ≥1 for alli= 0,...,n−1, while/bardblan/bardblρ= 1, since\nan∈K. Our hypothesis on deg ajreads\n/bardblaj/bardblρ>max\ni/negationslash=j/bardblai/bardblρ,\nwhich shows that fsatisfies the hypothesis of Theorem 11. Therefore fhasjzeros lying\nin the disk /bardbly/bardblρ<1, and the remaining n−jzeros lying outside the disk /bardbly/bardblρ≤1 in the\nalgebraic closure K(x).\nSuppose now that f(x,y) =f1(x,y)f2(x,y)···fr(x,y) is the product of rirreducible\npolynomials f1,...,f roverK[x], for some integer rwith 1≤r≤n. Sincea0/ne}ationslash= 0 we must\n9have\n1≤ /bardbla0/bardblρ=/bardblf(x,0)/bardblρ=/bardblf1(x,0)/bardblρ···/bardblfr(x,0)/bardblρ,\nwith/bardblfi(x,0)/bardblρ≥1 for each i= 1,...,r. Assume on the contrary that r > n−j≥1.\nThenr >1 and there exists at least one index twith 1≤t≤rfor which all the zeros\nofftlie in the disk /bardbly/bardblρ<1 (for otherwise each of the rfactorsf1,...,f rwould have at\nleast one zero outside the disk /bardbly/bardblρ≤1, and hence fwould be forced to have more than\nn−jsuch zeros). Recall now that the leading coefficient of fbelongs to K, so the leading\ncoefficient αtofftmust also belong to K, implying that /bardblαt/bardblρ= 1. Therefore, if we write\nft(x,y) =αt/producttext\nθ(y−θ), with the product running over all the zeros θofftinK(x), we\ndeduce that\n1≤ /bardblft(x,0)/bardblρ=/bardblαt/bardblρ/productdisplay\nθ/bardblθ/bardblρ<1,\nwhich is a contradiction. This completes the proof of the theorem. /square\nProof of Theorem 6.Sincea0∈K, the reciprocal ynf(x,1/y) off(x,y) with respect\ntoysatisfies the hypotheses in Theorem 5withn−jinstead of j(sinceynf(x,1/y) =\nb0(x) +b1(x)y+···+bn(x)ynwithbj=an−jfor each j), so it is the product of at most j\nirreducible factors over K[x]. Since the irreducible factors of ynf(x,1/y) are precisely the\nreciprocals with respect to yof the irreducible factors of f, we conclude that ftoo is the\nproduct of at most jirreducible factors over K[x]. /square\nProof of Theorem 7.Letg(x,y) =an−1\nnf(x,y/a n). We observe that\ng(x,y) =a0(x)an(x)n−1+···+aj(x)an(x)n−1−jyj+···+an−1(x)yn−1+yn,\nwhich shows that g∈K[x][y] andgis monic. In particular, the leading coefficient of g\nbelongs to K, and moreover, one may easily check that fandghave the same number of\nirreducible factors over K[x], counted with their multiplicities. Thus, it will be sufficient to\nprove our result for ginstead of f. Using our hypothesis on the degree of aj, we observe\nthat the coefficients of gsatisfy\ndeg(ajan−1−j\nn) = deg( aj)+(n−1−j)deg(an)\n>max\ni/negationslash=j{deg(ai)+(j−i)deg(an)}+(n−1−j)deg(an)\n= max\ni/negationslash=j{deg(ai)+(n−1−i)deg(an)}\n= max\ni/negationslash=jdeg(aian−1−i\nn).\nConsequently, by Theorem 5, the polynomial g(and hence ftoo) is a product of at most\nn−jirreducible polynomials over K[x], and this completes the proof of the theorem. /square\nProof of Theorem 8.By Theorem 7we know that fis a product of at most n−jirre-\nducible polynomials over K[x], if\ndegaj>max\ni/negationslash=j{degai+(j−i)degan}. (1)\nLet us consider now the reciprocal of fwith respect to y, that is\nynf(x,1/y) =an(x)+an−1(x)y+···+a0(x)yn.\n10By Theorem 7withynf(x,1/y) instead of fwe see that ynf(x,1/y) (and hence ftoo) is a\nproduct of at most jirreducible polynomials over K[x], if\ndegaj>max\ni/negationslash=n−j{degan−i+(n−j−i)dega0},\nor equivalently, after relabelling the indices, if\ndegaj>max\ni/negationslash=j{degai+(i−j)dega0}. (2)\nObserve now that conditions ( 1) and (2) will both hold if and only if deg ajsatisfies the pair\nof inequalities\ndegaj>max\ni<j{degai+(j−i)degan}and\ndegaj>max\ni>j{degai+(i−j)dega0}.\nThis completes the proof of the theorem. /square\n4.Examples\nWe will provide here some examples of infinite families of polynomials in Z[x,y] whose\nfactorization properties can be deduced using the results proved in the preceding section. In\nall our examples, we will assume that n≥2.\nExample 1. For any polynomials a1,...,a n∈Z[x] with max {dega1,...,degan} ≤2n−1\nandan/ne}ationslash= 0, the polynomial\nf1= 1+x2n+a1(x)y+···+an(x)yn∈Z[x,y]\nsatisfies the hypotheses of Corollary 3witha0= 1 +x2n, which is irreducible, being a\ncyclotomic polynomial. Therefore f1must be irreducible over Z[x].\nExample 2. For any positive integer k, and any nonzero polynomial b(x)∈Z[x] with\ndegb≤2k+1 and such that 1+ x2kis not a factor of b, consider the bivariate polynomial\nf2= (1+x2k)(1+x2)+b(x)(y+y2+···+yn−1)+(1+x2k)yn∈Z[x,y].\nIt is easy to check that f2satisfies the hypothesis of Corollary 3witha0= (1+x2k)(1+x2),\nq= 1+x2,a1=...=an−1=b, andan= 1+x2k, which is irreducible over Zand satisfies\nthe condition that deg an≥dega0−degq. We conclude that f2is irreducible over Z[x].\nExample 3. For any fixed index jwith 0≤j≤n−1, and any fixed, arbitrarily chosen\nnonzero integers c0,...,c n, the polynomial\nf3=c0x+c1xy+···+cj−1xyj−1+cjx2yj+cj+1yj+1+···+cnyn∈Z[x,y],\nsatisfies the hypothesis of Theorem 5with\nai=cixfor 0≤i≤j−1, aj=cjx2,andai=ciforj+1≤i≤n,\nsof3must be a product of at most n−jirreducible polynomials over Z[x].\n11Example 4. For any fixed index jwith 1≤j≤n−1, and any fixed, arbitrarily chosen\nnonzero integers c0,...,c n, the polynomial\nf4=c0+c1xy+···+cj−1xj−1yj−1+cjxj+1yj+cj+1xyj+1+···+cnxyn∈Z[x,y],\nsatisfies the hypothesis of Theorem 7with\nai=cixifor 0≤i≤j−1, aj=cjxj+1,andai=cixforj+1≤i≤n,\nsince we have\ndegaj=j+1> j= degaj−1+(j−(j−1))\n= max\n0≤i≤n, i/negationslash=j{degai+(j−i)degan}.\nTherefore, f4is a product of at most n−jirreducible polynomials over Z[x]. On the other\nhand, since deg a0= 0 and deg aj>maxi/negationslash=jdegai, we conclude by Theorem 8that actually\nf4is a product of at most min {j,n−j}irreducible polynomials over Z[x].\nExample 5. For a more general example that shows that the bound in Theorem 6is best\npossible, let nbe a positive integer, j < na divisor of n, and consider the polynomial\nf5(x,y) = (1+ xy+yn\nj)j∈Z[x,y],\nwritten also as f5=a0(x)+a1(x)y+···an(x)yn, say, with ai∈Z[x], anda0=an= 1. It is\neasy to see that f5has a unique coefficient of maximal degree, namely aj, and deg aj=j. If\nweignorethefactthat f5isthejthpowerof1+ xy+yn\nj,andweareonlylookingatthedegrees\nof its coefficients, we conclude by Theorem 6thatf5is a product of at most jirreducible\nfactors over Q[x]. This is indeed the case, since the polynomial 1 + xy+yn\njis irreducible\noverQ[x]. To see this, we will actually prove the more general fact that all th e polynomials\nof the form 1+ xy+ynwithn≥2 are irreducible over Q[x]. Denote 1+ xy+ynbygand\nconsider the Newton polygon NP(g) ofgwith respect to the valuation −deg(·) onQ[x].\nIt is easy to see that NP(g) has only two segments, the left-one with endpoints A= (0,0)\nandB= (1,−1), and the one on the right with endpoints B= (1,−1) andC= (n,0).\nNone of these two segments contain lattice points other than their endpoints, so by Dumas’\nTheorem, gcan have at most two irreducible factors, say h1andh2, with degyh1= 1 and\ndegyh2=n−1, sayh1=b0(x) +b1(x)yandh1=c0(x) +c1(x)y+···cn−1(x)yn−1, and\nwhose Newton polygons NP(h1) andNP(h2) should be translates of the two segments AB\nandBC. On the other hand the equality g=h1h2would imply b0c0= 1, so both b0and\nc0should have degree zero, meaning that NP(h1) should be precisely the segment AB, thus\nforcingb1to have degree 1. This obviously can not hold, since b1cn−1should be also equal\nto 1. Thus gmust be irreducible over Q[x], as claimed.\nAcknowledgments. Ms. Rishu Garg is thankful to the Council of Scientific and\nIndustrial Research (CSIR) for providing her Junior Research Fe llowship (JRF) wide grant\nno. CSIRAWARD/JRF-NET2022/11769 for carrying the present r esearch. This paper is a\npart of the research work undertaken by her for her Ph.D. thesis .\nThepresentresearchisalsosupportedbyScienceandEngineering ResearchBoard(SERB),\na statutory body of Department of Science and Technology (DST) , Government of India\nthrough the project grant no. MTR/2017/000575 awarded to th e last author under the\nMATRICS Scheme.\n12Disclosure statement. Theauthorsreportthattherearenocompetingintereststodec lare.\nReferences\n[1] A.J. Bevelacqua, Another irreducibility criterion. Amer. Math. Monthly. 120:7 (2013), 648–650.\n[2] A.I. Bonciocat, N.C. Bonciocat, Y. Bugeaud, and M. Cipu, Apollonio us circles and irreducibility criteria\nfor polynomials, Indag. Math. 33(2) (2022), 421–439.\n[3] A.I. Bonciocat, N.C. Bonciocat, Y. Bugeaud, M. Cipu, M. Mignotte , Apollonius circles and the number\nofirreduciblefactorsofpolynomials, Bull. Math. Soc. Sci. Math. Roumanie Tome66(114), no.2(2023),\n119–138.\n[4] N.C. Bonciocat, On an irreducibility criterion of Perron for multivar iate polynomials, Bull. Math. Soc.\nSci. Math. Roumanie, Nouvelle S´ erie 53(101), no. 3 (2010), 213–217.\n[5] N.C. Bonciocat, Y. Bugeaud, M. Cipu, M. Mignotte, Some P´ olya–t ype irreducibility criteria for multi-\nvariate polynomials, Comm. Algebra 40(2) (2012), 3733–3744.\n[6] N.C. Bonciocat, A. Zaharescu, Irreducible multivariate polynomia ls obtained from polynomials in fewer\nvariables, J. Pure Appl. Algebra 212(2008), 2338–2343.\n[7] J. Brillhart, M. Filaseta, and A. Odlyzko, On an irreducibility theore m of A. Cohn, Canad. J. Math.\n33(1981), no. 5, 1055–1059.\n[8] M. Cavachi, On a special case of Hilbert’s irreducibility theorem, J. Number Theory 82(2000), no. 1,\n96–99.\n[9] M. Cavachi, M. Vˆ ajˆ aitu, and A. Zaharescu, A class of irreducib le polynomials, J. Ramanujan Math.\nSoc.17(2002), no. 3, 161–172.\n[10] M. Cavachi, M. Vˆ ajˆ aitu, A. Zaharescu, An irreducibility criter ion for polynomials in several variables,\nActa Math. Univ. Ostrav. 12(2004), 13–18.\n[11] M. Cole, S. Dunn, and M. Filaseta, Further irreducibility criteria f or polynomials with non-negative\ncoefficients, Acta Arith. 175(2016), no. 2, 137–181.\n[12]H.L. Dorwart , Irreducibility of polynomials, Amer. Math. Monthly 42(1935), no. 6, 369–381.\n[13] G. Dumas, Sur quelques cas d’irr´ eductibilit´ e des polynomes ´ a c oefficients rationnels, J. Math. Pure\nAppl.2(1906), 191–258.\n[14] G. Eisenstein, ¨Uber die irreduzibilitat und einige andere eigenschaften der gleichung en,J. Reine Angew.\nMath.39(1850), 160–179.\n[15] A.J. Engler and A. Prestel, Valued Fields , Springer, 2000.\n[16] M. Filaseta, Irreducibility criteria for polynomials with non-negat ive coefficients, Canad. J. Math. 40\n(1988), no. 2, 339–351.\n[17] M. Filaseta, S. Gross, 4959866698915122609810424451291 8,J. Number Theory 137(2014), 16–49.\n[18] M. Filaseta, T. Luckner, On nth order Euler polynomials oforder nthat are Eisensteinian, Indag. Math.\nNew Ser. 35no. 1 (2024), 76–86.\n[19] K. Girstmair, On an Irreducibility Criterion of M. Ram Murty, Amer. Math. Monthly 112(2005), no.\n3, 269–270.\n[20] N.H. Guersenzvaig,Simple arithmeticalcriteriaforirreducibilityo fpolynomialswith integercoefficients,\nIntegers 13(2013), 1–21.\n[21] S. Kumar and J. Singh, A study of some recent irreducibility crite ria for polynomials having integer\ncoefficients, arXiv Preprint: https://arxiv.org/abs/2310.02860 pp 1–20.\n[22] D.-E. Mayer, Sur les e´ quations alg´ ebriques, Nouvelles Annales de Math´ ematiques 10(1891), 111–124.\n[23]O. Ore , Einige Bemerkungen ¨ uber Irreduzibilit¨ at, Jahresbericht der Deutschen Mathematiker-\nVereinigung 44(1934), 147–151.\n[24] O. Perron, Neue kriterien f¨ ur die irreduzibilit¨ at algebraischer gleichungen, J. Reine Angew. Math. 132\n(1907), 288–307.\n[25] G. P´ olya, Verschiedene Bemerkungen zur Zahlentheorie, Jahresber. Deutschen Math. Ver. 28(1919),\n31–40.\n[26] G. P´ olya and G. Szeg¨ o, Aufgaben und Lehrs¨ atze aus der An alysis, Springer-Verlag, Berlin, 1964.\n[27] M. Ram Murty, Prime numbers and irreducible polynomials, Amer. Math. Monthly 109(2002), no. 5,\n452–458.\n13[28] T. Sch¨ onemann, Von denjenigen Moduln, welche Potenzen von Primzahlen sind, J. Reine Angew. Math.\n32(1846), 93–105.\n[29] J. Singh and R. Garg, A note on Perron’s irreducibility criterion, Arch. Math. 121(2023), 33–38.\n[30] J. Singh and R. Garg, Some factorization results on polynomials h aving integer coefficients, Comm.\nAlgebra, https://doi.org/10.1080/00927872.2023.2301530\n[31] J. Singh and S. Kumar, A Generalization of Bevelacqua’s Irreduc ibility Criterion, Amer. Math. Monthly\n127:5 (2020), 456–459.\n[32] J.Singh andS. Kumar, Anote on Girstmair’sirreducibilitycriterion ,Bull. Aust. Math. Soc. 106(2022),\n62–66.\n[33] P. St¨ ackel, Arithmetischen Eigenschaften ganzer Funktione n,Journal f¨ ur Mathematik 148(1918), 101–\n112.\n[34] S.H. Weintraub, A mild generalization of Eisenstein’s criterion, Proc. Amer. Math. Soc. 141(4) (2013),\n1159–1160.\n[35] L. Weisner, Criteria for the irreducibility of polynomials, Bull. Amer. Math. Soc. 40(1934), 864–870.\n(1)Simion Stoilow Institute of Mathematics of the Romanian Aca demy, Research Unit 7,\nP.O. Box 1-764, Bucharest 014700, Romania\nnicolae.bonciocat@imar.ro\n(2,3)Department of Mathematics, Guru Nanak Dev University, Amri tsar-143005, India\nrishugarg128@gmail.com ,jitender.math@gndu.ac.in\n14This figure \"ORCID.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/2402.02324v1"    },    {        "title": "2402.02358v1.Repairing_Reed_Solomon_Codes_over_Prime_Fields_via_Exponential_Sums.pdf",        "content": "arXiv:2402.02358v1  [cs.IT]  4 Feb 2024Repairing Reed-Solomon Codes over Prime Fields via\nExponential Sums\nRoni Con∗Noah Shutty†Itzhak Tamo‡Mary Wootters§\nFebruary 6, 2024\nAbstract\nThis paper presents two repair schemes for low-rate Reed-So lomon (RS) codes over prime\nfields that can repair any node by downloading a constant number of bits from each surviving\nnode. The total bandwidth resulting from these schemes is gr eater than that incurred during\ntrivial repair; however, this is particularly relevant in t he context of leakage-resilient secret\nsharing. In that framework, our results provide attacks showing tha tk-out-of-nShamir’s\nSecret Sharing over prime fields for small kisnotleakage-resilient, even when the parties leak\nonly aconstant number of bits. To the best of our knowledge, these are the firs t such attacks.\nOur results are derived from a novel connection between expo nential sums and the repair of\nRS codes. Specifically, we establish that non-trivial bound s on certain exponential sums imply\nthe existence of explicit nonlinear repair schemes for RS co des over prime fields.\n∗Department of Computer Science, Tel Aviv University. Email : roni.con93@gmail.com\n†Stanford Institute for Theoretical Physics, Stanford Univ ersity, noaj@alumni.stanford.edu\n‡Department of Electrical Engineering-Systems, Tel Aviv Un iversity. Email: tamo@tauex.tau.ac.il\n§Department of Computer Science and Electrical Engineering , Stanford University, Email: marykw@stanford.edu\nThe work of Itzhak Tamo and Roni Con was supported by the Europ ean Research Council (ERC grant number\n852953). Noah Shutty was supported in part by NSF DGE-165651 8. Mary Wootters was supported in part by NSF\ngrants CCF-2133154 and CCF-1844628.\nA preliminary version of this work was presented at the IEEE I nternational Symposium on Information Theory\n(ISIT) 2023.\n11 Introduction\nReed-Solomon (RS) codes are a widely-used family of codes, both in t heory and in practice. Among\ntheir many applications, RS codes are utilized in distributed storage s ystems, as evidenced in\nsystems like Facebook, IBM, Google, etc. (refer to Table 1 in [5]). I n such systems, a large file\nis encoded using an erasure-correcting code and then distributed across multiple nodes. Upon the\nfailure of a node, it is desirable to efficiently set up a replacement node using information from the\nremaining nodes. In this work, we focus on the repair bandwidth —the total amount of information\ndownloaded—as our metric of efficiency. The challenge of recovering a failed node with minimal\nrepair bandwidth was first addressed in the seminal paper by Dimakis et al. [4] and has since been\na significant topic of research.\nWe begin by defining RS codes.\nDefinition 1.1. Letα1,α2,...,α nbe distinct points of the finite field Fqof orderq. Fork < nthe\n[n,k]qRS code defined by the evaluation set {α1,...,α n}is the set of codewords\n{(f(α1),...,f(αn))|f∈Fq[x],degf < k}.\nWhenn=q, the resulting code is called a full-length RS code .\nWhen the code used in the storage system is an RS code, the repair p roblem can be viewed\nas a variation of the standard polynomial interpolation problem: Rep airing, say, the ith node, is\nequivalent to recovering an evaluation f(αi) using minimal information from the evaluations f(αj)\nforj/\\⌉}atio\\slash=i. Formally, a repair scheme for an[n,k]qRS code with evaluationpoints α1,...,α nconsists\nof functions τj:Fq→ {0,1}mfor each j/\\⌉}atio\\slash=i; and a repair function G:{0,1}m×(n−1)×[n]→Fq,\nfor some parameter m. We note that the function τj,Gmay depend on the identity of the failed\nnodeiand should perhaps be called τj,iandGirespectively; however, for the same of notational\nclarity, we omit the isubscript.\nFor a failed node i∈[n], and any polynomial fof degree less than k(representing the stored\ndata), each surviving node j/\\⌉}atio\\slash=isends a message τj(f(αj)). The repair scheme then takes in the\nmessages τj(f(αj)), along with i, and outputs the missing information f(αi) usingG. Our goal is\nto minimize m, the number of bits sent by each node.1\nVia standard polynomial interpolation, it is clear that any kvalues of f(αj) suffice to recover\nthe polynomial f, and in particular, to recover f(αi). This requires klog2(q) bits of information;\nwe refer to this as trivial repair . However, as was shown in a line of work including [26, 9, 28, 3], it\nis possible to do better! That is, it is possible to recover f(αi) using strictly less than klog(q) bits\nfrom the other nodes!\n1.1 Repairing Reed-Solomon Codes over Prime Fields\nAll of the known RS repair schemes but the one in [3] require that the underlying field be an\nextension field. In contrast, our work focuses on prime fields . Our main motivation stems from\napplications in secret sharing .\n1We note that traditionally, the definition of a repair scheme permits contacting only a subset of the nodes, with\neach node potentially sending varying amounts of informati on. In the definition provided above, we assume the\nscenario where all surviving nodes transmit the same amount of information. Later, we will relax this definition to\nallow for some of the surviving nodes to abstain from sending any information.\n2RS Code Bandwidth per node Remarks\n[3, Theorem 3.3] [n,2]plog(p)\nn−2+On(1) Non-explicit construction.\n[3, Theorem 4.3] [n,k]plog(p)\nn−k+On(1) Repair only possible for\na particular failed node\n(rather than any failed\nnode)2using all the re-\nmaining nodes.\nTheorem 3.3 [pδ,3]p 3(lnlnp)1\n3\n(lnp)2\n3≤δ≤1\n2\nTheorem 4.5 [p,ΘB(√p)]p B≥3 Recovers the entire poly-\nnomial\nTable 1: Our result compared to [3]. If not stated in the Remarks colu mn, all of the results\npresented are explicit constructions, and all the results give repa ir schemes that repair any single\nfailed node using all the remaining nodes. For all results, pis a sufficiently large prime. For the\nresults in [3], nandkare assumed to be constants relative to p.\nIn secret sharing, the well-known Shamir’s secret sharing scheme (Shamir’s SSS) is analogous\nto RS codes. Informally speaking (see Section 1.3 for more details), repair schemes for RS codes\nin which every node transmits a small number of bits are equivalent to attacks on Shamir’s SSSs\nwhen the parties may each leak a small amount of adversarially select ed information about their\nshares. In essence, a repair scheme for an RS code implies that an in stance of Shamir’s SSS is not\nleakage-resilient .\nSincetheschemesof[26,9,28]andothersrequireextensionfields , anaturalhopeforconstructing\nShamir’s SSSs that are leakage-resilient is to work over prime fields, a nd indeed several works [1,\n21, 18, 19, 14] have shown that, when the dimension kof the code is large, Ω( n), Shamir’s SSS over\nprime fields is leakage resilient. In particular, their results imply that f or any constant m, there is\nsome constant α∈(0,1) so that any [ n,k]pRS code over a prime field Fpwithk≥αndoesnot\nadmit a repair scheme that downloads mbits from each surviving node (see Theorem 1.2).\nOn the other hand, [3] showed the existence of asymptotically optim al repair schemes for RS\ncodes of dimension k= 2 over sufficiently large prime fields Fp. In their work, each party leaks a\nnon-constant number of bits, specifically, m=1\nn−2log(p)+On(1) bits (see Table 1 for their results\nfor larger k). Our results continue the line of work initiated by [3]. We make progre ss by presenting\nschemes where each node transmits a constant number of bits, that is, m=O(1); equivalently, for\nShamir’s SSS, each party leaks a constant number of bits. Therefo re, our results have implications\nboth for attacks on Shamir’s SSS and for distributed storage syst ems.\n1.2 Our Results\nIn this paper, we present two schemes where every node of an RS c ode transmits a constant number\nof bits. Both schemes are explicit, meaning that the functions τjcomputed by the nodes as well\nas the reconstruction function Gare explicitly defined. It is noteworthy that the first scheme is a\n2This is achieved by puncturing the code in [3, Theorem 4.3].\n3repair scheme, while the second scheme not only enables the repair o f a failed node but is, in fact,\na decoding scheme that facilitates the recovery of the original cod eword.\nThis latter point might raise some red flags for the reader familiar with regenerating codes, as\nit implies that the scheme downloads enough information to recover t he entire codeword, while the\ngoal in regenerating codes is to generally download less information t han that. In fact, both of our\nschemes download moreinformation than k·log(p) bits, the amount required to recover the entire\ncodeword.\nThe fact that our schemes have such high download costs means th at they are not competitive\nwith the “trivial” repair scheme (which downloads log( p) bits from each of knodes) if one cares\nonly about total bandwidth. However, our schemes are useful in s ettings where each node can only\ntransmit very few bits; in such a setting, the trivial repair is not an o ption. This could be the case\nin distributed storage models (e.g., in a model where each link can only t ransmit a few bits). It is\nalso the case in the model of leakage-resilient secret sharing. In pa rticular, our results give attacks\non Shamir’s SSSs over prime fields, where each leaking party need only send a constant number of\nbits. To the best of our knowledge, these are the first such attac ks with a constant number of bits\nper leaking party.\nWith that in mind, we present our two main results below. Throughout this paper, let p\nbe a prime number. We proceed to present our first result: a repair scheme for an RS code of\ndimension 3).\nTheorem (informal, see Theorem 3.3) .Forexp/parenleftbig\n(lnp)2/3(lnlnp)1/3/parenrightbig\n≤n≤√p, there exists an\n[n,3]pRS code where any node can be repaired by downloading threebits from each of the n−1\nremaining nodes.\nOur second result is a decoding scheme for a full-length RS code, whic h enables the recovery\nof the original codeword using a constant number of bits from a sub set of the nodes. This implies\nthat not all nodes are required to participate in the decoding proce ss. We prove the following.\nTheorem (informal, see Theorem 4.5) .LetB≥3be an integer. The full length [p,k]pRS\ncode admits a decoding scheme by downloading Bbits from any p−mnodes where k+m≤\ncos/parenleftBig\n2π\n2B+2π\np/parenrightBig√p.\n1.3 Related Work\nLow-bandwidth repair for distributed storage. As mentioned above, the low-bandwidth\nrepair problem was introduced in [4], and since then there have been m any code constructions and\nrepair schemes aiming at optimal repair, for example [6, 7, 23, 27, 29 , 24, 30, 31, 8]. The study of\nrepairing Reed-Solomon codes was introduced in [26], and the works [9, 28], among others, have\nconstructed repair schemes for RS codes over extension fields wit h total bandwidth that is much\nsmaller than the trivial bandwidth; in particular, the work [28] showe d how to construct RS codes\nthat achieve the cut-set bound proved in [4], i.e., with the minimal possible bandwidth.\nOurworkfocuses onprime fields. To the best ofourknowledge, the onlyworkthat givespositive\nresults for repairing RS codes over prime fields is [3], discussed above and summarized in Table 1.\nLeakage-Resilient Secret Sharing. Shamir’sSSS, whichwasfirstintroducedin [25], is afunda-\nmental cryptographicprimitive that providesa secure method to d istribute a secret among different\nparties, so that any kcan recover the secret but no k−1 learn anything about it. Formally, given\n4a secrets∈F, areconstruction threshold k >0 and a number of parties n, Shamir’s SSS works as\nfollows. A dealer chooses a random polynomial fof degree at most k−1, so that f(0) =s. Then\npartyiis given the share f(αi), where α1,...,α n∈Fare distinct, pre-selected points. It is easy\nto verify that any kparties can reconstruct the polynomial f(x) and therefore recover the secret\nf(0) =s, while the shares of any k−1 parties reveal no information about the secret.\nBenhamouda, Degwekar, Ishai, and Rabin [1] considered the quest ion oflocal leakage-resilience\nof secret sharing schemes over prime fields and, in particular, Sham ir’s SSS over prime fields. In\nthis setting, the model is that each party imay adversariallyleak some function τi(f(αi))∈ {0,1}m\nof their share, for some (small) m. A scheme is m-local leakage resilient if for any two secrets s,s′,\nthe total variation distance between the leaked messages under sand the leaked messages under s′\nis negligible. The work of Benhamouda et al. established the following.\nTheorem 1.2. [1, Corollary 4.12] Let mbe a constant positive integer and let ngo to infinity.\nThere exists an αm<1, for which Shamir’s SSS with nplayers and threshold k=αmnism-local\nleakage resilient.\nWe also note that Benhamouda et al. [1] phrased a conjecture tha t for large enough n, Shamir’s\nSSS with threshold αnis 1-bit local leakage resilient for any constant α >0. Despite considerable\nprogress [18, 19, 14], this conjecture is still open.\nGiven Theorem 1.2, it is not feasible to repair RS codes over prime fields with a rate arbitrarily\nclose to 1 while only leaking a constant number of bits from each party . In [21, Lemma 2], it is\nshown that Shamir’s SSS are not leakage resilient for sublinear thres hold. Specifically,\nLemma 1.3. [21, Lemma 2] Let pbe the smallest prime larger than n. Then, for any constant\nc∈(0,1)and large enough n, it holds that Shamir’s SSS over Fpwithnplayers and threshold\nk=cn/lognis not1-local leakage resilient.\nHowever, prior to this work, no explicit local attacks were known. B y explicit attacks, we mean\nexplicit small image functions that adversary applies on the shares. Our results explicit state such\nfunctions and we also show that adversary not only gains significant insight into the secret but, in\nfact, can completely recover it.\n1.4 Organization\nThe preliminaries are given Section 2. The repair scheme is given in Sect ion 3 and the decoding\nscheme is given in Section 4. In Section 5, we summarize and present s ome open questions.\n2 Preliminaries\nWe begin with some needed notations. For integers a < blet [a,b] ={a,a+ 1,...,b}and [a] =\n{1,2,...,a}. An arithmetic progression in some field Fof length Nand a step s∈Fis a set of the\nform{a,a+s,...,a+(N−1)s}for some a∈F. Throughout, let p/\\⌉}atio\\slash= 2 be a prime number and Fp\nbe the finite field of size p. We will realize Fpas the set [ −p−1\n2,p−1\n2], where each element of Fpis\nviewed as an element of [ −p−1\n2,p−1\n2], and addition and multiplication are computed modulo p.\nFor an element γ∈Fpletγ·A:={γ·a:a∈A}be all the possible products of γwith\nelements in A, hence, the notation α∈γ·[−t,t] forα,γ,t∈Fpimplies that there exists j∈[−t,t]\nfor which α≡γ·jmodp. Finally, let ep(x) :Fp→Cbe the standard additive character of\nFp,ep(x) := exp/parenleftBig\n2π√−1\np·x/parenrightBig\n.\n52.1 Exponential sum bounds\nExponential sums have been a subject of extensive study in numbe r theory and have found nu-\nmerous applications in coding theory. In this work, we explore anoth er application of these sums.\nSpecifically, we demonstrate how bounds such as the Weil bound and those on Kloosterman sums\ncan be leveraged to develop repair schemes for Reed-Solomon (RS) codes over prime fields, where\neach node transmits a constant number of bits. In this section, we introduce several key bounds,\nbeginning with the Weil bound .\nTheorem 2.1. [17, Theorem 5.38] Let f∈Fp[X]be a non-constant polynomial of degree at most\nk. It holds that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nα∈Fpep(f(α))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(k−1)·√p .\nAs one can see, the bound is non-trivial only when deg( f)≤√p. The Weil bound has applica-\ntions throughout mathematics, theoretical computer science, a nd information theory. Motivated by\nthese applications, there are extensions to the Weil bound that imp rove the bound in certain classes\nof polynomials, see e.g., [2, 13]. Another important family of exponent ial sums are Kloosterman\nsums, which first appeared in [15] are sums of the form\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nα∈Aep/parenleftBiga\nα+bα/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhereA⊆F∗\npand gcd( a,p)/\\⌉}atio\\slash= 0. Bounds on these sums also have broad applications, mainly in\nanalytical number theory [10, 12], but also in coding theory [11, 20, 3 2].\n2.2 Repair Schemes via Arithmetic Progressions\nIn this section, we revisit the framework established in [3] that utilize s arithmetic progressions to\nconstruct repairschemes. For simplicity, we assume the failed node is thenth one in this discussion,\nthough this assumption is relaxed in our formal treatment.\nA repair scheme for the nth node consists of n−1 functions τi:Fp→[s] and a function\nG: [s]n−1→Fp. These functions satisfy the following condition for any codeword ( c1,...,c n)∈ C:\nG(τ1(c1),...,τ n−1(cn−1)) =cn. (1)\nIn the event of a failure at the nth node, each ith node (holding the symbol ci) computes τi(ci) and\ntransmitsit, requiring ⌈logs⌉bits. Therepairprocessiscompleteduponreceivingthe n−1messages\nτi(ci), with the nth symbol reconstructed using (1). The total bandwidth of the r epair scheme,\ndefined as the aggregate number of bits transmitted during the re pair, amounts to ( n−1)⌈log(s)⌉\nbits. This framework is applicable to any repair scheme, with the challe nge primarily lying in the\nidentification of suitable functions τi. These functions must be sufficiently informative to allow the\ncollective computation of the symbol cnfrom the information about ci. However, they should not\nbe excessively informative to ensure that the size of the image, den oted bys, remains small. This\nbalance is crucial for minimizing the total bandwidth used in the repair process.\nTo define the functions τi, we employ partitions in the following manner. It is important to\nnote that each function τiinduces a partition {τ−1\ni(a) :a∈[s]}ofFp. Conversely, any partition of\n6Fpintossubsets can define a function, where the function value at a point a∈Fpcorresponds to\nthe index of the subset containing a. Therefore, in our subsequent discussion, we will specify the\nfunctions τiby constructing partitions of Fpintossubsets, utilizing arithmetic progressions.\nThe work in [3] demonstrates the use of arithmetic progressions fo r creating partitions that lead\ntoefficient nonlinearrepairschemesforRS codesoverprimefields. T he specifics ofthis construction\nare detailed next.\nFix an integer 1 ≤t≤p, sets=⌈p/t⌉, and define A0,...,A s−1as the partition of Fpintos\narithmetic progressions of length tand step 1:\nAj=/braceleftBigg\n{jt,jt+1,...,jt+t−1}for 0≤j≤s−2,\n{(s−1)t,...,p−1} forj=s−1.(2)\nFor a nonzero γ∈Fp, it can be verified that γ·A0,...,γ·As−1also forms a partition of Fpinto\narithmetic progressions of length t(except for the last set γ·As−1), with step γ. Each function\nτi, fori∈[n−1], in the repair scheme is defined by a partition γi·A0,...,γ i·As−1, with an\nappropriately chosen γi. It is important to note that distinct i’s will have distinct γi’s, ensuring the\nuniqueness of each τi. As observed in [3], these partitions defined by the γi’s lead to a valid repair\nschemeif and only if for any two codewords c,c′∈ Cthat belong to the same set in all of the n−1\ndifferent partitions (i.e., ci,c′\ni∈γi·Ajifor alli∈[n−1]), it holds that cn=c′\nn.\nIn [3], the authors provided a simple yet sufficient condition for a linear code to have a valid\nrepair scheme. We adapt their condition specifically for RS codes, wh ich is the primary focus of\nthis paper.\nProposition 2.2. [3, Proposition 2.2] Consider an [n,k]pRS code defined with the evaluation\npointsα1,...,α n. Letℓ∈[n]be the index of the failed node. Let t < pbe an integer and for i∈\n[n]\\{ℓ}, letγi∈F∗\np. If for any polynomial f(x)∈Fp[x]of degree less than kwithf(αi)∈γi·[−t,t]\nfor alli∈[n]\\{ℓ}, it holds that f(αℓ) = 0, then, the γi’s define a valid repair scheme for the ℓth\nnode with a total bandwidth of (n−1)·log⌈p/t⌉bits.\nIn [3], the authors concentrated on the regime where t= Θ/parenleftBig\np1−1\nn−k+1/parenrightBig\n, withnandkbeing\nconstants relative to p. In this setting, each surviving node transmits Θ(log( p)) bits. Since pis\nconsidered to be growing in these results, the number of bits trans mitted is not constant.\nIn contrast, our work explores the regime where t= Θ(p). In this scenario, every node sends\naconstant number of bits, specifically log ⌈p/t⌉bits, to the replacement node. This approach\nsignificantly differs from the previous work by reducing the transmis sion to a constant bit size,\nregardless of the growth of p.\n3 Repair schemes using bounds on Kloosterman sums\nIn this section, we present our scheme for repairing single failed nod es for RS codes over prime\nfields. We shall extensively use the following simple lemma.\nLemma 3.1. Leta1,...,a n,t∈Fpsuch that ai∈[−t,t]for alli∈[n]. Then, |/summationtextn\ni=1ep(ai)| ≥\nn·cos/parenleftBig\n2πt\np/parenrightBig\n.\n7Proof.Let Re{x}denote the real part of the complex number x, then\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\ni=1ep(ai)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥Re/braceleftBiggn/summationdisplay\ni=1ep(ai)/bracerightBigg\n=n/summationdisplay\ni=1Re{ep(ai)} ≥n·cos/parenleftbigg2πt\np/parenrightbigg\n,\nwhere the second inequality follows since for all i∈[n], we have ai∈[−t,t].\nThe next theorem due to Korolev establishes a nontrivial upper on s hort Kloosterman sums.\nTheorem 3.2. [16, Theorems 2, 3] Let pbe a prime and let nbe an integer such that\ne(lnp)2/3·(lnlnp)1/3≤n≤√p .\nLetD=(lnn)3/2\n(lnp)(lnlnp)2. Then, it holds\n1. For any asuch that gcd(a,p) = 1,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\n1≤ν≤nep/parenleftBiga\nν/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n·260lnD\nD=o(n).\n2. For any a,bsuch that gcd(ab,p) = 1\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\n1≤ν≤nep/parenleftBiga\nν+bν/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n·222lnD\nD3/4=o(n).\nNext, we present our first repair scheme that is capable of repairin g every node in an [ n,3]pRS\ncode for nthat is small compared to p.\nTheorem 3.3. Letpbe a large enough prime and nbe an integer such that\n2exp/parenleftBig\n(lnp)2/3·(lnlnp)1/3/parenrightBig\n≤n≤√p .\nThen, the [n+1,3]pRS code defined by the evaluation points αi=ifori∈[0,n]admits a repair\nof any node by downloading three bits from all the other nodes .\nProof.Sett:=⌈p/8⌉and assume that we wish to repair the ℓth node. We will prove that the\ncondition in Proposition 2.2 holds with γi=i−ℓfor every i∈[0,n]\\ {ℓ}. Assume towards a\ncontradiction that it does not hold. Thus, there exists a polynomial f(x) =f2(x−ℓ)2+f1(x−ℓ)+\nf0∈Fp[x] such that f(i)∈(i−ℓ)·[−t,t] for alli∈[0,n]\\{ℓ}, butf(ℓ)/\\⌉}atio\\slash= 0. Define,\nS:=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\ni=0\ni/negationslash=ℓep/parenleftbiggf(i)\ni−ℓ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n8and note that by Lemma 3.1 and our choice of t, it holds that S≥n·cos/parenleftBig\n2πt\np/parenrightBig\n>0.7n. On the\nother hand,\nS=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\ni=0\ni/negationslash=ℓep/parenleftbigg\nf2·(i−ℓ)+f1+f0\ni−ℓ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\ni=0\ni/negationslash=ℓep/parenleftbigg\nf2·(i−ℓ)+f0\ni−ℓ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleℓ/summationdisplay\ni=1ep/parenleftbigg\n−f2·i−f0\ni/parenrightbigg\n+n−ℓ/summationdisplay\ni=1ep/parenleftbigg\nf2·i+f0\ni/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleℓ/summationdisplay\ni=1ep/parenleftbigg\nf2·i+f0\ni/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−ℓ/summationdisplay\ni=1ep/parenleftbigg\nf2·i+f0\ni/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nClearly, max( ℓ,n−ℓ)≥n/2. Assume without loss of generality that ℓ≥n/2 and let S′=/vextendsingle/vextendsingle/vextendsingle/summationtextℓ\ni=1ep/parenleftBig\nf2·i+f0\ni/parenrightBig/vextendsingle/vextendsingle/vextendsingle.\nRecall that by our assumption f(ℓ) =f0/\\⌉}atio\\slash= 0, which implies that gcd( f0,p) = 1. Consider\nthe following two options. First, if f2/\\⌉}atio\\slash= 0 then gcd( f0f2,p) = 1 and by the second bound in\nTheorem 3.2 S′=o(n). Second, if f2= 0 then by the first bound in Theorem 3.2 S′=o(n). In\neither case, we conclude that S≤0.5n+o(n) and we arrive at a contradiction for large enough p,\nwhich implies a large enough n.\nLastly, the claim about the bandwidth follows by noting that by Propo sition 2.2, each node\nsends/ceilingleftbig\nlog/parenleftbig/ceilingleftbigp\nt/ceilingrightbig/parenrightbig/ceilingrightbig\n≤ ⌈log(8)⌉= 3 bits.\nRemark 3.4. From a cryptographic perspective, Theorem 3.3 provides an e xplicit leakage attack\non the secret in a 3-out-of-nShamir’s Secret Sharing Scheme (SSS) over prime fields. This attack\nis significant as it requires leaking as little as 3bits from each share to completely reveal the secret.\nSpecifically, we construct deterministic leakage function s that enable the complete recovery of the\nsecret value, f(0). To the best of our knowledge, the only other known attacks on Shamir’s SSS are\nprobabilistic in nature, where the adversary randomly choo ses the leakage functions [21]. The above\nresult therefore shows a significant vulnerability of Shami r’s SSS to targeted attacks.\nRemark 3.5. In this section, we construct a repair scheme that downloads a constant number\nof bits from any node. The total bandwidth is larger than k·log(p), the amount of information\nneeded to learn the polynomial. One might ask, does the infor mation received from the nodes tell us\nthe entire polynomial? We would like to stress that this is no t the case here. We show that there\nare two distinct polynomials on which the scheme transmits t he same information. Consider the\npolynomials f(x) =xandg(x) = 2xand assume that we want to repair the node 0. A node that\nstoresf(α)org(α)will transmits the same value, since f(α),g(α)∈α·[0,t−1]for every α∈F∗\np.\nThus, we cannot distinguish between these two polynomials u pon receiving the information from the\nnodes.\n94 Repair scheme using the Weil bound\nIn Section 3, we developed a repair scheme leveraging bounds on Kloo sterman sums. This section\nintroduces a new scheme for full-length RS codes, utilizing the Weil bo und. This scheme can also\nrepair any node by downloading a constant number of bits from the m ajority of the other functional\nnodes. Remarkably, it facilitates the recovery of the entire polyno mial (codeword), and not just an\nindividual node.\nTo recover the entire polynomial, it is necessary to download at least klog(p) bits in total from\nthe nodes. A straightforward approach would be to retrieve the s ymbols of any knodes and then\nsolve the classical interpolation problem to reconstruct the polyno mial. Our proposed scheme,\nhowever, involves downloading a constant number of bits from more thanknodes. We demonstrate\nthat this information is sufficient to deduce the entire polynomial.\nIn order to show that the scheme indeed recovers the entire polyn omial, we will first introduce\nsome definitions and later present a stronger condition than the on e presented in Proposition 2.2.\nAdecoding scheme for a code Cconsists of nfunctions τi:Fp→A, where Ais some set,\nand a function G:An→Fn\np. This scheme must satisfy the condition that for any codeword\n(c1,...,c n)∈ C,\nG(τ1(c1),...,τ n(cn)) = (c1,...,c n). (3)\nThe following observation provides a condition for the existence of a decoding scheme.\nObservation 4.1. LetC ⊆Fn\npbe a code. A set of nfunctions τi:Fp→Adefine a valid decoding\nscheme, (i.e., there exists a function Gsatisfying (3)) if and only if the mapping\nc∈ C /ma√sto→(τ1(c1),...,τ n(cn))\nis injective.\nTo present the trivial approachsuggested above to retrieve the polynomial as a decoding scheme\nof an [n,k] RS code, let A=Fpand set, say the first kfunctions, τ1,...,τ k:Fp→Fp, as identity\nfunctions, and the remaining n−kfunctions, τi, as constant zero functions.\nOur goal is to construct decoding schemes non-explicitly and explicit ly from a set of nfunctions\nwhere each function outputs only a constant number of bits. We be gin with a non-explicit decoding\nscheme derived by a simple application of the probabilistic method, the proof of which is relegated\nto Appendix A.1. We then proceed with the explicit scheme derived fro m the Weil bound.\nProposition 4.2. Consider an [n,k]RS code over Fpwithk <(n+1)log( s)\n2log(p)+log(s), for some integer s.\nAssume further that the code is defined by the evaluation poin tsα1,...,α n∈Fp. Then, there are\nnfunctions τ1,...,τ n:Fp→[s]such that for any f∈Fp[x]of degree less than k, the mapping\nf/ma√sto→(τ1(f(α1)),...,τ n(f(αn)))is injective and therefore it is sufficient to retrieve ffrom it.\nProposition 4.2 shows that for a constant s, any [n,k]pRS code with k=O(n/log(p)) has a\ndecoding scheme where the functions τihave a constant-size image. This means it is feasible to\nrecover (decode) any polynomial of degree less than kby transmitting a constant number of bits\nfrom each node. However,the result is existential and not explicit. On the other hand, the following\nsimple claim shows that for kwhich is slightly larger than the bound in Proposition 4.2, a decoding\nscheme does not exist for an [ n,k]pRS code.\nClaim 4.3. Consider an [n,k]RS code over Fpwithk >nlogs\nlogp. Then, for any nfunctions\nτ1,...,τ n:Fp→[s], the mapping c∈ C →(τ1(c1),...,τ n(cn))is not injective.\n10Proof.Thereare pkcodewordsand nsvectors[s]n. Thus, for k >nlogs\nlogp, bythepigeonholeprinciple,\nthere are two codewords that map to the same element under τ1,...,τ n.\nBy combining the above results, we conclude that the maximum dimens ion of RS codes that\npermits a decoding scheme with a constant number of bits from each node isk= Θ(n/logp). Our\nnext result, while falling short ofthis benchmark, presentsan explic it decoding scheme. Specifically,\nwe introduce an explicit decoding scheme for a full-length RS code, i.e., of length pwithk=O(√p).\nThisdimensionissmallerthanthemaximumpossibledimensionwhichisΩ( p/logp)forafull-length\nRS code. An intriguing open question is the construction of an explicit decoding scheme for a full-\nlength RS code with dimension Ω( p/logp) that requires only a constant number of bits from each\nnode.\nAs in the case of the repair schemes, we construct the function τiby partitioning Fpinto\narithmetic progressions with step γi∈Fp. The following proposition provides a condition similar\nto that in Proposition 2.2 for the functions τidefined by the γi’s to define a valid decoding scheme.\nFurthermore, we allow also that some of the nodes do not send any in formation; equivalently, the\nfunction they compute on their data is a constant function.\nProposition 4.4. Consider an [n,k]pRS code defined by the evaluation points α1,...,α n. Let\nt < pbe an integer, M⊂[n]of sizem, andγi,i∈[n]\\M, be nonzero elements of Fp. If the only\npolynomial f∈Fp[x]of degree less than kfor which f(αi)∈γi[−t,t]for alli∈[n]\\Mis the zero\npolynomial, then the γi’s define a valid decoding scheme with a total bandwidth of (n−m)·log/parenleftbig\n⌈p\nt⌉/parenrightbig\nbits.\nNote that in the scheme in Proposition 4.4, the nodes with indices i∈Mare the nodes that do\nnot send any information.\nProof.Lets=⌈p\nt⌉and define the sets Aj, 0≤j≤s−1, as in (2). For each i∈[n]\\M, define the\nfunction τiaccording to the partition γi·A0,...,γ i·As−1, i.e.,τi(a) =jif and only if a∈γi·Aj.\nBy Observation 4.1, it suffices to show that the mapping for c∈ C,c/ma√sto→(τ1(c1),...,τ n(cn)), is\ninjective. Let c,c′∈ Cbe two codewords that agree on the nvaluesτi(c) =τi(c′) for alli∈[n]\\M.\nThen,c−c′is a codeword such that its ith symbol belongs to the set γi·[−t,t] for all i∈[n].\nTherefore, by the condition, c−c′must be the zero codeword, i.e., c=c′. The claim about the\nbandwidth follows since each partition consists of exactly ssets.\nNext, we introduce our decoding scheme that utilizes the Weil bound . The scheme facilitates\ndecoding even in scenarios where a subset of the nodes does not tr ansmit any information.\nTheorem 4.5. LetB≥3be a positive integer. Let kandmbe positive integers and pbe a prime\nsuch that k+m≤cos/parenleftBig\n2π\n2B+2π\np/parenrightBig√p. Then, the full-length [p,k]pRS code admits a decoding scheme\nby downloading Bbits from any p−mnodes.\nBefore delving into the proof of the theorem, it is important to note the impact of increasing B,\nthe number of bits transmitted by the nodes. By transmitting more information, we can increase\nthe sum of k+m. This implies that the decoding scheme can be adapted either to RS co des with a\nlarger dimension or to scenarios where a greater number of nodes d o not transmit any information.\nProof.Sett:=/ceilingleftbig\np/2B/ceilingrightbig\nand assume that the nodes ℓ1,...,ℓ m∈Fpdo not transmit any information\nduring the scheme. The result will follow by showing that the condition in Proposition 4.4 holds\nwithγi=/producttextm\nj=1(i−ℓj)−1for alli∈Fp\\ {ℓ1,...,ℓ m}. Assume towards a contradiction that\n11it does not hold. Then, there exists a nonzero polynomial f(x) =/summationtextk−1\nj=0ajxjsuch that f(i)∈/parenleftBig/producttextm\nj=1(i−ℓj)−1/parenrightBig\n[−t,t] for every i∈Fp\\{ℓ1,...,ℓ m}. Thus, by Lemma 3.1\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈Fpep\nf(i)·m/productdisplay\nj=1(i−ℓj)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥pcos(2πt/p)> pcos/parenleftbigg2π\n2B+2π\np/parenrightbigg\n.\nwhere the last inequality follows 2 πt/p≤2π\n2B+2π\np< π. On the other hand, by assumption f/\\⌉}atio\\slash≡0,\nand thus f(x)·/producttextm\nj=1(x−ℓj) is a non-constant polynomial of degree at most k+m−1 and therefore\nby the Weil bound (Theorem 2.1)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈Fpep\nf(i)·m/productdisplay\nj=1(i−ℓj)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(k+m−1)√p,\nand we arrive at a contradiction by the choice of kandm. The claim on the bandwidth follows\nfrom Proposition 4.4.\nRemark 4.6. We note that the decoding scheme presented in Theorem 4.5 is i n particular, also a\nrepair scheme. Indeed, any node of the mnodes that does not send any information can be viewed\nas the failed node.\nRemark 4.7. In Theorem 4.5, we utilized the classical Weil bound on expon ential sums, which\nprovides an upper bound for complete sums, i.e., sums that range over all elements in the prime\nfield. However, it is important to note that there are also upp er bounds for incomplete sums (see,\nfor example, [22] that bounds sums ranging over small subgro ups). These bounds on incomplete\nsums, can be employed to construct decoding schemes for RS.\nRecall that Theorem 1.2 implies that repairing the 0th node in an RS cod e is not feasible when\nthe rate is sufficiently high. Specifically, for any positive integer m, there exists a constant αmsuch\nthat an [ p,αmp]pRS code does not admit a repair of the 0th node using mbits of information from\neach of the other nodes.\nOn the other hand, Theorem 4.5 allows for the repair of the 0th node in a full-length RS code\nwith dimension O(√p). An interesting question arises: how far can one push the dimensio n of the\ncode while using this scheme? In other words, what is the largestdime nsion of a full-length RS code\nthat permits decoding (or repairing) via the scheme given in Theorem 4.5? The following example\nillustrates some of the limitations of this scheme.\nExample. Consider the [p,p−1\n2]pRS code over Fp. Our goal is to show that the decoding scheme\ngiven in Theorem 4.5 is not valid, even if all of the nodes exce pt the0th send almost all their\ninformation. Consider the codeword defined by the polynomia l\nf(x) =−p−3\n2/summationdisplay\ni=0(x+1)i=1−(x+1)p−1\n2\nx,\nof degreep−3\n2. Since we assume that all the nodes send their information ex cept for the 0th node,\nthen by the definition of γiin the proof of Theorem 4.5, we have that γi= 1/ifor each i∈F∗\np.\n12Next, let t= 3and define the function τias in Section 2.2. Since t= 3, the information τi(f(i))\nreceived from the ith node suffices to determine f(i)up to3possible values. Furthermore, τiis\ndefined by the partition γiAj,j= 0,...,⌊p/3⌋ofFpas in(2). Sincef(i)∈i−1·{0,1,2}for every\ni∈F∗\np, we have τi(f(i)) = 0. On the other hand, for g(x)≡0, alsoτi(g(i)) = 0for alli∈F∗\npand\ng(0) = 0/\\⌉}atio\\slash=−p−1\n2=f(0). We conclude that the received information does not suffice to recover the\nvalue of fat0, let alone decode f.\n5 Summary and Open Problems\nThis paper extends the work initiated in [3], focusing on the repair of R S codes over prime fields.\nWe advance this line of work by developing nonlinear repair schemes fo r RS codes, where each node\ntransmits only a constant number of bits. While the total bandwidth of our schemes exceeds that of\nthe trivial repair scheme, they offer a significant advantage in scen arios where each node is limited\nin power or communication and is thus constrained to transmit only a c onstant number of bits—a\ncondition under which the trivial scheme is not feasible.\nFurthermore, we demonstrate that our schemes can be interpre ted as explicit local leakage\nattacks on Shamir’s SSS. To our knowledge, these are the first inst ances of such explicit attacks.\nThe main contribution of this work is establishing connections betwee n bounds on exponential\nsums and the problem of repairing RS codes. By applying some known b ounds, we obtain results\nthat imply both a repair and a decoding scheme for certain RS codes.\nWe conclude with open questions for future research:\n1. Develop efficient algorithms to compute the repair function Gas defined in (1), i.e., efficiently\ncompute the lost data upon receiving inputs from the functioning no des.\n2. Determine the maximal rate of an RS code over prime fields for whic h there is a repair scheme\nthat downloadsaconstantnumberofbits fromeachnode. Constr uctingsuchschemesremains\nan interesting open question.\nReferences\n[1] Fabrice Benhamouda, Akshay Degwekar, Yuval Ishai, and Tal R abin. On the local leakage\nresilience of linear secret sharing schemes. Journal of Cryptology , 34(2):1–65, 2021.\n[2] Todd Cochrane and Christopher Pinner. An improved mordell typ e bound for exponential\nsums.Proceedings of the American Mathematical Society , 133(2):313–320, 2005.\n[3] Roni Con and Itzhak Tamo. Nonlinear repair of reed-solomon cod es.IEEE Transactions on\nInformation Theory , 68(8):5165–5177, 2022.\n[4] Alexandros G Dimakis, P Brighten Godfrey, Yunnan Wu, Martin J Wa inwright, and Kan-\nnan Ramchandran. 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Explicit minimum storage regenerating\ncodes.IEEE Transactions on Information Theory , 62(8):4466–4480, 2016.\n[30] Min Ye and Alexander Barg. Explicit constructions of high-rateM DS arraycodes with optimal\nrepair bandwidth. IEEE Transactions on Information Theory , 63(4):2001–2014, 2017.\n[31] Min Ye and Alexander Barg. Explicit constructions of optimal-acc ess MDS codes with nearly\noptimalsub-packetization. IEEE Transactions on Information Theory , 63(10):6307–6317,2017.\n[32] VictorAZinoviev. Onclassicalkloostermansums. Cryptography and Communications , 11:461–\n496, 2019.\nA Appendix\nA.1 Proof of Proposition 4.2\nFor the reader’s convenience we restate the proposition.\nProposition A.1. Consider an [n,k]RS code over Fpwithk <(n+1)log( s)\n2log(p)+log(s), for some integer s.\nAssume further that the code is defined by the evaluation poin tsα1,...,α n∈Fp. Then, there are\nnfunctions τ1,...,τ n:Fp→[s]such that for any f∈Fp[x]of degree less than k, the mapping\nf/ma√sto→(τ1(f(α1)),...,τ n(f(αn)))is injective and therefore it is sufficient to retrieve ffrom it.\n15Before delving into the proof of Proposition 4.2, it is important to not e its resemblance to\nthe proof of Theorem 2 in [21]. Both proofs employ a similar strategy : they randomly select the\nfunctions τiand then apply the union bound. However, a key distinction lies in the o bjective of the\nrecovery process. While our approach aims to recover the entire p olynomial, the focus of Theorem\n2 in [21] is solely on recovering the secret, i.e., the value of the polynom ial at zero.\nProof.Pick the functions τi:Fp→[s] uniformly at random. Namely, for each i∈[n] andα∈Fp,\nτi(α) is a uniformly distributed random element of [ s]. Fix two distinct polynomials f,gof degree\nless than k, then it holds that\nPr\nτ1,...,τn[(τ1(f(α1)),...,τ n(f(αn))) = (τ1(g(α1)),...,τ n(g(αn)))] =n/productdisplay\ni=1Pr[τi(f(αi)) =τi(g(αi))].\nIff(αi) =g(αi), then necessarily Pr[ τi(f(αi)) =τi(g(αi))] = 1, and since fandgare polynomials\nof degree less than kthey agree on at most k−1 of theαi’s. Iff(αi)/\\⌉}atio\\slash=g(αi), then by the definition\nofτi, we have that Pr[ τi(f(αi)) =τi(g(αi))] = 1/s. Thus,\nPr\nτ1,...,τn[(τ1(f(α1)),...,τ n(f(αn))) = (τ1(g(α1)),...,τ n(g(αn)))]≤/parenleftbigg1\ns/parenrightbiggn−k+1\n.\nNow, by the union bound, the probability that there exist two polyno mialsfandgof degree\nless than ksuch that ( τ1(f(α1)),...,τ n(f(αn))) = (τ1(g(α1)),...,τ n(g(αn))) is at most p2k·\n(1/s)n−k+1<1 by the choice of k. Thus with positive probablity the required functions τiex-\nist, and the result follows.\n16"    },    {        "title": "2402.02390v1.Improved_Upper_Bound_for_the_Size_of_a_Trifferent_Code.pdf",        "content": "arXiv:2402.02390v1  [cs.IT]  4 Feb 2024Improved Upper Bound for the Size of a Trifferent Code\nSiddharth Bhandari *, Abhishek Khetan†\nAbstract\nA subset C ⊆ { 0, 1, 2}nis said to be a trifferent code (of block length n) if for every three distinct\ncodewords x,y,z∈ C, there is a coordinate i∈ {1, 2, . . . , n}where they all differ, that is, {x(i),y(i),z(i)}\nis same as {0, 1, 2}. Let T(n)denote the size of the largest trifferent code of block lengt hn. Un-\nderstanding the asymptotic behavior of T(n)is closely related to determining the zero-error capac-\nity of the (3/2)-channel defined by Elias [ Eli88 ], and is a long-standing open problem in the area.\nElias had shown that T(n)≤2×(3/2)nand prior to our work the best upper bound was T(n)≤\n0.6937×(3/2)ndue to Kurz [ Kur23 ]. We improve this bound to T(n)≤c×n−2/5×(3/2)nwhere c\nis an absolute constant.\n1 Introduction\nLetqbe a positive integer and let Σ={0, 1, 2, . . . , q−1}be a finite alphabet. We use the notation [n]to\ndenote the set {1, 2, . . . , n}when nis a positive integer.\nDefinition 1.1 (q-perfect hash codes & Trifferent codes) .For positive integers q ≥2and n, a code C ⊆ Σn\nis said to be a q- perfect hash code ofblock length n if for any q distinct codewords x 1,x2, . . . , xqinCwe have\na coordinate i ∈[n]such that {xj(i)|1≤j≤q}=Σ, where x(i)denotes the ithcoordinate of x. When q =3,\na q-perfect hash code is also referred to as a trifferent code.\nWe will write T (q,n)to denote the maximum size a q-perfect hash code of block leng th n attains.\nUnderstanding the asymptotics of T(q,n)asnincreases is an important question, both in infor-\nmation theory and computer science. In this paper we will foc us solely on the case of q=3. As\nmentioned in Definition 1.1 , in this case a q-perfect hash code is popularly referred to as a ‘trifferent ’\ncode and examining the growth of T(3,n)is referred to as the ’trifference problem’, a long-standin g\nopen problem that has garnered considerable attention. (Se e for instance the 2014 Shannon Lecture:\n‘On The Mathematics of Distinguishable Difference’ by János Körner.) Since we concern ourselves only\nwith the case of q=3 in the remainder of the paper, we refer to T(3,n)asT(n). In a seminal work\nElias [ Eli88 ] showed that T(n)≤2×(3/2)n. Prior to our work the best upper bound on T(n)was\nT(n)≤0.6937×(3/2)nforn≥10, due to Kurz [ Kur23 ]. We improve this bound for sufficiently large\nnin the following result.\nTheorem 1.1 (Main theorem) .There exists a universal constant c with the following prope rty. LetC ⊆\n{0, 1, 2}nbe a trifferent code of block length n as defined in Definition 1.1 . Then,|C| ≤ c×n−2/5×(3/2)n.\nThus, T(n)≤c×n−2/5×(3/2)n.\nBefore delving into the trifference problem, we elucidate h owq-perfect hash codes are connected\nto perfect hashing and how they simultaneously serve as erro r-correcting codes for a classical channel\nin information theory. Subsequently, we highlight notable findings in the estimation of T(q,n)for the\ncases when q>3.\n*Siddharth Bhandari is at the Toyota Technological Institut e at Chicago. Email: siddharth@ttic.edu\n†Abhishek Khetan is a post-doctoral researcher at the Tata In stitute of Fundamental Research, CAM, Bangalore. Email:\nkhetan21@tifrbng.res.in\n1General q-perfect Hash Codes\nAq-perfect hash code C={x1,x2, . . . , xs}of block length nand size sis readily seen to be a family\n{h1, . . . , hn}ofnhash functions from [s]toΣwhere the ithhash function himaps j∈[s]toxj(i), i.e.,\ntheithcoordinate of the codeword xj. The family of hash functions {hi}has the property that any\nsubset Qof size qof the domain [s]is perfectly hashed by at least one of the hash functions hi, i.e.,\n{hi(j):j∈Q}=Σ. Alternatively, a q-perfect hash code, say Cas described above, can also be cast as a\ncover of the q-uniform complete hypergraph on the vertex set [s], say Ks(q), using nhypergraphs which\nareq-uniform and q-partite. Specifically, we think of each hash function hias a hypergraph Hiwhose\nvertex set is [s]and the edge set is {Q⊆[s]:|Q|=q∧{hi(j)|j∈Q}=Σ}. See the excellent survey of\nRadhakrishnan [ Rad01 ] for more details.\nAq-perfect hash code also serves as an error-correcting code f or a classical channel studied in zero-\nerror information theory: the q/(q−1)channel. The input and output alphabets of this channel are a\nset of qsymbols, namely Σ; when the channel receives the symbol i∈Σas input, the output symbol\ncan be anything other than iitself. For the q/(q−1)channel it is impossible to recover the message\nwithout error if the code has at least two codewords: in fact, no matter how large the block length, for\nevery set of up to q−1 input codewords, one can construct an output word that is co mpatible with\nallof them. However, there exist codes with positive rate where on receiving an output word from\nthe channel, one can narrow down the possibilities for the in put message to a set of size at most q−1,\nthat is, we can list-decode with lists of size q−1. Such codes are called (q−1)-list-decoding codes for\ntheq/(q−1)channel. It is well known that a q-perfect hash code Cof block length nand size sis\nequivalent to a (q−1)-list-decoding code for the q/(q−1)channel with block length n(for instance see\nthe introduction of Bhandari and Radhakrishnan [ BR22 ]).\nDefinition 1.2 (Rate & Capacity) .For positive integers q ≥2and n, let Cbe a q-perfect hash code of block\nlength n. Following Elias [ Eli88 ], we define the rate ofCas RC:=1\nnlog2(|C|/(q−1)). We define the q-\ncapacity as\ncap(q) =lim sup\nn→∞1\nnlog2T(q,n)\nq−1.\nRemark 1.2. It is not known if ‘ lim sup ’ can be replaced by ‘ lim’ in the definition of capacity; see [ Ari94 ,\nFootnote 1].\nMany significant improvements have been made in understandi ngcap(q)and related quantities for\nq>3. We list some of them below and refer the reader to the work of Bhandari and Radhakrishnan\n[BR22 ] for a more detailed survey. Fredman and Komlós’ seminal wor k [FK84 ] established cap(q)≤\nexp(−B1q)for a constant B1>0, independent of q. Guruswami and Riazanov [ GR19 ] demonstrated\nthe non-optimality of the Fredman-Komlós upper bound for q≥4 and provided explicit improvements\nforq=5, 6. Costa and Dalai [ CD20 ] resolved a conjecture by Guruswami and Riazanov, completi ng\nthe explicit computation for improving the Fredman-Komlós bound across all q, and introduced an\nalternative method yielding substantial enhancements for q=5, 6. For q=4, Dalai, Guruswami, and\nRadhakrishnan [ DGR20 ] improved the upper bound to cap(4)≤6/19≈0.3158, surpassing Arikan’s\nprevious bound of 0.3512 [ Ari94 ], while Körner and Marton [ KM88 , eq (1.2)] established a lower bound\nofcap(4)≥(1/3)lg(32/29)≈0.0473. Additionally, Xing and Yuan [ XY19 ] extended Körner and\nMarton’s concatenation technique, demonstrating improve d lower bounds on capacity for q=4, 8, all\nodd integers greater than 3 and less than 25, and sufficiently large qnot congruent to 2 (mod 4).\nThe Trifference Problem\nDespite receiving considerable attention, progress for th e trifference problem has been relatively mod-\nest when compared to the situation for q>3. Elias [ Eli88 ] showed that 0.08 ≈lg(3)−1.5≤cap(3)≤\nlg(3)−1≈0.58; Körner and Marton [ KM88 ] improved the lower bound above to 0.212 ≈(1/4)lg(9/5)≤\ncap(3)via code concatenation. Under the further assumption of lin earity, i.e., if we think of Σas\nF3and assume that the trifferent code C ⊆ Fn\n3is a linear subspace of Fn\n3, some improvements have\nbeen obtained in the upper bound on cap(3). Pohoata and Zakharov [ PZ22 ] obtained linear -cap(3)≤\n(1/4−ǫ)×log3(2)≈0.3962−ǫfor some absolute constant ǫ>0 following which Bishnoi, D’haeseleer,\nGijswijt and Potukuchi [ BDGP23 ] obtained linear -cap(3)≤(1/4.55)×log3(2)≈0.3483.\n2Notwithstanding the above results, the current best upper b ound on cap(3)for general trifferent\ncodes remains the one given by Elias [ Eli88 ], up to a constant factor. As such, there has been an impetus\nto view T(n), the largest size of a trifferent code of block length n, with a more refined lens. Elias’ upper\nbound and Körner and Marton’s lower bound can be recast in ter ms of T(n)asT(n)≤2×(3/2)nand\nT(n)≥(9/5)n/4respectively. Recently, via a computer search for a large tr ifferent code of block length\nup to n≤6, combined with a number theoretic argument, Fiore, Gnutti and Polak [ FGP22 ] showed that\nT(n)≤1.09×(3/2)nforn≥12. Even more recently Kurz [ Kur23 ] extended the computer search for\ntrifferent codes of block lengths up to n≤7 and obtained T(n)≤0.6937×(3/2)nforn≥10.\nWhat makes studying T(n)intriguing is the fact that the upper bound of 2 ×(3/2)nis obtained\nvia a relatively simple pruning argument (described below) which has proved difficult to improve.\n(See for instance the work of Costa and Dalai [ CD21 ] as to the limits of the ‘slice rank’ method for the\ntrifference problem, which, however was successful in boun ding the largest size of a 3-AP free set in\nFn\n3). Additionally, the lower bound of (9/5)n/4is obtained not via a purely random construction, but\nvia concatenating a random outer code and an algebraic inner code (known as the Tetra code). The\npruning argument for the upper bound of 2 ×(3/2)nis as follows: let Cbe a trifferent code of block\nlength nand let a1∈ {0, 1, 2}be a least occurring symbol in the first coordinate of all the c odewords.\nThen, let C1be the code obtained by deleting those codewords from Cthat have a1in the first coordinate.\nObserve that |C1| ≥(2/3)|C|. Now since Cwas trifferent, same is true for C1. But any three distinct\nstrings from C1cannot exhibit the trifference property in the first coordin ate and hence for any three\ndistinct codewords x,y,zinC1there must exist a coordinate i>1 such that {x(i),y(i),z(i)}={0, 1, 2}.\nProceeding iteratively in this manner we let a2be a least occurring symbol in the second coordinate of\ncodewords in C1, and then obtain C2fromC1by deleting those codewords which have a2in their second\ncoordinate, and so on, till we obtain Cn. Thus,|Cn| ≥(2/3)n×|C| . But observe that Cnis a trifferent\ncode where three distinct strings cannot exhibit the triffe rence property in anycoordinate. Therefore\n2≥ |C n|, which leads to |C| ≤2×(3/2)n.\nWe restate our main result below, which is an improved upper b ound on T(n).\nTheorem 1.1 (Main theorem) .There exists a universal constant c with the following prope rty. LetC ⊆\n{0, 1, 2}nbe a trifferent code of block length n as defined in Definition 1.1 . Then,|C| ≤ c×n−2/5×(3/2)n.\nThus, T(n)≤c×n−2/5×(3/2)n.\nTo prove Theorem 1.1 we first introduce a close variant of trifferent codes which w e call ‘bounded\ntrifferent’ codes.\nDefinition 1.3 (r-bounded trifferent codes) .LetC ⊆ { 0, 1, 2}nbe a trifferent code of block length n. For an\ninteger r≥0, we callCan r- bounded trifferent code if for all codewords x ∈ C we have that the number of 2’s\nin x is r, i.e., |{i∈[n]:x(i) =2}|=r. Further, for n ≥r let T b(n,r)denote the maximum size an r-bounded\ntrifferent code of block length n attains.\nIn the remainder of the paper, when we talk about Tb(n,r)it is to be understood that n≥ras\nanr-bounded trifferent code of block length n<rhas size 0. Note that Tb(n,r)≤2×(n\nr)as for a\ngiven subset of coordinates S⊆[n], in any trifferent code of block length nthere can be at most two\ncodewords, say xand y, such that Sis precisely the location of 2’s for both xand y, i.e., S={i∈[n]|\nx(i) =2}={i∈[n]|y(i) =2}. If there were three, they would be a counter-example to the t rifference\nproperty. Next, we prove a simple lemma relating T(n)and Tb(n,r), thus, highlighting the importance\nof studying r-bounded trifferent codes.\nLemma 1.3 (Size of trifferent codes in terms of r-bounded trifferent codes) .\nT(n)≤2(r−n)×Tb(n,r)/parenleftbiggn\nr/parenrightbigg×3n\nProof. It will be convenient to think of Σ:={0, 1, 2}asF3. LetCbe a trifferent code of block length n.\nLetArdenote the set of all those x∈Σnsuch that xwitnesses exactly r2s. More precisely\nAr={x∈Fn\n3:|{i∈[n]:x(i) =2}|=r}.\nFor strings xand vinFn\n3letx+vdenote addition in Fn\n3. Observe that the code C+v:={x+v|x∈ C}\nis also trifferent for any string v∈Fn\n3. Hence,|(C+v)∩Ar| ≤Tb(n,r).\n3Letvbe a uniformly random element of Fn\n3. We write C+vto denote the random subset C+v\nwhere vis picked according to the distribution of v. Note that for any x∈Fn\n3we have:\nEv[1[x∈ C+v]] = Pv(x−v∈ C) =|C|\n3n.\nTherefore,\nTb(n,r)≥Ev[|(C+v)∩Ar|] =∑\nx∈ArEv[1[x∈ C+v]] =|Ar|×|C|\n3n.\nNow using |Ar|=(n\nr)2n−r, we get the desired result.\nRemark 1.4. Even more generally, the above lemma shows that for any S ⊆ {0, 1, 2}nif T b(S)denotes that\nlargest size of a trifferent code contained in S, then we have T(n)≤Tb(S)\n|S|×3n. Hence, it might also be interesting\nto look at sets S which are not of the form {x∈ {0, 1, 2}n: #2′s in x=r}for some integer r.\nIn light of the above lemma it is natural to define the notion of anr-bounded density.\nDefinition 1.4 (r-bounded density) .Recall that T b(n,r)denotes the maximum size an r-bounded trifferent\ncode of block length n attains. The r- bounded density at block length n, denoted as ρb(n,r), is defined as\nρb(n,r) =2(r−n)×Tb(n,r)\n(n\nr)\nHence, Lemma 1.3 can be cast as T(n)≤ρb(n,r)×3nfor all r≥0. The bound obtained using the\npruning argument, that is T(n)≤2×(3/2)n, can be obtained as an instantiation of Lemma 1.3 with\nr=0. In this case, it is clear that Tb(n,r) = 2 (there can be at most two codewords in a trifferent\ncode if the symbol 2 does not appear in any codeword) and hence ρb(n,r) = 21−n. It turns out that\nρb(n, 1) = 22−nas we show that Tb(n, 1) = 2n(see Lemma 3.1 ). This bound is worse than what we\nobtained on ρb(n, 0). However, the situation improves when we consider r≥2. Our main contribution\nis showing that for r=3,ρb(n,r)≤c×n−2/5×2−nfor some absolute constant c. More precisely, we\nhave the following result.\nTheorem 1.5 (Bounding r-bounded density) .Let T b(n,r)andρb(n,r)be as defined in Definitions 1.3 and1.4\nrespectively. Then, we have constants c′and c such that\na) T b(n, 2)≤c′×n5/3and hence ρb(n, 2)≤c×n−1/3×2−nand\nb) T b(n, 3)≤c′×n13/5and hence ρb(n, 3)≤c×n−2/5×2−n.\nWe describe the proof idea of Theorem 1.5 for the case when r=2. We proceed via constructing\na graph related to an r-bounded trifferent code, say Cb, with block length n. This graph has roughly\nas many edges as |Cb|. The crucial observation is that certain bipartite structu res are forbidden in this\ngraph. Then, an application of the famous K˝ ovári–Sós–Turá n (KST) theorem yields a bound on the\nnumber of edges in this graph which also serves as a bound on th e size ofCb. We give the details below.\nRecall that each codeword in Cbhas exactly two 2’s. Now, consider the graph GCbon the vertex set\n[n]where for each codeword x∈ C ban edge{i,j}with i/\\e}atio\\slash=jis added to GCbifx(i) = x(j) = 2, i.e., i\nand jare the locations of 2’s in x. Note that an edge {i,j}can be added by at most 2 codewords in Cb.\nHence, GCbhas at least half as many edges as |Cb|. Next, we show via the PHP and trifference property\nofCbthat K3,9—the complete bipartite graph with the partite sets having s izes 3 and 9 respectively—is\nforbidden in GCb. Applying the KST theorem yields an upper bound on the number of edges in GCbas\nc′′×n5/3for some constant c′′. Hence, we obtain our desired bound on |Cb|and Tb(n, 2).\nThe proof for when r=3 proceeds along similar lines but we define the graph GCbmore prudently.\nThe detailed proof of Theorem 1.5 appears in Section 2 .\nArmed with Theorem 1.5 and Lemma 1.3 we easily obtain the proof of Theorem 1.1 .\n4Proof of Theorem 1.1 .With r=3 we have the following.\nT(n)≤2(r−n)×Tb(n,r)\n(n\nr)×3nbyLemma 1.3\n=ρb(n,r)×3n\n≤c×n−2/5×2−nbyTheorem 1.5 .\ngiving the desired result.\nFrom the above discussion it is tempting to analyze ρb(n,r)when both rand nare growing with n\ngrowing much faster than r. As such, we define a notion of r-bounded deficit and sup-bounded deficit\nwhich serve to get a sense of the speed at which ρb(n,r)decays.\nDefinition 1.5 (r-bounded deficit & sup-bounded deficit) .For an integer r ≥1, let T b(n,r)denote the\nmaximum size an r-bounded trifferent code of block length n a ttains. Let ∆r(n) = r−logTb(n,r)\nlogn. Then, the\nr-bounded deficit is defined as\n∆r:=lim sup\nn→∞/parenleftbigg\nr−logTb(n,r)\nlogn/parenrightbigg\n=lim sup\nn→∞∆r(n).\nand the sup-bounded deficit is defined as\n∆∞:=limr→∞∆r.\nRemark 1.6. 1. To be more precise we should have defined ∆∞aslim supr→∞∆r. However, Corollary 1.6.1\nshows that ∆ris increasing in r.\n2. If we unpack the above definition in terms of T b(n,r), then Lemma 1.3 yields that\nT(n)≤cr×n−∆r(n)×(3/2)n\nwhere c ris a constant depending only on r. Hence, studying ∆ras r grows is helpful in proving better\nupper bounds on T (n).\nSince, Tb(n,r)is at most 2 ×(n\nr), the r-bounded deficit, ∆r, is always non-negative. In fact, by\nTheorem 1.5 we have shown that ∆2≥1/3 and ∆3≥2/5. Via a simple application of the PHP we\nshow that ∆ris increasing in r.\nCorollary 1.6.1. ∆r+1≥∆rfor any integer r ≥1. Hence, for all integers r ≥3, there are constants c rand c′\nr\nsuch that we have: T b(n,r)≤c′\nr×nr−2/5and hence ρb(n,r)≤cr×n2/5×2−n.\nProof. LetCb⊆ {0, 1, 2}nbe an r+1-bounded trifferent code of block length n. By double-counting we\ncan find a coordinate i∈[n]such that there is a subset C′\nb⊆ C bof size at least r/n×|C b|and every\ncodeword of C′\nbhas a 2 at coordinate i. Notice that C′\nbcan be equivalently thought of as an r-bounded\ntrifferent code with block length n−1 since the i-th coordinate is same (namely 2) for each codeword.\nHence, Tb(n,r+1)≤n/r×Tb(n−1,r)from where it follows that ∆r+1≥∆r.\nNext, we turn our attention to establish upper bounds on ∆r, i.e., prove lower bounds on Tb(n,r).\nOur constructions are based on thinking about the codewords as point-line incidences in an appropriate\nfinite-dimensional vector space over a finite field. See Section 3 for the detailed constructions.\nTheorem 1.7 (Upper bounds on ∆r).Tb(n, 1) = 2n and hence ∆1=0. Also, ∆3≤3/2, i.e., T b(n, 3)≥\nc1×n3/2for some constant c 1>0: further, for every positive integer r, a power of 3, we have ∆r≤r−rαwhere\nα=1−log3(2)≈0.369 .\n5Further Questions\nA few interesting questions which arise are as follows. Is ∆∞=∞? If yes, this immediately shows that\nfor any constant dwe have T(n)≤n−d×(3/2)n, i.e., the size of a family of trifferent codes of growing\nblock lengths, say n→∞, decays faster than (3/2)ndivided by any polynomial factor. A more nuanced\nunderstanding in terms of ρb(n,r)with growing rmight also lead to an improved upper bound on\ncap(3), hence making progress on the long-standing open problem. F urther, it is also interesting to\nunderstand ∆rmore precisely for small values of rsuch as 2 and 3: is ∆2=1/2?\nAcknowledgements\nWe are highly indebted to Prof. Jaikumar Radhakrishnan for v arious insightful discussions and for\npainstakingly proofreading the manuscript. We thank Prof. Alexander Razborov for helpful discus-\nsions. We also acknowledge Aakash Bhowmick for helping us wi th experiments related to constructing\ntrifferent codes. Lastly, we thank Prof. Nishant Chandgoti a for his inputs and encouragement.\n2 Upper bounds on the size of r-bounded trifferent codes\nIn this section we prove Theorem 1.5 . We restate the theorem below for convenience.\nTheorem 1.5 (Bounding r-bounded density) .Let T b(n,r)andρb(n,r)be as defined in Definitions 1.3 and1.4\nrespectively. Then, we have constants c′and c such that\na) T b(n, 2)≤c′×n5/3and hence ρb(n, 2)≤c×n−1/3×2−nand\nb) T b(n, 3)≤c′×n13/5and hence ρb(n, 3)≤c×n−2/5×2−n.\nTo prove Theorem 1.5 we will need to apply the famous result of K˝ ovári, Sós and Tur án, known\npopularly as the KST theorem, or rather a version of it due to H yltén-Cavallius.\nTheorem 2.1 (KST theorem due to Hyltén-Cavallius [ HC58 ]).The Zarankiewicz function z (u,v;s,t)de-\nnotes the maximum possible number of edges in a bipartite gra ph G= (U∪V,E)for which |U|=u and\n|V|=v, but which does not contain a subgraph of the form K s,twhere s vertices come from U and t from V (here\nKs,tdenotes the complete bipartite graph with s and t vertices in the two partite sets). Then,\nz(u,v;s,t)<(t−1)1\ns(u−s+1)v1−1\ns+(s−1)v.\nProof of Theorem 1.5 .We first proceed with the proof when r=2. LetCbbe an r-bounded trifferent code\nof block length nwith r=2. Thus, every codeword of Cbhas two 2’s. As discussed previously, we\nconstruct a graph GCbusing the code Cb. The graph GCbhas the vertex set [n]and the set of edges Eis\nE={{i,j} |i/\\e}atio\\slash=j∧∃x∈ C b:x(i) =x(j) =2}.\nIn other words, for each codeword x∈ C b, an edge, say xe={i,j}, is added to GCb, where iand jare\nthe locations of 2’s in x.\nAs we have argued previously, for any subset S⊆[n]of coordinates there can be at most two\ncodewords, say xand y, such that Sis precisely the location of 2’s for both xand y, i.e., S={i∈[n]|\nx(i) = 2}={i∈[n]|y(i) = 2}. (If there were three, they would contradict the trifferenc e property.)\nThis shows that an edge {i,j}can be added by at most 2 codewords in Cb. Hence,\n|E| ≥ |C b|/2,\ni.e., there are at least half as many edges as |Cb|. At the cost of another factor of 1/2 on |E|we can\nassume that GCbis bipartite of the form (U∪V,E′), where V= [n]\\Uand|E′| ≥ | E|/2. (A random\nequi-bipartition of [n]will have this property on expectation.)\nNext, we show that K3,9is forbidden in GCb. To see this suppose there exist distinct i1,i2,i3inUand\ndistinct j1,j2, . . . , j9inVsuch that subgraph induced by GCbon these vertices is K3,9. We denote the\nedge{ik,jℓ}byek,ℓ. Further, let xk,ℓdenote a codeword in Cbcorresponding to which the edge ek,ℓwas\n6added to GCb: if there are two such codewords then choose one arbitrarily and fix it. By the PHP there\nis a subset T⊆[9]of size at least 3 such that all codewords in {x1,ℓ|ℓ∈T}have the same value on the\ncoordinates i2and i3, i.e.,|{x1,ℓ(i2)|ℓ∈T}|=1 and|{x1,ℓ(i3)|ℓ∈T}|=1. Again, by the PHP we\ncan find a subset T′⊆Tof size at least 2 such that all codewords in {x2,ℓ|ℓ∈T′}have the same value\non the coordinate i3, i.e.,|{x2,ℓ(i3)|ℓ∈T′}|=1. WLOG let us assume that T′={j1,j2}(see Figure\nFig. 1 ).\nThen, by the PHP there must exist two vertices icand id(with c<d) in{i1,i2,i3}such that xc,1(j2) =\nxd,1(j2). But then the three codewords xc,1,xd,1and xc,2are a counter-example to the trifference of Cb. To\nsee this let w∈[n]. Ifw/∈ {ic,id,j1,j2}, then none of the three codewords have the symbol 2 at wand\nhence xc,1,xd,1and xc,2cannot witness trifference at w. Now if w=ic, then both xc,1and xc,2have the\nsymbol 2 at w; ifw=id, then, because of our definition of T′we have xc,1(w) =xc,2(w); ifw=j1, then\nboth xc,1and xd,1have the symbol 2 at w; finally if w=j2, then by our assumption xc,1(j2) = xd,1(j2).\nSo no matter what wis, some two of xc,1,xd,1and xx,2agree on w, and hence these three codewords\ncontradict the trifference property. Hence, GCbisK3,9-free.\nApplying Theorem 2.1 with u,v=n/2 and s=3,t=9, yields c′′×n5/3as and upper bound on the\nsize of E′for some constant c′′. Hence, we obtain our desired bound on |Cb|and Tb(n, 2).\nNext, we turn our attention to the case when r=3. Again let Cbbe an r-bounded trifferent code with\nr=3 and block length n. Recall that each codeword in Cbnow has three 2’s. This time we construct a\nbi-partite graph GCb= (U,V,E)with U= [n],V=([n]\n2)and edge set Edescribed as follows. For each\ncodeword x∈ C bwe add the edge xe= (i,{j,k})toEwhere x(i) = x(j) = x(k) =2 and i<j<k. As\nargued previously, edge e∈Ecan be added by at most 2 codewords in Cb: therefore, |E| ≥0.5×|C b|.\nNow, we will show that GCbisKs=5,t=221-free.\nSuppose not, and that there exist vertices i1,i2, . . . , i5inUand S1,S2, . . . , St(note that each Siis a\n2-subset of coordinates) in Vsuch that GCbinduces a K5,ton these vertices. We denote the edge (ik,Sℓ)\nbyek,ℓand the codeword corresponding to it with xk,ℓ: if there are two such codewords then choose\none arbitrarily. By the PHP there is a subset T1⊆[t]of size at least t/24such that all codewords in\n{x1,ℓ|ℓ∈T}have the same value on the coordinates i2,i3,i4,i5, i.e.,|{x1,ℓ(i2)|ℓ∈T}|=1 and so\non for i3,i4,i5. Again, by the PHP there is a subset T2⊆T1of size at least |T1|/24=t/28such that\nall codewords in {x2,ℓ|ℓ∈T}have the same value on the coordinates i1,i3,i4,i5. Continuing this\nway for the remaining coordinates i3,i4,i5, we obtain a subset T5of size at least t/220=2: WLOG\nsayT5={S1,S2}, thus, for any c,d∈[5]we have xc,1(id) = xc,2(id). Further, let S=S2\\S1, then\n|S| ≤2. By the PHP there exists c,d∈[5]with c<dsuch that xc,1(S) =xd,1(S)where x(S)denotes the\ntuple(x(i)|i∈S). However, now the three codewords xc,1,xc,2and xd,1are a counter-example to the\ntrifference of Cb. This is because for any coordinate w∈[n]ifwis not ic,idor in S1,S2, then none of the\nthree codewords have the symbol 2 at w: further, if w=ic, then both xc,1and xc,2have the symbol 2 at\nw; ifw=id, then, because of our definition of T5we have xc,1(w) = xc,2(w); ifw∈S1, then both xc,1\nand xd,1have the symbol 2 at w; finally if w=S2, then by our assumption xc,1(S2) = xd,1(S2). Hence,\nGCbisK5,221-free.\nApplying Theorem 2.1 with u=nand v=(n\n2)and s=5,t=221, yields a bound on the number of\nedges in Easc′′×n3−2/5=c′′×n13/5for some constant c′′. Hence, we obtain our desired bound on\n|Cb|and Tb(n, 3).\nRemark 2.2. We can improve the (huge) constant 221appearing in the above proof to some extent by following\na strategy similar to the one employed in the case when r =2. However, it is easier to work with the current\nnumbers and this doesn’t seem to hurt the bounds of Theorem 2.1 beyond a constant factor.\n3 Lower Bounds on the size of r-bounded trifferent codes\nIn this section we will prove Theorem 1.7 which we restate below for convenience.\nTheorem 1.7 (Upper bounds on ∆r).Tb(n, 1) = 2n and hence ∆1=0. Also, ∆3≤3/2, i.e., T b(n, 3)≥\nc1×n3/2for some constant c 1>0: further, for every positive integer r, a power of 3, we have ∆r≤r−rαwhere\nα=1−log3(2)≈0.369 .\nWe will first focus on the case of r=1.\n79rest resti1i2i3j1j2··· j9\n2...22 2\n2...\n2x1,9...x1,2x1,1\n9rest resti1i2i3j1j2··· j9\n2...22 2\n2...\n2x2,9...x2,2x2,1\n9rest resti1i2i3j1j2··· j9\n2...222\n2...\n2x3,9...x3,2x3,1PHPx29rest restT′\ni1i2i3j1j2··· j9\n2...22 2\n2...\n2α\nαβ\nβ\n9rest resti1i2i3j1j2··· j9\n2...22 2\n2...\n2γ\nγ\n9rest resti1i2i3j1j2··· j9\n2...222\n2...\n2\ni1i2i3\nj1j2j3j4j5j6j7j8j9\nFigure 1: Illustrating the proof for r=2. The collection of codewords before the application of the\nPHP is displayed in the left column of the figure: they corresp ond to the edges of the K3,9supposed\nto exist for the sake of contradiction. For example, the code word x3,2corresponds to the highlighted\nedge{i3,j2}, and the first block of codewords on the left corresponds to th e edges incident to i1. After\napplying the PHP we are left with the non-grayed codewords in the right column corresponding to the\nsetT′={j1,j2}. Eventually, we find three codewords from the non-grayed one s which do not exhibit\nthe trifference property.\n8Lemma 3.1 (Maximum size of trifferent codes where each codeword has on e 2).For each integer n ≥1\nwe have T b(n, 1) =2n.\nProof. As we have argued previously, in any trifferent code C, for any subset S⊆[n]of coordinates\nthere can be at most two codewords, say xand y, such that Sis precisely the location of 2’s for both x\nand y, i.e., S={i∈[n]|x(i) =2}={i∈[n]|y(i) =2}. Hence, Tb(n, 1)≤2n. Next, we construct an\nr-bounded trifferent code ( r=1) with block length nand size 2 n. Let A1={x∈ {0, 1, 2}n|#2’s in x=\n1}. Further, for each i∈[n], letui,vi∈ {0, 1, 2}nbe defined as\nui(j) =\n\n2 if j=i,\n1 if i<j,\n0 otherwise,vi(j) =\n\n2 if j=i,\n1 if i>j,\n0 otherwise.\nConsider the code Cb⊆A1defined as ∪i∈[n]{ui,vi}. Clearly, |Cb|=2n. We claim the Cbis a trifferent\ncode. Consider any three different codewords x,yand z: there are two cases of interest to check (other\ncases can be reduced to these): (a) x=ui,y=vi,z=ujorvjwhere i/\\e}atio\\slash=j, in which case we have\n{x(j),y(j),z(j)}={0, 1, 2}and (b) x=ui,y=uj,z=ukorvkwhere i<j<k, in which case either\n{x(j),y(j),z(j)}={0, 1, 2}or{x(i),y(i),z(i)}={0, 1, 2}respectively.\nNow, we turn to the general case when ris a power of 3.\nLemma 3.2 (Maximum size of trifferent codes where each codeword has 3tmany 2’s) .Let t≥0be an\ninteger and let r =3t. Suppose that for all positive integers n ≥r we have T b(n,r)≥ct×n(3/2)tfor some\nconstant c t>0depending only on t. Then, for all positive integers n ≥3r we have T b(n, 3r) =ct+1×n(3/2)t+1\nfor some constant c t+1>0depending only on t +1.\nProof. LetFqbe a large enough finite field. Let P=F2\nqandLbe the set of all the affine lines in F2\nq.\nThus,\n|P|=q2,|L|=q2+q.\nLetS={(p,ℓ)∈ P ×L | p∈ℓ}and note that |S|=qand|L|=q3+q2. For each line ℓ∈ L choose\na permutation σℓof the points of ℓsuch that σℓhas no fixed points. Further, let f:S → P be defined\nasf(p,ℓ) =σℓ(p). Notice that f(p,ℓ)is a point on ℓbut is different from p. Let ϕ:P → { 0, 1, 2}nand\nψ:L → { 0, 1, 2}nbe injective functions whose images are 3t-bounded trifferent codes where nis the\nsmallest integer such that ct×n(3/2)t≥q2+q. The maps ϕand ψexist by the inductive hypothesis for\nt.\nDefine a map θ:S → { 0, 1, 2}nasθ(p,ℓ) = ϕ(f(p,ℓ)). Finally, define τ:S → { 0, 1, 2}n×\n{0, 1, 2}n×{0, 1, 2}nas\nτ(p,ℓ) = ( ϕ(p),ψ(ℓ),θ(p,ℓ)).\nWe claim that the image of τ, denoted by τ(S), is a 3 r=3t+1-bounded trifferent code.\nTowards this end, let e1= (p1,ℓ1),e2= (p2,ℓ2)and e3= (p3,ℓ3)be three pairwise distinct elements\nofSand consider their encodings τ(p1,ℓ1),τ(p2,ℓ2)and τ(p3,ℓ3). Ifp1,p2and p3are pairwise distinct,\nthen ϕ(p1),ϕ(p2)and ϕ(p3)will exhibit the trifference property as the image of ϕis a trifferent code by\nconstruction. Similarly, if ℓ1,ℓ2andℓ3are pairwise distinct then ψ(ℓ1),ψ(ℓ2)and ψ(ℓ3)will exhibit the\ntrifference property. So we may WLOG assume that p1=p2=p(say) and ℓ2=ℓ3=ℓ(say). Write\np′in place of p3andℓ′in place of ℓ1. See Fig. 2 . We show that θ(p,ℓ′),θ(p,ℓ)and θ(p′,ℓ)exhibit the\ntrifference property to finish the proof of the claim.\nll′\npp′ y zx\nFigure 2: The case when there are only two points and two lines .\n9Note that p/\\e}atio\\slash=p′andℓ/\\e}atio\\slash=ℓ′for otherwise e1,e2and e3would not be pairwise distinct. Let f(p,ℓ′) =\nσ′\nℓ(p) = x,f(p,ℓ) = σℓ(p) = yand f(p′,ℓ) = σℓ(p′) = z. We observe that x,yand zare pairwise\ndistinct. Indeed, the points yand zlie onℓand are different from p: hence, x/\\e}atio\\slash=yand x/\\e}atio\\slash=zasxlies\nonℓ′and the only common point between ℓandℓ′isp. Further, y/\\e}atio\\slash=zbecause σℓis permutation of the\npoints of ℓ.\nTherefore ϕ(x),ϕ(y)and ϕ(z)exhibit the trifferent property. But\nθ(e1) =ϕ(x),θ(e2) =ϕ(y),θ(e3) =ϕ(z)\nshowing that τ(S)is a trifferent code. It is also clear that the number of 2’s in each codeword of τ(S)is\n3rand the block length of the code is 3 n. Finally, since τis injective, the size of the code is |S|=q3+q2.\nSetting nto be the smallest integer larger than/parenleftBigq2+q\nct/parenrightBig(2/3)t\n, and n′=3n, we obtain a 3 r-bounded\ntrifferent code τ(S)with size at least ct+1×n′(3/2)t+1. (For smaller values of n′>3rwe can just adjust\nthe constant ct+1to make hypothesis true.)\nProof of Theorem 1.7 .Lemma 3.1 proves the first claim. For the next claim: Tb(n, 3t)≥ct×n(3/2)tis\nobtained directly from Lemma 3.2 with the base case of t=0 being Lemma 3.1 . Setting r=3tgives\n(3/2)t=rαwith α=1−log3(2)≈0.369. Hence, ∆r≤r−rα.\nReferences\n[Ari94] Erdal Arikan. An upper bound on the zero-error list- coding capacity. IEEE Trans. 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New bounds for perfe ct hashing via information theory.\nEuropean J. Combin. , 9(6):523–530, 1988. 2\n[Kur23] Sascha Kurz. Trifferent codes with small lengths, 2 023. 1,3\n[PZ22] Cosmin Pohoata and Dmitriy Zakharov. On the triffere nce problem for linear codes. IEEE\nTransactions on Information Theory , 68(11):7096–7099, 2022. 2\n[Rad01] Jaikumar Radhakrishnan. Entropy and counting. 200 1.2\n[XY19] Chaoping Xing and Chen Yuan. Beating the probabilist ic lower bound on perfect hashing.\n(manuscript), 2019. 2\n11"    },    {        "title": "2402.02406v1.Conformal_vector_fields_on_compact_connected_homogeneous_Finsler_manifolds.pdf",        "content": "arXiv:2402.02406v1  [math.DG]  4 Feb 2024CONFORMAL VECTOR FIELDS ON COMPACT CONNECTED\nHOMOGENEOUS FINSLER MANIFOLDS\nMING XU\nAbstract. Let (M,F) be a compact connected homogeneous non-Riemannian Finsle r man-\nifold with dim M >1. We prove that any conformal vector field on ( M,F) is a Killing vector\nfield. Further more, we prove that ρFis a homogeneous Finsler metric on Mif and only if\nρis a positive constant function.\nMathematics Subject Classification(2010): 53B40, 53C30, 5 3C60\nKeywords: conformal transformation group, conformal vect or field, homogeneous Finsler\nmanifold, isometry group, Killing vector field\n1.Introduction\nLet (M,g) be a connected Riemannian manifold. We denote by C(M,g) andI(M,g) the\ngroups of all conformal transformations and all isometries, and b yC0(M,g) andI0(M,g) their\nidentity components. Roughly speaking, they are Lie transformat ion groups (acting smoothly\nonM), and their Lie algebras are the spaces of all complete conformal v ector fields and all\ncomplete Killing vector fields respectively [ 12][17]. Notice C(M,g) may have an infinite dimen-\nsion when dim M= 2. The conformality in Riemannian geometry has been studied for ma ny\ndecades [ 1][7][9][13][19][22]. Using warped product or other techniques, it is easy to construc t\nnon-Killing conformal vector fields, which reveals C0(M,g)/\\e}atio\\slash=I0(M,g) in general. However,\nwhen (M,g) is a compact connected homogeneous Riemannian manifold with dim M >1,\nC0(M,g)/\\e}atio\\slash=I0(M,g) implies the following rigidity.\nTheorem 1.1. Let(M,g)be a compact connected homogeneous Riemannian manifold wit h\ndimM >1. Suppose that C0(M,g)/\\e}atio\\slash=I0(M,g), then(M,g)is isometric to a sphere with\nconstant curvature.\nTheorem 1.1is an immediate corollary of Lichnerowicz Theorem when dim M≥3 [14], and\nit follows follows after some simple observations when dim M= 2 (see Section 3.2). See also [ 9]\nwhere Theorem 1.1is proved when dim M >3.\nThe purpose of this paper is to generalize Theorem 1.1to Finsler geometry. On a Finsler\nmanifold ( M,F), the conformal transformation group C(M,F), the isometry group I(M,F),\nand their identity components, C0(M,F) andI0(M,F) respectively, can be similarly defined\n[4][8][11]. Therearevariousaverageprocesses[ 6][15][18]whichcanproduceaRiemannianmetric\nfrom a Finsler one while preserve or even enlarge the conformal tra nsformation group and the\nisometry group. Using this average technique, we can quickly verify that most C(M,F) are Lie\ntransformation groups (see Lemma 3.3and Lemma 3.4), and it follows naturally that their Lie\nalgebras C(M,F) are the spaces of complete conformal vector fields. The main the orem of this\npaper is\nTheorem 1.2. Let(M,F)be a compact connected homogeneous non-Riemannian Finsler man-\nifold with dimM >1. ThenC0(M,F) =I0(M,F), i.e., any conformal vector field on (M,F)\nis a Killing vector field.\nTheorem 1.2can also be formulated as\n12 MING XU\nTheorem 1.3. Let(M,F)be a compact connected non-Riemannian homogeneous Finsler man-\nifold with dimM >1, andρa positive smooth function on M. ThenρFis homogeneous if and\nonly ifρis constant.\nThe analytic method proving Theorem 1.1is not valid in the Finsler context. The proof\nof Theorem 1.2has three other ingredients, i.e., the average process, the Finsler Confromal\nLichnerowicz-Obata Conjecture (proved by V.S. Matveev et al [ 15]), and the Lie theory. In\nSection 2, we summarize some necessary knowledge in Finsler geomet ry. In Section 3, we\nintroduce the average process and use it to discuss the Finslerian c onformal transformation\ngroup. In Section 4, we prove Theorem 1.2and Theorem 1.3.\n2.Preliminaries\n2.1.Minkowski norm and Finslermetric. AMinkowski norm Fonafinitedimensionalreal\nvector space Vis a continuous function F:V→[0,+∞) satisfying the following requirements:\n(1) Regularity: Fis positive and smooth on V\\{0};\n(2) Positive 1-homogeneity: F(λy) =λF(y),∀λ≥0,y∈V;\n(3) Strong convexity: given any basis {e1,···,en}ofVand the corresponding linear co-\nordinate y=yiei⇔(y1,···,yn), the Hessian matrix ( gij(y)) = ([1\n2F2]yiyj) is positive\ndefinite when y/\\e}atio\\slash= 0, or equivalently, the fundamental tensor\ngy(u,v) =1\n2∂2\n∂s∂t|s=t=0F2(y+su+tv),∀u,v∈V,\nis an inner product on Vfor anyy∈V\\{0}.\nWe call a Minkowski norm FEuclidean whengy(·,·) in (3) is independent of y.\nA Finsler metric Fon a smooth manifold Mis a continuous function F:TM→[0,+∞)\nsuch that F|TM\\0is smooth and F|TxMis a Minkowski norm for each x∈M[3]. We usually\nuse the pair ( M,F) to denote a Finsler manifold . We call a Finsler metric FRiemannian if it\nis Euclidean in each tangent space. In this case, we also use g=F2to denote the metric.\n2.2.Finslerian conformality and isometry. It is obvious to see that, for any Finsler metric\nFand positive smooth function ρonM,ρFis also a Finsler metric on M. Here we call F\nandρFconformally equivalent . A diffeomorphism fon a Finsler manifold ( M,F) is called a\nconformal transformation ifFandf∗Fare conformally equivalent. In particular, we call fan\nisometry iff∗F=F.\nConformal transformations and isometries can be infinitesimal gen erated by conformal and\nKilling vector fields respectively. A smooth tangent vector field Von (M,F) is called conformal\nif there exists a smooth function ρonM, such that LVF=ρF(hereLis the Lie derivative).\nIn particular, a conformal vector field Vis aKilling vector field whenLVF= 0. A conformal\nor Killing vector field can generate a globally defined one-parameter s ubgroup of conformal\ntransformationsorisometriesonlywhenit iscomplete. Since weonlyd iscusscompactmanifolds\nin this paper, the completeness is automatically satisfied.\nWe denote by C(M,F),I(M,F),C0(M,F),I0(M,F),C(M,F) andI(M,F) the group of\nall conformal transformations, the group of all isometries, the id entity component of C(M,F),\nthe identity component of I(M,F), the space of all complete conformal vector fields, the space\nof all complete Killing vector fields on ( M,F), respectively.\n3.Conformality for a homogeneous Finsler manifold\n3.1.Average process and group of conformal transformations. LetFbe a Minkowski\nnorm on V. Averaging its fundamental tensor along the indicatrix S={y|y∈V,F(y) = 1},\nwe can get an Euclidean norm F. Here we only recall the construction in Section 2.3 of [ 6]. See\n[15][18] for more constructions which can also meet our needs.CONFORMAL VECTOR FIELDS ON COMPACT CONNECTED HOMOGENEOUS F INSLER MANIFOLDS 3\nLetgbe the Riemannian metric on V\\{0}determined by the fundamental tensor of F(we\ncall it the Hessian metric ofF), and denote by dµthe volume measure of ( S,g|S). Then\nF(u) =/radicalbigg/integraltext\ny∈Sgy(u,u)dµ/integraltext\ny∈Sdµ,∀u∈V,\nis theaveraged Euclidean norm induced by F.\nLemma 3.1. LetFibe a Minkowski norm on Vi,Fithe averaged Euclidean norm induced by\nFi, andl:V1→V2a real linear isomorphism satisfying F1=cF2◦lfor some positive constant\nc. ThenF1=cF2◦l.\nProof.\nFirst, we prove the lemma when c= 1, i.e., lis a linear isometry. Let gi,Sianddµibe the\nHessian metric of Fi, the indicatrix of Fi, and the volume measure of ( Si,gi|Si) respectively.\nThenlinduces an isometry between ( Si,gi|Si) (see Page 1488 of [ 21] or Lemma 4.1 in [ 20]), i.e.,\ng2,l(y1)(l(u),l(u)) =g1,y1(u,u),∀y1∈S1,u∈V1. Moreover, lpreserves the volume measure,\ni.e.,l∗dµ2=dµ1. So for any u∈V,\nF2(l(u))2=/integraltext\ny2∈S2g2,y2(l(u),l(u))dµ2/integraltext\ny2∈S2dµ2=/integraltext\ny1∈S1g2,l(y1)(l(u),l(u))l∗dµ2/integraltext\ny1∈S1l∗dµ2\n=/integraltext\ny1∈S1g1,y1(u,u)dµ1/integraltext\ny1∈S1dµ1=F1(u)2,\ni.e.,F1=F2◦l.\nSecond, we prove the lemma when l= id and F1=cF2. Sincec·id is a linear isometry\nbetween F1andF2, above discussion indicates that F1(u) =F2(cu) =cF2(u).\nSince any lin Lemma 3.1is the composition of the maps in above two cases, the proof is\nfinished.\nLetFbe a Finsler metric on M. We can average Fin each tangent space, and get a\nRiemannian metric g=F2onM. We call this gtheaveraged Riemannian metric induced by\nF. Using Lemma 3.1, we can immediately observe\nLemma 3.2. For the averaged Riemannian metric ginduced by a Finsler metric FonM, we\nhave the following:\n(1)whenf∈C(M,F)satisfies f∗F=ρF, thenf∗g=ρ2g;\n(2)whenV∈C(M,F)satisfies LVF=ρF, thenLVg= 2ρg;\n(3)C(M,F)⊂C(M,g),C0(M,F)⊂C0(M,g), andC(M,F)⊂C(M,g);\n(4)I(M,F)⊂I(M,g),I0(M,F)⊂I0(M,g), andI(M,F)⊂I(M,g).\nThis average process can help us quickly see that I(M,F) is a Lie transformation group (See\nSection 2.3 in [ 6]). Using a similar argument, we can prove\nLemma 3.3. For a connected Finsler manifold (M,F)withdimM >2,C(M,F)is a Lie\ntransformation group.\nProof.Letgbe the averaged Riemannian metric induced by F. By Theorem 1 in Page 310 of\n[12],C(M,g) is a Lie transformation group, i.e., C(M,g) is a Lie group, and the map Φ( f,x) =\nf(x) :C(M,g)×M→Mis smooth. It is easy to check that C(M,F) is a transformation group\nacting on M. Lemma 3.2tells us that C(M,F) is contained in C(M,g). To prove Lemma 3.3,\nwe only need to prove C(M,F) is a closed subgroup of C(M,g). Letfnbe a sequence in\nC(M,F), then we have\nF(x,y)\nF(x,y′)=F(fn(x),(fn)∗(y))\nF(fn(x),(fn)∗(y′)),∀x∈M,y,y′∈TxM\\{0},n∈N. (3.1)\nThe smoothness of Φ implies\nlim\nn→∞fn(x) =f(x),lim\nn→∞(fn)∗(y) =f∗(y)∈TM\\0,∀x∈M,y∈TxM\\0.4 MING XU\nSo we can take the limit of ( 3.1) forn→ ∞and get\nF(x,y)\nF(x,y′)=F(f(x),f∗(y))\nF(f(x),f∗(y′)),∀x∈M,y,y′∈TxM\\{0},\nwhich guarantees f∈C(M,F). This argument verifies the closeness of C(M,F) inC(M,g)\nand ends the proof.\nIt isa slightdifference between C(M,F) andI(M,F) that all I(M,F) areLie transformation\ngroups [5], but not all C(M,F). For those Lie transformation groups C(M,F) andI(M,F),\ntheir Lie algebras can be naturally identified as C(M,F) andI(M,F) respectively.\n3.2.Conformal fields on a compact homogeneous Finsler manifold. Let (M,F) be a\ncompact connected homogeneous Finsler manifold. The homogeneit y here means that I0(M,F)\nacts transitively on M. When dim M >2,C0(M,F) is a Lie transformation group by Lemma\n3.3. Now we discuss C0(M,F) case by case when dim M≤2.\nCase 1: dimM= 1. In this case, M=S1, and any Finsler metric FonMis homo-\ngeneous if and only if λ(x) = supy∈TxM\\{0}F(x,y)\nF(x,−y)is constant for all x∈M. In this case,\nC0(M,F) = Diff+(S1) and any smooth vector field is conformal. This explains the necessit y\nfor the dimension assumptions in Theorem 1.2and Theorem 1.3.\nWhen dim M= 2,I0(M,F) is a compact connected Lie group with dim I0(M,F)≤3. So\nThere are only three possibilities, a Riemannian S2with constant curvature, a Riemannian\nRP2with constant curvature, or a locally Minkowskian T2.\nCase 2: (M,F) is the Riemannian S2with constant curvature. In this case C0(M,F) =\nPGL(2,C) can be identified as the group of holomorphic diffeomorphisms on CP1=C/coproducttext{∞},\ni.e., each f∈C0(M,F) is represented by the map z/mapsto→az+b\ncz+dwithad−bc/\\e}atio\\slash= 0.\nCase 3: (M,F) is a Riemannian RP2with constant curvature. We can identify RP2=\nCP2/Z2, in which CP1=C/coproducttext{∞}andZ2generated by the antipodal map is represented by\nθ(z) =−z−1. Consider any V∈C(M,F). It one-to-one induces a θ-invariant holomorhpic field\nV′onCP1, which generates a one-parametersubgroup in H={f|f∈PGL(2,C),f◦θ=θ◦f}.\nThis observation identifies the Lie group C0(M,F) with the identity component of H. Direct\ncalculation shows dim H= 3, soC0(M,F) =I0(M,F) =SO(3) in this case.\nCase 4: (M,F) is a locally Minkowskian T2. Letgbe the averaged Riemannian metric\ninduced by F. By Lemma 3.3, (M,F) is a flat Riemannian torus. We may present MasM=\nC/Λ and assume that gis the standard Euclidean metric. Using Lemma 3.2, anyV∈C(M,F)\nis a conformal vector field on ( M,g). Letft(z) =ϕ(t,z) be the one-parameter subgroup\nof conformal transformations generated by V, in which ϕ:R×R/Λ→R/Λ is a smooth\nmap, then Vis represented byd\ndt|t=0ft. Since ϕ(t,·) is holomorphic for each fixed t, i.e.,\n∂\n∂zϕ(t,z) = 0, we have∂\n∂z(d\ndt|t=0ft) =d\ndt|t=0∂\n∂zϕ(t,z) = 0. So V=d\ndt|t=0ftis aC-valued\nholomorphic function on C/Λ, which must be constant. That implies dim C(M,F)≤2, so\nC0(M,F) =I0(M,F) =U(1)×U(1). Notice that when Fis non-Riemannian, one may\nalternatively use Theorem 4.1 in [ 18] to argue that any f∈C(M,T) must be homothetic. By\nthe compactness of M, all homothetic transformations must be isometries.\nTo summarize, we have the following lemma which explains the validity of T heorem1.1when\ndimM= 2.\nLemma 3.4. Let(M,F)be a compact connected homogeneous Finsler manifold with dimM >\n1. ThenC0(M,F)is a Lie transformation group and Lie(C0(M,F)) =C(M,F).\n4.Proof of the main results\nLet (M,F) be a compact connected homogeneous non-Riemannian Finsler man ifold with\ndimM >1. By Lemma 3.4, we can present Mas the homogeneous manifold M=G/H=\nK/K∩H, in which G=C0(M,F) andK=I0(M,F). Suppose C0(M,F)/\\e}atio\\slash=I0(M,F), or\nequivalently, g=C(M,F)/\\e}atio\\slash=k=I(M,F), i.e., (M,F) admits a non-Killing conformal vector\nfieldV.CONFORMAL VECTOR FIELDS ON COMPACT CONNECTED HOMOGENEOUS F INSLER MANIFOLDS 5\nLemma 4.1. Keeping all assumptions and notations in this section, then Mmust be a homo-\ngeneous sphere, and there exists a compact connected semi si mple normal subgroup K′ofK\nwhich acts transitively on M.\nProof. Letgbe the averaged Riemannian metric induced by F. By Lemma 3.2,gis\nhomogeneous Riemannian metric on M, andVis a non-Killing conformal vector field on\n(M,g). Theorem 1.1tells us that Mis a sphere. By the classification for homogeneous\nspheres [ 2][16],SO(n),SU(n),Sp(n) orSpin(9) acts transitively on the homogeneous Finsler\nsphereM=SO(n)/SO(n−1),SU(n)/SU(n−1) =U(n)/U(n−1),Sp(n)/Sp(n−1) =\nSp(n)U(1)/Sp(n−1)U(1) =Sp(n)Sp(1)/Sp(n−1)Sp(1)orSpin(9)/Spin(7)respectively. Since\nthe connected isometry groups of S6=G2/SU(3) andS7=Spin(7)/G2areSO(7) andSO(8)\nrespectively, G2andSpin(7) do not need to appear. The proof is finished.\nLemma 4.2. Keeping all assumptions and notations in this section, then for any element\nU∈g,ad(U) = [U,·] :g→gis semi simple, and B(U,U) = tr(ad( U)2)≤0, in which Bis the\nKilling form of g.\nThe proofofLemma 4.2needs the followingtheorem (Finsler ConfromalLichnerowicz-Obat a\nConjecture).\nTheorem 4.3. ([15]) Suppose that Uis a complete conformal vector field on a connected Finsler\nmanifold (M,F)withdimM >1. Then one of the following must happen:\n(1)Uis a Killing vector field on (M,ρF)for some positive smooth function ρonM;\n(2) (M,F)is conformally equivalent to a Riemannian sphere with const ant curvature;\n(3) (M,F)is conformally equivalent to a Minkowski space and Uis homothetic.\nProof of Lemma 4.2.Since (M,F) is compact and non-Riemannian, only (1) in Theorem\n4.3can happen, i.e., we can find a positive smooth function ρonM, such that the conformal\nvectorfield U∈gisa Killingvectorfield for( M,ρF). ThenUiscontainedin the Lie subalgebra\ng′which generates G′=I0(M,ρF). Since Mis compact, I0(M,ρF) is a compact subgroup in\nG=C0(M,F). So we can find an Ad( G′)-invariant inner product on g, for which ad( U) is anti\nself adjoint. The semi simpleness of ad( U) follows immediately. Moreover, the square of ad( U)\nis semi negative definite, so B(U,U) = tr(ad( U)2)≤0. This ends the proof.\nDenote by rthe radical (i.e., the maximal solvable ideal) of g. Levi’s Theorem (see Theorem\n5.6.6 in [ 10]) provides a linear decomposition g=s+r, in which sis a semi simple subalgebra\n(called a Levi subalgebra ). Notice that the Levi subalgebra sofgis not unique, because it can\nbe changed by any Ad( G)-action.\nLemma 4.4. Keeping all above assumptions and notations in this section , thensis compact.\nProof.The Killing form Bsatisfies the following associative property,\nB([W1,W2],W3)+B(W1,[W2,W3]) = 0,∀W1,W2,W3∈g, (4.2)\nso it is well known that r′={U|U∈g,B(U,g) = 0}is an ideal of g. By Cartan’s Criterion\n(see Theorem 5.4.20 in [ 10]),r′is solvable. Let π:g→g/r′be the quotient endomorphism.\nThenBinduces a negative definite bilinear form B(π(W1),π(W2)) =B(W1,W2) ong/r′, which\nsatisfies the associative property (see ( 4.2)). Sog/r′is compact and g/r′= [g/r′,g/r′]⊕c(g/r′).\nObviously, r=π−1(c(g/r′)) and then πinduces a Lie algebra isomorphism from sto [g/r′,g/r′].\nThe compactness of sis proved.\nProof of Theorem 1.2.We keep all assumptions and notations in this section, and look for\na contradiction from g=C(M,F)/\\e}atio\\slash=I=k. LetSbe the Lie subgroup sgenerates in Gand\nK′the subgroup in Lemma 4.2. BothSandK=I0(M,F) are compact and connected. By\nTheorem 14.1.3 in [ 10], i.e., each compact subgroup of Gis contained in a maximal compact\nsubgroup and all maximal compact subgroups of Gare conjugate to each other, we may assume\nthatSandKare contained in the same maximal compact subgroup LofG.\nClaim A :K′⊂S⊂K, i.e.,Sacts transitively and isometrically on ( M,F).6 MING XU\nDenoteby ltheLiealgebraof L, whichisacompactLiesubalgebraof g. Because l=s+(l∩r),\nin which l∩ris a solvable ideal of landsis semi simple, we have l∩r=c(l) and [l,l] =s.\nMoreover, k′= Lie(K′) is semi simple, so we have k′= [k′,k′]⊂[l,l] =s, which implies K′⊂S.\nSinceK′acts transitively on M, so does S. We can present MasM=S/S∩H, where\nS∩His the compact isotropy subgroup at o∈M. The isotropy action of S∩His contained\nin (0,+∞)×O(ToM,F(o,·)), in which O(ToM,F(o,·)) is the group of all linear isometries on\nthe Minkowski norm space ( ToM,F(o,·)). The compactness of S∩Himplies that the image\nof its isotropic action is contained in O(ToM,F(o,·)), i.e., for any y∈ToMandf∈S∩H,\nF(o,y) =F(o,f∗(y)). Now we consider any f∈Sandx∈M. The transitiveness of the\nK′-action on Mimplies a decomposition f=f1◦f2◦f3, such that f1,f3∈K′andf2∈S∩H\nsatisfyf3(x) =oandf1(o) =f(x). TheK′-action on Mis isometric, so for any y∈TxM,\nF(x,y) =F(o,(f3)∗(y)) =F(o,(f2)∗((f3)∗(y)))\n=F(f1(o),(f1)∗((f2)∗((f3)∗(y)))) =F(f(x),f∗(y)).\nSo we have S⊂I0(M,F) =K, which proves Claim A.\nBecause g/\\e}atio\\slash=k, Claim A implies that ris not contained in k. We have deduced l∩r=c(l), so\n[s,r∩k]⊂[l,l∩r] = 0. Further more, we claim\nClaim B : AnyU∈rcommuting with sis contained in k.\nLetftbe the conformal transformations generated by U. For each t∈R, (ft)∗U=U. The\nassumption [ U,s] = 0 implies f∗U=Ufor anyf∈S. By Claim A, Sacts isometrically and\ntransitively on ( M,F), soUis of constant length. If U= 0, then it is obviously a Killing vector\nfield. Otherwise ( ft)∗U=Uimplies each conformal ftis in fact isometric. Claim B is proved.\nBy Claim B, we has an ad( s)-invariant decomposition r=r0+···+rmwithm >0, such that\nthe ad(s)-action is trivial on r0=r∩k, and nontrivial irreducible on each other ri. For each\n0< i≤m, [ri,s]⊂riis ad(s)-invariant, so the irreducibility of riprovides ri= [ri,s]. Choose\nU∈ri\\{0}, then we have U /∈k. Because U∈[ri,s]⊂[r,g], ad(U) :g→gis nilpotent by\nCorollary 5.4.15 in [ 10]. On the other hand, ad( U) is semi simple by Lemma 4.2. So ad(U)=0\nong. In particular, we have [ U,s] = 0, which contradicts Claim B and ends the proof.\nProof of Theorem 1.3.Whenρisapositiveconstantfunction, ρFisobviouslyhomogeneous.\nAssuming that ρis non-constant and ρFis homogeneous. Then there exist x,x′∈Mwith\nρ(x1)/\\e}atio\\slash=ρ(x2), andf1∈I0(M,F),f2∈I0(M,ρF) withf1(x) =f2(x) =x′. Obviously\nf1/∈I0(M,ρF) andf2/∈I0(M,F). SoI0(M,F)/subsetnoteqlI0(M,F)∪I0(M,ρF)⊂C0(M,F), which\ncontradicts Theorem 1.2and ends the proof.\nAcknowledgement . ThispaperissupportedbyBeijingNaturalScienceFoundation(12 22003)\nand National Natural Science Foundation of China (12131012, 118 21101). The author sincerely\nthank Shaoqiang Deng, Xiaobo Liu and Mengke Wu for helpful discuss ions.\nReferences\n[1] D.V. Alekseevskii, Groups of conformal transformation s of Riemannian spaces, Math. USSR Sbornik 18 (2)\n(1972), 285-301. 1\n[2] A. Borel, Some remarks about Lie groups transitive on sph eres and tori, Bull. Amer. Math. Soc. 55 (1940),\n580-587. 5\n[3] D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann -Finsler Geometry, G.T.M. 200, Springer,\nNew York, 2000. 2\n[4] S. Deng, Homogeneous Finsler spaces, Springer Monogr. M ath., Springer - New York (2012). 1\n[5] S. Deng and Z. Hou, The group of isometries of a Finsler spa ce, Pacific J. Math. 207 (1) (2002), 149-155. 4\n[6] S. Deng and M. Xu, ( α1,α2)-metrics and Clifford-Wolf homogeneity, J. Geom. Anal 26 (2 016), 2282-2321.\n1,2,3\n[7] J. Ferrand, The action of conformal transformations on a Riemannian manifold, Math. Ann. 304 (1996),\n277-291. 1\n[8] J. 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Lichnerowicz, Sur les transformations conformes d’ une vari´ et´ e riemannienne compacte, Compt. Rend.\n259 (1964), 697-700. 1\n[15] V.S. Matveev, H.B. Rademacher, M. Troyanov and A. Zeghi b, Finsler conformal Lichnerowicz-Obata Con-\njecture, Ann. Inst. Fourier, Grenoble 59 (3) (2009), 937-94 9.1,2,5\n[16] D. Montgomery and H. Samelson, Transformation groups o f spheres, Ann. Math. 44 (1943), 454-470. 5\n[17] S.B. Myers and N. Steenrod, The group of isometries of a R iemannian manifold, Ann. Math. 40 (1939),\n400-416. 1\n[18] V.S. Matveev and M. Troyanov, The Binet-Legendre metri c in Finsler geometry, Geom. Topo. 16 (2012),\n2135-2170. 1,2,4\n[19] M. Obata, The conjectures on conformal transformation s of Riemannian manifolds, J. Diff. Geom. 6 (1971),\n247-258. 1\n[20] M. Xu, The Minkowski norm and Hessian isometry induced b y an isoparametric foliation on the unit sphere,\nSci. China Math. 65 (7) (2022), 1485-1516. 3\n[21] M. Xu and V.S. Matveev, Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski\nnorms with SO(k)×SO(n−k)-symmetric, Canad. J. Math. 74 (5) (2022), 1486-1516. 3\n[22] K. Yano and T. Nagano, Einstein spaces admitting a non-p arameter group of conformal transformations,\nAnn. Math. 69 (1959), 451-461. 1\n(Ming Xu) School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R.\nChina\nEmail address :mgmgmgxu@163.com"    },    {        "title": "2402.02424v1.Influence_of__f__mathcal_R___mathcal_T___mathcal_Q____Gravity_on_Cylindrical_Collapse.pdf",        "content": "arXiv:2402.02424v1  [gr-qc]  4 Feb 2024Influence of f(R,T,Q)Gravity on\nCylindrical Collapse\nM. Sharif1∗and Tayyab Naseer1,2†\n1Department of Mathematics and Statistics, The University o f Lahore,\n1-KM Defence Road Lahore, Pakistan.\n2Department of Mathematics, University of the Punjab,\nQuaid-i-Azam Campus, Lahore-54590, Pakistan.\nAbstract\nThis article examines the dynamics of gravitational collap se in\nf(R,T,Q) gravity, where Q=RabTab. We consider self-gravitating\nanisotropic cylindrical geometry whose interior is filled w ith dissipa-\ntive matter configuration andmatch it with exterior cylindr ically sym-\nmetric spacetime at the hypersurfacethrough junction cond itions. We\nemploy the Misner-Sharp and M¨ uler-Israel Stewart formali sms to de-\nrive the dynamical as well as transport equations correspon ding to\nthe model R+ Φ√\nT+ ΨQ, where Φ and Ψ are arbitrary coupling\nconstants. We then establish some relations between these e quations\nthrough which the impact of effective matter variables, heat d issipa-\ntion and the bulk viscosity on the collapse rate is studied. F urther,\nwe express the Weyl scalar in terms of the effective matter sect or. We\nalso obtain the conformal flatness by applying some restrict ions on\nthe considered model and taking dust configuration into the a ccount.\nFinally, we investigate various cases to check whether the m odified\ncorrections increase or decrease the collapse rate.\nKeywords: f(R,T,RabTab)theory; Gravitationalcollapse; Self-gravitating\nstructures.\nPACS:04.40.Dg; 04.50.Kd.\n∗msharif.math@pu.edu.pk\n†tayyabnaseer48@yahoo.com; tayyab.naseer@math.uol.edu.pk\n11 Introduction\nCosmological observations reveal that our universe is originated b y the ex-\npansion of superheated matter and energy. Cosmologists explore d that a\nconsiderable portion of this unfathomable universe is made up of sta rs, plan-\nets and galaxies. The most appealing and promising phenomenon in the\nstructural formation of these celestial objects is the gravitatio nal collapse.\nThe pioneer work of Chandrasekhar [1] on this phenomenon helps s cientists\nto understand its importance in the field of relativistic astrophysics . He\nfound that a star remains stable until its external pressure and in ternal force\nof attraction (due to its mass) are counterbalanced by each othe r. The dy-\nnamics of dust collapse has been discussed by Oppenheimer and Snyd er [2],\nfrom which they found that such collapse eventually results in a black hole.\nMisner and Sharp [3] studied spherical geometry coupled with anisot ropic\nfluid and checked how the collapse rate is affected by pressure aniso tropy.\nThe Misner-Sharp technique has been employed by Herrera and San tos [4]\nto investigate the collapsing rate of a sphere and found that the en ergy dissi-\npatesintheformofheat/radiations. Herrera et al. [5]analyzed th eimpact of\nanisotropy on the collapse of cylindrically symmetric matter source. Sharif\nand his collaborators [6] studied the dynamics of uncharged and cha rged\nspherical/cylindrical systems and deduced that the collapse rate is reduced\nin the presence of electric charge.\nAs the process of gravitational collapse is highly dissipative, the effe cts\nof heat dissipation in this phenomenon cannot be ignored [7, 8]. Chan [9 ]\nexplored the collapsing phenomenon for a radiating compact object and re-\nvealed that the shear viscosity increases anisotropy of the fluid dis tribution.\nDi Prisco et al.[10] studied anisotropic matter configuration and disclosed\nthat the explosion in the internal region of spherical geometry cau ses the\nformation of singularity. Nath et al. [11] examined the collapsing rat e by\nemploying matching criteria between quasi-spherical Szekeres and charged\nVaidya spacetimes as interior and exterior geometries, respective ly. They\nconcluded that the formation of naked singularity is supported by e lectric\ncharge. Herrera et al. [12] discussed self-gravitating viscous diss ipative fluid\nand found that the dissipative parameters decrease the force of gravity that\neventually decreases the collapsing rate.\nCylindrical gravitational waves exist and support cylindrically symme t-\nric self-gravitating structures whose study yields significant cons equences.\nSuch geometrical objects prompted many researchers to explor e their differ-\n2ent fundamental characteristics. The study of these structur es was pioneered\nby Bronnikov and Kovalchuk [13]. Wang [14] studied four-dimension al cylin-\nder and determined exact solutions to the field equations correspo nding to\na massless scalar field. He found that collapse of such object may re sult in\nthe black hole. Guha and Banerji [15] studied the dynamics of cylindr ical\nanisotropic geometry, experiencing heat dissipation and undergoin g the grav-\nitational collapse, and derived the solutions for the matter source . In stellar\nevolution, the Weyl tensor plays a significant role that helps to meas ure the\ncurvature of geometrical structure. The gravitational collapse of a sphere\nhas been discussed by Penrose [16] by formulating a relation betwe en state\nvariables and the Weyl tensor. Sharif and Fatima [17] represented the Weyl\ntensor in terms of matter variables, anisotropy and coefficient of s hear vis-\ncosity for cylindrical configuration, and discussed the collapsing ra te. They\nfound conformal flatness condition corresponding to the homoge neous energy\ndensity.\nThe current accelerated expansion of the universe is considered a s the\nmost fascinating phenomenon in the field of cosmology and astrophy sics for\nthe last couple of years [18, 19]. This expansion was claimed to be trigg ered\nby an obscure form of force having immense repulsive effects, know n as dark\nenergy. The study of such cosmic nature in the theory of general relativity\n(GR)faces somedeficiencies like cosmic coincidence andfine-tuning prob elm.\nIn this context, scientists developed several modifications to GRto address\nsuch issues appropriately. To study cosmological outcomes at larg e scales,\nthe simplest possible generalization of GRwas attained by replacing the\nRicci scalar with its generic functional, named f(R) theory [20]. Different\nmodified f(R) models have been investigated through multiple approaches\nand the obtained results are found to be physically feasible [21]-[24].\nThe idea of matter-geometry coupling was initially presented by Bert o-\nlami et al. [25] to study the appealing nature as well as composition o f\nthe universe. They analyzed the impact of such interaction in f(R) frame-\nwork by engaging the geometrical term in the fluid Lagrangian. This in -\nteraction was recently generalized by Harko et al. [26] at action lev el, who\nintroduced f(R,T) gravity, Tbeing trace of the energy-momentum tensor\n(EMT). The gravitational theories involving such a matter term res ults in its\nnon-vanishing divergence. This theory produces several astonis hing results\ncorresponding to self-gravitating structures [27]-[30]. However , thef(R,T)\ngravity fails to entail the coupling effects on compact bodies at some point,\nthus one needs to overcome this issue. Haghani et al. [31], in this reg ard,\n3generalized the functional by inserting an additional term Q, that represents\ncontraction of EMT with the Ricci tensor Rab. They considered three differ-\nent models in f(R,T,Q) gravity and studied their respective cosmological\napplications for high density regime as well as pressureless matter fl uid case.\nSharif and Zubair [32] studied the thermodynamical laws for the blac k\nhole by adopting the models R+λQandR(1+λQ) along with matter La-\ngrangian in terms of energy density as well as pressure. The energ y bounds\narealso addressed in this scenario, from which they concluded that only posi-\ntive values of the model parameter λsatisfy weak energy conditions [33]. The\nflat FLRW spacetime was considered to check the behavior of this ex tended\ntheory for different cosmological models [34]. They reconstructed the modi-\nfied gravitational action and also described de Sitter universe solut ions cor-\nresponding to the perfect fluid distribution. Baffou et al. [35] perfo rmed sta-\nbility analysis in this modified gravity for two particular cases and conc luded\nthat both models present stability through some perturbation fun ctions.\nYousafet al. [36]performed orthogonaldecomposition of theRiem anntensor\non the effective EMT corresponding to static/non-static spherica l structures\nandcomputedsomescalarstodiscussthestructuralevolutionof thesebodies.\nThe evolutionary patterns for cylindrical spacetime have also been discussed\nin modified scenario [37]. We also studied charged/uncharged sphere and\nobtained several physically acceptable anisotropic solutions throu gh differ-\nent schemes [38, 39].\nThe extensive discussion on the collapsing phenomenon has been don e\nin various modified backgrounds. The numerical simulations were emp loyed\nto study the collapse of spherical body in f(R) framework from which an\nunusual increment in the density of fluid has been found [40]. In this c on-\ntext, Shamir and Fayyaz [41] analyzed the dynamics of a self-gravit ational\ncylindrical geometry whose interior is filled with anisotropic dissipative fluid.\nThey found the collapse as the crucial element to examine the rapid a ccel-\neration. The dynamical equations have been used to investigate th e behav-\nior of anisotropic gravitating source in f(R,T) scenario [42]. Bhatti et al.\n[43] discussed the collapsing rate of dissipative anisotropic matter d istribu-\ntion with/without involving the effects of radiation density and coeffic ient of\nshear viscosity in f(R,T,Q) framework. Sharif et al. [44] studied the celes-\ntialobjects coupled withperfect/anisotropicconfigurations with /without the\nheat dissipation in different modified theories, and examined the influe nce of\ncorrection terms on the collapsing rate.\nThis article addresses the dynamics of cylindrical geometry involving the\n4impact of principal stresses and the dissipation flux in f(R,T,RabTab) the-\nory. The paper is structured in the following format. We define some elemen-\ntary terms related to the collapse and the extended theory, and f ormulate\nthe corresponding equations of motion and non-vanishing dynamica l identi-\ntiesforR+Φ√\nT+ΨQinsection 2. Moreover, theC-energy andthejunction\nconditions are calculated through Darmois criteria. Section 3formulates the\ndynamical equations and then couple them with the acceleration of t he fluid.\nWe further construct several dynamical forces in modified gravit y in section\n4to study their impact on the collapsing rate. Section 5explores some inter-\nesting relations between the effective state parameters and the W eyl scalar.\nThe last section summarizes all of our findings.\n2f(R,T,RabTab)Theory\nThe modified Einstein-Hilbert action for the f(R,T,Q) gravity (with κ=\n8π) has the following form [34]\nS=/integraldisplay/braceleftbiggf(R,T,Q)\n16π+LM/bracerightbigg√−gd4x, (1)\nwhereLMis the Lagrangian corresponding to the matter density. Imple-\nmenting the principle of least-action on Eq.(1) provides\nGab=T(EFF)\nab=1\nfR−LMfQ/parenleftBig\n8πTab+T(D)\nab/parenrightBig\n, (2)\nwhere,T(EFF)\nabandTabare termed as the effective and usual anisotropic mat-\nter EMT. Also, Gabis the Einstein tensor. The modified corrections are\nrepresented by T(D)\nabwhich has the form\nT(D)\nab=/parenleftbigg\nfT+1\n2RfQ/parenrightbigg\nTab+/braceleftbiggR\n2/parenleftbiggf\nR−fR/parenrightbigg\n−1\n2∇c∇d/parenleftbig\nfQTcd/parenrightbig\n−LMfT}gab−1\n2/square/parenleftbig\nfQTab/parenrightbig\n−2fQRc(aTc\nb)+∇c∇(a[Tc\nb)fQ]\n−/parenleftbig\ngab/square−∇a∇b/parenrightbig\nfR+2/parenleftbig\nfQRcd+fTgcd/parenrightbig∂2LM\n∂gab∂gcd, (3)\nwherefR=∂f(R,T,Q)\n∂R,fT=∂f(R,T,Q)\n∂TandfQ=∂f(R,T,Q)\n∂Q. Also, the math-\nematical expression of the D’Alambert operator is /square≡1√−g∂a/parenleftbig√−ggab∂b/parenrightbig\n5and∇cindicates the covariant derivative. Generally, the matter Lagrang ian\ncan be taken in terms of energy density or pressure, thus we cons ider it as\nLM=−µfor the case of anisotropic matter distribution which results in\n∂2LM\n∂gab∂gcd= 0 [31].\nTo discuss the collapse of dynamical cylinder, we take line element rep -\nresenting the interior geometry as\nds2=−A2dt2+B2dr2+C2dφ2+dz2, (4)\nwhereA=A(t,r) andB=B(t,r) are dimensionless, while C=C(t,r) has\nthedimension of r. The EMT portraying anisotropic dissipative fluid isgiven\nas\nTab=/parenleftbig\nµ+Pr/parenrightbig\nUaUb+Prgab+/parenleftbig\nPφ−Pr/parenrightbig\nKaKb+/parenleftbig\nPz−Pr/parenrightbig\nSaSb\n+ςaUb+ςbUa−/parenleftbig\ngab+UaUb/parenrightbig\nαΩ, (5)\nwhere P r,Pφand P zare the principal pressures and µis the energy density.\nAlso,αand Ω are the coefficient of bulk viscosity and the expansion scalar,\nrespectively. The four velocity ( Ua), four vectors ( KaandSa), heat flux ςa\nand Ω are defined as\nUa=−Aδ0\na,Ka=Cδ2\na,Sa=δ3\na, ςa=ςBδ1\na,Ω =Ua\n;a,(6)\nsatisfying the following relations\nUaUa=−1,KaKa= 1,SaSa= 1,UaKa= 0 =SaKa=UaSa.(7)\nDue to the interaction of matter components and geometry in this e xtended\ntheory, the EMT has non-disappearing divergence, i.e., ∇aTab/ne}ationslash= 0. This\nexerts an extra force in the gravitational field that triggers the n on-geodesic\nmotion of test particles. Consequently, we have\n∇aTab=2\n2fT+RfQ+16π/bracketleftbigg\n∇a/parenleftbig\nfQRcaTcb/parenrightbig\n+∇b/parenleftbig\nLMfT/parenrightbig\n−Gab∇a/parenleftbig\nfQLM/parenrightbig\n−1\n2∇bTcd/parenleftbig\nfTgcd+fQRcd/parenrightbig\n−1\n2/braceleftbig\n∇a(RfQ)+2∇afT/bracerightbig\nTab/bracketrightbigg\n.(8)\nThe trace of modified field equations is given by\n3∇c∇cfR−T(fT+8π)+R/parenleftbigg\nfR−T\n2fQ/parenrightbigg\n+1\n2∇c∇c(fQT)\n6+∇a∇c(fQTac)−2f+(RfQ+4fT)LM+2RacTacfQ\n−2gbd(fTgac+fQRac)∂2LM\n∂gbd∂gac= 0.\nThe disappearance of fQfrom the field equations provides the gravitational\neffects of f(R,T) theory, whereas the f(R) gravity can be achieved for fT=\n0 =fQ.\nWe adopt a standard model of the form\nf(R,T,Q) =f1(R)+f2(T)+f3(Q) =R+Φ√\nT+ΨQ.(9)\nIt is noteworthy that the gravitational model produces physically acceptable\nresults by taking different choices of the model parameters (involv ing in that\nmodel) within their noticed range. ForΦ = 0, this model was used to an alyze\nisotropic systems and some acceptable values of Ψ have been acquir ed for\nwhich the systems show stable behavior [32, 33]. The quantities R,Tand\nQof the model (9) become\nR=−2\nA3B3C/bracketleftbigg\nA3BC′′−AB3¨C −AB2C¨B+A2BCA′′−A3B′C′+B3˙A˙C\n+A2BA′C′−AB2˙B˙C+B2C˙A˙B−A2CA′B′/bracketrightbigg\n,\nT=−µ+Pr+Pφ+Pz−3αΩ,\nQ=−1\nA3B3C/bracketleftbigg\nµ/braceleftbig\nAB2C¨B −A2BCA′′+A2CA′B′−A2BA′C′+AB3¨C\n−B2C˙A˙B−B3˙A˙C/bracerightbig\n+2ςAB/braceleftbig\nAB˙C′−BA′˙C −A˙BC′/bracerightbig\n+/parenleftbig\nPr−αΩ/parenrightbig\n/braceleftbig\nB2C˙A˙B −AB2C¨B+A2BCA′′−AB2˙B˙C −A3B′C′−A2CA′B′\n+A3BC′′/bracerightbig\n+/parenleftbig\nPφ−αΩ/parenrightbig/braceleftbig\nA3BC′′−A3B′C′+A2BA′C′−AB2˙B˙C\n−AB3¨C+B3˙A˙C/bracerightbig/bracketrightbigg\n,\nwhere.=∂\n∂tand′=∂\n∂r.\nThe corresponding field equations are\n1\n1+Ψµ/parenleftbigg\n8πµ+µ(D)\nA2/parenrightbigg\n=B′C′\nB3C−C′′\nB2C+˙B˙C\nA2BC, (10)\n71\n1+Ψµ/parenleftBigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightBigg\n=˙A˙C\nA3C−¨C\nA2C+A′C′\nAB2C, (11)\n1\n1+Ψµ/parenleftBigg\n8πPφ−ζΩ+P(D)\nφ\nC2/parenrightBigg\n=˙A˙B\nA3B+A′′\nAB2−¨B\nA2B−A′B′\nAB3,(12)\n1\n1+Ψµ/parenleftbig\n8πPz−ζΩ+P(D)\nz/parenrightbig\n=˙A˙C\nA3C−¨B\nA2B−¨C\nA2C+˙A˙B\nA3B+A′′\nAB2\n+C′′\nB2C+A′C′\nAB2C−A′B′\nAB3−C′B′\nB3C−˙B˙C\nA2BC,\n(13)\n1\n1+Ψµ/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n=˙C′\nABC−˙BC′\nAB2C−A′˙C\nA2BC, (14)\nwhereζ= 8πα. These equations describe how gravity and matter compo-\nnents bend spacetime. The second term on the left hand side of the above\nequations/parenleftbig\nµ(D),P(D)\nr,P(D)\nφ,P(D)\nzandς(D)/parenrightbig\nappear due to the modifica-\ntion of gravity and their values are provided in Appendix A. The quantities/parenleftBig\n8πµ+µ(D)\nA2/parenrightBig\n,/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n,/parenleftbigg\n8πPφ−ζΩ+P(D)\nφ\nC2/parenrightbigg\n,/parenleftBig\n8πPz−ζΩ+P(D)\nz/parenrightBig\nand/parenleftBig\n8πς−ς(D)\nAB/parenrightBig\ndepict the effective energy density, effective principal pres-\nsures and the effective heat flux, respectively.\nThe C-energy within the interior geometrical structure (4) can be deter-\nmined as [45]\n˜m(t,r) =LˆE =L\n8(1−L−2∇aˆr∇aˆr), (15)\nwhereˆE is the total gravitational energy per specific length of the cylin-\nder and ˆ r=̺Lsymbolizes the circumference radius. The terms ̺andL\nare the areal radius of the cylinder and specific length, respective ly, whose\nmathematical expressions are ̺2=η(1)bηb\n(1)andL2=η(2)bηb\n(2). Also, the\nKilling vectors are defined as η(1)=∂\n∂φ,η(2)=∂\n∂z. Equation (15), after some\nmanipulation, yields the mass as\n˜m=L\n8/bracketleftbigg\n1−/parenleftbiggC′\nB/parenrightbigg2\n+/parenleftbigg˙C\nA/parenrightbigg2/bracketrightbigg\n. (16)\nThe 3D hypersurface Σ splits the geometry into interior and exterio r regions.\nThe interior region is defined in Eq.(4) while the exterior spacetime is ta ken\n8as\nds2=2M(v)\nRdv2−2dvdR+R2/parenleftbig\ndφ2+λ2dz2/parenrightbig\n, (17)\nwherevis the retarded time. Also, MandRsymbolize the mass and radius\nof the exterior region. We utilize Darmois junction conditions [46] who se\nfundamental forms are given by\n•The continuity of the metric coefficients of the interior and exterior\nspacetimes holds at the hypersurface.\n•There is no difference between the extrinsic curvature correspon ding to\nboth geometries at Σ that equals the radial pressure to the heat fl ux\nfor the case of dynamical fluid distribution.\nSincethecollapseofaself-gravitatingobject isassociatedwiththe matter\nsector, thus we only require to employ the second fundamental fo rm that\nyields, after some manipulation, in this modified theory as\nM−˜mΣ\n=L\n8,8πPr−ζΩ+P(D)\nr\nB2Σ\n=8πς−ς(D)\nAB. (18)\nIt is observed that the least satisfactory definition of the C-ener gy produces\nthe difference between masses of both regions byL\n8, which disappears in the\ncase of spherical spacetime. The other equation equals the effect ive radial\npressure and effective heat flux at the boundary Σ. This guarante es the\nfulfillment of the condition of vanishing radial pressure at the bound ary only\niftheheatfluxalongwithmodifiedcorrectionsdisappears, i.e., 8 πς−ς(D)\nAB= 0.\n3 Dynamics of the Cylindrical Star\nInitially, Misner and Sharp formulated some dynamical quantities to s tudy\nthe evolution of spherical geometry. The proper radial and tempo ral deriva-\ntives were used to compute the velocity and acceleration of the con sidered\ncollapsing source. These equations havelaterbeenusedinthestud yofspher-\nical as well as cylindrical spacetimes [47]. The dynamical equations in this\nscenario are\nT(EFF)b\na;bUa=/parenleftbig\n8πTb\na+T(D)b\na/parenrightbig\n;bUa= 0, (19)\nT(EFF)b\na;bςa=/parenleftbig\n8πTb\na+T(D)b\na/parenrightbig\n;bςa= 0. (20)\n9Equations (19) and (20) yield, respectively, as\n1\nA2/parenleftbigg\n8πµ+µ(D)\nA2/parenrightbigg.\n+˙B\nA2B/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n+1\nAB/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg′\n+˙C\nA2C/parenleftbigg\n8πµ+µ(D)\nA2+8πPφ−ζΩ+P(D)\nφ\nC2/parenrightbigg\n+1\nAB/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg/parenleftbigg2A′\nA+C′\nC/parenrightbigg\n= 0, (21)\nB\nA/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg.\n+A′\nA/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n+/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg′\n+C′\nC/parenleftbigg\n8πPr+P(D)\nr\nB2−8πPφ−P(D)\nφ\nC2/parenrightbigg\n+B\nA/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg/parenleftbigg2˙B\nB+˙C\nC/parenrightbigg\n= 0. (22)\nThese equations play significant role in the study of variations arising in the\nstellar evolution. Our goalisto discuss thedynamics of thecollapsing source,\nthus the definitions of proper radial as well as temporal derivative s are [3, 47]\nDr=1\nC′∂\n∂r,Dt=1\nA∂\n∂t. (23)\nThe radius of an astrophysical object decreases continuously du ring the\ncollapse as gravity dominates the outward pressure. Consequent ly, the ve-\nlocity of interior fluid turns out to be negative, i.e.,\nU=Dt(C) =˙C\nA<0. (24)\nUsing this equation in C-energy (16), we have\nC′\nB=/parenleftbigg\n1+U2−8˜m\nL/parenrightbigg1\n2\n=ω. (25)\nThe C-energy of the current cylindrical configuration yields after applying\nthe definition of Dtas\nDt(˜m) =−CL\n4(1+Ψµ)/braceleftBigg/parenleftBigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightBigg\nU\n10+/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\nω/bracerightbigg\n, (26)\nwhichdemonstratesthathowthetotalenergyvarieswithtime. Th isequation\nalso indicates how the collapsing phenomenon is influenced by the radia l\npressure, the expansion scalar as well as heat flux and modified cor rections.\nAsUis negative, thus the factor/parenleftBig\n8πPr−ζΩ+P(D)\nr\nB2/parenrightBig\nUon the right hand\nside of the above equation becomes positive, which guarantees tha t the total\nenergy of the system increases. The other entity/parenleftBig\n8πς−ς(D)\nAB/parenrightBig\nωconfirms the\nreduction of total energy as heat dissipates from the source.\nNext, in order to discuss the variation of energy between the adjo ining\ncylindrical surfaces, we employ the definition of Dron Eq.(16) and combine\nit with Eqs.(10) and (14) as\nDr(˜m) =CL\n4(1+Ψµ)/braceleftbigg/parenleftbigg\n8πµ+µ(D)\nA2/parenrightbigg\n+/parenleftbigg\n8πς−ς(D)\nAB/parenrightbiggU\nω/bracerightbigg\n.(27)\nThe collapse rate of the current setup is also affected by the effect ive energy\ndensity. The term/parenleftBig\n8πµ+µ(D)\nA2/parenrightBig\nin the above equation ultimately increases\nthetotalenergyofthemattersource. Thenextentity,/parenleftBig\n8πς−ς(D)\nAB/parenrightBig\nU\nω,reveals\nthat the heat energy dissipates from the system, as the fluid has n egative\nvelocity. Theaccelerationofthecollapsingsourcecanbecalculated bytaking\nproper temporal derivative of Uas\nDt(U) =−C\n1+Ψµ/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n−˜m\nC2+ωA′\nAB+L\n8C2/parenleftbig\n1+U2−ω2/parenrightbig\n.(28)\nOne can get the value ofA′\nAfrom Eq.(22) as\nA′\nA=−1/parenleftBig\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightBig/braceleftbiggB\nA/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg.\n+/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg′\n+C′\nC/parenleftbigg\n8πPr+P(D)\nr\nB2−8πPφ−P(D)\nφ\nC2/parenrightbigg\n+B\nA/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg/parenleftbigg2˙B\nB+˙C\nC/parenrightbigg/bracerightbigg\n. (29)\n11Inserting this value in Eq.(28), we have\nDt(U)/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n=−/braceleftbigg˜m\nC2−L\n8C2/parenleftbigg\n1+U2/parenrightbigg\n+C\n1+Ψµ/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg/bracerightbigg/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n−ω2\nC/braceleftbigg/parenleftbigg\n8πPr+P(D)\nr\nB2−8πPφ−P(D)\nφ\nC2/parenrightbigg\n+L\n8C/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ\n+P(D)\nr\nB2/parenrightbigg/bracerightbigg\n−ω/braceleftbigg1\nB/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg′\n+Dt/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n+1\nA\n×/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg/parenleftbigg2˙B\nB+˙C\nC/parenrightbigg/bracerightbigg\n. (30)\nThe left side of this equation represents the Newtonian force as th e prod-\nuct of acceleration ( DtU) and the term/parenleftbigg\n8πµ+µ(D)\nA2+ 8πPr−ζΩ +P(D)\nr\nB2/parenrightbigg\n(refers to the inertial mass density) appears. On the other hand , the same\nterm also arises in the first term on the right side, that now present s the\ngravitational mass density. Thus, the equivalence of these both m asses leads\nto the fulfillment of the equivalence principle. The second curly brack et de-\ntermines the impact of gravitational mass density and effective str esses in r\nas well as φdirections on the collapse rate. The role of gradient of effective\nradial pressure and the expansion scalar in this scenario can be man ifested\nthrough the entity/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg′\n. Also, the last two terms containing\nthe heat flux and modified corrections can well describe hydrodyna mics of\nthe cylinder.\n4 Transport Equations\nAs the EMT (5) involves the heat flux, the transport equations in th is regard\nare very useful tool to analyze the structural evolution of comp act geometry.\nThey also disclose how some physical quantities such as mass, heat a nd mo-\nmentumareevaluatedduring thecollapse. Thediffusionprocessissu pported\n12by the following transport equation given as\n̺habUc¯ςb;c+ ¯ςa=−ηhab(τ,b+τab)−1\n2ητ2/parenleftbigg̺Ub\nητ2/parenrightbigg\n;b¯ςa,(31)\nwhere ¯ς=/parenleftBig\n8πς−ς(D)\nAB/parenrightBig\nand hab=gab+UaUbis the projection tensor. Also,\nη, ̺, τand ab/parenleftbig\ni.e., a1=A′\nA/parenrightbig\nare mathematical symbols of the thermal con-\nductivity, relaxation time, temperature and acceleration, respec tively. After\nsome simplification, Eq.(31) produces\nBDt/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n=−ητ′\n̺−ητ\n̺/parenleftbiggA′\nA/parenrightbigg\n−ητ2B\n2A̺/parenleftbigg̺\nητ2/parenrightbigg./parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n−B\n2A/parenleftBigg\n3˙B\nB+˙C\nC+2A\n̺/parenrightBigg/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n. (32)\nThis equation becomes after inserting the value ofA′\nAas\nBDt/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n=−ητ2B\n2A̺/parenleftbigg̺\nητ2/parenrightbigg./parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n−B\n2A/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n×/parenleftbigg3˙B\nB+˙C\nC+2A\n̺/parenrightbigg\n−ητB\n̺ω/braceleftbigg\nDt(U)−L\n8C2/parenleftbig\n1+U2−ω2/parenrightbig\n+˜m\nC2+C\n1+Ψµ/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg/bracerightbigg\n−ητ′\n̺,(33)\nwhich demonstrates how much variation takes place in heat energy w ith the\npassage of time. This equation also explains the impact of temperatu re,\nthermal conductivity and relaxation time on the self-gravitating sy stems.\nEliminating Dt/parenleftBig\n8πς−ς(D)\nAB/parenrightBig\nfrom Eqs.(30) and (33), we obtain\nDt(U)/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2−ητ\n̺/parenrightbigg\n=−/braceleftbigg˜m\nC2−L\n8C2/parenleftbigg\n1+U2/parenrightbigg\n+C\n1+Ψµ/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg/bracerightbigg/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2−ητ\n̺/parenrightbigg\n×/braceleftbigg\n1−ητ\n̺/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg−1/bracerightbigg\n+ω2/braceleftbiggητL\n8̺C2−L\n8C2\n13×/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n−1\nC/parenleftbigg\n8πPr+P(D)\nr\nB2−8πPφ−P(D)\nφ\nC2/parenrightbigg/bracerightbigg\n−ω/bracketleftbigg\n−ητ′\n̺B+1\nB/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg′\n+1\n2/braceleftbigg˙B\nAB+˙C\nAC−2\n̺−ητ2\nA̺/parenleftbigg̺\nητ2/parenrightbigg./bracerightbigg\n×/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg/bracketrightbigg\n. (34)\nWe can rearrange this equation as\nDt(U)/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg/parenleftbig\n1−H/parenrightbig\n=−Fgrav/parenleftbig\n1−H/parenrightbig\n+Fhyd+ω2/braceleftbiggητL\n8̺C2−L\n8C2/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n−1\nC/parenleftbigg\n8πPr+P(D)\nr\nB2−8πPφ−P(D)\nφ\nC2/parenrightbigg/bracerightbigg\n, (35)\nwhere\nH=ητ\n̺/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg−1\n, (36)\nFgrav=/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg/braceleftbigg˜m\nC2−L\n8C2/parenleftbig\nU2+1/parenrightbig\n+C\n1+Ψµ/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg/bracerightbigg\n, (37)\nFhyd=−ω/bracketleftbigg1\nB/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg′\n+1\n2/braceleftbigg˙B\nAB+˙C\nAC−2\n̺−ητ2\nA̺/parenleftbigg̺\nητ2/parenrightbigg./bracerightbigg\n×/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n−ητ′\n̺B/bracketrightbigg\n. (38)\nEquation (35) explains how the collapse rate is affected by different f orces,\ncomprising Newtonian ( Fnewtn), hydrodynamical ( Fhyd) and gravitational\n(Fgrav) forces. It is known that energy always dissipates (in the form of\nradiation, convection and conduction) from higher to lower energy state of\nthe system. The energy of a star is dissipated through radiations, if photons\nacquire it from the higher phase of that object. On the other hand , when\n14photons do not possess all the energy, it will be dissipated by conve ction.\nThe hot gasses in this phenomenon move to the upper zone and thus radiate\nenergy, whereas cooler gases attain energy by traveling towards the hot zone.\nThere occur continuous collisions of atoms inside an object due to wh ich ev-\nery atom transfers its energy to the nearest one, and thus ener gy dissipates\nin the form of conduction.\nEquation(35)involvesanentity(1 −H)thatacknowledgestheequivalence\nprinciple, while the gravitational mass density and the term ( H)/parenleftbig\ndefined in\nEq.(36)/parenrightbig\nare inversely proportional to each other. This relation provides th e\nfact that the gravitational forceand thequantity (1 −H) arestrongly affected\nby each other, leading to the following different cases.\n•The entity (1 −H) remains positive only for H<1, which results in\nthe negative gravitational force (i.e., repulsive force) due to the a p-\npearance of minus sign in the first term on the right side of Eq.(35).\nConsequently, the collapse rate diminishes.\n•The rate of the cylindrical collapse increases for the case when H>1,\ni.e., (1−H)<0.\n•Finally, if we consider H= 1, the gravitational as well as inertial forces\ndisappear and we have from Eq.(35) as\nω2/braceleftbiggητL\n8̺C2−L\n8C2/parenleftbigg\n8πµ+µ(D)\nA2+8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg\n−1\nC/parenleftbigg\n8πPr+P(D)\nr\nB2−8πPφ−P(D)\nφ\nC2/parenrightbigg/bracerightbigg\n=ω/bracketleftbigg\n−ητ′\n̺B\n+1\nB/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg′\n+1\n2/parenleftbigg\n8πς−ς(D)\nAB/parenrightbigg\n×/braceleftbigg˙B\nAB+˙C\nAC−2\n̺−ητ2\nA̺/parenleftbigg̺\nητ2/parenrightbigg./bracerightbigg/bracketrightbigg\n. (39)\nThis equation expresses the involvement of temperature, therma l conductiv-\nity, the bulk viscosity and the modified corrections in the collapsing ph e-\nnomenon. The equilibrium position of the collapsing cylinder is supporte d\nby the hydrodynamical force (given on left side of the above equat ion), and\nhence, the collapse rate is reduced.\n155 Relation between the Weyl Scalar and Ef-\nfective Physical Quantities\nIn this section, we develop some relations between effective physica l variables\nand the Weyl scalar ( C2=CcadbCcadb, whereCcadbis the Weyl tensor). This\nscalar can be expressed as a linear combination of the Kretchmann s calar\n(R=RcadbRcadb,Rcadbis the Riemann tensor), the Ricci tensor ( Rab) and\nthe Ricci scalar as [10]\nC2=R−2RabRab+1\n3R2. (40)\nThe scalar Rcan be manipulated as\nR=4\nA4B4C4/braceleftBigg\nC4/parenleftbig\nR0101/parenrightbig2+B4/parenleftbig\nR0202/parenrightbig2+A4/parenleftbig\nR1212/parenrightbig2−A2B2/parenleftbig\nR1202/parenrightbig2\n2/bracerightBigg\n.\n(41)\nFor the considered spacetime (4), the Ricci scalar, non-zero com ponents of\nthe Riemann tensor and the Ricci tensor in terms of the Einstein ten sor are\nR=−2/parenleftbiggG11\nB2+G22\nC2−G00\nA2/parenrightbigg\n,\nR0101=G22/parenleftbig\nABC/parenrightbig2,R0202=G11/parenleftbig\nABC/parenrightbig2,R1212=G00/parenleftbig\nABC/parenrightbig2,\nR0212=G01/parenleftbig\nABC/parenrightbig2,R00=A2/parenleftbiggG11\nB2+G22\nC2/parenrightbigg\n,R01=G01,\nR11=B2/parenleftbiggG00\nA2−G22\nC2/parenrightbigg\n,R22=C2/parenleftbiggG00\nA2−G11\nB2/parenrightbigg\n.\nThese values provide the scalar R(41) as\nR=4\nA4B4C4/braceleftbig\nB4C4G2\n00+A4C4G2\n11+A4B4G2\n22−4A2B2C4G2\n01/bracerightbig\n.(42)\nInserting these equations in Eq.(40), the Weyl scalar takes the fo rm as\nC2=4\n3A4B4C4/braceleftbig\nB4C4G2\n00+A4C4G2\n11+A4B4G2\n22+A2B2C4G00G11\n16+A2B4C2G00G22−A4B2C2G11G22/bracerightbig\n. (43)\nUsing this equation and the field equations (10)-(12) yields\n√\n3C\n2=/bracketleftbigg/braceleftbigg1\n1+Ψβ/parenleftbigg\n8πµ+µ(D)\nA2+8πPr+P(D)\nr\nB2−8πPφ−P(D)\nφ\nC2/parenrightbigg/bracerightbigg2\n−1\n1+Ψβ/braceleftbigg/parenleftbigg\n8πPr−ζΩ+P(D)\nr\nB2/parenrightbigg/parenleftbigg\n8πPφ−ζΩ+P(D)\nφ\nC2/parenrightbigg\n+/parenleftbigg\n8πPr+P(D)\nr\nB2+2ζΩ−24πPφ−3P(D)\nφ\nC2/parenrightbigg/parenleftbigg\n8πµ+µ(D)\nA2/parenrightbigg/bracerightbigg/bracketrightbigg 1\n2\n.(44)\nThe necessary and sufficient condition for a spacetime to be confor mally\nflat is the energy density homogeneity. We check the validity of this r esult\nin the present modified gravity. For this, we have considered the st andard\nmodel (9). We take the case when R=R0andf2(T) as well as f3(Q) are\ntreated as constants, thus Eq.(44) is left with\n√\n3C\n2=/bracketleftbigg/braceleftbigg\n8π/parenleftbigg\nµ+Pr−Pφ−C0\n16π/parenrightbigg/bracerightbigg2\n−/parenleftbigg\n8πPr+2ζΩ−24πPφ−C0/parenrightbigg\n×/parenleftbigg\n8πµ−C0\n2/parenrightbigg\n−/parenleftbigg\n8πPr−ζΩ+C0\n2/parenrightbigg/parenleftbigg\n8πPφ−ζΩ+C0\n2/parenrightbigg/bracketrightbigg1\n2\n,(45)\nwhereC0= Φ√T0+ ΨQ0is a constant term. This equation depicts that\ninhomogeneityintheenergydensityofthefluidisinducedduetothep resence\nof the bulk viscosity and theprincipal pressures. The above relatio n also says\nthat inhomogeneity in the system (during evolution) is increased by t he tidal\nforces [16]. The only possibility to obtain conformally flat spacetime is t he\nconsideration of dust matter distribution, that gives in the absenc e of bulk\nviscosity as\n√\n3C= 16π/parenleftbigg\nµ−C0\n16π/parenrightbigg\n⇒√\n3C′= 16πµ′. (46)\nWeobserve fromthisequation that homogeneous energy density ( i.e.,µ′= 0)\nimpliesconformallyflatspacetime( C= 0throughregularaxiscondition)and\nvice-versa, and hence the required condition is obtained.\n176 Conclusions\nOur cosmos comprises an abundance of astronomical systems who se struc-\ntural formation is highly influenced by an appealing phenomenon, nam ed as\nthe gravitational collapse. The study of gravitational waves thro ugh multiple\nobservations has prompted several astrophysicists to investiga te the collaps-\ning rate of self-gravitating geometries in GRand other extended theories\n[48]. This article is based on the formulation of dynamical description o f the\ncylindrical fluid distribution to investigate the changes that are gra dually\nproduced within the system in the background of f(R,T,RabTab) gravity.\nThe effect of heat dissipation and principal pressures (such as P r,Pφand P z)\non the interior geometry has been considered. We have formulated two dy-\nnamicalequationsthroughMisner-Sharpformalismtoexaminethev ariations\nin the total energy with respect to radial as well as temporal coor dinates.\nWe have constructed the transport equation as well as some fund amental\nforces (such as the gravitational, hydrodynamical and Newtonian ) and then\ncoupled them with dynamical equations to analyze the impact of modifi ed\ngravity on the collapse rate. The entity His found to be in direct relation\nwith temperature as well as thermal conductivity and inversely rela ted with\ngravitational mass density. In the following, we summarize our resu lts.\n•The entity Hwill be less as compared to GRfor positive effect of the\nmodified corrections. This leads to the increment in the term (1 −H)\nas well as the gravitational force. However, the appearance of m inus\nsign ultimately diminishes the collapse rate.\n•Therateofcylindrical collapsemayincreaseforthecasewhenthee ffect\nof correction terms is negative.\n•One cannot say anything about the decrement/increment in the co l-\nlapsing rate when the corrections of this modified gravity involved in\nHhave opposite signs.\nThe relevance of density inhomogeneity and the Weyl scalar has also been\ndeveloped. By implying some constraints on the considered modified m odel,\nit has been shown that the homogenous density and the conformal flatness\nof the current setup imply each other. It is worth mentioning here t hat the\ntidal forces involving in the Weyl tensor produce more inhomogeneit y in the\nfluid during the evolutionary process. All these results can be reco vered in\nGRfor Φ = 0 = Ψ.\n18Appendix A\nThe modified corrections in Eqs.(10)-(14) are\nµ(D)=−A2/parenleftbig\nΦ√\nT+ΨQ/parenrightbig\n2+Ψ/braceleftbigg\nµ/parenleftbigg4˙A2\nA2−A′2\nB2+AA′′\nB2+3˙A˙B\nAB−AA′B′\nB3\n+3˙A˙C\nAC+AA′C′\nB2C−2¨C\nC−2¨B\nB/parenrightbigg\n+ ˙µ/parenleftbigg˙C\n2C+˙B\n2B/parenrightbigg\n−µ′/parenleftbigg2AA′\nB2−A2B′\n2B3\n+A2C′\n2B2C/parenrightbigg\n−µ′′A2\n2B2+Pr/parenleftbigg4A2B′2\nB4−A2B′′\nB3+˙B2\nB2/parenrightbigg\n−˙Pr˙B\n2B−5P′\nrA2B′\n2B3\n+P′′\nrA2\n2B2−Pφ/parenleftbigg˙C2\nC2−A2C′2\nB2C2/parenrightbigg\n−˙Pφ˙C\n2C+P′\nφA2C′\n2B2C−ς/parenleftbigg2A˙C′\nBC−2˙AB′\nB2\n+2˙A′\nB−4˙AA′\nAB−2A′˙B\nB2−2A˙BC′\nB2C−2A′˙C\nBC−2A′˙B\nB2/parenrightbigg\n−2˙ςA′\nB−2ς′˙A\nB/bracerightbigg\n,\nP(D)\nr=B2\n2/braceleftbigg/parenleftbiggΦ√\nT+ΨR/parenrightbigg\nPr+Φ√\nT+ΨQ+Φµ√\nT/bracerightbigg\n+Ψ/braceleftbigg\nµ/parenleftbigg¨AB2\nA3−A′2\nA2\n−4˙A2B2\nA4/parenrightbigg\n+5˙µ˙AB2\n2A3+µ′A′\n2A−¨µB2\n2A2+Pr/parenleftbiggB˙A˙B\nA3−3A′B′\nAB−B˙B˙C\nA2C\n+˙B2\nA2−4B′2\nB2−B¨B\nA2−3B′C′\nBC+2A′′\nA+2C′′\nC/parenrightbigg\n+˙Pr/parenleftbiggB2˙C\n2A2C−B2˙A\n2A3\n+2B˙B\nA2/parenrightbigg\n−P′\nr/parenleftbiggA′\n2A+C′\n2C/parenrightbigg\n+¨PrB2\n2A2−Pφ/parenleftbiggB2˙C2\nA2C2−C′2\nC2/parenrightbigg\n−˙PφB2˙C\n2A2C\n−P′\nφC′\n2C+ς/parenleftbigg2˙B′\nA−2˙AB′\nA2−4A′˙B\nA2−4˙BB′\nAB+2B˙C′\nAC−2˙BC′\nAC−2BA′˙C\nA2C/parenrightbigg\n+2˙ςB′\nA+2ς′˙B\nA/bracerightbigg\n,\nP(D)\nφ=C2\n2/braceleftbigg/parenleftbiggΦ√\nT+ΨR/parenrightbigg\nPφ+Φ√\nT+ΨQ+Φµ√\nT/bracerightbigg\n+Ψ/braceleftbigg\nµ/parenleftbigg¨AC2\nA3−A′2C2\nA2B2\n−4˙A2C2\nA4/parenrightbigg\n+5˙µ˙AC2\n2A3+µ′A′C2\n2AB2−¨µC2\n2A2+Pr/parenleftbiggB′′C2\nB3−4B′2C2\nB4−˙B2C2\nA2B2/parenrightbigg\n+˙PrC2˙B\n2A2B+5P′\nrC2B′\n2B3−P′′\nrC2\n2B2+Pφ/parenleftbigg˙C2\nA2−C¨C\nA2+C˙A˙C\nA3−CB′C′\nB3−C˙B˙C\nA2B\n19−C′2\nB2−CA′C′\nAB2+CC′′\nB2/parenrightbigg\n+˙Pφ/parenleftbiggC2˙B\n2A2B−C2˙A\n2A3+2C˙C\nA2/parenrightbigg\n−P′\nφ/parenleftbiggC2A′\n2AB2\n−C2B′\n2B3+2CC′\nB2/parenrightbigg\n+¨PφC2\n2A2−P′′\nφC2\n2B2−ς/parenleftbigg3C2˙AA′\nA3B+3C2˙BB′\nAB3+3C2A′˙B\nA2B2\n−C2˙A′\nA2B−C2˙B′\nAB2+C2˙AB′\nA2B2/parenrightbigg\n+ ˙ς/parenleftbigg2C2A′\nA2B+C2B′\nAB2/parenrightbigg\n+ς′/parenleftbiggC2˙A\nA2B+2C2˙B\nAB2/parenrightbigg\n−˙ς′C2\nAB/bracerightbigg\n,\nP(D)\nz=1\n2/braceleftbigg/parenleftbiggΦ√\nT+ΨR/parenrightbigg\nPφ+Φ√\nT+ΨQ+Φµ√\nT/bracerightbigg\n+Ψ/braceleftbigg\nµ/parenleftbigg¨A\nA3−A′2\nA2B2\n−4˙A2\nA4/parenrightbigg\n+5˙µ˙A\n2A3+µ′A′\n2AB2−¨µ\n2A2+Pr/parenleftbiggB′′\nB3−4B′2\nB4−˙B2\nA2B2/parenrightbigg\n+˙Pr˙B\n2A2B\n+5P′\nrB′\n2B3−P′′\nr\n2B2+Pφ/parenleftbiggC′2\nB2C2−˙C2\nA2C2/parenrightbigg\n+˙Pφ˙C\n2A2C−P′\nφC′\n2B2C+˙Pz/parenleftbigg˙B\n2A2B\n−˙A\n2A3+˙C\n2A2C/parenrightbigg\n+P′\nz/parenleftbiggB′\n2B3−A′\n2AB2−C′\n2B2C/parenrightbigg\n+¨Pz\n2A2−P′′\nz\n2B2\n+ς/parenleftbigg˙A′\nA2B−3˙AA′\nA3B−˙AB′\nA2B2+˙B′\nAB2−3A′˙B\nA2B2−3˙BB′\nAB3/parenrightbigg\n+ ˙ς/parenleftbigg2A′\nA2B\n+B′\nAB2/parenrightbigg\n+ς′/parenleftbigg˙A\nA2B+2˙B\nAB2/parenrightbigg\n−˙ς′C2\nAB/bracerightbigg\n,\nς(D)=−ςAB\n2/parenleftbiggΦ√\nT+ΨR/parenrightbigg\n+Ψ/braceleftbigg\nµ/parenleftbiggA′˙C\nAC−˙C′\nC+˙BC′\nBC/parenrightbigg\n+˙µA′\n2A+µ′˙A\n2B\n−˙µ′\n2+Pr/parenleftbigg˙C′\nC−A′˙C\nAC−˙BC′\nBC/parenrightbigg\n−˙PrA′\n2A−P′\nr˙B\n2B+˙P′\nr\n2+ς/parenleftbigg2¨B\nA−¨AB\nA2\n−4˙A˙B\nA2+2˙A2B\nA3+A′2\nAB+4A′B′\nB2−2A′′\nB+AB′′\nB2−˙B2\nAB+B¨C\nAC−AC′′\nBC\n−2AB′2\nB3−3B˙A˙C\n2A2C+˙B˙C\n2AC−A′C′\n2BC+3AB′C′\n2B2C/parenrightbigg\n−˙ς/parenleftbigg2˙AB\nA2+B˙C\n2AC/parenrightbigg\n+ς′/parenleftbigg2AB′\nB2+AC′\n2BC/parenrightbigg/bracerightbigg\n.\n20Data Availability Statement\nThis manuscript has no associated data.\nReferences\n[1] S Chandrasekhar Mon. 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Domínguez \nRodríguez a \na Departamento Física de la Materia Condensada, Unive rsidad de Sevilla, Apdo. 1065, \n41080 Seville, Spain  \nb C2RMF, CNRS UMR 171, Palais du Louvre, 14 quai Fran çois Mi)errand 75001 Paris, \nFrance  \nc LEM, CNRS UMR 104, ONERA, BP 72, 92322 CHATILLON Ce dex, France \n \n \nAbstract  \nThe plas%c deforma%on of Cr3+ -doped a-Al2 O3  (rub y) with four dis%nct Cr concentra%ons has \nbeen studied at temperatures between 900 and 1500 ◦  C. The mechanical tests indicate that the \nprocesses of disloca%on mul%plica%on and the adjust ments of the slip velocity of these \ndisloca%ons to the imposed strain rate in undoped a -Al2 O3 are not significantly affected by the \nCr concentra%on. The yield stress increment is appr oximately constant with temperature and the \nhardening increases with the Cr concentra%on. The c ri%cal resolved shear stress (CRSS) is fi8ed \nto a model based on disloca%on glide controlled by the nuclea%on and propaga%on of kink pairs, \nmodified to include the hardening due to Cr. Sa%sfac tory fits are achieved with adjustable \nparameters close to those employed for undoped a-Al 2 O3 . \n \n1. Introduc\u0010on  \nDuring the last decades, the plas%c deforma%on of s apphire (a-Al2 O3 ) has been studied by \nmany authors because it is a crystalline material w ith low symmetry whose deforma%on can take \nplace by slip in the basal plane as much as in the prisma%c planes, and also by twinning along \nrhombohedral (pyramidal) planes. Pyramidal slip and  basal twinning can also be ac%vated in this \ncrystal. (0 0 0 1)1/3 2¯ 11 0) basal slip is the pr imary deforma%on system for temperatures above \n700 ◦C.1–7 Prism plane slip along {1 2¯ 1 0 } 1¯ 01 0) is the easiest slip system below 700 ◦C.3 ,4 \nThe ( 1¯ 01 0) Burgers vector for prism plane slip,  which is the fourth in length in sapphire, is \ndissociated into three par%al disloca%ons with coll inear 1/3 1¯ 01 0) Burgers vector.8 The Peierls \nmechanism has been frequently invoked to explain th e movement of  disloca%ons in sapphire, as \nmuch in the basal plane as in the prisma%c plane, a t intermediate temperatures.3–6  Rhombohe- \ndral twinning, the soCest deforma%on mode of sapphi re above 500 ◦C, is ac%vated under a shear \nstress of 10 MPa, independent of test temperature; this stress can, however, be considerably \nincreased, by a factor of 5–10, in the presence of a high density of disloca%ons in the material.9 \nDoping with small concentra%ons of elements of diffe rent atomic radii or different valence allows \nto create, in a controlled way, point defects ac%ng  as obstacles to disloca%on glide. Solid solu%on \nhardening in sapphire (a-Al2 O3 ) has been first obs erved by Wachtman et al.10 in Cr-doped specimens deformed in basal slip. Klassen-Neklyudov a et al.11  showed that the addi%on of Cr \nraises the yield point of specimens tested in tensi on between 1650 and 1900 ◦C. Radford et al.12 \nstudied the influence of var- ious impuri%es (Fe, Ni , Cr, Ti and Mg) in compression between 1300 \nand 1600 ◦C. They concluded that hardening is relat ed to either the difference in size of the ions \nisovalent to Al3+ , or to the valence of the solute  in the case of aliovalent ions. Pletka et al.13 \nstudied the hardening of sapphire doped with Cr3+ ,  Ti3+ and Ti4+ , deformed by basal slip, \nbetween 1300 and 1500 ◦C for low  Cr concentra%ons, between 1400 and 1500 ◦C for larg e Cr \ncon- centra%ons, and above 1500 ◦C for Ti-doped sap phire. As for the isovalent Cr3+ and Ti3+ \nions, it was concluded that the harden- ing is cons istent with the models of Fleischer14 and \nLabusch,15 i.e. the hardening is due to the elas%c strain field of the solute atom. Hardening by \nthe aliovalent Ti4+  is more effec%ve due to the add i%onal electrical interac%on between \ndisloca%ons and  charged defects. \n In the present work, we have studied the dependence  of the yield stress of Cr3+  containing \nsapphire (rubies) deformed in basal slip between 90 0 and 1500 ◦C. Four Cr concentra%ons have \nbeen inves%gated between 60 and 9540 mol ppm. In an  a8empt to inves%gate situa%ons where \nthe Peierls mechanism should take an important part , the temperature range has been expanded \nwell below temperatures explored so far. In order t o explain  the temperature dependence of the \ncri%cal resolved shear stress (CRSS) in sapphire, M itchell et al.16–18 have developed a model \nbased on that designed by Hirth and Lothe19  for di sloca%on glide by nuclea%on and propaga%on \nof kink pairs. This model has been adapted to rubie s to include Cr-induced hardening for basal \nslip. \n \n2. Experimental procedure  \nThe rubies were provided by R.S.A. Le Rubis (Jarrie , France) under the form of cylinders 1.5–3.5 \ncm in diameter, grown by the Verneuil technique wit h four different Cr3+ concentra%ons whose \nvalues were determined by PIXE (par%cle induced X-r ay emission) at the CNA (Seville-Spain) and \nthe C2RMF-CNRS (Paris-France). The Cr contents of t he four types of rubies are shown in Table \n1. Together with the concentra%on of Cr, the conten t of other elements, mainly Ti and Ni, has \nbeen determined by making use of two soCwares (AXIL  and Gupix-Win). As shown in Table 1, the \nresults lie within the uncertainty of the PIXE tech nique. Impuri%es other than Cr are present in \nrela%vely small concentra%ons. Composi%on measureme nts in different points of the surface of \nthe samples indicated rela%vely good homogeneity. \nThe crystals were oriented by Laue back reflec%on X- ray diffrac%on. They were cut in the form of \nparallelepipeds of 2 mm × 2 mm × 4 mm. The compress ion direc%on is at 57◦ from the [000 1] \naxis of sapphire, toward the [0 1 1¯ 0] direc%on (o rienta%on “aa” in Castaing et al.2 ) contained \nin the (2 1¯ 1¯ 0) plane parallel to two lateral fa ces. The orienta%on of the compression load \nfavours the 1/3[1¯ 1¯ 2 0] and 1/3[1¯ 2 1¯ 0] basal  slip with  a Schmid factor of about 0.4 and it \nprevents  rhombohedral twinning from being ac%vated.2  Furthe r structural informa%on can be \nfound in Refs [8,18,20,21]. In order to permit op%c al observa%ons of the slip traces leC at the \nsurface aCer straining, the specimens were given a final polish with a diamond paste less than 3 \nJ..m in grain size. The compression tests were perf ormed in air at temperatures between 900 \nand 1500 ◦C at a constant cross-head speed  of  5 J..m/mn,  corresponding to  an  ini%al strain  \nrate of ε˙ =2.0 × 10−5 s−1  in an Instron machine, model 1185. Th e load was transmi8ed to the \nspecimens via SiC rods. Slip lines were analyzed by  op%cal microscopy (Leica DMRE). Foils \nsuitable for transmission electron microscopy (TEM)   observa%ons  of  the  disloca%on  structure  were  pre- pared by mechanical polishing followed b y ion thinning. The TEM observa%ons were \nperformed in a JEOL 200CX electron microscope (Labo ratoire d’Etude des Microstructures, CNRS- \nONERA, France). \n \n \n \n3. Results  \nConsistent with the crystalline orienta%on of the s pecimens and with deforma%on being \nachieved by basal slip, Fig. 1 shows sharp slip lin es at approximately 30◦ to the compression \ndirec%on on the (2 1¯ 1¯ 0) faces (Fig. 1a). Fainte r steps, perpendicular to the compression axis, \ncan be observed in the orthogonal faces (Fig. 1b). According to previous works,2  this suggests \nthat basal slip can be a8ributed to the ac%va%on of  the two families of disloca%ons with 1/3[1¯ \n1¯ 2 0] and 1/3[1¯ 2 1¯ 0] Burgers vectors in equal  numbers. No rhombohedral twinning was \nobserved.  \n \nFig. 1. Op%cal micrographs of the slip lines in the  lateral faces of a sample of ruby containing \n9540 mol ppm of Cr and deformed by basal slip at T = 1000 ◦ C, with plas%c strain ε = 2.5% and \nstress σ = 415 MPa: (a) the (2 1¯ 1¯ 0) face and (b) the ort hogonal face. The compression \ndirec%on is ver%cal, parallel to the edges of the s ample faces visible in the micrographs. \nThe stress–strain curves for the different Cr conten ts and three temperatures are shown in Fig. \n2. Except for the highest temper- ature and Cr cont ent, they generally exhibit a strong drop aCer \nthe yield point defining an upper and a lower yield point. We take the cri%cal resolved shear \nstress (CRSS) from the flow stress at the lower yiel d point. The flow stress increase with Cr \ncontent is the most pronounced for the high Cr conc entra%ons. In the two low Cr-doped crystals, \ni.e. 60 and 725 mol ppm, the flow stresses are very close, actually within the margin of error of \nour experiments (±1–5 MPa). They are not significant ly higher than for undoped sapphire.2,7 For \nall Cr concentra%ons and temperatures, we did not o bserved any serrated flow that is typical of \ndynamic strain ageing due to mobile disloca%ons int erac%ng with diffusing impuri%es. \nAs observed in Fig. 2, the Cr content influences the  upper yield stress and the flow stress for \nfurther deforma%on, however, not the decrement of e ngineering stress !).σ beyond the upper \nyield point which varies from !).σ = 170 ± 30 MPa at T = 1000 ◦C to !).σ ≈ 0 at T = 1500 ◦C, \nindependent of Cr content. This indicates that the mul%plica%on of disloca%ons and the \nadjustment of their mean velocity to the imposed st rain rate ε˙ are not influenced  by the Cr \nconcentra%on. reduced with increasing temperature. Fig. 2 shows in addi%on that the work-\nhardening rate ( !).σ/!).ε) is li8le dependent on Cr The region of zero work hardening aCer the \nlower point is  content. It remains in the range μ/70– μ/100 ( μ = 156 GPa, elas%c shear modulus), \nand compares well to the work-hardening rate found for undoped  sapphire  deformed  along  a  \nsingle 1/3[1¯ 1¯ 2 0] direc%on in the basal plane.2 2 In consequence, the presence of impuri%es \nof Cr3+ does not seem to affect the work hardening m echanism.22  The values of the CRSS for all \nthe compression tests are given in Table 2 for the various Cr composi%ons and temperatures  \ntested (the Schmid factor is 0.4). They are in addi %on plo8ed in Fig. 3 as a func%on of \ntemperature, between 900 and 1500 ◦C,  together with measurements achieved by various \nauthors2,4,7 for basal slip of undoped sapphire bet ween 600 and 1800 ◦C. There is a clear \nincrease of the CRSS with Cr addi%on. Actually, the  temperature dependence of the yield stress \nfor sapphire and the rubies are much the same in th at the stress increment from one Cr \nconcentra%on to the next is fairly independent of t est temperature. This is more clearly visible \nwhen the CRSS is plo8ed  with a linear scale as shown in Fig. 4 where the CR SS of ruby with a Cr \nconcentra%on of 9540 ppm is about 30 MPa above that  of undoped sapphire. \n \n \n \nFig. 2Engineering stress–strain curves of sapphire doped with different Cr concentra%ons and \ndeformed by basal slip at different temperatures. In  each case, the Cr concentra%on is \nindicated: (a) T = 1000 ◦ C; (b) T = 1300 ◦ C and (c) T = 1500 ◦ C. \n \n \n \nFig. 3. Plot of log (CRSS) vs. temperature for basa l slip in rubies for the different concentra%ons \nof Cr. The experimental values obtained by various authors2,4,7 for basal slip of undoped \nsapphire have been included. The logarithmic scale makes that the hardening caused by the Cr \nis hardly visible at low temperatures. The solid cu rves correspond to the fits from the kink pair \nmodel to the experimental values of the CRSS. \n \nThe present results agree qualita%vely with those p ublished by Pletka et al.13 who concluded \nthat the Cr solu%on hardening rate is independent o f temperature for Cr concentra%ons between \n100 and 9400 mol ppm between 1300 and 1500 ◦C. The experimental points obtained by Pletka \net al.13 for a concentra%on of Cr+3 of 9400 mol ppm  are included in Fig. 4. It is noted though \nthat for all the Cr concentra%ons, the CRSS values obtained by Pletka et al.13 are slightly superior \nto those obtained in our work. The origin of this d ifference is twofold, as Pletka et al. have used  \na higher strain rate (2.6 × 10−5 s−1 ) and their sa mples, grown by  the Verneuil and the Czochralski \ntechniques, contained a higher concentra%on of othe r impuri%es (Si, S, Cl, Ti and Fe) besides Cr. \n \nFig. 4. Experimental values of CRSS vs. temperature  for basal slip in sapphire and in rubies \ncontaining 9540 mol ppm of Cr and in sapphire. The experimental values obtained by Pletka et \nal.13  in ruby containing 9400 mol ppm of Cr are sh own. \n \nTEM observa%ons of deformed rubies have been perfor med in order to assess the part taken by \nthe Peierls mechanism in con- trolling disloca%on m o%on. This role is exemplified in Fig. 5, taken \nin a sample deformed at 1000 ◦C to 2% (9540 mol ppm  of Cr), by the high density of disloca%ons \nwith long straight  segments oriented along (1¯ 1¯ 2 0) and  10 1¯ 0) d irec%ons. Other  TEM \nobserva%ons of similar disloca%on structures have b een performed; they will be published in a \nfuture publica%on. These disloca%on structures are consistent with previous TEM observa%ons \nof disloca%ons aCer deforma%on below 800 ◦C3  and w ith the suggested Peierls mechanisms for \nthe deforma%on of undoped sapphire.3–6 \n \nFig.  5. TEM  micrograph  of  a  disloca%on  struct ure  in  ruby  containing9540 mol ppm of Cr \nand deformed at 1000 ◦ C to 2%. The foil was cut pa rallel to the (000 1) slip plane. \n \n \n4. Discussion  \nIn  metallic  solid  solu%ons,  solu%on  hardening increases with decreasing temperature below \nroom temperature23 and is approximately independent  of temperature at higher tempera- \ntures. This so-called plateau region is therefore a nalogous to the temperature insensi%vity of Cr \nsolu%on hardening in ruby between 900 and 1500 ◦C. Mechanisms used to explain solu%on \nhardening of metallic systems in the plateau region  may therefore be applicable to solu%on \nhardening in rubies. This together with the present  TEM observa%ons of segmented dis- \nloca%ons suggest to make use of a model of disloca% on glide controlled by the nuclea%on and \npropaga%on of kink pairs (Peierls mechanism) in ord er to model the plas%c proper%es of ruby in \nthe ranges of temperatures and Cr contents inves%ga ted. \n \n4.1. Deforma;on of sapphire and ruby  \nRubies can be plas%cally deformed at temperatures a s low as 900 ◦C. The specific orienta%on \nemployed in the present  study prevents rhombohedral twinning from being ac% vated and \nfavours basal slip,2 the single deforma%on mechanis m. The stress strain curves for ruby with 60 \nppm Cr (Fig. 2) are iden%cal to those for undoped s apphire; solid solu%on hardening becomes \nno%ceable only for high chromium content (725 ppm a nd above). Solu%on hardening by Cr is \nmoderate. Even the samples with largest Cr concentr a%on, show limited solu%on hardening (Figs. \n2 and 3). The structure of disloca%ons is dominated  by segments along crystallographic \ndirec%ons (Fig. 5) typical of a Peierls mechanism; it does not reveal any feature that would have \nresulted from strong interac%ons between dis- loca% ons and Cr impuri%es. \nAthermal Cr solu%on hardening is observed essen%all y because plas%c deforma%on can be \nachieved only at high temperature (above 0.6 %mes t he mel%ng temperature) in sapphire2–5 \nwhich is bri8le below 900 ◦C. This explains the imp ossibility to observe in sapphire thermally-\nac%vated Cr hardening below 0.2–0.3 %mes the mel%ng  temperature, contrary to observa%ons \nin metals.23 \nAlthough the analysis of work-hardening is beyond t he scope of the present inves%ga%on, one \ncan make the following remarks. The fact that two s lip direc%ons are simultaneously ac%vated in \nthe basal plane does not result in a work-hardening  rate larger than in the case of sapphire \ndeformed in single basal slip.3,22 In this case, ha rdening has been ascribed to the impediment \nof 1/3[1¯ 1¯ 2 0] disloca%on slip by prisma%c loops , themselves origina%ng from dipole \nconstric%on.22 In this context, doping  with Cr does not seem to substan%ally affect the for ma%on \nof  loops and the interac%ons between these and basal d isloca%ons. \n \n4.2. Chromium-related obstacles to disloca;on glide  \nWe discuss the possible mechanisms of Cr hardening of sap- phire that a previous study had \nconcluded to be dominated by size effects.13 The phy sical proper%es of ruby have been exten- \nsively explored for a number of reasons, in par%cul ar for its remarkable red colour24 which \nmakes it a8rac%ve not only for jewels but also for its ability at measuring hydrosta%c stresses \nthrough the shiC of light emission lines.25 The X-r ay absorp%on fine structure (XAFS) in the \nvicinity of Cr K edge shows that when Cr (ionic rad ius 0.0615 nm) subs%tutes for Al (ionic radius \n0.0535 nm) in sapphire, the oxygen atoms forming th e first shell move apart to accommodate \nthe larger ionic radius (Cr–O dis- tance = 0.197 nm , as for Cr2 O3 , compared to Al–O = 0.186 \nnm), while the distance between a Cr atom and Al at oms belong- ing to the next shell is very \nclose to the Al–Al bond length in sapphire. Atoms f urther than 0.25 nm apart are almost undis- \nplaced. In other words, atom relaxa%on around Cr is  very local, almost en%rely relaxed by the \nfirst shell of oxygen atoms. In so far as plas%city is concerned, elas%c interac%ons between Cr \natoms and edge disloca%ons are cancelled beyond the  limited volume subject to Cr-induced \nla]ce distor%on, in such a way that hardening due t o size effect is expected to be substan%ally \nless than for an elas%c relaxa%on following a r−1 b ehaviour.23 \nAnother contribu%on to solu%on hardening arises fro m the dif- ference in elas%c modulii due to \nsubs%tu%onal Cr.23  Roughly speaking, it is for an impurity located at one or two interatomic  \ndistances from the disloca%on that the maximum elas %c energies for the volume interac%on and \nfor the elas%c misfit interac%on ( μCr2 O3 = 120 GPa) become comparable. Assuming that dislo-\nca%on pinning is efficient only at such small distanc es, it cannot be concluded on the hardening \nmechanism based on classical elas%c model that is c learly inadequate at atomic distances. \nIt is nevertheless interes%ng to consider the role possibly played by short distance interac%ons \nwith kinks moving in their Peierls poten%al. In thi s case, one would expect substan%al changes \nin the ac%va%on energy for kink diffusion along the disloca%on line. The use of the kink pair model (see following sec%on) does not bring any evi dence that Cr atoms at the core  of the \ndisloca%ons have such an influence on their glide. \nThe comparison with experimental data is made with the models of Fleicher14  and of \nLabusch.15  Fleischer’s model,14  using the differences in size and shear modulus betw een \nsolvent  and solutes, predicts a dependence of hardening wit h the atomic concentra%on of solute \nc according to the equa%on !).σ = Kc 1/2 while Labusch’s,15 using a different sta%s%cal t reatment, \nyields a slightly different expression, i.e. !).σ = Kc 2/3 . In the next sec%on, such equa%ons are \nused although the mechanical data can hardly allow checking the validity of the various \nmechanisms.23 \n \n4.3. A kink-pair mechanism for disloca;on glide  \nA kink pair model has been developed by Hirth and L othe19 to explain disloca%on glide \ncontrolled by a Peierls mechanism. In this well-kno wn model, disloca%on mo%on takes place by \njumps of a segment from a minimum of the la]ce pote n%al to the next. The model has been \nsubsequently adapted by Mitchell et al.16,17 to inc lude various situa%ons, such as the glide of \ndissociated disloca%ons and the effect of impuri%es.  \nIn the case of basal slip in ruby and sapphire, the  dissocia%on of the basal disloca%ons in two 1/3 \n01 1¯ 0) par%al disloca%ons  generates a stacking-fault with a very high energy, 27 so that basal  \ndisloca%ons had rather be regarded as perfect dislo ca%ons with an extended core.8 In this case, \nthe kink pair model refers to its version for undis sociated disloca%ons. According to Mitchell et \nal.,17 the model designed for sapphire postulates t hat in a high Peierls stress materials the kinks \nare abrupt with a square shape, the propaga%on of w hich along the disloca%on line is \ndetermined by a secondary Peierls barrier. \nIn the long-segment limit, a kink pair expands un%l each kink annihilat es with kinks of the \nopposite sign origina%ng from a neighbouring kink s ource along the line. Following Hirth and \nLothe,19 Mitchell et al.17 have derived the strain rate in the case  of an undissociated disloca%on \nas \n \nwhere ρ is the mobile disloca%on density, b the Burgers vector of the perfect disloca%on, σ the \nresolved shear stress for the glide of disloca%ons,  h the height of the kink (periodicity of the \nPeierls barriers in the glide plane), a the periodicity along the disloca%on line, υ the a8empt \nfrequency (typically the Debye frequency), QD the ac%va%on energy for a kink to overcome its  \nsecondary Peierls barrier, Fk is the free energy of a single kink on a perfect disloca%on, k is the \nBoltzman constant, T is the  absolute temperature, and α a factor given by \n \n \n  \napproximately equal to unity at high temperatures a nd greater than unity for high stress and low \ntemperature. \nIn the short-segment limit, the disloca%on por%on on which the kink pair  nucleates is sufficiently \nshort for the pair to expand to the full segment le ngth. In this case, the strain rate is given by \n \nwhere L is the length of the glide disloca%on segments arbi trarily taken as ρ−1/2 . \nIn order to incorporate the athermal hardening from  Cr impu- ri%es in the model, we proceed by \nanalogy with the theories developed for solu%on har dening of metallic systems in the plateau \nregions and we simply replace in the above men%oned  Eqs. (1)–(3) the shear stress σ by σl, where \n                                          σ l = σ − Δσ = σ − Kcn                                                                  (4) \nand n = 1/2 or n = 2/3 depending upon whether we make use of the Fle ischer14  or the Labusch15  \nmodel, respec%vely. Eq. (4) assumes that the drivin g force to nucleate and move kinks is not the \napplied stress σ because a part, !).σ, of this stress is used to overcome an average ela s%c field \ndue to solute atoms in con- centra%on c. The presence of the Cr3+ solute ions in sapphire is \nassumed to have no other influence on the kink pair mechanism. We now check the fit between \nthe experimental data and the predic%ons of the mod el to evaluate these assump%ons.  We  take \nε˙ = 2 .0 × 10−5 s−1 ,  υ = 1013 s−1 ,  b = 0.475 nm(Burgers vector of the perfect disloca%on ), h = \na = 0.275 nm (oxygen–oxygen distance in the basal pla ne). For undoped sap- phire data, we \nshowed that it is not possible to achieve a sa%sfac tory fit over the whole temperature range \nusing a constant disloca%on density ρ in Eqs. (1) and (3).28 Indeed, it is not physically  acceptable \nto take the same disloca%on density in samples defo rmed at T = 527 ◦C, with the applied stresses \nof the order of 103 MPa, and in samples deformed at  T = 1727 ◦C, as the applied stresses differ \nby two orders of magnitude. Instead we assume that at the yield point the disloca%on density is \npropor%onal to the square of the resolved shear str ess following: \n (5) \nwhere C is a constant. This expression has been previously  employed by Pletka et al. in their \ninves%ga%on of the rela%on  ship between the density of disloca%ons and the str ess applied on \ndeformed sapphire beyond the lower yield point. In expression (5) too, σl is subs%tuted for σ \nsince it is σl that is responsible for disloca%on mul%plica%on. A8empts at fi]ng !).σ = Kcn with n \n= 1/2 and n = 2/3 provide the best results with the la8er value  (Table 3), in agreement with the \nconclusions of Pletka et al.13 \nLeast  square  fits  between  the  experimental  CRS S  data (Figs. 3 and 4) and the theore%cal \npredic%ons of Eqs. (1) and (3) have been op%mized n umerically considering both the long- \nsegment and the short-segment limits. As expected, several combina%ons of the set  of \nparameters in Eqs. (1)–(5) give fits of comparable g ood quali%es but the values of the adjustable \nparameters ( μ, C, K, ρ, QD + Fk , QD + 2 Fk ) can vary greatly. In the short-segment approach (Eq. \n(3)), the values of the op%mized parameters are not  acceptable because they are too far from \nthe data for sapphire and rubies and/or have no rea sonable physical jus%fica%on, as already \nfound no%ced for undoped sapphire.28 The long-segment approach (Eq. (1)) turns out to be also \nthe most appropriate to describe the temperature de pendence of the CRSS in ruby. The best \nvalues of the adjustment parameters are shown in Ta ble 3. With the value of the elas%c shear \nmodulus, μ = 156 GPa, the long-segment limit fit (fit I, Table 3) yields values of C (Eq. (5)) \ncomprised between 0.054 and 0.039, associated to so mewhat high values of disloca%on \ndensi%es. It is noted that Pletka et al.22  obtaine d larger values, i.e. C = 0.34 at T = 1400 ◦C down \nto C = 0.18 at T = 1620 ◦C. With a value for C fixed in such a way that disloca%on densi%es in ruby  \ncompare with those in undoped sapphire,22 fit II (Ta ble 3) is also rather sa%sfactory implying, \nhowever, a value of μ slightly inferior to the litera- ture values29 (135  GPa at 1500 ◦C) for \nsapphire. In both fits (I and II in Table 3), the pa rameter K is in good agreement with K = 0.93 GPa \nobtained by Pletka et al.13 for ruby at T = 1500 ◦C, with ε˙ = 2.6 × 10−5 s−1 and n = 2/3. \n \n \n \n \n \n \n \n \nThe ac%va%on energy QD + Fk (Eq. (1)) is comprised between 3.6 and 3.7 eV for  fit I and between \n3.4 and 3.5 eV for fit II. These values can be consi dered as physically acceptable, in accordance \nwith the literature16–18  that indicates that value s of Fk  and QD  of the order of 1 eV are \nreasonable. The fact that the ac%va%on energy in ru by is independent of Cr con- centra%on, \nincluding sapphire,28  is consistent with our ini%a l assump%on that the presence of Cr3+ is at \nthe origin of the aver- age stress field and has no influence on the kink pair mechanism (Eqs. (4) \nand (1)). In fits I and II (Table 3) as well as in m any other cases not shown here, the CRSS \ncalculated from the model is always close to the co rresponding experimental value. The curves \nfor the fits  shown in Fig. 3 were obtained in the long-segment approxima- %on (fit II in Table 3) \nthat gives physically meaningful parameter values. \n \n \n5. Conclusions  \nThe yield stress of ruby (a-Al2 O3 doped with Cr3+ ) deformed by basal slip between 900 and \n1500 ◦C does not depend signifi- cantly on Cr conten t. Disloca%on dynamics does not seem to be \naffected by the presence of Cr which only influences the flow stress. The higher the Cr \nconcentra%on, the stronger the ruby. Solid solu%on hardening is approximately constant in the \n900–1500 ◦C  range.  The  athermal  nature  of  sol id  solu%on hardening can be explained via \nelas%c interac%ons between dis- loca%ons and Cr ato ms in the a-Al2 O3 la]ce. \nThe work-hardening rate does not depend substan%all y on Cr content either, which in the \ninterpreta%on provided by Pletka et al.22 suggests that these impuri%es affect neither the \nforma%on of dipoles and loops nor their interac%ons  with glide disloca%ons. \nTEM observa%ons of ruby samples are consistent with  disloca%on mo%on controlled by a Peierls \nmechanism. The experimental values of the CRSS have  been fi8ed on the basis of the kink pair \nmodel. The good quality of the fits between the expe rimental data and a Peierls model adapted \nto the present samples strongly suggests that the C r3+  ions do not take part to the kink pair \nmechanism, at least directly; these ions would cont ribute only to an average field due to elas%c \nproper%es of the impuri%es in the la]ce. \nAcknowledgement  \nThe authors acknowledge the financial support of the  Min- istry of Science and Technology \n(Government of Spain) through the project MAT2003-0 4199-CO2-02. \n \nReferences  \n1. Pletka, B. J., Mitchell, T. E. and Heuer, A. H.,  Disloca%on structure in sapphire deformed by \nbasal slip. J. Am. Ceram. Soc. , 1979, 57 , 388. \n2. Castaing, J., Muñoz, A., Gomez-García, D. and Do mínguez -Rodríguez, A., Basal Slip in Sapphire \n(a-Al2 O3 ). Mater. Sc. Eng. A , 1997, 223 , 121–125.   \n3. Lagerlof, K. L. D., Heuer, A. H., Castaing, J., Rivie`re, J. P . and Mitchell, T. E., Slip and twinn ing in \nsapphire (a-Al2 O3 ). J. Am. Ceram. 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Acta Metall. , 1996, 44 , \n2145. \n21. Lee, W. E. and Lagerlof, K. P . D., Structural a nd electron diffrac%on data for sapphire (a-Al2 \nO3 ). J. Electron Microsc. Techn. , 1985, 2, 247. \n22. Pletka, B. J., Heuer, A. H. and Mitchell, T. E. , Work-hardening in sapphire (a-Al2 O3 ). Acta \nMetall. , 1977, 25 , 25–33. \n23. Haasen, P ., Physical Metallurgy . Cambridge University Press, 1986, p. 333. \n24. Gaudry, E., Kira%sin, A., Sainctavit, P ., Broud er, C., Mauri, F. and Ramos, A., Structural and \nelectronic relaxa%ons around subs%tu%onal Cr3+ and Fe3+ ions in corundum. Phys. Rev. B , 2003, \n67 , 094108. \n25. Kitzler, P ., He, J., Clarke, D. R. and Kenway, P . R., Structural relaxa%on around subs%tu%onal \nCr3+ ions in sapphire. J. Am. Ceram. Soc. , 1996, 79 , 3–11. \n26. Bass, J. D., A Handbook of Physical Constants . The American Geophysical Union, 1995, p. 45. \n27. Kenway, P . R., Calculate stacking-fault energie s in a-Al2 O3 . Phil. Mag. B , 1993, 68 , 171–183. \n28. Cas%llo-Rodríguez,  M.,  Castaing,  J.,  Muñoz,   A.,  Veyssie`re,  P .  and Domínguez-Rodríguez, \nA., Cri%cal Resolved Shear Stress and Peierls Mecha nism for Basal and Prism Plane Slip in \nSapphire at Different Temperatures. In Interna;onal Workshop. Mechanical Proper;es in \nAdvanced Materials: Recent Insights , ed. D Gómez, A Muñoz, F Gu%érrez, A Gal- lardo, C entro \nMixto CSIC-Universidad de Sevilla, ISBN: 84-933135- 9-9, 2006, pp. 27–29. \n29. Munro, R. G., Evaluated material proper%es for a sintered a-alumina. J. Am. Ceram. Soc. , \n1997, 80 , 1919–1928. \n \n \n "    },    {        "title": "2402.02466v1.Microstructure_and_high_temperature_mechanical_properties_of_liquid_phase_sintered__α__SiC_with_Y__2_O__3__Al__2_O__3__additions.pdf",        "content": "278B O L E T I N  D E  L A  S O C I E D A D  E S P A Ñ O L A  D E\nA R T I C U L OCerámica y Vidrio\nMicroestructura y propiedades mecánicas a altas temperaturas del \nα-SiC sinterizado con fase líquida de Y\n2O\n3-Al\n2O\n3 \nM. C AStillO-ROdRíguez, A. MuñOz Y A. dOMínguez-ROdRíguez\nDepartamento de Física de la Materia Condensada. Universidad de Sevilla. \nApartado 1065. 41080-Sevilla. España\nMuestras de α-SiC han sido sinterizadas con fase líquida (LPS) a 1950 ºC, en atmósfera de argón y tiempos de procesado \nentre 1 y 7 horas. Mediante microscopia electrónica de barrido (SEM) se ha caracterizado la microestructura, obteniéndose \nun tamaño medio de grano que aumenta con el tiempo de procesado desde 0.64 a 1.61 µm.\nEstas muestras han sido deformadas en compresión a carga constante, a temperaturas entre 1450 y 1625 ºC, tensiones \nentre 25 y 450 MPa y velocidades de deformación entre 4.2·10-8 y 1.5·10-6 s-1. Se han determinado los parámetros de fluencia, \nobteniéndose para n valores entre 2.4     0.1 y 4.5      0.2 y Q=680     35 kJ.mol-1 en muestras sinterizadas durante 1 hora, y n entre \n1.2     0.1 y 2.4     0.1 y Q=710     90 kJ.mol-1 para muestras sinterizadas durante 7 horas. Estos resultados se han correlacionado \ncon la microestructura y se han propuesto como posibles mecanismos de deformación el deslizamiento de frontera de grano \n(GBS) acomodado por difusión en volumen, y el movimiento de dislocaciones,  actuando ambos mecanismos de forma \nsimultánea. \nPalabras clave: α-SiC, propiedades mecánicas, microestructura, sinterización con fase líquida, fluencia.\nMicrostructure and high temperatures mechanical properties of liquid-phase-sintered α-SiC with Y2O3-Al2O3 additions\nSamples of α-SiC have been sintered with liquid phase (LPS) to 1950 ºC, in atmosphere of argon and processing times \nbetween 1 and 7 hours. Using scanning electron microscopy (SEM) the microstructure of the samples has been characterized, \nbeing obtained average grain size that grows with the processing time from 0.64 to 1.61 µm. \nThese samples have been deformed in compression under constant load, to temperatures between 1450 and 1625 ºC, stresses \nbetween 25 and 450 MPa and strain rate between 4.2·10-8 and 1.5·10-6 s1. The creep parameters have been determined, \nobtaining for n values between 2.4    0.1 and 4.5    0.2 and Q=680     35 kJ.mol-1 in samples sintered during 1 hour and n between \n1.2        0.1 and 2.4     0.1 and Q=710      90 kJ.mol-1 for samples sintered during 7 hours. These results have been correlated \nwith the microstructure, being the grain boundary sliding (GBS) accommodated by lattice diffusion and the movement of \ndislocations the possible deformation mechanisms, operating both in a simultaneous way.  \nKeywords: α−SiC, mechanicals properties, microstructure, liquid phase sintered, creep.\n1. INTRODUCCIÓN\nEl carburo de silicio SiC es un cerámico avanzado de gran \ninterés para aplicaciones estructurales debido a su elevada \nresistencia a la corrosión, excelentes propiedades mecánica \ntanto a baja como a alta temperatura, buena conductividad \ntérmica y bajo coeficiente de dilatación térmica. En gran \nparte de estas aplicaciones el material se encuentra sometido \na elevadas tensiones de origen mecánico o térmico, como \nconsecuencia de gradientes de temperatura, por lo que el \nconocimiento de las propiedades mecánicas del material a \naltas temperaturas juega un papel importante para muchas de \nestas aplicaciones.\nEn la bibliografía existen diversos estudios sobre las \npropiedades mecánicas del SiC a alta temperatura, \nfundamentalmente estudios de deformación a carga constante, \nsin embargo existe cierta dispersión en los resultados debido \na que los mecanismos de deformación dependen en gran \nmedida de la microestructura de las muestras y ésta es función \nfundamentalmente del método de obtención y también de \notros parámetros como impurezas o aditivos de sinterización \n[1-9]. En el SiC policristalino el carácter covalente del enlace Si-C, el bajo coeficiente de difusión de las especies atómicas \ny la alta energía de las fronteras de grano, hacen necesario \ntemperaturas y presiones de sinterizado muy elevadas. Como \nconsecuencia de esta dificultad de procesado, distintas técnicas \nde fabricación del SiC policristalino se han desarrollado \ndurante la última década, dando lugar a materiales con \ncaracterísticas muy diversas.\nLane y col. [1] estudian un α-SiC policristalino de 3.7 µm \nde tamaño medio de grano, obtenido por sinterización a 2100 \nºC con pequeñas cantidades de C (0.5 % en peso), B (0.42 % en \npeso) y otras impurezas en menores cantidades (por debajo \nde 100 p.p.m.). El mecanismo de deformación propuesto por \nestos autores es el deslizamiento de frontera de grano (GBS) \nacomodado por difusión en volumen, para temperaturas por \ndebajo de 1650 ºC. A temperaturas superiores encuentran que \nla actividad de dislocaciones contribuye de forma significativa \na los mecanismos de deformación. Nixon y col. [2] amplían \nel estudio de Lane para diferentes tamaños de grano. Los \nresultados obtenidos son similares, encontrando que la mayor \no menor contribución del movimiento de dislocaciones a los \nBol. Soc. esp. Ceram. V., 44 [ 5] 278-285 (2005)\n279mecanismos de deformación es función del tamaño de grano. \nBackhaus-Ricoult y col. [3] estudian las propiedades \nmecánicas a alta temperatura de dos tipos de SiC policristalino \nobtenidos por prensado isostático en caliente (HIP) a \ntemperatura de 2000 ºC y presión de 2 kbar. Uno de ellos sin \nimpurezas y otro con impurezas de B, Si, Fe, O y C (por debajo \nde 2000 p.p.m.). Estos autores proponen como mecanismo de \ndeformación el GBS acomodado por difusión en la frontera de \ngrano y cavitación, para tensiones de deformación inferiores \na 500 MPa; a tensiones superiores la frontera de grano no \nsupone un obstáculo para el movimiento de dislocaciones a \ntravés de ella por lo que el movimiento de dislocaciones es \nel mecanismo que controla la deformación. Por otra parte, \nencuentran un incremento en la velocidad de deformación \ncomo consecuencia de la presencia de los aditivos.\nJou y col. [4] obtienen SiC policristalino, de tamaño medio \nde grano 2 µm, sinterizado con fase líquida de Al2OC (10 % en \npeso) a una temperatura de 1875 ºC. Como resultado de los \nensayos de deformación a carga constante en este material los \nautores concluyen que la deformación a temperaturas entre \n1500 y 1650 ºC es controlada simultáneamente por procesos de \ndifusión en volumen y movimiento de dislocaciones.\nHamminger y col. [5] realizan ensayos de deformación a \ncarga constante en diferentes SiC policristalinos dopados con \naluminio y carbón y con boro y carbon y concluyen que, a \ntemperaturas entre 1470 y 1660 ºC y tensiones entre 100 y 190 \nMPa, la deformación en estos materiales está controlada por \nprocesos de difusión en volumen. \nGallardo-López y col. [6] estudian el SiC policristalino \nsinterizado con pequeñas cantidades de fase líquida de Y2O3 \ny Al2O3 (inferior al 2% en volumen) y 1.2 µm de tamaño de \ngrano. El GBS acomodado por difusión en volumen  y la \nactividad de dislocaciones, actuando de forma simultánea, \nson los mecanismos de deformación propuestos por estos \nautores para temperaturas entre 1575 y 1700 ºC y tensiones de \ndeformación entre 90 y 500 MPa.\nPara el SiC policristalino con silicio libre (entre 10 y 12 % \nen volumen), obtenido por reacción del Si en estado gaseoso \ncon una preforma porosa de SiC que contiene carbono libre, \nCarter y col. [7] proponen la actividad dislocaciones como \nmecanismo que controla la deformación  a temperaturas entre \n1575 y 1650 ºC y tensiones comprendidas entre 110 y 220 \nMPa.\nMuñoz y col. [8] y Martínez-Fernández y col. [9] han \nestudiado las propiedades mecánicas a alta temperatura del \nSiC  policristalino obtenido por reacción (RFSC) con diferentes \ncantidades de Si remanente (9 y 20% en volumen) y con una \nfase remanente de Si + NbSi2 en una proporción del 9% en \nvolumen. El estudio se ha realizado mediante ensayos de \ncompresión a velocidad constante y a carga constante. En \nestos materiales la deformación a temperaturas entre 1300 y \n1350 ºC y carga constante tiene lugar por deformación de la \nfase intergranular remanente, de forma que la velocidad de \ndeformación disminuye con la deformación hasta detenerse \ncomo consecuencia del contacto entre los granos de SiC. Los \nresultados obtenidos se explican mediante un modelo de \nfluencia con presencia de fase viscosa intergranular.\nEn este trabajo se estudia el comportamiento mecánico a \nalta temperatura de un material policristalino de SiC obtenido \nmediante sinterización con fase líquida (LPS) de Y2O3 y Al2O3, \ncon una cantidad moderada de estos aditivos (10 % en peso). \nEl material se ha  sinterizado en atmósfera de argón durante \ndiferentes tiempos (1, 3, 5 y 7 horas) con el fin de estudiar la \ninfluencia del tiempo de procesado en la microestructura del \nmaterial. El efecto del tiempo de procesado en las propiedades \nmecánicas a alta temperatura se ha estudiado para las muestras \nsinterizadas durante 1 y 7 horas. Los resultados obtenidos son \ndiscutidos y comparados con los existentes en la bibliografía \npara diferentes SiC policristalinos.2. MÉTODO EXPERIMENTAL\nLos materiales estudiados en este trabajo se han obtenido \npor sinterización con fase líquida (LPS), a partir de polvo α-\nSiC y utilizando como aditivos Y2O3 y Al2O3 en la proporción \n3/5 para obtener como fase intergranular el granate de itrio \ny aluminio Y3Al5O12  (YAG). Las muestras están compuestas \npor un 90 % en peso de α-SiC  y un 10 % en peso de YAG. El \nprocesado se ha realizado a 1950 ºC, en atmósfera de argón \ndurante diferentes tiempos (1, 3, 5 y 7 horas), obteniéndose \nuna densificación próxima al 98 %. Estos materiales han sido \nfabricados en el Department of Metallurgy and Material \nScience Engineering, Institute of Materials Science, University \nof Connecticut, USA. Más detalles sobre el procesado pueden \nser consultados en los trabajos de Padture y col. [10] y Pujar \ny col. [11].\nEl estudio microestructural de las muestras se ha realizado \npor microscopia electrónica de barrido (SEM), utilizando \nelectrones secundarios y retrodispersados, en un microscopio \nelectrónico de barrido Philips XL-30 con tensiones de trabajo \nde hasta 30 kV . Las muestras han sido previamente pulidas \ncon papel abrasivo y pasta de diamante de tamaño de grano \ndecreciente hasta 1 µm de diámetro. Posteriormente se ha \nrevelado la frontera de grano mediante ataque con plasma \nde CF4 mezclado con O2 en la proporción 1/6, durante dos \nhoras aproximadamente (Plasma Asher K1050X, Emitech). \nPor último, en algunos casos, ha sido necesario depositar \nsobre la superficie una lámina delgada de Au para favorecer la \nconducción evitando efectos de carga durante las observaciones \nmicroscópicas. Los parámetros morfológicos de los granos, \npara los distintos tipos de muestras, se han determinado \nanalizando entre 350 y 450 granos, en micrografías de \ndiferentes regiones, mediante un analizador de imágenes \nsemiautomático (Videoplan MOP 30, Kontron Electronik).\nLos ensayos mecánicos se han realizado con muestras \nsinterizadas durante 1 y 7 horas. Estas muestras, de dimensiones \naproximadas 2.5 x 2.5 x 5 mm, han sido sometidas a ensayos de \ncompresión a carga constante en atmósfera de argón. Se han \nefectuado en total 6 experiencias de fluencia, cada una de ellas \ncon una duración aproximada de 350 horas. Los ensayos se \nhan realizado en una máquina prototipo [12], a temperaturas \nentre 1550 y 1625 ºC y tensiones entre 25 y 450 MPa. \n3. RESULTADOS Y DISCUSIÓN\n3.1. Caracterización microestructural de las muestras\nLa necesidad de correlacionar la microestructura de los \nmateriales utilizados en nuestro estudio con los resultados \nde los ensayos mecánicos a carga constante hace necesaria la \ncaracterización inicial de los mismos en función del tiempo de \nprocesado en atmósfera de argón. \nLas imágenes obtenidas mediante SEM, a bajos aumentos \n(figura 1), muestran un material bifásico constituido por \ngranos de SiC (contraste oscuro) y una fase secundaria (YAG) \nformada a partir de los aditivos de sinterización (contraste \nclaro). Para los diferentes materiales estudiados se observa \nuna distribución homogénea de la fase intergranular  así como \nla presencia de cierta porosidad en todos ellos que se hace más \nsignificativa al ir aumentando el tiempo de sinterización.\nEstas observaciones sobre la porosidad están de acuerdo \ncon las medidas de la densidad realizadas por el método \nBol. Soc. esp. Ceram. V., 44 [ 5] 278-285 (2005)MICROESTRUCTURA y PROPIEDADES MECáNICAS A ALTAS TEMPERATURAS DEL α-SIC SINTERIzADO CON FASE LíqUIDA DE y2O3-Al2O3280 Bol. Soc. esp. Ceram. V., 44 [ 5] 278-285  (2005)de Arquímedes, para los diferentes tipos de muestras. En \nla tabla I se indican los valores obtenidos y su comparación \ncon la densidad teórica 3.310 ± 0.002 g.cm-3. Se observa \nuna disminución de la densidad al aumentar el tiempo de \nprocesado en atmósfera de argón.\nA mayores aumentos, la figura 2 muestra para los diferentes \nmateriales una localización de la fase intergranular en los \npuntos triples de los granos de SiC y rellenando cavidades \nentre estos granos. En esta figura se observa también un \nTiempo de \nprocesado (h)1 3 5 7\nDensidad \nmedida \n(g/cm3)3.27 ± 0.02 3.22 ± 0.03 3.20 ± 0.02 3.14 ± 0.03\nDensidad \nmedida / \ndensidad \nteórica (%)98.8±0.6 97.3 ± 1.0 96.7 ± 0.6 94.9 ± 1.0TABLA  I. VALORES DE LA DENSIDAD DE LAS MUESTRAS Y SU COMPARACIóN CON \nLA DENSIDAD TEóRICA.Fig. 1- Imágenes SEM de muestras sinterizadas en atmósfera de argón \ndurante a) 1 hora y b) 7 horas. El contraste oscuro corresponde a los \ngranos de SiC y el claro a la fase intergranular (YAG).\naumento en el tamaño de grano al aumentar el tiempo de \nprocesado. Este efecto se hace especialmente significativo para \nlas muestras sinterizadas durante 7 horas, y va acompañado \nde un cambio importante en la forma de algunos granos que \npasan de equiaxiados a muy alargados, con un incremento \nimportante de tamaño. Esto da lugar a una distribución \nbimodal tanto en la forma como en el tamaño de los granos.\nIgualmente se observa (figura 2) la subestructura núcleo-\nanillo en los granos de SiC, lo que indica que el crecimiento \nde grano tiene lugar mediante la disolución de los granos \nmás pequeños en la fase líquida y la posterior precipitación \nde los átomos (C y Si) sobre las partículas de mayor tamaño. \nEl núcleo corresponde a la partícula del polvo de partida \n(contraste más oscuro en el interior del grano) y el anillo al \nnuevo material depositado [13]. La diferencia de contraste \nes consecuencia de que el anillo presenta trazas de Y, Al y O \nprocedentes de la fase líquida. En el caso de granos de tamaño \nelevado, esta subestructura es más difícil de observar ya que \nla partícula de partida es pequeña frente al tamaño del grano \ny por tanto difícil de interceptar al seccionar el grano.\nEl crecimiento anisótropo de algunos granos para tiempos \nde sinterizado de 7 horas puede justificarse por la presencia en \nel polvo α-SiC, utilizado en la sinterización, de cierta cantidad \nde fase β. En este caso, a la temperatura de procesado se \nproduce la transformación β→α  en el interior de estos granos, \nformando granos SiC bifásicos β/α de forma que los átomos \n(Si y C) disueltos tienden a precipitar como fase α sobre zonas \nα del grano (minimiza la energía) y esto hace que los granos \nbifásicos crezcan de manera altamente anisótropa, dando \nlugar a granos alargados. Este proceso de transformación \ndel β-SiC a la temperatura de procesado está ampliamente \ndocumentado en la bibliografía [13-17] y es utilizado con \ncierta frecuencia para obtener SiC con una microestructura \npredeterminada. Estos trabajos muestran como la cinética de \ntransformación está fuertemente influenciada por la densidad \nde faltas de apilamiento en los granos de β-SiC, por lo que es \nprevisible que la transformación no sea completa ni aún en el \ncaso de las muestras procesadas durante 7 horas.\nEl elevado tamaño de algunos granos junto a la forma \nalargada de los mismos dificulta la necesaria redistribución \nde la fase intergranular para que esta ocupe las cavidades que \nvan quedando entre los granos, y ello hace que la porosidad \nde las muestras con tiempos de sinterizado de 7 horas sea \nsuperior al resto de las muestras con las que hemos trabajado. \nEfectivamente, para este tipo de muestras, se observa (figura \n2d) que la mayor parte de la porosidad se encuentra localizada \nen las proximidades de granos alargados de elevado tamaño.       \nLa figura 3 muestra los histogramas de distribución de \ntamaño de grano (diámetro planar equivalente, d=(4·área/\nπ)1/2 ) para las muestras sinterizadas durante 1 y 7 horas. Se \nobserva, en el caso de las muestras sinterizadas durante 7 \nhoras, un desplazamiento de la distribución hacia mayores \ntamaños de grano y un ensanchamiento de la misma tendiendo \na una distribución bimodal debido a la presencia de los granos \nalargados de elevado tamaño.\nOtros parámetros morfológicos analizados han sido el \nfactor de forma de los granos, F=4 π·área/perímetro2, y el \nfactor de aspecto Fasp, cociente entre la longitud máxima y la \nlongitud mínima perpendicular a ésta, en la sección planar \nde los granos. En la tabla II se presentan los valores medios \nobtenidos para el tamaño de grano así como para el resto \nde parámetros morfológicos medidos. En ella se observa el \ncrecimiento de grano que tiene lugar al aumentar el tiempo \nde procesado y el alejamiento de la forma equiaxiada de M. C ASTILLO-RODRígUEz, A. MUÑOz y A. DOMíNgUEz-RODRígUEz281MICROESTRUCTURA y PROPIEDADES MECáNICAS A ALTAS TEMPERATURAS DEL α-SIC SINTERIzADO CON FASE LíqUIDA DE y2O3-Al2O3\nBol. Soc. esp. Ceram. V., 44 [ 5] 278-285  (2005)Fig. 2-  Imágenes SEM de muestras sinterizadas en atmósfera de argón  durante a) 1 hora , b) 3 horas , c) 5  horas y d) 7 horas. Las zonas oscuras \ncorresponden a los granos de SiC y las claras a la fase intergranular (YAG).\nFig. 3- Distribución del tamaño de grano en muestras sinterizadas en atmósfera de argón para diferentes tiempos de procesado. a) 1 hora, b) 7 \nhoras. (a) (b)Número de gramos (%)\nNúmero de gramos (%)\nDiámetro ( m m) Diámetro ( m m)282los granos para tiempos de procesado de 7 horas. En la \ndeterminación de las incertidumbres tanto de los valores \nmedios como de sus desviaciones estándar se ha considerado \nque en primera aproximación estas magnitudes presentan una \ndistribución gaussiana. \n3.2. Ensayos de compresión a carga constante\nLos ensayos de deformación a carga constante (fluencia) se \nhan realizado para muestras sinterizadas durante 1 y 7 horas. \nPara el análisis de los mismos se ha utilizado la ecuación \nfenomenológica: \n   = A         exp ( -        )                                                    [1]\ndonde A es un parámetro que recoge la dependencia de \nla velocidad de deformación con las características \nmicroestructurales del material (morfología de los granos, \npresencia y distribución de fases secundarias, porosidad,...), \nσ la tensión aplicada sobre la muestra, d el diámetro medio \nde los granos, k la constante de Boltzmann, T la temperatura \nabsoluta de la experiencia, n y p los exponentes de tensión \ny  tamaño de grano y Q la energía de activación del proceso \nresponsable de la deformación. El valor de los parámetros n, p y \nQ constituye una información importante en la determinación \nde los mecanismos responsables de la deformación y su valor \nse obtiene de los ensayos de fluencia. \nLa figura 4 muestra   en función de ε para una muestra \nsinterizada durante 1 hora y deformada entre 1600 y 1550 ºC. En \nella puede verse que después de un régimen transitorio inicial, \nen el que la velocidad de deformación cambia rápidamente, \nse alcanza un régimen estacionario, caracterizado por una \nvelocidad de deformación aproximadamente constante, \nfunción de la temperatura y de la tensión aplicada. Como \nse observa en la figura, la modificación de alguna de estas \nvariables, temperatura T o tensión aplicada σ, conduce a un \nnuevo estado estacionario. El valor de n o de Q se determina a \npartir del cambio en la velocidad estacionaria de deformación \nal modificar durante el ensayo el parámetro correspondiente \nσ o T, y utilizando la ecuación [1]. En la figura se incluyen los \nvalores obtenidos de n y Q en los diferentes saltos de tensión \ny temperatura.\nLa tabla III resume los parámetros de ensayo y los valores \nmedidos para los parámetros de fluencia n y Q en los dos tipos de muestras estudiadas. Los valores de n y Q  se han calculado \ncomo promedio de los obtenidos en los diferentes saltos de \ntensión, para cada temperatura de ensayo, y en los diferentes \nsaltos de temperatura, respectivamente. \nLos valores medidos de la energía de activación \nconcuerdan con gran parte de los valores obtenidos en \ntrabajos previos existentes en la bibliografía, referidos a \nmateriales policristalinos de SiC sinterizados por diferentes \ntécnicas. Así, Lane y col. [1] obtienen valores de la energía \nde activación entre 338 y 434 kJ.mol-1, para temperaturas de \nensayo por debajo de 1650 ºC y entre 802 y 914 kJ.mol-1 para \nmayores temperaturas. Nixon y col. [2] encuentran valores de \nla energía de activación entre 387 y 541 kJ.mol-1 para T<1650 \nºC y entre 838 y 877 kJ.mol-1 para T>1650 ºC. Backhaus-Ricoult \ny col. [3] obtienen una energía de activación de 364 kJ.mol-1 en \nmateriales sin impurezas y de 453 kJ.mol-1 en materiales con \nimpurezas de B, Si, Fe, O y C, para tensiones inferiores a 500 \nMPa y un valor de Q = 629 kJ.mol-1 para tensiones por encima \nde 500 MPa. Igualmente, Jou y col. [4] encuentran un valor de \nla energía de activación de 743 kJ.mol-1 para temperaturas de \nensayo entre 1500 y 1650 ºC, Hamminger y col. [5] obtienen \nun valor de Q= 796 kJ.mol-1, Gallardo-López y col. [6] obtienen \nun valor de Q = 840 kJ.mol-1 a temperaturas entre 1575 y 1700 \nºC y  Carter y col. [7] encuentran un valor de Q = 711 kJ.mol-1 \nentre 1575 y 1700 ºC. \nPor otro lado, los valores de Q obtenidos en nuestro trabajo \nestán de acuerdo con los valores que se encuentran en la \nbibliografía  para la energía de activación medida por difusión \nen volumen del carbono (715 – 840 kJ.mol-1) [18, 19] y del \nsilicio (695 – 910 kJ.mol-1) [20, 21] en el SiC, lo que sugiere que \nlos mecanismos responsables de la fluencia en este material \nestán controlados por procesos de difusión en el interior de \nlos granos. Dado que el coeficiente de difusión del Si es, al \nmenos, un orden de magnitud inferior al del C en el rango de \ntemperaturas de nuestras experiencias, debe ser la difusión \nen volumen del Si el proceso que controla la velocidad de \ndeformación. Los datos anteriores de la difusión en el SiC  han \nsido medidos en α−SiC monocristalino, puro y dopado con \nnitrógeno (620 p.p.m.), y en β−SiC policristalino dopado con \nCu (60 p.p.m.).\nEn relación con los valores obtenidos en nuestro trabajo \npara el exponente de tensión, n = 2.4 ± 0.1 y n = 4.5 ± 0.2 en \nmuestras sinterizadas durante 1 hora y n = 1.2 ± 0.1 y n = 2.4 \n±0.1 en muestras sinterizadas durante 7 horas, cabe indicar \nBol. Soc. esp. Ceram. V., 44 [ 5] 278-285  (2005)Tiempo de \nprocesado(h) Diámetro (µm) Factor de forma Factor de aspecto\ndσd FσF FaspσFasp\n1 0.64 ± 0.02 0.29± 0.01 0.82 ± 0.01 0.11 ± 0.01 1.52 ± 0.02 0.34 ± 0.01\n3 0.97 ± 0.04 0.82 ± 0.05 0.85 ± 0.01 0.09 ± 0.01 1.48 ± 0.02 0.37 ± 0.01\n5 1.21 ± 0.05 0.73 ± 0.03 0.85 ± 0.01 0.09 ± 0.01 1.54 ± 0.02 0.39 ± 0.01\n7 1.61 ± 0.05 0.99 ± 0.06 0.78 ± 0.01 0.12 ± 0.01 1.87 ± 0.04 0.66 ± 0.04TABLA  II. V ALORES DE LOS PARáMETROS MORFOLóGICOS MEDIDOS ANTES DE LOS ENSAYOS MECáNICOS, PARA  LOS DIFERENTES TIPOS DE MUESTRAS\nM. C ASTILLO-RODRígUEz, A. MUÑOz y A. DOMíNgUEz-RODRígUEz283MICROESTRUCTURA y PROPIEDADES MECáNICAS A ALTAS TEMPERATURAS DEL α-SIC SINTERIzADO CON FASE LíqUIDA DE y2O3-Al2O3\nBol. Soc. esp. Ceram. V., 44 [ 5] 278-285  (2005)que también son comparables con gran parte de los resultados \nrecogidos en la bibliografía para el SiC policristalino fabricado \npor diferentes técnicas. Así, Lane y col. [1] obtienen valores \nque aumentan con la temperatura desde 1.44 a 1.71, Nixon y \ncol. [2] obtienen valores entre 1.0 y 1.4 para T<1650 ºC y entre \n1.4 y 2.5 para T>1650 ºC, Jou y col. [4] encuentran un valor de \n1.74, Hamminger y col. [5] encuentran valores próximos a 1, \nGallardo-López y col. [6] obtienen el valor n = 1.6, Carter y col. \n[7] obtienen n = 5.7 y Backhaus–Ricoult y col. [3] encuentran \npara bajas tensiones (inferiores a 500 MPa) el valor n = 1.5 y, \npara tensiones superiores, valores entre 3.5 y 4.\nLos parámetros morfológicos obtenidos de las observaciones \nmicroscópicas (SEM) realizadas sobre las muestras deformadas \nindican que no se han producido durante la fluencia cambios \nsignificativos en la forma y tamaño de los granos. La tabla \nIV muestra los valores de los parámetros morfológicos para \nlos materiales sinterizados durante 1 y 7 horas y deformados \na la más alta temperatura (1625 ºC). Estos valores son muy \npróximos a los obtenidos para los mismos tipos de muestras \nsin deformar (tabla II) lo cual indica que la deformación \nha tenido lugar principalmente por cambio de posición de \nlos granos en el interior de la muestra, es decir, por GBS. \nEl valor obtenido para la energía de activación sugiere la \ndifusión en volumen del Si como proceso de acomodación \ndel deslizamiento de frontera de grano. Este mecanismo \nde deformación ha sido propuesto de forma general en la Fig. 4- Velocidad de deformación    frente a la deformación    en el \nensayo de deformación a carga constante de una muestra sinterizada \nen atmósfera de argón durante 1 hora. Las temperaturas de trabajo \nhan sido 1600 y 1550 ºC y las tensiones aplicada entre 50 y 150 MPa. Se \nincluyen los valores calculados para n y Q en los diferentes saltos de \ntensión y temperatura. \nTemperatura  (ºC) 1625 1600 1580 1550\nTiempo de procesado\n1 horaσ (MPa) 26 - 63 51 - 142 200 - 442\n(s-1) 9.6·10-8–7.4· 10-76.3·10-8–7.1·10-74.2·10-8–1.5·10-6\nn 2.4 ± 0.1 2.4 ± 0.1 4.5 ± 0.2\nQ 685 ± 35 kJ.mol-1\nTiempo de procesado\n7 horasσ (MPa) 66 - 204 157 - 318 211 - 384\n(s-1) 5.8·10-8–2.3·10-74.3·10-8–3.2·10-79.1·10-8–2.7· 10-7\nn 1.2 ± 0.1 2.4 ± 0.1 2.4± 0.2\nQ 710 ± 90 kJ.mol-1TABLA  III. V ALORES DE LOS PARáMETROS DE ENSAYO Y DE LOS PARáMETROS DE  FLUENCIA  n Y Q OBTENIDOS DE LAS DIFERENTES EXPERIENCIAS DE COMPRESIóN A \nCARGA  CONSTANTE.\nbibliografía [1-7] para explicar la fluencia del SiC obtenido por \ndiferentes técnicas.\nSin embargo los valores obtenidos del exponente de tensión \nn son superiores a la unidad que es el valor que corresponde al \nmodelo clásico de GBS acomodado por difusión en volumen \n[22] aunque, como hemos indicado anteriormente, están \ndentro del rango de valores experimentales encontrados en \nla bibliografía en trabajos previos sobre el SiC. Estos valores \nde n superiores a la unidad sugieren que junto a la difusión \nen volumen debe existir otro mecanismo con igual energía \nde activación y mayor exponente de tensión, de forma que \nambos mecanismos contribuyen de manera independiente a la \ndeformación o a la acomodación del deslizamiento de frontera \nde grano durante la deformación, permitiendo alcanzar valores \nde deformación relativamente altos (superiores al 20 %).\nEl hecho de que el exponente de tensión crece al aumentar \nla tensión a que se somete la muestra sugiere la posibilidad de \nque el movimiento de dislocaciones en el interior de los granos \nsea el segundo mecanismo responsable de la acomodación \ndel GBS o incluso sea el mecanismo primario de deformación \n(fluencia-restauración) de forma que la proporción en que este \nmecanismo contribuye al proceso global de deformación o \nacomodación es tanto mayor cuanto más elevada es la tensión \naplicada sobre la muestra y más baja la temperatura.\nEsta hipótesis ha sido verificada por otros autores. Así, \nLane y col. [1] observaron por microscopia electrónica de \ntransmisión (TEM) una elevada densidad de dislocaciones \ndistribuidas homogéneamente en el volumen del grano. \nSus micrografías muestran gran cantidad de faltas de \napilamiento, bandas de deslizamiento y movimientos de \nsubida de dislocaciones, por lo que el valor entre 1.44 y 1.71 \nque encuentran para n a temperaturas superiores a 1650 ºC lo \njustifican considerando el movimiento de dislocaciones como \nmecanismo de deformación que actúa de forma simultánea \ne independiente al deslizamiento de frontera de grano \nacomodado por procesos de difusión en volumen. Igual \njustificación dan Nixon y col. [2] a los exponentes de tensión \nsuperiores a la unidad (entre 1.4 y 2.5) encontrados a T>1650 \nºC y Jou y col. [4] al valor n = 1.74 medido para tensiones entre \n40 y 200 MPa. Backhaus-Ricoult y col. [3] también realizan \nun estudio por TEM de las muestras deformadas a altas \ntensiones, por encima de 500 MPa, observando una elevada \ndensidad de dislocaciones. Estos autores consideran que a \naltos valores de la tensión la frontera de grano no es obstáculo \npara el movimiento de dislocaciones a través de ella, siendo \neste movimiento el mecanismo que de forma dominante \n284controla la deformación para altas tensiones y esto justifica \nque el exponente de tensión alcance los valores entre 3.5 y 4 \nque ellos encuentran.\nPor ultimo, indicar que también en nuestro grupo Gallardo-\nLópez y col. [6] en un trabajo previo sobre el SiC sinterizado \ncon fase líquida, con una cantidad más pequeña de fase \nintergranular (inferior al 2% en volumen), han realizado \nun estudio mediante TEM del material deformado. Las \nmicrografías muestran también una elevada densidad de \ndislocaciones en el volumen del grano, observándose faltas \nde apilamiento, bandas de deslizamiento y movimientos \nde subida de dislocaciones entre planos de deslizamiento \nparalelos. Este movimiento de dislocaciones permitió justificar \nel valor  n=1.6 medido a tensiones entre 90 y 500 MPa.\nUn estudio por TEM de la microestructura de dislocaciones \nestá en curso para verificar en nuestras muestras la contribución \nde la actividad de dislocaciones a la deformación del material. \nDe acuerdo con los resultados de los ensayos mecánicos esta \ncontribución debe ser mayor en las muestras sinterizadas \ndurante una hora que en las sinterizadas durante 7 horas, \nposiblemente por su diferente microestructura.\nLas velocidades de deformación    obtenidas en nuestros \nensayos se encuentran entre 4.2·10-8 y 1.5·10-6  s-1  para muestras \nsinterizadas durante 1 hora y entre 4.3·10-8  y 3.2·10-7 s-1 para \nmuestras sinterizadas durante 7 horas. Para un mismo valor \nde la temperatura y tensión aplicada se observa una velocidad \nde deformación al menos un orden de magnitud inferior en \nel caso de las muestras sinterizadas durante 7 horas. Esta \ndiferencia es consecuencia de la distinta microestructura que \npresentan ambos tipos de muestras. La microestructura de \ngranos alargados y entrecruzados que presentan las muestras \nsinterizadas durante 7 horas y su mayor tamaño, d=1.6 µm, \nen comparación con la microestructura de granos equiaxiados \nde menor tamaño, d=0.6 µm, de las muestras sinterizadas \ndurante 1 hora (figura 2) justifica su diferente velocidad de \ndeformación durante los ensayos de fluencia.\n4. CONCLUSIONES\nLa caracterización microestructural del material antes de \nlos ensayos mecánicos muestra una distribución homogénea \nde fase intergranular y una densificación próxima al 99 % para \nmuestras sinterizadas durante 1 hora, que disminuye al 95 % \nal aumentar el tiempo de procesado hasta 7 horas. El tamaño \nmedio de grano crece desde 0.64 a 1.61 µm al aumentar \nel tiempo de procesado de 1 a 7 horas, produciéndose un \nalejamiento de la forma equiaxiada en parte de los granos. \nEste efecto es especialmente significativo para 7 horas de \nsinterización. La morfología de los granos no se altera de \nforma apreciable durante la deformación.Los valores de los parámetros de fluencia y los resultados \ndel estudio microestructural sugieren que la deformación se \nproduce por GBS acomodado por la difusión en volumen del Si \ny por activación de dislocaciones, actuando ambos mecanismos \nde forma simultánea. La contribución de la actividad de \ndislocaciones al proceso global es tanto mayor cuanto más \nelevada es la tensión aplicada y menor la temperatura.\nLas muestras sinterizadas durante 7 horas son más \nresistentes a la deformación, de forma que, para unas mismas \ncondiciones de ensayo, su velocidad de deformación es un \norden de magnitud inferior a la de las sinterizadas durante \n1 hora.\nAgRADECIMIENTOS\nLos autores desean manifestar su agradecimiento a los \nprofesores F. Guiberteau y A. L. Ortiz, de la Universidad de \nExtremadura, por la fabricación de los materiales con que se \nha realizado este trabajo. De igual forma desean expresar su \nagradecimiento al Ministerio de Ciencia y Tecnología por la \nfinanciación recibida a través del proyecto MAT2001-0799.\nREFERENCIAS\n1. J. E. Lane, C. H. Carter, Jr. y R. F. Davis. “Kinetics and Mechanisms of high-\ntemperature creep in silicon carbide: II, sintered α−SiC”. J. Am. Ceram. Soc., \n71 (4), 281-295 (1988).\n2. R. D. Nixon y R. F. Davis. “Diffusion accommodated grain boundary sliding \nand dislocation glide in the creep of sintered alpha silicon carbide”. J. Am. \nCeram. Soc., 75 (7), 1786-1795 (1992).\n3. M. Backhaus-Ricoult, N. Mozdzierz y P . Eveno. “Impurities in silicon \ncarbide ceramics and their role during high temperature creep”. J. Phys. III. \nFrance, 3, 2189-2210 (1993).\n4. Z.C. Jou y A.V . Virkar. “High temperature creep of SiC densified using a \ntransient liquid phase”. J. Mater. Res., 6 (9), 1945-1949 (1991).\n5. R. Hamminger, G. Grathwoht y F.Thuemmler. “Microanalytical investigation \nof sintered SiC Part I: Bulk material and inclusions”. J. Mater. Sci., 18, 353 \n(1983).\n6. A. Gallardo-López, A. Muñoz, J. Martínez-Fernández y A. Domínguez-\nRodríguez. “High temperature compressive creep of liquid phase sintered \nsilicon carbide”. Acta Mater., 47 (7), 2185-2195 (1999).\n7. C. H. Carter, Jr., R.F. Davis y J. Bentley. “Kinetics and Mechanisms of high-\ntemperature creep in silicon carbide: I, Reaction bounded”. J. Am. Ceram. \nSoc 67 (10), 409-417 (1984).\n8. A. Muñoz, J. Martínez-Fernández, A. Domínguez-Rodríguez y M. Singh. \n“High-temperature Compressive Strength of Reaction-formed Silicon \nCarbide (RFSC) Ceramics”. J. Eur. Ceram. Soc., 18, 65-68 (1998).\n9. J. Martínez Fernández, A. Muñoz, A. R. Arellano López, F. M. Valera \nFeria, A. Domínguez Rodríguez y M. Singh. “Microstructure-mechanical \nproperties correlation in siliconized silicon carbide ceramics”. Acta Mater., \n51, 3259-3275 (2003).\n10. N. P . Padture. “In situ-toughened silicon carbide”. J. Am. Ceram. Soc., \n77(2), 519-523 (1994). \n11. V . V . Pujar, R. P . Jensen y N. P . Padture. “Densification of liquid-phase-\nsintered silicon carbide”. J. Mater. Sci. Lett., 19[11], 1011-1014 (2000).\n12. H. Gervais, B. Pellissier y J. Castaing. “Machine de fluage pour essais en \ncompression à hautes températures de matériaux céramiques”. Rev. Int. \nHtes Temp. et  Refract., 15, 43-47 (1978). \n13. A.L. Ortiz. “Control microestructural de cerámicos avanzados SiC \nsinterizados con fase líquida Y2 O3 - Al2 O3”. Tesis Doctoral, Universidad de \nExtremadura  (2002).\n14. S. K. Lee y C. H. Lee. “Effects of α-SiC versus β-SiC starting powders on \nmicrostructure and fracture –toughness of SiC sintered with Al2O3-Y2O3 \nadditives”. J. Am. Ceram. Soc., 77, 1655-1658 (1994).\n15. M. Nader, F. Aldinger y M. J. Hoffmann. “Influence of the α/β-SiC phase \ntransformation on microstructural development and mechanical properties \nof liquid phase sintered silicon carbide”. J. Mater. Sci., 34, 1197-1204 (1999).\n16. Y. W. Kim, M. Mitomo y G. D. Zhan. “Mechanism of grain growth in liquid-\nphase-sintered β-SiC”. J. Mater. Res., 14, 4291-4293 (1999).\n17. H. Xu, T. Bhatia, S. A. Deshpande, N. P . Padture, A. L. Ortiz y F. L. \nCumbrera. “Microstructural evolution in liquid-phase-sintered SiC: I, effect \nof starting SiC powder”. J. Am. Ceram. Soc., 84, 1578-1584 (2001).\n18. M. H. Hon y R. F. Davis. “Self-diffusion of C-14 in polycrystalline β-SiC”. J. \nMater. Sci., 14, 2411-2421 (1979).Tiempo de \nprocesado \n(h)Diámetro (µm) Factor de forma Factor de aspecto\nσdσF aspσFasp\n10.72 ± \n0.02 0.35± \n0.010.82 ± \n0.010.09 ± \n0.011.57 ± \n0.020.39 ± \n0.01\n71.68 ± \n0.05 1.06 ± \n0.070.80 ± \n0.010.11 ± \n0.011.71 ± \n0.020.40 ± \n0.01TABLA  IV: V ALORES  DE LOS  PAR áMETROS  MORFOL óGICOS  DE MUESTRAS \nSOMETIDAS A ENSAYOS DE FLUENCIA  A 1625 ºC. \nd F F\nM. C ASTILLO-RODRígUEz, A. MUÑOz y A. DOMíNgUEz-RODRígUEz\nBol. Soc. esp. Ceram. V., 44 [ 5] 278-285  (2005)285MICROESTRUCTURA y PROPIEDADES MECáNICAS A ALTAS TEMPERATURAS DEL α-SIC SINTERIzADO CON FASE LíqUIDA DE y2O3-Al2O3\n19. M. H. Hon y R. F. Davis. “Self-diffusion of Carbon-14 in high purity and \nN-doped α-SiC  single crystals”. J. Am. Ceram. Soc., 63, 546-552  (1980).\n20. M. H. Hon, R. F. Davis y D. E. Newbury. “Self-diffusion of Si-30 in \npolycrystalline β-SiC”. J. Mater. Sci. 15, 2073-2080 (1980).21. M. H. Hon, R. F. Davis y D. E. Newbury. “Self-diffusion of Silicon-30 in \nα-SiC single crystals”. J. Mater. Sci., 16, 2485 (1981).\n22. M. F. Ashby, y R. A. Verrall. “Diffusional accommodated flow and \nsuperplasticity”. Acta Metall., 21, 149-163 (1973).\nRecibido:  09.01.05\nAceptado: 28.04.05\nBol. Soc. esp. Ceram. V., 44 [ 5] 278-285  (2005)"    },    {        "title": "2402.02467v2.Blow_up_analysis_of_Large_conformal_metrics_with_prescribed_Gaussian_and_geodesic_curvatures.pdf",        "content": "arXiv:2402.02467v2  [math.DG]  16 Feb 2024BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS WITH\nPRESCRIBED GAUSSIAN AND GEODESIC CURVATURES\nRAYSSA CAJU, TIARLOS CRUZ, AND ALMIR SILVA SANTOS\nAbstract. Consider a compact Riemannian surface ( M,g) with nonempty boundary\nand negative Euler characteristic. Given two smooth non-constan t functions finMand\nhin∂Mwith max f= maxh= 0, under a suitable condition on the maximum points\noffandh, we prove that for sufficiently small positive constants λandµ, there exist at\nleast two distinct conformal metrics gλ,µ=e2uµ,λgandgλ,µ=e2uµ,λgwith prescribed\nsign-changing Gaussian and geodesic curvature equal to f+µandh+λ,respectively.\nAdditionally, we employ the method used by Borer et al. (2015) to stu dy the blowing\nup behavior of the large solution uµ,λwhenµ↓0 andλ↓0. Finally, we derive a new\nLiouville-type result for the half-space, eliminating one of the poten tial blow-up profiles.\n1.Introduction\nThe problem of prescribing curvature has attracted much attent ion in the last few\ndecades due to its relevance and importance in various branches of mathematics. Up\nto our knowledge, M. Berger [ 8] was the first to study the problem of prescribing the\nGaussian curvature on a given closed surface with negative Euler ch aracteristic. Later, in\na series of papers [ 26,27,28,29,30], J. Kazdan and F. Warner have addressed the problem\nof prescribing the Gaussian and scalar curvature on a closed manifo ld, in the conformal\nand nonconformal setting. It turned out that conformal defor mation of the metric is an\nimportant tool for finding geometric properties by using analysis te chniques.\nLet (M,g) be a closed Riemannian surface. Through a conformal deformatio n of metric\ng=e2ug,for some smooth function u, it is well known that the Gaussian curvature Kg\nandKgofgandg, respectively, are related by the following PDE\n(1.1) −∆gu+Kg=K¯ge2u,\nwhere∆ gdenotestheLaplace-Beltrami operatorwithrespect to g.According totheGauss-\nBonnet theorem, the solvability of equation ( 1.1) depends on the sign of the Euler char-\nacteristicχ(M), which imposes necessary conditions on the function to be realized as the\nGaussiancurvatureof g. Acomplete understanding ofwhich functions canbetheGaussian\ncurvature of some conformal metric remains a challenging problem, especially in the case\n2020Mathematics Subject Classification. 35B44, 58J32, 35J20, 35J60.\nKey words and phrases. Prescribed curvature problem, conformal metric, blow-up analys is, variational\nmethods.\nRC was partially supported by FONDECYT grant number 11230872 an d by Centro de Modelamiento\nMatem´ atico (CMM) BASAL fund FB210005 for center of excellence from ANID-Chile. TC was partially\nsupported by CNPq grant number 307419/2022-3 and 405468/20 21-0. ASS was partially supported by\nCNPq grant number 403349/2021-4, 408834/2023-4 and 312027 /2023-0. The authors were supported by\nFAPEAL, Brazil (Process E:60030.0000002329/202) .\n12 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nof the two-sphere, commonly referred to as the Nirenberg proble m. We direct the reader\nto the works [ 1,23] and the references contained therein.\nNow, consider a compact Riemannian surface ( M,g) with nonempty boundary ∂M. If\ngis a conformal metric to gwith Gaussian and geodesic curvature equal to f∈C∞(M)\nandh∈C∞(∂M), respectively, then there exists a smooth function usuch thatg=e2ug,\nwhich satisfies the boundary value problem\n(1.2)\n\n−∆gu+Kg=fe2uinM,\n∂u\n∂νg+κg=heuon∂M,\nwhereνgis the outward unit normal to ∂Mandκgis the geodesic curvatures of the\nboundary with respect to g.\nAlthoughtheexistence ofsolutionsto( 1.2)isnot completely understood, therearemany\nworks dedicated to studying the existence and also the blow-up ana lysis of these solutions.\nSince the literature on this topic is extensive, we only refer to recen t works [34,35,45] and\nthe references therein.\nIf (1.2) has a solution, by the Gauss-Bonnet Theorem, we obtainˆ\nMfdvg+ˆ\n∂Mhdσg= 2πχ(M),\nwheredvg=e2udvganddσg=eudσgare the area and length elements, respectively, of\nthe metricg=e2ug. This means that the solubility of ( 1.2) is also subject to topological\nconstraints. For example, if χ(M)<0,fandhare non negative functions, then there is\nno solution.\nThe main goal of this work is to study the structure of the solution s et of (1.2) on the\ncaseχ(M)<0. We will extend the results for closed surfaces obtained in [ 11] to compact\nsurfaces with nonempty boundary, as well as the corresponding im provements provided in\n[43] of the asymptotic behavior of the solutions obtained in [ 11]. In a future work we will\naddress the case χ(M) = 0.\nWithout loss of generality, by the celebrated B. Osgood, R. Phillips an d P. Sarnak\nuniformization theorem for surfaces with boundary [ 36] we may assume that Kg=−1 and\nκg≡0 (see also [ 34, Proposition 3.1] and [ 38, Lemma 2.1]). Taking this into account,\nsolving equation ( 1.2) is equivalent to solving\n(1.3)\n\n−∆gu−1 =fe2uinM,\n∂u\n∂νg=heuon∂M.\nThe problem ( 1.3) is the Euler-Lagrange equation of the functional If,h:H1(M)→R\ngiven by\n(1.4) If,h(u) =1\n2ˆ\nM/parenleftbig\n|∇u|2−2u−fe2u/parenrightbig\ndvg−ˆ\n∂Mheudσg.\nFinding a critical point to If,h, for general functions fandh, is a challenging task. The\narea and boundary term compete, and maybe it is not possible to ver ify ifIf,his bounded\nfrom below or not. For non-positive functions fandhwith one of them non-identically\nzero, [14, Theorem 2] implies the existence of a solution to ( 1.3). Besides, in this case, it isBLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 3\nnot difficult to see that the functional ( 1.4) is strictly convex (See also [ 34, Theorem 1.1]).\nThus, it holds the following result.\nTheorem 1.1. Let(M,g)be a compact Riemannian surface with ∂M/\\e}atio\\slash=∅andX(M)<0.\nIff∈C∞(M)andh∈C∞(∂M)are nonpositive functions with f/\\e}atio\\slash≡0orh/\\e}atio\\slash≡0, then(1.2)\nadmits a unique solution.\nRecently, existence of solutions to ( 1.3) was studied by R. L´ opez-Soriano, A. Malchiodi,\nand D. Ruiz [ 34]. They proved the existence of solutions when X(M)<0 under the\nassumptions f <0 andh(p)//radicalbig\n|f(p)|<1 for allp∈∂M. Moreover, if h≤0, then\nthe solution is unique, see [ 34, Theorem 1.1]. Their proof consists in to show that the\nfunctional is coercive, weakly lower semicontinuous, and strictly co nvex whenh≤0. They\nalso studied existence results of a solution for ( 1.2) in the case X(M) = 0, under extra\nassumptions on the integral of h, and studied the blow-up phenomenon. See [ 35] for results\nin the disk case.\nWe are interested in studying the solution set of ( 1.3) whenf≤0 andh≤0, both are\nnon-identically zero. First, considering the variational approach, since we are interested in\ncritical points for If,h, it is natural to study nondegeneracy of its local minimizer, which is\ndetermined by the second derivative\n(1.5) d2If,h(u)(m,m) =ˆ\nM(|∇m|2−2fm2e2u)dvg−ˆ\n∂Mhm2eudσg.\nBuilding upon the ideas in [ 20] (see also [ 10]), we can prove our first result.\nTheorem 1.2. Let(M,g)be a compact Riemannian surface with ∂M/\\e}atio\\slash=∅andχ(M)<0.\nSuppose that for f∈C0,α(M)andh∈C1,α(∂M)withh≤0, for some 0< α <1,the\nfunctionalIf,hadmits a local minimizer uf,h∈H1(M). Then\n(a) there exists a constant c0>0such that\nd2If,h(uf,h)(m,m)≥c0/ba∇dblm/ba∇dbl2\nH1,\nfor allm∈H1(M). In particular, uf,his a non-degenerate critical point of If,h.\n(b) there exists a neighborhood U×W⊂C0,α(M)×C1,α(∂M)of(f,h)such that for any\n(z,k)∈U×W, there exists a strict local minimizer for Iz,kinC2,α(M)smoothly\ndependent on z,k. In particular, if fandhare smooth, then the minimizer is smooth\nas well.\nItem (a) can be viewed as the counterpart to some results in [ 2,11] in the context of\nsurfaces with boundary. It would be interesting to know whether t his result holds for\ngeneralh∈C∞(∂M). Item (b) follows as a consequence of the implicit function theorem .\nWe now describe an interesting phenomenon that occurs. Consider nonconstant func-\ntionsf∈C∞(M) andh∈C∞(∂M) with max f= maxh= 0. For real numbers µ >0\nandλ >0,definefµ=f+µandhλ=h+λ. First, we observe that if µandλare\nlarge enough it follows from Gauss-Bonnet Theorem that ( 1.3) has no solution. On the\nother hand, by Theorem 1.2, forµandλsmall enough, the equation ( 1.3) admits a smooth\nsolutionuµ,λ, which is a strictly local minimizer of Ifµ,hλ. But, since max fµ>0, it is\nnot difficult to see that the functional Ifµ,hλis no longer bounded from below, and we may\nexpect the existence of a further critical point uµ,λof saddle-type. In this regard, we obtain\nthe following result.4 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nTheorem 1.3. Let(M,g)be a compact Riemannian surface with ∂M/\\e}atio\\slash=∅andχ(M)<0.\nLetf∈C∞(M)andh∈C∞(∂M)be nonconstant functions satisfying maxf= maxh= 0.\nConsider the families of functions hλ=h+λandfµ=f+µ,whereλandµare two real\nnumbers. There exist real numbers λ0andµ0such that for almost every 0< λ < λ 0and\n0<µ<µ 0, the functional Ifµ,hλadmits a strict local minimizer uλ,µand a second critical\npointuµ,λ/\\e}atio\\slash=uµ,λof mountain-pass type.\nIn the context of closed surfaces, the existence of a second solu tion to ( 1.1) was first\nproved in [ 18], when X(M)<0 and the prescribed Gaussian curvature fλ=f+λ,\nwithλ >0 small enough, changes sign. However, the method used does not provide the\ngeometric shape of the solutions. Later, in [ 11], the authors gave a new proof using the\nso-called “monotonicity trick” (see [ 39,40] and also [ 46]), which allowed them to bound\nthe volume of the “large” solutions when λ→0 and then perform a blow-up analysis.\nThe blow-up limit is spherical, as proved in a further work by M. Struwe [43]. See also\n[17], where, via a refined analysis, the authors constructed “branch es” of metrics exhibiting\nmultiple spherical bubbling in the limit solution by matched asymptotical expansion. In\n[37] the authors studied large conformal metrics with prescribed sca lar curvature in a\nsimilar manner as in [ 11,17].\nIn contrast to what happens in the closed surface case, the pres ence of boundary cru-\ncially gives rise to new non-compactness phenomena, especially the e xistence of blow-up\nprofiles with infinite area, as proved in [ 34]. In this work, the authors exploit the varia-\ntional formulation of the problem to obtain existence results, emplo ying a combination of\nminimization and min-max techniques. We refer to [ 3,5,7,9,16,21,25,35,49,50] for\nmore references on blow-up analysis of solutions and to [ 22,34] and the references therein\nfor the most recent progress in the topic and for including new gene ral blow-up results.\nOur next goal is to understand what happens with the solution uµ,λ, given by Theorem\n1.3, whenλ↓0 andµ↓0. The main result in this respect is the following.\nTheorem 1.4. Let(M,g)be a compact Riemannian surface with ∂M/\\e}atio\\slash=∅andχ(M)<0.\nLetf∈C∞(M)andh∈C∞(∂M)be nonconstant functions with maxf= maxh= 0.\nSuppose the maximum points of fandhare the same, non-degenerate, and critical points\noff. For(µ,λ)∈R2,definefµ:=f+µandhλ:=h+λ. Then, there exist two sequences\nof real numbers, µn↓0andλn↓0, and a sequence of solutions unof\n(1.6)\n\n−∆gun+Kg=fµne2uninM,\n∂un\n∂νg=hλneun on∂M,\nwhich are non-minimizing critical points of Ifµn,hλn, such that the conformal metric gn=\ne2unghas total curvature uniformly bounded,ˆ\nM|Kgn|dvgn+ˆ\n∂M|κgn|dσgn<C <∞.\nHereKgn=fµnandκgn=hλnare the Gaussian and geodesic curvatures of the metric\ngn, respectively. Moreover, there exist at most three points {p(i)\n∞∈M:i= 1,...,N}with\nN∈ {1,2,3}andf(p(i)\n∞) =h(p(i)\n∞) = 0, and two sequences r(i)\nn↓0andp(i)\nn→p(i)\n∞inM,\nsuch thatBLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 5\n(a)un→u∞smooth locally on M∞:=M\\{p(i)\n∞: 1≤i≤N}, andu∞induces a complete\nmetricg∞:=e2u∞gonM∞of finite total curvature\nˆ\nM∞|Kg∞|dvg∞+ˆ\n∂M∞|κ∞|dσg∞<∞,\nwhereKg∞=fandκg∞=hare the Gaussian and geodesic curvatures of g∞, respec-\ntively.\n(b) For each i∈ {1...,N},there holds r(i)\nn/λn→0and in local conformal coordinates\naroundp(i)\n∞we have that\nwn(x) :=vn(xn+r(i)\nnx)−vn(xn)→w∞(x) = log2Λ\nc∞Λ2+(s−s0)2+(t−t0+d∞Λ)2,\nsmoothly locally in R2\nt0=R×(t0,+∞), for somet0≥0, wherex= (s,t). Heres0∈R,\nxncorresponds to p(i)\nnin these coordinates, Λ>0andc∞,d∞∈[0,1], withd∞= 1\nifc∞= 0.The function w∞induces a metric g∞=e2w∞geucof Gaussian curvature\nKg∞=c∞onR2\nt0and geodesic curvature κg∞=d∞on∂R2\nt0, wheregeucis the euclidean\nmetric in R2.\nMoreover, if c∞= 0, thenN= 1. Ifc∞∈(0,1]andd∞= 0, then1≤N≤2. If\nc∞,d∞∈(0,1],then1≤N≤3.\nAlthough the blow-up analysis follows a similar methodology to that of t he case of\nclosed surfaces, the presence of the boundary fundamentally ch anges the dynamics of our\nanalysis. Primarily, there is a deep interdependence between the pa rametersµandλ, as\none needs to be correlated almost quadratically with the other. Sec ondly, differently from\nthe closed case, by analyzing a volume estimate, we manage to exclud e the possibility\nof whole bubbles of large curvature being formed in blow-up points ha ppening in the\nboundary. Finally, following the ideas developed in the paper of [ 34], which majorly uses\nEkeland’s variational principle, we are able to use the techniques app lied in [11,32,43]\nto understand the possibility of slow blow-ups. For technical reaso ns, it was necessary to\nsuppose in Theorem 1.4that the functions fandhhave the same maximum points. It\nwould be nice to know what happens if this is not the case.\nWe would like to emphasize that, although there are characterizatio ns of blow-up so-\nlutions to the Neumann boundary value problem in the literature, as m entioned earlier,\nthere are some issues in applying these results to prove Theorem 1.4. The first issue is that\nto use many of these works (see again [ 3,32,49,50]), it is necessary to require that the\nfunctions which are expected to prescribe as the Gaussian and geo desic curvatures do not\nchange sign, a condition that clearly does not hold in our case. The se cond is that these\nworks does not allow to bound uniformly the area and the boundary le ngth of the “large”\nsolutionsuµ,λasµ,λ→0 appropriately. Fortunately, we overcome these obstacles by\nadapting the mountain pass technique used by Struwe [ 40,41] in a way similar to [ 11,43].\nSee [19] for a similar result for the torus context, and [ 20] for theQ-curvature context.\nTo the best of our knowledge, there are few results in the literatur e on large conformal\nmetrics that simultaneously address the Gaussian and geodesic cur vatures. Recently, a\nresult in the disk has appeared, see [ 6]. In this work, the authors proved the existence6 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nof bubbling type solutions under certain nondegeneracy assumptio ns on their curvatures\nusing a Lyapunov-Schmidt reduction.\nThe analysis done in [ 11] gives two possibilities for the blow-up: one for the bubble\nsolution and another one for a solution that gives rise to a metric g∞in allR2with finite\nvolume, finite total curvature and Gaussian curvature Kg∞= 1 + (Ax,x), whereAis a\nnegative definite 2 ×2 matrix. Later, M. Struwe [ 43] ruled out the second possibility by\nshowing a Liouville-type theorem in the plane. See [ 31,32,44] for an extension of this\ntheorem to higher order operator. The proof of Theorem 1.4follows the same line. To\nrule out a potential second scenario for the blow-up, we prove the following Liouville-type\ntheorem in the half-space R2\n+, which might be of independent interest.\nTheorem 1.5. Suppose that Ais a negative definite 2×2matrix. Then there is no solution\nw∈C∞(R2\n+)of the equation\n\n\n−∆w=Fe2winR2\n+,\n∂w\n∂ν=ewon∂R2\n+,\nwithw≤C, whereF(x) := 1+(Ax,x), such that Fe2w∈L1(R2\n+),\nˆ\nR2\n+e2wdx<∞,ˆ\n∂R2\n+ewdl<∞,andˆ\nR2\n+Fe2wdx∈R.\nThe paper is structured as follows. In Section 2, we investigate the stability and nonde-\ngeneracy properties of local minimizers concerning the energy fun ctional (1.4). Progressing\ntoSection 3, weestablishanexistenceresultofaclassoflargeconformalsolut ionsofsaddle-\ntype as described in Theorem 1.3. Moving forward to Section 4, in order to prove Theorem\n1.4, we conduct a blow-up analysis to study the behavior of these large solutions as the\nparameters approach zero suitable. In Section 4.2we prove Theorem 1.5, which finalizes\nthe proof of Theorem 1.4.\nAcknowledgments: This work was carried out while TC was visiting ICTP and ASS\nwas visiting University of Chile. ASS is very grateful for the hospitalit y of University of\nChile. TC is very grateful to Claudio Arezzo for the hospitality and als o he would like to\nacknowledge support from the ICTP through the Associates Prog ramme (2018-2023). The\nauthors would like to thank S. Almaraz and R. L´ opez-Soriano for us eful comments on an\nearlier version of this paper.\n2.Stability results\nThroughout this paper, ( M,g) will denote a compact, connected Riemannian surface\nwith nonempty boundary and negative Euler characteristic, i.e., χ(M)<0. In this case,\nthe Gauss-Bonnet formula leads toˆ\nMKgdvg+ˆ\n∂Mκgdσg= 2πχ(M)<0,\nwhereKgandκgare the Gaussian and geodesic curvature, respectively, dvganddσgare\nthe area and length element of g.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 7\n2.1.Nondegeneracy of local minimizers. We will guarantee the nondegeneracy of\nlocal minimizers for the energy If,hfor any arbitrary functions fandh≤0, in the case\nthatχ(M)<0. Next proposition proves item (a) of 1.2.\nProposition 2.1. Let(M,g)be a compact Riemannian surface with ∂M/\\e}atio\\slash=∅andχ(M)<\n0. Letf∈C0,α(M)andh∈C0,α(∂M)withh≤0. Suppose the functional If,hgiven by\n(1.4)admits a local minimizer uf,h∈H1(M). Thenuf,his a non-degenerate critical point\nofIf,hin the sense that d2If,h(uf,h)(m,m)≥c0/ba∇dblm/ba∇dbl2\nH1,for some positive constant c0and\nfor allm∈H1(M).\nProof.Definec0:= inf{d2If,h(uf,h)(m,m) :/ba∇dblm/ba∇dblH1= 1}. Sinceuf,h∈H1(M) is a local\nminimizer of If,h, it follows that d2If,h(uf,h)(m,m)≥0 for allm∈H1(M). Thusc0≥0.\nTo establish the desired result, we need to prove that c0>0. By contradiction, let us\nassume that c0= 0.\nClaim 1: There ism∈H1(M) with/ba∇dblm/ba∇dblH1= 1 such that d2If,h(uf,h)(m,m) = 0.\nLet{mk}k∈Nbe a sequence of functions in H1(M) with/ba∇dblmk/ba∇dblH1= 1 and such that\nd2If,h(uf,h)(mk,mk)→0 ask→ ∞. Since a ball in any Hilbert space is weakly compact,\nthere exists a function m∈H1(M) and a subsequence of {mk}, still denoted by {mk},\nconverging weakly to minH1(M) and strongly in Lp(M) for anyp <∞. This implies\nthat\n(2.1) /ba∇dblm/ba∇dblH1≤liminf/ba∇dblmk/ba∇dblH1.\nBy elliptic regularity, uf,his at least continuous. Thus, (3.2) and (3.3) from [ 15] implies\nthatfm2\nke2uf,h→fm2e2uf,hinL1(M) andhm2\nkeuf,h→hm2euf,hinL1(∂M). Using ( 1.5)\nwe get\n/ba∇dbl∇mk/ba∇dbl2\nL2=d2If,h(uf,h)(mk,mk)+2ˆ\nMfm2\nke2uf,hdvg+ˆ\n∂Mhm2\nkeuf,hdσg\n→2ˆ\nMfm2e2uf,hdvg+ˆ\n∂Mhm2euf,hdσg\n≤d2If,h(uf,h)(m,m)+2ˆ\nMfm2e2uf,hdvg+ˆ\n∂Mhm2euf,hdσg=/ba∇dbl∇m/ba∇dbl2\nL2.\nThis implies that\n/ba∇dblmk/ba∇dbl2\nH1=/ba∇dblmk/ba∇dbl2\nL2+/ba∇dbl∇mk/ba∇dbl2\nH1≤ /ba∇dblm/ba∇dbl2\nL2+/ba∇dbl∇m/ba∇dbl2\nH1=/ba∇dblm/ba∇dbl2\nH1.\nBy (2.1) it follows that mk→mstrongly in H1, concluding the proof of the claim.\nBy Claim 1 we deduce that the functional v/ma√sto→d2If,h(uf,h)(v,v) attains a minimum at\nm. It follows that\nd2If,h(uf,h)(m,w) = 0,for allw∈H1(M),\nwhich implies that m∈H1(M) is a weak solution of the equation\n(2.2)\n\n−∆gm= 2fe2uf,hminM,\n∂m\n∂νg=heuf,hm on∂M.8 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nClaim 2: Form∈H1(M) given by Claim 1, it holds\nd4If,h(uf,h)(m,m,m,m ) =−8ˆ\nMfm4e2uf,hdvg−ˆ\n∂Mhm4euf,hdσg<0.\nFirst, note that mis not constant, otherwise ( 2.2) would imply\nˆ\nMfe2uf,hdvg+ˆ\n∂Mheuf,hdσg= 0,\nwhich by Gauss-Bonnet Theorem we find a contradiction, since X(M)<0. By the first\nequation in ( 2.2) we obtain\n2fe2uf,hm4=−m3∆gm=−1\n4∆gm4+3|∇m|2\ngm2.\nThus, using that h≤0, the second equation in ( 2.2) and integrating by parts, we get\nd4If,h(uf,h)(m,m,m,m ) =−8ˆ\nMfm4e2uf,hdvg−ˆ\n∂Mhm4euf,hdσg\n= 4ˆ\nMm3∆gmdvg−ˆ\n∂Mhm4euf,hdσg\n=ˆ\nM(∆gm4−12|∇m|2\ngm2)dvg−ˆ\n∂Mhm4euf,hdσg\n=−12ˆ\nM|∇m|2\ngm2dvg+ˆ\n∂Mm3/parenleftbigg\n4∂m\n∂νg−hmeuf,h/parenrightbigg\ndσg\n=−12ˆ\nM|∇m|2\ngm2dvg+ˆ\n∂M3heuf,hm4σg<0.\nSinceuf,hisalocalminimizerand d2If,h(uf,h)(m,m) = 0, then d3If,h(uf,h)(m,m,m) = 0.\nAlso we obtain that dIf,h(uf,h) = 0 andd2If,h(uf,h)(m,m) = 0. From this, we obtain the\nexpansion\nIf,h(uf,h+tm) =If,h(uf,h)+t4\n4d4If,h(uf,h)(m,m,m,m )+O(t5)<If,h(uf,h),\nfor smallt >0, where we used Claim 2. But this leads to a contradiction, since uf,his a\nlocal minimizer. /square\n2.2.Existence of strict local minimizers. We conclude this section with the result\nthat completes the proof of Theorem 1.2. Following the idea in [ 20] (see also [ 10]) we prove\nthe next proposition.\nProposition 2.2. Let(M,g)be a compact Riemannian surface with ∂M/\\e}atio\\slash=∅andχ(M)<\n0. Suppose that for some f∈C0,α(M)andh∈C1,α(∂M), withh≤0, the functional\nIf,hadmits a local minimizer uf,h∈H1(M). Then there exist a neighborhood U×W⊂\nC0,α(M)×C1,α(∂M)of(f,h)and a smooth map T:U×W→H1(M)such that for any\n(z,k)∈U×Wthe function T(z,k)is a strict local minimizer of Iz,k.\nProof.First, we prove the following claim.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 9\nClaim:There exist two neighborhoods, U×W⊂C0,α(M)×C1,α(∂M) of (f,h) andW′\nof (0,0)∈C0,α(M)×C1,α(∂M), such that for any ( m,z)∈U×Wthere exists a function\num,z,which depends smoothly on ( m,z) and solves the boundary value problem:\n(2.3)\n\n−∆gum,z−1 =me2um,zinM,\n∂um,z\n∂νg=zeum,z on∂M.\nConsider the map Z:C2,α(M)×C0,α(M)×C1,α(∂M)→C0,α(M)×C1,α(∂M) given by\nZ(w,m,z) =/parenleftbigg\n−∆gw+(f−me2w)e2uf,h,∂w\n∂νg+(h−zew)euf,h/parenrightbigg\n.\nNote that Z(0,f,h) = (0,0),the map ZisC1and its partial derivative with respect to w,\nDwZ(0,f,h) :C2,α(M)→C0,α(M)×C1,α(∂M), is given by\nDwZ(0,f,h)·(k) =/parenleftbigg\n−∆gk−2fke2uf,h,∂k\n∂νg−hkeuf,h/parenrightbigg\n.\nBy Proposition 2.1, Lax-Milgram Theorem and standard elliptic regularity results, we\nobtain that DwZ(0,f,h) is an isomorphism. By Implicit Function Theorem there is a\nneighborhood U×W⊂C0,α(M)×C1,α(∂M) of (f,h), a neighborhood W′⊂C0,α(M)×\nC1,α(∂M) of (0,0), and a smooth map Z0:U×W→W′such that for any ( m,z)∈U×W\nit holds\nZ(Z0(m,z),m,z) = 0.\nSinceuf,hsolves (1.3), settingum,z=Z0(m,z)+uf,h,we arrive at the desired claim.\nDue toum,zsolves (2.3), it is a critical point of the functional Im,z. The second variation\nis given by ( 1.5). Additionally, we assume that uf,his a local minimizer of the functional\nIf,h, then by Proposition 2.1we haved2If,h(uf,h)(w,w)≥c0>0 for allw∈H1(M) with\n/ba∇dblw/ba∇dbl2\nH1= 1. Since um,zdepends smoothly on ( m,z), we have the convergence um,z→uf,h\ninC2,α(M) when (m,z)→(f,h) inC0,α(M)×C1,α(∂M). Choosing U×Wsmaller if\nnecessary, this implies, that\nd2Im,z(um,z)(w,w)≥c1>0,\nfor all (m,z)∈U×Wandw∈H1(M) with/ba∇dblw/ba∇dbl2\nH1= 1. Given that um,zis a critical point\nofIm,z, the Taylor expansion yields that um,zis a strict local minimizer for Im,z./square\n3.Existence of a further critical point of saddle-type\nLetf∈C∞(M) andh∈C∞(∂M) benonconstant functions so that max f= maxh= 0.\nConsider the perturbations fµ=f+µandhλ=h+λ,whereµandλare real numbers.\nSetIµ,λ(u) :=Ifµ,hλ(u),u∈H1(M). As consequence of Theorem 1.2, there areµ0>0 and\nλ0>0 such that for any µ∈(0,µ0) andλ∈(0,λ0), the functional Iµ,λadmits a smooth\nstrict local minimizer, denoted by uµ,λ, which depends smoothly on µandλ. Thus, as\nµ↓0 andλ↓0,uµ,λ→u0smoothly in H1(M), whereu0is the unique solution of ( 1.3).\nSince maxf= 0, thengiven µ>0thereexists apoint pµ∈M\\∂Msuchthatfµ(pµ)>0.\nThis implies that Iµ,λis no longer bounded from below, independently of λ. Indeed, let\nU⊂M\\∂Mbe a neighborhood of pµsuch thatfµ(p)>0 for allp∈M. Letwbe a smooth10 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nnonnegative function supported in Uand withw(pµ)>0. We can see that Iµ,λ(tw)→ −∞\nast→ ∞. Therefore, by Theorem 1.2,Iµ,λhas a mountain pass geometry for all µandλ\nsmall enough. Thus we may expect the existence of another critica l point of saddle-type.\nSinceuµ,λ→u0when (µ,λ)→(0,0), assuming µ0andλ0are small enough, we can find\nρ>0 such that\n(3.1) Iµ,λ(uµ,λ) = inf\n/bardblu−u0/bardblH1<ρIµ,λ(u)≤sup\nγ,ν∈(0,µ0];θ,ξ∈(0,λ0]Iγ,θ(uν,ξ)<βu0,\nuniformly for all µ∈(0,µ0] andλ∈(0,λ0], where\n(3.2) βu0= inf\n(µ,λ)∈(0,µ0]×(0,λ0]{Iµ,λ(u) :ρ/2</ba∇dblu−u0/ba∇dblH1<ρ}.\nUsing that the functional Iµ,λis unbounded from below, for each µ∈(0,µ0] andλ∈\n(0,λ0] considervµ,λ∈H1(M) such that\n(3.3) Iµ,λ(vµ,λ)<Iµ,λ(uµ,λ).\nIn this case, setting\n(3.4) Γ =/braceleftbig\np∈C0/parenleftbig\n[0,1];H1(M)/parenrightbig\n:p(0) =u0andp(1) =vµ,λ/bracerightbig\n,\nwe have\n(3.5) cµ,λ= inf\np∈Γmax\nt∈[0,1]Iµ,λ(p(t))≥βu0>Iµ,λ(uµ,λ).\nSinceuµ,λ→u0asµ↓0 andλ↓0, we can fix the initial point of comparison paths p∈Γ\nto beu0instead ofuµ,λ,providedµ0andλ0are sufficiently small.\nNote that for every µ,λ,γ∈Rand everyu∈H1(M) it holds\n(3.6) Iµ,λ(u)−Iµ,γ(u) =−(λ−γ)ˆ\n∂Meudσg\nand\n(3.7) Iµ,λ(u)−Iγ,λ(u) =−µ−γ\n2ˆ\nMe2udvg.\nThis and Lemma 4.1imply that the functions µ/ma√sto→cµ,λandλ/ma√sto→cµ,λare non-increasing,\nand therefore differentiable at almost every µ∈(0,µ0) andλ∈(0,λ0), respectively. Also,\nthe mapt/ma√sto→cµ+t,λ+tis non-increasing and differentiable at almost every tin its domain.\nTherefore, the derivative at t= 0 of the function\n(3.8) t/ma√sto→cµ,λ(t) :=cµ+t,λ+t\nexists for almost every ( µ,λ)∈(0,µ0)×(0,λ0).\nProposition 3.1. Assume the derivative c′\nµ,λ(0)exists for some (µ,λ)∈(0,µ0)×(0,λ0).\nThen there exist two sequences, {pn}inΓand{tn}in[0,1], such that\n(a)Iµ,λ(pn(tn))→cµ,λ,\n(b) max\n0≤t≤1Iµ,λ(pn(t))→cµ,λ,\n(c)/ba∇dbldIµ,λ(pn(tn))/ba∇dbl →0asn→ ∞.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 11\nMoreover, {pn(tn)}satisfies\n(3.9)1\n2ˆ\nMe2pn(tn)dvg+ˆ\n∂Mepn(tn)dσg≤ −c′\nµ,λ(0)+3.\nProof.Consider the decreasing sequences µn=µ+1/nandλn=λ+1/n. Sincecµ,λis a\nmin-max, see ( 3.5), we can find two sequences pn∈Γ andtn∈[0,1] such that\n(3.10) max\nt∈[0,1]Iµ,λ(pn(t))≤cµ,λ+1\nn\nand\n(3.11) cµn,λn−1\nn<Iµn,λn(un),\nwhereun=pn(tn). By (3.6), (3.7), (3.10) and (3.11), we get\n0≤1\n2ˆ\nMe2undvg+ˆ\n∂Meundσg=Iµ,λ(un)−Iµn,λn(un)\n1/n≤cµ,λ(0)−cµ,λ(1/n)\n1/n+2,\nwhich implies that\n(3.12)1\n2ˆ\nMe2undvg+ˆ\n∂Meundσg≤ −c′\nµ,λ(0)+3\nforn∈Nsufficiently large. Since cµ,λ(t) is non-increasing, it follows that c′\nµ,λ(0)≤0. By\n(3.12) and Jensen’s inequality, it holds\n(3.13) 2ˆ\nMundvg≤logˆ\nMe2undvg≤C(µ,λ)\nand ˆ\n∂Mundσg≤logˆ\n∂Meundσg≤C(µ,λ),\nfor alln∈Nsufficiently large and for some constant C(µ,λ)>0. By (3.10) and (3.11),\nwe obtain\ncµn,λn−1\nn≤Iµn,λn(un)≤Iµ,λ(un)≤max\nt∈[0,1]Iµ,λ(pn(t))≤cµ,λ+1\nn.\nThis proves items (a) and (b).\nWe use ( 1.4), (3.10), (3.12) and (3.13) to conclude:\n/ba∇dbl∇un/ba∇dbl2\nL2= 2Iµ,λ(un)+2ˆ\nMundvg+ˆ\nM(f+µ)e2undvg+ˆ\n∂M(h+λ)eundσg\n≤2cµ,λ+2\nn+C1≤C2,\nfor some positive constant C2=C2(µ,λ) independent of n. Therefore, by H¨ older’s inequal-\nity and ( 3.13) we obtain\n(3.14) /ba∇dblun/ba∇dblH1≤C3,\nfor some positive constant C3=C3(µ,λ,g) independent of n.12 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nTo prove item (c), assume by contradiction that /ba∇dbldIµ,λ(un)/ba∇dbl ≥2δfor someδ >0 and\nfor allnsufficiently large, where un=pn(tn) as above. By ( 3.14), the sequence {un}is\nuniformly bounded in H1(M). Thus,\n/a\\}b∇acketle{tdIµn,λn(un),dIµ,λ(un)/a\\}b∇acket∇i}ht=/ba∇dbldIµ,λ(un)/ba∇dbl2−/a\\}b∇acketle{tdIµ,λ(un)−dIµn,λn(un),dIµ,λ(un)/a\\}b∇acket∇i}ht\n≥ /ba∇dbldIµ,λ(un)/ba∇dbl2−/ba∇dbldIµ,λ(un)−dIµn,λn(un)/ba∇dbl/ba∇dbldIµ,λ(un)/ba∇dbl (3.15)\n≥1\n2/ba∇dbldIµ,λ(un)/ba∇dbl2−1\n2/ba∇dbldIµ,λ(un)−dIµn,λn(un)/ba∇dbl2,\nwhere we have used ab≤a2/2+b2/2.\nClaim 1: There exists a constant C >0, independent of n, such that for all u∈H1(M)\nsatisfying /ba∇dblu/ba∇dblH1≤c, the following inequality holds:\n/ba∇dbldIµn,λn(u)−dIµ,λ(u)/ba∇dblH−1≤C(|µn−µ|+|λn−λ|).\nFor everyv∈H1(M), by H¨ older’s inequality we have\n/a\\}b∇acketle{tdIµn,λn(u)−dIµ,λ(u),v/a\\}b∇acket∇i}ht=−(µn−µ)ˆ\nMe2uvdvg−(λn−λ)ˆ\n∂Meuvdσg\n≤ |µn−µ|/parenleftbiggˆ\nMe4udvg/parenrightbigg1/2\n/ba∇dblv/ba∇dblL2(M)+|λn−λ|/parenleftbiggˆ\n∂Me2udσg/parenrightbigg1/2\n/ba∇dblv/ba∇dblL2(∂M).\nBy a Moser-Trundinger type inequality, see for example [ 15, Section 3.5], and the fact that\nthe trace operator u/ma√sto→u|∂Mis continuous from H1(M) toL2(∂M), see [15, Section 3.2],\nwe obtain Claim 1.\nBy (3.15) we obtain\n/a\\}b∇acketle{tdIµn,λn(un),dIµ,λ(un)/a\\}b∇acket∇i}ht ≥2δ2−C(|µn−µ|+|λn−λ|)2≥δ2,\nfor all sufficient large n. Now, choose a smooth function φ:R→Rsuch that 0 ≤φ≤1,\nφ(s) = 1 fors≥ −1/2, andφ(s) = 0 fors≤ −1. For any n∈Nandw∈H1(M) define\nφn(w) :=φ/parenleftbiggIµn,λn(w)−cµn,λn\n1/n/parenrightbigg\n.\nNote that by ( 3.11) we haveφn(un)/\\e}atio\\slash= 0. Also, define\n(3.16) ˜ pn(t) :=pn(t)−1√nφn(pn(t))dIµ,λ(pn(t))\n/ba∇dbldIµ,λ(pn(t))/ba∇dbl,0≤t≤1,\nand ˜un= ˜pn(tn). By (3.1), (3.3) and (3.5), we obtain\nIµn,λn(pn(1))≤Iµn,λn(pn(0))≤cµn,λ−1/n,\nfornlarge enough. This implies that φn(pn(1)) =φn(pn(0)) = 0. Thus ˜ pn(1) =pn(1) and\n˜pn(0) =pn(0), which implies that ˜ pn∈Γ. Since 0 ≤φn≤1, then/ba∇dblun−˜un/ba∇dblH1≤1.\nClaim 2: For allnsufficient large and any w∈H1(M) with/ba∇dblw/ba∇dblH1≤1, we have\nIµn,λn(un+w)≤Iµn,λn(un)+/a\\}b∇acketle{tdIµn,λn(un),w/a\\}b∇acket∇i}htH−1×H1+C/ba∇dblw/ba∇dbl2\nH1.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 13\nBy Taylor’s expansion, given x∈Mthere exist t(x),s(x)∈(0,1) such that\nIµn,λn(un+w)−Iµn,λn(un)−/a\\}b∇acketle{tdIµn,λn(un),w/a\\}b∇acket∇i}htH−1×H1\n=1\n2ˆ\nM|∇w|2dvg−1\n2ˆ\nMfµn(e2(un+w)−e2un−2e2unw)dvg\n−ˆ\n∂Mhλn(eun+w−eun−eunw)dσg\n≤1\n2/ba∇dblw/ba∇dbl2\nH1−ˆ\nMfµne2(un+tw)w2dvg−1\n2ˆ\n∂Mhλneun+sww2dσg\n≤1\n2/ba∇dblw/ba∇dbl2\nH1+/ba∇dblfµn/ba∇dblL∞ˆ\nMe2(un+tw)w2dvg+/ba∇dblhλn/ba∇dblL∞ˆ\n∂Meun+sww2dσg.\nBy H¨ older’s inequality and Sobolev’s embedding, we obtain\nˆ\nMe2(un+tw)w2dvg≤/parenleftbiggˆ\nMe4(un+tw)dvg/parenrightbigg1/2\n/ba∇dblw/ba∇dbl2\nL4≤/parenleftbiggˆ\nMe8undvgˆ\nMe8|w|dvg/parenrightbigg1/4\n/ba∇dblw/ba∇dbl2\nH1.\nSince the trace operator u/ma√sto→u|∂Mis continuous from H1(M) toL4(∂M), see [15, Section\n3.2], we get\nˆ\n∂Meun+sww2dσg≤/parenleftbiggˆ\n∂Me2(un+tw)dσg/parenrightbigg1/2/parenleftbiggˆ\n∂Mw4dσg/parenrightbigg1/2\n≤C/parenleftbiggˆ\n∂Me4undσgˆ\n∂Me4|w|dσg/parenrightbigg1/4\n/ba∇dblw/ba∇dbl2\nH1.\nNow Claim 2 follows from the Moser-Trudinger inequality, see [ 15, Section 3.5].\nAs consequence of Claim 2, by using ( 3.16), we have\nIµn,λn(˜un)≤Iµn,λn(un)−φn(un)√n/ba∇dbldIµ,λ(un)/ba∇dbl/a\\}b∇acketle{tdIµn,λn(un),dIµ,λ(un)/a\\}b∇acket∇i}ht+Cφn(un)2\nn.\nUsing that /ba∇dbldIµ,λ(un)/ba∇dbl ≥2δ, (3.15) and Claim 1, we obtain\nIµn,λn(˜un)≤Iµn,λn(un)−1\n2φn(un)√n/ba∇dbldIµ,λ(un)/ba∇dbl+φn(un)\n2n2√n/ba∇dbldIµ,λ(un)/ba∇dbl+Cφn(un)2\nn\n≤Iµn,λn(un)−δφn(un)√n+φn(un)\n4n2√nδ+Cφn(un)2\nn≤Iµn,λn(un)−δ\n2√nφn(un),\nfor allnlarge enough. It follows that\ncµn,λn≤max\nt∈[0,1]Iµn,λn(˜pn(t))≤max\nt∈[0,1]/parenleftbigg\nIµn,λn(pn(t))−δ\n2√nφn(pn(t))/parenrightbigg\n.\nNote that if Iµn,λn(pn(t))<cµn,λn−1/2nfor allt∈[0,1], then this leads to a contradiction\nbased on the definition of cµn,λn. This implies that the maximum can only be achieved\nat pointst∈[0,1] such that Iµn,λn(pn(t))≥cµn,λn−1/2n. At these points, we have\nφn(pn(t)) = 1. Thus, using that Iµn,λn(u)≤Iµ,λ(u) and (3.10), we find\ncµn,λn≤max\nt∈[0,1]Iµn,λn(pn(t))−δ\n2√n≤max\nt∈[0,1]Iµ,λ(pn(t))−δ\n2√n≤cµ,λ+1\nn−δ\n2√n.14 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nFinally, since c′\nµ,λ(0) exists and is non positive, it follows that\ncµ,λ(0)−cµ,λ(1/n)\n1/n≤ −c′\nµ,λ(0)+1,\nfor allnsufficiently large. Thus, cµ,λ≤cµn,λn+(−c′\nµ,λ(0)+1)/n. Therefore,\ncµn,λn≤cµn,λn+1√n/parenleftbigg−c′\nµ,λ(0)+2√n−δ\n2/parenrightbigg\n<cµn,λn,\nfornlarge enough, which is a contradiction. This implies item (c). /square\n3.1.Proof of Theorem 1.3.In this subsetion we complete the Proof of Theorem 1.3.\nTheorem 3.2. Suppose the derivative c′\nµ,λ(0)exists for some (µ,λ)∈(0,µ0)×(0,λ0).\nThen the functional Iµ,λadmits a critical point uµ,λwith energy Iµ,λ(uµ,λ) =cµ,λand\nbounds on area and length\n(3.17)1\n2ˆ\nMe2uµ,λdvg+ˆ\n∂Meuµ,λdσg≤ −c′\nµ,λ(0)+3,\nand such that uµ,λis not a local minimizer of Iµ,λ.\nProof.The proof of Proposition 3.1gives us two sequences {pn}in Γ and {tn}in [0,1] such\nthatun=pn(tn)∈H1(M) satisfies items (a)–(c) of Proposition 3.1, (3.9) and (3.14). From\n(3.14), we can assume un⇀uµ,λweakly inH1(M) andun→uµ,λstrongly in L2(M), for\nsomeuµ,λ∈H1(M). By [15, Proposition 3.6], up to a subsequence, we can suppose that\ne2un→e2uµ,λinL2(M) andeun→euµ,λinL2(∂M). From this and ( 3.9) we obtain ( 3.17).\nNote that\n/vextenddouble/vextenddouble∇un−∇uµ,λ/vextenddouble/vextenddouble2\nL2(M)=/angbracketleftbig\ndIµ,λ(un),un−uµ,λ/angbracketrightbig\n−ˆ\nM/angbracketleftbig\n∇uµ,λ,∇un−∇uµ,λ/angbracketrightbig\ndvg\n+ˆ\nM/parenleftbig\nun−uµ,λ/parenrightbig\ndvg+ˆ\nMfµe2un/parenleftbig\nun−uµ,λ/parenrightbig\ndvg+ˆ\n∂Mhλeun/parenleftbig\nun−uµ,λ/parenrightbig\ndσg.\nThis implies that ∇un→ ∇uµ,λinL2(M). Thus, we have that un→uµ,λstrongly in\nH1(M). Consequently Iµ,λ(un)→Iµ,λ/parenleftbig\nuµ,λ/parenrightbig\nanddIµ,λ(un)→dIµ,λ/parenleftbig\nuµ,λ/parenrightbig\nasn→ ∞, and\nby Proposition 3.1the function uµ,λis a critical point for Iµ,λat levelIµ,λ/parenleftbig\nuµ,λ/parenrightbig\n=cµ,λ.\nReasoning by contradiction, assume that uµ,λis a local minimizer of Iµ,λforµ>0 and\nλ >0 small enough. By Theorem 1.2, the function uµ,λis a strict local minimizer. The\nsequences {pn}and (un), as before, satisfy max\n0≤t≤1Iµ,λ(pn(t))→cµ,λandIµ,λ(un)→cµ,λ.\nSinceun→uµ,λinH1(M), there exists a function varbitrarily close to uµ,λsuch that\nIµ,λ(v) is less than Iµ,λ(uµ,λ). This is a contradiction. /square\nFrom Theorem 3.2we obtain Theorem 1.3.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 15\n4.Blow-up analysis and Proof of Theorem 1.4\nIn this section, we will assume that the functions f∈C∞(M) andh∈C∞(∂M) are\nnonconstant with max f= maxh= 0, the maximum point of fandhare the same,\nnon-degenerate, and critical points of f.\nWe beginwith the following lemma, where we show anexplicit estimate oft he mountain-\npass energy level cµ,λassociated with the set Γ ,as defined in ( 3.4). First, for convenience\nwe fix some notations, as follows: The euclidean metric will be denoted bygeuc. Given\np∈R2andr>0,define\nBr(p) :={(s,t)∈R2:s2+t2<r2};\nB+\nr(p) :={(s,t)∈Br(p)⊂R2:t≥0};\n∂+B+\nr(p) :={(s,t)∈∂Br(p)⊂R2:t>0};\nΓr(p) :={(s,t)∈∂B+\nr(p) :t= 0}.\nAlso, this sets can be denoted by Br,B+\nr,∂+B+\nrand Γr. Remember from the introduction\nthat we can suppose Kg=−1 andκg= 0.\nLemma 4.1. For any constant κ>2πthere exists (µκ,λκ)∈(0,µ0/2]×(0,λ0/2]such that\nfor any(µ,λ)∈(0,µκ)×(0,λκ)there exists vµ,λ∈H1(M)so that choosing vγ,ν=vµ,λfor\nevery(γ,ν)∈[µ,2µ]×[λ,2λ]in(3.4), there holds Iγ,ν(vγ,ν) =Iγ,ν(vµ,λ)<Iγ,ν(uγ,ν),and\nthe number cγ,νis independent of µandλsuch that (γ,ν)∈[µ,2µ]×[λ,2λ]. Moreover,\nwe obtain the bound cγ,ν≤κlog(2/γ).\nProof.Letp0∈∂Mbe a maximum point of f. By assumption it holds f(p0) =h(p0) = 0.\nConsider a local conformal coordinates x= (x1,x2) aroundp0= 0 such that g=e2v0geuc,\nfor some smooth function v0withv0(0) = 0 and ∂Mcorrespond to x2= 0.\nSincep0is an isolated maximum point of fandh, we have that\nf(x)≥ −µ/2 andfµ(x) =f(x)+µ≥µ/2,∀x∈B+√µ/L(0)\nand\nh(x1)≥ −λ/2 andhλ(x) =h(x)+λ≥λ/2,∀x∈Γ√\nλ/L(0),\nfor all (µ,λ)∈(0,µ0]×(0,λ0], and for some constant L>max{√µ0,√λ0}which does not\ndepend onλandµ. Consider a function zµ∈H1(B+\n1(0)), where B+\n1(0)⊂R2, given by\nzµ(x) =/braceleftBigg\n−logµ,if|x| ≤µ,\n−log|x|,ifµ≤ |x| ≤1.\nUsing coordinates, we define a function wµ∈H1(M) aswµ(x) =zµ(Lx/√µ) inB+√µ/L(p0)\nandwµ≡0 outside of B+√µ/L(p0). This implies that\nˆ\nM|∇gwµ|2\ngdvg=ˆ\nB+√µ\nL(p0)|∇gwµ|2\ngdvg=ˆ\nB+√µ\nL(0)|∇wµ(x)|2dx\n=L2\nµˆ\nB+√µ\nL(0)|∇zµ(Lx/√µ)|2dx=ˆ\nB+\n1(0)|∇zµ(y)|2dy=πˆ1\nµ1\nrdr=−πlogµ.16 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nUsing Young’s inequality 2 ab≤δa2+b2/δ, withδ=κ−2π\n4π,a=s/ba∇dbl∇gwµ/ba∇dbl2\nL2(M)andb=\n/ba∇dbl∇gu0/ba∇dbl2\nL2(M), we get\n/ba∇dbl∇g(u0+swµ)/ba∇dbl2\nL2(M)≤(1+δ)s2/ba∇dbl∇gwµ/ba∇dbl2\nL2(M)+(1+1/δ)/ba∇dbl∇gu0/ba∇dbl2\nL2(M)\n=−(κ+2π)s2\n4logµ+C,\nwhereC=C(u0,κ)>0. Also, we have\nˆ\nMfµe2(u0+swµ)dvg≥µ\n2ˆ\nB+√µ\nL(p)e2(u0+swµ)dvg−C≥cµ\n2ˆ\nB+√µ\nL(p)e2swµdvg−C\n≥cµ2\n2L2ˆ\nB+\n1(0)e2szµ(x)dx−C≥cµ2\n2L2ˆ\nB+\nµ(0)e2szµ(x)dx−C≥π\n2cL−2µ4−2s−C\nand\nˆ\n∂Mhλeu0+swµdσg≥λ\n2ˆ√µ/L\n−√µ/Leu0+swµdt−C≥ −C.\nHence, using ( 1.4) and thatwµ≥0, we obtain\nIµ,λ(u0+swµ)≤ −κ+2π\n2s2\n4logµ−π\n2cL−2µ4−2s+C.\nNote that for µ,λ∈(0,1) it holdsIµ,λ(u0+swµ)→ −∞ass→+∞. By (3.6) and (3.7),\nwe may fix a a sufficiently large sµ,λ>0 anddefine vµ,λ:=u0+sµ,λwµ(see (3.3)), satisfying\nIµ,λ(vµ,λ)<inf\n(µ,λ)∈(0,µ0)×(0,λ0)Iµ,λ(uµ,λ).\nThus, ifp(t) :=u0+tsµ,λwµ, by (3.5), we obtain\ncµ,λ≤max\nt∈[0,1]Iµ,λ(p(t))≤sup\ns>0/parenleftbiggκ+2π\n2s2\n4log1\nµ−π\n2cL−2µ4−2s+C/parenrightbigg\n.\nFor anyµ∈(0,1) the supremum in the latter quantity is achieved for some s(µ)>2,\nwiths(µ)→2 asµ→0+. Thus, for all small enough µ>0 it holds\ncµ,λ≤κ+2π\n2log1\nµ≤κlog1\nµ.\nSinceIγ,ν(vµ,λ)≤Iµ,λ(vµ,λ) forγ >µandν >λ,the same comparison function vµ,λcan\nbe used for every ( γ,ν)∈(µ,2µ)×(λ,2λ)⊂(0,µ0/2]×(0,λ0/2], and for such ( γ,ν) we\nobtain the bound\nIγ,ν(vµ,λ)<Iγ,ν(uγ,ν)<sup\n(s,t)∈[µ,2µ]×[λ,2λ]Iγ,ν(us,t)<βu0≤cγ,ν<κlog1\nµ<log2\nγ,\nwhereβu0is defined in ( 3.2). Finally, since vµ,λdepends continuously on µandλ, by\nconstruction, with Iµ,λ(vµ,λ)<inf{Iγ,ν(uγ,ν) : (γ,ν)∈(0,µ0/2]×(0,λ0/2]},the number\ncγ,νis defined independently of µandλsuch that (γ,ν)∈(µ,2µ)×(λ,2λ). /square\nIt is important to mention that the Lemma 4.1holds iff(p0) = 0 andh(p0)<0, but\nsince we are not interested in this case, we omitted the proof.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 17\nLemma 4.2. Forcµ,λ(t)defined as in (3.8)it holds\n(4.1) liminf\n(µ,λ)↓(0,0)µ/vextendsingle/vextendsinglec′\nµ,λ(0)/vextendsingle/vextendsingle≤2πandliminf\n(µ,λ)↓(0,0)λ/vextendsingle/vextendsinglec′\nµ,λ(0)/vextendsingle/vextendsingle≤2π.\nProof.For the first inequality, suppose that for κ>κ1>2π,we have|c′\nµ,λ(0)| ≥κµ−1for\nalmost every µandλsmall enough. Then for any λ1> µ1> µ2>0 sufficiently small,\nusing thatcµ,λ(t) is non-increasing, we obtain\ncµ2,λ1−µ1+µ2−cµ1,λ1≥ˆµ1\nµ2/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndtct,λ1−µ1+t/vextendsingle/vextendsingle/vextendsingle/vextendsingledt=ˆµ1\nµ2|c′\nt,λ1−µ1+t(0)|dt≥ˆµ1\nµ2κ\ntdt=κlogµ1\nµ2.\nBy Lemma 4.1, we havecµ2,λ1−µ1+µ2≤κ1log(2/µ2) for all sufficiently small µ2. Thus\ncµ1,λ1≤(κ1−κ)log2\nµ2−κlogµ1+κ1log2,\nfor all small enough µ2>0, which is a contraction by ( 3.5), sinceκ1<κ.\nNow, suppose that for κ > κ 1>2πit holds|c′\nµ,λ(0)| ≥κλ−1for almost every λandµ\nsmall enough. Then for any µ1>λ1>λ2>0 sufficiently small, as before we have\ncµ1−λ1+λ2,λ2−cµ1,λ1≥ˆλ1\nλ2/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndtcµ1−λ1+t,t/vextendsingle/vextendsingle/vextendsingle/vextendsingledt=ˆλ1\nλ2|c′\nµ1−λ1+t,t(0)|dt≥ˆλ1\nλ2κ\ntdt=κlogλ1\nλ2.\nFixµ1and choose λ1=µ1−λ2. For all sufficiently small λ2>0, we have by Lemma 4.1\nthatcµ1−λ1+λ2,λ2≤κ1log(2/(µ1−λ1+λ2))≤ −κ1log(λ2). Thus,\ncµ1,λ1≤(κ1−κ)log1\nλ2−κlogλ1,\nfor all small enough λ2>0.Again, this leads to a contradiction by ( 3.5), sinceκ1<κ./square\nProposition 4.3. There exist a constant C >0, two sequences λn↓0andµn↓0, with\nλ2\nn−λ3\nn≤µn≤λ2\nn+λ3\nn, and a corresponding solutions un:=uµn,λn/\\e}atio\\slash=uµn,λnof(1.6)of\n“mountain pass” type inducing conformal metrics gn=e2ungwith total curvature satisfyng\n(4.2)ˆ\nM|Kgn|dvgn+ˆ\n∂M|κgn|dσgn≤C\nfor some constant C >0independently of n, and\n(4.3) limsup\nn→+∞/parenleftbigg1\n2µnˆ\nMe2undvg+λnˆ\n∂Meundσg/parenrightbigg\n≤2π.\nProof.Givenλ >0, the function t/ma√sto→cλ2+t,λ+tis differentiable at almost every tin its\ndomain. Besides, if µ=λ2+tandλ=λ+t, thenµ∈(λ2−λ3,λ2+λ3) for allt≥0 small\nenough. Thus, we can find two sequences λn↓0 andµn↓0, withµn/λ2\nn∈(1−λn,1+λn)\nsuch thatc′\nµn,λn(0) exists. Thus, for each n, Theorem 3.2gives us the desired solution un.\nBy (1.6) and the Gauss-Bonnet Theorem, it follows thatˆ\nMfµndvgn+ˆ\n∂Mhλndσgn=ˆ\nMKgndvgn+ˆ\n∂Mκgndσgn= 2πχ(M).\nTherefore\n−ˆ\nMfe2undvg−ˆ\n∂Mheundσg=µnˆ\nMe2undvg+λnˆ\n∂Meundσg−2πχ(M).18 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nBy (3.17) we have\n1\n2ˆ\nMe2undvg+ˆ\n∂Meundσg≤ |c′\nµn,λn(0)|+3.\nSince 0< µn<λ2\nn+λ3\nn≤λn, using (4.1) we obtain ( 4.3). Finally, for nlarge enough, it\nholdsˆ\nM|f|e2undvg+ˆ\n∂M|h|eundσg≤λn|c′\nµn,λn(0)|+C≤C <∞\nuniformly in n∈N. From this bound, we obtain ( 4.2). /square\nLemma 4.4. Letunbe the sequence of solutions of (1.6)given by Proposition 4.3. Con-\nsider an open set Ω⊂⊂M−={x∈M:f <0andh<0}.\n(a) There exists a constant C >0such that for all nit holdsun>−C.\n(b) Letu+\nn= max{un,0}. Then, for nsufficiently large it holds\nˆ\nΩ/parenleftBig/vextendsingle/vextendsingle∇u+\nn/vextendsingle/vextendsingle2+(u+\nn)2/parenrightBig\ndvg+sup\nΩ(−h)ˆ\n∂Ω∩∂M(u+\nn)2dσg≤C(Ω).\n(c) There exists a constant C(Ω)such thatun≤C(Ω), for all sufficiently large n.\nProof.First, let us prove item (a). The proof is based on ideas from [ 18] and [47]. Letf\nbe a solution of the problem\n(4.4)\n\n−∆gv=b1,inM,\n∂v\n∂νg=b2,on∂M,\nwhereb1∈(0,1) andb2=−b1vol(M)\nvol(∂M)<0. Note that due to the choices of b1andb2, (4.4)\nhas a solution v. Thus, there exists a constant a>0 such that for all c≥a, the function\nϕc=v−csatisfies\n\n\n−∆gϕc−1−fµne2ϕc=b1−1−fµne2ϕc<0,inM,\n∂ϕc\n∂νg−hλneϕc=b2−hλneϕc<0, on∂M.\nItem (a) will follow by the following claim.\nClaim:un>ϕa.\nIf this is not the case, there is c > asuch thatun≥ϕconM, andun(p) =ϕc(p) for\nsomep∈M. By (1.6) we find that\n(4.5)\n\n−∆g(un−ϕc)−fµn(e2un−e2ϕc)>0,\n∂\n∂νg(un−ϕc)−hλn(eun−eϕc)>0.\nIfp/\\e}atio\\slash∈∂M, by the maximum principle, we conclude that un≡ϕc, which is impossible. If\np∈∂M, Hopf’s Lemma implies that\n∂\n∂νg(un−ϕc)(p)<0.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 19\nBut, the boundary equation of ( 4.5) gives us\n∂\n∂νg(un−ϕc)(p)>0,\nwhich is a contradiction.\nLet us prove item (b). Given any domain Ω ⊂⊂M−, it is enough to prove the result\nfor any geodesic ball B+\nr⊂⊂M−, since afterward the estimate for Ω can be deduced by a\ncovering argument. Let d= dist(Ω,∂M−\\∂M)>0.\nLetB2r⊂M−be the geodesic ball centered at some point p∈Mwith radius 2 r >0.\nIfB2r∩∂M=∅, then the result follows from the proof of [ 18, Lemma 2], see also the\nappendix in [ 11]. Thus, let us suppose that p∈∂Mand denote the geodesic ball as B+\n2r.\nLetψbe a smooth cut-off function 0 ≤ψ≤1 supported in B+\n2rsuch thatψ≡1 inB+\nr\nand|∇ψ| ≤Ad−1, for some positive constant A. Multiply the first equation in ( 1.6) by\nψ4u+\nnand use integration by parts to obtain\n(4.6)ˆ\nB+\n2r/parenleftBig/angbracketleftbig\n∇u+\nn,∇/parenleftbig\nψ4u+\nn/parenrightbig/angbracketrightbig\n−ψ4u+\nn−fne2u+\nnψ4u+\nn/parenrightBig\ndvg−ˆ\nΓ2rhλneu+\nnψ4u+\nndσg= 0,\nwhere Γ 2r=∂M∩B+\n2r.Note that /a\\}b∇acketle{t∇u+\nn,∇(ψ4u+\nn)/a\\}b∇acket∇i}ht=|∇(ψ2u+\nn)|2− |∇ψ2|2(u+\nn)2.Using\nYoung’s inequality we get\n|∇ψ2|2(u+\nn)2= 4|∇ψ|2(ψu+\nn)2≤4A2d−2(ψu+\nn)2≤1\n2ε(ψu+\nn)4+C,\nwhereC=C(ε,d,A). Also, we estimate ψ4u+\nn≤1\n2ε(ψu+\nn)4+C. Since there exists ε>0\nsuch that we have fµn,hλn≤ −εonB+\n2rfor sufficiently large n,e2t≥t3andet≥tfor\nt∈R, we obtain from ( 4.6) the following inequalityˆ\nB+\n2r/parenleftBig/vextendsingle/vextendsingle∇/parenleftbig\nψ2u+\nn/parenrightbig/vextendsingle/vextendsingle2+εψ4/parenleftbig\nu+\nn/parenrightbig4/parenrightBig\ndvg+εˆ\nΓ2rψ4(u+\nn)2≤ˆ\nB+\n2r/parenleftbig\n/a\\}b∇acketle{t∇u+\nn,∇(ψ4u+\nn)/a\\}b∇acket∇i}ht\n+|∇ψ2|2(u+\nn)2−fµne2u+\nnψ4u+\nn/parenrightBig\ndvg−ˆ\nΓ2rhλneu+\nnψ4u+\nndσg\n=ˆ\nB+\n2r/parenleftBig\n|∇ψ2|2/parenleftbig\nu+\nn/parenrightbig2+ψ4u+\nn/parenrightBig\ndvg≤εˆ\nB+\n2r/parenleftbig\nψu+\nn/parenrightbig4dvg+C,\nwhich impliesˆ\nB+\n2r/vextendsingle/vextendsingle∇/parenleftbig\nψ2u+\nn/parenrightbig/vextendsingle/vextendsingle2dvg+εˆ\nΓ2rψ4(u+\nn)2dσg≤C.\nHence the claim follows from Poincar´ e’s inequality.\nIt remains to prove Item (c). Again, we can suppose that Ω ∩∂M/\\e}atio\\slash=∅. We notice that it\nis enough to prove the result on an arbitrary metric ball. Let B+\n2r⊂B+\n2r⊂M−be a fixed\nhalf-ball. From localized versions of the Moser-Trundiger [ 15, Proposition 3.9], together\nwith the fact that ( u+\nn)∈H1(B+\n2r) and (u+\nn)∈L2(B+\n2r∩∂M), it follows that fµne2unand\nhλneunareL2-bounded. Let wnbe the solution of\n\n−∆gwn−1 =fµne2un,inB+\n2r,\n∂wn\n∂ν=hλneun, on Γ2r,\nwn= 0, on∂+B+\n2r,20 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nby a standard elliptic theory involving a Harnack inequality, which has b een established in\n[25], it follows that\n/ba∇dblwn/ba∇dblL∞(B+\nr)≤C,\nsince/ba∇dblfµne2un/ba∇dblL2(B+\n2r)≤Cand/ba∇dblhλneun/ba∇dblL2(B+\n2r∩∂M)≤C. Note that sn=un−wnimplies\nthat∂sn/∂νvanishes in Γ r. By extending snevenly to whole ball Br,snbecomes a\nharmonic function. By items (a) and (b), and together with the mea n value theorem for\nharmonic functions, we shows that sn, and hence un, is locally uniformly bounded from\nabove inB+\nr.\n/square\n4.1.Proof of Theorem 1.4.Given a sequence {un}of solutions to ( 1.6), we say that a\npointp0∈Mis a blow-up point for {un}if for anyr>0 there holds supBr(p)un(x)→+∞\nasn→ ∞. In this case, Lemma 4.4implies that f(p0) =h(p0) = 0, since we are supposing\nthatfandhhave the same maximum points. For the sequence {un}given by Proposition\n4.3, there must exist at least one blow-up point, since otherwise by reg ularity theory we\ncould extract a subsequence converging smoothly to the global min imizer ofIµn,λn, which\nis a contradiction by ( 3.1) and (3.5).\nAs in the proof of Lemma 4.1, in local coordinates xnearp0= 0, namely B+\nr:=B+\nr(0),\nfor somer >0,there is a smooth function v0such thatg=e2v0geuc. Thus, if unis a\nsolution to ( 1.6), thenvn:=un+v0satisfies\n\n−∆vn=fµn(x)e2vninB+\nr,\n∂vn\n∂ν=hλn(x)evnon Γr,\nwhereνis the outward unit normal to Γ r.\nLemma 4.5. It holds\n(a)limsup\nn→∞/parenleftbiggˆ\nB+\nr(0)f+\nµne2vndx+ˆ\nΓrh+\nλnevndl/parenrightbigg\n≥π.\n(b) Up to subsequences, the blow-up set for the sequence of so lutions{un}to(1.6)\nsatisfying (4.2)is finite.\nThe proof of item (a) follows by contradiction using [ 12, Theorem 1] and [ 24, Lemma\n3.2], see the proof of [ 4, Lemma 2.4]. Item (b) is a consequence of item (a) and ( 4.2).\nLet{un}be the sequence of solution given by Proposition 4.3. Let{p(i)\n∞: 1≤i≤N}\nbe the set of blow-up points to {un}. Since{un}is locally uniform bounded on M−,up\nto a subsequence, un→u∞smoothly locally on M∞:=M\\{p(i)\n∞: 1≤i≤N}. Note that\n(4.2) gives that ( −∆gun)nand (∂un/∂ν)nare uniformly L1-bounded. Then, since we may\nassume that un→u∞weakly inW1,p(M),forp<2, we have\nπN/summationdisplay\ni=1aiφ(p(i)\n∞) =ˆ\nM(−u∞∆gφ−φ−fe2u∞φ)+ˆ\n∂M/parenleftbigg\nu∞∂φ\n∂ν−φheu∞/parenrightbigg\n,\nforallφ∈C∞(M).Notethatai≥1byLemma 4.5. Observethat {p(i)\n∞: 1≤i≤N} ⊂∂M,\nsince all zeros of fbelongs to∂M.\nProposition 4.6. There holdsBLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 21\n(a)ai∈[1,2]for every 1≤i≤N;\n(b)The metric g∞=e2u∞gonM∞is complete with finite total curvature.\nProof.Consider for each 1 ≤i≤Na local conformal chart Ψ : B+\nR(p(i)\n∞)→B+\nR=B+\nR(0)⊂\nR2\n+with coordinates x, such that (locally) g=e2v0geuc,wherev0is a smooth function\ndefined on B+\nR. Ifv∞(x) =u∞(x) +v0(x), thenv∞(x) =−ailog|x|+w∞(x),wherew∞\nsolves\n(4.7)\n\n−∆w∞=fe2v∞inB+\nR,\n∂w∞\n∂ν=hev∞on ΓR.\nBy (4.2) and Fatou’s Lemma, we have fe2v∞∈L1(B+\nR) andhev∞∈L1(ΓR). In view\nof [12, Theorem 1] and [ 24, Lemma 3.2], we can follow the argument from the proof of\n[25, Lemma 3.4] to show that e2|w∞|ande|w∞|areLp-bounded in a small neighborhood\naroundx= 0, for some p >1. By the assumption of the maximum point of f, we have\nα−1|x|2≤ |f(x)| ≤α|x|2,for someα>0.Therefore\n(4.8) α−1|x|2−2aie2w∞≤ |f(x)|e2v∞≤α|x|2−2aie2w∞.\nFix anyq∈(1,2). By H¨ older’s inequality and ( 4.2), as in (5.11) of [ 11], we have the\nfollowing estimateˆ\nB+|x|2−2ai\nqdx=ˆ\nB+(|x|2−2aie2w∞)1\nqe−2w∞\nqdx\n≤α1\nq/parenleftbiggˆ\nB+|f(x)|e2v∞dx/parenrightbigg1\nq/parenleftbiggˆ\nB+e2|w∞|\nq−1dx/parenrightbiggq−1\nq\n≤C(q),\nfor some small half ball B+aroundx= 0. Thus 1 ≤ai≤2.\nTo prove item (b), let us divide the analysis in two cases.\nCase 1:ai<2.\nFor someq >1 there holds ∆ w∞∈Lq(B+),whereB+is a small half-ball around x= 0.\nSince we also are assuming that x= 0 is a non-degenerate maximum point of h, we have\nβ−1|x|2≤ |h(x)| ≤β|x|2,for someβ >0.This implies that\n(4.9) β−1|x|2−aiew∞≤ |h(x)|ev∞≤β|x|2−aiew∞.\nSinceai<2,there is a certain small domain B+aroundx= 0 such that, for some q >1,\n∂w∞/∂ν∈Lq(Γ) and ∆w∞∈Lq(B+).By standard elliptic theory, we have /ba∇dblw∞/ba∇dblL∞(B+)≤\nCuniformly on a sufficiently small half-ball B+aroundx= 0. Then there exists a constant\nC >0 such that e2v∞≥C|x|−2ai≥C|x|−2nearx= 0 onB+. Consequently, the metric\ng∞=e2u∞g=e2v∞gR2onB+\\ {0}is complete. Since we may also assume pointwise\nconvergence almost everywhere, by Fatou’s Lemma and ( 4.2) we get\nˆ\nM|f|e2u∞dvg+ˆ\n∂M|h|eu∞dag<∞.\nTheng∞has finite total curvature.\nCase 2:ai= 2.22 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nFirst, similarly to (5.13) in [ 12], using ( 4.7) and (4.8), for anyd>0 we get\n−∆(|x|2de−2w∞)≤C|x|2d−2.\nAlso, using ( 4.9) and that /a\\}b∇acketle{tx,ν/a\\}b∇acket∇i}ht= 0 on Γ, we obtain\n∂\n∂ν/parenleftbig\n|x|de−w∞/parenrightbig\n=−|x|de−w∞∂w∞\n∂ν+d|x|d−2e−w∞/a\\}b∇acketle{tx,ν/a\\}b∇acket∇i}ht ≤C|x|d−1.\nThus, there holds\n∂\n∂ν/parenleftbig\n|x|2de−2w∞/parenrightbig\n=C|x|2d−1e−w∞≤C|x|2d−1e|w∞|,\nwhere the right hand side is in Lq(Γ) for some q=q(d)>1, provided ona sufficiently small\nhalf-ball around x= 0 we have e|w∞|∈Lq.Since we also have ∆( |x|2de−2w∞)∈Lp(B+) for\nsomep=p(d)>1 we have by elliptic estimates that |x|de−w∞≤C. Thus, for any kthere\nis a constant K >0 such that e2v∞=|x|−4e2w∞≥k|x|d−4, and hence g∞is complete on\nB+\\{0}. /square\nMotivated by [ 34, Proposition 5.1], in the proof of Theorem 1.4we will use Ekeland’s\nvariational principle, which we recall below for the sake of the reade r. See [42, Theorem\n5.1] pg. 51 and [ 51, Theorem 1.4.1].\nTheorem 4.7. Let(X,d)be a complete metric space and consider a function ϕ:X→\n(−∞,+∞]that is lower semi-continuous, bounded from below and not id entical to +∞.\nLetε >0andλ >0be given and let x∈Xbe such that ϕ(x)≤infXϕ+ε. Then there\nexistsxεXsuch that\n(a)ϕ(xε)≤ϕ(x),\n(b)d(xε,x)≤λ,\n(c)ϕ(xε)<ϕ(z)+ε\nλd(xε,z)for everyz/\\e}atio\\slash=xε.\nProof of Theorem 1.4.To conclude our analysis, our goal is to analyze the behavior of the\nsolutionasweapproacheachpoint pi\n∞, for1≤i≤I. Wewillconsider twosequences λn↓0\nandµn↓0, withλ2\nn−λ3\nn≤µn≤λ2\nn+λ3\nn, such that c′\nµn,λn(0) exists, and the corresponding\nsequence of solutions un, which are given by Proposition 4.3. As illustrated in the proof\nof Proposition 4.6, consider a local conformal chart Ψ : B+\nR(p(i)\n∞)→B+\nR:=B+\nR(0)⊂R2\n+\nwith coordinates x= (s,t) and Ψ(p(i)\n∞) = 0, and let vn(x) =un(x) +v0(x) be such that\n(locally)g=e2v0geuc,wherev0is a smooth function on B+\nR. HereRis chosen sufficiently\nsmall in order to guarantee that in B+\nR(p(i)\n∞) the only maximum point of fandhisp(i)\n∞.\nThe function vnsolves\n(4.10)\n\n−∆vn=fµn(x)e2vninB+\nR,\n∂vn\n∂ν=hλnevn on ΓR.\nSincep(i)\n∞is a non-degenerate maximum point of fandh, we have\n(4.11) f(x) =1\n2∇2f(p(i)\n∞)(x,x)+O(|x|3),for allx∈B+\nRBLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 23\nand\n(4.12) h(x) =1\n2∇2h(p(i)\n∞)(x,x)+O(|x|3),for allx∈ΓR,\nwith∇2f(p(i)\n∞) and∇2h(p(i)\n∞) are negative definite. We can choose Rsmall enough such\nthat\n(4.13) −α1|x|2≤f(x)≤ −α2|x|2,for allx∈B+\nR(0)\nand\n(4.14) −α3|x|2≤h(x)≤ −α4|x|2,for allx∈ΓR,\nwith 0< α2≤α1and 0< α4≤α3. DefineKn:={p∈M:f(p) +µn≥0} ∩B+\nR(p(i)\n∞)\nandQn:={p∈∂M:h(p) +λn≥0}.Sincef(p(i)\n∞) =h(p(i)\n∞) = 0, then KnandQnare\nnonempty sets. Now, for each nconsideran=/radicalbig\nµn/α1. Using that µn≤λ2\nn+λ3\nnand\nthe inequalities ( 4.13) and (4.14) we obtain B+\nan:=B+\nan(p(i)\n∞)⊂Knand Γ an⊂Qn, for\nnlarge enough. Consider a sequence ( xn) inMsuch thatvn(xn) = supB+\nanvn(x). Since\nf(xn)+µn≥0, by (4.13) there exists some constant C >0 independent of nsuch that\n(4.15) |xn|2≤Cµn.\nUp to a subsequence, we see that we have two mutually disjoint case s:\nCase 1:λ2\nnevn(xn)→+∞.\nIn this case, for x∈B+\nanrescale\nwn(x) =vn(xn+rnx)−vn(xn)≤wn(0) = 0,\nwhich is defined in Ω n:={x∈R2:xn+rnx∈B+\nan(0)}, wherern>0 is so that\nrnλnevn(xn)= 1.\nNote thatrn/λn→0, and using that λ2\nn−λ3\nn≤µnwe havean/rn→+∞. Passing to a\nsubsequence we can assume thatd(xn,Γn)\nrn→t0,with either t0≥0 ort0= +∞. Sincevn\nsatisfies ( 4.10) then\n\n−∆wn=r2\nnfµn(xn+rnx)e2wne2vn(xn)=Fne2wnin Ωn,\n∂wn\n∂ν=rnhλn(xn+rnx)ewnevn(xn)=Hnewnon Γn,\nwhere Γ nis the straight portion of ∂Ωn,Fn(x) :=fµn(xn+rnx)/λ2\nnandHn(x) :=hλn(xn+\nrnx)/λn. Using ( 4.11), (4.15),rn/λn→0,f(xn)+µn≥0 andµn=λ2\nn+O(λ3\nn) we obtain\nthat\nlim\nn→+∞Fn(x) = lim\nn→+∞µn\nλ2nfµn(xn+rnx)\nµn= lim\nn→+∞f(xn)\nµn+1 =c∞\nand\nlim\nn→+∞Hn(x) = lim\nn→+∞h(˜xn)\nλn+1 =d∞,\nwherec∞,d∞∈[0,1], and ˜xnis the first coordinate of xn. Besides\n(4.16)ˆ\nΩne2wn(x)dx≤r−2\nne−2vn(xn)ˆ\nMe2vndvg=λ2\nn\nµnµnˆ\nMe2vndvg24 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nand\n(4.17)ˆ\nΓnewn(x)=ˆ\nΓnevn(xn+rnx)−vn(xn)≤λnˆ\n∂Mevn.\nSinceλ2\nn/µn→1, it follows by ( 4.3) that\n1\n2ˆ\nΩne2wn+ˆ\nΓnewn≤2π.\nIft0= +∞, then Ω nexhausts R2. It follows from the proof of Theorem 1.4 of [ 11] that\nwn→w∞inC2\nloc(R2), wherew∞satisfies the Liouville equation\n−∆w∞=c∞e2w∞inR2,withˆ\nR2e2w∞dx≤4π.\nThis implies that c∞>0. After replacing the equality rnλnevn(xn)= 1 byrnλnevn(xn)=\n1/√c∞, we obtain that w∞solves\n−∆w∞=e2w∞inR2,withˆ\nR2e2w∞dx≤4π.\nFrom the classification result in [ 13], it follows that w∞(x) =−log/parenleftbig\n1+1\n8|x|2/parenrightbig\nwith\nˆ\nR2e2w∞dx= 4π.\nThus we have ˆ\nΩnFne2wndx= 4π+o(1),\nasn→+∞. On the other hand, after a change of variables, [ 22, Lema 3.7] implies that\nˆ\nΩnFne2wndx+ˆ\nΓnHnewnda≤2π+o(1),\nwhich is a contradiction. Therefore, t0<+∞. In this case, Ω nexhausts R2\nt0:=R×\n(t0,+∞). By standard elliptic theory, see [ 9], for instance, up to a subsequence, wn→w∞\ninC2\nloc(R2\nt0), wherew∞satisfies\n(4.18)\n\n−∆w∞=c∞e2w∞inR2\nt0,\n∂w∞\n∂ν=d∞ew∞on∂R2\nt0,with1\n2ˆ\nR2\nt0e2w∞dx+ˆ\n∂R2\nt0ew∞dl≤2π,\nandc∞,d∞∈[0,1]. Thanks to the classifications of solutions to the equation ( 4.18), see\n[33,48], we obtain the existence of Λ >0 ands0∈Rsuch that\nw∞(s,t) = log2Λ\nc∞Λ2+(s−s0)2+(t−t0+d∞Λ)2,\nwhered∞>0 ifc∞= 0. This finishes the analysis of blow-up in Case 1.\nFinally, we prove an estimate for the number of blow-up points. Befo re, recall that\nd∞ˆ\n∂R2\n+ew∞=β, c∞ˆ\nR2\n+e2w∞= 2π−β,BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 25\nwhereβ:= 2πd∞√\nd2∞+c∞≤2π(see for instance Section 2.2 of [ 22]). As in ( 4.16) and (4.17),\nat each blow-up point p(i)\n∞we must have\n2π−β=c∞ˆ\nR2\n+e2w∞dx≤c∞limsup\nn→∞ˆ\nΩne2wndx≤c∞limsup\nn→∞/parenleftBigg\nλ2\nn\nµnµnˆ\nB+\nan(p(i)\n∞)e2undvg/parenrightBigg\nand\nβ≤d∞limsup\nn→∞/parenleftbigg\nµnˆ\nΓan(p(i)\n∞)eundσg/parenrightbigg\n.\nSincep(i)\n∞is the only maximum point in B+\nR(p(i)\n∞), then\n(2π−β)N≤c∞limsup\nn→∞/parenleftbiggλ2\nn\nµnµnˆ\nMe2undvg/parenrightbigg\nandβN≤d∞limsup\nn→∞/parenleftbigg\nµnˆ\n∂Meundσg/parenrightbigg\n,\nwhereN≥1 is the number of blow-up point. Consider the following cases\n•Ifc∞= 0, thenβ= 2πandd∞= 1. In this case, N= 1. In fact, by ( 4.3), it holds\n2πN≤limsup\nn→∞/parenleftbigg\nµnˆ\n∂Meundσg/parenrightbigg\n≤2π.\n•Ifd∞= 0, thenβ= 0 andc∞∈(0,1]. In this case, N≤2. In fact, again by ( 4.3),\nit holds\n2πN≤c∞limsup\nk→∞/parenleftbiggλ2\nn\nµnµnˆ\nMe2undvg/parenrightbigg\n≤4πc∞.\n•Ifc∞,d∞∈(0,1],thenN≤3. In fact, as before\n2πN≤c∞limsup\nk→∞/parenleftbiggλ2\nn\nµnµnˆ\nMe2ukdvg/parenrightbigg\n+d∞limsup\nn→∞/parenleftbigg\nλnˆ\n∂Meukdσg/parenrightbigg\n≤6π.\nCase 2:λ2\nnevn(xn)≤C.\nIn view of ( 4.13) we obtain f(x) +µn≤0 ifµn≤c|x|2, for some positive constant c.\nThus, by Lemma 4.5we obtain\nπ≤ˆ\nB+\nr(0)f+\nµn(x)e2vn(x)dx=ˆ\nB+\nc√µn(0)fµn(x)e2vn(x)dx≤cµ2\nne2vn(xn),\nwhich implies that cµnevn(xn)≥1, for some constant c >0 independent of n. Since\nλ2\nn/µn→1, thence−vn(xn)< λ2\nn< Ce−vn(xn),for positive constants candC, which does\nnot depend on n. By continuity, for each nwe can find εn∈(0,λ2\nn) such that\n(4.19) ce−vn(z)≤λ2\nn≤Ce−vn(z),for allz∈B+\nR(0) with |z−xn|<εn\nBy Ekeland’s variational principle (Theorem 4.7), takingϕ=e−vn\n2,λ=εnandε=ε2\nn, we\ncan find a sequence yn∈B+\nRsuch that\n(a)vn(xn)≤vn(yn),\n(b)|xn−yn|<εn,\n(c)e−vn(yn)<e−vn(x)+εn|x−yn|, for everyx/\\e}atio\\slash=yn.26 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nBy (a) and (b), it holdsvn(yn)→+∞andyn→0. Letwn(y) =vn(λny) + 2logλn,\ndefined inBn:=B+\nR/λn(0). Note that wnsatisfies\n\n\n−∆wn=Fne2wninBn,\n∂wn\n∂ν=Hnewnon Γn,\nwhereFn(y) =f(λny)/λ2\nn+µn/λ2\nnandHn(y) =h(λny)/λn+ 1. Since µn/λ2\nn→1, by\n(4.11) and (4.12) we obtain that Fn(y)→1\n2∇2f(p(i)\n∞)(y,y)+1 andHn(y)→1.\nBy (4.15), we have |xn/λn|<L, for someL>0. Note that if |x|<λnr, then|x−yn| ≤\nλnL0, withL0:=r+L+1.\nClaim:Givenr >0, it holdswn(y)<c, for ally∈Bnwith|y| ≤r, fornlarge enough,\nwherec>0 does not depend on n.\nUsing (4.19), item (c) and that |xn/λn|<L, if|x| ≤λnr, we get\nλ2\nnevn(x)≤C(1−CL0λn)−1,\nwhich implies the claim.\nBy this claim and the Harnack type inequalities (see [ 25, Lemma A.2]), wnis uniformly\nbounded in L∞\nloc(R2\n+). Therefore, up to a subsequence,\nwn→w∞inC2\nloc(R2\n+),\nwhich is a solution of the equation\n\n\n−∆w∞= (1+1\n2∇2f(p(i)\n∞)(y,y))e2w∞inR2\n+,\n∂w∞\n∂ν=ew∞ on∂R2\n+.\nAs before and using ( 4.2) we have\nˆ\nR2\n+e2w∞+ˆ\n∂R2\n+ew∞<∞andˆ\nR2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+1\n2∇2f(p(i)\n∞)(y,y)/vextendsingle/vextendsingle/vextendsingle/vextendsinglee2w∞<∞.\nBy Theorem 1.5, whose proof we postponed to the next section, we conclude that this is\nimpossible. Therefore, we complete the proof of Theorem 1.4. /square\n4.2.Ruling out slow blow-up: Non-existence result. In this subsection, our primary\nobjective is to eliminate certain blow-up cases identified in the latter p art. We aim to\nachieve thisbyconducting ananalysis akinto theonepresented in[ 32]and[43], specifically\nexcluding cases where the blow-up occurs in a “slow” manner.\nTheorem 4.8 (Theorem 1.5).Suppose that Ais a negative definite 2×2matrix. Then\nthere is no solution w∈C∞(R2\n+)of the equation\n(4.20)\n\n−∆w=Fe2winR2\n+,\n∂w\n∂ν=ewon∂R2\n+,BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 27\nwithw≤C, whereF(x) := 1+(Ax,x), such that Fe2w∈L1(R2\n+),\nV0:=ˆ\nR2\n+e2wdx<∞, H0:=ˆ\n∂R2\n+ewdx<∞,andK0:=ˆ\nR2\n+Fe2wdx∈R.\nProof.By contradiction, suppose wis a solution to ( 4.20). Motivated by the proof of [ 32,\nLemma 2.1] and [ 43, Theorem 4.5], consider the function\n˜w(x) =−1\n2πˆ\nR2\n+(log|x−y|+log|x−y|−2log|y|)F(y)dy\n−1\n2πˆ\n∂R2\n+(log|x−y|+log|x−y|−2log|y|)ewdy,\nwherexis the reflection point of xabout the boundary ∂R2\n+, that is, if x= (s,t), then\nx= (s,−t). As in [ 33, Proposition 3.2], it holds\n(4.21) lim\n|x|→∞˜w(x)\nlog|x|=−1\nπˆ\nR2\n+F−1\nπˆ\n∂R2\n+ew=−1\nπ(K0+H0) =:−d.\nIt is easy to check that ˜ wis a solution of\n\n−∆˜w=Fe2winR2\n+,\n∂˜w\n∂ν=ewon∂R2\n+.\nIn this way, the function v:=w−˜wis harmonic in R2\n+with∂v/∂ν= 0 in∂R2\n+. We extend\nvtoR2by even reflection. Since |v| ≤C+Clog(2 +|x|) for some constant C(see [33,\nProposition 3.1]), we have that vis constant. Hence w= ˜w+Cfor some constant C.\nFrom (4.21), we have e˜w≥c|x|−d, for|x|>1 and some constant c >0. SinceAis a\nnegative definite matrix, we obtainˆ\nR2\n+\\B+\n1|x|2−2ddx≤cˆ\nR2\n+|x|2e2˜wdx+c≤cˆ\nR2\n+|(Ax,x)|e2wdx+c=c(V0−K0)+c.\nThusd>2 and consequently K0+H0>2π. Since∇˜w=∇w, we have\n/a\\}b∇acketle{tx,∇˜w/a\\}b∇acket∇i}ht=−1\n2πˆ\nR2\n+/parenleftbigg/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\nF(y)e2w(y)dy\n−1\n2πˆ\n∂R2\n+/parenleftbigg/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\new(y)dy.\nThus ˆ\nB+\nR(0)F(x)e2w(x)/a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}htdx+ˆ\nΓR/a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}htewdσ\n=ˆ\nB+\nR(0)F(x)e2w(x)/parenleftBigg\n−1\n2πˆ\nR2\n+/parenleftbigg/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\nF(y)e2w(y)dy\n−1\n2πˆ\n∂R2\n+/parenleftbigg/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\new(y)dy/parenrightBigg\ndx28 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\n+ˆ\nΓRew(x)/parenleftBigg\n−1\n2πˆ\nR2\n+/parenleftbigg/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\nF(y)e2w(y)dy\n−1\n2πˆ\n∂R2\n+/parenleftbigg/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\new(y)dy/parenrightBigg\ndx.\nFor the right hand side of the above identity, expressing xasx=1\n2((x+y)+(x−y))\nandx=1\n2((x+y)+(x−y)), and taking into account that |x−y|=|x−y|, we get\nRHS=ˆ\nB+\nR(0)F(x)e2w(x)/parenleftBigg\n−1\n2πˆ\nR2\n+F(y)e2w(y)dy−1\n2πˆ\n∂R2\n+ew(y)dy/parenrightBigg\ndx\n+1\n2ˆ\nB+\nR(0)F(x)e2w(x)/parenleftBigg\n−1\n2πˆ\nR2\n+/parenleftbigg/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\nF(y)e2w(y)dy\n−1\n2πˆ\n∂R2\n+/parenleftbigg/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\new(y)dy/parenrightBigg\ndx\n+ˆ\nΓRew(x)/parenleftBigg\n−1\n2πˆ\nR2\n+F(y)e2w(y)dy−1\n2πˆ\n∂R2\n+ew(y)dy/parenrightBigg\ndx\n+1\n2ˆ\nΓRew(x)/parenleftBigg\n−1\n2πˆ\nR2\n+/parenleftbigg/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\nF(y)e2w(y)dy\n−1\n2πˆ\n∂R2\n+/parenleftbigg/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2/parenrightbigg\new(y)dy/parenrightBigg\ndx\n=−1\n2d/parenleftBiggˆ\nB+\nR(0)F(x)e2w(x)dx+ˆ\nΓRewdσ/parenrightBigg\n−1\n4π(I1+I2+I3+I4).\nTo simplify, consider\nG(x,y) :=/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2+/a\\}b∇acketle{tx+y,x−y/a\\}b∇acket∇i}ht\n|x−y|2.\nUsing Fubini’s Theorem and changing variables xandy, sinceG(x,y) =−G(y,x) and\nG(−x,−y) =G(x,y),we get\nI1=ˆ\nB+\nR(0)ˆ\nR2\n+G(x,y)F(y)e2w(y)F(x)e2w(x)dydx\n=ˆ\nB+\nR(0)ˆ\nR2\n+\\B+\nR(0)G(x,y)F(y)e2w(y)F(x)e2w(x)dydx\n=ˆ\nB+\nR/2(0)ˆ\nR2\n+\\B+\nR(0)G(x,y)F(y)e2w(y)F(x)e2w(x)dydx\n+ˆ\nB+\nR(0)\\B+\nR/2(0)ˆ\nR2\n+\\B+\n2R(0)G(x,y)F(y)e2w(y)F(x)e2w(x)dydxBLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 29\n+ˆ\nB+\nR(0)\\B+\nR/2(0)ˆ\nB+\n2R(0)\\B+\nR(0)G(x,y)F(y)e2w(y)F(x)e2w(x)dydx=I11+I12+I13.\nIf either |x| ≤R/2 and|y| ≥RorR/2≤ |x| ≤Rand|y| ≥2R, it holds |G(x,y)| ≤6.\nThis implies that\n|I11| ≤6ˆ\nB+\nR/2(0)|F(y)|e2w(y)dyˆ\nR2\n+\\B+\nR(0)|F(x)|e2w(x)dx\nand\n|I12| ≤6ˆ\nB+\nR(0)\\B+\nR/2(0)|F(y)|e2w(y)dyˆ\nR2\n+\\B+\n2R(0)|F(x)|e2w(x)dx.\nSinceFe2w∈L1(R2\n+), thenI11andI12go to zero as R→ ∞.\nForI13, from (4.21) we obtain e2w(x)≤ |x|−2µfor anyµ<dwhen|x| ≥1 is sufficiently\nlarge. Also for |x| ≥1 we have |F(x)| ≤C|x|2. This implies that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nB+\n2R(0)\\B+\nR(0)G(x,y)F(y)e2w(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nB+\n2R(0)\\B+\nR(0)/parenleftbigg|x+y|\n|x−y|+|x+y|\n|x−y|/parenrightbigg\n|F(y)|e2w(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤CR3−2µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nB+\n2R(0)\\B+\nR(0)/parenleftbigg1\n|x−y|+1\n|x−y|/parenrightbigg\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CR3−2µˆ\nB3R(0)1\n|z|dz≤CR3−2µ.\nSinced>2 we can choose µ∈(3/2,2), which implies that I13goes to zero when R→ ∞.\nTherefore,I1goes to zero when R→ ∞.\nSimilarly, we obtain\nI4=ˆ\nΓRˆ\n∂R2\n+G(x,y)ew(y)ew(x)dydx=ˆ\nΓRˆ\n∂R2\n+\\ΓRG(x,y)ew(y)ew(x)dydx\n=ˆ\nΓR/2ˆ\n∂R2\n+\\ΓRG(x,y)ew(y)ew(x)dydx+ˆ\nΓR\\ΓR/2ˆ\n∂R2\n+\\Γ2RG(x,y)ew(y)ew(x)dydx\n+ˆ\nΓR\\ΓR/2ˆ\nΓ2R\\ΓRG(x,y)ew(y)ew(x)dydx\n=I41+I42+I43.\nThe estimates for I41andI42are analogous to the one to I11andI12, but this time we\nuse thatew∈L1(∂R2\n+). Note that\nI43=ˆR\nR/2ˆ2R\nRG(x,y)/parenleftbig\new(y)ew(x)+ew(−y)ew(−x)/parenrightbig\ndydx\n+ˆR\nR/2ˆ2R\nRG(−x,y)/parenleftbig\new(−y)ew(x)+ew(y)ew(−x)/parenrightbig\ndydx.\nAs before,e2w(x)≤ |x|−2µfor anyµ<dwhen|x| ≥1 is large enough. This implies that\n|I43| ≤CR1−2µˆR\nR/2ˆ2R\nR/parenleftbigg1\ny−x+1\nx+y/parenrightbigg\ndydx≤CR2−2µlnR.30 R. CAJU, T. CRUZ, AND A. SILVA SANTOS\nForµ= 3/2, this implies that I43goes to zero when R→ ∞, and therefore I4goes to zero.\nAgain, using Fubini’s Theorem and changing variables xandy, we get\nI2=ˆ\nB+\nR(0)ˆ\n∂R2\n+G(x,y)ew(y)F(x)e2w(x)dσdx=−ˆ\n∂R2\n+ˆ\nB+\nR(0)G(x,y)ew(x)F(y)e2w(y)dydσ\n=−ˆ\nΓRˆ\nB+\nR(0)G(x,y)ew(x)F(y)e2w(y)dydσ−ˆ\n∂R2\n+\\ΓRˆ\nB+\nR(0)G(x,y)ew(x)F(y)e2w(y)dydσ.\nI3=ˆ\nΓRˆ\nR2\n+G(x,y)F(y)e2w(y)ew(x)dydσ\n=ˆ\nΓRˆ\nB+\nR(0)G(x,y)F(y)e2w(y)ew(x)dydσ+ˆ\nΓRˆ\nR2\n+\\B+\nR(0)G(x,y)F(y)e2w(y)ew(x)dydσ.\nThus\nI2+I3=ˆ\nΓRˆ\nR2\n+\\B+\nR(0)G(x,y)F(y)e2w(y)ew(x)dydσ\n−ˆ\n∂R2\n+\\ΓRˆ\nB+\nR(0)G(x,y)ew(x)F(y)e2w(y)dydσ=II1−II2.\nUsing the same idea used to estimate the integrals I1andI4, we obtain that the integrals\nII1andII2go to zero as R→ ∞. Therefore, we conclude that\n(4.22) RHS→ −1\n2d/parenleftBiggˆ\nR2\n+F(x)e2w(x)dx+ˆ\nR2\n+ewdσ/parenrightBigg\n=−π\n2d2.\nIfx= (s,t), then we have /a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}ht=s∂w/∂sforx∈ΓR. Thus, using ( 4.20) we obtain\nˆ\nΓR/a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}ht∂w\n∂νdy=ˆ\nΓRs∂w\n∂sewds=ˆ\nΓR∂\n∂s(sew)ds−ˆ\nΓRewds.\nThus\nLHS=ˆ\nB+\nR(0)F(x)e2w(x)/a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}htdx+ˆ\nΓR/a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}htewdσ\n=1\n2ˆ\nB+\nR(0)F(x)/a\\}b∇acketle{tx,∇e2w(x)/a\\}b∇acket∇i}htdx+ˆ\nΓR/a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}htewdσ\n=−1\n2ˆ\nB+\nR(0)e2w(x)div(F(x)x)dx+1\n2Rˆ\n∂+B+\nR(0)F(x)e2w(x)dσ+ˆ\nΓR/a\\}b∇acketle{tx,∇w/a\\}b∇acket∇i}htewdσ\n=−1\n2ˆ\nB+\nR(0)(/a\\}b∇acketle{tx,∇F(x)/a\\}b∇acket∇i}ht+2F(x))e2w(x)dx+1\n2Rˆ\n∂+B+\nR(0)F(x)e2w(x)dσ\n+ˆ\nΓR∂\n∂s(sew)ds−ˆ\nΓRewds.BLOW-UP ANALYSIS OF LARGE CONFORMAL METRICS 31\nUsing that Fe2w∈L1(R2\n+) andew∈L1(∂R2\n+), we conclude the existence of a sequence\nRi→ ∞such thatRiˆ\nΓ2,RiFe2w→0 andRiew(±Ri,0)→0. 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Caju) Department of Mathematical Engineering, University of Chi le\nBeauchef 851, Edificio Norte, Santiago, Chile\nEmail address :rcaju@dim.uchile.cl\n(T. Cruz) Institute of Mathematics, Federal University of Alagoas\n57072-970, Macei ´o-AL, Brazil\nEmail address :cicero.cruz@im.ufal.br\n(A. Silva Santos) Department of Mathematics, Federal University of Sergipe\n49107-230, Sao Crist ´ov˜ao-SE, Brazil\nEmail address :almir@mat.ufs.br"    },    {        "title": "2402.02476v1.Constraints_on_Triton_atmospheric_evolution_from_occultations__1989_2022.pdf",        "content": "Astronomy &Astrophysics manuscript no. Triton_atmosphere_1989_2022 ©ESO 2024\nFebruary 6, 2024\nConstraints on Triton atmospheric evolution from occultations:\n1989-2022\nB. Sicardy1, A. Tej2, A. R. Gomes-Júnior3,4, F. D. Romanov5, T. Bertrand1, N. M. Ashok6, E. Lellouch1, B.\nE. Morgado7, M. Assafin7,4, J. Desmars8,9, J. I. B. Camargo10,4, Y . Kilic11, J. L. Ortiz12, R.\nVieira-Martins10,4,7, F. Braga-Ribas13,4, J. P. Ninan14, B. C. Bhatt15, S. Pramod Kumar15, V . Swain16, S.\nSharma17, A. Saha2, D. K. Ojha14, G. Pawar18,19, S. Deshmukh18, A. Deshpande18, S. Ganesh6, J. K. Jain6, S. K.\nMathew20, H. Kumar21,16, V . Bhalerao16, G. C. Anupama15, S. Barway15, A. Brandeker22, H. G. Florén22,\nG. Olofsson22, G. Bruno23, Y . M. Mao24,25, R. H. Ye24,25, Q. Y . Zou24,25, Y . K. Sun24,25, Y . Y . Shen26,25, J.\nY . Zhao27, D. N. Grishin28, L. V . Romanova29, F. Marchis30,31, K. Fukui32, R. Kukita32, G.\nBenedetti-Rossi33,4,1, P. Santos-Sanz12, N. Dhyani18, A. Gokhale18, and A. Kate18\n(Affiliations can be found after the references)\nReceived mm:dd, yyyy; accepted mm:dd, yyyy\nABSTRACT\nContext. Around the year 2000, Triton’s south pole experienced an extreme summer solstice that occurs every ∼650 years, when the subsolar\nlatitude reached about 50◦S. Bracketing this epoch, a few occultations probed Triton’s atmosphere in 1989, 1995, 1997, 2008 and 2017. A recent\nground-based stellar occultation observed on 6 October 2022 provides a new measurement of Triton’s atmospheric pressure which is presented\nhere.\nAims. The goal is to constrain the V olatile Transport Models (VTMs) of Triton’s atmosphere that is basically in vapor pressure equilibrium with\nthe nitrogen ice at its surface.\nMethods. Fits to the occultation light curves yield Triton’s atmospheric pressure at the reference radius 1400 km, from which the surface pressure\nis induced.\nResults. The fits provide a pressure p1400=1.211±0.039µbar at radius1400 km (47-km altitude), from which a surface pressure of psurf=\n14.54±0.47µbar is induced (1 σerror bars). To within error bars, this is identical to the pressure derived from the previous occultation of 5\nOctober 2017, p1400=1.18±0.03µbar and psurf=14.1±0.4µbar, respectively. Based on recent models of Triton’s volatile cycles, the overall\nevolution over the last 30 years of the surface pressure is consistent with N 2condensation taking place in the northern hemisphere. However,\nmodels typically predict a steady decrease in surface pressure for the period 2005-2060, which is not confirmed by this observation. Complex\nsurface-atmosphere interactions, such as ice albedo runaway and formation of local N 2frosts in the equatorial regions of Triton could explain the\nrelatively constant pressure between 2017 and 2022.\nKey words. Planets and satellites: atmospheres – Planets and satellites: individual: Triton – Techniques: stellar occultations\n1. Introduction\nTriton is the largest of Neptune’s satellites. With the Satur-\nnian satellite Titan, it is the only satellite known to possess\na global atmosphere. Triton’s atmosphere was revealed during\nthe NASA V oyager 2 (V2) flyby of Neptune’s main satellite in\nAugust 1989. The Radio Science Subsystem (RSS, Tyler et al.\n1989; Gurrola 1995) occultation provided a surface pressure of\npsurf=14±1µbar (1σ-level). Combined with other V2 data\nand subsequent ground-based observations, it was seen that Tri-\nton’s tenuous atmosphere is mainly composed of nitrogen, N 2,\nin vapor pressure equilibrium with the icy surface.\nSince 1989, a handful of Earth-based observations of stel-\nlar occultations monitored Triton’s atmosphere. The 15 August\n1995 (Olkin et al. 1997), 18 July 1997 (Elliot et al. 2000 and\nMarques Oliveira et al. 2022, MO22 hereafter), and 4 November\n1997 (Elliot et al. 2003) events indicated a significant increase of\npressure relative to the RSS measurement. No further constraints\non Triton’s atmospheric pressure could be achieved from the 21\nMay 2008 occultation, which had a grazing geometry (MO22).The 5 October 2017 ground-based occultation campaign pro-\nvided the first dense coverage of Triton’s atmosphere, with 90\noccultation chords scanning both hemispheres of the satellite\n(MO22). Combined with the good signal-to-noise ratio of some\nof the light curves, this event yielded strong constraints on the\nthermal, density and pressure profiles of this atmosphere be-\ntween the altitude levels ∼8 km and∼190 km (from∼9µbar\nand a few nanobars, respectively). In particular, the atmospheric\npressure was found to be back to the RSS value of 1989. More-\nover, the detection and structure of a central flash revealed an\nessentially spherical atmosphere with an apparent oblateness of\nless than 0.0011 at the 8-km altitude level.\nTriton has recently experienced a rare “extreme southern\nsolstice” that occurs every ∼650 years; see details in Bertrand\net al. (2022). The subsolar latitude on the satellite reached about\n50◦S in 2000. Seasonal variations in surface pressure can then\nconstrain V olatile Transport Models (VTMs) that account for\nvolatile transport induced by insolation changes. These models\nassume that Triton’s atmosphere has a negligible radiative ther-\nmal influence on the energy balance of its surface. They calcu-\nlate the local insolation at the surface and the thermal infrared\nArticle number, page 1 of 8arXiv:2402.02476v1  [astro-ph.EP]  4 Feb 2024A&A proofs: manuscript no. Triton_atmosphere_1989_2022\ncooling, the heat storage in regions covered by N 2ice and con-\nduction in the subsurface, and, in the presence of N 2ice, the\ncondensation-sublimation rates necessary to force the surface\ntemperature to remain at the nitrogen frost point, which depends\nin turn on surface pressure. These processes are su fficient to esti-\nmate to first order the temporal evolution of the surface temper-\nature, pressure and N 2transport.\nThe first numerical Triton’s VTMs emerged in the late 1980s,\nmotivated by the V2 Triton flyby (Spencer 1990, Hansen & Paige\n1992, Spencer & Moore 1992, Brown & Kirk 1994). Triton’s at-\nmosphere has similarities with that of Pluto’s (a N 2atmosphere\ncontrolled by vapor pressure equilibrium with the surface ices).\nHence, crucial insights were gained after the Pluto New Hori-\nzons flyby in 2015 which allowed the VTMs for Triton to be\nupdated in light of new observable constraints (MO22, Bertrand\net al. 2022).\nAfter the pressure increase noted in 1995 and 1997, the sur-\nface pressure in 2017 was found to be close to the V2 value\n(MO22). To first order, the volatile transport models have been\nable to reproduce this trend for a wide range of model parame-\nters. It stems from nitrogen ice sublimation peaking in the south-\nern hemisphere in ∼2000-2005 as the southern N 2ice cap is un-\nder maximum insolation (summer solstice). However, Bertrand\net al. (2022)’s models do not suggest a strong surge in surface\npressure (see their Figs. 9, 10 and 21), as claimed from stellar\noccultations for the period 1995–1997 (e.g. Elliot et al. 2000),\nbut instead predict a moderate increase, with a peak around\n15–19µbar for the surface pressure. However, other processes\nnot taken into account in these models (e.g. ice albedo feedback)\nmay have increased the peak amplitude during this period.\nHere, we present results obtained from a new stellar occul-\ntation event observed on 6 October 2022 from ground-based\n(China and India) and space (CHEOPS) facilities. These new\nresults help constrain the evolution of Triton’s atmosphere over\nthe period 1989-2022. This is a timely event, considering the rar-\nity of Triton occultations, owing to the depleted stellar fields that\nNeptune’s system is currently crossing as seen from Earth.\n2. Observations\nThe Triton occultation campaign of 6 October 2022 was orga-\nnized under the auspices of the Lucky Star project1. Comprehen-\nsive details regarding the event can be found in a dedicated web\npage2. The compilation and management of the observational\ndata are facilitated by Lucky Star’s Occultation Portal website3\n(Kilic et al. 2022). The Gaia DR3 position at the epoch of oc-\ncultation and the Triton’s ephemeris from previous occultation\nevents were used for the final prediction, see Fig. 1.\nThe occultation was successfully observed from space by\nthe CHaracterising ExOPlanet Satellite (CHEOPS4). Two suc-\ncessful observations were obtained in India at Mt. Saraswati\nin the Himalayan region, while one light curve was success-\nfully recorded at Yanqi Lake, China. Attempts from Devasthal,\nUttarakhand (India), Mt. Abu and Dhanari, Rajasthan (India),\nUdaipur (India), Nakhodka (Russia) and three sites in Japan\nwere clouded out, see details in Appendix A and Table A.1.\n1https: //lesia.obspm.fr /lucky-star /\n2https: //lesia.obspm.fr /lucky-star /occ.php?p =109326\n3https: //occultation.tug.tubitak.gov.tr\n4https: //www.esa.int /Science_Exploration /Space_Science /Cheops\nEN300 kmequatorprimemeridianSCHEOPSYanqiLakeMt. SaraswatiFig. 1. Upper panel: The dark blue lines (continued in red outside the\nEarth) delimit the predicted path of Triton’s shadow on Earth on 6 Octo-\nber 2022. The shadow motion is from right to left. The shadow centers\n(black dots) are spaced by one minute, the larger black dot marking\nthe geocentric closest approach near 14:49:46 UT (Table 1). Colors in-\ndicates stations where the occultation was successfully detected. The\nblue arc shows the motion of the CHEOPS spacecraft during the event,\nwhile the red and green dots are for Mt. Saraswati and Yanqi lake, re-\nspectively. The white dots are for the stations that were clouded out, see\nTable A.1. Lower panel: The reconstructed geometry of the occultation\nas seen in the sky plane. The J2000 celestial north (N) and east (E) direc-\ntions and the scale are displayed in the upper right corner. The grey ar-\nrow near the equator shows the direction of rotation of the satellite. The\n(Neptune-facing) prime meridian is drawn as a thicker line compared to\nthe other meridians, and the label S marks the south pole. The dotted cir-\ncle indicates the layer in Triton’s atmosphere that causes the half-light\nlevel (∼90-km altitude). The colored lines are the trajectories of the star\nrelative to Triton (or ‘occultation chords’) as observed from various sta-\ntions (see labels), with the black arrow indicating the direction of mo-\ntion. The open circle is Triton’s predicted center while the cross marks\nthe actual center derived from the atmospheric fit. The o ffset between\nthe two mainly stems from a correction on Triton’s ephemeris.\n3. Data Analysis\nOur analysis used tools developed in the “Platform for Reduc-\ntion of Astronomical Images Automatically\" (PRAIA, Assafin\n2023a,b)5and the packages of the “Stellar Occultation Reduc-\ntion and Analysis\" (SORA, Gomes-Júnior et al. 2022)6\nThe resulting occultation light curves have been fitted using\nthe ray-tracing code initially developed for Pluto’s atmosphere\nby Dias-Oliveira et al. (2015), see also Meza et al. (2019) and\nSicardy et al. (2021). This code has been adapted to Triton’s at-\nmosphere, using the parameters of MO22 (their Table 2) and in\nTable 1 of this paper. Triton’s atmosphere is assumed to be trans-\nparent and composed of pure N 2, noting that the CH 4abundance\nof 10−4is negligible for our purposes. Further, it is assumed to be\n5https: //ov.ufrj.br /en/PRAIA /\n6https: //sora.readthedocs.io /\nArticle number, page 2 of 8B. Sicardy et al.: Constraints on Triton atmospheric evolution from occultations: 1989-2022\nTable 1. Parameters of occulted star and Triton, and results of the atmospheric fits. Error bars are given at 1 σlevel.\nOcculted star\nIdentification (Gaia DR3) 2639239368824994944\nICRS position at occultation epoch α=23h36m52.45142s±0.7 mas,δ=−03◦50′09.7952′′±0.9 mas\nTriton’s parameters\nMass, radius1GM T=1.4279×1012m3sec−2,RT=1353 km\nGeocentric distance 4 .33409×109km\nPole position2(ICRS) αp=20h 13m 52.4s, δp=20◦32’ 38.2\"\nSub-solar latitude, sub-observer latitude, longitude 34.2◦S, 34.6◦S, 115.0◦E\nNorth pole position angle3298.2◦\nReconstructed geometry\nHalf-light layer in Triton’s atmosphere, in shadow plane 1442 km, 1420 km\nGeocentric closest approach distance and time for shadow center ρC/A,G=4067±3 km, tC/A,G=14:39:46.26±0.04 s UT\nAtmospheric results\nThis work\nPressure at reference radius 1400 km, at surface4p1400=1.211±0.039µbar,psurf=14.54±0.47µbar\nOther works\n25 August 1989 (RSS occultation)5psurf=14.1±1µbar\n25 August 1989 (RSS occultation)6p1400=1.0±0.2µbar\n14 August 1995 (ground-based occultation)7p1400=1.4±0.1µbar\n18 July 1997 (ground-based occultation)8p1400=2.23±0.28µbar\n18 July 1997 (ground-based occultation)6p1400=1.9+0.45\n−0.30µbar\n4 November 1997 (HST occultation)9p1400=1.759±0.016µbar\n21 May 2008 (ground-based occultation)6p1400=1.15+1.03\n−0.37µbar\n5 October 2017 (ground-based occultation)6p1400=1.18±0.03µbar\nNotes.(1)McKinnon et al. (1995), where Gis the constant of gravitation.(2)On 6 October 2022, using Archinal et al. (2018)(3)Position angle of\nTriton’s north pole projected in the sky plane. Counted positively from celestial north to celestial east.(4)Assuming psurf/p1400=12.01, see text.\n(5)Gurrola (1995).(6)MO22.(7)Olkin et al. (1997).(8)Elliot et al. (2000).(9)HST: Hubble Space Telescope, Elliot et al. (2003).\nspherical, as supported by the shape of the central flash observed\nduring the 5 October 2017 event (MO22).\nTriton’s atmospheric thermal profile T(r) – where ris the dis-\ntance to Triton’s center – is the same as the one used by MO22,\nsee their Figs. 10, 11 and B.1 and their Table B.1. It starts from\nthe surface with a strong positive thermal gradient of 5 K km−1.\nThis gradient decreases rapidly and the temperature reaches a\nlocal maximum value of about 50 K at r=1363 km (10 km al-\ntitude). This is followed by a mesosphere with a mild negative\ngradient of about -0.2 K km−1centered around r=1375 km (23\nkm altitude). This mesosphere finally connects itself with a ther-\nmosphere that has a positive gradient of about 0.1 K km−1above\nthe 50-km altitude level.\nAχ2-minimization procedure is applied by simultaneously\nfitting all the four light curves. The M=7 free parameters of the\nfit are the pressure p1400at the reference radius r=1400 km, the\noffset (fc,gc) to apply to Triton’s ephemeris along the celestial\neast and north directions, respectively, and the contributions of\nTriton’s flux to the total Triton +star flux in the four observed\nlight curves. These contributions are di fferent from station to sta-\ntion because the instruments used di fferent filters (Table A.1), in-\nducing variations in the magnitudes of both Triton and the star.\nThe o ffset ( fc,gc) is obtained by determining the shift in time\n∆tto apply to all the light curves and the cross-track o ffset to\nTriton’s ephemeris, ∆ρ, that minimize χ2.\nThe fit uses a total N=2398 data points, providing a χ2\nvalue per degree of freedom χ2\ndof=χ2\nmin/(N−M), whereχ2\nminis\nthe minimum value of χ2obtained in the fitting procedure. The\nbest fit to the data (Fig. 2) has a value χ2\ndof=0.84, indicating\na satisfactory modeling of the data. The figure 3 shows the map\nofχ2vs. the pressure p1400at radius 1400 km and the cross-track o ffset∆ρ. As summarized in Table 1, the fit provides a\nbest-fitting pressure of p1400=1.211±0.039µbar (1σlevel).\nOur adopted temperature profile implies that psurf/p1400=12.01,\nyielding psurf=14.54±0.47µbar.\n4. Pressure Evolution\nFigure 3 shows the measured values of Triton’s atmospheric\npressure over the period 1989-2022, together with various VTMs\noutputs of Bertrand et al. (2022). We consider here the simula-\ntions of these authors performed with unlimited and fixed N 2ice\nreservoirs (i.e. no short-term frosts are involved) in the southern\nand northern hemispheres (see their Figs. 9-10), with some cases\nshown as dotted, dashed-dotted and dashed lines in Fig. 3.\nThese simulations inevitably predict a steady decrease in\npressure between 2005 and 2022, with a pressure in 2022 that\nis at least 5% lower than that in 2017 (and 5% in the case of the\ndash-dotted line in Fig. 3), followed by a steady decrease that is\npredicted to last for many decades. This is because, as the subso-\nlar point moves from the southern latitudes toward the equator in\n2022, nitrogen sublimation becomes less intense over the south-\nern reservoir of N 2ice, while condensation in the northern hemi-\nsphere and in the equatorial regions dominate sublimation at the\nglobal scale. This trend is also obtained in their simulations per-\nformed with limited N 2ice reservoirs and with seasonal frosts,\nthat best match the observational constraints (see their Fig. 21),\nincluding among others the surface pressure in 1989 and 2017\nand the latitudinal extent of the ice as suggested from V2 im-\nages.\nArticle number, page 3 of 8A&A proofs: manuscript no. Triton_atmosphere_1989_2022\nCHEOPSYanqiLake\nMt SaraswatiHCTMt SaraswatiGrowth\nTime (UT)Flux (Triton + star)0011\np1400(µbar)∆ρ(km)1σ3σ\nFig. 2. Upper panel: the best fit to the data is shown as blue lines, using\na Triton surface pressure of psurf=14.54µbar and other parameters pro-\nvided in Table 1. The residuals (observations minus model) are shown\nin green. The lower and upper horizontal dotted lines mark the zero flux\nand the average flux of Triton plus the star, respectively. Each light curve\nis normalized to the total flux of Triton +the star and is plotted over\na time interval of 6 minutes, the red tick mark indicating 14:41:40 UT.\nThe HCT light curve has been binned over five data points (1.05-second\ntime resolution) for a better comparison with the other light curves. We\nnote that a faint central flash is present in the Yanqi Lake light curve.\nThe 1σerror bars are obtained from the PRAIA software that accounts\nfor the instrumental and photon noises, knowing that the error bars for\nthe CHEOPS data points are dominated by the photon noise (see Ap-\npendix A). They are not visible here as they are smaller than the plotted\ndots. Lower panel: Theχ2map of the simultaneous fits, with the 1 σand\n3σlevel curves shown in green.None of the simulations mentioned above can explain both\nthe pressure surge observed in the years 1990’s and the return to\nthe pressure of 1989 in 2017 and 2022 (Fig. 3).\nSome simulations of Bertrand et al. (2022) do obtain a rel-\natively constant surface pressure between 1989, 2017 and 2022\nthat is consistent with our present result (Fig. 3, solid line). In\nthese simulations, the N 2condensation is more intense in the\nnorthern hemisphere during this period because the seasonal\nnorthern cap is strongly extended equatorward, in the form of\ntens-of-cm-thick frosts, due to a cold bedrock (typically, the\nbedrock surface bolometric albedo in these simulations is ≥0.9).\nThis balances the sublimation in the southern hemisphere and\ncause the pressure to remain relatively constant between 1989\nand 2022. However, these simulations (1) include the presence of\nextended N 2ice frosts to low northern latitudes in 1989, which\nseems inconsistent with the V2 images, unless the blue fringe\nobserved in the equatorial regions correspond to a cm-thick N 2\nfrost (see discussions about the blue fringe in Bertrand et al.\n2022, their section 8.4), and in fact (2) they do not include a\npeak in pressure at all during the 1995-1997 period.\nNevertheless, this surge should be considered with caution.\nThe black points in Fig. 3, taken from MO22, are all obtained\nusing the same temperature profile T(r) and the same ray-\ntracing code. In contrast, the blue points have been obtained\nwith methods varying from each other and from our approach.\nFor instance, using the same data set stemming from the 18\nJuly 1997 occultation, Elliot et al. (2000) and MO22 derive\np1400=2.23±0.28µbar and p1400=1.90+0.45\n−0.30µbar, respec-\ntively. Although these two values agree to within their respective\nerror bars, they show that a consistent comparison between re-\nsults obtained by various teams remains problematic. In that con-\ntext, if we consider only the values p1400=1.0±0.2µbar and\np1400=1.90+0.45\n−0.30µbar derived by MO22 for the 25 August 1989\nand 18 July 1997 measurements, respectively, then the surge of\npressure between 1989 and 1997 reaches a moderate 2.5 σlevel.\nThis said, the formation of local N 2frosts in the equatorial\nregion during the period 1980-2020 – which is very sensitive to\nsurface properties or albedo feedback – may not be simulated\nin the VTM with su fficient details and could alter the pressure\ncycle. Another possibility is that albedos and ice composition\nfeedback may have a significant role on Triton as geysers or haze\nparticles could deposit dark material (resp. bright ice grain) on\ntop of the ice, and further darken (resp. brighten) the ice, and\nthus impact the N 2sublimation-condensation rates. These pro-\ncesses, not taken into account in the VTM, have been shown to\nbe efficient runaway forcing mechanisms on Pluto (Earle et al.\n2018; Bertrand et al. 2020). They could be responsible for sig-\nnificant sub-seasonal atmospheric pressure variations occurring\nin Pluto’s atmosphere on timescales ranging from a few to tens\nof Earth years, as suggested by stellar occultations, see Arimatsu\net al. (2020) and (Yuan et al. 2023). The latter authors note that\nshort-term changes in Pluto’s surface ices have been reported\nGrundy et al. (2014); Lellouch et al. (2022); Holler et al. (2022).\nThis could reconcile with the observations presented here, in par-\nticular with the 1995-1997 surge of pressure of Triton’s atmo-\nsphere. However, all the points evoked above remain speculative\nand should be explored in details with the models.\n5. Conclusions\nThe main constraint obtained here is that the Triton’s atmo-\nspheric pressure is essentially the same in 1989, 2017 and\n2022, whatever occurred in between. More specifically, the pres-\nsure changed very little between 2017 and 2022 , with a non-\nArticle number, page 4 of 8B. Sicardy et al.: Constraints on Triton atmospheric evolution from occultations: 1989-2022\nEarth yearsPressure at 1400 km (µbar)   \nSurface pressure ( µbar)   1020300E00E03O9712G95\nFig. 3. Evolution of Triton’s atmospheric pressure from occultation measurements taken between 1989 and 2022; see dates and values in Table 1.\nThe red point is the result of the 6 October 2022 occultation (this work). It has been obtained using the same method as adopted by MO22\n(black points). The green point (G95) is from the V2 RSS occultation of 25 August 1989 (Gurrola 1995). The blue points are respectively: O97,\nground-based stellar occultation, 14 August 1995 (Olkin et al. 1997); E03, ground-based stellar occultation, 18 July 1997 (Elliot et al. 2000); E00,\nHST-based stellar occultation, 4 November 1997 (Elliot et al. 2003). It should be noted that MO22 analyzed the RSS data of August 1989 and the\noccultation data of July 1997 independently of Gurrola (1995) and (Elliot et al. 2000), respectively. The vertical axes show the pressures at the\nreference radius 1400 km (left) and at the surface (right), assuming psurf/p1400=12.01, see text. The lines are examples of VTM simulations by\nBertrand et al. (2022). Dashed-dotted line: southern cap within 90◦S-30◦S, northern cap within 45◦N-90◦N; Dotted line: southern cap within 90◦S-\n30◦S, northern cap within 60◦N-90◦N; Dashed line: southern cap within 90◦S-30◦S, no northern cap (volatile-free northern hemisphere); solid\nline: a simulation where the N 2ice freely evolves over millions of years so that permanent and seasonal polar caps as well as local frosts deposits\ncan form self-consistently. In this simulation, formation of thin seasonal N 2frosts is predicted in the current southern summer in the equatorial\nregions. None of the simulations satisfactorily fits the data. The dashed line accounts for the pressure increase of 1995-1997 but overestimates the\npressures measured in 2017 and 2022. The dotted line marginally explains the increase of 1995-1997 and satisfactorily fit the 2022 measurement,\nbut fails to explain the 2017 point. The dashed-dotted line does not fit the increase of 1995-1997, while going through the 2017 point, but fails\nto explain the 2022 point. The solid line accounts for a basically constant pressure between 1989, 2017 and 2022, as well as the slight increases\nbetween 2017 and 2022. However, it is only marginally consistent with the 1995 result, and does not account for the pressure surge of 1997.\nsignificant increase of 0 .031±0.049µbar (resp. 0.38±0.59µbar)\nbetween the two dates at 1400 km (resp. the surface).\nIt remains di fficult to clearly assess the seasonal trend of the\nN2cycle on Triton, because the simulations from Bertrand et al.\n(2022) do not reconcile all the available observations. i.e all pres-\nsure measurements over the 1989-2022 period and the visual ap-\npearance of Triton’s surface in V oyager 2’s images. Another is-\nsue is that our 2022 measurement is close in time to the 2017\nmeasurement, compared to a seasonal timescale on Triton ( ∼40\nyears). Complex processes, not taken into account in the VTMs,\ncould occur and perturb the N 2cycle over a short timescale of 5\nyears, thus temporarily masking the seasonal trend, see the dis-\ncussion at the end of the previous Section.\nMeanwhile, the 2022 observation confirms the fact that the\nsouthern cap has not retreated below the 30◦S latitude level since\n1989 from its∼15◦S extent at that time. Had it been the case, the\nsurface pressure would have dramatically collapsed since 1989,\nand would be inconsistent with the 2017 and 2022 occultation\nresults.\nAcknowledgements. We thank N. Billot, M. Beck, M. Günther, K. Isaak, I.\nPagano and the CHEOPS Project Science O ffice (PSO) for help in planning\nobservations based on the position of CHEOPS in time. GBr acknowledges\nsupport from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. We thank the\nstaffof IAO, Hanle, and CREST, Hosakote that made these observations pos-sible. The facilities at IAO and CREST are operated by the Indian Institute\nof Astrophysics. The GROWTH India Telescope (GIT) is a 70-cm telescope\nwith a 0.7-degree field of view, set up by the Indian Institute of Astrophysics\n(IIA) and the Indian Institute of Technology Bombay (IITB) with funding from\nDST-SERB and IUSSTF. It is located at the Indian Astronomical Observatory\n(Hanle), operated by IIA. We acknowledge funding by the IITB alumni batch\nof 1994, which partially supports the operations of the telescope. J.P.N. and\nD.K.O. acknowledge the support of the Department of Atomic Energy, Govern-\nment of India, under project identification No. RTI 4002. This study was partly\nfinanced by the National Institute of Science and Technology of the e-Universe\nproject (INCT do e-Universo, CNPq grant 465376 /2014-2). This study was fi-\nnanced in part by CAPES – Finance Code 001. The authors acknowledge the\nrespective CNPq grants: B.E.M. 150612 /2020-6; F.B.R. 314772 /2020-0; R.V .M.\n307368 /2021-1; M.A. 427700 /2018-3, 310683 /2017-3, 473002 /2013-2; J.I.B.C.\nacknowledges grants 305917 /2019-6, 306691 /2022-1 (CNPq) and 201.681 /2019\n(FAPERJ). The predictions of this event benefited from unpublished obser-\nvations made at the Pico dos Dias Observatory (Brazil), P-LP23. J.L.O. ac-\nknowledges support by spanish project PID2020-112789GBI00 from AEI. P.S-\nS. acknowledges financial support from the Spanish I +D+i project PID2022-\n139555NB-I00 funded by MCIN /AEI/10.13039 /501100011033. J.L.O. and P.S-\nS. acknowledge financial support from the Severo Ochoa grant CEX2021-\n001131-S funded by MCIN /AEI/10.13039 /501100011033. R.H.Y . acknowl-\nedges the grants from National Natural Science Foundation of China (No.\n12073059 & No. U2031139) and the National Key R&D Program of China\n(No. 2019YFA0405501, 2022YFF0503402). Q.Y .Z. acknowledges the grants\nfrom National Natural Science Foundation of China (No. 11988101 & No.\n42075123) and the grant from Chinese Academy of Sciences Hundred Talents\nProject (E12501100C).\nArticle number, page 5 of 8A&A proofs: manuscript no. Triton_atmosphere_1989_2022\nReferences\nArchinal, B. A., Acton, C. H., A’Hearn, M. F., et al. 2018, Celestial Mechanics\nand Dynamical Astronomy, 130\nArimatsu, K., Hashimoto, G. L., Kagitani, M., et al. 2020, A&A, 638, L5\nAssafin, M. 2023a, Planet. Space Sci., 238, 105801\nAssafin, M. 2023b, Planet. Space Sci., 239, 105816\nBertrand, T., Forget, F., White, O., et al. 2020, Journal of Geophysical Research\n(Planets), 125, e06120\nBertrand, T., Lellouch, E., Holler, B. J., et al. 2022, Icarus, 373, 114764\nBrandeker, A., Heng, K., Lendl, M., et al. 2022, A&A, 659, L4\nBrown, R. H. & Kirk, R. L. 1994, J. Geophys. Res., 99, 1965\nDias-Oliveira, A., Sicardy, B., Lellouch, E., et al. 2015, ApJ, 811, 53\nEarle, A. M., Binzel, R. P., Young, L. A., et al. 2018, Icarus, 303, 1\nElliot, J. L., Person, M. J., McDonald, S. W., et al. 2000, Icarus, 148, 347\nElliot, J. L., Person, M. J., & Qu, S. 2003, AJ, 126, 1041\nGomes-Júnior, A. R., Morgado, B. E., Benedetti-Rossi, G., et al. 2022, MNRAS,\n511, 1167\nGrundy, W. M., Olkin, C. B., Young, L. A., & Holler, B. J. 2014, Icarus, 235,\n220\nGurrola, E. M. 1995, PhD thesis, Stanford University.\nHansen, C. J. & Paige, D. A. 1992, Icarus, 99, 273\nHoller, B. J., Yanez, M. D., Protopapa, S., et al. 2022, Icarus, 373, 114729\nHoyer, S., Bonfanti, A., Leleu, A., et al. 2022, A&A, 668, A117\nKilic, Y ., Braga-Ribas, F., Kaplan, M., et al. 2022, MNRAS, 515, 1346\nKumar, H., Bhalerao, V ., Anupama, G. C., et al. 2022, AJ, 164, 90\nLellouch, E., Butler, B., Moreno, R., et al. 2022, Icarus, 372, 114722\nMarques Oliveira, J., Sicardy, B., Gomes-Júnior, A. R., et al. 2022, A&A, 659,\nA136\nMcKinnon, W. B., Lunine, J. I., & Banfield, D. 1995, in Neptune and Triton,\n807–877\nMeza, E., Sicardy, B., Assafin, M., et al. 2019, A&A, 625, A42\nMorgado, B. E., Bruno, G., Gomes-Júnior, A. R., et al. 2022, Astronomy & As-\ntrophysics, 664, L15\nMorris, B. M., Delrez, L., Brandeker, A., et al. 2021, A&A, 653, A173\nNinan, J. P., Ojha, D. K., Ghosh, S. K., et al. 2014, Journal of Astronomical\nInstrumentation, 3, 1450006\nOlkin, C. B., Elliot, J. L., Hammel, H. B., et al. 1997, Icarus, 129, 178\nSicardy, B., Ashok, N. M., Tej, A., et al. 2021, ApJ, 923, L31\nSpencer, J. R. 1990, Geophys. Res. Lett., 17, 1769\nSpencer, J. R. & Moore, J. M. 1992, Icarus, 99, 261\nTyler, G. L., Sweetnam, D. N., Anderson, J. D., et al. 1989, Science, 246, 1466\nYuan, Y ., Li, F., Fu, Y ., et al. 2023, A&A, 680, A9\n1LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne\nUniversité, 5 place Jules Janssen, 92190 Meudon, France\ne-mail: Bruno.Sicardy@obspm.fr\n2Indian Institute of Space Science and Technology, Thiruvanantha-\npuram, 695547, Kerala, India\n3Federal University of Uberlândia (UFU), Physics Institute, Av. João\nNaves de Ávila 2121, Uberlândia, MG 38408-100, Brazil\n4Laboratório Interinstitucional de e-Astronomia - LIneA, Av. Pastor\nMartin Luther King Jr 126, Rio de Janeiro, RJ 20765-000, Brazil\n5American Association of Variable Star Observers (AA VSO), 185\nAlewife Brook Parkway, Suite 410, Cambridge, MA 02138, USA\n6Physical Research Laboratory, Ahmedabad, 380009, Gujarat, India\n7Universidade do Rio de Janeiro - Observatório do Valongo, Ladeira\ndo Pedro Antonio 43, Rio de Janeiro, RJ 20.080-090, Brazil\n8Institut Polytechnique des Sciences Avancées IPSA, F-94200 Ivry-\nsur-Seine, France\n9IMCCE, Observatoire de Paris, PSL Research University, CNRS,\nSorbonne Université, Univ. Lille, F-75014 Paris, France\n10Observatório Nacional /MCTIC, Rio de Janeiro, Brazil\n11TÜBITAK National Observatory, Akdeniz University Campus, An-\ntalya 07058, Turkey\n12Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la\nAstronomía s /n. 18008-Granada, Spain\n13Federal University of Technology-Paraná (UTFPR /PPGFA), Cu-\nritiba, PR, Brazil\n14Tata Institute of Fundamental Research, Homi Bhabha Road, Co-\nlaba, Mumbai 400 005, India\n15Indian Institute of Astrophysics, II Block, Koramangala, Bangalore,\n560034, India16Department of Physics, Indian Institute of Technology Bombay,\nPowai, 400 076, India\n17Aryabhatta Research Institute of Observational Sciences, Manora\nPeak, Nainital 263002, India\n18Akashmitra Mandal, Kalyan, 421301, Maharashtra, India\n19Nicolaus Copernicus Astronomical Center, Polish Academy of Sci-\nences, ul. Rabia ´nska 8, 87-100 Toru ´n, Poland\n20Udaipur Solar Observatory, Physical Research Laboratory, P.O. Box\n198, Badi Road, 313001, Udaipur, India\n21Harvard College Observatory, Harvard University, 60 Garden St.\nCambridge 02158 MA\n22Department of Astronomy, Stockholm University, AlbaNova Uni-\nversity Center, 10691 Stockholm, Sweden\n23INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123\nCatania, Italy\n24Key Laboratory for Research in Galaxies and Cosmology, Chinese\nAcademy of Sciences, 80 Nandan Rd., Shanghai 200030, China\n25School of Astronomy and Space Science, University of Chinese\nAcademy of Sciences, 1 East Yanqi Lake Rd., Beijing 100049, P.\nR. China\n26Key Lab of Space Astronomy and Technology, National Astronom-\nical Observatories, Beijing 100101, P.R. China\n27Shandong Astronomical Society, 180 West Wenhua Rd., Weihai,\nShandong 264209, P. R. China\n28Vygonnaia street, house 2 /1, flat 59, Ussuriysk, Primorsky Krai\n692527, Russia\n29Koshkina Street, house 19 k. 1, flat 210, Moscow, 115409, Russia\n30SETI Institute, 339 N Bernardo Ave Suite 200, Mountain View, CA\n94043, USA\n31Unistellar, 5 allée Marcel Leclerc, bâtiment B, Marseille, F-13008,\nFrance\n32Unistellar Citizen Scientist\n33UNESP-São Paulo State University, Grupo de Dinâmica Orbital e\nPlanetologia, CEP 12516-410, Guaratinguetá, SP, Brazil\nArticle number, page 6 of 8B. Sicardy et al.: Constraints on Triton atmospheric evolution from occultations: 1989-2022\nAppendix A: Observations\nCHEOPS Observation\nCHEOPS is a dedicated mission for observing exoplanet tran-\nsits. It is equipped with a 32 cm Ritchey-Chrétien telescope with\na single, frame-transfer, back-illuminated, 1024 ×1024 pixel\nCCD. There is no filter which results in a bandpass of 0.33-\n1.13µm. Following the successful CHEOPS observation of a\nstellar occultation by the trans-Neptunian object Quaoar (Mor-\ngado et al. 2022), predictions were made for various bodies, in-\ncluding Triton. Since CHEOPS’ predicted orbit is only available\nat most four months prior to the event date, the occultations were\npredicted statistically, contrarily to ground-based observations.\nCHEOPS is kept in a Sun-synchronous dusk–dawn orbit, 700 km\nabove Earth’s surface7. Considering the spacecraft and shadow\nvelocities, the probability of CHEOPS crossing Triton’s shadow\npath was first estimated to be about 13%. The prediction was\ncontinuously updated during the four months before the event\nusing new estimations of CHEOPS’ orbits. Finally, observations\nwere triggered two weeks prior to the event, which was success-\nfully recorded with an exposure time of 3 s.\nThe defocussed point-spread function (PSF) of CHEOPS re-\nsults in the nearby Neptune (Fig. A.1) strongly contaminating\nthe standard aperture photometry as derived from the CHEOPS\nData Reduction Pipeline (DRP, Hoyer et al. 2022). To disentan-\ngle the photometry and furthermore take advantage of the shorter\ncadence of the imagettes (3 s in contrast to the standard 42 s used\nfor the subarray photometry), we extracted PSF photometry from\nthe imagettes using the Python package PIPE8(PSF Imagette\nPhotometric Extraction, see Morris et al. 2021; Brandeker et al.\n2022 for more details).\nThe resulting imagette photometry clearly resolves the\ningress and egress. The photometric noise is estimated by PIPE\nto increase from 0.5% to 0.9% per exposure during the 15 min\ncentered on the occultation. This noise is mainly due to the pho-\nton counting statistics, and is dominated by scattered light from\nthe nearby Neptune, with negligible contribution from the de-\ntector. The noise varies in time due to the asymmetric PSF of\nCHEOPS in combination with field rotation changing with time,\nthat causes a variable contamination from the planet.\nIAO Observations\nSuccessful observations were carried out at the Indian Astro-\nnomical Observatory (IAO), atop Mt. Saraswati, Digpa Ratsa\nRi in Hanle, Ladakh using the 2-m Himalayan Chandra Tele-\nscope (HCT) and the 0.7-m robotic GROWTH-India Telescope\n(GIT, Kumar et al. 2022). HCT recorded the event in the J-band\nwith the TIRSPEC instrument. It has a Teledyne 1024 ×1024\npixel Hawaii-1 PACE array with four quadrants and field of view\n(FoV) of 307′′×307′′and plate scale of 0.3′′per pixel. TIRSPEC\noffers the flexibility of sub-array acquisition for faster readout\n(Ninan et al. 2014). For this event, a 307′′×20′′sub-array was\nused to accommodate the target and the reference star in adja-\ncent quadrants. The detector was readout non-destructively in\nthe up-the-ramp (UTR) mode. The exposure time of each UTR\ncycle was 32 s, with dead-times of 6.3 s between UTR cycles and\nintegration time between consecutive non-destructive readout of\n0.21 s. Dark and flat frames were obtained in the same configura-\ntion. Photometric calibration of the occulted star was carried out\n7https://www.esa.int/Science_Exploration/Space_\nScience/Cheops/Cheops_overview2\n8https://github.com/alphapsa/PIPE\nFig. A.1. An imagette from CHEOPS showing the star being occulted\nby Triton in the centre and the bright PSF of Neptune to the lower\nright. The peculiar shape of the PSF is due to the defocussed optics\nof CHEOPS. The diameter of the imagette is 60 arcsecs and the sepa-\nration between Neptune and Triton varied between 18 and 15 arcsecs\nduring the observation.\nunder similar observing conditions on 5 October 2022. However,\nwith GIT, only the egress portion of the event could be captured.\nObservations were carried out in R-band with the 4096 ×4108\npixel Andor iKon-XL 230 CCD camera, which has a FoV of\n0.7◦. The camera was operated in the fast readout (1.0778 s)\nmode with exposure time of 3 s.\nFor each frame of HCT and GIT observations, the photo-\nmetric error was computed by PRAIA (Assafin 2023b) using a\nstandard procedure based on the signal-to-noise ratio. Then, by\ncalibrating the Triton flux using a close-by star, the flux ratio er-\nror was obtained from the propagated individual errors as 9%\nand 5% for HCT and GIT, respectively.\nYanqi Lake Observation\nF. D. Romanov contacted J.Y . Zhao who sent an observation re-\nquest to observers at Yanqi Lake Observatory, pertaining to the\nUniversity of Chinese Academy of Sciences (UCAS, Beijing,\nChina). The event was successfully recorded at this station on\nbehalf of F. D. Romanov, using the 0.7-m f /6.5 Corrected Dall-\nKirkham (PlaneWave CDK700) telescope equipped with an An-\ndor iKon-L DZ936-BV CCD. The CCD was operated in the R-\nband, using the sub-sampling and fast readout mode (5 MHz),\nwith an exposure time of 1 s and a readout time of 0.8 s. Us-\ning the same approach as for the IAO observations, the flux ratio\nerror was estimated to be around 4%.\nArticle number, page 7 of 8A&A proofs: manuscript no. Triton_atmosphere_1989_2022\nTable A.1. Circumstances of observations.\nSite Coordinates Telescope aperture (m) Exp. time /Cycle (s) Observers\nAltitude (m) Instrument /filter or comments\nPositive observations\nCHEOPS See Fig. 1 0.32 3.0 /3.024 A. R. Gomes-Júnior\nSpace broadband 0.33-1.13 µm B. E. Morgado\nMt. Saraswati 78 57 49.8 E 2.0 0.21 /0.21 B. C. Bhatt\nHCT 32 46 46.4 N TIFR Near Infrared Spectrometer S. Pramod Kumar\nIndia 4520 and Imager (TIRSPEC) /J\nMt. Saraswati 78 57 52.6 E 0.7 3 /4.0778 V . Swain\nGROWTH 32 46 45.2 N CCD Andor iKon-XL /R\nIndia 4517\nYanqi Lake 116 40 14.0 E 0.7 1 /1.8 F. D. Romanov\nChina 40 24 29.3 N CCD Andor iKon-L Y . M. Mao\n96 DZ936-BV R. H. Ye\nJohnson-Cousins Rc Q. Y . Zou\nY . K. Sun\nY . Y . Shen\nJ. Y . Zhao\nObservations with weather problems\nDevasthal 79 41 03.6 E 3.6 & 1.3 Clouded out A. Tej\nIndia 29 21 39.4 N TIRCAM2 /H & S. Sharma\n2450 ANDOR DZ436 /I A. Saha\nNakhodka 132 39 24.2 E 0.355 Clouded out F. D. Romanov\nRussia 42 52 02.6 N CMOS Canon EOS 6D D. N. Grishin\n2 L. V . Romanova\nMt. Abu 72 46 45.2 E 1.2 Clouded out J. K. Jain\nIndia 24 29 17.3 N LISA CCD /white\n1680\nUdaipur 73 40 26.4 E 0.5 Clouded out S. K Mathew\nIndia 24 36 15.5 N Multi-Application\n610 Solar Telescope (MAST) /white\nDhanari 72 55 39.6 E 0.2 Clouded out A. Deshpande\nIndia 24 40 55.5 N QHY5L-ii Mono S. Deshmukh\n369 N. Dhyani\nA. Gokhale\nA. Kate\nza9pya 130 38 53.0 E 0.11 Clouded out R. Kukita\nJapan 31 52 04.4 N eVscope v1.0\n335\n26e7vr 139 18 49.7 E 0.11 Clouded out K. Fukui\nJapan 37 44 16.2 N eVscope v1.0\n115\nv8vjs9 139 18 58.7 E 0.11 Clouded out K. Fukui\nJapan 37 44 11.6 N eVscope v2.0\n0\nArticle number, page 8 of 8"    },    {        "title": "2402.02505v1.Multi_functional_oxidase_like_activity_of_praseodymia_nanorods_and_nanoparticles.pdf",        "content": "Applied Surface Science  \nMulti-functional oxidase-like activity of praseodymia nanorods and nanoparticles\n--Manuscript Draft-- \n \nManuscript Number: APSUSC-D-22-10755R2\nArticle Type: Full Length Article\nKeywords: praseodymia;  Nanorods;  nanoparticles;  artificial enzyme;  oxidase\nCorresponding Author: Xiaowei Chen, Ph.D.\nUniversity of Cadiz\nPuerto Real, Cadiz SPAIN\nFirst Author: Lei Jiang, Ph.D.\nOrder of Authors: Lei Jiang, Ph.D.\nYaning Han\nSusana Fernández-García, Ph.D.\nMiguel Tinoco, Ph.D.\nZhuang Li\nPengli Nan\nJingtao Sun\nJuan Delgado, Ph.D.\nHuiyan Pan, Ph.D.\nGinesa Blanco, Ph.D.\nJavier Martínez-López, Ph.D.\nAna Hungria, Ph.D.\nJose Calvino, Ph.D.\nXiaowei Chen, Ph.D.\nAbstract: The ability to mimic protein-based oxidase with multi-functional inorganic nanozymes\nwould greatly advance biomedical and clinical practices. Praseodymia (PrO  x  )\nnanorods (NRs) and nanoparticles (NPs) have been synthesized using hydrothermal\nand precipitation methods. Both PrO  x  catalysts with different morphologies exhibit\nsignificantly higher oxidase-like activities (Michaelis-Menten constant  K  m  £ 0.026\nmM) than commercial PrO  x  and most so-far-reported artificial enzymes. One of the\nsubstrates, dopamine, can be oxidized and further polymerized to generate\npolydopamine in acidic conditions. Akin to CeO  2  , which is a well-studied nanozyme,\na different mechanism involving holes  +  , oxygen vacancies and oxygen mobility over\nPrO  x  catalysts has been proposed in this work. However, fluoride ions were found to\nimpose opposite effects on the oxidase-mimicking activity of PrO  x  and CeO  2  ,\nimplying a promising path for the exploration of new nanozymes. In support of this, PrO\nx  was further applied in colorimetric sensing of L-cysteine and fluoride with high\nsensitivity.\nSuggested Reviewers: Mengfei Luo, PhD\nZhejiang Normal University\nmengfeiluo@zjnu.cn\nProfessor Luo has been working on lanthanide-based catalysts for a long time.\nMingyuan Zheng, PhD\nDalian Institute of Chemical Physics Chinese Academy of Sciences\nmyzheng@dicp.ac.cn\nProfessor Zheng has been working on heterogeneous catalysis.\nJuewen Liu, PhD\nUniversity of Waterloo\nPowered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporationliujw@uwaterloo.ca\nPowered by Editorial Manager® and ProduXion Manager® from Aries Systems CorporationHighlights  \n \n Synthesis of praseodymia nanorods and nanoparticles  \n Excellent multi -funtional oxidase -like activities of praseodymia nanomaterials  \n Colorimetric sensing of L -cysteine and fluoride  using praseodymia nanorods  Highlights (for review)Graphical abstract  \n \nGraphical Abstract (for review)\n Click here to access/download;Graphical Abstract (for \nreview);Graphical abstract.docx Multi-functional oxidase -like activity  of \npraseodymi a nano rods and nanoparticles  \n  \nLei Jiang  a, *, Yaning Han  a, Susana Fernández -García  b, Miguel Tinoco  b, c, Zhuang Li \na, Peng li Nan  a, Jingtao Sun  a, Juan J. Delgado  b, e, Hui yan Pan  b, d, Ginesa Blanco  b, e, \nJavier Martínez -López f, Ana B. Hungría  b, e, Jose J. Calvino  b, e, Xiao wei Chen b, e, * \n \na Heavy Oil State Laboratory and Center for Bioengineering and Biotechnology, China \nUniversity of Petroleum (East China), Qingdao, 266580, China  \n \nb Departamento de Ciencia de los Materiales, Ingeniería Metalúrgica y Química  \nInorgánica, Facultad de Ciencias,  Universidad de Cádiz, Campus Río San Pedro, Puerto \nReal (Cádiz), E -11510, Spain  \n \nc Departamento de Química Inorgánica, Facultad de Ciencias Químicas, Universidad \nComplutense de Madrid, Av. Complutense S/N, Madrid (Madrid) E -28040, Spain  \n \nd Henan Key Laboratory of Industrial Microbial Resources and Fermentation \nTechnology, College of Biological and Chemical Engineering, Nanyang Institute of \nTechnology, Nanyang, 473004, China  \n \ne Instituto Universitario de Investigación en Microscopía Electrónica y Mater iales \n(IMEYMAT), Universidad de Cádiz, Campus Río San Pedro, Puerto Real (Cádiz), E -\n11510, Spain  \n \nf Departamento de Ciencias de la Tierra, Universidad de Cádiz, Campus Río San Pedro, \nPuerto Real (Cádiz), E -11510, Spain  \n \n \n* Correspond ing authors : L. Jiang, tel: 0086 -532-86981568, fax: 0086 -532-86981569, \nEmail  address : leijiang@upc.edu.cn ; X. Chen, tel: 0034 -956-012741, Email  address : \nxiaowei.chen@uca.es ;  \n \n \n \n \n \n \n \n  Revised manuscript (unmarked)\n Click here to view linked References\n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 2 \n Abstract  \n   The ability to mimic protein -based oxidase  with multi -functional inorganic \nnanozyme s would greatly advance biomedical and clinical practices. Praseodymia \n(PrO x) nanorods  (NRs)  and nanoparticles (NPs) have been  synthe sized using \nhydrothermal and precipitation m ethod s. Both  PrO x catalysts with different \nmorphologies exhibit significan tly higher oxidase -like activities  (Michaelis -Menten \nconstant Km  0.026 mM) than commercial PrO x and  most so-far-reported artific ial \nenzymes. One of the substrates , dopamine , can be oxidized and fu rther polymerized to \ngenerate polydopamine  in acidic conditions. Akin to CeO 2, which is a well-studied  \nnanozyme, a different mechan ism involving holes+, oxygen vacancies  and oxygen \nmobility  over PrO x catalysts  has been proposed in this work . However, f luoride ions \nwere found  to impose opposite effects on the oxidase -mimicking activity of PrO x and \nCeO 2, implying a  promising path for the exploration of new nanozymes. In support of \nthis, PrO x was further applied in colorimetric sensing of L-cysteine  and flu oride with \nhigh sensitivity.  \n \nKeywords : praseodymia, nanorods, nanoparticles, artificial enzyme, oxidase  \n \n \n \n \n \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 3 \n 1. Introduction  \n   Oxidase is one of the most important multi -functional enzyme  family  in nature . \nThis kind of enzyme s has ability to  oxidiz e chemicals containing groups such as amino, \npolyph enol or sulfur [1]. Mimicking or replac ing protein -based oxidase would \nsignificantly advance both biomedical research and biochemical applications. As a \nmeans to an end, recent studies  have shown the dev elopment of many attractive \noxidase -mimicking nanozymes involving noble metals  (Pt, Au  and Ag) [2] or metal \noxides  (Fe 3O4, CeO 2, CuO, MnO 2 and Mn 3O4) [2] . A representative nanozyme is CeO 2, \na rare earth oxide (REO) with excellent oxygen storage capability , rich surface charges , \nand thus , multiple oxidase -like properties  [3]. Lately , Chen et al . reported the so -far most \nactive oxid e-mimicking Mn 2O3 (surface area  8.5 m2/g) with kinetic parameter Km of \n0.13 mM  [4]. Despite the tremendous efforts, the exploration of nanozymes with high \ncatalytic activity and selectivity remains a great challenge.  \n   Praseodymium is the neig hbor of cerium in the periodic table of elements . CeO 2 \nis the predominant stoichiometry  of cerium oxide , while the common  phase of \npraseodymium oxide under ambient condition is roughly Pr6O11 [5, 6]. Pr6O11 can be \nconsidered as an oxygen -deficient modification of cubic fluorite -like PrO 2 and contains \nalready one third of Pr  in its trivalent state and two thirds of Pr4+. Pr6O11 has been  \nreported to have the highest oxygen mobility of all undoped REOs  [7]. From a doping \npoint of view, it can be attributed to the fact that Pr4+ is naturally “doped” with Pr3+ in \nPr6O11. In this sense, praseodymia  has a promising potential to be a cataly tic artificial \nenzyme  because of its Pr3+/Pr4+ pair. It has been used  as catalyst additive or support in \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 4 \n cataly tic oxidation of CO, methanol, methane [8] and soot [9], decomposition reactions \nof organic molecules such as 2 -propanol, oxidative coupling of methane and liquid -\nphase benzylation . For soot oxidation, three -dimensionally ordered macro -porous \npraseodymia was significantly more active than ceria [9]. Our previous stud ies have \nshown that Pr -doped ceria nanocubes are very active as artificial enzymes, from oxidant \n(oxidase -like) to antioxidant (hydroxyl radical scavenging), as well as paraoxon \ndegradation (phosphatase -like) activities [3, 10]. However, application  of praseodymia  as \nan oxidase has not been ever studi ed yet.  \n    The preparation method can strongly affect textural,  structural and redox  \nfeatures of praseodymia , such as phase composition, surface area, particle morphology  \nor size and Pr3+/Pr4+ ratio on surface . These propertie s subsequently tune their catalytic \nactivities as nanozymes . Both preseodymia NRs and NPs can be synthesized via \nprecipitation, hydrothermal, solvothermal and microwave -assisted methods  [11], after \ntherm al annealing of Pr(OH) 3 or thermal decomposition of praseomymium  compounds \n[12]. Despite Pr 6O11 is one of well-known praseodymia phase s, it is true that Pr3+/Pr4+ \nratio can change easily, depending on the preparation or storage conditions of the oxide \n[13]. For this reason, we will henceforth use the PrO x formula to refer praseodymia.  \n   In this work, PrO x NRs and NPs , with relatively high surface areas , have been  \nsynthesized using hydrothermal and precipitation  methods. Their oxidase -like activities \nfor oxidation of  3,3’,5,5’ -tetramethylbenzidine (TMB) , 2,2-azinobis -(3-\nethylbenzothizoline -6-sulfonic acid (ABTS), o-phenylenediamine (OPD) , and \ndopamine (DOPA)  have  been studied  using commercial praseodymia as a reference \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 5 \n sample . The mechanism and key intermediates, as well as the effect s of anions and \namino acids have been  investigated to use PrO x as a new nanozyme  and promote  the \ndevelopment of biomarker sensors for L -cystein e and F- anions.  \n2. Materials and Methods   \n2.1. Synthesis of PrO x NRs and NPs \n   A hydrothermal method was used to synthesize PrO x NRs (PrO x-NR) [11, 14]. 240 \nmL mixture  of 6 M NaOH (Alfa Aesar, 98%) and 0.05 M Pr(NO 3)3·6H 2O (Aldrich, \n99.9%) aqueous solution was stirred in a 300 mL Teflon container for 30 min. Then, \nthis container was sealed and placed in a stainless steel autoclave. The autoclave was \nheated to 180 oC and maintained at this temperature for 24 h. Afterwards, the autoclave  \nwas cooled down to room temperature and the resulting suspension was centrifuged \nand washed with deionized water several times and then with ethanol once (Panreac, \nAbsolute Ethanol). Finally, sample w as dried at 80 oC for 24 h in an oven and calcined \nat 500 oC for 4 h in a muffle furnace.  \nFor the s ynthesis of PrO x NPs (PrO x-NP), a precipitation method  was used  [15]. \nAn aqueous solution of 0.067  M Pr(NO 3)3·6H 2O (Alfa Aesar, 99.9%) was mixed with \n4.9 M NH 3·H2O in 75 mL solution and stirred at 80 oC for 3  h. The  mixture was kept \nat room temperature for 48  h followed by centrifugation for 30 min. The obtained \nprecipitate  was rinsed with  ultrapure  water and ethanol over three times, dried at 100 \nºC in an oven for 10 h, and  finally  calcined at 500 oC for 3 h in a muffle furnace.  \nA commercial Pr6O11 (99.9 9%) sample ( PrO x-C) was purchased from Alfa \nAesar.   \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 6 \n 2.2. Physical and compositional characterization  \n  X-Ray diffraction (XRD) patterns were recorded on a D8 ADVANCE \ndiffractometer of Bruker using a Cu Kα radiation, with a range of 25-75o, a step of 0.02o \nand a step time of 1  s. Phase identification and composition calculation were performed \nusing TOPAS v5 software from Bruker.  Brunauer -Emmett -Teller (BET) surface areas \nof samples were determined in a Micromeritics TriStar - 3020 via N2 adsorption at -\n196.15  ºC. Prior to analysis, samples were degasified at 150 oC for 8 h under vacuum.  \nThe surface chemical composition and oxidation states of samples were \ncharacterized by X-ray photoelectron spectroscopy (XPS) . Analyses were performed \non an ESCALAB 250Xi instrument. Spectra were recorded using monochromatized  Al \nKα X-Ray (1486.6  eV), with an X -ray power of 150 W. The spectrometer  was operated \nin a constant analyzer energy mode, with pass energy of 30 eV. Powder samples were \npressed into a disk, which was stuck on a double -sided adhesive conducting polymer \ntape and analyzed without further treatment. The binding energy scale was calibrated \nwith resp ect to the C 1s signal at 284.8 eV.  CasaXPS software, version 2.3.23.PR1.0 \n(Casa Software Ltd., Devon, UK), was used for spectra processing.  \nThe morphology and size of PrO x samples were studied by Scanning Electron \nMicroscopy  (SEM). A  FEI Nova Nano SEM450 microscope was used at 3 to 10 kV \naccelerating voltage s with a working distance of 5 mm. PrO x samples were also \ncharacterized by High Resolution Transmission Electron Microscopy (HR TEM) and \nScanning Transmission Electron Microscopy -High Angle Annular Dark Field imaging \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 7 \n (STEM -HAADF) using JEOL 2010 -F and FEI Titan3 Themis 60 –300 aberration -\ncorrected microscope s. HR TEM images were obtained with 0.19 nm spatial resolution \nat Scherzer def ocus and STEM -HAADF images were acquired by using an electron \nprobe of 0.5 nm of diameter at a camera length of 8 cm.  \nTemperature Programmed Reduction with H 2 (H2-TPR) started with a \npretreatment consisting in an oxidation under a 5% O 2/He flow (60 mL/min) at 500 oC \nfor 1 h. After the oxidation pretreatment, samples were cooled in the same 5% O 2/He \nflow down to 150 oC, and then, the flow was switched to pure He cooling down to room \ntemperature. The samples were reduced in a 60 mL/min flo w of 5% H 2/Ar with a \nheating rate of 10 oC/min up to  a maximum reduction temperature of 950 oC, keeping \nthem at this final temperature for 1 h. The outlet of H 2-TPR equipment was connected \nto a Thermostar GSD301T1 or a PrismaPlusTM mass spectrometer of Pfeiffer Vacuum. \nThe evolution of the mass/charge ratio (m/ z) = 18 was  used to monitor the formation of \nH2O during the H2-TPR process . \n   UV-Vis specular and diffuse reflectance measurements were carried out using an \nSHIMADZU 2600i UV–Vis-NIR double -beam spectrophotometer . The spectra in 220  - \n1400 nm range  were registered in an integrating sphere. Diffuse reflectance spectra were \ntransformed into apparent absorption spectra using the Kubelka –Munk function ( F(R)). The \ndirect and indirect optical band gap of materials was determined through construction of \nTauc plots by plotting ( F(R)hν)n against (hν), with  n=2 or n =½, for direct and indirect \ntransitions, respectively.  The optical band gap was obtained by extrapolati ng the linear part \nof this plot to the energy axis.  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 8 \n Fourier transform infrared (FTIR) spectra of the samples were obtain ed on a \nNicolet 6700 instrument . Sample s suspended or diluted in water  were added dropwise \nto a compressed potassium bromide IR -transparent pellet, and then dried to remove all \nwater . FTIR spectra were registered in the range of 4000 -500 cm-1. \n  Electron Paramagnetic Resonance (EPR) spectr a were measured in a dark room on a  \nBruker  ELEXSYS E500  spectroscope. Samples were prepared by dispersing 3 mg/m L \nPrO x in pH  4 acetate buffer . 5,5-dimethyl -1-pyrroline -N-oxide (DMPO) was used as the spin \ntrap for oxygen vacancies while 2,2,6,6 -Tetramethylpiperidinooxy (TEMPO) was used  for \nphoto -induced electron, which in tu rn indicates the generation of hole+. A light source from \na 300 W xenon lamp (PLS-SXE300 , BOFEILAI ) was used to generate light at the \nwavelength range of 320 – 780 nm.   \n   The generation of HO• over PrO x-C sample was monitored by using a \nfluorescent probe terephthalic acid (TA) which can react with the radicals to form a \nhighly fluorescent product, 2 -hydroxy TA (TAOH). H 2O2 was added to generate HO• \nradicals as negative (without PrO x-C) and positive contr ol (with PrO x-C). The \nfluorescence at 435 nm of TAOH was measured with excitation at 315 nm via a \nfluorescent spectrometer (FluoroMax -4, Horiba Jobin Yvon, France).  \n2.3. Artificial enzyme activities  \n   The oxidase mimetic activit ies for the oxidation of colorless TMB, ABTS, \nDOPA  and OPD  were  studied over three kinds of PrO x catalysts . The oxidized TMB, \nABTS , DOPA and OPD have characteristic UV-vis absorbance peak s at 652, 405 , 450 \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 9 \n and 450  nm, respectively . All the se organic  chemicals were bought from Sigma \nAldrich. The PrO x suspension was prepared dispersing powder  sample s in Milli Q water \nafter washing and centrifuging at 12000 rpm for 10 min three times. All suspens ions \nwere sonicated in a 100 W w ater bath sonicator  (Kunshan, KQ -100) for 2 h before each \nmeasurement.  An extra -strong 500 W ultrasonic processor (VC505, Sonics & Materials, \nInc.) was also employed to examine the possible effect of aggregation.  Radical \nscavengers, including TEMPO and p -benzoquinone, isopropanol (hydroxyl HO \nscavenger) and sodium oxalate  (hole+ scavenger) , were purchased from Sigma  Aldrich .  \n   A 4.16 mM  TMB stock solution was prepared in dimethyl sulfoxide . Then, a  \n0.2 mM TMB solution was prepared by diluting the concentrated  stock solution with \n10 mM pH 4.0 of acetate buffer. Similarly, 0.088 mM ABTS , 0.31 mM DOPA  and 4 \nmM OPD solutions were prepared in 10 mM acetate buffer pH 4.0. The UV -Vis \nabsorption spectra of substra tes before and after praseodymia addition were record ed \nin a SHIMADZU UV -2450 spectrophotometer. To explore and compare the kinetic \nactivities  of different samples , praseodymia amount  was kept constant  at 20 µg/mL \nwith TMB at 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35 mM to calcu late Michaelis -Menten \nparameter Km and maximum reaction rate Vmax.  \n   Inhibition effect of L -cysteine  was measured in solution of TMB and 45  µg/mL \nPrO x-NR. Concentration of L -cysteine  was varied from 0 to 10 µM. Effects  of other \namino acids , including 4 µM lysine (Lys), phenylalanine (Phe), arginine (Arg), \nhistidine (His), tyrosine (Tyr) , Homocysteine (Hcy), glutathione (GSH) and glycine \n(Gly), w ere compared with 4 µM L-cysteine  at the same condition s.  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 10 \n 2.4. Fluoride effect and d etection  \n   To compare impact s of several anions , including F-, on the oxidase activity of \nPrO x-NR, 90 µM TMB and 90 µg/mL PrO x-NR suspension , were added to 5 mM NaF, \nNa2SO 4, NaCl, Na 3PO 4, NH 4Br and Na2CO 3 solution s. Absorbance changes @ 652 nm \nwere measured. For example, t he quantitative detection of fluoride in commercial \ntoothpaste (Oral B from supermarket) was measured as follows . Briefly, 1 g toothpaste \nwas vigorously mixed with 32 mL water and filtered through a Millipore Amicon Ultra -\n15 (3 kDa molecular weight cutoff) ultrafiltration membrane . The filtrate was \ncentrifuged at 15000 rpm for 10 min and the superna tant was collected after  \ncentrifugation. Standard addition method  [16] was used  to analyze fluoride concentration \nwhile minimiz ing the matrix effect . NaF  of 0, 10.7, 21.4, 32.2, 42.9, 53.9, and 64.3 µM \nconcentrations were added to 1 mL  solution of 90 µg/mL PrO x-NR in pH 4.0 10 mM \nacetate buffer  to form spiked samples . Then 30 µL of collected supernatant was added \nto the mixture and stirred for 10 min. Finally , 90 µM TMB was added and absorbance \nat 652 nm was measured .  \n \n3. Results and discussion  \n   Two types of PrO x samples, i.e. PrO x-NR and PrO x-NP were synthesized and \ncharacterized, together with a commercial PrO x-C sample. All of them are dark powder s \n(Fig. 1a) , but both PrO x-NR and PrO x-NP samples are more brown -colored  than PrO x-\nC. Their XRD p atterns  (Fig. 1b) show that a ll the diffraction peaks can be attributed to \na fluorite -like structure , face -centered cubic lattice  Pr6O11 (Fm-3m,JCPDS 42 -1121).  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 11 \n The diffraction peaks of commercial PrO x-C are much narrower than those of PrO x-NR \nand PrO x-NP, indicating that the crystallinity of PrO x-C is much better than those  of the \nsynthesized PrO x samples. Using diffraction peak at 2 8o, the average particle sizes of \nPrO x-NR, PrO x-NP and PrO x-C were calculat ed by Scherrer equation to be  7.1, 10.3 \nand 49. 8 nm, respectively.  \n \nFig. 1. (a) Photographs  of three types of PrO x samples in solid  state and suspen sion in water , \nincluding the PrO x-NR synthesized by hydrothermal method , PrO x-NP synthesized by \nprecipitation and commercial PrO x-C. (b) XRD patterns of three PrO x samples.  \n \n   Fig. 2 shows SEM images of the three PrO x samples at low and high \nmagnifications. These samples show different  morphologies. PrO x-NR sample prepared \nby the hydrothermal route presents uniform NR shape. The PrO x NRs, with diameters \nin the 20 to 65 nm range and length up to 10 µm, are entangled (Figs. 2a and 2b). PrO x-\nNP synthesized by precipitation features irregul ar morphology particles or \nagglomerates with size less than  5 µm (Fig. 2e). High resolution  SEM image (Fig. 2f) \nof this sample indicates  its surfaces of uneven roughness. Commercial PrO x-C also \nappears to be constituted by agglomeration of large  irregularly -shaped particles. PrO x-\nC particles (Figs. 2i -2j) are not only larger but also show  much  smoother surfaces than \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 12 \n those of PrO x-NP.  \n   TEM and STEM -HAADF images of the three PrO x samples  are also illustrated \nin Fig. 2. Low magnification TEM image of PrO x-NR (Fig. 2c) and STEM images of \nPrO x-NP (Fig. 2g) and PrO x-C (Fig. 2k) samples confirm the size and morphology \nresults observed by SEM. The crystalline nature of all the samples is clear appreciated \nin HRTEM and STEM -HAADF images at the bottom row of Fig. 2. Digital diffraction \npattern (DDP) analysis of the selected area of HRTEM image of PrO x-NR (inset in Fig. \n2d) shows the major reflections of the fluorite type structure , which  is characteristic of \nhigher praseodymiu m oxide (eg. PrO 2). Similar to PrO x-NR catalyst, DDP analysis of \nselected areas of PrO x-NP (Fig. 2h) and PrO x-C (Fig. 2l) catalysts clearly confirm their \nfluorite type phase. Interestingly, the DDP of PrO x-C shows the presence of additional \nspots that it is a clear indication of a superstructure formed by the ordering of oxygen \nvacancies. A more detailed explanation is included in the su pporting information (Fig. \nS1a). In addition, the PrO x-NP sample is comprised of small crystals in nanometer scale \nas sho wn in Figs S1 b, which is in good agreement with average particle size of 10.3 nm \nof PrO x-NP calculated using Scherrer equation .   \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 13 \n  \nFig. 2. SEM , TEM and STEM -HAADF images of (a-d) PrO x-NR, (e-h) PrO x-NP and (i-l) PrO x-\nC samples  \n \n   BET specific surface areas and total pore volumes of three samples were \nmeasured by N 2 physisorption (Table 1 ). PrO x-NP possesses  the highest surface area, \n65 m2/g, followed by PrO x-NR with 27 m2/g and, finally,  PrO x-C with just 4 m2/g, a \nvalue characteristic of typical bulk materials. The tota l pore volume data of these \nsamples  follow the same trend. These results are in good agreement with the above -\nmentioned XRD  and TEM results. Poor crystallinity and small size of PrO x-NR and \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 14 \n PrO x-NP lead to their high BET specific surface areas, while PrO x-C presents bigger \nparticle size, much better crystallinity and much lower surface area.  \n   XPS spectra of three PrO x samples are shown in Fig. 3 for Pr 3d, O 1s and C \n1s core levels. For Pr 3d core level (Fig. 3a), peak deco nvolution  was performed by \nfollowing the analysis reported by Sinev et al. [17]. All spectra could be fitted using 7 \npeaks , i.e. peaks labelled as a, b and c are for 3d 5/2 component, and a’, b’ and c’ peaks \nare their corresponding 3d 3/2 counterparts. An extra peak, existing only in the 3d 3/2 \ncomponent, and labelled as t’, was also included as reported previously  [17]. There is no \nagreement yet on the assignment of peaks to Pr3+ or Pr4+ oxidation states unequivocally  \nin the literature [17-19]. However, Borchert et al. [18] propose d a method to determine the \noxidation degree of Pr by using peaks a-a’, which are undoubtedly related with Pr4+, \nand b-b’, which appear for both oxidation states. Although the y warn ed that the \ncalculation could lead  to a rough approximation, this is one of the most  consistent \nattempts  to determine Pr oxidation state  using XPS. Here we use t his method to \ncalculate Pr3+ percentage  and results in Table 1  indicate the mixed valence nature of \nthree PrO x samples  with a ll surface Pr3+ percentage  at 60 – 70%.  \n \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 15 \n  \nFig. 3. XPS spectra of (a) Pr 3d, (b) O 1s and (c) C 1s of PrO x-NR (blue), PrO x-NP (green) and \nPrO x-C (red) samples .   \n \nTable 1  Physicochemical  properties  and catalytic activities  of PrO x samples  \nCatalyst  BET \nsurface \nareaa \n(m2g-1) Total \npore \nvolumea  \n(cm3g-1) Percentage \nof Pr3+ b \n(%) Ratio of \nOcarbonate /Olatticeb Indirect \nband \ngapc \n(eV) Direct \nband \ngapc \n(eV) Km d \n(mM)  \nPrO x-NR   27 0.062   66 3.4 1.10 2.06 0.026 \nPrO x-NP   65 0.530  68 4.5 0.79 2.12 0.018  \nPrO x-C 4 0.021  61 1.9 0.77 2.20 0.180  \na Obtained from N 2 physisorption  \nb Calculated using XPS data  \nc Determined using Tauc method  \nd Kinetic parameters  using TMB as sub strate \n   Further insight about the chemical composition of samples is given by \nanalyzing O  1s and C  1s core levels (Fig . 3b and  3c). It is well known that praseodymia \nreacts readily with atmospheric water and CO 2 when it is exposed to air [20, 21]. Analysis \nof O 1s and C  1s core levels confirms these phenomena . O 1s spectra in Fig. 3b can  be \nfitted using a combination of 3 peaks: one peak around 528.6 eV that could be attributed \nto lattice oxygen  (O2-); a second feature, the most intense in all the samples, around \n(a)\n (b)\n (c)\nc c’b\nb’\na a’t’O2-CO32-\nOH-, H2OCH,CCC-OC=OCO32-NRNPC\nNRNPC\nNRNPCIntensity (arb. units)Intensity (arb. units)\nIntensity (arb. units)\n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 16 \n 531.3 -531.4 eV that can be assigned to carbonates  (CO 32-); and a third weak peak at \n533.5 -533.7 eV, due to the presence of OH- or water groups. Relative intensity of \noxygen from carbonate and lattice after fitting is presented in Table 1. It shows that \nthere could be a correlation between surface area and O carbonate /Olattice, i.e. the higher the \nsurface area, the larger the carbonation extent.  \n   C 1s core level in Fig. 3c also confirms  the existen ce of carbonate on all \nsamples . Peaks at 284.8 eV, 286.2 eV and 288.0  eV correspond to C -C, C-O and C=O , \nrespectively, from adventitious carbon . The dominan t peak at 289.4 -289.5 eV is due to \ncarbonate. The presence of these carbonated species is strongly related to the high Pr3+ \ncontent of the se samples as determined by XPS. In short, the higher the surface area of \nthe PrO x samples, the higher the carbonate content, and thus the higher the reduction \ndegree of praseodymium.  \n   Fig. 4 shows UV -visible diffuse reflectance spectra of PrO x catalysts. \nAbsorbance spectra of PrO x-NP and  PrO x-C samples are very similar in 220 -1400 nm \nwavelength range . In comparison, PrO x-NR exhibits a quite strong absorbance in the \nvisible range between 220 to 640 nm , in agreement with previous reports [12]. This is \nprobably why PrO x-NR appears more brown -color ed than the other two  samples (Fig. \n1a). As shown in XRD results, the main phase of three PrO x catalysts is Pr 6O11, which \ncontains both Pr3+ and Pr4+ oxidation states. The contribution of Pr3+ and Pr4+ \nabsorbance in UV -visible range is still in debate. Kang et al. reported that the presence \nof oxidation state of Pr4+ may be the origin of the strong/broad absorption in the visible \nregion by charge transfer (O2− → Pr4+) absorption [12]. On the other hand, p revious  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 17 \n studies  also reported Pr4+ does not absorb in the UV -vis region  [22]. In other words, the \nabsorbance in the 430 – 610 range can be mainly attributed to Pr3+. Meanwhile, Pr3+ \nabsorption bands between 430 – 490 nm due to the 3H4 ground state to 3PJ transition \nand that the 560 – 610 nm band originated  from  a 3H4 to 1D2 transition  [22]. It should be \nhighlighted that t here is no big difference between Pr3+ percentages (Table 1) of PrO x-\nNR and PrO x-NP catalysts calculated using XPS results , and Pr3+ percentage of PrO x-\nC is slightly lower than PrO x with NR and NP morphologies.  The reason of UV-visible \nstrong absorption of PrO x-NR different with PrO x-NP and PrO x-C is not clear, which \nneeds further study  in future .      \n \nFig. 4. UV-Visible diffuse reflectance spectra of PrO x samples.  \n    Using the UV -Vis diffuse reflectance results, indirect and direct band gap \nenergies of PrO x samples can be  determined via the Kubelka -Munk function. As shown \nin Table 1, there is no significant difference in direct band gap  value s (2.06 – 2.20 eV) \n400 600 800 1000 1200 14000.00.20.40.60.81.01.21.41.6\n PrOx-NR\n PrOx-NP\n PrOx-CAbsorbance\nWavelength (nm)\n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 18 \n for three PrO x samples , in accordance with previous result s in the literature  [10, 23]. \nHoweve r, some researchers have reported higher  values for direct band gap of PrO x, \nfrom 3.15 to 4.45 eV [24]. The origin of such difference in optical properties and band \ngap energies could stem from the  synthesis methods  of PrO x. The band gap reduction \nmay be attributed to both the uplift of v alence band (due to surface disorder) and \nlowering of conducting band (due to oxygen vacancies and more defect centers)  [25]. \n   Fig. 5 displays the formation of H2O over all PrO x samples during H 2-TPR \nprocess. The redox behavior of PrO x-NR is better than those of the other two samples. \nIts dominant reduction peak appears at 480 oC, at lower temperature than in PrO x-NP \n(502 oC) and PrO x-C (509 oC). The reason might be due to its smaller average \ncrystalline size (7.1 nm) than those of other two samples ( 10.3 and 49.8 nm)  according \nto calculation using Scherrer equation. The main reduction peak corresponds to the \nreduction of Pr4+ in bulk of PrO x samples. TPR profiles of PrO x-NP and PrO x-C \ncatalysts ar e similar in that both have a minor reduction peak at 310 or 332 oC, which \ncan be assigned to the reduction of surface  Pr4+ and the active surface oxygen species \nreact with hydrogen to form H 2O [12, 26]. The reduction peak of PrO x-NR at low \ntemperature is merged into an asymmetric peak at 480 oC. This result is in good \naccordance with previous TPR data of PrO x-NR and bulk PrO x in the literature [26]. The \nfinal product of H 2-TPR of three PrO x samples is expected to be Pr 2O3, according to \nliterature [8, 21].     \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 19 \n                    \nFig. 5. H2O evolution (m/z=18) during H 2-TPR of PrO x-NR, PrO x-NP and PrO x-C catalysts . \n    \n   The multi -functional oxidase -like activity of PrO x was studi ed by their effects \nin oxidizing conventional substrate s including TMB, ABTS , OPD  and DOPA . TMB, \nABTS  and OPD are chromogenic  substrates with oxidable  amino groups, while DOPA \nrepresents substrates with polyphenol groups. Once PrO x samples were added to \nsubstrates in pH 4.0 acetate buffer s, individual characteristic color s quickly appear ed \n(Fig. 6a), suggest ing the occurrence of oxidation of substrate s. The corresponding UV-\nvis absorbance spectr a change s of all substrates are illustrated  in Fig. S2. Similar to \nother previously reported oxidase -like nanomaterials,  the catalytic activities  of PrO x \ncatalysts are pH-dependen t and pH 4.0 is the optimum condition for all substrates  (Fig. \n200 400 600 800Normalized intensity of m/z = 18\nTemperature (oC)PrOx-NR\nPrOx-NP\nPrOx-C\n480 502310332\n509\n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 20 \n 6b). In comparison, TMB and DOPA are more sensitive to pH changes than ABTS, \nprobably because they are positively charged while ABTS is negatively charged, \ninducing different adsorption and catalytic behaviors of nanozyme s as previously \nreported  [27]. \n   Kinetics  of three PrO x samples (Fig. S3) can be calculated by the Michaelis -\nMenten equation  as shown in Method in Supporting information using TMB as a \nsubstrate and their kinetic parameters Km are listed in Table 1. Lower  Km value means \nhigher oxidase activity of nanozymes. These results indicate  that the oxidase -like \nactivities follow the order of PrO x-NP > PrO x-NR > PrO x-C, exactly the same order of \ntheir BET  surface areas . More importantly, Km values of both PrO x-NR (0.026 mM) and \nPrO x-NP (0.018 mM) are an order of magnitude lower than that of PrO x-C (0.18 mM), \nand significantly lower than most  of the  reported oxidase -like nano zymes . Their \nmaximum reaction rates Vmax also show  same tendency, 0.031, 0.029 and 0.0013 \nmM/min , respectively.  Table S1 compare s Km and Vmax values from this study with \nthose of the so -far-reported nanozymes in TMB oxidatio n since 2007, including hig h \nactivity of Mn 2O3 nanoparticles [4]. The comparison suggest s that PrO x-NP and PrO x-\nNR exhibit  relatively higher activities than those nanozymes of similar levels of Vmax \n(Table S1).   \n   Fig. 6c compares steady -state catalytic activities of three PrO x catalysts, as \nwell as those of three compounds of praseodymium, Pr 2O3, Pr 2(CO 3)3, and Pr(OH) 3. \nThere is almost no oxidase activity of TMB oxidation over these three Pr -containing \ncompounds with solely Pr oxidation state of +3. At the beginning of reaction  (< 1 min) , \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 21 \n PrO x-NP perform s faster than PrO x-NR, but over longer time  (> 2 min) , PrO x-NR is \nbetter than PrO x-NP. This is probably because the aggregated state of PrO x-NP partly \ninhibits the following attachment of substrate s, which was further examined with the \nhelp of extra -strong ultrasonicator in Fig. S4. However, PrO x-NR is much better \ndispersed in solution; therefore, it exhibits better long -term catalytic activity. \nFurthermore, similar Km and Vmax values of PrO x-NR and PrO x-NP means that TMB \noxidation occurs with similar kinetic rate, i.e. similar amount of TMB is oxidized over \nboth samples. Often the catalytic activities of nanomaterials can be compared as activity \nper unit surface area  [28]. In this study, TMB oxidation takes place  on the surface of PrO x \nsamples, considering that the BET surface area of PrO x-NR is less than half of PrO x-\nNP, it means that the activity per unit area on PrO x-NR surface is higher than that of \nPrO x-NP. \n   As mentioned before, the  Pr3+/Pr4+ ratio of PrO x can change easily  with storage \nconditions  so that it bears a risk of phase change  upon exposure to air after long  time \nstorage  [21]. Fig. S5a shows XRD patterns of thr ee PrO x catalysts stored in air for years.  \nPr(OH) 3 phase can be observed on all PrO x samples. However, PrO x-NR stored for 9 \nyears and PrO x-NP stored for 3 years contain less Pr(OH) 3 phase and more PrO 2 and \nPr6O11 phases than PrO x-C stored for 5 years. Since calcination of Pr(OH)s is a step of \nsynthesis of PrO x, it is reasonable  to recalcine PrO x samples  exposed in air for a long \ntime at  500 °C to obtain pure PrO x phase again. This hypothesis is confirmed, which \ncalcination at 500 oC leads to the formation of pure Pr 6O11 phase again as shown in Fig. \nS5a. Fig. S5 b shows the TMB oxidation reaction s by PrO x-NP and PrO x-C after three -\n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 22 \n year storage . Both samples were r ecalcine d at 500 C for 4 h  to regenerate Pr 6O11 and \neffective TMB oxidation  is observed, suggesting the good stability of PrO x samples .  \n \nFig. 6. (a) Colorless TMB, ABTS, OPD and DOPA substrates can be cataly tically oxidized  by \nPrO x to generate  characteristic colors at pH 4.0. (b) pH-dependence of oxidase -like activities  \nof PrO x-NR sample s. Normalized activity of each reaction was obtained by normalizing  \nabsorbance of oxidized TMB at 652 nm, oxidized ABTS at 405 nm and oxidized DOPA at 450 nm \nby absorbance of un -oxidized substrates, respectively.  (c) Absorbance change s at 652 nm of \n0.2 mM TMB in pH 4.0 acetate buffer when 9 2 µg/mL catalyst s (PrO x, Pr 2O3, Pr 2(CO 3)3, and \nPr(OH) 3) were added . (d) FTIR spectra of 1 M DOPA  at conditions of pH 4.0, pH 10.0 and pH \n4.0 with 1.1 m g/mL PrO x-NR. (e) Optical  microscop ic image of DOPA  solution after addition \nof PrO x-NR in pH 4. 0 solution, forming dark polydopamine  spheres . Inset is the image of \nDOPA solution without PrO x-NR. (f) Photographs  of PrO x-NR oxidizing and polymerizing \ndopamine in acidic condition.   \n   \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 23 \n     Since PrO x-NR nanozyme presents the highest steady -state activity and similar \nKm value to  PrO x-NP, further stud ies to understand mechanism and i nfluence s of amino \nacids and anions focus  on this catalyst . The activities of PrO x-NR catalysts are  so high \nthat they can even enable the polymerization  of concentrated DOPA to polydopamine \n(PDA) in acidic solution. FTIR spectra of DOPA at pH 4.0 and  10.0 and pH 4.0 with \nPrO x-NR conditions are compared in Fig. 6d. The characteristic band s at 2746 , 2638, \n2538 and 2432  cm-1 of DOPA  disappear at pH of 10.0  and at pH of 4.0 with PrO x-NR, \nbecause the polymerization of DOPA chang es NH 2 to secondary amine [29]. Meanwhile, \nthe width of 1620 cm-1 peak increases after polymerization  at pH of 10.0 and at pH of \n4.0 with addition of PrO x-NR, probably due to C=C stretching vibration of indole  \nstructure formed by intramolecular cyclization reactions  [30]. Further, results in Fig. 6e \nand 6f also demonstrate  the gradual oxidation and induced -polymerization of DOPA.  \n    It is well known that DOPA can be oxidize d and self -polymerize d to PDA in \nbasic  (pH >8)  media , but not in neutral and acidic media  [31-34]. PDA is a wet bio-\nadhesive that has unique photo -physical properties for a wide range of biomedical \napplications. To oxidize and polymerize PDA in acidic condition usually involves \nunconventional tools such as UV -illumination  [31, 32], plasma treatment  [33], high \ntemperature (over 120 °C for 16  h) [34]. To the best of our knowledge, this is the first \ntime that oxidative self -polymerization of PDA  is achieved in an acidi c environment \nwith solely the help of inorganic nanoparticles  as catalysts . It is worth noting that an \nemployment  of PrO x-NR as oxidase to synthesize PDA m ay also be beneficial to widen \nthe pH range of application of PDA.  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 24 \n     The key reasons for such high catalytic performance of PrO x may be attributed \nto the important intermediate holes+, which is remarkably different from other \nnanozymes. So far most reported oxidase -like nanozymes , such as CeO 2, MnO 2, iron-\nbased materials  etc. [2], interact with molecular oxygen (or other oxidizing reagents) to \ngenerate intermediate superoxide radical anion (O 2) as the key oxidants in the \noxidation process . In order to elucidate the mechanism of oxidase -like property of PrO x-\nNR, several scavengers  corresponding to different reactive oxygen species  and charge \ncarriers  were applied to PrO x-TMB system.  For example, p-benzoquinone  [35] in Fig.  7a \nand TEMPO  [36] in Fig. 7b  were used to capture O 2, while  isopropanol  [37] is a hydroxyl \n(HO) scavenger and sodium oxalate  [38] was added to eliminate hole+ [39]. Results show \nthat only sodium oxalate (hole+ inhibitor)  can significantly affect the oxidase -like \nactivity  and the inhibition  level  is highly depend ent on the  amount of eliminator . On \nthe contrary, the other three inhibitors show no influence  within the  concentration  range \nstudied in this work . These results  imply that, in PrO x-catalyzed oxidation , the key \nintermediate is hole+, not the conventional radicals like O2 or HO. To the best of our \nknowledge, this is the first time that hole+ was reported for the activity of nanozyme. \nThis feature of PrO x is also in  clear contrast to its affiliated oxide  CeO 2, which is well \nknow n for generat ing O2. Such hole+-induced oxidation is also dependent on pH \nvalues (Fig. 6b), probably because that the surface hydroxyls can react with hole+ to \nform hydroxyl radicals and thus the total oxidative capacity decreases at higher pH \nconditions.   \n    The generation of holes+ was further tested by EPR spectra . It was known that \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 25 \n photoexcitation may lead to the concomitant generation of electron/hole+ pairs  [40]. The \nidentification of photoinduced electron s by EPR is a clear indication of the presence of \nhole+. The spin lab el TEMPO does not react with oxidative intermediate s such as O2 \nor HO [40]. Therefore , it is good capture probe for photoinduced electrons , as well as \nindirect indication of hole+. Fig. 7c shows that TEMP O signal (g value ~ 2.00 0) \nintensity reduced 14% after 5 min light illumination, and almost disappeared after 20 \nmin illumination, suggesting the generation of electrons and hence holes+. Interestingly, \nEPR measurements also demonstrate the involvement of oxygen vacan cies, VO, in \nPrO x-catalyzed oxidation. Using DMPO as spin label, either using ABTS or TMB \nsubstrate, Fig. 7d shows clear reduction of VO (g value ~ 2.003 ) signal s when sodium \noxalate, the hole+ inhibitor, w ere added to the mixture. The oxidation of TMB over \nPrO x-C in dark and under day light (Fig. S7) have been studied. Results showed that \nthe enzymatic activity for oxidation of TMB is lower in the dark compared with under \nday light, suggesting the possible influence of photo -induced hole+. \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 26 \n  \nFig. 7. (a) Normalized  catalytic  activity of PrO x-NR oxidizing TMB  after adding p-\nbenzoquinone  (O2 scavenger ) and sodium oxalate  (hole+ scavenger ). (b) Effect of isopropanol \n(HO scavenger ) and TEMPO (O2 scavenger ). (c) EPR spectra of spin lab el TEMPO for \nelectrons generated by 3 mg/m L PrO x-NR pH 4.0 in dark and after light illumination of 5 and \n20 min. (d) EPR spectra of spin lab el DMPO for  3 mg/mL PrO x-NR with 1 mM ABTS  and 0.1 \nmM TMB  before and after addition of 20  mM sodium oxalate.  \n  Taken together, the above -mentioned results suggest that the catalytic \noxidation, or the oxidase -like activity , of PrO x is closely related to h ole+ and VO. \nEnergy level diagram  (Fig. 8a) of PrO x-NR demonstrate s that the distance between O-\n2p Valence Band  (VB) (2.81 eV) and Pr 4f Conduction Band ( CB) is 2.06 eV, and VO \nis located between these two band gaps. The level of VO can be estimated from the \nphoton energy profile (Fig. 8b) based on UV -Vis diffuse reflectance data in Fig. 4. Each \nabsorption band  can be associated with an electron transition between two energy states \n(Fig. 8a). For example, the absorption bands at 1.29 and 1.55 eV  could be assigned to \nthe charge transfer transition from oxygen vacancies to Pr 4f CB. Fig. 8c shows VB  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 27 \n XPS of PrO x-NR in comparison with a 100% reduced praseodymium oxide  [3]. Their \nO-2p VB  maximum are similar at around 2.81 eV.  \n \nFig. 8. (a) Energy level diagram and excitation process of PrO x-NR. (b) Photon energy spectrum \nof PrO x-NR. (c) V alence band XPS data for a  100% reduced praseodymium oxide and PrO x-\nNR. \n    The mechanism for PrO x-catalyzed oxidation is propo sed to be initiated by \nelectron s transfer ring from O -2p VB to the VO band, leaving a h ole+ available to \noxidize substrates like TMB  and DO PA (Fig. 8a) . Meanwhile , the electron can be \nfurther transfer red to the conduct ion band and O2, which is eventually reduced to H 2O. \nThe formation of O2 from O 2 requires the potential of electron to be less than -0.16 \neV [40], while the s tandard reduction potential for the O 2/H2O couple is 1.23 V ( Fig. 8a ) \n[41]. It is likely that the electron can reduce O2 to H2O without generating O2. On the \nother side, although hole+ has a greater potential (2.81 eV) than HO (2.2 eV), the \ninhibition measurement (Fig. 7b) shows no presence of HO radicals. Therefore , we \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 28 \n propose that hole+ may probably oxidize the substrate directly. Mo reover, the presence \nof VO and high oxygen mobility of PrO x are able to facilitate the electron/hole+ \nseparation, which leads to the high oxidizing capability of PrO x catalyst.  \n Despite the proposed mechanism above, it is worth noting that the oxidase -like \nproperty of PrO x is probably not only the result of a single factor. The hole+ in PrO x \ncrystalline lattice can be also generated by various oxidative species (HO•, O 2•, etc), \ndirectly or indirectly participating in the oxidation (depicted as dotted arrow in Fig. 8a). \nFor example, although the addition of HO• inhibitor (isopropanol) shows no impact on \nthe oxidase -like activity of PrO x (Fig. 7b) at increasing concentration, fluorescent \nexperiments using HO• probe terephthalic acid (TA) indicate the presence of a small \namount  of HO• radical compared with negative control without PrO x-C and positive \ncontrol with PrO x-C experiments (Fig. S8). The TA aqueous solution and a mixture of \nTA and H 2O2 solution do not show a fluorescent peak at 435 nm, which is a \ncharacteristic peak of 2-hydroxy TA (TAOH)  generated from reaction between HO• \nradicals and TA. When PrO x-C was mixed with TA, there is a small fluorescent peak at \n435 nm. The addition of H 2O2 to the mixture of PrO x-C and TA generates more HO• \nradicals, thus a stronger peak appe ars at 435 nm. All these results suggest that the \nexistence of multiple routes for generation of oxidative species, such as the interaction \nof OH－ with hole+ to form HO• (OH－+ h+→HO•). The origin of oxidase -like activity \nof PrO x in the crystalline lattice may be eventually related to the reversible exchange \nbetween Pr3+ and Pr4+ which is associated with the formation and migration of oxygen \nvacancies. Pr 6O11 has the highest oxygen mobility among all rare earth oxides  [7], partly \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 29 \n because of its inherent structure of  naturally “doped” Pr3+ in Pr4+ stoichiometry. The \noxygen defects or vacancies, particularly abundant in PrO x-NR of this study, lead to its \nexcellent oxygen storage capacity, which is fundamental for Pr 6O11 in previous catalytic \nstudies  [9].   \n    PrO x as oxidase is sensitive to reducing agents, such as L -cysteine . Absorbance  \nof oxidized TMB decrease s with addition of L -cystein e (Fig . 9a). In fact, a bsorbance \nchange @652 nm decreases linear ly with L-cysteine  concentration  within the 210-7 ~ \n110-5 M range , with a limit of detection (LOD) of 2.310-9 M (Fig. 9b). In comparison \nwith other nanomaterials  employed in various L-cysteine  sensors (Table S2)  [42-44], \nPrO x-TMB platform appear s to be a good colorimetric  sensing probe  for cysteine  and \nsimilarly structured Hcy in blood or serum samples, with negligible influence  from salt \nor amino acids of similar structure s (Fig. 9c).  \n Oxidase a ctivity of PrO x-NR is also very sensitive to  the fluoride anion  (F–), a \nhigh dose of which is toxic to human health and affects the aqueou s conditions in the \nenvironment  [45]. As shown  in Fig . 9d, F– addition inhibit s TMB oxidation over PrO x-\nNR and the inhibition level  is in linear relationship to F– concentration  (Fig. 9e). \nMeanwhile , other halide s (Cl– and Br–) and anions (SO 42-, PO 43- and CO 32-) do not \ninduce such dramatic effect (Fig. 9f). This high selectivity for F– ions of PrO x-TMB \nsystem can be  used as a sens ing platform  for the excessively  harmful F– anions which \ncould cause  dental mottling and skeletal manifestation  [46]. For example, this method \ncan be employed to measure the fluoride content in a supermarket toothpaste  (Oral -B \nas in Fig. 9d). To minimize the matrix effect s, standard addition method was used  (Fig. \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 30 \n S6) [16]. Accordingly , the measure d F– content is 0.14%, which  is very close to  the \nlabelled value (0.15%) on the too thpaste, suggesting good accuracy and applicability \nof this method in ordinary F– detection.  \n \n \nFig. 9. (a) UV-Vis spectr a of 0.2 m M TMB oxidized by 0.02 mg/m L PrO x-NR in presence of \nL-cysteine  (0 - 10 M). (b) Correlation between A 652 nm with L -cysteine  concentration  with \nLOD of 2.3 10-9 M and R2 = 0.9 81. (c) Selectivity of L -cysteine  detection  in comparison with \nsalts and other amino acids . (d) UV-Vis spectra of 0.09 m M TMB oxidized by 0.1 mg/mL PrO x-\nNR at different F– concentration.  Inset is picture of an Oral-B toothpaste containing 0.15% \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 31 \n fluoride . (e) Relationship  between A652 nm  with F– concentration. (f) Selectivity  of F– detection  \nin comparison with other  anion s.  \n Another phenomenon  worth noting is that the fluoride -induced inhibition on \nPrO x is in contrast with the enhancement effect of fluoride  on the nei ghbor oxide CeO 2 \n[47]. Based on p revious EPR and Raman studies  [48, 49], F– anions are adsorb ed on surface \nof CeO 2, increasing the concentration of VO, but does not affect  the amount of \ngenerated O2 or HO in function of mim icking oxidase . However, i t can accelerate the \noxidation process by replacing the oxidation product quickly on CeO 2 surface . Oxidase -\nlike PrO x also functions through VO, but its interaction with F– is different. Fig. 10 \nshows XPS spectra of PrO x-C before and after mixing  with F–. Opposite to CeO 2, whose \nCe3+ concentration  increased significantly (from 38% to 45%) after F– immersion  [48, \n49], the percentage of Pr3+ on PrO x decreases  from 61% to 56%, while the ratio of \nOcarbonate /Olattice decreases from 1.9 to 1.45. Adsorption of F element is also confirmed on \nPrO x surface because  a new F 1s peak at 68 4 eV appears in XPS of F–-treated PrO x (Fig. \nS9) [50]. F– was reported to bind to hydroxyl groups on surface of metal oxides including \nMg-Al [50]. As a strong electron withdrawing ligand , the addition of F– anions leads to \noxidation of some Pr3+ to Pr4+ in order to keep the charge  balance in PrO x. Thus, the \nactive sites Pr3+ on PrO x surface  decrease and fluoride -induced inhibition occurs . \nSecondly, F– may interrupt  the electron transfer pathway and reduce  the formation of \nsurface hole+, which also causes  the inhibition of PrO x activity. These  are probably the \nreasons why F– has very opp osite effect on the oxidase -like activity of PrO x to CeO 2.  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 32 \n  \nFig. 10. XPS spectra of (a) Pr 3d and (b) O 1s of PrO x-C before and after treating with 10  mM \nF– solution.    \n     Fig. 11 illustrate s schematically the possible mechanism in the multi -functional \noxidase -like activity of PrO x. Praseodymium cations can exist with +3 and +4 valence \nstates  in PrO x samples.  PrO x has the highest oxygen mobility because it has a larger \nnumber of possible phases for the oxides , so that the variety of stable phases enable fast \nchanges in the oxidation state s of praseodymium . PrO x demonstrates high  multi -\noxidase activit ies which can be specifically tuned by its morphology , size , redox \nproperty , Pr3+/Pr4+ ratio and reducing inhibitor such as L -cysteine . Due to the  generated \nhole+, oxygen vacancies and high oxygen mobility, PrO x-NR is able to initiate the \noxidation and polymerization of DOPA in acidic condition. Moreover, adsorption of F– \ncan inhibit the activity of PrO x-NR, which conversely enables the linear detection of F–. \nThis effect  makes PrO x a good probe for F– sensing, but with completely different \nmechanis m to CeO 2 which detect s F– by the boosted activity. In an environment  of \nweak target presence , it is usually easier to detect reduction of a strong signal (signal -\noff) than enhancement of a weak signal  (signal -on) with possibly high background \nnoise . From this point of view, for F– detection normally  at low  concentrations  in most \ntesting environments, PrO x-NR is probably better than CeO 2 nanomaterial s as sensors .  \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 33 \n  \nFig. 11. Schematic illustration of how  PrO x-NR catalyze s the oxidation of several substrate s \nincluding TMB, ABTS, OPD and DOPA. The oxidizing activity is high enough to oxidize and \npolymeri ze DOPA into PDA  in acidic conditions. The key intermediate s and reactive species \nare hole+ and oxygen vacancies VO, instead of  O2 or HO, in the oxidation process.  \n \n4. Conclusions  \n     PrO x is firstly reported to exhibit multi -functional oxidase -like activit ies with \nsignificantly high  kinetic performance . Three types of PrO x samples with different \nmorpholog ies and surface areas  are compared in their cataly tic activities  of oxidiz ing a \nseries of amino and polyph enol substrate s, such as TMB, ABTS, OPD and  DOPA. The \npolymeriz ation of DOPA can be trigger ed by PrO x-NR under acidic condition to \nproduce  polydopamine, a highly useful  wet bioadhesive. Different from previou sly \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 34 \n reported oxidase -like nanozymes, h ole+ and oxygen vacancies  may play  important role s \nin the oxidation process by PrO x. Adsorption of F– on PrO x inhibits  its activity, in clear \ncontrast to CeO 2, probably because of the increased concentration  of Pr4+ on surface \nand reduced oxygen accessibility  by the F– blockage. These results not only shed some \nlight on the discovery of REO artificial enzym es with new mechanism, but also imply \na new sensor for F– detection in fields of contamination and overdose .  \n \nCRediT authorship contribution statement  \n   Lei Jiang:  Conceptualization, Investigation, Validation, Data curation, Writing \n– original draft, Methodology, Formal analysis, Writing – review & editing. Yaning \nHan:  Data curation, Formal analysis. Susana Fernández -García : Investigation , \nReview & editing . Miguel Tinoco : Investigation, Review & editing . Zhuang Li:  \nInvestigation.  Peng li Nan : Investigation . Jingtao Sun : Investigation . Juan J. \nDelgado : Investigation, Methodology, Review & editing, Funding acquisition, Project \nadministration . Huiyan Pan : Investigation . Ginesa Blanco : Investigation , Review & \nediting . Javier Martí nez-López : Investigation. Ana B. Hungría:  Supervision , Review \n& editing . Jose J. Calvino:  Supervision, Review & editing.  Xiao wei Chen : \nConceptualization, Methodology, Investigation, Funding acquisition, Project \nadministration, Resources, Supervision, Data curation, Writing – review & editing.   \n \nDeclaration of Competing Interest  \n   The authors declare that they have no known competing financial interests or \n 1 \n 2 \n 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n10 \n11 \n12 \n13 \n14 \n15 \n16 \n17 \n18 \n19 \n20 \n21 \n22 \n23 \n24 \n25 \n26 \n27 \n28 \n29 \n30 \n31 \n32 \n33 \n34 \n35 \n36 \n37 \n38 \n39 \n40 \n41 \n42 \n43 \n44 \n45 \n46 \n47 \n48 \n49 \n50 \n51 \n52 \n53 \n54 \n55 \n56 \n57 \n58 \n59 \n60 \n61 \n62 \n63 \n64 \n65 35 \n personal relationships that could have appeared to influence the work reported in this \npaper.  \n \nAcknowledgement s \n   This work has been supported by the Ministry of Science, Innovation and \nUniversities of Spain  with reference number of PID2020 -113809RB -C33 and Junta de \nAndalucía (Spain) with reference number of PY18 -2727 . This work has been co -\nfinanced by the 2014 -2020 ERDF Operational Program and by the Department of \nEconomy,  Knowledge, Business and University of the Regional Government of \nAndalusia with project reference FEDER -UCA18 -107316 . The research projects \nfunded by Natural Science Foundation of Shandong Province (ZR2017LB028) , Key \nR&D Program of Shandong Province (2018GSF118032),  and Fundamental Research \nFunds for the Central Universities (18CX02125A) in China are also acknowledged. M. \nTinoco thanks FPU scholarship program from Ministry of Science and Innovation of \nSpain. H. Pan is grateful for financial support fr om Chinese Scholarship Council to \naccomplish her PhD study in University of Cadiz (Spain).  TEM/STEM data were \nacquired at the DME -UCA node of the Unique Spanish Infrastructure for Electron \nMicroscopy of Materials (ICTS -ELECMI).  \n \n \nReferences  \n[1] Q. Liu, A. Zhang, R. Wang, Q. Zhang, D. 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At this end, the Cavity-\nMolecular Dynamics approach proposed recently (Li et al., Proc. Nat. Acad. Sci.\nUSA, 2020, 117, 18324–18331) is employed. This formalism uses a fully atomistic\nrepresentation of liquid water, which is able to mimic key features of light-matter hy-\nbridization under vibrational strong coupling (VSC). The cavity frequency was set to\ndifferent frequencies, and in particular, following the IR spectrum of liquid water: high\nfrequency (corresponding to O-H stretching modes), medium frequency (corresponding\nto water molecule bending) and low frequencies (corresponding to librational modes\nof liquid water). First, the splitting of the infrared spectrum and no visible effect of\nthe cavity on the radial distribution function are confirmed. Second, an effect on the\ndielectric constant is found when the cavity is resonant with librational modes. Finally,\nwe notice an interesting effect on some dynamical properties, like the diffusion coeffi-\n1arXiv:2402.02510v1  [physics.chem-ph]  4 Feb 2024cient and the hydrogen bond jumping time. Notably, this effect is strongly dependent\non which vibrational mode the cavity is in resonance with.\nIntroduction\nIn recent years, there has been a growing interest in vibrational strong coupling (VSC) and\nin particular on its impact on chemical reactivity.1–3VSC is obtained by placing a molecular\nsystem inside an optical cavity such as a Fabri-P´ erot cavity that is resonant with a selected\nvibrational mode of the matter, and then a hybrid light-matter state is generated. When\nthe exchange of (virtual) photons is faster than their dissipation, the strong coupling regime\nis achieved.\nEarly approaches focused on the coupling of optical cavity modes with electronic states.4\nHowever, subsequent research demonstrated the feasibility of coupling molecular vibrations\nwith cavity modes in the infrared (IR) range, a phenomenon referred to as vibrational strong\ncoupling (VSC).5,6This coupling induces a splitting called Rabi splitting of distance hΩR\nin the infrared (IR) spectrum if the frequency of the cavity ( ωc) is at resonance with the\nvibrational frequency ( ωm) of the molecular system. It has been demonstrated that VSC\ncan also influence chemical reactivity.1Specifically, it has been observed that kinetics are\naltered, as highlighted in previous studies,7–13and more recently, other equilibrium properties\nhave been found to undergo modifications. These include changes in solvent polarity,14ion\nconductivity,15dielectric constant,15and the equilibrium constant in the formation of charge-\ntransfer complexes.16\nWhile the Rabi splitting in IR spectrum is clearly explained,17–21it is much less clear how\nthese other properties are modified in VSC regime. At this end, the last years have been char-\nacterized by an important theoretical activity in the field.22–29The theoretical description is\nbased on the Pauli–Fierz (PF) quantum electrodynamics (QED) Hamiltonian:30\n2ˆH=ˆHM+ˆHF (1)\nwhere ˆHMis the typical Hamiltonian of a molecular system and ˆHFis the interaction with\nthe vacuum field, which can be written as:22,28,31–35\nˆHF=/summationdisplay\nk,λ1\n2ˆp2\nk,λ+1\n2ω2\nk,λ/parenleftigg\nˆqk,λ+N/summationdisplay\ns=1ˆ⃗ µs·⃗ξ\nωk,λ√V ϵ0/parenrightigg2\n, (2)\nwhere kdenotes the (virtual) photon with given frequency, ωk, position, ˆ q, and momentum,\nˆp, operators, λthe two polarization directions orthogonal to the cavity walls (so typically\nxandy),Vthe cavity volume, ϵ0the vacuum permittivity, ˆ⃗ µsthe dipole operator for the\nmolecular system and ⃗ξthe polarization direction of the (virtual) photons.\nVarious theoretical studies using this Hamiltonian have been performed in recent years.\nA first kind of approach focused on adapting the transition-state theory (TST) for the\npresence of the cavity and thus obtained rate constants with an effective friction, in analogy\nwith Kramers and Grote-Hynes theories.36,37They generally consider only one mode for the\nmatter coupled to the cavity with effective coupling parameters. For example, Lindoy et\nal.24used an analytical rate which shows that chemical kinetics are enhanced when the bath\nfriction is weak and suppressed when the bath friction is strong. A quantum description\nvia TST of cavity-catalysed adiabatic chemical reaction was then carried out by Yang and\nCao,25providing an explanation for the maximum modification when the cavity frequency\nnear the vibrational resonance frequency and also for the collective and selective aspects.\nIn a similar vein, quantum dynamics were performed using different models, but generally\nconsidering only one matter mode coupled with the cavity. Morse dissociation or double\nwell models were studied in this way, showing an effect on the rate.38,39Also quasi-classical\ntrajectories were employed to reveal the modification of water dimer dissociation due to\nVSC.40Notably, using the open quantum system formalism, Reichmann and co-workers\nshow that it is possible to obtain the highest enhancement when the cavity is in resonance\n3with the reactant mode.38\nAnother approach, developed in Rubio’s group, is based on the application of the cou-\npling to the full nuclear-electron Hamiltonian.26This method was used, for example, to\nexamine whether the collective coupling implies local modifications of chemical properties31\nand applied on a HD+molecule.27Furthermore, quantum dynamics have been employed\nto describe a model of a chemical reaction in the condensed phase within the cavity.33,34,38\nHowever, these methods based on quantum dynamics only allow the description of model\nsystems and most of the time, the matter is described with only one explicit mode.\nTo study how VSC modifies the properties of a condensed phase system, an approach\nbased on molecular dynamics simulations was developed by Li et al.23, called Cavity Molec-\nular Dynamics (CavMD). In this approach, the atoms of the system interact with classical\nforce fields and the cavity acts on the dipole of the molecules using a simple charge rep-\nresentation of the molecular charge distribution (as in classical molecular dynamics). This\napproach has previously been used to study liquid water23first and then extended to other\nsystems. For example, it is possible to study the nonlinear response of carbon dioxide under\nVSC,41revealing that a strong pulse can lead to an enhanced overall molecular nonlinear\nabsorption that causes an ultrashort lifetime (0.2 ps) of the lower hybrid light–matter state.\nFurthermore, it has been employed to show that intermolecular vibrational energy transfer\nis facilitated under VSC.42The same authors also numerically illustrate that IR pumping a\nliquid confined in an optical cavity under vibrational strong coupling can selectively excite a\nsmall concentration of solute molecules.43Notably, this approach can be extended to include\nphenomenologically the cavity loss.44The CavMD approach has also been applied, using\na QM/MM approach, to study Fe(CO) 5in n-dodecane45predicting a preferred vibrational\nenergy transfer to the equatorial CO vibrations rather than the axial CO vibrations when\nexciting the lower polariton. Recently, it has been shown that including quantum effects in\nthe molecular dynamics simulations via ring polymer molecular dynamics or quasi-centroid\nmolecular dynamics35,46has only a slight effect of the IR spectra or other properties under\n4VSC.\nIn the present work, our aim is to model and understanding the effect of VSC on liquid\nwater and in particular on its dynamical properties. At this end we have considered a simple\nmolecular dynamics simulation of liquid water by employing the well-known q-TIP4P/F\nflexible model47and, using the approach proposed by Li et al.,23we investigated how different\nproperties are modified when the VSC is turned on. In particular, based on the works of\nLaage, Hynes and co-workers, it is known that many dynamical processes in liquid water,\nincluding, e.g., translational diffusion and reorientation, are governed by hydrogen-bond\nexchanges.48–50The rate-limiting step of these hydrogen-bond exchanges has been shown\nto be the combination of the initial hydrogen-bond elongation and of the new hydrogen-\nbond contraction. This motion is a low-frequency motion and in previous theoretical studies\non VSC on liquid water, the cavity frequency was set on high frequency values. Here,\nwe have tuned the cavity frequency in different spectral regions corresponding not only\nto O–H stretching but also to the bending and to the librational modes. We have thus\nanalyzed how (and if) different properties are modified under VSC. In particular, dynamical\nproperties, related to hydrogen bond dynamics were investigated. Two quantities are central\nin this context: the water diffusion and the hydrogen bond life-time and we found that\nclassical CavMD simulations provide an effect of the cavity on them. This should promote\nfurther investigation using, likely, more developed (but also computationally more expensive)\nmodels. Presently, we restricted ourselves to classical evolution of particles, in order to\nstrengthen the importance of a large statistical sampling to minimize the uncertainties on\nthe calculated properties. It was shown that using Path-Integral based MD, the IR splitting\ndo not change much as well as other properties like H-bond jump time.35,46,50In the present\nwork, we are interested in investigating the effect on the cavity: once this is deeply studied\nand the crucial cavity frequencies are identified, nuclear quantum effects could be added\non a limited set of physical conditions, since (as we will show) a relatively large number of\ntrajectories is needed to obtain converged properties.\n5The remainder of the article is as follows: first we will describe the basic principles of the\nmethods and the simulations performed. Then we will show results obtained on equilibrium\nproperties (vibrational spectra, water structure, static dielectric constant) as a function of\nthe cavity frequency and after on the dynamical ones (diffusion coefficients and hydrogen\nbond life-times), still as a function of the cavity frequency. Last section provides conclusions\nand outlook for future studies.\nMethods\nMolecular Dynamics Simulations\nWe simulated a system composed by a box of 216 water molecules using the flexible q-\nTIP4P/F force field.47The box size was set at 18.64 ˚A aligning with the one employed by\nLi and co-workers using the same number of molecules and water model, providing the cor-\nrect water density at 300 K.23Liquid water was simulated by employing standard periodic\nboundary conditions using an Ewald summation for the intermolecular Coulombic interac-\ntions. Then for each cavity frequency and for the simulation without the cavity, we first run\na 200 ps NVT simulation at 300 K, using a Langevin thermostat with τ= 100 fs and an\nintegration time of 1 fs, using velocity Verlet scheme.51After an equilibration time of 50 ps,\nwe selected 30 snapshots to run NVE simulations of 100 ps with an integration time step of\n0.5 fs.\nThe inclusion of the cavity followed the methodology suggested by Li et al23using the\nCavMD software,52a modified version of i-PI.53This software is coupled here with the\nLAMMPS package to use classical force fields.54CavMD models the system through two\nequations of motions, one corresponding to the atoms and one corresponding to the virtual\nphotons of the cavity. The equations of motion used are obtained after reducing all of the\n6operators in Eq. 2 to the corresponding classical observables:\nMj¨Rj=F(0)\nj−/summationdisplay\nk,λ/parenleftigg\nϵk,λ˜qk,λ+ϵ2\nk,λ\nmk,λω2\nk,λN/summationdisplay\ns=1⃗ µs·ξ/parenrightigg\n∂⃗ µj·ξ\n∂Rj, (3)\nmk,λ¨˜qk,λ=−mk,λω2\nk,λ˜qk,λ−ϵk,λN/summationdisplay\ns=1⃗ µs·ξ, (4)\nwith the coupling constant: ϵk,λ=/radicalig\nmk,λω2\nk,λ/V ϵ 0. (5)\nIn Eq. 3 and 4, an auxiliary mass mk,λis introduced to align with standard MD simulations\nwhere particle mass information is required. Consequently, the positions representing the\ncavity are modified as follows: qk,λ= ˜qk,λ·√mk,λ.\nThe positions and velocities of the atoms obtained from the dynamics are used to de-\ntermine the properties of interest. Various values of the cavity frequency are chosen that\ncorrespond to frequencies where a signal is present in the IR spectrum, such as 3550, 1640,\n800, 600, 400 and 300 cm−1. For each cavity frequency, a different value of the coupling\nconstant is chosen. As shown in Eq. 5, the coupling constant is proportional to the cavity\nfrequency, thus to have the same strength of the cavity for each frequency, the coupling con-\nstant needs to be adapted linearly. For the cavity frequency at 3550 cm−1the same coupling\nconstant is chosen as Li et al.23: 4·10−4a.u. then the other values are adapted linearly from\nthis following Equation 5: 1 .8·10−4, 8·10−5, 6·10−5, 4·10−5, 3·10−5a.u. respectively.\nIt is well known that this water model does not reproduce accurately all the liquid water\nproperties, but it has the great advantage of being well-known and is one of the simplest\nflexible water models. It was used by Li et al.23in their study of water under VSC and thus\nit is a natural choice for an initial exploration of the impact of VSC on dynamical properties\nand with different cavity frequencies.\n7Calculation of properties\nFrom MD simulations we obtained, first, equilibrium and thermodynamic properties such\nas the infrared spectrum, the vibrational density of states (VDOS), the radial distribution\nfunction (RDF) – from which the free energy barrier ∆ G‡can be estimated – and the\ndielectric constant.\nThe infrared spectrum is determined as the Fourier transform of the autocorrelation of\nthe total electric dipole moment of the molecular system and the VDOS is the autocorrelation\nof the velocities. As in the model, the direction of the cavity is along the z-axis, the IR and\nVDOS spectra are considered only along x- and y-axis. The spectra along zare reported in\nFig S1 of the supporting information (SI).\nIt has been shown50that the free energy barrier ∆ G‡, corresponding to the switch of a\nwater molecule linked by a hydrogen bond from one molecule to another molecule, can be\napproximated as:\n∆G‡=−kBT/bracketleftbigg\nln/parenleftbiggg(rmin)\ng(rmax1)/parenrightbigg\n+ ln/parenleftbiggg(rmin)\ng(rmax2)/parenrightbigg/bracketrightbigg\n, (6)\nwhere g(r) corresponds to the O-O RDF and rmax1,rmax2andrminare the distance cor-\nresponding to the first two maxima and the first minimum respectively of the RDF. Thus,\nwe estimated the free energy barrier for the hydrogen-bond switch reaction simply from the\nRDF calculated from equilibrium simulations, as a function of the cavity frequency.\nThe dielectric constant, ε, is calculated55,56using the formula:\nε= 1 + 3 yDgK (7)\nwhere the Kirkwood factor gK=⟨M2⟩/(Nµ2) and the dipole parameter yD=ρµ2/9kBTε0.\nN is the number of molecules, µis the absolute value of the molecular dipole and M=\n/summationtext\niµi,ρis the number density and ε0is the vacuum permittivity. From 30 independent\n8NVE molecular dynamic trajectories, we obtain the mean of the dielectric constant and the\nstandard error of this mean. The equation used for the standard error is shown in the SI.\nIn a second part, dynamical properties such as the diffusion coefficient and the hydrogen-\nbond jump time, i.e.the inverse rate constant for stable exchanges of hydrogen-bond part-\nners, are examined. The diffusion coefficient57is obtained through the slope mean square\ndisplacement (MSD) of the oxygen atoms. Here we limited ourselves to the diffusion coef-\nficient corresponding to a given water box, since we are interested into the impact of VSC\nkeeping the same simulation conditions. It is well known that to recover the diffusion co-\nefficient at infinite box size, one should consider the evolution of it as a function of the\nbox length.58To determine the jump time, the cross-correlation function of the probability,\nCI,F(t), for the system to donate a stable hydrogen-bond to an initial acceptor Iat time 0\nnI(0) and to form a hydrogen-bond to a new stable partner Fat time t nF(t) is used:49\nCI,F(t) =⟨nI(0)nF(t)⟩. (8)\nThis correlation function gives the probability to have formed a stable final state at time t\nwhen the system was in the initial state at time 0. Thus the decay time of the complementary\nprobability, 1 −CI,F(t)) =exp(−t/τ0), provides the jump time τ0corresponding to the time\nfor a water molecule to switch.49For both diffusion coefficient ( D) and jump time ( τ0) the\nstandard error of the means are estimated considering the slight temperature dependence.\nResults and discussion\nEquilibrium Properties\nAs previously discussed, the CavMD approach is able to mimic the formation of a hybride\nstate in which the vibrational modes of liquid water are mixed with the resonant cavity\nmode (see a pictorial explanation in Figure 1a). The IR spectrum of liquid water, obtained\n9from NVE trajectory with the q-TIP4P/F model was calculated from the Fourier transform\nof the dipole-dipole correlation function (see SI for details) and it results in the well-known\n3 bands (as shown in the bottom of Figure 1b): a broad one around 600 cm−1and two\nnarrower ones at 1650 and 3600 cm−1. These peaks represent the water librational motions,\nO–H bending and O–H stretching modes, respectively. We chose to couple our cavity with\nall three of these peaks for which the coupling constant were adapted proportionally. Results\nare summarized in Figure 1b.\nWhen the cavity frequency is at resonance with a peak of the IR spectra, a splitting can\nbe seen in this spectra, corresponding to the formation of two polaritonic modes. The two\nnew peaks correspond to the P+and P−polaritonic modes that arise from the hybrid state\nas shown in Figure 1(b). On the other hand, for the VDOS spectra, shown in Figure 1(c),\nthere is no splitting seen, as here we have all the contributions and the dark states (DS) are\nnumerically dominant.\nThe water structure is typically represented by the oxygen-oxygen radial distribution\nfunction (RDF) which is reported in Figure 2(a). Here we report also the RDFs obtained\nwhen the cavity is resonant with different modes, obtaining a negligible effect, similarly to\nwhat was reported by Li et al.23when the cavity was set at 3550 cm−1.\nFrom the RDF, it is possible to estimate the free energy barrier for the switching of a\nhydrogen bond of a given molecule from one to another water molecule, as discussed by\nWilkins et al.50At this end, we can obtain the potential of mean force (PMF) simply as:\nW(r) =−kBTln[g(r)] and it is reported in Figure 2(b). Unsurprisingly, as there is no large\nchange when including the cavity on the RDF there is also no big effect on the PMF.\nWe consider thus the following reaction: a given water molecule, in liquid, that is initially\nH-bonded to a second water molecule then changes to be H-bonded to a third molecule. This\nis formally similar to a chemical reaction and can be treated consequently.50From equation 6,\nthe ∆ G‡corresponding to this reaction is obtained and reported in Table 1 with and without\nthe cavity (and as a function of cavity frequency). As expected from the shape of both RDF\n10V0(N modes) V1V2\nMolecular\nstatesHybride\nStateCavity\nmodeshωm hωchΩRN-1 DS\n012\nP+\nP-(a)\n01000 2000 3000 4000 5000\n01000 2000 3000 4000 5000\nFrequency (cm-1) Frequency (cm-1)3550     cm1\n1640cm1\n800cm1\n600cm1\n400cm1\nNo cavity3550cm1\n1640cm1\n800cm1\n600cm1\n400cm1\nNo cavity(b) IR (c) VDOSFigure 1: (a) Scheme of vibrational strong coupling (b) IR spectra with cavity frequencies\nof 400, 600, 800, 1650, 3550 cm−1and excluding the cavity (c) VDOS spectra with the same\ncavity frequencies. The black vertical lines represent the cavity frequency\n112 3 4 5 6 7 8 9123Radial distribution function\n2 3 4 5 6 7 8 9\nO-O distance (Angstrom)0.5\n0.00.5Potential mean force (kcal/mol)\nNo cavity\n300cm1\n400cm1\n600cm1\n800cm1\n1640cm1\n3550cm1\nrmax2 rmin rmax1(a)\n(b)Figure 2: (a) Radial distribution function (RDF) of the O-O distance averaged over 30\ntrajectory of 100 ps with a save time step of 10 fs. At this scale, the RDF from the simulations\nwith and without the cavity are indistinguishable. (b) The corresponding potential mean\nforce of a water molecule.\n12and PMF, the barriers are almost unchanged when the cavity is on. We should also notice\nthat our values are in agreement with the energy barrier obtained previously by Wilkins et\nal. (1.22 kcal/mol)50with the same water model but with no cavity.\nTo continue the study of water within the cavity, we look at the dielectric constant, ε,\nobtained as function of cavity frequency. Results are summarized in Table 1 which also\nincludes values reported in the literature using the same water model. For more detail,\nscatter plots illustrating the dielectric constant values for each of the 30 trajectories can be\nfound in Fig. S2 of the SI.\nTable 1: For different values of cavity frequency, the approximated free energy barrier, ∆ G‡,\nand the mean dielectric constant from 30 trajectories and the standard error are shown\nωC(cm−1) ∆ G‡(kcal/mol) Dielectric constant\nNo cavity 1.16 53 ±3\n52.7±6.5a\n54.5b\n300 1.18 53 ±2\n400 1.15 60 ±2\n600 1.14 61 ±2\n800 1.17 58 ±3\n1640 1.16 55 ±2\n3550 1.15 55 ±2\n54.5b\naFrom Eltareb et al.59bFrom Li et al.46\nWe should first notice that we obtained without the cavity a value in agreement with\nwhat is reported in the literature,46,59with smaller error bar due to the large statistical\nsampling we performed and a different definition of the error (standard error vs standard\ndeviation).\nLi et al. have studied the effect of the cavity on the dielectric constant only when the\ncavity frequency is at 3550 cm−1finding no effect,46as we do. However, when the cavity\nis set at low frequencies, corresponding to water librational modes, there is an increasing of\ndielectric constant up to 60 – 61.\n13Dynamical Properties\nWe now dive into dynamical properties of liquid water that can be extracted from molecu-\nlar dynamics simulations, such as the diffusion coefficient and the time it takes one water\nmolecule to switch his hydrogen bond from one molecule to another, named here the jump\ntime. As discussed previously, to obtain dynamical properties we averaged, for each value of\ncavity frequency ( ωc) over 30 NVE trajectories generated from a long NVT trajectory.\nWe first consider diffusion coefficients, shown in Figure 3(a) as a function of the cavity\nfrequency. The value without frequency, with the corresponding error bar, are shown as\nhorizontal lines. As we can notice, for some cavity values, and particularly 400 and 3550 cm−1\nthe VSC reduces the diffusion coefficient. This reduction is small, although it does surpass\nthe margin of error. In panel (b) of the same Figure 3, we report the jump time obtained from\nthe same simulations (and also in this case the result without the cavity with its error bars is\nreported as horizontal line). The numerical values of these plots are shown in Tables S1 and\nS2 of the SI. We now obtain a clear effect when the cavity is in resonance with librational\nmodes, and namely at 300, 400 and 800 cm−1. On the other hand, we do not see any effect\nwhen the cavity frequency is set to 3550 cm1. Finally, when ωc= 600 cm−1the effect is\nsmall but on the opposite direction with respect to what was obtained for ωc= 400 cm−1\nboth regarding Dandτ0.\nAs described in details by Laage and Hynes,48,49the mechanism responsible to H-bond\njump in liquid water is a combination of angular motion of the O–H moving from one to the\nother oxygen atom acceptor and displacement motion of the new acceptor water molecule\nwhich passes from second to first shell (and the original acceptor from first to second). Based\non this picture, we decided to investigate the effect of the cavity on the librational modes\nand intriguingly we found an impact on the jump time. Furthermore, the jump mechanism\nand the water motion (which is not a purely diffusive mechanism but is somehow related to\ndiffusion) can be related. In Figure 4, we plot the logarithm of Dandτ0for the different\ncavity frequencies. Notably, we found that when the cavity frequency corresponds to the\n14No cavity 300 400 600 800 1640 3550\nCavity frequency (cm1)\n0.18750.19000.19250.19500.19750.20000.20250.20500.2075Diffusion coefficent (A2 ps1)\n(a)\nNo cavity 300 400 600 800 1640 3550\nCavity frequency (cm1)\n3.853.903.954.004.054.100 (ps)\nNo cavity\n300 cm1\n400 cm1\n600 cm1\n800 cm1\n1640 cm1\n3550 cm1\n(b)Figure 3: (a) Diffusion coefficients and (b) H-bond jump characteristic times obtained from\nCavMD simulations as a function of cavity frequency. The value outside cavity and corre-\nsponding error bars are reported as horizontal lines.\n15frequencies of the broad librational mode, there is a direct linear (in logarithmic scale)\nrelation between the two quantities. On the other hand, when ωc= 3550 cm−1only Dis\naffected but not τ0and the corresponding point does not follow this dependency. When\nωc= 1640 cm−1the point is closed to the no cavity values and thus the ”vicinity” to the line\nfitting the values in the 300–800 cm−1range is just due to the fact that there is no effect.\nActually, the slope of the straight line turns out to be -1.2 which is close to the ”ideal”\nvalue of -1 which would correspond to a perfect matching between jumping mechanism and\ndiffusion (see SI for more details).\n1.35 1.36 1.37 1.38 1.39\nln( 0) (ps)\n1.67\n1.66\n1.65\n1.64\n1.63\n1.62\n1.61\n1.60\n1.59\nln(D) (A2 ps1)\nNo cavity\n300 cm1\n400 cm1\n600 cm1\n800 cm1\n1640 cm1\n3550 cm1\nFigure 4: Natural logarithm of the diffusion coefficient as a function of the natural logarithm\nof the fitted jump time at 300K with their standard errors. The dashed line corresponds to\nthe linear fit of the data points associated with a cavity frequency resonant with librational\nmodes.\n16Conclusions\nWe have shown that, using molecular dynamics simulations, it is possible to investigate the\neffect of VSC on liquid water as a function of cavity frequencies.\nConcerning equilibrium properties, we found that when the cavity is in resonance with\nO–H stretching and HOH bending, there is no effect on water structure or dielectric constant,\nconfirming what reported by Li et al.23,46On the other hand there is an effect on the dielectric\nconstant, if the cavity is in resonance with librational motion of water.\nDynamical properties, and in particular, diffusion coefficient and H-bond jump time, were\ninvestigated and we found that the cavity can modify them in different ways. When set to\nthe O–H stretching, only the diffusion coefficient is modified, while when set to librational\nmodes both diffusion coefficient and jump time are modified. This suggests that the two\nphenomena are related and the cavity, acting on librational modes, is able to point out it.\nIf we put together the equilibrium and the dynamical analysis, in particular focusing on\nthe H-bond jump mechanism, we can analyse the role played by the cavity. At this end, let\nus consider the simple expression of the rate of the jump process as:\nk(T) =κ(T)kTST(T) =κ(T)ωRe−∆G‡\nkBT (9)\nwhere κ(T) is the transmission coefficient, which measures the deviation of the rate con-\nstant from the transition state theory (TST) rate constant ( kTST(T)),ωRis the characteristic\nfrequency of the reactive mode, ∆ G‡the free energy barrier of the process and kBthe Boltz-\nmann constant. From the simulations, we obtained that ∆ G‡is only slightly modified by\nthe cavity. Previous studies on simple models have shown that the reaction barrier is not\nmodified by the cavity.60By inspecting the RDF, we can also suggest that ωRis not largely\naffected by VSC. Finally, a modification in the transmission coefficient could be at the origin\nof the difference in the jump time (which is the inverse of the rate constant of Equation 9)\nobserved in simulations.\n17In TST, the assumption that all the trajectories passing the TS with non-zero velocity\nshould end-up into products is taken. The deviation from this assumption is measured by\nthe recrossing factor. In classical mechanics (i.e. when there is no tunneling), the recrossing\nfactor corresponds to the transmission coefficient. The coupling with a cavity could act on\nthat factor which corresponds to adding a friction on the rate. We should notice that recent\ntheoretical studies have pointed out that VSC acts as a friction term modifying the TST\nrate constant.24,25,28,61,62\nIn conclusion, we have observed that by using a simple molecular dynamics approach,\nit is possible to observe an effect of vibrational strong coupling, in particular if the cavity\nfrequency is resonant with pertinent matter modes, in the present case the librational modes\nof liquid water. More developed models could then be used to understand the effect of VSC\non different kinetic properties directly on condensed matter simulations.\nAcknowledgement\nThe authors thank Dr. Tao Li for providing support in utilising the CavMD software. 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Nature Comm. 2021 ,12, 1315.\n26On the Modification of Liquid Water Properties\nunder Vibrational Strong Coupling\nSupporting Information\nJessica Bowles,†Damien Laage,‡Johannes Richardi,†Rodolphe Vuilleumier,‡\nand Riccardo Spezia∗,†\n†Laboratoire de Chimie Th´ eorique, Sorbonne Universit´ e, Paris, France.\n‡PASTEUR, D´ epartement de Chimie, ´Ecole Normale Sup´ erieure, PSL University,\nSorbonne Universit´ e, Paris, France\nE-mail: riccardo.spezia@sorbonne-universite.fr\nComputational Details\nHere we report some details on how properties are obtained from our trajectories.\nFrom the independent NVE trajectories (see the manuscript) we have extracted the\ndielectric constant of each one and obtained the average. The standard error of the mean of\ndielectric constant is then obtained as:\ns=/radicaligg/summationtextNT\ni(εi−¯ε)2\n(NT−1)NT. (1)\nTo retrieve the value of the diffusion coefficient ( D) and of the jump time ( τ0) at 300 K,\nwe collect a value from each of the 30 trajectories of 100 ps and, since at odds with dielectric\nconstant, they show a (slight) temperature dependence, we have performed a linear fit. From\n1arXiv:2402.02510v1  [physics.chem-ph]  4 Feb 2024it, we obtained the values at 300 K that we used to obtain the standard error of the mean (the\naverage corresponds to the fitted value at 300 K, confirming that the NVT sampling used\nto generate the NVE trajectories is sufficient). We then calculate the error as the difference\nbetween the value of D (or τ) from the simulation ( Dsimorτsim) and the value of Dfit(or\nτfit) corresponding to the fit at the value of the average temperature of the simulation:\nσD=/radicaligg/summationtextNtrajs\ni (Dsim−Dfit)2\nN−1(2)\nFor calculating the infrared (IR) spectrum and the velocity density of states (VDOS)\nspectrum, the following equations where used:\nIIR=/integraldisplay+∞\n−∞e−iωt⟨⃗ µs(0)·⃗ µs(t)⟩dt (3)\nIVDOS =/integraldisplay+∞\n−∞e−iωt⟨⃗ v(0)·⃗ v(t)⟩dt (4)\nwhere ⃗ µsis the total dipole moment and vthe velocities of the atoms.\n2IR and VDOS spectra along Z\nNo cavity\n01000 2000 3000 4000 5000\n01000 2000 3000 4000 5000\n3550cm1\n1640cm1\n800cm1\n600cm1\n400cm1\n3550cm1\n1640cm1\n800cm1\n600cm1\n400cm1\nNo cavityIR VDOS\nFrequency (cm-1) Frequency (cm-1)\nFigure S1: IR and VDOS spectra along Z with cavity frequencies of 400, 600, 800, 1650,\n3550 cm−1and excluding the cavity\nIndividual values of the dielectric constant\n0 5 10 15 20 25 30\nN trajs405060708090100Dielectric constanteNo cavity\n400cm1\n600cm1\n0 5 10 15 20 25 30\nN trajs30405060708090100Dielectric constanteNo cavity\n1640cm1\n3550cm1\nFigure S2: Values of the dielectric constant for each of the 30 trajectories and the average\nshown as a vertical line. (left) with cavity frequencies of 300 and 600 cm−1where there is\nan influence of the cavity (right) with cavity frequencies of 1640 and 3550 cm−1where there\nis not an influence of the cavity\n3Values of D and τat 300 K and their standard errors\nTable S1: Values of diffusion coefficient and their error corresponding to Fig. 3a\nωC(cm−1) Diffusion coefficient (A2·ps−1) error\nNo cavity 0.197 0.003\n300 0.193 0.003\n400 0.191 0.002\n600 0.200 0.003\n800 0.193 0.003\n1640 0.198 0.003\n3550 0.190 0.002\nTable S2: Values of the jump time and their error corresponding to Fig. 3b\nωC(cm−1) Jump time (ps) error\nNo cavity 3.90 0.02\n300 3.99 0.03\n400 4.00 0.03\n600 3.86 0.03\n800 3.97 0.03\n1640 3.90 0.03\n3550 3.93 0.02\nDependence of the diffusion coefficient and the jump\ntime\nIf one assumes the following relation between the diffusion coefficient (D) and the hydrogen\nbond jump time ( τ0):\nD=aτn\n0 (5)\nthen taking the logarithm it reads:\nln(D) = ln( a) +nln(τ0) (6)\nThis is at the basis of a linear relation of the logarithms of Dandτ0corresponding to\n4the fit plotted on Fig. 4. From the fit we obtained : n=−1.2 and ln(a) = 0 .00012 which\nmeans that a≃1 and finally D=τ(−1.2)\n0 .\n5"    },    {        "title": "2402.02512v2.Addressing_the_Hubble_tension_in_Yukawa_cosmology_.pdf",        "content": "Addressing the Hubble tension in Yukawa cosmology?\nKimet Jusufi,1,∗Esteban Gonz´ alez,2,†and Genly Leon3, 4,‡\n1Physics Department, State University of Tetovo,\nIlinden Street nn, 1200, Tetovo, North Macedonia\n2Departamento de F´ ısica, Universidad Cat´ olica del Norte,\nAvenida Angamos 0610, Casilla 1280, Antofagasta, Chile\n3Departamento de Matem´ aticas, Universidad Cat´ olica del Norte,\nAvenida Angamos 0610, Casilla 1280, Antofagasta, Chile\n4Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa\nIn Yukawa cosmology, a recent discovery revealed a relationship between baryonic matter and the\ndark sector. The relation is described by the parameter αand the long-range interaction parameter\nλ- an intrinsic property of the graviton. Applying the uncertainty relation to the graviton raises\na compelling question: Is there a quantum mechanical limit to the measurement precision of the\nHubble constant ( H0)? We argue that the uncertainty relation for the graviton wavelength λcan be\nused to explain a running of H0with redshift. We show that the uncertainty in time has an inverse\ncorrelation with the value of the Hubble constant. That means that the measurement of the Hubble\nconstant is intrinsically linked to length scales (redshift) and is connected to the uncertainty in\ntime. On cosmological scales, we found that the uncertainty in time is related to the look-back time\nquantity. For measurements with a high redshift value, there is more uncertainty in time, which\nleads to a smaller value for the Hubble constant. Conversely, there is less uncertainty in time for\nlocal measurements with a smaller redshift value, resulting in a higher value for the Hubble constant.\nTherefore, due to the uncertainty relation, the Hubble tension is believed to arise from fundamental\nlimitations inherent in cosmological measurements. Finally, our findings indicate that the mass of\nthe graviton fluctuates with specific scales, suggesting a possible mass-varying mechanism for the\ngraviton.\nI. INTRODUCTION\nThe “Hubble tension” - the discrepancy in H0mea-\nsurements from different cosmological probes at different\nredshifts - is a well-known problem in modern cosmology\n[1–6]. Nowadays, we have measurements for H0(which\ndescribes the rate at which the current Universe is ex-\npanding) that are based on observations of objects in\nour local Universe, say Cepheid variable stars and Type\nIa supernovae (SNe Ia) [7] and, on the other hand, we\nhave the measurements on H0based on the Cosmic Mi-\ncrowave Background (CMB) radiation left over from the\nBig Bang, which provides a snapshot of the early uni-\nverse [8]. The tension arises because these two methods\nyield different results for H0. Observations from the local\nUniverse give a higher value for the Hubble constant com-\npared to predictions based on the CMB and the standard\ncosmological model.\nBased on current observations, the prevailing cosmo-\nlogical model depicts our Universe as homogeneous and\nisotropic on large scales. A particularly intriguing reve-\nlation is the existence of cold dark matter, an enigmatic\nform of matter that interacts solely through gravitational\nforces [9–12]. Despite numerous attempts, the direct de-\ntection of dark matter particles continues to be challeng-\ning, and its existence is only inferred through the gravi-\n∗kimet.jusufi@unite.edu.mk\n†esteban.gonzalez@ucn.cl\n‡genly.leon@ucn.cltational effects it imparts on galaxies and broader cosmic\nstructures [13].\nIn addition to the mysteries surrounding dark matter,\ndark energy has been introduced to explain the observed\naccelerated expansion of the Universe, as highlighted by\nmultiple independent observations. This dark energy is\nclosely associated with the cosmological constant [14–\n17]. The theoretical framework that comprehensively\ndescribes the observed cosmological phenomena is the\nΛCDM paradigm, the most successful model in modern\ncosmology. However, fundamental questions persist re-\ngarding the nature and behavior of dark matter and dark\nenergy [18–20]. A deep understanding of the Universe’s\nphysical depiction remains elusive even though scalar\nfields are introduced, as in the inflationary scenario\n[21–25]. Recent inconsistencies among cosmological\ndatasets have underscored tensions within the standard\nΛCDM model, prompting questions about its accuracy\nin describing the entire evolution and dynamics of the\nuniverse [26–28]. Some authors suggest that it may be\npossible to interpret the H0-tension as a problem of\nunsynchronized calibrations between the early and late\nUniverse, along with degeneracy with other parameters\nof our cosmological model [29]. In [30–32], the authors\nalso analyze the H0tension in modified cosmology.\nRegarding the oddities in observables that seem to\nhint beyond ΛCDM cosmologies, with deviations at\nleast from homogeneous and isotropic cosmologies, and\npossibly even alternatives to General Relativity [33].\nModified gravity reviews discuss cosmological issues,\nsuch as inflation, bounce, singularities, and perturba-\ntions [34–37].arXiv:2402.02512v2  [astro-ph.CO]  6 Mar 20242\nUsing the Yukawa potential in galactic systems, one\ncan gain insights into the nature of dark matter. In\nthis model, dark matter is postulated to be explained\nthrough the coupling between baryonic matter mediated\nby a long-range force represented by the Yukawa grav-\nitational potential. The Yukawa model introduces two\nessential parameters: the coupling parameter αand the\neffective length parameter λ, linked to the graviton mass.\nThe Yukawa potential finds application in various scenar-\nios, including f(R) gravity [35, 36, 38–42]. As demon-\nstrated in recent studies [43, 44], the Yukawa potential\ncan yield the ΛCDM as an effective model using the rela-\ntion between baryonic matter (Ω B,0), dark energy (Ω Λ,0)\nand dark matter (Ω DM, 0), i.e. Ω DM, 0=p\n2 ΩB,0ΩΛ,0.\nA similar relation has been recently obtained using the\nemergent nature of gravity, yielding a correspondence be-\ntween emergent gravity and modified gravity [45, 46]. In\nthis framework, dark matter appears as an apparent ef-\nfect arising naturally from the long-range force associated\nwith the graviton, modifying Einstein’s gravity at large\ndistances. The distribution of baryonic matter under-\ngoing the Yukawa-like gravitational interaction dictates\nthe amount of dark matter in this perspective [43, 44, 47].\nReferences [48, 49] discusses the Yukawa potential in the\ngalactic center and the solar system.\nIn this paper, our primary focus is to investigate the\nHubble tension within the framework of Yukawa cosmol-\nogy. Specifically, our goal is to illustrate the potential\nconnection between the Hubble tension and the quantum\nlimitations of measurements associated with the gravi-\nton in cosmological contexts. We argue this is a nat-\nural consequence in Yukawa cosmology since the wave-\nlength λis inherent to the graviton and plays a fun-\ndamental role in describing the energy density λvia\nΩΛ,0=c2α/[λ2H2\n0(1 + α)2] [43]. Possible implications\nof uncertainty relations in the Hubble tension were pro-\nposed recently in [50–52], but applied for the case of mas-\nsive photons. Still, the novel idea in the present paper is\nthatλis linked to the dark sector.\nThe structure of the paper is as follows. Section II\ncomprehensively reviews the Yukawa potential derived\nfrom f(R) gravity. In section III, we examine the mod-\nified Friedmann equations within the context of Yukawa\ncosmology, shedding light on the cosmological implica-\ntions of the proposed modifications. In Section IV, we\naddress the Hubble tension. Section V tests our result\nusing observations. Finally, we conclude in section VI.\nII. YUKAWA POTENTIAL AND MODIFIED\nGRAVITY\nLet us briefly elaborate on the emergence of Yukawa’s\ngravitational potential. One possible way to obtain such\nYukawa-type modifications to the Newtonian potential\nis to consider the weak field limit of extended theories\nof gravity, exemplified by f(R) models [34, 53, 54]. Inthese modified models of gravity, we can write down the\ngravitational action for f(R) gravity plus matter term as\nI=1\n16πGZ\nd4x√−g f(R) +Imatter [gµν, ϕ]. (1)\nIn this context, Rrepresents the Ricci scalar, Gde-\nnotes the Newtonian gravitational constant, and gis\nthe determinant of the metric tensor, gµν. Notably, in\nthe limit f(R)→R, the action reverts to the Einstein-\nHilbert action of general relativity. By taking variations\nof (1) with respect to gµν, we derive the field equations\nf′(R)Rµν−1\n2f(R)gµν−f′(R);µν+gµν□f′(R) = 8 πG T µν,\n(2)\nwhere Tµνrepresents the matter energy-momentum ten-\nsor. The semicolon and prime symbols denote the co-\nvariant derivative and the derivative with respect to R,\nrespectively, while □represents the d’Alembert operator.\nTaking the trace of Eq. (2), one deduces\n3□f′(R) +f′(R)R−2f(R) = 8 πG T. (3)\nTo illustrate the emergence of the Yukawa-like potential\ninf(R) theories, we consider the associated field equa-\ntions in the presence of matter. In the weak field limit, we\ncan perturb the metric tensor as gµν=ηµν+hµν,where\n|hµν| ≪ηµνrepresents a small perturbation around the\nMinkowski spacetime, ηµν. Assuming the analyticity of\nf(R), one can express the Taylor series [39, 42, 55], say,\nf(R) =f(R0)+f′(R0)(R−R0)+f′′(R0)(R−R0)2/2+. . . .\nSubsequently, imposing spherical symmetry, one can ob-\ntaing00=−(1−2Φ(r)), with the emergence of the grav-\nitational Yukawa-like potential\nΦ(r) =−GM\nr\u0000\n1 +α e−r\nλ\u0001\n, (4)\nwhere λis the range of interaction due to the massive\ngraviton. The Newtonian potential is obtained in the\nlimit α= 0. The deep origin of this parameter is un-\nknown; however, later on, we shall focus on the physical\ninterpretation of the parameter α. In the last equation,\nthe rescaling has been used G→G(1+α), along with the\ndefinitions f′(R0) = 1 + α, and λ2=−1+α\n6f′′(R0). The ob-\nservational constraints on the parameters of the Yukawa\ngravity were studied extensively in the literature. We\ndirect the reader for a comprehensive discussion on the\nvalue and sign of the parameter αto align with observa-\ntions to [56–59].\nSpecific reasons for exploring modified gravity arise\nfrom recent findings indicating the breakdown of the\nNewton–Einstein law of gravity at low accelerations in\nwide binary systems [60–62]. In addition, we briefly\nreview [43, 44] concerning important implications of\nYukawa potential in cosmology, in particular on the role\nofαandλon the nature of dark matter and dark energy.3\nIII. YUKAWA COSMOLOGY AND ΛCDM\nMODEL\nThe Yukawa potential can be conveniently expressed in\nterms of an effective length, which may be considered the\nwavelength of a massive graviton. Thus, the gravitational\npotential adopts the following form [43, 63]\nΦ(r) =−GMm\nr\u0000\n1 +α e−r\nλ\u0001\n,where λ=ℏ\nmgc.(5)\nUtilizing the relation F=−∇Φ(r), we deduce the cor-\nrection to Newton’s law of gravitation\nF=−GMm\nr2\u0014\n1 +α\u0012r+λ\nλ\u0013\ne−r\nλ\u0015\n. (6)\nNow, let us consider the flat spacetime background char-\nacterized by the FLRW metric\nds2=−dt2+a2(t)\u0002\ndr2+r2(dθ2+ sin2θdϕ2)\u0003\n.(7)\nIn this metric, ˜ rA=a(t)rrepresents the apparent FRW\nhorizon radius, where a(t) denotes the normalized scale\nfactor as a function of cosmic time, t. Assuming the\nmatter source is modeled as a perfect fluid with energy\ndensity ρand pressure p, the energy-momentum tensor\ncan be expressed as\nTµν= (ρ+p)uµuν+pgµν, (8)\nalong with the continuity equation\n˙ρ+ 3H(ρ+p) = 0 , (9)\nwith H≡˙a/abeing the Hubble parameter. Therefore,\nthe first Friedmann equation for a flat universe reads [43]\nH2=8πGeff\n3X\niρi+4πGeff\n3˜r2\nAX\niΓ2(ωi)ρi,(10)\nwhere ˜ r2\nA= 1/H2,Geff≡G(1 +α) and\nΓ2(ωi)≡α(1 + 3 ωi)\nλ2(1 +α)(1−3ωi)\f\f\f\f\f\nωi=0, (11)\nwith ωi=pi/ρibeing the equation of state param-\neter of the i-th cosmic species. Observing an appar-\nent singularity in the last equation at ω= 1/3 (ra-\ndiation) adds an intriguing result to our understand-\ning, suggesting a potential phase transition in the early\nUniverse from a radiation-dominated state to a matter-\ndominated one. In [43], the argument is made for em-\nploying two Friedman equations. The revised Friedmann\nequations are proposed for distinct phases of the Uni-\nverse: one accounting for radiation and quantum effects\nduring the early Universe (radiation-dominated epoch),\nand the other applicable to the late-time Universe post-\nphase transition, characterized by a matter-dominated\nphase where Yukawa modifications to gravity assume a\nsignificant role. In what follows, we shall elaborate on\ntwo cases of Yukawa cosmology for the late-time Uni-\nverse: the full model and the approximated model.A. Full Yukawa model\nThe sum will be omitted hereafter since we consider\nonly the case with ωi= 0. After we use ρcrit=3\n8πGH2\n0,\nwe get two solutions\nE2(z) =(1 +α)\n2X\niΩi\n±p\n(P\niΩi)2(1 +α)2+ 2Γ 2(ωi)Ωi(1 +α)/H2\n0\n2,\n(12)\nwhere E(z) = H(z)/H0,Ωi= Ω i,0(1 + z)3(1+ωi)and\nΩi,0= 8πGρ i0/(3H2\n0).\nIn Yukawa cosmology, we have two matter componentsP\niΩi= Ω B,0(1 +z)3+ Ω Λ,0, where z≡a−1−1 is the\ncosmological redshift, and Ω B,0and Ω Λ,0are the density\nparameters related to matter and dark energy, respec-\ntively. As was argued in [43], the physical interpretation\nof the term 2Γ 2(ωi)Ωi(1 +α)/H2\n0is closely linked to the\npresence of dark matter, which appears as an apparent\neffect in Yukawa cosmology [we consider ωi= 0, i.e., the\neffect of cold dark matter]\nΩ2\nDM(1 +α)2\n(1 +z)3≡2Γ2Ωi(1 +α)\nH2\n0. (13)\nSpecifically, it has been shown that the dark matter den-\nsity parameter can be related to the baryonic matter as\n(here we shall introduce the constant c)[43]\nΩDM=cp\n2αΩB,0\nλH0(1 +α)(1 +z)3, (14)\nwhere the subscript ‘0’ denotes quantities evaluated at\npresent, specifically at z= 0. That suggests that dark\nmatter can be interpreted as an outcome of the modified\nNewton law, characterized by αand Ω B. Let us introduce\nthe constant cusing the following definition [43]\nΩΛ,0=c2α\nλ2H2\n0(1 +α)2. (15)\nSubsequently, one can derive an expression that estab-\nlishes a relation between baryonic matter, effective dark\nmatter, and dark energy\nΩDM(z) =p\n2 ΩB,0ΩΛ,0(1 +z)3. (16)\nConsidering as a physical solution only the one with the\npositive sign, we have\nE2(z) =(1 +α)\n2\u0000\nΩB,0(1 +z)3+ Ω Λ,0\u0001\n+(1 +α)\n2s\n(ΩB,0(1 +z)3+ Ω Λ,0)2+Ω2\nDM(z)\n(1 +z)3.\n(17)\nWe shall refer to the last equation as the full Yukawa\nmodel in cosmology.4\nB. Approximate solution: Recovering ΛCDM\nLet us elaborate more on the late-time Universe in\nYukawa cosmology. One can show how effectively the\nΛCDM can be obtained. Let us consider an expansion\naround Ω2\nDM/(1 +z)3; then in leading-order terms, one\nfinds\nE2(z) = (1 + α)h\n˜ΩB,0(1 +z)3+ Ω Λ,0i\n,(18)\nwhere\n˜ΩB,0= Ω B,0\n1 +1\n2\u0010\n1 +ΩB,0\nΩΛ,0(1 +z)3\u0011\n. (19)\nAs expected, the dark matter contribution is absorbed\nin the first ˜ΩB,0. That is expected since we have only\na baryonic matter and cosmological constant in Yukawa\ncosmology, and the dark matter appears only as an ap-\nparent effect as given by Eq. (14) or Eq. (16). To obtain\nthe ΛCDM, first, we need to make the transition from\nYukawa cosmology to ΛCDM by making the replacement\n(1 +α)˜ΩB,0= ΩΛCDM\nB,0+ ΩΛCDM\nDM, 0, (20)\nyielding [43, 44]\nE2(z) =\u0000\nΩΛCDM\nB,0+ ΩΛCDM\nDM, 0\u0001\n(1 +z)3+ ΩΛCDM\nΛ,0,(21)\nwhere H0=HΛCDM\n0 along with the definitions\nΩΛCDM\nDM, 0≡(1 +α)p\n2 ΩB,0ΩΛ,0=q\n2ΩΛCDM\nB,0ΩΛCDM\nΛ,0,\nΩΛCDM\nΛ,0≡(1 +α) ΩΛ,0=c2α\nλ2H2\n0(1 +α),\nΩΛCDM\nB,0≡(1 +α) ΩB,0. (22)\nAs discussed in the preceding section, the parameter\nαplays a crucial role in altering Newton’s law. How-\never, an intriguing question emerges regarding the fun-\ndamental origin of this parameter. As was pointed out\nin [45], in some deep sense, αcould potentially be in-\nfluenced by the entanglement entropy arising from the\nintricate interplay between baryonic matter and the fluc-\ntuations in the gravitational field (gravitons). A subtle\nindication of this possibility becomes apparent when we\nutilize ΩΛCDM\nΛ,0 =ρΛ,0/ρcrit, where ρcrit= 3H2\n0/(8πG)\nandρΛ,0= Λc2/(8πG). From Eq. (15) one can obtain\nthe cosmological constant in terms of α, as follows\nΛ∼3α\nλ2. (23)\nIn this perspective, the energy density can be interpreted\nas a holographic representation of dark energy, serving as\nan expression for the cosmological constant\nρΛ∼3α c2\n8πGλ2. (24)In the next section, we shall argue that there exists a\nrelation between the Hubble constant and λviaH0=\nc/λ, where the role of the Hubble scale Lis played by\nλ. The emergence of the parameter αin our model may\nbe attributed to the influence of entanglement entropy,\nprimarily due to the presence of gravitons. That suggests\na potential scenario where the entropy of the Universe\nundergoes modification by the gravitons, leading to the\nchanges observed in gravity.\nIV. ADDRESSING THE HUBBLE TENSION IN\nYUKAWA COSMOLOGY\nRecent studies [64, 65] suggest that H0decreases grad-\nually with redshift, as observed in the Pantheon sample\nand Baryon Acoustic Oscillations (BAO). This evolv-\ningH0could help resolve Hubble tension. In [66], the\nRisaliti-Lusso relation for Quasars (QSOs) was combined\nwith SNe Ia to determine that non-calibrated Quasars\ncombined with SNe Ia and the ones corrected for the red-\nshift evolution are compatible with a flat ΛCDM model\nwith Ω m,0= 0.3 and H0= 70 km/s/Mpc. The H0values\nfall between the one from SNe Ia and the CMB.\nThe paper [67] combines various cosmological probes\nto constrain the ΛCDM model. Non-Gaussian likelihoods\nwere used to measure the matter-density parameter Ω m\nandH0with small uncertainties. The authors found that\nH0values were compatible with the Tip of the Red Gi-\nant Branch. Calibration for SNe Ia was shown to impact\ncosmological results. In [68], the authors found that the\nnon-Gaussianity of residuals affects the distance moduli\nof SNe Ia and can reduce the uncertainties of the param-\neters H0and Ω m,0in the ΛCDM model by 40%. In the\nstudy [69], the authors used a combination of data from\ndifferent cosmological models to constrain cosmological\nparameters and found a significant deviation from the\nflat ΛCDM model towards a phantom scenario.\nIn the study conducted by Dainotti et al. [70], a\ncombination of data from QSOs, Gamma-Ray Bursts\n(GRBs), SNe Ia, and BAO was used to conclude the cos-\nmological parameters of non-flat ΛCDM and flat wCDM\nmodels. The researchers used traditional Gaussian and\nnew best-fit likelihoods to analyze the data. The results\nshowed that when the new likelihoods were used instead\nof the Gaussian ones, uncertainties were reduced by up\nto 35% for all parameters concerning H0. In Bargiacchi\net al.’s study [71], the researchers used the cosmographic\napproach to fit QSOs alone and QSOs combined with\nGRBs, SNe Ia, and BAO. The results showed that the\nflat ΛCDM model was in strong tension ( >4σ) with the\ndata sets, indicating the power of both the cosmographic\napproach and high-redshift sources.\nLet us discuss in more detail some of the implications\nof Yukawa cosmology on the Hubble tension. In doing so,\nwe will consider two cases: the full Yukawa cosmological\nmodel given by Eq. (17) and the approximated solution,\nwhich is effectively the ΛCDM, provided by Eq. (21).5\nA. Running Hubble tension and H0diagnostic\nAs a first model, we shall review a possible resolution\nof the Hubble tension via the running H0diagnostic re-\ncently proposed in [72]. According to this approach, the\nmismatch between the Hubble constant may be achieved\nthrough the H0diagnostic ( H0) defined by\nH0=H(z)r\u0010\nΩΛCDM\nB,0+ ΩΛCDM\nDM, 0\u0011\n(1 +z)3+ ΩΛCDM\nΛ,0.(25)\nThe above expression can explain the Hubble constant\nmismatch, which stems from a difference between the ef-\nfective equation of state of the Universe within the FLRW\ncosmology framework and the current ΛCDM model.\nThis paper will use the full Yukawa and approximated\nmodels, but it is argued that this approach generally\ndoes not yield satisfactory results for either model. The\nlast relation lacks physical motivation and may merely\napproximate a deeper relation for the Hubble constant.\nAccording to [73], all observables indicate that there is an\nevolution towards higher values of Ω mbetween low and\nhigh redshifts, which is consistent with the expectation\nthat they may deviate from Planck’s values. However, it\nis essential to note that this research does not consider\nselection effects and other systematics across multiple ob-\nservables such as SNe Ia, cosmic chronometers, BAO,\nand QSOs. The study supports the idea that the flat\nΛCDM model is a dynamic model where the fitting pa-\nrameters evolve with time, cautioning that cosmological\ntensions may arise from the assumption that ( H0,Ωm,0)\nare unique within the flat ΛCDM model. The result re-\nmains the same when upgrading the SNe to the Pan-\ntheon+ dataset [74]. Even after upgrading older Lyman-\nαBAO to newer Lyman- αBAO, the observed trends in\ndata persist [75].\nIn what follows, we will give another approach to run-\nning a Hubble constant based on quantum approaches.\nB. Running Hubble tension with uncertainty\nrelations\nAs we noted, the H0diagnostic defined by the last\nequation when applied to ΛCDM model fails to provide\nsatisfactory results. Therefore, one needs to adopt a spe-\ncific equation of state for the cosmological model. Bel-\nlow, we shall give another approach to obtain a running\nHubble constant with the redshift where a crucial role\nis played by the look-back time (and was also studied\nrecently in [76] concerning its’ possible role on the Hub-\nble tension). Here, the look-back time quantity can be\ninterpreted as the uncertainty in time in cosmological ob-\nservations. The wavelength λis intricately linked to the\ngraviton mass. However, applying the uncertainty prin-\nciple from quantum mechanics to the graviton introduces\nintriguing results, as we shall see. Consider the follow-ing: a large uncertainty in position corresponds to a re-\nduced uncertainty in momentum (or graviton mass), and\nconversely, less uncertainty in position leads to height-\nened uncertainty in mass. Varied graviton mass values\nshould naturally emerge from diverse observations due\nto the inherent uncertainty relation. In the case of cos-\nmological scales, the precision of the graviton mass can\nbe notably enhanced compared to studies focusing on\ngalactic distances. This improvement arises from using\ncosmological distances, aligning with the uncertainty re-\nlation’s principles. The key insight here is that the Hub-\nble constant ( H0) tension may find its roots in quantum\nmechanical constraints inherent in cosmological measure-\nments. There may not be a necessity for introducing new\nphysics or other exotic explanations. In particular, the\nwavelength’s dependence on specific measurements with\ndistinct redshifts. The quantum mechanical limitations\nmanifest differently based on the characteristics of the\nmeasurements, thereby influencing the derived proper-\nties of the graviton, such as its mass. With this in mind,\nlet us assume for a given observation that\nλ(z)∼vg∆t∼ℏ\nmgc. (26)\nHence, one can formulate the uncertainty relation in\nterms of position and momentum as\n∆x∆p∼ℏ,where λ(z)∼∆x,andmgc∼∆p.\nTake note of the expression vg∆t=dg, denoting the\ntravel distance for the graviton. Here, vgrepresents the\nspeed of the graviton, and our expectation aligns with\nvg∼c, where csymbolizes the speed of light. Interest-\ningly, Eq. (26) can be rephrased in the context of the\nenergy-time uncertainty relation\n∆Eg∆t∼ℏ, (27)\nwhere\nmgc2∼∆Eg. (28)\n∆tgives the time for the graviton to reach us on Earth\nfrom a given distance. One can link this time to the age\nof the Universe t0considering the difference between this\nquantity, evaluated today, and that computed at a given\nredshift t(z). Namely, we can write\n∆t=t0−t(z) =t0−a(t)t0=zt0\n1 +z. (29)\nOn the other hand, we can compute ∆ tusing the Hubble\nparameter\nH(z) =˙a\na=1\nada\ndt. (30)\nThis equation implies\n∆t=t0−t(z) =Zt0\nt(z)dt=1\nH0Zz\n0dz′\n(1 +z′)E(z′),(31)6\nwhere t0≃13.797Gyr that provides the present values\nof the critical density parameters for matter and a dark\nenergy component, respectively.\nFrom Eqs. (29), (30) and (31) we obtain\nH0(z) =1 +z\nzt0Zz\n0dz′\n(1 +z′)E(z′). (32)\nIn the present paper, we have two cases: the full Yukawa\nsolution where E(z) is given by Eq. (17) and the ap-\nproximated Yukawa solution where E(z) is given by Eq.\n(21).\nPut differently, Eq. (32) indicates that the Hubble\nconstant’s value is contingent on the particular measure-\nment associated with a redshift z. This equation was\nobtained from Eq. (31), which is also known as the look-\nback time and was studied recently in [76] concerning\nits’ possible role on the Hubble tension. In the present\npaper, we argued that the look-back time quantity can\nbe interpreted as the uncertainty in time by Eq. (27)\nwhen applied to the massive graviton. The current work\nalso demonstrates the inherent emergence of this equa-\ntion within the framework of Yukawa cosmology.\nAs evident from Eq. (31), there exists an inverse re-\nlation between the uncertainty in time and the value of\nthe Hubble constant\n∆t∼z t0\n1 +z∼k\nH0, (33)\nor, if we use Eq. (26) we can write alternatively,\n∆t∼∆x\nc∼k\nH0. (34)\nwhere kis some factor of proportionality. For exceed-\ningly large redshift values, such as those associated with\nthe CMB Radiation, we encounter a scenario with greater\nuncertainty in time and less uncertainty in the graviton\nmass. Given that the uncertainty in time is inversely\nproportional to the H0, this results in a situation where\nwe get a smaller value for the Hubble constant, which\nagrees with the H0= 67 .4±0.5 km/s/Mpc value re-\nported in [8]. Conversely, for smaller redshift values or lo-\ncal measurements, as observed in galactic scales or cases\nlike Cepheids and SNe Ia, there is reduced uncertainty\nin time, leading to heightened uncertainty in the gravi-\nton mass. Consequently, this effect results in a higher\nvalue for the Hubble constant, which also agrees with\ntheH0= 73.04±1.04 km/s/Mpc value reported in [7].\nRewriting Eq. (34) in terms of λand differentiating it,\nλ(z)∼c k\nH0(z)=⇒dλ=c k\f\f\f\f\fdH0\nH2\n0\f\f\f\f\f. (35)\nOne can also get a relation between the uncertainty in\nthe graviton’s mass and the uncertainty in the Hubble\nconstant given by\n∆λ=c k\nH0\f\f\f\f\f∆H0\nH0\f\f\f\f\f=λ\f\f\f\f\f∆H0\nH0\f\f\f\f\f, (36)where dλ≃∆λ, and dH0≃∆H0. Since λ=ℏ/(mgc),\nwe can finally write\n\f\f\f\f\f∆λ\nλ\f\f\f\f\f∼\f\f\f\f\f∆mg\nmg\f\f\f\f\f∼\f\f\f\f\f∆H0\nH0\f\f\f\f\f. (37)\nA similar result was obtained for the case of massive pho-\ntons in [50]. In the present paper, we argued that this\nintricacy in the Hubble constant can thus be explained\nthrough the lens of fundamental limitations stemming\nfrom the uncertainty relation when applied to the gravi-\nton. The measurement of the Hubble constant is intri-\ncately tied to length scales and is intricately linked to\nthe uncertainty in time. Hence, the challenges associ-\nated with the Hubble constant find their roots in these\nfundamental limitations dictated by the uncertainty rela-\ntion in the context of graviton measurements. According\nto (35), we can also say that the uncertainty in the gravi-\nton mass is proportional to the uncertainty in the Hubble\nconstant.\nV. OBSERVATIONAL CONSTRAINTS\nIn this section, we will study the possibility of the\nYukawa cosmology in addressing the H0tension by con-\nfronting the full Yukawa cosmological model and their\nΛCDM approximation with the SNe Ia data. For that\nend, we compute the best-fit parameters at 1 σ(68.3%)\nof confidence level (CL) with the affine-invariant Markov\nchain Monte Carlo (MCMC) method [77], implemented\nin the pure-Python code emcee [78]. In this procedure,\nwe have considered 100 chains or “walkers” and the au-\ntocorrelation time τcorr, provided by the emcee module,\nas a convergence test, computing at every 50 steps the\nvalue of τcorrof each free parameter. If the current step is\nlarger than 50 τcorrand the value of τcorrchanged by less\nthan 1%, then we will consider that the chains have con-\nverged and the constraint is stopped. This convergence\ntest is complemented with the mean acceptance fraction\n(MAF), which must have a value between 0 .2 and 0 .5\n[78], and can be modified by the stretch move provided\nby the emcee module. For the statistics, we discard the\nfirst 5 τcorrsteps as “burn-in” steps, thin by τcorr/2, and\nwe flatten the chains. Finally, since we are performing a\nBayesian statistical analysis, we construct the following\nGaussian likelihood function:\nLSNe∝exp\u0012\n−χ2\nSNe\n2\u0013\n, (38)\nwhere χ2\nSNeis the merit function for the SNe Ia data,\nwhose construction is described in subsection V A.\nA. Type Ia supernovae data\nWe will examine the SNe Ia data through the Pan-\ntheon+ sample [79], an enhanced version succeeding the7\noriginal Pantheon sample [80]. This dataset comprises\n1701 data points within the redshift range of 0 .001≤\nz≤2.26. The merit function can be briefly formulated\nin matrix notation to facilitate our analysis, denoted by\nbold symbols, as:\nχ2\nSNe=∆D(z, θ, M )†C−1∆D(z, θ, M ), (39)\nwith [ ∆D(z, θ, M )]i=mB,i−M−µth(zi, θ) and C=\nCstat+Csysthe total uncertainty covariance matrix,\nwhere the matrices CstatandCsysaccounts for the statis-\ntical and systematic uncertainties, respectively. In this\nequation, µi=mB,i−Mrepresents the observational\ndistance modulus for the Pantheon+ sample, which is\nderived from a modified version of Tripp’s formula [81]\nand the use of the BBC (BEAMS with Bias Corrections)\napproach [82]. Consequently, the Pantheon+ sample pro-\nvides the corrected apparent B-band magnitude mB,iof\na fiducial SNe Ia at redshift zi, with Mrepresenting the\nfiducial magnitude of an SNe Ia. This last nuisance pa-\nrameter must be performed jointly with the model’s free\nparameters θunder investigation, for which we consider\nin our MCMC analysis the flat prior −20< M < −18.\nOn the other hand, the theoretical distance modulus for\na spatially flat FLRW spacetime is given by\nµth(zi, θ) = 5 log10\u0014dL(zi, θ)\nMpc\u0015\n+ 25, (40)\nwith dL(zi, θ) the luminosity distance given by\ndL(zi, θ) =c(1 +zi)Zzi\n0dz′\nHth(z′, θ), (41)\ncdenotes the speed of light measured in units of km/s\nandHthcorrespond to the theoretical Hubble parameter\nof the model confronted with observations.\nLike the Pantheon sample, a degeneracy exists between\nthe nuisance parameter MandH0. To effectively con-\nstrain the free parameter H0solely through SNe Ia data,\nit becomes imperative to incorporate the SH0ES (Super-\nnovae and H0for the Equation of State of dark energy\nprogram) Cepheid host distance anchors, as follows:\nχ2\nCepheid = ∆DCepheid (M)†C−1∆DCepheid (M),(42)\nwhere [∆ DCepheid (M)]i=µi(M)−µCepheid\ni , with\nµCepheid\ni the Cepheid calibrated host-galaxy distance ob-\ntained by SH0ES [7]. Therefore, in establishing the corre-\nspondence, Cepheid distances are utilized as the “theory”\ninstead of employing the model under study to calibrate\nM. This approach accounts for the sensitivity of the dif-\nference µi(M)−µCepheid\ni to both MandH0, while being\nrelatively insensitive to other parameters. In this con-\ntext, the Pantheon+ sample supplies µCepheid\ni , with the\ntotal uncertainty covariance matrix for Cepheid encap-\nsulated within the overall uncertainty covariance matrix\nC. Consequently, we can formulate the merit function\nfor the SNe Ia data as follows:\nχ2\nSNe=∆D′(z, θ, M )†C−1∆D′(z, θ, M ), (43)where\n∆D′\ni=\n\nmB,i−M−µCepheid\ni , i∈Cepheid host\nmB,i−M−µth(zi, θ),otherwise.\n(44)\nAlso, including Cepheid data is crucial to the purposes of\nthis paper and the H0tension. Hence, from now on, we\nwill exclude in our results the consideration of the nui-\nsance parameter Mand concentrate solely on the analy-\nsis of the free parameters specific to each model. Addi-\ntionally, since the best-fit parameters minimize the merit\nfunction, χ2\nmin, we can utilize the evaluation of these best-\nfit parameters as an indicator of the goodness of the fit,\nwhere a smaller value of χ2\nmincorresponds to a better fit.\nIt is important to mention that the theoretical Hubble\nparameter is obtained from Eqs. (16) and (17) in the\ncontext of the full Yukawa cosmological model, where\nthe condition H(z= 0) = H0leads to\n1 +α=2\nΩB,0+ Ω Λ,0+q\nΩ2\nB,0+ Ω2\nΛ,0+ 4Ω B,0ΩΛ,0.\n(45)\nOn the other hand, for the ΛCDM approximation of the\nfull Yukawa model, the theoretical Hubble parameter is\ngiven by Eqs. (21) and (22), and the condition H(z=\n0) = H0leads to\n1 +α=h\nΩB,0+p\n2ΩB,0ΩΛ,0+ Ω Λ,0i−1\n. (46)\nHence, the free parameters for both cosmological mod-\nels are θ={H0; ΩB,0; ΩΛ,0}, for which we consider the\nfollowing flat priors: H0= 100km/s\nMpch, with 0 .55< h <\n0.85, 0 <ΩB,0<0.2, and 0 <ΩΛ,0<1, within the con-\ndition α >0. With these parameters, we can indirectly\ninfer the value of αfrom Eqs. (45) and (46), according\nto the model, λfrom Eq. (15), and the graviton mass\nfrom the expression mg=ℏ/(cλ). We also constrain the\nfree parameters of the ΛCDM model as a reference model,\nwhose Hubble parameter is given only by Eq. (21). How-\never, the SNe Ia data is not able to independently con-\nstraint ΩΛCDM\nB,0and ΩΛCDM\nDM, 0. Thus, for the ΛCDM model,\nwe define Ω m,0≡ΩΛCDM\nB,0+ ΩΛCDM\nDM, 0, where the condition\nH(z= 0) = H0leads to ΩΛCDM\nΛ,0= 1−Ωm,0, and the free\nparameters are θ={H0; Ωm,0}, for which we consider\nthe flat prior 0 <Ωm,0<1. For this paper, where the\nconstraint is made only with late-time data, it is enough\nto obtain a best-fit on Ω m,0in the ΛCDM model. Still,\nfor both Yukawa models, obtaining a best-fit on Ω B,0is\nnecessary because the DM is written in terms of Ω B,0.\nSo, in our study, we also consider a Gaussian prior on\nΩB,0according to the value Ω B,0h2= 0.0224±0.0001\nreported in the last Planck results [8], which is weakly\ndependent on the cosmological model and, therefore, can\nbe considered as a model-independent measure.8\nTotal steps a MAFτcorr\nh Ωm,0 ΩB,0 ΩΛ,0\nΛCDM model\n1250 5 .0 0 .35 24 .3 22 .7 · · · · · ·\nFull Yukawa model\n2600 3 .0 0 .38 35 .8 · · · 50.9 51 .8\nFull Yukawa model Gaussian prior\n3050 3 .0 0 .42 33 .6 · · · 27.4 60 .7\nApproximate Yukawa model\n2800 3 .0 0 .38 37 .2 · · · 55.9 55 .7\nApproximate Yukawa model Gaussian prior\n2500 3 .0 0 .43 29 .7 · · · 23.7 31 .6\nTABLE I. Total number of steps, value for the stretch\nmove (a), mean acceptance fraction (MAF), and autocorre-\nlation time τcorrfor the free parameters space of the ΛCDM,\nFull Yukawa, and approximate Yukawa models. These val-\nues were obtained (except for awhich is given beforehand)\nwhen the convergence test described in Section V is fulfilled\nfor an MCMC analysis with 100 chains and the flat priors\n0.55< h < 0.85, 0 <Ωm,0<1, 0 <ΩB,0<0.2, and\n0<ΩΛ,0<1, within the condition α > 0. Alternatively,\nwe consider for both Yukawa models a Gaussian prior accord-\ning to the value Ω B,0h2= 0.0224±0.0001 reported in [8]. This\ninformation is provided so that our results are replicable.\nB. Results and discussions\nIn Table I, we present the total number of steps, the\nvalue for the stretch move, the mean acceptance fraction,\nand the correlation time for the free parameters space of\nthe ΛCDM, Full Yukawa, and approximate Yukawa mod-\nels. In Table II, we present their respective best-fit values\nandχ2\nmincriteria, with the first ones at 1 σCL; while in\nTable III, we present the inferred values α,λ, and mgfor\nthe Yukawa models, also at 1 σCL. In Figs. 1, 2, and 3,\nwe depict the posterior distribution and joint marginal-\nized regions at 1 σ, 2σ(95.5%), and 3 σ(99.7%) CL, for the\nfree parameters space of the ΛCDM, Full yukawa, and\napproximate Yukawa models, respectively. These results\nwere obtained by the MCMC analysis described in Sec-\ntion V for the SNe Ia data with two different priors on\nΩB,0for the Yukawa models.\nFrom Figures 2 and 3, it can be seen that there is no\nbest-fit for Ω B,0at 3σCL in the Full Yukawa model, and\nwe have an upper bound for the approximate Yukawa\nmodel. On the other hand, at 1 σCL, there is a best-\nfit for Ω B,0in the approximate Yukawa model, and an\nupper bound in the Full Yukawa model. This fact is\nreflected in the best-fit parameters in Table II. In the ap-\nproximated model, the constraint on Ω B,0is exclusively\nderived from SNe Ia data due to the relation between\nΩDM, 0and Ω B,0, according to Eq. (22), effectively break-\ning the degeneracy between these two parameters. Ad-\nditionally, the plots and the best-fits parameters reveal\na difference when a Gaussian prior on Ω B,0is employed,\nleading to improved constraints on the free parameters\nspace of the Yukawa models, including the inferred pa-\nrameters presented in Table III. However, the values of\nFIG. 1. Posterior distribution and joint marginalized regions\nof the free parameter space of the ΛCDM model, obtained by\nthe MCMC analysis described in Section V for the SNe Ia\ndata. The admissible joint regions correspond to 1 σ, 2σ, and\n3σCL, respectively. The corresponding best-fit values are\nshown in Table II.\nχ2\nminremain very similar. On the contrary, in Figs. 4\nand 5, we present plots for the H0diagnostic at 1 σCL\nas a function of the redshift zfor both the approximated\nand full Yukawa solutions, according to Eq. (25) and\nusing the chains of our MCMC analysis described in sec-\ntion V. The H0diagnostic indicates that the dynamical\nvalue of the Hubble constant tends to be lower than 70\nkm/s/Mpc at high redshifts, potentially suggesting an\nalleviation of the Hubble tension. Notably, the running\nHubble constant only alleviates the Hubble tension for lo-\ncal and large redshift values in the case of the full Yukawa\nmodel, unlike the approximated Yukawa model, which is\neffectively very similar to the ΛCDM model, as evident\nfrom the plots. Therefore, this approach does not yield\na satisfactory result for the approximate Yukawa model.\nTo address the latter inconsistency, we will explore fur-\nther in more detail the role of uncertainty relations on the\nHubble tension. In Figs. 6 and 7, we present H0(z) as a\nfunction of the redshift zaccording to Eq. (32) for the\nFull and approximate Yukawa models, respectively. For\nthe plot, we have used the best-fit values presented in Ta-\nble II, taking into account for the numerical integration\nthe limit H(z→0)→1/t0≡H0. Initially, the Hubble\nconstant increases with the increase of z; however, at a\nspecific redshift, there is a turning point after which the\nHubble constant decreases with increasing z.\nAs we pointed out, the Hubble tension arises from a\ndiscrepancy in the measurements of the Hubble constant\nH0, based on observations of objects in our local Universe9\nBest-fit valuesχ2\nminh Ωm,0 ΩB,0 ΩΛ,0\nΛCDM model\n0.734±0.010 0 .333±0.018 · · · · · · 1523.0\nFull Yukawa model\n0.734±0.010 · · · 0.142+0.042\n−0.063 0.292+0.092\n−0.128 1522.6\nFull Yukawa model Gaussian prior\n0.733±0.010 · · · 0.042±0.001 0 .087±0.009 1522 .5\nApproximate Yukawa model\n0.734±0.010 · · · 0.040+0.013\n−0.017 0.470+0.141\n−0.204 1523.0\nApproximate Yukawa model Gaussian prior\n0.734±0.010 · · · 0.042±0.001 0 .496+0.075\n−0.063 1523.0\nTABLE II. Best-fit values at 1 σCL and χ2\nmincriteria for the ΛCDM model, and the full and approximate Yukawa models for\nflat and Gaussian priors on Ω B,0. These values were obtained by the MCMC analysis described in Section V for the SNe Ia data.\nNote that the Hubble parameter at the current time is written in terms of the reduced Hubble constant as H0= 100km/s\nMpch.\nFIG. 2. Posterior distribution and joint marginalized regions of the free parameters space of the full Yukawa model, obtained\nby the MCMC analysis described in Section V for the SNe Ia data, for a flat prior on Ω B,0(left panel) and a Gaussian prior\non Ω B,0(right panel). The admissible joint regions correspond to 1 σ, 2σ, and 3 σCL, respectively. The corresponding best-fit\nvalues are shown in Table II.\n(Cepheid variable stars and SNe Ia) and the measure-\nments based on the CMB radiation. As argued in the\npresent paper, this tension might result from the mea-\nsurements performed on different scales. Based on this,\nwe showed that the following relation should hold:\n\f\f\f\f\f∆λ\nλCMB\f\f\f\f\f∼\f\f\f\f\f∆mg\nmCMBg\f\f\f\f\f∼\f\f\f\f\f∆H0\nHCMB\n0\f\f\f\f\f, (47)\nwhere\n∆λ=λCMB−λSNeIa, (48)\n∆mg=mCMB\ng−mSNeIa\ng, (49)\n∆H0=HCMB\n0−HSNeIa\n0 . (50)Based on our analysis of the SNe Ia data, we have demon-\nstrated that the Yukawa cosmology’s Hubble constant\naligns with the Hubble constant value in the ΛCDM\nmodel. This convergence is detailed in Table II. Namely,\nwe obtained for the Hubble constant HSNeIa\n0 ∼73.4\nkm/s/Mpc. This alignment between the Hubble con-\nstant values in Yukawa cosmology and the ΛCDM model,\nas demonstrated in our analysis of SNe Ia data, prompts\nus to extend this expectation to the analysis of CMB\ndata. We anticipate this agreement will persist, provid-\ning further coherence between the two models. From a\ntheoretical point of view, this fact is shown in Figs. 6\nand 7, where one can see for both models that for high\nredshift ( z∼1000), it is possible to get a value for the10\nFIG. 3. Posterior distribution and joint marginalized regions of the free parameters space of the approximated Yukawa model,\nobtained by the MCMC analysis described in Section V for the SNe Ia data, for a flat prior on Ω B,0(left panel) and a Gaussian\nprior on Ω B,0(right panel). The admissible joint regions correspond to 1 σ, 2σ, and 3 σCL, respectively. The corresponding\nbest-fit values are shown in Table II.\nFIG. 4. H0diagnostic for the full Yukawa model as a function of the redshift z, according to Eq. (25), for a flat prior (left\npanel) and a Gaussian prior (right panel). These results were obtained using the chains of the best-fit values for our MCMC\nanalysis presented in Table II. By definition, the H0diagnostic for the ΛCDM model is constant and equal to the best-fit value\nobtained for H0.\nHubble constant that is close to the value obtained by\nPlanck collaboration [8]. Namely, let us take the value\nHCMB\n0 = 67.40±0.5 km/s/Mpc obtained from [8], to get\n\f\f\f\f\f∆H0\nHCMB\n0\f\f\f\f\f∼0.089. (51)\nNow, since we showed that according to Eqs. (29) and\n(30), there exists a relation between the Hubble constantand wavelength; one can compute:\nλCMB∼c kCMB\nHCMB\n0∼4451kCMBMpc, (52)\nwhere can set kCMB∼1 and, we also have,\nλSNeIa∼c kSNeIa\nHSNeIa\n0∼4087kSNeIaMpc. (53)11\nFIG. 5. H0diagnostic for the approximate Yukawa model as a function of the redshift z, according to Eq. (25), for a flat\nprior (left panel) and a Gaussian prior (right panel). These results were obtained using the chains of the best-fit values for\nour MCMC analysis presented in Table II. By definition, the H0diagnostic for the ΛCDM model is constant and equal to the\nbest-fit value obtained for H0.\nFIG. 6. Plot of H0(z) for the Full Yukawa model as a function of the redshift z, according to Eq. (32), in the low/high redshift\ninterval. For the plot, we have used the best-fit values presented in Table II and we also plot in the low redshift interval the\nvalue of H0obtained by the SH0ES program [7] for further comparison.\nFIG. 7. Plot of H0(z) for the approximate Yukawa model as a function of the redshift z, according to Eq. (32), in the low/high\nredshift interval. For the plot we have used the best-fit values presented in Table II and we also plot in the low redshift interval\nthe value of H0obtained by the SH0ES program [7] for further comparison.12\nInferred values\nα λ [Mpc] mg×10−42[GeV]\nFull Yukawa model\n1.09+1.62\n−0.48 3772+705\n−582 1.70+0.31\n−0.27\nFull Yukawa model Gaussian prior\n6.07+0.48\n−0.44 4833+129\n−121 1.32±0.03\nApproximate Yukawa model\n0.41+1.08\n−0.33 2707+1165\n−1290 2.36+2.15\n−0.71\nApproximate Yukawa model Gaussian prior\n0.35±0.16 2543+384\n−514 2.51+0.64\n−0.33\nTABLE III. Inferred values at 1 σCL for the full and approx-\nimate Yukawa model for flat and Gaussian priors on Ω B,0.\nThese results were obtained using the chains of the best-fit\nvalues for our MCMC analysis presented in Table II. αis\ncomputed from Eq. (45) in the case of the full Yukawa model\nand Eq. (46) in the case of the approximate Yukawa model;\nwhile λis computed using Eq. (15) and the graviton mass\nfrom the expression mg=ℏ/(cλ).\nIn this way we get\n\f\f\f\f\f∆λ\nλCMB\f\f\f\f\f∼0.082. (54)\nTherefore, an excellent agreement exists with the result\n(51). Let us, however, comment briefly on the fact that\nfrom observational constraints given in Table III, we see\nthat we got four values for λand, although all these are\nof Mpc order and close to the value 4087 Mpc, they are\nnot the same. That has to do with the fact that we have\nchains of data/measurements. Hence we can take as an\neffective value for λeffthe average value, which is close\ntoλeff∼3464 Mpc. Comparing it with the equation\n(53), we get kSNeIa∼0.85. On the other hand, for CMB,\nthe value of kCMBmight not be exactly one as we ap-\nproximated it, but close to one, say kCMB∼kSNeIa, and,\nagain, the final result will not change much and can be\nclose to (51). However, more work is needed to obtain\nbetter constraints for λ. One can also compute the corre-\nsponding mass for the graviton, but first let us see from\n(27) that for λCMB= ∆xCMBwe have more uncertainty\nin position but more precision on the momentum,\n∆pCMB∼ℏ\n∆xCMB∼7.69×10−61N s, (55)\ncompared to the case the case for λSNeIa= ∆xSNeIawhere\nwe have less uncertainty in position but more uncertainty\non the momentum,\n∆pSNeIa∼ℏ\n∆xSNeIa∼8.33×10−61N s. (56)\nWe can further compute also the mass for each case\nmCMB∼ℏ\nλCMBc∼2.56×10−69kg, (57)\nand\nmSNeIa∼ℏ\nλSNeIac∼2.79×10−69kg. (58)Finally, if we use (47) we get\n\f\f\f\f\f∆m\nmCMB\f\f\f\f\f∼0.089. (59)\nAgain, as was expected, this result is in excellent\nagreement with Eq. (51). The Hubble tension, there-\nfore, might be explained by the fundamental quantum\nmechanical limitations of the measurements in cosmol-\nogy. The relation between λand H0should be, in\nprinciple, universal and should hold for any measure-\nment to add some values for H0from different measure-\nments. Let us finally point out that the energy den-\nsity of dark energy also depends on λ, and this im-\nplies ρSNeIa\nΛ ∼3α c2/(8πGλ2\nSNeIa), which also implies\nfundamental quantum mechanical limitations in measur-\ningρΛ. On the other hand, from the relation λSNeIa∼\nc kSNeIa/HSNeIa\n0 for the dark energy density parameter\nwe get ΩΛCDM\nΛ,0∼α/[k2\nSNeIa(1 + α)]. Interestingly, the\nuncertainty principle implies a potential dependence of\nthe graviton’s mass on cosmological scales. This suggests\nan underlying mechanism where varying matter densities\nacross cosmological scales influence the mass of the gravi-\nton, akin to the chameleon mechanism [83]. This idea was\nrecently studied within the framework of massive bigrav-\nity theory [84, 85]. If the mass of the graviton indeed\nvaries with specific scales, it could explain observed fluc-\ntuations, such as those seen in the CMB and SNe Ia.\nVI. CONCLUSIONS\nIn the present paper, we used the Yukawa modified\ngravity, which has two parameters, αandλ, related to\nthe graviton mass, to address the Hubble tension. In\nparticular, we used two models: full and approximated\nYukawa models. To address the Hubble tension, first,\nwe used the H0diagnostic Hubble constant, a showdown\nthat it tends to be lower than 70 km/sh/Mpc at high\nredshifts, suggesting a possible alleviation of the Hub-\nble tension. However, the running Hubble constant only\ntends to alleviate the Hubble tension in the case of the full\nYukawa model, unlike the approximated Yukawa model,\nwhich is effectively very similar to the ΛCDM model, in-\ndicating that the resolution of the Hubble tension could\nbe more attainable through our approach.\nThis paper primarily focuses on investigating the im-\npact of uncertainty relations on the phenomenon of the\nHubble tension, constituting its main result. Since dark\nmatter and dark energy are related to each other in terms\nof gravitons’ wavelength λ, it is natural to apply the un-\ncertainty relations to the graviton where λis defined as\nλ=ℏ/(mgc). In doing so, we have obtained the energy-\ntime uncertainty relation, which shows that the uncer-\ntainty in time exhibits an inverse correlation with the\nvalue of the Hubble constant, meaning that the measure-\nment of the Hubble constant is intricately tied to length\nscales and is intricately linked to the uncertainty in time.13\nThe MCMC analyses using the SNe Ia data showed that\nthe Hubble constant in Yukawa cosmology is the same\nas in ΛCDM. In general, this is expected to be the case\nfor CMB data. As we have argued in the present pa-\nper, to resolve this inconsistency, in cosmological scales,\nit was found that the uncertainty in time coincides with\nthe look-back time quantity. Therefore, the Hubble con-\nstant, in general, depends on the measurements of the\nlength scale with a given redshift. This indicates that\nwe have a smaller value for the Hubble constant when\nthere is increased uncertainty in time, especially for mea-\nsurements with a high redshift value (such as the CMB).\nSimilarly, if there is less uncertainty in time, the case for\nlocal measurements with a smaller redshift value (such\nas the Cepheids and SNe Ia) implies a higher value for\nthe Hubble constant. These results show that the Hub-\nble tension arises due to fundamental limitations inherent\nin cosmology measurements of the graviton’s wavelength\n(scales) by the uncertainty relation.\nCREDIT AUTHORSHIP CONTRIBUTION\nSTATEMENT\nKimet Jusufi: Conceptualization, Methodology, For-\nmal analysis, Investigation, Writing – original draft,\nWriting – review & editing, Supervision. Esteban\nGonz´ alez: Conceptualization, Methodology, Software,\nVisualization, Validation, Formal analysis, Investigation,\nWriting – review & editing, Funding acquisition, Super-\nvision, Project administration. Genly Leon: Concep-\ntualization, Writing – review & editing, Investigation,Funding acquisition, Project administration.\nDECLARATION OF COMPETING INTEREST\nThe authors declare that they have no known com-\npeting financial interests or personal relations that could\nhave appeared to influence the work reported in this pa-\nper.\nACKNOWLEDGMENTS\nE.G. was funded by Vicerrector´ ıa de Investigaci´ on\ny Desarrollo Tecnol´ ogico (VRIDT) at Universidad\nCat´ olica del Norte (UCN) through Resoluci´ on VRIDT\nN°076/2023. 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Sumpter1,3\n1Department of Mathematics, Uppsala University, Uppsala, Sweden\n2Nordita, Stockholm University and KTH Royal Institute of Technology, Stockholm,\nSweden\n3Department of Information Technology, Uppsala University, Uppsala, Sweden\nAbstract\nBuilding mathematical models of brains is difficult because of the sheer complexity of\nthe problem. One potential approach is to start by identifying models of basal cognition,\nwhich give an abstract representation of a range organisms without central nervous systems,\nincluding fungi, slime moulds and bacteria. We propose one such model, demonstrating\nhow a combination of oscillatory and current-based reinforcement processes can be used to\ncouple resources in an efficient manner. We first show that our model connects resources in\nan efficient manner when the environment is constant. We then show that in an oscillatory\nenvironment our model builds efficient solutions, provided the environmental oscillations are\nsufficiently out of phase. We show that amplitude differences can promote efficient solutions\nand that the system is robust to frequency differences. We identify connections between our\nmodel and basal cognition in biological systems and slime moulds, in particular, showing\nhow oscillatory and problem-solving properties of these systems are captured by our model.\n1 Introduction\nUnderstanding cognitive abilities, such as perception, problem solving, and learning, is a cen-\ntral and longstanding question in neurobiology. One crucial aspect of cognition is the role of\noscillatory processes [1]. In the human brain, for example, theta oscillations are linked to active\nlearning in infants [2] and play an important role in memory formation [3]; beta oscillations have\nshown pivotal for multisensory learning [4]; alpha oscillations are central in temporal attention\n[5]; and cross-frequency coupling, i.e. coupling between neurons of different frequency, has been\nproposed as a mechanism for working memory [6, 7]. Another crucial aspect of cognitive abilities\nis synaptic plasticity, i.e. changing the strength of the connections between neurons based on\nprevious activity [8, 9]. Such reinforcement processes have been studied in both experimental\nand computational settings. For example, a study on rats shows that reinforcement determines\nthe timing dependence of corticostriatal synaptic plasticity in vivo [10], and dopamine-dependent\nsynaptic plasticity has been studied in mice [11].\nWhile the human brain serves as the primary model system for cognitive research, the field\nof minimal cognition (also referred to as basal cognition) aims to articulate the fundamental\n*These authors contributed equally to this work\n1arXiv:2402.02520v1  [q-bio.NC]  4 Feb 2024requirements necessary for the generation of cognitive phenomena [12, 13]. Cognitive-like abil-\nities are not limited to organisms with central nervous systems, but are also found in aneural\norganisms [14]. All living organisms employ sensory and information-processing mechanisms to\nassess and engage with both their internal environment and the world around them, and oscilla-\ntions often play a crucial role in their function [14]. For example, self-sustained oscillations that\nencode various types of information have been observed in cells [15, 16]. Oscillations are also\nfound in the root apex of many plants [17], and have been suggested to be a driving mechanism\nfor deciding where the plant should branch its roots [18].\nPhysarum polycephalum is a popular model organism for studying problem-solving in aneural\norganisms and better understanding basal cognition [19–21]. In its vegetative state, known as the\nplasmodium, this organism takes the form of a colossal, mobile cell. The plasmodium’s funda-\nmental structure encompasses a syncytium of nuclei and an intricate intracellular cytoskeleton,\nforming a complex network of cytoplasmic veins. It can expand to cover extensive areas, reach-\ning dimensions of hundreds of square centimetres, and can divide into smaller, viable subunits.\nWhen these subunits come into contact, they have the remarkable ability to fuse and share in-\nformation, leading to the reformation of a giant plasmodium. These slime moulds demonstrate\nhabituation [22] as well as anticipation [23]. Early experiments on these single cellular organisms\nshow that they can find the shortest paths between food sources through a labyrinth [24, 25].\nFurther experiments have shown that it can construct efficient and robust transport networks\nwith the same topology as the Tokyo rail network [26], can avoid already exploited patches [27],\nand solve a Towers of Hanoi maze [28].\nExperiments show that partial bodies in the plasmodium of the slime mould can also be\nviewed as nonlinear oscillators, where the interactions between the oscillators are strongly af-\nfected by the geometry of the tube network [29, 30]. Thus both oscillatory and reinforcement\nprocesses are involved in Physarum polycephalum cognitive abilities [14]. Oscillations in slime\nmould take place in different parts of the organism and on multiple time scales; from short-period\noscillations in the contraction cycle, to long period oscillations in the cell cycle [31].\nIn this article we look at how oscillators, coupled through current reinforcement, can exhibit\nboth problem solving (i.e. shortest path finding) abilities and long-range synchronisation ob-\nserved in many cognitive phenomena. Following Boussard et al., who emphasize the slime mould\nas an ideal model system for relating basal cognitive functions to biological mechanisms [14], it\nis such a coupling of feedback with oscillations which which we refer to as minimal cognition.\nThis definition is also in line with discussions around properties of minimal cognition for other\nbiological systems [32–34]. We can also see our work as contributing a model of basal cognition , a\nterm first introduced in a special double issue of Philosophical Transactions of the Royal Society\nB in March 2021 [13]. This emerging field, as elucidated by Lyon and others, underscores the\npresence of cognitive abilities predating the evolution of nervous systems. Indeed, figure 1 in\n[13] illustrates various organisms that have been studied to comprehend basal cognition, includ-\ningBacillus subtilis [35], the aneural placozoan Trichoplax adhaerens [36], and the planarian\nflatworm Platyhelminthes [37].\nAligning with the broader scope of basal cognition, we aim here to define a mathematical\nmodel that elucidates the role of oscillations and reinforcement mechanisms for cognitive phe-\nnomona in aneural settings. We apply Beer’s four steps for theory creation around cognitive\nphenomena [38] to basal cognition. In this introduction, as a first step, we have presented a\nconceptual framework, i.e. the idea that reinforcement and oscillations are enough to produce\nsome basal cognitive abilities. Beer’s second step, which we perform in the next section, is to\n2present a mathematical model based of oscillators and reinforcements on networks. As a third\nstep, in section 3, we study the mathematical properties of this model with a particular focus\non what type of oscillations produce basal cognitive phenomena in aneural organisms. Finally,\nthe fourth step discusses how our model might be further developed in light of the results.\nFrom a mathematical point of view, in this paper, we start by studying the dynamics of\na small network, a cycle graph with two oscillators. We show that this model can generate\nshortest path networks, localised oscillatory behaviour and global oscillatory dynamics. Then,\naddressing questions raised by Boussard et al. [14], we look at how phase, frequency and\namplitude differences can be used to induce dynamic problem solving. Finally, we look at how\nlarger networks of coupled oscillators can create complex networks of connections that mimic\nsome notion of basal cognition.\n2 Model\nOur aim is to define a model which has a combination of reinforcement and oscillatory processes,\nwhich can both ‘solve’ the problem of finding the shortest path between two or more points and\nexhibit long-range oscillations. We abstract away from the specific biological systems, and find a\nmodel of basal (or minimal) cognition [13]. To this end, we describe the motion of particles which,\ndepending on the system, might be nutrients, chemical signals (e.g. in slime moulds [14] and\nmany fungi [39]), or information within a decentralised system (e.g. macromolecular networks\nin microorganisms [40]). The system itself is modelled as a network or a graph over which the\nparticles move. The particles travel on the edges of the graph and in doing so produce current\nreinforcement. The coupling between the system and its environment is modelled through an in-\nand out-flow of particles at some of the points on the network. In order to capture oscillations,\nwhich are also widespread in systems of microbes [41], slime moulds [14], and fungi [39], we will\nmake these in- and out-flow of particles change over time.\n2.1 Previous models\nWe build primarily on the work of Tero et al. in modelling slime mould [42]. They model the\nmovement of particles, which are nutrient and chemical signals, on nodes on a graph, which can\nbe thought of as the cytoskeleton of the plasmodium. The dynamics weights on the edges of the\ngraph are the thickness of the tubes within that form the cytoskeleton. Similar analogies can\nbe made for other basal cogintive systems. For example, for fungi, the graph is the hyphae, and\nthe particles are nutrients [39].\nTero et al.’s model uses an analogy to electrical networks, to describe a flux of particles, Qij,\nbetween the nodes. Equations for evolution of the conductivity of the edges, Dij(t), (i.e. the\nthickness of the edges) are then proposed as follows:\ndDij(t)\ndt=f(|Qij|)−λDij, (1)\nwhere fis a non-decreasing function. The equation above states that edges in the tubular\nnetwork are thickened if there is a sufficient volume flow, and diminished otherwise. The model\nhas been proven to eventually converge to the shortest path between food sources, independently\nof the network structure [43, 44]. In this work, the flow of particles is assumed to be in steady\nstate, i.e., Qij=Ni−Nj= 0, where Niis the number of particles. Watanabe et al. have\nstudied an algorithm based on the current reinforcement model, but with oscillatory input and\n3outputs [45]. As for Tero et al., they assume that all flow between all nodes, except for the\ninput and output nodes, is zero. They also assume that the total flux of the system is constant,\nmeaning that the sum of the input and output is always zero. Another adaptation of the Tero et\nal.-model have been presented by Ma et el. who introduced a stochastic version of the current\nreinforcement model, where the flow of the particles between the nodes are not assumed to be\nin steady state [46], and (in simulations) the model also finds the shortest paths between food\nsources.\nMathematical models of coupled oscillators, such as the Kuramoto model [47], have played a\npivotal role in investigating the dynamics of neuronal processes within the brain [48, 49]. And,\nwhile it has been adapted to include synaptic plasticity, by evolving coupling strength based\non phase differences and Hebbian learning principles [50–52], these models do not incorporate\nthe current reinforcement mechanism, which is central to problem solving by slime moulds\noutlined in the previous paragraph. Indeed, while the Kuramoto model has been used to model\nthe anticipation of periodic events in slime moulds [23], there is a lack of models using coupled\noscillators to study other problem solving abilities of slime moulds, such as finding shortest paths.\nAlim et al. have made an interesting starting point into this, by studying the mechanism of\nsignalling propagation in slime moulds [53]. Using the Stokes equations to model the cytoplasmic\nflow velocity with the active tension of the walls of the slime mould oscillating, they argue that\ntheir model will obey similar dynamics as the Tero et al. model [42], and thus lead to shortest\npaths solutions.\nWhat is missing if we are to build a basal cognition model, however, is a model which\nexplicitly models the number of particles on a node and incorporates oscillations. It is such a\nmodel we now define.\n2.2 Model definition\nOur model consists of an undirected network, G(E, V), where V={1,2, . . . , n }is the node\nset, and an edge ( i, j)∈Eis an unordered pair of two distinct nodes in the set V, where each\nedge ( i, j) is associated with a length lij(lij=∞if there is no edge). We denote the number\nof particles at each node ibyNi. For each edge ( i, j), there is a current of particles inversely\nproportional to the length of the connection, |Ni−Nj|/lij. The edge is also characterised by\nits conductivity (corresponding to thickness of the tubes in slime mould), Dij, which will also\naffect the current. The number of particles at a node i, will change based on the flow of the\nparticles from the neighbouring nodes of i,j∈Γ(i) according to the following equation:\ndNi(t)\ndt=X\nj∈Γ(i)Nj(t)−Ni(t)\nlijDij(t), (2)\n(ignoring sources and sinks for now). The conductivity of each edge ( i, j)∈Echanges based on\nthe following equation:\ndDij(t)\ndt=q|Ni(t)−Nj(t)|\nlijDij(t)−λDij(t). (3)\nHere, qis the reinforcement strength and λis the decay rate. Note that, unlike previous\ndeterministic models of current-reinforcement [42, 54, 55], we give dynamic equations for the\nnumber of particles at each node, as we do not assume the flow of particles to be in equilibrium\non each edge as in Tero et al. [42]. This approach is similar to that taken by [46], who looked at\n41 234\n5\naaaaa−b·N3lijDij\n(a) Non-oscillating nodes1 234\n5\nA1(1 + sin(2 πθ1t+ϕ1))−bN1(t) A1(1 + sin(2 πθ1t+ϕ1))−bN1(t) A1(1 + sin(2 πθ1t+ϕ1))−bN1(t) A1(1 + sin(2 πθ1t+ϕ1))−bN1(t) A1(1 + sin(2 πθ1t+ϕ1))−bN1(t)A3(1 + sin(2 πθ3t+ϕ3))\n−bN3(t)lijDij\n(b) Oscillating nodes\nFigure 1: The model on a C5graph, where lijis the length of the edge between node iandj,Dij\nis the conductivity (thickness) of the edge between iandj. In (a) the nodes are non-oscillating,\nand node 1 is a source, with an inflow at rate aand node 3 is a sink with an outflow at rate bN3.\nIn (b) the nodes a oscillating as A1(1 + sin(2 πθ1t+ϕ1))−bN1(t) and A3(1 + sin(2 πθ3t+ϕ3))\n−bN3(t) at node 1 and 3 respectively.\nstochastic model of current reinforcement similar to the one we study here. Another difference\nto much of the work by Tero and co-workers is that we only consider the case where the increase\nin conductivity is proportional to (and not a non-linear function of) the current.\nIn the above description there is no in- or out-flow of particles in the system and thus no\ninteraction between the system and the environment. We incorporate such interactions in two\ndifferent cases: (i) the non-oscillatory case, where particles continuously enter the network at\nsources and leave at the sinks with fixed rates, and (ii) the oscillatory case, where the sources\nand sinks change over time, in the sense that sources can become sinks and vice versa. We now\ndescribe each of these in more detail.\n2.3 Non-oscillatory sources and sinks\nWe start by studying non-oscillatory sources and sinks. In this setting, particles enter the\nnetwork at some source(s) S={s1, . . .} ⊂Vwith a constant rate a, and leave the network at\nsome sink(s) T={t1, . . .} ⊂Vwith a rate bNifor each sink i, where bis a constant. Thus, the\nequations describing the number of particles at each node iare given by:\ndNi(t)\ndt=\n\na+P\nvj∈Γ(vi)Nj(t)−Ni(t)\nlijDij(t), ifi∈S,\n−bNi(t) +P\nvj∈Γ(vi)Nj(t)−Ni(t)\nlijDij(t),ifi∈T,\nP\nvj∈Γ(vi)Nj(t)−Ni(t)\nlijDij(t), otherwise .(4)\nWe start by studying these equations in the case of cyclic graph of five nodes, C5. Here, the\nnodes are labeled (1 ,2,3,4,5), and node 1 is a source, with an inflow a, while node 3 is a sink,\nwith outflow −bN3at that node. See Figure 1a.\n52.4 Oscillatory nodes\nTo incorporate oscillations, we consider nodes which alternate between being being sources and\nsinks. For these nodes, The rate at which particles enter or leave the network through the\nenvironment changes over time. To this end, we model the interaction with the environment as\nthe following function, for each node i∈O,\ngi(t) =Ai(1 + sin(2 πθit+ϕi))−bNi(t), (5)\nwhere Odenotes the set of oscillatory nodes, and Ai, θi, ϕicharacterising the features of the\nenvironment around node i. Specifically, Aiis the amplitude, θiis the frequency, ϕiis the\nphase, and bis the output rate. We choose parameters for the function gi, so it can have both\npositive and negative values at varying time t, to model particles that can both enter and leave\nthe network at node i. For C5, we let nodes 1 and 3 be the oscillatory nodes, i.e., O={1,3}\n(see Figure 1b).\nWhen the phase difference between two nodes is π, the sine waves corresponding to the two\nnodes have exactly the opposite sign. In this case, when one node has particles entering the\nnetwork, the other has particles leaving it, and vice versa. In this case, we set ϕ1= 0, ϕ3=π,\nwhile set other parameters to be identical for both nodes, i.e., A1=A3=Aandθ1=θ3=θ.\nHence, the evolution of N1andN3is\n(dN1(t)\ndt=A(1 + sin(2 πθ))−bN1(t) +(N5(t)−N1(t))\nl51D51+(N2(t)−N1(t))\nl12D12\ndN3(t)\ndt=A(1 + sin(2 πθ+π))−bN3(t) +(N2(t)−N3(t))\nl23D23+(N4(t)−N3(t))\nl34D34, (6)\nwhile others maintain the same as in Eqs. (4).\nBuilding on from this special case, we now allow the phase of node 3 to take values between\n−πandπ, i.e., ϕ3=ϕ∈(−π, π]. Similarly, the amplitude, which corresponds to the number of\nparticles waiting at one node to enter the network and also the capacity of the node for them\nto enter the network, is set to A1=AandA3=αAwhere α=A3/A1encodes the amplitude\nratio. Similarly, the frequency, which corresponds to how fast the environment changes around\neach node, is set to θ1=θandθ3=γθ, where γ=θ3/θ1encodes the frequency ratio. The\nevolution of N1andN3are then modified to be\n(dN1(t)\ndt=A(1 + sin(2 πθ))−bN1(t) +(N5(t)−N1(t))\nl51D51+(N2(t)−N1(t))\nl12D12\ndN3(t)\ndt=αA(1 + sin(2 πγθ+ϕ))−bN3(t) +(N2(t)−N3(t))\nl23D23+(N4(t)−N3(t))\nl34D34,(7)\nwhile the other equations maintain the same as in Eqs. (4).\n2.5 Large graphs\nThe simplest graph for two nodes to have at least two paths connecting them is a cycle graph,\nand the smallest one to have two paths of different lengths while the two nodes are not directly\nconnected is a cycle graph of length 5, thus C5is selected to illustrate our model in the previous\nsections. There are many ways to generalise it to larger graphs, where, e.g., we can consider\ncycle graphs of arbitrary sizes (larger than 5), but there are still only two paths connecting the\nnodes. To systematically incorporate more complexity to the graph, we consider regular graphs\nof degree 3, i.e., each node now has 3 neighbours instead of 2 in cycle graphs, and with clear\ngeometric meaning where nodes and edges are the hexagonal tiling of the plane, leading to the\n6hexagonal lattice graph. We have also included a small level of noise to the position of some\nnodes in the place, so that paths between the nodes are mostly of different lengths; see Figure\n2 for example. In this way, we have more geometrically meaningful paths connecting the nodes,\nwhich better models biological systems.\nFigure 2: Example of a large graph regular graph of degree 3, where the node in blue corresponds\nto the oscillatory node with phase 0 and the nodes in red correspond to the oscillatory nodes\nwith phase πin later simulations.\n3 Results\nWe now show (in section 3.1) that our proposed model can find the shortest path in the cycle\ngraph of size 5 for a fixed source and sink, in the sense that at steady state conductivity is\nnon-zero only on edges on the shortest path between the source and sink. We then show (in\nsection 3.2) that this feature is maintained in simulations in the oscillatory setting when phase\ndifference is maximised and further investigate the role of phase, amplitude and frequency.\nFinally, in section 3.3, we look at larger graphs with multiple oscillating nodes.\n73.1 Non-oscillatory sources and sinks\nOur model on C5(shown in Figure 1a) is described by the following set of equations:\ndN1(t)\ndt=a+(N2−N1)\nl12D12+(N5−N1)\nl51D51\ndN2(t)\ndt=(N1−N2)\nl12D12+(N3−N2)\nl23D23\ndN3(t)\ndt=−bN3+(N2−N3)\nl23D23+(N4−N3)\nl34D34\ndN4(t)\ndt=(N3−N4)\nl34D34+(N5−N4)\nl45D45\ndN5(t)\ndt=(N1−N5)\nl51D51+(N4−N5)\nl45D45\ndD12(t)\ndt=q|N1−N2|\nl12D12−λD12,\ndD23(t)\ndt=q|N2−N3|\nl23D23−λD23,\ndD34(t)\ndt=q|N3−N4|\nl34D34−λD34,\ndD45(t)\ndt=q|N4−N5|\nl45D45−λD45,\ndD51(t)\ndt=q|N5−N1|\nl51D51−λD51.(8)\nBy setting all equations in the system (8) to zero, we derive the equilibrium points E1and\nE2:\nE1= (N∗\n1, N∗\n2, N∗\n3, N∗\n4, N∗\n5, D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) =\na\nb+λl23\nq+λl12\nq,a\nb+λl23\nq,a\nb, N∗\n4, N∗\n5,aq\nλ,aq\nλ,0,0,0),\nE2= (N∗\n1, N∗\n2, N∗\n3, N∗\n4, N∗\n5, D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) =\n(a\nb+λl34\nq+λl45\nq+λl51\nq, N∗\n2,a\nb,a\nb+λl34\nq,a\nb+λl34\nq+λl45\nq,0,0,aq\nλ,aq\nλ,aq\nλ).\nThese correspond to flow on the paths 1-2-3 and 1-5-4-3, respectively. Detailed derivations are\nprovided in Appendix A. When l12+l23=l34+l45+l51there is an equilibrium point\nE3= (N∗\n1, N∗\n2, N∗\n3, N∗\n4, N∗\n5, D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) =\u0012a\nb+λl23\nq+λl12\nq,a\nb+λl23\nq,a\nb,a\nb+λl34\nq,a\nb+λl34\nq+λl45\nq,aq\n2λ,aq\n2λ,aq\n2λ,aq\n2λ,aq\n2λ\u0013\n.\nThe three different combinations of steady states are shown in Figure 5.\nIn Figure 3 we see the temporal dynamics of our model with non-oscillatory sources and\nsinks on the C5graph, for two different values of the parameter l12. In these simulations, we set\nthe inflow rate a= 1, the outflow rate b= 0.5, the parameter of reinforcement q= 0.1, and the\n8decay parameter λ= 0.01. In the left panel l12= 10, meaning that 1-2-3 is the shortest path\nbetween the source and the sink. Whereas in the right panel l12= 30, meaning that 1-5-4-3\nis the shortest path between the source and the sink. We observe that the conductivity of the\nedges in the shortest path increases over time until the steady state, while the conductivity of\nthe edges that are not in the shortest path can also increase at the beginning, but will eventually\nconverge to 0. For the number of particles, almost all nodes have an initial accumulating period\nbefore converging to the steady states.\n1\n01I / Onode 1\nnode 3\n010# Particlesnode 1\nnode 2\nnode 3\nnode 4\nnode 5\n510Conductivity(1,2)\n(2,3)(1,5)\n(3,4)\n(4,5)\n0 2000 4000\nTime02Conductivity(1,5)\n(3,4)\n(4,5)\n0 2000 4000\nTime(1,2)\n(2,3)\nFigure 3: Results from the model on C5graph, with node 1 as a source, node 3 as a sink, an\ninflow at rate a= 1, an outflow at rate b= 0.5, and the lengths of all edges are lij= 10 apart\nfrom (1 ,2). On the left, we set l12= 10, thus path going through 1-2-3, is the shortest one. On\nthe right, we set l12= 30, thus the path going through 1-5-4-3 is the shortest one. The first row\nshows the input at node 1 and output at node 3, the second row plots the number of particles on\neach node, Ni(t), the third row indicates the conductivity of the edges in the shortest path, and\nthe last row indicates the conductivity of the remaining edges in the graph, where the dashed\nlines in the corresponding color shows the analytical results of the steady state.\n9The Jacobian matrix of the system (8), is given by the following:\nJ=\n−D12\nl12−D51\nl51D12\nl120 0D51\nl51N2−N1\nl120 0 0N5−N1\nl51D12\nl12−D12\nl12−D23\nl23D23\nl230 0N2−N1\nl12N3−N2\nl230 0 0\n0D23\nl23−b−D23\nl23−D34\nl34D34\nl340 0N3−N2\nl23N4−N3\nl340 0\n0 0D34\nl34−D34\nl34−D45\nl45D45\nl450 0N3−N4\nl34N5−N4\nl450\nD51\nl510 0D45\nl45−D45\nl45−D51\nl510 0 0N4−N5\nl45N1−N5\nl51qsgn (N1−N2)D12\nl12qsgn (N1−N2)D12\nl120 0 0q|N1−N2|\nl12−λ 0 0 0 0\n0qsgn (N2−N3)D23\nl23qsgn (N2−N3)D23\nl230 0 0q|N2−N3|\nl23−λ 0 0 0\n0 0qsgn (N3−N4)D34\nl34qsgn (N3−N4)D34\nl340 0 0q|N3−N4|\nl34−λ 0 0\n0 0 0qsgn (N4−N5)D45\nl45qsgn (N4−N5)D45\nl450 0 0q|N4−N5|\nl45−λ 0\nqsgn (N5−N1)D51\nl510 0 0qsgn (N5−N1)D51\nl510 0 0 0q|N5−N1|\nl51−λ\n(9)\nIn the appendix A, we show that for all three steady states, this Jacobian matrix will have\nzero determinant, meaning that at least one eigenvalue of the Jacobian matrix will be 0. Thus,\nthe steady states will be non-hyperbolic, implying that studying the eigenvalues of the Jacobian\nis not enough to determine stability of the steady states. A full analysis of the steady states\nusing center manifold theory would be required to do this [56].\nNonetheless, we can use the Jacobian matrix to understand more about what happens to\nthe system. We note that when D∗\n12=D∗\n23= 0 or D∗\n34=D∗\n45=D∗\n51= 0, at least one row\nin the Jacobian matrix will contain only zero values, and thus span a slow manifold. For the\nJacobian matrix evaluated at E1, the eigenvectors corresponding to the zero eigenvalues will\nbe (0 ,0,0,1,0,0,0,0,0,0) and (0 ,0,0,0,1,0,0,0,0,0). The implication of this observation is\nthat there is a line of steady states for both N∗\n4andN∗\n5, which form the slow manifold. For\nthe Jacobian matrix evaluated at E2, the eigenvector corresponding to the zero eigenvalue will\nbe (0 ,1,0,0,0,0,0,0,0,0), corresponding to a line of steady states for N∗\n2which also is a slow\nmanifold. These slow manifolds corresponds to the nodes with no edges, meaning that there are\nparticles left behind at these edges, that are stuck because the conductivity converges to 0 for\nthe longer paths.\nThe numerical bifurcation diagram in Figure 4 looks at the role of l12as a bifurcation\nparameter. In Figure 4 (a) we see that the number of particles at node 3, N∗\n3is always a/b,\nindependent on the length l12. However, N∗\n1,N∗\n4andN∗\n5, increase linearly as l12approaches the\nequilibrium point 20, after which they all are constant. N∗\n2is constant (a\nb+λl23\nq= 3) until the\nbifurcation point l12= 20, where it instead decrease linearly. In Figure 4 (b) we see that the\nconductivity of edges on the shortest path is always aq/λ, and 0 for the edges that are not the\nshortest path. When the length of l12approaches the bifurcation point 20, the simulations has\nnot reached steady state after 100000 time steps, and is thus not 0 and aq/λ in the plot. When\nl12= 20 the two paths are of equal lengths and Dij=aq\n2λfor all ( i, j).\n10Figure 4: Bifurcation diagram with l12as bifurcation parameter. Here l23=l34=l45=l51= 10,\na= 1.0,b= 0.5,q= 0.1 and λ= 0.01. The diagram is generated by simulating the equation\nsystem (10) through 100,000 time steps, each time initiated with random initial conditions.\nThis process is repeated 10 times for each l12value, with the bifurcation diagram displaying the\nresults from the last 100 time steps of each simulation.\nFigure 5 illustrates the overall picture. In (a) we see that when 1-2-3 is the shortest path,\nthere is a non-hyperbolic equilibrium where the conductivity on the edges of the longest path\nis zero, but the conductivity of the edges of the shortest path isaq\nλ. The number of particles at\nthe nodes with no connections, N∗\n4andN∗\n5, spans the slow manifold. In (b) the paths 1-2-3 and\n3-4-5-1 are of equal lengths and the conductivity of all edges isaq\n2λ. When the path through nodes\n1-2-3 is longer than 1-5-4-3 (Figure 5 (c)), the non-hyperbolic equilibrium point corresponds to\nhaving zero-conductivity on the edges going through node 2, and a line of steady states for the\nnumber of particles on node 2, N∗\n2. The conductivity on the edges on the shortest path isaq\nλ.\n1 234\n5\naq\nλaq\nλ\na\nb+λl23\nq+λl12\nq a\nb+λl23\nqa\nbN∗\n4\nN∗\n5\n(a) Non-hyperbolic stable equi-\nlibrium when l12+l23< l 34+\nl45+l51(E1).1 234\n5aq\n2λaq\n2λ\naq\n2λaq\n2λaq\n2λ\na\nb+λl23\nq+λl12\nq a\nb+λl23\nqa\nba\nb+λl34\nq\na\nb+λl34\nq+λl45\nq\n(b) Non-hyperbolic stable equi-\nlibrium when l12+l23=l34+\nl45+l51(E3).1 234\n5aq\nλaq\nλ\naq\nλ\na\nb+λl34\nq+λl45\nqN∗\n2a\nba\nb+λl34\nq\na\nb+λl34\nq+λl45\nq\n(c) Non-hyperbolic stable equi-\nlibrium when l12+l23> l 34+\nl45+l51(E2).\nFigure 5: Different equilibrium points for the model on the C5with non-oscillatory sources and\nsinks, where node 1 (blue) is a source, and node 3 (red) is a sink. In (a) we see the stable\nequilibrium when 1-2-3 is the shortest path, in (b) we see the stable equilibrium when both\npaths have equal length, and in (c) we see the stable equilibrium when 1-5-4-3 is the shortest\npath.\n113.2 Oscillatory nodes\nWe now use simulations to explore the behaviour of the model with oscillatory nodes on a C5\ngraph. We start from the case when the two oscillatory nodes have phase difference π(i.e.\ncompletely out of phase) and then explore the behaviour of the model when we vary the phase\ndifference between −πandπ. Finally, we look at how the amplitude and frequency of the\noscillatory nodes affect the model. Throughout the section, we set nodes 1 and 3 to be the\noscillatory nodes, and the length of all edges to be lij= 10, thus the shortest path between the\ntwo oscillatory nodes is known to be e1e2where e1= (1,2) and e2= (2,3).\nPhase difference. Figure 6 shows the temporal dynamics of our model when the two\noscillatory nodes have phase difference π. The top two panels of the figure show the oscillating\ninput and outputs over two different time windows. The number of particles at node 1 and 3\nhave the same frequency as the inputs and outputs, but with a short delay as the particles flow\ninto and out of the nodes. Oscillations with similar frequency can be seen at nodes 4 and 5,\nwhich are on the longer path, but with much smaller amplitude and a different phase. Node 2,\nwhich is on the shortest path, reaches steady state.\nThe conductivity of edges oscillates over time, but they also change over a longer time scale.\nSpecifically, the median conductivity of the edges on the shortest path generally increases, but\nwith a rate of increase that tends very slowly towards 0. Interestingly, the number of particles\noscillates at a frequency that is almost half that of the conductivity on the edges. This is\npresumably because the flow shuttles backwards and forwards. Moreover, the amplitude of the\nconductivity changes is much smaller than that of the oscillations of particles at nodes 1 and 3\n(note the scale on the axis for conductivity in Figure 6). The median conductivity of the edges\nthat are not on the shortest path decrease over time and converge to values either 0 or very close\nto 0 (with no or very small oscillations) depending on the initial conditions (see the bottom row\nof Figure 6).\nWe observe similar features of the model on cycles of larger sizes and also graphs of more\ngeneral structure (see more details in Appendix B.2).\n120.00.5I / Onode 1\nnode 3\n123# Particlesnode 1\nnode 2\nnode 3\nnode 4\nnode 5\n1.001.25Conductivity(1,2)\n(2,3)\n0 2000 4000\nTime01Conductivity\n4700 4800 4900 5000\nTime(1,5)\n(3,4)\n(4,5)Figure 6: Results from the model on a C5graph, with nodes 1 and 3 as the oscillating nodes,\nrateb= 0.5, amplitude A1=A3=A= 1, frequency θ1=θ3=θ= 0.01, and phases ϕ1= 0\nandϕ3=π, with the same reinforcement parameter q= 0.1 and decay parameter λ= 0.01 as\nin Figure 3. The first row plots the actual input/output to the two nodes, the second row shows\nthe number of particles in each node, the third row indicates the conductivity of the edges in the\nshortest path and the last row indicates the conductivity of the remaining edges in the graph,\nwith the full profile on the left while the change in the long term on the right.\nIn Figure 7 we examine the change of conductivity over the change of phase difference ϕ3−ϕ1,\nfrom−πtoπ. When the phase difference is −π, the conductivity of edges on the shortest path\nis maximal (see the points corresponding to phase difference −πat the left most of Figure 7).\nAs the phase difference becomes closer to one, the conductivity of the edges on the shortest\npath decreases, with that of the edge (2 ,3) approaching 0. Correspondingly, the conductivity of\nedges (3 ,4) and (1 ,5) slowly increases. In this situation, the two oscillating nodes are connected\nvia the shortest paths, but have additional arms that reach out along the longer path but do\nnot connect.\nFor phase differences between around −0.65πand 0 the network becomes disconnected:\nthe conductivity of edges does not change with phase and is zero for the edge (1 ,2) (see the\ncorresponding region in Figure 7). When the phase difference is 0, the two edges on the shortest\npath have the same conductivity value, as do the other two disconnected arms on the longer\npath. As the phase difference increases from 0 up until around 0 .65π, the conductivity of edges\nstabilises again with (2 ,3) now having zero conductivity. The same pattern is repeated now in\nthe sense that the results are invariant if we change the label of node 1 with 3, and that of nodes\n134 and 5.\nFigure 7: Conductivity results from the model on a C5graph, from edges in the shortest path\n(top) and not (bottom) after sufficiently long time (where the oscillation repeats) subject to\nchanging phase difference between the two oscillatory nodes 3 and 1, where other parameters\nare the same as in Figure 6, the line with asterisk markers indicate the middle value, and the\nshade indicates the amplitude. The inserted subfigures indicate the change of conductivity of\nedges along time in sufficiently long time.\nAmplitude, frequency and phase differences. When the amplitude or the frequency of\nthe two oscillatory nodes are not exactly the same, the model exhibits a rich span of behaviours.\nIn Figure 8, we explore the resulting graphs as we change both phase difference ∆ ϕ=ϕ3−ϕ1and\namplitude ratio A3/A1. In these figures, the thickness of the edges indicates their conductivity\nafter the simulation has been run sufficiently long time. The middle row where A3/A1= 1\n14corresponds to the case presented in Figure 7, and discussed in detail in the last paragraph,\nwhere the graph changes from at first being connected via the shortest path (∆ ϕ=−3π/4)\nand having additional arms, to becoming disconnected (e.g. ∆ ϕ=−π/4), to connecting again\n(∆ϕ= 3π/4) and the majority of the flow following the shortest path (∆ ϕ=π).\nWhen A3/A1is close to 1, the results exhibit similar phase transitions to the case of A3/A1=\n1; see the rows corresponding to ∆ ϕ= 9/10,10/9 in Figure 8. As the amplitude ratio A3/A1\nincreases, edges generally have larger values of conductivity. For example, when A3/A1= 2,\nall edges have nonzero conductivity values for all possible values of the phase difference ∆ ϕ.\nMoreover, if ∆ ϕis close to −πorπ, the edges in the shortest path have higher conductivity\nvalues than others. When A3/A1= 3 (and for larger values) the edges on the shortest path\nalways have higher conductivity values than others, independent of ∆ ϕ. Amplitude differences\ncan thus lead to a greater concentration of flow on the shorter path.\nWhen the amplitude ratio A3/A1is smaller than one, edges generally have smaller values of\nconductivity. Specifically, when A3/A1= 1/2 there are no edges that are incident on oscillatory\nnode 3 with non-zero conductivity values when the phase difference ∆ ϕ≤0. There is also only\na weak connection between oscillatory node 3 and node 2 in the shortest path when ∆ ϕ >0.\nWhen A3/A1is 1/5 or 1 /10, this disconnected network is the outcome for all possible values of\n∆ϕ.\n15A3/A1\\∆ϕ −3π\n4−π\n2−π\n40π\n4π\n23π\n4π\n1/10\n1/3\n1/2\n4/5\n9/10\n1\n10/9\n5/4\n2\n3\nFigure 8: Conductivity results (median) from the model on a C5graph at long enough time,\nsubject to changing amplitude ratio α=A3/A1(A1= 1) and phase difference ∆ ϕ=ϕ3−ϕ1\n(ϕ1= 0), where other parameters are the same as in Figure 6, and the thickness of edges is\nproportional to the conductivity value.\nIn Figure 9, we examine the resulting graphs as we change both phase difference ∆ ϕ=ϕ3−ϕ1\nand frequency ratio θ3/θ1. In these figures, the thickness of the edges, again, indicates their\nconductivity after the simulation has been run sufficiently long time. The third row where\nθ3/θ1= 1 corresponds to the case presented in Figure 8, exhibiting different features as the\nphase difference changes, as in the case of A3/A1= 1 in the last paragraph.\nHowever, when the frequency ratio θ3/θ1̸= 1, the resulting graphs remain the same at\nall phase difference values. As the frequency ratio θ3/θ1increases, less edges have nonzero\n16conductivity in general. For example, when θ3/θ1= 2 or 5, only edges that are incident on\neither of the two oscillatory nodes have nonzero conductivity values. When θ3/θ1= 7, the edge\nthat is incident on oscillatory node 3 but not in the shortest path now has zero conductivity.\nIn these two cases, the two oscillatory nodes are then connected via the shortest path and have\nadditional arm(s). When θ3/θ1= 10, no edges that are incident on oscillatory node 3 have\nnonzero conductivity, thus the graph become disconnected. Hence, frequency can affect the\nexploration depth of the flow on the graph.\nAs the frequency ratio θ3/θ1decreases, more edges have nonzero conductivity on the whole.\nSpecifically, when θ3/θ1= 1/2, all edges apart from the one that is not incident on either of\nthe oscillatory nodes have zero conductivity. When θ3/θ1= 1/10, all edges now have nonzero\nconductivity values, even though the conductivity of the one that has zero conductivity in the\nprevious case is very small.\nWith all above, we conclude that amplitude, frequency and phase all contain important\ninformation for the model to learn interesting patterns in the graph and the change in the\nenvironment.\nθ3/θ1\\∆ϕ −3π\n4−π\n2−π\n40π\n4π\n23π\n4π\n1/10\n1/2\n1\n2\n5\n7\n10\nFigure 9: Conductivity results (median) from the model on a C5graph at long enough time,\nsubject to changing frequency ratio γ=θ3/θ1(θ1= 0.01) and phase difference ∆ ϕ=ϕ3−ϕ1\n(ϕ1= 0), where other parameters are the same as in Fig. 6, and the thickness of edges is\nproportional to the conductivity value.\n173.3 Large graphs\nWe now proceed to examine the behaviour of the model, with oscillatory nodes, on larger graphs\n(see section 2.5 for the setup). The purpose of this investigation was to see if the model could also\nproduce large-scale patterns with shortest path connections between the nodes in the presence\nof external oscillations.\nIn Figure 10, we explore the behaviour of our model after sufficiently long time in two\ndifferent cases: where there are three oscillatory nodes (the blue node in Figure 2 and the two\nred nodes horizontally in the middle of the figure) and where five oscillatory nodes (one blue\nand four red) are included1. In both these simulations, the flow is still focused primarily on the\nshortest paths between the oscillatory nodes with the largest phase differences (black connecting\nlines in Figure 10). The oscillations dominate the behaviour of the particles. Following Figure\n10a to 10b to 10c we see that the concentration of particles goes from the periphery to the centre\nand then out to the periphary again. Figure 10d to 10e to 10f shows the same pattern but for\nfive nodes.\nWhile identifying parameter values to produce this outcome, we noted that frequency of\noscillatory nodes is more important in large graphs than the smaller examples we have already\nseen. The oscillations should be sufficiently slow to allow the information from one oscillatory\nnode to communicate with another, before the particles go back along the paths they have trav-\nelled from. Meanwhile, the amplitude should be sufficiently large to allow such communication\nbefore the conductivity reduces to 0. Hence in the simulations on larger graphs, we set the\nfrequency to be θ= 0.001 and amplitude to be A= 5, while maintaining the parameter of\nreinforcement being q= 0.1 and the decay parameter λ= 0.01.\n1Please refer to https://github.com/yuoohmaths/MinimalCognition for more details of the simulations.\n18(a) (d)\n(b) (e)\n(c) (f)\nFigure 10: Snapshots of the graphs at three different representative time points in sufficiently\nlong time, with various number of oscillatory nodes. (a),(b),(c): 3 oscillatory nodes, with equal\ntime difference. (d),(e),(f): 5 oscillatory nodes, with equal time difference. The edge thickness\nin the graphs is proportional to the conductivity, and the node redness is proportional to the\nnumber of particles.\n4 Discussion\nGuided by Beer’s four steps [33, 38], we have provided and investigated a mathematical frame-\nwork inspired by organisms exhibiting basal cognition [13]. Under this framework, the formula-\n19tion of the model is itself a contribution to this research area: we have built upon previous models\nof current reinforcement to explicitly include oscillations, demonstrating that such systems often\nexhibit the greatest flow on the shortest path between the oscillators.\nWe can think of our model as a demonstration of how organisms can monitor a dynamic envi-\nronment, transporting particles (e.g. nutrients or signals) effectively between different locations.\nThese abilities existed well before the development of nervous systems, let alone central nervous\nsystems [13]. This makes our model a potential step towards a more realistic slime mould model\n[31], as it captures both synchronisation and network construction observed in experiments of\nthese organisms [23, 42]. It also mimics other types of basal cognition, such as macromolecular\nnetworks in microbes [40], and signalling in fungi [39].\nIn a stationary environment the current or flow of particles converges to an efficient transport\nnetwork on the shortest path. Previous work on current reinforced models has proven that a\nshortest path is established in a model where the flow of particles is assumed to be at steady state\n[43, 44]. We have found that if we explicitly model the dynamics of that flow then, while flow\nis still concentrated on the shortest path, there exist non-hyperbolic stable node configurations.\nThese correspond to particles which, might be said to be, ‘stuck’ on the longer path. We suggest\na connection between these stuck particles and the way in which many organisms with basal\ncognition have an external memory. By leaving external cues in the environment, they can react\nwhen the environment changes [57]. We have established that the stuck particles are associated\nwith a slow manifold. This means we have very slowly changing particle numbers on the longer\npath spanning the slow manifold, while flow on the network converges to the shortest path. In\nthis way, the model has an external memory: the particles on the nodes corresponding to the\nlonger path ‘remember’ failed solutions, and might be reactivated when conditions change.\nThe dynamic networks in our simulations produce both efficient flow and long-range oscilla-\ntory dynamics. While the nodes oscillate greatly in terms of number of particles at them, the\nconnecting edges of the graph experience only small oscillations (this can be seen by contrasting\nthe scale of the oscillations in conductivity and number of particles in Fig. 6). The resulting\npattern has low frequency responses to changes in formation over a long range combined with\na rapid flow of information across the system. We see this in Figure 10, which is reminiscent of\ncross-frequency coupling, i.e. coupling between neurons of different frequency, which has been\nproposed as a mechanism for working memory [6, 7]. This is in contrast to the Kuramoto model,\nwhich captures synchronisation, but does not capture information transfer in the same way as\nour model.\nComparing our model to recent slime mould models created by Alim et al. [53], our analysis\nreveals that reinforcement coupled with oscillators can indeed find the shortest path, as suggested\nby these authors, and similar to what is found by Watanabe et al. [45]. However, in contrast to\nWatanabe et al., the sum of the input and output rates in our model are not set to zero. Yet,\nafter reaching a stable limit cycle, the input and output rates in our model cancel out, suggesting\nthat the oscillators self-organize to have the total flux to/from the system balanced. Our work\nalso aligns with Reid’s perspective on slime mold as a connected mass of oscillating units [58].\nIt further echoes early experiments by Takamatsu et al., displaying similar oscillatory behavior\nin conductivity (i.e. thickness of the plasmodium network) in rings of slime mould oscillators\n[29, 30].\nExpanding our discussion to other species with basal cognition, we can draw parallels be-\ntween our model and experiments in bacterial colonies that employ in phase and anti-phase\noscillatory behaviors during resource scarcity[35]. Similarly, the in-phase and out-of-phase oscil-\n20latory behaviour in our model are akin to the emergence of in-phase and out-of-phase oscillatory\nbehaviour that is also found in plant shoots. For example, emergent maize plants grow in a\ngroup which exhibit synchronized oscillatory motions that may be in-phase or anti-phase [59].\nDrawing connections to fungal mycelia, our model resonates with recent studies highlighting\nthe bidirectional transport of signals and nutrients [39]. The flexibility of nodes to switch be-\ntween acting as sources and sinks in our model provides a promising foundation for modeling\nbidirectional transportation within fungal mycelia [39].\nOur analysis shows that not only phase, but also amplitude and frequency of oscillations\ncan induce the construction of efficient transport networks. This confirms (in a model) the\nhypothesis of Boussard et al. (2021), who suggested that all three variations could play a role\nin how slime moulds build networks [14]. Phase differences certainly play a relatively more\nimportant role, especially when the oscillatory nodes are close to being out-of-phase. In this\ncase, the model nearly always has almost all its flow on the shortest path between the out pf\nphase nodes. But variations in any one, or a combination of phase, amplitude and frequency\nare sufficient to produce the basal cognition we observe in our simulations.\nWe do observe that very big frequency differences inhibits shortest path formation. For\nexample, the oscillating nodes become disconnected when the frequency ratio is 10 (see bottom\nrow of Figure 9). This, along with the need for the oscillators to be (somewhat) out of phase,\ngives some restrictions on the types of oscillations needed in order to create a flow between\nnodes. Amplitude primarily effects how far the particles can explore in the graph: too small\namplitudes do not allow long range communication.\nOur model does not include feedback between oscillatory nodes, which we know are part of\nhow living organisms assess, engage and adapt to the world around them. Such feedback could\nalso potentially change the phase, amplitude and frequency of the oscillations towards values\nwhich better facilitate communications. Our next modelling steps would therefore be to incor-\nporate feedback mechanisms for the oscillators into the model (as suggested by [14]). The intro-\nduction of feedback could involve the consideration of diverse forms of local information, lci(t),\nsuch as the potential difference in particles (P\nj∈Γ(i)(Ni(t)−Nj(t))), the absolute potential dif-\nference, or a combination of potential difference and conductivity (P\nj∈Γ(i)(Ni(t)−Nj(t))Dij(t)).\nMoreover, an exploration could encompass various strategies for the phase transition, including\nthe option of direct proportionality to local information (dϕi(t)\ndt=lci(t)). An initial exploration\nusing smaller networks could be undertaken to discern the impact of network topology on feed-\nback forms, as done by Ma et al. [60]. Alternatively, an evolutionary adaptive approach, as\nadvocated by Beer [38], offers another way to learn a feedback scheme.\nIn conclusion, the model we propose here already has similar properties to many biological\nsystems which exhibit basal (or even more complex forms of) cognition. We see it as a promising\nstarting point for future simulation models of these phenomena.\n21A Non-oscillatory sources and sinks\nA.1 Steady states\nThis section of the appendix derives the results for the steady states as presented in Section\n3.1. The equilibrium points of the system are determined by the solutions to the following set\nof equations:\na+(N∗\n2−N∗\n1)\nl12D∗\n12+(N∗\n5−N∗\n1)\nl51D51= 0, (10)\n(N∗\n1−N∗\n2)\nl12D∗\n12+(N∗\n3−N∗\n2)\nl23D∗\n23= 0, (11)\n−bN∗\n3+(N∗\n2−N∗\n3)\nl23D∗\n23+(N∗\n4−N∗\n3)\nl34D∗\n34= 0, (12)\n(N∗\n3−N∗\n4)\nl34D∗\n34+(N∗\n5−N∗\n4)\nl45D∗\n45= 0, (13)\n(N∗\n1−N∗\n5)\nl51D∗\n51+(N∗\n4−N∗\n5)\nl45D∗\n45= 0, (14)\nq|N∗\n1−N∗\n2|\nl12D∗\n12−λD∗\n12= 0, (15)\nq|N∗\n2−N∗\n3|\nl23D∗\n23−λD∗\n23= 0, (16)\nq|N∗\n3−N∗\n4|\nl34D∗\n34−λD∗\n34= 0, (17)\nq|N∗\n4−N∗\n5|\nl45D∗\n45−λD∗\n45= 0, (18)\nq|N∗\n5−N∗\n1|\nl51D∗\n51−λD∗\n51= 0. (19)\nIn general, the steady states for the conductivity are given by the equation:\nq\f\f\fN∗\ni−N∗\nj\f\f\f\nlijD∗\nij−λD∗\nij= 0. (20)\nFor this to hold, either D∗\nij= 0 or\f\f\fN∗\ni−N∗\nj\f\f\f=λlij\nq. IfD∗\nij̸= 0, then\f\f\fN∗\ni−N∗\nj\f\f\f=λlij\nq, and\nthus, Equation (20) can be simplified to\n±λ\nqD∗\ni−1,i+±λ\nqD∗\ni,i+1= 0. (21)\nAsDij(t) denotes the conductivity, we know that Dij(t)≥0 and thus, the two addends must\nhave opposite signs. This gives the solution\nD∗\ni−1,i=D∗\ni,i+1. (22)\nThis also implies that for a node i, which is neither a source nor a sink, ( N∗\ni−1−N∗\ni) and\n(Ni+1−Ni) must have opposite signs. We can now consider the following cases:\n22Case 1: D∗\n51= 0 and D12̸= 0\nIfD∗\n51is zero and D12is non-zero, then |N∗\n2−N∗\n1|=λl12\nq. We thus have that\na±λl12\nl12qD∗\n12= 0. (23)\nWe know that the conductivity must be positive, so\nD∗\n12=aq\nλ, (24)\nasa,q, and λare positive. This means that\n(N∗\n2−N∗\n1) =−λl12\nq. (25)\nFrom Equation (22) we know that D∗\n12=D∗\n23andD∗\n34=D∗\n45=D∗\n51, so the steady states for\nthe conductivity will in this case be given by ( D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) = (aq\nλ,aq\nλ,0,0,0).\nNow consider the equation for the change of particles at node with a sink:\ndN3(t)\ndt=−bN3+(N2−N3)\nl23D23+(N4−N3)\nl34D34. (26)\nThe steady states will be given the solution to the equation\n−bN∗\n3+(N∗\n2−N∗\n3)\nl23D∗\n23+(N∗\n4−N∗\n3)\nl34D∗\n34= 0. (27)\nNow, if ( D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) = (aq\nλ,aq\nλ,0,0,0), then the equation can be simplified to\n−bN∗\n3+(N∗\n2−N∗\n3)\nl23D∗\n23= 0, (28)\nwhere N∗\n2−N∗\n3=±λl23\nqandD∗\n23=aq\nλ, which gives the following\n−bN∗\n3±λl23\nq\nl23aq\nλ= 0. (29)\nAsN3denotes the number of particles at node 3, N3must be positive and we get\nN∗\n3=a\nb. (30)\nThis implies that N∗\n2−N∗\n3= +λl23\nq, and thus\nN∗\n2=a\nb+λl23\nq. (31)\nWe know that ( N∗\n2−N∗\n1) =−λl12\nq, so\nN∗\n1=a\nb+λl23\nq+λl12\nq. (32)\nThere are no exact solutions for N∗\n4andN∗\n5, so for case 1, we have the following steady state:\n(N∗\n1, N∗\n2, N∗\n3, N∗\n4, N∗\n5, D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) =\na\nb+λl23\nq+λl12\nq,a\nb+λl23\nq,a\nb, N∗\n4, N∗\n5,aq\nλ,aq\nλ,0,0,0).\n23Case 2: D12= 0 and D∗\n51̸= 0\nIfD12is zero and D∗\n51is non-zero, then |N∗\n5−N∗\n1|=λl51\nq, and thus\nD∗\n51=aq\nλ, (33)\nand\nN∗\n5−N∗\n1=−λl51\nq. (34)\nAgain, consider the equation for the change of particles at node with a sink:\ndN3(t)\ndt=−bN3+(N2−N3)\nl23D23+(N4−N3)\nl34D34. (35)\nThe steady states will be given the solution to the equation\n−bN∗\n3+(N+\n2−N+\n3)\nl23D∗\n23+(N∗\n4−N∗\n3)\nl34D∗\n34= 0. (36)\nNow, if ( D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) = (0 ,0,aq\nλ,aq\nλ,aq\nλ), then the equation can be simplified to\n−bN∗\n3+(N∗\n4−N∗\n3)\nl34D∗\n34= 0, (37)\nwhere N∗\n4−N∗\n3=±λl34\nqandD∗\n34=aq\nλ, which gives the following\n−bN∗\n3±λl34\nq\nl34aq\nλ= 0. (38)\nAsN3denotes the number of particles at node 3, N3must be positive and thus N∗\n4−N∗\n3= +λl34\nq.\nHence, we get\nN∗\n3=a\nb. (39)\nWe know that N∗\n4−N∗\n3=λl34\nq, so\nN∗\n4=a\nb+λl34\nq. (40)\nAs 4 is neither a source, nor sink, we know that N∗\n4−N∗\n5andN∗\n4−N∗\n3have opposite signs.\nAsN∗\n4−N∗\n3is positive, this implies that\nN∗\n4−N∗\n5=−λl45\nq, (41)\nand thus\nN∗\n5=a\nb+λl34\nq+λl45\nq. (42)\nAsN∗\n4−N∗\n5is negative, N∗\n1−N∗\n5must be positive and thus\nN∗\n5−N∗\n1=λl51\nq, (43)\n24which gives\nN∗\n1−\u0012a\nb+λl34\nq+λl45\nq\u0013\n=λ\nl51q, (44)\nso\nN∗\n1=a\nb+λl34\nq+λl45\nq+λl51\nq. (45)\nAs for N∗\n4andN∗\n5in case 1, there are no exact solution for N∗\n2in case 2. Thus the steady states\nare given by\n(N∗\n1, N∗\n2, N∗\n3, N∗\n4, N∗\n5, D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) =\n(a\nb+λl34\nq+λl45\nq+λl51\nq, N∗\n2,a\nb,a\nb+λl34\nq,a\nb+λl34\nq+λl45\nq,0,0,aq\nλ,aq\nλ,aq\nλ).\nCase 3: D12̸= 0 and D∗\n51̸= 0\nNow, if neither D12norD51is non-zero we have that\na±λ\nqD∗\n12±λ\nqD∗\n51= 0, (46)\nwhich gives\n±D∗\n12±D∗\n51=aq\nλ. (47)\nAs the conductivity is the same on the edges going from a non-source and non-sink, we know\nthat\nD∗\n12=D∗\n23, (48)\nand\nD∗\n34=D∗\n45=D∗\n51. (49)\nIf we assume that the conductivity is the same on all edges, we get that\na±λ\nqD∗\n12±λ\nqD∗\n12= 0. (50)\nAs the conductivity must be positive, we have\na−λ\nqD∗\n12−λ\nqD∗\n12= 0, (51)\nand thus\nD∗\n12=aq\n2λ. (52)\nHence,\nD∗\n12=D∗\n23=D∗\n34=D∗\n45=D∗\n51=aq\n2λ. (53)\nThis also means that\n(N∗\n2−N∗\n1) =−λl12\nq, (54)\nand\n(N∗\n5−N∗\n1) =−λl51\nq. (55)\n25Now using that for a node, i, which is neither a source nor a sink, ( N∗\ni−1−N∗\ni) and ( Ni+1−Ni)\nmust have opposite signs, we know that\n(N∗\n3−N∗\n2) =−λl23\nq, (56)\n(N∗\n4−N∗\n5) =−λl45\nq, (57)\n(N∗\n3−N∗\n4) =−λl34\nq. (58)\nTo find the steady states of the number particles, we first look at the following equation:\n−bN∗\n3+(N∗\n2−N∗\n3)\nl23D∗\n23+(N∗\n4−N∗\n3)\nl34D∗\n34= 0. (59)\nInserting ( N∗\n2−N∗\n3) =λl23\nq, (N∗\n4−N∗\n3) =λl34\nq, and D∗\n23=D∗\n34=aq\n2λ, we obtain\n−bN∗\n3+a\n2+a\n2= 0, (60)\nand thus\nN∗\n3=a\nb. (61)\nThis means that\nN∗\n4=a\nb+λl34\nq, (62)\nN∗\n2=a\nb+λl23\nq, (63)\nN∗\n1=a\nb+λl23\nq+λl12\nq, (64)\nand\nN∗\n5=a\nb+λl34\nq+λl45\nq. (65)\nBut we also know that ( N∗\n5−N∗\n1) =−λl51\nq, so combining equations 64 and 65 we get\nN∗\n5−N∗\n1=a\nb+λl34\nq+λl45\nq−a\nb+λl23\nq+λl12\nq=−λl51\nq, (66)\nwhich implies that\nl12+l23=l34+l45+l51. (67)\nThus for the steady states to be\n(N∗\n1, N∗\n2, N∗\n3, N∗\n4, N∗\n5, D∗\n12, D∗\n23, D∗\n34, D∗\n45, D∗\n51) =\n(a\nb+λl23\nq+λl12\nq,a\nb+λl23\nq,a\nb,a\nb+λl34\nq,a\nb+λl34\nq+λl45\nq,aq\n2λ,aq\n2λ,aq\n2λ,aq\n2λ,aq\n2λ),\nthe different paths between the source and the sink must have the same total length.\n26A.2 Stability\nAs an initial step to investigate the stability of the system described by Equation (8), we analyze\nthe Jacobian matrix, denoted as J. The expression for Jis given by the following extensive\nmatrix:\nJ=\n−D12\nl12−D51\nl51D12\nl120 0D51\nl51N2−N1\nl120 0 0N5−N1\nl51D12\nl12−D12\nl12−D23\nl23D23\nl230 0N2−N1\nl12N3−N2\nl230 0 0\n0D23\nl23−b−D23\nl23−D34\nl34D34\nl340 0N3−N2\nl23N4−N3\nl340 0\n0 0D34\nl34−D34\nl34−D45\nl45D45\nl450 0N3−N4\nl34N5−N4\nl450\nD51\nl510 0D45\nl45−D45\nl45−D51\nl510 0 0N4−N5\nl45N1−N5\nl51qsgn (N1−N2)D12\nl12qsgn (N1−N2)D12\nl120 0 0q|N1−N2|\nl12−λ 0 0 0 0\n0qsgn (N2−N3)D23\nl23qsgn (N2−N3)D23\nl230 0 0q|N2−N3|\nl23−λ 0 0 0\n0 0qsgn (N3−N4)D34\nl34qsgn (N3−N4)D34\nl340 0 0q|N3−N4|\nl34−λ 0 0\n0 0 0qsgn (N4−N5)D45\nl45qsgn (N4−N5)D45\nl450 0 0q|N4−N5|\nl45−λ 0\nqsgn (N5−N1)D51\nl510 0 0qsgn (N5−N1)D51\nl510 0 0 0q|N5−N1|\nl51−λ\n.(68)\nUpon evaluating Jat equilibrium E1, the resulting matrix JE1is given by:\nJE1=\n−a q\nl12λa q\nl12λ0 0 0 −λ\nq0 0 0 −a\nb−N5+l12λ\nq+l23λ\nq\nl51\na q\nl12λ−a l23q\nλ−a q\nl12λa l23q\nλ0 0λ\nq−l232λ\nq0 0 0\n0a q\nl23λ−b−a q\nl23λ0 0 0λ\nqN4−a\nb\nl340 0\n0 0 0 0 0 0 0 −N4−a\nb\nl34−l45(N4−N5) 0\n0 0 0 0 0 0 0 0 l45(N4−N5)a\nb−N5+l12λ\nq+l23λ\nq\nl51\na q2\nl12λ−a q2\nl12λ0 0 0 0 0 0 0 0\n0a q2\nl23λ−a q2\nl23λ0 0 0 0 0 0 0\n0 0 0 0 0 0 0q|N4−a\nb|\nl34−λ 0 0\n0 0 0 0 0 0 0 0q|N4−N5|\nl45−λ 0\n0 0 0 0 0 0 0 0 0q\f\f\fa\nb−N5+l12λ\nq+l23λ\nq\f\f\f\nl51−λ\n.(69)\nAs we have two zero columns, there will be two zero-eigenvalues, and thus the determinant will\nalso be zero, meaning that the equilibrium is non-hyperbolic. The two eigenvectors corresponding\nto the two zero eigenvalues will be (0 ,0,0, N∗\n4,0,0,0,0,0,0) and (0 ,0,0,0, N∗\n5,0,0,0,0,0).\nEvaluating JatE2we obtain:\nJE2=\n−a q\nl51λ0 0 0a q\nl51λ−a\nb−N2+l34λ\nq+l45λ\nq+l51λ\nq\nl120 0 0 −λ\nq\n0 0 0 0 0a\nb−N2+l34λ\nq+l45λ\nq+l51λ\nq\nl12−l23\u0000\nN2−a\nb\u0001\n0 0 0\n0 0 −b−a q\nl34λa q\nl34λ0 0N2−a\nb\nl23λ\nq0 0\n0 0a q\nl34λ−a l45q\nλ−a q\nl34λa l45q\nλ0 0 −λ\nql452λ\nq0\na q\nl51λ0 0a l45q\nλ−a l45q\nλ−a q\nl51λ0 0 0 −l452λ\nqλ\nq\n0 0 0 0 0q\f\f\fa\nb−N2+l34λ\nq+l45λ\nq+l51λ\nq\f\f\f\nl12−λ 0 0 0 0\n0 0 0 0 0 0q|N2−a\nb|\nl23−λ 0 0 0\n0 0 −a q2sign\u0010l34λ\nq\u0011\nl34λa q2sign\u0010l34λ\nq\u0011\nl34λ0 0 0 0 0 0\n0 0 0 −a q2sign\u0010l45λ\nq\u0011\nl45λa q2sign\u0010l45λ\nq\u0011\nl45λ0 0 0 0 0\na q2sign\u0010l51λ\nq\u0011\nl51λ0 0 0 −a q2sign\u0010l51λ\nq\u0011\nl51λ0 0 0 0 0\n.(70)\nAs there is a column of zeros, there will be at least one zero eigenvalue, meaning that this\nequilibrium is also non-hyperbolic. The eigenvector corresponding to the zero eigenvalue will be\n(0, N∗\n2,0, N∗\n4,0,0,0,0,0,0).\nUpon evaluating the Jacobian at the steady state when l12+l23=l34+l45+l51,JE1the\nexpression is given by:\n27JE3=\n−a q\n2l12λ−a q\n2l51λa q\n2l12λ0 0a q\n2l51λ−λ\nq0 0 0 −l12λ\nq+l23λ\nq−l34λ\nq−l45λ\nq\nl51\na q\n2l12λ−a l23q\n2λ−a q\n2l12λa l23q\n2λ0 0λ\nq−l232λ\nq0 0 0\n0a q\n2l23λ−b−a q\n2l23λ−a q\n2l34λa q\n2l34λ0 0λ\nqλ\nq0 0\n0 0a q\n2l34λ−a l45q\n2λ−a q\n2l34λa l45q\n2λ0 0 −λ\nql452λ\nq0\na q\n2l51λ0 0a l45q\n2λ−a l45q\n2λ−a q\n2l51λ0 0 0 −l452λ\nql12λ\nq+l23λ\nq−l34λ\nq−l45λ\nq\nl51\na q2sign\u0010l12λ\nq\u0011\n2l12λ−a q2sign\u0010l12λ\nq\u0011\n2l12λ0 0 0 0 0 0 0 0\n0a q2sign\u0010l23λ\nq\u0011\n2l23λ−a q2sign\u0010l23λ\nq\u0011\n2l23λ0 0 0 0 0 0 0\n0 0 −a q2sign\u0010l34λ\nq\u0011\n2l34λa q2sign\u0010l34λ\nq\u0011\n2l34λ0 0 0 0 0 0\n0 0 0 −a q2sign\u0010l45λ\nq\u0011\n2l45λa q2sign\u0010l45λ\nq\u0011\n2l45λ0 0 0 0 0\na q2sign\u0010l12λ\nq+l23λ\nq−l34λ\nq−l45λ\nq\u0011\n2l51λ0 0 0 −a q2sign\u0010l12λ\nq+l23λ\nq−l34λ\nq−l45λ\nq\u0011\n2l51λ0 0 0 0q\f\f\fl12λ\nq+l23λ\nq−l34λ\nq−l45λ\nq\f\f\f\nl51−λ\n.(71)\nThis expression can be further simplified to:\nJE3=\n−a q\n2l12λ−a q\n2l51λa q\n2l12λ0 0a q\n2l51λ−λ\nq0 0 0 −λ\nq\na q\n2l12λ−a l23q\n2λ−a q\n2l12λa l23q\n2λ0 0λ\nq−l232λ\nq0 0 0\n0a q\n2l23λ−b−a q\n2l23λ−a q\n2l34λa q\n2l34λ0 0λ\nqλ\nq0 0\n0 0a q\n2l34λ−a l45q\n2λ−a q\n2l34λa l45q\n2λ0 0 −λ\nql452λ\nq0\na q\n2l51λ0 0a l45q\n2λ−a l45q\n2λ−a q\n2l51λ0 0 0 −l452λ\nqλ\nq\na q2\n2l12λ−a q2\n2l12λ0 0 0 0 0 0 0 0\n0a q2\n2l23λ−a q2\n2l23λ0 0 0 0 0 0 0\n0 0 −a q2\n2l34λa q2\n2l34λ0 0 0 0 0 0\n0 0 0 −a q2\n2l45λa q2\n2l45λ0 0 0 0 0\na q2\n2l51λ0 0 0 −a q2\n2l51λ0 0 0 0 0\n.(72)\nThe determinant of JE3is given by:\ndet (JE3) =−a4q3(b l23l45q|l12λ+l23λ−l34λ−l45λ| −b l23l45l51λ q)\n16l12l34l51. (73)\nAt the equilibrium point E3, we now that both paths possess equal lengths, implying that:\n|l12, λ+l23, λ−l34, λ−l45, λ|=l51. (74)\nSubstituting equality (74) into (73), we find that the determinant becomes zero, indicating that\nJE3also corresponds to a non-hyperbolic equilibrium.\nB Further results\n12345\n6\n7\n1\n23456\nFigure 11: A C7graph (left) and a C6graph plus an extra edge between nodes 4 and 6 (right).\n28B.1 Non-oscillatory sources and sinks\nIn this section, we include more results from the simulations on cycle graphs of larger sizes and\nmore general graph structure; see Figures 12 and 13. We observe that the slime mould model\ncan still find the shortest path in both cases.\nFigure 12: Results from the model on C7graph, with node 1 as a source, node 3 as a sink, an\ninflow at rate a= 1, an outflow at rate b= 0.5, and the lengths of all edges are lij= 10 apart\nfrom (1 ,2). On the left, we set l12= 10, thus path going through 1-2-3, is the shortest one.\nOn the right, we set l12= 50, thus the path going through 1-7-6-5-4-3 is the shortest one. The\nfirst row shows the input at node 1 and output at node 3, the second row plots the number\nof particles on each node, Ni(t), the third row indicates the conductivity of the edges in the\nshortest path, and the last row indicates the conductivity of the remaining edges in the graph,\nwhere the dashed lines in the corresponding color shows the analytical results of the steady\nstate.\n29Figure 13: Results from the model on C6graph plus an extra edge between nodes 4 and 6 as the\nright in Figure 11, with node 1 as a source, node 3 as a sink, an inflow at rate a= 1, an outflow\nat rate b= 0.5, and the lengths of all edges are lij= 10 apart from (1 ,2). On the left, we set\nl12= 10, thus path going through 1-2-3, is the shortest one. On the right, we set l12= 40, thus\nthe path going through 1-6-4-3 is the shortest one. The first row shows the input at node 1 and\noutput at node 3, the second row plots the number of particles on each node, Ni(t), the third\nrow indicates the conductivity of the edges in the shortest path, and the last row indicates the\nconductivity of the remaining edges in the graph.\n30B.2 Oscillating nodes\n0.00.5I / Onode 1\nnode 3\n123# Particlesnode 1\nnode 2\nnode 3\nnode 4\nnode 5\nnode 6\nnode 7\n1.001.25Conductivity(1,2)\n(2,3)\n0 2000 4000\nTime01Conductivity\n4700 4800 4900 5000\nTime(1,7)\n(3,4)\n(4,5)\n(5,6)\n(6,7)\nFigure 14: Results from the model on a C7graph, with nodes 1 and 3 as the oscillating nodes,\nrateb= 0.5, amplitude A1=A3=A= 1, frequency θ1=θ3=θ= 0.01, and phases ϕ1= 0\nandϕ3=π, with the same reinforcement parameter q= 0.1 and decay parameter λ= 0.01 as\nin Figure 3. The first row plots the actual input/output to the two nodes, the second row shows\nthe number of particles in each node, the third row indicates the conductivity of the edges in the\nshortest path and the last row indicates the conductivity of the remaining edges in the graph,\nwith the full profile on the left while the change in the long term on the right.\n310.00.5I / Onode 1\nnode 3\n123# Particlesnode 1\nnode 2\nnode 3\nnode 4\nnode 5\nnode 6\n1.001.25Conductivity(1,2)\n(2,3)\n0 2000 4000\nTime01Conductivity\n4700 4800 4900 5000\nTime(1,6)\n(3,4)\n(4,5)\n(4,6)\n(5,6)Figure 15: Results from the model on a C6graph plus an extra edge between nodes 4 and 6\nas the right in Figure 11, with nodes 1 and 3 as the oscillating nodes, rate b= 0.5, amplitude\nA1=A3=A= 1, frequency θ1=θ3=θ= 0.01, and phases ϕ1= 0 and ϕ3=π, with the same\nreinforcement parameter q= 0.1 and decay parameter λ= 0.01 as in Figure 3. 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Euler-Masch eroni’s and Stirling’s\nintegral formula are classical, but the integral formula for Glaisher -Kinkelin ap-\npears to be new, as well as the integral formulas for the higher Stir ling-Ramanujan\nconstants. The method presented generalizes naturally to prove that many other\nconstants are exponential periods over the field Q(t,e−t).\n1.Introduction\nA celebrated result of De Moivre [17] and Stirling [46] is the asymptot ic expansion\nof the logarithm of the factorial function,\nlogs! =s/summationdisplay\nk=1logk=/parenleftbigg\ns+1\n2/parenrightbigg\nlogs−s+S0+O(1/s).\nThe constant term in this asymptotic expansion is the Stirling consta nt\nS0=log2π\n2= lim\ns→+∞s/summationdisplay\nk=1logk−/parenleftbigg/parenleftbigg\ns+1\n2/parenrightbigg\nlogs−s/parenrightbigg\n.\nIt is well known that the coefficients of the decaying rest O(1/s) of the asymptotic\nexpansion in the bases of the ( s−k)k≥1are all rational numbers given by Bernoulli\nnumbers. Thus, except for the Stirling constant, all coefficients in the asymptotic\n2020Mathematics Subject Classification. 40A05, 40G99, 11J81, 33B15, 11B68.\nKey words and phrases. Euler-Mascheroni, Stirling, Glaisher-Kinkelin, transcendental con stants,\nexponential period.\n12 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nexpansion are rational numbers. It is not known if the Stirling const ant is transcen-\ndental, and not even known if it is irrational1.\nEuler defined the Gamma function that interpolates the factorial f unction. The\nEuler-Mascheroni constant γappears when looking at the logarithmic derivative of\nthe Gamma function. It can also be defined in a similar way as the Stirling constant\nby looking at the asymptotic expansion of the divergent harmonic se ries\ns/summationdisplay\nk=11\nk= logs+γ+O(1/s).\nSo we have\nγ= lim\ns→+∞s/summationdisplay\nk=11\nk−logs.\nAgain, in this asymptotic expansion all coefficients other than γarerational numbers,\nthe decaying part in O(1/s) is given by Bernoulli numbers. The arithmetic nature\nof the Euler-Mascheroni constant is completely mysterious, but it is conjectured to\nbe transcendental (it is not even known to be irrational). For the h istory of the\nEuler-Mascheroni constant we refer to the classical account by Glaisher [22] and the\nmodern one by Lagarias [30].\nIn general, using the Euler-McLaurin formula, Ramanujan studied t he asymptotic\nexpansion of the logarithm of the n-factorial function (see [11, Chapter 9, p. 273])\ns!(n)= 11n.22n.33n.44n...ssn.\nFor example, for n= 1,2,3 we have,\ns/summationdisplay\nk=1klogk=/parenleftbigg1\n2s2+1\n2s+1\n12/parenrightbigg\nlogs−1\n4s2+S1+O(1/s),\ns/summationdisplay\nk=1k2logk=/parenleftbigg1\n3s3+1\n2s2+1\n6s/parenrightbigg\nlogs−1\n9s3+1\n12s+S2+O(1/s),\ns/summationdisplay\nk=1k3logk=/parenleftbigg1\n4s4+1\n2s3+1\n4s2−1\n120/parenrightbigg\nlogs−1\n16s4+1\n12s2+S3+O(1/s).\nThe Euler-McLaurin formula shows that, except for the constant coefficient Sn, all\ncoefficients are rational numbers, in particular those of the remain ing part O(1/s)\nare given by Bernoulli numbers.\n1The constants eandπare conjectured to be algebraically independent over the rationals and\nthis easily implies that the Stirling constant is irrational. The algebraic in dependence of eandπ\nfollows from Schanuel’s conjecture that also implies that the Stirling c onstant is transcendental.STIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 3\nThe constants Sk,k≥0, are generalizations of the Stirling constant that we name,\nforhistorical reasons, Stirling-Ramanujan constants . TheStirling constant appearsas\nS0forn= 0. The Euler-Mascheroni constant γappears as γ=S−1and corresponds\nto the divergent harmonic series.\nThe constant S1, more precisely A=eS1, appears in the works of Glaisher [22] and\nKinkelin [29]. It is known as the Glaisher-Kinkelin constant that arises in the theory\nof the Barnes Gamma function [4].\nAgain, all the coefficients in the above asymptotic expansions other than the con-\nstant one are rational numbers. The arithmetic nature of the Stir ling-Ramanujan\nconstants is unknown. It is natural to conjecture:\nConjecture 1. The Stirling-Ramanujan constants are transcendental.\nThese generalized Stirling constants had a deep meaning for Ramanu jan transal-\ngebraic view. For him, they are the “sum” of the divergent infinite se ries by the\nresummation method he discovered based on the Euler-McLaurin fo rmula. As he\nputs it, these are “the barycenter of the divergent sum” that ba lances between the\ndivergent series and the divergent integral in the Euler-McLaurin s ummation formula\n[23]. Ramanujan even computes these constants using the derivat ive of the Riemann\nzeta function at integer values. Ramanujan resummation procedu re is described (in\nanimperfect form)intheclassical bookof Hardy[24]. The reader ca ndirectly consult\nBerndt volumes [11] for a more faithful description. A rigorous app roach is proposed\nin the monographs by Candelpergher [13], [14]. Another rigorous app roach based on\nthe Poisson-Newton formula [35] was found by the authors [34].\nThe Stirling-Ramanujan constants reappeared after Ramanujan in 1933 when Ben-\ndersky [10] defined a new hierarchy of higher Gamma functions differ ent from that\nof Barnes [4]. The Bendersky Gamma function interpolates the n-factorial function.\nContrary to the Barnes Gamma functions, higher Bendersky Gamm a functions are\nnot meromorphic since they have ramified singularities at negative int egers. Indeed,\nhistorically works of Kinkelin [27] and Alekseevsky [2] [3] preceded by m any decades\nthose of Barnes and Bendersky. See the recent historical accou nt by Neretin [37].\nThe Stirling-Ramanujan constants have been rediscovered multiple times by sev-\neral authors, unaware that they were already studied in Ramanuj an notebooks (for\nexample, see constants BandCin [45, p. 56] and the comments therein). In some\nrecent literature these constants are named Bendersky-Adamchik constants (in [1]\nAdamchik computes these constants in terms of values of the zeta function and its\nderivative at integer values). For example in the recent article by Co ppo [16] where\nhe gives remarkable converging series for the constants. Anothe r noteworthy series\nrepresentation of the Glaisher-Kinkelin constant is given by by Pain in [39].4 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nAll these higher Gamma functions and the associated Stirling-Raman ujan con-\nstants arise in multiple mathematical contexts, as for example in Shin tani’s work\nin Algebraic Number theory, in the computation of determinants of L aplacians in\nRiemannian geometry, and in the zeta-regularization theory in Phys ics. As a cu-\nriosity, in the article [25] in the Physics literature we find the comput ation of the\nStirling-Ramanujan constants up to S3(probably the record in the literature).\nThe main purpose of this article is to give a general whole integral for mulas for all\nthe Stirling-Ramanujan constants Sn.\nTheorem 2 (General Exponential Period Formula) .Forn≥0we have\nSn= (−1)n+1n!/integraldisplay+∞\n01\ntn/parenleftBigg\n1\n1−e−t−n/summationdisplay\nk=−1bktk−rntn+1/parenrightBigg\ne−tdt\nt,\nwherernis the rational number\nrn=n+1/summationdisplay\nk=1(−1)k+1\nk!bn−kHk,\nwhere the (Hk)are the harmonic numbers, Hk= 1+1\n2+...+1\nk, and the coefficients\n(bk)have the generating function\n1\n1−e−t=+∞/summationdisplay\nk=−1bktk,\nand they are given by bk= (−1)k+1Bk+1\n(k+1)!, whereBkis thek-th Bernoulli number.\nThere are multiple different expressions for the Euler-Mascheroni, Stirling and\nGlasher-Kinkelin constants in the literature (see [45] for example) . But starting from\nS1these integral expressions seem to be new.\nForn=−1 the formula makes sense and we recover the classical integral fo rmula\nforγ=S−1\nγ=/integraldisplay+∞\n0t/parenleftbigg1\n1−e−t−1\nt/parenrightbigge−t\ntdt.\nThis integral formula is due to Euler back in 1770 ([19] section 25), bu t Whittaker-\nWhatson [47, Example 2, p. 248] attributes this formula to Dirichlet ( presumably to\n[18], 1836). Moreover, Euler also gives this formula in [20] (1785) an d devotes a full\narticle to it ([21], 1789).\nForn≤ −2 the formula makes sense. In that case the series\nζ(n) =+∞/summationdisplay\ns=01\n(s+1)nSTIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 5\nis converging, the asymptotic expansion of the partial sum reduce s to a constant\n(its sum), and the integral formula is nothing else but the classical f amous formula\nfor the Riemann zeta function used by Riemann in his investigations of the complex\nextension of ζ\nζ(s) =1\nΓ(s)/integraldisplay+∞\n0ts\n1−e−te−tdt\nt.\nForn= 0, for the Stirling constant S0we recover the classical integral formula due\nto Pringsheim [42] (see also [47], [39, p. 288])\nS0=log(2π)\n2=−/integraldisplay+∞\n0/parenleftbigg1\n1−e−t−1\nt−1\n2−t/parenrightbigge−t\ntdt.\nForn= 1, for the Glaisher-Kinkelin constant log Awe get the following integral\nformula that appears to be new in the literature:\nCorollary 3 (Integral formula for Glaisher-Kinkelin constant) .\nS1= logA=/integraldisplay+∞\n01\nt/parenleftbigg1\n1−e−t−1\nt−1\n2−t\n12+t2\n4/parenrightbigge−t\ntdt.\nWe also get the following new integral formulas for the constants S2andS3.\nCorollary 4. We have\nS2=−2/integraldisplay+∞\n01\nt2/parenleftbigg1\n1−e−t−1\nt−1\n2−t\n12−1\n72t3/parenrightbigge−t\ntdt\nS3= 6/integraldisplay+∞\n01\nt3/parenleftbigg1\n1−e−t−1\nt−1\n2−t\n12+t3\n720−1\n288t4/parenrightbigge−t\ntdt\nAn interesting fact about all these new formulas is that all the Stirlin g-Ramanujan\nconstants appear as Exponential Periods with base field of functio nsQ(t,e−t). This\nreveals their transalgebraic nature. In general, we can consider a classKof transal-\ngebraic functions, as those appearing in Liouville towers of function s (see for example\n[26], [44]), obtained by algebraic, logarithmic, integral and exponent ial extensions. A\nTransalgebraic period overKis a number of the form\nω=/integraldisplay\nηf(s)ds\nwhere the integral is converging and taken over an open path ηwheref∈Kis\nholomorphicand ηhasend-pointsthataresingularitiesof f. Thisnotionisinteresting\nparticularly when the class of functions Kis a countable class, then most of complex\nnumbers are not periods over K. For example this occurs for a Liouville type of\nextension of the differential field of rational functions Q(s). When we adjoint a6 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nfinite number of exponential function K=Q(t,eω1t,...,eωnt), we obtain Exponential\nPeriods.\nAdifferentnotionofexponentialperiodsisconsideredbyKonstevic handZagier[28]\nby analogy with the more classical algebraic periods, where an algebr aic differential\nform is integrated over a cycle of an algebraic variety. Is easy to se e that these are\nmore restrictive than Transalgebraic Periods. Our point of view follo ws the classical\npath. Exponential periods were studied by Liouville, Picard, Fuchs,... already in the\n19th century. They appear in the natural context of log-Riemann surface theory as\nthe locationof the ramifications, andas asymptotic values of trans cendental functions\nassociated to log-Riemann surfaces. These log-Riemann surfaces are transalgebraic\ncurves that generalize classical algebraic curves by allowing a finite n umber of infinite\nramification points. The determinant of the matrix of periods which c omes from a\nbases of a vector space of transcendental functions associate d to the transalgebraic\ncurve is the Ramificant Determinant. This determinant is a fundamen tal object of\nthe transalgebraic algebra of transalgebraic curves (see the orig inal manuscript [6],\nand subsequent articles [7], [8], [9], and the more recent one [41]).\nIf we remove the combinatorial complexity in the definition and the co nstants,\nup to a rational affine combination, the family of Stirling-Ramanujan c onstants is\nrationally equivalent to the family of Upsilon constants considered by the authors in\na related context:\nDefinition 5 (Upsilon constants) .Forn≥ −1we define\nΥn=/integraldisplay+∞\n01\ntn/parenleftBigg\n1\n1−e−t−n/summationdisplay\nk=−1bktk/parenrightBigg\ne−tdt\nt.\nObviously we have\nSn= (−1)n+1n!(Υn−rn),\nand the transcendence conjecture is equivalent to the transcen dence conjecture for\nthe Upsilon constants.\nConjecture 6. Upsilon constants Υnare transcendental.\nIt is natural to raise the question of their algebraic independence f orn≥ −1.STIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 7\n2.Higher Frullani integrals\nThe classical Frullani integral expresses log sas an exponential period:\nlogs=/integraldisplay+∞\n01\nt(e−t−e−st)dt.\nThis formula takes a nicer form when we single out the differential elem ente−tdt\nt, that\nis the natural differential2inC∗for periods with an exponential singularity at ∞and\na polar singularity at 0. The expressions are also nicer when we work w ith functions\nof the variable s+ 1 instead of s. This is because log( s+ 1), contrary to log( s), is\nan LLD function (a Left Located Divisor function has all of its zeros and poles, or in\ngeneral singularities, in the left half plane {ℜs <0}). This is a relevant notion in the\ntheory of Poisson-Newton formula ([34], [35], [36]).\nBecause of these reasons, the natural way to write down the Fru llani integral is\nlog(s+1) =/integraldisplay+∞\n0(1−e−st)e−tdt\nt.\nObserve that the Frullani integral is obtained by integration on the variablesover\nthe interval [1 ,s] of the elementary integral\n1\ns+1=/integraldisplay+∞\n0te−ste−tdt\nt,\nthat can be thought of the primitive generator integral. Starting w ith the Frullani\nintegral and iterating integrations on the variable sover the interval [1 ,s], we obtain\nhigher Frullani integrals that express sklogsas an exponential integral.\nTheorem 7. We have\n(s+1)nlog(s+1) =n/summationdisplay\nk=1/parenleftbiggn\nk/parenrightbigg\n(Hn−Hn−k)sk+\n+(−1)n+1n!/integraldisplay+∞\n01\ntn/parenleftBigg\ne−st−n/summationdisplay\nk=0(−s)ktk\nk!/parenrightBigg\ne−tdt\nt,\nwhereHn= 1+1\n2+···+1\nnare the harmonic numbers for n≥1, andH0= 0.\nThis explicit formula can be found in the references [32] and [33]. The se higher\nFrullani integrals are relevant to explain the most popular BBP formu las as was\n2For the reader knowledgeable of the theory of log-Riemann surfac es, the geometry associated to\nthis basic differential ω=e−tdt\ntis the tube-log-Riemann surface that derives from the log-Riemann\nsurface of the logarithm replacing a planar sheet by a tube C/Z(see [6]).8 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nunveiled in the recent article by D. Barsky and the authors [5]. A com plete proof of\nTheorem 7 can be found in this reference (Prop. 2.1 and Prop. 2.2 of [5]).\nWe have a polynomial part, without constant term, and an integral part which is\nthe Hadamard regularization of the Laplace transform of1\ntn+1e−tas considered in [35].\n3.Technical preliminaries\n3.1.Technical normalizations. It appears combinatorially simpler to find the in-\ntegral formulas for the constants appearing in a slight different ex pansion base, which\nis closer to the Upsilon constants from the introduction.\nIn the proof it is natural to compute the expansion on the shifted v ariables+1,\nand this comes down to\ns/summationdisplay\nk=1knlogk=˜An(s)log(s+1)+˜Bn(s)+˜Sn+˜Rn(1/s),\nwhere˜An∈Q[s],˜Bn∈sQ[s], deg˜An= deg˜Bn=n+1, and ˜Rn(1/s)∈Q[[1/s]].\nFromthisitfollowstheexpansionintheformusedtodefinetheStirling -Ramanujan\nconstants\ns/summationdisplay\nk=1knlogk=An(s)logs+Bn(s)+Sn+Rn(1/s),\nwithAn∈Q[s],Bn∈sQ[s], andRn(1/s)∈Q[[1/s]].\nIndeed, we have\ns/summationdisplay\nk=1knlogk=˜An(s)log(s+1)+˜Bn(s)+˜Sn+˜Rn(1/s)\n=˜An(s)log(1+1 /s)+˜An(s)logs+˜Bn(s)+˜Sn+˜Rn(1/s)\n=˜An(s)+∞/summationdisplay\nk=1(−1)k+1\nksk+˜An(s)logs+˜Bn(s)+˜Sn+˜Rn(1/s).\nTherefore, if we write down the polynomial as ˜An(s) =n+1/summationtext\nj=0˜ajsj, then we have\nSn=˜Sn+n+1/summationdisplay\nk=1(−1)k+1\nk˜ak. (1)STIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 9\n3.2.Asymptotic decay. We use the following classical and well known fact from\nLaplace transform theory (see [15] for example).\nProposition 8. Letf:R+→Rdifferentiable, in particular at the right at t= 0.\nWe assume f′is bounded. Then the Laplace transform\nLf(s) =/integraldisplay+∞\n0f(t)e−stdt\ndecays as O(1/s)whens→+∞. Moreover, if we assume that fis infinitely differen-\ntiable and the derivatives are bounded, then for every n≥1, we have the asymptotic\nexpansion\nLf(s) =f(0)\ns+f′(0)\ns2+...+f(n−1)(0)\nsn+O(1/sn+1).\nProof.The proof is immediate by integration by parts:\nLf(s) =/bracketleftbigg\n−f(t)e−st\ns/bracketrightbigg+∞\n0+1\ns/integraldisplay+∞\n0f′(t)e−stdt\n=f(0)\ns+1\ns/integraldisplay+∞\n0f′(t)e−stdt\n=f(0)\ns+1\nsLf′(s).\nAsf′(s) is bounded, we have that Lf′(s) is bounded for s≥1. This implies that\nLf(s) =O(1/s), and the same will be true for f′, that is, Lf′(s) =O(1/s), hence\nLf(s) =f(0)\ns+O(1/s2). Working inductively, we get the result for any n≥1./square\n4.Derivation of the integral formulas\nConsider the n-th higher Frullani integrals given in Theorem 7 and we add from\ns= 0 tos−1 to get (we use the convention 00= 1)\ns/summationdisplay\nk=1knlogk=n/summationdisplay\nk=1/parenleftbiggn\nk/parenrightbigg\n(Hn−Hn−k)/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg\n+\n+(−1)n+1n!/integraldisplay+∞\n01\ntn/parenleftBigg\n1−e−st\n1−e−t−n/summationdisplay\nk=0(−t)k\nk!/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg/parenrightBigg\ne−tdt\nt.\nBy Faulhaber’s formula, for k≥1,\ns−1/summationdisplay\nj=0jk10 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nis a polynomial in the variable swithout constant term, whose coefficients are given\nby Bernoulli numbers (we remind this fact in the Appendix).\nTherefore, we have that the sum\nn/summationdisplay\nk=1/parenleftbiggn\nk/parenrightbigg\n(Hn−Hn−k)/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg\nis also a polynomial in the variable swithout constant term and we can disregard it\nwhen looking for the constant coefficient of the asymptotic expans ion fors→+∞.\nThus, we reduce the problem at looking for the constant term of th e asymptotic\nexpansion of the integral\nIn= (−1)nn!/integraldisplay+∞\n01\ntn/parenleftBigg\ne−st−1\n1−e−t+n/summationdisplay\nk=0(−t)k\nk!/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg/parenrightBigg\ne−tdt\nt\n= (−1)nn!/integraldisplay+∞\n0/parenleftBigg\ne−st1\ntn1\n1−e−t−1\ntn1\n1−e−t+1\ntnn/summationdisplay\nk=0(−t)k\nk!/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg/parenrightBigg\ne−tdt\nt.\nWe use the expansion\n1\n1−e−t=+∞/summationdisplay\nk=−1bktk=1\nt+1\n2+1\n12t−1\n720t3+...\nThen\n(−1)n\nn!In=/integraldisplay+∞\n0/parenleftBigg\ne−stn/summationdisplay\nk=−1bktk−n+e−st1\ntn/parenleftBigg\n1\n1−e−t−n/summationdisplay\nk=−1bktk/parenrightBigg\n−1\ntn1\n1−e−t+1\ntnn/summationdisplay\nk=0(−t)k\nk!/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg/parenrightBigg\ne−tdt\nt\n=n/summationdisplay\nk=−1bk/integraldisplay+∞\n01\ntn−k/parenleftBigg\ne−st−n−k/summationdisplay\nl=0(−st)l\nl!/parenrightBigg\ne−tdt\nt+ (2)\n+/integraldisplay+∞\n0e−st1\ntn/parenleftBigg\n1\n1−e−t−n/summationdisplay\nk=−1bktk/parenrightBigg\ne−tdt\nt+ (3)\n+/integraldisplay+∞\n0/bracketleftBiggn/summationdisplay\nk=−1bk\ntn−kn−k/summationdisplay\nl=0(−st)l\nl!−1\ntn1\n1−e−t+1\ntnn/summationdisplay\nk=0(−t)k\nk!/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg/bracketrightBigg\ne−tdt\nt.(4)STIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 11\nIn line (2) we recognize the first n+1 higher Frullani integrals:\nn/summationdisplay\nk=−1bk/integraldisplay+∞\n01\ntn−k/parenleftBigg\ne−st−n−k/summationdisplay\nl=0(−st)l\nl!/parenrightBigg\ne−tdt\nt\n=/parenleftBiggn/summationdisplay\nk=−1bk(−1)n−k+1\n(n−k)!(s+1)n−k/parenrightBigg\nlog(s+1) (mod sQ[s])\n=(−1)n\nn!˜An(s)log(s+1) (mod sQ[s]),\nforsome polynomial ˜An(s)∈Q[s]. Notethat thecoefficients ˜ akof thepolynomials ˜An\ncan be computed explicitly from the coefficients ( bk)k≥−1. Indeed for 0 ≤j≤n+1,\nwe have\n˜aj=n−j/summationdisplay\nk=−1bk(−1)k+1n!\n(n−k−j)!j!. (5)\nThe second integral (3) is O(1/s) whens→+∞using Proposition 8.\nFor the third integral (4), we replace the sum of k-powers of the first s−1 integers\nby the Faulhaber polynomial in sthat is given by the Bernoulli type coefficients ( bk)\n(use Proposition 11 from the Appendix),\n/integraldisplay+∞\n0/bracketleftBiggn/summationdisplay\nk=−1bk\ntn−kn−k/summationdisplay\nl=0(−st)l\nl!−1\ntn1\n1−e−t+1\ntnn/summationdisplay\nk=0(−t)k\nk!/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg/bracketrightBigg\ne−tdt\nt\n=/integraldisplay+∞\n0/bracketleftBiggn/summationdisplay\nh=−1bh\ntn−hn−h/summationdisplay\nl=0(−st)l\nl!−1\ntn1\n1−e−t+1\ntnn/summationdisplay\nk=0(−t)k\nk!/parenleftBigg\n(−1)k+1k!k+1/summationdisplay\nl=1(−1)lbk−lsl\nl!/parenrightBigg/bracketrightBigg\ne−tdt\nt\n=/integraldisplay+∞\n01\ntn/parenleftBiggn/summationdisplay\nk=−1bktk−1\n1−e−t/parenrightBigg\ne−tdt\nt.\nThe simplification to the last result comes from the observation that there is a can-\ncellation of the terms in the second line for the first summation when hequals to\nk−l. In the first sum, the condition is 0 ≤l≤n−h/ne}ationslash=n+1. In the second sum the\ncondition is 1 ≤l≤k+1≤n+1. This means that only the terms for l= 0 in the\nfirst sum remain. This massive cancellation is to be expected since the coefficients\nof the monomials on the variable swould give divergent integrals if they wouldn’t\nvanish.12 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nTherefore, this gives the modified Stirling-Ramanujan constant\n˜Sn= (−1)n+1n!/integraldisplay+∞\n01\ntn/parenleftBigg\n1\n1−e−t−n/summationdisplay\nk=−1bktk/parenrightBigg\ne−tdt\nt.\nHence using (1) and (5), we get\nSn=˜Sn+n+1/summationdisplay\nj=1(−1)j+1\nj˜aj=˜Sn+n+1/summationdisplay\nj=1(−1)j+1\njn−j/summationdisplay\nh=−1bh(−1)h+1n!\n(n−h−j)!j!,\nandSn=˜Sn+ ˆrnwhere ˆrnis the rational number\nˆrn=n+1/summationdisplay\nj=1n−j/summationdisplay\nh=−1(−1)j+h\njbhn!\n(n−h−j)!j!.\nWe can insert this rational number inside the integral and we finally ge t\nSn= (−1)n+1n!/integraldisplay+∞\n01\ntn/parenleftBigg\n1\n1−e−t−n/summationdisplay\nk=−1bktk−rntn+1/parenrightBigg\ne−tdt\nt,\nwhere the rational number rnis given by\nrn=(−1)n\nn!ˆrn=n+1/summationdisplay\nj=1n−j/summationdisplay\nh=−1(−1)n+j+h\njbh1\n(n−h−j)!j!.\nWe can simplify this expression:\nProposition 9. We have\nrn=n−1/summationdisplay\nh=−1bh(−1)n−h+1\n(n−h)!Hn−h.\nProof.We compute\nrn=n+1/summationdisplay\nj=1n−j/summationdisplay\nh=−1(−1)n+j+h\njbh1\n(n−h−j)!j!\n=n−1/summationdisplay\nh=−1bh(−1)n−h\n(n−h)!n−h/summationdisplay\nj=1(−1)j\nj/parenleftbiggn−h\nj/parenrightbigg\n=n−1/summationdisplay\nh=−1bh(−1)n−h+1\n(n−h)!Hn−h,STIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 13\nwhere we use the identityn/summationtext\nk=1(−1)k\nk/parenleftbign\nk/parenrightbig\n=−Hn. These type of identities of harmonic\nnumbers are immediate when we write them down as exponential perio ds:\nHn= 1+1\n2+...+1\nn=n/summationdisplay\nk=1/integraldisplay+∞\n0e−(k−1)te−tdt=/integraldisplay+∞\n01−e−nt\n1−e−te−tdt,\nand then\nHn=/integraldisplay+∞\n01−(1−(1−e−t))n\n1−e−te−tdt\n=−/integraldisplay+∞\n0n/summationdisplay\nk=1/parenleftbiggn\nk/parenrightbigg\n(−1)k(1−e−t)k−1e−tdt\n=−n/summationdisplay\nk=1/parenleftbiggn\nk/parenrightbigg\n(−1)k/integraldisplay+∞\n0(1−e−t)ke−tdt\n=−n/summationdisplay\nk=1/parenleftbiggn\nk/parenrightbigg(−1)k\nk+1dt.\n/square\nThis completes the proof of Theorem 2.\n5.Particular cases\nUsing Proposition 9, we obtain\nr0=b−1= 1,\nr1=−H2b−1\n2!+H1b0\n1!=−1\n4,\nr2=H3b−1\n3!−H2b0\n2!+H1b1\n1!=1\n72,\nr3=−H4b−1\n4!+H3b0\n3!−H2b1\n2!=1\n288.14 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nTherefore Theorem 2 gives\nS0=log(2π)\n2=/integraldisplay+∞\n0/parenleftbigg1\nt+1\n2+t−1\n1−e−t/parenrightbigge−t\ntdt,\nS1= logA=/integraldisplay+∞\n01\nt/parenleftbigg1\n1−e−t−1\nt−1\n2−t\n12+t2\n4/parenrightbigge−t\ntdt,\nS2=−2/integraldisplay+∞\n01\nt2/parenleftbigg1\n1−e−t−1\nt−1\n2−t\n12−1\n72t3/parenrightbigge−t\ntdt,\nS3= 6/integraldisplay+∞\n01\nt3/parenleftbigg1\n1−e−t−1\nt−1\n2−t\n12+t3\n720−1\n288t4/parenrightbigge−t\ntdt.\nThe numbers Snup ton= 4 are computed by Bendersky [10, p. 265]. Translating\ninto natural logarithm (Bendersky uses decimal logarithm), we hav e\nS0= log(10) ·0,399089...= 0.918938...\nS1= log(10) ·0.108032...= 0.248754...\nS2= log(10) ·0.013223...= 0.030448...\nS3= log(10) ·(−1+0.991029...) =−0.008971...\nThis agrees with the numerical values of the integrals in Theorem 2 giv en above.\nForn=−1, we also have that the above formulas give r−1= 0 and\nS−1=γ=/integraldisplay+∞\n0t/parenleftbigg1\n1−e−t−1\nt/parenrightbigge−t\ntdt.\n6.General scope of the method\nThe precedent procedure is very general and allows to find a multitu de of integral\nformulas for many constants defined as the constant term in asym ptotic expansions\nof arithmetic nature. Consider sums (typical divergent) of the ty pe\ns/summationdisplay\nk=0f(k+1),\nwheref(s+ 1) is an LLD function (LLD means Left Located Divisor) so that the\ngeneral divisor of fare in the left half plane. When fis meromorphic (for example as\nthe Euler Gamma function Γ), then the general divisor is only compos ed by zeros and\npoles. In general, we can have ramified singularities (as for the logar ithm function\nlogs).\nWe consider the asymptotic when s→+∞of the above sum. For natural arith-\nmetic functions, these sums are divergent and have a natural asy mptotic expansionSTIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 15\non a natural bases of functions and a decaying part in O(1/s). The constant term in\nthe expansion define generalized Stirling-Ramanujan constants.\nA particularly interesting case is when fadmits a Malmst´ en type formula. We\nrecall that Malmst´ en formula for the Euler Gamma function ([31]), f orℜs >0,\nlogΓ(s+1) =/integraldisplay+∞\n0/parenleftbigg\ns+e−st−1\n1−e−t/parenrightbigge−tdt\nt.\nA general Malmst´ en formula for g=efover the ring Q(t,e−t)[e−st] is, forℜs >0,\nlogg(s+1) =f(s+1) =/integraldisplay∞\n0G(t,e−t,e−st)e−tdt\nt.\nNote that this means that f(s) behaves as an exponential period (with parameter s).\nThe higher Frullani integrals are just Malmst´ en formulas for the f(s) =snlog(s), or\ng(s) =ssn.\nIn that case the general procedure presented will also work. The key property is\nthat the sum for which we study the asymptotic is also an an exponen tial period in\nthe same ring Q(t,e−t)[e−st].\n7.Appendix: Bernoulli computations.\nWe review in this Appendix the computations with Bernoulli numbers th at we use.\nThe classical Bernoulli polynomials ( Bk(s))k≥0can be defined via their generating\nfunction\ntest\net−1=+∞/summationdisplay\nk=0Bk(s)tk\nk!.\nMaking the change of variable t→ −tthis can also be written (in a form that is more\nappealing to us) as\nte−st\n1−e−t=+∞/summationdisplay\nk=0(−1)kBk(s)tk\nk!.\nThe Bernoulli numbers are defined as Bk=Bk(0).\nWe prefer to work with a related sequence of polynomials ( bk(s))k≥0defined via\ntheir generating function\ne−st\n1−e−t=1\nt++∞/summationdisplay\nk=0bk(s)tk.\nThus we have\nbk(s) = (−1)k+1Bk+1(s)\n(k+1)!,16 V. MU ˜NOZ AND R. P ´EREZ-MARCO\nand the related Bernoulli like numbers bk=bk(0), and defining b−1= 1, thus\n1\n1−e−t=+∞/summationdisplay\nk=−1bktk.\nWe can compute in two ways the expansion of1−e−st\n1−e−t,\n1−e−st\n1−e−t=1\n1−e−t−e−st\n1−e−t=+∞/summationdisplay\nk=0(bk−bk(s))tk\nand, for a positive integer s,\n1−e−st\n1−e−t=s−1/summationdisplay\nj=0e−jt=+∞/summationdisplay\nk=0(−t)k\nk!/parenleftBiggs−1/summationdisplay\nj=0jk/parenrightBigg\n.\nThe sum abote can be taken from j= 1 tos−1, except in the case k= 0 (in which\ncase, we set 00= 1).\nIt follows the form of Faulhaber’s formula for the sum of k-powers of the first s−1\nintegers.\nProposition 10. We have\ns−1/summationdisplay\nj=0jk= (−1)kk!(bk−bk(s)).\nAs it is well known we can compute the Bernoulli polynomials from the se quence\nof Bernoulli numbers:\nBk(s) =k/summationdisplay\nj=0/parenleftbiggk\nj/parenrightbigg\nBk−jsj.\nHence we have\nbk(s) =k+1/summationdisplay\nj=0(−1)jbk−jsj\nj!\nand we conclude:\nProposition 11. We have\ns−1/summationdisplay\nj=0jk= (−1)k+1k!k+1/summationdisplay\nj=1(−1)jbk−jsj\nj!.STIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 17\nAcknowledgements. The first author has been partially supported by Comunidad\nde Madrid R+D Project PID/27-29 and Ministerio de Ciencia e Innovac i´ on Project\nPID2020-118452GB-I00 (Spain).\nReferences\n[1] ADAMCHIK, V.S.; Polygamma functions of negative order , Journal of Computational and\nApplied Mathematics, 100, p. 191-199, 1998.\n[2] ALEKSEEVSKY, V.P.; On the functions that are similar to the Gamma function (Russian),\nCharkow Ges., 1:1, p. 169-192, 1888.\n[3] ALEKSEEVSKY, V.P.; On the functions that are similar to the Gamma function (Russian),\nCharkow Ges., 1:2, p. 193-238, 1889.\n[4] BARNES, E.W.; The theory of G-function , Quat. J. 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P ´EREZ-MARCO\n[21] EULER, L.; Evolutio formulae integralis/integraltext\ndx/parenleftBig\n1\n1−x+1\nlogx/parenrightBig\na termino x= 0adx= 1extensae,\nNova Acta Academiae Scientarum Imperialis Petropolitinae 4, p.3-16; Opera Omnia: Series 1,\nVolume 18, p.318-334; Euler Archive [E629], eulerarchive.maa.org, 17 87.,\n[22] GLAISHER, J.W.L.; On the Product 11.22.33...nn, Messenger Math., 7, p.43-47, 1878.\n[23] HARDY, G.H; Ramanujan: Twelve lectures on subjects suggested by his lif e and work , Chelsea,\n1940.\n[24] HARDY, G.H; Divergent series , American Mathematical Society, Chelsea, 1949.\n[25] KAMELA, M.; BURGUESS, C.P. ; Massive-scalar effective actions on anti-de Sitter spaceti me,\nCan.J.Phys., 77, p.85-99, 1999.\n[26] KHOVANSKII, A.; Topological Galois Theory , Springer Monographs in Mathematics, 2014.\n[27] KINKELIN, H.; Ueber der mit der Gammafunction verwandte Transcendente un d derenAnwer-\ndung auf die Integralrechung , J. Reine Angew. 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Iterated primitives of logarithmic powers ,\nInternational Journal of Number Theory, 7, 3, p.623-634, 2011.\n[34] MU ˜NOZ, V.; P ´EREZ-MARCO, R.; Poisson-Newton formulas and Dirichlet series ,\narXiv:1301.6511, 2013.\n[35] MU ˜NOZ, V.; P ´EREZ-MARCO, R.; Unified treatment of Explicit and Trace Formulas via\nPoisson-Newton Formula , Comm. Math. Physics, 334, 1, p.331-366, 2015.\n[36] MU ˜NOZ, V.; P ´EREZ-MARCO, R.; On the genus of meromorphic functions , Proceedings of the\nAmerican Mathematical Society, 143, p.341-351, 2015.\n[37] NERETIN, Y.; The double Gamma function and Vladimir Alekseevsky , arXiv:2402.07740, 2024.\n[38] PAIN, J.-Ch.; Series representation for the logarithm of the Glaisher-Ki nkelin constant ,\narXiv:2304.07629, 2023.\n[39] PATRA, B.; Complex variables and special functions , PHI Learning Private Ltd, 2014.\n[40] P´EREZ-MARCO, R.; On the definition of Euler Gamma function , L’Enseignement\nMath´ ematique, 68, 1/2, p.135-160, 2022 (see also ArXiv 2001.04445, hal-02437549) .\n[41] P´EREZ-MARCO, R.; Difference equations and Omega function , ArXiv 2301.04759, 2023.\n[42] PRINGSHEIM, A.; Zur Theorie des Gamma-Functionen , Math. Annalen, 31, 1888.\n[43] RAMANUJAN, S.; Notebooks , Tata Institute of Fundamental Research, Mumbai, 1957.\n[44] RITT, J.F.; Integration in finite terms. Liouville’s theory of elementa ry methods , Columbia\nUniversity Press, New York, 1948.\n[45] SRIVASTAVA, H.M.; CHOI, J. ; Zeta and q-zeta functions and associated series and integrals ,\nElsevier, 2012.\n[46] STIRLING,J.; Methodus differentialis sive tractatus de summatione et int erpola- tione serierum\ninfinitarum , London, 1730.(see also the annotated translation by I. Tweedle, Springer, 2003).STIRLING-RAMANUJAN CONSTANTS ARE EXPONENTIAL PERIODS 19\n[47] WHITTAKER, E.T.; WHATSON, G.N.; A course in modern analysis , Cambridge Univ. Press.,\n4th edition, 1927.\nInstituto de Matem ´atica Interdisciplinar and Departamento de ´Algebra, Geometr ´ıa\ny Topolog ´ıa, Universidad Complutense de Madrid, Spain\nEmail address :vicente.munoz@ucm.es\nCNRS, IMJ-PRG, Universit ´e Paris Cit ´e, France\nEmail address :ricardo.perez.marco@gmail.com"    },    {        "title": "2402.02720v1.Discounted_Adaptive_Online_Prediction.pdf",        "content": "Discounted Adaptive Online Prediction\nZhiyu Zhang\nHarvard University\nzhiyuz@seas.harvard.eduDavid Bombara\nHarvard University\ndavidbombara@g.harvard.edu\nHeng Yang\nHarvard University\nhankyang@seas.harvard.edu\nAbstract\nOnline learning is not always about memorizing everything. Since the future can be statistically very\ndifferent from the past, a critical challenge is to gracefully forget the history while new data comes in. To\nformalize this intuition, we revisit the classical notion of discounted regret using recently developed techniques\nin adaptive online learning. Our main result is a new algorithm that adapts to the complexity of both the\nloss sequence and the comparator, improving the widespread non-adaptive algorithm – gradient descent with\na constant learning rate. In particular, our theoretical guarantee does not require any structural assumption\nbeyond convexity, and the algorithm is provably robust to suboptimal hyperparameter tuning. We further\ndemonstrate such benefits through online conformal prediction, a downstream online learning task with\nset-membership decisions.\n1 Introduction\nA typical workflow of sequential decision making consists of two stages: offline design and online fine-tuning.\nThe offline stage gives us an initial policy to work with, utilizing a variety of priors such as domain knowledge,\noffline dataset and inductive bias. The online stage corrects the time-dependent imperfection of such priors, by\nrefining the policy using sequentially revealed data. Taking a convex stateless abstraction, we will study the\nproblem as Online Convex Optimization (OCO) [ Haz22 ,Ora23a ]. It is a two-person repeated game between us\n(the player) and an adversarial environment, with the following per-round interaction protocol.\n1. We make a prediction xt∈ Xusing past observations, where Xis a closed and convex subset of Rd.\n2.The environment picks a convex loss function lt:X →R, arbitrarily depending on our prediction history\nx1, . . . , x t.\n3. We observe a subgradient gt∈∂lt(xt) and suffer the loss lt(xt).\n4.The environment determines whether the game stops or not. If the game stops, let Tbe the total number\nof rounds.\nIntuitively, xtcorresponds to the parameter of a decision making policy, which might be thought as the weight\nof a machine learning model. The initial prediction x1encodes the offline priors, while the progression of the\nxtsequence captures the online fine-tuning process. Different from the standard settings of OCO, we do not\nexplicitly assume anything beyond convexity: the domain Xdoes not need to be bounded, and the loss functions\nare not necessarily Lipschitz/curved/smooth.\nOf particular importance is the performance metric, which ultimately determines the behavior of our prediction\nalgorithm. The default one is to upper-bound the static regret over the entire time horizon [Zin03],\nRegretT(l1:T, u):=TX\nt=1lt(xt)−TX\nt=1lt(u), (1)\n1arXiv:2402.02720v1  [cs.LG]  5 Feb 2024but quite clearly, its strength hinges on an implicit stationarity assumption on the environment: there has to be\na fixed prediction u∈ Xcalled a comparator that works well throughout the game,1which is often too good\nto be true in the online fine-tuning task we aim for. One could consider nonstationary performance metrics\ninstead, most notably the Zinkevich-style dynamic regret [Zin03 ] and the strongly adaptive regret [DGSS15 ].\nAlthough both of them are nontrivial problems extensively studied in recent theoretical works, associated optimal\nalgorithms require increasing computation per round. This poses a critical barrier for computation-constrained\nlong horizon applications.\nDiverging from these approaches, the present work revisits a “legacy” nonstationary performance metric\ncalled the discounted regret : with a discount factor λ∈(0,1], define\nRegretλ\nT(l1:T, u):=TX\nt=1λT−t[lt(xt)−lt(u)]. (2)\nIt is a generalization of the undiscounted Eq.(1), with λ= 1 recovering the latter. Since the weight of the past\ndecays quickly, it is possible to upper-bound such a performance metric while forgetting much of the history.\nThis controls the computational complexity, and meanwhile, the algorithm can stay sufficiently agile.\nTo further justify the importance of this approach, let us take a closer look at its background. Since\ndiscounting is such a natural idea, it is not surprising that the discounted regret was studied in a number of\nearly works, e.g., [ CBL06 ]. Although the community shifted away from it, the associated algorithmic insight has\nthrived on its own. For example, an extremely prevalent practice in nonstationary decision making (sometimes\ncalled continual learning [ABVR+23]) isOnline Gradient Descent (OGD) with constant2learning rate (LR).\nFrom a theoretical perspective (Section 3.1), if the loss functions l1:Tare Lipschitz and the domain Xis bounded,\nthen constant-LR OGD guarantees a minimax optimal upper bound on the discounted regret Eq.(2), rather\nthan the standard undiscounted regret Eq.(1). The latter is deeply associated to convergence , which in turn\nrequires the learning rate to shrink to zero.\nThe main goal of this paper is to design an adaptive algorithm that further improves this popular OGD baseline.\nIn standard online learning terms [ Ora23a , Chapter 4.2 and 9], an algorithm is adaptive if its regret bound\ndepends on the complexity of the actual problem instance, rather than its conservative overestimate. Precisely,\nwe study the discounted regret Eq.(2): instead of upper-bounding its worst case value supl1:T,uRegretλ\nT(l1:T, u)\nunder artificial assumptions, we will upper-bound Regretλ\nT(l1:T, u) itself by a function of both the observed\ngradients g1, . . . , g Tand the comparator u∈ X. Such a fine-grained approach ensures that the discounted regret\nbound of our algorithm is almost never worse than the minimax optimal bound of constant-LR OGD, and can\nimprove the latter when the environment is “easy”, e.g., when the offline priors are good. On the practical side,\nthere is another key strength: our algorithm gradually forgets the past online data but not the offline priors,\nwhereas constant-LR OGD forgets everything altogether.\n1.1 Contribution\nThis paper presents two types of new results, ( i) discounted online learning theory, and ( ii) applications in\nOnline Conformal Prediction (OCP).\n•First, we consider the OCO problem with discounted regret Eq.(2). Defining the effective time horizon\nH=PT\nt=1λ2(T−t)and the effective gradient variance V=PT\nt=1λ2(T−t)∥gt∥2, we develop a new algorithm\nthat virtually guarantees (see Theorem 4 for the details)\nRegretλ\nT(l1:T, u)≤˜O\u0010\n∥u−x1∥√\nV\u0011\n,\nmatching an instance-dependent Ω( ∥u−x1∥√\nV) lower bound. In comparison, constant-LR OGD achieves\nanO(DG√\nH) upper bound assuming D-bounded domain and G-Lipschitz losses – our result improves\nthat while removing both assumptions.\nThe key to our result is a simple rescaling trick , which converts scale-free undiscounted regret bounds to\ntheir discounted analogues. In this way, advances in stationary online learning can be exploited quite easily\n1Otherwise, even if the static regret is small, the algorithm’s total lossPT\nt=1lt(xt) is not necessarily small.\n2Non-annealing and horizon-independent.\n2in the general nonstationary setting, which provides a compelling alternative to the standard aggregation\napproach there (e.g., the meta-expert framework that minimizes the dynamic regret and the strongly\nadaptive regret, surveyed in Appendix A).\n•Second, we move to OCP, a downstream online learning task with confidence set predictions. To combat\nthe distribution shifts commonly found in practice, recent works [ GC21 ,GC22 ,BWXB23 ] developed the\napplication of nonstationary OCO algorithms (such as constant-LR OGD) to this setting. The twist is\nthat besides appropriate regret bounds, one has to establish coverage guarantees as well, i.e., bounding the\nmagnitude of the cumulative gradient sum over a sliding time window.\nFor this problem, we demonstrate various quantitative strengths of our discounted OCO approach. In\nparticular, the gradient adaptivity in OCO leads to the adaptivity to the targeted miscoverage rate in OCP,\nand the comparator adaptivity in OCO allows removing the “oracle tuning” assumption from existing\nworks. In terms of the techniques, since our algorithm is built on the Follow the Regularized Leader (FTRL)\nframework rather than OGD, the associated coverage guarantee follows naturally from the stability of its\niterates. Such an FTRL-based analytical strategy could be of independent interest.\nFinally, we demonstrate the practicality of our algorithm through OCP experiments.\n1.2 Notation\nThroughout this paper, ∥·∥denotes the Euclidean norm. Π X(x) is the Euclidean projection of xonto a closed\nconvex set X. The diameter of a set Xis supx,y∈X∥x−y∥.\nFor two integers a≤b, [a:b] is the set of all integers csuch that a≤c≤b. The brackets are removed when\non the subscript, denoting a tuple with indices in [ a:b]. Ifa > b , then the productQb\ni=aλi:= 1. 0 represents a\nzero vector whose dimension depends on the context. log means natural logarithm.\nWe define the imaginary error function aserfi(x):=Rx\n0exp(u2)du; this is scaled by√π/2 from the conventional\ndefinition, thus can also be queried from standard software packages like SciPy andJAX .\n2 Related work\nDiscounted regret Motivated by the intuition that “the recent history is more important than the distant past”,\nthe discounted regret has been studied by a rich line of works [ CBL06 ,FH08 ,CZ10 ,KP11 ,CBGLS12 ,BS19 ].\nMost of them were before the burst of adaptive online learning, although the under-appreciated [ KP11 ] presented\nideas that foreshadowed some of the later developments. Recently, the study of dynamic regret and strongly\nadaptive regret has taken over the field of nonstationary online learning, which we summarize in Appendix A.\nThese are indeed very valuable topics, but we believe that the discounted regret could still be a useful alternative,\nespecially due to the associated simplicity and computational efficiency. Through a modern adaptive treatment,\nwe hope to renew the interest of the field in this classical metric, which then benefits downstream applications\nsuch as OCP.\nAnother remark is that discounting on a data stream is tightly connected to filtering and sliding windows in\nsignal processing [ Opp99 ] and theoretical computer science [ DGIM02 ]. The relation of our results to such fields\ncould be investigated in future works.\nAdaptive online learning Adaptive OCO in the stationary setting has been mainly studied from two angles.\nThe famous AdaGrad [DHS11 ,McM17 ] achieves the adaptivity to the loss functions l1:T, while parameter-free\nonline learning [ SM12 ,MO14 ,OP16 ,FKMS17 ,ZCP22 ] concerns the adaptivity to the comparator u. A number\nof recent works achieve them both [CO18, Cut19, MK20, CLW21, JC22, ZYCP23], which our work builds on.\nComparison with DL optimizers As we will show, discounted OCO algorithms employ Exponential Moving\nAverage (EMA) under the hood. This is an important idea in deep learning (DL) optimization, as exemplified\nby the empirical success of RMSProp [TH+12] and Adam [KB14 ], and the enormous literature that tries to\nexplain it. A notable issue here is the appeared mismatch between the two intuitions: DL is usually modeled\nas a (stationary and non-convex) stochastic optimization problem that requires convergence [RM51 ,KW52 ],\nwhereas EMA is employed (at least in discounted OCO algorithms) for the “forgettive and non-convergent”\n3behavior. Reconciling this conflict is an important problem for DL optimization, which this work does not\nconsider. Instead, we focus on the well-posed problem of nonstationary online learning, where non-convergence\nis indeed necessary.\n3 Discounted adaptivity\nSetting We study the OCO problem defined at the beginning of the paper, with a slightly generalized discounted\nregret. Instead of having a fixed discount factor λ∈(0,1] throughout the game, let us consider a sequence of\ndiscount factors λ1, . . . , λ T∈(0,∞), and the t-th one is revealed at the end of the ( t+ 1)-th round3(for all t≤0,\nletλt= 1). In this setting, the weight of the past can possibly increase. Accordingly, we define the generalized\ndiscounted regret as\nRegretλ1:T\nT(l1:T, u):=TX\nt=1 T−1Y\ni=tλi!\n[lt(xt)−lt(u)], (3)\nwhich recovers Eq.(2) as its fixed- λ <1 special case. Throughout this paper, we will treat λ1:Tas part of the\nproblem description and focus on bounding Eq.(3), but in practice, a meta-learning approach might be applied\non top to select λ1:Tonline.\n3.1 Preliminary\nIntuitively, the main difference between the discounted regret Eq.(3) and the standard undiscounted regret Eq.(1)\nis the effective time horizon. Instead of the maximum length T, in the t-th round we care about an exponentially\nweighted look-back window of length\nHt:=tX\ni=1\nt−1Y\nj=iλ2\nj\n, (4)\nwhich is roughly min[(1−λ2)−1, T] in the fixed- λspecial case. For later use, we also define the discounted\ngradient variance Vtand the discounted Lipschitz constant Gt,\nVt:=tX\ni=1\nt−1Y\nj=iλ2\nj\n∥gi∥2, G t:= max\ni∈[1:t]\nt−1Y\nj=iλj\n∥gi∥. (5)\nA classical wisdom in online learning is that for gradient descent, the learning rates could be set proportional\nto the inverse square root of the time horizon. Combining it with Htabove, it is likely a folklore that non-adaptive\nOGD can be generalized easily to the discounted setting. Let us make it concrete.\nOnline Gradient Descent Consider the OGD update rule: after observing the loss gradient gt, we pick a\nlearning rate ηt, take a gradient step and project the update back to the domain X, i.e.,\nxt+1= ΠX(xt−ηtgt).\nThroughout this section, the initialization x1is fixed at the origin. This is without loss of generality, since a\ndifferent initialization can be implemented by simply shifting the coordinates. We call an algorithm non-adaptive\nif its regret bound is on the worst case, supl1:T,uRegretλ1:T\nT(l1:T, u). This implies that ηtonly depends on the\nhistorical discount factors, and our baseline (constant-LR OGD) is the most notable example.\nTheorem 1 (Abridged Theorem 6 and 7) .If the loss functions are all G-Lipschitz, the diameter of the domain\nis at most D, and the discount factor λt=λ∈(0,1), then OGD with a constant learning rate ηt=D\nG√\n1−λ2\nguarantees for all T= Ω(1\n1−λ),\nsup\nl1:T,uRegretλ\nT(l1:T, u) =O\u0010\nDGp\nHT\u0011\n.\n3This might seem a bit strange, but it is due to our regret definition: the loss function ltis undiscounted in the t-th round.\n4Conversely, fix any variance budget V∈(0, G2HT], and any comparator usuch that u,−u∈ X. For any\nalgorithm, there exists a loss sequence such that VTdefined in Eq.(5) satisfies VT=V, and\nmaxh\nRegretλ1:T\nT(l1:T, u),Regretλ1:T\nT(l1:T,−u)i\n= Ω\u0010\n∥u∥p\nVT\u0011\n.\nPicking the domain Xas a norm ball centered at the origin, one could see that the worst case bound of\nconstant-LR OGD is optimal under the Lipschitzness and bounded-domain assumptions. Such a result forms the\ntheoretical foundation of this common practice. Meanwhile, there is an instance-dependent gap that illustrates a\nnatural direction to improve the algorithm: removing the supremum, can we directly achieve\nRegretλ1:T\nT(l1:T, u) =O\u0010\n∥u∥p\nVT\u0011\n, (6)\nwithout any extra assumption at all? If successful, such a bound will never be worse than the O(DG√HT)\nbound of constant-LR OGD (under the same assumptions), while the algorithm could be agnostic to both Dand\nG. This is the key strength of adaptive online learning: without imposing any artificial structural assumption,\nthe algorithm performs as if it knows the “correct” assumption from the beginning.\n3.2 Main result\nTo pursue Eq.(6), our main idea is the following rescaling trick.\n•In the undiscounted setting ( λt= 1), Eq.(6) has been almost achieved by a number of recent works\n[Cut19 ,MK20 ,JC22 ,ZCP23 ] modulo necessary residual factors. For any algorithm A(possibly taken\nfrom these works), let RegretT,A(g1:T, u) denote its undiscounted static regret with respect to linear losses\n⟨gt,·⟩and comparator u, c.f., Eq.(1).\n•Next, consider the discounted regret Eq.(3). We take an aforementioned algorithm Aand apply it to a\nsequence of surrogate loss gradients ˆg1:T, where\nˆgt= t−1Y\ni=1λ−1\ni!\ngt. (7)\nIfλ1, . . . , λ T<1, this amounts to “upweighting” recent losses, or equivalently, “forgetting” older ones.4The\ngenerated prediction sequence x1:Tsatisfies a discounted regret bound that scales with RegretT,A(ˆg1:T, u),\nthe undiscounted regret of the base algorithm Aon ˆg1:T.\nRegretλ1:T\nT(l1:T, u) = T−1Y\nt=1λt!\n·TX\nt=1 t−1Y\ni=1λ−1\ni!\n[lt(xt)−lt(u)]\n≤ T−1Y\nt=1λt!\nRegretT,A(ˆg1:T, u). (8)\nThis might seem trivially simple, but we will show that using the right base algorithm Aleads to a range\nof practically desirable properties, which are in our opinion not obvious beforehand.\nWithin this trick, a particular challenge is that even if all the actual loss functions ltare Lipschitz in a\ntime-uniform manner ( ∃G, s.t., max t∥gt∥ ≤G), the surrogate loss functions ⟨ˆgt,·⟩are not. Therefore, the base\nalgorithm Acannot rely on any a priori knowledge or estimate of the time-uniform Lipschitz constant, similar to\nthe classical scale-free property [ OP18 ].5To make this concrete, we first forego the adaptivity to the comparator\nuand analyze an example based on the famous AdaGrad [DHS11 ]. The result is visually appealing, as all the\nannoying residual factors required by the full generality of our problem ( simultaneous adaptivity to both l1:T\nandu) are gone. The algorithm itself will be a building block of our main results.\n4Such an insight is similar to RMSProp [TH+12] and Adam [KB14 ] in DL optimization. Here we consider OCO instead, and the\nfocus is on proving rigorous discounted regret bounds rather than the EMA-type empirical update rules.\n5A scale-free algorithm generates the same predictions x1:Tif all the gradients g1:Tare scaled by an arbitrary c >0. However,\nwe need a bit more, since an algorithm can be scale-free even if it requires an estimate of the time-uniform Lipschitz constant at the\nbeginning [MK20, JC22].\n5Gradient adaptive OGD Proposed for the undiscounted setting, AdaGrad [DHS11 ] symbolizes OGD\nwith gradient-dependent learning rates . Using it as the base algorithm Aleads to the following RMSProp -like\nprediction rule and its discounted regret bound.\nTheorem 2. If the diameter of Xis at most D, then OGD with learning rate ηt=DV−1/2\nt guarantees for all\nT∈N+and loss sequence l1:T,\nsup\nuRegretλ1:T\nT(l1:T, u)≤3\n2Dp\nVT.\nAs one would hope for, the bound strictly improves constant-LR OGD while matching the lower bound on the√VTdependence. It is tempting to seek an even better learning rate ηtthat improves the remaining Dto∥u∥,\nbut such a direction leads to a dead end (Remark B.1). To solve this problem, we will resort to the Follow the\nRegularized Leader (FTRL) framework [ Ora23a , Chapter 7] instead of OGD. The key intuition [ FHPF22 ,JC22 ]\nis that, FTRL is stronger since it memorizes the initialization , whereas OGD (without extra regularization) does\nnot.\nSimultaneous adaptivity To achieve simultaneous adaptivity to both l1:Tandu, things can get subtle. As\nmentioned earlier, there are a number of choices for the base algorithm A. We will adopt the latest version\nfrom [ ZYCP23 ], which offers an important benefit (no explicit T-dependence, Remark B.4). Without loss of\ngenerality,6let us assume the domain X=Rd. The undiscounted algorithm from [ ZYCP23 ] is surveyed in\nAppendix B.2 for easier reference.\nOverall, our discounted algorithm employs the polar-decomposition technique from [ CO18 ]: using polar\ncoordinates, predicting xt∈Rdcan be decomposed into two independent tasks, learning the good direction\nu/∥u∥and the good magnitude ∥u∥. The direction is learned by the RMSProp -like algorithm from Theorem 2,\nwhile the magnitude is learned by a discounted variant of the erfi-potential learner [ZYCP23 , Algorithm 1] that\noperates on the positive real line [0 ,∞). This magnitude learner itself will be important for our downstream\napplication, therefore we present its pseudocode as Algorithm 1.\nAlgorithm 1 1D magnitude learner on [0 ,∞).\nRequire: Hyperparameter ε >0 (default ε= 1).\n1:Initialize parameters v1= 0,s1= 0,h1= 0.\n2:fort= 1,2, . . .do\n3: Ifht= 0, define the unprojected prediction ˜xt= 0. Otherwise, with erfi(x):=Rx\n0exp(u2)du(which can be\nqueried from SciPy andJAX ),\n˜xt=ε·erfi \nst\n2p\nvt+ 2htst+ 16h2\nt!\n−εhtp\nvt+ 2htst+ 16h2\ntexp\u0014s2\nt\n4(vt+ 2htst+ 16h2\nt)\u0015\n.\n4: Predict xt= Π [0,∞)(˜xt), the projection of ˜ xtto the domain [0 ,∞).\n5: Receive the 1D loss gradient gt∈Rand the discount factor λt−1∈(0,∞).\n6: Clipgtby defining gt,clip= Π [−λt−1ht,λt−1ht](gt), and update ht+1= max ( λt−1ht,|gt|).\n7: Ifgt,clip˜xt< gt,clipxt, define a surrogate loss gradient ˜ gt,clip= 0. Otherwise, ˜ gt,clip=gt,clip.\n8: Update vt+1=λ2\nt−1vt+ ˜g2\nt,clip,st+1=λt−1st−˜gt,clip.\n9:end for\nAlgorithm 1 has the following intuition. At its center is a special instance of FTRL, which generates the\nprediction xtusing the discounted gradient variance vt, the discounted gradient sum st, and the discounted\nLipschitz constant ht. Complementing this core component, two additional ideas are applied to fix certain\ntechnical problems: ( i) the unconstrained-to-constrained reduction from [ CO18 ,Cut20 ], and ( ii) the hint-and-\nclipping technique from [ Cut19 ]. The readers are referred to Appendix B.2 for a detailed explanation. The point\nis that understanding the inner workings of the algorithm is not strictly necessary to proceed: we treat the\n6Due to [ Cut20 , Theorem 2], given any unconstrained algorithm that operates on X=Rd, we can impose any closed and convex\nconstraint without changing its regret bound.\n6result from [ ZYCP23 ] as a black box and wrap it using the rescaling trick. In summary, this yields the following\ntheorem proved in Appendix B.3.\nTheorem 3. Given any hyperparameter ε >0, Algorithm 1 guarantees for all time horizon T∈N+, loss\nsequence l1:T, comparator u∈[0,∞)and stability window length τ∈[1 :T],\nRegretλ1:T\nT(l1:T, u)≤εq\nVT+ 2GTS+ 16G2\nT+u(S+GT)+\u0012\nmax\nt∈[T−τ+1:T]xt\u0013\nGT+ T−1Y\nt=T−τλt!\u0012\nmax\nt∈[1:T−τ]xt\u0013\nGT−τ,\nwhere\nS= 8GT\u0010\n1 +p\nlog(2uε−1+ 1)\u00112\n+ 2q\nVT+ 16G2\nT\u0010\n1 +p\nlog(2uε−1+ 1)\u0011\n.\nThis result might seem a bit intimidating, so let us take a few steps to interpret it. In particular, we want to\njustify the appropriate asymptotic regime to consider ( VT≫G2\nTandu≫ε), such that our main result on Rd\n(Theorem 4) can use the big-Oh notation to improve clarity.\n•First, one would typically expect VT≫G2\nT, since when λt=λ∈(0,1) and |gt|=Gfor all t, we have\nGT=GandVT=HTG2≈(1−λ2)−1G2. As long as the discount factor λtis close enough to 1, the\ncondition VT≫G2\nTis likely to hold for general/practical gradient sequences as well.\n•Second, the hyperparameter εserves as a prior guess of the comparator u. If the guess is correct ( ε=u),\nthen by assuming VT≫G2\nTandmax t∈[1:T]xt=O(u) (roughly speaking, the predictions are stable7), the\nregret bound becomes\nRegretλ1:T\nT(l1:T, u) =O\u0010\nup\nVT\u0011\n, (9)\nexactly matching the lower bound. Realistically such an “oracle tuning” is illegal, since εis selected at the\nbeginning of the game, while reasonable comparators uare hidden before all the loss functions are revealed.\nThis is where our algorithm truly shines: as long as εis moderately small, i.e., O(u), we would have the\nO(uS) term dominating the regret bound, which only suffers a multiplicative logarithmic penalty relative\nto the impossible oracle-optimal rate Eq.(9). In comparison, it is well-known that the regret bound of\nconstant-LR OGD with learning rate ηdepends polynomially onηandη−1(c.f., the proof of Theorem 6),\nwhich means our algorithm is provably more robust to suboptimal hyperparameter tuning.\n•Third, we explain the use of τ. In nonstationary environments, the range of xtcan vary significantly over\ntime, therefore a time-uniform characterization of its stability ( max t∈[1:T]xt, Remark B.2) could be overly\nconservative. We use a stability window of arbitrary length τto divide the time horizon into two parts: the\nearlier part is forgotten rapidly (due to theQT−1\nt=T−τλtmultiplier), so only the later part really matters.\nExample 1. Suppose again that λt=λ∈(0,1). Ifλ≈1, the “forgetting” multiplier can be approximated by\nT−1Y\nt=T−τλt=λτ=λ1\n1−λ·(1−λ)τ≈e(λ−1)τ. (Lemma B.1)\nConsider λ= 0.99for example: τ= 700 ensures λτ≈e−7<10−3. That is, if the past range of xtis negligible\nafter a 10−3-attenuation, then our regret bound only depends on the “localized stability” max t∈[T−699,T]xt\nevaluated in the recent 700 rounds, which is a small fraction in (let us say) T≈million . More intuitively, it\nmeans “past mistakes do not matter”.\nGiven the 1D magnitude learner and its guarantee, the extension to Rdis now standard [ CO18 ]. We defer\nthe pseudocode to Appendix B.3.\n7Although the comparator uis arbitrary, the most important one is arg minPT\nt=1\u0010QT−1\ni=tλi\u0011\nlt(u), i.e., the best-in-hindsight\nstatic comparator (if it exists). Therefore, essentially, the “stability” of an algorithm in our exposition depends on both the algorithm\nitself and the loss functions.\n7Theorem 4 (Main result) .Given any hyperparameter ε >0, Algorithm 3 in Appendix B.3 guarantees for all\nT∈N+, loss gradients g1:Tand comparator u∈Rd,\nRegretλ1:T\nT(l1:T, u)≤O\u0010\n∥u∥p\nVTlog(∥u∥ε−1)∨ ∥u∥GTlog(∥u∥ε−1)\u0011\n+\u0012\nmax\nt∈[1:T]xt\u0013\nGT,\nwhere O(·)is in the regime of large VT(VT≫GT)and large ∥u∥(∥u∥ ≫ε). Furthermore, for u= 0,\nRegretλ1:T\nT(l1:T,0)≤O\u0010\nεp\nVT\u0011\n+\u0012\nmax\nt∈[1:T]xt\u0013\nGT.\nSimilar to Theorem 3, the iterate stability term can be split into two parts using an arbitrary τ. The\nimportant message is that if the iterates are indeed stable ( max t∈[1:T]xt=O(∥u∥)) and we suppress all the\nlogarithmic factors with ˜O(·), then\nRegretλ1:T\nT(l1:T, u)≤˜O\u0010\n∥u∥p\nVT\u0011\n.\nIt matches the lower bound and improves all the aforementioned algorithms. Granted, there are several nuances\nin this statement, but they are in general necessary even in the undiscounted special case [Ora23a, Chapter 9].\nFinally, we summarize a range of practical strengths. As shown earlier, the proposed algorithm is robust to\nhyperparameter tuning. Observations and mistakes from the distant past (which are possibly misleading for the\nfuture) are appropriately forgotten, such that the algorithm runs “consistently” over its lifetime. Compared\nto the typical aggregation framework in nonstationary online learning [ DGSS15 ,ZLZ18 ], our algorithm runs\nfaster and never explicitly restarts (although we require given discount factors to “softly” restart). In addition,\ncompared to constant-LR OGD that also tries to minimize the discounted regret, our algorithm makes a better\nuse of the offline knowledge – this is so important that we have to discuss it further.\nOn the initialization So far we have fixed the initialization x1to 0. This makes the bounds cleaner but hides\na crucial consideration. Suppose we have a ML model that “almost” explains the environment, except just a small\namount of noise. Formulated as OCO, it could mean that the optimal prediction sequence x∗\nt∈arg minxlt(x)\nsatisfies\nx∗\nt= ¯x+δt,\nwhere the time-invariant ¯xcan be obtained through offline pre-training , and the noise δtis small and only\nrevealed online. Naturally, our algorithm could initialize at x1=¯x, such that the discounted regret bound\nbecomes ˜O\u0000\n∥u−¯x∥√VT\u0001\n. If the noise δt≈δover a sufficiently long look-back window at T, then with respect\ntou=¯x+δ(which approximates the optimal prediction sequence {x∗\nt}), our regret is simply ˜O\u0000\n∥δ∥√VT\u0001\n. In\nother words, our online prediction performance nicely scales with the accuracy of the pre-trained model over\nsliding time windows.\nIn contrast, even if initialized properly, constant-LR OGD does not exhibit solid theoretical improvement.\nThis is fundamentally related to FTRL versus OGD: through the incremental steps, OGD quickly loses track of\neverything in the past, while our algorithm has a “regularization module” that always remembers the initialization\n(even though past gradients are forgotten).\n4 Online conformal prediction\nNext, we consider an application that complements the theory. Conformal prediction [ VGS05 ] is a framework\nthat quantifies the uncertainty of black box ML models. We study its online version [ GC21 ,BGJ+22,GC22 ,\nZFG+22,BWXB23 ], where no statistical assumptions (e.g., exchangeability) are imposed at all. Our setting\nfollows the nicely written [ BWXB23 , Section 2], and the readers are referred to [ AB23 ,Rot22 ] for additional\nbackground.\n84.1 Preliminary\nSimilar to OCO, OCP is again a two-person repeated game. Let a constant α∈(0,1) be the targeted\nmiscoverage rate fixed before the game starts. At the beginning of the t-th round, we receive a set-valued function\nCt:R≥0→2Ymapping any radius parameter r∈[0,∞) to a subset Ct(r) of the label space Y. The Ctfunction\nisnested : for any r′> r, we have Ct(r)⊂ Ct(r′)⊂ Y. Then,\n1. We pick a radius parameter rt∈[0,∞) and output the prediction set Ct(rt).\n2.The environment reveals the optimal radius r∗\nt∈[0,∞). Intuitively, our prediction set Ct(rt) is “large\nenough” only if rt> r∗\nt.\n3. Our performance is evaluated by the pinball loss l(α,∗)\nt(rt), where for all r∈[0,∞),\nl(α,∗)\nt(r):=(\nα(r−r∗\nt), r > r∗\nt,\n(α−1)(r−r∗\nt),else.(10)\nHere is a simple 1D forecasting example. Suppose there is a base ML model that in each round makes a\nprediction ˆxt∈Rof the true time series x∗\nt∈R. On top of that, we “wrap” such a point prediction xtby a\nconfidence set prediction Ct(rt) = ( ˆxt−rt,ˆxt+rt). Ideally we want Ct(rt) to cover the true series x∗\nt, and this\ncan be checked after x∗\ntis revealed. That is, by defining the optimal radius r∗\nt=|x∗\nt−ˆxt|, we claim success if\nrt> r∗\nt.\nQuite naturally, since r∗\ntis arbitrary, it is impossible to ensure coverage unless rtis meaninglessly large. A\nreasonable objective is then asking our empirical (marginal) coverage rate to be approximately 1 −α, which\namounts to showing\f\f\f\f\f1\nTTX\nt=11[rt≤r∗\nt]−α\f\f\f\f\f=o(1), (11)\nand this is beautifully equivalent to characterizing the cumulative subgradients of the pinball loss,8\f\f\fPT\nt=1∂l(α,∗)\nt(rt)\f\f\f=\no(T). One catch is that there are trivial predictors satisfying Eq.(11) [ BGJ+22], so one needs an extra measure\nto rule them out. Such a “secondary objective” can be the regret bound on the pinball loss,9\nTX\nt=1l(α,∗)\nt(rt)−TX\nt=1l(α,∗)\nt(u) =o(T), (12)\nwhich motivates using OCO algorithms such as OGD to select rt.\nHow does nonstationarity enter the picture? Since the proposal of OCP in [ GC21 ], the main emphasis is\non problems with distribution shifts , which traditional conformal prediction methods based on exchangeability\nand data splitting have trouble dealing with. For example, the popular ACI algorithm [ GC21 ] essentially uses\nconstant-LR OGD – as we have shown, this is inconsistent with minimizing the standard regret Eq.(12), and the\n“right” OCO performance metric that justifies it could be a nonstationary one (e.g., the discounted regret). In a\nsimilar spirit, [ GC22 ,BWXB23 ] apply dynamic andstrongly adaptive OCO algorithms, effectively analyzing the\nsubinterval variants of Eq.(11) and Eq.(12).\nAnother key ingredient of OCP is assuming the optimal radius max tr∗\nt≤Dfor some D > 0, which is often\nreasonable in practice and important for the coverage guarantee. As opposed to prior works [ BGJ+22,BWXB23 ]\nthat require knowing Dat the beginning to initialize properly, we seek an adaptive algorithm agnostic to this\noracle knowledge.\nOur goal Overall, we aim to show that without knowing D, applying Algorithm 1 leads to discounted adaptive\nversions of the marginal coverage bound Eq.(11) and the regret bound Eq.(12). This offers advantages over\ntheACI-like approach that tackles similar sliding window objectives using constant-LR OGD. Along the way,\nwe demonstrate how the structure of OCP allows controlling the iterate stability of Algorithm 1 (or generally,\nFTRL algorithms), which then makes the proof of coverage fairly easy.\n8At the singular point r∗\nt, define the subgradient ∂l(α,∗)\nt(r∗\nt) =α−1.\n9Another choice [BGJ+22] is bounding the conditional coverage using calibration .\n94.2 Main result\nBeyond pinball loss From now on, define ACPas the OCP algorithm that uses Algorithm 1 to select rt(see\nAppendix C.1 for pseudocode). We make a major generalization: instead of using subgradients of the pinball loss\nEq.(10) to update Algorithm 1, we use subgradients g∗\nt∈∂f∗\nt(rt), where f∗\nt(r) is any convex function minimized\natr∗\nt, and right at rt=r∗\ntwe have g∗\nt≤0 without loss of generality. This includes the pinball loss l(α,∗)\nt(r) as\na special case. Notably, f∗\nt(r) does not need to be globally Lipschitz,10which unleashes the full power of our\nbase algorithm. Put together, ACPtakes a confidence hyperparameter εand a sequence of discount factors λ1:T\ndetermined by our objectives. We assume max tr∗\nt≤D, but Dis unknown by ACP.\nStrength of FTRL The key to our result is Lemma 4.1 connecting the prediction rt+1to the discounted\ncoverage metric\nS∗\nt:=−tX\ni=1\nt−1Y\nj=iλj\ng∗\ni. (13)\nIfλt= 1 for all tand we use the pinball loss to define g∗\n1:T, then |S∗\nT|/Trecovers Eq.(11). The point is that if\nλt=λ <1, then just like the intuition throughout this paper, Eq.(13) is essentially a sliding window coverage\nmetric. Associated algorithm would gradually forget the past, which intuitively counters the distribution shifts.\nLemma 4.1 (Abridged Lemma C.2) .ACPguarantees for all t∈N+,\n|S∗\nt| ≤O\u0010p\nV∗\ntlog(rt+1ε−1)∨G∗\ntlog(rt+1ε−1)\u0011\n,\nwhere V∗\ntandG∗\ntare defined on the OCP loss gradients using Eq.(5), and O(·)is in the regime of rt+1≫ε.\nThe proof of this lemma is a bit involved, but the high level idea is very simple: if we use a FTRL algorithm\n(rather than OGD) as the OCO subroutine for OCP, then the radius prediction is roughly rt+1≈ψ(S∗\nt/p\nV∗\nt)\nfor some prediction function ψ(if the algorithm is not adaptive enough then the denominator is√Htinstead),\nwhich means S∗\nt≈p\nV∗\ntψ−1(rt+1) =O(√Ht). Dividing both sides by Htyields the desirable coverage guarantee.\nIn comparison, the parallel analysis using OGD can be much more complicated (e.g., the grouping argument in\n[BWXB23 ]) due to the absence of S∗\ntin the explicit update rule. Such an analytical strength of FTRL seems to\nbe overlooked in the OCP literature.\nWe also note that although the above lemma depends only logarithmically on rt+1, without any problem\nstructure the latter could still be large (exponential in t), which invalidates this approach. The remaining step is\nshowing that if the underlying optimal radius r∗\ntis time-uniformly bounded by D, then even without knowing D,\nwe could replace rt+1in the above lemma by O(D). Intuitively it should make sense; materializing it carefully\ngives us the final result.\nTheorem 5. Without knowing D,ACPguarantees that for all T∈N+, we have the discounted coverage bound\n|S∗\nT| ≤O\u0012q\nV∗\nTlog(Dε−1)∨G∗\nTlog(Dε−1)\u0013\n,\nand the discounted regret bound from Theorem 3.\nTo interpret this result, let us focus on the coverage bound, since the discount regret bound has been discussed\nextensively in Section 3. First, consider the pinball loss and the undiscounted setting λt= 1, where our bound can\nbe directly compared to prior works. For OGD, [ GC21 , Proposition 1] shows that the learning rate η=D/√\nT(as\nsuggested by regret minimization) achieves |S∗\nT|=O(√\nT), while [ BWXB23 , Theorem 2] shows that ηt=D/√Vt\n(i.e.,AdaGrad ) achieves |S∗\nT|=O(α−2T3/4logT). Although the latter is empirically strong, the theory is a bit\nunsatisfying as one would expect the gradient adaptive approach to be a “pure upgrade” (plus, the bound blows\nup as α→0). Our Theorem 5 is in some sense the “right fix”, as essentially, |S∗\nT|=˜O(pV∗\nT)≤˜O(√\nT). This is\nprimarily due to the strength of FTRL over OGD, which we hope to demonstrate. Besides, our algorithm also\nimproves the regret bound of these baselines while being agnostic to D.\n10For example, one might use a “skewed quadratic function” f∗\nt(r) =1\n2(α−1[r≤r∗\nt])(r−r∗\nt)2to penalize the under/over-coverage\nmargin.\n10All these algorithms can be extended to the sliding window setting (e.g., the algorithm from [ GC21 ] becomes\nconstant-LR OGD, which is the version actually applied in practice), and the above comparison still roughly holds.\nThere is just one catch: OGD algorithms make coverage guarantees on “exact sliding windows” [ T−H+ 1 : T]\nof length H, whereas our algorithm bounds coverage on a slightly different “exponential window” of effective\nlength HT, Eq.(13). Nonetheless, all these algorithms also guarantee certain discounted regret bounds, which\nare still comparable like in Section 3.\nAdaptivity in OCP Next, we discuss the benefits of adaptivity more concretely in OCP. Suppose again that\nλt= 1 and we use the pinball loss. Then, instead of the non-adaptive bound |S∗\nT|=O(√\nT), we have\n|S∗\nT|=˜O\u0010p\nV∗\nT\u0011\n=˜O\nvuutα2TX\nt=11[rt> r∗\nt] + (α−1)2TX\nt=11[rt≤r∗\nt]\n.\nSince asymptotically the miscoverage rate is α, we havePT\nt=11[rt≤r∗\nt]≈αT, which means the bound is roughly\n˜O\u0010p\nα(1−α)T\u0011\n. That is, the gradient adaptivity in OCO translates to the target rate adaptivity in OCP.\nIn terms of the sample complexity, the OGD baseline requires at least ε−2rounds to guarantee an empirical\nmarginal coverage rate within [ α−ε, α+ε], while our algorithm requires at least αε2rounds. Very concretely, in\na typical setting of α= 0.1 (i.e., we want 90% confidence sets), our algorithm only requires1\n10as many samples\ncompared to the OGD baseline. Besides the coverage bound, such an α-dependent saving applies to the regret\nbound as well.\nInitialization in OCP Finally, we comment on the role of initialization in OCP, mirroring the discussion\nat the end of Section 3. There, the idea is that FTRL always remembers the initialization, therefore we might\ninject prior knowledge by choosing it properly. A possibly interesting observation is that initializing at r1= 0 is\na sensible choice in OCP. It always “drags” the radius prediction towards zero, such that intuitively, “within all\nthe candidate prediction sets satisfying the coverage and regret bounds, we pick the smallest one”.\n5 Experiment\nWe now demonstrate the practicality of our algorithm in OCP experiments. Our setup builds on the great work\nof [BWXB23 ]. Except our own algorithms, we adopt the implementation of the baselines and the evaluation\nprocedure from there. Details are deferred to Appendix D.\nSetup We consider image classification in a sequential setting, where each image is subject to a corruption\nof time-varying strength. Given a base machine learning model that generates the Ctfunction (by scoring all\nthe possible labels), the goal of OCP is to select the radius parameter rt, which yields a set of predicted labels.\nIdeally, we want such a set to contain the true label, while being as small as possible. The targeted miscoverage\nrateαis selected as 0 .1.\nWe test three versions of our algorithm: MagL-D is our Algorithm 1 with ε= 1 and λt= 0.999;MagL is its\nundiscounted version ( λt= 1); and MagDis is a much simplified variant of Algorithm 1 that basically sets ht= 0.\nWe are aware that a possible complaint towards our approach is that the algorithm is too complicated, but as\nwe will show, this simplified version presented as Algorithm 5 also demonstrates strong empirical performance\ndespite losing the performance guarantee.\nThe baselines we test are summarized in Table 1, following the implementation of [ BWXB23 ]. In particular,\nSf-Ogd [BWXB23 ] is equivalent to AdaGrad with oracle tuning : by definition it requires knowing the maximum\npossible radius Dto set the learning rate, and in practice, Dis estimated from an offline dataset (which is a\nform of oracle tuning from the theoretical perspective). To test the effect of such tuning, we create another\nbaseline called “Simple OGD”, which is simply Sf-Ogd with its estimate of Dset to 1. We emphasize that\ndespite its name, Simple OGD is still a gradient adaptive algorithm.\nFour metrics are evaluated, and we define them formally in Appendix D. First, the average coverage measures\nthe empirical coverage rate over the entire time horizon. Similarly, the average width refers to the average size\nof the prediction set, also over the entire time horizon. Ideally we want the average coverage to be close to\n11Method Avg. Coverage Avg. Width LCE 100 Runtime\nSimple OGD 0.899 125.0 0.11 1.00±0.03\nSf-Ogd 0.899 124.5 0.09 1.05±0.03\nSaocp 0.883 119.2 0.10 11.07±0.19\nSplitConformal 0.843 129.5 0.47 1.57±0.04\nNExConformal 0.892 123.0 0.14 2.22±0.02\nFaci 0.889 123.4 0.12 2.98±0.07\nFaci-S 0.892 124.7 0.11 2.17±0.07\nAlg. 1: MagL-D 0.884 118.5 0.08 1.06±0.03\nAlg. 2: MagL 0.894 122.1 0.09 1.05±0.02\nAlg. 5: MagDis 0.888 122.1 0.07 1.02±0.04\nTable 1: Performance of different methods. Baselines are colored in black, while our algorithms are colored in\nblue. The best performer in each metric is bolded and the second-best is underlined . The runtime, plus/minus\nits standard deviation, is normalized by the mean of Simple OGD’s runtime; averages are take over ten trials per\nalgorithm.\n1−α= 0.9, and if that is satisfied, lower average width is better. Different from these two, the local coverage\nerror (LCE 100) measures the deviation of the local empirical coverage rate (over the “worst” sliding time window\nof length 100) from the target rate 0 .9 – this is arguably the most important metric (due to the distribution\nshifts), and lower is better. Finally, we also test the runtime of all the algorithms, normalized by that of Simple\nOGD.\nResult The results are summarized in Table 1. Among all the algorithms tested, our algorithms achieve\nthe lowest local coverage error ( LCE 100). In terms of metrics on the entire time horizon, our algorithms also\ndemonstrate competitive performance compared to the baselines: although the average coverage is worse than\nthat of Simple OGD and Sf-Ogd ,11the average width is lower.12In addition, our algorithms run almost as fast\nas Simple OGD and Sf-Ogd , and importantly, they are significantly faster than Saocp [BWXB23 ] which is an\naggregation algorithm that minimizes the strongly adaptive regret.\nBy comparing the three versions of our algorithm, as one would expect, the discounted version ( MagL-D )\nimproves the undiscounted version ( MagL ). Remarkably, the much simplified MagDis achieves competitive\nperformance despite the lack of performance guarantees.\n6 Conclusion\nThis work revisits the classical notion of discounted regret using recently developed techniques in adaptive online\nlearning. In particular, we propose a discounted “simultaneously” adaptive algorithm (with respect to both\nthe loss sequence and the comparator), with demonstrated practical benefits in online conformal prediction.\nAlong the way, we propose ( i) a simple rescaling trick to minimize the discounted regret; and ( ii) a FTRL-based\nanalytical strategy to guarantee coverage in online conformal prediction. Such techniques could be of independent\ninterest.\nMoving forward, we hope the present work can help reviving the community’s interest in the discounted\nregret. For nonstationary online learning, it is a simpler metric to study than the alternatives (the dynamic\nregret [ Zin03 ] and the strongly adaptive regret [ DGSS15 ]), while offering certain computational and statistical\nadvantages (Appendix A). More broadly, there are recent works [ CMO23 ,AZKD24 ] suggesting its connection to\ndeep learning optimization (e.g., Adam ). 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Improving adaptive online\nlearning using refined discretization. arXiv preprint arXiv:2309.16044 , 2023.\n[ZYZ18] Lijun Zhang, Tianbao Yang, and Zhi-Hua Zhou. Dynamic regret of strongly adaptive methods. In\nInternational Conference on Machine Learning , pages 5882–5891. PMLR, 2018.\n[ZZZZ20] Peng Zhao, Yu-Jie Zhang, Lijun Zhang, and Zhi-Hua Zhou. Dynamic regret of convex and smooth\nfunctions. Advances in Neural Information Processing Systems , 33:12510–12520, 2020.\n16Appendix\nOrganization Appendix A surveys the related topic of dynamic and strongly adaptive regret. Appendix B\nand C contain the details of discounted online learning and the OCP application, respectively. Details of the\nexperiments are presented in Appendix D.\nA Dynamic and strongly adaptive regret\nThis section surveys the recent predominant approach in nonstationary online learning, namely the aggregation-\ntype algorithms that minimize either the dynamic regret or the strongly adaptive regret. We discuss the main\nidea behind these algorithms, as well as the possible practical concerns.\nDefinition For clarity, let us assume the diameter of the domain Xis at most D.\n•Generalizing the undiscounted static regret Eq.(1), the Zinkevich-style dynamic regret [Zin03 ,ZLZ18 ,\nZYZ18 ,ZZZZ20 ,BW21 ,BW22 ,JC22 ,LH23 ,ZCP23 ] allows the comparator u∈ X to be time-varying.\nFormally, the goal is to upper-bound\nRegretT(l1:T, u1:T):=TX\nt=1lt(xt)−TX\nt=1lt(ut),\nfor all comparator sequences u1, . . . , u T. If the losses are G-Lipschitz, then a typical dynamic regret bound\nhas the form\nRegretT(l1:T, u1:T)≤˜O\u0010\nGp\nDPuT\u0011\n,\nwhere Pu:=PT−1\nt=1∥ut−ut+1∥is called the path length of the u1:Tsequence.\n•Alternatively, the (strongly) adaptive regret [HS09 ,DGSS15 ,AKCV16 ,JOWW17 ,Cut20 ,LH23 ] generalizes\nthe undiscounted static regret Eq.(1) to subintervals of the time horizon. Formally, for a time interval\nI ⊂[1 :T], we define\nRegretI(lI, u):=X\nt∈Ilt(xt)−X\nt∈Ilt(u),\nand with G-Lipschitz losses, the typical goal is to show that simultaneously on all time intervals I,\nRegretI(lI, u)≤˜O\u0010\nDGp\n|I|\u0011\n,\nregardless of the specific loss sequence and the comparator. Note that the name “strongly adaptive” is\ndue to historical reasons; in general it is narrower that the recent concept of adaptive online learning\nin the literature. The latter (adopted in this work) refers to achieving instance-dependent performance\nguarantees.\nHidden in the above definitions is an implicit dependence on the nonstationarity of the environment. Taking\nthe dynamic regret for example,13the standard follow-up reasoning is bounding the total loss of the algorithm,PT\nt=1lt(xt), through the oracle inequality ,\nTX\nt=1lt(xt)≤inf\nu1:T\"TX\nt=1lt(ut) +˜O\u0010\nGp\nDPuT\u0011#\n.\nAlthough the dynamic regret bound holds for all comparator sequences u1:T, the only important ones are\nthose with low total loss,PT\nt=1lt(ut). In this way, the path length Puof the “important comparator sequence”\nessentially measures the variation of the loss sequence l1:T.\n13For the strongly adaptive regret, a similar argument can be made using the length |I|of “important time intervals”, where the\nenvironment is almost stationary and a time-invariant comparator u∈ X induces low loss.\n17Algorithm Due to the intricate connections between these two performance metrics [ZYZ18, Cut20, BW21],\nall known algorithms that minimize either of them share the same two-level compositional design philosophy:\n1. The low level maintains a class of “base online learning algorithms” in parallel, each targeting a different\n“nonstationarity level” of the environment that is possibly correct.\n2.The high level aggregates these base algorithms, in order to adapt to the true nonstationarity level unknown\nbeforehand.\nWithin this procedure, a particularly important consideration is the range of targeted nonstationarity levels.\nThis determines the amount of base algorithms maintained at the same time, thus affects the overall statistical\nand computational performance.\nPractical concern Following the above reasoning, we argue that minimizing either the dynamic regret or the\nstrongly adaptive regret requires targeting a wide range of nonstationarity levels, which is sometimes impractical.\nFor example, in the dynamic regret, the path length Puof the “important comparator sequence” can take any\nvalue from Θ(1) to Θ( T), whose ratio grows with T. From the computational perspective, the consequence is that\nthe standard model selection approach on an exponentially spaced grid of Purequires maintaining O(logt) base\nalgorithms in the t-th round,14making the entire algorithm slower over time. Furthermore, excessively conflicting\nbeliefs Pu= Θ(1) and Pu= Θ( T) are maintained simultaneously, with the former advocating convergence and\nthe latter on the exact contrary. As we demonstrate shortly, this may cause “statistical failures”: even in periodic\nenvironments where Pu= Θ( T) is the only “correct belief”, the algorithm may still be deceived by the possibility\nofPu= Θ(1) and converge to trivial time-invariant decisions.\nPossible example of failure Consider a 1D OCO problem inspired by time series forecasting, where the loss\nfunctions l1:Tare the quadratic loss with respect to a ground truth z1:Tsequence. That is,\nlt(x) = (x−zt)2.\nLet the ground truth be a unit square wave with period 2 H:zt= (−1)⌊(t−1)/H⌋. Furthermore, we set the domain\nX= [−2,2], such that the Lipschitz constant G= 6.\nConsider any algorithm with an optimal Pu-dependent dynamic regret bound, or an optimal strongly adaptive\nregret bound for 1D OCO (with bounded domain and Lipschitz losses). Then, as a consequence, the algorithm\nalso guarantees a sublinear static regret bound ,\nTX\nt=1lt(xt)≤min\nu∈XTX\nt=1lt(u) +o(T).\nSuppose the time horizon Tis a multiple of 2 H. Then,\nTX\nt=1lt(u) =T\n2(u−1)2+T\n2(u+ 1)2=T(u2+ 1),\nwhich means that the optimal fixed comparator is u= 0. It suggests that the xtsequence generated by the\nalgorithm converges to the trivial time-invariant prediction 0 in the long run, even though the ground truth\nztdoes not converge. (Technically, the algorithm can do better by “learning” the periodicity of the ground\ntruth and “changing the direction” accordingly, e.g., always predicting a nonzero xtof the same sign aszt.\nHowever, since the algorithm is designed for adversarial environments which are not necessarily periodic, such a\nstrategic behavior is unlikely to hold.) To be more specific, we expect the xtsequence to track the ground truth\nztsequence, but less and less responsively as tincreases (which could be called “degradation”), and eventually\nxtconverges to 0.\nWe performed preliminary numerical experiments to validate this hypothesis. The expected degradation\nis clearly observed on the algorithm from [ JC22 ], but for the structurally simpler meta-expert algorithm from\n[ZLZ18 ], the degradation of the xtsequence is not significant given the scale of our preliminary experiments. We\ndefer a thorough investigation of this problem to future works.\n14A notable exception is [ LH23 ], which shows that log log tcomputation per round is achievable at the expense of slightly larger\nregret bounds.\n18Takeaway Back to our main argument, we aim to show that for a nonstationary online learning algorithm, it\ncould be impractical to simultaneously maintain a wide range of beliefs on the correct nonstationarity level (of\nthe environment). In some sense, this is on the contrary of a common perception of the field (i.e., the adaptivity\nto the true nonstationarity level is always better). Our study of discounted algorithms is partially motivated\nby this idea: such discounted algorithms only target one nonstationarity level, which is specified by the given\ndiscount factor.\nB Detail of Section 3\nAppendix B.1 contains omitted proofs from Section 3, excluding the analysis of our main algorithm. Appendix B.2\nintroduces the undiscounted algorithm from [ ZYCP23 ] and its guarantees; these are applied to our reduction\nfrom Section 3.2. Appendix B.3 proves our main result, i.e., discounted regret bounds that adapt simultaneously\ntol1:Tandu.\nB.1 Preliminary proofs\nWe first prove an auxiliary lemma.\nLemma B.1. For all λ∈(0,1),λ1\n1−λ≤e−1. Moreover, limλ→1−λ1\n1−λ=e−1.\nProof of Lemma B.1. For the first part of the lemma, taking log on both sides, it suffices to show\n1\n1−λlogλ≤ −1.\nThis holds due to log λ≤λ−1. The second part is due to lim λ→1−(1−λ)−1logλ=−1.\nThe following theorem characterizes non-adaptive OGD.\nTheorem 6. If the loss functions are all G-Lipschitz and the diameter of the domain diam (X)is at most D,\nthen OGD with learning rate ηt=DG−1H−1/2\nt guarantees for all T∈N+,\nsup\nl1:T,uRegretλ1:T\nT(l1:T, u)≤3\n2DGp\nHT.\nFurthermore, if the discount factor λt=λfor some λ∈(0,1), then OGD with a time-invariant learning rate\nηt=DG−1√\n1−λ2guarantees for all T≥1\n2(1−λ)−1,\nsup\nl1:T,uRegretλ\nT(l1:T, u)≤3\n2DG√\n1−λ2≤3\n2√\n1−e−1DGp\nHT.\nProof of Theorem 6. We start with the first part of the theorem. The standard analysis of gradient descent\ncenters around the descent lemma [Ora23a, Lemma 2.12],\n⟨gt, xt−u⟩ ≤1\n2ηt∥xt−u∥2−1\n2ηt∥xt+1−u∥2+ηt\n2∥gt∥2.\nApplying convexity and taking a telescopic sum,\nRegretλ1:T\nT(l1:T, u)\n=TX\nt=1 T−1Y\ni=tλi!\n[lt(xt)−lt(u)]\n≤TX\nt=1 T−1Y\ni=tλi!\n⟨gt, xt−u⟩\n19≤TX\nt=1 T−1Y\ni=tλi!\u00121\n2ηt∥xt−u∥2−1\n2ηt∥xt+1−u∥2+ηt\n2∥gt∥2\u0013\n=1\n2η1∥x1−u∥2 T−1Y\ni=1λi!\n+1\n2TX\nt=2\u00121\nηt−λt−1\nηt−1\u0013 T−1Y\ni=tλi!\n∥xt−u∥2+TX\nt=1ηt\n2 T−1Y\ni=tλi!\n∥gt∥2.(14)\nNow, consider the second sum on the RHS. Notice that Ht=λ2\nt−1Ht−1+ 1,\n1\nηt−λt−1\nηt−1=G\nD\u0010p\nHt−λt−1p\nHt−1\u0011\n=G\nD\u0012q\nλ2\nt−1Ht−1+ 1−q\nλ2\nt−1Ht−1\u0013\n≥0.\nTherefore, we can apply the diameter condition ∥xt−u∥ ≤D(and the Lipschitzness ∥gt∥ ≤Gas well), and use\na telescopic sum again to obtain\nRegretλ1:T\nT(l1:T, u)≤D2\n2η1 T−1Y\ni=1λi!\n+D2\n2TX\nt=2\u00121\nηt−λt−1\nηt−1\u0013 T−1Y\ni=tλi!\n+G2\n2TX\nt=1ηt T−1Y\ni=tλi!\n=D2\n2ηT+G2\n2TX\nt=1ηt T−1Y\ni=tλi!\n=1\n2DGp\nHT+1\n2DGTX\nt=1H−1/2\nt T−1Y\ni=tλi!\n.\nTo proceed, define\nSum T:=TX\nt=1H−1/2\nt T−1Y\ni=tλi!\n.\nWe now show that Sum T≤2√HTby induction.\n•When T= 1, we have H1= 1 and Sum 1=H−1/2\n1 = 1, therefore Sum 1≤2√H1.\n•When T >1, starting from the induction hypothesis Sum T−1≤2p\nHT−1,\nSum T=H−1/2\nT +λT−1Sum T−1≤H−1/2\nT + 2λT−1p\nHT−1.\nApplying HT=λ2\nT−1HT−1+ 1,\nSum T≤(λ2\nT−1HT−1+ 1)−1/2+ 2q\nλ2\nT−1HT−1≤2q\nλ2\nT−1HT−1+ 1 = 2p\nHT,\nwhere the inequality is due to the concavity of square root.\nCombining everything above leads to the first part of the theorem.\nAs for the second part (time-invariant λ <1), we follow the same procedure until Eq.(14). The learning rate\nηtis independent of t; denote it as η=DG−1√\n1−λ2. Then,\nRegretλ\nT(l1:T, u)≤λT−1\n2η∥x1−u∥2+1−λ\n2ηTX\nt=2λT−t∥xt−u∥2+η\n2TX\nt=1λT−t∥gt∥2\n≤D2λT−1\n2η+D2(1−λ)\n2ηTX\nt=2λT−t+ηG2\n2TX\nt=1λT−t\n=D2λT−1\n2η+D2(1−λ)\n2η1−λT−1\n1−λ+ηG2\n21−λT\n1−λ\n≤DG\n2√\n1−λ2+DG√\n1−λ2\n2(1−λ)\n20≤3\n2DG√\n1−λ2. (1−λ2≤2(1−λ))\nFollowing the definition of HT, in the case of time-invariant λ <1,\nHT=1−λ2T\n1−λ2.\nDue to Lemma B.1, if T≥1\n2(1−λ)−1, we have λ2T≤e−1, therefore\n1√\n1−λ2≤r\nHT\n1−e−1.\nCombining the above completes the proof.\nAccompanying the upper bounds, the following theorem characterizes an instance-dependent discounted\nregret lower bound.\nTheorem 7. Consider any combination of time horizon T∈N+, discount factors λ1:T, Lipschitz constant\nG >0, and variance budget V∈\u0000\n0, G2HT\u0003\n, where HTis defined in Eq.(4). Furthermore, consider any nonzero\nu∈ X that satisfies −u∈ X. For any OCO algorithm that possibly depends on these quantities, there exists a\nsequence of linear losses lt(x) =⟨gt, x⟩such that ∥gt∥ ≤Gfor all t∈[1 :T],\nV=TX\nt=1 T−1Y\ni=tλ2\ni!\n∥gt∥2,\nand\nmaxh\nRegretλ1:T\nT(l1:T, u),Regretλ1:T\nT(l1:T,−u)i\n≥1√\n2∥u∥√\nV .\nProof of Theorem 7. The proof is a mild generalization of the undiscounted argument, e.g., [ Ora23a , Theorem 5.1].\nWe provide it here for completeness.\nWe first define a random sequence of loss gradients. Let ε1, . . . , ε Tbe a sequence of iid Rademacher random\nvariables: εtequals ±1 with probability1\n2each. Also, let\nL=s\nVPT\nt=1QT−1\ni=tλ2\ni.\nThen, we define the loss gradient sequence ˜ g1:Tas ˜gt=Lεtu\n∥u∥. Notice that L≤G.\nNow consider the regret with respect to ˜l1:T, the random linear losses induced by the random gradient\nsequence ˜ g1:T.\nRegretλ1:T\nT\u0010\n˜l1:T, u\u0011\n=TX\nt=1 T−1Y\ni=tλi!\n⟨˜gt, xt−u⟩=TX\nt=1 T−1Y\ni=tλi!\nL\n∥u∥⟨u, xt⟩εt−TX\nt=1 T−1Y\ni=tλi!\nL∥u∥εt.\nRegretλ1:T\nT\u0010\n˜l1:T,−u\u0011\n=TX\nt=1 T−1Y\ni=tλi!\n⟨˜gt, xt+u⟩=TX\nt=1 T−1Y\ni=tλi!\nL\n∥u∥⟨u, xt⟩εt+TX\nt=1 T−1Y\ni=tλi!\nL∥u∥εt.\nTherefore,\nEh\nmaxh\nRegretλ1:T\nT(˜l1:T, u),Regretλ1:T\nT(˜l1:T,−u)ii\n=E\"TX\nt=1 T−1Y\ni=tλi!\nL\n∥u∥⟨u, xt⟩εt#\n+E\"\nmax\"\n−TX\nt=1 T−1Y\ni=tλi!\nL∥u∥εt,TX\nt=1 T−1Y\ni=tλi!\nL∥u∥εt##\n=TX\nt=1 T−1Y\ni=tλi!\nL\n∥u∥E[⟨u, xt⟩εt] +L∥u∥E\"\f\f\f\f\fTX\nt=1 T−1Y\ni=tλi!\nεt\f\f\f\f\f#\n21=L∥u∥E\"\f\f\f\f\fTX\nt=1 T−1Y\ni=tλi!\nεt\f\f\f\f\f#\n≥1√\n2L∥u∥vuutTX\nt=1 T−1Y\ni=tλ2\ni!\n(Khintchine inequality [Haa81])\n=1√\n2∥u∥√\nV .\nFinally, note that the above lower-bounds the expectation. There exists a loss gradient sequence g1:Twith\ngt∈ {Lu/∥u∥,−Lu/∥u∥}, such that maxh\nRegretλ1:T\nT(l1:T, u),Regretλ1:T\nT(l1:T,−u)i\n≥1√\n2∥u∥√\nV.\nThe next theorem, presented in Section 3.2, analyzes gradient adaptive OGD.\nTheorem 2. If the diameter of Xis at most D, then OGD with learning rate ηt=DV−1/2\nt guarantees for all\nT∈N+and loss sequence l1:T,\nsup\nuRegretλ1:T\nT(l1:T, u)≤3\n2Dp\nVT.\nProof of Theorem 2. As shown in [ Ora23a , Chapter 4.2], applying OGD with the undiscounted gradient-\ndependent learning rate\nηt=DqPt\ni=1∥ˆgi∥2(15)\nto surrogate linear losses ⟨ˆgt,·⟩guarantees the undiscounted regret bound\nTX\nt=1⟨ˆgt, xt−u⟩ ≤3\n2DvuutTX\nt=1∥ˆgt∥2.\nFor the discounted setting, we follow the rescaling trick from Section 3.2. First, consider the effective\nprediction rule when ˆ gtis defined according to Eq.(7).\nxt+1= ΠX\nxt−DqPt\ni=1∥ˆgi∥2ˆgt\n\n= ΠX\nxt−DrPt\ni=1\u0010Qi−1\nj=1λ−2\nj\u0011\n∥gi∥2 t−1Y\ni=1λ−1\ni!\ngt\n\n= ΠX\nxt−DrPt\ni=1\u0010Qt−1\nj=iλ2\nj\u0011\n∥gi∥2gt\n,\nwhich is exactly the prediction rule this theorem proposes. As for the regret bound, we have\nTX\nt=1⟨ˆgt, xt−u⟩ ≤3\n2DvuutTX\nt=1∥ˆgt∥2=3\n2DvuutTX\nt=1 t−1Y\ni=1λ−2\ni!\n∥gt∥2.\nPlugging it into Eq.(8) and using the definition of VTfrom Eq.(5) complete the proof.\nRemark B.1 (Failure of OGD) .As mentioned in Section 3.2, one might suspect that OGD with an even better\nlearning rate than Theorem 2 (i.e., possibly using the oracle knowledge of ∥u∥) can further improve the regret\n22bound to O(∥u∥√VT), matching the lower bound (Theorem 7). We now explain where this intuition could come\nfrom, and why it is misleading.\nConsider the undiscounted setting ( λt= 1). For any scaling factor α, it is well-known that OGD with learning\nrateη=αG−1T−1/2achieves the undiscounted regret bound O\u0010\n(α+∥u∥2α−1)G√\nT\u0011\n[Ora23a , Chapter 2.1],\nwhich becomes O(∥u∥G√\nT)if the oracle tuning α=∥u∥is allowed. This might suggest that the remaining\nsuboptimality of Theorem 2 can be closed by a similar oracle tuning, i.e., setting\nηt=∥u∥√Vt. (16)\nHowever, such an analogy misses a key point: the aforementioned nice property of “ α-scaled OGD” only holds\nwith time-invariant learning rates, and therefore, it can be found that OGD with the time-varying learning rate\nEq.(16) does not yield the O(∥u∥G√VT)bound we aim for, let alone the impossibility of oracle tuning. Broadly\nspeaking, it is known (but sometimes overlooked) that OGD has certain fundamental incompatibility with time-\nvarying learning rates, but this can be fixed by an extra regularization centered at the origin [ OP18 ,FHPF22 ,JC22 ].\nSuch a procedure encodes the initialization into the algorithm, which essentially makes it similar to FTRL.\nB.2 Algorithm from [ZYCP23]\nIn this subsection we introduce the algorithm from [ ZYCP23 ] and its guarantee. Proposed for the undiscounted\nsetting ( λt= 1), it achieves a scale-free regret bound that simultaneously adapts to both the loss sequence and\nthe comparator.\nFollowing the polar-decomposition technique from [ CO18 ], the main component of this algorithm is a\nsubroutine (a standalone one dimensional OCO algorithm) that operates on the positive real line [0 ,∞). We\npresent the pseudocode of this subroutine as Algorithm 2. Technically it is a combination of [ ZYCP23 , Algorithm 1\n+ (part of) Algorithm 2] and [ Cut19 , Algorithm 2]. Except the choice of the Φ function which uses an unusual\ncontinuous-time analysis, everything else is somewhat standard in the community. Here is an overview of the\nidea.\n•The core component is the potential function Φh,ε(v, s) that takes two input arguments, the observed\ngradient variance vand the observed gradient sum s. For now let us briefly suppress the dependence on\nauxiliary parameters handε. At the beginning of the t-th round, with the observations of past gradients\ng1:t−1, we “ideally” (there are two twists explained later) would like to define its summaries (sometimes\ncalled sufficient statistics )\nvt=t−1X\ni=1g2\ni, s t=−t−1X\ni=1gi,\nand predict xt=∂2Φ(vt, st), i.e., the partial derivative of the potential function Φ with respect to its\nsecond argument, evaluated at the pair ( vt, st). This is the standard procedure of the “potential method”\n[CBL06 ,MO14 ,MK20 ] in online learning, which is associated to a well-established analytical strategy\n[MO14 ]. Equivalently, one may also interpret this procedure as Follow the Regularized Leader (FTRL)\n[Ora23a , Chapter 7.3] with linearized losses: the potential function Φ is essentially the convex conjugate of\na FTRL regularizer [ZCP22, Section 3.1].\nA bit more on the design of Φ: the intuition is that his typically much smaller than vands, so if we set\nh= 0 (rigorously this is not allowed since the prediction would be a bit too aggressive; but just for our\nintuitive discussion this is fine), then the potential function is morally\nΦ(v, s)≈ε√v \n2Z s\n2√v\n0erfi(u)du−1!\n,\nwhich is associated to the prediction function\n∂2Φ(v, s)≈ε·erfi\u0012s\n2√v\u0013\n=εZs/√\n4v\n0exp(u2)du.\n23Algorithm 2 ([ZYCP23, Algorithm 1 and 2] + [Cut19, Algorithm 2]) Undiscounted 1D magnitude learner on\n[0,∞).\nRequire: Hyperparameter ε >0 (default ε= 1).\n1:Initialize parameters v1= 0, s1= 0, h1= 0. Define the following ( h, ε)-parameterized potential function\nΦh,ε(v, s) with two input arguments vands.\nΦh,ε(v, s):=εp\nv+ 2hs+ 16h2 \n2Z s\n2√\nv+2hs+16h2\n0erfi(u)du−1!\n.\n2:fort= 1,2, . . .do\n3: Ifht= 0, define an unprojected prediction ˜ xt= 0; otherwise, define\n˜xt=∂2Φht,ε(vt, st)\n=ε·erfi \nst\n2p\nvt+ 2htst+ 16h2\nt!\n−εhtp\nvt+ 2htst+ 16h2\ntexp\u0014s2\nt\n4(vt+ 2htst+ 16h2\nt)\u0015\n.\n4: Predict xt= Π [0,∞)(˜xt), the projection of ˜ xtto the domain [0 ,∞).\n5: Receive the loss gradient gt∈R.\n6: Clip the gradient by defining gt,clip= Π [−ht,ht](gt), and let ht+1= max ( ht,|gt|).\n7: Define a surrogate loss gradient ˜ gt,clip∈R, where\n˜gt,clip=(\ngt,clip, gt,clip˜xt≥gt,clipxt,\n0, else.\n8: Update vt+1=vt+ ˜g2\nt,clip,st+1=st−˜gt,clip.\n9:end for\nThe use of the erfifunction might seem obscure, but essentially it is rooted in a recent trend [ DK20 ,\nZCP22 ,HLPR23 ] connecting online learning to stochastic calculus: we scale the OCO game towards its\ncontinuous-time limit and solve the obtained Backward Heat Equation . Prior to this trend, the predominant\nidea was using [MO14, OP16, MK20]\nΦ(v, s)≈ε√vexp\u0012s2\nconstant ·v\u0013\n, (17)\n∂2Φ(v, s)≈εconstant ·s\nv3/2exp\u0012s2\nconstant ·v\u0013\n.\nIt can be shown that the erfiprediction rule is quantitatively stronger, and quite importantly, it makes the\nhyperparameter ε“unitless” [ ZYCP23 , Section 5]. In contrast, in the more classical potential function\nEq.(17), εcarries the unit of “gradient squared”, which means the algorithm requires a guess of the time-\nuniform Lipschitz constant max t∥gt∥at the beginning of the game . Due to the discussion in Section 3.2\n(right after introducing the rescaling trick), this suffers from certain suboptimality (Remark B.4) when\napplied with rescaling.\n•Although most of the heavy lifting is handled by the choice of Φ, a remaining issue is that with h̸=0,\nthe prediction xt=∂2Φ(vt, st) is not necessarily positive, which violates the domain constraint [0 ,∞).\nTo fix this issue, we adopt the technique from [ CO18 ,Cut20 ] which has become a standard tool for the\ncommunity: predict xt= Π [0,∞)(˜xt) which is the projection of ˜xt=∂2Φ(vt, st) to the domain [0 ,∞), and\nupdate the sufficient statistics ( vt+1, st+1) using a surrogate loss gradient ˜gt,clip(Line 7 of Algorithm 2)\ninstead of gt,clip(the clipping will be explained later; for now we might regard gt,clip=gt, the actual\ngradient).\n24The intuition behind Line 7 is that, if the unprojected prediction ˜xt=∂2Φ(vt, st) is already negative and\nthe actual loss gradient gt,clipencourages it to be “more negative”, then we set ˜gt,clip= 0 in the update\nof the ( vt+1, st+1) pair to avoid this undesirable behavior (unprojected prediction drifting away from the\ndomain). In other situations, it is fine to use gt,clipdirectly in the update of ( vt+1, st+1), therefore we\nsimply set ˜ gt,clip=gt,clip.\nRigorously, [ Cut20 , Theorem 2] shows that for all comparator u∈[0,∞),⟨gt,clip, xt−u⟩ ≤ ⟨ ˜gt,clip,˜xt−u⟩.\nThat is, as long as the unprojected prediction sequence ˜x1:Tguarantees a good regret bound (in an improper\nmanner, i.e., violating the domain constraint), then the projected prediction sequence x1:Talso guarantees\na good regret bound, but properly.\n•Another twist is related to updating the auxiliary parameter h, which was previously ignored. Due to the\ntypical limitation of FTRL algorithms, we have to guess the range of gt(i.e., a time-varying Lipschitz\nconstant Gtsuch that ∥gt∥ ≤Gt)right before making the prediction xt, and htserves as this guess. [ Cut19 ]\nsuggests using the range of past loss gradients ht=max i∈[1:t−1]|gi|to guess that |gt| ≤ht. Surely this\ncould be wrong: in that case ( |gt|> ht), we clip gtto [−ht, ht] before sending it to the ( vt+1, st+1) update.\nWe also clarify a possible confusion related to “guessing the Lipschitzness”. We have argued that if an\nalgorithm requires guessing the time-uniform Lipschitz constant max t∥gt∥at the beginning of the game,\nthen it does not serve as a good base algorithm Ain our rescaling trick. The use of htis different: it is\nupdated online as a guess of ∥gt∥, which is fine (to apply with rescaling).\nIn terms of the concrete guarantee, the following theorem characterizes the undiscounted regret bound\nof Algorithm 2. The proof is a straightforward corollary of [ ZYCP23 , Lemma B.2] and [ Cut20 , Theorem 2],\ntherefore omitted.\nTheorem 8 ([Cut20 ,ZYCP23 ]).Algorithm 2 guarantees for all time horizon T∈N+, loss gradients g1:Tand\ncomparator u∈[0,∞),\nTX\nt=1gt(xt−u)≤εvuutTX\nt=1g2\nt+ 2GS+ 16G2+uS+TX\nt=1|gt−gt,clip||xt−u|,\nwhere G= max t∈[1:T]|gt|and\nS= 8G\u0010\n1 +p\nlog(2uε−1+ 1)\u00112\n+ 2vuutTX\nt=1g2\nt+ 16G2\u0010\n1 +p\nlog(2uε−1+ 1)\u0011\n.\nRemark B.2 (Iterate stability) .The above bound has an iterate stability termPT\nt=1|gt−gt,clip||xt−u|, which\ncan be further upper-bounded by (u+max t∈[1:T]xt)GT. Similar characterizations of stability have appeared\nbroadly in stochastic optimization before [ OP21 ,IHC23 ,Ora23b ]. For the general adversarial setting we consider,\n[Cut19 ] suggests using artificial constraints to trade max txtfor terms that only depend on g1:Tandu. We do\nnot take this route because ( i) it makes our discounted algorithm more complicated but does not seem to improve\nthe performance; and ( ii) even without artificial constraints, the prediction magnitude is indeed controlled in\nour main application (online conformal prediction), and possibly in the downstream stochastic setting as well\n(following [Ora23b]).\nGiven the above undiscounted 1D subroutine, we can extend it to Rdfollowing the standard polar-\ndecomposition technique [ CO18 ]. Overall, the algorithm becomes the λt= 1 special case of Algorithm 3\n(our main algorithm presented in Section B.3). This is as expected, since the discounted setting is a strict\ngeneralization. The following theorem is essentially [ZYCP23, Theorem 2 + the discussion after that].\nTheorem 9 ([ZYCP23 ]).With λt= 1for all t, Algorithm 3 from Section B.3 guarantees for all T∈N+, loss\ngradients g1:Tand comparator u∈Rd,\nTX\nt=1⟨gt, xt−u⟩ ≤O\u0010\n∥u∥p\nVTlog(∥u∥ε−1)∨ ∥u∥GTlog(∥u∥ε−1)\u0011\n+TX\nt=1∥gt−gt,clip∥∥xt−u∥,\n25where VTandGTare defined in Eq.(5), and O(·)is in the regime of large VT(VT≫GT)and large ∥u∥(∥u∥ ≫ε).\nFurthermore, if the comparator u= 0, we have\nTX\nt=1⟨gt, xt⟩ ≤O\u0010\nεp\nVT\u0011\n+TX\nt=1∥gt−gt,clip∥∥xt∥.\nB.3 Analysis of the main algorithm\nThis subsection presents our main discounted algorithm and its regret bound. We start from its 1D magnitude\nlearner.\nTheorem 3. Given any hyperparameter ε >0, Algorithm 1 guarantees for all time horizon T∈N+, loss\nsequence l1:T, comparator u∈[0,∞)and stability window length τ∈[1 :T],\nRegretλ1:T\nT(l1:T, u)≤εq\nVT+ 2GTS+ 16G2\nT+u(S+GT)+\u0012\nmax\nt∈[T−τ+1:T]xt\u0013\nGT+ T−1Y\nt=T−τλt!\u0012\nmax\nt∈[1:T−τ]xt\u0013\nGT−τ,\nwhere\nS= 8GT\u0010\n1 +p\nlog(2uε−1+ 1)\u00112\n+ 2q\nVT+ 16G2\nT\u0010\n1 +p\nlog(2uε−1+ 1)\u0011\n.\nProof of Theorem 3. The proof mostly follows from carefully checking the equivalence of the following two\nalgorithms: ( i) Algorithm 1 (the discounted algorithm) on a sequence of loss gradients g1:T, and ( ii) Algorithm 2\n(the undiscounted algorithm) on the sequence of scaled surrogate gradients,\u0010Qt−1\ni=1λ−1\ni\u0011\ngt;∀t∈[1 :T]. Since\nthe quantities in Algorithm 1 and 2 follow the same notation, we separate them by adding a superscript Don\nquantities in Algorithm 1, and NDin their Algorithm 2 counterparts.\nWe show this by induction: suppose for some t∈[1 :T], we have\nsD\nt= t−2Y\ni=1λi!\nsND\nt, vD\nt= t−2Y\ni=1λ2\ni!\nvND\nt, hD\nt= t−2Y\ni=1λi!\nhND\nt.\nSuch an induction hypothesis holds for t= 1. Then, from the prediction rules (and the fact that all the discount\nfactors are strictly positive), the unprojected predictions ˜ xD\nt= ˜xND\nt, and the projected predictions xD\nt=xND\nt.\nNow consider the gradient clipping. Since gND\nt=\u0010Qt−1\ni=1λ−1\ni\u0011\ngD\nt,\ngND\nt,clip= Π[−hND\nt,hND\nt]\u0000\ngND\nt\u0001\n=c−1Π[−chND\nt,chND\nt]\u0000\ncgND\nt\u0001\n(∀c >0)\n= t−1Y\ni=1λ−1\ni!\nΠ[−λt−1hD\nt,λt−1hD\nt]\u0000\ngD\nt\u0001\n(c=Qt−1\ni=1λi)\n= t−1Y\ni=1λ−1\ni!\ngD\nt,clip.\nThen, due to xD\nt=xND\ntand ˜xD\nt= ˜xND\nt, we have ˜ gND\nt,clip=\u0010Qt−1\ni=1λ−1\ni\u0011\n˜gD\nt,clip.\nFinally, consider the updates of sD\nt+1,vD\nt+1andhD\nt+1, as well as their Algorithm 2 counterparts.\nsD\nt+1=λt−1sD\nt−˜gD\nt,clip= t−1Y\ni=1λi!\nsND\nt− t−1Y\ni=1λi!\n˜gND\nt,clip= t−1Y\ni=1λi!\n\u0000\nsND\nt−˜gND\nt,clip\u0001\n= t−1Y\ni=1λi!\nsND\nt+1.\nSimilarly,\nvD\nt+1= t−1Y\ni=1λ2\ni!\nvND\nt+1,\n26hD\nt+1= max\u0000\nλt−1hD\nt,\f\fgD\nt\f\f\u0001\n= max\" t−1Y\ni=1λi!\nhND\nt, t−1Y\ni=1λi!\n\f\fgND\nt\f\f#\n= t−1Y\ni=1λi!\nhND\nt+1.\nThat is, the induction hypothesis holds for t+ 1, and therefore we have shown the equivalence of the considered\ntwo algorithms.\nAs for the regret bound of Algorithm 1, combining the reduction Eq.(8) and the regret bound of Algorithm 2\n(Theorem 8 from Appendix B.2) immediately gives us\nRegretλ1:T\nT(l1:T, u)≤εq\nVT+ 2GTS+ 16G2\nT+uS+ T−1Y\nt=1λt!TX\nt=1 t−1Y\ni=1λ−1\ni!\n|gt−gt,clip||xt−u|,\nwhere\nS= 8GT\u0010\n1 +p\nlog(2uε−1+ 1)\u00112\n+ 2q\nVT+ 16G2\nT\u0010\n1 +p\nlog(2uε−1+ 1)\u0011\n.\nThe remaining clipping error term can be bounded similarly as [Cut19, Theorem 2]. For any τ∈[1 :T],\nT−τX\nt=1 t−1Y\ni=1λ−1\ni!\n|gt−gt,clip||xt−u| ≤ max\nt∈[1:T−τ]|xt−u|T−τX\nt=1 t−1Y\ni=1λ−1\ni!\n|gt−gt,clip|\n= max\nt∈[1:T−τ]|xt−u|T−τX\nt=1 t−1Y\ni=1λ−1\ni!\n(ht+1−λt−1ht) (from Algorithm 1)\n= max\nt∈[1:T−τ]|xt−u|T−τX\nt=1\" t−1Y\ni=1λ−1\ni!\nht+1− t−2Y\ni=1λ−1\ni!\nht#\n= max\nt∈[1:T−τ]|xt−u|\" T−τ−1Y\ni=1λ−1\ni!\nhT−τ+1−h1#\n≤\u0012\nmax\nt∈[1:T−τ]xt+u\u0013 T−τ−1Y\ni=1λ−1\ni!\nhT−τ+1.\nSimilarly,\nTX\nt=T−τ+1 t−1Y\ni=1λ−1\ni!\n|gt−gt,clip||xt−u| ≤\u0012\nmax\nt∈[T−τ+1:T]xt+u\u0013\" T−1Y\ni=1λ−1\ni!\nhT+1− T−τ−1Y\ni=1λ−1\ni!\nhT−τ+1#\n.\nFinally, multiplyingQT−1\nt=1λtand using GT=hT+1andGT−τ=hT−τ+1complete the proof.\nAs for the extension to Rdfollowing [ Cut18 ], we present the pseudocode as Algorithm 3. There is a small\ntwist: when applying Algorithm 1 as the 1D subroutine, in its Line 6 we set gt,clip=gt, and ht+1is given by the\nmeta-algorithm. That is, the gradient clipping is handled on the high level (Algorithm 3) rather than the low\nlevel (Algorithm 1).\nAlgorithm 3 induces our main theorem (Theorem 4). The proof combines the undiscounted regret bound\n(Theorem 9) and our rescaling trick. It is almost the same as the above proof of Theorem 3, therefore omitted.\nRemark B.3 (Importance of forgetting) .Different from the undiscounted algorithm in [ ZYCP23 ], the above\ndiscounted generalization rapidly forgets the past observations (which could be misleading for the future). Such an\nidea might even be useful in stochastic convex optimization as well, in particular regarding functions with spatially\ninhomogeneous Lipschitzness. An example is the quadratic function, which is both strongly convex and smooth.\nGlobally it is not Lipschitz, but near its minimizer the Lipschitz constant is small. When optimizing such a\nfunction, if an optimization algorithm internally estimates the Lipschitz constant from its historical observations,\nthen this estimate could be overly conservative as the iterates move closer to the minimizer. An intuitive solution\nis to gradually forget the past, just like the idea of discounted online learning algorithms. Rigorously characterizing\nthis intuition could be an exciting direction, which we defer to future works.\n27Algorithm 3 Discounted adaptivity on Rd.\n1:Define A1das a minor variant of Algorithm 1 (with hyperparameter ε), where its Line 6 is replaced by: “Set\ngt,clip=gt, and receive a hint ht+1.”\n2:Define ABas the algorithm from Theorem 2, on the d-dimensional unit L2norm ball (with D= 2).\n3:Initialize h1= 0.\n4:fort= 1,2, . . .do\n5: Query A1dfor its prediction yt∈R.\n6: Query ABfor its prediction wt∈Rd;∥wt∥ ≤1.\n7: Predict xt=wtyt, receive the loss gradient gt∈Rdand the discount factor λt∈(0,∞).\n8: Update ht+1= max ( λt−1ht,∥gt∥), and define gt,clip=gtλt−1ht/ht+1(ifht+1= 0 then gt,clip= 0).\n9: Send⟨gt,clip, wt⟩andgt,clipas the surrogate loss gradients to A1dandAB, respectively.\n10: Send the discount factor λttoA1dandAB.\n11: Send the hint ht+1toA1d.\n12:end for\nRemark B.4 (Benefit of [ ZYCP23 ]).We used [ ZYCP23 ] as the base algorithm in our scaling trick, but there\nare other options [ MK20 ,JC22 ]. The problem of such alternatives is that, they still need an estimate of the\ntime-uniform Lipschitz constant at the beginning of the game, due to certain unit inconsistency (discussed in\nAppendix B.2, e.g., Eq.(17)). In the undiscounted setting, they guarantee\nTX\nt=1⟨gt, xt−u⟩ ≤O\u0010\n∥u∥p\nVTlog(∥u∥T)∨ ∥u∥GTlog(∥u∥T)\u0011\n+TX\nt=1∥gt−gt,clip∥∥xt−u∥,\nas opposed to Theorem 9 in this paper. After the scaling trick, the Tdependence in this bound will be transferred\nto the obtained discounted regret bound, such that the latter also depends on the end time T. In other words,\nsuch a discounted regret bound is not “lifetime consistent”.\nC Detail of Section 4\nThis section presents omitted details of our OCP application. First, we present the pseudocode of ACPin\nAppendix C.1, with the notations (relevant quantities are with the superscipt ∗) from the OCP setting. All the\nconcrete proofs are presented in Appendix C.2.\nC.1 Pseudocode of OCP algorithm\nThe pseudocode of ACPis Algorithm 4. This is equivalent to directly applying Algorithm 1, our main 1D OCO\nalgorithm, to the setting of Section 4.\nIn particular, the problem structure of OCP allows removing the surrogate loss construction (Line 7 of\nAlgorithm 1), which makes the algorithm slightly simpler. To see this, notice that the surrogate loss ˜gt,clipthere\nis only needed (i.e., does not equal gt,clip) when the unprojected prediction ˜xt<0 and the projected prediction\nxt= 0. In the OCP notation, this means that our radius prediction rt= 0≤r∗\nt, therefore the subgradient g∗\nt\nevaluated at rtsatisfies g∗\nt≤0. Back to the notation of Algorithm 1, we have gt≤0, therefore after clipping\ngt,clip≤0. Putting things together,\ngt,clip(˜xt−xt)≥0.\nThat is, the condition in Algorithm 1 that triggers ˜gt,clip= 0 is impossible, therefore ˜gt,clipalways equals gt,clip.\nC.2 Omitted proofs\nMoving to the analysis, we first prove the key lemma connecting the prediction magnitude of ACPto the coverage\nmetric S∗\nt, Eq.(13). This is divided into two steps.\n28Algorithm 4 The proposed OCP algorithm ACP.\nRequire: Hyperparameter ε >0 (default ε= 1).\n1:Initialize S∗\n0,clip= 0,V∗\n0,clip= 0,G∗\n0= 0.\n2:fort= 1,2, . . .do\n3: IfG∗\nt−1= 0, define the unprojected prediction ˜ rt= 0. Otherwise,\n˜rt=ε·erfi\nS∗\nt−1,clip\n2q\nV∗\nt−1,clip+ 2G∗\nt−1S∗\nt−1,clip+ 16\u0000\nG∗\nt−1\u00012\n\n−εG∗\nt−1q\nV∗\nt−1,clip+ 2G∗\nt−1S∗\nt−1,clip+ 16\u0000\nG∗\nt−1\u00012exp\n\u0010\nS∗\nt−1,clip\u00112\n4\u0010\nV∗\nt−1,clip+ 2G∗\nt−1S∗\nt−1,clip+ 16\u0000\nG∗\nt−1\u00012\u0011\n.\n4: Predict rt= Π [0,∞)(˜rt).\n5: Receive the OCP loss gradient g∗\nt∈Rand the discount factor λt−1∈(0,∞).\n6: Clipg∗\ntby defining\ng∗\nt,clip= Π [−λt−1G∗\nt−1,λt−1G∗\nt−1](g∗\nt).\n7: Compute running statistics of past observations,\nS∗\nt,clip=−tX\ni=1\nt−1Y\nj=iλj\ng∗\ni,clip, V∗\nt,clip=tX\ni=1\nt−1Y\nj=iλ2\nj\n\r\rg∗\ni,clip\r\r2, G∗\nt= max\ni∈[1:t]\nt−1Y\nj=iλj\n∥g∗\ni∥.\n8:end for\n•First (Lemma C.1), we approximate the prediction rule of ACP, such that rt+1characterizes the discounted\nsum of clipped gradients\nS∗\nt,clip=−tX\ni=1\nt−1Y\nj=iλj\ng∗\ni,clip, (18)\nwhich is an internal quantity of Algorithm 4.\n•The second step (Lemma C.2) is connecting S∗\nt,clipto the unclipped version S∗\nt, Eq.(13). This is the version\npresented in the main paper.\nLemma C.1. Consider S∗\nt,clipfrom Eq.(18). ACP(Algorithm 4) guarantees for all t,\n\f\fS∗\nt,clip\f\f≤2q\nV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u0011\n+ 13G∗\nt\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u00112\n,\nwhere V∗\nt,clipandG∗\ntare internal quantities of Algorithm 4.\nProof of Lemma C.1. Throughout this proof we will consider the internal variables of Algorithm 4. The\nsuperscript ∗are always removed to simplify the notation. Assume without loss of generality that internally\nin Algorithm 4, Gt̸=0 for the considered t. Otherwise, all the gradients g1, . . . , g tare zero, which makes the\nstatement of the lemma obvious.\nTo upper-bound St,clip, the analysis is somewhat similar to the control of Fenchel conjugate in [ ZYCP23 ,\nLemma B.1], although the latter serves a different purpose. Consider the input argument of the erfifunction in\nthe definition of rt+1. There are two cases.\n•Case 1: St,clip<2p\nVt,clip+ 2GtSt,clip+ 16G2\nt.\nWith straightforward algebra,\nS2\nt,clip−8GtSt,clip−4\u0000\nVt,clip+ 16G2\nt\u0001\n<0,\n29St,clip≤4Gt+q\n16G2\nt+ 4 (Vt,clip+ 16G2\nt)≤2p\nVt,clip+ 13Gt.\n•Case 2: St,clip≥2p\nVt,clip+ 2GtSt,clip+ 16G2\nt.\nSince Gt̸= 0, Algorithm 4 predicts rt+1= Π [0,∞)(˜rt+1), where\n˜rt+1=ε·erfi \nSt,clip\n2p\nVt,clip+ 2GtSt,clip+ 16G2\nt!\n−εGtp\nVt,clip+ 2GtSt,clip+ 16G2\ntexp\"\nS2\nt,clip\n4(Vt,clip+ 2GtSt,clip+ 16G2\nt)#\n.\nDue to a lower estimate of the erfifunction [ ZYCP23 , Lemma A.3], for all x≥1,erfi(x)≥exp(x2)/2x.\nThen,\nerfi \nSt,clip\n2p\nVt,clip+ 2GtSt,clip+ 16G2\nt!\n≥p\nVt,clip+ 2GtSt,clip+ 16G2\nt\nSt,clipexp\"\nS2\nt,clip\n4(Vt,clip+ 2GtSt,clip+ 16G2\nt)#\n,\nand simple algebra characterizes the multiplier on the RHS,\np\nVt,clip+ 2GtSt,clip+ 16G2\nt\nSt,clip−2Gtp\nVt,clip+ 2GtSt,clip+ 16G2\nt\n=Vt,clip+ 16G2\nt\nSt,clipp\nVt,clip+ 2GtSt,clip+ 16G2\nt≥0.\nTherefore,\nrt+1≥˜rt+1\n≥ε\n2erfi \nSt,clip\n2p\nVt,clip+ 2GtSt,clip+ 16G2\nt!\n+εexp\"\nS2\nt,clip\n4(Vt,clip+ 2GtSt,clip+ 16G2\nt)#\n× p\nVt,clip+ 2GtSt,clip+ 16G2\nt\n2St,clip−Gtp\nVt,clip+ 2GtSt,clip+ 16G2\nt!\n≥ε\n2erfi \nSt,clip\n2p\nVt,clip+ 2GtSt,clip+ 16G2\nt!\n.\nDue to another estimate of erfi−1[ZYCP23 , Lemma A.4], for all x≥0 we have erfi−1(x)≤1+p\nlog(x+ 1).\nThen,\nSt,clip≤2q\nVt,clip+ 2GtSt,clip+ 16G2\nt·erfi−1\u0000\n2rt+1ε−1\u0001\n≤2q\nVt,clip+ 2GtSt,clip+ 16G2\nt\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u0011\n.\nSimilar to the algebra of Case 1,\nSt,clip≤2p\nVt,clip\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u0011\n+ 13Gt\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u00112\n.\nCombining the two cases, we have\nSt,clip≤2p\nVt,clip\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u0011\n+ 13Gt\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u00112\n.\nThat is, St,clipis now bounded from the above by ˜O(p\nVt,clip).\nOn the other side, we now consider bounding St,clipfrom below. This is a classical induction argument similar\nto [ZYCP23, Lemma 4.1]. Consider any time index i∈[1 :t],\n30•IfSi−1,clip≤0, then according to the prediction rule of Algorithm 4, we have ri= 0 regardless of\nthe value of Gi−1. Then, due to the structure of OCP, we have gi≤0, thus gi,clip≤0 and Si,clip=\nλi−1Si−1,clip−gi,clip≥λi−1Si−1,clip.\n•IfSi−1,clip>0, then Si,clip≥λi−1Si−1,clip− |gi,clip| ≥ − Gi.\nCombining these two and using an induction from S0,clip= 0, we have St,clip≥ −Gt.\nSummarizing the above completes the proof.\nThe following lemma is the full version of Lemma 4.1 presented in Section 4; we further use V∗\nt,clip≤V∗\nt\nthere.\nLemma C.2. ACPguarantees for all t,\n|S∗\nt| ≤2q\nV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u0011\n+ 14G∗\nt\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u00112\n,\nwhere V∗\nt,clipandG∗\ntare internal quantities of Algorithm 4.\nProof of Lemma C.2. Given Lemma C.1, the remaining task is connecting |S∗\nt,clip|to|S∗\nt|. To this end,\n|S∗\nt| ≤\f\fS∗\nt,clip\f\f+tX\ni=1\nt−1Y\nj=iλj\n\f\fg∗\ni−g∗\ni,clip\f\f,\nand the sum on the RHS is at most G∗\ntfollowing the proof of Theorem 3.\nThe next lemma exploits the bounded domain assumption.\nLemma C.3. Ifmax tr∗\nt≤D, then without knowing D,ACPguarantees for all t,\n|S∗\nt| ≤2q\nV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 Dε−1)\u0011\n+ 15G∗\nt\u0010\n1 +p\nlog (1 + 2 Dε−1)\u00112\n,\nwhere V∗\nt,clipandG∗\ntare internal quantities of Algorithm 4.\nProof of Lemma C.3. Consider S∗\nt,clipdefined in Eq.(18). We now use induction to show that\n\f\fS∗\nt,clip\f\f≤2q\nV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 Dε−1)\u0011\n+ 14G∗\nt\u0010\n1 +p\nlog (1 + 2 Dε−1)\u00112\n,\nand after that, we complete the proof using |S∗\nt| ≤\f\f\fS∗\nt,clip\f\f\f+G∗\nt, just like Lemma C.2.\nConcretely, note that such a statement on\f\f\fS∗\nt,clip\f\f\ftrivially holds for t= 1. Then, suppose it holds for any t.\nIn the t+ 1-th round, there are two cases.\n•Case 1: rt+1≤D.\nDue to Lemma C.1, we have\n\f\fS∗\nt,clip\f\f≤2q\nV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u0011\n+ 13G∗\nt\u0010\n1 +p\nlog (1 + 2 rt+1ε−1)\u00112\n≤2q\nV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 Dε−1)\u0011\n+ 13G∗\nt\u0010\n1 +p\nlog (1 + 2 Dε−1)\u00112\n.\n\f\fS∗\nt+1,clip\f\f≤λt\f\fS∗\nt,clip\f\f+\f\fg∗\nt+1,clip\f\f\n= 2q\nλ2\ntV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 Dε−1)\u0011\n+ 13λtG∗\nt\u0010\n1 +p\nlog (1 + 2 Dε−1)\u00112\n+\f\fg∗\nt+1,clip\f\f\n≤2q\nV∗\nt+1,clip\u0010\n1 +p\nlog (1 + 2 Dε−1)\u0011\n+ 14G∗\nt+1\u0010\n1 +p\nlog (1 + 2 Dε−1)\u00112\n.\n31•Case 2: rt+1> D.\nIn this case, rt+1> r∗\nt+1so the OCP gradient g∗\nt+1≥0, and the clipped gradient g∗\nt+1,clip≥0. Meanwhile,\nin order to have rt+1> D≥0, we must have S∗\nt,clip≥0 from the prediction rule. Then, due to the same\nsigns of g∗\nt+1,clipandS∗\nt,clip, we have\n\f\fS∗\nt+1,clip\f\f=\f\fλtS∗\nt,clip−g∗\nt+1,clip\f\f\n≤λt\f\fS∗\nt,clip\f\f∨\f\fg∗\nt+1,clip\f\f\n≤2q\nλ2\ntV∗\nt,clip\u0010\n1 +p\nlog (1 + 2 Dε−1)\u0011\n+ 14λtG∗\nt\u0010\n1 +p\nlog (1 + 2 Dε−1)\u00112\n≤2q\nV∗\nt+1,clip\u0010\n1 +p\nlog (1 + 2 Dε−1)\u0011\n+ 14G∗\nt+1\u0010\n1 +p\nlog (1 + 2 Dε−1)\u00112\n.\nCombining the two cases, the induction statement holds in the t+ 1-th round. Finally we use |S∗\nt| ≤\f\f\fS∗\nt,clip\f\f\f+\nG∗\nt.\nWith everything above, Theorem 5 is a simple corollary.\nTheorem 5. Without knowing D,ACPguarantees that for all T∈N+, we have the discounted coverage bound\n|S∗\nT| ≤O\u0012q\nV∗\nTlog(Dε−1)∨G∗\nTlog(Dε−1)\u0013\n,\nand the discounted regret bound from Theorem 3.\nProof of Theorem 5. The regret bound trivially applies. The coverage bound follows from Lemma C.3 and the\nfact that V∗\nt,clip≤V∗\nt. Then we consider the asymptotic regime of D≫ε.\nD Detail of experiment\nThis section presents details of our experiment. Our setup builds on the great work of [BWXB23].\nSetup We test our OCP algorithm (Algorithm 4, which is based on OCO Algorithm 1 and 2) in the context\nof classifying altered images that arrive in a sequential manner. Given a parameterized prediction set Ct(·)\ndependent on the label provided by a base ML model, at each time step our algorithms predict the radius rt\nthat corresponds to the uncertainty of the base model’s prediction, resulting in the prediction set Ct(rt). We\nadopt the procedure, code, and base model from [ BWXB23 ]. Given a sequence of images, we expect that if\nimages are increasingly “corrupted” by blur, noise, and other factors, the prediction set size must increase to\naccount for the deviation from the base model’s training distribution. We hypothesize that the rate of such a\ndistribution shift also affects the OCP algorithms’ performance, thus we test the cases where the corruption\nlevels shift suddenly versus gradually.\nOur algorithms are specifically designed to not use knowledge of the maximum magnitude Dof the optimal\nradius r∗\nt(i.e., the maximum uncertainty level). In some prediction scenarios, it is conceivable that Dis\nimpossible to know a priori and thus cannot be used as a hyperparameter. In contrast, certain algorithms from\nthe literature use an empirical estimate of Dfrom an offline dataset, which amounts to “oracle tuning”. These\ninclude the Strongly Adaptive Online Conformal Prediction ( Saocp ) and Scale-Free Online Gradient Descent\n(Sf-Ogd ) proposed by [ BWXB23 ]. In our study, we create a modified version of Sf-Ogd called “Simple OGD”,\nthat does not use such an oracle tuning. The only hyperparameter that we set is the learning rate, which we set\nto 1 for Simple OGD. Note that despite the name, Simple OGD is also gradient adaptive, which differs from the\nACI algorithm from [GC21].\nBaselines We perform OCP for ten different algorithms:\n1.MagL-D : We test our Algorithm 1 with ε= 1 and discount factor λt= 0.999, which we name MagL-D\n(Mag nitude Learner with Lipschitz Constant Estimate and Discounting).\n32Algorithm 5 Modified 1D magnitude learner on [0 ,∞).\nRequire: Hyperparameter ε >0.\n1:Initialize parameters v1>0,s1= 0.\n2:fort= 1,2, . . .do\n3: Define the unprojected prediction ˜ xt,\n˜xt=ε·erfi\u0012st\n2√vt\u0013\n.\n4: Predict xt= Π [0,∞)(˜xt), the projection of ˜ xtto the domain [0 ,∞).\n5: Receive the 1D loss gradient gt∈Rand the discount factor λt−1∈(0,∞).\n6: Ifgt˜xt< gtxt, define a surrogate loss gradient ˜ gt= 0. Otherwise, ˜ gt=gt.\n7: Update vt+1=λ2\nt−1vt+ ˜g2\nt,st+1=λt−1st−˜gt.\n8:end for\n2.MagL : We test Algorithm 2, the undiscounted algorithm that uses the running estimate of the Lipschitz\nconstant, ht, with ε= 1. We name this algorithm MagL (Mag nitude Learner with Lipschitz constant\nestimate).\n3.MagDis : We also test Algorithm 5 with ε= 1 and λt= 0.999, which is a simplified version of Algorithm\n1 that essentially sets ht= 0, does not clip gt, and initializes vt>0. We name this algorithm MagDis\n(Mag nitude Learner with Discounting).\nUsing the implementation from [ BWXB23 ], we also obtain results for Saocp ,Sf-Ogd , Simple OGD,\nStandard Split Conformal Prediction (SCP) [ VGS05 ], Non-Exchangeable SCP (NExCP) [ BCRT23 ], Fully-\nAdaptive Conformal Inference ( Faci) [GC22], and Faci-S [BWXB23].\nMetrics We choose a targeted coverage rate of 90% for all experiments, which means α= 0.1. To quantify the\nalgorithms’ performance, we follow the definitions from [ BWXB23 ] to evaluate the coverage (local and average),\nprediction width (local and average), worst-case local coverage error ( LCE k), and runtime. They are defined as\nfollows.\n•First, let Ytbe the true label of the t-th image, and for brevity, let bCt← C t(rt) be the t-th prediction set\nover the labels. Then, err tis the indicator function of miscoverage at time t:\nerrt:=1h\nYt/∈bCti\n,\nwhere err t= 1 if the prediction set bCtdoes not include the true label Yt.\n•Local Coverage Over any sliding window, k, the local coverage is defined as:\nLocal Coverage( t):=1\nkt+k−1X\ni=t(1−erri).\nFor all trials, we used an interval length of k= 100.\n•Average Coverage The average coverage is similarly defined, but averaged over the total time steps T.\nFor all experiments, T= 6011.\nAvg.Coverage :=1\nTTX\ni=0(1−erri).\n•Local Width The local width is the cardinality of bCt, averaged over the length ktime window:\nLocal Width( t):=1\nkt+k−1X\ni=t\f\f\fbCt\f\f\f.\n33It is compared to the “best fixed” local width defined as follows. If we let C∗\nt← C t(r∗\nt) denote the\noptimal prediction set had we known the optimal radius r∗\ntbeforehand, then the best fixed local width\nLocal Width∗(t) is defined as the 1 −αquantile of {|C∗\ni|;i∈[t:t+k−1]}.\n•Average Width Similarly, the average width is defined as:\nAvg.Width :=1\nTTX\nt=1\f\f\fbCt\f\f\f.\n•Local Coverage Error (LCE) The LCE over the sliding window of length kis defined as:\nLCE k:= max\nτ,τ+k−1⊆[1,T]\f\f\f\f\fα−1\nkτ+k−1X\nt=τerrt\f\f\f\f\f.\nEssentially, it means the largest deviation of the empirical miscoverage rate (evaluated over sliding time\nwindows of length k) from the targeted miscoverage rate α.\nMain results Our main results are shown in Figure 1 and Table 1 (Section 5). The purpose of Figure 1 is to\ndemonstrate the dependence of the local coverage and the local width on (1) sudden and (2) gradual distribution\nshifts (i.e., the time-varying corruption level).15The purpose of Table 1 is to summarize the performance of our\nalgorithms and compare them to the baselines; the results there correspond to the case of sudden distribution\nshift.\n0.80.9Local Coverage\nLocal Coverage\n100200Local Width\nLocal Width\n0 1000 2000 3000 4000 5000 6000\nTime05Corruption Level 0 1000 2000 3000 4000 5000 6000\nTimeCorruption LevelBest Fixed MagL-D MagL MagDis\nFigure 1: The local coverage (first row), local width (second row), and corresponding corruption level (third row)\nof our algorithms. Results are obtained using corrupted versions of TinyImageNet, with time-varying corruption\nlevel (distribution shift). (Left) Results for sudden changes in corruption level. (Right) Results for gradual\nchanges in corruption level. The distribution shifts every 500 steps. Moving averages are plotted with a window\nsize of 100 time steps ( k= 100).\nFigure 1 justifies the validity of our algorithms. Specifically, the local coverage of our algorithms fluctuate\naround the target coverage of 0.9 for both sudden and gradual distribution shifts. Similarly, the local width\napproximately replicates the best fixed local width, Local Width∗(t), as shown in Figure 1 (middle row).\n15For better visibility, a 1D Gaussian filter is applied to the local width and the local coverage plots, same as in [BWXB23].\n34Hyperparameter sensitivity We also test the algorithms’ sensitivities to the offline estimate of D, which we\ndenote as Dest. The baselines Saocp andSf-Ogd require this parameter to initialize, while Simple OGD and\nall three of our algorithms do not. To this end, we rerun the experiments above and change the ratio of Destto\nthe true value D. The following settings are tested:\nDest\nD=\b10−3,10−2,10−1,100,101,102,103\t\n.\nWe plot the influence of Deston the average coverage and the average width in Figure 2. For these experiments,\nwe use the case when image corruptions are sudden. Coverage being much larger or much less than 0.9 is not a\ndesirable behavior; given satisfactory coverage, lower width is desirable.\n10−310−210−1100101102103\nDest/D0.20.40.60.8Average Coverage\n10−310−210−1100101102103\nDest/D050100150Average Width\nSimple OGD SF-OGD SAOCP MagL-D MagL MagDis\nFigure 2: The average coverage (first row) and average width (second row) as a function of the estimated\nmaximum radius, Destrelative to the true radius D. The performance of Sf-Ogd andSaocp [BWXB23 ] are\nsensitive to Dest/D. Averages are taken over the entire time horizon, where the total time steps T= 6011.\nAsDest/Dincreases, both the average coverage and average width increase for Saocp andSf-Ogd . In\nthe opposite direction, the average coverage and average width also decrease as Dest/Ddecrease for Saocp\nandSf-Ogd . When Dest=D, Simple OGD and Sf-Ogd have approximately equivalent performance. This is\nnot to say that using an offline estimate of Dest/Ddoes not improve performance. The catch is that, for the\nparticular experimental dataset, setting the learning rate to 1 in Simple OGD coincidentally just happens to be a\nwell-picked learning rate, given the true value of the maximum radius magnitude D. In contrast, our magnitude\nlearners remain fixed under variations in Dest/D, since they do not need Destto initialize. The baselines SCP,\nNExCP, Faci, and Faci-S are not presented in Figure 2 as to better highlight the performance of Saocp , OGD,\nand our magnitude learners. Benchmarks on these additional baselines can be found in [BWXB23].\nSimpleOGDMagDis SF-OGDMagLMagL-D\nSplitConformalFACI-S\nNExConformalFACISAOCP051015Runtime\n1.0 1.02 1.05 1.05 1.06 1.572.17 2.222.9811.07\nFigure 3: The runtime per time step of each algorithm, normalized to the runtime of Simple OGD . The runtime\nofSaocp is longest due to it being a meta-algorithm that initializes Sf-Ogd on each time step.\n35Runtime We also measure the time for each algorithm to complete a single prediction. Runtime results are\nprovided in Figure 3. The results are normalized relative to the runtime of Simple OGD. Note the runtime\nstandard deviations in Table 1. Accounting for standard deviations, the runtime differences between Simple OGD,\nSf-Ogd ,MagL , and MagDis are negligible. The runtime of Saocp is longest due to it being a meta-algorithm\nthat initializes Sf-Ogd on each time step, c.f., Appendix A.\n36"    },    {        "title": "2402.02722v1.Dilatonic_Geometrodynamics_of_a_Two_Dimensional_Curved_Surface_due_to_a_Quantum_Mechanically_Confined_Particle.pdf",        "content": "Dilatonic Geometrodynamics of a Two-Dimensional Curved Surface due to a\nQuantum Mechanically Confined Particle\nLeo Rodriguez,1, 2,∗Shanshan Rodriguez,1,†Zhenzhong Xing,3,‡and L. R. Ram-Mohan2, 4,§\n1Department of Physics, Grinnell College, Grinnell, IA 50112\n2Department of Physics, Worcester Polytechnic Institute, Worcester, MA 01609\n3Department of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14850\n4Departments of Electrical & Computer Engineering and Mechanical Engineering,\nWorcester Polytechnic Institute, Worcester, MA 01609\n(Dated: February 6, 2024)\nWe provide a unique and novel extension of da Costa’s calculation of a quantum mechanically\nconstrained particle by analyzing the perturbative back reaction of the quantum confined particle’s\neigenstates and spectra upon the geometry of the curved surface itself. We do this by first for-\nmulating a two dimensional action principle of the quantum constrained particle, which upon wave\nfunction variation reproduces Schr¨ odinger’s equation including da Costa’s surface curvature induced\npotentials. Given this action principle, we vary its functional with respect to the embedded two\ndimensional inverse-metric to obtain the respective geometrodynamical Einstein equation. We solve\nthis resulting Einstein equation perturbatively by first solving the da Costa’s Schr¨ odinger equation\nto obtain an initial eigensystem, which is used as initial-input data for a perturbed metric inserted\ninto the derived Einstein equation. As a proof of concept, we perform this calculation on a two-\nsphere and show its first iterative perturbed shape. We also include the back reaction of a constant\nexternal magnetic field in a separate calculation. The geometrodynamical analysis is performed\nwithin a two dimensional dilation gravity analog, due to several computational advantages.\nPACS numbers: 11.25.Hf, 04.60.-m, 04.70.-s\nI. INTRODUCTION\nSince da Costa’s, Jensen and Koppe’s seminal inde-\npendent calculations1,2demonstrating the emergence of\ncurvature induced surface potentials within Schr¨ odinger’s\nequation, the importance of differential geometric consid-\nerations in quantum semiconductor systems has spawned\na wide plethora of exciting research areas. These\ncurvature-induced surface potentials arise while expand-\ning and separating the Laplace-Beltrami operator over\nR3and confining a quantum particle to an embedded\ncurved two dimensional surface M2. Taking M2to the\nnanoscale opens a wide area of curvature induced quan-\ntum phenomena ranging from curved-spintronics, geo-\nmetric nonlinear Hall effects, curvilinear magnetism, to\nstrain-induced geometric potentials, to name a few3. Ad-\nditionally, the curvature of the nano surfaces, such as\ngraphene nanotori, Buckyballs, diffusive ferromagnetic\nnanowires, etc., feeds back onto the quantum electronic\nmaterial properties leading to an optimization and or\ntuning question/problem4–9.\nNow, since the da Costa paradigm (which generalizes\nJensen and Koppe’s work to arbitrary M2) necessitates\ndifferential geometric considerations in quantum systems,\nit sets the stage for holographic, analog-gravity, worm-\nhole and black hole dual description extensions for two di-\nmensional nanoscale semiconductors and materials10–18.\nThis stringy-pathway also provides a unique, novel and\npreviously unexplored way to address an important afore-\nmentioned question about geometric shape optimization\nfor curved two dimensional semiconductor devices3. This\nis the premise of our work in this manuscript.The important question of shape optimization and per-\nturbative shape evolution of semiconductor devices due\nto external fields and/or quantum mechanical processes\non their surfaces has been of great interest in the above\nmentioned and many other related research fields19–25.\nThe specific proposed premise of this paper is to think\nabout confining a quantum mechanical particle to the two\ndimensional surface of a curved semiconductor or nano\ndevice (such as a Buckyball/carbon-nanotube, etc.) and\nanalyze how the particles’ resulting eigensystem feeds\nback onto the geometry of the curved surface. In other\nwords, is there an optimized nano-geometry for a given\nquantum particle to be confined to, which would repre-\nsent the total system’s lowest eigenenergy configuration\nand therefore minimize any externally required work for\nconfinement. Knowledge of such optimized geometries\nwould then have significant application in device design\nand manufacturing.\nOur preliminary results demonstrate there is a quan-\ntum confined eigensystem feedback which generates geo-\nmetric deformations (a geometric strain of some kind) of\nthe original chosen two dimensional semiconductor sur-\nface. It thus becomes an important task to compute\nthese deformations in order to discover, or perturbatively\nnarrow down, the ideal optimized geometric shape for\nthe two dimensional curved surface. Additionally, in the\npresence of external fields, such as electromagnetic fields,\nthe geometric deformations on the semiconductor attain\nadditional complications with rich physics. The full ge-\nometrodynamics of this rich physical scenario turns out\nto be fully captured by a dual two dimensional dilaton\ngravity, nearly identical to the two dimensional quantumarXiv:2402.02722v1  [hep-th]  5 Feb 20242\ngravities that arise in the near horizon of four dimen-\nsional classical black holes.\nOur starting point is the seminal work of da Costa1,\nwho showed that for a particle quantum mechanically\nconfined to a two dimensional curved surface embedded\ninR3, the pullback of Schr¨ odinger’s equation results in\nthe appearance of surface potentials that depend on the\ntwo dimensional curved surface’s Ricci scalar and also\nthe extrinsic curvature. Thus, motivated by the finite ele-\nment method, formulating a two dimensional action prin-\nciple of the quantum constrained particle, which upon\nwave function variation reproduces Schr¨ odinger’s equa-\ntion according to da Costa, provides us with a founda-\ntional starting point to a full geometrodynamical descrip-\ntion of this physical scenario.\nLet⃗Rbe the position vector of any point Q∈R3,\nsuch that ⃗R=⃗R(x, y, z ). Next, ⃗ r=⃗ r(θ, ϕ) is the lo-\ncus of points of a curved two dimensional surface M2,\nembedded in R3. To reach QfromM2we choose a per-\npendicular distance ρfromM2toQ. Thus ⃗R=⃗ r+ρˆn,\nwhere ˆ nis the unit normal to M2. We will define the\nthree dimensional Euclidean metric in the usual way:\ngµν=\u0010\n∂µ⃗R\u0011\n·\u0010\n∂ν⃗R\u0011\n, (1)\nand the metric on M2(first fundamental form) given by:\nhab= (∂a⃗ r)·(∂b⃗ r). (2)\nHere, greek indices range over the three cartesian coor-\ndinates and latin indices range over the two embedded\ncurved coordinates. We also define the extrinsic curva-\nture (second fundamental form) given by:\nkab= (∂a⃗ r)·(∂bˆn), (3)\nwhich depends on how ⃗ ris embedded in R3and informs\nabout the rate of change of ˆ nas it is parallel transported\nacross M2.\nNow, given the above definitions and quoting results\nfrom [1], the determinant g= det ( gµν) can be shown to\ntake the form:\ng=h\u0000\n1 +ρK+ρ2R/2\u00012, (4)\nwhere his the determinant of hab,Kis the trace of the\nextrinsic curvature and Ris the Ricci scalar curvature\n(twice the Gaussian curvature) of M2. Now consider-\ning this result and the three dimensional steady state\nSchr¨ odinger equation:\n−ℏ2\n2m1√g∂µ(√ggµν∂νψ) +V0˜δ(ρ)ψ=Eψ, (5)\nwhere ˜δ(r) is an anti-Dirac delta-function such that\nV(ρ) =V0˜δ(ρ) confines the particle to the two dimen-\nsional surface, i.e.\n˜δ(ρ) =(\n0ρ= 0\n∞ρ̸= 0, (6)we are able to separate the three dimensional equation\n(5) into two equations via the ansatz ψ=ψ2D(θ, ϕ)R(ρ)\nyielding1:\n−ℏ2\n2m1√\nh∂a\u0010√\nhhab∂bψ2D\u0011\n+\n−ℏ2\n2mn\n(K/2)2− R/2o\nψ2D=E2Dψ2D (7)\n−ℏ2\n2mR′′+V0˜δ(ρ)R=EρR. (8)\nII. TWO DIMENSIONAL GRAVITY DUALITY\nThe main computational tool we implement to solve\nderived and given field equations is rooted in the finite\nelement analysis within the framework of the action in-\ntegral principle26. In this paradigm, device geometry is\ndiscretized into refined finite element meshes. Within\neach element, potential functions are expressed as linear\ncombination of Hermite interpolation polynomials mul-\ntiplied by as-yet undetermined coefficients. Employing\nthe principle of stationary action, variation of the action\nintegral with respect to these undetermined coefficients\nresults in a set of linear equations, which are then solved\nusing sparse matrix solvers.\nThus, our first task is to construct an action whose\nequation of motion is given by (7) after functional vari-\nation with respect to ψ2D. Assuming, without loss of\ngenerality, that ψ2Dis real as this does not affect the fi-\nnal version of Schr¨ odinger’s equation in (7), we have the\nobvious choice given by:\nS=ℏ2\n4mZ\nd2x√\nh\nn\n∂aψ2D∂aψ2D+h\nR/2−(K/2)2i\nψ2\n2D−ϵψ2\n2Do\n,\n(9)\nwhere ϵis the scaled dimensionless energy E2D. The\nabove action can now be employed via the finite element\nanalysis method to solve the Schr¨ odinger equation (7)\nin order to compute eigenstates and eigenvalues for a\nplethora of differing curved two dimensional geometries.\nHowever, what is previously unexplored and one of the\nmajor research insights of this paper, is that the above\naction can be utilized to explore shape optimization of\nthe two dimensional embedded semiconductor surface via\na metric variation with respect to hab. In other words,\nwe are the first to formulate a full geometrodynamical\ndescription of the back reaction of the confined quantum\nparticle’s eigensystem onto the embedded curved surface,\nby computing the respective Einstein equation of the ac-\ntion given above in (9).\nThe respective Einstein equation of (9) will be given\nby the variation with respect to the embedded curved\nsurface’s inverse metric (for canonical reasons) hab. How-\never, before performing this variation, it is a very useful3\ntask to rescale the wave function in terms of a dilaton\nfield φ, given by ψ=e−φ. This rescaling improves and\nsimplifies the very lengthy calculation of the respective\nEinstein equation. Given this rescaling, the action (9)\ntakes the form:\nS=ℏ2\n4mZ\nd2x√\nhe−2φ\b\n∂aφ∂aφ+R/2− K2/4−ϵ\t\n.\n(10)\nAt this point we should pause and comment on how con-\nveniently and surprisingly the above action for a quan-\ntum particle confined to a two dimensional curved surface\nis analogously described by a two dimensional dilaton\ngravity, which traditionally arises when describing hori-\nzon degrees of freedom of four dimensional black holes.\nFor comparison, the two dimensional theory that arises\nin the near horizon of a charged spherically symmetric\n(Reissner-N¨ ordstrom) black hole is given by:\nSRN\n2D=\nℓ2\n4GZ\nd2x√−ge−2φ\u001a\n2∂aφ∂aφ+R −GF2+2e2φ\nℓ2\u001b\n,\n(11)\nwhere Gis Newton’s gravitational constant, ℓis the two\ndimensional AdS radius and Fis the two dimensionally\nreduced electromagnetic curvature two form.\nWhen comparing the above two actions, (10) and (11),\nan immediate duality is evident which we summarize in\nthe ‘da Costa/2D dilaton Gravity’ dictionary of Table I.\nNext, given the above duality we proceed to calculate the\ngeometrodynamical Einstein equation of (10) via varia-\ntion with respect to the inverse two dimensionally curved\ninverse metric hab. This derivation of the Einstein equa-\ntion is in general non-trivial, mainly due to the overall\ne−2φdilatonic weight in the action. This weight in con-\njunction with Hamilton’s principle does not allow us to\ndrop Ricci tensor variations as boundary terms, as one\nwould in the Einstein-Hilbert action principle of four di-\nmensional general relativity. In stead, terms of the order\nδRabcontribute to the final Euler-Lagrange equations of\nmotion. Specifically, our Einstein equation of motion re-\nlies on the following properties of the variation of the\nRiemann tensor and Levi-Civita connection in any di-\nmension:\nδRa\nbmn=∇mδΓa\nbn− ∇ nδΓa\nbm\nδΓa\nmn=1\n2\u0000\n−hdn∇mδhad−hdm∇nδhad\n+hsmhdn∇aδhsd\u0001\n.(12)\nAdditionally we note that the Einstein tensor:\nRab−1\n2gabR= 0 (13)\nis identically zero for any Riemann surface in two dimen-\nsions, thus all embedded hab’s are Einstein manifolds.After very extensive and lengthy calculation, we obtain\nthe following final result for the variation of (10) with\nrespect to hab:\n\u001a\n−1\n2hab\u0012\n∂cφ∂cφ−K2\n4−ϵ\u0013\n+∂aφ∂bφ−1\n2Kkab\u001b\ne−2φ\n+1\n2\u0000\nhab□e−2φ− ∇ a∇be−2φ\u0001\n= 0.\n(14)\nThe above can be simplified without loss of generality in\ntwo dimensions by tracing and thus yielding one single\ngeometrodynamical Einstein equation, given by:\n−K2/4 +ϵ+ 2∇aφ∇aφ−□φ= 0. (15)\nThis result (15) in conjunction with (7) provides a full\nquantum mechanical and geometrodynamical description\nof the da Costa’s quantum confined particle.\nIII. GEOMETRIC OPTIMIZATION: A SINGLE\nITERATIVE PERTURBATIVE SOLUTION\nAs proposed above we are interested in using (7) and\n(15) as a powerful tool for the study of shape optimiza-\ntion in this particular physical scenario. A full itera-\ntive perturbative approach using finite element analysis\nand including external fields, is outlined in detail in the\nflowchart depicted in Fig. 1. However as a proof of con-\nStart by Choos-\ning a 2D Met-\nric Ansatz hab\nPerturb the Ge-\nometry hPert\nab =\nhab+fab(λ, θ, ϕ )Solve (7) Us-\ning FEM\nObtain Initial\nUnperturbed\nEigensystem ( φ0\nn, ϵ0\nn)\nGoal 1: Solve for\nMetric Perturbation\nfab(λ, θ, ϕ ) by Feed-\ninghPert\nab,(φ0\nn, ϵ0\nn)\ninto (15)\nDetermine the Full\nPerturbed Geometry\nhPert\naband Treat\nas a New AnsatzGoal 3: Add\nExternal\nE&M Fields\nDetermine\nInduced Surface\nPotential if\nGoal 3 addedGoal 2: Reevaluate\nuntil a Converging\nOptimum Shape\nFIG. 1: Flowchart of an iterative perturbative shape\noptimization using (7) and (15), assuming convergence.\ncept and for the sake of analyticity, our initial approach\nhere will follow a first iterative perturbative outline given\nby:\n•First we select a specific two dimensional curved\nsurface. Surfaces that are of interest in this area of4\nTABLE I: Dictionary of da Costa’s 2D quantum confined objects (quantities) and their corresponding duals in 2D\ndilaton gravity.\nda Costa’s 2D Curved Confined Quantum Particle Two Dimensional Near-Horizon Black Hole Dilaton Gravity\nℓ two dimensional AdS radius ℏ Planck’s constant\nm confined particle mass G Newton’s Gravitational Constant\nφ logarithm of the quantum wave function φ gravitational dilaton scale of the S2factor\nhab(h) Euclidian 2D curved embedded metric gab(−g)s-wave near-horizon dimensionally reduced black hole\nR semiconductor surface Ricci scalar curvature R near-horizon 2D spacetime Ricci scalar curvature\nK2/4 modulo semiconductor surface intrinsic curvature squared GF2modulo internal Maxwell tensor squared\nϵ unitless energy eigenvalue2e2φ\nℓ2 cosmological term\nstudy include the two sphere and the two torus27–29\namong others. We will chose the two sphere for\nanalytical reasons.\n•We then proceed to solve (7) thereby obtaining an\ninitial eigensystem, i.e.\u0000\nφ0\nn, ϵ0\nn\u0001\n. As aforemen-\ntioned for the two sphere, the initial eigensystem\nis obtained analytically and given by spherical har-\nmonics.•Next, we introduce a perturbation to the initially\nchosen geometry. For example, choosing a two\nsphere with metric given by:\nhab= (∂a⃗ r)·(∂b⃗ r) =\u00121 0\n0 sin2θ\u0013\n(16)\na simple and general perturbation, to O(λ3), in the\nradial direction is given by:\nhpert\nab=\u0012\n(α+λF)2+λ2(∂θF)2λ2(∂θF) (∂ϕF)\nλ2(∂θF) (∂ϕF) ( α+λF)2sin2θ+λ2(∂ϕF)2\u0013\n. (17)\nfor some small perturbation parameter λ, arbitrary\nscale parameter αand unknown function F(θ, ϕ).\n•The next step is to take\u0000\nφ0\nn, ϵ0\nn\u0001\nandhpert\naband in-\nsert them into the traced Einstein equation (15)\nfor a given nvalue, thereby obtaining a partial dif-\nferential equation for F(θ, ϕ) and foraging for an\nanalytical solution for the given perturbation.\n•We then end by demonstrating a plot of the this\nfirst iterative perturbative shape optimization of\nM2, due to a quantum confined particle.\nIn addition to the above outline and flow chart, a full\nsimultaneous numerical solution via the finite element\nanalysis paradigm of (7) and (15) is currently under in-\nvestigation by our research collaboration for near future\nforthcoming work, to provide an instant numerical geo-\nmetric optimization for M2due to quantum confinement.\nA. The s-wave Case\nAs mentioned above, choosing the two sphere as our\ninitial geometry for M2and setting ϵ=−l(l+ 1) in\n(9), results in spherical harmonics, Ym\nl(θ, ϕ) where n→\n(l, m), as our initial eigensystem. Feeding the above into\n(15) and solving in the s-wave scenario ( l=m= 0) andassuming only θdependence in the perturbation func-\ntionFin (17), yields the following analytical solution to\nO(λ2):\nF(θ) =α\u00141\n2+βcosθ\n+γ\u0012\n−1 + cos θ\u0012\n−1\n2ln (1−cosθ) +1\n2ln (1 + cos θ)\u0013\u0013\u0015\n,\n(18)\nwhere βandγare integration constants. The integra-\ntion constants should be dependent upon semiconductor\nmaterial properties, such as modulus, stress, strain, con-\nductive properties, etc. We have not included those type\nof contributions in our theoretical frame work as of yet,\nand thus, we will choose values for α,βandγthat pro-\nvide finite solutions to first iteration with respect to the\nflow chart in Fig. 1, that are scalable to a perturbed unit\nsphere for proof of concept.\nFor the s-wave case, we have identified three finite in-\nteresting cases depicted in Fig. 2-4 below. For each fig-\nure, we implemented the perturbation solution (18) and\nplotted the resulting perturbed two sphere. The three\ncases included values for α,βandγthat range between\n0 and 1. The specific choice of αis determined in order\nto compare our plots to that of a unit two sphere for vi-\nsualization purposes. The first case in Fig. 2, βandγ\nare positive and equal; for the second case in Fig. 3, β5\nandγare of opposing sign and equal in magnitude; and\nin the third case, Fig. 4, the values for βandγdiffer by\na factor of 10.\n(a)\n (b)\nFIG. 2: Perturbed geometry 2a due to a quantum\nconfined particle a lada Costa, and comparison to\ninitial unperturbed geometry (green) 2b. The values for\nβandγare positive, equal and bounded between 0 and\n1.\n(a)\n (b)\n(c)\nFIG. 3: Perturbed geometry 3a and its cross sectional\nplot 3b (displaying the interior) due to a quantum\nconfined particle a lada Costa, and comparison to\ninitial unperturbed geometry (green) 3c. The values for\nβandγare opposite in sign, equal in magnitude and\nbounded between 0 and 1.\nFrom the above mentioned figures it is apparent that\nthere is a geometrodynamical feedback from the quan-\ntum confined particle’s eigensystem onto the initially cho-\nsen geometry of M2. We have made specific choices\nto make the geometric deformations visible, but it is\nstill an ongoing investigation on tuning these parame-\nters to actual material science properties of specific nano-\nsemiconductors.\n(a)\n (b)\n(c)\nFIG. 4: Perturbed geometry 4a and its cross sectional\nplot 4b (displaying the interior) due to a quantum\nconfined particle a lada Costa, and comparison to\ninitial unperturbed geometry (green) 4c. The values for\nβandγare opposite in sign, differ by a factor of 10 in\nmagnitude and bounded between 0 and 1.\nB. The l= 1and m=±1Cases\nTwo of the l= 1 cases are another scenario where\nwe are able to obtain analytical solutions for the per-\nturbation function F, assuming only θdependence and\nfollowing the same procedures from the previous subsec-\ntion III A. For the m= 0 case, we were not able to find a\nclosed form analytical solution, but the other two cases\nm=±1, there exists one common and unique analyti-\ncal closed form solution in terms of Legendre polynomi-\nals (Pn(x)) and Legendre functions of the second kind\n(Qn(x)), given by:\nF(θ) =αh\n2 +α2+βP(−1+√\n33)/2(cosθ)\n+γQ(−1+√\n33)/2(cosθ)i\n.(19)\nFor this specific case, we display two sets of plots for\nchoices of the scale parameter αand integration con-\nstants βandγ. In the first plot, displayed in Fig. 5,\nthe integration constants are equal and between 0 and 1.\nVarying the constants in sign only affects the direction\nof the polar peaks along the z-axis. The overall pumpkin\nshape seems universally unaffected. In Fig. 6 the integra-6\ntion constants differ by a factor of 10, but still sampled\nbetween 0 and 1. Both perturbed geometries are com-\npared to a green unit two sphere in the second plot of\neach case.\n(a)\n (b)\nFIG. 5: Perturbed geometry 5a due to a quantum\nconfined particle a lada Costa, and comparison to\ninitial unperturbed (green) geometry 5b. The values for\nβandγare positive, equal and bounded between 0 and\n1.\n(a)\n (b)\nFIG. 6: Perturbed geometry 6a due to a quantum\nconfined particle a lada Costa, and comparison to\ninitial unperturbed geometry 6b. The values for βand\nγare positive, differ by a factor of 10 and bounded\nbetween 0 and 1.\nIV. EXTERNAL FIELDS\nFinally and as proposed above, we will detail and out-\nline how to incorporate the presence of external electro-\nmagnetic fields in our formalism of geometric shape op-\ntimization. The presence of external fields is very impor-\ntant, since they can be used to tune desired outcomes and\nor appear naturally due to electronic devices/designs.\nHowever, our unique dual description allows us to un-\nveil new phenomena and rich physics, that has to-date\neluded the majority of researchers in this field. This ismainly due to our action integral approach, motivated\nfrom the finite element analysis paradigm, and resulting\ngravity analog detailed in Table I. To begin, let us start\nby outlining a simple example of a unit two sphere cen-\ntered at the origin with an external constant magnetic\nfield along the z-axis, i.e.:\nhab=\u00121 0\n0 sin2θ\u0013\nand\n⃗B=B0ˆk.(20)\nOne choice for a three vector potential whose curl yields\nthe above magnetic field is given by:\n⃗A=−B0yˆı. (21)\nNow, to incorporate this external magnetic field in da\nCosta’s (7), we need to first augment the operator ∂aby\nthe pullback of ⃗Agiven by Aa=∂a⃗ r·⃗Aand thus defining\nthe gauge covariant derivative, in the usual way, on the\ntwo dimensional curved surface: Da=∇a−ieAa.\nThis is standard practice in the respective research\nfield, but what has not been addressed before (until now\nin this work), is the fact that pulling down the three vec-\ntor⃗Ato the two dimensional curved surface also induces\na circulating surface magnetic field sourced by antipar-\nallel surface currents, projected and distributed nontriv-\nially on the respective embedded curved surface. Let us\ndemonstrate; first, the pullback of the three vector po-\ntential in the above simple example gives rise to the em-\nbedded electromagnetic field tensor (curvature two form)\ngiven by:\nFab=∂aAb−∂bAa=\u0012\n0−Bθ\nBθ0\u0013\n=\u0012\n0 −B0cosθsinθ\nB0cosθsinθ 0\u0013 (22)\nwhich (keeping in mind that Fabis purely spacial) re-\nveals the induced embedded two dimensional specially\ndynamic magnetic field. The source of this induced mag-\nnetic field is a current density nontrivially distributed\nacross the curved two dimensional surface, depicted in\nFig. 8, which can be determined by the Maxwell equa-\ntion:\n∇bFab=−Ja, (23)\nwhere ∇ais the Levi-Civita covariant derivative and\nJa=\u0000\nJθ, Jϕ\u0001\nis the conserved U(1) N¨ other current den-\nsity on the curved embedded surface.\nA simple calculation following from the above yields\nthe surface current density:\nJa=\u0000\n0, B0\u0000\n2 cos2θ−sinθ\u0001\u0001\n, (24)\nwhose normalized plot is given in Fig 7. We see that\nin the presence of a constant external magnetic field, the\nquantum confined particle (to a two dimensional curved7\n0 1 2 3−1012Induced Current Density, Jϕ\nθin radians\nFIG. 7: Plot of the induced surface current density Jϕ\nover the range of θ. We see that the charge density has\nits maximum at the poles where the external magnetic\nfield pierces the two sphere, it has a negative value on\nthe equator and is zero at ∼π/3.3 and ∼π+π/3.3.\nFIG. 8: Color-coded surface density plot of Jϕover the\ntwo sphere.\nsurface) experiences both attractive and repulsive dy-\nnamical Lorenz force interactions, which will feed back\nupon the geometrodynamics of the curved surface. To\nincorporate these new features, the action principle of\n(10) needs to be augmented in the following form:\nS=ℏ2\n2mZ\nd2x√\nh\ne−(φ∗+φ)\u001a\nD∗\naφ∗Daφ+R/2− K2/4−ϵ−1\n4ΦF2\u001b\n,\n(25)\nwhere we now consider a complex dilaton, Φ =\n2me(φ∗+φ)/ℏ2andF2=FabFab. This necessary aug-mentation will alter the resulting geometrodynamical\nEinstein equation of motion (15), given the U(1) energy\nmomentum tensor in terms of its functional generator:\n−2√\nhδSU(1)\nδhab≡TU(1)\nab=−1\n4habF2+hcdFacFbd,(26)\nand adds additional new features to the shape optimiza-\ntion procedures outlined in previous sections. Addition-\nally, the variation of the above total action (25) with\nrespect to the pullback field Aawill give rise to an addi-\ntional equation of motion or constraint, given the func-\ntional generator equation for U(1) symmetry:\nδS\nδAa≡Ja. (27)\nThis is in part due to the gauge covariant derivative Da\nexhibiting Aadependence and where\nΦJa=ie(φ∗∇aφ−φ∇aφ∗) + 2e2φ∗Aaφ (28)\nis the previously encountered conserved induced surface\nN¨ other current in terms of the complex dilaton. This\nnew feature, which is also missing in previous research\nendeavors, will help and allow for more numerical con-\nstraints in the shape optimization endeavor. To summa-\nrize without loss of generality, let us consider the simpli-\nfied case for a real dilaton φ∗=φ. In this case the total\ngeometrodynamical shape optimization problem reduces\nto the simultaneous solution of the following equations of\nmotion:\n\n\n−ℏ2\n2m1√\nh∂a\u0010√\nhhab∂bψ2D\u0011\n+\n−ℏ2\n2mn\n(K/2)2− R/2o\nψ2D=E2Dψ2D daCSEq\n−K2/4 +ϵ+ 2∇aφ∇aφ−□φ+m\n4ℏ2F2= 0 EEq\nand∇bFab=−JaMEq,\n(29)\nwhich are the da Costa Schr¨ odinger equation with in-\nduces curvature surface potentials, the Einstein equa-\ntion and the pulled back induced Maxwell equation. We\nshould note here that despite starting this section with a\nsimplistic example, the above shape optimization equa-\ntions are not specific to any given example of a quantum\nconfined particle within a given external field configura-\ntion, but instead it is a recipe for any general curved\ntwo dimensional surface in any arbitrary external field\nconfiguration.\nV. CONCLUSION\nThe seminal work and contribution of R. C. T. da\nCosta cannot be credited enough and overstated in initi-\nating a near revolution before their time. This pioneer-\ning work combined the arts of differential geometry and\nquantum mechanics before the advent of string theory,8\nstring theory’s AdS/CFT correspondence, AdS/CMT\ncorrespondence and the general gauge gravity correspon-\ndence. In fact, the majority of research explosion based\nupon this paradigm has spawned only in the last decade,\nas discussed in the introduction.\nHowever, the majority or even entirety of this explo-\nsion revolves around carbon copies of solving, numerically\nor analytical limiting cases of the da Costa Schr¨ odinger\nequation in (7) only. What we have done in this article\nis formulate the first novel extension by showing how to\nimplement the da Costa paradigm as a basis for a ge-\nometrodynamical shape optimization of the two dimen-\nsional curved surfaces with an initial quantum confined\neigensystem, a fundamental question and problem in the\ncurrent semiconductor (device manufacturing) field of\nresearch3.\nIn this work, we have performed an initial proof of con-\ncept perturbative calculation of how the two dimensional\ncurved geometry would be deformed given the feedback-\nwork exhibited by the initial confined eigensystem. Addi-\ntionally, we have laid the basis ground work formulation\nfor how to extend our work within a full numerical iter-\native solution approach with any given external electro-\nmagnetic field configuration. In doing so, we have relied\non the finite element analysis method rooted within a\nstationary action principle.\nSome theoretical considerations and comments are\ndue. First, the action in (9) is not a classical one, but\nshould be interpreted as a quantum effective one, since\nthe variation of (9) with respect to the wave function\nψyields the quantum mechanical Schr¨ odinger equation.\nIn other words, (9) is the effective action defend in the\ncanonical way:\nS= Γeff(ψ) =−iln (Z), (30)\nwhere\nZ=Z\nDxeiScl(x)(31)\nis the partition function of the classical action, Scl(x).\nNow, this implies that the Einstein equation in (14) is aquantum mechanical energy momentum tensor:\n⟨Tab⟩=\u001a\n−1\n2hab\u0012\n∂cφ∂cφ−K2\n4−ϵ\u0013\n+∂aφ∂bφ+\n−1\n2Kkab\u001b\ne−2φ+1\n2\u0000\nhab□e−2φ− ∇ a∇be−2φ\u0001,\n(32)\nsince\n⟨Tab⟩=Z\nDx(Tab)eiScl(x)=Z\nDx\u0012−2√\nhδ\nδhab\u0013\neiScl(x)\n=−2√\nhδ\nδhabZ\nDxeiScl(x)=−2√\nhδ\nδhabZ\n=iZ−2√\nhδ\nδhabΓeff(ψ),\n(33)\nwhere we have used the functional generator of Taband\nthe wave function ψacts like an auxiliary field for the\nsake of locality restoration. The above yields the equa-\ntion of motion in (14) modulo an arbitrary normalization\nofiZ. Additionally, ⟨Tab⟩above is a renormalized energy\nmomentum tensor (EMT), since it satisfies a Bianchi\nidentity. However, what is an important question for\nthe semiconductor and or materials science researchers\nis, what combination of the above renormalized stress\ntensor is conserved and or not. In other words, what\ncombination of the above EMT contributes to the two\ndimensional gravitation anomaly cancelation. Dissecting\nthis detail in (14) would be a first step towards isolat-\ning and incorporation materialistic properties of the two\ndimensional curved surface into the shape optimization\nprocedure. This is a matter for future research and work.\nACKNOWLEDGMENTS\nLR would like to thank the Grinnell College Harris\nFellowship Foundation for supporting this work and also\nWPI for their hospitality where this work was initially\nconceived. LR and SR would like to thank Vincent\nRodgers for support, encouragement and very enlight-\nening discussions. LRR thanks PK Aravind for enlight-\nening discussions.\n∗rodriguezl@grinnell.edu\n†rodriguezs@grinnell.edu\n‡zx296@cornell.edu\n§lrram@wpi.edu\n1R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).\n2H. Jensen and H. Koppe, Annals of Physics 63, 586 (1971).\n3P. Gentile, M. Cuoco, O. M. Volkov, Z.-J. Ying, I. J. Vera-\nMarun, D. Makarov, and C. Ortix, Nature Electronics 5,\n551 (2022).\n4T. Salamone, H. G. Hugdal, M. Amundsen, and S. H.Jacobsen, Phys. Rev. B 105, 134511 (2022).\n5V. Atanasov and A. Saxena, Phys. Rev. B 81, 205409\n(2010).\n6A. Iorio and G. Lambiase, Phys. Rev. D 90, 025006 (2014).\n7J. D. M. de Lima, E. Gomes, F. F. da Silva Filho,\nF. Moraes, and R. Teixeira, The European Physical Jour-\nnal Plus 136, 551 (2021).\n8J. Gomes Silva, J. Furtado, and A. C. A. Ramos, The\nEuropean Physical Journal B 93, 225 (2020).\n9Y. Ma, B. Ding, Y. Chen, and D. Wen, International9\nJournal of Mechanical Sciences 232, 107621 (2022).\n10A. G. Schmidt and M. E. Pereira, Annals of Physics 458,\n169465 (2023).\n11V. Atanasov, Physics Letters A 384, 126042 (2020).\n12A. Duenas-Vidal and J. Segovia, Phys. Rev. D 108, 086006\n(2023).\n13G. Alencar, V. B. Bezerra, and C. R. Muniz, The Euro-\npean Physical Journal C 81, 924 (2021).\n14T. F. de Souza, A. C. A. Ramos, R. N. Costa Filho, and\nJ. Furtado, Phys. Rev. B 106, 165426 (2022).\n15D. Biswas and S. Ghosh, EPL 132, 10004 (2020),\narXiv:1908.06423 [hep-th].\n16R. N. Costa Filho, S. Oliveira, V. Aguiar, and D. da Costa,\nPhysica E: Low-dimensional Systems and Nanostructures\n129, 114639 (2021).\n17Z. Zali and J. Sadeghi, International Journal of Geo-\nmetric Methods in Modern Physics 17, 2050135 (2020),\nhttps://doi.org/10.1142/S0219887820501352.\n18L. Meschede, B. Schwager, D. Schulz, and J. Berakdar,\nPhys. Rev. A 107, 062806 (2023).\n19L. Rodriguez, S. Rodriguez, S. Bharadwaj, and\nL. R. Ram-Mohan, Phys. Rev. B 101, 174417 (2020),\narXiv:1912.05780 [hep-th].20S. A. Solin, T. Thio, D. R. Hines, and J. J. Heremans,\nScience 289, 1530 (2000).\n21J. Moussa, L. R. Ram-Mohan, A. C. H. Rowe, and S. A.\nSolin, Journal of Applied Physics 94, 1110 (2003).\n22L. M. Pugsley, L. R. Ram-Mohan, and S. A. Solin, Journal\nof Applied Physics 113, 064505 (2013).\n23J. Moussa, L. R. Ram-Mohan, J. Sullivan, T. Zhou, D. R.\nHines, and S. A. Solin, Phys. Rev. B 64, 184410 (2001).\n24M. Otomori, T. Yamada, K. Izui, and S. Nishiwaki, Struc-\ntural and Multidisciplinary Optimization 51, 1159 (2015).\n25T. Yamada, K. Izui, S. Nishiwaki, and A. Takezawa, Com-\nputer Methods in Applied Mechanics and Engineering 199,\n2876 (2010).\n26L. R. Ram-Mohan, Finite Element and Boundary Element\nApplications in Quantum Mechanics (Oxford, 2002).\n27L. C. da Silva, C. C. Bastos, and F. G. Ribeiro, Annals of\nPhysics 379, 13 (2017).\n28G. Ferrari and G. Cuoghi, Phys. Rev. Lett. 100, 230403\n(2008).\n29A. G. Schmidt, Physica E: Low-dimensional Systems and\nNanostructures 106, 200 (2019)."    },    {        "title": "2402.02725v2.Cybersickness_Detection_through_Head_Movement_Patterns__A_Promising_Approach.pdf",        "content": "*These authors contributed equally to this work.  Cybersickness Det ection through Head Movement   \nPatterns : A Promising Approach  \nMasoud Salehi1*, Nikoo Javadpour1*, Brietta Beisner1, Mohammadamin Sanaei1, \nStephen B. Gilbert1 \n \n1 Iowa State University, Ames IA 50011, USA  \n \nAbstract. Despite the widespread adoption of Virtual Reality (VR) technol-\nogy, cybersickness  remains a barrier for some users. This research investigates \nhead movement patterns as a novel physiological marker for  cybersickness de-\ntection. Unlike traditional markers, head movements provide a continuous, non -\ninvasive measure that can be easily captured through the sensors embedded in all \ncommercial VR headsets.  \nWe used a publicly available dataset from a VR experime nt involving 75 par-\nticipants and analyzed head movements across six axes (up/down, left/right, for-\nward/backward, and rotational movements). An extensive feature extraction pro-\ncess was then performed on the head movement dataset and its derivatives, in-\ncludi ng velocity, acceleration, and jerk. Three categories of features were ex-\ntracted, encompassing statistical, temporal, and spectral features. Subsequently, \nwe employed the Recursive Feature Elimination method to select the most im-\nportant and effective featu res.  \nIn a series of experiments, we trained a variety of machine learning algo-\nrithms. The results demonstrate a 76% accuracy and 83% precision in predicting \ncybersickness in the subjects based on the head movements. This study’s contri-\nbution to the cybers ickness literature lies in offering a preliminary analysis of a \nnew source of data and providing insight into the relationship of head movements \nand cybersickness.  \nKeywords:  Cybersickness, Machine Learning, Postural Sway, Head Move-\nments, Windowing, Fourier  Transform, Wavelet Transform, Recursive Feature \nElimination, Time Series  \n \n1 Introduction  \nIn recent years, virtual reality (VR) has gained popularity as a technology that can pro-\nvide users with immersive, interactive experiences. Various fields, such as ent ertain-\nment [1], education [2], training  [3], and healthcare [4], have found applications in vir-\ntual reality due to its ability to i simulate realistic environments and provide a sense of \npresence. Despite the benefits of VR, it has some limitations incl uding the experience \nof cybersickness, a form of motion sickness, that may occur in virtual environments [5, 2  Masoud Salehi, Nikoo  Javadpour, et al.  \n \nMasoud Salehi  | DATASPECT IT -SERVICE S, NECKARGEMUEND, GE RMANY  6]. Cybersickness prevents people from widely embracing and enjoying VR experi-\nences due to symptoms such as nausea, dizziness, disorientation, and fatigue [7]. Addi-\ntionally, research has shown differences between VR and face -to-face (F2F) conditions \nacross various parameters, highlighting the importance of understanding these distinc-\ntions in optimizing user experiences [8, 9, 10, 11]. \nCybersickness can be caused by a variety of factors, including those from hardw are \n[12], software  [13,14], users’ individual differences, and the user’s task [5, 15, 16]. \nWhile the exact mechanism is not yet fully modelled, it includes factors such cue con-\nflict (when the sensory cues received by the brain from the virtual environment contra-\ndict the cues from the body's own sens es) [17] and postural instability (how well a \nperson can maintain balance) [1, 18]. This present research focuses on a specific form \nof postural sway which is head movements. Research shows that individuals  who ex-\nperience dizziness and patients with a vestibular deficiency show different patterns of \nhead movements  [19, 33]. Therefore, in this research we aim to use the head movements \nsignals as a potential identifier for cybersickness.  Head movements data can be a reli-\nable, cheap, and easy to access source of data coming from accelerometer and gyro-\nscope sensors embedded in a ll commercial VR headsets.  \nSeveral researchers have explored methods of predicting cybersickness using ma-\nchine learning  [19], which has shown promising results. By identifying patterns and \nappropriate identifiers extracted from physiological data; machine learning models can \nbe trained to predict cybersickness [20, 21]. However, to the best of our knowledge, \nthere has not been a study that utilizes and processes head movements data to train a \nmachine learning model to make predictions for cybersickness.  \nThis study develops a predictive model that provides predictive capability for cyber-\nsickness symptoms. This approach use s head movements data and its derivatives: ve-\nlocity, acceleration, and jerk. Extensive feature extraction techniques are then applied \nto extract a wide range of statistical features, temporal features (time domain), and \nspectral features (frequency domain). To mitigate the problem of having a relatively \nsmall data sample, a feature selection technique called Recursive Feature Elimination \n[22] was u sed to choose only the most useful features in each experiment. A variety of \nmachine learning models were run based on these features to test against a hold -out test \nset. \nThe next section of this paper provides an overview of related work on machine \nlearni ng to identify cybersickness . The experimental methodology for the prepro-\ncessing, and extraction of features from this Polhemus dataset is described  in [23]. A \nPolhemus is a 3D tracking device worn by the participants of the study that captured \nthe dataset. The machine learning algorithms used for prediction are desc ribed, as well \nas their strengths and weaknesses. The paper then assesses the performance of the pre-\ndictive model and discusses the implications of our findings, potential applications, and \nfuture research directions.     \n Innovative Cybersickness Detection: Exploring Head Movement Patterns in Virtual Reality  \n2 Related Work  \n \nMany researchers have explored methods of measuring cybersickness. While SSQ [24] \nhas frequently been used as a subjective measure, self -reported by participants, re-\nsearchers have widely explored the possibility of objective physiological measures, in-\ncludi ng EEG [25] ; EDA  [26] ; heart rate, heart rate varia bility, blood pressure [27], \n[28], and respiration [27]. Unfortunately, results have been mixed. Since sensors for \nthese methods can be expensive a nd feel invasive, popular headsets are not equipped \nwith them. On the other hand, movement accelerometers are already included in most \nVR headsets. Previous research in the field has attempted to use postural or head move-\nment data collected by the headset to identify cybersickness  [20], but results have been \ninconsistent.  \nVarious machine learning methods have been employed to predict cybersickness, \nand these approaches incorporated different kind of variables into their models. Certain \nstudies have focused on internal physiological measurements, including factors such as \nheart rate [9], and muscle activity [28]. In addition, some investigations emphasize eye -\nrelated variables, including measures such as eye blinks and tracking eye movements \n[29, 32, 33]. Also, a subset of research concentrates on neural activities, employing \nelectroencephalogram (EEG) data to drive their models  [28]. Studies utilizing EEG data \nhave reported promising results [34]. \nAmong factors influencing cybersickness, body posture could be a significant con-\ntributor  [1]. A sensory mismatch between what the eyes see and what  the body feels in \nterms of balance and movement can disrupt the sensory integration process, causing the \nsymptoms of cybersickness [35, 36]. Several studies have focused on this connection \nindicating that ind ividuals with greater postural instability are more prone to experienc-\ning severe cybersickness  [1]. For example, adopting a standing posture, characterized \nby decreased stability, has been associated with a higher likelihood of severe sickness \ncompared to a sitting posture. Also, regarding the locomotion methods, when the body \nmovement is not well alig ned with the movement in the VR environment, it can lead to \ncybersickness.  \nEarly investigations by researchers like [37] and [1] highlighted a potential link, sug-\ngesting that both increases and decreases in body sway could be indicators of how in-\ndividuals react to VR environments. These studies put forward the idea that monitoring \nour body's adjustments in VR could be key to predicting occurrences of cybersickness, \nhighlighting the significan ce of postural changes in understanding our interactions with \nvirtual spaces.  \nContrasting viewpoints from studies such as those by  [20] and [21] introduced a \nlayer of skepticism, ques tioning the straightforward relationship between postural sway \nand cybersickness. These pieces of research pointed out instances where posture \nchanges did not consistently match up with cybersickness episodes, raising doubts \nabout the reliability of postur al data as a catch -all predictor for VR -induced discomfort.  \nThe debate extends further when focusing on specific movements, like those of the \nneck and back, with some researchers proposing a strong connection to cybersickness. \nHowever, findings from [38] countered this by showing that detailed analyses on these 4  Masoud Salehi, Nikoo  Javadpour, et al.  \n \nMasoud Salehi  | DATASPECT IT -SERVICE S, NECKARGEMUEND, GE RMANY  body parts failed to conclusively link them to cybersickness, illustrating the complexity \nof establishing a direct correlation. This diversity in research outcomes underscores the \nchallenge in achieving a unified understanding of how our physical responses are inter-\ntwined with our virtual exper iences.  \nThe discussion around postural sway and cybersickness has focused mainly on using \ntools like force plates or analyzing overall body movements to measure postural stabil-\nity. These approaches have provided valuable insights but also reveal a gap in o ur un-\nderstanding of how specific parts of the body, particularly the head, contribute to the \nexperience of cybersickness in virtual reality (VR) environments. Until now, the spe-\ncific impact of head movements on cybersickness has not been extensively explor ed in \nthe realm of cybersickness research.  \nThis research introduces a novel perspective by specifically examining head move-\nment data to understand its role in predicting cybersickness. This approach diverges \nfrom previous studies that have not fully consid ered the potential of head movements \nas a predictive measure for VR -induced discomfort. By shifting the focus to the head, \nwe aim to uncover new insights into how the orientation and motion of the head could \nbe closely linked with the onset of cybersicknes s. This novel approach not only fills a \nsignificant gap in the current research landscape but also opens up new avenues for \ndeveloping more effective predictive models of cybersickness, potentially leading to \nimproved VR design and user experiences.       \n3 Method  \nFigure 1 provides an overall view of the methods and approach in this study. Section \n3.1 describes the dataset used in this research and discusses some details about the col-\nlection process and the experiment. Section 3.2 describes different forms of  kinematics \ndata that are calculated from the raw movement data and discusses the logic for incor-\nporating these forms of signals. Section 3.3 explains the approach in windowing the \ndata and discuss the benefits provided by it for our modeling performance. Section 3.4 \ndiscusses the feature extraction approach and explain the logic for choosing the wide \nrange of features. And finally, Section 3.5 describes the modelling pipeline.   \n Innovative Cybersickness Detection: Exploring Head Movement Patterns in Virtual Reality  \n \n \n \n \n \nFig. 1. Overall view of the study with representative diagrams of data; these are not real data.  \n \n3.1 Dataset  \nThis study uses a secondary, publicly available dataset titled \"APAL 2019: Postural \nData, Game Performance, and Subjective Responses of Cybersickness in Virtual Real-\nity Head -Mounted Displays\" published by  [16]. It en compasses data from 79 partici-\npants (41 women and 38 men) aged between 18 to 49, with an average height of 1.72 \nmeters and weight of 71.58 kilograms. The study utilized the Simulator Sickness Ques-\ntionnaire (SSQ) for assessing motion sickness, categorizing participants into \"Well\" or \n\"Sick\" groups based on their responses. The recorded variables include positional data \n6  Masoud Salehi, Nikoo  Javadpour, et al.  \n \nMasoud Salehi  | DATASPECT IT -SERVICE S, NECKARGEMUEND, GE RMANY  (X, Y, Z in centimeters) and attitude data (pitch, roll, yaw in degrees) for the head.  Due \nto the variability of SSQ data and its subjective nature, a different variable was chosen \nas dependent variable in our study. The last 10s of the movement signals of the partic-\nipants who either indicated they were sick at the end of the experiment or chose to stop \nthe experiment early because of severe si ckness served as the “Sick” labeled samples. \nThe first 10  seconds of all participants  served  as “not Sick” samples. Using this ap-\nproach, the modeling task become a binary classification task. For more details about \nthe dataset, please take a look at  [16].  \n3.2 Kinematics Data  \nThe recorded head movements signal in this study records six variables. It includes \nhead disposition in centimeter along the X, Y, and Z axes and head rotations in degrees \nalong three axes; pitch (flexion/extension), roll (lateral flexion) , and yaw (axial rota-\ntion) (see Figure 2).  \n \n \nFig. 2. Axes of the head movements in our study  \n \nIn addition to the movements data, in the study of biomechanics and postural sway, \nresearchers also consider the velocity and acceleration of the movements signa l when \ndoing kinematic analysis .[39, 40]  \nJerk has also been used in several research as an indicator of postural sway smooth-\nness, and postural instability [41, 42]. Jerk describes how smoothly or abruptly a move-\nment changes in terms of acceleration. Higher values of jerk indicate more abrupt, and \nlower values of jerk indicate smoother movements.  \n \nInnovative Cybersickness Detection: Exploring Head Movement Patterns in Virtual Reality  \n  \n(a)  \n (b) \n \nFigure 3.  Derivatives of the original signals. (a) X position and its derivatives: Velocity (cm/s), Acceleration (cm/s^2), \nand Jerk (cm/s^3). (b) Yaw in degrees and its derivatives: Velocity (cm/s), Acceleration (cm/s^2), and Jerk (cm/s^3).  \n \n3.3 Windowing the Signals  \nAs the first step to start processing the data, we perform a signal windowing task using \na sliding window. Sliding window is a popular technique when dealing with time -series \ndata. In this technique, the signals are divided into fixed length intervals, cho sen by the \nuser. This technique provided several benefits in the context of our research.   \nFirst, it provides practicality. Participants have participated in a 15 -minute study. If \none had chosen to use of the whole data to train the sickness detection mod el, the model \ncould have made predictions only after at least after 15 minutes of observation, limiting \nthe practicality and near -real time use cases of the model. Windowing the data into \nsmaller lengths (1s, 2s, 3s, 5s, etc.) resulted in a model that made  estimations of the \nsickness more frequently.   \nSecond, windowing helps reduce the dimensionality and increasing a smaller sample \nsize. When using machine learning algorithms, a large number of features and small \nnumber of samples can lead the model into i ssues like overfitting and poor generaliza-\ntion. In this present context, the original signal was a 15 -minute time series of six var-\niables recorded with 60hz sampling, resulting a dimensionality of 324,000 (15 mins x \n60s x 60Hz x 6 variables) for only 75 ob servations. When considering a window size \nlike 1s, the dimensionality would be reduced to 360 and number of observations would \nincrease to 67,500.   \n8  Masoud Salehi, Nikoo  Javadpour, et al.  \n \nMasoud Salehi  | DATASPECT IT -SERVICE S, NECKARGEMUEND, GE RMANY  Third, windowing improves the quality of the feature extraction methods. In this \ncontext, participants’ mo vement signals, including temporal trend trajectories and spec-\ntral characteristics, may change over time, which is associated with the transition of the \npostural sway from a healthy state to the sickness state. By segmenting the signal into \nsmaller, tempor ally localized intervals, windowing not only facilitates a more granular \nanalysis of time -varying trends, but also significantly enhances frequency resolution.  \n \n3.4 Feature Extraction  \nHandling time series data in machine learning can be challenging as there ar e not any \nexplicit feature coming with the raw data. As a result, a meticulous feature extraction \nprocess is needed to identify a set of appropriate features. In the literature of postural \nsway and Center of Position analysis using force plate data, resear chers have put for-\nward a vast set of variables that can be extracted from the data  [43], [44], [45] , [46]. \nThese features can be mainly categorized in to three categories: statistical, temporal \n(time domain), and spectral (frequency domain).  \nWhile these lists are extensive and have been s hown to be very effective in measur-\ning a subjects’ postural stability and balance, they are not directly applicable to the \ncontext of head movement data. To the best of our knowledge, there is only a limited \nbody of literature discussing the processing and  feature extraction of head movements \ndata. These papers usually focus on a set of simple statistical features such as average \nacceleration, average velocity, and frequency of movements  [39, 47]. As a result of the \nlack of literature in head movements signal processing, in this paper, a brute force ap-\nproach was used. Influenced by postural sway literature, a large number of features in \nthe three statistical, temporal, and spectral domains were used for the four types of \nkinematic data mentioned in Section 3.2.  \nIt should be noted that while these features are statistically and mathematically \nmeaningful, not all of them are good predict ors of cybersickness. In addition, having a \nlarge number of these features can reduce the performance of our machine learning \nmodel. Therefore, we employ a feature selection technique, we select a handful of most \nmeaningful features to train our model.  \n \nStatistical Features  \nWhen analyzing head movement data for detecting cybersickness, it was crucial to ex-\ntract features that capture the complexity and nuances of such movements across dif-\nferent domains. Statistical features like kurtosis, skewness, mean, standard deviation, \nand histogram are fundamental as they provide insights into the distribution and varia-\nbility of head movement data, enabling the identification of outliers or patterns indica-\ntive of cybersickness. These features help in understanding the basic structure and dis-\npersion of the data, which are essential in distinguishing normal head movements from \nthose affected by cybersickness.    \n \nTemporal Features  Innovative Cybersickness Detection: Exploring Head Movement Patterns in Virtual Reality  \nTime domain features, including autocorrelation, zero crossing rate, and mean absolute \ndifference, were vital fo r capturing the temporal characteristics of head movements. \nThese features allow for the analysis of how head movement changes over time, iden-\ntifying specific patterns or irregularities that occur during episodes of cybersickness. \nFor instance, autocorrela tion helps in understanding the periodicity of movements, \nwhile neighborhood peaks can indicate sudden changes in movement direction or \nspeed, which are common in cybersickness episodes.  \n \nSpectral Features  \nTime series data may include spectral and frequenc y information. A common method \nto look into spectral domain of the data employed by researchers in the study of Center \nof Position (COP) and postural sway is Fast Fourier Transform (FFT)  [48, 49] . We \nnote that applying FFT in the context of postural sway should  be done with caution, as \nthe postural  sway and COP signals usually demonstrate nonstationary characteristics  \n[49, 50]. However, windowing the data into smaller intervals can make the data closer \nto the stationary assumption.  \nIn addition to Fourier transform techniques, the literature on postural sway analysis \nalso recommends employing wavelet analysis for its distinct advantages. Wavelet trans-\nforms offer a more fitting approach for analyzing non -stationary signals due to their \ncapability to capture intermittent, time -localized dynamics. This attribute makes them \nparticularly effective for examining nonlinear systems tha t exhibit time delays  [50]. \nTable 1 shows a list of all of the extracted features. We used a time series analysis \nPython package TSFEL to compute these features for each window [51]. \n \nTable 1.  List of extracted features  \nFeature Category  List of Features  \nStatistical Features  Absolute energy, Average power, ECDF, ECDF Percentile, \nECDF Percentile Count, Entropy, Histogram, Interquartile \nrange, Kurtosis, Max, Mean, Mean absolute deviation, Me-\ndian, Median absolute deviation, Min, Peak to peak distance, \nRoot mean square, Skewness, Standard deviation, Variance  \nTemporal Features  Area under the curve, Autocorrelation, Centroid, Mean abso-\nlute diff, Mean diff, Median absolute diff, Median diff, Neg-\native turning points, Neighborho od peaks, Positive turning \npoints, Signal distance, Slope , Sum absolute diff, Zero cross-\ning rate  \nSpectral Features  FFT mean coefficient, Fundamental frequency, Wavelet ab-\nsolute mean, Wavelet energy, Wavelet entropy, Wavelet \nstandard deviation, Wavelet variance, Human range energy, \nLPCC, MFCC, Max power spectrum , Maximum frequency, \nMedian frequency, Power bandwidth, Spectral centroid, \nSpectral decrease, Spectral distance, Spectral entropy, Spec-\ntral kurtosis, Spectral positive turning points, Spectral roll -\noff, Spectral roll -on, Spectral skewness, Spectral slope, Spec-\ntral spread, Spectral variation  10  Masoud Salehi, Nikoo  Javadpour, et al.  \n \nMasoud Salehi  | DATASPECT IT -SERVICE S, NECKARGEMUEND, GE RMANY  3.5 Modeling  \nThe modeling process began by dividing the dataset into training and testing sets. To \nprevent data leakage from the training set to the test set, the participants were segre-\ngated into training and test groups before shuffling all windowed data. This approach \nguaranteed that no participant's data appears in both training and test sets, thus avoiding \nthe potential for the model to learn participant -specific movement patterns. A stratified \nrandom sampling method  [52] for the train/test split to maintain the proportionality of \ndifferent classes in both sets.  \nIn the second step, we u sed Recursive Feature Elimination  [53] to identify the top \n50 most informative and relevant features from ou r original dataset, which may contain \nup to 984 features per kinematic signal (i.e., 984, 1968, 2,952, or 3,936 features in each \nof the four experimental setting). RFE systematically assesses the importance of each \nfeature, removing those with minimal cont ribution to the model's predictive accuracy. \nGiven the small sample size in this study, RFE was particularly beneficial as it reduces \nmodel complexity and improves the performance  [22, 53] . We trained a random forest -\nbased RFE mo del on the training dataset and applied the same feature selection process \nto the test set.  \nIn the third phase, the class imbalance issue was addressed within the training set \nusing the Synthetic Minority Over -sampling Technique (SMOTE) [54] This method \nmitigates model bias towards the majority class by artificially augmenting the minority \nclass.  \nFinally, to ensure the robustness and reliability of our results , a Monte Carlo Cross -\nValidation method  [55] was used , repeating each experiment 50 times. This approach \nhelps mitigate potential biases arising from random train/test splits, ensuring the stabil-\nity and generali zability of our findings.  \n4 Results  \nSix machine learning models were trained on the processed training data and evaluated \nagainst the test set. These models include Logistic Regression, Random Forest, SVM, \nK-nearest Neighbors, Decision Tree, and an ensemble model Gradient Boosting. the \nmodeling pipeline was tested in four settings, adding kinematics signals one by one. \nTable 2 shows the average results of the 50 repetitions of Monte Carlo cross validation \nexperiments. Numbers in parentheses reflect the standa rd deviations. In all settings, \nGradient Boosting outperformed other models in terms of accuracy, precision, and F1 \nScore.   \nComparing different settings’ results uncovered an interesting pattern. The perfor-\nmance of the best predictive model increased when  more variations of the original \nmovement signal were provided to the pipeline. Since all of the models in all the set-\ntings used the same number of 50 features selected by RFE, it could be assumed that \nthe RFE choice of the most useful features should be d ifferent in different settings.  \nTable 3 shows evaluation of this assumption. Also, it was observed that the standard \ndeviation of the results decreased when more kinematic signals were added into our \nbase feature set.  Innovative Cybersickness Detection: Exploring Head Movement Patterns in Virtual Reality  \nTable 2.  Mean Results of Monte Carlo CV (50 repetition)  \nExperiment  Models  Accuracy  \n(SD)  Precision  \n(SD)  Recall  \n(SD)  F1 Score  \n(SD)  \nMovement  \n Logistic Regression  54.3% \n(8.6%)  53.1% \n(11.3%)  48.5% \n(19.2%)  49.8% \n(14.7%)  \nRandom Forest  58.0% \n(9.6%)  64.0% \n(19.4%)  30.5% \n(18.3%)  39.7% \n(19.2%)  \nGradient Boosting  63.4% \n(9.3%)  69.5% \n(13.2%)  44.6% \n(18.6%)  58.0%  \n(16.7%)  \nSVM  58.8% \n(8.0%)  60.3% \n(9.6%)  47.7% \n(17.2%)  52.2% \n(13.3%)  \nK-Nearest Neigh-\nbors 59.6% \n(7.9%)  60.1% \n(9.0%)  54.0%  \n(16.7%)  56.0% \n(12.6%)  \nDecision Tree  61.5% \n(7.1%)  64.9% \n(11.0%)  47.0% \n(14.8%)  53.8% \n(13.1%)  \nMovement  \n+ \nVelocity  Logistic Regression  53.3% \n(9.3%)  51.4% \n(14.2%)  47.4% \n(20.1%)  48.4% \n(16.7%)  \nRandom Forest  64.7% \n(8.8%)  76.6% \n(14.2%)  41.6% \n(14.8%)  52.9% \n(14.9%)  \nGradient Boosting  72.7% \n(7.8%)  80.8% \n(9.9%)  60.1%  \n(13.9%)  68.0%  \n(10.9%)  \nSVM  59.5% \n(8.0%)  61.8% \n(12.6%)  45.8% \n(15.3%)  51.9% \n(13.7%)  \nK-Nearest Neigh-\nbors 56.3% \n(6.3%)  55.3% \n(5.7%)  64.8% \n(13.6%)  59.2% \n(8.3%)  \nDecision Tree  62.1% \n(6.8%)  65.1% \n(11.9%)  48.6% \n(13.2%)  55.2% \n(12.7%)  \nMovement  \n+ \nVelocity  \n+ \nAccelera-\ntion Logistic Regression  54.3% \n(8.0%)  53.7% \n(10.5%)  44.0% \n(18.2%)  47.4% \n(14.6%)  \nRandom Forest  66.1% \n(7.9%)  77.7% \n(11.0%)  44.3% \n(14.6%)  55.4% \n(13.8%)  \nGradient Boosting  74.5% \n(6.6%)  81.5% \n(7.0%)  63.6% \n(12.6%)  70.8% \n(9.4%)  \nSVM  60.5% \n(7.7%)  65.5% \n(10.6%)  42.1% \n(16.5%)  50.0% \n(14.2%)  \nK-Nearest Neigh-\nbors 56.9% \n(6.1%)  55.4% \n(5.1%)  67.5% \n(14.0%)  60.5% \n(8.3%)  \nDecision Tree  63.2% \n(6.5%)  66.7% \n(7.2%)  51.3% \n(13.1%)  57.4% \n(10.5%)  \nMovement  \n+ \nVelocity  \n+ \nAccelera-\ntion \n+ \nJerk Logistic Regression  55.6% \n(9.0%)  55.0% \n(12.3%)  46.6% \n(18.0%)  49.6% \n(15.3%)  \nRandom Forest  68.2% \n(8.1%)  80.2% \n(10.5%)  47.9% \n(14.6%)  59.1% \n(13.5%)  \nGradient Boosting  76.0% \n(5.9%)  82.7% \n(6.9%)  66.3% \n(11.5%)  73.0%  \n(8.1%)  \nSVM  62.0% \n(7.7%)  66.5% \n(10.8%)  46.3% \n(16.2%)  53.5% \n(14.1%)  \nK-Nearest Neigh-\nbors 58.3% \n(5.6%)  56.3% \n(5.1%)  72.0%  \n(13.0%)  62.8% \n(8.1%)  \nDecision Tree  64.3% \n(6.7%)  67.4% \n(7.3%)  54.0% \n(12.9%)  59.5% \n(10.7%)  12  Masoud Salehi, Nikoo  Javadpour, et al.  \n \nMasoud Salehi  | DATASPECT IT -SERVICE S, NECKARGEMUEND, GE RMANY   \nTo examine the validity of our assumption that “RFE choice of the most useful features \nshould be different in different settings” we look into the importance of the features of \ngradient boosting model which was the best model. Looking at the Table 3, we ca n see \nthat indeed the RFE model is picking different set of features in different experiments \nwhen presented with additional, possibly more informative set of features.  \n \nTable 3.  Results  \nExperiment  Features Importance  \nMovement  ('movement_Roll  _Neighborhood peaks', 0.101), ('movement_Roll  _Fun-\ndamental frequency', 0.065), ('movement_Roll  _Entropy', 0.062), ('move-\nment_Roll  _ECDF Percentile_1', 0.032), ('movement_Z _ECDF Percen-\ntile_1', 0.03), ('movement_X _Max', 0.03), ('movement_Z _ECDF Percen-\ntile_0', 0.0241), ('movement_Yaw  _Neighbourhood peaks', 0.0199), \n('movement_Z _Median', 0.016), ('movement_X _ECDF Percentile_1', \n0.015), ('movement_Z _FFT mean coefficient_0', 0.013), ('move-\nment_Roll  _Mean', 0.013), ('movement_Roll  _Max', 0.011), ('move-\nment_Pitch _Spectral distance', 0.011), ('movement_Pitch _Wavelet \nstandard deviation_6', 0.011), ('movement_Z _Mean', 0.01), ('move-\nment_Y _Spectral distance', 0.01022Mean absolute diff', 0.0014), ('move-\nment_Yaw  _Autocorrelation', 0.0013), ('movement_Roll  _Wavelet vari-\nance_3', 0.0004), ('movement_Y _Spectral entropy', 0.00041)  \nMovement  \n+ \nVelocity  \n ('Velocity_Z _Histogram_6', 0.0783), ('movement_Roll  _Fundamental \nfrequency', 0.0737), ('movement_Roll  _Neighbourhood peaks', 0.0439), \n('Velocity_Pitch _Histogram_6', 0.0255), ('movement_Roll  _ECDF Per-\ncentile_1', 0.0228), ('movement_X _Max', 0.0201), ('movement_Z _ECDF \nPercentile_1', 0.017994609145126163), ('movement_Roll  _Entropy', \n0.0179), ('movement_Z _ECDF Percentile_0', 0.0170), ('movement_Pitch \n_Wavelet variance_6', 0.0146), ('movement_Z _Median', 0.0145), ('Veloc-\nity_Z _Histogram_8', 0.0110), ('Velocity_Z _Histogram_4', 0.0107), \n('movement_Z _Absolute energy', 0.0104), ('movement_Z _FFT mean co-\nefficient_0', 0.0102), ('Velocity_Roll  _Histogram_9', 0.0102), ('move-\nment_Pitch _Wavelet standard deviation_6', 0.0098), ('Velocity_Pitch \n_Histogram_4', 0.0097), ('Velocity_Pitch _Spectral roll -on', 0.0091), \n('movement_X _ECDF Percentile_1', 0.0089), ('Velocity_Roll  _Histo-\ngram_0', 0.0085)  \n \nMovement  \n+ \nVeloci ty \n+ \nAcceleration  'movement_Roll  _Fundamental frequency', 0.0726), ('Velocity_Z _Histo-\ngram_6', 0.057), ('movement_Roll  _Entropy', 0.0440), ('movement_Roll  \n_Neighbourhood peaks', 0.037), ('acceleration_X _Neighbourhood peaks', \n0.0313), ('acceleration_Z _ Entropy', 0.0309), ('Velocity_Pitch _Spectral \nroll-on', 0.0273), ('movement_Roll  _ECDF Percentile_1', 0.021), ('move-\nment_Z _ECDF Percentile_1', 0.0157), ('movement_X _Max', 0.0125), \n('movement_Yaw  _Fundamental frequency', 0.0112), ('movement_Pitch \n_Wavel et standard deviation_6', 0.0108), ('acceleration_Pitch _Power \nbandwidth', 0.0105), ('movement_Roll  _Max power spectrum', 0.0101), \n('movement_Z _Median', 0.0098), ('movement_X _ECDF Percentile_1', \n0.0097), ('movement_Z _ECDF Percentile_0', 0.0096), ('Velo city_Z _His-\ntogram_4', 0.0094), ('movement_Pitch _Wavelet variance_6', 0.0087), Innovative Cybersickness Detection: Exploring Head Movement Patterns in Virtual Reality  \n('Velocity_Yaw  _Spectral roll -on', 0.0086), ('movement_Y _Power band-\nwidth', 0.0085), ('Velocity_Pitch _Histogram_1', 0.0084)  \nMovement  \n+ \nVelocity  \n+ \nAcceleratio  \n+ \nJerk \n ('moveme nt_Roll  _Fundamental frequency', 0.0870), ('Velocity_Z _His-\ntogram_6', 0.0518), ('movement_Roll  _Neighbourhood peaks', 0.0423), \n('acceleration_Z _Entropy', 0.0276), ('Velocity_Pitch _Spectral roll -on', \n0.0275), ('movement_Roll  _Entropy', 0.0213), ('jerk_ _Y _Neighbourhood \npeaks', 0.0181), ('acceleration_X _Neighbourhood peaks', 0.01690), \n('jerk__X _Neighbourhood peaks', 0.0157), ('movement_Roll  _ECDF \nPercentile_1', 0.0151), ('movement_Yaw  _Fundamental frequency', \n0.0148), ('movement_Z _ECDF Percentile_1' , 0.0141), ('movement_X \n_Max', 0.0130), ('movement_Z _Average power', 0.0128), ('move-\nment_Pitch _Wavelet standard deviation_5', 0.0107), ('movement_Z _Me-\ndian', 0.0106), ('acceleration_Z _Neighbourhood peaks', 0.0103), ('move-\nment_Y _Power bandwidth', 0.0102 ), ('movement_Roll  _Max power \nspectrum', 0.0097)  \n \n5 Discussion  \n \nThis study has introduced a novel approach to addressing the persistent issue of cyber-\nsickness in Virtual Reality (VR) environments. While cybersickness has been exten-\nsively studied through traditional physiological markers, this research focused on the \nanalysis of head movement patterns as a promising marker for cybersickness detection. \nThe study leveraged a publicly available dataset from a VR experiment involving 75 \nparticipants [23] and meticulously analyzed head movements and their derivatives \nacross multiple axes and extracted statistical, temporal, and spectral  features. Results \nincluded 76% accuracy and 83% precision in predicting cybersickness, showcasing the \npotential effectiveness of head movement patterns as an essential contributor to the un-\nderstanding and mitigation of cybersickness in VR applications.   \n \nThis study contributes to the field of cybersickness by providing an analysis of a new \nsource of physiological marker that can be used as an indicator of cybersickness. The \nresults also identify most valuable set of head movement features that are worth b eing \nextracted. 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Lab. \nSyst., vol. 56, no. 1, pp. 1 –11, Apr. 2001, doi: 10.1016/S0169 -7439(00)00122 -2. \n \n "    },    {        "title": "2402.02742v1.Bifurcation_to_complex_dynamics_in_largely_modulated_voltage_controlled_parametric_oscillator.pdf",        "content": "Bifurcation to complex dynamics in largely\nmodulated voltage-controlled parametric oscillator\nTomohiro Taniguchi1*\n1National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Emerging\nComputing Technologies, Tsukuba, Ibaraki 305-8568, Japan\n*tomohiro-taniguchi@aist.go.jp\nABSTRACT\nAn experimental demonstration of a parametric oscillation of a magnetization in a ferromagnet was performed recently by\napplying a microwave voltage, indicating the potential to be applied in a switching method in non-volatile memories. In the\nprevious works, the modulation of a perpendicular magnetic anisotropy field produced by the microwave voltage was small\ncompared with an external magnetic field pointing in an in-plane direction. A recent trend is, however, opposite, where an\nefficiency of the voltage controlled magnetic anisotropy (VCMA) effect is increased significantly by material research and\nthus, the modulated magnetic anisotropy field can be larger than the external magnetic field. Here, we solved the Landau-\nLifshitz-Gilbert equation numerically and investigated the magnetization dynamics driven under a wide range of the microwave\nVCMA effect. We evaluated bifurcation diagrams, which summarize local maxima of the magnetization dynamics. For low\nmodulation amplitudes, the local maximum is a single point because the dynamics is the periodic parametric oscillation. The\nbifurcation diagrams show distributions of the local maxima when the microwave magnetic anisotropy field becomes larger than\nthe external magnetic field. The appearance of this broadened distribution indicates complex dynamics such as chaotic and\ntransient-chaotic behaviors, which were confirmed from an analysis of temporal dynamics.\nV oltage controlled magnetic anisotropy (VCMA) effect1modulates a perpendicular magnetic anisotropy at a ferromagnetic\nmetal/nonmagnetic insulating layer interface by modulating of electron states near the interface2–4and/or inducing magnetic\nmoments5. It enables us to manipulate the direction of the magnetization in a ferromagnet electrically without the Joule heating,\nand thus, is expected to be a new writing method in magnetoresistive random access memory (MRAM), whose writing scheme\ncurrently relies on spin-transfer torque6, 7. The material researches for highly efficient VCMA effect has been reported, where\nthe perpendicular magnetic anisotropy is largely modulated by a small voltage8–17. The efficiency recently achieved reaches to\nabout 300 fJ/(Vm)18, which corresponds to a magnetic anisotropy field on the order of kilo-Oersted for typical VCMA-based\nMRAM. At the same time, analyses on the magnetization dynamics driven by the VCMA effect have been investigated both\nexperimentally and numerically19–28. It has been revealed that the switching is unstable when the pulse width of the voltage is\nshort because the dynamics becomes very sensitive to the pulse shape in such condition24. For a long-pulse regime, however,\nthe switching also becomes unstable due to noise23.\nTo overcome the issue, a parametric oscillation of the magnetization by applying microwave voltage was proposed in\nRef.29. For the magnetization switching in the VCMA-based MRAM using a perpendicularly magnetized free layer, an\nexternal magnetic field Happlpointing in an in-plane direction is applied and induces the magnetization precession around Happl,\nwhich eventually relaxes to the direction of Happl. When the microwave voltage witn an oscillation frequency f=2fL, where\nfL=γHappl/(2π)(γis the gyromagnetic ratio) is the Larmor frequency, is applied, however, a sustainable oscillation of the\nmagnetization is excited. This oscillation is classified into the parametric oscillation, and for simplicity, we call the oscillator as\nvoltage-controlled parametric oscillator. Since this oscillation is stable, it can be used to manipulate the magnetization direction\neven in a long-pulse regime, which results in a reliable magnetization switching. Note that the previous work29focuses on a\nparameter region of HKa/Happl≪1, where HKais the amplitude of the modulated magnetic anisotropy field generated by the\nmicowave VCMA effect. The recent progress of the VCMA efficiency, however, makes the opposite limit, HKa/Happl≫1,\navailable because the value of HKais growing rapidly, as mentioned above. The dynamical behavior of the magnetization in\nthis limit has not been investigated yet.\nIn this work, we study the magnetization dynamics driven by the microwave VCMA effect by solving the Landau-Lifshitz-\nGilbert (LLG) equation numerically. We change the value of HKain a wide range including the limit of HKa/Happl≫1. To\nclarify the change of the dynamical behavior, bifurcation diagrams are evaluated, which summarize local maxima of the\noscillating magnetization. When the magnitude of HKais small, the parametric oscillation is induced, and the bifurcation\ndiagram becomes single points because the dynamics is periodic. When HKabecomes larger than Happl, however, the bifurcation\n1arXiv:2402.02742v1  [cond-mat.mes-hall]  5 Feb 2024Figure 1. (a) Schematic illustration of magnetization oscillation in a ferromagnetic/nonmagnetic trilayer. The unit vector m\npointing in the magnetization direction in free layer shows a sustainable oscillation around an external magnetic field Happlin\nthe in-plane direction when the frequency of a microwave voltage is twice the Larmor frequency fL. (b) Dynamical trajectory\nof the magnetization in a parametric oscillation state. Parameters are Happl=300 Oe and HKa=100 Oe. The black triangle\nindicates the direction of the magnetization motion. (c) Time evolution of mzin the parametric oscillation state. (d) Fourier\nspectrum of mz. The inset shows the spectrum around the main peak in a linear scale.\ndiagrams show broad distributions. It is revealed that the appearance of these complex, broadened structures indicated chaotic\nor transient-chaotic behavior, which are confirmed from evaluations of the Lyapunov exponent and analyses on temporal\ndynamics.\nSystem description\nIn Fig. 1(a), we show a schematic illustration of a ferromagnetic/nonmagnetic/ferromagnetic trilayer. The top and bottom\nferromagnets correspond to free and reference layer, respectively. We apply a macrospin assumption in the free layer, whose\nvalidity in dynamical state driven by the VCMA effect has been confirmed experimentally29. Thus, the unit vector mpointing\nin the magnetization direction in the free layer conserves its magnitude, i.e., |m|=1andd|m|/dt=0. An external magnetic\nfield Happlis applied to an in-plane direction. We assume that the shape of the free layer is a cylinder, and therefore, the free\nlayer does not have an in-plane magnetic anisotropy. For convenience, we use a Cartesian coordinate, where the xaxis is\nparallel to Happlwhile the zaxis is normal to the film plane. The magnetic field in the free layer H, in the absence of an applied\nvoltage, is given by\nH=Happlex+[HK−4πM(Nz−Nx)]mxez, (1)\nwhere ek(k=x,y,z) is the unit vector in the kdirection. The perpendicular magnetic anisotropy field includes the contribution\nfrom the interfacial magnetic anisotropy field HK30–32and the shape magnetic anisotropy field 4πM(Nz−Nx)with the\ndemagnetization coefficients Nk(Nx=Nydue to the cylindrical symmetry). The net perpendicular magnetic anisotropy field,\nHK−4πM(Nz−Nx), determines the retention time of MRAM. In the conventional scheme of the writing in the VCMA-based\nMRAM, the direct voltage modulates this net perpendicular magnetic anisotropy close to zero to excite the magnetization\nprecession around the external magnetic field33. If the perpendicular component largely remains finite, the magnetization just\nmoves its direction to the direction of the external field with the angle sin−1(Happl/H′\nK)and the precession cannot be excited,\nwhere H′\nKis the reduced perpendicular magnetic anisotropy field by the VCMA effect. In fact, the numerical simulation in\nRef.33assumes that the net perpendicular magnetic anisotropy field is completely canceled by the VCMA effect. Note that\nthis assumption is important not only for making the simulation simple but also for experiments. If the direct (or intrinsic)\ncomponent of the perpendicular magnetic anisotropy field remains finite during the precession, an instantaneous frequency\nbecomes nonuniform. Such a nonuniform frequency will increase the switching error because the pulse width of the voltage for\nwriting the bit is determined as a half of the Larmor precession period for VCMA-based MRAM. Therefore, it is preferable\nto make the direct component of the perpendicular magnetic anisotropy field zero during the switching for the conventional\nswitching scheme. In the parametric oscillation state, both direct and microwave voltages are applied to the trilayer, and the\ndirect voltage modulates the total perpendicular magnetic anisotropy so that it becomes close to zero29, while the microwave\nvoltage provides an oscillating magnetic anisotropy field. Accordingly, the magnetic field used in the following calculation\nbecomes,\nH=Happlex+HKasin(2πft)mzez, (2)\nwhere HKaand fare the magnitude and frequency of the magnetic anisotropy field due to the microwave voltage. The\nmagnetization dynamics driven by this magnetic field is described by the LLG equation,\ndm\ndt=−γm×H+αm×dm\ndt, (3)\n2/14Figure 2. (a) Dynamical trajectory of the magnetization in a parametric oscillation state. Parameters are Happl=300Oe and\nHKa=620 Oe. The black triangle indicates the direction of the magnetization motion. (b) Time evolution of mzin the\nparametric oscillation state. (c) Fourier spectrum of mz.\nwhere αis the damping constant. Throughout this paper, we use the values of γ=1.764×107rad/(Oe s) and α=0.005. The\npreparation of the initial state of mby investigating thermal equilibrium is explained in Methods34, 35. Recall that the LLG\nequation conserves the norm of mas|m|=1. Therefore, although mis a three-dimensional vector, its dynamical degree of\nfreedom is two due to this constraint. In fact, if we use a spherical coordinate, for example, the dynamics of mis described by\ntwo variables (zenith and azimuth angles). It should be noted that dynamical systems described by differential equations cannot\nshow chaotic behavior when the dynamical degree of freedom is less than or equal to two, according to the Poincaré-Bendixson\ntheorem36. The presence of the microwave voltage, however, makes the present system non-autonomous and provides a\npossibility to excite chaotic behavior, as shown below.\nResults\nHere, we study the change of the magnetization dynamics for various magnitude of HKa.\nParametric oscillation\nLet us first start by confirming the parametric oscillation studied previously29. It was shown in Ref.29that a sustainable\noscillation of the magnetization is excited when the frequency of the microwave voltage, f, is twice the Larmor frequency,\nfL=γHappl/(2π). Therefore, in the following, we fix the value of fto be f=2fL. Figure 1(b) shows the dynamical trajectory\nof the magnetization in a steady state, where Happl=300Oe while HKa=100Oe, i.e., HKa/Happl≪1, as in the case of the\nprevious work29. Since |m|=1is satisfied in the LLG equation, it is useful to draw the dynamical trajectory on a unit sphere,\nas shown in this figure. Time evolution of mzin a steady state is also shown in Fig. 1(c). These results indicate an appearance\nof the sustainable oscillation of the magnetization mentioned above. In Fig. 1(d), the Fourier spectrum of mzis shown, where\nthe inset shows it around the main peak in a linear scale. Its main peak appears at 0.84GHz, which is the same with fLwith\nHappl=300Oe. These results are consistent with the previous works29. Since the magnetization switches its direction between\nmz≃+1andmz≃ −1periodically with the period 1/(2fL), this parametric oscillation can be used as a switching scheme in\nVCMA-based MRAM29. Recall that this oscillation is sustained by the microwave modulation of the magnetic anisotropy; if\nthis time-dependent modulation is absent, the magnetization monotonically relaxes to the direction of the external magnetic\nfield. Spin-wave propagation through a parametric excitation is another example of the magnetization dynamics caused by\nmicrowave voltage, which has been studied previously37–40.\nAppearance of complex dynamics and bifurcation diagram\nWhen the value of HKafurther increases, the magnetization dynamics becomes complex. In Fig. 2(a), we show the dynamical\ntrajectory of the magnetization for HKa=620Oe, while Happl=300Oe is the same with that used in Fig. 1(b). We observe a\nclear change of the magnetization dynamics by comparing Figs. 1(b) and 2(a). The trajectory is not a simple circle in Fig. 2(a).\nThe time evolution of mzand its Fourier transformation are shown in Figs. 2(b) and 2(c), respectively. Figure 2(b) indicates that\nthe magnetization dynamics is still periodic, while Fig. 2(c) indicates the appearance of multipeak structure. These results\nindicate that the application of the microwave voltage is no longer applicable to the switching method for the VCMA-based\nMRAM when the modulation of the magnetic anisotropy field becomes larger than the external magnetic field due to the\nbreakdown of the simple parametric oscillation.\nA way to distinguish these dynamics, such as the difference between Figs. 1(b) and 2(a), qualitatively is to draw a bifurcation\ndiagram36. In Fig. 3(a), we summarize local maxima of mzfor various HKa, where Happl=300Oe. Recall that the parametric\n3/14Figure 3. (a) A bifurcation diagram summarizing local maxima of mz(t)and (b) Lyapunov exponent for various HKa, where\nHappl=300 Oe.\nFigure 4. (a) Dynamical trajectory of the magnetization in a parametric oscillation state. Parameters are Happl=300Oe and\nHKa=1200 Oe. (b) Time evolution of mzin the parametric oscillation state. (c) Fourier spectrum of mz.\noscillation is excited when HKais small. In this case, the dynamics is periodic and mzis similar to a simple trigonometric\nfunction, as shown in Fig. 1(c). The local maxima of mzfor this case, thus, saturate to a single point. When the dynamics\nbecome complex, the bifurcation diagram shows broadened structure. For example, there are three points at HKa=620 Oe in\nFig. 3(a), the validity of which is confirmed from Fig. 2(b). When the value of HKafurther increases, the bifurcation diagram\nshows largely broadened structure and finally shows simplified structure again. As discussed below, these correspond to chaotic\nand transient-chaotic behaviors41–43. Note that the appearance of complex but still periodic structure might weakly depend on\nthe initial state (see Methods) while the region of the broadened structure might depend on the measurement time, as will be\nmentioned below. It is difficult to analytically estimate the value of HKaat which the bifurcation from a simple parametric\noscillation to a complex oscillation occurs; see Methods. However, the numerical simulations for various parameters imply\nthat the complex oscillation appears when HKabecomes larger than Happl, as discussed below. We also evaluated Lyapunov\nexponent36by Shimada-Nagashima method44, as shown in Fig. 3(b). The Lyapunov exponent is an inverse of a time scale\ncharacterizing an expansion of a difference between two solutions of the LLG equation with an infinitesimally different initial\nconditions; see Methods explaining the evaluation method of the Lyapunov exponent. A negative Lyapunov exponent means\nthat the magnetization saturates to a fixed point. The Lyapunov exponent is zero when the magnetization dynamics are periodic,\nwhile it becomes positive when the dynamics are chaotic. We notice that the Lyapunov exponent in the present system is zero\nor positive, depending on the value of HKa. Thus, the Lyapunov exponent can be used as an indicator to distinguish between\nperiodic oscillation and chaotic dynamics. For example, we can conclude that the dynamics in Fig. 2(a) is periodic not only\nfrom the temporal dynamics shown in Fig. 2(b) but also from the fact that the Lyapunov exponent for HKa=620 Oe is zero.\nChaotic dynamics\nIn Fig. 4(a), we show the dynamical trajectory of the magnetization for HKa=1200 Oe. The dynamical trajectory covers\nalmost the whole region of the unit sphere. The time evolution of mzbecomes non-periodic, as shown in Fig. 4(b), and the\nFourier spectrum shows a broad structure having several peaks. These results imply chaotic dynamics of the magnetization41, 42.\n4/14Figure 5. (a) Dynamical trajectory of the magnetization in a parametric oscillation state. Parameters are Happl=300Oe and\nHKa=1600 Oe. The black triangle indicates the direction of the magnetization motion. (b) Time evolution of mzin the\nparametric oscillation state. (c) Fourier spectrum of mz.\nThe appearance of chaos for this parameter can also be concluded from the fact that the Lyapunov exponent shown in Fig. 3(b)\nis positive.\nAs mentioned above, chaotic dynamics in the present system are excited because of the presence of the microwave voltage.\nSimilar examples in spintronics devices have been found in spin-torque oscillators with time-dependent inputs45–47. The\ndifferences of the phenomena observed between the voltage-controlled parametric oscillator studied here and spin-torque\noscillator are as follows. In the spin-torque oscillators, a sustainable oscillation of the magnetization is driven by direct currents,\nand an injection of time-dependent inputs is not a necessary condition for the oscillation. Spin-torque oscillator cannot show\nchaotic behavior when only the direct current is injected due to the constraint by the Poincaré-Bendixson theorem. A way to\nexcite chaos in spin-torque oscillator is to inject time-dependent inputs such as alternating current and/or magnetic field45–47.\nWhen the magnitudes of these time-dependent inputs are relatively small, synchronization may occur. When their magnitudes\nare further increased, chaos might be induced. On the other hand, the sustainable oscillation of the magnetization in the present\nvoltage-controlled parametric oscillator is driven by microwave voltage. In other words, this time-dependent input is a necessary\ncondition for the oscillation. Chaotic dynamics appear when the magnitude of the microwave voltage becomes relatively large,\nas shown in Fig. 4.\nLet us briefly mention applicability of the chaotic dynamics for practical devices. Chaos in spintronics devices has been\nstudied both experimentally and theoretically using various methods48–57. An excitation of chaos in spintronics devices may\nevoke interest from a viewpoint of brain-inspired computing58, 59. For example, it was found that a computational capability of\nphysical reservoir computing is enhanced when spin-torque oscillators are near the edge of chaos58, where chaos was excited\nby adding another ferromagnet to the oscillator. An excitation of chaos in spin-torque oscillator by time-dependent inputs,\nhowever, seems to require large power consumption. For example, the amplitude of the alternating current density assumed\nin the numerical simulations in Refs.45–47is on the order of 107−108A/cm2. The value is larger than the switching current\ndensity of the state of the art spin-transfer torque driven MRAM60, and thus, is hardly desirable. The large current also causes\nlarge power consumption, which is also unsuitable for practical applications. The voltage-controlled parametric oscillator, on\nthe other hand, ideally reduces the power consumption significantly. In fact, this point has been a motivation for developing\nVCMA-based MRAM. However, these VCMA-based devices often require an external magnetic field for both the switching\nand parametric oscillation, which is not preferable in practical applications. A way to solve the issue might be to use an\neffective field, instead of applying external magnetic field, such as an interlayer exchange coupling field, as investigated in the\nstudy of spin-orbit torque driven MRAM61. Physical reservoir computing by the VCMA-based MRAM without an external\nmagnetic field was investigated recently62, however, switching nor parametric oscillation was used there. A development of the\nvoltage-controlled parametric oscillator without requiring an external magnetic field will be of interest as a future work for\napplying it to practical applications.\nTransient chaos\nIn Fig. 5(a), we show the dynamical trajectory of the magnetization for HKa=1600 Oe. The time evolution of mzand its\nFourier transformation are also shown in Figs. 5(b) and 5(c), respectively. We note that the dynamics shown here corresponds to\nmz(t)in a long-time limit, i.e., a steady state. The results look similar to those shown in Figs. 1 and 2. Also, the local maxima\nofmzin the bifurcation diagram in Fig. 3(a) concentrates on a single point. However, there is a critical difference between the\nmagnetization dynamics shown in Fig. 5 and those shown in Figs. 1 and 2.\nTo clarify their differences, it is necessary to focus on a process to reach the steady state. In Figs. 6(a) and 6(b), we show\nthe time evolution of mz(t)from t=0to the steady states for HKa=620Oe and 1600 Oe, respectively. When HKais relatively\n5/14Figure 6. Time evolution of mznear t=0 for (a) HKa=620 Oe and (b) HKa=1600 Oe.\nFigure 7. Time evolution of mzfor (a) HKa=1900 Oe, (b) HKa=1980 Oe, and (c) HKa=2000 Oe.\n6/14Figure 8. Bifurcation diagrams summarizing the local maxima of mzfor (a) Happl=200 Oe, (b) Happl=300 Oe, and (c)\nHappl=400Oe. Note that (b) is identical to Fig. 3(a) but is shown here for comparison. The value of the damping constant αis\n0.005. Those for α=0.020 are shown in (d)-(f).\nsmall [see Fig. 6(a)], the magnetization immediately reaches to the steady state shown in Fig. 2(b). When HKais relatively\nlarge [see Fig. 6(b)], on the other hand, chaotic behavior appears initially, and it suddenly changes to the steady state shown\nin Fig. 5(b). The phenomenon shown in Fig. 5(b) corresponds to a transient chaos43, where dynamical systems initially\nshow chaotic behavior but it suddenly disappears. Time necessary to move from chaos to a steady state is sensitive to various\nparameters and initial conditions of systems43. For example, in Figs. 7(a), 7(b), and 7(c), we show time evolution of mzfor\nHKa=1900 Oe, 1980 Oe, and 2000 Oe, respectively, where the initial conditions are common. The results indicate that the\ntime necessary to realize the steady state depends on the parameter HKaand it does not show, for example, a monotonic change\nwith respect to the change of the parameter. The transient chaos in spintronics devices was predicted in a spin-torque oscillator\nwith a delayed-feedback circuit54. The phenomenon has not been verified experimentally yet, although chaos was confirmed\nrecently57.\nWe note that the classification of chaos and transient chaos depends on a measurement time43. For example, in the present\nwork, we solve the LLG equation from t=0tot=5µs and classify the magnetization dynamics. If we change this maximum\ntime ( 5µs) to, for example 2µs, the dynamics for HKa=2000 Oe, shown in Fig. 7(c), will be classified as chaos. Another\nexample can be seen in the bifurcation diagram and the Lyapunov exponent shown in Fig. 3, where a broadened structure in\nthe bifurcation diagram and a positive Lyapunov for HKa=1920 Oe, indicating chaos for this parameter. If we measure the\ndynamics for this parameter longer, however, the dynamics might change to a steady state; in such a case, the dynamics will be\nclassified as transient chaos. As a general knowledge on transient chaos43, we should remember that the classification of chaos\nand transient chaos has such an arbitrary property.\nBifurcation diagrams for various applied fields\nAs mentioned above, a bifurcation diagram is useful to classify the magnetization dynamics, although, for example, the\ndifference between a simple steady oscillation and a transient chaos should be verified from temporal dynamics, as mentioned\nabove. We also notice that the whole structure of the bifurcation diagram does not change qualitatively even when the initial\ncondition is slightly changed. Recall that the present system includes only three parameters, HKa,Happl, andα. Therefore, in\nFigs. 8(a)-8(c), we show the bifurcation diagrams for (a) Happl=200Oe, (b) Happl=300Oe, and (c) Happl=400Oe. These\nfigures indicate that the local maxima of mzfor relatively small HKaconcentrate on single points, which indicate the excitation\nof the parametric oscillation. As HKaincreases, broadened structures appear, i.e., the local maxima of mzhave various values,\n7/14implying the appearance of complex oscillation and chaos. The figures also imply that this boundary between the parametric\noscillation and the complex dynamics, i.e., the boundary between the single and multiple points in the bifurcation diagram,\nlocates near HKa/Happl≃1.5−2.0. We examine similar calculations for different αbut the boundary seems to be unchanged;\nsee Figs. 8(d)-8(f), which are the bifurcation diagrams for α=0.020. Because of high nonlinearity in the LLG equation, it is\ndifficult to analytically verify these indications; however, it might be useful to design, for example, the modulation voltage for\nVCMA-based MRAM utilizing the parametric oscillation. We keep this question as a future work.\nWe note that the value of HKa/Happl≃1.5−2.0is available by current technology. The typical value of Happlis on the\norder of 100Oe; for example, it is 250Oe in Ref.23and720Oe in Ref.29. Although the value of Happlcan be further large\nexperimentally, a large Happlmight be unsuitable for practical applications; see Methods for analytical treatment of the LLG\nequation. On the other hand, the value of HKarelates to the VCMA efficiency η, the saturation magnetization M, the applied\nvoltage Vappl, and the thicknesses of the free and insulating layers, dFanddI, via HKa= (2ηVappl)/(MdFdI). Substituting their\ntypical values [ ηis about 300fJ/(Vm)18,Vapplis about 0.5V at maximum, Mis about 1000 emu/cm3,dFis about 1nm, and dI\nis about 2.5nm), HKacan be on the order of kilo Oersted at maximum, as written in the introduction. Therefore, we believe\nthat the value of HKa/Happl≃1.5−2.0 is experimentally achievable.\nConclusion\nIn summary, the magnetization dynamics in the voltage-controlled parametric oscillator for a large microwave voltage limit were\ninvestigated by numerical simulation of the LLG equation. As the modulation increases, the dynamical trajectory changes from\na simple parametric oscillation to complex oscillations, which are still periodic but have several local maxima in the amplitude.\nSuch dynamics will be of little preference in a switching scheme for VCMA-based MRAM. The evaluation of the bifurcation\ndiagrams for various values of the external magnetic field and the damping constant indicated that the simple parametric\noscillation is broken when the amplitude of the modulated magnetic anisotropy field becomes larger than the external magnetic\nfield, and this bifurcation point seems to be hardly sensitive to the value of the damping constant. A further enhancement of the\nmicrowave modulation leads to chaotic and transient-chaotic behaviors, which might make the voltage-controlled parametric\noscillator applicable to other usage in electric devices. These dynamics were classified from the bifurcation diagrams, Lyapunov\nexponent, and analyses on temporal dynamics.\nMethods\nPreparation of initial state\nWe prepare natural initial states by solving the LLG equation with thermal fluctuation34. For this purpose, we use Eq. (1) as the\nmagnetic field. We add a torque, −γm×h, due to thermal fluctuation to the right-hand side of Eq. (3). Here, the components of\nhsatisfy the fluctuation-dissipation theorem63,\n⟨hk(t)hℓ(t′)⟩=2αkBT\nγMVδkℓδ(t−t′). (4)\nIn the numerical simulation, the random field is given by\nhk(t) =s\n2αkBT\nγMV∆tξk(t), (5)\nwhere the time increment ∆tis 1 ps in this work. White noise ξkis defined from two random numbers, ζaandζb, in the range\nof0<ζa,ζb<1by the Box-Muller transformation as ξa=p\n−2lnζasin(2πζb)andξb=p\n−2lnζacos(2πζb). We added\nEq. (5) to the magnetic field and solved the LLG equation numerically using the Runge-Kutta method for the investigation\nof thermal equilibrium before applying microwave voltage. The value of the net perpendicular magnetic anisotropy field,\nHK−4πM(Nz−Nx), in the absence of microwave voltage is assumed to be 6.283kOe, while the saturation magnetization\nis set to be 955emu/cm329. The temperature Tis300K, and the volume is V=π×50×50×1.1nm3, which is typical for\nVCMA experiments. The thermal fluctuation excites a small-amplitude oscillation of the magnetization around the energetically\nminimum state with the ferromagnetic resonance frequency. We pick the temporal directions of the oscillating magnetization\nand use them as the natural initial states. It should be noted that the value of HKaat which the complex but still periodic\noscillation appears weakly depends on the choice of the initial state, despite the structure of the bifurcation diagram is not\nchanged qualitatively. It might relates to the presence of two stable phases of the parametric oscillation with respect to the\nmicrowave voltage35.\n8/14Analytical treatment of the LLG equation\nHere, we discuss an analytical treatment of the parametric oscillation, which provides a condition to excite the oscillation. It is,\nhowever, not applicable to investigate the bifurcation from the simple parametric oscillation to the complex but still periodic\noscillation shown in Figs. 1(b) and 2(a).\nSince we are interested in the oscillation around the xaxis, we introduce angles ΘandΦasm= (cosΘ,sinΘcosΦ,sinΘsinΦ).\nFor simplicity, we introduce notations k=γHKaandh=γHappl. The LLG equation for ΘandΦare given as dΘ/dt=\n−(1/sinΘ)(∂ε/∂Φ)−α(∂ε/∂Θ)and sin Θ(dΦ/dt) = (∂ε/∂Θ)−α(1/sinΘ)(∂ε/∂Φ), where\nε=−hcosΘ−k\n2sin(2πft)sin2Θsin2Φ, (6)\ni.e.,\ndΘ\ndt=ksin(2πft)sinΘsinΦcosΦ−α\u0002\nh−ksin(2πft)cosΘsin2Φ\u0003\nsinΘ, (7)\ndΦ\ndt=h−ksin(2πft)cosΘsin2Φ+αksin(2πft)sinΦcosΦ. (8)\nSince we are interested in an oscillation where myandmzoscillates with the frequency fL=f/2, we introduce Ψ=Φ−2πfLt,\nwhich obeys\ndΨ\ndt=h−2πfL−ksin(2πft)cosΘsin2(Ψ+2πfLt)+αksin(2πft)sin(Ψ+2πfLt)cos(Ψ+2πfLt). (9)\nIn the parametric oscillation states, the tilted angle Θof the magnetization from the xaxis and the phase difference Ψare\napproximately constants35. Therefore, after averaging Eqs. (7) and (9) over a period 1 /fL, we obtain,\ndΘ\ndt=k\n4sinΘcos2Ψ−α\u0012\nh−k\n4cosΘsin2Ψ\u0013\nsinΘ, (10)\ndΨ\ndt=σ−k\n4cosΘsin2Ψ+αk\n4cos2Ψ, (11)\nwhere σ=h−2πfL. Although the present work focuses on the case of σ=0only, we keep σas finite here, for generality.\nThe steady state solutions of ΘandΨafter averaging are\ncosΘ=±4(σ+α2h)p\n(1+α2)k2−[4α(h−σ)]2, (12)\ntan2Ψ=±p\n(1+α2)k2−[4α(h−σ)]2\n4α(h−σ). (13)\nThese solutions imply, for example, that (1+α2)k2>[4α(h−σ)]2for exciting the parametric oscillation, which can easily\nbe satisfied when α≪1. These solutions, however, cannot be used to discuss, for example, the bifurcation from the simple\nparametric oscillation to the complex oscillation because Φabove is assumed to oscillates with only a single frequency fL,\nwhile the complex oscillation is a superposition of multiple frequencies. Even for the parametric oscillation state, the above\nsolutions may not be quantitative due to the fact that, strictly speaking for example, Θis not constant.\nAlthough a reliability of the magnetization switching by the parametric oscillation is not the main scope of this work,\nlet us briefly provide a comment on the relationship between the switching reliability and the value of HKa/Happlmentioned\nin the introduction. First, the value of Happlshould be carefully chosen. For a large Happl, the precession frequency of the\nmagnetization becomes high. Such a fast precession makes the switching unstable because the dynamics becomes sensitive to\nthe pulse shape, as mentioned in the introduction. When Happlis small, however, the switching is also unstable because the\ndynamics is highly affected by thermal fluctuation, which is also written in the introduction. Summarizing them, the value of\nHapplshould be carefully determined, depending on the various factors, such as circuit quality controlling the pulse shape and\nthe volume of the ferromagnetic layer. Second, a large HKais considered to be preferable for a reliable switching due to the\n9/14following reasons. First, as mentioned below Eqs. (12) and (13), there is a threshold value of HKa[(1+α2)k2>[4α(h−σ)]2]\nto excite the parametric oscillation. Second, Eq. (12) indicates that the cone angle Θof the magnetization precession around\nHapplbecomes close to 90◦when HKais large. In other words, the cone angle decreases as HKadecreases. A precession with a\nsmall cone angle is unstable because such a small oscillation is easily disturbed by thermal fluctuation. Regarding these points,\nit will be preferable to increase HKa/Happl, orHKa, for a reliable switching because there is a threshold of HKaof the parametric\noscillation and it is necessary to make the oscillation robust against thermal fluctuation. However, as revealed in the main text,\nthe precession becomes complex when HKa/Happlbecomes greatly large, which is a main message in this work.\nEvaluation method of Lyapunov exponent\nThe evaluation method of the Lyapunov exponent is as follows34, 64. We denote the solution of the LLG equation at a certain time\ntasm(t). We introduce the zenith and azimuth angles, θandϕ, asm= (mx,my,mz) = ( sinθcosϕ,sinθsinϕ,cosθ). We also\nintroduce m(1)(t) = ( sinθ(1)cosϕ(1),sinθ(1)sinϕ(1),cosθ(1)), where θ(1)andϕ(1)satisfy ε=p\n[θ−θ(1)]2+[ϕ−ϕ(1)]2.\nNote that εcorresponds to the distance between m(t)andm(1)(t)attin the spherical space. Since the Lyapunov exponent\ncharacterizes the sensitivity of the dynamical system to the initial state, we study an expansion of εwith time. For this purpose,\nwe assume a small value of ε, which is ε=1.0×10−5in this work. For convenience, we introduce a notation,\nD[m(t),m(1)(t)] =q\u0002\nθ(t)−θ(1)(t)\u00032+\u0002\nϕ(t)−ϕ(1)(t)\u00032. (14)\nSolving the LLG equations of m(t)andm(1)(t), we obtain m(t+∆t)andm(1)(t+∆t), where ∆tis time increment. From them,\nwe evaluate the distance between m(t+∆t)andm(1)(t+∆t)as\nD[m(t+∆t),m(1)(t+∆t)] =q\u0002\nθ(t+∆t)−θ(1)(t+∆t)\u00032+\u0002\nϕ(t+∆t)−ϕ(1)(t+∆t)\u00032. (15)\nThen, a temporal Lyapunov exponent at t+∆tis given as\nΛ(1)=1\n∆tlnD(1)\nε, (16)\nwhereD(1)=D[m(t+∆t),m(1)(t+∆t)].\nNext, we introduce m(2)(t+∆t) = ( sinθ(2)cosϕ(2),sinθ(2)sinϕ(2),cosθ(2)), where θ(2)andϕ(2)are defined as\nθ(2)(t+∆t) =θ(t+∆t)+εθ(1)(t+∆t)−θ(t+∆t)\nD[m(t+∆t),m(1)(t+∆t)], (17)\nϕ(2)(t+∆t) =ϕ(t+∆t)+εϕ(1)(t+∆t)−ϕ(t+∆t)\nD[m(t+∆t),m(1)(t+∆t)]. (18)\nAccording to these definitions, m(t+∆t)andm(2)(t+∆t)satisfy\nD[m(t+∆t),m(2)(t+∆t)] =ε. (19)\nIt means that m(2)(t+∆t)is defined by moving m(t+∆t)to the direction of m(1)(t+∆t)with a distance εin the (θ,ϕ)phase\nspace. Solving the LLG equations for m(t+∆t)andm(2)(t+∆t), we obtain m(t+2∆t)andm(2)(t+2∆t). The temporal\nLyapunov exponent at t+2∆tis\nΛ(2)=1\n∆tlnD(2)\nε, (20)\nwhereD(2)=D[m(t+2∆t),m(2)(t+2∆t)].\nThese procedures are generalized. At t+n∆t, we have m(t+n∆t) = ( sinθ(t+n∆t)cosϕ(t+n∆),sinθ(t+n∆t)sinϕ(t+\nn∆t),cosθ(t+n∆t))andm(n)(t+n∆t) = ( sinθ(n)(t+n∆t)cosϕ(n)(t+n∆),sinθ(n)(t+n∆t)sinϕ(n)(t+n∆t),cosθ(n)(t+\nn∆t)). Then, we define m(n+1)(t+n∆t) = ( sinθ(n+1)(t+n∆t)cosϕ(n+1)(t+n∆),sinθ(n+1)(t+n∆t)sinϕ(n+1)(t+n∆t),cosθ(n+1)(t+\nn∆t))by moving m(t+n∆t)to the direction of m(n)(t+n∆t)with a distance εin the phase space as\nθ(n+1)(t+n∆t) =θ(t+n∆t)+εθ(n)(t+n∆t)−θ(t+n∆t)\nD[m(t+n∆t),m(n)(t+n∆t)], (21)\nϕ(n+1)(t+n∆t) =ϕ(t+n∆t)+εϕ(n)(t+n∆t)−ϕ(t+n∆t)\nD[m(t+n∆t),m(n)(t+n∆t)]. (22)\n10/14Note that m(t+n∆t)andm(n+1)(t+n∆t)satisfy D[m(t+n∆t),m(n+1)(t+n∆t)] =ε. Then, solving the LLG equations of\nm(t+n∆t)andm(n+1)(t+n∆t), we obtain m(t+ (n+1)∆t)andm(n+1)(t+ (n+1)∆t). A temporal Lyapunov exponent at\nt+(n+1)∆tis\nΛ(n+1)=1\n∆tlnD(n+1)\nε, (23)\nwhereD(n+1)=D[m(t+(n+1)∆t),m(n+1)(t+(n+1)∆t)]. The Lyapunov exponent is defined as a long-time average of the\ntemporal Lyapunov exponent as\nΛ=lim\nN→∞1\nNN\n∑\ni=1Λ(i). (24)\nIn the present study, we solve the LLG equation from t=0totmax=5µs with ∆t=1ps. Therefore, there are tmax/∆t=5×106\nsteps during the evaluation of the magnetization dynamics. We use the last 1×106steps for the evaluation of the Lyapunov\nexponent. Note that this method is an application of Shimada-Nagashima method44for the evaluation of the Lyapunov exponent\nfrom numerical simulation of an equation of motion to the LLG equation. 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This work was supported by a JSPS KAKENHI Grant, Number\n20H05655.\nAuthor contributions statement\nT.T. designed the project, performed the numerical simulations, prepared figures, and wrote the manuscript.\nCompeting interests\nThe authors declare no competing interests.\nData availability\nThe datasets used and/or analyses during the current study available from the corresponding author on reasonable request.\n13/14Additional information\nCorrespondence and requests for materials should be addressed to T.T.\n14/14"    },    {        "title": "2402.02743v1.A_Grammar_of_Dumont_and_a_Theorem_of_Diaconis_Evans_Graham.pdf",        "content": "arXiv:2402.02743v1  [math.CO]  5 Feb 2024A Grammar of Dumont and\na Theorem of Diaconis-Evans-Graham\nWilliam Y .C. Chen1and Amy M. Fu2\n1Center for Applied Mathematics\nTianjin University\nTianjin 300072, P.R. China\n2School of Mathematics\nShanghai University of Finance and Economics\nShanghai 200433, P.R. China\nEmails:1chenyc@tju.edu.cn,2fu.mei@mail.shufe.edu.cn\nAbstract\nWe come across an unexpected connection between a remarkabl e grammar of Du-\nmont for the joint distribution of (exc,fix)overSnand a beautiful theorem of Diaconis-\nEvans-Graham on successions and fixed points of permutation s. With the grammar in\nhand, we demonstrate the advantage of the grammatical calcu lus in deriving the gener-\nating functions, where the constant property plays a substa ntial role. In consideration\nof left successions of a permutation, we present a grammatic al approach to the joint\ndistribution investigated by Roselle. Moreover, we obtain a left succession analogue,\nor a shifted version, of the Diaconis-Evans-Graham theorem , exemplifying the idea of\na grammar assisted bijection. The grammatical labelings gi ve rise to an equidistribu-\ntion of(jump,des) and(exc,drop) restricted to the set of left successions and the set\nof fixed points, where jump is the statistic defined to be the number ascents minus the\nnumber of left successions.\nKeywords: Context-free grammars, increasing binary trees, the Diaco nis-Evans-Graham\ntheorem, successions, fixed points.\nAMS Classification: 05A15, 05A19\n1 Introduction\nThis paper is concerned with a beautiful theorem of Diaconis -Evans-Graham [5] on the cor-\nrespondence between successions and fixed points of permuta tions. Unlike a typical equidis-\ntribution property, this theorem possesses an attractive f eature that the bijection can be re-\n1stricted to a specific set of successions and the same set of fix ed points, and so it says more\nthan just an equidistribution.\nThe topic of the enumeration of successions of permutations has a rich history. Dumont\nreferred to the work of Roselle [9] on the joint distribution of the number of ascents and the\nnumber of successions. In fact, the grammar proposed by Dumo nt [6] is meant to deal with\nthe joint distribution of the number of excedances, the numb er of drops and the number of\nfixed points of a permutation. His argument can be rephrased i n the language of a grammat-\nical labeling of complete increasing binary trees. We will s how that this grammar is related\nto the Diaconis-Evans-Graham theorem, even though it does n ot look so at first sight. It is\nworth mentioning that Dumont’s citation to Roselle was not a ccurate, nevertheless such an\nincident was somehow just to the point. Indeed, this work wou ld not have come into being\nwithout the lucky pointer of Dumont.\nFirst, we come to the realization that the grammar of Dumont c an be adapted to a problem\nof Roselle. We need to pay attention to the notion of a left suc cession, analogous to that of\na left peak of a permutation. As remarked by Roselle, the cons ideration of a left succession\nat position 1was regarded convenient for the computation of the generati ng function for\ninterior successions. The definition of a left succession as sumes that a zero is patched at\nthe beginning of a permutation and that it should be taken int o account for the counting. In\ncontrast to a left succession, a usual succession is called a n interior succession.\nOnce the grammar is proposed, a formal justification is neede d to endow it with a combi-\nnatorial significance. This is usually done by a grammatical labelings. Then there is no fear\nin performing the grammatical calculus for the purpose of de riving the generating functions.\nWe will how the grammar of Dumont works for the joint distribu tion of(exc,fix). Further,\nwe give a different labeling scheme for permutations which s hows that the same grammar\nof Dumont suits equally well for the joint distribution of (jump,lsuc) , wherejump andlsuc\ndenote the number of jumps and the number of left successions of a permutation, respec-\ntively, whose definitions will be given later. It is no surpri se that the constant property plays\na substantial role in the computation.\nWhile the grammar is instrumental in establishing an equidi stribution, it is not clear\nwhether one can take a step forward in obtaining a Diaconis-E vans-Graham type theorem\nin regard with a given set left successions and the same set of fixed points. Fortunately, the\nanswer is yes. In fact, it is exactly where the idea of a gramma r assisted bijection comes on\nthe scene.\nLet us see what will evolve then.\n22 A grammar of Dumont\nIn this section, we recall a remarkable grammar of Dumont [6] for the joint distribution of\nthe statistics (exc,drop,fix)overSn, the set of permutations of [n] ={1,2,...,n}, where\nn≥1. For a permutation σ∈Sn, an index 1≤i≤nis called an excedance if σi> i, or\na drop ifσi< i, or a fixed point if σi=i. The number of excedances, the number of drops\nand the number of fixed points of σare denoted by exc(σ),drop(σ)andfix(σ), respectively.\nA drop of a permutation is also called an anti-excedance.\nThe joint distribution of (exc,fix)was determined by Foata-Sch¨ utzenberger [7], see also\nShin-Zeng [10]. For n≥1, define\nFn(x,z) =/summationdisplay\nσ∈Snxexc(σ)zfix(σ).\nThen∞/summationdisplay\nn=0Fn(x,z)tn\nn!=(1−x)ezt\next−xet. (2.1)\nPutting\nFn(x,y,z) =/summationdisplay\nσ∈Snxexc(σ)ydrop(σ)zfix(σ),\nthe formula (2.1) can be converted into the homogeneous form\n∞/summationdisplay\nn=0Fn(x,y,z)tn\nn!=(y−x)ezt\nyext−xeyt. (2.2)\nThe grammar of Dumont reads\nG={a→az, z→xy, x→xy, y→xy}. (2.3)\nLetDbe the formal derivative with respect to G.\nTheorem 2.1 (Dumont) .The following relation is valid for n≥0,\nDn(a) =aFn(x,y,z). (2.4)\nDumont’s argument can be understood as a description of the p rocedure of recursively\ngenerating permutations in the cycle notation. Recall that each cycle is written with the\nsmallest element at the beginning and the cycles are arrange d in the increasing order of the\nsmallest elements. Here we give an explanation in the langua ge of a grammatical labeling,\nwhich we call the (a,x,y,z)-labeling, both for permutations and for increasing binary trees.\nGiven a permutation σof[n], represent it in the cycle notation. Use ato signify a position\nwhere a new cycle may be formed. If iis in a1-cycle, we label it by z. If(i,j)is an arc in the\ncycle notation, that is, σ(i) =j, we label it by xifi < j=σ(i), that is,iis an excedance, or\n3byyifi > j , that is,iis a drop. Then an insertion of n+1intoσcan be formally described\nwith the aid of the grammar rules.\nWith the above grammar at disposal, one can build a complete i ncreasing binary tree to\nrecord the insertion process of generating a permutation of [n+1] from a permutation of [n],\nin the cycle notation, of course. With the help of the corresp ondence between permutations\nand complete increasing binary trees, see, Stanley [11], he re comes a labeling scheme: The\nrightmost leaf is labeled by a, each fixed point has a left leaf z, anx-leaf corresponds to an\nexcedance and a y-leaf corresponds to a drop.\nBelow is a permutation in the cycle notation with labels, cor responding to the tree in\nFigure 1,\n(y1x8y4x9y6)(z2) (y3x5)(z7)a.\n1\n4\n8 6\n92\nz3\n5 7\naxy y\nxyxyza\nFigure 1: The (a,x,y,z)-labeling for (exc,drop,fix).\nTo recover a permutation σfrom a complete increasing binary tree T, we may decompose\nTinto a forest of planted increasing binary trees by removing the edges from the root to the\na-leaf and deleting the a-leaf. Each component is either a single root or a tree for whi ch the\nroot has only one child. Such a tree is called a planted increa sing binary tree. For example,\nthe tree in Figure 1 has the decomposition as given in Figure 2 .\nClearly, a planted increasing binary tree can be regarded as a representation of a cycle\nsince a cycle can be expressed in the form of the minimum eleme nt followed by a permu-\ntation. But the permutation after the minimum element in tur n corresponds to an increasing\nbinary tree. In this way, the permutation\n(18496)(2)(35)(7)\ncan be recovered from the tree in Figure 1.\nA grammatical derivation of the generating function of the E ulerian polynomials An(x,y)\nwas given in [4]. The same reasoning can be carried over to the computation of the generating\n41\n4\n8\nxy6\n9\nxyy2\nz3\n5\nxy7\nz\nFigure 2: A forest of planted increasing binary trees.\nfunction of Fn(x,y,z). Bear in mind that the generating function with respect to th e formal\nderivative Dis defined by\nGen(w,t) =∞/summationdisplay\nn=0Dn(w)tn\nn!,\nwithwis a Laurent polynomial in the variables a,x,y,z . Note that the generating functions\nwith respect to Dpermits the multiplication property, which is equivalent t o the Leibniz rule,\nsee [4] and references therein.\nTheorem 2.2. We have\nGen(a,t) =a(y−x)ezt\nyext−xeyt. (2.5)\nProof. In virtue of the rules\nx→xy, y→xy,\nwe have the generating function\nGen(x,t) =x−y\n1−yx−1e(x−y)t,\nsee [4]. Since D(z−y) =xy−xy= 0, i.e.,z−yis a constant relative to D, we find that\nDn(ax−1) =Dn−1/parenleftbig\nax−1(z−y)/parenrightbig\n=ax−1(z−y)n,\nand hence\nGen(ax−1,t) =∞/summationdisplay\nn=0Dn(ax−1)tn\nn!=ax−1e(z−y)t. (2.6)\nBy the Leibniz rule or the product rule, we infer that\nGen(a,t) = Gen( x·ax−1,t) = Gen( x,t)Gen(ax−1,t) =a(y−x)ezt\nyext−xeyt,\nas required.\nSettinga= 1, we arrive at the relation (2.2). Setting z= 0yields the generating function\nof the derangement polynomials, see Brenti [1].\n53 The joint distribution of Roselle\nIn this section, we give an account of the generating functio n of Roselle [9] for the joint\ndistribution of the number of ascents and the number of succe ssions in a nutshell. Starting\nwith recurrence relations, Roselle employed the symbolic m ethod to accomplish the task of\ncomputation. Such an antiquate mechanism is rarely in deman d these days, but perhaps it\nshould not be completely forgotten, even though it might see m obscure or dubious and even\nif its extinction is inevitable.\n3.1 The formulas of Roselle\nLet us recall some definitions. Let n≥1, and letσbe a permutation of [n]. We assume\nthatσ0= 0. An ascent or a rise of σis an index 0≤i≤n−1such that σi< σi+1.\nThe number of ascents of σis denoted by asc(σ). An index i(1≤i≤n−1)is called a\ndescent of σifσi> σi+1. In this definition, the index nis not counted as a descent. The\nnumber of descents of σis denoted by des(σ). An index i(1≤i≤n−1)ofσis called a\nsuccession, or an interior succession, if σi+1 =σi+1. We call an index i(1≤i≤n)a left\nsuccession if σi−1+1 =σi. Mind the subtlety here concerning the choice of the index fo r a\nleft succession.\nIn order to single out ascents that are not left successions, we say that an index 1≤i≤n\nofσis a jump if i−1is an ascent but iis not a left succession, that is, σi≥σi−1+2. The\nnumber of jumps of σis denoted by jump(σ).\nFor2≤i≤n, ifiis jump, then i−1is called a big ascent by Ma-Qi-Yeh-Yeh [8], and\nthe number of big ascents of σis denoted by basc(σ). It should be noted that the position 1\nis not considered a big ascent in any event.\nLetP(n,r,s)denote the number of permutations of [n]withrascents and s(interior)\nsuccessions. For example, P(3,2,0) = 2 . The two permutations of {1,2,3}with two\nascents and no successions are 132,213. Nevertheless, 132has a left succession. While the\nterm left succession is not manifestly put to use, one can find a clue through the generating\nfunction for the number of permutations of [n]withrascents and with no left successions,\nsee Roselle [9]. As a matter of fact, he introduced the polyno mial\nP∗\nn(x) =n−1/summationdisplay\nr=1P∗(n,r)xr,\nwhereP∗(n,r)is number of permutations on [n]with the first element greater than 1, and\nwithrrises, no successions. Notice that P∗(n,r)is the number of permutations on [n]with\nrrises and no left successions. By means of P∗\nn(x), he defined the polynomial\nP∗\nn(z,x) =n/summationdisplay\nj=0/parenleftbiggn\nj/parenrightbigg\nP∗\nj(x)zn−j,\n6and derived the generating function for the joint distribut ion of number of jumps and number\nof left successions.\nTheorem 3.1 (Roselle) .We have\n∞/summationdisplay\nn=0P∗\nn(z,x)tn\nn!=(1−x)ezt\next−xet. (3.1)\nNotice that this formula coincides with (2.1) for the joint d istribution of (exc,fix). But\nthis is by no means a coincidence. As it happens, we will have t he same grammar and so we\nought to have the same story.\nDefine\nPn(z,x) =n/summationdisplay\nr=1r−1/summationdisplay\ns=0P(n,r,s)xrzs.\nRoselle showed that\nPn(z,x) =P∗\nn(xz,x)+x(1−z)P∗\nn−1(xz,x). (3.2)\nCombining the formula (3.1) and the relation (3.2) yields th e generating function of\nPn(z,x)in the following form, see [8].\nCorollary 3.2.\n∞/summationdisplay\nn=0Pn+1(z,x)tn\nn!=x(1−x)2e(xz+1)t\n(ext−xet)2. (3.3)\nProof. In light of (3.1), we find that\n∞/summationdisplay\nn=9P∗\nn(xz,x)tn\nn!=(1−x)exzt\next−xet,\n∞/summationdisplay\nn=0P∗\nn+1(xz,x)tn\nn!=/parenleftbigg(1−x)exzt\next−xet/parenrightbigg′\n.\nOwing to (3.2), an easy computation reveals (3.3).\n3.2 A grammatical labeling for left successions\nAs alluded by the grammar of Dumont, we tend to believe that th e notion of a left succession\nshould be considered as a legitimate object of the subject, b ut it does not seem to have gained\nenough recognition.\nDefine\nLn(x,y,z) =/summationdisplay\nσ∈Snxjump(σ)ydes(σ)zlsuc(σ).\n7Forn= 0, setL0(x,y,z) = 1 .\nThe following theorem shows that the polynomials Ln(x,y,z)can be generated by the\ngrammar Gin (2.3) of Dumont, that is,\nG={a→az, z→xy, x→xy, y→xy}.\nTheorem 3.3. LetDbe the formal derivative with respect to G. Forn≥0, we have\nDn(a) =aLn(x,y,z). (3.4)\nThe above theorem can be confirmed by a labeling scheme of perm utations. Assume that\nσis a permutation of [n]. Consider the position after each element σifori= 0,1,...,n ,\nwithσ0= 0. First of all, label the position after the maximum element nbya. For the\nremaining positions, if iis a jump, then label the position on the right of σibyx; ifiis a left\nsuccession, then label the position on the left of σibyz, ifiis a descent and σi/ne}ationslash=n, label\nthe position on the right of σibyy. Below is an example for such a labeling:\nx2x6y3z4y1x5x8z9a7y. (3.5)\nWrite∗for the element n+1to be inserted into σ. The change of labels can be described\nas follows. Assume that ∗is to be inserted at the position between σiandσi+1, where\n0≤i≤n.\n1. If∗is inserted at a position a, that is,σi=n, then we get nz∗aσi+1in the neighbor-\nhood, this operation is captured by the rule a→az.\n2. If∗is inserted at a position x, then we see the update of σ:σixσi+1→σix∗aσi+1.\nIn the meantime, the label aafterninσ, wherever it is, will be switched to y, because\n∗is not inserted after n. This change of labels is reflected by the rule x→xy.\n3. If∗is inserted at a position y, sinceσi/ne}ationslash=n, the update of σcan be described by\nσiyσi+1→σix∗aσi+1. In the meantime, the label aafternin the labeling of σ,\nwherever it is, will be switched to y. This change of labels is governed by the rule\ny→xy.\n4. If∗is inserted at a position z, then we have the update σizσi+1→σix∗aσi+1. In\nthe meantime, the label aafternin the labeling of σ, wherever it is, will be switched\ntoy. This change of labels is in compliance with rule z→xy.\nWe now have the same grammar for two occasions. Thus we reach a n equidistribution.\nTheorem 3.4. Forn≥1, the statistics (jump,des,lsuc) and the statistics (exc,drop,fix)\nare equidistributed over the set of permutations of [n].\n8In fact, we are going to pursue a stronger version of the above theorem, that is, a left suc-\ncession analogue of the Diaconis-Evans-Graham theorem. As can be seen, while a grammar\nmight be sufficient to guarantee an equidistribution of two s ets of statistics, it does not tell us\nexplicitly how to form a bijection. Nevertheless, there are occasions that the grammar could\nbe helpful in establishing a correspondence even with a spec ific constraint.\n3.3 Back to interior successions\nReturning to the original formulation of the joint distribu tion of Roselle, let Rn(x,y,z)de-\nnote the homogeneous form of Pn(z,x), that is,\nRn(x,y,z) =/summationdisplay\nσ∈Snxjump(σ)ydes(σ)zsuc(σ), (3.6)\nwhich we call the Roselle polynomials.\nUsing the same reasoning for the grammatical labeling for le ft successions together with\na slight alteration of the grammar, a grammatical calculus c an be carried out for the Roselle\npolynomials. Suppose that we are working with the grammar fo r left successions, but we\nwould like to avoid 1being counted as a left succession, which is labeled by z. This require-\nment can be easily met by turning to an additional label bas a substitute of the label z. That\nis to say, the rule z→xyshould be recast as b→xy. For example, we should start with\nthe initial labeling 0b1ainstead of 0z1a. As for the original labels a,x,y,z , their roles will\nremain unchanged. Thus we meet with the mended grammar\nG={a→az, b→xy, x→xy, y→xy,z→xy}. (3.7)\nLetDbe the formal derivative of Gin (3.7). We have\nD(ab) =abz+axy,\nwhich is the sum of weights of the two permutations\n0b1z2a0,0x2a1y0.\nIn general, for n≥0, the following relation holds\nRn(x,y,z) =Dn−1(ab)|a=1,b=1. (3.8)\nThe grammatical calculus shows that the generating functio n for the Roselle polynomials\nis essentially a product of the generating function of Ln(x,y,z)and the generating function\nof the bivariate Eulerian polynomials.\nTheorem 3.5. We have\nGen(ab,t) =a(y−x)ezt\nyext−xeyt/parenleftbiggx−y\n1−yx−1e(x−y)t−x+b/parenrightbigg\n. (3.9)\n9Proof. By the Leibniz rule, we get\nGen(ab,t) =∞/summationdisplay\nn=0Dn(ab)tn\nn!= Gen(a,t)Gen(b,t).\nSinceD(b) =D(x) =xy, it follows that\nGen(b,t) = Gen( x,t)−x+b=x−y\n1−yx−1e(x−y)t−x+b,\nwhich, together with Theorem 2.2, yields (3.9).\nSettinga= 1,y= 1,b=x,z=xz, and treating the initial two elements 01of a\npermutation as an ascent, we see that\nDn−1(ab)|a=1,y=1,b=x,z=xz=x/summationdisplay\nσ∈Snxjump(σ)(xz)suc(σ)=/summationdisplay\nσ∈Snxasczsuc, (3.10)\nwhich is the generating function for the joint distribution of(asc,suc) over permutations on\n[n], that is,\nPn(z,x) =Dn−1(ab)|a=1,y=1,b=x,z=xz.\nMoreover,\n∞/summationdisplay\nn=0Pn+1(z,x)tn\nn!= Gen(ab,t)|a=1,y=1,b=x,z=xz=x(1−x)2e(z+1)t\n(ext−xet)2,\nwhich is in accordance with (3.3).\nWe finish this section with a relation between Rn(x,y,z)andLn(x,y,z).\nTheorem 3.6. Forn≥0, we have\nRn+1(x,y,z) =Ln(x,y,z)+n/summationdisplay\nk=1/parenleftbiggn\nk/parenrightbigg\nAk(x,y)Ln−k(x,y,z), (3.11)\nwhere for k≥1,Ak(x,y)are the bivariate Eulerian polynomials.\nThis relation admits a combinatorial interpretation. Let Tbe a complete increasing bi-\nnary tree on [n+ 1]. Suppose that we wish to interpret Rn+1(x,y,z)in terms of complete\nincreasing binary trees. We may adopt the following labelin g forLn+1(x,y,z), except that\nif the root of Thas az-leaf, we should label it by 1rather than z. If this is the case, then the\nsubtree of Tcan be viewed a complete increasing tree on [n]with the labeling for Ln(x,y,z).\nIf the root of Thas a nonempty left subtree, then this subtree does not have a nyz-leaves,\nwhich can be reckoned as a labeling for the Eulerian polynomi als, and so we are done.\n104 A analogue of the Diaconis-Evans-Graham theorem\nThe main result of this paper is a left succession analogue of the Diaconis-Evan-Graham\ntheorem on successions and fixed points of permutations. The grammar of Dumont can be\nutilized to produce a bijection, which implies an equidistr ibution of the statistics (jump,des)\nand(exc,drop) restricted to a given set of left successions and the same set of fixed points.\nConsequences and applications will be discussed.\nFor a permutation σ∈Sn, define\nM(σ) ={i|1≤i≤n−1, σi+1 =σi+1},\nG(σ) ={i|1≤i≤n−1, σi=i},\nF(σ) ={i|1≤i≤n, σi=i}.\nIt should be stressed that the index nis excluded in the definition of G(σ). Given a subset\nI⊆[n−1], denote by Mn(I)the set of permutations of [n]withIbeing the set of (interior)\nsuccessions, and denote by Gn(I)the set of permutations σ∈Snsuch that G(σ) =I.\nSimilarly, Fn(I)denotes the set of permutations σof[n]such that F(σ) =I.\nTheorem 4.1 (Diaconis-Evans-Graham) .Letn≥1andI⊆[n−1]. Then there is a bijection\nbetweenMn(I)andGn(I).\nFor the special case I=∅, a permutation without successions is called a relative der ange-\nment. Let Dndenote the number of derangements of [n], and letQndenote the number of\nrelative derangements of [n]. Roselle [9] and Brualdi [2] deduced that\nQn=Dn+Dn−1. (4.1)\nA bijective proof of this relation was given in [3], appealin g to the first fundamental transfor-\nmation. Taking I=∅, a permutation in Gn(I)may or may not have nas a fixed point. The\npermutations in these two cases are counted by Dn−1andDn, respectively. Thus the I=∅\ncase of the proof of the Diaconis-Evans-Graham theorem yiel ds a combinatorial interpreta-\ntion of (4.1).\nHere comes the question of what happens for left successions . Define the set of left\nsuccessions of a permutations as\nL(σ) ={σi|1≤i≤n, σi−1+1 =σi}.\nFor anyI⊆[n], denote by Ln(I)the set of permutations σ∈Snsuch that L(σ) =I. We\nestablish the following correspondence, and the proof is a g rammar assisted bijection.\nTheorem 4.2. Forn≥1and anyI⊆[n], there is a bijection from Ln(I)toFn(I)that maps\nthe pair of statistics (jump,des) to the pair of statistics (exc,drop) .\n11Proof. Given a permutation σon[n], we wish to construct a complete increasing binary T\nwith the(a,x,y,z)-labeling. The map can be described as a recursive procedure . Forn= 1,\nthe permutation z1ais mapped to the complete increasing tree with one internal v ertex1.\nAssume that σ=σ1σ2···σnis a permutation of [n]and thatTis the tree corresponding\ntoσ. For1≤i≤n, the position iis referred to the position immediately before σi, whereas\nthe position n+1is meant to be the position after σn.\nTo build a bijection, we need to inductively maintain a coord inated property of σandT.\nBesides having the same weight, they should be synchronized in a certain sense to keep the\nprocess running till the completion of the task. More precis ely, we say that the labeling of\nσis coherent with the labeling of Tprovided that the following conditions are satisfied. In\nfact, these properties should be preserved after each updat e.\n• If the position iinσis labeled by x, then the vertex σiinThas ax-leaf;\n• If the position iinσis labeled by y, then the vertex σi−1+1inThas ay-leaf;\n• If the position iinσis labeled by z, then the vertex σiinThas az-leaf.\nSuppose that ∗=n+1is to be inserted into σ. It is necessary to find out how to update\nthe treeTaccordingly. Now that there are n+ 1(insertion) positions for σand there are\nn+1leaves for T, it suffices to define a map from the set of positions to the set o f leaves of\nTwith the understanding that when ∗is inserted at position, say i,Twill be updated to T′by\nturning the corresponding leaf of Tinto an internal vertex ∗. Denote by σ′the permutation\nproduced from σby inserting ∗at the position i. There are four cases on the ground of the\nfour rules of the grammar.\n1. If∗is inserted at a position labeled by a, we add ∗toTat the position of the a-leaf.\nThis operation is consistent with the rule a→az.\n2. For a label zat the position i, by the induction hypothesis, we know that the vertex σi\ninT′has az-leaf, so we apply the rule z→xyto thisz-leaf to update T. Notice that\nwhen∗is inserted, the label aon the right of ninσwill be switched to y. In view of\nthe construction, this y-label corresponds to the y-leaf of∗inT′. By inspection, we\nfind that the labeling of σ′is coherent with the labeling of T′.\n3. When the insertion occurs at position ilabeled by x, by the induction hypothesis, we\nknow that the vertex σiinThas ax-leaf. Then we apply the rule x→xyto this\nleaf. Notice that the y-leaf of∗inT′corresponds to the y-label on the right of nby\nthe construction. Again, it can be checked that the labeling ofσ′is coherent with the\nlabeling of T′.\n124. For a position ilabeled by y, by the induction hypothesis, we know that the vertex\nσi−1+1inThas ay-leaf. Then we apply the rule y→xyto this leaf. In this case, the\nlabeling of σ′remains coherent with the labeling of T′.\nSo far we have provided a procedure to update Tdepending on where the element ∗is\ninserted into σ. Moreover, every stage of this procedure is reversible. The detailed examina-\ntion is omitted. Notice that the grammar ensures that the map is weight-preserving, that is,\nthe weight of σequals that of T.\nIt should be added that a left succession iis created in σwhenever a vertex σiwith a left\nz-leaf is created in T. Meanwhile, a left succession iis destroyed in σwhenever a z-leaf\nwith parent σiis destroyed. This completes the proof.\nTaking the following permutation σwith2jumps and 1descent as an example,\nz1x3y2x4z5a,\nwhereL(σ) ={1,5}. Figure 3 illustrates how to build the corresponding trees s tep by step,\nwhere a label in boldface indicates where an insertion takes place.\nWe remark that in principle one can reformulate the above bij ection in a way without\nresorting to increasing binary trees as an intermediate str ucture. In doing so, all we need is\na comparative scrutiny of the two recursive procedures to ge nerate permutations. One relies\non the linear representation and the other on the cycle notat ion.\nForn= 3, the correspondence is given in the table below. The cases wh enF(I) =∅are\nnot listed, such as I={1,2}.\nI⊆[n]Ln(I)→Fn(I)(jump,des)→(exc,drop)\n∅2 1 3→(123)\n3 2 1→(132)(2,1)\n(1,2)\n{1} 1 3 2→(1)(23) (1,1)\n{2} 3 1 2→(13)(2) (1,1)\n{3} 2 3 1→(12)(3) (1,1)\n{1,2,3}1 2 3→(1)(2)(3) (0,0)\nWe conclude with discussions about consequences of the abov e correspondence. Given a\nsubsetIof[n], denote by Fn(I)the set of permutations of [n]withIbeing the set of fixed\npoints. Let Ln(I)stands for the set of permutations of [n]withIbeing the set of elements\nwhose indices are left successions.\nThe special case I=∅of Theorem 4.1 is related to the notion of a skew derangement a s\ntermed in [3]. Returning to the definition of F(σ), as it is not required that σ(n)/ne}ationslash=n, we\n13Permutations Trees\nz1a1\nza\nz1z2a1\nz2\nza\nz1x3a2y1\nz2\n3\nxya\nz1x3y2x4a1\nz2\n3\nx4\nxya\nz1x3y2x4z5a1\nz2\n3\nx4\nxy5\nza\nFigure 3: An example.\nmay regard σas a one-to-one map ffrom[n]to the set {0,1,...,n−1}. In the case I=∅,\na permutation σis inFn(I)if and only if f(i)/ne}ationslash=ifor anyi, that is,fis a skew derangement.\nSpecializing Theorem 4.2 to I=∅, we obtain another combinatorial interpretation of (4.1),\nalong with the following property.\nCorollary 4.3. There is a one-to-one correspondence between the set of perm utations on\n[n]without left successions and the set of permutations withou t fixed points such that the\nstatistics (jump,des) are transformed into the statistics (exc,drop) .\nNotice that a relative derangement has a left succession if a nd only if it begins with the\nelement1. In this case, we may treat the element 1as zero and treat the rest of the permutation\nas a permutation of [n−1]. This yields a permutation of [n−1]without left successions.\n14Thus Theorem 4.2 does offer an alternative way to establish ( 4.1) combinatorially. Strictly\nspeaking, the set of relative derangements of [n]can be divided into two classes. The first\nconsists of those with no left successions, as counted by Dn, and the second consists of those\nbeginning with 1, as counted by Dn−1.\nBy comparing Theorem 4.1 with Theorem 4.2, we are led to a corr espondence between\nleft successions and interior successions.\nCorollary 4.4. LetI⊆[n−1]. WriteLn(I)for the set of permutations of [n]withI\nbeing the set of elements whose indices are left successions and write Mn(I)for the set of\npermutations of [n]withIbeing the set of interior successions. Then there exists a bi jection\nbetweenLn(I)andMn(I).\nAcknowledgment. This work was supported by the National Science Foundation o f\nChina.\nReferences\n[1] F. Brenti, Unimodal polynomials arising from symmetric functions, Proc. Amer. Math.\nSoc., 108 (1990) 1133–1141.\n[2] R.A. Brualdi, Introductory Combinatorics, 5th ed., Pea rson/Prentice Hall, 2009.\n[3] W.Y .C. Chen, The skew, relative, and classical derangem ents, Discrete Math., 160\n(1996) 235–239.\n[4] W.Y .C. Chen and A.M. Fu, A Context-free grammar for the e-positivity of the trivariate\nsecond-order Eulerian polynomials, Discrete Math., 345 (2 022) 112661.\n[5] P. Diaconis, S.N. Evans and R. Graham, Unseparated pairs and fixed points in random\npermutations, Adv. in Appl. Math., 61 (2014) 102–124.\n[6] D. Dumont, Grammaires de William Chen et d´ erivations da ns les arbres et arbores-\ncences, S´ em. Lothar. Combin., 37 (1996) Art. B37a.\n[7] D. Foata and M.-P. Sch¨ utzenberger, Th´ eorie g´ eom´ etr ique des polynˆ omes eul´ eriens,\nLecture Notes in Mathematics, V ol. 138, Springer-Verlag, B erlin, 1970.\n[8] S.-M. Ma, H. Qi, J. Yeh and Y .-N. Yeh, On the joint distribu tions of succession and\nEulerian statistics, arXiv:2401.01760.\n[9] D.P. Roselle, Permutations by number of rises and succes sions, Proc. Amer. Math. Soc.,\n19 (1968) 8–16.\n15[10] H. Shin and J. Zeng, The q-tangent and q-secant numbers via continued fractions, Eu-\nropean J. Combin., 31 (2010) 1689–1705.\n[11] R.P. Stanley, Enumerative Combinatorics, V ol. I, seco nd ed., Cambridge Univ. Press,\nCambridge, 2012.\n16"    },    {        "title": "2402.02753v1.Crosstalk_Attacks_and_Defence_in_a_Shared_Quantum_Computing_Environment.pdf",        "content": "Crosstalk Attacks and Defence in a Shared Quantum Computing Environment\nBenjamin Harper,1, 2Behnam Tonekaboni,2Bahar Goldozian,2Martin Sevior,1and Muhammad Usman1, 2\n1School of Physics, The University of Melbourne, Parkville, 3010, Victoria Australia\n2Quantum Systems, Data61, CSIRO, Clayton, 3168, Victoria Australia\nQuantum computing has the potential to provide solutions to problems that are intractable on\nclassical computers, but the accuracy of the current generation of quantum computers suffer from\nthe impact of noise or errors such as leakage, crosstalk, dephasing, and amplitude damping among\nothers. As the access to quantum computers is almost exclusively in a shared environment through\ncloud-based services, it is possible that an adversary can exploit crosstalk noise to disrupt quan-\ntum computations on nearby qubits, even carefully designing quantum circuits to purposely lead to\nwrong answers. In this paper, we analyze the extent and characteristics of crosstalk noise through\ntomography conducted on IBM Quantum computers, leading to an enhanced crosstalk simulation\nmodel. Our results indicate that crosstalk noise is a significant source of errors on IBM quantum\nhardware, making crosstalk based attack a viable threat to quantum computing in a shared environ-\nment. Based on our crosstalk simulator benchmarked against IBM hardware, we assess the impact\nof crosstalk attacks and develop strategies for mitigating crosstalk effects. Through a systematic\nset of simulations, we assess the effectiveness of three crosstalk attack mitigation strategies, namely\ncircuit separation, qubit allocation optimization via reinforcement learning, and the use of spectator\nqubits, and show that they all overcome crosstalk attacks with varying degrees of success and help\nto secure quantum computing in a shared platform.\nI. INTRODUCTION\nIn the pursuit of achieving scalability and performance\nin quantum computing, efficient control and mitigation\nof noise are essential for the high-fidelity implementa-\ntion of quantum algorithms of practical interest. Noise\nin quantum computers may take several forms including\nenvironmental noise that causes decoherence in qubits via\ndephasing and relaxation processes; also, computational\ngates and measurement operations on qubits may result\nin imperfect fidelities [1, 2]. In addition to the aforemen-\ntioned sources of noise, qubits connectivity—which is es-\nsential in executing multi-qubit gates—could lead to the\nemergence of unwanted correlated dynamics. These cor-\nrelated dynamics give rise to an important type of noise\nknown as crosstalk noise [3–6], which results in a decline\nin quantum coherence and the introduction of errors in\nquantum computations.\nIt is noted that the word crosstalk is often used in the\nfields of electrical engineering and communication theory\nto describe the phenomenon of unwanted signal coupling\nacross different pathways [7]. However, in quantum com-\nputing, the term is adapted to describe a variety of phys-\nical phenomena when an undesired influence is exerted\nby one subsystem of an experimental apparatus, such as\na qubit or control pulse, on another qubit. Consider su-\nperconducting qubits, such as those used in IBM devices,\nas an example. The qubits in these devices are placed in\na certain topology, with each qubit connected to one,\ntwo, or more neighbor qubits. A cross-resonance pulse\nwill then be utilized to apply a two-qubit gate to a pair\nof linked qubits [5, 6]. While this always-on coupling,\nutilises the two-qubit gates, they also result in an unde-\nsirable crosstalk effect. This is depicted in the top image\nof FIG. 1(a), where a single qubit gate pulse targets only\nthe center qubit, but due to physical coupling, there isundesired crosstalk on the neighbor qubits. Even with\nquantum platforms without always-on coupling, such as\natomic qubits, the profile of the laser gate pulse may leak\nto the side atoms and adversely affect them [8]. The bot-\ntom image of FIG. 1(a) illustrates this mechanism.\nWhile crosstalk is expected to be present in all quan-\ntum devices and suppressing crosstalk in quantum com-\nputers is an active topic of research (see [9–11] as exam-\nples), here we investigate crosstalk as a substantial secu-\nrity concern. It is almost certain that functional quantum\ncomputers will be operated as multi-user facilities, exe-\ncuting circuits for multiple users simultaneously, includ-\ning public access through cloud platforms. Since the op-\nerations of neighboring circuits affect each other through\ncrosstalk, it is possible to craft an algorithm that can in-\nterfere with nearby normal operations, causing programs\nto fail, provide faulty results, or leak information. This\ncould be exploited by an adversary or deployed against\nan adversary’s quantum computer operations [12]. An\nillustration of one such scenario is shown in FIG. 1(b) ex-\nhibiting a malicious actor interfering with an algorithm\nemployed on a multi-user quantum computer.\nTo demonstrate a crosstalk attack, we executed a\nthree-qubit Grover algorithm with a target answer of\n‘110’, on an IBM device. The experiment encompassed\ntwo scenarios: one without any external interference\nand the other with the imposition of a crosstalk attack.\nThe crosstalk attack was implemented through a series\nof CNOT gates on two proximate qubits. Top row of\nFIG. 1(c) shows the circuit diagram of the experiment\nand the histogram in this figure shows the results of the\nexperiment. In the absence of any attack, the majority\nof counts are the target answer. In contrast, when sub-\njected to the attack, the target result is wiped out and the\nhistogram shows an approximately uniform distribution\nof counts.arXiv:2402.02753v1  [quant-ph]  5 Feb 20242\nGrover's Algorithm\n(a) (b) (c)\nShared\nQC\nCtrl Pulse\nTwo-qubit gate pulse\nPulse leakageTarget\nFIG. 1. (a) Illustration of crosstalk effect mechanism. (Top) shows the crosstalk effect due to the always on coupling between\nqubits, the coupling is demonstrated with solid lines. Although the control pulse (red wavy arrow) aims only the target\nqubit (green) in the middle, neighbour qubits (blue) are affected. (Bottom) depicts the leakage of a two-qubit gate that aims\nonly two middle qubits (green). Although there is no link between the qubits, the leakage of the two-qubit gate affects the\nneighbour qubits. (b) Illustration of a future shared Quantum Computer with multiple users each running their own programs.\nA malicious actor has gained access and can employ crosstalk attacks against other programs (background is IBM 433-Qubit\nOsprey device). (c) Top: We implement Grover’s algorithm with the target of “6” on a real 5-qubit IBM Quantum computer.\nQubits 0-2 implement Grover’s algorithm while on qubits 3 and 4 we implement a series of CNOT gates. (Bottom) When the\nqubits 3 and 4 are not operating, the circuit returns the blue histogram, showing normal operation. When we also apply CNOT\noperations on qubits 3 and 4, we instead observe the orange histogram showing crosstalk from the CNOT gates completely\noverwhelming the operation of the circuit.\nIt is important to note that crosstalk attacks are dis-\ntinctly different from malware-based attacks on conven-\ntional computers. Crosstalk noise is inherent in the ac-\ntual physics of the operation of the quantum computer\nand this kind of attack vector is simply not possible in\nconventional computers. Therefore, defending against\nthese attacks requires strategies in addition to normal\nsecurity procedures.\nDespite significant evidence highlighting the potential\nsecurity implications of crosstalk attacks on quantum\nprocessors, there is currently a lack of quantitative mod-\nelling and practical mitigation strategies for real devices.\nAddressing this gap is the primary objective of this pa-\nper. The structure of the paper is as follows. First in\nSec. II, we characterize crosstalk noise on an IBM device.\nThis characterization will be the framework for simulat-\ning circuits under the effects of crosstalk noise in the rest\nof the paper. Then, in Sec. III, we present three tech-\nniques to detect and mitigate the effects of a crosstalk at-\ntack on a circuit. The first technique (Sec. IIIA) employs\nunused buffer qubits to distance the circuits from poten-\ntial crosstalk attacks. This technique also provides an\nindication of the extent of crosstalk noise in an IBM de-\nvice, laying the foundation for basic understanding of the\nseverity of such noise mechanism. The second technique(Sec. IIIB) employs a reinforcement machine learning ap-\nproach and uses the noise model of the device to position\ncircuits such that the potential for crosstalk noise to im-\npact their performance is minimised. The final technique\n(Sec. IIIC) employs extra qubits, but in contrast to first\ntechnique these extra qubits are used as spectator qubits.\nBy measuring the spectator qubits at the correct inter-\nval, it is possible to detect the presence of an attack with\na high level of confidence. Finally, in Sec. IV, we discuss\nthe importance of the security implications of crosstalk\nattacks and our successful strategies to mitigate them.\nII. CHARACTERISING AND SIMULATION OF\nCROSSTALK ON IBM DEVICES\nEssential to our work in this paper is a realistic char-\nacterisation of the noise due to crosstalk in a physical\ndevice. For this, we opted to use idle tomography [13].\nThe Python library PyGSTi [14] is used to perform the\nidle tomography calculations, and also simulate circuits\nwith crosstalk noise. We benchmark crosstalk on a real\nquantum device, ibm hanoi, and use this data to pre-\npare the crosstalk simulator. We then run validation\ncircuits on the crosstalk simulator, the real device, and3\nother existing quantum software simulators to verify that\nthe crosstalk simulator is an accurate representation of\nreal devices.\nA. Idle Tomography\nIdle tomography (IDT) [13], as the name suggests,\nseeks to characterise the idle error rates of qubits on a\nphysical device. To do this, a family of tomography cir-\ncuits is prepared. The circuits we used consist of three\nparts; preparation, idling and un-preparation, as shown\nin FIG. 2(a). In the preparation step, each circuit ini-\ntialised in the ∣0⟩state and is prepared in a unique Pauli\nbasis. In the second phase, the qubits are left to idle\nfor a variable period of time (measured in gate lengths).\nFinally, the state preparation is undone. In a noise-free\nquantum computer, this returns the state to ∣0⟩n. In a\nnoisy device, however, the measurement results will be\nperturbed away from the target distribution. A large\nnumber of circuits in this form are generated, where only\nthe Pauli state preparation is changed.\nThe noise channel modelled by idle tomography in\nPyGSTi is the reduced HSA model [13],\nL=H+S+A. (1)\nThis model has three components; Hamiltonian errors\n(H) are the rate of over (under) rotation induced on a\nspecific Pauli axis. The stochastic error rate ( S) is the\nprobability of a random Pauli X,YorZerror. Finally\naffine errors ( A) correspond to non-unitary noise pro-\ncesses.\nThe information contained in the probability distribu-\ntion of a circuit’s output scales exponentially with the\nnumber of qubits — as there are an exponential num-\nber of outcomes. Conversely, the number of parameters\nin the reduced HSA model scales polynomially with the\nnumber of qubits in the device. As such, the number\nof experiments which need to be performed to bench-\nmark a device increases logarithmically with the number\nof qubits [13], though the number of shots for each exper-\niment to obtain a good probability distribution increases\nexponentially.\nWhen considering the noise due to gates on a quantum\ncomputer, we typically look at the qubits on which the\ngate is applied. In crosstalk noise, we are interested in\nthe noise that a gate causes on other qubits around it.\nIn effect, we want to know the idle error rate of qubits in\nthe device when a gate is being applied to neighbouring\nqubits. Thus, to quantify the crosstalk error rates due\nto an m-qubit gate Uon an n-qubit device, we choose m\nqubits on which we want to benchmark the gate. On the\nremaining n−mqubits, idle tomography is performed.\nWhile the IDT qubits are idling, the gate Uis performed\nin parallel. The presence of crosstalk in the device means\nthatUwill cause errors in the IDT qubits.\nIn this work, we benchmark a 27-qubit IBM device —\nibmhanoi. We benchmark crosstalk due to the 2-qubitCNOT gate, which requires running 165 benchmark cir-\ncuits per CNOT benchmarked. These results were then\npostprocessed classically to produce a realistic simulator\nwith crosstalk noise.\nB. Simulation of Crosstalk\nHaving created a crosstalk-aware classical simulator,\nit is necessary that we validate that its performance\nmatches that of the real device it is based on. To per-\nform this validation of the crosstalk simulator, we ran\nfive benchmark circuits and compared the results to the\nreal quantum device, as well as other simulation meth-\nods. Ideally, our simulator will be a better model of the\nreal device in circuits with a large amount of crosstalk,\ninduced by CNOT gates. The output of these experi-\nments is a set of counts, which is used to approximate\nthe probability distribution of that quantum circuit. A\ngood simulator should have a similar probability distri-\nbution to the output distribution of the real device. The\nmetric used to measure performance is the Total Varia-\ntional Distance (TVD) [15], which measures the distance\nbetween probability distributions in the L∞metric. The\nTVD is explained further in appendix A.\nIn FIG. 2(b) we show TVD relative to the real device\nfor three scenarios: 1) No Noise , a noise free simula-\ntion of the circuit, 2) Dephasing Noise , simulation using\ndephasing noise from the builtin noise model in IBM’s\nQISKIT package which does not include crosstalk, and\n3)Crosstalk Noise , the noisy simulation that we devel-\noped for this paper. The TVD is measured relative to the\nresults from running each circuit on the real device. As\nsuch, a lower TVD means that the simulator is a better\nrepresentation of the real device. In this case, crosstalk\nnoise is a significant source of noise in the device, hence\nthe crosstalk simulator is closest in TVD to the real de-\nvice. We then compare these three scenarios on five dif-\nferent test cases: 1) The syn10 and 2) syn30 test cases\nare synthetic circuits of a repeated CNOT next to an\nidling qubit — similar to the idle tomography circuits\nused to derive the noise models, 3) the grover3 test is\na 3-qubit Grover’s algorithm with two iterations, 4) the\ntoffoli test is a sequence of repeated Toffoli gates, which\nis itself decomposed into CNOT gates and single qubit\nrotations, 5) the attack test case is a Grover’s algorithm\ncircuit in the presence of a CNOT attack. Circuit di-\nagrams for these test circuits are given in appendix B.\nFIG. 2(c) and (d) show the simulated probability distri-\nbution for two of these test cases, illustrating how the\ncrosstalk simulator counts are much closer to the real\ndevice than other simulation techniques.4\n(a)\nIdle Tomography Circuit\n...\n...n−2Pauli Pauli−1\nqa\nqb\n(b)\n(c)\n(d)\nFIG. 2. (a) Example of an idle tomography circuit for benchmarking crosstalk. Here we are benchmarking the crosstalk due\nto a CNOT between qubits qaandqb, with idle tomography being performed on the remaining n−2 qubits on the device. The\ntomography qubits are prepared into a Pauli basis state, then left to idle while CNOTs are applied between qubits qaandqb.\nAt the end of the circuit, the state preparation is undone, leaving the qubits in the ∣000⟩state in a noise-free environment. (b)\nWe now investigate the performance of our simulator through benchmarking. A variety of benchmark circuits are run on a real\ndevice, and three simulators. A description of the simulation techniques and benchmark circuits is provided in the main text.\nPerformance of the simulation techniques is judged using Total Variational Distance (see text), with the device noise at 0 used\nas a reference point. Lower is better. (c) and (d) present the measured counts in two of the test cases, (c) Toffoli, (d) Grover.\nIn both cases the target circuit output is ∣110⟩. The target output distribution for each simulation technique is to match the\nexperiment results.\nIII. CROSSTALK DETECTION AND\nMITIGATION\nIn this section we explore various techniques to detect\nand/or mitigate a crosstalk attack. For all the techniques\ndiscussed, we use the same model of attack; that is, a\nhostile user with the ability to craft a repeated CNOT\ncircuit (illustrated in FIG. 1(c)) maximising crosstalk.\nWe note that in principle a variety of attacks can be\ndesigned by using different types of quantum gates or\ntheir combinations, as many quantum gates can induce\ncrosstalk [16]. However, in this work we use only CNOT\ngates as a proof-of-concept demonstration.\nA. Mitigation by Circuit Separation\nWe begin with a study of how the strength of crosstalk\nchanges across the device. While generally crosstalk is\na non-local effect, such that quantum gates may affect\nneighbouring qubits, we expect it to be relatively lo-\ncal, i.e. the qubits most affected by crosstalk from anoperation are the ones physically closest to that opera-\ntion. Here we define the radius of separation, r, to be\nthe number of idle qubits on the shortest path between\ntwo algorithms on a device’s graph of connectivity. Note\nthat this may not correspond perfectly to the physical\ndistance between qubits, it is a useful approximation. A\nrecent study [16] has found that while adjacent computa-\ntions can adversely affect each other, only a small radius\nof separation is required to mitigate this. Our work ex-\ntends further, by exploring the effect when one of the\ncomputations is not useful, but in fact a CNOT-dense\nattack designed specifically to disrupt neighbouring com-\nputations.\nTo test this, we created a circuit containing Grover’s\nalgorithm, and placed next to it a repeated CNOT attack\nas described above, where every layer in the Grover cir-\ncut is accompanied by a CNOT gate on the neighbouring\nattack qubits. The distance of separation between the al-\ngorithm and the attack was then increased progressively,\nand the resulting probability distributions are compared\nin FIG. 3. The results show a high level of variance in the\neffect of crosstalk on the device — while some configura-5\ntions with a separation of zero between the circuit and the\nattack show a significant deterioration in performance,\nsome configurations are nearly unaffected. Increasing\nthe radius of separation between circuits increases the\naverage fidelity, and also reduces the variance between\noutcomes. This indicates that the radius of separation\nis unnecessarily heavy handed — FIG. 3(a) shows how a\nsignificant fraction of the device must be left idle for even\na small radius of separation, while FIG. 3(b) shows that\nthat for some configurations, only a small or zero sepa-\nration is necessary to prevent interference. This implies\nthat intelligent selection of qubits could allow circuits to\nbe placed close to each other without crosstalk interfer-\nence between them. To address these issues, a machine\nlearning approach is developed below, which will scale\nto larger devices and automate the process of identifying\ngood qubits.\nB. Mapping Circuits with Reinforcement Learning\nAs discussed in the introduction, crosstalk error arises\nfrom the unwanted effects of controls on nearby qubits.\nConsequently, the allocation of physical qubits for a given\nalgorithm will influence the impact of crosstalk error.\nThus, finding the optimal qubit allocation is a crucial\ntask to achieve the optimal accuracy of computation. In\nthis section, we utilize reinforcement learning (RL) to\nmitigate crosstalk noise by optimizing qubit allocation.\nThe proposed method improves circuit fidelity by identi-\nfying and implementing practical mapping computations\nonto qubits that show reduced crosstalk errors.\nOur method uses RL, a process in which an algorithm\niteratively refines its decision-making through feedback\nfrom a structured training regimen [17]. In our specific\ncontext, the algorithm learns to map a target circuit to\nphysical qubits, taking into account the quantum com-\nputer’s noise profile, particularly crosstalk, and identify-\ning the mapping that minimizes errors. Previous stud-\nies [18, 19] have demonstrated the effectiveness of RL\nin reducing such errors. The proposed method will be\ndiscussed in more detail in the following section.\n1. Methodology and Definition of Framework Components\nOur framework aims to develop a trained policy net-\nwork capable of effectively mapping quantum circuits to\nphysical qubits, which is illustrated in FIG. 4. The main\ncomponents of an RL method are action ,RL-state and\npolicy . In our model these components are as follows.\nThe “action” in our RL model represents the allocation\nof qubits to physical qubits on the quantum computer.\nThese actions are constrained by the hardware’s topo-\nlogical limitations, dictating the possible interactions be-\ntween qubits. The number of actions is a hyperparameter\nof our model. Central to our RL model’s framework is the\n“RL-state,” which represents the current configuration ofr=0\nr=1\nr=2Algorithm\nAttack\nIdle\na)\nb)\nFIG. 3. a) Circuits may be placed on the device with a\nseparation radius between them. Here we place an circuit\nperforming Grover’s algorithm in green, while in red is a\nrepeated CNOT attack. Attacks may be placed anywhere\noutside the radius of separation. b) When the circuits are\ndirectly adjacent, the output fidelities of Grover’s algorithm\nare largely noise. If the separation radius between the circuits\nis increased, the output fidelities return towards the ideal dis-\ntribution.\nthe quantum circuit. The RL state includes factors such\nas the number of qubits and the sequence of quantum\ngates.\nThe policy network acts as the decision-making compo-\nnent within the reinforcement learning framework. Our\npolicy is represented by a neural network with two lay-\ners, which through the use of gradient descent, learns to\noptimize the mapping of qubits to their physical counter-\nparts. Through an iterative process shown in FIG. 4, or\nas psuedocode in algorithm 1, the neural network contin-\nuously learns by processing the current state of the quan-\ntum circuit and evaluating each mapping’s effectiveness.\nThe effectiveness of each action is assessed through a\ndesigned reward function. The reward is calculated in6\nInitial \nPolicy \nnetwork\n3H\nHH\nH\nX\nHH\nH 21\n4HH\nH\nHH\nHH XLearning Loop over 500,000 circuits\nChoose\nanactionEvaluate \nRewardUpdate\nPolicyChoose \na circuitApply \ntheActionTrained \nPolicy \nnetwork\n  \n \n  \n  \n \n \n   \n    \n  \n    \n    \n  \n  \n    \n    \n  \n    \nq1\nq2\nq3\nq4\nSet of actions\nSet of target circuitsLoop over the set of actions on each circuit\nFIG. 4. The flowchart shows the learning cycle for optimizing quantum circuit mapping. It begins with an initial, untrained\npolicy network. In each iteration, the network picks a circuit, and for each circuit, it evaluates different mappings (actions).\nThe number of actions is defined as a hyperparameter within the policy network. The efficacy of each action is assessed through\na reward system, which informs the policy network. After training, the trained policy network can be applied to a new circuit,\nusing its learned strategies to predict the optimal qubit mapping. Psuedocode describing the learning cycle is also shown in\nalgorithm 1\nterms of the fidelity of the executed circuit. This reward\nguides the policy network to identify the optimal map-\npings. Over time, the network learns to choose mappings\nthat enhance circuit performance consistently. After the\npolicy network has completed a set of actions and re-\nceived the corresponding feedback, it updates its policy\nusing gradient descent. Once the policy is updated, the\nnetwork proceeds to the next quantum circuit. The dark\nblue loop shown in the FIG. 4 encircles the set of tar-\nget circuits, which are the training data for our model.\nIn each loop a new quantum circuit is introduced to the\npolicy network. The network processes this circuit to de-\ncide which action to take. Once an action is taken, the\nnetwork reviews the results to learn from its decision.\nThis review process is critical because it provides feed-\nback that the policy network uses to refine its parameters.\nAs a result, the policy network’s ability to select the most\nrewarding actions is progressively enhanced through this\niterative learning.\nOnce training is complete, the trained policy network\ncan receive any quantum circuit and efficiently predict its\noptimal mapping. This is a critical feature of the model,\nas it allows for practically applying the policy network’s\nlearned strategies to new, unseen quantum circuit con-\nfigurations.\nAnother significance of our method is its adaptabil-\nity to varying noise. Since noise landscape in quantum\ncomputers changes on daily basis adaptability to is essen-\ntial. Therefore, the ability to update mappings without\ncomplete re-training is crucial. This process begins with\na trained reinforcement learning model familiar with a\nspecific noise environment. The model then undergoespartial re-training using data that reflects the latest noise\nprofiles, which could influence qubit allocation.\n2. Results\nTo achieve the objective of having a robust model, we\nneed a large set of data. Note that covering every pos-\nsible permutation of circuits is time consuming and im-\npractical due to the massive variety of potential quantum\ncircuit combinations. Despite this barrier, our data set\nis sufficient for our object as we discuss below.\nThe number of circuits that the model trains are de-\nfined as a hyperparameter, we trained on a dataset con-\ntaining 500 ,000 different circuits. These training circuits\nwere randomly generated, with a uniform distribution of\ngates, and a uniform random number of up to 5 qubits\nand 20 gates chosen. We included various circuit con-\nfigurations with different numbers of qubits, gates, and\ntypes of gates to ensure our dataset was comprehensive.\nIt allows our model to learn how to handle a wide selec-\ntion of quantum circuit setups effectively. A repository\nof these circuits is available upon request.\nExamples of our trained model are illustrated in\nFIG. 5. In this figure, we compare the performance of\nthe RL-based circuit mapping to that of an initial map-\nping. The initial mapping we refer to is a linear mapping\napproach where the first qubit is mapped onto the first\nphysical qubit (qubit 0), the second qubit to the second\nphysical qubit (qubit 1), and so on sequentially. This\ncomparison highlights the effectiveness of RL-based cir-\ncuit mapping over initial mapping under various noise7\n(a)\nTarget Circuit Landscape of the crosstalk noise -\nCase1\n  \n \n  \n  \n \n \n   \n    \n  \n    \n    \n  \n  \n    \n    \n  \n    \nLow noise\nHigh noiseHH\nHX\nHH\nHHLandscape of the crosstalk noise -\nCase2\n  \n \n  \n  \n \n \n   \n    \n  \n    \n    \n  \n  \n    \n    \n  \n    \nLow noise\nHigh noise\n(b)\n  \n \n  \n  \n \n \n   \n    \n  \n    \n    \n  \n  \n    \n    \n  \n    \n  \n \n  \n  \n \n \n   \n    \n  \n    \n    \n  \n  \n    \n    \n  \n    Landscape of the crosstalk noise -\nCase1\nFidelity : 0.49\nFidelity : 0.99Crosstalk Noise\nCase1\nTarget Circuit\nRL Based \nImplementationInitial\nImplementation\nLandscape of the crosstalk noise -\nCase1HH\nH\nHH\nH1\n320\nH X\n3 H\nHH\nH\nX\nHH\nH 21\n4Trained Policy \nnetwork\n(c)\n  \n \n  \n  \n \n \n   \n    \n  \n    \n    \n  \n  \n    \n    \n  \n    Landscape of the crosstalk noise -\nCase2\nCrosstalk Noise\nCase2\nTarget CircuitHHH\nX\nHH\n7\n12106\nH HRL Based \nImplementation\nFidelity : 0.97\nFIG. 5. Comparison of circuit mapping strategies in the presence of two different noise landscapes. (a) Displays the target\ncircuit with two noise scenarios, Case1 depicting an intrinsic noise landscape and Case2 an altered noise profile. (b) A target\ncircuit is introduced, and an initial mapping shows poor fidelity. In contrast, utilizing the reinforcement learning (RL) method,\nthe circuit is strategically placed on the device, causing a significantly improved fidelity. (c) Demonstrates RL’s robustness in\nachieving optimal fidelity despite varied noise conditions in Case 2.\nconditions. FIG. 5(a) presents the target circuit for im-\nplementation on a physical quantum computer. The\nstudy employs two distinct noise scenarios: Case1 is the\nintrinsic noise environment of an IBM quantum com-\nputer, while Case2 introduces an intentionally modified\nnoise framework, where the noise between qubits 1 and\n4 is intentionally increased. This modification was de-\nsigned to test the robustness of our RL model.\nHere, the fidelity is the probability of the circuit pro-\nducing the correct output. In contrast to the initial map-ping, which causes low fidelity, the RL-based strategy in\nFIG. 5(b) demonstrates a significant improvement in fi-\ndelity. The improvement in fidelity is achieved through\nthe RL algorithm’s strategic circuit placement, enhancing\noverall performance. FIG. 5(c) further investigates the\nresilience of the RL method by introducing an artificial\nvariation in the noise landscape, represented by Case2 in\nFIG. 5(a). Despite the altered conditions, the RL method\nconsistently maintains high fidelity. This consistency un-\nderscores the method’s adaptability to the noise.8\nAlgorithm 1 Reinforcement training loop\nPolicy Network ←initial\nforcircuit∈training data do\nformapping ∈actions do\nmapped circuit ←mapping(circuit)\nfidelity ←reward(mapped circuit)\nPolicy Network ←update(Policy network, fidelity)\nend for\nend for\nC. Crosstalk Attack detection via a Spectator\nQubit\nThe use of Spectator Qubit (SQ) in quantum control is\na novel idea that has been studied theoretically [20, 21]\nand experimentally [22]. A SQ refers to a qubit that\ncoexists inside the same physical space as a data qubit\n(DQ). The SQ should be positioned in close proximity\nto the data qubit in order to effectively capture the same\nnoise. However, it should also be positioned at a sufficient\ndistance to avoid any potential interaction with the data\nqubit. The objective is to perform frequent measure-\nments on the SQ in order to gain knowledge about the\neffect of noise on the data qubits. Subsequently, equipped\nwith the acquired knowledge, we have the choice of cor-\nrecting the imprecision in the data qubit to a certain\ndegree or declaring that the data qubit has been con-\nsiderably affected, resulting in the loss of its quantum\ninformation. Despite recent research on the SQ idea, its\ndirect use on quantum computing devices has not been\ndemonstrated. The main obstacle to implementing SQ on\nquantum computing devices is the lack of multiple mea-\nsurements, resets, and feedback loops inside a running\ncircuit in current devices. Instead, we use the results of\nthe IBM quantum device characterization from Sec. II\nto replicate the SQ notion for detecting the presence of\nnearby crosstalk attacks.\n1. Dynamical correlation between the spectator and the\ndata qubits\nThe key assumption underlying the SQ concept is that\nSQ and DQ exhibit correlated dynamics when subjected\nto noise. In previous studies on SQ, this correlation has\nbeen artificially introduced either by choosing a similar\nHamiltonian for both SQ and DQ in theoretical studies\n[20, 21] or by using engineered noise in an experimental\nsetting [22].\nIn contrast, we intend to implement the SQ concept on\nan IBM device without engineering any noise or Hamil-\ntonian. Thus, in the first stage, we analyse the correlated\ndynamics of two qubits, one as the DQ and one as the SQ,\nunder the influence of a crosstalk attack. To accomplish\nthis, we simulate the dynamics of the SQ and DQ for var-\nious settings on the 27-qubit IBMQ using the crosstalknoise model characterised from the real 27-qubit device\ndescribed in Sec. II.\nFIG. 6(a) shows an example of the aforementioned sim-\nulation. Here, we choose qubits 11 and 14 as the DQ and\nthe SQ respectively, as depicted by the solid green(red)\nsquare for DQ(SQ) on the topology of the device. Also,\nwe choose qubits 12 and 13 as attack qubits (AQ) de-\npicted via dash black squared on the topology. Further-\nmore, the quantum circuit in FIG. 6(a) shows our ap-\nproach to simulating the dynamics of the SQ and DQ\nunder the effect of a crosstalk attack. At time t=t0, we\ninitialize the DQ in ∣+⟩=(∣0⟩+∣1⟩)/√\n2 via a Hadamard\ngate and leave the SQ in ∣0⟩state. We then apply a series\nof CNOT gates on AQ.\nOur simulation confirms the correlated dynamics of the\nSQ and DQ. In FIG. 6(a), we demonstrate this corre-\nlated dynamics by picturing the state of the DQ and SQ\nat different times via Bloch sphere. The top row shows\nthe state of DQ at different times, while the bottom row\nshows the state of SQ. The states of the SQ and the DQ\natt=t0are∣0⟩and∣+⟩respectively, as expected. As\ntime passes, at t1, t2,andt3, we observe that the states\nof both SQ and DQ rotate in the same direction (the\nrotation trajectories are shown by points). The synchro-\nnised rotations are caused by the crosstalk attack which\nconfirms that SQ and DQ’s dynamics are correlated.\nWe emphasise here that this correlated dynamics is\nsimulated using the data from the characterization of the\nreal IBM device (see Section II), which represents the\ncrosstalk effect in the real device and is not engineered\ncorrelated noise.\n2. Measurement Strategy for the Spectator Qubit\nWe showed the correlated dynamics of SQ and DQ.\nThe idea is to gather information about noise by mea-\nsuring the SQ. However, measuring the spectator qubit\ncauses its state to be projected to either ∣0⟩or∣1⟩, so\nit might not be a useful SQ after a measurement. To\novercome this, following a similar setup as in [20], we re-\npeatedly measure the SQ, record the result, and reset the\nSQ to ∣0⟩state; we then repeat the measurement/reset\nprocess many times. The challenge is to find out how fre-\nquently we must measure and reset the SQ. If we measure\nthe SQ often, it will not have enough time to be impacted\nby the crosstalk attack; if we wait too long before measur-\ning the SQ, its state may rotate back to its initial value\nor be influenced by other noise rather than the crosstalk\nthat we are interested in.\nTo address the above question, we consider a scenario\nwhere we would like to detect the presence of a crosstalk\nattack. We use a similar setup as before, i.e., qubit 14\nas SQ and qubits 12 and 13 as AQ (note that we do not\nneed DQ in this scenario). The goal is to identify if a\ncrosstalk attack is present by measuring the SQ. To do\nthat, we measure the SQ repeatedly and reset it after\neach measurement. In the absence of crosstalk, the spec-9\nFIG. 6. (a) Demonstration of correlated dynamics of data qubit and spectator qubit under the effect of crosstalk attack. Here,\nqubit 11 of the IBM device, is chosen as data qubit and initialised at ∣+⟩using a Hadamard gate. Qubit 14 plays the role\nof the spectator and its initial state is ∣0⟩. Crosstalk attack is performed via a series of CNOT gates on qubits 12 and 13.\nBloch spheres of data qubit (top row) and spectator qubit (bottom row) at different times demonstrate the correlated dynamics\nbetween the spectator and the data qubit. (b) Schematic for finding the optimum waiting time to detect the presence of the\ncrosstalk attack. If the waiting time is chosen to be too short or too long, the spectator qubit would not detect the presence\nof the crosstalk attack. (c) By removing shots with detected crosstalk we mitigate the effect of the crosstalk.\ntator qubit stays at state ∣0⟩so the measurement result\nwould always be ‘0’. Thus, if the measurement results in\n‘1’, it would be a flagfor the presence of an attack. To\ncontinue, we count the number of flags and store them\ninf; then, if the number of flags is greater than a prede-\nfined threshold f0, we indicate that the shot was under\nattack. Note that, in an ideal simulation where the noise\nis due to only crosstalk attacks, f0=0, but if we con-\nsider other possible noise, especially measurement noise,\nwe may consider f0>0 to be more relaxed in deciding\nthe presence of an attack.\nFurthermore, we run Nshots of the setup (all shots\nare under attack) and we define a success rate η=Nd/N\nwhere Ndis number of detected shots under attack.\nWe repeat the procedure for different waiting time and\ncalculate the success rate ηfor each waiting time to\nfind the optimum waiting time. In our simulation weuseN=1000 shots and waiting time defined in units of\nnumber of CNOT gates. In FIG. 6(b), we illustrated\nthree highlighted scenarios: measuring the SQ too\nfrequently (orange), optimal waiting time (green) and\nlong waiting time (red). Moreover, the plot shows our\nsimulation result. As we expected, if we measure the\nspectator qubit after only one gate, its state has not\nevolved enough to detect the attack. On the other hand,\nif we wait too long, we see the state rotates back to ∣0⟩\nand we get low success rate. Our simulation shows that\noptimum waiting time for our setup is approximately 7\nCNOT gates.10\n3. Detection of the crosstalk attack\nNow that we have a protocol to detect the presence\nof a crosstalk attack, we can use it to post-select our\nshots. To demonstrate this idea, we simulate the follow-\ning scenario: Consider two data qubits initialized in an\nentangled state, which after some time we would like to\nuse. Also, consider that some but not all of the shots are\nunder crosstalk attack. Although we do not know which\nshots are under attack, we use the spectator qubit to de-\ntect the impacted shots and discard them, i.e., post select\non the shots that are not under attack. In our simulation,\nat the end of the circuit, we disentangle the qubits be-\nfore measurement, so the measurement result of the data\nqubits should be 00. Also, in the simulation, we choose\nthat 20% of the shots are under crosstalk attack. This\npercentage is arbitrary, but because we are discarding\nthe attacked shots, we choose a relatively small percent-\nage (below 50%). FIG. 6(c) shows our circuit as well as\nthe normalized histogram of the final count of the data\nqubits. Blue represents all shots, while green shows the\npost-selected shots. From the histogram, it is obvious we\nhave successfully discarded many of the impacted shots.\nAlthough we successfully flagged the presence of the\nattack, discarding the shots is wasteful. Correcting the\nimpacted shots using the SQ has been done in [20, 21] for\ndephasing noise. Correcting the qubit state that is im-\npacted by a crosstalk attack is more complicated, and we\nleave it for our future work. Also, we did not consider the\ncrosstalk effect of SQ measurement/reset on DQ. Because\nthis method is not now available on IBM devices, incor-\nporating it into our model would be a future work. More-\nover, for a larger circuit, we probably need more than one\nspectator qubit. So, optimizing the multi-spectator qubit\nis another objective for future work.\nIV. CONCLUSION AND DISCUSSION\nCrosstalk noise is a serious issue for near term and\nfuture large scale quantum devices. In this paper we\nhave demonstrated the adverse effects of crosstalk on a\ncircuit, particularly in a device shared by many users.\nWe developed a full map of the crosstalk on a real IBM\ndevice, and used this to create a high fidelity simulator\nfor the device.\nWe then developed a number of techniques to detect\nand/or mitigate the effects of a hostile crosstalk attack\non a shared device. The first technique, circuits separa-\ntion, was effective, but sees a substantial amount of the\ndevice left to idle, while scaling poorly to devices with\nlarge numbers of qubits. We then developed a reinforce-\nment learning model to learn the noise on the device,\nand placed circuits in such a way that minimises the po-\ntential for crosstalk interference on the device. The fi-\nnal technique was based on the idea of spectator qubits,\nwhich have already been used for detecting and mitigat-\ning general noise models. We applied the spectator qubittechnique to the problem of crosstalk noise, and found\nthat it was highly effective at detecting the presence of\na crosstalk attack. Further work on the spectator qubit\ntechnique includes correcting for a crosstalk induced er-\nror, rather than simply discarding erronous shots. It is\nalso necessary to further explore the optimal placement\nof spectator qubits around a circuit.\nThe crosstalk attacks studied here, and elsewhere in\nthe literature, are simple repeated CNOTs, designed to\nmaximise the disrupting effect of crosstalk. It may be\npossible to create an attack which disrupts a computa-\ntion in a targeted way, for example, by causing Grover’s\nalgorithm to select an incorrect answer with high con-\nfidence. Future work can explore such possibility and\ndevelop methods to detect and mitigate such scenarios.\nThere is the potential to flip the crosstalk attacks —\nand instead use crosstalk to extract information. Quan-\ntum algorithms require classical data, which may be sen-\nsitive. This data is encoded in state preparation circuits,\nand a third party may be able to derive some knowledge\nof the secret data through exploiting crosstalk. While\nchallenging, the potential impact of this could be signif-\nicant, justifying further work.\nFinally, the crosstalk benchmarking described here can\nbe applied to a variety of hardware architectures, such as\nsilicon qubits, trapped ions, or others. These are very\ndifferent physical systems, which will have very differ-\nent properties. It is important to properly explore the\neffect of crosstalk on other hardware, to see how bad\n(or not) it is. In summary, we have laid the foundation\nto exploit crosstalk noise as a mechanism to attack and\ndisrupt quantum computations in a shared environment.\nWe have also reported mitigation strategies and demon-\nstrated their working on proof of concept examples. In\nnear to medium term era when quantum computing is\nanticipated to be based on shared environment through\ncloud platforms, securing it will be of paramount impor-\ntance in particular for security sensitive data and com-\nputations.\nACKNOWLEDGMENT\nBH acknowledges the support of the CSIRO Research\nTraining Program Scholarship. The work was partially\nsupported by the Australian Army through Quantum\nTechnology Challenge 2023. The computational re-\nsources were provided by the National Computing In-\nfrastructure (NCI) and Pawsey Supercomputing Cen-\nter through National Computational Merit Allocation\nScheme (NCMAS). The research was supported by the\nUniversity of Melbourne through the establishment of the\nIBM Quantum Network Hub at the University.\nData availability: The data that support the find- ings\nof this study are available within the article.\nCompeting financial interests: The authors declare\nno competing financial or non-financial interests.11\n[1] D. Suter and G. A. ´Alvarez, Rev. Mod. Phys. 88, 041001\n(2016).\n[2] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt,\nand R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).\n[3] M. Sarovar, T. Proctor, K. Rudinger, K. Young,\nE. Nielsen, and R. Blume-Kohout, Quantum 4,\n10.22331/Q-2020-09-11-321 (2020).\n[4] G. White, K. Modi, and C. Hill, Physical Review Letters\n130, 160401 (2023).\n[5] E. Magesan and J. M. Gambetta, Phys. Rev. A 101,\n052308 (2020).\n[6] Y. Sung, L. Ding, J. Braum¨ uller, A. Veps¨ al¨ ainen, B. Kan-\nnan, M. Kjaergaard, A. Greene, G. O. Samach, C. Mc-\nNally, D. Kim, A. Melville, B. M. Niedzielski, M. E.\nSchwartz, J. L. Yoder, T. P. Orlando, S. Gustavsson,\nand W. D. Oliver, Phys. Rev. X 11, 021058 (2021).\n[7] F. Mazda, Telecommunications engineer’s reference book\n(Butterworth-Heinemann, 2014) p. xi.\n[8] C. Fang, Y. Wang, S. Huang, K. R. Brown, and J. Kim,\nPhysical Review Letters 129, 240504 (2022).\n[9] Z. Zhou, R. Sitler, Y. Oda, K. Schultz, and G. Quiroz,\nPhys. Rev. Lett. 131, 210802 (2023).\n[10] P. Parrado-Rodr´ ıguez, C. Ryan-Anderson, A. Bermudez,\nand M. M¨ uller, Quantum 5, 487 (2021).\n[11] D. Buterakos, R. E. Throckmorton, and S. Das Sarma,\nPhys. Rev. B 97, 045431 (2018).\n[12] A. A. Saki, M. Alam, K. Phalak, A. Suresh, R. O.\nTopaloglu, and S. Ghosh, A Survey and Tutorial on Se-curity and Resilience of Quantum Computing (2021),\narXiv:2106.06081 [quant-ph].\n[13] R. J. Blume-Kohout, Idle Tomography. , Tech. Rep.\nSAND2019-0010PE (Sandia National Lab. (SNL-NM),\nAlbuquerque, NM (United States), 2019).\n[14] Erik and Kmrudin, pyGSTi: Version 0.9.1 Alpha (2016).\n[15] A. Ash-Saki, M. Alam, and S. Ghosh, IEEE Transactions\non Quantum Engineering 1, 1 (2020), conference Name:\nIEEE Transactions on Quantum Engineering.\n[16] Y. Ohkura, T. Satoh, and R. Van Meter, IEEE Trans-\nactions on Quantum Engineering 3, 1 (2022), conference\nName: IEEE Transactions on Quantum Engineering.\n[17] N. Quetschlich, L. Burgholzer, and R. Wille, in 2023\n60th ACM/IEEE Design Automation Conference (DAC)\n(IEEE, 2023) pp. 1–6.\n[18] Y. Baum, M. Amico, S. Howell, M. Hush, M. Liuzzi,\nP. Mundada, T. Merkh, A. R. Carvalho, and M. J. Bier-\ncuk, PRX Quantum 2, 040324 (2021).\n[19] H. P. Nautrup, N. Delfosse, V. Dunjko, H. J. Briegel, and\nN. Friis, Quantum 3, 215 (2019).\n[20] B. Tonekaboni, A. Chantasri, H. Song, Y. Liu, and H. M.\nWiseman, Phys. Rev. A 107, 032401 (2023).\n[21] H. Song, A. Chantasri, B. Tonekaboni, and H. M. Wise-\nman, Phys. Rev. A 107, L030601 (2023).\n[22] K. Singh, C. Bradley, S. Anand, V. Ramesh, R. White,\nand H. Bernien, Science 380, 1265 (2023).\nAppendix A: Total Variational Distance\nFor two probability distributions, P,Qover the same space of outcomes Ω, the Total Variational Distance (TVD)\nis the maximum difference between the distributions over all events,\nTVD(P, Q)=supE∈Ω∣P(E)−Q(E)∣ (A1)\nThe TVD measures a distance between two distributions — so when comparing the output fidelities of various\nsimulators, a reference distribution is required. Previous work benchmarking the performance of a crosstalk simula-\ntor [15] has chosen to use an ideal noise-free simulation as the reference distribution, with the crosstalk simulator and\nreal device measured relative to this. However, this may be misleading, as the real device and the crosstalk simulator\nmay be a similar distance away from an ideal simulation, but not close to each other at all. As such, in this paper\nwe use the real device probability distribution as reference, and so a better simulation of that real device has a lower\nTVD. For a circuit with real counts cd, and simulated counts cs, we find that the distance is,\ndist(cs)=TVD(cs, cd) (A2)\nAppendix B: Benchmarking Circuits\nBelow are circuit diagrams for the circuits used in benchmarking the crosstalk simulator used in section II B.12\n−6 −4 −2 0 2 4 600.10.20.30.4\nTVD\nFIG. 7. The total variational distance is the greatest distance between two probability distributions.\n1. syn n\nThis is a synthetic circuit where two idle qubits are prepared into a superposition.\n. . .\n. . .\n⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟨⟨⟪rl⟫m⟩⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪\nntimes∣0⊗2⟩ H H\nqa\nqb\nThe CNOT gate is repeated ntimes.\n2. toffoli\nThis is a simple repeated Toffoli gate test. The Toffoli gates are decomposed into 1-qubit gates and the 2-qubit\nCNOT gate.\n. . .\n. . .\n. . .\n⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟨⟨⟪rl⟫m⟩⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪\nntimesqa X H H\nqb X H H\nqc H H\nThe Toffoli gate is repeated ntimes.13\n3. grover\nThe grover circuit is a simple 3-qubit Grover’s algorithm circuit with two iterations.\n∣0⟩\n∣0⟩\n∣0⟩Oracle Diffusion Operator Oracle Diffusion Operator\n4. attack\nThe attack test circuit is identical to the Grover’s algorithm circuit, with the addition of a crosstalk attack on\nadjacent qubits;\n∣0⟩\nOracle Diffusion Operator Oracle Diffusion Operator ∣0⟩\n∣0⟩\nCNOT gates are repeated during the execution of the circuit."    },    {        "title": "2402.02787v1.Ab_initio_Investigation_of_Thermal_Transport_in_Insulators__Unveiling_the_Roles_of_Phonon_Renormalization_and_Higher_Order_Anharmonicity.pdf",        "content": "Ab initio Investigation of Thermal Transport in Insulators: Unveiling the Roles of\nPhonon Renormalization and Higher-Order Anharmonicity\nSoham Mandal, Manish Jain, and Prabal K. Maiti∗\nCentre for Condensed Matter Theory, Department of Physics,\nIndian Institute of Science, Bangalore-560012, India\n(Dated: February 6, 2024)\nThe occurrence of thermal transport phenomena is widespread, exerting a pivotal influence on\nthe functionality of diverse electronic and thermo-electric energy-conversion devices. The traditional\nfirst-principles theory governing the thermal and thermodynamic characteristics of insulators relies\non the perturbative treatment of interatomic potential and ad-hoc displacement of atoms within\nsupercells. However, the limitations of these approaches for highly anharmonic and weakly bonded\nmaterials, along with discrepancies arising from not considering explicit finite temperature effects,\nhighlight the necessity for a well-defined quasiparticle approach to the lattice vibrations. To ad-\ndress these limitations, we present a comprehensive numerical framework in this study, designed\nto compute the thermal and thermodynamic characteristics of crystalline semiconductors and in-\nsulators. The self-consistent phonon renormalization method we have devised reveals phonons as\nquasiparticles, diverging from their conventional characterization as bare normal modes of lattice vi-\nbration. The extension of the renormalization impact to interatomic force constants (IFCs) of third\nand fourth orders is also integrated and demonstrated. For the comprehensive physical insights, we\nemployed an iterative solution of the Peierls-Boltzmann transport equation (PBTE) to determine\nthermal conductivity and carry out Helmholtz free energy calculations, encompassing anharmonicity\neffects up to the fourth order. In this study, we utilize our numerical framework to showcase its\napplicability through an examination of phonon dispersion, phonon linewidth, anharmonic phonon\nscattering, and temperature-dependent lattice thermal conductivity in both highly anharmonic ma-\nterials (NaCl and AgI) and weakly anharmonic materials (cBN and 3C-SiC). Our study finds that\nignoring higher-order (4-phonon) phonon scattering processes is not viable for materials with strong\nanharmonicity in their interatomic potential. Meanwhile, renormalization demonstrates negligible\nimpact on materials characterized by weak anharmonicity. The investigation also explores the effect\nof fourth-order Helmholtz free energy and pressure-dependent thermal conductivity of NaCl crystal.\nThe theoretical framework developed in this work will help us to understand the physical insight of\nphonon and phonon-driven thermal and thermodynamic properties of materials and offer valuable\nguidance for the strategic development of efficient thermal management techniques.\nI. INTRODUCTION\nLattice vibrations are the primary heat carriers in\ncrystalline insulators and semiconductors. The quasi-\nparticles of quantized lattice vibration is known as\na phonon. Understanding phonon and phonon-driven\nproperties in materials are of immense importance in the\ndomain of material science, solid-state physics, and chem-\nistry. This is due to the evergrowing technological appli-\ncation of such materials. Phonons play a role in several\nphysical phenomena, including thermal transport, ther-\nmal expansion of solids, the stability of crystalline solids,\nand structural phase transitions, either directly or in-\ndirectly. Phonons and several phonon-assisted physical\nphenomena have been theoretically analyzed and well-\nestablished for more than a century with their experimen-\ntal validations. Rapid increase of high performance com-\nputational power has enhanced our ability to use first-\nprinciple computation to predict phonon-driven physi-\ncal phenomena through numerical techniques. These\ntraditional ab initio schemes range from simple mod-\n∗maiti@iisc.ac.inels taking into account only the harmonic approxima-\ntion [1, 2] to predict phonon dispersions, the density\nof states, and heat capacity to more sophisticated ap-\nproaches to include anharmonicity in the interatomic po-\ntential to predict thermal expansion and thermal con-\nductivity of materials[3, 4]. These harmonic and quasi-\nharmonic approximations are excellent in capturing the\naforementioned phonon-driven properties of weakly an-\nharmonic strong covalently bonded materials, like silicon\nand diamond[5].\nIn the case of materials with a significant degree of\nanharmonicity in their interatomic potentials, such as al-\nkali halides and heavy-metal chalcogenides, simple mod-\nels of calculating the normal modes of lattice vibration\nare inadequate. Materials with extreme anharmonici-\nties, such as solid helium, typically have bare phonon\nfrequencies having imaginary values[6]. Renormalization\nmakes these squared frequencies positive as anharmonic-\nity is sufficiently large. Furthermore, at a higher temper-\nature (depending on the Debye temperature of a specific\nmaterial), conventional ab initio approaches fail to com-\nply with experimental measurements. Therefore, a well-\ndefined and robust quasi-particle methodology is crucial\nto cover weakly bonded, highly anharmonic materials and\nto fit measured data over the entire experimental temper-arXiv:2402.02787v1  [cond-mat.mtrl-sci]  5 Feb 20242\nature ranges.\nThe interatomic force constants (IFCs) calculation in\ntraditional ab initio approaches involves the density-\nfunctional perturbation (DFPT)[7, 8] method or the\nfinite-difference supercell[3, 9–11] method. The deter-\nmination of IFCs in DFPT relies on analyzing the an-\nalytical derivatives of the potential energy. It can effi-\nciently determine harmonic IFCs and related properties\nof materials. While the utilization of DFPT has been re-\nported for calculating third-order anharmonic IFCs[12],\nthe computational cost of determining higher-order an-\nharmonic IFCs is generally prohibitive. Furthermore,\nDFPT does not explicitly incorporate finite tempera-\nture effects. The finite-difference supercell method ap-\nproximates the derivative of the potential energy by em-\nploying finite differences, and by calculating the forces\non atoms within the perturbed supercell at 0K, the\nIFCs can be determined. The perturbation is accom-\nplished by ad hoc displacement of atoms within the su-\npercell. To compute higher-order anharmonic IFCs (be-\nyond 3rd order), force calculation for a considerable num-\nber of perturbed snapshots is necessary for the finite-\ndifference supercell method[13]. Recent studies[14–17]\nhave explored the incorporation of fourth-order IFCs and\nfour-phonon scattering processes across various materi-\nals to reduce discrepancies between theoretical predic-\ntions and experimental observations of the thermal trans-\nport properties. Classical force field (FF) based molecu-\nlar dynamics-based approaches for thermal conductivity\ncalculation, using Green-Kubo response function-based\nformalism[18, 19] or the non-equilibrium heating-cooling\nmethod[20] can take into account finite temperature ef-\nfects and the inherent anharmonicity. In our current\nwork, we will employ the temperature-dependent effec-\ntive potential (TDEP)[21, 22] method, which is a compu-\ntationally efficient framework capable of addressing theaforementioned challenges. TDEP has been used in re-\ncent studies to determine the phonons and associated\nphysical properties of materials[23–25].\nIn this work, we have developed a unified numerical\nscheme to calculate IFCs up to the fourth order utiliz-\ning the temperature-dependent force-displacement data\nfrom ab initio molecular dynamics (AIMD). We have also\ndeveloped a Thermal Stochastic Snapshot (TSS) tech-\nnique to avoid the computational overhead of AIMD.\nOur current study also involves a numerical approach\nfor the self-consistent phonon renormalization and calcu-\nlation of thermal transport properties of the insulating\ncrystalline solids. In this renormalization method, bare\nsecond-order IFCs are renormalized only up to first-order\nusing the statistical operator renormalization technique.\nThis renormalization process can also be realized through\nthe one-phonon propagator approach in the many-body\ntheory[26]. In the results section, we will demonstrate\nthe phonon-driven physical properties of materials with\npronounced anharmonicity (NaCl and AgI) and those\nwith weak anharmonicity (cBN and 3C-SiC) in their in-\nteratomic potentials. This work thoroughly investigates\nthe impact of self-consistent renormalization of harmonic\nand anharmonic IFCs and higher-order phonon scatter-\ning processes on thermal transport properties. Finally,\nour investigation will examine the influence of fourth-\norder Helmholtz free energy on the lattice constant and\npressure-dependent thermal conductivity of crystalline\nNaCl.\nII. METHODS\nThe potential energy of a crystal due to the interaction\namong the constituent atoms, U, can be expanded in the\ndisplacements from the initial configuration, assuming U\nis an analytic function of the positions of the atoms, as:\nU=U0+X\nN1µX\nαΦα\n1(N1µ)uα(N1µ) +X\nN1µ,N 2νX\nα,β1\n2!Φαβ\n2(N1µ, N 2ν)uα(N1µ)uβ(N2ν)+\nX\nN1µ,N 2ν,N 3πX\nα,β,γ1\n3!Φαβγ\n3(N1µ, N 2ν, N 3π)uα(N1µ)uβ(N2ν)uγ(N3π)+\nX\nN1µ,N 2ν,N 3π,N 4ρX\nα,β,γ,δ1\n4!Φαβγδ\n4(N1µ, N 2ν, N 3π, N 4ρ)uα(N1µ)uβ(N2ν)uγ(N3π)uδ(N4ρ) +. . .(1)\nHere U0is the potential energy at the equilibrium. Φ 1,\nΦ2, Φ 3, and Φ 4are first-, second-, third-, and fourth-\norder interatomic force constants (IFCs), respectively.\nN1,N2,N3, and N4denote the lattice points of the crys-\ntal. µ,ν,π, and ρrun over the basis atoms; α,β,γ,\nandδare cartesian indices. The first-order derivative ofcrystal potential Ugives the force on atoms:\nFα(N1µ) =−∂U\n∂uα(N1µ)(2)\nOur calculation of the interatomic force constants (up\nto the fourth order) involved conducting ab initio molec-\nular dynamics (AIMD) simulations at a designated tem-\nperature on a supercell of the crystal under considera-3\nMaterial Order of IFC Nearest neighbour Cutoff distance( ˚A) Total IFCs Independent IFCs\n2nd 11 9.88 2790 55\nNaCl 3rd 5 6.55 175446 1240\n4th 2 4.40 1111158 2627\n2nd 11 10.75/6.10/7.43 2700 74\nAgI/cBN/3C-SiC 3rd 6 8.05/4.56/5.56 272214 3860\n4th 2 4.90/2.79/3.39 795906 3758\nTABLE I. Number of independent IFCs up to 4th order after applying space group symmetry, permutation symmetry, and\nacoustic sum rule of the IFCs.\ntion. Unless explicitly stated otherwise, all MD simula-\ntions conducted in this study utilize a 5 ×5×5 super-\ncell configuration. MD trajectories provide us with the\ndisplacements ( uα(N1µ)) and forces ( Fα(N1µ)) on each\natom in Eq. 2 at every time step. Eq. 2 can be rewritten\nas a matrix equation, Dx=f, with these displacement\nand force datasets. Here, the matrix elements of Dare\nconstructed from the atomic displacements; fis a row\nof atomic forces; xis the row of IFCs to calculate. The\nmatrix equation is usually an overdetermined system. A\nleast-squares technique is used to calculate the IFCs by\ninverting the displacement matrix Dthrough the Moore-\nPenrose pseudoinverse[27].\nA. Symmetry\nIn general, the number of IFCs for the d-th order is\n(3N)d, where Nis the total number of atoms in the crys-\ntal. As such, solving the matrix equations for determin-\ning higher order IFCs (beyond fourth order) is computa-\ntionally prohibitive. To date even calculations including\nfourth-order IFCs have only been attempted for simplest\nmaterials like Silicon and NaCl. We make several approx-\nimations to make these calculations tractable. While in\nprinciple, one should include all higher-order terms in\nEq. 1, we include the terms till fourth order. To fur-\nther reduce the number of IFCs, we use cutoff distances,\nas the third-, fourth- and non-polar contribution of the\nsecond-order IFCs fall rapidly with distance. Ewald-\nSummation technique is used to separate the long-range\npart of second-order IFCs caused by dipole-dipole inter-\nactions in polar materials, as explained in a later section.\nThis allows us to identify a suitable set of IFCs for a spe-\ncific temperature that can accurately match the forces\nobtained from molecular dynamics simulations. We re-\nduce the computational cost further by exploiting per-\nmutation symmetry, point-group symmetry, and acoustic\nsum rule (ASR) of the IFCs. The specifics of these sym-\nmetries are discussed in Ref[9]. These symmetries pro-\nvide constrained equations among the IFCs, from which\nwe extract a set of independent IFCs. After determin-\ning the independent IFCs, the rest of the IFCs can be\nreconstructed as their linear combinations. We recon-\nstruct the whole set of IFCs for the calculation of mate-rial properties. For determining point-group symmetries,\nwe employ the SPGLIB[28] crystal-symmetry library to\nfind the transformation matrices of the point group oper-\nations. The number of independent IFCs drastically re-\nduces when we combine these symmetries, as illustrated\nin Table I, making it more computationally feasible to\ncalculate up to the fourth order.\nB. Long-range correction\nThe vibration-induced dipole-dipole interaction contri-\nbution to the second-order IFCs diminishes at a rate of\n∼1/r3with distance. It poses a problem in the case\nof polar materials, as we cannot set a cutoff for second-\norder IFCs and, in principle, must extend them to in-\nfinity. Most of the available packages address this issue\nby incorporating a non-analytic component into the Dy-\nnamical matrix[2, 4]. In this mixed-space approach[29],\nthe short-range second-order IFCs are calculated in real\nspace separately. To incorporate long-range correction,\na direction-dependent non-analytical term (in reciprocal\nspace) is introduced to the Dynamical matrix. This cor-\nrection is applicable only for the long wavelength ( q→0)\nlimit. With a reasonable level of accuracy, this method\nillustrates the LO-TO splitting of polar materials along\nthe high symmetry path. In our case, we used the Ewald\nSummation technique, as outlined in Ref.[30, 31], to com-\npute the harmonic IFCs in reciprocal space from the Born\neffective charge and dielectric tensor of the material. We\nthen use an inverse Fourier transform to convert the re-\nciprocal space IFCs resulting from the dipole-dipole inter-\naction into real-space IFCs. The long-range contribution\nto the force is then calculated by multiplying these newly\ncreated real-space IFCs with the displacement dataset al-\nready obtained from MD. We eliminated this long-range\nforce component from the MD force to fit only the short-\nrange component of the IFCs. Subsequently, the har-\nmonic IFCs (in reciprocal space) obtained through the\nEwald Summation technique, which includes the ionic\ncontribution, are incorporated into the Dynamical matrix\nto accurately determine the phonon dispersion, group ve-\nlocity, and density of states. Therefore, the complete dy-4\nnamical matrix is defined as:\nDαβ(µν, q) =1p\nMµMνX\nNΦαβ\n2(short −range )(0µ, Nν )eiq·R(N)\n+Ewald term.\n(3)\nwhere Mµis the mass of the basis atom type µ. The\nphonon frequency ( ωqs) and eigenvector ( ϵ(qs)) are ob-\ntained by performing diagonalization of the dynamical\nmatrix.\nC. Thermal Stochastic Snapshots\nThe method we discussed is based on the Temperature-\ndependent Effective Potential (TDEP) method described\nby Hellman et al. [21] TDEP requires expensive ab ini-\ntiomolecular dynamics (AIMD) on a supercell at a finite\ntemperature to map it into the model potential energy\ngiven by Eq. 1. AIMD is cumbersome computation-\nally as it requires a sufficiently long run to achieve equi-\nlibrium at a specific temperature, followed by the selec-\ntion of well-spaced snapshots for proper sampling of the\nBorn-Oppenheimer (BO) surface of the supercell. To cir-\ncumvent the computation-intensive nature of this method\nwhile still retaining all finite temperature effects, we opt\nfor the thermal stochastic snapshot (TSS) technique, as\ndescribed in Ref[32–34]. In the TSS scheme, the atoms\n(Nv) in the supercell are displaced following the equa-\ntion:\nuα(Nν) =r\nℏ\nMνN0X\nqss\n2nqs+ 1\nωqscos(2 πξ(1,qs))\n×q\n−ln(1−ξ(2,qs))ϵα(qs, ν)eiq·R(N)(4)\nWhere ξ(1,qs)andξ(2,qs)are two mode-dependent random\nnumbers generated from a uniform distribution between\n0 and 1, with ( qs) referring to phonon modes. To ensure\nthat the displacement uα(Nν) is a real value, the random\nnumbers are restricted such that ξ(1,qs)=ξ(1,−qs)and\nξ(2,qs)=ξ(2,−qs).ωqsandϵα(qs, ν) are the frequency and\neigenvector of the phonon mode ( qs), respectively. N0is\nthe number of q-points in the Brillouin zone (BZ) sam-\npled. nqsis the Bose-Einstein distribution corresponding\nto the temperature T. The displacement is consistent\nwith the canonical ensemble, which takes into account\ncomplete quantum statistics, thereby including all finite\ntemperature effects. Additionally, it also accounts for\nthe zero-point motions of atoms, which is not achievable\nin AIMD. These supercells of displaced atoms are input\ninto an ab initio force calculator to calculate the force\non each atom. We employed the SIESTA[35] package to\ncarry out the force computation at the Gamma point for\neach stochastically perturbed supercell. A limitation of\nutilizing Eq. 4 is that it necessitates prior knowledge\nof phonon frequencies ( ωqs)and eigenvectors ( ϵα(qs, ν)),which are the calculations we aim to perform. As such,\na self-consistent approach is utilized. The procedure is\ninitiated using the IFCs obtained through TDEP by con-\nducting AIMD at a very low temperature (80K for NaCl,\nAgI, cBN and 3C-SiC studied here). We initiate TSS\ncalculations from the TDEP-generated IFCs to obtain\nan initial set of IFCs. By using these IFCs as feedback,\nwe generate a new set of a force-displacement dataset\nthrough TSS and obtain a new set of IFCs. This proce-\ndure is repeated until the desired accuracy is achieved self\nconsistently. To determine the IFCs of the next higher\ntemperature, we utilized the IFCs from the previous tem-\nperature and subsequently executed the self-consistent\ncycle. By doing so, we only had to run AIMD once to\ninitiate the process.\nD. Renormalization\nIn the self-consistent phonon renormalization ap-\nproach, an effective harmonic IFC is obtained by com-\nbining unrenormalized (bare) second-order IFCs and\nfourth-order anharmonic IFCs. The statistical oper-\nator renormalization technique produces the equation\nfor the renormalized harmonic IFC, which is stated as\nfollows[6, 34, 36]:\nΨαβ\n2(0µ, Nν ) = Φαβ\n2(0µ, Nν ) +\nℏ\n4N0X\nN3πX\nN4ρX\nγδX\nqsΦαβγδ\n4(0µ, Nν, N 3π, N 4ρ)\n×eiq·(R(N3)−R(N4))Eγ(qs, ρ)Eδ(qs, π)2nqs+ 1\nΩqsp\nMρMπ(5)\nWhere Φαβ\n2(0µ, Nν ) and Φαβγδ\n4(0µ, Nν, N 3ρ, N 4π)\nstands for bare second- and fourth-order IFCs, and\nΨαβ\n2(0µ, Nν ) is the renormalized second-order IFCs. The\nrenormalization process can also be realized by utilizing\nthe one-phonon propagator approach[26]. The expres-\nsion mentioned above is obtained by truncating an infi-\nnite series that includes contributions from all even-order\nIFCs. As the statistical average is calculated over the\neffective harmonic phonon states, Ω qsandEα(qs, ν) in\nEq. 5, represent the eigenvalue and eigenvector of renor-\nmalized phonon, which are unknown initially. As such,\nthe problem is solved self-consistently. We start the self-\nconsistent loop from the eigenvalue and eigenvectors of\nbare second-order IFCs (Φαβ\n2(0µ, Nν )). During the fol-\nlowing iterations, the second term in Eq. 5 is revised us-\ning the eigenvalues and eigenvectors of the renormalized\nharmonic IFCs (Ψαβ\n2(0µ, Nν )) from the previous itera-\ntion. The loop is repeated until the desired level of pre-\ncision is achieved. It is important to note that, thus far,\nno interaction among the phonon states has been consid-\nered. The present study does not delve into the impact of\nthe interaction among self-consistent phonon states in the\nenergy levels, mediated by the odd terms in the crystal5\npotential, as it is outside the scope of this work. Follow-\ning the procedure described in Ref.[34], we extend the\nrenormalization effect to anharmonic IFCs of third and\nfourth order. The initial step to achieve this involves\nthe multiplication of recently constructed renormalized\nsecond-order IFCs (Ψαβ\n2(0µ, Nν )) with the available dis-placement dataset ( uα(Nν)) from TSS (or AIMD), which\nenables the calculation of the force ( f2nd\nren.) attributed to\nthe renormalized harmonic IFCs. Afterward, we sub-\ntract the calculated force from the actual ab initio force\n(fabinitio ), which has already been obtained from TSS (or\nAIMD). The rest, which remains in Eq. 2, can be written\nas:\nfabinitio −f2nd\nren.=−X\nN2ν,N 3πX\nβ,γ1\n2!Ψαβγ\n3(0µ, N 2ν, N 3π)uβ(N2ν)uγ(N3π)−X\nN2ν,N 3π,N 4ρX\nβ,γ,δ1\n3!Ψαβγδ\n4(0µ, N 2ν, N 3π, N 4ρ)\n×uβ(N2ν)uγ(N3π)uδ(N4ρ)\n(6)\nBy fitting Eq. 6 with the displacement dataset already obtained from TSS (or AIMD), we can determine the renor-\nmalized third- (Ψαβγ\n3(0µ, N 2ν, N 3π)) and fourth-order (Ψαβγδ\n4(0µ, N 2ν, N 3π, N 4ρ)) interatomic force constants.\nE. Free Energy\nThe expression of Helmholtz free energy up to fourth\norder anharmonicity is given by[6]:\nF=U0+FH+FA, (7)where harmonic free energy,\nFH=X\nq,s\u00141\n2ℏωqs+kBTln\u0012\n1−e−ℏωqs\nkBT\u0013\u0015\n,(8)\nand the anharmonic contribution to free energy:\nFA= 12X\nqs,q′s′ϕ(qs,−qs, q′s′,−q′s′)\u0012\nnqs+1\n2\u0013\u0012\nnq′s′+1\n2\u0013\n−\n18\nℏX\nqs,q′s′,q′′s′′\"\n|ϕ(qs, q′s′, q′′s′′)|2(\nnqsnq′s′+nqs+1\n3\n(ωqs+ωq′s′+ωq′′s′′)p+2nqsnq′′s′′−nqsnq′s′+nq′′s′′\n(ωqs+ωq′s′−ωq′′s′′)p)\n+\n2ϕ(qs,−qs, q′′s′′)ϕ(q′s′,−q′s′,−q′′s′′)nqsnq′s′+nqs+1\n4\n(ωq′′s′′)p#\n.(9)\nHere ϕ(qs, q′s′, q′′s′′) and ϕ(qs, q′s′, q′′s′′, q′′′s′′′) are scattering matrix elements related to third and fourth-order\nIFCs:\nϕ(qs, q′s′, q′′s′′) =1\n3!\u0012ℏ\n2\u00133/21√N01\n√ωqsωq′s′ωq′′s′′X\nµ,N 2ν,N 3πX\nα,β,γΦαβγ\n3(0µ, N 2ν, N 3π)ei⃗q′·⃗R(N2)ei⃗q′′·⃗R(N3)×\nϵα(qs, µ)ϵβ(q′s′, ν)ϵγ(q′′s′′, π)p\nMµMνMπ,(10)\nand,\nϕ(qs, q′s′, q′′s′′, q′′′s′′′) =1\n4!\u0012ℏ\n2\u001321\nN01\n√ωqsωq′s′ωq′′s′′ωq′′′s′′′X\nµ,N 2ν,N 3π,N 4ρX\nα,β,γ,δΦαβγδ\n4(0µ, N 2ν, N 3π, N 4ρ)×\nei⃗q′·⃗R(N2)ei⃗q′′·⃗R(N3)ei⃗q′′′·⃗R(N4)×ϵα(qs, µ)ϵβ(q′s′, ν)ϵγ(q′′s′′, π)ϵδ(q′′′s′′′, ρ)p\nMµMνMπMρ.(11)\nThe given expressions for scattering matrix elements in-\ncorporate quasi-momentum conservation, where the re-lations q+q′+q′′=Gandq+q′+q′′+q′′′=Ghold,\nwith Grepresenting the reciprocal lattice vector of the6\ncrystal.\nF. Thermal Conductivity\nThe non-equilibrium distribution function obtained\nfrom the solution of the linearized Peierls-Boltzmanntransport equation (PBTE) is used to calculate thermal\nconductivity ( κ). The sources of the scattering term in\nPBTE are currently limited to three-phonon (3-ph), four-\nphonon (4-ph), and isotopic impurity[37] scattering in\nour implementation. The three-phonon scattering rate,\nas given by Fermi golden rule:\n1\nτ3−ph(qs)=2π\nℏ2X\nq1s1,q2s2\"\n|ϕ(qs, q 1s1,−q2s2)|2(nq1s1−nq2s2)δ(ωqs+ωq1s1−ωq2s2) +1\n2|ϕ(qs,−q1s1,−q2s2)|2×\n(1 +nq1s1+nq2s2)δ(ωqs−ωq1s1−ωq2s2)#\n,(12)\nthe four-phonon scattering rate:\n1\nτ4−ph(qs)=2π\nℏ2X\nq1s1,q2s2,q3s3\"\n1\n6|ϕ(qs,−q1s1,−q2s2,−q3s3)|2nq1s1nq2s2nq3s3\nnqsδ(ωqs−ωq1s1−ωq2s2−ωq3s3)+\n1\n2|ϕ(qs, q 1s1,−q2s2,−q3s3)|2(1 +nq1s1)nq2s2nq3s3\nnqsδ(ωqs+ωq1s1−ωq2s2−ωq3s3)+\n1\n2|ϕ(qs, q 1s1, q2s2,−q3s3)|2(1 +nq1s1)(1 + nq2s2)nq3s3\nnqsδ(ωqs+ωq1s1+ωq2s2−ωq3s3)#\n,(13)\nand the phonon-isotopic impurity scattering rate:\n1\nτph−iso(qs)=πω2\nqs\n2N0X\nq1s1X\nµg2(µ)|ϵ(qs, µ)∗·ϵ(q1s1, µ)|2δ(ωqs−ωq1s1), (14)\nwhere g2(µ) =P\nifµi(1−Mµi/¯Mµ)2, with Mµiandfµi\nbeing the mass and fractional concentration of the ith\nisotope of µth basis atom and ¯Mµ=P\nifµiMµiis the\naverage mass. The total scattering rate, as given by\nMatthiessen’s rule:\n1\nτ(qs)=1\nτ3−ph(qs)+1\nτ4−ph(qs)+1\nτph−iso(qs).(15)\nThe expression for thermal conductivity tensor ( κ) in\nrelaxation time approximation (RTA) is given by:\nκRTA=1\nN0V0X\nq,s∂nqs\n∂Tℏωqsτ(qs)⃗ vqs⊗⃗ vqs. (16)\nThe thermal conductivity tensor from the iterative solu-\ntion of PBTE is represented by the following expression:\nκ=kBT2\nN0V0X\nq,s∂nqs\n∂T⃗ vqs⊗⃗∆qs (17)\nHere N0is the number of q-points in the BZ sampled, V0\nis the unit cell volume, and ⃗ vqs=1\n2ωqs⟨ϵ(qs)|∂D\n∂⃗ q|ϵ(qs)⟩is phonon group velocity with Dbeing dynamical ma-\ntrix (Eq. 3). The self-consistent equation governing\nthe vector function ( ⃗∆qs) is interconnected with the\nphonon scattering rate and 3-ph, 4-ph, and phonon-\nisotope scattering probabilities. For further details re-\ngarding these terms, readers can consult the following\nreferences[34, 38–40]. The novel aspects of our theo-\nretical framework are as follows: (1) The iterative so-\nlution of PBTE[41], which differentiates the momentum-\nconserving normal processes as non-resistive. (2) The\ncalculation of phonon group velocity involves the ex-\nact analytical differentiation of the long-range Ewald\nterm[31] for polar materials. This distinguishes our im-\nplementation from the others, which typically differen-\ntiate direction-dependent non-analytic terms for deter-\nmining phonon group velocity in polar materials. (3)\nIn the treatment of energy conservation within the Bril-\nlouin zone (BZ) sum, the tetrahedron method[42] is em-\nployed. The tetrahedron method offers several advan-\ntages over the Gaussian or Lorentzian approximation of\nthe Dirac-delta function. One notable advantage is that\nit does not rely on any adjustable parameters and ex-\nhibits slightly improved convergence with the number of\nq-points. In this study, the convergence of BZ integra-7\ntion for thermal conductivity is achieved using a q-mesh\ngrid of 17 ×17×17 for the 3-phonon scattering processes.\nThe phonon linewidth (Γ( qs)) discussed in this paper is\ndefined as:\nΓ(qs) =1\n2τ(qs). (18)\nFor all the ab initio calculations, we utilized SIESTA[35],\na code based on atomic-orbital basis. To expand the\nwave functions, we employ the polarized double-zeta ba-\nsis set. Norm-conserving pseudopotentials are utilized,\nand exchange-correlation treatment follows the GGA-\nPBE[43] approach.\nIII. RESULTS\nA. High κMaterial: cBN and 3C-SiC\nWe selected cubic-Boron Nitride (cBN) and cubic-\nSilicon Carbide (3C-SiC) as materials characterized by\nlow anharmonicity in their interatomic potential. High\nthermal conductivity in these materials is attributed to\nthe combination of weak anharmonicity and the light\natomic mass of their constituent atoms. Figure 1(a) and\nFigure 2(a) display phonon dispersion for cBN and 3C-\nSiC, respectively, presenting results from both the QHA\nand renormalized IFCs at 300K. Note that in QHA, the\ntemperature effect on the lattice constant is taken into\naccount, and the displacement-force dataset (Eq. 2) is\nfitted up to the fourth order. There is little to no no-\nticeable difference between the phonon frequencies ob-\ntained through the QHA and the renormalized approach.\nThis signifies the existence of weak higher-order anhar-\nmonic terms in these materials, which appear in the\nrenormalization process. The figures also include the\nexperimental phonon frequency for cBN[44–46] and 3C-\nSiC[49], demonstrating a close agreement with our cal-\nculations. Figure 1(b-e) and Figure 2(b-e) illustrate the\ntemperature-dependent phonon linewidth for one acous-\ntic branch and optic branches along the high-symmetry\nΓ→Xpath of cBN and 3C-SiC, respectively. As de-\npicted in the figures, the phonon linewidth exhibits a\nmonotonic increase with temperature, resulting in re-\nduced thermal transport as the temperature rises. Ob-\nserve that the phonon linewidth for the acoustic branch\nis significantly smaller compared to that of the optic\nbranches, suggesting lower scattering for the acoustic\nphonons. This difference contributes to the high ther-\nmal conductivity of acoustic phonons, given the inverse\nrelationship between thermal conductivity and linewidth.\nMore than 90% of the total κvalue for both materi-\nals is contributed by three acoustic modes, as depicted\nin the inset plot of Figure 1(f) and Figure 2(f). Also,\nobserve that certain peak positions occur within the\nlinewidth curve along the high-symmetry path. These\npeaks emerge as both momentum conservation (Eq. 10)\nand energy conservation (Eq. 12) are met to a greaterextent, leading to heightened scattering for these phonon\nmodes ( qs). Consequently, the linewidth of these phonon\nmodes becomes more pronounced, manifesting as peaks\nwithin the linewidth curve. The temperature-dependent\nthermal conductivity of cBN and 3C-SiC is presented in\nFigure 1(f) and Figure 2(f) across the 100K to 500K\nrange. Note that for these materials characterized by\nhigh thermal conductivity, renormalization of the IFCs\nhas minimal to no impact on the κvalues. For instance,\ncBN exhibits a thermal conductivity of 648.31 W/m.K\nat 300K, according to the QHA. Meanwhile, the renor-\nmalization of IFCs predicts κat 666.97 W/m.K, repre-\nsenting a roughly 3% increase compared to the QHA-\nderived value. The experimental determination of the\nκof cBN is 648 ±52 W/m.K as extracted from Ref.[48]\nand 850 ±90 W/m.K as extracted from Ref.[47] Note\nthat in our calculation of phonon-isotope scattering, we\nconsidered the natural abundance of the isotopes of the\nconstituent atoms, and the experimental values extracted\nalso correspond to the natural isotopic concentration in\nthe sample. For 3C-SiC, our calculation using the QHA\nyields a thermal conductivity value of 364.25 W/m.K,\nwhile the renormalization predicts a slightly elevated κ\nof 378.68 W/m.K at 300K. The experimental κof single-\ncrystalline 3C-SiC as extracted from the Ref.[50] is 437\nW/m.K. Figure 2(f) also includes the thermal conduc-\ntivity of polycrystalline 3C-SiC, as extracted from the\nRef.[51], which tends to match our calculation at higher\ntemperatures. The bar plots inset in Figure 1(f) and\nFigure 2(f) demonstrate the contribution of the three\nacoustic phonon branches to thermal conductivity at var-\nious temperatures. With increasing temperature, higher-\nfrequency phonon branches and optic branches tend to\ncontribute more to thermal conduction. Figure 1(g) and\nFigure 2(g) illustrate the three-phonon + phonon-isotope\nscattering rates at temperatures of 300K and 500K for\ncBN and 3C-SiC, respectively. The elevation in scatter-\ning rate as temperature increases accounts for the reduc-\ntion in thermal conductivity. To translate the scattering\nrate plot into thermal conductivity, we computed spec-\ntral thermal conductivity, as shown in Figure 1(h) and\nFigure 2(h). Observe that the spectral thermal conduc-\ntivity tends to decrease as temperature rises, resulting\nin a reduction in the overall thermal conductivity. Ad-\nditionally, note that the spectral thermal conductivity\nfor renormalized IFCs is slightly higher than that of the\nQHA for all the temperatures. This accounts for the\nmarginal increase in thermal conductivity observed with\nthe renormalized approach in our calculations.\nB. Low κMaterial: NaCl and AgI\nTo demonstrate the practicality of our numerical\nframework for materials with low thermal conductiv-\nity and significant anharmonicity, we selected sodium\nchloride (NaCl) and silver iodide ( γ-AgI, space group\n-F¯43m) as representative materials. Being a member8\n(a)                                            (b)                                   (d)\n(c)                                     (e)\n(f)                                                        (g)                          (h)\nFIG. 1. (a) Phonon dispersion of cBN along the high symmetry path calculated from QHA and renormalized IFCs at 300K.\nObserve that there is no significant difference between QHA and renormalization. The experimental data are extracted from\nExp.1[44], Exp.2[45], and Exp.3[46]. (b) - (e) Variation in phonon line width along the Γ →Xhigh-symmetry path for\none acoustic branch (TA) and three optic branches (TO, ZO, and LO) in cBN across temperatures ranging from 100K to\n500K. (f) Thermal conductivity of cBN calculated across the temperature range of 100K to 500K. Renormalization results\nin approximately a 2% to 3% difference in the κvalue compared to the QHA. Experimental data are obtained from Exp.\n(2020)[47] and Exp. (2018)[48]. Inset plot: Contribution from the three acoustic phonon branches to the thermal conductivity\nat different temperatures. (g) Phonon scattering rates, considering 3-phonon and phonon-isotope scattering, calculated for cBN\nat 300K and 500K using renormalized IFCs. The elevated scattering rates at higher temperatures contribute to a reduction in\nthermal conductivity. (h) Spectrally resolved thermal conductivity of cBN at various temperatures, providing insights into the\nscattering rate plot. Notably, the renormalized spectral κexhibits a slight increase across all temperatures.\nof the alkali halides family, NaCl exhibits high anhar-\nmonicity in its inter-atomic potential due to the weakly\nbonded, highly ionic nature of its constituent atoms. Fur-\nthermore, it displays remarkably low thermal conductiv-\nity, making it an ideal candidate for studying the an-\nharmonicity within our current framework. Figure 3(a)\npresents the phonon dispersion of NaCl crystal at 300K,\ncalculated from QHA and through the renormalization of\nthe phonon quasiparticle as described in the method sec-\ntion. Our findings indicate that the renormalization pro-\ncess significantly increases the phonon frequency of the\noptical modes of NaCl. Such behavior is not observed in\ncBN and 3C-SiC. The acoustic branches remain largely\nunaffected after renormalization. The experimentally\nmeasured data[52] through the inelastic neutron scat-\ntering demonstrates a close agreement with the renor-malized phonon frequency. This emphasizes the signifi-\ncance of treating phonons as quasiparticles through the\nrenormalization process, rather than considering them\nsolely as normal modes of lattice vibrations, especially\nfor highly anharmonic materials. Figure 3(b) to Fig-\nure 3(d) show the dependence of phonon dispersion with\ntemperature from 100K to 400K, both for bare phonon\nwithin QHA and renormalized phonon. The outcome re-\nveals that, in comparison to the renormalized phonon,\nthe bare phonon experiences a higher degree of softening\nin the optical phonon branches as the temperature in-\ncreases. In Figure 3(d), the plot illustrates the variation\nof phonon frequency at the Γ-point of the Brillouin zone\n(BZ) with temperature for both the transverse optical\n(TO) and longitudinal optical (LO) branches. In contrast\nto QHA, the renormalized phonon demonstrates a mini-9\n(a)                                       (b)                                   (d)\n(c)                                     (e)\n(f)                                                        (g)                               (h)\nFIG. 2. (a) Phonon dispersion of 3C-SiC along the high symmetry path calculated from QHA and renormalized IFCs at 300K.\nObserve that there is no significant difference between QHA and renormalization. The experimental data are extracted from\nRef.[49] (b) - (e) Variation in phonon linewidth along the Γ →Xhigh-symmetry path for one acoustic branch (LA) and\nthree optic branches (TO, ZO, and LO) in 3C-SiC across temperatures ranging from 100K to 500K. (f) Thermal conductivity\nof 3C-SiC calculated across the temperature range of 100K to 500K. Renormalization results in a marginally higher κvalue\ncompared to the QHA. Experimental data are obtained from Exp.1[50] and Exp.2[51]. Note that Exp.2[51] corresponds to\na polycrystalline sample, which tends to match better with our calculation at higher temperatures. Inset plot: Contribution\nfrom the three acoustic phonon branches to the thermal conductivity at different temperatures. (g) Phonon scattering rates,\nconsidering 3-phonon + phonon-isotope scattering, calculated for 3C-SiC at 300K and 500K using renormalized IFCs. The\nelevated scattering rates at higher temperatures contribute to a reduction in thermal conductivity. (h) Spectrally resolved\nthermal conductivity of 3C-SiC at various temperatures. Notably, the renormalized spectral κexhibits a slight increase across\nall temperatures.\nmal temperature dependence while remaining consistent\nwith the experimental data. Figure 3(e) illustrates the\ndistribution of phonon linewidth or inverse phonon life-\ntime along the Γ-X high-symmetry path in the BZ due to\nthe three-phonon scattering process at 300K. The quan-\ntitative value of the linewidth can be obtained from the\nvertical extent of the highlighted area around the phonon\nbranch, with the unit being the same as the phonon fre-\nquency (THz). In Figure 3(f), it can be observed that\nfor the temperature range of 100K-400K, the linewidth\nof the LO branch at the Γ-point is greater in the case\nof the renormalized phonon. Conversely, the renormal-\nized linewidth of the TO branch consistently remains\nlower compared to that of the bare phonon. It should\nbe noted that, in our approach, anharmonic third-order\nIFCs are also subject to renormalization, according toEq. 5. The increased linewidth of the LO branch indi-\ncates a higher rate of 3-phonon scattering, resulting in\na reduced contribution to thermal conduction. Figure\n4(a) presents the thermal conductivity ( κ) in the 100K to\n400K temperature range calculated from bare and renor-\nmalized IFCs. The thermal conductivity calculated from\nthe three-phonon scattering process of the renormalized\nphonon consistently overestimates the values obtained\nfrom QHA and experimental data. For example, the\nrenormalized κat 100K is 28.78 W/mK, exhibiting an\napproximately 17% increase compared to the unrenor-\nmalized κof 24.62 W/mK. At 300K, the renormalization\nresults in an overprediction of about 25%. Importantly,\nour implementation of the tetrahedron method for en-\nergy conservation delta function does not involve any ad-\njustable parameters for the thermal conductivity calcu-10\n(a) (b) (c)\n(d)\n(e) (f)\nFIG. 3. (a) Phonon dispersion of NaCl along the high-symmetry path at 300K. The renormalization leads to an upward shift in\nthe frequency of the optical branches, aligning with the experimental data.[52] (b) Temperature dependence of the bare phonon\ndispersion within the 100K-400K range. Observe the pronounced softening of optical branches with increasing temperature.\n(c) The renormalization leads to weaker temperature dependence phonon dispersion consistent with the experimental data.\n(d) Temperature dependence of the phonon frequency at the Γ point for LO and TO branches. Notice the reduced extent\nof softening with increasing temperature due to renormalization, closely matching the experimental data.[53] (e) Distribution\nof phonon linewidth along the Γ-X path at 300K. The vertical spread of the shaded region surrounding each phonon branch\ndetermines the linewidth. (f) Temperature dependence of the phonon linewidth at the Γ point for LO and TO branches.\n(a) (b) (c)\nFIG. 4. (a) Thermal conductivity of NaCl calculated using the QHA and renormalization. Notably, renormalization consistently\noverestimates the thermal conductivity across all temperatures. Experimental values are extracted from Ref.[54–56] (b) Three-\nphonon scattering rate calculated from bare and renormalized IFCs at 300K. (c) Spectral distribution of thermal conductivity\nat 100K and 300K calculated through QHA and renormalization.\nlation, distinguishing it from the Gaussian or Lorentzian\napproximations. On the other hand, the thermal conduc-\ntivity obtained from the bare IFCs exhibits a close agree-ment with the experimental data, even in the scenario\ninvolving minimal three-phonon scattering processes. At\na temperature of 300K, the κderived from the QHA is11\n(a)(b)                                        (d)\n(c)                                         (e)\n(f)                                                           (g)              (h)\nFIG. 5. (a) Phonon dispersion of AgI along the high symmetry path calculated from QHA and renormalized IFCs at 300K.\n(b) - (e) Variation in phonon linewidth for one acoustic branch (ZA) and three optic branches (TO, ZO, and LO) in AgI across\ntemperatures ranging from 100K to 450K. (f) Thermal conductivity of AgI calculated over the temperature range of 100K to\n450K using 3-phonon scattering. Results from four-phonon scattering processes are presented for the 113q-grid for 200 K, 300\nK, and 400 K temperatures. Experimental data are sourced from Exp. (1982)[57] and Exp. (2023)[58]. It is noteworthy that\nboth the QHA and renormalized values tend to over-predict the measured κvalues in three-phonon, whereas the four-phonon\nprediction closely aligns with the experimental data. Inset plot: Convergence of κwith q-grid in BZ for three-phonon and\nfour-phonon processes. (g) Phonon scattering rates, taking into account 3-phonon + phonon-isotope scattering at 113and\n173q-grid, and 3-phonon + 4-phonon + phonon-isotope scattering processes at 113q-grid. The highlighted regions indicate a\nsignificant increase in the scattering rate, approximately an order of magnitude higher for four-phonon processes compared to\nthree-phonon processes. Notably, the two dips in the scattering rate in the highlighted region correspond to the two peaks in\nthe spectral κ. (h) Spectrally resolved thermal conductivity of AgI for three-phonon and four-phonon scattering, illustrating\nthe notable decrease in κwhen fourth-order processes are considered.\nin close proximity to the experimental results, with a de-\nviation of less than 7% from the findings in Ref[55] and\nless than 16% from Ref[54]. To interpret this anoma-\nlous behavior of renormalized phonon, we calculated the\nthree-phonon scattering rate from renormalized and bare\nIFCs, as presented in Figure 4(b). As illustrated in the\nfigure, the scattering rate derived from the bare IFCs\nsurpasses the scattering rate of the renormalized IFCs.\nFurthermore, the bare phonon exhibits a greater extent\nof temperature-induced softening, as depicted in Figure\n3(a-d). By combining these two factors, the thermal con-\nductivity predicted from the unrenormalized IFCs con-\nsistently remains lower than that from the renormalized\nIFCs, since κis directly proportional to the phonon fre-\nquency and inversely proportional to the scattering rate.This can be observed in Figure 4(c), which illustrates\nthe spectral distribution of κat 100K and 300 K. At\nboth temperatures, the renormalized spectral κconsis-\ntently surpasses the values obtained from the unrenor-\nmalized spectral κacross all phonon frequency ranges,\nreinforcing the higher thermal conductivity exhibited by\nthe renormalized phonons. The peculiar characteristics\nof the renormalized phonon can be addressed by incorpo-\nrating higher-order phonon scattering processes, as elab-\norated in the Ref.[34], especially for highly anharmonic\nmaterials like NaCl. The inclusion of four-phonon scat-\ntering processes provides a means to address this issue.\nFigure 5 presents the thermal transport-related phys-\nical properties of AgI. Research interest has recently\ngrown in the thermal transport of materials exhibit-12\ning ultralow thermal conductivity[59, 60], primarily due\nto their potential applications in thermoelectric devices.\nAmong these materials, AgI stands out as a promising\ncandidate. Understanding the ultralow thermal trans-\nport in AgI has been enhanced by recent studies[58, 61]\nhighlighting the significance of higher-order phonon scat-\ntering processes. Figure 5(a) presents the phonon disper-\nsion of AgI along the high symmetry path of BZ at 300K\ncalculated from both the unrenormalized and renormal-\nized IFCs. Note that the phonon frequencies experience\na slight stiffening across both the acoustic and optical\nbranches when considering renormalized IFCs. The in-\ncrease in phonon frequency in AgI resulting from renor-\nmalization is of a smaller magnitude than that observed\nin NaCl. Figure 5(b-e) depicts temperature-dependent\nphonon linewidth for one acoustic branch and three op-\ntic branches along the high symmetry Γ →Xpath\nfor the 100K to 450K temperature range. The mono-\ntonic increase of the linewidth with temperature implies\nhigher phonon scattering as temperature rises. Figure\n5(f) presents the thermal conductivity of AgI for the\ntemperature range 100K to 450K, calculated using both\nQHA and renormalized IFCs. The thermal conductiv-\nity calculations, relying solely on anharmonic 3-phonon\nand phonon-isotope scattering as the primary sources of\nscattering in PBTE, are represented by solid red and blue\nlines for QHA and renormalized IFCs, respectively. The\nexperimental values, as extracted from the Ref.[57, 58],\nare also presented in the figure. Observe that our cal-\nculated κvalues from three-phonon scattering processes\nconsistently exceed the experimental values across all\ntemperature ranges, and the process of renormalization\nleads to an additional overprediction compared to the\nQHA values. For example, our calculated κof AgI at\n300K is 1.03 W/m.K, whereas the experimental κis\n0.36 W/m.K, as reported in Ref.[58] and 0.41 W/m.K\nas extracted from the Ref[57]. The renormalization pre-\ndicts κas 1.17 W/m.K, which is about 13% higher than\nthe QHA value. The discrepancy between the calcu-\nlated thermal conductivity and experimental observa-\ntions highlights the significance of higher-order phonon\nscattering processes (specifically, 4-phonon interactions)\nfor highly anharmonic materials such as AgI. To enhance\nthe robustness of our numerical framework in handling\nmaterials with extreme anharmonicity, we incorporated\nthe computation of four-phonon scattering processes for\nAgI. Figure 5(f) displays the computed thermal conduc-\ntivity, taking into account four-phonon scattering, for\ntemperatures of 200 K, 300 K, and 400 K. Take note of\nthe impressive alignment between the computed κand\nexperimental values achieved simply by integrating the\nfourth-order scattering process. Our calculation predicts\ntheκof AgI to be 0.41 W/m.K at 300 K, closely re-\nsembling the experimental values of 0.41 W/m.K[57] and\n0.36 W/m.K[58]. The computational expense increases\nsignificantly by several orders of magnitude when incor-\nporating four-phonon scattering. To address this compu-\ntational bottleneck, we mitigated it by leveraging symme-tries in the calculation of fourth-order scattering matrix\nelements (Eq. 11). In Eq. 11, the first index ( qs) is con-\nstrained to the irreducible Brillouin Zone (BZ), and the\nscattering matrix elements for permutations among the\nremaining three indices ( q′s′, q′′s′′, q′′′s′′′) are computed\nonly once, as they yield the same values. We also uti-\nlized specific computer hardware acceleration techniques\nto enhance the speed of the computation. To provide\na comparison of computational costs, our three-phonon\ncalculation for a 17 ×17×17 (173) q-grid in the BZ\nrequired approximately 6 minutes, utilizing 240 CPU\ncores distributed across 5 compute nodes. In contrast,\nour four-phonon calculation for an 11 ×11×11 (113) q-\ngrid took 15 hours, employing 960 CPU cores distributed\nacross 20 compute nodes. The convergence of κwith q-\ngrid is depicted in the inset plot in Figure 5(f). While\nit requires a 173q-grid for three-phonon convergence,\nno significant difference in κvalues for four-phonon has\nbeen observed between 73and 113q-grid. The conver-\ngence of κat a lower q-grid for four-phonon processes\nhas been well-established in previous studies[13, 16]. In\nour calculation, we address energy conservation using the\nparameter-free tetrahedron method[42], which achieves\nconvergence with a smaller q-grid compared to Gaussian\nor Lorentzian approximations. In Figure 5(g), the scat-\ntering rate is depicted for both three-phonon+phonon-\nisotope scattering processes and the additional contribu-\ntion from fourth-order scattering mechanisms calculated\nthrough the Fermi golden rule. Note that the scatter-\ning rates, encompassing four-phonon processes, are com-\nputed with an 113q-grid. To facilitate a direct compari-\nson with three-phonon scattering, we have also included\nthe three-phonon scattering rate for both 113q-grid and\n173q-grid. Note that the inclusion of four-phonon scat-\ntering results in an increased scattering rate compared to\nthe conventional three-phonon scattering. In the high-\nlighted region of Figure 5(g), the four-phonon scattering\nrate is approximately an order of magnitude greater than\nthat of the three-phonon. The rise in phonon scattering\nrates results in a decrease in the thermal transport coeffi-\ncient of AgI during the four-phonon process, as depicted\nin Figure 5(f). For a more in-depth exploration of the\nimpact of higher-order four-phonon scattering on ther-\nmal conductivity, we computed the spectral distribution\nofκ, as depicted in Figure 5(h). Observe that the two\npeaks in the spectral κalign with the two dips in the\nhighlighted region of the phonon scattering rate in Fig-\nure 5(g). The spectral κexhibits a notable reduction\nin the four-phonon process across all phonon frequency\nranges, leading to an overall decline in the total thermal\nconductivity compared to the three-phonon scenario.\nC. Free energy and pressure dependent thermal\nconductivity\nFigure 6(a) depicts the per unit cell free energy varia-\ntion as a function of unit cell volume at 300K. The figure13\n(d)                                          (e)                               (f)                           (g)\n(a)                                                                     (b)                                                     (c)\nFIG. 6. (a) Variation of harmonic and fourth-order free energy with unit cell volume at 300K. Observe the overlap between\nF2ndandF4thfor QHA, resulting from the third-order and fourth-order term cancellation. The inset plot illustrates pressure\ncalculated using the volumetric derivative of free energy. (b) Thermal conductivity as a function of pressure at 300K, computed\nusing QHA and renormalized IFCs. Experimental data extracted from Ref[54, 62]. (c) Phonon dispersion of NaCl at 0 GPa and\n3 GPa pressure, computed at 300K using renormalized Interatomic Force Constants (IFCs). Observe the significant phonon\nstiffening, particularly in the optical branches, under increased pressure. (d) Three-phonon scattering rate computed at 0 GPa\nand 3 GPa pressure. The reduction in anharmonic scattering rate with increased pressure results in higher thermal conductivity\nat elevated pressures. (e) Spectral decomposition of thermal conductivity at 0 GPa and 3 GPa pressure. Notice that the spectral\nκrises due to the lower scattering rates of phonon modes as pressure increases. (f) and (g) Three-phonon scattering phase\nspace (AAA, AA0, and AOO) of NaCl at two different pressures. Note that there is no significant change in AAA and AOO\nphase space as pressure increases. The AAO phase space is enhanced for higher-frequency phonon modes due to the widening\nof the acoustic-optic gap in the phonon dispersion, enabling more higher-frequency modes to participate in scattering events\nwhile energy conservation is maintained.\nillustrates that the free energy calculated up to the sec-\nond and fourth order in QHA is nearly identical across all\nvolumes. In our observations using QHA, we find that\nthe contribution from the third-order and fourth-order\nanharmonic terms to the free energy mutually nullifies.\nConversely, in the renormalized case, such cancellation\ndoes not occur. The points of lowest free energy on the\ncurve correspond to the equilibrium lattice constant of\nthe crystal at zero pressure and the specific tempera-\nture. By minimizing the fourth-order free energy using\nthe renormalized IFCs, the lattice constant of NaCl at\n300K is determined to be 5.637 ˚A. The lattice constant\nobtained from the renormalized harmonic free energy is\nat 5.624 ˚A. In contrast, the QHA predicts the lattice con-\nstant to be 5.632 ˚A. The experimentally measured lattice\nconstant at NaCl at 300K is 5.645 ˚A[63]. While both the\nQHA and renormalization techniques underpredict the\nlattice constant, the renormalized fourth-order approach\nshows better agreement with the measured data. In Fig-ure 6(a), the inset plot illustrates the variation of unit\ncell volume with pressure for NaCl at 300K. The pres-\nsure ( P) is determined using the thermodynamic relation\ninvolving the free energy: P=−\u0000∂F\n∂V\u0001\nT. The pressure-\ndependent thermal conductivity of NaCl at 300K is pre-\nsented in Figure 6(b). As the pressure rises from 0 to 3.4\nGPa, the thermal conductivity obtained from QHA ex-\nperiences an increase, going from 7.53 W/mK to 11.12\nW/mK. As explained in the earlier section, the ther-\nmal conductivity obtained from renormalized IFCs tends\nto overpredict, primarily because we considered only up\nto three-phonon scattering processes. The experimental\ndata[54, 62] also exhibit a similar increasing trend for\nthermal conductivity, as seen in the QHA values. The\nphonon modes stiffen with the increase in pressure, as\ndepicted in Figure 6(c). It is important to note that the\nrate of phonon stiffening is more significant in the optic\nbranches compared to the acoustic branches. In order to\ncomprehend the rise in thermal conductivity with pres-14\nsure, we calculated three-phonon scattering rates at 0\nGPa and 3GPa pressure, as depicted in Figure 6(d). As\nthe pressure increases, the total three-phonon scattering\nrate decreases, leading to higher thermal conductivity at\nelevated pressure levels. The spectral thermal conduc-\ntivity, as shown in Figure 6(e), also shifts to higher val-\nues as pressure increases due to reduced scattering rates\nof phonon modes. To investigate the anharmonic three-\nphonon scattering processes further, we calculated the\nscattering phase space involving different polarizations\nof phonon modes. The scattering phase space considered\nhere is defined as[64]:\nP(ωqs) =2\n3N2pN0X\nq′,s′X\nq′′,s′′h\nδ(ωqs+ωq′s′−ωq′′s′′) +\n1\n2δ(ωqs−ωq′s′−ωq′′s′′)i\n,\n(19)\nwhere Npis the number of polarizations. The above\nexpression implicitly incorporates quasi-momentum con-\nservation: q±q′+q′′=G, which leads to considering\nonly the summation over q′. Multiple scattering pro-\ncesses are feasible, involving various acoustic (A) and\noptic (O) phonon branch combinations. Figures 3(f) and\n3(g) illustrate the three-phonon scattering phase space\nfor AAA, AAO, and AOO scattering processes. Due to\nenergy conservation constraints, scattering processes in-\nvolving three optical branches (OOO) are not permitted.\nAs depicted in Figure 6(f), the AAA scattering phase\nspace remains relatively constant as pressure increases.\nAAA scattering phase space at 0 GPa also reveals more\nfeasible scattering events conserving momentum and en-\nergy than at 3 GPa. As depicted in Figure 6(g), the\nAOO scattering phase space demonstrates minimal to no\nvariation with the increase in pressure. Conversely, the\nAAO scattering phase space displays an increment with\nincreasing pressure. This is attributed to the stiffening\nof phonon dispersion, which results in an increased gap\nbetween acoustic and optic phonons. According to en-\nergy conservation principles, only acoustic phonons with\nfrequencies higher than this are allowed to take part in\nAAO scattering processes[65, 66]. So, a greater num-\nber of higher frequency phonon modes participate in the\nscattering process, as evident from Figure 6(g).\nIV. CONCLUSIONS\nIn summary, we have developed a comprehensive\nnumerical framework for understanding phonon and\nphonon-driven physical properties of crystalline semi-\nconductors and insulators. The phonon quasiparticle\napproach forms the foundation for our self-consistent\nphonon renormalization technique, which effectively\ncharacterizes both the vibrational attributes of strongly\nanharmonic solids and those of harmonic crystals. Ourrenormalization technique also extends to the anhar-\nmonic third- and fourth-order IFCs. Both the TDEP\nand TSS utilized in this study include explicit variations\nin temperature when computing IFCs. The investiga-\ntion of the NaCl crystal in our study demonstrates a no-\ntable alignment between the phonon dispersion obtained\nfrom the renormalized IFCs and the experimental data.\nOn the other hand, the thermal conductivity values of\nNaCl exhibit a close correspondence with the experimen-\ntal measurements when considering the IFCs obtained\nthrough the quasiharmonic approximation (QHA). Our\ninvestigation reveals that both the QHA and the renor-\nmalization approach consistently overestimate the exper-\nimentally measured thermal conductivity of AgI when\nconsidering three-phonon scattering processes. This im-\nplies that higher-order phonon scattering processes are\ncrucial considerations in materials with ultralow thermal\nconductivity, like AgI. Consequently, we undertook the\ncomputationally demanding task of calculating fourth-\norder scattering processes to address these discrepan-\ncies. We observe that materials exhibiting high κex-\nperience minimal impact on thermal conductivity and\nphonon-driven physical properties through the renormal-\nization of the IFCs. Both cBN and 3C-SiC, investigated\nin this study as materials with high thermal conductiv-\nity, demonstrate a 2-3% difference in κvalues between\nQHA and renormalization. These results closely match\nexperimental observations, even when considering only\nup to 3-phonon scattering processes. In our current in-\nvestigation, the Helmholtz free energy takes into account\nanharmonicity up to the fourth order and demonstrates\na close agreement with the experimental lattice constant\nof the NaCl crystal. Additionally, we have examined how\nthe thermal conductivity of NaCl changes under varying\npressure conditions. Anharmonic three-phonon scatter-\ning rates and scattering phase space are also analyzed\nto elucidate the behavior of the thermal transport coef-\nficient with temperature and pressure.\nWhile the phonon dispersion and its associated proper-\nties exhibit favorable alignment with experimental data\nthrough the renormalization approach, anharmonic char-\nacteristics like lattice thermal conductivity consistently\nexhibit an overprediction compared to experimental mea-\nsurements when IFCs are subjected to renormalization,\nboth for NaCl and AgI. The primary reason behind these\ndiscrepancies is that we have considered only up to the\nthree-phonon scattering processes. The significance of\nhigher-order anharmonicity cannot be overlooked in the\ncase of strongly anharmonic materials such as NaCl and\nAgI. While our current exploration focuses solely on the\nfour-phonon scattering processes for AgI, we have omit-\nted the consideration of higher-order scattering processes\nfor other materials in this study due to computational\nlimitations. The findings of our present investigation are\nsummarized in Table II. The effects of electron-phonon\nscattering processes[67] are not accounted for in our cur-\nrent study. For certain semiconductors with higher car-\nrier concentrations at standard room temperature, ignor-15\nMaterial Temperature (K) κ(QHA) ( Wm−1K−1)κ(Wm−1K−1)\n(Renormalized)κ(Exp.) ( Wm−1K−1)\n300K 648.31 666.97 ∼648±52[48], 850 ±90[47]\ncBN 400K 512.95 527.42 ∼460±40[48], ∼603±70[47]\n500K 423.64 437.94 ∼360±42[48], ∼483±62[47]\n300K 364.25 378.68 ∼437[50]\n3C-SiC 200K 677.71 692.97 ∼768[50]\n250K 484.02 499.28 ∼552[50]\n300K 7.53 9.95 6.02[54], ∼5.8[56], ∼8.1[55]\nNaCl 200K 10.90 13.75 ∼10.27[54], ∼11.35[55]\n100K 24.62 28.78 ∼22.47[54], ∼22.79[55]\n300K 1.03 1.17, 0.41(4-ph) 0.36[58], ∼0.41[57]\nAgI 200K 1.54 1.74, 0.77(4-ph) ∼0.61[58], ∼0.72[57]\n400K 0.78 0.88, 0.25(4-ph) ∼0.21[57]\nTABLE II. Comparison of thermal conductivity ( κ) at various temperatures for the materials examined in this study, calculated\nvia the QHA and renormalized IFCs, alongside experimental ( κ) values for reference. The symbol ∼denotes data extracted\nfrom corresponding references’ figures, which may contain inherent errors.\ning the influence of electron-phonon scattering processes\nis not viable.\nIn conclusion, we offer an extensive numerical frame-\nwork that enables the exploration of mechanical, ther-\nmodynamic, and thermal properties driven by crystalline\nvibrations in insulators and semiconductors. This frame-\nwork is built upon a sturdy foundation of interatomic\nforce constant (IFC) calculation and phonon renormal-\nization techniques. The present development enables\nthe efficient and higher-accuracy numerical and theoret-\nical study for understanding phonon and phonon-driven\nphysical properties for better and more efficient applica-\ntion of materials.V. ACKNOWLEDGMENTS\nWe thank TUE-CMS, IISc, funded by DST, India,\nas well as Param-Pravega at SERC, IISc, for providing\nthe computational facilities. 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Ordej´ on, The elphbolt ab initio solver for the cou-\npled electron-phonon boltzmann transport equations, npj\nComputational Materials 8, 28 (2022)."    },    {        "title": "2402.02796v1.Wavelet_characterizations_of_the_Sobolev_wavefront_set__bandlimited_wavelets_and_compactly_supported_wavelets.pdf",        "content": "arXiv:2402.02796v1  [math.FA]  5 Feb 2024WAVELET CHARACTERIZATIONS OF THE SOBOLEV WAVEFRONT SET:\nBANDLIMITED WAVELETS AND COMPACTLY SUPPORTED WAVELETS\nHARTMUT F ¨UHR, MAHYA GHANDEHARI\nAbstract. We consider the problem of characterizing the Sobolev wavef ront set of a tempered\ndistribution u∈ S′(Rd) in terms of its continuous wavelet transform, with the latt er being\ndefined with respect to a suitably chosen dilation group H⊂GL(Rd). We derive necessary and\nsufficient criteria for elements of the Sobolev wavefront set , formulated in terms of the decay\nbehaviour of a given generalized continuous wavelet transf orm. It turns out that the character-\nization of directed smoothness of finite order can be perform ed in the two important cases: (1)\nbandlimited wavelets, and (2) wavelets with finitely many va nishing moments (e.g. compactly\nsupported wavelets).\nThe main results of this paper are based on a number of fairly t echnical conditions on the\ndilation group. In order to demonstrate their wide applicab ility, we exhibit a large class of\ngeneralized shearlet groups in arbitrary dimensions fulfil ling all required conditions, and give\nestimates of the involved constants.\nKeywords: wavefront set; square-integrable group representation; cont inuous wavelet transform; aniso-\ntropic wavelet systems; shearlets\nAMS Subject Classification: 42C15; 42C40; 46F12\n1.Introduction\nA recurring theme in wavelet analysis and its various genera lizations is the relationship be-\ntween smoothness of the analyzed function on the one hand, an d the decay of its wavelet co-\nefficients on the other. Global smoothness, as quantified by no rms of smoothness spaces such\nas Besov spaces, is related to global decay properties, wher eas local smoothness is related to a\nmore localized decay behaviour of the coefficients. This pape r is concerned with the later class\nof questions, for a large class of generalized continuous wa velet transforms.\nOur aim is to give criteria for N-regular directed points of a tempered distribution u∈ S′(Rd)\nin terms of continuous wavelet transform decay. A wavelet tr ansform is defined with respect\nto a suitably chosen dilation group H⊂GL(Rd) and a wavelet vector φ∈ S(Rd). We obtain\nsuch criteria for two important cases: (1) when the wavelet v ector is compactly supported in\nfrequency, and (2) when the wavelet vector has a certain mini mal number of vanishing moments;\nthis case allows to use wavelet vectors that are compactly su pported.\nTheN-regular directed points, formally defined in equation (4) b elow, describe the oriented\nlocal regularity of a tempered distribution u: if (x,ξ) is anN-regular directed point of u, then\nucan be considered smooth of order Natx, when viewed in direction ξ. TheN-wavefront set\nWFN(u), a.k.a. the Sobolev wavefront set of uof orderN, consists of all directed points ( x,ξ)\nthat are not N-regular directed points of u. We typically describe the wavefront set WFN(u)\nby giving criteria for elements in its complement.\nIdeally, we would like to achieve results of the following fo rm:\n(x,ξ) is anN-regular directed point of u⇔ ∃neighborhood Uofx∀y∈U(1)\n∀h∈K:|Wψu(y,h)| ≤C/bardblh/bardblN,\n12 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nwhereK⊂His a suitable subset of dilations which explicitly depends o nψand a certain\n(sufficiently small) cone containing directions near ξ. Intuitively, Kcontains those h∈Hsuch\nthatπ(0,h)ψis a small-scale wavelet oscillating in direction ξ. A characterization for wavefront\nsets of infinite order - similar to the one in (1) - was given in [ 5]; here the decay of a fixed order\nNon the right-hand side is replaced by decay of arbitrary degr ee. ForN-regular directed points,\nwe will settle for more modest statements, of the following t ypes:\n(x,ξ) is anN-regular directed point of u (2)\n⇒ ∃neighborhood Uofx∀y∈U∀h∈K1:|Wψu(y,h)| ≤C/bardblh/bardblN−δ1,\ntypically called a “direct” theorem, as well as an “inverse” theorem (both in the parlance of\n[11]), namely\n∃neighborhood Uofx∀y∈U∀h∈K2:|Wψu(y,h)| ≤C/bardblh/bardblαN+δ2(3)\n⇒(x,ξ) is anN-regular directed point of u\nwith suitable constants α≥1,δ1,δ2≥0 and suitably defined subsets K1,K2⊂Hof dilations.\nFor the case of the shearlet transform in dimension two, a sim ilar characterization had been\nobtained by Grohs [11]. As related work, discussing this que stion with varying degrees of\ngenerality, we mention [13, 16, 12]. We point to the gap betwe en the exponents of the direct\nand the inverse statements in (2) and (3). To our knowledge, t his gap cannot be closed; see [11]\nfor results of similar type on shearlets. Consequently, the results of our paper provide a near\ncharacterization ofN-regular directed points. In order to extend such character izations to a\nmoregeneralsetup, inparticularforexpansiontoawidevar iety ofgeneralized wavelets, weadopt\nthe point of view from [5], with the aim of developing a compre hensive and unified approach.\nJust as in the mentioned paper, a recurrent theme in our work i s the need to understand the\ninfluence of structural properties of the underlying group o n the problem at hand.\nThis article contains several substantial generalization s and adaptations of the setup from [5].\nMost importantly, the investigation of finite degrees of smo othness makes it possible to include\nlarger classes of analyzing wavelets. The results of [5] rel ied on using bandlimited wavelets,\ni.e. Schwartz functions whose Fourier support is compactly supported, well away from the zero\nfrequency. This leads to wavelet coefficient decay of arbitra ry orders, provided the analyzed\nsignal is sufficiently smooth. Restricting the investigatio n of smoothness and decay behaviour\nto finite orders allows to replace the bandlimited analyzing wavelets by a more general class of\nwavelets, and thus to employ wavelets that are compactly sup ported in space rather than in\nfrequency. The decisive feature of the wavelet which enable s this replacement is the number of\nvanishing moments (see Definition 4.5 for a precise formulat ion of vanishing moments).\nUsing the vanishing moment criteria results in significant c onsequences in our theory. Firstly,\nthe formulation of direct and inverse theorems for the (near ) characterization of N-regular\ndirected points in this setting requires lower bounds for th e number of vanishing moments on\nthe analyzing wavelet. The recurrent theme of the dependenc e of the developed criteria on the\nunderlying group becomes visible here, as well. Secondly, w avelets with finitely many vanishing\nmoments can be real-valued. The symmetry properties of thes e wavelets in the Fourier domain\nmake them essentially incapable of distinguishing the freq uency direction ξfrom its opposite\n−ξ. In order to address these issues, we replace the wavefront s et by the so-called unsigned\nwavefront set , and reformulate the direct and inverse theorems according ly. We emphasize that\nthese issues, in particular the inability of the wavelet tra nsform criteria to distinguish opposite\ndirections of the wavefront set, are also present in the prec ursor papers [15, 11] (albeit not\naddressed explicitly), but not in [5]. We point to subsectio n 6.1 for an extended comparison of\nour results to the predecessor papers [15, 11].SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 3\n1.1.Overview of the paper and main results. The chief purpose of this paper is to adapt\nand extend the previously developed techniques for wavelet characterization of singularities, in\nordertoallow thetreatment offinitedegrees ofsmoothness, usingcompactly supportedwavelets.\nThe progress of this endeavour can be traced through the main results of our paper, which are\nthe following:\n•Theorem 3.4, which establishes criteria for points in the So bolev wavefront set, using\nwavelet coefficient decay with respect to bandlimited wavele ts;\n•Theorem 5.4, showing that these criteria can be transferred between different wavelets,\nas soon as the cross-kernel associated to the wavelets decay s sufficiently quickly;\n•Theorem 5.8 and Corollary 5.9, which establish Sobolev wave front set criteria using\ncompactly supported wavelets;\n•Theorem 6.12, which guarantees the applicability of the pre vious results for a large class\nof shearlet dilation groups.\nThe remainder of the paper is divided into 5 sections. Sectio n 2 reviews the basic definitions\nregarding the Fourier and wavelet transforms, ( N−)regular directed points and similar matters.\nSubsection 2.3 recalls the conditions on the dilation group needed to characterize elements of\nthe wavefront set in terms of wavelet coefficient decay, and re calls related results from [5].\nThis section also contains the definition of the cone-affiliat ed subsets Ko/i(W,V,R)⊂Hthat\nare instrumental in the precise formulation of the local wav elet coefficient decay conditions on\nwhich the wavelet based criteria for regular directed point s are based. This section prepares\nthe necessary background for both the formulation and the pr oof of Theorem 3.4, presented in\nthe subsequent section. This result formulates necessary a nd sufficient criteria for N-regular\ndirected points formulated in terms of wavelet coefficient de cay. Even though its proof is largely\nan adaptation of [5, Theorem 3.5], we include a sketch of the p roof of Theorem 3.4, as it will be\nused in the main result of the subsequent section.\nTheorem 3.4 only allows to use analyzing wavelets that are co mpactly supported in frequency,\nwhichraisesthequestionwhetherasimilarcharacterizati on forothertypesofwavelets ispossible\nas well. The extension to more general wavelets requires the introduction of various auxiliary\nnotions, definitions and results. Introducing these is the m ain purpose of Section 4. To begin\nwith, since vanishing moment criteria open the door to the us e of real-valued wavelets, we\nintroduce the unsigned wavefront set and exhibit its basic p roperties in Subsection 4.1. We then\nreview the important definitions and results in connection w ith vanishing moments, which were\nintroduced in [7, 8, 10], in Subsection 4.2. The chief purpos e of these results is to guarantee\na certain decay of wavelet coefficients, once both the wavelet and the analyzed function are\nsufficiently smooth, with sufficiently many vanishing moments . We then set out to write an\nexplicit cross-kernel convolution formula (Proposition 4 .10) which will be used later on.\nIn Section 5, we use the theory developed in Sections 3 and 4 to extend our scheme to\ncompactly supported wavelets. Theorem 5.4 is the central to ol for the transfer of wavelet\ncoefficient decay statements between pairsof wavelets with s ufficiently many vanishingmoments.\nAs such it is of considerable independent interest, since it establishes that the local wavelet\ncoefficient decay of a given signal genuinely reflects propert ies of the signal, once the analyzing\nwavelet has a certain oscillatory behaviour. As a surprisin g byproduct of Theorem 5.4, we are\nable to close a gap in Theorem 3.4; this result is stated in Cor ollary 5.6.\nWe subsequently use Theorem 5.4 to prove Theorem 5.8, showin g that wavelets that are\ncompactly supported in space (rather than in frequency) can also be used for the near charac-\nterization of unsigned regular directed points of finite ord er. As a further application we note in\nCorollary 5.9 that real-valued wavelets with vanishing mom ents of arbitrary order can be used\nto characterize unsigned regular directed points of infinit e order.4 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nThe closing section 6 verifies the extensive list of conditio ns imposed on the dilation group\nthroughout sections 3 – 5 for the class of shearlet dilation groups . We thereby exhibit a large\nclass of dilation groups in arbitrary dimensions to which th e main results of this paper are\napplicable, with concrete estimates for all constants invo lved in the criteria. These observations\nare summarized in Theorem 6.12. This result also fosters an u nderstanding of the influence of\nvarious features of the underlying group on properties of th e wavelet transform. We close by\ncomparing our results for generalized shearlet groups to th e main results of [15, 11] in Subsection\n6.1.\n2.Basic definitions regarding regular directed points and wav elet transforms\n2.1.N-Regular directed points and the wavefront set. First let us introduce some nota-\ntion. Given R>0 andx∈Rd, we letBR(x) andBR(x) denote the open/closed ball with radius\nRand center x, respectively. We let Sd−1⊂Rddenote the unit sphere. By a neighborhood of\nξ∈Sd−1, we will always mean a relatively open setW⊂Sd−1withξ∈W. GivenR >0 and\nan open set W⊂Sd−1, we define the (truncated) cone generated by Was\nC(W) :=/braceleftbig\nrξ′/vextendsingle/vextendsingleξ′∈W, r>0/bracerightbig\n=/braceleftbigg\nξ∈Rd\\{0}/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ\n|ξ|∈W/bracerightbigg\nandC(W,R) :=C(W)\\BR(0).\nBoth sets are clearly open subsets of Rd\\{0}and thus of Rd.\nGiven a tempered distribution uand an integer N∈N, we call (x,ξ)∈Rd×Sd−1anN-\nregular directed point of uif there exists ϕ∈C∞\nc(Rd), identically one in a neighborhood of\nx, as well as a ξ-neighborhood W⊂Sd−1and a constant CN>0 with\n∀ξ′∈C(W) :/vextendsingle/vextendsingle/hatwiderϕu(ξ′)/vextendsingle/vextendsingle≤CN(1+|ξ′|)−N. (4)\nNote that this condition effectively only concerns the behavi our at large frequencies: we may\nreplaceC(W) in inequality (4) by C(W,R) for anyR>0, without changing the definition of an\nN-regular point. In practice, the suitable Rwill be chosen depending on ξ. TheN-wavefront\nsetWFN(u) consists of all directed points ( x,ξ) that are not N-regular directed points of u.\n2.2.Continuous wavelet transforms in higher dimensions. We now introduce the neces-\nsary notions pertaining to continuous wavelet transforms i n some detail. We fix a closed matrix\ngroupH <GL(d,R), the so-called dilation group , and letG=Rd⋊H. This is the group\nof affine mappings generated by Hand all translations. Elements of Gare denoted by pairs\n(x,h)∈Rd×H, and the product of two group elements is given by ( x,h)(y,g) = (x+hy,hg).\nThe left Haar measure of Gis given by d( x,h) =|det(h)|−1dxdh, where dxand dhare the left\nHaar measures of RdandHrespectively. The group Hacts onRdby matrix multiplication.\nThe dual action is just the (right) action Rd×H∋(ξ,h)/mapsto→hTξ∈Rd.\nThe group Gacts unitarily on L2(Rd) by the quasi-regular representation defined by\n[π(x,h)f](y) =|det(h)|−1/2·f/parenleftbig\nh−1(y−x)/parenrightbig\n. (5)\nThe dilation group His called irreducibly admissible , if the associated quasi-regular rep-\nresentation πis irreducible and square-integrable. By [6], this propert y has a characterization\nin terms of the dual action: His irreducibly admissible if and only if there is a unique con ull\nopen orbit O=HTξ0of the dual action, with the additional property that the sta bilizer groups\nassociated to Oare compact.\nGiven functions ψ,f∈L2(Rd), thecontinuous wavelet transform of fwith respect to\nψis the function Wψf:G→C,Wψf(x,h) =/a\\}bracketle{tf,π(x,h)ψ/a\\}bracketri}ht. In this context, ψis called the\nanalyzing wavelet . Moreover, πinduces an action of GonS(Rd), the space of Schwartz func-\ntions. We write /a\\}bracketle{t· | ·/a\\}bracketri}ht:S′(Rd)×S/parenleftbig\nRd/parenrightbig\n→Cfor the natural extension of the L2-scalar productSOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 5\n/a\\}bracketle{t·,·/a\\}bracketri}ht, which means /a\\}bracketle{tu|ψ/a\\}bracketri}ht:=u/parenleftbig\nψ/parenrightbig\n. For future reference, let us observe that a straightforwar d\ncalculation yields\n[F(π(x,h)f)](ξ) =|det(h)|1/2·e−2πi/a\\}bracketle{tx,ξ/a\\}bracketri}ht·/hatwidef/parenleftbig\nhTξ/parenrightbig\n(6)\nforf∈L1/parenleftbig\nRd/parenrightbig\n+L2/parenleftbig\nRd/parenrightbig\n. Here, as in the remainder of the paper, we use the convention\nFf(ξ) =/hatwidef(ξ) =ˆ\nRdf(x)·e−2πi/a\\}bracketle{tx,ξ/a\\}bracketri}htdx\nfor the Fourier transform of f∈L1/parenleftbig\nRd/parenrightbig\n.\nUsing a Schwartz function ψas analyzing wavelet allows to extend the continuous wavele t\ntransform to the space of tempered distributions in a straig htforward manner. Namely, given a\ntempered distribution u∈ S′/parenleftbig\nRd/parenrightbig\nand someψ∈ S(Rd), we define the wavelet transform of\nuwith respect to ψby\nWψu:G→C,(x,h)/mapsto→ /a\\}bracketle{tu|π(x,h)ψ/a\\}bracketri}ht.\nThroughout this paper, wavelets ψwill always come from the Schwartz class, and accordingly\nthe wavelet transform is defined on S′(Rd).\nA waveletψ∈ S(Rd) is called admissible if it fulfills the admissibility condition\nCψ=ˆ\nH|/hatwideψ(hTξ)|2dh<∞for allξ∈ O.\nThe most important consequence of the admissibility condit ion is the inversion formula\nf=1\nCψˆ\nGWψf(x,h)π(x,h)ψd(x,h)\nvalid (in the weak operator sense) for all f∈L2(Rd). Note that this inversion formula does not\nnecessarily apply to tempered distributions.\n2.3.Technical conditions on the dual action. We will write V⋐Oto indicate that V,O ⊂\nRdare open sets and that the closure V⊂ Ois a compact subset of O. We assume that\nthe dilation group is irreducibly admissible, and so the ass ociated quasi-regular representation\nis irreducible and square-integrable. As a consequence of t his assumption, all the properties\nrequired in [5, Assumption 2.1] are guaranteed; we spell thi s out in the following lemma. We\nrefer to [5, Lemma 2.2] and the text leading up to it for a proof . Observe that part (b) of the\nfollowing lemma is slightly stronger than the correspondin g part (b) from [5, Assumption 2.1],\nbut the reasoning from [5] in fact shows the stronger stateme nt. In particular, observe that\nξ∈ Oentails−ξ∈ O.\nLemma 2.1. Suppose the dilation group is irreducibly admissible. Then t here exists an open,\nconull,HT-invariant subset O ⊂Rdwith the following properties:\n(a) For each ξ∈ O, we have R∗ξ⊂ O, whereR∗:=R\\{0}.\n(b) Any Schwartz function ψsuch that/hatwideψis compactly supported inside Ois admissible, i.e.,\nsatisfies\n∀ξ∈ O:ˆ\nH|/hatwideψ(hTξ)|2dh<∞.\nWe next formally define the sets KiandKowhich associate dilations to directions.\nDefinition 2.2. Let∅/\\e}atio\\slash=W⊂Sd−1be open with W⊂ O(which implies C(W)⊂ O).\nFurthermore, let ∅/\\e}atio\\slash=V⋐OandR>0. We define\nKi(W,V,R) :=/braceleftbig\nh∈H/vextendsingle/vextendsingleh−TV⊂C(W,R)/bracerightbig6 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nas well as\nKo(W,V,R) :=/braceleftbig\nh∈H/vextendsingle/vextendsingleh−TV∩C(W,R)/\\e}atio\\slash=∅/bracerightbig\n.\nIf the parameters are provided by the context, we will simply w riteKiandKo. Here, the sub-\nscriptsi/ostand for “inner/outer”.\nThese two types of sets are the central tool of our analysis. T he intuition behind their\ndefinition is that Kicontains all dilations hwith the property that the wavelets π(y,h)ψonly\n“see” directions in the cone C(W,R), as long as supp( /hatwideψ)⊂Vholds. Here, we used the property\nsupp(F[π(y,h)ψ])⊂h−Tsupp(/hatwideψ)⊂h−TVwhich is immediate from equation (6). As a result,\nlocal regularity in these directions should entail a decay e stimate for the wavelet coefficients\n(Wψu)(y,h) withh∈Ki.\nOn the other hand, Kocontains all those dilations that contribute to the (formal )wavelet\nreconstruction\n/hatwiderϕu(ξ) =ˆ\nRdˆ\nH(Wψϕu)(y,h)·(F[π(y,h)ψ])(ξ)dh\n|det(h)|dy (7)\nof the frequency content of a (localized) tempered distribu tionϕu, forξ∈C(W,R), again under\nthe assumption supp( /hatwideψ)⊂V. Thus, decay estimates for wavelet coefficients with dilatio ns in\nh∈Koshould allow to predict local regularity of uin these directions.\nNext, we show how the sets KiandKoare related to each other when moving along the\npoints of the orbit.\nLemma 2.3. Letξ0,ξ1∈ O∩Sd−1, such that hTξ0=ξ1.\n(a) For every R1>0andξ1-neighborhood W1inSd−1, there exists R0>0and aξ0-\nneighborhood W0such that\nhTC(W0,R0)⊆C(W1,R1).\nFor allR0,R1>0and neighborhoods W0,W1satisfying the above inclusion, we have\nKi(W1,V,R1)⊇h−1Ki(W0,V,R0).\n(b) For every R0>0andξ0-neighborhood W0inSd−1, there exists R1>0and aξ1-\nneighborhood W1such that\nC(W1,R1)⊆hTC(W0,R0).\nFor allR0,R1>0and neighborhoods W0,W1satisfying the above inclusion, we have\nKo(W1,V,R1)⊆h−1Ko(W0,V,R0).\nProof.To prove (a), let W1,R1begiven. Pick δ>0 such that for all v∈Bδ(ξ1) one hasv\n|v|∈W1\n(see the proof of [5, Lemma 4.5]). Let R0=R1·/bardblh−1/bardblandW0= (h−TBδ(ξ1))∩Sd−1, which is\nindeed a neighborhood of ξ0inSd−1. To prove that R0,W0are as desired, let x∈C(W0,R0).\nHencex=rw, withr > R 0,w∈W0. Thenv=hTw∈Bδ(ξ1), and by choice of δ, one has\nv\n|v|∈W1. It follows that\nhTx=rhTw=r|v|·v\n|v|∈R+·W1.\nFinally, by choice of x,\nR1=R0/bardblh−1/bardbl−1<|x|/bardblh−1/bardbl−1=|h−ThTx|/bardblh−1/bardbl−1≤ |hTx|,\nwhich establishes that hTx∈C(W1,R1). This proves the first inclusion in (a). Next, assume\nhTC(W0,R0)⊆C(W1,R1), and pick g∈Ki(W0,V,R0). So we have\ng−TV⊆C(W0,R0)⊆h−TC(W1,R1),SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 7\nand thush−1g∈Ki(W1,V,R1).\nTo prove (b), let W0,R0be as given in the assumption. We apply (a) for h−Tξ1=ξ0\nto findR1,W1such thath−TC(W1,R1)⊆C(W0,R0). Multiplying both sides by hTyields\nthe desired inclusion. To prove the next inclusion, assume C(W1,R1)⊆hTC(W0,R0), and\nletg∈Ko(W1,V,R1). By definition, this means g−TV∩C(W1,R1)/\\e}atio\\slash=∅. AsC(W1,R1)⊆\nhTC(W0,R0), this implies\ng−TV∩hTC(W0,R0)/\\e}atio\\slash=∅,\nor (hg)−TV∩C(W0,R0)/\\e}atio\\slash=∅, which entails hg∈Ko(W0,V,R0). /square\nWe now come to an important technical assumption concerning the dual action. Given a\nmatrixh, we let/bardblh/bardbldenote the operator norm of the induced linear map with respe ct to the\neuclidean norm.\nDefinition 2.4. Letξ∈ O∩Sd−1and∅/\\e}atio\\slash=V⋐O. The dual action is called V-microlocally\nadmissible in direction ξif there exists a ξ-neighborhood W0⊂ O ∩Sd−1and someR0>0\nsuch that the following hold:\n(a) There exist α1>0andC >0such that\n/bardblh−1/bardbl ≤C·/bardblh/bardbl−α1\nholds for all h∈Ko(W0,V,R0).\n(b) There exists α2>0such that\nˆ\nKo(W0,V,R0)/bardblh/bardblα2dh<∞.\nThe dual action is called microlocally admissible in direction ξif it isV-microlocally ad-\nmissible in direction ξfor some ∅/\\e}atio\\slash=V⋐O. It is called globallyV-microlocally admissible\nif there is ∅/\\e}atio\\slash=V⋐Osuch that the dual action is V-microlocally admissible in direction ξfor\nallξ∈ O∩Sd−1.\nRemark 2.5. We note that under the prevailing assumption that Ois a single orbit with compact\nstabilizers, microlocal admissibility in direction ξ∈ Oalready entails global microlocal admissi-\nbility. Under further additional assumptions such as the co ne approximation property, condition\n(a) entails condition (b); see [5, Lemma 6.1] .\nThe main purposeof microlocal admissibility is to establis h a systematic relationship between\nthe norm of h∈Hand the norms of the frequencies contained in the support of F(π(y,h)ψ).\nSuch norm estimates, which we list below, will be instrument al in our proof. See [5, Lemma 2.8]\nfor a proof of the following.\nLemma 2.6. Assume that the closed subgroup H≤GL/parenleftbig\nRd/parenrightbig\nis irreducibly admissible, and\nconsequently, the properties listed in Lemma 2.1 hold. Then 0/∈ O. Furthermore, the following\nhold:\n(a) Assume that ∅/\\e}atio\\slash=V⋐O. Then there exists a constant C1=C1(V)>0such that, for\nallh∈Hand allξ′∈V:\n/vextendsingle/vextendsingleh−Tξ′/vextendsingle/vextendsingle−1≤C1·/bardblh/bardbl.\n(b) Assume that the dual action fulfills condition 2.4(a), fo r some∅/\\e}atio\\slash=V⋐O, a suitable\nξ-neighborhood W0⊂Sd−1and someR0>0. Then there exists C2>0such that\n/bardblh/bardbl ≤C2·/vextendsingle/vextendsingleh−Tξ′/vextendsingle/vextendsingle−1\nα18 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nholds for all h∈Ko(W0,V,R0)andξ′∈V, withα1as in Definition 2.4. Consequently,\nwe have:\nsuph∈Ko(W0,V,R0)/bardblh/bardbl<∞.\n(c) Withα1as in Definition 2.4, for all h∈Ko(W0,V,R0)\n|det(h)|−1≤C3·/bardblh/bardbl−dα1(8)\nand\n1+/vextenddouble/vextenddoubleh−1/vextenddouble/vextenddouble≤C4·/bardblh/bardbl−α1, (9)\nhold, for constants C4=C4(W0,V,R0)>0andC3=C3(W0,V,R0)>0.\n3.Criteria for N-regular directed points: Bandlimited wavelets\nLetN0=N∪{0}. In the following, we use the Schwartz norms\n|ψ|N:= max\nα∈Nd\n0,|α|≤Nsup\nz∈Rd(1+|z|)N|∂αψ(z)|.\nNote that these norms are invariant under complex conjugati on. Letu∈ S′(Rd) denote a\ntempered distribution. We say uis of orderN(u) if for allf∈ S(Rd), we have\n|/a\\}bracketle{tu|f/a\\}bracketri}ht| ≤C|f|N(u).\nIt is well-known that every tempered distribution uis of finite order. In the next lemma, we\ncollect various known decay behaviors which will be used in o ur proofs later on. We refer to [5,\nLemma 3.1].\nLemma 3.1. Letψ,ϕ∈ S(Rd)andu∈ S′(Rd).\n(a) For all N∈N0and(x,h)∈G, the following inequality holds:\n|π(x,h)ψ|N≤CN|ψ|N·|det(h)|−1/2(1+/bardblh−1/bardbl)N·max/braceleftbig\n1,/bardblh/bardblN/bracerightbig\n·(1+|x|)N,\nwith a constant CNindependent of ψ,x,h.\n(b) LetN(u)denote the order of u. Then for all (x,h)∈G, the following inequality holds:\n|Wψu(x,h)| ≤C·|det(h)|−1/2(1+/bardblh−1/bardbl)N(u)·max/braceleftig\n1,/bardblh/bardblN(u)/bracerightig\n·(1+|x|)N(u),\nwhereC >0is a constant depending on ψ,uandN(u)but not on x,h.\n(c) For all N∈Nand(x,h)∈G, we have\n|Wψϕ(x,h)| ≤CN|ϕ|d+N+1|ψ|N·|det(h)|−1/2(1+/bardblh−1/bardbl)N·max/braceleftbig\n1,/bardblh/bardblN/bracerightbig\n·(1+|x|)−N\nwithCNindependent of ϕ,ψ,h,x .\n(d) Assume that the dual action is V-microlocally admissible at ξfor some ∅/\\e}atio\\slash=V⋐Oand\nthatψ∈ S/parenleftbig\nRd/parenrightbig\nwithsupp(/hatwideψ)⊂V. ChooseR0>0and aξ-neighborhood W0⊂Sd−1as\nin Definition 2.4. Then\n|Wψϕ(x,h)| ≤CM,N,ψ,W 0,V,R0·|ϕ|M+N·|det(h)|−1/2·(1+|x|)−N/bardblh/bardblM\nholds for all x∈Rd,h∈Ko(W0,V,R0)andM,N∈N, where the constant CM,N,ψ,W 0,V,R0\nis independent of x,handϕ.\nWe next address how wavelet coefficient decay and regular dire cted points are affected by\ncertain multiplication operators. We refer to [5, Lemma 3.2 ] for the proof of (i) and [5, Lemma\n3.4] for the proof of (ii).\nLemma 3.2. Letu∈ S′(Rd)and(x,ξ)∈Rd×Sd−1. Letψ∈C∞\nc(Rd)be identically one in\nsome neighborhood of x. Then we have:SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 9\n(i)(x,ξ)is anN-regular directed point of uiff it is anN-regular directed point of ψu.\n(ii) Letϕ∈ S(Rd)be such that supp(/hatwideϕ)⊂Vfor some ∅/\\e}atio\\slash=V⋐O. Moreover, assume that\nthe dual action is V-microlocally admissible in direction ξ. Then for all N∈N, there\nexist constants CN>0andǫ>0, such that the estimate\n|Wϕu(y,h)−(Wϕ(ψu))(y,h)| ≤CN/bardblh/bardblN\nholds for all y∈Bǫ(x)and allh∈Ko(W0,V,R0), as soon as R0>0and theξ-\nneighborhood W0⊂Sd−1satisfy part (a) of the definition of V-microlocal admissibility\n(Definition 2.4).\nRemark 3.3. We can now formulate wavelet criteria for N-regular directed points in the case\nof bandlimited wavelets. Theorem 3.4 is an adaptation of [5, Theorem 3.5] to the case of finite\nsmoothness. There are twonoteworthy gaps between the necessary part (a) and the suffici ent\npart (b): One concerns the set of matrices h∈Hfor which the norm estimate holds. For the\nnecessary condition the pertinent set is Ki(W,V,R)(which may well be empty), whereas the\nsufficient condition requires a decay estimate for the typica lly larger set Ko(W,V,R). This gap\nhas already been observed for the criteria in [5]. There exist criteria, the so-called (weak) cone\napproximation condition(s) , on the dual action that allow to determine when this gap can b e\nclosed. These are fulfilled, in particular, for the shearlet d ilation group and its relatives. In\nSection 5, we prove Theorem 5.4, which allows to derive a decay statement for sets of the type\nKo(W,V,R)from decay assumptions for sets of the type Ki(W′,V,R′). As a result, we are able\nto close this gap in Corollary 5.6 without assuming the cone ap proximation conditions. Instead,\nwe required that our analyzing wavelet has vanishing moment s of sufficiently large order.\nThe second gap concerns the exponents in the decay estimates f or wavelet coefficients: For\nN-regular directed points (x,ξ), the necessary criterion notes a decay of order N−δ1, whereas\nthe sufficient criterion requires order α1N+δ2, withα1,δ1,δ2>0. To our current knowledge,\nthis gap can not be closed (although possibly reduced); see t he rather similar results for shearlets\nobtained in [11]. We refer to subsection 6.1 for a more detailed discussion.\nTheorem 3.4. Assume that the dual action is V-microlocally admissible in direction ξ, with\nparameters α1,α2as in the definition. Let u∈ S′(Rd)and(x,ξ)∈Rd×(Sd−1∩O).\n(a) If(x,ξ)is anN-regular directed point of u, then there exists a neighborhood Uofx,\nsomeR>0and aξ-neighborhood W⊂Sd−1∩Osuch that for alladmissibleψ∈ S(Rd)\nwithsupp(/hatwideψ)⊂V, the following estimate holds:\n∃C >0∀y∈U∀h∈Ki(W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblN−α1d/2.\nFor each such ψ, we even have\n∃C >0∀y∈U∀h∈Ki(W,/hatwideψ−1(C\\{0}), R) :|Wψu(y,h)| ≤C·/bardblh/bardblN−α1d/2.\n(b) Letψ∈ S(Rd)be admissible with supp(/hatwideψ)⊂V. Assume that there exist a neighborhood\nUofx, a number R>0and aξ-neighborhood W⊂Sd−1∩Osuch that\n∃C >0∀y∈U∀h∈Ko(W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblα1N+3\n2α1d+α2\nThen(x,ξ)is anN-regular directed point of u.\nProof.Part (a) is provided by a verbatim reproduction of the proof o f [5, Theorem 3.5(a)].\nIn a similar way, part (b) is obtained by a straightforward ad aptation of the argument for [5,\nTheorem 3.5(b)]. Since this argument is more involved, we pr ovide a somewhat more detailed\nsketch.10 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nFor the proof of part (b), observe that we may assume uto be compactly supported by\nLemma 3.2. Let ε1>0 be such that Bε1(x)⊆U, and assume that ε1satisfies the statement in\nLemma 3.2(ii). By assumption,\n|Wψu(y,h)| ≤C/bardblh/bardblα1(N+3\n2d)+α2(10)\nholds for all y∈Bε1(x) andh∈Ko(W,V,R). WithR0>0 andW0⊂Sd−1∩Oas in Definition\n2.4, we may assume W⊂W0⊂ OandR>R 0. In particular, this implies C(W,R)⊂ O, thanks\nto Lemma 2.1(a).\nLet\nη:=ˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)·π(y,h)ψdh\n|det(h)|dy, (11)\nwhere the integral defining ηis to be understood in the weak sense, i.e., given ϕ∈ S(Rd), we let\n/a\\}bracketle{tη|ϕ/a\\}bracketri}ht=ˆ\nRdˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)[π(y,h)ψ](z)ϕ(z)dh\n|det(h)|dydz\n=ˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)·(Wψϕ)(y,h)dh\n|det(h)|dy. (12)\nUsing parts (b) and (d) of Lemma 3.1 together with Lemma 2.6, i t can be shown that ηis\na well-defined element of S′(Rd), and the integral in equation (12) converges absolutely fo r\nallϕ∈ S/parenleftbig\nRd/parenrightbig\n. Next, we show that u=η. Indeed, for the compactly supported tempered\ndistribution u, its Fourier transform /hatwideucan be represented as a continuous function on Rd, which\nhas an explicit formula on Ogiven as follows:\n/hatwideu(ξ′) =ˆ\nRdˆ\nHWψu(y,h)F[π(y,h)ψ](ξ′)dh\n|deth|dy,\nfor allξ′∈ O. SinceOis a co-null set in Rd, for everyϕ∈ S(Rd) we have\n/a\\}bracketle{tu|ϕ/a\\}bracketri}ht=/a\\}bracketle{t/hatwideu| F−1ϕ/a\\}bracketri}ht=ˆ\nO/hatwideu(ξ′)F−1ϕ(ξ′)dξ′\n=ˆ\nOˆ\nRdˆ\nHWψu(y,h)F[π(y,h)ψ](ξ′)F−1ϕ(ξ′)dh\n|deth|dydξ′\n=ˆ\nOˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)F[π(y,h)ψ](ξ′)F−1ϕ(ξ′)dh\n|deth|dydξ′\n=ˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)/parenleftbiggˆ\nRdF[π(y,h)ψ](ξ′)F−1ϕ(ξ′)dξ′/parenrightbiggdh\n|deth|dy\n=ˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)/parenleftbiggˆ\nRd[π(y,h)ψ](z)ϕ(z)dz/parenrightbiggdh\n|deth|dy\n=ˆ\nRdˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)[π(y,h)ψ](z)ϕ(z)dh\n|deth|dydz\n=/a\\}bracketle{tη|ϕ/a\\}bracketri}ht.\nToobtainthedesiredresult, wewillshowthat inasuitablen eighborhoodof x, thedistribution\nuis given by a CN-function. Consider the auxiliary function κobtained by pointwise evaluation\nof the integral defining ηweakly in equation (11), i.e.\nκ:Bε1/2(x)→C, z/mapsto→ˆ\nRdˆ\nKo(W,V,R)Wψu(y,h)[π(y,h)ψ](z)dh\n|det(h)|dy.SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 11\nNote that the convergence that is crucial to equate pointwis e and weak definitions is guaranteed\nby the assumptions on wavelet coefficient decay. We then write κ=κ1+κ2, where\nκ1(z) :=ˆ\nBε1(x)ˆ\nKo(W,V,R)Wψu(y,h)[π(y,h)ψ](z)dh\n|det(h)|dy.\nNote thatκ2is obtained by integration over the complement of the ball Bε1(x), where we make\nno specific assumptions on the wavelet coefficient decay. Usin g the proof of Theorem 3.5 of [5],\nwe have that κ2isC∞insideBε1/2(x), andκ1isN-fold differentiable as soon as the decay\norderMof the wavelet coefficients fulfills M≥α1(N+3\n2d)+α2. HenceκisCN, and for every\nϕ∈ S(Rd) with supp( ϕ)⊂Bǫ1/2(x), we clearly have\n/a\\}bracketle{tη|ϕ/a\\}bracketri}ht=ˆ\nRdκ(z)ϕ(z)dz .\n/square\n4.Towards an extension to compactly supported wavelets\nAs demonstrated in [11] for shearlets in two dimensions, it i s possibleto sometimes replace the\nassumption that /hatwideψis compactly supported by suitable finite vanishing moment c onditions, and\nstill obtain near characterizations of N-regular directed points. This allows to use wavelets that\nare compactly supported in space rather than in frequency. The chief purpose of this section\nis to prepare the ground for the desired generalization, by h ighlighting and clarifying various\ntechnical issues that arise in the process.\nThe idea underlying the extension is reasonably simple: Ass ume that we have an estimate\n|Wψu(y,h)| ≤C/bardblh/bardblN, (13)\nvalid on a set Bǫ(x)×Ki/o(W,V,R), with a wavelet ψsatisfying supp( /hatwideψ)⊂V⋐O. Assume\nthatη∈ S(Rd) is a second admissible wavelet. Under suitable convergenc e assumptions, it is\npossible to plug the inversion formula\nη=1\nCψˆ\nGWψη(x,h)π(x,h)ψd(x,h)\ninto the definition of the wavelet transform Wηu, and obtain that the wavelet transforms Wηu\nandWψuare related by a convolution product:\nWηu=1\nCψWψu∗Wηψ .\nSubsection 4.2 will be devoted to formulating the explicit c riteria needed for the above identity\nto hold. Hence in order to transfer decay estimates from WψutoWηu, we need to understand\nwhich conditions on the cross-kernel Wηψallow to transfer the wavelet coefficient decay from\nWψutoWηu, possibly on a slightly smaller set K′. Note also that the same reasoning, with Wψη\nreplacingWηψ, would also allow to transfer decay estimates for WηutoWψu. This allows us to\nextend both necessary and sufficient wavelet coefficient decay conditions for N-regular directed\npoints.\nOur proof strategy also requires the use of methods to estima te and quantify the decay of\nthe cross-kernel Wηψ; here we will take advantage of previous work developed in [7 , 8, 10].\nHowever, one further aspect that needs to be addressed in con nection with vanishing moment\nconditions is the fact that these conditions can be fulfilled byreal-valued wavelets . Due to\nthe symmetry condition that real-valuedness of ψimposes on/hatwideψ, these wavelets are generally\nunable to distinguish between directions ξ,−ξ, i.e., our wavelet-based near characterizations will12 HARTMUT F ¨UHR, MAHYA GHANDEHARI\ndescribe unsignedN-regular directed points, in the sense of Definition 4.1 belo w. We emphasize\nthat this restriction is not due to a general shortcoming of o ur approach, and that it is also\npresent (rather more implicitly, but not less stringently) in the precursor papers [15, 11]. We\nrefer to Subsection 6.1 for a more detailed explanation. But now we start the discussion of\nunsigned regular directed point.\n4.1.Unsigned regular directed points and real-valued distributions. As already men-\ntioned, attempting to derive decay statements from vanishi ngmoment conditions opens the door\nto theuseof real-valued wavelets. For thesewavelets ψhowever, onehas /hatwideψ(−ξ) =/hatwideψ∗(ξ) :=/hatwideψ(ξ).\nHence by construction, such a wavelet will not be able to dist inguish the decay behaviour inside\na coneCcontaining ξand the cone −C, containing −ξ. As a consequence, we will restrict\nattention to real-valued wavelets in the following, and wil l pay the price that the behaviour in\ndirectionsξand−ξis indistinguishable. The following definition adapts the n otion ofN-regular\ndirected points to this setting. While we expect that these n otions have already been studied in\nsome detail elsewhere, we have not been able to locate a conve nient source.\nDefinition 4.1. For(x,ξ),(x′,ξ′)we write (x,ξ)∼(x′,ξ′)iffx=x′,ξ=±ξ′, and use Rd×\nSd−1/±to denote the set of equivalence classes.\nGivenN∈N,u∈ S′(Rd)and(x,±ξ)∈Rd×Sd−1/±, we call (x,±ξ)anunsigned N-\nregular directed point of uif both(x,ξ)and(x,−ξ)areN-regular directed points of u. We\ncall(x,±ξ)unsigned regular directed point if(x,±ξ)is an unsigned N-regular directed\npoint for all N, or equivalently, if (x,ξ)and(x,−ξ)are regular directed points of u.\nIt turns out that this notion is closely related to the N-regular directed points of the real and\nimaginary parts of u. We first recall the necessary terminology: Given u∈ S′(Rd), we define\nu∈ S′(Rd) as/a\\}bracketle{tu|f/a\\}bracketri}ht=/a\\}bracketle{tu|f/a\\}bracketri}ht; recall that we use L2-notation here. The realandimaginary parts\nofuare then defined in the usual manner as Re( u) =1\n2(u+u) and Im(u) =1\n2i(u−u).\nIt follows that u= Re(u)+iIm(u), and for real-valued functions f, one has\n/a\\}bracketle{tRe(u)|f/a\\}bracketri}ht= Re/a\\}bracketle{tu|f/a\\}bracketri}ht,/a\\}bracketle{tIm(u)|f/a\\}bracketri}ht= Im/a\\}bracketle{tu|f/a\\}bracketri}ht.\nWe callureal-valued ifu= Re(u). It is easy to see that a tempered distribution is real-valu ed\niff it maps real-valued Schwartz functions to real values.\nIn order to express the just-describedoperations on temper ed distributions in Fourier domain,\nwe need to extend the involution f∗(x) :=f(−x) from functions to tempered distributions, by\nletting\n/a\\}bracketle{tu∗|f/a\\}bracketri}ht=/a\\}bracketle{tu|f∗/a\\}bracketri}ht.\nPart (b) of the following lemma makes clear that for real-val ued distributions, the distinction\nbetween signed and unsigned N-regular directed points is moot, and part (c) relates unsig ned\nN-regular directed points of utoN-regular directed points of Re( u) and Im(u).\nLemma 4.2. Letu∈ S′(Rd).\n(a)/hatwideu= (/hatwideu)∗.\n(b) Ifuis real-valued, then its Fourier transform /hatwideufulfills/hatwideu= (/hatwideu)∗. As a consequence,\n(x,−ξ)is a regular directed point of uiff(x,ξ)is.\n(c)(x,±ξ)is an unsigned N-regular directed point of uiff(x,ξ)is anN-regular directed\npoint of Re(u)andIm(u), or equivalently, iff (x,±ξ)is an unsigned N-regular directed\npoint ofRe(u)andIm(u).SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 13\nProof.Part (a) is straightforward. Part (b) follows from the fact t hat the cut-out function ϕ\nin the definition of N-regular directed points can be chosen real-valued, and the n (a) applies to\nthe product ϕu, and yields the desired statement.\nFor part (c), assume that ( x,±ξ) is an unsigned N-regular directed point of u, i.e., we have\ndecay of/hatwiderϕuof orderNinsideC∪−C, for a suitable cone Cin direction ξ. Again we choose the\ncut-out function ϕto be real-valued, and then ϕu=ϕu. So Re(ϕu) =ϕ(Re(u)), and the same\nholds for the imaginary part as well. Using this fact, togeth er with part (a), we get\n(ϕRe(u))∧(ξ′) = (Re(ϕu))∧(ξ′) =1\n2/parenleftig\n/hatwiderϕu+/hatwiderϕu/parenrightig\n(ξ′) =1\n2/parenleftig\n/hatwiderϕu(ξ′)+/hatwiderϕu(−ξ′)/parenrightig\n.\nSo, the same decay behaviour for ( ϕRe(u))∧, and similarly for ( ϕIm(u))∧, holds inC∪ −C.\nThis establishes that ( x,ξ) and (x,−ξ) areN-regular directed points of Re( u) and Im(u); or\nequivalently, that ( x,±ξ) is an unsigned N-regular directed point of Re( u) and Im(u).\nThe converse direction follows directly from part (b) and u= Re(u)+iIm(u). /square\nThe next somewhat technical notion formulates a fairly subt le structural assumption on the\ndilation group Hthat initially seems a bit opaque. We will repeatedly commen t on the role of\nthis assumption to make its impact on our arguments clear.\nDefinition 4.3. We callH0<Hahalf-space adapted subgroup if it is an open subgroup\nwith the property that, for any ξ0∈ O, one has O= (HT\n0ξ0)∪ −(HT\n0ξ0), where the union is\ndisjoint.\nAs a pertinent class of examples with this property we mentio n the generalized shearlet\ndilation groups featured in Section 6. On the other hand, it i s clear that not all admissible\ndilation groups have a half-space adapted subgroup. Note th at by definition H0is necessarily\na proper open subgroup of index 2. The standard higher-dimen sional wavelet dilation groups\nH=R+SO(Rd) are connected, and therefore do not possess a half-space ad apted subgroup.\nThe chief purpose of introducing the notion of half-space ad apted subgroups is that in certain\ncontexts, the integration domain Hmay be swapped for the smaller subset H0, and this will\nbe instrumental in guaranteeing the necessary decay condit ion. The following lemma spells this\nphenomenon out for the inversion formula associated to real -valued wavelets.\nLemma 4.4. LetH0<Hdenote a half-space adapted subgroup, and let ψdenote a real-valued\nadmissible wavelet for H. Then, for all f∈L2(Rd), one has\nf=2\nCψˆ\nG0Wψf(x,h)π(x,h)ψd(x,h),\nwhereG0=Rd⋊H0.\nProof.WritingO+=HT\n0ξ0, one has O=O+∪−O+. The admissibility assumption on ψentails\nthat\nCψ=ˆ\nH|/hatwideψ(hTξ0)|2dh<∞,\nand accordingly, that /bardblWψf/bardbl2\nL2(G)=Cψ/bardblf/bardbl2, as well as\nf=1\nCψˆ\nGWψf(x,h)π(x,h)ψd(x,h).\nBy assumptions on ψandH0, we haveˆ\nH0|/hatwideψ(hTξ0)|2dh=ˆ\nH0|/hatwideψ(−hTξ0)|2dh=Cψ\n2.14 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nSinceO+∪(−O+) =Ois conull, the admissibility condition for non-irreducibl e quasi-regular\nrepresentations (see e.g. Theorem 1 of [9]) allows to conclu de for the restriction of WψftoG0\nthat/bardblWψf/bardbl2\nL2(G0)=Cψ\n2/bardblf/bardbl2, and accordingly\nf=2\nCψˆ\nG0Wψf(x,h)π(x,h)ψd(x,h).\n/square\n4.2.Vanishing moments and wavelet coefficient decay. Our methods and techniques\nbuildon previous work on wavelet coefficient decay estimates fromvanishingmoment conditions.\nIn this subsection, we shall roughly sketch the pertinent re sults from [7, 8, 10].\nThe first question that arises concerns the precise nature of the vanishing moment criteria.\nIn higher dimensions, these depend on the dilation group, or more precisely, on the open dual\norbitO:\nDefinition 4.5. Letr∈Nbe given. We say ψ∈ S(Rd)has vanishing moments in Ocof\norderrif all derivatives ∂α/hatwideψof degree |α|<rare vanishing on Oc.\nClearly, anybandlimitedwavelet, i.e., anySchwartzfunct ionψwith/hatwideψ∈C∞\nc(O) hasvanishing\nmoments of all orders. Note that the vanishing moment condit ions are equivalent to\n∀|α|<r,∀ξ∈ Oc:ˆ\nRdxαψ(x)e−2πi/a\\}bracketle{tξ,x/a\\}bracketri}htdx= 0.\nWe next define an auxiliary function A:O →R+that is used in the decay estimates. Given\nany pointξ∈ O, let dist(ξ,Oc) denote the distance of ξto the set Oc, and define\nA(ξ) := min/parenleftigg\ndist(ξ,Oc)\n1+/radicalbig\n|ξ|2−dist(ξ,Oc)2,1\n1+|ξ|/parenrightigg\n. (14)\nBy definition, Ais a continuous function with A(·)≤1. Ifη∈ Ocdenotes an element of minimal\ndistance to ξ, the fact that R+·η⊂ Octhen entails that ηandξ−ηare orthogonal with respect\nto the standard scalar product on Rd, and we obtain the more transparent expression\nA(ξ) = min\nη∈Oc/parenleftbigg|ξ−η|\n1+|η|,1\n1+|ξ|/parenrightbigg\n. (15)\nWe define a further family of auxiliary functions Φ ℓ:H→R+∪{∞}, forℓ∈N, via\nΦℓ(h) =ˆ\nRdA(ξ)ℓA(hTξ)ℓdξ . (16)\nThe motivation for the definitions of Aand Φℓis best explained as follows: Two functions\nη,ψ∈ S(R) are admissible wavelets if and only if /hatwideη(0) =/hatwideψ(0) = 0. The usual arguments relating\nsmoothness of a function and the decay of its Fourier transfo rm (and vice versa), together with\nthe convolution theorem, allow us to estimate\n|Wψη(x,h)|=|η∗(π(0,h)ψ∗)(x)| ≤C(1+|x|)−N|h|1/2N/summationdisplay\nj=0/vextenddouble/vextenddouble/vextenddouble∂j/parenleftig\n/hatwideη/hatwideψ(h·)/parenrightig/vextenddouble/vextenddouble/vextenddouble\n1.(17)\nHere the sum\nN/summationdisplay\nj=0/vextenddouble/vextenddouble/vextenddouble∂j/parenleftig\n/hatwideη/hatwideψ(h·)/parenrightig/vextenddouble/vextenddouble/vextenddouble\n1SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 15\ncan be understood as a term measuring the overlap between /hatwideηand the dilate of /hatwideψ, and it goes\nto zero ash→0,h→ ∞, with a speed depending on the number of vanishing moments of η\nandψ. The analog of this overlap term in the higher-dimensional s etting is essentially provided\nby the auxiliary function Φ ℓ.\nThe following result guarantees that the set of compactly su pported Schwartz functions with\nvanishing moments of a given order is nonempty. It follows fr om Lemma 4.1 from [7].\nLemma 4.6. For anyr >0there exists a nonzero compactly supported ψ∈ S(Rd)having\nvanishing moments in Ocof orderr.\nWe next quote the following general estimate [8, Lemma 3.7], which can be viewed as a\ngeneralization of (17):\nLemma 4.7. Let0<m<r , and letη,ψ∈ S(Rd)denote functions with vanishing moments of\norderrinOc. Then there exists C >0independent of x,h,η,ψ such that\n|Wψη(x,h)| ≤C|/hatwideη|r|/hatwideψ|r(1+|x|)−m|det(h)|1/2(1+/bardblh/bardbl)mΦr−m(h).\nNext, we use the above lemma to obtain estimates on Wψη. Recall that Ois an open orbit in\n/hatwiderRd, with the property that the stabilizer groups associated wi th theH-action on Oare compact.\nThis property allows us to transfer the Lebesgue measure on/hatwiderRdto the Haar measure on H, i.e.\nfor fixedξ0∈ Oand everyF∈Cc(O) we have\nˆ\nOF(ξ)dξ=ˆ\nHF(hTξ0)|det(h)|\n∆H(h)dh.\nFix an element ξ0∈ O, and define AH:H→[0,1] to beAH(h) =A(hTξ0). Our computations\nin the following proposition are inspired by techniques dev eloped in [10].\nProposition 4.8. Assume that there exists a positive number ℓ1∈Rand a constant Cso that\nfor everyh∈H,\n/bardblh±1/bardblAH(h)ℓ1≤C. (18)\nLetβ1,β2,β3be positive numbers with β1∈N, and consider functions ψ1,ψ2∈ S(Rd)with\nvanishing moments of order rinOc. Ifr>d/2+ℓ1(β1+β2+β3)+β1then for every (x,h)∈G,\n|Wψ1ψ2(x,h)| ≤D(1+|x|)−β1(1+/bardblh/bardbl)−β2(1+/bardblh−1/bardbl)−β3,\nwhere the constant Ddepends on ψ1,ψ2,β1,β2,β3, andr,ℓ1, and does not depend on x,h.\nProof.Without loss of generality, we can assume that C≥1. By Lemma 4.7, we have\n|Wψ1ψ2(x,h)| ≤C1(1+|x|)−β1|det(h)|1/2(1+/bardblh/bardbl)β1Φr−β1(h).\nHence it remains to show that the map h/mapsto→ |det(h)|1/2(1+/bardblh/bardbl)β1+β2(1+/bardblh−1/bardbl)β3Φr−β1(h) is\nbounded. To do so, define m(h) =|det(h)|1/2(1+/bardblh/bardbl)β1+β2(1+/bardblh−1/bardbl)β3, and observe that mis\nsubmultiplicative, as β1,β2,β3>0. Our goal is then to prove that mΦr−β1belongs toL∞(H).16 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nForℓ∈N, using the submultiplicativity of m, we have\nm(h)Φℓ(h) =m(h)ˆ\nRdA(ξ)ℓA(hTξ)ℓdξ\n=m(h)ˆ\nHA(kTξ0)ℓA(hTkTξ0)ℓ|det(k)|\n∆H(k)dk\n=m(h)ˆ\nHAH(k)ℓAH(kh)ℓ|det(k)|\n∆H(k)dk\n=m(h)ˆ\nH|det(k)|−1AH(k−1)ℓAH(k−1h)ℓdk,\n≤ˆ\nH/parenleftig\n|det(k)|−1m(k)AH(k−1)ℓ/parenrightig/parenleftig\nm(k−1h)AH(k−1h)ℓ/parenrightig\ndk\n= (B1∗B2)(h),\nwhereweusetheauxiliaryfunctions B1,B2:H→[0,∞)definedas B1(k) =|det(k)|−1m(k)AH(k−1)ℓ\nandB2(k) =m(k)AH(k)ℓ. Now, it is only left to show that if ℓ > ℓ1(β1+β2+β3) +d\n2then\nB1,/tildewiderB2∈L2(H), where we use the notation /tildewiderB2(k) =B2(k−1) whenever k∈H. Indeed, if that\nis the case, we will have /bardblmΦℓ/bardbl∞≤ /bardblB1/bardbl2/bardbl/tildewiderB2/bardbl2<∞.\nNote that (1+ /bardblh±1/bardbl)AH(h)ℓ1=AH(h)ℓ1+/bardblh±1/bardblAH(h)ℓ1≤2C. Ifℓ>d/2+ℓ1(β1+β2+β3),\nwe have\n/bardblB1/bardbl2\n2=ˆ\nH|det(k)|−2m(k)2AH(k−1)2ℓdk\n=ˆ\nH|det(k)|−2|det(k)|(1+/bardblk/bardbl)2β1+2β2(1+/bardblk−1/bardbl)2β3AH(k−1)2ℓdk\n≤(2C)2(β1+β2+β3)ˆ\nH|det(k)|−1AH(k−1)2ℓ−2ℓ1(β1+β2+β3)dk\n= (2C)2(β1+β2+β3)ˆ\nHAH(k)2ℓ−2ℓ1(β1+β2+β3)|det(k)|\n∆H(k)dk\n= (2C)2(β1+β2+β3)ˆ\nOA(ξ)2ℓ−2ℓ1(β1+β2+β3)dξ\n≤(2C)2(β1+β2+β3)ˆ\nO/parenleftbigg1\n1+|ξ|/parenrightbigg2ℓ−2ℓ1(β1+β2+β3)\ndξ,\nwhich is a bounded integral provided that 2 ℓ−2ℓ1(β1+β2+β3)>d.SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 17\nAlso,\n/bardbl/tildewiderB2/bardbl2\n2=ˆ\nHm(k−1)2AH(k−1)2ℓdk\n=ˆ\nH|det(k)|(1+/bardblk/bardbl)2(β1+β2)(1+/bardblk−1/bardbl)2β3AH(k)2ℓ1\n∆H(k)dk\n≤(2C)2(β1+β2+β3)ˆ\nHAH(k)2ℓ−2ℓ1(β1+β2+β3)|det(k)|\n∆H(k)dk\n= (2C)2(β1+β2+β3)ˆ\nOA(ξ)2ℓ−2ℓ1(β1+β2+β3)dξ\n≤(2C)2(β1+β2+β3)ˆ\nO/parenleftbigg1\n1+|ξ|/parenrightbigg2ℓ−2ℓ1(β1+β2+β3)\ndξ,\nwhich is a bounded integral provided that 2 ℓ−2ℓ1(β1+β2+β3)> d. Finally, letting D=\nC1/bardblmΦr−β1/bardbl∞finishes the proof. /square\nThe estimates in the previous proposition motivate the intr oduction of a further family of\nauxiliary functions.\nLemma 4.9. Fors,t∈R+andp∈R, letΘs,t:H→(0,∞)be defined as\nΘs,t(h) = (1+ /bardblh/bardbl)−s(1+/bardblh−1/bardbl)−t.\nIfs≥dimH+d+1andt≥dimH+d+dpthen|det(·)|−pΘs,t∈L1(H).\nProof.We first recall the inequalities |det(h)| ≤(1 +/bardblh/bardbl)dand ∆H(h)≤((1 +/bardblh/bardbl)(1 +\n/bardblh−1/bardbl))dimH. The first one is known as Hadamard’s inequality; for the seco nd one we refer\nto the proof of Lemma 2.8 of [10].\nThese inequalities allow to obtain the following useful est imate:\n|det(h)|−pΘs,t(h)∆H(h)\n|det(h)|=|det(h−1)|p+1Θs,t(h)∆H(h)\n≤(1+/bardblh−1/bardbl)d(p+1)(1+/bardblh/bardbl)−s(1+/bardblh−1/bardbl)−t((1+/bardblh/bardbl)(1+/bardblh−1/bardbl))dimH\n= Θs−dimH,t−dimH−d−dp(h). (19)\nFix an arbitrary ξ0∈ O, and define/tildewideΘs,t:H→(0,∞) as/tildewideΘs,t(h) =´\nHξ0Θs,t(kh)dHξ0k, where\nHξ0is the stabilizer subgroup at ξ0with Haar measure d Hξ0k. Note that the integral defining\n/tildewideΘs,tis bounded, as Hξ0is compact. Next, we observe that /tildewideΘs,tcan be bounded from above and\nbelow by multiples of Θ s,t, i.e., there exist positive constants as,t,bs,tsuch that\nas,tΘs,t≤/tildewideΘs,t≤bs,tΘs,t.\nIndeed, the inequality /bardblgh/bardbl ≤ /bardblg/bardbl /bardblh/bardblentails Θ s,t(gh)≥Θs,t(g)Θs,t(h). Hence we get\n/tildewideΘs,t(h) =ˆ\nHξ0Θs,t(kh)dHξ0k≥Θs,t(h)ˆ\nHξ0Θs,t(k)dHξ0k=as,tΘs,t(h),\nwhereas,t:=´\nHξ0Θs,t(k)dHξ0k. On the other hand, the inequality /bardblh/bardbl ≤ /bardblkh/bardbl/bardblk−1/bardblimplies\nthat (1+ /bardblkh/bardbl)−1≤(1+/bardblh/bardbl)−1(1+/bardblk−1/bardbl), which in turn implies that\nˆ\nHξ0Θs,t(kh)dHξ0k≤Θs,t(h)ˆ\nHξ0(1+/bardblk−1/bardbl)s(1+/bardblk/bardbl)tdHξ0k=bs,tΘs,t(h),18 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nwherebs,t:=´\nHξ0(1+/bardblk−1/bardbl)s(1+/bardblk/bardbl)tdHξ0k. Note that both as,t,bs,tare finite, as integrals\nof continuous functions on a compact group, and that as,t>0.\nThe function /tildewideΘs,tcan be regarded as a well-defined function on O, which we denote by Fs,t,\nin the following manner:\nFs,t:O →(0,∞), Fs,t(hTξ0) =/tildewideΘs,t(h),\nsince/tildewideΘs,t(lh) =/tildewideΘs,t(h) for every h∈Handl∈Hξ0. We will use Fs,tto relate the 1-norm of\nΘs,tto an integral on the orbit O. First observe that for ξ=hTξ0, we clearly have|ξ|\n|ξ0|≤ /bardblh/bardbl,\nwhich implies that (1+ /bardblh/bardbl)−1≤(1+|ξ|\n|ξ0|)−1. Hence, as t,s≥0 we have\nFs,t(ξ) =Fs,t(hTξ0) =/tildewideΘs,t(h)≤bs,t(1+/bardblh/bardbl)−s(1+/bardblh−1/bardbl)−t≤bs,t/parenleftbigg\n1+|ξ|\n|ξ0|/parenrightbigg−s\n.\nLettings′=s−dimHandt′=t−d−dp−dimH, from the inequality (19) we get\nˆ\nH|det(h)|−pΘs,t(h)dh≤ˆ\nHΘs′,t′(h)|det(h)|\n∆H(h)dh\n≤1\nas′,t′ˆ\nOFs′,t′(ξ)dξ\n≤bs′,t′\nas′,t′ˆ\nO/parenleftbigg\n1+|ξ|\n|ξ0|/parenrightbigg−s′\ndξ,\nwhich is a bounded integral if s′≥d+1 andt′≥0. /square\nAs a first application of these estimates, we can formulate ex plicit criteria for the formula\nWηu=Wψu∗Wηψto hold.\nProposition 4.10. Assume that there exist positive numbers ℓ1,C∈R+so that for every\nh∈H,\n/bardblh±1/bardblAH(h)ℓ1≤C . (20)\nLetu∈ S′(Rd)denote a tempered distribution of order N(u). Letη,ψ∈ S(Rd)denote functions\nwith vanishing moments of order rinOc, where\nr>2d+N(u)+ℓ1(5+2dim(H)+6d+3N(u))+2.\nThen\nWηu=1\nCψWψu∗Wηψ\nholds with absolute convergence of the convolution integra l. IfH0< His a half-space adapted\nsubgroup and ψis real-valued, convolution over Gcan be replaced by convolution over G0=\nRd⋊H0, with twice the normalization factor.\nProof.We first recall the wavelet inversion formula\nη=1\nCψˆ\nGWψη(x,h)π(x,h)ψd(x,h), (21)\nwhich is valid in the weak L2-sense. The proof is based on an alternative interpretation of the\nright hand side as a Bochner integral converging in | · |M, for a suitable choice of M. More\nprecisely, we let\nM=/braceleftbigg\nN(u)+d\n2+1deven\nN(u)+d+1\n2dodd.SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 19\nTo see that the desired convergence holds, we introduce the q uantities\nβ1=M+d+1, β2= 2+dim(H)+3\n2d+N(u), β3= 1+dim(H)+3d+N(u),(22)\nand observe that\nr−d/2−ℓ1(β1+β2+β3)−β1>0.\nThis implies, for sufficiently small ǫ>0 that\nr−d/2−ℓ1(β1+ǫ+β2+β3)−β1−ǫ>0. (23)\nWe can then estimate the norm of the integrand in (21) as follo ws:\n|(Wψη)(x,h)π(x,h)ψ|M\n≤CM|ψ|M|Wψη(x,h)|(1+|x|)M|det(h)|−1/2(1+/bardblh/bardbl)M(1+/bardblh−1/bardbl)M\n≤C′(1+|x|)M−β1−ǫ|det(h)|−1/2(1+/bardblh/bardbl)M−β2(1+/bardblh−1/bardbl)M−β3\n=C′(1+|x|)M−β1−ǫ|det(h)|−1/2Θβ2−M,β3−M(h),\nwhere the constant C′does not depend on x,h. Here the first inequality is due to Lemma 3.1(a),\nand the second one follows from Proposition 4.8, which is app licable thanks to (23). Recall that\nd(x,h) =dxdh\n|det(h)|. Now Lemma 4.9 and our choice of β2,β3imply (via M≤N(u) + 1 +d\n2)\nthat|det(h)|−3/2Θβ2−M,β3−Mis integrable over H. Also, since β1+ǫ−M > d, the function\n(1+|x|)M−β1−ǫis integrable over Rd, and thus the right-hand side of (21) is a Bochner integral\nconverging in the completion XofS(Rd) with respect to |·|M. In fact, the inequality d<2M\nentails/bardblf/bardbl2≤ |f|M, which implies that Xembeds continuously into L2(Rd). Thus the limit of\nthe Bochner integral coincides with the weak L2-limit, i.e., with η. So the equation given in (21)\nholds inXas well. Since N(u)≤M,uextends to a continuous functional on X, which entails\nthat we can interchange evaluation of uand Bochner integration, giving for all ( x′,h′)\nWηu(x′,h′) =/a\\}bracketle{tπ(x′,h′)−1u,η/a\\}bracketri}ht\n=1\nCψˆ\nGWψη(x,h)/a\\}bracketle{tπ(x′,h′)−1u,π(x,h)ψ/a\\}bracketri}htd(x,h)\n=1\nCψˆ\nG/a\\}bracketle{tu,π((x′,h′)(x,h))ψ/a\\}bracketri}htWηψ((x,h)−1)d(x,h)\n=1\nCψˆ\nGWψu(x,h)Wηψ((x,h)−1(x′,h′))d(x,h)\n=1\nCψ(Wψu∗Wηψ)(x′,h′).\nHere absolute convergence of the convolution integral is gu aranteed by Bochner convergence.\nThe statement for real-valued wavelets ψthen follows from Lemma 4.4. /square\n5.Criteria for N-regular directed points: Compactly supported wavelets\nAfter the groundwork laid in the previous section, we can now establish the first major result\nof our paper, namely Theorem 5.4. It shows that the decay orde r of wavelet coefficients is stable\nunder exchange of wavelets, as long as both wavelets in the ex change have a certain minimal\nnumber of vanishing moments. As an auxiliary notion to guara ntee this transfer, one further\nsomewhat technical condition needs to be formulated. The in tuition behind this definition and\nits necessity is discussed in Remark 5.5.20 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nDefinition 5.1. LetV⋐O. We say that Hfulfills the V-overlap control condition with\nparameters γ0≥0,γ1,γ2>0in direction ξ∈ O∩Sd−1if there exist\n•a half-space adapted subgroup H0<H,\n•a neighborhood W′ofξinSd−1andR′>0;\nsuch that the following holds: For every pair (W′′,R′′)consisting of a ξ-neighborhood W′′⊆W′\nandR′′≥R′there exists a ξ-neighborhood WandR>0such that for all L≥0and all\ns≥γ0+γ1L , t≥γ0+γ2L\nthe estimate\n∀h∈Ko(W,V,R) :ˆ\nH0\\h−1Ki(W′′,V,R′′)Θs,t(g)dg≤C/bardblh/bardblL\nholds with a constant Cindependent of h∈Ko(W,V,R).\nClearly, ifHsatisfiesthe V-overlapcontrolconditionwith W′andR′asintheabovedefinition,\nthen it would satisfy the same condition if we replace W′by/tildewiderW′⊆W′, andR′by/tildewiderR′≥R′. We\nnext want to establish that the overlap condition only needs to beverified for one direction. This\nargument is somewhat akin to the proof of the analogous state ment for the cone approximation\nproperty [5, Lemma 4.5].\nLemma 5.2. IfHfulfills theV-overlap control condition with parameters γ0≥0,γ1,γ2>0at\nξ0∈ O, then it fulfills this condition at all directions ξ1∈ O ∩Sd−1with the same parameters\nγ0,γ1,γ2.\nProof.Givenξ1∈ O ∩Sd−1, we pickh∈HwithhTξ0=ξ1. Assume that W′\n0andR′\n0are\ngiven to guarantee the V-overlap control condition at ξ0. By appealing to Lemma 2.3(b) we find\nW′\n1,R′\n1so thatKo(W′\n1,V,R′\n1)⊆h−1Ko(W′\n0,V,R′\n0). Next, we show that W′\n1andR′\n1satisfy the\nrequirements of the V-overlap control condition with parameters γ1,γ2>0 in direction ξ1.\nLetW′′\n1⊂W′\n1andR′′\n1> R′\n1be given. Applying Lemma 2.3(a), we can then find a ξ0-\nneighborhood /tildewiderW0and/tildewiderR0>0 with\nKi(W′′\n1,V,R′′\n1)⊇h−1Ki(/tildewiderW0,V,/tildewiderR0).\nThenW′′\n0=W′\n0∩/tildewiderW0⊂W′\n0andR′′\n0= max(R′\n0,/tildewiderR0)≥R′\n0fulfill\nh−1Ki(W′\n0,V,R′\n0)⊇h−1Ki(W′′\n0,V,R′′\n0),\nas well. By V-overlap control condition at ξ0, givenW′′\n0,R′′\n0, there exist a ξ0-neighborhood W0\nandR0>0 such that for all L≥0 and alls≥γ0+γ1Landt≥γ0+γ2Lthe estimate\n∀l∈Ko(W0,V,R0) :ˆ\nH0\\l−1Ki(W′′\n0,V,R′′\n0)Θs,t(g)dg≤C/bardbll/bardblL(24)\nholds with a constant Cindependent of l∈Ko(W0,V,R0). Applying Lemma 2.3(b) again, there\nexists aξ1-neighborhood W1andR1>0 such that Ko(W1,V,R1)⊆h−1Ko(W0,V,R0).\nGivenL, picks≥γ0+γ1Landt≥γ0+γ2L. Leth1∈Ko(W1,V,R1) be arbitrary,\nand denote hh1=h0. Nowh−1Ki(W′′\n0,V,R′′\n0)⊆Ki(W′′\n1,V,R′′\n1) impliesh−1\n0Ki(W′′\n0,V,R′′\n0)⊆\nh−1\n1Ki(W′′\n1,V,R′′\n1), and thus\nˆ\nH0\\h−1\n1Ki(W′′\n1,V,R′′\n1)Θs,t(g)dg≤ˆ\nH0\\h−1\n0Ki(W′′\n0,V,R′′\n0)Θs,t(g)dg . (25)SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 21\nClearly,h0∈Ko(W0,V,R0), ash1∈Ko(W1,V,R1). So combining (24) and (25), we get\nˆ\nH0\\h−1\n1Ki(W′′\n1,V,R′′\n1)Θs,t(g)dg≤C/bardblh0/bardblL≤C/bardblh/bardblL\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=C′/bardblh1/bardblL,\nwhich shows the desired property. /square\nThe lemma shows that the V-overlap condition is a global property on O ∩Sd−1, once it is\nestablished for a single direction. Accordingly, we can dro p the reference to the direction ξin\nthe subsequent usage of that condition.\nRemark 5.3. The following theorem is central to our paper, as it weakens th e requirements on\nthe analyzing wavelet from compact support in frequency to fi nite numbers of vanishing moments.\nBut besides this rather direct benefit, the theorem is also of independent interest for generalized\nwavelet analysis, since it expresses that, as soon as the ana lyzing wavelet is chosen from a suitable\nclass, a certain type of wavelet coefficient decay is independ ent of the precise choice of wavelet.\nIn other words, this decay is recognized as an intrinsic prope rty of the analyzed function. This is\nakin to a fundamental observation from coorbit theory [2, 3,4]. Recall that coorbit spaces quantify\nintegrability properties of wavelet coefficients, and there by provide a global measure of wavelet\ncoefficient decay. An important foundational aspect of this t heory is consistency , i.e. the fact\nthat (up to equivalence) this global decay behaviour is inde pendent of the analyzing wavelet, see\n[2]. The more recent paper [10]provides guarantees for this independence in terms of vanis hing\nmoment conditions on wavelets, for the wavelet transforms o f the type studied in our paper.\nIn this context, Theorem 5.4 can be seen as an analog to the coorb it space results in [10],\nby guaranteeing a similar type of consistency of wavelet coe fficient decay localized inside cone-\naffiliated sets, which holds for all wavelets with sufficiently many vanishing moments.\nTheorem 5.4. Letx0∈Rdandξ0∈ O ∩Sd−1be fixed. Assume that the dual action is V-\nmicrolocally admissible in direction ξ0, with associated ξ0-neighborhood W0andR0>0, and\nwith exponent α1. LetN∈Nbe fixed.\nLetu∈ S′(Rd)be a tempered distribution of order N(u). Letψ1,ψ2∈ S(Rd)be wavelets of\nvanishing moments of order rinOc, withψ1being real-valued. Assume that the parameters\nβ1,β2,β3andrfulfill the following inequalities:\n(i)β1≥N+N(u)+α1(N(u)+d/2)+d+1,\n(ii)β2≥dimH+N+d+1, andβ3≥dimH+2d(which implies Θβ2−N,β3−d∈L1(H)),\n(iii)β1≥2+3\n2d+N(u),β2≥2+dim(H)+3\n2d+N(u)andβ3≥1+dim(H)+3d+N(u),\n(which implies |det(·)|−pΘβ2−N(u),β3−N(u)∈L1(H)forp=1\n2,3\n2),\n(iv)Hfulfills theV-overlap control condition with parameters γ0≥0,γ1,γ2>0, and\nβ2≥γ0+γ1α1/parenleftbigg3\n2d+N(u)/parenrightbigg\n+N(u)+γ1N\nβ3≥γ0+γ2α1/parenleftbigg3\n2d+N(u)/parenrightbigg\n+N(u)+3\n2d+γ2N\n(v)r>d/2+ℓ1(β1+β2+β3)+β1, whereℓ1>0satisfies (18) of Proposition 4.8.\nLetǫ>0be given, and assume that there exist suitable choices of ξ0-neighborhood W′,R′with\n|Wψ1u(y,h)| ≤C/bardblh/bardblNfor everyy∈Bǫ(x0)and everyh∈Ki(W′,V,R′), where the constant C\nis independent of h,y. Then there exist ξ0-neighborhood W,Rsuch that |Wψ2u(y,h)| ≤C′/bardblh/bardblN\nfor everyy∈Bǫ/2(x0)and everyh∈Ko(W,V,R), where the constant C′is independent of the\nchoice ofh,y.22 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nProof.Throughout this proof note that assumption ( v) guarantees that Proposition 4.8 is avail-\nable to estimate Wψ1ψ2.\nSuppose that W′,R′are as provided by the premise of the theorem, that is,\n|Wψ1u(y,g)| ≤C/bardblg/bardblNfor everyy∈Bǫ(x0) and every g∈Ki(W′,V,R′),(26)\nwhere the constant Cis independent of yandg. We first note that the premise remains true if\nwe replace W′by sufficiently small W′′⊂W′andR′by sufficiently large R′′>R′to make the\noverlap control condition applicable, and we choose W,Ras provided by that condition so that\nfors,tas in Definition 5.1, we have\n∃C1>0∀h∈Ko(W,V,R) :ˆ\nH0\\h−1Ki(W′′,V,R′′)Θs,t(g)dg≤C1/bardblh/bardblL. (27)\nSince replacing WandRbyW∩W0and max {R,R0}does not disturb the conclusion of the\noverlap control condition, we can also assume that W⊆W0andR≥R0.\nWe start out by verifying that, undertheassumptions of thet heorem, the convolution relation\nWψ2u=2\nCψ1(Wψ1u∗Wψ2ψ1)\nholds with convolution taken over Rd⋊H0. This is done most easily by comparing the assump-\ntions (iii) and (v) with those of Proposition 4.10, and with ( 22) and (23).\nThroughoutthefollowing computations, let h∈Ko(W,V,R), anddenote K=Ki(W′′,V,R′′).\nWe now split the convolution integral into Wψ2u(y,h) =2\nCψ1(I1+I2), where\nI1=ˆ\nh−1Bǫ/2(0)ˆ\nH0Wψ1u(y+hx,hg)Wψ2ψ1(−g−1x,g−1)dxdg\n|det(g)|,\nI2=ˆ\nRd\\h−1Bǫ/2(0)ˆ\nH0Wψ1u(y+hx,hg)Wψ2ψ1(−g−1x,g−1)dxdg\n|det(g)|.\nLety∈Bǫ/2(x0) andh∈Ko(W,V,R) be given. We prove appropriate upper bounds for I1and\nI2separately.\n|I1|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nh−1Bǫ/2(0)ˆ\nH0Wψ1u(y+hx,hg)Wψ2ψ1(−g−1x,g−1)dxdg\n|det(g)|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ˆ\nh−1Bǫ/2(0)ˆ\nH0|Wψ1u(y+hx,hg)Wψ1ψ2(x,g)|dxdg\n|det(g)|\n=I′\n1+I′′\n1,\nwhere we split the integral into I′\n1andI′′\n1by taking the inner integral over h−1KandH0\\h−1K\nrespectively. Thepoint of this separation of integrals is t hat for all (x,g)∈h−1(Bǫ/2(0))×h−1K,\none has (y+hx,hg)∈Bǫ(x0)×K, and thus the decay assumptions on Wψ1ucan be used toSOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 23\ncontinue\nI′\n1=ˆ\nh−1Bǫ/2(0)ˆ\nh−1K|Wψ1u(y+hx,hg)Wψ1ψ2(x,g)|dxdg\n|det(g)|\n≤C/bardblh/bardblNˆ\nh−1Bǫ/2(0)ˆ\nh−1K/bardblg/bardblN|Wψ1ψ2(x,g)|dxdg\n|det(g)|\n≤CD/bardblh/bardblNˆ\nh−1Bǫ/2(0)ˆ\nh−1K(1+|x|)−β1(1+/bardblg/bardbl)N−β2(1+/bardblg−1/bardbl)−β3dxdg\n|det(g)|\n≤CD/bardblh/bardblN/parenleftiggˆ\nh−1Bǫ/2(0)(1+|x|)−β1dx/parenrightigg/parenleftbiggˆ\nh−1K(1+/bardblg/bardbl)N−β2/bardblg−1/bardbld(1+/bardblg−1/bardbl)−β3dg/parenrightbigg\n≤CD/bardblh/bardblN/parenleftbiggˆ\nRd(1+|x|)−β1dx/parenrightbigg/parenleftbiggˆ\nH(1+/bardblg/bardbl)−(β2−N)(1+/bardblg−1/bardbl)−(β3−d)dg/parenrightbigg\n≤D1/bardblh/bardblN,\nwhere the two integrals in the penultimate line are bounded, sinceβ1> dand Θβ2−N,β3−d\nbelongs toL1(H) thanks to condition (ii). Also note that D1is a product of the constant from\nProposition 4.8, the constant Cfrom the assumption, and terms depending on β1,β2, andβ3.\nTo boundI′′\n1, we use (b) of Lemma 3.1 and Proposition 4.8. To avoid clutter of notation, we\nwill denote N(u) byN′.\nI′′\n1=ˆ\nh−1Bǫ/2(0)ˆ\nH0\\h−1K|Wψ1u(y+hx,hg)Wψ1ψ2(x,g)|dxdg\n|det(g)|\n≤ˆ\nh−1Bǫ/2(0)ˆ\nH0\\h−1KC2|det(hg)|−1/2(1+/bardblg−1h−1/bardbl)N′max{1,/bardblhg/bardblN′}(1+|y+hx|)N′\n·(1+|x|)−β1(1+/bardblg/bardbl)−β2(1+/bardblg−1/bardbl)−β3dxdg\n|det(g)|\n=C2J1J′\n1,\nwhereC2is the product of the constant terms from part (b) of Lemma 3.1 and Proposition 4.8,\nand\nJ1=ˆ\nh−1Bǫ/2(0)(1+|x|)−β1(1+|y+hx|)N′dx,\nJ′\n1=ˆ\nH0\\h−1K|det(hg)|−1/2(1+/bardblg−1h−1/bardbl)N′(1+/bardblhg/bardbl)N′\n(1+/bardblg/bardbl)−β2(1+/bardblg−1/bardbl)−β3dg\n|det(g)|.\nIt is easy to obtain a bound for J1. Indeed, we note that |y+hx| ≤ |y|+|hx| ≤ |x0|+ǫ≤2|x0|,\nasx∈h−1Bǫ/2(0),y∈Bǫ/2(x0), and without loss of generality we can assume ǫ≤ |x0|. So,\nJ1≤(1+2|x0|)N′|det(h)|−1ˆ\nBǫ/2(0)(1+|h−1x|)−β1dx≤(1+2|x0|)N′|det(h)|−1λRd(Bǫ/2(0)),\nwhereλRd(Bǫ/2(0)) is the volume of theǫ\n2-ball.24 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nWe next estimate the various factors in the integrand of J′\n1as follows:\n|det(hg)|−1/2\n|det(g)|≤ |det(h)|−1/2(1+/bardblg−1/bardbl)3\n2d,\n(1+/bardblg−1h−1/bardbl)N′≤(1+/bardblh−1/bardbl)N′(1+/bardblg−1/bardbl)N′,\n(1+/bardblhg/bardbl)N′≤(1+/bardblh/bardbl)N′(1+/bardblg/bardbl)N′.\nHere we used Hadamard’s inequality to estimate the determin ant against the norm, and sub-\nmultiplicativity of the norm. These estimates lead to\nJ′\n1≤ |det(h)|−1/2(1+/bardblh/bardbl)N′(1+/bardblh−1/bardbl)N′ˆ\nH0\\h−1KΘβ2−N′,β3−N′−3\n2d(g)dg\n≤C1|det(h)|−1/2(1+/bardblh/bardbl)N′(1+/bardblh−1/bardbl)N′/bardblh/bardblL.\nHere the second inequality used the overlap control conditi on, applied with\ns=β2−N′, t=β3−N′−3\n2d , L=α1/parenleftbigg3\n2d+N′/parenrightbigg\n+N .\nThe applicability of this condition is guaranteed by assump tion (iv), as well as h∈Ko(W,V,R),\nK=Ki(W′′,V,R′′) and the choice of W,Rdepending on W′′,R′′according to the mentioned\ncondition (see (27)).\nUsing Hadamard’s inequality, followed by property (a) of mi crolocal admissibility (Defini-\ntion 2.4) for h∈Ko(W,V,R), we have\n|det(h−1)|3/2≤ /bardblh−1/bardbl3\n2d/lessorsimilar/bardblh/bardbl−3\n2α1d,\nwhich allows to estimate I′′\n1:\nI′′\n1≤C3|det(h)|−3/2(1+/bardblh/bardbl)N′(1+/bardblh−1/bardbl)N′/bardblh/bardblN+α1(3\n2d+N′)\n≤C4(1+/bardblh/bardbl)N′(1+/bardblh−1/bardbl)N′/bardblh/bardblN+α1N′.\nNow, if/bardblh−1/bardbl ≤1, then 1+ /bardblh−1/bardbl ≤2; otherwise, 1+ /bardblh−1/bardbl ≤2/bardblh−1/bardbl/lessorsimilar/bardblh/bardbl−α1by microlocal\nadmissibility. In any case, we obtain\n(1+/bardblh−1/bardbl)N′/bardblh/bardblN+α1N′≤C5·max{/bardblh/bardblN,/bardblh/bardblN+α1N′}.\nFurthermore, Lemma 2.6 supplies that /bardblh/bardblis bounded on Ko(W,V,R), whence we finally arrive\nat\nI′′\n1≤C6/bardblh/bardblN.\nThis also concludes the estimate of I1.\nWe now estimate I2, as follows:\n|I2| ≤ˆ\nRd\\h−1Bǫ/2(0)ˆ\nH0|Wψ1u(y+hx,hg)|/vextendsingle/vextendsingleWψ2ψ1(−g−1x,g−1)/vextendsingle/vextendsingledxdg\n|det(g)|\n=ˆ\nRd\\h−1Bǫ/2(0)ˆ\nH0|Wψ1u(y+hx,hg)||Wψ1ψ2(x,g)|dxdg\n|det(g)|\n≤ˆ\nRd\\h−1Bǫ/2(0)ˆ\nH0C2|det(hg)|−1/2(1+/bardblg−1h−1/bardbl)N′max{1,/bardblhg/bardblN′}(1+|y+hx|)N′\n·(1+|x|)−β1(1+/bardblg/bardbl)−β2(1+/bardblg−1/bardbl)−β3dxdg\n|det(g)|\n=C2I′\n2I′′\n2,SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 25\nwhereC2is the product of the constant terms from part (b) of Lemma 3.1 and Proposition 4.8,\nand\nI′\n2=ˆ\nRd\\h−1Bǫ/2(0)(1+|x|)−β1(1+|y+hx|)N′dx,\nI′′\n2=ˆ\nH0|det(hg)|−1/2(1+/bardblg−1h−1/bardbl)N′max{1,/bardblhg/bardblN′}(1+/bardblg/bardbl)−β2(1+/bardblg−1/bardbl)−β3dg\n|det(g)|.\nNote that for every z∈Rd, we have\n(1+|z|)N′= (1+|h(h−1z)|)N′≤(1+/bardblh/bardbl|h−1z|)N′≤max{1,/bardblh/bardbl}N′(1+|h−1z|)N′,\nand sinceǫ≤ |x0|, for everyx∈Rd\n(1+|y+x|)N′≤(3\n2|x0|+1)N′(1+|x|)N′fory∈Bǫ/2(x0).\nSo, lettingωd−1denote the surface area of the ( d−1)-sphere of radius 1, we get\nI′\n2=ˆ\nRd\\h−1Bǫ/2(0)(1+|x|)−β1(1+|y+hx|)N′dx\n=|det(h)|−1ˆ\nRd\\Bǫ/2(0)(1+|h−1x|)−β1(1+|y+x|)N′dx\n≤(3\n2|x0|+1)N′|det(h)|−1ˆ\nRd\\Bǫ/2(0)(1+|h−1x|)−β1(1+|x|)N′dx\n≤(3\n2|x0|+1)N′|det(h)|−1max{1,/bardblh/bardbl}N′ˆ\nRd\\Bǫ/2(0)(1+|h−1x|)−β1+N′dx\n≤(3\n2|x0|+1)N′max{1,/bardblh/bardbl}N′ˆ\nRd\\h−1Bǫ/2(0)(1+|x|)−β1+N′dx\n≤(3\n2|x0|+1)N′max{1,/bardblh/bardbl}N′ˆ\nRd\\Bǫ\n2/bardblh/bardbl(0)1\n(1+|x|)β1−N′dx\n=ωd−1(3\n2|x0|+1)N′max{1,/bardblh/bardbl}N′ˆ∞\nǫ\n2/bardblh/bardblrd−1\n(1+r)β1−N′dr\n≤ωd−1(3\n2|x0|+1)N′max{1,/bardblh/bardbl}N′\nβ1−N′−d(ǫ\n2)d+N′−β1/bardblh/bardblβ1−N′−d\n≤C6max{/bardblh/bardblβ1−N′−d,/bardblh/bardblβ1−d},\nwhere we used the fact that Bǫ\n2/bardblh/bardbl(0)⊆h−1Bǫ/2(0), andβ1−N′> d. Here,C6is a constant\nthat only depends on x0,β1,N(u), andd.26 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nFinally, using the submultiplicativity of h/mapsto→(1+/bardblh/bardbl)N′, we have\nI′′\n2=ˆ\nH0|det(hg)|−1/2(1+/bardblg−1h−1/bardbl)N′max{1,/bardblhg/bardblN′}(1+/bardblg/bardbl)−β2(1+/bardblg−1/bardbl)−β3dg\n|det(g)|\n≤(1+/bardblh/bardbl)N′(1+/bardblh−1/bardbl)N′\n|det(h)|1/2ˆ\nH0|det(g)|−1/2(1+/bardblg−1/bardbl)N′−β3(1+/bardblg/bardbl)N′−β2dg\n|det(g)|\n≤C7/bardblh/bardbl−α1(N′+d/2)/bardbl|det(·)|−3/2Θβ2−N′,β3−N′/bardblL1(H),\nwhere in the last inequality, we used parts (b) and (c) of Lemm a 2.6. Putting all these together,\nwe get\n|I2| ≤D2max{/bardblh/bardblβ1−N′−d−α1(N′+d/2),/bardblh/bardblβ1−d−α1(N′+d/2)}\nwhereD2=C2C6C7/bardbl|det(·)|−3/2Θβ2−N′,β3−N′/bardblL1(H), which is finite thanks to condition (iii).\nUsing condition (i) on β1and the fact that /bardblh/bardblis bounded on Ko(W,V,R), we obtain |I2| ≤\nD3/bardblh/bardblN, whereD3is clearly a uniform bound independent of the choice of h∈Ko(W,V,R) and\nx∈Bǫ/2(0). /square\nRemark 5.5. With hindsight on the proof of Theorem 5.4, we can provide some more intuition\nregarding the overlap control condition. The initial idea of the proof is to split the integration\ndomain in the convolution formula\nWψ2u=1\nCψ1(Wψ1u∗Wψ2ψ1),\ninto the part where the decay assumptions on Wψ1uapply, and its complement. On the latter,\nwe can only combine moderate growth of Wψ1uwith rapid decay of Wψ2ψ1to achieve the desired\nestimates. This rationale motivated the introduction of the auxiliary functions Θs,tand the\nvarious related results. The overlap control condition is in tended to provide the right type of\nestimate that makes the general approach work.\nHowever, in the presence of a half-space adapted subgroup H0(e.g., in the shearlet case), rapid\ndecay ofWψ2ψ1will not be enough, as long as we integrate over the whole grou p. To see this,\nrecall that O= (HT\n0ξ0)∪−(HT\n0ξ0), and for simplicity consider a direction ξ∈HT\n0ξ0in the proof\nof Theorem 5.14. Assuming further that V,W⊂HT\n0ξ0, it follows that Ki/o(W,V,R)⊂H0. As\na consequence, it turns out that the part of the integration d omain where we solely rely on the\ndecay estimates for Wψ2ψ1to estimate Wψ2u(x,h)contains the full coset H\\H0, independent\nof the choice of h∈Ko(W,V,R). As a consequence, the contribution of the integral over the\ncoset is a constant.\nNow it is important to remember that the right-hand side of th e desired inequality is a positive\npower ofh, which becomes arbitrarily small. Hence any nontrivial con tribution from the coset\nH\\H0is essentially fatal to our efforts. In particular, an analog o f Definition 5.1 with the\nwhole group HreplacingH0would not make sense. This is why we replace integration over H\nby integration over H0, as in our formulation of the overlap control condition.\nThe only apparent alternative to this choice appears to make a ssumptions guaranteeing that\neither the wavelets or the signal have Fourier transforms wh ich vanish on a half-space. However,\nthese assumptions are incompatible with compact support of the involved functions.\nNote that the conclusion of the theorem allows to derive a dec ay statement for outercone-\naffiliated sets of the type Ko(W,V,R) from decay assumptions for innercone-affiliated sets ofSOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 27\nthe typeKi(W′,V,R′). Thus, as stated in the following corollary, Theorem 5.4 al lows to close\none of the two gaps mentioned in Remark 3.3.\nCorollary 5.6. Letu∈ S′(Rd)and(x,ξ)∈Rd×(O ∩Sd−1). Assume that the dual action is\nV-microlocally admissible in direction ξwith parameters α1,α2. Also assume that Hfulfills the\nV-overlap control condition, and there exists ℓ1>0satisfying (18) of Proposition 4.8. Then,\nwe have\n(a) If(x,ξ)is anN-regular directed point of u, then there exists a neighborhood Uofx,\nsomeR >0and aξ-neighborhood W⊂Sd−1such that for allreal-valued admissible\nψ∈ S(Rd)withsupp(/hatwideψ)⊂V, the following estimate holds:\n∃C >0∀y∈U∀h∈Ko(W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblN−α1d/2.\n(b) Letψ∈ S(Rd)be admissible with supp(/hatwideψ)⊂V. Assume there exists a neighborhood U\nofx, someR>0and aξ-neighborhood W⊂Sd−1such that\n∃C >0∀y∈U∀h∈Ko(W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblα1N+3\n2α1d+α2\nThen(x,ξ)is anN-regular directed point of u.\nProof.Sinceψis bandlimited, it satisfies vanishing moments condition fo r anyr. So, conditions\n(i), (ii), (iii) and (v) of Theorem 5.4 are automatically sat isfied for the case ψ1=ψ2=ψ.\nCombining Theorem 5.4 and Theorem 3.4 finishes the proof of (a ). Part (b) follows directly\nfrom Theorem 3.4. /square\nRemark 5.7. Another condition that allows to close the gap between inner and outer cone\nconditions is provided by the cone approximation property , which guarantees inclusions of the\ntypeKo(W,V,R)⊂Ki(W′,V,R′), for suitable choices of W,Rdepending on W′,R′.\nIt is currently not clear how the cone approximation property is related to the assumptions\nunderlying Theorem 5.8 and its corollary. We note that both co nditions are fulfilled for our chief\nclass of examples, the shearlet dilation groups covered in S ection 6.\nWe now present a near characterization of unsigned regular d irected points in the case where\nthe wavelet is real-valued and has sufficiently many vanishin g moments. Note that there exist\ncompactly supported wavelets fulfilling these conditions. Observe also that this theorem uses\nthe same cone-related subsets for both implications.\nTheorem 5.8. Fix(x,ξ)∈Rd×(O ∩Sd−1). LetVbe a relatively compact subset of O,\nand assume that the dual action is V-microlocally admissible in direction ξ, with associated\nparameters α1,α2>0. Assume that Hfulfills theV-overlap condition, as well as inequality\n(18). Then there exist constants s0,s1,s2≥0, depending only on HandV, such that the\nfollowing holds:\nLetu∈ S′(Rd)denote a tempered distribution of order N(u). Letψ∈ S(Rd)denote a wavelet\nwith vanishing moments in Ocof order\nr>s0+s1N+s2N(u).\n(a) If(x,±ξ)is an unsigned N-regular directed point of u, then there exists a neighborhood\nUofx, someR >0and aξ-neighborhood W⊂Sd−1such that the following estimate\nholds:\n∃C >0∀y∈U∀h∈Ko(W,V,R)∪Ko(−W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblN−α1d/2.28 HARTMUT F ¨UHR, MAHYA GHANDEHARI\n(b) Let the wavelet ψbe real-valued. Assume that Uis a neighborhood of x, and that there\nareR>0and aξ-neighborhood W⊂Sd−1such that\n∃C >0∀y∈U∀h∈Ko(W,V,R)∪Ko(−W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblα1N+3\n2α1d+α2\nThen(x,±ξ)is an unsigned N-regular directed point of u.\nProof.Clearly, the lower bound imposed on rvia assumptions (i)-(v) of Theorem 5.4 can be\nguaranteed to hold by a lower estimate of the type\nr>s0+s1N+s2N(u),\nwith suitably chosen s0,s1,s2depending only on HandV.\nFor the proof of (a), pick a wavelet η∈ S(Rd) that fulfills supp( /hatwideη)⊂V0⊂V, w.l.o.g.\nV0∩ −V0=∅. The latter assumption guarantees that Re( η),Im(η) are both nonzero, with\nFourier transforms supported in V0∪−V0⊂ O. Hence they are admissible wavelets as well, with\nall moments in Ocvanishing. By Lemma 4.2, ( x,±ξ) is an unsigned N-regular directed point\nof both Re( u) and Im(u). By Theorem 3.4, there exist a ξ-neighborhood W′inO ∩Sd−1, a\nneighborhood Uofx, andR′>0 such that\n|Wη(Reu)(y,h)| ≤C·/bardblh/bardblN−α1d/2,|Wη(Imu)(y,h)| ≤C·/bardblh/bardblN−α1d/2(28)\nholds forh∈Ki(W′,V,R′)∪Ki(−W′,V,R′) andy∈U. Here, we used the fact that the\ndual action is V-microlocal admissible in any direction, in particular in t he direction of −ξ(see\nRemark 2.5). Writing\nη1= Re(η),η2= Im(η),\nwe have that WηRe(u) =Wη1Re(u)−iWη2Re(u), withWη1Re(u) andWη2Re(u) real-valued.\nHence the analog of (28) also holds for WηiRe(u), fori= 1,2.\nNow Theorem5.4, appliedto η1, allows tochoose W,Rdependingon W′,R′insuchaway that\nthe same type of wavelet coefficient decay holds for WψRe(u) insideKo(W,V,R)∪Ko(−W,V,R).\nThe analogous reasoning can be applied to WψIm(u), possibly at the expense of further restrict-\ningW,R. But then u= Re(u)+iIm(u) implies the same decay for Wψu.\nFor the proof of part (b) we pick a wavelet η∈ S(Rd) that fulfills supp( /hatwideη)⊂V⊆ O. Then\nηhas vanishing moments of all orders on O. Sinceψis real-valued, we can use Theorem 5.4\nto transfer this decay to Wη(u) for allh∈Ko(W,V,R)∪Ko(−W,V,R). Thus, Theorem 3.4\nprovides that both ( x,ξ) and (x,−ξ) areN-regular directed points of u. /square\nNote that the slight detour via Re( u) and Im(u) is necessitated by the fact that for complex-\nvalued wavelet ηand distribution u, there is no simple estimate of |WRe(ψ)u|,|WIm(ψ)u|against\n|Wψu|available, which would allow the transfer of decay statemen ts from one side to the other.\nThe same observation applies to |Wψ(Re(u))|and|Wψ(Im(u))|.\nAs an easy application of the theorem, we note an adaptation o f the main result from [5], from\nbandlimited Schwartz wavelets to real Schwartz wavelets wi th vanishing moments of arbitrary\norder.\nCorollary 5.9. Fix(x,ξ)∈Rd×(O ∩Sd−1). LetVbe a relatively compact subset of O, and\nassume that the dual action is V-microlocally admissible in direction ξ. Further assume that H\nfulfills theV-overlap condition, as well as inequality (18). Let u∈ S′(Rd), and letψ∈ S(Rd)\ndenote a real-valued wavelet with vanishing moments in Ocof arbitrary order. Then the following\nare equivalent:\n(a)(x,±ξ)is an unsigned regular directed point of u.SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 29\n(b) There exists a neighborhood Uofx, someR>0and aξ-neighborhood W⊂Sd−1such\nthat the following holds:\n∀N >0∃C >0∀y∈U∀h∈Ko(W,V,R)∪Ko(−W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblN.\n6.Shearlet dilation groups\nGiven that thelists of(partly implicit) assumptionsofthe Theorems 3.4and5.8 aresomewhat\nlong and technical, it is legitimate to ask whether there are examples fulfilling all of them. It is\nthe main purpose of this section to demonstrate that there ex ists a fairly large example class for\nwhich all the assumptions can be verified, with reasonably ex plicit expressions for the involved\nconstants.\nWe letT(d,R) denote the matrix group of upper triangular matrices of siz edwith ones on the\ndiagonal. Elements of T(d,R) are called unipotent .t(d,R) refers to the Lie algebra of T(d,R),\nwhich consists of the upper triangular matrices with zero di agonal.\nDefinition 6.1. LetH <GL(d,R)denote an irreducibly admissible dilation group. His called\nageneralized shearlet dilation group , if there exist two closed subgroups S,D < H with the\nfollowing properties:\n(i)Sis a connected abelian Lie subgroup of T(d,R);\n(ii)D={exp(rY) :r∈R}is a one-parameter group, where Yis a diagonal matrix;\n(iii) Every h∈Hcan be written uniquely as h=±ds, withd∈Dands∈S.\nSis called the shearing subgroup ofH, andDis called the scaling subgroup ofH.\nRemark 6.2. The two-dimensional shearlet groups are given by\nHc=/braceleftbigg\n±/parenleftbigg\na t\n0ac/parenrightbigg\n:a>0,t∈R/bracerightbigg\n<GL(2,R).\nHerec∈Rcan be chosen arbitrarily.\nRemark 6.3. For dimension d≥3, emphasis of research so far has been focused on standard\nandToeplitz shearlet groups, which we define below. We remark that with gr owing dimension d,\nthe number of possible choices distinct from these two class es of groups grows considerably.\n(i) Forλ= (λ2,...,λd)∈Rd−1, we define the standard shearlet group in ddimensions Hλ\nas the set\n\n\nǫdiag/parenleftig\na,aλ2,...,aλd/parenrightig\n1t1... td−1\n1 0...0\n...0\n1\n|a>0,\nti∈R,\nǫ∈ {±1}\n\n.\n(ii) Forδ∈R, we define the Toeplitz shearlet group in ddimensions Hδas the set\n\n\nǫdiag/parenleftig\na,a1−δ,...,a1−(d−1)δ/parenrightig\n·T(1,t1,...,td−1)|a>0,\nti∈R,\nǫ∈ {±1}\n\n,30 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nwhere the matrix T(1,t1,...,td−1)is defined by\nT(1,t1,...,td−1) :=\n1t1t2... td−2td−1\n1t1t2... td−2\n............\n1t1t2\n1t1\n1\n.\nBelow we will establish that the results of the previous sect ions, in particular Theorems 3.4\nand 5.8, are applicable to shearlet groups from a large examp le class, including the above-listed\nfamilies, provided that the scaling subgroup fulfills an add itional condition. Thankfully, the\nmajority of the necessary technical conditions have alread y been verified in previous papers.\nThe following lemma already guarantees the applicability o f Theorem 3.4.\nLemma 6.4. LetHdenote a shearlet dilation group with scaling subgroup D= exp(RY), where\nw.l.o.g.Y= diag(1,λ2,...,λd). Let\nλmax= max{λi: 2≤i≤d}, λmin= min{λi: 2≤i≤d}.\nAssume that 0<λmin≤λmax<1. Then\n(a) There exists an open ∅/\\e}atio\\slash=V⋐Hsuch thatHfulfills theV-cone approximation property;\nthat is, for every ξ∈ O ∩Sd−1we have the following: for all ξ−neighborhoods W⊆\nSd−1and allR >0, there exists a ξ-neighborhood W′⊆Sd−1andR′>0satisfying\nKo(W′,V,R′)⊆Ki(W,V,R).\n(b)His microlocally admissible, with\nα1=1\nλmin, α2=d\nλmin+ǫ , (29)\nwhereǫ>0is arbitrary.\nProof.The cone approximation property is shown in Proposition 30 o f [1], and microlocal ad-\nmissibility is stated in Proposition 32 of that paper. The co nstants are not explicitly stated\nin this result, but they can be found in the proof: Part (a) of D efinition 2.4 is shown to hold\nwithα1=1\nλmin. For the existence of α2, the reader is referred to Lemma 4.7 of [5], which\nappeals to the existence of α1and the cone approximation property. For an actual formula f or\nα2, the reader should consult the proofs of [5, Lemma 2.8 and Lem ma 4.7] to find the condition\nα2>dα1. /square\nWe next establish the condition allowing the application of the decay estimates from Propo-\nsition 4.8. Before we formulate the statement, we cite a stat ement concerning the structure of\nthe shearing subgroup and its Lie algebra.\nLemma 6.5 ([1] Lemma 5. and Lemma 6.) .LetSbe the shearing subgroup of a generalized\nshearlet dilation group H⊂GL(Rd). Then the following statements hold:\n(a) There exists a unique basis X2,...,Xdof the Lie algebra sofSwithXT\nie1=eifor\n2≤i≤d, called the canonical basis of s.\n(b) We have S={Id+X|X∈s}.\n(c) Letsk= span{Xj:j≥k}, fork∈2,...,d. These are associative matrix algebras\nsatisfying sksℓ⊂smax{k,ℓ}+1, where we write sm={0}form>d.\n(d)His the inner semidirect product of the normal subgroup Swith the closed subgroup\nD∪−D.SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 31\nNote that since sconsists of strictly upper triangular matrices, it is nilpo tent. We let n(s)\ndenote the nilpotency degree of s, i.e.\nn(s) = min{k∈N:Xk= 0∀X∈s}.\nAs will be seen more clearly in our computations below, n(s) enters quite naturally in compu-\ntations with inverse matrices in H.\nLemma 6.6. LetHdenote a shearlet dilation group with scaling subgroup D= exp(RY), where\nw.l.o.g.Y= diag(1,λ2,...,λd). Letsdenote the Lie algebra of the shearing subgroup, and n(s)\nits nilpotency degree. Assume that λmin,λmax, defined in Lemma 6.4, fulfill 0<λmin≤λmax<1.\nThen\n/bardblh±1/bardblAH(h)ℓ1≤1\nholds for all h∈H, with\nℓ1=n(s)+1. (30)\nProof.We refer to [10], Theorem 4.10. Note that the cited theorem ma kes no assumptions on\nY, and gives the more general formula\nℓ1=n(s)−1+2/bardblY/bardbl.\nHere we restrict attention to the case 0 <λmin≤λmax<1, which entails /bardblY/bardbl= 1. /square\nThus the only remaining obstacle to the applicability of The orem 5.8 is the overlap control\ncondition from Definition 5.1. Unfortunately, this conditi on requires somewhat lengthy compu-\ntations, which we now prepare. Recall that we need to estimat e the integralˆ\nH0\\h−1Ki(W,V,R)Θs,t(g)dg .\nWe will do this using fairly explicit computations in partic ular coordinates for H. With respect\nto these coordinates, the following challenges present the mselves:\n(1) Estimate norms of products and inverses in H;\n(2) explicitly determine the integration domain H0\\h−1Ki(W,V,R) for certain hinH;\n(3) determine the Haar measure d g.\nWe start by introducing the coordinates we will use: Using th e canonical basis X2,...,Xdof\ns, we define, for t∈Rd−1anda/\\e}atio\\slash= 0,\nh(t,a) =/parenleftigg\nId+d/summationdisplay\ni=2tiXi/parenrightigg−1\nsgn(a)diag(|a|,|a|λ2,...,|a|λd)∈H .\nHereId= diag(1,...,1) denotes the d×didentity matrix. It is easy to see that ( t,a)∋\nRd−1×R×→His a bijection, and that restricting this mapping to Rd−1×{1}yields a bijection\nontoS. The inversion of the first factor is the cause of some awkward ness later on, but it will\nallow a fairly simple description of cone-affiliated subset Ki/o⊂H.\nNote that we can write\nId+d/summationdisplay\ni=2tiXT\ni=/parenleftbigg\n1 0T\nt Id−1+A(t)T/parenrightbigg\n, (31)\nwithA(t) being a (d−1)×(d−1) strictly upper-triangular matrix satisfying\n/bardblA(t)/bardbl ≤C|t| (32)\nwith a constant Cdepending only on H.\nWe next describe group operations with respect to these coor dinates:32 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nLemma 6.7. LetHdenote a shearlet dilation group, with canonical basis X2,...,Xdfors.\n(a) There exists a bilinear map M:Rd−1×Rd−1→Rd−1such thath(t,1)−1h(t′,1)−1=\nh(t+t′+M(t,t′),1)−1. For alli= 2,...,d, theith entryM(t,t′)ionly depends on the\nentriest2,...,ti−1,t′\n2,...,t′\ni−1.\n(b) There exists a map B:Rd−1→Rd−1, polynomial of degree n(s)−1, such that for all\nt∈Rd−1,\nh(t,1)−1=h(−t+B(t),1).\nHereB(t)ionly depends on the entries t2,...,ti−1.\n(c) For all a>0andt= (t2,...,td)T∈Rd−1, we have\nh(0,a)h(t,1)h(0,a)−1=h((a1−λ2t2,...,a1−λdtd),1)\nProof.For part (a), we have by definition\nh(t,1)−1h(t′,1)−1=/parenleftigg\nId+d/summationdisplay\ni=2tiXi/parenrightigg/parenleftigg\nId+d/summationdisplay\ni=2t′\niXi/parenrightigg\n=Id+d/summationdisplay\ni=2(ti+t′\ni)Xi+d/summationdisplay\ni,j=2tit′\njXiXj\n=Id+d/summationdisplay\ni=2(ti+t′\ni)Xi+/summationdisplay\niM(t,t′)iXi,\nwhereM(t,t′)idenotes the coefficient of the Xi-coordinate of/summationtextd\nℓ,k=2tkt′\nℓXkXℓ. Clearly this\ncoordinate depends bilinearly on the tk,t′\nℓ, and only on those with ℓ,k < i, by part (c) of\nLemma 6.5.\nThe fact that t/mapsto→h(t,1) induces a bijection between Rd−1andSentails the existence of a\nunique map B′withh(t,1)−1=h(B′(t),1). Furthermore, every unipotent matrix is inverted by\na finite Neumann series, and for elements X∈s, the number of terms in the series is at most\nn(s)−1. Now part (a) of the current lemma implies that B′is a polynomial of degree ≤n(s)−1.\nFurthermore, part (c) of Lemma 6.5 implies that B′(t) =−t+B(t), andBhas the described\nproperties.\nFor part (c) we first note that by definition of the canonical ba sis, the vector t∈Rd−1such\nthath=h(t,1)−1is just given by the first row of h. Computing the conjugation action of the\ndiagonal matrix on the row entries then gives the equation\nh(0,a)h(t,1)−1h(0,a)−1=h((a1−λ2t2,...,a1−λdtd),1)−1.\nInverting both sides of the equation yields (c). /square\nLemma 6.8. In the(t,a)-coordinates, left Haar measure on His determined as\nˆ\nHf(h)dh=ˆ\nR׈\nRd−1f(h(t,a))dtda\n|a|d−tr(Y),\nwhere dtis the Lebesgue measure of Rd−1and dais the Haar measure of the multiplicative group\nR×.\nProof.It is easy to see from part (a) of the previous lemma that Haar i ntegration on Sis given\nbyˆ\nRd−1f(h(t,1)−1)dt .SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 33\nSinceSis an abelian group, this implies that\nˆ\nRd−1f(h(t,1))dt\nyields the same Haar integral. Furthermore, one can use part (c) of the previous lemma to\ndetermine the impact of the conjugation action of Don Haar measure of S. Now a standard\nformulafor theHaar integral of semidirect products, i.e. [ 14, 15.29], yields thedesired result. /square\nWe next provide approximate descriptions of the cone-affilia ted sets with respect to the co-\nordinates that we just introduced. In order to do this, we firs t parameterize the open orbit O\nby the global chart provided by affine coordinates\nΩ :R××Rd−1→ O Ω(τ,v) =τ(1,vT)T,\nand consider the corresponding diffeomorphism to its image\nω:Rd−1→Sd−1∩O, ω(v) =(1,vT)T\n/radicalbig\n1+|v|2.\nGivenǫ>0, we set\nWǫ={v∈Rd−1:|v|<ǫ}=Bǫ(0).\nClearly,{Wǫ:ǫ>0}is a neighbourhood basis of the origin in Rd−1, and so {ω(Wǫ) :ǫ>0}is\na neighbourhood basis of ξ0= (1,0,...,0)∈Sd−1∩O. Furthermore, for fixed 0 <τ1<τ2and\nǫ0>0 the set\nV= Ω((τ1,τ2)×Wǫ0)\nis an open subset with V⋐O.\nLemma 6.9. Assume that λmin,λmax, defined in Lemma 6.4, fulfill 0<λmin≤λmax<1. Let\nV= Ω((τ1,τ2)×Wǫ0),R>1, where0<τ1<1<τ2andW=ω(Wǫ), withǫ<1. SupposeR\nis sufficiently large, meaning,\nR>2τ2max/braceleftbigg\n1,(2ǫ0\nǫ)1\n1−λmax,(2Cǫ0)1\n1−λmax/bracerightbigg\n, (33)\nwithCfrom(32). Then, the following inclusions hold:\n{h(t,a) :|t|<δ0,0<a<a 0} ⊆Ki(W,V,R)⊆Ko(W,V,R)⊆ {h(t,a) :|t|<δ1,0<a<a 1},\nwherea0,a1,δ0,δ1are given by\na0=τ1\nR, δ0=ǫ\n3\nand\na1=2τ2\nR, δ1= 3ǫ .\nProof.Our argument is inspired by the proof of Proposition 30 in [1] , so we will not expand on\nits details here.\nGiven the definition of V, we have\n(h−1)T(V) = Ω/parenleftbig\n(a−1τ1,a−1τ2)×(t+(Id−1+A(t)T)Wa\nǫ0)/parenrightbig\n, (34)\nwhereWa\nǫ0:= diag(a1−λ2,...,a1−λd)(Wǫ0). It is also easy to see that for every 0 < Rand\n0<ǫ≤1, we have\n(R,∞)×Wǫ⊆Ω−1(C(ω(Wǫ),R))⊆(R\n2,∞)×Wǫ. (35)34 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nBy (34) and (35), if h=h(t,a) belongs to Ko(W,V,R), then we must have\n(i) 0<a<2τ2\nR\n(ii)∃ξ∈Bǫ0(0) s.t.|t+(Id−1+A(t)T)diag(a1−λ2,...,a1−λd)ξ|<ǫ.\nSinceR>2τ2, we havea<1, and consequently, we get\n|t|<ǫ+(1+C|t|)a1−λmaxǫ0≤ǫ+(1+C|t|)(2τ2\nR)1−λmaxǫ0.\nThe above inequality, together with R>max/braceleftig\n2τ2(2ǫ0\nǫ)1\n1−λmax,2τ2(2Cǫ0)1\n1−λmax/bracerightig\n, gives|t|<3ǫ.\nFor theother inclusion, note that by (35), h∈Kiis guaranteed if Ω−1((h−1)T(V))⊆(R,∞)×\nWǫ. For the latter inclusion to hold, it is enough to have\n(i) 0<a<τ1\nR,\n(ii)|t|+(1+C|t|)a1−λmaxǫ0<ǫ.\nCombining (33) and 0 <a<τ1\nR, one can verify that (ii) holds whenever |t|<ǫ\n3. /square\nFor estimates of the function Θ r,s, we need to estimate norms of products of the type\nh(t,a)−1h(t′,a′). This is provided by the next lemma:\nLemma 6.10. Let(t,a),(t′,a′)∈Rd−1×R+. Assume that λmin,λmax, defined in Lemma 6.4,\nfulfill0<λmin≤λmax<1.\n(a) LetV= Ω((τ1,τ2)×Wǫ0),Rsatisfying (33), where0<τ1<1<τ2andW=ω(Wǫ),\nwithǫ<1. Then there exists a constant C1such that for all h=h(t,a)∈Ko(W,V,R),\nthe following inequalities hold:\naλmin≤ /bardblh(t,a)/bardbl ≤C1aλmin\n(b)/bardblh(t,a)−1h(t′,a′)/bardbl ≥a−1a′.\n(c) Assume that a,a′<1and|t|<δ1. There exist constants Cs,CM>0depending only on\nHsuch that\n/bardblh(t,a)−1h(t′,a′)/bardbl ≥a−1(a′)λmax\n−δ1+(1−CMδ1)\n\n|t′|\nCs|t′|<Cs\n|t′|1/(n−1)\nC1/(n−1)\ns|t′| ≥Cs\n,\nwheren=n(s).\nProof.For the proof of ( a), we note that the diagonal entries of h(t,a) area,aλ2,...,aλd. Since\nh∈Ko(W,V,R) andRsatisfies (33), by Lemma 6.9 we know a<1. So,\n/bardblh(t,a)/bardbl ≥aλmin.\nFor the upper bound, first note that by Lemma 6.9, we have |t|<3ǫ. Now, submultiplicativity\nof the norm yields\n/bardblh(t,a)/bardbl ≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg\nId+d/summationdisplay\ni=2tiXi/parenrightigg−1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublediag(a,aλ\n2,...,aλd)/vextenddouble/vextenddouble/vextenddouble\n≤C1aλmin,\nwhereC1is the maximum attained by the continuous map t/mapsto→/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftig\nId+/summationtextd\ni=2tiXi/parenrightig−1/vextenddouble/vextenddouble/vextenddouble/vextenddoubleon the\ncompact set B3ǫ(0).SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 35\nFor part (b), we simply use that a−1a′is the upper left entry of h(t,a)−1h(t′,a′).\nFor part (c), we first compute\nh(t,1)−1h(t′,1) =h(t,1)−1h(−t′+B(t′),1)−1\n=h(t−t′+B(t′)+M(t,−t′+B(t′)),1)−1\n=Id+d/summationdisplay\ni=2t′′\niXi,\nwith\nt′′=t−t′+B(t′)+M(t,−t′+B(t′)),\nwhereBis the polynomial map from Lemma 6.7(b). Hence\nh(t,a)−1h(t′,a′) = diag( a−1,a−λ2,...,a−λd)/parenleftigg\nId+d/summationdisplay\ni=2t′′\niXi/parenrightigg\ndiag(a′,(a′)λ2,...,(a′)λd).\nThe norm of that product is greater or equal to the norm of its fi rst row, which is\n(a−1a′,a−1(a′)λ2t′′\n2,...,a−1(a′)λdt′′\nd),\nhence/vextenddouble/vextenddoubleh(t,a)−1h(t′,a′)/vextenddouble/vextenddouble≥a−1(a′)λmax|t′′|.\nFor the estimate of |t′′|, note that by Lemma 6.7(b), −t′+B(t′) is a polynomial of order n−1.\nThe polynomial has zero constant term because h(0,1)−1=h(0,1), hence we have\n|−t′+B(t′)| ≤n−1/summationdisplay\nk=1ck|t′|k≤Cs/braceleftbigg\n|t′| |t′|<1\n|t′|n−1|t′| ≥1, (36)\nwith positive constants ck,Csdependingon the generators X2,...,Xdof the Lie algebra s. Since\nthe inversion map on His its own inverse, we have t′=t′−B(t′)+B(−t′+B(t′)). This implies,\nvia (36), the estimate\n|t′| ≤Cs/braceleftbigg\n|−t′+B(t′)| |−t′+B(t′)|<1\n|−t′+B(t′)|n−1|−t′+B(t′)| ≥1.\nDistinguishing the cases |−t′+B(t′)| ≥1 and|−t′+B(t′)|<1 allows to derive\n|−t′+B(t′)| ≥min/braceleftigg\n|t′|1/(n−1)\nC1/(n−1)\ns,|t′|\nCs/bracerightigg\n,\nor equivalently,\n|−t′+B(t′)| ≥\n\n|t′|\nCs|t′|<Cs\n|t′|1/(n−1)\nC1/(n−1)\ns|t′| ≥Cs.\nIt follows, using |t|<δ1and the constant CMbounding the bilinear map M,\n|t′′| ≥ |−t′+B(t′)|(1−CMδ1)−δ1≥ −δ1+(1−CMδ1)\n\n|t′|\nCs|t′|<Cs\n|t′|1/(n−1)\nC1/(n−1)\ns|t′| ≥Cs.\n/square36 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nLemma 6.11. LetH=DS∪−DSdenote a shearlet dilation group, with 0<λmin≤λmax<1.\nLetH0=DS. ThenHfulfills the overlap control condition, with\nγ0= 2d+1, γ1=λmin\n1−λmax, γ2=λminλmax\n1−λmax\nProof.By Lemma5.2, it sufficestoconcentrate onthedirection ξ0= (1,0,...,0)T. LetV,W′,R′\nbe given as in Lemma 6.9, i.e. V= Ω((τ1,τ2)×Wǫ0),W′=ω(Wǫ′) withǫ′<1, andR′satisfying\n(33). In addition, assume that ǫ′<7Cs, whereCsdenotes the constant from Lemma 6.10(c).\nLetW′′⊂W′andR′′>R′be arbitrary, and w.l.o.g. assume that W′′=ω(Wǫ′′) withǫ′′≤ǫ′.\nWe pickR=R′′andW=ω(Wǫ), where\nǫ<min/braceleftbigg\nǫ′′,1\n6CM,ǫ′′\n21Cs/bracerightbigg\n. (37)\nWe need to estimate, for h∈Ko(W,V,R) the integral\nˆ\nH0\\Ki(W′′,V,R′′)Θr,s(h−1g)dg;\nthisisthesameintegralasinthedefinitionoftheoverlapco ntrolcondition, since h∈Ko(W,V,R)⊆\nH0and dgis just Haar integration. Using the parameterization g=h(t′,a′), the inclusion rela-\ntions from Lemma 6.9 with a′′\n0=τ1\nR′′,δ′′\n0=ǫ′′\n3imply that it is sufficient to estimate\nˆ\na′≥a′′\n0ˆ\nRd−1Θr,s(h−1h(t′,a′))d(h(t′,a′))\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1+ˆ\na′≤a′′\n0ˆ\n|t′|≥δ′′\n0Θr,s(h−1h(t′,a′))d(h(t′,a′))\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI2\nagainst asuitable power of /bardblh/bardbl, whereh=h(t,a), and|t|<δ1,a<a 1withδ1= 3ǫanda1=2τ2\nR.\nHere d(h(t′,a′)) denotes the left Haar integration over H, as computed in Lemma 6.8\nFor the estimate of I1, we observe that a′≥a′′\n0implies via Lemma 6.10(b) that\n/bardblh(t,a)−1h(t′,a′)/bardbl ≥a−1a′′\n0,\nhence\n(1+/bardblh(t,a)−1h(t′,a′)/bardbl)−1≤a\na′′\n0\nand thus for every 0 <α<r\nΘr,s(h(t,a)−1h(t′,a′))≤CaαΘr−α,s(h(t,a)−1h(t′,a′)).\nTherefore, as soon as r−α≥dim(H) +d+ 1 = 2d+ 1 ands≥dim(H) +d= 2d, which\nguarantees the integrability of Θ r−α,svia Lemma 4.9, we have that\nI1≤Caαˆ\nHΘr−α,s(h(t,a)−1g)dg\n≤C′/bardblh(t,a)/bardblα/λmin,\nwith the last inequality due to Lemma 6.10(a). Here, C′only depends on τ1,R′′,α,r,s, and not\nona,t.\nFor the estimate of I2, note that Ko(W,V,R)⊆Ko(W′,V,R′) and conditions of Lemma 6.9\nforW′,V,R′are satisfied. So, the assumption h(t,a)∈Ko(W,V,R) and the integral bound\n|t′| ≥δ′′\n0entail\n|t′| ≥ǫ′′\n3,|t| ≤δ1= 3ǫ<ǫ′′\n7Cs.SOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 37\nIt follows successively from (37) that\n1−CMδ1≥1\n2,(1−CMδ1)|t′|\nCs≥ǫ′′\n6Cs,\nand finally, via Lemma 6.10(c),\n/bardblh(t,a)−1h(t′,a′)/bardbl ≥a−1(a′)λmaxC′\nwith a global constant C′=ǫ′′\n42Cs. It follows for 0 <β <min(r,s) that\nI2≤(C′)−βˆ\na′≤a′′\n0ˆ\n|t′|≥δ′′\n0aβ(a′)−βλmaxΘr−β,s(h−1h(t′,a′))d(h(t′,a′))\n= (C′)−βaβ(1−λmax)ˆ\na′≤a′′\n0ˆ\n|t′|≥δ′′\n0aβλmax(a′)−βλmaxΘr−β,s(h−1h(t′,a′))d(h(t′,a′))\n≤C′′aβ(1−λmax)ˆ\nHΘr−β,s−βλmax(h−1g)dg ,\nwhere the last inequality used\n|a||a′|−1≤ /bardblh(t′,a′)−1h(t,a)/bardbl,\nsee Lemma 6.10(b), as well as Θ r−β,s−βλmax≥0. Thus, again by Lemma 4.9 and Lemma 6.10(a),\nI2≤C′′′/bardblh/bardblβ(1−λmax)/λmin\nwith a finite constant C′′′as soon as r≥2d+ 1+βands≥2d+βλmax. Here,C′′′depends\nonly onǫ′′,Cs,β,r,s,H , and is independent of h.\nFor a given L>0, substituting α=Lλminandβ=Lλmin\n1−λmax, we find the desired estimate\nˆ\nH0\\Ki(W′′,V,R′′)Θr,s(h−1g)dg≤˜C/bardblh/bardblL,\nfor allh∈Ko(W,V,R), as soon as\nr≥2d+1+Lλmin\n1−λmax, s≥2d+Lλminλmax\n1−λmax.\n/square\nWith all conditions for the applicability of our main result s to shearlet dilation groups being\nchecked, the following theorem spells out the resulting ver sion of Theorem 5.8 for this class of\ngroups.\nTheorem 6.12. LetHdenote a shearlet dilation group. Let Ydenote the infinitesimal generator\nof the scaling subgroup of H, with eigenvalues λ1,...,λd, normalized such that λ1= 1. Let\nλmin= min{λ2,...,λd},λmax= max{λ2,...,λd}, and assume that 0< λmin≤λmax<1and\nλmin+λmax≥1. Letn0denote the nilpotency degree of the shearing subgroup of H.\nDefine\ns0= 1+/parenleftbigg3\n2+1\n2λmin/parenrightbigg\nd+(n0+1)/parenleftbigg13\n2d+(1\n2λmin+3+3λmax\n2−2λmax)d+3/parenrightbigg\n(38)\ns1= 1+(n0+1)/parenleftbigg\n1+λmin\n1−λmax+λminλmax\n1−λmax/parenrightbigg\n(39)\ns2= 1+1\nλmin+(n0+1)/parenleftbigg\n2+1\nλmin+2\n1−λmax/parenrightbigg\n. (40)38 HARTMUT F ¨UHR, MAHYA GHANDEHARI\nLetu∈ S′(Rd)denote a tempered distribution of order N(u). Letψ∈ S(Rd)denote a\nreal-valued wavelet with vanishing moments in R×{0}d−1of order\nr>s0+s1N+s2N(u). (41)\n(a) If(x,±ξ)is an unsigned N-regular directed point of u, then there exists a neighborhood\nUofx, someR >0and aξ-neighborhood W⊂Sd−1such that the following estimate\nholds:\n∃C >0∀y∈U∀h∈Ko(W,V,R)∪Ko(−W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblN−d\n2λmin.\n(b) Assume that Uis a neighborhood of x, and that there are R >0and aξ-neighborhood\nW⊂Sd−1such that\n∃C >0∀y∈U∀h∈Ko(W,V,R)∪Ko(−W,V,R) :|Wψu(y,h)| ≤C·/bardblh/bardblN\nλmin+3d\n2λmin+α2.\nThen(x,±ξ)is an unsigned N-regular directed point of u.\nProof.We firstcollect the various constants requiredfor the appli cation of Theorem 5.8, starting\nwith dim(H) =d. Recall from Lemma 6.4 that His microlocally admissible, with constant\nα1=1\nλmin>1 andα2=d\nλmin+ǫ∗,for anyǫ∗>0.\nLemma 6.6 contributes the constant ℓ1=n0+ 1 for which inequality (18) holds. By Lemma\n6.11,Hfulfills the overlap control condition with constants\nγ0= 2d+1, γ1=λmin\n1−λmax, γ2=λminλmax\n1−λmax.\nClearly,γ1α1≥1. Moreover, the assumption λmin+λmax≥1 implies that\nγ1≥1,γ2α1≥1. (42)\nAs a consequence, the sharpest lower bound for β1contained in the conditions (i) through (iv)\nof Theorem 5.4 is provided by condition (i), whereas the rele vant lower bounds for β2,β3are\nprovided by (iv), as a consequence of (42). Thus the conditio ns of Theorem 5.4 (i) through (iv)\nare fulfilled by\nβ1= 1+/parenleftbigg\n1+1\n2λmin/parenrightbigg\nd+N+/parenleftbigg\n1+1\nλmin/parenrightbigg\nN(u)\nβ2= 1+/parenleftbigg\n2+3\n2−2λmax/parenrightbigg\nd+λmin\n1−λmaxN+/parenleftbigg\n1+1\n1−λmax/parenrightbigg\nN(u)\nβ3= 1+/parenleftbigg7\n2+3λmax\n2−2λmax/parenrightbigg\nd+λminλmax\n1−λmaxN+1\n1−λmaxN(u).\nBy Theorem 5.4, the relevant lower bound for the number of van ishing moments is given by the\ninequality\nr>d\n2+ℓ1(β1+β2+β3)+β1.\nPlugging in the values for β1,β2andβ3that we just obtained, and sorting terms, results in the\nequivalent inequality\nr>s0+s1N+s2N(u),\nwiths0,s1,s2given by equations (38) through (40). /squareSOBOLEV WAVEFRONT SET AND COMPACTLY SUPPORTED WAVELETS 39\nNotethatTheorem5.8isapplicabletotheshearletdilation groupHassoonasλmin+λmax≥1\nand 0<λmin≤λmax<1 hold. It is worthwhile noting that these conditions are als o sufficient\nto guarantee that Hcharacterizes the wavefront set of infinite order, by Theore m 28 of [1].\nFurthermore, Corollary 5.9 also applies under these condit ions.\nWe note that both the standard and the Toeplitz shearlet grou ps from Remark 6.3 can fulfill\nthese conditions, when suitable parameters are chosen. For the standard case this is obvious,\nfor the Toeplitz case any parameter 0 <δ≤1/dwill work.\n6.1.Discussion/comparison/future directions for results on generalized shearlets. It\nis interesting to note that there is essentially only one con dition required to make a shearlet\ndilation group a useful tool for the characterization of the Sobolev wavefront set. In particular,\nthe structure of the shearing subgroup only influences the in volved constants.\nTheorem 6.12 can be briefly summarized by the statement that t here is an easily computed\nlower bound on the number rof vanishing moments that allows a near characterization of\nelements of the unsignedwavefront set, valid for any Schwar tz wavelet with rvanishingmoments\ninR×{0}d−1. It establishes an affine dependence of the required number of vanishing moments\non the degree of smoothness and/or the order of the distribut ion under consideration. The main\ninformation provided by Theorem 6.12, on top of guaranteein g the applicability of Theorem 5.8,\nis the concrete expressions for the involved constants, and their dependence on the structure of\nthe underlying group.\nUltimately only very limited information on the group is req uired. Note that s0,s1,s2only\nneed to be determined once for each group, and they depend onl y on the eigenvalue range for\nthe infinitesimal generator of the scaling subgroup and the n ilpotency degree n0of the shearing\nsubgroup. As the dimension dincreases, the number of relevant groups grows rather quick ly.\nFor example, it is known that for dimensions d≥7, there exists an uncountably infinite family\nof shearing subgroups that are not related by conjugation (s ee e.g. Remark 15 of [1]), and we\nexpect that many of these subgroups can be combined with a sca ling subgroup fulfilling the\neigenvalue condition of Theorem 6.12. These observations e mphasize that Theorem 6.12 applies\nto a large variety of groups.\nIt is reasonable to expect that the generality and scope of th ese results comes at the price of\nsuboptimal estimates and constants. Hence for specific choi ces of the dilation group HTheorem\n6.12 should rather be seen as a proof of principle, with consi derable room for improvement.\nOne instance where currently known results indicate such a p ossibility concerns the gap\nbetween the exponents in the decay conditions of the direct a nd inverse theorems. In the\nshearlet case the requirement 0 < λmin<1 implies that α1=1\nλmin>1. This entails that the\ngap between the exponents, namely N−α1d/2 andα1N+3\n2α1d+α2grows linearly with N.\nThis can be contrasted to the results in [11]. The setting con sidered there is essentially that of\na shearlet dilation group in dimension two, with λmin= 1/2 andn0= 1. A detailed comparison\nof our results with the direct and inverse theorems (i.e., 3. 1 and 5.5) of [11] cannot be easily\nperformed, due to the various differences of notations and defi nitions. In spirit however, the\nresults are very similar, relying on lower bounds for number s of vanishing moments that are\naffine functions of the smoothness degree N. A closer inspection indicates that the gap between\nthe decay orders in the direct and inverse theorems, which gr ows linearly with Nin our setting,\ncan be kept under control in [11]. It is currently unclear whe ther this difference in the results\ncan be addressed within our framework by the use of more refine d and possibly more specific\nestimates than the ones that we are currently using.\nIn other respects, our results are stronger and/or more wide ly applicable than the ones in\n[11]. To begin with, the results in the cited paper only apply to shearlets in dimension two. As40 HARTMUT F ¨UHR, MAHYA GHANDEHARI\na further important difference we note that [11] imposes the ad ditional constraint u∈L2(Rd).\nThis assumption implies that the order N(u) is bounded by d/2, which is the main reason why\nN(u) systematically features in our assumptions, but not in [11 ]. As a consequence of its focus\nonL2, [11] can work with a somewhat extended reservoir of admissi ble wavelets, which explains\nadditional assumptions in the direct theorem of [11] on the d ecay properties of ψ, that are\ntrivially fulfilled by Schwartz functions, and hence do not a ppear in our results.\nA final difference between our paper and the predecessor papers [15, 11] concerns the role of\ntheunsignedwavefront set. Whileboth[15,11]formulateth eirresultsforregulardirectedpoints,\nin both sources the definition of such a point actually refers to Fourier decay inside C∪−C, for a\nsuitably chosen cone C⊂R2(see beginning of Section 5 in [15], and [11, Definition 2.1]) . Hence\nthey effectively refer to unsigned regular directed points in the terminology of our Definition 4.1,\nwithout mentioning this fact explicitly. And this restrict ion is not coincidental; it results from\nthe fact that the shearlet groups in [15, 11] only use positiv e scalingsa>0. In the terminology\nof our paper, this corresponds precisely to replacing Hby the half-space adapted subgroup H0.\nThis choice requires a more restrictive admissibility cond ition being used in [15, 11], which is\nultimately responsible for the inability of the wavelet tra nsform to distinguish directions ξand\n−ξin the wavefront set analysis.\nWe emphasize that, in contrast to the mentioned sources, our Theorem 3.4 does apply to\nregular directed points proper, as do the results of the prec ursor paper [5].\nAcknowledgements\nThispaperwas initiated duringavisitof thesecondauthort otheDepartment ofMathematics\nat RWTH Aachen University in 2018. The authors thank the depa rtment for the support and\nhospitality duringthisvisit. Thesecondauthoralsoackno wledges supportfromNational Science\nFoundation Grant DMS-1902301 during the preparation of thi s article.\nReferences\n[1] G. S. Alberti, S. Dahlke, F. De Mari, E. De Vito, and H. F¨ uh r. Recent progress in shearlet theory: sys-\ntematic construction of shearlet dilation groups, charact erization of wavefront sets, and new embeddings.\nInFrames and other bases in abstract and function spaces , Appl. Numer. Harmon. Anal., pages 127–160.\nBirkh¨ auser/Springer, Cham, 2017.\n[2] H. G. Feichtinger and K. Gr¨ ochenig. A unified approach to atomic decompositions via integrable group\nrepresentations. In Function spaces and applications (Lund, 1986) , volume 1302 of Lecture Notes in Math. ,\npages 52–73. Springer, Berlin, 1988.\n[3] H. G. Feichtinger and K. Gr¨ ochenig. Banach spaces relat ed to integrable group representations and their\natomic decompositions. I. J. Funct. 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Kutyniok and D. Labate. Resolution of the wavefront s et using continuous shearlets. Trans. Amer. Math.\nSoc., 361(5):2719–2754, 2009.\n[16] G. Kutyniok and P. Petersen. Classification of edges usi ng compactly supported shearlets. Appl. Comput.\nHarmon. Anal. , 42(2):245–293, 2017.\nH. F¨uhr, Lehrstuhl f ¨ur Geometrie und Analysis, RWTH Aachen University, D-52056 Aachen,\nGermany\nM. Ghandehari, Department of Mathematical Sciences, Unive rsity of Delaware, Newark, DE\n19716, USA"    },    {        "title": "2402.02819v1.Striking_the_right_tone__towards_a_self_consistent_framework_for_measuring_black_hole_ringdowns.pdf",        "content": "Striking the right tone: towards a self-consistent framework for measuring black hole\nringdowns\nTeagan A. Clarke ,1, 2,∗Maximiliano Isi ,3Paul D. Lasky ,1, 2Eric Thrane ,1, 2\nMichael Boyle ,4Nils Deppe ,4, 5Lawrence E. Kidder ,4Keefe Mitman ,6\nJordan Moxon ,6Kyle C. Nelli ,6William Throwe ,4and Nils L. Vu6\n1School of Physics and Astronomy, Monash University, VIC 3800, Australia\n2OzGrav: The ARC Centre of Excellence for Gravitational-Wave Discovery, Clayton, VIC 3800, Australia\n3Center for Computational Astrophysics, Flatiron Institute, New York NY 10010, USA\n4Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA\n5Department of Physics, Cornell University, Ithaca, NY, 14853, USA\n6Theoretical Astrophysics, Walter Burke Institute for Theoretical Physics,\nCalifornia Institute of Technology, Pasadena, California 91125, USA\nThe ringdown portion of a binary black hole merger consists of a sum of modes, each containing\nan infinite number of tones that are exponentially damped sinusoids. In principle, these can be\nmeasured as gravitational-waves with observatories like LIGO/Virgo/KAGRA, however in practice\nit is unclear how many tones can be meaningfully resolved. We investigate the consistency and\nresolvability of the overtones of the quadrupolar ℓ=m= 2 mode by starting at late times when\nthe gravitational waveform is expected to be well-approximated by the ℓmn = 220 tone alone.\nWe present a Bayesian inference framework to measure the tones in numerical relativity data. We\nmeasure tones at different start times, checking for consistency: we classify a tone as stably recovered\nif and only if the 95% credible intervals for amplitude and phase at time toverlap with the credible\nintervals at all subsequent times. We test the first four overtones of the fundamental mode and find\nthat the 220 and 221 tones can be measured consistently with the inclusion of additional overtones.\nThe 222 tone measurements can be stabilised when we include the 223 tone, but only in a narrow time\nwindow, after which it is too weak to measure. The 223 tone recovery appears to be unstable, and\ndoes not become stable with the introduction of the 224 tone. Within our framework, the ringdown\nof the fundamental mode of binary black hole waveforms can be self-consistently described by three\nto four tones, which are stable from 10 Mafter the peak strain. However, additional tones are not\nobviously required because the fit amplitudes are consistent with zero. We conclude that recent\nclaims of overtone detection are not necessarily an exercise in over-fitting; the observed tones can be\nself-consistently modelled, although, not until ≈10Mfrom peak strain with our four-tone model.\nI. INTRODUCTION\nThe final stage of a binary black hole coalescence,\ncalled the ringdown, consists of a perturbed remnant\nblack hole emitting gravitational waves. In general rel-\nativity, the gravitational waves from the ringdown can\nbe decomposed using spin-weighted spheroidal harmon-\nics into quasi-normal modes [QNMs, see 1–5]. Each\nspheroidal harmonic mode is labeled with indices ℓ≥2\nand|m| ≤ℓ. There are an infinite number of tones asso-\nciated with each angular mode, each denoted with n≥0.\nThe frequency and damping time of each tone depends\nonly on the mass and spin of the remnant black hole\n(assuming zero charge) according to the no-hair theorem\n[e.g., 6].\nUnderstanding how to measure black hole tones can al-\nlow us to undertake tests of general relativity and the no-\nhair theorem for black holes [7–12]. However, the start\nof the ringdown—defined as the time when the signal\ncan be described with black hole perturbation theory—is\nambiguous [e.g., 13–16]. Understanding how early the\nperturbative prescription can be applied to the signal\n∗teagan.clarke@monash.eduis key to correctly performing tests of general relativ-\nity and the no-hair theorem. Beginning the analysis too\nearly could result in over-fitting to non-linear features in\nthe signal [e.g., 17–19]. If one waits too long to begin\nthe analysis, the strain amplitude will have decayed ex-\nponentially, making spectroscopic tests impossible given\nthe finite sensitivity of our instruments [14, 20].\nNumerical relativity simulations provide numerical so-\nlutions to the field equations [21–23], and are the state of\nthe art for investigating the QNM decomposition of the\nringdown. However, the components contained in these\nsolutions remain unclear. Black hole perturbation theory\nprovides predictions for the frequency and damping times\nof each tone, but the optimal time to fit them and how\nmany tones can be meaningfully extracted remains un-\nclear due to, e.g., non-linearities, non-orthogonal QNM\ndecomposition, and error in the simulations themselves.\nMuch effort has been devoted to determining how many\ntones measured at what time provide the best fit to the\nnumerical simulations.\nMany studies [e.g., 24–29] find that several overtones\nof the ℓm= 22 mode can be extracted at early times\nafter the merger and that they are needed to infer the\ncorrect mass and spin of the black hole. Ref. [30] sug-\ngests that the linear QNM model may be valid as earlyarXiv:2402.02819v1  [gr-qc]  5 Feb 20242\nas the strain peak time. They show that higher over-\ntones of the 22 mode up to n= 7 improve fits to simu-\nlated gravitational-waveforms from the strain peak time.\nRef. [31] fit away these seven tones from the NR data\nusing frequency domain filters, uncovering evidence for\nsecond-order effects and spherical-spheroidal mode mix-\ning.\nRefs. [32, 33] find that seven tones of the ℓ=m= 2\nas well as many tones from additional QNMs can be in-\ncluded to improve the fits. Tones up to n= 9 have also\nbeen explored for their potential to further improve fits\nat the strain peak time [34]. The apparent linearity of\nthe signal at such early times could be explained if the\nnon-linear effects are hidden behind the apparent hori-\nzon of the black hole [e.g., 35, 36], unable to reach the\nobserver at infinity. On the other hand, nonlinear effects\nlike second order QNMs, may be required to accurately\nmodel the ringdown of higher harmonics [e.g., 37, 38], al-\nthough the magnitude of such second-order contributions\nare by far subdominant for existing detections.\nWhile it seems clear that the early ringdown can be\nmodeled with a large number of overtones, there is de-\nbate as to whether the associated fits are physical and to\nwhich extent [e.g., 27, 39], i.e., are we overfitting tones\nto produce what amounts to a phenomenological model?\nAnd how does one even answer this question?\nRefs. [40, 41] construct analytic fits for stable tones as\na function of binary parameters. They point out that in\nthe purely perturbative QNM regime, tone amplitudes\nand phases should be measured consistently at different\nstart times. Ref. [41] find that the amplitudes for tones\nwithn >0 are not consistent in time and omit these from\ntheir analytic fits. However, [42] finds that analytic fits\nfor tones are biased when fitting at early times, possibly\nindicating the presence of higher tones and non-linearities\nnear the strain peak.\nRef. [19] further explore whether or not the ringdown\ntones are consistent when measured at different start\ntimes. They argue: if the overtone parameters do not\nyield consistent fits at different start times, they are\nnot physical, but rather show evidence of overfitting.\nRef. [43] finds that overtones beyond 223 are always in-\nconsistent by this metric when fitting in the time do-\nmain, while [44] use analytic modelling to caution that\nincreasing the number of tones may end up overfitting\nto a misspecified model. Ref. [19] similarly finds that\nadding tones improves the fits, even to unphysical hybrid\nwaveforms, which they point out as a sign of overfitting.\nRef. [45] attempts to resolve this issue by measuring the\nstability of the tones as they are being fitted, iteratively\nremoving inconsistent tones until only stably recovered\ntones are left. For the 22 extraction of a numerical-\nrelativity waveform, they find that five tones are stably\nrecovered, with 221 being the highest stably measured\novertone. The other robust tones consist of retrograde\nmodes or different ℓmmodes due to other angular QNMs\nmixing into the 22 spherical harmonic.\nIn parallel to the debate about the physicality of thelinear perturbation fits, a number of studies have at-\ntempted to measure black hole overtones in gravitational-\nwave data with mixed results. Assuming a QNM decom-\nposition, Refs. [17, 46] search for overtones in the late-\ntime ringdown of GW150914, only finding evidence of\nthe fundamental 220 tone. By studying at earlier times\nin the ringdown, Refs. [9, 16, 47, 48] find evidence for the\n221 tone in GW150914. Refs. [49–51] find only weak sup-\nport for the 221 tone in GW150914. However, [11, 52]\ndo not find evidence for any higher tones beyond the\nfundamental in GW150914 (although see Refs. [53, 54]\nfor further discussion on these results). Additionally,\nRefs. [55, 56] find evidence for higher harmonics in the\nsignal of GW190521 [57].\nIn this paper, we seek to arrive at a self-consistent , per-\nturbative model of the post-merger ringdown signal. We\npresent a Bayesian-inference, forward-modelling proce-\ndure as a method for extracting the ringdown QNMs from\nnumerical-relativity simulations. Using a numerical-\nrelativity waveform, we start at the end of the ringdown\nwhen the strain should be dominated by the 220 tone\nand use this to fit the n= 0 amplitude and phase.\nWe also fit a noise amplitude—a phenomenological\ntool, which we introduce in order to account for vari-\nous unmodeled physics—to cover anything that causes\nthe late-time ringdown to deviate from a pure 220 tone,\napart from some small mode-mixing contribution from\nthe 320 tone. Having established the asymptotic be-\nhaviour of the ringdown signal, we work backward, car-\nrying out Bayesian inference to see if there is support for\nadditional tones at earlier times.\nWe assess whether each tone is stably recovered by\nchecking whether the fits obtained from different times\nare consistent. In doing so, we aim to determine if\na numerical-relativity waveform can be self-consistently\nmodeled using a superposition of tones, and how many\ntones can be said to be present.\nThe remainder of this paper is organised as follows. In\nSection II we describe our model and fitting procedure.\nIn Section III we present our results before discussing\ntheir implications in Section IV.\nII. FRAMEWORK\nWe model the ringdown strain as a sum of damped\nsinusoids, writing each spin-weighted spherical harmonic\nmode of the strain as a complex-valued time-series: h=\nh+−ih×as\nhNtones\nℓm(t) =NtonesX\nn=0Aℓmne−i[ωℓmn(t−t0)+ϕℓmn],(2.1)\nwhere ωℓmn(Mf, χf) are the complex frequencies deter-\nmined by the remnant mass and spin through the no-hair\ntheorem [58, 59]. The imaginary component of ωℓmnis\nthe inverse tone damping time τℓmn. The tone ampli-\ntudes Aℓmnand phases, ϕℓmndepend on the nature of3\nthe perturbations on the black hole. The variable t0is\nthe reference time of the fit, which we take as the peak\ntime of the 22 mode strain, t= 0M. We use geometric\nunits in this study and measure tin units of the initial\nbinary mass M, which is set to unity.\nWe fit the ℓ=m= 2 mode alone and vary Ntones\nto change the number of tones in the fit. We order the\ntones in descending order of damping time: 220, 221,\n..., 22 Ntones. We fit numerical relativity data that has\nbeen decomposed into spin-weighted spherical-harmonics\n−2Yℓm. However, the basis for a perturbed black hole is\nactually described by spin-weighted spheroidal harmonics\n−2Sℓm[e.g., 2, 60]. Because of this, mode-mixing occurs\nbetween QNMs with the same mbut different ℓ. For\nthe fundamental 22 mode, the dominant source of mode-\nmixing comes from the 320 [e.g., 24], which has a higher\nfrequency than the 220 and a comparable damping time.\nTo account for this, we also include the 320 in our fits of\nthe 22 mode.\nThe data for our fit is from the Simulating eXtreme\nSpacetimes (SXS) catalogue [61, 62]. We use the Cauchy\ncharacteristic evolved (CCE) waveform simulation [63–\n65], which was generated using the SpECTRE code’s CCE\nmodule [66]. The waveform is mapped to the super-\nrest frame of the remnant black hole with Bondi-van\nder Burg-Metzner-Sachs (BMS) frame fixing [33, 67, 68].\nThis is the correct frame mapping for QNM extraction\nand eliminates unphysical frame effects. The frame fixing\nis performed using the scri python module [69–72]. We\nuse the SXS:0305 waveform, which corresponds to the\nmost likely parameters for the GW150914 LIGO-Virgo\nobservation [73]. The simulation produces a remnant\nmass Mf= 0.952Mand dimensionless spin χf= 0.692.\nWe use the highest available resolution for this study,\nwhich has a time resolution of ≈0.1Maround the strain\npeak time. We do not interpolate to a uniform time grid\nfor this study.\nWe use the MCMC sampler emcee [74] to fit a damped\nsinusoid model according to Eq. (2.1), fitting the tone\namplitude and phase as a function of time. We fix the\nfrequency and damping time of each tone as a function\nof the known asymptotic remnant mass and spin from\nthe numerical relativity simulation waveform metadata.\nIt has been pointed out [e.g., 24, 75] that the mass and\nspin of the black hole are still evolving at early times\nin the ringdown, potentially impacting QNM fits. We\ndo not consider this effect in this study. We calculate\nthe frequency and damping times using the qnmpython\npackage for Kerr black holes [76]. We use uniform priors\nfor all parameters.\nIn order to carry out Bayesian inference, we must intro-\nduce some notion of uncertainty. One path forward could\nbe to set the noise at the level of estimated numerical-\nrelativity precision. However, this numerical noise does\nnot capture the systematic uncertainty in our calcula-\ntion. Our premise is that the very end of the numerical-\nrelativity waveform should be consistent with a pure 220\ntone with small contributions from the 320 tone and late-\n0.5\n0.00.5h+data\nfit\n102\n101\n100|et/220 h+|\n0 20 40 60 80 100\nt-tpeak [M]104\n102\n100| residual |FIG. 1. Ringdown time series. The top panel shows the 22\nmode numerical-relativity data h+(t), which is the real part of\nthe 22 mode strain, in solid blue alongside a fit obtained using\njust the 220 and 320 tones predicted by perturbation theory.\nThe dashed vertical line represents the start time for this\nfit (50M). The grey band represents the artificial noise σsys.\nThe second panel shows the same timeseries, but multiplied\nbyet/τ220in order to counteract the exponential decay of the\n220 tone. The horizontal line represents our model for the\nuncertainty σsysin this regime, which in this representation\nis constant. The bottom panel shows the difference between\nthe numerical-relativity waveform and our fit as a function of\ntime.\ntime polynomial tails [e.g., 77, 78], although the latter are\nyet to be seen in numerical simulations from the SXS cat-\nalogue. We use this late-time fit as a point of reference\nfor understanding the earlier part of the waveform. If\nthe end of the waveform is inconsistent with a pure 220\ntone, this additional structure represents a systematic er-\nror within our framework.\nIn order to quantify this systematic uncertainty, we fit\nthe late waveform (starting from t= 100 M), allowing\nfor a mixture of the 220, 320 and 221 tones. We choose\n100M as the earliest time when the 221 is beginning to\nbecome unresolved for an analytic waveform test consist-4\ning of three tones with the same frequency and damping\ntimes of the 220, 320 and 221 in the numerical-relativity\nsimulation and amplitudes and phases consistent with\nour fits to the numerical-relativity data.\nIdeally one would wait longer than 100M. However, the\nnumerical-relativity error starts to increase from 100M,\nsuggesting the numerical relativity error starts to become\nmore relevant from this point.1We model the systematic\nuncertainty so that the artificial noise on the strain is\ndrawn from a Gaussian distribution with width σsys. The\nartificial noise is implemented by dividing the data by\nthe exponential term of the 220 tone, e−t/τ220, before\nperforming our fits. τ220is the damping time of the 220\ntone≈11.74M, calculated using the simulation remnant\nmass and spin in the qnmpackage [76].\nWe vary σsysuntil the posterior for the amplitude\nof the 221 tone A221is consistent with zero at one-\nsigma credibility. This enforces our requirement that\nthe late-time ringdown should not contain any contri-\nbution from overtones higher than the 220. We obtain a\nvalue of σsys= 0.06, which is ≈70 times higher than\nthe expected numerical-relativity error at t= 100 M.\nThis suggests that our ability to understand the late-\ntime behaviour of the ringdown is likely limited not by\nnumerical-relativity noise, but by theoretical uncertainty\nabout the relative contribution of various tones and/or\nnon-linearities.2The quantity σsyshas no physical mean-\ning; it is a purely phenomenological tool that we intro-\nduce in order to quantify our present theoretical uncer-\ntainty about the behavior of the late-time ringdown.\nOur decision to model σsyswith an artificial uncer-\ntainty (rather than an absolute uncertainty) is motivated\nby experimentation. We find that our fits are relatively\nconsistent in size and magnitude and that our sampling\nis better behaved when we assume an artificial uncer-\ntainty as a function of time, while they become difficult\nto compare assuming a fixed absolute uncertainty. Our\nmodel for σsysis shown in Fig. 1. The top panel shows\nthe real part of the 22 mode numerical-relativity wave-\nform h+(t) and fit. We include a grey band representing\nour choice for the artificial noise, while the middle panel\nshows h+(t) multiplied by et/τ220; the artificial noise σsys\ncan be visualised as a horizontal line at 0.06 on this panel.\nThe bottom panel is a time series showing the difference\nbetween the numerical-relativity waveform and our fit.\nWe provide more details on this approach in Appendix\nA.\n1The numerical relativity error is the difference between the two\nhighest resolution waveforms after they are mapped to the same\nBMS frame.\n2Presumably, if one could obtain a sufficiently long and accurate\nnumerical-relativity waveform, σsyswould approach the level of\nthe numerical-relativity noise.III. RESULTS\nA. Consistency of tones with time\nWe seek to determine how many and which tones are\nrequired to produce a self-consistent measurement of the\nringdown signal across time. To this end, it is useful to\nintroduce a criterion to determine if each tone is stably\nrecovered. For a tone to be recovered stably at time t0, we\nrequire that the 95% credible intervals for the amplitude\nand phase of that tone overlap with the 95% credible in-\ntervals of allsubsequent fits (corresponding to larger val-\nues of t0). Thus, e.g., the fit at 10 Mmust be consistent\nwith all the fits between 10 Mand 70 Mto be considered\nstable at 10 M. We are interested in tones that are both\nconsistent and resolved. In order for a tone to be con-\nsidered stable and resolved at time t0, we further require\nthat the 95% credible interval for the amplitude excludes\nzero. This creates three possible classifications:\n1.Stable recovery. The amplitude of the tone is\nnon-zero and the amplitude and phase of the tone\nare consistent with subsequent fits.\n2.Unstable recovery. The amplitude of the tone\nis non-zero, but the amplitude and/or phase are\ninconsistent with subsequent fits.\n3.Unresolved. The amplitude of the tone is consis-\ntent with zero.\nThis classification method is illustrated in Fig. 2. Each\npanel shows amplitude of a different tone as a function\nof start time t0. There are five tones in our fit: 220,\n320, 221, 222 and 223. This amounts to four tones of\ntheℓ=m= 2 QNM plus a contribution from the 320\ndue to mode-mixing. At regularly spaced values of t0,\nwe plot the posterior for each amplitude in teal. The\namplitudes are all measured at a reference time of t=\ntpeakwhile the fit is performed at different start times t0.\nThe background is shaded according to the stability of\nthe tone. The light-gray regions indicate that the tone is\nunstably recovered, the white regions indicate it is stably\nrecovered, and the dark-gray regions indicate that the\ntone is unresolved. To avoid clutter, we do not show the\naccompanying plots of phase, although we also require\nphase consistency in order for a tone to be classified as\nstable. We provide the corresponding plot of the phase\nposteriors in Appendix B.\nFrom the first two panels, we see that the 220 and\n320 tones are stably recovered from t0= 0M. In Ap-\npendix C we compare the 220 posteriors for the lowest\nand highest nfits, demonstrating how the 220 stabilises\nwith the inclusion of more tones. The 221 tone is initially\nunstably recovered, stabilises at 10 M, and then becomes\nunresolved at 30 M. The 222 tone is stably recovered\nwithin a narrow window around 5 −15Mwhile the 223\ntone is unresolved. With our framework, the ℓ=m= 2\nringdown signal is well described by four tones which5\nTABLE I. Summary of when each tone is stably recovered\nfor different values of Ntones. The variable tstable indicates\nthe time at which the amplitude and phase of each tone be-\ncome consistent with subsequent fits using later parts of the\nwaveform (and which the amplitude of the tone is inconsistent\nwith zero). If the tone is not stably recovered for any time,\nthis cell is marked “ −”. The variable tunresolved indicates the\ntime at which the amplitude becomes consistent with zero. If\nthe tone is never consistent with zero, this cell is marked “ −”.\nτdenotes the damping time for each tone.\nNtones tone τ[M] t stable [M] tunresolved [M]\n1 220 11.74 30 −\n2 220 11.74 30 −\n320 11.27 0 −\n3 220 11.74 15 −\n320 11.27 5 −\n221 3.88 30 45\n4 220 11.74 5 −\n320 11.27 0 −\n221 3.88 20 40\n222 2.30 − 20\n5 220 11.74 0 −\n320 11.27 0 −\n221 3.88 10 35\n222 2.30 5 15\n223 1.62 − 15\nare stable from approximately 10 M, when considering a\nmodel allowing 220, 320, 221, 222 and 223 contributions.\nWe investigate how the stability changes with the num-\nber of tones. First we investigate the 220 alone. We find\nthat the 220 is not stably recovered until approximately\n30Munless other tones are included in the fit. This is\nconsistent with the expectation that the 220 should be-\ncome dominant at late times. Adding the 320 does not\nimprove the stability of the recovery significantly. When\nwe add the 221, the 220 becomes stably recovered at\n15M. The earliest stable time for the 220 changes to 5 M\nwhen we add the 222 and 0 Mwhen we add the 223.3\nThis analysis suggests that the 22 mode ringdown sig-\nnal is well described by four tones. Adding the 224 tone\nmay slightly improve the recovery stability of the 222,\nbut does not improve that of the 223 and decays too\nquickly to be stably recovered itself. The 221 behaviour\nremains unchanged with the inclusion of the 224. Ad-\nditional tones produce fits consistent with zero ampli-\ntude. Table I summarises the recovery stability of differ-\nent tones as we vary the number of tones N.\nWe test the impact of reducing the spacing in start\ntimes t0on our results. We test the five-tone fit using\na spacing of 2 Mbetween 0 and 20 Mand find the tones\n3The fits in Section III are all carried out assuming the frequency\nand damping times for each tone predicted by perturbation the-\nory using the remnant mass and spin of the simulation. In Ap-\npendix D, we provide the results of investigations where we treat\nthe tone frequency as a free parameter.\n0.96500.96750.9700A220\n0.0020.004A320\n024A221\n01020A222\n0 20 40 60\nt0 [M]0102030A223stable\nunstable\nunresolvedFIG. 2. Posterior distributions for amplitude as a function\noft0. Each panel represents a different tone. We show the\nregimes where each fit is stably recovered (white), unstably\nrecovered (light gray) and unresolved (dark gray). We also fit\nthe phase of each tone and require the phase to also be stable\nfor that tone to be considered stable. For simplicity we only\nshow the amplitude posteriors here.We show the correspond-\ning phase posteriors in Fig. 6. The 221, 222 and 223 tones\nare unstably recovered at early times and become unresolved\nat late times, which implies that they do not significantly\nimprove the fit at late times. The 223 tone is not stably re-\ncovered at any time but may stabilise with additional tones.6\n0 20 40 60\nt0 [M]3.0\n3.2\n3.4\n3.6\n3.8\n4.0log10 ( - ln max)\nn = 0\nn = 1\nn = 2\nn = 3\nFIG. 3. Maximum natural log likelihood as a function of\nt0for different values of Ntones. (To handle the large dy-\nnamic range of likelihood values, we actually plot the log10of\n−lnLmax.) All four models include the 320 tone. At early\ntimes the fits improve as we add tones to the model before\nsaturating at around 30 M, lending further support to to the\nconclusion that the fit is dominated by the 220 past 30 M.\nbehave consistently with the more coarsely spaced test.\nHowever, the 221 recovery is slightly improved, stabilis-\ning at 6 M, rather than 10 M. The 222 recovery stabilises\nat 8M— slightly later than the previous test— and be-\ncomes unresolved at 14 M. The 223 recovery remains\nunstable at all times and becomes unresolved at 12 M.\nAs a consistency check, we assess the goodness-of-fit\nfor each of our models by comparing the maximum log\nlikelihood of each model as a function of time. Figure 3\nshows the log likelihood of each model as a function of\nstart time t0. The likelihood increases as the number of\ntones is increased, suggesting that the addition of tones\nserves to improve the fit, although the improvements are\nminimal from 30 −40Mafter the strain peak.\nB. How many tones can be resolved at the\nwaveform peak?\nRef. [30] suggests that the perturbative regime can be\napplied as early as the waveform peak ( t0= 0M), and\nthat seven tones produce the best fit at this time. We\ninvestigate the number of tones that can be resolved at\n0M, when the strain amplitude is maximal. We fit our\nnumerical-relativity data with seven tones consisting of\nthe first six (over)tones of the 22 mode and the 320 tone.\nWe find that overtones higher than 224 are not resolved\nbecause their amplitude posteriors are consistent with\nzero. This appears to be consistent with Fig. 9 in [30].\nWe also test with six tones and find that when the 225\nis the highest tone, the 225 can be resolved away from\nzero, but is not resolved with the addition of the 226. It\nbecomes consistent with zero at the next time step. Thissuggests that at least some over-fitting is likely occurring\nforn= 7 within the framework of our phenomenological\nnoise model, since those higher overtones are not resolved\nfor this fit. Figure 4 shows the posteriors for the ampli-\ntudes of each tone included in this fit.\nIV. DISCUSSION AND CONCLUSIONS\nIn this work we introduce a framework to determine\nunder what circumstances a perturbative description can\nbe self-consistently applied to a binary black hole ring-\ndown obtained from numerical relativity simulations. We\nstart near the end of the waveform where the 220 tone is\nexpected to be dominant and work backward, using the\nlate-time waveform to set the scale of systematic uncer-\ntainty in our model. We propose a criterion for stability\nto ensure that the fits obtained at different start times\nare consistent. We find that it is possible to achieve a\nself-consistent perturbative description within our frame-\nwork from ≈10Mafter peak strain. The perturbative\ndescription does not appear to be stable back to the peak\nstrain.\nThe 220 is not stably recovered without the addition\nof the 221 until late times. However, the 221 is only\nstably recovered and resolvable with this method for a\nshort time interval between 10 −35M. Including the 222\nappears to improve the fits at early times and allows the\n220 and 221 to be recovered stably at earlier times. The\n222 recovery can be briefly stabilised with the addition of\nthe 223. However the 223 is not stably recovered at any\nof the times we investigate and does not seem to stabilise\nwith the addition of the 224.\nThe 320 tone, which we add as the dominant contribu-\ntion from mode mixing, is always stable from the wave-\nform peak. Ref. [30] suggests that the perturbative model\ncan be applied at or before t0=tpeak. We show that at\nt0=tpeak = 0Mthe numerical-relativity data can be\nexplained with no more than five tones (four tones of\nthe 22 mode along with the 320 tone). Any higher tones\nincluded in the fit are consistent with zero amplitude.\nWe have shown that a superposition of quasinormal\ntones can be used to produce a self-consistent description\nof the ringdown. However, this does not prove that these\nfits are “physical” (as opposed to phenomenological) or\nthat the signal is consistent with a perturbed black hole.\nThis study brings into focus a great challenge at the heart\nof the black-hole spectroscopy program: it is unclear how\none can even answer the question of whether the pertur-\nbative description is physical or phenomenological. The\nstability of the quasinormal tone fits is a requirement\nfor their physical interpretability, but further work is re-\nquired to understand the relation of these observables to\nthe underlying nature of the spacetime and our ability to\nprobe it.7\nA220 = 0.97+0.00\n−0.00\n0.00300.0036A320A320 = 0.00+0.00\n−0.00\n4.054.20A221A221 = 4.11+0.05\n−0.03\n10.012.515.0A222A222 = 11.29+1.45\n−0.55\n2550A223A223 = 23.08+12.60\n−4.04\n060120A224A224 = 28.69+41.41\n−12.83\n080160A225A225 = 16.50+49.56\n−15.03\n0.96700.96760.9682A22003060A226\n0.0030 0.0036A3204.05 4.20\nA22110.012.515.0\nA22225 50\nA223060120\nA224080160\nA225030 60\nA226A226 = 5.93+16.37\n−3.77\nFIG. 4. The amplitude posteriors for a seven-tone fit at the strain peak ( t0= 0M). Tones up to n= 4 are measured confidently\nwith amplitudes consistent with [30], higher overtones return posteriors that support zero. This suggests that including tones\nhigher than the 224 may be over-fitting. We notice that the correlation between tones increases as nincreases. This is consistent\nwith the findings of [18], and reflects the increasing difficulty to distinguish between higher tones that decay quickly and may\nnot be stably recovered.8\nAddendum\nAs we were preparing this manuscript, we became\naware of work by [79] and [80]. Ref. [79] also attempts\nto extract tones in a self-consistent framework—by itera-\ntively subtracting away the tone with the longest damp-\ning time. They find that this technique improves the sta-\nbility of the extracted amplitude and phases for up to five\ntones of SXS:0305 . This is broadly consistent with our\nresult that only a limited number of tones can be fitted\nin a self-consistent framework, although we find this is\nlimited to four tones rather than five. They also include\nthe 320 contribution from mode-mixing and find that this\nimproves the stability of the early tones but introduces\nmore instability of tones higher than n= 2. Ref. [80]\nuses Bayesian inference to compare the performance of\nnon-linear inspiral-merger-ringdown (IMR) models with\nlinear QNM ringdown models. They find that IMR mod-\nels produce better fits with higher Bayes factors at early\ntimes. They find that overtones may be measurable in\nhigh SNR events consistent with third-generation obser-\nvatories. They caution that the instability of tones may\nmake linear QNM models less reliable than non-linear\nmodels for performing tests of general relativity.\nACKNOWLEDGMENTS\nWe thank Dana Jones and Gregorio Carullo for their\nhelpful comments on this work. We also thank Saul\nTeukolsky, Vishal Baibhav, Will Farr, Harrison Siegel\nand Ben Farr for helpful advice and discussions. This\nwork is supported through Australian Research Coun-\ncil (ARC) Centre of Excellence CE170100004, Discov-\nery Projects DP220101610 and DP230103088, and LIEF\nProject LE210100002. This work was supported in part\nby the Sherman Fairchild Foundation and NSF Grants\nNo. PHY-2011968, PHY-2011961, PHY-2309211, PHY-\n2309231, OAC-2209656 at Caltech., as well as NSF\nGrants No. PHY-2207342 and OAC-2209655 at Cornell.\nT. A. C. receives support from the Australian Govern-\nment Research Training Program. The authors are grate-\nful for for computational resources provided by the LIGO\nLaboratory computing cluster at California Institute of\nTechnology supported by National Science Foundation\nGrants PHY-0757058 and PHY-0823459, and the Ngar-\nrgu Tindebeek / OzSTAR Australian national facility at\nSwinburne University of Technology.\nAppendix A: Noise Model\nIn the body of this manuscript, we employ an “artifi-\ncial noise” model in which the systematic uncertainty is\na constant error relative to the 220 tone; see Section II.\nWe also test a “flat noise” model in which the system-\natic uncertainty is constant at all times. In Fig. 5 we\n0.9700.975A220flat noise\nartificial noise\n0 20 40 60\nt0 [M]4.8104.815220\nFIG. 5. Posterior for amplitude and phase as a function of\nstart time t0for the 220 tone. (The fits also include the 320\ntone; not shown here.) The “flat-noise” model (teal) assumes\nthe systematic error does not change with respect to t0. The\n“artificial-noise” model (orange) assumes that systematic er-\nror is a constant fraction of the 220 amplitude. The artificial-\nnoise model produces more consistent fits than the flat-noise\nmodel.\nshow how the flat-noise fits compare to the artificial-\nnoise fits. (For this analysis, we include the 220 and\n320 tones.) We find that the posteriors are more compa-\nrable in magnitude and size over time using the artificial\nnoise approach. We also find better sampler convergence,\nwhich is likely due to the SNR being comparable at early\nand late times rather than orders of magnitude differ-\nent. Thus, in order to facilitate a self-consistent overtone\nmodel, we employ the artificial noise model in the main\nbody of the manuscript. However we emphasise that this\nnoise model is entirely phenomenological and not physi-\ncally motivated.\nAppendix B: Phase posteriors\nIn Fig. 6 we plot the phase posteriors of the five-tone\nfit we present in the main body of the manuscript. The\ncorresponding posteriors for the tone amplitudes are pre-\nsented in Fig. 2.\nAppendix C: 220 stability\nFigure 7 shows the posterior distributions recovered\nfor the 220 tone with the lowest and highest dimensional9\n4.8064.808220\n3.54.04.5320\n0.02.55.0221\n0.02.55.0222\n0 20 40 60\nt0 [M]0.02.55.0223\nstable\nunstable\nunresolved\nFIG. 6. Posterior distributions for phase as a function of t0.\nEach panel represents a different tone. We show the regimes\nwhere each fit is stably recovered (white), unstably recovered\n(light gray) and unresolved (dark gray). The 221, 222 and\n223 tones are unstably recovered at early times and become\nunresolved at late times, which implies that they do not sig-\nnificantly improve the fit at late times. The 223 tone is not\nstably recovered at any time but may stabilise with additional\ntones. We show the corresponding amplitude posteriors in\nFig. 2.\nmodels included in the study. We also show the median\nvalues that are common to all posteriors in the 5-tone fit.\n0.9660.9680.970A220\n0 20 40 60\nt0 [M]4.8064.8084.810220\n220 only\n5 tonesFIG. 7. The amplitude and phase posteriors of the 220 tone\nwhen measured with a 1 and 5-tone model. The 5-tone model\nis stable at all times, while the 1-tone model becomes stable\nat 30M. The dashed horizontal lines show the median of the\noverlap at 95% confidence for the 5-tone model.\nAppendix D: Varying the frequency of overtones\nIn the main body of this manuscript, we assume that\nthe frequency and damping time of each tone are fixed to\nthe values expected from black-hole perturbation theory.\nHere, we relax this assumption so that the frequency can\nbe treated as a free parameter. We test a three-tone fit\nincluding the 220, 320 and 221 tones and allowing the\nfrequency of the 221 to vary. We use a uniform prior be-\ntween 0.5 and 0.6 for the 221 frequency. Figure 8 shows\nthe result of this fit. The predicted frequency value is\nshown as a red line in the bottom panel. The predicted\nfrequency is consistent with the posterior only for start\ntimes t0>15M. The posteriors for other parameters\ndo not change significantly when f221is treated as a free\nparameter. The fact that the numerical-relativity data\nprefer the wrong value of f221before 15 Mcould mean\nthat non-linearities are present in the data at early times\nthat are influencing the fit. Alternatively, it may be that\nmore tones are required at early times to avoid this be-\nhavior, which is what [e.g., 30] would imply.\nWe also measure the frequency of the 220 tone in a\ntwo-mode fit with the 220 and 320 tones. We find that\nthe posterior moves away from the predicted frequency\nvalue while the phase becomes unstable. The frequency\npredicted by general relativity is only recovered for three\ntime steps between 35 and 45 M. In the three- and four-\ntone models, varying the 221 frequency slightly improves\nthe stability of the first three tones. The general relativ-10\nTABLE II. The stability of our tones when we vary either\nthef220orf221as free parameters. In the two-tone fit, we\nallow f220to vary. In the three- and four-tone fits, we allow\nf221to vary.\nNtones tone τ[M] t stable [M] tunresolved [M]\n2 220 11.74 60 -\n320 11.27 5 -\n3 220 11.74 10 -\n320 11.27 0 -\n221 3.88 20 45\n4 220 11.74 5 -\n320 11.27 0 -\n221 3.88 15 35\n222 2.30 - 20ity frequency for the 221 is recovered at relatively late\ntimes: 15 Mfor a three-mode fit and 10 Mfor a four-\nmode fit. This further highlights that caution is required\nwhen fitting the tone frequencies due to the uncertain\nnumber of tones required, potential non-linearities and\nnon-optimal fits, especially when doing so to test the no-\nhair theorem as shown in [e.g., 16, 30].\nTable II summarises the stability of the recovery of\neach of the tones when we sample the frequency as well\nas the amplitude and phase. Treating f221as a free pa-\nrameter helps stabilise the fits at earlier times than when\nwe sample in the amplitude and phase alone. 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In this paper, we argue that these tasks suffer\nfrom a common “interest amnesia\" problem, and a solution exists\nto mitigate it simultaneously. We figure that long-term cues can be\nthe cornerstone since they reveal multi-interest and clarify long-\ntail interest. Inspired by the observation, we propose a novel and\nunified framework in the retrieval stage, “Trinity\", to solve interest\namnesia problem and improve multiple interest modeling tasks. We\nconstruct a real-time clustering system that enables us to project\nitems into enumerable clusters, and calculate statistical interest\nhistograms over these clusters. Based on these histograms, Trinity\nrecognizes underdelivered themes and remains stable when facing\nemerging hot topics. Trinity is more appropriate for large-scale\nindustry scenarios because of its modest computational overheads.\nIts derived retrievers have been deployed on the recommender\nsystem of Douyin, significantly improving user experience and\nretention. We believe that such practical experience can be well\ngeneralized to other scenarios.\nKEYWORDS\ninterest modeling, interest amnesia, statistical method\n1 INTRODUCTION\nLarge-scale industrial Recommender Systems (RS) confront numer-\nous requests daily, and are supposed to capture each user’s interest.\nWith the growth of daily active users (DAU) as well as their time\nspending on the platform, interest modeling becomes more and\nmore challenging and diversified. The emerging problems can be\ncategorized as (1) multi-interest, which means users may be inter-\nested in many themes and topics (In Douyin, most users have more\nthan 10interested topics based on our proposed clustering system).\n(2) long-tail interest, referring to unpopular themes that only at-\ntract niche audience. (3) long-term interest, which mainly describes\nthemes that users are willing to watch consistently, rather than\nrecent hot topics or emerging trend. For example, in Fig.3 one user\nwatched many game and talk show videos about six months ago. In\nthe last month, she was obsessed with dancing and fitness themes.\nNowadays she watches some research progress, while she has been\nfollowing economic news for years. Here all topics mentioned above\ncan be defined as multi-interest. Games/talk shows/dancing/fitness\nare long-term interest topics that may occasionally be “forgotten\"\nby our system, but still attract her if delivered. Research progress is\nMulti-interest\nLong-terminterestLong-tailinterestLong-terminterestrevealsmulti-interestMulti-interestisvaluedforlong-tailinterest\nLong-terminterestclarifieslong-tailinterestFigure 1: The interest reciprocity relationship and inspiration\nof Trinity. Multi-/long-term/long-tail interests are mutually\ndependent and reinforcing. See text for detailed examples.\na typical long-tail theme that is hardly appealing for everyone, but\ncan be enjoyed by considerable users (that is why we also focus on\nlong-tail interest). In this case, research progress has a chance to\nbecome multi-interest.\nThe key idea of this paper is that we believe the multi-/long-\ntail/long-term interests are mutually dependent and reinforcing.\nTherefore, we aim to improve these interest modeling tasks simul-\ntaneously. As illustrated in Fig.1, we have some insightful observa-\ntions. First, a user’s recent behavior sequence can be easily domi-\nnated by several hot topics, so we need long-term cues to identify\nthe rest topics of multi-interest. Then, in a completed industry sys-\ntem, popular themes are already well-delivered and multi-interest\nwe are considering essentially refers to global long-tail themes. For\nexample, in Douyin most users accept news and snapshots while\nwe have to match calligraphy and its audience. Finally long-tail\ninterest also relies on long-term cues since whether one is truly\ninterested in a certain topic can only be confirmed by long obser-\nvations. Inspired by these observations, we consider improving\nmultiple interest modeling tasks simultaneously.\nHowever, Modeling these interests is critical but quite hard: in\ncurrent system we employ online learning models to rank can-\ndidates, which are trained by streaming paradigm and naturally\ntend to newcoming samples. So they may “forget\" some topics of\ninterest due to occasional misses of the corresponding trainingarXiv:2402.02842v1  [cs.IR]  5 Feb 2024Conference’17, July 2017, Washington, DC, USA Yan and Jiang, et al.\nhardsearch+\nKey=itemIDValue=clusterIDsprimaryclusterssecondaryclustersbehaviorsequence\ntrainingphasenearestneighborsearch123451123454lossfunction\nstoreIDs\nKey=itemIDValue=clusterIDsread\n[5,8,20,…99][105,337,512,…1010]arrangestatisticalhistogramsitem\nMLPMLP\nselectclustersformulti-interestre-rankmodelglobalpopularity\nselectclustersforlong-tailinterestre-rankmodelservingphase(a)\n(b)(c)(d)ANNsearchsystemi2isearch\nre-rankmodeldumpselectseedsforlong-terminterest\nmulti-interestlong-tailinterestlong-terminterest\nFigure 2: The framework of the proposed Trinity: (a) The training procedure. (b) By projecting user’s long-term behavior\nsequence into histograms and customized strategy, multi-interest can be captured. (c) By comparing user’s behavior with global\ncluster popularity, interests on long-tail themes can be clarified. (d) Time-agnostic embeddings produced in the training phase\nare applied for an i2i search retrieval.\nsamples, leading to the “interest amnesia\" problem, and worsening\nthe delivery. Existing methods focus on modifying online learning\nframework, and introduce additional predictive heads and featuresthat record users’ past behavior. However, such methods pay ex-\ncessive computational overheads and they only consider one of\ninterest modeling tasks while neglecting their mutual relationship.\nIn this paper, we figure that the interest amnesia problem is es-\nsentially caused by online learning framework itself, and modifyingTrinity: Syncretizing Multi-/Long-tail/Long-term Interests All in One Conference’17, July 2017, Washington, DC, USA\nwatchhistorytimeline\n6monthsago1monthago\nrecently\nFigure 3: Watch history of a sampled user which reveals her\nshort-term interests like games/talk shows and long-term\ninterest (connected by dashed lines) on economics.\nit may be ineffective. On the contrary, we propose a statistic-based\nframework named “Trinity\". Trinity comprehends long-term cues\nfirst and summarizes user interest into statistical histograms, by\nwhich we can easily distinguish multi-/long-tail/long-term topics.\nThen, the underdelivered topics are selected by customized strate-\ngies to improve user experience. Since Trinity is modeled based on\nlong-term statistical information, it can be impregnable to represent\nmultiple interests while withstanding interest amnesia problem.\nTrinity is composed of three individual retrievers: Trinity-M for\nmulti-interest, Trinity-LT for long-tail interest and Trinity-L for\nlong-term interest, as demonstrated in Sec.4. To our best knowl-\nedge, Trinity is the first work that syncretizes multiple interest\nmodeling into a unified framework and innovatively introduces\nlong sequence cues (length ≥1000) into retrieval stage.\nCompared with existing methods which focus on one of interest\nmodeling tasks, Trinity has some remarkable advantages: (1) Based\non statistical histogram, each interested cluster is clearly evaluated\nand will not be forgotten or neglected. (2) Since items are assigned\nto indices exclusively, the computational overheads with interested\ntopic growth are linearly increased, which outperforms existing\nmethods. (3) Based on its clustering system, the whole delivery\nprocedure is more explainable and manageable. We could work out\nmore advanced strategies to reach better user experience. Trinity\nexcellently benefits industrial recommender system, and we have\nlaunched the mentioned retrievers in Douyin and Douyin Lite.\nIn summary, the contributions of this paper are:\n(1)We reveal the relationship among multi-/long-tail/long-term\ninterests, and demonstrate their mutual relationship.\n(2)We propose the novel and unified framework of Trinity,\nwhich improves multiple interest modeling tasks simultane-\nously.\n(3)We demonstrate detailed implementation of Trinity with\naffordable computational overheads, which advances large-\nscale industrial recommender system.\n2 RELATED WORK\nIn this section we summarize existing works into multi-interest,\nlong-tail interest and long-term interest fields, and discuss how our\nwork is different from them.Recently, one of the popular methods considering multi-interest\nmodeling is MIND [ 6]. It proposes multiple user representations\n(in this paper they are also denoted as “heads\") to retrieve candi-\ndates from multiple fields, and has been the mainstream approach\nin industry applications. ComiRec [ 1] further considers diversity\nfor improving system performance. MVKE [ 15] uses sub-expert-\nnetworks called “virtual kernels\" to capture user preference on var-\nious tasks and topics, and combine them to output comprehensive\nresults. Some works consider user interest from different aspects,\ne.g. Jiang [ 4] et al. explore multi-scale time effects on user inter-\nests. [ 11,12] introduce similar ideas into sequence recommendation\nscenarios. Xie et al. [ 14] improve negative mining and routing algo-\nrithms to address training issues in multi-interest learning. Since\nthese methods train multiple user representations to cover various\nuser preference, we summarize them as “Multi-U\" class methods.\nAll the methods above focus on modifying existing online learning\nframework, while our proposed work relies on long-term statistics.\nLong-tail items lack of user feedbacks or collaborative data so\nthere are many methods focusing on multi-modal or graph meth-\nods [ 5,7,17] to improve the representations and prediction. Huang\nand Wu [ 3] extract product profiles and similar products in a topic\nmodel. Kumar et al. [ 10] realize long-tail recommendation using\nfew shot learning, instead. Note in this paper, we are not focusing\nexactly on long-tail items. Long-tail interest mainly refers to niche\ntopics but their items are unnecessary to be obscure.\nIn the area of long-term interest, Pi et al. [ 8] directly improve long\nsequence modeling by decoupling the most resource-consuming\npart from the entire model to a separate module. They update user\nrepresentations by trigger events rather than requests and support\nthousands-length sequence. The most well-known method of cap-\nturing long-term interest is Search-based Interest Model (SIM [ 9]).\nIt searches past topic-relevant behavior sequence to supplemen-\ntarily measure user preference, but is mostly adopted in ranking\nmodels. Our approach avoids directing involving long sequence in\nmodels, which may be not well scalable. Instead, we exploit clusters\nand general interest distributions to mine long-term cues.\nOur work employs VQ-VAE [ 13] in the training procedure to as-\nsign items to indices, and construct a hierarchical clustering system\nmeanwhile. It allocates cluster embeddings and permits items to\nfind their nearest neighbors. By that we can project untold items\ninto enumerable clusters, and cope with user behavior sequence.\nIts varieties [ 2,18] have been widely used in information retrieval.\n3 THE BASIC TRINITY FRAMEWORK\nIn this section, we first demonstrate the key idea of Trinity, and\nintroduce its basic training framework. The details of improving\nmulti-/long-tail/long-term interests with Trinity framework are\norganized in Sec.4.\nAs illustrated in Fig.1, we consider multiple interests simulta-\nneously. Different from existing work, we argue that modeling\ninterests are not isolated tasks, on the contrary, these interests are\nmutually dependent and reinforcing. So taking the whole situa-\ntion into consideration greatly benefits the system. We have three\nperspectives:\n1) Long-term interest reveals multi-interest. Users’ short-term\ninterest can be dominated by emerging trends and unpredictable hotConference’17, July 2017, Washington, DC, USA Yan and Jiang, et al.\ntopics, resulting in homogeneous items in their behavior sequences.\nBy involving longer behavior sequences, we can figure out which\ntopics form multi-interest and obtain more comprehensive user\npreference.\n2) Multi-interest is valued for long-tail interest. Multi-interest\noffers a wide collection of content themes, in which hot topics\nand niche themes can be both captured. However, items of hot\ntopics (e.g. news, games) have been well estimated and delivered by\nexisting retrievals and ranking stages in a completed recommender\nsystem. When we are thinking of multi-interest, we are actually\nfocusing on some uncommon themes, such as science and poems.\n3) Long-term interest clarifies long-tail interest. Long-tail themes\nattract only a narrow audience, while many plays are just made\nby casual interest. To find out the real audience who consistently\nconsume these themes we need to backtrack long-term behavior\nrecords.\nFor now, the interrelationships of interests are coherent, and\ninspire us to model them simultaneously. But it is still hard to im-\nplement it in the online learning framework. Existing works have\nbeen struggling to solve the interest amnesia problem. However,\nonline learning models are forced to fit current samples so they\ninevitably tend to newcoming samples and popular items (occupy\nthe most impressions). If samples of some topics become infrequent,\nthe corresponding interests will be gradually forgotten. We con-\nclude this phenomenon as “interest amnesia problem\", which needs\nenormous labour to be mitigated in online learning framework.\nInstead of modifying the online learning framework, in this pa-\nper, we try to employ another instrument: statistics . Statistical\ncues extracted on long-term behavior can clearly reveal every inter-\nested topics, and remains stable when facing emerging trends. So\nit effectively mitigates interest amnesia problem and distinguishes\nmulti-/long-tail/long-term interests. Specifically, we setup an inter-\nest clustering system. Then, users’ past behaviors are projected into\nthe system, resulting in long-term behavior histograms. Finally, we\ncan implement customized delivery strategies from the histograms.\nAs long as we search long-term behaviors, long-term interest mod-\neling is improved, and we improve the others as following:\n(1) Sort the histogram descendingly by behavior count, choose\nthe top-k clusters and disperse them, ensuring that items of middle\nclusters are not dominated by head clusters. This provides compre-\nhensive results and improves multi-interest.\n(2) For each global long-tail cluster, if the current user frequently\nconsume items from this theme, we enhance its delivery. This\nmatches niche themes with their audience, and thus improves long-\ntail interest.\nConsidering the implementation, we need the following instru-\nments: a clustering system which can be constructed by embeddings\nand evenly measures long-term and recent behaviors, a storage sys-\ntem by which we can map behaviors into clusters and thus form\nusers’ histograms in real time, and a series of delivery strategies.\nIn Fig.2(a) we illustrate the basic framework of Trinity. The\ntraining phase is mainly composed of a SIM (Search-based Interest\nModel [ 9]) head and a VQ-VAE [ 13] structure. For each item, we\nfirstly search its SIM sequence, and extract embeddings for both\n(denoted as green cubes) as xand[b1,b2,...b𝑖], for𝑖≤𝑁𝑏. Be-\nhavior sequence embeddings are mean-pooled to form user-side\nrepresentation as b=Í\n𝑖b𝑖/𝑁𝑏. Meanwhile we prepare primaryand secondary learnable clusters 𝑐1\n𝑗,𝑐2\n𝑘where𝑗in1...𝐽,𝑘in1...𝐾\n(denoted as yellow cubes), and allocate their embeddings e1\n𝑗and\ne2\n𝑘.𝐽and𝐾are set to be 128 and 1024. We search the top-1 nearest\nneighbor clusters for the current item (e.g. ID=4 and 1 in Fig.2) as\nc1\nˆ𝑗andc2\nˆ𝑘and apply Binary Cross Entropy (BCE) loss functions for\n(b,x), (b,e1\nˆ𝑗) and ( b,e2\nˆ𝑘) pairs:\n𝐿=∑︁\n𝑝∑︁\nA=x,e1\nˆ𝑗,e2\nˆ𝑘𝑦𝑝log(𝜎(b𝑇\n𝑝A𝑝))+( 1−𝑦𝑝)log(1−𝜎(b𝑇\n𝑝A𝑝)),\n(1)\nwhere subscript 𝑝denotes sample index in a batch. here 𝑦=1if the\nuser finishes the video, or interacts (upvote/share/follow/comment)\nwith it, or watches it for more than 10 seconds. Moreover, we add\nsome random negative samples to obtain more discriminative em-\nbeddings. Note the assignment relationship between items and\nclusters are determined by training progress in real time by nearest\nneighbor search, and we dump such information into a key-value\nstorage (key=item ID and value=[primary cluster ID, secondary\ncluster ID]). Our implementation follows the same design in VQ-\nVAE where gradients for clusters are directly cast on items while\ncluster embeddings are updated by moving average of their belong-\ning item embeddings. Here we use a hierarchical clustering system\nof two levels. The fine-grained secondary level is designed to se-\nlect items while the coarse-grained primary level avoids repetitive\ntopics and acts as dispersion (see Sec.4.1 for detailed example).\nSince item embeddings are trained by a SIM head, which involves\nbehaviors record in a very wide time range. Recent items and very\n“old\" items are optimized simultaneously and evenly measured.\nThis affords us a collaboration-based, time-variant and temporal-\nunbiased clustering system (which can not be achieved by multi-\nmodal features). More importantly, infinite items are discretized\ninto enumerable cluster set, which makes cluster-level statistics\npossible.\n4 INTEREST MODELING IN LARGE-SCALE\nRECOMMENDER SYSTEM\nIn the section above, we have demonstrated the training procedure.\nIn this section, we introduce specific and practical strategies to solve\ninterest amnesia problem and improve multi-/long-tail/long-term\ninterest modeling respectively. Note in this paper, we only focus\non the retrieval stage. The reason is two-fold: on one hand, retriev-\ners determine candidates for ranking models. If we lose topics of\ninterest at this stage, ranking models can do nothing to improve\ninterest modeling. On the other hand, retrievers always employ\nApproximate Neighbor Nearest (ANN) search system for computa-\ntional efficiency. In accurate calculation scenarios where ranking\nmodels evaluate each candidate exhaustively, we can boost some\ntopics by dispersion to “remember\" them. However, we can hardly\nemploy the same idea because approximate calculation discards\nsome candidates at the very beginning.\n4.1 Multi-interest\nThe key idea of improving multi-interest modeling is selecting\nclusters with moderate but significant intensity in user’s statistical\nhistograms, which have already been consumed by users but areTrinity: Syncretizing Multi-/Long-tail/Long-term Interests All in One Conference’17, July 2017, Washington, DC, USA\neasy to be forgotten by models. Given behavior counts on each\ncluster, well-delivered, underdelivered and uninterested topics are\nclearly revealed.\nAs illustrated in Fig.2(b), in the serving stage, for each request\nwe first extract user’s long-term behavior sequence (length=2500).\nNote the sequence only stores item IDs that 𝑝𝑙𝑎𝑦𝑡𝑖𝑚𝑒≥10𝑠or have\nbeen finished or interacted (upvote/follow/share/comment). Then,\nthe corresponding cluster IDs are read from the key-value storage\nmentioned in Sec.3 in real time, and are organized into primary and\nsecondary histograms h1=[ℎ1\n1,ℎ1\n2,...ℎ1\n𝐽]andh2=[ℎ2\n1,ℎ2\n2,...ℎ1\n𝐾]\nwhere each bar refers to behavior count of the 𝑗/𝑘-th cluster.\nBy simply sorting the histograms descendingly, user’s inter-\nest distribution is clearly revealed. For example, if a sorted his-\ntogram is[50,20,20,4,2,0,0,0]and the corresponding indices are\n[10,33,100,91,62,21,5,83], cluster whose ID=10 is the user’s ma-\njor interest (we define “major interest\" clusters which own the\nstrongest response). However, there also exists significant response\nat index=33 and 100, which constitutes “multi-interest\" we concen-\ntrate in this paper (major interests are also part of multi-interest,\nbut they always have been well-delivered by existing retrievers).\nClusters of 91 and 62 are treated as exploring interests that may be-\ncome user’s multi-interest or global long-tail interest. Other clusters\nare ignored.\nInspired by such observation, we introduce a strategy-based re-\ntriever to improve multi-interest modeling. First, we read user’s\npast behavior cluster IDs: (𝑐1\n𝑖,𝑐2\n𝑖)𝐿\n𝑖=1where𝐿≤2500. In each pair,\nsecondary cluster ID is attached as primary cluster’s child node.\nIt results in structural data like 𝑐1\n𝑗[ℎ1\n𝑗] ⇒𝑐2\n𝑗𝑘[ℎ2\n𝑗𝑘]where “⇒\"\ndenotes one-to-many mapping relationship. Note in the training\nstage items individually search their nearest clusters so primary\nand secondary clusters are only statistically related. There may\nexist same secondary clusters under different primary clusters and\nwe need dual subscripts in ℎ2\n𝑗𝑘. In the following procedures, pri-\nmary clusters are used as dispersion method, and we only leverage\nsecondary clusters to deliver items.\nFirst, we select primary clusters where ℎ1\n𝑗is greater than 𝑇𝑝=30\nand each of whose secondary cluster has ℎ2\n𝑗𝑘greater than 𝑇𝑠=10.\nThen we randomly choose one secondary cluster for each primary\ncluster. If there are not enough clusters, we select the secondary\ncluster from primary clusters above with the greatest ℎ2\n𝑗𝑘. If there is\nstill not enough clusters, we collect secondary clusters one-by-one\nfrom largest to smallest. In each step above, we remove duplicated\nclusters. The algorithm is summarized in Alg.1, which forms an\nadditional retriever named “Trinity-M\" by adding a re-rank model\nto evaluate the candidates of selected clusters.\nSome middle clusters can be dominated by major interested\ntopics since model bias (model believes most items of cluster A\noutperforms ones of cluster B), so in an unitary two-tower retriever,\nthey cannot obtain enough impressions. However, in Trinity-M\nwe only select one secondary cluster in each primary cluster. This\nacts like dispersion and we can collect complementary cluster set,\nwhich are not forgotten. By “awakening\" these topics, users get part\nof their interest back, and feel more satisfied with our system. In\nSec.5.3 we show the complementariness and comprehensiveness of\nTrinity-M, which leads to significantly improved user experience aswell as more diversified system. Note we involve long sequence cues\nin Trinity framework by projecting behaviors into clusters, while\navoiding straight employing in online learning models (retrievers\nhave to manage billions of candidates, we can hardly use SIM [ 9]\nlike ranking models). To our best knowledge, Trinity is the first\nwork that successfully employs long sequence cues ( ≥1000) in the\nretrieval stage.\nAlgorithm 1: Trinity retriever for multi-interest\nData: Structural data 𝑐1\n𝑗[ℎ1\n𝑗]⇒𝑐2\n𝑗𝑘[ℎ2\n𝑗𝑘], threshold𝑇𝑝=30,\n𝑇𝑠=10,𝑁𝑀=10\nResult: Secondary cluster set 𝑅𝑀\n𝑅𝑀←∅;\n𝑝←0;\nforeach𝑐1\n𝑗do\nifℎ1\n𝑗≥𝑇𝑝and∀𝑐2\n𝑗𝑘,ℎ2\n𝑗𝑘≥𝑇𝑠then\nrandomly choose one secondary cluster 𝑐2\n𝑗ˆ𝑘;\nif𝑐2\n𝑗ˆ𝑘∉𝑅𝑀then\n𝑅𝑀←𝑅𝑀∪{𝑐2\n𝑗ˆ𝑘};\nend\nend\nend\nif|𝑅𝑀|≥𝑁𝑀then\n𝑅𝑀←{𝑟𝑎𝑛𝑑𝑜𝑚𝑐ℎ𝑜𝑜𝑠𝑒(𝑅𝑀,𝑁𝑀)};\nreturn𝑅𝑀;\nend\nforeach𝑐1\n𝑗do\nifℎ1\n𝑗≥𝑇𝑝then\n𝑐2\n𝑗ˆ𝑘←𝑎𝑟𝑔𝑚𝑎𝑥(ℎ2\n𝑗𝑘), where𝑐2\n𝑗𝑘∉𝑅𝑀;\n𝑅𝑀←𝑅𝑀∪{𝑐2\n𝑗ˆ𝑘};\nend\nend\nif|𝑅𝑀|≥𝑁𝑀then\n𝑅𝑀←{𝑟𝑎𝑛𝑑𝑜𝑚𝑐ℎ𝑜𝑜𝑠𝑒(𝑅𝑀,𝑁𝑀)};\nreturn𝑅𝑀;\nend\nsort all𝑐2byℎ2from largest to smallest;\nwhile|𝑅𝑀|<𝑁𝑀do\nif𝑐2𝑝∉𝑅𝑀then\n𝑅𝑀←𝑅𝑀∪{𝑐2𝑝};\n𝑝←𝑝+1;\nend\nend\nreturn𝑅𝑀;\nWe compare advantages and disadvantages of Trinity-M with the\n“Multi-U\" class methods in Table.1. Here we use “Multi-U\" to sum-\nmarize most existing works that train several user representations\n(heads) and search multiple candidates results for capturing multi-\ninterest. The most well-known Multi-U methods are MIND [ 6],\nComiRec [1] and MVKE [15].Conference’17, July 2017, Washington, DC, USA Yan and Jiang, et al.\nAspect Multi-U Trinity-M\nFlexibility ○\nEfficiencydeteriorates with consistently keeps\nmany heads linear increasing\nExplainabilitysemantically semantically\nambiguous meaningful\nExtensibilityhas not has been achieved\nbeen validated in this paper\nFungibility feasible only complementary\nTable 1: Comparison between Multi-U class methods and\nTrinity-M.\n(1)Flexibility . It measures whether we can adjust our inter-\nested topics flexibly. In Trinity-M the strategy is customized\nso extending considered topics, skipping head clusters and\nimplementing topic sampling would be easy. However, Multi-\nU methods provide no effortless solution without training\nnew models.\n(2)Efficiency . The efficiency describes how serving cost in-\ncreases with more considered topics. For Multi-U methods,\nwhen we introduce an additional head, we execute once more\nu2i search. However, as illustrated in Fig.4, items can be re-\ndundantly searched by multiple heads. So the more heads we\nhave, the less unique candidates we can retrieved by adding\nheads. On the contrary, in Trinity-M items are assigned to\nclusters deterministically and exclusively. When we add ad-\nditional clusters, they only bring supplementary candidates.\nFinding more interested clusters consistently costs linearly\nadditional overheads.\n(3)Explainability . The explainability describes semantic mean-\ning of specific heads/clusters. In Multi-U methods heads are\nrestrained to learn different samples, however, we can hardly\ncontrol the training procedure or appoint specific topics to\nbe optimized. In trinity-M, clusters themselves are semanti-\ncally meaningful (see Sec.5.1), and we know what topics we\nare considering.\n(4)Extensibility . The extensibility measures whether the method\ncan be extended to other scenarios. Multi-U methods propose\nseldom solution for long-tail or long-term interest modeling\ntasks. While Trinity simultaneously improves these interest\nmodeling problems, as demonstrated below.\n(5)Fungibility . The fungibility refers to whether we can use the\nproposed method to replace existing retrievers. For this as-\npect, Multi-U methods are alternative because their multiple\nheads easily cover candidates of a single u2i model. However,\nsince Trinity is modeled based on long-term statistic, emerg-\ning items/topics are hard to be prominent in time. So we use\nTrinity-M as an additional retriever and only consider its\nsupplementary function.\n4.2 Long-tail Interest\nFor long-tail interest, our idea is to compare personal interest distri-\nbution against global interest distribution based on clusters. First,\nwe need to automatically recognize/define global long-tail clusters.\nuserrepresentationitemrepresentationMulti-UmethodsTrinity\nclustercentroidFigure 4: In Multi-U methods, each user representation (yel-\nlow dashed circle) searches candidates individually, so some\nitems may be retrieved (solid hollow circles) more than once\n(deeper gray refers to more retrievals). With the head count\nincreasing, redundant computational overheads grow heav-\nily. However, in Trinity items are assigned exclusively, each\nitem can be retrieved by at most once.\nThen, we check if there exists significant response in user’s statisti-\ncal histogram on these clusters. Clusters meeting this criteria are\ncollected to form an additional retriever.\nInspired by [ 16], we employ the similar streaming frequency\nestimation, but apply it on clusters rather than items. Specifically,\ncluster𝑐𝑘(for long-tail interest we only consider secondary clusters)\nis mapped by a hash function H, and when it occurs in the training\nstreaming, its occurrence interval is calculated by\n𝐵[H(𝑐𝑘)]←( 1−𝛼)·𝐵[H(𝑐𝑘)]+𝛼·(𝑡−𝐴[H(𝑐𝑘)]),(2)\nwhere𝐴[H(𝑐𝑘)]records the last occurrence timestamp of the\ncluster.𝑡−𝐴[H(𝑐𝑘)]denotes the latest occurrence interval and\n𝐵[H(𝑐)]is its global moving average. This is illustrated in Fig.2(c).\nIn general, we can recognize global long-tail themes from clusters\nof large occurrence intervals. However, large occurrence interval\nclusters are not only composed by long-tail themes. There are also\nsome “juxtapose\" clusters which represent hot topics, but seize few\nitems because of training occasionality. We remove clusters that\nhave less than 𝑇𝑖=3items and effectively avoid juxtapose clusters.\nIn our implementation, we define top 600 clusters (cover about 22%\nimpressions) ranked by occurrence intervals as long-tail clusters\nwithout juxtapose clusters.\nAfter defining long-tail clusters, we check user’s statistical his-\ntogram. If a cluster has more than 𝑇𝑙=3behavior count, it is\nregarded as “convinced\" response on long-tail interest, and is in-\nvolved. We introduce a sampler to select 𝑁𝐿𝑇=20clusters where\neach cluster has probability of\n𝑃𝑟(𝑐𝑘)=(𝛽+ℎ𝑘)𝛼\nÍ𝐾\n𝑞=1(𝛽+ℎ𝑞)𝛼(3)\nto be sampled. The hyper parameter 𝛼adjusts the distribution (for\nexample,𝛼=0forms uniform distribution) and we use 𝛽as a\nsmooth term. In our implementation we set 𝛼=0.75and𝛽=0.1.Trinity: Syncretizing Multi-/Long-tail/Long-term Interests All in One Conference’17, July 2017, Washington, DC, USA\nThe designed sampler outperforms a uniformly random sampler\non user experience as well as diversity since items belonging to\ntop clusters are more probable to be consumed (see Sec.5.2). Note\nduplicated clusters with Trinity-M have been removed at first.\nThe algorithm above is summarized in Alg.2. After collecting\nsecondary clusters, we feed their items into a re-rank model to\nobtain retrieval results. This retriever is denoted as Trinity-LT.\nAlgorithm 2: Trinity retriever for long-tail interest\nData:𝑐2\n𝑘[ℎ2\n𝑘], global occurrence intervals 𝐵[H(𝑐)],\nthreshold𝑇𝑖=3,𝑇𝑙=3,𝑁𝐶=600,𝑁𝐿𝑇=20\nResult: Secondary cluster set 𝑅𝐿𝑇\n𝑅𝐿𝑇←∅;\n𝐶←∅;\nforeach𝑐𝑘do\nif𝑐𝑘has equal or more than 𝑇𝑖items then\n𝐶←𝐶∪{𝑐𝑘};\nend\nend\nsort clusters 𝑐𝑘∈𝐶by𝐵[H(𝑐𝑘)], from largest to smallest;\nchoose top𝑁𝐶clusters as𝐶;\nforeach𝑐2\n𝑘do\nif𝑐2\n𝑘∈𝐶andℎ2\n𝑘≥𝑇𝑙and𝑐2\n𝑘∉𝑅𝐿𝑇then\n𝑅𝐿𝑇←𝑅𝐿𝑇∪{𝑐2\n𝑘};\nend\nend\n𝑅𝐿𝑇←{𝑠𝑎𝑚𝑝𝑙𝑒𝑟(𝑅𝐿𝑇,𝑁𝐿𝑇)};\nreturn𝑅𝐿𝑇;\n4.3 Long-term Interest\nAs illustrated in Sec.3, SIM head optimizes user’s long past behavior\nitems and current items simultaneously. So we believe the generated\nitem embeddings represent long-term cues, and construct an i2i\nretriever to capture user’s long-term interest.\nAs illustrated in Fig.2(d), we train a pre-rank model to select\nseeds from user’s past behavior sequence. To capture compre-\nhensive interest, any 𝑝𝑙𝑎𝑦𝑡𝑖𝑚𝑒≥10𝑠, finish or interaction (up-\nvote/share/comment/follow) feedback is treated as positive label.\nThe model follows conventional two-tower architecture, and in-\nvolves random negative samples.\nAfter ranking user’s behavior items (length ≤2500), we project\nthem into Trinity clustering system, and disperse by limiting that\ncandidates of the same cluster can not exceed 𝑇𝑐. Then we randomly\nselect𝑁𝐿from top𝑁𝑠ranked items as seeds, and search items with\nsimilarity measured by Trinity embeddings. Similar with Trinity-M\nand Trinity-LT, searched items are fed into the re-rank model, and\nform the retriever of Trinity-L. Note long-term cues have already\nbeen leveraged by Trinity-M and Trinity-LT, and we introduce\nTrinity-L as a lightweighted and supplementary extension.\n4.4 Implement Details\nThe proposed three retrievers select clusters and aggregate their\nbelonging items, while remaining 10K-100K candidates, which isstill too large for the following ranking stages. So we employ re-\nrank models to shrink candidates to about 1000. The re-rank models\nfollow two-tower architecture, but precisely predict ranks without\nANN in the serving phase.\nThe target of re-rank models is to predict video play time. Specif-\nically, samples that are watched within 2 seconds are negative. For\nthe rest positive samples, we use their play time (clipped to be at\nmost 5 minutes) to weight the in-batch Softmax loss [ 16]. In fact\nsuch a model has been deployed as a major retriever called “stay-\ntime retriever\", and we just reuse it as re-rank models for additional\nretrievers.\n5 EXPERIMENTS\nIn this section, we present the performance of Trinity. First, we\nshow visualized analysis on its clustering system, demonstrate\nsome interesting properties of the proposed model. Then, we show\nonline performance of the additional retrievers on our large-scale\nindustry scenarios. We also analyze them in detail to figure how\nthey improve interest modeling tasks as well as user experience.\n5.1 Clustering System of Trinity\nIn Fig.5 we visualize some items belonging to Trinity clusters. Items\nof the same row share the same secondary cluster ID, and we use\n(cluster ID, item ID) to refer them. The most important difference be-\ntween Trinity clustering system and an intuitive clustering system\nfrom human perspective (denoted as label system), is its clustering\nmanners. For example, in the first row, all items are about parent-\nchild relationship rather than parenting, which is uncommon from\nhuman perspective. These items come from multiple tags such as\nnews and laws, even the last example (A,c) talks about imaginary\nparent-child relationship that actually doesn’t happen. The similar\nphenomenon can be observed in the cluster of B, where items are\nall about handsome men despite he is a real person or an animation\nmodel. The theme of a cluster focuses on an unconventional topic,\nand covers multiple tag aspects.\nThe third row shows a much common taxonomy example where\nthree items are all about education, and each of them explains\nknowledge of a sub-topic (career, economics and parenting). The\nmost surprising example comes from the last row. The items show\nscenic spots and foods in three places, while these places are in\nthe same province. So we can conclude another property of Trinity\nclustering: its taxonomy is varying with general topics. In the ed-\nucation topic, it categorizes items by subjects, while in the travel\ntopic, it divides items into geographical bins.\n5.2 Online Experiments\nWe conduct online A/B experiments and show the results of large-\nscale industrial recommender system on Douyin and Douyin Lite.\nAs mentioned above, we simply launch three additional retrievers:\nTrinity-M Trinity-LT and Trinity-L onto the whole system, and\nreport the performance.\nMetrics . The ultimate goal of recommendation algorithm is im-\nproving user experience, which is widely measured by Daily Active\nUsers (DAU). However, DAU can hardly be observed in A/B experi-\nments so we need alternative metrics. In this paper, we use AverageConference’17, July 2017, Washington, DC, USA Yan and Jiang, et al.\n爸爸扣碗给女儿剪刘海，女儿崩溃大哭。/Daddycoveredabowltocuthisdaughter’sbangsandshebrokedownandcried.新闻/News90后父母培养2岁孩子学做饭。/Parentsborninthe1990strainedtheir2-year-oldchildrentolearntocook.新闻/News强迫别人叫自己爸爸违法吗？/Isitillegaltoforcesomeonetocallyoudaddy?法律/Laws心碎到奈何。/Heartbrokentonoavail.电视剧/TVdrama原来只有关系好的人才让人拍肩膀啊。/Onlyclosepeopleareallowedtobepattedontheshoulder.动画/Animation就好像真的成婚了一样。/It’slikewearetrulygettingmarried.电视剧/TVdrama人生最大的奢侈是浪费时间。/Thegreatestluxuryinlifeisawasteoftime.教育-职业/Education-Career为什么有钱人都喜欢贷款？/Whydorichpeoplelikeloans?教育-经济/Education-Economics一岁之内的哭闹不要抱。/Don‘tcompromisetogiveahugforbabiesunder1yearold.教育-育儿/Education-Parenting潮州古城欢迎您。/WelcometoancientcityofChaozhou.旅游/Travel汕尾八大景区。/8majorsceneryinShanwei.旅游/Travel每次来都爱吃的潮汕甜面。/ChaoshansweetnoodlesthatIlovetoeat,everytimeIcomehere.美食/FoodsabcABCD\nFigure 5: The visualization of items belonging to Trinity clusters. Items of the same row share the same secondary cluster ID.\nDouyin Douyin LiteRetrieversWatch Time AAD AAH AT Watch Time AAD AAH AT\nMIND [6] (6 heads*) +0.093% +0.009% +0.033% -0.472% +0.176% - +0.051% -0.270%\nTrinity-M +0.118% +0.008% +0.046% +0.153% +0.178% +0.018% +0.078% +0.038%\nTrinity-LT +0.069% - +0.019% +0.546% +0.060% - - -\nTrinity-LT (uniform sampling) - - - +0.154% - - - -\nTrinity-L +0.051% +0.009% +0.020% -0.080% +0.069% +0.014% +0.029% -0.081%\nTable 2: Results of online A/B experiments, measured by Watch Time, Average Active Days (AAD), Average Active Hours (AAH)\nand Average Tags (AT).\nActive Days (AAD) as a surrogate metric to describe user experi-\nence. When a user first arrives the platform, he/she is randomly\ngrouped into control or experimental group. Then we count his/heractive days during the experimental period, and calculate the av-\nerage difference between the two groups. For most applications,\nDAU is the most critical metric. However, in Douyin, there are\nmany users who log in almost everyday, so AAD can not describeTrinity: Syncretizing Multi-/Long-tail/Long-term Interests All in One Conference’17, July 2017, Washington, DC, USA\ntheir experience. To tackle this problem, we propose an additional\nmetric Average Active Hours (AAH) which calculates users’ active\nhours similar with AAD. For multi-interest and long-tail interest\nmodeling, diversity is expected to be improved. So we also involve\nAverage Tags (AT) which counts how many tags are consumed over\nall items. In addition, we also employ Watch Time as an auxiliary\nmetric.\nIn Table.2 we show the online performance, where only statisti-\ncally significant metrics are listed. We have exhaustedly verified\nMulti-U methods, none of them obtains significant improvement\nwithin acceptable computational overheads. We report the results\nof MIND [ 6] here with 6 serving heads and omit others for con-\ncise. Such cost far exceeds our budget and the method is infeasible.\nOne of unexpected observations is that MIND diminishes diversity,\nwhich implies all these heads repetitively retrieve some hot topics.\nOn the contrary, Trinity-M significantly improve AAD and AAH\non Douyin and Douyin Lite under limited overheads, and is success-\nfully deployed. We provide an ablation study on Trinity-LT where\nuniform sampling means using a uniformly distributed sampler to\nselect clusters. The group of uniform sampler has no positive impact\non user experience and less diversity improvement, which verifies\nthat clusters with larger response of a user are more likely to be\nconsumed. Trinity-L also significantly improves AAD and AAH\non Douyin and Douyin Lite. One interesting observation is that\nTrinity-L contributes much less AAH but reaches competitive AAD\nimprovement compared with Trinity-M. This implies that multi-\ninterest modeling mainly benefits high-active users (active more\nthan 20 days in the past month). It consists with our expectation\nsince high-active users are more likely to discover new interested\ntopics.\n5.3 The Comprehensiveness and\nComplementariness of Trinity-M\nIn Fig.6 we visualize topics that can be retrieved by stay-time re-\ntriever, MIND and Trinity-M for 2 users. Here we show top-ranked\nresults since only these candidates can be fed into the following\nranking stages. Note MIND and Trinity-M can retrieve their left\ntopics, but we omit them for concise visualization.\nFor the first user, stay-time retriever only provides popular topics\nsuch as sports and games. MIND retrieves several additional topics\nlike TV drama and travel. Here calligraphy is a pretty positive case\nsince it is a long-tail topic. Trinity-M provides more topics (history,\nastronomy and agriculture) in addition. Trinity-M’s retrieved clus-\nters are not only more comprehensive, but also tend to long-tail\ntopics.\nThe example of the second user is much different, however. This\nuser’s interest mainly comes from popular topics such as games,\nparenting and news. Surprisingly, even hot topics can not be entirely\nretrieved by the unitary stay-time retriever. And Trinity-M provides\ncomplementary results of news and animation. Note the “Games\" in\nthe last column is different from the one in the first column, since\nit is a sub-category describing console games. In Douyin, most\nusers have more than 10 interested secondary clusters measured by\nTrinity’s clustering system, so only Trinity-M meets our demand\nof modeling multi-interest.\nStay-timeretrieverMINDTrinity-MSportsGamesFoodsTVdramaCalligraphyTravelHistoryAgricultureAstronomy\nGamesGamesAnimationNewsSportsSnapshotsMusicJoking\nParenting\nFigure 6: Top retrieved results of multiple retrievers. Two\nusers are divided by boxes, and we only show supplementary\ntopics of MIND and Trinity-M.\nThemes Finance Laws Photography Career\nVV +0.904% +0.478% +0.389% +0.230\nThemes Movie Games Variety TV drama\nVV -0.520% -0.349% -0.349% -0.322%\nTable 3: Impression distribution change caused by Trinity-\nLT.\n5.4 What contents benefit from Trinity-LT?\nThe motivation behind Trinity-LT is improving global long-tail\nthemes delivery, and in our implementation we use taxonomy pro-\nvided by Trinity itself. In this part we verify long-tail themes deliv-\nery based on label system instead, which mainly comes from human\nperspective. From our practical experience, when items estimation\nis improved, their long-term impressions obtain significant increase,\nvice versa. So we can simply check impression distribution change\ncaused by Trinity-LT to validate its influence on long-tail interest\nmodeling.\nIn Table.3 we show the impression distribution change men-\ntioned above where VV (Video View) refers to impressions of the\nspecific theme. Some typical boosted themes are finance (+0.904%),\nlaws (+0.478%), photography (+0.389%) and career (+0.230%). In our\nplatform they are all recognized as niche topics. Note the results are\ncalculated across the whole market, in Trinity-LT these themes areConference’17, July 2017, Washington, DC, USA Yan and Jiang, et al.\nboosted even more significantly. On the contrary, hot topics occupy\nless impressions thus Trinity-LT matches niche themes with their\naudience and generally deboosts popular themes, which meets our\nexpectation.\n5.5 Seed Distribution of Trinity-L\nHere we analyze the properties of Trinity-L. Since each item in\nuser’s 2500-length past behavior sequence can probably be selected\nas a seed, we expect that Trinity-L will retrieve earlier seeds. Com-\npared with the existing i2i retriever whose seeds are selected by\nrules on recent behavior and employs conventional online learn-\ning model embeddings, Trinity-L has more seeds of 7-15 days ago\n(+67%) and 15-30 days ago (+28%). In the experiment Trinity-L deliv-\ners the earliest seeds for mid-active users (active more than 10 but\nless than 20 days in the past 30 days) and obtain significant AAD\nimprovement on this group. It suggests that the refined long-term\ninterest modeling leads to better user experience, and Trinity-L\nindeed captures long-term cues to improve user experience. Note\n\"refined\" here needs Trinity embeddings since without them we\ncannot obtain significant AAD improvement even on long-term\nseeds.\n6 CONCLUSION\nIn this paper we focus on multi-/long-tail/long-term interest mod-\neling at retrieval stage. We figure that online learning framework\nhardly solves the widely existing interest amnesia problem, and pro-\npose a unified and statistic-based framework to tackle the problem:\nTrinity. Based on a collaborative and time-variant clustering system,\nTrinity derives three individual retrievers (Trinity-M, Trinity-LT\nand Trinity-L) for the corresponding interest aspects. Trinity-M\nand Trinity-LT support customized strategies and retrieve targeted\ntopics. While Trinity-L recovers some interested topics that have\nbeen forgotten. All of the retrievers have been deployed on Douyin\nand Douyin Lite, and have significantly improved user experience.\nWe believe long-term statistic directly mitigates interest amnesia,\nand is tailored for large-scale industry systems.\nREFERENCES\n[1]Yukuo Cen, Jianwei Zhang, Xu Zou, Chang Zhou, Hongxia Yang, and Jie Tang.\n2020. Controllable Multi-Interest Framework for Recommendation. In Proceedings\nof the 26th ACM SIGKDD International Conference on Knowledge Discovery &\nData Mining (Virtual Event, CA, USA) (KDD ’20) . Association for Computing\nMachinery, New York, NY, USA, 2942–2951. https://doi.org/10.1145/3394486.\n3403344\n[2]Rong Huang, Danfeng Zhang, Weixue Lu, Han Li, Meng Wang, Daiting Shi,\nJun Fan, Zhicong Cheng, Simiu Gu, and Dawei Yin. 2023. Learning Discrete\nDocument Representations in Web Search. In Proceedings of the 29th ACM SIGKDD\nConference on Knowledge Discovery and Data Mining (, Long Beach, CA, USA,)\n(KDD ’23) . 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Practice\non Long Sequential User Behavior Modeling for Click-Through Rate Prediction.\nInProceedings of the 25th ACM SIGKDD International Conference on Knowledge\nDiscovery & Data Mining (Anchorage, AK, USA) (KDD ’19) . Association for\nComputing Machinery, New York, NY, USA, 2671–2679. https://doi.org/10.1145/\n3292500.3330666\n[9]Qi Pi, Xiaoqiang Zhu, Guorui Zhou, Yujing Zhang, Zhe Wang, Lejian Ren, Ying\nFan, and Kun Gai. 2020. Search-based User Interest Modeling with Lifelong Se-\nquential Behavior Data for Click-Through Rate Prediction. CoRR abs/2006.05639\n(2020). arXiv:2006.05639 https://arxiv.org/abs/2006.05639\n[10] Yaman Singla, Dhruva Sahrawat, Shubham Maheshwari, Debanjan Mahata,\nAmanda Stent, Yifang Yin, Rajiv Shah, and Roger Zimmermann. 2020. Harnessing\nGANs for Zero-Shot Learning of New Classes in Visual Speech Recognition. Pro-\nceedings of the AAAI Conference on Artificial Intelligence 34 (04 2020), 2645–2652.\nhttps://doi.org/10.1609/aaai.v34i03.5649\n[11] Qiaoyu Tan, Jianwei Zhang, Ninghao Liu, Xiao Huang, Hongxia Yang, Jingren\nZhou, and Xia Hu. 2021. Dynamic Memory based Attention Network for Sequen-\ntial Recommendation. Proceedings of the AAAI Conference on Artificial Intelligence\n35, 5 (May 2021), 4384–4392. https://doi.org/10.1609/aaai.v35i5.16564\n[12] Qiaoyu Tan, Jianwei Zhang, Jiangchao Yao, Ninghao Liu, Jingren Zhou, Hongxia\nYang, and Xia Hu. 2021. Sparse-Interest Network for Sequential Recommendation.\nInProceedings of the 14th ACM International Conference on Web Search and Data\nMining (Virtual Event, Israel) (WSDM ’21) . Association for Computing Machinery,\nNew York, NY, USA, 598–606. https://doi.org/10.1145/3437963.3441811\n[13] Aäron van den Oord, Oriol Vinyals, and Koray Kavukcuoglu. 2017. Neural\nDiscrete Representation Learning. CoRR abs/1711.00937 (2017). arXiv:1711.00937\nhttp://arxiv.org/abs/1711.00937\n[14] Yueqi Xie, Jingqi Gao, Peilin Zhou, Qichen Ye, Yining Hua, Jae Boum Kim,\nFangzhao Wu, and Sunghun Kim. 2023. Rethinking Multi-Interest Learn-\ning for Candidate Matching in Recommender Systems. In Proceedings of the\n17th ACM Conference on Recommender Systems (Singapore, Singapore) (Rec-\nSys ’23) . Association for Computing Machinery, New York, NY, USA, 283–293.\nhttps://doi.org/10.1145/3604915.3608766\n[15] Zhenhui Xu, Meng Zhao, Liqun Liu, Lei Xiao, Xiaopeng Zhang, and Bifeng\nZhang. 2022. Mixture of Virtual-Kernel Experts for Multi-Objective User Profile\nModeling. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge\nDiscovery and Data Mining (Washington DC, USA) (KDD ’22) . Association for\nComputing Machinery, New York, NY, USA, 4257–4267. https://doi.org/10.1145/\n3534678.3539062\n[16] Xinyang Yi, Ji Yang, Lichan Hong, Derek Zhiyuan Cheng, Lukasz Heldt, Adi-\ntee Ajit Kumthekar, Zhe Zhao, Li Wei, and Ed Chi (Eds.). 2019. Sampling-Bias-\nCorrected Neural Modeling for Large Corpus Item Recommendations .\n[17] Hongzhi Yin, Bin Cui, Jing Li, Junjie Yao, and Chen Chen. 2012. Challenging\nthe Long Tail Recommendation. Proc. VLDB Endow. 5, 9 (may 2012), 896–907.\nhttps://doi.org/10.14778/2311906.2311916\n[18] Jingtao Zhan, Jiaxin Mao, Yiqun Liu, Jiafeng Guo, Min Zhang, and Shaoping Ma.\n2022. Learning Discrete Representations via Constrained Clustering for Effective\nand Efficient Dense Retrieval. In Proceedings of the Fifteenth ACM International\nConference on Web Search and Data Mining (Virtual Event, AZ, USA) (WSDM\n’22). Association for Computing Machinery, New York, NY, USA, 1328–1336.\nhttps://doi.org/10.1145/3488560.3498443"    },    {        "title": "2402.02844v1.Comparing_Knowledge_Sources_for_Open_Domain_Scientific_Claim_Verification.pdf",        "content": "Comparing Knowledge Sources for Open-Domain\nScientific Claim Verification\nJuraj Vladika and Florian Matthes\nDepartment of Computer Science\nTechnical University of Munich\nGarching, Germany\n{juraj.vladika, matthes}@tum.de\nAbstract\nThe increasing rate at which scientific knowl-\nedge is discovered and health claims shared\nonline has highlighted the importance of devel-\noping efficient fact-checking systems for scien-\ntific claims. The usual setting for this task in\nthe literature assumes that the documents con-\ntaining the evidence for claims are already pro-\nvided and annotated or contained in a limited\ncorpus. This renders the systems unrealistic for\nreal-world settings where knowledge sources\nwith potentially millions of documents need to\nbe queried to find relevant evidence. In this pa-\nper, we perform an array of experiments to test\nthe performance of open-domain claim verifi-\ncation systems. We test the final verdict predic-\ntion of systems on four datasets of biomedical\nand health claims in different settings. While\nkeeping the pipeline’s evidence selection and\nverdict prediction parts constant, document re-\ntrieval is performed over three common knowl-\nedge sources (PubMed, Wikipedia, Google)\nand using two different information retrieval\ntechniques. We show that PubMed works bet-\nter with specialized biomedical claims, while\nWikipedia is more suited for everyday health\nconcerns. Likewise, BM25 excels in retrieval\nprecision, while semantic search in recall of rel-\nevant evidence. We discuss the results, outline\nfrequent retrieval patterns and challenges, and\nprovide promising future directions.\n1 Introduction\nThe fast promulgation of knowledge in the digital\nworld has made keeping track of information trust-\nworthiness a challenging endeavor. In particular,\nscience and health have become popular talking\npoints and brought with it an abundance of medi-\ncal advice that permeate online resources (Swire-\nThompson et al., 2020). A report by the Pew Re-\nsearch Center (Fox and Duggan, 2013) found that\nover one-third of American adults have searched\nthe Internet for medical conditions and asked it\nFigure 1: The experimental setup of the study: scientific\nclaims are passed through a fixed pipeline using three\ndifferent knowledge sources, resulting in different final\nprediction performance (as measured by F1 score).\nmedical questions before going to a medical pro-\nfessional. The sought information ranged from\nself-diagnosis to finding medications.\nAutomated solutions for claim verification based\non Natural Language Processing (NLP) have\nemerged as a potential aid to help with bringing\nlight into the information overload (Nakov et al.,\n2021). While most work in the automated fact-\nchecking domain is concerned with claims related\nto politics, society, rumors, and general misinfor-\nmation, there has been an increasing interest in\nfact-checking of scientific and biomedical claims\n(Kotonya and Toni, 2020; Wright et al., 2022b).arXiv:2402.02844v1  [cs.CL]  5 Feb 2024The task of automated claim verification consists of\nretrieving evidence for a claim being checked and\nthen predicting a veracity label based on the discov-\nered evidence. The most common setting for this\ntask either already provides the source document\nthat will contain evidence for the claim or works\nover a limited, manually constructed collection of\ndocuments (Saakyan et al., 2021). While this is\nan important step in developing models capable of\nreading comprehension and detecting which spans\nprovide evidence in a given context, this is not a\nrealistic setting for automated claim verification\nsystems deployed in the real world. In such a sce-\nnario, the documents containing evidence are not\nknown and knowledge bases containing them can\npossibly contain millions of documents. Moreover,\nwith the rise of medical assistants and conversa-\ntional agents in healthcare, many users are turning\nto these systems as a source of health-related infor-\nmation and medical support (Valizadeh and Parde,\n2022).\nTo address these research gaps, we perform an\narray of experiments that test the performance of\nNLP systems for claim verification in the open do-\nmain. In the experiments, we keep the parts of\nthe verification pipeline concerned with evidence\nsentence selection and verdict prediction fixed,\nand vary the knowledge source being used and\ninformation retrieval techniques being deployed to\nquery the databases. Since the final goal of fact-\nverification systems is to provide a verdict on the\ncorrectness of a claim, we measure the usefulness\nof knowledge sources and retrieval techniques by\nlooking at verdict prediction scores. For this pur-\npose, we leverage four English datasets of biomedi-\ncal and health claims that contain gold annotations\nstemming from domain experts. We use the verac-\nity labels of claims in datasets as ground truth.\nWe opt for three large-scale knowledge sources:\nPubMed, the cardinal collection of biomedical re-\nsearch publications; Wikipedia, as the largest pub-\nlicly curated encyclopedia of human knowledge;\nand Google search results (representing \"the whole\nweb\"), which is a straightforward and intuitive way\nhow users seek information. Finally, we perform a\nqualitative analysis of retrieved evidence for some\ninteresting example claims, present the insights\nfrom results, and provide future directions.\nOur contributions are as follows:\n1.We test the claim verdict prediction perfor-\nmance of a fixed fact-verification systemon four biomedical fact-checking datasets\nby using three different knowledge sources\n(PubMed, Wikipedia, Google Search).\n2.We compare the final label prediction perfor-\nmance by retrieving evidence using different\ntechniques (sparse retrieval with BM25 and\nsemantic search with dense vectors).\n3.We provide a qualitative error analysis of re-\ntrieved evidence for different types of claims\nand provide insights and future directions for\nopen-domain claim verification.\nWe make the data and code of the experiments\navailable in a GitHub repository.1\n2 Foundations\n2.1 Pipeline for Automated Claim Verification\nThe systems for automated claim verification and\nautomated fact-checking are usually modeled as\na framework with three components, where each\ncomponent is a well-established NLP task (Zeng\net al., 2021). This framework is a three-component\npipeline (Figure 1) consisting of (1) document\nretrieval; (2) evidence selection; (3) verdict pre-\ndiction. We are mostly concerned with how the\ndocument retrieval part affects the further entail-\nment process. That is why we fix the evidence-\nsentence selection model and the entailment predic-\ntion model. This way, the quality of the data source\nand the retrieval technique are the most important\nvariables being tested. Two of these subtasks, or\neven all three (Zhang et al., 2021), can be learned\ntogether in a joint system with a shared represen-\ntation. For the sake of simplicity of testing, we\nchoose a pipeline system that performs each task\nsequentially.\nIn document retrieval, given a claim cand a\ncorpus of ndocuments D={d1, d2, ..., d n}, the\ntask is to select top kmost relevant documents\ng1, ..., g kfor the claim (query), with a function\nw(c, d). After the documents are retrieved, the\nnext step is to select evidence sentences that serve\nas rationale in making a decision regarding the\nclaim’s veracity. From mcandidate sentences\ns1, s2, ..., s mcomprising the selected documents,\ntopjsentences are selected as evidence sentences\n⃗ e={e1, e2, ..., e n}with a function z(c, s). Finally,\na verdict prediction function is trained to predict\n1https://www.github.com/jvladika/\ncomparing-knowledge-sourcesy(c,⃗ e)∈ {SUPPORTED ,REFUTED }. This means\nthe final task is essentially a binary classification\ntask with two classes, which makes it suitable for\nevaluation with standard classification metrics: pre-\ncision, recall, and binary F1.\nSince we are focusing on testing the influence of\nthe knowledge source on the final claim verdict pre-\ndiction, we experiment with different knowledge\nsources Dand retrieval functions w(c, d). Other\ncomponents of the pipeline are fixed to make a fair\ncomparison. After testing the values of kandjwith\ndifferent values in the set of {1,3,5,10,20}, we\nset both to be 10since it provided the best F1 per-\nformance and the best trade-off between covering\nenough content while not cluttering with too much\nnoise. This means we retrieve the top 10 documents\nand then select the top 10 sentences from them. For\nz(c, s), we select the model SPICED (Wright et al.,\n2022a), which is a sentence similarity model that\ncatches paraphrases of scientific claims well and re-\ncently set state-of-the-art performance in evidence\nselection on a couple of scientific claim-verification\ndatasets. For the verdict predictor y(c,⃗ e), we first\nconsidered specialized biomedical language mod-\nels (Vladika et al., 2023a). In the end, we chose the\nDeBERTa-v3 model (He et al., 2021), since it had\nthe best performance and was shown to be an ex-\nceptionally powerful model for textual entailment\nrecognition on the GLUE benchmark. We use a ver-\nsion of DeBERTa-v3 additionally fine-tuned on var-\nious Natural Language Inference (NLI) datasets.2\nIt should be noted that we use these two models out-\nof-the-box and do not fine-tune them on any of our\ndatasets in any experiment. This is an intentional\nzero-shot setting that aims to verify the real-world\nsituation of using a system on yet unknown claims.\n2.2 Datasets\nWe choose four English datasets of biomedical and\nhealth claims, built for different purposes.\nSCIFACT (Wadden et al., 2020) is a dataset of\n1,109 claims (in test and dev set) which were expert-\nwritten from citation sentences found in biomedi-\ncal research publication abstracts. These publica-\ntions originate from PubMed, which is one of the\ndatabases queried in our paper.\nPUBMEDQA(Jin et al., 2019) is a dataset of\n1,000 labeled claims that were generated from\nabstract of biomedical papers originating from\n2https://huggingface.co/MoritzLaurer/\nDeBERTa-v3-large-mnli-fever-anli-ling-wanliPubMed. Even though more of a question-\nanswering dataset in nature, it also provides\nyes/no/maybe labels which make it usable as a fact-\nchecking dataset.\nHEALTH FC(Vladika et al., 2023b) is a dataset\nof 750 claims concerning everyday health and span-\nning various topics like nutrition, immune system,\nmental health, and physical activity. The claims\noriginate from user inquiries and they were checked\nby a team of medical experts using clinical trial re-\nports and systematic reviews as the main evidence\nsource. All the claim verdict explanations are de-\nscribed in a user-friendly language.\nCOVERT(Mohr et al., 2022) is a dataset of 300\nclaims related to health and medicine, which are\nall causative in nature (such as \" vaccines cause\nautism \"). All the claims originate from Twitter,\nwhich means some claims are written informally\nand thus make an additional challenge by providing\na real-world scenario of misinformation checking.\nDataset DomainP\nSCIFACT biomedical\nresearch456 237 693\nPUBMEDQA biomedical\nresearch552 338 890\nHEALTH FC consumer\nhealth202 125 327\nCOVERT health misin-\nformation198 66 264\nTable 1: The four datasets used in the experiments,\nincluding their domain and label distribution.\nFor all of the datasets, we leave out any claims\nlabeled with NOTENOUGH INFORMATION (NEI)\nlabel. This is because some datasets do not in-\nclude this label, and those that do include it de-\nfine it differently. For SCIFACT, NEI means no\nevidence documents are present in their internal\ncorpus. For HEALTH FC, NEI means no conclusive\nevidence for the claim was found in any clinical\ntrials. Table 1 shows the final distribution of labels\nfor each dataset, after leaving out the claims with\nnot enough information.\n3 Experiment Setup\n3.1 Knowledge Sources\nFor testing on Wikipedia, we used the latest avail-\nable dump of English Wikipedia that we found,\nfrom May 20th, 2023, containing 6.6 million ar-ticles.3For PubMed, the US National Library of\nMedicine provides MEDLINE, a snapshot of cur-\nrently available abstracts in PubMed that is updated\nonce a year. We used the 2022 version found at\ntheir website.4While this yields 33.4M abstracts,\nwe pre-processed the data following González-\nMárquez et al. (2023) and removed non-English\npapers, papers with no abstracts, and papers with\nunfinished abstracts, which yields 20.6M abstracts.\nFor Google results, we used Google’s publicly\navailable Custom Search JSON API.5\n3.2 Document Retrieval Techniques\nWe test the performance of two different document\nretrieval techniques, namely a sparse one and a\ndense one. Since both types of approaches are\ndeployed in modern search systems, we want to\nsee how much of a difference they make in find-\ning appropriate documents that can verify a claim.\nAs a representative sparse technique, we opt for\nBM25, an improvement over TF-IDF that takes\ninto account term frequency, document length, and\ninverse document frequency. Despite its simplic-\nity, it has proven to be a cornerstone of informa-\ntion retrieval approaches due to comparative per-\nformance to more sophisticated neural approaches\n(Kamphuis et al., 2020).\nRecently, with the advance of large language\nmodels, encoding both the claim and documents\nwith dense vector embeddings and then search-\ning most similar vectors with cosine similarity\nhas proven to be a powerful retrieval method\n(Karpukhin et al., 2020). A particularly success-\nful recent approach is SimCSE (Gao et al., 2021),\nwhich uses contrastive learning and entailment-\nbased training to enhance similarity scoring.\nWe chose a biomedical variation BioSimCSE\n(Kanakarajan et al., 2022) which fits our use case.\nFor dense retrieval, we encode the entirety of\nour PubMed corpus and Wikipedia corpus with\nBioSimCSE and store the embeddings. For sparse\nretrieval, we construct an inverted index out of\nWikipedia and PubMed corpora and later query it\nusing BM25 metrics. After selecting the top 10\ndocuments in each method, the top 10 most similar\nsentences were taken and jointly with claim the ver-\ndict was predicted based on the entailment relation.\n3https://dumps.wikimedia.org/enwiki/20230520/\n4https://www.nlm.nih.gov/databases/download/\npubmed_medline.html\n5https://developers.google.com/custom-search/\nv1/overviewFor Google Search, we took the top 10 returned\nGoogle snippets as \"evidence sentences\" that we\nthen concatenate and use as the evidence block for\nlabel prediction. All the experiments were run on a\nsingle Nvidia V100 GPU card, in a single run. One\nrun on one dataset, one knowledge source, and one\nretrieval technique costed one computation hour.\n3.3 Baseline\nTo establish a baseline, we first run the system with\nthe gold evidence provided with each of the four\ndatasets. These are the sentences or snippets that\nwere given by the annotators or creators of the re-\nspective datasets. The performance is shown in\nTable 2. It should be noted that this performance\nis different from those reported in papers introduc-\ning these datasets because we remove the claims\nlabeled with NEI and we also did not fine-tune the\nmodel on the datasets. This is an intentional choice\nbecause the idea is to test the systems in a zero-\nshot / off-the-shelf setting. We expect the results in\nour experiments to be lower than this because hav-\ning annotated evidence is an easier setting than the\nopen-domain claim verification where the evidence\nneeds to be discovered.\nGold Evidence\nDataset Precision Recall F1 Score\nSCIFACT 77.9 88 .4 82.8\nPUBMEDQA 74.4 80 .4 77.3\nHEALTH FC 80.5 83 .4 81.9\nCOVERT 80.7 86 .4 83.4\nTable 2: Results of final verdict prediction over four\ndatasets using the gold evidence sentences provided\nwith the datasets.\n4 Results\nTable 3 shows the performance of the claim veri-\nfication system using evidence retrieved with two\ndifferent techniques from two different knowledge\nsources, PubMed and Wikipedia. As expected, the\nF1 scores are lower than the oracle setting of us-\ning gold evidence from Table 2. Still, they come\nremarkably close to it, taking into account the com-\nplexity of finding relevant documents in a sea of\n6 and 20 million articles, and further selection of\nrelevant sentences from them, to produce a final\nverdict. This indicates the open-domain setting is a\npromising endeavor in scientific claim verification.BM25 Semantic Search\nSource Dataset Precision Recall F1 Macro Precision Recall F1 Macro\nPUBMEDSCIFACT 79.9 72 .6 76 .1 73 .7 80 .0 76.8\nPUBMEDQA 70.0 70 .6 70 .3 66 .7 84 .4 74.5\nHEALTH FC 62.7 78 .7 69 .7 62 .6 84 .6 72 .0\nCOVERT 76.0 83 .3 79 .5 75 .6 76 .8 76 .2WIKIPEDIASCIFACT 67.9 83 .3 74 .8 68 .8 83 .6 75 .4\nPUBMEDQA 65.3 83 .0 73 .1 68 .3 78 .5 73 .2\nHEALTH FC 62.9 87 .4 73 .1 65 .2 92 .6 76.5\nCOVERT 72.4 78 .3 75 .2 78 .5 86 .8 82.5\nTable 3: Results of final verdict prediction over four datasets using evidence retrieved from PubMed and Wikipedia.\nFor both knowledge sources, the evidence from\ndocuments retrieved using semantic search outper-\nformed the standard sparse metric BM25. Still,\nBM25 fares well compared to the relatively recent\nsemantic approaches. It is also notable to observe\nin Table 3 that BM25 excels in precision more so\nthan recall, always beating semantic retrieval in\nthis metric. This is not too surprising consider-\ning BM25 relies on exact keyword matching and\nis better suited for this use case. While BM25\nslightly beats the dense BioSimCSE in precision,\nit is significantly outperformed in recall in the first\nthree datasets. In deeper analysis, as we will show\nin the next section, we saw the dense technique\nwould more often retrieve articles talking about the\nclaim content using alternate naming for diseases,\nwhich led to picking up more supporting arguments\nfor positive claims. COVERT is the only dataset\nfor which BM25 performed better in the PubMed\nsetting, in both precision and recall. Considering\nthe noisy nature of this dataset (tweets and infor-\nmal language), the inverse document frequency\n(idf) feature of BM25 was better at finding exact\nmatches for important but rare keywords mentioned\nin the tweets and ignoring the more common but\nunimportant words. On the other hand, the poorer\nperformance of the dense technique could be be-\ncause the embedding was swayed in vector space\ndue to noisy irrelevant topics from tweets.\nWhen looking at the performance of claim ver-\nification systems over Wikipedia in Table 3, it is\nonce again apparent that dense retrieval found more\nrelevant documents with better evidence and out-\nperformed the sparse retrieval. Nevertheless, in\nthis case, the precision of BM25 was worse thanBioSimCSE. In general, recall in all settings was\nhigher than the ones from PubMed and precision\nlower, which shows better prediction of the positive\n(supported) class but also its over-prediction.\nGoogle Snippets\nDataset Precision Recall F1 Score\nSCIFACT 75.5 91 .5 82 .7\nPUBMEDQA 66.7 95 .6 78 .5\nHEALTH FC 62.3 92 .6 74 .5\nCOVERT 76.4 68 .7 72 .3\nTable 4: Results of final verdict prediction over four\ndatasets using evidence retrieved from \"the whole web\"\n(using Google).\nAnother experiment consisted of querying \"the\nwhole web\" in order to find relevant evidence for\na verdict. This is a common setting explored as\none of the straightforward baselines in some fact\nverification papers (Gupta and Srikumar, 2021; Hu\net al., 2022) and it mimics how humans would be-\ngin the process of a claim checking. Table 4 reports\non this performance. Considering the short nature\nof Google snippets, they usually do not actually\nprovide \"evidence\" but commonly the verdict on\nthe claim itself as reported on the source website\ncontaining the snippet.\nAt first glance, the performance with Google\nsearch seems impressive, especially considering\nthat for the two most challenging datasets, SCI-\nFACT andPUBMEDQA, the performance is im-\nproved when compared to the first two tables. A\nmore careful look reveals this to be an artefactClaim PubMed (semantic) Wikipedia (semantic)\nCan regular intake\nof vitamin C prevent\ncolds? (Refuted )Nevertheless, given the consistent effect of vitamin C on the\nduration and severity of colds in the regular supplementation\nstudies, and the low cost and safety, it may be worthwhile for\ncommon cold patients to test on an individual basis whether\ntherapeutic vitamin C is beneficial for them. (Hemilä and Chalker,\n2013) ( Supported )According to the most recently published Cochrane review on\nvitamin C and the common cold, one gram per day or more\nof vitamin C does not influence common cold incidence in the\ngeneral community, i.e., it does not prevent colds. ( en wiki:\nVitamin C and the common cold ) (Refuted )\nCan lung cancer\nscreening by com-\nputed tomography\n(CT) also do harm?\n(Supported )Lung cancer screening with low dose computed tomography (ct)\nis the only method ever proven to reduce lung cancer-specific\nmortality in high-risk current and former cigarette smokers. We\naim to explain why the risks associated with radiation exposure\nfrom lung cancer screening are very low and should not be used\nto avoid screening or dissuade... (Frank et al., 2013) ( Reftued )Low-dose CT screening has been associated with falsely positive\ntest results which may result in unneeded treatment. In a series\nof studies assessing the frequence of false positive rates, results\nreported that rates ranged from 8-49%.( en wiki: Lung cancer\nscreening ) (Supported )\nCan ginkgo extract\nrelieve the symp-\ntoms of tinnitus?\n(Refuted )Ginkgo biloba is a plant extract used to alleviate symptoms associ-\nated with cognitive deficits, e.g., decreased memory performance,\nlack of concentration, decreased alertness, tinnitus, and dizziness.\nPharmacologic studies have shown that the therapeutic effect of\nginkgo... (Søholm, 1998) ( Supported )Ginkgo leaf extract is commonly used as a dietary supplement,\nbut there is no scientific evidence that it supports human health or\nis effective against any disease. Systematic reviews have shown\nthere is no evidence for effectiveness of ginkgo in treating high\nblood pressure, menopause-related cognitive decline, tinnitus,\npost-stroke recovery, or altitude sickness. ( en wiki: Gingko\nBilboa ) (Refuted )\nThe most prevalent\nadverse events to\nSemaglutide are\ngastrointestinal.\n(Supported )We evaluated gastrointestinal (GI) adverse events (AEs) with\nonce-weekly Semaglutide 2.4 mg in adults with overweight or\nobesity and their contribution to weight loss (WL). GI AEs were\nmore common with semaglutide 2.4 mg than placebo, but typ-\nically mild-to-moderate and transient. (Wharton et al., 2022)\n(Supported )Possible side effects include nausea, diarrhea, vomiting, consti-\npation, abdominal pain, headache, fatigue, indigestion/heartburn,\ndizziness, abdominal distension, belching, hypoglycemia (low\nblood glucose) in patients with type 2 diabetes, flatulence, gas-\ntroenteritis, and gastroesophageal reflux disease (GERD) ( en\nwiki: Semaglutide ) (Refuted )\nMacrolides protect\nagainst myocardial\ninfarction. (Refuted )Our findings indicate that macrolide antibiotics as a group are\nassociated with a significant risk for MIbut not for arrhythmia\nand cardiovascular mortality. (Gorelik et al., 2018) ( Refuted )Macrolides are a class of natural products that consist of a large\nmacrocyclic lactone ring to which one or more deoxy sugars,\nusually cladinose and desosamine, may be attached. ( en wiki:\nMacrolide ) (Supported )\nTable 5: Example claims and retrieved evidence from the two different knowledge bases, where only one of them\nprovided a correct final verdict.\nof data leakage and the way these two datasets\nwere constructed (a similar phenomenon already\nobserved in fact-checking datasets, Glockner et al.\n(2022)). Considering that in both of them the\nclaims originate from sentences actually contained\nin PubMed abstracts, Google Search is powerful\nenough to be able to find the exact sentence that was\nthe origin of these claims. The two other datasets,\nHEALTH FCandCOVERT, give a more realistic\npicture of the performance of Google snippets con-\nsidering they contain organic claims that originated\nfrom online users. It is interesting to see that for\nthese two datasets Google beats both settings of\nPubMed but succumbs to Wikipedia as the knowl-\nedge source. This can be attributed to the fact that\nthe simple language of claims in these two datasets\ncan be easier to verify with Google results like\nblogs and news portals, as opposed to the complex\nlanguage found in PubMed research publications.\n5 Discussion\nIn this section, we provide further insights and a\ndeeper look into the performance of our pipeline for\nopen-domain claim verification of scientific claims\nin large knowledge sources. We do this with a qual-\nitative analysis where we looked at what kind of\ndocuments and sentences are retrieved from dif-ferent knowledge sources with different retrieval\ntechniques and outline some common patterns with\nrepresentative examples.\n5.1 Popular and Specialized Claims\nThe performance of Wikipedia and PubMed as a\nknowledge source is considerably close to each\nother when looking at Table 3. Nonetheless, there\nare differences with respect to the claim’s do-\nmain and popularity. It is evident from the tables\nthat Wikipedia slightly outperformed PubMed for\nHealthFC, the dataset dealing with everyday con-\nsumer health questions, and CoVert, with social\nmedia claims related to the COVID-19 pandemic.\nThe simple language in which these claims are\nposed (e.g., Does regular consumption of coffee in-\ncrease the risk of heart disease such as heart attack\nor stroke? as opposed to Omnivores produce less\ntrimethylamine N-oxide from dietary I-carnitine\nthan vegetarians ) corresponds to the more user-\nfriendly language of Wikipedia, when compared\nto the often highly technical language of medical\nresearch found at PubMed.\nOther than the simpler language of claims, an-\nother factor for using Wikipedia as a knowledge\nsource is the claim’s popularity and established re-\nsearch on it. Most claims in HealthFC are commonClaim BM25 (PubMed) Semantic (PubMed)\nDo heat patches\ncontaining capsaicin\nhelp with neck pain?\n(Refuted )The objective of this study was to evaluate the efficacy of a hy-\ndrogel patch containing capsaicin 0.1% compared with a placebo\nhydrogel patch without capsaicin to treat chronic myofascial\nneck pain (...) There was no significant difference between the\ntwo groups in any of the outcome measures. ( Refuted )In two randomized trials, a single 60-min application of the cap-\nsaicin 8% patch reduced pain scores significantly more than a\nlow-concentration (0.04%) capsaicin control patch in patients\nwith PHN . (Supported )\nDoes a herbal com-\nbination of rosemary,\nlovage, and centaury\nrelieve symptoms of\nuncomplicated cysti-\ntis? (Supported )The herbal medicinal product Canephron ®N contains BNO\n2103, a defined mixture of pulverized rosemary leaves, cen-\ntaury herb, and lovage root(...) When given orally, BNO 2103\nreduced inflammation and hyperalgesia in experimental cystitis\nin rats. ( Supported )Rosmarinus officinalis l., rosemary, is traditionally used to treat\nheadache and improve cardiovascular disease partly due to its\nvasorelaxant activity, while the vasorelaxant ingredients remain\nunclear. ( Refuted )\nThe extracellular\ndomain of TMEM27\nis cleaved in hu-\nman beta cells.\n(Supported )Here, we report the identification and characterization of trans-\nmembrane protein 27 ( TMEM27 , collectrin) in pancreatic beta\ncells. ( Supported )We also show that TMEM2 is strongly expressed in endothelial\ncells in the subcapsular sinus of lymph nodes and in the liver\nsinusoid, two primary sites implicated in systemic HA turnover.\n(Refuted )\nNormal expression\nof RUNX1 causes\ntumorsupressing\neffects. (Supported )RUNX1 is a well characterized transcription factor essential for\nhematopoietic differentiation and RUNX1 mutations are the cause\nof leukemias. runx1 is highly expressed in normal epithelium of\nmost glands and recently has been associated with solid tumors.\n(Refuted )RUNX gene over-expression inhibits growth of primary cells but\ntransforms cells with tumor suppressor defects , consistent with\nreported associations with tumor progression. ( Supported )\nTable 6: Example claims and retrieved evidence from PubMed, using the two different retrieval techniques, where\nonly one of them provided a correct final verdict.\nhealth concerns people search for on the Internet.\nThis means there is often systematic reviews done\non these claims and Wikipedia encourages citing\nsystematic reviews in its articles when available.\nWe noticed our system often retrieved sentences\nmentioning reviews. Table 5 shows in first three\nrows how this led to the correct verdict prediction\nfor Wikipedia, but incorrect for PubMed, since\nthe PubMed retriever found standalone studies that\nmight disagree from the research consensus. For\n327 claims in HealthFC, combined evidence re-\ntrieved from Wikipedia mentions \"systematic re-\nview\" 89 times, while \"Cochrane review\"6is men-\ntioned 60 times (for 1000 claims in PubMedQA,\nthe number is 29 and 11). On the other hand, row\n4 of Table 5 shows an example of evidence from\nWikipedia being too broad and generalized, while\nrow 5 shows a claim for which there was simply no\nrelevant evidence in the Wikipedia article. For spe-\ncialized claims concerning deeper medical knowl-\nedge or specific research hypotheses, PubMed is a\nsuperior knowledge base.\n5.2 Precision and Coverage\nThe comparison between the two retrieval tech-\nniques in Table 3 shows that semantic search out-\nperforms BM25 in all cases, except for CoVERT on\nPubMed (F1: 79.5). Considering that most systems\nfrom existing work on automated fact-checking use\nonly BM25 in their pipelines, these results can mo-\ntivate future research towards deploying semantic\n6Cochrane is an international organization formed to syn-\nthesize medical research findings.search with different sentence embedding models.\nDense retrieval’s ability to deal with synonyms and\nparaphrases is especially important in the medical\nfield where numerous diseases, drugs, chemical\ncompounds can have multiple names and symbols.\nWhile semantic search provides higher coverage,\nBM25 offers better precision. Table 3 shows that\nfor PubMed, using BM25 as a retrieval technique\nachieves higher precision for all datasets, with an\nespecially high improvement for SciFact. The exact\nmatch of words posed in the query helps retriev-\ning studies and documents that deal with concepts\nmentioned in the claim. When looking at Table 6,\nthe first three rows show examples of claims for\nwhich dense retrieval got swayed into similar but\nirrelevant documents, while BM25 managed to un-\ncover the correct ones. In the first row, capsaicin is\nmentioned in both, but only the one from BM25 is\nabout neck pain. In the second row, the exact drug\nwith specified ingredients is uncovered by BM25,\nwhile semantic search did not retrieve it. The third\nrow shows an example of when an exact match\ncan be important (TMEM27 vs. TMEM2). On\nthe other hand, the fourth row shows an example\nof a common use case where semantic matching\nis beneficial – for this claim to be matched with\nBM25, \"tumorsuppressing\" and \"effects\" would\nhave to be mentioned, but dense retrieval can catch\nparaphrases like \"tumor suppressor defects\".\n5.3 Future Directions\nBased on our findings and discussion, we see the\nfuture work could focus on these direction:•Modeling disagreement. We observed how\ndifferent studies and sources can come to dif-\nfering conclusions regarding a claim. In this\npaper, we chose the majority vote among the\ntop 10 documents as the final decision, but\nthis diminishes the information about the pre-\ndiction uncertainty. This is part of the broader\nML problem of learning with disagreements\n(Leonardelli et al., 2023). The end users of\nfact-checking systems could appreciate the\nadded interpretability of seeing the level of\ndisagreement among different sources.\n•Assessing evidence quality. When it comes\nto medical research articles from PubMed, not\nall of them hold the same weight, considering\nthe research relevance. While it is hard to\nassess their validity of results automatically,\nmodeling metadata aspects could give a hint\non how to differently weight certain publica-\ntions. Parameters such as the number of cita-\ntions, the impact score and reputation of the\nsource journal, the institutions of the authors\ncould lead to more trustworthy results. Sim-\nilarly, the sources of Wikipedia articles and\nGoogle search results contain website of dif-\nfering reputation and credibility – filtering to\ntrusted domains such as university websites or\nacademic publishers could enhance the level\nof trust and performance (Kotonya and Toni,\n2020). Lastly, temporal aspect (date of publi-\ncation) is very important since research on cer-\ntain topics advances and changes with time.\n•Retrieval-augmented generation (RAG) for\nverifying claims. Modern generative large\nlanguage models (LLMs) have shown the\npower to both exhibit reasoning capabilities\nand generate coherent text for users. They\nalready possess learned medical knowledge\nin their internal weights, but are prone to hal-\nlucinations. Therefore, a promising research\navenue is to amplify LLMs by passing the re-\ntrieved evidence passages from sources like\nWikipedia and PubMed to them (Pan et al.,\n2023). How to effectively combine this, while\nbalancing the trade-off of readability and fac-\ntuality, is an open challenge.6 Related Work\n6.1 Automated Fact-Checking\nThe task of automated fact-checking refers to veri-\nfying the truthfulness of a given claim using back-\nground knowledge and relevant evidence (Guo\net al., 2022). It is still mostly done manually by\ndedicated experts, but ongoing research efforts try\nto automate parts of it with NLP methods. For\nthis purpose, many datasets have been constructed.\nThey contain either synthetic claims generated\nfrom Wikipedia (Thorne et al., 2018; Schuster et al.,\n2021) or real-world claims found on portals dedi-\ncated to fact-checking of trending political and so-\ncietal claims (Augenstein et al., 2019) or appearing\nin social media (Nielsen and McConville, 2022).\nScientific fact-checking is a variation of the task\nthat is concerned with assessing claims rooted in\nscientific knowledge (Vladika and Matthes, 2023a).\nThe most prominent domains are health (Sarrouti\net al., 2021) and climate science (Diggelmann et al.,\n2021).\n6.2 Open-domain Claim Verification\nClaim verification is similar to the NLP task of\nquestion answering, where the goal is to either re-\ntrieve or generate an answer to a question based on\ndiscovered evidence (Rogers et al., 2023), and it\ncan also be analyzed in a closed domain or open\ndomain. In the closed domain, the evidence comes\nfrom an already provided source document. This\nsetting is also called Machine Reading Comprehen-\nsion (MRC) since the goal is to build models that\nare efficient in recognizing which parts of text cor-\nrespond to a given query (Baradaran et al., 2022).\nIn the open domain, only the final answer is known\nand it is the goal of a system to find appropriate ev-\nidence in a large corpus of documents or other type\nof resources (Chen and Yih, 2020). Other related\ntasks include Natural Language Inference (Vladika\nand Matthes, 2023b) and Evidence Retrieval (Wad-\nhwa et al., 2023).\nThere have also been efforts in open-domain\nscientific fact verification. Wadden et al. (2022) ex-\npand the corpus of evidence research documents for\nthe dataset SCIFACT of biomedical claims, from\nthe original 5k to about 500k. In such a setting,\nthey discovered significant performance drops in\nF1 scores of final verdict predictions. This work\nanalyzed only one knowledge source (biomedical\nabstracts) and focused on data annotation in such\na setting, while our paper expands the researchpaper corpus even further to 20 million abstracts,\nand analyzes other knowledge sources and retrieval\nmethods. In Pugachev et al. (2023), the authors\ntake consumer-health question datasets and test the\npredictive performance of a system using PubMed\nand Wikipedia. A bigger focus was put on fine-\ntuning the models on different datasets and testing\nthe efficiency of built-in searche engines of the re-\nspective databases. Furthermore, Stammbach et al.\n(2023) test the performance of six fact-checking\ndatasets from different domains (including encyclo-\npedic, political) using evidence retrieved from three\ndifferent knowledge bases, while looking solely at\none biomedical dataset and one retrieval technique\n(BM25). Also related is the work by Sauchuk et al.\n(2022), which shows the clear importance of the\ndocument-retrieval component of the fact-checking\npipeline on the performance of the whole system.\nTo the best of our knowledge, our paper features\nthe biggest corpora (using the entirety of available\nPubMed and Wikipedia dumps, with 20.6M and\n6.6M articles), searches \"the whole web\", analyzes\ndifferent retrieval techniques (BM25 and semantic),\nand analyzes datasets of different type and purpose:\nexpert-geared research claims ( SCIFACT andPUB-\nMEDQA), and organic user-posed health claims\n(HEALTH FC and C OVERT).\n7 Conclusion\nIn this paper, we conducted a number of ex-\nperiments assessing the performance of a fact-\nverification system in an open-domain setting.\nMoving away from the standard setup of a\nsmall evidence corpus, we expand the knowledge\nsources to two large document bases (PubMed and\nWikipedia), searching the whole web via Google\nSearch API, and experiment with two retrieval tech-\nniques (sparse and dense). We measured the verdict\nprediction performance over four established fact-\nchecking datasets. Our results show that search-\ning for evidence in the open domain provides sat-\nisfyingly high F1 performance, not far from the\nclosed-domain setting, with a room for further im-\nprovement. We conclude that the knowledge source\nperform comparably, with Wikipedia being better\nfor popular and trending claims and PubMed for\ntechnical inquiries. We demonstrate the general\nsuperiority of dense retrieval techniques, with ex-\namples of where it falls short and BM25 retrieval\nwould be beneficial. We hope our research will\nencourage more exploration of the open-domainsetting in the NLP fact-checking community and\naddressing real-world misinformation scenarios.\nLimitations\nIn this study, we performed automatic assessment\nof claims related to medicine and health. These are\ntwo sensitive fields where misinformation, model\nhallucination, and incorrect evidence retrieval can\nlead to harmful consequences, misinformation\nspread, and societal effects. The automated sci-\nentific fact-verification system described in this\nwork is still far from being safe and consistent for\nadoption in the real world, due to imperfect per-\nformance and drawbacks that arise. In case such\nan automated fact-verification system would be de-\nployed and produce misleading verdicts, this could\ndecrease the trust in the potential use and develop-\nment of such solutions.\nIn our work, for easier comparison we disregard\nclaims annotated with NOTENOUGH INFORMA -\nTION due to different definitions of this label across\ndifferent datasets and also absence of it in some\ndatasets. This is an important label in claim veri-\nfication, since not all claims can be conclusively\nassessed for their veracity. Future work should find\na way to effectively include this label into model\npredictions. This is especially important in the sci-\nentific domain considering the constantly evolving\nnature of scientific knowledge, and sometimes con-\nflicting evidence from different research studies.\nLastly, the fact-checking pipeline used in this pa-\nper is a complex system with multiple factors – the\nchoice of the retrieval method, of the sentence se-\nlection model, the top k value, the NLI model, and\nthe prediction threshold. Some incorrect predic-\ntions could have come from, e.g., faulty entailment\nprediction of the NLI model or other factors that do\nnot necessarily stem from the choice of the knowl-\nedge base. Still, we put strict attention to keeping\nall the factors constant and frozen, to ensure a com-\nparable setup. We focused on reporting only those\nphenomena and patterns that we observed were\ncommonly occurring, after a thorough analysis of\nretrieved evidence for each claim.\nAcknowledgements\nThis research has been supported by the Ger-\nman Federal Ministry of Education and Research\n(BMBF) grant 01IS17049 Software Campus 2.0\n(TU München). We would like to thank the anony-\nmous reviewers for helpful feedback.References\nIsabelle Augenstein, Christina Lioma, Dongsheng\nWang, Lucas Chaves Lima, Casper Hansen, Chris-\ntian Hansen, and Jakob Grue Simonsen. 2019. Mul-\ntiFC: A real-world multi-domain dataset for evidence-\nbased fact checking of claims. In Proceedings of\nthe 2019 Conference on Empirical Methods in Natu-\nral Language Processing and the 9th International\nJoint Conference on Natural Language Processing\n(EMNLP-IJCNLP) , pages 4685–4697, Hong Kong,\nChina. Association for Computational Linguistics.\nRazieh Baradaran, Razieh Ghiasi, and Hossein\nAmirkhani. 2022. 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Fact or fiction: Verifying\nscientific claims. In Proceedings of the 2020 Con-\nference on Empirical Methods in Natural Language\nProcessing (EMNLP) , pages 7534–7550, Online. As-\nsociation for Computational Linguistics.\nDavid Wadden, Kyle Lo, Bailey Kuehl, Arman Cohan,\nIz Beltagy, Lucy Lu Wang, and Hannaneh Hajishirzi.\n2022. SciFact-open: Towards open-domain scientific\nclaim verification. In Findings of the Association\nfor Computational Linguistics: EMNLP 2022 , pages\n4719–4734, Abu Dhabi, United Arab Emirates. As-\nsociation for Computational Linguistics.\nSomin Wadhwa, Vivek Khetan, Silvio Amir, and By-\nron Wallace. 2023. RedHOT: A corpus of annotated\nmedical questions, experiences, and claims on social\nmedia. In Findings of the Association for Compu-\ntational Linguistics: EACL 2023 , pages 809–827,\nDubrovnik, Croatia. 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In Proceedings of the\n2021 Conference on Empirical Methods in Natural\nLanguage Processing , pages 3580–3586."    },    {        "title": "2402.02845v1.A_general_integral_identity_with_applications_to_a_reverse_Serrin_problem.pdf",        "content": "arXiv:2402.02845v1  [math.AP]  5 Feb 2024A GENERAL INTEGRAL IDENTITY WITH\nAPPLICATIONS TO A REVERSE SERRIN PROBLEM\nROLANDO MAGNANINI, RICCARDO MOLINAROLO, AND GIORGIO POGGE SI\nAbstract. We prove a new general differential identity and an asso-\nciated integral identity, which entails a pair of solutions of the Poisson\nequation with constant source term. This generalizes a form ula that\nthe first and third authors previously proved and used to obta in quan-\ntitative estimates of spherical symmetry for the Serrin ove rdetermined\nboundary value problem. As a first application, we prove a qua ntitative\nsymmetry result for the reverse Serrin problem , which we introduce for\nthe first time in this paper. In passing, we obtain a rigidity r esult for\nsolutions of the aforementioned Poisson equation subject t o a constant\nNeumann condition.\n1.Introduction\nLet Ω be a bounded domain in the Euclidean space RN,N≥2, with suf-\nficiently regular boundary Γ. We consider solutions of the Po isson equation:\n(1.1) ∆ u=Nin Ω.\nIn [19, 20], the first and third author of this paper stated and then proved\nthe following integral identity:\n(1.2)ˆ\nΩ(−u)/braceleftbigg\n|∇2u|2−(∆u)2\nN/bracerightbigg\ndx=1\n2ˆ\nΓ/parenleftbig\nu2\nν−R2/parenrightbig\n(uν−qν)dSx.\nHere,uis a solution of (1.1) such that\n(1.3) u= 0 on Γ\nandq=qzis a quadratic polynomial of the form\n(1.4) qz(x) =1\n2|x−z|2+aforx∈RN,\nfor somez∈RNanda∈R. Moreover, νis the exterior unit normal on Γ\nand the constant Rhas the value\n(1.5) R=N|Ω|\n|Γ|.\nAlso, notice that it is easy to see that\n|∇2u|2−(∆u)2\nN= ∆P,\n2020Mathematics Subject Classification. Primary 35N25, 35B35, 35M12; Secondary\n35A23.\nKey words and phrases. Serrin overdetermined problem, integral identities, stab ility,\nquantitative estimates.\n12 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nwhen we set:\nP=1\n2|∇u|2−u.\nThis is a standard P-function for (1.1).\nThefunction ∆ Pis important. Itis non-negative, by theCauchy-Schwarz\ninequality and vanishes if and only if uis a quadratic polynomial as in (1.4).\nThus, it gives a quadratic-radiality test foru. By this important feature,\n(1.2) yields the Serrin Symmetry Theorem (see [30] for the ce lebrated proof\nby the method of moving planes and [31] for the first proof base d on integral\nidentities). This asserts that, if thesolution uof (1.1) and (1.3) has constant\nnormal derivative uνon Γ, then Ω must be a ball. This is now easily seen\nfrom (1.2), since it turns out that the constant value of uνon Γ must equal\nR, by the divergence theorem. Thus, (1.2) gives that ∆ P≡0 (and hence\nradial symmetry), being as −u>0 on Ω, by the strong maximum principle.\nThe use of the volume integral in (1.2) to test the radiality o fuwas first\nnoticed in [27].\nBesides giving rigidity, (1.2) is even more important, beca use it allows\nto construct sharp quantitative estimates on how the releva nt domain Ω\ndeviates from being a ball (in the Hausdorff topology of domai ns) in terms\nof some norm of the deviation of uνfrom a constant. This theme was\ninaugurated in [1], by using a quantitative version of the me thod of moving\nplanes. There, a quantitative symmetry estimate is obtaine d in terms of a\nlogarithmic profile of a C1(Γ)-norm of the aforementioned deviation. It is\nworth noticing that this approach also works for positive so lutions of quite\ngeneral semilinear Poisson equations. That strategy was im proved in [10]\nthus obtaining a quantitative symmetry estimate in terms of a polynomial\nprofileof the Lipschitz seminorm of uν. (See also [8, 6, 7] for related research\nand [5, 11, 12] for recent generalizations to the fractional setting.)\nAn approach based on a quantitative version of the arguments contained\nin [31] was initiated in [3]. The method only works for the Poi sson equation\n(1.1). However, the measure of the relevant deviation of uνfrom a constant\nis greatly relaxed by using a Lebesgue norm on Γ. (We refer the interested\nreader to [18, 23, 29] for a survey on the several stability re sults available in\nthe literature; see also [9, 13] for related results in differe nt settings.)\nIdentity (1.2) was used by the first and third author to obtain optimal\n(for low dimension) and allegedly1optimal (for large dimension) quantitative\nbounds. This was the theme of a series of papers ([20, 21, 23]) culminating\nwith the proof of the inequality\n(1.6) ρe−ρi≤cψ(/ba∇dbluν−R/ba∇dbl2,Γ).\n(See also [24, 28, 26] for recent extensions to mixed boundar y value prob-\nlems.) In (1.6), we mean that\n(1.7) ρe= max\nx∈Γ|x−z|, ρi= min\nx∈Γ|x−z|,\nfor some point z∈Ω. Hence, ρiandρeare the radii of the largest ball\ncontained in Ω and the smallest ball containing Ω, both cente red atz.2\n1Here, we mean that we conjecture that the bounds obtained in [ 23] are indeed optimal.\n2In [23], it is also shown that, besides ρe−ρi, theL2-deviation of the Gauss map of Γ\nfrom that of a sphere can be estimated by the same profile ψ.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 3\nThe profile ψ: (0,∞)→(0,∞) changes with the dimension. In fact, by\n[21] (and the improvement provided by [23] in the case N= 3), we have\nthat\n(1.8) ψ(t) =\n\nt ifN= 2,\ntmax[log(1/t),1] ifN= 3,\nt2/(N−1)ifN≥4,\nfort >0. The constant cin (1.6) only depends on the dimension N, the\ndiameterdΩof Ω, and the radii riandreof the uniform interior and exterior\nsphere conditions of Ω, as defined in Section 2. (We recall tha t those con-\nditions are equivalent to the C1,1regularity of Γ as shown in [2, Corollary\n3.14].) In [23] improvements of the profile are also provided forN≥4. In\nparticular, if Γ is of class C2,α, then [23, Theorem 4.4] ensures that (1.6)\nholds true with\n(1.9) ψ(t) =t4/(N+1)forN≥4,\nprovidedcdependson the C2,α-regularity of Γ, instead of its C1,1-regularity.\nIn this paper, we derive an integral identity, which general izes (1.2). It\nsimply involves any two solutions uandvof classC2(Ω) of (1.1), and reads\nas:\n(1.10)ˆ\nΩ(u−u)∆Pdx+ˆ\nΩ/a\\}b∇acketle{t(I−∇2v)∇u,∇u/a\\}b∇acket∇i}htdx=\nˆ\nΓ(u−u)/a\\}b∇acketle{t∇2u∇u,ν/a\\}b∇acket∇i}htdSx+1\n2ˆ\nΓ|∇u|2(uν+vν)dSx+\n−ˆ\nΓ/a\\}b∇acketle{t∇v,∇u/a\\}b∇acket∇i}htuνdSx−ˆ\nΓ(u−u)uνdSx+Nˆ\nΓ(u−u)(uν−vν)dSx.\nTheproofisasimpleapplication ofthedivergencetheoremt oanappropriate\ndifferential identity (see Theorem 3.1 for details).\nIn (1.10), we mean that\nu= max\nΩu= max\nΓu,\nso thatu−u>0 in Ω, by the strong maximum principle. Of course, if we\nrequire that usatisfies (1.3) or, more generally,\n(1.11) u=u0on Γ,\nfor someu0∈R, (1.10) will simply return (1.2) (in the latter case with −u\nreplaced by u−u=u0−u).\nThe significant generality of (1.10) opens the path to its app lication to\nold and new rigidity results for solutions of the Poisson equ ation (1.1). In\nthis paper, we shall test the effectiveness of (1.10) by analys ing the stability\nof what we shall call the reverse Serrin problem , as opposed to the classical\nSerrinproblem. Theformerconsistsinconsideringasoluti onof (1.1)subject\nto the Neumann condition\n(1.12) uν=Ron Γ,\nand then imposing that ualso satisfies (1.11).4 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nOf course, the rigidity part of this problem gives the same (s pherical)\nsolution of the classical Serrin problem. The stability par t is however not\nexplored.\nWe shall see in Section 6 that one can derive stability estima tes for the\nreverse Serrin problem by exploiting those already obtaine d for the classical\nSerrin problem. However, this procedure leads to estimates which require a\nstrong measure of the deviation of ufrom being constant on Γ. In fact, we\nneed to use the norm /ba∇dblu−u/ba∇dblC1,α(Γ), as shown in Theorem 6.2.\nInstead, the full power of (1.10) (in place of (1.2)) allows t o weaken such\na stringent deviation. In fact, in the two stability results we shall describe\nnext, the deviation of the function ufrom being a constant on Γ will be\nmeasured in terms of its oscillation ,\nosc\nΓu= max\nΓu−min\nΓu,\nand some suitable norm of its tangential gradient ∇Γu(see Section 2 for its\ndefinition). Our first stability result reads as follows.\nTheorem 1.1 (The reverse Serrin problem with uniform deviation) .Let\nΩ⊂RN,N≥2, be a bounded domain with boundary Γof classC2.\nSupposeu∈C2(Ω)is a solution of (1.1),(1.12)and letzbe any minimum\npoint ofuonΩ. Then it holds that\nρe−ρi≤cψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ/parenrightbigg\n,\nwhereψis given in (1.8). The constant c>0depends on N,dΩ,ri, andre.\nWe singled out this theorem since it is relatively easy to obt ain from our\nnew identity. Moreover, the dependence of the constant con the mentioned\nparameters can be written explicitly.\nIn the next result, we show how the deviation of ufrom being a constant\ncan be further weakened.\nTheorem 1.2 ( The reverse Serrin problem with weak deviation) .LetΩ⊂\nRN,N≥2, be a bounded domain with boundary Γof classC2,α.\nSupposeu∈C2(Ω)is a solution of (1.1),(1.12)and letzbe any minimum\npoint ofuonΩ. Then it holds that\nρe−ρi≤cψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl2,Γ/parenrightbigg\n,\nwhereψis given in (1.8). The constant c>0depends on N,dΩ,ri, andre.\nMoreover, for N≥4the profile ψcan be improved to (1.9). In this\ncase, the dependence of conriandremust be replaced with that on the\nC2,α-regularity of Γ.\nWe shall now describe the key points of the proofs of Theorems 1.1 and\n1.2 and highlight with the corresponding results in classic al Serrin problem.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 5\nThefirststep is torewrite (1.10) conveniently, when ualso satisfies (1.12).\nIn fact, in this case we obtain the following identity:\n(1.13)ˆ\nΩ(u−u)|∇2h|2dx=1\n2ˆ\nΓ|∇Γu|2hνdSx+\nˆ\nΓ(u−u)/bracketleftbig\n(N−1)RH−N/bracketrightbig\nhνdSx−ˆ\nΓ(u−u)/a\\}b∇acketle{t(∇ν)∇Γu,∇Γu/a\\}b∇acket∇i}htdSx.\nHere,h=q−u, whereqis given in (1.4). Also, His the mean curvature\nof Γ and ∇νis the Jacobian of a suitable extension to a neighborhood of\nΓ of the unit normal vector field νon Γ. Since up to suitable rotations\n∇νdepends on the principal curvatures of Γ, we realize that eac h surface\nintegral at the right-hand side of (1.13) contains a quantit y which is locally\ndefined on Γ and can be made small in terms of the relevant devia tions used\nin Theorems 1.1 and 1.2.\nIn order to obtain the new version (1.13) from (1.10), terms l ikeuνoruνν\nshould be replaced in (1.10). The latter term pops up when we c onsider the\nfirstsurfaceintegral in(1.10). Inorderto see that, weexte nd thevector field\nνto a tubular neighborhood of Γ and observe that ∇(uν) =/a\\}b∇acketle{t∇(uν),ν/a\\}b∇acket∇i}htν,\nsince Γ is a level surface of uν. This procedure leads to discover that\n/angbracketleftbig\n(∇2u)∇u,ν/angbracketrightbig\n=−/angbracketleftbig\n(∇ν)∇Γu,∇Γu/angbracketrightbig\n+uνuννon Γ,\nsince we also know that ( ∇ν)ν= 0 on Γ (see Section 2 for details).\nFurthermore, the term uννis treated by using the well-known formula for\nthe Laplace operator of a smooth function von a closed surface Σ:\n∆v=vνν+∆Σv+(N−1)Hvν.\nHere, ∆ ΣistheLaplace-Beltrami operatoronΣand Histhemeancurvature\nof Σ. (All these notations and computations are collected in Section 2.)\nThus, since usatisfies (1.1) and (1.12), we obtain that\nuνuνν=NR−(N−1)R2H−R∆Γuon Γ.\nFinally, if we want obtain the desired dependence on ∇Γu, we must use\nintegration by parts on Γ, i.e. apply the formula:ˆ\nΓ(u−u)∆ΓudSx=ˆ\nΓ|∇Γu|2dSx.\nThis last step and the pointwise analysis on Γ are peculiar of the reverse\nSerrin problem and were not needed in the classical counterp art.\nSince we assume that uν=Ron Γ the remaining integrals in (1.10) are\nthen easily treatable.\nNow, we focus our attention on the left-hand side of (1.10). N otice that,\nif we choose v=q, withqas in (1.4), the second volume integral in (1.10)\ndisappears (see also Theorem 3.2). Hence, the left-hand sid e of (1.13) is\neasily obtained by observing that |∇2h|2= ∆P. The harmonic function h\nwas also used in the treatment of (1.6) and can be viewed as a de viation of\nufromq, thus measuring how far the solution (and the domain) is from a\nspherical configuration.\nTherefore, in order to obtain the estimate in Theorem 1.2, th e plan is\nto show that the volume integral in (1.13) is small provided t he right-hand6 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nside of (1.13) becomes small in terms of some deviation of uon Γ from a\nconstant.\nNotice that if that integral is zero, then |∇2h|= 0, being as u−u >0\nby the strong maximum principle. Thus, his an affine function. As a\nconsequence, we obtain in passing a fairly general rigidity result, which\nholds in the case of fairly general domains.\nTheorem 1.3. LetΩ⊂RN,N≥2, be a bounded domain with boundary Γ\nof classC2. Letu∈C2(Ω)be a solution of (1.1),(1.12).\nIf the right-hand side of (1.13)is non-positive, then his affine, and hence\nuis a quadratic polynomial. As a result, Ωis a ball.\nA version of this theorem for the solution of (1.1), (1.3) was proved in\n[20]. Of course, the condition on the sign of the right-hand s ide of (1.13) in\nTheorem 1.3 is satisfied if uis also constant on Γ.\nFinally, in order to treat the stability issue, one observes that the affine\nfunctionhin Theorem 1.3 can be further normalized to a constant by an\nappropriate choice of the center zof the paraboloid q. With such a choice,\nif the volume integral (1.13) is small, then his close to a constant. Heuris-\ntically, this means that the oscillation of hon Γ can be controlled by a\nweighted norm of |∇2h|. Now, since we have that\n1\n2(ρ2\ne−ρ2\ni) = osc\nΓq≤osc\nΓh+osc\nΓu,\nthedesiredestimate for ρe−ρican thenbederivedfromoneof theoscillation\nofhon Γ in terms of the aforementioned weighted integral, which can be\nderived from some inequalities proved in [23].\nThe structure of this paper is as follows. In Section 2, we col lect some\npreliminary formulas and results for functions defined on su rfaces. Section 3\ncontains ourgeneral identity anditscorollaries, togethe r withrigidity results\nrelated to the reverse Serrin problem, including the proof o f Theorem 1.3.\nIn Section 4, we gather some estimates for the Neumann proble m, which are\ninstrumental to trace the dependence of the constant cof Theorem 1.2 on\nthe relevant parameters. The proofs of Theorems 1.1 and 1.2 a re contained\nin Section 5. Section 6 contains the alternative stability r esult for the Serrin\nreverse problem (Theorem 6.2) obtained via existing result s for the classical\nSerrin problem.\n2.Notations and preliminary formulas\nLet Σ⊂RNbe a surface of class C2, which is the boundary of a bounded\ndomainD⊂RNand consider a function v∈C2(D).\nWe use the following decomposition for the gradient ∇vofvon Σ:\n(2.1) ∇v=vνν+∇Σv.\nHere,νdenotes the exterior unit normal vector field to Σ. This decom posi-\ntion defines what we shall call the tangential gradient ∇Σvofvon Σ. It is\nclear that, if wis another function in C2(D), then\n/a\\}b∇acketle{t∇v,∇w/a\\}b∇acket∇i}ht=vνwν+/a\\}b∇acketle{t∇Σv,∇Σw/a\\}b∇acket∇i}ht.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 7\nWe recall the well-known formula (see [15]):\n(2.2) ∆ v=vνν+∆Σv+(N−1)Hvνon Σ.\nHere, ∆ Σis the Laplace-Beltrami operator on Σ and Hdenotes the mean\ncurvature of Σ defined by\nH=1\nN−1N−1/summationdisplay\nj=1κj,\nwhereκ1,...,κ N−1are the principal curvatures of Σ with respect to the\ninterior normal.\nWe also recall from [15] the following formula of integratio n by parts on\nΣ, which holds for any v,w∈C2(D):\n(2.3)ˆ\nΣ/a\\}b∇acketle{t∇Σv,∇Σw/a\\}b∇acket∇i}htdSx=−ˆ\nΣv∆ΣwdSx.\nIn particular, we have that´\nΣ∆ΣwdSx= 0.\nWe can always extend the vector field νsmoothly to a tubular neighbor-\nhood of Σ in D. For instance, we can set ν=−η∇δΓ, whereδΣis the\ndistance function , defined by\nδΣ(x) = dist(x,Σ) forx∈D,\nandηis a cut-off function which equals 1 in a neighborhood of Γ. In f act,\nwe know from [16] that there is neighborhood Uof Σ inDsuch thatδΓ∈\nC2,α(U) and we have that\n|∇δΣ|= 1 inU.\nHence,ηis chosen as a smooth function with support contained in the s et\ninUand such that η≡1 near Σ.\nAt any point in Σ, we can choose coordinates (see, e.g., [14, A ppendix\n14.6]) so that\n∇ν=−∇2δΣ=\nκ1···0 0\n............\n0···κN−10\n0···0 0\n.\nThroughout the rest of the paper, unless explicitly indicat ed,\nΩ is a bounded domain in RN,N≥2, with boundary Γ\nsatisfying the uniform interior and exterior sphere condit ions.(2.4)\nWe denote by dΩthe diameter of Ω. Also, we recall the standard defi-\nnitions of uniform interior and exterior sphere conditions , whose respective\nradii will be denoted by riandre. Namely, there exists ri>0 (resp.re>0)\nsuch that for each p∈Γ it holds that B∩Γ ={p}for some ball B⊂Ω\n(resp.B⊂RN\\Ω) of radius ri>0 (resp.re>0). As already mentioned,\nthe two properties are equivalent to the C1,1-regularity of Γ, as shown in [2].\nThe following lemma will be useful. For simplicity of notati on, we do not\nusean explicit sign convention for themeancurvature H. Inother words, we\nmean that, on each component Σ of Γ, His defined as just above described.\nLemma 2.1. Letu∈C2(Ω)be a solution of (1.1), andc∈R. It holds that8 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\n(i) ifu=conΓ, then\n/angbracketleftbig\n∇2u∇u,ν/angbracketrightbig\n=uν[N−(N−1)Huν]onΓ;\n(ii) ifuν=conΓ, then\n/angbracketleftbig\n∇2u∇u,ν/angbracketrightbig\n=−/a\\}b∇acketle{t∇ν∇Γu,∇Γu/a\\}b∇acket∇i}ht+uν[N−∆Γu−(N−1)Huν]onΓ.\nProof.(i) Since Γ is a level surface for u, by (2.1) we infer that ∇u=uνν,\nand hence/angbracketleftbig\n∇2u∇u,ν/angbracketrightbig\n=uνuνν.\nMoreover, (2.2) with v=ugives that\n(2.5) uνν=N−∆Γu−(N−1)Huνon Γ.\nSince ∆ Γu= 0 on Γ, the desired identity ensues at once.\n(ii) We proceed as already described by extending νto a neighborhood\nU. The ensuing computations are generally performed in U, and hence\nevaluated on Γ.\nSince Γ is a level surface of /a\\}b∇acketle{t∇u,ν/a\\}b∇acket∇i}ht, we have that\n∇/a\\}b∇acketle{t∇u,ν/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t∇/a\\}b∇acketle{t∇u,ν/a\\}b∇acket∇i}ht,ν/a\\}b∇acket∇i}htν\non Γ. Next, we compute that\n∇/a\\}b∇acketle{t∇u,ν/a\\}b∇acket∇i}ht= (∇2u)ν+∇ν∇u=\n(∇2u)ν+uν(∇ν)ν+∇ν∇Γu= (∇2u)ν+∇ν∇Γu.\nHere, we have used that ( ∇ν)ν= 0 inU, sinceνis unitary.\nThus, we infer that\n(∇2u)ν+(∇ν)∇Γu=\n/bracketleftbig\nuνν+/a\\}b∇acketle{t(∇ν)∇Γu,ν/a\\}b∇acket∇i}ht/bracketrightbig\nν=/bracketleftbig\nuνν+/a\\}b∇acketle{t(∇ν)ν,∇Γu/a\\}b∇acket∇i}ht/bracketrightbig\nν=uννν.\nHence, multiplying by ∇ugives that\n/angbracketleftbig\n(∇2u)∇u,ν/angbracketrightbig\n=\n−/angbracketleftbig\n(∇ν)∇Γu,∇u/angbracketrightbig\n+uνuνν=−/angbracketleftbig\n(∇ν)∇Γu,∇Γu/angbracketrightbig\n+uνuνν.\nHere, we have again used (2.1) and ( ∇ν)ν= 0. Therefore, the desired\nidentity follows by (2.5). /square\nRemark 2.2. From the aforementioned properties of ∇ν, we easily infer\nthat\nκ|∇Γu|2≤/angbracketleftbig\n(∇ν)∇Γu,∇Γu/angbracketrightbig\n≤κ|∇Γu|2on Γ,\nwhere\nκ= min/bracketleftbig\nκ1,...,κ N−1/bracketrightbig\nandκ= max/bracketleftbig\nκ1,...,κ N−1/bracketrightbig\nat each point of Γ. We also recall that\n−1\nre≤κ≤κ≤1\nrion Γ,\nunder our assumption (2.4).AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 9\n3.Integral identities and rigidity results\nIn this section, we shall derive the identities (1.10) and (1 .13). We will\nalso prove Theorem 1.3.\nTheorem 3.1 (General identity) .Suppose Ω⊂RN,N≥2, is a bounded\ndomain as defined in (2.4). Letu,v∈C2(Ω)be solutions of (1.1).\nSet\nu= max\nΩu= max\nΓu,\nand\nP=1\n2|∇u|2+(u−u)onΩ.\nThen the identity (1.10)holds, that is:\nˆ\nΩ(u−u)∆Pdx+ˆ\nΩ/a\\}b∇acketle{t(I−∇2v)∇u,∇u/a\\}b∇acket∇i}htdx=\nˆ\nΓ(u−u)/a\\}b∇acketle{t∇2u∇u,ν/a\\}b∇acket∇i}htdSx+1\n2ˆ\nΓ|∇u|2(uν+vν)dSx+\n−ˆ\nΓ/a\\}b∇acketle{t∇v,∇u/a\\}b∇acket∇i}htuνdSx−ˆ\nΓ(u−u)uνdSx+Nˆ\nΓ(u−u)(uν−vν)dSx.\nProof.The proof proceeds by direct computations and an applicatio n of the\ndivergence theorem. By simply taking derivatives and by str aightforward\nalgebraic manipulations, we first infer that the following k ey differential\nidentity holds:\n(u−u)∆P+/a\\}b∇acketle{t(I−∇2v)∇u,∇u/a\\}b∇acket∇i}ht=\ndiv/braceleftbigg\nP∇u+(u−u)∇P+1\n2|∇u|2∇v−/a\\}b∇acketle{t∇v,∇u/a\\}b∇acket∇i}ht∇u/bracerightbigg\n+div{(N−1)(u−u)∇u−N(u−u)∇v}.\nThus, an application of the divergence theorem yields:\nˆ\nΩ(u−u)∆Pdx+ˆ\nΩ/a\\}b∇acketle{t(I−∇2v)∇u,∇u/a\\}b∇acket∇i}htdx=\nˆ\nΓ/braceleftBig\nPuν+(u−u)Pν+1\n2|∇u|2vν−/a\\}b∇acketle{t∇v,∇u/a\\}b∇acket∇i}htuν/bracerightBig\ndSx+\nˆ\nΓ/braceleftbig\n(N−1)(u−u)uν−N(u−u)vν/bracerightbig\ndSx.\nHence, we infer that\nˆ\nΩ(u−u)∆Pdx+ˆ\nΩ/a\\}b∇acketle{t(I−∇2v)∇u,∇u/a\\}b∇acket∇i}htdx=ˆ\nΓ/braceleftBig1\n2|∇u|2+(u−u)/bracerightBig\nuνdSx+\nˆ\nΓ/braceleftBig\n/a\\}b∇acketle{t∇2u∇u,ν/a\\}b∇acket∇i}ht−uν/bracerightBig\n(u−u)dSx+ˆ\nΓ/braceleftBig1\n2|∇u|2vν−/a\\}b∇acketle{t∇v,∇u/a\\}b∇acket∇i}htuν/bracerightBig\ndSx+\nˆ\nΓ/braceleftbig\n(N−1)(u−u)uν−N(u−u)vν/bracerightbig\ndSx.\nIdentity (1.10) then ensues by rearranging the terms. /square10 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nWe now present a corollary of the previous result, when vis a quadratic\nsolutionqof (1.1).\nCorollary 3.2 (Mother identity for the quadratic case) .Suppose Ω⊂RN,\nN≥2, is a bounded domain as defined in (2.4).\nLetu,u, andPbe as in Theorem 3.1 and let qbe a quadratic polynomial\nof the form (1.4). Then, the following identity holds:\n(3.1)ˆ\nΩ(u−u)∆Pdx=\n1\n2ˆ\nΓ|∇u|2/parenleftbig\nuν+qν/parenrightbig\ndSx−ˆ\nΓ/a\\}b∇acketle{t∇q,∇u/a\\}b∇acket∇i}htuνdSx+\nˆ\nΓ(u−u)/a\\}b∇acketle{t∇2u∇u,ν/a\\}b∇acket∇i}htdSx+ˆ\nΓ(u−u)/bracketleftbig\n(N−1)uν−Nqν/bracketrightbig\ndSx.\nAn alternative formula is:\n(3.2)ˆ\nΩ(u−u)∆Pdx=1\n2ˆ\nΓu2\nν/parenleftbig\nuν−qν/parenrightbig\ndSx+\n1\n2ˆ\nΓ|∇Γu|2/parenleftbig\nuν+qν/parenrightbig\ndSx−ˆ\nΓ/a\\}b∇acketle{t∇Γq,∇Γu/a\\}b∇acket∇i}htuνdSx+\nˆ\nΓ(u−u)/a\\}b∇acketle{t∇2u∇u,ν/a\\}b∇acket∇i}htdSx+ˆ\nΓ(u−u)/bracketleftbig\n(N−1)uν−Nqν/bracketrightbig\ndSx.\nProof.Weapply (1.10)with v=q, sinceqisclearly asolution of (1.1) onthe\nwholeRN. We have that ∇2q=I. As a result, the second volume integral\nat the left-hand side of (1.10) vanishes, and hence (3.1) ens ues. Formula\n(3.2) then simply follows by applying (2.1) and rearranging some terms. /square\nNext, we consider the case in which uνis constant on Γ with noconstraint\non the values of uon Γ.\nCorollary 3.3 (Constant Neumann condition) .Suppose Ω⊂RN,N≥2,\nis a bounded domain as defined in (2.4).\nLetqbe as in (1.4)and seth=q−u, whereu∈C2(Ω)is a solution\nof(1.1). Further, suppose that usatisfies (1.12). Then the identity (1.13)\nholds, that is:\nˆ\nΩ(u−u)|∇2h|2dx=1\n2ˆ\nΓ|∇Γu|2hνdSx+\nˆ\nΓ(u−u)/bracketleftbig\n(N−1)RH−N/bracketrightbig\nhνdSx−ˆ\nΓ(u−u)/a\\}b∇acketle{t(∇ν)∇Γu,∇Γu/a\\}b∇acket∇i}htdSx.\nProof.We work on (3.2). As already observed, we compute that ∆ P=\n|∇2h|2. Next, we compute the surface integrals at the right-hand si de. We\nbegin withˆ\nΓu2\nν/parenleftbig\nuν−qν/parenrightbig\ndSx=R2ˆ\nΓ/parenleftbig\nuν−qν/parenrightbig\ndSx= 0,AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 11\nthanks to the divergence theorem. Next, we compute that\n1\n2ˆ\nΓ|∇Γu|2/parenleftbig\nuν+qν/parenrightbig\ndSx−ˆ\nΓ/a\\}b∇acketle{t∇Γq,∇Γu/a\\}b∇acket∇i}htuνdSx=\n1\n2ˆ\nΓ|∇Γu|2/parenleftbig\nR+qν/parenrightbig\ndSx−Rˆ\nΓ(u−u)∆ΓqdSx.\nHere, we used (1.12) and (2.3), in the second integral.\nMoreover, we infer that\nˆ\nΓ(u−u)∆ΓqdSx=−(N−1)ˆ\nΓ(u−u)HqνdSx+(N−1)ˆ\nΓ(u−u)dSx.\nThis follows from the fact that taking v=qin (2.2) gives:\n∆Γq= (N−1)(1−Hqν/parenrightbig\non Γ.\nNow, we apply item (ii) of Lemma 2.1 and get:\nˆ\nΓ(u−u)/a\\}b∇acketle{t∇2u∇u,ν/a\\}b∇acket∇i}htdSx=−ˆ\nΓ(u−u)/a\\}b∇acketle{t(∇ν)∇Γu,∇Γu/a\\}b∇acket∇i}htdSx+\nRˆ\nΓ(u−u)[N−∆Γu−(N−1)RH]dSx.\nMoreover, thanks to (2.3) we compute thatˆ\nΓ(u−u)∆ΓudSx=ˆ\nΓ|∇Γu|2dSx.\nThus, we can write that\nˆ\nΓ(u−u)/a\\}b∇acketle{t∇2u∇u,ν/a\\}b∇acket∇i}htdSx=−ˆ\nΓ(u−u)/a\\}b∇acketle{t(∇ν)∇Γu,∇Γu/a\\}b∇acket∇i}htdSx+\n−(N−1)R2ˆ\nΓ(u−u)HdSx−Rˆ\nΓ|∇Γu|2dSx+NRˆ\nΓ(u−u)dSx.\nSumming up all these formulas then gives (1.13), after strai ghtforward\nalgebraic manipulations. /square\nWe now present a quadratic-radial symmetry test. Its analog ous for the\nsolution of (1.1), (1.3) essentially provides the spherica l detector used in\n[31, 27, 3, 19, 20] (see also [29, Lemma 1.9] for additional de tails). Here, we\nrather focus on solutions of (1.1), (1.12). For comparison, we also provide\n(in item (i)) a version of its analogous for the solution of (1.1), (1.11 ).\nLemma 3.4 (Radial symmetry test) .LetΩbe a bounded domain in RNas\ndefined in (2.4).\nLetu∈C2(Ω)be a solution of (1.1)satisfying either (1.11)or(1.12). If\n∆P≡0inΩ, thenuis a quadratic polynomial and Γis a sphere of radius\nRgiven by (1.5).\nProof.We know that\n∆P=|∇2u|2−(∆u)2\nN=|∇2u|2−/a\\}b∇acketle{t∇2u,I/a\\}b∇acket∇i}ht2\nN≥0.\nHere,/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htdenotes the inner product in RN2and hence the last inequality\nfollows by the Cauchy-Schwarz inequality. Thus, the fact th at ∆P≡012 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\ngives that the N2-vectors ∇2uandIcoincide in Ω, being as (1.1) in force.\nTherefore,uis a quadratic polynomial of the form (1.4) for some z∈RN\nanda∈R(see also [19, 20] and [29, Lemma 1.9]).\n(i) If (1.11) holds then we have that |x−z|2= 2(u0−a) for every\nx∈Γ. Hence, Γ must be a sphere of radius/radicalbig\n2(u0−a) and we have that\nx−z=/radicalbig\n2(u0−a)ν(x). Thus, we obtain that\n2(u0−a)|Γ|=ˆ\nΓ|x−z|2dSx=\n/radicalbig\n2|u0−a|ˆ\nΓ/a\\}b∇acketle{tx−z,ν(x)/a\\}b∇acket∇i}htdSx=N|Ω|/radicalbig\n2|u0−a|.\nConsequently, (1.5) gives that 2( u0−a) =R2,\nu(x) =1\n2(|x−z|2−R2)+u0,\nand Γ is a sphere centered at zof radiusR.\n(ii) If (1.12) holds, since uis a quadratic polynomial as in (1.4), we infer\nthat\n(3.3) /a\\}b∇acketle{tx−z,ν(x)/a\\}b∇acket∇i}ht=uν(x) =Rforx∈Γ.\nNext, letxmandxMbe points in Γ such that\n|xm−z|= min\nx∈Γ|x−z|and|xM−z|= max\nx∈Γ|x−z|.\nIt is clear that xm−zandxM−zare parallel to ν(xm) andν(xM). Hence,\n(3.3) gives that |xm−z|=|xM−z|=R, i.e.|x−z|=Rfor everyx∈Γ.\nThus, again, Γ must be the sphere centered at zof radiusR. /square\nA consequence of item (ii) of this lemma and Corollary 3.3 is T heorem\n1.3, the general rigidity result stated in the Introduction .\nProof of Theorem 1.3. Theassumptionontheright-handsideof (1.13)forces\nthe function |∇2h|2= ∆Pto be identically zero, being as u−u>0 by the\nstrong maximum principle. Our claim then follows from Lemma 3.4./square\n4.Estimates for solutions of the Neumann problem\nIn this section, we derive some geometric and spectral estim ates for so-\nlutions of (1.1), (1.12), which will be useful to trace the de pendence of the\nconstantcin Theorems 1.1 and 1.2.\nWe begin with a geometric inequality that adapts [20, Lemma 3 .1], ob-\ntained for the solution of (1.1), (1.11).\nLemma 4.1. LetΩbe a bounded domain in RN. Letu∈C2(Ω)be a\nsolution of (1.1). Then, it holds that\nu−u(x)≥1\n2δΓ(x)2for everyx∈Ω.\nMoreover, if Γis of classC1and satisfies the uniform interior sphere con-\ndition with radius ri, then\nu−u(x)≥1\n2riδΓ(x)for everyx∈Ω.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 13\nProof.Let us define by fthe unique solution of\n∆f=Nin Ω, f= 0 on Γ.\nThen, by comparison u−u≥ −finΩ, and the conclusion follows by [20,\nLemma 3.1]. /square\nWe shall now derive a priori estimates on the harmonic functi onh=q−u\nanditsderivatives, intermsofgeometric, spectral, andre gularityparameters\nof the domain Ω.\nIn what follows, we shall use the fact that a function usatisfying (1.1)\nand (1.12) minimizes the functional E:W1,2(Ω)→Rdefined by\nE(v) =1\n2ˆ\nΩ|∇v|2dx+Nˆ\nΩvdx−Rˆ\nΓvdSx,\nfor anyv∈W1,2(Ω). In the following result, ν2(Ω),σ2(Ω), denote the\nsecond (non-trivial) Neumann and Steklov eigenvalues. Res pectively, they\nare defined by\nν2(Ω) = min/braceleftbiggˆ\nΩ|∇v|2dx:v∈W1,2(Ω),ˆ\nΩv2dx= 1,ˆ\nΩvdx= 0/bracerightbigg\n.\nand\nσ2(Ω) = min/braceleftbiggˆ\nΩ|∇v|2dx:v∈W1,2(Ω),ˆ\nΓv2dSx= 1,ˆ\nΓvdSx= 0/bracerightbigg\n.\nProposition 4.2 (L2bound for oscillation) .LetΩbe a bounded domain in\nRNas defined in (2.4). Letu∈C2(Ω)be a solution of (1.1),(1.12)and set\nh=q−u, withqgiven by (1.4). Then, it holds that\n(4.1) /ba∇dblh−hΩ/ba∇dbl2,Ω≤2/ba∇dblR−qν/ba∇dbl2,Γ/radicalbig\nν2(Ω)σ2(Ω).\nHere,hΩdenotes the mean value of honΩ.\nProof.Sincehis harmonic in Ω and hν=qν−Ron Γ, it is clear that h\nminimizes in W1,2(Ω) the functional defined by\nW1,2(Ω)∋v/ma√sto→1\n2ˆ\nΩ|∇v|2dx−ˆ\nΓv(qν−R)dSx.\nBy takingv≡1, the minimality of hthen gives thatˆ\nΩ|∇h|2dx≤2ˆ\nΓh(qν−R)dSx= 2ˆ\nΓ(h−hΓ)(qν−R)dSx,\nwherehΓis the mean value of hon Γ. Here, we used that qν−Rhas a\nzero mean value on Γ. The Cauchy-Schwarz inequality and the d efinition of\nσ2(Ω) then give that\nˆ\nΩ|∇h|2dx≤2/ba∇dblqν−R/ba∇dbl2,Γ/ba∇dblh−hΓ/ba∇dbl2,Γ≤2/ba∇dblqν−R/ba∇dbl2,Γ/radicalbig\nσ2(Ω)/ba∇dbl∇h/ba∇dbl2,Ω,\nand hence\n/ba∇dblh−hΩ/ba∇dbl2\n2,Ω≤ν2(Ω)−1ˆ\nΩ|∇h|2dx≤4/ba∇dblqν−R/ba∇dbl2\n2,Γ\nν2(Ω)σ2(Ω),\nby the Poincar´ e inequality. Thus, (4.1) ensues. /square14 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nRemark 4.3. Thanks to [4, Theorems 1 and 2], we have that, within the\nclass of uniformly bounded and uniformly Lipschitz domains ,ν2(Ω) and\nσ2(Ω) are bounded from below by a positive constant independen t of Ω.\nTheorem 4.4 (Uniform bound for the derivatives of h).Setm= 1or\nm= 2. LetΩbe a bounded domain in RNof classCm,α,0≤α<1. Letu\nbe a solution of (1.1),(1.12)and seth=q−u, withqgiven by (1.4). Then,\nthere exists a constant Cthat depends on N,dΩ, and theCm,αregularity of\nΓ, such that\n/ba∇dblh−hΩ/ba∇dblm,α;Ω≤C,\nwere/ba∇dbl·/ba∇dblm,α;Ωdenotes the norm in Cm,α(Ω).\nProof.For simplicity of notation, in this proof by the same letter cwe shall\ndenote a generic constant, whose dependence will be specifie d if needed.\nWe have that his harmonic in Ω and hν=qν−Ron Γ. We thus can\napply a standard a priori estimate for the Neumann problem (s ee e.g., [17,\nChapter 7]; see also[14, Theorem 6.30] for m= 2) to obtain that\n(4.2) /ba∇dblh/ba∇dblm,α;Ω≤c/parenleftbigg\nmax\nΩ|h−hΩ|+/ba∇dblqν−R/ba∇dblm−1,α;Γ/parenrightbigg\n.\nHere,cdepends on N,m,α,dΩ, and theCm,αregularity of Γ. By arguing\nas in [24, Lemma 4.9] (see also [25]) we can find a constant csuch that\n(4.3) max\nΩ|h−hΩ|= max\nΓ|h−hΩ| ≤c/braceleftbigg/ba∇dblh−hΩ/ba∇dbl2,Ω\nσN/2+σ/ba∇dbl∇h/ba∇dbl∞,Ω/bracerightbigg\n,\nfor any 0<σ < σ 0. Here,candσ0only depend on N,dΩ, and theCm,α-\nregularity of Γ.3\nNow, if we choose σ= min{1/(2c2),σ0}(here,cis the maximum of the\ntwo constants appearing in (4.2) and (4.3)) and we plug (4.3) into (4.2), we\neasily find that\n/ba∇dblh/ba∇dblm,α;Ω≤c(/ba∇dblh−hΩ/ba∇dbl2,Ω+/ba∇dblqν−R/ba∇dblm−1,α;Γ),\nbyusingthetrivial inequality /ba∇dbl∇h/ba∇dbl∞,Ω≤ /ba∇dblh/ba∇dblm,α;Ω.Here,ccanbeexplicitly\nderived in terms of σ0and the constants in (4.2) and (4.3).\nTheterm /ba∇dblqν−R/ba∇dblm−1,α;Γcan beeasily boundedby a constant depending\non the same relevant parameters. Finally, Proposition 4.2 a nd Remark 4.3\nensure that the norm /ba∇dblh−hΩ/ba∇dbl2,Ωcan be bounded by a constant depending\non the same relevant parameters. The desired conclusion the n ensues. /square\n5.Quantitative symmetry results\nIn this section, based on the identity (1.13), we shall provi de our stabil-\nity estimates for the radial symmetry in the reverse Serrin o verdetermined\nproblem (Theorems 1.1 and 1.2). To this aim, for a point z∈Ω, we recall\nthe definition (1.7):\nρi= min\nx∈Γ|x−z|andρe= max\nx∈Γ|x−z|.\n3This argument gives a constant depending on Nand the parameters related to a\nuniform interior cone condition. This condition is equival ent to a Lipschitz-type regularity\nof Γ (see [24] for details). Of course, the relevant paramete rs can be bounded in terms of\ntheC1,αregularity of Γ.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 15\nIt is clear that ρiis the radius of the largest ball contained in Ω and ρeis\nthe radius of the smallest ball that contains Ω, both balls ce ntered atz.\nThus, for a solution uof (1.1), (1.12), we want to bound the difference\nρe−ρiin terms of the sum of the oscillation of uon Γ and a suitable Lebegue\nnorm of |∇Γu|on Γ.\nRemark 5.1. Letube a solution of (1.1) , (1.12). A convenient choice for\nthe pointzin (1.7) is any minimum point of uinΩ. In fact, such a point\nmust fall in Ω, since otherwise uν≤0 at that point, contrary to (1.12).\nWe shall first adapt to our case some estimates on the oscillat ion of a\nharmonic function, which were derived in [23].\n5.1.Interpolating estimates for Sobolev functions. We now recall\nthree key inequalities for the oscillation of a harmonic fun ction. These de-\nscend from some interpolation inequalities proved in [23, T heorem 4.1] (see\nalso [24, Appendix]). Here, we present these inequalities f orC2domains, as\nit is enough for the purposes of the present paper. Neverthel ess, the orig-\ninal theorem was stated and proved for a bounded domain satis fying the\n(θ,a)-uniform cone condition (see [24, Appendix] for a definitio n).\nLemma 5.2. LetΩ⊂RN,N≥2, be a bounded domain with boundary Γ\nof classC2, and letube solution of (1.1),(1.12).\nLeth=qz−u, whereqzis that in (1.4)andzis any global minimum\npoint ofuinΩ. Then, we have that\nosc\nΓh≤c\n\n/ba∇dbl√δΓ∇2h/ba∇dbl2,Ω ifN= 2,\n/ba∇dbl√δΓ∇2h/ba∇dbl2,Ωmax/bracketleftBig\nlog/parenleftBige/ba∇dbl∇h/ba∇dbl∞,Ω\n/ba∇dbl√δΓ∇2h/ba∇dbl2,Ω/parenrightBig\n,1/bracketrightBig\nifN= 3,\n/ba∇dbl∇h/ba∇dblN−3\nN−1\n∞,Ω/ba∇dbl√δΓ∇2h/ba∇dbl2\nN−1\n2,ΩifN≥4,\nwherecis an explicit constant that only depends on N,dΩ,ri, and a lower\nbound forδΓ(z).\nProof.By definition of hand the choice of z, we have that his harmonic\nand that ∇h(z) = 0 at the point z∈Ω.\nThe desired result is essentially contained in [23, Theorem 4.1]. To be\nprecise, that theorem gives an estimate for the difference ρe−ρi, which is\nthere related to the oscillation of the particular harmonic functionv=q−f,\nwherefis the solution of the Dirichlet problem (1.1), (1.3) and qis given\nin (1.4). Indeed there holds that\nmax\nΓv−min\nΓv=1\n2(ρ2\ne−ρ2\ni)\nand\n(5.1) ρ2\ne−ρ2\ni≥/parenleftbigg|Ω|\n|B|/parenrightbigg1\nN\n(ρe−ρi).\nNow, ifuis a solution of the Neumann problem (1.1), (1.12), instead, u\nis no longer constant on Γ. Nevertheless, a careful inspecti on reveals that\nthe inequalities contained in [23, Theorem 4.1] concern the oscillation on Γ\nof any harmonic function in Ω, whose gradient vanishes at a gi ven pointz16 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nin Ω, and that the constant conly depends on N,dΩ,ri, and a lower bound\nof the distance δΓ(z) ofzto Γ. Thus, the desired result follows. /square\nRemark 5.3. By adapting the arguments of [21, Remark 2.9] (see also [22]\nfor generalizations) to the present setting, we can choose x∈Ω such that\nδΓ(x) =ri,y∈Γ such that |y−z|=δΓ(z) and, recalling Lemma 4.1,\ncompute that\nr2\ni\n2≤u−u(x)≤u+max\nΩ(−u) =u−u(z) =\nu−u(y)+u(y)−u(z)≤osc\nΓu+/ba∇dbl∇u/ba∇dbl∞,ΩδΓ(z),\nfrom which we get that\nδΓ(z)≥r2\ni−2osc\nΓu\n2/ba∇dbl∇u/ba∇dbl∞,Ω.\nIn the following lemma, we use the tangential norm defined by\n/ba∇dblu−u/ba∇dbl2\n1,2,τ,Γ=/ba∇dblu−u/ba∇dbl2\n2,Γ+/ba∇dbl∇Γu/ba∇dbl2\n2,Γ.\nLemma 5.4. LetΩ⊂RN,N≥2, be a bounded domain with boundary Γ\nof classC2, and letube a solution of (1.1),(1.12).\nLeth=q−u, whereqis that in (1.4)andzis any global minimum point\nofuinΩ. Then, we have thatˆ\nΩ(u−u)|∇2h|2dx≤c/bracketleftBig\n1+osc\nΓu/bracketrightBig\n/ba∇dblu−u/ba∇dbl2\n1,2,τ,Γ,\nwherecis an explicit constant that only depends on N,dΩ,ri,re, and a lower\nbound forδΓ(z).\nProof.For simplicity of notation, in this proof by the same letter cwe shall\ndenote a generic constant, whose dependence will be specifie d if needed.\nWe start working on the right-hand side (1.13). By Remark 2.2 and\nthe Holder inequality, we see that there is a constant c, only depending on\nN,dΩ,reandri, such that\nˆ\nΩ(u−u)|∇2h|2dx≤\nc/bracketleftbiggˆ\nΓ|∇Γu|2dSx+/ba∇dblhν/ba∇dbl2,Γ/ba∇dblu−u/ba∇dbl2,Γ+/ba∇dblu−u/ba∇dbl∞,Γˆ\nΓ|∇Γu|2dSx/bracketrightbigg\n.\nHere, we have used the fact that |hν| ≤dΩ+Ron Γ.\nNext, let us define by fthe unique solution of\n∆f=Nin Ω, f= 0 on Γ.\nBy [21, Lemma 2.5] applied with v=h, we obtain thatˆ\nΓh2\nνdSx≤ˆ\nΓ|∇h|2dSx≤cˆ\nΩ(−f)|∇2h|2dx\nwherecis an explicit constant that only depends on N,dΩ,ri,and a lower\nbound forδΓ(z). Since, by comparison, u−u≥ −fonΩ, we then infer thatˆ\nΓh2\nνdSx≤cˆ\nΩ(u−u)|∇2h|2dx.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 17\nHence, this formula and the trivial inequalities\n/ba∇dblu−u/ba∇dbl∞,Γ≤osc\nΓu,/ba∇dblu−u/ba∇dbl2,Γ≤ /ba∇dblu−u/ba∇dbl1,2,τ,Γ,/ba∇dbl∇Γu/ba∇dbl2,Γ≤ /ba∇dblu−u/ba∇dbl1,2,τ,Γ\ngive that\nˆ\nΩ(u−u)|∇2h|2dx≤\nc/ba∇dblu−u/ba∇dbl1,2,τ,Γ/bracketleftBigg/parenleftBig\n1+osc\nΓu/parenrightBig\n/ba∇dblu−u/ba∇dbl1,2,τ,Γ+/radicalBiggˆ\nΩ(u−u)|∇2h|2dx/bracketrightBigg\n.\nThis inequality easily gives the desired bound. /square\nWe are now in position to prove our quantitative estimates fo r the radial\nsymmetry in the reverse Serrin problem.\nProof of Theorem 1.1. Set\n(5.2) σ= min/braceleftbigg\n1,r2\ni\n4/bracerightbigg\n.\nWe first notice that if\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ≥σ,\nthen the conclusion trivially holds true. In fact, by recall ing the definition\nofψin (1.8), we easily get that\nρe−ρi≤dΩ≤/braceleftBigg\ndΩ\nσψ(oscΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ) ifN= 2,3,\ndΩ\nσ2/(N−1)ψ(oscΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ) ifN≥4.\nThus, it only remains to prove the result when\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ<σ.\nUnder this assumption and recalling (1.1) and (1.12), it is c lear that\n(5.3) /ba∇dbl∇u/ba∇dbl∞,Ω=/ba∇dbl∇u/ba∇dbl∞,Γ=/ba∇dbl∇Γu+uνν/ba∇dbl∞,Γ<σ+R≤σ+dΩ.\nThelastinequalityfollowsfrom(1.5)andbyputtingtogeth ertheisoperimet-\nric inequality |Γ| ≥N|B|1/N|Ω|(N−1)/Nand the trivial bound |Ω| ≤ |B|dN\nΩ.\nAs usual, we consider the harmonic function h=q−uand choose the\nminimum point of qto coincide with the point zin our assumptions. Since\nzis a minimum point for both qandu, we have that ∇h(z) = 0.\nThanks to Lemma 4.1, the left-hand side of identity (1.13) ca n be esti-\nmated from below as\n(5.4)1\n2riˆ\nΩδΓ|∇2h|2dx≤ˆ\nΩ(u−u)|∇2h|2dx.\nNow, from (5.1) we infer that\n(5.5)1\n2/parenleftbigg|Ω|\n|B|/parenrightbigg1\nN\n(ρe−ρi)≤osc\nΓq≤osc\nΓh+osc\nΓu.18 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nCombiningLemma5.2, Lemma5.4,(5.4),andthetrivialinequ ality√\na2+b2≤\n|a|+|b|, we infer that\n(5.6) osc\nΓh≤cψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ/parenrightbigg\n,\nfor some explicit constant cthat only depends on N,dΩ,re, andri. Here,\nwe used two facts. Firstly, (5.3), (5.2) and the trivial boun d/ba∇dbl∇q/ba∇dbl∞,Ω≤dΩ\ngive the explicit upper bound for |∇h|:\n/ba∇dbl∇h/ba∇dbl∞,Ω≤ /ba∇dbl∇q/ba∇dbl∞,Ω+/ba∇dbl∇u/ba∇dbl∞,Ω≤min/braceleftbigg\n1,r2\ni\n4/bracerightbigg\n+2dΩ.\nSecondly, combining Remark 5.3, (5.3), (5.2), and the curre nt smallness\nassumption on thedeviation of ufrombeingconstant on Γ, gives theexplicit\nlower bound for δΓ(z):\nδΓ(z)≥r2\ni\n2/parenleftBig\nmin/braceleftBig\n1,r2\ni\n4/bracerightBig\n+dΩ/parenrightBig.\nFinally, by the trivial inequality osc Γu≤oscΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γand the\ndefinition of ψ, we easily obtain that\nψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ/parenrightbigg\n+osc\nΓu≤Cψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl∞,Γ/parenrightbigg\n,\nwith\nC=/braceleftBigg\n2 if N= 2,3,\n1+σ(N−3)/(N−1)ifN≥4,\nwhereσis that defined in (5.2).\nTheconclusion easily follows byputtingtogether thelast i nequality, (5.6),\nand (5.5). /square\nProof of Theorem 1.2. We shall adapt the argument used in the proof of\nTheorem 1.1 to the setting with the current deviation.\nLetσbe given by (5.2). As before, it is clear that it is enough to ch eck\nthe case in which\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl2,Γ<σ.\nThis time, we observe that\n(5.7) /ba∇dbl∇u/ba∇dbl∞,Ω≤ /ba∇dbl∇q/ba∇dbl∞,Ω+/ba∇dbl∇h/ba∇dbl∞,Ω≤dΩ+/ba∇dbl∇h/ba∇dbl∞,Ω.\nAn explicit bound for /ba∇dbl∇h/ba∇dbl∞,Ωwill be obtained at the end of the proof.\nAlso, as before we can use (5.4). Similarly, by proceeding as in the proof\nof Theorem 1.1, with osc Γu+/ba∇dbl∇Γu/ba∇dbl∞,Γreplaced by osc Γu+/ba∇dbl∇Γu/ba∇dbl2,Γ, we\narrive at the inequalities\n(5.8) ψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl2,Γ/parenrightbigg\n+osc\nΓu≤Cψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl2,Γ/parenrightbigg\nand\n(5.9) osc\nΓh≤cψ/parenleftbigg\nosc\nΓu+/ba∇dbl∇Γu/ba∇dbl2,Γ/parenrightbigg\n.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 19\nHere,ψis that in (1.8), Cis the same constant appearing in the proof of\nTheorem 1.1, and cis an explicit constant that only depends on N,dΩ,ri,\nre, a lower bound for δΓ(z), and an upper bound for /ba∇dbl∇h/ba∇dbl∞,Ω.\nThe desired conclusion will easily follow from (5.5), (5.9) , and (5.8), after\nobtaining suitable bounds for δΓ(z) and/ba∇dbl∇h/ba∇dbl∞,Ω.\nBy putting together (5.2), Remark 5.3, (5.7), and the curren t smallness\nassumption on the deviation of ufrom being constant on Γ, we easily get\nthat\nδΓ(z)≥r2\ni\n4(dΩ+/ba∇dbl∇h/ba∇dbl∞,Ω).\nWe are thus left to obtain an upper bound for /ba∇dbl∇h/ba∇dbl∞,Ω.\nThanks to Theorem 4.4 (with m= 1) and recalling [2, Corollary 3.14],\n/ba∇dbl∇h/ba∇dbl∞,Ωcan be clearly bounded by a constant that depends on N,dΩ,ri,\nandre.\nWhen Γ is of class C2,α, we can obtain an improvement of the profile ψ,\nby arguing as in [23, Lemma 4.2 and Corollary 4.3] (with p= 2,q= +∞).\nIn fact, we infer that\n/ba∇dbl∇h/ba∇dbl∞,Ω≤c/ba∇dbl∇2h/ba∇dbl(N−1)/(N+1)\n∞,Ω/ba∇dblδ1/2\nΓ∇2h/ba∇dbl2/(N+1)\n2,Ω,\nfor somecdependingon N,dΩ,ri,re. Hence, by recalling Theorem 4.4 (with\nm= 2), we easily get that\n/ba∇dbl∇h/ba∇dbl∞,Ω≤c/ba∇dblδ1/2\nΓ∇2h/ba∇dbl2/(N+1)\n2,Ω,\nwherecdepends on N,dΩ,and theC2,α-regularity of Γ.\nPlugging the last inequality in Lemma 5.2 and using Lemma 5.4 and (5.4)\nas before, gives (5.9) with the improved profile shown in (1.9 ). Clearly, for\nthis profile, (5.8) remains true with Cgiven by\nC=/braceleftBigg\n2 if N= 2,3,\n1+σ(N−3)/(N+1)ifN≥4.\nThus, combining (5.5), (5.9), and (5.8) gives the improved s tability for\nN≥4. /square\n6.Stability via the classical Serrin problem\nWe now present a quantitative bound for the reverse Serrin pr oblem that\ncan be obtained exploiting the existing stability results f or the classical\nSerrin problem. The proof is based on a simple trick. As discu ssed in the\nIntroduction, we get a poorer estimate than those obtained i n Theorems 1.1\nand1.2. Infact, themain drawback is that thedeviations osc Γu+/ba∇dbl∇Γu/ba∇dbl∞,Γ\nand osc Γu+/ba∇dbl∇Γu/ba∇dbl2,Γused in Theorems 1.1 and 1.2 must be replaced by\nthe much more stringent deviation /ba∇dblu−u/ba∇dblC1,α(Γ).\nWe start with an estimate for an auxiliary function.\nProposition 6.1. LetΩ⊂RN,N≥2, be a bounded domain with boundary\nΓof classC1,α,0<α<1. Letube a solution of (1.1),(1.12).\nSetf=u−w, wherewis the solution of the Dirichlet problem\n∆w= 0inΩ, w=uonΓ.20 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\nThen, we have that\n/ba∇dblR−fν/ba∇dblC0,α(Γ)≤c/ba∇dblu−u/ba∇dblC1,α(Γ).\nHere,conly depends on Nand theC1,α-regularity of Γ.\nProof.We notice that\n/ba∇dblR−fν/ba∇dblC0,α(Γ)=/ba∇dblwν/ba∇dblC0,α(Γ)≤ /ba∇dblu−w/ba∇dblC1,α(Ω).\nThe desired conclusion then follows from [14, Theorem 8.33] . /square\nTheorem 6.2 (The reverse Serrin problem with strong deviation) .LetΩ\nbe a bounded domain in RN,N≥2, with boundary Γof classC1,1.\nLetube solution of (1.1),(1.12), and letzbe any global minimum point\nofuonΩ. Then, we have that\nρe−ρi≤cψ/parenleftbig\n/ba∇dblu−u/ba∇dblC1,α(Γ)/parenrightbig\n,\nwhereψis the profile defined in (1.8). The constant conly depends on N,\ndΩ,ri, andre.\nForN≥4the profileψcan be improved to obtain (1.9), at the cost of\nreplacing the dependence of con theriandrewith that on the C2,α-regularity\nofΓ.\nProof.It is clear that the function fdefined in Proposition 6.1 satisfies:\n∆f=Nin Ω, f= 0 on Γ.\nBy [21, Theorem 3.1] (for N/\\e}atio\\slash= 3) and [23, Theorem 4.4] (for N= 3), we\nhave that\nρe−ρi≤cψ/parenleftbig\n/ba∇dblfν−R/ba∇dblL2(Γ)/parenrightbig\n,\nwhereconly depends on N,dΩ,ri, andre. Here,ψis as in (1.8).\nFurthermore, the improvement of the profile ψto (1.9) for N≥4 is\nobtained from [23, Theorem 4.4], at the aforementioned cost .\nThe conclusion then follows from Proposition 6.1 and by noti ng that\n/ba∇dblfν−R/ba∇dblL2(Γ)≤ |Γ|1/2/ba∇dblfν−R/ba∇dblL∞(Γ)≤ |Γ|1/2/ba∇dblfν−R/ba∇dblC0,α(Γ).\nAsusual, |Γ|canbeestimated intermsofthedesiredparametersbyrecall ing\n[23, Remark 3.1]. /square\nRemark 6.3. Theargument may be adjusted to work also in the semilinear\ncase, at the cost of replacing the application of [21, Theore m 3.1] and [23,\nTheorem 4.4] by the results in [1, 10], and hence obtaining a m ore general\nbut poorer stability estimate.\nAcknowledgements\nThe authors wish to thank S. Dipierro, M. Onodera, and E. Vald inoci for\nuseful discussions on Section 6.\nR. Magnanini is partially supported by the PRIN grant n. 2017 58MTR2\nof the Ministero dell’Universit` a e della Ricerca (MUR).\nR. Molinarolo is partially supported by the SPIN Project “DO Main per-\nturbation problems and INteractions Of scales - DOMINO” of t he Ca’ Fos-\ncari University of Venice.AN INTEGRAL IDENTITY AND THE REVERSE SERRIN PROBLEM 21\nG. 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Pierre, Shape Variation and Optimization . EMS Press, European\nMathematical Society, Zurich, 2018.\n[16] S. G. Krantz, H. R. Parks, Distance to Ckhypersurfaces , J. Differential Equations 40\n(1981), 116–120.\n[17] G. M. Lieberman, Second order parabolic differential eq uations, World Scientific Pub-\nlishing Co., Inc., River Edge, NJ, 1996. xii+439 pp.\n[18] R. Magnanini, Alexandrov, Serrin, Weinberger, Reilly: symmetry and stab ility by\nintegral identities , Bruno Pini Mathematical Seminar (2017), 121–141.\n[19] R. Magnanini, G. Poggesi, On the stability for Alexandrov’s Soap Bubble theorem , J.\nAnal. Math. 139 (2019), 179–205.22 MAGNANINI, MOLINAROLO, AND GIORGIO POGGESI\n[20] R. Magnanini, G. Poggesi, Serrin’s problem and Alexandrov’s Soap Bubble Theorem:\nenhanced stability via integral identities , Indiana Univ. Math. J., 69 (2020), 1181–\n1205.\n[21] R. Magnanini, G. Poggesi, Nearly optimal stability for Serrin’s problem and the soap\nbubble theorem , Calc. Var. (2020), 59:35.\n[22] R. Magnanini, G. Poggesi, The location of hot spots and other extremal points , Math.\nAnn. 384 (2022), pp. 1–39.\n[23] R. Magnanini, G. Poggesi, Interpolating estimates with applications to some quanti-\ntative symmetry results , Math. in Engin. 5 (2023), 1–21.\n[24] R. Magnanini, G. Poggesi, Quantitative symmetry in a mixed Serrin-type\nproblem for a constrained torsional rigidity , Calc. Var. 63, 23 (2024).\nhttps://doi.org/10.1007/s00526-023-02629-w. Preprint arxiv:2210.10288.\n[25] R. Magnanini, G. Poggesi, An interpolating inequality for solutions of uniformly ell ip-\ntic equations , Geometric properties for parabolic andelliptic PDEs, 233 –245. Springer\nINdAM Ser., 47 Springer, Cham, [2021], ©2021.\n[26] F. Pacella, G. Poggesi, A. Roncoroni, Optimal quantitative stability for a Serrin-type\nproblem in convex cones , preprint (2023) arXiv:2309.02128\n[27] L. E. Payne, P. W. Schaefer, Duality theorems in some overdetermined boundary value\nproblems , Math. Methods Appl. Sci.11 (1989), 805–819.\n[28] G. Poggesi, Soap bubbles and convex cones: optimal quantitative rigidi ty, preprint\n(2022) arXiv:2211.09429.\n[29] G. Poggesi, The Soap Bubble Theorem and Serrin’s problem: quantitative sym-\nmetry, PhD Thesis, Universit` a di Firenze, defended on February 2 019, preprint\narxiv:1902.08584.\n[30] J. Serrin, A symmetry problem in potential theory , Arch. Rational Mech. Anal. 43\n(1971), 304–318.\n[31] H. F. Weinberger, Remark on the preceding paper of Serrin , Arch. Rational Mech.\nAnal. 43 (1971), 319–320.\nDipartimentodi Matematicaed Informatica “U. Dini”, Unive rsit`adiFirenze,\nviale Morgagni 67/A, 50134 Firenze, Italy.\nEmail address :rolando.magnanini@unifi.it\nURL:https://people.dimai.unifi.it/magnanini/\nDipartimentodi Matematicaed Informatica “U. Dini”, Unive rsit`adiFirenze,\nviale Morgagni 67/A, 50134 Firenze, Italy.\nCurrent address : Dipartimento di Scienze Molecolari e Nanosistemi, Univer sit` a Ca’\nFoscari Venezia, Via Torino 155, 30172 Venezia Mestre, Ital y.\nEmail address :riccardo.molinarolo@unive.it\nDepartmentof Mathematics andStatistics, The Universityo f Western Aus-\ntralia, 35 Stirling Highway, Crawley, Perth, WA 6009, Austr alia\nEmail address :giorgio.poggesi@uwa.edu.au"    },    {        "title": "2402.02875v2.Low_energy__α__harmonic_maps_into_the_round_sphere.pdf",        "content": "arXiv:2402.02875v2  [math.AP]  6 Feb 2024Low-energy α-harmonic maps into the round sphere\nBen Sharp\nWednesday 7thFebruary, 2024\nAbstract\nWe classify low-energy α-harmonic maps from a closed non-spherical Riemannian surface Σ o f\nconstant curvature to the round sphere via their bubble scales an d centres. In particular we show\nthat as 1<α↓1 and assuming Eαis close to |Σ|+ 4πthen degree-one α-harmonic maps blow a\nbubble based at a critical point acof a an explicit function Jand at scale/radicalbig\n|J(ac)|−1(α−1). Up\nto a constant, Jis the sum of the squares of any L2-orthonormal basis of holomorphic one-forms\non the domain.\n1 Introduction\nOne of the founding pillars of geometric analysis concerns t he existence and regularity of harmonic\nmaps. Given two Riemannian manifolds M,N , a harmonic map u∈W1,2(M,N ) is a critical point\n(in some suitable sense) of the Dirichlet energy\nE(u) =1\n2/integraldisplay\nM|∇u|2dM.\nWhenM= Σ is two-dimensional this functional is invariant under co nformal deformations of the\ndomain which leads to a wonderfully rich theory from both an a nalytic and geometric viewpoint. In\nbroad strokes this functional does not enjoy straightforwa rd regularity properties, nor does it satisfy\na Palais-Smale condition. These difficulties lead to both int eresting and challenging mathematical\nproblems. Remarkably critical points are always smooth in t wo dimensions thanks to H´ elein’s moving\nframe approach [6, 7]; the resulting estimates hold only up t o a potentially small regularity scale\nr(u,N, Σ)>0 whereupon the Dirichlet energy on any ball of radius ris below some threshold. Along\na sequence of smooth maps uk(for instance) it is possible for rkto shrink to zero (i.e. for the Dirichlet\nenergy to concentrate) and all the derivatives to blow up. Su itably re-scaling the domain to reverse\nthe energy concentration leads to a sequence of harmonic map s defined on larger and larger regions of\nR2which converge to a harmonic sphere ω:R2∪{∞} =S2→N.\nStarting with the famous work of Sacks-Uhlenbeck [22] α-harmonic maps have been used to study\nexistence theory for harmonic maps in two-dimensions via a r egularised functional Eαandα > 1.\nGiven a closed Riemannian surface Σ and Nsome closed Riemannian manifold, the α-energyEαof a\nmapu∈W1,2α(Σ,N) is defined as follows\nEα(u) :=1\n2/integraldisplay\nΣ(2 +|∇u|2)αdΣ so that Eα(u)α↓1− − →E(u) +|Σ|.\nForα > 1 fixed,Eαcrucially satisfies the Palais-Smale condition, whilst all finite-energy critical\npoints (α-harmonic maps) are uniformly regular, up to a regularity sc aler=r(α,Eα(u),N,Σ) with\nsmooth estimates which potentially degenerate ( r→0), asα↓1, on regions where the Dirichlet\nenergy concentrates. Nevertheless, it is well-known that h armonic maps appear as weak W1,2-limits of\nα-harmonic maps, as α↓1, modulo the appearance of singularities, again taking the form of harmonic\nspheres (bubbles) and geodesics of length ℓ∈[0,∞] (necks) which connect the weak limit with the\nbubbles. An ideal outcome would be that all the Dirichlet ene rgy is captured by the limiting harmonic\nmap and bubbles (an energy identity) or further that all neck s have zero length (a no-neck result).\nIn general both of these ideals are known to fail during the li miting process for α-harmonic maps:\n1examples of (infinite-length) necks appearing and energy-l oss can be found in [15]. Still, one can use\nanalytical methods to find α-harmonic maps (direct minimisation, or min-max procedure s) and in\nturn hope to recover relevant information on harmonic maps, asα↓1 see e.g. the seminal works of\nSacks-Uhlenbeck [22] and Micallef-Moore [17].\nNatural and important questions present themselves: how pr ecisely does this singular convergence\nhappen and can one recover all harmonic maps from surfaces as a limit ofα-harmonic maps?\nWhen the target has positive Ricci curvature and the sequenc e ofα-harmonic maps has finite\nMorse index, it is known that only finite-length geodesic nec ks appear and all energy is accounted for\nin the limiting process [14]. For arbitrary closed targets a nd certain sequences of α-harmonic maps\nobtained by min-max methods, similarly, all energy is accou nted for in the limit by the work of Tobias\nLamm [10], although it is not known whether or not necks might appear. If the target is a round\nsphereSn, thanks to Jiayu Li and Xiangrong Zhu we know more: all energy is accounted for and the\nconvergence happens in L∞(no necks appear at all) [13].\nWhen considering maps between round two-spheres u:S2→S2it is well-known that, up to\nreversing orientations, the only harmonic maps are holomor phic and therefore rational maps from the\nextended complex plane to itself (see e.g. the Theorem of Woo d-Lemaire [4, 11.3] for a more general\nresult). Thus if we restrict to degree one harmonic maps, we a re left only with the M¨ obius maps\nand their complex conjugates. Remarkably the question “are all harmonic maps attained as limits\nofα-harmonic maps” has only recently received attention. Lamm , Malchiodi and Micallef show that\nall degree ±1α-harmonic maps u:S2→S2withα-energy lower than 8α2π(Dirichlet energy lower\nthan 12π), andαclose to 1, must be rotations [11, 12]. Thus in particular the majority of deg ree-one\nharmonic maps between spheres are not reached by smooth limi ts ofα-harmonic maps. Indeed they\nessentially classify all low-energy α-harmonic maps u:S2→S2of degree 0 or ±1, see [12].\nIn this article we study α-harmonic maps u: Σ→S2, from non-spherical closed surfaces Σ, which\nhaveα-energy bounded above by 4 π+|Σ|+δforδ>0 small (in particular Dirichlet energy bounded\nabove by 4π+δ) andαclose to 1. The energy bound immediately tells us that the top ological degrees\nof these maps are either 0 or ±1.\nExamples of such degree ±1 low-energy α-harmonic maps are guaranteed to exist by direct methods\nand we show that any such map must be quantitatively C1-close, with bounds purely in terms of α, to\na finite-dimensional space of singularity models Zwhere each z∈ Z approximates a single degree ±1\nbubble and constant-body map configuration. In particular w e obtain estimates on both the bubble\nscale and bubble proximity purely in terms of ( α−1) and geometric data J: Σ→R.\nDefinition 1.1. Let (Σ,g) be a closed Riemannian surface of genus γ≥1 equipped with a metric g\nof constant curvature zero when γ= 1 and constant curvature −1 otherwise. In the flat case we also\nimpose Area g(Σ) = 1. Let {φj}be an arbitrary L2-orthonormal basis of holomorphic one-forms on Σ\nand define\nJ(a) :=−2πcγ/summationdisplay\nj|φj(a)|2\ng<0\nwherec1= 1 and when γ≥2,cγ= 4. One may check directly that Jis independent of the choice of\nbasis.\nRemark 1.2. Whenγ= 1we have J ≡ − 2πwhilst in general we always know that J<0by\nRiemann-Roch. Jis directly related to the regular part of the Green’s functi on on Σ, see e.g. Appendix\nB and [16].\nWithout going into fine details, we let Z ⊂W1,2(Σ,S2) denote the space of maps approximating\na single degree ±1 bubble at small scale, so topologically Z∼=Σ×[λ0,∞)×O(3) and given a∈Σ,\nλ≥λ0andR∈O(3) the corresponding z=zR\nλ,a∈ Zsatisfiesz∼Rπ(λ·)nearaand in suitable local\ncoordinates, otherwise z∼R(0,0,−1) on the rest of Σ .Here and throughout π(x) =/parenleftBig\n2x\n1+|x|2,1−|x|2\n1+|x|2/parenrightBig\nis\nthe inverse stereographic projection from the South Pole an d in particular for an almost-singular limit\nwe haveλ≫1, in which case π(λ·) maps a tiny disc (e.g of radius λ−1\n2) into most of S2\\(0,0,−1).\nSpaces closely related to Zhave appeared elsewhere in [16, 20] where they are used in the study\nofH-surfaces in R3and harmonic maps from surfaces into arbitrary analytic tar gets respectively; we\n2recall the precise definition in Section 2.1. For now it is suffi cient to note that any map which is (e.g.\nW1,2∩L∞) close to a single degree ±1 bubble and constant-body map configuration must be close to\nZ.\nIn [16] the authors show that J(a) appears as the highest order term in an expansion in λof the\nH-energyEHwhen restricted to Z. The space of singularity models Zappearing in [16] is slightly\ndifferent to the one appearing here and the H-energyEHis the Dirichlet energy plus an enclosed\nvolume term. In any case it is shown in that:\nλ3∂λEH(z) = 8πJ(a) +O(λ−1) where a=a(z) andλ=λ(z).\nOn the other hand, if uis sufficiently close to Zandz∈ Zis its closest-point projection (in an appro-\npriate norm), one finds, after carefully analysing the highe r-order derivatives of EHin a neighbourhood\nofZ, that forλsufficiently large:\nλ3|∂λEH(z)| ≤Cλ(logλ)1\n2/ba∇dbldEH(u)/ba∇dblL2+λ2/ba∇dbldEH(u)/ba∇dbl2\nL2+O(λ−1)(logλ)1\n2.\nThese estimates in particular heavily rely on a suitable non -degeneracy of the second variation in nor-\nmal (toZ) directions, along with some careful estimates on d EHand d2EHin a small neighbourhood\nofZ.\nComparing the two facts above, one guarantees a “quantitati ve non-existence result” for almost-\ncritical points close to Zin that setting, since crucially J<0, so/ba∇dbldEH(u)/ba∇dblmust be bounded away\nfrom 0 by some suitable function of λ- see [16]. As appears in the work of Rupflin [20] this expansio n\nremains morally true (with some serious additional complic ations) also for the Dirichlet energy when\nstudying almost-harmonic maps into general closed analyti c targetsu: Σ→Nwhich are sufficiently\nclose to a zero-body one-bubble singularity based upon any u n-branched bubble ˆ ω:S2→N. The\nsituation in [20] is more subtle as the space of nearby harmon ic maps (to ˆ ω) is not restricted to\npre-composition by M¨ obius maps, moreover any space of sing ularity models Zmust also take into\naccount any non-integrable Jacobi fields. Nevertheless it i s still the case that J(a) appears as the\ndominating term in the Dirichlet energy expansion (in λ), when restricted to Z. These facts yield deep\nimplications for the harmonic map flow from non-spherical su rfaces into analytical target manifolds,\nand in particular on discreteness of the low-end of the Diric hlet energy spectrum in this case. On the\nother hand recent progress concerning low-energy discrete ness of the Dirichlet energy spectrum for\nmaps from spherical domains to analytic targets can be found in [21].\nIn this article we restrict N=S2and study the behaviour of Eαin a neighbourhood of Z- the\nrestriction of the target to a sphere is necessary in order to have the favourable bubble-convergence\nresults of J. Li and X. Zhu [13], which are unavailable (and un true) for arbitrary targets. This then\nprovides the technical benefit that the elements of Zare totally explicit. The flavour of the main\nresults are rather different in the sense that α-harmonic maps are guaranteed to exist, so our focus is\nvery much on gleaning their properties as opposed to a quanti tative non-existence result.\nFollowing [20], it is necessary to define the following bubbl e-weighted inner product and norm for\nV,W∈W1,2(Σ,R3) whenz∈ Z\n/a\\}b∇acketle{tV,W/a\\}b∇acket∇i}htz=/integraldisplay\nΣ/parenleftbig\n∇V·∇W+ρ2\nzV·W/parenrightbig\ndΣ,with/ba∇dblV/ba∇dblz=/a\\}b∇acketle{tV,V/a\\}b∇acket∇i}ht1\n2\nwhere, forλ=λ(z)\nρz(p) :=\n\nλ\n1+λ2dg(p,a)2ifp∈Bι(a)\nλ\n1+λ2ι2 ifp∈Σ\\Bι(a)\nandιis half the injectivity radius of Σ.\nOur results are guided by the properties of Eαwhen restricted to Z. Whenλis bounded Eα(z) is\ncomparable to the Dirichlet energy, so we expect to pick up si milar terms in our expansion of Eαas\nfor the Dirichlet energy, ∂λEα∼8πJλ−3<0. However, for each fixed α>1, it is straightforward to\nsee thatEα(z)→ ∞ asλ→ ∞ (indeed we have λ2α−2≤C(Eα(z) + 1) from Lemma 2.5) so for large\n3λwe expect∂λEα>0. These facts together suggest that ∂λEαmust vanish somewhere on Z. Here,\nwe show that (Proposition 4.1), for a=a(z) andλ=λ(z),\nλ3−2α∂λEα(z) = 8π/parenleftbig\nJ(a)λ−2+ (α−1)/parenrightbig\n+O(λ−1(α−1)) +O(λ−3) +O((α−1)2).\nThis expansion indicates that ∂λEα(z) will vanish for a z∈ Z whenλ(z) anda(z) approximately\nsatisfyJ(a)λ−2+ (α−1) = 0. Thus for any α-harmonic map uwhich is close to z∈ Z, we should\nalso expect the first (highest-order) term in this expansion to be close to zero. This would force\n−J(a)λ−2∼(α−1) and suggest the appropriate bubble scale λofuin terms of ( α−1). Our first\nresult confirms this line of reasoning, and since uisα-harmonic we are able to also prove quantitative-\ncloseness to the space of singularity models ZinC1. We remind the reader that degree ±1α-harmonic\nmaps satisfying the hypotheses of the below Theorem are guar anteed to exist by direct methods.\nTheorem 1.3. Let(Σ,g)be as in Definition 1.1. There exist α0=α0(Σ)>1,δ=δ(Σ)>0,C=\nC(Σ)<∞so that if 1<α≤α0andu: Σ→S2isα-harmonic with Eα(u)≤4π+|Σ|+δthen either\n1.uhas degree zero and /ba∇dbl∇u/ba∇dblL∞(Σ)≤C, or\n2.uhas degree ±1and there is z(u)∈ Zrealising inf{/ba∇dblu−z/ba∇dblz:z∈ Z}. Moreover for any such\nz(u), settinga=a(z(u))andλ=λ(z(u))we have\n|(α−1) +λ−2J(a)| ≤C(α−1)3\n2|log(α−1)|.\nFurthermore\n/ba∇dblu−z(u)/ba∇dblW1,2α(Σ)≤C(α−1)|log(α−1)|.\nFinally, for any s∈[1,∞)there isK=K(s,Σ)so that\n/ba∇dblu−z(u)/ba∇dblL∞(Σ)≤K(α−1)1−1\ns|log(α−1)|and/ba∇dbl∇u−∇z(u)/ba∇dblL∞(Σ)≤K(α−1)1\n2−1\ns|log(α−1)|.\nWe furthermore study the dependence of Eα|Zon the gluing point a. Of course a flat torus is\ninvariant under Euclidean translations in the domain: give nA∈TaΣ,\n∇A(Eα(zR\nλ,a)) =∂\n∂ss=0(Eα(zR\nλ,as)) = 0 where∂as\n∂ss=0=A, a0=a\nin this case and there is no restriction on the location of a bu bble. This is no longer true for higher-\ngenus surfaces and we show that (see Proposition 4.2):\nλ4−2α∇A(Eα(z)) = 4π∇AJ(a) +O(λ−1) +O(λ(α−1))a=a(z),λ=λ(z).\nTheorem 1.3 tells us that when uisα-harmonic and close to z∈ Z then (α−1)∼λ−2, so the first\nterm above is of highest order. Once again, our rule of thumb s uggests that ∇AJ(a) must be close to\nzero, which we confirm next.\nTheorem 1.4. Let(Σ,g)be as in Definition 1.1 with genus γ≥2. There exist α0=α0(Σ)>1,δ=\nδ(Σ)>0,C=C(Σ)<∞so that if 1< α≤α0andu: Σ→S2isα-harmonic of degree ±1with\nEα(u)≤4π+|Σ|+δthen ifz(u)∈ Zachieves inf{/ba∇dblu−z/ba∇dblz:z∈ Z}(as guaranteed by Theorem 1.3\npart 2) then for a=a(z):\n|∇J(a)| ≤C(α−1)1\n2|log(α−1)|.\nWe immediately obtain the following compactness result:\nCorollary 1.5. Let(Σ,g)be as in Definition 1.1 and {uk: Σ→S2}a sequence of αk-harmonic maps\nwith 1<αk↓1andEαk(uk)→Λ≤4π+|Σ|then, up to subsequence, we either have\n1.ukall have degree zero and they converge smoothly to a degree-z ero harmonic map u: Σ→S2\nor\n42.ukall have degree 1or all have degree −1and they bubble-converge to a single-bubble-constant-\nbody configuration where the bubble is blown at a critical poi ntacofJand at scale\nrk=/radicalbig\n|J(ac)|−1(αk−1).\nRemark 1.6. As stated previously Jcan be defined purely via mixed second-order derivatives of t he\nregular part of the Green’s function of the Laplacian on Σ. For simply connected domains in the plane\nthe equivalent function is known to be closely related to the Robin function - see [2]. The critical points\nof the Robin function, and related functions are known to be s ignificant in other critical variational\nproblems see e.g. [1, 3, 5, 8, 9, 19].\nOutline of the paper The proofs of the main Theorems (see Section 3) rely heavily u pon a careful\nanalysis of the behaviour of Eαwhen restricted to the manifold Z(and in small neighbourhoods\nthereof). In Section 2 we recall the space of singularity mod elsZand expand the α-energy along Z\nforλlarge. We recall also the crucial background results concer ningα-harmonic maps into round\nspheres [13] which we combine with suitable elliptic estima tes to prove first that the α-harmonic maps\nsatisfying our hypotheses are either analytically well-co ntrolled or lie close to Z, (Lemmata 2.10 –\n2.12). We also recall the desired non-degeneracy of energy i n normal directions which essentially\nfollows from a similar estimate in [20] (cf [16]): this compa rison relies heavily on the known theory for\nα-harmonic maps into spheres, see Lemma 2.13. In Section 4 we d erive an expansion in λand (α−1)\nfor∂λEα(z) and∇A(Eα(z)) along the submanifold Z. The key point of these expansions are that\nproperties of Jdetermine the behaviour of the Eα-energy profile when restricted to Z. In Section 5\nwe obtain control on d Eαand d2Eαwhen we are on and nearby Z- these estimates are crucially of\n“lower-order” than those appearing in Section 4 which allow s us to pass information on the behaviour\nofEαonZto the behaviour of Eαin a neighbourhood of Z. We provide the proofs of our main\nTheorems in Section 3 assuming the technical expansions/es timates from Sections 4 and 5.\nAcknowledgements\nI would like to thank Mario Micallef and Tobias Lamm for their interest, encouragement and inspiring\nconversations. I would also like to thank Melanie Rupflin and Andrea Malchiodi for their helpful\ncomments on, and interest in, the final drafts of this work. Th e author was supported by the EPSRC\ngrant EP/W026597/1.\n2 Preliminaries\n2.1 Definition of the set of adapted bubbles\nLet Σ be as in Definition 1.1. Here we will define the set of adapt ed bubbles, denoted by Z={Rzλ,a:\nΣ→S2}whereλ≫1 is the bubble scale, a∈Σ is where the bubble is attached and R∈O(3). Notice\nthis follows exactly along the lines of [16] and more relevan tly [20] - indeed it corresponds to the same\nspace in the latter case when N=S2and ˆω:S2→S2is the identity. We recall the full details for our\nsetting:\nDenote by πthe inverse stereographic projection π(x) =/parenleftBig\n2x\n1+|x|2,1−|x|2\n1+|x|2/parenrightBig\nand we want to define\nzλ,ain a way that we have zλ,a≈πλ(x) :=π(λx) in oriented isothermal coordinates x=Fa(p) on the\ngeodesic ball Bι(a)⊂(Σ,g),ι:=1\n2inj(Σ,g), which are as follows:\nRemark 2.1. Ifgis hyperbolic we set ρ= tanh(ι)and use that for any a∈Σthere exists an\norientation preserving isometric isomorphism\nFa: (B2ι(a),g)→/parenleftBig\nDρ,4\n(1−|x|2)2gE/parenrightBig\nwithFa(a) = 0\nwhereDρ:={x∈R2:|x|<ρ}andgEis the Euclidean metric. In the flat case we will always a-priori\npick a tiling of R2which represents Σ, setρ= 2ιand use the resulting euclidean translations Fato\nthe origin as coordinates.\n5Whenγ≥2we note that Fais determined only up to a Euclidean rotation on Dρ, but that this\nwill not affect the definition of the bubble set since this rota tion can be undone by choosing an ambient\nrotationR∈O(3)(see Remark 2.3 and the following definition of Z).\nIn the coordinates introduced in Remark 2.1\nFa:B2ι(a)×B2ι(a)→Dρ×Dρ, F a(p,q) = (Fa(p),Fa(q)) = (x,y) (2.1)\nwe write the Green’s function GviaGa(x,y) =−log|x−y|+Ja(x,y) for some smooth function J.\nHence we have for x/\\e}atio\\slash=y\n∂yiGa(x,y) =xi−yi\n|x−y|2+∂yiJa(x,y). (2.2)\nThe function Jdepends on the choice of coordinates used and is only defined w hen both arguments\nare close to a. We express Jain terms which clearly show dependence on the choice of coord inates via\n(forp,q∈B2ι(a)):\nJa(p,q) :=G(p,q) + log|Fa(p)−Fa(q)|so thatJa(p,q) =Ja(Fa(p),Fa(q))\nand setting ∂qi= (F−1\na)∗∂yiwe have by definition\n∂qiJa(p,q) =∂yi(Ja(p,F−1\na(y))) =∂yiJa(Fa(p),y).\nDefinervia 4r= tanh(ι) in the hyperbolic case and 4 r=ιin the flat case. Let φ∈C∞\nc(D2r,[0,1])\nbe radial with φ≡1 onDr. Given any a∈Σ andFa:B2ι(a)→Dρa choice of isometry as in Remark\n2.1, we define\n˜zλ,a(p) =\n\nˆzλ,a(Fa(p)) if p∈Bι(a)\n(2\nλ(∂q1G(p,a)−∂q1Ja(a,a)),2\nλ(∂q2G(p,a)−∂q2Ja(a,a)),−1) ifp /∈Bι(a)(2.3)\nwhere ˆzλ,ais given in the local coordinates x=Fa(p) by (with ∇y= (∂y1,∂y2))\nˆzλ,a(x) :=φ(x)/parenleftbig\nπλ(x) +/parenleftbig2\nλ∇yJa(x,0)−2\nλ∇yJa(0,0),0/parenrightbig/parenrightbig\n+ (1−φ(x))/parenleftbig2\nλ∇yGa(x,0)−2\nλ∇yJa(0,0),−1/parenrightbig\n.(2.4)\nLettingj(x) =jλ,a(x) :=/parenleftbig2\nλ∇yJa(x,0)−2\nλ∇yJa(0,0),0/parenrightbig\n, ˜za,λis described in these local coordi-\nnates onBι(a) by\nˆzλ,a(x) =πλ(x) +j(x) + (φ(x)−1)/parenleftbig/parenleftbig\n−2\nλ∇yGa(x,0) +2\nλ∇yJa(x,0),1/parenrightbig\n+πλ(x)/parenrightbig\n=πλ(x) +j(x) + (φ(x)−1)O(λ−2).(2.5)\nThe error in (2.5) follows from the definition of πλand (2.2):\n(−2\nλ∇yGa(x,0) +2\nλ∇yJa(x,0),1) +πλ(x) =2\n1 +λ2|x|2/parenleftbigg\n−x\nλ|x|2,1/parenrightbigg\n=O(λ−2) onD2r\\Dr.\nRemark 2.2. The conformal covariance of the Laplacian implies that ∂yiJa(x,0)is harmonic in x,\nso in particular jis also harmonic.\nOn an unrelated note, for symmetry reasons it is possible to c heck that ∇yJa(0,0) = 0whenγ= 1,\nhowever it is unclear whether this remains true for γ≥2.\nRemark 2.3. Whenγ≥2we note the dependence on the choice of Favia (forpclose toa):\n˜zFa(p) = ˜zλ,a(p) =πλ(Fa(p)) +jFa(p) + (φ(Fa(p))−1)O(λ−2),\nwithjFa(p) =ja\nλ(p) =2\nλ/parenleftbig\n∂q1Ja(p,a)−∂q1Ja(a,a),∂q2Ja(p,a)−∂q2Ja(a,a),0/parenrightbig\n. Thus if we pick an-\nother isometry as per Remark 2.1 ˇFawe have ˇR=Fa◦ˇF−1\na∈SO(2)and now let Rbe the obvious\nextension of ˇRtoO(3)(as the identity in the third component). Following the defin itions above we\nhave\n˜zFa(p) =R˜zˇFa(p).\nThus a different choice of Fain Remark 2.1 has the effect of an ambient rotation and does not change\nthe eventual bubble set Zdefined below.\n6We then set\nzλ,a=P(˜zλ,a) :=˜zλ,a\n|˜zλ,a|and define Z:={z=zR\nλ,a=Rzλ,a|a∈Σ,R∈O(3),λ> 1}.\nSee Appendix A for further expansions and estimates for z∈ Z and their derivatives. Here are\nthroughout we will also use the notation\nZα,δ:={z∈ Z:Eα(z)−|Σ|−4π≤δ}.\nRemark 2.4 (Notation ).The notation f=O(g)means that there exists some C <∞so that\n|f| ≤Cgandf≃gmeans there is C <∞so thatC−1|g| ≤ |f| ≤C|g|. The constant Cmay change\nfrom one line to the next but in any given statement Cis fixed and depends at most upon the injectivity\nradius of, and the Green’s function on, Σ(sometimes neither). In particular for any fixed Σas in\nDefinition 1.1, a-posteriori there is some uniform Cwhich works for all statements in this paper.\nWe will use the notation ∆,∇,·,dΣto denote these objects with respect to the metric gon\nΣunlesswe are inx=Fa(p)coordinates at which point we will use the notation ∇g,∆g,·g,dxg\netc to denote these objects. In these coordinate only, the sam e symbols without a gare the standard\nEuclidean ones (as in e.g. (2.4)above).\nIn the first instance we will determine the behaviour of EαonZ- whereas in Sections 4 and 5 we\nwill more carefully analyse the behaviour of d Eαand d2Eαon and close to Z.\nLemma 2.5. Letα≤2,δ≤1andz∈ Zα,δ, then forλ=λ(z),\nEα(z)−|Σ|−4π= (λ2α−2−1)4π+O(λ2α−4) +O((1 +λ2α−2)(α−1)).\nCorollary 2.6. Let1< α≤2,δ≤1andz∈ Zα,δ, then there exists C=C(Σ)<∞so that\nλ(z)2α−2≤C. Furthermore for any ε>0there isα0=α0(Σ)>1andδ=δ(Σ)>0so thatα≤α0\nandz∈ Zα,δimplies\n|λ(z)2α−2−1|<ε.\nProof.\nEα(z) = 2α−1|Σ|+1\n2/integraldisplay\nΣ/parenleftBig\n(2 +|∇z|2)α−2α/parenrightBig\ndΣ\n= 2α−1|Σ|+O(λ−2) +1\n2/integraldisplay\nDr/parenleftBig\n(2 +|∇gz|2\ng)α−2α/parenrightBig\ndxg\nwheregdenotes the metric in x-coordinates. With cγ∈ {1,4}defined as in Definition 1.1 we have\ndxg= (cγ+O(|x|2)) dxand|∇gz|2\ng= (c−1\nγ+O(|x|2))|∇z|2, noting that when γ= 1 we have cγ= 1\nand no error (i.e. everything is Euclidean).\nOnDrwe have from (A.1) and (A.3),\n∇z=∇πλ+∇j⊤\nλ,a+∇Kλ,a\n=∇πλ+O(λ−1). (2.6)\nUsing that, for x∈Dρ,O(λ−2) +O(λ−1)∇πλ=O(|∇πλ|2(λ−2+|x|2)) we obtain\n|∇gz|2\ng=|∇πλ|2(c−1\nγ+χ(λ,|x|)) (2.7)\nwhereχ(λ,|x|) =O(λ−2+|x|2).\nBy the mean value formula, we note that (given a,b> 0) there is some c∈(0,a) so that\n(a+b)α−1−bα−1= (α−1)a(c+b)α−2≤(α−1)abα−2. (2.8)\nEmploying this twice, with the roles of aandbare reversed, we may write\n(2 +|∇gz|2\ng)α= (a+b)α=a(a+b)α−1+b(a+b)α−1=aα+bα+ (α−1)e0\n7wherea= 2 andb=|∇πλ|2(c−1\nγ+χ) ande0≤(abα−1+aα−1b).\nThis yields\n/parenleftbig\n(2 +|∇gz|2\ng)α−2α/parenrightbig\ndxg=|∇πλ|2α(c−1\nγ+χ)αdxg+ (α−1)(O(|∇πλ|2α−2) +O(|∇πλ|2)) dx.\nUsing that ( c−1\nγ+χ)α=c−α\nγ+χforχ=O(λ−2+|x|2) we now have\nEα(z) = 2α−1|Σ|+O(λ−2)\n+1\n2/integraldisplay\nDrc1−α\nγ|∇πλ|2α+χ|∇πλ|2α+ (α−1)(O(|∇πλ|2α−2) +O(|∇πλ|2)) dx.\nFrom e.g. (A.2) we have\n/integraldisplay\nDr|∇πλ|2α=πλ2α−28α\n2α−1+O(λ−2α)/integraldisplay\nDr|∇πλ|2α−2=O(λ2α−4)\n/integraldisplay\nDr|∇πλ|2=O(1)/integraldisplay\nDrχ|∇πλ|2α=O(λ2α−4)\nwhich gives\nEα(z)−2α−1|Σ|= 4πλ2α−28α−1c1−α\nγ\n2α−1+O(λ2α−4) +O((α−1))\nfrom which the result follows.\n2.2 Analytical set up and further background results\nWe crucially require the energy identity and no-neck result of Jiayu Li and Xiangrong Zhu for α-\nharmonic maps into round spheres [13, Theorem 1.1, Lemma 4.1 ]. In particular their results straight-\nforwardly imply:\nTheorem 2.7. [13, Theorem 1.1, Lemma 4.1]\nIf{uk} ∈W1,2αk(Σ,S2)is a sequence of αk-harmonic maps with 1< αk↓1andEαk(uk)≤Λthen\nthere exist a finite number of bounded-energy smooth harmoni c mapsu∞: Σ→S2,{ψj}:R2→S2\nand accompanying distinct point-scale sequences (aj\nk,rj\nk)so thataj\nk→ak∈Σ,rj\nk→0and:\n1. For alljwe have (rj\nk)1−αk→1, and/ba∇dbl(2 +|∇uk|2)αk−1/ba∇dblL∞(Σ)→1\n2.Eαk(uk)→ |Σ|+E(u∞) +/summationtext\njE(ψj)\n3. /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleuk−u∞−/summationdisplay\njφj\nk(·)/parenleftBig\nψj((rj\nk)−1Faj\nk(·))−ψj(∞)/parenrightBig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nW1,2α(Σ)→0,\nand /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleuk−u∞−/summationdisplay\njφj\nk(·)/parenleftBig\nψj((rj\nk)−1Faj\nk(·))−ψj(∞)/parenrightBig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞(Σ)→0,\nwhereφj\nk∈C∞(Σ)is defined by φj\nk(p) =φ(Faj\nk(p))onspt(φj\nk)⊂Bι(aj\nk)for some fixed φ∈\nC∞\nc(Dρ,[0,1])withφ≡1onDρ/2andι,ρare defined as in Remark 2.1.\nRemark 2.8. Notice that in the scenario that one bubble forms and u∞is a constant, we must have\nψ(∞) =u∞and in the above convergence we may write\n/ba∇dbluk−φk(·)(ψ(r−1\nkFak(·))−(1−φk(·))ψ(∞)/ba∇dbl → 0\nfor both norms.\n8Weighted inner product: Forv∈C∞(Σ,S2) we denote by\nΓ2(v) :={V∈W1,2(Σ,R3) :V(x)∈Tv(x)S2,a.e. x∈Σ}.\nFollowing [20], it is necessary to define the following inner product for V,W∈W1,2(Σ,R3) whenz∈ Z\n(so in particular also for V,W∈Γ2(z))\n/a\\}b∇acketle{tV,W/a\\}b∇acket∇i}htz=/integraldisplay\nΣ/parenleftbig\n∇V·∇W+ρ2\nzV·W/parenrightbig\ndΣ,with/ba∇dblV/ba∇dblz=/a\\}b∇acketle{tV,V/a\\}b∇acket∇i}ht1\n2\nwhere\nρz(p) :=\n\nλ\n1+λ2dg(p,a)2≃ |∇z|ifp∈Bι(a)\nλ\n1+λ2ι2 ifp∈Σ\\Bι(a)\nso in particular |∇z| ≤Cρzon Σ.\nRemark 2.9. We have the following useful estimate from [20, (2.28) and Ap pendix B]: For any\ns∈[1,∞)there exists K=K(s,Σ)so that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n∂Bι(a)vdΣ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/ba∇dblv/ba∇dblLs(Σ)≤K(s,Σ)(logλ)1\n2/ba∇dblv/ba∇dblz.\nLemma 2.10. Ifu: Σ→S2is anα-harmonic map with /ba∇dblu−z/ba∇dbl2\nz<ε0\n4(withε0from Theorem C.1),\nα>1and andz∈ Zthen there is some C=C(Σ)<∞so that/ba∇dbl∇u/ba∇dblL∞≤Cλ(z)and we have: for\nanyp∈Σands∈[1,∞)there exists K(s,Σ)<∞\n|u(p)−z(p)|+λ−1|∇u(p)−∇z(p)| ≤K(s,Σ)(λ2\ns(logλ)1\n2/ba∇dblu−z/ba∇dblz+ (α−1) +λ−2).\nProof. Forz∈ Z with/ba∇dblu−z/ba∇dbl2\nz<ε\n4letλandabe defined by z.\nFixµ=1\n2ιand notice first that for any pwith d Σ(p,a)> µ there is some uniform radius r >0\nso that/ba∇dblz/ba∇dblC2(Br(p))≤Cλ−2. Thus, there is some uniform radius rso that/ba∇dbl∇u/ba∇dbl2\nL2(Br(p))≤2/ba∇dblu−\nz/ba∇dbl2\nz+Cr2λ−4<ε0(ε0from Coroallry C.2) meaning that /ba∇dbl∇u/ba∇dblL∞(Br/2)≤C(/ba∇dblu−z/ba∇dblz+λ−2) by the\nregularity of α-harmonic maps. Thus the estimates trivially hold on Σ \\Bι\n2(a).\nOnBι(a) our usual coordinates xare defined on DRwhereR= tanh(ι/2) whenγ≥2 elseR=ι.\nLet ˆz(x) =z(λ−1x), which satisfies /ba∇dblˆz/ba∇dblC2(DλR)≤Cfor some uniform constant C. In particular there\nis some uniform radius rso that /integraldisplay\nDr(x)|∇ˆz|2<ε0\n4\nfor anyDr(x)⊂DλR. Defining ˆ u(x) =u(λ−1(x)) in these coordinates, and the scale-invariance of\n/ba∇dbl·/ba∇dblzgive that/integraldisplay\nDr(x)|∇ˆu|2≤2/ba∇dblu−z/ba∇dbl2\nz+ 2/integraldisplay\nDr(x)|∇ˆz|2<ε0\nand sinceuisα-harmonic ˆusolves, defining f(x) =4\n(1−|x|2)2whenγ≥2 elsef≡1, and ˆf(x) =\nf(λ−1x):\n−∆ˆu= ˆu|∇ˆu|2+ (α−1)∇(ˆf−1|∇ˆu|2)·∇ˆu\nλ−2+ˆf−1|∇ˆu|2\nmeaning that ∇ˆusatisfies uniform L∞-estimates at this scale by Theorem C.1. In particular defin-\ningµ=tanh(ι\n4)\ntanh(ι\n2)whenγ≥2 orµ=3\n4otherwise, /ba∇dblˆu/ba∇dblC1(DµλR)≤Cand by undoing the scaling\n/ba∇dbl∇u/ba∇dblL∞(DµR)≤Cλand/ba∇dbl∇u/ba∇dblL∞(Bι\n2)≤Cλ.\nUsing Theorem C.1 again tells us that\n−∆ˆu= ˆu|∇ˆu|2+ (α−1)gwhere/ba∇dblg/ba∇dblL4(D2(x))≤C\n9for any disc of radius 2 in DλR.\nNow, in our coordinates we have that −∆ˆz= ˆz|∇ˆz|2+O(λ−2) onDλR, as can be checked directly\nfrom the definition. Thus\n−∆(ˆu−ˆz) = (ˆu−ˆz)|∇ˆz|2+ ˆu(∇ˆu−∇ˆz)·(∇ˆu+∇ˆz) +O(λ−2) + (α−1)g.\nStandard elliptic estimates, using the scale-invariance o f the/ba∇dbl · /ba∇dblz-norm, first give for all s≥1 (on\nballs with arbitrary centre x∈DµλR) the existence of a constant Ksso that\n/ba∇dblˆu−ˆz/ba∇dblW2,2(D3\n2(x))≤Ks(/ba∇dblu−z/ba∇dblz+λ−2+ (α−1) +/ba∇dblˆu−ˆz/ba∇dblLs(D2(x)))\nand then by Sobolev embedding\n/ba∇dblˆu−ˆz/ba∇dblW2,4(D1(x))≤Ks(/ba∇dblu−z/ba∇dblz+λ−2+ (α−1) +/ba∇dblˆu−ˆz/ba∇dblLs(D2(x)))\nBy undoing the scaling and using Remark 2.9 we have\n/ba∇dblˆu−ˆz/ba∇dblLs≤λ2\ns/ba∇dblu−z/ba∇dblLp(Σ)≤K(s,Σ)λ2\ns(logλ)1\n2/ba∇dblu−z/ba∇dblz.\nWe are left with, in particular\n/ba∇dblˆu−ˆz/ba∇dblL∞(DµλR)+/ba∇dbl∇ˆu−∇ˆz/ba∇dblL∞(DµλR)≤K(s,Σ)(λ2\ns(logλ)1\n2/ba∇dblu−z/ba∇dblz+ (α−1) +λ−2)\nand therefore\n/ba∇dblu−z/ba∇dblL∞(Bι/2(a))+λ−1/ba∇dbl∇u−∇z/ba∇dblL∞(Bι/2(a))≤K(p,Σ)(λ2\np(logλ)1\n2/ba∇dblu−z/ba∇dblz+ (α−1) +λ−2).\nLemma 2.11. There exist α0=α0(Σ)>1,δ=δ(Σ)>0andC=C(Σ)<∞so that if 1<α≤α0,\nandu: Σ→S2is a degree zero α-harmonic map with Eα(u)≤4π+|Σ|+δthen/ba∇dbl∇u/ba∇dblL∞(Σ)≤C.\nProof. Suppose not, then there is a sequence ukofαkharmonic maps of degree zero with αk↓1,\nEα(uk)→4π+|Σ|andukbubble converges to a limit with at least one bubble. In fact i t must have\nexactly one bubble of degree ±1 and energy 4 πsou∞is a constant map. Since bubble convergence\npreserves homotopy type in this case we have contradicted th e fact that each ukhas degree zero.\nUsing the above, along with a direct application of Lemma 4.2 in [20] we are able to prove\nLemma 2.12. For allε1>0there exist α0=α0(Σ)>1,δ=δ(Σ)>0so that if 1< α≤α0and\nu: Σ→S2is a degree-one α-harmonic map with Eα(u)≤4π+|Σ|+δ, then\n1./ba∇dbl∇u/ba∇dblL∞>ε−1\n1\n2.1≤ /ba∇dbl(2 +|∇u|2)α−1/ba∇dblL∞(Σ)<1 +ε1\n3. There exists z∈ Zso that/ba∇dblu−z/ba∇dblW1,2α(Σ)+/ba∇dblu−z/ba∇dblL∞(Σ)<ε1.\n4. (from [20, Lemma 4.2]) There exists z∈ Zso that/ba∇dblu−z/ba∇dblz= infζ∈Z{/ba∇dblu−ζ/ba∇dblζ}and for this z\nwe may still assume that /ba∇dblu−z/ba∇dblL∞<ε1\nProof. Suppose not, then in particular we know that there is some ε1>0 and sequences αk↓1,\nδk→0 and maps ukas in the statement for which one of 1 −3 fails for infinitely many kand for this\nε1>0.\n2.cannot fail for infinitely many ksince either /ba∇dbl∇uk/ba∇dblremains uniformly bounded (in which case\nit is trivially true) or ukconverges as in Theorem 2.7, for which we have that 2 .holds eventually.\nIf 1.fails for infinitely many kthen the regularity theory of α-harmonic maps implies that uk\nmust converge smoothly to a harmonic map u∞: Σ→S2which itself has degree one and E(u∞) =\nE1(u∞)−|Σ|= 4π. Since this map has degree one we know that the area of its imag e is at least 4 π,\n10thus its Dirichlet energy equals its area and it must be confo rmal. We have proved the existence of a\ndegree one holomorphic map from Σ to S2, contradicting the fact that Σ is not a sphere.\nThus in particular we cannot have smooth convergence to a lim it, and we must have convergence\nto a non-trivial bubble configuration of total energy 4 πand degree one. Since all harmonic spheres\nare conformal and therefore rational maps and their complex conjugates there is exactly one bubble\nψof degree one (or minus one) and u∞must be a constant which we write in the form R(0,0,−1) for\nsome appropriate R∈O(3). Thusψis of the form Rπλ(x−x0) for some fixed x0. By a translation\nin the domain and rotation in the target we may assume that ψ=πλ0and we have that ukconverges\nas in Theorem 2.7. Letting zkbe defined via λ=r−1\nkλ0→ ∞ we see immediately that 3 .cannot fail\nfor infinitely many k(see also Remark 2.8).\nWe have proven the required result for the first three points.\nFor 4.: it follows directly from Lemma 4.2 in [20] that for any ν1>0 there isν2>0 so that if\nthere isz0∈ Z with\n/ba∇dblu−z0/ba∇dblL∞+/ba∇dblu−z0/ba∇dblW1,2≤ν2\nthen there is z∈ Z realising dist z(u,Z)≤Cν2so that\n/ba∇dblu−z/ba∇dblL∞≤ν1,\nthus by picking α0−1 andδsufficiently small we can ensure 4 .holds also.\nIt is crucial that the energy is non-degenerate in normal dir ections (to Z) in our setting. The proof\nof this fact follows from the analogous results for the Diric hlet energy [20] (cf [16] for H-surfaces) -\nsee the claim and its proof in the below.\nLemma 2.13. There exist α0=α0(Σ)>1,δ=δ(Σ)>0,c0>0so that for any α≤α0andu: Σ→\nS2α-harmonic, degree one, Eα(u)−4π−|Σ|<δ, and ifz∈ Zrealises/ba∇dblu−z/ba∇dblz= infζ∈Z{/ba∇dblu−ζ/ba∇dblζ},\nthen setting W= dP(z)[u−z]andw=u−zwe have\nd2Eα(z)[W,W ]≥c0/ba∇dblw/ba∇dbl2\nz.\nProof. WriteTzZ⊥zto mean the /a\\}b∇acketle{t,/a\\}b∇acket∇i}htz-orthogonal complement of TzZ ⊂W1,2(Σ,R3). We first\nClaim: IfV∈Γ2(z)∩TzZ⊥zthen there is ˜ c0>0 so that\nd2Eα(z)[V,V]≥˜c0/ba∇dblV/ba∇dbl2\nz.\nAssuming the Claim, it follows from [20, Lemma 4.3], that if Y=W⊤TzZ∈Γ2(z)∩TzZand\nV=W⊥TzZ∈Γ2(z)∩TzZ⊥zthen there exists C <∞so that\n/ba∇dblY/ba∇dbl2\nz≤C/ba∇dblw/ba∇dbl2\nL∞/ba∇dblw/ba∇dbl2\nz\nand therefore\n/ba∇dblV/ba∇dbl2\nz≥ /ba∇dblW/ba∇dbl2\nz−C/ba∇dblw/ba∇dbl2\nL∞/ba∇dblw/ba∇dbl2\nz.\nSinceW= dP(z)[u−z] we have (by checking directly, or see (5.5)) /ba∇dblW/ba∇dblz≥1\n2/ba∇dblw/ba∇dblzso that the\nabove becomes, after choosing α0,δappropriately and applying Lemma 2.12\n/ba∇dblV/ba∇dbl2\nz≥1\n4/ba∇dblw/ba∇dbl2\nz.\nFor anyε>0, once again ensuring that α0,δare sufficiently small, we may ensure from Lemmata\n2.10 and 2.12 λ(z)>Cε−1and from Corollary 2.6 that\n1≤(2 +|∇z|2)α−1≤1 +Cε (2.9)\nwhich in turn, amongst other things yields that\nd2Eα(z)[A,B] =/integraldisplay\nΣ(2 +|∇z|2)α−1(∇A·∇B−|∇z|2A·B)\n11+2α(α−1)/integraldisplay\nΣ(2 +|∇z|2)α−1(∇z·∇A)(∇z·∇B)\n2 +|∇z|2\n≤C/ba∇dblA/ba∇dblz/ba∇dblB/ba∇dblz.\nPutting the above threads together, along with some straigh tforward application of Young’s inequality\nwith aνand possibly after lowering α0andδwe have some c0>0 so that\nd2Eα(z)[W,W ] = d2Eα(z)[V,V] + d2Eα(z)[Y,Y] + 2d2Eα(z)[V,Y]\n≥˜c0/ba∇dblV/ba∇dbl2\nz−Cν/ba∇dblY/ba∇dbl2\nz−ν/ba∇dblV/ba∇dbl2\nz\n≥c0/ba∇dblw/ba∇dbl2\nz\nas required.\nIt remains to prove the Claim . From (2.9)\nd2Eα(z)[V,V] =/integraldisplay\nΣ(2 +|∇z|2)α−1(|∇V|2−|∇z|2|V|2)\n+2α(α−1)/integraldisplay\nΣ(2 +|∇z|2)α−1(∇z·∇V)2\n2 +|∇z|2\n≥/integraldisplay\nΣ|∇V|2−|∇z|2|V|2−Cε/ba∇dblV/ba∇dbl2\nz\n= d2E(z)[V,V]−Cε/ba∇dblV/ba∇dbl2\nz.\nWe now appeal directly to Lemma 3.2 in [20] which tells us that all eigenvalues of d2E(z) are uniformly\nbounded away from zero on ( TzZ)⊥\nz, forλsufficiently large, which for us is ensured by Lemmata 2.10\nand 2.12 implying λ(z)> Cε−1. In our case all eigenvalues must be non-negative since the l imiting\nbubble (the identity map on the sphere) is stable, so all eige nvalues are uniformly positive and bounded\naway from zero: there exists ˆ c0so that\nd2E(z)[V,V]≥ˆc0/ba∇dblV/ba∇dbl2\nz.\nBy potentially lowering α0,δto makeεsufficiently small, we have proved the claim.\n3 Proof of the main theorems assuming further technical deta ils\nProof of Theorem 1.3. Letube anα-harmonic map as per the hypotheses. For energy reasons umust\nhave degree zero or one. If it has degree zero then Lemma 2.11 p roves the first part of the Theorem.\nIfuhas degree one, we may choose δ,α0as in Lemma 2.12 associated with some ε<ε0\n4and letz\nbe a resulting element in Z. Settingw=u−z,ut=P(z+t(u−z)) whereP(q) =q\n|q|is the closest\npoint projection to S2. We letWt=∂tut∈Γ2(ut) withW0=W= dP(z)[w].\nThen\n0 = dEα(u)[W1] =/integraldisplay1\n0∂\n∂tdEα(ut)[Wt] dt+ dEα(z)[W]\n=/integraldisplay1\n0d2Eα(ut)[Wt,Wt]−d2Eα(z)[W,W ] + dEα(ut)[dP(ut)∂tWt] dt\n+d2Eα(z)[W,W ] + dEα(z)[W].\nNow by potentially lowering α0,δand utilising Lemma 2.13 there is some c0>0 so that\nd2Eα(z)[W,W ]≥c0/ba∇dblw/ba∇dbl2\nz.\nBy Lemmata 5.2 and 2.12 along with Corollary 2.6 we may lower α0,δsufficiently so that\n|d2Eα(ut)[Wt,Wt]−d2Eα(z)[W,W ]|+|dEα(ut)[dP(ut)∂tWt]| ≤c0\n2/ba∇dblw/ba∇dbl2\nz.\n12Lemma 5.1 combined with (5.5) gives\n|dEα(z)[W]| ≤C(logλ)1\n2(λ−2+ (α−1))/ba∇dblw/ba∇dblz.\nPutting the above estimates together gives\n/ba∇dblw/ba∇dblz≤C(logλ)1\n2(λ−2+ (α−1)). (3.1)\nNow, forT∈TzZwe setTt= dP(ut)[T] and note that\n0 = dEα(u)[T1] =/integraldisplay1\n0∂\n∂tdEα(ut)[Tt] dt+ dEα(z)[T]\n=/integraldisplay1\n0d2Eα(ut)[Tt,Wt]−d2Eα(z)[T,W ] + dEα(ut)[dP(ut)∂tTt]\n+d2Eα(z)[T,W ] + dEα(z)[T]. (3.2)\nWe will apply this formula for Tλ=∂λzwhich gives, first using Lemma 5.1, (5.5) and then (3.1):\n|d2Eα(z)[Tλ,W]| ≤C(logλ)1\n2(λ−2+λ−1(α−1))/ba∇dblw/ba∇dblz≤Cλ−1(logλ)(λ−3+λ−1(α−1) + (α−1)2)\nand Lemma 5.3 with (3.1):\n|d2Eα(z)[Tλ,W]−d2Eα(ut)[Tλ\nt,Wt] + dEα(ut)[dP(ut)∂tTλ\nt]| ≤Cλ−1/bracketleftbig\n/ba∇dblw/ba∇dbl2\nz+ (α−1)/ba∇dblw/ba∇dblz/bracketrightbig\n≤Cλ−1(logλ)(λ−4+ (α−1)2).\nTherefore from (3.2) and (3.1) we have,\nλ3−2αdEα(z)[Tλ]≤C(logλ)(λ−3+λ−1(α−1) + (α−1)2).\nBy Proposition 4.1 we have\nλ3−2α∂\n∂λEα(z) = 8π/parenleftbig\nJ(a)λ−2+ (α−1)/parenrightbig\n+ (α−1)O(λ−1) +O(λ−3) + (α−1)2O(1),\nwhich means\nJ(a)λ−2+ (α−1) =O(λ−3logλ) + (α−1)/bracketleftbig\n(O((α−1) logλ) +O(λ−1logλ)/bracketrightbig\n+O((α−1)2)).\nNow Corollary 2.6 gives 1 ≤λ2α−2≤1 +εso in particular ( α−1) logλ≤Cεand we can assume first\nO((α−1) logλ) +O(λ−1logλ)≤1\n2\nwhich readily gives the existence of some C <∞so that\nC−1λ−2≤(α−1)≤Cλ−2\nand then\nJ(a)λ−2+ (α−1) =O(λ−3logλ) =O((α−1)3\n2|log(α−1)|)\nwhich is the first desired estimate in the degree one case. The final three estimates follow directly\nfrom this combined with (3.1), Lemma 2.10 and Remark 2.9.\nProof of Theorem 1.4. Since we know that |J(a)|λ−2≃(α−1) applying (3.2) to T=TAand using\nLemmata 5.1 and 5.3 combined with (3.1) we have first\n|d2Eα(z)[TA,W]| ≤C(logλ)λ−3\n13and second\n|d2Eα(z)[TA,W]−d2Eα(ut)[TA\nt,Wt] + dEα(ut)[dP(ut)∂tTA\nt]|\n≤C(logλ)λ−3\nmeaning that\nλ4−2α∇A(Eα(z)) =λ4−2αdEα(z)[∇Az]≤Cλ−1(logλ).\nNow by Proposition 4.1 we have\nλ4−2α∇A(Eα(z)) = 4π∇AJ(a) +O(λ−1),\nwhich means that\n∇AJ(a) =O((α−1)1\n2|log(α−1)|)\nas required.\n4 Energy expansions when restricted to Z\nRecalling the notation introduced in Remark 2.4:\nProposition 4.1. Letα≤2,δ≤1andz∈ Zα,δ. Forλ=λ(z)anda=a(z)we have\nλ3−2α∂\n∂λEα(z) = 8π/parenleftbig\nJ(a)λ−2+ (α−1)/parenrightbig\n+O(λ−1(α−1)) +O(λ−3) +O((α−1)2).\nProof. Using (A.11) and integrating (C.3)\n∂\n∂λEα(z) =−α/integraldisplay\nΣ(2 +|∇z|2)α−1∆z·∂z\n∂λdΣ\n−α(α−1)/integraldisplay\nΣ(2 +|∇z|2)α−2∇|∇z|2·∇z·∂z\n∂λdΣ\n=−α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1∆gz·∂z\n∂λdxg\n−α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2∇g|∇gz|2\ng·g∇gz·∂z\n∂λdxg+O(λ−4).\n=−α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1∆z·∂z\n∂λdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI\n−α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2∇|∇gz|2\ng·∇z·∂z\n∂λdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nII+O(λ−4)\nwhere we have used that ∆ gfdxg= ∆fdxand furthermore ∇gf1·g∇gf2dxg=∇f1·∇f2dx.\nDealing with I:For the first term above we have from (A.1),\n∆z= ∆πλ+ ∆j⊤+ ∆Kand∂z\n∂λ=∂λπλ+∂λj⊤+∂λK.\nBy (A.2)\n∆z·∂λz= ∆πλ·∂λj⊤+ ∆j⊤·∂λπλ+ ∆j⊤·∂λj⊤+ (∆z−∆K)·∂λK+ ∆K·∂λz\n= ∆πλ·∂λj⊤+ ∆j⊤·∂λπλ+O/parenleftbigg\nλ−3λ|x|\n1 +λ2|x|2/parenrightbigg\n, (4.1)\nusing (A.4), (A.6), (A.7) and (A.9).\n14The first term above may be written\n∆πλ·∂λj⊤=−|∇πλ|2πλ·/bracketleftBig\n(∂λj)⊤−(j·πλ)∂λπλ−(j·∂λπλ)πλ/bracketrightBig\n=|∇πλ|2(j·∂λπλ)\nand the second term, since jis harmonic (see Remark 2.2)\n∆j⊤·∂λπλ=−∆((j·πλ)πλ)·∂λπλ\n=−2∇(j·πλ)·∇πλ·∂λπλ\n=−|∇πλ|2(j·∂λπλ)−λ−1[(x1∂1j+x2∂2j)·πλ]|∇πλ|2\nusing thatπλis conformal (see (A.2)) and ∂λπλ=λ−1x·∇πλ(x).\nThus (4.1) becomes\n∆z·∂λz=−λ−1[xi∂ij·πλ]|∇πλ|2+O/parenleftbigg\nλ−3λ|x|\n1 +λ2|x|2/parenrightbigg\n.\nIn particular using (2.7), (A.2) and Corollary 2.6:\n(2 +|∇gz|2\ng)α−1≤C(1 +|∇πλ|2α−2)≤C(1 +λ2α−2)≤C (4.2)\ngiving\nI =−α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1∆z·∂z\n∂λdx\n=α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1[λ−1xi∂ij·πλ]|∇πλ|2dx+O(λ−4)(4.3)\nas can be checked directly.\nRecall again from (2.7) that |∇gz|2\ng=|∇πλ|2(c−1\nγ+χ) whereχ=O(λ−2+|x|2) which gives (by a\nfew applications of the mean value formula (2.8)), when |x| ≤λ−1\n2\n(2 +|∇gz|2\ng)α−1=|∇πλ|2α−2(c−1\nγ+χ)α−1(1 + 2(c−1\nγ+χ)−1|∇πλ|−2)α−1\n=c1−α\nγ|∇πλ|2α−2+O((α−1)|∇πλ|2α−4)\nsinceχ≤ |∇πλ|−2.\nThus\n(2 +|∇gz|2\ng)α−1=\n\nc1−α\nγ|∇πλ|2α−2+e1if|x| ≤λ−1\n2\ne1 if|x|>λ−1\n2(4.4)\nwhere\ne1=\n\nO((α−1)|∇πλ|2α−4) if|x| ≤λ−1\n2\n1 +O(α−1) if |x|>λ−1\n2.(4.5)\nNow (4.3) and (4.4) give\nI =α/integraldisplay\nD\nλ−1\n2c1−α\nγ|∇πλ|2α[λ−1xi∂ij·πλ] dx+/integraldisplay\nDre1[λ−1xi∂ij·πλ]|∇πλ|2dx+O(λ−4).(4.6)\nRecall that j(x) =jλ,a(x) =/parenleftbig2\nλ∇yJa(x,0)−2\nλ∇yJa(0,0),0/parenrightbig\n, andπ1,2\nλ=2λx\n1+λ2|x|2so, first by Taylor\nexpansion about x= 0 we have\n(x1∂1j+x2∂2j) =x1∂1j(0) +x2∂2j(0) +λ−1O(|x|2)\n15and then\nλ−1(x1∂1j+x2∂2j)·πλ=4λ−1\n1 +λ2|x|2/parenleftbig\n(x1)2∂x1∂y1J(0,0) + (x2)2∂x2∂y2J(0,0)/parenrightbig\n+4λ−1x1x2f0\n1 +λ2|x|2+O/parenleftbiggλ−1|x|3\n1 +λ2|x|2/parenrightbigg (4.7)\nwheref0is the constant f0=∂x2∂y1J(0,0) +∂x1∂y2J(0,0).\nIn particular\nλ−1xi∂ij·πλ=O(λ−3)(4.5)=⇒/integraldisplay\nDre1[λ−1xi∂ij·πλ]|∇πλ|2dx=O(λ−4(α−1)) +O(λ−6) (4.8)\nand /integraldisplay\nD\nλ−1\n2|∇πλ|2αλ−1|x|3\n1 +λ2|x|2dx=O(λ−4) (4.9)\nas can be checked directly.\nUsing (4.7)-(4.9), (4.6) becomes\nI =α/integraldisplay\nD\nλ−1\n2c1−α\nγ|∇πλ|2α4λ−1\n1 +λ2|x|2/parenleftbig\n(x1)2∂x1∂y1J(0,0) + (x2)2∂x2∂y2J(0,0)/parenrightbig\ndx\n+α/integraldisplay\nD\nλ−1\n2c1−α\nγ|∇πλ|2α4λ−1x1x2f0\n1 +λ2|x|2dx+O(λ−4).\nUsing the facts that, for radial fand a disc of any radius centred at the origin\n/integraldisplay\nDx1x2f(|x|) dx= 0,and/integraldisplay\nD(x1)2f(|x|) dx=/integraldisplay\nD(x2)2f(|x|) dx=1\n2/integraldisplay\nD|x|2f(|x|) dx\nwe see that, setting J(a) := (∂x1∂y1Ja(0,0) +∂x2∂y2Ja(0,0)),which coincides with Definition 1.1 (see\nAppendix B)\nI = 2αc1−α\nγ8αλ2α−3J(a)/integraldisplay\nD\nλ−1\n2(λ|x|)2\n(1 +λ2|x|2)2α+1dx+O(λ−4)\n= 2αc1−α\nγ8αλ2α−5J(a)/integraldisplay\nD\nλ1\n2|x|2\n(1 +|x|2)2α+1dx+O(λ−4)\n=8απc1−α\nγ\n2α−1J(a)λ2α−5+O(λ−4)\n= 8πJ(a)λ2α−5+O(λ−4) +O(λ−3(α−1)).\nDealing with II:From (2.6) we have, on Dr\n∇z=∇πλ+O(λ−1) (4.10)\nand from (A.1), (A.6)\n∂λz=∂λπλ+∂λj⊤+∂λK\n=∂λπλ+O(λ−2|x|).(4.11)\nFrom (A.1) and the definition of |∇gz|2\ngit follows that\n∇|∇gz|2\ng=c−1\nγ∇|∇z|2+O(|x|2)∇|∇z|2+O(|x|)|∇z|2\n=c−1\nγ∇|∇πλ|2+O(|x|2)∇|∇πλ|2+O(1)/braceleftBig\n∇2j⊤·∇j⊤+∇2K·∇K\n+∇2πλ·∇j⊤+∇πλ·∇2j⊤+∇2πλ·∇K+∇πλ·∇2K+∇2j⊤·∇K\n+∇j·∇2K/bracerightbig\n+O(|x||∇z|2).\n16After applying (A.2)-(A.4),\n∇|∇gz|2\ng=c−1\nγ∇|∇πλ|2+O/parenleftBigg\nλ\n(1 +λ2|x|2)3\n2/parenrightBigg\n. (4.12)\nUsing (4.10)-(4.12), along with (A.2) and ∂λπλ=λ−1x·∇πλwe recover\nII =−α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2∇|∇gz|2\ng·∇z·∂λzdx\n=−α(α−1)c−1\nγ/integraldisplay\nDr(2 +|∇gz|2\ng)α−2/bracketleftBigg\n∇|∇πλ|2·∇πλ·∂λπλ+O/parenleftBigg\nλ2|x|\n(1 +λ2|x|2)7\n2/parenrightBigg/bracketrightBigg\ndx.(4.13)\nSimilarly to (4.4) and since χ|∇πλ|2α−4≤C|∇πλ|2α−6we have\n(2 +|∇gz|2\ng)α−2=\n\nc2−α\nγ|∇πλ|2α−4+e2if|x| ≤λ−1\n2\ne2 if|x|>λ−1\n2(4.14)\nwhere\ne2=\n\nO(|∇πλ|2α−6) if|x| ≤λ−1\n2\nO(1) if |x|>λ−1\n2.\nThus (4.14) gives\nα(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2λ2|x|\n(1 +λ2|x|2)7\n2dx= (α−1)O(λ−3logλ)\nand therefore (4.13) becomes\nII =−α(α−1)c−1\nγ/integraldisplay\nDr(2 +|∇gz|2\ng)α−2∇|∇πλ|2·∇πλ·∂λπλdx+ (α−1)O(λ−3logλ). (4.15)\nBy direct computation\n∇|∇πλ|2·∇πλ·∂λπλ=−128λ5|x|2\n(1 +λ2|x|2)5,\nso by (4.14) again it can be checked that:\nII =−α(α−1)c1−α\nγ/integraldisplay\nD\nλ−1\n2∇|∇πλ|2·∇πλ·∂λπλ|∇πλ|2α−4dx+ (α−1)O(λ−2),\nand (4.15) becomes\nII = 23α+1α(α−1)λ2α−1c1−α\nγ/integraldisplay\nD\nλ−1\n2λ2|x|2\n(1 +λ2|x|2)2α+1dx+ (α−1)O(λ−2)\n=8απc1−α\nγ\n2α−1(α−1)λ2α−3+ (α−1)O(λ−2)\n= 8π(α−1)λ2α−3+ (α−1)O(λ−2) + (α−1)2O(λ−1).\nProposition 4.2. Letα≤2,δ≤1andz∈ Zα,δ. Then for a surface of genus γ≥2, forλ=λ(z)\nanda=a(z)we have\nλ4−2α∇A(Eα(z)) = 4π∇AJ(a) +O(λ−1) +O(λ(α−1)).\n17Proof. From (A.11) and (A.18), similarly to the beginning of the pro of of Proposition 4.1 we have\n∇A(Eα(z)) =−α/integraldisplay\nΣ(2 +|∇z|2)α−1∆z·∇AzdΣ\n−α(α−1)/integraldisplay\nΣ(2 +|∇z|2)α−2∇|∇z|2·∇z·∇AzdΣ\n=−α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1∆z·∇Azdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nIII\n−α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2∇|∇gz|2\ng·∇z·∇Azdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nIV+O(λ−3).\nDealing with III:For the first term above we have\n∆z= ∆πλ+ ∆j⊤+ ∆K\nand trivially\n∇Az=∇Aπλ+∂Aj⊤\na+∂AK\ngiving\n∆z·∇Az= ∆πλ·∇Aj⊤+ ∆j⊤·∇Aπλ+ ∆j⊤·∇Aj⊤+ (∆z−∆K)·∇AK+ ∆K·∇Az\n= ∆πλ·∇Aj⊤−∆j⊥·∇Aπλ+O/parenleftbigg\nλ−1λ|x|\n(1 +λ2|x|2)2/parenrightbigg\n(4.16)\nsincejis harmonic (Remark 2.2)and from (A.4), (A.9), (A.15), (A.1 7).\nThe first term above may be written\n∆πλ·∇Aj⊤=−πλ|∇πλ|2·/bracketleftBig\n(∇Aja)⊤−(j·πλ)∇Aπλ−(j·∇Aπλ)πλ/bracketrightBig\n=|∇πλ|2(j·∇Aπλ)\n=−|∇πλ|2(j·Ai∂iπλ) +O/parenleftbigg\nλ−1λ|x|\n(1 +λ2|x|2)2/parenrightbigg\nfrom (A.14), the definition of πλand that |j| ≤C|x|λ−1.\nAgain using (A.14) the second term becomes\n∆j⊤·∇Aπλ=−∆((j·πλ)πλ)·∇Aπλ\n=−2∇(j·πλ)·∇πλ·∇Aπλ\n= 2∇(j·πλ)·∇πλ·Ai∂iπλ+O/parenleftbigg\nλ−1λ|x|\n(1 +λ2|x|2)2/parenrightbigg\n=|∇πλ|2(j·Ai∂iπλ+Ai∂ij·πλ) +O/parenleftbigg\nλ−1λ|x|\n(1 +λ2|x|2)2/parenrightbigg\nsince∂1πλ·∂2πλ= 0,|∂1πλ|2=|∂2πλ|2=1\n2|∇πλ|2.\nThus (4.16) may be written\n∆z·∇Az=|∇πλ|2Ai∂ij·πλ+O/parenleftbigg\nλ−1λ|x|\n(1 +λ2|x|2)2/parenrightbigg\n.\nUsing again (2 + |∇gz|2\ng)α−1≤C(see (4.2)) we see that\n−α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1λ−1λ|x|\n(1 +λ2|x|2)2dx=O(λ−3)\n18and therefore that\nIII = −α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1∆zλ,a·∇Azdx\n=−α/integraldisplay\nDr(2 +|∇gz|2\ng)α−1|∇πλ|2Ai∂ij·πλdx+O(λ−3).\nRecall that j(x) =jλ,a(x) =/parenleftbig2\nλ∇yJa(x,0)−2\nλ∇yJa(0,0),0/parenrightbig\n, andπ1,2\nλ=2λx\n1+λ2|x|2so. Combining this\nwith (4.4)-(4.5), after a short calculation we obtain\nIII =−41−αα/integraldisplay\nD\nλ−1\n2|∇πλ|2αAi∂ij·πλdx−α/integraldisplay\nDr\\D\nλ−1\n2|∇πλ|2Ai∂ij·πλdx\n+O((α−1)λ−2).(4.17)\nBy Taylor expansion about x= 0 we have\nAi∂ij=Ai∂ij(0) +x1A1∂2\n11j(0) + (x1A2+x2A1)∂2\n12j(0) +x2A2∂2\n22j(0) +O(λ−1|x|2)\nwhich gives\nAi∂ij·πλ=4\n1 +λ2|x|2(x1)2(A1∂x1∂x1∂y1J(0,0) +A2∂x2∂x1∂y1J(0,0))\n+4\n1 +λ2|x|2(x2)2(A1∂x1∂x2∂y2J(0,0) +A2∂x2∂x2∂y2J(0,0))\n+f0(|x|)x1x2+fi(|x|)xi+O/parenleftbigg|x|3\n1 +λ2|x|2/parenrightbigg\nfor some suitable radial functions f0,f1,f2.\nUsing also the facts:/integraltext\nDrxif(|x|) = 0 for each i,/integraltext\nDrx1x2f(|x|) = 0,/integraltext\nDr(x1)2f(|x|) =/integraltext\nDr(x2)2f(|x|)\nand from Lemma B.1\n−1\n2∇AJ(a) =Ai∂xi[∂x1∂y1Ja(0,0) +∂x2∂y2Ja(0,0)]\nwe are left with (after a series of direct calculations from ( 4.17))\nIII = 4π∇AJ(a)λ2α−4+O((α−1)λ−2) +O(λ−3).\nDealing with IV:Combining (4.10) and (4.12) with\n∇Az=∇Aπλ+∇Aj⊤+∇AK\n=∇Aπλ+O(λ−1)\n=−Ai∂iπλ+O(λ−1)\nwe obtain\n∇|∇gz|2\ng·∇z·∇Az=−1\n4∇|∇πλ|2·∇πλ·(Ai∂iπλ) +O/parenleftBigg\nλ3\n(1 +λ2|x|2)7\n2/parenrightBigg\n=1\n8|∇πλ|2Ai∂i|∇πλ|2+O/parenleftBigg\nλ3\n(1 +λ2|x|2)7\n2/parenrightBigg\nsinceπλis conformal.\nUsing (4.14) we have that\n−α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2λ3\n(1 +λ2|x|2)7\n2dx= (α−1)O(λ−1),\n19giving\nIV =−1\n8α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2|∇πλ|2Ai∂i|∇πλ|2dx+O(λ−1(α−1)). (4.18)\nNow, since\nAi∂i|∇πλ|2=f(|x|)(A1x1+A2x2)\nfor some radial function fwe note that\n/integraldisplay\nDF(|x|)|∇πλ|2Ai∂i|∇πλ|2dx= 0\nagain for any radial function F. Thus it remains to suitably estimate only the non-radial te rms of\n(2 +|∇gz|2\ng)α−2in (4.18). Instead of (4.14) we again use |∇gz|2\ng=|∇πλ|2(1\n4+χ) and the mean value\ntheorem to give\n(2 +|∇gz|2\ng)α−2−(2 +1\n4|∇πλ|2)α−2=1\n4|∇πλ|2α−4/parenleftbig\n(2|∇πλ|−2+ 1 +χ)α−2−(2|∇πλ|−2+ 1)α−2/parenrightbig\n=χ|∇πλ|2α−4.\nSince (2 + |∇πλ|2)α−2is radial we are left with\n−1\n8α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2|∇πλ|2Ai∂i|∇πλ|2dx= (α−1)/integraldisplay\nDrχ|∇πλ|2α−2Ai∂i|∇πλ|2dx\n=O(λ−1(α−1))\nas can be checked directly using e.g. (A.2). Thus we have IV = O(λ−1(α−1)) and we are done.\n5 Estimates for dEαandd2Eαclose to Z\nHere we prove that the desired ‘error’ terms appearing in the comparisons of 0 = d Eα(u)[·] and\ndEα(z)[·] are lower order in relation to the expansions of Section 4. W e first prove the required\nestimates on Zitself:\n5.1 Estimates on Z\nLemma 5.1. Letα≤2,δ≤1andz∈ Zα,δ, then there exists C=C(Σ)<∞so that for any\nV∈Γ2(z)we have\n|dEα(z)[V]| ≤C(logλ)1\n2(λ−2+ (α−1))/ba∇dblV/ba∇dblz.\nFurthermore\n|d2Eα(z)[∂λz,V]| ≤C(logλ)1\n2(λ−2+λ−1(α−1))/ba∇dblV/ba∇dblz\nand\n|d2Eα(z)[∇Az,V]| ≤C(logλ)1\n2(λ−1+λ(α−1))/ba∇dblV/ba∇dblz.\nProof. From (4.2) we know (2 + |∇z|2)α−1≤C. Thus\n|dEα(z)[V]|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle−α/integraldisplay\nΣ(2 +|∇z|2)α−1∆z·VdΣ\n−α(α−1)/integraldisplay\nΣ(2 +|∇z|2)α−2∇|∇z|2·∇z·VdΣ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/integraldisplay\nΣ|∆z·V|+ (α−1)|V||∇|∇z|2·∇z|\n2 +|∇z|2dΣ\n≤C/integraldisplay\nDr|∆z·V|+ (α−1)|V||∇|∇gz|2\ng·∇z|\n2 +|∇gz|2gdx+Cλ−2/integraldisplay\nΣ\\Ur(a)|V|dΣ\n20≤C/integraldisplay\nDr|∆(πλ+j⊤)·V|+ (α−1)|V||∇|∇gz|2\ng·∇z|\n2 +|∇gz|2gdx+Cλ−2/integraldisplay\nΣ|V|dΣ\n≤C/integraldisplay\nDr|∆(πλ+j⊤)·V|+ (α−1)|V||∇|∇gz|2\ng·∇z|\n2 +|∇gz|2gdx+C(logλ)1\n2λ−2/ba∇dblV/ba∇dblz(5.1)\nusing (A.10) and (A.11) and Remark 2.9.\nUsing (A.9) we note that\n|∇|∇gz|2\ng·∇z|\n2 +|∇gz|2g≤C|∇2z||∇z|2+|x||∇z|3\n2 +|∇z|2≤C(|∇2z|+|x||∇z|)\n≤Cρzλ\n(1 +λ2|x|2)1\n2\nwhich yields\n/integraldisplay\nDr(α−1)|V||∇|∇gz|2\ng·∇z|\n2 +|∇gz|2gdx≤C(α−1)/parenleftbigg/integraldisplay\nDrρ2\nz|V|2/parenrightbigg1\n2/parenleftbigg/integraldisplay\nDrλ2\n(1 +λ2|x|2)/parenrightbigg1\n2\n≤C(logλ)1\n2(α−1)/ba∇dblV/ba∇dblz. (5.2)\nFinally since V∈Γ2(z) we need only estimate (∆ π+∆j⊤)⊤zwhich we do now. From (A.2), (A.8)\nand (A.5) we have\n∆π+ ∆j⊤=O/parenleftbiggλ2\n(1 +λ2|x|2)2/parenrightbigg\nπλ+O/parenleftbigg|x|\n1 +λ2|x|2/parenrightbigg\n∇πλ\nand also, since πλ·j⊤= 0\nπ⊤z=πλ−(πλ·z)z=−j⊤−K−(πλ·K)(πλ+j⊤+K) =O(λ−1|x|)\nwhich gives\n(∆π+ ∆j⊤)⊤z=O/parenleftbigg|x|\n1 +λ2|x|2/parenrightbigg\nρz.\nWe thus have\nC/integraldisplay\nDr|∆(πλ+j⊤)·V| ≤C/parenleftbigg/integraldisplay\nDrρ2\nz|V|2/parenrightbigg1\n2/parenleftbigg/integraldisplay\nDr|x|2\n(1 +λ2|x|2)2/parenrightbigg1\n2\n≤Cλ−2(logλ)1\n2/ba∇dblV/ba∇dblz. (5.3)\nPutting together (5.1), (5.2) and (5.3) gives the first estim ate.\nFor the second one we use (A.11) to give\n|d2Eα(z)[∂λz,V]|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplay\nΣ(2 +|∇z|2)α−1(∇∂λz·∇V−|∇z|2∂λz·V)\n+α(α−1)/integraldisplay\nΣ(2 +|∇z|2)α−2(∇z·∇∂λz)(∇z·∇V)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(∇∂λz·∇V−|∇z|2∂λz·V) dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+C(α−1)/integraldisplay\nDr|∇∂λz||∇V|+Cλ−2/integraldisplay\nΣ\\U(r)|∇V|+ρz|V|\nwhere the last term is bounded above by λ−2|Σ|/ba∇dblV/ba∇dblzby Cauchy-Schwartz.\nOnDrwe have∂λz=∂λπλ+O(|x|λ−2),|∇z|2=|∇πλ|2(1 +χ) and\n∇∂λz=∇∂λπ+∇∂λj⊤+∇∂λK=∇∂λπλ+O(λ−2)\n21so the above becomes, using also (A.9) and (A.10)\n|d2Eα(z)[∂λz,V]| ≤α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(∇∂λπλ·∇V−|∇πλ|2∂λπλ·V) dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+C/integraldisplay\nDr(λ−2+|x|2)|∇πλ|2|∂λz||V|+|x|λ−2|∇πλ|2||V|\n+C(α−1)/integraldisplay\nDr(1 +λ2|x|2)−1|∇V|+Cλ−2/ba∇dblV/ba∇dblz\n≤α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(∇∂λπλ·∇V−|∇πλ|2∂λπλ·V) dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+C(λ−2+ (α−1)λ−1)/ba∇dblV/ba∇dblz\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(−∆∂λπλ·V−|∇πλ|∂λπλ·V)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+α(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2|∇|∇gz|2\ng||∇∂λπλ||V|\n+α/integraldisplay\n∂Dr(2 +|∇gz|2\ng)α−1|∇∂λπλ||V|+C(λ−2+ (α−1)λ−1)/ba∇dblV/ba∇dblz.\nWe estimate using (A.2) (A.9) and (A.10)\n(2 +|∇gz|2\ng)α−2|∇|∇gz|2\ng||∇∂λπλ| ≤C|∇∂λπλ||∇2z||∇z|+|x||∇z|2\n2 +|∇z|2≤C(1 +λ2|x|2)−1\n2ρz\nand also ∇∂λπλ|∂Dr=O(λ−2). Combining this with Remark 2.9) the above reduces to\n|d2Eα(z)[∂λz,V]| ≤C/integraldisplay\nDr|(∆∂λπλ+|∇πλ|∂λπλ)·V|\n+C(logλ)1\n2(λ−2+ (α−1)λ−1)/ba∇dblV/ba∇dblz.\nOne checks, by direct calculation, that\n∆∂λπλ+|∇πλ|2∂λπλ=−2\nλ|∇πλ|2π3\nλπλ.\nThus the remaining term to estimate has an integrand of the fo rm, using that V·z= 0 a.e.:\n2\nλ|π3\nλ||∇πλ|2|πλ·V|=2\nλ|π3\nλ||∇πλ|2||(j+K)||V| ≤Cλ−2ρz|V|\nwhich finishes the second estimate.\nFor the third estimate we do the same calculations for the sec ond but using instead (A.13)-(A.18)\nto give first\n|d2Eα(z)[∇Az,V]| ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(∇∇Az·∇V−|∇z|2∇Az·V) dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+C(α−1)/integraldisplay\nDr|∇∇Az||∇V|+Cλ−1/integraldisplay\nΣ\\U(r)|∇V|+ρz|V|\nand then, on Drwe have ∇Az=−Aj∂jπλ+O(|x|2)|∇πλ|,|∇z|2=|∇πλ|2(1 +χ) and\n∇∇Az=−Aj∇∂jπλ+O(λ−1) +O(1)(1 +λ2|x|2)−1\nmeaning that (since |∇Aj∂jπλ| ∼λ−1on∂Dr),\n|d2Eα(z)[∇Az,V]| ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(∇(Aj∂jπλ)·∇V−|∇πλ|2(Aj∂jπλ)·V) dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n22+C/integraldisplay\nDrχ|∇πλ|3|V|+C(α−1)/integraldisplay\nDr|∇2πλ||∇V|+Cλ−1/ba∇dblV/ba∇dblz\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(∇(Aj∂jπλ)·∇V−|∇πλ|2(Aj∂jπλ)·V) dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+C(λ−1+λ(logλ)1\n2(α−1))/ba∇dblV/ba∇dblz\n≤α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDr(2 +|∇gz|2\ng)α−1(−∆(Aj∂jπλ)·V−|∇πλ|2(Aj∂jπλ)·V)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+C(α−1)/integraldisplay\nDr(2 +|∇gz|2\ng)α−2|∇|∇gz|2\ng||∇(Aj∂jπλ)||V|\n+C/integraldisplay\n∂Dr|∇(Aj∂jπλ)||V|\n+C(λ−1+λ(logλ)1\n2(α−1))/ba∇dblV/ba∇dblz\n≤C/integraldisplay\nDr|(∆(Aj∂jπλ) +|∇πλ|2(Aj∂jπλ))·V|\n+C(logλ)1\n2(λ−1+λ(α−1))/ba∇dblV/ba∇dblz\nOnce again by direct calculation we have\n∆(Aj∂jπλ) +Aj∂jπλ|∇πλ|2=−πλAj∂j|∇πλ|2\nand using again that V·z= 0 a.e.:\n|Aj∂j|∇πλ|2πλ·V| ≤λ2|x|2\n(1 +λ2|x|2)2ρz|V|.\nWe leave the final details to the reader since they are analogo us to the previous case.\n5.2 Difference estimates for dEαandd2EαnearZ\nHere we will consider u∈W1,∞(Σ,S2) which are close to a specific region of Zin the sense that there\nis somez∈ Z and Λ<∞so that\n/ba∇dblu−z/ba∇dblL∞(Σ)≤Λ and max {(2 +|∇u|2)α−1,(2 +|∇z|2)α−1} ≤1 + Λ.\nWe will apply the below results in the setting that uisα-harmonic and satisfies the hypotheses of the\nmain Theorems, so in particular Λ can be thought of as small.\nFor future reference we first note, setting ut=P(z+tw) =z+tw\n|z+tw|and forA,B∈Γ2(z),C,D∈\nΓ2(ut):\nd2Eα(z)[A,B]−d2Eα(ut)[C,D] =\nα/integraldisplay\nΣ(2 +|∇ut|2)α−1(∇A·(∇B−∇D) + (∇A−∇C)·∇D)\n−α/integraldisplay\nΣ(2 +|∇ut|2)α−1|∇z|2(A·(B−D) + (A−C)·D)\n+α/integraldisplay\nΣ((2 +|∇z|2)α−1−(2 +|∇ut|2)α−1)/bracketleftbig\n∇A·∇B−|∇z|2A·B/bracketrightbig\n+α/integraldisplay\nΣ(2 +|∇ut|2)α−1(|∇ut|2−|∇z|2)C·D\n+2α(α−1)/integraldisplay\nΣ(2 +|∇z|2)α−1(∇z·∇A)(∇z·∇B)\n2 +|∇z|2\n+2α(α−1)/integraldisplay\nΣ(2 +|∇ut|2)α−1(∇ut·∇C)(∇ut·∇D)\n2 +|∇ut|2. (5.4)\n23Lemma 5.2. LetΛ>0,u∈W1,∞(Σ,S2)andz∈ Z. Suppose max{(2+|∇u|2)α−1,(2+|∇z|2)α−1} ≤\n1 + Λand/ba∇dblu−z/ba∇dblL∞≤1\n2. Settingw:=u−z,ut=P(z+tw),Wt= dP(z+tw)[w]andW=W0,\nthere exists C=C(Σ)<∞so that\n|d2Eα(z)[W,W ]−d2Eα(ut)[Wt,Wt]| ≤C(1 + Λ)(Λ + /ba∇dblw/ba∇dblL∞+ (α−1))/ba∇dblw/ba∇dbl2\nz,\nand\n|dEα(ut)[dP(ut)[∂tWt]]| ≤C(1 + Λ)/ba∇dblw/ba∇dblL∞/ba∇dblw/ba∇dbl2\nz.\nProof. First that since w⊥z= (w·z)z=−1\n2|w|2zwe haveW=w+1\n2|w|2zgiving, using very rough\nestimates\n1\n2|w| ≤ |W| ≤ |w|,|∇W| ≤2(|∇w|+|w|2|∇z|),and|∇w| ≤2(|∇W|+|W|2|∇z|).\nThus there is some Cso that\n/ba∇dblW/ba∇dblL∞≃ /ba∇dblw/ba∇dblL∞and/ba∇dblW/ba∇dblz≃ /ba∇dblw/ba∇dblz. (5.5)\nStraight from the definition:\n|Wt−W|=|dP(z+tw)[w]−dP(z)[w]| ≤C|W|2,and\n|∇(Wt−W)| ≤C(|W||∇W|+|W|2|∇z|)(5.6)\nso in particular using (5.5)\n/ba∇dblW−Wt/ba∇dblz≤C/ba∇dblw/ba∇dblL∞/ba∇dblw/ba∇dblz,/ba∇dblWt/ba∇dblz≃ /ba∇dblw/ba∇dblzand/ba∇dblWt/ba∇dblL∞≃ /ba∇dblw/ba∇dblL∞. (5.7)\nWe also have ∇ut=∇z(1 +O(t|w|)) +t∇w(1 +O(|w|)) which gives\n|∇ut| ≤C(|∇z|+|∇w|),|∇ut−∇z| ≤Ct|∇w|+Ct|w||∇z|,\n||∇z|2−|∇ut|2| ≤C(|∇w|2+|∇z||∇w|+|∇z|2|w|)(5.8)\nand\n|∇ut| ≤(1−t)|∇z|(1 +O(|w|)) +t|∇u|(1 +O(|w|)). (5.9)\nBy the hypotheses on uandz, using (5.9) we have\n1≤(2 +|∇ut|2)α−1≤1 +CΛ for some C <∞ (5.10)\nand then\n|(2 +|∇ut|2)α−1−(2 +|∇z|2)α−1| ≤CΛ.\nCombining this with (5.5)-(5.8) and (5.4) give for A=B=WandC=D=Wt, recalling that\n|∇z| ≤Cρzon Σ:\n|d2Eα(z)[W,W ]−d2Eα(ut)[Wt,Wt]| ≤\nC(1 + Λ)/integraldisplay\nΣ(|∇W|+|∇Wt|)|∇(W−Wt)|+ρ2\nz(|W|+|Wt|)|W−Wt|\n+CΛ/integraldisplay\nΣ|∇W|2+ρ2\nz|W|2+C(1 + Λ)/integraldisplay\nΣ(|∇w|2+ρz|∇w|+ρ2\nz|w|)|Wt|2\n+C(1 + Λ)(α−1)/integraldisplay\nΣ|∇W|2+|∇Wt|2\n≤C(1 + Λ)(/ba∇dblw/ba∇dblL∞/ba∇dblw/ba∇dbl2\nz+ Λ/ba∇dblw/ba∇dbl2\nz+ (α−1)/ba∇dblw/ba∇dbl2\nz).\nNow,\ndP(ut)[∂tWt] = dP(ut)(d2P(z+tw)[w,w])\n= dP(z)[d2P(z)[w,w]] + dP(z)[(d2P(z+tw)−dP2(z))[w,w]]\n+ (dP(ut)−dP(z))[d2P(z+tw)[w,w]].\n24Thus by (C.2), using that A(z)(V,W ) =−z(V·W) for the round sphere, this gives\ndP(ut)[∂tWt] =|w|2W+ dP(z)[(d2P(z+tw)−dP2(z))[w,w]]\n+ (dP(ut)−dP(z))[d2P(z+tw)[w,w]]\nwhich in particular yields\n|∇(dP(ut)[∂tWt])| ≤C(|∇W||W|2+|∇z||W|3).\nThis and (5.8) give\n|dEα(ut)[dP(ut)[∂tWt]]| ≤C(1 + Λ)/integraldisplay\nΣ|∇ut||∇W||W|2+|∇ut||∇z||W|3\n≤C(1 + Λ)/parenleftbigg/integraldisplay\nΣ|∇z||∇W||W|2+|∇w||∇W||W|2+|∇z|2|W|3\n+/integraldisplay\nΣ|∇w||∇z||W|3/parenrightbigg\n≤C(1 + Λ)/ba∇dblw/ba∇dblL∞/ba∇dblw/ba∇dbl2\nz\nas required.\nLemma 5.3. Letu∈W1,∞(Σ,S2)andz∈ Z. Suppose max{(2 +|∇u|2)α−1,(2 +|∇z|2)α−1} ≤1 + Λ\nand/ba∇dblu−z/ba∇dblL∞≤Λfor some Λ<∞. Setw:=u−z,ut=P(z+tw)andWt= dP(z+tw)[w]and\ngivenT∈Γ2(z)letTt= dP(ut)[T]. Then there exists C=C(Σ)<∞so that:\n1. WhenTλ=∂λz\n|d2Eα(z)[Tλ,W]−d2Eα(ut)[Tλ\nt,Wt]| ≤Cλ−1(1 + Λ)/bracketleftbig\n/ba∇dblw/ba∇dbl2\nz+ (α−1)/ba∇dblw/ba∇dblz/bracketrightbig\nand\n|dEα(ut)[dP(ut)∂tTλ\nt]| ≤Cλ−1(1 + Λ)/ba∇dblw/ba∇dbl2\nz.\n2. WhenTA=∇Az,\n|d2Eα(z)[TA,W]−d2Eα(ut)[TA\nt,Wt]| ≤Cλ(1 + Λ)(/ba∇dblw/ba∇dbl2\nz+ (α−1)/ba∇dblw/ba∇dblz)\nand\n|dEα(ut)[dP(ut)∂tTA\nt]| ≤Cλ(1 + Λ)/ba∇dblw/ba∇dbl2\nz.\nProof. Given 0≤x≤y∈Rassume that (2 + y2)α−1≤1 + Λ then the mean value formula straight-\nforwardly gives the existence of some ξ∈[x,y] so that:\n(2 +y2)α−1−(2 +x2)α−1= (α−1)2ξ\n(2 +ξ2)(2 +ξ2)α−1(y−x)\n≤4(1 + Λ)(α−1)1√\n2 +ξ(y−x). (5.11)\nCombining this with (5.8) gives\n|(2 +|∇z|2)α−1−(2 +|∇ut|2)α−1| ≤C(1 + Λ)(α−1)|∇w|+|w||∇z|√\n2 +|∇z|if|∇u| ≥1\n2|∇z|(5.12)\nwhilst we trivially note for later that\n|∇z| ≤2|∇w|if|∇u|<1\n2|∇z|. (5.13)\nDirectly from (5.11) and (5.8) we also see that\n|(2 +|∇z|2)α−1−(2 +|∇ut|2)α−1| ≤C(1 + Λ)(α−1)(|∇w|+|w||∇z|). (5.14)\n25Using|∇z| ≃ρzonBι(a) together with (5.4), (5.8), (5.10) and (5.12)-(5.14) give s, withU1={|∇u| ≥\n1\n2|∇z|}andU2= Σ\\U1\n|d2Eα(z)[Tλ,W]−d2Eα(ut)[Tλ\nt,Wt]| ≤\nC(1 + Λ)/integraldisplay\nΣ|∇Tλ||∇(W−Wt)|+|∇(Tλ−Tλ\nt)||∇Wt|\n+C(1 + Λ)/integraldisplay\nΣρ2\nz(|Tλ||W−Wt|+|Wt||Tλ−Tλ\nt|)\n+C(1 + Λ)(α−1)/integraldisplay\nBι(a)∩U1|∇w|+|w|ρz√\n2 +ρz/bracketleftBig\n|∇Tλ||∇W|+ρ2\nz|Tλ||W|/bracketrightBig\n+CΛ/integraldisplay\nBι(a)∩U2/bracketleftBig\n|∇Tλ||∇W|+ρ2\nz|Tλ||W|/bracketrightBig\n+C(1 + Λ)(α−1)/integraldisplay\nΣ\\Bι(a)(|∇w|+|w|ρz)/bracketleftBig\n|∇Tλ||∇W|+ρ2\nz|Tλ||W|/bracketrightBig\n+C(1 + Λ)/integraldisplay\nΣ(|∇w|2+|∇z||∇w|+|∇z|2|w|)|Tλ\nt||Wt|\n+C(1 + Λ)(α−1)/integraldisplay\nΣ|∇Tλ||∇W|+|∇Tλ\nt||∇Wt|.\nWe collect the following readily-checked facts concerning Ttwhich follow from the definition:\n|Tt−T| ≤C|W||T| |∇ (Tt−T)| ≤C(|W||∇T|+|W||∇z||T|+|∇W||T|). (5.15)\nNow forT=Tλ=∂λzwe have|Tλ| ≤Cλ−1and by (5.15) we have the same bound for Tλ\nt. Also,\n|∇Tλ| ≤Cλ−1ρzand by (5.15) |∇Tλ\nt| ≤C(λ−1ρz+λ−1∇W).\nNote that both Tλand∇Tλare bounded by λ−2on Σ\\Bι(a). The above now becomes, using also\n(5.6), (5.13) and again with U1={|∇u| ≥1\n2|∇z|}andU2= Σ\\U1:\n|d2Eα(z)[Tλ,W]−d2Eα(ut)[Tλ\nt,Wt]| ≤\nCλ−1(1 + Λ)/integraldisplay\nΣρz(|W||∇W|+|W|2ρz) +ρz|W||∇Wt|+|∇W||∇Wt|\n+Cλ−1(1 + Λ)/integraldisplay\nΣρ2\nz(|W|2+|Wt||W|)\n+Cλ−1(1 + Λ)(α−1)/integraldisplay\nBι(a)∩U1(|∇w|+|w|ρz)(|∇W|+ρz|W|)\n+Cλ−1Λ/integraldisplay\nBι(a)∩U2|∇W||∇w|+|W|ρz|∇w|\n+Cλ−2(1 + Λ)(α−1)/integraldisplay\nΣ\\Bι(a)(|∇w|+|w|ρz) [|∇W|+ρz|W|]\n+Cλ−1(1 + Λ)(α−1)/integraldisplay\nΣρz(|∇W|+|∇Wt|) +|∇W||∇Wt|\nwhich readily gives the required estimate by (5.5) and (5.7) .\nBy definition ∂tTt= d2P(ut)[Tt,Wt], so by (C.1) and (C.2), d P(z)((d2P(z)[W,T ])) = 0 giving\ndP(ut)∂tTt= (dP(ut)−dP(z))[d2P(z)[T,W ]] + dP(ut)[(d2P(ut)−d2P(z))[T,W ]]\n+ dP(ut)[d2P(ut)[Tt−T,W ]] + dP(ut)[d2P(ut)[Tt,Wt−W]].\nThis means in particular using (5.6) – (5.8) and (5.15)\n|∇dP(ut)∂tTt| ≤C(|∇z||W|2|T|+|W||∇W||T|+|∇T||W|2)\n26and forT=Tλ\ndEα(ut)[dP(ut)∂tTλ\nt]≤Cλ−1/integraldisplay\nΣ(ρz+|∇w|)(ρz|W|2+|W||∇W|+ρz|W|2)≤Cλ−1/ba∇dblw/ba∇dbl2\nz.\nWhenT=TAthe proof follows exactly the same lines expect that we use |TA| ≤Cλand\n|∇TA| ≤Cλρz, whilst both TAand∇TAare bounded by λ−1on Σ\\Bι(a). These give precisely the\nsame estimates with a factor of λ2the only final difference.\nA Initial expansions\nInUr(a) =F−1\na(Dr), ˜zλ,a=πλ(Fa(p)) +jλ,a(Fa(p)) meaning that in Fa-coordinates x∈Drwe have\nz(x) =zλ,a(x) =P(πλ(x) +jλ,a(x)) =πλ+/integraldisplay1\n0∂\n∂tP(πλ+tjλ,a) dt\n=πλ+ dP(πλ)[jλ,a] +/integraldisplay1\n0(dP(πλ+tjλ,a)[jλ,a]−dP(πλ)[jλ,a]) dt\n=πλ+j⊤\nλ,a+Kλ,a (A.1)\nwherej⊤\nλ,a= dP(πλ)[jλ,a] =jλ,a−(jλ,a·πλ)πλ. Mostly we will drop the explicit λ,adependencies\nfromz,jandK, writing simply z(x) =πλ+j⊤+K.\nWe will frequently use standard properties of πλ(x) =/parenleftBig\n2λx\n1+λ2|x|2,1−λ2|x|2\n1+λ2|x|2/parenrightBig\n:\n−∆πλ=|∇πλ|2πλ, π λ·∂iπλ= 0, i= 1,2\n∂1πλ·∂2πλ= 0,|∂1πλ|2=|∂2πλ|2, ∂ λπλ=λ−1x·∇πλ(x),\n|∇πλ|2=8λ2\n(1 +λ2|x|2)2, |∇2πλ|=O/parenleftBigg\nλ2\n(1 +λ2|x|2)3\n2/parenrightBigg\n.(A.2)\nOnUr(a)∼=Drwe have the following rough estimates as can be checked direc tly: From the definition\nofj=jλ,awe have |∇j|=O(λ−1) and since j(0) = 0 we have\n|j|=O(|x|λ−1) giving |∇K|=O(|x|λ−2),and|∇j⊤|=O(λ−1) (A.3)\nafter a short calculation.\nUsing (A.2) and the properties of PalongS2we also have\n|∇2j⊤|=O/parenleftbig\nλ−1+ (1 +λ2|x|2)−1/parenrightbig\n,|∇2K|=O(λ−2) and |(∆K)⊤|=O/parenleftbiggλ−1|x|\n1 +λ2|x|2/parenrightbigg\n.(A.4)\nRecall that the third component of jis zero, and π1,2\nλ=2λx\n1+λ2|x|2, which gives\nj·πλ=O/parenleftbigg|x|2\n1 +λ2|x|2/parenrightbigg\nand|∇(j·πλ)|=O/parenleftbigg|x|\n1 +λ2|x|2/parenrightbigg\n. (A.5)\nA.1 Expansions involving ∂λz\nAfter further estimates one can check that\n|∂λj|=O(|x|λ−2) giving |∂λj⊤|=O(|x|λ−2),and|∂λK|=O(|x|2λ−3). (A.6)\nFor our purposes later we note that\nLemma A.1.\n∆j⊤·∂λj⊤=O/parenleftbiggλ−1|x|2\n(1 +λ2|x|2)2/parenrightbigg\n. (A.7)\n27Proof. Sincejis harmonic we have\n∆j⊤=−∆[(j·πλ)πλ]\n=−(∆(j·πλ))πλ−(j·πλ)∆πλ−2∇(j·πλ)·∇πλ\n=−(2∇j·∇πλ−2|∇πλ|2j·πλ)πλ−2∇(j·πλ)·∇πλ. (A.8)\nWe also have\n∂λj⊤=∂λ(j−(j·πλ)πλ)\n= [(∂λj)⊤−(j·πλ)∂λπλ]−(j·∂λπλ)πλ.\nThese together give\n∆j⊤\nλ,a·∂λj⊤\nλ,a=−(2∇j·∇πλ−2|∇πλ|2j·πλ)(j·∂λπλ)\n−2∇(j·πλ)·∇πλ·[(∂λj)⊤−(j·πλ)∂λπλ].\nUsing∂λπλ=λ−1x· ∇πλand∇j· ∇πλ=O(1)\n(1+λ2|x|2)and the aforementioned estimates we see\ntherefore that\n|∆j⊤\nλ,a·∂λj⊤\nλ,a| ≤ |(2∇j·∇πλ−2|∇πλ|2j·πλ)(j·∂λπλ)\n+ 2∇(j·πλ)·∇πλ·[(∂λj)⊤−(j·πλ)∂λπλ]|\n≤Cλ−3λ2|x|2\n(1 +λ2|x|2)2.\nWe finally have, on Ur(a):\n|∇zλ,a| ≃λ\n1 +λ2|x|2,|∆zλ,a| ≃λ2\n(1 +λ2|x|2)2,|∇2z| ≃λ2\n(1 +λ2|x|2)3\n2, (A.9)\nalong with\n|∂λz| ≃λ−1λ|x|\n1 +λ2|x|2and|∇∂λz| ≃1\n(1 +λ2|x|2). (A.10)\nOn the other hand, in Σ \\Ur(a) it is straightforward to check, using (2.3), and (2.5) that we have\nthe following estimates on zλ,a:\n|∇zλ,a|=O(λ−1) and/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂zλ,a\n∂λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle,|∆zλ,a|,/vextendsingle/vextendsingle∇|∇zλ,a|2/vextendsingle/vextendsingle,|∇∂λz|=O(λ−2). (A.11)\nA.2 Expansions involving ∇Azwhenγ≥2\nLetA∈TaΣ∼=(Fa)∗T0Dand for 0 ≤s≪1,as:=ExpΣ\na(sA) and we wish to compute\n∇Azλ,a(p) =∂\n∂szλ,as(p)\ns=0.\nGiven our choice of local coordinates Faatawe chooseFas:= (τA\ns)−1◦FawhereτA\nsis a Hy-\nperbolic translation along the geodesic ExpD\n0(sA) by distance s|A|g(0). Specifically ExpD\n0(sA) =\n2Atanh(s/2) :=γ(s) andτA\nsis given explicitly by (see e.g. [18])\n(τA\ns)−1(x) =(1−|γ(s)|2)x−(|x|2−2x·γ(s) + 1)γ(s)\n|γ(s)|2|x|2−2x·γ(s) + 1\nwhere all norms and inner-products appearing above are Eucl idean. In particular\n∂\n∂s(τA\ns)−1(x)\ns=0=−A−A|x|2+ 2(x·A)x. (A.12)\n28Thus forpclose toawe have\n∇Az(p) =∇Aπλ(p) +∇A(ja)⊤(p) +∇AK(p) (A.13)\nwhere we abuse notation by writing\n∇Aπλ=∂\n∂ss=0πλ(Fas(p)),\n∇A(ja\nλ)⊤=∂\n∂ss=0(jas\nλ(p)−(jas\nλ(p)·πλ(Fas(p)))πλ(Fas(p)))\netc.\nIn particular by (A.12) we see that for Facoordinates x∈Dr,\n∇Aπλ=−Ai∂iπλ−|x|2Ai∂iπλ+ 2(A·x)xi∂iπλ, (A.14)\nnoting that we have ∇Aπλ·πλ≡0.\nForAa unit vector we collect the following rough estimates on Drwhich can be checked by direct\ncalculation - see also A.2.1. First of all |∇Ajλ,a| ≤Cλ−1which yields that\n|∇Aj⊤|=O(λ−1),|∇AKλ,a|=O(|x|λ−2) and |∇Aj⊤\nλ,a·∆j⊤\nλ,a|=O/parenleftbigg|x|\n(1 +λ2|x|2)2/parenrightbigg\n(A.15)\nwhere the final estimate above follows by suitably adapting t he proof of Lemma A.1 where the only\ndifference is that ∇Aj∼λ|x|−1∂λjand∇Aπλ∼λ|x|−1∂λπλ.\nBy appendix A.2.1 we also have |∇∇Aj| ≤Cλ−1which gives\n|∇∇Aj⊤|=O/parenleftbig\nλ−1+ (1 +λ2|x|2)−1/parenrightbig\nand|∇∇AK|=O(λ−2). (A.16)\nThus we have\n|∇Az| ≃λ\n1 +λ2|x|2and|∇∇Az| ≃λ2\n(1 +λ2|x|2)3\n2(A.17)\nonUr(a).\nOn the other hand, in Σ \\Ur(a) it is straightforward to check,\n|∇Azλ,a|,|∇∇Azλ,a|=O(λ−1). (A.18)\nA.2.1 Computing ∇Aj\nHere we expand on the computations for ∇Aj- we note that the precise form of the derivation below\nis not crucial for our needs, but we nevertheless give full de tails for the sake of the reader.\nWe have∂\n∂ss=0τA\ns(x) =A+A|x|2−2(x·A)x\nnoting for later that,\n∂\n∂ss=0∇ylog|τ−A\ns(x)−y|\ny=0=−Aj∂xj∇ylog|x−y|\ny=0+A\n=Aj∂yj∇ylog|x−y|\ny=0+A (A.19)\nand\nλ\n2(jλ,as)1,2(p) =∇yJas(Fas(p),y)\ny=0−∇yJas(Fas(as),y)\ny=0\n=/parenleftbig\n∇y(G(p,F−1\nas(y))) +∇ylog|Fas(p)−y|/parenrightbig\ny=0\n−/parenleftbig\n∇y(G(F−1\nas(0),F−1\nas(y))) +∇ylog|y|/parenrightbig\ny=0. (A.20)\n29By definition we have F−1\nas=F−1\na◦τA\nsand in the below we set x=Fa(p) so thatFas(p) =τ−A\ns(x).\nThus we have, by (A.19) and (A.20):\n∂\n∂ss=0λ\n2(jλ,as)1,2(p) =/parenleftbig\nAj∂yj∇y(G(p,F−1\na(y))) +Aj∂yj∇ylog|x−y|/parenrightbig\ny=0+A\n−/parenleftbig\nAj∂xj∇y(G(F−1\na(x),F−1\na(y))) +Aj∂yj∇y(G(F−1\na(x),F−1\na(y)))/parenrightbig\nx,y=0\n=Aj∂yj∇yJa(x,0) +A−Aj∂xj∇yJa(0,0)−Aj∂yj∇yJa(0,0)\n+/parenleftbig\nAj∂xj∇ylog|x−y|+Aj∂yj∇ylog|x−y|/parenrightbig\nx,y=0\n=Aj∂yj∇yJa(x,0)−Aj∂xj∇yJa(0,0)−Aj∂yj∇yJa(0,0) +A.\nThus in particular\n∇Aj=2\nλ(Aj∂yj∇Ja(x,0),0) +1\nλ(C,0) (A.21)\nfor a constant vector C ∈R2.\nB Derivatives of J\nWe work with the Green’s function which solves\n−∆pG(p,a) = 2πδa−2π\nVolg(Σ). (B.1)\nIfx,yare local coordinates near a∈Σ as in (2.1), then for z=x1+ix2,ζ=y1+iy2\nJ(a) := (∂x1∂y1Ja(0,0) +∂x2∂y2Ja(0,0)) = 4 lim\nx→0Re/parenleftbigg∂2Ga\n∂¯ζ∂z(x,0)/parenrightbigg\nwhere, from [16, Proposition 6.2, Remark 6.3], in any holomo rphic coordinates ( z,ζ) we have\n∂2G\n∂¯ζ∂z(z,ζ)dz⊗dζ=−π/summationdisplay\njφj(z)⊗φj(ζ)\nand{φj}jis anL2-orthonormal basis of holomorphic one-forms on Σ. In partic ular when Σ has genus\nγwe have\nJ(a) =−2πcγ/summationdisplay\nj|φj(a)|2\ng\nwherec1= 1 and when γ≥2,cγ= 4.\nLemma B.1. With the set-up as above we have\n−1\n2∇AJ(a) =Ai∂xi[∂x1∂y1Ja(0,0) +∂x2∂y2Ja(0,0)].\nProof. We first note that\n−1\n2∇AJ(a) =πcγ∇Σ\nA/summationdisplay\nj/a\\}b∇acketle{tφj,φj/a\\}b∇acket∇i}htg=πcγ/summationdisplay\nj/a\\}b∇acketle{t∇Σ\nAφj,¯φj/a\\}b∇acket∇i}htg+/a\\}b∇acketle{tφj,∇Σ\nAφj/a\\}b∇acket∇i}htg\n= 2πcγRe\n/summationdisplay\nj/a\\}b∇acketle{t∇Σ\nAφj(a),¯φj(a)/a\\}b∇acket∇i}htg\n\nwhere/a\\}b∇acketle{t,/a\\}b∇acket∇i}htgand∇Σ\nAare the metric and connection on Σ extended complex linearly toTCΣ respectively.\n30In our local coordinates xwe writeφj=˜φjdzandA=Ai∂xi. Since∇Σdz(0) = 0 we have that\n∇Σ\nAφj(0) = (Ai∂xi˜φj(0))dz. Noting also that Ai∂xi∂2Ga\n∂¯ζ∂z(x,0) =−π(Ai∂xi˜φj(x))˜φj(0) and|dz(0)|2\ng=\n2c−1\nγ, gives\n−1\n2∇AJ(a) = 4πRe\n/summationdisplay\nj(Ai∂xi˜φj(0))˜φj(0)\n= 4 lim\nx→0Re/parenleftbigg\nAi∂xi∂2Ga\n∂¯ζ∂z(x,0)/parenrightbigg\n=Ai∂xi[∂x1∂y1+∂x2∂y2]Ja(0,0),\nsince forx/\\e}atio\\slash=y, by (2.2)∂2Ga\n∂¯ζ∂z(x,y) =1\n4(∂x1−i∂x2)(∂y1+i∂y2)Ja(x,y).\nC First and second variation of Eα\nWe briefly derive the first and second variations of Eαalong maps u∈W1,2α(Σ,N) assuming N ֒→Rd\nis closed and isometrically embedded and Σ is any closed Riem annian manifold.\nLet Ων(N) denote a small ν-neighbourhood of NinRdwhereν > 0 is small enough so that\nP: Ωδ(N)→N, the nearest point projection, is well-defined and smooth. W e note the following easy\nto check facts concerning the behaviour of PonN. Specifically given anyX,Y∈Rdandk∈N:\ndP(k)[X] =X⊤whereX⊤is the projection onto TkN (C.1)\nd2P(k)[X,Y ] =A(k)(X⊤,Y⊤) +/a\\}b∇acketle{tA(k)(·,X⊤),Y⊥/a\\}b∇acket∇i}ht♭+/a\\}b∇acketle{tA(k)(·,Y⊤),X⊥/a\\}b∇acket∇i}ht♭(C.2)\nwhere of course X⊥=X−X⊤is the projection onto VkNandA(k)(X⊤,Y⊤) :=/parenleftBig\n∇Rd\nX⊤Y⊤/parenrightBig⊥\n. For a\nround sphere A(k)(X⊤,Y⊤) =−k/a\\}b∇acketle{tX⊤,Y⊤/a\\}b∇acket∇i}ht.\nC.1 First variation and regularity of solutions\nForu∈W1,2α(Σ,Sn),V∈W1,2α(Σ,Rd) so thatV(x)∈Tu(x)Nalmost everywhere we have\ndEα(u)[V] =α/integraldisplay\nΣ(2 +|∇u|2)α−1∇u·∇VdΣ (C.3)\nmeaning that α-harmonic maps are weak solutions to div Σ((2 +|∇u|2)α−1∇u)⊤= 0 where ‘⊤’ is\nprojection onto Tu(x)N. Sinceα-harmonic maps are always smooth ([22, Proposition 2.3]) in particular\nthey are solutions to\n−∆u=−A(u)(∇u,∇u) + (α−1)∇|∇u|2·∇u\n2 +|∇u|2.\nLetD1⊂R2be a unit Euclidean ball equipped with a metric g=fg0andg0is the standard Eu-\nclidean metric. If u∈W1,2α(D,N) isα-harmonic then (with all operators below now being Euclidea n)\n−∆u=−A(u)(∇u,∇u) + (α−1)∇(f−1|∇u|2)·∇u\n2 +f−1|∇u|2.\nWe first recall the standard ε-regularity estimate for systems like the above when α > 1. For\nconvenience we introduce an extra constant ν, reflecting that we will want to apply these estimates\nfor a re-scaled version of an α-harmonic map, ˆ u(x) =u(λ−1x), however this has no consequence on\nthe result which can be easily adapted from [22, Section 3]:\nTheorem C.1. Letu∈W1,2α(D,N),ν > 0,f∈C∞(D,R>0)andN ֒→Rda smooth closed\nRiemannian manifold isometrically embedded in some Euclid ean space.\nThere exists ε0=ε0(N)>0,α0=α0>1andK(N,/ba∇dbllogf/ba∇dblC1)<∞so that for all 1<α≤α0if\nuweakly solves\n−∆u=−A(u)(∇u,∇u) + (α−1)∇(f−1|∇u|2)·∇u\n2ν+f−1|∇u|2(C.4)\n31and/integraltext\nD|∇u|2<ε0then\n/ba∇dbl∇2u/ba∇dblL4(D1\n2)+/ba∇dbl∇u/ba∇dblL∞(D1\n2)≤K/parenleftbigg/integraldisplay\nD|∇u|2/parenrightbigg1\n2\n.\nIn particular the above implies:\nCorollary C.2. LetΣbe as in Definition 1.1 and N ֒→Rda smooth closed Riemannian manifold\nisometrically embedded in some Euclidean space.\nLetr <inj(Σ). There exists ε0=ε0(N)>0,α0>1andK(N)<∞so that ifu: Σ→Nis\nα-harmonic, 1<α<α 0and/integraltext\nBr(a)|∇u|2<ε0then\n/ba∇dbl∇u/ba∇dblL∞(Br\n2(a))≤Kr−1/parenleftBigg/integraldisplay\nBr(a)|∇u|2/parenrightBigg1\n2\n.\nC.2 Second variation\nGivenu∈W1,2α(Σ,N) and perturbations V,W∈W1,2α(Σ,Rd) so thatV(x),W(x)∈Tu(x)Nalmost\neverywhere we consider us,t=P(u+sV+tW) and do a Taylor expansion about s,t= 0 using\n(C.1)-(C.2): us,t=u+tW+sV+stA(u)(V,W ) +O(t2) +O(s2).Thus\nd2Eα(u)[V,W ] =∂2\n∂s∂ts,t=0Eα(us,t)\n=α/integraldisplay\nΣ(2 +|∇u|2)α−1(∇V·∇W+∇u·∇(A(u)(V,W )))\n+2α(α−1)/integraldisplay\nΣ(2 +|∇u|2)α−2(∇u·∇V)(∇u·∇W).\nUsing∇u·A(u)(V,W ) = 0 we see ∇u·∇(A(u)(V,W )) = div( ∇u·A(V,W ))−∆u·A(u)(V,W ) and\nsince we have (∆ u)⊥=A(u)(∇u,∇u) weakly, we have\nd2Eα(u)[V,W ] =α/integraldisplay\nΣ(2 +|∇u|2)α−1(∇V·∇W−A(u)(∇u,∇u)·A(u)(V,W ))\n+2α(α−1)/integraldisplay\nΣ(2 +|∇u|2)α−2(∇u·∇V)(∇u·∇W).\nOf course when N=Snis a round sphere and u∈W1,2α(Σ,Sn),V,W∈W1,2α(Σ,Rd) so that\nV(x),W(x)∈Tz(x)Nalmost everywhere:\nd2Eα(u)[V,W ] =α/integraldisplay\nΣ(2 +|∇u|2)α−1/parenleftbig\n∇V·∇W−|∇u|2(V·W)/parenrightbig\n+2α(α−1)/integraldisplay\nΣ(2 +|∇u|2)α−2(∇u·∇V)(∇u·∇W). 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Foundations of hyperbolic manifolds , volume 149 of Graduate Texts in Mathe-\nmatics . Springer, New York, second edition, 2006.\n[19] Olivier Rey. The role of the Green’s function in a nonlin ear elliptic equation involving the critical\nSobolev exponent. J. Funct. Anal. , 89(1):1–52, 1990.\n[20] Melanie Rupflin. /suppress Lojasiewicz inequalities for almost harmonic maps near simple bubble trees.\narXiv:2101.05527v2, 2021.\n[21] Melanie Rupflin. Low energy levels of harmonic maps into analytic manifolds. arXiv:2303.00389v1,\n2023.\n[22] J. Sacks and K. Uhlenbeck. The existence of minimal imme rsions of 2-spheres. Ann. of Math.\n(2), 113(1):1–24, 1981.\nB. Sharp: School of Mathematics, University of Leeds, Leeds LS2 9JT, UK\nb.g.sharp@leeds.ac.uk\n33"    },    {        "title": "2402.02895v1.Noisy_group_testing_via_spatial_coupling.pdf",        "content": "NOISY GROUP TESTING VIA SPATIAL COUPLING\nAMIN COJA-OGHLAN, MAX HAHN-KLIMROTH, LUKAS HINTZE, DOMINIK KAASER, LENA KRIEG, MAURICE ROLVIEN,\nOLGA SCHEFTELOWITSCH\nABSTRACT . We study the problem of identifying a small set k∼nθ, 0<θ<1, of infected individuals within a large pop-\nulation of size nby testing groups of individuals simultaneously. All tests are conducted concurrently. The goal is to\nminimise the total number of tests required. In this paper we make the (realistic) assumption that tests are noisy, i.e. that\na group that contains an infected individual may return a negative test result or one that does not contain an infected\nindividual may return a positive test results with a certain probability. The noise need not be symmetric. We develop an\nalgorithm called SPARC that correctly identifies the set of infected individuals up to o(k) errors with high probability with\nthe asymptotically minimum number of tests. Additionally, we develop an algorithm called SPEX that exactly identifies\nthe set of infected individuals w.h.p. with a number of tests that matches the information-theoretic lower bound for the\nconstant column design, a powerful and well-studied test design. MSc: 05C80, 62B10, 68P30, 68R05\n1. I NTRODUCTION\n1.1.Background and motivation. Few mathematical disciplines offer as abundant a supply of easy-to-state but\nhard-to-crack problems as combinatorics does. Group testing is a prime example. The problem goes back to the\n1940s [15]. Within a population of size nwe are to identify a subset of kinfected individuals. To this end we test\ngroups of individuals simultaneously. In an idealised scenario called ‘noiseless group testing’ each test returns a\npositive result if and only if at least one member of the group is infected. All tests are conducted in parallel. The\nproblem is to devise a (possibly randomised) test design that minimises the total number of tests required.\nNoiseless group testing has inspired a long line of research, which has led to optimal or near-optimal results\nfor several parameter regimes [3, 10]. But the assumption of perfectly accurate tests is unrealistic. Real tests are\nnoisy [26]. More precisely, in medical terms the sensitivity of a test is defined as the probability that a test detects\nan actual infection, viz. that a group that contains an infected individual displays a positive test result. Moreover,\nthe specificity of a test refers to the probability that a healthy group returns a negative test result. If these accura-\ncies are reasonable high (say 99%), one might be tempted to think that noiseless testing provides a good enough\napproximation. Yet remarkably we will discover that this is far from correct. Even a seemingly tiny amount of noise\nhas an enormous impact not only on the number of tests required, but also on the choice of a good test design; we\nwill revisit this point in Section 1.5. Hence, in group testing, like in several other inference problems, the presence\nof noise adds substantial mathematical depth. As a rough analogy think of error-correcting linear codes. In the\nabsence of noise the decoding problem just boils down to solving linear equations. By contrast, the noisy version,\nthe closest vector problem, is NP-hard [13].\nIn the present paper we consider a very general noise model that allows for arbitrary sensitivities and specifici-\nties. To be precise, we assume that if a test group contains an infected individual, then the test displays a positive\nresult with probability p11and a negative result with probability p10=1−p11. Similarly, if the group consists of\nhealthy individuals only, then the test result will display a negative outcome with probability p00and positive result\nwith probability p01=1−p00. Every test is subjected to noise independently.\nUnder the common assumption that the number kof infected individuals scales as a power k∼nθof the pop-\nulation size nwith an exponent 0 <θ<1 we contribute new approximate and exact recovery algorithms SPARC\nandSPEX . These new algorithms come with randomised test designs. We will identify a threshold mSPARC =\nmSPARC (n,k,p) such that SPARC correctly identifies the set of infected individuals up to o(k) errors with high prob-\nability over the choice of the test design, provided that at least (1 +ε)mSPARC tests are deployed. SPARC is efficient,\nAmin Coja-Oghlan and Lena Krieg are supported by DFG CO 646/3 and DFG CO 646/5. Max Hahn-Klimroth and Olga Scheftelowitsch are\nsupported by DFG CO 646/5.\n1arXiv:2402.02895v1  [cs.DM]  5 Feb 2024i.e. has polynomial running time in terms of n. By contrast, we will prove that with (1 −ε)mSPARC tests it is impos-\nsible to identify the set of infected individuals up to o(k) errors, regardless the choice of test design or the running\ntime allotted. In other words, SPARC solves the approximate recovery problem optimally.\nIn addition, we develop a polynomial time algorithm SPEX that correctly identifies the status of allindividuals\nw.h.p., provided that at least (1 +ε)mSPEX tests are available, for a certain mSPEX (n,k,p). Exact recovery has been\nthe focus of much of the previous literature on group testing [3]. In particular, for noisy group testing the best\nprevious exact recovery algorithm is the DDalgorithm from [16]. DDcomes with a simple randomised test design\ncalled the constant column design . Complementing the positive result on SPEX , we show that on the constant\ncolumn design exact recovery is information-theoretically impossible with (1 −ε)mSPEX tests. As a consequence,\nthe number mSPEX of tests required by SPEX is an asymptotic lower bound on the number of tests required by any\nalgorithm on the constant column design, including DD. Indeed, as we will see in Section 1.5, for most choices of\nthe specificity/sensitivity and of the infection density θ,SPEX outperforms DDdramatically.\nThroughout the paper we write p=(p00,p01,p10,p11) for the noisy channel to which the test results are sub-\njected. We may assume without loss that\np11>p01, (1.1)\ni.e. that a test is more likely to display a positive result if the test group actually contains an infected individual; for\notherwise we could just invert the test results. A test design can be described succinctly by a (possibly random)\nbipartite graph G=(V∪F,E), where Vis a set of nindividuals and Fis a set of mtests. Write σ∈{0,1}Vfor the\n(unknown) vector of Hamming weight kwhose 1-entries mark the infected individuals. Further, let σ′∈{0,1}Fbe\nthe vector of actual test results , i.e.σ′\na=1 if and only if σv=1 for at least one individual vin test a. Finally, let\nσ′′∈{0,1}Fbe the vector of displayed tests results , where noise has been applied to σ′, i.e.,\nP£\nσ′′\na=σ′′|G,σ′\na=σ′¤\n=pσ′σ′′ independently for every a∈F. (1.2)\nThe objective is to infer σfromσ′′given G. As per common practice in the group testing literature, we assume\nthroughout that k= ⌈nθ⌉and that kand the channel pare known to the algorithm [3].1\n1.2.Approximate recovery. The first main result provides an algorithm that identifies the status of all but o(k)\nindividuals correctly with the optimal number of tests. For a number z∈[0,1] let h(z)= −zln(z)−(1−z)ln(1−z).\nFurther, for y,z∈[0,1] let DKL¡\ny∥z¢\n=yln(y/z)+(1−y)ln((1 −y)/(1−z)) signify the Kullback-Leibler divergence.\nWe use the convention that 0ln0 =0ln0\n0=0. Given the channel pdefine\nφ=φ(p)=h(p00)−h(p10)\np00−p10, cSh=cSh(p)=1\nDKL¡\np10∥(1−tanh( φ/2))/2¢. (1.3)\nThe value 1/ cSh=DKL¡\np10∥(1−tanh( φ/2))/2¢\nequals the capacity of the p-channel [16, Lemma F .1]. Let\nmSPARC (n,k,p)=cShkln(n/k).\nTheorem 1.1. For any p,0<θ<1andε>0there exists n 0=n0(p,θ,ε)such that for every n >n0there exist\na randomised test design Gscwith m ≤(1+ε)mSPARC (n,k,p)tests and a deterministic polynomial time inference\nalgorithm SPARC such that\nPh\n∥SPARC (Gsc,σ′′\nGsc)−σ∥1<εki\n>1−ε. (1.4)\nIn other words, once the number of tests exceeds mSPARC=cShkln(n/k),SPARC applied to the test design Gsc\nidentifies the status of all but o(k) individuals correctly w.h.p. The test design Gscemploys an idea from coding\ntheory called ‘spatial coupling’ [32, 34]. As we will elaborate in Section 2, spatial coupling blends a randomised\nand a topological construction. A closely related design has been used in noiseless group testing [10].\nThe following theorem shows that Theorem 1.1 is optimal in the strong sense that it is information-theoretically\nimpossible to approximately recover the set of infected individuals with fewer than (1 −ε)mSPARC tests on any test\ndesign. In fact, approximate recovery w.h.p. is impossible even if we allow adaptive group testing where tests are\nconducted one-by-one and the choice of the next group to be tested may depend on all previous results.\n1These assumption could be relaxed at the expense of increasing the required number of tests (details omitted).\n2Theorem 1.2. For any p,0<θ<1andε>0there exist δ=δ(p,θ,ε)>0and n 0=n0(p,θ,ε)such that for all n >n0,\nall adaptive test designs with m ≤(1−ε)mSPARC (n,k,p)tests in total and any function A: {0,1}m→{0,1}nwe have\nP£\n∥A(σ′′)−σ∥1<δk¤\n<1−δ. (1.5)\nTheorem 1.2 and its proof are a relatively simple adaptation of [16, Corollary 2.3], where cShkln(n/k) was estab-\nlished as a lower bound on the number of tests required for exact recovery.\n1.3.Exact recovery. How many tests are required in order to infer the set of infected individuals precisely, not\njust up to o(k) mistakes? Intuitively, apart from an information-theoretic condition such as (1.3), exact recovery\nrequires a kind of local stability condition. More precisely, imagine that we managed to correctly diagnose all\nindividuals y̸=xthat share a test with individual x. Then towards ascertaining the status of xitself only those tests\nare relevant that contain xbut no other infected individual y; for the outcome of these tests hinges on the status\nofx. Hence, to achieve exact recovery we need to make certain that it is possible to tell the status of xitself from\nthese tests w.h.p.\nThe required number of tests to guarantee local stability on the test design Gscfrom Theorem 1.1 can be ex-\npressed in terms of a mildly involved optimisation problem. For c,d>0 and θ∈(0,1) let\nY(c,d,θ)=©\ny∈[0,1] : cd(1−θ)DKL¡\ny∥exp(−d)¢\n<θª\n. (1.6)\nThis set is a non-empty interval, because y7→DKL¡\ny∥exp(−d)¢\nis convex and y=exp(−d)∈Y(c,d,θ). Let\ncex,0(d,θ)=(\ninf©\nc>0 : inf y∈Y(c,d,θ)cd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\np01∥p11¢¢\n≥θª\nifp11<1,\ninf{c>0 : 0̸∈Y(c,d,θ)} otherwise.(1.7)\nIfp11=1 letz(y)=1 for all y∈Y(c,d,θ). Further, if p11<1 then the function z7→DKL¡\nz∥p11¢\nis strictly decreasing\non [ p01,p11]; therefore, for any c>cex,0(d,θ) and y∈Y(c,d,θ) there exists a unique z(y)=zc,d,θ(y)∈[p01,p11] such\nthat\ncd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡z(y)∥p11¢¢\n=θ. (1.8)\nIn either case set\ncex,1(d,θ)=(\ninf©\nc>cex,0(d,θ) : inf y∈Y(c,d,θ)cd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡z(y)∥p01¢¢\n≥1ª\nifp01>0,\ncex,0(d,θ) otherwise.(1.9)\nFinally, define\ncex,2(d)=1±¡\nh(p00exp(−d)+p10(1−exp(−d)))−exp(−d)h(p00)−(1−exp(−d))h(p10)¢\n, (1.10)\ncex(θ)=inf\nd>0max{ cex,1(d,θ),cex,2(d)}, (1.11)\nmSPEX (n,k,p)=cex(θ)kln(n/k).\nTheorem 1.3. For any p,0<θ<1andε>0there exists n 0=n0(p,θ,ε)such that for every n >n0there exist\na randomised test design Gscwith m ≤(1+ε)mSPEX (n,k,p)tests and a deterministic polynomial time inference\nalgorithm SPEX such that\nPh\nSPEX (Gsc,σ′′\nGsc)=σi\n>1−ε. (1.12)\nLike SPARC from Theorem 1.1, SPEX uses the spatially coupled test design Gsc. Crucially, apart from the num-\nbers nand mof individuals and tests, the value of dat which the infimum (1.11) is attained also enters into the\nconstruction of that test design. Specifically, the average size of a test group equals dn/k. Remarkably, while the\noptimal value of dfor approximate recovery turns out to depend on the channel ponly, a different value of dthat\nalso depends on kmay be the right choice to facilitate exact recovery. We will revisit this point in Section 1.5.\n1.4.Lower bound on the constant column design. Unlike in the case of approximate recovery we do not have a\nproof that the positive result on exact recovery from Theorem 1.3 is optimal for any choice of test design. However,\nwe can show that exact recovery with (1 −ε)cexkln(n/k) tests is impossible on the constant column design Gcc.\nUnder Gcceach of the nindividuals independently joins an equal number ∆of tests, drawn uniformly without\nreplacement from the set of all mavailable tests. Let Gcc=Gcc(n,m,∆) signify the outcome. The following theorem\nshows that exact recovery on Gccis information-theoretically impossible with fewer than mSPEX tests.\n3Theorem 1.4. For any p,0<θ<1andε>0there exists n 0=n0(p,θ,ε)such that for every n >n0and all m ≤\n(1−ε)mSPEX (n,k,p),∆>0and any AGcc: {0,1}m→{0,1}nwe have\nP£\nAGcc(σ′′\nG)=σ¤\n<ε. (1.13)\nAn immediate implication of Theorem 1.4 is that the positive result from Theorem 1.3 is at least as good as the\nbest prior results on exact noisy recovery from [16], which are based on running a simple combinatorial algorithm\ncalled DDonGcc. In fact, in Section 1.5 we will see that the new bound from Theorem 1.3 improves over the bounds\nfrom [16] rather significantly for most θ,p.\nThe proof of Theorem 1.4 confirms the combinatorial meaning of the threshold cex(θ). Specifically, for c=\nm/(kln(n/k))<cex,2(d) from (1.10) a moment calculation reveals that w.h.p. Gcccontains ‘solutions’ σof at least\nthe same posterior likelihood as the true σsuch that σandσdiffer significantly, i.e. ∥σ−σ∥1=Ω(k). By contrast,\nthe threshold cex,1(d,θ) marks the onset of local stability. This means that for c<cex,1(d,θ) there will be numerous\nσclose to but not identical to σ(i.e. 0< ∥σ−σ∥1=o(k)) of the at least same posterior likelihood. In either case any\ninference algorithm, efficient or not, is at a loss identifying the actual σ.\nIn recent independent work Chen and Scarlett [8] obtained Theorem 1.4 in the special case of symmetric noise\n(i.e. p00=p11). While syntactically their expression for the threshold cex(θ) differs from (1.6)–(1.11), it can be\nchecked that both formulas yield identical results (see Appendix E). Apart from the information-theoretic lower\nbound (which is the part most relevant to the present work), Chen and Scarlett also proved that it is information-\ntheoretically possible (by means of an exponential algorithm) to infer σw.h.p. on the constant column design with\nm≥(1+ε)cex(θ)kln(n/k) tests if p00=p11. Hence, the bound cex(θ) is tight in the case of symmetric noise.\n1.5.Examples. We illustrate the improvements that Theorems 1.1 and 1.3 contribute by way of concrete examples\nof channels p. Specifically, for the binary symmetric channel and the Z-channel it is possible to obtain partial\nanalytic results on the optimisation behind cex(θ) from (1.11) (see Appendices C–D). As we will see, even a tiny\namount of noise has a dramatic impact on both the number of tests required and the parameters that make a good\ntest design.\n1.5.1. The binary symmetric channel. Although symmetry is an unrealistic assumption from the viewpoint of ap-\nplications [26], the expression (1.3) and the optimisations (1.9)–(1.11) get much simpler on the binary symmetric\nchannel, i.e. in the case p00=p11. For instance, the value of dthat minimises cex,2(d) from (1.10) turns out to be\nd=ln2. The parameter denters into the constructions of the randomised test designs GscandGccin a crucial\nrole. Specifically, the average size of a test group equals dn/k. In effect, any test is actually negative with proba-\nbility exp( −d+o(1)) (see Proposition 2.3 and Lemma 5.1 below). Hence, if d=ln2 then w.h.p. about half the tests\ncontain an infected individual. In effect, since p11=p00, after the application of noise about half the tests display\na positive result w.h.p.\nIn particular, in the noiseless case p11=p00=1 a bit of calculus reveals that the value d=ln2 also minimises\nthe other parameter cex,1(d,θ) from (1.9) that enters into cex(θ) from (1.11). Therefore, in noiseless group testing\nd=ln2 unequivocally is the optimal choice, the optimisation on d(1.11) effectively disappears and we obtain\ncex(θ)=max½θ\n(1−θ)ln22,1\nln2¾\n, (1.14)\nthereby reproducing the optimal result on noiseless group testing from [10].\nBut remarkably, as we verify analytically in Appendix C at positive noise1\n2<p11=p00<1 the value of dthat\nminimises cex,1(d,θ) does not generally equal ln2. Hence, if we aim for exact recovery, then at positive noise it is no\nlonger optimal for all 0 <θ<1 to aim for about half the tests being positive/negative. The reason is the occurrence\nof a phase transition in terms of θwhere the ‘local stability’ term cex,1(d,θ) takes over as the overall maximiser in\n(1.10). Consequently, the objective of minimising cex,1(d,θ) and the optimal choice d=ln2 for cex,2(d) clash. In\neffect, the overall minimiser dforcex(d,θ) depends on both pand the infection density parameter θin a non-trivial\nway. Thus, the presence and level of noise has a discernible impact on the choice of a good test design.2\nFigure 1 displays the performance of the algorithms SPARC andSPEX from Theorems 1.1 and 1.3 on the binary\nsymmetric channel. For the purpose of graphical representation the figure does not display the values of cex(θ),\nwhich diverge as θ→1, but the value 1/ cex(θ). This value has a natural information-theoretic interpretation: it is\n2This observation confirms a hypothesis stated in [16, Appendix F]. As mentioned in Section 1.4, independent work of Chen and Scarlett [8]\non the case of symmetric noise reaches the same conclusion.\n4cSh\ncSh\n0.0 0.2 0.4 0.6 0.8 1.0\nθ0.00.10.20.30.40.50.60.7rate\np11= 0.90p11= 0.99\np11= 0.90p11= 0.99\n0.69310.9619d(A) Rates on the binary symmetric channel for\np00=p11=0.99 and p00=p11=0.9.\ncSh\ncSh\n0.0 0.2 0.4 0.6 0.8 1.0\nθ0.00.10.20.30.40.50.60.7rate\np11= 0.50p11= 0.90p11= 0.50p11= 0.90\n0.51080.9094d(B) Rates on the Z-channel ( p00=1) with p11=0.9 and\np11=0.5.\nFIGURE 1. Information rates on different channels in nats. The horizontal axis displays the in-\nfection density parameter 0 <θ<1. The colour indicates the optimal value of dfor a given θ.\nthe average amount of information that a test reveals about the set of infected individuals, measured in ‘nats’ . In\nother words, the plots display the information rate of a single test (higher is better). The optimal values of dare\ncolour-coded into the curves. While in the noiseless case d=ln2 remains constant, in the noisy cases dvaries\nsubstantially with both θand p00=p11.\nFor comparison the figure also displays the rate in the noiseless case (dashed line on top) and the best previous\nrates realised by the DDalgorithm on Gccfrom [16] (dotted lines). As is evident from Figure 1, even noise as small\nas 1% already reduces the rate markedly: the upper coloured curve remains significantly below the noiseless black\nline. That said, Figure 1 also illustrates how the rate achieved by SPEX improves over the best previous algorithm DD\nfrom [16]. Somewhat remarkably, the 10%-line for cex(θ) intersects the 1%-line for DDfor an interval of θ. Hence,\nfor these θthe algorithm from Theorem 1.3 at 10% noise requires fewer tests than DDat 1%.\nFigure 1 also illustrates how approximate and exact recovery compare. Both coloured curves start out as black\nhorizontal lines. These bits of the curves coincide with the rate of the SPARC algorithm from Theorem 1.1. The rate\nachieved by SPARC , which does not depend on θ, is therefore just the extension of this horizontal line to the entire\ninterval 0 <θ<1 at the height cShfrom (1.3). Hence, particularly for large θapproximate recover achieves much\nbetter rates than exact recovery.\n1.5.2. The Z -channel. In the case p00=1 of perfect specificity, known as the Z-channel, it is possible to derive\nsimple expressions for the optimisation problems (1.6)–(1.11) (see Appendix D):\ncex,1(d,θ)= −θ\nd(1−θ)ln(1−exp(−d)p11), (1.15)\ncex,2(d)=¡\nh(p10+(1−p10)exp(−d))−(1−exp(−d))h(p10)¢−1. (1.16)\nAs in the symmetric case, there is a tension between the value of dthat minimises (1.15) and the objective of\nminimising (1.16). Recall that since the size of the test groups is proportional to d, the optimiser dhas a direct\nimpact on the construction of the test design.\nFigure 1b displays the rates achieved by of SPEX (solid line) and, for comparison, the DDalgorithm from [16]\n(dotted grey) on the Z-channel with sensitivities p11=0.9 and p11=0.5. Additionally, the dashed red line indicates\nthe noiseless rate. Once again the optimal value of dis colour-coded into the solid SPEX line. Remarkably, the SPEX\nrate at p11=0.5 (high noise) exceeds the DDrate at p11=0.9 for a wide range of θ. As in the symmetric case the\nhorizontal cSh-lines indicate the performance of the SPARC approximate recovery algorithm.\n51.5.3. General (asymmetric) noise. While in the symmetric case the term cex,2(d) from (1.10) attains its minimum\nsimply at d=ln2, with φfrom (1.3) the minimum for general pis attained at\nd=dSh=ln(p11−p01)−ln¡\n(1−tanh( φ/2))/2 −p10¢\n[16, Lemma F .1]. (1.17)\nOnce again the design of Gsc(as well as Gcc) ensures that w.h.p. a exp( −d)-fraction of tests are actually negative\nw.h.p. The choice (1.17) ensures that under the p-channel the mutual information between the channel input and\nthe channel output is maximised [16, Lemma F .1]:\n1\ncSh=1\ncex,2(dSh)=DKL¡\np10∥(1−tanh( φ/2))/2¢\n, (1.18)\nAs can be checked numerically, the second contributor cex,1(d,θ) to cex(θ) may take its minimum at another d.\nHowever, we are not aware of a simple explicit expressions for cex,2(θ) from (1.10) for general noise.\n1.6.Related work. The monograph of Aldridge, Johnson and Scarlett [3] provides an excellent overview of the\ngroup testing literature. The group testing problem comes in various different flavours: non-adaptive (where all\ntests are conducted concurrently) or adaptive (where tests are conducted in subsequent stages such that the tests\nat later stages may depend on the outcomes of earlier stages), as well as noiseless or noisy. An important result of\nAldridge shows that noiseless non-adaptive group testing does not perform better than plain individual testing if\nk=Ω(n), i.e. if the number of infected individuals is linear in the size of the population [5]. Therefore, research\non non-adaptive group testing focuses on the case k∼nθwith 0 <θ<1. For non-adaptive noiseless group testing\nwith this scaling of ktwo different test designs (Bernoulli and constant column) and various elementary algorithms\nhave been proposed [7]. Among these elementary designs and algorithms the best performance to date is achieved\nby the DDgreedy algorithm on the constant column design [21]. However, the DDalgorithm does not match the\ninformation-theoretic bound on the constant column design for all θ[9].\nCoja-Oghlan, Gebhard, Hahn-Klimroth and Loick proposed a more sophisticated test design for noiseless group\ntesting based on spatial coupling [10], along with an efficient inference algorithm called SPIV . Additionally, they\nimproved the information-theoretic lower bound for non-adaptive noiseless group testing. The number of tests\nrequired by the SPIV algorithm matches this new lower bound. In effect, the combination of SPIV with the spa-\ntially coupled test design solves the noiseless non-adaptive group testing problem optimally both for exact and\napproximate recovery.\nThe present article deals with the noisy non-adaptive variant of group testing. A noisy version of the efficient DD\nalgorithm was previously studied on both the Bernoulli and the constant column design [16, 31]. The best previous\nexact recovery results for general noise were obtained by Johnson, Gebhard, Loick and Rolvien [16] by means of\nDDon the constant column design (see Theorem 2.1 below). Theorem 1.4 shows in combination with Theorem 1.3\nthat the new SPEX algorithm performs at least as well as any algorithm on the constant column design, including\nand particularly DD.\nApart from the articles [16, 31] that dealt with the same general noise model as we consider here, several con-\ntributions focused on special noise models, particularly symmetric noise ( p00=p11). In this scenario Chen and\nScarlett [8] recently determined the information-theoretically optimal number of tests required for exact recovery\non the Bernoulli and constant column designs. The constant column design outperforms the Bernoulli design.\nThe information-theoretic threshold identified by Chen and Scarlett matches the threshold cex(θ) from (1.10) in\nthe special case of symmetric noise (see Appendix E). However, Chen and Scarlett do not investigate the issue of\nefficient inference algorithms. Instead, they pose the existence of an efficient inference algorithm that matches\nthe information-theoretic threshold as an open problem. Theorem 1.3 applied to symmetric noise answers their\nquestion in the affirmative.\nA further contribution of Scarlett and Cevher [29] contains a result on approximate recovery under the assump-\ntion of symmetric noise. In this case Scarlett and Cevher obtain matching information-theoretic upper and lower\nbounds, albeit without addressing the question of efficient inference algorithms. Theorem 1.1 applied to the spe-\ncial case of symmetric noise provides a polynomial time inference algorithm that matches their lower bound.\nFrom a practical viewpoint non-adaptive group testing (where all tests are conducted in parallel) is preferable\nbecause results are available more rapidly than in the adaptive setting, where several rounds of testing are required.\nThat said, adaptive schemes may require a smaller total number of tests. The case of noiseless adaptive group\ntesting has been studied since the seminal work of Dorfman [15] from the 1940s. For the case k∼nθa technique\nknown as generalized binary splitting gets by with the optimal number of tests [19, 6]. Aldridge [4] extended this\n6approach to the case k=Θ(n), obtaining near-optimal rates. Recently there has been significant progress on upper\nand lower bounds for noisy and adaptive group testing, although general optimal results remain elusive [30, 33].\nBeyond group testing in recent years important progress has been made on several inference problems by\nmeans of a combination of spatial coupling and message passing ideas. Perhaps the most prominent case in point\nis the compressed sensing problem [14, 22]. Further applications include the pooled data problem [1, 2, 17] and\nCDMA [32], a signal processing problem. The basic idea of spatial coupling, which we are going to discuss in some\ndetail in Section 2.3, goes back to work on capacity-achieving linear codes [23, 24, 25, 32, 34]. The SPIV algorithm\nfrom [10] combines a test design inspired by spatial coupling with a combinatorial inference algorithm. A nov-\nelty of the present work is that we replace this elementary algorithm by a novel variant of the Belief Propagation\nmessage passing algorithm [27, 28] that lends itself to a rigorous analysis.\n1.7.Organisation. After introducing a bit of notation and recalling some background in Section 1.8 we given an\noutline of the proofs of the main results in Section 2. Subsequently Section 3 deals with the details of the proof of\nTheorem 1.1. Moreover, Section 4 deals with the proof of Theorem 1.3, while in Section 5 we prove Theorem 1.4.\nThe proof of Theorem 1.2, which is quite short and uses arguments that are well established in the literature,\nfollows in Appendix A. Appendix B contains the proof of a routine expansion property of the test design Gsc. Finally,\nin Appendices C and D we investigate the optimisation problems (1.9)–(1.11) on the binary symmetric channel and\nthe Z-channel, and in Appendix E we compare our result to the recent result of Chen and Scarlett [8].\n1.8.Preliminaries. As explained in Section 1.1, a test design is a bipartite graph G=(V∪F,E) whose vertex set\nconsists of a set Vindividuals and a set Fof tests. The ground truth, i.e. the set of infected individuals is encoded\nas a vector σ∈{0,1}Vof Hamming weight k. Since we will deal with randomised test designs, we may assume that\nσis auniformly random vector of Hamming weight k(by shuffling the set of individuals). Also recall that for a test\nawe let σ′\na∈{0,1} denote the actual result of test a(equal to one if and only if acontains an infected individual),\nwhile σ′′\na∈{0,1} signifies the displayed test result obtained via (1.2). It is convenient to introduce the shorthands\nV0={x∈V:σx=0}, V1={x∈V:σx=1}, F0=©\na∈F:σ′\na=0ª\n, F1=©\na∈F:σ′\na=1ª\n,\nF−=©\na∈F:σ′′\na=0ª\n, F+=©\na∈F:σ′′\na=1ª\n, F±\n0=F±∩F0, F±\n1=F±∩F1\nfor the set of infected/healthy individuals, the set of actually negative/positive tests, the set of negatively/positively\ndisplayed tests and the tests that are actually negative/positive and display a positive/negative result, respectively.\nFor each node uofGwe denote by ∂u=∂Guthe set of neighbours of u. For an individual x∈Vwe also let\n∂±x=∂±\nGx=∂Gx∩F±be the set of positively/negatively displayed tests that contain x.\nWe need Chernoff bounds for the binomial and the hypergeometric distribution. Recall that the hypergeometric\ndistribution Hyp( L,M,N) is defined by\nP£\nHyp( L,M,N)=k¤\n=Ã\nM\nk!Ã\nL−M\nN−k!Ã\nL\nN!−1\n(k∈{0,1,...,min{ M,N}}). (1.19)\n(Out of a total of Litems of which Mare special we draw Nitems without replacement and count the number of\nspecial items in the draw.) The mean of the hypergeometric distribution equals M N /L.\nLemma 1.5 ([20, Equation 2.4] ) .LetXbe a binomial random variable with parameters N ,p. Then\nP£\nX≥qN¤\n≤exp¡\n−N D KL¡\nq∥p¢¢\nfor p<q<1, (1.20)\nP£\nX≤qN¤\n≤exp¡\n−N D KL¡\nq∥p¢¢\nfor0<q<p. (1.21)\nLemma 1.6 ([18]) .For a hypergeometric variable X∼Hyp( L,M,N)the bounds (1.20) –(1.21) hold with p =M/L.\nThroughout we use asymptotic notation o(·),ω(·),O(·),Ω(·),Θ(·) to refer to limit n→ ∞ . It is understood that\nthe constants hidden in, e.g., a O(·)-term may depend on θ,por other parameters, and that a O(·)-term may have\na positive or a negative sign. To avoid case distinctions we sometimes take the liberty of calculating with the values\n±∞ . The usual conventions ∞+∞ = ∞·∞ = ∞ and 0 ·∞ = 0 apply. Furthermore, we set tanh( ±∞ )= ±1. Also\nrecall that 0ln0 =0ln0\n0=0. Additionally, ln0 = −∞ and1\n0=ln1\n0= ∞ .\nFinally, for two random variables X,Ydefined on the same finite probability space ( Ω,P[·]) we write\nI(X,Y)=X\nω,ω′∈ΩP£\nX=ω,Y=ω′¤\nlnP£\nX=ω,Y=ω′¤\nP[X=ω]P[Y=ω′]\n7for the mutual information ofX,Y. We recall that I(X,Y)≥0.\n2. O VERVIEW\nWe proceed to survey the functioning of the algorithms SPARC andSPEX . To get started we briefly discuss the best\npreviously known algorithm for noisy group testing, the DDalgorithm from [16], which operates on the constant\ncolumn design. We will discover that DDcan be viewed as a truncated version of the Belief Propagation (BP) mes-\nsage passing algorithm. BP is a generic heuristic for inference problems backed by physics intuition [27, 35]. Yet\nunfortunately BP is notoriously difficult to analyse. Even worse, it seems unlikely that full Belief Propagation will\nsignificantly outperform DDon the constant column design; evidence of this was provided in [11] in the noiseless\ncase. The basic issue is the lack of a good initialisation of the BP messages.\nTo remedy this issue we resort to the spatially coupled test design Gsc, which combines a randomised and a\nspatial construction. The basic idea resembles domino toppling. Starting from an easy-to-diagnose ‘seed’ , the\nalgorithm works its way forward in a well-defined direction until all individuals have been diagnosed. Spatial\ncoupling has proved useful in related inference problems, including noiseless group testing [10]. Therefore, a\nnatural stab at group testing would be to run BP on Gsc. Indeed, the BP intuition provides a key ingredient to the\nSPARC algorithm, namely the update equations (see (2.30) below), of which the correct choice of the weights (Eq.\n(2.29) below) is the most important ingredient. But in light of the difficulty of analysing textbook BP , SPARC relies\non a modified version of BP that better lends itself to a rigorous analysis. Furthermore, the SPEX algorithm for\nexact recovery combines SPARC with a clean-up step.\nSPARC andSPEX can be viewed as generalised versions of the noiseless group testing algorithm called SPIV\nfrom [10]. However, [10] did not exploit the connection with BP . Instead, in the noiseless case the correct weights\nwere simply ‘guessed’ based on combinatorial intuition, an approach that it seems difficult to generalise. Hence,\nthe present, systematic derivation of the weights (2.29) also casts new light on the noiseless case. In fact, we expect\nthat the paradigm behind SPARC andSPEX , namely to use BP heuristically to find the correct parameters for a\nsimplified message passing algorithm, potentially generalises to other inference problems as well.\n2.1.TheDDalgorithm. TheDDalgorithm from [16] utilises the constant column design Gcc.3Thus, each individual\nindependently joins an equal number ∆of random tests. Given the displayed test results, DDfirst declares certain\nindividuals as uninfected by thresholding the number of negatively displayed tests. More precisely, DDdeclares as\nuninfected any individual that appears in at least α∆negatively displayed tests, with αa diligently chosen thresh-\nold. Having identified the respective individuals as uninfected, DDlooks out for tests athat display a positive result\nand that only contain a single individual xthat has not been identified as uninfected yet. Since such tests ahint\natxbeing infected, in its second step DDdeclares as infected any individual xthat appears in at least β∆positively\ndisplayed tests awhere all other individuals y∈∂a\\xwere declared uninfected by the first step. Once again βis a\ncarefully chosen threshold. Finally, DDdeclares as uninfected all remaining individuals.\nTheDDalgorithm exactly recovers the infected set w.h.p. provided the total number mof tests is sufficiently\nlarge such that the aforementioned thresholds α,βexist. The required number of tests, which comes out in terms\nof a mildly delicate optimisation problem, was determined in [16]. Let\nq−\n0=exp(−d)p00+(1−exp(−d))p10, q+\n0=exp(−d)p01+(1−exp(−d))p11. (2.1)\nTheorem 2.1 ([16, Theorem 2.2]) .Letε>0and with α∈(p10,q−\n0)andβ∈(0,exp( −d)p11)let\ncDD=min\nα,β,dmax©\ncDD,1(α,d),cDD,2(α,d),cDD,3(β,d),cDD,4(α,β,d)ª\n, where\ncDD,1(α,d)=θ\n(1−θ)DKL¡\nα∥p10¢, cDD,2(α,d)=1\ndD KL¡\nα∥q−\n0¢, cDD,3(β,d)=θ\nd(1−θ)DKL¡\nβ∥p11exp(−d)¢,\ncDD,4(α,β,d)= max\n(1−α)∨β≤z≤1µ\nd(1−θ)µ\nDKL¡\nz∥q+\n0¢\n+1½\nβ>zexp(−d)p01\nq+\n0¾\nzD KL¡\nβ/z∥exp(−d)p01/q+\n0¢¶¶−1\n.\nIf m≥(1+ε)cDDkln(n/k), then there exists ∆>0and 0≤α,β≤1such that the DDalgorithm outputs σw.h.p.\nThe distinct feature of DDis its simplicity. However, the thresholding that DDapplies does seem to leave some-\nthing on the table. For a start, whether DDidentifies a certain individual xas infected depends only on the results of\n3The article [16] also investigates the performance of DDon the Bernoulli design, which turns out to be inferior.\n8tests that have distance at most three from xin the graph Gcc. Moreover, it seems wasteful that DDtakes only those\npositively displayed tests into consideration where all but one individual were already identified as uninfected.\n2.2.Belief Propagation. Belief Propagation is a message passing algorithm that is expected to overcome these\ndeficiencies. In fact, heuristic arguments suggests that BP might be the ultimate recovery algorithm for a wide\nclass of inference algorithms on random graphs [35]. That said, rigorous analyses of BP are few and far between.\nFollowing the general framework from [27], in order to apply BP to a group testing design G=(V∪F,E) we equip\neach test a∈Fwith a weight function\nψa=ψG,σ′′,a: {0,1}∂a→R≥0, σ∂ai7→(\n1{∥σ∥1=0}p00+1{∥σ∥1>0}p10 ifa∈F−,\n1{∥σ∥1=0}p01+1{∥σ∥1>0}p11 ifa∈F+.(2.2)\nThus, ψatakes as argument a {0,1}-vector σ=(σx)x∈∂aindexed by the individuals that take part in test a. The\nweight ψa(σ) equals the probability of observing the result that adisplays if the infection status were σxfor every\nindividual x∈∂a. In other words, ψaencodes the posterior under the p-channel. Given G,σ′′the weight functions\ngive rise to the total weight of σ∈{0,1}Vby letting\nψ(σ)=ψG,σ′′(σ)=1{∥σ∥1=k}Y\na∈Fψa(σ∂a). (2.3)\nThus, we just multiply up the contributions (2.2) of the various tests, and add in the prior assumption that precisely\nkindividuals are infected. The total weight (2.3) induces a probability distribution\nµG,σ′′(σ)=ψG,σ′′(σ)/ZG,σ′′, where ZG,σ′′=X\nσ∈{0,1}VψG,σ′′(σ). (2.4)\nA simple application of Bayes’ rule shows that µGmatches the posterior of the ground truth σgiven the test results.\nFact 2.2. For any test design G we have µG,σ′′(σ)=P£\nσ=σ|σ′′¤\n.\nBP is a heuristic to calculate the marginals of µG,σ′′or, in light of Fact 2.2, the posterior probabilities P[σxi=\n1|σ′′]. To this end BP associates messages with the edges of G. Specifically, for any adjacent individual/test pair\nx,athere is a message µx→a,t(·) from xtoa, and another one µa→x,t(·) in the reverse direction. The messages are\nupdated in rounds and therefore come with a time parameter t∈Z≥0. Moreover, being probability distributions\non {0,1}, the messages always satisfy\nµx→a,t(0)+µx→a,t(1)=µa→x,t(0)+µa→x,t(1)=1. (2.5)\nThe intended semantics is that µx→a,t(1) estimates the probability that xis infected given all known information\nexcept the result of test a. Analogously, µa→x,t(1) estimates the probability that xis infected if we disregard all\nother tests b∈∂x\\ {a}.\nThis slightly convoluted interpretation of the messages facilitates simple heuristic formulas for computing the\nmessages iteratively. To elaborate, in the absence of a better a priori estimate, at time t=0 we simply initialise in\naccordance with the prior, i.e.,\nµx→a,0(0)=1−k/n µx→a,0(1)=k/n. (2.6)\nSubsequently we use the weight function (2.2) to update the messages: inductively for t≥1 and r∈{0,1} let\nµa→x,t(r)∝\n\np11 ifr=1,a∈F+,\np11+(p01−p11)Q\ny∈∂a\\{x}µy→a,t−1(0) if r=0,a∈F+,\np10 ifr=1,a∈F−,\np10+(p00−p10)Q\ny∈∂a\\{x}µy→a,t−1(0) if r=0,a∈F−,(2.7)\nµx→a,t(r)∝µk\nn¶rµ\n1−k\nn¶1−rY\nb∈∂x\\{a}µb→x,t−1(r). (2.8)\nThe∝-symbol hides the necessary normalisation to ensure that the messages satisfy (2.5). Furthermore, the ( k/n)r\nand (1 −k/n)1−r-prefactors in (2.8) encode the prior that precisely kindividuals are infected. The expressions\n(2.7)–(2.8) are motivated by the hunch that for most tests athe values ( σy)y∈∂ashould be stochastically dependent\nprimarily through their joint membership in test a. An excellent exposition of BP can be found in [27].\n9How do we utilise the BP messages to infer the actual infection status of each individual? The idea is to perform\nthe update (2.7)–(2.8) for a ‘sufficiently large’ number of rounds, say until an approximate fixed point is reached.\nThe (heuristic) BP estimate of the posterior marginals after trounds then reads\nµx,t(r)∝µk\nn¶rµ\n1−k\nn¶1−rY\nb∈∂xµb→x,t−1(r) ( r∈{0,1}). (2.9)\nThus, by comparison to (2.8) we just take the incoming messages from alltests b∈∂xinto account. In summary,\nwe ‘hope’ that after sufficiently many updates we have µx,t(r)≈P[σx=r|σ′′]. We could then, for instance, declare\nthe kindividuals with the greatest µx,t(1) infected, and everybody else uninfected.\nFor later reference we point out that the BP updates (2.7)–(2.8) and (2.9) can be simplified slightly by passing to\nlog-likelihood ratios. Thus, define\nηx→a,t=lnµG,σ′′,x→a,t(1)\nµG,σ′′,x→a,t(0), ηG,σ′′,a→x,t=lnµG,σ′′,a→x,t(1)\nµG,σ′′,a→x,t(0), (2.10)\nwith the initialisation ηG,σ′′,x→a,0=ln(k/(n−k))∼(θ−1)ln nfrom (2.6). Then (2.7)–(2.9) transform into\nηa→x,t=(\nlnp11−ln£\np11+(p01−p11)Q\ny∈∂a\\{x}1\n2¡\n1−tanh¡1\n2ηy→a,t−1¢¢¤\nifa∈F+,\nlnp10−ln£\np10+(p00−p10)Q\ny∈∂a\\{x}1\n2¡\n1−tanh¡1\n2ηy→a,t−1¢¢¤\nifa∈F−,(2.11)\nηx→a,t=(θ−1)ln n+X\nb∈∂x\\{a}ηb→x,t,,ηx,t=(θ−1)ln n+X\nb∈∂xηb→x,t. (2.12)\nIn this formulation BP ultimately diagnoses the kindividuals with the largest ηx,tas infected.\nUnder the assumptions of Theorem 2.1 the DDalgorithm can be viewed as the special case of BP with t=2\napplied to Gcc. Indeed, the analysis of DDevinces that on Gccthe largest kBP estimates (2.9) with t≥2 correctly\nidentify the infected individuals w.h.p.4\nIt is therefore an obvious question whether BP on the constant column fits the bill of Theorem 1.1. Clearly,\nBP remedies the obvious deficiencies of DDby taking into account information from a larger radius around an\nindividual (if we iterate beyond t=2). Also in contrast to DD’s hard thresholding the update rules (2.7)–(2.8) take\ninformation into account in a more subtle, soft manner. Nonetheless, we do not expect that BPapplied to the\nconstant column design meets the information-theoretically optimal bound from Theorem 1.1. In fact, there is\nstrong evidence that BPdoes not suffices to meet the information-threshold for all θeven in the noiseless case [11].\nThe fundamental obstacle appears to be the ‘cold’ initialisation (2.6), which (depending on the parameters) can\ncause the BP messages to approach a meaningless fixed point. Yet for symmetry reasons on the constant column\ndesign no better starting point than the prior (2.6) springs to mind; after all, Gccis a nearly biregular random graph,\nand thus all individuals look alike. To overcome this issue we will employ a different type of test design that enables\na warm start for BP . This technique goes by the name of spatial coupling.\n2.3.Spatial coupling. The thrust of spatial coupling is to blend a randomised construction, in our case the con-\nstant column design, with a spatial arrangement so as to provide a propitious starting point for BP . Originally\nhailing from coding theory, spatial coupling has also been used in previous work on noiseless group testing [10].\nIn fact, the construction that we use is essentially identical to that from [10] (with suitably adapted parameters).\nTo set up the spatially coupled test design Gscwe divide the set V=Vn={x1,..., xn} of individuals into\nℓ= ⌈p\nlnn⌉ (2.13)\npairwise disjoint compartments V [1],..., V[ℓ]⊆Vsuch that ⌊n/ℓ⌋ ≤ | V[i]| ≤ ⌈ n/ℓ⌉. We think of these compart-\nments as being spatially arranged so that V[i+1] comes ‘to the right’ of V[i] for 1 ≤i<ℓ. More precisely, we\narrange the compartments in a ring topology such that V[ℓ] is followed again by V[1]. Hence, for notational con-\nvenience let V[ℓ+j]=V[j] and V[1−j]=V[ℓ−j+1] for 1 ≤j≤ℓ. Additionally, we introduce ℓcompartments\nF[1],..., F[ℓ] of tests arranged in the same way: think of F[i] as sitting ‘above’ V[i]. We assume that the total num-\nber mof tests in F[1]∪...∪F[ℓ] is divisible by ℓand satisfies m=Θ(kln(n/k)). Hence, let each compartment F[i]\ncontain precisely m/ℓtests. As in the case of the individuals we let F[ℓ+j]=F[j] for 0 <j≤ℓ. Additionally, let\ns= ⌈lnln n⌉ and ∆=Θ(lnn) (2.14)\n4In the noiseless case DDis actually a special case of a discrete message passing algorithm called Warning Propagation [27].\n10be integers such that ∆is divisible by s. Construct Gscby letting each x∈V[i] join precisely ∆/stests from F[i+j−1]\nfor j=1,..., s. These tests are chosen uniformly without replacement and independently for different xand j.\nAdditionally, Gsccontains a compartment F[0] of\nm0=2cDDks\nℓln(n/k)=o(kln(n/k)) (2.15)\ntests. Every individual xfrom the first scompartments V[1],..., V[s] joins an equal number ∆0of tests from F[0].\nThese tests are drawn uniformly without replacement and independently. For future reference we let\nc=cn=m\nkln(n/k), d=dn=k∆\nm; (2.16)\nthe aforementioned assumptions on m,∆ensure that c,d=Θ(1) and the total number of tests of Gsccomes to\nℓX\ni=0|F[i]| =(c+o(1))kln(n/k). (2.17)\nIn summary, Gscconsists of ℓequally sized compartments V[i],F[i] of tests plus one extra serving F[0] of tests.\nEach individual V[i] joins random tests in the sconsecutive compartments F[i+j−1] with 1 ≤j≤s. Additionally,\nthe individuals in the first scompartments V[1]∪···∪ V[s], which we refer to as the seed , also join the tests in F[0].\nWe will discover momentarily how Gscfacilitates inference via BP . But first let us make a note of some basic\nproperties of Gsc. Recall that σ∈{0,1}V, which encodes the true infection status of each individual, is chosen\nuniformly and independently of Gscfrom all vectors of Hamming weight k. Let V1={x∈V:σx=1},V0=V\\V1\nand let Vr[i]=Vr∩V[i] be the set of individuals with infection status r∈{0,1} in compartment i. Furthermore,\nrecall that σ′\na∈{0,1} denotes the actual result of test a∈F=F[0]∪···∪ F[ℓ], and that σ′′\nasignifies the displayed\nresult of aas per (1.2). For r∈{0,1} and 0 ≤i≤ℓlet\nFr[i]=Fr∩F[i], F+\nr[i]=Fr[i]∩F+, F−\nr[i]=Fr[i]∩F−.\nThus, the subscript indicates the actual test result, while the superscript indicates the displayed result. In Sec-\ntion 3.1 we will prove the following.\nProposition 2.3. The test design Gscenjoys the following properties with probability 1−o(n−2).\nG1: The number of infected individuals in the various compartments satisfy\nk\nℓ−s\nk\nℓlnn≤min\ni∈[ℓ]|V1[i]| ≤max\ni∈[ℓ]|V1[i]| ≤k\nℓ+s\nk\nℓlnn. (2.18)\nG2: For all i ∈[ℓ]the numbers of tests that are actually/displayed positive/negative satisfy\nm\nℓexp(−d)p00−p\nmln3n≤¯¯F−\n0[i]¯¯≤m\nℓexp(−d)p00+p\nmln3n, (2.19)\nm\nℓexp(−d)p01−p\nmln3n≤¯¯F+\n0[i]¯¯≤m\nℓexp(−d)p01+p\nmln3n, (2.20)\nm\nℓ(1−exp(−d))p10−p\nmln3n≤¯¯F−\n1[i]¯¯≤m\nℓ(1−exp(−d))p10+p\nmln3n, (2.21)\nm\nℓ(1−exp(−d))p11−p\nmln3n≤¯¯F+\n1[i]¯¯≤m\nℓ(1−exp(−d))p11+p\nmln3n. (2.22)\n2.4.Approximate recovery. We are going to exploit the spatial structure of Gscin a manner reminiscent of domino\ntoppling. To get started we will run DDon the seed V[1]∪···∪ V[s] and the tests F[0] only; this is our first domino.\nThe choice (2.15) of m0ensures that DDdiagnoses all individuals in V[1]∪···∪ V[s] correctly w.h.p. The seed could\nthen be used as an informed starting point from which we could iterate BP to infer the status of the individuals in\nV[s+1]∪...∪V[ℓ]. However, this algorithm appears to be difficult to analyse. Instead we will show under that the\nassumptions of Theorem 1.1 a modified, ‘paced’ version of BP that diagnoses one compartment (or ‘domino’) at\na time and then re-initialises the messages ultimately classifies all but o(k) individuals correctly. Let us flesh this\nstrategy out in detail.\n11Input: G,σ′′\nOutput: an estimate of σ\n1Let (τx)x∈V[1]∪···∪ V[s]∈{0,1}V[1]∪···∪ V[s]be the result of applying DDtoV[1]∪···∪ V[s] and F[0];\n2Setτx= ∗ for all individuals x∈V\\ (V[1]∪···∪ V[s]);\nAlgorithm 1: SPARC , steps 1–2\n2.4.1. The seed. Recall that each individual x∈V[1]∪···∪ V[s] independently joins ∆0random tests from F[0]. In its\nthe initial step SPARC runs DDon the test design G0comprising V[1]∪···∪ V[s] and the tests F[0] only. Throughout\nSPARC maintains a vector τ∈{0,1,∗}Vthat represents the algorithm’s current estimate of the ground truth σ, with\n∗indicating ‘undetermined as yet’ .\nSince Proposition 2.3 shows that the seed contains (1 +o(1))ks/ℓinfected individuals w.h.p., the choice (2.15)\nofm0and Theorem 2.1 imply that DDwill succeed to diagnose the seed correctly for a suitable ∆0.\nProposition 2.4 ([16, Theorem 2.2]) .There exist ∆0=Θ(lnn)such that the output of DDsatisfies τx=σxfor all\nx∈V[1]∪···∪ V[s]w.h.p.\n2.4.2. A combinatorial condition. To simplify the analysis of the message passing step in Section 2.4.3 we observe\nthat certain individuals can be identified as likely uninfected purely on combinatorial grounds. More precisely,\nconsider x∈V[i] for s<i≤ℓ. Ifxis infected, then any test a∈∂xis actually positive. Hence, we expect that xap-\npears in about p11∆tests that display a positive result. In fact, the choice (2.14) of sensures that w.h.p. even within\neach separate compartment F[i+j−1], 1≤j≤sthe individual xappears in about p11∆/spositively displayed\ntests. Thus, let\nV+[i]=(\nx∈V[i] :sX\nj=1¯¯|∂x∩F+[i+j−1]|−∆p11/s¯¯≤ln4/7n)\n. (2.23)\nThe following proposition confirms that all but o(k/s) infected individuals x∈V[i] belong to V+[i]. Additionally,\nthe proposition determines the approximate size of V+[i]. For notational convenience we define\nV+\n0[i]=V+[i]∩V0, V+\n1[i]=V+[i]∩V1, V+=[\ns<i≤ℓV+[i].\nRecall q+\n0from (2.1). The proof of the following proposition can be found in Section 3.2.\nProposition 2.5. W.h.p. we have\nℓX\ni=s+1|V1[i] \\V+[i]| ≤kexp(−Ω(ln1/7n)) and (2.24)\n¯¯V+[i] \\V1[i]¯¯≤n\nℓexp¡\n−∆DKL¡\np11∥q+\n0¢\n+O(ln4/7n)¢\nfor all s +1≤i≤ℓ. (2.25)\n2.4.3. Belief Propagation redux. The main phase of the SPARC algorithm employs a simplified version of the BP\nupdate rules (2.7)–(2.8) to diagnose one compartment V[i],s<i≤ℓ, after the other. The textbook way to employ\nBP would be to diagnose the seed, initialise the BP messages emanating from the seed accordingly and then run BP\nupdates until the messages converge. However, towards the proof of Theorem 1.1 this way of applying BP seems\ntoo complicated both to run and to analyse. Instead, SPARC relies on a ‘paced’ version of BP . Rather than updating\nthe messages to convergence from the seed, we perform one round of message updates, then diagnose the next\ncompartment, re-initialise the messages to coincide with the newly diagnosed compartment and proceed to the\nnext undiagnosed compartment.\nWe work with the log-likelihood versions of the BP messages from (2.10)–(2.12). Suppose we aim to process\ncompartment V[i],s<i≤ℓ, having completed V[1],..., V[i−1] already. Then for a test a∈F[i+j−1],j∈[s], and\na adjacent variable x∈V[i+j−s]∪V[i+j−1] we initialise\nηx→a,0=\n\n−∞ ifτx=0,\n+∞ ifτx=1,\nln(k/(n−k)) if τx= ∗.(2.26)\n12The third case above occurs if and only if x∈V[i]∪ ··· ∪ V[i+j−1]; that is, if xbelongs to an as yet undiag-\nnosed compartment. For the compartments that have been diagnosed already we set ηx→a,0to±∞ , depending on\nwhether xhas been classified as infected or uninfected.\nLet us now investigate the ensuing messages ηa→x,1forx∈V[i] and tests a∈∂x∩F[i+j−1]. A glimpse at\n(2.11) reveals that for any test athat contains an individual y∈V[i+j−s]∪···∪ V[i−1] with τy=1 we have\nηa→x,1=0. This is because (2.26) ensures that ηy→a,0= ∞ and tanh( ∞)=1. Hence, the test acontains no further\ninformation as to the status of x. Therefore, we call a test a∈F[i+j−1]informative towards V[i] ifτy=0 for all\ny∈∂a∩(V[i+j−s]∪···∪ V[i−1]).\nLetWi,j(τ) be the set of all informative a∈F[i+j−1]. Then any a∈Wi,j(τ) receives ηy→a,0= −∞ from all\nindividuals ythat have been diagnosed already, i.e. all y∈V[h] with i+j−s≤h<i. Another glance at the up-\ndate rule shows that the corresponding terms simply disappear from the product on the r.h.s. of (2.11), because\ntanh(−∞ )= −1. Consequently, only the factors corresponding to undiagnosed individuals y∈V[i]∪···∪ V[i+j−1]\nremain. Hence, with r=1{a∈F+} the update rule (2.11) simplifies to\nηa→x,1=lnp1r−lnh\np1r+(p0r−p1r)(1−k/n)|∂a∩(V[i]∪···∪ V[i+j−1])|−1i\n. (2.27)\nThe only random element in the expression (2.27) is the number¯¯∂a∩(V[i]∪···∪ V[i+j−1])¯¯of members\nof test afrom compartments V[i]∪···∪ V[i+j−1]. But by the construction of Gscthis number has a binomial\ndistribution with mean\nE¯¯∂a∩(V[i]∪···∪ V[i+j−1])¯¯=j∆n\nms+o(1)=d j n\nks+o(1) [using (2.16)].\nSince the fluctuations of¯¯∂a∩(V[i]∪···∪ V[i+j−1])¯¯are of smaller order than the mean, we conclude that w.h.p.\n(2.27) can be well approximated by a deterministic quantity:\nηa→x,1=(\nw+\nj+o(1), if a∈F+,\n−w−\nj+o(1), if a∈F−, where (2.28)\nw+\nj=lnp11\np11+(p01−p11)exp(−d j/s)≥0, w−\nj= − lnp10\np10+(p00−p10)exp(−d j/s)≥0. (2.29)\nNote that in the case p10=0, the negative test weight W−\njevaluates to w−\nj= ∞ , indicating that individual contained\nin negative test definitely are uninfected. Finally, the messages (2.28) lead to the BP estimate of the posterior\nmarginal of xvia (2.12), i.e. by summing on all informative tests a∈∂x. To be precise, letting\nW±\nx,j(τ)=∂x∩Wi,j(τ)∩F±\nbe the positively/negatively displayed informative tests adjacent to xand setting\nW+\nx(τ)=sX\nj=1w+\nj¯¯¯W+\nx,j(τ)¯¯¯, W−\nx(τ)=sX\nj=1w−\nj¯¯¯W−\nx,j(τ)¯¯¯, (2.30)\nwe obtain\nηx,1=W+\nx(τ)−W−\nx(τ)+‘lower order fluctuations’. (2.31)\nOne issue with the formula (2.28) is the analysis of the ‘lower order fluctuations’ , which come from the random\nvariables |∂a∩(V+[i]∪···∪ V+[i+s−1])|. Of course, one could try to analyses theses deviations caefully by resorting\nto some kind of a normal approximation. But for our purposes this is unnecessary. It turns out that we may simply\nuse the sum on the r.h.s. of (2.31) to identify which individuals are infected. Specifically, instead of computing the\nactual BP approximation ηx,1after one round of updating, we just compare W+\nx(τ) and W−\nx(τ) with the values that\nwe would expect these random variables to take if xwere infected. These conditional expectations work out to be\nW+=p11∆sX\nj=1exp( d(j−s)/s)w+\nj, W−=(\np10∆Ps\nj=1exp( d(j−s)/s)w−\njifp10>0\n0 otherwise.(2.32)\nThus, SPARC will diagnose V[i] by comparing W±\nx(τ) with W±. Additionally, SPARC takes into account that infected\nindividuals likely belong to V+[i], as we learned from Proposition 2.5.\nLet\nζ=(lnlnln n)−1(2.33)\n133fori=s+1,...,ℓdo\n4 forx∈V[i]do\n5 ifx̸∈V+[i]orW+\nx(τ)<(1−ζ)W+orW−\nx(τ)>(1+ζ)W−then\n6 τx=0// classify as uninfected\n7 else\n8 τx=1// classify as infected\n9return τ\nAlgorithm 2: SPARC , steps 3–9.\nbe a term that tends to zero slowly enough to absorb error terms. The following proposition, which we prove in\nSection 3.3, summarises the analysis of phase 3. Recall from (2.16) that c=m/(kln(n/k)).\nProposition 2.6. Assume that for a fixed ε>0we have\nc>cex,2(d)+ε. (2.34)\nThen w.h.p. the output τofSPARC satisfies\nX\nx∈V+1{τx̸=σx}≤kexpµ\n−Ωµlnn\n(lnln n)5¶¶\n.\nThe proof of Proposition 2.6, which can be found in Section 3.3, is the centerpiece of the analysis of SPARC . The\nproof is based on a large deviations analysis that bounds the number of individuals x∈V+[i] whose corresponding\nsums W±\nx(τ) deviate form their conditional expectations given σx. We have all the pieces in place to complete the\nproof of the first theorem.\nProof of Theorem 1.1. Setting d=dShfrom (1.17) and invoking (1.18), we see that the theorem is an immediate\nconsequence of Propositions 2.4, 2.5 and 2.6. □\n2.5.Exact recovery. As we saw in Section 1.3 the threshold cex,1(d,θ) encodes a local stability condition. This con-\ndition is intended to ensure that w.h.p. no other ‘solution’ σ̸=σwith∥σ−σ∥1=o(k) of a similarly large posterior\nlikelihood exists. In fact, because Gscenjoys fairly good expansion properties the test results provide sufficient\nclues for us to home in on σonce we get close, provided the number of tests is as large as prescribed by Theo-\nrem 1.3. Thus, the idea behind exact recovery is to run SPARC first and then apply local corrections to fully recover\nσ; a similar strategy was employed in the noiseless case in [10].\nThough this may sound easy and a simple greedy strategy does indeed do the trick the noiseless case [10], in the\npresence of noise it takes a good bit of care to get the local search step right. Hence, as per (1.11) suppose that c,d\nfrom (2.16) satisfy c>max{ cex,2(d),cex,1(d,θ)}+ε. Also suppose that we already ran SPARC to obtain τ∈{0,1}Vwith\n∥τ−σ∥1=o(k) (as provided by Proposition 2.6). How can we set about learning the status of an individual xwith\nperfect confidence?\nAssume for the sake of argument that τy=σyfor all ythat share a test a∈∂xwith x. If acontains another\ninfected individual y̸=x, then unfortunately nothing can be learned from aabout the status of x. In this case we\ncall the test a tainted . By contrast, if τy=0 for all y∈∂a\\ {x}, i.e. if aisuntainted , then the displayed result σ′′\na\nhinges on the infection status of xitself. Hence, the larger the number of untainted positively displayed a∈∂x,\nthe likelier xis infected. Consequently, to accomplish exact recovery we are going to threshold the number of\nuntainted positively displayed a∈∂x. But crucially, to obtain an optimal algorithm we cannot just use a scalar,\none-size-fits-all threshold. Instead, we need to carefully choose a threshold function that takes into account the\ntotal number of untainted tests a∈∂x.\nTo elaborate, let\nYx=¯¯©\na∈∂x\\F[0] :∀y∈∂a\\ {x} :σy=0ª¯¯ (2.35)\nbe the total number of untainted tests a∈∂x; to avoid case distinctions we omit seed tests a∈F[0]. Routine calcu-\nlations reveal that Yxis well approximated by a binomial variable with mean exp( −d)∆. Therefore, the fluctuations\nofYxcan be estimated via the Chernoff bound. Specifically, the numbers of infected/uninfected individuals with\n140.2 0.3 0.4 0.5 0.6 0.70.50.60.70.80.9\n123456FIGURE 2. The threshold function z(·) (red) on the interval Y(cex,1(d,θ),d,θ) and the resulting\nlarge deviations rate cex,1(d,θ)d(1−θ)¡\nDKL¡\nα∥exp(−d)¢\n+αDKL¡z(α)∥p01¢¢\n(black) with θ=1/2,\np00=0.972, p11=0.9 at the optimal choice of d.\nYx=α∆can be approximated as\nE|{x∈V1:Yx=α∆}|=kexp¡\n−∆DKL¡\nα∥exp(−d)¢\n+o(∆)¢\n, (2.36)\nE|{x∈V0:Yx=α∆}|=nexp¡\n−∆DKL¡\nα∥exp(−d)¢\n+o(∆)¢\n. (2.37)\nConsequently, since k= ⌈nθ⌉ = o(n) ‘atypical’ values of Yxoccur more frequently on healthy than on infected\nindividuals. In fact, recalling (1.6), we learn from a brief calculation that for α̸∈Y(c,d,θ) not a single x∈V1with\nYx=α∆exists w.h.p. Hence, if Yx/∆̸∈Y(c,d,θ) we deduce that xis uninfected.\nFor xsuch that Yx/∆∈Y(c,d,θ) more care is required. In this case we need to compare the number\nZx=¯¯©\na∈∂+x\\F[0] :∀y∈∂a\\ {x} :σy=0ª¯¯ (2.38)\nofpositively displayed untainted tests to the total number Yxof untainted tests. Since the test results are put\nthrough the p-channel independently, Zxis a binomial variable given Yx. The conditional mean of Zxequals\np11Yxifxis infected, and p01Yxotherwise. Therefore, the Chernoff bound shows that\nP£\nZx=αβ∆|Yx=α∆¤\n=exp¡\n−α∆DKL¡\nβ∥pσx1¢\n+o(∆)¢\n. (2.39)\nIn light of (2.39) we set up the definition (1.9) of cex,1(d,θ) so that z(·) can be used as a threshold function to\ntell infected from uninfected individuals. Indeed, given Yx,Zxwe should declare xuninfected if either Yx/∆̸∈\nY(c,d,θ) orZx/Yx<z(Zx/∆), and infected otherwise; then the choice of z(·) and cex(θ) would ensure that all\nindividuals get diagnosed correctly w.h.p. Figure 2 displays a characteristic specimen of the function z(·) and the\ncorresponding rate function from (1.9).\nYet trying to distil an algorithm from these considerations, we run into two obvious obstacles. First, the thresh-\noldz(·) may be hard to compute precisely. Similarly, the limits of the interval Y(c,d,θ) may be irrational (or worse).\nThe following proposition, which we prove in Section 4.1, remedies these issues.\nProposition 2.7. Letε>0and assume that c >cex(d,θ)+ε. Then there exist δ>0and an open interval ; ̸=I=\n(l,r)⊆[δ,1−δ]with endpoints l ,r∈Qsuch that for any ε′>0there exist δ′>0and a step function Z:I→\n(p01,p11)∩Qsuch that the following conditions are satisfied.\nZ1: cd(1−θ)DKL¡\ny∥exp(−d)¢\n>θ+δfor all y ∈(0,1) \\ ( l+δ,r−δ).\nZ2: cd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nZ(y)∥p11¢¢\n>θ+δfor all y ∈I.\nZ3: cd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nZ(y)∥p01¢¢\n>1+δfor all y ∈I.\nZ4: If y,y′∈Isatisfy |y−y′| <δ′, then |Z(y)−Z(y′)| <ε′.\nThe second obstacle is that in the above discussion we assumed that τy=σyfor all y∈∂a\\ {x}. But all that the\nanalysis of SPARC provides is that w.h.p. τy̸=σyfor at most o(k) individuals y. To cope with this issue we resort\nto the expansion properties of Gsc. Roughly speaking, we show that w.h.p. for any small set Sof individuals (such\nas {y:τy̸=σy}) the set of individuals xthat occur in ‘many’ tests that also contain a second individual from Sis\n15significantly smaller than S. As a consequence, as we apply the thresholding procedure repeatedly the number of\nmisclassified individuals decays geometrically. We thus arrive at the following algorithm.\nInput: G sc,σ′′\nOutput: an estimate of σ\n1Letτ(1)be the output of SPARC (Gsc,σ′′);\n2for i=1,...,⌈lnn⌉do\n3 For all x∈V[s+1]∪···∪ V[ℓ] calculate\n4 Yx(τ(i))=X\na∈∂x\\F[0]1n\n∀y∈∂a\\{x}:τ(i)\ny=0o\n,Zx(τ(i))=X\na∈∂x\\F[0]:σ′′a=11n\n∀y∈∂a\\{x}:τ(i)\ny=0o\n;\n5 Letτ(i+1)\nx=(\nτ(i)\nx ifx∈V[1]∪···∪ V[s],\n1©\nYx(τ(i))/∆∈Iand Zx(τ(i))/∆>Z(Yx(τ(i))/∆)ª\notherwise;\n6return τ(⌈lnn⌉)\nAlgorithm 3: TheSPEX algorithm.\nProposition 2.8. Suppose that c >cex(d,θ)+εfor a fixed ε>0. There exists ε′>0, a rational interval Iand a\nrational step function Zsuch that w.h.p. for all 1≤i<lnn we have\n∥σ−τ(i+1)∥1≤1\n3∥σ−τ(i)∥1.\nWe prove Proposition 2.8 in Section 4.2.\nProof of Theorem 1.3. Since Proposition 2.8 shows that the number of misclassified individuals decreases geomet-\nrically as we iterate Steps 3–5 of SPEX , we have τ(⌈lnn⌉)=σw.h.p. Furthermore, thanks to Theorem 1.1 and Propo-\nsition 2.7 SPEX is indeed a polynomial time algorithm. □\nRemark 2.9. In the noiseless case p00=p11=1 Theorem 1.3 reproduces the analysis of the SPIV algorithm\nfrom [10]. One key difference between SPARC andSPEX on the one hand and SPIV on the other is that the for-\nmer are based on Belief Propagation, while the latter relies on combinatorial intuition. More precisely, the SPIV\nalgorithm infers from positive tests by means of a weighted sum identical to W+\nx(τ) from (2.30) with the special\nvalues p00=p11=1 and d=ln2. In the noiseless case the weights w+\njwere ‘guessed’ based on combinatorial intu-\nition. Furthermore, in noiseless case we can be certain that any individual contained in a negative test is healthy,\nand therefore the SPIV algorithm only takes negative tests into account in this simple, deterministic manner. By\ncontrast, in the noisy case the negative tests give rise to a second weighted sum W−\nx(τ). An important novelty is\nthat rather than ‘guessing’ the weights w±\nj, here we discovered how they can be derived systematically from the BP\nformalism. Apart from shedding new light on the noiseless case as well, we expect that this type of approach can\nbe generalised to quite a few other inference problems as well. A second novelty in the design of SPEX is the use of\nthe threshold function Z(·) that depends on the number untainted tests. The need for such a threshold function\nis encoded in the optimisation problem (1.9) that gives rise to a non-trivial choice of the coefficient dthat governs\nthe size of the tests. This type of optimisation is unnecessary in the noiseless case, where simply d=ln2 is the\noptimal choice for all θ∈(0,1).\n2.6.The constant-column lower bound. Having completed the discussion of the algorithms SPARC andSPEX for\nTheorems 1.1 and 1.3, we turn to the proof of Theorem 1.4 on the information-theoretic lower bound for the con-\nstant column design. The goal is to show that for m<(cex(θ)−ε)kln(n/k) no algorithm, efficient or otherwise, can\nexactly recover σonGcc.\nTo this end we are going to estimate the probability of the actual ground truth σunder the posterior given the\ntest results. We recall from Fact 2.2 that the posterior coincides with the Boltzmann distribution from (2.4). The\nfollowing proposition, whose proof can be found in Section 5.1, estimates the Boltzmann weight ψGcc,σ′′(σ) of the\nground truth. Recall from (2.16) that d=k∆/m. Also recall the weight function from (2.3).\nProposition 2.10. w.h.p. the weight of σsatisfies\n1\nmlnψGcc,σ′′(σ)= − exp(−d)h(p00)−(1−exp(−d))h(p11)+O(m−1/2ln3). (2.40)\n16We are now going to argue that for c<cex(θ) the partition function ZGcc,σ′′dwarfs the Boltzmann weight (2.40)\nw.h.p. The definition (1.11) of cex(θ) suggests that there are two possible ways how this may come about. The first\noccurs when c=m/(kln(n/k)) is smaller than the local stability bound cex,1(d,θ) from (1.9). In this case we will\nessentially put the analysis of SPEX into reverse gear. That is, we will show that there are plenty of individuals x\nwhose status could be flipped from infected to healthy or vice versa without reducing the posterior likelihood. In\neffect, the partition function will be far greater than the Boltzmann weight of σ.\nProposition 2.11. Assume that there exists y ∈Y(c,d,θ)such that there exists z ∈(p01,p11)such that\ncd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p11¢¢\n<θ and (2.41)\ncd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p01¢¢\n<1. (2.42)\nThen w.h.p. there exist nΩ(1)pairs (v,w)∈V1×V0such that for the configuration σ[v,w]obtained from σby inverting\nthe v ,w-entries we have\nµGcc,σ′′(σ[v,w])=µGcc,σ′′(σ). (2.43)\nWe prove Proposition 2.11 in Section 5.2.\nThe second case is that cis smaller than the entropy bound cex,2(d) from (1.10). In this case we will show by way\nof a moment computation that ZGcc,σ′′exceeds the Boltzmann weight of σw.h.p. More precisely, in Section 5.3 we\nprove the following.\nProposition 2.12. Letε>0. If c<cex,2(d)−ε, then\nP£\nlnZGcc,σ′′≥kln(n/k)£\n1−c/cex,2(d)+o(1)¤¤\n>1−ε+o(1).\nProof of Theorem 1.4. Proposition 2.11 readily implies that µGcc,σ′′(σ)=o(1) w.h.p. if c<cex,1(d,θ). Furthermore,\nin the case c<cex,2(d) the assertion follows from Propositions 2.10 and 2.12. □\n3. A NALYSIS OF THE APPROXIMATE RECOVERY ALGORITHM\nIn this section we carry out the proofs of the various proposition leading up to the proof of Theorem 1.1 (Propo-\nsitions 2.3, 2.5 and 2.6). Throughout we work with the spatially coupled test design Gscfrom Section 2.3. Hence,\nwe keep the notation and assumptions from (2.13)–(2.17). We also make a note of the fact that for any x∈V[i] and\nany a∈F[i+j−1],j=1,..., s,\nP[x∈∂a]=1−P[x̸∈∂a]=1−Ã\n|F[i+j−1]|−1\n∆/s!Ã\n|F[i+j−1]|\n∆/s!−1\n=∆ℓ\nms+Oµµ∆ℓ\nms¶2¶\n; (3.1)\nthis is because the construction of Gscensures that xjoins∆/srandom tests in each of F[i+j−1], drawn uniformly\nwithout replacement.\n3.1.Proof of Proposition 2.3. Property G1is a routine consequence of the Chernoff bound and was previously\nestablished as part of [10, Proposition 4.1]. With respect to G2we may condition on the event Ethat the bound\n(2.18) from G1is satisfied. Consider a test a∈F[i]. Recall that acomprises individuals from the compartments\nV[i−s+1],..., V[i]. Since the probability that a specific individual joins a specific test is given by (3.1) and since\nindividuals choose the tests that they join independently, on Efor each i−s+1≤h≤iwe have\n|V1[h]∩∂a| ∼Bin\nk\nℓ+O\ns\nk\nℓlnn\n,∆ℓ\nms+Oµµ∆ℓ\nms¶2¶\n. (3.2)\nCombining (2.16) and (3.2), we obtain\nE[|V1[h]∩∂a| |E]=∆k\nms+O\ns\nℓ\nklnn\n=d\ns+O¡\nk−1/2ln3/2n¢\n. (3.3)\nFurther, combining (3.2) and (3.3), we get\nP[V1[h]∩∂a= ; |E]=exp\n\nk\nℓ+O\ns\nk\nℓlnn\n\nlnµ\n1−∆ℓ\nms+Oµµ∆ℓ\nms¶2¶¶\n=expµ\n−d\ns+O¡\nk−1/2ln3/2n¢¶\n.\n17Multiplying these probabilities up on i−s+1≤h≤i, we arrive at the estimate\nP[V1[h]∩∂a= ; |E]=exp¡\n−d+O¡\nk−1/2ln8/5n¢¢\n.\nHence,\nE[|F0[i]||E]=m\nℓexp(−d)+O(p\nmln2n). (3.4)\nTo establish concentration, observe that the set ∂xof tests that a specific infected individual x∈V1[h] joins can\nchange |F0[i]|by∆at the most. Moreover, changing the neighbourhood ∂xof a healthy individual cannot change\nthe actual test results at all. Therefore, by Azuma–Hoeffding,\nP£\n||F0[i]|−E[|F0[i]| |E]|≥p\nmln2n|E¤\n≤2expµ\n−mln4n\n2k∆2¶\n=o(n−3). (3.5)\nThus, (3.4), (3.5) and G1show that with probability 1 −o(n−2) for all 1 ≤i≤ℓwe have\n|F0[i]|=m\nℓexp(−d)+O(p\nmln2n), and thus |F1[i]|=m\nℓ(1−exp(−d))+O(p\nmln2n). (3.6)\nSince the actual test results are subjected to the p-channel independently to obtain the displayed test results, the\ndistributions of |F±\n0/1[i] given F0/1[i] read\n¯¯F−\n0[i]¯¯=Bin(|F0[i]|,p00),¯¯F+\n0[i]¯¯=Bin(|F0[i]|,p01), (3.7)\n¯¯F−\n1[i]¯¯=Bin(|F1[i]|,p10),¯¯F+\n1[i]¯¯=Bin(|F1[i]|,p11). (3.8)\nThus, G2follows from (3.6), (3.7) and Lemma 1.5 (the Chernoff bound).\n3.2.Proof of Proposition 2.5. Any x∈V1[i] has a total of ∆/sneighbours in each of F[i],..., F[i+s−1]. Moreover,\nall tests a∈∂xare actually positive. Since the displayed result is obtained via the p-channel independently for\nevery a, the number of displayed positive neighbours |∂x∩F+[i+j−1]|is a binomial variable with distribution\nBin(∆/s,p11). Since ∆=Θ(lnn) and s=Θ(lnln n), the first assertion (2.24) is immediate from Lemma 1.5.\nMoving on to the second assertion, we condition on the event Ethat the bounds (2.19)–(2.22) hold for all i∈[ℓ].\nThen Proposition 2.3 shows that P[E]=1−o(n−2). Given Ewe know that\n|F+[i]| =q+\n0m\nℓ+O(p\nmln3n), |F−[i]| =(1−q+\n0)m\nℓ+O(p\nmln3n). (3.9)\nNow consider an individual x∈V0[i]. Also consider any test a∈F[i+j−1] for j∈[s]. Then the actual result σ′\na\nofais independent of the event { x∈a}, because xis uninfected. Since the displayed result σ′′\nadepends solely on\nσ′\na, we conclude that σ′′\nais independent of { x∈a} as well. Therefore, (3.9) shows that on the event Ethe number\nof displayed positive tests that xis a member of has conditional distribution\n|∂x∩F+[i+j−1]| ∼Hypµm\nℓ,q+\n0m\nℓ+O(p\nmln3n),∆\ns¶\n. (3.10)\nSince the random variables ( |∂x∩F+[i+j−1]|)1≤j≤sare mutually independent, (2.25) follows from Lemma 1.6.\n3.3.Proof of Proposition 2.6. We reduce the proof of Proposition 2.6 to three lemmas. The first two estimate the\nsums from (2.30) when evaluated at the actual ground truth σ.\nLemma 3.1. Assume that (2.34) is satisfied. w.h.p. we have\nX\ns<i≤ℓX\nx∈V+\n1[i]1©\nW+\nx(σ)<(1−ζ/2)W+orW−\nx(σ)>(1+ζ/2)W−ª\n≤kexpµ\n−Ωµlnn\n(lnln n)4¶¶\n. (3.11)\nLemma 3.2. Assume that (2.34) is satisfied. w.h.p. we have\nX\ns≤i<ℓX\nx∈V+\n0[i]1©\nW+\nx(σ)≥(1−2ζ)W+andW−\nx(σ)≤(1+2ζ)W−ª\n≤k1−Ω(1). (3.12)\nWe defer the proofs of Lemmas 3.1 and 3.2 to Sections 3.4 and 3.5. While the proof of Lemma 3.1 is fairly routine,\nthe proof of Lemma 3.2 is the linchpin of the entire analysis of SPARC , as it effectively vindicates the BP heuristics\nthat we have been invoking so liberally in designing the algorithm.\nAdditionally, to compare W±\nx(σ) with the algorithm’s estimate W±\nx(τ) we resort to the following expansion prop-\nerty of Gsc.\n18Lemma 3.3. Let0<α,β<1be such that α+β>1. Then w.h.p. for any T ⊆V of size |T| ≤exp(−lnαn)k,\n¯¯¯¯¯(\nx∈V:X\na∈∂x\\F[0]1{T∩∂a\\{x}̸= ;}≥lnβn)¯¯¯¯¯≤|T|\n8lnln n.\nIn a nutshell, Lemma 3.3 posits that for any ‘small’ set Tof individuals there are even fewer individuals that share\nmany tests with individuals from T. Lemma 3.3 is an generalisation of [10, Lemma 4.16]. The proof, based on a\nroutine union bound argument, is included in Appendix B for completeness.\nProof of Proposition 2.6. Proposition 2.4 shows that for all individuals xin the seed V[1],..., V[s] we have τx=σx\nw.h.p. Let M[i]=©\nx∈V+[i] :τx̸=σxª\nbe the set of misclassified individuals in V+[i]. We are going to show that\nw.h.p. for all 1 ≤i≤ℓand for large enough n,\n|M[i]|≤kexpµ\n−lnn\n(lnln n)5¶\n. (3.13)\nWe proceed by induction on i. As mentioned above, Proposition 2.4 ensures that M[1]∪···∪M[s]= ; w.h.p.\nNow assume that (3.13) is correct for all i<h≤ℓ; we are going to show that (3.13) holds for i=has well. To this\nend, recalling ζfrom (2.33) and W±from (2.32), we define for p10>0\nM1[h]=©\nx∈V+\n1[h] :W+\nx(σ)<(1−ζ/2)W+orW−\nx(σ)>(1+ζ/2)W−ª\n,\nM2[h]=©\nx∈V+\n0[h] :W+\nx(σ)>(1−2ζ)W+andW−\nx(σ)<(1+2ζ)W−ª\n,\nM3[h]=©\nx∈V+[h] :|W+\nx(σ)−W+\nx(τ)|+|W−\nx(σ)−W−\nx(τ)| >ζ(W+∧W−)/8ª\nand further for p10=0\nM1[h]=©\nx∈V+\n1[h] :W+\nx(σ)<(1−ζ/2)W+ª\n,\nM2[h]=©\nx∈V+\n0[h] :W+\nx(σ)>(1−2ζ)W+andW−\nx(σ)=0ª\n,\nM3[h]=©\nx∈V+[h] :|W+\nx(σ)−W+\nx(τ)| >ζ(W+)/8ª\n.\nWe claim that M[h]⊆M1[h]∪M2[h]∪M3[h]. To see this, assume that x∈M[h] \\ (M1[h]∪M2[h]). Then for\nSPARC to misclassify x, it must be the case that\n|W+\nx(τ)−W+\nx(σ)|+|W−\nx(τ)−W−\nx(σ)| >ζ\n8(W+∧W−),\nand thus x∈M3[h].\nThus, to complete the proof we need to estimate |M1[h]|,|M2[h]|,|M3[h]|. Lemmas 3.1 and 3.2 readily show\nthat\n|M1[h]|+|M2[h]| ≤kexpµ\n−Ωµlnn\n(lnln n)4¶¶\n. (3.14)\nFurthermore, in order to bound |M3[h]|we will employ Lemma 3.3. Specifically, consider x∈M3[h]. Since W+∧\nW−=Ω(∆)=Ω(lnn) for p10>0 by (2.29) and (2.32), there exists j∈[s] such that\n||W+\nx,j(τ)|−|W+\nx,j(σ)|| > ln1/2n or ||W−\nx,j(τ)|−|W−\nx,j(σ)|| > ln1/2n.\nSince W+=Ω(lnn) for p10=0 there exists j∈[s] such that\n||W+\nx,j(τ)|−|W+\nx,j(σ)|| > ln1/2n\nAssume without loss that the left inequality holds. Then there are at least ln1/2ntests a∈∂x∩F+[h+j−1] such\nthat∂a∩(M[1]∪···∪M[h−1])̸= ;. Therefore, Lemma 3.3 shows that\n|M3[h]| ≤|M[h−s]∪···∪M[h−1]|\n8lnln n.\nHence, using induction to bound |M[h−s]|,...,|M[h−1]|and recalling from (2.14) that s≤1+lnln n, we obtain\n(3.13) for i=h. □\n193.4.Proof of Lemma 3.1. The thrust of Lemma 3.1 is to verify that the definition (2.23) of the set V+[i] faithfully\nreflects the typical statistics of positively/negatively displayed tests in the neighbourhood of an infected individual\nx∈V1[i] with s<i≤ℓ. Recall from the definition of Gscthat such an individual xjoins tests in F[i+j−1] for j∈[s].\nMoreover, apart from xitself a test a∈F[i+j−1]∩∂xrecruits from V[i+j−s],..., V[i+j−1]. In particular, a\nrecruits participants from the s−jcompartments V[i+j−s],..., V[i−1] preceding V[i]. Let\nUi,j=©\na∈F[i+j−1] : ( V1[1]∪···∪ V1[i−1])∩∂a= ;ª\n=(\na∈F[i+j−1] :i−1[\nh=i+j−sV1[h]∩∂a= ;)\n(3.15)\nbe the set of tests in F[i+j−1] that do not contain an infected individual from V[i+j−s],..., V[i−1].\nClaim 3.4. With probability 1−o(n−2)for all s ≤i<ℓ, j∈[s]we have\n|Ui,j| =¡\n1+O(n−Ω(1))¢m\nℓ·exp( d(j−s)/s).\nProof. We condition on the event Ethat G1from Proposition 2.3 holds. Then for any a∈F[i+j−1] and any\ni+j−s≤h<ithe number V1[h]∩∂aof infected individuals in afrom V[h] is a binomial variable as in (3.2). Since\nthe random variables ( V1[h]∩∂a)hare mutually independent, we therefore obtain\nP£\n(V1[i+j−s]∪··· V1[i−1])∩∂a= ; |E¤\n=exp( d(j−s)/s)+O(n−Ω(1)). (3.16)\nHence,\nE£\n|Ui,j| |E¤\n=m\nℓexp£\nd(j−s)/s+O(n−Ω(1))¤\n. (3.17)\nFurther, changing the set ∂xof tests that a single x∈V1joins can alter |Ui,j|by∆at the most, while changing the\nset of neighbours of any x∈V0does not change |Ui,j|at all. Therefore, Azuma–Hoeffding shows that\nP£¯¯|Ui,j|−E[|Ui,j| |E]¯¯>p\nmln2n|E¤\n≤2expµ\n−mln4n\n2k∆2¶\n=o(n−2). (3.18)\nThe assertion follows from (3.17)–(3.18). □\nLet\nq−\n1,j=p01exp( d(j−s)/s), q+\n1,j=p11exp( d(j−s)/s). (3.19)\nClaim 3.5. For all s <i≤ℓ, x∈V1[i]and j ∈[s]we have\nP·\n|W+\nx,j| <(1−ζ/2)q+\n1,j∆\ns¸\n+P·\n|W−\nx,j| >(1+ζ/2)q−\n1,j∆\ns¸\n≤expµ\n−Ωµlnn\n(lnln n)4¶¶\n.\nProof. Fix s≤i<ℓ,1≤j≤sand x∈V1[i]. In light of Proposition 2.3 and Claim 3.4 the event Ethat G1from\nProposition 2.3 holds and that |Ui,j| =exp( d(j−s)/s)m\nℓ(1+O(n−Ω(1))) has probability\nP[E]=1−o(n−2). (3.20)\nLetUi,j= |∂x∩Ui,j|. Given Ui,jwe have\nUi,j∼Hypµm\nℓ+O(1),expµd(j−s)\ns¶m\nℓ(1+O(n−Ω(1))),∆\ns¶\n.\nTherefore, Lemma 1.6 shows that the event\nE′=½¯¯¯¯Ui,j−∆\nsexpµd(j−s)\ns¶¯¯¯¯≥lnn\n(lnln n)2¾\nhas conditional probability\nP£E′|E¤\n≤expµ\n−Ωµlnn\n(lnln n)4¶¶\n. (3.21)\nFinally, given Ui,jthe number |W+\nx,j|of positively displayed a∈∂x∩Ui,jhas distribution Bin( Ui,j,p11); sim-\nilarly, |W−\nx,j|has distribution Bin( Ui,j,p01). Thus, since ∆=Θ(lnn) while s=O(lnln n) by (2.14), the assertion\nfollows from (3.20), (3.21) and Lemma 1.5. □\nProof of Lemma 3.1. The lemma follows from Claim 3.5 and Markov’s inequality. □\n203.5.Proof of Lemma 3.2. To prove Lemma 3.2 we need to estimate the probability that an uninfected individual\nx∈V[i],s<i≤ℓ, ‘disguises’ itself to look like an infected individual. In addition to the sets Ui,jfrom (3.15),\ntowards the proof of Lemma 3.2 we also need to consider the sets\nPi,j=F1[i+j−1]∩Ui,j, Ni,j=F0[i+j−1]∩Ui,j,\nP±\ni,j=F±\n1[i+j−1]∩Ui,j, N±\ni,j=F±\n0[i+j−1]∩Ui,j.\nIn words, Pi,jandNi,jare the sets of actually positive/negative tests in Ui,j, i.e. actually positive/negative tests\na∈F[i+j−1] that do not contain an infected individual from V[i+j−s]∪···∪ V[i−1]. Additionally, P±\ni,j,N±\ni,j\ndiscriminate based on the displayed tests results. We begin by estimating the likely sizes of these sets.\nClaim 3.6. Let s<i≤ℓand j ∈[s]. Then with probability 1−o(n−2)we have\n|Pi,j| =¡\n1+O(n−Ω(1))¢m\nℓ·exp( d j/s)−1\nexp( d), |Ni,j| =¡\n1+O(n−Ω(1))¢m\nℓ·exp(−d), (3.22)\n|P+\ni,j| =¡\n1+O(n−Ω(1))¢m\nℓ·p11(exp( d j/s)−1)\nexp( d),|P−\ni,j| =¡\n1+O(n−Ω(1))¢m\nℓ·p10(exp( d j/s)−1)\nexp( d), (3.23)\n|N+\ni,j| =¡\n1+O(n−Ω(1))¢m\nℓ·p01exp(−d), |N−\ni,j| =¡\n1+O(n−Ω(1))¢m\nℓ·p00exp(−d). (3.24)\nProof. SinceNi,j=F0[i+j−1], the second equation in (3.22) just follows from Proposition 2.3, G2. Furthermore,\nsincePi,j=Ui,j\\Ni,j, the first equation (3.22) is an immediate consequence of Claim 3.4. Finally, to obtain\n(3.23)–(3.24) we simply notice that given |Pi,j|,|Ni,j|we have\n|P+\ni,j| =Bin(|Pi,j|,p11),|P−\ni,j| =Bin(|Pi,j|,p10),|N+\ni,j| =Bin(|Ni,j|,p01),|N−\ni,j| =Bin(|Ni,j|,p00).\nHence, (3.23)–(3.24) just follow from (3.22) and Lemma 1.5. □\nLetUbe the event that G1–G2 from Proposition 2.3 hold and that the estimates (3.22)–(3.24) hold. Then by\nProposition 2.3 and Claim 3.6 we have\nP[U]=1−o(n−2). (3.25)\nTo facilitate the following computations we let\nq−\n0,j=exp(−d)p00+(exp( d(j−s)/s)−exp(−d))p10,q+\n0,j=exp(−d)p01+(exp( d(j−s)/s)−exp(−d))p11. (3.26)\nAdditionally, we introduce the shorthand λ=(lnln n)/ln3/7nfor the error term from the definition (2.23) of V+[i].\nOur next step is to determine the distribution of the random variables |W±\nx,j|that contribute to W±\nx(σ) from (2.30).\nClaim 3.7. Let s<i≤ℓand j ∈[s]. Given Ufor every x ∈V+\n0[i]we have\n|W−\nx,j| ∼Hypµ¡\n1+O(n−Ω(1))¢mq−\n0\nℓ,¡\n1+O(n−Ω(1))¢mq−\n0,j\n2ℓ,(1+O(λ))∆\nsp10¶\n, (3.27)\n|W+\nx,j| ∼HypÃ\n¡\n1+O(n−Ω(1))¢mq+\n0\nℓ,¡\n1+O(n−Ω(1))¢mq+\n0,j\n2ℓ,(1+O(λ))∆\nsp11!\n. (3.28)\nProof. The definition (2.23) of V+[i] prescribes that for any x∈V+\n0[i] we have ∂x∩F+[i+j−1]=(p11+O(λ))∆/s.\nThe absence or presence of x∈V0[i] in any test adoes not affect the displayed results of a, because xis uninfected.\nTherefore, the conditional distributions of |W±\nx,j|read\n|W±\nx,j| ∼Hypµ\n|F±[i+j−1]|,|N±\ni,j|+|P±\ni,j|,(1+O(λ))∆\nsp10¶\n.\nSince on Uthe bounds (2.19)–(2.22) and (3.22)–(3.24) hold, the assertion follows. □\nWe are now ready to derive an exact expression for the probability that for x∈V0[i] the value W+\nxgets (almost)\nas large as the value W+that we would expected to see for an infected individual. Recall the values q±\n1,jfrom (3.19).\n21Claim 3.8. Let\nM+=min1\nssX\nj=1DKLÃ\nzj∥q+\n0,j\nq+\n0!\ns.t.sX\nj=1Ã\nzj−q+\n1,j\np11!\nw+\nj=0, z1,..., zs∈(0,1). (3.29)\nThen for all s <i≤ℓand all x ∈V[i]we have\nP£\nW+\nx≥(1−2ζ)W+|U,x∈V+\n0[i]¤\n≤exp(−(1+o(1))p11∆M+).\nProof. Since∆=O(lnn) and s=O(lnln n) by (2.14) we can write\nP£\nW+\nx(σ)≥(1−2ζ)W+|U,x∈V+\n0[i]¤\n≤X\n0≤v1,...,vs≤∆/s1(\nnX\nj=1vjw+\nj≥(1−2ζ)W+)\nPh\n∀j∈[s] :W+\nx,j(σ)=vj|U,x∈V+\n0[i]i\n≤exp( o(∆)) max\n0≤v1,...,vs≤∆/s1(\nnX\nj=1vjw+\nj≥(1−2ζ)W+)\nPh\n∀j∈[s] :W+\nx,j(σ)=vj|U,x∈V+\n0[i]i\n. (3.30)\nFurther, given Uand x∈V+\n0[i] the random variables ( W+\nx,j)j∈[s]are mutually independent because xjoins tests in\nthe compartments F[i+j−1],j∈[s], independently. Hence, Claim 3.7 shows that for any v1,..., vs,\nPh\n∀j∈[s] :W+\nx,j(σ)=vj|U,x∈V+\n0[i]i\n=sY\nj=1Ph\nW+\nx,j(σ)=vj|U,x∈V+\n0[i]i\n. (3.31)\nThus, consider the optimisation problem\nM+\nt=min1\nssX\nj=1DKLÃ\nzj∥q+\n0,j\nq+\n0!\ns.t.sX\nj=1zjw+\nj≥(1−t)W+,z1,..., zs∈[q+\n0,j/q+\n0,1]. (3.32)\nThen combining (3.30) and (3.31) with Claim 3.7 and Lemma 1.6 and using the substitution zj=vj/(p11∆), we\nobtain\nP£\nW+\nx≥(1−2ζ)W+|U,x∈V+\n0[i]¤\n≤exp(−∆p11M+\n2ζ+o(∆)). (3.33)\nWe claim that (3.33) can be sharpened to\nP£\nW+\nx≥(1−2ζ)W+|U,x∈V+\n0[i]¤\n≤exp(−∆p11M+\n0+o(∆)). (3.34)\nIndeed, consider any feasible solution z1,..., zstoM+\nζsuch thatP\nj≥szjw+\nj<W+. Obtain z′\n1,..., z′\nsby increasing\nsome of the values z1,..., zssuch thatP\nj≤sz′\njw+\nj=W+. Then because the functions z7→DKL(z∥q+\n0,j/q+\n0) with\nj∈[s] are equicontinuous on the compact interval [0,1], we see that\n1\nssX\nj=1DKL³\nzj∥q+\n0,j/q+\n0´\n≥1\nssX\nj=1DKL³\nz′\nj∥q+\n0,j/q+\n0´\n+o(1)\nuniformly for all z1,..., zsand z′\n1,..., z′\ns. Thus, (3.34) follows from (3.33).\nFinally, we notice that the condition zj≥q+\n0,j/q+\n0in (3.32) is superfluous. Indeed, since DKL(q+\n0,j/q+\n0∥q+\n0,j/q+\n0)=\n0 there is nothing to be gained from choosing zj<q+\n0,j/q+\n0. Furthermore, since the Kullback-Leibler divergence is\nstrictly convex and (1.1) ensures that q+\n0,j/q+\n0<q+\n1,j/p11for all j, the optimisation problem M+\n0attains a unique\nminimum, which is not situated at the boundary of the intervals [ q+\n0,j/q+\n0,1]. That said, the unique minimiser\nsatisfies the weight constraintP\nj≥szjw+\njwith equality; for otherwise we could reduce some zj, thereby decreasing\nthe objective function value. In summary, we conclude that M+\n0=M+. Thus, the assertion follows from (3.34). □\nClaim 3.9. Let\nM−=min1\nssX\nj=1DKLµ\nzj∥q−\n0,j\nq−\n0¶\ns.t.sX\nj=1µ\nzj−q−\n1,j\np01¶\nw−\nj=0, z1,..., zs∈(0,1). (3.35)\nThen for all s <i≤ℓand all x ∈V[i]we have\nP£\nW−\nx≤(1+2ζ)W−|U,x∈V+\n0[i]¤\n≤exp(−(1+o(1))p10∆M−).\n22Recall that by convention 0 ·∞ = 0. Thus for p10=0 the condition of (3.35) boils down to zj=q−\n1,j/p01and the\noptimisation becomes trivial.\nProof. Analogous to the proof of Claim 3.8. □\nClearly, as a next step we need to solve the optimisation problems (3.29) and (3.35). We defer this task to Sec-\ntion 3.6, where we prove the following.\nLemma 3.10. LetXhave distribution Be(exp( −d))and let Ybe the (random) channel output given input X. Then\np11M++p10M−=I(X,Y)\nd−DKL¡\np11∥q+\n0¢\n.\nProof of Lemma 3.2. In light of Claims 3.8 and 3.9 and Lemma 3.10 to work out that all but o(k) positive individuals\nare identified correctly, using Markov’s inequality we need to verify that\n¯¯V+\n0¯¯exp¡\n−∆¡\np11M++p10M−¢¢\n<k (3.36)\nTaking the logarithm of (3.36) and simplifying, we arrive at\nln(n)¡\n1−cd(1−θ)¡\nDKL¡\np11∥(1−exp(−d))p11+exp(−d)p01¢\n+p11M++p10M−¢¢\n<θln(n). (3.37)\nThus we need to show that\ncd£\nDKL¡\np11∥(1−exp(−d))p11+exp(−d)p01¢\n+p11M++p10M−¤\n>1. (3.38)\nThis boils down to cI(X,Y)>1, which in turn is identical to (2.34). □\n3.6.Proof of Lemma 3.10. We tackle the optimisation problems M±via the method of Lagrange multipliers. Since\nthe objective functions are strictly convex, these optimisation problems possess unique stationary points. As the\nparameters from (3.19) satisfy q+\n1,j/p11=exp( d(j−s)/s), the optimisation problem (3.29) gives rise to the following\nLagrangian.\nClaim 3.11. The Lagrangian\nL+=sX\nj=1DKLÃ\nzj∥q+\n0,j\nq+\n0!\n+λw+\nj¡\nzj−exp(−d(s−j)/s)¢\nhas the unique stationary point z j=exp(−d(s−j)/s),λ= −1.\nProof. Since the objective functionPs\nj=1DKL³\nzj∥q+\n1,j/p11´\nis strictly convex, we just need to verify that λ= −1 and\nzj=exp(−d(s−j)/s) is a stationary point. To this end we calculate\n∂L+\n∂zj=lnz+\nj\n1−z+\nj−lnq+\n1,j\np11−q+\n1,j+λw+\nj,∂L+\n∂λ=sX\nj=1¡\nzj−exp¡\n−d(s−j)/s¢¢\nw+\nj. (3.39)\nSubstituting in the definition (2.29) of the weights w+\njand the definitions (3.19) of p11,q+\n1,jand simplifying, we\nobtain\n∂L+\n∂zj¯¯¯zj=exp(−d(s−j)/s)\nλ=−1=lnexp(−d(s−j)/s)\n1−exp(−d(s−j)/s)−lnp11(exp¡\nd j/s¢\n−1)+p01\np11(exp( d)−1)+p01\n+lnµ\n1−p11(exp( d j/s)−1)+p01\np11(exp( d)−1)+p01¶\n−lnp11\np11+(p01−p11)exp(−d j/s)=0.\nFinally, (3.39) shows that setting zj=exp(−d(s−j)/s) ensures that ∂L+/∂λ=0 as well. □\nAnalogous steps prove the corresponding statement for M−.\nClaim 3.12. Assume p 10>0. The Lagrangian\nL−=sX\nj=1DKLµ\nzj∥q−\n0,j\np01¶\n+λw−\nj¡\nzj−exp(−d(s−j)/s)¢\nhas the unique stationary point z j=exp(−d(s−j)/s),λ= −1.\n23Having identified the minimisers of M±, we proceed to calculate the optimal objective function values. Note\nthat for M−the minimisers zjfor the cases p10>0 and p10=0 coincide.\nClaim 3.13. Let\nλ+=ln(q+\n0)=ln(p01exp(−d)+p11(1−exp(−d))), λ−=ln(q−\n0)=ln(p00exp(−d)+p10(1−exp(−d))).\nThen\np11dexp( d)M+=(λ++d)¡\np11¡\n(d−1)exp( d)+1¢\n−p01¢\n+p01ln(p01)\n−p11¡\n(d−1)exp( d)+1¢\nln(p11)−d(d−1)exp( d)p11+O(1/s),\np10dexp( d)M−=(λ−+d)¡\np10¡\n(d−1)exp( d)+1¢\n−p00¢\n+p00ln(p00)\n−p10¡\n(d−1)exp( d)+1¢\nln(p10)−d(d−1)exp( d)p10+O(1/s)if p 10>0.\nProof. We perform the calculation for M+in detail. Syntactically identical steps yield the expression for M−, the\nonly required change being the obvious modification of the indices of the channel probabilities. Substituting the\noptimal solution zj=exp(−d(s−j)/s) and the definitions (3.19) and (2.1), (3.19) and (3.26) of q+\n1,j,q+\n0,q+\n0,jinto\n(3.29), we obtain\nM+=1\nssX\nj=1DKLµ\nexp(−d(s−j)/s)∥p01+(exp( d j/s)−1)p11\np01+(exp( d)−1)p11¶\n=I++O(1/s), where (3.40)\nI+=Z1\n0DKLµ\nexp( d(x−1))∥p11(exp( d x)−1)+p01\np11(exp( d)−1)+p01¶\ndx;\ntheO(1/s)-bound in (3.40) holds because the derivative of the integrand x7→DKL³\nexp( d(x−1))∥p11(exp( d x)−1)+p01\np11(exp( d)−1)+p01´\nis bounded on [0,1]. Replacing the Kullback-Leibler divergence by its definition, we obtain I+=I+\n1+I+\n2, where\nI+\n1=Z1\n0exp (d(x−1))lnexp (d(x−1))(p11(exp (d)−1)+p01)\np11(exp (d x)−1)+p01dx,\nI+\n2=Z1\n0(1−exp (d(x−1)))ln\n1−exp (d(x−1))\n1−p11(exp (d x)−1)+p01\np11(exp (d)−1)+p01\ndx.\nSplitting the logarithm in the first integrand, we further obtain I+\n1=I+\n11+I+\n12, where\nI+\n11=Z1\n0exp (d(x−1))ln£\nexp (d(x−1))(p11(exp (d)−1)+p01)¤\ndx,\nI+\n12= −Z1\n0exp (d(x−1))ln£\np11(exp (d x)−1)+p01¤\ndx.\nSetting Λ+=ln(p11(exp (d)−1)+p01)=λ++dand introducing u=exp (d(x−1)), we calculate\nI+\n11=1\nd·Z1\nexp(−d)ln(u)du+Z1\nexp(−d)Λ+du¸\n=1\nd£\n(d+1)exp( −d)−1+(1−exp (−d))Λ+¤\n. (3.41)\nConcerning I+\n12, we once again substitute u=exp (d(x−1))to obtain\nI+\n12= −1\ndZ1\nexp(−d)ln¡\np11exp( d)u+p01−p11¢\ndu\n= −1\nd·µp01\np11exp (−d)−exp (−d)+1¶\nΛ+−exp (−d)p01\np11ln(p01)+exp (−d)−1¸\n(3.42)\nProceeding to I2, we obtain\nI+\n2=Z1\n0(1−exp (d(x−1)))lnp11(exp (d)−1)+p01\np11exp( d)dx\n=Z1\n0(1−exp (d(x−1)))ln¡\np11(exp( d)−1)+p01¢\ndx−Z1\n0(1−exp (d(x−1)))ln( p11exp( d))dx\n=(Λ+−ln(p11)−d)(d−1+exp (−d))\nd. (3.43)\n24Finally, recalling that I+=I1+I2=I+\n11+I+\n12+I+\n2and combining (3.40)–(3.43) and simplifying, we arrive at the\ndesired expression for M+. □\nProof of Lemma 3.10. We have\nI(X,Y)=H(Y)−H(Y|X)=h(p00exp(−d)+p10(1−exp(−d)))−exp(−d)h(p00)−(1−exp(−d))h(p11).\nHence, Claim 3.13 yields\np11M++p10M−=−h(p00)\ndexp( d)−(1−exp(−d))h(p11)\nd+h(p11)\n+(p11−p01)λ++(p10−p00)λ−\ndexp( d)+d−1\nd£\np11λ++p10λ−¤\n=−h(p00)\ndexp( d)−(1−exp(−d))h(p11)\nd−DKL¡\np11∥q+\n0¢\n−λ+\ndexp(λ+)−λ−\ndexp(λ−)\n=I(X,Y)\nd−DKL¡\np11∥q+\n0¢\n,\nas desired. □\n4. A NALYSIS OF THE EXACT RECOVERY ALGORITHM\nIn this section we establish Propositions 2.7 and 2.8, which are the building blocks of the proof of Theorem 1.3. We\ncontinue to work with the spatially coupled design Gscfrom Section 2.3 and keep the notation and assumptions\nfrom (2.13)–(2.17).\n4.1.Proof of Proposition 2.7. Assume that c>cex,1(d,θ)+εand let c′=cex,1(d,θ)+ε/2. Since c>c′+ε/2 the\ndefinitions (1.11) of cex,1(d,θ) and (1.6) of Y(c′,d,θ) ensure that for small enough δ>0 we can find an open interval\nI⊆Y(c′,d,θ) with rational boundary points such that Z1is satisfied.\nLet ¯Ibe the closure of I. Then by the definition of cex,1(d,θ) there exists a function y∈¯I7→zy∈[p01,p11]\nsuch that for all y∈¯Iwe have\ncd(1−θ)£\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nzy∥p11¢¤\n=θ. (4.1)\nIn fact, because the Kullback-Leibler divergence is strictly convex the equation (4.1) defines zyuniquely. The in-\nverse function theorem implies that the function y7→zyis continuous and therefore uniformly continuous on ¯I.\nAdditionally, once again because the Kullback-Leibler divergence is convex and c>cex,1(d,θ), for all y∈¯Iwe have\ncd(1−θ)£\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nzy∥p01¢¤\n>1.\nTherefore, there exists ˆδ=ˆδ(c,d,θ)>0 such that for all y∈¯Iwe have\ncd(1−θ)£\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nzy∥p01¢¤\n>1+ˆδ. (4.2)\nCombining (4.1) and (4.2), we find a continuous y∈¯I7→ˆzysuch that for small enough δ>0 for all y∈[0,1] we\nhave\ncd(1−θ)£\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nˆzy∥p11¢¤\n>θ+2δ, and (4.3)\ncd(1−θ)£\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nˆzy∥p01¢¤\n>1+2δ. (4.4)\nAdditionally, by uniform continuity for any given 0 <ε′<δ/2 (that may depend arbitrarily on δandI) we can\nchoose δ′>0 small enough so that\n|ˆzy−ˆzy′| <ε′/2 for all y,y′∈¯Iwith|y−y′| <δ′. (4.5)\nFinally, let y0,..., yνwithν=ν(δ′,ε′)>0 be a large enough number of equally spaced points in ¯I=[y0,yν].\nThen for each iwe pick Z(yi)∈[p01,p11]∩Qsuch that |ˆzyi−Z(yi)|is small enough. Extend Zto a step function\n25¯I→Q∩[0,1] by letting Z(y)=Z(yi−1) for all y∈(yi−1,yi) for 1 ≤i≤ν. Since y7→ˆzyis uniformly continuous, we\ncan choose νlarge enough so that (4.3)–(4.5) imply\ncd(1−θ)£\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nZ(y)∥p11¢¤\n>θ+δ for all y∈¯I,\ncd(1−θ)£\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nZ(y)∥p01¢¤\n>1+δ for all y∈¯I,\n|Z(y)−Z(y′)| <ε′for all y,y′∈¯Isuch that |y−y′| <δ′,\nas claimed.\n4.2.Proof of Proposition 2.8. As in the proof of Proposition 2.6 in Section 3.3 we will first investigate an idealised\nscenario where we assume that the ground truth σis known. Then we will use the expansion property provided by\nLemma 3.3 to connect this idealised scenario with the actual steps of the algorithm.\nIn order to study the idealised scenario, for x∈V[i] and j∈[s] we introduce\nYx,j=¯¯©\na∈F[i+j−1]∩∂x:V1∩∂a⊆{x}ª¯¯, Zx,j=¯¯©\na∈F+[i+j−1]∩∂x:V1∩∂a⊆{x}ª¯¯,\nYx=sX\nj=1Yx,j, Zx=sX\nj=1Zx,j.\nThus, Yx,jis the number of untainted tests in compartment F[i+j−1] that contain x, i.e. test that do not contain\nanother infected individual. Moreover, Zx,jis the number of positively displayed untainted tests. Finally, Yx,Zx\nare the sums of these quantities on j∈[s]. The following lemma provides an estimate of the number of individuals\nxwith a certain value of Yx.\nLemma 4.1. w.h.p. for all 0≤Y≤∆and all i ∈[ℓ]we have\nX\nx∈V0[i]1{Yx=Y}≤nexp¡\n−∆DKL¡\nY/∆∥exp(−d)¢\n+o(∆)¢\n, (4.6)\nX\nx∈V1[i]1{Yx=Y}≤kexp¡\n−∆DKL¡\nY/∆∥exp(−d)¢\n+o(∆)¢\n. (4.7)\nProof. Let 1≤i≤ℓand consider any x∈V[i]. Further, obtain Gsc−xfrom Gscby deleting individual x(and, of\ncourse, removing xfrom all tests). Additionally, obtain G′\nscfrom Gsc−xby re-inserting xand assigning xto∆/s\nrandom tests in the compartments F[i+j−1] for j∈[s] as per the construction of the spatially coupled test design.\nThen the random test designs GscandG′\nscare identically distributed.\nLetEbe the event that Gscenjoys properties G1andG2from Proposition 2.3. Then Proposition 2.3 shows that\nP[E]=1−o(n−2). (4.8)\nMoreover, given Efor every j∈[s] the number of tests in F[i+j−1] that contain no infected individual aside from\nxsatisfies\nX\na∈F[i+j−1]1{∂a∩V1\\ {x}= ;}=(1+O(n−Ω(1)))m\nℓexp(−d); (4.9)\nthis follows from the bounds on F0[i+j−1] provided by G2and the fact that discarding xcan change the numbers\nof actually positive/negative tests by no more than ∆.\nNow consider the process of re-inserting xto obtain G′\nsc. Then (4.9) shows that given Ewe have\nYx,j∼Hypµm\nℓ+O(1),(1+O(n−Ω(1)))m\nℓexp(−d),∆\ns¶\n(j∈[s]).\nThese hypergeometric variables are mutually independent given Gsc−x. Therefore, Lemma 1.6 implies that on E,\nP[Yx=Y|Gsc−x]≤exp¡\n−∆DKL¡\nY/∆∥exp(−d)¢\n+o(∆)¢\n. (4.10)\nThis estimate holds independently of the infection status σx. Thus, the assertion follows from (4.8), (4.10) and\nMarkov’s inequality. □\nAs a next step we argue that for cbeyond the threshold cex,1(d,θ) the function Zfrom Proposition 2.7 separates\nthe infected from the uninfected individuals w.h.p.\nLemma 4.2. Assume that c >c∗(d,θ)+ε. LetI=(l,r),δ>0be the interval and the number from Proposition 2.7,\nchoose ε′>0sufficiently small and let δ′,Zbe such that Z1–Z4are satisfied. Then w.h.p. the following statements\nare satisfied.\n26(i) For all x ∈V1we have Yx/∆∈(l+ε′,r−ε′)andZx/∆>Z(Yx/∆)+3ε′.\n(ii) For all x ∈V0with Yx/∆∈Iwe have Zx/∆<Z(Yx/∆)−3ε′.\nProof. LetEbe the event that the bounds (4.6)–(4.7) hold for all 0 ≤Y≤∆. Then (4.7) and Proposition 2.7, Z1show\nthat w.h.p. Yx/∆∈(l+ε′,r−ε′) for all x∈V1, provided ε′>0 is small enough. Moreover, for a fixed 0 ≤Y≤∆such\nthat Y/∆∈Iand i∈[ℓ] letX1(Y) be the number of variables x∈V1[i] such that Yx=YandZx≤∆Z(Y/∆)+3ε′∆.\nSince xitself is infected, all tests a∈∂xare actually positive. Therefore, ais displayed positively with probability\np11. As a consequence, Lemma 1.5 shows that\nP£\nZx≤∆Z(Yx/∆)+3ε′∆|Yx=Y¤\n≤exp¡\n−Y D KL¡\nZ(Y/∆)+3ε′∆/Y∥p11¢\n+o(∆)¢\n. (4.11)\nCombining (4.7) and (4.11), recalling that k= ⌈nθ⌉and choosing ε′>0 sufficiently small, we obtain\nE\"\nX\nx∈V1[i]1©\nYx=Y,Zx≤∆Z(Yx/∆)+3ε′∆ª\n|E#\n≤kexp¡\n−∆DKL¡\nY/∆∥exp(−d)¢\n−Y D KL¡\nZ(Y/∆)+3ε′∆/Y∥p11¢\n+o(∆)¢\n(4.12)\n≤n−Ω(1)[due to Proposition 2.7, Z2]. (4.13)\nTaking a union bound on the O(ln2n) possible combinations ( i,Y), we see that (i) follows from (4.13). A similar\nargument based on Proposition 2.7, Z3yields (ii). □\nProof of Proposition 2.8. For t=1...⌈lnn⌉consider the set of misclassified individuals after t−1 iterations:\nMt=©\nx∈V[s+1]∪··· V[ℓ] :τ(t)\nx̸=σxª\n.\nPropositions 2.5 and 2.6 show that w.h.p. the size of the initial set satisfies\n|M1|≤kexp¡\n−Ω(ln1/8n)¢\n. (4.14)\nWe are going to argue by induction that |Mt|decays geometrically. Apart from the bound (4.14), this argument\ndepends on only two conditions. First, that the random graph Gscindeed enjoys the expansion property from\nLemma 3.3. Second, that (i)–(ii) from Lemma 4.2 hold. Let Ebe the event that these two conditions are satisfied,\nand that (4.14) holds. Propositions 2.5 and 2.6 and Lemmas 3.3 and 4.2 show that P[E]=1−o(1).\nTo complete the proof we are going to show by induction on t≥2 that on E,\n|Mt| ≤ |Mt−1|/3. (4.15)\nIndeed, consider the set\nM∗\nt=(\nx∈V[s+1]∪··· V[ℓ] :X\na∈∂x\\F[0]|∂a∩Mt−1\\{x}|≥∆/lnln n)\n.\nSince by (4.14) and induction we know that |Mt−1|≤kexp¡\n−Ω(ln1/8n)¢\n, the expansion property from Lemma 3.3\nimplies that M∗\nt≤Mt−1/3. Therefore, to complete the proof of (4.15) it suffices to show that Mt⊆M∗\nt.\nTo see this, suppose that x∈Mt.\nCase 1: x∈V1but Yx(τ(t−1))/∆̸∈I:Lemma 4.2 (i) ensures that Yx/∆∈(l+ε′,r−ε′). Therefore, the case\nYx(τ(t−1))/∆̸∈Ican occur only if at least ε′∆tests a∈∂xcontain a misclassified individual x′∈Mt−1.\nHence, x∈M∗\nt.\nCase 2: x∈V1and Yx(τ(t−1))/∆̸∈Ibut Zx(τ(i))/∆≤Z(Yx(τ(i))/∆):by Lemma 4.2 (i) we have Zx/∆>Z(Yx/∆)+\n2ε′. Thus, if Zx(τ(t−1))/∆≤Z(Yx(τ(t−1))/∆), then by the continuity property Z4we have |Yx(τ(t−1))−Yx| >\nε′∆. Consequently, as in Case 1 we have x∈M∗\nt.\nCase 3: x∈V0:as in the previous cases, due to Z4and Lemma 4.2 (ii) the event x∈Mtcan occur only if\n|Yx−Yv(τ(t−1))| >ε′∆. Thus, x∈M∗\nt.\nHence, Mt⊆M∗\nt, which completes the proof. □\n275. L OWER BOUND ON THE CONSTANT -COLUMN DESIGN\n5.1.Proof of Proposition 2.10. The following lemma is an adaptation of Proposition 2.3 ( G2) toGcc.\nLemma 5.1. The random graph Gccenjoys the following properties with probability 1−o(n−2):\nmexp(−d)p00−p\nmln3n≤¯¯F−\n0¯¯≤mexp(−d)p00+p\nmln3n, (5.1)\nmexp(−d)p01−p\nmln3n≤¯¯F+\n0¯¯≤mexp(−d)p01+p\nmln3n, (5.2)\nm(1−exp(−d))p10−p\nmln3n≤¯¯F−\n1¯¯≤m(1−exp(−d))p10+p\nmln3n, (5.3)\nm(1−exp(−d))p11−p\nmln3n≤¯¯F+\n1¯¯≤m(1−exp(−d))p11+p\nmln3n. (5.4)\nThe proof of Lemma 5.1 is similar to that of Proposition 2.3 (see Section 3.1).\nProof of Proposition 2.10. The definition (2.2) of the weight functions ensures that\nlnψGcc,σ′′(σ)= |F−\n0|lnp00+|F+\n0|lnp01+|F−\n1|lnp10+|F+\n1|lnp11.\nSubstituting in the estimates from (5.1)–(5.4) completes the proof. □\n5.2.Proof of Proposition 2.11. LetXr(Y) be the set of individuals x∈Vrsuch that\nX\na∈∂x1{∂a\\ {x}⊆V0}=Y.\nHence, xparticipates in precisely Ytests that do not contain another infected individual.\nLemma 5.2. Let y∈Y(c,d,θ)be such that y ∆is an integer . Then w.h.p. we have\n|X0(y∆)| =nexp¡\n−∆DKL¡\ny∥exp(−d)¢\n+o(∆)¢\n, (5.5)\n|X1(y∆)| =kexp¡\n−∆DKL¡\ny∥exp(−d)¢\n+o(∆)¢\n. (5.6)\nProof. Let Y=y∆and let Ebe the event that the bounds (5.1)–(5.4) hold. We begin by computing E|X1(Y)|. By\nexchangeability we may condition on the event S={σx1= ··· = σxk=1}, i.e. precisely the first kindividuals are\ninfected. Hence, by the linearity of expectation it suffices to prove that\nP[x1∈X1(Y)|E,S]=exp¡\n−∆DKL¡\ny∥exp(−d)¢\n+o(∆)¢\n. (5.7)\nLetG′=Gcc−x1be the random design without x1and let F′\n0be the set of actually negative tests of G′. Given\nm′\n0= |F′\n0|the number of tests a∈∂x1such that ∂a\\ {x1}⊆V0has distribution Hyp( m,m′\n0,∆), because x1joins\nprecisely ∆tests independently of all other individuals. Hence, (1.19) yields\nP£\nx1∈X1(Y)|S,m′\n0¤\n=Ã\nm′\n0\nY!Ã\nm−m′\n0\n∆−Y!Ã\nm\n∆!−1\n. (5.8)\nExpanding (5.8) via Stirling’s formula and using the bounds (5.1)–(5.2), we obtain (5.7), which implies that\nE£\n|X1(y∆)| |E¤\n=kexp¡\n−∆DKL¡\ny∥exp(−d)¢\n+o(∆)¢\n. (5.9)\nSince the above argument does not depend on the infection status of x1, analogously we obtain\nE£\n|X0(y∆)| |E¤\n=(n−k)exp¡\n−∆DKL¡\ny∥exp(−d)¢\n+o(∆)¢\n. (5.10)\nTo turn (5.9)–(5.10) into “with high probability”-bounds we resort to the second moment method. Specifically,\nwe are going to show that\nE£\n|X1(y∆)|(|X1(y∆)|−1)|E¤\n∼E£\n|X1(y∆)| |E¤2, (5.11)\nE£\n|X0(y∆)|(|X0(y∆)|−1)|E¤\n∼E£\n|X0(y∆)| |E¤2. (5.12)\nThen the assertion is an immediate consequence of (5.9)–(5.12) and Lemma 5.1.\nFor similar reasons as above it suffices to prove (5.11). More precisely, we merely need to show that\nP[x1,x2∈X1(Y)|E,S]∼P[x1∈X1(Y)|E,S]2. (5.13)\n28To compute the probability on the l.h.s. obtain G′′=Gcc−x1−x2by removing x1,x2. Let m′′\n0be the number of\nactually negative tests of G′′. We claim that on E,\nP£\nx1,x2∈X1(Y)|S,m′′\n0¤\n=∆−YX\nI=0Ã\nm′′\n0\nI!Ã\nm′′\n0\nY!Ã\nm′′\n0−Y\nY!Ã\nm−m′′\n0\n∆−Y−I!Ã\nm−m′′\n0\n∆−Y−I!Ã\nm\n∆!−2\n. (5.14)\nIndeed, we first choose 0 ≤I≤∆−Ytests that are actually negative in G′′that both x1,x2will join. Observe that\nthese tests adonot satisfy ∂a\\ {x1/2}⊆V0. Then we choose Ydistinct actually negative tests for x1and x2to join.\nFinally, we choose the remaining ∆−Y−Itests for x1,x2among the actually positive tests of G′′.\nSince on Ethe total number m′′\n0is much bigger than ∆, it is easily verified that the sum (5.14) is dominated by\nthe term I=0; thus, on Ewe have\nP£\nx1,x2∈X1(Y)|S,m′′\n0¤\n=(1+O(∆2/m))Ã\nm′′\n0\nY!Ã\nm′′\n0−Y\nY!Ã\nm−m′′\n0\n∆−Y!2Ã\nm\n∆!−2\n. (5.15)\nFurthermore, a careful expansion of the binomial coefficients from (5.15) shows that uniformly for all m′\n0,m′′\n0=\nmexp(−d)+O(pmln3n) we have\nP£\nx1,x2∈X1(Y)|S,m′′\n0=m′′\n0¤\nP£\nx1∈X1(Y)|S,m′\n0=m′\n0¤2∼1,\nwhence we obtain (5.13). A similar argument applies to |X0(Y)|. □\nAs a next step consider the set Xr(Y,Z) of all x∈Xr(Y) such that\nX\na∈∂x∩F+1{∂a\\ {x}⊆V0}=Z.\nCorollary 5.3. Let y∈Y(c,d,θ)be such that y ∆is an integer and let z ∈(p01,p11)be such that z ∆is an integer and\nsuch that (2.41) –(2.42) are satisfied. Then w.h.p. we have\n|X0(y∆,z∆)| =nexp¡\n−∆¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p01¢¢\n+o(∆)¢\n, (5.16)\n|X1(y∆,z∆)| =kexp¡\n−∆¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p11¢¢\n+o(∆)¢\n. (5.17)\nProof. Let Y=y∆and Z=z∆. We deal with |X0(y∆,z∆)|and|X1(y∆,z∆)|by two related but slightly different\narguments. The computation of |X1(y∆,z∆)|is pretty straightforward. Indeed, Lemma 5.2 shows that w.h.p. the\nsetX1(Y) has the size displayed in (5.6). Furthermore, since (1.2) provides that tests are subjected to noise inde-\npendently, Lemma 1.5 shows that\nE[|X1(Y,Z)| | |X1(Y)|]= |X1(Y)|exp¡\n−∆¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p11¢¢\n+o(∆)¢\n. (5.18)\nMoreover, we saw in the proof of Lemma 5.2, any x1,x2∈X1(Y) have disjoint sets of untainted tests. Hence, in\nperfect analogy to (5.18) we obtain\nE[|X1(Y,Z)|(|X1(Y,Z)|−1)| |X1(Y)|]= |X1(Y)|(|X1(Y)|−1)exp¡\n−2∆¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p11¢¢\n+o(∆)¢\n.\n(5.19)\nThus, (5.17) follows from (5.18)–(5.19) and Chebyshev’s inequality.\nLet us proceed to prove (5.16). As in the case of |X1(y∆,z∆)|we obtain\nE[|X0(Y,Z)| | |X0(Y)|]= |X0(Y)|exp¡\n−∆¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p01¢¢\n+o(∆)¢\n, (5.20)\nHence, as in (5.10) from the proof of Lemma 5.10,\nE[|X0(Y,Z)| |E]=nexp¡\n−∆¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p01¢¢\n+o(∆)¢\n. (5.21)\nWith respect to the second moment calculation, it is not necessarily true that xi,xj∈X0(Y,Z) with k<i<j≤n\nhave disjoint sets of untainted tests. Thus, as in the expression (5.14) let m′′\n0be the number of actually negative tests\n29ofG′′=Gcc−xi−xjand introduce 0 ≤I≤∆to count the untainted tests that xi,xjhave in common. Additionally,\nwrite 0 ≤I1≤min{ I,Z} for the number of common untainted tests that display a negative result. Then\nP£\nxi,xj∈X0(Y,Z)|S,m′′\n0¤\n=X\nI,I1Ã\nm′′\n0\nI!Ã\nm′′\n0−I\nY−I!Ã\nm′′\n0−Y\nY−I!Ã\nm−m′′\n0\n∆−Y!2Ã\nm\n∆!−2\n·Ã\nI\nI1!\npI−I1\n00pI1\n01\"Ã\nY−I\nZ−I1!\npY−Z−I+I1\n00pZ−I1\n01#2\n. (5.22)\nAs in the proof of Lemma 5.2 it is easily checked that the summand I=I1=0 dominates (5.22), and that therefore\nE[|X0(Y,Z)|(|X0(Y,Z)|−1)|E]∼E[|X0(Y,Z)| |E]2. (5.23)\nThus, (5.16) follows from (5.21), (5.23) and Chebyshev’s inequality. □\nProof of Proposition 2.11. By continuity we can find y,zthat satisfy (2.41)–(2.42) such that y∆,z∆are integers,\nprovided that nis large enough. Now, if (2.41)–(2.42) are satisfied, then Corollary 5.3 shows that\n|X1(y∆,z∆)×X0(y∆,z∆)| =nΩ(1).\nHence, take any pair ( v,w)∈X1(y∆,z∆)×X0(y∆,z∆). Then { a∈∂v:∂a\\{v}⊆V0} and { a∈∂w:a∈F0} are disjoint,\nbecause v∈V1. Therefore, any such pair ( v,w) satisfies (2.43). □\n5.3.Proof of Proposition 2.12. We are going to lower bound the partition function ZGcc,σ′′by way of a moment\ncomputation. To this end we are going to couple the constant column design ( Gcc,σ′′) with the displayed test\nresults σ′′with another random pair ( Gcc,σ′′′) where the test results indicated by the vector σ′′′are purely random,\ni.e. do not derive from an actual vector σof infected individuals. One can think of ( Gcc,σ′′′) as a ‘null model’ .\nConversely, in the language of random constraint satisfaction problems [12], ultimately ( Gcc,σ′′) will turn out to\nbe the ‘planted model’ associated with ( Gcc,σ′′′).\nHence, let m+be the number ∥σ′′∥1of positively displayed tests of ( Gcc,σ′′). Moreover, for a given integer 0 ≤\nm+≤mletσ′′′∈{0,1}Fbe a uniformly random vector of Hamming weight m+, drawn independently of Gcc,σ,σ′′.\nIn other words, in the null model ( Gcc,σ′′′) we simply choose a set of uniformly random tests to display positively.\nLet ˆF+={a∈F:σ′′′\na=1} and ˆF−=F\\F+. Moreover, just as in (2.2) define weight functions\nψGcc,σ′′′,a: {0,1}∂a→R≥0, σ∂ai7→(\n1{∥σ∥1=0}p00+1{∥σ∥1>0}p10 ifσ′′′\na=0,\n1{∥σ∥1=0}p01+1{∥σ∥1>0}p11 ifσ′′′\na=1.(5.24)\nIn addition, exactly as in (2.3)–(2.4) let\nψGcc,σ′′′(σ)=1{∥σ∥1=k}Y\na∈FψGcc,σ′′′,a(σ∂a), ZGcc,σ′′′=X\nσ∈{0,1}VψGcc,σ′′′(σ),µGcc,σ′′′(σ)=ψGcc,σ′′′(σ)/ZGcc,σ′′′.\n(5.25)\nWe begin by computing the mean of the partition function (aka. the ‘annealed average’).\nLemma 5.4. For any 0≤m+≤m we have E£\nZGcc,σ′′′¤\n=¡n\nk¢¡m\nm+¢−1P£\nm+=m+¤\n.\nProof. Writing out the definitions of Gcc,σ′′′, we obtain\nE£\nZGcc,σ′′′¤\n=Ã\nm\nm+!−1X\nGX\nσ′′∈{0,1}F:∥σ′′∥1=m+\nσ∈{0,1}:∥σ∥1=kP[Gcc=G]ψGcc,σ′′(σ)\n=Ã\nm\nm+!−1X\nG,σ′′,σ:∥σ′′∥1=m+,∥σ∥1=kP[Gcc=G]P£\nσ′′=σ′′|Gcc=G,σ=σ¤\n[by (5.24)–(5.25)]\n=Ã\nn\nk!Ã\nm\nm+!−1X\nG,σ′′,σ:∥σ′′∥1=m+,∥σ∥1=kP[Gcc=G]P[σ=σ]P£\nσ′′=σ′′|Gcc=G,σ=σ¤\n=Ã\nn\nk!Ã\nm\nm+!−1X\nσ′′:∥σ′′∥1=m+P£\nσ′′=σ′′¤\n=Ã\nn\nk!Ã\nm\nm+!−1\nP£\nm+=m+¤\n,\nas claimed. □\n30As a next step we sort out the relationship of the null model and of the ‘real’ group testing instance.\nLemma 5.5. Let0≤m+≤m be an integer . Then for any G and any σ′′∈{0,1}Fwith∥σ′′∥1=m+we have\nP£\nGcc=G,σ′′=σ′′|m+=m+¤\n=P£\nGcc=G,σ′′′=σ′′¤\nZG,σ′′\nE£\nZGcc,σ′′′¤ . (5.26)\nProof. We have\nP£\nGcc=G,σ′′=σ′′|m+=m+¤\n=X\nσ:∥σ∥1=kP[Gcc=G]P£\nσ′′=σ′′|Gcc=G,σ=σ¤\n¡n\nk¢\nP[m+=m+]\n=X\nσ:∥σ∥1=kP[Gcc=G]ψG,σ′′(σ)¡n\nk¢\nP[m+=m+][by (2.2)–(2.3)]\n=P[Gcc=G]ZG,σ′′¡n\nk¢\nP[m+=m+][by (2.4)]\n=P[Gcc=G]P£\nσ′′′=σ′′¤\nZG,σ′′\n¡n\nk¢¡m\nm+¢−1P[m+=m+][asσ′′′is uniformly random]\n=P£\nGcc=G,σ′′′=σ′′¤\nZG,σ′′\nE£\nZGcc,σ′′′¤ [by Lemma 5.4],\nas claimed. □\nCombining Lemmas 5.4–5.5, we obtain the following lower bound on ZGcc,σ′′.\nCorollary 5.6. Let0≤m+≤m be an integer . For any δ>0we have\nP\"\nZGcc,σ′′<δÃ\nn\nk!Ã\nm\nm+!−1\nP£\nm+=m+¤\n|m+=m+#\n<δ.\nProof. Lemmas 5.4 and 5.5 yield\nP\"\nZGcc,σ′′<δÃ\nn\nk!Ã\nm\nm+!−1\nP£\nm+=m+¤\n|m+=m+#\n=X\nG,σ′′:∥σ′′∥1=m+1{ZG,σ′′<δ¡n\nk¢¡m\nm+¢−1P£\nm+=m+¤\n}ZG,σ′′P£\nGcc=G,σ′′′=σ′′|m+=m+¤\n¡n\nk¢¡m\nm+¢\nP[m+=m+]<δ,\nas desired. □\nProof of Proposition 2.12. Since Proposition 2.3 shows that P£\nm+=mq+\n0+O(pmln3n)¤\n=1−o(1), the proposition\nfollows immediately from Corollary 5.6. □\nREFERENCES\n[1] A. Alaoui, A. Ramdas, F . Krzakala, L. 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Proc. IEEE International\nSymposium on Information Theory Proceedings (2011) 1489–1493.\n[33] B. Teo, J. Scarlett: Noisy adaptive group testing via noisy binary search: IEEE Transactions in Information Theory, 68(2022) 3340–3353.\n[34] L. Wang, X. Li, Y. Zhang, K. Zhang: Evolution of scaling emergence in large-scale spatial epidemic spreading. PLoS ONE 6(2011).\n[35] L. Zdeborová, F . Krzakala: Statistical physics of inference: thresholds and algorithms. Advances in Physics 65(2016) 453–552.\nAPPENDIX A. P ROOF OF THEOREM 1.2\nThe basic idea is to compute the mutual information of σandσ′′. What makes matters tricky is that we are dealing\nwith the adaptive scenario where tests may be conducted one by one. To deal with this issue we closely follow\nthe arguments from [16, 3]. Furthermore, the displayed test results are obtained by putting the actual test results\nthrough the noisy channel.\nAs a first step we bound the mutual information between σandσ′′from above under the assumption that\nthe statistician applies a adaptive scheme where the next test to be conducted depends deterministically on the\npreviously displayed test results. Let mbe the total number of tests that are conducted.\nLemma A.1. For a deterministic adaptive algorithm we have I (σ,σ′′)≤m/cSh.\nProof. Letσ′be the vector of actual test results. Then\nI(σ,σ′′)=X\ns,s′′P£\nσ′′=s′′|σ=s¤\nP[σ=s]lnP£\nσ′′=s′′|σ=s¤\nP[σ′′=s′′]\n=X\ns,s′,s′′P£\nσ′′=s′′|σ′=s′¤\nP£\nσ′=s′|σ=s¤\nP[σ=s]lnP£\nσ′′=s′′|σ′=s′¤\nP£\nσ′=s′|σ=s¤\nP[σ′′=s′′]\n=I(σ′′,σ′)−H(σ′|σ)≤I(σ′′,σ′).\n32Furthermore,\nI(σ′,σ′′)=X\ns′′,s′P£\nσ′′=s′′,σ′=s′¤\nlnP£\nσ′′=s′′,σ′=s′¤\nP[σ′′=s′′]P[σ′=s′].\nSince the tests are conducted adaptively, we obtain\nP£\nσ′′=s′′|σ′=s′¤\n=mY\ni=1Ph\nσ′′\ni=s′′\ni| ∀j<i:σ′′\nj=s′′\nj,σ′\ni=s′\nii\n.\nHence,\nI(σ′,σ′′)=mX\ni=1X\ns′′\n1,...,s′′\ni,s′\niPh\n∀j<i:σ′′\nj=s′′\nj,σ′\ni=s′\nii\n·Ph\nσ′′\ni=s′′\ni| ∀j<i:σ′′\nj=s′′\nj,σ′\ni=s′\nii\nlnPh\nσ′′\ni=s′′\ni| ∀j<i:σ′′\nj=s′′\nj,σ′\ni=s′\nii\nPh\nσ′′\ni=s′′\ni| ∀j<i:σ′′\nj=s′′\nji .\nIn the last term σ′\niis a Bernoulli random variable (whose distribution is determined by σ′′\njforj<i), and σ′′\niis the\noutput of that variable upon transmission through our channel. Furthermore, the expression in the second line\nabove is the mutual information of these quantities. Hence, the definition of the channel capacity implies that\nI(σ′,σ′′)≤m/cSh. □\nProof of Theorem 1.2. As a first step we argue that it suffices to investigate deterministic adaptive group testing\nalgorithms (under the assumption that the ground truth σis random). Indeed, a randomised adaptive algorithm\nA(·) can be modeled as having access to a (single) sample ωfrom a random source that is independent of σ. Now,\nif we assume that for an arbitrarily small δ>0 we have\nE°°A(σ′′,ω)−σ°°\n1<δk,\nwhere the expectation is on both ωandσ, then there exists some outcome ωsuch that\nE°°A(σ′′,ω)−σ°°\n1<δk,\nwhere the expectation is on σonly.\nThus, assume that A(·) is deterministic. We have I(σ,σ′′)=H(σ)−H(σ|σ′′). Furthermore, H(σ)∼kln(n/k).\nHence, Lemma A.1 yields\nH(σ|σ′′)=H(σ)−I(σ,σ′′)≥H(σ)−m/cSh,\nwhich implies the assertion. □\nAPPENDIX B. P ROOF OF LEMMA 3.3\nThis proof is a straightforward adaption of [10, proof of Lemma 4.16]. Fix T⊆Vof size t= |T| ≤exp(−lnαn)kas\nwell as a set R⊆Vof size r= ⌈t/λ⌉withλ=8lnln n. Let γ= ⌈lnβn⌉. Further, let U⊆F[1]∪···∪ F[ℓ] be a set of\nsizeγr≤u≤∆t. Additionally, let E(R,T,U) be the event that every test a∈Ucontains two individuals from R∪T.\nThen\nP\"\nR⊆(\nx∈V:X\na∈∂x\\F[0]1{T∩∂a\\{x}̸= ;}≥γ)#\n≤P[E(R,T,U)]. (B.1)\nHence, it suffices to bound P[E(R,T,U)].\nFor a test a∈Uthere are no more than¡r+t\n2¢\nways to choose distinct individuals xa,x′\na∈R∪T. Moreover, (3.1)\nshows that the probability of the event { xa,x′\na∈∂a} is bounded by (1 +o(1))(∆ℓ/(ms))2; in fact, this probability\nmight be zero if we choose an individual that cannot join adue to the spatially coupled construction of Gsc. Hence,\ndue to negative correlation\nP[E(R,T,U)]≤\"Ã\nr+t\n2!µ(1+o(1))∆ℓ\nms¶2#u\n.\n33Consequently, by the union bound the event E(r,t,u) that there exist sets R,T,Uof sizes |R| =r,|T| =t,|U| =u\nsuch that E(R,T,U) occurs has probability\nP[E(r,t,u)]≤Ã\nn\nr!Ã\nn\nt!Ã\nm\nu!\"Ã\nr+t\n2!µ(1+o(1))∆ℓ\nms¶2#u\n.\nHence, the bounds γt/λ≤γr≤u≤∆tyield\nP[E(r,t,u)]≤Ã\nn\nt!2Ã\nm\nu!\"Ã\n2t\n2!µ(1+o(1))∆ℓ\nms¶2#u\n≤³en\nt´2tµ2e∆2ℓ2t2\nms2u¶u\n≤·³en\nt´λ/γ2eλ∆2ℓ2t\nγms2¸u\n≤·³en\nt´λ/γ\n·tln4n\nm¸u\n[due to (2.13), (2.14)].\nFurther, since γ=Ω(lnβn) and m=Ω(klnn) while t≤exp(−lnαn)kandα+β>1, we obtain\nP[E(r,t,u)]≤exp(−ulnΩ(1)n).\nThus, summing on 1 ≤t≤exp(−lnαn)k,γr≤u≤∆tand recalling r= ⌈t/λ⌉, we obtain\nX\nt,uP[E(r,t,u)]≤X\nu≥1uexp(−ulnΩ(1)n)=o(1). (B.2)\nFinally, the assertion follows from (B.1) and (B.2).\nAPPENDIX C. P ARAMETER OPTIMISATION FOR THE BINARY SYMMETRIC CHANNEL\nLetD(θ,p)={d>0 : max{ cex,1(d,θ),cex,2(d)}=cex(θ)} be the set of those dwhere the minimum in the optimisation\nproblem (1.10) is attained. The goal in this section is to show that for the binary symmetric channel this minimum\nis not always attained at the information-theoretic value dSh=ln2 that minimises the term cex,2(d) from (1.10).\nProposition C.1. For any binary symmetric channel pgiven by 0<p01=p10<1/2 there is ˆθ(p01)such that for all\nθ>ˆθ(p01)we have d Sh=ln(2)̸∈D(θ,p).\nIn order to show that dShis suboptimal, we will use the following analytic bound on cex,1(d,θ) for binary sym-\nmetric channels.\nLemma C.2. For a binary symmetric channel pgiven by p 01<1/2 we have\nθ\n−(1−θ)dln(1−(1−a)exp(−d))<cex,1(d,θ)≤1\n−(1−θ)dln(1−(1−a)exp(−d)), where (C.1)\na=exp¡\n−DKL¡\n1/2∥p01¢¢\n=2p\np01(1−p01). (C.2)\nThe proof of Lemma C.2 uses the following fact, which can be verified by elementary calculus.\nFact C.3. For all d >0,0<p<1and 0≤z≤1we have\nargmin\n0≤y≤1©\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p¢ª\n=aexp(−d)\n1−(1−a)exp(−d), where a=exp¡\n−DKL¡\nz∥p¢¢\n,and\nmin\n0≤y≤1©\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p¢ª\n= − ln(1−(1−a)exp(−d)).\nProof of Lemma C.2. Fact C.3 shows that\nmin\n0≤y≤1©\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\n1/2∥p01¢ª\n= − ln(1−(1−a)exp(−d)).\nwith aas in (C.2). Hence, it is sufficient to show that\nθ<min\n0≤y≤1©\ncex,1(d,θ)·d(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\n1/2∥p01¢¢ª\n≤1. (C.3)\nLet us first prove the upper bound. Choose ˆc(d,θ) such that\nmin\n0≤y≤1©\nˆc(d,θ)d(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\n1/2∥p01¢¢ª\n=1;\nwe need to show that cex,1(d,θ)≤ˆc(d,θ). By the channel symmetry and because p01<1/2<p11, we have\nDKL¡\n1/2∥p01¢\n=DKL¡\n1/2∥p11¢\n≤DKL¡\np01∥p11¢\n.\n34Therefore, the definition of ˆc(d,θ) ensures that for all y∈[0,1] we have\nˆc(d,θ)·d(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\np01∥p11¢¢\n≥ˆc(d,θ)·d(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\n1/2∥p01¢¢\n≥1>θ,\nand thus ˆc(d,θ)≥cex,0(d,θ). Once again by channel symmetry and the definitions of ˆc(d,θ) and z(y) (see (1.8)),\nwe see that DKL¡z(y)∥p11¢\n≤DKL¡\n1/2∥p11¢\n=DKL¡\n1/2∥p01¢\nand hence DKL¡z(y)∥p01¢\n≥DKL¡\n1/2∥p01¢\nfor all y.\nConsequently, the definition of ˆc(d,θ) ensures we see that for all y∈[0,1],\nˆc(d,θ)·d(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡z(y)∥p01¢¢\n≥ˆc(d,θ)·d(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\n1/2∥p01¢¢\n≥1.\nHence, ˆc(d,θ)≥cex,1(d,θ), which is the right inequality in (C.3).\nMoving on to the lower bound, choose ˇc(d,θ) such that\nmin\n0≤y≤1©\nˇc(d,θ)d(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\n1/2∥p11¢¢ª\n=θ; (C.4)\nwe need to show that cex,1(d,θ)>ˇc(d,θ). The y=ˆywhere the minimum (C.4) is attained satisfies ˆy∈Y(c,d,θ), be-\ncause DKL¡\n1/2∥p11¢\n>0 (due to p11>1/2). Moreover, z(ˆy)=1/2 by the definition of z(·). But since DKL¡\n1/2∥p01¢\n=\nDKL¡\n1/2∥p11¢\nby symmetry of the channel, we have\nˇc(d,θ)·d(1−θ)¡\nDKL¡\nˆy∥exp(−d)¢\n+ˆyD KL¡z(ˆy)∥p01¢¢\n=θ<1.\nHence, we obtain ˇc(d,θ)<cex,1(d,θ), which is the left inequality in (C.3). □\nTo complete the proof of Proposition C.1 we need a second elementary fact.\nFact C.4. The function f (x,p)=ln(x)ln(1−px)is concave in its first argument for x ,p∈(0,1) , and for any given\np∈(0,1) , any x maximizing f (x,p)is strictly less than1\n2.\nProof of Proposition C.1. We reparameterise the bounds on cex,1(d,θ) from Lemma C.2 in terms of exp( −d), ob-\ntaining\nθ\n−(1−θ)f(exp(−d),1−a)<cex,1(d,θ)≤1\n−(1−θ)f(exp(−d),1−a), where\nf(x,p)=ln(x)ln(1−px), a=exp¡\n−DKL¡\n1/2∥p01¢¢\n.\nAs 0<a<1 Fact C.4 shows that any xmaximizing f(x,1−a) is strictly less than 1/2. Hence, for ˆd>ln(2) minimizing\nf(exp(−d),1−a), we have f(exp(−d),1−a)<f(1/2,1 −a)=f(exp(−dSh),1−a). In particular, the value\nˆθ(p01)=inf(\n0<θ<1 :cex,2(ˆd)\nθ<1\n−ˆd(1−θ)ln(1−(1−a)exp(−ˆd))<θ\n−dSh(1−θ)ln(1−(1−a)exp(−dSh)))\nis well defined. Hence, for all θ>ˆθ(p01) we have\nmax©\ncex,1(ˆd,θ),cex,2(ˆd)ª\n=cex,1(ˆd,θ)<cex,1(dSh,θ)=max©\ncex,1(dSh,θ),cex,2(dSh)ª\n,\nand thus dSh̸∈D(θ,p). □\nAPPENDIX D. P ARAMETER OPTIMISATION FOR THE Z-CHANNEL\nMuch as in Appendix C the goal here is to show that also for the Z-channel the value dShfrom (1.17) at which\ncex,2(d) from (1.10) attains its minimum is not generally the optimal choice to minimise max{ cex,1(d,θ),cex,2(d)}\nand thus obtain the optimal bound cex(θ). To this end we derive the explicit formula (1.15) for cex,1(d,θ); the\nderivation of the second formula (1.16) is elementary.\nProposition D.1. For a Z -channel pgiven by p 01=0and 0<p11<1we have c ex,1(d,θ)=θ\n−(1−θ)dln(1−exp(−d)p11).\nProof. We observe that for a Z-channel we have cex,1(d,θ)=cex,0(d,θ). Indeed, fix any c>cex,0(d,θ). Then by\nthe definitions (1.7) of cex,0(d,θ) and of z(y) we have z(y)>p01. Since the Z-channel satisfies p01=0, the value\nDKL¡z(y)∥p01¢\ndiverges for all c>cex,0(d,θ), rendering the condition in the definition of cex,1(d,θ) void for all y>0.\nMoreover, since c>cex,0(d,θ), we also have 0 ̸∈Y(c,d,θ), and thus c≥cex,1(d,θ). Since cex,1(d,θ)≥cex,0(d,θ) by\ndefinition, this implies that cex,1(d,θ)=cex,0(d,θ) on the Z-channel.\nHence, it remains to verify that cex,1(d,θ)=cex,0(d,θ) has the claimed value. This is a direct consequence of\nFact C.3 (with z=p01and p=p11) in combination with the fact that DKL¡\np01∥p11¢\n=DKL¡\n0∥p11¢\n= − ln(p10) and\nthus 1 −exp(−DKL¡\np01∥p11¢\n)=1−p10=p11. □\n35The following proposition shows that indeed d=dShis not generally the optimal choice. Recall that D(θ,p) is\nthe set of dwhere the minimum in the optimisation problem defining cex(θ) is attained for a given channel p.\nProposition D.2. For a Z -channel pgiven by p 01=0and any 0<p11<1there is a ˆθ(p11)<1such that for all\nθ>ˆθ(p11)we have d Sh̸∈D(θ,p).\nTowards the proof of Proposition D.2 we state the following fact whose proof comes down to basic calculus.\nFact D.3. For a Z -channel pwhere p 01=0and 0<p10<1, we have exp(−dSh(p))>1\n2.\nProof of Proposition D.2. First we show that d=dShdoes not minimize cex,1(d,θ) for all 0 <p11<1. We reparame-\nterise the expression for cex,1(d,θ) from Proposition D.1 in terms of exp( −d), obtaining\ncex,1(d,θ)=θ\n(1−θ)f(exp(−d),p11), where f(x,p11)=ln(x)ln(1−p11x).\nHence, for any given p11the value of cex,1(d,θ) is minimized when f(exp(−d),p11) is minimized. Using elementary\ncalculus we check that fis concave in its first argument on the interval (0,1), and that for all 0 <p11<1 the value\nofxmaximizing f(x,p11) is strictly smaller than1\n2(see Fact C.4). Now for any Z-channel with 0 <p10<1 we have\nexp(−dSh)>1\n2(using Fact D.3). Hence, for the Z-channel, dShdoes not minimize cex,1(d,θ).\nNow let d1be a dminimizing cex,1(d,θ); in particular, d1̸=dSh. Sinceθ\n1−θis increasing in θand unbounded as\nθ→1, the same holds for cex,1(d1,θ) and cex,1(dSh,θ). Hence, we may consider\nˆθ(p11)=inf©\n0<θ<1 :cex,1(d1,θ)>cex,2(d1),cex,1(dSh,θ)>cex,2(dSh)ª\n,\ncheck that it is strictly less than 1, and that by definition of ˆθ(p11), it holds for all θ>ˆθ(p11) that\nmax©\ncex,1(d1,θ),cex,2(d1)ª\n=cex,1(d1,θ)<cex,1(dSh,θ)=max©\ncex,1(dSh,θ),cex,2(dSh)ª\n.\nConsequently, dSh̸∈D(θ,p). □\nAPPENDIX E. C OMPARISON WITH THE RESULTS OF CHEN AND SCARLETT ON THE SYMMETRIC CHANNEL\nChen and Scarlett [8] recently derive the precise information-theoretic threshold of the constant column design\nGccfor the symmetric channel (i.e. p11=p00). The aim of this section is to verify that their threshold coincides with\nm∼cex(θ)kln(n/k), with cex(θ) from (1.11) on the symmetric channel. The threshold quoted in [8, Theorems 3\nand 4] reads m∼min d>0cCS(d,θ)kln(n/k), where cCS(d,θ) is the solution to the following optimisation problem:\ncCS(d,θ)=max{ cex,2(d,θ),cls(d,θ)}, where (E.1)\ncls(d,θ)=·\n(1−θ)d min\ny∈(0,1), z∈(0,1)max½1\nθ³\nDKL³\ny∥e−d+yD KL³\nz∥e−d´´´\n,\nmin\ny′∈[|y(2z−1)|,1]³\nDKL³\ny′∥e−d´\n+y′DKL¡\nz′(z,y,y′)∥p01¢´¾¸−1\n(E.2)\nz′(z,y,y′)=1\n2+y(2z−1)\n2y′.\nLemma E.1. For symmetric noise (p 00=p11) and any d >0we have c CS(d,θ)=max{ cex,1(d,θ),cex,2(d)}.\nProof. The definition (E.2) of cls(d) can be equivalently rephrased as follows:\ncls(d,θ)=inf©\nc>0 :∀y,z∈[0,1]∀y′∈£¯¯y(2z−1)¯¯,1¤\n:\n¡\nf1(c,d,θ,y,z)≥θ∨f2(c,d,θ,y,z,y′,z′(z,y,y′))≥1¢ª\n, where\nf1(c,d,θ,y,z)=cd(1−θ)¡\nDKL¡\ny∥exp(−d)¢\n+yD KL¡\nz∥p11¢¢\n,\nf2(c,d,θ,y,z,y′,z′)=cd(1−θ)¡\nDKL¡\ny′∥exp(−d)¢\n+y′DKL¡\nz′(z,y,y′)∥p01¢¢\n.\nConsequently, c<cls(d,θ) iff\n∃y,z,y′s.t.f1(c,d,θ,y,z)<θand f2(c,d,θ,y,z,y′,z′(z,y,y′))<1.\nRecall that c<cex,1(d,θ) iff\n∃y,zs.tf1(c,d,θ,y,z)<θand f2(c,d,θ,y,z,y,z)<1.\n36Since z′(z,y,y)=zwe conclude that if c<cex,1(d,θ), then c<cls(d,θ). Hence, cex,1(d,θ)≤cls(d,θ).\nTo prove the converse inequality, we are going to show that any c<cls(d,θ) also satisfies c<cex,1(d,θ). Hence,\nassume for contradiction that c<cls(d,θ) and c≥cex,1(d,θ). Then the following four inequalities hold:\ncd(1−θ)³\nDKL³\ny∥e−d´\n+yD KL¡\nz∥p11¢´\n<θ, cd(1−θ)³\nDKL³\ny∥e−d´\n+yD KL¡\nz∥p01¢´\n≥1, (E.3)\ncd(1−θ)³\nDKL³\ny′∥e−d´\n+y′DKL¡\nz′(z,y,y′)∥p01¢´\n<1, cd(1−θ)³\nDKL³\ny′∥e−d´\n+y′DKL¡\nz′∥p11¢´\n≥θ. (E.4)\nThe two inequalities on the left are a direct consequence of c<cls(d,θ). Note that if one of the right two inequalities\nis violated then c<cex,1(d,θ). Combining the inequalities of (E.3) leads us to\ncd(1−θ)³\nDKL³\ny∥e−d´\n+yD KL¡\nz∥p01¢´\n−cd(1−θ)³\nDKL³\ny∥e−d´\n+yD KL¡\nz∥p11¢´\n=cd(1−θ)y¡\nDKL¡\nz∥p01¢\n−DKL¡\nz∥1−p01¢¢\n=cd(1−θ)y(1−2z)lnµp01\n1−p01¶\n>1−θ. (E.5)\nThe remaining inequalities given by (E.4) imply\ncd(1−θ)³\nDKL³\ny′∥e−d´\n+y′DKL¡\nz′(z,y,y′)∥p01¢´\n−cd(1−θ)³\nDKL³\ny′∥e−d´\n+y′DKL¡\nz′∥p11¢´\n=cd(1−θ)y′¡\nDKL¡\nz′(z,y,y′)∥p01¢\n−DKL¡\nz′(z,y,y′)∥1−p01¢¢\n=cd(1−θ)y′y\ny′(1−2z)lnµp01\n1−p01¶\n<1−θ,\nwhich contradicts (E.5). □\nAMIN COJA-OGHLAN ,amin.coja-oghlan@tu-dortmund.de , TU D ORTMUND , FACULTY OF COMPUTER SCIENCE AND FACULTY OF MATH -\nEMATICS , 12 O TTO -HAHN -ST, D ORTMUND 44227, G ERMANY .\nMAXHAHN -KLIMROTH ,maximilian.hahnklimroth@tu-dortmund.de , TU D ORTMUND , FACULTY OF COMPUTER SCIENCE , 12 O TTO -\nHAHN -ST, D ORTMUND 44227, G ERMANY .\nLUKAS HINTZE ,lukas.rasmus.hintze@uni-hamburg.de , UNIVERSITÄT HAMBURG , FAKULTÄT FÜR MATHEMATIK , INFORMATIK UND NATUR -\nWISSENSCHAFTEN , INFORMATIK , VOGT -KÖLLN -STR. 30, 22527 H AMBURG , GERMANY .\nLENA KRIEG ,lena.krieg@tu-dortmund.de , TU D ORTMUND , FACULTY OF COMPUTER SCIENCE , 12 O TTO -HAHN -ST, D ORTMUND 44227,\nGERMANY .\nMAURICE ROLVIEN ,maurice.rolvien@tu-dortmund.de , TU D ORTMUND , FACULTY OF COMPUTER SCIENCE , 12 O TTO -HAHN -ST, D ORT-\nMUND 44227, G ERMANY .\nDOMINIK KAASER ,dominik@kaaser.at , TU H AMBURG , INSTITUTE FOR DATA ENGINEERING , E-19, B LOHMSTRASSE 15, 21079 H AMBURG ,\nGERMANY\nOLGA SCHEFTELOWITSCH ,olga.scheftelowitsch@tu-dortmund.de , TU D ORTMUND , FACULTY OF COMPUTER SCIENCE , 12 O TTO -HAHN -\nST, D ORTMUND 44227, G ERMANY .\n37"    },    {        "title": "2402.02901v1.Revisiting_the_role_of_cosmic_ray_driven_Alfvén_waves_in_pre_existing_magnetohydrodynamic_turbulence__I__Turbulent_damping_rates_and_feedback_on_background_fluctuations.pdf",        "content": "Astronomy &Astrophysics manuscript no. C2024 ©ESO 2024\nFebruary 6, 2024\nRevisiting the role of cosmic-ray driven Alfvén waves in\npre-existing magnetohydrodynamic turbulence\nI. Turbulent damping rates and feedback on background fluctuations\nSilvio Sergio Cerri\nUniversité Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France\ne-mail: silvio.cerri@oca.eu\nFebruary 6, 2024\nABSTRACT\nContext. Alfvén waves (AWs) excited by the cosmic-ray (CR) streaming instability (CRSI) are a fundamental ingredient for CR\nconfinement. The e ffectiveness of such “self-confinement” relies on a balance between CRSI growth rate and the damping mechanisms\nacting on quasi-parallel AWs excited by CRs. One relevant mechanism is the so-called “turbulent damping”, in which an AW packet\ninjected in pre-existing turbulence undergoes a cascade process due to its nonlinear interaction with fluctuations of the background.\nAims. The turbulent damping of an AW packet in pre-existing magnetohydrodynamic (MHD) turbulence is re-examined, revised, and\nextended to include most-recent theories of MHD turbulence that account for dynamic alignment and reconnection-mediated regime.\nThe case in which the role of feedback of CR-driven AWs on pre-existing turbulence is important will also be discussed.\nMethods. The so-called Elsässer formalism is employed. Particular attention is given to the role of a “nonlinearity parameter” χwthat\nestimates the strength of the nonlinear interaction between CR-driven AW packets and the background fluctuations. We point out the\ndifference between χwand the parameter χzthat instead describes the intrinsic strength of nonlinear interactions between pre-existing\nfluctuations. Turbulent damping rates of quasi-parallel AW packets and cosmic-ray feedback (CRF) are derived within this formalism.\nResults. When the strength of nonlinear interaction is properly taken into account, one finds that (i) the turbulent damping rate\nof quasi-parallel AWs in sub-Alfvénic turbulence depends on the background-fluctuations’ amplitude to the third power, hence is\nstrongly suppressed, and (ii) the dependence on the AW’s wavelength (and thus on the CR gyro-radius from which it is excited) is\ndifferent from what has been previously obtained. Finally, (iii) when dynamic alignment of cascading fluctuations and the possibility\nof a reconnection-mediated range is included in the picture, the turbulent damping rate exhibits novel regimes and breaks. Finally, a\ncriterion for CRF is derived and a simple phenomenological model of CR-modified scaling of background fluctuations is provided.\nKey words. turbulence – magnetohydrodynamics – cosmic rays\n1. Introduction\nTurbulent, magnetized plasmas permeates a wide range of space\nand astrophysical environments (e.g., Quataert and Gruzinov\n1999; Schekochihin and Cowley 2006; Brandenburg and Lazar-\nian 2013; Bruno and Carbone 2013; Ferrière 2020). Understand-\ning the properties of the turbulent cascade – how fluctuations’\nenergy is transferred from injection to dissipation scales, thus\nheating the plasma and also producing non-thermal particles in\nthe process – is a relevant task by itself, since it can elucidate the\nrole that turbulence plays in the dynamics and thermodynamics\nof several astrophysical systems. In fact, inspired by the semi-\nnal work of Kolmogorov (1941) in hydrodynamics, turbulence\nin magnetized plasmas has been the object of several theoretical\nefforts aimed at obtaining universal scaling for its fluctuations at\nlarge (“fluid”) magnetohydrodynamic (MHD) scales (e.g., Irosh-\nnikov 1963; Kraichnan 1965; Goldreich and Sridhar 1995; Ng\nand Bhattacharjee 1997; Galtier et al. 2000; Cho and Lazar-\nian 2002; Boldyrev 2006; Lazarian et al. 2012; Chandran et al.\n2015; Mallet et al. 2015; Boldyrev and Loureiro 2017; Mallet\net al. 2017; Cerri et al. 2022; Schekochihin 2022). At the same\ntime, these astrophysical environments are also populated with\ncosmic rays (CRs), i.e., charged particles with supra-thermal\n(relativisitc) energies that pervades the interstellar, intergalac-tic, and intracluster media (e.g., Brunetti and Jones 2014; Am-\nato and Blasi 2018; Faucher-Giguère and Oh 2023; Ruszkowski\nand Pfrommer 2023) and get scattered by magnetic-field fluctu-\nations (Ginzburg and Syrovatskii 1964; Berezinsky et al. 1990).\nWhile cosmic-ray transport partly depends upon the properties\nof pre-existing turbulence (e.g., Schlickeiser and Miller 1998;\nChandran 2000; Lerche and Schlickeiser 2001; Yan and Lazar-\nian 2002, 2008; Teufel et al. 2003; Shalchi and Schlickeiser\n2004; Fornieri et al. 2021; Lazarian and Xu 2021; Lemoine\n2023; Kempski et al. 2023), CRs can also generate their own\nscattering fluctuations through streaming instabilities (e.g., Kul-\nsrud and Pearce 1969; Lee 1972; Skilling 1975; Gary 1993; Bell\n2004; Amato 2011; Weidl et al. 2019a,b; Marcowith et al. 2021).\nThese cosmic-ray driven Alfvén-wave (AW) packets, however,\nwill also interact with pre-existing fluctuations of the turbulent\nenvironment in which they are generated. This interaction has\nbeen thought to represent one of their damping mechanisms,\ni.e., the so-called “turbulent damping”, a process for which a\nCR-generated AW packet is cascaded to dissipation by its non-\nlinear interaction with background fluctuations. This damping\nmechanism was originally proposed by Farmer and Goldreich\n(2004), and subsequently extended by Lazarian (2016) to ac-\ncount for di fferent regimes of background turbulence. However,\nan important parameter that has not been taken into account in\nArticle number, page 1 of 17arXiv:2402.02901v1  [physics.plasm-ph]  5 Feb 2024A&A proofs: manuscript no. C2024\nthese previous works is the strength of the nonlinear interaction\n(usually referred to as “nonlinearity parameter” χ) between the\nAW packet and pre-existing turbulent fluctuations. This parame-\nter is indeed di fferent from (and typically much smaller than) the\nnonlinear parameter describing the regime of background turbu-\nlence, which also needs to be taken into account (as done by\nLazarian 2016). We will show that taking this di fference into ac-\ncount completely changes the estimate of the turbulent damping\nrate—which is indeed almost always much lower than any rate\nderived previously. Moreover, such damping rate and its scal-\ning strongly depends upon the properties of background turbu-\nlence. In the previous literature, only what we can call “classic”\ntheories of MHD turbulence has been taken into account, i.e.,\nisotropic Kolmogorov-like turbulence (Kolmogorov 1941, here-\nafter “K41”), a weak cascade of Alfvénic fluctuations (Ng and\nBhattacharjee 1997; Galtier et al. 2000, hereafter “W0”), and\na critically balanced Alfvénic cascade à la Goldreich and Srid-\nhar (1995) (hereafter, “GS95”). However, advanced theories of\nMHD turbulence that extend the above “classic” picture have\nbeen formulated in the past years. This is the case, for instance,\nof a theory that incorporates “dynamic” (i.e., scale-dependent)\nalignment of fluctuations in a critically balanced Alfvénic cas-\ncade (Boldyrev 2006, hereafter “B06”), which can intrinsically\nlead at even smaller scales to a regime usually referred to as\n“tearing-mediated turbulence” (i.e., a regime where magnetic\nreconnection mediates the generation of smaller-scale fluctua-\ntions, Boldyrev and Loureiro 2017; Mallet et al. 2017, hereafter\n“TMT”). It is also worth mentioning that the conditions under\nwhich critical balance and the associated cascades develop (i.e.,\nGS95, B06, and TMT regimes) may not cover all the possible\nscenarios in MHD turbulence (e.g., see discussion in Oughton\nand Matthaeus 2020). However, analytical (phenomenological)\nscaling of turbulent fluctuations and their anisotropy can be only\nderived for these cases. Moreover, several numerical simulations\nandin-situ measurements in the solar wind have provided solid\nevidences for these regimes (e.g., Chen 2016; Sahraoui et al.\n2020; Schekochihin 2022, and references therein). Therefore, it\nis of interest to derive turbulent damping rates for all these the-\nories. The results obtained here have indeed implications for the\nso-called cosmic-ray “self-confinement”, since its e ffectiveness\nfor CR scattering is the result of a competition between di fferent\ndamping mechanisms and a balance between the most-relevant\ndamping rate and the growth rate of the CR streaming instabil-\nity (e.g., Farmer and Goldreich 2004; Blasi et al. 2012; Lazarian\n2016; Kempski and Quataert 2022; Xu and Lazarian 2022).\nThe paper is organized as follows. In Section 2 the Elsässer\nformalism and the definitions of timescales and nonlinear param-\neter are introduced. In Section 3 such formalism is employed\nto derive general expressions for the turbulent damping rates,\nwhose scaling are then explicitly derived in Section 4 for various\nturbulent regimes and within di fferent theories of MHD turbu-\nlence. Additionally, some considerations about the feedback of\nCR-driven AWs on pre-existing fluctuations and possible phe-\nnomenological models for the CR-modified background turbu-\nlent spectrum are discussed in Section 5. Finally, a summary\nand discussion of the results is provided in Section 6. It is worth\nstressing that this work is meant to primarily provide a rigorous,\ngeneral formalism for deriving the turbulent damping rates of\nCR-driven AW packets, as well as some criterion for the possible\nrelevance of CR feedback on pre-exsiting fluctuations. A more\nextensive discussion about di fferent damping rates, the role of\ncoherent structures and compressible turbulence, as well as the\nimplications for specific astrophysical systems will be the focus\nof a following paper (hereafter “Paper II”).2. Setting the stage: Elsässer formalism\nThe usual formulation of magnetohydrodynamic (MHD) equa-\ntions for an incompressible plasma with mass density ρ0, viscos-\nityνand resistivity η, are (e.g., Kulsrud 2005)\n∂u\n∂t+(u·∇)u=(B·∇)B\n4πρ0−∇Ptot\nρ0+ν∇2u\u0000∇·u=0\u0001,(1)\n∂B\n∂t+(u·∇)B=(B·∇)u+η∇2B\u0000∇·B=0\u0001,(2)\nwhere uis the fluid velocity, Bis the magnetic field, and Ptot=\nPth+B2/8πis the sum of the thermal and magnetic pressure.\nEquations (1)-(2) can be reformulated as\n∂z±\n∂t+(z∓·∇)z±=−∇Ptot\nρ0+µ+∇2z±+µ−∇2z∓, (3)\nwhere z±=u±B/p\n4πρ0=u±vAare the Elsässer variables (El-\nsässer 1950), which satisfy ∇·z±=0,vAis the Alfvén-speed\nvector associated to B, andµ±=(ν±η)/2. Here we assume ν=η\nfor simplicity1, so thatµ+=ηandµ−=0. By splitting the vari-\nables into a background quantity (denoted by a “0” subscript2)\nand purely transverse fluctuations, i.e., z±=z±\n0+δz±\n⊥where\nz±\n0=±B0/p\n4πρ0=±vA,0is the Alfvén speed associated to the\nbackground magnetic field B0andδz±\n⊥=δu⊥±δB⊥/p\n4πρ0the\nfluctuating Elsässer fields, equation (3) rewrites as\n\u0012∂\n∂t∓vA,0∇∥| {z }\nω±\nlin∼k±\n∥vA,0+δz∓\n⊥·∇⊥|    {z    }\nω±\nnl∼k±\n⊥δz∓\n⊥−η∇2\n|{z}\nω±\ndiss∼ηk±2\u0013\nδz±\n⊥=−∇δPtot\nρ0,(4)\nwhere the parallel and perpendicular directions are defined with\nrespect to B0for the global equations (but will be later defined\nwith respect to a scale-dependent mean field ⟨B⟩kin a turbulent\nenvironment); we also mention that the term ∇δPtot/ρ0in prac-\ntice contributes just as a multiplicative factor (in Fourier space)\nto the nonlinear term (and associated timescale) on the left-hand\nside.3One important feature of the formulation in (4) is that it ex-\nplicitly shows that the nonlinear term ( δz∓\n⊥·∇⊥)δz±\n⊥is due only to\nthe interaction of counter-propagating Alfvén-wave packets, δz+\n⊥\nbeing transverse fluctuations propagating at the Alfvén speed\nvA,0in the direction of B0, whileδz−\n⊥are fluctuations propagating\nat the same speed in the direction of −B0.\nFrom (4) one can define a nonlinear parameter χ, which mea-\nsures the strength of nonlinear e ffects with respect to the linear\npropagation term, namely\nχ±∼|(δz∓\n⊥·∇⊥)δz±\n⊥|\n|(vA,0∇∥)δz±\n⊥|∼ω±\nnl\nω±\nlin∼τ±\nA\nτ±\nnl∼k±\n⊥δz∓\nk\nk±\n∥vA,0, (5)\nand involves the wave-vector components ( k±\n∥,k±\n⊥) of the evolv-\ning fluctuation δz±\n⊥and the amplitude δz∓\n⊥of the counter-\npropagating fluctuation that induces the nonlinearities on δz±\n⊥.\n1The ratio Pm =ν/ηis called “magnetic Prandtl number”. Regimes\nwith Pm ,1 may nevertheless be relevant in some astrophysical envi-\nronments (e.g., for dynamo e ffects, see Rincon 2019).\n2The subscript “0” formally implies a “large-scale” average procedure\nB0=⟨B⟩L. In the latter, L∼ℓ0will be the injection scale of turbulence.\nIn general, a “mean” quantity can be defined with respect to any given\nscaleλ, i.e., as the result of a scale-dependent average, e.g, ⟨B⟩λ; if\nλ≪ℓ0, then⟨B⟩λcan significantly di ffer from B0(see later).\n3By taking the divergence of (4) and using the incompressiblity condi-\ntion∇·δz±\n⊥=0, one finds that pressure fluctuations satisfy the condition\n∇·[∇δPtot]=∇2δPtot=ρ0∇⊥·[(δz∓\n⊥·∇⊥)δz±\n⊥] (Schekochihin 2022).\nArticle number, page 2 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\nIn the following, this parameter will play a central role to esti-\nmate the nonlinear cascade rate (or, “turbulent damping”) of an\nAlfvén-wave packet interacting with background fluctuations. In\nparticular, to obtain the correct turbulent damping rate it is nec-\nessary to make a careful distinction between (i) the nonlinear pa-\nrameterχzcharacterizing counter-propagating pre-existing fluc-\ntuations, and (ii) the nonlinear parameter χwdescribing the in-\nteraction between the AW packet and background turbulence.\n3. Turbulent damping of an Alfvén-wave packet\nConsider an Alfvén-wave packet that is injected in an environ-\nment filled with pre-existing Alfvénic turbulence. Let δwbe\nthe initial Elsässer variable of the packet, and λw\n⊥andλw\n∥its\nwavelengths perpendicular and parallel to a mean-magnetic field\n⟨B⟩λwat such scales4, respectively (the corresponding wave vec-\ntors being kw\n⊥∼1/λw\n⊥andkw\n∥∼1/λw\n∥). The Alfvénic fluctua-\ntions populating the turbulent background will be characterized\nby certain scale-dependent relations for their Elsässer amplitude\nδz±\nλz\n⊥, their wavelength anisotropy λz,±\n∥/λz,±\n⊥(for which the cor-\nresponding wave vectors can be denoted as kz,±\n⊥∼1/λz,±\n⊥and\nkz,±\n∥∼1/λz,±\n∥), and, if allowed, for the alignment angle between\nδz+\nλz\n⊥andδz−\nλz\n⊥, i.e., sinθz\nλz,±\n⊥. It is now instructive to derive the\ngeneral relations first, leaving the explicit scaling belonging to\ndifferent turbulence theories for later. Hereafter we consider the\ncase of balanced background turbulence, and thus drop the ±su-\nperscript everywhere for simplicity of notation.\nWhile propagating, the AW packet will interact nonlinearly\nonly with counter-propagating Alfvénic fluctuations of the back-\nground. In terms of Elsässer variables, the nonlinear interaction\nis indeed described by the term ( δz·∇)δw. The strength of this\nnonlinear interaction can then be determined by comparing the\nabove nonlinear term with the term describing its linear propaga-\ntion, ( vA,0·∇)δw. The interaction between the AW packet and the\npre-existing fluctuations is described by the nonlinear parameter\nof the packet, i.e.,\nχw∼τw\nA\nτw\nnl∼(λw\n∥/vA,0)\n(λw\n⊥/δzλw\n⊥)∼ λw\n∥\nλw\n⊥! δzλw\n⊥\nvA,0!\n, (6)\nwhere “local-in-scale interactions” have been assumed, so that\nδzλz\n⊥has been substituted with δzλw\n⊥in the timescale associated\nto the nonlinear interaction between the AW packet and pre-\nexisting turbulence, i.e., τw\nnl∼λw\n⊥/δzλz\n⊥∼λw\n⊥/δzλw\n⊥. A parameter\nχw≳1 means strong nonlinear interactions, while χw<1 de-\nnotes the weakly nonlinear regime. We stress that the parameter\nin (6) is di fferent from the nonlinear parameter that characterizes\nbackground turbulence, i.e., χz∼τz\nA/τz\nnl∼(λz\n∥/λz\n⊥)(δzλz\n⊥/vA,0).5\n4One can think of each component iof this mean field at scale λas\ndefined, for instance, by ⟨Bi⟩λ∼\u0012Rk′<1/λ\n1/ℓ0B2\ni,k′dk′\u00131/2\n, i.e., a magnetic\nfield that is the result of the contribution from the background field B0\nplus all the fluctuations δBλ′at scalesλ′> λ such that the nonlinear\ntimescaleτnl,λ′over which the associated “turbulent eddy” evolves is\nmuch longer than the nonlinear evolution timescale τnl,λof fluctuations\nat the scaleλ(viz.,τnl,λ′≫τnl,λ, so that turbulent eddies on scales λ′ap-\npear as “frozen” over the turnover time of turbulent eddies at scale λ).\nOperationally, this mean field can be defined in di fferent ways (e.g., Cho\nand Vishniac 2000; Cho and Lazarian 2004; Horbury et al. 2008; Wicks\net al. 2010; Chen et al. 2011; Matthaeus et al. 2012; Mallet et al. 2016;\nCerri et al. 2019), but what is the most-appropriate operational defini-\ntion is still matter of debate (e.g., see Oughton and Matthaeus 2020).\n5Or, if scale-dependent (“dynamic”) alignment is taken into account,\nit isχz∼sinθz\nλz\n⊥(λz\n∥/λz\n⊥)(δzλz\n⊥/vA,0) (see Appendix A).And we point out that, while background fluctuations can have\nχz≳1 at scaleλz\n⊥∼λw\n⊥(strong pre-existing turbulence), the\nconditionχw≳1 does not necessarily hold at those same scales.\nIn fact, one should note that χwis not only proportional to\nthe amplitude of background fluctuations at the scale λw\n⊥, i.e.,\nδzλw\n⊥/vA,0, but it depends also on the AW-packet’s propagation\nangle with respect to the mean magnetic field ⟨B⟩λwat that scale:\nλw\n∥/λw\n⊥∼kw\n⊥/kw\n∥=tanΘw\nkB, where Θw\nkBis the angle between kw\nand⟨B⟩λw∼1/kw. As a result, if the background fluctuations are\nsuch that MA,0≲1 at injection scale ℓ0(MA,0≈δz0/vA,0is the\nAlfvénic Mach number of pre-existing turbulence for isotropic\ninjection at scales ℓ0), then strong nonlinear interactions at scales\nλw\n⊥≪ℓ0(whereδzλw\n⊥≪δz0) requireλw\n∥/λw\n⊥∼vA,0/δzλw\n⊥≫1,\nand thus this regime is achieved only by quasi-perpendicular AW\npackets. For critically balanced pre-existing fluctuations, for in-\nstance, this indeed means that an AW packet undergoes strong\nnonlinear interactions (and thus severe turbulent damping) only\nif its wavevector matches the anisotropy of background turbu-\nlence, i.e., if λw\n∥∼λz\n∥(λw\n⊥). Therefore the nonlinear interaction\nbetween a quasi-parallel AW, λw\n∥/λw\n⊥≪1, and anisotropic pre-\nexisting turbulence ( λz\n⊥/λz\n∥≪1) is always weak, χw≪1.\nThis is shown explicitly by considering the quasi-parallel\npropagation limit, which is the case of interest for CR-generated\nAW packets. In this case, the propagation can only be as par-\nallel as the external turbulence allows, i.e., the propagation an-\ngle cannot be smaller than the amount δbλw\n⊥/⟨B⟩λw∼δzλw\n⊥/vA,0\nbecause of the field-line distortions induced by pre-existing tur-\nbulent fluctuations δbλw\n⊥over the wavelength λw\n⊥of the packet.\nTherefore, the quasi-parallel (q ∥) propagation limit is set by\nλw\n∥\nλw\n⊥\f\f\f\f\f\fmin=λw,q∥\n∥\nλw,q∥\n⊥∼δzλw,q∥\n⊥\nvA,0. (7)\nHence, the associated nonlinear parameter in this limit is\nχw,q∥∼δzλw,q∥\n⊥\nvA,02\n. (8)\nThe strongly nonlinear regime can thus be achieved only if\nthe AW packet interacts with pre-existing super-Alfvénic turbu-\nlence ( MA,0>1). In this case, the concept of quasi-parallel prop-\nagation in (7) does not apply at scales where δbλw/⟨B⟩λw>1,\ni.e., the distinction between λw\n⊥andλw\n∥is lost andλw\n⊥∼λw\n∥∼λw\nholds; hence, χw∼δzλw/vA,0≳1. However, even for super-\nAflvénic injection of external turbulence, the nonlinear interac-\ntion between the AW and external fluctuations will become weak\nat “small-enough” scales, i.e. at scales λw\n⊥<ℓ A=ℓ0/M3\nA,0≪ℓ0\nfor which the initially super-Alfvénic fluctuations become sub-\nAlfvénic,δzλw\n⊥/vA,0<1, and anisotropic (see Appendix A). An-\nother way to visualize this is by rewriting (8) as\nχw,q∥∼λz\n⊥\nλz\n∥2\n(χz)2, (9)\nwhich is≪χzfor anisotropic background fluctuations, λz\n⊥≪λz\n∥,\nand thusχw,q∥≪1 even if pre-existing turbulence is critically\nbalanced (χz∼1).\nHaving clarified the packet’s nonlinear regimes that have\nto be considered in terms of background turbulence, we can\nnow estimate the associated rate of “turbulent damping”, i.e.\nthe timescale over which such AW packet undergoes a cascade\nprocess due to its nonlinear interaction with pre-existing tur-\nbulent fluctuations. The cascade time of the packet is given by\nArticle number, page 3 of 17A&A proofs: manuscript no. C2024\nτw\ncasc∼(τw\nnl)2/τw\nA∼τw\nnl/χwfor the weak regime ( χw<1), while\nit isτw\ncasc∼τw\nnlin the strongly nonlinear case ( χw≳1, which we\nrecall also implies that λw\n∥∼λw\n⊥∼λw; see paragraph below (8)).\nAs a result, the turbulent damping rate is:\nΓw\nturb∼1\nτwcasc∼\u0012λw\n∥\nλw\n⊥\u0013\u0010λw\n⊥\nℓ0\u0011−1\u0012δzλw⊥\nvA,0\u00132vA,0\nℓ0ifχw<1\n\u0010λw\nℓ0\u0011−1\u0010δzλw\nvA,0\u0011vA,0\nℓ0ifχw≳1(10)\nwhich reduces to\nΓw,q∥\nturb∼λw,q∥\n⊥\nℓ0−1δzλw,q∥\n⊥\nvA,03vA,0\nℓ0(11)\nfor quasi-parallel propagation and χw,q∥<1. (We recall that χw\nshould not be identified with the nonlinear parameter χzthat de-\nscribes the strength of background turbulence). Therefore, when\nχw<1 the turbulent damping rate of an AW packet is non-\nlinear with respect to the background-fluctuations’ amplitude\nand depends on the propagation angle of the wave, becoming\na third-order quantity of the pre-existing turbulent amplitude in\nthe quasi-parallel limit.\nWe conclude by highlighting that in (11) there is a λ−1\n⊥\nin front of the δz3\nλ⊥term. Therefore, a turbulent perpendicular\nscaling for δzλ⊥∝λα\n⊥with a spectral index α > 1/3 (e.g.,\nweak Alfvénic turbulence as in Galtier et al. (2000) or the\ntearing-mediated regime in Boldyrev and Loureiro (2017); Mal-\nlet et al. (2017)) will produce a turbulent damping rate of quasi-\nparallel AW packets that decreases with decreasing scale, while\nfluctuations that scale with α < 1/3 (e.g., critically balanced\nstrong Alfvénic turbulence with scale-dependent alignment as\nin Boldyrev (2006)) will result in a damping rate that increases\nwith decreasing λw\n⊥. For a Kolmogorov-like perpendicular scal-\ningδzλ⊥∝λ1/3\n⊥as in critically balanced strong Alfvénic turbu-\nlence without dynamic alignment (Goldreich and Sridhar 1995),\nwe can expect that the turbulent damping of quasi-parallel AW\npackets becomes scale-independent, i.e., Γw,q∥\nGS95∼const.\n4. Turbulent damping with explicit MHD scalings\nThe Alfvénic Mach number MA,0∼δz0/vA,0defines the cas-\ncading regimes of background fluctuations. (We recall that ℓ0is\nthe injection scale and δz0the fluctuations’ amplitude at such\nscale.) If we assume isotropic injection, the nonlinear param-\neter of background turbulence at scale ℓ0indeed corresponds\nto the Alfvénic Mach number at injection, i.e., χz\n0≈MA,0. If\nMA,0<1 turbulence is called “sub-Alfvénic”: it starts as an\nanisotropic weak cascade that transitions into critically balanced,\nstrong turbulence at smaller scales (still anisotropic, but in a dif-\nferent fashion). “Trans-Alfvénic” turubulence ( MA,0≈1) con-\nsists of an anisotropic strong cascade of critically balanced fluc-\ntuations at all scales. When MA,0>1 (large-amplitude injec-\ntion), turbulence is called “super-Alfvénic” and fluctuations ini-\ntially undergo an isotropic (“hyrdodynamic-like”) cascade un-\ntil sub-Alfvénic amplitudes are attained at smaller scales, and\nturbulence becomes critically balanced and anisotropic. Then,\nif scale-dependent (“dynamic”) alignment of fluctuations is al-\nlowed in the critical-balance range, an additional transition to a\ndifferent regime of anisotropic, strong turbulence can occur at\neven smaller scales due to magnetic reconnection. In the follow-\ning we only summarize the relevant scaling of background turbu-\nlent fluctuations in the di fferent ranges. A more detailed deriva-\ntion of these ranges and of the associated scaling is provided in\nAppendix A.The turbulent damping rates of this Section are derived as\nfollows:\n1. We employ the known perpendicular scaling of background\nturbulence, δzλ⊥, and locality of interactions ( λz\n⊥∼λw,q∥\n⊥) in\n(11) to obtain the turbulent damping rate as a function of the\npacket’s perpendicular wavelength, Γw,q∥\nturb(λw,q∥\n⊥).\n2. We use the quasi-parallel condition in (7) to retrieve the scal-\ning of Γw,q∥\nturbwith respect to the parallel wavelength λw,q∥\n∥.\nThe behaviour of the turbulent damping rates derived in this\nSection for di fferent regimes of MA,0are shown in Figure 1,\nalong with dashed lines showing previous estimates presented in\nexisting literature (Farmer and Goldreich 2004; Lazarian 2016).\nWe stress that previous results not only overestimate such damp-\ning rate by a factor that could be several orders of magnitude,\nbut in most cases they also obtain a completely di fferent result\non how this damping rate depends upon the packet’s wavelength.\n4.1. MHD turbulence without scale-dependent alignment\nWe first consider the “classic” picture in which dynamic align-\nment of turbulent fluctuations does not occur. In this case, we\nhave three possible regimes for background turbulence:\n[W0] A weak anisotropic cascade with fluctuations’ scaling\nδz(W0)\nλz\n⊥/vA,0∼MA,0(λz\n⊥/ℓ0)1/2that only generates smaller\nperpendicular scales (i.e., λz\n∥∼ℓ0≈const.) and transitions\ninto a critically balanced cascade at λz\n⊥,CB∼M2\nA,0ℓ0. This\ncascade is realized in the range of scales λz\n⊥,CB≲λz\n⊥≲ℓ0,\nand only for sub-Alfvénic injection ( MA,0<1).\n[K41] A strong, isotropic (“hydrodynamic-like”) cascade\ncharacterized by the scaling δz(K41)\nλz/vA,0∼MA,0(λz/ℓ0)1/3.\nThese fluctuations attain sub-Alfvénic amplitudes, becoming\nanisotropic and critically balanced, at a scale ℓA∼M−3\nA,0ℓ0.\nThis cascade is realized at scales ℓA≲λz≲ℓ0, and only for\nsuper-Alfvénic injection ( MA,0>1).\n[GS95] A strong anisotropic cascade of critically balanced\nfluctuations with perpendicular scaling δz(GS95)\nλz\n⊥∝(λz\n⊥)1/3.\nThis type of cascade is realized either for trans-Alfvénic\ninjection ( MA,0≈1), or when cascading fluctuations of\nthe two above regimes reach the scale λz\n⊥,CBorℓA, re-\nspectively. For trans /sub-Alfvénic injection ( MA,0≲1),\nthe dependence on MA,0of the scaling is δz(GS95)\nλz\n⊥/vA,0∼\nM4/3\nA,0(λz\n⊥/ℓ0)1/3, and the cascade achieves dissipation at a\nscaleλz(subA)\n⊥,min/ℓ0∼M−1\nA,0S−3/4\n0. For super-Alfvénic injection\n(MA,0>1), the fluctuations’ scaling with MA,0is linear, i.e.,\nδz(GS95)\nλz\n⊥/vA,0∼MA,0(λz\n⊥/ℓ0)1/3, and the dissipation scale is\ngiven byλz(supA)\n⊥,min/ℓ0∼(MA,0S0)−3/4. Here S0=ℓ0vA,0/ηis\nthe Lundquist number at injection scale.\nBy using (8) one can verify that for both (W0) and (GS95)\nrange of scales the nonlinear interaction between quasi-parallel\nAW packets with wavelength λw,q∥\n⊥< ℓ 0(λw,q∥\n⊥< ℓ A) and back-\nground turbulence with MA,0≲1 (MA,0>1) is weak, i.e.,\nχw,q∥≪1. Hence, the cascade time is not just the nonlin-\near timeτw\nnl, and the turbulent damping rate is given by (11).\nThis changes significantly the e ffectiveness of turbulent damp-\ning in pre-existing turbulence with respect to previous esti-\nmates (Farmer and Goldreich 2004; Lazarian 2016).\nArticle number, page 4 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\n10−1310−1110−910−710−510−310−1\nλw,q/bardbl\n/bardbl//lscript010−310−1101103105Γw,q/bardbl\nsubA·/lscript0/vA,0\nMA,0= 0.1\nS0= 1014(W0) range\neq.(46) in Lazarian (2016)\n(GS95) range\neq.(34) in Lazarian (2016)\n(B06) range\n(TMT) rangeS−1\n0S−5/7\n0M8/7\nA,0 M4\nA,0 MA,0\nM4\nA,0S1/7\n0M32/7\nA,0M2\nA,01S1/2\n0M2\nA,0\nM3\nA,0\n10−1310−1110−910−710−510−310−1\nλw,q/bardbl\n/bardbl//lscript0102104106108Γw,q/bardbl\nsupA·/lscript0/vA,0\nMA,0= 10\nS0= 1014(K41) range\n(GS95) range\neq.(52) in Lazarian (2016)\n(B06) range\n(TMT) rangeS−1\n0S−5/7\n0M−6/7\nA,0 M−3\nA,0 1\nMA,0M3\nA,0S1/7\n0M18/7\nA,0S1/2\n0M3/2\nA,0\nFig. 1: Normalized turbulent damping rateℓ0\nvA,0Γw,q∥\nturbfor quasi-parallel AW packets with normalized parallel wavelength λw,q∥\n∥/ℓ0,\nin a background plasma with Lunquist number S0=1014and di fferent turbulent regimes (see Appendix A). Solid lines represent\ndamping rates derived in this work (equations (12), (14), (15), and (17)), while dashed lines report the damping rates in Lazarian\n(2016) for reference. General expressions for transition scales and damping-rate values are reported on the right and upper axis.\nLeft: damping rates in sub-Alfvénic turbulence ( MA,0=0.1). Right: damping rates in super-Alfvénic turbulence ( MA,0=10).\n4.1.1. sub- and trans-Alfvénic turbulence ( MA,0≤1) without\ndynamic alignment\nIn sub-Alfvénic background turbulence without dynamic align-\nment, a quasi-parallel AW packet with (normalized) parallel\nwavelength ˆλw\n∥=λw,q∥\n∥/ℓ0is subject to the following turbulent\ndamping rate:\n[MA,0<1, no dynamic alignment]\nΓw,q∥\nsubA∼M8/3\nA,0\u0010ˆλw\n∥\u00111/3vA,0\nℓ0M4\nA,0≲ˆλw\n∥≲MA,0\nM4\nA,0vA,0\nℓ0ˆλw(subA)\n∥,min≲ˆλw\n∥≲M4\nA,0(12)\nwhere ˆλw(subA)\n∥,min∼(MA,0ˆλz(subA)\n⊥,min)4/3∼S−1\n0is the minimum\npacket’s wavelength that is e ffectively subject to turbulent damp-\ning, with ˆλz(subA)\n⊥,min∼M−1\nA,0S−3/4\n0being the (normalized) dissipa-\ntion scale of the turbulence. The ranges of ˆλw\n∥in (12) have been\ndetermined accordingly to the quasi-parallel condition in (7) and,\nassuming local interactions λw,q∥\n⊥∼λz\n⊥, employing the λz\n⊥range\nof validity for each turbulent regime (see Appendix A).\nThe trans-Alfvénic regime is trivially obtained from (12)\nwhen the initial weak cascade does not occur:\n[MA,0≃1, no dynamic alignment]\nΓw,q∥\ntransA∼vA,0\nℓ0(ˆλz\n⊥,min)4/3≲ˆλw\n∥≲1 (13)\nwhere ˆλz\n⊥,min∼S−3/4\n0is the (normalized) dissipation scale of\nGS95 turbulence in the trans-Alfvénic regime.\nThe damping rates in (12) di ffer from what previously de-\nrived in the literature because here the nonlinear parameter of\nthe AW packet has been properly taken into account. In fact, one\ncan verify that the turbulent-damping rate in the (W0) range of\nsub-Alfvénic turbulence, i.e., equation (46) of Lazarian (2016),\ncan be recovered if the nonlinear parameter χzof background\nturbulence is employed instead of the nonlinear parameter χw\nof the AW packet. Analogously, the result in the (GS95) range\nof sub-Alfvénic turbulence in equation (34) of the same paperis recovered by assuming strong interactions between the quasi-\nparallel AW packet and background fluctuations, i.e., identify-\ningχwwithχz∼1 at those scales. However, given the ex-\npression for χwin (6), the assumption χw∼1 would require\nλw\n∥/λw\n⊥∼vA,0/δzλw\n⊥≫1, which is inconsistent with the quasi-\nparallel limit λw\n∥/λw\n⊥≪1. The same argument applies when\ncomparing the damping rate for the trans-Alfvénic case given\nin (13) with equation (9) of Farmer and Goldreich (2004).\nIt is important to stress that the results obtained here\nstrongly change the e ffectiveness of the turbulent damping of\nCR-generated Alfvén-wave packets. In fact, depending on the\nAlfvénic Mach number MA,0and on the Lunquist number S0, the\ndamping rates in (12)-(13) can be several orders of magnitude\nsmaller than the ones previously derived and usually employed\nin CR studies (see, e.g., left panel in Figure 1). In fact, the damp-\ning rate of an AW packet interacting with pre-existing weak tur-\nbulence is at least a factor MA,0smaller that what previously esti-\nmated (i.e., at scale λw,q∥\n∥,max∼MA,0ℓ0, when this di fference is at its\nminimum, then it increases even further due to the di fferent de-\npendence on λw,q∥\n∥). When the packet starts to interact with strong\nturbulence, i.e. for λw,q∥\n∥,CB∼M4\nA,0ℓ0, the damping rate becomes at\nleast a factor M4\nA,0≪1 smaller than what has been derived in\nthe literature (a di fference that, again, increases even further with\ndecreasing packet’s parallel wavelength due to the radically dif-\nferentλw,q∥\n∥-dependence of Γw,q∥\nGS95in (12)-(13) with respect to the\nresults in Farmer and Goldreich (2004) and in Lazarian (2016)).\nThis is indeed true also for trans-Alfvénic injection ( MA,0≃1),\nin which case the damping rate in (13) would be the same as the\none in the literature only at scales λw,q∥\n∥∼ℓ0, and then the two\nresults would rapidly diverge with decreasing wavelength of the\nquasi-parallel AW packet at λw,q∥\n∥< ℓ 0. Finally, the maximum\ndifference between the damping rate obtained here and the ones\nfound in the literature is achieved at the minimum wavelength\nfor which this damping mechanism is e ffective: atλw,q∥\n∥∼λw\n∥,min\nthe actual damping rate is a factor ∼S−1/2\n0M2\nA,0≪1 smaller\nthan the results in Farmer and Goldreich (2004) and in Lazar-\nian (2016); in astrophysical systems this factor can represents\nArticle number, page 5 of 17A&A proofs: manuscript no. C2024\nmany orders of magnitude, since S0can be extremely large, e.g.,\nlarger than 1020(Priest and Forbes 2007). (We also point out that\nin sub-Alfvénic turbulence the ordering S0≫M−4\nA,0is implied in\norder to have a significant (GS95) range; see Appendix A).\n4.1.2. super-Alfvénic turbulence ( MA,0>1) without dynamic\nalignment\nWhen the injection regime of background fluctuations is super-\nAlfvénic, an AW packet is instead subject to a turbulent damping\ngiven by:\n[MA,0>1, no dynamic alignment]\nΓw,q∥\nsupA∼MA,0\u0010ˆλw\u0011−2/3vA,0\nℓ0M−3\nA,0≲ˆλw≲1\nM3\nA,0vA,0\nℓ0ˆλw(supA)\n∥,min≲ˆλw\n∥≲M−3\nA,0(14)\nwhere the damping rate in the range M−3\nA,0≲ˆλw≲1 is obtained\nfrom theχw≳1 part of (10), and the shortest wavelength af-\nfected by turbulent damping is ˆλw(supA)\n∥,min∼MA,0(ˆλz(supA)\n⊥,min)4/3∼\nS−1\n0. We also point out that the isotropic normalized wavelength\nˆλw=λw/ℓ0enters the damping rate in the range ℓA≲λw≲ℓ0,\ni.e., where the AW packet interacts with hydrodynamic-like pre-\nexisting turbulence.\nIn the range M−3\nA,0≲ˆλw≲1, the packet’s nonlinear parameter\nis larger than unity (i.e., χw>1), and thus our result agrees with\nthe one provided in equation (55) of Lazarian (2016). This is\na consequence of the fact that the quasi-parallel condition (7)\ndoes not apply in this range and so, for local interactions, there\nis no di fference between the expressions for χwand forχz. On\nthe other hand, at smaller scales ( λw,q∥\n∥≲ℓA), the quasi-parallel\ncondition does apply and the damping rate depends explicitly on\nthe normalized parallel wavelength ˆλw\n∥=λw,q∥\n∥/ℓ0. Hence, in this\nrange of scales our result is again di fferent from equation (52)\nof Lazarian (2016) for the same reason discussed in the sub-\nAlfvénic case, i.e., that χw,q∥\nλ⊥≪χz\nλ⊥∼1 (cf. equation (9) in\nSection 3), and therefore the two nonlinear parameters need not\nto be confused at scales below ℓA.\nAlso for this regime, we stress that the result in (14) implies a\ndrastic change in the e ffectiveness of turbulent damping for CR-\ngenerated Alfvén-wave packets. In fact, while our result agrees\nwith the turbulent damping rate found in the literature for the\nrange of scales ℓA≲λw,q∥\n∥≲ℓ0, the corresponding rate at smaller\nscales,λw,q∥\n∥< ℓ A, can be several orders of magnitude smaller\nthan the one usually employed in CR studies (see, e.g., left panel\nin Figure 1). The damping rate in (14) indeed rapidly diverges\nfrom the one given in Lazarian (2016) with decreasing wave-\nlength of the quasi-parallel AW packet when λw,q∥\n∥< ℓ A, reach-\ning its maximum di fference atλw,q∥\n∥∼λw\n∥,min, where the actual\ndamping rate in (14) is a factor ∼S−1/2\n0M3/2\nA,0≪1 smaller than\nthe one provided in the literature. (We also point out that for\nsuper-Alfvénic injection, the ordering S0≫M3\nA,0is implied in\norder to have a significant (GS95) range; see Appendix A).\n4.2. MHD turbulence with dynamic alignment\nThe “classic” picture presented above is now extended to the\ncase in which counter-propagating Elsässer fields δz+\nλ⊥andδz−\nλ⊥(or, in a similar way, velocity and magnetic-field fluctuations,δuλ⊥andδbλ⊥) tend to align with each other in a scale-dependent\nfashion (Boldyrev 2006). This “dynamic alignment” not only\ninduces a weakening of the nonlinear interaction (and thus\nmodifies the fluctuations’ scaling and anisotropy), but can also\nopen the possibility of a reconnection-mediated regime at small\nscales—still within MHD scales (Boldyrev and Loureiro 2017;\nMallet et al. 2017). In this section, we consider the case when\nsuch a scale-dependent alignment occur only in critically bal-\nanced turbulent fluctuations6,χz∼1. In this case, in addition to\nthe (W0) and (K41) regimes of the previous Section 4.1, one can\nhave two additional regimes for background turbulence:\n[B06] An anisotropic, strong cascade of critically balanced\nand dynamically aligned fluctuations that replaces the\n(GS95) regime. In this case, fluctuations’ alignment angle\ndecreases with decreasing scale so that sin θz\nλ⊥∝(λz\n⊥/ℓ0)1/4.\nFor sub- and trans-Alfvénic injection ( MA,0≤1), the perpen-\ndicular scaling of turbulent fluctuations at scales λz\n⊥≲λz\n⊥,CB,\nturns out to be δz(B06)\nλz\n⊥/vA,0∼M3/2\nA,0(λz\n⊥/ℓ0)1/4. This cascade\ncan further turn into a reconnection-mediated regime\nbelow a transition scale λz(subA)\n⊥,∗/ℓ0∼M−2/7\nA,0S−4/7\n0. For\nsuper-Alfvénic injection ( MA,0>1), turbulent fluctuations\nat scalesλz\n⊥≲ℓAfollow instead a perpendicular scaling\ngiven byδz(B06)\nλz\n⊥/vA,0∼M3/4\nA,0(λz\n⊥/ℓ0)1/4. In this super-\nAlfvénic regime, a transition to reconnection-mediated\nturbulence may occur at a scale λz(supA)\n⊥,∗/ℓ0∼M−9/7\nA,0S−4/7\n0.\nWhen the dissipation scale for a given regime is larger\nthan the corresponding transition scale, then this (B06)\ncascade does not transition into the tearing-mediated regime.\nWhen this is case, the dissipation scale is achieved at\nλz(subA)\n⊥,min/ℓ0∼(MA,0S0)−2/3in the trans /sub-Alfvénic regime,\nor atλz(supA)\n⊥,min/ℓ0∼M−1\nA,0S−2/3\n0for super-Alfvénic injection.\n[TMT] A strong anisotropic cascade of critically balanced\nand dinamically (mis-)aligned fluctuations that are generated\nby magnetic-reconnection processes. In this case, fluctua-\ntions scale as δz(TMT)\nλz\n⊥∝S1/5\n0(λz\n⊥)3/5and are subject to a scale-\ndependent mis-alignment given by sin θz\nλz\n⊥∝(λz\n⊥/ℓ0)−4/5.\nFor sub- and trans-Alfvénic injection ( MA,0≤1), the\nperpendicular scaling of tearing-mediated turbulent fluctu-\nations is given by δz(TMT)\nλz\n⊥/vA,0∼S1/5\n0M8/5\nA,0(λz\n⊥/ℓ0)3/5, while\nin the super-Alfvénic regime ( MA,0>1) they scale as\nδz(TMT)\nλz\n⊥/vA,0∼S1/5\n0M6/5\nA,0(λz\n⊥/ℓ0)3/5. In this regime, the dis-\nsipation scale is the same as for the GS95 cascade, i.e.,\nλz(subA)\n⊥,min/ℓ0∼M−1\nA,0S−3/4\n0for trans /sub-Alfvénic turbulence,\norλz(supA)\n⊥,min/ℓ0∼(MA,0S0)−3/4for super-Alfvénic injection.\nOne can verify that the nonlinear interaction between a\nquasi-parallel AW packet and the anisotropic turbulent fluctua-\ntions populating the background is weak also for these cascades,\ni.e.,χw,q∥\nλ⊥<1, except for the case of super-Alfvénic injection at\nscalesλw≳ℓA, where instead χw\nλ∼χz\nλ>1 holds.\n6Addressing the case in which alignment could occur also at weak\nnonlinearities (Cerri et al. 2022) may be still premature at this point, and\nit requires to account for the alignment induced by background fluctua-\ntions on the AW packet itself. For the sake of simplicity, this case will\nnot be treated here and will be addressed separately in a following work.\nArticle number, page 6 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\n4.2.1. sub- and trans-Alfvénic turbulence ( MA,0≤1) with\ndynamic alignment\nA quasi-parallel AW packet with (normalized) parallel wave-\nlength ˆλw\n∥=λw,q∥\n∥/ℓ0injected in pre-existing sub-Alfvénic turbu-\nlence for which dynamic alignment of critically balanced fluctu-\nations occurs is subject to the following turbulent damping rate:\n[MA,0<1, with dynamic alignment]\nΓw,q∥\nsubA∼M8/3\nA,0\u0010ˆλw\n∥\u00111/3vA,0\nℓ0M4\nA,0≲ˆλw\n∥≲MA,0\nM24/5\nA,0\u0010ˆλw\n∥\u0011−1/5vA,0\nℓ0ˆλw(subA)\n∥,∗≲ˆλw\n∥≲M4\nA,0\nM4\nA,0\u0010\nS0ˆλw\n∥\u00111/2vA,0\nℓ0ˆλw(subA)\n∥,min≲ˆλw\n∥≲ˆλw(subA)\n∥,∗\n(15)\nwhere ˆλw(subA)\n∥,∗∼S1/5\n0(MA,0ˆλz(subA)\n⊥,∗)8/5∼S−5/7\n0M8/7\nA,0is the\nwavelength below which the AW packet interacts with back-\nground fluctuations in the TMT regime, while ˆλw(subA)\n∥,min∼\nS1/5\n0(MA,0ˆλz(subA)\n⊥,min)8/5∼S−1\n0is the shortest wavelength at which\nthe turbulent damping is e ffective.\nThe trans-Alfvénic regime is obtained from the above case,\ni.e., when there is no (W0) range:\n[MA,0≃1, with dynamic alignment]\nΓw,q∥\ntransA∼\u0010ˆλw\n∥\u0011−1/5vA,0\nℓ0S−5/7\n0≲ˆλw\n∥≲1\n\u0010\nS0ˆλw\n∥\u00111/2vA,0\nℓ0S−1\n0≲ˆλw\n∥≲S−5/7\n0(16)\nwhere we have explicitly written the transition and dissipation\nscales, i.e., ˆλw(transA)\n∥,∗∼S1/5\n0(ˆλz(transA)\n⊥,∗ )8/5∼S−5/7\n0andˆλw(transA)\n∥,min∼\nS1/5\n0(ˆλz(transA)\n⊥,min)8/5∼S−1\n0, respectively.\nOne can see that when it comes to the interaction of the AW\npacket with anisotropic background fluctuations, including dy-\nnamic alignment in the picture changes the behaviour of the\nturbulent damping rate significantly with respect to the “clas-\nsic” scenario. In the range of scales for which the packet inter-\nacts with critically balanced turbulence (i.e., λw,q∥\n∥≲λw,q∥\n∥,CB), the\ndamping rate due to this nonlinear interaction is always larger\nthan the corresponding rate obtained without dynamic alignment\n(cf. equations (12)-(13) and see the left panel of Figure 1). This\ncan be understood by considering that dynamic alignment means\na shallower perpendicular spectrum of background fluctuations\n(−3/2 instead of−5/3), and thus, at any scale λz\n⊥<λz\n⊥,CB, there\nis more turbulent power to nonlinearly damp the AW packet.\nIn general, if a CR-driven Alfvén-wave is injected in a back-\nground of sub-Alfvénic turbulence with dynamic alignment, now\nthe damping rate interestingly exhibits two breaks that separate\nthe three distinct regimes available in this scenario (contrary to\nthe single break that is present without dynamic alignment). A\nfirst break occurs for wavelengths interacting with background\nfluctuations at the transition scale between weak and strong\nturbulence ( λw,q∥\n∥∼λw,q∥\n∥,CB∼M4\nA,0ℓ0), while a second break\nemerges when the wavelength corresponds to a scale for which\nthe AW packet starts to interact with tearing-mediated turbulence\n(λw,q∥\n∥∼λw(subA)\n∥,∗∼S−5/7\n0M8/7\nA,0ℓ0). In astrophysical situations\nfor which this damping mechanism is the main process that de-\ntermines the e fficiency of CR confinement, these breaks could\nleave a signature at the corresponding energies in the propagated\nspectrum of these cosmic particles.4.2.2. super-Alfvénic turbulence ( MA,0>1) with dynamic\nalignment\nWhen background fluctuations are injected with MA,0>1 and\ndynamic alignment of critically balanced turbulent fluctuations\ntakes place, an AW packet undergoes a turbulent damping with\nthe following rate:\n[MA,0>1, with dynamic alignment]\nΓw,q∥\nsupA∼MA,0\u0010ˆλw\u0011−2/3vA,0\nℓ0M−3\nA,0≲ˆλw≲1\nM12/5\nA,0\u0010ˆλw\n∥\u0011−1/5vA,0\nℓ0ˆλw(supA)\n∥,∗≲ˆλw\n∥≲M−3\nA,0\nM3\nA,0\u0010\nS0ˆλw\n∥\u00111/2vA,0\nℓ0ˆλw(supA)\n∥,min≲ˆλw\n∥≲ˆλw(supA)\n∥,∗\n(17)\nwhere ˆλw(supA)\n∥,∗∼S1/5\n0M6/5\nA,0(ˆλz(supA)\n⊥,∗)8/5∼S−5/7\n0M−6/7\nA,0and the\nshortest wavelength for turbulent damping to be e ffective is\nˆλw(supA)\n∥,min∼S1/5\n0M6/5\nA,0(ˆλz(supA)\n⊥,min)8/5∼S−1\n0.\nAgain, we stress that the isotropic normalized wavelength\nˆλw=λw/ℓ0enters the damping rate in the range ℓA≲λw≲ℓ0,\ni.e., where the AW packet interacts with hydrodynamic-like pre-\nexisting turbulence. In this range of scales, the result is un-\nchanged with respect to the turbulent damping rate obtained\nwithout dynamic alignment (Section 4.1.2). At smaller scales\nλw,q∥\n∥≲ℓA, the quasi-parallel condition in (7) applies again\nand the damping rate depends explicitly on the normalized par-\nallel wavelength ˆλw\n∥=λw,q∥\n∥/ℓ0. In this regime, the turbulent\ndamping rate di ffers significantly from the one obtained with-\nout dynamic alignment. When a scale-dependent alignment of\nfluctuations is taken into account, the turbulent damping is al-\nways much more e ffective than in the case obtained without dy-\nnamic alignment (leading to a damping rate that can be larger\nby orders of magnitude with respect to the one in (14), de-\npending on the Alfvénic Mach nubmer MA,0and on the Lun-\nquist number S0at injection scales; e.g., see right panel Fig-\nure 1). Finally, analogously to the case of sub-Alfvénic injec-\ntion, also when a CR-driven Alfvén-wave packet is injected in\na background of super-Alfvénic turbulence for which dynamic\nalignment can occur, the damping rate exhibits two breaks. The\nfirst break emerges for wavelengths comparable to the transi-\ntion scale between hydrodynamic-like and critically balanced\nturbulence ( λw∼ℓA∼M−3\nA,0ℓ0), while a second break occurs\nat wavelengths corresponding to the scale marking the transition\nbetween a dynamically aligned cascade and the tearing-mediated\nrange (λw,q∥\n∥∼λw(supA)\n∥,∗∼S−5/7\n0M−6/7\nA,0ℓ0).\n5. Feedback on background fluctuations\nWhen deriving the scaling of Γw,q∥\nturbin the previous Section, we\nhave neglected the feedback that the injected Alfvén-wave pack-\nets could have on pre-existing turbulent fluctuations. In general,\nbackground fluctuations could also be a ffected by their nonlinear\ninteraction with these AW packets—let us call it “feedback”. It\nis therefore instructive to understand when such e ffect has to be\ntaken into account. The relevance of this feedback can be esti-\nmated by comparing two timescales:\n1. The intrinsic cascade time of background fluctuations, i.e.,\nthe timescale of the cascade process induced by pre-existing\nArticle number, page 7 of 17A&A proofs: manuscript no. C2024\n(counter-propagating) fluctuations on themselves, τ(z|z)\ncasc∼\nτ(z|z)\nnl/χ(z|z)(or justτ(z|z)\ncasc∼τ(z|z)\nnl, ifχ(z|z)≳1).\n2. The CR-induced cascade time, i.e., the timescale of the cas-\ncade that would be induced by the injected AW packets\non the background fluctuations, τ(z|w)\ncasc∼τ(z|w)\nnl/χ(z|w)(or just\nτ(z|w)\ncasc∼τ(z|w)\nnl, ifχ(z|w)≳1).\nHereafter, the simpler super-script “ z” will be used instead of\n“(z|z)” for the sake of homogeneity of notation with the previ-\nous Sections. Moreover, for the sake of clarity in the qualitative\ndiscussion that will follow, the e ffect of dynamic alignment will\nnot be taken into account in this Section. The nonlinear param-\neterχ(z|w)describing the interaction of background fluctuations\nwith CR-driven AW packets is χ(z|w)\nλ⊥∼(λz\n∥,λ⊥/λz\n⊥)(δwλ⊥/vA,0)∼\n(δwλ⊥/δzλ⊥)χz\nλ⊥, while the timescale associated to such nonlin-\near interaction is τ(z|w)\nnl,λ⊥∼λ⊥/δwλ⊥∼(δzλ⊥/δwλ⊥)τz\nnl,λ⊥. The ratio\nof the two cascade timescales is thus given by:\nτ(z|w)\ncasc\nτz\ncasc∼\u0000δzλ⊥/δwλ⊥\u00012ifχz\nλ⊥<1 andχ(z|w)\nλ⊥≲1\nor\nχz\nλ⊥∼1 andχ(z|w)\nλ⊥<1\n\u0000δzλ⊥/δwλ⊥\u0001χz\nλ⊥ifχz\nλ⊥<1 andχ(z|w)\nλ⊥>1\n\u0000δzλ⊥/δwλ⊥\u0001ifχz\nλ⊥∼1 andχ(z|w)\nλ⊥≳1\n(δzλ/δwλ)(δwλ/vA,0)−1ifχz\nλ>1 andχ(z|w)\nλ<1\n(δzλ/δwλ) ifχz\nλ>1 andχ(z|w)\nλ≳1\n(18)\nFeedback e ffects shall be taken into account if the nonlinear cas-\ncade process that would be induced by the injected Alfvén-wave\npackets becomes faster than the intrinsic cascading process of\nbackground fluctuations, i.e., at scales where τ(z|w)\ncasc/τz\ncasc≲1.\nDeciding whether or not “cosmic-ray feedback” (CRF) has to be\ntaken into account then requires to compare the scale-dependent\namplitudes of both the CR-driven AW packets ( δwλ) and the pre-\nexisting background fluctuations ( δzλ). A precise estimate of the\nscales where such feedback is important thus requires a detailed\nknowledge of how the CRSI saturation level in pre-existing tur-\nbulence depends upon plasma parameters and background con-\nditions, which is not yet achieved. At this stage, we provide only\na general, qualitative discussion.\nLet us consider cases in which the CRSI saturates at a level\n(δw/vA,0)2∼(δB(CRSI)/B0)2≪1. Then, from (18) one can\nsee that when the CR-driven AW packets interact with super-\nAlfvénic turbulence ( MA,0>1), and at scales where the cascade\nis hydrodynamic-like ( λw≳ℓA), the ratioτ(z|w)\ncasc/τz\ncascis typically\nmuch larger than unity. Therefore, cosmic-ray feedback is likely\nnegligible at all scales belonging to the (K41) regime, when the\nCR-driven instability produces fluctuations at a δB(CRSI)/B0≪1\nlevel (which in turn depends also on the presence of a mean\nfield B0). On the other hand, for trans- and sub-Alfvénic injec-\ntion ( MA,0≲1), or for super-Alfvénic injection at scales below\nwhich the hydrodynamic-like cascade transitions to the critically\nbalanced regime ( λw\n∥<ℓ A), the CR feedback on pre-existing tur-\nbulence is negligible only as long as the packets’ amplitudes are\nsmaller than the background fluctuations’ level at that scale. As a\nconsequence, cosmic-ray feedback should be taken into account\nat scalesλ⊥≲λCRF\n⊥, where the (perpendicular) CR-feedbackscaleλCRF\n⊥is defined as the scale at which δwλCRF\n⊥∼δzλCRF\n⊥holds.\nIfδB(CRSI)/B0is sufficiently small, such scale may fall below the\nturbulent dissipation scale and the feedback can be neglected at\nall the scales relevant for CR transport. However, the growth and\nsaturation level of the CRSI depends upon the CR-to-thermal\ndensity ratio nCR/nth. In the Galactic halo this ratio is negligibly\nsmall, and the above reasoning applies. This may not be the case\nnear CR sources, where such density ratio is not so small. We\nthus expect this feedback to be relevant in these environments.\nThis issue will be addressed in more detail and quantitatively in\nthe accompanying Paper II.\nFinally, we remark that the CR-feedback scale discussed\nabove is a perpendicular scale, λCRF\n⊥. However, the CRSI growth\nrate is such that quasi-parallel Alfvén waves λw\n∥≪λw\n⊥are mainly\nproduced. This means that the scale λCRF\n⊥at which the CR-driven\nfluctuations’ amplitude becomes comparable to the amplitude\nof background fluctuations is related to the injected wavelength\nλw,q∥\n∥by the condition (7), i.e.,\nλCRF\n∥∼δzλw,q∥\n⊥\nvA,0λCRF\n⊥≪λCRF\n⊥. (19)\nHence, a quasi-parallel Alfvén waves δwλ∥driven by CRs at\nscalesλw,q∥\n∥≲λCRF\n∥can actually a ffect pre-existing turbulent\nfluctuations δzλ⊥on much larger scales λCRF\n⊥≳λ⊥≫λw,q∥\n∥.\n(We recall that we assume that nonlinear interactions are local\nin perpendicular scale, so in what follows we do not need to dis-\ntinguish between the two scales λw\n⊥andλz\n⊥, i.e.,λw\n⊥∼λz\n⊥∼λ⊥).\nIn the following, we attempt to provide two phenomenolog-\nical models for the CR-modified cascade of background fluctu-\nations. However, these are just plausible models at this stage.\nFocused numerical investigations will be necessary in order to\nverify if (and under which circumstances) they can be realized.\n5.1. A phenomenological model for CR-modified scaling of\npre-existing turbulent fluctuations\nA simple phenomenological model for the cascade modified by\nthe CR-generated AW packets can be constructed as follows.\nLet assume that, at scales where χz\nλ⊥≲1, pre-exiting fluc-\ntuations and their anisotropy follow the scaling δz(0)\nλ⊥∝λαz\n⊥\n⊥and\nλ∥,λ⊥∝λδz\n⊥, respectively (with αz\n⊥>0 andδz=1−αz\n⊥if critical\nbalance holds, χz∼1, orδz=0 for weak turbulence, χz<1).\nSuch scaling for the fluctuations corresponds to a perpendicular\npower spectrum E(0)\nδz(k⊥)∝k−ξz\n⊥\n⊥, withξz\n⊥=1+2αz\n⊥. At scales\nwhereχz\nλ⊥>1, the scaling is isotropic (i.e., δz=1), and so\nδz(0)\nλ∝λαzandE(0)\nδz(k)∝k−ξz, withξz=1+2αz. These are the\n“unperturbed” properties of background fluctuations, i.e., with-\nout CR feedback, and are thus denoted by a “(0)” superscript.\nThen let assume a scaling δwλ∥∝(λw\n∥)−αw\n∥for the CR-driven\n(quasi-parallel) fluctuations, corresponding to a parallel power\nspectrum ECRSI(kw\n∥)∝(kw\n∥)ξw\n∥, whereξw\n∥=2αw\n∥−1. Due to the\nquasi-parallel condition (7) (valid if χz\nλ⊥≲1, i.e. at scales where\nbackground fluctuations are, or have become, sub-Alfvénic and\nanisotropic) the corresponding perpendicular scaling for these\nfluctuations is typically steeper than its parallel counterpart, and\nis related to the perpendicular scaling of pre-existing fluctua-\nArticle number, page 8 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\ntions, namely7δwλ⊥∝λ−αw\n⊥\n⊥withαw\n⊥=αw\n∥(1+αz\n⊥); the cor-\nresponding perpendicular spectrum is then ECRSI(k⊥)∝(k⊥)ξw\n⊥,\nwithξw\n⊥=2αw\n∥(1+αz\n⊥)−1=ξw\n∥+(ξw\n∥+1)(ξz\n⊥−1)/2. On the\nother hand, at scales where χz\nλ>1 this distinction between αw\n⊥\nandαw\n∥is lost, and we just assume a scaling δwλ∝λ−αwand a\n(isotropic) power spectrum ECRSI(k)∝(k)ξwwithξw=2αw−1.\nWe now want to know how the scaling of δz(0)are modi-\nfied by the presence of δw, knowing that we are in a regime in\nwhich such feedback is faster that the intrinsic cascade time of\nδz(0)(viz.,τ(z|w)\ncasc≪τz\ncasc). In this limit, one can verify that the CR\nfeedback on pre-existing fluctuations is also faster than the tur-\nbulent damping of CR-driven fluctuations ( viz.,Γw,q∥\nturbτ(z|w)\ncasc≪1).8\nThis condition allows to neglect the mutual feedback between\nbackground fluctuations and CR-driven Alfvén waves, i.e., to\nconsider only the modification to the scaling of pre-existing tur-\nbulence induced by the (undamped) AW packets generated by\nCRs. Within these assumptions, we can derive the scaling of CR-\nmodified turbulence by replacing the cascade timescale,\nτz\ncasc→τ(z|w)\ncasc, (20)\nby rescaling the unperturbed background fluctuations into a “1st-\norder modified” fluctuations (denoted by a “(1)” superscript),\nδz(0)\nλ⊥→δz(1)\nλ⊥=ζλ⊥δz(0)\nλ⊥, (21)\nand by requiring that the cascade rate is still scale-inependent,\ni.e., (δz(1)\nλ⊥)2/τ(z|w)\ncasc∼ε∼const. This procedure readily provides\nthe scaling factor\nζλ⊥∼τ(z|w)\ncasc\nτz\ncasc1/2\n, (22)\nwhich can be estimated using (18) for the various χzandχw\nregimes, and, as a first approximation, by employing the unper-\nturbed scaling δz(0)\nλ⊥; the rescaling factor computed in this way\nwill be denoted as ζ(0)\nλ⊥.9The resulting CR-modified perpendicu-\nlar power spectrum of background fluctuations is then given by\n7The quasi-parallel condition, i.e., kw\n∥∼(δzk⊥/vA,0)−1k⊥, and condition\non the total energy, i.e., ( kw\n∥)−1(δwkw\n∥)2dkw\n∥∼k−1\n⊥(δwk⊥)2dk⊥, have been\nemployed to derive the perpendicular scaling of the fluctuations, δwk⊥.\n8The condition τ(z|w)\ncasc≪τz\ncascindeed implies Γw,q∥\nturbτ(z|w)\ncasc≪Γw,q∥\nturbτz\ncasc,\nand one can show that Γw,q∥\nturbτz\ncasc≲1 for any value of χwandχz. In\ntheχw<1 regime, Γw,q∥\nturbτz\ncasccan be estimated using (11), obtaining\nthatΓw,q∥\nturbτz\ncasc∼(λ/ℓ 0)(δzλ⊥/vA,0) in theχz<1 regime, or Γw,q∥\nturbτz\ncasc∼\n(δzλ⊥/vA,0)2ifχz≥1. Thus, Γw,q∥\nturbτz\ncasc≪1 holds for any χzat scales\nλ⊥≪ℓ0, which further implies the inequality Γw,q∥\nturbτ(z|w)\ncasc≪1. The case\nχw≥1, on the other hand, is attained only in the χz>1 regime and at\nscalesλ⊥≥ℓA. In this case, one should use (10) and would find that\nΓw,q∥\nturbτz\ncasc∼1. Thus, the ordering Γw,q∥\nturbτ(z|w)\ncasc≪1 still holds for χw≥1.\n9One can imagine to perform an expansion of the rescaling factor\nbased on iteratively modified timescales τz(n)\ncasc,λ⊥, and rewrite it as a series\nζλ⊥=(τ(z|w)\ncasc,λ⊥)1/2\u0010P\nn1/τz(n)\ncasc,λ⊥\u00111/2=ζ(0)\nλ⊥\u0010\n1+P\nn>0τz(0)\ncasc,λ⊥/τz(n)\ncasc,λ⊥\u00111/2\n(recall that we have assumed that the CR-driven fluctuations δwλ⊥are\nunaffected by background fluctuations). The ratio of 0th to nth timescale\nisτz(0)\ncasc,λ⊥/τz(n)\ncasc,λ⊥∝δz(n)\nλ⊥/δz(0)\nλ⊥(or even∝(δz(n)\nλ⊥/δz(0)\nλ⊥)2). Then, if the\nCR-induced modification of the pre-existing spectrum is a steepening,\nthe cascade timescale would significantly increase at increasing n, i.e.,\nτz(n)\ncasc,λ⊥≫τz(n−1)\ncasc,λ⊥≫···≫τz(1)\ncasc,λ⊥≫τz(0)\ncasc,λ⊥. So, if the series converges\nand its contribution is negligible (which shall be verified), ζλ⊥≈ζ(0)\nλ⊥.E(1)\nδz(k⊥)∼(ζ(0)\nλ⊥)2E(0)\nδz(k⊥), i.e., E(1)\nδz(k⊥)∝k−(ξz\n⊥+ ∆ξCRF\n⊥)\n⊥ , where\nthe CR-induced modification of the spectral index is\n∆ξCRF\n⊥∼1\n2(ξz\n⊥+1)(ξw\n∥+3)−2 ifχz\nλ⊥<1 andχ(z|w)\nλ⊥≲1\nor\nχz\nλ⊥∼1 andχ(z|w)\nλ⊥<1\n1\n4(ξz\n⊥+1)(ξw\n∥+5)−3 ifχz\nλ⊥<1 andχ(z|w)\nλ⊥>1\n1\n4(ξz\n⊥+1)(ξw\n∥+3)−1 ifχz\nλ⊥∼1 andχ(z|w)\nλ⊥≳1\n1\n2(ξz+2ξw+1) if χz\nλ>1 andχ(z|w)\nλ<1\n1\n2(ξz+ξw) if χz\nλ>1 andχ(z|w)\nλ≳1\n(23)\nWe remind that this result is only valid at scales where ζλ⊥<1,\nand only if Γw,q∥\nturbτ(z|w)\ncasc≪1 (i.e., if the CR-driven Alfvén-wave\npackets are una ffected by pre-existing fluctuations). It is interest-\ning to note that the anisotropy of the pre-existing cascade would\nbe una ffected ifχ(z|w)<1 (leavingℓ∥,λ⊥≈const.ifχz<1,\norℓ∥,λ⊥∝λδz\n⊥ifχz∼1), or ifχz>1 (leavingℓ∥,λ⊥∝λ⊥).\nOn the other hand, if χ(z|w)∼1, a critical balance between the\nAlfvén time τz\nAand the nonlinear time τ(z|w)\nnlwould be estab-\nlished, leading to a modified anisotropy ℓ∥,λ⊥∝λ1+αw\n⊥\n⊥=λδz+δCRF\n⊥\nwithδCRF=αw\n∥+αz\n⊥(αw\n∥−1). This means that the anisotropy of\npre-existing fluctuations would be reduced if αw\n∥< αz\n⊥/(1+αz\n⊥)\n(e.g., for GS95 turbulence, this means αw\n∥<1/4, corresponding\nto a CR-driven spectrum ∝k−1/2\n∥or steeper), and it would be\ninstead increased otherwise.\n5.1.1. Over-critical interaction ( χ(z|w)>1) and alternative\nCR-modified scaling of pre-existing fluctuations\nWhen nonlinear interactions between the CR-driven Alfvén-\nwave packets and background turbulence are “over-critical”, i.e.,\nχ(z|w)>1, it is reasonable to consider that pre-existing scal-\ning are not just perturbatively modified. Therefore, we present\nan alternative model in which the intrinsic cascade time of pre-\nexisting fluctuations τz\ncascis completely replaced by the nonlinear\ntimescaleτ(z|w)\nnl, without further re-scaling of δz(0)as in (22). This\nallows to directly derive the CR-modified scaling of background\nfluctuations δzCRFby requiring ( δzCRF)2/τ(z|w)\nnl∼ε=const.\nIfχz≲1, pre-existing scaling are anisotropic, and the condi-\ntion above yields the perpendicular scaling\nδwλ⊥(δzCRF\nλ⊥)2\nλ⊥∼ε=const.⇒δzCRF\nλ⊥∝λ(1+αw\n⊥)/2\n⊥. (24)\nThis corresponds to a modified perpendicular spectrum\nECRF\nδz(k⊥)∝k−(ξz\n⊥+∆ξCRF\n⊥)\n⊥ with\n∆ξCRF\n⊥=ξw\n∥+ξz\n⊥(ξw\n∥−3)+9\n4, (25)\nwhere the link to the the original scaling of pre-existing fluctua-\ntions is a consequence of the quasi-parallel condition (7).\nIfχz>1, CR-modified fluctuations would follow the\nisotropic scaling δzCRF\nλ∝λ(1+αw)/2, corresponding to a spectrum\nECRF\nδz(k)∝k−(ξw+5)/2, (26)\nwhich does not depend on the original scaling of pre-existing\nfluctuations due to the loss of “quasi-parallel” concept.\nArticle number, page 9 of 17A&A proofs: manuscript no. C2024\n6. Discussion and conclusions\nThe turbulent damping of an Alfvén-wave (AW) packet excited\nby cosmic-rays (CRs) in pre-existing incompressible magneto-\nhydrodynamic (MHD) turbulence is re-examined by carefully\ntaking into account the role of the “nonlinearity parameter” χw\nthat quantifies the strength of the nonlinear interaction between\nthe packet and background fluctuations. In particular, the dif-\nference between χwand the nonlinear parameter χzthat instead\ndescribes the regime of background turbulence (i.e., the intrinsic\nstrength of nonlinear interactions between pre-existing fluctua-\ntions) has been elucidated. The derivation of turbulent damping\nrates in a “classic” MHD turbulence scenario (i.e., without the\nso-called “dynamic alignment”) has thus been revised taking into\naccount such di fference between χwandχz, and new scaling re-\nlations for the damping rates have been obtained. Furthermore,\nby considering most-recent theories of MHD turbulence that ac-\ncount for a scale-dependent (“dynamic”) alignment of fluctu-\nations and the possibility of a reconnection-mediated regime,\ncompletely new damping rates have been also obtained for the\nfirst time. Finally, the role of cosmic-ray feedback (CRF) on\npre-existing turbulence is also examined and a simple criterion\nfor CRF e ffects is derived. Two very simple phenomenological\nmodels of CR-modified scaling of background fluctuations are\nalso obtained. Taking into account the feedback of CR-generated\nfluctuations on pre-existing turbulence may be relevant in astro-\nphysical environments where the density of cosmic-ray nCRis\nnot negligible with respect to the density of the background ther-\nmal plasma nth(e.g., near CR sources). The issue of CRF ef-\nfects, as well as the role of other damping mechanisms, will be\naddressed in more detail in the following “Paper II”.\nThe main features of the new turbulent damping rates ob-\ntained in this work can be summarized as follows:\n•The nonlinear interaction between a quasi-parallel (q ∥) AW\npacket and pre-existing anisotropic turbulence is always\nweak (equations (8)-(9)). As a result, the turbulent damping\nrate of the packets depends on the background-fluctuations’\namplitude to the third power (equation (11)), and thus\nis strongly suppressed with respect to what previously\nestimated. This is true at any wavelength when the AW\npacket interacts with sub- and trans-Alfvénic turbulence,\nand also for those packets whose wavelength interacts with\nfluctuations at scales where they become critically balanced\nin the case of super-Alfvénic injection.\n•How the turbulent damping rate Γw,q∥\nturbdepends on (i)\nthe AW-packet’s parallel wavelength λ∥(and thus on the\nCR gyro-radius from which it is excited) and on (ii) the\ninjection-scale Alfvénic Mach number MA,0, in the “classic”\nMHD turbulence scenario is significantly di fferent from\nwhat is present in the existing literature (equations (12)-\n(14)). In fact, the damping rate agrees with the literature only\nwhen the AW packet interacts with isotropic (“K41”) turbu-\nlence, namely Γw\nK41∝MA,0λ−2/3. On the contrary, when the\npacket interacts with weak turbulence (“W0”), the damping\nrate scales as Γw,q∥\nW0∝M8/3\nA,0λ1/3\n∥(instead of∝M8/3\nA,0λ−2/3\n∥\npreviously obtained), while when it interacts with critically\nbalanced turbulence (“GS95”), turbulent damping does not\ndepend on the wavelength, i.e., Γw,q∥\nGS95∼const.(and it is\n∝M4\nA,0forMA,0<1 or∝M3\nA,0ifMA,0>1, instead of\n∝M2\nA,0λ−1/2\n∥or∝MA,0λ−1/2\n∥, respectively, as reported in\nexisting literature).•Including dynamic alignment of pre-existing fluctuations in\nthe picture, and thus allowing also for the possibility of a\nreconnection-mediated range, introduces novel regimes and\nbreaks in the turbulent damping rate (equations (15)-(17)).\nWhen a quasi-parallel AW packet interacts with critically\nbalanced and dynamically aligning anisotropic turbulence\n(“B06”), it is subject to a damping rate Γw,q∥\nB06∝M24/5\nA,0λ−1/5\n∥\nifMA,0<1 orΓw,q∥\nB06∝M12/5\nA,0λ−1/5\n∥ifMA,0>1. Alfvén-wave\npackets that interact with tearing-mediated turbulence\n(“TMT”) are instead subject to a damping rate that is now\nsensitive also to the injection-scale Lundquist number S0,\nand it scales as Γw,q∥\nTMT∝M4\nA,0(S0λ∥)1/2ifMA,0<1 or\nΓw,q∥\nTMT∝M3\nA,0(S0λ∥)1/2ifMA,0>1.\n•Accounting for dynamic alignment (and tearing-mediated\nturbulence) introduces two breaks in the turbulent damping\nrate, instead of the single one that is present in the “classic”\npicture. For sub-Alfvénic turbulence, the first break corre-\nsponds to the transition scale between weak and strong tur-\nbulence (i.e., λ∥∼M4\nA,0ℓ0in terms of parallel wavelength\nof the AW packet), while it corresponds to the transition\nscale between isotropic and anisotropic turbulence for super-\nAlfvénic injection (i.e., λ∼M−3\nA,0ℓ0). This is the same type\nof break that one would have in “classic” MHD turbulence.\nA second break, on the other hand, emerges due to the tran-\nsition to tearing-mediated turbulence (which is only possi-\nble if dynamic alignment occurs), i.e., at a packet’s paral-\nlel wavelength λ∥∼S−5/7\n0M8/7\nA,0ℓ0in sub-Alfvénic turbu-\nlence, or atλ∥∼S−5/7\n0M−6/7\nA,0ℓ0for super-Alfvénic injection.\n(We recall that MA,0andS0are the Alfvénic Mach number\nof turbulent fluctuations and Lunquist number of the back-\nground plasma at injection scale ℓ0, respectively). Since CR\n“self-confinement” relies on a balance between the growth\nof these CR-driven Alfvén waves and their damping, it is\nreasonable to imagine that in astrophysical situations where\nturbulent damping is the most-relevant damping mechanism,\nthese breaks would emerge also in the propagated CR spec-\ntrum. It is thus interesting to mention that, assuming a Galac-\ntic magnetic field B∼1-3µG and an injection scale of back-\nground turbulence ℓ0∼30-100 pc, the above breaks in the\ndamping rate could be translated to CR energies ECR(assum-\ningλ∥∼rL, where rLis the Larmor radius of the cosmic ray).\nA first break at ECR,1∼10 TeV would indeed emerge if the\ninjection-scale Alfvénic Mach number MA,0of pre-existing\nturbulence is in the range 0 .07≲MA,0≲0.14 (i.e., sub-\nAlfvénic injection with MA,0of order∼0.1) or in the range\n15≲MA,0≲30 (i.e., super-Alfvénic injection with MA,0of\norder∼10). Additionally, for both MA,0regimes determined\nabove, a second break at CR energies ECR,2∼300 GeV\nwould be consistently recovered if the Lundquist number of\nthe background plasma is of order S0∼105–107.\nIt is worth concluding by stressing once more that the\nturbulent damping rates obtained in this work di ffer dramatically\nfrom the ones that are present in the literature even for “classic”\nMHD turbulence due to the confusion between χwandχz, and\nthat these previous results available in the literature are the\none that have been considered in existing CR studies. Hence, a\nnumber of previous works on CR self-confinement may need to\nbe revised in view of the results presented here.\nArticle number, page 10 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\nAcknowledgements. The author warmly acknowledges the kind hospitality of\nthe Gran Sasso Science Institute (GSSI) in L’Aquila, and, in particular, Pasquale\nBlasi and Ottavio Fornieri, who introduced him to the paper by Farmer & Gol-\ndreich on turbulent damping during his first visit in November 2022. He also\ngreatly acknowledges Alexandre Marcowith, Thierry Passot, and Steve Shore\nfor helpful comments and feedback on the manuscript. The author is grateful\nto Stas Boldyrev, Ben Chandran, Matt Kunz, Alex Lazarian, Bill Matthaeus,\nThierry Passot, Alex Schekochihin, and Pierre-Louis Sulem for useful conver-\nsations (and di fferent opinions) about MHD turbulence theory over the years.\nThis work has been partially supported by the French government, through the\nUCAJEDIInvestments in the Future project managed by the National Research\nAgency (ANR) with the reference number ANR-15-IDEX-01. 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Scattering of Cosmic Rays by Magnetohydro-\ndynamic Interstellar Turbulence. Phys. Rev. Lett. , 89(28):281102.\nYan, H. and Lazarian, A. (2008). Cosmic-Ray Propagation: Nonlinear Dif-\nfusion Parallel and Perpendicular to Mean Magnetic Field. Astrophys. J. ,\n673(2):942–953.\nArticle number, page 12 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\nAppendix A: Scaling of turbulent fluctuations at\nmagnetohydrodynamic (MHD) scales\nIn this Appendix we briefly review the turbulent scaling\nof Alfvénic fluctuations at “fluid” (MHD) scales (see, e.g.,\nSchekochihin 2022, for a more detailed review on the topic).\nFirst, we present the standard scenario without scale-dependent\nalignment of turbulent fluctuations. In the second part, an al-\nternative scenario in which such “dynamic alignment” is taking\nplace in the critically balanced regime is presented.\nIn the following, isotropic injection will be assumed and ℓ0\nwill denote the injection scale (i.e., the injection properties will\nnot be a ffected by the presence of a mean magnetic-field direc-\ntionB0at that scale). Balanced injection will also be assumed,\ni.e., that the same amount of energy is injected in the Elsässer\nfields atℓ0,|δz+\n0|2≈|δz−\n0|2=δz2\n0. We thus define an injection-\nscale Alfvénic Mach number MA,0=δz0/vA,0≈δb0/B0, where\nvA,0=B0/p\n4πρ0is the Alfvén speed associated to the back-\nground plasma (having mass density ρ0and being embedded in\na mean field B0). For isotropic injection, the nonlinear parameter\nat injection scales χ0=(k⊥,injδz0)/(k∥,injvA,0) (see (5)) identifies\nwith the injection-scale Alfvénic Mach number, i.e., χ0=MA,0.\nAnalogously, we define the injection-scale Lundquist number\nS0=ℓ0vA,0/η, whereηis the resistivity of the background\nplasma. The Lunquist and the Alfvénic Mach numbers can be\ncombined to provide the (injection-scale) magnetic Reynolds\nnumber Rm 0=ℓ0δz0/η=MA,0S0.\nAn example of the resulting turbulent spectra and fluctua-\ntions’ anisotropy for di fferent injection regimes and type of cas-\ncades are summarized in Figures A.1 and A.2.\nA.1. MHD turbulence without dynamic alignment\nIn this Section, we present what can be called as the “classic\ncascade” of Alfvénic fluctuations, i.e., the standard scenario in\nwhich a scale-dependent alignment of turbulent fluctuations is\nnot taken into account. In this case, depending on the large-\nscale regime of injection, turbulence can start as either fluid-like\n(Kolmogorov 1941) or wave-like (Ng and Bhattacharjee 1997;\nGaltier et al. 2000), until the point at which the cascade tran-\nsitions into a critically balanced state (Goldreich and Sridhar\n1995) and eventually reaches dissipation.\nSub- and Trans-Alfvénic injection ( MA,0≤1).When the in-\njection conditions are isotropic and sub-Alfvénic, i.e., such that\nχ0=MA,0<1, then fluctuations initially cascade in a weakly\nnonlinear regime. During that weak cascade only smaller per-\npendicular scales λ⊥<ℓ 0are generated, while λ∥∼ℓ0∼const.10\n10This result is formally obtained through wave-turbulence theory (Ng\nand Bhattacharjee 1997; Galtier et al. 2000). In weak MHD turbulence,\nthe main contribution to the cascade is the three-wave interaction, where\nan Alfvén wave with frequency ω±\n1and wave-vector k±\n1nonlinearly in-\nteracts with a counter-propagating Alfvén wave having frequency ω∓\n2\nand wave-vector k∓\n2in order to generate a third wave with ω3and\nk3. The resonance conditions for this process essentially correspond to\n“momentum” and “energy” conservation laws, namely k±\n1+k∓\n2=k3\nandω±\n1+ω∓\n2=ω3. Since for Alfvén waves these conditions on parallel\nwave-vectors become k±\n1,∥−k∓\n2,∥=±k3,∥andk±\n1,∥+k∓\n2,∥=k3,∥, the only\nnon-trivial solution requires that either k∓\n2,∥=0 and k3,∥=k±\n1,∥, ork±\n1,∥=0\nandk3,∥=k∓\n2,∥. This means that the parallel wave-vector does not change\nduring the three-wave interaction and only smaller perpendicular scales\nwith k3,⊥=k1,⊥+k2,⊥are generated by the weak cascade.The cascade timescale in such weak regime is\nτ(subA)\ncasc,λ⊥∼τ(subA)\nnl,λ⊥\nχ(subA)\nλ⊥∼vA,0\nℓ0λ⊥\nδz(subA)\nλ⊥2\n, (A.1)\nfrom which the fluctuations’ scaling in the inertial range are ob-\ntained by requiring a constant energy cascading rate εthrough\nscales:\n(δz(subA)\nλ⊥)2\nτ(subA)\ncasc,λ⊥∼ε=const.⇒δz(subA)\nλ⊥\nvA,0∼MA,0 λ⊥\nℓ0!1/2\n,(A.2)\nwhere we have used the fact that the cascading rate is constant\nthrough scales and thus it is the same as the injection rate, i.e.,\nε∼ε0∼δz2\n0/τ(subA)\ncasc,0∼M4\nA,0v3\nA,0/ℓ0. The fluctuations’ power\nspectrum is obtained as Eδz∼(δzk⊥)2/k⊥, and thus the one asso-\nciated to the weak cascade is E(subA)\nδz(k⊥)∝k−2\n⊥(here and in the\nfollowing, we explicitly employ the more familiar wave-vector\nnotation k⊥∼λ−1\n⊥for the spectrum).\nThe weak cascade would reach a dissipation scale λ(subA)\n⊥,dissif the\nnonlinear timescale τnl,λ⊥∼λ⊥/δzλ⊥becomes comparable to the\ncharacteristic dissipation time τdiss,λ⊥∼λ2\n⊥/η, i.e.,\nλ(subA)\n⊥,diss\nδz(subA)\nλ⊥,diss∼(λ(subA)\n⊥,diss)2\nη⇒λ(subA)\n⊥,diss\nℓ0∼(MA,0S0)−2/3. (A.3)\nHowever, the scaling in (A.2) implies that the nonlinear param-\neterχλ⊥increases with decreasing scales, i.e.,\nχ(subA)\nλ⊥∼ℓ0/vA,0\nλ(subA)\n⊥/δz(subA)\nλ⊥∼MA,0 λ⊥\nℓ0!−1/2\n, (A.4)\nand will thus achieve critical balance at a perpendicular scale\nχ(subA)\nλ⊥,CB∼1⇒λ⊥,CB\nℓ0∼M2\nA,0. (A.5)\nA transition to strong turbulence occurs only if λ⊥,CB≫λ(subA)\n⊥,diss,\nand comparing (A.3) and (A.5), this means only if S0≫M−4\nA,0.\nAt scales below λ⊥,CBturbulence stays critically balanced and\nthe cascade timescale identifies with the nonlinear time, i.e.,\nτ(subA)\ncasc,λ⊥<λ⊥,CB∼τ(subA)\nnl,λ⊥∼λ⊥\nδz(subA)\nλ⊥(A.6)\nAs a result, the scaling of turbulent fluctuations at λ⊥<λ⊥,CBis\nsuch that\n(δz(subA)\nλ⊥<λ⊥,CB)2\nτ(subA)\nnl,λ⊥<λ⊥,CB∼ε=const.⇒δz(subA)\nλ⊥<λ⊥,CB\nvA,0∼M4/3\nA,0 λ⊥\nℓ0!1/3\n,\n(A.7)\nwhile the critical-balance condition τA,λ⊥∼τnl,λ⊥sets the scale-\ndependent anisotropy of turbulent fluctuations,\nλ∥,λ⊥<λ⊥,CB\nvA,0∼λ⊥\nδz(subA)\nλ⊥<λ⊥,CB⇒λ∥,λ⊥<λ⊥,CB\nℓ0∼M−4/3\nA,0 λ⊥\nℓ0!2/3\n(A.8)\nEquation (A.8) implies that below λ⊥,CBthe critically balanced\ncascade starts to generate also smaller parallel scales. From (A.7)\nArticle number, page 13 of 17A&A proofs: manuscript no. C2024\none can obtain a reduced (one-dimensional) perpendicular spec-\ntrumE(subA)\nδz(k⊥λ⊥,CB>1)∝k−5/3\n⊥ and a reduced parallel spec-\ntrumE(subA)\nδz(k∥)∝k−2\n∥.11\nThe above cascade eventually reaches dissipation at a scale\nλ(subA)\n⊥,dissfor which the nonlinear and dissipation timescales be-\ncome comparable, τ(subA)\nnl,λ⊥∼τdiss,λ⊥, i.e.,12\nλ(subA)\n⊥,diss\nδz(subA)\nλ⊥,diss∼(λ(subA)\n⊥,diss)2\nη⇒λ(subA)\n⊥,diss\nℓ0∼M−1\nA,0S−3/4\n0. (A.9)\nSuper-Alfvénic injection ( MA,0>1).When fluctuations are\ninjected (isotropically) with χ0=MA,0>1, the resulting tur-\nbulence starts as a strong “hydrodynamic-like” cascade, i.e., tur-\nbulence is isotropic and nearly insensitive to the presence of a\nbackground magnetic field for as long as δb/B0>1 holds (that\nis, until the presence of a mean field starts to play a role in the\ncascade, at smaller scales). In the hydro-like range, fluctuations\ncascade with the nonlinear characteristic timescale,\nτ(supA)\ncasc,λ∼τ(supA)\nnl,λ∼λ\nδz(supA)\nλ, (A.10)\nwhereλis the isotropic wavelength of the fluctuations. The scal-\ning for the fluctuating Elsässer variable immediately follow from\nthe constancy of the energy cascade rate ε, i.e.,\n(δz(supA)\nλ)2\nτ(supA)\nnl,λ∼ε=const.⇒δz(supA)\nλ\nvA,0∼MA,0 λ\nℓ0!1/3\n,(A.11)\nwhich corresponds to a Kolmogorov-like, isotropic fluctuations’\npower spectrumE(supA)\nδz(k)∝k−5/3. This cascading regime goes\non until it reaches dissipation, i.e., λ(supA)\ndiss∼(MA,0S0)−3/4ℓ0.\nHowever, there is another important scale usually referred to as\nthe “Alfvén scale” ℓAfor whichδz(supA)\nλ∼vA,0, given by\nℓA\nℓ0∼M−3\nA,0, (A.12)\nwhich is attained well before dissipation (i.e., ℓA≫λ(supA)\ndiss) only\nifS0≫M3\nA,0, and below which the cascade becomes critically\nbalanced (χ(supA)\nℓA∼1) and thus anisotropic. Turbulent fluctua-\ntions at scales λ⊥<ℓ Athus follow the (GS95) scaling, i.e.,\nδz(supA)\nλ⊥<ℓA\nvA,0∼ λ⊥\nℓA!1/3\n∼MA,0 λ⊥\nℓ0!1/3\n, (A.13)\nwith a fluctuations’ wavelength anisotropy that now follows the\nrelationλ∥,λ⊥<ℓA/ℓ0∼M−1\nA,0(λ⊥/ℓ0)2/3. This corresponds to re-\nduced perpendicular and parallel power spectra at k⊥ℓA≳1 that\n11There are di fferent ways to obtain the reduced parallel spectrum,\ngiven the anisotropy in (A.8). One option is to invert the anisotropy\nrelation to obtain the scaling of δzk∥∝k−1/2\n∥, and then use the critical-\nbalance condition to employ τA,k∥∼(k∥vA,0)−1instead ofτnlin the\ncondition (δzk∥)2/τA,k∥∼ε. Another way is to use the condition that\nthe total energy must be obtained by integrating both one-dimensional\nspectra independently, i.e.,R\ndk∥E(k∥)=Etot=R\ndk⊥E(k⊥), and using\nthe anisotropy relation to rewrite E(k⊥) and d k⊥.\n12Here we implicitly assume that the condition λ(subA)\n⊥,diss≪λ⊥,CBholds\nalso when the dissipation scale is computed using (A.9), which is indeed\nautomatically fulfilled as long as S0≫M−4\nA,0.are∝k−5/3\n⊥ and∝k−2\n∥, respectively.\nThe dissipation scale λ(supA)\n⊥,dissin the super-Alfvénic regime\nis given again by matching the scale-dependent nonlinear\ntimescaleτ(supA)\nnl,λ⊥and the dissipation timescale τdiss,λ⊥for this\ntype of cascade, i.e.,\nλ(supA)\n⊥,diss\nδz(supA)\nλ⊥,diss∼(λ(supA)\n⊥,diss)2\nη⇒λ(supA)\n⊥,diss\nℓ0∼(MA,0S0)−3/4. (A.14)\nThe condition S0≫M3\nA,0ensures that the above scale is well\nbelow the Alfvén scale, i.e., λ(supA)\n⊥,diss≪ℓA(cf. (A.12) and (A.14)).\nA.1.1. Summary of scaling for MHD turbulence without\ndynamic alignment\nIf one puts all the relations of Section A.1 back together, then the\nscaling for the (normalized) fluctuation amplitudes δˆz=δz/vA,0\nat MHD scales are the following. (We remind the reader that\nˆλ⊥=λ⊥/ℓ0is the normalized perpendicular wavelength).\nMA,0≤1regime (no dynamic alignment, S0≫M−4\nA,0):\nδˆz(subA)\nˆλ⊥∼MA,0ˆλ1/2\n⊥ˆλ⊥,CB<ˆλ⊥≤1 [W0]\nM4/3\nA,0ˆλ1/3\n⊥ˆλ(subA)\n⊥,diss<ˆλ⊥≤ˆλ⊥,CB [GS95]\n(A.15)\nwhere ˆλ⊥,CB∼M2\nA,0andˆλ(subA)\n⊥,diss∼M−1\nA,0S−3/4\n0, while the fluctua-\ntions’ anisotropy is given by\nˆλ(subA)\n∥,ˆλ⊥∼const. ˆλ⊥,CB<ˆλ⊥≤1 [W0]\nM−4/3\nA,0ˆλ2/3\n⊥ˆλ(subA)\n⊥,diss<ˆλ⊥≤ˆλ⊥,CB [GS95]\n(A.16)\nwhere ˆλ∥,ˆλ⊥=λ∥,ˆλ⊥/ℓ0is the normalized parallel wavelength of\nturbulent fluctuations.\nMA,0>1regime (no dynamic alignment, S0≫M3\nA,0):\nδˆz(supA)\nˆλ⊥∼MA,0ˆλ1/3 ˆℓA<ˆλ≤1 [K41]\nMA,0ˆλ1/3\n⊥ˆλ(supA)\n⊥,diss<ˆλ⊥≤ˆℓA [GS95]\n(A.17)\nwhere ˆℓA∼M−3\nA,0andˆλ(supA)\n⊥,diss∼(MA,0S0)−3/4, while fluctuations\nexihbit an anisotropy\nˆλ(supA)\n∥,ˆλ⊥∼ˆλ⊥∼ˆλ ˆℓA<ˆλ≤1 [K41]\nM−1\nA,0ˆλ2/3\n⊥ˆλ(supA)\n⊥,diss<ˆλ⊥≤ˆℓA [GS95]\n(A.18)\nwith ˆλ∥,ˆλ⊥=λ∥,ˆλ⊥/ℓ0.\nSee Figures A.1 and A.2 for the resulting spectra and fluctu-\nations’ anisotropy versus perpendicular wavenumber k⊥.\nArticle number, page 14 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\nA.2. MHD turbulence with scale-dependent alignment\nIn this Section, we present an alternative model to the “classic\npicture” of the Alfvénic cascade presented in Section A.1. In\nthis case, after a hydrodynamic- or wave-like range, the cas-\ncade transitions into a critically balanced state in which fluc-\ntuations undergo a “dynamic” (i.e., scale-dependent) alignment\nprocess (Boldyrev 2006). Such dynamically aligned, critically\nbalanced cascade can further transition into a so-called “tearing-\nmediated regime” at MHD scales (Boldyrev and Loureiro 2017;\nMallet et al. 2017), before reaching the actual dissipation scales.\nSub- and Trans-Alfvénic injection ( MA,0≤1) with dynamic\nalignment. For isotropic and sub-Alfvénic injeciton, i.e., such\nthatχ0=MA,0<1, fluctuations initially develop a weak cas-\ncade following the same scaling as in (A.2). Dynamic alignment\nenters the picture only as soon as critical balance is reached, i.e.,\nat scalesλ⊥≤λ⊥,CB∼M2\nA,0ℓ0: the idea behind this e ffect is that\nElsässer fields tend to align in order to reduce the strength of\nnonlinearities.13Such process produces a scale-dependent angle\nbetweenδz+\nλ⊥andδz−\nλ⊥that scales as\nsinθ(subA)\nλ⊥<λ⊥,CB∼M−1/2\nA,0 λ⊥\nℓ0!1/4\n, (A.19)\nwhich in turns appears explicitly in the nonlinear-time scaling,\nnamely,\nτ(subA)\nnl,λ⊥<λ⊥,CB∼λ⊥\nδz(subA)\nλ⊥<λ⊥,CBsinθ(subA)\nλ⊥<λ⊥,CB∼M1/2\nA,0λ3/4\n⊥ℓ1/4\n0\nδz(subA)\nλ⊥<λ⊥,CB(A.20)\nwhich decreases slower than the corresponding timescale when\ndynamic alignment is not present (cf. (A.6)). As a result, in the\npresence of a scale-dependent alignment, turbulent fluctuations\natλ⊥<λ⊥,CBscale as\nδz(subA)\nλ⊥<λ⊥,CB\nvA,0∼M3/2\nA,0 λ⊥\nℓ0!1/4\n(A.21)\nwhile the critical-balance condition τA,λ⊥∼τnl,λ⊥<λ⊥,CBsets\nthe scale-dependent anisotropy of dinamically aligned turbulent\nfluctuations,\nλ∥,λ⊥<λ⊥,CB\nℓ0∼M−1\nA,0 λ⊥\nℓ0!1/2\n. (A.22)\nEquation (A.22) implies that below λ⊥,CBa dynamically aligned,\ncritically balanced cascade exhibits a stronger anisotropy than\nthe corresponding cascade without a scale-dependent align-\nment.14Using (A.21) one obtains the reduced perpendicular\n13Another e ffect of dynamic alignment is that fluctuations exhibit\nthree-dimensional anisotropy: if we call λthe length-scale of these\n3D-anisotropic turbulent eddies in the direction perpendicular to both\na mean-field⟨B⟩λand magnetic-field fluctuations δB⊥,λat such scale\n(δB⊥,λbeing perpendicular to ⟨B⟩λ), thenℓλandξλdenote the length-\nscales along⟨B⟩λandδB⊥,λ, respectively (Boldyrev 2006). In the fol-\nlowing k⊥∼λ−1\n⊥refers to the shortest length-scale λ, and we neglect the\ndistinction between the two transverse directions kλ∼λ−1andkξ∼ξ−1;\nindeed an angular average of fluctuations’ properties in a wave-vector\nplane transverse to ⟨B⟩λwould be dominated by the scaling with kλ.\n14This scaling involves the parallel length-scale ℓλand the shortest\nperpendicular length-scale λ. However, fluctuations are 3D-anisotropic.\nSinceξλ∝λ3/4(Boldyrev 2006), the anisotropy scales as ℓξ∝ξ2/3,\nwhen considering the longest perpendicular length-scale ξ.spectrumE(subA)\nδz(k⊥>k⊥,CB)∝k−3/2\n⊥, which is slightly shal-\nlower that the−5/3 obtained without dynamic alignment; on the\nother hand, the parallel spectrum is still E(subA)\nδz(k∥)∝k−2\n∥.\nAt this point, if the Lundquist number is “not large enough” (i.e.,\nsuch that S0≲M−4\nA,0; see later), this dynamically aligned, crit-\nically balanced cascade will reach the dissipation scale λ(subA)\n⊥,diss\nwhenτnl,λ⊥∼τdiss,λ⊥. Using (A.20), this means\nλ(subA)\n⊥,diss\nℓ0∼(MA,0S0)−2/3. (A.23)\nHowever, in most cases of interest, the Lundquist number S0is\nlarge enough that this critically balanced cascade of dynamically\naligning fluctuations transitions to a “tearing-mediated” cascade.\nSuch transition occurs at a perpendicular scale λ⊥,∗for which\nthe timescale associated to the (linear) growth rate of the tear-\ning instability, γt\nλ⊥∼S−1/2\n0(λ⊥/ℓ0)−3/2(δzλ⊥/vA,0)1/2(vA,0/ℓ0),\nbecomes comparable to the eddy turnover time at that scale\nτ(subA)\nnl,λ⊥∼λ⊥/δz(subA)\nλ⊥, i.e.,γt\nλ⊥,∗τ(subA)\nnl,λ⊥,∗∼1, yielding\nλ(subA)\n⊥,∗\nℓ0∼M−2/7\nA,0S−4/7\n0. (A.24)\nComparing (A.24) and (A.23), one finds that a tearing-mediated\nrange emerges only if S0≫M−4\nA,0, so thatλ(subA)\n⊥,∗≫λ(subA)\n⊥,diss.\nIn this regime, the generation of turbulent fluctuations at scales\nλ⊥≲λ⊥,∗is due to the disruption of the (dynamically aligning15)\nturbulent eddies by magnetic reconnection; hence, the scale λ⊥,∗\nis usually referred to as the “disruption scale”. Thus the tearing\ninstability timescale τt\nλ⊥∼1/γt\nλ⊥is the “cascade time” in this\nrange of scales16, and assuming a constant energy flux through\nscales, (δz(subA)\nλ⊥<λ⊥,∗)2/τt\nλ⊥∼ε=const., provides with the fluctua-\ntions’ scaling in the tearing-mediated regime:\nδz(subA)\nλ⊥<λ⊥,∗\nvA,0∼S1/5\n0M8/5\nA,0 λ⊥\nℓ0!3/5\n, (A.25)\ncorresponding to a reduced spectrum E(subA)\nδz(k⊥>k⊥,∗)∝k−11/5\n⊥ .\nDue to the nonlinear stage of the tearing instability, turbulent\nfluctuations in the reconnection-mediated range tend to mis-\nalign in a scale-dependent fashion, following the scaling17\nsinθ(subA)\nλ⊥<λ⊥,∗∼S−3/5\n0M−4/5\nA,0 λ⊥\nℓ0!−4/5\n, (A.26)\nwhile fluctuations’ anisotropy in this range is obtained from the\nCB-like condition γt\nλ⊥τA,λ⊥∼1, i.e.,\nλ∥,λ⊥<λ⊥,∗\nℓ0∼S2/5\n0M−4/5\nA,0 λ⊥\nℓ0!6/5\n. (A.27)\nThe tearing-mediated cascade eventually dissipates at a scale\nwhere the characteristic dissipation time becomes comparable\nwith the tearing timescale, namely,\nγt\nλ⊥τdiss,λ⊥∼1⇒λ(subA)\n⊥,diss\nℓ0∼M−1\nA,0S−3/4\n0. (A.28)\n15We remark that a tearing-mediated regime fundamentally relies on\nthe fact that turbulent fluctuations develop anisotropy in the plane per-\npendicular to a mean field ( viz.,λ≪ξλ). Hence, tearing-mediated tur-\nbulence only exists if fluctuations do align in a scale-dependent fashion.\n16One can verify a posteriori that in this regime fluctuations’ scaling\nindeed preserves the condition τnl,λ⊥∼1/γt\nλ⊥at all scales below λ⊥,∗.\n17This is obtained as the ratio between the resistive inner scale δand the\nlongitudinal scale ζof the current layer (Boldyrev and Loureiro 2017).\nArticle number, page 15 of 17A&A proofs: manuscript no. C2024\nwhich, interestingly enough, is exactly the same dissipation scale\n(A.9) that was found for the (GS95) cascade.\nSuper-Alfvénic injection ( MA,0>1) with dynamic alignment.\nIn this regime, the cascade develops in a hydrodynamic-like\nfashion until the Alfvén scale ℓA∼M−3\nA,0ℓ0, i.e., without be-\ning affected by dynamic alignment. Thus fluctuations follow the\nscaling in (A.11) down to ℓA, and only below such scale the cas-\ncade becomes critically balance and dynamic alignment plays a\nrole. Atλ⊥<ℓ A, the fluctuations’ alignment angle scales as\nsinθ(supA)\nλ⊥<ℓA∼M3/4\nA,0 λ⊥\nℓ0!1/4\n, (A.29)\nand the nonlinear time at such scales is thus given by\nτ(supA)\nnl,λ⊥<ℓA∼λ⊥\nδz(supA)\nλ⊥<ℓAsinθ(supA)\nλ⊥<ℓA∼M−3/4\nA,0λ3/4\n⊥ℓ1/4\n0\nδz(supA)\nλ⊥<ℓA. (A.30)\nAs a result, in the presence of a scale-dependent alignment, tur-\nbulent fluctuations at λ⊥<ℓ Ascale as\n(δz(supA)\nλ⊥<ℓA)2\nτ(supA)\nnl,λ⊥<ℓA∼ε=const.⇒δz(supA)\nλ⊥<ℓA\nvA,0∼M3/4\nA,0 λ⊥\nℓ0!1/4\n(A.31)\ncorresponding to a ∝k−3/2\n⊥spectrum for k⊥ℓA>1. Fluctuations’\nscale-dependent anisotropy is obtained via the critical-balance\nconditionτA,λ⊥<ℓA∼τnl,λ⊥<ℓA, i.e.,\nλ∥,λ⊥<ℓA\nℓ0∼M−3/2\nA,0 λ⊥\nℓ0!1/2\n. (A.32)\nThe above cascade of critically balanced, dynamically aligned\nfluctuations can either reach actual dissipation at a scale\nλ(supA)\n⊥,diss/ℓ0∼M−1\nA,0S−2/3\n0, or, if S0≫M3\nA,0holds, will instead\ntransition to the tearing-mediated regime at a (disruption) scale18\nλ(supA)\n⊥,∗\nℓ0∼M−9/7\nA,0S−4/7\n0. (A.33)\nAt scalesλ⊥<λ(supA)\n⊥,∗, fluctuations will then follow the scaling\nδz(supA)\nλ⊥<λ⊥,∗\nvA,0∼S1/5\n0M6/5\nA,0 λ⊥\nℓ0!3/5\n, (A.34)\ncorresponding to a ∝k−11/5\n⊥ spectrum at k⊥λ(supA)\n⊥,∗>1. In this\nrange, fluctuations develop a scale-dependent (mis-)alignment\nangle\nsinθ(supA)\nλ⊥<λ⊥,∗∼S−3/5\n0M−3/5\nA,0 λ⊥\nℓ0!−4/5\n, (A.35)\nand an aniosotropy given by\nℓ∥,λ⊥<λ⊥,∗\nℓ0∼S2/5\n0M−3/5\nA,0 λ⊥\nℓ0!6/5\n. (A.36)\nFinally, this tearing-mediated regime reaches dissipation at\nλ(supA)\n⊥,diss\nℓ0∼η3/4\nε1/4ℓ0∼(MA,0S0)−3/4. (A.37)\n18We remind the reader that the transition scale in (A.33) is obtained\nusing the condition γt\nλ⊥,∗τ(supA)\nnl,λ⊥,∗∼1, where the growth rate of the tearing\ninstability is given by γt\nλ⊥∼S−1/2\n0(λ⊥/ℓ0)−3/2(δzλ⊥/vA,0)1/2(vA,0/ℓ0).A.2.1. Summary of scaling for MHD turbulence with dynamic\nalignment\nWe summarize here all the scaling of Section A.2 for the (nor-\nmalized) fluctuation amplitudes δˆz=δz/vA,0with respect to the\n(normalized) perpendicular wavelength ˆλ⊥=λ⊥/ℓ0.\nMA,0≤1regime (with dynamic alignment, S0≫M−4\nA,0):\nδˆz(subA)\nˆλ⊥∼MA,0ˆλ1/2\n⊥ˆλ⊥,CB<ˆλ⊥≤1 [W0]\nM3/2\nA,0ˆλ1/4\n⊥ˆλ(subA)\n⊥,∗<ˆλ⊥≤ˆλ⊥,CB [B06]\nS1/5\n0M8/5\nA,0ˆλ3/5\n⊥ˆλ(subA)\n⊥,diss<ˆλ⊥≤ˆλ(subA)\n⊥,∗ [TMT]\n(A.38)\nwith ˆλ⊥,CB∼M2\nA,0,ˆλ(subA)\n⊥,∗∼M−2/7\nA,0S−4/7\n0, and ˆλ(subA)\n⊥,diss∼\nM−1\nA,0S−3/4\n0, while the fluctuations’ anisotropy is given by\nˆλ(subA)\n∥,ˆλ⊥∼const. ˆλ⊥,CB<ˆλ⊥≤1 [W0]\nM−1\nA,0ˆλ1/2\n⊥ˆλ(subA)\n⊥,∗<ˆλ⊥≤ˆλ⊥,CB [B06]\nS2/5\n0M−4/5\nA,0ˆλ6/5\n⊥ˆλ(subA)\n⊥,diss<ˆλ⊥≤ˆλ(subA)\n⊥,∗ [TMT]\n(A.39)\nwhere ˆλ∥,ˆλ⊥=λ∥,ˆλ⊥/ℓ0is the normalized parallel wavelength of\nturbulent fluctuations.\nMA,0>1regime (with dynamic alignment, S0≫M3\nA,0):\nδˆz(supA)\nˆλ⊥∼MA,0ˆλ1/3 ˆℓA<ˆλ≤1 [K41]\nM3/4\nA,0ˆλ1/4\n⊥ˆλ(supA)\n⊥,∗<ˆλ⊥≤ˆℓA[B06]\nS1/5\n0M6/5\nA,0ˆλ3/5\n⊥ˆλ(supA)\n⊥,diss<ˆλ⊥≤ˆλ(supA)\n⊥,∗ [TMT]\n(A.40)\nwhere ˆℓA∼M−3\nA,0,ˆλ(supA)\n⊥,∗∼M−9/7\nA,0S−4/7\n0, and ˆλ(supA)\n⊥,diss∼\n(MA,0S0)−3/4, while the fluctuations’ anisotropy is given by\nˆλ(supA)\n∥,ˆλ⊥∼ˆλ⊥∼ˆλ ˆℓA<ˆλ≤1 [K41]\nM−3/2\nA,0ˆλ1/2\n⊥ˆλ(supA)\n⊥,∗<ˆλ⊥≤ˆℓA[B06]\nS2/5\n0M−3/5\nA,0ˆλ6/5\n⊥ˆλ(supA)\n⊥,diss<ˆλ⊥≤ˆλ(supA)\n⊥,∗ [TMT]\n(A.41)\nwith ˆλ∥,ˆλ⊥=λ∥,ˆλ⊥/ℓ0.\nSee Figures A.1 and A.2 for the resulting spectra and fluctu-\nations’ anisotropy versus perpendicular wavenumber k⊥.\nArticle number, page 16 of 17Silvio Sergio Cerri: Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence\n101103105107109\nk⊥/lscript010−1910−1610−1310−1010−710−4Eδz(k⊥)//lscript0v2\nA,0\nMA,0= 0.1\nS0= 1014∝k−2\n⊥\n∝k−5/3\n⊥∝k−3/2\n⊥\n∝k−11/5\n⊥(W0) range\n(GS95) range\n(B06) range\n(TMT) range1 M−2\nA,0 M2/7\nA,0S4/7\n0MA,0S3/4\n0\nM2\nA,0\nM6\nA,0\nM−15/14\nA,0S−6/7\n0\nMA,0S−5/4\n0\n1011031051071091011\nk⊥/lscript010−1810−1510−1210−910−610−3100Eδz(k⊥)//lscript0v2\nA,0\nMA,0= 10\nS0= 1014∝k−5/3\n∝k−5/3\n⊥∝k−3/2\n⊥\n∝k−11/5\n⊥ (K41) range\n(GS95) range\n(B06) range\n(TMT) range1 M3\nA,0 M9/7\nA,0S4/7\n0(MA,0S0)3/4\nM2\nA,0\nM−3\nA,0\nM−3/7\nA,0S−6/7\n0\nM3/4\nA,0S−5/4\n0\nFig. A.1: Normalized reduced spectrum, Eδz(k⊥)/ℓ0v2\nA,0, versus fluctuations’ perpendicular wave-vector, k⊥ℓ0, in a plasma with\nLunquist number S0=1014and sub-Alfvénic ( MA,0=0.1, left) or super-Aflvénic ( MA,0=10, right) injection regimes. Di fferent\ncolors represent di fferent cascading regimes (see legend), and general expressions for transition scales and fluctuations’ power level\nare reported on the right and upper axis. Solid lines show ideal scaling from (A.15), (A.17), (A.38), and (A.40) for the nominal range\nℓ−1\n0≲k⊥≲λ−1\n⊥,diss, while dashed lines represent their extension in the dissipation range with a damping factor ∼exp(−k2\n⊥λ2\n⊥,diss).\n101103105107109\nk⊥/lscript010−510−410−310−210−1100k/bardbl/k⊥\nMA,0= 10\nS0= 1014k/bardbl∝const.\nk/bardbl∝k2/3\n⊥\nk/bardbl∝k1/2\n⊥\nk/bardbl∝k6/5\n⊥(W0) range\n(GS95) range\n(B06) range\n(TMT) range1 M−2\nA,0 M2/7\nA,0S4/7\n0MA,0S3/4\n0\n1\nM2\nA,0\nM6/7\nA,0S−2/7\n0MA,0S−1/4\n0\n1011031051071091011\nk⊥/lscript010−310−210−1100k/bardbl/k⊥\nMA,0= 10\nS0= 1014k/bardbl∝k⊥\nk/bardbl∝k2/3\n⊥\nk/bardbl∝k1/2\n⊥\nk/bardbl∝k6/5\n⊥(K41) range\n(GS95) range\n(B06) range\n(TMT) range1 M3\nA,0 M9/7\nA,0S4/7\n0(MA,0S0)3/4\n1\nM6/7\nA,0S−2/7\n0M3/4\nA,0S−1/4\n0\nFig. A.2: Wave-vector anisotropy of fluctuations, k∥/k⊥, versus normalized fluctuations’ perpendicular wave-vector, k⊥ℓ0, for the\ncascades shown in Figure A.1 in the nominal range ℓ−1\n0≲k⊥≲λ−1\n⊥,diss(cf. equations (A.16), (A.18), (A.39), and (A.41)).\nArticle number, page 17 of 17"    },    {        "title": "2402.02919v1.Attractors_in_STGR_with_Boundary_Correction.pdf",        "content": "Attractors in STGR with Boundary Correction\nAndronikos Paliathanasis1, 2,∗\n1Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa\n2Departamento de Matem´ aticas, Universidad Cat´ olica del Norte,\nAvda. Angamos 0610, Casilla 1280 Antofagasta, Chile\nWe investigate the asymptotic behavior of the cosmological field equations in Sym-\nmetric Teleparallel General Relativity, where a nonlinear function of the boundary\nterm is introduced instead of the cosmological constant to describe the accelera-\ntion phase of the universe. Our analysis reveals constraints on the free parameters\nnecessary for the existence of an attractor that accurately represents acceleration.\nHowever, we also identify asymptotic solutions depicting Big Rip and Big Crunch sin-\ngularities. To avoid these solutions, we must impose constraints on the phase-space,\nrequiring specific initial conditions.\nPACS numbers:\nKeywords: Cosmology; Symmetric teleparallel; boundary term; f(Q, C)-theory; attractors\n1. INTRODUCTION\nRecent cosmological observations [1–3] challenge Einstein’s General Relativity (GR). In\nrecent years, many cosmologists have proposed various modified gravitational models to\nexplain these observations [4–20]. Although these models can be examined using numeri-\ncal techniques, analytical treatment is necessary to derive constraints on the models’ free\nparameters and draw conclusions about their cosmological viability.\nIn gravitational physics, the field equations are nonlinear differential equations, making\nthe derivation of analytic solutions a challenging task. Certain approaches employed in\nthe literature for constructing analytic solutions rely on symmetry analysis [21–24] and\nthe Painlev´ e algorithm [25–27]. However, in order to understand the global behaviour of\n∗Electronic address: anpaliat@phys.uoa.grarXiv:2402.02919v1  [gr-qc]  5 Feb 20242\nthe physical properties in a given gravitational model, we can study the behaviour of the\nfield equations in the long run. The analysis of the asymptotics has been widely used in\ngravitational theories, yielding many interesting results [28–37].\nIn this work, we investigate the impact of nonlinear boundary corrections in Symmetric\nTeleparallel General Relativity (STGR) [38] on cosmological solutions. In particular we\nassume the gravitational Lagrangian to be L=Q+f(B), where Qis the nonmetricity\nscalar, which is the fundamental scalar of STGR, and f(B) is an arbitrary function of the\nboundary term Bwhich gives the difference of Qwith the Ricci scalar of the background\nspacetime. This model belongs to the the family of f(Q, B)-gravity, which proposed recently\nin [39–41] and it is a generalization of f(Q) theory [42, 43]. The boundary term has been\nintroduced before in the gravitational Action Integral previously within the framework of\nteleparallelism [44] with various applications in cosmological studies, see for instance [45–49].\nThe boundary term introduces higher-order derivatives into the field equations, which\ncan be attributed to scalar fields [50]. These newly introduced scalar fields play a role in the\nfield equations, imparting dynamic behavior to dark energy. In the following sections, we\nutilize asymptotic analysis to establish constraints on the free parameters of the gravitational\ntheory and discuss the model’s viability based on initial conditions [36, 37]. The paper is\nstructured as follows.\nIn Section 2 we briefly discuss the basic definitions of STGR and we introduce the bound-\nary correction term. We consider a Friedmann–Lemaˆ ıtre–Robertson–Walker (FLRW) ge-\nometry and in Section 3 we present the field equations for the gravitational model of our\nconsideration. Section 4 includes the main results of this study where we present a de-\ntailed analysis of the dynamics for the cosmological field equations. Finally, in Section 5 we\nsummarize our results.\n2. SYMMETRIC TELEPARALLEL GENERAL RELATIVITY\nConsider a four-dimensional (non-Riemannian) manifold characterized by the metric ten-\nsorgµνand the symmetric and flat connection Γκ\nµν, defining the covariant derivative ∇µ.\nFurthermore, we assume that the metric tensor and the connection Γκ\nµνpossess identical\nsymmetries.\nBecause Γκ\nµνdiffers from the Levi-Civita connection it follows ∇λgµν̸= 0. The tensor3\nfield\nQλµν≡ ∇ λgµν=∂gµν\n∂xλ−Γσ\nλµgσν−Γσ\nλνgµσ\nis called the nonmetricity tensor and it is the essential for the STGR.\nBy definition, the connection is symmetric and flat, implying that both the curvature\ntensor and the torsion tensor vanish.\nFrom the nonmetricity tensor we can construct the scalar Qas [38]\nQ=QλµνPλµν, (1)\nwhich is the Lagrangian function of STGR.\nIn particular, in STGR the Action Integral is defined as\nSSTGR =Z\nd4x√−gQ. (2)\nTensor Pλµνis given by the following expression [38]\nPλ\nµν=−1\n4Qλ\nµν+1\n2Qλ\n(µ ν)+1\n4\u0000\nQλ−¯Qλ\u0001\ngµν−1\n4δλ\n(µQν), (3)\nand\nQµ=Qν\nµν,¯Qµ=Qν\nµν. (4)\nLet˜Γκ\nµνbe the Levi-Civita connection for the metric tensor gµν, that is ˜∇λgµν= 0,\nand ˜Ris the Ricciscalar defined by the Levi-Civita connection. Then by definition [43]\nR−Q=BQwhere Bis a boundary given by the expression [43] BQ=−˜∇µ\u0000\nQµ−¯Qµ\u0001\n.\nConsequently it follows\nZ\nd4x√−gQ≃Z\nd4x√−gR+ boundary terms. (5)\nand the theory is equivalent to GR.4\n2.1. Boundary corrections\nThe influence of the boundary term on gravitational phenomena has been previously\ninvestigated, particularly in the context of teleparallel gravity [44].\nIn this study we consider the modified gravitational Action Integral\nˆSSTGR =Z\nd4x√−g(Q+f(B)), (6)\nwhere we introduce a nonlinear function fwhich depend on the boundary B. Action ˆSSTGR\nbelongs to the family of f(Q, B) (also known as f(Q, C)) theory [39–41]. In our consideration\nwe assume the dynamical degrees of freedom provided by the correction term f(B) to play\nthe role of a dynamical dark energy.\nThe gravitational field equations are\n0 =2√−g∇λ\u0000√−gPλ\nµν\u0001\n−1\n2(Q+B)gµν+\u0000\nPµρσQρσ\nν−2QρσµPρσ\nν\u0001\n+\u0012B\n2gµν− ∇ µ∇ν+gµνgκλ∇κ∇λ−2Pλ\nµν∇λ\u0013\nf,B, (7)\nor equivalent\nGµν=Tf(B)\nµν, (8)\nwhere now Gµνis the Einstein tensor and Tf(B)\nµνattributes the dynamical degrees of freedom\nof the boundary term, that is,\nTf(B)\nµν=\u0000\n∇µ∇ν−gµνgκλ∇κ∇λ+ 2Pλ\nµν∇λ\u0001\nf,B+1\n2(f(B)−Bf,B)gµν. (9)\nWe introduce the scalar field ζ=f,B; thus, the energy momentum tensor Tf(B)\nµνbecomes\nTf(B)\nµν=\u0000\n∇µ∇νζ−gµνgκλ∇κ∇λζ+ 2Pλ\nµν∇λζ\u0001\n+1\n2V(ζ)gµν, (10)\nwhere V(ζ) = (f(B)−Bf,B).5\n3. FLRW COSMOLOGY\nOn very large scales, the universe is isotropic and homogeneous, described by the spatially\nflat FLRW geometry with the line element\nds2=−N(t)dt2+a2(t)\u0000\ndx2+dy2+dz2\u0001\n, (11)\nwhere a(t) is the scale factor and H(t) =1\nN˙a\nais the Hubble function and N(t) is the lapse\nfunction, where without loss of generality we assume N(t) = 1.\nThe FLRW geometry admits six isometries consisted by the three translation symmetries\n∂x, ∂y, ∂z,\nand the three rotations\ny∂x−x∂y, z∂ x−x∂z, z∂ y−y∂z.\nThe requirement for the connection Γκ\nµνto be symmetric, flat, and to inherit the symme-\ntries of the background geometry leads to three distinct families of connections [51, 52]. The\ncosmological field equations for these three families of connections were derived previously\nin [50].\nFor the first connection, namely Γ 1, the modified Friedmann equations are\n−3H2= 3˙ζH+1\n2V(ζ), (12)\n−2˙H−3H2=¨ζ+1\n2V(ζ), (13)\nin which the scalar field ζsatisfy the Klein-Gordon equation\n˙H+ 3H2+1\n6V,ζ= 0. (14)\nFor connection Γ 2the cosmological field equations are\n−3H2= 3˙ζH−3\n2˙ζ˙ψ+1\n2V(ζ), (15)\n−2˙H−3H2=3\n2˙ζ˙ψ+¨ζ+1\n2V(ζ), (16)6\nwhere the scalar fields satisfy the equations motion\n¨ζ+ 3H˙ζ= 0, (17)\n6˙H+ 18H2−9H˙ψ−3¨ψ+V,ζ= 0. (18)\nFinally, for the third connection, i.e. Γ 3, the field equations are\n−3H2= 3H˙ζ+3\n2˙ζ\na2˙Ψ+1\n2V(ζ), (19)\n−2˙H−3H2=V(ζ)\n2+˙ζ\n8a2˙Ψ, (20)\nand the equations of motion for the scalar fields are\n3˙Ψ¨ζ+˙ζ\u0010\n˙ΨH−2¨Ψ\u0011\n= 0, (21)\n6˙H+ 18H2+V,ζ−3\na2˙Ψ2\u0010\nH˙Ψ−¨Ψ\u0011\n= 0. (22)\nWe emphasize that for connections Γ 2and Γ 3, scalars ψand Ψ play crucial roles in the\ndynamics’ evolution. In contrast, in the case of connection Γ 1, scalar γserves as a gauge\nfunction. Connection Γ 1is defined in the coincidence gauge, whereas connections Γ 2and Γ 3\nare defined in the noncoincidence gauge.\nWe continue our study with the phase-space analysis of the field equations corresponding\nto the three connections. For the potential function V(ζ) we consider the exponential\nfunction V(ζ) =V0eλζwhich corresponds to the function\nf(B) =−B\nλln\u0012\n−B\nλV0\u0013\n−B\nλ. (23)\n4. ANALYSIS OF ASYMPTOTICS\nIn the following, we conduct an analysis of the asymptotics for the considered cosmological\nmodel. Specifically, we introduce dimensionless variables and express the field equations as\na set of algebraic-differential equations. We calculate the stationary points and investigate\ntheir stability properties. Each stationary point corresponds to an asymptotic solution with\nspecific physical properties. Finally, based on the stability properties, we can establish7\nconstraints on the free parameters of the models and discuss the initial value problem. This\nanalysis is applied to the three different families of connections.\n4.1. Connection Γ1\nTo examine the asymptotic evolution of the field equations for the first connection, we\nintroduce the new variables\nz=˙ζ\nH, y=V(ζ)\n6H2, λ=V,ζ\nV, τ= lna, (24)\nwith inverse transformation\n˙ζ=zH , V (ζ) = 6 yH2, V,ζ=λV , a =eτ. (25)\nThus, the field equations transform into\ndz\ndτ= 3 (1 + z) +y(λ(2 +z)−3), (26)\ndy\ndτ=y(6 +λ(2y+z)), (27)\ndλ\ndτ=λ2z(Γ (λ)−1),Γ (λ(ζ)) =V,ζζV\n(V,ζ)2, (28)\nand\n1 +y+z= 0. (29)\nMoreover, the equation of state parameter wΓ1\neffis expressed as follows\nwΓ1\neff= 1 +2\n3λy. (30)\nFor the exponential potential, where λis always a constant and with the application of\nthe latter constraint equation we reduce the field equations to the single differential equation\ndz\ndτ= (1 + z) (6−λ(2 +z)). (31)\nThe stationary points A= (z(A)) of the latter equation are two (in the finite and infinity8\nregimes), point A1=−1 and point A2=6\nλ−2. Point A1corresponds to a stiff fluid solution\nwith wΓ1\neff(A1) = 1, while for point A2we calculate wΓ1\neff(A2) =−3 +2λ\n3, where it follows\nthat the de Sitter universe is recovered for λ= 3, and the solution describes acceleration for\nλ <4. Finally, for λ >6 point A1is an attractor while for λ <6, the attractor is point A2.\n4.2. Connection Γ2\nWe introduce the dimensionless variables\nx=˙ψ\n2H, z=˙ζ\nH, y=V(ζ)\n6H2, λ=V,ζ\nV, τ= lna, (32)\nthat is,\n˙ψ= 2xH , ˙ζ=zH , V (ζ) = 6 yH2, V,ζ=λV , a =eτ. (33)\nIn terms of the new variables the field equations are\ndx\ndτ=1\n2\u0000\n3x(y−2z−1) + 3 x2z+ 3 (1 + z) + (2 λ−3)y\u0001\n, (34)\ndz\ndτ=3\n2z(y−1 +z(x−1)), (35)\ndy\ndτ=y(3 (1 + y) + (λ−3 (1−x))z), (36)\ndλ\ndτ=λ2z(Γ (λ)−1),Γ (λ(ζ)) =V,ζζV\n(V,ζ)2. (37)\nFriedmann’s first equation yields the constraint\n1 +z(1−x) +y= 0, (38)\nwhile the equation of state parameter reads\nwΓ2\neff=y−z(1−x). (39)\nFor the exponential potential, i.e. λis a constant, the stationary points B=9\n(B(x), B(z), B(y)) are\nB1=\u0012\nx,−1\n1−x,0\u0013\n, B2=\u0012\n1−λ\n3,0,−1\u0013\n. (40)\nB1describes a family of points with correspond to stiff fluid solutions, that is, wΓ2\neff(B1) =\n1. On the other hand, B2describes the de Sitter universe with wΓ2\neff(B2) =−1.\nAs far as the stability is concerned the eigenvalues of the two-dimensional system in the\nspace of variables {x, z}, around the stationary points B1are\b\n0,6−λ\n1−x\t\n, while around\nthe point B2are{−3,−3}. Thus, point B2is always an attractor, while for the stability\nproperties of B1we should employ the center manifold theorem (CMT).\nWe introduce the new variable ¯ z=z+1\n1−x, such that the coordinates of points B1to be\nB1= (x,0,0). Then, we assume ¯ z=h(x) in order to determine the center manifold. In\norder a stable manifold to exist it should hold h(x1) = 0 anddh\ndx|x→x1= 0. We calculate\nh(x) =1\n1−x−h0(1−x−λ); thus, points B1describe always unstable solutions.\n4.2.1. Poincare variables\nBecause the dynamical variables are not constraint, they can take values at the infinity.\nHence, in order to study the analysis at the infinity we introduce the Poincare variables\nx=X√\n1−X2−Z2, z=Z√\n1−X2−Z2, dT =√\n1−X2−Z2dτ,\nwhere {X2, Z2} ≤1.\nIn terms of the new variables the field equations are expressed as\ndX\ndT=F1(X, Z),dZ\ndT=F2(X, Z), (41)\nwhile the equation of state parameter is\nwΓ2\neff=−1−2ZX−√\n1−X2−Z2\n1−X2−Z2. (42)10\nThe stationary points B∞= (B∞(X), B∞(Z)) at the infinity are\nB∞\n1±= (±1,0), B∞\n2±= (0,±1), (43)\nand for λ= 3, there exist the family of points\nB∞\n3±=\u0010\nX,±√\n1−X2\u0011\n. (44)\nStationary points B∞\n1±describe de Sitter solutions, i.e. w(Γ2)\neff\u0000\nB∞\n1±\u0001\n=−1, while the\nasymptotic solutions at points B∞\n2±describe Big Crunch or Big Rip singularities, that is\nw(Γ2)\neff\u0000\nB∞\n2±\u0001\n=∓∞. Similarly, the family of points B∞\n3±describe Big Crunch and Big Rip\nsingularities.\nAs far as the stability is concerned, the eigenvalues for the linearized system around\npoints B∞\n1±are{0,0}, from where we infer that the stationary points describe unstable\nsolutions. For points B∞\n2±the eigenvalues are {±6,±(λ−3)}, which means that the Big\nCrunch solution B∞\n2−is an attractor for λ >3. Finally, for λ= 3 the stability of the points\nB∞\n3±depend on the sing of the dynamical variable X.\nIn Fig. 1 we present phase-space portraits for the dynamical system, where it is clear for\nλ <3, the unique attractor is the de Sitter solution described by point B2. Moreover, in\nFig. 2 we present qualitative evolution of the equation of state parameter for various sets of\ninitial conditions.\n4.3. Connection Γ3\nFor the set of field equations related to the connection Γ 3we introduce the dimensionless\nvariables\n¯x=1\n2a2H˙Ψ, z=˙ζ\nH, y=V(ζ)\n6H2, λ=V,ζ\nV, τ= lna, (45)\nthat is,\n1\n˙Ψ= 2a2¯xH , ˙ζ=zH , V (ζ) = 6 yH2, V,ζ=λV , a =eτ. (46)11\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nXZPhase-space forλ=1\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nXZPhase-space forλ=3\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nXZPhase-space forλ=4\nFIG. 1: Phase-space portraits for the cosmological field equations of connection Γ 2in the Poincare\nvariables (41). The phase-space portraits are for λ= 1, λ= 3 and λ= 4. With dots are the\nstationary points and red lines correspond to the family of points B1. We observe that for λ <3,\nthe unique attractor is the de Sitter solution described by point B2.\n-1-1\n301\n31\nTweffΓ2(T)λ=1\n-1-1\n301\n31\nTweffΓ2(T)λ=3\n-1-1\n301\n31\nTweffΓ2(T)λ=4\nFIG. 2: Qualitative evolution of the equation of state parameter w(Γ2)\nefffor different values of\nλ= (1,3,4) and for various initial conditions ( X0, Z0). Solid lines are for (0 .95,0.1), dashed lines\nare for (0 .9,0.19), dotted lines are for (0 .8,0.3) and dash-dotted lines are for ( −0.8,0.3).\nThe field equations read\nd¯x\ndτ=¯x\n2z(z(1 + ¯x)−3 +y(3−2λ(1−z))), (47)\ndz\ndτ= 3 + (3 −¯x)z+y(λ(2 +z)−3), (48)\ndy\ndτ=y(6 +λ(2y+z)), (49)\ndλ\ndτ=λ2z(Γ (λ)−1),Γ (λ(ζ)) =V,ζζV\n(V,ζ)2, (50)12\nand constraint\n1 +y+ (1 + ¯ x)z= 0. (51)\nMoreover, the equation of state parameter is expressed as\nwΓ3\neff= 1 +2\n3λy. (52)\nWe remark that for the exponential potential λis always constant. Hence the stationary\npoints C= (C(¯x), C(z), C(y)) for the latter algebraic-differential system are\nC1= (0,−1,0), C2=\u0012\n0,2\nλ(3−λ),1−6\nλ\u0013\n. (53)\nPoint C1describes a stiff fluid solution, with wΓ3\neff(C1) = 1. On the other hand, the\nasymptotic solution at point C2describes an ideal gas with equation of state parameter\nwΓ3\neff(C2) =−3 +2\n3λ. The stationary point describes acceleration for λ < 4, while the\ncosmological constant is recovered for λ= 3.\nWe make use of the constraint equation(51) and we reduce by one the dimension of the\ndynamical system. The eigenvalues of the linearized system around the stationary points C1\nandC2are{2,6−λ};\b\nλ−6,1\n2(3λ−14)\t\nrespectively. Therefore, point C1is a saddle\npoint when λ > 6, otherwise is a source; while point C2is an attractor for λ <14\n3. We\nremark that when C2describes acceleration it is always attractor.\n4.3.1. Poincare variables\nFor the analysis at the infinity regime we work in the two-dimensional space defined by\nthe dynamical variables {¯x, z}.\nThe Poincare variables are defined as\n¯x=¯X√\n1−¯X2−Z2, z=Z√\n1−¯X2−Z2, dT =p\n1−¯X2−Z2dτ,\nwhere\b¯X2, Z2\t\n≤1.13\nThe field equations are reduced to the system of the form\ndX\ndT=F1(X, Z),dZ\ndT=F2(X, Z), (54)\nand the equation of state parameter is expressed as\nwΓ3\neff= 1−2\n3λ \n1 +Z\u0000¯X+√\n1−X2−Z2\u0001\n1−X2−Z2!\n. (55)\nThe stationary points C∞= (X(C∞), Z(C∞)) at the infinity; that is, 1 −(X(C∞))2−\n(Z(C∞))2= 0, are\nC∞\n1±= (0,±1).\nWe calculate wΓ3\neff\u0000\nC∞\n1±\u0001\n=∓λ∞. Hence, for λ > 0,C∞\n1+corresponds to a Big Rip\nsingularity, and C∞\n1−to a Big Crunch; nevertheless for λ < 0,C∞\n1+corresponds to a Big\nCrunch singularity, and C∞\n1−to a Big Rip singularity.\nThe eigenvalues of the linearized system around the stationary points are {±2λ,0}. Be-\ncause the second eigenvalue is zero, we employ the CMT and we found that the stationary\npoints does not posses any submanifold where the solutions are stable. Thus, the stationary\npoints are saddle points or sources.\nIn Fig. 3 we present phase-space portraits for the dynamical system in Poincare variables.\nFurthermore, in Fig. 2 we present qualitative evolution of the equation of state parameter\nfor various sets of initial conditions.\n5. CONCLUSIONS\nWe conducted a detailed analysis of the asymptotic dynamics for an extension of STGR\nin which nonlinear components of the boundary term are introduced in the gravitational\nintegral. In STGR, the definition of the connection is not unique, and for the spatially\nflat FLRW, there are three families of different connections. Although the selection of the\nconnection does not affect the gravitational model in STGR when nonlinear terms of the\nboundary scalar are introduced, new dynamical degrees of freedom appear. The new degrees\nof freedom can be attributed to scalar fields, leading to three different sets of gravitational\nfield equations, corresponding to the families of connections.14\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nXZPhase-space forλ=2\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nXZPhase-space forλ=3\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nXZPhase-space forλ=4\nFIG. 3: Phase-space portraits for the cosmological field equations of connection Γ 3in the Poincare\nvariables (41). The phase-space portraits are for λ= 2, λ= 3 and λ= 4. With dots are the\nstationary points. We observe that the unique attractor is point C2, while the two stationary\npoints at the infinity regime are saddle points.\n-1-1\n301\n31\nTweffΓ3(T)λ=1\n-1-1\n301\n31\nTweffΓ3(T)λ=2.5\n-1-1\n301\n31\nTweffΓ3(T)λ=3\nFIG. 4: Qualitative evolution of the equation of state parameter w(Γ3)\nefffor different values of\nλ= (2,2.5,3) and for various initial conditions ( X0, Z0). Solid lines are for ( −0.95,0.05), dashed\nlines are for ( −0.8,0.2), dotted lines are for (0 .8,−0.2) and dash-dotted lines are for (0 .5,0.7).\nFor the three different models, we employed dimensionless variables and determined the\nstationary points. We reconstructed the asymptotic solutions at the stationary points and\ninvestigated their stability properties. The results are summarized in Table I.\nThe three gravitational models admit asymptotic solutions that can describe late-time\nacceleration. However, for connections Γ 2and Γ 3, asymptotic solutions appear that can\ndescribe Big Rip or Big Crunch singularities. For connection Γ 2, a Big Crunch singularity is\nstable when the parameter λ >3. Thus, in this case, for the unique attractor to be the de\nSitter solution, it follows that λ≤3. However, for these two specific models, the phase-space\nanalysis leads to the derivation of sets for the initial conditions to avoid the appearance of15\nTABLE I: Asymptotic solutions in STGR with boundary correctons\nPoint w effAcceleration Stability\nConnection Γ 1\nA1 1 No Stable λ >6\nA2 −3 +2\n3λ λ < 4 Stable λ <6\nConnection Γ 2\nB1 1 No Unstable\nB2 −1 Yes Stable\nB∞\n1± −1 No Unstable\nB∞\n2± ∓∞ Big Rip B∞\n2+B∞\n2−Stable λ >3\nB∞\n3±(λ= 3) ∓∞ Yes Stable\nConnection Γ 3\nC1 1 No Unstable\nC2 −3 +2\n3λ λ < 4 Stable λ <14\n3\nC∞\n1± ∓λ∞ Big Rip Unstable\nsuch types of singularities.\nTheQ+f(B) theory provides a mechanism to explain the late-time acceleration phase\nof the universe. Nevertheless, it is not obvious from this analysis if this theory can describe\nother eras of the cosmological history and if it solves the H0-tension. 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Our findings show that for certain values of\nthe fractional order parameter, the damping parameter, and the forcing amplitude high oscillations\namplitude can be induced. This phenomenon is due to the appearance of a resonance in the Duffing\noscillator only when the damping term is fractional.\n1arXiv:2402.02940v1  [nlin.CD]  5 Feb 2024I. INTRODUCTION\nNonlinear oscillators are dynamical systems that exhibit oscillatory behaviors but do not\nfollow a linear relationship between the restoring force and the displacement. These sys-\ntems often arise in various scientific and engineering fields, and their study has profound\nimplications in understanding complex phenomena and designing efficient devices. When\nconsidering damping in nonlinear oscillators, a commonly used approach is to introduce a\nfractional derivative in the damping term. The use of fractional derivatives in damping terms\nprovides a powerful tool to capture the memory effects and non-local behavior observed in\ncertain physical systems. Unlike the traditional integer-order derivatives that describe in-\nstantaneous rates of change, fractional derivatives involve a non-integer order and account\nfor past history in the damping process. This allows us a more accurate representation of\ndamping in systems with memory and hysteresis as occurs in some properties of complex\nmaterials. The presence of a fractional derivative damping term in nonlinear oscillators can\nlead to novel behaviors such as sub-diffusive or super-diffusive damping, where the decay\nof oscillations is slower or faster than in traditional damping scenarios. Moreover, the in-\nclusion of the fractional derivatives can alter the stability regions, bifurcation boundaries,\nand resonance responses of the oscillatory system, introducing rich and intricate dynamics.\nThe study of nonlinear oscillators with fractional derivative in the damping term finds ap-\nplications in disciplines such as mechanical engineering, physics, robotics, control systems,\nand even social sciences [1–10]. Understanding the behavior and control of these oscillators\nis crucial for optimizing energy dissipation, enhancing stability, and designing systems with\nspecific response characteristics.\nOn the other hand, nonlinear resonances [11] is a fascinating phenomenon that occurs\nwhen a system, subjected to an external force or perturbation, exhibits a response that\nis not proportional to the input. Unlike linear resonance, where the response is directly\nproportional to the forcing frequency, nonlinear resonances arise due to the intricate interplay\nbetween multiple frequencies and energy transfer mechanisms within the system, as in our\ncase the fractional order parameter, the damping parameter and the forcing amplitude. In\nnonlinear resonances, the system’s response can display a wide range of complex behaviors,\nincluding the generation of harmonics, sub-harmonics, chaotic oscillations, and bifurcations.\nAs stated before, the fractional derivatives are the key to explore and modeling materials\n2with memory effects and fractal geometry or diffusion processes, among others. In fact,\nfractional derivatives can play a crucial role on them. Therefore, it is relevant to analyze\nwhen the fractional order parameter can trigger complex behaviors and nonlinear resonance\nphenomena.\nHere, we analyze the Duffing oscillator with a fractional derivative in its damping term\nas in Ref. [12]. Specifically, we focus our study in the numerical analysis of the fractional\norder parameter effects on the oscillations amplitude of the system. We have found that the\nfractional order parameter can induce resonance phenomena. Firstly, we analyze, for smaller\nforcing amplitude values, beyond a certain threshold, the appearance of resonance peaks that\nare triggered by the high damping parameter values and specific values of the fractional\norder parameter. These high amplitude oscillations are related with a complex dynamics of\nthe oscillator in the phase space. Then, we find a region of resonances for higher forcing\namplitude values. Consequently, we characterize and analyze the appearance of resonance\nphenomena induced by the effect of the three main ingredients: high forcing amplitude and\ndamping parameter values, and some specific values of the fractional order parameter. It\nis relevant to emphasize that these complex behaviors and resonance phenomena are an\neffect of the fractional order parameter, and they only appear when a fractional derivative\ndamping term exists.\nThe organization of this paper is as follows. In Sec. II, we introduce the fractional\nDuffing oscillator and its dynamics. In Sec. III, we numerically present the effects on the\noscillations amplitude of the fractional order parameter in which a resonant-like behavior\nis uncovered. The impact of the forcing amplitudes due to the fractional order parameter\nis analyzed in Sec. IV, where we numerically characterize the birth of this resonance-like\nbehavior. Conclusions and a discussion of the main results of this paper are presented in\nSec. V.\nII. MODEL DESCRIPTION\nWe study a Duffing oscillator with a fractional derivative of fractional order parameter α\ninstead of the first derivative, i.e., ˙ x(t)→Dαx(t) =dαx(t)/dtαas a damping term. Then,\nthe Duffing oscillator with a fractional damping term is given by\nd2x\ndt2+µdαx\ndtα−x+x3=Fcosωt, (1)\n3where µis the damping parameter, Fandωare the forcing amplitude and forcing frequency,\nrespectively, and αis the fractional order parameter. In order to integrate the system, we\nuse the Gr¨ unwald-Letnikov fractional derivative, as in previous works [12, 13]. Consequently,\nthe system is composed by a set of three fractional differential equations that yield\n\n\nDαx=y,\nD1−αy=z,\nDz=µy+Fcos (ωt) +x−x3,(2)\nwhere zis an auxiliary component coming from the transformation into a fractional order\nsystem. The integration step used is π/2000, and the maximum time value is t= 300 that\nis large enough for the system to reach the steady state. We have carried out numerical\nsimulations with several initial conditions and, although the results are numerically different,\nthe general trends of the curves are qualitatively similar. So, we carry on the study with\nthe initial condition ( x0, y0, z0) = (0 .2,0.3,0.0), as in Ref. [12, 14]. We expect that our\nconclusions are of general validity and not specific for the considered boundary conditions.\nIn Fig. 1, we show some examples of trajectories of the system and the impact of the α\nvalue on its dynamics, when the µparameter varies. In fact, according to Fig. 1(a), where\nµ= 0.6, and Fig. 1(b), where µ= 0.9, we have to change the αparameter from α= 0.3\ntoα= 0.1, respectively, if we want to observe similar oscillations amplitudes. In Fig. 1(c),\nwe compare trajectories for µ= 0.9 and α= 0.3 (in blue) and for µ= 0.9 and α= 0.1 (in\nred). Having traced more trajectories than those shown, we have seen that the phenomenon\nis complex and worthy of study.\nIII. FRACTIONAL ORDER PARAMETER EFFECTS ON THE OSCILLATIONS\nAMPLITUDE\nAs we previously mentioned, some preliminary computations about the resonance induced\nby the fractional derivative in the Duffing damping term have been done in [12]. The authors\nhave disclosed that for a high value of the damping parameter µ= 0.8 and for distinctive\nvalues of the fractional order parameter α, some interesting peaks pop up when the forcing\namplitude is large enough, as shown in Fig. 2. Also, for higher αvalues we have seen that\nthe oscillations amplitude do not show those kind of peaks and that the curve stays smooth\n4FIG. 1: In this figure, we have plotted the asymptotic behaviors of the system trajectory in phase\nspace for (a) µ= 0.6 and α= 0.3, (b) µ= 0.9 and α= 0.1 and (c) µ= 0.9 and α= 0.1 (in red)\nandα= 0.3 (in blue). We fix the forcing amplitude F= 1.5 for all cases. Note the key role of the\nαandµparameters in the amplitude of the oscillations.\nunder an oscillations amplitude the of value 0 .5. We can see that two particular peaks stand\nout in the amplitude and the Q−factor plots, Fig. 2(a) and Fig. 2(c), respectively. However,\nthis fact was just mentioned, though not analyzed in detail. As stated before, the Q−factor\nprovides an idea of how much the signal is amplified by a certain parameter, in our case α.\nWe calculate it by computing the sine and cosine components:\nBs=2\nnTZnT\n0x(t) sinωtdt and Bc=2\nnTZnT\n0x(t) cosωtdt, (3)\nwhere T= 2π/ω andnis an integer. Then, we can find the dependence on αwith\nQ=p\nB2\ns+B2\nc\nα. (4)\nThose peaks are related with fractional order parameter values that suddenly give birth\nto aperiodic orbits, as we can see in Fig. 2(b). They are resonance-like peaks for the α\nparameter, as confirmed by the Q−factor, but it is worth of study the conditions for them\nto appear and the parameter values for which the gain in amplitude is higher. Our objective\n5now is to analyze in depth this resonant behavior. For this purpose, we plot the gradient of\nthe amplitude in the parameter set α−µfor two values of the forcing amplitude F= 0.01\nandF= 0.1 in Fig. 3. In the first case, Fig. 3(a), we can see that the gradient in the\nparameter set is smooth, the oscillations amplitude is not that high and the higher oscil-\nlations amplitude zone is very easy to spot. The only interesting thing is the intermediate\noscillations amplitude smudge for µ≳0.6. Even more interesting are the other two panels\nwhere Figs. 3(c) is a zoom of Fig. 3(b). When we reach the forcing amplitude value F= 0.1,\nthe parameter set becomes more complicated and the high and low oscillations amplitude\nintermingle.\nFrom values µ≳0.4, the structure that can be spotted in Fig. 2(a) and that ends at\nα≈0.15, starts to take form with lots of the resonance peaks, in which the oscillations span\nboth potential wells. In fact, in Fig. 4 we show the oscillations amplitude plots for different\nµvalues as a section of the 3 −Dfigure. There, we can see that for µ= 0.35 there is no\nsign of the structure, while it is visible for µ= 0.55. Clearly, for higher µvalues it becomes\neven more visible. In the 3 −Dfigure, it is possible to discern the built-up of such structure\nand how the higher oscillations amplitude peaks diminish when µgrows. This structure is\nthe result of the higher oscillations amplitude being attenuated by the dissipation term, so\nthat the peaks are the consequence of the fractional order parameter feat interacting with\nthe forcing amplitude. Hence, it seems that they are fractional-induced amplitude peaks\nthat behave like a resonance when only the fractional order parameter varies. In fact, if we\ncompute the Q−factor for the parameter µthe peaks do not show up in the plot.\nIn Fig. 4(e), we can see that the trajectories become bounded to one well when the dis-\nsipation is too strong, µ= 1, for the fractional order parameter to maintain the interwell\noscillations. Figure 3(c) has been plotted to have a better understanding of the higher am-\nplitude oscillations zone. In particular of the cross like patch centered at µ= 0.6. Here, we\ncan find a region of small oscillations amplitude embedded inside a high oscillations ampli-\ntude region. It is precisely there, where the already mentioned structure, well recognisable\nin Fig. 2(a), starts to take form.\n6FIG. 2: We plot here (a) the oscillations amplitude, (b) the asymptotic behaviors diagram and (c)\ntheQ−factor for F= 0.1, µ= 0.8. The two resonance peaks can be recognise in the panels.\nIV. THE FORCING AMPLITUDE EFFECT ON THE OSCILLATIONS AMPLI-\nTUDE\nTo carry on our analysis, we have decided to study the effect of the forcing amplitude F\non the oscillations amplitude since it will provide more easily clear light on the resonance-like\nphenomenon. The main result of this manuscript is shown in Fig. 5, where we have depicted\nthe gradient parameter set F−αof the oscillations amplitude for different values of µ. As\nexpected, the higher the damping term the lower the oscillations amplitude. Interestingly,\na high amplitude oscillations region starts to appear when µ >0.5, in particular it becomes\nmore evident at µ≈0.6. This region is centered around F= 1.5 and spans all or part\nof the parameter set, depending on the damping parameter. In fact, we can see there\nthat all along the panels of Fig. 5 that region shrinks along the αaxis together with the\noscillations amplitude. That means that for a high enough damping parameter value, the\nhigh oscillations amplitude is confined around a certain forcing amplitude value, that is,\nF= 1.5 and it is induced by a certain range of fractional order parameter values. These\nresults corroborate what was observed in Fig. 1.\n7FIG. 3: The figure shows the oscillations amplitude for F= 0.01 (a) and F= 0.1 (b). Panel (c) is\na zoom of panel (b). Here it is possible to appreciate how the difference in the amplitude forcing\ncomplicates the topology of the parameter set.\nBesides, the region shrinks spanning smaller values of the fractional order parameter\nwhen the dissipation grows. This phenomenon can be explained as in the previous section.\nThe interaction among high dissipation, certain values of the fractional order parameter and\nhigh forcing amplitude triggers a resonant behavior. The interesting fact is that the gain in\namplitude is confined to certain αvalues, so we can define the phenomenon as a resonance\ninduced by the fractional order parameter. The difference with F= 0.1 is that now the\nforcing amplitude is large enough to sustain the high oscillations amplitude inside a large\nregion of the parameter set and not just for some particular values of α.\nFor a better understanding and the characterization of the phenomenon, we have deep-\nened its analysis with the other figures below. We want to stress out that all the figures\nfrom now on are related with Fig. 5 and they provide more specific details of the features\n8FIG. 4: In panel (a), we show the 3 Dversion of Fig. 3(b). The other panels display 4 slices of the\npanel (a) along the αaxis and for (b) µ= 0.35, (c) µ= 0.55, (d) µ= 0.76, and (e) µ= 1. Here,\nthe formation of the central structure is clear. In fact, in panel (c) it is possible to see how the\nhigh oscillations amplitude of panel (b) decreases and start to form the central structure of panels\n(d) and (e).\nof the resonance phenomenon. Therefore, we start by showing in Fig. 6(a) the oscillations\namplitude versus the forcing amplitude Fforµ= 0.6. Every curve is associated with a\nfractional order parameter value as indicated there. It is possible to see that the amplitude\ncurves corresponding with the two values α= 0.1 and α= 0.3, for this damping parame-\nter, cross the high oscillations amplitude region of Fig. 5(b) and present the corresponding\npeaks, near F= 1.5. On the other hand, the curves related with the other two αvalues are\nsmooth after a first jump, at α≈0.5. In Fig. 6(b), we show the frequencies of the oscilla-\ntions, calculated with the Fast Fourier transform. It is possible to see that from F= 0.8\nthe calculated frequencies tend naturally to the forcing frequency ω= 1, but not near the\nforcing amplitude for which the resonance peaks are produced for the αvalues that cross\nthe high amplitude region of Fig. 5(b).\nIn order to have a deeper insight on the dynamics of the system, we plot Fig. 7 in which\nwe show the maxima-minima diagram for µ= 0.6 and for 4 values of the αparameter, along\n9FIG. 5: We show the oscillations amplitude in the parameter set F−α, for different µvalues. Here,\nit appears along the panels the formation of a well located region of high oscillations amplitude\nnearF= 1.5\nwith two orbits in the case of F= 1.5 and α= 0.1 (Fig. 7(e)) and α= 0.3 (Figs. 7(f)). These\nfractional order parameter values are the ones that cross the high oscillations amplitude in\nFig. 5(b). The other two values, see Figs. 7(c) and (d), do not cross that region as indicated\nby the maxima-minima diagram smoothness. One consideration can be made, we can see\nthat there is an important difference between the two plotted orbits, as seen in Figs. 7(e)\nand (f). The first one is not periodic, while the second one is periodic of period 3. It seems\nthat when the fractional order parameter grows, the large amplitude aperiodic trajectories\ndisappear becoming periodic as if the fractional order parameter contributed to introduce\norder into chaos.\nNow, we check the robustness of the resonance induced by the fractional order parameter\nwhen the forcing frequency ωchanges. For that purpose, we plot the Q−factor in Fig. 8,\nwhich shows the oscillations amplitude and the maxima-minima diagram in function of the\nforcing frequency and for F= 1.5 and µ= 0.6. In Figs. 8(a) and (b) it is possible to notice\nsome peaks in the Q−factor curve that assure us that they are related with the fractional\norder parameter induced resonance. This resonance phenomenon needs an interaction among\n10FIG. 6: The figure shows four slices of Fig. 5(b) overlapped in panel (a) and the oscillations\nfrequencies calculated with the Fast Fourier transform in panel (b). In the first panel, the curves\nrelated to fractional order values, α= 0.1 and α= 0.3, that cross the high amplitude oscillations\nregion are recognisable. The other two, after they grow at F∼0.5, become smooth. The frequen-\ncies in panel (b) for F≳0.9 tend to the forcing frequency ω= 1, except for the ones related with\nthe resonance peak.\nhigh forcing amplitude, high damping parameter and small values of the fractional order\nparameter. In Fig. 8(a), we have highlighted three peaks at α= 0.54, α= 1 and α= 2.01\nthat will be discussed later when we analyze higher µvalues. The peaks are corroborated\nby the maxima-minima diagram, as seen in Fig. 8(c). In this last figure, it is interesting to\nnotice that the higher peaks are related with aperiodic trajectories.\nNow, we analyze the situation in which the damping parameter has a large value and,\ntherefore, we are in the overdamped regime. In Fig. 9, we can find a similar plot as Fig. 6\nfor the damping parameter value µ= 0.9. Here, in Fig. 9(a), we can see that only for\nα= 0.1 there is a resonance peak, as expected from Fig. 5(e). The computed oscillations\nfrequencies, Fig. 9(b), behave as in the previous case. This result is confirmed by Fig. 10,\nwhere we can see that the resonance amplitude can be found only for α= 0.1, Fig. 10(a),\nwhile for the others fractional values the maxima-minima diagram is smooth for large forcing\n11FIG. 7: We show the behavior of the system for fixed values of αandµ. Comparing panel (a) and\n(b), we can observe that the system behaviors at F= 1.5 are different. For α= 0.1, in panel (e),\nthe oscillations are clearly not periodic, while for α= 0.3, in panel (f), the oscillations are periodic\nof period 3. In the other two panels, the higher oscillations amplitude enlargement is lacking.\namplitudes, Figs. 10(b)-(d). Interestingly, the high oscillations amplitude is related with\nperiodic trajectories, as we can see in Fig. 10(e), differently from what was happening in\nFig. 7(e). We also compare the trajectories for F= 1.5 at α= 0.1 and at α= 0.3 in\nFig. 10(f) and we can appreciate the gain in amplitude, due to the resonance. Then, we\nplot Fig. 11, by varying the forcing frequency ω. Here, we can spot the three peaks already\nmentioned before, as depicted in Fig. 8(a). Again, the interaction between high damping\nparameter, forcing and fractional order parameter induces the resonance. In this figure, we\ncan see that the Q−factor curves, Fig. 11(a) and Fig. 11(d), become smoother and that the\nthree peaks at ω= 0.5, ω= 1 and ω= 2 are more recognizable. Of course, those peaks are\nrelated with higher oscillations amplitude, as shown in Fig. 11(b) and Fig. 11(e). Then, in\nFig. 11(c) and Fig. 11(f), we can appreciate that the peaks match regions of non periodic\ndynamics of the system. Contrarily to the µ= 0.6 situation, in the µ= 0.9 and µ= 1\ncases, the resonance peaks are centered to specific values of the forcing frequency ω, related\namong them as ω0= 1,ω1/2=ω0/2 and ω2= 2ω0. So, we obtained the resonance frequency\n12FIG. 8: We show the Q−factor (a), the oscillations amplitude (b) and the maxima-minima diagram\nin function of the forcing frequency ωfor the following parameters: µ= 0.6,α= 0.1 and F= 1.5.\nIn the first panel, three peaks are emphasized with three vertical lines in order to compare them\nwith similar peaks in other forthcoming figures. The peaks in the panels are the result of the\nequilibrium reached by the high oscillations suppression due to the damping term, the effect of the\nforcing amplitude and the action of the fractional order parameter.\nand its harmonic and subharmonic. Also, in Fig. 11(c) and Fig. 11(f), we can see that the\ndissipation keeps doing its job by bringing more order into chaos.\nFinally, we present the oscillations amplitude gradient for F= 1 in the parameter set\nµ−α, Fig. 12(a) and for F= 2 in Fig. 12(b). As expected, in the both cases, the high\noscillations amplitudes are common in all the parameter set, although the higher ones are\nconcentrated at small µandαvalues, like in Fig. 3(b). Then, in the first case, F= 1.5,\nthe lower amplitude oscillations can be found in the exact opposite corner of the figure and\nscattered all around its upper center. Otherwise, in the second case, F= 2, as the damping\nparameter reaches the value µ= 0.6, the oscillations amplitude diminishes abruptly and for\nthe parameter values for which the resonance is triggered the oscillations amplitude becomes\nsmaller than in the F= 1.5 case. This outcome is corroborated by Fig. 12(c), where the sign\nof the difference between the oscillations amplitude at F= 2 and F= 1.5 is shown. The\n13FIG. 9: Panel (a) shows four slices of Fig. 5(e) overlapped. Panel (b) shows the oscillations\nfrequencies calculated with the fast Fourier transform. In panel (a), it is possible to see that the\ncurve for α= 0.1 is the only one showing the peak due to the high oscillations amplitude region of\nFig. 7. Also here, the oscillations frequencies tend to the forcing frequency ω= 1, except for the\nones related with the oscillations amplitude peak.\nyellow points indicate a positive difference AF=2> A F=1.5, the blue ones a negative difference\nAF=1.5> A F=2. The points in this plot are clearly intermingled, but the blue points generate\na more defined region for higher µvalues. This figure shows well that, although the forcing\namplitude is bigger in the F= 2 case, the interaction between the damping parameter and\nthe fractional order parameter can trigger trajectories with a higher oscillations amplitude,\ngiving birth to a resonance for F= 1.5. Then, in the other panels, we show three slices\nof the two gradient plots, shown in Fig. 12(a) and Fig. 12(b). Here, we can see that, for\nsmall damping parameter values ( µ= 0.18) the oscillations amplitude for F= 2 is generally\ncomparable with the oscillations amplitude for F= 1.5. When the damping parameter\ngrows, µ= 0.5 the oscillations amplitude for F= 1.5 are a little bit higher than the\nF= 2 case, but not for all the αvalues. Then, when the damping parameter grows further,\nµ= 0.8, the oscillations amplitude for F= 1.5 is definitively higher. Figures 12(d-f) show\nthe tendency of the oscillations amplitude for both forcing amplitudes. All of this corroborate\n14FIG. 10: The panels show the behavior of the system for fixed αandµ. Analyzing panel (a), we\ncan see that the system behavior at F= 1.5 is periodic of period 3, as shown in panel (e), unlike\nin Fig. 7(e). In the other three panels, the higher oscillations amplitude enlargement is lacking. In\nfact, in panel (f) we compare the orbits for α= 0.1 and α= 0.3 and we can appreciate the red\ntrajectory broadening with respect to the blue orbit, due to the resonance effect.\nthe results shown in Fig. 12(c). Again, at a high damping parameter value the F= 1.5 case\noscillates at higher amplitudes due to the resonance induced by the interaction among the\nhigh damping parameter, the fractional order parameter and the forcing amplitude.\n15FIG. 11: The panels show for µ= 0.9,α= 0.1 and F= 1.5 (a) the Q−factor, (b) the oscillations\namplitude and (c) the maxima-minima diagram in function of the forcing frequency ω. Panels\n(d), (e) and (f) show respectively the same plots for µ= 1. In panels (a) and (d) the already\ncommented three peaks are emphasized with three vertical lines in order to compare them with\nsimilar peaks in Fig. 8. Those peaks can be seen more clear, and others, that can be seen in Fig. 8,\nare disappearing as an effect of the dissipation growth. Also they are centered to certain values of\nthe forcing frequency ω= 0.5, ω= 1 and ω= 2.\n16FIG. 12: The figure shows (a) the gradient of the oscillations amplitude in the parameter set α−µ\nforF= 1, (b) the same gradient plot for F= 2, and (c) the sign of the difference AF=2−AF=1.5. In\npanel (c) the yellow points show the positive sign of the difference, while the blue points represent\nthe negative sign. The panels (d-f) show the oscillations amplitude at different values of the\ndamping parameter µand for the forcing amplitudes: F= 2 in blue and F= 1.5 in red. We can\nsee that as the damping parameter grows, the oscillations amplitude for F= 1.5 becomes higher\nthan in the case of F= 2.\n17V. CONCLUSIONS AND DISCUSSION\nWe have studied the fractional Duffing oscillator with a damping fractional term, using\nthe Gr¨ unwald-Letnikov integrator. For the parameter values used, we have find that the\nfractional order parameter can induce a resonance. When the forcing is relatively small, at\nF= 0.1, the peaks related with the resonance appear for particular values of the fractional\norder parameter, values that change in function of the damping parameter. For larger values\nof the forcing amplitude, say F= 1.5, the higher oscillations amplitude due to the resonance\nform a region in the parameter set ( F−α). In both cases the phenomenon is clear, the high\noscillations amplitude, that the oscillator exhibits at low damping parameter values, are\nattenuated at higher dissipation but the interaction between a particular forcing amplitude\nand certain low fractional order parameter values maintain them, generating a resonance.\nWe think that this phenomenon can be relevant to better understand the relation between\nthe damping parameter and the fractional order parameter. Besides, it shows the richness in\nthe dynamical behavior due to the presence of a fractional term in the nonlinear oscillator.\nThis last point can easily be observed in the overdamped case, where the dynamics of the\nsystem becomes still complex. Finally, we expect that this study can be useful to predict\nthe birth of complex or resonant behaviors in viscoelastic and porous materials or diffusion\nprocesses, that can be only modeled by using fractional derivatives.\nVI. ACKNOWLEDGMENTS\nThis work has been supported by the Spanish State Research Agency (AEI) and the\nEuropean Regional Development Fund (ERDF, EU) under Project No. PID2019-105554GB-\nI00 (MCIN/AEI/10.13039/501100011033).\n[1] Borowiec M, Litak G, Syta A. Vibration of the Duffing oscillator: effect of fractional damping.\nShock Vib. 2007; 14: 29-36.\n[2] Dafermos C. Dissipation in materials with memory. In: Lodge AS, Renardy M, Nohel JA,\neditors. Viscoelasticity and Rheology. Proceedings of a Symposium Conducted by the Math-\nematics Research Center, Wisconsin–Madison: Academic Press; 1985, p. 221-234.\n18[3] Lu Z, Wang Z, Zhou Y, Lu X. Nonlinear dissipative devices in structural vibration control: A\nreview. J. Sound Vib. 2018; 423: 18-49.\n[4] Chellaboina V, Haddad WM. Exponentially dissipative nonlinear dynamical systems: a non-\nlinear extension of strict positive realness. In: Jeltsema D, Scherpen J, editors. Proceedings\nof the 2000 American Control Conference, Chicago: IEEE Xplore; 2000, p. 3123-3127.\n[5] De S, Kunal K, Aluru NR. Nonlinear intrinsic dissipation in single layer MoS 2 resonators.\nRSC Adv. 2017; 7: 6403-6410.\n[6] Elliott SJ, Tehrani MG, Langley RS. Nonlinear damping and quasi-linear modelling. Philos.\nTrans. R. Soc. A 2015; 373: 20140402.\n[7] Horr AM, Schmidt LC. A fractional-spectral method for vibration of damped space structures.\nEng. Struct. 1996; 18: 947-956.\n[8] Ding Y, Liu X, Chen P, Luo X, Luo Y. Fractional-Order Impedance Control for Robot Ma-\nnipulator. Fractal fract. 2022; 6: 684.\n[9] Li X, Liu Z, Li J, Tisdell C. Existence and controllability for nonlinear fractional control\nsystems with damping in Hilbert spaces. Acta Math. Sci. 2019; 39; 229-242.\n[10] Strogatz SH. Nonlinear dynamics and chaos with student solutions manual: With applications\nto physics, biology, chemistry, and engineering. Boca Raton: CRC press; 2018.\n[11] Rajasekar S, Sanju´ an MAF. Nonlinear resonances. Switzerland: Springer International Pub-\nlishing; 2016.\n[12] Coccolo M, Seoane JM, Lenci S, Sanju´ an MAF. Fractional damping effects on the transient\ndynamics of the Duffing oscillator. Commun. Nonlinear Sci. Numer. Simul. 2023; 117: 106959.\n[13] Ortiz A, Yang J, Coccolo M, Seoane JM, Sanju´ an MAF. Fractional damping enhances chaos\nin the nonlinear Helmholtz oscillator. Nonlinear Dyn. 2020; 102: 2323-2337.\n[14] Syta A, Litak G, Lenci S, Scheffler M. Chaotic vibrations of the duffing system with fractional\ndamping. Chaos 2014; 24: 013107.\n19"    },    {        "title": "2402.02977v3.Variational_Flow_Models__Flowing_in_Your_Style.pdf",        "content": "Variational Flow Models: Flowing in Your Style\nKien Do, Duc Kieu, Toan Nguyen, Dang Nguyen, Hung Le, Dung Nguyen, Thin Nguyen\nApplied Artificial Intelligence Institute (A2I2), Deakin University, Australia\n{k.do, v.kieu, k.nguyen, d.nguyen, thai.le, dung.nguyen, thin.nguyen}@deakin.edu.au\nAbstract\nWe introduce “posterior flows” - generalizations of “probability flows” to a broader\nclass of stochastic processes not necessarily diffusion processes - and propose a sys-\ntematic training-free method to transform the posterior flow of a “linear” stochastic\nprocess characterized by the equation Xt=atX0+σtX1into a straight constant-\nspeed (SC) flow, reminiscent of Rectified Flow. This transformation facilitates fast\nsampling along the original posterior flow without training a new model of the\nSC flow. The flexibility of our approach allows us to extend our transformation\nto inter-convert two posterior flows of two distinct “linear” stochastic processes.\nMoreover, we can easily integrate high-order numerical solvers into the trans-\nformed SC flow, further enhancing the sampling accuracy and efficiency. Rigorous\ntheoretical analysis and extensive experimental results substantiate the advantages\nof our framework.\n1 Introduction\nDiffusion models [ 13,42,46,47] have recently achieved remarkable results across diverse machine\nlearning tasks, spanning image generation [ 8,39,40], audio synthesis [ 20], image editing [ 33,6],\nand video generation [14].\nThe strength of diffusion models mainly stems from their iterative sampling process which gradually\nremoves noise from a noisy latent sample to recover the clean sample. This iterative process can\nbe viewed as an advanced version of the generation process in hierarchical V AEs [31, 49], allowing\ndiffusion models to approximate the data distribution more accurately. Intriguingly, in the context of\ncontinuous time, this process becomes the diffusion process - a fundamental mechanism underpinning\nvarious natural phenomena in our physical world [ 42,48]. In this continuous-time setting, diffusion\nmodels can be formulated using stochastic differential equations (SDEs) [ 48]. However, this sampling\nprocess tends to be slow, typically requiring hundreds to thousands of function evaluations to generate\nhigh-quality images [ 13,48]. This problem is mainly caused by the random noise present in each\nSDE-based update step. It can be mitigated by leveraging the “probability flow” ODE, which has\nthe same marginal distribution of samples as the diffusion SDE, for sampling [ 32,48]. This class\nofstatistically equivalent ODEs provides an interesting connection between diffusion models and\nneural ODEs [7].\nRecent studies have extended diffusion SDEs to a general class of “linear” stochastic processes\ndefined by the equation Xt=atX0+σtX1, which bridge two arbitrary distributions p0andp1,\ncorresponding to the random variables X0andX1, respectively. The statistically equivalent ODEs of\nthese processes are modeled by minimizing the “flow matching” loss [ 22] - a variant of the “score\nmatching” loss in diffusion models [ 15,50]. Since the non-parametric formulas of these ODEs\ninvolve the posterior distribution p(x0, x1|xt), we term these ODEs “posterior flow” ODEs. We\ninterpret their parameterized models through the lens of variational inference when p1is a Gaussian\ndistribution, coining these models as “Variational Flow Models” (VFMs).\nPreprint. Under review.arXiv:2402.02977v3  [cs.LG]  29 Mar 2024Numerical solversStraight constant-speed (SC) flow\nPosteriorflowTo SC flowTo posterior flowFigure 1: An illustration of our method that transforms a non-straight posterior flow into a straight\nconstant-speed one and performs updates on it.\nOne notable VFM is Rectified Flow which approximates the posterior flow of the stochastic process\nXt= (1−t)X0+tX1[25,24]. Demonstrated to exhibit straightness and constant speed, this flow\nsignificantly reduces the number of sampling steps even with the use of simple numerical methods\nlike the Euler method. Currently, obtaining a Rectified Flow involves either training it from scratch\n[25] or distilling it from an existing posterior flow [ 26]. These approaches are resource-intensive,\nespecially for large models like Stable Diffusion [ 39]. Moreover, they lack flexibility as new training\nare required when converting other posterior flows into Rectified Flow and vice versa.\nTo overcome this challenge, in this work we propose a systematic training-free method to transform\nthe posterior flow (or its VFM) of any“linear” stochastic process into a straight constant-speed\n(SC) flow akin to Rectified Flow, and vice versa. Our key idea is to convert the original process\nXt=atX0+σtX1into the process ¯Xt= (1−t)X0+tX1associated with Rectified Flow (or the\nprocesses associated with other SC flows) via some clever change-of-variables operations which\nallow us to apply rules of calculus to easily compute the velocity of the SC flow from the velocity of\nthe original (posterior) flow. With our proposed transformation, instead of directly moving from a\nsample xtat time tto a sample xt′at time t′along the original flow, we first map xtto the sample ¯xt\nin the transformed SC flow. Then, we transition from ¯xtto¯xt′along the SC flow and finally recover\nxt′from ¯xt′. Consequently, we can make use of straight constant-speed flows to speed up sampling\neven when the original flow is strongly curved. The essence of this approach lies in opting for a faster\nparallel path, akin to a straight constant-speed highway, instead of navigating a designated curved\nroad when traveling from one place to another as illustrated in Fig. 1.\nOur method is versatile, applicable to all existing diffusion models, including DDPM / VP SDE\n[13,48], SMLD / VE SDE [ 46,48] as the equation Xt=atX0+σtX1encompasses the SDE\nforms of these models [ 25]. Moreover, our framework is adaptable and can readily accommodate\nadditional functionalities. Firstly, beyond converting a posterior flow into a SC flow, we can extend the\ntransformation to two posterior flows of two different “linear” stochastic processes. This is particularly\nimpactful in ODE trajectory simulation, where learning a single velocity model for one ODE is\nsufficient to simulate the trajectories of an infinite number of other ODEs. Secondly, our framework\nfacilitates the seamless adoption of high-order numerical ODE solvers for the transformed SC flow\nwith minimal adjustments compared to working on the original flow. These solvers significantly\nenhance sample quality and reduce the number of sampling steps, positioning them on par with\ncurrent state-of-the-art training-free samplers for diffusion models like DPM-Solver++ [ 28] and\nUniPC [57] (Fig. 2) which implicitly employ a special variant of our proposed SC transformations.\nComprehensive mathematical derivations, coupled with extensive experiments on a 2D toy dataset\nand large-scale image generation tasks, validate the correctness of our proposed transformation\nmethod and showcase the effectiveness of our high-order numerical solvers.\n2Real Image DDIM (NFE=999) DDIM DEIS-1 DPM-Solver++ (1M) SC-Euler\nDEIS-2 DPM-Solver++ (2M) UniPC-1 SC-AB2 SC-AB1AM2 SC-AB2AM2\nDEIS-3 DPM-Solver++ (3M) UniPC-2 SC-AB3 SC-AB2AM3 SC-AB3AM3\nFigure 2: Real images and images generated by our proposed numerical solvers ( SC-* ) and baseline\nsolvers with NFE=10. The model is Stable Diffusion v1.4 and the text prompt is “a kitchen with\nwhite counter tops and cabinets. ” .\n2 Background\nBelow we provide a brief overview of diffusion models [ 13,46] and rectified flows [ 25,24]. To\nsimplify our presentation, we will focus on continuous-time models, as the discrete-time counterparts\ncan be easily obtained by selecting an appropriate discretization scheme.\n2.1 Diffusion Models\nDiffusion models (DMs) [ 13,42,46,47] are a class of generative models that iteratively add Gaussian\nnoise to input data to transform the data distribution p0to a Gaussian distribution p1. In DDPM\n[13], given an input image x0∼p0, a noisy image xtat time t(0< t≤1) can be sampled from\np(xt|x0). This distribution is the Gaussian distribution of the form N\u0000\nxt;√αtx0,(1−αt) I\u0001\nwhere\nαt≡α(t)is a positive, decreasing function of twithα0≈1andα1≈0. This means xtcan be\ncomputed as:\nxt=√αtx0+√\n1−αtϵ (1)\nwhere ϵ∼ N (0,I). Song et al. [ 48] have shown that the diffusion process in Eq. 1 can be expressed\nas the variance-preserving stochastic differential equation (VP SDE):\ndxt=−βt\n2xtdt+p\nβtdwt (2)\nwhere wtdenotes the Wiener process, and βt≡β(t)is a function of tderived from αt. The\nassociated probability flow ordinary differential equation (PF ODE) of this SDE, which has the same\nmarginal distribution of samples pt(xt), is expressed as:\ndxt=−βt\n2(xt+∇logp(xt))dt (3)\nwhere ∇logp(xt)is the score function , approximated as−ϵθ(xt,t)√1−αt.ϵθis a noise network trained by\nminimizing the noise matching (NM) loss:\nLNM(θ) =Eth\nλtEx0,ϵh\n∥ϵθ(xt(x0, ϵ), t)−ϵ∥2\n2ii\n(4)\nwhere λtdenotes the weight at time tand is set to 1 by default. LNMis a variant of the score matching\n(SM) loss [ 15,50]. In Eq. 4, t∼ U([0,1]),x0∼p0,ϵ∼ N (0,I), andxtis computed from x0andϵ\nvia Eq. 1.\n3Another well-known diffusion model is SMLD [46, 47], characterized by the diffusion process:\nxt=x0+σtϵ (5)\nEq. 5 is just a reparameterization of Eq. 1 by regarding xtandσtin Eq. 5 asxt√αtandq\n1−αt\nαtin\nEq. 1, respectively. The diffusion process has been shown to be equivalent to the variance exploding\nSDE (VE SDE) [48, 16]:\ndxt=r\nd(σ2\nt)\ndtdwt (6)\nThe corresponding PF ODE of this SDE is:\ndxt=−σt∇xlogpt(xt)dσt (7)\n≈ϵθ(xt, t)dσt (8)\nwhere ϵθis the noise network identical to the one in DDPM.\n2.2 Generalized Rectified Flows\nLiu et al. [ 25] have explored a broader class of stochastic processes between the two boundary\ndistributions p0andp1, characterized by a general formula:\nXt=ϕ(X0, X1, t) (9)\nIn this equation, X0,X1,Xtrepresent random variables with marginal distributions p0,p1,pt,\nrespectively. The function ϕ:Rd×Rd×[0,1]→Rdmaps a coupling (X0, X1)(w.r.t. the joint\ndistribution p0,1) toXtat time t∈[0,1]. Their goal is to find a deterministic mapping from X0to\nX1governed by the ODE: dxt=vθ(xt, t)dt, such that the marginal distribution of Xtobtained\nfrom the ODE matches that of Xtderived from the stochastic process in Eq. 9. This statistically\nequivalent ODE is referred to as a Generalized Rectified Flow (GRF) and vθis learned by minimizing\nthevelocity matching (orflow matching ) lossLVM[22, 25] below:\nLVM(θ) =EtEx0,x1h\n∥vθ(xt, t)−vϕ(x0, x1, t)∥2\n2i\n(10)\nwhere t∼ U([0,1]),x0, x1∼p0,1,xt=ϕ(x0, x1, t), and vϕ(x0, x1, t) := ˙ xt=dϕ(x0,x1,t)\ndtdenotes the target velocity at time tfor the trajectory between x0,x1. Sometimes, we express the\ntarget velocity as vϕ(xt, t) := ˙xtinstead of vϕ(x0, x1, t) :=dϕ(x0,x1,t)\ndtto emphasize that it is a\nfunction of xt=ϕ(x0, x1, t)rather than x0,x1.\nA remarkable variant of Eq. 9 is:\nXt=atX0+σtX1 (11)\nwhere at≡a(t),σt≡σ(t)are continuously differentiable functions of tsatisfying that a0= 1,\na1= 0, andσ0= 0. When X1follows the standard distribution (i.e., p1=N(0,I)), Eq. 9 describes\nvarious diffusion processes including VP SDE and VE SDE. This implies that probability flow ODEs\ncan be regarded as specific instances of GRFs. Eq. 9 also has an important special case:\nXt= (1−t)X0+tX1 (12)\nwhich constitutes a linear interpolation between X0andX1. The statistically equivalent ODE of this\nstochastic process is called Rectified Flow (RF).\n3 Variational Flow Models\nBy introducing the Generalized Rectified Flow (GRF) framework (Eqs. 9, 10), deriving a special\nvariant of the framework (Eq. 11) that encompasses various diffusion processes and their correspond-\ning PF ODEs, and providing theoretical analysis for this special variant, Liu et al. [ 25] have done\nan inspiring work. In this section, we aim to (i) convey their findings using simpler notations , (ii)\ninterpret the GRF framework from the perspective of variational inference , and (iii) derive explicit\nformulas for the velocities of well-known diffusion models . (ii) and (iii) are our novel contributions.\nDetailed proofs are available in the appendix, with some not covered in [ 25]. The content in this\nsection will greatly assist in presenting our method in the next section.\n4Model at σt vϕ(x0, x1, t)v∗(xt, t)\nxt,x0|t xt,x1|t\nVP SDE√αt√1−αtβtαt\n2\u0010\nx1√1−αt−x0√αt\u0011\nβtαt\n2(1−αt)\u0010\nxt−x0|t√αt\u0011\n−βt\n2\u0010\nxt−x1|t√1−αt\u0011\nsub-VP SDE√αt1−αtβt√αt\n2\u0000\n2√αtx1−x0\u0001βt√αt\n1−αt\u0000√αtxt−1+αt\n2x0|t\u0001\n−βt\n2\u0000\nxt−(1 +αt)x1|t\u0001\nVE SDE 1 σt ˙σtx1˙σt\nσt\u0000\nxt−x0|t\u0001\n˙σtx1|t\nRect. Flow 1−t t x 1−x01\nt\u0000\nxt−x0|t\u00011\n1−t\u0000\nx1|t−xt\u0001\nTable 1: The coefficients at,σt, target velocities vϕ(x0, x1, t), and posterior velocities v∗(xt, t)\ncorresponding to well-known diffusion models [ 48] and Rectified Flow (Rect. Flow) [ 25]. In the case\nof (sub-)VP SDE, βt=t\u0000\nβmax−βmin\u0001\n+βminandαt=e−1\n2t(βt+βmin).\nIf we regard LVMin Eq. 10 as a function of a non-parameteric velocity vrather than the parameters θ,\nwe can prove that this loss attains its global minimum of 0 at v∗(xt, t) :=Ex0,x1|xt[vϕ(x0, x1, t)]\n(Appdx. A.1). Furthermore, the ODE characterized by v∗exhibits the same marginal distribution of\nsamples (i.e., p(xt)) as the stochastic process in Eq. 9 at every time t∈[0,1](proof in Appdx. A.2).\nThe intuition for this is that given either of the boundary distributions p0,p1is fixed, the match\nin velocity results in a corresponding match in probability. This alignment is substantiated by the\ncontinuity equation∂pt(xt)\n∂t=∇·(pt(xt)v(xt, t)), where the temporal evolution of pt(xt)primarily\nrelies on the velocity v(xt, t). We refer to v∗as the posterior velocity based on its formula and the\nODE characterized by v∗as the posterior flow ODE or simply posterior flow of the given stochastic\nprocess. Since vθcan be regarded as a variational approximation ofv∗withvθ=v∗leading to the\nmaximum likelihood when p1is a Gaussian distribution (details in Appdx. A.3), we refer to vθas a\nvariational velocity and its associated ODE as a variational flow ODE or aVariational Flow Model\n(VFM).\nFor the class of “linear” stochastic processes defined in Eq. 11, the corresponding target velocities\nvϕ(x0, x1, t)and posterior velocities v∗(xt, t)of these processes are represented as follows:\nvϕ(x0, x1, t) :=dϕ(x0, x1, t)\ndt= ˙atx0+ ˙σtx1 (13)\nv∗(xt, t) :=Ex0,x1|xt[vϕ(x0, x1, t)] (14)\n=Ex0,x1|xt[˙atx0+ ˙σtx1] (15)\nInterestingly, v∗(xt, t)can also be expressed in the following forms (details in Appdx. A.1):\nv∗(xt, t) =˙σt\nσtxt+at\u0012˙at\nat−˙σt\nσt\u0013\nx0|t (16)\n=˙at\natxt+σt\u0012˙σt\nσt−˙at\nat\u0013\nx1|t (17)\nwhere x0|t:=Ex0|xt[x0]andx1|t:=Ex1|xt[x1]. The reason behind Eqs. 16, 17 is that given xt\nandx0(x1),x1(x0) can be computed with certainty via Eq. 9.Thus, the stochasticity in p(x0, x1|xt)\ncan be fully transformed into the stochasticity in either p(x0|xt)orp(x1|xt). Eqs. 16, 17 suggest\nthatv∗(xt, t)can be modeled indirectly via modeling either x0|torx1|t. In the context of diffusion\nmodels, x0|tandx1|tare modeled using a denoising autoencoder x0,θ(xt, t)[16,51,50] and a noise\nnetwork x1,θ(xt, t)(i.e.,ϵθ(xt, t)) [46, 13], respectively.\nIn Table 1, we provide explicit formulas for the target and posterior velocities of well-known diffusion\nmodels. The detailed derivations can be found in Appdx. A.4.\n4 Method\n4.1 Transforming posterior flows into straight constant-speed flows\nIn this section, we establish theoretically that the posterior flow of any linear stochastic process\ndefined by Eq. 11, no matter how intricately curved, can be transformed into straight constant-speed\nflows reminiscent of Rectified Flow [ 25]. Our investigation carries substantial implications in practical\napplications by offering a systematic training-free method to perform efficient sampling with minimal\nupdates for a wide range of stochastic processes, including the diffusion processes in Table 1.\n5Let us begin by examining the stochastic processes described by the equations1:\nXt= (1−φt)X0+φtX1and (18)\nXt=X0+φtX1 (19)\nwhere φtis a continuously differentiable function of tsatisfying that φ0= 0. This condition ensures\nthat the distribution at t= 0aligns with p0. The posterior velocities corresponding to these processes\nare computed as follows:\nv∗(xt, t) = ˙φt\u0000\nx1|t−x0|1,t\u0001\nand (20)\nv∗(xt, t) = ˙φtx1|t (21)\nIn Eq. 20 and Eq. 21, the two vectors x1|t−x0|1,tandx1|trepresent the direction ofv∗, respectively\nwhile ˙φtrepresents the scale in length ofv∗.\nSince these two direction vectors depend on the estimated boundary points x1|t,x0|1,twhich are\nexpected to be invariant w.r.t. t(Appdx. A.6), the velocities of samples in the same trajectory should\nhave almost the same direction . This implies that the flows characterized by either Eq. 20 or Eq. 21\narenearly straight . If˙φtvaries across time, these flows experience varying speeds. By contrast, when\n˙φtis constant w.r.t. t, these flows maintain a constant speed. The case where ˙φt= 1, or equivalently\nφt=t2, is of special interest because in this case, the instantaneous velocity v∗(xt, t)becomes the\naverage velocity (i.e., x1|t−x0|1,tfor Eq. 20 and x1|tfor Eq. 21). This implies that in ideal cases\nwe only need one Euler update to generate samples of a boundary distribution given samples of the\nother. When φt=t, Eqs. 18, 19 become:\nXt= (1−t)X0+tX1and (22)\nXt=X0+tX1 (23)\nEq. 22 aligns with the stochastic process corresponding to Rectified Flow [ 25], while Eq. 23 resembles\nVE SDE (Table 1), differing only in the coefficient of X1, which is t∈[0,1]rather than some function\nσtoft[16]. Given the correspondence between a stochastic process and its posterior flow, we term the\nstochastic process associated with a straight (constant-speed) posterior flow as astraight (constant-\nspeed) stochastic process . The transformation of a posterior flow into a straight constant-speed one\nis tantamount to converting its associated stochastic process into a straight constant-speed process.\nBelow, we present in detail how this can be accomplished in two steps. Initially, a non-straight flow\n(process) is transformed into a straight flow (process). Subsequently, the straight flow (process)\nobtained in the first step is converted into a straight constant-speed flow (process). This procedure\nturns a sample xtof the original flow into a sample ¯xtof the straight constant-speed flow. Following\nthe straight constant-speed flow, ¯xtseamlessly transitions to ¯xt′and is then mapped back to xt′.\n4.1.1 Non-straight flows →Straight flows\nWe can transform a stochastic process of the form Xt=atX0+σtX1into a straight one of the form\nXt= (1−φt)X0+φtX1(Eq. 18) by reformulating the original stochastic process as follows:\nXt=atX0+σtX1 (24)\n⇔Xt\nat+σt=at\nat+σtX0+σt\nat+σtX1 (25)\n⇔˜Xt= (1−φt)X0+φtX1 (26)\nwhere ˜Xt:=Xt\nat+σt,φt:=σt\nat+σt.φ0= 0because a0= 1andσ0= 0. Possible numerical problems\nrelated to this transformation are discussed in Section 4.2.\nLetxtbe a sample in the original posterior flow associated with the process in Eq. 24, then ˜xt=xt\nat+σtbecomes a sample in the straight posterior flow associated with the process in Eq. 26. From ˜xt, we\nperform an Euler update with the velocity v∗(˜xt, t)and step size t′−tto obtain ˜xt′:\n˜xt′= ˜xt+ (t′−t)v∗(˜xt, t) (27)\n1We don’t include the stochastic process Xt=φtX0+X1here because all the results w.r.t. this process\ncan be easily obtained from the results w.r.t. the process Xt=X0+φtX1by swapping the role of at,X0and\nσt,X1, respectively.\n2˙φt= 1implies that φt=t+cwithcis a constant. Since φ0= 0, we have c= 0.\n6We then recover xt′from ˜xt′as:\nxt′= (at′+σt′) ˜xt′ (28)\nSince ˜Xtis a scaled version of Xt(by a factor of1\nat+σt), we refer to the transformation from Xtto\n˜Xtin Eqs. 24-26 as “variable scaling” . The remaining question revolves around computing v∗(˜xt, t).\nIntuitively, v∗(˜xt, t)can be approximated by vθ(˜xt, t)ifpt(xt)has support over the entire sample\nspace. However, in practice, this approximation is unrealiable. It is because at time t,vθonly yields\naccurate outputs for samples from the original stochastic process (i.e., xt∼pt(xt)) on which vθwas\ntrained, while ˜xtandpt(˜xt)significantly differ from xtandpt(xt), respectively. Fortunately, we\ncan overcome this problem by expressing v∗(˜xt, t)viav∗(xt, t)based on the following results:\nProposition 1 (Velocities of a scaled process ).LetXt=atX0+σtX1be a stochastic process\ndefined in Eq. 11 and let ˜Xt=ktXtbe another stochastic process derived from Xtwithktbeing\na differentiable function of t. The target velocity vϕ(˜xt, t)and posterior velocity v∗(˜xt, t)w.r.t.\nthe stochastic process ˜Xtcan be computed via the target velocity vϕ(xt, t)and posterior velocity\nv∗(xt, t)w.r.t. the stochastic process Xt, respectively, as follows:\nvϕ(˜xt, t) =ktvϕ(xt, t) +˙ktxt\nv∗(˜xt, t) =ktv∗(xt, t) +˙ktxt\nProof. By definition, vϕ(˜xt, t) :=d˜xt\ndtwhere ˜xt=ktxt=kt(atx0+σtx1)for some x0,x1. Thus,\nwe can compute vϕ(˜xt, t)as follows:\nvϕ(˜xt, t) =d\ndt(ktxt) (29)\n=˙ktxt+kt˙xt (30)\n=˙ktxt+ktvϕ(xt, t) (31)\nRecalling that vϕ(xt, t) = ˙atx0+ ˙σtx1(Eq. 13), vϕ(˜xt, t)can be expressed as a function of x0,x1\nandtas follows:\nvϕ(˜xt, t) =˙kt(atx0+σtx1) +kt(˙atx0+σtx1)\n=\u0010\n˙ktat+kt˙at\u0011\nx0+\u0010\n˙ktσt+kt˙σt\u0011\nx1\n=d(ktat)\ndtx0+d(ktσt)\ndtx1\nThe posterior velocity v(˜xt, t)can be computed as follows:\nv∗(˜xt, t) :=Ex0,x1|˜xt[vϕ(˜xt, t)] (32)\n=Ex0,x1|xth\n˙ktxt+ktvϕ(xt, t)i\n(33)\n=˙ktxt+ktEx0,x1|xt[vϕ(xt, t)] (34)\n=˙ktxt+ktv∗(xt, t) (35)\nIn Eq. 33, p(x0, x1|xt) =p(x0, x1|˜xt)because ˜xt=ktxtis adeterministic function of xtand map\nto the same (x0, x1)pair as xt.\nApplying Eq. 35 to the process in Eq. 26 with a note that kt=1\nat+σt, we have:\nv∗(˜xt, t) =(at+σt)v∗(xt, t)−(˙at+ ˙σt)xt\n(at+σt)2(36)\nGiven the relationships between v∗(xt, t),xt,x1|tandx0|tin Eqs. 16, 17, we can also express\nv∗(˜xt, t)in terms of xt,x1|torxt,x0|torx1|t,x0|1,t. Details about these expressions of v∗(˜xt, t)\nare provided in Appdx. A.7.1. In the last case, v∗(˜xt, t) = ˙φt\u0000\nx1|t−x0|1,t\u0001\nand Eq. 27 can be\nexpressed as follows:\n˜xt′= ˜xt+ (t′−t) ˙φt\u0000\nx1|t−x0|1,t\u0001\n(37)\n7It is interesting to see that since ˜xtrepresents a sample from the straight flow ˜Xt= (1−φt)X0+\nφtX1,v∗(˜xt, t)equals ˙φt\u0000\n˜x1|t−˜x0|1,t\u0001\n, as per Eq. 20. It might suggest that ˜x1|t=x1|tand\n˜x0|1,t=x0|1,t, and this impression is correct because ˜xtis a deterministic transformation of xt.\nWe can also transform the original stochastic process into the straight process ˜Xt=X0+σt\natX1by\nsetting ˜XttoXt\nat. In this case, the posterior velocity v∗(˜xt, t)can be computed as follows:\nv∗(˜xt, t) =atv∗(xt, t)−˙atxt\na2\nt(38)\nAfter computing ˜xt′via the Euler update in Eq. 27, we retrieve xt′from ˜xt′asxt′=at˜xt′.\n4.1.2 Straight flows →Straight constant-speed flows\nA straight flow often requires fewer Euler updates than the original one, especially when the original\nflow is strongly curved. However, if a straight flow has varying speeds over time, using large step\nsizes will lead to inaccurate updates. Therefore, it is desirable to make a straight flow constant-\nspeed. Due to space limit, we will only present the transformation procedure for the straight process\n˜Xt= (1−φt)X0+φtX1here and provide the discussion for ˜Xt=X0+φtX1in Appdx. A.7.2.\nThere are two ways to make ˜Xt= (1−φt)X0+φtX1withφt=σt\nat+σtconstant-speed. The first\napproach is based on the observation that the posterior flow corresponding to this process has the\nformd˜xt=v∗(˜xt, t)dtwithv∗(˜xt, t) = ˙φt\u0000\nx1|t−x0|1,t\u0001\n, which allows us to rewrite the posterior\nflow as follows:\nd˜xt= ˙φt\u0000\nx1|t−x0|1,t\u0001\ndt (39)\n=\u0000\nx1|t−x0|1,t\u0001\ndφt (40)\nEqs. 39, 40 suggest that instead of moving with non-constant speed when the change of time is\nconstant (i.e.,dt), we can move with constant speed when the change of time is non-constant (i.e.,\ndφt). Therefore, we refer to this approach as “time adjustment” . We can compute x1|t−x0|1,tas\nv∗(˜xt,t)\n˙φtwithv∗(˜xt, t)given in Eq. 36. We can also compute x1|t,x0|1,tdirectly via v∗(xt, t)and\nxtand then take the difference between the two terms. Both ways lead to the same result:\nx1|t−x0|1,t=(at+σt)v∗(xt, t)−(˙at+ ˙σt)xt\nat˙σt−˙atσt(41)\nEq. 41 is preferred if we model v∗(xt, t)asvθ(xt, t). On the other hand, if we model x1|tas\nx1,θ(xt, t)(x1,θis the noise network ϵθin diffusion models), it will be better to compute x1|t−x0|1,t\nas follows:\nx1|t−x0|1,t=(at+σt)x1|t−xt\nat(42)\nOnce x1|t−x0|1,thas been computed via either Eq. 41 or Eq. 42, we can perform an Euler update\nwith the constant-speed velocity x1|t−x0|1,tto transition ˜xtto˜xt′:\n˜xt′= ˜xt+ (φt′−φt)\u0000\nx1|t−x0|1,t\u0001\n(43)\nand then recover xt′from ˜xt′via Eq. 28.\nIt’s intriguing to observe the similarity between the Euler updates in Eqs. 43 and 37, with the\ndistinction lying in the coefficient of the constant velocity x1|t−x0|1,t, where it is (φt′−φt)in\nthe former and ˙φt(t′−t)in the latter. The former discretizes dφt, while the latter is merely the\nfirst-order Taylor approximation of the former. To highlight that Eq. 43 is an update on a straight\nconstant-speed flow, we present it with new notations as follows:\n¯xt′= ¯xt+ (φt′−φt)\u0000\nx1|t−x0|1,t\u0001\n(44)\n= ¯xt+ (φt′−φt)v∗(¯xt, t) (45)\nwhere ¯xtdenotes a sample from the straight constant-speed flow d¯Xt= (X1−X0)dφtwith\n¯Xt=˜Xt.\n8Type Stochastic process Sample Posterior velocity\nOri. Xt=atX0+σtX1 xt v∗(xt, t)\nSN˜Xt= (1−φt)X0+φtX1, with˜xt=xt\nat+σtv∗(˜xt, t) =(at+σt)v∗(xt,t)−(˙at+ ˙σt)xt\n(at+σt)2 , or\nφt=σt\nat+σt,˙φt=at˙σt−˙atσt\n(at+σt)2 v∗(˜xt, t) =at˙σt−˙atσt\n(at+σt)2\u0010(at+σt)x1|t−xt\nat\u0011\nSCd¯Xt= (X1−X0)dφt ¯xt= ˜xt v∗(¯xt, t) =(at+σt)x1|t−xt\nat, or\n¯Xt= (1−t)X0+tX1 ¯xt= ˜xt+ (t−φt) (x1−x0) v∗(¯xt, t) =(at+σt)v∗(xt,t)−(˙at+ ˙σt)xt\nat˙σt−˙atσt\nTable 2: Summary of the transformations from a (stochastic) process of the form Xt=atX0+σtX1\nto the corresponding straight process of the form ˜Xt= (1−φt)X0+φtX1and straight constant-\nspeed process of either the form d¯Xt= (X1−X0)dφtor the form ¯Xt= (1−t)X0+tX1.\nType Stochastic process Sample Posterior velocity\nOri. Xt=atX0+σtX1 xt v∗(xt, t)\nSN˜Xt=X0+φtX1, with˜xt=xt\natv∗(˜xt, t) =atv∗(xt,t)−˙atxt\na2\nt, or\nφt=σt\nat,˙φt=at˙σt−˙atσt\na2\ntv∗(˜xt, t) =at˙σt−˙atσt\na2\ntx1|t\nSCd¯Xt=X1dφt ¯xt= ˜xt v∗(¯xt, t) =x1|t, or\n¯Xt=X0+tX1 ¯xt= ˜xt+ (t−φt)x1 v∗(¯xt, t) =atv∗(xt,t)−˙atxt\nat˙σt−˙atσt\nTable 3: Summary of the transformations from a (stochastic) process of the form Xt=atX0+σtX1\nto the corresponding straight process of the form ˜Xt=X0+φtX1and straight constant-speed\nprocess of either the form d¯Xt=X1dφtor the form ¯Xt=X0+tX1.\nThe second approach is to transform ˜Xt= (1−φt)X0+φtX1into the process ¯Xt= (1−t)X0+\ntX1by setting ¯Xt=˜Xt+ (t−φt) (X1−X0). Since ¯Xtis shifted from ˜Xtwith the offset\n(t−φt) (X1−X0), we name this transformation “variable shifting” . In this approach, v∗(¯xt, t)\nalso equals x1|t−x0|1,t(details in Appdx. A.7.2) and can be approximated via Eq. 41 or Eq. 42.\nWith ¯xtandv∗(¯xt, t)in hand, we perform the following Euler update to obtain ¯xt′:\n¯xt′= ¯xt+ (t′−t)v∗(¯xt, t) (46)\n= ¯xt+ (t′−t)\u0000\nx1|t−x0|1,t\u0001\n(47)\nSubsequently, we retrieve ˜xt′from ¯xt′as follows:\n˜xt′= ¯xt′−(t−φt)\u0000\nx1|t′−x0|1,t′\u0001\n(48)\nFinally, we restore xt′from ˜xt′using Eq. 28.\nHowever, the computation of x1|t′−x0|1,t′in Eq. 48 necessitates knowledge of xt′, which is presently\nunknown. To circumvent this, we can instead compute x1|t−x0|1,t. Given that xtandxt′belong\nto the same trajectory, their estimated boundary pairs\u0000\nx1|t, x0|1,t\u0001\nand\u0000\nx1|t′, x0|1,t′\u0001\nare expected\nto be similar. Therefore, x1|t−x0|1,tserves as an approximation for x1|t′−x0|1,t′. Eq. 48 can be\nreformulated as follows:\n˜xt′≈¯xt′−(t−φt)\u0000\nx1|t−x0|1,t\u0001\n(49)\nInterestingly, when the Euler method is used, Eq. 49 renders the “variable shifting” approach\nequivalent to the “time adjustment” approach, which is proven in Appdx. A.7.2.\nWe summarize our proposed transformations in Tables 2 and 3. To enhance readability in subsequent\nsections, we will use the term “ SNflow/process ”to denote a straight non-constant-speed flow/process\nand “ SCflow/process” to indicate a straight constant-speed flow/process.\n4.2 Ensuring the numerical robustness of our method\nOne factor affecting the quality of samples generated by our method is the numerical robustness\nof some operations in our proposed SN and SC transformations. As shown in Tables 2, 3, the\ncomputations of ˜xt,v∗(˜xt, t),¯xt, and v∗(¯xt, t)from xt,v∗(xt, t)(orx1|t) often involves division\nby some time-dependent terms, which can lead to a severe numerical problem if these terms approach\n0 at a specific time t. We can mitigate this problem by clipping the denominators in the formulas of\n˜xt,v∗(˜xt, t),¯xtandv∗(¯xt, t)to a small value. For example, v∗(¯xt, t)computed via Eq. 42 has atin\n9its denominator, making it undefined at t= 1since a1= 0by design. To ensure numerical stability,\nwe adjust attosign(at)∗max (|at|, ϵ)where sign(at)= 1 if at≥0and -1 otherwise and ϵis a\nsmall positive number. Through our experiments, we have observed that using this simple clipping\ntechnique is sufficient to maintain numerically stable updates along SN and SC flows in most practical\nscenarios. It is worth noting that constraining the time twithin the range [0,1−ϵ]does notalways\nguarantee robustness, particularly in cases where a1, as a function of ϵ, can become exceedingly small\nwithout reaching precisely 0. For example, if at= (1−t)5andϵ= 1e−3thena1=ϵ5= 1e−15,\nwhich, although not exactly 0, is extremely small and can cause numerical instability.\n4.3 DDIM as a special case of our method\nIt is worth noting that DDIM [ 43] can be viewed as a special case of our method, which transforms the\nprobability flow associated with VP SDE / DDPM [ 13,48] into a straight constant-speed counterpart\nby applying a “variable scaling” transformation followed by a “time adjustment” transformation to\nthe original flow. A DDIM-styled Euler update for a general class of stochastic processes Xt=\natX0+σtX1is given as follows:\n¯xt′= ¯xt+ (φt′−φt)x1|t (50)\n⇔xt′\nat′=xt\nat+\u0012σt′\nat′−σt\nat\u0013\nx1|t (51)\n⇔xt′=at′\u0012xt−σtx1|t\nat+σt′\nat′x1|t\u0013\n(52)\n⇔xt′=at′x0|1,t+σt′x1|t (53)\nwhere ¯xt:=xt\natandφt:=σt\nat. When the original stochastic process is VP SDE (i.e., at=√αt\nandσt=√1−αt),tis converted to an integer in [1, T],t′=t−1, and x1|tis approximated by\nϵθ(xt, t), Eq. 52 can be written as follows:\nxt−1=p\nαt−1\u0012xt−√1−αtϵθ(xt, t)√αt\u0013\n+p\n1−αt−1ϵθ(xt, t) (54)\nwhich matches the deterministic sampling step in DDIM [43].\n4.4 Using higher-order numerical solvers on straight constant-speed flows\nIn Section 4.1, we have discussed how to transform a non-straight flow into a straight constant-\nspeed (SC) flow and perform Euler updates on the SC flow. While achieving a perfectly straight\nconstant-speed flow is ideal, it is seldom realized in practice due to the inherent randomness of the\nindependent coupling typically used in training the velocity/noise model (Eqs. 4, 10). The straightness\nquality of an SC flow can be improved by “reflowing” the original flow [ 25], i.e., retraining the\nvelocity network vθwith the deterministic coupling defined by vθ. Unfortunately, this approach is\nresource-intensive, particularly with large models and extensive training data. A more cost-effective\nstrategy is to devise better numerical solvers to enhance accuracy and speed of transitioning along SC\nflows. This initiative begins by exploring well-established higher-order ODE solvers such as Runge\nKutta and linear multi-step methods as alternatives to the first-order Euler solver. Notably, integrating\nthese high-order solvers into our framework requires little yet non-trivial modification to the update\nformula in Eq. 45.\nIn this paper, we exclusively focus on SC flows obtained via the “time adjustment” transformation\nbecause applying high-order numerical solvers to this type of SC flows is more straightforward\ncompared to the counterpart obtained through the “variable shift” transformation. The primary\nadvantage of the former is the exact back-and-forth mapping between xt′and¯xt′without the need\nforv∗(¯xt′, t′). The exploration of high-order numerical solvers for the latter type of SC flows is\nreserved for future work.\nTo streamline our presentation, we adopt the notations tiandti+1to denote the current and next time\nsteps, respectively, instead of tandt′. We also use ti−kto indicate the k-th step before the current\nstepti. The posterior velocity v∗(¯xti, ti)of a sample ¯xtiin a SC flow will be denoted as ¯v∗\nti.\n10Our proposed high-order solvers on SC flows adhere to the general update formula below:\n¯xti+1= ¯xti+ ∆φti+1¯v◦\nti:ti+1(55)\nwhere ∆φti+1:=φti+1−φti, and ¯v◦\nti:ti+1represents the average velocity along SC flows within the\ntime interval from titoti+1. Different numerical methods yield distinct approximations of ¯v◦\nti:ti+1.\n4.4.1 Runge-Kutta methods\n0 0 0 ··· 0\nc2a21 0··· 0\n...............\nckak1ak2··· 0\nb1 b2··· bk\nTable 4: The Butcher tableau for explicit Runge-Kutta methods\nThe general update formula of explicit Runge-Kutta (RK) methods on SC flows is given by:\n¯xti+1= ¯xti+ ∆φti+1kX\nj=1bjfj (56)\nwhere f1,f2,...,fkare computed as follows:\nf1=v∗(¯xti, ti) (57)\nf2=v∗\u0012\n¯xti+ ∆φτ2\u0012a2,1\nc2f1\u0013\n, τ2\u0013\n;τ2=ti+c2∆ti+1 (58)\n...\nfk=v∗\n¯xti+ ∆φτk\nk−1X\nj=1ak,j\nckfj\n, τk\n;τk=ti+ck∆ti+1 (59)\nHere, ∆φti+1:=φti+1−φtiand∆φτk:=φτk−φtifor all k≥2. The coefficients b1,...,bk,c2,...,\nck,a2,1,...,ak,k−1should satisfyPk\nj=1bj= 1,Pk−1\nj=1ak,j=ck, and other conditions which ensure\nthat Runge-Kutta methods of order kclosely replicate the Taylor series method of the same order.\nTypically, these coefficients are structured within the Butcher tableau, as illustrated in Table 4, for\nconvenient reference.\nWhen comparing Eq. 56 with Eq. 55, it becomes evident that the average velocity ¯v◦\nti:ti+1can be\ninterpreted as the linear interpolation of intermediate velocities f1,f2,...,fkdue to the conditionPk\nj=1bj= 1. Moreover, fkcan be seen as the linear interpolation of f1,...,fk−1sincePk−1\nj=1ak,j\nck=\n1.\nThe main advantage of the update formulas presented in Eqs. 57-59 lies in the separation of sample\nand time updates intotwo distinct time spaces . Specifically, the sample update is performed in the\nscaled time space (i.e., φ-space) while the time update is conducted in the original time space (i.e.,\nthet-space).\nIf both the sample and time updates are executed in the φ-space, explicit Runge-Kutta methods will\nexhibit the following general formulas:\n¯xφi+1= ¯xφi+ ∆φi+1kX\nj=1bjfj (60)\n11where f1,f2,...,fkare computed as follows:\nf1=v∗(¯xφi, φi) (61)\nf2=v∗\u0012\n¯xφi+c2∆φi+1\u0012a2,1\nc2f1\u0013\n, φi+c2∆φi+1\u0013\n(62)\n...\nfk=v∗\n¯xφi+ck∆φi\nk−1X\nj=1ak,j\nckfj\n, φi+ck∆φi+1\n (63)\nHere, ∆φi+1:=φi+1−φi. In the above equations, we delibrately use the notation φiinstead of\nφtito emphasize that we explicitly specify φiduring sampling, rather than specifying tiand then\ncomputing φtias a function of ti.\nHowever, since the velocity/noise network vθ/x1,θis trained in the t-space rather than the φ-space,\nwe need to map φi+ck∆φi+1back to the t-space for computing fk. This requires knowing the\nanalytical form of the inverse mapping t(φ) :=φ−1(t), which is often challenging to obtain for\nmany stochastic processes. Moreover, t(φi+ck∆φi+1)is usually a real number different from the\ninteger time steps used in discrete-time models like DDPM [ 13] or Stable Diffusion [ 39]), leading\nto the approximation error when t(φi+ck∆φi+1)is approximated with the closest discrete time\nstep. By performing time updates in the t-space as shown in Eqs. 57-59, these challenges can be\neffectively mitigated.\nIn Appdx. B.1, we delve into the specific constraints governing the coefficients of explicit Runge-\nKutta (RK) methods of order 2 and 3. These contraints have some degrees of freedom that give us\nflexibility in applying RK methods to discrete-time models. Specifically, we can tailor ckto ensure\nthatti+ck∆ti+1aligns with a predefined discrete time step. Additionally, we provide the update\nformulas for several well-known RK methods such as the midpoint method, Kutta’s third-order\nmethod (RK3), and the classic Runge-Kutta method (RK4) on SC flows.\nBelow is the update formula for Heun’s method, a second-order RK method, on SC flows:\n¯xa\nti+1= ¯xti+ ∆φti+1¯v∗\nti(64)\n¯xti+1= ¯xti+ ∆φti+1v∗\u0010\n¯xa\nti+1, ti+1\u0011\n+ ¯v∗\nti\n2(65)\nWe note that ¯v∗\ntiequals x1|ti−x0|tiand can be computed directly via Eq. 41 or Eq. 42. To compute\nv∗\u0010\n¯xa\nti+1, ti+1\u0011\nin Eq. 65, we first recover xa\nti+1from ¯xa\nti+1asxa\nti+1=\u0000\nati+1+σti+1\u0001\n¯xa\nti+1or\nxa\nti+1=ati+1¯xa\nti+1depending on whether the straight process is ˜Xt= (1−φt)X0+φtX1(Eq. 18)\nor˜Xt=X0+φtX1(Eq. 19). Then, we feed the tuple ( xa\nti+1,ti+1) to the velocity/noise network\nvθ/x1,θto compute v∗\u0010\n¯xa\nti+1, ti+1\u0011\n.\n4.4.2 Linear multi-step methods\nHigh-order RK methods typically compute intermediate velocities f1,f2,... at intermediate steps\nbetween tiandti+1to derive the final velocity ¯v◦\nti:ti+1for update. However, they do not reuse\nthese intermediate velocities in subsequent updates, resulting in computational inefficiency. To\naddress this limitation of RK methods, linear multi-step methods such as Adams-Bashforth (AB)\nand Adams-Moulton (AM) [ 3,11] utilize cached velocities ¯v∗\nti−1,¯v∗\nti−2,... from prior updates. This\nallows them to achieve a high-order approximation of ¯xti+1with minimal or no additional model\nevaluations. For both AB and AM updates on SC flows, we propose the following approximations of\n¯v◦\nti:ti+1:\n¯v◦\nti:ti+1≈Rti+1\nti¯v(t)dt\n∆ti+1(66)\nand\n¯v◦\nti:ti+1≈Rφti+1\nφti¯v(φt)dφt\n∆φti+1(67)\n12where ∆ti+1:=ti+1−ti,∆φti+1:=φti+1−φti;¯v(t)and¯v(φt)areLagrange interpolating\npolynomial functions oftandφt, respectively.\nThe main difference between Eq. 66 and Eq. 67 is that in the former, velocity interpolation is\nconducted in the t-space with evenly-spaced time steps while in the latter, interpolation is performed\nin the φ-space with unevenly-spaced time steps. Given that φtcan be extremely large at t= 0\nandt= 1 when the straight process is ˜Xt=φtX0+X1(i.e., φ0=a0\nσ0with σ0= 0) and\n˜Xt=X0+φtX1(i.e., φ1=σ1\na1witha1= 0), respectively, interpolation in the φ-space with\nunequal time gaps tends to yield inaccurate results in these cases. By contrast, this problem does not\noccur with the interpolation in the t-space (Eq. 66) thanks to the well normalization of twithin [0,1]\nand the use of uniform time intervals.\nThis problem can alternatively be addressed if we perform linear multi-step updates at evenly-spaced\ntime steps in the φ-space as follows:\n¯xφi+1= ¯xφi+ ∆φi+1¯v◦\nφi:φi+1(68)\n¯v◦\nφi:φi+1≈Rφi+1\nφi¯v(φ)dφ\n∆φi+1(69)\nwhere ∆φi+1:=φi+1−φiis assumed to be (approximately) the same for different i. The notation\nφiin Eqs. 68, 69 has the same interpretation as the courterpart in Eqs. 60-63.\nBelow, we delve into the details of the Adams-Bashforth (AB) and Adams-Moulton (AM) methods\ncorresponding to Eq. 66. At the core of these methods lie the interpolating coefficients (L0,L1,\n...) and velocities (¯v∗\nti,¯v∗\nti−1, ...). By examining the formulas of the AB and AM methods w.r.t\nEq. 66, one can easily derive the formulas of the counterparts related to Eqs. 67 and 69. Specifically,\nthe interpolating velocities w.r.t Eq. 67 mirror those associated with Eq. 66, while the interpolating\ncoefficients w.r.t. Eq. 67 can be obtained from the counterparts corresponding to Eq. 66 by replacing ti\nwithφtiin the formulas. The interpolating coefficients w.r.t Eq. 69 closely resemble their counterparts\nw.r.t. Eq. 66 as both cases involve evenly-spaced time steps, albeit in different time spaces. As for\nthe interpolating velocities w.r.t Eq. 69, although they differ from those w.r.t. Eq. 66, they can be\ncomputed by mapping φiback to tφiin the t-space.\nAdams-Bashforth methods In an Adams-Bashforth method of order k(ABk),¯v(t)is a Lagrange\ninterpolating polynomial of order kpassing through kvelocities ¯v∗\nti,¯v∗\nti−1, ...,¯v∗\nti−(k−1)at time steps\nti,ti−1,...,ti−(k−1), respectively. The general update formula of an AB kmethod is expressed as\nfollows:\n¯xti+1= ¯xti+ ∆φti+1\nk−1X\nj=1Lj¯v∗\nti−j\n (70)\nwhere L0, ..., L k−1are referred to as interpolating coefficients.\nIn particular, the AB2 method has the following update:\n¯xti+1= ¯xti+ ∆φti+1\u0010\nL0¯v∗\nti+L1¯v∗\nti−1\u0011\n(71)\nHere, L0=ti+1+ti−2ti−1\n2(ti−ti−1)andL1=−(ti+1−ti)\n2(ti−ti−1). Meanwhile, the AB3 update is given by:\n¯xti+1= ¯xti+ ∆φti+1\u0010\nL0¯v∗\nti+L1¯v∗\nti−1+L2¯v∗\nti−2\u0011\n(72)\nwhere L0,L1, and L2are computed as follows:\nL0=2\u0000\nt2\ni+1+ti+1ti+t2\ni\u0001\n−3 (ti−1+ti−2) (ti+1+ti) + 6ti−1ti−2\n6 (ti−ti−1) (ti−ti−2)(73)\nL1=(ti+1−ti) (2ti+1+ti−3ti−2)\n6 (ti−1−ti) (ti−1−ti−2)(74)\nL2=(ti+1−ti) (2ti+1+ti−3ti−1)\n6 (ti−2−ti) (ti−2−ti−1)(75)\n13Detailed mathematical derivations of these interpolating coefficients are provided in Appdx. B.2.\nIf time steps are evenly spaced (i.e., ∆ti+1= ∆ti=...= ∆t), the AB2 and AB3 updates can be\nwritten in simpler forms below:\n¯xti+1= ¯xti+ ∆φti+1\u00123\n2¯v∗\nti−1\n2¯v∗\nti−1\u0013\n,and (76)\n¯xti+1= ¯xti+ ∆φti+1\u001223\n12¯v∗\nti−16\n12¯v∗\nti−1+5\n12¯v∗\nti−2\u0013\n(77)\nEqs. 76, 77 resemble the update formulas of the well-known second- and third-order AB methods\n[11].\nIn an AB method of order k, we should use a numerical method of order at least k−1to perform\nits initial k−1updates to ensure its order of accuracy is k[3]. Consequently, we can use either\nthe Euler solver or Heun’s solver for the first update of the AB2 method. The latter option incurs\n1 additional model evaluation. Similarly, for the first two updates of the AB3 method, we can use\neither the AB2 method, Heun’s method, or the RK3 method, incurring at most 1, 2, and 4 additional\nmodel evaluations, respectively. In general, to maintain the order of accuracy with minimal additional\nmodel evaluations, we should conduct the first k−1updates of an AB method of order kusing AB\nmethods of increasing orders from 1 to k−1.\nAdams-Moulton methods Unlike explicit AB methods, AM methods are implicit . Therefore, in an\nAM method of order k(AMk),¯v(t)is a Lagrange interpolating polynomial of order kgoing through\nkvelocities ¯v∗\nti+1,¯v∗\nti, ...,¯v∗\nti+1−(k−1)at time steps ti+1,ti−1,...,ti+1−(k−1), respectively.\nThe AM2 update is expressed as:\n¯xti+1= ¯xti+ ∆φti+1v∗\u0000\n¯xti+1, ti+1\u0001\n+ ¯v∗\nti\n2(78)\nwhich is essentially the trapezoidal method. Instead of solving Eq. 78 using a root finding algorithm\n(e.g., Newton’s method), we can approximate v∗\u0000\n¯xti+1, ti+1\u0001\nbased on a reasonable prediction of\n¯xti+1. If we use the Euler method as the predictor, we obtain Heun’s method. On the other hand, if\nwe use the AB2 method as the predictor, we get the following predictor-corrector method, referred to\nas AB2AM2:\n¯xa\nti+1= ¯xti+ ∆φti+1\u0010\nL0¯v∗\nti+L1¯v∗\nti−1\u0011\n(79)\n¯xti+1= ¯xti+ ∆φti+1v∗\u0010\n¯xa\nti+1, ti+1\u0011\n+ ¯v∗\nti\n2(80)\nwithL0,L1similar to those in Eq. 71.\nIn general, we can combine an AB method of either order k−1orkwith an AM method of order k\nto create a predictor-corrector method of order k[11, 34] as follows:\n¯xa\nti+1= AB\u0010\n¯xti,¯v∗\nti, ...,h\n¯v∗\nti−(k−1)i\u0011\n(81)\n¯xti+1= AM\u0010\n¯xti,¯v∗,a\nti+1,¯v∗\nti, ...,¯v∗\nti+1−(k−1)\u0011\n(82)\nHere, ¯v∗,a\nti+1in Eq. 82 denotes v∗\u0010\n¯xa\nti+1, ti+1\u0011\nand the notionh\nv∗\nti−(k−1)i\nin Eq. 81 implies that\nv∗\nti−(k−1)can be either provided or not. If v∗\nti−(k−1)is available, the predictor is an AB method of\norder k; otherwise it is an AB method of order k−1.\nHowever, the aforementioned predictor-corrector method requires 2 function evaluations per update:\none for computing the current velocity v∗\ntiand the other for computing the predicted future velocity\n¯v∗,a\nti+1. Therefore, compared to an AB method of the same order with a similar number of function\nevaluations (NFE), the predictor-corrector method exhibits almost no gain in accuracy. Fortunately,\nwe can reduce the NFE to 1 per update by reusing ¯v∗,a\ntifrom the previous step ti−1in place of ¯v∗\ntito\npredict ¯xa\nti+1in Eq. 81. This technique draws inspiration from UniPC [ 57]. Although it introduces a\nsmall error when predicting ¯xa\nti+1since ¯v∗,a\nti̸= ¯v∗\nti, this error will be corrected in the correction step\nin Eq. 82.\n144.5 A general formula for straight stochastic processes\nWe can generalize our analysis in Section 4.1 to derive a general formula for straight flows (processes).\nClearly, the velocity v∗of a straight flow should have its direction unchanged over time. This implies\nthatv∗should have the form v∗(xt, t) = ˙φtEx0,x1|xt[f(x0) +h(x0, x1) +g(x1)]where f,h,g\nare arbitrary functions. This posterior velocity corresponds to the following stochastic process:\nXt=φt⊕(f(X0) +h(X0, X1) +g(X1)) (83)\nwhere the operator ⊕denotes the sum of the products between φtwith added constants and all\nadditive terms in f,h, andg. For example, the expression φt⊕\u0000\nX1−2X2\n0X1−3X3\n0\u0001\nis equivalent\nto(φt+b)X1−2 (φt+c)X2\n0X1−3 (φt+d)X3\n0where b,c,dare constants w.r.t. t. Normally,\nwe want that at t= 0,Xtdoes not depend on X1and at t= 1,Xtdoes not depend on X0. Therefore,\nto simplify our formula, we discard the term h(X0, X1)in Eq. 83 and rewrite it as:\nXt=φt⊕(f(X0) +g(X1)) (84)\nEq. 84 provide a general formula for straight stochastic processes. If we further assume that\nf(X0)≡ −X0andg(X1)≡X1, Eq. 84 becomes:\nXt=φt⊕(−X0+X1) (85)\n= (c−φt)X0+φtX1 (86)\nwhere cis a constant w.r.t. t. It is reasonable to set cto 1 because at t= 0,φ0= 0and the initial\ndistribution aligns with p0. This means Eq. 86 can be written as:\nXt= (1−φt)X0+φtX1 (87)\nwhich give us the same formula as Eq. 18. We can also derive the formula in Eq. 19 by considering\nv∗(xt, t) = ˙φtEx0,x1|xt[g(x1)]withg(X1)≡X1.\n4.6 Flowing along arbitrary paths\nAn important extension of our proposed transformations in Section 4.1 is that if we have learned the\nposterior flow of a linear stochastic process Xt=atX0+σtX1, we can simulate the posterior flow\nof adifferent linear stochastic process described by Xt=btX0+ζtX1without training a new model .\nThis can be done in two ways. In the first approach, we simply perform updates along the posterior\nflow of the original process and transform samples xtin this flow into samples ˆxtin the posterior\nflow of the target process. This transformation aligns with the transformation of the original process\nXt=atX0+σtX1into the target process ˆXt=btX0+ζtX1which is described by the equation:\nˆXt=bt\natXt+\u0012\nζt−σtbt\nat\u0013\nX1 (88)\nEq. 88 suggests that ˆxtcan be computed from xtas follows:\nˆxt=bt\natxt+\u0012\nζt−σtbt\nat\u0013\nx1|t (89)\nwhere x1|tcan be either modeled as x1,θ(xt, t)or computed from vθ(xt, t)andxtusing Eq. 17. The\nmain drawback of this approach is sampling inefficiency if sampling via the original flow requires\nmuch more steps than via the target flow. The second approach can overcome this limitation by first\ntransforming the original flow into a straight constant-speed (SC) flow, performing updates along the\nSC flow and then mapping the updated samples ¯xt′in the SC flow into samples ˆxt′in the target flow.\nA general formula for computing ˆxt′from ¯xt′is:\nˆxt′≈kt′\u0012\n¯xt′−\u0012\nt′−ζt′\nkt′\u0013\nv∗(¯xt, t)\u0013\n(90)\nwhere ktisbt+ζtfor the SC process ¯Xt= (1−t)X0+tX1andbtfor the SC process ¯Xt=\nX0+tX1.\nOur study above also suggests that the SC process Xt= (1−t)X0+tX1should be regarded as the\nstandard form of the class of linear stochastic processes Xt=atX0+σtX1because this process\ncan easily be converted into other processes in the class via the combination of “variable scaling”\nand “variable shifting” transformations, in the same manner as the standard Gaussian distribution\nN(0,I)is transformed into other Gaussian distributions of the form N\u0000\nµ, σ2I\u0001\n. The SC process\nXt=X0+tX1isnotqualified for being standard because the coefficient of X0is not 0 at t= 1.\n15p1 p0 posterior SN SC\nposterior SN SC\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530N= 1 N= 2 N= 3 N= 5 N= 10 N= 20 N= 50\nFigure 3: Starting from samples of p1, we generate samples of p0by performing NEuler updates\nalong the posterior flow and its transformed SN, SC flows of the “third-degree” stochastic process.\nThe top and middle rows show generated samples and xt-trajectories of the three flows, respectively.\nThe bottom row shows the ˜xt-trajectories of the SN flow and the ¯xt-trajectories of the SC flow.\n5 Experiments\nOur experiment encompasses both toy and real image datasets. The former serves to illustrate and\nvalidate our theoretical analyses presented in Section 4, while the latter demonstrates the effectiveness\nof our proposed high-order solvers on SC flows. Due to space constraint, we only provide a selection\nof results here and defer the remaining results in Appdx. E.\n5.1 2D Toy Dataset\nWe created a simple 2D toy dataset in which p0andp1are mixtures of three and two Gaussian\ndistributions respectively as shown in Fig. 3 (top row). Details about the means and covariance\nmatrices of these Gaussian distributions can be found in Appdx. D.1. Our experiment involves\nthree stochastic processes of the form Xt=atX0+σtX1, where the distributions of X0and\nX1correspond to p0andp1, respectively. For clarity, we coin these processes as “VP-SDE-like”,\n“third-degree”, and “fifth-degree” processes. The “VP-SDE-like” process share the same atandσtas\nVP-SDE (refer to Table 1). The “third-degree” process features atandσtas third degree polynomials\noft, with at= 3 (1 −t)3−6 (1−t)2+4 (1 −t)andσt= 2t3−3t+2. Similarly, the “fifth-degree”\nprocess features atandσtas fifth degree polynomials of t, with at= (1−t)5andσt=t5. We\nmodel the posterior flow of these processes by learning either vθorx1,θ. The training details are\nprovided in Appdx. D.1. After training, we generate samples of X0by starting with samples of\nX1and performing Euler updates along the posterior , the transformed straight (SN), and straight\nconstant-speed (SC) flows of the original process. Unless explicitly stated otherwise, the SN and\nSC flows are characterized by the equations ˜Xt= (1−φt)X0+φX1andd¯Xt= (X1−X0)dφt,\nrespectively, with φt=σt\nat+σt. The small number ϵfor ensuring numerical stability (Section 4.2) is\nset to 1e−3.\n5.1.1 Empirical analysis on straight and straight constant-speed flows\nFig. 3 displays the generated samples and trajectories associated with the posterior, SN, and SC flows\nof the third-degree process. It is evident that sampling along the SC flow significantly outperforms\nsampling along the posterior and SN flows in terms of accuracy and speed. The former requires\napproximately 5 Euler updates to generate good-quality samples while the latter two need at least 20\nEuler updates. We observe similar phenomena for the fifth-degree and VP-SDE-like processes, as\ndepicted in Figs. 4 and 5. For the third-degree process, the SN flow does not exhibit a discernible\nimprovement over the posterior flow. However, for the fifth-degree process, where the posterior flow\nis strongly curved, the SN flow demonstrates clear enhancement.\n16p1 p0 posterior SN SC\nposterior SN SC\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530N= 1 N= 2 N= 3 N= 5 N= 10 N= 20 N= 50\nFigure 4: Generated samples (top), xt-trajectories (middle), and ˜xt/¯xt-trajectories (bottom) of the\nposterior flow and its transformed SN, SC flows of the fifth-degree process , with varying numbers of\nupdate steps N.\np1 p0 posterior SN SC\nposterior SN SC\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\nN= 1 N= 2 N= 3 N= 5 N= 10 N= 20 N= 50\nFigure 5: Generated samples (top), xt-trajectories (middle), and ˜xt/¯xt-trajectories (bottom) of the\nposterior flow and its transformed SN, SC flows of the VP-SDE-like process , with varying numbers\nof update steps N.\nFurthermore, the middle row of Fig. 3 illustrates that the xt-trajectories recovered from the SN and\nSC flows closely align with the xt-trajectories of the posterior flow when the number of updates is\nsubstantial. This result empirically validates the correctness of our transformations from the posterior\nflow to the SN, SC flows and vice versa.\nComparing trajectories in the middle and bottom rows of Figs. 4, 5 (and also Fig. 3), it is apparent that\nthe SN and SC flows exhibit straighter paths than the posterior flow. However, the SN and SC flows\narenotperfectly straight due to randomness caused by the independent coupling used in training.\nTo enhance the straightness of the SN and SC flows, we learn a newvelocity network vθusing the\ndeterministic coupling induced by either the posterior flow or the SC flow. This technique is inspired\nby [25] where it is termed “reflowing”. As illustrated in Fig. 6 (a), the “reflowed” SC flow is almost\nperfectly straight and can generate good-quality samples with just 1 Euler update. Its trajectories still\nexhibit crossovers because the original posterior flow is not an optimal (transport) map between p0\nandp1.\nTo empirically confirm that the stochastic process associated with the SN flow indeed follows the\nequation ˜Xt= (1−φt)X0+φtX1, we directly learn the posterior flow of this process and show\nthe resemblance between the learned flow and the SN flow in Fig. 6 (b).\n17posterior SN SC\nposterior SN SC\nSN (learned)\nSN (third-degree)\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530N= 1(Reflowed) N= 50 (Reflowed) N= 1000\n(a) (b)\nFigure 6: (a)Generated samples (left), xt-trajectories (middle) and ˜xt/¯xt-trajectories (right) of the\nposterior, SN, and SC flows of the third-degree process after being “reflowed” withN=1 and N=50\nEuler updates. (b)Thext-trajectories of the posterior flow of the process Xt= (1−φt)X0+φtX1\nand the ˜xt-trajectories of the SN flow of the third-degree process with N=1000 Euler updates.\nEuler Heun RK4 SC-Euler\nEuler Heun RK4 SC-Euler\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\nN= 1 N= 2 N= 3 N= 5 N= 10 N= 20 N= 50\nFigure 7: Generated samples (top) and xt-trajectories (bottom) obtained via using the Euler solver on\nthe SC flow (SC-Euler) and other numerical solvers on the posterior flow of the third-degree process\nwith different numbers of update steps N.\n5.1.2 Comparison with high-order numerical solvers on posterior flows\nIn Fig. 7, we show the generated samples and xt-trajectories obtained via using the first-order Euler\nsolver on the SC flow (SC-Euler) and other numerical solvers such as Heun’s solver (second-order)\nand the RK4 solver (fourth-order) on the posterior flow of the third-degree process. The results reveal\nthat our method performs comparably to the RK4 solver and slightly outperforms Heun’s solver\non the posterior flow, even though our method requires 4 and 2 times fewer calls to the velocity\ncomputation per update, respectively. We observe similar outcomes for the fifth-degree processes, as\nillustrated in Fig. 8. This suggests that SC-Euler is more preferable than other high-order solvers\non the posterior flow, particularly in scenarios where calculating the velocity is computationally\nexpensive.\nEuler Heun RK4 SC-Euler\nEuler Heun RK4 SC-Euler\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\nN= 1 N= 2 N= 3 N= 5 N= 10 N= 20 N= 50\nFigure 8: Generated samples (top) and xt-trajectories (bottom) obtained via using the Euler solver on\nthe SC flow (SC-Euler) and other numerical solvers on the posterior flow of the fifth-degree process\nwith different numbers of update steps N.\n18N\ntarget (learned) target (from source)\ntarget (learned) target (from source)\n5\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n50\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n1000\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\nthird-degree →fifth-degree third-degree →VPSDE-like fifth-degree →third-degree\nFigure 9: Given the learned velocity corresponding to a source process, we simulate the flow of a\ntarget process (labeled as target (from source)) and compare it with the learned flow of the target\nprocess (labeled as target (learned)).\nN\ntarget (learned) target (from source)\ntarget (learned) target (from source)\n5\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n50\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n1000\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\nVP-SDE-like →fifth-degree VP-SDE-like →third-degree fifth-degree →VP-SDE-like\nFigure 10: Given the learned velocity corresponding to a source process, we simulate the flow of\na target process (labeled as target (from source)) and compare it with the learned flow of the target\nprocess (labeled as target (learned)).\n5.1.3 Flowing along arbitrary paths\nBy utilizing the second approach in Section 4.6, given the trained velocity corresponding to the third-\ndegree or the VP-SDE-like process, we can simulate the posterior flows of the other two processes with\na fair quality, as depicted in Figs. 9, 10. However, when the fifth-degree process is used as source, we\nare unable to simulate the posterior flows of the third-degree and VP-SDE-like processes even when\nthe number of updates is set to 1000. We guess a possible reason is that for the fifth-degree process\nwithv∗(xt, t)being modeled, v∗(¯xt, t)is computed as either(at+σt)v∗(xt,t)−(˙at+ ˙σt)xt\nat˙σt−˙atσt(Eq. 41) or\natv∗(xt,t)−˙atxt\nat˙σt−˙atσt(Eq. 42) and has extremely large values at t= 1andt= 0since at˙σt−˙atσt= 0\nfor this process (see the image at row 3 column 4 in Fig. 15), as shown in rows 1, 2 column (e) in\nFig. 17. Consequently, the computation of samples in the target flow via Eq. 89 tend to be incorrect.\nThe third-degree and VP-SDE-like process do not exhibit this problem as the velocities of their\ncorresponding SC flows well behave (rows 1, 2 column (e) in Figs. 16, 18). By using the “standard”\nprocess Xt= (1−t)X0+tX1as the source process, we can simulate the posterior flows of all the\nabove processes better, which is shown in Fig. 11.\n19N\ntarget (learned) target (from source)\ntarget (learned) target (from source)\n5\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n50\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n1000\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\nrectified →third-degree rectified →fifth-degree rectified →VPSDE-like\nFigure 11: Given the learned velocity of a rectified flow, we simulate the flow of a target process\n(labeled as target (from source)) and compare it with the learned flow of the target process (labeled as\ntarget (learned)).\ns= 0\n1020 50 100 200\nNFE0.0000.0020.0040.006MSEDPM-1S\nSC-EulerDPM-1M\n1020 50 100 200\nNFE0.0000.0010.0020.0030.004MSEDPM-2S\nUniPC-1\nSC-AB2DPM-2M\nSC-AB1AM2\nSC-AB2AM2\n1020 50 100 200\nNFE0.0000.0010.0020.0030.004MSEDPM-3S\nUniPC-2\nSC-AB3DPM-3M\nSC-AB2AM3\nSC-AB3AM3\ns= 8\n1020 50 100 200\nNFE0.0000.0050.0100.0150.0200.025MSEDPM-1S\nSC-EulerDPM-1M\n1020 50 100 200\nNFE0.000.010.020.030.040.05MSEDPM-2S\nUniPC-1\nSC-AB2DPM-2M\nSC-AB1AM2\nSC-AB2AM2\n1020 50 100 200\nNFE0.000.010.020.030.040.05MSEDPM-3S\nUniPC-2\nSC-AB3DPM-3M\nSC-AB2AM3\nSC-AB3AM3\n(a) First-order (b) Second-order (c) Third-order\nFigure 12: MSEs between 1000 ImageNet 256 ×256 samples generated by a 1000-step DDIM and\nthose generated by our proposed solvers (SC-*) and baselines. The solvers are categorized according\nto their accuracy orders. The number of function evaluations varies from 10 to 200. DPM stands for\nDPM-Solver++. Top row: Classifier-guidance is not applied. Bottom row: Classifier-guidance with\nthe guidance scale of 8 is applied.\ns= 0\n5 6 7 8 9\nNFE35404550FIDDPM-1S\nSC-EulerDPM-1M\n5 6 7 8 9\nNFE2530354045FIDDPM-2S\nUniPC-1\nSC-AB2DPM-2M\nSC-AB1AM2\nSC-AB2AM2\n5 6 7 8 9\nNFE304050FIDDPM-3S\nUniPC-2\nSC-AB3DPM-3M\nSC-AB2AM3\nSC-AB3AM3\ns= 8\n5 6 7 8 9\nNFE20304050FIDDPM-1S\nSC-EulerDPM-1M\n5 6 7 8 9\nNFE20406080100FIDDPM-2S\nUniPC-1\nSC-AB2DPM-2M\nSC-AB1AM2\nSC-AB2AM2\n5 6 7 8 9\nNFE50100150FIDDPM-3S\nUniPC-2\nSC-AB3DPM-3M\nSC-AB2AM3\nSC-AB3AM3\n(a) First-order (b) Second-order (c) Third-order\nFigure 13: FIDs of 5000 ImageNet 256 ×256 samples generated by our proposed solvers (SC-*) and\nbaselines with varying accuracy orders (1-3) and numbers of function evaluations (5-9). DPM stands\nfor DPM-Solver++. Top row: Classifier-guidance is not applied. Bottom row: Classifier-guidance\nwith the guidance scale of 8 is applied.\n205.2 Large-scale Image Generation\nIn this section, we compare the performances of our proposed high-order ODE solvers (Section 4.4)\nwith current state-of-the-art ODE solvers for diffusion models [ 27,28,57] in the context of large-scale\nimage generation tasks. To streamline presentation, we focus on showcasing the generation results\nspecifically for ImageNet 256 ×256.\nOur implementation builds upon the open-source code3provided by the authors of the paper [ 8]. In\naddition, we utilize their pretrained class-conditional DDPM on ImageNet 256 ×256 images for image\ngeneration. The baselines are DPM-Solver++ [ 28] and UniPC [ 57]. They are run using their official\nsource codes45. We keep all default settings of the baselines except disabling dynamic thresholding\n[40] since it has not been implemented in our methods.\nWe conduct two different experiments. In the first experiment, we aim to examine the convergence\ncapability and speed of our methods and the baselines. We vary the number of function evaluations N\nfrom 10 to 200 and generate 1000 samples for each value of N. The mean square error (MSE) between\nthese samples and 1000 samples generated by a 1000-step DDIM [ 43] serves as the convergence\nerror. To ensure a fair play ground, numerical methods are categorized according to their order\nof accuracy and only methods in the same category are compared. As illustrated in Fig. 12, our\nmethods as well as the baselines converge as Nincreases. Our SC-Euler also performs similarly to\nthe first-order variants of DPM-Solver++ as they all resemble DDIM. When classifier guidance is\nnot applied ( s= 0), higher-order solvers tend to converge faster than lower-order ones (i.e., requires\nfewer steps to reach the same MSE value). However, when the classifier guidance is applied with s\n= 8, third-order solvers seem to converge slower than second-order solvers. In addition, at N= 10,\nfirst-order solvers achieve better MSE values than second- and third-order solvers (about 0.025 vs\n0.04, 0.05). The main reason is that when the classifier guidance is not applied, the posterior flow\nremains unchanged and updates along this flow or its associated SC flow should follow the theoretical\nanalysis on convergence rate. By contrast, when the classifier guidance is applied, the posterior flow\nis modified and the theory may not be applicable. In this case, lower-order solvers, which are more\nrobust to errors, tend to perform better than higher-order solvers, especially when Nis small.\nTherefore, in the second experiment, our target is to compare the quality of images generated by\nour methods and the baselines when using a small number of function evaluations. We generate\n5000 images for each value of Nfrom 5 to 9 and compute the Frechet Inception Distance (FID)\n[12] between these images and those in the ImageNet training set using the pytorch-fid library6. The\nImageNet training images are resized and center-cropped to the size 256 ×256. Fig. 13 shows the\nresults of this trial. It is clear that the performances of our methods follow the increasing order\nfrom bad to good: AB1-AM2 (AB2-AM3) < AB2 (AB3) < AB2-AM2 (AB3-AM3) when there\nis no classifier guidance and follow the reverse order when classifier guidance is employed. It\ncould possibly because the convergence rates of these methods follow this order while the levels of\nrobustness follow the reverse order. For the baseline solvers, we also observe the same phenomenon.\nThe second- and third-order variants of UniPC perform better than the second- and third-order variants\nof DPM-Solver++ M/S when s= 0 but worse when s= 8. These results suggest that there is no\nwinning solver in all cases but should be customized for certain scenarios. In this experiment, we\nsee that when Nis small, SC-AB1AM2 (SC-AB2AM3) is more accurate and robust than DPM-\nSolver++ 2S (3S), SC-AB2 is on par with DPM-Solver++ 2M regardless of classifier guidance, and\nSC-AB2AM2 (SC-AB3AM3) has quite similar level of accuracy to UniPC-1 (2) but is slightly more\nrobust.\n5.3 Text-to-Image Generation with Stable Diffusion\nWe continue comparing our proposed multi-step solvers with baseline solvers [ 28,56,57] on the\ntext-to-image generation task with Stable Diffusion. We utilize the pretrained Stable Diffusion from\nthe Hugging Face repository CompVis/stable-diffusion-v1-47, which can also be found in the Github\n3https://github.com/openai/improved-diffusion\n4https://github.com/LuChengTHU/dpm-solver\n5https://github.com/wl-zhao/UniPC\n6https://github.com/mseitzer/pytorch-fid\n7https://huggingface.co/CompVis/stable-diffusion-v1-4\n218 9 10 11 12\nNFE789101112MSE (latent)DDIM\nDPM-1S\nSC-EulerDEIS-1\nDPM-1M\n8 9 10 11 12\nNFE78910MSE (latent)DPM-2S\nDEIS-2\nSC-AB1AM2\nSC-AB2AM2DPM-2M\nUniPC-1\nSC-AB2\n8 9 10 11 12\nNFE68101214MSE (latent)DPM-3S\nDEIS-3\nSC-AB2AM3\nSC-AB3AM3DPM-3M\nUniPC-2\nSC-AB3(a) First-order (b) Second-order (c) Third-order\nFigure 14: MSEs between 1000 latent samples generated by a 1000-step DDIM and those generated\nby our proposed solvers (SC-*) and baseline solvers. The model is Stable Diffusion and the text\nprompts are COCO2014 captions. DPM stands for DPM-Solver++.\nlink8. The baseline solvers are taken from the Diffusers library [ 52] with the default configurations.\nFollowing [ 28,57], we randomly sample 1000 image-caption pairs from the COCO2014 validation\nset [21] and use the captions as text prompts for the pretrained Stable Diffusion to generate 1000\nimages of size 512 ×512. The default guidance scale s= 7.5 is applied. The mean square error (MSE)\nbetween the latent samples generated by a 1000-step DDIM and those generated by each numerical\nsolvers serves as a convergence error.\nFrom Fig. 14 (a), it is expected that our Euler solver performs similarly to the first-order variants\nof DPM-Solver++ and DEIS as they are all equivalent to DDIM. However, the considerably worse\nresults of DDIM surprises us since in theory, they should be the same as ours and the baselines’. Our\nhypothesis is that the implementation of DDIM in the Diffusers library may not be the best version.\nIn terms of second- and third-order solvers, our SC-AB1AM2 (-AB2AM3) and SC-AB2 (-AB3) are\ncomparable to DPM-Solver++ 2M (3M) and UniPC-1 (-2) and slightly better than DPM-Solver++\n2S (3S) as shown in Fig. 14 (b, c). It is because both DPM-Solver++ and its derivative, UniPC,\nimplicitly perform an SC transformation in their implementations (Appdx. C). Among all high-order\nsolvers, DEIS achieves the worst performance, possibly because of it operates on the posterior flow\nrather than the SC flow. These results reflect the qualities of images generated by our solvers and the\nbaseline solvers in Figs. 24, 25, 26.\n6 Related Work\n6.1 Diffusion and Flow Models\nDiffusion models were proposed in [ 13,42,46] under the variational inference and score matching\nframeworks. Originally, these models only supported discrete time and their sampling was primarily\nbased on Markov Chain Monte Carlo (MCMC) methods such as Langevin dynamics [ 55]. The\nresearch by Song et al. [ 48] established an interesting connection between diffusion models and\nstochastic differential equations (SDEs), allowing to represent diffusion models in the continuous-\ntime form. In addition, this work introduced probability flow ordinary differential equations (PF\nODEs), which share the same marginal distributions of samples as the original diffusion SDEs.\nThese PF ODEs are easier to solve due to their deterministic update steps, thereby accelerating the\nsampling process of diffusion models without compromising the sample quality. Consequently, this\nadvancement has driven subsequent researches on a general class of ODE-based flows connecting two\narbitrary distributions [ 2,22,24,25]. Among various flow models, those that enable fast sampling are\nof special interest. They include straight flows (e.g., Rectified Flow) [ 24,25,38] and flows derived\nfrom optimal transport maps [ 10,22,37]. However, these methods necessitate training a flow model\nwith the desired straightness property, which can be very expensive especially for large models like\nStable Diffusion. By contrast, our method can straighten any pretrained non-straight flow models\nwithout training a new one.\n6.2 Variational Interpretation of Diffusion and Flow Models\nAs far as we are aware, the connection between score matching and maximum likelihood was explored\nearly by Lyu [ 30]. This study interprets the score matching loss as the Fisher divergence and links it\n8https://github.com/CompVis/stable-diffusion\n22to the KL divergence in the case the diffusion process is Xt=X0+√\ntX1withX1∼ N (0,I). [42]\nproposed discrete-time diffusion models within the variational inference framework. Ho et al. [ 13]\nutilized the noise matching loss LNMto train diffusion models, interpreting it as a weighted variational\nbound. Kingma et al. [ 18] extended the variational analysis to a general class of diffusion processes\ncharacterized by the equation Xt=atX0+σtX1withX1∼ N (0,I). They formally defined the\nsignal-to-noise ratio (SNR) at each time step tof these processes and reformulated the variational\nbound in terms of the SNR. An important finding in their work is that in the continuous-time setting,\nthe variational bound remains invariant to the noise schedule (i.e., the choice of at,σt), except for\nthe SNR values at t= 0 andt= 1. This result complements our work by ensuring that given a\nlearned flow, we can simulate any flow as long as the SNRs of the two flows at the end points are the\nsame. Besides, their transformation of time from ttoSNR(t)relates to our transformation from tto\nφtand was utilized to derive fast numerical solvers in [ 27,28]. Song et al. [ 45] directly derived the\nvariational bound for diffusion models in their continuous-time SDE forms rather than generalizing\nfrom the discrete-time setting as in [ 18]. Their approach involves computing the KL divergence\nbetween the ground-truth and parameterized SDEs, which is upper-bounded by the KL divergence\nbetween the two corresponding path measures. This KL divergence is then transformed into the\nscore matching loss using the Girsanov theorem [ 36]. Kim et al. [ 17] expanded the result in [ 45]\nto nonlinear diffusion processes. Albergo et al. [ 1] derived the variational bound for general flow\nmodels using the termdD KL\ndt, which differs from previous approaches [ 18,45] that utilize DKL. The\nauthors showed thatdD KL\ndtbetween two marginal distributions of samples at time tcan be expressed\nanalytically by applying the continuity equation (Lemma 2.21 in [ 1]). However, there are some\nambiguities in their proof. For example, it is unclear how\u0010\nρ\nˆρ\u0011\n∇ ·\u0010\nˆbˆρ\u0011\ncan be transformed into\n−∇\u0010\nρ\nˆρ\u0011\n·ˆbˆρ.\n6.3 Fast Sampling Methods for Diffusion Models\nImproving the sampling process of diffusion models has been an important research topic since the\ntime this kind of models was proposed [ 19,35,43,53,54]. Improved samplers can generally be\ncategorized as either training-based [ 29,41,44] or training-free methods [ 4,23,27,28,43,56,57].\nTraining-based methods typically involve in training another network that directly transforms the\nlatent sample x1to the data sample x0in a single step via distilling knowledge from a pretrained\ndiffusion model. Some methods like [ 29,41] need to generate (x0, x1)pairs for distillation while\nothers avoid this by leveraging the consistency loss [ 44]. By contrast, training-free methods usually\nexploit special structures in the generation process of diffusion models to reduce the number of\nsteps. Analytic-DDPM [ 4] estimates the optimal variance in the denoising distribution q\u0000\nxti−1|xti\u0001\nof DDPM. DDIM [ 43] uses a different denoising distribution q\u0000\nxti−1|xti, x0|ti\u0001\nthat utilizes the\ninformation in x0|tito improve sample quality. The variance of this distribution can vary and when it\nequals 0, the sampling process becomes deterministic. Karras et al. [ 16] introduced several important\nmodifications to the design of diffusion models to improve the sampling speed and the quality of\ngenerated images. These changes include training a “straight” process Xt=X0+σtX1instead\nof the original diffusion process, modeling the denoised sample (i.e., x0|t) instead of noise (i.e.,\nx1|t), and employing second-order Heun’s solver for fast sampling. DEIS [ 56] and DPM-Solver [ 27]\nexploit the semi-linear structure of PF ODEs to apply exponential integrators. However, DPM-Solver\nemploys an additional change-of-variables operation that links to our SC transformation (Appdx. C.1)\nwhile DEIS does not. DPM-Solver++ [ 28] extends DPM-Solver by supporting the model of x0|t\n([16]) besides the noise model as well as multi-step methods. UniPC [ 57] integrates a corrector into\nDPM-Solver++, making it a predictor-corrector method. Our work takes a different approach from\nthese works by leveraging straight, constant-speed flows to perform updates, which results in a more\ngeneralizable framework that can be applied to a larger class of stochastic processes rather than just\ndiffusion processes.\n7 Conclusion\nWe have introduced a variational inference perspective on models that approximate the posterior flows\nof a general class of stochastic processes defined by Xt=atX0+σtX1. Besides, we have presented\na training-free approach to convert these flows into straight constant-speed flows that facilitate fast\nand more accurate sampling. 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Unipc: A unified predictor-\ncorrector framework for fast sampling of diffusion models. arXiv preprint arXiv:2302.04867 ,\n2023.\n27Table of Content for Appendix\nA Theoretical Results on Posterior Flows and Straight Constant-speed Flows 28\nA.1 Derivation of the posterior velocity v∗(xt, t). . . . . . . . . . . . . . . . . . . . . 28\nA.2 Proof of the statistical equivalence between the stochastic process Xt=atX0+σtX1\nand its corresponding posterior flow ODE . . . . . . . . . . . . . . . . . . . . . . 30\nA.3 Variational inference interpretation of the velocity matching loss . . . . . . . . . . 30\nA.4 Explicit formulas of vϕ(x0, x1, t)andv∗(xt, t)for diffusion models and Rectified\nFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\nA.5 Tweedie’s formula for posterior Gaussian mean estimation . . . . . . . . . . . . . 38\nA.6 Explanation of why x1|tandx0|1,tshould be invariant w.r.t. t. . . . . . . . . . . . 39\nA.7 Mathematical derivations of the transformed velocities v∗(˜xt, t)andv∗(¯xt, t). . . 39\nB High-order Numerical Solvers on Straight Constant-speed Flows 41\nB.1 Runge-Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41\nB.2 Linear multi-step methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44\nC Analysis of Existing High-order Solvers for Diffusion Models 48\nC.1 DPM-Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48\nD Experimental Settings 50\nD.1 2D Toy Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50\nE Additional Experimental Results 51\nE.1 2D Toy Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51\nE.2 Text-to-Image Generation with Stable Diffusion . . . . . . . . . . . . . . . . . . . 54\nATheoretical Results on Posterior Flows and Straight Constant-speed Flows\nA.1 Derivation of the posterior velocity v∗(xt, t)\nLet us recall that the posterior velocity v∗is the solution of minimizing the velocity matching loss\nLVM(Eq. 10):\nv∗:= min\nvLVM(v) (91)\n28where LVM(v)can be written as follow:\nLVM(v) =Eth\nEx0,x1Ext|x0,x1h\n∥vϕ(x0, x1, t)−v(xt, t)∥2ii\n(92)\n=Eth\nExtEx0,x1|xth\n∥v(xt, t)∥2−2v(xt, t)·vϕ(x0, x1, t)\n+∥vϕ(x0, x1, t)∥2ii\n(93)\n=EtExth\n∥v(xt, t)∥2−2v(xt, t)·Ex0,x1|xt[vϕ(x0, x1, t)]\n+Ex0,x1|xth\n∥vϕ(x0, x1, t)∥2ii\n(94)\n=EtExth\n∥v(xt, t)∥2−2v(xt, t)·Ex0,x1|xt[vϕ(x0, x1, t)]\n+\r\rEx0,x1|xt[vϕ(x0, x1, t)]\r\r2+consti\n(95)\n=EtExth\r\rv(xt, t)−Ex0,x1|xt[vϕ(x0, x1, t)]\r\r2i\n+const (96)\nEq. 96 implies that:\nv∗(xt, t) =Ex0,x1|xt[vϕ(x0, x1, t)] (97)\nIf the stochastic process is Xt=atX0+σtX1, we have:\nvϕ(x0, x1, t) = ˙atx0+ ˙σtx1 (98)\nv∗(xt, t) =Ex0,x1|xt[˙atx0+ ˙σtx1] (99)\nIn this case, we can further decompose v∗(xt, t)as follows:\nv∗(xt, t) =Ex0,x1|xt[˙atx0+ ˙σtx1] (100)\n=Ex0|xtEx1|x0,xt[˙atx0+ ˙σtx1] (101)\n=Ex0|xt[˙atx0] +Ex0|xtEx1|x0,xt[ ˙σtx1] (102)\n= ˙atEx0|xt[x0] + ˙σtEx0|xtEx1|x0,xt[x1] (103)\n= ˙atEx0|xt[x0] + ˙σtEx0|xt\u0014xt−atx0\nσt\u0015\n(104)\n=˙σt\nσtxt+\u0012\n˙at−˙σtat\nσt\u0013\nEx0|xt[x0] (105)\n=˙σt\nσtxt+\u0012\n˙at−˙σtat\nσt\u0013\nx0|t (106)\n=˙σt\nσtxt+at\u0012˙at\nat−˙σt\nσt\u0013\nx0|t (107)\nwhere x0|t:=Ex0|xt[x0]. Eq. 104 is equivalent to Eq. 103 because given xtandx0,x1is determin-\nistically identified asxt−atx0\nσtaccording to Eq. 9. v∗(xt, t)can also be expressed as:\nv∗(xt, t) =˙at\natxt+\u0012\n˙σt−σt˙at\nat\u0013\nEx1|xt[x1] (108)\n=˙at\natxt+\u0012\n˙σt−σt˙at\nat\u0013\nx1|t (109)\n=˙at\natxt+σt\u0012˙σt\nσt−˙at\nat\u0013\nx1|t (110)\ndue to the interchangeability between x0andx1.\n29A.2 Proof of the statistical equivalence between the stochastic process Xt=atX0+σtX1and\nits corresponding posterior flow ODE\nFor the stochastic process Xt=atX0+σtX1,xtis sampled from pt(xt) =\nEp0,1(x0,x1)[pt(xt|x0, x1)]where p0,1denotes the distribution of the coupling (X0, X1). The in-\nfinitesimal change of pt(xt)can be formulated as follows:\n∂pt(xt)\n∂t=∂\n∂tEp0,1(x0,x1)[pt(xt|x0, x1)] (111)\n=Ep0,1(x0,x1)\u0014∂pt(xt|x0, x1)\n∂t\u0015\n(112)\n=−Ep0,1(x0,x1)[∇ ·(pt(xt|x0, x1)vϕ(x0, x1, t))] (113)\n=− ∇ ·Ep0,1(x0,x1)[pt(xt|x0, x1)vϕ(x0, x1, t)] (114)\n=− ∇ ·Z\np0,1(x0, x1)pt(xt|x0, x1)vϕ(x0, x1, t)dx0x1 (115)\n=− ∇ ·Z\np0,1(x0, x1, xt)vϕ(x0, x1, t)dx0x1 (116)\n=− ∇ ·Z\npt(xt)pt(x0, x1|xt)vϕ(x0, x1, t)dx0x1 (117)\n=− ∇ ·\u0012\npt(xt)Z\npt(x0, x1|xt)vϕ(x0, x1, t)dx0x1\u0013\n(118)\n=− ∇ ·\u0000\npt(xt)Ep(x0,x1|xt)[vϕ(x0, x1, t)]\u0001\n(119)\n=− ∇ · (pt(xt)v∗(xt, t)) (120)\nwhere Eq. 113 is equivalent to Eq. 112 because∂pt(xt|x0,x1)\n∂t=−∇ · (pt(xt|x0, x1)vϕ(x0, x1, t))\nis the continuity equation for the ODE ˙xt=vϕ(xt(x0, x1, t), t)that connects two boundary points\nx0,x1; Eq. 114 is equivalent to Eq. 113 due to the linearity of the divergence operator; Eq. 120 is the\ncontinuity equation for the posterior flow ODE.\nThe above result implies that the changes in probability over time for the stochastic process and its\nposterior flow ODE are identical. Therefore, given the same initial distribution (either p0orp1), the\nstochastic process and its posterior flow ODE share the same marginal distribution of samples. In\nother words, they are statistically equivalent.\nA.3 Variational inference interpretation of the velocity matching loss\nA.3.1 Variational inference interpretation for discrete-time diffusion processes\nIf we assume our stochastic process is a discrete-time diffusion process, we can base on the theoretical\nanalyses in [ 13,18,43] to show that the velocity matching loss in Eq. 10 serves as a variational upper\nbound on the negative log-likelihood.\nGiven a general diffusion process defined by Xt=atX0+σtX1withX1∼ N (0,I), we have\np(xt|x0) =N\u0000\nxt|atx0, σ2\ntI\u0001\nandp(xs|x0) =N\u0000\nxs|asx0, σ2\nsI\u0001\nfor any two time steps 0≤s <\nt≤1. Since p(xt|x0)andp(xs|x0)are both Gaussian distributions, p(xs|xt, x0)is also a Gaussian\ndistribution characterized by the following equation [5, 43]:\np(xs|xt, x0) =N \nxs\f\f\f\fp\nσ2s−γ2s\nσtxt+ \nas−atp\nσ2s−γ2s\nσt!\nx0, γ2\nsI!\n(121)\n=N\u0012\nxs\f\f\f\fas\nat(xt−σtx1) +p\nσ2s−γ2sx1, γ2\nsI\u0013\n(122)\nwhere γscan take any value in the range [0, σs].\nAccording to [ 13,18], the variational upper bound on the negative log-likelihood of a data sample x0\nis represented as follows:\n30−logp(x0)≤VUB (x0) (123)\n=Ep(xt1:1|x0)\u0014\nlogp(xt1:1|x0)\npθ(x0:1)\u0015\n(124)\n=Ep(x1|x0)[DKL(p(x1|x0)∥p(x1))]| {z }\nPrior loss\n+Ep(xt1|x0)[−logpθ(x0|xt1)]\n| {z }\nReconstruction loss\n+NX\ni=2Ep(xt|x0)\u0002\nDKL\u0000\np\u0000\nxti−1|xti, x0\u0001\n∥pθ\u0000\nxti−1|xti\u0001\u0001\u0003\n| {z }\nDiffusion loss(125)\nwhere 0 =t0< t1< ... < t N= 1.\nHere, we focus on the KL divergence term in the diffusion loss in Eq. 125. If we model pθ\u0000\nxti−1|xti\u0001\nasp\u0000\nxti−1|xti, x0,θ(xti, ti)\u0001\nsuch that its formula is identical to the formula of p\u0000\nxti−1|xti, x0\u0001\nin Eq. 121 but with x0replaced by x0,θ(xti, ti), we can express the KL divergence in Eq. 125\nanalytically as follows:\nDKL\u0000\np\u0000\nxti−1|xti, x0\u0001\n∥pθ\u0000\nxti−1|xti\u0001\u0001\n=DKL\u0000\np\u0000\nxti−1|xti, x0\u0001\n∥p\u0000\nxti−1|xti, x0,θ(xti, ti)\u0001\u0001\n(126)\n=\u0012\nas−at√\nσ2s−γ2s\nσt\u00132\n2γ2s∥x0,θ(xti, ti)−x0∥2\n2+const (127)\nFollowing [ 18], we set γ2\ns=σ2\ns\u0010\n1−a2\nt\nσ2\ntσ2\ns\na2s\u0011\n=σ2\ns\u0010\n1−SNR(t)\nSNR(s)\u0011\n. The coefficient in Eq. 127\nsimplifies to1\n2\u0010\na2\ns\nσ2s−a2\nt\nσ2\nt\u0011\n=1\n2(SNR(s)−SNR(t))and Eq. 127 becomes:\nDKL\u0000\np\u0000\nxti−1|xti, x0\u0001\n∥pθ\u0000\nxti−1|xti\u0001\u0001\n=1\n2(SNR(s)−SNR(t))∥x0,θ(xti, ti)−x0∥2\n2+const (128)\nWhen the time is continuous, Kingma et al. [18] have shown that Eq. 128 can be written as:\n1\n2dSNR(t)\ndt∥x0,θ(xti, ti)−x0∥2\n2+const (129)\nwheredSNR(t)\ndt=2at(˙atσt−at˙σt)\nσ3\nt.\nIt is worth noting that when the time is continuous, the KL divergence in Eq. 128 degenerates and\nEq. 129 may no longer represent a KL divergence. Nevertheless, for a very tiny discrete time step dt,\nEq. 129 can still serve as an estimate for the KL divergence. Therefore, in practice, Eq. 129 remains\nviable for deriving the variational bound for the continuous-time case.\nSince in the continuous-time case, x0,θ(xt, t)andx0can be transformed into vθ(xt, t)andvϕ(xt, t)\nrespectively using Eq. 107, we can reformulate Eq. 129 to incorporate vθ(xt, t)andvϕ(xt, t)as\n31follows:\n1\n2dSNR(t)\ndt∥x0,θ(xti, ti)−x0∥2\n2+const\n=1\n2dSNR(t)\ndt\na2\nt\u0010\n˙at\nat−˙σt\nσt\u00112∥vθ(xti, ti)−vϕ(xti, ti)∥2\n2+const (130)\n=1\n22at(˙atσt−at˙σt)\nσ3\nt\n(˙atσt−at˙σt)2\nσ2\nt∥vθ(xti, ti)−vϕ(xti, ti)∥2\n2+const (131)\n=at\nσt(˙atσt−at˙σt)∥vθ(xti, ti)−vϕ(xti, ti)∥2\n2+const (132)\nEq. 132 implies that the weighted velocity matching loss can be considered as a variational upper\nbound on the negative log-likelihood.\nInterestingly, if we convert vθandvϕintox1,θandx1, respectively using Eq. 16, we can rewrite\nEq. 132 as:\n1\n2dSNR(t)\ndt∥x0,θ(xti, ti)−x0∥2\n2+const\n=1\n2at\u0010\n˙σtat−˙atσt\nat\u00112\nσt(˙atσt−at˙σt)∥x1,θ(xti, ti)−x1∥2\n2+const (133)\n=1\n2˙atσt−at˙σt\natσt∥x1,θ(xti, ti)−x1∥2\n2+const (134)\n=1\n2at\nσt(−˙φt)∥x1,θ(xti, ti)−x1∥2\n2+const (135)\n=−1\n2˙φt\nφt∥x1,θ(xti, ti)−x1∥2\n2+const (136)\n=1\n2d(−logφt)\ndt∥x1,θ(xti, ti)−x1∥2\n2+const (137)\nwhere φt=σt\natand˙φt=˙σtat−˙atσt\na2\nt. It is straightforward to show that:\nφt=s\n1\nSNR(t)(138)\n−logφt= logat\nσt=1\n2logSNR(t) (139)\nThis implies that Eq. 137 is equivalent to:\n1\n4dlogSNR(t)\ndt∥x1,θ(xti, ti)−x1∥2\n2+const (140)\nSimilar to SNR(t),−logφtis a monotonic function and can be used in the change-of-variables\nformula, leading to the following expression of the variational bound:\nVUB (x0) =Z1\n0Ex1\u00141\n2dSNR(t)\ndt∥x0,θ(xti, ti)−x0∥2\n2\u0015\ndt+const (141)\n=Z1\n0Ex1\u00141\n2d(−logφt)\ndt∥x1,θ(xti, ti)−x1∥2\n2\u0015\ndt+const (142)\n=1\n2Ex1\u0014Z1\n0dψt\ndt∥x1,θ(xti, ti)−x1∥2\n2dt\u0015\n+const (143)\n=1\n2Ex1\"Zψ1\nψ0∥ˆx1,θ(xψt, ψt)−x1∥2\n2dψt#\n+const (144)\nwhere ψt≡ −logφt=1\n2logSNR(t),ψ1,ψ0are the two values of ψtatt= 1 and 0, respectively.\n32Model at σt PF ODE Forward SDE\nVP SDE√αt√1−αt dxt=−βt\n2(xt+∇logpt(xt))dt dx t=−βt\n2xtdt+√βtdwt\nsub-VP SDE√αt1−αtdxt=−βt\n2\u0000\nxt+\u0000\n1−α2\nt\u0001\n∇logpt(xt)\u0001\ndt dx t=−βt\n2xtdt+q\nβt\u0000\n1−α2\nt\u0001\ndwt\nVE SDE 1 σt dxt=−˙σtσt∇logpt(xt)dt dx t=√2 ˙σtσtdwt\nRect. Flow∗1−t t dx t=−1\n1−t(xt+t∇logpt(xt))dt dx t=−1\n1−txtdt+q\n2t\n1−tdwt\nGeneral at σt dxt=˙at\natxt−gtσt∇logpt(xt) dxt=˙at\natxtdt+√2gtσtdwt\nTable 5: The coefficients at,σt, PF ODE formulas, and forward SDE formulas corresponding to\nwell-known diffusion processes [ 18,48] and Rectified Flow (Rect. Flow) [ 25]. In the case of (sub-)VP\nSDE, βt=t\u0000\nβmax−βmin\u0001\n+βminandαt=e−1\n2t(βt+βmin). For Rectified Flow, the latent distribution\np1is assumed to be N(0,I). For general diffusion processes (General), gt=σt\u0010\n˙σt\nσt−˙at\nat\u0011\nsimilar\nto the coefficient of x1|tin Eq. 110.\nA.3.2 Variational proxy interpretation for general continuous-time stochastic processes\nWhen the stochastic process is continuous-time and non-diffusion, it is not straightforward to apply\nthe above technique because p(xs|xt, x0)is not a Gaussian distribution and the KL divergence is not\nwell defined. Informally, minimizing LVMenhances the alignment between the parameterized flow\nrepresented by vθand the posterior flow characterized by v∗, thereby narrowing the gap between p0,θ\nandp0. In this sense, LVMcan be viewed as a variational proxy for the negative log-likelihood.\nA.4 Explicit formulas of vϕ(x0, x1, t)andv∗(xt, t)for diffusion models and Rectified Flow\nBelow, we derive the explicit formulas of the target and posterior velocities for well-known diffusion\nmodels [ 18,48] and Rectified Flow [ 25] with the summary provided in Table. 1. In addition, we also\npresent the explicit formulas of the PF ODEs and SDEs of these models in Table. 5. For Rectified\nFlow, we assume that the latent distribution p1isN(0,I)so that we can treat its corresponding\nstochastic process as an SDE.\nA.4.1 VP SDE\nThe (forward) VP SDE [48] is an Ornstein-Uhlenbeck process characterized by the equation:\ndxt=−βt\n2xtdt+p\nβtdwt (145)\nwhere wtis the Wiener process at time t, βt=t\u0000\nβmax−βmin\u0001\n+βmin. Its corresponding Fokker-\nPlanck equation (aka Kolmogorov forward equation) is:\n∂pt\n∂t=βt\n2(∇ ·(ptxt) + ∆ pt) (146)\nwhere ∇·and∆denote the divergence and Laplace operators, respectively. When the initial distri-\nbution is p0=δx0where δx0denotes the Dirac delta distribution which has value zero everywhere\nexcept at x0, this Fokker-Planck equation has the following solution:\np(xt|x0) =N\u0010\nx0e−1\n2Rt\n0βτdτ,\u0010\n1−e−Rt\n0βτdτ\u0011\nI\u0011\n(147)\nThe closed-form expression ofRt\n0βτdτexists and is given by:\nZt\n0βτdτ=Zt\n0\u0000\nτ\u0000\nβmax−βmin\u0001\n+βmin\u0001\ndτ (148)\n=\u0012τ2\n2\u0000\nβmax−βmin\u0001\n+τβmin\u0013\f\f\f\fτ=t\nτ=0(149)\n=t2\n2\u0000\nβmax−βmin\u0001\n+tβmin (150)\n=t\n2\u0000\nβt+βmin\u0001\n(151)\n33Thus, p(xt|x0)can be expressed as:\np(xt|x0) =N\u0010\nx0e−1\n2t\n2(βt+βmin),\u0010\n1−e−t\n2(βt+βmin)\u0011\nI\u0011\n(152)\n=N\u0000√αtx0,(1−αt) I\u0001\n(153)\nwhere αt:=e−t\n2(βt+βmin). Eq. 152 is the same as Eq. 33 in [ 48] while Eq. 153 is similar to Eq. 4 in\n[13]. According to [48], the probability flow ODE of the VP SDE is:\ndxt=−βt\n2(xt+∇logp(xt))dt (154)\nEq. 153 implies that we can represent the VP SDE as:\nXt=√αtX0+√\n1−αtX1 (155)\nwhere X1is the standard Gaussian. Eq. 155 is an instance of the general class of stochastic processes\ndescribed by Xt=atX0+σtX1(Eq. 11) with at=√αtandσt=√1−αt.\nTaking the derivatives of αtw.r.t. t, we have:\ndαt\ndt=d\ndt\u0010\ne−1\n2t(βt+βmin)\u0011\n(156)\n=e−1\n2t(βt+βmin)d\ndt\u0012\n−1\n2t\u0000\nβt+βmin\u0001\u0013\n(157)\n=αt\u0012\n−1\n2\u0012\u0000\nβt+βmin\u0001\n+tdβt\ndt\u0013\u0013\n(158)\n=−αt\n2\u0000\nβt+βmin+t\u0000\nβmax−βmin\u0001\u0001\n(159)\n=−αt\n2(2βt) (160)\n=−αtβt (161)\n˙atand˙σtcan be derived fromdαt\ndtas follows:\n˙at=d\u0000√αt\u0001\ndt=1\n2√αtdαt\ndt=−βt√αt\n2(162)\n˙σt=d\u0000√1−αt\u0001\ndt=1\n2√1−αt−dαt\ndt=βtαt\n2√1−αt(163)\nThus, we can express vϕ(x0, x1, t)as:\nvϕ(x0, x1, t) = ˙atx0+ ˙σtx1 (164)\n=−βt√αt\n2x0+βtαt\n2√1−αtx1 (165)\n=βtαt\n2\u0012x1√1−αt−x0√αt\u0013\n(166)\nv∗(xt, t)can be represented based on Eq. 108 as:\nv∗(xt, t) =˙at\natxt+\u0012\n˙σt−˙atσt\nat\u0013\nEx1|xt[x1] (167)\n=−βt√αt\n21√αtxt+\u0012βtαt\n2√1−αt+βt√αt\n2√1−αt√αt\u0013\nx1|t (168)\n=−βt\n2xt+\u0012βtαt\n2√1−αt+βt(1−αt)\n2√1−αt\u0013\nx1|t (169)\n=−βt\n2xt+βt\n2√1−αtx1|t (170)\n=−βt\n2\u0012\nxt−x1|t√1−αt\u0013\n(171)\n34It is not straightforward to obtain an analytical formula of x1|t. However, if we assume that Eq. 171\nis correct and compare it with Eq. 154, we can guess ∇logp(xt) =−x1|t√1−αt. In DDPM [ 13],x1|tis\nmodeled by a noise network ϵθ(xt, t), which suggests that we can approximate the score function in\nDDPM as−ϵθ(xt,t)√1−αt.\nOn the other hand, if we express v∗(xt, t)according to the formula in Eq. 106, we have:\nv∗(xt, t) =˙σt\nσtxt+\u0012\n˙at−˙σtat\nσt\u0013\nEx0|xt[x0] (172)\n=βtαt\n2√1−αt1√1−αtxt (173)\n+\u0012−βt√αt\n2−βtαt\n2√1−αt√αt√1−αt\u0013\nx0|t (174)\n=βtαt\n2 (1−αt)xt−βt√αt\n2\u0012\n1 +αt\n1−αt\u0013\nx0|t (175)\n=βtαt\n2 (1−αt)xt−βt√αt\n2 (1−αt)x0|t (176)\n=βtαt\n2 (1−αt)\u0012\nxt−x0|t√αt\u0013\n(177)\nSince xtis a Gaussian sample given x0,Ex0|xt[x0]can be interpreted as the posterior estimation of\nthe “mean” x0given its sample xt. The analytical expression of Ex0|xt[x0]exists an is given by the\nso-called “Tweedie’s formula” [9] (Appdx. A.5) as follows:\nx0|t:=Ex0|xt[x0] =xt+ (1−αt)∇logpt(xt)√αt(178)\nApplying this formula of x0|tto Eq. 177, we have:\nv∗(xt, t) =βtαt\n2 (1−αt)\u0012\nxt−xt+ (1−αt)∇logpt(xt)\nαt\u0013\n(179)\n=βtαt\n2 (1−αt)\u0012\n−1−αt\nαtxt−1−αt\nαt∇logpt(xt)\u0013\n(180)\n=−βt\n2(xt+∇logpt(xt)) (181)\nwhich exactly matches the formula of the PF ODE velocity in Eq. 154.\nA.4.2 VE SDE\nThe (forward) VE SDE [48] is represented as follows:\ndxt=p\n2 ˙σtσtdw (182)\nIts corresponding probability flow ODE is characterized by the equation:\ndxt=−˙σtσt∇logp(xt)dt (183)\n=−σt∇logp(xt)dσt (184)\n≈ϵθ(xt, t)dσt (185)\nSolving the Fokker-Planck equation of the VE SDE, we get p(xt|x0) =N\u0000\nx0, σ2\ntI\u0001\n. This means\nthe VE SDE can be written in the following form:\nXt=X0+σtX1 (186)\nwhere the distribution of X1isN(0,I). Eq. 186 is a variant of Eq. 11 with at= 1∀t∈[0,1].\n35The formula of the target velocity is:\nvϕ(x0, x1, t) = ˙σtx1 (187)\nIf we compute v∗(xt, t)based on Eq. 108, we have:\nv∗(xt, t) =˙at\natxt+\u0012\n˙σt−˙atσt\nat\u0013\nEx1|xt[x1] (188)\n= 0 + ( ˙ σt−0)x1|t (189)\n= ˙σtx1|t (190)\nOn the other hand, if Eq. 106 is utilized, v∗(xt, t)can be represented as follows:\nv∗(xt, t) =˙σt\nσtxt+\u0012\n˙at−˙σtat\nσt\u0013\nEx0|xt[x0] (191)\n=˙σt\nσtxt+\u0012\n0−˙σt\nσt\u0013\nx0|t (192)\n=˙σt\nσt\u0000\nxt−x0|t\u0001\n(193)\nAccording to the Tweedie’s formula [9] (Appdx. A.5), we have:\nx0|t=Ex0|xt[x0] =xt+σ2\nt∇logpt(xt) (194)\nTherefore, Eq. 193 is equivalent to:\nv∗(xt, t) =−˙σtσt∇logpt(xt) (195)\nwhich is identical to the velocity of the probability flow ODE in Eq. 183.\nA.4.3 Sub-VP SDE\nThe (forward) sub-VP SDE [48] has the following formula:\ndxt=−βt\n2xtdt+r\nβt\u0010\n1−e−2Rt\n0βτdτ\u0011\ndwt (196)\nThe solution of the Fokker-Planck equation corresponding to the sub-VP SDE is p(xt|x0) =\nN\u0010√αtx0,(1−αt)2I\u0011\n. This allows us to reformulate the sub-VP SDE as:\nXt=√αtX0+ (1−αt)X1 (197)\nwhere αt:=e−t\n2(βt+βmin)is the same as in the VP SDE. In the above process, at=√αtand\nσt= 1−αt. Their derivatives w.r.t. tcan be expressed as follows:\n˙at=d\u0000√αt\u0001\ndt=1\n2√αtdαt\ndt=−βt√αt\n2(198)\n˙σt=d(1−αt)\ndt=−dαt\ndt=αtβt (199)\nwheredαt\ndt=−αtβtaccording to Eq. 161.\nThe target velocity is represented as:\nvϕ(x0, x1, t) =−βt√αt\n2x0+αtβtx1 (200)\n36The posterior velocity v∗(xt, t), according to Eq. 108, is expressed as:\nv∗(xt, t) =˙at\natxt+\u0012\n˙σt−˙atσt\nat\u0013\nEx1|xt[x1] (201)\n=−βt√αt\n2√αtxt+\u0012\nαtβt+βt√αt\n21−αt√αt\u0013\nx1|t (202)\n=−βt\n2xt+\u0012\nαtβt+βt(1−αt)\n2\u0013\nx1|t (203)\n=−βt\n2xt+βt(1 +αt)\n2x1|t (204)\n=−βt\n2\u0000\nxt−(1 +αt)x1|t\u0001\n(205)\nMeanwhile, if Eq. 105 is used, the posterior velocity has the following analytical expression:\nv∗(xt, t) =˙σt\nσtxt+\u0012\n˙at−˙σtat\nσt\u0013\nEx0|xt[x0] (206)\n=αtβt\n1−αtxt+\u0012−βt√αt\n2−αtβt√αt\n1−αt\u0013\nx0|t (207)\n=βtαt\n1−αtxt−βt√αt(1−αt) + 2βtαt√αt\n2 (1−αt)x0|t (208)\n=βtαt\n1−αtxt−βt√αt(1 +αt)\n2 (1−αt)x0|t (209)\n=βt√αt\n1−αt\u0012√αtxt−1 +αt\n2x0|t\u0013\n(210)\nA.4.4 Rectified Flow\nRectified Flow [25] is the posterior flow of the following stochastic process:\nXt= (1−t)X0+tX1 (211)\nHere, at= 1−tandσt=t. The target velocity is vϕ(x0, x1, t) =x1−x0.\nThe closed-form expression of v∗(xt, t)according to Eq. 108 is:\nv∗(xt, t) =˙at\natxt+\u0012\n˙σt−˙atσt\nat\u0013\nEx1|xt[x1] (212)\n=−1\n1−txt+\u0012\n1 +t\n1−t\u0013\nx1|t (213)\n=−1\n1−txt+1\n1−tx1|t (214)\n=1\n1−t\u0000\nx1|t−xt\u0001\n(215)\nBy contrast, if we use Eq. 105, v∗(xt, t)can be expressed as:\nv∗(xt, t) =˙σt\nσtxt+\u0012\n˙at−˙σtat\nσt\u0013\nEx0|xt[x0] (216)\n=1\ntxt+\u0012\n−1−1−t\nt\u0013\nx0|t (217)\n=1\ntxt−1\ntx0|t (218)\n=1\nt\u0000\nxt−x0|t\u0001\n(219)\n37If we assume that X1is the standard Gaussian, we can represent x0|tusing the Tweedie’s formula\n(Appdx. A.5) as follows:\nx0|t=xt+t2∇logpt(xt)\n1−t(220)\nConsequently, the posterior velocity in Eq. 219 can be written as:\nv∗(xt, t) =1\nt\u0012\nxt−xt+t2∇logp(xt)\n1−t\u0013\n(221)\n=1\nt\u0012−t\n1−txt−t2\n1−t∇logp(xt)\u0013\n(222)\n=−1\n1−t(xt+t∇logp(xt)) (223)\nThis implies that the probability flow ODE is defined by the equation:\ndx=\u0012\n−1\n1−txt−t\n1−t∇logp(xt)\u0013\ndt (224)\nThis probability flow ODE corresponds to the following diffusion process:\ndx=−1\n1−txtdt+r\n2t\n1−tdwt (225)\nA.5 Tweedie’s formula for posterior Gaussian mean estimation\nAssume that xis a sample from an isotropic Gaussian distribution with an unknown mean µand\naknown standard deviation σ, i.e., x∼ N\u0000\nµ, σ2I\u0001\n. The posterior expectation of µgiven xcan\ncomputed as follows:\nˆµ=Ep(µ|x)[µ] =x+σ2∇logp(x) (226)\nwhere p(x) =P\nµp(x, µ) =Ep(µ)[p(x|µ)]. Eq. 226 is known as Tweedie’s formula [9].\nTo prove the correctness of Eq. 226, we begin with the score function ∇logp(x)and transform it as\nfollows:\n∇logp(x) =1\np(x)∇p(x) (227)\n=1\np(x)∇X\nµp(µ)p(x|µ) (228)\n=1\np(x)X\nµp(µ)∇p(x|µ) (229)\n=X\nµp(µ)p(x|µ)\np(x)∇logp(x|µ) (230)\n=X\nµp(µ|x)\u0012µ−x\nσ2\u0013\n(231)\n=Ep(µ|x)\u0014µ−x\nσ2\u0015\n(232)\n=Ep(µ|x)[µ]−x\nσ2(233)\nEq. 230 uses the log trick ∇p(x|µ) =∇logp(x|µ)\np(x|µ), and Eq. 231 makes use of the fact that if\nx∼ N\u0000\nµ, σ2I\u0001\nthen∇logp(x|µ) =µ−x\nσ2. We can rearrange Eq. 233 to obtain Eq. 226.\n38A.6 Explanation of why x1|tandx0|1,tshould be invariant w.r.t. t\nBecause two samples xt,xsin the same ODE trajectory have the same (unknown) boundary points\nx0,x1, the distributions p(x0|xt)andp(x1|xt), characterized by the ODE, tend to be sharp and\nsimilar to p(x0|xs)andp(x1|xs), respectively. This means Ep(x0|xt)[x0]≈Ep(x0|xs)[x0]and\nEp(x1|xt)[x1]≈Ep(x1|xs)[x1], or equivalently, x0|t≈x0|sandx1|t≈x1|s. Given this result, we\ncan apply the same reasoning to show that x1|0,t≈x1|0,sandx0|1,t≈x0|1,s. We note that\nA.7 Mathematical derivations of the transformed velocities v∗(˜xt, t)andv∗(¯xt, t)\nBelow, we derive the velocities of the transformed straight (SN) and straight constant-speed (SC)\nflows corresponding to the stochastic process Xt=atX0+σtX1. Our results are summarized in\nTables 2 and 3.\nA.7.1 Non-straight flows →Straight flows\nApplying the results in Proposition 1 to the SN process ˜Xt=X0+φtX1with ˜Xt=Xt\natandφt=σt\nat,\nwe can compute vϕ(˜xt, t)andv∗(˜xt, t)as follows:\nvϕ(˜xt, t) =atvϕ(xt, t)−˙atxt\na2\nt(234)\nv∗(˜xt, t) =atv∗(xt, t)−˙atxt\na2\nt(235)\n≈atvθ(xt, t)−˙atxt\na2\nt(236)\nMoreover, in the case of the SN process ˜Xt= (1−φt)X0+φtX1, we can represent vϕ(˜xt, t)and\nv∗(˜xt, t)viax0,x1andx0|1,t,x1|trespectively as follows:\nvϕ(˜xt(x0, x1, t), t) :=d˜xt(x0, x1, t)\ndt(237)\n=d\ndt((1−φt)x0+φtx1) (238)\n= ˙φt(x1−x0) (239)\nv∗(˜xt(x0, x1, t), t) :=Ex0,x1|˜xt[vϕ(˜xt(x0, x1, t), t)] (240)\n=Ex0,x1|˜xt[ ˙φt(x1−x0)] (241)\n= ˙φtEx0,x1|˜xt[x1−x0] (242)\n= ˙φtEx0,x1|xt[x1−x0] (243)\n= ˙φt\u0000\nx1|t−x0|1,t\u0001\n(244)\nwhere the analytic formula of ˙φtis:\n˙φt=d\ndt\u0012σt\nat+σt\u0013\n(245)\n=(at+σt) ˙σt−(˙at+ ˙σt)σt\n(at+σt)2(246)\n=at˙σt−˙atσt\n(at+σt)2(247)\n39This means besides Eq. 36, v∗(˜xt, t)can be computed as:\nv∗(˜xt, t) =at˙σt−˙atσt\n(at+σt)2\u0000\nx1|t−x0|1,t\u0001\n(248)\n=at˙σt−˙atσt\n(at+σt)2\u0012\nx1|t−xt−σtx1|t\nat\u0013\n(249)\n=at˙σt−˙atσt\n(at+σt)2\u0012(at+σt)x1|t−xt\nat\u0013\n(250)\n≈at˙σt−˙atσt\n(at+σt)2\u0012(at+σt)x1,θ−xt\nat\u0013\n(251)\nApplying the same line of reasoning from Eqs. 237-244 to the SN process ˜Xt=X0+φtX1, we\nhavevϕ(˜xt(x0, x1, t), t) = ˙φtx1andv∗(˜xt(x0, x1, t), t) = ˙φtx1|twith ˙φt=at˙σt−˙atσt\na2\nt.\nA.7.2 Straight flows →Straight constant-speed flows\nTime adjustment transformation Since the posterior velocity corresponding to the straight (SN)\nprocess ˜Xt= (1−φt)X0+φtX1isv∗(˜xt, t) = ˙φt\u0000\nx1|t−x0|1,t\u0001\n(Eq. 244), we can compute\nthe posterior velocity v∗(¯xt, t)corresponding to the straight constant-speed (SC) process d¯Xt=\n(X1−X0)dφtderived from this SN process as follows:\nv∗(¯xt, t) =x1|t−x0|1,t (252)\n=v∗(˜xt, t)\n˙φt(253)\n=(at+σt)v∗(xt, t)−(˙at+ ˙σt)xt\nat˙σt−˙atσt(254)\nwhere v∗(˜xt, t) =(at+σt)v∗(xt,t)−(˙at+ ˙σt)xt\n(at+σt)2 and˙φt=at˙σt−˙atσt\n(at+σt)2are given in Eq. 36 and Eq. 247,\nrespectively.\nSimilarly, the posterior velocity v∗(¯xt, t)associated with the SC process d¯Xt=X1dφtderived\nfrom the SN process ˜Xt=X0+φtX1can be expressed as follows:\nv∗(¯xt, t) =x1|t (255)\n=v∗(˜xt, t)\n˙φt(256)\n=atv∗(xt, t)−˙atxt\nat˙σt−˙atσt(257)\nwhere v∗(˜xt, t) =atv∗(xt,t)−˙atxt\na2\ntand˙φt=at˙σt−˙atσt\na2\nt.\nVariable shifting transformation If we transform the SN process ˜Xt= (1−φt)X0+φtX1\ninto the SC process ¯Xt= (1−t)X0+tX1using the “variable shifting” transformation ¯Xt=\n˜Xt+ (t−φt) (X1−X0), we can represent the target and posterior velocities of this SC process as\nfollows:\nvϕ(x0, x1, t) :=d¯xt(x0, x1, t)\ndt(258)\n=d\ndt(˜xt(x0, x1, t) + (t−φt) (x1−x0)) (259)\n=d˜xt(x0, x1, t)\ndt+ (1−˙φt) (x1−x0) (260)\n=vϕ(˜xt, t) + (1 −˙φt) (x1−x0) (261)\n= ˙φt(x1−x0) + (1 −˙φt) (x1−x0) (262)\n=x1−x0 (263)\n40v∗(¯xt, t) :=Ex0,x1|¯xt[vϕ(x0, x1, t)] (264)\n=Ex0,x1|¯xt[x1−x0] (265)\n=Ex0,x1|˜xt[x1−x0] (266)\n=Ex0,x1|xt[x1−x0] (267)\n=x1|t−x0|1,t (268)\nwhere Eq. 266 is equivalent to Eq. 265 because ¯xtand˜xtare in the same straight trajectory and have\nthesame boundary points x0andx1att= 0andt= 1, respectively.\nSimilarly, if we transform the SN process ˜Xt=X0+φtX1into the SC process ¯Xt=X0+tX1\nusing the transformation ¯Xt=˜Xt+ (t−φt)X1, we can express the target and posterior velocities\nof this SC process as follows:\nvϕ(x0, x1, t) =x1 (269)\nv∗(¯xt, t) =x1|t (270)\nEquivalence between the “time adjustment” and “variable shifting” transformations Here,\nwe prove that applying the “time adjustment” (TA) and “variable shifting” (VS) transformations to\nan SN flow leads to equivalent SC flows when the Euler method is used for sampling. Our main\nidea is to show that the SN samples recovered from the SC flows corresponding to the TA and VS\ntransformations at step t′(i.e.,˜xTA\nt′and˜xVS\nt′), are the same. Without loss of generality, we assume the\nSN flow corresponds to the process ˜Xt= (1−φt)X0+φtX1. Then, we have:\n˜xTA\nt′ (271)\n= ˜xTA\nt+ (φt′−φt)\u0000\nx1|t−x0|1,t\u0001\n(272)\n= ˜xt+ (t−φt)\u0000\nx1|t−x0|1,t\u0001\n+ (t′−t)\u0000\nx1|t−x0|1,t\u0001\n−(t′−φt′)\u0000\nx1|t−x0|1,t\u0001\n(273)\n= ¯xVS\nt+ (t′−t)\u0000\nx1|t−x0|1,t\u0001\n+ (φt′−t′)\u0000\nx1|t−x0|1,t\u0001\n(274)\n= ¯xVS\nt′−(t′−φt′)\u0000\nx1|t−x0|1,t\u0001\n(275)\n= ˜xVS\nt′ (276)\nwhich completes the proof.\nB High-order Numerical Solvers on Straight Constant-speed Flows\nB.1 Runge-Kutta methods\nB.1.1 Second-order Runge-Kutta methods\nBelow, we explicitly derive the relationships between the coefficients b1, b2, c2, a2,1in the formulas\nof the second-order RK methods.\nThe Taylor series of order 2 for x(ti+1)is given by:\nx(ti+1) =x(ti+h) (277)\n=x(ti) +hx′(ti) +h2\n2x′′(ti) +O\u0000\nh3\u0001\n(278)\nwhere h:= ∆ti+1=ti+1−ti.O\u0000\nh3\u0001\nmeans that the truncation error is a third-degree polynomial\nfunction of h.\n41We can express x′′(t)as follows:\nx′′(t) =d\ndt(x′(t)) (279)\n=d\ndtv(x, t) (280)\n=∂v(x, t)\n∂xv(x, t) +∂v(x, t)\n∂t(281)\n=vx(x, t)v(x, t) +vt(x, t) (282)\nwhere vx(x, t)andvt(x, t)denote∂v(x,t)\n∂xand∂v(x,t)\n∂t, respectively. Applying the formula of x′′(t)\nin Eq. 282 to Eq. 278, we have:\nx(ti+1) =x(ti) +hv(xti, ti)\n+h2\n2(vx(xti, ti)v(xti, ti) +vt(xti, ti))\n+O\u0000\nh3\u0001\n(283)\n=x(ti) +h\u0012\nv(xti, ti) +h\n2vx(xti, ti)v(xti, ti) +h\n2vt(xti, ti)\u0013\n+O\u0000\nh3\u0001\n(284)\nThe Taylor series (TS) method of order 2 requires knowing vxandvt, which are usually unavailable\nin practice. To overcome this problem, we approximate it with Runge-Kutta methods of the same\norder described below:\nx(ti+1) =x(ti) +hV(xti, ti, h) (285)\nwhere V(xti, ti, h)is defined as:\nV(xti, ti, h) :=b1v(xti, ti) +b2v(xti+a2,1hv(xti, ti), ti+c2h) (286)\nThe second term in the RHS of Eq. 286 can be approximated using the first-order Taylor expansion\nfor functions of two variables x,tas follows:\nv(xti+a2,1hv(xti, ti), ti+c2h)\n=v(xti, ti) +a2,1hv(xti, ti)vx(xti, ti) +c2hvt(xti, ti) +O\u0000\nh2\u0001\n(287)\nFrom Eqs. 285, 286, 287, we have a Runge-Kutta method of order 2 can be written as:\nx(ti+1) =x(ti) +h\u0010\n(b1+b2)v(xti, ti)\n+b2a2,1hvx(xti, ti)v(xti, ti)\n+b2c2hvt(xti, ti)\u0011\n+O\u0000\nh3\u0001\n(288)\nFor Eq. 288 to be similar to Eq. 284, the coefficients b1, b2, c2, a2,1should satisfy the following\nconditions:\na2,1=c2 (289)\nb1+b2= 1 (290)\nb2c2= 1/2 (291)\nSince there are 4 variables but only 3 equations, we can let c2vary arbitrarily in the range (0,1]and\ncompute other coefficients based on c2as follows:\na2,1=c2 (292)\nb2=1\n2c2(293)\nb1= 1−1\n2c2(294)\n42Heun’s method In Heun’s method, c2= 1, which means τ2=ti+ ∆ti+1=ti+1. Since ti+1is\nalways specified before sampling, this method has noproblem working with discrete-time models.\nExplicit midpoint method The explicit midpoint method corresponds to c2=1\n2. Its coefficients\nare given in Table 6 and its update on SC flows is represented as follows:\n¯xti+1= ¯xti+ ∆φti+1f2 (295)\nwhere f1,f2are given by:\nf1=v∗(¯xti, ti) (296)\nf2=v∗(¯xti+ ∆φτ2f1, τ2) ;τ2=ti+1\n2∆ti+1 (297)\n0 0 0\n1/2 1/2 0\n0 1\nTable 6: The Butcher tableau for the explicit midpoint method\nDifferent from Heun’s method, the explicit midpoint method is not always applicable to discrete-time\nmodels when c2isfixed at1\n2. This is because τ2can be a non-integer iftiandti+1have different\nparities (i.e.,tiis odd and ti+1is even, or vice versa), resulting in an unreliable computation of f2.\nWe can overcome this problem by ensuring that all Ndiscrete sampling time steps t1,...,tNhave the\nsame parity but this is very tedious. A much easier way is rounding τ2to the closest integer, then\nrecomputing c2asc2=τ2−tiand other coefficients via Eqs. 292-294.\nB.1.2 Third-order Runge-Kutta methods\nFor the third-order RK methods, we present the set of constraints governing their coefficients without\nexplicit derivation. They are as follows:\na2,1=c2 (298)\na3,1+a3,2=c3 (299)\nb1+b2+b3= 1 (300)\nb2c2+b3c3= 1/2 (301)\nb2c2\n2+b3c2\n3= 1/3 (302)\nc2a3,2b3= 1/6 (303)\nThis system comprises 8 variables but only 6 equations, which means two variables can be chosen\narbitrarily. Since our focuses are c2andc3, we set c3= 1and let c2vary. Other variables can be\ncomputed based on c2as follows: a3,1=3c2−3c2\n2−1\nc2(2−3c2),a3,2=1−c2\nc2(2−3c2),a2,1=c2,b3=2−3c2\n6(1−c2),\nb2=1\n6c2(1−c2), and b1=3c2−1\n6c2.\nKutta’s third-order method (RK3) The RK3 method corresponds to c3= 1andc2= 1/2. Its\ncoefficients are shown in Table 7 and its update on SC flows is given below:\n¯xti+1= ¯xti+ ∆φti+1\u0012f1+ 4f2+f3\n6\u0013\n(304)\nwhere f1,f2,f3are computed as:\nf1=v∗(¯xti, ti) (305)\nf2=v∗(¯xti+ ∆φτ2f1, τ2) ;τ2=ti+1\n2∆ti+1 (306)\nf3=v∗(¯xti+ ∆φτ3(2f2−f1), τ3) ;τ3=ti+ ∆ti+1 (307)\n430 0 0 0\n1/2 1/2 0 0\n1 -1 2 0\n1/6 2/3 1/6\nTable 7: The Butcher tableau for the RK3 method\nB.1.3 Fourth-order Runge-Kutta methods\nSince it is challenging to derive the formulas of the coefficients of fourth-order RK methods with\none free variable. Below, we only provide the coefficients for the well-known classic Runge-Kutta\nmethod (RK4).\nClassic Runge-Kutta method (RK4) The classic Runge-Kutta method (RK4) on SC flows is\ncharacterized by the following equation:\n¯xti+1= ¯xti+ ∆φti+1\u0012f1+ 2f2+ 2f3+f4\n6\u0013\n(308)\nwhere f1,f2,f3are expressed as:\nf1=v∗(¯xti, ti) (309)\nf2=v∗(¯xti+ ∆φτ2f1, τ2) ;τ2=ti+1\n2∆ti+1 (310)\nf3=v∗(¯xti+ ∆φτ3f2, τ3) ;τ3=ti+1\n2∆ti+1 (311)\nf4=v∗(¯xti+ ∆φτ4f3, τ4) ;τ4=ti+ ∆ti+1 (312)\nThe Butcher tableau for the RK4 method is shown in Table 8.\n0 0 0 0 0\n1/2 1/2 0 0 0\n1/2 0 1/2 0 0\n1 0 0 1 0\n1/6 1/3 1/3 1/6\nTable 8: The Butcher tableau for the RK4 method\nB.2 Linear multi-step methods\nIn an AB method of order k(ABk),¯v(t)is characterized by the following equation:\n¯v(t) =k−1X\nj=0Q0≤m≤k−1\nm̸=j(t−ti−m)\nQ0≤m≤k−1\nm̸=j(ti−j−ti−m)¯v∗\nti−j(313)\n=k−1X\nj=0ℓj(t) ¯v∗\nti−j(314)\nIntegrating both sides of Eq. 314, we have:\nRti+1\nti¯v(t)dt\n∆ti+1=1\n∆ti+1Zti+1\nti\nk−1X\nj=0ℓj(t) ¯v∗\nti−j\ndt (315)\n=k−1X\nj=0 Rti+1\ntiℓj(t)dt\n∆ti+1!\n¯v∗\nti−j(316)\n=k−1X\nj=0Lj¯v∗\nti−j(317)\n44where Lj:=Rti+1\ntiℓj(t)dt\n∆ti+1. This implies that the update formula of an AB method of order kis:\n¯xti+1= ¯xti+ ∆φti+1\nk−1X\nj=0Lj¯v∗\nti−j\n (318)\nOn the other hand, in an AM method of order k(AMk),¯v(t)is defined as follows:\n¯v(t) =k−1X\nj=0Q0≤m≤k−1\nm̸=j(t−ti+1−m)\nQ0≤m≤k−1\nm̸=j(ti+1−j−ti+1−m)¯v∗\nti+1−j(319)\nwhich leads to the following update formula:\n¯xti+1= ¯xti+ ∆φti+1\nk−1X\nj=0˜Lj¯v∗\nti+1−j\n (320)\nwhere ˜ℓj(t) :=Q0≤m≤k−1\nm̸=j(t−ti+1−m)\nQ0≤m≤k−1\nm̸=j(ti+1−j−ti+1−m)and˜Lj:=Rti+1\ntiℓj(t)dt\n∆ti+1.\nAB2 In the AB2 method, ¯v(t)is expressed as follows:\n¯v(t) =ℓ0(t) ¯v∗\nti+ℓ1(t) ¯v∗\nti−1(321)\nwhere ℓ0(t) =t−ti−1\nti−ti−1andℓ1(t) =t−ti\nti−1−ti. The formula of L0can be derived as follows:\nL0=1\n∆ti+1Zti+1\ntiℓ0(t)dt (322)\n=1\nC0∆ti+1Zti+1\nti(t−ti−1)dt (323)\n=1\nC0∆ti+1 \nt2\n2−ti−1t\f\f\f\ft=ti+1\nt=ti!\n(324)\n=1\nC0∆ti+1\u0012t2\ni+1−t2\ni\n2−ti−1(ti+1−ti)\u0013\n(325)\n=1\n2C0∆ti+1(ti+1−ti) (ti+1+ti−2ti−1) (326)\n=ti+1+ti−2ti−1\n2 (ti−ti−1)(327)\nwhere C0=ti−ti−1. By replacing C0withC1=−C0andti−1withtiin Eq. 326, we obtain the\nformula of L1, which is:\nL1=ti+1−ti\n2C1(328)\n=−(ti+1−ti)\n2 (ti−ti−1)(329)\nTherefore, the AB2 update on SC flows is:\n¯xti+1= ¯xti+ ∆φti+1\u0010\nL0¯v∗\nti+L1¯v∗\nti−1\u0011\n(330)\nwhere L0andL1are provided in Eqs. 327 and 329, respectively.\n45AM2 The velocity ¯v(t)in the AM2 method is represented as:\n¯v(t) =˜ℓ0(t)v∗\nti+1+˜ℓ1(t)v∗\nti(331)\nwhere ˜ℓ0(t) =t−ti\nti+1−tiand˜ℓ1(t) =t−ti+1\nti−ti+1.\nWe can compute ˜L0by replacing C1in Eq. 328 with ˜C0=ti+1−ti, leading to the formula:\n˜L0=ti+1−ti\n2 (ti+1−ti)=1\n2(332)\nThe formula of ˜L1can be obtained by replacing ti−1in Eq. 327 with ti+1and is given below:\n˜L1=ti−ti+1\n2 (ti−ti+1)=1\n2(333)\nThus, the AM2 update on SC flows is:\n¯xti+1= ¯xti+ ∆φti+1\u0012¯v∗\nti+1+ ¯v∗\nti\n2\u0013\n(334)\nAB3 In the AB3 method, ¯v(t)is represented as follows:\n¯v(t) =ℓ0(t) ¯v∗\nti+ℓ1(t) ¯v∗\nti−1+ℓ2(t) ¯v∗\nti−2(335)\nwhere ℓ0(t),ℓ1(t),ℓ2(t)are given by:\nℓ0(t) =(t−ti−1) (t−ti−2)\n(ti−ti−1) (ti−ti−2)(336)\nℓ1(t) =(t−ti) (t−ti−2)\n(ti−1−ti) (ti−1−ti−2)(337)\nℓ2(t) =(t−ti) (t−ti−1)\n(ti−2−ti) (ti−2−ti−1)(338)\nThe coefficient L0has the following formula:\nL0=Zti+1\ntiℓ0(t)dt (339)\n=1\nC0∆ti+1Zti+1\nti(t−ti−1) (t−ti−2)dt (340)\n=1\nC0∆ti+1Zti+1\nti\u0000\nt2−(ti−1+ti−2)t+ti−1ti−2\u0001\ndt (341)\n=1\nC0∆ti+1 \nt3\n3−(ti−1+ti−2)t2\n2+ti−1ti−2t\f\f\f\fti+1\nti!\n(342)\n=1\nC0∆ti+1\u0012t3\ni+1−t3\ni\n3−(ti−1+ti−2)t2\ni+1−t2\ni\n2+ti−1ti−2(ti+1−ti)\u0013\n(343)\n=(ti+1−ti)\nC0∆ti+1\u0012t2\ni+1+ti+1ti+t2\ni\n3−(ti−1+ti−2) (ti+1+ti)\n2+ti−1ti−2\u0013\n(344)\n=1\n6C0\u0000\n2\u0000\nt2\ni+1+ti+1ti+t2\ni\u0001\n−3 (ti−1+ti−2) (ti+1+ti) + 6ti−1ti−2\u0001\n(345)\n=1\n6C0\u0010\n2 (ti−ti−2) (ti+1+ti−2ti−1) + 2 ( ti+1−ti−1) (ti+1−ti−2)\n−(ti+1−ti) (ti−1−ti−2)\u0011\n(346)\nwhere C0= (ti−ti−1) (ti−ti−2).\n46We can quickly compute L1by replacing ti−1andC0in Eq. 345 with tiandC1=\n(ti−1−ti) (ti−1−ti−2), respectively. This leads to the formula:\nL1=1\n6C1\u0000\n2\u0000\nt2\ni+1+ti+1ti+t2\ni\u0001\n−3 (ti+ti−2) (ti+1+ti) + 6titi−2\u0001\n(347)\n=1\n6C1\u0000\n2t2\ni+1−ti+1ti−t2\ni−3ti+1ti−2+ 3titi−2\u0001\n(348)\n=(ti+1−ti) (2ti+1+ti−3ti−2)\n6C1(349)\n=(ti+1−ti) (2ti+1+ti−3ti−2)\n6 (ti−1−ti) (ti−1−ti−2)(350)\nSimilarly, to compute L2, we replace ti−2andC0in Eq. 345 with tiandC2=\n(ti−2−ti) (ti−2−ti−1), respectively. The resulting formula is given below:\nL2=1\n6C2\u0000\n2\u0000\nt2\ni+1+ti+1ti+t2\ni\u0001\n−3 (ti+ti−1) (ti+1+ti) + 6titi−1\u0001\n(351)\n=1\n6C2\u0000\n2t2\ni+1−ti+1ti−t2\ni−3ti+1ti−1+ 3titi−1\u0001\n(352)\n=(ti+1−ti) (2ti+1+ti−3ti−1)\n6C2(353)\n=(ti+1−ti) (2ti+1+ti−3ti−1)\n6 (ti−2−ti) (ti−2−ti−1)(354)\nConsequently, the AB3 update on SC flows is:\n¯xti+1= ¯xti+ ∆φti+1\u0010\nL0¯v∗\nti+L1¯v∗\nti−1+L2¯v∗\nti−2\u0011\n(355)\nwhere L0,L1,L2are provided in Eqs. 346, 350, 354, respectively\nAM3 ¯v(t)of the AM3 method is expressed as:\n¯v(t) =˜ℓ0(t) ¯v∗\nti+1+˜ℓ1(t) ¯v∗\nti+˜ℓ2(t) ¯v∗\nti−1(356)\nwhere ˜ℓ0(t),˜ℓ1(t),˜ℓ2(t)have the following expressions:\n˜ℓ0(t) =(t−ti) (t−ti−1)\n(ti+1−ti) (ti+1−ti−1)(357)\n˜ℓ1(t) =(t−ti+1) (t−ti−1)\n(ti−ti+1) (ti−ti−1)(358)\n˜ℓ2(t) =(t−ti+1) (t−ti)\n(ti−1−ti+1) (ti−1−ti)(359)\n˜L0can be computed by replacing C2in Eq. 353 with ˜C0= (ti+1−ti) (ti+1−ti−1). This results in\nthe formula:\nL0=(ti+1−ti) (2ti+1+ti−3ti−1)\n6 (ti+1−ti) (ti+1−ti−1)(360)\n=2ti+1+ti−3ti−1\n6 (ti+1−ti−1)(361)\n47To compute ˜L1, we replace ti−2andC0in Eq. 345 with ti+1and˜C1= (ti−ti+1) (ti−ti−1),\nrespectively. The analytical expression is given below:\n˜L1=1\n6C1\u0000\n2\u0000\nt2\ni+1+ti+1ti+t2\ni\u0001\n−3 (ti+1+ti−1) (ti+1+ti) + 6ti+1ti−1\u0001\n(362)\n=1\n6C1\u0000\n−t2\ni+1−ti+1ti+ 2t2\ni+ 3ti+1ti−1−3titi−1\u0001\n(363)\n=1\n6C1(−(ti+1−ti) (ti+1+ti)−ti(ti+1−ti) + 3ti−1(ti+1−ti)) (364)\n=(ti+1−ti) (3ti−1−2ti−ti+1)\n6C1(365)\n=(ti+1−ti) (3ti−1−2ti−ti+1)\n6 (ti−ti+1) (ti−ti−1)(366)\n=ti+1+ 2ti−3ti−1\n6 (ti−ti−1)(367)\nWe compute ˜L2by replace ti−1,ti−2, and C0in Eq. 345 with ti+1,ti, and ˜C2=\n(ti−1−ti+1) (ti−1−ti)respectively, leading to the following equation:\n˜L2=1\n6C2\u0000\n2\u0000\nt2\ni+1+ti+1ti+t2\ni\u0001\n−3 (ti+1+ti) (ti+1+ti) + 6ti+1ti\u0001\n(368)\n=1\n6C2\u0000\n−t2\ni+1+ 2ti+1ti−t2\ni\u0001\n(369)\n=−(ti+1−ti)2\n6C2(370)\n=−(ti+1−ti)2\n6 (ti−1−ti+1) (ti−1−ti)(371)\n=−(ti+1−ti)2\n6 (ti+1−ti−1) (ti−ti−1)(372)\nThus, the AM3 update on SC flows is:\n¯xti+1= ¯xti+ ∆φti+1\u0010\n˜L0¯v∗\nti+1+˜L1¯v∗\nti+˜L2¯v∗\nti−1\u0011\n(373)\nwhere ˜L0,˜L1,˜L2are given in Eqs. 361, 367, 371, respectively.\nC Analysis of Existing High-order Solvers for Diffusion Models\nIn this section, we aim to analyze current state-of-the-art (SOTA) high-order solvers for diffusion\nmodels based on our framework and notations. Our analysis will focus exclusively on DPM-Solver\n[27], as other solvers either have connections to or derive from this solver [28, 56, 57].\nC.1 DPM-Solver\nThe main idea behind DPM-Solver and DEIS is to leverage the semi-linear structure in the probability\nflow ODEs of diffusion models to reduce the approximation error. Here, we consider the posterior\nflow ODE of a general stochastic process Xt=atX0+σtX1, represented as follows:\n˙xt=v∗(xt, t) (374)\n=˙at\natxt+σt\u0012˙σt\nσt−˙at\nat\u0013\nx1|t (375)\n=f(t)xt+g(t)x1,θ(xt, t) (376)\nwhere f(t) :=˙at\nat,g(t) :=σt\u0010\n˙σt\nσt−˙at\nat\u0011\n.\n48Eq. 376 is a semi-linear ODE because the velocity v∗(xt, t)is the sum of a linear term f(t)xtand a\nnon-linear term g(t)x1,θ(xt, t). The exact solution for this type of ODEs is:\nxt′=eRt′\ntf(τ)dτxt+Zt′\nt\u0010\neRt′\nτf(r)drg(τ)x1,θ(xτ, τ)\u0011\ndτ (377)\nSince f(t) =˙at\nat=dlogat\ndt, we have eRt′\ntf(τ)dτ=elogat′−logat=at′\natandeRt′\nτf(r)dr=at′\naτ. The\nabove equation can be rewritten as:\nxt′=at′\natxt+at′Zt′\ntg(τ)\naτx1,θ(xτ, τ)dτ (378)\nInterestingly, we can representg(τ)\naτas follows:\ng(τ)\naτ=στ\naτ\u0012˙στ\nστ−˙aτ\naτ\u0013\n(379)\n=˙στ\naτ−˙aτστ\na2τ(380)\n=aτ˙στ−˙aτστ\na2τ(381)\n= ˙φτ (382)\nwhere φt=σt\nat.\nTherefore, Eq. 378 is equivalent to:\nxt′=at′\natxt+at′Zt′\nt˙φτx1,θ(xτ, τ)dτ (383)\n⇔xt′\nat′=xt\nat+Zt′\nt˙φτx1,θ(xτ, τ)dτ (384)\n⇔¯xt′= ¯xt+Zφt′\nφtˆx1,θ(xφτ, φτ)dφτ (385)\nwhere ¯xt=xt\natand¯xt′=xt′\nat′. In Eq. 385, we change the time variable from τtoφτ.\nEq. 385 is exactly an update on the SC flow obtained by applying the “variable scaling” transformation\n˜Xt=Xt\natfollowed by the “time adjustment” transformation d¯Xt=X1dφt(Table 3). The termRφt′\nφtˆx1,θ(xφτ, φτ)dφτcan be approximated using different numerical methods such as Taylor\nexpansion, Runge-Kutta, or linear multi-step methods.\nThe authors of DPM-Solver, however, use a different transformation of tinspired by the work [ 18].\nThey set λt:= logat\nσt, which leads to a different expression ofg(τ)\naτgiven below:\ndλt\ndt=σt\nat˙atσt−at˙σt\nσ2\nt=˙at\nat−˙σt\nσt(386)\ng(τ)\naτ=˙στ\naτ−˙aτστ\na2τ=−dλτ\ndτστ\naτ(387)\nConsequently, in their work, Eq. 378 is reformulated as follows:\nxt′=at′\natxt−at′Zt′\ntdλτ\ndτστ\naτx1,θ(xτ, τ)dτ (388)\n=at′\natxt−at′Zλt′\nλte−λˆx1,θ(xλ, λ)dλ (389)\nwhere ˆx1,θ(xλ, λ)is equivalent to x1,θ(xτ, τ)but with the time variable changed from τtoλ. The\nintegral in Eq. 389 is similar to the counterpart in Eq. 385 except for the presence of the coefficient\n49e−λdue to using a different change of variables. However, this coefficient does not cause any trouble\nwhen approximating the integral using numerical methods.\nTo simplify our presentation, we rewrite Eq. 389 with tandt′replaced with tiandti+1, respectively:\nxti+1=ati+1\natixti−ati+1Zλti+1\nλtie−λˆx1,θ(xλ, λ)dλ (390)\nIn the DPM-Solver paper, the authors approximate ˆx1,θ(xλ, λ)using the (k−1)-th order Taylor\nexpansion as follows:\nˆx1,θ(xλ, λ) =k−1X\nj=0(λ−λti)j\nj!ˆx(j)\n1,θ\u0000\nxλti, λti\u0001\n+O\u0010\n(λ−λti)k\u0011\n(391)\nwhere ˆx(j)\n1,θ\u0010\nxλti−1, λti−1\u0011\ndenotes the j-th derivative of ˆx1,θ(xλ, λ)atλ=λti−1. Substituting this\nexpression of ˆx1,θ(xλ, λ)into Eq. 390, we have:\nxti+1=ati+1\natixti−ati+1k−1X\nj=0ℓj(λ) ˆx(j)\n1,θ\u0010\nxλti−1, λti−1\u0011\n+O\u0010\u0000\nλti+1−λti\u0001k+1\u0011\n(392)\nwhere ℓj(λ) =Rλti+1\nλtie−λ(λ−λti)j\nj!which can be expressed in closed-form [ 27]. To approximate\nˆx(j)\n1,θ\u0010\nxλti−1, λti−1\u0011\n, the authors leverage Runge-Kutta methods with the time λ. For example, the\nDPM-Solver-2 update (Algorithm 1 in [27]) is given below:\nτ2=tλ\u0012λti+1+λti\n2\u0013\n(393)\nxτ2=aτ2\natixti−στ2\u0012\ne∆λti+1\n2−1\u0013\nˆx1,θ(xti, ti) (394)\nxti+1=ati+1\naτ2xti−σti+1\u0010\ne∆λti+1−1\u0011\nˆx1,θ(xτ2, τ2) (395)\nThis is indeed the explicit midpoint method in which the intermediate time step is computed in the\nλ-time space and then mapped back to the original time space using the time inverse function tλ(·)\n(Eq. 393). This update can be view as a special case of the general update in Eqs. 60-63.\nD Experimental Settings\nD.1 2D Toy Dataset\nIn this dataset, p0is a mixture of three 2D Gaussians with the mean vectors and covariance matrices\ngiven below:\nµ0=\u0014\n20\n20\u0015\n,\u0014\n25\n10\u0015\n,\u0014\n10\n26\u0015\nΣ0=\u0014\n0.620.72\n0.721.42\u0015\n,\u0014\n1.32−0.92\n−0.921.02\u0015\n,\u0014\n1.220.0\n0.0 1 .22\u0015\np1is a mixture of two 2D Gaussian with the mean vectors and covariance matrices given below:\nµ1=\u0014\n5\n−5\u0015\n,\u0014\n−5\n3\u0015\nΣ1=\u0014\n1.020.0\n0.0 1 .02\u0015\n,\u0014\n1.020.0\n0.021.02\u0015\n50Concat( [xt, t])\nLinear( dx+ 1, 100, bias=True)\nTanh()\nLinear(100, 100, bias=True)\nTanh()\nLinear(100, dx, bias=True)\nTable 9: Architecture of the velocity network for experiments on the 2D toy dataset.\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\nat σt at σt at σt\n0.0 0.2 0.4 0.6 0.8 1.04\n3\n2\n1\n0\n0.0 0.2 0.4 0.6 0.8 1.00.51.01.52.0\n0.0 0.2 0.4 0.6 0.8 1.05\n4\n3\n2\n1\n0\n0.0 0.2 0.4 0.6 0.8 1.0012345\n0.0 0.2 0.4 0.6 0.8 1.02.0\n1.5\n1.0\n0.5\n0.0\n0.00 0.25 0.50 0.75 1.0001000020000300004000050000\n˙at ˙σt ˙at ˙σt ˙at ˙σt\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.0\n0.0 0.2 0.4 0.6 0.8 1.001234\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.00 0.25 0.50 0.75 1.000.0000.0050.0100.0150.020\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.0\n0.00 0.25 0.50 0.75 1.0001000020000300004000050000\nat+σt at˙σt−˙atσt at+σt at˙σt−˙atσt at+σt at˙σt−˙atσt\nthird-degree fifth-degree VP-SDE-like\nFigure 15: Coefficients associated with the third-degree (left), fifth-degree (middle), and VP-SDE-like\n(right) processes.\nWe employ a simple three-layer MLP, similar to the one in [ 25], to model either the posterior velocity\nv∗(xt, t)or the predicted latent sample x1|tof each process. The network architecture is outlined in\nTable 9. During training, we minimize the velocity matching loss if the network models v∗(xt, t),\nand the noise matching loss if it models x1|t. Training is conducted on 30,000 samples drawn from\np0andp1for 20000 iterations, utilizing a batch size of 2048. We employ the Adam optimizer with a\nlearning rate of 0.003.\nIn Fig. 15, we plot the coefficients associated with the third-degree, fifth-degree, and VP-SDE-like\nprocesses as functions of time. Besides a1andσ0equal 0 for all processes, at˙σt−˙atσtis 0 at t= 0\nandt= 1for the fifth-degree process.\nE Additional Experimental Results\nE.1 2D Toy Dataset\nE.1.1 Empirical analysis on the numerical robustness of our method\nDuring our experiments, we have observed that if the model is vθ(xt, t), using the simple clamping\ntechnique presented in Section 4.2 with ϵas small as 1e-6 is enough to ensure the numerical stability\nof sampling along the SC flow for all the three processes (third-degree, fifth-degree and VP-SDE-like)\nas illustrated in Figs. 16, 17, 18. On the other hand, if the model is x1,θ(xt, t), we achieve numerical\nstability only for the VP-SDE-like process when using this clamping technique with ϵbeing 1e-6,\nwhich can be seen from the average maximum absolute values of xtfor the VP-SDE-like process (the\nred curves in rows 3, 4 column (b) of Fig. 18) at t= 0 being normal (about 30) while the counterparts\nfor the third-degree and fifth-degree process (the red curves in rows 3, 4 column (b) of Figs. 16, 17)\nbeing much larger than normal. The numerical instability for the fifth-degree process only happens\nfor a small fraction of generated samples as the majority can still be visualized in rows 3, 4 column (a)\nof Fig. 17, yet for the third-degree process, it happens for almost all samples. This numerical problem\ncan only be partially alleviated for the third-degree process by increasing the clipping threshold ϵand\nthe number of sampling steps N.\n51model SC type ϵ N samples xt ¯xt (φt′−φt)vθ(¯xt, t) vθ(¯xt, t) φt′−φt\nvθ(xt, t)Eq. 22 1e-6 20\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.21234\n1.0 0.8 0.6 0.4 0.223242526\n1.0 0.8 0.6 0.4 0.20.050.100.15\nEq. 23 1e-6 20\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.20241e6\n1.0 0.8 0.6 0.4 0.20241e6\n1.0 0.8 0.6 0.4 0.2345\n1.0 0.8 0.6 0.4 0.20.000.250.500.751.001e6\nx1,θ(xt, t)Eq. 22 1e-3 20\n1.0 0.8 0.6 0.4 0.20100200300400\n1.0 0.8 0.6 0.4 0.20100200300\n1.0 0.8 0.6 0.4 0.220406080\n1.0 0.8 0.6 0.4 0.2400420440460\n1.0 0.8 0.6 0.4 0.20.050.100.15\n1e-3 20\n1.0 0.8 0.6 0.4 0.20100200300400\n1.0 0.8 0.6 0.4 0.210002000300040005000\n1.0 0.8 0.6 0.4 0.2020004000\n1.0 0.8 0.6 0.4 0.2681012\n1.0 0.8 0.6 0.4 0.202004006008001000\nEq. 23 1e-3 1000\n1.0 0.8 0.6 0.4 0.2204060\n1.0 0.8 0.6 0.4 0.2010002000300040005000\n1.0 0.8 0.6 0.4 0.201000200030004000\n1.0 0.8 0.6 0.4 0.25.05.56.06.5\n1.0 0.8 0.6 0.4 0.20200400600\n1e-6 1000\n1.0 0.8 0.6 0.4 0.20241e5\n1.0 0.8 0.6 0.4 0.2241e6\n1.0 0.8 0.6 0.4 0.20241e6\n1.0 0.8 0.6 0.4 0.2681012\n1.0 0.8 0.6 0.4 0.20.000.250.500.751.001e6\n(a) (b) (c) (d) (e) (f)\nFigure 16: Generated samples (a) and the evolution of the maximum absolute values of different\ntracked variables over time (b-f) when sampling along the SC flow of the third-degree process, with\ndifferent types of models, types of SC flows, values of clipping threshold ϵ, and numbers of steps N.\nmodel SC type ϵ N samples xt ¯xt (φt′−φt)vθ(¯xt, t) vθ(¯xt, t) φt′−φt\nvθ(xt, t)Eq. 22 1e-6 20\n1.0 0.8 0.6 0.4 0.2051015\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.20246\n1.0 0.8 0.6 0.4 0.20102030401e3\n1.0 0.8 0.6 0.4 0.20.000.050.100.150.20\nEq. 23 1e-6 20\n1.0 0.8 0.6 0.4 0.2051015\n1.0 0.8 0.6 0.4 0.20241e6\n1.0 0.8 0.6 0.4 0.201231e6\n1.0 0.8 0.6 0.4 0.2051015201e3\n1.0 0.8 0.6 0.4 0.202461e5\nx1,θ(xt, t)Eq. 22 1e-6 20\n1.0 0.8 0.6 0.4 0.2024681e3\n1.0 0.8 0.6 0.4 0.20.000.250.500.751.001e4\n1.0 0.8 0.6 0.4 0.20121e3\n1.0 0.8 0.6 0.4 0.25101e4\n1.0 0.8 0.6 0.4 0.20.000.050.100.150.20\nEq. 23 1e-6 20\n1.0 0.8 0.6 0.4 0.2024681e3\n1.0 0.8 0.6 0.4 0.20241e6\n1.0 0.8 0.6 0.4 0.201231e6\n1.0 0.8 0.6 0.4 0.2456\n1.0 0.8 0.6 0.4 0.202461e5\n(a) (b) (c) (d) (e) (f)\nFigure 17: Generated samples (a) and the evolution of the maximum absolute values of different\ntracked variables over time (b-f) when sampling along the SC flow of the fifth-degree process, with\ndifferent types of models, and types of SC flows.\nmodel SC type ϵ N samples xt ¯xt (φt′−φt)vθ(¯xt, t) vθ(¯xt, t) φt′−φt\nvθ(xt, t)Eq. 22 1e-6 20\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.20123\n1.0 0.8 0.6 0.4 0.223.524.024.525.025.5\n1.0 0.8 0.6 0.4 0.20.000.050.100.15\nEq. 23 1e-6 20\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.20200400600800\n1.0 0.8 0.6 0.4 0.20100200300\n1.0 0.8 0.6 0.4 0.23.54.04.55.0\n1.0 0.8 0.6 0.4 0.20204060\nx1,θ(xt, t)Eq. 22 1e-6 20\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.20123\n1.0 0.8 0.6 0.4 0.2232425\n1.0 0.8 0.6 0.4 0.20.000.050.100.15\nEq. 23 1e-6 20\n1.0 0.8 0.6 0.4 0.25101520\n1.0 0.8 0.6 0.4 0.20200400600800\n1.0 0.8 0.6 0.4 0.20100200300\n1.0 0.8 0.6 0.4 0.24.04.55.0\n1.0 0.8 0.6 0.4 0.20204060\n(a) (b) (c) (d) (e) (f)\nFigure 18: Generated samples (a) and the evolution of the maximum absolute values of different\ntracked variables over time (b-f) when sampling along the SC flow of the VP-SDE-like process, with\ndifferent types of models, and types of SC flows.\n52N\nSC (v)\nSC (x1,)\nSC (v)\nSC (x1,)\n5\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n50\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n1000\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n¯Xt= (1−t)X0+tX1¯Xt=X0+tX1\nFigure 19: Generated samples (left), xt-trajectories (middle) and ¯xt-trajectories (right) of the SC\nflows of the third-degree process, with different number of updates N, different types of SC flows\n(two equations in the bottom), and different types of parameterized models ( vθvsx1,θ).\nN\nSC (v)\nSC (x1,)\nSC (v)\nSC (x1,)\n5\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n50\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n1000\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n¯Xt= (1−t)X0+tX1¯Xt=X0+tX1\nFigure 20: Generated samples (left), xt-trajectories (middle) and ¯xt-trajectories (right) of the SC\nflows of the fifth-degree process, with different numbers of update steps N, different SC types (two\nequations in the bottom), and different types of models ( vθvsx1,θ).\nE.1.2 Comparison between modeling v∗(xt, t)and modeling x1|t\nAs discussed in Section 4.1, we can learn a posterior flow by modeling either v∗(xt, t)orx1|t. A\nmodel of v∗(xt, t)is trained via the velocity matching loss in Eq. 10 while a model of x1|tis trained\nvia the noise matching loss in Eq. 4. Fig. 19 shows the generation results w.r.t. two implementations\nof the SC flow of the third-degree process, one using a model vθ(xt, t)ofv∗(xt, t)(orange) and the\nother using a model x1,θ(xt, t)ofx1|t(red). It is evident that the SC flow using vθ(xt, t)greatly\noutperforms the counterpart using x1,θ(xt, t)for either of the two formulas ¯Xt= (1−t)X0+tX1\nand¯Xt=X0+tX1and any number of update steps N. Our experiments with the fifth-degree and\nthe VP-SDE-like processes also demonstrate that using vθ(xt, t)leads to better results than using\nx1,θ(xt, t), as depicted in Figs. 20, 21, although the performance gaps in these cases are not as\nsignificant as in the case of the third-degree process. These results suggest that it is better to learn\nvθ(xt, t)than learn x1,θ(xt, t). A possible reason is that for different samples xtcomputed from\nthe same x1and different x0during training, vθ(xt, t)will be matched to different target velocities\nvϕ(x0, x1, t) = ˙atx0+ ˙σtx1while x1,θ(xt, t)will be matched to the same target sample x1. This\nmeans vθ(xt, t)tends to be more adaptive to the change of xtand produce more accurate outputs\nthanx1,θ(xt, t). We also observed the same phenomenon for the SN flow, i.e., modeling the SN\nflow using vθ(xt, t)achieves better results than using x1,θ(xt, t), as shown in Fig. 22.\n53N\nSC (v)\nSC (x1,)\nSC (v)\nSC (x1,)\n5\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n50\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n1000\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n¯Xt= (1−t)X0+tX1¯Xt=X0+tX1\nFigure 21: Generated samples (left), xt-trajectories (middle) and ¯xt-trajectories (right) of the SC\nflows of the VP-SDE-like process, with different numbers of update steps N, different SC types (two\nequations in the bottom), and different types of models ( vθvsx1,θ).\nN\nSN (v)\nSN (x1,)\nSN (v)\nSN (x1,)\n5\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n50\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n1000\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n10\n 5\n 0 5 10 15 20 25 3010\n5\n051015202530\n¯Xt= (1−t)X0+tX1¯Xt=X0+tX1\nFigure 22: Generated samples (left), xt-trajectories (middle) and ˜xt-trajectories (right) of the SN\nflows of the third-degree process, with different numbers of update steps N, different SC types (two\nequations in the bottom), and different types of models ( vθvsx1,θ).\nE.2 Text-to-Image Generation with Stable Diffusion\nFig. 23 shows the convergence errors of our solvers and baselines when the number of function\nevaluations (NFE) increases from 10 to 200. It is evident that all solvers converge as the NFE\nincreases and higher-order solvers tend to converge faster than lower-order counterparts.\nIn Figs. 24, 25, 26, we display images generated by our solvers alongside those generated by the\nbaseline solvers, using COCO captions as input prompts. Notably, images generated by first-order\nsolvers, except DDIM, are identical. However, there are slight content variations in images generated\nby our high-order solvers compared to those from the baseline solvers, though their overall quality\nremains quite similar.\n541020 50 100 200\nNFE0.010.020.030.04MSEDDIM\nDPM-1S\nSC-EulerDEIS-1\nDPM-1M\n1020 50 100 200\nNFE0.000.010.020.03MSEDPM-2S\nDEIS-2\nSC-AB1AM2\nSC-AB2AM2DPM-2M\nUniPC-1\nSC-AB2\n1020 50 100 200\nNFE0.000.010.020.030.040.05MSEDPM-3S\nDEIS-3\nSC-AB2AM3\nSC-AB3AM3DPM-3M\nUniPC-2\nSC-AB3(a) First-order (b) Second-order (c) Third-order\nFigure 23: MSEs between 1000 512 ×512 images generated by a 1000-step DDIM and those generated\nby our proposed solvers (SC-*) and baseline solvers. The model is Stable Diffusion and the text\nprompts are COCO2014 captions. DPM stands for DPM-Solver++.\nReal Image DDIM* DDIM DEIS-1 DPM-Solver++ (1M) SC-Euler\n“Two large colorful birds touch their beaks together.”\n“A man jumping a brown horse over an obstacle.”\n“A kitchen with a center island next to a refrigerator freezer.”\nFigure 24: Real images (first column) and images generated by first-order numerical solvers with\nNFE=10. The model is Stable Diffusion and the text prompts are COCO2014 captions.\nDEIS-2 DPM-Solver++ (2M) UniPC-1 SC-AB2 SC-AB1AM2 SC-AB2AM2\n“Two large colorful birds touch their beaks together.”\n“A man jumping a brown horse over an obstacle.”\n“A kitchen with a center island next to a refrigerator freezer.”\nFigure 25: Images generated by second-order numerical solvers with NFE=10. The model is Stable\nDiffusion and the text prompts are COCO2014 captions.\n55DEIS-3 DPM-Solver++ (3M) UniPC-2 SC-AB3 SC-AB2AM3 SC-AB3AM3\n“Two large colorful birds touch their beaks together.”\n“A man jumping a brown horse over an obstacle.”\n“A kitchen with a center island next to a refrigerator freezer.”\nFigure 26: Images generated by third-order numerical solvers with NFE=10. The model is Stable\nDiffusion and the text prompts are COCO2014 captions.\n56"    },    {        "title": "2402.03030v1.Rejection_Sampled_Universal_Quantization_for_Smaller_Quantization_Errors.pdf",        "content": "1\nRejection-Sampled Universal Quantization for\nSmaller Quantization Errors\nChih Wei Ling and Cheuk Ting Li\nDepartment of Information Engineering\nThe Chinese University of Hong Kong, Hong Kong, China\nEmail: chihweiLing@link.cuhk.edu.hk, ctli@ie.cuhk.edu.hk\nAbstract\nWe construct a randomized vector quantizer which has a smaller maximum error compared to all known lattice quantizers\nwith the same entropy for dimensions 5, 6, ..., 48, and also has a smaller mean squared error compared to known lattice quantizers\nwith the same entropy for dimensions 35, ..., 48, in the high resolution limit. Moreover, our randomized quantizer has a desirable\nproperty that the quantization error is always uniform over the ball and independent of the input. Our construction is based\non applying rejection sampling on universal quantization, which allows us to shape the error distribution to be any continuous\ndistribution, not only uniform distributions over basic cells of a lattice as in conventional dithered quantization. We also characterize\nthe high SNR limit of one-shot channel simulation for any additive noise channel under a mild assumption (e.g., the AWGN\nchannel), up to an additive constant of 1.45 bits.\nI. I NTRODUCTION\nEntropy-constrained quantization has been extensively studied in the past decades. For scalar quantization, it was shown by\nGish and Pierce [1] that the uniform quantizer, followed by entropy coding, is asymptotically optimal under certain assumptions.\nThey also compared the entropy of the uniform quantizer with the rate-distortion function [2] for some difference distortion\nmeasures. In [3] (also see [4]), Zador studied the trade-off between the distortion and the rate for vector quantizers, for\nboth fixed-rate quantizers and entropy-constrained quantizers. Gersho [5] generalized the study in [1] to vector quantizers and\nconjectured that the tessellating quantizers, which include lattice quantizers, are asymptotically optimal. Chou et al. [6] studied\nentropy-constrained vector quantizers by formulating it as a Lagrangian model similar to Lloyd’s [7] and designed a greedy\nalgorithm to find the local optimal vector quantizer.\nA randomized quantizer is a quantizer that depends on a random state independent of the input signal. A popular example\nissubtractive dithering [8]–[11]. Entropy-constrained quantization is often used in conjunction with randomized quantizers,\nsuch as dithered quantization. Notably, it was shown in the works on universal quantization by Ziv [12] and Zamir and Feder\n[13] that dithered quantization has a (conditional) entropy within a constant redundancy away from the rate-distortion function\nfor difference distortion measures. See [14], [15] for further analyses on universal quantization. Dithered quantization was\ncombined with channel simulation techniques to create a nonuniform error distribution in [16].\nThe maximum error (or maximum distortion) of a quantizer (i.e., the largest possible magnitude of the error) has also\nbeen studied. For lattice quantizers, the maximum error is given by the covering radius [17]. The maximum error has also\nbeen studied in epsilon entropy [18], [19] and D-semifaithful codes [20]–[23]. Notably, D-semifaithful codes are usually\nentropy-constrained (variable-length) instead of fixed-length to ensure that the distortion is bounded almost surely.\nIn this paper, we construct a randomized vector quantizer, called rejection-sampled universal quantizer (RSUQ), based on\napplying rejection sampling on top of a universal quantizer, in order to guarantee that the error follows an arbitrary uniform\ndistribution or an arbitrary continuous distribution (using the layered construction in [24], [25]). Main contributions:\n•The RSUQ, with an error distribution uniform over the n-ball, has a smaller maximum error compared to all known lattice\nquantizers with the same entropy for dimensions n= 5,6, . . . , 48, showing that RSUQ performs better than known lattice\nquantizers as a D-semifaithful code, and it also has a smaller mean squared error compared to known lattice quantizers\nwith the same entropy for dimensions n= 35, . . . , 48, in the high resolution limit.\n•The RSUQ has a redundancy at most (log2e)/nfrom the rate-distortion lower bound for a given maximum error in the\nhigh resolution limit, smaller than the O((log2n)/n)redundancy for conventional lattice quantizers [17].\n•We also prove that RSUQ has a similar universal quantization property as [12], [13], in the sense that it has an entropy\nwithin a constant gap away from the rate-distortion function regardless of the source distribution.\n•We also study the entropy required to ensure that the quantization error is precisely a prescribed continuous distribution\n(i.e., we are simulating an additive noise channel [25]–[27]), where we characterize the optimal entropy in the “high SNR\nlimit” (the range of the input signal tends to infinity) within a constant gap of log2e.arXiv:2402.03030v1  [cs.IT]  5 Feb 20242\nA. Related Works\nRandomized compression where the reconstruction follows a prescribed conditional distribution given the source is referred\naschannel simulation [26], [28], [29]. RSUQ in this paper can be regarded as a one-shot channel simulation scheme for\nsimulating an additive noise channel with an arbitrary continuous noise distribution. Compared to existing one-shot channel\nsimulation schemes [27], [30] with a logarithmic optimality gap, RSUQ only has a constant optimality gap (Corollary 14).\nAdditive noise channel simulation has applications to machine learning [31]–[33] and differential privacy [34]–[38], and RSUQ\nmay be applied to these areas.\nA related work is [16], where dithered quantization was combined with ordered random coding (which combines two channel\nsimulation techniques: minimal random coding [31] and Poisson functional representation [27], [39]) to create a nonuniform\nerror distribution. Our construction combines dithered quantization with rejection sampling andlayered quantization [24], [25]\ninstead. Unlike [16] which focuses on channel simulation, our proposed scheme also works well as a compression scheme,\nensuring lower error than known lattice quantizers for some situations.\nA recent work [40] also studied a construction, called shift-periodic quantizers , which can ensure that the error is uniform\nover any prescribed set. The main difference between [40] and our construction is that, to ensure a uniform error over A ⊆Rn,\n[40] decomposes a basic cell of a lattice of the same volume as Ainto infinitely many pieces to form A, whereas the RSUQ\nin this paper starts with a basic cell containing Aand performs rejection sampling. Compared to [40], RSUQ is significantly\nsimpler to implement,1and requires a smaller entropy for every dimension n≥2(see Figures 1 and 2).\nNotations\nEntropy is in bits, and logarithms are to the base 2. Write H(X)for the entropy, and h(X)for the differential entropy. For\nA,B ⊆Rn,γ∈R,G∈Rn×n,z∈Rnwrite γA:={γx:x∈ A} ,GA:={Gx:x∈ A} ,A+z:={x+z:x∈ A} ,\nA+B={x+y:x∈ A,y∈ B} for the Minkowski sum, A − B ={x−y:x∈ A,y∈ B} , and µ(A)for the Lebesgue\nmeasure of A. Let Bn:={x∈Rn:∥x∥ ≤1}be the unit n-ball, and let κn:=µ(Bn) =πn/2/Γ(n/2 + 1) be the volume\nof the n-ball.\nII. R ANDOMIZED QUANTIZERS\nThe definition of a randomized quantizer, which includes dithered quantization [8], [9], [12], [13] and distribution preserving\nquantization [41] as special cases, is given below.\nDefinition 1: Arandomized quantizer is a pair (PS, Q), where PSis a distribution over S(the state distribution ),Q:\nRn× S → Rnis the quantization function satisfying that the set Q(Rn, s) ={Q(x, s) :x∈Rn}is finite or countable for\nevery s∈ S. In particular, if PSis a degenerate distribution, we call this a deterministic quantizer .\nOperationally, we first generate the random state S∼PSwhich is available to both the encoder and the decoder. The\nencoder observes Sand a random input X∈Rn, computes Y=Q(X, S), and compresses Yin a variable-length sequence\nof bits and sends it to the decoder, which is possible since Y∈Q(Rn, S)has at most countably many choices given S. For\nexample, the encoder can compress Yusing a Huffman code [42] conditional on S. The decoder can then use Sto decode\nthe sequence of bits and recover Y. One can also view (Q(·, s))s∈Sas an ensemble of quantizer (each Q(·, s) :Rn→Rnis\na quantizer), and one of the quantizers is selected at random (see [41]).\nThe Huffman encoding of Q(X, S)conditional on Srequires an expected length approximately H(Q(X, S)|S). Therefore,\nas in the works on universal quantization [12], [13], the conditional entropy H(Q(X, S)|S)measures the average number of\nbits needed. We are particularly interested in the growth of H(Q(X, S)|S)when Xis uniform over a large set. The following\nis a straightforward generalization of the quantity studied in [25] to Rn, which also coincides with the normalized entropy\nconsidered in [40] for the special case of shift-periodic quantizers.\nDefinition 2: The normalized entropy of a randomized quantizer (PS, Q)is defined as\nH(Q) := lim sup\nτ→∞\u0000\nH(Q(Xτ, S)|S)−logµ(τBn)\u0001\n,\nwhere Xτ∼Unif( τBn)is independent of S∼PS.\nThe normalized entropy can also be regarded as a high resolution limit [5], [13]. If we quantize X∼Unif( aBn)by a\nscaled version of the quantizer, i.e., Yτ=τ−1Q(τX, S), then lim supτ→∞\u0000\nH(Yτ|S)−logµ(τaBn)\u0001\n=H(Q).\nNote that the normalized entropy may be negative. Loosely speaking, when X∼Unif(X)is uniform over a large set X,\nwe expect H(Q(X, S)|S)≈H(Q) + log µ(X).\n1The construction in [40], even for n= 2, is already quite hard to characterize.3\nThe error of the quantizer is Q(X, S)−X, and the magnitude of the error is ∥Q(X, S)−X∥. The maximum error of a\nrandomized quantizer is the largest possible (magnitude of) error it can make:\nsup\nx∈Rn, s∈S\r\rQ(x, s)−x\r\r. (1)\nThe following lower bounds of the normalized entropy can be obtained easily from standard rate-distortion arguments. The\nproofs are included in Appendix A for completeness.\nProposition 3 (Rate-distortion lower bounds): For any randomized quantizer (PS, Q), we have the following lower bounds\non the normalized entropy:\n•If the maximum error (1) is at most r, then H(Q)≥ −nlogr−logκn. (Recall that κn:=µ(Bn).)\n•If the mean squared error satisfies (let Xτ∼Unif( τBn))\nlim supτ→∞E[∥Q(Xτ, S)−Xτ∥2]≤D,\nthenH(Q)≥ −(n/2) log(2 πeD/n ).\nGiven the rate-distortion lower bounds, we can define the redundancy of a randomized quantizer Qin a similar manner\nas [13]. The redundancy with respect to maximum error (per dimension) is\n(1/n)H(Q) + log r+ (1/n) logκn, (2)\nwhich must be nonnegative, where ris the maximum error (1) of Q. The redundancy w.r.t. MSE (per dimension) is\n(1/n)H(Q) + (1 /2) log(2 πeD/n ), (3)\nwhere D= lim supτ→∞E[∥Q(Xτ, S)−Xτ∥2].\nIII. L ATTICE QUANTIZERS\nWe review the basics of lattices, which can be found in [43], [44]. Given a full-rank generator matrix G∈Rn×nof a lattice\nGZn={Gj:j∈Zn}, its packing radius is\nλ(G) := sup\b\nλ≥0 : (λBn+Gj)j∈Znare disjoint\t\n=1\n2min\nj∈Zn\\{0}∥Gj∥, (4)\nand its covering radius is\nλ(G) := inf\b\nλ≥0 :λBn+GZn=Rn\t\n. (5)\nItspacking density δ(G)andcovering density Θ(G)are\nδ(G) :=(λ(G))nκn\n|detG|,Θ(G) :=(λ(G))nκn\n|detG|. (6)\nA bounded set P ⊆Rnis called a basic cell of the lattice GZnif(P+Gj)j∈Znforms a partition of Rn[43], [44]. This\nimplies that µ(P) =|detG|. In particular, the V oronoi cell V:={x∈Rn: arg min j∈Zn∥x−Gj∥=0}(break ties in\nlexicographical order of j) is a basic cell. Given a basic cell P, we can define a deterministic quantizer QP:Rn→Rn,\nQP(x) =ywhere y∈GZnis the unique lattice point such that x∈ −P +y. The normalized entropy (Definition 2) of\nQPis−log|detG|=−logµ(P).2Conventionally, we usually take Pto be the V oronoi cell V. The resultant deterministic\nquantizer QVhas a maximum error λ(G). The redundancy w.r.t. maximum error (2) is3\n−1\nnlogµ(V) + log λ(G) +1\nnlogκn\n=1\nnlog Θ( G)bits / dimension . (7)\nIt was shown in [17], [45] that there is a sequence of lattices with\n1\nnlog Θ( G)≤logn\nn+ (log2√\n2πe)log log n\nn+O\u00101\nn\u0011\n, (8)\nand hence the redundancy w.r.t. maximum error for the optimal lattice quantizer is O((log n)/n). This scaling order is precise\nfor the optimal lattice quantizer since it was shown in [46] that the smallest possible covering density Θ(G)is at least\n(1−o(1))e−3/2nasn→ ∞ , implying that (1/n) log Θ( G)is at least in the order (logn)/n.\n2This follows from Theorem 6 by substituting A=P.\n3The redundancy w.r.t. maximum error n−1log Θ( G)for a lattice quantizer coincides with the logarithm of the covering efficiency [44].4\nWe also study the mean squared error (MSE). For a random signal Xuniform over a large set, the MSE E[∥QV(X)−X∥2]\nis close to the second moment of the lattice [3]\n1\nµ(V)ˆ\nV∥x∥2dx. (9)\nThe normalized second moment [44] is often studied:\nGn(V) :=´\nV∥x∥2dx\nn(µ(V))1+2/n, (10)\nwhich is invariant under scaling ( Gn(V) =Gn(γV)forγ >0). The redundancy w.r.t. MSE (3) can be computed as\n1\n2log\u0000\n2πeG n(V)\u0001\nbits / dimension , (11)\nwhich is also the high resolution limit of the redundancy of universal quantization [13]. Zador’s bound [3], [4] implies that\nthe optimal lattice quantizer has a redundancy w.r.t. MSE\nlogn\n2n+O\u00101\nn\u0011\nbits / dimension . (12)\nGiven a basic cell P, we can also construct the subtractively dithered quantizer [13], which is a randomized quantizer with\nrandom state S=V∼Unif(P), and quantization function Q(x,v) =QP(x−v) +v, where QPis the deterministic lattice\nquantizer described earlier. A useful property of dithered quantizer is that the quantization error is always uniform over Pand\nindependent of the input, i.e., Q(x,V)−x∼Unif(P)for every x∈Rn(see [44], [47]). Therefore, when Pis the V oronoi\ncellV, the MSE E[∥Q(X,V)−X∥2]exactly equals the second moment (9) regardless of the distribution of X. Nevertheless,\nthe dithered quantizer still has a normalized entropy −log|detG|, and a maximum error λ(G)whenPis the V oronoi cell\nV. The dithered quantizer does not provide any advantage in terms of normalized entropy and maximum error compared to\ndeterministic lattice quantizers.\nIV. R EJECTION -SAMPLED UNIVERSAL QUANTIZERS\nWe introduce a randomized quantizer with an error uniformly distributed over a set Awhich is a subset of a basic cell,\nbased on applying rejection sampling on top of universal quantization [13]. Intuitively, we keep generating new dither signals\nuntil the error falls in A.\nDefinition 4: Given a basic cell Pof the lattice GZn, and a subset A ⊆ P , the rejection-sampled universal quantizer\n(RSUQ) for Aagainst Pis the randomized quantizer (PS, QA,P), where S= (Vi)i∈N+,V1,V2, . . .iid∼Unif(P)are i.i.d.\ndither signals, and the quantization function is\nQA,P(x,(vi)i) :=QP(x−vk) +vk, (13)\nwhere\nk:= min\b\ni:QP(x−vi) +vi−x∈ A\t\n, (14)\nandQPis the lattice quantizer for basic cell P(Section III).\nNote that subtractive dithering [12], [13] is a special case of RSUQ where A=P.4Although RSUQ requires a sequence\nof dithers V1,V2, . . ., they can practically be generated by sharing a small random seed among the encoder and the decoder,\nwhich can be used to generate the sequence using a pseudorandom number generator (PRNG).\nTo carry out the RSUQ in practice, the encoder and the decoder each has a PRNG initialized with the same seed, so the\noutputs of the two PRNGs will be the same. The encoder observes X∈Rn, keeps generating V1,V2, . . .iid∼Unif(P)from the\nPRNG until iteration Kwhere QP(X−VK)+VK−X∈ A, computes M:=QP(X−VK)∈GZn, and encodes and transmits\n(K,M). The decoder observes (K,M), generates V1, . . . ,VKiid∼Unif(P)from the PRNG, and outputs Y=M+VK. It\nis straightforward to check that the error is uniform over A. The proof is given in Appendix B.\nProposition 5: For any x∈Rn, the quantization error Z:=QA,P(x, S)−xof the RSUQ (PS, QA,P)(where S∼PS)\nfollows the uniform distribution over the set A, i.e., Z∼Unif(A).\nWe now bound the entropy of RSUQ. The proof is given in Appendix C.\n4Technically, RSUQ requires a sequence of dithers V1,V2, . . ., whereas subtractive dithering only requires one dither, though when A=P, only the first\nV1is used.5\nTheorem 6: Consider the RSUQ QA,PforAagainst P. For a random input Xsatisfying X∈ X almost surely for some\nX ⊆Rnwithµ(X)<∞, independent of S∼PS, the conditional entropy of the quantizer satisfies\nH(QA,P(X, S)|S)\n≤logµ(X+A+P − P )−logµ(A)−1−p\nplog(1−p)\n≤logµ(X+A+P − P )−logµ(A) + log e, (15)\nwhere p:=µ(A)/µ(P). Hence, the normalized entropy H(QA,P)is upper-bounded by\n−logµ(A)−1−p\nplog(1−p)≤ − logµ(A) + log e.\nThe proof suggests that, operationally, if we know that X∈ X, then we should compress Kusing the optimal prefix-free\ncode for Geom( p), and then compress M:=QP(X−VK)∈ M := (X+A − P )∩GZnusing⌈log|M|⌉ number of bits.\nThis encoding scheme has an advantage that the construction does not depend on the distribution of X, and has the same\nexpected length regardless of the distribution of X.\nIn particular, we are interested in the RSUQ for the ball rBn. In this case, the MSE is exactly nr2/(n+ 2) regardless\nof the distribution of X, unlike deterministic lattice quantizers where the MSE is the second moment (9) only for Xthat is\nsufficiently spread out. We have the following corollary of Theorem 6. The proof is in Appendix D.\nCorollary 7: Fix any r >0and generator matrix G∈Rn×n. There exists an RSUQ for the ball rBnagainst the V oronoi\ncellVof some lattice, with normalized entropy\nH(Q)≤ −nlogr−logκn−1−δ(G)\nδ(G)log(1−δ(G)) (16)\n≤ −nlogr−logκn+ log e. (17)\nwhere δ(G)is the packing density (6). Since this RSUQ has an error distribution Unif( rBn), it has a maximum error rand\nMSEE[∥Q(X, S)−X∥2] =nr2/(n+ 2) .\nWhile the upper bound (16) depends on the packing density δ(G)of the lattice, the upper bound (17) does not depend on\nthe lattice. Therefore, the tradeoff between maximum error (or MSE) and normalized entropy of RSUQ is not sensitive to the\nchoice of the lattice, in stark contrast to conventional lattice quantizers where the lattice must be designed carefully. Moreover,\nfor RSUQ, we can choose a lattice with a fast and simple quantization algorithm, e.g., a root lattice [48]. In contrast, for a\ngood conventional lattice quantizer, a fast algorithm is often not easy to construct (e.g., [49]).\nComparing the upper bound in (17) and the lower bounds in Proposition 3, we can see that the RSUQ has a redundancy\nw.r.t. maximum error (2) (per dimension) at most\nloge\nnbits / dimension ,\nsignificantly smaller than the O((log n)/n)redundancy for the optimal lattice quantizer in (8). It has a redundancy w.r.t.\nMSE (3) at most\nlogn\n2n+1.4\nnbits / dimension ,\n(see Appendix E), the same scaling as the optimal lattice quantizer (12), but is significantly easier to construct due to the\ninsensitivity to the choice of the lattice.\nFigure 1 plots the following for dimension n= 2, . . . , 48:\n•The redundancy w.r.t. maximum error (2) of lattice quantizer, using the best known sphere covering lattice (with Θ(G)\ngiven in [50]), computed using (7).5\n•The redundancy of shift-periodic quantizers [40], which can also ensure an error distribution Unif( Bn).6\n•The upper bound on the redundancy w.r.t. maximum error of RSUQ with error distribution Unif( Bn)computed using\n(16) and the best known lattice for sphere packing (with δ(G)given in [51]).\n•The general upper bound (loge)/non the redundancy w.r.t. maximum error of RSUQ with error distribution Unif( Bn)\nby (17), which applies to any lattice.\nFigure 1 highlights two advantages of RSUQ:\n•RSUQ (with the best lattice for sphere packing) achieves a smaller redundancy (and hence a smaller normalized entropy\nfor the same maximum error) compared to the best known lattice quantizer for n∈ {5, . . . , 48}.7\n5Note that [50] only gave Θ(G)up to n= 24 . Forn= 25, . . . , 48,Θ(G)is computed using the optimal product lattice among the lattices in [50] which,\nto the best of the authors’ knowledge, are the best known Θ(G).\n6Forn= 2,3, we use the bounds computed in the examples in [40]. For n≥4, we use the general bound in [40, Thm 11 part 1].\n7One objection to this comparison is that if we actually encode the description into bits, then there is an at most 1bit penalty incurred by Huffman coding\n[42] (or a 1/nbit penalty on the redundancy). Note that both the lattice quantizer and RSUQ are subject to this penalty. Moreover, even if we impose this\npenalty on RSUQ and do not impose this penalty on lattice quantizers, RSUQ still outperforms lattice quantizers for n∈ {11, . . . , 48}.6\nFigure 1. Log-scale plot of the redundancy w.r.t. maximum error (2) of the best known lattice quantizer (red line, using (7), with the best known Θ(G)from\n[50]), the shift-periodic quantizer (green line), the RSUQ with error distribution Unif( Bn)constructed using the best known sphere packing lattice (blue\nline), and the RSUQ constructed using an arbitrary lattice (black line).\nFigure 2. Log-scale plot of the redundancy w.r.t. MSE (3) of the best known lattice quantizer (red line), the shift-periodic quantizer (green line), the RSUQ\nwith error distribution Unif( Bn)constructed using the best known sphere packing lattice (blue line), and the RSUQ constructed using an arbitrary lattice\n(black line).\n•Even with an arbitrary lattice, RSUQ achieves a smaller redundancy compared to the best known lattice quantizer for\nn∈ {7, . . . , 48}. RSUQ is insensitive to the choice of lattice, since the gap between the (per dimension) redundancy\nimplied by (16) (which depends on the lattice) and the redundancy implied by (17) (which holds for every lattice) is\nO(1/n)asn→ ∞ . This makes the construction of RSUQ simpler than that of lattice quantizer, where the performance\nof the latter is highly sensitive to the construction of the lattice, which is often not easy.\nWe also carry out the same comparison for mean squared error instead of maximum error in Figure 2, i.e., we consider the\nredundancy w.r.t. MSE (3) instead of maximum error. We consider the dimensions n= 2, . . . , 48. The redundancy of lattice\nquantizers is obtained from the formula (11), with the best known Gn(V)from [52]. The bounds for RSUQ are computed\nusing (16) and the MSE in Corollary 7. We can see that RSUQ achieves a smaller redundancy compared to the best known\nlattice quantizer (for the same MSE) for n∈ {35, . . . , 48}. While the advantage of RSUQ is less pronounced when MSE is\nconsidered, we emphasize the aforementioned advantage that RSUQ works for any choice of lattice, whereas a lattice with a\nsmall second moment (e.g., [52]) is not easy to construct.7\nV. RSUQ FOR UNIVERSAL QUANTIZATION\nThe RSUQ retains the universal quantization property of [12], [13], i.e., it is an almost-optimal lossy compressor for any\ndistribution of the input X. Now, instead of the squared 2-norm distortion, we may consider a more general difference distortion\nmeasure. Let d:Rn→[0,∞)be a distortion function. The distortion between the input Xand output Yisd(Y−X).\nThe following result is a generalization of the universal quantization result in [13] to RSUQ, showing that the conditional\nentropy of RSUQ is close to the rate-distortion function, regardless of the input distribution. It is proved using similar techniques\nas in [13]. The proof is given in Appendix F\nTheorem 8: Consider the rejection-sampled universal quantizer QA,PforAagainst P. For a random input Xfollowing any\ndistribution, independent of S∼PS, the conditional entropy of the quantizer satisfies\nH(QA,P(X, S)|S)≤R(D)−logp+C+ log e,\nwhere p:=µ(A)/µ(P),D:=E[d(Q(X, S)−X)]is the expected distortion of the RSUQ, R(D) := infPˆX|X:E[d(ˆX−X)]≤DI(X;ˆX)\nis the rate-distortion function, and\nC:= sup\nPT:E[d(−T)]≤DI(T;T+Z),\nwhere the supremum is over PT(letT∼PT) satisfying E[d(−T)]≤D, andZ∼Unif(A).\nFor squared error distortion d(x) =∥x∥2,Cis bounded in terms of the normalized second moment (10) of A(see [13]):\nC≤n\n2log\u0000\n4πeG n(A)\u0001\n.\nVI. N ONUNIFORM ERROR DISTRIBUTION\nWe have presented the RSUQ construction that guarantees a uniform error distribution over a bounded set A. In this section,\nwe generalize the construction to general (uniform or nonuniform) continuous error distributions. This is based on the idea\nin [24], [25] that any continuous distribution can be expressed as a mixture of uniform distributions. Consider a probability\ndensity function f:Rn→[0,∞). Write its superlevel set as\nL+\nt(f) :={x∈Rn:f(x)≥t}.\nLetfT(t) :=µ(L+\nt(f))fort >0, which is also a probability density function. If we generate T∼fT, and then X|{T=\nt} ∼Unif( L+\nt(f)), then we have X∼f.8Therefore, we can express the distribution fas a mixture of uniform distributions\nUnif( L+\nt(f)). We now generalize RSUQ to allow a nonuniform error distribution f. Intuitively, we first generate T∼fT\nrandomly as a part of the random state, and then apply RSUQ for L+\nT(f). This is a generalization of the layered randomized\nquantizer in [25] (which focuses on scalar quantization, i.e., n= 1).\nDefinition 9: Given a basic cell Pof the lattice GZn, and a probability density function f:Rn→[0,∞)where L+\nt(f)\nis always bounded for t >0, and β: (0,∞)→[0,∞)satisfying L+\nt(f)⊆β(t)Pfort >0, the layered rejection-sampled\nuniversal quantizer (LRSUQ) for fagainst Pis the randomized quantizer (PS, Qf,P), where S= (T,(Vi)i∈N+),T∼fT\nwhere fT(t) :=µ(L+\nt(f)), andV1,V2, . . .iid∼Unif(P)is a sequence of i.i.d. dither signals. The quantization function is\nQf,P(x, t,(vi)i) :=β(t)·\u0000\nQP(x/β(t)−vk) +vk\u0001\n,\nwhere\nk:= min\b\ni:β(t)·\u0000\nQP(x/β(t)−vi) +vi\u0001\n−x∈L+\nt(f)\t\n, (18)\nandQPis the lattice quantizer for basic cell P(Section III).9\nFirst, we show that the LRSUQ indeed gives the desired error distribution.\nProposition 10: Consider the layered rejection-sampled universal quantizer Qf,P. For any random input X, the quantization\nerrorZ:=Qf,P(X, T,(Vi)i)−Xfollows the pdf f, independent of X.\nProof: IfT=tis fixed and (Vi)i∈N+is still random, then the LRSUQ become the RSUQ in Definition 4 with the basic\ncellβ(t)P. We have Z|{T=t} ∼Unif( L+\nt(f))independent of Xby Proposition 5. Hence, Z∼fby the fundamental\ntheorem of simulation [53] (footnote 8).\nThe LRSUQ can also be regarded as a one-shot channel simulation scheme [26], [27], [30], whether the encoder observes\nX, and sends a sequence of bits to the decoder (which can share common randomness with the encoder) to allow it to generate\n8The fundamental theorem of simulation [53] states that if we generate (X, T)∼Unif({(x, t) : 0≤t≤f(x)}), then X∼f. Here T∼fT, and the\nconditional distribution of Xgiven T=tisUnif( L+\nt(f)). Hence, we can generate Tfirst, and then generate Xgiven T.\n9Since β(T)>0almost surely, division by zero will not occur in x/β(t).8\nYfollowing a prescribed conditional distribution PY|Xgiven X. LRSUQ is a one-shot channel simulation scheme for the\nadditive noise channel Y=X+Zwhere Z∼f.\nTo study the normalized entropy of LRSUQ, we utilize the following quantity introduced in [25].\nDefinition 11 ( [25]): Given a probability density function f:Rn→[0,∞), its layered entropy is\nhL(f) :=ˆ∞\n0µ(L+\nt(f)) log µ(L+\nt(f))dt.\nThe layered entropy has the following alternative characterization (the one-dimensional case has been used as an intermediate\nstep in [25]). The proof is given in Appendix G.\nTheorem 12: We have\nhL(f) = sup h(X|T)\nwhere X∼f, and the supremum is over conditional distributions PT|X(letT|X∼PT|X) satisfying that the conditional\ndistribution PX|T(·|t)is uniform (i.e., PX|T(·|t)isUnif(At)for some At⊆Rnwith0< µ(At)<∞) for PT-almost all t.\nMoreover, the supremum is attained when (X, T)∼Unif({(x, t) : 0≤t≤f(x)}). As a result, hL(f)≥h(f).\nIt was shown in [25] that, for a one-dimensional error distribution f:R→[0,∞)that is unimodal (i.e., L+\nt(f)is always\nconnected), among randomized quantizers that guarantees an error distribution f, the smallest possible normalized entropy is\ngiven by −hL(f). The optimum is achieved by the layered randomized quantizer [25]. In this paper, we will show that for\na general number of dimensions n, and a general continuous distribution foverRn(not necessarily unimodal), the smallest\npossible normalized entropy for the error distribution fis still given by −hL(f)within a constant gap of loge. The proof is\ndivided into two parts: the achievability part showing that LRSUQ has a normalized entropy upper-bounded by −hL(f)+log e,\nand the converse part showing that no randomized quantizer with error distribution fcan have a normalized entropy smaller\nthan−hL(f). The proof is given in Appendix H.\nTheorem 13: Consider the layered rejection-sampled universal quantizer Qf,P. For a random input Xsatisfying X∈ X\nalmost surely for some X ⊆Rnwithµ(X)<∞, independent of S∼PS, the conditional entropy of the quantizer satisfies\nH(Qf,P(X, S)|S)\n≤ET\u0002\nlogµ(X+ 3ηβ(T)Bn)\u0003\n−hL(f) + log e, (19)\nwhere T∼fTwithfT(t) :=µ(L+\nt(f)),η:= supx∈P∥x∥, and β(t)is given in Definition 9. Hence, under the mild assumption\nthatE[log(1 + β(T))]<∞, the normalized entropy is upper-bounded by\n−hL(f) + log e.\nNote that we can also study the worst-case normalized entropy\nlim sup\nτ→∞\u0010\nsup\nPX:∥X∥≤τH(Q(X, S)|S)−logµ(τBn)\u0011\n, (20)\nwhich concerns the worst-case conditional entropy over all distributions of XoverτBn(instead of only X∼Unif( rBn)as\nin the original normalized entropy in Definition 2). Theorem 13 establishes that the worst-case normalized entropy of LRSUQ\nis also at most −hL(f) + log e(under the assumption E[log(1 + β(T))]<∞).\nAs a result, we can construct an LRSUQ for an arbitrary continuous error distribution fwith normalized entropy at most\n−hL(f) + log e, under a mild assumption on f.10The proof is given in Appendix I.\nCorollary 14: If a probability density function foverRnsatisfies\nˆ∞\n0\u0010\nsup\nz:∥z∥≥xf(z)\u0011\nxn−1log(1 + x)dx <∞, (21)\nthen there exists an LRSUQ Qf,Pforfwith normalized entropy upper-bounded by −hL(f) + log e.\nWe then show the converse part, which generalizes the scalar randomized quantization converse result in [25] to vector\nquantization. The proof is given in Appendix J.\nTheorem 15: If a randomized quantizer satisfies that the error Q(X, S)−Xhas a probability density function fwhen\nX∼Unif(X)for a certain set X ⊆Rn,0< µ(X)<∞, then we have the following bound on the conditional entropy\nH(Q(X, S)|S)≥logµ(X)−hL(f).\n10The authors are unaware of any naturally occurring distribution fthat violates condition (21).9\nAs a result, if the error Q(x, S)−x∼ffor every x∈Rn, then the normalized entropy of the randomized quantizer is\nlower-bounded by −hL(f).\nCorollary 14 and Theorem 15 establish that, letting H∗(f)be the smallest11normalized entropy among randomized quantizers\nwith error distribution f(i.e.,Q(x, S)−x∼ffor every x), under the assumption (21), we have\n−hL(f)≤H∗(f)≤ −hL(f) + log e. (22)\nUnlike channel coding where the asymptotic capacity of the additive noise channel X→X+Z(with noise Z∼f, under\nthe constraint ∥X∥ ≤r) is approximately logµ(rBn)−h(f)under the “high SNR limit” r→ ∞ , (22) establishes that the\ncommunication cost for simulating the channel X→X+Z(where X∼Unif( rBn))12is approximately logµ(rBn)−hL(f)\nunder the high SNR limit, which is given in terms of the layered entropy hL(f)instead of the differential entropy h(f).\nVII. C ONCLUSION AND DISCUSSIONS\nIn this paper, we studied the rejection-sampled universal quantizer (RSUQ), which has smaller maximum and mean-squared\nerrors compared to currently known lattice quantizers [50], [52] with the same entropy for moderately large dimensions in the\nhigh-resolution limit. Moreover, RSUQ possesses the desirable property, similar to the universal quantizer [12], [13], that the\nquantization error is always uniform over a prescribed set (e.g., a ball) and independent of the input distribution. Furthermore,\ngiven a prescribed nonuniform noise distribution under a regularity condition, RSUQ can be applied to simulate the additive\nchannel with the noise distribution by utilizing the layered construction techniques described in [24], [25], where the entropy\nis shown to be within a constant logeaway from the optimum in the high SNR limit.\nFor potential future directions, it may be of interest to investigate the performance of RSUQ in machine learning [31]–[33]\nand differential privacy [34]–[38] using some particular lattices with fast decoding and quantization algorithms.\nVIII. A CKNOWLEDGEMENTS\nThis work was partially supported by two grants from the Research Grants Council of the Hong Kong Special Administrative\nRegion, China [Project No.s: CUHK 24205621 (ECS), CUHK 14209823 (GRF)].\nAPPENDIX\nA. Proof of Proposition 3\nConsider a random X, and let Y=Q(X, S). Since Xis independent of S, we have\nH(Y|S)≥I(X;Y|S)\n=I(X;Y, S)\n≥I(X;Y)\n=h(X)−h(X|Y)\n≥h(X)−h(X−Y)\n=h(X)−h(Y−X).\nIf the maximum error is at most r(i.e.,∥Y−X∥ ≤r), then\nH(Y|S)≥h(X)−h(X−Y)\n≥h(X)−logµ(rBn)\n=h(X)−nlogr−logκn.\nSubstituting X∼Unif( τBn)gives the lower bound of the normalized entropy for fixed maximum error. If the mean squared\nerror is at most D, since the Gaussian distribution maximizes the differential entropy for a fixed mean squared norm,\nH(Y|S)≥h(X)−h(X−Y)\n≥h(X)−n\n2log2πeD\nn.\nSubstituting X∼Unif( τBn)and taking τ→ ∞ gives the lower bound of the normalized entropy for a given bound on the\nMSE.\n11Technically, H∗(f)is the infimum of the set of normalized entropies of randomized quantizers with error distribution f.\n12We can also study the worst-case cost for simulating the channel X→X+Z, where Xfollows any distribution with ∥X∥ ≤r. The construction in\nTheorem 13 establishes that the cost is still approximately logµ(rBn)−hL(f).10\nB. Proof of Proposition 5\nIn the definition of RSUQ, the dither signals are uniformly distributed over the basic cell P, i.e., Vi∼Unif(P)for every\ni∈N+. Fix any x∈Rn. LetY:=QA,P(x,(Vi)i) =QP(x−VK) +VKas in (13). Let Zi:=Q−P(x−Vi) +Vi−x.\nBy [44, Lemma 4.1.1 and Theorem 4.1.1], we have Zi∼Unif(P)i.i.d. for i∈N+. From (14), K= min {i:Zi∈ A} , and\nhence ZK=Y−x∼Unif(A)by the standard rejection sampling argument.\nC. Proof of Theorem 6\nLetm:=QP(x−vk)∈GZn. By (14), m+vk−x∈ A , and hence m∈ A − vk+x⊆ A − P +X. LetM:=\n(X+A − P )∩GZn. We know m∈ M . We have P+M ⊆ X +A+P − P , and µ(P+M) =|M|µ(P).\nConsider a random X∈ X. Let Kbe defined according to (14), and M:=QP(X−VK). Note that the output QP(X−\nVK) +VKis a function of ((Vi)i, K,M). Therefore H(QA,P(X,(Vi)i)|(Vi)i)≤H(K) +H(M). We have\nH(M) =H(GQP(X−VK))\n≤log|M|\n≤logµ(X+A+P − P )−logµ(P).\nForK, note that K∼Geom( p), where p:=µ(A)/µ(P)is the acceptance probability of the rejection sampling (see the proof\nof Proposition 5). By the entropy of geometric random variable,\nH(K) =−logp−1−p\nplog(1−p)\n=−logµ(A)−1−p\nplog(1−p) + log µ(P)\n≤ −logµ(A) + log e+ log µ(P)\nsince1−p\nplog(1−p)is decreasing for p∈(0,1). The result (15) follows from H(QA,P(X,(Vi)i)|(Vi)i)≤H(K) +H(M).\nFor the bound on the normalized entropy, assume supx∈P∥x∥ ≤η(recall that a basic cell must be bounded) and substitute\nX=τBn, we have µ(X+A+P − P )≤µ((τ+ 3η)Bn) = (1 + o(1))µ(τBn)asτ→ ∞ , completing the proof.\nD. Proof of Corollary 7\nWe construct an RSUQ for the ball rBnagainst the V oronoi cell Vof some lattice γGZn(where γ >0is some scaling\nfactor). To ensure rBn⊆ V , we need the radius of the ball to be not greater than the packing radius of the lattice, i.e.,\nr≤λ(γG) =γλ(G). Hence we can take γ=r/λ(G). This RSUQ guarantees an error distribution Y−x∼Unif( rBn),\nand hence has a maximum error r.\nInvoking Theorem 6, the normalized entropy is upper-bounded by\n−nlogr−logκn−1−δ(G)\nδ(G)log(1−δ(G)).\nThis is because µ(rBn) =rnκnand\nµ(rBn)\nµ(V)=rnκn\n|det(r/λ(G))G|=δ(G).\nNote that the MSE E[∥QA,P(X, S)−X∥2](for any distribution of X) of the RSUQ for the ball rBnis\n1\nµ(rBn)ˆ\nrBn∥x∥2dx\n=r21\nκnˆ\nBn∥x∥2dx\n=r21\nκnˆ\nSn−1ˆ1\n0tn−1∥tx∥2dtdx\n=r21\nκnˆ\nSn−1ˆ1\n0tn+1dtdx\n=nr2\nn+ 2, (23)\nwhere Sn−1:={x∈Rn:∥x∥= 1}is the (n−1)-dimensional sphere. An upper bound of the normalized entropy for unit\nMSE can be obtained by substituting r=p\n1 + 2 /n(so that the MSE is 1) into (16).11\nE. Bounding the Redundancy w.r.t. MSE\nFrom Corollary 7, the MSE of RSUQ isnr2\nn+2. Fixing the MSE to D, we have\nr=r\nD(n+ 2)\nn.\nBy Corollary 7, the normalized entropy is\n−nlogr\nD(n+ 2)\nn−logκn+ log e\n=−n\n2logD(n+ 2)\nn−logπn/2\nΓ(n/2 + 1)+ log e\n=−n\n2log(πD)−n\n2log\u0000\n1 +2\nn\u0001\n+ log Γ( n/2 + 1) + log e\n≤ −n\n2log(πD)−n\n2log\u0000\n1 +2\nn\u0001\n+ log Γ( n/2 + 1) + log e\n(a)\n≤ −n\n2log(πD)−n\n2log\u0000\n1 +2\nn\u0001\n+ log√πn+n\n2logn\n2e\n+1\n6nloge+ log e\n=−n\n2log2πeD\nn−n\n2log\u0000\n1 +2\nn\u0001\n+ log√πn\n+1\n6nloge+ log e\n≤ −n\n2log2πeD\nn+1\n2logn+ 1.4\nforn≥2, where (a) is by Stirling’s approximation [54]. Comparing with the −(n/2) log(2 πeD/n )lower bound in Proposi-\ntion 3, we know that the redundancy w.r.t. MSE (per dimension) is at most (1/2)(log n)/n+ 1.4/n.\nF . Proof of Theorem 8\nThe arguments in the proof are mostly similar to [13], which are included here for the sake of completeness. Let Y=\nQP(X−VK) +VKbe the output. Let Z=Y−X. Note that Z∼Unif(A)is independent of X. Fix any PˆX|Xsatisfying\nE[d(ˆX−X)]≤D. Assume ˆX|X∼PˆX|X, and (X,ˆX)is independent of Z. We have\nH(Y|(Vi)i)≤H(Y|VK) +H(VK|(Vi)i)\n(a)\n≤H(Y|VK) +H(K)\n(b)=I(X;Y|VK) +H(K)\n≤I(X;Y,VK) +H(K)\n(c)=I(X;Y) +H(K)\n≤I(X;Y,ˆX) +H(K)\n=I(X;ˆX) +I(X;Y|ˆX) +H(K),\nwhere (a) is because H(VK|(Vi)i, K) = 0 , (b) is because H(Y|X,VK) = 0 , (c) is because H(VK|Y) = 0 sinceVKis the\nunique element in (Y+GZn)∩ P. We have\nI(X;Y|ˆX) =h(Y|ˆX)−h(Y|X,ˆX)\n=h(Y−ˆX|ˆX)−h(Z)\n≤h(Y−ˆX)−h(Z)\n=h(X−ˆX+Z)−h(Z)\n=I(X−ˆX;X−ˆX+Z)\n≤C\nsinceE[d(ˆX−X)]≤D. Also note that H(K) =−logp−1−p\nplog(1−p)≤ − logp+ log eas shown in the proof of\nTheorem 6. This completes the proof.12\nG. Proof of Theorem 12\nWe show that for all jointly distributed (Z, W)where Z∼fandPZ|W(·|w) = Unif( Aw)for some Aw⊆Rnwith\n0< µ(Aw)<∞, we have\nh(Z|W) =E[logµ(AW)]≤hL(f).\nLet(Z, T)∼Unif({(z, t) : 0≤t≤f(z)}). We have PZ|T(·|t) = Unif( L+\nt(f))andh(Z|T) =hL(f).\nWe first show the following claim: for every γ≥0,\nEh\nmaxn\n1−γ\nµ(AW),0oi\n≤Eh\nmaxn\n1−γ\nµ(L+\nT(f)),0oi\n. (24)\nTo show this claim, let C ⊆Rnbe such that µ(C) =γandinfz∈Cf(z)≥supz∈Rn\\Cf(z).13We have\nE[max{1−γ/µ(AW),0}]\n=E\u0014max{µ(AW)−γ,0}\nµ(AW)\u0015\n(a)\n≤E\u0014µ(AW\\C)\nµ(AW)\u0015\n=E\u0002\nPZ|W(AW\\C |W)\u0003\n= 1−E\u0002\nPZ|W(C |W)\u0003\n= 1−PZ(C).\nEquality in (a) holds if µ(AW\\C) = max {µ(AW)−γ,0}, or equivalently, µ(AW\\C) = 0 orµ(C\\A W) = 0 almost surely.\nThis holds when AW=L+\nt(f)for some tsince infz∈Cf(z)≥supz∈Rn\\Cf(z), which is the case for T. Therefore, E[max{1−\nγ/µ(L+\nT(f)),0}] = 1−´\nCf(z)dz, and the claim (24) is proved.\nFor0< β < α ,\n(loge)ˆ∞\nβ1\nγmax{1−γ/α, 0}dγ\n= (log e)ˆ∞\nβmax{1/γ−1/α,0}dγ\n= (log e)ˆα\nβ(1/γ−1/α)dγ\n= (log e)\u0010\nlnα\nβ+β\nα−1\u0011\n= logα\nβ−(loge) maxn\n1−β\nα,0o\n.\nHence, for any α, β > 0,\nmax{log(α/β),0}\n= (log e)\u0012\nmaxn\n1−β\nα,0o\n+ˆ∞\nβ1\nγmaxn\n1−γ\nα,0o\ndγ\u0013\n.\nApplying this to (24), we have, for any β >0,\nEh\nmaxn\nlogµ(AW)\nβ,0oi\n≤Eh\nmaxn\nlogµ(L+\nT(f))\nβ,0oi\n,\nand hence\nEh\nmaxn\nlogµ(AW),logβoi\n≤Eh\nmaxn\nlogµ(L+\nT(f)),logβoi\n.\nTaking β→0, we have E[logµ(AW)]≤E[logµ(L+\nT(f))].\n13We can find Cas follows. Let t0= inf{t:µ(L+\nt(f))≤γ}. Although taking C=L+\nt0(f)might not work since t7→µ(L+\nt(f))might be discontinuous,\nsince µ(S\nt>t0L+\nt(f))≤γandµ(L+\nt0(f))≥γ, we can still take C=S\nt>t0L+\nt(f)∪(L+\nt0(f)∩rBn), where r≥0is chosen such that µ(C) =γ.13\nH. Proof of Theorem 13\nSince LRSUQ can be regarded as applying RSUQ for L+\nT(f)against β(t)Pwith a random T, we can invoke Theorem 6\nto obtain\nH(Qf,P(X, S)|S)\n=Eh\nlogµ(X+L+\nT(f) +β(t)P −β(t)P)\n−logµ(L+\nT(f)) + log ei\n≤E[logµ(X+ 3ηβ(t)Bn)]−hL(f) + log e\nsince L+\nT(f)⊆β(t)P ⊆ηβ(t)Bn. For the normalized entropy, assume E[log(1 + β(T))]<∞and take X=τBn. We have\nE\u0002\nlogµ(X+ 3ηβ(T)Bn)\u0003\n−logµ(τBn)\n=E\u0002\nlogµ((τ+ 3ηβ(T))Bn)\u0003\n−logµ(τBn)\n=nE\u0002\nlog(1 + 3 ηβ(T)/τ)\u0003\n→0\nasτ→ ∞ by the dominated convergence theorem since E[log(1 + β(T))]<∞.\nI. Proof of Corollary 14\nLetPbe a basic cell such that Bn⊆ P . Let g(x) := supz:∥z∥≥xf(z). Taking β(t) = sup {x≥0 :g(x)≥t}, we have\nL+\nt(f)⊆ {z:g(∥z∥)≥t} ⊆β(t)Bn⊆β(t)P. Hence,\nE[log(1 + β(T))]\n=ˆ∞\n0µ(L+\nt(f)) log(1 + β(t))dt\n≤ˆ∞\n0µ(β(t)Bn) log(1 + β(t))dt\n=κnˆ∞\n0(β(t))nlog(1 + β(t))dt\n(a)\n≤κn(n+ log e)ˆ∞\n0ˆβ(t)\n0xn−1log(1 + x)dxdt\n=κn(n+ log e)ˆ∞\n0ˆg(x)\n0xn−1log(1 + x)dtdx\n=κn(n+ log e)ˆ∞\n0g(x)xn−1log(1 + x)dx,\nwhere (a) is because\ndxnlog(1 + x)\ndx\n=nxn−1log(1 + x) +xnloge\n1 +x\n=xn−1\u0010\nnlog(1 + x) +xloge\n1 +x\u0011\n≤(n+ log e)xn−1log(1 + x).\nThe result follows from Theorem 13.\nJ. Proof of Theorem 15\nWe use a simplified version of the arguments in [25, Theorem 4]. Let X∼Unif(X). LetY:=Q(X, S). Write Q−1(y, s) :=\n{x∈Rn:Q(x, s) =y}. 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We consider\ntwo different strategies to define the threshold depending on the city size. In strategy ( P) a\nproportion pof citizens need to get infected before isolation, while in strategy ( U) an uniform\nthreshold Lis chosen for all cities. If city sizes were constant both strategies would coincide.\nHowever, in reality city size distributions are heavy-tailed. In this case in strategy ( P) the\ninfection of a large city can lead to a much larger number of secondary infections of cities\nthan in strategy ( U). To quantify the extend to which these infections fuel the epidemic we\nbenchmark the two strategies against each other by comparing the number of individuals\nthat get isolated when the number of individuals that get infected is under both strategies\nthe same. Furthermore, we define and determine a basic reproduction numbers R(U)\n0and\nR(P)for variant (P) and (U) that represent the initial strengths at which the infection\nspreads between cities . We fit our model to mobility data from France, Japan and Poland\nto identify the parameter regimes of our model that are empirically relevant. Furthermore,\nwe investigate by means of simulations to which extend the geographic structure of the city\ndistribution (that we neglect in our model) influences the infection dynamics between cities.\n∗University of G¨ ottingen, Institute for Mathematical Stochastics, Goldschmidtstrasse 7, 37077 G¨ ottingen,\nGermany; E-mail: viktor.bezborodov@uni-goettingen.de\n†Wroc law University of Science and Technology, Faculty of Information and Communication Technology,\nul. Janiszewskiego 11/17, 50-372 Wroc law, Poland; E-mail: tyll.krueger@pwr.edu.pl\n‡University of L¨ ubeck, Institute for Mathematics, Ratzeburger Allee 160, D-23562, L¨ ubeck, Germany;\nE-mail: cornelia.pokalyuk@uni-luebeck.de\n§Wroc law University of Science and Technology, Department of Computational Intelligence, Wroc law,\nPoland; E-mail: piotr.szymanski@pwr.edu.pl\n¶Universit´ e Paris-Saclay, INRAE, MaIAGE, F-78350 Jouy-en-Josas, France; E-mail:\naurelien.velleret@nsup.org , corresponding author\n1arXiv:2402.03100v1  [physics.soc-ph]  5 Feb 20241 Introduction\nThe spread of Sars-CoV-2 in early 2020 and the subsequent invasions of more pathogenic\nmutants of the original strain made the necessity of effective methods to control a pandemic\nstrikingly clear. Since neither drugs nor vaccines were available at the beginning of the\npandemic, non-pharmaceutical interventions were arranged to restrict contacts between\ninfected and uninfected individuals and slow down the pace of the epidemic.\nSome countries, e.g. China, New Zealand and Australia, followed so called zero-COVID\nstrategies where a lockdown is imposed when only a small number of individuals gets in-\nfected. In Europe many countries coupled the level of non-pharmaceutical interventions to\nrelative incidences calculated from the daily number of individuals tested positive in coun-\nties. In general population sizes differ greatly between counties. Heavy-tailed distributions,\nlike the Zipf law or a log-normal distribution, fit very well the distribution of the counties\npopulation sizes for many countries, for France, Poland and Japan see e.g. Figure 1. As a\nconsequence in counties with large population sizes many individuals need to be tested pos-\nitive until measures are applied, while in counties with small population sizes thresholds for\ninterventions are already exceeded when only a relatively small number of individuals gets\ntested positive. This is particularly problematic, since pathogens often start to spread first\nwithin metropolitan areas and only afterwards hit the rural countryside, see for example\nfor the initial spread of COVID-19 in the US [SBB20].\nTo quantify this issue (and its consequences) we consider in this paper two different\ncriteria at which interventions are imposed, either when a certain fixed number (later called\nstrategy (U)) or when a certain proportion of individuals (later called strategy (P)) gets\ntested positive in an urban unit, from now on called city. We assume that the cities make\nup the vertices of a random graph and we only model the successful infections of cities,\ninstead of the complete infection process between individuals. We implicitly assume that\ncities represent well-mixed populations and within all cities the infection spreads equally\nfast. The infection spreads according to a SIR model between cities in the sense that each\ncity can participate in the epidemic only once and is after a successful infection isolated\n(put under lockdown) till the end of the epidemic. Even though complete isolation of a\ncity is a rather extreme scenario, it is reasonable to assume that the spread of an arising\npathogen occurs in waves if preventive measures are applied that slow down the spread of\nthe pathogen and/or if the newly arising pathogen causes at least temporal immunity. Here,\nwe consider a single wave and for simplicity approximate this wave by a SIR-process. We\nassume that the probability of a city to be infected depends (only) on its size. We introduce\ntwo parameters that reflect the city’s risk of infection due to infected individuals visiting\nthe city and due to visits of the city’s inhabitants to infected cities.\nWe define, for both model variants, basic reproduction numbers R(U)\n0andR(P)\n0that\nmeasure the strength at which the infection spreads between cities at the beginning of the\nepidemic. Moreover, we determine the asymptotic probability of a city to be infected when\nthe number of cities is large. We benchmark the two containment strategies (U) and (P)\nagainst each other analytically when it is equally likely for a city to get infected by infected\nvisitors or by inhabitants that got infected during travel. In this case we can show that given\nthat under both model variants the same number of individuals get infected more individuals\n2Figure 1: Heavy-tailed distributions were fitted to the size distributions of areas of attraction\nin France and Poland as well as of prefectures in Japan.\n3are affected by the containment strategies in variant (P), if and only if infections of cities\nare positively correlated with the city size. In the general case we calculate numerically the\ntheoretical infection probabilities to compare the two strategies.\nWith mobility data of commuters from France, Japan and Poland we evaluate the fit\nof our assumptions on mobility between cities and analyse which parameter regimes of our\nmodel are empirically relevant.\n2 Model\nThe underlying mathematical model in our analysis is the inhomogeneous random directed\ngraph from [COC20], see also [BJR07] for the undirected case. The nodes of the random\ngraph will be interpreted as cities, and the type of a node is the corresponding city size.\nEach city is thought to be a large and well-mixed entity. To generate a graph of size Nwe\nassociate to each vertex ia city size xi(which could be either be deterministic or random),\nfori∈ {1, ..., N}= [ [1, N] ].\nThe disease can be transmitted between two cities along edges of the random graph.\nThese edges are formed by means of two kernels κA, κB: R2\n+→ R+on R+with\nκA(x, y) :=kIL(x)·xb·ya(1)\nand\nκB(x, y) :=kIL(x)·xa−1·y1+b. (2)\nfor some constants kI>0,a, b∈ Rand some (non-decreasing) function L: R+→ R+.\nThe kernels κAandκBrepresent two types of disease transmission between cities. While\nthe first kernel represents the contribution of infections initiated from outside , i.e. occurring\nin a city of size ydue to visiting individuals residing in an infected city of size x, the second\nkernel represents infections initiated from inside , i.e. occurring due to inhabitants of a city of\nsizeyvisiting an infected city of size xand bringing the infection back. After the successful\ninfection of a city we assume that the number of infected individuals grows deterministically\nat exponential speed (representing the expected number of inhabitants that get infected).\nWhen in a city of size xthe number of infected individuals crosses the threshold L(x) the\ncity gets isolated. From this time point on we assume that no infections can leak outside\nand we do not track further the infection process within the city. Furthermore, we assume\nfor simplicity that all transmissions occur immediately before isolation of the city, since the\ngrowth is strongest at this time.\nWe consider two alternative choices of the function Lthat correspond to two differ-\nent strategies for the containment of the epidemic. Strategy ( U) consists in prescribing a\nthreshold L∨for isolation that is independent of the city size, i.e. LU(x) :=L∨. Strategy\n(P) consists in prescribing a threshold for isolation that is proportional to the city size, thus\ngiven through a proportion p∨, i.e. LP(x) :=p∨·x.\nIn reality, Lis random, e.g. for instance due to the time-shift between the infection of\nan individual and detection of an infection. We neglect these fluctuations here. The pa-\nrameter kIrepresents the rate of travel, contact and (successful) transmission of the disease.\n4For the first kernel we assume that each citizen of a city of size xvisits a city of size y\nat a rate proportional to xbya. While yacorresponds to the bias towards a visit of a city of\nsizey,xbrefers to mobility of citizens of a city of size x. Ifa >0 there is a bias of travels\nbeing more often directed into large cities, while if b >0 there is a bias of inhabitants of\nlarge cities to travel more often abroad.\nSecondly, a city vof size ymay get infected because some of its citizens have visited\na city uand brought back the disease, i.e. the infection of vis initiated from inside. We\nassume that these infections happen at a rate proportional to ( L(x)/x)·xa·y1+b. The factor\ny1+bcorresponds to the rate of outwards visits from city v, while xarefers to the likelihood\nthat uis the destination. For the probability that the contacts made by citizens of city v\nvisiting city ulead to an infection of city v, we choose a rate that is proportional to L(x)/x,\ni.e. proportional to the incidence rate.\nThis definition implicitly assumes that there is independence between the way a city\ngets infected (from outside or inside), whose citizens get infected and whose citizens are\ntraveling abroad. For simplicity we have chosen a power-law relation between the city size\nand mobility as well as attractiveness. We find a (surprisingly) good fit of a power-law\ndependence between inbound as well as outbound travel counts and city sizes, in particular\nfor French areas of attraction, see Figure 2 as well as Section 4.2 for more details.\nFor a graph of size Nwith vertices of sizes x1, ..., x Na directed edge from itojis build\nwith probability\npij:= 1−exp\u0012\n−1\nN(κA(xi, xj) +κB(xi, xj))\u0013\n(3)\nWe can justify these probabilities when the city sizes xi, xjare large. Assume that with\nprobability\n \n1−kIL(xi)\nxixb\nixa\nj\nNxixj!xixj\nthere is no contact between (any pair of) individuals living in city iandjdue to individuals\ntravelling from city ito city j. As N→ ∞ this probability is equivalent to\n1−exp\u0012\n−κA(xi, xj)\nN\u0013\n.\nSetting pA(x, y) = 1−exp(−κA(x,y)\nN) and analogously pB(x, y) = 1−exp(−κB(x,y)\nN) we arrive\nat\npij=pA(xi, xj) +pB(xi, xj)−pA(xi, xj)·pB(xi, xj)\n2.1 Interaction kernel\n2.1.1 The limit of a large number of cities\nWe are interested in a scaling of a sparse graph limit, when the number of vertices Ncon-\nverges to ∞and each city infects a.s. asymptotically a (random) finite number of cities\n5according to some law that depends only on the sizes of the currently susceptible cities and\nthe infecting city. In this setting we will compare our model with a branching process that\ndepicts the initial transmission between cities. Locally with high probability the random\ngraph looks like a random tree generated by a branching process. This gives a link between\nlarge random graphs and branching processes [BJR07].\nWe assume that in the case of deterministic city sizes the empirical distributionPN\ni=1δxi\nconverges to some probability distribution βon ( R+,B) with Bthe Borel- σ-algebra. Al-\nternatively city sizes are drawn independently according to such a probability distribution\nβ.\nEmpirical size distributions of large cities are generally very well approximated by heavy-\ntailed distributions, like the log-normal distribution which density takes the form\nβ(x) :=1{x>0}√\n2πσxexp\u0012\n−(ln(x)−µ)2\n2σ2\u0013\n,for some µ∈ R, σ > 0,\nor power law distribution with a density of the form\nβ(x) :=1{x>xL}\nZx−ϕ,for some xL>0 and ϕ >1, (4)\nwith an appropriate normalizing constant Z.\nWe found that the (tail of the) size distribution of French areas of attraction is well\napproximated by a log-normal distribution, while a power law distribution with ϕ= 2.81\n(respectively ϕ= 1.85) can be fitted well to the size distribution of Polish (respectively\nJapanese) areas of attraction, see Figure 1. For these figures, the Python package “power-\nlaw” has been used with a fit according to the Vuiper distance [ABP14].\nThanks to (1), (2) and (3), for large Nthe probability that there is an edge from itoj\nis equivalent to κ(xi, xj)/N, where\nκ(x, y) :=kI·(L(x)·xb·ya+L(x)·xa−1·y1+b) (5)\nWe assume that β, a, b andLare chosen appropriately, such that κ∈L1( R+× R+).\nFor instance, if βis a power-law distribution with exponent ϕ, we require for strategy ( U)\nthat\n1 +b−ϕ <−1 and a−ϕ <−1,\nwhile we require for strategy ( P) that\n2 +b−ϕ <−1 and 1 + a−ϕ <−1.\nFurthermore, we require the following convergence property\n1\nN2E\nX\ni,j∈[ [1,N] ]κ(xi, xj)\n−→\nN→∞Z∞\n0Z∞\n0κ(x, y)β(dx)β(dy)<∞,\n6see also Assumption 3.1.d. in [COC20] or (2.11) in [BJR07].\nWe write io− →jif there is an edge between iandjcorresponding to an infection of j\nfrom outside generated by a visitor from city i, and by iι− →jif there is an edge between\niandjcorresponding to an infection of jfrom inside initiated by a visit in city i. We\nconsider the following random point process on {A, B} ×R+\nξN\ni,→:=X\nio− →jδ(A,xj)+X\niι− →jδ(B,xj).\nThe asymptotic property of ξN\ni,→is expressed in terms of the following functions KA, KB\nand probability measures νA, νB. To simplify notations, we abbreviate Zγ:=R\nyγβ(dy), for\nγ >0.\nKA(x) :=kIL(x)xbZa, K B(x) :=kIL(x)\nx·xaZ1+b,\nνA(dy) :=yaβ(dy)\nZa, ν B(dy) :=y1+bβ(dy)\nZ1+b. (6)\nThe independence properties and (5) yield that ξN\ni,→converges weakly in law as Ntends\nto infinity to a Poisson random measure (PRaMe) Mi,→on the state space {A, B} ×R+\nwith intensity KA(xi)·δA⊗νA+KB(xi)·δB⊗νB.\nThe number of cities that get infected from outside by citizens of city iis Poisson\ndistributed with mean KA(xi). In particular, this distribution depends only on xi. The size\ndistribution of any city infected from outside is independently prescribed by the probability\nmeasure νA, which happens not to depend on i. Because of this independence property,\nwe may refer to a city typically infected from outside when we consider the randomness of\nsampling the city size according to νA.\nSymmetrically, the number of cities that get infected from inside by citizens of iis\nPoisson distributed with mean KB(xi) and the size distribution of any city infected from\ninside is independently prescribed by the probability measure νB, which happens also not\nto depend on i. Because of this independence property, we may refer to a city typically\ninfected from inside when we consider the randomness of sampling the city size according\ntoνB.\nWe can iteratively proceed this process and arrive at a discrete-time branching process\nV∞associated to the infection process. Even though the branching process is not evolving\non the actual graph of cities, we can view it as evolving on a random tree. We will call\nthe nodes of the branching process infected cities, as we did for the infection process. The\nnext generation of infected cities is generated by means of PRaMes with intensity measures\nKA·δA⊗νA+KB·δB⊗νBthat are drawn independently for each infected city (of the\nprevious generation). Above we have argued about the convergence as Ntends to infinity of\nthe law of ξN\ni,→, that is in particular we consider the very beginning of the infection process.\nWhether an approximation of the infection process by a branching process is reasonable\nalso at later time points, we evaluate by means of simulations in Section 4.3 and in Section\n4.6. In particular, we show simulation results that assess the impact of isolating (large)\ncities during the epidemic.\n7We are primarily interested in the number and sizes of the infected cities. However,\ndistinguishing cities that get infected from outside vs those that get infected from inside\nwill be simplifying the analysis. With this respect the following proposition will prove useful\nlater. It is a direct consequence of the above conclusions.\nProposition 2.1.1. The projection V(2)on the two-type space {A, B}of the process V∞\nyields a two-type branching process with offspring generated as follows. Assume ng−1\nAcities\nare infected from outside and ng−1\nBcities are infected from inside in generation g−1. Then\nfirst city sizes (xA\ni)i∈ng−1\nAand(xB\nj)j∈ng−1\nBare drawn independently according to the measure\nνAandνB,resp., for these cities. In the next generation gthe number ng\nAof cities infected\nfrom outside is Poisson distributed with parameterP\ni∈ng−1\nAKA(xA\ni) +P\nj∈ng−1\nBKA(xB\nj)\nand the number ng\nBof cities infected from inside is Poisson distributed with parameterP\ni∈ng−1\nAKB(xA\ni) +P\nj∈ng−1\nBKB(xB\nj).\nWe consider two different kinds of initial conditions. Either only the numbers n0\nAand\nn0\nBof cities infected from outside and inside are specified or in addition to the numbers n0\nA,\nn0\nBalso the city sizes (xA,0\ni)i∈n0\nAand(xB,0\nj)j∈n0\nBare given in generation 0. In the latter\ncase in generation 1 obviously city sizes are not resampled and the number n1\nAof cities in-\nfected from outside is Poisson distributed with parameterP\ni∈n0\nAKA(xA,0\ni)+P\nj∈n0\nBKA(xB,0\nj)\nand the number n1\nBof cities infected from inside is Poisson distributed with parameterP\ni∈n0\nAKB(xA,0\ni) +P\nj∈n0\nBKB(xB,0\nj).\n2.1.2 Backward in time branching process\nSimilar as in the previous section we can follow backwards in time potential infection routes\nalong which a city can get infected. To infer (approximately) the infection probability for a\ncityjof size xjwe approximate the backward infection chains also by a branching process.\nFor a target city jwe say that j′is an infector of jfrom outside (resp. from inside) if j′o− →j\n(resp. if jι− →j′). Then the random variable\nξN\nj,←:=X\nj′o− →jδ(C,xj′)+X\nj′ι− →jδ(D,xj′).\ngives the possible infectors of j, where CandD, resp., indicate if the infecting city intro-\nduced the infection from outsite or from inside and xj′denotes the size of the infecting\ncity. In analogy to the process ξN\nj,←and thanks to (5) and the independence properties\nin the large population limit ξN\nj,→converges weakly to a PRaMe Mj,←on the state space\n{C, D} × R+with intensity KC(xj)·δC⊗νC+KD(xj)·δD⊗νD, where\nKC(x) :=kIxaZ∞\n0L(z)·zbβ(dz), KD(x) :=kIx1+bZ∞\n0L(z)·za−1β(dz) (7)\nνC(dy) :=L(y)·ybβ(dy)R∞\n0L(z)·zbβ(dz), νD(dy) :=L(y)·ya−1β(dy)R∞\n0L(z)·za−1β(dz)(8)\nIn the limit the number of infectors of jfrom outside (resp. from inside) is Poisson\ndistributed with mean KC(xj) (resp. KD(xj)). The sizes of the infectors of jfrom outside\n8(resp. from inside) are independently prescribed by the probability νC(resp. νD), which\nhappens not to depend on j.\nThe corresponding backward branching process we denote by Λ∞. In this discrete-time\nmultitype branching process an element with type x≥0 is in the next generation replaced by\nelements drawn according to a Poisson random measure with intensity KC(x)νC+KD(x)νD.\nIn analogy to Proposition 2.1.1 we have the following result.\nProposition 2.1.2. The projection Λ(2)on the two-type space {C, D}of the process Λ∞\nstarted at the first generation generates a two-type branching process. Provided Λ∞starts\nwith city iof size xi, the initial condition of Λ(2)is given by independent Poisson random\nnumbers of cities infectors from outside and from inside respectively, with averages KC(xi)\nandKD(xi)respectively.\n3 Analysis of the branching approximation\n3.1 Probability of generating an outbreak\nIn this section we aim at an approximate formula for the probability that from a city of size\nxan outbreak is generated. As we approximated the infection process by the branching\nprocess V∞, it is natural to approximate the outbreak probability by the probability of\nexplosion of V∞started with a single infected city of size x(see e.g. Theorem 3.11 in\n[COC20] for a rigorous convergence result in this direction). It is well known that the\nexplosion probability is equal to the survival probability for branching processes. Let us\ndenote this probability by η(x). According to Proposition 2.1.1 η(x) coincides with the\nprobability of survival of the two-type branching process V2.\nThanks to Lemma 5.4 in [BJR07] ηis the solution the following equation\nη(x) = 1−exp\u0002\n−KA(x)R∞\n0η(y)νA(dy)−KB(x)R∞\n0η(y)νB(dy)\u0003\n. (9)\nAccording to Lemma 5.6 in [BJR07] we can estimate ( η(x))xiteratively by setting\nηk+1(x) = 1−exp\u0014\n−KA(x)Z∞\n0ηk(y)νA(dy)−KB(x)Z∞\n0ηk(y)νB(dy)\u0015\n(10)\nand starting with η0≡1.\nThe expression for ηon the r.h.s. of (9) is a function of the two unknown parameters\nηA=Z∞\n0η(y)νA(dy) (11)\nand\nηB=Z∞\n0η(y)νB(dy). (12)\nWe check the performance of the iterative procedure through the convergence of the\nassociated sequences ( ηk\nA)kand ( ηk\nB)kwith\nηk\nA=Z∞\n0ηk(y)νA(dy) (13)\n9and\nηk\nB=Z∞\n0ηk(y)νB(dy), (14)\nwhich fulfill the recursions\nηk+1\nA= 1−Z∞\n0exph\n−KA(x)·ηk\nA−KB(x)·ηk\nBi\nνA(dx),\nηk+1\nB= 1−Z∞\n0exph\n−KA(x)·ηk\nA−KB(x)·ηk\nBi\nνB(dx).\nFor our data these two sequences appear to converge quickly, see Figure 26. This suggests\nthat a very accurate estimation of ( η(x))xis rapidly obtained.\nFurthermore, the numerically calculated values of ( η(x))xappear to be close to the\noutbreak probabilities for simulated epidemics for strategy (P) for France, see Figure 5. For\nJapan and for Poland and for strategy (U) theoretical outbreak probabilities fit less well,\nsee Figure 6 as well as Figures 16 - 22 in the appendix.\n3.2 Backward in time process and probability of infection\nIn this section we give an approximation of the probability that a city (of size x) eventually\ngets infected during an outbreak. Similarly as for the probability of an outbreak we approx-\nimate the probability of a city of size xeventually to be infected by the probability that\nthe associated backward branching process introduced in Section 2.1.2 with initial state δx\nsurvives.\nLet us denote by π(x) the probability of survival of the backward process Λ∞starting\nfrom a city of size x. According to Proposition 2.1.2 it coincides with the survival probability\nof the two-type branching process Λ2. Thanks to Lemma 5.4 in [BJR07] and similarly as\nforη,πis the solution the following equation.\nπ(x) = 1−exp\u0014\n−KC(x)Z∞\n0π(y)νC(dy)−KD(x)Z∞\n0π(y)νD(dy)\u0015\n. (15)\nWe estimate π(x) iteratively, in analogy to the estimation procedure for η(x). The per-\nformance of the iterative procedure is again checked through the convergence of the two\nsummary parameters πCandπD, i.e. defined with π0\nC=π0\nD= 1 and at step k≥0:\nπk+1\nC:= 1−Z∞\n0exph\n−KC(x)·πk\nC−KD(x)·πk\nDi\nνC(dx),\nπk+1\nD:= 1−Z∞\n0exph\n−KC(x)·πk\nC−KD(x)·πk\nDi\nνD(dx).\nIn a few steps, it provides a very accurate estimation of π(see Figure 26).\nBy means of simulations we evaluate the suitability of the approximation with ( π(x))\nin our context, in particular when the goodness of the branching approximation is at stake,\nsee Section 4.3 and Section 4.6.\n103.3 Comparison of the two strategies\nIn this section we aim to compare the efficiency of the two strategies (U) and (P). We recall\nthat strategy ( U) consists in prescribing a threshold L∨for isolation that is independent\nof the city size while under strategy ( P) a city gets isolated when a certain proportion p∨\nof inhabitants of the city gets infected. To arrive at a comparison of the efficiency of the\ntwo strategies, we assume that after isolation of a city no further inhabitants get infected.\nIn particular, at the end of an epidemic in an isolated city of size xunder strategy (U)\n(under strategy (P), resp.) there are L∨(p∨x, resp.) individuals that have been infected.\nFurthermore, we approximate the infection probabilities by the probabilities introduced in\nSection 3.2. Depending on the strategies, we denote these probabilities by {πU(x)}x>0and\n{πP(x)}x>0, resp. We measure the burden of the epidemic by the number I⋆of eventually\ninfected people. Under our assumption for strategy ( U) the number of eventually infected\nindividuals is\nIU:=Z∞\n0L∨·πU(x)β(dx), (16)\nwhile for strategy ( P) the number of eventually infected individuals is\nIP:=Z∞\n0p∨·x·πP(x)β(dx). (17)\nSimilarly, the number of individuals eventually under isolation is (under our assump-\ntions) given by\nQU:=Z∞\n0x·πU(x)β(dx), Q P:=Z∞\n0x·πP(x)β(dx). (18)\nAssume that we adjust the parameters L∨andp∨such that I⋆is the same under both\nstrategies. Then, we regard the strategy for which less people need to be isolated as the\nmore efficient strategy . In Subsection 3.3, we consider the case of a rank-one kernel where\na= 1 + band show that strategy ( P) is more efficient than strategy ( U) iffa < 1, see\nProposition 3.3.1. If a= 1, both strategies have the same efficiency. In the general case,\nfor which ais not necessarily equal to 1 + b, we do not have an analytical comparison of\nthe two strategies. To arrive still at a comparison of the efficiency of strategy (U) and (P),\nwe compare them by means of simulations, see Subsection 4.4.\nComparison for rank-one kernels\nIn this subsection we assume that a= 1 + b, in particular the kernel of the (two-type)-\nbranching process Λ(2)is actually of rank 1.\nThe probabilities πUandπPdepend on kI, a,β, and resp. L∨andp∨. To compare the\ntwo strategies, we fix kI, a, β and a value I⋆for the number of people that eventually get\ninfected. We adjust the parameters L∨andp∨such that IU=IP=I⋆.\nThe next proposition states which strategy is more efficient in the case where a=b+ 1\ndepending on the value of a.\n11Proposition 3.3.1. Assume βis a non-Dirac probability measure, a,kIandI⋆are given\nand assume that a=b+ 1.\ni) Ifa= 1, then QU=QP, i.e. both strategies are equally efficient.\nii) If a <1, then QU> Q P, i.e. strategy (P) is more efficient.\niii) If a >1, then QU< Q P, i.e. strategy (U) is more efficient.\nProof of Proposition 3.3.1 Instead of prescribing the values for IUandIP(toI⋆), it\nis more convenient to prescribe the same value Q⋆for the numbers QUandQPof people\neventually under isolation. Since IU, IP, QUandQPare increasing functions of the threshold\nvalues, proving Proposition 3.3.1 is equivalent to showing that whatever Q⋆, it holds:\ni)IU=IPifa= 1,\nii)IU< IPifa <1,\niii)IU> IPifa >1.\nIn the rank-one kernel situation, (15) simplifies into:\nπ(x) = 1−exp\u0014\n−2KC(x)Z∞\n0π(y)νC(dy)\u0015\n.\nHence, we deduce from (8) that πUandπPcan be expressed in the form\nπγ(x) := 1 −exp(−γ·xa),\nthat is πU=πγUwith\nγU:= 2kIZ∞\n0L∨·ya−1·πU(y)β(dy) = 2 kIZ∞\n0L∨·ya−1·πγU(y)β(dy), (19)\nand similarly, πP=πγPwith\nγP:= 2kIZ∞\n0p∨·ya·πP(y)β(dy) = 2 kIZ∞\n0p∨·ya·πγP(y)β(dy). (20)\nSince the function : γ7→πγis increasing, by (18) QU=QPimplies γU=γP. Further-\nmore, by exploiting (19) and (20), we can express L∨andp∨in terms of γUandγPresp.\nWe obtain the ratio IU/IPas a function of γU,γP,βanda.\nLet us first consider the case where a≥1. For ease of notations, we denote for any\nζ >0:\nZ⋆\nζ:=Z∞\n0xζ·πγ⋆(x)β(dx).\nSince γU=γP, by combining (19) and (20) we get L∨Z⋆\na−1=p∨Z⋆\na. Then, the ratio IU/IP\nis expressed as follows:\nIU\nIP=L∨Z⋆\n0\np∨Z⋆\n1=Z⋆\n0\nZ⋆\na−1·Z⋆\na\nZ⋆\n1.\n12When a= 1, we directly obtain that IU=IP.\nOtherwise, note that 0 <1∧(a−1)<1∨(a−1)< a. This ratio is compared to 1\nthanks to H¨ older’s inequality:\n\u0012Z⋆\na−1\nZ⋆\n0\u00131/(a−1)\n>\u0012Z⋆\na\nZ⋆\n0\u00131/a\n·1,\u0012Z⋆\n1\nZ⋆\n0\u0013\n>\u0012Z⋆\na\nZ⋆\n0\u00131/a\n,\nwhich entails\nZ⋆\na−1· Z⋆\n1\n(Z⋆\n0)2>\u0012Z⋆\na\nZ⋆\n0\u0013a−1\na+1\na\n=Z⋆\na\nZ⋆\n0.\nThis inequality is equivalent to IP> IU, which concludes the proof in the case where a >1.\nWhen a <1, we look for the shifted moments:\nZ⋉\nζ:=Z∞\n0xζ·xa−1πγ⋆(x)β(dx).\nThen, the ratio IU/IPis expressed as follows:\nIU\nIP=Z⋉\n1−a\nZ⋉\n0·Z⋉\n1\nZ⋉\n2−a.\nNote that 0 <1∧(1−a)<1∨(1−a)<2−a. Thanks to H¨ older’s inequality, with a\nsimilar reasoning as before, we deduce this time that IP< IUwhatever Q⋆. This concludes\nthe proof of Proposition 3.3.1.\n3.4 Basic reproduction number\nIn this section we define a basic reproduction number R0for the branching process V∞\n(that approximates the transmission process between cities). We set\nR0:= lim\nk→∞Ex[Xk]1/k, (21)\nwhere Xkis the number of infected cities at the k-th generation.\nWe will deduce from Proposition 3.5.2 below that this definition actually does not depend\nonx. Indeed, Proposition 3.5.2 states that R0is in fact the principal eigenvalue of the\nintegral operator Tassociated to the PRaMes MAandMB, which is defined as follows. We\nrefer to Subsection 4.6 for the numerical study of this quantity for our datasets.\nWe denote by B(R+) the set of bounded measurable functions on R+and write ⟨µ\f\ff⟩:=R∞\n0f(x)µ(dx) for any µ∈ M 1(R+) and f∈ B(R+). Then, for any x∈R+andf∈ B(R+)\nwe define\nTf(x) :=⟨δx\f\fTf⟩:=KA(x)· ⟨νA\f\ff⟩+KB(x)· ⟨νB\f\ff⟩ (22)\n=Ex\u0000\n⟨MA+MB\f\ff⟩\u0001\n=Z\nR+κ(x, y)f(y)β(dy),\n13where the subscript xat the expectation Emeans that the infection process is started from\na city of size xandκ(x, y) =kI·L(x)·(xb·ya+xa−1·y1+b).T:=KA⟨νA\f\f·⟩+KB⟨νB\f\f·⟩\nis another equivalent writing of the same definition. In this notation it is clear, that Tfis\na measurable function (a priori not bounded) whereas δxTis a positive measure on R+.\nRecalling (21), we can interpret R0in terms of Tk:\nR0= lim\nk→∞⟨δx\f\fTk1⟩1/k,\nwhere 1is the function uniformly equal to 1. Indeed, denote by M(u;k)the point measure\nonR+which points (counted with multiplicity) correspond to the sizes of the cities that get\ninfected from city uafter kgenerations. By considering the conditional expectation with\nrespect to the city sizes of the first generation of cities infected by u, one shows by means\nof the branching property the following equality:\nE\u0010\n⟨M(u;2)\f\ff⟩\f\fM(u;0)=x\u0011\n=E\u0010\n⟨M(u;1)\f\fTf⟩\f\fM(u;0)=x\u0011\n=⟨δx\f\fT2f⟩.\nBy induction, for any k∈N,Tkcorresponds to the expectation of M(u;k), i.e.:\n⟨δx\f\fTkf⟩=E\u0010\n⟨M(u;k)\f\ff⟩\f\fM(u;0)=x\u0011\n. (23)\nRemarks 3.4.1. The operator Tis a kernel operator. If the kernel κwould be symmetric\nand would satisfy Z\nR+Z\nR+k(x, y)2β(dx)β(dy)<∞, (24)\nthen our definition would coincide with the definition of R0as in [BJR07] as\nsup{∥Tf∥L2(β);f∈L2(β),∥f∥L2(β)≤1}.\nThis follows from the following Proposition 3.5.2 and essentially Lemma 5.15 in [BJR07].\nCondition (24) ensures the compactness of the operator, as proved in Lemma 5.15 of\n[BJR07] and noted in Remark 3.12 of [COC20]. Without symmetry nor condition (24),\nR0is defined as the spectral radius of T, see in particular Theorem 3.10 in [COC20].\nThe spectral radius of Tis identified in Proposition 3.5.2. Our rank-two kernel has\nexactly two real eigenvalues values, the leading eigenvalue is R0.\n3.5 Spectral analysis of T\nTo analyse the long time behavior of T, the following lemma is helpful in that it relates T\nto a matrix operation on two-dimensional space. For this, we define the matrix W:\nW=\u0012⟨νA\f\fKA⟩ ⟨νA\f\fKB⟩\n⟨νB\f\fKA⟩ ⟨νB\f\fKB⟩\u0013\n. (25)\nThe entry ⟨νA\f\fKA⟩can be interpreted as the average number of cities that get infected\nfrom outside from a typical city that got infected from outside. A similar interpretation\nholds for the other entries. Wis classically related to the long-time behavior of V2, cf\n[KPS66].\n14Lemma 3.5.1. For any k∈N,x∈R+andfa non-negative measurable function:\n⟨δx\f\fTkf⟩=\u0000\nKA(x)KB(x)\u0001\n·Wk−1·\u0012⟨νA\f\ff⟩\n⟨νB\f\ff⟩\u0013\n.\nProof. The equality follows by induction and the definition of Tgiven in (22).\nThe projection property given in Lemma 3.5.1 is strongly connected to the projection\nof V∞onto V2given in Proposition 2.1.1. It greatly simplifies the spectral analysis of T,\nas we can see thanks to the following proposition.\nIf any of the entries of Wis infinite, then T21≡ ∞ because\n⟨δx\f\fT21⟩=KA(x)(⟨νA\f\fKA⟩+⟨νA\f\fKB⟩) +KB(x)(⟨νB\f\fKA⟩+⟨νB\f\fKB⟩).\nThis directly implies that R0=∞in this case. Therefore, in the following, we assume that\nall entries are finite.\nIfβis a power-law distribution with exponent ϕ, this assumption translates to\n1 + 2 b−ϕ <−1 and 2 a−1−ϕ <−1, (26)\nin the case of strategy ( U) and to\n2 + 2 b−ϕ <−1 and 2 a−ϕ <−1 (27)\nin the case of strategy ( P). Recall that a bounded measurable function fis called an eigen-\nfunction of Tif there exists some value λsuch that Tf=λf. Similarly, a signed measure µis\ncalled an eigenmeasure of Tif there exists some value λsuch that µT=λµ.λis then called\nan eigenvalue of T. If an eigenmeasure qofTis a probability distribution, it satisfies the\nproperty for being a quasi-stationary distribution (QSD), namely qT(dx)/⟨qT1⟩=q(dx).\nDenote by vTthe transposition of a vector v. We have the following relationships between\nthe eigenvalues and eigenvector of Wand the eigenvalues and eigenmeasures of T.\nProposition 3.5.2. Assume that the entries of Ware finite. Then Thas two distinct\nand real eigenvalues λ0andλ1, such that λ0> λ 1∨0and that coincide with the ones\nofW. The leading eigenmeasure q0ofTcan be chosen as a probability measure, thus as\na QSD. Similarly, the leading left eigenvector of Wcan be chosen as\u0000\nqA\n0qB\n0\u0001\nsuch that\nqA\n0+qB\n0= 1,qA\n0∧qB\n0≥0. The following relation holds between them:\nq0=qA\n0·νA+qB\n0·νB. (28)\nOn the other hand, the leading eigenfunction h0ofTcan be chosen as a positive measurable\nfunction h0such that ⟨q0\f\fh0⟩= 1. Similarly, the leading right eigenvector of Wcan be\nchosen as\u0000\nhA\n0hB\n0\u0001Tsuch that qA\n0hA\n0+qB\n0hB\n0= 1,hA\n0∧hB\n0≥0. The following relationship\nholds between the function h0and the vector\u0000\nhA\n0hB\n0\u0001T:\nh0=hA\n0\nλ0·KA+hB\n0\nλ0·KB. (29)\n15In addition, there exists a function h1, a measure q1and constant C > 0such that we\nhave the following exact result of exponential convergence at rate λ1/λ0:\n(λ0)−k⟨δx\f\fTkf⟩ −h0(x)· ⟨q0\f\ff⟩= (λ1/λ0)k·h1(x)· ⟨q1\f\ff⟩, (30)\nwith the following bounds:\n|h1| ≤C·(KA∨KB),|q1|(dx)≤C·(νA∨νB)(dx).\nIn particular,\nR0= lim\nk⟨δx\f\fTk1⟩1/k=λ0.\nBefore we proceed with the proof of Proposition 3.5.2, let us interpret the quantities h0\nandq0.\nIt follows from Proposition 3.5.2 that Ex[Xk] =⟨δx\f\fTk1⟩is asymptotically equivalent\nto (R0)k·h0(x). This property is why it is reasonable to call h0the survival capacity and\nwhat makes h0relevant as an indicator of network centrality, see Section 3.7. Furthermore,\nit motivates to consider the eigenmeasure q0because q0is involved in the normalization\ncondition on h0.\nActually, the density of q0at value xcan be interpreted as the likelihood for a city\nrandomly chosen among the infected cities at generation gto have size x, for large gin\nthe branching approximation. More precisely, for any initially infected city uand on the\nevent of survival of the branching approximation, we can demonstrate that the sequence\nof normalized random measures M(u;g)(dx)/⟨M(u;g)|1⟩converges to q0(dx) asgtends to\ninfinity. In the above expression, we recall that M(u;g)is the point measure on R+which\npoints (counted with multiplicity) correspond to the sizes of the cities that get infected from\ncityuafter ggenerations.\nIndeed, the city sizes at generation gare prescribed by independent sampling with dis-\ntributions νAandνBresp., conditionally on the numbers of cities infected from outside\nand of those infected from inside, at generation g. According to Proposition 2.1.1, these\ntwo numbers can be inferred by studying the process V2. In such setting of a discrete-type\nGalton-Watson processes, [KPS66] provides a description of the relative proportions of the\ndifferent types. In our case, it implies that asymptotically a proportion qA\n0of cities are\ninfected from inside. We also recall that on the event of survival, the number of infected\ncities at generation gtends a.s. to infinity with g. We then account for the next sampling\nof cities sizes, and recall that q0(dx) =qA\n0νA(dx) + (1 −qA\n0)νB(dx) to deduce the above\nclaim of convergence thanks to the law of large number.\nThe study of the backward-in-time process is analogous. Recall that the eigenvalues of\nan adjoint operator are complex conjugates of the eigenvalues of the original operator. In\nour case, the eigenvalues of the backward operator ˆTsimply coincide with the ones of T,\nnamely R0andR. On the other hand, the QSDs and the eigenvectors do not generally\ncoincide, except in the specific case of the strategy P.\n16Proof of Proposition 3.5.2 Since Whas positive entries, the Perron-Frobenius theorem\nensures that Whas two distinct real eigenvalues λ0, λ1of the form λ0>max{λ1,0}. Also,\nthe leading left and right eigenvectors have necessarily entries of the same sign, contrary to\nthe corresponding eigenvectors of the second eigenvalue.\nAssume that qis an eigenmeasure of Twith eigenvalue λ, (when Tis treated as an\nadjoint operator on measures).\nUnder T, any non-negative measure µsuch that ⟨µ\f\fKA⟩and⟨µ\f\fKB⟩are both finite is\nmapped to a measure uniquely prescribed as a linear combination of νAandνB. If either\n⟨q\f\fKA⟩or⟨q\f\fKB⟩would be infinite, then qT=∞which would contradict the fact that q\nis an eigenmeasure of T. This implies that qcan necessarily be expressed as follows:\nq=\u0000\nqAqB\u0001\n·\u0012νA\nνB\u0013\n.\nFrom this representation and since νAandνBare not colinear, with Lemma 3.5.1 it follows\nthat qT=λqis equivalent to\u0000\nqAqB\u0001\nbeing a left eigenvector of Wwith eigenvalue λ.\nIn particular, recalling the Perron-Frobenius theorem the leading left eigenvector ( qA\n0qB\n0)\nofWand the leading eigenmeasure q0ofT(with leading eigenvalue λ0) can be chosen\nto be non-negative. By assuming further that qA\n0andqB\n0sum up to one and that q0is\na probability measure and since νAandνBare probability measures, relationship (28) is\nfulfilled.\nSimilarly, under T, any non-negative function fsuch that ⟨νA\f\ff⟩and⟨νB\f\ff⟩are both\nfinite is mapped to a function uniquely prescribed as a linear combination of KAandKB.\nLethbe an eigenfunction of T. If either ⟨νA\f\fh⟩or⟨νB\f\fh⟩would be infinite, then Th=∞\nwhich would contradict the fact that his an eigenfunction of T. This implies that hcan\nnecessarily be expressed as follows:\nh=\u0000\nhAhB\u0001\n·\u0012KA\nKB\u0013\n.\nFrom this representation and since KAandKBare not colinear, with Lemma 3.5.1 it follows\nthatTh=λhis equivalent to\u0000\nhAhB\u0001Tbeing a right eigenvector of Wwith eigenvalue\nλ. Recalling Perron-Frobenius theorem, the leading eigenfunction h0ofTand the leading\nright eigenvector\u0000\nhA\n0hB\n0\u0001Tcan be chosen non-negative and such that ⟨q0\f\fh0⟩= 1 and\nqA\n0hA\n0+qB\n0hB\n0= 1. Let us denote by c >0 the constant such that h0(x) =c·(hA\n0KA(x) +\nhB\n0KB(x)). Then with Lemma 3.5.1 ⟨q0\f\fh0⟩= 1 translates into\nc·\u0000\nqA\n0qB\n0\u0001\n·W·\u0012hA\n0\nhB\n0\u0013\n= 1.\nRecalling that\u0000\nhA\n0hB\n0\u0001Tis a right eigenvector of Wwith eigenvalue λ0and that qA\n0hA\n0+\nqB\n0hB\n0= 1, it implies that c= 1/λ0so that relation (29) fulfilled.\nFinally, let\u0000\nqA\n1qB\n1\u0001\nbe a left eigenvector of Tcorresponding to the eigenvalue λ1. It\nmust satisfy that qA\n1·hA\n0+qB\n1·hB\n0= 0, because when eigenvectors of the different eigenvalues\nλ0> λ 1are considered, we have\nλ1·(qA\n1·hA\n0+qB\n1·hB\n0) =\u0000\nqA\n1qB\n1\u0001\n·W·\u0012hA\n0\nhB\n0\u0013\n=λ0·(qA\n1·hA\n0+qB\n1·hB\n0).\n17Since hA\n0andhB\n0are positive, the signs of qA\n1andqB\n1are necessarily different. Similarly,\nany right eigenvector of Wwith eigenvalue λ1has entries of opposite signs. Therefore, by\nrescaling appropriately, we can define\u0000\nhA\n1hB\n1\u0001Tas the unique left eigenvector of Wsuch\nthatqA\n1·hA\n1+qB\n1·hB\n1= 1. Define similarly as for the leading eigenvectors:\nq1:=qA\n1·νA+qB\n1·νB, h 1:=hA\n1\nλ1·KA+hB\n1\nλ1·KB. (31)\nThe spectral decomposition of Wimplies that for any reals ℓA, ℓB, fA, fBandk∈N:\n\u0000\nℓAℓB\u0001\n·Wk·\u0012fA\nfB\u0013\n=h\n(λ0)k·(ℓAhA\n0+ℓBhB\n0)·(qA\n0fA+qB\n0fB) +λk\n1·(ℓAhA\n1+ℓBhB\n1)·(qA\n1fA+qB\n1fB)i\nCombined with Lemma 3.5.1, (28), (29) and (31), it directly entails (30) and concludes the\nproof of Proposition 3.5.2 with C:= max {qA\n1, qB\n1, hA\n1, hA\n1}. □\n3.6 Computation of the basic reproduction number\nThis subsection is dedicated to the computation of R0, under the two strategies (P) and\n(U). The corresponding eigenvectors are given in Subsection 3.7 dedicated to eigenvector\ncentrality. Recall, that under strategy (U) L(x)≡L∨and under strategy (P) L(x) =p∨·x.\nA crucial role is played by the moments Zγwith exponent γ >0, cf (6) and just above.\nUnder strategy ( U) the entries of the two-type transmission matrix Wdefined in (25),\nhere abbreviated as WU, are the following:\nWU\nA,A:=Z∞\n0νA(dx)kIL∨xbZ∞\n0yaβ(dy) =kIL∨Za+b,\nWU\nA,B:=kIL∨·Z2a−1· Z1+b\nZa, WU\nB,A:=kIL∨·Za· Z1+2b\nZ1+b, WU\nB,B:=kIL∨Za+b.\nIt is easily deduced that the largest eigenvalue under Scenario Uis\nR(U)\n0=kIL∨rU\n0,\nwhere\nrU\n0=Za+b+p\nZ2a−1Z2b+1. (32)\nUnder strategy ( P) the entries of the corresponding two-type transmission matrix WP\nare\nWP\nA,A=WP\nB,B:=kIp∨Z1+a+b, WP\nA,B:=kIp∨·Z2a· Z1+b\nZa, WP\nB,A:=kIp∨·Za· Z2+2b\nZ1+b.\nIt is easily deduced that the largest eigenvalue under strategy ( P) is\nR(P)\n0=kIp∨rP\n0,\nwhere\nrP\n0=Z1+a+b+p\nZ2aZ2+2b. (33)\n183.6.1 Elementary adjustment of the two strategies\nA possibility to adjust the two strategies ( U) and ( P) to each other is to put R(U)\n0andR(P)\n0\nat the same level R⋆and compare the corresponding threshold values L∨andp∨. This is\nparticularly useful, if one knows parameter regimes for which one of the two strategies leads\nto a subcritical epidemic and one wants to choose the threshold of the other strategy such\nthat the corresponding epidemic is also subcritical. In the following proposition, we give\nconditions on the parameters aandbthat enable to upper-bound p∨depending on L∨.\nProposition 3.6.1. Let us define γ:= (2 a)∨(2 + 2 b)andδ≤(2a−1)∧(2b+ 1). For any\nβsuch that the moment Zγis finite, it holds that Zδ+1·rU\n0≤ Z δ·rP\n0. Consequently, when\nR(U)\n0andR(P)\n0are put at the same level R⋆, it implies that p∨≤L∨· Zδ/Zδ+1.\nIn particular, it implies that\np∨≤L∨/Z1 (34)\nprovided a≥1/2andb≥ −1/2, and\np∨≤L∨· Z1/Z2 (35)\nprovided a≥1andb≥0.\nIn the case a= 1 + bwe have\np∨=L∨·Z1+2b/Z2+2b. (36)\nThe first inequality (34) means that the threshold for strategy ( P) must be smaller than\nL∨divided by the expected city size. For the second inequality (35) note that\np∨Z2\nZ1=ZxR\nyβ(dy)xp∨β(dx),\nwhich can be interpreted as the average threshold number of infected people in the city of\na randomly chosen individual. So Inequality (35) means that p∨must be chosen such that\nthis average threshold number is smaller than L∨.\nIn practice, βis often heavy-tailed. In this case the two upper bounds can be very\nfar apart. For the data from France (D30+, see Subsection 4.1 for the description of the\ndataset), the ratio Z2/(Z1)2is actually close to 36, while for the Japanese data (also D30+)\nthe ratio Z2/(Z1)2is even larger than 80. Since a≈1 and b≈0 in both cases, only the\nsecond estimate of pVbyL∨· Z2/Z1is thus relevant.\nFor heavy-tailed distributions βthe right-hand side of (36) is very much affected by the\nvalue of b.\nProof of Proposition 3.6.1 Thanks to (32) and (33), it is enough to show that:\nZa+bZδ+1≤ Z δZa+b+1,\nZ2a−1Z2b+1(Zδ+1)2≤(Zδ)2Z2aZ2b+2(37)\n19The inequalities (37) follow from H¨ older’s inequality. In general for γ > δ , choosing\nq=γ−δ+ 1,p=q\nq−1, we note:\n1/p+ 1/q= 1, xδ+1=xδ/p·x(γ+1)/q, xγ=x(γ+1)/p·xδ/q\nwhich entails\nZδ+1≤(Zδ)1\np·(Zγ+1)1\nq,Zγ≤(Zγ+1)1\np·(Zδ)1\nq.\nThe inequalities (37) are deduced from the particular cases of γin{a+b,2a−1,2b+ 1}.\nδ <(2a−1)∧(2b+ 1) entails γ > δ in these three cases.\nEquality (36) follows from (32) and (33).\nThis ends the proof of Proposition 3.6.1.\n3.7 Eigenvector centrality\nIn terms of the approximation by the forward in time branching process, one can get explicit\nformulas for a classical notion of centrality for epidemics evolving on a network, namely\neigenvector centrality, which assigns to each city size xthe value of the leading eigenfunction\nh0(x), see [New10, Chapter 7], [Bon87]. It is a measure for the number of infections that are\ntriggered when node xgets infected and hence, allows to compare the relative importance\nof the different cities during the course of an epidemic. The value is primarily of interest\nin situations where the goal of containing the epidemic is no longer within reach and the\naim is instead to delay its progression. By targeting strict measures on cities with a high\neigenvector centrality value, one wishes to target cities with a high potential for additional\ninfections.\nAccording to Proposition 3.5.2, the leading eigenfunction h0takes the following form:\nh0(x) = (R0)−1·(KA(x)·hA\n0+KB(x)·hB\n0), (38)\nwhere the 2 ×1 vector ( hA\n0hB\n0)Tis the eigenvector of the 2 ×2 transmission matrix W. Since\nWA,A=WB,Bwe arrive at the following expression for ( hA\n0hB\n0)T(with similar expressions\nfor (qA\n0qB\n0), the left eigenvector of W):\nhA\n0:=pWA,B+pWB,A\n2pWB,A, hB\n0:=pWA,B+pWB,A\n2pWA,B, (39)\nWith these expressions one observes that the normalisation is such that ( hA\n0hB\n0)Tis ac-\ntually independent of the value of the thresholds p∨andL∨under strategy ( P) and ( U),\nrespectively.\nThe eigenvector centrality value h0(x) can be decomposed into the two factors fA(x) =\n(hA\n0/R0)·KA(x) and fB(x) = ( hB\n0/R0)·KB(x). Since by definition KAandKB(cf (6))\ndepend polynomially on x, both factors are on a log scale linear functions in the city size.\nWe refer to Section 4.7, in particular to Figures 14, for numerical calculations of the\nvalues of h0.\n204 Data analysis and simulations\n4.1 Datasets\nTo evaluate at least to some extend the eligibility of our model to map a real epidemic spread\nof a pathogen between cities and to infer which size distributions and parameter combina-\ntions of aandbare empirically relevant we analysed mobility data from France, Poland\nand Japan. The datasets were derived from the census based estimation of work-related\nmobility obtained from statistical offices in France (2017 data, INSEE1), Japan (2015,\nMinistry for Transportation2) and Poland (2016, GUS3), in particular, the datasets were\ngathered several years ago before any Sars-Cov2 containment regulations were arranged.\nThe interviewed people specified their place of residence and their work place. Importantly,\nthis information allows to distinguish between (work-related) inbound and outbound trav-\nels. We assume that people are commuting regularly between the place of residence and\nthe work place, since remote work was not very common before the Sars-Cov2 pandemic.\nSince mobility between nearby cities might be stronger influenced by proximity of the cities\ninstead of the city sizes and we aim to depict with our model rather the spread of a disease\non a nationwide level rather than on a regional level, we filtered the traveler counts by ex-\ncluding mobility between municipalities closer to each other than 30 km. The rationale for\nthis threshold is that the mean commuting time in European countries is roughly about 40\nto 45 minutes per day, see [GNMV22]. Travelling within this time frame back and forth up\nto 30 km should be common. For France and Japan the data was extracted on a municipal\nlevel and reaggregated after filtering to urban areas (sourced by INSEE for France4and\nOECD for Japan5). For Poland the data was reaggregated to powiats. Municipalities not\nreferenced among the list of proposed urban units were left isolated, as they mostly are\nsmall and isolated thus contributing little to the core attraction effect. To evaluate the\neffect of this filtering, we performed our data analysis also for non-filtered data as well as\nfor data with a filtering that takes into account travels of distances of at least 50 km. In\nthe following we will abbreviate the corresponding datasets by D1+, D30+ and D50+. The\nresults of the D1+ and D50+ analysis can be found (mainly) in the appendix, because our\nmodel generally fits worse to this data (see Figures 17-22).\n4.2 Estimating cities attractiveness and inhabitants mobility\nWe used the empirical number of inbound and outbound travels of work-related mobility\nin France, Japan and Poland to estimate the values of aandbin the different countries.\nAccording to our model we assume that the probability pi(x) that a city of size xis\nchosen as a travel target is proportional to xa. Based on the travel counts to a city we\nempirically estimate this probability by the ratio of the ”number of travels to the city” to\n1https://www.insee.fr/fr/statistiques/4509353\n2https://www.mlit.go.jp/sogoseisaku/soukou/sogoseisaku_soukou_fr_000018.html\n3https://stat.gov.pl/obszary-tematyczne/rynek-pracy/opracowania/przeplywy-ludnosci-zwiazane-z-\nzatrudnieniem-w-2016-r-,20,1.html\n4https://www.insee.fr/fr/information/4803954\n5https://www.oecd.org/regional/regional-statistics/functional-urban-areas.htm\n21the ”total number of travels”. In case our model is good, we consequently assume that\ncxa≈# travels to the city\ntotal number of travels\nand hence log(number of travels to the city) ≈a·log(x) +c′for some appropriate constant\nc′. To arrive at an estimate of a, we fit a linear regression with least squares to the data\npoints (log( xi),log(number of travels to city i))i, where xidenotes the size of city iand set\nˆaequal to the slope of this regression line.\nSimilarly we estimate b. According to our model we assume that the probability that an\nindividual of a city of size xis travelling is proportional to xb. Hence on average we assume\nthat x·cxbmany individuals will leave a city of size xfor travel for some c >0. Based\non the outbound travel counts we estimate this number by the number of travels from the\ncity of size x. In particular, we assume log(number of outbound travels from the city) ≈\n(1 +b) log( x) +c′for some appropriate constant c′. Consequently, we fit a linear regression\nwith least squares to the data points (log( xi),log(number of outbound travels from city i))i\nto arrive at an estimate of b, and set ˆb+ 1 equal to the slope of this regression line.\nUsing the census data filtered according to D30+ described in Section 4.1 we estimate\nˆa= 0.96 and ˆb=−0.09 for France ,\nˆa= 1.05 and ˆb= 0.05 for Japan,\nand\nˆa= 1.95 and ˆb=−0.11 for Poland,\nsee Figure 2. These estimates (as well as the estimates that are obtained when filtering the\nmobility data according to D1+ and D50+) are also added in the Figure series9, where the\nefficiency of strategy (U) and (P) are compared for a range of aandbvalues.\n4.3 Infection probabilities and probabilities to trigger an outbreak\nIn Sections 3.1 and 3.2 we derive (implicit) formulas that allow for numerical calculations\nof infection probabilities of cities of size xas well as outbreak probabilities for epidemics\nstarting in a city of size xin the limit of an infinitely large graph. By means of simulations\nwe evaluate to which extend these asymptotic probabilities give a good approximation\nof the corresponding probabilities for infection processes on finite graphs in empirically\nrelevant parameter regimes. We compared the analytical results with infection and outbreak\nprobabilities that were derived by simulating epidemics. We simulated infection processes\non two different random graphs, that are abbreviated in the figures by KG for kernel graph\nand TG for transportation graph. For both graphs the number of vertices and the city sizes\ncorresponding to the vertices are taken from the data sets for France, Poland and Japan\nthat fulfill the distance restrictions D1+, D30+ or D50+, see Section 4.1. For the kernel\ngraph simulations edge probabilities are calculated according to Formula (3) (for strategy\n(P) and (U), abbreviated as strP and strU in the figure legend) based on the estimates of\nthe ”best fit” values of aandb, see Section 4.2. For the transportation graph simulations\n22Figure 2: Inbound travel counts (left column) and outbound travel counts for the estimation\nofaandb, resp., for France, Poland and Japan.\n23edge probabilities are determined by the empirical mobility matrices (based on commuters\ndata) given by\np(m)\ni,j= 1−exp\u0012\n−kI\nN\u0012tij\nxiL(xi) +tjiL(xi)\nxi\u0013\u0013\n.\nwhere tijdenotes the number of travels from city ito city j,L(x) =L∨for strategy (U) and\nL(x) =p∨xfor strategy (P). We adjusted the free parameter kI, such that the theoretical\ninfection probability / outbreak probability (given by (9) and (15)) is equal to 0.5 for cities\nof size 105for France and Japan and of size 105for Poland (in the figures these two versions\nfor adjustment are abbreviated as pI and pO, resp.). Note that adjusting kIis equivalent to\nadjusting R0. After the simulation of infection and outbreak probabilities for any city of the\n(finite) graph, we collect cities of similar size into 20 bins (such that every bin contains about\n30 cities for France, 15 cities for Poland and 40 cities for Japan). For each bin we calculate\nthe average infection and outbreak probability. The values are mapped as blue diamonds.\nThe plots concerning the infection probabilities can be found in Figure 3 and 4, as well\nas the plots concerning the outbreak probabilities can be found in Figure 5 and Figure 6.\nAdditional figures (Figures 15-22) can be found in the Appendix. To illustrate the variation\nof these probabilities within a bin, standard deviation (in the figures in red), minimal and\nmaximal values of the simulated probabilities (in the figures in violet) are plotted as well\nas the coefficient of determination abbreviated as R2in the figures is calculated6.\nFor a broad range of scenarios (varying city size heterogeneity, emmissivity and at-\ntractiveness coefficients as well as containment strategies) we observe that the analytical\ninfection probability function is in very good agreement with the empirical infection prob-\nabilities derived from simulations on the kernel graph. Furthermore infection probabilities\ncalculated from simulations based on the transportation graph are also in very good agree-\nment with the theoretical ones for France. For Poland the correspondence is worse and\nfor Japan it is not good. As the infection probabilities calculated on KG graphs do nicely\nagree with the analytical prediction, we suppose that larger deviations from the analytical\nprobability are neither due to the finite number of cities (in relation to the heterogeneity\nlevel in city sizes) nor due to the values of aandb. We rather hypothesize the reason for the\ndiscrepancy is the strong spatial structure in Poland and Japan. In Poland the level of the\ninfrastructure of the Western part can be clearly distinguished from the one of the Eastern\npart of the country. In Japan (work related) mobility depends strongly on the pathway\nof the Shinkansen. The centrality in France to Paris makes mobility between cities more\ndependent on city sizes than on geographical distances between cities.\nFor the outbreak probabilities we observe a very good agreement between simulated\nvalues and theoretical predictions under strategy (P) for both France and Poland. Given the\nsymmetry under strategy (P) between infection and outbreak probability for this strategy\nthis is expected.\nOn the other hand, we see that under strategy (U), the empirical outbreak probabilities\ntypically do not agree well with the analytic relation, and part of this effect can already be\nobserved for the simulations on the kernel graph.\n6In our context R2is not necessarily ≥0 since we compare theoretical outbreak/infection probabilities\nwith simulated outbreak/infection probabilities and do not perform a regression analysis\n24Figure 3: Comparison for France of simulated and theoretical infection probabilities, on the\nleft for strategy (P) (abbreviated as strP in the figures) and on the right for strategy (U)\n(abbreviated as strU in the figures). For the figures in the upper/lower row the R0value\nis adjusted such that for cities of size 105the theoretical outbreak/infection probability is\n0.5, see Section 3.2 for more details.\nFigure 4: Comparison of simulated and theoretical infection probabilities, on the left for\nstrategy (P) and on the right for strategy (U) for mobility data from Poland and Japan in\nthe upper and lower row, resp. The R0value is adjusted such that for cities of size 105the\ntheoretical infection probability of infection is 0.5, see Section 3.2.\n25Figure 5: Comparison of empirical and theoretical outbreak probabilities, on the left for\nstrategy (P) and on the right for strategy (U) for mobility data from France. For the\nfigures in the upper row/lower row the R0value is adjusted such that for cities of size 105\nthe outbreak probability/ probability of infection is 0.5, see Section 3.2 for more details.\nFigure 6: Comparison of empirical and theoretical outbreak probabilities, on the left for\nstrategy (P) and on the right for strategy (U) for mobility data from Poland and Japan in\nthe upper and lower row, resp. For the figures the R0value is adjusted such that for cities\nof size 105for Poland and of size 105fo Japan the probability of infection is 0.5, see Section\n3.2 for more details.\n26Figure 7: Comparison of the theoretical and simulated proportion of infected cities and iso-\nlated persons depending on the initial reproduction using the city size distribution filtered\naccording to D30+ from France and assuming either mobility according to the transporta-\ntion graph (TG) or according to the kernel graph (KG). For the simulated proportion in the\nlower row, we conditioned on outbreak sizes larger than 20 infected cities. The theoretical\nproportions (called KB in the legend) are calculated from analytical infection probabilities\nin the lower row. In the upper row an additional factor corresponding to the probability of\na successful outbreak is taken into account.\n27Figure 8: Comparison of the theoretical and simulated proportions of infected cities and\nisolated persons depending on the initial reproduction number for Poland, as in previous\nfigure for France, and for Japan (for Japan only without conditioning on an outbreak, since\nfor Japan outbreak sizes are typically smaller than 20 for the range of initial reproduction\nnumbers considered in the plot).\n28To assess further the influence of the geographical structure we plotted for a range of\nreproduction numbers the proportion of cities as well as people under isolation calculated\naccording to the theoretical probabilities as defined in (15) and the corresponding propor-\ntions when epidemics are simulated with respect to the transportation graph as described in\nthe beginning of this section, see Figures 7 and 8, where we can observe a good agreement\nfor France and Poland, but not for Japan.\nMore precisely, the simulated values of proportions of cities as well as people under\nisolation are taken as the averages over replicates (in the upper-panel) and as the averages\nover the subset of replicates with (relatively) large outbreaks of size at least 20. For each\nreplicate, the infection starts with a single infected city randomly chosen according to the\ndistribution νA.\nConcerning the analytical formulas, recall that for each considered initial reproduction\nnumber ra unique value of kIis associated to, from which the corresponding infection\nprobability function πr:R+7→[0,1] (according to (15)) and outbreak probability func-\ntionηr:R+7→[0,1] (according to (9)) can be deduced. Conditionally on an outbreak,\nthe expected proportion of cities (respectively of people) is given asR\nR+πr(x)β(dx) (re-\nspectively asR\nR+x·πr(x)β(dx)) and compared (as the KB curves) to the averages over\nthe replications with large outbreaks (the TG and KG curves). The unconditioned aver-\nages on the other hand are compared toR\nR+ηr(y)νA(dy)·R\nR+πr(x)µ(dx) (respectively toR\nR+ηr(y)νA(dy)·R\nR+x·πr(x)µ(dx)), where the factorR\nR+ηr(y)νA(dy) expresses the prob-\nability that an outbreak starts from a single city whose size is randomly chosen according\ntoνA.\n4.4 Comparison of strategies (U) and (P) in the general case\nIn the general case, where ais not necessarily equal to 1 + b, we do not have analytical\nresults on the efficiency of strategy (U) in comparison to strategy (P) and vice-versa as in\nProposition 3.3.1. Instead we compare strategy (U) and (P) numerically. We consider the\ncity size distributions (with a filtering according to D30+) from France, Japan and Poland\nand a parameter range for aandb, that covers the values of aandbwe estimated for\nmobility data from France, Japan and Poland.\nFurthermore, we consider three different degrees of severity of outbreaks, with the frac-\ntion of the population in quarantined cities set to 10%, 50% or 90%. Fixing this fraction\ndetermines uniquely the value Q⋆and as a consequence also the product mp:=kIp∨for\nstrategy ( P) and the product mL:=kIL∨for strategy ( U). It then follows from (17)\nand (16) that the ratio IP/IUdoes not depend on kI. Hence, in the numerical calcula-\ntions kIcan be chosen as an arbitrary positive real number, that fulfillsmp\nkI∈[0,1] and\nmL\nkI≤minx{β(x)}.\nGiven the values of p∨,L∨,kIas well as aandbwe calculate numerically the values of\nIPandIQ, i.e. the expected (asymptotic) proportion of infected individuals under strategy\n(P) and ( U) conditioned on Q⋆(which implies a large outbreak ), and plotted log( IP/IU)\nin Figure 9.\nIn these figures we also added the estimated values of aandbfor France, Japan and\nPoland, respectively. According to the estimated values of aandbboth strategies perform\n29log(IP)−log(IU)France\n Poland\n Japan\nFigure 9: Logarithm of the ratio of the expected number of isolated individuals under\nstrategy (U) vs. strategy (P) for a fixed number of infected individuals. The colour code\nfor the values of log( IP)−log(IU) is given on the right of the figures. Green points represent\nestimates of ( a, b) for each country, ×,•,⋆, resp. give estimated values of (a,b) for D1+,\nD30+ and D50+, resp. For the black line IP=IUand for the dashed black line a=b+ 1.\napproximately equally well in France and Japan, while for the estimated aandbvalues of\nPoland strategy Uperforms better. Concretely, the number of infected persons is roughly\nthree times larger under strategy Uif the fraction of quarantined cities is 90% and roughly\n1.5 times larger, if the fraction of quarantined cities is only 10%.\nWe can observe that the function IP/IUexhibits intriguing patterns and is strongly\ninfluenced by value of the fraction of quarantined cities as well as the city size distribution,\nsee Figure 9. For example for Poland the curve at which both strategies perform equally\nwell, i.e. at which IP=IU, turns from a concave curve for a large fraction of quarantined\ncities into a convex curve for a small fraction of quarantined cities. For France this curve\nis for two of three scenarios neither concave nor convex in the considered parameter regime\nofaandb. However, we can observe for all considered city size distributions that we\nhave a transition from a more concave to a more convex curve for decreasing fractions of\nquarantined cities.\n30Figure 10: Comparison for France (D30+), pI of aggregated (TG) and theoretical (KG)\ndegree distributions, on the left for strategy (P) and on the right for strategy (U). Inde-\ngrees/outdegrees are plotted in the upper/lower row respectively.\n31Figure 11: Comparison for Poland (D30+), pI of aggregated (TG) and theoretical (KG)\ndegree distributions, on the left for strategy (P) and on the right for strategy (U). Inde-\ngrees/outdegrees are plotted in the upper/lower row respectively.\nFigure 12: Comparison for Japan (D30+), pI of aggregated (TG) and theoretical (KG)\ndegree distributions, on the left for strategy (P) and on the right for strategy (U). Inde-\ngrees/outdegrees are plotted in the upper/lower row respectively.\n324.5 Indegree and Outdegree\nTo assess the fit of a (finite) graph generated by means of the kernel model with edge\nprobabilities given in (3) to the model based on the empirical mobility matrix we compared\nthe corresponding indegree and outdegree distribution, see Figure 10-12 for a scatter plot of\nindegrees vs outdegrees for (empirical) city size distributions (and mobility matrices) from\nFrance.\nSince a city can get infected from inside or outside, the indegrees i(P)\niandi(U)\nifor city i\nunder strategy (P) and (U), resp., are given in the model based on the empirical mobility\nmatrix a) for strategy ( P)\ni(P)\ni=X\njtjip∨+X\njtijp∨,\nwhere tklis the travel count from city kto city land b) for strategy ( U)\ni(U)\ni=X\njtjiL∨\nxj+X\njtijL∨\nxj.\nSimilarly, the outdregees o(P)\niando(U)\nifor city iunder strategy (P) and (U), resp., for\ncityiis given in the model based on the empirical mobility matrix a) for strategy (P)\no(P)\ni=X\njtjip∨+X\njtijp∨,\nand b) for strategy (U)\no(U)\ni=X\njtjiL∨\nxi+X\njtijL∨\nxi.\nNote that the indegree and outdregree vectors ( i(P)\ni)iand ( o(P)\ni)icoincide in the case of\nstrategy (P), be it for the transportation graph TG, the kernel graph KG(and the kernel\nbranching process KBthe two latter being identical by construction). This is not the case\nfor strategy ( U). Here we observe that the outdegrees are nearly constant (i.e. almost not\ndependent on the city size), while indegrees are clearly increasing with city sizes.\nIt is striking that the degrees in the transportation graph are much more variable than\nin the kernel graph, for cities of similar sizes. Nonetheless, for France and Poland (see\nFigures 10-11), the fit of the degrees in the transportation graph by the one in the kernel\ngraph is rather good in average over the bins (recall that there are 20 bins for each country,\nthus 15 to 40 cities per bin). The fit is a bit worse for the outdegree under strategy ( U), with\nthe kernel graph outdegree consistently overestimating the other. For Japan (see Figure 12),\nwe see that the fit of the degrees is rather good for small city sizes yet worsens as city sizes\nincrease, except for the outdegree under strategy ( U) where there is a strong discrepancy\n(still with the kernel graph estimate constantly larger than the other).\n33Figure 13: R0-expected value and variations calculated from simulated epidemics, see\nSection 3.4, as compared to the analytical value. The considered city size distribution is the\none of France filtered according to D30+. For the left plots epidemics have been simulated\nunder strategy Uand for the right plots under strategy P. For the figures on top epidemics\nhave been simulated with mobility according to the TG matrix, in the bottom according to\nthe KG matrix.\n4.6 Validity of the estimation of R0\nThe basic reproduction number is an important characteristic of an epidemic process. In\na branching process (approximation) it is the expected offspring number, i.e. the expected\nnumber of infections caused by a typical infected entity. In SIR models, like the one we\nare considering here, only in the very beginning a branching process approximates well the\nepidemic process, because the number of susceptible entities decreases over time and also\nthe distribution of the characteristic of infected entities, which is in our model the city size\ndistribution, changes over time. This makes it difficult to estimate the basic reproduction\nnumber.\nTo illustrate this we calculated estimates for the basic reproduction number from sim-\nulated epidemics. Since the beginning of an epidemic strongly depends on the city size of\nthe primarily infected city we start estimating the R0-value only when the epidemic has\nbeen run for several generations. For this purpose we filtered the simulations for epidemics\ngenerating a relatively large outbreak with at least 10 cities infected within one generation.\nFor the i-th such simulated epidemic let Z(i)\nnbe the number of infected cities in generation\n34nand and define T(i)as the first generation at which this process reaches at least 10 (i.e.\nZ(i)\nn≥10>max ℓ≤n−1Z(i)\nℓ). Given some N∈ Nassume that the simulations i1, ..., i kfor\nsome k∈ Nare still ongoing in generation N+T(ij)forj= 1, ..., k . We base the estimation\nofR0in generation Nafter the random time at which the threshold value 10 was reached\non the trajectories ( Z(ij)\nT(ij)+n)n=0,...,N forj= 1, ..., k .\nAssuming branching process dynamics, according to the theorem of Heyde-Seneta for\neach trajectory there exists a value W(i)with\nZn+T(i)≈W(i)·(R0)n+T\nand hence, we have\nln(Z(i)\nn+T)≈ln(W(i)) +Tln(R0) +nln(R0).\nThis motivates to infer the logarithm of the basic reproduction number on the basis of the\ni-th trajectory by a least-square regression in n∈[0, N] for a given N. If the outbreak has\nstopped before generation T(i)+N(or before T(i)is reached), the i-th trajectory is not\naccounted for. The inferred value is then denoted R(i)\n0(N) and we look how the distribution\nofR(i)\n0(N) over ivaries with increasing Nup to 9.\nIn Figure 13 the average over iof these estimates of R0are depicted for various scenarios,\ntogether with intervals corresponding to one standard deviation on both side and the 5 and\n95% quantiles. With values of Nup to generation 7, we always keep more than 10%\ntrajectories accounted for, (the evaluation is put to 0 in generations 8 and 9 in the left plots\nfor strategy (U) due to this lack of trajectories).\nFor the different panels of the figure, we used the city size distribution of France filtered\naccording to D30+ and adjusted kI( thus the expected R0value) according to rule pI, that\nis so that the infection probability obtained from the branching approximation for a city\nsize of 105is 0.5. The left panel shows the estimation under strategy ( P), the right panel\nunder strategy ( U). For the top panel we used the empirical mobility matrix (TG), and for\nthe bottom panel the kernel graph (KG, with the French D30+-city size distribution).\nIn any cases, the procedure for estimating the value of R0does not produce satisfying\noutcomes. The inferred values do not agree with the expected one, and the inferred values\nare rapidly declining with increasing N(though it reduces variability in the estimation). We\ncan also observe that the results for the top and bottom panels are extremely close, which\ndemonstrates that the reason for the bad estimation of R0lies in the city size distribution\nrather than more intricate aspects of the connection graph. This is discussed in more details\nin appendix Section 7.1.\n4.7 Eigenvector centrality\nComplementary to the outbreak probability the eigenvector centrality value provides ad-\nditional insight into which cities mainly drive the epidemic. For large city sizes and large\nenough infection rate, outbreak probabilities cannot be distinguished. Eigenvector central-\nities of these cities are nonetheless still different, especially under strategy (P), and higher\ncentrality values actually correspond to higher risk factors (figures not shown). This lends\nsupport to the incentive to restrict the cities according to the full order of size.\n35Figure 14: Eigenvector centrality for France (upper row) and Poland (lower row) in log-log\nplot, on the left for strategy (P) and on the right for strategy (U).\nAdmittedly, the largest cities are in any case most likely to become infected early without\nstrong prior regulation, so their contribution to the growth rate should not be sustained for\nlong. However, this is all the more reason to put in place preventive restrictions in order to\navoid secondary cases coming out of these cities.\nIn Section 3.7 we argued that in our analytical model (with a kernel of the form (6))\nthe eigenvector centrality value is composed of the two factors fAandfBwhich depend\non a log scale linearly on the city size. While for the French best fit kernel both factors\nare of almost of identical size, for the Polish best fit kernel the eigenvector centrality value\nis dominated for large cities by the factor fB, i.e. by infections imported by people the\nvisiting large cities from other cities, and for small Polish cities (below 105inhabitants) the\neigenvector centrality value is dominated by the factor fA, i.e. by infections imported into\nthe small cities by inhabitants visiting other cities and bringing back the disease.\nAs expected, the heterogeneity in eigenvector centrality is much more pronounced for\nstrategy ( P) than for strategy ( U), where the centrality value is close to a constant for\nPoland, slightly decreasing with population size for France, and slightly increasing for Japan,\ncf Fig 25. Since in France the estimated value of ais less than 1 and of bis less than 0, both\nattractiveness and emissivity reduce the role of large cities. Furthermore, under strategy\n(U) the number of infected citizens does not increase with the city size.\n36As noted in Subsection 3.7, eigenvector centrality values are independent of the threshold\nvalues, as would be outdegree with any natural normalisation. This is in stark contrast with\noutbreak probability, for which the inflection point (where the sigmoid function changes from\nbeing concave to convex) moves up with increasing threshold values (figures not shown).\nContrary to outdegrees and eigenvector centrality values, outbreak probability makes visible\nthe distinction between cities that are not likely to produce any outbreak and the others. On\nthe other hand, it might be misleading in suggesting that all cities well above the inflection\npoint are contributing equally. For strategy ( P), this suggestion is efficiently corrected by\nother measures like eigenvector centrality or outdegree.\n5 Discussion\nIn this manuscript we analyse a simplified but analysable model that mimics the spread of\nan infection between cities. We eased epidemic dynamics with respect to several aspects,\nwhich we will discuss next. A key tool in our analysis are approximations with branching\nprocesses. We use them to derive approximate formula for the probability of an city to get\ninfected or for the probability to trigger an outbreak. Furthermore, we base our definition\nof a basic reproduction number R0on an approximation with branching processes.\nThe city size distribution, however, is in the branching process time homogeneous, in\nparticular the tail of the distribution is not altered over time even though typically rather\nlarge cities get infected soon. While for outbreak and infection probabilities this assumption\ndoes not have such a big effect, one observes that the analytically calculated basic repro-\nduction number R0quickly deviates from the simulated one. In particular, for strategy ( P)\ntheR0value drops down quickly.\nFurthermore, we assume discrete generations. In particular, this means that indepen-\ndently of the size of an infected city it is assumed that once the city is infected it takes one\ngeneration in the branching process until the threshold number of individuals necessary for\nisolation gets infected. Due to the exponential speed at which infections within well-mixed\nentities generally expand, this assumption should be reasonable.\nIn addition the considered kernels are at most of rank two and polynomial. We see a\nrelatively good fit for these kernels, but other kernels could give a better fit to the actual\ntransmission patterns. In particular, we ignore completely that nearby cities might be more\nlikely to infect each other than cities that are far apart. Such models are substantially\nmore difficult to analyse. In particular, because dependencies between cities arise, so that\nbranching process approximations are no longer valid. With this respect it is surprising\nthat simulations based on the transportation graphs (which do not ignore spatial structure)\nagree well with the simulations based on a kernel graph. The fit is particularly good for\nFrench data. A possible explanation for the good fit could be the centralist transportation\nstructure of France which diminishes the impact of the geographic distance between cities.\nIn Poland the transportation network is in the east less developed than in the west and\nnorth. This implies that in particular in the eastern part of Poland geographical distances\nare more relevant. In Japan the linear island structure of the country influences strongly the\ntransportation network, even though cities located along the railway of the same Shinkansen\nline are effectively very close. These two effects generate a particular geometric structure.\n37To evaluate the fit of our model to actual transmission patterns we evaluate commuting\ndata retrieved from census data. An alternative source of mobility data is given by GPS\ndata obtained from mobile phones. An obstacle for the analysis of this kind of data are\nprivacy restrictions as well as their frequent only commercial availability. With this kind of\ndata one could also investigate the effect of mobility variability in time, e.g. in winter and\nin summer during holidays.\nGiven that during a pandemic many countries try to contain epidemic waves it would\nbe reasonable to investigate our model in more general settings, e.g. to model successive\nepidemic waves, e.g. in terms of an SIRS-like epidemic between cities, and to include\nvaccination.\n6 Summary and conclusion\nIn this study we were interested in the performance of containment regulations, that shall\nprevent the initial spread of a pathogen in a population within a country. For analytical\ntractability we considered a simple toy model, where the pathogen is assumed to spread on\na sparse random graph. The nodes of the graph represent cities or similar extended units\nthat get attacked by the pathogen. Inhabitants of the cities propagate the pathogen further\nby traveling between cities. We assume that the strength of mobility between cities only\ndepends on the sizes of the cities (and no other parameters, like geographical proximity or\nthe like). The probability to travel to and to travel from, resp., a city of size xis proportional\ntoxaandxb, resp., that is the parameter areflects the attractiveness of a cities and the\nparameter bthe travelling habit of inhabitants depending on the size of the city they are\nliving in.\nWithin a successfully infected city the infection is assumed to spread quickly until a\ncertain threshold value L(x), that depends (solely) on the city size x, of individuals get\ninfected. Afterwards the city gets isolated.\nWe consider two different functions L. In the proportional variant , that we call variant\n(P) for short, we set LP(x) =p∨x, i.e. a proportion p∨of individuals needs to get infected\ntill the city gets isolated, in the uniform variant , variant ( U) for short, we set LU(x)≡L∨,\ni.e. the number of individuals that need to get infected is L∨independent of the city size.\nTo analyse the infection process we approximate it by appropriate (forward and back-\nward) branching processes. Building on these approximations and on methods developed\nin [COC20] we identify (approximate) probabilities, that a city of size xgets infected and\ntriggers an outbreak, respectively.\nFor a comparison of the two strategies we say that strategy ( P) is more efficient than\nstrategy ( U), if under strategy ( U) less people need to get isolated, given that under both\nstrategies on average the same number of individuals get infected. In Proposition 3.3.1 we\nshow that in the case a= 1+ bstrategy P(asymptotically) outcompetes strategy U, iffa <1\nand both strategies perform equally well, if a= 1. This means that if the attractiveness\nof a city is growing stronger than linear in the city size, strategy ( U) should be preferred,\nbecause under strategy ( P) large cities would fuel the epidemic disproportionately high.\nTo infer which size distributions and parameter combinations of aandbare empirically\nrelevant we analysed mobility data from three data sets representing France, Poland and\n38Japan. They were derived from the census based estimation of work-related mobility ob-\ntained from statistical offices in France in 2017 in Poland in 2016 and in Japan in 2015.\nThe data sets were gathered several years ago before any Sars-Cov2 containment regulations\nwere arranged. For France and Japan, the data was extracted for municipal level of mobil-\nity and reaggregated to urban areas (sourced by INSEE for France) and OECD for Japan.\nFor Poland, the data was aggregated to powiats. Mobility for municipalities closer to each\nother than 30 km was ignored before the data got aggregated into larger units because our\nproposed model does not capture the very local diffusion effects due to travels of relatively\nshort duration.\nThe tails of the empirical city size distributions fit very well to the tails of power-law\ndistributions with coefficients that reflect very high levels of heterogeneity in city sizes.\nThe estimation of aandbwas conducted independently, building upon the projection of\nrespectively the total influx and total outflux as a function of the city size.\nInterestingly, the parameter combination for France and Japan happen to be very sim-\nilar, with a power law coefficient close to 1.8, an attractiveness a≈1 and an emissiveness\nb≈0.These values of aandbare all the more surprising that they correspond to a kind of\nneutral case: All contact pairs are equally likely, i.e. all individuals have roughly the same\nlikelihood to travel independent of the size of the city they are living in (i.e. b= 0) and\nthe target city is chosen proportional to its number of citizens ( a= 1). On the other hand\nfor Poland the power law coefficient is larger (close to 3) meaning that the population is\nmore evenly distributed between powiats. While the estimated emissiveness coefficient bis\nstill quite close to 0, the attractivity coefficient ais close to 2, meaning that attractivity of\npowiats is significantly biased towards larger powiats.\nThese inferences demonstrate how various the level of city heterogeneity can be. As\ncompared to administrative spatial units, we expect units of the form of Functional Urban\nAreas to reflect more accurately the heterogeneity in social contacts. This should help both\nto improve the reliability of the simplifications we made in the design of our non-spatial\nmodel and the effectiveness of targeted strategies. While our criterion predicts similar\nperformances of strategy ( U) over strategy ( P) for both France and Japan, it underlines\nthe potential interest of strategy ( U) for Poland: by restricting outbreaks also in large\npowiats early on, the diffusion of the disease is reduced in a more effective manner as\ncompared to a containment strategy that allows for longer growth in larger powiats which\nwould happen under strategy ( P).\nTo check the relevance of these predictions that rely solely on the contribution of popu-\nlation size, without any specific reference to the spatial distribution of cities, we conducted\nnumerical simulations of epidemics. The city size distributions are directly taken from the\nthree considered datasets (French Areas of Attraction, Polish Powiats and Japanese Urban\nAreas). Two different random graphs were considered to obtain the probabilities of trans-\nmission between cities. For the kernel graph (abbreviated by KG), this probability simply\ndepends on the following quantities: a scaling factor kIfor the stringency of the regulation,\nthe sizes of the target and source city, attractivity coefficient a, the emissiveness coefficient\nband the strategy. For the transportation graph (abbreviated by TG), the empirical matrix\nof connections between cities is directly weighted by the scaling factor kIfor the stringency\nof the regulation, after the effect of each strategy is taken into account. Beyond others\n39we calculated the infection probability (abbreviated as pI) and the outbreak probability\n(abbreviated as pO), as a function of city size based on a branching approximation and\nthen compare these theoretical probabilities to simulated probabilities that are obtained\nwith averages of infection outcomes for 20 to 40 cities of similar size over many epidemic\nruns, either for the kernel graph or for the transportation graph. We are led to distinguish\nmany situations, according to the country of reference, the regulation strategy ( PorU),\nand to the rule for adjusting the scaling factor kI.\nFor France and to a lesser extend for Poland, we observe a remarkably good fit by the\ntheoretical value of the simulated infection probabilities, even for TG although the fit is\nof course better for KG, both with rules pIandpO. By comparing the simulated and\ntheoretical proportions of infected cities and people for varying reproduction numbers, so\nvarying scaling factor kI, we observe that this fit is robust to the stringency of the regulation,\nwhich makes it a reasonable choice of optimization. The quality of the fit is particularly\nnoticeable given the level of simplifications and the heterogeneity in the data. The much\npoorer fit of the TG simulated infection probabilities for the Japanese data indicates that the\nspatial structure of the country is not well-captured by the dependency on city size, which\nis possibly linked to the more linear shape of migrations, structured along the Shinkansen.\nIt demonstrates at least that the fit is also very dependent on the specificity of the country\nand not simply on the coefficients ( φ,aandb), which are for France and Japan similar.\nFor outbreak probabilities, the fit does appear satisfying only under strategy ( P) or\nfor values very close to 1, again for France and to a lesser extent to Poland. Given that\nsymmetries in the models under strategy Pmake infection and outbreak probabilities to be\ndefined as the same quantity in the branching approximation, that the fit is much better in\nthis case is not so surprising. Under strategy ( U) for any country however, the theoretical\nvalues of outbreak probability largely overcome the proportion of runs for which significant\noutbreak are indeed produced. This is all the more distinct when these probabilities are not\ntoo close to either 0 or 1. This effect can partly be explained by the discrepancy between\nthe finite population model and the branching approximation.\nReferences\n[ABP14] J. Alstott, E. Bullmore, and D. Plenz. powerlaw: A python package for analysis\nof heavy-tailed distributions. PLOS ONE , 9(1):1–11, 2014.\n[BJR07] B. Bollobas, S. Janson, and O. Riordan. The phase transition in inhomogeneous\nrandom graphs. Random Structures & Algorithms , 31(1):3–122, 2007.\n[Bon87] P. F. Bonacich. Power and centrality: A family of measures. Am. J. Sociol. ,\n92:1170–1182, 1987.\n[COC20] J. Cao and M. Olvera-Cravioto. Connectivity of a general class of inhomoge-\nneous random digraphs. Random Structures & Algorithms , 56:722–774, 2020.\n[GNMV22] J.I. Gim´ enez-Nadal, J.A. Molina, and J. Velilla. Trends in commuting time\nof european workers: A cross-country analysis. Transport Policy , 116:327–342,\n2022.\n40[KPS66] H. Kesten and B. P. Stigum. A limit theorem for multidimensional galton-\nwatson processes. Ann. Mathem. Stat. , 37(5):1211–1223, 1966.\n[New10] M.E.J. Newman. Networks : an introduction . Oxford University Press, 2010.\n[SBB20] A. J. Stier, M. G. Berman, and L. M. A. Bettencourt. Covid-19 attack rate\nincreases with city size. Health Economics eJournal , 2020.\n417 Supplementary material\nHere, we provide additional figures to analyse the robustness of our results. In particular,\nwe investigate another method to adjust the value of kI(based on the outbreak probability),\ndifferent filtering methods of our datasets (regarding the travel distances, see Subsection\n4.1), and the role of the city size distribution for the quality of R0estimation.\n42Figure 15: Comparison of simulated and theoretical infection probabilities, on the left for\nstrategy (P) and on the right for strategy (U) for mobility data from Japan and Poland.\nTheR0value is adjusted such that for cities of size 105the theoretical outbreak probability\nis 0.5 (thus pO), see Section 3.2.\nFigure 16: Comparison of simulated and theoretical outbreak probabilities, on the left for\nstrategy (P) and on the right for strategy (U) for mobility data from Japan and Poland.\nTheR0value is adjusted such that for cities of size 105the theoretical outbreak probability\nis 0.5, see Section 3.2\n43Figure 17: Comparison of simulated and theoretical infection (top row) and outbreak prob-\nabilities (middle row) and of infected cities and isolated cities as a function of the theoretical\nR0 value (bottom row), on the left for strategy (P) and on the right for strategy (U), for\nFrance without restriction on the distance. The R0value is adjusted such that for cities of\nsize 105the theoretical infection outbreak probability is 0.5, see Section 3.2.\n44Figure 18: Comparison of simulated and theoretical infection (top row) and outbreak prob-\nabilities (middle row) and of infected cities and isolated cities as a function of the theoretical\nR0 value (bottom row), on the left for strategy (P) and on the right for strategy (U), for\nFrance with a restriction on the distance of 50km. The R0value is adjusted such that for\ncities of size 105the theoretical infection outbreak probability is 0.5, see Section 3.2.\n45Figure 19: Comparison of simulated and theoretical infection (top row) and outbreak prob-\nabilities (middle row) and of infected cities and isolated cities as a function of the theoretical\nR0 value (bottom row), on the left for strategy (P) and on the right for strategy (U), for\nPoland without restriction on the distance. The R0value is adjusted such that for cities of\nsize 105the theoretical infection outbreak probability is 0.5, see Section 3.2.\n46Figure 20: Comparison of simulated and theoretical infection (top row) and outbreak prob-\nabilities (middle row) and of infected cities and isolated cities as a function of the theoretical\nR0 value (bottom row), on the left for strategy (P) and on the right for strategy (U), for\nPoland with a restriction on the distance of 50km. The R0value is adjusted such that for\ncities of size 105the theoretical infection outbreak probability is 0.5, see Section 3.2.\n47Figure 21: Comparison of simulated and theoretical infection (top row) and outbreak prob-\nabilities (middle row) and of infected cities and isolated cities as a function of the theoretical\nR0 value (bottom row), on the left for strategy (P) and on the right for strategy (U), for\nJapan without restriction on the distance. The R0value is adjusted such that for cities of\nsize 105the theoretical infection outbreak probability is 0.5, see Section 3.2.\n48Figure 22: Comparison of simulated and theoretical infection (top row) and outbreak prob-\nabilities (middle row) and of infected cities and isolated cities as a function of the theoretical\nR0 value (bottom row), on the left for strategy (P) and on the right for strategy (U), for\nJapan with a restriction on the distance of 50km. The R0value is adjusted such that for\ncities of size 105the theoretical infection outbreak probability is 0.5, see Section 3.2.\n497.1R0estimation: number of cities, size heterogeneity, graph structures\nRecall from Section 4.6 that the R0value could not get reliably estimated from the dynamics\nin the number of cities infected at each generation (even when we average over replications of\noutbreaks). So we investigate the effects responsible for this bad quality of the R0estimation\npossibly due to ( i) the too small number of cities, ( ii) a tail of city size distributions that\nis too heavy and ( iii) the specific graph structure and check the validity of our proposed\nprocedure, for a range of powerlaw distributions.\nConcerning (i), we anticipate that large cities are quickly hit and then isolated. This\nstrongly reduces the number of secondary infections.To reduce this effect we increase the\nnumber of cities by a factor 10 (without changing the city size distribution), by replicating\neach city size ten times. For both clarity and computational efficiency, we produce a kernel\ngraph structure on this extended dataset, which we abbreviate as “10KG”, and compare\nin Figure 23 the results of the R0estimation between the original kernel graph and the\nextended kernel graph. The R0values is in both cases adjusted such that for cities of size\n105the theoretical infection outbreak probability is 0.5, see Section 3.2, which leads to\nidentical values between these two graphs (and for the kIvalues as well). There is a clear\nimprovement with the extended graph, yet the estimations come close to the theoretical\nvalue only under strategy ( U) for a few generations (then still with large variations between\nruns). Furthermore, the decline of the estimated value with the number of generations\nconsidered is more strikingly visible. This confirms that the rapid establishment of immunity\nhinders the estimation of R0.\nConcerning (i) and (ii) we then consider other size distributions of 10 ,000 cities (of the\norder associated to 10 KG) that are randomly sampled according to powerlaw distributions\nwith various coefficients, namely ϕ= 2,3,4 and 5. In Figure 24, the corresponding estima-\ntions of R0are compared between these different distributions, both under strategy ( P) (on\nthe left) and under strategy ( U) (on the right) in the upper-panel with the original threshold\nof 10 infected cities (in a single generation, for the start of the estimation interval) and in\nthe lower-panel with a larger threshold of 100 infected cities. As in Figure 23, the R0values\nis adjusted such that for cities of size 105the theoretical infection outbreak probability is\n0.5. Since the corresponding values differ between the various sampled distributions, the\nestimations are rescaled by the expected value to ease the comparison. Furthermore we set\nthe estimated R0-value to 0 for those generations for which less than 20 replicates (out of\nthe 200 produced) keep a persistent outbreak over the whole interval of estimation.\nWe observe that the very low quality of R0-estimation is also present with the most\nheterogeneous distribution, sampled according to a power-law coefficient of ϕ= 2, yet\nmuch more suitable for the other distributions. The fluctuations between replicates are\nstill very large when estimation starts with 10 infected cities. With the larger threshold\nof 100 infected cities, these fluctuations are largely reduced, which improves the quality\nof estimation though the estimator is significantly biased downwards. The increase in the\nnumber of generations produces similar effects as the one of the threshold, though not as\nlarge for the considered values.\nFinally, concerning (iii), simplifying the graph structure does not lead to a significant\nimprovement of the estimation, as can be seen in the comparison between the transportation\ngraph and the kernel graph (see Figure 13).\n50Figure 23: R0 estimation for the kernel graphs derived from the values of aandbof France\n(D30+) with the associated city size distributions (KG, top tow) compared to the graph\nwhere these nodes are duplicated 10 times (10KG, bottom row), on the left for strategy (P)\nand on the right for strategy (U).\nFigure 24: R0 estimation for the kernel graphs derived from the values of aandbof France\n(D30+) comparing city size distributions of 104cities (of the order of 10KG) given with\npower-laws of coefficient 2, 3 4 and 6, on the left for strategy (P) and on the right for strategy\n(U). For the top row, the original threshold of 10 cities currently infected is considered, while\nfor the bottom row, this threshold is set to 100 cities.\n517.2 Eigenvector centrality for Japan\nThe plots display very similar patterns as for France, cf Section 4.7. Due to the lower\nquality of the kernel graph approximation for Japan, the exact values are not very reliable,\nyet the general trend still deserves to be remarked.\nFigure 25: Eigenvector centrality for Japan, on the left for strategy (P) and on the right\nfor strategy (U).\n7.3 Infection/outbreak probabilities: quality of the analytical estimations\nWe present in Figure 26 the convergence rate of the iterated procedures towards the infec-\ntion/outbreak probabilities, see Section 3.1 and 3.2. For both strategies ( U) and ( P), the\nplots display a very clear trend of convergence (linear in log-scale) for both ( ηk\nA),(ηk\nB), see\n(13) and (14), towards ηAandηB, see (11) and (12), as well as of (the analogously defined\nquantities) ( πk\nC) and ( πk\nD) towards the analogously defined probabilities πCandπD.\n52Figure 26: Convergence of ( ηk\nA),(ηk\nB), see (13) and (14), towards ηAandηB, see (11)\nand (12), as well as of (the analogously defined quantities) ( πk\nC) and ( πk\nD) towards the\nanalogously defined probabilities πCandπD, resp. for strategy (P) and (U), resp., on the\nleft and right. As a city size distribution βthe empirical size distribution from France\n(D30+) is chosen. The upper row shows log10(|ηk\nA−η30\nA|/η30\nA) as well as the analogous\nquantity relative to ηBat the different iterations k= 0, ...,25, the lower row shows the\nanalogous values for πCandπD. The constant kIis chosen such that the theoretical\nprobability of infection/outbreak (approximated at a large enough iteration) is 0.5. In the\nfigure legend these two versions are abbreviated by pI and pO.\n53"    },    {        "title": "2402.03160v2.Equations_of_genus__4__curves_from_their_theta_constants.pdf",        "content": "arXiv:2402.03160v2  [math.AG]  18 Mar 2024EQUATIONS OF GENUS 4CURVES FROM THEIR THETA\nCONSTANTS\nJEROEN HANSELMAN, ANDREAS PIEPER, SAM SCHIAVONE\nAbstract. In this article we give explicit formulas for the equations o f\na generic genus 4 curve in terms of its theta constants. The me thod uses\nthe Prym construction and the beautiful classical geometry around it.\nContents\n1. Introduction 1\n1.1. Motivation and applications 3\n1.2. Previous work 4\n1.3. Structure of the article 5\n1.4. Acknowledgements 5\n2. Preliminaries 6\n2.1. The Prym canonical map 6\n2.2. Two linear maps 7\n2.3. Milne’s bijection 10\n2.4. The bielliptic case 11\n2.5. Tritangent planes and theta derivatives 12\n2.6. The generalized Jacobi derivative identity 13\n3. Reconstructing the curve 14\n3.1. An auxiliary set of odd theta characteristics 14\n3.2. Finding the quadric 16\n3.3. Finding a cubic 19\n3.4. Main result 24\n4. Examples 26\n5. Future work 27\n5.1. Constructing CM curves. 27\n5.2. Higher genus 28\nReferences 29\n1.Introduction\nRecall that over k=Cthe theta constants of a principally polarized\nabelian variety (p.p.a.v.) of dimension gwith (small) period matrix τ∈Hg\nare given by the 2g−1(2g+ 1) numbers ϑ/bracketleftbiga\nb/bracketrightbig\n(0,τ) witha,b∈1\n2Zgrunning\n12 HANSELMAN, PIEPER, SCHIAVONE\nthrough a set of representatives of pairs in1\n2Zg/Zgsatisfying 4 atb≡0\nmod 2. Here ϑ/bracketleftbiga\nb/bracketrightbig\nis the Riemann theta function with characteristics, given\nby\nϑ/bracketleftbiga\nb/bracketrightbig\n(z,τ) =/summationdisplay\nn∈Zgexp/parenleftbig\nπi(n+a)tτ(n+a)+2πi(n+a)t(z+b)/parenrightbig\n.\nMumford [Mum66] gave a purely algebraic definition of the the ta constants\nthat works over any algebraically closed field kof characteristic /\\e}atio\\slash= 2. He also\nshowed that the theta constants determine the principally p olarized abelian\nvariety uniquely.\nThe goal of this article is to provide closed formulas for rec overing explicit\nequations of a generic genus 4 curve C, given the values of its algebraic theta\nconstants. The outcome is a quadric Qand a cubic Γ in P3such thatQ∩Γ\nis equal to the image of the canonical embedding of C. Our main result is\nthe following (Theorem 3.11 below).\nTheorem. LetCbe a generic genus 4curve over an algebraically closed\nfieldkof characteristic 0 or of characteristic pwithplarge enough. There\nare explicit formulas for the equations of Cin terms of its theta constants.\nThey use only the following algebraic operations: Elementar y arithmetic,\ntaking square roots, and solving linear systems of equation s.\nThe problem of recovering the equations of a curve from its th eta con-\nstants was classically studied by Rosenhain (genus 2) and Ar onhold-Weber\n(plane quartics). Later Takase [Tak96] generalized Rosenh ain’s work to ar-\nbitrary hyperelliptic curves. In all these cases they explo it the fact that\nthe moduli space of these curves equipped with a full level 2 s tructure is\nrational. In the hyperelliptic case this rationality follo ws from the existence\nof Weierstrass equations, and for plane quartics the formul as of Aronhold\ngive an explicit rational parametrization.\nIn genus 4 we are faced with a different situation, as the moduli space of\ngenus 4 curves with a full level 2 structure is not unirationa l. (This will be\nproved in a later article [HPS24a].) This fact may explain wh y the general\nproblem has remained open, only garnering results in specia l cases. For\ninstance, Schottky [Sch88] managed to solve the problem in t he case where\none of the theta constants vanishes, which is equivalent to t he canonical\nquadric being a cone.\nAnother phenomenon not appearing in the theory of plane quar tics is\nthat not every p.p.a.v. is a Jacobian when g/greaterorequalslant4. In fact, a 4-dimensional\n(indecomposable) p.p.a.v. is the Jacobian of a smooth curve if and only if\nthe Schottky modular form vanishes [FR70, Chapter 5][Igu81 ][Fre83].\nRecall from [FR70, Chapter 5] that the Schottky modular form is related\nto the Prym construction as follows. Given an ´ etale double c over˜C→\nC, the Prym variety Prym( ˜C/C) = ker(Nm : Jac( ˜C)→Jac(C))◦is in a\nnatural way a principally polarized abelian variety. In par ticular, the theta\nconstants of Prym( ˜C/C) must satisfy the quartic Riemann identities. TheEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 3\nSchottky modular form is derived from these identities by ex pressing the\ntheta constants of Prym( ˜C/C) in terms of the theta constants of Jac( C).\nThis suggests that the Prym construction could play a role in the problem\nofreconstructingthecurvefromthethetaconstants, andth isistheapproach\nwe pursue here. Furthermore, we will use a beautiful classic al geometric\nconstruction concerning the Prym due to Caporali, Wirtinge r, P. Roth (see\n[Cob29] for a classical textbook reference), and W. P. Milne [Mil23]. The\nmodern development was pioneered by Recillas [Rec71] and he nceforth it is\ncommonly referred to as Recillas’s trigonal construction. In this article we\nmainly follow the articles of Catanese [Cat81] and Bruin–Se rt¨ oz [BS20] that\nare closer to the classical works.\n1.1.Motivation and applications. In addition to the intrinsic value of\nrecovering a genus 4 curve from its theta constants (or its pe riod matrix),\nour method also has several applications. It is interesting in arithmetic and\ncomplex geometry to be able to find higher genus curves with sp ecial prop-\nerties. Being able to reconstruct a curve from the theta cons tants (or period\nmatrix) of its Jacobian allows us to construct Jacobians wit h interesting\nproperties and find corresponding curves that inherit those properties. Be-\nlow we list a couple of examples that illustrate this strateg y. Theseexamples\nwere computed using our Magma[BCP97] implementation of our algorithm\n[HPS24c].\n•Gluing. In [HSS21], the first and third authors, together with Sijs-\nling, describedseveral methods for gluing genus 1 and genus 2 curves\nalong their torsion. More precisely, let X1be a curve of genus 1 and\nletX2be a curve of genus 2 with Jacobian varieties J1andJ2, re-\nspectively. Gluing X1andX2along their 2-torsion means finding a\ncurveX3with Jacobian J3and an isogeny φ:J1×J2→J3such\nthat kerφis contained in the 2-torsion of J1×J2.\nThe analytic method described in [HSS21, §2] for computing the\nperiod matrices of such gluings easily generalizes to highe r genera.\nHowever, after constructing the relevant period matrix, it still re-\nmains to (a) determine whether the corresponding abelian va riety is\na Jacobian; and (b) if so, recover the equation of the corresp onding\ncurve. In Example 4.1, we show how our algorithm can be used to\naccomplish this last step, allowing one to glue two genus 2 cu rves\nalong their 2-torsion.\n•Constructing modular abelian varieties . Shimura associates to\na Hecke newform fwith level Γ 1(N) a subvariety Afof the modular\nJacobianJ1(N). ThisAfis simple and End Q(Af)⊗Qcontains the\nfieldKgenerated bythecoefficients ofthe q-expansionof f. Further-\nmore,Kis totally real and one has [ K:Q] = dim(Af). Conversely,\nKhare and Wintenberger [KW09a, Corollary 10.2], [KW09b] ha ve\nshown that every simple abelian variety AoverQwith the property\nthat End Q(A)⊗Qcontains an RM field Ksuch that [ K:Q] =4 HANSELMAN, PIEPER, SCHIAVONE\ndim(A), must be modular. This means that there exists a Hecke\nnewformfwith level Γ 1(N) such that Ais isogenous to Af. Fur-\nthermore,Nis then equal to the conductor of A.\nJohn Cremona [Cre97] gave a method for computing the period\nmatrix of such modular abelian varieties and used this to con struct\nmany examples of modular elliptic curves. Since the computa tion of\nthe period matrix works for general g, one can use the method of the\npresent article to compute curves of genus 4 whose Jacobian i s iso-\nmorphic to Af. Although there is no general reason why Afshould\nbe a Jacobian, Noam Elkies (private communication) provide d an\nexample of an eigenform fsuch thatAfis the Jacobian of a genus\n4 curve. In Example 4.2 we recover this genus 4 curve over Qwhose\nJacobian is a modular abelian variety with real multiplicat ion by the\nmaximal totally real subfield of Q(ζ15).\n1.2.Previous work. Lehavi [Leh15] gave an effective method for recon-\nstructing a non-hyperelliptic genus 4 curve from its tritan gent planes, which\npartially inspired this paper. Lehavi’s method is based on t he cartesian\ndiagrams\nS2H0(ΩC(η))×H0(Ω⊗2\nC)S2H0(ΩC)S2H0(ΩC(η))\nS2H0(ΩC) H0(Ω⊗2\nC)ϕ\n(1.1)\nwhereηrunsthroughallthe255non-trivialtwo-torsionpointsinJ ac(C)[2]\\{0}\nandS2denotes the symmetric square. He focuses on the vector space\nS2H0(ΩC(η))×H0(Ω⊗2\nC)S2H0(ΩC)\nratherthanthemap ϕ. Thenovelty ofthepresentpaperisalinkbetweenthe\nmapϕand the Prym construction. This allows us prove that the know ledge\nofϕfor just one two-torsion point ηis sufficient to recover the curve C. In\nthis way our method is more efficient than Lehavi’s, but our res ult does not\nlogically imply his main theorem. The reason is that we do not only use\nthe tritangent planes, but in addition information coming f rom the Prym\nvariety.\nVarious mathematicians have written articles about recons tructing genus\n4 curves using theta functions (starting from the period mat rices of their\nJacobians). Agostini, C ¸elik, and Eken [AC ¸E22] describe a method using\nDubrovin threefolds to numerically reconstruct a curve of a rbitrary genus\nfrom the period matrix their Jacobian. They use theta deriva tives up to\norder 4 to compute degree 4 polynomials vanishing on the curv e. In genus\n4 they require Gr¨ obner basis computations to write the equa tions as the\nintersection of a quadric and a cubic.\nIn[Kem86]Kempfdescribesamethodtoreconstructagenus4c urveusing\nthe singular point of the theta divisor. This was implemente d in the workEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 5\nof Chua, Kummer, and Sturmfels [CKS18]. In order to find this s ingular\npoint they need to solve a highly non-linear system for compu ting the point\nwhere both a theta function and all its derivatives vanish.\nOur method is different in that we give explicit formulas for th e equations\nof the reconstructed curve Cin terms of its theta constants. In order to\ndo this we use only algebraic operations, namely elementary arithmetic,\nextraction of square roots, and solving linear systems of eq uations. We\ndirectly obtain a model for Cas the intersection of a quadric and cubic in\nP3without the need for further computation.\nThere are several advantages of using just theta constants a nd no deriva-\ntives. First, it is much better understood how to compute the ta constants\nefficiently; for example, Labrande and Thom´ e’s algorithm [L T16]. In a\nforthcoming article, Elkies and Kieffer [EK24] will describe an algorithm\nfor computing theta constants with quasi-linear complexit y. Second, when\nusing derivatives up to order n, the total number of evaluations is multiplied\nby/parenleftbigg+n\nn/parenrightbig\n; the number of possibilities of taking a partial derivative .\n1.3.Structure of the article. The article is organized as follows. In Sec-\ntion 2 we begin by defining the objects that will play key roles in our con-\nstruction. In Section 3 we describe our main result: an algor ithm that takes\ntheta constants of the Jacobian of a genus 4 curve as its input and, using\nonly elementary arithmetic, extraction of square roots, an d solving linear\nsystems of equations, outputs the corresponding quadric an d cubic whose\nsolution set is equal to the image of the canonical embedding of the corre-\nsponding curve. In Section 4 we present two examples as an app lication of\nour method. First, we construct a gluing with an interesting endomorphism\nalgebra, and then the curve whose Jacobian is the modular abe lian variety\ndiscussed above, i.e., with real multiplication by the maxi mal real subfield\nofQ(ζ15). Finally, in Section 5 we briefly discuss how our method can b e\napplied to find more interesting curves, as well as how it can b e generalized\nto higher genus.\nFor an outline of the various steps in our method, see Algorit hm 3.12.\n1.4.Acknowledgements. TheauthorswouldliketothankThomasBouchet,\nNils Bruin, Edgar Costa, Igor Dolgachev, Noam Elkies, Gavri l Farkas, Sam\nGrushevsky, Avi Kulkarni, Elisa Lorenzo Garc´ ıa, David Lub icz, Martin\nRaum, and Christophe Ritzenthaler for helpful conversatio ns during the\nwriting of this article. Furthermore, we thank Brendan Hass ett, David\nLehavi, and Jeroen Sijsling for reading a previous version o f the manuscript\nand giving suggestions to improve the article.\nThefirstauthorissupportedbyMaRDI,fundedbytheDeutsche Forschungs-\ngemeinschaft (DFG), project number 460135501, NFDI 29/1. T he third au-\nthor was supported by the Simons Collaboration in Arithmeti c Geometry,\nNumber Theory, and Computation via Simons Foundation grant 550033.6 HANSELMAN, PIEPER, SCHIAVONE\n2.Preliminaries\n2.1.The Prym canonical map. This section summarizes results from\nCatanese’s article [Cat81]. Let Cbe a non-hyperelliptic genus 4 curve,\nη∈Jac(C)[2]\\ {0}be a non-trivial 2-torsion point. The Prym canonical\nmap is the map\nφη:C−→P(H0(ΩC(η)))∼=P2\nassociated to the linear system |ΩC(η)|. The map φηis either a ramified\ndegree 2 map onto a smooth cubic (the bielliptic case) or a bir ational map\nonto a singular plane sextic. (For details, see [Cat81, §1].) We will discuss\nthe bielliptic case in Section 2.4 below. Thus assume now we a re in the\nnon-bielliptic case and hence the Prym canonical curve im( φη) is a singular\nplane sextic embedded in P2. Letb1:S→P2be the repeated blow-up of P2\nin the singular points of the Prym canonical curve. Denote by E1,...,Ek\nthe total transform of the exceptional divisors of the succe ssive blow-ups.\nOnSwe can consider the line bundle L=b∗\n1OP2(3)⊗O(−/summationtextk\nj=1(rj−1)Ej)\nwhererjis the multiplicity of the singular point blown up in the jth step.\nThe line bundle Linduces a map b2:S→P(H0(ΩC))∼=P3whose image,\nwhich we denote by Γ η, is a cubic symmetroid [Cat81, p. 37]. We recall the\ndefinition of a cubic symmetroid.\nDefinition 2.1. Acubic symmetroid is a cubic surface Vwhich is the\nvanishing scheme of the determinant of a symmetric 3 ×3 matrix of linear\nforms. That is, there exists a symmetric 3 ×3 matrixAwith entries aij∈\nk[x0,...,x 3] homogeneous of degree 1 such that V=V(det(A)).\nWe define a rational map c:P2/axisshort/axisshort/arrowaxisrightΓηas the composition of b−1\n1with the\nmapb2:S→Γη. Thesituationissummarizedbythefollowingcommutative\ndiagram.\nS\nC P2Γη⊆P3b2 b1\nφη c(2.1)\nThe mapb2:S→Γηis a blow-up; this follows from the observations in\n[Cat81, p. 37]. Furthermore, the composition c◦φη:C→P3is equal to\nthe canonical embedding and hence independent of η. This will be shown\nin the proof of Lemma 2.6 below.\nWe will describe the situation for a generic C. In this case, the Prym\ncanonical image im( φη) has six nodes that are the pairwise intersection of\nfour lines, say n1,...,n 4. The surface Sis the blow-up in these six nodes.\nThe strict transforms of the niunderb1are (−2)-curves. These are con-\ntracted byb2toA1singularities on Γ η.\nThere are two possible degenerate cases. First, it could hap pen that two\nof the lines nicoincide. In this case, the two corresponding A1-singularities\ncombine to form a single A3-singularity. Then Γ ηhas three singular points:\nTwo of type A1and one of type A3.EQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 7\nSecond, three of the lines nicould coincide. In that case, Γ ηhas one\nsingularity of type A1and one of type A5. For more details, see [Cat81, p.\n33].\nThemain consequence of Catanese’s construction is thefoll owing theorem\nthat he attributes to Wirtinger–Coble–Recillas.\nTheorem 2.2. The mapη/ma√sto→Γηgives a bijection between:\n•Jac(C)[2]\\{0}.\n•Irreducible cubic symmetroids containing the canonical ima ge ofC\ninP3.\nProof.See [Cat81, Theorem 1.5]. /square\n2.2.Two linear maps. We will now define a linear map ϕthat maps\ncertain quadratic forms on the right P3to quadratic forms on the left P2.\nWe begin by considering the following cartesian diagram.\nS2H0(ΩC(η))×H0(Ω⊗2\nC)S2H0(ΩC)S2H0(ΩC(η))\nS2H0(ΩC) H0(Ω⊗2\nC)(2.2)\nwhere the maps S2H0(ΩC(η))→H0(Ω⊗2\nC), S2H0(ΩC)→H0(Ω⊗2\nC) are given\nby multiplication. The injectivity of the right vertical ar row expresses the\nfact that there are no quadrics vanishing on im( φη).\nDefinition 2.3. Following Lehavi [Leh15, p. 2] we define VC,η⊂S2H0(ΩC)\nas the image of the embedding\nS2H0(ΩC(η))×H0(Ω⊗2\nC)S2H0(ΩC)֒→S2H0(ΩC).\nIdentifying S2H0(ΩC(η))×H0(Ω⊗2\nC)S2H0(ΩC) with its image VC,η, we define\nLehavi’s map ϕto be the projection map onto the first factor\nϕ:VC,η→S2H0(ΩC(η)).\nFrom the facts that dim( S2H0(ΩC)) = 10,dim(S2H0(ΩC(η))) = 6 and\ndim(H0(Ω⊗2\nC)) = 9, and that the bottom map S2H0(ΩC)→H0(Ω⊗2\nC) is\nsurjective, then dim( VC,η) = 10 + 6 −9 = 7 by elementary properties of\ncartesian diagrams. Furthermore, it will later turn out to b e useful that\nker(ϕ) is one-dimensional and generated by the quadratic form Qvanishing\nonC.\nNext, we define a right inverse ψforϕusing the following lemma.\nLemma 2.4. Assume that Γηis generic. Let us denote by Ni⊂S, i=\n1,...,4the exceptional divisors of b2. Thenb∗\n1O(2)∼=b∗\n2O(2)⊗OS(−/summationtextNi).\nIfΓηis not generic, then the Nihave to be taken with appropriate multi-\nplicities.8 HANSELMAN, PIEPER, SCHIAVONE\nProof.Consider the curves ni=b1(Ni). Then, by the discussion following\nDefinition 2.1, the ni∈P2are lines such that the 6 singular points of im( φη)\nare the intersection points of pairs from {ni|i= 1,...,4}.\nBy definition of b2we have an isomorphism\nb∗\n2O(1)∼=L=b∗\n1OP2(3)⊗OS\n−6/summationdisplay\nj=1Ej\n.\nThese two facts imply the lemma. /square\nWe define\nWη:= H0/parenleftig\nS,b∗\n2O(2)⊗OS/parenleftig\n−/summationdisplay\nNi/parenrightig/parenrightig\nand consider it as a vector subspace of H0(S,b∗\n2O(2))∼=S2H0(C,ΩC). In\nthe generic case, Wηis the vector space of quadrics vanishing at the nodes of\nΓη. Inanycaseonehasdim( Wη) = 6. Thiscanbeseenfromanisomorphism\nwhich we will give now:\nψ:S2H0(C,ΩC(η))∼−→Wη\nwhich is defined to be the composition of isomorphisms\nS2H0(C,ΩC(η))∼=H0(P2,O(2))b∗\n1∼→H0(S,b∗\n1O(2))∼=H0(S,b∗\n2O(2)⊗OS(−/summationtext\niNi)).\nGeometrically this means the following: for any q∈S2H0(C,ΩC(η)), the\nimageψ(q) is the unique (up to scaling) quadratic form that cuts out th e\ncurve\nc(V(q)) =b2(b−1\n1(V(q)))⊂Γη.\nRemark 2.5. The mapψis only well-defined up to multiplication by a\nscalar because of the arbitrary choice of an isomorphism b∗\n1O(2)∼=b∗\n2O(2)⊗\nOS(−/summationtextNi). We resolve this ambiguity in the proof of the following lem ma.\nThe following lemma proves that ψis a right-inverse of ϕ.\nLemma 2.6.\n(i)Wηis a subspace of VC,η.\n(ii) There exists a natural way of removing the ambiguity by a s calar in\nthe definition of ψ. Furthermore, with this choice one has\nϕ◦ψ= id.\nProof.We start by considering the diagram\nC\nP2P3φη\ncEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 9\nwhere the right diagonal map is the canonical embedding. Thi s diagram is\ncommutative as a consequence of [Cat81, Equation 1.13]. Com bining this\nwith (2.1), we see that the diagram\nC\nS\nP2P3φη\nb1b2\nc(2.3)\ncommutes. Now recall from Lemma 2.4 that the definition of ψis based on\nthe isomorphism\nb∗\n1O(2)∼=b∗\n2O(2)⊗OS/parenleftigg\n−/summationdisplay\niNi/parenrightigg\nwhich has the ambiguity by a scalar. Restricting this isomor phism toC\ngives an isomorphism\nΩC(η)⊗2∼=Ω⊗2\nC⊗OC.\n(Notice that Ccannot go through the nodes of Γ ηbecause then Q∩Γη\nwould be singular.) We can now remove the ambiguity in the defi nition ofψ\nby requiring that the latter isomorphism is induced from the isomorphism\nOC(2η)∼=OC. This, together with the commutativity of (2.3), implies th at\nthe diagram\nS2H0(ΩC(η))\nS2H0(ΩC) H0(Ω⊗2\nC)ψ\ncommutes. From the definition of a fiber product this implies t hatWη=\nim(ψ)⊆VC,η, proving (i).\nConsider now the diagram defining ϕ.\nVC,η S2H0(ΩC(η))\nS2H0(ΩC) H0(Ω⊗2\nC)ϕ\nSince this diagram also commutes and the map S2H0(ΩC(η))→H0(Ω⊗2\nC)\nis injective, we conclude that\nϕ◦ψ= id,\nwhich proves the lemma. /square10 HANSELMAN, PIEPER, SCHIAVONE\nThe situation is summarized by the following diagram.\nVC,η S2H0(ΩC(η))\nS2H0(ΩC) H0(Ω⊗2\nC)ϕ\nψ\n2.3.Milne’s bijection. Let/tildewideC→Cbe the´ etale double cover associated to\nη∈Jac(C)[2]\\{0}, so/tildewideChas genus 7. Since the Prym variety Prym( /tildewideC/C) is\nprincipally polarized, then, up to quadratic twist, Prym( /tildewideC/C) is isomorphic\nto the Jacobian of some genus 3 curve X(see [OU73] and [BR11]). (The\nSchottky-Jung relations imply that the Prym is indecomposa ble.) Since we\nhave an isomorphism\nH0(X,ΩX)∼=H0/parenleftbig\nJac(X),ΩJac(X)/parenrightbig\n∼=H0/parenleftig\nPrym(/tildewideC/C),ΩPrym(/tildewideC/C)/parenrightig\n∼=H0(C,ΩC(η))\nby the definition of the Prym variety, the curve Xcanonically maps to\nP(H0(C,ΩC(η)))∼=P2.\nRemark 2.7. TheRecillas trigonal construction (see [Cob29, §50] or [BS20,\nLemma 5.6]) gives an explicit geometric construction of Xas a plane quartic\n(generic case), but this is not used in the present article.\nIn order to state the main theorem of this section we need the f ollowing\ndefinition. Given a tritangent plane HofC, thenH.C= 2DwhereD\nis a divisor of degree 3. We say that a pair of tritangents H,H′differ by\nη∈Jac(C)[2] ifD−D′is linearly equivalent to η, whereH.C= 2Dand\nH′.C= 2D′.\nTheorem 2.8. IfXis a plane quartic, there is a bijection\n/braceleftbig\nPairs of tritangent planes of Cdiffering by η/bracerightbig∼→/braceleftbig\nBitangents of X/bracerightbig\nsuch that for any pair H,H′viewed as elements of H0(C,ΩC)mapping to\na bitangent ℓ⊂P(H0(C,ΩC(η)))viewed as an element of H0(C,ΩC(η)), we\nhave:\n(i) The product HH′∈S2H0(C,ΩC)lies inVC,η.\n(ii) There exists λ∈k×such that\nϕ(HH′) =λℓ2. (2.4)\nProof.The bijection between pairs of tritangent planes differing by ηand\nbitangents of Xwas discovered by W. P. Milne [Mil23] (see [BS20, Theorem\n7.1] for a modern proof). The assertion (i) is due to Lehavi [L eh15, p. 2]. To\nprove(ii) onecan usetheargument from[BS20, Theorem 7.1], thegeometric\ninterpretation of the map ψ, and Lemma 2.6. /squareEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 11\nRemark 2.9. A similar statement holds when Xis a hyperelliptic curve.\nThe main difference is that one has to replace the bitangents of Xwith the\nlines through the pairs of Weierstrass points of Xin thebijection (see [BS20,\nSection 7.2]).\nMilne’s theorem will play a key role in reconstructing the ge nus 4 curve\nbecause it links the map ϕto the Prym. Since the bitangent lines of Xcan\nbe computed with classical formulas, namely the Schottky-J ung relations\n[FR70, Theorem 1], and the Aronhold-Weber formulas [Dol12, Theorem\n6.1.9] [Fio16, Theorem 2], the information on the righthand side of the\nidentity\nϕ(HH′) =λℓ2\ncan be readily computed, except for the unknown constant λ. A central\nstep of our method is to compute the map ϕby interpolating this identity\nthrough ten pairs of tritangent planes and their correspond ing bitangents.\nRemark 2.10. The formulas in [Dol12, Theorem 6.1.9, Equations (1) - (7)]\ncontain typos that will be corrected in a future edition. We t hank Igor\nDolgachev for providing us with these corrections.\n2.4.The bielliptic case. We will explain now the modifications that have\nto be made in the bielliptic case. All the facts in this sectio n are due to\nCatanese and Bruin–Sert¨ oz; see [Cat81, (1.9)], [BS20, §4.2] for proofs. Let\nCbe a smooth non-hyperelliptic genus 4 curve and η∈Jac(C)[2]\\ {0}.\nWe will say that the pair ( C,η) isbielliptic if there is a degree two map\nπ:C→Eonto a smooth genus 1 curve and a two-torsion point η0∈Jac(E)\nsuch thatπ∗η0=η.\nThen (C,η) is bielliptic if and only if the Prym canonical map φη:C→\nP2factors through a degree two map onto a smooth plane cubic E⊂P2.\nIn this situation, one can still obtain Γ ηas the cone over Ewith vertex\ncorresponding to the one-dimensional subspace\nπ∗H0(E,ΩE)⊂H0(C,ΩC).\nOne still has C⊂Γηand the map π:C→Eis induced by projecting away\nfrom the vertex of Γ η.\nOn the other hand, Γ ηis a cubic symmetroid since the two-torsion point\nη0defines a symmetric determinantal equation for E⊂P2[Dol12, Section\n4.1.3]. But this cubic symmetroid is degenerate in the follo wing sense: when\nwriting\nΓη=V/parenleftigg\ndet/parenleftigg3/summationdisplay\ni=0Aixi/parenrightigg/parenrightigg\nwithAi∈Mat3,3(k) symmetric, the matrices Aiare linearly dependent.\nAlso, the map cdoes not exist in the bielliptic case because Γ ηis not a\nrational surface.12 HANSELMAN, PIEPER, SCHIAVONE\nNevertheless, the linear maps ϕ,ψare still defined. Indeed, ϕwas defined\nunconditionally. On the other hand, the map\nψ:S2H0(C,ΩC(η))→Wη⊂S2H0(C,ΩC)\ncan be defined by taking Wηto be the set of quadrics that are singular at\nthe vertex of Γ η. The map ψis then the map sending a conic to the affine\ncone over it.\nAll the results from the previous sections still hold in the b ielliptic case.\nHowever, the proof of Lemma 2.6 needs a separate argument whi ch we give\nnow. Indeed, instead of diagram 2.3 we consider the diagram\nC\nP2P3φη\nwhere the map P3/axisshort/axisshort/arrowaxisrightP2is the projection away from the vertex of Γ η.\nThe diagram commutes because, as noted above, this projecti on induces the\nmapC→E. The rest of the proof follows the same line of reasoning as in\nLemma 2.6.\n2.5.Tritangent planes and theta derivatives. In order to compute of\nthe tritangent planes on the left-hand side of Equation (2.4 ), we require\nformulas that express them purely in terms of the theta const ants. To derive\nsuch formulas one begins with the following well-known conn ection between\ntritangent planes and theta derivatives.\nTheorem 2.11. LetCbe a smooth curve of genus 4overCwith small\nperiod matrix τ.\nFor any odd theta characteristic/bracketleftbigg\na\nb/bracketrightbigg\n∈1\n2Z8/Z8, the equation\n4/summationdisplay\ni=1∂ϑ/bracketleftbiga\nb/bracketrightbig\n∂zi(0,τ)xi= 0 (2.5)\ndefines a tritangent plane for the canonical image of Cin theP3with coor-\ndinatesx1,...,x 4. Here the basis for H0(C,ΩC)must be the basis induced\nby the isomorphism\nJac(C)∼=Cg/(Zg+τZg).\nProof.See, e.g., [Fio16, p. 5]. /square\nNext, we want to replace the theta derivatives by theta const ants. This is\nachieved with the same approach as in the genus 3 case [Fio16] : one chooses\n5 tritangent planes and applies a coordinate transform that puts them into\nthe standard form\nxi= 0, x 1+x2+x3+x4= 0.EQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 13\nThen Cramer’s rule expresses the equations for the other tri tangent planes\nin terms of determinants of Jacobi matrices of theta functio ns (see [Fio16,\nEquation (17)] or Equation (3.1) in the present article). Th e next section\nexplains how these determinants can be expressed in terms of theta con-\nstants.\n2.6.The generalized Jacobi derivative identity. In this section, we\nintroduceFay’sgeneralization ofJacobi’sderivativefor mula. Itexpressesthe\nJacobian determinant of anazygetic system of oddtheta func tionsevaluated\nat 0 as a polynomial in the theta nullvalues.\nFirst, we will recall some definitions from the theory of thet a characteris-\ntics. A system of theta characteristics c1,c2,c3∈1\n2Z2gis called an azygetic\ntriple if\ne∗(c1+c2+c3) =−e∗(c1)e∗(c2)e∗(c3),\nwheree∗:1\n2Z2g→ {±1}is the parity map/bracketleftbigg\na\nb/bracketrightbigg\n/ma√sto→(−1)4atb. More generally,\nanarbitrary systemof characteristics c1,...,cnis calledazygeticif any triple\ncontained in c1,...,cnis an azygetic triple.\nA system of theta characteristics is called essentially independent if every\nsum of a subset of even cardinality of the system is non-zero.\nDefinition 2.12. Aspecial fundamental system is a system of 2 g+2 char-\nacteristicsm1,...,mg,n1,...,ng+2∈1\n2Zgsuch that:\ni)m1,...,mg,n1,...,ng+2is azygetic.\nii) The characteristics m1,...,mgare odd.\niii) The characteristics n1,...,ng+2are even.\nWe will now define the left-hand side of the generalized Jacob i derivative\nidentity.\nDefinition 2.13. Letm1,...,mg∈1\n2Z2gbeasystem of oddcharacteristics.\nTheJacobian nullvalue ofm1,...,mgis defined to be the function on the\nSiegel upper half space\nD(m1,...,mg) :Hg−→C\ngiven by the formula\nD(m1,...,mg)(τ) =π−gdet/parenleftigg/parenleftbigg∂ϑ[mi]\n∂zj/parenrightbigg\ni,j=1,...,g/parenrightigg\n(0,τ).\nNow ifm1,...,mgis azygetic and essentially independent then, for g/lessorequalslant5,\nD(m1,...,mg) can be expressed as the following polynomial in the theta\nnullvalues.\nTheorem 2.14. Letg/lessorequalslant5andM={m1,...,mg}be an azygetic essentially\nindependent system of odd theta characteristics. Then for al lτ∈Hg,\nD(m1,...,mg)(τ) =/summationdisplay\nN±g+2/productdisplay\ni=1ϑ[ni](0,τ) (2.6)14 HANSELMAN, PIEPER, SCHIAVONE\nwhere the sum runs over all sets N={n1,...,ng+2}such thatM∪Nis a\nspecial fundamental system.\nFurthermore, the signs ±are explicit, unique and independent of τ.\nProof.See [Fro85] for g= 4 and [Fay79] for g/lessorequalslant5. /square\nRemark 2.15. The number of terms on the right of Fay’s generalized Ja-\ncobi derivative identity is given by the following table.\ng 12345\nno. of terms 11128\nRemark 2.16. Igusa[Igu80][Igu83] hasaconjectural generalization of F ay’s\ngeneralized Jacobi identity to arbitrary g/greaterorequalslant6. However, as it is known that\nin this case a Jacobian nullvalue D(m1,...,mg) cannot be a polynomial in\nthe theta constants [Fay79, p. 12], Igusa’s conjectural for mula has a sum of\nJacobian nullvalues on the left-hand side.\nIt would be an interesting problem to use Igusa’s conjectura l identity\nto express Jacobian nullvalues as rational functions in the theta constants.\nSuch expressions must exist by general principles [Igu72, T heorem V.9], but\nthey seem to be unknown.\n3.Reconstructing the curve\n3.1.An auxiliary set of odd theta characteristics. In order to use\nthe generalized Jacobi identity (Theorem 2.14) for the comp utation of the\npairs of tritangent planes in Equation (2.4) we will need a se t of odd theta\ncharacteristics satisfying certain properties. Indeed, o n the left-hand side\nof the identity we have the Jacobian nullvalue of a set of four azygetic\nessentially independentoddtheta characteristics. Tryin gtooptimize theuse\nof the generalized Jacobi identity leads us to choose the fol lowing auxiliary\nset of odd theta characteristics.\nLemma 3.1. Letη∈1\n2Z8/Z8be arbitrary. There exist odd theta character-\nisticsξ1,...,ξ5, χ1,χ′\n1,...,χ 10,χ′\n10∈1\n2Z8/Z8such that\ni)ξ1,...,ξ5is azygetic and essentially independent.\nii) For alli∈ {1,...,10}the systems ξ1,...,ξ4,χiandξ1,...,ξ4,χ′\niare\nazygetic and essentially independent.\niii) For all ione hasχi−χ′\ni=η.\nProof.Without loss of generality we can assume that η=1\n2/bracketleftbigg\n0 0 0 0\n1 0 0 0/bracketrightbigg\n.\n(This is the convention used in, e.g., [FR70].) An explicit a nswer is then\ngiven by:EQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 15\nξ1=1\n2/bracketleftbigg\n1 1 1 0\n1 1 1 0/bracketrightbigg\n, ξ2=1\n2/bracketleftbigg\n1 0 1 0\n0 0 1 0/bracketrightbigg\n, ξ3=1\n2/bracketleftbigg\n1 1 1 0\n0 0 1 0/bracketrightbigg\n,\nξ4=1\n2/bracketleftbigg\n1 0 1 0\n0 1 1 0/bracketrightbigg\n, ξ5=1\n2/bracketleftbigg\n0 1 1 0\n0 1 0 0/bracketrightbigg\n,\nχ1=1\n2/bracketleftbigg\n0 1 1 0\n0 1 0 0/bracketrightbigg\n, χ2=1\n2/bracketleftbigg\n0 1 0 0\n0 1 0 0/bracketrightbigg\n, χ3=1\n2/bracketleftbigg\n0 1 0 1\n0 1 0 0/bracketrightbigg\n,\nχ4=1\n2/bracketleftbigg\n0 1 1 1\n0 1 0 0/bracketrightbigg\n, χ5=1\n2/bracketleftbigg\n0 1 0 1\n0 1 1 0/bracketrightbigg\n, χ6=1\n2/bracketleftbigg\n0 1 0 0\n0 1 1 0/bracketrightbigg\n,\nχ7=1\n2/bracketleftbigg\n0 1 0 0\n0 1 1 1/bracketrightbigg\n, χ8=1\n2/bracketleftbigg\n0 1 1 1\n0 1 1 1/bracketrightbigg\n, χ9=1\n2/bracketleftbigg\n0 1 0 0\n0 1 0 1/bracketrightbigg\n,\nχ10=1\n2/bracketleftbigg\n0 1 1 0\n0 1 0 1/bracketrightbigg\nandχ′\ni=χi+ηfori∈ {1,...,10}. /square\nRemark 3.2. Theanswerinthepreviousproofisfoundbychoosing ξ1,...,ξ5\nsuitablyandthensolvingasystemofequationsover F2forχ1,χ′\n1,...,χ 10,χ′\n10\nallowingηtobearbitrary. Intheend onetransformsthecharacteristi cs such\nthatηbecomes1\n2/bracketleftbigg\n0 0 0 0\n1 0 0 0/bracketrightbigg\n.\nOne can still see the (affine) linear structure in the shape of t he charac-\nteristicsχi,χ′\ni. Indeed, the χiare all the odd theta characteristics whose\nleft 2×2 block is /bracketleftbigg\n0 1\n0 1/bracketrightbigg\n.\nFrom this observation one sees that one cannot add further pa irsχi,χ′\niwith\nthe required properties to the system.\nFor the rest of the article, we define Hi,H′\ni∈H0(C,ΩC) to be the linear\nforms cutting out the tritangent planes corresponding to χi,χ′\ni. By con-\nstruction, for any i= 1,...,10 the tritangent planes HiandH′\nidiffer byη.\nWe denote by ℓi∈H0(C,ΩC(η)) the linear form cutting out the bitangent\nofXthat corresponds to ( Hi,H′\ni) under Milne’s bijection (Theorem 2.8).\nThe main point of the special conditions in Lemma 5.1 is that t hey guar-\nantee that the tritangent planes Hi,H′\nican be written with simple formulas\nin terms of the theta constants, as we now explain. Indeed, we choose the\ncoordinate system for P3such that the tritangent planes corresponding to\nthe odd characteristics ξi, i= 1,...,5 are in normal form\nxi= 0, i∈ {0,...3},\nx0+x1+x2+x3= 0.\nThen, with the same argument as in [Fio16, Equation (17)] one sees that\nfor anyi∈ {1,...,10}the tritangent plane Hiis given by the equation16 HANSELMAN, PIEPER, SCHIAVONE\nD(χi,ξ2,ξ3,ξ4)\nD(ξ5,ξ2,ξ3,ξ4)x0+D(ξ1,χi,ξ3,ξ4)\nD(ξ1,ξ5,ξ3,ξ4)x1+\nD(ξ1,ξ2,χi,ξ4)\nD(ξ1,ξ2,ξ5,ξ4)x2+D(ξ1,ξ2,ξ3,χi)\nD(ξ1,ξ2,ξ3,ξ5)x3= 0(3.1)\nin this coordinate system. An analogous formula with χireplaced by χ′\ni\ngives the tritangent planes H′\ni.\nBy Lemma 3.1 the set ξ1,...,ξ5and all the sets ξ1,...,ξ4,χias well\nasξ1,...,ξ4,χ′\nifori= 1,...,10 are azygetic and essentially independent.\nThis implies that the Jacobian nullvalue in our equations fo rHi,H′\nican be\nexpressed in terms of theta constants via Theorem 2.14.\nRemark 3.3. ThedenominatorsinEquation(3.1) arenon-zeroforageneri c\ncurve. But it can happen that they vanish: such curves must ex ist because\nthe corresponding locus in the Satake compactification of th e Siegel mod-\nuli space is given by the vanishing locus of a non-zero polyno mial in the\ntheta constants on the (open) Torelli locus. By [FC14, Theor em 2.3] such\na polynomial is a section of an ample line bundle. Furthermor e, since the\nboundary of the Torelli locus in the Satake compactification has codimen-\nsion two, this locus must be non-empty. Nevertheless, the au thors do not\nknow an explicit example of this phenomenon.\n3.2.Finding the quadric. In this section, we describe how to recover\nthe quadric Qcontaining the canonical embedding of C. The first step is\nobtaining the equation for the map ϕdefined in the diagram (2.2). To do\nthis, we construct and solve a linear system as follows.\nRecall from Theorem 2.8 that\nϕ(HH′) =λℓ2\nfor any pair of tritangent planes ( H,H′) that maps to a bitangent ℓunder\nMilne’s bijection.\nLemma 3.4. For a generic genus 4curve, the elements HiH′\ni∈VC,η, for\ni= 1,...,10form a generating set of VC,η.\nProof.Since the property is open in the moduli space and the moduli s pace\nisirreducible, it sufficestoshowthat onegenus4curvesatis fiesthecondition\nof the lemma. We will give an example in the proof of Lemma 3.5 b elow./square\nUsing the knowledge of the equations of the pairs of tritange nt planes\nHi,H′\niand the corresponding bitangents ℓifori= 1,...,10, we know ev-\nerything in the equations\nϕ(HiH′\ni) =λiℓ2\ni, i= 1,...,10 (3.2)\nexcept forthe scalars λi. Thelinear dependenciessatisfied by the HiH′\niyield\na homogeneous linear system of equations with unknowns λi. (Recall that\nHiH′\ni∈VC,ηand dim(VC,η) = 7, as mentioned in Section 2.2.) Solving thisEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 17\nlinear system, we recover the linear map ϕ, and thus the quadric Qfrom\nker(ϕ).\nLemma 3.5. For a generic genus 4curve the vector (λ1,...,λ 10)is uniquely\ndetermined from this linear system up to a scalar.\nProof.With the same argument as in Lemma 3.4 it suffices to check one\nexample.\nWe will now construct an example over F37and verify the required prop-\nerties with the aid of Magma. (See the files Lemma-Section-3.2-proofs.m\nandconstruct-Lemma-Section-3.2-proofs.m in [HPS24c].) Consider the\nplane quartic X⊂P2with equation\nX:t4+14t3u+16t3v+32t2u2+26t2uv+18t2v2+29tu3+4tu2v+11tuv2\n+2tv3+26u4+16u3v+27u2v2+22uv3+11v4= 0.\nThe curveXwas chosen in such a way that all bitangents are rational. Thi s\nwas achieved using the Aronhold-Weber formulas and as a cons equence, we\nexplicitly know equations for all bitangents.\nNext, using Recillas’s trigonal construction [Rec93, Theo rem 2.15] with\nthe explicit geometric version of [BS20, Theorem 1.5] one ca n compute a\ngenus 4 curve Cwith a two-torsion point ηsuch that Jac( X) is isomorphic\nto the Prym. All the explicit properties of the curve Cthat we claim below\nwill be a consequence of this construction.\nWe find that the canonical embedding of the curve CintoP3is cut out\nby the equations\n0 = 11xw+yz,\n0 =x3+3x2y+9x2z+15x2w+23xy2+10xyw+12xz2+6xzw+12xw2\n+8y3+y2w+17yw2+11z3+14z2w+7zw2+28w3.\nChas the following 10 products of pairs of tritangent planes ( Hi,H′\ni) dif-\nfering byη. (Note that many of these are irreducible quadrics over F37, but\nfactor as a product of two linear forms over F372.)\nH1H′\n1= 2x2+9xy+18xz+6xw+8y2+20yz+5yw+15z2+23zw+33w2,\nH2H′\n2= 21x2+25xy+20xz+29xw+2y2+32yz+34yw+16z2+5zw+36w2,\nH3H′\n3= 4x2+13xy+28xz+9xw+7y2+28yz+22yw+19z2+20zw+33w2,\nH4H′\n4= 30x2+27xy+6xz+29xw+28y2+4yz+16yw+25z2+4zw+13w2,\nH5H′\n5= 24x2+22xy+2xz+32xw+22y2+4yz+27yw+31zw+21w2,\nH6H′\n6= 9x2+16xy+19xz+6xw+17y2+12yz+14yw+23z2+15zw+32w2,\nH7H′\n7= 29x2+26xy+30xz+36xw+22y2+34yz+11yw+12z2+35zw+23w2,\nH8H′\n8= 25x2+26xy+30xz+24xw+32y2+16yz+2yw+28z2+3zw+33w2,\nH9H′\n9= 8x2+30xz+17xw+19y2+31yz+3yw+23z2+17w2,\nH10H′\n10= 13x2+34xy+28xz+14y2+13yz+22yw+29z2+16zw+14w2.18 HANSELMAN, PIEPER, SCHIAVONE\n(It is important to ensure that the odd theta characteristic s corresponding\nto (Hi,H′\ni) are given by the ( χi,χ′\ni) from Lemma 3.1.)\nByexplicitcomputationonechecksthatthe7quadraticform sH3H′\n3,...,H 10H′\n10\nare linearly independent and thus form a basis of VC,η. This proves Lemma\n3.4.\nNext, the bitangent lines of Xcorresponding to ( Hi,H′\ni) under Milne’s\nbijection are given by\nℓ1= 12t+16u+23v,\nℓ2= 3t+19u+10v,\nℓ3=t+u+v,\nℓ4= 10t+35u+23v,\nℓ5= 7t+17u+24v,\nℓ6= 23t+27u+30v,\nℓ7= 24t+27u+31v,\nℓ8=t,\nℓ9= 35t+19u+v,\nℓ10= 13t+17u+23v.\nTo recoverϕ, we now use the Equations (3.2) ϕ(HiH′\ni) =λiℓ2\ni, which imply\nthat the row vector Λ = ( λi) must satisfy the system of linear equations\nΛM= 0 whereMis the matrix below.\n\n33 14 34 34 33 11 0 0 0 0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 9 3 23 28 10 26 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 1 2 1\n19 22 6 20 21 18 2 14 24 6 10 35 10 33 9 30 13 27\n29 14 35 17 11 23 19 13 14 29 34 24 31 29 17 22 36 8\n11 21 11 26 29 12 10 9 10 27 23 21 4 11 4 33 24 1\n26 3 24 4 27 34 35 14 1 31 15 23 36 7 19 34 26 30\n21 0 0 0 0 0 22 0 0 0 0 0 29 0 0 0 0 0\n4 35 33 28 1 1 21 8 16 36 33 33 32 21 5 2 8 8\n8 1 34 22 16 13 1 14 32 12 2 34 7 24 2 10 14 16\n\nOne readily verifies that the left kernel of Mis one-dimensional and gener-\nated by (1,21,13,32,8,33,17,3,19,18). This proves Lemma 3.5. /square\nExample 3.6. For illustrative purposes we continue to explain how the\nquadric containing Cis recovered. The matrix for\nϕ:VC,η−→S2H0(C,ΩC(η))EQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 19\nis given by\n18 22 30 24 3 2 8\n15 17 27 17 0 36 1\n31 24 30 25 0 35 34\n17 18 7 35 0 14 22\n16 16 32 5 0 19 16\n19 20 26 20 0 19 13\n\nwith respect to thebasis H3H′\n3,...,H 10H′\n10ofVC,ηand thestandard (mono-\nmial) basis of S2H0(C,ΩC(η)). By computing the kernel of ϕone recovers\nthe quadratic form\n27xw+26yz= 26(11xw+yz)\nwhich vanishes on C.\n3.3.Finding a cubic. In this section, we explain how the knowledge of\nthe map\nϕ:VC,η−→S2H0(C,ΩC(η))\ncan be used to recover the equations of the curve C. As we know that ϕ\nrecoversQ, it remains to explain the calculation of a cubic containing C.\nWe begin by giving a connection between the natural right-in verseψofϕ\nand the Cayley cubic Γ η. For this purpose, we will introduce a sextic form\nGwhich vanishes with multiplicity two on Γ η. In this definition we use the\nabbreviations V= H0(C,ΩC), W= H0(C,ΩC(η)).\nLetGbethehomogeneous formthat isgiven bythefollowing compos ition\nof polynomial maps\nV∨→S2(V∨)∼=(S2V)∨→V∨\nC,ηψ∨\n−→(S2W)∨∼=S2(W∨)disc−→k,(3.3)\nwhere the first map is x/ma√sto→x⊗xand the map ( S2V)∨→V∨\nC,ηis dual to the\ndefining inclusion.\nIt is easy to see that Ghas degree 6 because the first map has degree 2,\nthe map disc has degree dim( W) = 3, and all the other maps are linear.\nThenextpropositionwillshowthatthehomogeneousform Gisthesquare\nof a cubic form cutting out the cubic symmetroid.\nRemark 3.7. The intuition behind considering the composition\nV∨→S2(V∨)∼=(S2V)∨→V∨\nC,ηψ∨\n−→(S2W)∨∼=S2(W∨)\nis as follows: we want to understand the image Γ ηof the map\nc:P2/axisshort/axisshort/arrowaxisrightP3.\nBut we do not have access to the map cexplicitly; we only have the map\nϕwhich is linked to the way ctransforms quadrics. The best we can do\nstarting from a point x∈P3is to form the vector subspace of quadrics\nvanishing at xand, then, apply the map ϕ. This gives us a linear system of\nconics on P2from which we can hope to detect whether or not xis in the20 HANSELMAN, PIEPER, SCHIAVONE\nimage ofc. Dualizing this consideration shows that it is natural to st udy\nthe composition\nV∨→S2(V∨)∼=(S2V)∨→V∨\nC,ηψ∨\n−→(S2W)∨∼=S2(W∨).\nProposition 3.8. One has V(G) = 2Γη.\nProof.Assume first that ( C,η) is not bielliptic. We begin by showing that\nGvanishes on Γ η. LetU⊂Γηbe the complement of the base locus of the\nbirational map c−1: Γη/axisshort/axisshort/arrowaxisrightP2. We will show that Gvanishes on the open\nsubsetU. To that end, let x∈Ube arbitrary and let y=c−1(x)∈P2\ndenote its image. Then x(resp.,y) gives a non-zero vector in V∨(resp.,\nW∨). We claim that under the composition\nγ:V∨→S2(V∨)∼=(S2V)∨→V∨\nC,ηψ∨\n−→(S2W)∨∼=S2(W∨),\nxmaps to a multiple of y⊗y.\nSinceWηis the image of ψ, the mapγfactors as\nV∨→S2(V∨)∼=(S2V)∨→W∨\nηψ∨\n−→(S2W)∨∼=S2(W∨).\nConsider now any quadratic form q∈S2Wvanishing at y. We denote\nby/a\\}bracketle{t·,·/a\\}bracketri}htWthe natural pairing between S2WandS2(W∨) and define /a\\}bracketle{t·,·/a\\}bracketri}htV\nsimilarly. Then we see that\n/a\\}bracketle{tq,γ(x)/a\\}bracketri}htW=/a\\}bracketle{tq,ψ∨(x⊗x)/a\\}bracketri}htW=/a\\}bracketle{tψ(q),x⊗x/a\\}bracketri}htV. (3.4)\nNext, we know that the map ψis given by pull-pushing quadrics through\nthe diagram\nS\nP(H0(C,ΩC(η)))∼=P2Γη,c\ni.e., the quadratic form ψ(q) is the unique (up to scalar) quadratic form\nvanishing on the curve c(V(q))⊂Γη.\nSincey=c−1(x) andqvanishes at y, this implies that ψ(q) vanishes at\nx, i.e.,\n/a\\}bracketle{tψ(q),x⊗x/a\\}bracketri}htV= 0.\nBy3.4γ(x)mustthereforebeorthogonaltoallthequadraticformsvan ishing\natyand thus is a multiple of y⊗y. (It could a priori be zero.) This proves\nthe claim.\nNext, the observation disc( y⊗y) = 0 shows that Gvanishes on Γ η. To\nprove that Gvanishes with multiplicity two, one can use the same argu-\nment with adjugate matrices as in the discussion preceding T heorem 4.1.4\nin [Dol12] to show that γ3\nη|G2whereγηis a cubic form cutting out Γ η.\nThis implies that γ2\nη|Gbecause Γ ηis irreducible.\nIt remains to show that G/\\e}atio\\slash= 0. One possible argument would be to go\nthrough the classification of symmetroid cubic surfaces on [ Cat81, §1, p. 33]EQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 21\nand explicitly verify that G/\\e}atio\\slash= 0 in every case. Alternatively, we propose\nthe following general proof. Choose a plane H⊂P3such that Γ η∩His\nsmooth. Then Γ η∩His a smooth plane curve and the restriction γη|H\ngives a determinantal equation for it. Hence, the general th eory of deter-\nminantal equations for smooth plane curves from Dolgachev’ s book [Dol12,\nSection 4.1.2] applies. The degree six form G|Happears as a special case of\nthe construction preceding [Dol12, Theorem 4.1.4] (where i t is denoted by\ndet(N)). In loc. cit. he proves that this polynomial is non-zero. T herefore,\nwe conclude that G|H/\\e}atio\\slash= 0 and thus G/\\e}atio\\slash= 0. This proves the non-bielliptic\ncase.\nAssume now that ( C,η) is bielliptic. Then Γ ηis a cone over a smooth\nplane cubic E⊂P2. Furthermore, after applying a coordinate transforma-\ntion that maps the vertex to (0 : 0 : 0 : 1) it is easy to see that V(G) is a\ncone over the samevertex by looking at the resulting equatio n. (This follows\nfrom the fact that in the bielliptic case, ψmaps a conic to the cone over it,\nas mentioned in Section 2.4.) Thus we are reduced to a stateme nt about\nthe symmetric determinantal representation of EinP2. Then Dolgachev’s\nargument applies directly and the proposition follows. /square\nHowever, we do not have direct access to the map ψin our reconstruction\nmethod. The next lemma shows how to find a cubic vanishing on Cfrom\nthe knowledge of ϕalone.\nLemma 3.9. Let/tildewideψbe any right inverse of ϕ. Define/tildewideGusing the same\nformula as for Gin (3.3) but with ψreplaced by/tildewideψ. Then\nG≡/tildewideGmodQ.\nProof.By Lemma 2.6, the map ψsatisfiesϕ◦ψ= id after fixing the ambi-\nguity in the definition. Then ψ,/tildewideψare two right inverses of ϕ. Therefore the\ndifferenceψ−/tildewideψsatisfies\nim(ψ−/tildewideψ)⊆ker(ϕ) = span(Q).\nDually, the difference ψ∨−/tildewideψ∨:V∨\nC,η−→(S2W)∨vanishes on span( Q)⊥⊂\nV∨\nC,η.\nWe are now ready to show that G≡/tildewideGmodQ. We claim that the two\npolynomial maps defining G,/tildewideGagree when restricted to the vanishing locus\nofQ. Indeed, let x∈V∨be an arbitrary vector satisfying Q(x) = 0. The\nlatter means that the element x⊗x∈S2(V∨)∼=(S2V)∨is orthogonal to\nQ. Therefore under the first arrows in the composition defining G,˜G\nV∨→S2(V∨)∼=(S2V)∨→V∨\nC,η\nxmaps into span( Q)⊥. Above we noted that ψ∨and/tildewideψ∨agree on span( Q)⊥.\nThis is sufficient to prove the claim. Therefore the two polyno mial maps\ndefiningG,/tildewideGagree when restricted to the vanishinglocus of Q. This implies22 HANSELMAN, PIEPER, SCHIAVONE\nthat\nG≡/tildewideGmodQ./square\nUsingthepreviouslemmawecanfindacubicΓvanishingon Cbyextract-\ning a square root of /tildewideGmodQ. Furthermore, one has C=Q∩Γη=Q∩Γ.\nExample 3.10. For illustrative purposes, we continue the calculation fro m\nExample 3.6 and compute the equation of a cubic cutting out th is curve.\nDefine/tildewideψ:S2W→VC,ηto be the linear map represented by\n\n0 14 1 35 34 17\n0 5 23 1 34 3\n0 28 31 14 11 36\n0 12 11 23 23 30\n25 24 28 3 5 8\n0 33 24 16 0 12\n0 0 0 0 0 0\n\nwith respect to the basis H3H′\n3,...,H 10H′\n10ofVC,ηand the standard mono-\nmial basis of S2W(as in Example 3.6). Then the composition\nV∨→S2(V∨)∼=(S2V)∨→V∨\nC,η/tildewideψ∨\n−→(S2W)∨\nis the map\n\nx\ny\nz\nw\n/ma√sto→\n33x2+21xy+10xz+8xw+23y2+30yz+13yw+34z2+zw+11w2\n6x2+22xy+32xz+9xw+10y2+7yz+31yw+16z2+13zw\n8xy+15xz+22xw+28y2+36yz+25yw+23z2+23zw+2w2\n35x2+17xy+36xz+9xw+10yz+17yw+z2+11zw+11w2\n26x2+17xy+26xz+17xw+8yz+29yw+2z2+29zw+19w2\n17xy+35xz+24xw+4y2+34yz+18yw+4z2+36zw+4w2\n.\nTo realize the final isomorphism ( S2W)∨∼=S2(W∨), we form the symmetric\nmatrix\n\nv1v2v3\nv2v4v5\nv3v5v6\n\nwherev1,...,v6are the entries of this resulting vector above. We then take\nthe discriminant of the quadratic form associated to this sy mmetric matrix,EQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 23\nobtaining the sextic form\n˜G= 3x6+18x5y+17x5z+28x5w+17x4y2+5x4yz+2x4yw+19x4z2\n+27x4w2+18x3y3+8x3y2z+34x3y2w+18x3yz2+31x3yzw\n+31x3yw2+11x3z3+2x3z2w+18x3zw2+31x3w3+29x2y4+8x2y3z\n+21x2y3w+17x2y2z2+12x2y2zw+22x2y2w2+36x2yz3+24x2yz2w\n+17x2yzw2+33x2yw3+27x2z4+x2z3w+24x2z2w2+32x2zw3\n+x2w4+31xy5+28xy4z+30xy3z2+5xy3zw+17xy3w2+34xy2z3\n+28xy2z2w+18xy2zw2+20xy2w3+14xyz4+14xyz3w+xyz2w2\n+7xyzw3+10xyw4+15xz5+12xz4w+33xz3w2+29xz2w3+29xzw4\n+35xw5+7y6+y5z+11y5w+29y4z2+13y4zw+5y4w2+32y3z3\n+6y3z2w+5y3zw2+3y3w3+29y2z4+23y2z3w+36y2z2w2\n+11y2zw3+36y2w4+8yz5+21yz4w+33yz3w2+15yz2w3+26yzw4\n+7yw5+30z6+36z5w+14z4w2+31z3w3+20z2w4+29zw5+21w6.\nDividing out the leading coefficient and taking the square roo t of this sextic\nform modulo Q, we recover the cubic equation\nΓ :x3+3x2y+9x2z+15x2w+23xy2+10xyw+12xz2+6xzw+12xw2\n+8y3+y2w+17yw2+11z3+14z2w+7zw2+28w3= 0.\nTogether with the quadric Q: 11xw+yz= 0 found above in Example 3.6,\nthis gives the image of the canonical embedding.24 HANSELMAN, PIEPER, SCHIAVONE\n3.4.Main result. Summarizing the discussion of the previous sections, we\nhave proven the following theorem.\nTheorem 3.11. LetCbe a generic genus 4curve over an algebraically\nclosed field kof characteristic 0 or of characteristic pwithplarge enough.\nThere are explicit formulas for the equations of Cin terms of its theta\nconstants. They use only the following algebraic operations : Elementary\narithmetic, taking square roots, and solving linear system s of equations.\nWe now give an overview of these formulas:\nAlgorithm 3.12: Reconstruct a genus 4 curve from its theta con-\nstants\nInput : Theta constants ϑC/bracketleftbiga\nb/bracketrightbig\nof a smooth non-hyperelliptic genus\n4 curveCdefined over k, considered as a point in P135\nk.\nOutput: A quadricQand a cubic Γ in P3\nksuch thatC∼=Q∩Γ or\nan error ifCis not generic enough.\n1Choose the two-torsion point given by η=1\n2/bracketleftbigg\n0 0 0 0\n1 0 0 0/bracketrightbigg\n.\n2Compute the theta constants for the plane quartic Xvia the\nSchottky-Jung relations\nϑX/bracketleftbiga\nb/bracketrightbig2=ϑC/bracketleftbig0a\n0b/bracketrightbig\nϑC/bracketleftbig0a\n1b/bracketrightbig\n.\nCompute the 10 bitangent lines ℓivia the Aronhold-Weber formulas\n[Dol12, Theorem 6.1.9] [Fio16, Theorem 2] from the theta con stants\nϑX/bracketleftbiga\nb/bracketrightbig\n. (Make sure to take the bitangents with the correct odd\ntheta characteristics.)\n3Try to compute the tritangent planes Hi,H′\nifrom the Equations\n(3.1). If there is a division by zero, give an error.\n4Verify that the HiH′\nigenerate a linear space of dimension 7 or throw\nan error.\n5From the Equations (3.2)\nϕ(HiH′\ni) =λiℓ2\ni\nwe get a homogeneous linear system of equations for the λi.\n6Verify that the kernel of this linear system of equations is\none-dimensional or give an error.\n7From theλicompute a matrix for the linear map ϕ.\n8Compute a generator for ker( ϕ) and call it Q.\n9Choose a linear map /tildewideψsuch thatϕ◦/tildewideψ= id.\n10Compute the degree six polynomial /tildewideGas in Lemma 3.9.\n11Compute a square root Γ of /tildewideGmodQ.\n12ReturnQand Γ.\nRemark 3.13. Although our main result Theorem 3.11 requires the base\nfieldkto be algebraically closed, with additional information on e can re-\ncover equations over arithmetic base fields. Suppose we are g iven the thetaEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 25\nconstants of a curve Cas complex numbers, but we expect that Cis defined\nover a number field K. If in addition to these theta constants, we are also\ngiven a big ( g×2g) period matrix for Cwith respect to a K-rational basis\nfor H0(C,ΩC), we can recover equations for Cdefined over Kas follows.\nFirst applying our theorem, we obtain approximate equation s forCwith\ncoefficients in C. Denote by /tildewideCthis complex curve, which is isomorphic to\nCoverC.\nComputing the big period matrix for /tildewideCand comparing it with that of C,\nwe can determine the PGL 4transformation that changes coordinates on P3\nto make the equations of /tildewideCdefined over K. We then use LLL to recognize\nthe resulting coefficients as elements of K. See Section 4 below for concrete\nexamples of this process.\nRemark 3.14. When using the Schottky-Jung relations one takes a square\nroot to compute the ϑX/bracketleftbiga\nb/bracketrightbig\n. This leads to a sign choice that has to be made\ncorrectly. For this one can use the discussion following [Gl a80, Theorem 3.1]\nwhere Glass makes use of the quartic Riemann identities to fin d a correct\nchoice of signs. At this point, the assumption that the ϑC/bracketleftbiga\nb/bracketrightbig\nare the theta\nconstants of a genus 4 curve is used. Indeed, the latter impli es that the\nSchottky modular form vanishes. By construction of this mod ular form,\nthis is equivalent to requiring that the ϑX/bracketleftbiga\nb/bracketrightbig\nsatisfy the quartic Riemann\nidentities.\nRemark 3.15. If one of the ϑC/bracketleftbig0a\n0b/bracketrightbig\norϑC/bracketleftbig0a\n1b/bracketrightbig\nvanish, then the Schottky-\nJung relations imply that Xhas a vanishing even theta-null. This means\nthat the curve Xbecomes hyperelliptic. Our method still works in this\ncase by making the following adjustment: the canonical map X→P2is a\ndegree two cover of a smooth conic ramified in eight points. Re place the\n28 bitangent lines by the/parenleftbig8\n2/parenrightbig\n= 28 lines through pairs of ramification points\n(see also Remark 2.9).\nNotice that in this situation, the curve Chas a vanishing theta null and\nthus the canonical quadric Qbecomes singular. Conversely, if Chas a\nvanishing theta null, we can achieve that ϑC/bracketleftbig0\n0/bracketrightbig\n= 0 by choosing the level\nstructure appropriately. Thus we can arrange for the curve Xto be hyper-\nelliptic.\nIn computations over Cexploiting this is advantageous because using a\nhyperelliptic curve Xwill increase the speed and minimize the numerical\nerror.\nRemark 3.16. Therearethreepossibilitiesforthecurve Ctonotbegeneric\nenough: (i) there is a division by zero in step 3, (ii) the prod uctsHiH′\nido\nnot generate VC,η, or (iii) the λiare not uniquely determined by the linear\nsystem of equations in step 6. The first situation can occur; s ee Remark 3.3.\nHowever, the authors do not know if there exist curves exhibi ting either of\ntheother two phenomena. Usingthesame strategy as inthepro ofof Lemma\n3.4, 3.5, i.e., by producing an example over a finite field, we o btained that26 HANSELMAN, PIEPER, SCHIAVONE\nthe open locus in the moduli space given by conditions (ii) an d (iii) has\nnon-empty intersection with the following loci:\n•Chas a vanishingeven theta constant, i.e., the quadric Qis singular;\n•Any degeneration of the cubic symmetroid Γ η;\n•In particular, the bielliptic locus.\nNevertheless, if the reconstruction fails for one of the thr ee reasons above,\nall hope is not lost—one can choose a different 2-torsion point ηand try\nagain. As there are 255 possibilities, and every 2-torsion p oint will only\ngive a closed locus (of expected codimension 1) parametrizi ng curves for\nwhich the construction can go wrong, we conjecture that for e very non-\nhyperelliptic genus 4 curve over a field of characteristic /\\e}atio\\slash= 2 there exists a\ntwo-torsion point η∈Jac(C)[2]\\{0}such that our reconstruction method\nworks.\n4.Examples\nExample 4.1. Gluing. Consider the hyperelliptic genus 2 curve\nC1:y2= 24x5+36x4−4x3−12x2+1\nwhich has LMFDB label 20736.l.373248.1 . From the information on\nthe curve’s homepage, we see that the geometric endomorphis m algebra\nof Jac(C1) is the quaternion algebra B2,3, i.e., the unique quaternion alge-\nbra over Qramified at 2 and 3. We look for interesting genus two curves\nC2such that the quotient of A= Jac(C1)×Jac(C2) by a maximal isotropic\nsubgroup of A[2] is the Jacobian of a smooth genus 4 curve C(possibly up\nto a quadratic twist). Using the criterion of [BK23, Theorem 1.2] one finds\nthat, for example, the curve\nC2:y2= 3x5−68x4+159x3+232x2−132x+16\nhas this property. Using the methods of Costa–Mascot–Sijsl ing–Voight\n[CMSV19], we verify that End0\nQ(Jac(C2))∼=Q(√\n5).\nOur method produces the following equations for the genus 4 c urveC:\n0 = 10x2+8xy−9y2−33z2−30zw−40w2\n0 =−6x3−2x2y−xy2+5xz2−22xzw−5xw2+3y3+11yz2\n+10yzw−11yw2.\nBy construction we have that End0\nQ(Jac(C))∼=B2,3×Q(√\n5).\nA priori the curve will come out in a coordinate system where t he equa-\ntions are not defined over Q. Using the knowledge of a big period matrix for\nJac(C) associated to a Q-rational basis for H0(C,ΩC), we found the PGL 4\ntransformation that changes coordinates into this Q-rational basis.\nExample 4.2. Modular Jacobians. Let fbe the modular form orbit with\nLMFDB label 778.2.a.a. We useour methodto computethe abeli an fourfold\nthat corresponds to it via modularity for RM abelian varieti es overQ. ThisEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 27\nabelian variety Ais the subvariety of J0(778) = Jac( X0(778)) corresponding\ntoa4-dimensional Hecke-invariant subspace. IthasRM byth efieldofHecke\neigenvalues whichisthetotally real subfieldof Q(ζ15); thisisthequarticfield\nwith LMFDB label 4.4.1125.1. We begin by computing its perio d matrix\nusingMagma’s command Periods. However, this period matrix need not\ncorrespond to a principally polarized abelian variety: the pullback of the\nprincipal polarization on J0(778) is not in general principal.\nTo remedy this, we call the command SomePrincipalPolarizations\nfrom the ModularCurves GitHub repository [S+23]. This produces several\nbig period matrix candidates. Evaluating the Schottky modu lar form on the\nfirst of these yields a complex number with absolute value 6 .2124·10−300\nwhencalculated withprecision 300, indicating that this pr incipallypolarized\nabelian variety is likely a Jacobian. Applying our method th en produces the\ncorrespondinggenus 4 curve, which, after a change of variab le, is isomorphic\nto the curve CinP3given by the following equations.\n0 =x2−xz−xw−y2−yw+2z2+zw−4w2\n0 = 2xyw−2xz2−12xzw−10xw2−y2z−2y2w+yzw+4yw2+2z3\n−20zw2−18w3\nAs a sanity check for this heuristic example, we compute the p laces of bad\nreduction of C. Using the methods from [Bou23], we compute the invariants\nofCand in particular the discriminant. The result is 29·11330·3894and\ntherefore candidates for the primes of bad reduction are 2 ,113, and 389.\nHowever, modulo 113 the given equations reduce to a smooth ge nus 4 curve,\nalthough the canonical quadric degenerates to a cone. Thus t he primes of\nbad reduction are 2 and 389 as one would expect, since the leve l offis\n778 = 2·389. As a further sanity check, we have computed and compared\nthe localL-factors offandCfor primes up to 1000 and verified that they\nmatch. A priori the Jacobian of Ccould have been a quadratic twist of the\nabelian variety corresponding to f, but our computations also showed this\nis not the case.\nWethankNoamElkiesforsuggestingthisexample, aswell asE dgarCosta\nfor his help in computing polarizations.\n5.Future work\n5.1.Constructing CM curves. Abelian varieties with complex multipli-\ncation are another source of examples where the period matri x is known,\nbut the actual equations defining the varieties are not. Furt hermore, there\nare interesting conjectures about when a CM p.p.a.v. variet y is a Jacobian.\nIndeed, for certain values of n, a CM p.p.a.v. is automatically the Jaco-\nbian of a degree nsuperelliptic curve if the CM order contains a primitive\nnthroot of unity and certain conditions on the CM-type are satis fied, as\nshown in the work of de Jong and Noot [DJN91]. For example, if n= 3\nandg= 4, then they require that the CM order contains a primitive t hird28 HANSELMAN, PIEPER, SCHIAVONE\nroot of unity, say ζ3. Furthermore, the action of ζ3on the differential forms\nof the abelian variety must be the same as the action of the aut omorphism\n(x,y)/ma√sto→/parenleftbig\nx,exp/parenleftbig2πi\n3/parenrightbig\ny/parenrightbig\non the differential forms of a superelliptic curve of\nthe form\ny3=x(x−1)(x−λ)(x−µ)(x−ν). (5.1)\nThis can be translated into requiring that the CM type Φ = {ϕ1,...,ϕ 4}\nsatisfies\nϕ1(ζ3) = exp/parenleftbigg2πi\n3/parenrightbigg\nϕ2(ζ3) = exp/parenleftbigg2πi\n3/parenrightbigg2\nϕ3(ζ3) = exp/parenleftbigg2πi\n3/parenrightbigg2\nϕ4(ζ3) = exp/parenleftbigg2πi\n3/parenrightbigg2\n.(5.2)\nThe family 5.1 and the other families of superelliptic curve s from [DJN91]\nhave the special property that the dimension of the family (3 , in the above\nexample) matches the dimension of a PEL-Shimurasubvariety wherethe en-\ndomorphism structure is given by Z[ζn] with a certain tangent space action.\n(In our example this action would be given by the same conditi ons as in\n(5.2).) From this they conclude that the CM-points of that Sh imura variety\ngive rise to infinitely many curves with CM Jacobian. Moonen [ Moo10] has\nshown that the collection of special (in the above sense) fam ilies of superel-\nliptic curves is finite and listed all of them.\nTo rule out CM fields containing primitive roots of unity, Moo nen and\nOort [MO11] defined the notion of a Weyl CM field. These are CM fie lds\nKof degree [K:Q] = 2gsuch that the Galois group of the splitting field is\nas large as possible, i.e., ( Z/2)g⋊Sg. They put forward the conjecture that\nfor a given fixed g/greaterorequalslant4 there are finitely curves of genus gwhose Jacobian\nhas CM by a Weyl CM field.\nFromthisperspectiveitwouldbeinterestingtocomputeexp licitexamples\nof genus 4 curves with CM.\n5.2.Higher genus. In work in progress [HPS24b] we plan to generalize the\nMilne correspondence to arbitrary g. This would give a bijection between\npairs of odd theta hyperplanes in Pg−1forCand odd theta hyperplanes in\nPg−2for the Prym variety Prym( C,η). Notice that in this context the Prym\nvariety is no longer automatically a Jacobian. Nevertheles s, one can define\nodd theta hyperplanes for arbitrary p.p.a.v.s; however, th ese will lack the\ngeometric interpretation as multitangents of a canonicall y embedded curve\n(unless the p.p.a.v. happens to be a Jacobian).\nThis generalization would give formulas for the equations o f a generic\ngenus 5 curve in terms of its theta constants. Indeed, for g/greaterorequalslant5 the image ofEQUATIONS OF GENUS 4 CURVES FROM THEIR THETA CONSTANTS 29\nthecanonical embeddingof ageneric curveof genus gis cut out by quadratic\nequations. For g= 5, one should be able to compute these equations with\na suitable adaption of the results in Section 3.2.\nForg= 6 we also expect that our method works. However, in order to\nget a formula purely in terms of theta constants, it remains t o finda formula\nfor the Jacobian nullvalues as a rational function in the the ta constants; see\nalso Remark 2.16.\nHere we use that in the cartesian diagram\nS2H0(ΩC(η))×H0(Ω⊗2\nC)S2H0(ΩC)S2H0(ΩC(η))\nS2H0(ΩC) H0(Ω⊗2\nC)(5.3)\nthe vertical maps are injective for a generic curve of genus g= 5,6 which\nwas proven by Donagi [Don92, Theorem 5.1] for g= 5 (here one interprets\nthe right vertical map as the codifferential of the Prym map) an d Beauville\n[Bea77, Proposition 7.10] for g= 6.\nForg/greaterorequalslant7 it is not longer true that\nS2H0(ΩC(η))→H0(Ω⊗2\nC)\nisinjective (forsimpledimensionreasons). Thismeanstha t wecan nolonger\nviewVC,ηas a linear subspace of S2H0(C,ΩC) because here the injectivity\nof the map\nS2H0(C,ΩC(η))→H0(C,Ω⊗2\nC)\nin (2.2) was crucially used. Nonetheless, one can still use o dd theta data\nto produce a system of equations satisfied by the analogues of the unknown\nconstantsλifrom Equation (3.2), but these equations will be quadratic\ninstead of linear. We conclude by pointing out that the map\nmη:S2H0(ΩC(η))→H0(Ω⊗2\nC)\nis in fact known to be surjective for a generic curve of genus g/greaterorequalslant7 by [LS96,\nLemma 2.1] and it is an isomorphism for a generic curve of g= 6 by the\nresult of Beauville [Bea77, Proposition 7.10] cited above.\nRemark 5.1. Lehavi’s method [Leh15] generalizes to genus 5 (see [Leh22] )\nbut not to genus 6 because the fact that mηis an isomorphism causes prob-\nlems. This is not a restriction for the method of the present p aper.\nIn the recent preprint [C ¸L24], C ¸elik and Lehavi propose an alternative\nway of extending Lehavi’s technique to genus 6 and 7. Their ap proach is\ndifferent from ours and uses equations of degree 4.\nReferences\n[AC ¸E22] Daniele Agostini, T¨ urk¨ u ¨Ozl¨ um C ¸elik, and Demir Eken. Numerical reconstruc-\ntion of curves from their Jacobians. In Proceedings of the Conference “Arith-\nmetic, Geometry, Cryptography, and Coding Theory 2021” , volume779of Con-\ntemporary Mathematics , pages 1–12, 2022. 430 HANSELMAN, PIEPER, SCHIAVONE\n[BCP97] Wieb Bosma, John Cannon, and Catherine Playoust. Th e Magma algebra\nsystem. I. The user language. J. Symbolic Comput. , 24(3-4):235–265, 1997.\nComputational algebra and number theory (London, 1993). UR L:http://dx.\ndoi.org/10.1006/jsco.1996.0125 ,doi:10.1006/jsco.1996.0125 . 3\n[Bea77] Arnaud Beauville. Vari´ et´ es de Prym et jacobienne s interm´ ediaires. Ann. Sci.\n´Ecole Norm. Sup. (4) , 10(3):309–391, 1977. URL: http://www.numdam.org/\nitem?id=ASENS_1977_4_10_3_309_0 . 29\n[BK23] Nils Bruin and Avinash Kulkarni. 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Enriques surfaces and\ngenus 4 curves: A mysterious remark from Coble’s book. in preparation , 2024.\n2\n[HPS24b] JeroenHanselman, AndreasPieper, andSamSchiavo ne.Reconstructingcurves\nfrom theta constants. in preparation , 2024. 28\n[HPS24c] Jeroen Hanselman, Andreas Pieper, and Sam Schiavo ne. reconstructing-g4.\nGitHub repository, https://github.com/JHanselman/reconstructing-g4 ,\n2024. 3, 17\n[HSS21] Jeroen Hanselman, Sam Schiavone, and Jeroen Sijsli ng. Gluing curves of\ngenus 1 and 2 along their 2-torsion. Math. Comp. , 90(331):2333–2379, 2021.\ndoi:10.1090/mcom/3627 . 3\n[Igu72] Jun-ichi Igusa. Theta functions , volume 194 of Die Grundlehren der mathema-\ntischen Wissenschaften . Springer-Verlag, New York-Heidelberg, 1972. 14\n[Igu80] Jun-ichi Igusa. On Jacobi’s derivative formula and its generalizations. Ameri-\ncan Journal of Mathematics , 102(2):409–446, 1980. 14\n[Igu81] Jun-ichi Igusa. On the irreducibility of Schottky’ s divisor. J. Fac. Sci. Univ.\nTokyo Sect. 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Effective reconstruction of generic g enus 5 curves\nfrom their theta hyperplanes. Experimental Mathematics , 0(0):1–\n15, 2022. arXiv:https://doi.org/10.1080/10586458.2022.2041133 ,\ndoi:10.1080/10586458.2022.2041133 . 29\n[LS96] Herbert Lange and Edoardo Sernesi. Quadrics contain ing a Prym-canonical\ncurve.J. Algebraic Geom. , 5(2):387–399, 1996. 29\n[LT16] Hugo Labrande and Emmanuel Thom´ e. Computing theta f unctions in quasi-\nlinear time in genus 2 and above. Cryptology ePrint Archive, Paper 2016/179,\n2016. URL: https://eprint.iacr.org/2016/179 . 5\n[Mil23] William P. Milne. Sextactic Cones and Tritangent Pl anes of the Same System\nof a Quadri-Cubic Curve. Proc. London Math. Soc. (2) , 21:373–380, 1923.\ndoi:10.1112/plms/s2-21.1.373 . 3, 10\n[MO11] Ben Moonen and Frans Oort. The Torelli locus and speci al subvarieties. In\nHandbook of moduli. Vol. II , pages 549 – 594, 2011. URL: https://arxiv.\norg/abs/1112.0933 . 2832 HANSELMAN, PIEPER, SCHIAVONE\n[Moo10] Ben Moonen. Special subvarieties arising from fami lies of cyclic covers of the\nprojective line. Doc. Math. , 15:793 – 819, 2010. 28\n[Mum66] David Mumford. On the equations defining abelian var ieties. I. Invent. Math. ,\n1:287–354, 1966. doi:10.1007/BF01389737 . 2\n[OU73] FransOortandKenjiUeno.Principally polarized abe lianvarieties ofdimension\ntwo or three are Jacobian varieties. J. Fac. Sci. Univ. Tokyo Sect. IA Math. ,\n20:377–381, 1973. 10\n[Rec71] Sevin Recillas. A Relation Between Curves Of Genus 3 And Curves Of\nGenus 4 . ProQuest LLC, Ann Arbor, MI, 1971. Thesis (Ph.D.)–Brandei s\nUniversity. URL: http://gateway.proquest.com/openurl?url_ver=Z39.\n88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertati on&res_dat=xri:\npqdiss&rft_dat=xri:pqdiss:7130147 . 3\n[Rec93] SevinRecillas. Symmetriccubicsurfaces andcurve sofgenus3and4. Bollettino\ndella Unione Matem` atica Italiana. Serie VII. B , 01 1993. 17\n[S+23] Andrew V. Sutherland et al. ModularCurves. GitHub repos itory,https://\ngithub.com/AndrewVSutherland/ModularCurves , 2023. 27\n[Sch88] Friedrich Schottky. Ueber specielle Abelsche Func tionen vierten Ranges. Jour-\nnal f¨ ur die reine und angewandte Mathematik , 1888(103):185–203, 1888. URL:\nhttps://doi.org/10.1515/crll.1888.103.185 . 2\n[Tak96] Koichi Takase. A generalization of Rosenhain’s nor mal form for hyperellip-\ntic curves with an application. Proceedings of the Japan Academy, Series A,\nMathematical Sciences , 72(7):162 – 165, 1996. doi:10.3792/pjaa.72.162 . 2\nJeroen Hanselman, RPTU Kaiserslautern-Landau\nEmail address :hanselman@mathematik.uni-kl.de\nAndreas Pieper, Universit ´e de Rennes\nEmail address :andreas.pieper@univ-rennes1.fr\nSam Schiavone, Massachusetts Institute of Technology\nEmail address :sam.schiavone@gmail.com"    },    {        "title": "2402.03189v1.Nagata_Dimension_and_Lipschitz_Extensions_Into_Quasi_Banach_Spaces.pdf",        "content": "NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO\nQUASI-BANACH SPACES\nJAN B ´IMA\nAbstract. Given two metric spaces N⊆M in inclusion and 0<p≤\n1, we wish to determine the smallest constant tp(N,M)such that any\nLipschitz map f:N→Zinto anyp-Banach space Zcan be extended\nto a Lipschitz map f′:M→Zsatisfying Lipf′≤tp(N,M)·Lipf.\nIn this article, we prove that if Nhas finite Nagata dimension at most\ndwith constant γ, then tp(N,M)≲pγ·(d+ 1)1/p−1·log(d+ 2) for\nall0< p≤1. We show that examples of spaces with finite Nagata\ndimension include doubling spaces, as well as minor-excluded metric\ngraphs. We also establish that the constant tp(N,M)generally increases\naspapproaches zero.\n1.Introduction\nSuppose thatMandTare metric spaces and Nis a non-empty subset\nofM. A classical question in metric space theory, known as the Lipschitz\nextension problem , asks what the smallest possible constant Cis, such that\nany Lipschitz map f:N →T can be extended to˜︁f:M→T , where\nLip˜︁f≤CLipf. Owing to its fundamental importance and widespread\napplication in geometry and approximation theory, this problem has received\nsignificant attention. The literature on the subject is extensive, with notable\ncontributions including those by Kirszbraun [30], Johnson, Lindenstrauss,\nand Schechtman [27], Ball [8], and Lee and Naor [33], to name a few.\nIn this paper, our focus is on two specific kinds of Lipschitz extension\nproblems, namely the trace andabsolute extendability problems . For any\ntwo metric spaces N ⊆M in inclusion and a family of metric space F,\nwe define the trace tF(M,N)as the infimum over constants C∈(0,∞]\nwhere any Lipschitz map f:N →T for anyT ∈F , can be extended\nto˜︁f:M → T such that Lip˜︁f≤CLipf. Subsequently, the absolute\nextendability constant aeF(N)is the supremum of traces aeF(M,N)across\nall metric spaces M⊇N . That is, aeF(N) = sup{aeF(N,M) :N⊆M} .\nThe family of all Banach spaces is most commonly taken as F. In this\nsetting, the classes of absolute extendable metric spaces have been identi-\nfied and extensively investigated, along with the associated trace problems,\n2010 Mathematics Subject Classification. 54C20, 46A16 (Primary), 54F45, 05C63,\n05C83 (Secondary).\nKey words and phrases. Lipschitz extension problem, absolute extendability, Whitney\ncover, Nagata dimension, doubling space, graph minor, Lipschitz free space.\nI would like to express my gratitude to M. C´ uth for his numerous valuable insights into\nmy research efforts. I acknowledge the support of GA ˇCR 23-04776S and of the Charles\nUniversity project GA UK No. 138123. The work was completed while the author was an\nemployee at MSD Czech Republic, Prague, Czech Republic.\n1arXiv:2402.03189v1  [math.FA]  5 Feb 20242 JAN B ´IMA\nby Johnson, Lindenstrauss, and Schechtman [27], Matouˇ sek [36], Lee and\nNaor [33], Lang and Schlichenmaier [32], Brudnyi and Brudnyi [18], Naor\nand Rabani [38], and Basso [9]. More recent research has explored exten-\nsions of Lipschitz maps ranging into quasi-metric and quasi-Banach spaces,\nas detailed by Basso [11] and Albiac et al. [4], respectively. This paper\nalso contributes to this ongoing line of research. Interestingly, establishing\nLipschitz extendability results in the quasi-metric setting often necessitates\ninnovative proof techniques. These novel approaches, in turn, provide a new\nperspective on the case where Banach spaces are considered.\nBefore presenting our results in greater detail, we recall the concept of\nquasi-Banach spaces and properly adopt the definitions of trace and absolute\nextendability. In what follows, the notation A≲Bmeans that A≤CBfor\nsome universal constant C≥0.\nTo that end, we recall (X,∥·∥)is called a quasi-normed space ifXis a\nvector space and ∥·∥is a quasi-norm, that is, (i) ∥x∥>0for anyx̸= 0, (ii)\n∥αx∥=|α|∥x∥for any scalar αandx∈X, (iii) and∥x+y∥≲∥x∥+∥y∥\nfor anyx, y∈X. We callXaquasi-Banach space if it is complete with\nrespect to the linear metric topology induced by ∥·∥.\nIt turns out that every quasi-Banach space is isomorphic to a p-Banach\nspace for some 0< p≤1, as shown in [29, Theorem 1.2]. The converse is\ntrivially true.\nDefinition 1. LetXbe a vector space and 0< p≤1. We say that a map\n∥·∥:X→[0,∞)is ap-norm onXif, in addition to conditions (ii) and (iii),\n(iii’)∥x+y∥p≤∥x∥p+∥y∥pfor anyx, y∈X.\nWe then call (X,∥·∥)ap-normed space. IfXis complete with respect to\nthe metricd(x,y) =∥x−y∥p, wherex, y∈X, we say (X,∥·∥)is ap-Banach\nspace .\nDeveloping extendability results for maps ranging into any general quasi-\nBanach space would be overly ambitious. Therefore, we adopt the following\ndefinition.\nDefinition 2. For a metric space Nand each 0<p≤1, we define the p-trace\ntp(N,M)ofNinM⊇N to be the infimum over all C∈(0,∞]such that\nfor anyp-Banach space Z, any Lipschitz map f:N →Zhas a Lipschitz\nextension˜︁f:M→Zwith Lip˜︁f≤CLipf.\nWe define the absolutep-extandability constant aep(N) = sup{tp(N,M) :\nM⊇N} . If aep(N)<∞, we say thatNisabsolutelyp-extendable .\nTo better motivate the notions we just introduced, particularly in relation\nto thep-Banach setting where 0< p≤1, we can consider extensions from\nfinite subsets. Note that it is easy see that finite metric spaces are absolutely\np-extendable. However, determining the lower and upper estimates on the\nextendability constant ae1(n) = sup{ae1(N) :|N|≤n}is a significant open\nproblem, see Lee and Naor [34] and Naor and Rabani [38]. In general, the\nproblem becomes even more challenging for 0<p< 1. While it can easily\nbe observed that aep(N) = 1 for any two-point metric space Nand0<p<\n1, the absolute extendability constant typically increases as papproaches\nzero. In particular, we have the following result, which is intriguing whenNAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 3\ncompared to [11, Theorem 1.1], asserting that tp(N,M)≤m+ 1whenever\n|M\\N|≤ m, for anym∈Nand all 0<p≤1.\nTheorem I (cf. Theorem 30) .LetN={0,1,2}⊆(R,|·|)andM=N∪\n{3/2}. Then t1(N,M) = 1 but tp(N,M)>1for any 0<p< 1. Moreover,\nwe have tp(N,M)→2asp→0.\nHaving motivated the notion of absolute extandability, let us review the\nfollowing series of results on non-trivial families of absolutely 1-extendable\nspaces, which will have a significant role in the sequel.\nTheorem 3 (Johnson, Lindenstrauss, and Schechtman [27]) .IfNis a subset\nof ann-dimensional normed vector space Y, then t1(N,Y)≲n.\nObserve that, as a straightforward corollary, we have ae(n)≲nfor all\nn∈N. Indeed, it suffices to observe that for any n-point metric space\nNandM⊇N , there exists a non-expansive map of Mintoℓn\n∞which,\nmoreover, is an isometry on N.\nThe proof presented in [27] relies on a specific Whitney-type cover of the\nambient space Y⊇N, with its existence being facilitated by the presence of\nthe Lebesgue measure on Y. Abstracting this approach, the only necessary\ncondition for the proof to pass through is that the space Mfulfills the\ndoubling property.\nDefinition 4. We say that a metric space (N,ρ)isdoubling if there exists\nλN∈N, called the doubling constant , such that any closed ball in Nof\nradius 2r>0can be covered by λN-many closed balls of radius r.\nExamples of doubling metric spaces are provided by subsets of Carnot\ngroups, see [22], and, in particular, subsets of finite-dimensional spaces. To\nthat end, we recall that whenever Nis a subset of an n-dimensional normed\nspace, we have logλN≲n, as shown in [41, Theorem 3].\nAs an application of general extension results obtained using the method\nofstochastic metric decomposition , Lee and Naor [33] achieved the following.\nTheorem 5 (Lee and Naor [33]) .IfNis a doubling metric space, then\nae1(N)≲logλN.\nRecently, a generalization of the result was addressed for maps ranging\nintop-Banach spaces by Albiac et al. [4].\nTheorem 6 (Albiac et al. [4]) .IfNis a doubling metric space, then aep(N)≲\n(15λ4\nN)1/pfor all 0<p≤1.\nThe proof of Theorem 6 uses a distinct method in comparison to that\nof Theorem 5. Specifically, the technique of stochastic decompositions as\napplied in [33] results in an extension map being defined by a Bochner inte-\ngral, specifically over a distributional partition of unity. This approach does\nnot readily adapt to the quasi-metric setting. Meanwhile, the estimate from\nTheorem 6 is evidently suboptimal for p= 1in light of Theorem 5, raising\nthe question of whether the approach can be improved. Revisiting the ideas\nused in the original result due to Johnson, Lindenstrauss, and Schechtman\n[27], we show that this indeed is the case.4 JAN B ´IMA\nNotation 7. In what follows, the notation A≲pBmeans that A≤C(p)·B\nfor some constant C(p)≥0dependent only on p.\nTheorem II (for doubling spaces ; cf. Corollary 16) .IfNhas doubling con-\nstantλN>1, then aep(N)≲pλ3/p−3\nN·logλNfor all 0<p≤1.\nImportantly, we show that the assumptions on N, necessary for the exis-\ntence of a specific Whitney-type cover, can actually be further generalized.\nMore precisely, it suffices to assume that Nhas a finite Nagata dimen-\nsion. Notably, the resulting quantitative estimate is a new result even in\nthe Banach setting where p= 1.1To this end, we recall an earlier estimate\nshowing that ifNhas Nagata dimension at most dwith constant γ, then\nae1(N)≲γd3, see [39, Corollary 5.2].\nDefinition 8. Let(N,ρ)be a metric space. Given γ≥1andd∈N0, we\nsay thatNhasNagata dimension at most dwith constant γ, if for every\ns>0, there exists a family Cof non-empty subsets in Nwith the following\nproperties:\n(a)CcoversN, i.e.,⋃︁\nC∈CC=N,\n(b) for every C∈C,diamC≤γs,\n(c) for every A⊆N with diamA≤s, we have|{C∈C:C∩A̸=∅}|≤\nd+ 1.\nIt is known that if a metric space Nhas a doubling constant λN, it has\nNagata dimension at most λ3\nN−1with constant 2 (see Lemma 15). The\nNagata dimension can in fact be bounded by the logarithm of the doubling\nconstant (see [23], where an alternative definition of the doubling dimension,\nreferred to as the Assouad dimension , is considered). In our applications,\nhowever, this gain would be compensated by increased values of the constant\nγ.\nTheorem II (for spaces with finite Nagata dimension ; cf. Theorem 14) .If\nNhas Nagata dimension at most dwith constant γ, then aep(N)≲pγ·(d+\n1)1/p−1·log(d+ 2) for all 0<p≤1.\nAn important class of spaces with finite Nagata dimension is constituted\nbymetric graphs , which are one-dimensional simplicial complexes induced\nby weighted graphs (see Definition 18). It can be shown that if a weighted\ngraphG= (V,E)excludes the complete graph on mvertices,Km, as a\nminor, then the Nagata dimension, together with constant γof the metric\ngraph Σ(G), can be bounded in terms of m. We note that for minor-excluded\nunweighted graphs, the finiteness of the Nagata dimension was established\nin [40, Theorem 2.2]. In Proposition 20, we generalize this result to the class\nof all metric graphs.\nWe remark that a trace theorem for subsets of metric trees, which are\nmetric graphs induced by weighted trees, was established in Matouˇ sek [36].\nMore recently, this result has been generalized to an absolute extendability\nresult for subsets of metric graphs, induced by minor-excluded weighted\ngraphs, in Lee and Naor [33, Theorem 5.1]. By aplying Proposition 20 in\n1We remark that this result has been recently independently discovered by Basso [10],\nin the narrower context of the Banach setting with p= 1. See also Remark 34.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 5\nconjunction with the general extension result for spaces with finite Nagata\ndimensions, as established in Theorem II, we present an alternative proof\nfor this generalized result and adapt it to the p-Banach setting.\nTheorem III (cf. Theorem 23) .IfΣ(G)is a metric graph induced by a count-\nable and connected weighted graph Gwhich excludes the complete graph Km\nas a minor, then aep(S)≲pm2·9m(1/p−1)for any subset S⊆Σ(G)and any\n0<p≤1.\nTheorem II has an important application to the theory of Lipschitz free\np-spaces . We note that these spaces were first considered by Albiac and\nKalton [2], and their systematic study was initiated by Albiac et al. [3] and\nhas continued through [4, 5, 14]. For a given metric space Mand0<p≤1,\nthere exists a unique, up to an isomoprhism, p-Banach spaceFp(M), called\ntheLipschitz free space over M, such thatMembeds isometrically into\nFp(M)via a mapδ:M→Fp(M), and for every p-Banach space Yand a\nLipschitz map f:M→Ywhich vanishes at the origin, fextends uniquely\nto a linear operator Tf:Fp(M)→Ysuch that Lipf=∥Tf∥.\nIt was an open question whether, for metric spaces N⊆M and0<p≤1,\nthe linearization Ti:Fp(N)→Fp(M)of the canonical inclusion i:N→M\nis an isomorphism (see [3, Question 6.2]). As an essential part, Theorem II\n(see also Corollary 17 or apply Theorem III with m= 3) was used in [20,\nTheorem 3.21] to show that this is indeed the case. Specifically, the value\n∥T−1\ni∥can be bounded by the absolute p-extendability constant of a metric\ntree, that is, a graph which excludes K3as minor.\nThis paper is structured as follows. In Section 2, we lay out a general\nextension result which only relies on the existence, and quality, of a partic-\nular Whitney-type cover on the space M\\N . The presence of such covers\nfor spaces with finite Nagata dimension is established in Section Section 3.\nSection 4 introduces the counterexample from Theorem I. The proof will\nrely on the properties of Lipschitz free p-spaces, and in this context, we will\ndiscuss the general connection of the Lipschitz extension problem to Lips-\nchitz freep-spaces. In Section 5, we formulate two open questions related\nto the dependence of the Lipschitz extension constant on p.\n2.Covers of the Whitney Type and Lipschitz Extensions\nIn order to extend Lipschitz maps defined on Nto the ambient space\nM, we consider a partition of unity {ϕi}i∈Iinduced by a cover {Ki}i∈Ion\nM\\N . For this purpose, we adopt the general concept of a Whitney-type\ncover; thus, we cover the space M\\N in a way that the diameters of the\ncovering sets are proportional to their distance from the set N.\nDefinition 9. LetNbe a closed subset of a metric space (M,ρ). We say\nthat a family{Ki}i∈Iof open sets inM\\N is aWhitney cover of M\\N\nwith parameters (o,s,d,a ), whereo∈Nands, d, a> 0, provided\n(i) for any x∈M\\N , it holds|{i∈I:x∈Ki}|≤o,\n(ii) for any x∈M\\N , there exists i∈I such thatρ(x,M\\Ki)≥\ns·ρ(x,N),\n(iii) it holds that diamKi≤d·ρ(Ki,N)for anyi∈I,\n(iv) for any i∈Iandx, y∈Ki, we haveρ(x,N)/ρ(y,N)<a.6 JAN B ´IMA\nObserve that by condition (ii), {Ki}i∈Iis, in particular, a cover of M\\N .\nNote that we do not require the covering sets to be disjoint. However,\ntheoverlapping constant proves to be a significant factor for the quantita-\ntive estimates on the Lipschitz constant of resulting extensions. This is so\nbecause, for a given point y∈M\\N , the extension of a map fwill be\ndefined as∑︁\ni∈Iϕi(y)f(xi), for some predetermined xi∈N. Note that in\nthe setting where p <1, there arises an additional quasimetric factor (see\ncondition (iii’)), which grows as n1/p−1fornsummands. For brevity, we will\nintroduce the following notation.\nNotation 10. We setC(p,n) =n1/p−1where 0< p≤1andn∈N. Note\nthat for any p-normed space (X,∥·∥p)andxi∈X, wherei∈{1,...,n}, we\nhave∥∑︁n\ni=1xi∥p≤C(p,n)∑︁n\ni=1∥xi∥p.\nThis reasoning also explains why we introduce another type of cover that\nis distinct from the one recently considered by Lee and Naor [33]. Mainly, the\nauthors examine distributions over covers and derive ’well-behaved’ parti-\ntions of unity where desirable properties are typically attained ”on average.”\nWhile it is often possible to pass to locally finite partitions, we find it more\nconvenient for our purposes to construct the covers in a way that lets us\nmore precisely track the overlapping constant.\nAssuming the existence of Whitney cover on M\\N , we can now con-\nstruct a Lipschitz extension to the space M. Let us note that the adapted\ndefinition of Whitney-type cover, yet without the condition (iv), appeared\nin [4, Proposition 5.2], and a related result on Lipschitz extension for maps\nranging into p-Banach spaces was given in [4, Theorem 5.1]. We remark\nthat in the referenced paper, the extension problem was examined within\nthe context of Lipschitz free p-spaces. Here we achieve to refine the exten-\nsion result further by employing an optimization trick originating from the\nwork of Johnson, Lindenstrauss, and Schechtman [27].\nTheorem 11. LetN ⊂ (M,ρ)be non-empty and such that M\\N ad-\nmits a Whitney cover with parameters (o,s,d,a ), whereo∈N,o >1, and\ns, d > 0. Then for any 0< p≤1, thep-trace tp(N,M)ofNinMis at\nmostD(p,s,d,a )·C(p,o) log2(2o), whereD(p,s,d,a )is a universal constant,\nquantified in (6).\nLet us first note the following elementary inequality.\nLemma 12. For anym≥1,n∈N, andai≥0, wherei∈{1,...,n}and at\nleast oneaiis non-zero, it holds that\nn∑︂\ni=1am−1\ni/n∑︂\ni=1am\ni≤n1/m/(︄n∑︂\ni=1am\ni)︄1/m\n.\nProof. Note thatx↦→x(m−1)/mis concave on R+. Consequently, we get∑︁n\ni=1am−1\ni/n≤(∑︁n\ni=1am\ni/n)(m−1)/m. Rearranging the terms, the claim is\nestablished. □\nProof of Theorem 11. Let{Ki}i∈Ibe a Whitney cover for M\\N with pa-\nrameters (o,s,d,a ). We can assume that each set Ki, wherei∈I, is non-\nempty. Also, we pick 0< p≤1and consider a parameter m≥1, to be\noptimized later.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 7\nWe construct a partition of unity {ϕi}i∈IonM\\N as follows. For every\ni∈I, we define a map ϕ′\ni:M\\N → Rasx↦→ρm(x,M\\Ki), where\nx∈M\\N . It follows from conditions (i) and (ii) that for any x∈M\\N ,\nthe set{i∈ I :ϕ′\ni(x)>0}is finite non-empty. Consequently, we set\nϕi:M\\N→ [0,1]by lettingϕi=ϕ′\ni/∑︁\nj∈Iϕ′\nj, for eachi∈I.\nNote that for each i∈I, we haveρ(N,Ki)>0. Indeed, each Ki⊆M\\N\nis non-empty, so ρ(N,Ki)>0is trivially true if Kiis a singleton. Otherwise,\nifKicontains at least two points, then 0<diamKi≤d·ρ(Ki,N)as per\ncondition (iii).\nConsequently, we can pick xi∈N such thatρ(xi,Ki)≤2ρ(N,Ki). By\ncondition (iii), it follows that for any i∈Iandx∈Ki,\n(1) ρ(x,xi)≤ρ(xi,Ki) + diamKi≤(2 +d)·ρ(N,Ki).\nConsider a Lipschitz map f:N→Zinto somep-Banach space Z. We\nmay, without loss of generality, assume that Lipf= 1.\nWe define an extension f′:M→Zasf′(x) =f(x)if and only if x∈N\nandf(x) =∑︁\ni∈Iϕi(x)f(xi)otherwise. In the rest of the proof, we will\nestimate the Lipschitz constant of f′. Note that since the Lipschitz constant\noff′, when restricted to N, is the same as that of f, we can consider the\nfollowing three cases.\nCase 1. Assume that x∈N andy∈M\\N .\nNote that for all i∈ I withϕi(y)>0, we haveρ(y,xi)≤(2 +d)·\nρ(N,Ki)≤(2 +d)·ρ(x,y)by (1). Consequently, ρ(x,xi)≤ρ(x,y) +\nρ(y,xi)≤(3 +d)ρ(x,y), and\n⃦⃦f′(x)−f′(y)⃦⃦\np=⃦⃦⃦⃦⃦∑︂\ni∈I(f′(x)−f′(xi))ϕi(y)⃦⃦⃦⃦⃦\np\n≤C(p,o) max\ni∈I\nϕi(x)>0ρ(x,xi)\n≤C(p,o)(3 +d)·ρ(x,y).\nCase 2. Letx, y∈M\\N be such that ϕi(x)ϕi(y) = 0 for alli∈I.\nRecall that by condition (ii), there exists i∈ I withρ(x,M\\Ki)≥\ns·ρ(x,N), so that, in particular, ϕi(x)>0. It follows that ϕi(y) = 0 , i.e.,\ny∈M\\Ki, andρ(x,y)≥ρ(x,M\\Ki)≥s·ρ(x,N).\nPickx′∈N such thatρ(x′,x)≤2ρ(x,N). By the previous paragraph,\nρ(x′,x)≤2/s·ρ(x,y)andρ(x′,y)≤ρ(x′,x) +ρ(x,y)≤(2/s+ 1)·ρ(x,y).\nIt follows from Case 1 that\n⃦⃦f′(x)−f′(y)⃦⃦\np≤(︂⃦⃦f′(x)−f′(x′)⃦⃦p\np+⃦⃦f′(x′)−f′(y)⃦⃦p\np)︂1/p\n≤C(p,o)(3 +d)(︁ρp(x,x′) +ρp(x′,y))︁1/p\n≤C(p,o)(3 +d)(︂\n21+p/sp+ 1)︂1/p·ρ(x,y),\nwhere, for the third inequality, we used the fact that (2/s+ 1)p≤2p/sp+ 1\nby subadditivity of x↦→xp.\nCase 3. Finally, let x, y∈M\\N andi∈Ibe such that{x,y}⊆Ki.8 JAN B ´IMA\nNote that whenever x∈Kjfor somej∈I, then according to (1), we\nhave\nρ(xi,xj)≤ρ(xi,x) +ρ(x,xj)\n≤(2 +d)(ρ(N,Ki) +ρ(N,Kj))\n≤2(2 +d)·ρ(x,N),(2)\nand similarly for y.\nWe may rewrite f′(x)−f′(y) =∑︁\nj∈I(f(xj)−f(xi))(ϕj(x)−ϕj(y)). In\nlight of (2), we will now estimate the sum∑︁\nj∈I|ϕj(x)−ϕj(y)|.\nTo that end, letI′={j∈I:ϕj(x)>0orϕj(y)>0}, where|I′|≤2o\nby condition (i). Consequently, we enumerate the set I′asI′={i1,...,in}\nfor somen≤2o. Also, we define a= (ak)n\nk=1, b= (bk)n\nk=1∈Rnby letting\nak=ρ(x,M\\Kik),bk=ρ(y,M\\Kik)for eachk∈{1,...,n}.\nFor everyk∈{1,...,n}, we define dk: [0,1]→Ras\nλ↦→(λak+ (1−λ)bk)m\n∑︁n\nl=1(λal+ (1−λ)bl)m, λ∈[0,1].\nNote thatdk(1) =ϕik(x)anddk(0) =ϕik(y), for eachk∈{1,...,n}.\nPickk∈{1,...,n}. A short computation shows that for any λ∈(0,1),\nd′\nk(λ) =m(λak+ (1−λ)bk)m−1\n∑︁n\nl=1(λal+ (1−λ)bl)m(ak−bk)\n−m(λak+ (1−λ)bk)mn∑︂\nl=1(λal+ (1−λ)bl)m−1\n(∑︁n\nl=1(λal+ (1−λ)bl)m)2(al−bl).\nTaking the absolute value and summing across k, we obtain\n(3)n∑︂\nk=1|d′\nk|(λ)≤2m∑︁n\nl=1(λal+ (1−λ)bl)m−1\n∑︁n\nl=1(λal+ (1−λ)bl)m·ρ(x,y), λ∈(0,1),\nwhere we used the fact that |al−bl|=|ρ(x,M\\Kil)−ρ(y,M\\Kil)|≤ρ(x,y)\nfor anyl∈{1,...,n}.\nBy Lemma 12 with n≤2o, we deduce for every λ∈(0,1),\n(4)n∑︂\nk=1|d′\nk|(λ)≤2m(2o)1/m(︄n∑︂\nk=1(λak+ (1−λ)bk)m)︄−1/m\n·ρ(x,y).\nAlso, recall that by condition (ii), there exist k′, k′′∈{1,...,n}with\nρ(x,M\\Kik′)≥s·ρ(x,N)andρ(y,M\\Kik′′)≥s·ρ(y,N). Consequently,\nwe get for any λ∈(0,1),\nn∑︂\nk=1(λak+ (1−λ)bk)m≥∑︂\nk∈{k′,k′′}(λak+ (1−λ)bk)m\n≥(min{aik′,bik′′}/2)m\n≥(s/2)m(min{ρ(x,N),ρ(y,N)})m.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 9\nHence, continuing (4), we obtain\n∑︂\nj∈I|ϕj(x)−ϕj(y)|=n∑︂\nk=1|dk(1)−dk(0)|\n≤n∑︂\nk=1∫︂1\n0|d′\nk|(λ)dλ\n≤4/s·m(2o)1/m\n·(min{ρ(x,N),ρ(y,N)})−1·ρ(x,y).(5)\nConsequently,\n∥f′(x)−f′(y)∥p=⃦⃦⃦⃦⃦⃦∑︂\nj∈I(f(xj)−f(xi))(ϕj(x)−ϕj(y))⃦⃦⃦⃦⃦⃦\np\n≤C(p,2o) max\nj∈I′∥f(xj)−f(xi)∥p\n·∑︂\nj∈I′|ϕj(x)−ϕj(y)|\n≤ρ(x,y)·C(p,2o)(8/s)(2 +d)m(2o)1/m\n·max{ρ(x,N),ρ(y,N)}\nmin{ρ(x,N),ρ(y,N)},\nwhere the third inequality follows from (2) and (5).\nWe note that max{ρ(x,N),ρ(y,N)}/min{ρ(x,N),ρ(y,N)}<aby con-\ndition (iv). Taking m= log2(2o), we get that\nm(2o)1/m= log2(2o)·2log2(2o)/log2(2o)≤2 log2(2o).\nAltogether, it follows that Lipf′≤D·C(p,o) log2(2o), whereDdepends\nonly onp,s,d, anda. Quantitatively, we may set\n(6)D(p,s,d,a ) = max{︃\n32/s·C(p,2)(d+ 2)a,(3 +d)(︂\n2p+1/sp+ 1)︂1/p}︃\n.\nThe proof is complete. □\nWe remark that the proof actually shows a stronger claim. Specifically,\nit establishes the existence of simultaneous Lipschitz extension fromN\ntoM(see [17]). That is, there is a bounded linear extension operator\nLip(N,Z)→Lip(M,Z), wherein the spaces of Lipschitz maps are endowed\nwith the Lipschitz semi-norm. Also note, by the method of proof, that\nRngf′⊆conv Rngf.\n3.Covers for Metrics with Finite Nagata Dimension\nWe recall that the concept of Nagata dimension is closely related to that of\nthe asymptotic dimension (consider Gromov [26] and Bell and Dranishnikov\n[13]). The Nagata dimension was first considered by Nagata [37], and its\ncontemporary formulation is due to Assouad [7]. As we will show, doubling\nspaces and metric graphs provide examples of spaces with a finite Nagata\ndimension. Intriguingly, for a metric space (X,ρ)with a Nagata dimension\nnot exceeding d, its corresponding metric snowflake (X,ρp)(for sufficiently10 JAN B ´IMA\nsmall exponents 0<p< 1), can be embedded into a product of d+ 1metric\ntrees (see [32, Theorem 1.3]). Furthermore, we recall that ultrametric spaces\nhave Nagata dimension zero with constant 1 (also see [12]).\nThe aim of this section is to establish a general theorem affirming the\nexistence of Whitney covers for spaces Nwith finite Nagata dimension.\nSubsequently, we examine particular distinguished classes of such spaces,\nalong with estimates on their dimension.\nThe essence of the proof showing the existence of Whitney covers involves\npartitioningM\\N into a collection of expanding metric annuli within M\\N .\nThese annuli increase in size directly proportional to their distance from\nN, and the corresponding covering sets of these annuli are then induced\nby Nagata covers of N, associated with various s > 0. In this context,\ncondition (c) controls the overlapping constant of the cover. The details\nfollow.\nProposition 13. LetNbe a closed subset of a metric space (M,ρ). IfNhas\nNagata dimension at most dwith constant λ, then there exists a Whitney\ncover ofM\\N with parameters (3(d+ 1),ϵ/2,4(1 +γ)(ϵ+ 2),5), for each\n0<ϵ< 1/2.\nProof. For eachj∈Z, we consider the metric annuli\nAj={x∈M\\N : 2j≤ρ(x,N)<2j+1}.\nAlso, for every j∈Zwe fix a coverCj, associated with sj>0(to be\noptimized later), that demonstrates that Nhas Nagata dimension at most\ndwith coefficient λ. We then set\nBj\nC={x∈Aj:ρ(x,N) =ρ(x,C)}, whereC∈Cjandj∈Z.\nWe assert that for any j∈Z, ifsjis such that sj≥2j+2, then{Bj\nC:\nC∈Cj}is a cover of Aj. For this, note that for any x∈Aj, the open\nballU(x,2j+1)meetsN, and, moreover, U(x,2j+1)∩N has diameter at\nmost 2·2j+1. Consequently, {C∈Cj:U(x,2j+1)∩C̸=∅}is finite by\ncondition (c). It follows that\nρ(x,N) = inf{ρ(x,y) :y∈N}\n= inf{ρ(x,y) :y∈U(x,2j+1)∩N}\n= min{ρ(x,C) :C∈{C∈Cj:U(x,2j+1)∩C̸=∅}},\nand the assertion is established.\nIn the next step, we define a Whitney cover of M\\N as follows. We pick\n0< ϵ < 1/2and for each j∈ZandC∈Cj, we setKj\nC=⋃︁{U(x,ϵ·2j) :\nx∈Bj\nC}.\nNote that for any x∈M\\N andj∈Z, ifx∈Kj\nCfor someC∈Cj, then\nU(x,(ϵ+ 2)·2j)∩C̸=∅. Consequently, provided that sj≥2j+1(ϵ+ 2), we\nget that|{C∈Cj:U(x,(ϵ+ 2)·2j)∩C̸=∅}|≤d+ 1by condition (c).\nRecall that Bj\nC⊆Ajfor anyj∈ZandC∈Cj, so thatKj\nC⊆{x∈\nM\\N : 2j−1≤ρ(x,N)<5/2·2j}. This establishes condition (iv) with\na= 5. Moreover, for every j∈Zwe have\n(7){i∈Z:Aj∩Ki\nC̸=∅for someC∈Ci}⊆{j−1,j,j+ 1}.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 11\nTogether with the previous part, it follows that the overlapping constant\nofrom condition (i) is at most 3(d+ 1), provided that sj≥2j+1(ϵ+ 2) for\neachj∈Z.\nTo adress condition (ii), we assert that for any x∈M\\N , ifx∈Bj\nCfor\nsomej∈ZandC∈Cj, thenρ(x,M\\Kj\nC)≥ϵ/2·ρ(x,N). Indeed, note\nthatρ(x,M\\Kj\nC)≥ϵ·2jby the definition of Kj\nC, and 2j≥1/2·ρ(x,N)as\nx∈Aj.\nFinally, for condition (iii), we estimate the diameter of Kj\nCfor any given\nj∈ZandC∈Cj. To that end, note that by the definition, there exists\ny∈Bj\nCsuch thatρ(x,y)< ϵ·2j, and, consequently, ρ(x,C)≤ρ(x,y) +\nρ(y,C)<(ϵ+ 2)·2j. It now follows that for any x, x′∈Kj\nC, we have\nρ(x,x′)≤ρ(x,C)+diamC+ρ(x′,C)<2(ϵ+2)·2j+γsj. That is, diamKj\nC≤\n(ϵ+ 2)·2j+1+γsj.\nChoosingsj= 2j+1(ϵ+ 2) for eachj∈Z, we get diamKj\nC<2j+1(ϵ+\n2)(1 +γ)≤4(1 +γ)(ϵ+ 2)·ρ(N,Kj\nC). Here, we used that 2j−1≤ρ(N,Kj\nC)\nby the definition of Kj\nC.\nIt follows that the sets Kj\nC, wherej∈ZandC∈Cj, form a Whitney\ncover ofM\\N with parameters (3(d+ 1),ϵ/2,4(1 +γ)(ϵ+ 2),5). □\nOverall, in light of Theorem 11, we have established the following.\nTheorem 14. IfNhas Nagata dimension at most dwith constant γ, then\naep(N)≲pγ·(d+ 1)1/p−1·log(d+ 2) for all 0<p≤1.\nA similar argument to the one used in the proof of Proposition 13 can be\nused to construct Whitney covers when Nis doubling, as demonstrated, for\nexample, in [4, Proposition 5.2]. However, to the best of our knowledge, this\napproach would still provide an estimate that is quantitatively the same (in\nterms ofλN) as the one we obtain by estimating the Nagata dimension of\ndoubling metric spaces, followed by applying Proposition 13.\nLemma 15. Let(N,ρ)be a doubling metric space with doubling constant\nλN>1. ThenNhas Nagata dimension at most λ3\nN−1with constant 2.\nProof. Lets>0be given. By the Zorn lemma, we obtain a maximal subset\nSofNthat iss-separated, i.e., ρ(x,x′)>sfor everyx, x′∈S.\nWe claim that for any x∈N, the closed ball B(x,3s)contains no more\nthanλ3\nNelements of S. Given thatNis doubling, we can cover B(x,3s)\nwithλ3\nNclosed balls of radius s/2. Each of these balls contains a maximum\nof one element of S, thus proving our claim.\nConsequently, by [6, Lemma 2.4] applied to the subspace S, there is a\ncoloringk:S→{1,...,λ3\nN}such thatk(x)̸=k(x′)wheneverρ(x,x′)≤3s,\nfor anyx, x′∈Swithx̸=x′.\nIt is easy to see that for any subset A⊆N with diamA≤sand each\ni∈Rngk, there exists at most one x∈k−1(i)for which the closed ball\nB(x,s)intersectsA.\nTherefore, if we denote Ci={B(x,s) :x∈k−1(i)}, wherei∈Rngk,\nand defineC=⋃︁{Ci:i∈Rngk}, thenCsatisfies conditions (a) to (c) with\nd=λ3\nN−1andλ= 2. □12 JAN B ´IMA\nConsequently, we obtain the following corollary.\nCorollary 16. IfNhas doubling constant λN>1, then aep(N)≲pλ3/p−3\nN·\nlogλNfor all 0<p≤1.\nContinuing the applications of Theorem 14, we note that an important\nclass of spaces with finite Nagata dimension is provided by metric trees ,\nwhich are geodesic metric spaces where all geodesic triangles are degenerate.\nGiven that every metric tree has Nagata dimension at most 1 with constant\n6 (see [32, Proposition 3.2], where the result is formulated without stating\nthe exact value of the constant 6; however, this value can be recovered from\nthe proof), we derive the following generalization of a classical extension\nresult due to Matouˇ sek [36]. Note that we implicitly use the fact that if a\nspaceNhas a Nagata dimension of at most dwith a constant γ, then any\nof its subspaces does as well.\nCorollary 17. IfTis a metric tree and S⊆Tany its subset, then aep(S)≲p1\nfor any 0<p≤1.\nAs we will see, Corollary 17 is in fact a specific corollary of a more gen-\neral result showing that a metric graph induced by a weighted graph , which\nexcludes the complete graph Kmas a minor , has finite Nagata dimension.\nIn particular, this dimension, along with the associated constant γ, can be\nbounded in terms of m. The connection to metric trees is that they are in-\nduced by weighted graphs which do not contain cycles, meaning they exclude\nK3, the complete graph on three vertices, as a minor.\nBefore we generalize this result to graphs that exclude minors of higher\ndegrees, we recall the following definition.\nDefinition 18 ([33]) .LetG= (V,E)be a connected undirected graph and\nletϕ:E→[0,∞)be a weight function that assigns weights to the edges in\nE. Consider the one-dimensional simplicial complex Σ(G), which is formed\nby replacing every edge e∈Ewith a segment of length equal to ϕ(e).\nThe metric graph (Σ(G),ρϕ)induced by Gandϕdenotes Σ(G)with its\ninherent Riemannian semimetric structure, which restricts to the shortest\npath semimetric on the vertices of G.\nNote that even when all the weights are positive, if the set of edges is infi-\nnite, the resulting space (Σ(G),ρϕ)can fail to satisfy the separation axiom.\nIn other words, it becomes a semimetric space. However, in what follows, we\ncan readily adapt the metric notions to the semimetric setting, or implicitly\ntransition to an induced quotient space (Σ(G)/∼,ρ′\nϕ), where we set vtow\nif and only if ρϕ(v,w) = 0 . There will be no loss of generality in the present\ncontext due to these considerations.\nDefinition 19. Given a connected graph G= (V,E), we say that a graph\nG′is aminor ofGif it can be derived from Gin a finite sequence of the\nfollowing operations:\n(i) removing an edge {v,w}∈E, which results in a graph G′= (V,E′),\nwhereE′=E\\{e},\n(ii) contracting an edge {v,w}=e∈E, which results in a graph G′=\n(V′,E′), whereV′=V\\{w}andE′=E\\{e∈E:w∈e}∪{{v,u}:\n{w,u}∈E, u̸=v}.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 13\nThe concept of minor exclusion is of significant importance in the theory\nof topological and geometric properties of graphs. For example, it is closely\nlinked to the problem about embeddability of graphs into Banach spaces.\nWe recall that the Kuratowski Theorem characterizes planar graphs only in\nterms of minor exclusion (refer also to [42]).\nMore relevant to the topic of this paper, there has also been considerable\ninterest in the study of dimensionality invariants of graphs, focusing par-\nticularly on the asymptotic and Nagata dimensions of semimetrics induced\nby minor-excluded graphs and minor-closed families of graphs. Consider\nthe contributions by Gromov [26], Ostrovskii and Rosenthal [40], Liu [35],\nBonamy et al. [15], and Distel [21], to name a few.\nSpecifically, Ostrovskii and Rosenthal [40] established that the Nagata\ndimension of unweighted graphs that exclude the complete graph on mver-\ntices,Km, is at most 4m−1with a constant that depends only on m. In this\npaper, we generalize their result to the context of weighted graphs. That is,\nwe show the following.\nProposition 20. IfG= (V,E)is a countable and connected weighted graph\nwhich excludes the complete graph Kmas a minor, then the metric graph\n(Σ(G),ρ)has a Nagata dimension at most 32m−2−1with constant γ≲m.\nOur approach is based on the general Klein–Plotkin–Rao decomposition\ntechnique for unweighted graphs, introduced by Klein, Plotkin, and Rao [31]\nand later refined by Fakcharoenphol and Talwar [24] and Abraham et al. [1].\nThis method also surfaced by Ostrovskii and Rosenthal [40]. Furthermore,\nit was applied to metric graphs by Lee and Naor [33], where the technique\nwas employed to establish the existence of stochastic decompositions, as to\naddress the Lipschitz extension problem.\nThe very general idea of the proof is to decompose the metric graph into\na set of metric annuli and consider the connected components within these\nannuli. Then, we inductively apply the decomposition algorithm on each of\nthese connected components.\nNotation 21. In what follows, if Cis a connected subset of Σ(G), we letρC\ndenote the weak distance onC, i.e.,ρC(w,w′)forw, w′∈Cis defined as\nthe infimum over lengths of paths in Cconnecting wtow′.\nIn the sequel, we will require the following technical result. Its proof is\nhighly technical and relies on the decomposition technique borrowed from\nFakcharoenphol and Talwar [24], which has been adapted here to the setting\nof metric graphs. Consequently, the proof will be presented in the Appendix.\nLemma 22. LetGbe a countable and connected weighted graph such that\nthere arer > 0,m∈Nwithm≥3,δ∈ {0,1,2}m−1,Σ(G) =C1⊇\n...⊇Cm, and points si∈Cifori∈{1,...,m}satisfying that for every\ni, i′∈ {1,...,m}, wherei < i′, we have ρ(si,si′)>24mrand the set\nCi+1is a pathwise connected component of Ani={v∈Ci: 3r(ni−1)≤\nρCi(v,si)−δir<3rni}, for someni∈N0.\nThenGcontainsKmas a minor.\nWe are now ready to give the proof of Proposition 20.14 JAN B ´IMA\nProof. Letr>0. In what follows, we can assume that diam Σ(G)>24mr,\nsince otherwise{Σ(G)}would be a Nagata cover with the desired properties.\nWe setA∅={Σ(G)},I∅=∅, andF∅(Σ(G)) = (v,)(which is a sequence of\nlength one) for some v∈Σ(G). For eachi∈{1,..., 2m−2}, we inductively\ndefine setsAδ, Iδ⊆P(Σ(G))and a mapFδ:Aδ→(Σ(G)∪{∅} )i+1(without\nloss of generality, assume that ∅/∈Σ(G)), whereδ∈{0,1,2}i, with the\nfollowing properties.\nSpecifically, we will assume that for each i∈{0,..., 2m−2}andδ=\n(δ1,...,δi)∈{0,1,2}i(where we use the convention that δ=∅ifi= 0),\n(i)Aδ∪Iδis a partition of Σ(G),\n(ii) for any C∈Aδ, there exists a unique C′∈A (δ1,...,δi−1)such that\nC⊆C′. Moreover, Fδ(C)⊃F(δ1,...,δi−1)(C′),\n(iii) for any C∈Iδ, we have diamC≤(48m+ 6)r,\n(iv) for any C∈Aδ, if we denote Fδ(C) = (sj)i+1\nj=1, then either si+1∈C\norsi+1=∅. Moreover, if lis the greatest l∈{1,...,i + 1}such that\nsl′̸=∅for eachl′≤l, thensl′′=∅for everyl′′>l,\n(v) ifC,(sj)i+1\nj=1, andlare as above and, moreover, l < i + 1, then for\neveryv∈Cthere exists j∈{1,...,l}such thatρ(sj,v)≤24rm,\n(vi) ifC,(sj)i+1\nj=1, andlare as above, and if j∈{1,...,l}is such that l+\nj≤i, thenρ(sj,v)>24rmfor anyv∈C. Hence, by condition (v),\nwe have that i<2l,\n(vii) ifC,(sj)i+1\nj=1, andlare as above, then ρ(sj,sj′)>24rmfor every\nj, j′≤l, wherej̸=j′,\n(viii) ifCand(sj)i+1\nj=1are as above, and if si̸=∅andC′∈A (δ1,...,δi−1)\nis such that C⊆C′, thenCis a pathwise connected component of\n{v∈C′: 3r(n−1)≤ρC′(v,si)−δir<3rn}for somen∈N0.\nMoreover, we will assume that\n(ix) for any v∈Σ(G), there exists δ∈{0,1,2}iandC∈Aδ∪Iδsuch\nthat the open ball of radius rcentered at vis contained in C, i.e.,\nU(v,r)⊆C.\nLeti∈{1,..., 2m−2}be such that for any δ∈{0,1,2}i−1, the sets\nAδ,Iδ, and the associated map Fδwere defined, satisfying conditions (i)\nand (iii) to (vii).\nPickδ= (δ1,...,δi−1,δi)∈{0,1,2}i. IfA(δ1,...,δi−1)=∅, we putAδ=∅\nandIδ=I(δ1,...,δi−1). Otherwise ifA(δ1,...,δi−1)̸=∅, we define for each\nC∈A (δ1,...,δi−1)the setsaCandiCas follows. Let (sj)i\nj=1=F(δ1,...,δi−1)(C)\nand pick the greatest l∈{1,...,i}such that for any l′≤l, we havesl′̸=∅.\nAssume that si̸=∅. We letiC=∅. As foraC, we consider the annuli\n(8)An={v∈C: 3r(n−1)≤ρC(v,si)−δir<3rn}, n∈N0.\nSubsequently, if A′\nndenotes the set of disjoint pathwise connected topological\ncomponents of Anfor eachn∈N0, we define aC=⋃︁\nn∈N0A′\nn.\nIfsi=∅, we consider the following three cases. Let B={v∈C:\nρ(si−l,v)≤24mr}, where we use that i−l≤lby condition (vi). If Bis\nempty, we put aC={C}andiC=∅. IfBis non-empty and B3r={v∈C:\nρC(B,v)≤3r}contains all C, we putaC=∅andiC={C}.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 15\nIn the remaining case, when BandC\\B3rare non-empty, we put Bδir=\n{v∈C:ρC(B,v)≤δir}and letiCandaCbe the sets of connected\ncomponents in BδirandC\\Bδir, respectively.\nHaving defined aCandiCfor eachC∈A (δ1,...,δi−1), we put\nAδ=⋃︂\n{aC:C∈A (δ1,...,δi−1)}\nand\nIδ=I(δ1,...,δi−1)∪⋃︂\n{iC:C∈A (δ1,...,δi−1)}.\nNote that by the induction hypothesis, it is easy to see that Aδ∪Iδis\na partition of Σ(G), which establishes condition (i). Also, the first part of\ncondition (ii) clearly is satisfied.\nTo address condition condition (iii), by the inductive assumption is suffices\nto pickC∈iC′for someC′∈A (δ1,...,δi−1). Note that for any C∈I(δ1,...,δi−1),\nwe can find C′∈A (δ1,...,δi−1)such thatC∈iC′. Also, ifiandlare as in\nthe definition of iC′, it is easy to see that, indeed, ρ(v,si−l)≤24mr+ 3r\nfor anyv∈C. Consequently, it follows from the triangle inequality that\ndiamC≤(48m+ 6)r.\nFor eachC∈ Aδ, we define Fδ(C)as follows. First, we pick C′∈\nA(δ1,...,δi−1)such thatC′⊇C, i.e.,C∈aC′. If we denote (s′\nj)i\nj=1=\nF(δ1,...,δi−1)(C′), we letFδ(C) = (sj)i+1\nj=1be such that sj=s′\njfor anyj∈\n{1,...,i}. Also, ifsi̸=∅and there exists a point v∈Cat distance greater\nthan 24mrfrom{sj:j∈{1,...,i}}, we letsi+1=v. Otherwise we put\nsi+1=∅. It is easy to see that Fδsatisfies the remaining part of condition (ii)\nand conditions (iv), (v) and (vii). Also condition (viii) clearly follows from\nthe inductive assumption and the construction.\nAs for condition (vi), let C,(sj)i+1\nj=1, andlbe as in the statement. By\nthe inductive hypothesis, we can assume that j=i−l≥1. But then, if\nC′∈A (δ1,...,δi−1)is such that C′⊇C, it follows from the construction that\nanyv∈C′withρ(sj,v)≤24mris now contained in some component within\nIδ. In other words, ρ(sj,v)>24mrfor anyv∈C.\nTo establish condition (ix), pick v∈Σ(G)and assume that for some\n(δ1,...,δi−1)∈{0,1,2}i−1andC∈A (δ1,...,δi−1)∪I(δ1,...,δi−1), it holds that\nU(v,r)⊆C. Assume that Fδ(C) = (sj)i\nj=1satisfies that si̸=∅. Then, ifδi\nis such that ρC(v,si)−δir∈[3r(n−1) +r,3rn−r)for somen∈N0, then\nalsoU(v,r)⊆An={w∈C: 3r(n−1)≤ρC(w,si)−δir<3rn}. It is easy\nto see that U(v,r)remains within a single connected component in An.\nAs for the remaining cases, note that if si=∅andBandC\\B3rare as\nin the part where we defined the corresponding sets aCandiC, it suffices\nto consider the case when B̸=∅andC\\B3r̸=∅. IfρC(B,v)≤r, we put\nδi= 2and thenU(v,r)clearly remains within a single connected component\ninBδir={u∈C:ρC(B,u)≤2r}. Otherwise if ρC(B,v)> r, we let\nδi= 0, in which case U(v,r)remains within a single connected component\ninC\\Bδir={u∈C:ρC(B,u)>0}.\nIn the following part, we will use the sets Iδ⊆ P(Σ(G)), whereδ∈\n{0,1,2}2m−2, to construct the actual Nagata cover of Σ(G).\nTo that end, let us first verify that Aδ=∅for eachδ= (δ1,...,δ 2m−2)∈\n{0,1,2}2m−2. Indeed, let us assume for a contradiction that there exists C∈16 JAN B ´IMA\nA(δ1,...,δ 2m−2). It follows from conditions (vii) and (viii) and Lemma 22 that if\nwe putF(δ1,...,δ 2m−2)(C) = (sj)2m−1\nj=1, thensm=∅, i.e.,l<m . Consequently,\nconditions (ii) and (vi) show that 2m−2<2l≤2(m−1), which is absurd.\nIn particular, it follows that each set Iδis a partition of Σ(G).\nSimilarly, condition (iii) shows that the diameter of each set in Iδis\nbounded by (48m+ 6)r.\nFor everyδ∈{0,1,2}2m−2and each element C∈Iδ, we consider the set\n(9) C′={x∈C:ρ(x,⋃︂\n{S:S∈Iδ\\{C}})≥r},\nand putI′\nδ={C′:C∈Iδ}.\nIt is easy to see from (9) that for any subset A⊆Σ(G)with diamA≤r/2,\nwe have\n|{C′∈I′\nδ:C′∩A̸=∅}|≤ 1, δ∈{0,1,2}2m−2.\nConsequently,\n(10)|{C′∈⋃︂\n{I′\nδ:δ∈{0,1,2}2m−2}:C′∩A̸=∅}|≤ 32m−2.\nAt the same time, for each v∈Σ(G), ifδ∈{0,1,2}2m−2is such that\nU(v,r)⊆Cfor someC∈Iδ(see condition (ix)), then also v∈C′. This\nshows that⋃︁\nδ∈{0,1,2}2m−2I′\nδis a cover of Σ(G).\nTherefore, for any r >0, we have constructed a cover Cwhich satisfies\nconditions (a) to (c) with s=r/2,d= 32m−2−1(see (10)), and γ≲m,\nthus proving the claim. □\nAs a consequence, we obtain the following extension result for subset of\nmetric graphs.\nTheorem 23. IfΣ(G)is a metric graph induced by a countable and connected\nweighted graph Gwhich excludes the complete graph Kmas a minor, then\naep(S)≲pm2·9m(1/p−1)for any subset S⊆Σ(G)and any 0<p≤1.\n4.Lipschitz Extensions and Lipschitz Free Spaces\nRecall that in the definition of p-trace and of absolute p-extendability,\nwe considered the existence of Lipschitz extensions for each Lipschitz map\nf:N →Zthat ranges into any p-Banach space Z. Nevertheless, we can\ndevelop these notions for a single, canonical choice of Zalong with the map\nf. This is facilitated by a universal object associated with N, referred to as\ntheLipschitz free p-space overN.\nTheorem 24 (cf. [3, Theorem 4.5]) .Let(N,ρ)be a pointed metric space.\nGiven 0<p≤1, there exists a p-Banach space (Fp(N),∥·∥), called the Lip-\nschitz freep-space overN, and a map δN:N→Fp(N)such that\n(i)δNis an isometric embedding with δN(0N) = 0Fp(N),\n(ii)Fp(N) =span{δN(x) :x∈N} ,\n(iii) for any Lipschitz map f∈Lip0(N,Y), whereYis ap-Banach space,\nthere is a unique bounded linear operator Lf:Fp(N)→Ysuch that\nLf◦δN=f. This operator is called the canonical linearization of f.\nMoreover, it satisfies that ∥Lf∥= Lipf.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 17\nA distinctive feature of Lipschitz free spaces is that they relate the clas-\nsical linear theory to the non-linear geometry of Banach spaces. This was\noriginally observed in the seminal paper by Godefroy and Kalton [25], and\nsubsequently pursued in works including [5, 16, 28], to name a few.\nIn particular, the problem about the p-trace ofNinM, a nonlinear\nphenomenon, is equivalent to the problem about existence of a specific linear\noperator between the Lipschitz free p-spacesFp(M)andFp(N). We note\nthat the connection between the Lipschitz extension problem and the theory\nof Lipschitz free spaces was recently explored by Albiac et al. [4].\nDefinition 25 (cf. [4, Definition 2.6]) .IfNis a subspace of a pointed metric\nspaceMand if 0< p≤1, we sayNiscomplementably p-amenable in\nMwith constant C <∞provided there exists a bounded operator T:\nFp(M)→Fp(N)satisfyingT◦Li= IdFp(N)and∥T∥≤C. Here,Liis the\ncanonical linearization of the inclusion i:N →M . In other words, Liis\nthe unique linear map Li:Fp(N)→Fp(M)such that for each x∈N, we\nhaveLi(δN(x)) =δM(x).\nThe exact relation between the constant of complementable p-amenability\nand thep-trace ofNinMis described below.\nProposition 26. LetNbe a subspace of a pointed metric space M. Then,\nfor each 0<p≤1and0<C <∞, the following are equivalent:\n(i)Nis complementably p-amenable inMwith constant less than C,\n(ii) the inclusion map i:N→Fp(N)extends to a map i′:M→Fp(N)\nsuch that Lipi′<C,\n(iii) thep-trace ofNinMis less than C, i.e., tp(N,M)<C.\nProof. In order to prove that condition (i) implies condition (ii), observe\nthat ifT:Fp(M)→Fp(N)is the bounded operator from the definition\nof complementable p-amenability with constant 0<C′<C, we can define\ni′=T◦δM, where Lipi′≤∥T∥≤C′<C becauseδMis an isometry.\nAlso, condition (ii) implies condition (iii), as the canonical linearization\nLi′:Fp(M)→Fp(N)ofi′, given by property (iii) of the Lipschitz free\np-spaceFp(M), satisfiesLi′◦Li= IdFp(N)with∥Li′∥<C.\nTo show that condition (ii) implies condition (iii), note that for any p-\nBanach space Zand a Lipschitz map f:N →Z, ifLf:Fp(N)→Z\nis the canonical linearization of fgiven by property (iii) of the Lipschitz\nfreep-spaceFp(N), we may take f′=Lf◦i′. From this, it follows that\nLipf′≤Lipi′·Lipfand, consequently, tp(N,M)≤Lipi′<C.\nMoreover, condition (iii) clearly implies condition (ii), thereby proving\nthe claim. □\nLet us remark that in [4], numerous results, related solely to the structural\nproperties of Lipschitz free p-spaces over doubling metrics, were derived from\nthe absolute p-extendability of this class of metrics. Having generalized these\nextendability results to the class of spaces with finite Nagata dimension in\nSection 3, we can restate some of these structural results accordingly. For\ninstance, consider the following.\nCorollary 27 (for spaces with finite Nagata dimension ; cf. [4, Corollary 5.3]) .\nIfMis a metric space with finite Nagata dimension , then there exists a net18 JAN B ´IMA\n(Ti)i∈Iof finite-rank projections on Fp(M), uniformly bounded in norm,\nthat converges uniformly to the identity map on compact sets. In particular,\nFp(M)has theπ-property .\nWe note that the same proof as in [4, Corollary 5.3] is applicable here.\nFor the proof of the following corollary, in contrast to its counterpart in\ndoubling spaces, we additionally assume that Mis aproper metric space .\nThat is, all closed, bounded subspaces of Mare compact. Observe that not\nall countable spaces of finite Nagata dimension are proper. For example,\nany countably infinite set Mequipped with the discrete metric ρ(x,y) = 1\nif, and only if, x, y∈M andx̸=y, is not proper, although its Nagata\ndimension is 0.\nCorollary 28 (for spaces with finite Nagata dimension ; cf. [4, Corollary 5.4]) .\nIfMis a complete countable proper metric space with finite Nagata dimen-\nsion, thenF1(M)has the finite dimensional decomposition property . In\nparticular, it has the metric approximation property , after suitable renorm-\ning (for the definition of approximation properties, see [19]).\nLastly, we would like to highlight the following important application\nof the absolute p-extendability result for subsets of metric trees (refer to\nCorollary 17).\nTheorem 29 (cf. [20, Theorem 3.21]) .IfN⊂M are metric spaces in inclu-\nsion and 0< p≤1, then the canonical linearization Ti:Fp(N)→Fp(M)\nof the inclusion i:N→M is an isomorphism. Moreover, ∥T−1\ni∥≲p1.\nTo conclude this section, we show that the p-trace ofNinMgenerally\nincreases as papproaches zero. Based on the results outlined above, it will\nsuffice to consider extensions of the canonical inclusion i:N→Fp(N).\nTheorem 30. LetN={0,1,2}⊆ (R,|·|)andM=N∪{ 3/2}. Then\nt1(N,M) = 1 but tp(N,M)>1for any 0< p < 1. Moreover, we have\ntp(N,M)→2asp→0.\nIn the proof, we will frequently need to estimate the p-norm of an element\nm∈Fp(N). Let us note that while [20, Theorem 2.2] provides a finite algo-\nrithm for the computation of the p-norm in Lipschitz free spaces over finite\nmetrics, expressing the norm explicitly can still pose a significant challenge.\nHowever, when applied to the three-point space Nunder consideration, it\nyields the following formula.\nFact 31 (cf. [20, Corollary 2.7]) .For any 0<p≤1andx, y∈R, we have\n∥xδ(1) +yδ(2)∥p\nFp(N)= min{|x|p+ 2p|y|p,2p|x+y|p+|x|p,|x+y|p+|y|p}.\nWe are now ready to give the proof of the theorem.\nProof. In line with Proposition 26, we will examine extensions i′:M→\nFp(N)of the canonical embedding i:N→Fp(N). To this end, we develop\nestimates on the Lipschitz constant Lipi′, with respect to the image of 3/2\nunder this map.\nObserve that if a, b∈Rare given and i(a,b):M→Fp(N)is an extension\nof the canonical isometric embedding i, defined by i(a,b)(3/2) =aδ(1)+bδ(2),NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 19\nthen the Lipschitz constant of i(a,b)equals\n(11) Lipi(a,b)= max{1,2/3·∥aδ(1) +bδ(2)∥Fp(N),\n2∥(a−1)δ(1) +bδ(2)∥Fp(N),2∥aδ(1) + (b−1)δ(2)∥Fp(N)}.\nThep-norm of∥aδ(1) +bδ(2)∥Fp(N), as shown in Fact 31, is given by\n(12)∥aδ(1)+bδ(2)∥p\nFp(N)= min{|a|p+2p|b|p,2p|a+b|p+|a|p,|a+b|p+|b|p}.\nSimilarly, the expression for ∥aδ(1) + (b−1)δ(2)∥p\nFp(N)is\n(13) min{|a|p+ 2p|b−1|p,2p|a+b−1|p+|a|p,|a+b−1|p+|b−1|p}.\nSpecifically, we deduce that ∥aδ(1) + (b−1)δ(2)∥Fp(N)is bounded from\nbelow by|1−b|. Likewise, the value of ∥(a−1)δ(1) +bδ(2)∥Fp(N)has a\nlower bound of|1−a|. As a result, we establish that\n(14) Lipi(a,b)≥2 max{|1−b|,|1−a|}.\nIn particular, this implies that Lipi(a,b)>2, unless both 0≤a, b≤2.\nNote that Lipi(1,0)= 2 for any 0< p < 1. This is evident upon not-\ning that, by property (i) of the Lipschitz free p-spaceFp(N), we have\n∥i(1,0)(3/2)−i(1,0)(x)∥Fp(N)=∥δ(1)−δ(x)∥Fp(N)= 1for allx∈N\\{ 1}.\nTherefore, we deduce that tp(N,M)≤Lipi(1,0)= 2.\nIt follows that for any 0< p≤1, there exists an extension i(a∗,b∗)with\nthe smallest Lipschitz constant, which satisfies 0≤a∗, b∗≤2. Indeed, this\nextension emerges as a minimum of (11), a continuous expression by Fact 31,\nover a compact interval {(a,b) : 0≤a, b≤2}. Furthermore, given that the\nminimum is attained, Proposition 26 yields tp(N,M) = Lipi(a∗,b∗).\nWe assert that Lipi(a∗,b∗)>1for any given 0< p < 1. For the sake\nof contradiction, let us assume that Lipi(a∗,b∗)= 1. Once again, referring\nto (14), we deduce that min{a∗,b∗}≥ 1/2. But then, (12) implies that\n∥a∗δ(1) +b∗δ(2)∥Fp(N)≥(2p+ 1)1/p/2>3/2, and, hence, Lipi(a∗,b∗)>1by\n(11), a contradiction.\nLastly, we wish to show that tp(N,M)→2asp→0. To that end, we\nrelabel the coefficients a∗andb∗, associated with the optimal extensions, as\napandbpfor each 0<p< 1.\nWe assert that min{ap,bp}→0asp→0. To prove this, suppose, on the\ncontrary, that there exists ϵ>0and a sequence (pi)i∈N∈(0,1]N, satisfying\npi→0asi→∞ , such that min{api,bpi}> ϵfor alli∈N. However, then\n(12) reveals that ∥apiδ(1) +bpiδ(2)∥pi> ϵ(2pi+ 1)1/pi, with the right-side\nconverging to infinity as i→∞ . Consequently, by (11), we deduce that\nLipi(api,bpi)→∞ asi→∞ . However, by our choice of apandbp, we also\nknow that Lipi(api,bpi)≤2for anyi∈N, a contradiction.\nAltogether, having established that min{ap,bp}→ 0asp→0, we note\nthat (14) implies that Lipi(ap,bp)→2asp→0. This proves the final part,\nand the claim follows. □\n5.Open Problems\nIn Section 3, we observed many times that the estimates on the absolute\np-extendability constant increase as papproaches zero, and specifically, they20 JAN B ´IMA\ngrow exponentially in terms of 1/p. Subsequently, in Section 4, we estab-\nlished a basic counterexample demonstrating that the p-trace typically does\nindeed depend on p. Consequently, we are curious to see how this behavior\nis exhibited across the different classes of spaces that we have considered in\nthis paper, and whether we can derive lower estimates on this.\nIn the following discussion, whenever Nis a metric space with ae1(N)<\n∞, we denote q(N,p) = aep(N)/ae1(N)for each 0<p≤1. It is easy to see\nthat limp→0q(N,p)exists asp↦→aep(N)does not increase in p.\nQuestion 32. Is it true that for each n≥2, we have sup{limp→0q(N,p) :\nNis a doubling with λN≤n}=∞? More specifically, can it be shown that\nfor each fixed n≥2,p↦→log sup{q(N,p) :λN≤n}grows proportionally\nto1/pasp→0? What about if we consider Nto only have a finite Nagata\ndimension at most dwith constant γ?\nIn relation to the example of Theorem 30, we would like to know whether,\nif the answer to the very first question in Question 32 is positive, it could\npotentially be identified within examples involving finite metric spaces.\nRecall that by Corollary 16, we have aep(n)<∞for everyn∈Nand\n0<p≤1(we have a trivial estimate λN≤nfor everyn-point metric space\nN). Interestingly, observe also that in the proof of Theorem 30, while the\noptimal projection was identified as a weighted sum of two points in the\ntarget Lipschitz free p-spaceFp(N)forp= 1, the norm of that particular\nextension would become excessively large as p→0. In particular, as p→0,\nthe Lipschitz norm of the optimal extension converged to that of a ”trivial”\nextension that projected the additional point to a single point in Fp(N). It is\nclear that such a projection does not require the underlying linear structure\nof the target space.\nAt the same time, note that a remark to [11, Theorem 1.1] shows that for\neachm∈N, there exist spaces N ⊂M with|N|= 2and|M\\N| =m,\nsuch that if fis the identity map f:N→N , then Lipf′≥(m+ 1) Lipf\nfor any extension f′:M→N off. Consequently, we pose the following\nquestion.\nQuestion 33. Is it true that limp→0aep(n) =∞for eachn≥2? Additionally,\nfor anyn, m∈N, is it true that sup{limp→0tp(N,M) :N⊂M with|N|≤\nnand|M\\N|≤m}=m+1? What if we relax the condition that |N|≤n?\nRemark 34.We wish to acknowledge that a generalization of the extension\ntheorem from doubling spaces to spaces with finite Nagata dimension has\nbeen recently independently discovered by Basso [10], in the narrower con-\ntext of the Banach setting with p= 1. This is the content of Theorem II and,\nspecifically, Theorem 14. Interestingly, both methods of proof share similari-\nties. Our result is more broadly applicable, dealing with p-Banach spaces for\np<1. Conversely, additional classes of spaces are considered in [10], specif-\nically Lipschitz n-connected spaces (refer to Lang and Schlichenmaier [32]),\nand the author provides estimates on the absolute 1-extendability constant\nfor finite metric spaces.NAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 21\nAppendix\nWe have adapted the results of Fakcharoenphol and Talwar [24] to the\nsetting of metric graphs. It is important to note that, unlike in [24] where the\nauthors deal with the unweighted graph G, we cannot generally assume the\nexistence of shortest-length paths in Σ(G)and hence have to approximate\nthese accordingly. Thus, if elements u, v⊆Cof some connected subset C\nofΣ(G)andϵ >0are given, we define a p: [0,1]→C⊆Σ(G)to be an\nϵ-path inCfromutovifpis continuous and injective, p(0) =u,p(1) =v,\nand its length is less than ρC(v,w) +ϵ. It is evident that every path can be\nwritten as a union of subpaths contained within the edges of G. Moreover,\nin what follows, it will be sufficient to fix any 0<ϵ<r .\nLemma 22. LetGbe a countable and connected weighted graph such that\nthere arer > 0,m∈Nwithm≥3,δ∈ {0,1,2}m−1,Σ(G) =C1⊇\n...⊇Cm, and points si∈Cifori∈{1,...,m}satisfying that for every\ni, i′∈ {1,...,m}, wherei < i′, we have ρ(si,si′)>24mrand the set\nCi+1is a pathwise connected component of Ani={v∈Ci: 3r(ni−1)≤\nρCi(v,si)−δir<3rni}, for someni∈N0.\nThenGcontainsKmas a minor.\nProof. We wish to construct for each i∈{2,...,m−1}setsAm−i+1,...,Am\n(called supernodes ) such that for each j, j′∈{m−i+1,...,m}, wherej̸=j′,\n(i)sj∈Aj⊆Cj,\n(ii) the intersection of AjandAj′is a singleton,\n(iii)Aj∩Aj′∩Aj′′=∅for any given j′′∈{m−i+ 1,...,m}\\{j,j′},\n(iv) ifTis the set of all points xsuch thatx∈Al∩Al′for somel, l′∈\n{m−i+ 1,...,m}, wherel̸=l′, thenAj\\Tis pathwise connected.\nIn addition, for each j∈{m−i+ 1,...,m}, we construct a non-self-\nintersecting ϵ-pathPjconnecting uj∈ Ajtosm−i, a subpath Tj⊂Pj\n(called a tail) of length 24rconnecting ujto a pointtj(called the tipofTj)\nand a point hj∈Tj(called the middle point ofTj), which is at distance 12r\nfromujalong the path Pj.\nWe will assume that for every j, j′∈{m−i+ 1,...,m}, wherej̸=j′,\n(v)Tj∩(Aj′∪Tj′) =∅,\n(vi) the distance between the tails TjandTj′inΣ(G)is greater than\n24r(m−i). Moreover, ρ(hj,hj′)>24r(m−i+ 1),\n(vii)ρ(hj,sm−i)>24r(m−i+ 1),\n(viii)ρ(sj,hj)≤12r(i−1).\nMoreover, we claim that the following properties are then always satisfied.\nClaim 35.For everyj, j′∈{m−i+ 1,...,m}, wherej̸=j′, it follows from\nconditions (i) to (viii) that\n(ix)((Pj\\Tj)∪{tj})∩Tj′=∅, andAj∩((Pj\\Tj)∪{tj}) =∅,\n(x) for any w∈Cm−i+1, we haveρCm−i(sm−i,w)−ρCm−i(sm−i,hj)>\n9r−ϵandρCm−i(sm−i,w)>21r−ϵ. Also, for every w′∈((Pl\\\nTl)∪{tl}), wherel∈{m−i+ 1,...,m}, we haveρCm−i(sm−i,hj)−\nρCm−i(sm−i,w′)>6r−ϵ.\nProof of Claim 35. To address condition (ix), let us assume for a contradic-\ntion thatv∈((Pj\\Tj)∪{tj})∩Tj′for somej, j′∈{m−i+ 1,...,m},22 JAN B ´IMA\nwherej̸=j′. We recall that by condition (vi), v∈Tj′andtjare at distance\nmore than 24r(m−i)from each other.\nAlso, we have\nρCm−i(sm−i,uj′)≤ρCm−i(sm−i,v) +ρCm−i(v,uj′)\n≤ρCm−i(sm−i,v) + 24r.\nSimilarly, decomposing the ϵ-pathPjinto segments from sm−iovervandtj\nup touj, we get\nρCm−i(sm−i,uj)>ρCm−i(sm−i,v) +ρCm−i(v,tj) + 24r−ϵ\n>ρCm−i(sm−i,v) + 24r(m−i+ 1)−ϵ.\nConsequently, we see that |ρCm−i(sm−i,uj′)−ρCm−i(sm−i,uj)|≥24r(m−\ni)−ϵ >3r. At the same time, there exists nm−i∈N0such thatuj, uj′∈\nCm−i+1⊆Anm−i={v∈Cm−i: 3r(nm−i−1)≤ρCm−i(v,sm−i)−δm−ir <\n3rnm−i}. It follows that|ρCm−i(sm−i,uj)−ρCm−i(sm−i,uj′)|<3r, which is\nabsurd.\nAs for the second part of condition (ix), we can similarly verify that for any\nv∈(Pj\\Tj)∪{tj}, we haveρCm−i(sm−i,uj)−ρCm−i(sm−i,v)>24r−ϵ. As in\nthe already proven part, we recall that |ρCm−i(sm−i,uj)−ρCm−i(sm−i,w)|<\n3rfor anyw∈Cm−i+1by the assumption. That is, for any v∈(Pj\\Tj)∪{tj}\nandw∈Cm−i+1we get\n(15) ρCm−i(sm−i,w)−ρCm−i(sm−i,v)>21r−ϵ.\nIn particular, we deduce that (Pj\\Tj)∪{tj}andAj⊆Cm−i+1are disjoint.\nIn order to establish condition (x), we can verify as above that for any\nw∈Cm−i+1andj∈{m−i+ 1,...,m}, we have\nρCm−i(sm−i,w)−ρCm−i(sm−i,hj)\n>(ρCm−i(sm−i,uj)−3r)−(ρCm−i(sm−i,uj)−12r+ϵ) = 9r−ϵ.\nClearly also ρCm−i(sm−i,w)>21r−ϵby (15).\nAt the same time, we have ρCm−i(sm−i,uj)−ρCm−i(sm−i,hj)≤12rfor\neachj∈{m−i+1,...,m}. Consequently, we obtain that ρCm−i(sm−i,w)−\nρCm−i(sm−i,hj)<15rfor anyw∈Cm−i+1. Similarly, it follows from (15)\nthatρCm−i(sm−i,w)−ρCm−i(sm−i,w′)>21r−ϵfor anyw′∈(Pj′\\Tj′)∪{tj′},\nwherej′∈ {m−i+ 1,...,m}. Subtracting the inequalities, we obtain\nρCm−i(sm−i,hj)−ρCm−i(sm−i,w′)>6r−ϵ. This establishes condition (x).\n□\nHaving established Claim 35, we proceed to the inductive construction.\nLeti= 2. We consider an ϵ-pathpinCm−1connecting sm−1andsm.\nWe can easily see that pcan be cut into two subpaths of equal lengths,\ncorresponding to sets Am−1andAmsatisfying conditions (i) to (iv). Also,\nfor eachj∈ {m−1,m}, we construct the tail Tj∋sjsuch that it is\nsubpath of length 24rof someϵ-pathPjinCm−2connecting sjandsm−2.\nAdditionally, we let tj∈Tjdenote the other endpoint of Tj, other than sj,\nand let the middle point hj∈Tjbe a point at distance 12rfromsj, along\nthe tailTj.\nIt is easy to check that the tails Tm−1andTmare disjoint. Specifically,\nnote that if u∈Tm−1andv∈Tm, then max{ρ(sm−1,u),ρ(sm,v)}≤24r, soNAGATA DIMENSION AND LIPSCHITZ EXTENSIONS INTO QUASI-BANACH SPACES 23\nthatρ(u,v)≥ρ(sm−1,sm)−2·24r>24r(m−2). Similarly, ρ(hm−1,hm)>\n24r(m−1)and for each j∈{m−1,m}, we haveρ(sm−2,hj)≥ρ(sm−2,sj)−\nρ(sj,hj)>24r(m−1). This establishes conditions (vi) and (vii).\nTo adress condition (v), it is easy to see that since smandAm−1are at\ndistance at least 24rm−ϵfrom each other in Σ(G), the tailTmof length\n24rfromsmdoes not meetAm−1(similarly for the role of mandm−1\ninterchanged). We can also easily verify the validity of condition (viii).\nAltogether, we have verified that also the paths Pj, tailsTjand the tips tj\nand middle points hj, wherej∈{m−1,m}, satisfy conditions (v) to (viii).\nThis concludes the proof of the basis step.\nLet nowi∈{2,...,m−2}be such that there exist supernodes Aj, paths\nPj, tailsTj, and tipstjand middle points hj, wherej∈{m−i+ 1,...,m},\nsatisfying conditions (i) to (viii) (and thus also conditions (ix) and (x)).\nFor eachj∈{m−i+1,...,m}, we letA′\nj=Aj∪Tj. We also defineA′\nm−i\nto be the union of all paths (Pj\\Tj)∪{tj}, wherej∈{m−i+ 1,...,m}.\nBy the induction hypothesis, it is easy to see that A′\nm−i,...,A′\nmsatisfy\nconditions (i) to (iii).\nTo address condition (iv), we let T′denote the set of all xsuch that\nx∈A′\nl∩A′\nl′for somel, l′∈{m−i,...,m}, wherel̸=l′. Also, let Tbe\nas in condition (iv). Note that for any j∈{m−i+ 1,...,m}, we have\nT′∩A′\nj= (T∩Aj)∪{tj}by the construction. Moreover, T∩Tj=∅by\ncondition (v). Consequently, it follows that A′\nj\\T′= (Aj∪Tj)\\T′= (Aj\\\nT)∪(Tj\\{tj})is pathwise connected. Similarly, it follows from condition (v)\nthatA′\nm−i\\T′=A′\nm−i\\{tj:j∈{m−i+1,...,m}}=⋃︁\nj∈{m−i+1,...,m}Pj\\Tj\nis pathwise connected.\nFor eachj∈ {m−i,...,m}, we denote uj=hj(oruj=sm−ifor\nj=m−i) and construct an ϵ-pathP′\njinCm−i−1, connecting ujandsm−i−1,\nsuch thatP′\nj∩Cm−iis a subpath of P′\nj. In other words, there exists v0∈P′\nj\nsuch thatP′\njis the union of two paths: p1inCm−iconnecting ujandv0,\nandp2fromv0tosm−i−1. Moreover, p2\\{v0}⊂Cm−i−1\\Cm−i.\nNote that we impose an additional condition on P′\nj, compared to the basis\nstep, to ensure that T′\njis disjoint fromA′\nj′, wherej′̸=j. Indeed, this is\nbecause we generally know less about the distance from ujto the setA′\nj′.\nClaim 36.There exists a path P′\njwith the desired properties.\nProof of Claim 36. To construct such a path Pj′, we consider nm−i−1∈N0\nsatisfying that ρCm−i−1(sm−i−1,uj)−δm−i−1r∈[3(nm−i−1−1)r,3nm−i−1r),\nand putϵ′= min{ϵ,3nm−i−1r+δm−i−1r−ρCm−i−1(sm−i−1,uj)}. Subse-\nquently, we let pbe anϵ′-path from ujtosm−i−1. Note that by the choice\nofϵ′, we have\n(16)ρCm−i−1(sm−i−1,v)−δm−i−1r<3nm−i−1r, wherev∈p.\nWe claim that sm−i−1/∈Cm−i. Indeed, note that by the assumption,\nwe haveρ(sm−i−1,sm−i)>24rm, and at the same time, for any v∈\nCm−iit holds that|ρCm−i−1(sm−i−1,sm−i)−ρCm−i−1(sm−i−1,v)|<3rby\nthe assumption that Cm−iis a connected component in Anm−i−1={v∈\nCm−i−1: 3r(nm−i−1−1)≤ρCm−i−1(v,sm−i−1)−δm−i−1r<3rnm−i−1}.24 JAN B ´IMA\nAlong the path p, starting from ujand going to sm−i−1/∈Cm−i, letebe\nthe first edge such that for some v∈e∩p, we havev /∈Cm−i. In particular,\nit follows from (16) that ρCm−i−1(sm−i−1,v)−δm−i−1r<3(nm−i−1−1)r. By\nthe assumption that Cm−iis a pathwise connected component of Anm−i−1=\n{v∈Cm−i−1: 3r(nm−i−1−1)≤ρCm−i−1(v,sm−i−1)−δm−i−1r<3rnm−i−1},\nwe see that e∩Cm−isplits into at most two pathwise connected components.\nHence, going from ujtovalongp, we see that this path further splits into\ntwo pathwise connected subsets, the first of which is contained in Cm−iand\nthe other is not.\nWe denote ϵ′′= min{ϵ,3(nm−i−1−1) +δm−i−1r−ρCm−i−1(sm−i−1,v)}\nand modify pso that from v, it continues as some ϵ′′-pathp′inCm−i−1from\nvtosm−i−1. Without loss of generality, we can assume that the modified\npath does not intersect itself. Moreover, by the choice of ϵ′′, the pathp′does\nnot intersect Cm−i. Hence, by the construction, the modified path phas the\nproperty that p∩Cm−iis a subpath of p. We letP′\nj=p. □\nHaving established the existence of a path P′\njwith the desired properties,\nwe let the tail T′\nj∋ujbe a subpath of length 24rofP′\nj. Additionally, we let\nt′\nj∈T′\njdenote the other endpoint of T′\nj, other than uj, and the middle point\nh′\nj∈T′\njbe the point at distance 12rfromuj, along the tail T′\nj. It remains\nto verify conditions (v) to (viii).\nUsing the induction hypothesis that the middle points hj, wherej∈\n{m−i+ 1,...,m}, andsm−iare more than 24r(m−i+ 1) far from each\nother inCm−i(see conditions (vi) and (vii)), we observe similarly as in the\nbasis step that condition (vi) holds for the new tails T′\njand midpoints h′\nj\n(in particular, the new tails T′\njare disjoint).\nSimilarly, we obtain that for each j∈{m−i,...,m}, the tailT′\njis disjoint\nfromTj′for anyj′∈{m−i+ 1,...,m}, wherej′̸=j.\nMore generally, to establish condition (v), we wish to show that A′\njandT′\nj′\nare disjoint for each j, j′∈{m−i,...,m}, wherej̸=j′. To that end, assume\nfor a contradiction that A′\njandT′\nj′meet at some point point v. As before,\nwe denoteuj′=hj′(orsm−iforj′=m−i). We recall that uj′, v∈Cm−i\nand, consequently, ρCm−i−1(sm−i−1,uj′)−ρCm−i−1(sm−i−1,v)<3r.\nAlso, ifℓdenotes the length of the subpath of T′\nj′connecting uj′andv\n(this subpath lies in Cm−iby the construction of P′\nj′, i.e.,ℓ≥ρCm−i(uj′,v)≥\n|ρCm−i(sm−i,hj′)−ρCm−i(sm−i,v)|), we have that ρCm−i−1(sm−i−1,uj′)>ℓ+\nρCm−i−1(sm−i−1,v)−ϵ, that is,ρCm−i−1(sm−i−1,uj′)−ρCm−i−1(sm−i−1,v) +\nϵ>ℓ . Altogether, we get that\n(17) 3r+ϵ>ℓ≥|ρCm−i(sm−i,hj′)−ρCm−i(sm−i,v)|.\nWe note that if j < m−i, then necessarily v∈Aj⊆Cm−i+1, because\nA′\nj=Aj∪TjandT′\nj′∋vandTjare disjoint by the already proven part.\nOtherwise if j=m−i, thenv∈(Pl\\Tl)∪{tl}for somel∈{m−i+1,...,m}\nby the construction. Consequently, in both cases ( j <m−iandj=m−i),\ncondition (x) shows that |ρCm−i(sm−i,hj′)−ρCm−i(sm−i,v)|>6r−ϵ, which\nis impossible by (17). This contradiction finishes the proof that condition (v)\nholds.REFERENCES 25\nAs for condition (viii), we claim that ρ(sj,h′\nj)≤12rifor eachj∈{m−\ni,...,m}. Indeed, for j >m−ithis follows from inductive hypothesis and\nthe fact that ρ(h′\nj,hj)≤12r. Moreover, we clearly have ρ(sm−i,h′\nm−i)≤\n12r.\nAs it also holds that ρ(sm−i−1,sj)>24rmfor anyj∈{m−i,...,m}by\nthe assumption, we deduce that ρ(sm−i−1,h′\nj)>24r(m−i)for any such j,\nwhich verifies condition (vii).\nAltogether, we have verified that also the paths Pj, tailsTjand the tips\ntjand middle points hj, wherej∈{m−i,...,m}, satisfy conditions (v)\nto (viii). This concludes the proof of the inductive step.\nConsequently, we can find supernodes Ai, pathsPi, tailsTi, and tipsti\nand middle points hi, wherei∈{2,...,m}, satisfying conditions (i) to (x).\nThen, as in the first part of the proof of the induction step, we construct\nsupernodesAi, wherei∈{1,...,m}, satisfying conditions (i) to (iv).\nWe claim that Kmis a minor of G. Indeed, if m= 3, it is clear that we\nfind a cycle in G. Form> 3, we inductively define A′\ni=Ai∩V\\⋃︁i−1\nj=1A′\nj,\nwherei∈{1,...,m}. Note it follows from condition (iv) that the vertex sets\nA′\niare non-empty connected subgraphs in Gand moreover, by the definition\nofA′\ni, any two vertex sets are connected by an edge in E(consider the cases\nwhen two distinct supernodes meet at a vertex of Gor within the interior\nof some edge). The claim follows. □\nReferences\n[1] I. Abraham et al. “Cops, robbers, and threatening skeletons: Padded\ndecomposition for minor-free graphs”. In: Proceedings of the forty-sixth\nannual ACM symposium on Theory of computing . 2014, pp. 79–88.\n[2] F. Albiac and N. Kalton. Lipschitz structure of quasi-Banach spaces.\nIsrael Journal of Mathematics 170 (Mar. 2009), pp. 317–335.\n[3] F. Albiac et al. Lipschitz free p-spaces for 0<p< 1.Israel Journal of\nMathematics 240.1 (Sept. 2020), pp. 65–98.\n[4] F. Albiac et al. Lipschitz free spaces isomorphic to their infinite sums\nand geometric applications. Transactions of the American Mathemat-\nical Society 374.10 (July 2021), pp. 7281–7312.\n[5] F. Albiac et al. Structure of the Lipschitz free p-spacesFp(Zd)and\nFp(Rd)for0< p≤1.Collectanea Mathematica 73.3 (July 2022),\npp. 337–357.\n[6] P. Assouad. Plongements lipschitziens dans Rn.Bulletin de la Soci´ et´ e\nMath´ ematique de France 111 (1983), pp. 429–448.\n[7] P. Assouad. Sur la distance de Nagata. C. R. Acad. Sci. Paris S´ er. I\nMath. 294.1 (1982), pp. 31–34.\n[8] K. Ball. Markov Chains, Riesz Transforms and Lipschitz Maps. Geo-\nmetric and functional analysis 2.2 (1992), pp. 137–172.\n[9] G. Basso. Absolute Lipschitz extendability and linear projection con-\nstants. Studia Mathematica 264.3 (2022), pp. 335–359.\n[10] G. Basso. Lipschitz extension theorems with explicit constants. arXiv\npreprint arXiv:2310.13554 (2023).\n[11] G. Basso. Lipschitz Extensions to Finitely Many Points. Analysis and\nGeometry in Metric Spaces 6.1 (2018), pp. 174–191.26 REFERENCES\n[12] G. Basso and H. Sidler. Approximating spaces of Nagata dimension\nzero by weighted trees. Illinois Journal of Mathematics 67.1 (Apr.\n2023).\n[13] G. Bell and A. Dranishnikov. Asymptotic dimension. Topology and its\nApplications 155.12 (2008), pp. 1265–1296.\n[14] J. B´ ıma. Lipschitz-free spaces and subsets of finite-dimensional spaces.\nProceedings of the Royal Society of Edinburgh: Section A Mathematics\n(2023), pp. 1–30.\n[15] M. Bonamy et al. Asymptotic dimension of minor-closed families and\nAssouad–Nagata dimension of surfaces. Journal of the European Math-\nematical Society (2023).\n[16] L. Borel-Mathurin. Approximation properties and nonlinear geometry\nof Banach spaces. Houston journal of mathematics 38 (Jan. 2012),\npp. 1135–1148.\n[17] A. Y. Brudnyi and Y. A. Brudny˘ ı. Simultaneous Lipschitz extensions.\nRussian Mathematical Surveys 60.6 (2005), p. 1057.\n[18] A. Brudnyi and Y. Brudnyi. Metric Spaces with Linear Extensions Pre-\nserving Lipschitz Condition. American Journal of Mathematics 129.1\n(2007), pp. 217–314.\n[19] P. G. Casazza. “Chapter 7 - Approximation Properties”. In: Handbook\nof the Geometry of Banach Spaces . Ed. by W. Johnson and J. Linden-\nstrauss. Vol. 1. Handbook of the Geometry of Banach Spaces. Elsevier\nScience B.V., 2001, pp. 271–316.\n[20] M. C´ uth and T. Raunig. Canonical embedding of Lipschitz-free p-\nspaces . 2023.\n[21] M. Distel. Proper minor-closed classes of graphs have Assouad-Nagata\ndimension 2. arXiv preprint arXiv:2308.10377 (2023).\n[22] E. L. Donne. Analysis and Geometry in Metric Spaces 5.1 (2017),\npp. 116–137.\n[23] E. L. Donne and T. Rajala. Assouad Dimension, Nagata Dimension,\nand Uniformly Close Metric Tangents. Indiana University Mathemat-\nics Journal 64.1 (2015), pp. 21–54.\n[24] J. Fakcharoenphol and K. Talwar. “An Improved Decomposition The-\norem for Graphs Excluding a Fixed Minor”. In: RANDOM-APPROX .\n2003.\n[25] G. Godefroy and N. J. Kalton. Lipschitz-free Banach spaces. Studia\nMathematica 159.1 (2003), pp. 121–141.\n[26] M. Gromov. “Asymptotic Invariants of Infinite Groups”. In: Geometric\nGroup Theory . Ed. by G. A. Niblo and M. A. Roller. Vol. 2. London\nMath. Soc. Lecture Note Series 182. Cambridge Univ. Press, 1993,\npp. 1–295.\n[27] W. B. Johnson, J. Lindenstrauss, and G. Schechtman. Extensions of\nLipschitz maps into Banach spaces. Israel Journal of Mathematics 54.2\n(June 1986), pp. 129–138.\n[28] N. J. Kalton. Approximation properties and nonlinear geometry of\nBanach spaces. Mathematische Annalen 354 (2012), pp. 1247–1288.REFERENCES 27\n[29] N. J. Kalton, N. T. Peck, and J. W. Roberts. An F-space Sampler . Lon-\ndon Mathematical Society Lecture Note Series. Cambridge University\nPress, 1984.\n[30] M. Kirszbraun. ¨Uber die zusammenziehende und Lipschitzsche Trans-\nformationen. ger. Fundamenta Mathematicae 22.1 (1934), pp. 77–108.\n[31] P. Klein, S. A. Plotkin, and S. Rao. “Excluded Minors, Network\nDecomposition, and Multicommodity Flow”. In: Proceedings of the\nTwenty-Fifth Annual ACM Symposium on Theory of Computing .\nSTOC ’93. San Diego, California, USA: Association for Computing\nMachinery, 1993, pp. 682–690.\n[32] U. Lang and T. Schlichenmaier. Nagata dimension, quasisymmetric\nembeddings, and Lipschitz extensions. International Mathematics Re-\nsearch Notices 2005.58 (Jan. 2005), pp. 3625–3655.\n[33] J. Lee and A. Naor. Extending Lipschitz functions via random metric\npartitions. Inventiones mathematicae 160 (Feb. 2004), pp. 59–95.\n[34] J. R. Lee and A. Naor. Absolute Lipschitz extendability. Comptes Ren-\ndus Mathematique 338.11 (2004), pp. 859–862.\n[35] C.-H. Liu. Assouad-Nagata dimension of minor-closed metrics. arXiv\npreprint arXiv:2308.12273 (2023).\n[36] J. Matouˇ sek. Extension of Lipschitz mappings on metric trees. eng.\nCommentationes Mathematicae Universitatis Carolinae 031.1 (1990),\npp. 99–104.\n[37] J. Nagata. Note on dimension theory for metric spaces. Fundamenta\nMathematicae 45 (1958), pp. 143–181.\n[38] A. Naor and Y. Rabani. On Lipschitz extension from finite subsets.\nIsrael Journal of Mathematics 219.1 (Apr. 2017), pp. 115–161.\n[39] A. Naor and L. Silberman. Poincar´ e inequalities, embeddings, and wild\ngroups. Compositio Mathematica 147.5 (2011), pp. 1546–1572.\n[40] M. I. Ostrovskii and D. Rosenthal. Metric dimensions of minor ex-\ncluded graphs and minor exclusion in groups. International Journal of\nAlgebra and Computation 25.04 (2015), pp. 541–554.\n[41] C. A. Rogers. Covering a sphere with spheres. Mathematika 10.2\n(1963), pp. 157–164.\n[42] H. Tverberg. “A Proof of Kuratowski’s Theorem”. In: Graph Theory in\nMemory of G.A. Dirac . Ed. by L. D. Andersen et al. Vol. 41. Annals\nof Discrete Mathematics. Elsevier, 1988, pp. 417–419.\nEmail address :jan.bima@mff.cuni.cz\nCharles University, Faculty of Mathematics and Physics, Department of\nMathematical Analysis, Sokolovsk ´a 83, 186 75 Prague 8, Czech Republic"    },    {        "title": "2402.03248v5.On_a_question_of_Gary_G__Gundersen_concerning_meromorphic_functions_sharing_three_distinct_values_IM_and_a_fourth_value_CM.pdf",        "content": "arXiv:2402.03248v5  [math.CV]  26 Feb 2024ON A QUESTION OF GARY G. GUNDERSEN\nCONCERNING MEROMORPHIC FUNCTIONS SHARING\nTHREE DISTINCT VALUES IM AND A FOURTH VALUE\nCM\nXIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nAbstract. In1992, Gundersen[11]proposedthefollowing famousopen\nquestion: if two non-constant meromorphic functions share three values\nIM and share a fourth value CM, then do the functions necessar ily share\nall four values CM? The open question is a long-standing ques tion in\nthe studies of the Nevanlinna′s value distribution theory of meromorphic\nfunctions, and has not been completely resolved by now. In th is paper,\nwe prove the following result: suppose that fandgare two distinct\nnon-constant meromorphic functions, and one of fandghas finite or-\nder. Iffandgsharea1, a2, a3IM anda4CM, where a1, a2, a3, a4are\nfour distinct complex values in the extended complex plane, thenfand\ngsharea1, a2, a3anda4CM. Applying the main result obtained in this\npaper, we completely resolve a question proposed by Gary G. G undersen\nin [8, p.458] concerning the nonexistence of two distinct no n-constant\nmeromorphic functions sharing three distinct values DM and a fourth\nvalue CM. The obtained result also improves the correspondi ng result\nin Mues [30, pp.109-117] concerning the nonexistence of two distinct\nnon-constant entire functions that share three distinct fin ite values DM.\nExamples are provided to show that the main results obtained in this\npaper, in a sense, are best possible.\n1.Introduction and main results\nIn this paper, by meromorphic functions we will always mean m eromor-\nphic functions in the complex plane. We adopt the standard no tations in\nthe Nevanlinna theory of meromorphic functions as explaine d in [13, 18, 48].\nThroughout this paper, we denote by E⊂[0,+∞) a set of finite linear mea-\nsure. For a meromorphic function fin the complex plane we denote by\nS(r,f) any quantity satisfying S(r,f) =o(T(r,f)),asr/\\e}atio\\slash∈Eandr→ ∞.\nCorresponding author: Xiao-Min Li .\n2010 Mathematics Subject Classification. Primary 30D35; Secondary 30D30.\nProject supported in part by the NSF of Shandong Province, Ch ina (No.\nZR2019MA029), the FRFCU(No. 3016000841964007), the NSF of Shandong Province,\nChina (No. ZR2014AM011) and the NSFC (No.11171184).\nKeywords. Nevanlinna′s theory; Meromorphic functions; Shared values; Uniquenes s\ntheorems.\n12 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nFollowing [12, p.466], we define the notion of the small funct ion of a non-\nconstant meromorphic function in the complex plane as follo ws: letfbe a\nnon-constant meromorphic function. Next we denote by S(f) the set of all\nmeromorphicfunctions αinthecomplexplanesuchthat T(r,α) =o(T(r,f))\nfor allr∈[0,+∞) possibly outside a set E⊂[0,+∞) of finite linear mea-\nsure. Functions in the set S(f) are called small compared to f,or slowly\nmoving with respect to f,or a small function of ffor short.\nLetfandgbe two non-constant meromorphic functions in the complex\nplane, and let abe a value in the extended plane. We say that fandgshare\nthe valueaCM, provided that fandghave the same set of a-points, and\neach common a-point of fandghas the same multiplicities as the a-point\noffandg.We say that fandgshare the value aIM, provided that fand\nghave the same set of a-points ignoring multiplicities. We say that fandg\nshare the value aDM, provuded that fandgshareaIM, whilef−aand\ng−ahave no common zeros of the same multiplicity (cf.[8, 48]). F or a non-\nconstant meromorphic function f,we denote by µ(f) andρ(f) respectively\nthe lower order and the order of f,the relevant definitions can be found, for\nexample, in [13, 18, 48]. For convenience, we recall them bel ow:\nDefinition 1. For a non-constant meromorphic function f,the lower order\nand the order of f,denoted as µ(f) andρ(f) respectively are defined as\n(1)µ(f) = liminf\nr→∞logT(r,f)\nlograndρ(f) = limsup\nr→∞logT(r,f)\nlogrrespectively .\nFollowing [5, p.293], we say that ameromorphicfunction fin the complex\nplaneis of regular growth, provided µ(f) =ρ(f).Here we mention that both\nofµ(f) andρ(f) may be + ∞.\nIn 1926, Nevanlinna [33] proved the following theorems that are the fa-\nmous five-value theorem and the four-value theorem respecti vely:\nTheorem 1. ([33, p.109]) If fandgare non-constant meromorphic func-\ntions that share five values IM, then f=g.\nTheorem 2. ([33, p.122]) Let fandgbe distinct non-constant meromor-\nphic functions, and let a1, a2, a3, a4be four distinct complex values in the\nextended complex plane. If fandgsharea1, a2, a3, a4CM, thenfis a\nM¨ obius transformation of g, two of the shared values, say a1anda2, are\nPicard exceptional values, and the cross ratio ( a1,a2,a3,a4) =−1.\nIn1976, L. Rubelposedthefollowingquestion(cf.[8]): whe therdothefour\ndistinct CM shared values be replaced with four distinct IM s hared values or\nnotinTheorem2? Inthisdirection, Gundersen[8]andGunder sen[9]proved\nthe following results respectively, where Theorem 4 improv ed Theorems 2\nand 3, and Theorem 3 improved Theorem 2:\nTheorem 3. ([8, Theorem 1]) If fandgshare three values CM and share\na fourth value IM, then they share all four values CM.MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 3\nTheorem 4. ([9, Theorem 1]) If two non-constant meromorphic functions\nshare two values CM and share two other values IM, then the fun ctions\nshare all four values CM.\nGundersen [8] gave the following famous example that shows t hat the\nL.Rubel’s question above is negative:\nExample 1. ([8, pp.458-459]) Let f=eh+1\n(eh−1)2andg=(eh+1)2\n8(eh−1),wherehis\na non-constant entire function, we can easily verify that fandgshare 0,1,\n∞,−1/8 DM, and fis not a M¨ obius transformation of g.\nBased upon Example 1 and Theorem 4, Gundersen [11] proposed t he\nfollowing famous question:\nQuestion 1. ([11, p.100]) If two non-constant meromorphic functions sh are\nthree values IM and share a fourth value CM, then do the functi ons neces-\nsarily share all four values CM?\nQuestion 1 is the main open question in the theory of meromorp hic func-\ntions that share four values. This open question appears to b e difficult and\nhas not been completely resolved by now.\nIn the past several decades, there are many partial results o n Question\n1. For the ease of the following narrations. We introduce the following\nnotation: let fandgbe two non-constant meromorphic functions, and let\nabe a value in the extended complex plane. Suppose that fandgsharea\nIM. Next we denote by NE(r,a;f,g) the counting function of those common\na-points offandgin|z|<rwhere each point in NE(r,a;f,g) has the same\nmultiplicities regard to fandg,and each point in NE(r,a;f,g) is counted\nonly once. We define\nτ(a) = liminf\nr→∞NE(r,a;f,g)\nN/parenleftig\nr,1\nf−a/parenrightigifN/parenleftbigg\nr,1\nf−a/parenrightbigg\n/\\e}atio\\slash= 0\nfor the large positive number r,whereN/parenleftig\nr,1\nf−a/parenrightig\n=N(r,f) ifa=∞.From\nthe definition we see that 0 ≤τ(a)≤1.We mention that τ(a) is defined as\nτ(a) = 1,whenfandgshareaCM. We recall the following results due to\nMues [31] and Wang [46] respectively that improved Theorem 4 :\nTheorem 5. ([31, Theorem 1]) Let fandgbe two distinct non-constant\nmeromorphic functions in the complex plane that share four d istinct values\nin the extended complex plane. Suppose that one of the four di stinct values\nis shared by fandgCM, and that τ(b)>2\n3for another value bof the four\ndistinct shared values. Then fandgshare all the four distinct values CM.\nTheorem 6. ([46, Theorem 1]) Let fandgbe two distinct non-constant\nmeromorphic functions in the complex plane that share four d istinct values4 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nin the extended complex plane. Supposethat two of the four di stinct values,\nsayaandbsatisfyτ(a)>4\n5andτ(b)>4\n5.Thenfandgshare all the four\ndistinct values CM.\nAs far as we know, there are other research works concerning Q uestion 1\nand the uniqueness question of meromorphic functions that s hare four dis-\ntinct values in the complex plane, such as Mues [30, 32], Gund ersen [11],\nReinders [38], Ueda [44], Wang[45], Song-Chang [41], Stein metz [42, 43],\nHuang[20] and Huang-Du[21], etc. In addition, Ishizaki[17 ], Li [23] and Yao\n[50] studied the uniqueness question of non-constant merom orphic functions\nsharing four distinct small functions. We recall the follow ing partial results\nin Gundersen [11, Theorem 1] and Li-Yi [24] on the open questi on respec-\ntively:\nTheorem 7. ([11, Theorem 1]) Let fandgbe two non-constant meromor-\nphic functions that share al, a2, a3IM anda4CM, where a1, a2, a3anda4\nare four distinct complex values in the extended complex pla ne. Suppose\nthat there exist some real constant λ >4/5 and some set I⊂(0,∞) that\nhas infinite linear measure such that N/parenleftig\nr,1\nf−a4/parenrightig\n/T(r,f)≥λfor allr∈I.\nThenfandgshare all four values CM. Here N/parenleftig\nr,1\nf−a4/parenrightig\n=N(r,f) when\na4=∞.\nTheorem 8. ([11, Theorem 1]) Let fandgbe two distinct nonconstant\nmeromorphic functions, and let a1, a2, a3, a4be four distinct values in the\nextended complex plane. If fandgsharea1, a2, a3IM anda4CM, then\nfandgare functions of normal growth, fandghave the same order, and\nthe order of fandgis a positive integer or infinite.\nNext we consider the following special case of Question 1:\nQuestion 2. If two non-constant meromorphic functions share three valu es\nIM and share a fourth value CM, where one of the two non-consta nt mero-\nmorphic functions has finite order, then do the two functions necessarily\nshare all four values CM?\nIn this paper, we give an affirmative answer to Question 2 for tw o distinct\nnon-constant meromorphic functions satisfying the assump tions in Question\n2. Indeed, we prove the following theorem in this paper:\nTheorem 9. Suppose that fandgare two distinct non-constant meromor-\nphic functions, and one of fandghas finite order. If fandgsharea1, a2,\na3IM anda4CM, where a1, a2, a3, a4are four distinct complex values in\nthe extended complex plane, then fandgsharea1, a2, a3anda4CM.\nRemark 1. Theorem 9 gives an affirmative answer to Question 1 for non-\nconstant meromorphic functions fandgof finite order sharing three values\nIM and a fourth value CM.MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 5\nFrom Theorem 9 we get the following result:\nCorollary 1. Suppose that fandgare two distinct non-constant entire\nfunctions, and one of fandghas finite order. If fandgsharea1, a2, a3\nIM, wherea1, a2, a3are three distinct finite complex values, then fandg\nsharea1, a2anda3CM.\n2.Preliminaries\nIn this section, we will introduce some results that play an i mportant\nrole in proving the main result in this paper. First of all, we introduce the\nfollowing result due to Adams-Straus[1]:\nLemma 1. ([1, Theorem 3]). Let fandgbe two nonconstant rational\nfunctions. If fandgshare four distinct values a1,a2,a3,a4IM, thenf=g.\nThe following result was proved by R.Nevanlinna in [33] orig inally:\nLemma 2. ([33, p.373, Satz 3] or[8, Theorem 2]) Let fandgbe two dis-\ntinct non-constant meromorphic functions, and let a1,a2,a3anda4be four\ndistinct complex values in the extended complex plane. If fandgsharea1,\na2, a3anda4IM, then\n(i)T(r,f) =T(r,g)+S(r,f) and (ii) 2T(r,f) =4/summationtext\nj=1N/parenleftig\nr,1\nf−aj/parenrightig\n+S(r,f).\nRemark 2. A proof of Lemma 2 can be in Gundersen [11, p.101, proof of\nLemma 1]. From [13, Theorem 2.1] and the proof of [11, Lemma 1] we see\nthat the error term S(r,f) in Lemma 2 is expressed as\n(2)\nS(r,f) =\n\nm/parenleftig\nr,f′\nf/parenrightig\n+m/parenleftigg\nr,4/summationtext\nj=1f′\nf−aj/parenrightigg\n+O(1),when{a1,a2,a3,a4} ⊂C;\nm/parenleftig\nr,f′\nf/parenrightig\n+m/parenleftigg\nr,3/summationtext\nj=1f′\nf−aj/parenrightigg\n+O(1),whena4=∞.\nThe following result and its proof can be found in Gundersen [ 9]:\nLemma 3. ([9, p.550, Lemma2] and[9, p.549, Corollary 1(II)]). Let fandg\nbetwo diatinct non-constant meromorphic functions that sh are four distinct\nvaluesa1, a2, a3anda4IM, wherea4=∞.Then the following statement\nholds:\n(i)N0/parenleftig\nr,1\nf′/parenrightig\n+N0/parenleftig\nr,1\ng′/parenrightig\n=S(r,f),whereN0/parenleftig\nr,1\nf′/parenrightig\nandN0/parenleftig\nr,1\ng′/parenrightig\n“count” respectively only those points in N0/parenleftig\nr,1\nf′/parenrightig\nandN0/parenleftig\nr,1\ng′/parenrightig\nwhich\ndo not occur when f(z) =g(z) =ajfor somej= 1,2,3,4.6 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\n(ii)4/summationtext\nj=1N∗(r,aj,f,g) =S(r,f),whereN∗(r,aj,f,g) with 1 ≤j≤4\nandj∈Zdenotes the counting function of those common aj-points off\nandgin|z|<rthat are multiple for both fandg,and each such point in\nN∗(r,aj,f,g) is counted according to the number of the times of the smalle r\nof the two multiplicities.\nRemark 3. A proof of Lemma 3 can be found in the proof of Lemma 3\nin [11, p.102]. Following the expression of the Mues′s function Ψ( f) in [6,\npp.176-177] and the proof of Lemma 3 in [11, p.102], we can see that the\nerror termS(r,f) in Lemma 3 is expressed as\n(3)S(r,f) =m(r,φ)+O(1) =m/parenleftigg\nr,3/summationdisplay\ns=13/summationdisplay\nt=1cstf′\nf−asg′\ng−at/parenrightigg\n+O(1),\nwherecstwiths,t∈ {1,2,3}is a finite complex constant that depends only\nona1, a2anda3,andφis called the E.Mues’s function that is an entire\nfunction defined as\nφ=f′g′(f−g)2\n(f−a1)(f−a2)(f−a3)(g−a1)(g−a2)(g−a3).\nThis can be found in Mues [31, p.171, proof of Lemma 1]. Here we mention\nthatbasedupontheassumptionthattwodistinctnon-consta ntmeromorphic\nfunctionsfandgshare 0,1,cand∞IM, wherecis a finite complex number\nsuch thatc/\\e}atio\\slash∈ {0,1,∞},the Mues′s functionφin Mues [31, p.171, proof of\nLemma 1] is written into\nφ=f′g′(f−g)2\n(f)(f−1)(f−c)(g)(g−1)(g−c).\nThe following result is due to Yang [49]:\nLemma 4. ([49, Theorem 1.6]). Suppose that fis a non-constant mero-\nmorphic function in the complex plane and kis a positive integer. Then,\none of the following two cases holds:\n(i) If the order of fis finite, then m/parenleftig\nr,f(k)\nf/parenrightig\n=O(logr),asr→ ∞.\n(ii) If the order of fis infinite, then\nm/parenleftigg\nr,f(k)\nf/parenrightigg\n=O(logT(r,f)+logr),asr/\\e}atio\\slash∈Eandr→ ∞.\nHereE⊂(0,+∞) is a set of finite linear measure.\nRemark 4. Based upon the assumptions of Lemma 2 or the assumptions\nof Lemma 3, we additionally suppose that fhas finite order. Then, from\nLemma 4(i), the formula (2) in Remark 2, the formula (3) in Rem ark 3 weMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 7\ndeduce that the error term S(r,f) in Lemma 2 and the error term S(r,f)\nin Lemma 3 can be estimated as S(r,f) =O(logr),asr→ ∞.\nThe following result is from Yang-Yi[48]:\nLemma 5. ([48, p.11, Theorem 1.5]). If fis a transcendental meromorphic\nfunction in the complex plane, then lim\nr→∞T(r,f)\nlogr=∞.\nFrom Lemma 2(i), Lemma 3(ii), Lemma 4(i), Remark 2-Remark 4 w e get\nthe following result:\nLemma 6. ([24, proof of Lemma 9]). Let fandgbe two distinct non-\nconstant meromorphic functions, and one of fandghas finite order. Sup-\npose thatfandgsharea1,a2,a3,a4IM, wherea1,a2,a3are three distinct\nfinite values and a4=∞.Then\n1\n7T(r,f)≤3/summationdisplay\nj=16/summationdisplay\nl=1N(1,l)(r,aj;f,g)+3/summationdisplay\nj=16/summationdisplay\nm=2N(m,1)(r,aj;f,g)+O(logr),\nasr→ ∞.Here and in what follows, N(1,l)(r,aj;f,g) denotes the reduced\ncounting function of those common zeros of f−ajandg−ajin|z|< r\nthat are simple zeros of f−ajand are zeros of g−ajof multiplicity lfor\n1≤l≤6 withl∈Z,and 1≤j≤3 withj∈Z,whileN(m,1)(r,aj;f,g)\ndenotes the reduced counting function of those common zeros off−ajand\ng−ajin|z|< rthat are zeros of f−ajof multiplicity mand are simple\nzeros ofg−ajform≥2 andm∈Z.\nThe following result is from Li-Yang [22]:\nLemma 7. ([22, Lemma 7]). Let f1andf2be two non-constant meromor-\nphic functions such that\nN(r,fj)+N/parenleftbigg\nr,1\nfj/parenrightbigg\n=S(r) forj∈ {1,2}.\nIffs\n1ft\n2−1 is not identically zero for all integers sandtsuch that |s|+|t|>0,\nthen for any positive number ε,we have\nN0(r,1;f1,f2)≤εT(r)+S(r),\nwhereN0(r,1;f1,f2) denotes the reduced counting function of the common\n1-points in |z|< r,andS(r) is any quantity such that S(r) =o(T(r)),as\nr/\\e}atio\\slash∈Eandr→ ∞,whereT(r) =T(r,f1)+T(r,f2),andE⊂R+denotes a\nset of finite linear measure.\nIn 1999, Zhang [53] proved the following result that improve d Lemma 7:8 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nLemma 8. ([53, Lemma 6]). Let f1andf2be two non-constant meromor-\nphic functions such that\nN(r,fj)+N/parenleftbigg\nr,1\nfj/parenrightbigg\n=S(r) forj∈ {1,2}.\nThen, either N0(r,1;f1,f2) =S(r),or there exist two integers sandtwith\n|s|+|t|>0,such thatfs\n1ft\n2= 1.Here and in what follows, N0(r,1;f1,f2)\ndenotes the reduced counting function of the common 1-point s in|z|< r,\nandS(r) is any quantity such that S(r) =o(T(r)),asr/\\e}atio\\slash∈Eandr→ ∞,\nwhereT(r) =T(r,f1)+T(r,f2),andE⊂R+denotes a set of finite linear\nmeasure.\nThe following result is due to Markushevich [27]:\nLemma 9. ([27]). LetQ(z) =qnzn+qn−1zn−1+···+q1z+q0,wherenis\na positive integer and qn=|qn|eiθnwith|qn|>0 andθn∈[0,2π).For any\ngiven positive number εsatisfying 0 <ε<π\n4n,we consider 2 nangles:\nSj:−θn\nn+(2j−1)π\n2n+ε<θ<−θn\nn+(2j+1)π\n2n−ε,\nwherejis an integer satisfying 0 ≤j≤2n−1.Then, there exists a positive\nnumberR=R(ε) such that Re( Q(z))>|qn|(1−ε)rnsin(nε) forz∈Sj\nwith|z|=r>Rwherejis even, and Re( Q(z))<−|qn|(1−ε)rnsin(nε) for\nz∈Sjwith|z|=r>Rwherejis odd.\nNext we introduce the Nevanlinna theory of meromorphic func tions in\none angular domain that will play a key role in the proof of the main result:\nLetfbe a meromorphic function on the angular domain Ω(α,β) ={z:\nα≤argz≤β},where 0< β−α≤2π.Following Goldberg-Ostrovskii [7,\np.25], we define\nAα,β(r,f) =ω\nπ/integraldisplayr\n1/parenleftbigg1\ntω−tω\nr2ω/parenrightbigg\n{log+|f(teiα)|+log+|f(teiβ)|}dt\nt,\nBα,β(r,f) =2ω\nπrω/integraldisplayβ\nαlog+|f(reiθ)|sinω(θ−α)dθ,\nCα,β(r,f) = 2/summationdisplay\n1<|bm|<r/parenleftbigg1\n|bm|ω−|bm|ω\nr2ω/parenrightbigg\nsinω(θm−α),\nwhereω=π\nβ−αand{bm}+∞\nm=1⊂Ω(α,β) withbm=|bm|eiθmandm∈Z+\nis the sequence of poles of fonΩ(α,β),and each distinct point in the\nsequence {bm}+∞\nm=1is repeated as many times as its multiplicity of a pole\noffonΩ(α,β).We denote by Cα,β(r,f) the reduced form of Cα,β(r,f).In\notherword, ifeach point intheabove expressionof Cα,β(r,f) is counted only\nonce for each distinct point in the sequence {bm}+∞\nm=1,we denoteCα,β(r,f)\ninstead ofCα,β(r,f).We callCα,β(r,f) the Nevanlinna′s angular counting\nfunction of poles of fonΩ(α,β).The Nevanlinna′s angular characteristicMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 9\nfunction is defined as Sα,β(r,f) =Aα,β(r,f) +Bα,β(r,f) +Cα,β(r,f).We\nrecall the following two results are due to Zheng [54]:\nLemma 10. ([54, Lemma 3]). Let fbe meromorphic on Ω(α,β).Then for\narbitrary complex number a∈C,we have\nSα,β/parenleftbigg\nr,1\nf−a/parenrightbigg\n=Sα,β(r,f)+O(1).\nLemma 11. ([54, Lemma 4]). Let fbe meromorphic on Ω(α,β).Then for\narbitraryqdistincta1,a2,···,aqinC∪{∞},we have\n(q−2)Sα,β(r,f)≤q/summationdisplay\nj=1Cα,β/parenleftbigg\nr,1\nf−aj/parenrightbigg\n+Rα,β(r,f),\nwhere the term Cα,β/parenleftig\nr,1\nf−aj/parenrightig\nwill be replaced by Cα,β(r,f) when some\naj=∞.\nFrom Lemma 2.5.3 in [55, p.81] and the formula\nmα,β(r,f) =1\n2π/integraldisplayβ\nαlog+|f(reθ)|dθwith 0≤α<β≤2π\nin [55, p.56], we get the following result:\nLemma 12. ([55, Lemma 2.5.3]). Let fbe a meromorphic function in the\ncomplex plane. For any positive number rsuch that 0 <r<R, we have\nRα,β(r,f)\n≤K/parenleftbigg/parenleftbiggR\nr/parenrightbiggω/integraldisplayR\n1logT(t,f)\nt1+ωdt+logr\nR−r+logR\nr+1\nrωm/parenleftbigg\nr,f′\nf/parenrightbigg\n+1/parenrightbigg\n,\nwhereω=π\nβ−αwith 0≤α<β≤2π,andKis a constant independent of\nrandR.\nFrom Remark 4, Lemma 4 (i) and Lemma 12 with r∈(0,+∞) and\nR= 2r,we deduce the following result:\nRemark 5. Based upon the assumptions of Lemma 12, we additionally\nsuppose that fis a non-constant meromorphic function of finite order in\nthe complex plane. Then, we have Rα,β(r,f) =O(logr) whenr∈(0,+∞),\nR= 2randr→ ∞.\nNextweintroducetheNevanlinnatheoryinasimplyconnecte ddomainon\nC(cf.[55, pp.26-38]): let D⊂Cbe a simply connected domain surrounded\nby finitely many piecewise analytic curves. Then for any a∈D,there exists\na Green function, denoted as GD(z,a),forDwith singularity at a∈D\nwhich is uniquely determined by the following conditions:\n(1)GD(z,a) is harmonic in D\\{a};10 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\n(2) in a neighborhood of a,GD(z,a) = log1\n|z−a|+ωD(z,a) for some func-\ntionωD(z,a) harmonic in D;\n(3)GD(z,a)≡0 on the boundary of D.\nWe denote by Γ = ∂Dthe positive boundary of Dandnthe inner normal\nof Γ with respect to D.Since forz∈D, GD(z,a)>0 and forz∈Γ,\nGD(z,a) = 0,from the definition of directional derivative it follows tha t the\ndirectional derivative of GD(z,a) on Γ in the inner normal is non-negative,\nthat is to say,∂G\n∂n≥0 withG=GD(z,a).From the Green formula, in\nview of the Green function, we can establish the following fo rmula, which is\nan extension of the Poisson formula for a disk. For a generali zation of the\nformula, the reader is referred to [7, p.4, Theorem 1.1]. Fol lowing [55, p.28],\nwe define the counting function of fwith the center at aforDas follows:\n(4) N(D,a,f) =/summationdisplay\nbn∈DGD(bn,a)+n(0,a,f)ωD(a,a),\nwhereais a point such that a∈D,andbnwithn∈Z+is a pole of f\nappearingoften according toits multiplicities of apoleof f(z),andn(0,a,f)\nis the multiplicity of a pole of f(z) ata,whileN(D,a,f) is the sum in (4)\ncounting all distinct bn∈Dwithn∈Z+and withn(0,a,f) replaced by 1\nwhenf(a) =∞.Following Zheng [55, p.28], we also define\n(5) m(D,a,f) =1\n2π/integraldisplay\nΓlog+|f(ζ)|∂GD(ζ,a)\n∂nds\nand\n(6) T(D,a,f) =m(D,a,f)+N(D,a,f),\nwhich are called the proximity function and the Nevanlinna c haracteristic\nfunction of fwith the center at aforDrespectively. For pmeromorphic\nfunctionsf1,f2,...,f pon the a simply connected domain Dsurrounded by\nfinitely many piecewise analytic curves, where pis a positive integer such\nthatp≥2,wefollow Zheng[55, pp.28-29] tohave thefollowing basicpr oper-\nties concerning the proximity functions, Nevanlinna count ing functions and\nthe Nevanlinna characteristic functions of meromorphic fu nctions onDwith\nthe center at aforD:\n(i)m\nD,a,p/summationdisplay\nj=1fj\n≤p/summationdisplay\nj=1m(D,a,f j)+logp;\n(ii)m\nD,a,p/productdisplay\nj=1fj\n≤p/summationdisplay\nj=1m(D,a,f j);MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 11\n(iii)N\nD,a,p/summationdisplay\nj=1fj\n≤p/summationdisplay\nj=1N(D,a,f j)whenfj(a)/\\e}atio\\slash=∞for 1≤j≤p−1\nandj∈Z;\n(iv)N\nD,a,p/productdisplay\nj=1fj\n≤p/summationdisplay\nj=1N(D,a,f j)whenfj(a)/\\e}atio\\slash=∞for 1≤j≤p−1\nandj∈Z;\n(v)T\nD,a,p/summationdisplay\nj=1fj\n≤p/summationdisplay\nj=1T(D,a,f j)+logpwhenfj(a)/\\e}atio\\slash=∞for\n1≤j≤p−1 andj∈Z;\n(vi)T\nD,a,p/productdisplay\nj=1fj\n≤p/summationdisplay\nj=1T(D,a,f j) whenfj(a)/\\e}atio\\slash=∞for 1≤j≤p−1\nandj∈Z.\nWe recall the following result that is called the Nevanlinna second funda-\nmental theorem with the center at aforD(cf.[55, p.32]):\nLemma 13. (cf.[55, p.32, Theorem 2.1.4]) Let f(z) be a meromorphic func-\ntion onD∪∂D,whereDis a simply connected domain such that D⊂C,\nand leta1, a2, ..., a qbeqdistinct finite complex numbers, where qis a\npositive integer such that q≥2,Then fora∈Dsuch thatf(a)/\\e}atio\\slash∈ {0,∞}\nandf(a)/\\e}atio\\slash=ajwith 1≤j≤qandj∈Z,we have\n(7)\n(q−1)T(D,a,f)≤N(D,a,f)+q/summationdisplay\nj=1N/parenleftbigg\nD,a,1\nf−aj/parenrightbigg\n−N1(D,a,f)+S(D,a,f),\nwhere\n(8)S(D,a,f) =m/parenleftbigg\nD,a,f′\nf/parenrightbigg\n+q/summationdisplay\nj=1m/parenleftbigg\nD,a,f′\nf−aj/parenrightbigg\n+q/parenleftbigg\nlog+2q\nδ+log+δ\n2q+log2/parenrightbigg\n+logq−log|f′(a)|\n+q/summationdisplay\nj=1(log|f(a)−aj|+ε(aj,D))\nand\nN1(D,a,f) = 2N(D,a,f)−N(D,a,f′)+N/parenleftbigg\nD,a,1\nf′/parenrightbigg\n.\nHereδ= min\n1≤j<k≤q|aj−ak|andε(aj,D)≤log+|aj|+log2 for 1 ≤j≤qwith\nj∈Z.12 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nRemark 6. Following Zheng [55, pp.35-37], we have a discussion as foll ows:\nletfbea non-constant meromorphic function on D=D∪∂D,whereD⊂C\nis a simple connected domain that is surrounded by finitely ma ny piecewise\nanalytic curves, in other words, the boundary Γ = ∂Dof the simple con-\nnected domain Dconsists of finitely many piecewise analytic curves. Then,\nfor a point a∈Dsuch thatf(a)/\\e}atio\\slash= 0,∞,there exists the corresponding\nRiemann mapping φa(z) :D→ {w∈C:|w|<1}such thatφa(a) = 0.\nThen, it is easy for us to see that GD(z,a) =−log|φa(z)|withz∈D.\nFor detailed reasonings, we refer to Zheng [55, p.35]. In thi s way, we can\nobtain the Green functions for some special simple connecte d domains on C\nas follows:\n(6.1) ForD={z∈C:|z|<R}and a point asuch that |a|<R\nandf(a)/\\e}atio\\slash= 0,∞,whereRis some positive number, we follow Zheng [55,\npp.35-36] to have\n(9)φa(z) =R(z−a)\nR2−azandGD(z,a) =−log|φa(z)|= log/vextendsingle/vextendsingle/vextendsingle/vextendsingleR2−az\nR(z−a)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nforz∈D∪∂D.Noting that φa(z) is analytic on the positive boundary\nΓ =∂DofD,we see that the tangent vector sat any point along the\npositive boundary Γ = ∂DofDbecomes the inner normal nof Γ at the\ncorresponding point on Γ after sis rotatedπ/2 anticlockwise. Therefore,\nfor eachζ∈Γ we have\n∂log|φa(ζ)|\n∂s=∂argφa(ζ)\n∂nand∂log|φa(ζ)|\n∂n=−∂argφa(ζ)\n∂s\nThis implies that\ndlogφa(ζ) =∂logφa(ζ)\n∂sds=i∂argφa(ζ)\n∂sds=−i∂log|φa(ζ)|\n∂nds\n=i∂GD(ζ,a)\n∂nds(10)\nfor eachζ∈Γ.Here and in what follows, for each ζ∈Γ we denote by sthe\ntangent vector along the positive boundary Γ of Dat the point ζ∈Γ,and\ndenote by nthe corresponding inner normal of the positive boundary Γ of\nDat the point ζ∈Γ.\n(6.2) ForD={z∈C:|z|<R}and the point a∈D,whereRis\nany given positive number, we follow Zheng [55, p.36] to writ e the prox-\nimity function m(D,a,f),the Nevanlinna counting function N(D,a,f),\nthe Nevanlinna characteristic function T(D,a,f) offwith the center at\naforD={z∈C:|z|<R}intom(R,a,f), N(R,a,f) andT(R,a,f)\nrespectively. In particular, the proximity function m(R,0,f),the Nevan-\nlinna counting function N(R,0,f),the Nevanlinna characteristic function\nT(R,0,f) offwith the center at 0 for D={z∈C:|z|<R}are reduced\nintom(R,f), N(R,f) andT(R,f) respectively. Indeed, from (9) and (10)MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 13\nwe deduce\n(11)∂GD(ζ,z)\n∂nds=R2−|z|2\n|ζ−z|2dθandωD(z,a) = log/vextendsingle/vextendsingle/vextendsingle/vextendsingleR2−az\nR/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nfor any complex number ζ=Reiθwithθ∈[0,2π) and any given complex\nnumberzsuch that |z−a|<R.In particular, we have from (11) that\n(12)∂GD(ζ,0)\n∂nds=dθandωD(z,0) = logR\nfor any complex number ζ=Reiθwithθ∈[0,2π).Therefore, it follows\nfrom (4)-(6) and (11) that the proximity function m(D,0,f),the Nevan-\nlinna counting function N(D,0,f),the Nevanlinna characteristic function\nT(D,0,f) offwith the center at 0 for D={z∈C:|z|<R}are reduced\ninto\n(13) m(R,f) =1\n2π/integraldisplay2π\n0log+|f(Reiθ)|dθ\n(14) N(R,f) =/summationdisplay\n0<|bn|<RlogR\n|bn|+n(0,f)logR\nand\n(15) T(R,f) =m(R,f)+N(R,f).\nrespectively.\n(6.3)Based upontheassumptionsofLemma13, wefollow Zheng[55, p.35]\nto see thatN(D,a,f) andT(D,a,f) are non-negative for the case of f(a)/\\e}atio\\slash=\n∞,whileforthe case of f(a) =∞,T(D,a,f) may benegative. For example,\nwe consider the function f(z) =1\n2zpwithD=/braceleftbig\nz∈C:|z|<1\n2/bracerightbig\n,wherep\nis a positive integer. In view of (4), (13)-(15) and Theorem 2 .16 in Zheng\n[55, p.34] we deduce T(D,0,f) =T/parenleftbig1\n2,f/parenrightbig\n=−log2 andN/parenleftig\nD,0,1\nf−eiθ/parenrightig\n=\nN/parenleftig\n1\n2,1\nf−eiθ/parenrightig\n= 0 withθ∈[0,2π] forD={z∈C:|z|<1\n2}.It is obvious\nthatN(D,a,f)≥0 and soT(D,a,f)≥0,whenωD(a,a)≥0 fora∈D=\n{z∈C:|z|<1\n2}.\nFor convenience in stating the following result, we shall us e the following\nnotation. We shall let ( f,H) denote a pair that consists of a transcendental\nmeromorphic function fand a finite set H={(k1,j1),(k2,j2),...,(kq,jq)}\nof distinct pairs of integers that satisfy kl> jlforl= 1,2...,q.We recall\nthe following result due to Gundersen [10]:\nLemma 14. (cf.[10, p.89, Corollary 1]) Let ( f,H) be a given pair where f\nhas finite order ρ,and letεbe a given constant. Then, there exists a set\nE⊂[0,2π) that has linear measure zero, such that if ψ0∈[0,2π)\\E,then14 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nthere is a constant R0=R0(ψ0)>1 such that for all zsatisfying arg z=ψ0\nand|z| ≥R0,and for all ( k,j)∈H,we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(k)(z)\nf(j)(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |z|(k−j)(ρ−1+ε).\nNext we introduce the notion of the spherical derivative of a meromor-\nphic function in the complex plane (cf.[13, 49]): let fbe a non-constant\nmeromorphic function. The spherical derivative of fatz∈Cis given as\nf#(z) =|f′(z)|\n1+|f(z)|2.We recall the following result from Chang-Zalcman [3]:\nLemma 15. ([3, Lemma 1]). Let fbe a meromorphic function on C.Iff\nhas bounded spherical derivative on C,thenfis of order at most 2 .If, in\naddition,fis an entire function, then the order of fis at most 1 .\nRemark 7. Following He-Xiao [19, pp.53-55], we introduce the Ahlfors -\nShimizu’s characteristic function of a meromorphic functi on: suppose that\nfis a meromorphic function in the complex plane. Then, the Ahl fors-\nShimizu′s characteristic function of f,denoted as ˚T(r,f),is defined as\n˚T(r,f) =/integraldisplayr\n01\nt\n1\nπ/integraldisplay/integraldisplay\n|z|≤t(f#(z))2dxdy\ndt,\nwherez=x+yiwithx,y∈R.The Nevanlinna’s characteristic function\nT(r,f) and the Ahlfors-Shimizu’s characteristic function ˚T(r,f) differ by a\nbounded quantity that is independent of r∈(0,+∞) (cf.[19, p.56]). This\nimplies the first part of Lemma 15. The result of Lemma 15 for en tire\nfunctions is much subtler, which is a special case of Theorem 3 in [4].\nWe also need some notions of normal families of meromorphic f unctions\nin a domain of the complex plane. Following Montel [29], we ca ll a class Fof\nmeromorphic functions in a domain D⊆Cto be normal in the domain D,\nprovidedthatforany given sequence {fn(z)}ofmeromorphicfunctionsin F,\nwe can find a subsequence {fnp(z)} ⊆ {fn(z)}which converges everywhere\ninDand uniformly on compact subsets of Dwith respect to the chordal\nmetric on Riemann sphere. We then say that {fnp(z)}converges locally\nuniformly in D.An equivalent statement is that for every z0inDthere\nexists a neighbourhood |z−z0|<δin which {fnp(z)}or/braceleftig\n1\nfnp(z)/bracerightig\nconverges\nuniformly as p→ ∞.This implies that if w1andw2are two points in the\nw-plane, their distance in the chordal metric of Riemann sphe re is\nk(w1,w2) =|w1−w2|/radicalbig\n(1+|w1|2)(1+|w2|2)≤ |w1−w2|MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 15\nAlso\nk(w1,w2) =|1\nw1−1\nw2|\n/radicalig\n(1+|1\nw1|2)(1+|1\nw2|2)≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nw1−1\nw2/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nThus, if either fn(z)→f(z) or1\nfn(z)→1\nf(z)uniformly in a set E⊆C∪\n{∞},thenk(fn(z),f(z))→0,uniformly in E.Thus, if one of these two\nconditions holds uniformly in some neighbourhood of every p oint ofD,then\nk(fn(z),f(z))→0 uniformly in some neighbourhood of every point of D,\nand hence by the Heine-Borel theorem uniformly on every comp act subset\nofD; see, for example, Hayman [13, pp.157-160]. We recall the fo llowing\nresult that is the well-known Pang-Zalcman’s lemma:\nLemma 16. (Pang-Zalcman’s Lemma, [35, 36, 52]). Let Fbe a family of\nmeromorphic functions in the unit disc △={z∈C:|z|<1}andαbe a\nreal number satisfying −1< α <1.IfFis not normal at a point z0∈ △,\nthen for each −1<α<1,there exist:\n(i) pointszm∈ △, zm→z0,(ii) positive numbers ρm, ρm→0+and\n(iii) functions fm∈ Fsuch thatfm(zm+ρmζ)\nραm→g(ζ) spherically uniformly on\nany compact subset of C,wheregis a non-constant meromorphic function\nin the complex plane such that its order ρ(g)≤2.The function gmay be\ntaken to satisfy the normalization g#(ζ)≤g#(0) = 1 for each ζ∈C.\nFrom Lemma 16 for α= 0 we get the following result:\nLemma 17. ([43, Rescaling Lemma 7.1.]). Let fandgbetwo non-constant\nmeromorphic functions in the extended complex plane, and le ta1,a2,a3,a4\nbe four distinct complex values in the extended complex plan e such that\nfandgsharea1, a2, a3, a4IM. Then, either the spherical derivatives\nf#(z) andg#(z) are uniformly bounded on C,or else there exist sequences\n{zm}+∞\nm=1⊂Cand{ρm}+∞\nm=1⊂(0,+∞) withρm→0,asm→+∞,such\nthat thesequences {fm(z)}+∞\nm=1and{gm(z)}+∞\nm=1withfm(z) andgm(z) being\ndefined asfm(z) =f(zm+ρmz) andgm(z) =g(zm+ρmz) respectively for\neachm∈Z+,simultaneously tend to non-constant meromorphic function s\nˆf(z) and ˆg(z) in the complex plane respectively, such that ˆf(z) and ˆg(z)\nhave uniformly bounded spherical derivatives on C,and such that ˆf(z) and\nˆg(z) share the four complex values a1, a2, a3, a4IM.\nRemark 8. From Lemma 15 and Lemma 17 we deduce that the orders of\nthe non-constant meromorphic functions ˆf(z) and ˆg(z) in Lemma 17 satisfy\nρ(ˆf)≤2 andρ(ˆg)≤2.\nFrom Theorem 9, Lemma 17, Remark 8 and [39, p.9, Hurwitz theor em],\nwe get the following result:16 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nLemma 18. There do not exist two distinct transcendental meromorphic\nfunctionsfandgin the complex plane such that one of the spherical deriva-\ntivesf#(z) andg#(z) are unbounded on C,and such that fandgsharea1,\na2, a3DM anda4CM, where a1, a2, a3, a4are four distinct values in the\nextended complex plane.\nProof.On the contrary, we suppose that there exist two distinct tra nscen-\ndental meromorphic functions fandgin the complex plane such that one\nof the spherical derivatives f#(z) andg#(z) is unbounded on C,and such\nthatfandgsharea1, a2, a3DM anda4CM. Without loss of general-\nity, we suppose that the spherical derivatives f#(z) is unbounded on C,\nand suppose that a1, a2, a3are three distinct finite values, while a4=∞.\nThen, from Lemma 17 we see that there exist sequences {zm}+∞\nm=1⊂C\nand{ρm}+∞\nm=1⊂(0,+∞) withρm→0,asm→+∞,such that the se-\nquences {fm(z)}+∞\nm=1and{gm(z)}+∞\nk=1withfm(z) andgm(z) being defined\nasfm(z) =f(zm+ρmz) andgm(z) =g(zm+ρmz) respectively for each\nm∈Z+,simultaneously tend to some two non-constant meromorphic f unc-\ntions, say ˆf(z) and ˆg(z) in the complex plane respectively, such that the\norders of ˆf(z) and ˆg(z) satisfyρ(ˆf)≤2 andρ(ˆg)≤2.Moreover, from [39,\np.9, Hurwitz theorem] and the assumption that fandgsharea1, a2, a3\nDM anda4CM, we deduce that ˆfand ˆgalso sharea1, a2, a3DM and\na4CM. This together with Lemma 1 implies that either ˆfand ˆgare two\nnon-constant rational functions such that ˆf= ˆgor that ˆfand ˆgare two\ndistinct transcendental meromorphic functions in the comp lex plane. We\nhave a discussion as follows:\nCase 1. Suppose that ˆfand ˆgare two non-constant rational functions\nsuch that ˆf= ˆg.First of all, from the second fundamental theorem we have\n(16) T(r,ˆf)≤3/summationdisplay\nj=1N/parenleftigg\nr,1\nˆf−aj/parenrightigg\n+O(1).\nFrom(16)weseethatatleastoneof N/parenleftig\nr,1\nˆf−a1/parenrightig\n,N/parenleftig\nr,1\nˆf−a2/parenrightig\nandN/parenleftig\nr,1\nˆf−a3/parenrightig\n,\nsayN/parenleftig\nr,1\nˆf−a1/parenrightig\n,satisfies\n(17) N/parenleftbigg\nr,1\nˆf−a1/parenrightbigg\n/\\e}atio\\slash=O(1).\nFrom (17) and the known result that ˆfand ˆgsharea1DM, we see that\nthere exists at least one common a1-point ˆz0∈Cofˆfand ˆgsuch that ˆz0\nhas different multiplicities related to ˆfand ˆg.However, from the known\nresultˆf= ˆgwe see that ˆfand ˆgsharea1CM, and so ˆ z0has the same\nmultiplicities related to ˆfand ˆg.This is a contradiction.\nCase 2. Suppose that ˆfand ˆgare two distinct transcendental mero-\nmorphic functions. First of all, from Remark 2, Lemma 4(i) an d the secondMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 17\nfundamental theorem we have\n(18) T(r,ˆf)≤3/summationdisplay\nj=1N/parenleftigg\nr,1\nˆf−aj/parenrightigg\n+O(logr).\nFrom(18)weseethatatleastoneof N/parenleftig\nr,1\nˆf−a1/parenrightig\n,N/parenleftig\nr,1\nˆf−a2/parenrightig\nandN/parenleftig\nr,1\nˆf−a3/parenrightig\n,\nsayN/parenleftig\nr,1\nˆf−a1/parenrightig\n,satisfies\n(19) N/parenleftbigg\nr,1\nˆf−a1/parenrightbigg\n/\\e}atio\\slash=O(logr).\nFrom (19) and the known result that ˆfand ˆgsharea1DM, we see that there\nexist infinitely many common a1-points of ˆfand ˆgin the complex plane, and\neach such common a1-point of ˆfand ˆghas different multiplicities related\ntoˆfand ˆg.However, from Theorem 9, the known results of ρ(ˆf)≤2 and\nρ(ˆg)≤2,and the known result that ˆfand ˆgsharea1, a2, a3DM anda4\nCM, we see that ˆfand ˆgshare the four values a1, a2, a3, a4CM, and so\nwe see that each common a1-point of ˆfand ˆghas the same multiplicities\nrelated to ˆfand ˆg.This is a contradiction. Lemma 18 is thus completely\nproved. /square\n3.Proof of Theorem 9\nFirst, before the formal beginning of the proof of Theorem 9, we want\nto point out that the lines from the next formal beginning of t he proof of\nTheorem 9 to the end of Case 1 of the next formal beginning of th e proof\nof Theorem 9 are covered essentially by the proof of Theorem 6 .2 in [43]\neven without any assumption on the order of growth. For the co nvenience\nof the readers, we give the detailed proof for this part on the assumption\nof finite order of growth as follows: first of all, we suppose, w ithout loss\nof generality, that fhas finite order ρ(f) =:ρ <∞.Combining this with\nLemma 1, Lemma 2 (i) and the assumption that fandgare two distinct\nnon-constant meromorphicfunctions that share a1,a2,a3IM anda4CM, we\ndeduce that fandgare two distinct transcendental meromorphic functions\nin the complex plane. From Lemma 2(i), Remark 4 and the assump tion\nρ(f)<∞,we have\n(20) T(r,f) =T(r,g)+O(logr),asr→ ∞.\nFrom (20), Theorem 8 and Definition 1, we deduce that the order s offand\ngare positive integers such that\n(21) 1 ≤ρ(f) =ρ(g) =:ρ<∞.18 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nWithout loss of generality, we assume that a1= 0, a2= 1, a3=cand\na4=∞,and set\n(22)ϕ:=f′′\nf′−f′\nf−f′\nf−1−f′\nf−c−g′′\ng′+g′\ng+g′\ng−1+g′\ng−c.\nBy (21) and Whittaker [47, p. 82] we have\n(23) ρ(f) =ρ(f′) =ρ(g) =ρ(g′)<∞.\nBy (22), (23) and Lemma 4(i) we have\n(24) m(r,ϕ) =O(logr),asr→ ∞.\nSincefandgsharea1, a2, a3IM anda4CM, by simple calculating we\ncan see that φdefined as (22) is analytic at any point z∈Csuch that\nf(z) =g(z) =ajfor somej= 1,2,3,4,and so from Lemma 3(i) and\nRemark 4 we have\n(25) N(r,ϕ)≤N0/parenleftbigg\nr,1\nf′/parenrightbigg\n+N0/parenleftbigg\nr,1\ng′/parenrightbigg\n=O(logr).\nFrom (24) and (25) we have\n(26) T(r,ϕ) =m(r,ϕ) +N(r,ϕ) =O(logr).\nFrom (26) and Lemma 5 we see that φis a rational function. Combining\nthis with (22), we deduce that (22) can be rewritten into\n(27) ϕ=P1+q1/summationdisplay\nj=1mj\nz−zj,\nwhereP1is reduced to zero or a non-vanishing polynomial, q1is a non-\nnegative integer such thatq1/summationtext\nj=1mj\nz−zjin the right hand side of (27) is reduced\nto zero when q1= 0,andmjwith 1≤j≤q1andj∈Zis some non-zero\ninteger when q1≥1 andq1∈Z,whilez1, z2, ..., z q1are those distinct\npoints in Csuch thatf′(zj) = 0 and g′(zj)(f(zj)−al)(g(zj)−al)/\\e}atio\\slash= 0\nwith 1≤j≤q1andj∈Zfor somel∈ {1,2,3,4},org′(zj) = 0 and\nf′(zj)(f(zj)−al)(g(zj)−al)/\\e}atio\\slash= 0 with 1 ≤j≤q1andj∈Zfor somel∈\n{1,2,3,4},orzjwith1≤j≤q1andj∈Zissuchacommonzeroof f′andg′\ninCwith different multiplicities for f′andg′,but (f(zj)−al)(g(zj)−al)/\\e}atio\\slash= 0\nwith 1≤j≤q1andj∈Zfor alll∈ {1,2,3,4}whenq1≥1 andq1∈Z.By\n(22) and (27) we have\n(28)f′′\nf′−f′\nf−f′\nf−1−f′\nf−c−g′′\ng′+g′\ng+g′\ng−1+g′\ng−c=P1+q1/summationdisplay\nj=1mj\nz−zj.\nWe integrate both sides of (28), and then we have\n(29)f′g(g−1)(g−c)\ng′f(f−1)(f−c)=ReP,MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 19\nwhereRis reduced to the constant 1 when q1= 0,andR(z) =q1/producttext\nj=1(z−zj)mj\nwhenq1≥1,whilePis a complex constant or a non-constant polynomial\nsuch that\n(30) P=/integraldisplayz\n0P1(η)dη+A,\nwhereAis a finite complex constant. Next we rewrite (29) into\n(31)f′\nf(f−1)(f−c)=RePg′\ng(g−1)(g−c).\nOn the other hand, from (21), Lemma 2(i), Lemma 3(ii), Lemma 4 (i),\nRemark 2-Remark 4 and the assumption that fandgare non-constant\nmeromorphic functions that share 0 ,1, cIM and∞CM, we have from we\nhave\n(32)\n2T(r,f) =N(r,f)+N/parenleftbigg\nr,1\nf/parenrightbigg\n+N/parenleftbigg\nr,1\nf−1/parenrightbigg\n+N/parenleftbigg\nr,1\nf−c/parenrightbigg\n+O(logr),\nand\n(33)\nN∗(r,0;f,g)+N∗(r,1;f,g)+N∗(r,c;f,g)+N(2(r,f) =O(logr),asr→ ∞.\nHere and in what follows, N(2(r,f) denotes the counting function of the\nmultiple poles of fin|z|<r.From (32), (33) and Lemma 6 we have\n(34)\n1\n7T(r,f)≤3/summationdisplay\nj=16/summationdisplay\nl=1N(1,l)(r,aj;f,g)+3/summationdisplay\nj=16/summationdisplay\nm=2N(m,1)(r,aj;f,g)+O(logr),\nasr→ ∞.From (31) we consider the following two cases:\nCase 1. Suppose that P1is reduced to the constant 0 .Then, it follows\nfrom (30) that Pis reduced to the finite complex constant A.Therefore,\n(31) can be rewritten into\n(35)f′\nf(f−1)(f−c)=ReAg′\ng(g−1)(g−c).\nAccording to (31), (34) and the assumption that fandgshare 0,1, cIM\nand∞CM, we have a discussion as follows:\nSubcase 1.1. Suppose that\n(36)\nN(1,1)(r,0;f,g)+N(1,1)(r,1;f,g)+N(1,1)(r,c;f,g)/\\e}atio\\slash=O(logr),asr→ ∞.\nHere and in what follows, N(1,1)(r,a;f,g) denotes the reduced counting\nfunction of the common simple zeros of f−aandg−ain|z|< rfor the\nvaluea∈ {0,1,c}.From (36) we suppose, without loss of generality, that\n(37) N(1,1)(r,0;f,g)/\\e}atio\\slash=O(logr),asr→ ∞.20 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nFrom (37) we see that there exists an a infinite sequence {z1,j}∞\nj=1of the\ncommon simple zeros of fandgin the complex plane, such that z1,j→ ∞,\nasj→ ∞.Then, it follows by (35) that R(z1,j)eA= 1 for each j∈Z+.\nCombining this with the fact that Ris a non-vanishing rational function,\nwededucethat ReAisreducedtotheconstant1 ,andso(35)canberewritten\ninto\n(38)f′\nf(f−1)(f−c)=g′\ng(g−1)(g−c).\nFrom (38) and the assumption that fandgshare 0,1,cIM we deduce that\nfandgshare 0,1, cCM. This together with the assumption that fandg\nshare∞CM, we get the assertion of Theorem 9.\nSubcase 1.2. Supposethat thereexists somepositive integer ksatisfying\n2≤k≤6,such that\n(39)\nN(1,k)(r,0;f,g)+N(1,k)(r,1;f,g)+N(1,k)(r,c;f,g)/\\e}atio\\slash=O(logr),asr→ ∞,\nwhere and in what follows, N(1,k)(r,a;f,g) denotes the reduced counting\nfunction of those common zeros of f−aandg−ain|z|<rthat are simple\nzeros off−a,and are zeros of g−aof multiplicity kfora∈ {0,1,c}.From\n(39) we suppose, without loss of generality, that\n(40) N(1,k)(r,0;f,g)/\\e}atio\\slash=O(logr),asr→ ∞.\nFrom (40) we see that there exists an infinite sequence {z2,j}∞\nj=1of those\ncommon zeros of fandgin the complex plane that are simple zeros of f,\nand are zeros of gof multiplicity k.Combining this with (35), we deduce\n(41) R(z2,j)eA=1\nkfor eachj∈Z+.\nSinceRis a non-vanishing rational function, we deduce from (41) th atReA\nis reduced to the constant 1 /k,and so (35) can be rewritten into\n(42)kf′\nf(f−1)(f−c)=g′\ng(g−1)(g−c).\nFrom (42), theassumptionthat fandgshare0,1,cIM, andthesupposition\nthatkis a positive integer satisfying 2 ≤k≤6,we deduce that if z0,jis a\nzero off−ajof multiplicity ν0,jfor 1≤j≤3 andj∈Z,thenz0,jis a zero\nofg−ajof multiplicity kν0,jfor 1≤j≤3 andj∈Z.This implies that the\nfunctions\n(43) ϕ1=:kf′\nf(f−1)−g′\ng(g−1)\nand\n(44) ϕ2=:kf′\nf(f−c)−g′\ng(g−c)MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 21\nare two entire functions. Therefore, from (21), (43) and Lem ma 4 (i), we\ndeduce\n(45)\nT(r,ϕ1) =m(r,ϕ1) =m/parenleftbigg\nr,kf′\nf(f−1)−g′\ng(g−1)/parenrightbigg\n=m/parenleftbigg\nr,kf′\nf−1−kf′\nf−g′\ng−1+g′\ng/parenrightbigg\n≤m/parenleftbigg\nr,f′\nf−1/parenrightbigg\n+m/parenleftbigg\nr,f′\nf/parenrightbigg\n+m/parenleftbigg\nr,g′\ng−1/parenrightbigg\n+m/parenleftbigg\nr,g′\ng/parenrightbigg\n+O(1)\n=O(logr)\nand\n(46)\nT(r,ϕ2) =m(r,ϕ2) =m/parenleftbigg\nr,kf′\nf(f−c)−g′\ng(g−c)/parenrightbigg\n=m/parenleftbigg\nr,1\nc/parenleftbiggkf′\nf−c−kf′\nf−g′\ng−c+g′\ng/parenrightbigg/parenrightbigg\n≤m/parenleftbigg\nr,f′\nf−c/parenrightbigg\n+m/parenleftbigg\nr,f′\nf/parenrightbigg\n+m/parenleftbigg\nr,g′\ng−c/parenrightbigg\n+m/parenleftbigg\nr,g′\ng/parenrightbigg\n+O(1)\n=O(logr),asr→ ∞.\nFrom (45) and (46) we deduce that either ϕj= 0 withj∈ {1,2}or that\nϕjwithj∈ {1,2}is reduced to a non-vanishing polynomial. Suppose that\nϕ1= 0.Then, it follows by (43) that\n(47)kf′\nf(f−1)=g′\ng(g−1).\nBy (42) and (47) we deduce f=g.This contradicts the assumption that\nfandgare two distinct non-constant meromorphic functions. Simi larly,\nwe get a contradiction from (42) and (44) provided that ϕ2= 0.Next we\nsuppose that ϕ1andϕ2are non-vanishing polynomials. Integrating both\nsides of (43), we get\n(48) logg(f−1)k\nfk(g−1)=/integraldisplayz\n0ϕ1(η)dη+c1,\nwherec1is a finite complex constant. By (48) we have\n(49)\ng(f−1)k\nfk(g−1)=A1eϕ3=:f1withA1=ec1andϕ3(z) =/integraldisplayz\n0ϕ1(η)dη.\nNext we rewrite (44) into\n(50)kf′\nf−c−kf′\nf−g′\ng−c+g′\ng=cϕ2.22 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nIntegrate both sides of (50) and we get\n(51) logg(f−c)k\nfk(g−c)=c/integraldisplayz\n0ϕ2(η)dη+c2,\nwherec2is a finite complex constant. By (51) we have\n(52)\ng(f−c)k\nfk(g−c)=A2eϕ4=:f2withA2=ec2andϕ4(z) =c/integraldisplayz\n0ϕ2(η)dη.\nFrom (42) and the assumption that fandgshare 0,1,∞IM, we deduce\nthat ifz0,j∈Cwith 1≤j≤3 andj∈Zis a common zero of f−ajand\ng−ajwith 1≤j≤3 andj∈Z,thenz0,jis a zero of g−ajof multiplicity\nlarger than or equal to kwith 2≤k≤6 andk∈Zfor 1≤j≤3 and\nj∈Z.Therefore, from (21), Lemma 2(ii), Remark 2, Lemma 4 (i) and t he\nfirst fundamental theorem, we deduce\n2T(r,g) =N(r,g) +N/parenleftbigg\nr,1\ng/parenrightbigg\n+N/parenleftbigg\nr,1\ng−1/parenrightbigg\n+N/parenleftbigg\nr,1\ng−c/parenrightbigg\n+O(logr)\n≤N(r,g) +N/parenleftbigg\nr,1\ng′/parenrightbigg\n+O(logr)\n=N(r,g) +m(r,g′)+N(r,g′)−m/parenleftbigg\nr,1\ng′/parenrightbigg\n+O(logr)+O(1)\n≤2N(r,g)+m(r,g)+N(r,g)−m/parenleftbigg\nr,1\ng′/parenrightbigg\n+m/parenleftbigg\nr,g′\ng/parenrightbigg\n+O(logr)+O(1)≤2N(r,g)+T(r,g)+O(logr),\nand so we have\n(53) T(r,g)≤2N(r,g)+O(logr),asr→ ∞.\nSincefandgare transcendental meromorphic functions such that fandg\nshare∞CM, we deduce by (20), (53) and Lemma 5 that\n(54) lim\nr→∞N0(r,∞;f,g)\nT(r,f)= lim\nr→∞N0(r,∞;f,g)\nT(r,g)= lim\nr→∞N(r,g)\nT(r,g)≥1\n2,\nwhereandinwhatfollows, N0(r,∞;f,g) denotesthereducedcountingfunc-\ntion of the common poles of fandgin|z|<r.\nBy (49) and (52) we deduce that if z12∈Cis a common pole of fandg,\nthenz12is a common zero of f1−1 andf2−1.Combining this with (54)\nand the assumption that fandgshare∞CM, we deduce\n(55)lim\nr→∞N0(r,1;f1,f2)\nT(r,f)= lim\nr→∞N0(r,1;f1,f2)\nT(r,g)≥lim\nr→∞N0(r,∞;f,g)\nT(r,f)\n= lim\nr→∞N0(r,∞;f,g)\nT(r,g)= lim\nr→∞N(r,g)\nT(r,g)≥1\n2.MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 23\nFrom (49), (52) and the Valiron-Mokhon′ko lemma (cf.[28]) we deduce\n(56) N(r,f1)+N(r,f2)+N/parenleftbigg\nr,1\nf1/parenrightbigg\n+N/parenleftbigg\nr,1\nf2/parenrightbigg\n= 0,\n(57)T(r,f1) =T/parenleftbigg\nr,g(f−1)k\nfk(g−1)/parenrightbigg\n≤T/parenleftbigg\nr,g\ng−1/parenrightbigg\n+T/parenleftbigg\nr,(f−1)k\nfk/parenrightbigg\n=kT(r,f)+T(r,g) +O(1)\nand\n(58)T(r,f2) =T/parenleftbigg\nr,g(f−c)k\nfk(g−c)/parenrightbigg\n≤T/parenleftbigg\nr,g\ng−c/parenrightbigg\n+T/parenleftbigg\nr,(f−c)k\nfk/parenrightbigg\n=kT(r,f)+T(r,g)+O(1),\nasr→ ∞.By (57) and (58) we have\n(59)T(r,f1)+T(r,f2)≤2kT(r,f)+2T(r,g) +O(1),asr→ ∞.\nBy (20), (55) and (59) we have\n(60)\nlim\nr→∞N0(r,1;f1,f2)\nT(r,f1)+T(r,f2)≥lim\nr→∞N(r,g)\n2kT(r,f)+2T(r,g) +O(1)≥1\n4k+4.\nBy (56), (60) and Lemma 8, we see that there exist two integers sandt\nwith|s|+|t|>0,such thatfs\n1ft\n2= 1.This together with the definitions of\nf1andf2in (49) and (52) respectively gives\n(61)/parenleftbiggg(f−1)k\nfk(g−1)/parenrightbiggs/parenleftbiggg(f−c)k\nfk(g−c)/parenrightbiggt\n= 1\nFrom (61) we have a discussion as follows:\nSuppose that one of sandtis equal to zero, say s= 0.Thent/\\e}atio\\slash= 0.\nTherefore, (61) can be rewritten into\n(62)/parenleftbiggf−c\nf/parenrightbiggkt\n=/parenleftbiggg−c\ng/parenrightbiggt\n.\nWe take the Nevanlinna′s characteristic functions on both sides of (62), and\nthen use the Valiron-Mokhon′ko lemma (cf.[28]) to get\n(63) k|t|T(r,f) =|t|T(r,g) +O(1),asr→ ∞.\nBy (20), (63) and the supposition that kis a positive integer such that\nk≥2,we haveT(r,f) =O(logr).Combining this with Lemma 5 and the\nsupposition that fis a transcendental meromorphic function in the complex\nplane, we get a contradiction.\nSuppose that s+t= 0.Thens=−t.Moreover, from |s|+|t|>0 we see\nthatsandtare two non-zero inters. Therefore, (61) can be rewritten in to\n(64)/parenleftbiggf−c\nf−1/parenrightbiggkt\n=/parenleftbiggg−c\ng−1/parenrightbiggt\n.24 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nWe take the Nevanlinna′s characteristic functions on both sides of (64) and\nthen use the Valiron-Mokhon′ko lemma (cf.[28]) to get (63). By (20), (63)\nand the supposition that kis a positive integer such that k≥2,we have\nT(r,f) =O(logr).Combining this with Lemma 5 and the supposition that\nfis a transcendental meromorphic function in the complex pla ne, we get a\ncontradiction.\nSuppose that s+t/\\e}atio\\slash= 0.We first rewrite (61) into\n(65)/parenleftbigg(f−1)s(f−c)t\nfs+t/parenrightbiggk\n=(g−1)s(g−c)t\ngs+t.\nWe take the Nevanlinna′s characteristic functions on both sides of (65) and\nthenusethesupposition s+t/\\e}atio\\slash= 0andtheValiron-Mokhon′kolemma(cf.[28])\nto get\n(66) k|s+t|T(r,f) =|s+t|T(r,g)+O(1),asr→ ∞.\nBy (20), (66) and the supposition that kis a positive integer such that\nk≥2,we haveT(r,f) =O(logr).Combining this with Lemma 5 and the\nsupposition that fis a transcendental meromorphic function in the complex\nplane, we get a contradiction.\nSubcase 1.3. Supposethat thereexists somepositive integer ksatisfying\n2≤k≤6,such that\n(67)\nN(k,1)(r,0;f,g)+N(k,1)(r,1;f,g)+N(k,1)(r,c;f,g)/\\e}atio\\slash=O(logr),asr→ ∞.\nwhere and in what follows, N(k,1)(r,a;f,g) denotes the reduced counting\nfunction of those common zeros of f−aandg−ain|z|<rthat are zeros\noff−aof multiplicity k,and are simple zeros of g−afora∈ {0,1,c}.From\n(67) we suppose, without loss of generality, that\n(68) N(k,1)(r,0;f,g)/\\e}atio\\slash=O(logr),asr→ ∞.\nFrom (68) we see that there exists an infinite sequence {z3,j}∞\nj=1of those\ncommon zeros of fandgin the complex plane that are zeros of fof multi-\nplicityk,and simple zeros of g.Combining this with (35), we deduce\n(69) R(z3,j)eA=kfor eachj∈Z+.\nSinceRis a non-vanishing rational function, and Ais a complex constant,\nwe deduce from (69) that ReAis reduced to the constant k,and so (35) can\nbe rewritten into\n(70)f′\nf(f−1)(f−c)=kg′\ng(g−1)(g−c).\nFrom (70), theassumptionthat fandgshare0,1,cIM, andthesupposition\nthatkis a positive integer satisfying 2 ≤k≤6,we deduce that if ˜ z0,jis a\nzero ofg−ajof multiplicity ˜ ν0,jwith 1≤j≤3 andj∈Z,then ˜z0,jis aMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 25\nzero off−ajof multiplicity k˜ν0,jwith 1≤j≤3 andj∈Z.This implies\nthat the functions\n(71) ϕ5=:f′\nf(f−1)−kg′\ng(g−1)\nand\n(72) ϕ6=:f′\nf(f−c)−kg′\ng(g−c)\nare two entire functions. Next we use (21), (71), (72), Lemma 4 (i), the lines\nof (45) and (46) in Subcase 1.2 to deduce that either ϕj= 0 withj∈ {5,6}\northatϕjwithj∈ {5,6}isreducedtoanon-vanishingpolynomial. Suppose\nthat one of ϕ5= 0 andϕ6= 0 holds. Then, we use (70)-(72) and the similar\nreasonings as in Subcase 1.2 to get a contradiction. Next we s uppose that\nϕ5andϕ6are non-vanishing polynomials. Then, we use (71), (72), the the\nsimilar reasonings from the line before (48) to the line of (5 2) to deduce\n(73)(f−1)gk\nf(g−1)k=A3eϕ7=:f3withϕ7(z) =/integraldisplayz\n0ϕ5(η)dη.\nand\n(74)(f−c)gk\nf(g−c)k=A4eϕ8=:f4withcϕ8(z) =/integraldisplayz\n0ϕ6(η)dη,\nwhere and in what follows, A3andA4are two finite and non-zero complex\nconstants. From (70) and the assumption that fandgshare 0,1, cIM,\nwe deduce that if ˜ z0,j∈Cwith 1≤j≤3 andj∈Zis a common zero of\nf−ajandg−ajwith 1≤j≤3 andj∈Z,then ˜z0,jis a zero of f−ajof\nmultiplicity larger than or equal to kwith 2≤k≤6 andk∈Zfor 1≤j≤3\nandj∈Z.Therefore, from (21), Lemma 2(ii), Remark 2, Lemma 4 (i) and\nthe first fundamental theorem, we deduce\n2T(r,f) =N(r,f)+N/parenleftbigg\nr,1\nf/parenrightbigg\n+N/parenleftbigg\nr,1\nf−1/parenrightbigg\n+N/parenleftbigg\nr,1\nf−c/parenrightbigg\n+O(logr)\n≤N(r,f)+N/parenleftbigg\nr,1\nf′/parenrightbigg\n+O(logr)\n=N(r,f)+m(r,f′)+N(r,f′)−m/parenleftbigg\nr,1\nf′/parenrightbigg\n+O(logr)+O(1)\n≤2N(r,f)+m(r,f)+N(r,f)−m/parenleftbigg\nr,1\nf′/parenrightbigg\n+m/parenleftbigg\nr,f′\nf/parenrightbigg\n+O(logr)+O(1)≤2N(r,f)+T(r,f)+O(logr),\nand so we have\n(75) T(r,f)≤2N(r,f)+O(logr),asr→ ∞.26 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nSincefandgare transcendental meromorphic functions such that fandg\nshare∞CM, we deduce by (20), (75) and Lemma 5 that\n(76) lim\nr→∞N0(r,∞;f,g)\nT(r,f)= lim\nr→∞N0(r,∞;f,g)\nT(r,g)= lim\nr→∞N(r,f)\nT(r,f)≥1\n2,\nFrom (73) and (74) we deduce that if z34∈Cis a common pole of fand\ng,thenz34is a common zero of f3−1 andf4−1.Combining this with (76)\nand the assumption that fandgshare∞CM, we deduce\n(77)lim\nr→∞N0(r,1;f3,f4)\nT(r,f)= lim\nr→∞N0(r,1;f3,f4)\nT(r,g)≥lim\nr→∞N0(r,∞;f,g)\nT(r,f)\n= lim\nr→∞N0(r,∞;f,g)\nT(r,g)= lim\nr→∞N(r,f)\nT(r,f)≥1\n2.\nBy (73), (74) and the Valiron-Mokhon′ko lemma (cf.[28]) we deduce\n(78) N(r,f3)+N(r,f4)+N/parenleftbigg\nr,1\nf3/parenrightbigg\n+N/parenleftbigg\nr,1\nf4/parenrightbigg\n= 0,\n(79)T(r,f3) =T/parenleftbigg\nr,(f−1)gk\nf(g−1)k/parenrightbigg\n≤T/parenleftbigg\nr,f−1\nf/parenrightbigg\n+T/parenleftbigg\nr,gk\n(g−1)k/parenrightbigg\n=T(r,f)+kT(r,g) +O(1)\nand\n(80)T(r,f4) =T/parenleftbigg\nr,(f−c)gk\nf(g−c)k/parenrightbigg\n≤T/parenleftbigg\nr,f−c\nf/parenrightbigg\n+T/parenleftbigg\nr,gk\n(g−c)k/parenrightbigg\n=T(r,f)+kT(r,g) +O(1),\nasr→ ∞.By (79) and (80) we have\n(81)T(r,f3)+T(r,f4)≤2T(r,f)+2kT(r,g) +O(1),asr→ ∞.\nBy (20), (77), (81), Lemma 5 and the obtained result that fandgare\ntranscendental meromorphic functions we deduce\n(82) lim\nr→∞N0(r,1;f3,f4)\nT(r,f3)+T(r,f4)≥lim\nr→∞N(r,f)\n2T(r,f)+2kT(r,g)≥1\n4(k+1).\nBy (73), (74), (78), (82) and Lemma 8, we see that there exist s ome two\nintegerssandtwith|s|+|t|>0,such thatfs\n3ft\n4= 1.This together with\nthe definitions of f3andf4in (73) and (74) respectively gives\n(83)/parenleftbigg(f−1)gk\nf(g−1)k/parenrightbiggs/parenleftbigg(f−c)gk\nf(g−c)k/parenrightbiggt\n= 1.\nNext we use (83) and the similar reasonings from the line afte r (61) to the\nend of Subcase 1.2 to get a contradiction.MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 27\nCase 2. Suppose that P1is a non-zero polynomial, then it follows by the\ndefinition of Pin (30) we see that Pis a non-constant polynomial. Next we\nlet\n(84) P(z) =pnzn+pn−1zn−1+···+p1z+p0,\nwherenis a positive integer, and pn, pn−1, ..., p 1, p0withpn/\\e}atio\\slash= 0 are\nfinite complex constants. Next we let pn=|pn|eiθnwith|pn|>0 and\nθn∈[0,2π),and consider the following 2 nangles for any given positive\nnumberεsatisfying 0 <ε<π\n8n:\nSj:−θn\nn+(2j−1)π\n2n+ε<θ<−θn\nn+(2j+1)π\n2n−ε,\nwherejis an integer satisfying 0 ≤j≤2n−1.Then, it follows from (84)\nand Lemma 9 that there exists a positive number R=R(ε) such that for\n|z|=r>R,we have\n(85)\nRe(P(z))>|pn|(1−ε)rnsin(nε),whenz∈Sjandjis an even integer ,\nwhile\n(86)\nRe(P(z))<−|pn|(1−ε)rnsin(nε),whenz∈Sjandjis an odd integer .\nFor convenience we next let\n(87) Ω( αj,ε,βj,ε) ={z∈C:αj,ε<argz <βj,ε}\nand\n(88) Ω(αj,2ε,βj,2ε) ={z∈C:αj,2ε≤argz≤βj,2ε}\nwith\n(89)αj,ε=−θn\nn+(2j−1)π\n2n+εandβj,ε=−θn\nn+(2j+1)π\n2n−ε\nand\n(90)αj,2ε=−θn\nn+(2j−1)π\n2n+2εandβj,2ε=−θn\nn+(2j+1)π\n2n−2ε\nfor each integer jsatisfying 0 ≤j≤2n−1.Then, we have the following\nclaim:\nClaim 2I. Based upontheassumptionsof Theorem9andthesupposition\nof Case 2, there are at most finitely many common zeros of f−alandg−al\ninΩ(αj,2ε,βj,2ε) for each integer l∈ {1,2,3}and each integer jsatisfying\n0≤j≤2n−1.\nWe prove Claim 2I: on the contrary, we suppose that Claim 2I is not\nvalid. Then, there are infinitely many zeros of f−alinΩ(αj,2ε,βj,2ε) for\neach integer l∈ {1,2,3}and each integer jsatisfying 0 ≤j≤2n−1.\nNext we let {z2,l,m}+∞\nm=1⊂Ω(αj,2ε,βj,2ε) be the infinite sequence of all the\nzeros off−alonΩ(αj,2ε,βj,2ε) forl∈ {1,2,3}and the integer jsatisfying\n0≤j≤2n−1,where each distinct point in {z2,l,m}+∞\nm=1is repeated as28 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nmany times as its multiplicity of a zero of f−alwithl∈ {1,2,3},and the\ninfinite sequence of {z2,l,m}+∞\nm=1is arranged according to increasing moduli,\nsuch that\n(91) lim\nm→∞|z2,l,m|=∞\nand\n(92) |z2,l,1| ≤ |z2,l,2| ≤ ··· ≤ |z2,l,m| ≤ ···\nforl∈ {1,2,3}.\nWe consider the following two subcases:\nSubcase 2.1. Suppose that jwith 0≤j≤2n−1 andj∈Zis an\neven integer. Then, by (31), (85), (91), (92) and (33) we dedu ce that there\nexist some positive integer Nl,jand a large positive number Rl,jsatisfying\nRl,j≥ Rforl∈ {1,2,3}and the even integer jsatisfying 0 ≤j≤2n−1,\nsuch that\n(93) |z2,l,m|>Rl,j,asm≥Nl,jandm∈Z,\nand such that each point z2,l,msatisfying (93) is not a common multiple zero\noff−alandg−alwithl∈ {1,2,3},and satisfies\n(94)n(z2,l,m) =|R(z2,l,m)eRe(P(z2,l,m))| ≥e|pn|(1−2ε)|z2,l,m|nsin(nε)≥4,\nasm≥Nl,jandm∈Zforl∈ {1,2,3}and the even integer jsatisfying\n0≤j≤2n−1.Here and in what follows, n(z2,l,m) denotes the multiplicity\nofz2,l,mof a zero of f−alforl∈ {1,2,3},the positive integer msatisfying\nm≥Nl,jand the even integer jsatisfying 0 ≤j≤2n−1.Therefore,\nfrom (31), (33)-(94) and the assumption that fandgshare 0,1 andcIM,\nwe deduce that z2,l,mis also a simple zero of g−alforl∈ {1,2,3},the\npositive integer msatisfyingm≥Nl,j,and the even integer jsatisfying\n0≤j≤2n−1.Next we use Lemma 11 for Ω(αj,2ε,βj,2ε) to deduce\n(95)Sαj,2ε,βj,2ε(r,f)≤Cαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−a1/parenrightbigg\n+Cαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−a2/parenrightbigg\n+Cαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−a3/parenrightbigg\n+Rαj,2εβj,2ε(r,f)\nfor the positive number rsuch thatr>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.From\n(93), (94) and Lemma 10 we have\n(96)Cαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−al/parenrightbigg\n=/parenleftbigg\nCαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−al/parenrightbigg\n−Cαj,2ε,βj,2ε/parenleftbigg\nRl,j,1\nf−al/parenrightbigg/parenrightbigg\n+Cαj,2ε,βj,2ε/parenleftbigg\nRl,j,1\nf−al/parenrightbigg\n≤1\n4Cαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−al/parenrightbigg\n+O(1)MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 29\n≤1\n4Sαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−al/parenrightbigg\n+O(1) =1\n4Sαj,2ε,βj,2ε(r,f)+O(1)\nfor thel∈ {1,2,3},the even integer jsatisfying 0 ≤j≤2n−1,and the\npositive number rsuch thatr >max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.From (21)\nand Remark 5 we have\n(97) Rαj,2εβj,2ε(r,f) =O(logr)\nfor the even integer jsatisfying 0 ≤j≤2n−1,and the positive number r\nsuch thatr>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.From (95)-(97) we deduce\n(98) Sαj,2ε,βj,2ε(r,f) =O(logr)\nfor the even integer jsatisfying 0 ≤j≤2n−1,and the positive number\nrsuch thatr >max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.Next we use Lemma 10 and\nthe reasonings in the proof of Lemma 2.2.2 in [55, pp.53-54] t o deduce\n(99)\nSαj,2ε,βj,2ε(r,f)+O(1) =Sαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−al/parenrightbigg\n≥Cαj,2ε,βj,2ε/parenleftbigg\nr,1\nf−al/parenrightbigg\n≥2sin/parenleftbig\n2εωαj,2ε/parenrightbig/summationdisplay\n|z2,l,m|∈Ω(αj,2ε,βj,2ε)/parenleftbigg1\n|z2,l,m|ωj,2ε−|z2,l,m|ωj,2ε\nr2ωj,2ε/parenrightbigg\n+O(1)\n≥2sin(2εωj,2ε)/summationdisplay\n|z2,l,m|>Rl,j/parenleftbigg1\n|z2,l,m|ωj,2ε−|z2,l,m|ωj,2ε\nr2ωj,2ε/parenrightbigg\n+O(1)\n= 2sin(2εωj,2ε)/integraldisplayr\n1/parenleftbigg1\ntωj,2ε−tωj,2ε\nr2ωj,2ε/parenrightbigg\ndn0(t)+O(1)\n= 2ωj,2εsin(2εωj,2ε)/integraldisplayr\n1n0(t)/parenleftbigg1\nt1+ωj,2ε+tωj,2ε−1\nr2ωj,2ε/parenrightbigg\ndt+O(1)\n≥2ωj,2εsin(2εωj,2ε)/integraldisplayr\n11\ntωj,2εdN0(t)+O(1)\n= 2ωj,2εsin(2εωj,2ε)N0(r)\nrωj,2ε+2ω2\nj,2εsin(2εωj,2ε)/integraldisplayr\n1N0(t)\nt1+ωj,2εdt+O(1)\n≥2ωj,2εsin(2εωj,2ε)N0(r)\nrωj,2ε+O(1),\nforl∈ {1,2,3},the even integer jsatisfying 0 ≤j≤2n−1,and the\npositive number rsuch thatr >max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.Here and\nin what follows, ωj,2ε=π\nβj,2ε−αj,2ε=nπ\nπ−4nεfor the integer jsatisfying 0 ≤\nj≤2n−1,whileN0(t) is defined as N0(t) =/integraltextt\n1n/parenleftBig\nu,Ω(αj,2ε,βj,2ε),1\nf−al/parenrightBig\nuduwith\nn/parenleftig\nu,Ω(αj,2ε,βj,2ε),1\nf−al/parenrightig\nbeingthenumberofzerosof f−alforl∈ {1,2,3}\nin Ω(αj,2ε,βj,2ε)∩{z: 1<|z| ≤u}counted with multiplicities. From (98)30 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nand (99) we have\n(100) 2 ωj,2εsin(2εωj,2ε)N0(r)\nrωj,2ε≤O(logr)\nforl∈ {1,2,3}and the positive number rsuch that\nr>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.\nNow we use (93) and (94) to deduce |z2,l,m|>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R\nfor the positive integer msatisfyingm>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZNl,2˜l.Combin-\ning this with (91), we deduce that for any given positive numb errsatisfying\nr >max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈Z≥ R,there exists some positive integer N2,l,m(r)\nwithl∈ {1,2,3}that satisfies\n(101) N2,l,m(r)>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZNl,2˜l,\nsuch that for the positive integer msatisfyingm > N 2,l,m(r) withl∈\n{1,2,3},we have\n(102) |z2,l,m| ≥r>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.\nFrom (102) we have\n(103)N0(2|z2,l,m|) =/integraldisplay2|z2,l,m|\n1n/parenleftig\nt,Ω(αj,2ε,βj,2ε),1\nf−al/parenrightig\ntdt\n≥/integraldisplay2|z2,l,m|\n|z2,l,m|n/parenleftig\nt,Ω(αj,2ε,βj,2ε),1\nf−al/parenrightig\ntdt\n≥n/parenleftbigg\n|z2,l,m|,Ω(αj,2ε,βj,2ε),1\nf−al/parenrightbigg/integraldisplay2|z2,l,m|\n|z2,l,m|1\ntdt\n=n/parenleftbigg\n|z2,l,m|,Ω(αj,2ε,βj,2ε),1\nf−al/parenrightbigg\nlog2≥n(z2,l,m)log2\n≥e|pn|(1−2ε)|z2,l,m|nsin(nε)log2\nfor the positive integer msatisfyingm>N 2,l,m(r) withl∈ {1,2,3}.From\n(100)-(103) we have\n(104)2ωj,2εsin(2εωj,2ε)e|pn|(1−2ε)|z2,l,m|nsin(nε)log2\n(2|z2,l,m|)ωj,2ε\n≤2ωj,2εsin(2εωj,2ε)N0(2|z2,l,m|)\n(2|z2,l,m|)ωj,2ε≤O(log(2|z2,l,m|))\nfor the positive integer msatisfyingm>N 2,l,m(r) withl∈ {1,2,3}.From\n(91), (104) and the known equality ωj,2ε=π\nβj,2ε−αj,2ε=nπ\nπ−4nεwith the evenMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 31\nintegerjsatisfying 0 ≤j≤2n−1,we have\n(105)\nO(1)≥2ωj,2εsin(2εωj,2ε)e|pn|(1−2ε)|z2,l,m|nsin(nε)log2\n(2|z2,l,m|)ωj,2εlog(2|z2,l,m|)→+∞,asm→ ∞\nforl∈ {1,2,3},the even integer jsatisfying 0 ≤j≤2n−1,and the given\npositive number εsatisfying 0 <ε<π\n8n.This is a contradiction.\nSubcase 2.2. Suppose that jwith 0≤j≤2n−1 andj∈Zis an\nodd integer. Then, by (31), (86), (91), (92) and (33) we deduc e that there\nexist some positive integer Nl,jand a large positive number Rl,jsatisfying\nRl,j≥ Rforl∈ {1,2,3}and the odd integer jsatisfying 0 ≤j≤2n−1,\nsuch that (93) holds and each point z2,l,msatisfying (93) is not a common\nmultiple zero of f−alandg−alforl∈ {1,2,3},and each point z2,l,m\nsatisfying (93) satisfies\n(106) ˜ n(z2,l,m) =e−Re(P(z2,l,m))\n|R(z2,l,m)|≥e|pn|(1−2ε)|z2,l,m|nsin(nε)≥4,\nasm≥Nl,jforl∈ {1,2,3}and the odd integer jsatisfying 0 ≤j≤2n−1.\nHere and in what follows, ˜ n(z2,l,m) denotes the multiplicity of z2,l,mof a\nzero ofg−alforl∈ {1,2,3},the positive integer msatisfyingm≥Nl,j,\nand the odd integer jsatisfying 0 ≤j≤2n−1.From (31), (33), (106) and\nthe assumption that fandgshare 0,1 andcIM, we deduce that z2,l,mis\nalso a simple zero of f−alforl∈ {1,2,3},the positive integer msatisfying\nm≥Nl,j,and the odd integer jsatisfying 0 ≤j≤2n−1.Next we use\nLemma 11 for Ω(αj,2ε,βj,2ε) to deduce\n(107)Sαj,2ε,βj,2ε(r,g)≤Cαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−a1/parenrightbigg\n+Cαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−a2/parenrightbigg\n+Cαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−a3/parenrightbigg\n+Rαj,2εβj,2ε(r,g)\nforr>0 and the odd integer jsatisfying 0 ≤j≤2n−1,From (93), (106)\nand Lemma 10 we have\n(108)Cαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−al/parenrightbigg\n=/parenleftbigg\nCαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−al/parenrightbigg\n−Cαj,2ε,βj,2ε/parenleftbigg\nRl,j,1\ng−al/parenrightbigg/parenrightbigg\n+Cαj,2ε,βj,2ε/parenleftbigg\nRl,j,1\ng−al/parenrightbigg\n≤1\n4Cαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−al/parenrightbigg\n+O(1)\n≤1\n4Sαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−al/parenrightbigg\n+O(1) =1\n4Sαj,2ε,βj,2ε(r,g)+O(1)\nfor thel∈ {1,2,3},the odd integer jsatisfying 0 ≤j≤2n−1,and the\npositive number rsuch thatr >max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.From (21)32 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nand Remark 5 we have\n(109) Rαj,2ε,βj,2ε(r,g) =O(logr)\nfor the odd integer jsatisfying 0 ≤j≤2n−1,and the positive number r\nsuch thatr>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.From (107)-(109) we deduce\n(110) Sαj,2ε,βj,2ε(r,g) =O(logr)\nfor the odd integer jsatisfying 0 ≤j≤2n−1,and the positive number r\nsuch thatr>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.Next we use Lemma 10 and the\nreasonings in the proof of [55, pp.53-54, Lemma 2.2.2] to ded uce\n(111)\nSαj,2ε,βj,2ε(r,g)+O(1) =Sαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−al/parenrightbigg\n≥Cαj,2ε,βj,2ε/parenleftbigg\nr,1\ng−al/parenrightbigg\n≥2sin/parenleftbig\n2εωαj,2ε/parenrightbig/summationdisplay\n|z2,l,m|∈Ω(αj,2ε,βj,2ε)/parenleftbigg1\n|z2,l,m|ωj,2ε−|z2,l,m|ωj,2ε\nr2ωj,2ε/parenrightbigg\n+O(1)\n≥2sin(2εωj,2ε)/summationdisplay\n|z2,l,m|>Rl,j/parenleftbigg1\n|z2,l,m|ωj,2ε−|z2,l,m|ωj,2ε\nr2ωj,2ε/parenrightbigg\n+O(1)\n= 2sin(2εωj,2ε)/integraldisplayr\n1/parenleftbigg1\ntωj,2ε−tωj,2ε\nr2ωj,2ε/parenrightbigg\nd˜n0(t)+O(1)\n= 2ωj,2εsin(2εωj,2ε)/integraldisplayr\n1˜n0(t)/parenleftbigg1\nt1+ωj,2ε+tωj,2ε−1\nr2ωj,2ε/parenrightbigg\ndt+O(1)\n≥2ωj,2εsin(2εωj,2ε)/integraldisplayr\n11\ntωj,2εd˜N0(t)+O(1)\n= 2ωj,2εsin(2εωj,2ε)˜N0(r)\nrωj,2ε+2ω2\nj,2εsin(2εωj,2ε)/integraldisplayr\n1˜N0(t)\nt1+ωj,2εdt+O(1)\n≥2ωj,2εsin(2εωj,2ε)˜N0(r)\nrωj,2ε+O(1)\nforl∈ {1,2,3},the odd integer jsatisfying 0 ≤j≤2n−1,and the\npositive number rsuch thatr >max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.Here and\nin what follows, ωj,2ε=π\nβj,2ε−αj,2ε=nπ\nπ−4nεfor the integer jsatisfying 0 ≤\nj≤2n−1,while˜N0(t) is defined as ˜N0(t) =/integraltextt\n1n/parenleftBig\nu,Ω(αj,2ε,βj,2ε),1\ng−al/parenrightBig\nuduwith\nn/parenleftig\nu,Ω(αj,2ε,βj,2ε),1\ng−al/parenrightig\nbeingthenumberof zeros of g−alforl∈ {1,2,3}\nin Ω(αj,2ε,βj,2ε)∩{z: 1<|z| ≤u}counted with multiplicities. From (110)\nand (111) we have\n(112) 2 ωj,2εsin(2εωj,2ε)˜N0(r)\nrωj,2ε≤O(logr)MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 33\nfortheoddinteger jsatisfying0 ≤j≤2n−1,andthepositivenumber rsuch\nthatr>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l≥ R.Now we use (93) and (106) to deduce\n|z2,l,m|>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈Z≥ Rfor the positive integer msatisfyingm>\nmax\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZNl,2˜l.Combining this with (91), we deduce that for the\npositive number rsatisfyingr>max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈Z≥ R,there exists some\nlarge positive integer N2,l,m(r) withl∈ {1,2,3}that satisfies (101), such\nthat for the positive integer msatisfyingm>N 2,l,m(r) withl∈ {1,2,3}we\nhave (102). From (102) we have\n(113)\n˜N0(2|z2,l,m|) =/integraldisplay2|z2,l,m|\n1n/parenleftig\nt,Ω(αj,2ε,βj,2ε),1\ng−al/parenrightig\ntdt\n≥/integraldisplay2|z2,l,m|\n|z2,l,m|n/parenleftig\nt,Ω(αj,2ε,βj,2ε),1\ng−al/parenrightig\ntdt\n≥n/parenleftbigg\n|z2,l,m|,Ω(αj,2ε,βj,2ε),1\ng−al/parenrightbigg/integraldisplay2|z2,l,m|\n|z2,l,m|1\ntdt\n=n/parenleftbigg\n|z2,l,m|,Ω(αj,2ε,βj,2ε),1\ng−al/parenrightbigg\nlog2≥˜n(z2,l,m)log2\n≥e|pn|(1−2ε)|z2,l,m|nsin(nε)log2\nforl∈ {1,2,3}and the positive integer msatisfyingm > N 2,l,m(r).From\n(101), (102), (112), (113) we have\n(114)2ωj,2εsin(2εωj,2ε)e|pn|(1−2ε)|z2,l,m|nsin(nε)log2\n(2|z2,l,m|)ωj,2ε\n≤2ωj,2εsin(2εωj,2ε)˜N0(2|z2,l,m|)\n(2|z2,l,m|)ωj,2ε≤O(log(2|z2,l,m|))\nfor the odd integer jsatisfying 0 ≤j≤2n−1,and the positive integer\nmsatisfyingm > N 2,l,m(r) withl∈ {1,2,3}.From (91), (114) and the\nknown equality ωj,2ε=π\nβj,2ε−αj,2ε=nπ\nπ−4nεwith the odd integer jsatisfying\n0≤j≤2n−1,we deduce (105) for l∈ {1,2,3},the odd integer jsatisfying\n0≤j≤2n−1,and the given positive number εsatisfying 0 <ε<π\n8n.From\n(105) we get a contradiction. This proves Claim 2I.\nLast, we use Lemma 13, Lemma 14 and Claim 2I to complete the pro of of\nTheorem 9. For this purpose, we have a discussion as follows: suppose that\nlimsup\nr→∞N(r,f)/T(r,f)>4\n5.Then, it follows from Lemma 2.2 in [16] that\nthere exist some real constant λ>4/5 and some set I⊂(0,∞) that has in-\nfinite linear measure such that N(r,f)/T(r,f)≥λfor allr∈I.Combining\nthis with Theorem 7, we see that fandgshare all four values al,a2,a3,a434 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nCM, and so Theorem 9 is valid. Next we suppose that limsup\nr→∞N(r,f)\nT(r,f)≤4\n5.\nThen, it follows from the known equality T(r,f) =m(r,f)+N(r,f) that\n(115) liminf\nr→∞m(r,f)\nT(r,f)≥1\n5.\nFrom (115) we see that there exists some positive number R0satisfying\nR0≥max\nl∈{1,2,3}max\n0≤˜l≤n−1,˜l∈ZRl,2˜l,\nsuch that\n(116) m(r,f)≥/parenleftbigg1\n5−ε0/parenrightbigg\nT(r,f) forr≥R0,\nwhere and in what follows, ε0is a positive constant such that 0 <2ε0<1\n5.\nNext we define the following directional arc:\n(117) γj=/braceleftig\nz∈C:z=reiθwithθ∈[αj,βj] fromαjtoβj/bracerightig\nwith\n(118) αj=−θn\nn+(2j−1)π\n2nandβj=−θn\nn+(2j+1)π\n2n\nfor each integer jsatisfying 0 ≤j≤2n−1.From (116)-(118) we have\n(119)m(r,f) =1\n2π/integraldisplay2π\n0log+|f(reiθ)|dθ=1\n2π2n−1/summationdisplay\nj=0/integraldisplayβj\nαjlog+|f(reiθ)|dθ\n≥/parenleftbigg1\n5−ε0/parenrightbigg\nT(r,f) forr≥R0.\nFrom (117)-(119) we see that there exists some integer jsatisfying 0 ≤j≤\n2n−1,sayj= 0 such that\n(120)1\n2π/integraldisplayβ0\nα0log+|f(reiθ)|dθ≥1\n2n/parenleftbigg1\n5−ε0/parenrightbigg\nT(r,f) forr≥R0.\nNext we set the simply connected domain\n(121)D0,2ε={z∈C:α0+2ε<argz<β0−2εwith 0<|z|<r}\nwith the positive boundary Γ 0,2ε=:L0,2ε+γ0,2ε+L−\n1,2ε,whereL0,2εand\nL1,2εare two oriented segments such that\n(122)L0,2ε:z=tei(−θn\nn−π\n2n+2ε)witht∈[0,r] fromt= 0 tot=r\nand\n(123)L1,2ε:z=tei(−θn\nn+π\n2n−2ε)witht∈[0,r] fromt= 0 tot=r.\nrespectively, and γ0,2εis a directional arc such that\n(124) γ0,2ε:z=reiθwithθ∈[α0,2ε,β0,2ε] fromα0,2εtoβ0,2ε,MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 35\nwhere and in what follows, α0,2εandβ0,2εare defined as (90) for j= 0.Next\nwe also define GD0,2ε(z,a0) withz∈D0,2εas\n(125) GD0,2ε(z,a0) = log1\n|z−a0|+ωD0,2ε(z,a0)\nforz∈D0,2ε.Here and in what follows, εis a small positive number\nsatisfying 0 < ε <1\n8n,anda0∈D0,2εis a complex number such that\nf(a0)/\\e}atio\\slash∈ {a1,a2,a3,∞}.Then, from Remark 6.3 and the definition of the\nproximity function m(D0,2ε,a0,f) offwith the center at a0forD0,2εwe\nhave\n(126)m(D0,2ε,a,f) =1\n2π/integraldisplay\nΓ0,2εlog+|f(ζ)|∂GD0,2ε(ζ,a0)\n∂nds\n=1\n2π/integraldisplay\nL0,2ε+γ0,2ε+L−1\n1,2εlog+|f(ζ)|∂GD0,2ε(ζ,a0)\n∂nds\n≥1\n2π/integraldisplay\nγ0,2εlog+|f(ζ)|∂GD0,2ε(ζ,a0)\n∂nds\nfor the small positive number εsatisfying 0 <ε<1\n8n.Next we also set the\nfollowing transformation\n(127) z=eiα0,2ε˜zβ0,2ε−α0,2ε\n2πwith ˜z∈˜D0,\nwhere and in what follows, ˜D0is a simply connected domain such that\n(128) ˜D0=:{˜z∈C: 0<|˜z|<˜Rand 0<arg˜z<2π},\nwhere˜Ris a positive number satisfying\n(129) ˜R=r2π\nβ0,2ε−α0,2ε,\nand the positive boundary of the simply connected domain ˜D0is˜Γ0=\n˜L0+ ˜γ0+˜L−\n1,where and in what follows, ˜L0and˜L−1\n1are two oriented\nsegments, and ˜ γ0is a directional arc. For detail, we define them as follows:\nThe oriented segment ˜L0is such an oriented segment that for each ˜ζ∈˜L0\nwe have ˜ζ=twitht∈[0,˜R],the starting point of ˜L0is˜ζ= 0,the end point\nof˜L0is˜ζ=˜Rwith arg ˜ζ= 0 and for each ˜ζ∈ {˜z∈R: 0<˜z≤˜R}we have\narg˜ζ= 0.\nThe oriented segment ˜L−1\n1is such an oriented segment that for each ˜ζ∈\n˜L−1\n1we have ˜ζ=twitht∈[0,˜R],the starting point of ˜L−1\n1is˜ζ=˜Rwith\narg˜ζ= 2π,the end point of ˜L−1\n1is˜ζ= 0 and for each ˜ζ∈ {˜z∈R: 0<˜z≤\n˜R}we have arg ˜ζ= 2π.\nThe directional arc ˜ γ0is such a directional arc that for each ˜ζ∈˜γ0we\nhave˜ζ=˜Rei˜θwith˜θ∈[0,2π],the starting point of ˜ γ0is˜ζ=˜Rwith\narg˜ζ= 0,and the end point of ˜ γ0is˜ζ=˜Rwith arg ˜ζ= 2π.36 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nIn addition, from (90) with j= 0 we have\n(130) 0 <β0,2ε−α0,2ε\n2π<1\n2n.\nFrom (127) and (130) we can easily verify that the transforma tion of\n(127) maps the simply connected domain ˜D0of (128) onto the simply con-\nnected domain D0,2εof (121) in a one-to-one mapping manner, and maps\nthe positive boundary ˜Γ0=˜L0+ ˜γ0+˜L−\n1of˜D0into the positive boundary\nΓ0,2ε=L0,2ε+γ0,2ε+L−\n1,2εofD0,2εin a one-to-one mapping manner, where\nthe oriented segment ˜L0,oriented segment ˜L−1\n1and the directional arc ˜ γ0\nare mapped into the oriented segment L0,2ε,the oriented segment L−1\n1,2εand\nthe directional arc γ0,2εrespectively by the transformation of (127) in a one-\nto-one mapping manner. Next we prove that the transformatio n of (127)\nis also an analytic transformation in the simply connected d omain˜D0of\n(128). For this purpose we write the polar form of each point ˜ z∈˜D0into\n(131) ˜ z= ˜rei˜θwith 0<˜r<˜Rand 0<˜θ<2π.\nAccordingly the pole form of the region ˜D0can be rewritten into\n(132) ˜D0={(˜r,˜θ) : 0<˜r<˜Rand 0<˜θ<2π}.\nNext we substitute (131) into (127), and then use (132) to hav e\n(133) z=x(˜r,˜θ)+iy(˜r,˜θ) with (˜r,˜θ)∈˜D0.\nwherex(˜r,˜θ) = Re(z) andy(˜r,˜θ) = Im(z) are two bivar functions in the\nsimply connected domain ˜D0of (128) of the real variables ˜ rand˜θ,such that\n(134) x(˜r,˜θ) = ˜rβ0,2ε−α0,2ε\n2πcos/parenleftbiggβ0,2ε−α0,2ε\n2π˜θ+α0,2ε/parenrightbigg\nand\n(135) y(˜r,˜θ) = ˜rβ0,2ε−α0,2ε\n2πsin/parenleftbiggβ0,2ε−α0,2ε\n2π˜θ+α0,2ε/parenrightbigg\nfor each pair of real numbers (˜ r,˜θ)∈˜D0of (132). Next we can easily verify\nthat the bivar functions x(˜r,˜θ) andy(˜r,˜θ) in (134) and (135) respectively\nhave continuous first-order partial derivatives at each poi nt (˜r,˜θ)∈˜D0of\n(132), and satisfy the polar form of the Cauchy-Riemann part ial differential\nequations\n(136)∂x\n∂˜r=1\n˜r∂y\n∂˜θand∂y\n∂˜r=−1\n˜r∂x\n∂˜θ\nfor each pair of real numbers (˜ r,˜θ)∈˜D0of (132). From (136), the analysis\nabove and the polar form of the sufficient conditions for an ana lytic function\n(cf.[40]), we deduce that the function of (127) is also an ana lytic transforma-\ntion in the simply connected domain ˜D0of (128). Combining this with the\nknown result that the transformation of (127) maps the simpl y connectedMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 37\ndomain ˜D0of (128) onto the simply connected domain D0,2εof (121) in a\none-to-one mapping manner, we see that the univalent transf ormation (127)\non the simply connected domain ˜D0is a univalent and analytic map on\nthe simply connected domain, and maps conformally the simpl y connected\ndomain ˜D0of (128) with the positive boundary ˜Γ0=˜L0+ ˜γ0+˜L−\n1onto\nthe simply connected domain D0,2εof (121) with the positive boundary\nΓ0,2ε=L0,2ε+γ0,2ε+L−\n1,2ε.For convinience of the following discussion, we\ndenote the univalent and analytic transformation (127) as\n(137) z=˜φ˜a0(˜z) with ˜z∈˜D0,\nand there exists a unique point ˜ a0∈˜D0satisfying\n(138) a0=˜φ˜a0(˜a0),\nsuch that the univalent and analytic mapping (137) maps conf ormally the\nsimply connected domain ˜D0of (128) with the positive boundary ˜Γ0=\n˜L0+ ˜γ0+˜L−\n1onto the simply connected domain D0,2εof (121) with the\npositive boundary Γ 0,2ε=L0,2ε+γ0,2ε+L−\n1,2ε.\nOntheotherhand,fromtheRiemannmappingtheorem(cf.[2, p .230,Theorem\n1]) we also see that there exists a unique univalent and analy tic function on\nthe simply connected domain D0,2εof (121), say\n(139) w=φa0(z) forz∈D0,2εwithφa0(a0) = 0,\nsuch that the univalent and analytic transformation (139) m aps conformally\nthe simply connected domain D0,2εof (121) with the positive boundary\nΓ0,2ε=L0,2ε+γ0,2ε+L−\n1,2εonto the open unit disk D={w∈C:|w|<1}\nwiththepositiveboundary ∂D:w=eiΨwithΨ∈[0,2π],wherethestarting\npoint of∂Disw= 1 with arg w= 0,and the end point of ∂Disw= 1 with\nargw= 2π.From (137)-(139) we deduce that the composition\n(140) w= Φ˜a0(˜z) =:φa0◦˜φ˜a0(˜z) with Φ ˜a0(˜a0) = 0 for ˜z∈˜D0\nof the univalent and analytic transformation in (139) and th e univalent and\nanalytic transformation in (137) maps conformally the simp ly connected\ndomain ˜D0of (128) with the positive boundary ˜Γ0=˜L0+ ˜γ0+˜L−\n1onto\nthe open unit disk D={w∈C:|w|<1}with the positive boundary ∂D\nmentioned above, and maps the positive boundary ˜Γ0=˜L0+ ˜γ0+˜L−\n1of\n˜D0into the positive boundary ∂Dof the open unit disk Dmentioned above\nin a one-to-one mapping manner. As in (9) of Remark 6.1, we ded uce from\n(128) and (140) that the univalent and analytic transformat ion of (140) is\nsuch that\n(141)\nΦ˜a0(˜z) =˜R(˜z−˜a0)\n˜R2−˜a0˜zandG˜D0(˜z,˜a0) =−log|Φ˜a0(˜z)|= log/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜R2−˜a0˜z\n˜R(˜z−˜a0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle38 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nfor ˜z∈˜D0∪∂˜D0=/braceleftig\n˜z∈C:|˜z| ≤˜R/bracerightig\n.As mentioned as above, ˜ γ0is such a\ndirectional arc that for each ˜ζ∈˜γ0we have ˜ζ=˜Rei˜θwith˜θ∈[0,2π],the\nstarting point of ˜ γ0is˜ζ=˜Rwith arg ˜ζ= 0,and the end point of ˜ γ0is˜ζ=˜R\nwith arg ˜ζ= 2π.Furthermore, the univalent and analytic transformation of\n(137) maps the directional arc ˜ γ0into the directional arc γ0,2εof (124).\nTherefore, we use (124), (137), (138), (139), (140), (141), the invariance of\nfirst order differential form, and the similar reasonings of (1 0) in Remark\n6.1 to have\n(142)\ndlogφa0(ζ) =∂logφa0(ζ)\n∂sds=−i∂log|φa0(ζ)|\n∂nds=i∂GD0,2ε(ζ,a0)\n∂nds=\ndlogΦ˜a0/parenleftig\n˜ζ/parenrightig\n=∂logΦ˜a0/parenleftig\n˜ζ/parenrightig\n∂˜sd˜s=−i∂log/vextendsingle/vextendsingle/vextendsingleΦ˜a0/parenleftig\n˜ζ/parenrightig/vextendsingle/vextendsingle/vextendsingle\n∂˜nd˜s=i∂G˜D0/parenleftig\n˜ζ,˜a0/parenrightig\n∂˜nd˜s\nfor each ˜ζ∈˜γ0\\{˜R}withζ=˜φ˜a0(˜ζ).Here and in what follows, for each\n˜ζ∈˜γ0\\ {˜R},we denote by ˜sthe tangent vector along the part positive\nboundary ˜γ0\\ {˜R}of˜D0at the point ˜ζ∈˜γ0\\ {˜R},and denote by ˜nthe\ncorresponding inner normal of the part positive boundary ˜ γ0\\{˜R}of˜D0at\nthepoint ˜ζ∈˜γ0\\{˜R}.Accordingly, foreach ζ=˜φ˜a0(˜ζ)∈γ0,2ε\\{α0,2ε,β0,2ε},\nwe denote by sthe tangent vector along the part positive boundary γ0,2ε\\\n{α0,2ε,β0,2ε}ofD0,2εat the point ζ=˜φ˜a0(˜ζ)∈γ0,2ε\\ {α0,2ε,β0,2ε},and\ndenote by nthe corresponding inner normal of the part positive boundar y\nγ0,2ε\\{α0,2ε,β0,2ε}ofD0,2εat the point ζ=˜φ˜a0(˜ζ)∈γ0,2ε\\{α0,2ε,β0,2ε}.\nFurthermore, from (141) and (142) we have\n(143)\n∂GD0,2ε(ζ,a0)\n∂nds=∂G˜D0/parenleftig\n˜ζ,˜a0/parenrightig\n∂˜nd˜s=˜R2−|˜a0|2\n|˜ζ−˜a0|2d˜θ= Re/parenleftigg˜Re˜θ+˜a0\n˜Re˜θ−˜a0/parenrightigg\nd˜θ\nandforeachcomplexnumber ˜ζ=˜Rei˜θ∈˜γ0\\{˜R}with˜θ∈(0,2π).Therefore,\nfrom (120), (126), (129), (137), (138), (143), (90) and (118 ) withj= 0,\nRemark 6.3, the supposition f(a0)/\\e}atio\\slash∈ {a1,a2,a3,∞},the definition of the\ndirectional arc γ0,2εof (124), the definition of the the directional arc ˜ γ0\nfrom Line 12 to Line 14 after (129), and the definitions of the p roximity\nfunction and the Nevanlinna characteristic function of fwith the center at\na0forD0,2ε,we deduce\n(144)\nT(D0,2ε,a0,f)≥m(D0,2ε,a0,f) =1\n2π/integraldisplay\nΓ0,2εlog+|f(ζ)|∂GD0,2ε(ζ,a0)\n∂nds\n≥1\n2π/integraldisplay\nγ0,2εlog+|f(ζ)|∂GD0,2ε(ζ,a0)\n∂ndsMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 39\n=1\n2π/integraldisplay\n˜γ0log+|f/parenleftig\n˜φ˜a0(˜ζ)/parenrightig\n|∂G˜D0(˜ζ,˜a0)\n∂˜nd˜s\n=1\n2π/integraldisplay2π\n0log+/vextendsingle/vextendsingle/vextendsinglef/parenleftig\n˜φ˜a0/parenleftig\n˜Rei˜θ/parenrightig/parenrightig/vextendsingle/vextendsingle/vextendsingle˜R2−|˜a0|2\n|˜Rei˜θ−˜a0|2d˜θ\n≥1\n2π˜R2−|˜a0|2\n(˜R+|˜a0|)2/integraldisplay2π\n0log+/vextendsingle/vextendsingle/vextendsinglef/parenleftig\n˜φ˜a0/parenleftig\n˜Rei˜θ/parenrightig/parenrightig/vextendsingle/vextendsingle/vextendsingled˜θ\n=1\n2π˜R2−|˜a0|2\n(˜R+|˜a0|)2/integraldisplayβ0,2ε\nα0,2εlog+/vextendsingle/vextendsingle/vextendsinglef/parenleftig\nreiθ/parenrightig/vextendsingle/vextendsingle/vextendsingledθ\n≥˜R2−|˜a0|2\n(˜R+|˜a0|)21\n2n/parenleftbigg1\n5−2ε0/parenrightbigg\nT(r,f)\nfor the small positive number εsatisfying 0 <ε<1\n8n,the positive number\nε0satisfying 0 <2ε0<1/5,and the large positive number rsatisfying\nr≥R0.On the other hand, from Lemma 13, Claim 2I and the definition of\nthe counting function of fwith the center at a0forD0,2ε,we have\n(145)T(D0,2ε,a0,f)≤3/summationdisplay\nl=1N/parenleftbigg\nD0,2ε,a0,1\nf−al/parenrightbigg\n+S(D0,2ε,a0,f)\n≤S(D0,2ε,a0,f)+O(1),\nwhere\n(146)\nS(D0,2ε,a0,f) =m/parenleftbigg\nD0,2ε,a0,f′\nf/parenrightbigg\n+3/summationdisplay\nl=1m/parenleftbigg\nD0,2ε,a0,f′\nf−al/parenrightbigg\n+3/parenleftbigg\nlog+6\nδ0+log+δ0\n6+log2/parenrightbigg\n+log6−log|f′(a0)|\n+3/summationdisplay\nl=1(log|f(a0)−al|+ε(al,D0,2ε))\nwithδ0= min\n1≤j<k≤3|aj−ak|andε(al,D0,2ε)≤log+|al|+ log2 for l∈\n{1,2,3}.From (146), Claim 2I and Lemma 14 we see that there exists a\nsetEθ⊂[0,2π)\\2n−1/uniontext\nj=0[αj,ε,βj,ε] that has linear measure zero, such that\nifψ0∈2n−1/uniontext\nj=0[αj,ε,βj,ε]⊂[0,2π)\\Eθ,then there is a constant, say R0=\nR(ψ0,a1,a2,a3)>1 that depend only upon ψ0,a1,a2anda3,such that for\nallzsuch that arg z=ψ0and|z| ≥R0,and for each l∈ {1,2,3},we have\n(147)/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′(z)\nf(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |z|ρ−1+εand/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′(z)\nf(z)−al/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |z|ρ−1+ε.40 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nFrom the formula (2.1.1) in [55, p.27], we have\n(148)1\n2π/integraldisplay\n∂D0,2ε∂GD0,2ε(ζ,a0)\n∂nds= 1.\nFrom (147), (148) and the definitions of the proximity functi ons and the\nNevanlinna characteristic functions off′\nfandf′\nf−alforl∈ {1,2,3}with the\ncenter ata0forD0,2ε,we have\n(149)m/parenleftbigg\nD0,2ε,a0,f′\nf/parenrightbigg\n+3/summationdisplay\nl=1m/parenleftbigg\nD0,2ε,a0,f′\nf−al/parenrightbigg\n≤4(ρ−1+ε)logr\nfor|z|=r≥R0with allzsuch that arg z=ψ0∈2n−1/uniontext\nj=0[αj,ε,βj,ε].From\n(146) and (149) we deduce\n(150) S(D0,2ε,a0,f)≤4(ρ−1+ε)logr+O(1)\nfor|z|=r≥R0with allzsuch that arg z=ψ0∈2n−1/uniontext\nj=0[αj,ε,βj,ε].From\n(145) and (150) we have\n(151) T(D0,2ε,a0,f)≤4(ρ−1+ε)logr+O(1)\nfor|z|=r≥R0with allzsuch that arg z=ψ0∈2n−1/uniontext\nj=0[αj,ε,βj,ε].From\n(144) and (151) we have\n(152)˜R2−|˜a0|2\n(˜R+|˜a0|)21\n2n/parenleftbigg1\n5−2ε0/parenrightbigg\nT(r,f)≤4(ρ−1+ε)logr+O(1)\nfor|z|=r≥R0with allzsuch that arg z=ψ0∈2n−1/uniontext\nj=0[αj,ε,βj,ε].From\n(152), Lemma 5, the supposition that ε0is a positive number satisfying\n0<2ε0<1/5 and the known result that fandgare transcendental mero-\nmorphic functions, we get a contradiction. This completes t he proof of\nTheorem 9.\n4.On a question of Gary G. Gundersen concerning the\nnonexistence of two distinct non-constant meromorphic\nfunctions sharing three distinct values DM and a fourth\nvalue CM\nIn 1979, Gary G. Gundersen proposed the following two questi ons in [8]:\nQuestion 3. ([8, p.458]) Do there exist two distinct non-constant mero-\nmorphic functions that share two distinct values CM and shar e the other\ntwo distinct values DM, where the four values are four distin ct values in the\nextended complex plane?MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 41\nQuestion 4. ([8, p.458]) Do there exist two distinct non-constant merom or-\nphic functions that share one value CM and share the other thr ee distinct\nvalues DM, where the four values are four distinct values in t he extended\ncomplex plane?\nIn1983, Gundersen[9, Theorem1]provedthatiftwodistinct non-constant\nmeromorphic functions share two values CM and share two othe r values IM,\nwherethe four values are four distinct values in the extende d complex plane,\nthen the two meromorphic functions share all four values CM. This implies\nthat Gundersen[9, Theorem 1] gave a negative answer to Quest ion 3. But\nQuestion 4 is still open by now. The following question is a sp ecial case of\nQuestion 4:\nQuestion 5. Do there exist two distinct non-constant entire functions t hat\nshare three distinct finite values DM?\nInthisdirection, werecallthefollowingresultinMues[30 ]thatcompletely\nresolved Question 5:\nTheorem 10. ([30, pp.109-117]) Theredonotexisttwodistinctnon-cons tant\nentire functions that share three distinct finite values a1, a2, a3DM.\nFor the existence of three distinct non-constant meromorph ic functions\nsharing four distinct values IM, where the four distinct sha red values are\nin the extended complex plane that are neither CM shared valu es nor DM\nshared values, we recall the following result due to Steinme tz[42]:\nTheorem 11. ([42, Theorem 2]) Let abe a cube root of unity such that\na/\\e}atio\\slash= 1,and letγbe a non-constant entire function. Then, every solution of\nthe differential equation\n(153)/parenleftbiggdU\ndz/parenrightbigg2\n= 4(γ′)2U(U+1)(U−a)\nisameromorphicfunctioninthecomplex plane, andif Uisanarbitrarynon-\nconstant solution of the differential equation (153), then th ere exist three\ndistinct meromorphic functions in the complex plane which a re solutions of\nthe algebraic equation P(U(z),w) = 0 with\nP(x,y) =y3−3((a−1)x2−2x)y2−3(2x2−(a−1)x)y−x3,\nand share the four values 0 ,1,−a,∞IM.\nFor the existence of two distinct non-constant meromorphic functions\nsharing four distinct values DM, where the four distinct sha red values are in\ntheextendedcomplexplane, werecallthefollowingresultd uetoReinders[37]:\nTheorem 12. ([37, Theorem 1]) let\nF=U′\n8√\n3·U\nU+1andG=U′\n8√\n3·U+4\n(U+1)2,42 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nwhereUis a non-constant solution of the differential equation\n(U′)2= 12U(U+1)(U+4).\nThenFandGshare the four values 0 ,1,−1,∞,and everybj-point ofF\nandGfor eachj∈ {1,2,3,4}withb1= 0, b2= 1, b3=−1 andb4=∞is\neither simple for Fand triple for Gor triple for Fand simple for G.\nIn thispaper, we will completely resolve Question 4. Indeed , wewill prove\nthe following result that gives a negative answer to Questio n 4:\nTheorem 13. There do not exist two distinct non-constant meromorphic\nfunctions that share a1,a2,a3DM anda4CM, where a1,a2,a3,a4are four\ndistinct complex values in the extended complex plane.\nRemark 9. Theorem 13 directly improves Theorem 10.\nFrom Example 1 above and the following example we see that the number\nof the three distinct DM shared values and the number of one CM shared\nvalue in Theorem 13, in a sense, are best possible:\nExample 2. ([42, p.94]) Let Pdenote the Weierstrass P-function with a\npair of primitive periods 2 ω−ω′andω+ω′.Then\nF(z) =(P(z)−P(ω/2))(P(z)−P(3ω/2))2\nP(z)−P(ω), G(z) =F(z+ω)\nandH(z) =F(z+ω′) share the value 0 , a1, a2,∞IM, where a1and\na2are two distinct complex numbers such that a1/a2witha1/a2/\\e}atio\\slash=−1\nis a cube root of unity, while ωandω′are two complex numbers such\nthat 2ω−ω′andω+ω′are linearly independent on R.That is to say, for\neach pair of real numbers λ1andλ2such that |λ1|+|λ2| /\\e}atio\\slash= 0,we have\nλ1(2ω−ω′)+λ2(ω+ω′)/\\e}atio\\slash= 0.Moreover, we can verify that the four values\n0, a1, a2,∞are neither CM shared values nor DM shared values of F(z),\nG(z) andH(z).\n5.Proof of Theorem 13\nSuppose that the conclusion of Theorem 13 is not valid. Then, there\nexist two distinct non-constant meromorphic functions fandgthat share\na1,a2,a3DM anda4CM. Without loss of generality, we supposethat a1,a2,\na3are three distinct finite values and a4=∞.Combining this with Lemma\n1, we deduce that fandgare two distinct transcendental meromorphic\nfunctions. Next we consider the following two cases:\nCase 1. Supposethat one of fandghas finiteorder, say ρ(f) =ρ<∞.\nThen, from Lemma 2 (i), Remark 2 and Lemma 4 (i) we deduce (20). From\n(20)andthedefinitionoftheorderofameromorphicfunction inthecomplex\nplane in Definition 1 we get (21). Without loss of generality, we also suppose\nthata1= 0, a2= 1, a3=canda4=∞,wherecis a finite value such thatMEROMORPHIC FUNCTIONS SHARING FOUR VALUES 43\nc/\\e}atio\\slash∈ {0,1}.Then, from (21) and the conclusion of Theorem 9, we see that f\nandgshare all the four values a1, a2, a3, a4CM. On the other hand, from\n(21), Remark 2, Lemma 4 (i) and the second fundamental theore m we have\n(154) T(r,f)≤3/summationdisplay\nj=1N/parenleftbigg\nr,1\nf−aj/parenrightbigg\n+O(logr).\nFrom (154), Lemma 5 and the known result that fis a transcendental\nmeromorphic function in the complex plane, we deduce that at least one of\nN/parenleftig\nr,1\nf−a1/parenrightig\n,N/parenleftig\nr,1\nf−a2/parenrightig\nandN/parenleftig\nr,1\nf−a3/parenrightig\n,sayN/parenleftig\nr,1\nf−a1/parenrightig\n,satisfies\n(155) N/parenleftbigg\nr,1\nf−a1/parenrightbigg\n/\\e}atio\\slash=O(logr).\nFrom (155) and the assumption that fandgsharea1DM, we see that there\nare infinitely many common a1-points offandgin the complex plane, and\neach such common a1-point offandghas different multiplicities related to\nfandg.This contradicts the known result that fandgsharea1CM.\nCase 2. Suppose that fandghave infinite order. Then, from Remark\n7 and the definition of the order of a meromorphic function in t he complex\nplane in Definition 1 we deduce that both f#(z) andg#(z) are unbounded\nonC.Next, from Lemma 18 and the supposition that fandgsharea1, a2,\na3DM anda4CM, we get a contradiction. This proves Theorem 13.\n6.Concluding remarks\nFrom Question 1 and Theorem 9 we propose the following conjec ture:\nConjecture 1. Suppose that fandgare two distinct transcendental mero-\nmorphic functions of infinite order, if fandgsharea1, a2, a3IM anda4\nCM, where a1, a2, a3, a4are four distinct complex values in the extended\ncomplex plane, then fandgshare the four values a1, a2, a3, a4CM.\nAcknowledgements. Firstly, the authors wish to express their thanks\nto Prof. Dr. Norbert Steinmetz for spending his valuebale ti me to read the\npresent paper throughly and pose several serious problems c oncerning the\nexistence of the classification for subcase 1.1-Subcase 1.3 and the estimation\nofthedirectionalderivativeoftheGreenfunction GD(z,a)alongthepositive\nboundaryof the simply connected domain in Case 2 of the proof of Theorem\n9, and to communicate with the first author of this paper for se veral times\nto discuss the serious problems. This speeds up the process o f the revision\nof the present paper. Secondly, the authors wish to express t heir thanks to\nProf. Gary G. Gundersenfor spendinghis valuebale timeto re ad thepresent\npaper throughly and propose some constructive opinions and suggestions.\nThis also speeds up the process of the present revised versio n of this paper,\nand causes me to profit life-long.44 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\nReferences\n[1] W. W. 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Reinders, A new characterisation of Gundersen′s example of two meromorphic\nfunctions sharing four values, Results Math. 24(1993) no. 1-2, 174-179.\n[39] J. Schiff, Normal Families, Springer-Verlag, New York, 1993.\n[40] H. Silverman, Polar form of the Cauchy-Riemann equatio ns, PRIMUS: Problems,\nResources, and Issues in Mathematics Undergraduate Studie s, 10(2000), no.3, 241-\n245\n[41] G. D. Song and J.M.Chang, Meromorphic functions sharin g four values, Southeast\nAsian Bull. Math. 26(2002), no.4, 629-635.\n[42] N. Steinmetz, A uniqueness theorem for three meromorph ic functions, Ann. Fenn.\nMath.13(1988), no. 1, 93-110.\n[43] N. Steinmetz, Reminiscence of an open problem: remarks on Nevanlinna′s four-value-\ntheorem, Southeast Asian Bull. Math. 36(2012), no.3, 399-417.\n[44] H. Ueda, Some estimates for meromorphic functions shar ing four values, Kodai Math.\nJ.17(1994), no. 2, 329-340.\n[45] J. P. Wang, Meromorphic functions sharing four values, Indian J. Pure Appl. Math.\n32(2001), no. 1, 37-46.\n[46] S. P. wang, On meromorphic functions that share four val ues, J. Math. Anal. Appl.\n173(1993), no. 2, 359-369.\n[47] J. M. Whittaker, The order of the derivative of a meromor phic function, J. London\nMath. Soc. s1-11(1936), no. 2, 82-87.\n[48] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer\nAcademic Publishers Dordrecht/Boston/London, 2003.46 XIAO-MIN LI1*, QING-FEI ZHAI2, HONG-XUN YI3\n[49] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin Heidelberg, 1993.\n[50] W. H. Yao, Meromorphic functions sharing four small fun ctions on 2CM + 2IM,\nArch. Math. 81(2003), no. 6, 666-677.\n[51] H. X. Yi, On a question of Gross, Sci. China Ser. A, 38(1995), no. 1, 8-16.\n[52] L. Zalcman, A heuristic principle in complex function t heory, Amer. Math. Monthly\n82(1975), no. 8, 813-817.\n[53] Q. C. Zhang, Meromorphic functions sharingthree value s, IndianJ. Pure Appl. Math.\n30(1999), no.7, 667-682.\n[54] J. H. Zheng, On uniqueness of meromorphic functions wit h shared values in one\nangular domain, Complex Var. Theory Appl. 48(2003), no. 9, 777-785.\n[55] J. H. Zheng, Value Distribution of Meromorphic Functio ns, Tsinghua University\nPress, Beijing, 2009.\n1,2Department of Mathematics, Ocean University of China, Qing dao, Shan-\ndong 266100, P. R. China\nEmail:lixiaomin@ouc.edu.cn(X.-M. Li), 1476157537@qq.com(Q.- F. Zhai).\n3Department of Mathematics, ShandongUniversity, Jinan, Sh andong250199,\nP. R. China\nEmail:hxyi@sdu.edu.cn(H.-X. Yi)."    },    {        "title": "2402.03250v2.The_Fefferman_Phong_uncertainty_principle_for_representations_of_Lie_groups_and_applications.pdf",        "content": "arXiv:2402.03250v2  [math.CA]  2 Mar 2024THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE\nFOR REPRESENTATIONS OF LIE GROUPS AND\nAPPLICATIONS\nFABIO NICOLA\nAbstract. We prove a new uncertainty principle for square-integrable\nirreducible unitary representations of connected Lie groups. The con-\ncentrationofthe matrix coefficientsis measured in terms ofweighte dLp\nnorms, with weights in the local Muckenhoupt class A∞,locassociated\nwith a subRiemannian left-invariant metric and a relatively invariant\nmeasure. The result is reminiscent of the Fefferman-Phong uncert ainty\nprinciple, and is new even for the Schr¨ odinger representation of t he re-\nduced Heisenberg group, which corresponds to the short-time Fo urier\ntransform.\nAs an application, we give an optimal estimate of the order of mag-\nnitude of the bottom of the spectrum and of the essential spectr um of\nsemiclassical anti-Wick operators in Rdwith a nonnegative symbol a\nin the class A∞(in particular, for polynomial symbols). Precisely, we\nshow that the infimum inf (x0,ω0)∈R2d−/integraltext\nB((x0,ω0),h1/2)a(x,ω)dxdωrepre-\nsents (up to multiplicative constants) both a lower bound and an upp er\nbound for the bottom of the spectrum, uniformly with respect to h >0.\nSimilarly the quantity\nliminf\n(x0,ω0)→∞−/integraldisplay\nB((x0,ω0),h1/2)a(x,ω)dxdω\nrepresents both a lower bound and an upper bound for the bottom of\nthe essential spectrum, uniformly with respect to h >0. Similar results\nare proved for semiclassical symbol classes.\n1.Introduction and discussion of the main results\n1.1.The Fefferman-Phong uncertainty principle for representa-\ntions.Leth >0 andϕ(x) =π−d/4e−1\n2|x|2,x∈Rd. Consider the so-called\n“coherent states”\n(1.1) ϕh\n(x0,ω0)(x) :=h−d/4ei\nhω0xϕ(h−1/2(x−x0))x,x0,ω0∈Rd.\nIt iswell known that thecelebrated Heisenberg uncertainty princip le canbe\nrephrased in several ways as an inequality involving the functions ϕh\n(x0,ω0).\nFor example, one can show that the following inequality holds true, fo r\nDate: March 5, 2024.\n2010Mathematics Subject Classification. 22E30, 35J10, 47B35, 58J50.\nKey words and phrases. Uncertainty principle, Lie group, representation, anti-Wick\noperator, spectrum, essential spectrum, semiclassical analysis .\n12 FABIO NICOLA\nf∈L2(Rd):\n(1.2) (2 πh)−d/integraldisplay\nR2d(|x|2+|ω|2)|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω≥2dh/ba∇dblf/ba∇dbl2\nL2(Rd).\nMoreover equality occurs if and only if f=cϕh\n(0,0)for some c∈C(see e.g.\n[17, 25, 32] and Remark 4.2 below).\nThe above expression /a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htcan be regarded as a matrix coefficient\nof the Schr¨ odinger representation of the reduced Heisenberg g roup (see\ne.g. [59] and Section 2.6 below). This suggests that similar estimates c an\nhold for other representations of Lie groups, where the concent ration of the\nmatrix coefficients is measured in terms of weighted Lpnorms.\nThis note is devoted precisely to this problem. We will prove (under a\ncertain mild assumption) a similar lower bound for every square-integ rable\nirreducible unitary representation of a connected Lie group G. The role of\nthe weight |x|2+|ω|2in (1.2) will be played by any function in the local\nMuckenhoupt class A∞,locassociated with a subRiemannian left-invariant\nmetric of Gand a relatively invariant measure. In particular, the group G\nitself will play the role of the phase space (we refer to the recent co ntribu-\ntion by Gr¨ ochenig [33] for suggestive speculations about the poss ibility of\nregarding any nilpotent Lie group as a phase space).\nTo state our result precisely, let us introduce some notation and te rmi-\nnology. Let Gbe a connected Lie group, and let µbe a right Haar measure\nofG. Letχ:G→(0,+∞) be a continuous character and let µχbe the\nabsolutely continuous measure that has density χwith respect to µ. In\nparticular, if χis the modular function, µχis a left Haar measure, which\nwill be also denoted by λ.\nLetπ:G→ U(H) be a square-integrable strongly continuous irreducible\nunitaryrepresentationonaHilbertspace H. Hence, thereexists φ∈ H\\{0}\nsatisfying the “admissibility condition”\n(1.3) cφ:=/integraldisplay\nG|/a\\}b∇acketle{tφ,π(x)φ/a\\}b∇acket∇i}ht|2dλ(x)<∞.\nFor such a φ, we define the corresponding generalized wavelet transform\n(also named “voice transform” in the literature)\n(1.4) Vφf(x) =1√cφ/a\\}b∇acketle{tf,π(x)φ/a\\}b∇acket∇i}ht, x∈G\nforf∈ H. It turns out that Vφ:H →L2(G,λ) is an isometry, and that the\nfunction Vφfis continuous on G, for every f∈ H. We observe that many\nof the transforms arising in applied harmonic analysis (e.g. the short -time\nFourier transform, the wavelet transform and the shearlet tran sform) fall\nwithin this general framework (see e.g. [8, 24, 28, 59] and the ref erences\ntherein).\nFor 1≤p <∞, we are interested in the following minimization problem:\n(1.5)I(w) := inf/braceleftBig/integraldisplay\nGw|Vφf|pdµχ:f∈ Hand/ba∇dblVφf/ba∇dblLp(G,µχ)= 1/bracerightBig\n,THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 3\nwherew:G→[0,+∞) is a Borel function.\nLetX1,...,X ℓbe a family of left-invariant linearly independent vec-\ntor fields which satisfy H¨ ormander’s condition, and denote by d C(·,·) the\nassociated left-invariant Carnot–Carath´ eodory distance. Acc ordingly, we\nassume that the weight wbelongs to the local Muckenhoupt class A∞,loc\nassociated with the metric measure space ( G,dC,µχ) (see [9, 30, 55, 56]\nand Section 2 below). Let us briefly recall its definition.\nDefinition 1.1 (Local Muckenhoupt class A∞,loc).Letα∈(0,1). We\ndenote by Aα\n∞,loc(G,dC,µχ)the set of all functions w∈L1\nloc(G,µχ), with\nw >0almost everywhere, such that for all balls Bof radius r∈(0,1]and\nall Borel subsets E⊂Bin the metric measure space (G,dC,µχ),\n(1.6) µχ(E)≥(1−α)µχ(B) =⇒/integraldisplay\nEwdµχ≥α/integraldisplay\nBwdµχ.\nWe also set A∞,loc(G,dC,µχ) =∪0<α<1Aα\n∞,loc(G,dC,µχ).\nHere and in all the following by “ball” we will always mean “open ball”.\nComing back to the problem (1.5), notice that we have the trivial lowe r\nboundI(w)≥infw. We will improve this estimate by replacing the infi-\nmum ofwby the infimum of the averages of wover unit balls, that is\n(1.7) IUP(w) := inf/braceleftBig\n−/integraldisplay\nBwdµχ:Bis a ball of radius 1/bracerightBig\n.\nTo this end, we will assume a mild horizontal regularity and decay con-\ndition on Vφφ, namely that\n(1.8) C0(φ) :=1√cφℓ/summationdisplay\nj=1/integraldisplay\nGχ−1/p|XjVφφ|dλ <∞.\nTheorem 1.2. Let1≤p <∞,α∈(0,1),η >0. There exists a constant\nC=C(α,p,η)>0such that, for every φ∈ H \\ {0}satisfying (1.3)and\n(1.8)withC0(φ)≤η, and every weight w∈Aα\n∞,loc(G,dC,µχ)we have\nI(w)≥CIUP(w).\nThe lower bound in Theorem 1.2 is sharp in general, in the sense that in\ncertain special cases one can prove that the upper bound I(w)≤CIUP(w)\nholds too, as we will see in a moment.\nTheorem 1.2 is reminiscent of the Fefferman-Phong lower bound [21,\n22] for second order pseudodifferential operators with a nonneg ative real\nsymbola(x,ω) (see also Fefferman andS´ anchez-Calle [23], Nagel, Stein and\nWainger [43], Parmeggiani [47], S´ anchez-Calle [51]). Roughly speaking ,\ntheir (deep) result states that the lowest eigenvalue of such an op erator is\nbounded below by\n(1.9) inf\nBsup\n(x,ω)∈Ba(x,ω),\nwhere the infimum is taken over a family of testing boxes Bof measure\nunit in phase space, conveniently built from a. Hence, the geometry of the4 FABIO NICOLA\nphase space is tailored to the given symbol. Notice that in Theorem 1.2\nthe metric is, instead, fixed in advance, which makes things simpler. O n\nthe other hand, since wcould be singular, in (1.7) the average of wappears\n(instead of the supremum in (1.9)).\nAs strategy, we will prove a suitable lower bound for the p-subLaplacian\nwith potential w, at sufficiently small scales, and then we will come back\nto the original problem by using the reproducing property of Vφ. We will\nuse ideas from [22] and also from Jerison [35], where lower bounds we re\nproved for partial differential equations with constant coefficient s (see also\nH¨ ormander [34, Theorem 3.2.5] for an early instance of this philoso phy).\nNotice that the use of weighed Lpnorms as a measure of the time-\nfrequency concentration dates back at least to Cowling and Price [1 4] and\nDaubechies [17]. In particular, in the case of the short-time Fourier trans-\nform and the wavelet transform, and when φis a Gaussian function or a\nCauchy wavelet, optimization problems similar to (1.5) have been rece ntly\ninvestigated, among others, by Abreu, Frank, Galbis, Knutsen, K ulikov,\nOrtega-Cerd` a, Ramos, Riccardi, Romero, Speckbacher, Tilli, Tra passo and\nthe author (see [1, 26, 29, 37, 38, 44, 45, 46, 48, 49] and also Seip [52] for\nearly work in this connection). We emphasize that Theorem 1.2 is new\neven for these particular transforms. Also, we point out that unc ertainty\ninequalities of a different kind, on unimodular Lie groups, were addres sed,\namong others, by Astengo, Ciatti, Cowling, Dall’Ara, Di Blasio, Martin i,\nSundari, Suragan, Ricci, Ruzhansky and Trevisan (see [4, 11, 12, 1 6, 40, 50]\nand the references therein).\n1.2.Applications to Anti-Wick operators. As an application of The-\norem 1.2 (in the case of the Schr¨ odinger representation of the re duced\nHeisenberg group) we provide an optimal estimate of the order of m ag-\nnitude of the bottom of the spectrum and of the essential spectr um of\nanti-Wick operators with nonnegative symbols. This represented, in fact,\nthe original motivation for this note.\nAnti-Wick operators (also named localization, or Toeplitz, or FBI ope r-\nators in the literature) were introduced by Berezin [6], motivated by the\nmathematicalformalismofquantumfieldtheory(secondquantizat ion), and\ncorrespond to a certain ordering of the creation and annihilation op erators\n(see Remark 4.2). Roughly speaking, to a function a(x,ω) in phase space\none associates an operator Aa,honL2(Rd) given by\nAa,hf= (2πh)−d/integraldisplay\nR2da(x,ω)/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htϕh\n(x,ω)dxdω\n(see Section 4 for the precise definition, also for more general coh erent\nstates). Anti-Wick operators appear naturally in semiclassical ana lysis,\ncomplex analysis, mathematical physics and mathematical signal pr ocess-\ning [7, 17, 18, 57, 60, 61], but they play also a fundamental role in the\nstudy of lower bounds of pseudodifferential operators in view of th e factTHE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 5\nthat nonnegative symbols give rise to nonnegative operators; see the pi-\noneering paper by C´ ordoba and Fefferman [13] and Lerner’s book [3 9].\nHowever, in view of the uncertainty principle, we expect that the sp ectrum\nis in fact separated from 0 -for example, although the symbol |x|2+|ω|2\nvanishes at the origin, in (1.2) we have a lower bound of order h. Here\nwe are interested in estimating, in general, the order of magnitude of this\nspectral gap . We briefly summarize our results as follows (see Theorem 5.1\nand Corollary 5.2).\nSuppose that the symbol abelongs to the (Euclidean) Muckenhoupt class\nA∞(R2d)(see Definition 2.8) and let σ(Aa,h)andσess(Aa,h)be the spectrum\nand essential spectrum of Aa,h, respectively. Let\nλ(a,h) := inf/braceleftbigg\n−/integraldisplay\nBa(x,ω)dxdω:Bis a ball of radius h1/2/bracerightbigg\nand\nλess(a,h) := liminf\n(x0,ω0)→∞−/integraldisplay\nB((x0,ω0),h1/2)a(x,ω)dxdω.\nThen we have\nC−1λ(a,h)≤infσ(Aa,h)≤Cλ(a,h)\nand\nC−1λess(a,h)≤infσess(Aa,h)≤Cλess(a,h),\nfor a suitable constant C >0depending only on the A∞bound of a.\nIn particular, this encompasses the case of nonnegative polynomia l sym-\nbols, which was of special interest in [6] (see Remark 4.2). For examp le,\nifa(x,ω) =|x|2+|ω|2we have λ(a,h) =Cdh, for a constant Cddepend-\ning on the dimension, and the above result agrees with (1.1) (see also the\ndiscussion at the end of Remark 4.2).\nAs a consequence, Aa,hhas purely discrete spectrum (i.e. consisting of\nisolated eigenvalues of finite multiplicity) if and only if\nlim\n(x0,ω0)→∞/integraldisplay\nB((x0,ω0),h1/2)a(x,ω)dxdω= +∞\n(here one could equivalently consider balls of any fixed radius R >0).\nThisresultisarefinementofthewellknownsufficientcondition“ a(x,ω)→\n+∞as (x,ω)→ ∞” in Shubin [58] (which is valid, however, for more gen-\neral symbols).\nInSection6wewillshowthatsimilarresultsholdforsemiclassical symb ol\nclasses, modulo an error hN(h∈(0,1]), with Narbitrarily large. We\nemphasize that no assumption is made on the zero set of the symbol . Also,\none could consider a real-valued symbol ain these classes, bounded below,\nand apply the results to a−infa(see Remark 6.4).6 FABIO NICOLA\n1.3.Connection withSchr¨ odinger operators. Theaboveproblemsare\nrelated, at least in spirit, to the positivity and discreteness of the spectrum\nof Schr¨ odinger operators, which have a distinguished tradition go ing back\nat least to the work by Molˇ canov [42] (where the average of the p oten-\ntial over cubes going to infinity already appeared); see also the alre ady\ncited work by Fefferman and Phong [21, 22], Maz’ya and Shubin [41] an d\nSimon [54]. In the case of Muckenhoupt potentials the issue of the dis cret-\nness of the spectrum was addressed by Dall’Ara [15] (see also Ausch er and\nBen Ali [5]), whereas the phase space localization of the eigenfunctio ns is\ncurrently object of study by Decio, De Dios Point, Malinnikova and Ma y-\nboroda [19] (see also Arnold, David, Filoche, Jerison and Mayboroda [3]\nand the references therein). Also, recently Bruno and Calzi [9] co nsidered\na left-invariant subLaplacian on a noncompact connected Lie group with\na potential V∈A∞,loc(G,dC,λ) (with the above notation, where λis a\nleft Haar measure). A characterization of the discreteness of th e spectrum\nwas provided under the hypothesis that the absolutely continuous measure\nthat has density Vwith respect to λis locally doubling. As a byproduct\nof the results in Section 2 we will see that, in fact, this condition is alwa ys\nverified for such a potential (Corollary 2.6). Hence their characte rization\nholds for every V∈A∞,loc(G,dC,λ); we will briefly comment on this in the\nAppendix.\nWe finally observe that it would be natural, at this point, to extend th e\naboveinvestigation torougher symbols, inthespirit oftheresults in [41]for\nSchr¨ odinger operators, also as regards the asymptotics of the eigenvalues.\nWe decided to postpone to a subsequent work a systematic investig ation of\nthese issues, which would take us too far.\n2.Preliminaries results\n2.1.Left-invariant subRiemannian structures. We use the same no-\ntationasintheintroduction. Hence, GisaconnectedLiegroup, X1,...,X ℓ\nare linearly independent left-invariant vector fields satisfying H¨ or mander’s\ncondition and d C(·,·) is the corresponding Carnot–Carath´ eodory distance,\ndefined as follows.\nApiecewise C1curveγ: [0,T]→Gissaidhorizontal ifγ′(t) =/summationtextℓ\nj=1cj(t)Xj\nfor a.e.t, for some functions cj∈L∞([0,T]),j= 1,...,ℓ. We define the\nlength of such a curve as\nL(γ) =/integraldisplayT\n0/parenleftBigℓ/summationdisplay\nj=1cj(t)2/parenrightBig1/2\ndt.\nForx,y∈G, by the H¨ ormander’s condition and the connectedness of G\nthere is at least one horizontal curve γjoiningxandy(γ(0) =x,γ(T) =y)\nand we define\ndC(x,y) = inf\nγL(γ)THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 7\nwhere the infimum is taken over all these horizontal curves. It tur ns out\nthat d C(·,·) is a left-invariant distance on G-the Carnot-Carath´ eodory\ndistance associated with the vector fields X1,...,X ℓ- and is continuous as\na function on G×G. Indeed, it induces on Gthe same manifold topology\nand the balls are relatively compact. Moreover, for any x,y∈Gthere\nexists a length minimizing horizontal curve γjoinigxandy, namely with\nL(γ) = dC(x,y). Such a curve will be called a geodesic. We refer to [2] for\na proof of all these facts.\nWe denote by µa right Haar measure of G, and we write µχfor the\nabsolutely continuous measure that has density χwith respect to µ, where\nχ:G→(0,+∞) is any continuous character. We have\n0< µχ(B(x,r))<∞, x∈G, r >0,\nwhereB(x,r) :={y∈G: dC(x,y)< r}.\nWhileµχis not left-invariant, in general, it is relatively invariant in the\nsense that, for every Borel subset E⊂Gandx∈G,\nµχ(xE) =χ(x)∆(x−1)µχ(E)\nwhere ∆( x) is the modular function.\nIt follows from [43] that µχis locally doubling, in the sense that, for\neveryR >0,\n(2.1) D(R) := sup\nBµχ(2B)\nµχ(B)<∞,\nwhere the supremum is taken over the balls of radius r∈(0,R], and 2B\ndenotes the ball with the same center as Band double radius.\n2.2.Reproducing kernel property. We have defined in (1.4) the gener-\nalized wavelet transform Vφassociated with a square-integrable irreducible\nunitary representation π:G→ U(H), where His a Hilbert space. The\nvectorφ∈ H \\ {0}satisfies the admissibility condition (1.3), where λis\nthe left Haar measure. As already observed there, Vφ:H →L2(G,λ) is an\nisometry ([59, Theorem 7.2]). Hence, writing\nVφf=VφV∗\nφVφf,\nand using that\nV∗\nφF=1√cφ/integraldisplay\nGF(x)π(x)φdλ\nforF∈L2(G,λ) (where cφis defined in (1.3)), we obtain the reproducing\nproperty\n(2.2) Vφf(x) =1√cφVφf∗Vφφ(x),\n(cf. [59, Theorem 7.6] andalso [24]), where theconvolution oftwo fu nctions\nu,vonGis defined as\nu∗v(x) =/integraldisplay\nGu(y)v(y−1x)dλ(y) =/integraldisplay\nGu(xy)v(y−1)dλ(y).8 FABIO NICOLA\nAlso, observe that if Xis a left-invariant vector field,\nX(u∗v) =u∗Xv.\nWe also observe that\nVφφ(x−1) =Vφφ(x).\nHence, if Xis a (real) left-invariant vector field on G,\n(2.3) X(Vφf∗Vφφ)(x) =/integraldisplay\nGVφf(xy)XVφφ(y)dλ(y).\n2.3.Poincar´ e inequality. We will need the local Poincar´ e inequality in\nthe following form. For any ball B⊂Gandu∈L1\nloc(G), we write\n(2.4) uB=−/integraldisplay\nBudµχ:=1\nµχ(B)/integraldisplay\nBudµχ\nfor the average of uoverB. Moreover, let ∇u= (X1u,...,X ℓu) be the\nhorizontal gradient, and set |∇u|= (/summationtextℓ\nj=1|Xju|2)1/2.\nTheorem 2.1 (Poincar´ e inequality) .Let1≤p <∞. For every R >0\nthere existsa constant CR>0suchthat, for everyball Bof radius r∈(0,R]\nand every u∈Lp\nloc(G,µχ), with∇u∈Lp\nloc(G,µχ),\n(2.5)/integraldisplay\nB|∇u|pdµχ≥CRr−p/ba∇dblu−uB/ba∇dblp\nLp(B,µχ).\nProof.The inequality (2.5) was proved in [10, Theorem 3.1] for a general\nnoncompact connected Lie group G, assuming u∈C∞(G) (see also [16] for\na new, beautiful approach to Poincar´ e inequalities on groups). Th e same\nproof in fact holds for Gcompact (and connected).\nForuas in the statement, when proving (2.5) on a given ball B, we can\nsuppose that uhas compact support, hence u,∇u∈Lp(G,µχ). Now, (2.5)\nholds for uε=ϕε∗u, ifϕεis asmoothapproximate identity for L1(G,λ)\n(see e.g. [9, Section 2.2]). On the other hand, as ε→0,\n(uε)B=uB/integraldisplay\nGϕε(y)χ(y)∆(y−1)dλ(y)→uB\n(where ∆ is the unimodular function) because χ(e) = ∆(e) = 1 (where eis\nthe unit of G). Moreover uε→uinLp(G,µχ) and∇uε=ϕε∗∇u→ ∇u\ninLp(G,µχ). Therefore we have the desired conclusion. /square\n2.4.Local doubling property. We collect here some auxiliary results\nrelated to the local doubling property of µχ(cf. (2.1)). We begin with the\nfollowing fact.\nProposition 2.2. For every x0∈G, the function r/ma√sto→µχ(B(x0,r))is\ncontinuous on (0,∞).\nProof.By monotone convergence and since the balls have finite measure,\nthepossiblejumpofthefunctioninthestatement, at r=R, equalsµχ(ER),\nwhereER:={y∈G: dC(x0,y) =R}. Hence we have to prove that\nµχ(ER) = 0.THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 9\nTo this end, we first observe that inthe present setting the differe ntiation\ntheorem holds true: if f∈L1\nloc(G,µχ),\n(2.6) lim\nr→0−/integraldisplay\nB(x,r)f dµχ=f(x)a.e. x∈G.\nThis can be seen by a standard argument. Namely, it is clear that this\nholds iffis continuous. The extension to f∈L1\nloc(G,µχ) follows by the\ndensity of the continuous functions with compact support in L1(G,µχ) (µχ\nis a Radon measure), and the weak type inequality\nµχ({x∈G:M1f(x)> t})≤C\nt/ba∇dblf/ba∇dblL1(G,µχ)t >0\nfor therestricted centered maximal operator\nM1f(x) := sup\n0<r≤1−/integraldisplay\nB(x,r)|f|dµχ.\nThe latter inequality isin turna consequence ofthe local doubling pro perty\nofµχand the Vitali covering argument; see e.g. [56, Theorem 3].\nNow suppose, by contradiction, that µχ(ER)>0. By applying (2.6) to\nf=χERwe see that there exists y0∈ERwith the following property: for\neveryδ∈(0,1) there is 0 < rδ<min{R,1}such that\n(2.7) µχ(ER∩B(y0,rδ))≥δµχ(B(y0,rδ)).\nLetγby a geodesic joining x0andy0, henceL(γ) =R. By the continuity\nof the Carnot–Carath´ eodory distance there exists a point z0on this curve\nsuch that d C(z0,y0) =rδ/2. Clearly, d C(x0,z0) =R−rδ/2, because γis a\ngeodesic. Now, we have B(z0,rδ/4)⊂B(y0,rδ) and\n(2.8) µχ(B(z0,rδ/4))≥C0µχ(B(z0,2rδ))≥C0µχ(B(y0,rδ))\nwhereC0=D(1)−3, cf. (2.1).\nChoosing δ >1−C0weseefrom(2.7)and(2.8)that ER∩B(z0,rδ/4)/\\e}atio\\slash=∅.\nLetv0∈ER∩B(z0,rδ/4). Then\nR= dC(x0,v0)≤dC(x0,z0)+dC(z0,v0)< R−rδ\n2+rδ\n4< R,\nwhich is a contradiction. /square\nProposition 2.3. For every R >0there exists a constant N=NR>0\nsuch that, for every r∈(0,R], there exists a covering of Gmade of balls\nBjof radius rand such that/summationtext\njχBj≤N.\nProof.Consider a maximal family of pairwise disjoint balls B(xj,r/2).\nThen the balls Bj:=B(xj,r) coverG. Ifxbelongs to, say, B(xj,r), with\nj= 1,...,N, thenB(xj,r/2)⊂B(x,2r). On the other hand, since the\nleft Haar measure λis locally doubling ( λ= ∆µ, where ∆ is the modular\nfunction) we have\nλ(B(xj,r/2))≥cRλ(B(xj,2r)) =cRλ(B(x,2r)),\nand therefore NcR≤1. /square10 FABIO NICOLA\n2.5.Muckenhoupt property. We prove some properties of weights in\nthe local Muckenhoupt class Aα\n∞,loc(G,dC,µχ), as defined in Definition 1.1.\nThis complements the analysis in [9].\nThe following basic fact will be crucial in the proof of our main result.\nLemma 2.4. Letα∈(0,1). For every w∈Aα\n∞,loc(G,dC,µχ)we have\nµχ({x∈B:w(x)≥αwB})≥αµχ(B)\nfor every ball Bof radius r∈(0,1].\nProof.If the set\nE:={x∈B:w(x)< αwB}\nhad measure µχ(E)>(1−α)µχ(B), by (1.6) we would have\nα≥αµχ(E)\nµχ(B)>/integraltext\nEwdµχ/integraltext\nBwdµχ≥α,\nwhich is a contradiction. /square\nThe following result completes the analysis in [9] about the relationship s\nbetween several definitionsofthelocalMuckenhoupt class A∞,loc(G,dC,µχ)\nand provides, indeed, a characterization analogous to that valid in t he\nEuclidean case (cf. [30, Theorem 9.3.3]).\nForR >0, letBRbe the collection of open balls of radius Rin (G,dC).\nProposition 2.5. Letw∈L1\nloc(G,µχ), withw >0almost everywhere. Let\nR >0. The following conditions are equivalent.\n(1)There are ε,δ∈(0,1)such that for every ball B∈ BRand every\nBorel subset F⊂B,\nµχ(F)≤εµχ(B) =⇒/integraldisplay\nFwdµχ≤δ/integraldisplay\nBwdµχ.\n(2)There are p∈(1,∞)andC >0such that, for every ball B∈ BR\nand every Borel subset F⊂B,/integraltext\nFwdµχ/integraltext\nBwdµχ≥C/parenleftBigµχ(F)\nµχ(B)/parenrightBigp\n.\n(3)There are p∈(1,∞)andC >0such that, for every ball B∈ BR,\n/parenleftBig\n−/integraldisplay\nBwdµχ/parenrightBig/parenleftBig\n−/integraldisplay\nBw−p′/pdµχ/parenrightBigp/p′\n≤C.\n(4)There are δ,c >0such that, for every ball B∈ BR,\nµχ/parenleftBig/braceleftBig\nx∈G:w(x)≥δ−/integraldisplay\nBwdµχ/bracerightBig/parenrightBig\n≥cµχ(B).\n(5)w∈A∞,loc(G,dC,µχ).\nMore precisely, in the implication (1)⇒(2)one can choose the constants\np,Cin(2)depending only on Rand the constants ǫ,δin(1), and similarly\nfor all the other implications.\nIn particular, if one of the conditions (1),(2),(3),(4)holds for some\nRthen all these conditions hold for every R.THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 11\nProof.It was proved in [9, Proposition 6.1] that (3) ⇒(2)⇒(1). On\nthe other hand, the proof of Proposition 2.4 (taking B∈ BR) shows that\n(1)⇒(4). It follows from [9, Propositions 6.3 and 6.5] that (4) implies\nthat (3) holds for allR, under the additional assumption that the function\nr/ma√sto→µχ(B(x0,r)) is continuous on (0 ,R′] for some R′>0. On the other\nhand we know that this is always the case, by Proposition 2.2. Hence,\nif one of the conditions (1), (2), (3), (4) holds for some Rthen all those\nconditions hold for every R >0.\nFinally, (5) is clearly equivalent to (1) for R= 1.\nThe remark about the dependence of the constants follows by an in spec-\ntion of the proofs.\nActually the above mentioned results in [9] were proved when χis the\nmodular function of G, henceµχis a left Haar measure. However, an\ninspection of those proofs reveals that they extend with minor cha nges to\nevery measure of the type µχ. /square\nCorollary 2.6. Letα∈(0,1). For every R >0there exists a constant\nC=C(α,R)such that, for every w∈Aα\n∞,loc(G,dC,µχ)we have\n/integraldisplay\n2Bwdµχ≤C/integraldisplay\nBwdµχ\nfor all balls Bof radius r∈(0,R].\nProof.This result follows from Proposition 2.5. Alternatively, it follows by\ncombining Proposition 2.2 with [9, Propositions 6.5 and 6.3]. /square\nForr∈(0,1], let\n(2.9) Cr(w) = inf/braceleftBig\n−/integraldisplay\nBwdµχ:Bis a ball of radius r/bracerightBig\n.\nObserve that C1(w) =IUP(w) as defined in (1.7).\nProposition 2.7. Letα∈(0,1). There exist positive constants M1=\nM1(α),κ1=κ1(α)andM2,κ2(depending only on the constant D(1/2)in\n(2.1)) such that, for 0< r′≤r≤1andw∈Aα\n∞,loc(G,dC,µχ),\nM1/parenleftBigr\nr′/parenrightBig−κ1Cr(w)≤Cr′(w)≤M2/parenleftBigr\nr′/parenrightBigκ2Cr(w)\nProof.The desired result follows from (2.1) with R= 1/2 and Corollary\n2.6. /square\n2.6.The Schr¨ odinger representation of the reduced Heisenberg\ngroup.ThereducedHeisenberggroupcanberealizedastheproduct Hd:=\nRd×Rd×R/2πZendowed with the product\n(x,ω,t)·(x′,ω′,t′) = (x+x′,ω+ω′,t+t′−xω′).\nIt turns out to be a connected unimodular Lie group, and a Haar mea sure\nis just the Lebesgue measure (identifying HdwithRd×Rd×[0,2π)). A12 FABIO NICOLA\nbasis of the vector space of left-invariant vector fields is given by\nXj=∂\n∂xj,Ωj=∂\n∂ωj−xj∂\n∂t, T=∂\n∂t\nforj= 1,...,d.\nThe representation πofHdonL2(Rd) given by\n(2.10) π(x,ω,t)f(y) =eiteiωyf(y−x),\nis theSchr¨ odinger representation ofHd. It is unitary, irreducible and\nsquare-integrable. Indeed, every ϕ∈L2(Rd)\\{0}is admissible (cf. (1.3)),\nsince\n(2π)−(d+1)/integraldisplay\nHd|/a\\}b∇acketle{tϕ,π(x,ω,t)ϕ/a\\}b∇acket∇i}ht|2dxdωdt=/ba∇dblϕ/ba∇dbl4\nL2.\nWe refer to [59, Chapter 17] for details.\nInthefollowingwe willconsider theclass A∞,loc(Hd)definedinDefinition\n1.1, with respect to the left-invariant Riemannian distance induced b y the\nabove vector fields, and the Haar measure.\n2.7.Euclidean Muckenhoupt class A∞.We recall the following defi-\nnition of the A∞class inRd, cf. [30, Chapter 9]. We denote by |E|the\nLebesgue measure of a set E.\nDefinition 2.8. Letα∈(0,1). We denote by Aα\n∞(Rd)the set of functions\nw∈L1\nloc(Rd), withw >0almost everywhere, such that for all balls B⊂Rd\nand all Lebesgue-measurable subsets E⊂Bwe have\n|E| ≥(1−α)|B|=⇒/integraldisplay\nEw(x)dx≥α/integraldisplay\nBw(x)dx.\nWe also set A∞(Rd) =∪0<α<1Aα\n∞(Rd).\nWe will also use the notation Aα\n∞,loc(Rd) according to Definition 1.1,\nwhereRdis regarded a metric measure space with the Euclidean distance\nand the Lebesgue measure. The following result will be useful later.\nLemma 2.9. For every α∈(0,1)there exists β∈(0,1)such that, ev-\neryw∈Aα\n∞,loc(R2d), regarded as a function on Hdconstant on the fibers,\nbelongs to Aβ\n∞,loc(Hd).\nMoreover, for every r >0there exists C=C(α,r)>0such that for\neverywas above, and (x0,ω0,t0)∈Hd,\nC−1−/integraldisplay\nB((x0,ω0),r)w(x,ω)dxdω≤ −/integraldisplay\nB((x0,ω0,t0),r)w(x,ω)dxdωdt (2.11)\n≤C−/integraldisplay\nB((x0,ω0),r)w(x,ω)dxdω\nProof.The first statement follows by the following observations. By Propo -\nsition 2.5 we can limit ourselves to check that the condition (3) in Propo si-\ntion 2.5 holds for balls of sufficiently small radius. It is also clear, since t he\nclassAα\n∞,loc(R2d) is translation invariant and wis constant on the fibers,THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 13\nthat by a left translation on Hdwe can reduce ourselves to the case of balls\ncentered at e= (0,0,0) (the unit of Hd).\nNow, setting Br=B(e,r)⊂HdandB′\nr=B((0,0),r)⊂R2dwe easily\nsee that the exist r0>0,c,C >0 such that, for r < r0,\nB′\ncr×(−cr,cr)⊂Br⊂B′\nCr×(−Cr,Cr).\nHence it is enough to check the condition (3) in Proposition 2.5 with the\nballsB=Brreplaced by B′\nr×(−r,r), withrsmall enough, and the\nconclusion is then clear.\nThe proof of the inequalities (2.11) is similar (again it suffices to conside r\nballs of radius rsufficiently small, by Proposition 2.7). /square\nRemark 2.10.For future reference we notice the following elementary re-\nmarks, that are a consequence of the fact that the topology indu ced by the\naboveRiemannianmetricon Hd=R2d×R/2πZistheproducttopologyand\nthe projection Hd→R2dis a continuous and open group homomorphism.\n(a) There exists r0∈(0,1] such that the projection on R2dof ev-\nery ballB((x0,ω0,t0),r0)⊂Hdis contained in the Euclidean ball\nB((x0,ω0),1).\n(b) There exists r1∈(0,1] such that the projection on R2dof every ball\nB((x0,ω0,t0),1)⊂Hdcontains the Euclidean ball B((x0,ω0),r1).\n3.Proof of Theorem 1.2\nThe following result can be regarded as a generalization of the “Main\nLemma” in [22, page 146], and reduces to it when p= 2,G=Rnwith\nthe Euclidean metric and Lebesgue measure, and wis a (nonnegative)\npolynomial.\nLemma 3.1. Let1≤p <∞,α∈(0,1). There exists C=C(p,α)>0\nsuch that, for every every ball Bof radius r∈(0,1], every weight w∈\nAα\n∞,loc(G,dC,µχ)withwB≥r−p, and every u∈Lp\nloc(G,µχ), with∇u∈\nLp\nloc(G,µχ)we have\n(3.1)/integraldisplay\nB(|∇u|p+w|u|p)dµχ≥Cr−p/ba∇dblu/ba∇dblp\nLp(B,µχ).\nProof.By the Poincar´ e inequality (Theorem 2.1) we have\n/integraldisplay\nB|∇u|pdµχ≥Cr−p/ba∇dblu−uB/ba∇dblp\nLp(B,µχ).\nHence it is sufficient to prove that\n(3.2)/integraldisplay\nB(|∇u|p+w|u|p)dµχ≥Cr−p|uB|pµχ(B).\nTo this end, again by the Poincar´ e inequality we have\n(3.3)/integraldisplay\nB(|∇u|p+w|u|p)dµχ≥Cr−p/ba∇dblu−uB/ba∇dblp\nLp(B,µχ)+/integraldisplay\nBw|u|pdµχ.14 FABIO NICOLA\nHence, setting E:={x∈B:w(x)≥αwB}, by Lemma 2.4 we deduce\nthat/integraldisplay\nB(|∇u|p+w|u|p)dµχ≥/integraldisplay\nE(Cr−p|u−uB|p+αwB|u|p)dµχ\n≥/integraldisplay\nE(Cr−p|u−uB|p+αr−p|u|p)dµχ\n≥min{C,α}r−p/integraldisplay\nE|u−uB|p+|u|p)dµχ\n≥2−p+1min{C,α}r−p|uB|pµχ(E)\n≥2−p+1αmin{C,α}r−p|uB|pµχ(B),\nwhich is (3.2). /square\nWe can now prove Theorem 1.2.\nProof of Theorem 1.2. We can suppose IUP(w)>0. LetCr(w) be the\nfunction defined in (2.9). Then by Proposition 2.7 we have Cr(w)>0 for\neveryr∈(0,1].\nWe apply Lemma 3.1 to the weight r−pw/Cr(w), which belongs to the\nclassAα\n∞,loc(G,dC,µχ) for every r∈(0,1]. Hence, for every ball Bof radius\nr∈(0,1] andu∈Lp\nloc(G,µχ), with∇u∈Lp\nloc(G,µχ) we have\n/integraldisplay\nB/parenleftBig\n|∇u|p+r−pw\nCr(w)|u|p/parenrightBig\ndµχ≥Cr−p/ba∇dblu/ba∇dblp\nLp(B,µχ).\nByProposition 2.3there exists a covering of Gmade ofballs ofradius rand\nwith a number of overlaps uniformly bounded with respect to r. Applying\nthe above estimate to each of these balls and summing up we obtain\n(3.4)/integraldisplay\nG/parenleftBig\n|∇u|p+r−pw\nCr(w)|u|p/parenrightBig\ndµχ≥Cr−p/ba∇dblu/ba∇dblp\nLp(G,µχ)\nfor a new constant C >0.\nNow, we apply this estimate with u=Vφffor anyf∈ Hwith\n/ba∇dblVφf/ba∇dblLp(G,dµχ)= 1.\nLet us verify that ∇u∈Lp(G,dµχ).\nUsing (2.2), (2.3) and Minkowski’s inequality, together with\n/integraldisplay\nG|u(xy)|pdµχ(x) =χ(y−1)/ba∇dblu/ba∇dblp\nLp(G,µχ),\nwe see that, for j= 1,...,ℓ,\n/ba∇dblXju/ba∇dblLp(G,µχ)=1√cφ/ba∇dblXj(Vφf∗Vφφ)/ba∇dblLp(G,µχ)≤C′\nj/ba∇dblVφf/ba∇dblLp(G,µχ)=C′\nj,\nwhere\nC′\nj:=1√cφ/integraldisplay\nGχ−1/p|XjVφφ|dλ <∞THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 15\nby the assumption (1.8). As a consequence,/integraldisplay\nG|∇u|pdµχ≤C′′,\nwhich combined with (3.4) yields\n/integraldisplay\nBr−pw\nCr(w)|u|pdµχ≥C′′\nifris chosen small enough, so that Cr−p≥2C′′(andr≤1). Fixing rin\nthis way, we have /integraldisplay\nBw|u|pdµχ≥C′′Cr(w)rp.\nBy Proposition 2.6 we know that\nCr(w)≥C′′′C1(w) =C′′′IUP(w),\nfor some constant C′′′>0. This concludes the proof of Theorem 1.2. /square\n4.Anti-Wick operators\nIn this section we recall the definition and basic properties of anti-W ick\noperators.\n4.1.Anti-Wick operators with a distribution symbol. Leth >0and\nϕ∈ S(Rd), with/ba∇dblϕ/ba∇dblL2(Rd)= 1. Consider the generalized coherent states\nϕh\n(x0,ω0)(x) :=h−d/4ei\nhω0xϕ(h−1/2(x−x0))x,x0,ω0∈Rd.\nObserve that /ba∇dblϕh\n(x0,ω0)/ba∇dblL2(Rd)= 1 for every h >0,x0,ω0∈Rd. When\nϕ(x) =π−d/4e1\n2|x|2we recapture the particular case considered in Introduc-\ntion; cf. (1.1).\nIt is well known that the Parseval formula\n(4.1) (2 πh)−d/integraldisplay\nR2d|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω=/ba∇dblf/ba∇dbl2\nL2\nholds true for every f∈L2(Rd). It is equivalent to the resolution of the\nidentityformula:\nf= (2πh)−d/integraldisplay\nR2d/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htϕh\n(x,ω)dxdω,\nto be understood weakly in L2(Rd); see [18, Proposition 235] and [31, The-\norem 3.2.1]. This motives the following definition.\nDefinition 4.1. Leta∈ S′(R2d). The anti-Wick operator OpAW\nh(a) :\nS(Rd)→ S′(Rd)withh-symbolais given by\n(4.2) OpAW\nh(a)f= (2πh)−d/integraldisplay\nR2da(x,ω)/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htϕh\n(x,ω)dxdω,\nto be understood weakly in the sense of distributions, namel y (with abuse\nof notation) for f,g∈ S(Rd),\n/a\\}b∇acketle{tOpAW\nh(a)f,g/a\\}b∇acket∇i}ht= (2πh)−d/integraldisplay\nR2da(x,ω)/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht/a\\}b∇acketle{tg,ϕh\n(x,ω)/a\\}b∇acket∇i}htdxdω.16 FABIO NICOLA\nIt is easy to see that if f∈ S(Rd) the function /a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htis Schwartz, so\nthat the above definition makes sense.\nRemark4.2.To give the flavor of this quantization procedure, following [6]\nwe compute the anti-Wick operator associated with a polynomial sym bol\na(z) :=/summationdisplay\n|α|+|β|≤mcα,βzαzβ, z:=x+iω.\nConsider the annihilation and creation operators ajanda+\nj,j= 1,...,d,\nrespectively, given (in S(Rd) or even S′(Rd)) by\naj=h∂\n∂xj+xj,a+\nj=−h∂\n∂xj+xj.\nLet us check that\nOpAW\nh(a) =/summationdisplay\n|α|+|β|≤mcα,βaα(a+)β.\nWe observe that, for f∈ S(Rd),\n(xj+iωj)/a\\}b∇acketle{tϕh\n(x,ω),f/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tϕh\n(x,ω),a+\njf/a\\}b∇acket∇i}ht\nand taking the conjugate of both sides,\n(xj−iωj)/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{ta+\njf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht.\nHence, for f,g∈ S(Rd),\n/a\\}b∇acketle{tOpAW\nh(a)f,g/a\\}b∇acket∇i}ht= (2πh)−d/summationdisplay\n|α|+|β|≤mcα,β/integraldisplay\nR2dzαzβ/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht/a\\}b∇acketle{tϕh\n(x,ω),g/a\\}b∇acket∇i}htdxdω\n= (2πh)−d/summationdisplay\n|α|+|β|≤mcα,β/integraldisplay\nR2d/a\\}b∇acketle{t(a+)βf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht/a\\}b∇acketle{tϕh\n(x,ω),(a+)αg/a\\}b∇acket∇i}htdxdω\n=/summationdisplay\n|α|+|β|≤mcα,β/a\\}b∇acketle{t(a+)βf,(a+)αg/a\\}b∇acket∇i}ht,\nwhich gives the desired conclusion.\nIn particular, to the symbol |x|2+|ω|2=|z|2appearing in (1.2) there\ncorresponds the operator\nd/summationdisplay\nj=1aja+\nj=−h2∆+|x|2+dh.\nSince the lowest eigenvalue of the harmonic oscillator −h2∆ +|x|2isdh,\nthis agrees with the lower bound (1.2).\nWe also observe that the class of anti-Wick operators with polynomia l\nsymbols coincides with that of differential operators with polynomial coef-\nficients.THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 17\n4.2.Self-adjoint realizations via quadratic forms. Suppose now that\na(x,ω)is a nonnegative measurable function in R2dwith at most polyno-\nmial growth, in the sense that for some C,N >0,\n(4.3)/integraldisplay\nB(0,r)a(x,ω)dxdω≤C(1+r)N∀r >0.\nConsider the set\nDa,h:={f∈L2(Rd) :/integraldisplay\nR2da(x,ω)|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω <∞}.\nLemma 4.3. The setDa,his a vector space that contains the Schwartz\nclassS(Rd). Moreover Da,his complete with respect to the norm\n/ba∇dblf/ba∇dbl2\na,h:= (2πh)−d/integraldisplay\nR2d(a(x,ω)+1)|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω.\nProof.It is clear that Da,his a vector space. If f∈ S(Rd) we have that\nthe function ( x,ω)/ma√sto→ /a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htis Schwartz; in particular, for every M >0\nthere exists a constant CMsuch that\n|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2≤CM(1+|x|2+|ω|2)−M/2.\nSetting for brevity Φ( x,ω) :=a(x,ω)|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2, we therefore have, for\neveryM >0,\n/integraldisplay\nR2dΦ(x,ω)dxdω=/integraldisplay\nB(0,1)Φ(x,ω)dxdω+∞/summationdisplay\nk=0/integraldisplay\nB(0,2k+1)\\B(0,2k)Φ(x,ω)dxdω\n≤CM/parenleftBig/integraldisplay\nB(0,1)a(x,ω)dxdω+∞/summationdisplay\nk=02−kM/integraldisplay\nB(0,2k+1)\\B(0,2k)a(x,ω)dxdω/parenrightBig\n≤CM/parenleftBig/integraldisplay\nB(0,1)a(x,ω)dxdω+∞/summationdisplay\nk=02−kM/integraldisplay\nB(0,2k+1)a(x,ω)dxdω/parenrightBig\n.\nThis expression is certainly finite if we choose M > N, whereNis the\nconstant in (4.3).\nConcerning the second statement, let fnbe a Cauchy sequence; hence,\nfor every ε >0,\n(4.4)/integraldisplay\nR2d(a(x,ω)+1)|/a\\}b∇acketle{tfn−fm,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω < ε\nforn,m > N ε. Then by (4.1) we see that fnconverges in L2(Rd) to some\nfand therefore /a\\}b∇acketle{tfn,ϕh\n(x,ω)/a\\}b∇acket∇i}ht → /a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htpointwise. Letting m→+∞in\n(4.4) and using the Fatou Lemma we obtain\n/integraldisplay\nR2d(a(x,ω)+1)|/a\\}b∇acketle{tfn−f,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω≤ε\nforn > N ε, namely f∈ Da,hand/ba∇dblfn−f/ba∇dbla,h→0. /square18 FABIO NICOLA\nAs a consequence of the previous lemma, the quadratic form\n(4.5) Qa,h(f) := (2πh)−d/integraldisplay\nR2da(x,ω)|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω,\nwith domain Da,h, is a density defined nonnegative closed form (notice that\n/ba∇dblf/ba∇dbl2\na,h=Qa,h(f)+/ba∇dblf/ba∇dbl2\nL2). Hence it is canonically associated with a unique\nself-adjoint operator Aa,hinL2(Rd) with domain dom Aa,h⊂ Da,hand such\nthat\n(4.6) /a\\}b∇acketle{tAa,hf,g/a\\}b∇acket∇i}ht=Qa,h(f,g)f∈domAa,h, g∈ Da,h,\nwhereQa,h(·,·) is the sesquilinear formcorresponding to the quadratic form\nQa,h(·) (see e.g. [27, Theorem 1.18]). Moreover, denoting by σ(Aa,h) its\nspectrum, we have\ninfσ(Aa,h) = inf\nf∈Da,h\\{0}Qa,h(f)\n/ba∇dblf/ba∇dbl2\nL2\n= inf\nf∈L2(Rd)\\{0}(2πh)−d/integraltext\nR2da(x,ω)|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω\n/ba∇dblf/ba∇dbl2\nL2, (4.7)\nwhere in the last equality we used the fact that if f∈L2(Rd)\\ Da,hthe\nabove integral is + ∞.\nRemark4.4.We claim that S(Rd)⊂domAa,handAa,hf= OpAW\nh(a)ffor\nf∈ S(Rd), where OpAW\nh(a) is defined in Definition 4.1.\nIndeed, we know that the domain dom Aa,hconsists of the functions\nf∈ Da,hsuch that\n|Qa,h(f,g)| ≤C/ba∇dblg/ba∇dblL2∀g∈ Da,h\nfor some C >0. Since|/a\\}b∇acketle{tg,ϕh\n(x,ω)/a\\}b∇acket∇i}ht| ≤ /ba∇dblg/ba∇dblL2by the Cauchy-Schwarz inequal-\nity, we see that the latter estimate is certainly satisfied when the fu nction\na(x,ω)/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}htbelongs to L1(R2d). By arguing as in the proof of Lemma\n4.3, we see that this is the case if f∈ S(Rd). Hence S(Rd)⊂domAa,hand\nthe claim then follows from (4.6).\nHenceAa,his a self-adjoint extension of OpAW\nh(a), regarded as a densely\ndefined symmetric operator in L2(Rd) with domain S(Rd). One could sim-\nilarly consider any closed subspace ˜D⊂ Da,h(with respect to the norm\n/ba∇dbl·/ba∇dbla,h), withS(Rd)⊂˜D, and consequently the selfadjoint operator ˜Aas-\nsociated with the quadratic form Qa,hwith form domain ˜D. In particular,\nif˜Dis the closure of S(Rd) inDa,hthe operator ˜Ais the Friedrichs exten-\nsion. In the subsequent sections we will provide lower and upper bou nds\nfor the bottom of the spectrum and of the essential spectrum of Aa,h, but\nthe results will apply equally to these realizations ˜A. This is true for the\nlower bounds, since the quadratic form associated with Aa,hextends that\nassociated with ˜A, and, as regards the upper bounds, because only the\nvaluesQa,h(f) withf∈ S(Rd) will enter in their proofs.THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 19\nRemark4.5.The space Da,h, endowed with the norm /ba∇dbl·/ba∇dbla,h(andh= 1),\ncoincides with the so-called weighted modulation space denoted in the liter-\nature by M2,2\na+1; see [31, Chapter 11]. However, in the theory of modulation\nspaces the weight (the function 1+ a, in this case) is usually required to sat-\nisfy certain further conditions (see [31, Definition 11.1.1]), whereas here we\nonly assume that ais nonnegative and satisfies the growth condition (4.3).\nUnder those extra assumptions it follows from [31, Proposition 11.3.4 ] that\nS(Rd) is in fact dense in Da,h. However we will not need this property.\n5.The bottom of the spectrum - Muckenhoupt symbols\nWe use the same notation as in Section 4.2. We are interested in esti-\nmating the bottom of the spectrum and of the essential spectrum of the\noperator Aa,h(defined in Section 4.2 via the quadratic form Qa,hin (4.7)\nwith form domain Da,hand a window ϕ∈ S(Rd) with/ba∇dblϕ/ba∇dblL2= 1) in terms\nof the nonnegative symbol a. Here we will consider symbols in the Muck-\nenhoupt class A∞(R2d). We will use (4.7) and Theorem 1.2 applied to\nthe Schr¨ odinger representation πof the reduced Heisenberg group Hd, as\ndefined in Section 2.6.\n5.1.The bottom of the spectrum. We observe that, by rescaling, for\nf∈L2(Rd),\n(2πh)−d/integraldisplay\nR2da(x,ω)|/a\\}b∇acketle{tf,ϕh\n(x,ω)/a\\}b∇acket∇i}ht|2dxdω\n= (2π)−d/integraldisplay\nR2da(h1/2x,h1/2ω)|/a\\}b∇acketle{tDhf,ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω, (5.1)\nwhere we have set Dhf(x) :=hd/4f(h1/2x) for the unitary dilation and\n(5.2) ϕ(x0,ω0)(x) :=eiω0xϕ(x−x0)x,x0,ω0∈Rd,\ni.e.ϕ(x0,ω0)=ϕh\n(x0,ω0)withh= 1. Also observe that for the representation\nπin (2.10) we have π(x,ω,t)ϕ=eitϕ(x,ω). Hence, by (4.7) we can write\ninfσ(Aa,h) (5.3)\n= inf\nf∈L2(Rd)\\{0}(2π)−d/integraltext\nR2da(h1/2x,h1/2ω)|/a\\}b∇acketle{tf,ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω\n/ba∇dblf/ba∇dbl2\nL2\n= inf\nf∈L2(Rd)\\{0}(2π)−(d+1)/integraltext\nHda(h1/2x,h1/2ω)|/a\\}b∇acketle{tf,π(x,ω,t)ϕ/a\\}b∇acket∇i}ht|2dxdωdt\n/ba∇dblf/ba∇dbl2\nL2.\nAlso, let\nλ(a,h) : = inf/braceleftbigg\n−/integraldisplay\nBa(x,ω)dxdω:Bis a ball of radius h1/2/bracerightbigg\n(5.4)\n= inf/braceleftbigg\n−/integraldisplay\nBa(h1/2x,h1/2ω)dxdω:Bis a ball of radius 1/bracerightbigg\n. (5.5)20 FABIO NICOLA\nTheorem 5.1. Letα∈(0,1). There exists C=C(α,ϕ)>0such that,\nfor every a∈Aα\n∞(R2d)andh >0,\n(5.6) C−1λ(a,h)≤infσ(Aa,h)≤Cλ(a,h).\nProof of Theorem 5.1. First of all we notice that asatisfies (4.3) for some\nC,N >0,becausethemeasure a(x,ω)dxdωisdoubling. Moreprecisely, we\nobserve for future reference (see e.g. [30, Theorem 9.3.3]) that there exists\na constant C >0 depending on αanddsuch that for every a∈Aα\n∞(R2d)\nandx0,ω0∈Rd,k≥0,\n(5.7)/integraldisplay\nB((x0,ω0),2k)a(x,ω)dxdω≤Ck/integraldisplay\nB((x0,ω0),1)a(x,ω)dxdω.\nAlso, observe that the function a(h1/2x,h1/2ω) belongs to Aα\n∞(R2d) for\neveryh >0, by the dilation invariance of this class.\nThe first inequality in (5.6) then follows by applying (5.3) and Theorem\n1.2 with p= 2 where:\n• H=L2(Rd),G=Hdis the reduced Heisenberg group, and πis the\nSchr¨ odinger representation of Hd(cf. Section 2.6).\n•φ=ϕ.\n•χ= 1 and µχis therefore the Haar measure of Hd.\n•Theleft-invariantRiemannianmetricon Hdisinduced bythevector\nfieldsXj, Ωj,j= 1,...,dandTin Section 2.6.\n•The weight wonHdis given by w(x,ω,t) =a(h1/2x,h1/2ω).\nIndeed, since a∈Aα\n∞(R2d)⊂Aα\n∞,loc(R2d), by Lemma 2.9 we have w∈\nAβ\n∞,loc(Hd), for some β∈(0,1) depending only on αandd. Also, (1.8) is\nsatisfied because\nVφφ(x,ω,t) = (2π)−(d+1)/2/a\\}b∇acketle{tϕ,π(x,ω,t)ϕ/a\\}b∇acket∇i}ht= (2π)−(d+1)/2e−it/a\\}b∇acketle{tϕ,ϕ(x,ω)/a\\}b∇acket∇i}ht,\nthe function /a\\}b∇acketle{tϕ,ϕ(x,ω)/a\\}b∇acket∇i}htis Schwartz, and Xj,ΩjandThave polynomial\ncoefficients.\nThe last sentence in Lemma 2.9 and (5.5) allow us to transfer to R2dthe\nlower bound on Hdobtained from Theorem 1.2.\nConcerning the second inequality in (5.6), by (5.3) and (5.5) we see\nthat it suffices to prove that, given α∈(0,1), there exists a constant\nC=C(α,ϕ)>0 such that for every x0,ω0∈Rd,a∈Aα\n∞(R2d) andh >0,\n/integraldisplay\nR2da(h1/2x,h1/2ω)|/a\\}b∇acketle{tϕ(x0,ω0),ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω (5.8)\n≤C/integraldisplay\nB((x0,ω0),1)a(h1/2x,h1/2ω)dxdω.\nNamely, it is enough to prove that for every a∈Aα\n∞(R2d)\n/integraldisplay\nR2da(x,ω)|/a\\}b∇acketle{tϕ,ϕ(x−x0,ω−ω0)/a\\}b∇acket∇i}ht|2dxdω\n≤C/integraldisplay\nB((x0,ω0),1)a(x,ω)dxdω (5.9)THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 21\nwith a constant Cdepending only on αandϕ.\nSince the expression /a\\}b∇acketle{tϕ,ϕ(x,ω)/a\\}b∇acket∇i}htis a Schwartz function, the latter estimate\nfollows as in the proof of Lemma 4.3 (with h= 1), considering annuli\ncentered at ( x0,ω0), and using (5.7) to conclude. /square\n5.2.The bottom of the essential spectrum. We now estimate the\nbottom of the essential spectrum σess(Aa,h). We recall its characterization\nin terms of the associated quadratic form (cf. [27, Lemma 1.20]):\ninfσess(Aa,h) = inf{liminf\nn→∞Qa,h(fn) : (fn)⊂ Da,h, (5.10)\n/ba∇dblfn/ba∇dblL2= 1, fn→0 weakly in L2(Rd)}.\nLet\nλess(a,h) : = liminf\n(x0,ω0)→∞−/integraldisplay\nB((x0,ω0),h1/2)a(x,ω)dxdω\n= liminf\n(x0,ω0)→∞−/integraldisplay\nB((x0,ω0),1)a(h1/2x,h1/2ω)dxdω.\nCorollary 5.2. Letα∈(0,1). There exists C(α,ϕ)>0such that, for\neverya∈Aα\n∞(R2d)andh >0,\n(5.11) C−1λess(a,h)≤infσess(Aa,h)≤Cλess(a,h).\nIn particular, σ(Aa,h)is purely discrete if and only if\nlim\n(x0,ω0)→∞/integraldisplay\nB((x0,ω0),h1/2)a(x,ω)dxdω= +∞.\nProof.Let us prove the first inequality in (5.11). We suppose λess(a,h)<\n+∞(which holds for some h >0 if and only if it holds for every h >0,\nbecause the measure a(x,ω)dxdωis doubling). The case λess(a,h) = +∞\nrequires only minor modifications.\nLetR >0 such that\n(5.12) −/integraldisplay\nB((x0,ω0),h1/2)a(x,ω)dxdω≥1\n2λess(a,h)\nfor|x0|2+|ω0|2> R2. Consider the function\nb(x,ω) :=C(1+|x|+|ω|)−1,\nwhere the constant C >0 is chosen so that\n(5.13) b(x,ω)≥1\n2λess(a,h) for |x|2+|ω|2≤(R+h1/2)2.\nObserve that Cwill depend on aandh, butb∈Aβ\n∞(R2d) for some β∈\n(0,1) depending only on d. Indeed, the weight |z|−1,z= (x,ω)∈R2d\nbelongs to A∞(R2d) (see e.g. [55, page 218]) and therefore the same holds\nfor (1 +|x|+|ω|)−1≍(1 +|z|)−1≍min{1,|z|−1}becauseA∞(R2d) is a\nlattice; see e.g. [36, Proposition 4.3]). Hence a+b∈Amin{α,β}\n∞(R2d) and\nDa+b,h=Da,hbecausebis bounded.22 FABIO NICOLA\nWe will check in a moment that\n(5.14) σess(Aa,h) =σess(Aa+b,h).\nApplying Theorem 5.1 to Aa+b,hwe see that, for f∈ Da+b,h=Da,hand\n/ba∇dblf/ba∇dblL2= 1,\n(5.15) Qa+b,h(f)≥Cλ(a+b,h),\ncf. (5.4), for a constant Cdepending only on αandϕ.\nOn the other hand we have\nλ(a+b,h)\n= inf\n(x0,ω0)∈R2d/parenleftBig\n−/integraldisplay\nB((x0,ω0),h1/2)a(x,ω)dxdω+−/integraldisplay\nB((x0,ω0),h1/2)b(x,ω)dxdω/parenrightBig\n≥1\n2λess(a,h),\nby (5.12) and (5.13). Hence, by (5.14), (5.10) (with a+bin place of a) and\n(4.7) (since inf σ(Aa+b,h)≤infσess(Aa+b,h)) we obtain the first inequality\nof (5.11).\nIt remains to check (5.14). This follows from Weyl’s theorem for qua-\ndratic forms (cf. [27, Theorem 1.51]) if we prove that the (nonnega tive)\nquadratic form Qb,h(cf. (4.5)) is compact in Da,h, i.e. that for some C >0,\n(5.16) Qb,h(f)≤C/ba∇dblf/ba∇dbl2\na,h∀f∈ Da,h,\nand that the bounded operator in Da,hinduced by the form Qb,hvia the\nRiesz representation theorem is compact. Now, since bis bounded, the\nquadratic form Qb,hcomes in fact from a bounded operator in L2(Rd) and\nit suffices to prove that this operator is compact in L2(Rd) (cf. [27, page\n56]). But this is well known, since b→0 at infinity (cf. [59, Theorem\n14.5]).\nLet us now prove the second inequality in (5.11). By (5.10), (4.5) and\n(5.1) we see that it suffices to prove that there exists C >0 depending\nonαandϕsuch that, for every a∈Aα\n∞(R2d),h >0 andeverysequence\n(xn,ωn)→ ∞, we have\n/integraldisplay\nR2da(h1/2x,h1/2ω)|/a\\}b∇acketle{tfn,ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω≤C/integraldisplay\nB((xn,ωn),1)a(h1/2x,h1/2ω)dxdω\nforsomesequence fn∈ S(Rd),/ba∇dblfn/ba∇dblL2= 1, with fn→0 weakly in L2(Rd).\nNow, this is true with fn=ϕ(xn,ωn)by (5.8) and, moreover, fn→0 weakly\ninL2(Rd). Indeed, for g∈L2(Rd),/a\\}b∇acketle{tg,ϕ(x,ω)/a\\}b∇acket∇i}ht →0 as (x,ω)→ ∞because\nthis holds if g∈ S(Rd) (/a\\}b∇acketle{tg,ϕ(x,ω)/a\\}b∇acket∇i}htis a Schwartz function, in that case) and\nwe have the uniform bound |/a\\}b∇acketle{tg,ϕ(x,ω)/a\\}b∇acket∇i}ht| ≤ /ba∇dblg/ba∇dblL2by the Cauchy-Schwarz\ninequality. /square\nCorollary 5.3. For every polynomial P(x,ω)inR2dof degree N≥1, and\nβ >−1/N, the conclusions of Theorem 5.1 and Corollary 5.2 hold true f or\na=|P|β, with a constant Cdepending only on N,β,dandϕ.THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 23\nProof.Thisfollowsbyobserving thatsuchafunction abelongsto Aα\n∞(R2d),\nfor some α∈(0,1) depending only on N,dandβ(see [55, page 219]). /square\n6.The bottom of the spectrum - semiclassical symbols\nIn this section we continue the study of the bottom of the spectru m of\nthe anti-Wick operator Aa,hwith a nonnegative symbol ain certain symbol\nclasses arising in semiclassical analysis (cf. [20, page 81], [53, Definit ion\nA.2.2] and [61, Section 4.3]). We moreover suppose h∈(0,1].\nDefinition 6.1. Form∈R,ρ∈[0,1/2), we denote by Sm\nρthe space of the\nfamilies of smooth functions a(·,h)inR2d, depending on a parameter h∈\n(0,1]such that for some N0∈Nand every α,β∈Nd, with|α|+|β| ≥N0,\n(6.1) sup\n(x,ω,h)∈R2d×(0,1]|∂α\nx∂β\nωa(x,ω,h)|hm+ρ(|α|+|β|)<∞.\nWe observe that a finite number of derivatives ∂α\nx∂β\nωaare allowed to be\nundounded.\nNotice that a nonnegative symbol a∈Sm\nρcertainly satisfies the growth\ncondition (4.3). Hence, as in Section 4.2 we can consider the corresp onding\nanti-Wick operator Aa,hdefined via the quadratic form Qa,hin (4.5), for\na window ϕ∈ S(Rd), with/ba∇dblϕ/ba∇dblL2= 1. Observe that Aa,hwill therefore\ndepend on heven through the symbol a.\nLet\n(6.2) λ′(a,h) := inf\n(x0,ω0)∈R2dsup\n(x,ω)∈B((x0,ω0),h1/2)a(x,ω,h).\nTheorem 6.2. Leta∈Sm\nρfor some m∈R,ρ∈[0,1/2), witha≥0.\nFor every N∈Nthere exists CN>0such that, for every h∈(0,1],\n(6.3) C−1\nNλ′(a,h)−CNhN≤infσ(Aa,h)≤CN(λ′(a,h)+hN).\nProof.Let us prove the first inequality in (6.3). By (5.3) we have to prove\nthat, for f∈L2(Rd),/ba∇dblf/ba∇dblL2= 1,\n(6.4)/integraldisplay\nR2da(h1/2x,h1/2ω,h)|/a\\}b∇acketle{tf,ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω≥C−1\nNλ′(a,h)−CNhN,\nwhere the functions ϕ(x,ω)are defined in (5.2).\nWe know that asatisfies (6.1) for |α|+|β| ≥N0, for some N0∈N. For\nN≥0, letM=MN≥0 be the minimum natural number such that/braceleftBigg\nM+1≥N0\n−m+(1\n2−ρ)(M+1)≥N.\nThen\n|∂α\nx∂β\nω(a(h1/2x,h1/2ω,h))| ≤C′\nNhNfor|α|+|β|=M+1.\nLetP(x0,ω0),Mbe the Taylor polynomial of a(h1/2x,h1/2ω,h) of order M=\nMNcentered at ( x0,ω0)∈R2d. We have\n(6.5)\n|a(h1/2x,h1/2ω,h)−P(x0,ω0),M(x,ω,h)| ≤C′′\nNhN,(x,ω)∈B((x0,ω0),1)24 FABIO NICOLA\nfor a constant C′′\nNdepending on N.\nNow, for ε >0 andr∈(0,1] let\nΓε,r,N:=ε+ inf\n(x0,ω0,t0)∈Hd−/integraldisplay\nB((x0,ω0,t0),r)|P(x0,ω0),M(x,ω,h)|dxdωdt,\nwhereB((x0,ω0,t0),r) is a ball in the reduced Heisenberg group Hd(see\nSection 2.6 for the relevant definitions). Clearly Γ ε,r,N≥ε.\nForε >0, (x0,ω0,t0)∈Hdandr∈(0,1], consider the weight on Hd,\nconstant on the fibers, given by\nwε,r,N,x 0,ω0(x,ω,t) :=r−2Γ−1\nε,r,N(ε+|P(x0,ω0),M(x,ω,h)|).\nLetusobservethatthefunctions |P(x0,ω0),M|belongtotheclass Aα\n∞(R2d)⊂\nAα\n∞,loc(R2d) for some α∈(0,1) depending only on M, i.e. on N(see\ne.g. [55, page 219]). The same holds, for the same α, for the functions\nr−2Γ−1\nε,r,N(ε+|P(x0,ω0),M|). Hence, by Lemma 2.9, the functions wε,r,N,x 0,ω0\nbelong to Aβ\n∞,loc(Hd) for some β∈(0,1) depending only on N. Moreover\n−/integraldisplay\nB((x0,ω0,t0),r)wε,r,N,x 0,ω0(x,ω,t)dxdωdt≥r−2.\nAs a consequence of Lemma 3.1 applied to Hd, and with p= 2, denoting\nby∇the gradient with respect to the basis of left-invariant vector field s on\nHdas in Section 2.6, we have, for r∈(0,1],\n/integraldisplay\nB((x0,ω0,t0),r)/parenleftbig\n|∇u(x,ω,t)|2+wε,r,N,x 0,ω0(x,ω,t)|u(x,ω,t)|2/parenrightbig\ndxdωdt\n≥C′′′\nNr−2/integraldisplay\nB((x0,ω0,t0),r)|u(x,ω,t)|2dxdωdt (6.6)\nfor every u∈L2\nloc(Hd), with∇u∈L2\nloc(Hd).\nBy Remark 2.10 (a) there exists r0∈(0,1] such that the projection on\nR2dof every ball B((x0,ω0,t0),r0)⊂Hdis contained in the Euclidean ball\nB((x0,ω0),1). Hence, by (6.5), if r∈(0,r0] we have, on B((x0,ω0,t0),r)\n|P(x0,ω0),M(x,ω,h)| ≤a(h1/2x,h1/2ω,h)+C′′\nNhN.\nTherefore, if r∈(0,r0],\n/integraldisplay\nB((x0,ω0,t0),r)/parenleftBig\n|∇u(x,ω,t)|2\n+a(h1/2x,h1/2ω,h)+C′′\nNhN+ε\nr2Γε,r,N|u(x,ω,t)|2/parenrightBig\ndxdωdt\n≥C′′′\nNr−2/integraldisplay\nB((x0,ω0,t0),r)|u(x,ω,t)|2dxdωdt. (6.7)\nArguingasintheproofofTheorem1.2,with u(x,ω,t) =/a\\}b∇acketle{tf,π(x,ω,t)ϕ/a\\}b∇acket∇i}ht=\ne−it/a\\}b∇acketle{tf,ϕ(x,ω)/a\\}b∇acket∇i}ht(whereπis the Schr¨ odinger representation of Hd; cf. SectionTHE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 25\n2.6) we obtain, for some r∈(0,r0] (small enough, depending on ϕ),\n/integraldisplay\nR2da(h1/2x,h1/2ω,h)+C′′\nNhN+ε\nΓε,r,N|/a\\}b∇acketle{tf,ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω\n≥C′′′′\nN/integraldisplay\nR2d|/a\\}b∇acketle{tf,ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω.\nSinceεwas arbitrary, using (4.1) (with h= 1) we see that this implies (6.4)\nif we prove that, for some constant cr,N>0,\n(6.8) Γ ε,r,N≥cr,N(λ′(a,h)−C′′\nNhN), r∈(0,1], h∈(0,1].\nTo this end, observe that, by Lemma 2.9,\n−/integraldisplay\nB((x0,ω0,t0),r)|P(x0,ω0),M(x,ω,h)|dxdωdt\n≥cr,N−/integraldisplay\nB((x0,ω0),r)|P(x0,ω0),M(x,ω,h)|dxdω\n≥c′\nr,Nsup\n(x,ω)∈B((x0,ω0),1)|P(x0,ω0),M(x,ω,h)|.\nOn the other hand, by (6.5) we have, for ( x,ω)∈B((x0,ω0),1),\n|P(x0,ω0),M(x,ω,h)| ≥a(h1/2x,h1/2ω,h)−C′′\nNhN.\nHence\n−/integraldisplay\nB((x0,ω0,t0),r)|P(x0,ω0),M(x,ω,h)|dxdωdt\n≥c′\nr,N/parenleftBig\nsup\n(x,ω)∈B((x0,ω0),1)a(h1/2x,h1/2ω,h)−C′′\nNhN/parenrightBig\n.\nTaking the infimum over ( x0,ω0,t0)∈Hdgives (6.8) and therefore con-\ncludes the proof of the first inequality in (6.3).\nFinally let us prove the second inequality in (6.3). By (5.3) we see that\nit suffices to show that, for every ( x0,ω0)∈R2d,h∈(0,1],\n/integraldisplay\nR2da(h1/2x,h1/2ω,h)|/a\\}b∇acketle{tϕ(x0,ω0),ϕ(x,ω)/a\\}b∇acket∇i}ht|2dxdω\n≤CNsup\n(x,ω)∈B((x0,ω0),1)a(h1/2x,h1/2ω,h)+CNhN, (6.9)\nwhere the functions ϕ(x,ω)are defined in (5.2).\nWith the same notation as above, by (6.1) we have\n|a(h1/2x,h1/2ω,h)−P(x0,ω0),M(x,ω,h)| ≤C′′′′′\nNhN(1+|x−x0|+|ω−ω0|)M+1.\nSince the function |/a\\}b∇acketle{tϕ(x0,ω0),ϕ(x,ω)/a\\}b∇acket∇i}ht|=|/a\\}b∇acketle{tϕ,ϕ(x−x0,ω−ω0)/a\\}b∇acket∇i}ht|in (6.9) has rapid\ndecayinx−x0, ω−ω0, weseethatitsufficestoverify(6.9)withthefunction\na(h1/2x,h1/2ω,h) replaced by |P(x0,ω0),M(x,ω,h)|. The desired conclusion is\nthen clear by (5.9). /square26 FABIO NICOLA\nArguing as in the proof of Corollary 5.2 we also obtain the following\nresult concerning the essential spectrum.\nLet\nλ′\ness(a,h) := liminf\n(x0,ω0)→∞sup\n(x,ω)∈B((x0,ω0),h1/2)a(x,ω,h).\nCorollary 6.3. Leta∈Sm\nρfor some m∈R,ρ∈[0,1/2), witha≥0. For\neveryN∈Nthere exists CN>0such that, for every h∈(0,1],\n(6.10) C−1\nNλ′\ness(a,h)−CNhN≤infσess(Aa,h)≤CN(λ′\ness(a,h)+hN).\nIn particular, σ(Aa,h)is purely discrete if and only if\nlim\n(x0,ω0)→∞sup\n(x,ω)∈B((x0,ω0),h1/2)a(x,ω,h) = +∞.\nProof.The proof is similar to that of Corollary 5.2, so that we only focus\non the needed modifications.\nLet us prove the first inequality in (6.10) for all h∈(0,1] such that\nλ′\ness(a,h)<+∞. An easy modification of the argument below will show\nthat the estimate holds for the h’s such that λ′\ness(a,h) = +∞as well.\nSimilarly to the proof of Corollary 5.2 one considers R >0 (depending\nonaandh) such that\nsup\n(x,ω)∈B((x0,ω0),h1/2)a(x,ω,h)≥1\n2λ′\ness(a,h)\nfor|x0|2+|ω0|2> R2, and threrefore the symbol b(x,ω,h) :=C(1+|x|+\n|ω|)−1where the constant C >0 is chosen (depending on aandh) so that\nb(x,ω,h)≥1\n2λ′\ness(a,h) for |x|2+|ω|2≤(R+h1/2)2.\nHence\nλ′(a+b,h)≥1\n2λ′\ness(a,h),\nwhere the function λ′(·,·) was defined in (6.2). Hence, the first inequality\nin (6.10) will follow as in the proof of Corollary 5.2 if we prove the lower\nbound\n(6.11) Qa+b,h(f)≥C−1\nNλ′(a+b,h)−CNhN,\nfor allf∈ Da+b,h=Da,hwith/ba∇dblf/ba∇dblL2= 1, and h∈(0,1].\nThis can be obtained by following a pattern similar to the proof of Theo -\nrem6.2. Namely, onestillconsiderstheTaylorpolynomial P(x0,ω0),M(x,ω,h)\nofa(h1/2x,h1/2ω,h), centeredat( x0,ω0)andofdegree M=MN, whereMN\nis defined as at the beginning of the proof of Theorem 6.2. As in that p roof,\nwe define the quantity Γ ε,r,Nand the weight wε,r,N,x 0,ω0but with the func-\ntion|P(x0,ω0),M(x,ω,h)|replaced by |P(x0,ω0),M(x,ω,h)|+b(h1/2x,h1/2ω,h).\nThe key point is that this function still belongs to Aα\n∞(R2d) for a constant\nα∈(0,1) depending only on M=MN, i.e. on N, by the closure of this\nclasswith respect tosums, scaling andmultiplication by apositive cons tant\n(the above constant Cappearing in the expression of b, in this case).THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 27\nHence, now\nΓε,r,N:=ε\n+ inf\n(x0,ω0,t0)∈Hd−/integraldisplay\nB((x0,ω0,t0),r)/parenleftbig\n|P(x0,ω0),M(x,ω,h)|+b(h1/2x,h1/2ω,h)/parenrightbig\ndxdωdt\nand\nwε,r,N,x 0,ω0(x,ω,t) :=r−2Γ−1\nε,r,N/parenleftbig\nε+|P(x0,ω0),M(x,ω,h)|+b(h1/2x,h1/2ω,h)/parenrightbig\n.\nWith these adjustments one can follow the same argument as in the p roof\nofTheorem 6.2. Only theproofof theinequality analogousto (6.8), n amely\n(6.12) Γ ε,r,N≥cr,N(λ′(a+b,h)−C′′\nNhN), r∈(0,1], h∈(0,1].\nrequires some comments. In this case, we observe that, by Lemma 2.9 and\nCorollary 2.6,\n−/integraldisplay\nB((x0,ω0,t0),r)/parenleftbig\n|P(x0,ω0),M(x,ω,h)|+b(h1/2x,h1/2ω,h)/parenrightbig\ndxdωdt\n≥cr,N−/integraldisplay\nB((x0,ω0),r)/parenleftbig\n|P(x0,ω0),M(x,ω,h)|+b(h1/2x,h1/2ω,h)/parenrightbig\ndxdω\n≥c′\nr,N−/integraldisplay\nB((x0,ω0),1)|P(x0,ω0),M(x,ω,h)|dxdω\n+c′\nr,N−/integraldisplay\nB((x0,ω0),1)b(h1/2x,h1/2ω,h)dxdω.\nThe first integral can be estimated from below as in the proof of (6.8 ). For\nthe second integral we observe that, for some constant cddepending only\nond,\n−/integraldisplay\nB((x0,ω0),1)b(h1/2x,h1/2ω,h)dxdω≥cdsup\n(x,ω)∈B((x0,ω0),1)b(h1/2x,h1/2ω,h),\nas one sees from the expression of b, using Peetre’s inequality: for z,w∈\nB((x0,ω0),1),\n(1+|h1/2z|)−1≥c′\nd(1+|h1/2w|)−1(1+h1/2|z−w|)−1≥c′′\nd(1+|h1/2w|)−1.\nHence one arrives at\n−/integraldisplay\nB((x0,ω0,t0),r)/parenleftbig\n|P(x0,ω0),M(x,ω,h)|+b(h1/2x,h1/2ω,h)/parenrightbig\ndxdωdt\n≥c′′′\nr,N/parenleftBig\nsup\n(x,ω)∈B((x0,ω0),1)(a(h1/2x,h1/2ω,h)−C′′\nNhN)\n+ sup\n(x,ω)∈B((x0,ω0),1)b(h1/2x,h1/2ω,h)/parenrightBig\n≥c′′′\nr,N/parenleftBig\nsup\n(x,ω)∈B((x0,ω0),1)/parenleftbig\na(h1/2x,h1/2ω,h)+b(h1/2x,h1/2ω,h)/parenrightbig\n−C′′\nNhN/parenrightBig\n.\nTaking the infimum over ( x0,ω0,t0)∈Hdyields the inequality in (6.12)\nand therefore concludes the proof of the lower bound (6.11).\nThe second inequality in (6.10) follows from (6.9). /square28 FABIO NICOLA\nRemark6.4.Ifa∈Am\nρisreal-valuedandboundedbelow, namelyinf a(·,h)>\n−∞forevery h∈(0,1], we canapply theresults of Theorem 6.2andCorol-\nlary 6.3 to the nonnegative symbol a−infa(replacing abya−infain\nthe definition of the form domain Da,hand norm /ba∇dbl·/ba∇dbla,hin Section 4.2). In\nparticular, one obtains\nC−1\nNλ′(a−infa,h)−CNhN≤infσ(Aa,h)−infa≤CN(λ′(a−infa,h)+hN)\nfor every h∈(0,1].\nAppendix\nIn this appendix we comment on a characterization, recently prove d in\n[9], of Schr¨ odinger operators with purely discrete spectrum asso ciated with\na left-invariant subLaplacian on a noncompact connected Lie group , with\na potential in A∞,loc.\nLetGbe a noncompact connected Lie group. Consider a left-invariant\nsubRiemannian structure induced by the linearly independent vecto r fields\nX1,...,X ℓsatisfying H¨ ormander’s condition. Let d Cthe the correspond-\ning Carnot–Carath´ eodory distance, as in Section 2. Let λbe a left Haar\nmeasure of GandV∈A∞,loc(G,dC,λ) (cf. Definition 1.1). Consider the\nvector space\nD′\nV:={u∈L2(G,λ) :|∇u| ∈L2(G,λ),√\nVu∈L2(G,λ)}\nwhere∇u= (X1u,...,X ℓu) is the horizontal gradient, understood in the\nsense of distributions. Then\nQ′\nV(u) :=/integraldisplay\nG(|∇u|2+V|u|2)dλ, u ∈ D′\nV\nisadensitydefinednonnegativeclosedquadraticform,andtheref oredefines\na self-adjoint operator on L2(G,λ) that we will denote by LV(cf. [9]).\nNow, it was proved in [9, Theorem 6.6] that if V∈A∞,loc(G,dC,λ) and\nif the absolutely continuous measure that has density Vwith respect to λ\nis locally doubling, in the sense that there exist C,R′>0 such that\n/integraldisplay\n2BV dλ≤C/integraldisplay\nBV dλ\nfor every ball Bof radius r∈(0,R′], thenLVhas purely discrete spectrum\nif and only if\n(6.13) lim\nx→∞/integraldisplay\nB(x,R)V dλ= +∞\nfor some (equivalently for all) R >0. Hence, combining this result with\nCorollary 2.6 we obtain the following characterization.\nTheorem 6.5. LetV∈A∞,loc(G,dC,λ). The operator LVhas purely\ndiscrete spectrum if and only if (6.13)holds for some (equivalently for all)\nR >0.THE FEFFERMAN-PHONG UNCERTAINTY PRINCIPLE 29\nAcknowledgments\nWe would like to thank Tommaso Bruno, Mattia Calzi, Gian Maria\nDall’Ara and Eugenia Malinnikova for interesting comments on a prelimi-\nnary version of the manuscript.\nReferences\n[1] L. D. Abreu and M. Speckbacher. Donoho-Logan large sieve prin ciples for modu-\nlation and polyanalytic Fock spaces. Bull. Sci. Math. , 171:Paper No. 103032, 25,\n2021.\n[2] A. Agrachev, D. Barilari, and U. Boscain. A comprehensive introduction to sub-\nRiemannian geometry , volume 181 of Cambridge Studies in Advanced Mathematics .\nCambridge University Press, Cambridge, 2020. From the Hamiltonian viewpoint.\nWith an appendix by Igor Zelenko.\n[3] D. N. Arnold, G. David, M. Filoche, D. Jerison, and S. Mayboroda. Localization\nof eigenfunctions via an effective potential. Comm. 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Pseudodifferential operators and spectral theory . Springer-Verlag,\nBerlin, second edition, 2001. Translated from the 1978 Russian orig inal by Stig I.\nAndersson.\n[54] B. Simon. Schr¨ odinger operators with purely discrete spectr um.Methods Funct.\nAnal. Topology , 15(1):61–66, 2009.\n[55] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality , and oscilla-\ntory integrals , volume 43 of Princeton Mathematical Series . Princeton University\nPress, Princeton, NJ,1993.With the assistanceofTimothyS. Mur phy, Monographs\nin Harmonic Analysis, III.\n[56] J.-O. Str¨ omberg and A. Torchinsky. Weighted Hardy spaces , volume 1381 of Lecture\nNotes in Mathematics . Springer-Verlag, Berlin, 1989.\n[57] N. Teofanov, J. Toft, and P. Wahlberg. Pseudo-differential o perators with isotropic\nsymbols, Wick and anti-Wick operators, and hypoellipticity. J. Math. Pures Appl.\n(9), 167:48–100, 2022.\n[58] M. A. ˇSubin. Conditions for the discreteness of the spectrum of certain operators.\nMat. Zametki , 11:233–240, 1972.\n[59] M.W.Wong. Wavelet transforms and localization operators , volume136of Operator\nTheory: Advances and Applications . Birkh¨ auser Verlag, Basel, 2002.\n[60] K. Zhu. Analysis on Fock spaces , volume 263 of Graduate Texts in Mathematics .\nSpringer, New York, 2012.\n[61] M. Zworski. Semiclassical analysis , volume138of Graduate Studies in Mathematics .\nAmerican Mathematical Society, Providence, RI, 2012.\nDipartimento di Scienze Matematiche, Politecnico di Torin o, Corso Duca\ndegli Abruzzi 24, 10129 Torino, Italy.\nEmail address :fabio.nicola@polito.it"    },    {        "title": "2402.03265v1.Symmetries_and_conservation_laws_of_a_fifth_order_KdV_equation_with_time_dependent_coefficients_and_linear_damping.pdf",        "content": "arXiv:2402.03265v1  [math.AP]  5 Feb 2024Symmetries and conservation laws of a fifth-order\nKdV equation with time-dependent coefficients and\nlinear damping\nR. de la Rosaa, M.L. Gandariasb, M.S. Bruz ´onc\naUniversidad de C ´adiz, Spain (e-mail: rafael.delarosa@uca.es).\nbUniversidad de C ´adiz, Spain (e-mail: marialuz.gandarias@uca.es).\ncUniversidad de C ´adiz, Spain (e-mail: m.bruzon@uca.es).\nAbstract\nA fifth-order KdV equation with time dependent coefficients a nd linear damping has\nbeen studied. Symmetry groups have several different appli cations in the context of nonlin-\near differential equations. For instance, they can be used t o determine conservation laws.\nWe obtain the symmetries of the model applying Lie’s classic al method. The choice of\nsome arbitrary functions of the equation by the equivalence transformation enhances the\nstudy of Lie symmetries of the equation. We have determined t he subclasses of the equa-\ntion which are nonlinearly self-adjoint. This allow us to ob tain conservation laws by using\na theorem proved by Ibragimov which is based on the concept of adjoint equation for non-\nlinear differential equations.\nKeywords : Classical Symmetries; Equivalence Transformations; Par tial Differential Equa-\ntions; Conservation laws.\nPACS : 02.20.Hj; 02.20.Sv; 02.30.Em; 02.30.Jr; 47.10.ab.\nMSC : 35C07; 35Q53.\n1 Introduction\nLie symmetry groups play a fundamental role in the study of di fferential equations, particularly\nfor nonlinear equations. Lie’s theory provides a useful, po werful tool when analysing partial\ndifferential equations. Moreover, it has numerous well kno wn applications, prominent among\nthese are the obtaining of exact solutions of partial differ ential equations, directly or via simi-\nlarity solutions [7, 17], clasify invariant equations, red uce the number of independent variables\nor determine conservation laws [1, 2, 5, 6, 14].\nOver the last years, nonlinear equations with variable coef ficients have become increasingly\nimportant due to these describe many nonlinear phenomena mo re realistically than equations\n1with constant coefficients. For instance, the variable coef ficient Gardner equation [4, 10], the\nnonlinear Schr¨ odinger equation [11, 15, 18] or the variabl e coefficient reaction-diffusion equa-\ntion [3, 16]. The problem lies in the fact that the analysis of Lie symmetries of equations in-\nvolving arbitrary functions seems rather difficult. Equiva lence transformations allow to perform\nfurther study in a simpler way.\nAn equivalence transformation is a non-degenerate transfo rmation acting on dependent and\nindependent variables which maps the considered equation t o another equation with the same\ndifferential structure, except maybe the form of the arbitr ary functions. Most important ad-\nvantage of equivalence transformations is that they permit an analysis for complete equivalent\nclasses instead of considering individual equations. In pa rticular, they allow us to reduce of a\nclass to its subclass with fewer number of arbitrary functio ns.\nThe study of nonlinear evolution equations (NLEEs) is one of the most important areas of\nresearch in the field of nonlinear dynamics. There is a large n umber of nonlinear evolution\nequations which are being studied nowadays. Many of these eq uations are generalizations or\ncombinations of the KdV equation, Boussinesq equation and m any others.\nThe KdV equation is established as a simple mathematical mod el by which it can be de-\nscribed the physics of a shallow layer of fluid subject, for in stance, it is an analytical model of\ntsunami generation by sub-marine landslides. On the other h and, the dynamics of shallow water\nwaves can also be described by using a KdV equation. Generali zations of KdV equation which\nhave more than one nonlinear term allow a better analysis of t hese phenomena.\nThe concept of conservation law, more specifically, of a cons erved quantity, has its origin\nin physics. Conservation laws are very useful when it comes t o analyse or know of physical\nproperties of concrete systems, overall physical aspects o f NLEEs. NLEEs would be difficult\nto understand without having deepened into the study of its c onservation laws.\nIn this paper, we consider a family of fifth-order KdV equatio ns with time-dependent coef-\nficients and linear damping term of the form\nut+A(t)uxxxxx+B(t)uxxx+C(t)uuxxx\n+E(t)uux+F(t)uxuxx+Q(t)u=0,(1)\nwhere A(t)/ne}ationslash=0,B(t),C(t)/ne}ationslash=0,E(t),F(t)andQ(t)are arbitrary smooth functions of t. We\nperform an analysis of Lie symmetries of the equation. We obt ain the continuous equivalence\ntransformations of class (1). This allow us to reduce the num ber of arbitrary elements of class\n(1) which will result in a complete study of Lie symmetries of class (1). Moreover, we determine\nthe subclasses of the equation which are quasi self-adjoint and nonlinear self-adjoint. From\nthese concepts we construct conservation laws by using a the orem proved by Ibragimov.\n2 Equivalence Transformations\nIn this section, we search for equivalence transformation o f class (1). An equivalence transfor-\nmation of class (1) is a nondegenerate point transformation ,(t,x,u)to(˜t,˜x,˜u)which remains\nthe differential structure but with different arbitrary fu nctions, ˜A(˜t),˜B(˜t),˜C(˜t),˜E(˜t),˜F(˜t)and\n2˜Q(˜t)from the original ones. In order to obtain the equivalence tr ansformation for our equation\nwe apply Lie’s infinitesimal criterion [13]. In this case, we require not only the invariance of\nclass (1) but also the invariance of the auxiliary system\nAx=Au=Bx=Bu=Cx=Cu=\n=Ex=Eu=Fx=Fu=Qx=Qu=0(2)\nin the augmented space (t,x,u,A,B,C,E,F,Q).\nWe consider the one-parameter group of equivalence transfo rmations in (t,x,u,A,B,C,E,F,Q)\ngiven by\n˜t=t+ε τ(t,x,u)+O(ε2),\n˜x=x+ε ξ(t,x,u)+O(ε2),\n˜u=u+ε η(t,x,u)+O(ε2),\n˜A=A+ε ω1(t,x,u,A,B,C,E,F,Q)+O(ε2),\n˜B=B+ε ω2(t,x,u,A,B,C,E,F,Q)+O(ε2),\n˜C=C+ε ω3(t,x,u,A,B,C,E,F,Q)+O(ε2),\n˜E=E+ε ω4(t,x,u,A,B,C,E,F,Q)+O(ε2),\n˜F=F+ε ω5(t,x,u,A,B,C,E,F,Q)+O(ε2),\n˜Q=Q+ε ω6(t,x,u,A,B,C,E,F,Q)+O(ε2),\nwhere εis the group parameter. In this case, the vector field takes th e following form\nY=τ∂t+ξ ∂x+η∂u+ω1∂A+ω2∂B\n+ω3∂C+ω4∂E+ω5∂F+ω6∂Q.(3)\nInvariance of system (1)-(2) under a Lie group of point trans formations with infinitesimal gen-\nerator (3) leads to a system of determining equations.\nAfter having solved the determining system, omitting tedio us calculations, we obtain that\nthe equivalence generator of class (1) is spanned by\nY1=x∂x+1\n2u∂u+5A∂A+3B∂B+5\n2C∂C\n+1\n2E∂E+5\n2F∂F,\nY2=∂x,\nYα=α∂t−αtA∂A−αtB∂B−αtC∂C\n−αtE∂E−αtF∂F−αtQ∂Q,\nYr=ru∂u−rC∂C−rE∂E−rF∂F−rt∂Q,\n3where α=α(t),r=r(t)are arbitrary smooth functions with αt/ne}ationslash=0. We obtain that the equiv-\nalence group of class (1) consists of the transformations\n˜t=α(t),˜x=(x+k2)ek1,˜u=er(t)kr+k1\n2u,\nwhere k1,k2andkrare arbitrary constants, and the arbitrary functions are gi ven by\n˜A=e5k1\nαtA, ˜B=e3k1\nαtB,\n˜C=e5k1\n2−rkr\nαtC, ˜E=ek1\n2−rkr\nαtE,\n˜F=e5k1\n2−rkr\nαtF, ˜Q=Q−rtkr\nαt.\nAs there are two arbitrary functions, α(t)andr(t), in the group of transformations we can es-\ntablish two of the arbitrary functions of class (1). Thus, we can set ˜A=˜C=1 by the equivalence\ntransformation\n˜t=/integraldisplay\ne5k1Adt,˜x=(x+k2)ek1,˜u=e−2k1C\nAu, (4)\nwhich changes equation (1) into the equation\n˜u˜t+˜u˜x˜x˜x˜x˜x+˜B(˜t)˜u˜x˜x˜x+˜u˜u˜x˜x˜x\n+˜E(˜t)˜u˜u˜x+˜F(˜t)˜u˜x˜u˜x˜x+˜Q(˜t)˜u=0,\nwhere the new arbitrary functions are related with the old on es by\n˜B=e−2k1B\nA, ˜E=e−2k1E\nC,\n˜F=F\nC, ˜Q=e−5k1/parenleftbig\nQ+/parenleftbig\nLog/parenleftbigA\nC/parenrightbig/parenrightbig\nt/parenrightbig\nA.\nThus, we can restrict without loss of generality our study to the class\nut+uxxxxx+B(t)uxxx+uuxxx\n+E(t)uux+F(t)uxuxx+Q(t)u=0,(5)\ndue to symmetries and conservation laws obtained for class ( 5) can be extended to class (1)\nusing transformation (4).\n43 Lie classical symmetries of class (5)\nLie’s classical method is based on the determination of the s ymmetry group of a differential\nequation, i.e., the largest group of transformations actin g on dependents and independents vari-\nables of the equation so that transforms solutions of the equ ation into other solutions.\nWe consider the one-parameter Lie group of infinitesimal tra nsformations in (t,x,u)given by\nt∗= t+ετ(t,x,u)+O(ε2),\nx∗=x+εξ(t,x,u)+O(ε2),\nu∗=u+εη(t,x,u)+O(ε2),\nwhere εis the group parameter.\nThe infinitesimal operator of the local Lie group of point tra nsformations which are admitted\nby equation (5) is given by\nv=τ(t,x,u)∂t+ξ(t,x,u)∂x+η(t,x,u)∂u. (6)\nThe fifth prolongation of the vector field (6) has the followin g form\npr(5)v=v+ζt∂ut+ζx∂ux+ζxx∂uxx\n+ζxxx∂uxxx+ζxxxxx∂uxxxxx,(7)\nwhere\nζJ(t,x,u(5))=DJ(η−τut−ξux)+τuJt+ξuJx\nwith J= (j1,..., jk), 1≤jk≤2 and 1≤k≤5, and u(5)denote the sets of partial derivatives\nup to fifth order [12]. Invariance criterion implies that the 5th-prolongation of the vector field v\n(7) acting on the equation (5) must be zero when the equation h olds. This leads to a set of 17\ndetermining equations. By simplifying the system we obtain that the infinitesimals are given by\nτ=k2t+k3,ξ=k2\n5x+δ,η=ρ−2k2\n5u,\nwhere δ=δ(t),ρ=ρ(t,x),B=B(t),E=E(t),F=F(t)andQ=Q(t)must satisfy the fol-\nlowing conditions:\n5(k2t+k3)Bt+2k2B+5ρ=0,\nρxF=0,\n(k2t+k3)Et+2k2\n5E=0,\nρxxF+ρE−δt=0,\n(k2t+k3)Ft=0,\nρQ+ρxxxB+ρxxxxx+ρt=0,\n(k2t+k3)Qt+k2Q+ρxE+ρxxx=0.(8)\n5From determining system (8) if the functions B,E,F,Q,δandρare arbitrary we obtain v=∂x.\nFor the following cases we obtain new symmetries:\n3.1 Case 1: F=0\nIn this case, we get the following symmetries\nτ=k2t+k3,ξ=k2\n5x+δ,η=ρ−2k2\n5u,\nwhere δ=δ(t),ρ=ρ(t,x),B=B(t),E=E(t)andQ=Q(t)must satisfy the following con-\nditions:\n5(k2t+k3)Bt+2k2B+5ρ=0, (9)\n(k2t+k3)Et+2k2\n5E=0, (10)\nρE−δt=0, (11)\nρQ+ρxxxB+ρxxxxx+ρt=0, (12)\n(k2t+k3)Qt+k2Q+ρxE+ρxxx=0. (13)\n3.2 Case 2: F=constant /ne}ationslash=0\nNow, we obtain the following symmetries\nτ=k2t+k3,ξ=k2\n5x+δ,η=σ−2k2\n5u,\nwhere δ=δ(t),σ=σ(t),B=B(t),E=E(t),F=F(t)andQ=Q(t)are related by the fol-\nlowing conditions:\n5(k2t+k3)Bt+2k2B+5σ=0, (14)\n(k2t+k3)Et+2k2\n5E=0, (15)\nσE−δt=0, (16)\nσQ+σt=0, (17)\n(k2t+k3)Qt+k2Q=0. (18)\nIn the previous cases, k2andk3are arbitrary constants.\n64 Formal Lagragian, adjoint equation and nonlinearly self-\nadjointness\nIn this section we proceed to give some prior concepts necess ary to construct conservation laws.\nTo achieve this objective, we use a theorem on conservation l aws proved by Ibragimov [8].\nThe theorem is valid for any system of differential equation s where the number of equations\nis equal to the number of dependent variables, and does not re quire existence of a classical\nLagrangian. This theorem is based on the concept of adjoint e quation for nonlinear equations\nwhich is defined as\nDefinition 1 Consider an sth-order partial differential equation\nF(x,u,u(1),..., u(s))=0 (19)\nwith independent variables x= (x1,..., xn)and a dependent variable u ,where u (1)={ui},\nu(2)={ui j},...denote the sets of the partial derivatives of the first, secon d, etc. orders, u i=\n∂u/∂xi, ui j=∂2u/∂xi∂xj.The formal Lagrangian is defined as\nL=vF/parenleftbig\nx,u,u(1),..., u(s)/parenrightbig\n, (20)\nwhere v=v(x)is a new dependent variable. The adjoint equation to (19) is\nF∗(x,u,v,u(1),v(1),..., u(s),v(s))=0, (21)\nwith\nF∗(x,u,v,u(1),v(1),..., u(s),v(s))=δ(vF)\nδu,\nwhere\nδ\nδu=∂\n∂u+∞\n∑\ns=1(−1)sDi1···Dis∂\n∂ui1···is\ndenotes the variational derivatives (the Euler-Lagrange o perator). Here\nDi=∂\n∂xi+ui∂\n∂u+ui j∂\n∂uj+···\nare the total differentiations.\nTheorem 1 The adjoint equation to class (5) is\nvQ+(uxvxx+uxxvx)F−uvxE−vxxxB\n−vxxxxx−uvxxx−3uxvxx−3uxxvx−vt=0(22)\nNow, we use the following definition given in [9]\n7Definition 2 Equation (19) is said to be nonlinearly self-adjoint if the equation obtained from\nthe adjoint equation (21) by the substitution\nv=ϕ(x,u), (23)\nwith a certain function ϕ(x,u)/ne}ationslash=0is identical with the original equation (19), i.e.,\nF∗|v=ϕ=λ(x,u,...)F, (24)\nfor some differential function λ=λ(x,u,...). Ifϕ=u or ϕ=ϕ(u)andϕ′(u)/ne}ationslash=0, equation\n(19) is said self-adjoint orquasi self-adjoint , respectively.\nGiven equation (5) we apply Definition 2. Taking into account expression (22) and using (23)\nand its derivatives, equation (24) can be written as\n−uQλ+ϕQ−ϕxuE−ϕxxxB−ϕxxxu−ϕxxxxx\n−ϕt−(λ+ϕu)ut−ϕuuuuu u5\nx−5ϕuuuux u4\nx\n+(ϕuuF−ϕuuuB−ϕuuuu−10ϕuuuxx−3ϕuu)u3\nx\n+(2ϕuxF−3ϕuuxB−3ϕuuxu−10ϕuuxxx\n−6ϕux)u2\nx−(uEλ−ϕxxF+ϕuuE+3ϕuxxB\n+3ϕuxxu+3ϕxx+5ϕuxxxx)ux−15ϕuuxu2\nxx\n−(Bλ+uλ+ϕuB+ϕuu+10ϕuxx)uxxx\n+(ϕxF−3ϕuxB−3ϕuxu−3ϕx−10ϕuxxx)uxx\n−10ϕuuuxxuxxx−10ϕuuuuu3\nxuxx−10ϕuuuu2\nxuxxx\n−30ϕuuuxu2\nxuxx−(λ+ϕu)uxxxxx−5ϕuxuxxxx\n−5ϕuuuxuxxxx−15ϕuuuuxu2\nxx−20ϕuuxuxuxxx\n−(Fλ−2ϕuF+3ϕuuB+3ϕuuu+30ϕuuxx\n+6ϕu)uxuxx=0.(25)\nFrom the ut,ux,uxx,... coefficients we obtain\nTheorem 2 Equation (5) with B (t), E(t), F(t)and Q(t)arbitrary functions is nonlinearly self-\nadjoint and\nϕ=α(t)u+β(t,x) (26)\nin the following cases:\n•If F(t)=2, we obtain\nα(t)=c1e2/integraltextQ(t)dt,β(t)=c2e/integraltextQ(t)dt.\n•If F(t)=3, we obtain\nα(t)=0,β=β(t,x),\nwhere β(t,x)must verify the following equation\nβQ+βxBE−βxxxxx−βt=0. (27)\n8•If F(t)/ne}ationslash=2,3, we obtain\nα(t)=0,β(t)=c2e/integraltextQ(t)dt.\nHere, c 1and c 2are arbitrary constants.\n5 Conservation laws\nConservation laws appear in many of physical, chemical and m echanical processes, such laws\nenable us to solve problems in which certain physical proper ties do not change in the course of\ntime within an isolated physical system. The importance of c onservation laws also embraces\nmathematics, for instance, the integrability of a partial d ifferential equation (PDE) is strongly\nrelated with the existence of conservation laws. Furthermo re, they can be used to obtain exact\nsolutions of a PDE.\nGiven equation (5), a conservation law is a space-time diver gence such that\nDtC1(t,x,u,ux,ut,...)+DxC2(t,x,u,ux,ut,...)=0. (28)\nIn order to obtain conservation laws we use the following the orem given in [8]\nTheorem 3 Any Lie point, Lie-B ¨acklund or non-local symmetry\nv=ξi(x,u,u(1),...)∂\n∂xi+η(x,u,u(1),...)∂\n∂u, (29)\nof equation (19) provides a conservation law D i(Ci)=0for the simultaneous system (19),(21).\nThe conserved vector is given by\nC1=τL+W/bracketleftbigg∂L\n∂ut/bracketrightbigg\n,\nC2=ξL+W/bracketleftbigg∂L\n∂ux−Dx/parenleftbigg∂L\n∂uxx/parenrightbigg\n+D2\nx/parenleftbigg∂L\n∂uxxx/parenrightbigg\n+D4\nx/parenleftbigg∂L\n∂uxxxxx/parenrightbigg/bracketrightbigg\n+Dx(W)/bracketleftbigg∂L\n∂uxx\n−Dx/parenleftbigg∂L\n∂uxxx/parenrightbigg\n−D3\nx/parenleftbigg∂L\n∂uxxxxx/parenrightbigg/bracketrightbigg\n+D2\nx(W)/bracketleftbigg∂L\n∂uxxx+D2\nx/parenleftbigg∂L\n∂uxxxxx/parenrightbigg/bracketrightbigg\n−D3\nx(W)/bracketleftbigg\nDx/parenleftbigg∂L\n∂uxxxxx/parenrightbigg/bracketrightbigg\n+D4\nx(W)∂L\n∂uxxxxx,(30)\n9where Lis given by (20) and W is defined as follows:\nW=η−ξjuj. (31)\nIn order to apply Theorem 3 to our equation we perform the foll owing change of notation:/parenleftbig\nξ1,ξ2/parenrightbig\n=(τ,ξ).\nNow, we proceed to construct conservation laws for the cases considered in Section 3. We omit\nagain the case in which the functions are arbitrary due to thi s leads to a trivial conservation law.\n5.1 Case 1.\nIn this case, we obtain conservation laws for F(t)=0 and F(t) =constant/ne}ationslash=2,3. For these\ncases, the equation admits the following generator\nv=(k2t+k3)∂t+/parenleftbiggk2\n5x+δ/parenrightbigg\n∂x+/parenleftbigg\nρ−2k2\n5u/parenrightbigg\n∂u,\nwhere δ=δ(t)andρ=ρ(t,x)satisfy the conditions (9)-(13) for F(t) =0,δ=δ(t)and\nρ=σ(t)verify the conditions (14)-(18) for F(t)=constant/ne}ationslash=2,3. From Theorem 2, for both\ncases, we have\nϕ=c2e/integraltextQ(t)dt\nThus, we obtain the conservation law (28) with the conserved vector\nC1=c2eH/parenleftbigg\nρ−k2\n5u+(k2t+k3)Qu/parenrightbigg\n,\nC2=c2\n2(k2t+k3)eHQ/parenleftbig\nEu2+Fu2\nx−2Buxx−2uxxxx\n−u2\nx−2uuxx/parenrightbig\n−c2k2\n10eH/parenleftbig\n2Buxx+Fu2\nx+2uxxxx/parenrightbig\n+2uuxx−u2\nx+c2eH(ρxxB−ρxux+ρxxu+ρxxxx),\nwhere H(t) =/integraltextQ(t)dt. Remember that for case F(t) =constant/ne}ationslash=2,3 the terms in which\nappear ρx,ρxx,... are eliminated.\n5.2 Case 2.\nNow, we obtain conservation laws of equation (5) for F(t)=2 and F(t)=3. For these cases,\nthe equation admits the following generator\nv=(k2t+k3)∂t+/parenleftbiggk2\n5x+δ/parenrightbigg\n∂x+/parenleftbigg\nσ−2k2\n5u/parenrightbigg\n∂u,\nwhere δ=δ(t)andσ=σ(t)must verify the conditions (14)-(18).\n105.2.1 F(t)=2\nWe focus on case F(t)=2. From Theorem 2, we have\nϕ=c1e2/integraltextQ(t)dtu+c2e/integraltextQ(t)dt.\nBy using Theorem 3, we obtain the conserved vector\nC1=/parenleftbig\nc1eHu+c2/parenrightbig\n(k2t+k3)eHQu\n+c1e2Hu/parenleftbigg\nσ−3k2\n10u/parenrightbigg\n+c2eH/parenleftbigg\nσ−k2\n5u/parenrightbigg\n,\nC2= ( k2t+k3)eHQ/parenleftbig/parenleftbig\n2uuxxxx−2uxuxxx+u2\nxx\n+2u2uxx−Bu2\nx+2Buu xx+2\n3Eu3/parenrightbigg\nc1eH\n+/parenleftbigg1\n2Eu2+Buxx+uxxxx+uuxx+u2\nx/parenrightbigg\nc2/parenrightbigg\n+c1\n2eHσ/parenleftbig\nEu2+2Buxx+2uxxxx+2uuxx+u2\nx/parenrightbig\n−3\n10/parenleftbigg2\n3Eu3+2Buu xx−Bu2\nx+2uuxxxx+u2\nxx\n−2uxuxxx+2u2uxx/parenrightbig\n−3\n10/parenleftbig\nEu2+2Buxx+u2\nx\n+2uxxxx+2uuxx),\nwhere H(t)=/integraltextQ(t)dt.\n5.2.2 F(t)=3\nNow, we consider case F(t)=3. From Theorem 2, we have\nϕ=β(x,t),\nwhere βmust verify (27). By using Theorem 3, we obtain the conserved density\nC1= ( k2t+k3)uβt+/parenleftbiggk2x\n5+δ/parenrightbigg\nβx+/parenleftbigg\nσ−k2u\n5/parenrightbigg\nβ,\nWe do not show the conserved flux C2due to its length but we can state that the conserved\nvector obtained verifies (28).\n116 Conclusions\nIn this paper, Lie group analysis has been successfully appl ied to study the symmetries of a fifth-\norder KdV equation with time dependent coefficients and line ar damping. We have obtained\nthe infinitesimal generators of the equivalence transforma tion of the equation. It is shown that\nthe equivalence group obtained is an infinite dimensional gr oup. Moreover, the usage of equiv-\nalence groups allow an exhaustive study of the equation tran sform class (1) into subclass (5)\nof simpler structure. We have obtained classical symmetrie s of class (5) involving the different\narbitrary functions. Nonlinear self-adjointness of equat ion (5) has been carried out. This allow\nus to obtain new conservation laws by using Ibragimov’s meth od.\nAcknowledgements\nWe would like to thank the Editor and Referees for their timel y and valuable comments and\nsuggestions. The authors acknowledge the financial support from Junta de Andalucia group\nFQM-201. The first author express his sincerest gratitude to the Universidad Polit´ ecnica de\nCartagena for support him. The second and third authors also acknowledge the support of\nDGICYT project MTM2009-11875 with the participation of FED ER.\nReferences\n[1] A. Biswas et al., Solitons and conservation laws of Klein –Gordon equation with power law\nand log law nonlinearities, Nonlinear Dyn, 73, 2191–2196 (2 013).\n[2] M.S. Bruz´ on, M.L. Gandarias, N.H. Ibragimov, Self-adj oint sub-classes of generalized thin\nfilm equations, J. Math. Anal. Appl., 357, 307–313 (2009).\n[3] M.S. Bruz´ on, M.L. 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Ibragimov, Nonlinear self-adjointness and conser vation laws, J. Phys. A: Math.\nTheor., 44, 432002–432010 (2011).\n[10] E.V . Krishnan et al., A study of shallow water waves with Gardner’s equation, Nonlinear\nDyn, 66, 497–507 (2011).\n[11] H. Liu, F. Yan, C. Xu, The bifurcation and exact travelli ng wave solutions of (1+2)-\ndimensional nonlinear Schr¨ odinger equation with dual-po wer law nonlinearity, Nonlinear\nDyn, 67, 465–473 (2012).\n[12] P. Olver, Applications of Lie groups to differential eq uations. Springer-Verlag, New York\n(1993).\n[13] L.V . Ovsyannikov, Group analysis of differential equa tions. Academic, New York, (1982).\n[14] P. Razborova, A.H. Kara, A. Biswas, Additional conserv ation laws for Rosenau–KdV–\nRLW equation with power law nonlinearity by Lie symmetry, No nlinear Dyn, 79, 743–748\n(2015).\n[15] M. Senthilvelan, M. Torrisi, A. Valenti, Equivalence t ransformations and differential in-\nvariants of a generalized nonlinear Schr¨ odinger equation , J. Phys. A: Math. Gen., 39, 3703–\n3713 (2006).\n[16] O. Vaneeva et al., Enhanced group analysis and conserva tion laws of variable coefficient\nreaction-diffusion equations with power nonlinearities, J. Math. Anal. Appl., 330, 1363–\n1386 (2007).\n[17] G.W. Wang, T.Z. Xu, Invariant analysis and exact soluti ons of nonlinear time fractional\nSharma–Tasso–Olver equation by Lie group analysis, Nonlin ear Dyn, 76, 571–580 (2014).\n[18] H.A. Zedan, E. Aladrous, S. Shapll, Exact solutions for a perturbed nonlinear Schr¨ odinger\nequation by using B¨ acklund transformations, Nonlinear Dy n, 74, 1145–1151 (2013).\n13"    },    {        "title": "2402.03267v1.Sojourns_of_locally_self_similar_Gaussian_processes.pdf",        "content": "arXiv:2402.03267v1  [math.PR]  5 Feb 2024Sojourns of locally self-similar Gaussian processes\nSvyatoslav M. Novikov\nDepartment of Actuarial Science, University of Lausanne, Unil-Do rigny, 1015 Lausanne,\nSwitzerland\nARTICLE HISTORY\nCompiled February 6, 2024\nABSTRACT\nGiven a Gaussian risk process R(t) =u+c(t)−X(t),t≥0, the cumulative Parisian\nruin probability on a finite time interval [0 ,T] with respect to L≥0 is defined\nas the probability that the sojourn time that the risk proces sRspends under the\nlevel 0 on this time interval [0 ,T] exceeds L. In this contribution we derive exact\nasymptotic approximations of the cumulative Parisian ruin probability for a general\nclass of Gaussian processes introduced in [9] assuming that Xis locally self-similar.\nWe illustrate our findings with several examples. As a byprod uct we show that\nBerman’s constants can be defined alternatively by a self-si milar Gaussian process\nwhich could be quite different to the fractional Brownian mot ion.\nKEYWORDS\nGaussian process; self-similar process; ruin probability ; sojourn time; occupation\ntime; Berman constants\nAMS CLASSIFICATION\nPrimary 60G15; Secondary 60G70\n1. Introduction\nThe Gaussian risk model R(t) =u+c(t)−X(t),t≥0 is a classical object in modeling\nthe risk of a certain insurance portfolio, see e.g., [7, 8, 11 ] . Typically, Xis a Brownian\nmotion modelling the total claim amount, ustands for the initial capital and c(t)\nmodels the cumulative premium income up to time t≥0. The classical ruin time is\nthe first time when the risk process Rdrops below zero in a given interval, say [0 ,T].\nThe cumulative Parisian ruin, also known as the sojourn ruin is recently considered\nin the literature, see e.g., [4, 12–14] and the references th erein. By definition, the\ncumulative Parisian ruin in the time interval [0 ,T] occurs when the total sojourn time\n(occupation time) that Rspend below zero in the time interval [0 ,T] exceeds a certain\npredefined threshold, say L≥0. More specifically, since the sojourn time that R\nspends below 0 is given by\n/integraldisplayT\n0I(R(t)<0)dt,\nCONTACT Svyatoslav M. Novikov. Email: Svyatoslav.Novikov @unil.chwithI(·) the indicator function, the sojourn probability is thus de fined by\npX(T,L) =P/braceleftbigg/integraldisplayT\n0I(X(t)−c(t)> u)dt > L/bracerightbigg\n.\nClearly, when L= 0, weretrievetheclassical ruinprobabilityonthefinitet imehorizon\n[0,T].\nThe asymptotics of sojourn times has been the main topic of ma ny research papers by\nBerman, see e.g., [1, 2] where the stationary and locally sta tionary cases are discussed.\nRecently, [3, 6] derived results for the tail asymptotics of the sojourn times of general\nGaussian processes and fields.\nAtypical assumptionintheliteratureontheGaussianproce ssXisthatthevariance\nfunction σ2(t),t∈[0,T] attains its maximum at some unique point, say t0= 0 and\nfurther\nσ(t0)−σ(t)∼btβ, t↓0, (1)\nwith some b,βpositive constants; here ∼stands for asymptotic equivalence when the\nargument tends to 0 or ∞(depending on the context).\nTheassumption on the variance function is accompanied by an asymptotic assumption\non the correlation function [15]. Specifically, it is an asym ptotic condition on the\ncorrelation rXofXat zero, saying that\n1−rX(t,s)∼aVY(t,s), s,t↓0 (2)\nfor some a >0,κ∈(0,2], where\nVY(t,s) = Var(Y(t)−Y(s)) =|t−s|κ, s,t≥0\nis the variogram of a centered fractional Brownian motion (f Bm).\nIn this contribution, we shall consider the locally self-si milar risk model, i.e., Xis\nlocally self-similar as introduced in [9, 16]. This definiti on allows for a more general\nY(t),t≥0, which is again a centered Gaussian process with a.s. conti nuous sample\npaths satisfying (set below VY(t) =VY(t,1) = Var( Y(1)−Y(t)))\nS1)Yis self-similar with index α/2>0 andσ2\nY(1) = 1;\nS2) there exist κ∈(0,2] andcY>0 such that VY(1−h)∼cY|h|κash→0.\nFor notational symplicity, Y∈S(α,κ,cY) shall stand as an abbreviation that Y\nsatisfies Item S1)-Item S2).\nDefinition 1.1. A centered Gaussian process X(t),t≥0is said to be locally self-\nsimilar at 0with parameters aandY, if(2)holds for some a >0,Y∈S(α,κ,cY).\nBrief organisation of thepaper: Inour firstresultwe show th at theBerman constant\ndefined with respect to a standard Brownian motion, can be alt ernatively defined with\nrespect to a self-similar process, see Thm 2.2. Then we obtai n the tail asymptotics of\nthe cumulative Parisian ruin (sojourn time) for locally sel f-similar Gaussian processes.\nOur findings are illustrated by several examples. Auxiliary results are presented in\nSection 4 followed by the proofs postponed to Section 5.\n22. Main Results\nIn this section we shall investigate the cumulative Parisia n ruin considering as trend\nfunction c(t) =dtγfor some d,γpositive constants. More general trends can be dealt\nwith similar techniques and will therefore not be considere d here. More specifically,\nwe shall investigate the asymptotics of\npX(T,Lu) =P\n\nT/integraldisplay\n0I(X(t)−dtγ> u)dt > L u\n\n\nas the initial capital u→ ∞for constant Ludependent on uandXbeing a locally\nself-similar Gaussian process.\nIn [9, Thm 3.4] under some conditions a relation between the P ickands constant of\na self-similar process Yand the Pickands constant of a fractional Brownian motion\nBκwas proved.\nRecently the properties of Berman constants were investiga ted in [10]. As therein,\nthe Berman constant with respect to a Gaussian process ζcan be defined as\nBh\nζ(x) = lim\nT→∞1\nT/integraldisplay\nRP/braceleftbigg/integraldisplayT\n0I/parenleftBig√\n2ζ(t)−Var(ζ(t))−h(t)+y >0/parenrightBig\ndt > x/bracerightbigg\ne−ydy.\nFor convenience we will denote B0\nζ(x) byBζ(x). We also introduce\nBh\nζ(x,E) =/integraldisplay\nRP/braceleftbigg/integraldisplay\nEI/parenleftBig√\n2ζ(t)−Var(ζ(t))−h(t)+y >0/parenrightBig\ndt > x/bracerightbigg\ne−ydy.\nWe show below that as in [9, Thm 3.4] the Berman constants can b e defined directly\nwith respect to a standard fractional Brownian motion, whic h is a remarkable result.\nThereis an additional technical difficulty, compared to [9, T hm 3.4] because we cannot\napply the Slepian inequality.\nTheorem 2.1. If/hatwideY∈S/parenleftbig\nκ,κ,c/hatwideY/parenrightbig\nandx≥0, then\nB/hatwideY(x) =c1/κ\n/hatwideYBBκ/parenleftBig\nc1/κ\n/hatwideYx/parenrightBig\n∈(0,∞). (3)\nRemark that Theorem 2.1 does not extend to BY(x) withY∈S(α,κ,cY) in the\nsame way as [9, Thm 3.4], since [9, Lm 4.1 (ii)] cannot begener alized to BY(x). Indeed,\nsetting/hatwideY(t) =Y(tκ/α) as in [9], one obtains\nBY(x)\n= lim\nT→∞1\nT/integraldisplay\nRP/braceleftbigg/integraldisplayT\n0I/parenleftBig√\n2/hatwideY(tα/κ)−Var/parenleftBig\n/hatwideY(tα/κ)/parenrightBig\n+y >0/parenrightBig\ndt > x/bracerightbigg\ne−ydy\n= lim\nT→∞1\nT/integraldisplay\nRP/braceleftBigg/integraldisplayTα/κ\n0I/parenleftBig√\n2/hatwideY(t)−Var/parenleftBig\n/hatwideY(t)/parenrightBig\n+y >0/parenrightBigκ\nαtκ/α−1dt > x/bracerightBigg\ne−ydy,\nwhich is not expressible in terms of B/hatwideY(·).\n3We extend [16, Thm 4.2] to the case of sojourn functional. Set in the following\nˆβ=βκ\nα,ˆγ=γκ\nα.\nWrite next Ψ for the survival function of a standard normal ra ndom variable.\nTheorem 2.2. IfX(t),t≥0is locally self-similar at 0with parameters aandY\nwhich satisfies (1)then we have\nP\n\nT/integraldisplay\n0I(X(t)−dtγ> u)dt > L u\n\n∼cupΨ(u), u→ ∞, (4)\nwhere\n(i) Ifα <min(β,2γ), α≤κ, then for Lu=Lu−2/α+(κ−α\nκ)(2\nα−max(1/γ,2/β)),L≥0\nit holds that\np= 2/κ−max/parenleftBig\n2/ˆβ,1/ˆγ/parenrightBig\n,\nc= (acY)1/κ∞/integraldisplay\n0exp/parenleftBig\n−dI(2γ≤β)zγ−bI(β≤2γ)zβ/parenrightBig\n×zα/κ−1BBκ/parenleftBig\nL(acY)1/κzα/κ−1/parenrightBig\ndz.\n(ii) Ifα <min(β,2γ), α > κ, then for Lu=Lu−2/α−ǫ,\nǫ∈(0,α−κ\nα/parenleftBig\n2/κ−max/parenleftBig\n2/ˆβ,1/ˆγ/parenrightBig/parenrightBig\n],L >0it holds that p=ǫα\nα−κ,\nc= (acY)1/κ∞/integraldisplay\n0exp/parenleftBigg\n−dI/parenleftbigg\nǫ=α−κ\nα(2/κ−1/ˆγ)/parenrightbigg\nzγ\n−bI/parenleftbigg\nǫ=α−κ\nα/parenleftBig\n2/κ−2/ˆβ/parenrightBig/parenrightbigg\nzβ/parenrightBigg\nzα/κ−1BBκ/parenleftBig\nL(acY)1/κzα/κ−1/parenrightBig\ndz.\nIf further ǫ=α−κ\nα/parenleftBig\n2/κ−max/parenleftBig\n2/ˆβ,1/ˆγ/parenrightBig/parenrightBig\n, then the same holds with L= 0.\n(iii) Ifα≥min(β,2γ)orα > κ, then for Lu=Lumin(−2/α,−2/β,−1/γ),L >0it holds\nthatp= 0,\nc=Bh\nYI(α≤min(β,2γ))/parenleftBig\nLa1/κ,[0,∞)/parenrightBig\nwith\nh(t) =a−β/αbtβI(β≤min(α,2γ))+a−γ/αdtγI(2γ≤min(α,β)).\nIfα≥min(β,2γ), then the same holds for L= 0.\n4In particular, the corresponding Berman constants are posit ive and finite.\nCorollary 2.1. IfY∈S(α,κ,cY),VY(x)is decreasing in some neighbourhood of 0,\nVY(x)attains its maximum on [0,1]at the unique point x= 0and1−VY(x)∼Rxβ\nwithβ≥1,β > α/2, and in addition, the function∂\n∂xVY(x)\nxβ−1is bounded on (0,1], then\nthe asymptotics from Theorem 2.2 hold for X(t) =Y(1)−Y(t)fora= 1/2,b=R/2.\n3. Examples\nIn this section we consider different possible examples of Yin Corollary 2.1.\nRemark that if α >min(β,2γ), then\nc=Bh\n0/parenleftBig\nLa1/κ,[0,∞)/parenrightBig\n= exp/parenleftBig\n−h/parenleftBig\nLa1/κ/parenrightBig/parenrightBig\n=aβ/κ−β/αbtβI(β≤min(α,2γ)) +aγ/κ−γ/αdtγI(2γ≤min(α,β)).\nBelowRY(t,s),t,s∈[0,T] stands for the covariance function of Y(t),t∈[0,T].\nExample 3.1. Forα∈(1,2) consider the covariance function\nRY(t,s) =(t+s)α−|t−s|α\n2α.\nIn view of [16] β= 1,R=α22−α,Y∈S(α,α,21−α). Letγ=β/2 = 1/2, then (4)\nholds with L≥0,Lu=Lu−2,p= 0,c=e−α21−αL−dL1/2.\nExample 3.2. Sub-fractional Brownian motion with parameter α∈(0,2) is a cen-\ntered Gaussian process with covariance function\nRY(t,s) =1\n2−2α−1(tα+sα−(t+s)α+|t−s|α\n2).\nIn view of [16] β=α,R=2α−1\n2−2α−1,Y∈S(α,α,(2−2α−1)−1). Letγ=β/2, then\n(4) holds with L≥0,Lu=Lu−2/α,p= 0 and c=Bh\nY/parenleftbig\n2−1/αL,[0,∞)/parenrightbig\n, where\nh(t) =2α−1\n2−2α−1tα+√\n2dtα/2.\nExample 3.3. Negative sub-fractional Brownian motion Ywith parameter α∈(2,4]\nis a centered Gaussian process with covariance function\nRY(t,s) =1\n2α−1−2/parenleftbigg(t+s)α+|t−s|α\n2−tα−sα/parenrightbigg\n.\nIn view of [16] β= 2,R=α(α−1)\n2α−1−2,Y∈S(α,2,α(α−1)2α−3\n2α−1−2). Letγ=β/2 = 1, then (4)\nholds with L≥0,Lu=Lu−1,p= 0 and\nc= exp/parenleftbigg\n−22/αα(α+1)\n2α+1−8L2−21/α−1/2dL/parenrightbigg\n.\nExample 3.4. Weighted fBm Ywith parameters κ∈(0,2] anda >1 is a centered\n5Gaussian process with covariance function\nRY(t,s) =Γ(a+κ)\n2Γ(a)Γ(κ)/integraldisplaymin(t,s)\n0ua−1/parenleftbig\n(t−u)κ−1+(s−u)κ−1/parenrightbig\ndu.\nIn view of [16] β=a,R=Γ(a+κ)\nΓ(a+1)Γ(κ),Y∈S(a+κ−1,κ,Γ(a+κ)\nΓ(a)Γ(κ+1)). Letγ=β\n2=\na\n2,α=a+κ−1.\nIfκ >1, thenα > βand (4) holds with Lu=Lu−2/a,p= 0 and\nc= exp/parenleftbigg\n−21−κ\na+κ−1−a\nκΓ(a+κ)\nΓ(a+1)Γ(κ)La−2a\n2(a+κ−1)−a\n2κdLa/2/parenrightbigg\n.\nIfκ= 1, then Y(t) has the same distribution as B1(ta), and (4) holds with L≥0,\nLu=Lu−2/a,p= 0 and c=Bh\nY/parenleftbig\n2−1/κL,[0,∞)/parenrightbig\n, whereh(t) =ta+√\n2dta/2.\nIfκ <1, thenα < β,α > κ, hence, (4) holds with\nLu=Lu−2\na+κ−1−ǫ, ǫ∈/parenleftBig\n0,2(a−1)(1−κ)\nκa(a+κ−1)/bracketrightBig\n, p=ǫa+κ−1\na−1\nand\nc=/parenleftbiggΓ(a+κ)\n2Γ(a)Γ(κ+1)/parenrightbigg1/κ∞/integraldisplay\n0exp/parenleftbigg\n−dza/2−Γ(a+κ)\n2Γ(a+1)Γ(κ)za/parenrightbigg\n×za−1\nκBBκ/parenleftBigg\nL/parenleftbiggΓ(a+κ)\n2Γ(a)Γ(κ+1)/parenrightbigg1/κ\nza−1\nκ/parenrightBigg\ndz,\nwithL≥0 ifǫ=2(a−1)(1−κ)\nκa(a+κ−1),\nc=/parenleftbiggΓ(a+κ)\n2Γ(a)Γ(κ+1)/parenrightbigg1/κ∞/integraldisplay\n0za−1\nκBBκ/parenleftBigg\nL/parenleftbiggΓ(a+κ)\n2Γ(a)Γ(κ+1)/parenrightbigg1/κ\nza−1\nκ/parenrightBigg\ndz,\nwithL >0 if 0< ǫ <2(a−1)(1−κ)\nκa(a+κ−1),\nc=Bh\nY/parenleftBig\n2−1/κL,[0,∞)/parenrightBig\n,\nwhereh(t) =−21−κ\na+κ−1Γ(a+κ)\nΓ(a+1)Γ(κ)ta−2a\n2(a+κ−1)dta/2withL >0 ifǫ= 0.\nExample 3.5. Integrated fBm Ywith parameter α∈(0,2) is a Gaussian process\ndefined as\nY(t) =√\nα+2/integraldisplayt\n0Bα(s)ds.\n6Its covariance function is of the form\nRY(t,s) =(α+2)/parenleftbig\nsα+1t+stα+1/parenrightbig\n+|t−s|α+2−tα+2−sα+2\n2(α+1).\nIn view of [16] Y∈S(α+2,2,α+2) and β=α+1 ifα≤1 andβ= 2, ifα≥1. Let\nγ=β/2, then for some b >0 (4) holds with Lu=Lu−max(1,2\nα+1),p= 0 and\nc= exp/parenleftBig\n−d·2min((α+1)/2,1)·(1\nα+2−1\n2)Lmin((α+1)/2,1)−bLmin(α+1,2)/parenrightBig\n.\nExample 3.6. Time-average offBm Ywithparameter α∈(0,2] isaGaussianprocess\ndefined as\nY(t) =√\nα+21\nt/integraldisplayt\n0Bα(s)ds.\nIn view of [16] β= 1,Y∈S(α,2,1) andR=α/2 + 1, if α >1,R= 2, ifα= 1.\nLetγ=β/2 = 1/2. Ifα >1, then (4) holds with Lu=Lu−2,p= 0 and c=\ne−21\n2αdL1/2−21\nα(α+2\n4)L. Ifα= 1, then (4) holds with p= 0 and c=Bh\nY/parenleftBig\nL√\n2,[0,∞)/parenrightBig\n,\nwhereh(t) =t+√\n2dt1/2.\nExample 3.7. Dual fBm Ywith parameter α∈(0,2) a centered Gaussian process\nwith covariance function\nRY(t,s) =tαs+sαt\nt+s.\nIn view of [16] Y∈S(α,2,α/2) andβ= 1 and R= 2, ifα >1,R= 3, ifα= 1. Let\nγ=β/2. Ifα >1, then (4) holds with Lu=Lu−2,p= 0,\nc= exp/parenleftBig\n−21\n2α−1/4dL1/2−21\nα−1/2L/parenrightBig\n.\nIfα= 1, then (4) holds with Lu=Lu−2,p= 0,c=Bh\nY/parenleftBig\nL√\n2,[0,∞)/parenrightBig\n, whereh(t) =\n3t+√\n2dt1/2.\n4. Auxiliary lemmas\nWe have to shorten the segment [0 ,T], since for t≫umin(−1/ˆγ,−2/ˆβ,−2/κ)+pin [6, Lm\n4.1] we are not able to get a limiting process.\nRecall that ˆβ=βκ\nα,ˆγ=γκ\nα,/hatwideY(t) =Y/parenleftbig\ntκ/α/parenrightbig\n. Under the notation of Theorem 2.2\ndenoteσX(t) = Var(X(t)). For a random variable Z, we setZ=Z√\nVar(Z)\nif Var(Z)>0.\nLemma 4.1. Suppose that liminf u→∞f(u)/u,limsupu→∞f(u)/u∈(0,∞). There\nexist absolute constants δ,F,G > 0such that\n7P\n\nsup\nt∈[A,A+T]u−2/κX(tκ/α)> f(u),\nsup\nt∈[t0,t0+T]u−2/κX(tκ/α)> f(u)\n\n≤FT2exp(−G(t0−(A+T))κ)Ψ(f(u))\nfor allT≥1,t0> A+T > A > 0and any u≥u0= (4δ)−κ/2(t0+T)κ/2, i.e.\n(t0+T)u−2/κ≤4δ.\nLemma 4.2. There exist absolute constants δ,F,G > 0such that\nP\n\nsup\nt∈[A,A+T]u−2/κX(tκ/α)> u,\nsup\nt∈[A+T,A+2T]u−2/κX(tκ/α)> u\n\n≤F/parenleftBig\nT2exp/parenleftBig\n−G√\nTκ/parenrightBig\n+√\nT/parenrightBig\nΨ(u)\nfor allA >0,T≥1and any u≥u0= (4δ)−κ/2(A+2T)κ/2, i.e.(A+2T)u−2/κ≤4δ.\nLemma 4.3. (i) Ifα≤β, then for all small enough δ >0\nlim\nM→∞limsup\nu→∞1\nu2/κ−2/ˆβΨ(u)P/braceleftBigg\nsup\nt∈[Mu−2/ˆβ,δ]X(tκ/α)> u/bracerightBigg\n= 0; (5)\n(ii) Ifα > β, then for all small enough δ >0\nlim\nM→∞limsup\nu→∞1\nΨ(u)P/braceleftBigg\nsup\nt∈[Mu−2/ˆβ,δ]X(tκ/α)> u/bracerightBigg\n= 0. (6)\nLemma 4.4. i) Ifα≤2γ, then for all small enough δ >0\nlim\nM→∞limsup\nu→∞1\nu2/κ−1/ˆγΨ(u)P/braceleftBigg\nsup\nt∈[Mu−1/ˆγ,δ]/parenleftBig\nX(tκ/α)−u−dtˆγ/parenrightBig\n>0/bracerightBigg\n= 0; (7)\nii) Ifα >2γ, then for all small enough δ >0\nlim\nM→∞limsup\nu→∞1\nΨ(u)P/braceleftBigg\nsup\nt∈[Mu−1/ˆγ,δ]/parenleftBig\nX(tκ/α)−u−dtˆγ/parenrightBig\n>0/bracerightBigg\n= 0.(8)\nLemma 4.5. Ifα > κ, letǫ∈[0,2/κ−2/α), then for Lu=Lu−2/α−ǫwithL >0for\nall small enough δ >0\nlim\nM→∞limsup\nu→∞1\nΨ(u)uǫα\nα−κP\n\nδ/integraldisplay\nMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ntκ/α−1dt > L u\n\n= 0.\n8Lemma 4.6. For each p∈(0,2/κ)we have\nlim\nM→∞limsup\nu→∞1\nupΨ(u)P/braceleftBigg\nsup\nt∈[0,1\nMu−2/κ+p]X(tκ/α)> u/bracerightBigg\n= 0.\nNext, we will need a Pickands lemma for the sojourn functiona l (which is a slightly\nmodified version of [6, Lm 4.1]).\nConsider ξu,j(t),t∈E1, j∈Su, u≥0 a family of centered Gaussian random fields\n(GRF’s) with continuous sample paths and variance function σ2\nu,j.\nSuppose in the following that Suis a countable set for all ularge.\nFor simplicity in the following we assume that 0 ∈E1.\nAs in [6] we shall impose the following assumptions:\nC1{gu,j,j∈Su}is a sequence of deterministic functions of usatisfying\nlim\nu→∞inf\nj∈Sugu,j=∞.\nC2 Var( ξu,j(0)) = 1 for all large uand any j∈Suand there exists some bounded\ncontinuous function honE1such that\nlim\nu→∞sup\ns∈E1,j∈Su/vextendsingle/vextendsingleg2\nu,j(1−σu,j(s))−h(s)/vextendsingle/vextendsingle= 0.\nC3 There exists a centered GRF ζ(s),s∈Rkwith a.s. continuous sample paths\nsuch that\nlim\nu→∞sup\ns,s′∈E1,j∈Su/vextendsingle/vextendsingleg2\nu,j/parenleftbig\nVar/parenleftbig\nξu,j(s)−ξu,j/parenleftbig\ns′/parenrightbig/parenrightbig/parenrightbig\n−2Var/parenleftbig\nζ(s)−ζ/parenleftbig\ns′/parenrightbig/parenrightbig/vextendsingle/vextendsingle= 0.(9)\nC4 There exist positive constants C,ν,u0such that\nsup\nj∈Sug2\nu,jVar/parenleftbig\nξu,j(s)−ξu,j/parenleftbig\ns′/parenrightbig/parenrightbig\n≤C/bardbls−s′/bardblν\nholds for all s,s′∈E1,u≥u0.\nDenote by C(Ei),i= 1,2 the Banach space of all continuous functions f:Ei/ma√sto→R,\nwithEi⊂Rki,ki≥1,i= 1,2 being compact rectangles equipped with the sup-norm.\nLet Γ :C(E1)→C(E2) be a continuous functional satisfying\nF1) For any f∈C(E1), anda >0,b∈R, Γ(af+b) =aΓ(f)+b;\nF2) There exists c >0 such that\nsup\nt∈E2Γ(f)(t)≤csup\ns∈E1f(s),∀f∈C(E1).\nDefine below for given x≥0 and for given positive σ-finite measure ηonE2\nBΓ,h,η\nζ(x,E2) =/integraldisplay\nRP/braceleftbigg/integraldisplay\nE2I/parenleftBig\nΓ/parenleftBig√\n2ζ−Var(ζ)−h/parenrightBig\n(t)+y >0/parenrightBig\nη(dt)> x/bracerightbigg\ne−ydy.\n9Lemma 4.7. Let{ξu,j(s),s∈E1,j∈Su,u≥0}be a family of centered GRF’s\ndefined as above satisfying Item C1-Item C4 and let ΓsatisfyF1-F2. Letηbe a\npositiveσ-finite measure on E2being equivalent with the Lebesgues measure on E2. If\nfor all large uand allj∈Su\nP/braceleftbigg\nsup\nt∈E2Γ(ξu,j)(t)> gu,j/bracerightbigg\n>0,\nthen for all x∈[0,η(E2))\nlim\nu→∞sup\nj∈Su/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/braceleftBig/integraltext\nE2I(Γ(ξu,j)(t)> gu,j)η(dt)> x/bracerightBig\nΨ(gu,j)−BΓ,h,η\nζ(x,E2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0,(10)\nand the constant BΓ,h,η\nζ(x,E2)is continuous at x∈(0,η(E2)).\nThe convergence (10)is uniform in xin compact subsets of (0,η(E2)).\nIfE1=E2and Γ is the identity functional, that is, Γ( f) =ffor each f∈C(E1),\nthen we suppress the superscript Γ and write simply Bh,η\nζinstead of BΓ,h,η\nζ.\nFor a given level u∈Rdefine the excursion set of Xabove the level uby\nAu(X) ={t∈E:X(t)> u}.\nConsider the following convergence\nlim\nu→∞P/braceleftbigg\nµ(Au(X))> v(u)z/vextendsingle/vextendsingle/vextendsinglesup\nt∈EX(t)> u/bracerightbigg\n=¯F(z) (11)\nTheorem 4.8. LetEu,u > 0be compact set of Rksuch that\nlimu→∞P/braceleftbig\nsupt∈EuX(t)> u/bracerightbig\n= 0. Suppose that there exist collections of Lebesgue\nmeasurable disjoint compact sets Ik(u,n),k∈Ku,nwithKu,nnon-empty countable\nindex sets such that\nE(u,n) =/uniondisplay\nk∈Ku,nIk(u,n)⊂Eu,\nthen(11)holds with E=Eu, µ=µuif the measures µuhave no atoms and the\nfollowing three conditions are satisfied:\nA1) (Reduction to relevant sets)\nlim\nn→∞limsup\nu→∞P/braceleftBig\nsupt∈Eu\\E(u,n)X(t)> u/bracerightBig\nP/braceleftBig\nsupt∈E(u,n)X(t)> u/bracerightBig= 0.\nA2) (Uniform single-sum approximation) There exists v(u)>0and¯Fu,n,k,n≥1such\nthat\nlim\nu→∞sup\nk∈Ku,n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleP{µu({t∈Ik(u,n) :X(t)> u})> v(u)x}\nP/braceleftBig\nsupt∈Ik(u,n)X(t)> u/bracerightBig −¯Fu,n,k(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0,(12)\n10wherex≥0, n≥1, and for all x≥0\nlim\nn→∞limsup\nu→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationtext\nk∈Ku,nP/braceleftBig\nsupt∈Ik(u,n)X(t)> u/bracerightBig\n¯Fu,n,k(x)\n/summationtext\nk∈Ku,nP/braceleftBig\nsupt∈Ik(u,n)X(t)> u/bracerightBig−¯F(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0,(13)\nwhere¯F(x)∈(0,1].\nA3) (Double-sum negligibility) For all large nand large u,♯Ku,n≥2and\nlim\nn→∞limsup\nu→∞/summationtext\ni/\\egatio\\slash=j,i,j∈Ku,nP/braceleftBig\nsupt∈Ii(u,n)X(t)> u,supt∈Ij(u,n)X(t)> u/bracerightBig\n/summationtext\nk∈Ku,nP/braceleftBig\nsupt∈Ik(u,n)X(t)> u/bracerightBig = 0.\n5. Proofs\nHereafter, Qi,i∈Nare some positive constants which might be different from line to\nline and f(u,n)∼g(u),u→ ∞,n→ ∞means that\nlim\nn→∞lim\nu→∞f(u,n)\ng(u)= 1.\nWe shall give first the proofs of the auxiliary results and the n present the proofs of\nthe main results.\n5.1. Proofs of auxiliary results\nRemark that by the Borell inequality (see, e.g., [15])\nP\n\nT/integraldisplay\n0I(X(t)−dtγ> u)dt >/parenleftBigκ\nα/parenrightBig\nLu\n\n\n=P\n\nTα/κ/integraldisplay\n0I/parenleftBig\n/hatwideX(t)−dtˆγ> u/parenrightBig\ntκ/α−1dt > L u\n\n\n=P\n\nδ/integraldisplay\n0I/parenleftBigg/hatwideX(t)\n1+dtˆγ\nu> u/parenrightBigg\ntκ/α−1dt > L u\n\n+o(Ψ(u))\n=P\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n+o(Ψ(u)).(14)\nasu→ ∞for an arbitrary fixed δ.\nDenotec1= inf\nx∈[0,1]V/hatwideY(1,x)\n|1−x|κ, c2= sup\nx∈[0,1]V/hatwideY(1,x)\n|1−x|κ.\nIn view of S2 we have\n0< c1,c2<∞.\n11Pickδpositive such that 4 δ </parenleftBig\n1\n8ac2/parenrightBig1/κ\nand for all t,s∈[0,4δ] we have\n1−r/hatwideX(t,s)∈/bracketleftBiga\n2V/hatwideY(t,s),2aV/hatwideY(t,s)/bracketrightBig\n.\nThen\n1−r/hatwideX(t,s)∈/bracketleftBigac1\n2|t−s|κ,2ac2|t−s|κ/bracketrightBig\nfor allt,s∈[0,4δ]. (15)\nWe claim that for such δ >0 the conclusions of Lemmas 4.1-4.6 hold.\nProof of Lemma 4.1. The proof is analogous to the proof of [16, Lm 3.12]. But for\nclarity we provide a complete proof below.\nLetu0= (4δ)−κ/2(t0+T)κ/2and{Zu(t1,t2) : (t1,t2)∈[A,A+T]×[t0,t0+T]},\nwhere\nZu(t1,t2) =X((t1u−2/κ)κ/α)+X((t2u−2/κ)κ/α). Note that\nP/braceleftBigg\nsup\nt∈[A,A+T]u−2/κX(tκ/α)> f(u),sup\nt∈[t0,t0+T]u−2/κX(tκ/α)> f(u)/bracerightBigg\n≤P/braceleftBigg\nsup\n(t1,t2)∈[A,A+T]×[t0,t0+T]Zu(t1,t2)>2f(u)/bracerightBigg\n. (16)\nHence, using the fact that x≤1−e−2xforx∈[0,1/2], by (15), as\n4ac2u−2|t2−t1|κ≤4ac2(u−2/κ(t0+T))κ≤4ac2(4δ)κ<1/2,\nwe obtain\nVX/parenleftBig\n(t1u−2/κ)κ/α,(t2u−2/κ)κ/α/parenrightBig\n= 2/parenleftBig\n1−r/hatwideX(t1u−2/κ,t2u−2/κ)/parenrightBig\n≥ac1u−2|t2−t1|κ; (17)\nVX/parenleftBig\n(t1u−2/κ)κ/α,(t2u−2/κ)κ/α/parenrightBig\n= 2/parenleftBig\n1−r/hatwideX(t1u−2/κ,t2u−2/κ)/parenrightBig\n≤/parenleftbig\n1−exp/parenleftbig\n−8ac2u−2|t2−t1|κ/parenrightbig/parenrightbig\n(18)\nfor allt1,t2≤t0+T. Since\nσ2\nZu(t1,t2) = 2/parenleftBig\n1+r/hatwideX(t1u−2/κ,t2u−2/κ)/parenrightBig\n= 4−VX/parenleftBig\n(t1u−2/κ)κ/α,(t2u−2/κ)κ/α/parenrightBig\n,\nthen from (17) for any ( t1,t2)∈[A,A+T]×[t0,t0+T],\n2≤σ2\nZu(t1,t2)≤4−ac1u−2(t0−(A+T))κ. (19)\n12Now observe that\nP/braceleftBigg\nsup\n(t,s)∈[A,A+T]×[t0,t0+T]Zu(t1,t2)>2f(u)/bracerightBigg\n≤P/braceleftBigg\nsup\n(t1,t2)∈[A,A+T]×[t0,t0+T]Zu(t1,t2)>2f(u)/radicalbig\n4−ac1u−2(t0−(A+T))κ/bracerightBigg\n.(20)\nRemark that\nVZu((t1,t2),(s1,s2)) =VZu((t1,t2),(s1,s2))−(σZu(t1,t2)−σZu(s1,s2))2\nσZu(t1,t2)σZu(s1,s2).(21)\nNote that (using (21) and (19)) for any ( t1,t2),(s1,s2)∈[A,A+T]×[t0,t0+T], we\nhave\nVZu((t1,t2),(s1,s2))≤VZu((t1,t2),(s1,s2))\nσZu(t1,t2)σZu(s1,s2)\n≤1\n2E/braceleftBigg/parenleftBig/parenleftBig\nX((t1u−2/κ)κ/α)−X((s1u−2/κ)κ/α)/parenrightBig\n+/parenleftBig\nX((t2u−2/κ)κ/α)−X((s2u−2/κ)κ/α)/parenrightBig/parenrightBig2/bracerightBigg\n≤VX/parenleftBig\n(t1u−2/κ)κ/α,(s1u−2/κ)κ/α/parenrightBig\n+VX/parenleftBig\n(t2u−2/κ)κ/α,(s2u−2/κ)κ/α/parenrightBig\n(22)\n≤/parenleftbig\n1−exp/parenleftbig\n−8ac2u−2|t1−s1|κ/parenrightbig/parenrightbig\n+/parenleftbig\n1−exp/parenleftbig\n−8ac2u−2|t2−s2|κ/parenrightbig/parenrightbig\n,(23)\nwhere (22) follows from the inequality ( x+y)2≤2(x2+y2) and (23) follows from\n(18).\nConsider two independent, identically distributed center ed stationary Gaussian\nprocesses {Z1,u(t1) :t1≥0},{Z2,u(t2) :t2≥0}withRZ1,u(t1,s1) =\nexp/parenleftbig\n−8ac2u−2|t1−s1|κ/parenrightbig\nand letZu(t1,t2) =1√\n2(Z1,u(t1)+Z2,u(t2)). Hence, by (23),\nfor any ( t1,t2),(s1,s2)∈[A,A+T]×[t0,t0+T],\nVZu((t1,t2),(s1,s2))\n≤/parenleftbig\n1−exp/parenleftbig\n−8ac2u−2|t1−s1|κ/parenrightbig/parenrightbig\n+/parenleftbig\n1−exp/parenleftbig\n−8ac2u−2|t2−s2|κ/parenrightbig/parenrightbig\n=VZu((t1,t2),(s1,s2))\nand due to Slepian’s inequality, we obtain that (with u∗=2f(u)√\n4−ac1u−2(t0−(A+T))κ)\nP/braceleftBigg\nsup\n(t1,t2)∈[A,A+T]×[t0,t0+T]Zu(t1,t2)> u∗/bracerightBigg\n≤P/braceleftBigg\nsup\n(t1,t2)∈[A,A+T]×[t0,t0+T]Zu(t1,t2)> u∗/bracerightBigg\n=P/braceleftBigg\nsup\n(t1,t2)∈[0,T]2Zu(t1,t2)> u∗/bracerightBigg\n, (24)\n13where the last equality follows from stationarity of Zu(·,·). Now we can apply [5, Thm\n2.2] to (24), which combined with (16) and (20) gives\nP\n\nsup\nt∈[A,A+T]u−2/κX(tκ/α)> f(u),\nsup\nt∈[t0,t0+T]u−2/κX(tκ/α)> f(u)\n\n\n≤[T]/summationdisplay\nk=0[T]/summationdisplay\nl=0P\n\nsup\nt∈[A+k,A+(k+1)]u−2/κX(tκ/α)> f(u),\nsup\nt∈[t0+l,t0+(l+1)]u−2/κX(tκ/α)> f(u)\n\n\n≤4T2/parenleftBig\nHBκ/parenleftBig\n(4ac2c2)1/κ/parenrightBig/parenrightBig2\nΨ(u∗)(1+o(1)), (25)\nuniformly in T≥1 asu→ ∞. Since\n(u∗)2=4f2(u)\n4−ac1u−2(t0−(A+T))κ≥f2(u)+ac1\n4/parenleftbiggf(u)\nu/parenrightbigg2\n(t0−(A+T))κ\n≥f2(u)+c2ac1\n4(t0−(A+T))κ,\nwherec>0 is a constant such thatf(u)\nu≥cfor allu≥u0, then, since\nΨ(u)≤exp(−u2/2)\nu√\n2πfor anyu >0,\nΨ(u∗)≤exp/parenleftBig\n−1\n2/parenleftBig\nf2(u)+c2ac1\n4(t0−(A+T))κ/parenrightBig/parenrightBig\n√\n2π/radicalBig\nf2(u)+c2ac1\n4(t0−(A+T))κ\n≤exp/parenleftbig\n−1\n2f2(u)/parenrightbig\n√\n2πf(u)exp/parenleftbigg\n−c2ac1\n8(t0−(A+T))κ/parenrightbigg\n. (26)\nBy (25) and (26) we obtain that\nP/braceleftBigg\nsup\nt∈[A,A+T]u−2/κX(t)> f(u),sup\nt∈[t0,t0+T]u−2/κX(t)> f(u)/bracerightBigg\n≤4F1(4c2ac2)2/κ(HBκ(1))2T2exp/parenleftbigg\n−c2ac1\n8(t0−(A+T))κ/parenrightbigg\nΨ(f(u)),(27)\nfor anyu≥u0and some positive constant F1, such thatexp(−1\n2f2(u))√\n2πf(u)≤F1Ψ(f(u))\nforu > u 0. This completes the proof with F= 4F1(4c2ac2)2/κ(HBκ(1))2andG=\nc2ac1\n8.\nProof of Lemma 4.2. The proof is almost the same as the proof of [16, Lm 3.13],\nbut for completeness we provide it below.\n14Letu0= (4δ)−κ/2(A+2T)κ/2andXu(t) =X((tu−2/κ)κ/α). We have\nP/braceleftBigg\nsup\nt∈[A,A+T]Xu(t)> u, sup\nt∈[A+T,A+2T]Xu(t)> u/bracerightBigg\n=P\n\nsup\nt∈[A,A+T]Xu(t)> u,\n/braceleftBigg\nsup\nt∈[A+T,A+T+√\nT]Xu(t)> u∨sup\nt∈[A+T+√\nT,A+2T]Xu(t)> u/bracerightBigg\n\n\n≤P/braceleftBigg\nsup\nt∈[A,A+T]Xu(t)> u, sup\nt∈[A+T+√\nT,A+2T+√\nT]Xu(t)> u/bracerightBigg\n+P/braceleftBigg\nsup\nt∈[A+T,A+T+√\nT]Xu(t)> u/bracerightBigg\n≤F1T2exp/parenleftBig\n−G√\nTκ/parenrightBig\nΨ(u) +P/braceleftBigg\nsup\nt∈[A+T,A+T+√\nT]Xu(t)> u/bracerightBigg\n,\nwhere the last inequality follows from Lemma 4.1 with t0=A+T+√\nT. Let/tildewideX\nbe a stationary centered GRF with R/tildewideX(t,s) = exp( −8ac2|t−s|κ). Due to Slepian’s\ninequality and the stationarity of /tildewideX\nP/braceleftBigg\nsup\nt∈[A+T,A+T+√\nT]Xu(t)> u/bracerightBigg\n≤P/braceleftBigg\nsup\nt∈[A+T,A+T+√\nT]u−2/κ/tildewideX(t)> u/bracerightBigg\n≤[√\nT]/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[A+T+k,A+T+(k+1)]u−2/κ/tildewideX(t)> u/bracerightBigg\n= 2√\nTP/braceleftBigg\nsup\nt∈[0,1]u−2/κ/tildewideX(t)> u/bracerightBigg\nWe can apply [5, Thm 2.2] to the right side of the inequality ab ove. We obtain that\nfor sufficiently large u≥u0,\nP/braceleftBigg\nsup\nt∈[A,A+T]Xu(t)> u, sup\nt∈[A+T,A+2T]Xu(t)> u/bracerightBigg\n≤F1T2exp/parenleftBig\n−G√\nTκ/parenrightBig\nΨ(u)+4HBκ(a1/κ)√\nTΨ(u)(1+o(1))\n≤F/parenleftBig\nT2exp/parenleftBig\n−G√\nTκ/parenrightBig\n+√\nT/parenrightBig\nΨ(u)\nfor some constant F >0. This completes the proof.\nProof of Lemma 4.3. The proof could be obtained by a slight modification of the\nproofof [16, Thm4.2]. But for clarity we providea complete p roofbelow. Fix arbitrary\nǫ∈(0,b).Setting q= min/parenleftBig\n−2/ˆβ,−2/κ/parenrightBig\n,Mu=Mu−2/ˆβandN= [δ−Mu\nuq],ifasuitably\nsmallδis chosen, we obtain\n15P/braceleftBigg\nsup\nt∈[Mu,δ]/hatwideX(t)> u/bracerightBigg\n≤N/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[Mu+kuq,Mu+(k+1)uq]X(tκ/α)> u(1+(b−ǫ)(Mu+kuq)ˆβ)/bracerightBigg\n.\nRemark that by (15), if a suitably small δis chosen, then\n1−r/hatwideX(t,s)≤1−exp(−4ac2|t−s|κ),∀t,s∈[0,4δ].\nConsequently, we can apply the Slepian inequality. More spe cifically, let\n{Y(t),t≥0}be a centered Gaussian process with covariance function RY(t,t′) =\nexp(−4ac2|t−t′|κ).\nThen we have\nN/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[Mu+kuq,Mu+(k+1)uq]X(tκ/α)> u(1+(b−ǫ)(Mu+kuq)ˆβ)/bracerightBigg\n≤N/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[Mu+kuq,Mu+(k+1)uq]Y(t)> u(1+(b−ǫ)(Mu+kuq)ˆβ)/bracerightBigg\n=N/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[0,uq]Y(t)> u(1+(b−ǫ)(Mu+kuq)ˆβ)/bracerightBigg\n.\nBy [5, Thm 2.2] (which gives an asymptotic approximation uni form ink) for some\nC >0, where Cis independent of M, the above expression is equal to\n(1+o(1))C∞/summationdisplay\nk=0Ψ(u(1+(b−ǫ)(Mu−2/ˆβ+kuq)ˆβ)).\nSince there exists u0>0 such that for all u≥u0andc≥1 we have\nΨ(cu)\nΨ(u)≤2(cu√\n2π)−1e−c2u2/2\n(u√\n2π)−1e−u2/2=2\ncexp/parenleftbig\n−u2(c2−1)/2/parenrightbig\n≤2exp/parenleftbig\n−(c−1)u2/parenrightbig\n,(28)\nwe obtain\n∞/summationdisplay\nk=0Ψ(u(1+(b−ǫ)(Mu+kuq)ˆβ))\n≤Ψ(u)u−q−2/ˆβ∞/summationdisplay\nk=0exp/parenleftBig\n−(b−ǫ)(M+kuq+2/ˆβ)ˆβ/parenrightBig\nuq+2/ˆβ\n≤Ψ(u)u−q−2/ˆβ\nexp/parenleftBig\n−(b−ǫ)Mˆβ/parenrightBig\n+∞/integraldisplay\n0exp/parenleftBig\n−(b−ǫ)(M+x)ˆβ/parenrightBig\ndx\n\n16asu→ ∞. Since further lim M→∞∞/integraltext\n0exp/parenleftBig\n−(b−ǫ)(M+x)ˆβ/parenrightBig\ndx= 0, then the claim\nfollows.\nProof of Lemma 4.4. The proof is almost the same as the proof of Lemma 4.3, but\nfor completeness we provide it below.\nFix arbitrary ǫ∈(0,d). Setting q= min(−2/κ,−1/ˆγ),N= [δ−Nu\nuq] andNu=\nMu−1/ˆγwe can write further\nP/braceleftBigg\nsup\nt∈[Nu,δ]/hatwideX(t)−dtˆγ> u/bracerightBigg\n≤N/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[Nu+kuq,Nu+(k+1)uq]X(tκ/α)> u+(d−ǫ)(Nu+kuq)ˆγ/bracerightBigg\n.\nApplying again the Slepian inequality we have for Y(t),t≥0 a centered Gaussian\nprocess with covariance function RY(t,t′) = exp( −4ac2|t−t′|κ)\nN/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[Nu+kuq,Nu+(k+1)uq]X(tκ/α)> u+(d−ǫ)(Nu+kuq)ˆγ/bracerightBigg\n≤N/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[Nu+kuq,Nu+(k+1)uq]Y(t)> u+(d−ǫ)(Nu+kuq)ˆγ/bracerightBigg\n=N/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈[0,uq]Y(t)> u+(d−ǫ)(Nu+kuq)ˆγ/bracerightBigg\n.\nBy [5, Thm 2.2] for some C >0, where Cis independent of M, the above expression\nis equal to\n(1+o(1))C∞/summationdisplay\nk=0Ψ(u+(d−ǫ)(Nu+kuq)ˆγ).\nHence utilising (28) yields\n∞/summationdisplay\nk=0Ψ(u+(d−ǫ)(Nu+kuq)ˆγ)\n≤Ψ(u)u−q−1/ˆγ∞/summationdisplay\nk=0exp/parenleftBig\n−(d−ǫ)(M+kuq+1/ˆγ)ˆγ/parenrightBig\nuq+1/ˆγ\n≤Ψ(u)u−q−1/ˆγ\nexp/parenleftBig\n−(d−ǫ)Mˆγ/parenrightBig\n+∞/integraldisplay\n0exp/parenleftBig\n−(d−ǫ)(M+x)ˆγ/parenrightBig\ndx\n\nasu→ ∞. Since∞/integraltext\n0exp/parenleftbig\n−(d−ǫ)(M+x)ˆγ/parenrightbig\ndx→0 asM→ ∞the proof is complete.\n17Proof of Lemma 4.5. Remark that\nP\n\nδ/integraldisplay\nMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ntκ/α−1dt > L u\n\n\n≤Nu/summationdisplay\nj=0P\n\n2j+1Mu−2/κ+ǫα\nα−κ/integraldisplay\n2jMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ntκ/α−1dt >1\n2(j+1)2Lu\n\n\n≤Nu/summationdisplay\nj=0P\n\n2j+1Mu−2/κ+ǫα\nα−κ/integraldisplay\n2jMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ndt >/parenleftBig\n2jMu−2/κ+ǫα\nα−κ/parenrightBig1−κ/α1\n2(j+1)2Lu\n\n\n=Nu/summationdisplay\nj=0P\n\n2j+1Mu−2/κ+ǫα\nα−κ/integraldisplay\n2jMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ndt >/parenleftbig\n2jM/parenrightbig1−κ/α1\n2(j+1)2Lu−2/κ\n\n,\nwhereNu=/bracketleftBig\nlog2(δ)−log2/parenleftBig\nMu−2/κ+ǫα\nα−κ/parenrightBig/bracketrightBig\n.\nBut if\n2j+1Mu−2/κ+ǫα\nα−κ/integraldisplay\n2jMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ndt >/parenleftbig\n2jM/parenrightbig1−κ/α1\n2(j+1)2Lu−2/κ,\nthen setting\nAk=\n\nsup\nt∈[/parenleftBig\nk+2jMuǫα\nα−κ/parenrightBig\nu−2/κ,/parenleftBig\nk+1+2jMuǫα\nα−κ/parenrightBig\nu−2/κ]X(tκ/α)> u\n\n\none obtains\n/vextendsingle/vextendsingle/vextendsingle/braceleftBig\nk∈Z: 0≤k≤2jMuǫα\nα−κ, Akholds/bracerightBig/vextendsingle/vextendsingle/vextendsingle>/parenleftbig\n2jM/parenrightbig1−κ/α1\n2(j+1)2L,\nwhich implies that there exist integers 0 ≤k≤l≤2jMuǫα\nα−κsuch that\nl−k >/parenleftbig\n2jM/parenrightbig1−κ/α 1\n2(j+1)2L−1. ForM >(8/L)1\n1−κ/αsupj≥02−j(j+1)2\n1−κ/αwe obtain\nl−k−1>/parenleftbig\n2jM/parenrightbig1−κ/α 1\n4(j+1)2L.\n18Therefore,\nP\n\n2j+1Mu−2/κ+ǫα\nα−κ/integraldisplay\n2jMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ndt >/parenleftbig\n2jM/parenrightbig1−κ/α1\n2(j+1)2Lu−2/κ\n\n\n≤/summationdisplay\n0≤k<k+1+(2jM)1−κ/α 1\n4(j+1)2L<l≤2jMuǫα\nα−κP(Ak∪Al)\n≤/summationdisplay\n0≤k≤2jMuǫα\nα−κ−1∞/summationdisplay\nl=0Fexp/parenleftbigg\n−G/parenleftbigg\nl+/parenleftbig\n2jM/parenrightbig1−κ/α1\n4(j+1)2L/parenrightbiggκ/parenrightbigg\nΨ(u)\n≤2jMuǫα\nα−κ∞/summationdisplay\nl=0Fexp/parenleftbigg\n−G/parenleftbigg\nl+/parenleftbig\n2jM/parenrightbig1−κ/α1\n4(j+1)2L/parenrightbiggκ/parenrightbigg\nΨ(u)\nforsomeconstants F,G >0,wherethepenultimateinequality followsfromLemma4.1,\nifδis chosen small enough so that the conclusion of Lemma 4.1 hol ds, since one can\ncheck that/parenleftBig\nl+1+2jMuǫα\nα−κ/parenrightBig\nu−2/κ≤4δ≤(2ac2)1/κ.\nFinally, we have\nP\n\nδ/integraldisplay\nMu−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ntκ/α−1dt > L u\n\n\n≤uǫα\nα−κ∞/summationdisplay\nj=0∞/summationdisplay\nl=02jMFexp/parenleftbigg\n−G/parenleftbigg\nl+/parenleftbig\n2jM/parenrightbig1−κ/α1\n4(j+1)2L/parenrightbiggκ/parenrightbigg\nΨ(u) (29)\nSince the series\nF(M) :=∞/summationdisplay\nj=0∞/summationdisplay\nl=02jMFexp/parenleftbigg\n−G/parenleftbigg\nl+/parenleftbig\n2jM/parenrightbig1−κ/α1\n4(j+1)2L/parenrightbiggκ/parenrightbigg\nconverges for each M, we obtain\nF/parenleftBig\n2j′M/parenrightBig\n=∞/summationdisplay\nj=j′∞/summationdisplay\nl=02jMFexp/parenleftbigg\n−G/parenleftbigg\nl+/parenleftbig\n2jM/parenrightbig1−κ/α1\n4(j+1)2L/parenrightbiggκ/parenrightbigg\n→0 (30)\nasj′→ ∞, but due to (29)\nlimsup\nu→∞1\nΨ(u)uǫα\nα−κP\n\nδ/integraldisplay\n(2j′M)u−2/κ+ǫα\nα−κI/parenleftBig\nX(tκ/α)> u/parenrightBig\ntκ/α−1dt > L u\n\n≤F/parenleftBig\n2j′M/parenrightBig\n,\nwhich, combined with (30), finishes the proof of Lemma 4.5.\n19Proof of Lemma 4.6. From [16, Thm 4.1] it follows that there exists a constant\nC >0 such that for all M >0\nP/braceleftBigg\nsup\nt∈[0,1\nMu−2/κ+p]X(tκ/α)> u/bracerightBigg\n∼C\nMupΨ(u), u→ ∞,\nwhich implies the conclusion of Lemma 4.6.\nProof of Lemma 4.7. This is the same as [6, Lm 4.1]; the uniform convergence in\nxon compact sets follows from the continuity of BΓ,h,η\nζ(x,E2) combined with the\nmonotonicity of BΓ,h,η\nζ(x,E2) and\nP/braceleftBig/integraltext\nE2I(Γ(ξu,j)(t)> gu,j)η(dt)> x/bracerightBig\ninx.\nProof of Theorem 4.8. This is a minor extension of [6, Thm 1.1] and therefore we\nomit the details.\n5.2. Proof of main results\nProof of Theorem 2.1. LetX(t), t∈[0,T] be a Gaussian process with unit vari-\nance such that rX(t,s) = exp/parenleftbig\n−V/hatwideY(t,s)/parenrightbig\nand set\nXu(t) =X/parenleftBig\ntu−2/κ/parenrightBig\n.\nForT >0 letu(T) be so large (we can find such an u(T) by Lemma 4.7) that\nu(T)−2/κT→0\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nΨ(u(T))P/braceleftbigg/integraldisplayT\n0I/parenleftbig\nXu(T)(t)> u(T)/parenrightbig\n> x/bracerightbigg\n−B/hatwideY(x,[0,T])/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1. (31)\nNext fix Sandntwo positive integers and denote\nΣΣn,S=n2−1/summationdisplay\nk=nn2−1/summationdisplay\nl=k+1P\n\nsup\n[kS,(k+1)S]Xu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig\n,\nsup\n[lS,(l+1)S]Xu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig\n\n.\n20We have the following bound\nP/braceleftBigg/integraldisplayn2S\n0I/parenleftbig\nXu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/parenrightbig\n> x/bracerightBigg\n≤P/braceleftBigg\nsup\n[0,nS]Xu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/bracerightBigg\n+P/braceleftBigg/integraldisplayn2S\nnSI/parenleftbig\nXu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/parenrightbig\n> x/bracerightBigg\n≤P/braceleftBigg\nsup\n[0,nS]Xu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/bracerightBigg\n+n2−1/summationdisplay\nk=nP/braceleftBigg/integraldisplay(k+1)S\nkSI/parenleftbig\nXu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/parenrightbig\n> x/bracerightBigg\n+ΣΣn,S. (32)\nOn the other hand\nP/braceleftBigg/integraldisplayn2S\n0I/parenleftbig\nXu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/parenrightbig\n> x/bracerightBigg\n≥P/braceleftBigg/integraldisplayn2S\nnSI/parenleftbig\nXu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/parenrightbig\n> x/bracerightBigg\n≥n2−1/summationdisplay\nk=nP/braceleftBigg/integraldisplay(k+1)S\nkSI/parenleftbig\nXu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/parenrightbig\n> x/bracerightBigg\n−ΣΣn,S.(33)\nWrite next for S >0\nf(S) = limsup\nn→∞ΣΣn,S\nn2SΨ(u(n2S)).\nSince lim n→∞u(n2S)−2/κn2S= 0, by Lemma 4.1 and Lemma 4.2\nlim\nS→∞f(S) = 0.\nConsequently, letting n→ ∞in (32) and (33), divided by n2S, using (31) for the\nleft-hand side and the fact that (due to [16, Thm 4.1])\nP/braceleftBigg\nsup\n[0,nS]Xu(n2S)(t)> u/parenleftbig\nn2S/parenrightbig/bracerightBigg\n=O(n)\ncombined with Lemma 4.7 for the right-hand sides we obtain\nBBκ(c1/κ\n/hatwideY·)(x,[0,S])\nS−f(S)≤liminf\nnB/hatwideY/parenleftbig\nx,[0,n2S]/parenrightbig\nn2S\n≤limsup\nnB/hatwideY/parenleftbig\nx,[0,n2S]/parenrightbig\nn2S≤BBκ(c1/κ\n/hatwideY·)\nS+f(S),\n21which implies\nBBκ(c1/κ\n/hatwideY·)(x,[0,S])\nS−f(S)≤liminf\nTB/hatwideY(x,[0,T])\nT\n≤limsup\nTB/hatwideY(x,[0,T])\nT≤BBκ(c1/κ\n/hatwideY·)(x,[0,S])\nS+f(S).\nLettingS→ ∞, we obtain B/hatwideY(x) =BBκ(c1/κ\n/hatwideY·)(x) =c1/κ\n/hatwideYBBκ/parenleftBig\nc1/κ\n/hatwideYx/parenrightBig\n.\nProof of Theorem 2.2. In the cases when p(α,β,γ,κ)>0 (that is, cases (i) and\n(ii)), we can fix M′>0 and apply Theorem 4.8 to the process/hatwideX(t)\n1+dtˆγ\nuwith\nEu=/bracketleftbigg1\nM′u−2/κ+p,M′u−2/κ+p/bracketrightbigg\n, Ik(u,n) = [tu,n,k, tu,n,k+1],\ntu,n,k=1\nM′u−2/κ+p+kn/parenleftbig\nac/hatwideY/parenrightbig−1/κu−2/κ,\nwhere 0≤k≤N(u,n) with\nN(u,n) =/bracketleftBigg/parenleftbig\nM′−1\nM′/parenrightbig\nup/parenleftbig\nac/hatwideY/parenrightbig1/κ\nn/bracketrightBigg\n−1.\nAlso, substitute µu(dt) =tκ/α−1dt,x=L,v(u) =u−2+p(α−κ)\nα. Remark that in the\ncases when p(α,β,γ,κ)>0 we have Lu=v(u)x.\nCondition A1 Remark that\nP/braceleftBigg\nsup\nt∈[tu,n,N (u,n),M′u−2/κ+p]/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n≤P/braceleftBigg\nsup\nt∈[tu,n,N (u,n),M′u−2/κ+p]X(tκ/α)> u/bracerightBigg\n≤P/braceleftBigg\nsup\nt∈[tu,n,N (u,n),M′u−2/κ+p]X2(t)> u/bracerightBigg\nfor large uby (15) and the Slepian lemma, where X2is a centered stationary Gaussian\nprocess with unit variance such that RX2(t,s) = exp( −4ac2|t−s|κ). Therefore, since\nM′u−2/κ+p−tu,n,N(u,n)≤n/parenleftbig\nac/hatwideY/parenrightbig−1/κu−2/κ\nby [5, Thm 2.2] we obtain\nP/braceleftBigg\nsup\nt∈[tu,n,N (u,n),M′u−2/κ+p]/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n=O(Ψ(u)), u→ ∞.\n22As we will see later\nP/braceleftBigg\nsup\nt∈E(u,n)/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n≥cΨ(u)up(34)\nfor allu > un,n > n 0, if we properlychoose un>0,c >0,n0>0. Therefore, condition\nA1 is proven.\nCondition A3 We have\n/summationdisplay\ni<j,i,j∈Ku,nP/braceleftBigg\nsup\nt∈Ii(u,n)/hatwideX(t)\n1+dtˆγ\nu> u,sup\nt∈Ij(u,n)/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n≤/summationdisplay\ni<j,i,j∈Ku,nP/braceleftBigg\nsup\nt∈Ii(u,n)X(tκ/α)> u,sup\nt∈Ij(u,n)X(tκ/α)> u/bracerightBigg\n≤Σ1,n+Σ2,n,\nwhere\nΣ1,n=N(u,n)−2/summationdisplay\nk=0N(u,n)−k/summationdisplay\nl=2P/braceleftBigg\nsup\nt∈Ik(u,n)X(tκ/α)> u,sup\nt∈Ik+l(u,n)X(tκ/α)> u/bracerightBigg\n,\nΣ2,n=N(u,n)−1/summationdisplay\nk=0P/braceleftBigg\nsup\nt∈Ik(u,n)X(tκ/α)> u,sup\nt∈Ik+1(u,n)X(tκ/α)> u/bracerightBigg\n.\nBy applying [16, Lm 4.3] after Lemma 4.1 for large enough uandnand some positive\nconstants F1,G1we have\nΣ1,n≤(acˆY)−1/κN(u,n)−2/summationdisplay\nk=0N(u,n)−k/summationdisplay\nl=2F1n2exp(−G1((l−1)n)κ)Ψ(u)\n≤(acˆY)−1/κN(u,n)−2/summationdisplay\nk=0∞/summationdisplay\nl=1F1n2exp(−G1(ln)κ)Ψ(u)\n≤2(acˆY)−1/κN(u,n)F1n2exp(−G1nκ)Ψ(u)\n≤2\nnM′upF1n2exp(−G1nκ)Ψ(u).\nSettu,n,k,l:=l\nntu,n,k+1+n−l\nntu,n,kand remark that tu,n,k,0=tu,n,k,tu,n,k,n=tu,n,k+1\nandtu,n,k,l+1−tu,n,k,l= (ac/hatwideY)−1/κu−2/κfor all 0≤k≤N(u,n)−1 and 0≤l≤n−1.\nTherefore,\nlimsup\nu→∞sup\n0≤k≤N(u,n)−1,\n0≤l≤n−1sup\nt,s∈[tu,n,k,l,tu,n,k,l +1]/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−r/hatwideX(t,s)\n|t−s|κ−cˆY/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\nBy applying [16, Lm 4.3] after Lemma 4.2 for uandnlarge enough and some positive\n23constants F2,G2by Lemma 4.7 with x= 0 we have\nΣ2,n≤(acˆY)−1/κN(u,n)F2/parenleftBig\nn2exp/parenleftBig\n−G2√\nnκ/parenrightBig\n+√n/parenrightBig\nΨ(u)\n≤M′up1\nnF2/parenleftBig\nn2exp/parenleftBig\n−G2√\nnκ/parenrightBig\n+√n/parenrightBig\nΨ(u)\nimplying further\nlim\nn→∞limsup\nu→∞1\nΨ(u)up/summationdisplay\ni/\\egatio\\slash=j, i,j∈Ku,nP/braceleftBigg\nsup\nt∈Ii(u,n)X(tκ/α)> u,sup\nt∈Ij(u,n)X(tκ/α)> u/bracerightBigg\n= 0.\nAs we will see later from (36), we also have\nliminf\nn→∞liminf\nu→∞1\nΨ(u)up/summationdisplay\ni∈Ku,nP/braceleftBigg\nsup\nt∈Ii(u,n)/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n>0 (35)\nand thus condition A3 follows.\nCondition A2 Lett=1\nM′up−2/κ+(τ+kn)/parenleftbig\nac/hatwideY/parenrightbig−1/κu−2/κ,τ∈[0,n]. Then\nP/braceleftBigg\nµu/parenleftBigg\n{t∈Ik(u,n) :/hatwideX(t)\n1+dtˆγ\nu> u}/parenrightBigg\n> v(u)x/bracerightBigg\n=P/braceleftBigg/integraldisplay\nI(n,k)tκ/α−1I/parenleftBigg/hatwideX(t)\n1+dtˆγ\nu> u/parenrightBigg\ndt > Lu−2+p(α−κ)\nα/bracerightBigg\n=P\n\n/integraldisplay\n[0,n]/parenleftBigg\n1\nM′+(τ+kn)/parenleftbig\nac/hatwideY/parenrightbig−1/κ\nup/parenrightBiggκ/α−1\nI/parenleftBigg/hatwideX(t)\n1+dtˆγ\nu> u/parenrightBigg\ndτ > L/parenleftbig\nac/hatwideY/parenrightbig1/κ\n\n\n=P\n\n/integraldisplay\n[0,n]I/parenleftBigg\nX(tκ/α)>u+dtˆγ\n∗\nσ/hatwideX(t∗∗)/parenrightBigg\ndτ\n> L/parenleftbig\nac/hatwideY/parenrightbig1/κ/parenleftBigg\n1\nM′+(τ∗+kn)/parenleftbig\nac/hatwideY/parenrightbig−1/κ\nup/parenrightBigg1−κ/α\n\n\nfor some τ∗∈[0,n];t∗, t∗∗∈Ik(u,n).\nInturn,if nislargeenough,thenbyLemma4.7, sinceinLemma4.7theconv ergence\nis uniform in x, uniformly in kwe obtain (Item C1, Item C2 hold trivially, Item C3\n24follows from the definition of c/hatwideY, Item C4 follows from (15))\nP\n\n/integraldisplay\n[0,n]I/parenleftBigg\nX(tκ/α)>u+dtˆγ\n∗\nσ/hatwideX(t∗∗)/parenrightBigg\ndτ\n> L/parenleftbig\nac/hatwideY/parenrightbig1/κ/parenleftBigg\n1\nM′+(τ∗+kn)/parenleftbig\nac/hatwideY/parenrightbig−1/κ\nup/parenrightBigg1−κ/α\n\n\n∼Ψ/parenleftBigg\nu+dtˆγ\n∗\nσ/hatwideX(t∗∗)/parenrightBigg\nBBκ\nL/parenleftbig\nac/hatwideY/parenrightbig1/κ/parenleftBigg\n1\nM′+(τ∗+kn)/parenleftbig\nac/hatwideY/parenrightbig−1/κ\nup/parenrightBigg1−κ/α\n,[0,n]\n\n∼Ψ(u)BBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κτ1−κ/α\nu,n,k,[0,n]/parenrightBig\n×exp/parenleftBig\n−du1+(p−2/κ)ˆγτˆγ\nu,n,k−bu2+(p−2/κ)ˆβτˆβ\nu,n,k/parenrightBig\nuniformly in k, where we define τu,n,kastu,n,ku2/κ−p.\nBy letting x=L= 0 in the above asymptotics we obtain\nP/braceleftBigg\nsup\nt∈Ik(u,n)/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n∼Ψ(u)BBκ(0,[0,n])\n×exp/parenleftBig\n−du1+(p−2/κ)ˆγτˆγ\nu,n,k−bu2+(p−2/κ)ˆβτˆβ\nu,n,k/parenrightBig\nuniformly in k. Therefore\n¯Fu,n,k(x) =BBκ/parenleftBig\nx/parenleftbig\nac/hatwideY/parenrightbig1/κτ1−κ/α\nu,n,k,[0,n]/parenrightBig\nBBκ(0,[0,n])\nsatisfies (12). Also we have as u→ ∞\n/summationdisplay\nk∈Ku,nP/braceleftBigg\nsup\nt∈Ik(u,n)/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n∼/summationdisplay\nk∈Ku,nΨ(u)BBκ(0,[0,n])exp/parenleftBig\n−du1+(p−2/κ)ˆγτˆγ\nu,n,k−bu2+(p−2/κ)ˆβτˆβ\nu,n,k/parenrightBig\n∼up\nn/parenleftbig\nac/hatwideY/parenrightbig−1/κΨ(u)BBκ(0,[0,n])\n×/summationdisplay\nk∈Ku,nexp/parenleftBig\n−du1+(p−2/κ)ˆγτˆγ\nu,n,k−bu2+(p−2/κ)ˆβτˆβ\nu,n,k/parenrightBig\n(τu,n,k+1−τu,n,k)\n∼up\nn/parenleftbig\nac/hatwideY/parenrightbig−1/κΨ(u)BBκ(0,[0,n])\n×M′/integraldisplay\n1\nM′exp/parenleftBig\n−du1+(p−2/κ)ˆγzˆγ−bu2+(p−2/κ)ˆβzˆβ/parenrightBig\ndz (36)\n25and\n/summationdisplay\nk∈Ku,nP/braceleftBigg\nsup\nt∈Ik(u,n)/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n¯Fu,n,k(x)\n∼/summationdisplay\nk∈Ku,nΨ(u)BBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κτ1−κ/α\nu,n,k,[0,n]/parenrightBig\n×exp/parenleftBig\n−du1+(p−2/κ)ˆγτˆγ\nu,n,k−bu2+(p−2/κ)ˆβτˆβ\nu,n,k/parenrightBig\n∼up\nn/parenleftbig\nac/hatwideY/parenrightbig−1/κΨ(u)\n×/summationdisplay\nk∈Ku,nBBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κτ1−κ/α\nu,n,k,[0,n]/parenrightBig\n×exp/parenleftBig\n−du1+(p−2/κ)ˆγτˆγ\nu,n,k−bu2+(p−2/κ)ˆβτˆβ\nu,n,k/parenrightBig\n(τu,n,k+1−τu,n,k)\n∼up\nn/parenleftbig\nac/hatwideY/parenrightbig−1/κΨ(u)\n×M′/integraldisplay\n1\nM′exp/parenleftBig\n−du1+(p−2/κ)ˆγzˆγ−bu2+(p−2/κ)ˆβzˆβ/parenrightBig\n×BBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κz1−κ/α,[0,n]/parenrightBig\ndz. (37)\nTherefore, (13) is satisfied with\n¯F(x) =M′/integraltext\n1\nM′exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\nA(z)dz\nM′/integraltext\n1\nM′exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\nA(0)dz,\nwhereA(z) =BBκ/parenleftBig\nx/parenleftbig\nac/hatwideY/parenrightbig1/κz1−κ/α/parenrightBig\n.\nIndeed, the existence, positivity and finiteness of BBκ(x) was proved in [6]. In par-\nticular,\nsup\nn>1BBκ(0,[0,n])\nn<∞;\nhence, we can swap the integral and the limit in nin (36) and (37), since\nsup\nn>1,x≥0BBκ(x,[0,n])\nn≤sup\nn>1BBκ(0,[0,n])\nn<∞.\nTherefore, condition A2 is also satisfied.\nThe condition (35) now follows from (36), so, the proof of A3 i s finished, hence, (36)\n26implies\nP/braceleftBigg\nsup\nt∈E(u,n)/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n∼up\nn/parenleftbig\nac/hatwideY/parenrightbig−1/κΨ(u)BBκ(0,[0,n])\n×M′/integraldisplay\n1\nM′exp/parenleftBig\n−du1+(p−2/κ)ˆγzˆγ−bu2+(p−2/κ)ˆβzˆβ/parenrightBig\ndz, (38)\nwhich implies (34), so, the proof of A1 is also finished, and (3 8) implies\nP/braceleftBigg\nsup\nt∈Eu/hatwideX(t)\n1+dtˆγ\nu> u/bracerightBigg\n∼up\n/parenleftbig\nac/hatwideY/parenrightbig−1/κΨ(u)BBκ(0)M′/integraldisplay\n1\nM′exp/parenleftBig\n−du1+(p−2/κ)ˆγzˆγ−bu2+(p−2/κ)ˆβzˆβ/parenrightBig\ndz\n∼up\n/parenleftbig\nac/hatwideY/parenrightbig−1/κΨ(u)BBκ(0)\n×M′/integraldisplay\n1\nM′exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\ndz\nOn combining the above with Theorem 4.8 and (36), we obtain\nP\n\nM′up−2/κ/integraldisplay\n1\nM′up−2/κI\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > Lu−2+p(α−κ)\nα\n\n\n∼upΨ(u)\n/parenleftbig\nac/hatwideY/parenrightbig−1/κ\n×M′/integraldisplay\n1\nM′exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\n×BBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κz1−κ/α/parenrightBig\ndz\n27for allL≥0. Fixǫ∈[0,L\n2], then\nP\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n\n≤P\n\nM′up−2/κ/integraldisplay\n1\nM′up−2/κI\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt >(L−2ǫ)u−2+p(α−κ)\nα\n\n\n+P\n\n1\nM′up−2/κ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > ǫu−2+p(α−κ)\nα\n\n\n+P\n\nδ/integraldisplay\nM′up−2/κI\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > ǫu−2+p(α−κ)\nα\n\n.\nSince for now we consider only the cases when p >0, by Lemmas 4.3-4.6 for all M′\nwe obtain\nP\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n\n≤/parenleftbig\n1+c/parenleftbig\nM′/parenrightbig\n+o(1)/parenrightbigupΨ(u)\n/parenleftbig\nac/hatwideY/parenrightbig−1/κ\n×M′/integraldisplay\n1\nM′exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\n×BBκ/parenleftBig\n(L−2ǫ)/parenleftbig\nac/hatwideY/parenrightbig1/κz1−κ/α/parenrightBig\ndz,\nwherec(M′)→0 asM′→ ∞.\nTherefore, letting M′→ ∞and then letting ǫ↓0 by the continuity of BBκ(x) inx\nand by the monotone convergence theorem we obtain\nP\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n\n≤(1+o(1))upΨ(u)\n/parenleftbig\nac/hatwideY/parenrightbig−1/κ\n×∞/integraldisplay\n0exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\n×BBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κz1−κ/α/parenrightBig\ndz.\n28On the other hand, for all M′\nP\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n\n≥P\n\nM′up−2/κ/integraldisplay\n1\nM′up−2/κI\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n\n≥(1+o(1))upΨ(u)\n/parenleftbig\nac/hatwideY/parenrightbig−1/κ\n×M′/integraldisplay\n1\nM′exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\n×BBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κz1−κ/α/parenrightBig\ndz,\nso, finally, if p >0, then (the first asymptotics follows from (14))\nP\n\nT/integraldisplay\n0I(X(t)−dtγ> u)dt >/parenleftBigκ\nα/parenrightBig\nLu\n\n\n∼P\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n\n∼upΨ(u)\n/parenleftbig\nac/hatwideY/parenrightbig−1/κ\n×∞/integraldisplay\n0exp/parenleftBig\n−dI(1+(p−2/κ)ˆγ= 0)zˆγ−bI/parenleftBig\n2+(p−2/κ)ˆβ= 0/parenrightBig\nzˆβ/parenrightBig\n×BBκ/parenleftBig\nL/parenleftbig\nac/hatwideY/parenrightbig1/κz1−κ/α/parenrightBig\ndz,\nwhich, combined with the fact that c/hatwideY=cY(κ/α)κ, finishes the proof for p >0.\nIn the cases when p(α,β,γ,κ) = 0 (that is, case (iii)), again, fix M′>0. Let\nq= min/parenleftBig\n−2/κ,−2/ˆβ,−1/ˆγ/parenrightBig\n, then if p= 0, then Lu=Luq(κ/α). Remark that\n29P\n\nM′uq/integraldisplay\n0I/parenleftBigg/hatwideX(t)\n1+dtˆγ\nu> u/parenrightBigg\ntκ/α−1dt > L u\n\n\n=P\n\nM′/integraldisplay\n0I/parenleftBigg/hatwideX(tuq)\n1+dtˆγ\nu1−qˆγ> u/parenrightBigg\ntκ/α−1dt > L uu−q(κ/α)\n\n\n=P\n\nM′/integraldisplay\n0I\n/hatwideX(tuq)σ/hatwideX(tuq)\n1+dtˆγ\nu1−qˆγ> u\ntκ/α−1dt > L\n\n,\nso, we can apply Lemma 4.7 to the process/hatwideX(tuq)σ/hatwideX(tuq)\n1+dtˆγ\nu1−qˆγwithη(dt) =tκ/α−1dt,gu,j=\nu.\nCondition Item C1 will hold obviously.\nLet us check condition Item C2:\ng2\nu,j/parenleftBigg\n1−σ/hatwideX(tuq)\n1+dtˆγ\nu1−qˆγ/parenrightBigg\n∼u2b(tuq)ˆβ+ud(tuq)ˆγ→btˆβI/parenleftBig\nq=−2/ˆβ/parenrightBig\n+dtˆγI(q=−1/ˆγ)\nasu→ ∞.\nAs for Item C3:\ng2\nu,jVar/parenleftBig\n/hatwideX(tuq)−/hatwideX(suq)/parenrightBig\n∼2aV/hatwideY(t,s)u2+qκ→2aV/hatwideY(t,s)I(q=−2/κ)\nasu→ ∞.\nCondition Item C4, as before, follows from (15).\nTherefore, for ˆh(t) =btˆβI/parenleftBig\nq=−2/ˆβ/parenrightBig\n+dtˆγI(q=−1/ˆγ) and for a Gaussian cen-\ntered process ζ(t) =/hatwideY/parenleftbig\na1/κt/parenrightbig\nI(q=−2/κ) on [0,∞) we obtain\nP\n\nM′uq/integraldisplay\n0I/parenleftBigg/hatwideX(t)\n1+dtˆγ\nu> u/parenrightBigg\ntκ/α−1dt > L u\n\n\n∼Ψ(u)Bˆh,η\nζ/parenleftbig\nL,[0,M′]/parenrightbig\n= Ψ(u)Bˆh(a−1/κ·),η\n/hatwideYI(q=−2/κ)/parenleftBig\nLa1/κ,[0,M′a1/κ]/parenrightBig\n(39)\nasu→ ∞.\n30By (14) and Lemmas 4.3-4.5, letting M′→ ∞in the above asymptotics, we obtain\nP\n\nT/integraldisplay\n0I(X(t)−dtγ> u)dt >/parenleftBigκ\nα/parenrightBig\nLu\n\n\n∼P\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n\n∼Ψ(u)Bˆh(a−1/κ·),η\n/hatwideYI(q=−2/κ)/parenleftBig\nLa1/κ,[0,∞)/parenrightBig\n= Ψ(u)Bh\nYI(q=−2/κ)/parenleftBig\nL/parenleftBigκ\nα/parenrightBig\na1/κ,[0,∞)/parenrightBig\n,\nwithh(t) =a−β/αbtβI/parenleftBig\nq=−2/ˆβ/parenrightBig\n+a−γ/αdtγI(q=−1/ˆγ), and Theorem 2.2 is\nproven. We do not need to check the finiteness of this Berman co nstant, since\nP\n\nδ/integraldisplay\n0I\nX(tκ/α)>u/parenleftBig\n1+dtˆγ\nu/parenrightBig\nσ/hatwideX(t)\ntκ/α−1dt > L u\n\n=O(Ψ(u))\nby Lemmas 4.3-4.5 and (39).\nAcknowledgements\nFinancial support by SNSF Grant 200021–196888 is kindly ack nowledged.\nDisclosure statement\nThe author has no conflicts of interest to declare that are rel evant to the content of\nthis article.\nFunding\nThis work was supported by the SNSF under Grant 200021–19688 8.\nReferences\n[1] S.M. Berman, Sojourns above a high level for a Gaussian process with a poin t of\nmaximum variance , Comm. Pure Appl. Math. 38 (1985), pp. 519–528. Available at\nhttp://dx.doi.org/10.1002/cpa.3160380505. MR 803245\n[2] S.M. Berman, Sojourns and Extremes of Stochastic Processes , The Wadsworth &\nBrooks/Cole Statistics/Probability Series, Wadsworth & Brooks/C ole Advanced Books\n& Software, Pacific Grove, CA, 1992. MR 1126464 (93a:60055)\n[3] K. D¸ ebicki, Z. 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This configuration is sin-\ngular in the two-derivative theory. We show that the addition of higher-derivative terms in\nfour-dimensional N= 2 supergravity resolves the singularity. In doing so, we extend the\nrecently-developed “new attractor mechanism” to include the effect of higher-derivative\nterms. Remarkably, the one-loop, four-derivative F-term contribution to the prepotential\nleads to a precise match of the gravitational and microscopic index. We also comment,\nusing the effective theory near the horizon, on the possibility of a string-size near-extremal\nblack hole. Our results clarify the meaning of different descriptions of this system in the\nliterature. The thermal state transitions to a winding condensate and a gas of strings\nwithout ever reaching a small black hole, while the index is captured by the rotating Eu-\nclidean black hole solution and is constant and thus smoothly connected to the microscopic\nensemble.arXiv:2402.03297v1  [hep-th]  5 Feb 2024Contents\n1 Introduction 1\n2 The two-charge black hole and its transition to free strings 4\n2.1 Review of the two-charge black hole 5\n2.2 Transition to free strings via the charged Horowitz-Polchinski solution 10\n3 Gravitational configurations that contribute to the index 13\n3.1 Imposing supersymmetry 14\n3.2 Explicit form of the solution 16\n3.3 Summary of properties 18\n4 The index and new attractor mechanism in 4d N= 2supergravity 19\n4.11\n2-BPS solutions of N= 2 supergravity 20\n4.2 Euclidean black holes and near-horizon behavior 21\n4.3 Two-charge solution 24\n5 Higher derivative corrections to the gravitational index 28\n6 Can the low-temperature physics be nearly conformal? 33\n7 Discussion 36\nA Solution generation for rotating two-charge systems 37\nA.1 Generating α′corrections in various cases 40\n1 Introduction\nConsider a fundamental heterotic string of energy M, with wunits of winding and nunits\nof momentum around a circle of finite size. In the weakly-coupled theory the string has a\ntower of excited states whose degeneracy grows exponentially in the excitation numbers.\nIn the strongly-coupled theory, the effective description of the string is as a source for\nthe metric and other low-energy fields of supergravity, and the solution of the low-energy\ntheory is generically a black hole in the ( d+ 1)-dimensional non-compact spacetime whose\nentropy corresponds to the exponentially growing degeneracy of states [1–3].\nWhen all the right-movers are in their ground states, the heterotic string configuration\npreserves half the supersymmetry of the theory and obeys M2=Q2\nR/2 where QRis the\nright-moving momentum. The resulting tower of1\n2-BPS Dabholkar-Harvey states has a\ndegeneracy which grows as exp(4 π√nw) [4]. What is the strongly-coupled description\nof this ensemble of supersymmetric states? One natural possibility is that they form a\n– 1 –supersymmetric black hole. Such a black hole would be extremal with AdS 2×Sd−1near-\nhorizon geometry. In this paper we mainly discuss four non-compact directions ( d= 3),\nalthough we make a few comments on other dimensions as well.\nAs is well-known, this simple picture runs into various problems.1The zeroth-order\nproblem, which was already discussed in the early days, is that the two-derivative super-\ngravity solutions with a fundamental string source is singular, in that the event horizon\nlies on top of the curvature singularity [6]. A more sophisticated analysis suggested that\nincluding higher-derivative terms in the low-energy supergravity stretches the horizon to\nstring scale, thus resolving the singularity [7], [8–10].\nHowever, there are still problems with this picture. Firstly, if we think of obtaining\nthe extremal black hole solution by lowering the mass to its extremal value, there is good\nevidence that there is a transition at low temperatures to the so-called Horowitz-Polchinski\nsolution [11] instead of obtaining an extremal black hole [12].2Secondly, there is no known\nAdS 2×Sd−1solution in supergravity which admits 16 Killing spinors. This could be an\nissue of simply not having found such a solution but, in fact, we can sharpen this argument\nby analyzing possible superconformal groups in the near-horizon region. As we explain in\nSection 6, by studying the quantum super Schwarzian theory near the horizon, one can rule\nout the existence of a decoupled near-horizon AdS 2region with 16 or more supercharges\ndescribing the heterotic string with all its symmetries.\nIn order to clarify the situation, it is useful to focus on particular observables that are\namenable to precise calculation. In particular, the supersymmetric index [14] is indepen-\ndent of coupling, and can be precisely compared in the two descriptions. For the1\n2-BPS\nstates that we are considering, the relevant index is a helicity supertrace in flat space-\ntime (see e.g. [15]). This quantity is protected against changes of coupling, for the usual\nreason of bosonic and fermionic supermultiplets appearing in pairs. In the free theory, the\ngenerating function of this index is the chiral partition function of 24 free bosonic fields\ni.e. 1/η(τ)24[4], whose coefficient at level nwgrows as exp (4 π√nw) as mentioned above.\nThe question we would like to address is: what is the saddle-point of the dual gravitational\npath integral of this index?\nThe aim of this paper is to present and analyze this gravitational saddle-point. Here,\nby saddle-point we mean a Euclidean solution of the low-energy supergravity, treating GN\nin perturbation theory. The main subtlety—and difference from previous treatments—is\nthe inclusion of ( −1)Fin the gravitational functional integral. We use the construction of\nthe saddle-point for the gravitational index discussed in [16, 17]. The basic idea is that\n(−1)Fcan be included in the gravitational functional integral by adding an imaginary part\nto a background bosonic field that couples to an R-symmetry. This implies that the solution\nis necessarily complex. The Euclidean solution is non-extremal and has the topology of a\ncigar times Sd−1, with the horizon at the tip of the cigar.\nIn asymptotic flat space, the relevant background field that implements ( −1)Fis the\n1Notably, this picture does hold in three- and four-charge systems, and leads to the famous agreement\nof statistical and thermodynamic entropy of black holes in string theory [5].\n2See also [13] that studies the validity of the small black hole analysis.\n– 2 –angular velocity Ω which obeys βΩ = 2 πi. The solution has non-zero rotation along one\naxis, which defines a north and south pole at the horizon. Applying this idea to four-\ndimensional N= 2 supergravity coupled to an arbitrary number of vector multiplets leads\nto the new attractor mechanism [18]. The metric of these solutions can be written in the\nIsrael-Wilson-Perj´ es (IWP) form [19, 20], which is sourced by harmonic functions with two\nsimple poles at the north and south poles, respectively. Near the poles, the values of the\nscalars of supergravity get fixed in terms of the charges of the black hole. The complex\nnature of the solution is reflected in that the holomorphic scalars are fixed at the north pole\nand vanish at the south pole. (The anti-holomorphic scalar has the opposite behavior.)\nIn this paper we construct the rotating Euclidean black hole as a solution to the\neffective four-dimensional supergravity describing the heterotic string theory on T6. We\nalso write the solution in the variables of the ten-dimensional heterotic theory. As it turns\nout, the solution is singular at the two-derivative level. In the Einstein frame in four\ndimensions, the metric is smooth but the string coupling goes to zero and so the string-\nframe metric is singular. This opens the possibility that higher-derivative effects can resolve\nthis singularity.\nIn order to resolve this singularity, we study the effect of higher-derivative corrections\nin the supergravity action on the Euclidean rotating black hole solution. We discuss a\nclass of higher-derivative corrections that are summarized in the holomorphic prepotential\nof four-dimensional N= 2 supergravity, and can be calculated using topological string\nmethods in the dual Type II theory. In particular, the one-loop term in the prepotential\nleads to the supersymmetric completion of a particular four-derivative term proportional to\nthe square of the Weyl tensor. This leads to an extension of the new attractor mechanism\nto include higher-derivative terms.\nIn the two-charge system, we find that the singularity is resolved in the higher-\nderivative theory and, in particular, the string coupling at the tip is now non-zero. Note\nthat the resolution involves a play-off between 2-derivative and 4-derivative effects. This\nwould lead to an uncontrollable approximation for generic observables. However, the cal-\nculation for supersymmetric quantities such as the index has a better chance of leading to\na precise result, along the lines of the non-renormalization theorem discussed in [21, 22].\nIn particular, if D-terms vanish on our supersymmetric configurations, the fact that the\ntopological string expansion for Type II on K3×T2is one-loop exact implies that our\nresult would be rigorous. Quite remarkably we find that, with these assumptions, the\ngravitational free-energy of the index agrees precisely with that of the microscopic index\nto give the entropy 4 π√nwincluding the numerical coefficient3.\nNow we briefly summarize the overall picture that we propose. First consider the\nthermal partition function and states in the Lorentzian theory. Away from extremality\nthe fundamental heterotic string is described by the two-charge black hole solution with\n3To evaluate the action we assume it is temperature independent and compute it in the large βlimit.\nAt two derivative level we have proved this earlier [18]. The calculation of the on-shell action (the Gibbons-\nHawking free energy of the system) at finite βincluding the effect of 4-derivative terms is complicated and\nwe postpone it to future work.\n– 3 –a macroscopic horizon. As we lower the mass towards extremality there is a transition\nto a gas of free strings, without ever reaching the extremal black hole solution [23]. Now\nconsider the gravitational path integral for the supersymmetric index as in the present\npaper. The saddle-point is a supersymmetric non-extremal Euclidean geometry valid at all\ntemperatures. The saddle-point is really like an instanton rather than describing Lorentzian\nstates. At the two-derivative level the solution has a large horizon area, but contains\ncurvature singularities near the poles of the horizon. These singularities are resolved by\nthe inclusion of string-scale higher-derivative corrections in the effective action coming from\nstring theory. The index calculated in this manner is independent of coupling, and agrees\nwith the index of a gas of strings at weak-coupling.4In this respect, the index behaves in\na manner similar to the supersymmetric indices corresponding to big black holes5[16–18].\nThey do not depend on coupling, and have continuously connected dual pictures in the\ngravitational and microscopic regimes, rather than a sharp transition.\nThe plan of paper is as follows. In Section 2 we review the family of two-charge black\nhole solutions in heterotic string theory, their thermodynamics, and the transition to the gas\nof free strings via the Horowitz-Polchinski solution. In Section 3 we discuss the gravitational\nsaddle-point for the index, i.e. the Euclidean rotating black hole, in the heterotic string\nvariables. In Section 4 we switch to the four-dimensional effective supergravity description\nat the two-derivative level. We review the new attractor mechanism and the embedding\nof the Euclidean rotating black hole in this description. In Section 5 we include a four-\nderivative term in the 4d supergravity coming from the string compactification. We show\nhow it desingularizes the Euclidean rotating black hole, and analyze its effect on the entropy.\nIn Section 6 we analyze the superconformal groups that can possible arise in the near-\nhorizon geometry and the quantum effects of the respective super-Schwarzians. We show\nthat AdS 2cannot arise at the quantum level in the two-charge system. In Section 7 we\ndiscuss further possible directions. In the appendix we discuss how the solution generating\ntechnique [24, 25] relates the rotating charged and uncharged solutions.\nNote added While this paper was in preparation we received [26], which contains overlap\non the two-derivative solutions that calculate the index.\n2 The two-charge black hole and its transition to free strings\nIn this section we review the properties of the two-charge black hole in the heterotic string\ntheory on T6. In particular, we discuss some peculiar features of the naive extremal limit of\nthis solution. Then we review the construction of the charged Horowitz-Polchinski solution\nand how it modifies the approach to extremality.\n4There may even exist a Horowitz-Polchinski-type supersymmetric solution at intermediate coupling.\nThe details of this would be interesting and may involve a perhaps suitably complexified sigma-model. We\ncomment further on this possibility in Section 3.1.\n5Big black holes refer to black holes with with horizon curvatures parameterically smaller than the\nstring and Planck scale.\n– 4 –2.1 Review of the two-charge black hole\nWe begin by reviewing the two-charge black hole, constructed by Sen in [27], as a solution\nto the four dimensional supergravity theory arising from a toroidal compactification of\nheterotic string theory.\n2.1.1 Heterotic string on T6\nThe four dimensional theory has N= 4 supersymmetry and contains the supergravity mul-\ntiplet, and 22 N= 4 vector multiplets. The supergravity multiplet has six graviphotons,\nresulting in a total gauge group U(1)28. We collectively denote the potentials by Aawith\na= 1, . . . , 28. The moduli space consists of the dilaton Φ and 132 scalar fields organized\nin a symmetric 28 ×28 dimensional matrix Mthat satisfies MLMT=L.Lis a diagonal\nmatrix Laa=−1 for a= 1, . . . , 22 and Laa= +1 for a= 23, . . . , 28. The scalars Mencode\ndeformations of the torus, of the background B-field, and of the Wilson lines. Besides these\nfields, we also have a four dimensional B-field Bµν, which is dual to a scalar.\nThe action, in Lorentzian signature, is given by\nI=1\n16πZ\nd4x√−g e−Φh\nR+ (∂Φ)2+1\n8Tr [∂µM L ∂µM L]\n−1\n12HµνρHµνρ−Fa\nµν(LML )abFbµνi\n+ (fermions) + (bdy terms) ,(2.1)\nwhere we denote the U(1)28field strength by Fa\nµν=∂µAa\nν−∂νAa\nµand the B-field field\nstrength by Hµνρ=∂µBνρ+ 2Aa\nµLabFb\nνρ+ (cyclic). The action is invariant under an\nO(6,22,R) transformation which acts on a, bindices only, so it does not act on the dilaton,\nmetric or Bfield. Consistency with quantization of electromagnetic charge restricts it to\nO(6,22;Z).\nThe solutions we discuss in the following have boundary conditions at infinity such\nthat locally\nMab|∞=δab, Φ|∞= 0, g µν|∞=ηµν. (2.2)\nFor the gauge fields, both the electromagnetic ones and the B-field, we can choose to fix\ntheir holonomies or their charges at infinity.\n2.1.2 Sen’s black hole solution\nThe procedure to find the electrically charged black holes described in [27] is the following.\nFirst, one realizes that for metrics with a U(1) isometry (corresponding to Euclidean time),\nthe symmetry of the action gets enhanced to O(7,23). One can think about this in terms\nof aT7=T6×S1compactification of the heterotic string, but [27] verifies it by explicit\nconstruction of the O(7,23) action. We discuss such transformations more generally in\nAppendix A, not necessarily restricting our attention to black hole solutions.\n– 5 –Let us present the final results. The metric in the Einstein frame gE\nµν=e−Φgµνand\nLorentzian signature is given by\ne−Φds2= ∆1/2h\n−r2+a2cos2θ−2mr\n∆dt2+dr2\nr2+a2−2mr+ dθ2\n+sin2θ\n∆[∆ + a2sin2θ(r2+a2cos2θ+ 2mrcoshαcoshβ)] dϕ2\n−2mrasin2θ(coshα+ cosh β)\n∆dtdϕi\n, (2.3)\nwhere we define the function\n∆ = ( r2+a2cos2θ)2+ 2mr(r2+a2cos2θ)(cosh αcoshβ−1) +m2r2(coshα−coshβ)2.\n(2.4)\nThe dilaton is given by\nΦ =1\n2log(r2+a2cos2θ)2\n∆. (2.5)\nWe can decompose the vector fields into a 22-dimensional vector of one-forms ⃗AL, made\nup of the first 22 components of A, and a 6-dimensional vector of one-forms ⃗ARmade of\nthe last 6 components of A. These gauge potentials and the two-form are given by\n⃗AL=−⃗ n√\n2mrsinhα((r2+a2cos2θ) cosh β+mr(coshα−coshβ))\n∆dt\n+⃗ n√\n2mrasinhαsin2θ(r2+a2cos2θ+mrcoshβ(coshα−coshβ))\n∆dϕ , (2.6)\n⃗AR=−⃗ p√\n2mrsinhβ((r2+a2cos2θ) cosh α+mr(coshβ−coshα))\n∆dt\n+⃗ p√\n2mrasinhβsin2θ(r2+a2cos2θ+mrcoshα(coshβ−coshα))\n∆dϕ , (2.7)\nB=mrasin2θ(coshα−coshβ)\n∆(r2+a2cos2θ+mr(coshαcoshβ−1)) dtdϕ .(2.8)\nThe scalar moduli are given by\nM=I28+ \nPnnTQnpT\nQpnTPppT!\n, (2.9)\nwhere\nP=2m2r2sinh2αsinh2β\n∆, (2.10)\nQ=−2mrsinhαsinhβ\n∆(r2+a2cos2θ+mr(coshαcoshβ−1)). (2.11)\nHere ⃗ nand⃗ pare arbitrary 22 and 6 dimensional unit vectors. It will be important to\nknow the charges and chemical potentials as functions of the parameters appearing in the\nsolution, which are ( m, a,α,β,⃗ n, ⃗ p). Define ⃗QLto be a 22-dimensional vector made out\nof the first 22 components of Qa. Similarly define ⃗QRto be the last 6 components. To\n– 6 –compute the charges we can evaluate the large rlimit of the gauge potentials and read\nthem off from A∼Q/rdt. Using that ∆ ∼r4asr→ ∞ we obtain\n⃗QL=m√\n2sinhαcoshβ⃗ n , ⃗QR=m√\n2sinhβcoshα⃗ p . (2.12)\nWe will also use QL, QRto denote the modulus of the respective charge vectors. Since the\nblack holes are electrically charged and rotating there will also be a non-trivial magnetic\ndipole moment, but it will not be important for the discussion here.\nLet us now move on to the other charges encoded in the metric. The mass and angular\nmomentum of the black hole can be read off from the large rbehavior of gttandgtϕand\nare given by\nM=m(1 + cosh αcoshβ)\n2, J =ma(coshα+ cosh β)\n2. (2.13)\nThe metric has an outer horizon r+at the locations where g−1\nrrvanishes\nr2−2mr+a2= 0 ⇒ r±=m±p\nm2−a2. (2.14)\nThe temperature and angular velocities are\nβ=2πm(coshα+ cosh β)(m+√\nm2−a2)√\nm2−a2, (2.15)\nΩ =a\nm(coshα+ cosh β)(m+√\nm2−a2). (2.16)\nWe see that for the Lorentzian solution to make sense we need m≥aand the limit m→ais\nan extremal limit (in the sense that inner and outer horizons have the same area) although\nthe temperature does not vanish. We will expand on this later. When a→0 and α,β→0\nthen the solution becomes the Schwarszchild black hole. We can check this by looking at\nthe potentials, for example β→8πm.\nHaving found the location of the horizon and the angular velocity, we can evaluate the\nelectric chemical potentials. These quantities are given by µ=iVA|r→∞−iVA|hor.where\nV=∂t+ Ω∂ϕ. This is a gauge invariant statement, but to be more rigorous we should\napply a large gauge transformation such that iVA|hor.= 0. The electric potentials are\n⃗ µL=⃗ n√\n2sinhα\ncoshα+ cosh β, ⃗ µ R=⃗ p√\n2sinhβ\ncoshα+ cosh β. (2.17)\nFinally we can compute the area of the event horizon, with respect to the Einstein frame\nmetric,\nA= 4πm(coshα+ cosh β)(m+p\nm2−a2). (2.18)\nThe on-shell action of the black hole is, in the grand canonical ensemble where we set\nDirichlet boundary conditions for all fields, given by\nlogZ=−Ion−shell=A\n4−βM+β⃗ µL·⃗QL+β⃗ µR·⃗QR+βΩJ . (2.19)\n– 7 –We can compare the supersymmetric limit with the extremal limit. We already men-\ntioned the extremal limit is m=a. The solution is supersymmetric when\nSUSY : M2=⃗Q2\nR\n2. (2.20)\nCombining with (2.13) and (2.12) we see that it implies\nm2(1 + cosh αcoshβ)2=m2cosh2αsinh2β. (2.21)\nIf all parameters are finite and real we cannot satisfy (2.21) as we need either (i) m= 0,\nimplying M= 0 and ⃗QR=⃗0 and, moreover, A= 0—implying a singularity at the horizon;\nor (ii) we have m̸= 0 but the solution is complex, for example we can see that if we solve\nfor cosh αas a function of cosh βthen cosh α<1 for all cosh β∈(+1,∞). As explained by\nSen the supersymmetric solutions have m→0 and either βgoing to infinity or α,βgoing\nto infinity together, but it is not always extremal in the sense of being zero temperature.\n2.1.3 Thermodynamics without (−1)Finsertion\nWe want to describe the termodynamics of non-rotating electric black holes in this setup,\nwithout any insertion of ( −1)F(which we will discuss in later sections). We then set a= 0\nand the expressions in the previous section simplify. The mass and entropy of the black\nhole are given by\nM=m(1 + cosh αcoshβ)\n2, S = 2πm2(coshα+ cosh β), (2.22)\nand the temperature\nβ= 4πm(coshα+ cosh β). (2.23)\nThe expressions for the charges and electric potentials are unmodified.\nThere are several distinct limits one can consider. In this section we assume the validity\nof (2.22) and (2.23), and discuss the thermodynamics implied by them. However, we should\nkeep in mind that these relations have α′corrections which can become important when\nthe black hole becomes small and, in fact, as we will discuss in Section 2.2, the correct\ndescription in certain regime is in fact given by horizonless geometries.\nLow temperature limit The simplest limit one can consider is the low temperature\nlimit, i.e. β→ ∞ , while holding the charges fixed. In this limit we have\nsinhα∼√\n28πQL\nβ∼0, coshα∼1, (2.24)\nsinhβ∼√\n28πQR\nβ∼0, coshβ∼1. (2.25)\nFor this reason, if we compute the mass and entropy as functions of temperature we get\nM∼β\n8π, S ∼β2\n16π. (2.26)\nWe simply recover the behavior of a regular Schwarszchild black hole, the bigger the black\nhole the bigger the mass and the colder it gets. Since the black hole becomes large and\nweakly curved, the black hole description is trustworthy even though the ensemble is ther-\nmodynamically unstable.\n– 8 –The naive extremal limit Since we are considering here the special case a= 0, what\nwe mean by the extremal limit is m→0. However, to keep the charges (2.12) fixed,\nthe limit is not so straightforward. The only solution to this problem is to take either\n|α| → ∞ or|β| → ∞ . For simplicity, here we will focus on the cases where α,β>0.\nThe generalization of the discussion to the cases where one or both of them are negative\nis straightforward.\nWe will consider first the case of β→ ∞ , which as it turns out corresponds to the case\nofQR> Q L. To keep the charges fixed, we take β→ ∞ andm→0 while keeping\nmcoshβ→m0 (2.27)\nfinite. The first surprise, under the assumption that we can trust the black hole solution,\nis that this extremal limit does not imply vanishing temperature. Instead\nβext= 4πm0, Q L=m0√\n2sinhα, Q R=m0√\n2coshα. (2.28)\nFrom the last two equations of (2.28) we see that tanh α=QL/QRandm0=√\n2q\nQ2\nR−Q2\nL.\nTherefore the extremal limit has a finite, charge dependent, temperature\nTmax=1\n4√\n2πq\nQ2\nR−Q2\nL. (2.29)\nThis solution is precisely Sen’s small black hole since the area of the event horizon now\nvanishes\nA∼m2coshβ∼m0m→0. (2.30)\nWe saw earlier that at low temperature the black hole is very large since it behaves as the\nSchwarszchild solution. As the temperature increases the mass decreases until it reaches ex-\ntremality at βext. This is the situation if QR> Q L. At this point the entropy vanishes since\nthe area of the horizon vanishes. If we further extrapolate the expressions to temperature\nhigher than Tmax, the mass parameter becomes negative m < 0, leaving a naked singularity.\nMoreover, the parameter βgoes to the complex plane. At the temperature Tmaxthe value\nof the mass is given by M2\next=Q2\nR/2 which is precisely the BPS mass which is the lowest\npossible from the dual quantum mechanical point of view. In Figure 1, we show how the\nmass and entropy behave as functions of temperature. We should however stress that we\nshould not trust the curves when the temperature becomes too close to Tmaxand the black\nhole becomes string scale. We will discuss the correct behavior in Section 2.2.\nAbove we considered the case of β→ ∞ and therefore QR> Q L. The discussion of\nthe case QL> Q Ressentially mirrors the previous case, but the interpretation is slightly\ndifferent. In this case as the black hole temperature rises, the solution eventually has\nα→ ∞ instead. The expressions above are still true, but now the lowest mass black hole\nhasM=QL/√\n2 which is not supersymmetric. Since QL> Q Rthere is now a range\nof masses, namely Q2\nL/2≥M2≥Q2\nR/2 which is allowed by unitarity in the microscopic\ntheory, but where black holes do not exist. Therefore depending on the sign of QR−QL\nthe ground state might be BPS or not.\n– 9 –(a)\n (b)\nFigure 1 :(a) An illustration of the black hole mass (divided by its extremal value Mext) as a function of\ntemperature (divided by Tmax= 1/(4π√\n2p\n|Q2\nR−Q2\nL|)). The mass decreases until the temperature reaches\nmaximum. The dashed part of the blue line means that the black hole is of string scale and the curve is just\nan extrapolation. (b) An illustration of the black hole entropy (divided by the extremal entropy Sextfrom\nmicrostate counting) as a function of temperature. The naive extrapolation of the entropy simply vanishes\nwhen we reach the maximum temperature.\nFinally, if QL=QRthenα=β. We will not go into the details of this special case,\nbut just note that we need to take both of them to infinity, and the extrapolation of (2.23)\nsuggests that the extremal limit corresponds to infinite temperature βext= 0.\nWe note that the black hole solution in the two derivative theory (2.1) is symmetric\nwith respect to turning on the left or right moving charges. However, from the microscopic\ntheory point of view, we wouldn’t expect such symmetry. For one, the solution should be\nsupersymmetric only when M=QR/√\n2 but not when M=QL/√\n2. As it turns out, this\nsymmetry is broken once one considers the leading α′corrections to the black hole [28].\n2.2 Transition to free strings via the charged Horowitz-Polchinski solution\nAs we saw in the previous section, the extrapolation of the gravity solution (2.3) to the\nextremal limit has some peculiar features. The black hole solution becomes singular as the\nhorizon shrinks to zero size. Of course, once the size of the horizon becomes comparable\nto string scale, the higher curvature corrections would become important and there is no\nreason to trust the solution (2.3). In this section, we will review the result in [12], which\nstates that the approach to extremality of the two-charge system is in fact described by a\ncharged analogue of the Horowitz-Polchinski solution [11].\nHere we will only sketch the idea behind the construction of the solution as details can\nbe found in [12]. The original uncharged Horowitz-Polchinski solution can be constructed\nexplicitly by considering the effective action of the thermal winding modes coupled to\ngravity and dilaton. The uncharged solution is trustworthy when the temperature is below\n– 10 –but very close to the Hagedorn temperature, ( ˜β−βH)/βH≪1,6where we can focus on only\nthe winding modes with winding number ±1 and the fluctuation of the gttcomponent of\nthe metric. The solution carries a classical entropy, which agrees with the Hagedorn density\nof states for a gas of strings close to the Hagedorn temperature. This suggests that the\nphysical interpretation of the solution is describing a gas of highly excited strings, weakly\ninteracting via gravity (see also [29]). It is further suggested in [12] that in Heterotic string\ntheory, this classical solution could potentially be smoothly connected to the Schwarzschild\nblack hole solution as we lower the temperature.7\nAn important property of the solution which differs from the Schwarzschild black hole\nis that as we increase the temperature, the physical size ℓsizeof the solution expands as\n1/q\n˜β−βH. Physically, this means the classical solution is describing a more and more\ndilute string gas. When the temperature is very close to the Hagedorn temperature, the\nclassical solution breaks down and the correct description is simply the quantum description\nof free string gas. In four dimensions, the solution is only valid when\n˜g4\n3s≲(˜β−βH)/βH≪1. (2.31)\nGiven the uncharged solution, [12] performed a solution generating transformation to\nfind the charged solution. The explicit form of the solution can be found in [12], while here\nwe will only mention a few main properties. It turns out that the ˜β→βHlimit of the “seed”\nsolution is mapped to the extremal limits M∼QR/√\n2 orM∼QL/√\n2 of the generated\nsolutions. Differing from the naive extrapolation of the black hole results in Section 2.1.3,\nwhere the temperature rises and reaches a maximum, the temperature of the generated\nHorowitz-Polchinski solution goes to zero, i.e. β→ ∞ in the extremal limit. Furthermore,\nthe solution correctly reproduces the entropy as expected from counting microstates of long\nstrings carrying momentum and winding in the extra directions\nS∼√\n2πℓsq\nQ2\nR−Q2\nL, M ∼QR√\n2,\nS∼πℓsq\nQ2\nL−Q2\nR, M ∼QL√\n2.(2.32)\nThe thermodynamics of the charged solution can be determined purely from the ther-\nmodynamics of the seed solution plus the form of the solution generating transformation at\nspatial infinity. We will discuss the derivation of such relations in Appendix A, extending\nthe analysis in [12] to the cases where the seed solution has rotation. Specifying (A.29)–\n(A.33) to the case without rotation, and expanding the temperature of the seed solution\n6We will use tilded quantities to denote the quantities in the uncharged solution, which are distinct\nfrom those of the charged solution but related via the solution generating procedure, see Appendix. A for\na more detailed discussion.\n7While in the Type II theories, a worldsheet index argument suggests that the two solutions cannot be\nsmoothly interpolated.\n– 11 –around βH= 2πRH= 2π(1 + 1 /√\n2)ℓs, we get\nM=˜S′\n2πRHcoshαcoshβ,\nS=˜S′\n2\u0014\u0012\n1−α′\n2R2\nH\u0013\ncoshα+\u0012\n1 +α′\n2R2\nH\u0013\ncoshβ\u0015\n,\nβ=βH\n2\u0014\u0012\n1 +α′\n2R2\nH\u0013\ncoshα+\u0012\n1−α′\n2R2\nH\u0013\ncoshβ\u0015\n,\nQL=˜S′\n√\n2πRHcoshβsinhα,\nQR=˜S′\n√\n2πRHcoshαsinhβ,(2.33)\nwhere ˜S′is the entropy of the seed solution after stripping off the dependence on the string\ncoupling, and is given by\n˜S′≈c0q\nℓ3s(˜β−βH). (2.34)\nHere c0≈2.41 is an order one number coming from the properties of the seed solution.\nIn summary, (2.33) and (2.34) describe the thermodynamic quantities of the charged\nHorowitz-Polchinski solution, parameterized by three parameters {˜β,α,β}. They are only\nvalid when we satisfy (2.31). Note that the lower end of the inequliaty in (2.31) is expressed\nin terms of the string coupling constant in the seed solution, rather than the generated\nsolution.\nWith (2.33), we can reexamine how the naive approach to extremality in section 2.1.3\ngets modified. As we mentioned, now the extremal limit corresponds to ˜β→βH, and to\nfix the charges, we need to combine it with the limits α→ ∞ orβ→ ∞ . For the case\nofα→ ∞ , we have M→QL/√\n2, and for β→ ∞ , we have M→QR/√\n2. From the\nexpression of βin (2.33), we clearly see that both would correspond to the limit where\nβ→ ∞ , i.e. the temperature goes to zero in the extremal limit.\nWe can consider some special cases for which we can write down simpler formulas\nfor how the energy and entropy depends on the temperature. Consider the case QL= 0\n(α= 0,β→ ∞ ), at low temperature we have\nM−Mext\nMext∼S−Sext\nSext∼2e−2β∼(πℓsT)2, (2.35)\nwhere Mext=QR/√\n2 and Sext=√\n2πℓsQR. Translating (2.31) into the parameters of the\ngenerated solution, we find that the solution is trustworthy when\n\u0012ℓs\nQR\u00131\n2\n≪QRT≪1. (2.36)\nOn the other hand, for the case of QR= 0 (β= 0,α→ ∞ ), at low temperature we have\nM−Mext\nMext∼S−Sext\nSext∼2e−2α∼2(πℓsT)2, (2.37)\n– 12 –(a)\n (b)\nFigure 2 :(a) The blue curves are from Figure 1 for comparison. The orange curve is the mass of the\ncharged Horowitz-Polchinski solution as a function of the temperature. It is trustworthy whenp\nℓs/|Q| ≪\nT/T max≪1, where |Q|can be taken to be the greater of |QL|,|QR|. The dashed lines are extrapolations.\nWe see that the temperature goes to zero in the extremal limit. (b) The orange curve is the entropy of\nthe Horowitz-Polchinski as a function of the temperature. We see that S/S extgoes to one in the extremal\nlimit. The precise shape of the curves depends on the values of QRandℓsin Planck units. In the\nmicrocanonical ensemble, where we vary the mass, we expect the two curves to be joined together near\nwhere their extrapolations cross.\nwhere Mext=QL/√\n2 and Sext=πℓsQL. The region of validity of the solution is similar\nto (2.36), but with QRreplaced by QL. Note that we have δS∝δM, which is expected\nfrom the microscopic entropy formula\nS= 2πℓs\"r\nM2−Q2\nL\n2+1√\n2r\nM2−Q2\nR\n2#\n(2.38)\nwhen we expand around extremality.\nWe note that the low temperature thermodynamics is qualitatively different from what\none would expect if there were an AdS 2region in the geometry. In Section 6 we will argue\nthat the fact that the low energy physics is not given by AdS 2could be argued from more\ngeneral principle. In Figure 2 we illustrate that after considering the charged Horowitz-\nPolchinski solution, the naive extremal limit in Figure 1 is modified significantly.\nWe close this section by mentioning that other than the thermodynamic quantities,\none can also look at the physical size of the Horowitz-Polchinski solution, and it matches\nwith the size of the BPS string expected from a random walk picture [12],\nℓphys∼(nw)1\n4ℓs. (2.39)\n3 Gravitational configurations that contribute to the index\nWe now initiate our study of the gravitational path integral that evaluates the index of\ntwo-charge states. We first construct supersymmetric solutions by taking a non-trivial limit\n– 13 –of the two-charge black hole analyzed in the previous section. The situation is more subtle\nthan its ‘big black hole’ counterpart and we emphasize some puzzling features. Those\nfeatures will go away later when we include higher derivative corrections.\n3.1 Imposing supersymmetry\nWe want to construct supersymmetric finite temperature black hole solutions that con-\ntribute to the gravitational path integral for the index. We actually need to consider the\nhelicity supertrace, but one can simply absorb the fermion zero modes and we will not\nworry about this here. Imposing supersymmetry at the boundary (i.e. the asymptotic re-\ngion of flat space) requires fermions to be periodic. As explained in [17] this constrains the\nangular velocity Ω. Supersymmetry in the bulk requires the existence of a globally defined\nKilling spinor. For the black hole geometries presented in Section 2 this constrains the\nmass parameter M. Explicitly these two conditions are given by\nBoundary Supersymmetry : βΩ = 2 πimod 4 πZ,\nBulk Supersymmetry : M=QR/√\n2.\nThe first condition fixes the fermion periodicity conditions because this choice of angular\nvelocity amounts to an insertion of eβΩJ→e2πiJ= (−1)F. Since all excitations have\nintegral or half-integral angular momentum this condition defines the angular velocity\nonly up to a shift multiple of 4 π/β. Bulk and boundary supersymmetry are in principle\nindependent. For example, in the case of Reissner-Nordstr¨ om, if βΩ = 2 πithen this implies\nthat M=Qand a globally defined Killing spinor exists, but if βΩ = 6 πior any other\nodd multiple of 2 πi, then M̸=Qand supersymmetry is not preserved in the bulk. One\nunexpected feature of the two-charge system is that even for βΩ = 2 πibulk supersymmetry\nis not guaranteed.\nThe discussion here will assume that in the presence of ( −1)Fthe gravitational path\nintegral is dominated by a black hole-like saddle. Can there be another saddle similar to\nthe Horowitz-Polchinski one that contributes to the index? One option, since the thermal\ncircle is non-contractible, is to take the Horowitz-Polchinski solution of the previous section\nand impose fermions to be periodic. The problem with this proposal is that the solution\nis not supersymmetric and therefore does not contribute to the index due to fermion zero-\nmodes.8Note also that the Horowitz-Polchinski solution also breaks down before we reach\nexact extremality due to quantum corrections. Even if this had worked, the problem is\nthat the equations of motion that Horowitz-Polchinski solves are not correct in the presence\nof (−1)F. At least in flat space, periodic fermions lead to the standard GSO projection\nand therefore the light winding mode that condenses in the Horowitz-Polchinski solution\nis not part of the low energy spectrum [30]. It is therefore not clear which equations\nwe are supposed to solve to find such a solution. It is possible that by turning on an\nimaginary angular potential, similar to the black hole case, one can evade the flat space\nGSO projection argument and arrive at a supersymmetric solution, but we are not sure\n8Even if it did, it produces a contribution that depends on temperature in the free energy, inconsistent\nwith the behavior of an index.\n– 14 –how to construct such a solution explicitly (see related discussion in [31]). We should\nemphasize that even if such a solution was found and produces the correct index, it would\nnot invalidate our discussion below. On the contrary, it would be in harmony with the\nconjecture that there is no phase transition between the Horowitz-Polchinski solution and\nthe black hole in the heterotic theory. Since the supergravity sector is part of the low\nenergy spectrum when fermions are periodic, and since there are black hole solutions in\nthis sector, this will be our focus in the rest of the paper.\nAfter these introductory remarks let us impose βΩ = 2 πiand see what the implications\nare for the two-charge black hole. Since the angular velocity is imaginary it is convenient\nto introduce a=ia, soawill then be real. The condition becomes then\nβΩ\n2πi=a√\nm2+a2= 1. (3.1)\nIt is convenient sometimes to replace the variables ( m,a) by ( m, x =a/m). Then the\nboundary supersymmetry condition becomes x=√\n1 +x2, and the only solution is to take\nx→ ∞ . We are working in the thermal ensemble and the black hole temperature, in the\nlimit x=a/m→ ∞ , becomes\nβ=2πm(coshα+ cosh β)(m+√\nm2+a2)√\nm2+a2\f\f\f\na/m→∞= 2πm(coshα+ cosh β).(3.2)\nSince the temperature is a free parameter set by our choice of boundary conditions at\nasymptotic flat space, the right hand side of this expression should remain finite. We\nanalyze below two ways to implement this condition.\nSetting a→ ∞ and mfinite As a first possibility we can keep m,αandβfinite,\nwith one combination of them set by the temperature, and send a→ ∞ to implement\nboundary supersymmetry. We will see now this option is ruled out after imposing bulk\nsupersymmetry. For mfinite, combining (2.13) with (2.12) one can show that setting\nM2=Q2\nR/2 implies the following relation between the parameters αandβ\n(1 + cosh αcoshβ)2= cosh2αsinh2β. (3.3)\nThere is no real solution of this equation for finite αandβ, since the constraint implies\neα+β<0. It is not clear that the black hole solves the correct equations of motion since\nfor complex values of αandβthe transformation used to generate the solution is not in\nO(7,23) any longer. There is one more problem with this scenario that is worth mentioning.\nWe evaluate the area of the event horizon, given in (2.18), in the large alimit\nA→4πm(coshα+ cosh β)mx ⇒ A= 2βmx . (3.4)\nWe see that at fixed temperature and keeping mfinite, the area is A∝a→ ∞ . Now\ninstead of vanishing, the area actually diverges! It is not obvious that this feature is\nnecessarily problematic, as explained in [23], but we would like to avoid it if possible since\nthe interpretation of such a saddle is unclear. For these reasons we will not consider this\ndirection any further.\n– 15 –Setting m→0and afinite Another way of implementing the bulk supersymmetry\ncondition is to keep afinite while taking the mass parameter mto zero. This is the regime\nconsidered in [27], in the absence of the ( −1)Finsertion. Looking at the expression for the\ntemperature given in (3.2) we see that this limit requires taking (cosh α+ cosh β)→ ∞ .\nHow this limit is taken is fixed by bulk supersymmetry. For this purpose let us take the\nm→0 limit of the charges\n⃗QL=β\n2π√\n2sinhαcoshβ\ncoshα+ cosh β⃗ n , ⃗QR=β\n2π√\n2sinhβcoshα\ncoshα+ cosh β⃗ p . (3.5)\nand similarly the mass\nM=β\n4π1 + cosh αcoshβ\ncoshα+ cosh β. (3.6)\nOne can check that β→ ∞ while keeping αfinite does provide a solution of M2=Q2\nR/2,\nas pointed out in [27]. To guarantee both boundary and bulk supersymmetry, as well as a\nfinite temperature, we then take\nm→0,β→ ∞ , such that mcoshβ→m0, (3.7)\nwhere we introduced the finite parameter m0. In terms only of parameters that are finite\nin the limit, the temperature and charges are\nβ= 2πm0, Q L=m0√\n2sinhα, Q R=m0√\n2coshα. (3.8)\nWe can use the second and third equations to solve for m0andαin terms of the charges.\nExplicitly, we obtain\nm0=q\n2(Q2\nR−Q2\nL), tanhα=QL\nQR. (3.9)\nUsing these values, we see that the temperature is fixed in terms of the charges! Moreover,\nthe parameter ais undetermined. We will comment more on this later. This solution\nonly exists if QR> Q L. No supersymmetric solution of this kind exists if QL> Q R\ninstead.9This is consistent with the spectrum of heterotic strings compactified on a torus,\nand consistent with the spectrum of Lorentzian black holes in the absence of the ( −1)F\ninsertion discussed in Section 2. Finally, the chemical potentials of the left and right charges\nhave magnitudes µL= 0 and µR= 1/√\n2, independent of the parameters m0andα.\n3.2 Explicit form of the solution\nNow that we have determined the scaling of all parameters in the supersymmetric limit we\ncan write down explicitly how the solution looks like. The same solution was written down\n9What happens with the supergravity solution when QL=QR? In this case α=βand bulk supersym-\nmetry demands that α=β→ ∞ . But it is evident from (3.5) that in this case keeping the temperature\nfinite leads to QL, QR→ ∞ , therefore no solution exists with finite QL=QR. Another option is to take\nβ→0 such that βeβis finite. This implies now that m∼e−2β, vanishing faster than before. To keep the\nblack hole area finite we need to send a→ ∞ . Since its not clear whether this is a physical solution or not\nwe will not consider it further and restrict ourselves to QR> Q L.\n– 16 –in [6, 32] although it was interpreted differently. In the supersymmetric limit the metric\nsimplifies slightly,\ne−Φds2= ∆1\n2nr2−a2cos2θ\n∆dt2\nE+dr2\nr2−a2+ dθ2\n+sin2θ\n∆[∆−a2sin2θ(r2−a2cos2θ+ 2m0rcoshα)] dϕ2−2m0rasin2θ\n∆dtEdϕo\n,\n(3.10)\nwhere the supersymmetric limit of the function ∆ becomes\n∆ = ( r2−a2cos2θ)2+ 2m0r(r2−a2cos2θ) cosh α+m2\n0r2. (3.11)\nHere, we remind the reader that d s2is defined in the string frame. The dilaton is still\ngiven by Φ = log\u0000\nr2−a2cos2θ\u0001\n/∆1/2. The solution is real in Euclidean space. From the\nsecond term in the first line we see that the horizon is located at\nr+=a, (3.12)\ngiving a geometric interpretation to a. One can verify directly with the supersymmetric\nsolution that smoothness of the metric at the horizon imposes values of the temperature and\nangular velocity consistent with the previous analysis, namely β= 2πm0and Ω = i/m 0,\nsuch that βΩ = 2 πi. In fact, the solution is smooth everywhere on the sphere except at\nthe poles, which we comment on later. The area of the event horizon in Einstein frame is\nA=Z\ndθdϕq\ngE\nθθgE\nϕϕ=Z\ndθdϕ m0asinθ= 4πm0a, (3.13)\nwhich is finite and non-vanishing! The rotation necessary to implement the ( −1)Fhas\nturned the small black hole (at extremality) into a macroscopic one at finite temperature\nfor the index. This seems to suggest we are now able to use the gravitational path integral\nto evaluate the index of the two-charge state avoiding small black holes, but this is too\nnaive. Even though the solution has a macroscopic horizon, the black hole geometry is\nactually singular at the poles of the sphere θ= 0, πso the rotation alone has not fixed all\nissues. This is clear when looking in the Einstein frame metric which is given by the string\nframe one multiplied by eΦ. Near the horizon\neΦ\f\f\nr=a= ∆−1\n2a2sin2θ , (3.14)\nand ∆ |r=ais a non-vanishing finite function of θ. Therefore we see that eΦvanishes at the\ntwo poles of the sphere. A more detailed analysis shows that the string frame metric is\nsingular, while the vanishing of eΦat the poles makes the Einstein frame metric smooth,\nalthough higher derivative corrections in the Einstein frame can be large due to dilaton\ndiverging.\n– 17 –The solution is specified by the gauge potentials besides the metric. These also have\na finite limit in the supersymmetric limit given by\n⃗AL=−⃗ n√\n2m0rsinhα(r2−a2cos2θ)\n∆dt−i⃗ n√\n2m2\n0r2asinh 2αsin2θ\n2∆dϕ ,\n⃗AR=−⃗ p√\n2m0r((r2−a2cos2θ) cosh α+m0r)\n∆dt\n+i⃗ p√\n2m0rasin2θ(r2−a2cos2θ+m0rcoshα)\n∆dϕ ,\nB=−im0rasin2θ\n∆(r2−a2cos2θ+m0rcoshα) dtdϕ . (3.15)\nThese gauge potentials are smooth and one can extract from them the chemical potentials,\nverifying that µL= 0 and µR= 1/√\n2. Finally, the functions appearing in the moduli\nscalar profiles Malso simplify\nP=2m2\n0r2sinh2α\n∆, Q=−2m0rsinhα\n∆(r2−a2cos2θ+m0rcoshα). (3.16)\nThese functions are smooth and finite at the horizon.\nIn general, the Euclidean action of the two-charge heterotic black hole is given by\nIon−shell=βM−A/4−βΩJand comes solely from the boundary terms in the action.\nThis is the result in an ensemble of fixed electric charges. If we have a supersymmetric\nsolution then the first term is βQ2\nR/2. In the rest of the paper we will always include an\nappropriate boundary term that has the effect of shifting the zero-point energy such that\nI→I−βQ2\nR/2. After this modification, −Ion−shellsimplifies to A/4 +βΩJ. We can\nshow now that whenever a supersymmetric saddle exists, then these two terms cancel each\nother and the classical two-derivative contribution to the on-shell action, and therefore\nto log Zindex vanishes. The area and the angular momentum can be written as\nA\n4=βa\n2, J =iβa\n4π. (3.17)\nWe already discussed the area. The angular momentum can be read off directly from the\nsupersymmetric solution or taking the appropriate limit of (2.13). Using that βΩ = 2 πi\nwe get the desired cancellation\nA\n4+βΩJ= 0,⇒ logZindex =−Ion−shell= 0. (3.18)\nWe found a black hole with a macroscopic area that nevertheless gives a vanishing contri-\nbution to the index to leading order. This is the desired conclusion, since a non-zero action\nat the two-derivative level would, by dimensional analysis, grow too fast with the charges\nand would not reproduce the microscopic index.\n3.3 Summary of properties\nTo recap, we have found a supersymmetric finite temperature black hole with two-charges\nthat evaluates the index avoiding the small black hole regime. Even after including the\n– 18 –rotation necessary to implement the ( −1)Fsome puzzles remain to be resolved. We finish\nthis section by summarizing and expanding on some of them. In the next sections we will\nresolve them by including higher derivative corrections.\n1. The solution we found has a curvature singularity at the poles of the S2horizon\nr=r+=aatθ= 0 and π. This forces us to include higher derivative corrections\neven though the event horizon is macroscopic.\n2. For a given set of charges, the solution only exists for a unique value of the tem-\nperature. This is surpring since charges and temperatures can be independently\ndetermined by boundary conditions at infinity. After taking the supersymmetric\nlimit we found\nβ= 2πq\n2(Q2\nR−Q2\nL).\nFor comparison, in the Reissner-Nordstr¨ om case analyzed in [17] the inverse tem-\nperature is β=πQ(Q+ 2a)/a, while r+=Q+a. A solution therefore exists for\nany temperature and the horizon radius depends on its value. For the case consid-\nered here, the singular feature is that the dependence on adisappeared completely\nfrom β, making it a function only of the charge. What solution contributes to the\ngravitational path integral when β̸= 2πq\n2(Q2\nR−Q2\nL)?\n3. In the Reissner-Nordstr¨ om case quoted in the previous item, not only is the tempera-\nture a tunable parameter but it also determines a unique value of a. In the two-charge\ncase another consequence of the fact that adrops out of the temperature10is that\nthere is a moduli space of solutions labeled by afrom 0 to ∞. All values of acon-\ntribute equally and have to be integrated over. This is a disturbing state of affairs\nand we will see that higher derivative corrections will determine a unique value of a.\nBefore addressing the effect of higher derivative corrections, we will show in the next\nsection that these features are common to a class of two-charge black holes in N ≥ 2\nsupergravity.\n4 The index and new attractor mechanism in 4d N= 2supergravity\nIn this section we move to the N= 4 four-dimensional effective theory that describes the\ncompactification of the heterotic string on T6. From the string-string duality, this is equiva-\nlent to Type II theory on K3×T2. More generally, we can consider four-dimensional N= 2\ntheories that arise from the compactification of Type II on a Calabi-Yau 3-fold. These the-\nories are described by N= 2 supergravity coupled to vector and hyper multiplets. The\nfields that participate in the black hole solutions live in the graviton multiplet and the\n10The fact that adrops out, and the temperature is fixed in terms of the charges, is special to compacti-\nfications to four dimensions. Similar black hole solutions in toroidal compactifications to D > 4 dimensions\ndo not have this problem [26]. The reason we insist on working in D= 4 is the fact that supersymmetric\nhigher derivative solutions are known and we use them in Section 5.\n– 19 –vector multiplet, and we focus on this sector. In this section we discuss the theory with a\ntwo-derivative action, and in the next section we consider higher-derivative corrections.\nThe conformal supergravity formalism is an elegant and powerful formalism to describe\nthe theory of N= 2 supergravity coupled to a number of vector multiplets [33, 34]. In\nthis formalism, local conformal invariance is a gauge symmetry of the theory. Accordingly,\neach field is assigned a conformal or Weyl weight. Eventually, the conformal symmetry\nis gauge-fixed by assigning a dimensionful number like mPlto a field with non-zero Weyl\nweight. This gauge-fixing leads to Poincar´ e supergravity.\nThe bosonic physical field content of each vector multiplet consists of a vector field AI\nand a complex scalar XI,I= 0,1, . . . , n v. The scalars have Weyl weight 1. Solutions\nof the theory carry electric and magnetic charges ( QI, PI) under these gauge fields. The\ntwo-derivative action is completely specified by the prepotential F(X), which is a homo-\ngeneous function of degree two and has Weyl weight 2. An important role is played by\nthe first derivatives FI=∂F/∂XIwhich have Weyl weight 1. At the two-derivative level\nthe equations of motion of the theory have Sp(2nv+ 2;R) symmetry, under which the\ncharge vector Γ = ( PI, QI) and the period vector Ωhol= (XI, FI) transform as 2( nv+ 1)-\ndimensional vectors.\nTwo important invariants of the symplectic symmetry are11the generalized K¨ ahler\npotential (of Weyl weight 2)\ne−K(X,X)=i\u0000\nXIFI−XIFI\u0001\n=i⟨Ωhol,Ωhol⟩, (4.1)\nand the central charge function (of Weyl weight 0) ,\nZ(Γ; Ω) = ⟨Γ,Ω⟩=e1\n2K(X,X)(PIFI−QIXI). (4.2)\nThe conjugate central function Z(Γ; Ω) = Z(Γ;Ω) is also used below.\n4.11\n2-BPS solutions of N= 2supergravity\nThe general1\n2-BPS stationary solution of this supergravity theory is described as fol-\nlows [35, 36]. The input to the solution the vector H(x) =\u0000eHI(x), HI(x)\u0001\nof harmonic\nfunctions on base space. These functions are sources of electric and magnetic fields.\nThe metric takes the form\nds2=e2U(dtE+iω)2+e−2Udxmdxm, (4.3)\nwhere U(x) is a function of the base space coordinates xm,m= 1,2,3, and ω=ωmdxmis\na one-form on the base space that governs the angular rotation of the solution. In order to\nhave asymptotically flat space, one has e−2U(∞)= 1. Note that we have directly described\nthe Euclidean solution with compact time coordinate tE. The warping of the base space\nand the connection ωin (4.3) are given by\ne−2U=i\u0000\nYIGI−YIGI\u0001\n, ⋆ dω=i⟨dH, H⟩, (4.4)\n11Here, and below, the symplectic product of two vectors A= (eAI, AI) and B= (eBI, BI) is given\nby⟨A, B⟩=eAIBI−eBIAI.\n– 20 –where the weight-0 normalized fields ( YI,GI) are given by\n\u0000\nYI,GI\u0001\n=e1\n2K(X,X)Z(H; Ω)\u0000\nXI, FI\u0001\n(4.5)\nand similarly YI,GIby the complex conjugate equations. The scalar fields obey the\ngeneralized stabilization equations12\nYI−YI=ieHI,GI−GI=iHI, (4.6)\nThe above equations of supersymmetry for the ( YI,GI) fields can be presented in terms\nof a prepotential function G(Y) with GI=∂G/∂YI, where Gis the same mathematical\nfunction as F.13\nBesides the metric and the scalars, the solution is specified by the nv+1 vector multiplet\nU(1) gauge fields. These are also determined by the harmonic functions in a specific way,\npresented for example in equation (3.21) of [18].\n4.2 Euclidean black holes and near-horizon behavior\nIn this subsection we review the “new attractor mechanism” that was found in [18]. Here\nwe present all the equations in coordinate language. We refer the reader to [18] for details\nof the coordinate-invariant equations.\nExtremal black holes Supersymmetric extremal solutions are obtained by choosing the\nharmonic functions in the above discussion to have electric and magnetic sources at one\npoint, say the origin, i.e.,\nH(x) = h+Γ\nr, r =|x|. (4.7)\nThe equations (4.4) show that the solution is spherically symmetric and static, i.e. ω= 0\nin (4.3). The vector h≡H(∞) is the asymptotic value of sources. The generalized\nstabilization equations (4.6) imply that the asymptotic behavior of the scalars as r→0 is\nYI∼YI\n∗\nr,GI∼GI∗\nr, (4.8)\nwhere the constant values YI\n∗, GI∗(and therefore Z∗) are determined by the extremal\nattractor equations\nYI\n∗−YI\n∗=iPI, G I∗−GI∗=iQI. (4.9)\nThe behavior of the metric as r→0 is\ne−2U∼i\u0000\nYI\n∗GI∗−YI\n∗GI∗\u0001\nr2. (4.10)\nFrom the form of the metric (4.3) we see that the Bekenstein-Hawking entropy of the black\nhole is given by SBH=πi\u0000\nYI\n∗GI∗−YI\n∗GI∗\u0001\n.\n12Note that the signs in these equations are opposite to the one in [18].\n13One should note that the rescaling between XIandYIis not holomorphic, but one can still apply the\nconcept of holomorphic and anti-holomorphic to the Yvariables directly. Indeed, we encounter holomorphic\n(and anti-holomorphic) functions of YIin the discussion of the entropy.\n– 21 –Non-extremal supersymmetric solutions In order to obtain non-extremal solutions\nrelevant for the index, we replace the double pole behavior (4.10) of the extremal metric\nby two single poles. Accordingly, we split the sources into a pair of sources at two different\npoints called the north and south pole. The total charge is split as\nγN=1\n2\u0000\nPI+inI, QI+imI\u0001\n, γ S=1\n2\u0000\nPI−inI, QI−imI\u0001\n, Γ = γN+γS,\n(4.11)\nand the corresponding harmonic functions are given by\nH(x) = h+γN\n|x−xN|+γS\n|x−xS|. (4.12)\nCompared to the extremal solution we have introduced electric and magnetic dipole charges\n(mI, nI) and the distance |xN−xS|between the north and south poles. The equation for ω\nin (4.4) shows that the metric necessarily has angular momentum. All these parameters of\nthe solution are determined by the new attractor mechanism [18] that we now recall.\nThe main point is that the spinning non-extremal solution near one of the poles has a\nsimple scaling behavior, and the values of the scalars at the poles are therefore determined\ncompletely by the charges, similar to the extremal solution. There is, however, an important\ndifference in that now we only have simple poles. As we briefly review below, one of the\nconsequences is that the solution is necessarily complex.\nThe full solution has a symmetry which exchanges the north with the south pole and\nthe scalars with their complex conjugates. Therefore we focus on the north pole and let\nthe parameter ρ=|x−xN|be very small. From (4.12) we have, as ρ→0,\nH(x)∼γN\nρ+H(0)+ O( ρ), H(0)=h+γS\n|xN−xS|. (4.13)\nThe generalized stabilization equations (4.6) then imply that\nYI∼YI\nN\nρ+YI(0)+ O( ρ),GI∼GNI\nρ+G(0)\nI+ O( ρ), (4.14)\nand similarly for the conjugate scalars, with\n\u0000\nYI\nN−YI\nN, GIN−GIN\u0001\n=iγN,\u0000\nYI(0)−YI(0), G(0)\nI−G(0)\nI\u0001\n=iH(0).(4.15)\nNote the Laurent expansion (4.14) combined with the definition of the scalar fields (4.5)\nand the homogeneity property of the prepotential leads to the following identity that we\nuse below,\nG(0)\nI=∂GI\n∂YJ(xN)YJ(0)⇒ YI\nNG(0)\nI=GJNYJ(0). (4.16)\nNow, the equation (4.4) for the metric implies that\ne−2U∼i\u0010YI\nN\nρ+YI(0)\u0011\u0010GIN\nρ+G(0)\nI\u0011\n−i\u0010YI\nN\nρ+YI(0)\u0011\u0010GIN\nρ+G(0)\nI\u0011\n+ O( ρ).(4.17)\n– 22 –Demanding a simple pole at xNimplies that\n\u0000\nYI\nN,GIN\u0001\n= 0 ⇒\u0000\nYI\nN, GIN\u0001\n=iγN, (4.18)\nwhere we use the first equation of (4.15) to obtain the second equality. The real part of\nthese equations,\nYI\nN−(YI\nN)∗=iPI, G IN−(GIN)∗=iQI, (4.19)\nare recognized as the extremal attractor equations (4.9). Assuming a unique solution, we\nobtain ( YI\nN, GIN) = (YI\n∗, GI∗). The scalars at the north pole are thus fixed by the solution\nto the extremal attractor equations. Further, the dipole charges can then be read off to be\n−nI=YI\nN+ (YI\nN)∗,−mI=GIN+ (GIN)∗. (4.20)\nThe metric now only has a simple pole,\ne−2U∼i\nρ\u0010\nYI\nNG(0)\nI−GINYI(0)\u0011\n+ O(1) ,\n=−1\nρ\u0010\nYI\nNH(0)\nI−GINeHI(0)\u0011\n+i\nρ\u0010\nYI\nNG(0)\nI−GINYI(0)\u0011\n+ O(1) ,\n=−1\nρ\u0010\nYI\nNH(0)\nI−GINeHI(0)\u0011\n+ O(1) .(4.21)\nHere, we use the second equation in (4.15) to reach the second line, and the identity (4.16)\nin order to obtain the final equality.\nFinally, the behavior of the connection ωnear the poles is given by expanding the\nequation (4.4). Using (4.14) we obtain\n⋆dω=bn\nρ2⟨γN, H(0)⟩+ O(1 /ρ), (4.22)\nwhere bn= (x−xN)/|x−xN|, the three-dimensional unit vector with the north pole as\norigin. This equation implies that there is a three-dimensional Dirac string associated\nwith ωemanating from the north pole. There is a similar condition at the south pole. The\ncondition of smoothness implies that this Dirac string should end smoothly at the poles.\nThis regularity condition leads to (see the discussion in [37, 38] for more details)\ni⟨γN, γS⟩\n|xN−xS|+i⟨γN, h⟩=β\n4π. (4.23)\nThis equation fixes the distance between the north and south poles in terms of the tem-\nperature, charges at the poles, and the asymptotic values of the moduli. As β→ ∞ , we\nobtain the attractor solution with Ω →0,|xN−xS| →0, and an infinite throat.\nTo summarize, we obtain a Euclidean metric with a cigar-like topology. The consis-\ntency of the periods of angular and time coordinates, leads to the condition βΩ = 2 πi. The\nvalues of the scalars at the tip of the cigar at the north and south poles are fixed by the\ncharges in a manner similar to the extremal attractor mechanism, but now all the equations\nare split into a holomorphic piece at the north pole and an anti-holomorphic piece at the\nsouth pole. The holomorphic scalar fields take their attractor values at the north pole, and\nthe anti-holomorphic scalars vanish. At the south pole we have the conjugate equations,\nand the anti-holomorphic scalars take their attractor values and the holomorphic scalars\nvanish.\n– 23 –4.3 Two-charge solution\nWith this set-up, we study the behavior of these1\n2-BPS rotating Euclidean solutions with\ntwo-charges. We begin by making some general comments that apply to any theory of\nN= 2 supergravity. We will make contact with the black hole studied in sections 2 and 3\nat the end. The classical on-shell action of the non-extremal supersymmetric solution\nreviewed above is given, after removing a factor of βMBPSby a boundary term, by\n−Ion−shell =π(qIYI+ 2iG)\f\f\nN+π(qI¯YI−2i¯G)\f\f\nS, (4.24)\n=πi⟨γN, γS⟩, (4.25)\nwhere Gis the same function of YIasF(XI). The result in the first line comes from\nshowing that the on-shell action is a total derivative up to delta functions localized at the\npoles [18]. Using the new attractor equations it can be put in the form of the second line14.\nThis quantity πi⟨γN, γS⟩isnotthe area of the event horizon of the non-extremal solution,\nwhich is instead equal to\nA=β|xN−xS|. (4.26)\nThis can be easily derived (see [18]) from the metric (4.3) together with the equation for\nωin (4.4) and we will not repeat it here. The on-shell action and the area only coincide\nat zero temperatures, such that log Zindex is proportional to the extremal area. This is\ndue to the fact that these black holes satisfy the quantum statistical relation, i.e. the on-\nshell action is equal to A/4 + 2 πiJ(since βΩ = 2 πi) and at low temperatures the angular\nmomentum vanishes.\nGiven the discussion in the previous paragraph, in a general N= 2 theory of super-\ngravity, solutions that have vanishing area at extremality are characterized by\n⟨γN, γS⟩= 0. (4.27)\nThis condition guarantees that the extremal area of the black hole is zero at the two-\nderivative level, and the index of the black hole is small. Since γN/Sare determined\nin terms of the solution to the extremal attractor equations, this can be rephrased as a\ncondition on the choice of charges that the black hole carries. Finally, as emphasized above,\nthe condition (4.27) still allows for the finite temperature supersymmetric solution of [18]\nto have a finite size event horizon with area (4.26).\nFor the choice of charges that satisfy ⟨γN, γS⟩= 0, this might suggest that we have\nsucceeded in finding a smooth solution that contributes to the gravitaitonal path integral\nfor the index. However, this is too quick. Let us recall the regularity condition at the\nhorizon (4.23). When charges are chosen to satisfy ⟨γN, γS⟩= 0, the condition degenerates\nto:\nβ= 4πi⟨γN, h⟩. (4.28)\nThis is problematic for the following reasons\n14The rewriting in the second line is only valid at the two-derivative level.\n– 24 –1. First, the temperature (4.28) is fixed in terms of the black hole charges and asymptotic\nmoduli, while we should be free to set them independently.\n2. The second problem is that |xN−xS|has disappeared completely from the equation.\nThere is nothing fixing the separation between the two poles and therefore there is a\nmoduli space of solutions, characterized for example by the area of the event horizon.\nIn particular |xN−xS|= 0 is part of the moduli space which gives rise to the original\nsingular small black hole we were trying to avoid.\n3. The extremal area is also proportional to Z∗(Γ)Z∗(Γ), which implies Z∗(Γ) = 0 for\nthe small black hole. In the non-extremal solution Z= 0 and ¯Z= 0 at the north and\nsouth poles. The solution is then singular whenever ⟨γN, γS⟩= 0.\nThese are precisely the shortcomings enumerated in Section 3.3 for the two-charge black\nhole in the heterotic string. We see here all those features are universal to N= 2 super-\ngravity with vanishing extremal area at the two-derivative level.\nTo provide some more intuition we will consider some special cases, and connect to the\nblack hole in Section 3. To leading order, compactifications of type II or heterotic string\ntheory to four dimensions can be described by the following prepotential\nF(X) =−1\n6CIJKXIXJXK\nX0. (4.29)\nThe value of CIJKdepends on the model. For example for type IIA compactified on K3×T2\nthe coefficients CIJKare the intersection numbers of a basis of 4-cycles. This case is related\nby duality to the toroidally compactified heterotic string theory in Sections 2 and 3. Instead\nof studying this theory we will analyze a simpler version where only X1, X2, X3are non-\nzero and moreover X2=X3. Therefore consider a theory with nv= 2 and a prepotential\nof the form\nF(X) =−X1(X2)2\nX0. (4.30)\nThis theory is equivalent to a sector of (2.1), as explained in [8]. We make more comments\non this below. A concrete solution with vanishing extremal area involves turning on only\nthe following two charges\nQI= (Q0,0,0), PI= (0 , P1,0). (4.31)\nWe take Q0<0 and P1>0. In the context of the toroidally compactified heterotic\nstring these charges have the interpretation of momentum nand winding walong a circle\nS1⊂T6, namely\nQ0=−n , P1=w . (4.32)\nMoreover the interpretation of the moduli is X1/X0=ie−Φwhile X2/X0parametrizes\nthe volume of a cycle inside T6. The solution to the extremal attractor equations (4.9) is\ngiven by\nYI\n∗=\u0010\n0,iP1\n2,0\u0011\n, G I∗=\u0010iQ0\n2,0,−p\nP1|Q0|\u0011\n. (4.33)\n– 25 –This result should be surprising. Since G2∗=−2Y1\n∗Y2\n∗/Y0\n∗, how come X0=X2= 0 and\nG2∗is finite? What is the physical value of the moduli X2/X0that describes the geometry\nofT6? To see how this comes about let us turn a small magnetic charge P2. (Physically\nthe way this is regulated is by turning on eH2=h2̸= 0 and going slightly away from the\nhorizon, keeping P2= 0. In practice it is easier to turn on a small P2at the attractor\npoint, both lead to the same answer.) Then the attractor equation becomes\nYI\n∗=\u0010P1P2\n2p\nP1|Q0|,iP1\n2,iP2\n2\u0011\n. (4.34)\nWe see that as P2→0 the ratio between Y2\n∗andY0\n∗is finite\nY2\n∗\nY0∗→ir\n|Q0|\nP1=irn\nw. (4.35)\nIn the application to the heterotic string this guarantees that near the horizon (or near xN\nandxSfor the non-extremal metric) the volume of T6is finite, and also explains why G2∗\nis non-vanishing. The other moduli controlling the string coupling does diverge\nY1\n∗\nY0∗=ip\nP1|Q0|1\nP2→i√nw× ∞ . (4.36)\nThis divergence will be regulated by higher derivative corrections. In string theory this\nimplies that the string coupling vanishes at the horizon (extremal) or at the poles xN/S\n(non-extremal).\nThe new attractor equations imply that ( YI\n∗, GI∗) and ( ¯YI\n∗,¯GI∗) are precisely the local\ncharge at the north and south poles respectively\nγN=\u0010\n0,P1\n2,0;Q0\n2,0, ip\nP1|Q0|\u0011\n, γ S= (γN)∗. (4.37)\nWe can replace these local charges into equation (4.12). Once these harmonic functions\nare known, the full non-extremal solution can be constructed. Let us assume for simplicity\nthat the moduli at infinity are chosen such that H1=eH0= 0 and eH2=h2. We also choose\nh2= 0, although this does not imply H2= 0 since γN/Sinclude a dipole in this direction.\nFor concreteness we also pick X1/X0|∞=i2 and X2/X0|∞=i, implying h1=−h0= 1\nandh2= 1/2.15The emblackening factor is\ne−2U= 2 ˜H2q\n−eH1H0 (4.38)\n=n\u0010\n1 +1\n2P1\n|x−xN|+1\n2P1\n|x−xS|\u0011\u0010\n1 +1\n2|Q0|\n|x−xN|+1\n2|Q0|\n|x−xS|\u0011o1\n2. (4.39)\nNote that e−2U=−iY1/Y0=e−Φ. Curiously this is independent of H2, which carries\nthe information of the dipole. (Of course the dipole does appear in the scalars and gauge\nfields.) One can readily verify that the charge vector at the poles satisfies ⟨γN, γS⟩= 0.\n15This specific choice of asymptotic moduli is done mainly to connect later to the solution in Section 3.\nThe solution can be easily generalied to arbitrary X1,2/X0|∞.\n– 26 –All the previous considerations then apply to this black hole. A solution only exist with\na unique temperature, it has a moduli space of solutions labeled by the horizon area, and\nthe scalars are singular at the poles of the horizon.\nLet us now show that this solution is precisely the same as the one analyzed in Section 3\nearlier16. This has to be true since it was proven in [8] that the action (2.1) can be obtained\nfromN= 2 supergravity with prepotential (4.30), so we will be brief. Consider the metric\ngiven in (3.10) and change variables ( tE, r, θ, ϕ ) to ( tE, x1, x2, x3) according to\nx1=p\nr2−a2sinθcosϕ , (4.40)\nx2=p\nr2−a2sinθsinϕ , (4.41)\nx3=rcosθ . (4.42)\nIn this coordinates the metric (3.10) has the form [26]\nds2=e2Φ(dtE+iω)2+ dxmdxm. (4.43)\nwhere iω=am0(r2−a2cos2θ)−1rsin2θdϕand the dilaton becomes\ne−2Φ=\u0010\n1 +m0eα\n2|x−xN|+m0eα\n2|x−xS|\u0011\u0010\n1 +m0e−α\n2|x−xN|+m0e−α\n2|x−xS|\u0011\n, (4.44)\nwithxN= (0,0,a) and xS= (0,0,−a). This metric looks different than the one in (4.3).\nThe solution is to recall (3.10) is given in string frame while the formalism used here for\nN= 2 supergravity generates an action with canonical Einstein-Hilbert term. Therefore\nwe should translate (3.10) into Einstein frame, which gives\ne−Φds2=eΦ(dtE+iω)2+e−Φdxmdxm. (4.45)\nIt is possible now to put this metric in the form (4.3) with 2 U= Φ for a suitable choice of\nthe constants h. This also implies that P1=m0eαwhile Q0=−m0e−α. This is consistent\nwith the interpretation of P1as winding and Q0as momentum. More importantly, the\nseparation between the poles is identified with\n|xN−xS|= 2a. (4.46)\nFor example, (4.26) implies that the horizon metric in IWP form is A= 2βawhich matches\nthe result in Section 3. Moreover, for our choice of handγNwe obtain i⟨γN, h⟩=p\nP1|Q0|/2 = m0/2 which implies from (4.28) a value of the temperature β= 2πm0,\nmatching Section 3 as well.\n16The reader might wonder why we used two different formalism to describe the solution. The presen-\ntation in Section 3 is useful since it can be extended outside of the supersymmetric regime as in Section 2.\nThe advantage of the formalism in this section is that, within supersymmetric solutions, it will allow us to\nincorporate higher derivative corrections, as in Section 5.\n– 27 –5 Higher derivative corrections to the gravitational index\nIn this section we present the rotating black hole solution including higher-derivative cor-\nrections in the N= 2 supergravity coming from string theory. The higher-derivative\ncorrections in N= 2 supergravity are of two types: chiral superspace integrals of the holo-\nmorphic prepotential function, and full superspace integral, sometimes called F-terms and\nD-terms, respectively, in the N= 2 context. We only consider the first type of terms, which\nare summarized by corrections to the tree-level holomorphic prepotential function G(YI)\ndiscussed in the previous section. We then make comments on D-terms and the controlla-\nbility of the solution.\nThe treatment begins by introducing another chiral field bAof weight 2, whose top\ncomponent is proportional to the Weyl-squared tensor in four-dimensions. The prepotential\nis extended to a function F(XI,bA) and, in addition to the first derivatives FI=∂F/∂XI,\nwe have FbA=∂F/∂bA. The particular form of the prepotential that arises in Calabi-Yau\nthree-fold compactifications of Type II string theory is\nF(XI,bA) =−1\n6CIJKXIXJXK\nX0−cbA\n64X1\nX0+. . . . (5.1)\nHerebAeffectively counts the loops in the topological string expansion. As bAcontains four\nderivatives, we can also think of the series as an expansion in the number of derivatives.\nThe cubic or tree-level term was discussed in the previous section, and we have shown\nthe one-loop term. Here c=c2/24 where c2is the second Chern class of the Calabi-Yau.\nIn addition there are typically an infinite series of terms given by the higher loops of the\ntopological string expansion. In Type II theory on K3×T2, the term X1/X0above is\ncompleted by worldsheet instanton effects to a modular function of X1/X0. Importantly,\nthe expansion terminates at this order due to the fermion zero modes on T2.\nIn our case of interest here, i.e. the two-charge black hole, the two-derivative result\nis singular, so that in the full solutions the correction terms are as important as the two-\nderivative term. This means that, a priori, one does not have a controllable expansion.\nNevertheless it is useful to see explicitly that including higher-derivative terms can lead to\na smooth solution, as we show below.\nA proof that D-terms vanish on configurations that preserve supersymmetry would\njustify the approach where we only keep F-terms.17In some situations, like the attractor\nblack hole entropy, one can indeed show that D-terms do not affect the quantity of inter-\nest [21, 22], and therefore one obtains an exact formula [39–42]. In our present context, as\nwe see below, the entropy arising from the prepotential (5.1) agrees with the microscopic\nstring theory index including the numerical coefficient, pointing to a non-renormalization\nresult.\nThe analysis of supersymmetry equations and solutions of the higher-derivative theory\nwas performed in the impressive set of papers [36, 43, 44]. Just as we define the rescaled\n17As opposed to previous work [8–10], we do not expect the generic expansion to be controllable for the\nthermal partition function which does not include an insertion of ( −1)F.\n– 28 –fields\u0000\nYI,GI\u0001\nin (4.5), we also define a rescaled field Υ = eK(X,X)Z2bA. The higher-\nderivative equations are, generally speaking, more complicated than the two-derivative\ntheory. The exception is the equations governing the scalar fields, which are still given by\nthe generalized stabilization equations (4.6), although they now depend on Υ.\nNew attractor with higher-derivative corrections\nThe solution is specified by a vector of harmonic functions H= (eHI, HI). The generalized\nstabilization equations are now\nYI−YI=ieHI,GI(Y,Υ)−GI(Y,Υ) = iHI, (5.2)\nand they determine YIin terms of Hand Υ. The metric is of the IWP form as in the\ntwo-derivative theory\nds2=e2U(dtE+iωmdxm)2+e−2Udxmdxm. (5.3)\nThe functions Uandωare now governed by\ne−2U=i\u0000\nYIGI− GIYI\u0001\n+ 128 i eU∇p\u0000\n(∇pe−U)(GΥ−GΥ)\u0001\n−32i e4U(R(ω)p)2(GΥ−GΥ)−64e2UR(ω)p∇p(GΥ+GΥ).(5.4)\n−iR(ω)p=HI↔\n∇pHI−128i∇q\u0002\n∇[p(e2UR(ω)q](GΥ−GΥ))\u0003\n−128∇q\u0000\n2∇[pU∇q](GΥ+GΥ)\u0001\n.(5.5)\nIn the two-derivative theory GΥ= 0 and the above equations reduce to the simpler equa-\ntions (4.4). Finally, the value of Υ is\nΥ = −64\u0010\n∇pU−i\n2e2UR(ω)p\u00112\n. (5.6)\nHere we follow the notation of [43] with ∇athe derivative in the three-dimensional flat\nbase-space, and R(ω)p=εpqs∇qωs. For the non-extremal rotating saddle we set, as in\nSection 4\nH(x) = h+γN\n|x−xN|+γS\n|x−xS|, γ N+γS= Γ. (5.7)\nTherefore part of the problem is to determine γN,γSand|xN−xS|using smoothness, just\nlike we did at the two derivative level [18].\nAs an exercise we can study the behavior of all the fields in an extremal attractor\nbackground as in the discussion near (4.7), as r→0,\nExtremal: R(ω) = 0 ,YI∼Y∗\nr+. . . , e−2U∼a\nr2+. . . , Υ∼Υ∗\nr2+. . . .\n(5.8)\nThe fact that U∼logrtogether with (5.6) implies that Υ ∗=−64. The scalars YI\n∗are\ndetermined by YI\n∗−¯YI\n∗=iPIandGI∗−¯GI∗=iQI. The scalar fields therefore behave as\nin the two-derivative theory. The scaling behavior of the other fields can be read off from\n– 29 –their conformal weights—this can be explained either by using a conformal compensator, or\nby thinking of the other fields as functions of the scalar fields. The only thing to be careful\nof is that coefficients can be zero, as happens for R(ω) because of the spherical symmetry.\nThe value Υ ∗=−64 is precisely the one required for the Wald entropy of the extremal\nblack hole to equal the logarithm of the growth of microscopic states of the fundamental\nheterotic string [45].\nNow we repeat the analysis for the rotating non-extremal solution, and determine the\nvalue of γN/S. We focus on the behavior of the solution near the north pole, since the\nbehavior at the south pole is related by complex conjugation. Recall that the idea in\nthe rotating non-extremal solution is to demand that the metric factor e−2Uhas only a\nsingle pole, leading to a finite temperature horizon. Accordingly, we assume the following\nbehavior for the non-extremal solution, as ρ=|x−xN| →0,\nYI∼YN\nρ+. . . , e−2U∼a1\nρ+. . . , R (ω)p=a2\nρ2bn+. . . , Υ∼Υ∗\nρ2+. . . ,\n(5.9)\nwherebn= (x−xN)/|x−xN|. Now we analyze the equations (5.4)–(5.6). The upshot is that\nthe solution to the extremal attractor ( YI\n∗, GI∗) together with Υ ∗=−64 determines γN.\nTogether with the smoothness condition, this completely determines the sources H. Now\nthe full solution can be determined in terms of the boundary data (Γ and β)18.\nWe begin with (5.5). GΥhas Weyl weight 0 and so GΥ∼O(1) as ρ→0. The leading be-\nhavior of the second term on the right hand side is therefore proportional to ∇[p(e2UR(ω)q]).\nNow, e2UR(ω)q∼a2\nρbnq+. . ., and therefore the anti-symmetry of the action of the deriva-\ntives annihilates this term so as to give no contribution at O(1 /ρ2). A similar analysis\napplies to the last term in (5.5). This shows that the value of R(ω) at the pole in the\nhigher-derivative theory is the same as its value given in (4.22) in the two-derivative theory.\nWe conclude that a2=i⟨γN, H(0)⟩. In particular, this means that the condition of there\nbeing no Dirac-string singularity implies the same relation (4.23) as in the two-derivative\ntheory, which we repeat here for convenience\ni⟨γN, γS⟩\n|xN−xS|+i⟨γN, h⟩=β\n4π. (5.10)\nNow we move to (5.4). Let us start by considering the first term on the right hand side.\nThis is the value of e−2Uin the two-derivative theory. We consider a solution that cancels\nthe double pole coming from this term by imposing ¯YI\nN=¯GIN= 0 (and YI\nS=GIS= 0 for\nthe south pole). The stabilization equation, together with the old attractor, would then\nimply\niγN= (YI\n∗, GI∗). (5.11)\nThis solution is not yet complete, as we have not yet determined Υ ∗which appears im-\nplicitly in GI. We will come back to this point shortly. First, we explain how the other\n18We assume that, given H, a solution to the partial differential equations (5.4) and (5.5) exists and is\nunique. This has been verified in the low temperature limit numerically.\n– 30 –terms in (5.4) lead to a vanishing double pole. As ρ→0, calling the constant GΥ=C1,\nthe leading behavior of the second and third terms is governed by\nC1\u0000\n128i eU∇2e−U−32i e4U(R(ω)p)2\u0001\n∼ −32i C1\u0010\n1 +a2\n2\na2\n1\u00111\nρ2+. . . . (5.12)\nWe see that in order to have a vanishing double pole we must have a2=±ia1. In the\ntwo-derivative theory we have, from (4.21) and (4.18), the value a1=−⟨γN, H(0)⟩so\nthat a2=ia1. We choose this solution also in the higher-derivative theory. This implies\na condition of vanishing of other O(1 /ρ) terms of the right-hand side of (5.4) involving\nhigher derivative terms. From the equation (5.6), we deduce that\nΥ∼ −64\nρ2\u00101\n2−i\n2a2\na1\u0011\n=−64\nρ2. (5.13)\nThis completes the derivation since now we determined Υ ∗=−64. As in the two-derivative\nsolution, we see that the holomorphic (anti-holomorphic) scalar fields take the attractor\nvalues at the north (south) pole. Furthermore they determine the dipole charges γN(γS)\nthrough the relation (5.11).\nSo far we described the metric and scalar profiles. The gauge fields are also determined\nby the harmonic sources through the equations FI\n0p=−∇p[e2U(YI+¯YI)] and GI\n0p=\n−∇p[e2U(GI+¯GI)] where FI\nµνandGI\nµνdetermine the field strength and its dual. These\nequations are the same as in the two-derivative theory.\nTo conclude the characterization of the solution, we propose a formula for the on-\nshell action of such configurations, including boundary terms that remove the zero-point\nenergy βMBPS, to be\n−Ion−shell=π(qIYI+ 2iG)|N+π(qI¯YI−2i¯G)|S. (5.14)\nWe have two reasons that justify this expression. First, it reduces to the two-derivative\naction (4.24) when Gis independent of Υ, for any temperature. (Note that when Gdepends\non Υ (4.25) is different from (4.24).) Second, one can prove this formula in the presence\nof arbitrary higher-derivative corrections, under the assumption that the on-shell action is\ntemperature independent. The action in the zero-temperature limit was explicitly evaluated\nin [39] and matches (5.14) (as well as the Wald entropy derived in [43]).\nThe reason we have not attempted an explicit evaluation of the action is that it requires\nsolving the differential equations (5.4) and (5.5). Nevertheless, there might be a more\nelegant way to obtain (5.14). The split form of the action in (5.14) is similar to observations\nof the factorization of the partition function in [46, 47]. An efficient way to prove this could\nbe to apply the techniques of [48], we leave this for future work.\nThe corrected two-charge solution\nWe now come back to the specific theory relevant to the two-charge black hole in heterotic\nstring theory, with only X1, X2=X3, with prepotential\nF(XI,bA) =−X1(X2)2\nX0−cbA\n64X1\nX0. (5.15)\n– 31 –For the heterotic string c= 1 but we find it useful to keep it general to keep track of\nthe effect of higher derivative corrections. The non-zero monopole charges in the solution\nare taken to be Q0, P1and recall that Q0<0 and P1>0. According to the discussion\nin Sections 4.2, 4.3, we choose sources ( eH0,eH1,eH2, H0, H1, H2). Let us begin by solving\nthe new attractor equations which determine γN/S. The starting point is the extremal\nattractor solution\nYI\n∗= s\ncP1\n|Q0|,iP1\n2,0!\n, G I∗= \niQ0\n2,−s\ncP1\n|Q0|,0!\n, Υ∗=−64,(5.16)\nwhich is equal to YNat the north pole while ¯YN= 0. The divergence in the dilaton (near\nthe north and south pole) at the horizon is now regulated since Y0\n∗is finite. The fact that\nnow G2∗= 0 and G1∗̸= 0 might seem like a major modification compared to the two-\nderivative level discussion. To see how this comes about one can turn a small charge P2\n(or, equivalently, go slightly away from the pole, as explained in Section 4.3). For example,\nthis produces a term G2∗∼p\nP1|Q0|P2\n4c+(P2)2. We see explicitly that the limits c→0 and\nP2→0 do not commute and this is the source of discrepancy. This is clearly due to the\nsingular nature of the two-derivative solution and would not happen for a big black hole.\nAccording to the new attractor solution (5.11) the charge vectors associated to the\nnorth and south pole contributions are\nγN= \nis\ncP1\n|Q0|,P1\n2,0,Q0\n2,−is\ncP1\n|Q0|,0!\n(5.17)\nandγS= (γN)∗. Notice again that this is not a small correction to the two-derivative\nsolution for γN! One can check that now we have\ni⟨γN, γS⟩= 2p\ncP1|Q0|. (5.18)\nThe final ingredient to determine the sources (5.7) and therefore the full solution is to\nchoose h. At infinity we have Υ →0 and therefore we can repeat the same analysis as in\nthe two-derivative theory to establish the values hI=eHI|∞andhI=HI|∞that allow us\nto compare with the solution in Sections 2 and 3. The answer found in Section 4.3 is hI=\n(−1,0,0) and hI= (0,1,1\n2). The solution to the generalized stabilization equations (4.6)\nwith prepotential (5.1) becomes then an algebraic equation that relates ( YI,GI) to (eHI, HI)\nand Υ, which can be solved easily19.\nThe relation (4.23) fixes the parameter 2 a=|xN−xS|in terms of β, as anticipated\nearlier in Section 3. Since, in the language of that section, i⟨γN, γS⟩= 2p\ncP1|Q0|= 2√cm0\nandi⟨γN, h⟩=√c(P1−Q0)√\nP1|Q0|= 2√ccoshα, the relation (4.23) gives\n2√cm0\n2a+ 2√ccoshα=β\n4π,⇒a=4πm0\nβ√c−8πcoshα. (5.19)\n19It is important to emphasize that the stabilization equations are not enough to determine Yin terms\nofHin the higher-derivative theory. The reason is that they involve Υ, which depends on Uandω.\nTherefore the stabilization equations are not decoupled from (5.4) and (5.5).\n– 32 –This relation resolves the problem appearing in the two-derivative expression (4.28), namely\nthat the temperature was apparently fixed in terms of the charges and asymptotic mod-\nuli. Now we see that the temperature and the moduli are independent quantities. The\nrelation (5.19) shows the solution is only valid as long as the denominator is positive. Curi-\nously, the denominator changes sign at a finite value of β. (The same phenomenon is found\neven for the big black hole where |xN−xS|becomes negative above a charge-dependent\ntemperature. Explaining this remains an open question.)\nIn summary, we find a solution that is smooth with the dilaton finite everywhere,\nall the parameters of the solution are determined in terms of boundary data which are\nindependent values of the temperature and moduli.\nTo conclude this section we evaluate the on-shell action of the solution we constructed,\nusing (5.14). Since Zindex∼e−Ion−shellwe find that it makes the following contribution to\nthe gravitational index\nlogZindex =π(qIYI+ 2iG)\f\f\nN+π(qI¯YI−2i¯G)\f\f\nS, (5.20)\n= 4πp\ncP1|Q0|. (5.21)\nThis is now finite. To compare with the heterotic string action we recall that c= 1.\nTherefore we find a gravitational index given, in terms of the charges n=−Q0and\nw=P1, by\nlogZindex =−Ion-shell = 4π√nw . (5.22)\n(In terms of the parameter m0of Section 3 this is 4 πm0.) This result precisely matches\nthe microscopic evaluation of the index counting BPS fundamental heterotic strings, as\nreviewed in the introduction.\nNote that the same phenomenon is known to occur if one assumes an attractor AdS 2\ngeometry [45]. What is remarkable here is that one obtains the same result even though\nthe rotating geometry for the index preserves much less symmetry, and is compatible with\nthe discussion in Section 2 on the partition function.\n6 Can the low-temperature physics be nearly conformal?\nIn Section 2 we studied two-charge black holes in string theory. At low temperatures the\nblack hole becomes small, i.e. it has a horizon area that vanishes at the two-derivative\nlevel. Following [12], we explained how a transition takes place to a winding condensate\nwithout a horizon, before we reach the small black hole, and ultimately the ground state\nis described by a gas of strings and not a black hole.\nIs the behavior described above universal? Could there be a string theory solution that\ndoes not transition to a gas of strings or other stringy objects without a horizon, such that\nthe ground state is really described by a small black hole? In this section we use effective\ntheory arguments to comment on the possibility of a string-size near-extremal black hole.\nOur arguments also apply for big black holes with 16 or more supercharges.\nBy a small black hole we mean a gravitational solution with a finite (string scale) area\nthat exists at very low temperatures, which develops an AdS 2throat. This assumption can\n– 33 –be justified in some cases where there is a smooth limit to the two-derivative theory [49, 50].\nWe also assume that the index is a function that grows exponentially with the charges.\nThe latter requirement is satisfied by most examples in string theory (an exception being\ne.g. the D0 brane black hole). Finally, we assume that the extremal limit of the small black\nhole preserves at least 16 supercharges.\nThe near-extremal AdS 2phase of such black holes implies the presence of a nearly-\nconformal symmetry of the quantum system describing the black hole microstates. Its\nlow energy description is given by the Schwarzian theory. Since we assume the system\npreserves a certain number of supercharges, we have a supersymmetric generalization of\nthe Schwarzian theory.\nIt is possible to label the different supersymmetric extensions of the Schwarzian theory\nby the superconformal group of symmetries broken by the near-AdS 2solution. For example\nthe bosonic Schwarzian theory would be labeled by PSL(2 ,R), the one describing the near-\nBPS limit of the 1/16-BPS black holes in AdS 5would be labeled by SU(1 ,1|1) and the one\ndescribing the near-BPS limit of large black holes in flat space by PSU(1 ,1|2). Therefore\nwe can address this problem by analyzing the possible superconformal groups. To perform\nour analysis we use that the supersymmetric Schwarzian action acts as a U(1) generator\nin the space of super-reparametrization, such that we can apply the localization principle\nof [51]. We assume that the R-symmetry of the superconformal group is realized in a\ngeometric manner in spacetime.20We now show that no Schwarzian theory exists with\nthese properties.\nGiven the choice of a superconformal group we can derive the partition function of\nsuch a theory including quantum corrections. These corrections are one-loop exact and\nhave the following form. Denote by ⃗ µthe vector of chemical potentials that couple to the\nr= rank( G) Cartan generators of the R-symmetry group G, with the periodicity µ∼µ+1.\nThe partition function then takes the form\nZ=X\n⃗ n∈ZrβNF−NB\n2Zone−loop(⃗ µ+⃗ n)eS0+1\nβI(⃗ µ+⃗ n), (6.1)\nwhere Zone−loop(⃗ µ+⃗ n), and I(⃗ µ+⃗ n), are functions of chemical potentials only arising from\nthe one loop determinant, and on-shell action, respectively. The sum over integers ⃗ nis a\nsum over saddles related by large gauge transformations. S0is the extremal entropy while\nNFandNBcorrespond to the number of fermionic generator and bosonic generators in\nthe superconformal group. The overall prefactor of βNF−NB\n2in the one-loop determinant\ncomes from subtracting a contribution from zero-modes.\nWhen NF≥NBthe theory can predict the presence of a large number of ground states\nof order eS0. For example for SU(1 ,1|1),NB−NF= 0 so that the one-loop determinant\nis temperature independent and in the zero-temperature limit the action becomes tem-\nperature independent, resulting in Z(β→ ∞ )∼eS0+O(e−β). For PSU(1 ,1|2), instead,\nNF= 8> N B= 6 and the one-loop determinant diverges as β→ ∞ . In this case the\n20These assumptions do not rule out the possibility of a sigma model description. See [52–56] for a\nrelated discussion on the AdS 3near-horizon configurations of a fundamental string.\n– 34 –sum over saddles is important in order to regulate this divergence and once again we find\nZ(β→ ∞ )∼eS0.\nWhen NB> N Fthe one-loop determinant vanishes in the β→ ∞ limit and its\nunavoidable to obtain Z(β→ ∞ ) = 0. This implies that the quantum corrections due to\nthe Schwarzian mode would destroy the classical degeneracy of ground states. In other\nwords log Z∼S0−NF−NB\n2logβ→ −∞ in the large βlimit. Since the index gives a lower\nbound on the number of ground states, we can use this criterion to rule out the existence\nof a small black hole horizon with an AdS 2factor. In other words, there is no mass-gap\nand therefore no decoupled AdS 2geometry in such theories.\nBy the arguments in the previous paragraph we reduced the problem to an analysis of\nthe number of bosonic vs fermionic generators of superconformal groups [57]. The list of\nsuch groups can be found e.g. in Table 1 of the review [58], see [59] for an extensive list.\nLet us begin with superconformal groups with more than 16 supercharges. There are\nthree possible families. The first family are SU(1 ,1|n) with n >4, which has a SU( n)×U(1)\nR-symmetry. This group has NB=n2+ 3 and NF= 4n. For all n >1 one has NB> N F\nand therefore the ground states do not survive quantum corrections. The same is true for\nOSp( n|2) for n >8 since NB=1\n2(n2−n) + 3 and NF= 2nand for OSp(4∗|2n) with n >2\nwhich has NB= 2n2+n+ 6 and NF= 8n. Therefore, if an index predicts a large ground\nstate degeneracy, those states cannot describe small black holes preserving more than 16\nsupercharges.\nNext, we consider the case of systems with 16 supercharges. In this case there are four\nsuperconformal groups, which are\nSuperconformal Group NB/NF R-symmetry\nOSp(4∗|4) 16/16 SU(2) ×Spin(5)\nSU(1,1|4) 19/16 SU(4) ×U(1)\nF(4) 24/16 SO(7)\nOSp(8 |2) 31/16 SO(8)(6.2)\nThe last three of these groups have NB> N Fand therefore do not have a ground state\ndegeneracy. The case OSp(4∗|4) is special since NB=NF. Here, the Spin(5) R-symmetry\ndoes not act as a geometric rotation of any subspace of T6.21\nThus, there is no superconformal group, and therefore no Schwarzian theory, which\ncould describe the low energy phase of such small black holes. This argument can be ex-\ntended to theories with 14 supercharges. Below 14 there are multiple possibilities of nearly-\nconformal theories with a large number of ground states such as SU(1 ,1|1) or PSU(1 ,1|2),\nand other superconformal groups which haven’t found an application yet. Another theory\nthat deserves further study along these lines is D(2 ,1;α).\n21It is possible that the supercharges transform as a spinor of the tangent space Spin(5). This possibility\nwas discussed in related AdS 3near-horizon configurations in [53].\n– 35 –7 Discussion\nWe would like to conclude with some comments and future directions.\nWe have evaluated the gravitational path integral that corresponds to the index of\nthe two-charge states in heterotic string theory. This calculation involved constructing a\nsolution that combines two-derivative with four-derivative interactions. An obvious future\ndirection is to make this construction more precise. The advantage of the present paper\ncompared with previous attempts is that we turned this question into a concrete program,\nstarting with proving that D-terms vanish due to supersymmetry. Besides this, it would be\nnecesasry to prove that the action coming from F-terms is temperature independent and\nreproduces (5.14) for any prepotential.\nAnother natural direction is to compute quantum corrections around the black hole\nsolution evaluating the index. We hope that one can use the recent techniques in [60]\nand [61] to reproduce the logarithmic correction to the index in the large N=√nwlimit.\nAn even more ambitious goal can be to use supergravity localization to reproduce the exact\ngenerating function 1 /η24(τ).\nFinally, it is interesting to consider whether this approach can be extended to black\nholes with one charge. An example is a 10 dimensional type IIA string theory black hole\nwith D0-charge N. This leads to a small black hole in the extremal limit and its near-\nextremal excitations are dual to the BFSS matrix model. In this case even a microscopic\ncalculation of the index is hard and has only been completed for the N= 2 theory [62],\nalthough there is a simple M-theory prediction for it for all N. To attack this problem using\nthe gravitational path integral, one can easily find a rotating D0-brane black hole starting\nwith the uncharged Myers-Perry solution in 10 dimensions ( t, x1, . . . , x9), uplifting to 11\ndimensions by adding an extra circle x10, performing a boost in the t−x10coordinates, and\nfinally dimensionally reducing back to 10 dimensions. Imposing the boundary conditions\nrelevant for the index one can find, at the two-derivative level, a supersymmetric rotating\nnon-extremal black hole with a macroscopic area and zero on-shell action, similar to its\ntwo-charge counterpart of Section 3 [63]. The solution is singular at the poles of the horizon\nand the string coupling becomes large at those locations. An evaluation of the index of the\nBFSS model using the gravitational path integral for type IIA seems to require a new idea\nand cannot be computed by a simple extension of the techniques used here.\nAcknowledgements\nIt is a pleasure to thank Pietro Benetti-Genolini, Jan Boruch, Matt Heydeman, Luca Iliesiu,\nJuan Maldacena, Ashoke Sen, and Edward Witten for interesting and useful discussions.\nY.C. acknowledges the KITP program “What is String Theory? Weaving Perspectives\nTogether”, during which this work is completed. S.M. acknowledges the support of the\nJ. Robert Oppenheimer Visiting Professorship at the Institute for Advanced Study, Prince-\nton. The work of GJT is supported by the University of Washington and the DOE award\nDE-SC0024363.\n– 36 –A Solution generation for rotating two-charge systems\nIn this appendix, we discuss how one can use the solution generating procedure to work out\nthe properties of the rotating two-charge system from the uncharged (but rotating) solution.\nWe will derive the thermodynamic quantities of the charged solution with the information\nof the thermodynamic quantities of the uncharged solution as well as the transformation\nat infinity. The derivation applies to general classical solutions in string theory, including\nblack hole solutions as well as potential rotating Horowitz-Polchinski solutions that have\nnot yet been constructed. One application of our derivation, which we discuss in appendix.\nA.1, is that one can work out the α′corrections to the rotating two-charge black hole in\n[27], without needing to find the corrected geometry explicitly. The discussion here mostly\nfollows the derivation in [12], with the extra ingredient that the seed solution can carry\nrotation.\nThe starting point is a seed solution in Ddimensions which has an asymptotic time\ncircle with length ˜β= 2π˜R. For the purpose of applying to the discussion in the main text,\nwe have D= 4, though the discussion is more general. We will assume the existence of a\nU(1) isometry, time translation, everywhere in the solution. For simplicity, we consider the\ntransformations that generate charges along one extra circle, and the relevant symmetry\ngroup is O(2,2) which acts on fields in d=D−1 dimensional space once we dimensional\nreduce on both the time circle and the extra circle. This is a subgroup of the O(7,23)\ntransformation of heterotic string theory compactified on T6we considered in the main\ntext, but it also applies to the Bosonic/Type II theories compactified on a torus. We\nparametrize the transformation by the parameters ( α,β).\nThe main equation, which we refer to [12] for justification, is the invariance of the\naction under the solution generating transformation.\nlogZ(R,Ω, r, µ n, µw,ΦD) = log ˜Z(˜R,˜Ω,˜ΦD). (A.1)\nLet’s unpack this equality by explaining the notations. All the tilded quantities refer to the\nseed solution, which has an angular potential ˜Ω other than the inverse temperature (over\n2π)˜Rand asymptotic value of the dilaton ˜ΦD. The quantities without tildes are those for\nthe generated solution, which also has rbeing the size of the extra circle, and µn, µwbeing\nthe chemical potentials for the momentum and winding charges QnandQw. We will fix\nr= 1 in the following discussion.22We can factor out the dependence of the dilaton and\ndefine the primed quantities23\ne−ΦDlogZ′(R,Ω, µn, µw)≡logZ(R,Ω, µn, µw, ϕD), e−˜ΦDlog˜Z′(˜R,˜Ω)≡log˜Z(˜R,˜Ω,˜ΦD)\n(A.2)\nUsing the invariance of the d=D−1 dimensional dilaton, the asymptotic value of the\ndilaton transforms as\ne−˜ΦD˜R=e−ΦDR (A.3)\n22This is only valid since we are using rto denote the asymptotic size of the circle. Inside the spacetime\nit will be varying.\n23For the dilaton, here we follow the convention used in the main text, see (2.1).\n– 37 –and therefore from (A.1) we have\nlogZ′=R\n˜Rlog˜Z′. (A.4)\nFor the generated solution, we have the general statistical expression for the action\nlogZ=e−ΦDlogZ′=S−2πRM + 2πRΩJ+ 2πRµ nQn+ 2πRµ wQw (A.5)\nwhere various charges and entropy can be computed as\n2πRQ n=e−ΦD∂µnlogZ′,2πRQ w=e−ΦD∂µwlogZ′,2πRJ =e−ΦD∂ΩlogZ′,\n2πM=e−ΦD\u0012\n−∂R+µn\nR∂µn+µw\nR∂µw+Ω\nR∂Ω\u0013\nlogZ′, S =e−ΦD(1−R∂R) logZ′.\n(A.6)\nOn the other hand, for the seed solution, we have\nlog˜Z′=˜S′−2π˜R˜M′+ 2π˜R˜Ω˜J′,\n2π˜R˜J′=∂˜Ωlog˜Z′,2π˜M′= \n−∂˜R+˜Ω\n˜R∂˜Ω!\nlog˜Z′,˜S′=\u0010\n1−˜R∂˜R\u0011\nlog˜Z′.(A.7)\nNote that here similar to ˜Z′we have also defined primed thermodynamic quantities which\nare simply themselves but without the dilaton dependence (for example, ˜S=e−˜ΦD˜S′).\nNow, combining the expression for Qnin (A.6) with (A.7), we get\n2πRQ n=e−ΦD∂µn\u0012R\n˜Rlog˜Z′\u0013\n=e−ΦDR\"\n∂˜R\n∂µn∂˜R\u00121\n˜Rlog˜Z′\u0013\n+1\n˜R∂˜Ω\n∂µn∂˜Ω\u0010\nlog˜Z′\u0011#\n=e−ΦDR\"\n−1\n˜R2∂˜R\n∂µn˜S′+ 2π∂˜Ω\n∂µn˜J′#\n.\n(A.8)\nSimilar derivations lead to\n2πQw=e−ΦD\"\n−1\n˜R2∂˜R\n∂µw˜S′+ 2π∂˜Ω\n∂µw˜J′#\n, (A.9)\n2πJ=e−ΦD\"\n−1\n˜R2∂˜R\n∂Ω˜S′+ 2π∂˜Ω\n∂Ω˜J′#\n, (A.10)\nS=e−ΦD\"\nR2\n˜R2∂˜R\n∂R˜S′−2πR2∂˜Ω\n∂R˜J′#\n, (A.11)\nand finally, for the energy, we have\n2πM= 2πe−ΦD˜M′+e−ΦD\"\n−1 +R\n˜R∂˜R\n∂R−µn\n˜R∂˜R\n∂µn−µw\n˜R∂˜R\n∂µw−Ω\n˜R∂˜R\n∂Ω#˜S′\n˜R\n+ 2πe−ΦD\"\n−1−R\n˜Ω∂˜Ω\n∂R+Ω\n˜Ω∂˜Ω\n∂Ω+µn\n˜Ω∂˜Ω\n∂µn+µw\n˜Ω∂˜Ω\n∂µw#\n˜Ω˜J′.(A.12)\n– 38 –(A.8) - (A.11) constitute as generalizations of the formulas in [12]. These relations are\ngeneral and we didn’t use the specific form of the solution generating transformation yet.\nAll the information of the specific solution generation is contained in the map between\n{˜R,˜Ω} ↔ { R,Ω, µn, µw}. (A.13)\nTo find these relations, only the asymptotic form of the solution generating transformation\nis needed. To zeroth order in α′, the transformation is the same in the bosonic/Type II\nand heterotic case, while they are different at order α′, which we defer to A.1. Following\nthe derivation of [12] (which uses the formalism of [64]), we have24\n˜R=Rp\n1−µ2np\n1−µ2w, ˜Ω =Ωp\n1−µ2np\n1−µ2w. (A.14)\nAn easy way to understand the relation between Ω and ˜Ω is the following. Let’s say that\nthe rotation is in φdirection, then in the seed solution we have periodic identification\n(tE, φ)∼(tE+1, φ+2πi˜R˜Ω), where we’ve chosen the Euclidean time coordinate such that\nits periodicity is one. This identification should remain invariant under the transforma-\ntion, meaning that ˜R˜Ω should be invariant, which explains (A.14). To connect with the\ndiscussion in the main text, we can further parameterize µnandµwin terms of α,β\nµn= tanhα+β\n2, µ w= tanhα−β\n2. (A.15)\nUsing (A.14) and (A.15), as well as QL,R=Qn±Qw, we can simplify (A.8) - (A.11) and\nget\n2πQL=e−ΦDcoshβsinhα\"˜S′\n˜R+ 2π˜Ω˜J′#\n, (A.16)\n2πQR=e−ΦDcoshαsinhβ\"˜S′\n˜R+ 2π˜Ω˜J′#\n, (A.17)\n2πJ= 2πe−ΦDcoshα+ cosh β\n2˜J′, (A.18)\nS=e−ΦD˜S′coshα+ cosh β\n2, (A.19)\n2πM= 2πe−ΦD˜M′+e−ΦD(coshαcoshβ−1) ˜S′\n˜R+ 2π˜Ω˜J′!\n. (A.20)\nThe point of (A.16) - (A.20) is that once we know {˜M′,˜S′,˜J′}of the uncharged solution,\nthe thermodynamic quantities of the charged solution will be completely determined. For\nexample, a consistency check of these expressions is that if we plug in the {˜M′,˜S′,˜J′}\nfor a four dimensional Kerr black hole, we correctly reproduce the thermodynamics of the\ntwo-charge black hole in [27].\nAs an aside, it is interesting to note from the expressions that the combination of\n˜S′\n˜R+ 2π˜Ω˜J′is the “charge” being transformed.\n24Here we follow the convention of [12] for the normalization of the gauge fields and charges, which leads\nto a√\n2 factor difference compared to the main text, i.e. Qappendix =Qmain text /√\n2.\n– 39 –A.1 Generating α′corrections in various cases\nA practical application of (A.16) - (A.20) is that one could use them to work out the α′\ncorrection to the rotating two-charge system using the knowledge of the α′correction to\nthe seed solution, without needing to find the explicit corrected solution. Of course, this\nis assuming that we don’t encounter any singular solutions along the solution generating\ntransformation, so perturbative α′corrections are under control. For this reason, one\ncannot apply the results here directly to the solution we studied in the main text, which\nis singular at the two derivative level. Nonetheless, the formulas would apply when we are\nfar away from the singular limit Ω = 2 πi/β.\nThe starting point of the derivation is the α′correction to the thermodynamics of the\nKerr black hole (focusing on D= 4). This can be dervied by simply evaluating the leading\nα′correction to the action on the uncorrected black hole background. The end results, in\nthe bosonic/heterotic cases, are\n˜β=4πr+(a2+r2\n+)\nr2\n+−a2,˜Ω =a\nr2\n++a2, (A.21)\n˜M′=r2\n++a2\n2r+,˜S′=π(r2\n++a2) + 2πλ, ˜J′=a˜M′. (A.22)\nHere λ=α′/2, α′/4 for bosonic and heterotic string theory, respectively. λvanishes for\nType II theories and the leading order correction happens at α′3, therefore we won’t discuss\nit here. We emphasize that r+, ain (A.21) and (A.22) are only parameters to parametrize\n˜βand˜Ω and they differ from the corresponding geometric quantities by α′corrections.\nFor bosonic string theory, we can then plug (A.21), (A.22) into (A.16) - (A.20) can get\nthe leading α′corrections to the rotating two charge system:\n1\nT=4πr+(r2\n++a2)\nr2\n+−a2coshα+ cosh β\n2,Ω =a\nr2\n++a22\ncoshα+ cosh β, (A.23)\nQL= sinh αcoshβ\u0014π(r2\n++a2)\n2r++λπ(r2\n+−a2)\nr+(r2\n++a2)\u0015\n, (A.24)\nQR= cosh αsinhβ\u0014π(r2\n++a2)\n2r++λπ(r2\n+−a2)\nr+(r2\n++a2)\u0015\n, (A.25)\nJ=coshα+ cosh β\n2ar2\n++a2\n2r+, (A.26)\n2πM=π(r2\n++a2)\n2r+(1 + cosh αcoshβ) +λπ(r2\n+−a2)\nr+(r2\n++a2)(coshαcoshβ−1), (A.27)\nS=coshα+ cosh β\n2(π(r2\n++a2) + 2πλ), (A.28)\nwhere we’ve set GN= 1 in the above expressions.\nFor the case of the heterotic string theory, the correct solution generating transforma-\ntion in fact differs from the bosonic and type II theory by terms proportional to α′. The\n– 40 –correct transformation was motivated by studying the Horowitz-Polchinski solution in [12],\nbut was later independently verified by studying the α′corrections to the non-rotating two-\ncharge system [28]. The derivation here follows that of [12], with the main new ingredient\nbeing that the angular potential Ω is related to ˜Ω of the seed by RΩ = ˜R˜Ω. We omit the\nintermediate steps and present the results analogous to (A.16) to (A.20).\nR=˜R\n2\u0014\u0012\n1 +α′\n2˜R2\u0013\ncoshα+\u0012\n1−α′\n2˜R2\u0013\ncoshβ\u0015\n,Ω =˜R\nR˜Ω, (A.29)\n2πQL=e−ΦDcoshβsinhα\"˜S′\n˜R+ 2π˜Ω˜J′#\n,2πQR=e−ΦDcoshαsinhβ\"˜S′\n˜R+ 2π˜Ω˜J′#\n,\n(A.30)\nJ=e−ΦD˜J′\n2\u0014\u0012\n1 +α′\n2˜R2\u0013\ncoshα+\u0012\n1−α′\n2˜R2\u0013\ncoshβ\u0015\n, (A.31)\nS=e−ΦD\u0014coshα+ cosh β\n2˜S′−α′(coshα−coshβ)\n4˜R2(˜S′+ 2π˜R˜Ω˜J′)\u0015\n, (A.32)\n2πM= 2πe−ΦD˜M′+e−ΦD(coshαcoshβ−1) ˜S′\n˜R+ 2π˜Ω˜J′!\n. 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Surya Prasath2,3,4,5* \n \n1School of Computer Science and Engineering (SCOPE), Vellore Institute of Technology, \nVellore, India  \nshreyash.rawat@gmail.com , vijayarajan.v@vit.ac.in  \n2Division of Biomedical Informatics, Cincinnati Children’s Hospital Medical Center, \nCincinnati, OH 45229 USA  \n3Department of Pediatrics, University of Cincinnati, OH USA  \n4Department of Biomedical Informatics, College of Medicine, University of Cincinnati, OH \n45267 USA  \n5Department of Computer Science, University of Cincinnati, OH 45221 USA  \n*prasatsa@uc.edu  \n \n \nAbstract  \n \nText extraction is a highly subjective problem which depends on the dataset that one is working \non and the kind of summarization details  that need s to be extracted out . All the steps ranging \nfrom preprocessing of the data , to the choice of an optimal model for prediction s, depends on \nthe problem  and the corpus  at hand . In this paper, w e describe a text extraction model where \nthe aim is to extract word specified information relating to the semantics such that we can get \nall related and meaningful information about that word  in a succinct format . This model can \nobtain  meaningful results and can augment  ubiquitous search model or a normal clustering or \ntopic modelling algorithm s. By utilizing  new technique called two cluster assignment \ntechnique  with K -means model, we  improved  the ontology of the retrieved text. We further \napply the vector average damping technique for flexible movement of clusters. Our \nexperimental results on a recent corpus of Covid -19 shows that we obtain good results based \non main  keywords.   \n \nKeywords: Clustering, text mining, cluster assignment, modified K -means, COVID -19. \n \n \n1. Introduction  \n \nWe live in a n information rich world where we are inundated  with a large number  of ambient \nresources like text data and we typically fail to leverag e all of it. There is a gargantuan amount \nof information present on the world wide web whether it be the books or articles. There is a \nhuge amount of gap between the skills we possess to make use of the information and the \namount of information available. Large amounts of data tacitly lead to large amounts of \ninformation that may or may not be useful for real wo rld insights , hence we need tools and \ntechniques to refine the data available to us so that we can exactly get what we want. Scientific \nresearch  articles in journals and books are one major area where a lot of potential information exists. However,  due to the large number  of papers on each topic it is almost impossible for \none person to go through all of them and get the information that she was looking for about a \nspecific technique or a new invention.  Effective text extraction is one of the solutions to this \nimportant problem . There has been a tremendous amount of research on text extraction and we \nhave surely come a long way from the simple keyword search algorithm to the various complex \nalgorithms for effective topic modelling. Th e one area that has been left scarce is word \nweighted clustering where you get the power of clustering algorithms along with the topics that \nyou want to extract information about  [17]. Normal clustering such as the ones based on K -\nmeans or topic modelling algorithms take each input token (derived from sentence) as an equal \nentry and clusters the points closest in the region [7]. This method does give you the most \nfrequently occurring  topics but might well miss out on topics that are very important but less \nfrequently occurring . In this work, we consider the classical K -means clustering as the basis \nfor text mining from  scientific literature.  \n \nWith the recent coronavirus disease (COVID -19), there is a deluge of scientific articles that are \nbeing released and effective summarization and clustering are important in distilling the salient \ninformation  [5]. The Kaggle COVID -19 open research dataset (CORD) [ 16], indexed more \nthan 63000 scholarly articles including over 51000 with full text containing information about \nthe COVID -19, SARS -CoV -2 and related coronaviruses. There is an immense need for \neffective clustering of these scientific articles to ex tract important details for the medical and \nresearch community in tackling this ongoing pandemic. Using clustering models within  the \ntext mining literature  we propose  to create a n efficient  model wherein information about \nCOVID -19 related  topics can be extracted by giving importance to that word in determining \nthe cluster centroids in the algorithm. Leveraging the K -means clustering model along with our \nnewly introduced two cluster assignment technique here, we devise a keyword s-based \nsummarization of COVID -19 data.  We further utilize the classical vector averaging damping \nfactor [2] for free movement of the cluster centroids we obtain accurate results for not just the \nfrequent topics but also for the important ones that the user wishes to search for.  Our \nexperimental results indicate that we obtain good results based on the modified clustering \napproach considered here. Further, relevant keywords -based text mining on  the recent COVID -\n19 Open Research Dataset (CORD -19) dataset indicate that the two-cluster  assignment along \nwith K -means provided succinct  and salient  summarized information from the large amount \ntext. \n \nWe organized the rest of the paper  as follows. Section 2 introduces our modified K -means \nmodel for clustering the new COVID -19 dataset based on important keywords. Section 3 \nprovides experimental results and analysis,  and Section 3 concludes the work . \n \n2. Methodology  \n \n2.1 Dataset  \n \nThe dataset we used here  consists of  2GB text data on coronavirus (COVID -19, SARS -CoV -\n2) COVID -19 Open Research Dataset (CORD -19) [6], which has over 5 9000 scholarly articles \nand includes over 41000 with full text  availability . This is an ideal data to test our algorithm as \nthe dataset is fresh due to the velocity of the amount of research done in a very short period of \ntime. We have taken 100/200 research papers from 3 different corpus namely Biorxiv_medxriv, \ncomm_use_subset and pmc_custom_license. We obta ined similar results with batch sizes 400 \nand above . The subsets allow us to easily visualize the data points without crowding and helps \nus visualize the clustering process from a fundamental view. Hence the results can be used by the researchers to extract information rather than the prosaic keyword search algorithms  [16]. \nThis dataset contains information not only about coronavirus but also every related article \nrelating to the coronaviruses  and therapeutics. Due to the rapid research perpetuating, these \ntechniques will be in demand and can be  required to extract meaningful information  as quickly \nas possible . \n \n2.2 Preprocessing  \n \n \nFig.1. Flowchart of the pre -processing steps utilized in our clustering approach for Covid -19 \nscholarly articles.  \n \n \n \nPreprocessing steps are important as they set up the data appropriately for stages of text mining \nand information extraction . The cleaner the data the better will be the results predicted by the \nmodel. This is also one of the most obscure steps among all as there is no specified steps or \nframework to carry out preprocessing. It largely depends on the context of the problem and the \ndataset that is being analyzed  for insights  [8]. The overall steps we have utilized here are shown \nin Figure 1. In our work,  the preprocessing starts by chunking the research paper into a group \nof two to three sentences. Taking the assumptions that two sentences will probably be stating \nsimilar topics, this technique has two benefits, the first being the reduction of time complexity \nas instead of assigning each new sentence we will assign chunks that will probably give similar \noutputs. Effective clusters due to absence of redundant data p oints is the second benefit that \nthis method possesses.  Then these chunks have to be arranged in a structured format to feed \nthem into machine learning models. This is done by first tokenizing each chunk and removing \nthe stop words and punctuations from th e sentences. In research articles we can find a lot of \nwords that do not require attention when it comes to text extraction like numbers, websites, \ncitations, figures and graphs. Since we are dealing with scholarly articles there were a lot of \nspecific kin ds of words that were to be kept and many that were removed. Regular expressions \nwere used for this purpose wherein certain patterns were compiled,  and the tokens were then \nmatched with these expressions [9].  \n \nApart from removing the regular stop words, there were still many words like articles or \nnumbers that have no use to cluster sentences but will unnecessarily increase the dimensions \nof the vector embeddings of these sentences. Hence to dispel such words we  used a POS \neliminator wherein we remove words with certain parts of speech to generate accurate vectors \naccording to the main topics. This played an important part in removing the redundant pronouns \nand articles.  After getting the tokens cleaned,  we now h ave to create vectors for these tokens \nso that they can be fed into the predictive models. There are various kinds of approaches present \nto facilitate the token to vector transformation like the Count Vectorizer, tf-idf and the word to \nvec. We have incorporated the tf -idf approach [10] for two important reasons:  \n1. It reduces the importance of the frequency for a given word. If we had used \nCountVectorizer , it would have given huge importance for frequently occurring  words \nwhich is exactly what we do not want. Hence, it reduces the weight factor of frequency \nwhile determining the vector values.  \n2. It tf-idf is a normalized model and obtains the values within a range so that the model \ndoes not categorize the important words with a high weight vector as outliers.  \n \n2.3 Dimensionality Reduction and Batching  \n \nWhile generally converting sentences into tf -idf vectors the number of columns represent the \nvocabulary of the training set which is usually very large as it takes into account all the cleaned \ntokens of all the sentences. This is not favorable  to the clustering algorithms as it increases the \ntime required to calculate the distance between each point and the cluster centroid [11]. Another \ndrawback of a large number of dimensions is the presence of redundant or similar words that \nare not lemmatiz ed to the sam e root word. It takes words like these as separate columns which \neventually mean the same thing and unnecessarily increases complications. Dimensionality \nreduction is also required to visualize the data as it is elusive to make sense of data by just \nlookin g at their vectors. The data visualization for the keywords “ Vaccine” and “ Transmission” \nare shown in Figure 2 . \n \nThere are plenty of dimensionality reduction algorithms present for this purpose like principal \ncomponent analysis (PCA), linear discriminant analysis (LDA) and auto encoders [12]. While \nautoencoders are generally suited for image and audio data, LDA explo res the relationship \nbetween dependent and independent variables. In other words, it finds the direction that \nmaximizes the difference between two classes which is ideal for prediction purposes. But what \nwe want here is the distinction between independent variables for clustering purposes and PCA \ndoes exactly that. PCA finds the direction which maximizes the variance in the data thus \ndiscovering the relationship between independent variables.  \n It is cumbersome  to store and process all the available data in the memory as it exhausts the \nRAM and no further processing is possible on data. To overcome this obstacle,  we have used  \na batching technique wherein the entire dataset is stored in the memory rather than the RAM \nand we access the memory in batches, process it and then store it back to the memory. This \nprevents the RAM memory from being exhausted although it may increase the processing time \nto some extent. We have utilized batch sizes of 50 ful l text scholarly articles which will be \nprocessed at one batch. Another major advantage of our modified K Means algorithm is the \ntime complexity. The additional processing steps coalesced into the algorithm does not increase \nthe time complexity by polynomi al factors. The time complexity of our algorithm turns out to \nbe O(n^2 + n) which approximates to the time complexity of the normal K -Means which is \nO(n^2).   \n \n  \nFig 2 . Distribution of COVID -19 scholarly  articles d ata with search query  keywords  as \n“Vaccine” and “ Transmission” . \n \n \n2.4 Clustering  \n \nFor forming effective clusters from the normalized tf -idf vectors as inputs we used a variation \nof K -means algorithm. The standard K -means algorithm [13] assigns K number of random \npoints as clusters having equal weights and then finds the distance of each point from the \nclusters and then the new cluster centroi ds are derived by taking the average of all the points \nbelonging to that particular cluster. This process goes on until the average of the cluster points \nis the same as the previous clusters. This method works well for most common cases and gives \nyou the m ost commonly occurring  topics. Hence , if we want a cluster with a specific topic,  we \nneed to keep increasing the number of clusters until we get that cluster.  \n \nTo overcome this obstacle we have come up with a modified K -means model that has three \namendments to improve performance. The major one being giving explicit weights to each tf -\nidf vector [15]. The weights that will be assigned to each vector will depend on the search query \nor the word that one is interested in. The chunks where this word is syntactically and \nsemantically significant gets a higher score compared to others. The rest of the chunks that are \nnot related to the search query get a default score o f 0 which causes quite a few problems. The \ncloistered chunks are given a weight of 0.01 in our case and may vary depending upon the \nnonzero weights of the other chunks. We need to ensure that the weighted chunks that are \nrelated to the search term should d iscern from the others.  \n \nIf we give all the cloistered chunks a weight of zero,  then the clusters obtained will all be \nresolutely focused  on the search query which will lead to redundant and repeating clusters \ncontaining the same information. Hence the chunk with a higher weight pulls the centroid of \nthat cluster more strongly than the rest which leads to clusters biased towards these vecto rs, \nsee Fig ure 3 for the visualization. If we assign the unrelated chunks with a weight of 0, then \nthey will not contribute anything towards the calculation of new centroids for that epoch. This \nwill lead to all the clusters being crowded near the search term and will lea d to inaccurate \nresults. To overcome this impediment the unrelated chunks were assigned a weight of 0.01 so \nthat they have a minimal say during the centroid calculation but do not influence the centroids \nto a large extent.  The other amendment to the K -means we proposed includes a novel technique \ncalled “two cluster assignment”. The motivation behind this technique was the inclusion of \nchunks under more than one topic or cluster. Since we are using chunks that greatly reduces \nthe computation power and time complexity, the re is a possibility that a group of sentences \nbelong to multiple clusters. Therefore,  while assigning a chunk to a cluster, if the distance \nbetween the chunk and the two centroids is less than a threshold then we assign that chunk  to \nboth the clusters. This leads to more accurately extracted text for both the topics. Another \nchange that we have incorporated is the usage of centroid damping while calculating the mean \nof cluster centroids. This technique is similar to the average damping factor [1] but instead of \nassigning a weight of zero to the current centroid, it giv es it a weight that has been assigned to \nthe cloistered chunks which is 0.01 in our case. This neither devalues the current cluster \ncentroid nor exalts it, making it i deal for the calculation of the new cluster centroid. Algorithm \n1 describes the steps of our two-cluster  assignment -based  K-means model.  \n \nIteration -wise clustering for the words “ Vaccine” , “Transmission” , “Incubation”, and \n“Surveillance”  are shown in Figure  4 to 7 respectively.  The black points shown in these images  \nrepresent those points which were assigned to two clusters simultaneously in that iteration. \nAfter the centroids stabilize , the highlighted common points were usually those which fit well \nin both cluster topics hence making the cluster content more accurate than before. After \nobtaining the cluster centroids, we now need to assign each chunk to a cluster so that we can \nretrieve the full text of the required cluster. The results obtained were quite accurate stating \nresults about the search query in great depth. We note that clusters about the same topic have \nbeen well separated expla ining different things about the same topic.  \n \n \n \nFig 3. Weights assignment for two cluster assignment -based  K-means model.  We assign two \nclusters instead of one.  \n \n \nAlgorithm 1  \n \n1. t = 0 \n2. Set {c1^t, c2^t, …. , ck^t} to be distinct points from input {x1, x2, ….. , xn}  \n3. Repeat  \n  t = t + 1  \n  d = []  \n  for i = 1….n do  \nfor j = 1.….k do  \nd.append (|| cj^t, xi ||, cj^t)  \nx = sort(d[0])  \nj* = x[0][1]  \ncj* = cj* U {xi}  \nIf (x[1][0] - x[0][0] < 0.01) #threshold  \ni* = x[1][1]  \nci* = ci* U {xi}  \n4. #Cluster assignment  \n Foreach i = 1….k do  \n  ri^t = (1 / |Ci| ) * sum( xj * wj ) for every xj in Ci  \n  Until sum from 1 ...k  ( || ri^t - ri^(t-1)) < epsilon  \nend \n \n \nFig 4.  Iteration -wise clustering for the word “ Vaccine ”. We show for iterations 1, 3, 7, 9, 11, \n15. The black points represent the common cluster points through the iterations.  \nFig 5. Iteration -wise clustering for the word “Transmission”. We show for iterations 1, 2, 3, \n6, 9, and 13. The black points represent the common cluster points through the iterations.  \n \nFigure 6. Search query - “Incubation”. We show for iterations 1, 3, 7, 9, 11 and 15. The black \npoints represent the common cluster points through the iterations.  \n \n \nFigure 7. Search query - “Surveillance ”. We show for iterations 1, 3, 7, 9, and 11. The black \npoints represent the common cluster points through the iterations.  \n \n \n2.5 Fine Tuning  \n \nSelecting the ideal value of K for the algorithm has always been a tedious task. There are few \nmethods like the elbow method [14] that plot the distortion score against the number of clusters \nlike those shown in Fig ure 8. The ideal number of clusters turns out to be the one from where \nthe graph starts to flatten out. The distortion score is a metric which is calculated by how far \neach point is from its assigned cluster. While following the “two cluster assignment” technique \nthere can be one point that is a part of two clusters. In that case the point will be counted in \nboth the clusters separately to determine the distortion factor.  \n \nTable 1. Results obtained for word “ Vaccine”  \nCorpus  Total \nno. of \npara # of related para \nobtained by \nsearch algorithm  # of related para \nobtained by \nnormal K -Means  # of related para \nobtained by \nmodified K -\nmeans  \nBiorxiv_medxriv  3785  1758  1150  1156  \ncomm_use_subset  2115  948 474 479 \npmc_custom_license  2012  1023  452 461 \n \nTable 2. Results obtained for word “ Transmission”  \n                                                \nCorpus  Total \nno. of \npara # of related para \nobtained by \nsearch algorithm  # of related para \nobtained by \nnormal K -Means  # of related para \nobtained by \nmodified K -\nmeans  \nBiorxiv_medxriv  3785  1615  898 915 \ncomm_use_subset  2115  1024  732 733 \npmc_custom_license  2012  893 391 396 \n \nTable 3. Results obtained for word “ Incubation”  \n \nCorpus  Total \nno. of \npara # of related para \nobtained by \nsearch algorithm  # of related para \nobtained by \nnormal K -Means  # of related para \nobtained by \nmodified K -\nmeans  \nBiorxiv_medxriv  3785  899 710 717 \ncomm_use_subset  2115  561 337 370 \npmc_custom_license  2012  566 289 311 \n \nTable 4. Results obtained for word “Surveillance”  \n \nCorpus  Total \nno. of \npara # of related para \nobtained by \nsearch algorithm  # of related para \nobtained by \nnormal K -Means  # of related para \nobtained by \nmodified K -\nmeans  \nBiorxiv_medxriv  3785  153 79 83 \ncomm_use_subset  2115  264 115 127 \npmc_custom_license  2012  248 164 164 \n \n  \nFig 8.  Trend of distortions values as the K -values increase, distortions versus K values for \nkeyword s: (a)  “vaccine” , (b) “transmission”, (c) “incubation” , (d) “surveillance” . \n \nThe number of dimensions the tf -idf vector should be reduced to is again an empirical decision \nwhere we can only get the answer by implementing with different values and checking the \ndistortion values of the clusters. The max number of iterations of the al gorithm is another factor \nto be looked upon. It is usually set to a higher value to be safe about the right clusters being \nformed. Partial clustering [2] is one powerful method which can be a substitute to random \nseeding of the centroids. In this method , K-means is performed on a subset of data and then \nthose obtained clusters act as the initial clusters for the entire data. This might be more effective \nthan random seeding but there will be a huge difference in the clustering time as you perform \nthe algorit hm 1.5 times instead of 1.  \n \n3. Experimental Results and Discussion  \n \nAfter obtaining the clusters, we need to check if the results obtained corroborate with the \nrequired information. In our experiments, we have chosen some example weighted words , such \nas “Vaccine” and “ Transmission” , “Incubation”, and “Surveillance”  as we wanted some useful \ninformation about them.  About vaccines, the first task consists of finding all possible things \nthat we know about the vaccines for coronavirus and the second being the methods or \nprocedures that can be applied for making the future vaccin e that can cure this disease. \nSimilarly,  for the transmission we were looking for information about its transmission rate and \nmeans of transmission. We can judge our accuracy of the clusters by checking the ten most \ncommon words occurring in each cluster which will give us a good idea as to what  kind of \nsentences the clusters accommodate.  \n \nWe got good  results wherein two clusters had the most important word as vaccine. One which \ngave insights about the vaccine and what was currently being used and the second cluster \nconsisted of the potential combinations that could form the vaccine for the coronavirus. The \nonly way to ju dge the accuracy of the clustering algorithm in this kind of problem is by trying \nout different K values until you get the required clusters.  The threshold value to be selected in \nthe two-cluster  assignment technique is another empirical decision to be made. We tried the \nmodel for various threshold values according to the distances obtained for each point from each \ncluster centroid. The most vigilantly separated results were obtained for a thresho ld value of \n0.01 where the points were neatly clustered,  and the common points were important to both the \nclusters.  \n \nThe number of related paragraphs obtained for the sample queries have been portrayed in Table \n1 to 4 . There is a conspicuous difference between the modified K -means and the search \nalgorithm, where the obtained information in case of K -means is concise and well related to \nthe cluster hence facilitating accurate text extraction by winnowing out the junk which does \ncontain that word but is not related to the context.  The Figure 9 shows the explanation obtained \nfor the invention of the new vaccine that can be p roposed. As evident from the Fig ure 9 and \nFigure 10, there are numerous chunks that are close to each other and that cluster constitutes \ncommon information regarding vaccines. As the points deter,  they form two to three  other \nclusters which contain specific information or topics on vaccines like scope for invention of \nnew vaccines or current knowledge of vaccines to fight different kinds of viruses. This is clearly \nevident as the results obtained in Fig ure 9 states about the possible new structure for inven tion \nof vaccines and Fig ure 10 gives us information about the vaccines and technology currently in \npractice . Figure 11 to 13 shows the content obtained by taking the word s “Transmission” , \n“Incubations”, and “Surveillance” . Figure 11 shows results  for “Transmission” as the search \nquery and we note that  it provides  relevant information about the transmission information both \nexperimental and academic about the novel coronavirus.  \n \n \nFig 9. Gives information about the potential new vaccines  \n \n \nFig 10. Gives information on current knowledge of vaccines  \n \n \nFig 11. Results for cluster with most important word as “ Transmission”  \n \n \nFigure 12. Results obtained for the keyword “ Incubation”  \n \n \nFig 13. Results obtained for search term as “ Surveillance”  \n \n \n4. Conclusion  \n \nWe present ed a word weighted information extraction algorithm that retrieves information not \nonly syntactically but also semantically giving much more accurate results than the prosaic \nsearch algorithm. We introduced quite a few modifications in the existing K -Means a lgorithm \nwhich in turn proved out to be quite effective.  The introduction of chunking for large datasets \nproved out to be a tenable step which not only helped in reduction of time complexity but also \nmade predictions much accurate simultaneousl y reducing computing power. The two-cluster  \nassignment technique made sure that group of sentences is present in all the clusters that it \nshould be in rather than the one it is closest to , making our approach efficient . The tweaked \nword weight vector that assigns the non -related vectors a partial weight rather than zero \nprovided good  results by not undermining the non -related vectors completely.  The performance \nof our proposed  clustering method  can further be improved . We can  use other effective seeding \nmethods that reduces the clustering time. Another proposed information retrieval technique can \nbe retrieval of information using semantic similarity. If we can represent a sentence into a \nvector that shows its semantic properties,  then that can be used to get sentences similar to a \nsearch query. The only difference will be the input entered will be a sentence instead of a word.  \n \n \nReferences  \n \n1. Larsen B,  Aone  C (1999) Fast and effective text mining using linear -time document \nclustering. In Proceedings of the fifth ACM SIGKDD international conference on \nKnowledge discovery and data mining  (pp. 16 -22). \n2. Jain AK (20 10) Data clustering: 50 years beyond K -means. Pattern Recognition \nLetters , 31(8), 651 -666. \n3. Blömer J, Lammersen C, Schmidt M , Sohler C  (2016 ) Theoretical analysis of the k -\nmeans algorithm  - a survey. In Algorithm Engineering  (pp. 81 -116). Springer, Cham.  \n4. Han J, Pei J, Kamber M (2011) Data mining: Concepts and techniques . Elsevier.  \n5. Brainard  J (2020)  Scientists are drowning in COVID -19 papers. Can new tools keep \nthem afloat? Science Magazine, May 2020. Available online \nhttps://www.sciencemag.org/news/2020/05/scientists -are-drowning -covid -19-papers -\ncan-new-tools -keep-them -afloat . \n6. COVID -19 Open Research Dataset (CORD -19). https://www.kaggle.com/allen -\ninstitute -for-ai/CORD -19-research -challenge . \n7. Jing L, Ng MK, Xu J, Huang JZ (2005) Subspace clustering of text documents with \nfeature weighting k -means algorithm. In Pacific -Asia Conference on Knowledge \nDiscovery and Data Mining  (pp. 802 -812). Springer, Berlin, Heidelberg.  \n8. Vijayarani S, Ilamathi MJ,  Nithya M (2015) Preprocessing techniques for text mining -\nan overview. International Journal of Computer Science and Communication \nNetworks , 5(1), 7 -16. \n9. Bille P, Thorup M (2010). Regular expression matching with multi -strings and \nintervals. In Proceedings of the Twenty -first annual ACM -SIAM Symposium on \nDiscrete Algorithms  (pp. 1297 -1308). Society for Industrial and Applied Mathematics.  \n10. Ramos J (2003) Using tf -idf to determine word relevance in document queries. In \nProceedings of the First Instructional Conference on Machine Learning  (Vol. 242, pp. \n133-142).  11. Paukkeri MS, Kivimäki I, Tirunagari S, Oja E, Honkela T (2011) Effect of \ndimensionality reduction on different distance measures in document clustering. In \nInternational Conference on Neural Information Processing  (pp. 167 -176). Springer, \nBerlin, Heidelberg.  \n12. Van Der Maaten L, Postma E, Van den Herik J (2009) Dimensionality reduction: a \ncomparative. Journal of  Mach ine Learn ing Research , 10(66-71), 13.  \n13. Likas A, Vlassis N, Verbeek J J (2003). The global k -means clustering algorithm. \nPattern Recognition , 36(2), 451 -461. \n14. Kodinariya TM, Makwana PR (2013). Review on determining number of cluster in K -\nmeans clustering. International Journal  Advance Research in  Computer Science and \nManagement Studies , 1(6), 90-95. \n15. Kerdprasop K, Kerdprasop N, Sattayatham P (2005) Weighted k -means for density -\nbiased clustering. In International Conference on Data Warehousing and Knowledge \nDiscovery  (pp. 488 -497). Springer, Berlin, Heidelberg.  \n16. Agrawal S, Chaudhuri S, Das G (2002). DBXplorer: enabling keyword search over \nrelational databases. In Proceedings of the 2002 ACM SIGMOD International \nConference on Management of data  (pp. 627 -627).  \n17. Shi H, Wang P, Yang X, Yu H.  (2020)  An improved mean imputation clustering \nalgorithm for incomplete data. Neural Processing Letters . 2:1-4. \n \n \n \n "    },    {        "title": "2402.03415v1.Cutoff_for_mixtures_of_permuted_Markov_chains__general_case.pdf",        "content": "Cutoff for mixtures of permuted Markov chains:\ngeneral case\nBastien Dubail∗\nKTH, Stockholm, Sweden\nAbstract\nWe investigate the mixing properties of a finite Markov chain in random environment\ndefined as a mixture of a deterministic chain and a chain whose state space has been permuted\nuniformly at random. This work is the counterpart of a companion paper where we focused\non a reversible model, which allowed for a few simplifications in the proof. We consider here\nthe general case. Under mild assumptions on the base Markov chains, we prove that with\nhigh probability the resulting chain exhibits the cutoff phenomenon at entropic time logn/h,\nhbeing some constant related to the entropy of the chain, when the chain is started from a\ntypical state. However contrary to the reversible case uniform cutoff at entropic time does not\nhold, as we provide an example where the worst-case mixing time has at least polylogarithmic\norder. Wealsoprovideapolylogarithmicupperboundontheworst-casemixingtime, whichin\nfact plays a crucial role in deriving the main result for typical states. Incidentally, our proof\ngives a clear picture of when to expect uniform cutoff at entropic time: it appears as the\nconsequence of a uniform transience property for the covering Markov chain used throughout\nthe proof, which lies on an infinite state space and projects back onto the initial chain.\n1 Introduction\n1.1 Cutoff for mixtures of permuted Markov Chains\nThis paper establishes a cutoff phenomenon at entropic time for a model of Markov chain in\nrandom environment. Let n≥1be an integer and P1, P2twon×nstochastic matrices. This\npaper is basically concerned with the mixing time of the Markov chain with transition matrix\np1P1+p2SP2S−1\nwhere p1∈(0,1), p2= 1−p1andSis a uniform n×npermutation matrix. This model is inspired\nby the work [14] of Hermon, Sousi and Sly, who proved the cutoff phenomenon at entropic time\nfor the simple random walk on a sequence of deterministic graphs to which is added a random\nuniform matching of the vertices. The model above aims to look at more general Markov chains,\npossibly non reversible, obtained by superposing two deterministic ones with a layer of randomness\nin between taking the form of a permutation of the states. Our model will in fact allow the\nprobabilities p1, p2to depend on the current state of the chain and on the permutation S, which\n∗Correspondence to be sent to: bastdub@kth.se\n1arXiv:2402.03415v1  [math.PR]  5 Feb 2024is necessary to recover the case of the simple random walk on the superposition of graphs if these\nare not regular. For this case however, and more generally for reversible chains, there is a more\nadapted model tailored to preserve reversibility, corresponding to a specific choice of p1, p2. This\nmodel is studied in the companion paper [11]. We focus here on the most general case.\nLet us first introduce general notations that will be used throughout the paper. Given two\nmeasures µ, νon a countable set S, their total variation distance is defined as\n∥µ−ν∥TV:= sup\nA⊆S|µ(A)−ν(A)|=1\n2X\nx∈S|µ(x)−ν(x)|.\nIfPis the transition kernel of a positive recurrent, irreducible and aperiodic Markov chain on S,\nit admits a unique invariant measure π. In that case, given a starting vertex x∈Sandε∈(0,1),\nthe mixing time is defined as\ntmix(x, ε) := inf {t≥0 :\r\rPt(x,·)−π\r\r\nTV< ε}.\nand the worst-case mixing time is\ntmix(ε) := sup\nx∈Stmix(x, ε).\nIf the chain is not irreducible or aperiodic, we consider the mixing time to be infinite. Given two\nfunctions f, g:N→R,f(n) =O(g(n))if there exists a constant C >0such that |f(n)| ≤Cg(n)\nfor all n,f=o(g)iff(n)/g(n)− − − − →\nn→∞0andf= Θ( g)iff=O(g)andg=O(f). An event\nA=A(n)is said to occur with high probability, if P(A) = 1−o(1). Given an integer n≥1, we\nwrite [n] :={1, . . . , n }.\nTheorem 1.1. LetP1, P2be two n×nstochastic matrices, p1, p2∈Mn([0,1])ben×nmatrices\nwith entries in [0,1]such that p1+p2= 1entry-wise, σa uniform permutation of nelements and\nconsider the Markov chain with transition probabilities\n∀x, y∈[n] :P0(x, y) :=p1(x, σ(x))P1(x, y) +p2(x, σ(x))P2(σ(x), σ(y)). (1)\nSuppose\n(H1) for each i∈ {1,2}the non-zero entries of P1, P2areΘ(1)andmax y∈VP\nx∈VPi(x, y) =\nO(1).\n(H2) the entries of p1, p2areΘ(1).\n(H3) For all x∈[n]andi= 1,2:\n\f\f{y∈V:∃k≥0 :Pk\ni(x, y)>0}\f\f≥(\n3ifi= 1\n2ifi= 2,\n(H4) There exists a constant l≥1independent of nsuch that\n\f\f{x∈V| ∀y∈V:P(x, y)>0⇔(I+P)l(y, x)>0}\f\f=n−o(n)\nThen the chain defined by (1)is irreducible, aperiodic and there exists h= Θ(1)for which the\nfollowing holds. For all ε∈(0,1), there exists C(ε)such that for all x∈[n]with high probability,\ntmix(x, ε)≤logn\nh+C(ε)p\nlogn,\ntmix(x,1−ε)≥logn\nh−C(ε)p\nlogn.\n2Note that in Theorem 1.1, the starting state of the chain is fixed before generating the random\nenvironment. This implicitely makes the state xtypical: from the result for a fixed xone can\nfor instance replace xwith a state chosen uniformly at random. However it cannot be chosen\narbitrarily after the permutation σis fixed. Consequently our result does not establish uniform\ncutoff in the sense that with high probability tmix(x, ε) = log n/h+o(logn)simultaneously for\nallx∈[n]. This makes a strong difference with the reversible case studied in [11] in which we\nwere able to prove uniform cutoff at entropic time. It turns out that with the level of generality\nconsidered here, the lack of a uniform result is not an artifact of the proof but is characteristic\nof the model. In whole generality, there is no uniform cutoff at entropic time, as proved by the\nfollowing result.\nTheorem 1.2. For all a >1, there exist transition matrices P1, P2satisfying Assumptions (H1)\n- (H4) such that with high probability, the Markov chain P0= 1/2(P1+SP2S−1)has worst-case\nmixing time max xtmix(x,1/2)≥(logn)a.\nOur model is not the first instance which exhibits a non-uniform cutoff phenomenon. It was\nalready observed for instance in the case of simple random walks on the giant component of Erdös-\nRenyi graphs: with high probability these contain segments of length of order logn, resulting in\nthe worst-case mixing time being of order (logn)2[4, 12], while the typical mixing time is O(logn)\n[5].\nIn our model, the potential slowdown of the chain is a consequence of irreversibility: having too\nmuch bias on the transition probabilities, the environment can with positive probability contain\ntrapping neighbourhoods , from which the chain escapes in a time that scales exponentially with\nthe size of the neighbourhood. The previous proposition is based on an example where these\nneighbourhoods are of size O(log log n), hence the polylogarithmic order in the result. Our last\nresult establishes a matching upper bound of polylogarithmic order on the worst-case mixing time.\nTheorem 1.3. In the setting of Theorem 1.1 under the same assumptions there exists a≥1such\nthat for all ε∈(0,1), with high probability\ntmix(ε)≤(logn)a.\nRemark 1.1.Let us comment on the assumptions made above. Hypotheses (H1) and (H2) bound\nuniformly in nthe transition probabilities and the number of ingoing and outgoing transitions at\nevery state, while (H3) imposes a minimal size on the set of states reachable by a given state.\nThese are not restrictive: it is easy to find examples where these assumptions are not satisifed and\nthe mixing time scales differently as Θ(log n), without cutoff. The most restrictive assumption\nis thus (H4). It can be interpreted as a constraint on the degree of irreversibility of the chains\nP1, P2, which for instance prevents them from having too many transient states. It is not clear\nwhether it is really needed for our result but from the proof it does not seem superfluous either, as\nhavingmoreirreversibilitymayamplifythetrappingphenomenonmentionnedabove. Nonetheless,\nirreversibility is still permitted to a considerable extent and appears as a genuine limitation rather\nthan a technical one.\n31.2 Related works\nThis paper is the direct extension of [11] where we analyse a specific case of the Markov chain (1)\ntailored to preserve reversibility, for which cutoff at entropic time can be proved uniformly over\nthe starting state. We tried to propose there a thorough overview of the most recent works about\ncutoff in random and generic settings, so we will remain brief in this paper and refer the reader\nto the \"Related works\" section of [11] for a more comprehensive list of references.\nAs we mentionned already, our model is directly inspired from that of Hermon, Sly and Sousi\n[14] who proved cutoff for the simple random on a graph with an added uniform matching, as\nwell as related models: [2] in which Baran, Hermon, Šarković and Sousi prove a phase transition\nwhen weights are added on the random matching, and [15] in which Hermon, Šarković and Sousi\nstudy the simple random walk on graphs with community structures. The absence of uniform\ncutoff was already observed for the simple random walk on the giant component of a supercritical\nErdös-Renyi graph, studied in the work of Berestycki, Lubetzky, Peres and Sly [5] from which\nwe re-used a few arguments. Generally, our proof stems from the strategy initially used by Ben\nHamou and Salez [3], then by Bordenave, Caputo and Salez [6, 7] for non-backtracking random\nwalks.\n1.3 Proof outline\nWe first mention that the results presented in this paper were obtained in the same time as\nthose of [11] for the reversible model. As noted there, the arguments barely used the reversibility\nassumption, which was made mainly to avoid the technical difficulties that eventually make cutoff\nonly valid for typical starting states. Thus the key arguments of the proof already appear in [11]\nand we will in this paper mostly emphasize on what needs to be adapted.\nDealingwithanunknowninvariantmeasure: firsttheabsenceofareversibilityassumption\nprevents any knowledge of the invariant measure. To establish mixing, we follow the strategy of\n[6, 7], already used in the reversible case, of proving convergence towards an approximate invariant\nmeasure ˆπ. If∥Pt\n0(x,·)−ˆπ∥TV≤εholds uniformly in xfor a given time t, then the invariance\nproperty implies that a true invariant measure πsatisfies ∥π−ˆπ∥TV≤ε, establishing in particular\nthe uniqueness of the invariant measure. The additional issue here is that Theorem 1.1 is valid\nonly for typical starting vertices, so we will have to prove that mixing towards ˆπoccurs both at\nt≃logn/hfor a typical starting state and at larger times for an arbitrary starting state, namely\nt= (log n)a. Note that since ˆπis not a priori an invariant measure, there is no monotonicity in\nthe distance to equilibrium that would maintain the chain distributed as ˆπonce it has become\nclose to it. We note that Proposition 3.2 below gives an expression for the proxy measure ˆπwe\nuse in our proofs.\nThe entropic method: the proof is based on the entropic method, which consists in two main\narguments: first an entropic concentration is shown to occur at the entropic time logn/h, which\nis shown to imply cutoff in the second part of the proof. These arguments have come so far with\ntwo different variations, depending on whether reversibility is supposed or not. Since we do not\nassume reversibility, we use the strategy that was originally designed for non-backtracking random\n4walks [3, 6, 7]. First we prove a quenched entropic concentration property for trajectories, namely\nwe prove a statement of the form P0(X0···Xt)≃e−th+O(√\nt), where we write P(X0···Xt) =\nQt−1\ni=0P0(Xi, Xi+1). This statement is given here as a rough heuristic, which would be correct\nin the non-backtracking case. In general, what we prove in practice is rather concentration of\nthe probability to follow the loop-erased trace ofX, so we are back at studying non-backtracking\ntrajectories. This concentration phenomenon is proved by a coupling with another Markov chain\nlyingonaninfiniterandomstatespace, whichwecallaquasi-treeinreferenceto[14], asthisobject\nis similar to theirs. Basically, it is designed to be a stationary approximation of the universal cover\nof the finite chain, in the same way Erdös-Renyi random graphs can be approximated by infinite\nGalton-Watson trees for instance.\nAnalysis on the quasi-tree: all the differences with the reversible case are contained in this\npart of the argument. At the basis of the analysis for reversible chains in [14, 15, 2, 11] is the\nproperty of uniform transience, which asserts that at any vertex, the probability to escape to\ninfinity along the neighbouring \"branch\" of the quasi-tree, conditional on the environment, is\nlower bounded away from 0uniformly in n. Such a uniform lower bound may not be achieved\nin the general model of Theorem 1.1 and is ultimately the reason why cutoff cannot occur for a\nuniform starting state. For instance our counter example to uniform cutoff (Theorem 1.2) stems\ndirectly from a quasi-tree which is close to being recurrent, resulting in the chain to be potentially\nstuck in trapping neighbourhoods. The lack of uniform transience is bypassed by proving that the\nprobability to escape at infinity along is in expectation lower bounded uniformly in n. This will be\nproved thanks to a comparison with a reversible chain, which is the main reason why Hypothesis\n(H4) comes into play. The lower bound on the expected escape probabilities will then imply that\nsuch a lower bound is also achieved when restricting to typical points. Very roughly, the difference\nwith the reversible case will thus consist in arguing that the trajectories of the chain remain up to\nthe mixing time on typical parts of the quasi-tree and consequently avoid traps. While uniform\ntransience allows to directly use \"quenched\" statements, ie conditional on the environment, we\nwill have to replace these by \"annealed\" statements, which average over the environment. Our\nanalysis in that regards is thus largely inspired by the paper [5] and the references cited therein,\nwhere the same issue was present.\nArgument for the lower bound: the lower bound in the mixing time is based on a simple\ncoverageargument. Fromtheconcentrationofentropy, attime t=O(logn)thechainisnecessarily\nconfined in a set of size eth+O(√\nt)at most. This becomes o(n)fort≤logn/h−C√lognand\na large constant C > 0, which is sufficient to conclude provided the invariant measure is well\nspread-out across the state space. Since the latter is unknown, we use the proxy measure ˆπand\nits expression which is explicit enough for our purpose.\nConcentration of nice trajectories: for the upper bound, the second part of the argument\nuses the entropic concentration to show a second concentration phenomenon. Specifically, for all\nx, y∈V, we define a set Nt(x, y)of \"nice\" trajectories between x, y, of length t, which will have\nthe property that: 1. they are typical, that is the trajectory of the chain is likely to be nice,\n2. the probability Pt\nN(x, y)of following a nice trajectory concentrates around its mean, which\n5is shown to be independent of the starting point. From these properties we deduce that Pt(x,·)\nmixes towards a proxy stationary distribution ˆπ. Concentration around the mean will be proved\nusing a concentration inequality for \"low-degree functions\" on the symmetric group, developped\nspecifically for this purpose and stated as Theorem 1.2 in our work for the reversible case [11].\nThese arguments are for the most part identical to the reversible case.\nGetting a bound on the worst-case mixing time : nice paths are in fact really defined\nonly from typical states, so the previous arguments will ensure Pt\n0(x,·)≃ˆπat time t= log n/h+\nO(√logn)only for a typical x. To obtain an upper bound on the worst-case mixing time, which\nas remarked in the first paragraph above is crucial to justify ˆπis close to being invariant, we will\nessentiallyarguethatforsome a≥1, attime s= (log n)athechainislikelytoreachatypicalpoint\nand have its tsubsequent steps follownig a nice trajectory, regardless of its starting state. Thus\nmixing at time t=O(logn)from typical states implies mixing at time s+t=O((log n)a)from any\nstarting state. This argument was already apparent in the reversible case, where s= Θ(log log n)\nsteps were enough, which was again the consequence of the quenched uniform lower bound on\nescape probabilities in quasi-trees. In the general case, a heuristic is as follows: we will get from\nthe estimates on escape probabilities (Prop. 5.1) that these are so unlikely to fall below (logn)−b\nfor some b >0that we can basically take (logn)−bas a quenched lower bound. Thus we can expect\nit takes (logn)blsteps at worst for the chain to escape potentially trapping neighbourhoods of size\nl, after which the chain is likely to be back in a typical neighbourhood if lis chosen adequatly.\n1.4 Organisation of the paper\nIn Section 2 we give material that will be used throughout the paper, including the definition of\nquasi-trees. In Section 3, we give in detail the main technical arguments of the paper that sum\nup the arguments of the entropic method, from which we will be able to deduce Theorems 1.1\nand 1.3. In Section 4, we provide a counter-example to uniform entropic mixing time, proving\nTheorem 1.2. The three next sections deal with the analysis of the quasi-tree: Section 5 establish\nthe lower bound on escape probabilities, Section 6 proves technical results about the regeneration\nstructure of quasi-trees while Section 7 basically establishes \"nice properties\" in the quasi-tree\nsetting, including in particular the concentration of the speed and entropy. These nice properties\nare then transferred back to the finite setting in Section 8, which also contains the arguments that\nwill lead to the upper bound on the worst-case mixing time. Finally, these properties are used in\nSection 9 where nice paths are properly defined and shown to have their probability concentrate\naround the mean.\n2 First moment argument, quasi-trees\nBasicnotations: throughoutthepaper,allquantitiesinvolvedmayhaveanimplicitdependency\ninnandthetermconstantwillrefertoquantitiesthatareindependentof nandoftherandomness.\nGiven a set S,|S|denotes its cardinality, 1Sis the indicator of S. Given x, y∈R, we write\nx∧y:= min( x, y),x∨y:= max( x, y). We use the standard Landau notations o, Ofor deterministic\nsequences: f(n) =O(g(n))if there exists a constant C > 0such that |f(n)| ≤Cg(n)for all n,\n6f=o(g)iff(n)/g(n)− − − − →\nn→∞0. We may write Oε(·)to precise a potential dependence of the\nimplicit constant in ε. Given two functions, f= Θ( g)iff=O(g)andg=O(f), and f≫gif\ng=o(f). All the random variables considered in this paper are defined on an implicit probability\nspace with measure P. IfYn, Zn, Zare random variables, we write ZnP− →Zfor convergence in\nprobability and Zn=oP(Yn)ifZn/YnP− →0. In particular Zn=oP(1)ifZnP− →0. An event\nA=A(n)is said to occur with high probability, if P(A) = 1−o(1).\n2.1 Quenched vs annealed probability, first moment argument\nConsider (Xt)t≥0a Markov chain in random environment, ie with random transition probabilities,\nsuch as the one defined by (1). There are two laws naturally associated with the process (Xt)t≥0.\nThe probability P, which averages over both the random walk and the environment, gives rise to\ntheannealed law of the process (Xt)t≥0under which it is not a Markov chain. It is however a\nMarkov chain under the quenched law, which conditions on the environment. To emphasize the\ndistinction, it iswrittenusing adifferent font, namely Pwill denotethe quenched distribution. For\nall state x, we write Px:=P[· |X0=x],Px:=P[· |X0:=x]. Of course taking the expectation\nof the quenched law gives back the annealed law, which is the basis of the following first moment\nargument that will be used used throughout the article. To prove a trajectorial event Ahas\nquenched probability vanishing to 0in probability as n→ ∞, it suffices to prove Ahas annealed\nvanishing probability, as Markov’s inequality implies for all ε >0,\nP[P[A]≥ε]≤P[A]\nε. (2)\nIn most of this paper we will first prove statements valid for fixed starting states, chosen inde-\npendantly of the environment, such as Px[A] =o(1), to obtain Px[A] =oP(1). These can be\ninterpreted as conditional statements for the case where the starting state is also random, with\na law which independant of the environment. Results for fixed starting states thus extend auto-\nmatically to typical states , for instance taken uniformly at random in the case the state space is\n[n]. To obtain stronger results valid for all states simultaneously, union bound shows it suffices to\nimprove the annealed error bounds to Px[A] =o(1/n). This strategy will be used in Section 8 to\nextend results from typical to all states.\n2.2 The two-lift chain and half-integer time steps\nLet us start by rewriting and actually generalizing our model further, which lacks symmetry in\nthe roles of P1andP2. To obtain a model that is more symmetric, we construct the chain as a\nprojection of its two-lift: this consists in seeing P1, P2as two disjoint components of one Markov\nchain on a twice larger state space, which can be projected back to obtain the original model.\nThis is essentially a matter of unifying notation and recover a setting which is somewhat similar\nto that of [14], but it can also prove useful on its own. All in all, the main model we will work\nwith in this paper is the following.\nLetV:= [2 n], V1:= [n], V2:= [n+ 1,2n]. Let σbe now a uniform bijection from V1toV2,\nwhich can be extended as a matching, or involution, ηonVby\nη|V1:=σ, η |V2:=σ−1.\n7This matching defines an equivalence relation, namely x∼η(x). For all x∈V, define V(x) =\nV1 1x∈V1+V2 1x∈V2. Let Pbe a stochastic matrix on Vwith block form P:=\u0000P10\n0P2\u0001\nandp:\nV×V→[0,1]such that p(x, y)+p(y, x) = 1for all x, y∈Vand let q(x, y) := 1−p(x, y) =p(y, x).\nConsider the Markov chain on Vgiven by the transition probabilities:\n∀x, y∈V:P(x, y) :=(\np(x, η(x))P(x, y)ifV(x) =V(y)\nq(x, η(x))P(η(x), y)ifV(x)̸=V(y).(3)\nWe will work with this Markov chain most of the time, but the main object of interest of this\npaper is rather its projection to the quotient V/∼= [n]. As is easily checked, the condition\nq(x, y) =p(y, x)implies that P(x, y) +P(x, η(y)) =P(η(x), y) +P(η(y), η(y)). This condition\nensures the projection is itself a Markov chain, with transition matrix given by\n∀x, y∈[n],¯P(x, y) :=P(x, y) +P(x, η(y)) (4)\nThis model contains (1) as a particular case: taking p(x, y) :=p1(x, y−n)for all x∈V1, y∈V2\nsuffices to define pcompletely. Furthermore σ−nis a uniform permutation on [n]. From these it\nis easily checked that the projected chain ¯Pcoincides with P0.\nGiven a probability measure µonV, its projection ¯µon[n]is defined by ¯µ(x) =µ(x)+µ(η(x))\nfor all x∈[n]. Notice that if πis invariant for P, then ¯πis invariant for ¯P. Furthermore, total\nvariation distance is non-increasing under projections, so\n\r\r¯Pt(x,·)−¯π\r\r\nTV≤\r\rPt(x,·)−π\r\r\nTV. (5)\nTherefore to obtain the upper bound in Theorem 1.1, which is the most difficult part of the\nargument, it suffices to prove an upper bound for the two-lift. In fact, our results hold for both\nchains.\nNow let us introduce another characteristic of the two-lift. The two-lift is by construction\nmade of two disjoint subspaces V1, V2: when at xthe Markov chain stays in the same subspace\nwith probability p(x, η(x)), and change with probability q(x, η(x)), after which it takes a step\nindependent of η. Because of this, it may be convenient to actually distinguish between these two\nsteps, adding transitions at half integer times, defining transition probabilities\nP\u0002\nXt+1/2=y|Xt=x\u0003\n=\n\np(x, η(x))ify=x\nq(x, η(x))ify=η(x)\n0 otherwise\nP\u0002\nXt+1=y|Xt+1/2=x\u0003\n=P(x, y)(6)\nfor all t∈N. As is easily checked, the Markov chain (Xt)t∈Nevaluated at integer times exactly\nhas the transition matrix P(3).\n2.3 The graph point of view\nTo emphasize the underlying tree structure of the Markov chains considered in this paper, these\ncan be visualized as evolving on directed graphs.\nDefinition 2.1. IfKis a Markov kernel on a countable state space S, the underlying graph\nG= (S, E)is the oriented graph with vertex set Sand edge set\nE:={(x, y)∈S|K(x, y)>0}.\n8We consider in fact multi-graphs, so that loops and multi-edges are allowed in G. In quasi-trees,\nwe will use graphs of Markov chains as parts of a larger structure. To that end, define a morphism\nof directed graphs as a map between two graphs that preserves the adjacency relations as well\nas the orientations of the edges. If weighted graphs are considered, such as underlying graphs of\nMarkov chains, we may also implicitely require that weights are preserved. An isomorphism is as\nusual a bijective morphism.\nGiven an oriented edge e= (x, y)∈E, write e−:=xande+:=yfor its initial and terminal\nvertex respectively. A path pinGcan be made either of vertices or oriented edges, which can\nbe identified. Given a path p, we write K(p)for the product of transition probabilities along the\nedges of this path. The set Sis equipped with the graph metric provided by G: for all x, y∈,\ndK(x, y)is the minimal number of edges of a path joining xtoy. It is not symmetric in general\nbecause of the orientation of the edges. It thus gives rise to several notions of balls, depending on\nthe orientation: given x∈V, r∈N∪ {+∞}, the forward and backward (closed) K-balls of radius\nraround xare respectively the sets\nB+\nP(x, r) :={y∈S|dK(x, y)≤r} B−\nP(x, r) :={y∈S|dK(y, x)≤r}.\nWithout further precision, the K-ball of radius raround xisBK(x, r) := B−\nK(x, r)∪B+\nK(x, r),\nwhich forgets about the orientation of edges. Note that for any x∈S,B+\nK(x,∞)is the set that\ncan be reached by the chain from the state x. Ifxis recurrent, this is the communicating class of\nx.\nApply the previous definition to the case at hand: from now on, let G= (V, E)be the\nunderlying graph of the Markov chain P: the graphs G1= (V1, E1), G2= (V2, E2)spanned by V1\nandV2respectively are the underlying graphs of P1andP2and form two disconnected components\nofG. Adding the edges (x, η(x))of the uniform matching yields a random graph G∗:= (V, E∗),\nwhich can be seen as the underlying graph of the Markov chain (Xt)t≥0, provided we take into\naccount half-integer time steps (6). We introduce a terminology inspired by [14] to distinguish\nbetween these edges: an edge (x, y)∈E∗is called a small-range edge if (x, y)∈E, that is if\nP(x, y)>0orP(y, x)>0. An edge (x, η(x))added by the random matching will be called a\nlong-range edge. Note that long-range edges appear in Ewith both orientations. When restricting\nto integer times only, the Markov chain Xmoves along a long-range and a small-range edge at\nonce or along a small-range edge only. Thus two vertices at P-distance 1from each other may in\nfact be separated by two edges in G∗. Similarly, a path in G∗without further precision will be\nrefering to a possible trajectory of the chain P.\n2.4 Quasi-trees\nWe now define quasi-trees, which are designed to be an infinite approximation of the graph G∗.\nThe same terminology and notation as in the finite setting will be used to emphasize analogies.\nThis abuse is justified by the fact that later, both settings will be coupled so that quantities with\nmatching terminology or notation will be equal. If necessary, we will introduce distinct notation.\nDefinition 2.2. A rooted quasi-tree is a 4-tuple (G, O, η, ι )where G= (V,E)is an oriented graph,\nO∈ Vis a distinguished vertex and ιis a map ι:V → Vthat labels vertices of Gwith states in\nV. We call ι(x)the type of the vertex x∈V. Finally ηis a map η=V → V, which satisfies:\n9(i)ηis an involution either of Vor ofV∖{O}with no fixed point. The quasi-tree is called\nrespectively two-sided orone-sided in these cases.\n(ii) For all x∈ V, the edge (x, η(x))is present in Ewith both orientations. Such edges are called\nlong-range edges, others are called small-range edges.\n(iii) for all x, y∈ V, there is a unique family, possibly empty, of long-range edges e1, . . . , e ksuch\nthat all paths between xandyof minimal length contain the edges e1, . . . , e k.\nIn particular, there must exist a unique sequence of long-range edges joining the root Otox.\nIf this sequence is non-empty let x◦be the the endpoint of these long-range edges which is the\nfurthest from O. We call such a vertex a center. If this sequence is empty, let x◦:=O, however\nthe root is not considered a center.\nThe two types of edges lead in turn to two additional types of paths and distances.\n1. Asmall-range path is a path made exclusively of small-range edges. Given x, y∈V, the\nsmall-range distance dSR(x, y)is the minimal number of edges in small-range path from x\ntoy. The corresponding small-range balls are written BSR(x, r).\n2. Along-range path is a sequence e= (ei)k\ni=1of long-range edges such that for all i≤k−1\ndSR(ei, ei+1)<∞(soecould be completed with small-range paths between long-range edges\nto obtain a genuine path in G). The length |e|ofeis the number of edges it contain. It joins\ntwo vertices x, y∈ VifdSR(x, e1)<∞anddSR(ek, y)<∞. Given x, y∈V, thelong-range\ndistance dLR(x, y)is the minimal length of a long-range path between xandy. We write\nBLR(x, r)for the corresponding long-range balls .\nAs for the distance in Markov chain underlying graphs, these distances are not symmetric, so we\nmay add +or−sign in exponent to denote forward and backward balls. From the definition,\nfor any x∈ V,B+\nSR(x,∞) =B+\nLR(x,0)is the set of vertices which can be joined from xby a\nsmall-range path. We call the subgraph spanned by this set the small-range component ofx. We\ncan now state a fourth property we require for quasi-trees:\n(iv) for all x∈ V,B+\nLR(x,0)is isomorphic to B+\nP(ι(x◦),∞),\nthat is, the small-range component of xis the whole set of vertices in Vthan can be reached from\nι(x◦)in the graph G.\nIn the sequel we will refer to quasi-trees by their graph component only while keeping the other\nparameters implicit.\nProperty (iii) above clearly gives quasi-trees a tree structure. For this reason, let us introduce\na few extra notations. Given k≥0, the k-th level Vkis the set of vertices which are at long-range\ndistance kfrom the root. Set also V≤k:=S\ni≤kVi. The requirement that ηhas no fixed point\nimplies that every vertex xis joined to vertex η(x)in another level. If xis a center, η(x)is in a\nlevel below, whereas if xis not a center η(x)is in a level above.\nOne-sided quasi-trees and subquasi-trees We introduced one-sided quasi-trees to consider\nsubquasi-trees. If xis a center, the subquasi-tree Gxofxis the graph spanned by the vertices\nyfor which all paths between Oandypass through x. Ifxis not a center, η(x)is and we set\n10Gx:=Gη(x), soGxdoes not in fact contain xin that case. Finally the complement subgraph G∖Gx\nis the graph spanned by vertices which can be reached from Oby a long-range path that does not\nuse the long-range edge (x, η(x)).\nNon-backtracking paths, loop-erased paths, deviation, regeneration One main inter-\nest of having a genuine tree structure is the consideration of loop-erased trajectories. We thus\nintroduce the following definitions.\nDefinition 2.3. LetGbe a quasi-tree. A long-range path e= (e1, . . . , e k)backtracks over a\ndistance l≥1if it contains a subpath of distinct edges and of length limmediately followed by\nthe reversed path, that is there exists i≤k−2l+1such that ei+l+j=ei+l−1−jfor all j∈[0, l−1],\nwhere ejdenotes the edge ejwith reversed orientation. The loop-erased path ξ(e)is the path\nobtained after erasing all backtracking steps. The long-range path eis called non-backtracking if\nξ(e) =e.\nTo previous definition is extended to general paths by extracting the long-range path: given\na general path p, letξ(p)denote the loop-erased path formed by the long-range edges crossed by\np. This is a non-backtracking path, which we call the loop-erased path or loop-erased trace of p.\nThe long-range distance crossed by pis the length of ξ(p), or equivalently the long-range distance\nbetween its endpoints.\nThe last two definitions require integer parameters. Given R≥1, letG(R)be the connected\ncomponent of Oof the subgraph of Gspanned by the set\nV(R):={x∈ V | dSR(x◦, x)< R}. (7)\nA path pis said to deviatefrom a small-range distance Rif it is not included in G(R).\nFinally, the following notion of regeneration edges will be used in several places: let L≥1be\nan integer and consider a path p. A long-range edge ecrossed by pis said to be a regeneration\nedge for pwith horizon Lif after the first time going through ethe path crosses a long-range\ndistance Lor ends before going back to the endpoint of ewhich was first visited by p.\nMarkov chains on quasi-trees LetGbe a quasi-tree. It is naturally the underlying graph of\nthe Markov chain (Xt)t≥0onGwhich has transition probabilities:\nP\u0002\nXt+1/2=y\f\fXt=x\u0003\n=\n\np(ι(x), ι(η(x)))ify=x\nq(ι(x), ι(η(x)))ify=η(x)\n0 otherwise\nP\u0002\nXt+1=y| Xt+1/2=x\u0003\n=(\nP(ι(x), ι(y))ify◦=x◦\n0 otherwise(8)\nfor all t∈N. The kernel of this Markov chain will be written P.\n2.5 The covering quasi-tree\nAmong all quasi-trees, one is very natural to consider: this is the quasi-tree obtained by using the\nrandom matching η:V→Vto define the matching on the quasi-tree. Given x∈V, this is the\nquasi-tree (G∗(x), O, ι, ˜η)defined by ι(O) =xand\n∀y∈ V:ι(˜η(y)) =η(ι(y)).\n11The fact that this defines a unique quasi-tree is the consequence from property (iv) in Definition\n2.2, which imply the quasi-tree can be built iteratively. Starting from the small-range of O, which\nis necessarily B+\nP(x,∞), the previous equation uniquely determines the long-range edges at long-\nrange distance 0from Oand thus the whole ball BLR(O,1). The process iterates to infinity to\nyield an infinite quasi-tree G∗.\nThis quasi-tree has the property that it covers exactly G∗: the map ιis a surjective morphism\nofG∗onto G∗which preserves the transition probabilities, so the Markov chain Xdefined above\nprojects exactly onto the chain X: for all t≥0ι(Xt) = Xtin distribution, conditional on\nX0=x,X0=O.\nThe chain XonG∗will not be studied per se. We introduced it mainly to obtain easier\ndefinitions in the finite setting of objects and quantities that are naturally considered in the\nidealized setting of quasi-trees. First notice that the notions of long-range and small-range edges\ndefined in the finite setting coincide with the projections under ιof the corresponding edges\nin the covering quasi-tree. Thus we can extend the notions of small-range, long-range paths,\nbacktracking, etc. defined above to G∗by taking their projections under ι. An exception is the\nlong-range distance, which in the finite setting will make sense only if restricting the quasi-tree:\nletR≥1and recall the definition of the restricted quasi-tree G(R)(7). Given x∈Vandl≥0,\nwe define B(G,R)\nLR (x, r)inG∗as the projection\nB(G,R)\nLR (x, r) :=ι\u0010\nBLR(O, r)∩ G(R)(x)\u0011\n.\nThe exponent (G, R)is used to distinguish between the two settings. When not necessary, we may\ndrop it from notation. Note from the definition that in G∗, for any vertices x, y∈V,dLR(x, y) = 0\nif and only if dSR(x, y)< R.\nFinally, we introduce a last definition which is specific to G∗: along-range cycle is a non-\ndeviating non-backtracking long-range path whose starting and ending point are at small-range\ndistance at most Rfrom each other. A subgraph of G∗is said to be quasi-tree-like if it does not\ncontain any long-range cycle. A quasi-tree-like subgraph can thus be identified with a neighbour-\nhood of the root in the covering quasi-tree. Lemma 2.1 below will establish that most vertices\nhave in fact a quasi-tree-like neighbourhood. This relies on a bounded degree property, which is\nthe object of the following paragraph.\nBounded degrees and transition probabilities: Assumption (H1) imply that the graph G\nand consequently G∗,Gas well, have degrees bounded uniformly in n. Together with Assumption\n(H2) it also implies all transition probabilities are lower bounded uniformly in n. Consequently, let\n∆denoteauniformboundonallthedegreesand δ >0auniformboundontransitionprobabilities,\nwhich will serve throughout the paper. It gives in particular the growth of balls in G∗and any\nquasi-tree G: for all x∈Vandl≥0\n|BSR(x, l)| ≤ |BP(x, l)| ≤∆l+1−1\n∆−1,\f\f\fB(G,R)\nLR (x, l)\f\f\f≤∆R∆R(l+1)−1\n∆R−1. (9)\nIn particular, ∆Rcan be thought of as the \"long-range\" degree.\n122.6 Random setting\nTheabovedefinitionsshouldmakeprettyclearthattheforthcomingproofsarebasedonacoupling\nbetween the finite chain Xand the chain Xon an infinite quasi-tree. The quasi-tree in question\nis similar to the covering quasi-tree but contains much more independence, allowing basically to\nresample the matching ηat each new long-range edge. It can be constructed iteratively as follows:\nι(O)is taken uniformly at random in V, which determines the small-range component around\nOby point (iv) of the definition. Then to every vertex xwhose type ι(x)is known, adjoin a\nlong-range edge (x, η(x))tox, with ι(η(x))being taken uniformly in V∖V(ι(x)).\nThe following formalisation of random quasi-trees will be useful to deal with subquasi-trees\nand their independency properties. It is analog to the Ulam-Harris labelling for Galton-Watson\ntrees. Let\nU:=[\nk≥0{\u0000\n(ui)k\ni=1, v\u0001\n|(u, v)∈Vk+1,∀i∈[2, k], ui/∈V(ui−1), v /∈V(uk)}.(10)\nThe elements in Ucorresponding to k= 0have their first coordinate empty and are identified\nwith V. Given two sequences u, v, write uvfor their concatenation. If u= (ui)k\ni=1is non-empty,\nwrite u♭:= (ui)k−1\ni=1and¯u:=uk. A rooted quasi-tree can then be represented as a pair (V, O)\nwhere V ⊂U,O∈Vsatisfy:\n(i)V ∩V=B+\nSR(O,∞)\n(ii)∀(u, v)∈ V,(u♭,¯u)∈ V.\n(iii) there exists an involution η:V → Vsuch that for all (u, v)∈ V,\n•either η(u, v) = (u♭,¯u), ((u, v)is then a center),\n•orη(u, v) = (uv, ρ(u, v))for some ρ(u, v)∈V∖V(v).\nIn the second case, for all w∈V,(uv, w )∈ Vif and only if w∈B+\nSR(ρ(u, v),∞).\nIt is easily checked that with this definition the set Videntifies with the vertex set of a quasi-tree\nas defined in 2.2. Note the type of a vertex (u, v)∈Uisι(u, v) =v.\nConsider now a family (ζ(u,v))(u,v)∈Uof independent random variables, where for every (u, v)∈\nU,ζ(u,v)is uniform in V∖V(v). Letting Ube independent and uniform in V, letVbe the random\nsubset of Uobtained by taking O=Uandρ(u, v) := ζ(u, v)in the case (u, v)is not a center.\nThis yields the same random quasi-tree as in the iterative definition above.\nThe previous construction is a bit heavy and can be avoided most of the time, however it has\nthe advantage that it allows to consider subquasi-trees in a rigourous way. For every x∈U,Gx\nis well-defined if x∈ V, otherwise we set it to be empty. Then recall that in the case x∈ V G x\ncontains xif and only if it is a center. Thus the events x∈ Vandxbeing a center are both\nmeasurable with respect to G∖Gx. Ifx∈ Vis not a center, the knowledge of G∖Gxdoes not\nreveal the value of η(x), thus conditional on G∖Gxandx∈ Vnot being a center, Gxhas the\nlaw of G′\nO′where (G′, O′)is an independent copy of (G, O). Ifxis a center, the second coordinate\nofx= (u, v)reveals ι(x)and thus reveals a part of the subquasi-tree Gx. Thus conditional and\nG∖Gxandx∈ Vbeing a center, Gxhas the law of G′\nO′conditional on ι(η(O′)) =ι(x).\n132.7 Coupling and sequential generation\nWe now explain how one can couple the Markov chain (Xt)t≥0onG∗with the Markov chain\n(Xt)t≥0on the random infinite quasi-tree Gdefined in the previous section.\nThe following procedure generates the neighbourhood of a vertex in either G∗orGup to some\ngiven long-range distance.\nLetx∈Vbe the point whose long-range neighbourhood is to be explored up to long-range\ndistance L≥0. For t≥1we write EQtfor the exploration queue, that is the set of vertices which\nremain to be explored, and Dtfor the set of explored vertices. The initiation is similar for G∗and\nquasi-trees: D0:=∅;EQ0:=B+\nSR(x, R)inG,EQ0:=B+\nSR(x,∞)inG. The procedure repeats\nthen the following steps. If one wants to explore a neighbourhood in G∗sampling is performed\nwithout replacement:\nSequential generation of G∗, sampling without replacement: fort≥0,\n1. pick y∈EQt: sample η(y)uniformly at random in (V∖V(y))∖Dt,\n2. add y, η(y)toDt+1and remove them from EQt+1,\n3. add all vertices z∈B+\nSR(η(y), R)∖{η(y)}such that z /∈DtanddLR(x, z)< LtoEQt+1.\nIf performed with replacement, each new value η(y)is picked uniformly in V∖V(y)inde-\npendently of previous draws and considered a new vertex in a set V. Specifically the procedure\nbecomes:\nSequential generation of the quasi-tree G, sampling with replacement: fort≥0,\n1’. pick y∈EQt: sample η(y)uniformly at random in V∖V(y),\n2’. add y, η(y)toDt+1and remove yfrom EQt+1,\n3’. add all vertices z∈B+\nSR(η(y), R)∖{η(y)}such that dLR(x, z)< LtoEQt+1.\nSince the constraint z /∈Dthas been removed in step 3’, vertices which would in Gbe already\nexplored are here added to the exploration queue and thus considered new vertices. The environ-\nment explored is the Llong-range ball of a quasi-tree as in Definition 2.2. Taking L=∞would\nthus yield a realization of the infinite random quasi-tree G. Finally, the procedures can be adapted\nto explore ball for the graph distance, that is balls BPinG∗.\nSequential generation along trajectories Under the annealed law, the two processes can be\ngenerated together with the environment. Later our goal will be to couple weights, which require\nthe exploration of the whole L-long range neighbourhood around the trajectory. The sequential\ngeneration of the environment along trajectories thus consists in the following steps. Consider for\ninstance the finite setting G: let the chain be started at X0:=x∈V, and D0:=∅. Then for all\nt≥0:\na. Explore the long-range L-neighbourhood of Xtwith the first procedure described above.\n14b. This determines completely the transition probabilities (6) at the state Xt, allowing to\nsample Xt+1.\nIf one considers the second procedure one obtains instead (Xt)t≥0. One can very naturally couple\nthe two procedures and hence both processes using rejection sampling: for each y∈EQtsample\nη(y)uniformlyin V∖V(y)anduseitforstep1’inthegenerationof G. Ifinaddition BSR(η(y), R)∩\nDt=∅, it can also be used for step 1in the generation of G∗. Otherwise, make a second draw with\nthe first procedure. This rejection sampling scheme yields a coupling of (Xt)t≥0and(Xt)t≥0until\nthe first time a newly revealed small-range balls BSR(η(y), R)contain vertices that have already\nbeen explored, that is when a long-range cycle appears around the trajectory:\nτcoup:= inf(\nt≥0|t[\nk=0BLR(Xk, L)contains a long-range cycle)\n. (11)\nAs a first application of the coupling and the first moment argument, we give the following\nLemma.\nLemma 2.1. There exists CR, CLlarge enough such that for R=CRlog log n, L=CLlog log n,\nfor all x∈V, ε∈(0,3/10)andt=O(n3/10−ε),\n(i)Px[∃s≤t:BP(Xs,⌊logn/(5 log ∆) ⌋)is not quasi-tree-like ] =oP(n−ε) =oP(1).\n(ii)Pxh\n∃s≤t:B(G,R)\nLR (Xs, L)is not quasi-tree-likei\n=oP(n−ε) = oP(1). As a consequence\nPx[τcoup≤t] =oP(n−ε) =oP(1).\nProof.Let the chain be started at x∈Vandt=o(n3/10−ε)forε∈(0,3/10). Write k:=\n⌊logn/(5 log ∆) ⌋. By (9), the number of vertices contained in a P-ball of radius kisO(∆k). This\nimpliesS\ns≤tBP(Xs, k)contains m=O((t+1)∆k)vertices. Therefore, the exploration procedure\nalong the path X0···Xtrequires at most mdraws of values η(y), each having probability at\nmost m∆R/nthat the small-range ball BSR(η(y), R)contains already explored vertices. Hence\nthe number of long-range cycles inS\ns≤tBP(Xs, k)is stochastically upper bounded by a binomial\nBin(m, m ∆R/n)so that by Markov’s inequality\nPx[∃s≤t:BP(Xs, k)is not quasi-tree-like ]≤m2∆R\nn=O\u0012(t+ 1)2∆R∆2k\nn\u0013\n=O\u0010\nt2n−3/5+o(1)\u0011\n=o(n−2ε+o(1)) =o(n−ε)\nby the choice of t=O(n3/10−ε)andR, L =O(log log n). This bound on the annealed probability\nimpliesthequenchedresult(i)bythefirstmomentargument(2). Theresult(ii)isprovedsimilarly,\nnoting that ∆RLis still no(1)(in this case we can take tup to n1/2−ε).\n3 The entropic method: main arguments\n3.1 Nice trajectories\nOur application of the entropic method consists in finding a definition of nice trajectories designed\ntobetypicaltrajectoriesandhavetheirprobabilityconcentratingaroundthemean. Thelatterwill\ncome from a constraint about an entropy-like quantity for the chain, which arises from comparing\n15the trajectories of the finite chain Xwith the loop-erased trace of Xin the (infinite) quasi-tree\nsetting. Other properties will thus be required from nice paths, whose goal is basically to make\nthem close to being non-backtracking trajectories in a quasi-tree. All in all, the defining properties\nof a nice trajectory will essentially be that:\n(i) it is contained in a quasi-tree-like portion of the graph, the endpoint having a quasi-tree-like\nneighbourhood up to P-distance ⌊αlogn⌋for some α >0(Lemma 3.1),\n(ii) it does not deviate or backtrack too much, and contains sufficiently many regeneration edges\n(Lemma 3.2),\n(iii) the drift, ie the long-range distance travelled, and entropy concentrate on this trajectory\n(Proposition 3.1).\nAs mentionned in the proof outline, nice trajectories will be likely to be followed by the chain\nonly if it is started at a typical state. To derive the upper bound on the worst-case mixing time,\nwe will thus establish that for an arbitrary starting state the trajectory becomes nice only after\nsome time, namely after s= (log n)asteps for some a >0. This is why in the sequel we state\ndifferent results with a \"typical, non-shifted\" version valid from time 0and a \"worst-case, shifted\"\nversion valid for arbitrary starting states from time s. For the first important property of nice\ntrajectories, the typical version has in fact already been proved in Lemma 2.1. The worst-case\nversion is given in the following.\nLemma 3.1. There exists a≥1such that for all s≥(logn)aandt=o(n1/16),\n(i) for all CR, CL>0, letting R:=CRlog log n, L:=CLlog log n,\nmax\nx∈VPxh\n∃t′∈[s, s+t] :B(G∗,R)\nLR (Xt′, L)is not quasi-tree-likei\n=oP(1),\n(ii)\nmax\nx∈VPx\u0014\nBP\u0012\nXs,\u0016logn\n10 log ∆\u0017\u0013\nis not quasi-tree-like\u0015\n=oP(1). (12)\nFor the second property we recall the notions of deviation, backtracking and regeneration edges\nare given in Definition 2.3.\nLemma 3.2. LetΓ(R, L, M )denote the set of paths pinG∗such that pdoes not deviate from a\nsmall-range distance more than R, backtrack over a long-range distance Lor contain a subpath of\nlength Mwithout a regeneration edge. There exist constants κ≥1,a≥1,CR, CL>0such that\nforL=CLlog log n,R=CRlog log nandM= (log log n)κ, for all t=O(logn), the following\nholds:\n(i) for all x∈V:\nPx[(X0···Xt)∈Γ(R, L, M )] = 1 −oP(1)\n(ii) for all s≥(logn)a:\nmin\nx∈VPx[(Xs···Xs+t)∈Γ(R, L, M )] = 1 −oP(1).\n16Remark 3.1.Notice that for any trajectorial event A\nPx[(Xs, Xs+1, . . .)∈A] =X\ny∈VPx[Xs=y]Py[(X0, X1, . . .)∈A]\n≤max\ny∈VPy[(X0, X1, . . .)∈A].\nThus if we prove Aholds from time twith probability 1−oP(1)uniformly over the starting state\nthis automatically extends to larger times t≥s. In the above lemmas for instance it will be\nsufficient to prove the case s= (log n)ato establish the case s≥(logn)a.\n3.2 Concentration of drift and entropy\nThe third and main property of nice trajectories consists in the concentration of an entropy-like\nquantity for the finite chain. To that end, we define weights as follows.\nThroughout the paper, we will use the letter τfor a variety of stopping times. It should be\nclear from the context what the notation refers to. If xis an element or a set τxwill generally\ndenote the hitting time of x. Ifl≥1,τlwill generally denote the time a certain distance lis\nreached. For all l≥0, consider in this section\nτl:= inf{t≥0| |ξ(X0···Xt)|=l}\nand fix R, L≥1for the rest of this section. Let\nτ(R)\nSR:= inf{t≥0|(X0···Xt)deviates from a small-range distance R} (13)\nτ(L)\nNB:= inf{t≥0|(X0···Xt)backtracks over a long-range distance L}. (14)\nThese are stopping times which depend on R, L. By construction if pis a path in Γ(R, L, M )as\ndefined in Lemma 3.2, the stopping time τ(R)\nSR∧τ(L)\nNBdoes not occur on the trajectory p. Given a\nlong-range edge eandx∈Vsuch that dSR(x, e−)< R, define the weights\nwx,R,L(e) :=Pxh\nξ(X0···XτL)1=e, τL< τ(R)\nSRi\nwR,L(e|x) :=Pxh\nξ(X0···XτL)1=e|τL< τη(x)∧τ(R)\nSRi (15)\nHere ξ(X0···XτL)1denotes the first edge of ξ(X0···XτL). Then for a non-backtracking long-\nrange path ξ=ξ1···ξk, set\nwx,R,L(ξ) :=wx,R,L(ξ1)kY\ni=2wR,L(ξi|ξ+\ni−1) (16)\nwhere empty products are by convention equal to 1. The notation is consistent with the the\nidentification of edges with paths of length 1.\nRemark 3.2.Note that for fixed x∈V,P\newx,R,L(e)≤1andP\newR,L(e|x)≤1where the\nsum is over all long-range edges. By extension the sum of weights over all non-backtracking paths\nstarting from xis at most 1.\nGiven a sequence u= (ui)i≤lof length landk≥1, we write (u)≤k:= (ui)i≤kfor the sequence\ntruncated at length k. The following lemma show that weights are good proxies for measuring the\nprobability that the loop-erased trace follows a given non-backtracking path. It is the same as for\nthe reversible case as there is here no difference.\n17Lemma 3.3 ([11, Lemma 3.3]) .Letx∈Vandk≥1be an integer. Suppose that B(G,R)\nLR (x, k)is\nquasi-tree-like. Then for all non-backtracking long-range path ξof length k, started in B+\nSR(x, R)\nPx\nξ(X0···Xτk+L−1)≤k=ξ,\nτk+L−1< τ(R)\nSR∧τ(L)\nNB\n≤wx,R,L(ξ)≤Px\nξ(X0···Xτk+L−1)≤k=ξ,\nτk+L−1< τ(R)\nSR∧τ(L)\nNB\n+u(ξ)(17)\nwhere u(ξ)≥0is such that\nX\nξu(ξ)≤Pxh\nτ(R)\nSR∧τ(L)\nNB≤τk+L−1i\n,\nthe sum being over non-backtracking long-range paths of length kfrom x.\nThe first part of the proof of Theorem 1.1 consists in proving the following quenched concen-\ntration phenomenon.\nProposition 3.1. There exist d, h= Θ(1) and constants a≥1, CR, CL>0for which the\nfollowing holds. Letting R:=CRlog log n, L:=CLlog log n, for all ε∈(0,1), there exist constants\nCLR(a, ε), Ch(a, ε)such that for all t≫1,t=O(logn):\n(i) for all x∈V, with high probability\nPx\"\n||ξ(X0···Xt)| −dt| ≤CLR√\nt\n|−logwX0,R,L(ξ(X0···Xt))−ht| ≤Ch√\nt#\n≥1−ε.\n(ii) for all s≥(logn)a, with high probability,\nmin\nx∈VPx\"\n||ξ(Xs···Xs+t)| −dt| ≤CLR√\nt\n|−logwXs,R,L(ξ(Xs···Xs+t))−ht| ≤Ch√\nt#\n≥1−ε.\n3.3 Concentration of nice paths\nThesecondpartoftheargumentconsistsinfindingasetof\"nicetrajectories\", definedinparticular\nso that they satisfy the entropic and drift requirements given by Proposition 3.1. Eventually the\ndrift and entropic concentration are used to show that the probability of following a nice trajectory\nconcentrates around its mean. The latter can be computed, providing an approximate stationary\ndistribution ˆπforPonV.\nGiven u∈V,L≥1consider again τL:= inf{t≥0| |ξ(X0···Xt)|=L}and\nQ(L)\nu:=P\u0002\n· |X1/2=u, τη(u)> τL\u0003\n(18)\nIn words, this measure considers trajectories immediately after a regeneration time. By Lemma\n3.2, if one takes L= Θ(log log n), the conditionning by τL< τX0essentially forbids the chain\nto come back at all to uon a time scale O(logn). Ifνis a probability measure on V, write\nQ(L)\nν:=P\nu∈Vν(u)Q(L)\nuandE(L)\nQνfor the expectation with respect to this measure. All in all, the\nsecond part of the argument is summarized in the following proposition.\nProposition 3.2. There exist a deterministic probability measure νonV, a deterministic s0=\nΘ(log n), constants CL>0, a, κ≥1and for all x, y∈V, t∈Na set Nt(x, y)of length tpaths\n18between xandyfor which the following holds. Let L:=CLlog log n, M := (log log n)κand write\nPt\nN(x, y) :=P\np∈Nt(x,y)P(p). Consider the random probability measure\nˆπ(v) :=1\nEQ(L)\nν[T1∧M]MX\nr=0Q(L)\nν[Xr+s0=v, r < T 1≤M] (19)\nwhere T1denotes the first regeneration time with horizon L=CLlog log n. For all ε∈(0,1)there\nexists C(ε)>0such that for t= log n/h+C(ε)√logn,\n(i) for all x∈V, with high probability,\nX\ny∈VPt\nN(x, y)≥1−ε,\n(ii) for all s≥(logn)a, with high probability\nmin\nx∈VX\ny∈VPsPt\nN(x, y)≥1−ε.\n(iii) there exists c= (cv)v∈Vsuch thatP\nv∈Vcv=oP(1)and with high probability, for all x, y∈V,\nPt\nN(x, y)≤(1 +ε)ˆπ(y) +c(y) +ε\nn,\n(iv) the measure ˆπcan be decomposed as ˆπ= ˆπ1+ ˆπ2with\n∥ˆπ1∥2\n2=oP((log n)b/n), ˆπ2(V) =oP(1)\nfor some b >0.\nProof of Theorem 1.1 and 1.3. RecallPis the transition matrix of the two-lift chain on V, which\nprojects to a transition matrix ¯Pon[n].\nStart with the upper bound on the mixing time. Let ε >0andt:= log n/h+C(ε)√logn.\nSinceP≥PNentry-wise, for all x∈Vands≥0,\n\r\rˆπ−Ps+t(x,·)\r\r\nTV=X\nz∈V\u0002\nˆπ(z)−Ps+t(x, z)\u0003\n+≤X\ny,z∈VPs(x, y)\u0002\nˆπ(z)−Pt(y, z)\u0003\n+\n≤X\ny,z∈VPs(x, y)h\n(1 +ε)ˆπ(z) +c(z) +ε\nn−Pt\nN(y, z)i\n+.\nPoint (iii) of the above proposition implies that with high probability, the right hand side sum-\nmands are non-negative for all x∈V, so the sum can be computed to obtain that with high\nprobability,\n\r\rˆπ−Ps+t(x,·)\r\r\nTV≤1−PsPt\nN(x, y) + 3ε.\nUsing point (ii), if s≥(logn)awith high probability\nmax\nx∈V\r\rPs+t(x,·)−ˆπ\r\r\nTV≤4ε\n19which projects by (5) to\nmax\nx∈V\r\r¯Ps+t(x,·)−¯ˆπ\r\r\nTV≤4ε.\nwith ¯ˆπthe projection onto [n]ofˆπ. Since this estimate is uniform in the starting state, it extends\nto any starting distribution and particularly to a stationary distribution. Thus for any stationary\ndistribution πof¯P,\r\rπ−¯ˆπ\r\r\nTV≤4ε. (20)\nand from triangular inequality we obtain that with high probability\nmax\nx∈[n]\r\r¯Ps+t(x,·)−π\r\r\nTV≤8ε.\nSince this is valid for any invariant measure, the latter must be unique. This proves in particular\nthe irreducibility and aperiodicity of the chain and establishes Theorem 1.3.\nOn the other hand for a fixed x∈V, taking s= 0in the above bounds yields and using point\n(i) that with high probability,\r\r¯Pt(x,·)−π\r\r\nTV≤8ε,\nproving the upper bound in Theorem 1.1.\nLet us prove the lower bound. For all t≥0, θ > 0andx, y∈[n],\n¯Pt(x, y) =Pt(x, y) +Pt(x, η(y))\n≥Px[Xt∈ {y, η(y)}, wx,R,L(ξ(X0···Xt))≤θ].\nIf equality holds, then\n¯ˆπ(y)−Px[Xt∈ {y, η(y)}, wx,R,L(ξ(X0···Xt))≤θ]≤\u0002¯ˆπ(y)−¯Pt(x, y)\u0003\n+\nIf equality does not hold, there must exist a non-backtracking long-range path ξbetween xandy\norxandη(y)for which w(ξ)> θ, in which case\n¯ˆπ(y)−Px[Xt∈ {y, η(y)}, wx,R,L(ξ(X0···Xt))≤θ]≤ˆπ(y) 1∃ξ:wx,R,L (ξ)>θ\n+ ˆπ(η(y)) 1∃ξ:wx,R,L (ξ)>θ.\nWe did not precise where ξlies to ease notation in the indicator functions but it depends on y\nandη(y)respectively. Combining the two inequalities we obtain that in either case\n¯ˆπ(y)−Px[Xt∈ {y, η(y)}, wx,R,L(ξ(X0···Xt))≤θ]≤\u0002¯ˆπ(y)−¯Pt(x, y)\u0003\n++ ˆπ(y) 1∃ξ:wx,R,L (ξ)>θ\n+ ˆπ(η(y)) 1∃ξ:wx,R,L (ξ)>θ.\nDecompose now ˆπ= ˆπ1+ˆπ2as in Proposition 3.2. From (iv), summing over y∈[n]in the previous\ninequality yields\nPx[wx,R,L(ξ(X0···Xt))> θ]≤\r\r¯Pt(x,·)−¯ˆπ\r\r\nTV+X\ny∈Vˆπ1(y) 1∃ξ:wx,R,L (ξ)>θ+oP(1).\nBy Cauchy-Schwarz inequality,\nX\ny∈Vˆπ1(y) 1(∃ξ:wx,R,L(ξ)> θ)≤\nX\ny∈Vˆπ1(y)2\n1/2\nX\nξ1(wx,R,L(ξ)> θ)\n1/2\n,\n20where the second sum is over non-backtracking long-range paths from x. By Remark 3.2 weights\nsum up to at most 1hence so this sum contains at most θ−1positive terms and\nPx[Xt=y, wx,R,L(ξ(X0···Xt))> θ]≤\r\r¯Pt(x,·)−¯ˆπ\r\r\nTV+s\n1\nθX\ny∈Vˆπ1(y)2+oP(1).(21)\nTo complete the proof, let ε∈(0,1)and specialize to t:= log n/h−C1√lognandθ:=\nn−1exp(C2√logn)for some for some C1(ε),C2(ε)>0. Choosing the constant C1large enough,\nexp(−th−Ch(ε)√\nt) =n−1exp(( C1−Ch/h)√logn−o(√logn))> θfor large enough n, hence\nProposition 3.1 implies that the left hand-side of (21) is at least 1−ϵwith high probability. On\nthe other hand, (iv) shows that the square-root in the right hand side is oP(1). All in all, this\nproves that with high probability\n\r\r¯Pt(x,·)−¯ˆπ\r\r\nTV≥1−2ϵ.\nFinally by (20) we deduce\r\r¯Pt(x,·)−¯π\r\r\nTV≥1−6ϵ\nwith high probability.\nRemark 3.3.From the previous proof, the decomposition of ˆπgiven by (iv) also extends to\nthe unique invariant measures of Pand ¯P:π=π1+π2with∥π1∥2\n2=oP((log n)b/n)and\nπ2([n]) =oP(1).\n4 Counter-example to uniform entropic time\nIn this section we prove Theorem 1.2. Let n≥1andδ∈(0,1). Consider the reversible Markov\nchains Q1, Q2on the segments {0,1,2}and{0,1}which go from left to right with probability δ,\nfrom right to left with probability 1−δand stay at the extremities with remaining probability. Let\nP1bethetransitionmatrixon [6n]correspondingto 2ndisjointcopiesof Q1, thatis P1(i, j) =Q1(i\nmod 3 , jmod 3)forall i, j∈[6n]. Similarlylet P2bethematrixcorrespondingto 3ncopiesof Q2.\nOur counter-example to uniform mixing time is obtained with the matrix ¯P:= 1/2(P1+SP2S−1)\nand small δ. On this example, it will be exceptionally be clearer to forget about the two-lift and\nhalf-integer time transitions. We thus identify each xwith η(x), both in the finite and quasi-\ntree setting, which has the effect of pruning the long-range edges in the graphs, and we use the\nnotations G∗,G,Xto refer to the objects obtained after this operation.\nConsider first the quasi-tree Gthat would be obtained from P1andP2: because the underlying\ngraphs G1, G2are forests, all realizations of quasi-trees are in fact genuine trees. On the other\nhand, if δis small, the Markov chains on segments are heavily biased towards 0. Say a vertex\nx∈ Ghas type 0if the map ιidentifies xwith the leftmost vertex of a segment. Then consider\nthe quasi-tree obtained Tif all centers in Ghave type 0. As one can check, this is a periodic\nweighted tree Twhere except the root all vertices have their transition probabilities depending\nonly on whether the degree is 2or3, see Figure 1. Notice that for sufficiently small δ, the chain\nonTbecomes recurrent. Of course this configuration occurs with probability 0, however a finite\nportion of Tcan arise with positive probability. Coming back to the finite graph G∗, which\nresembles locally a quasi-tree, this probability is in fact sufficiently large that G∗also contains\nsuch neighbourhoods, which trap the chain for a long time.\n21......δ\n1−δδ\n1−δ 1−δδ\nP1 P21−δ δδ1−δType 0 Type 0\n1−δ\n2δ\n21−δ\n2δ\n2δ\n2Figure 1: The finite chains P1,P2and the quasi-tree Twith all centers of type 0, after identifying\neach x≃η(x). The probability to remain at a vertex is not represented.\nLemma 4.1. There exists β > 0such that for all δ∈(0,1), with high probability there exists\nx∈[6n]for which\nPx[Xt/∈B¯P(x, βlog log n)] =o(1)\nfor all t≤(logn)βlog(2δ/(1−δ)).\nProof.Letl:=βlog log n. In the present setting, the degrees of G∗andGare clearly bounded\nby∆ = 3. Consequently the exploration of a P-ball of radius laround any x∈Vreveals at\nmost O(∆l) =O((log n)βlog 3)vertices. Suppose we reveal such neighbourhoods around vertices\nx0, . . . , x k, each time picking a vertex disjoint from the previously explored neighbourhoods. As\nexplained in Section 2.7, this exploration can be performed with replacement at a total variation\ncost which is o(1)as long as k∆l=o(√n). If this holds we can thus make the neighbourhoods of\nallxkcoincide with trees with probability 1−o(1). Now in the quasi-tree G, each center has type\n0with probability at least 1/3, so the probability that all centers up to depth lhave type 0is at\nleast 3−∆l+1. Consequently, theprobabilitythatnovertexamong x0, . . . , x khasitsneighbourhood\nisomorphic to T≤l, the subtree of Tup to depth l, is upper bounded by\n(1−3−∆l+1)k≤exp(−k3−∆l+1) = exp( −k9−(logn)βlog 3)\nForβ >0small enough, we can take for instance k=n1/4so that the previous probability is o(1)\nand in the same time k∆l=o(√n)so the approximation with the quasi-tree is justified. This\nproves that with high probability, there exists x0∈[6n]such that B¯P(x0, l)coincides with T≤l.\nItremainstoshowthattheMarkovchain (Xt)t≥0isveryunlikelytoescape B¯P(x, l)bytime ecl.\nIt suffices to prove the statement for the Markov chain (Xt)t≥0onT. This is done by considering\nthe maximal drift of the chain: the vertices which maximize the probability to go away from the\nroot are vertices of degree 3. IfXtis at such a vertex, it goes away from the root with probability\nδ, moves towards the root with probability with probability (1−δ)/2and stays put with remaining\nprobability. Consequently, the distance between Xtand the root is stochastically upper bounded\nby a lazy random walk on the segment [0, l]which goes left with probability (1−δ)/2and right\nwith probability δ. By the classical Gambler’s ruin problem, this random walk has probability at\nmost O((2δ/(1−δ))l)to go from 1tolbefore returning to 0. In turn this implies that the hitting\n22time τlof level lbyXtdominates stochastically a geometric random variable with parameter\nr:= (2 δ/(1−δ))l. In the end we deduce that if t=t(n) =o(1/r)\nP[Xt/∈B¯P(x0, l)]≤P[τl≤t]\n≤1−(1−r)t\n=rt+o(rt) =o(1).\nSince 1/r= (log n)βlog(2δ/(1−δ)), this proves the result.\nProof of Theorem 1.2. Leta > 1,t:= (log n)aandε∈(0,1). By Theorem 1.1 the chain is\nirreducible and aperiodic with high probability. Let πbe the unique stationary measure of ¯P.\nThanks to the lemma, we can find β >0andδ >0small enough for which there exists with high\nprobability a state x∈[n]satisfying Px[Xt/∈B] =o(1), where B:=B¯P(x, βlog log n). Then\nrecall the decomposition of π=π1+π2given by Remark 3.3. Since the ball Bcontains at most\nO(∆βlog log n) =no(1)vertices, Cauchy-Schwarz inequality implies\nπ1(B)≤s\n|B|X\nu∈Bπ1(u)2=n−1/2+oP(1),\nwhile π2(B)≤π2([n]) =oP(1). Consequently\n¯Pt(x, B)−π(B) = 1−oP(1)\nwhich implies tmix(x, ε)≥(logn)a.\nRemark 4.1.As was noticed above for δsufficiently small the Markov chain (Xt)t≥0on the tree\nTbecomes sufficiently biased towards the root to be recurrent. On the other hand, the Borel-\nCantelli lemma implies that the quasi-tree Gcontains a.s. infintely many copies of T≤lfor any l.\nThe existence of such \"trapping neighbourhoods\" which could potentially slow down the chain by\nan important amount is a problem that will arise in the next section where we study asymptotic\nproperties of (Xt)t≥0. In particular the spectral radius of this Markov chain may be equal to 1,\ndespite the tree-like aspect of the graph.\n5 Analysis on the quasi-tree on quasi-trees I: escape proba-\nbilities\nThe objective of the three following sections is to prove the concentration of the drift and entropy\nalong with the other nice properties of Section 3 for the Markov chain (Xt)t≥0on a random\ninfinite quasi-tree G. This will allow us later to deduce the corresponding statements for the chain\nXthanks to the coupling presented in Section 2.7. As in [5, 14], the argument is based on the\nexistence of regeneration times, which are times at which Xtvisits a long-range edge for the first\nand last time. A first step towards this objective is to lower bound the probability of escaping to\ninfinity in the quasi-tree. Because of the potential trapping phenomenon mentionned in Remark\n4.1, this requires more involved arguments than in the reversible vase.\n235.1 Escape probabilities\nWerecall Gdenotesarandomrootedquasi-treewhichunder Phasthelawoftherandomquasi-tree\ndescribed in Section 2.6. Its vertex set is V.\nDefinition 5.1. Given a non-center vertex x∈ V, let\nqEsc(x) :=Px[∀t≥1 :Xt∈ Gx].\nbe the quenched probability that the chain enters the subquasi-tree of xand never leaves it. If x\nis a center, it is useful to also consider starting at time 1/2and let\nqEsc(x) :=Px[∀t≥0 :Xt∈ Gx]∧P\u0002\nτη(x)=∞\f\fX1/2=x\u0003\n.\nWe call these quantities the escape probability atx.\nRemark 5.1.Note that if xis not a center,\nqEsc(x)≥q(x, η(x))P\u0002\nτx=∞ | X 1/2=η(x)\u0003\n.\nSimilarly, if xis a center, Assumption (H3) asserts there exists y̸=xin the same small-range\ncomponent as x, which is thus not a center, so that\nqEsc(x)≥p(x, η(x))P(x, y)qEsc(y).\nBy Assumption (H2) the entries of pandqare bounded. Thus to lower bound escape probabilities,\nit matters little to consider a center vertex or not, and to start at integer or half-integer time.\nFurthermore, recall the Ulam labelling of Section 2.6. For all x∈U, ifx∈ Vis not a center\nthen the probability qEsc(x)is measurable with respect to Gxandι(x)only. Hence\nP[qEsc(x)∈ · | x∈ V,G∖Gx] =P[qEsc(O)∈ · | ι(O) =ι(x)]. (22)\nIfxis a center,\nP[qEsc(x)∈ · | x∈ V,G∖Gx] =P[qEsc(η(O))∈ · | ι(O) =ι(η(x)), ι(η(O)) =ι(x)].\nThus the law of escape probabilities is entirely determined by the case of the root, conditional\nonι(O)andι(η(O)). To ease the notation, we will somes times keep ιimplicit in the sequel and\nwrite only O, η(O)when conditionning on the types of these vertices.\nIn the reversible model, a uniform lower bound remains satisfied as proved by Proposition 4.1\nin [11]. However, in the general model, a uniform lower bound on escape probabilities may not\nbe achievable. The counter example of Section 4 and Remark 4.1 show that in the corresponding\nquasi-tree,\ninf\nx∈VqEsc(x) = 0 .\nThis phenomenon, which was already present in [5], is ultimately what prevents uniform cutoff.\nCutoff from typical vertices remains possible as we will argue that \"trapping neighbourhoods\",\nwhere escape probabilities tend to 0, are sufficiently rare that they are essentially ignored from the\nchain, provided it is started at a typical vertex. The starting point is the following lemma, which\nwill be proved with comparison arguments. The result is stated for the root but from Remark 5.1\nit extends to other subquasi-trees.\n24Lemma 5.1. For all types of O, η(O),\nE[qEsc(O)|O, η(O)] = Θ(1) .\nEquivalently, there exists a constant q0>0such that for all types of O, η(O),\nP[qEsc(O)≥q0|O, η(O)]≥q0. (23)\nEstablishing this lemma is the main difficulty so the proof is deferred to the next sections.\nUsing the tree structure of G, the previous lemma can be boostrapped to yield a much better\ncontrol on the behaviour of escape probability at typical vertices.\nUsing the tree structure of G, the previous lemma can be boostrapped to yield a much better\ncontrol on the behaviour of the escape probability at typical vertices.\nProposition 5.1. There exist constants q0, δ∈(0,1)such that for all types of O, η(O)andk≥0,\nP\u0002\nqEsc(x)< q0δ4k\f\fO, η(O)\u0003\n≤q2k\n0. (24)\nProof of Proposition 5.1. Recall that from assumptions (H1) and (H2) we suppose all transition\nprobabilities are lower bounded by a constant δ. This is also the constant δof the proposition.\nBy Assumption (H3), every vertex y∈ Vhas at least one vertex at small-range distance 1and\ntwo vertices if ι(y)∈V1. Consequently, for any realization of Gxandk≥0, there are at least\n2kcenters at distance at long-range distance 2kfrom xinGxwhich can be joined from xin less\nthan 4ksteps by Xt. Since transition probabilities are bounded below by δ, it suffices that one\nof these vertices yhasqEsc(y)≥q0to obtain that qEsc(O)≥q0δ4k. Now since these vertices\nare centers and at the same long-range distance from xtheir respective subtrees are disjoint and\nthus independent. The probability that all yhave qEsc(y)< q0is thus upper bounded by q2k\n0by\nLemma 5.1.\nFrom the previous proposition, we also deduce the following. Below, a path refers to a possible\ntrajectory of the chain Pand its P-length is the corresponding number of transitions.\nProposition 5.2. For all C >0there exist constants q0, α∈(0,1]such that the following holds:\nfor all r≥0, with probability at least 1−exp(−Cr)conditional on the types of O, η(O), every\npath in GofP-length rstarting from Ocontains a proportion at least αof vertices xsuch that\nqEsc(x)≥q0.\nThe proof is concluded with an argument similar in spirit to Lemma 2.2 in [10] (initially due\nto Grimmett and Kesten [13]) and used to prove Lemma 2.3 in [5].\nProof of Prop 5.2. We reason conditional on ι(O)andι(η(O))so we can assume these are fixed.\nWe distinguish the case of large and small paths, for which it suffices to find one state with lower\nbounded escape probability . Namely if r <10, Lemma 5.1 shows that qEsc(O)≥q0with positive\nprobability, in which case every path of length at most 10contains a proportion at least 1/10\nvertices with escape probability lower bounded by q0.\nFor larger paths, since Gis random, we use the Ulam labelling to consider vertices indepen-\ndently of the realization of G. Given q0>0to be determined and x∈U, letAx= 1x∈V,qEsc(x)≥q0.\nConsider now a self-avoiding path pinUstarted at ι(O)which can be the realization of a path\n25ofP-length r≥10inG, starting at O. Ifpis indeed a path in G, letKpbe the set of vertices\nwhich are at small-range distance at most 1from pbut are not endpoints of a long-range edge\ncrossed by p. Let us give an example for clarity: a path in Umay for instance contain vertices\nu0, u1,(u1, v1),(u1, v2)∈Usuch that u0=ι(O),P(u0, u1)>0,V(v1)̸=V(u2)andP(v1, v2)>0.\nIfη(u1)is such that P(η(u1), v1)>0, this path becomes a genuine path in GofP-length 3,\nwith K(p) ={u0,(u1, v1),(u1, v2)}unless η(u1)∈ {v1, v2}in which case one needs to discard the\ncorresponding vertex. Define A(p) :=P\nx∈KpAx. Then by construction vertices which are in Kp\nmust have disjoint subquasi-trees, so conditional on p∈ GA(p)is a sum of independent Bernoulli\nvariables. Let us lower bound the number of these variables. Since the path is self-avoiding p\ncontains r+1vertices. Supposing kis the number of long-range edges crossed by p,r+1−2kver-\ntices do not belong to a long-range edge of p.k+ 1is also the number of small-range components\ncrossed by p. Now every other of these small-range component is isomorphic to a component of\nV1, which from assumption (H3) contains at least three vertices. By the self-avoiding property,\nthere is at least one vertex of such a small-range component which is in Kp. Hence\nm:=|Kp| ≥max( r+ 1−2k,⌊(k+ 1)/2⌋)≥r\n5−1≥1.\nOn the other hand, for all q∈(0,1)Proposition 5.1 and Remark 5.1 imply the existence of\na value of q0>0, independent of n, such that conditional on x∈ V, the Bernoulli variable Ax\nhas parameter lower bounded by q. Thus for any path pinU, conditional on it being in G,A(p)\ndominates a Binomial (m, q). From Chernoff’s bound, if Zis such a binomial variable, for any\nα, θ > 0\nP[Z < αm ] =P\u0002\ne−θZ> e−θαm\u0003\n≤eθαm+logE[e−θZ]\n=em(θα+log(1 −q+qe−θ)).\nThe base of the exponential can be made arbitrarily small by letting q→1,θ→ ∞andα→0,\nnamely for all ε∈(0,1), for all q >1−ε, there exists α∈(0,1)such that for all m≥0,\nP[Z < αm ]≤εm. (25)\nHence, combining the previous observations, for all ε >0one can find q0>0andα >0, all\nindependent of n, so that for all self-avoiding path pof length rstarting from ι(O)inU.\nP[A(p)< αr|ι(O), ι(η(O)),p∈ G]≤εr. (26)\nSince the number of self-avoiding paths of P-length rinGis bounded by O(∆r)thanks to (9), we\ncan bound\nP[∃p⊂ G:|p|P=r, A(p)< αr|O, η(O)]\n=E\nX\np⊂U1p⊂G,|p|P=rP[A(p)< αr|ι(O), ι(η(O)),p∈ G]\f\f\f\f\f\fO, η(O)\n\n≤O((∆ε)r)\nwhere p⊂U,Gdenotes a self-avoiding path of UorGrespectively. Thus given C >0,A(p)≥αr\nholds simultaneously for all paths with probability 1−(ε∆)r= 1−e−Crchoosing εand then\n26q0, αaccordingly. This almost proves the result, except that the vertices considered in the random\nvariables A(p)may not be on the path p. However even if the proportion αrof vertices with lower\nbounded escape probability may lie partly outside p, this consists of vertices which can be reached\nin at most one step from p. Hence up to modifying the constant q0we can in the end deduce that\nevery path pcontains a proportion αrof vertices with lower bounded escape probability.\nRemark 5.2.The proof actually shows a further property: with probability 1−e−Crevery path\npcontains a constant proportion of vertices for which the probability to escape to infinity outside\npis lower bounded. This can be useful to argue that the chain is unlikely to follow a given path.\n5.2 Lower bound on escape probabilities\nThe proof relies on a comparison with a reversible chain, for which we have a uniform lower bound\non escape probabilities thanks to the analysis done in [11] (see Prop. 4.1 and Remark 4.3). For\nthis comparison to be valid, we will need Assumption (H4). We start with the simpler case where\nthe property of (H4) is satisfied by all vertices, ie there exists l≥1such that\n∀x, y∈V P (x, y)>0⇔(I+P)l(y, x)>0. (H4’)\nNote this implies in particular that all states are recurrent.\n5.2.1 Comparison of Green Functions\nThe comparison will only be possible on a random subgraph of G. The starting point is Theorem\n2.25 in [19], which proves that transience of a non-reversible Markov chain can be deduced from\nthat of a reversible chain, provided their transition probabilities and their invariant measures can\nbe compared up to multiplicative factors. This qualitative result can be made quantitative as it\nbased on a comparison between Green functions, which is the consequence of the following lemma.\nLemma 5.2 ([19, Lemma 2.24]) .LetHbe a Hilbert space with inner product ⟨·,·⟩and let T1, T2\nbe two invertible linear operators on Hsuch that:\n(i)T1is self-adjoint,\n(ii)⟨T2f , f⟩ ≥ ⟨T1f , f⟩ ≥0for all f∈H.\nThen for all f∈H,\nT−1\n1f , f\u000b\n≥\nT−1\n2f , f\u000b\n.\nBy Remark 5.1 it suffices to lower bound P\u0002\nτO=∞ | X 1/2=η(O)\u0003\n. Consider ˜G={O} ∪ G O\nandlet P0denotethetransitionkernelofthechainon ˜Gwhichgoesfrom Otoη(O)withprobability\n1at integer time steps, stays at Oat half-integer time steps, and otherwise has the transitions of\nX. Let\nGP0(x, y) :=X\nt∈NPt\n0(x, y)\nbe the Green function of P0.GP0(O, O)is the expected number of visits to O, which has a\ngeometric distribution of parameter qEsc(η(O))since the chain automatically goes from Oto\nη(O). Hence\nGP0(O, O) =P\u0002\nτO=∞ | X 1/2=η(O)\u0003−1(27)\n27soitisequivalenttolowerboundescapeprobabilitiesandupperboundtheGreenfunction. Lemma\n5.2 will allow to do so by comparison, expressing the Green function as GP0= lim z→1(I−zP0)−1.\nWe will apply the lemma to compare the Markov chain (Xt)t≥0with basically the simple\nrandom walk on a subgraph of the quasi-tree. To this end, we will need to show that the transition\nprobabilities as well as an invariant measure can be lower bounded pointwise by those of the\nsimple random walk, up to multiplicative factor. Since the graph Ghas bounded degrees by (9),\nthe transition probabilities and invariant measure if its simple random walk are both of constant\norder. By Assumption (H1) transition probabilities of (Xt)t≥0are also of constant order however\nthese transitions may occur in only one direction. From the hypothesis done in this section (H4’),\nthis issue is solved by considering a bounded power of a lazy version of the chain. However when\nit comes to the invariant measure, the counter example from Section 4 shows that we can expect\nthe infimum of an invariant measure to be 0, which is why comparison is only possible on a\nsubgraph. Since the invariant measure can be taken up to multiplicative constant, comparison\nwith the simple random walk requires finding a subgraph where the invariant measure is of the\nsame order or larger than the measure of the root. In particular we need to find explicitly one\nsuch invariant measure for (Xt)t≥0. This can be done thanks to the tree structure of G.\n5.2.2 Invariant measure of (Xt)t≥0\nThe idea is based on Kolmogorov’s criterion for reversibility: for a reversible chain, an invariant\nmeasure can be determined explicitely from the detailed balance equation, which allows to propa-\ngate the value of the measure from a state xto another state yby considering any path between x\nandy. Since the invariant measure, which does not need to be a probability, is determined up to\nmultiplicative constant anyway, this gives an algorithmic way to compute an invariant measure.\nFor a non-reversible chain this procedure does not work as multiple paths between vertices may\ngive different values of the measure (the absence of this phenomenon is precisely Kolmogorov’s\ncondition for reversibility). However if there are no multiple paths, in other words if the underlying\ngraph is a tree, and transitions can be reversed, then we can find such a reversible measure. Now\nin our setting, quasi-trees have a large-scale tree structure arising from the long-range edges, while\ncycles can only be present in small-range components. We can thus combine invariant measures of\nsmall-range components to obtain invariance with respect to transitions inside a component with\nthe previous idea to obtain invariance along long-range edges as well.\nLetπPbe an invariant measure of Pwhich puts positive mass on every communicating class.\nAssumption (H4’) implies that Phas no transient states so πP(x)>0for all x∈V. Extend πP\ntoGby setting πP(x) :=πP(ι(x))for all x. The invariant measure νis described in terms of the\nfollowing weights: for every center x, set\nax:=πP(η(x))q(η(x), x)\nπP(x)q(x, η(x))(28)\nThen define\nZ(x) :=kY\ni=1axi (29)\nwhere x1, . . . x k=xis the unique sequence of centers that lie on any path in Gfrom Otox, and\nZ(O) := 1. We extend Zto non-center vertices by Z(x) :=Z(x◦).\n28Lemma 5.3. Under Assumption (H4’)the measure νonVdefined by\nν(x) :=Z(x)πP(x) (30)\nis an invariant measure of the Markov chain (Xt)t≥0.\nProof.Recall Pdenotes the transition kernel of the Markov chain (Xt)t≥0and consider x, y∈ V.\nConsider x, y∈ V. Having P(x, y)>0requires that either xis in the same small-range component\nasy, which occurs if and only if x◦=y◦, orx=η(y◦)is in a lower level, or xis in a higher level,\nthat is x=η(z)forz̸=y◦in the small-range component of y. In the second case, (28) implies\nν(η(y◦))P(η(y◦), y) =Z(η(y◦))πP(η(y◦))q(η(y◦), y◦)P(y◦, y)\n=Z(y◦)πP(y◦)q(y◦, η(y◦))P(y◦, y)\nSimilarly if z̸=y◦is in the small-range component of y,\nν(η(z))P(η(z), y) =Z(y◦)aη(z)πP(η(z))q(η(z), z)P(z, y)\n=Z(y◦)πP(z)q(z, η(z))P(z, y).\nThus these two cases can be regrouped together as the sumP\nx◦=y◦Z(y◦)πP(x)q(x, η(x))P(x, y).\nThen using that p+q≡1and the invariance of πPwith respect to P, we deduce\nX\nxν(x)P(x, y) =X\nx:x◦=y◦Z(y◦)πP(x)p(x, η(x))P(x, y) +Z(y◦)πP(x)q(x, η(x))P(x, y)\n=Z(y◦)X\nx∈V(y)πP(x)P(x, y)\n=Z(y◦)πP(y)\n=ν(y).\n5.2.3 Lower bounding the invariant measure\nLet us rearrange weights in the product (29) to obtain ratios that involve vertices in the same\nsmall-range component. Let x∈ Vand(xi)k\ni=1be the unique sequence of centers joining Otox,\npossibly empty. Then set\nZ′(x) :=k−1Y\ni=1πP(η(xi+1))q(η(xi+1), xi+1)\nπP(xi)q(xi, η(xi)). (31)\ntaking by convention empty products to be equal to 1. By construction\nZ(x) =πP(η(x1))q(η(x1), x1)\nπP(xk)q(xk, η(xk))Z′(x) (32)\nThus to lower bound Z(x)is essentially suffices to bound Z′(x). The idea now is that the process\nZ′(x)can be described as a branching random walk on R∗\n+with symmetric increments. We do not\nneed to dive to much into the theory of branching random walks, so we only define a branching\nrandom walk informally as a Markov field defined on a Galton-Watson tree.\n29In our case, the underlying tree Tcan be obtained by collapsing all vertices of a small-range\ncomponent into one node and keeping only long-range edges. We call this tree the skeleton tree of\nG. This tree has naturally a root which identifies with OinG, while other vertices identify with\ncenters. Thanks to this identification, Z′can be seen as a random field on T. Write TOfor the\nsubtree of Tspanned by centers of GO.\nLemma 5.4. Suppose Psatisfies (H4’)with constant l≥1. Then for all realization of Gthere\nexists a connected subgraph G′⊂ GOcontaining η(O)such that\n(i) all vertices of G′are at P-distance at most 2lfrom their center,\n(ii)Z′(x) = Θ(1) for every center x∈ G′,\n(iii) with probability Θ(1), any reversible chain with underlying graph G′has uniformly lower\nbounded escape probabilities.\nProof.The starting point is that the tree Tand the identification with centers of the quasi-tree\ncan be generated in two steps. Suppose for instance that P1andP2are irreducible. Then all\nsmall-range components of Gare isomorphic to the unique communicating class of either P1or\nP2, so the tree Tis just a n-regular tree. In general the tree Tcan be described as a two-type\nGalton-Watson tree. For i= 1,2andk≥0, letMi(k)be the number of communicating classes\ninViwith exactly kvertices. The tree Thas the law of the random tree where every vertex of\ntype i∈ {1,2}gives rise to k−1children with probability kM¯i(k)/n, all of types ¯i, where ¯i= 2\nifi= 1and vice versa. By Assumption (H3), M1(1) = M1(2) = M2(1) = 0, soTalways contains\nthe deterministic tree that alternates between degree 2and degree 3vertices.\nSuppose now Tis a tree generated according to the previous law. Suppose the type of the\nroot is fixed, which determines the types of all other vertices. Then assign labels to edges of\nT, consisting of pairs (u, v)such that: v∈B+\nP(u,∞), v̸=u. They need to satisfy the two\nconstraints: 1. the labels on edges incident to a same vertex xshare the same first coordinate;\n2. if this coordinate is u, then for all v∈B+\nP(u,∞), v̸=uthere is an edge incident to xwith\nlabel (u, v). Because of the first constraint, we extend labels to vertices, calling uthe label of\nx. In addition, call the label (u, v)a backbone label if dSR(u, v)≤2ordSR(v, u)≤2, which is\nsymmetric in (u, v). Remark that every vertex xhas always one backbone label, nay two if xhas\ntype i= 1. IfTis a tree with such a labelling of the edges, we can construct a quasi-tree Gsuch\nthatTis the skeleton tree of G: labels of vertices define the type ιof centers, while labels of edges\ndetail how the corresponding small-range components are linked to each other. Backbone labels\ncorrespond to vertices which are close to their center: if dSR(u, v)≤2then (H4’) implies that\ndSR(v, u)≤2lsou, vare at bounded distance from each other in both directions.\nWe then define weights on labelled edges. If ehas label (u, v), set\nbe=b(u,v):=πP(v)q(v, η(v))\nπP(u)q(u, η(u)). (33)\nNote that b(u,v)=b−1\n(v,u). Given a vertex x∈ T, we have by construction that Z′(x) :=Qk\ni=1bei\nwhere (ei)k\ni=1is the unique path of edges from the root to xand empty product is equal to 1.\nFrom what precedes, if (u, v)is a backbone label then P(u, v)k>0for some k∈[1,2l]. Recall that\n30from Assumption (H1) and (H2) there is a constant lower bound δon all transition probabilities.\nThus\nb(u,v)≥δ2l+1=:bmin\nis bounded below uniformly in n, using that πP(v)≥πP(u)Pk(u, v). Consequently backbone\nlabels imply bounded weights uniformly in n.\nNow labels can be assigned randomly: suppose x∈ Thas type i∈ {1,2}andk−1vertices\ndescendant vertices. Pick a communicating class Cof size kinViuniformly at random, pick a\nuniform u∈ Cand distribute labels ((u, v))v∈C∖{u}uniformly over the descendant edges of x.\nWith this procedure, the quasi-tree obtained from the random labels has the distribution of G.\nFurthermore, labels are independent over edges that are not incident to a same vertex.\nIn particular Z′\nxis a product of independent weights over the path from the root to x. Fur-\nthermore, since every choice in this construction is uniform, the labels are also uniform over all\npossible pairs (u, v). In particular an edge is given a label (u, v)or(v, u)with the same probability,\nwhich implies it is given a weight b(u,v)orb(v,u)with same probability. Since the latter are inverse\nof each other we deduce that the distribution of the weight assigned to an edge is symmetric, that\nisbehas the same law as b−1\ne. Then for all x∈ T,Z′\nxneeds also be symmetric as it is a product\nof independent weights.\nWe conclude the proof by a comparison with a branching process. We can reason conditional\nonT, so only labels and the associated weights are random. Consider the power tree T(4)\nOobtained\nby keeping only the vertices of TOwhich are at depth a multiple of kfrom the root, and where\nvertices are joined by an edge if one is the ancestor of the other in T. By definition an edge of\nT(4)\nOidentifies to a 4-tuple of edges in TO. LetT′′be the tree defined as the connected component\nofη(O)of all edges in T(4)\nOwhich identify to 4-tuple of edges with backbone labels, and such that\nthe product of weights is above 1. What precedes shows that except η(O), every vertex in T(4)\nO\nhas always 4edges with backbone labels, and the product of weights is above 1with probability\n1/2. Thus the tree T′′dominates stochastically a Galton Watson tree whose offspring law is\nbinomial Bin(4 ,1/2)and is thus supercritical. This gives back a subtree T′⊂ TOof the original\ntree, which is the one we are looking for. The labels on these edges give rise to a set of vertices\nwhich span the desired graph G′. Since all labels on T′are backbone labels, these vertices are all\nat distance at most 2lfrom their center, and Z′(x)≥b4\nminfor all x∈ T′. It remains to prove the\nlast statement. This is the consequence of the arguments used in the reversible model: from the\nabove comparison with a supercritical Galton-Watson tree, if the latter is infinite, which occurs\nwith probability Θ(1), then the skeleton tree T′is sufficiently branching to make any reversible\nchain on G′have uniformly lower bounded escape probabilities. We refer the reader to Proposition\n4.1 and Remark 4.3 in [11] for details.\nProof of Lemma 5.1 with assumption (H4’).By (27), it suffices to bound the Green function GP0\natO. The measure νis easily tweaked to give an invariant measure for P0: the graph G∗\nOis still\na quasi-tree which can be seen as having CO:={O} ∪SR(η(O))as the small-range component of\nthe root. Then replace πPlocally on that small-range component by the invariant measure π∗\nPof\nCO. This modification does not affect the field Z′(x)(31) thus from (32) Z(x)gets only modified\nby one factor. In the end we obtain an invariant measure ν0forP0given by νO(x) :=π∗\nP(x)if\n31x∈ COand otherwise\nν0(x) :=π∗\nP(η(x1))q(η(x1), x1)\nπP(xk)q(xk, η(xk))Z′(x)πP(x)\nif(xi)k\ni=1is the sequence of centers from Otox. In particular, η(x1)∈ COandxk∈BSR(x,+∞).\nConsider now the graph G′= (V′,E′)of Lemma 5.4. For all x∈ V′,Z′(x)≥cand all vertices\nofV′are at bounded distance from their centers. As was argued in the proof of Lemma 5.4, this\nimplies that the ratios between the measures of vertices inside the same small-range component\ncan be bounded uniformly in n. Thus\nπ∗\nP(η(x1))\nν0(O)πP(x)\nπP(xk)= Θ(1) .\nSince the invariant measure is defined up to multiplicative constant, let us suppose π∗\nPwas taken\nin order to have ν0(O) = 1. Using also the fact that the probabilities qare bounded by (H2), we\ndeduce that there exist a constant cindependent of nsuch that for all x∈ V′\nν0(x)≥c.\nOn the other hand, let Kdenote the transition kernel of the simple random walk on G′. The\nmeasure µdefined by µ(x) := degG′(x)for all x∈ V′in an invariant measure for K. Since the\ngraph Ghas degrees bounded by ∆, the entries of both Kandµcan be lower bounded by a\nconstant depending only on ∆. We can thus bound\nsup\nx∈V′µ(x)/ν0(x)≤C\nfor some other constant C > 0independent of n. Consider now the transition kernels ¯P:=\n(I+P0)l+1/2l+1and ¯K:= (1−1\n2Cµ\nν0)I+1\n2Cµ\nν0K, where lis given by the assumption (H4’). The\nmeasure ν0is still invariant for ¯Pbut now one also has that ¯Kis reversible with respect to ν0.\nLetz∈(0,1). We apply Lemma 5.2 to operators T1:= (I−z¯K)andT2:= (I−z¯P)on the\nHilbert space\nℓ2(V′) :={f:V′→R| ⟨f , f⟩ν0<∞},⟨f , g⟩ν0:=X\nx∈V′ν0(x)f(x)g(x).\nBy (H4’) every small-range edge of G′, regardless of the orientation, has positive transition prob-\nability under ¯P, bounded below by (δ/2)l+1. Similarly, if x, y∈ V′andP(η(x), y)>0, then\n¯P(x, η(x))≥2−l−1P0(x, y)P0(y, η(x))≥(δ/2)l+1so long-range edges too have a lower bounded\nprobability. Thus for all x, y∈ V′\n¯P(x, y)≥ϵ¯K(x, y)\nfor some constant ϵ. As a consequence1\n1−ϵ(¯P −ϵ¯K)defines a transition kernel with invariant\nmeasure ν0. This defines thus a contracting operator on ℓ2(V′), which implies\n\n(¯P −ϵ¯K)f , f\u000b\nν0≤(1−ϵ)⟨f , f⟩ν0\nand\n⟨T2f , f⟩ν0≥ϵ⟨T1f , f⟩ν0\nforall f∈ℓ2(V′). Thereversibilityof ¯Kimplyingalsotheself-adjointnessof T2alltheassumptions\nof Lemma 5.2 are satisfied, hence\n\nT−1\n2f , f\u000b\nν0≤ϵ−1\nT−1\n1f , f\u000b\nν0.\n32Taking fas the indicator function at O∈ V′we obtain\nG¯P(O, O|z)≤ϵ−1G¯K(O, O|z)\nwith G¯P(O, O|z) =P\nt≥0¯Pt(O, O)ztand similarly for ¯K. Take the limit z→1from below to\nobtain\nG¯P(O, O)≤ϵ−1G¯K(O, O)\nNow we claim that Green functions of ¯PandP0can be related as\nGPO(O, O)≤2l−1(l+ 1) G¯P(O, O).\nIndeed, ¯Pis obtained by applying the successive operations of \"lazification\" and taking the power\nl+1. It can be checked that the \"lazification\" multiplies the Green function by 2. Then let (Yt)t≥0\nbe the lazy chain with matrix (I+P0)/2. Taking the power l+ 1of this chain comes down to\nconsider the chain only at times which are a multiple of l+ 1. Now for all m∈[0, l], for all t≥1,\nlaziness implies\nPO\u0002\nY(l+1)t=O\u0003\n≥PO\u0002\nY(l+1)t−m=O\u0003\n/2m\nso we can average and get\nPO\u0002\nY(l+1)t=O\u0003\n≥lX\nm=01\n2m(l+ 1)PO\u0002\nY(l+1)t−m=O\u0003\nThen summing over tyields:\nG¯P(O, O)≥lX\nm=01\n2m(l+ 1)X\nt=−mmod l+1PO[Yt=O]\n≥1\n2l(l+ 1)G(I+PO)/2(O, O)\n=1\n2l−1(l+ 1)GPO(O, O).\nFinally it remains to upper bound the Green function of ¯K. Since it is a reversible chain,\nPoint (iii) of Lemma 5.4 shows there is an event of probability Θ(1)on which it has uniformly\nlower bounded escape probabilities which from (27) implies its Green function is uniformly upper\nbounded at the root.\n5.3 General case: o(n)transient or close to transient states\nWe now consider the general case where we suppose only (H4). In this case there can be o(n)\nvertices around which it not possible to revert the transitions in lsteps. Transient states are such\nvertices, so we call these vertices close to transient . There are two issues that prevent the previous\nproof to apply directly: first if Pcontains transient states, πPtakes the value 0so the measure ν\nas defined in Lemma 5.3 cannot be invariant, and the chain (Xt)t≥0is incidentally not irreducible.\nThis problem is fixed by the fact that we need only a \"sub-invariant\" measure, satisfying νP ≤ν,\nto apply the comparison argument, which allows for a loss of mass at transient states. Secondly,\neven without genuine transient states, the comparison with a reversible chain fails at close to\ntransient states. We will argue that since these are in number o(n), they are sufficiently rare to\nbe discarded.\n33Quasi-stationary measures and transient classes Our goal is to define a sub-invariant\nmeasure which puts mass on transient states if there are any. We call such measures quasi-\nstationary, byanalogytoclassicalquasi-stationarymeasures, whichdescribethelimitingbehaviour\nof absorbing Markov chains before absorption. Transitionning along a non-backtracking edge, that\nis an edge that can only be crossed in one direction, is indeed very similar to absorption.\nLet(Yt)t≥0be a Markov chain on a countable state space Wwith transition kernel Q. A set\nA⊂Wis called absorbing if for all x∈A,Q(x, A) = 1. Suppose the chain contains a non-empty\nabsorbing set Aand write Bfor its complement. A measure µonBis called quasi-stationary if\nfor all t≥0, for all y∈B,\nµ(y) =X\nx∈Bµ(x)Px[Yt=y, τA> t]\nAs for classical stationary measures, quasi-stationary measures can be obtain from the Perron-\nFrobenius theorem. Let QBthe restriction of QtoB: this is a sub-stochastic matrix, with\nnon-negative entries, hence it admits a maximal positive eigenvalue λ≤1, which in turn admits\nleftandrighteigenvectorswithallpositiveentries. Since Qt\nB(x, y) =Px[Yt=y, τA> t], wededuce\nthat left eigenvectors µwith positive entries define quasi-stationary measures on B. Note also that\nµQB=λµ≤µis a sub-invariant measure.\nRemark 5.3.Our notion of quasi-stationary measures differs slightly from the classical one which\nnot only intersects but conditions by the event that τA> t.\nNow, ifwesupposethat Wcontainstransientandrecurrentstates, thelatterformanabsorbing\nset. A quasi-stationary measure in this setting consists thus of a quasi-stationary measure on the\nset of transient states, with respect to the set of recurrent states.\nEscape probability Let us move back to lower bounding the escape probability. We can in fact\nprove a slightly stronger result, namely allowing for a positive fraction αnof close to transient\nstates, with αsmall enough. Pcontains in particular at most αngenuine transient states. If\nthere are such transient states, there exist edges (x, y)between the transient and recurrent classes\nsuch that P(x, y)>0butPk(y, x) = 0for all k≥0. Call these non-backtracking edges. Non-\nbacktracking edges in Gobviously give non-backtracking edges in G. We apply the argument with\na version of the Green function GP0(27) killed at non-backtracking edges. Les G′denote the\nconnected component of the root in Gafter deletion of the non-backtracking edges and consider\nthe transition kernel P′\n0given by the restriction of P0toG′, which is a sub-Markov operator. The\ncorresponding Green function GP′\n0(O, O)counts the average number of visits to Obefore crossing\na non-backtracking edge. Crossing a non-backtracking edge of GOobviously prevents the chain\nfrom going back to O, hence the number of visits to Oremains a geometric variable of parameter\nqEsc(η(O))and thus qEsc(η(O)) =GP′\n0(O, O)−1.\nWe thus aim to upper bound this Green function uniformly in n. After truncation, the remain-\ning long-range edges in G′connect small-range components which are either both isomorphic to a\ncommunicating class of G, or the component furthest from the root is isomorphic to a transient\nclass. We can apply the same idea as in Lemma 5.3 to find a sub-invariant measure, considering\nquasi-stationary measures on transient classes. Let πPbe a positive combination, with bounded\ncoefficients, of invariant measures on communicating classes and quasi-stationary measures on\ntransient states. Then the exact same formula as in Lemma 5.3 yields a sub-invariant measure.\n34Next we truncate G′a second time to get rid of close to transient states as well. Let G′′be\nthe subgraph of G′spanned by the set V′′of vertices which do not identify under ιto such states.\nBy construction, the operator (I+P′)l/2lcan be compared edge-wise with the simple random\nwalk on G′′. We can thus apply Lemma 5.2 provided we have a comparison between sub-invariant\nmeasures. This can be done similarly by studying the random field Z, the only difference with the\nprevious section is that we need we only look at vertices in V′′. This can be done by modifying\nthe definition of the \"backbone labels\" in the proof of Lemma 5.4, adding the requirement that\na label (u, v)is a backbone label only if uandvare not close to transient. Provided αis small\nenough, the expected number of backbone labels can be made strictly larger than one, so that the\ncomparison with a supercritical Galton-Watson tree still holds. One last think to be aware of is\nthat we want a result conditional on ι(η(O)), which can be close to transient and thus η(O)/∈ G′′.\nIn this case the argument is to be applied to another subquasi-tree of GOat bounded distance\nfrom η(O).\n6 Analysis on the quasi-tree II: Markovian regeneration struc-\nture\nProposition5.1impliesinparticularthat qEsc(O)isa.s. positiveandthustheMarkovchain (Xt)t≥0\na.s. transient. As a consequence, the shortest path from OtoXteventually has to go through\na unique sequence of long-range edges (ξi)∞\ni=1, which is called the loop-erased chain . Among the\nedges of the loop-erased chain, some have the property to be crossed only once. Thanks to this\nproperty, these so-called regeneration edges yield a Markov decomposition of the quasi-tree which\nwe will use in the next section to prove concentration of the drift and entropy. The regeneration\nedges will also be used later to compute the approximate invariant measure ˆπ.\n6.1 Markov renewal processes\nWe start with general results about Markov renewal processes that will be necessary in the sequel.\nWith the exception of Lemma 6.1, the entirety of this section was already present in [11] so we\nwill not provide the proofs that can already be found there.\nDefinition 6.1. LetSbe a countable state space and Ea Polish space. Consider a process\n(Y, Z) = (Yk, Zk)k≥0taking values in S×E, satisfying for all k≥1,x, y∈S, z∈E,\nP[Yk=y, Zk=z|Yk−1=x, Zk−1, . . . , Y 0, Z0] =P[Y′\n1=y, Z′\n1=z|Y0=x],(M1)\nwhere (Y′\n1, Z′\n1)is an independent copy of (Y1, Z1). This process is thus a Markov chain with a\nstronger Markov property that the usual one, in that the time dependence occurs only through\ntheY-coordinate. In particular Y= (Yk)k≥0is a Markov chain on S.\nAMarkov renewal process is a process (Yk, Tk)k≥0is a process taking values in S×Nsuch\nthat (Yk, Tk−Tk−1)k≥0satisfies (M1) and Tk−Tk−1≥1a.s. for all k≥1, taking T−1:= 0. The\ndelay T0can have arbitrary distribution. We call transition kernels of (Y, T)the family of kernels\n(Qt)t≥1, where for each t≥1,\nQt(x, y) :=P[Y1=y, T1=t|Y0=x, T0= 0] = P[Y1=y, T1−T0=t|Y0=x].\n35Notice then that Yhas transition kernel Q:=P\nt≥1Qt.\nRemark 6.1.If(Y, Z)satisfies (M1), the initial pair (Y0, Z0)can an have arbitrarily law. As usual,\nifνis a probability measure on S×Ewe write\nPν:=X\ny∈SZ\nP[· |Y0=y, Z=z]ν(y, dz).\nOn the other hand, the (M1) implies that (Yk, Zk)k≥1conditional on Y0is independent of Z0.\nThus if we are interested in a quantity that is measureable only with respect to (Yk, Zk)k≥1, we\nwill slightly abuse notation by writing Pu:=P[· |Y0=u](and write similarly for expectation),\nand by extension Pµ:=P\ny∈Sµ(y)Pyfor a measure µonS. In particular, note that if µis an\ninvariant measure for Y, the process (Yk, Zk)k≥1becomes stationary under Pµ. In the sequel if Y\nis positive recurrent, we only consider invariant measures which are probability distributions.\nLemma 6.1. LetSbe a countable set and consider a process (Yk, Zk)k≥0taking values in S×\nR+which satisfies (M1). Suppose there exists α∈(0,1], θ > 0such that E\u0002\neθZα\n0\u0003\n<∞and\nmax u∈SEu\u0002\neθZα\n1\u0003\n<∞. Then for all k≥0,\nEh\neθ(Pk\ni=0Zi)αi\n≤\u0012\nmax\nu∈SEuh\neθZα\n1i\u0013k\n. (34)\nAs a consequence, there exists C, C′>0depending on θ,E\u0002\neθZα\n0\u0003\nandmax u∈SEu\u0002\neθZα\n1\u0003\nsuch\nthat for all k≥0\nP\"kX\ni=0Zi> Ck1/α#\n≤e−C′k. (35)\nProof.Letα∈(0,1], θ > 0such that E\u0002\neθZα\n0\u0003\n<∞andmax u∈SEu\u0002\neθZα\n1\u0003\n<∞. Let (Fk)k≥0\ndenote the standard filtration of (Y, Z). Since α≤1,(x+y)α≤xα+yαfor all x, y≥0.\nConsequently Markov’s property used with (M1) implies\nEh\neθ(Pk\ni=0Zi)αi\n=Eh\neθ(Pk−1\ni=0Zi)α\nEh\neθZα\nk\f\f\fFk−1ii\n=Eh\neθ(Pk−1\ni=0Zi)α\nEYk−1h\neθ(Z′\n1)αii\n,\nwhere Z′\n1denotes an independent copy of Z1. Thus\nEh\neθ(Pk\ni=0Zi)αi\n≤max\nu∈SEuh\neθZα\n1i\nEh\neθ(Pk−1\ni=0Zi)αi\nfrom which the first inequality follows by induction. The second is an application of Chernoff’s\nbound.\nThe next results specify to the setting where the process Ztakes integer values. In particular\nwe state analogs of classical renewal theorems in the context of a Markov renewal process (Y, T).\nThe following generalizes the so-called elementary renewal theorem.\nProposition6.1. Let(Y, T)be a Markov renewal process with state space Ssuch that Yis positive\nrecurrent with stationary distribution µandmax u∈SEu[T1]<∞. Given t≥0let\nNt:= sup {k≥0, Tk≤t}.\n36Then a.s.\nlim\nk→∞Tk\nk=Eµ[T1] (36)\nand\nlim\nt→∞Nt\nt=1\nEµ[T1]. (37)\nThe following Proposition is an analog of the classical renewal theorem, with a quantitative\nbound on the speed of convergence. It is not necessary for the asymptotic analysis on the quasi-\ntree but will be used to essentially compute the annealed laws of XandXand obtain the value\nof the limiting measure ˆπin Proposition 3.2.\nProposition6.2. Let(Y, T)bea Markovrenewalprocess withstate space S, suchthat max u∈SEu[T1]<\n∞andYis positive recurrent with stationary distribution µ, irreducible and aperiodic. Let\nQt(x, y) :=P[Y1=y, T1−T0=t|Y0=x]for all t∈Zand suppose\nα:= min\nx∈SX\nt∈Z\ny∈S(Qt(x, y)∧Qt+1(x, y))>0.\nThen for all probability distribution νonS×N,\nX\ny∈S\f\f\f\fPν[∃k≥0 :Yk=y, Tk=t]−µ(y)\nEµ[T1]\f\f\f\f− − − →\nt→∞0. (38)\nMore precisely, given ε∈(0,1), letKν(ε)be the minimal integer such that\nPν[T0> K ν(ε)]≤ε.\nThere exists C(ε)>0such that for all probability distribution νonS×N,\nX\ny∈S\f\f\f\fPν[∃k≥0 :Yk=y, Tk=t]−µ(y)\nEµ[T1]\f\f\f\f≤ε\nfor all\nt≥C(ε)\nα \nt(Y)\nmix(ε) max\nu∈SEu[T1] +Kν(ε) +Eµ\u0002\nT2\n1\u0003\nEµ[T1]!2\nmax\nvEv[T1] (39)\nwhere t(Y)\nmix(ε)denotes the ε-mixing time of Y.\nThelastresultweneed isa variancebound forMarkov chains. Sincewe willonly dealhere with\nMarkov chains with constant order mixing time, what we require is the following. It is obtained\nby combining Propositions 5.4 and 5.5 in [11].\nProposition 6.3. LetS=S(n)be countable and Ea Polish space. Let (Yk, Zk)k≥0be a process\nonS×Esatisfying (M1). Suppose that Yis irreducible, positive recurrent, with invariant measure\nµand has mixing time tmix(1/4) = O(1)asn→ ∞. Let (fi)i≥1be a family of functions such that\nfor all i≥1,fi:S×E→Randmax u∈SEu\u0002\nfi(Y1, Z1)2\u0003\n<∞. Then there exists a constant\nC >0such that for all k≥1\nVarµ\"kX\ni=1fi(Yi, Zi)#\n≤CkX\ni=1Varµ[fi(Y1, Z1)].\nThe previous result can be applied to the case of a Markov renewal process (Yk, Tk)k≥0, to get\nthat\nVarµ[Tk] =O(kVarµ[T1]). (40)\n376.2 Regeneration structure\nLet us come back to the setting of quasi-trees.\nDefinition 6.2. A time t∈N/2is called a regeneration time if (Xt−1/2,Xt)is a long-range edge\ncrossed for the first and last time at time t.\nLemma 6.2. A.s. there exists an infinity of regeneration times.\nThe proof is similar to that of Lemma 3.3 in [18]. It can also be deduced from the tail estimates\nproved later on.\nFrow now on, let T0:= 0, Y0:=ι(O), L0:= 0and(Tk)k≥1be the sequence of successive\nregeneration times of X. For all k≥1, define\nYk=ι(XTk−1/2) Lk:=d(O,XTk).\n(Yk)k≥1is the Markov chain that dictates the law of the environments between successive regen-\neration times. To describe this Markov chain, introduce the measure\nQu=P\u0002\n· | X 1/2=η(O), ι(O) =u, τO=∞\u0003\n. (41)\nRemark 6.2.By Lemma 5.1, there exists a constant q0>0such that for all u∈V,\nP\u0002\nτO=∞ | X 1/2=η(O), ι(O) =u\u0003\n≥E[qEsc(η(O))|ι(O) =u]≥q0.\nwhich implies\nP\u0002\n·, τO=∞ | X 1/2=η(O), ι(O) =u\u0003\n≤Qu[·]≤q−1\n0P\u0002\n· | X 1/2=η(O), ι(O) =u\u0003\n.\nConsequentlythelawsofthequasi-tree GunderP[· |ι(O) =u]andQuarewithinconstantfactors\nof each other for events that do not contradict τO=∞. In particular, up to a change of constant\nthe uniform lower bounds of Proposition 5.1 and 5.2 for the random variables qEsc(O), qEsc(η(O))\nalso hold under the probability Qu, for any u∈V. Furthermore, the law of the centers added to\nthe quasi-trees remains essentially uniform in the sense that for all v∈Vsuch that V(v)̸=V(u)\nQu[ι(η(O)) =v] = Θ(1 /n). (42)\nIndeed by what precedes\nQu[ι(η(O)) =v]≤q−1\n0P[η(O) =v|ι(O) =u] =q−1\n0/n\nand\nQu[ι(η(O)) =v]≥E\u0002\n1ι(η(O))=v 1τO=∞\f\fι(O) =u,X1/2=η(O)\u0003\n≥E\u0002\n1ι(η(O))=vE[qEsc(η(O))|ι(O), ι(η(O))]\f\fι(O) =u\u0003\n≥q0/n.\nRemark 6.3.Section 6.1 only considered integer valued processed for the time component of\nMarkov renewal processes. To apply the results of this section we will thus implicitely identify\nregeneration times with an integer-valued process.\n38The following lemma is the analog of Lemma 3.6 in [14] and is proved in a similar way.\nLemma 6.3. •The sequences (Yk, Tk)k≥1and(Yk, Lk)k≥1are Markov renewal processes.\n•The sequence\u0010\nYk+1,GXTk∖GXTk+1,(Xt)Tk≤t<Tk+1\u0011\nk≥0is a Markov chain whose transition\nprobabilities only depend on the first coordinate Yk, ie the law of this triplet at time k+ 1\nconditional on time kis only measurable with respect to Yk.\n•For all k≥1, conditional on Yk, the pair (GXTk,(Xt)t≥Tk)has the law of (GO,X)under the\nprobability QYk.\nRemark 6.4.Note that although (Y, T)starts at time 0, it is a Markov renewal process from time\n1, and the delay is thus T1. However by the lemma, under the measure Qu,v, for any u, v∈Vsuch\nthatV(u)̸=V(v),T1has distribution given by a transition probability and thus (Y, T)becomes\na Markov renewal process already from time 0, with 0delay. We will use this observation often\nin the sequel.\nA non-negative random variable Zhasstretched exponential tail if there exists α∈(0,1)such\nthatE[expZα]<∞. A stretched exponential tail implies the existence of polynomial moments\nof all orders.\nLemma 6.4. For all u∈V, under both measures QuandPO[· |ι(O) =u]:\n•L1has exponential tail,\n•T1has stretched exponential tail.\nFurthermore, given L, t≥0, let ˜T(L)\n1denote the first time a regeneration occurs outside the ball\nBLR(O, L)andUt(L) =P\ns≤t1Xs∈BLR(O,L)the total time spent by the chain in BLR(O, L)before\nt. There exists α∈(0,1]such that for all L, m, k ∈N, m≥1, for all x∈BLR(O, L)such that\ndLR(O, x) =L, conditional on either X0=xorX1/2=x\nPh\n˜T(L)\n1> k+m\f\f\fBLR(O, L), Uk+m(L)≤mi\n≤e−kα.\nRemark 6.5.Proposition 2.2 in [1] shows that in the case of a randomly biased random walk on\na Galton-Watson tree, stretched exponential can be the right order of magnitude for the tail of\nthe first regeneration time. From the proof, it can be seen that this slowdown is precisely caused\nby the same trapping phenomenon as mentionned in Remark 4.1, when the walk can move from a\npart of the tree where it goes very fast to infinity to a part where it tends to go back to the root.\nThis suggests that for some choices P1, P2andp, stretched exponential tails give the right tail for\nthe regeneration times in our model too.\nProof.For the tails, Remark 6.2 shows it suffices to prove the result for the law PO[· |ι(O)].\nSuppose ι(O)is fixed. The proof of the exponential tail for regeneration levels is taken from that\nof Lemma 4.2 in [10]. It actually proves exponential tails for a stronger notion of regeneration,\nwhich looks at times when a long-range distance is reached for the first and last time. Namely let\n˜T1:= inf{t∈N/2|∀s < t :dLR(O,Xs)< dLR(O,Xt),∀s≥t:dLR(O,Xs)≥dLR(O,Xt)}.\nand˜L1:=dLR(O,X˜T1). Obviously T1≤˜T1andL1≤˜L1, so it suffices to prove an exponential\ntail for ˜L1.\n39Suppose the chain is started at O. Consider the \"ladder times\" (Si)K\ni=0, where K∈N∪{∞}is\na random integer, possibly infinite, defined recursively as follows. Let S0:= 0, and define τ0as the\nfirst return time to O. Ifτ0=∞setK:= 0. Otherwise, let M0:= max {dLR(O,Xt),0≤t≤τ0}\nbe the maximal distance reached until τ0andS1:= min {t∈N/2 :dLR(0,Xt)> M 0}. Define the\nrest of the sequence recursively: let τithe first return time to level dLR(0,XSi)−1. Ifτi=∞,\nK:=i, and Sj:=∞for all j≥i+ 1, otherwise set Mi:= max {dLR(O,Xt),0≤t≤τi}and\nSi+1:= min {t∈N/2 :dLR(O,Xt)> M i}. Notice that SK=˜T1ifK∈(0,∞)and ˜T1= 1/2if\nK= 0. Consequently for any C >0, ifl≥1is large enough,\nPOh\n˜L1> Cl\f\f\fι(O)i\n≤PO[K <∞, d(O,XSK)> Cl|ι(O)]\n=PO\"\nK <∞,KX\ni=1\u0000\ndLR(O,XSi)−dLR(0,XSi−1)\u0001\n> Cl\f\f\f\f\fι(O)#\n=∞X\nk=1PO\"kX\ni=1\u0000\ndLR(O,XSi)−dLR(0,XSi−1)\u0001\n> Cl,{τi<∞}k−1\ni=1, τk=∞\f\f\f\f\fι(O)#\n≤PO[K > l |ι(O)] +PO\"l−1X\ni=0Zi> Cl\f\f\f\f\fι(O)#\n.\nwhere we write Z0:=dLR(O,XS1) 1τ0<∞,Zi:=\u0000\ndLR(O,XSi+1)−dLR(0,XSi)\u0001\n1τi<∞for all i≥1.\nTo prove this is exponentially small in l, it thus suffices to establish that Khas exponential tail\nand that for fixed l≥1, the second probability in the right-hand side is also exponentially small\ninl.\nIt was already observed in Remark 6.2 that P\u0002\nτO=∞ | X 1/2=η(O), ι(O)\u0003\n= Θ(1). We\ndeduce that Kis stochastically dominated by a geometric variable of constant parameter and\nthus has exponential tail, even conditional on ι(O).\nOn the other hand, by construction, for all i≥1ifSi<∞the trajectory (Xt)Si≤t<τiis\nentirely contained in the subquasi-tree GXSi. Furthermore, dLR(O,XSi+1) =Mi+ 1is determined\nby this trajectory. Thus for i≥1,Zi= (Mi−Mi−1) 1τi<∞depends only on η(XSi−1/2)andGXSi.\nLetting ˜Yi:=ι(η(XSi−1/2))fori≥0, we can deduce that if i≥1, l≥1andSi<∞,\nPO\u0002\nZi=l| G∖GXSi,(Xs)s≤Si\u0003\n=Ph\nτO′<∞, M′\n0=l| X1/2=η(O′), ι(O′) =˜Yii\nwith (G′, O′, M′\n0)an independent copy of (G, O, M 0). We do not prove this in detail, which can\nbe established as Lemma 6.3. See also the proof of Lemma 4.4 of [10]. If we extend the process\n(˜Y , Z)by considering that ˜Yitakes an arbitrary value †ifSi=∞and consider that Zi+1= 0a.s.\ngiven ˜Yi=†, the previous equation implies in particular that\u0010\n˜Yi, Zi\u0011\ni≥0is a Markov chain that\nsatisfies the (M1) property. Furthermore for all l≥1,\nP\u0002\nτO<∞, M0≥l| X1/2=η(O), ι(O)\u0003\n≤P\u0002\n∃t > s≥l:dLR(η(O),Xs) =l−1,Xt=O| X1/2=η(O), ι(O)\u0003\n.\nThisisanannealedprobabilityofbacktrackingoveralong-rangedistance L. Letushereanticipate\non the sequel and use Lemma 7.3 which will prove this probability is exponentially small in L.\nWith what precedes, this proves that there exists constants C′, θ > 0such that for any possible\n40state uof the chain ˜Y,Eh\neθZ1\f\f˜Y0=ui\n< C′. Exponential moments of Z0can be bounded\nsimilarly. The reader will then have recognized the condition to apply Lemma 6.1, which proves\nthat for some constant C > 0,POhPl−1\ni=0Zi> Cl\f\f\fι(O)i\nis exponentially small in l, with all\nconstants being independent of n.\nLet us move to the proof of stretched exponential tails for T1. It is inspired from the proof\nof Prop. 2.2 in [1]. For ease of notation we will here keep the conditionning by ι(O), but all\nprobabilities below should be considered conditionning on it. Given k, t≥0, let τkdenote the\nhitting time of level kandKtdenote the time spent before tat a center, at half-integer times\nsteps. For any C >0andk≥0,\nPO\u0002\nT1> Ck3\u0003\n≤PO[T1> τk] +PO\u0002\nCk3< T1≤τk\u0003\n≤PO[T1> τk] +PO\u0002\nKCk3≤k3\u0003\n+PO\u0002\nKτk> k3\u0003\n.\nThe first term can be rewritten as P[T1> τk] =P[L1> k]which is exponentially small by the\nfirst part. For the second term, note that when the chain is at a non-center vertex xat integer\ntimes, it has probability Θ(1)to go to the center η(x)by (H2). Thus the first hitting time time\nof a center is stochastically dominated by a geometric variable, even under the quenched law. In\nturn we deduce\nPO\u0002\nKCk3≤k3\u0003\n≤P\nk3X\ni=1Zi≥Ck3\n,\nwhere the Ziare independent geometric variables of positive constant parameter. By Chernoff’s\ninequality, the right-hand side is exponentially small in k3for a good choice of C >0. The third\nterm can be further decomposed as follows. Given a center x∈ V, letN(x)denote the number of\nvisits to xat half-integer times. Given i≥1, write zifor the i-th distinct center visited by the\nchain and Mifor the number of distinct centers visited in Viat half-integer times. Then\nPO\u0002\nKτk> k3\u0003\n≤PO\"\nmore than k2distinct centers are visited\nbefore τkat half-integer times#\n+PO\u0002\n∃i≤k2:N(zi)> k\u0003\n.\nBy union bound\nPO\"\nmore than k2distinct centers are visited\nbefore τkat half-integer times#\n≤kX\ni=1PO[Mi≥k].\nLetτ(i)\nkdenote here the hitting time of the k-th distinct center of Vi. By the strong Markov\nproperty\nPO[Mi≥k] =POh\nτ(i)\nk<∞i\n≤EOh\n1τ(i)\nk−1<∞(1−qEsc(Xτ(k)\nk−1))i\n.\nSince the centers visited at τ(i)\nk−1andτ(i)\nkare distinct and at the same level, their subtrees are\ndisjoint and independent. Hence\nPO[Mi≥k]≤E\u0014\nPOh\nτ(i)\nk−1<∞i\u0012\n1−min\nu,vE[qEsc(η(O′))|ι(O′) =u, ι(η(O′) =v)]\u0013\u0015\n,\n41where (G′, O′)is as usual an independent copy of (G, O)andV(u)̸=V(v). By Lemma 5.1, the\nexpected escape probability can be lower bounded by some constant q0∈(0,1]hence by induction\nP[Mi≥k]≤(1−q0)k−1\nand\nPO\"\nmore than k2distinct centers are visited\nbefore τkat half-integer times#\n≤k(1−q0)k−1\nis exponentially small in k.\nTo bound the last term, let\nq′\nEsc(x) :=P\u0002\nτx=τη(x)=∞\f\fX1/2=x\u0003\n.\nfor all center x∈ V. Like qEsc(x), this quantity depends only on Gxandη(x). Furthermore, if y\nis a vertex at small-range distance 1from x, then\nq′\nEsc(x)≥p(x, η(x))P(x, y)qEsc(y)\n≥δqEsc(y),\nthus lower bounding q′\nEscis essentially the same as lower bounding qEsc. For all i≥1,\nPO[N(zi)> k] =X\nx∈VE\u0002\n1zi=x,N(x)>k\u0003\n≤X\nx∈VE\u0002\n1zi=x(1−q′\nEsc(x))k\u0003\n=X\nx∈VPO[zi=x] (1−q′\nEsc(x))k\nTo average over the environment, use the \"Ulam labelling\" of Section 2.6:\nPO[N(zi)> k]≤X\nx∈UE[ 1x∈VPO[zi=x]]E\u0002\n(1−q′\nEsc(x))k\f\fx∈ V,G∖Gx\u0003\nFrom Proposition 5.1, one can easily obtain that\nmax\nx∈UE\u0002\n(1−q′\nEsc(x))k\f\fx∈ V,G∖Gx\u0003\n≤e−kc\nfor some constant c >0, hence by union bound\nPO\u0002\n∃i≤k2:N(zi)> k\u0003\n≤k2e−kc,\nwhich is stretched exponential, as desired.\nFinally, for the last statement note that if the chains is started precisely at the boudnary of\nthe ball BLR(O, L), then a regeneration can occur within a number of steps which is bounded\ninL, thus from the lower bound on escape probabilities (Lemma 5.1) it is easy to lower bound\nPx[Uk+m(L)≤m|BLR(O, L)]uniformly in n, L, m. Thus the conditionning by Uk+m(L)≤m\nonlymodifiesupperboundsundertheusuallawofthequasi-treebyafactorofconstantorder. The\nupper bound for the conditional probability then follows from the same arguments as above.\n426.3 Mixing of the regeneration chain\nWe now investigate the mixing properties of the Markov chain Yunderlying the regeneration\nprocess, which we call the regeneration chain. Lemma 6.5 below will establish that Yhas constant\nmixing time, allowing us later to get moment bounds similar to the iid case. The lemma actually\nproves a stronger mixing property of both the chain Yand the time process Tthat will be used\nto derive an approximation of the invariant measure on X.\nLemma 6.5. Let(Qt)t≥1denote the transition kernels of the Markov renewal process (Y, T)and\nQ:=P\nt≥1Qt, after identification of Twith a process in N. Recall that from Hypothesis ((H4))\nthere exists a constant l≥1such that the set\nS=S(l) :={x∈V| ∀y∈V:P(x, y)>0⇔(I+P)l(y, x)>0}.\nsatisfies |S|= 2n−o(n).\n(i) For all u∈Vandv∈S,\nQ(u, v) = Θ(1 /n). (43)\nAs a consequence for all ε∈(0,1), the chain Yhas mixing time Oε(1)asn→ ∞.\n(ii) for all u∈V,X\nt≥1X\nv∈VQt(u, v)∧Qt+1(u, v) = Θ(1) . (44)\nProof of Lemma 6.5. Letε∈(0,1). The statement about the mixing time of Yis easily deduced\nfrom (43) and the fact that |S|= 2n−o(n). Summing on vyields the following Doeblin’s condition\nforY: there exists a constant c >0such that\n∥Q(u,·)−Q(u′,·)∥TV≤1−c−o(1),\nfor all u, u′∈V. It is then easy to obtain from Doeblin’s condition that Yhas mixing time\nt(Y)\nmix(ε)≤(c−1+o(1)) log ε−1.\nLet us now prove (43). Consider u, v∈Vwith v∈S. By definition for all t≥1\nQt(u, v) =Qu[Y1=v, T1=t] =P\u0002\nT1=t, ι(Xt−1/2) =v\f\fτO=∞, ι(O) =u,X1/2=η(O)\u0003\n.\nProbabilistic statements below will thus be made with respect to Quunless stated otherwise.\nObserve that if V(u)̸=V(v), it is possible to realize Y1=vwith regeneration occurring at\nthe first long-range edge crossed by the chain, which can be reached in half a step from η(O)\nby Assumption (H3). Let w∈Vsuch that P(w, v)>0. From Remark 6.2, Qu[ι(η(O)) =w] =\nΘ(1/n). Using that transition probabilities are bounded uniformly in nby (H1), (H2), we deduce\nthatQu[Y1=v]≥Θ(1/n). On the other hand, if V(u) =V(v)the alternation between V1andV2\nforbids a regeneration at the first long-range edge. This is why we need to suppose v∈Sto make\nsure the chain can backtrack and prevent regeneration at the first long-range. Since Soccurs with\nprobability 1−o(1)under the uniform law on V, centers of the quasi-tree GOalso have probability\nΘ(1)to be in SunderQu. Thus using the above arguments we can lower bound by Θ(1/n)the\nprobability that the chain goes from η(O)to a vertex ywith ι(y) =vusing one long-range edge,\ncomes back to η(O), which ensures no regeneration occurred on the long-range edge, and returns\n43η(O)η(y)\nη(x)y\ny′z\nz′η(y′)s1s3\ns3s2−1\nx\nOι(O) =uι(z) =v\nι(z′) =vFigure 2: Argument of the proof of Lemma 6.5\ntoy, all that within a bounded number of steps. The chain can then escape in Gywith probability\nΘ(1)by Lemma 5.1, which will imply Y1=v.\nThe proof of (44) is similar but requires taking time into account. Suppose u∈V1and let\nv∈S∩V2. Our argument is illustrated in Figure 2. The idea is to use an intermediary component\nofV1to make the regeneration time shift by 1. Consider the set\nS1:={x∈V∩S|∃y̸=x∈V, s≤l:P2(x, y)>0andPs(y, x)>0}.\nFrom Assumption (H3), n−o(n)vertices of S∩V1are recurrent and contained in a communicating\nclass of size at least 3. Consequently if x∈S∩V1, either there exists y̸=xon a loop around\nxof length less than lsuch that P2(x, y)>0, in which case x∈S1, orxis in a communicating\nclass{x, y, z}of size exactly 3with P2(x, x) = 1. However in that case P2(y, z), P2(z, y)>0, so\ny, z∈S1. Thus S1has size at least 2n/3−o(n)and probability Θ(1)under the uniform law on\nV1orQwfor any w∈V2.\nFor any realization of G, there exists a path (η(O), x, η(x), y, η(y), z)with two long-range edges\n(x, η(x))and(y, η(y))whereas (η(O), x),(η(x), y),(η(y), z)are small-range edges with distinct\nendpoints. If ι(η(O))∈S, which occurs with probability Θ(1), we can suppose that xwas chosen\nso that Ps1(ι(x), ι(x))>0, for some s1≤l+ 1. Similarly, ι(η(x))∈S1with probability Θ(1), in\nwhich case we can suppose ywas chosen so that there exists a path between yandη(x)of length\ns2≤land a vertex y′̸=xon this path with P2(ι(η(x)), ι(y′))>0. Then if v∈Sthere exists\nw∈V2such that P(w, v)>0andP(v, w)s3>0for some s3≤l. Conditional on the previous\nvalues of ι,ι(η(y)) = wwith probability at least Θ(1/n). Gathering what precedes, make the\nfollowing observation: from η(O)the chain can go to xin half a step and then to zin two more\nsteps, after which it can come back to xins3+s2+s1steps going through y′, to eventually return\natzin two steps and escape to infinity through η(z). We will then have Y1=v, T1=t+ 1/2with\nt= 4 + s1+s2+s3.\nNow instead of y, we can apply the same arguments with y′, supposing that the chain reaches\na vertex z′with ι(η(y′)) = wandι(z′) =v. By construction z′is reachable in just one more\nstep from x, but the loop around xwhich passes through z′takes the same number of steps.\nThus the previous argument can be applied similarly with just one time step difference to obtain\n44Y1=v, T1=t+ 1 + 1 /2. Summing on the values of ι(η(O)), ι(η(x)), we obtain that\nX\nt≥1Qt(u, v)∧Qt+1(u, v)≥Θ(1/n).\nThe case where u∈V2is similar, with one less necessary long-range edge to cross to reach a\ncomponent of V1∩S1, hence the bound applies to all u∈V. Then as |V2∩S|=n−o(n)summing\noverv∈V2∩Syields (44).\nLetµdenote the stationary distribution of the Markov chain (Yk)k≥0and\nQµ:=X\nu∈Vµ(u)Qu.\nIn the sequel EQµdenotes the expectation with respect to µ. Later, we use similar notations for\nvariance, covariance, etc.\nThefactthat Ymixesin O(1)stepswillbesufficientfortherestoftheanalysisofthequasi-tree.\nPoint (ii) in Lemma 6.5 will then be used to obtain the expression of the approximate invariant\nmeasure in (19). The mixing of the regeneration chain will be used conditional on some already\nrevealed parts of the environment, which in turn requires conditionning by a neighbourhood of\nthe root. We did something similar in the reversible case (see Prop. 5.6 in [11]) however in the\nmore general case, the neighbourhood can trap the chain. This should not be the case however\nif the neighbourhood is typical. To implement this idea we condition further by the fact that the\nchain rapidly escapes this neighbourhood.\nProposition 6.4. Letνbe the law of ι(η(O))underQµ. For all v∈V,\nν(v) = Θ(1 /n). (45)\nFurthermore given L, t≥0, let Ut(L) =P\ns≤t1Xs∈BLR(O,L)denote the total time spent by the\nchain in BLR(O, L)before t. Consider here ˜T1=˜T1(L)to be the first regeneration time outside\nBLR(O, L), while for k≥2, let ˜Tkbe the first regeneration time after ˜Tk−1. For all ε∈(0,1)\nthere exists C(ε)such that for all L, m≥0andt∈N/2, for all x∈BLR(O, L),dLR(O, x) =L,\nift≥C(ε)m2then\nX\nv∈V\f\f\f\fPxh\n∃k≥0 :˜Tk=t, ι(Xt) =v\f\f\fBLR(O, L), Ut(L)≤mi\n−ν(v)\nEQµ[T1]\f\f\f\f≤ε.\nand\nX\nv∈V\f\f\f\fPh\n∃k≥0 :˜Tk=t, ι(Xt) =v\f\f\fX1/2=x, B LR(O, L), Ut(L)≤mi\n−ν(v)\nEQµ[T1]\f\f\f\f≤ε.\nProof.The first statement is a direct consequence of (42).\nLet us prove the second statement. Let ε∈(0,1), m, t≥1. For all k≥1let˜Yk:=ι(X˜Tk−1/2).\nApply Proposition 6.2 with the Markov renewal process of regeneration times considered here.\nNote that by the Markov property of (Y, T)conditionning by BLR(O, L), Ut(L)≤myields the\nsame transition kernels as (Y, T) and only affects the law of (˜Y1,˜T1). Lemmas 6.5 and 6.4 imply\nthat the two quantities αandmax u∈SEu[T1]in this Proposition are bounded uniformly in n, as\nis the mixing time t(Y)\nmix(ε)ofYfor all ε∈(0,1). Finally, the last statement of Lemma 6.4 implies\nthat\nPxh\n˜T1≥m+C1(ε)\f\f\fBLR(O, L), Um+C1(ε)(L)≤mi\n≤ε\n45for some C1(ε)>0. This gives the value of the quantity Kν(ε)considered in Proposition 6.2,\nwhich thus proves that there exists C(ε)such that for t≥C(ε)m2,\nX\nu∈V\f\f\f\fPxh\n∃k≥0 :˜Tk=t−1/2,˜Yk=u\f\f\fBLR(O, L), Ut(L)≤mi\n−µ(u)\nEQµ[T1]\f\f\f\f≤ε.\nThen note that conditional on ˜Tk=t−1/2,˜Yk=u,ι(Xt)is distributed as ι(η(O))underQu.\nFinally the arguments apply in the same way if the chain is started at time xat time 1/2instead\nof0.\n7 Analysis on the quasi-tree III: concentration of drift and\nentropy\nIn this section we establish \"nice properties\" for the chain X, proving in particular concentration\nof the drift and entropy.\n7.1 Typical paths in quasi-trees\nWe start with the following lemma, which will basically ensure the chain Xonly visits typical\nvertices by time t=O(logn), allowing to use the uniform lower bound on escape probabilities.\nLemma 7.1. LetAbe a measurable set of quasi-trees. For all t≥0,\nmax\nu,v∈VPO\n[\ns≤t{GXs∈ A}\f\f\f\f\f\fι(O) =u, ι(η(O)) =v\n\n≤tmax\nu,v∈VPO[GO∈ A | ι(O) =u, ι(η(O)) =v].\nProof.Fix the types of O, η(O). Let z0:=Oand for i≥1letzibe the i-th distinct vertex visited\nby the chain. There are at most tdistinct vertices visited up to time t, hence\nPO\n[\ns≤t{GXs∈ A}\f\f\f\f\f\fO, η(O)\n≤P\n[\ni≤t{Gzi∈ A}\f\f\f\f\f\fO, η(O)\n\n≤tmax\ni≥0P[Gzi∈ A | O, η(O)].\nNote that for x∈Uthe probability PO[zi=x]is measurable with respect to G∖Gx. Thus for\nalli≥0,\nPO[Gzi∈ A | ι(O), ι(η(O))] =X\nx∈UPO[zi=x,Gx∈ A | O, η(O)]\n=X\nx∈UE[PO[zi=x]PO[Gx∈ A | G ∖Gx]|O, η(O)].\nNow the second factor in this sum can be bounded as\nPO[Gx∈ A | G ∖Gx]≤max\nu,v∈VP[G′\nO′∈ A | ι(O′) =u, ι(η(O′)) =v]\nwith (G′, O′)an independent copy of (G, O). The remaining terms then sum up to 1, yielding the\nresult.\n46The previous Lemma will be combined with the following.\nLemma 7.2. Given α, q0∈(0,1]andl≥0, letAl=Al(q0, α)be the set of quasi-trees for which\nevery path of length lstarting from Ocontains a proportion at least αof vertices xsatisfying\nqEsc(x)≥q0. There exists a constant Csuch that for all l≥0, on the event {G ∈ A l},\nPO[∃t > s≥0 :Xs∈ Gx, dLR(O,Xs)≥l, dLR(O,Xt) = 0] ≤e−Cl\nPO[∃t≥0 :dSR(O,Xt)≥l]≤e−Cl.\nSimilarly, for all x∈ V, on the event {Gx∈ Al},\nPx[∃t > s≥0 :Xs∈ Gx, dLR(x,Xs)≥l, dLR(x,Xt) = 0] ≤e−Cl\nPx[∃t≥0 :Xt∈ Gx, dSR(X◦\nt,Xt)≥l]≤e−Cl.\nProof.Given q0>0, consider the set Esc = Esc( q0) :={x∈ V | qEsc(x)≥q0}. For all x∈\nG ∩Esc∖{O},Px[τO<∞]≤1−qEsc(x)<1−q0.\nLetl≥0andτlbe the first time t∈Nsuch that d(O,Xt) =landτOthe hitting time of O.\nBy strong Markov’s property,\nPO[∃t > s≥0 :dLR(O,Xs)≥l, dLR(O,Xt) = 0] = EOh\n1τl<∞PXτl[τO<∞]i\n≤ max\nx1:dLR(O,x1)=lPx1[τO<∞].\nLetτ(k)\nEscbe the k-th hitting time of Esc∖{O}and suppose x1∈ Gis at long-range distance l\nfrom O. Then x1must also be at P-distance at least lfrom O, thus if G ∈ A landX0=x1\nit is impossible that τO< τ(⌊αl⌋)\nEscas this would imply the existence of a path contradicting the\ndefinition of Al. Thus when G ∈ A l, applying Strong Markov’s property at the successive stopping\ntimes τ(k)\nEscyields\nPx1[τO<∞] =Px1h\nτ(⌊αl⌋)\nEsc≤τO<∞i\n≤(1−q0)⌊αl⌋.\nThe argument for the second bound is similar. Let τ(k)\nEscbe now the k-th hitting time of Esc,\nso we do not discard Oanymore, and let τldenote now the hitting time of BSR(O, l). As before,\nobserve it is impossible to have τl< τ(⌊αl⌋)\nEscifG ∈ A l. However, to reach BSR(O, l)there must be\nno escape to infinity before τl. Hence on the event {G ∈ A l},\nPO[∃t≥0 :dSR(O,Xt)≥l] =PO[τl<∞]\n=POh\nτ(⌊αl⌋)\nEsc≤τl<∞i\n≤(1−q0)⌊αl⌋,\nthe last line being obtained by applying the strong Markov property at times τ(k)\nEsc.\nFinally the two last statements for a starting state x∈ Vare proved with the exact same\nreasoning.\nWe now establish an analog of Lemma 3.2 in the quasi-tree setting.\n47Lemma 7.3. LetΓ(R, L, M )denote the set of paths pinGsuch that pdoes not deviate from a\nsmall-range distance more than R, backtrack over a long-range distance Lor contain a subpath\nof length Mwithout a regeneration edge. There exists C > 0andα∈(0,1]such that for all\nR, L, M ≥0, for all t≥0, for any types of O, η(O),\nPO[Xs···X s+t/∈Γ(R, L, M )|O, η(O)]≤(s+t)e−C(R∧L∧Mα).\nProof.FixR, L, M,s, t≥0and the types of O, η(O). Let us start with the deviation property:\nPO[∃t′∈[s, s+t] :dSR(X◦\nt′,Xt′)≥R|O, η(O)]\n≤PO[∃t′≤s+t:dSR(O,Xt′)≥R|O, η(O)]\n+P\u0002\n∃t1≤t2≤s+t:Xt2∈ GXt1, dSR(X◦\nt2,Xt2)≥R\f\fO, η(O)\u0003\n(46)\nGiven q0>0,α∈(0,1), letAR=AR(q0, α)be the set of quasi-trees, for which every path of\nlength Rstarting from Ocontains a proportion at least αof vertices xsuch that qEsc(x)≥q0.\nThen the first term can be bounded as\nPO[∃t′≤s+t:dSR(O,Xt′)≥R|O, η(O)]≤P[G/∈ AR|O, η(O)]\n+E[ 1G∈A RPO[∃t′≤s+t:dSR(O,Xt′)≥R]|O, η(O)].\nBy Proposition 5.2 there exist constants q0, α > 0such that the first term can be made exponen-\ntially small in R, while Lemma 7.2 shows the second term is also exponentially small. The second\nterm of (46) can be bounded similarly:\nP\u0002\n∃t1≤t2≤s+t:Xt2∈ GXt1, dSR(X◦\nt2,Xt2)≥R\f\fO, η(O)\u0003\n≤P\u0002\n∃t1≤s+t:GXt1/∈ AR\f\fO, η(O)\u0003\n+X\nt1≤s+tEh\n1GXt1∈ARPXt1\u0002\n∃t2≥0 :Xt2∈ GXt1, dSR(X◦\nt2,Xt2)≥R\u0003\f\f\fO, η(O)i\n≤(s+t)e−CR\nfor some constant C >0, using Lemma 7.1 and Proposition 5.2 to bound the first term and Lemma\n7.2 for the second.\nThe probability that the trajectory backtrack over a long-range distance Lbetween sands+t\nis established with the same arguments. It then remains to handle the regeneration requirement.\nBy union bound for all t≥0andu∈V\nPO[∃s′∈[s, s+t] : [s′, s′+M]∩ {Tk, k≥1}=∅ |O, η(O)]\n≤PO[∃k≤s+t:Tk+1−Tk≥M|O, η(O)]\n≤(s+t) max\nk≤s+tPO[Tk+1−Tk≥M|O, η(O)].\nThe stretched exponential tails of regeneration times (Lemma 6.4) conclude the proof.\n7.2 Concentration of the drift\nProposition 7.1. Letd:=EQµ[L1]\nEQµ[T1]. Then for all s≥0, a.s.\ndLR(Xs,Xs+t)\nt− − − →\nt→∞d. (47)\n48Furthermore, there exists α >0for which the following holds. For all ε >0there exists C=C(ε)\nsuch that for all s, t≥0, for all types of O, η(O),\nPOh\n|dLR(Xs,Xs+t)−dt|> C√\nt\f\f\fO, η(O)i\n≤ε+C√se−tα. (48)\nProof.For notational simplicity we omit writing the conditionning by ι(O)andι(η(O)). As can\nbe checked this conditionning does not affect the proof as the technical results that will be used\nhold even conditional on the long-range edge at the root. For all t≥0, let\nNt:= max {k≥0|Tk≤t}.\nThen\nLNt≤dLR(O,Xt)≤LNt+TNt+1−TNt.\nIt is easy to prove that (TNt+1−TNt)/t→0, hence the law of large numbers (47) follows from\nLemma 6.3 and Proposition 6.1 which prove Nt/t− − − →\nt→∞1/Eµ[T1]andLk/k− − − − →\nk→∞Eµ[L1]a.s..\nThen we establish (48) in the case s= 0. It suffices to prove that for all ϵ >0there exists\nC >0such that for all t, k≥0\nPO\u0014\f\f\f\fNt−t\nEQµ[T1]\f\f\f\f> C√\nt\u0015\n≤ε (49)\nPOh\f\fLk−EQµ[L1]k\f\f> C√\nki\n≤ε.\nIndeed if this holds using that regeneration levels are non-decreasing we deduce that |LNt−dt|>\nC√\ntwith probability at most εfor some C=C(ε), whereas union bound and Markov’s inequality\nshow that for all C′>0\nPO\u0014\nTNt+1−TNt> C′√\nt,\f\f\f\fNt−t\nEQµ[T1]\f\f\f\f≤C√\nt\u0015\n≤PO\u0014\n∃k:\f\f\f\fk−t\nEQµ[T1]\f\f\f\f≤C√\nt, Tk+1−Tk> C′√\nt\u0015\n≤2C√\ntmax u∈VEQu[T1]∨EO[T1]\nC′√\nt\n≤ε\nfor large enough C′=C′(ε), using also Lemma 6.4 to argue the expectations are O(1).\nLet us now prove the claim. We only prove the upper tail of the first inequality as the other\nbounds are established similarly. Let\nk=\u0016t\nEQµ[T1]+C√\nt\u0017\n.\nThe objective is to apply the Bienaymé Chebychev inequality with the variance bound for Markov\nchains (40), which is valid only when started at equilibrium. Let ε∈(0,1/2)andt0=t(Y)\nmix(ε)\nbe the mixing time of (Yl)l≥0, then decompose Tk=Z1+Z2with Z1=Pt0\ni=1Ti−Ti−1and\nZ2=Pk\ni=t0+1Ti−Ti−1.\nWe write further Z′\n2:=Z2−EQµ[Z2]. Stationarity implies EQµ[Z2] = (k−t0)EQµ[T1]hence\nPO[Nt> k] =P[Tk< t]\n=PO\u0002\nZ1+Z′\n2< t−(k−t0)EQµ[T1]\u0003\n≤PO[Z1+Z′\n2< z]\n49where\nz:= (t0+ 1)EQµ[T1]−CEQµ[T1]√\nt.\nBy Lemmas 6.4 and 6.5, t0and the expectation above can be bounded by constants independent\nofn. Thus for some constant C′>0,\nPO[Z1+Z′\n2< z]≤POh\n|Z1+Z′\n2|>2C′√\nti\n≤POh\nZ1> C′√\nti\n+POh\n|Z′\n2|> C′√\nti\n.\nFrom Lemma 6.5 and the fact that t0=O(1), we also deduce that EO[Z1] =O(1)so Markov’s\ninequality ensures the first term is smaller than εfor all tby taking the constant C′large enough.\nFor the second term, since t0=t(Y)\nmix(ε)\nPOh\n|Z′\n2|> C′√\nti\n=X\nyPO[Yt0=y]Qyh\f\fTk−t0−EQµ[Tk−t0]\f\f> C′√\nti\n≤ ∥Yt0−µ∥TV+Qµh\f\fTk−t0−EQµ[Tk−t0]\f\f> C′√\nti\n≤ε+Qµh\f\fTk−t0−EQµ[Tk−t0]\f\f> C′√\nti\n.\nUsing the Bienaymé-Chebychev inequality with (40), we deduce\nQµh\f\fTk−t0−EQµ[Tk−t0]\f\f> C′√\nti\n≤VarQµ(Tk−t0)\n(C′)2t≤C′′(k−t0) VarQµ[T1]\n(C′)2t,\nfor some constant C′′>0, as the regeneration chain has constant mixing time by Lemma 6.5. By\nLemma 6.4, the variance of T1is of constant order while k=O(t), hence the probability above\ncan be made smaller than εfor all tby taking C′large enough.\nFor the general case s≥0, we apply the previous arguments with a shifted version of the\nprocesses. For s≥0fixed, consider\nT(s)\nk:=TNs+k−s, L(s)\nk:=LNs+k−dLR(O,Xs)\nifk≥1andT(s)\n0:= 0, L(s)\n0:= 0. Then let\nN(s)\nt:= max {k≥0|T(s)\nk≤t}.\nNotice that\nL(s)\nN(s)\nt≤dLR(Xs,Xs+t)≤L(s)\nN(s)\nt+T(s)\nN(s)\nt+1−T(s)\nN(s)\nt.\nThese processes still satisfy the conclusions of Lemma 6.3 and have the same increments as the\nusual regeneration times. Thus the only thing to be careful when applying the above arguments\nis the law of the first regeneration time that now depends on s. Using the concentration (49) for\nNs, union bound and Lemma 6.4, we can see that for all m≥0,\nPh\nT(s)\n1> mi\n=X\nk≥0P[Ns=k, Tk+1−s > m ]\n≤P\u0002\f\fNs−s/EQµ[T1]\f\f> C√s\u0003\n+P\u0002\n∃k:\f\fk−s/EQµ[T1]\f\f≤C√s, Tk+1−Tk> m\u0003\n≤ε+ 2C√se−mα.\nTaking m=√\ntyields the stretched exponential term of (48). Then for higher regeneration times\n(T(s)\nk)k≥2, the above arguments apply.\n507.3 Concentration of the entropy\nThe concentration of the entropy in Proposition 3.2 is based on the convergence of the entropy for\nthe loop-erased chain in the quasi-tree, that is the convergence of −logP[ξ′\nk=ξk|ξ]/ktowards\na constant, the entropic rate of the chain. Such convergence is well-known in the context of groups\nor random walks on Galton-Watson trees, see [16, 17]. We will however not prove this result but\nestablish concentration directly for a notion of weights similar to those of (15). Of course, these\nare designed to mimick the law of the loop-erased chain, so the argument is similar. In fact the\nfirst step is to prove the convergence and concentration of the loop-erased chain when restricted\nto regeneration steps.\nLemma 7.4. Letξ′be an independent copy of the loop-erased chain ξ. There exists h′= Θ(1)\nsuch that\nlim\nk→∞−logP\u0002\nξ′\nLk=ξLk\f\fX\u0003\nk=h′. (50)\nFurthermore, for all ε >0, there exists C(ε)>0such that for all k, l≥1, for all types of O, η(O),\nPh\f\f−logP\u0002\nξ′\nLk=ξLk\f\fX\u0003\n−h′k\f\f> C(ε)√\nk\f\f\fO, η(O)i\n≤ε,\nPh\f\f\f−logPh\nξ′\nLk+l=ξLk+l\f\f\fX, ξ′\nLk=ξLki\n−h′l\f\f\f> C√\nl\f\f\fO, ηi\n≤ε.(51)\nProof.Again let us omit the conditionning by O, η(O)which does not affect the argument. Given\nk≥1, we write G(k) :=GXTk, to simplify notations. Given u∈Vandga possible realization of\nthe subquasi-tree GO, let\nQu,g:=P\u0002\n· | X 1/2=η(O), τO=∞, ι(O) =u,GO=g\u0003\n.\nFinally let ξ′(k)denote a loop-erased chain on G(k−1), started at XTk−1, independent of X.\nHaving ξ′\nLk=ξLkimplies ξLm∈ξ′for all m≤k. Thus for k≥2\nP[ξLk∈ξ′| X] =Eh\n1(ξ′\nLk−1=ξLk−1)P\u0002\nξ′\nLk=ξLk\f\fX,(ξ′\ni)i≤Lk−1\u0003\f\f\fXi\n=Eh\n1(ξ′\nLk−1=ξLk−1)QYk−1,G(k−1)\u0002\nξLk∈ξ′(k)|(Xt)t≥Tk−1\u0003\f\f\fXi\n=Ph\nξ′\nLk−1=ξLk−1\f\f\fXi\nQYk−1,G(k−1)\u0002\nξLk∈ξ′(k)|(Xt)t≥Tk−1\u0003\n.\nLetting Z1:=−logP\u0002\nξ′\nL1=ξL1\f\fX\u0003\nand\nZi:=−logQYi−1,G(i−1)\u0002\nξLi∈ξ′(i)|(Xt)t≥Ti−1\u0003\nfori≥2, we just proved that\n−logP\u0002\nξ′\nLk=ξLk\f\fX\u0003\n=kX\ni=1Zi\nand\n−logPh\nξ′\nLk+l=ξLk+l\f\f\fX, ξ′\nLki\n=k+lX\ni=k+1Zi.\nUnderQµ, the sequence (Yk)k≥1is stationary. Lemma 6.3 then shows that (Zi)i≥2is stationary\nas well. Provided Z1<∞a.s.,EO[Z1]<∞andEQµ[|Z1|]<∞, which is proved afterwards,\nBirkhoff’s ergodic theorem implies the law of large numbers (50).\n51Let us now show that for all l≥1, there exists a constant Cl≥0such that\nEOh\n|Z1|li\n< C l\nand for all i≥2andu∈V\nEQuh\n|Zi|li\n≤Cl. (52)\nThis is sufficient to justify that Z1<∞a.s.,EQµ|Zi|<∞and will be used to prove (51). We\nonly treat the case i≥2as the case i= 1is similar. Let l≥1andu∈V. Let us identify here a\nlong-range edge (η(x), x)with its center vertex x. By Lemma 6.3\nEQuh\n|Zi|li\n=EQu\"X\nx∈VQu,G[XTi=x]\u0000\n−logQYi−1,G(i−1)[x∈ξ′(i)]\u0001l#\n=X\nv∈VQu[Yi−1=v]EQv\"X\nx∈VQv,Gh\nX′\nT′\n1=xi\n(−logQv,G[x∈ξ′])l#\nwhere X′is an independent copy of Xunder the law Qv,G,ξ′is its loop-erased trace and T′\n1its\nfirst regeneration time. Thus it suffices to prove the upper bound\nmax\nu∈VEQu\"X\nx∈VQu,G[XT1=x] (−logQu,G[x∈ξ])l#\n≤Cl.\nFixu∈V. For all k≥1, ifdP(η(O), x) =k, the chain goes from η(O)toxwith probability at\nleast δk, after which it escapes to infinity in Gxwith probability qEsc(x), hence\nX\nx∈VQu,G[XT1=x] (−logQu,G[x∈ξ])l≤X\nk≥1Qu,G[dP(η(O),XT1) =k]kl(−logδ)l\n+X\nx∈VQu,G[XT1=x] (−logqEsc(x))l\nAveraging on the environment yields\nEQu\"X\nx∈VQu,G[XT1=x] (−logQu,G[x∈ξ])l#\n≤(−logδ)lEQu\u0002\ndP(η(O),XT1)l\u0003\n+EQu\u0002\n(−logqEsc(XT1))l\u0003\nNow for any k≥1,\nQu[dP(η(O),XT1)≥k]≤Qu[T1≥k]\nwhich is stretched exponential by Lemma 6.4, hence EQu\u0002\ndP(η(O),XT1)l\u0003\n=O(1). On the other\nhand\nEQu\u0002\n(−logqEsc(XT1))l\u0003\n=X\nv∈VQu[Y1=v]EQv\u0002\n(−logqEsc(η(O)))l\u0003\n.\nBy Proposition 5.1 and Remark 6.2, EQv\u0002\n(−logqEsc(η(O)))l\u0003\n=O(1)for any v∈V, hence\nEu\u0002\n(−logqEsc(XT1))l\u0003\n=O(1)as well and we deduce (52) for some constant Cl.\nTo establish (51), we argue as in the end of the proof of Proposition 7.1 for the drift. The\nbound is eventually proved using the Bienaymé-Chebychev inequality. Since Ymixes in constant\ntime t0by Lemma 6.5, separating the sumPk\ni=1Ziinto two sums corresponding respectively\nto regeneration times before or after t0shows that it suffices to establish the variance bound\n52VarQµhPk\ni=2Zii\n=O(k)under the stationary measure. Let us modifiy the definition of Z1,\nsetting Z1:=−logQι(O),GO[ξL1∈ξ′| X]so that the whole sequence (Zi)i≥1is stationary. It\nthen suffices to prove that for all k≥1,CovQµ(Z1, Zk)is stretched exponential in k.\nLetk > 1. Let τ′\nk−1be the first time the chain X′reaches the Lk−1-th level, and con-\nsider the loop-erased trace ξ′(1, k)obtained from observing X′up to time τ′\nk−1. Define W1:=\nexp(−Z1), W1k:=Qι(O),GO[ξL1∈ξ′(1, k)| X],Z1,k:=−logW1k. Bytheinequality |logx−logy| ≤\n|x−y|/(x∧y),\n|Z1−Z1,k| ≤|W1−W1k|\nW1∧W1k\nObserve that if ξL1is exclusively in one of the two paths ξ′(1, k),ξ′then the chain X′backtracks\nfrom the Lk−1-th level to the (L1−1)-th. Letting A′denote this event, we thus have\n|W1−W1k| ≤Qι(O),GO[A′| X].\nLetAk(q0, α)be the set of quasi-trees for which every length kpath from the root contains\na proportion at least α > 0of vertices xsuch that qEsc(x)≥q0>0. For some constants\nq0, α, ε∈(0,1)andC1>0to determine, consider the events\nE0:={GO∈ Ak(q0, α)} ∩ {qEsc(ξL1)≥εk}\nE1={dP(O,XT1)≤k/C 1}\nIfGO∈ A k(q0, α), Lemma 7.2 implies Qι(O),GO[A′| X]is exponentially small in k. On E1, the\nchain X′goes from OtoξL1with probability at least δk/C 1, after which ξL1∈ξ′=ξ′(1, k)if the\nchain X′escapes to infinity. Thus on the event E0∩E1,\nW1∧W1k≥δk/C 1εk,\nhence choosing the constant C1large enough and εclose enough to 1yields eventually\n|Z1−Z1,k| 1E0∩E1≤e−c1k\nfor some constant c1>0. Next we claim that given this choice of C1andε, there is an appropriate\nchoice of the remaining parameters that ensures\nQµ[Ec\n0]≤e−c2k,Qµ[Ec\n1]≤e−c2kβ\nfor some c2>0andβ∈(0,1]. By Remark 6.2 and Proposition 5.2 there exist q0, α > 0such that\nGO/∈ Ak(q0, α)has exponentially small probability, whereas Proposition 5.1 implies qEsc(ξL1)< εk\nwithdoublyexponentiallysmallprobability. For E1observe dP(O,XT1)> k/C 1implies T1> k/C 1\nwhich has stretched exponentially small probability by Lemma 6.4. Then by the moment bounds\n(52) for some c3>0,\nCovQµ(Z1, Z2k) =EQµ\u0002\n(Z1−EQµ[Z1])(Z2k−EQµ[Z2k]) 1E1\u0003\n+O\u0010\ne−c3kβ\u0011\n=EQµ[(Z1−Z1,k)Z2k 1E1]−EQµ[(Z1−Z1,k) 1E1]EQµ[Z2k]\n+EQµ\u0002\n(Z1,k−EQµ[Z1])(Z2k−EQµ[Z2k]) 1E1\u0003\n+O\u0010\ne−c3kβ\u0011\n=EQµ[(Z1−Z1,k)Z2k 1E0∩E1]−EQµ[(Z1−Z1,k) 1E0∩E1]EQµ[Z2k]\n+EQµ\u0002\n(Z1,k−EQµ[Z1])(Z2k−EQµ[Z2k]) 1E1\u0003\n+O\u0010\ne−c3kβ\u0011\n=EQµ\u0002\n(Z1,k−EQµ[Z1])(Z2k−EQµ[Z2k]) 1E1\u0003\n+O\u0010\ne−c3kβ\u0011\n.\n53Note that by construction Z1,k 1E1is measurable with respect to G∖G(k)and(Xt)t≤Tk−1, whereas\nZ2kis measurable with respect to G(2k−1)and(Xt)T2k−1≤tonly. Thus by conditionning on\n{Yk−1, Y2k−1}Markov’s property implies\nEQµ\u0002\n(Z1,k−EQµ[Z1])(Z2k−EQµ[Z2k]) 1E1\u0003\n=EQµ\u0002\nEQµ\u0002\n(Z1,k−EQµ[Z1]) 1E1\f\fYk−1\u0003\nEQµ\u0002\nZ2k−EQµ[Z2k]\f\fY2k−1\u0003\u0003\nThe right-hand side can be written as EQµ[f(Yk−1)g(Y2k−1)]for some functions f, g. From the\nproof of Proposition 6.3, this expectation is exponentially small in k, provided fandghave finite\nmoments. This is the direct consequence of (52) for g, while for fwe need moment bounds on\nZ1k. These can be established with the same arguments as above.\nTo relate the previous concentration with the weights (15), we define similar weights in the\nquasi-tree. Let τldenote here the first time tsuch that dLR(X0,Xt) =l. For all long-range edge\ne∈ G, write Gefor the subquasi-tree at any endpoint of e(they give the same quasi-tree). Given\nR, L≥0,x∈ Vand a long-range edge eat long-range distance 0from x\nwx,R,L(e) :=Pxh\nXτL∈ Ge, τL< τ(R)\nSRi\nwR,L(e|x) :=Ph\nXτL∈ Ge, τL< τ(R)\nSR\f\f\fX1/2=x, τL< τη(x)i (53)\nThen if e= (ei)k\ni=1is a long-range non-backtracking path starting from BLR(x,0), set\nwx,R,L(e) :=wx,R,L(ξ1)kY\ni=2wR,L(ξi|ξ+\ni−1)\nwhere the product if taken equal to 1if empty.\nIn the proof of Lemma 7.4, we introduced the measure\nQu,g:=P\u0002\n· | X 1/2=η(O), ι(O) =u, τO=∞,GO=g\u0003\n.\nwhere u∈Vandgis a possible realization of the subquasi-tree GO. We use a notation which may\nbe reminiscent of (18) as these two measures are very similar, although note that here uis not\nthe type of the starting state of the chain but its long-range neighbour. There should be no risk\nof confusion as the measure of (18) will not be used until Section 9. Using the same arguments\nas for the lemma, we can see that\nPx[ξ1=e1, . . . , ξ k=ek] =Px[ξ1=e1]kY\ni=2Qι(e+\ni−1),Gei−1[ξ1=ei]. (54)\nThe following Lemma establishes thus a bound on individual weights.\nLemma 7.5. There exist constants C, C 0, C1, C2>0such that for all R, L > 0the following\nholds, conditional on the types of O, η(O):\n(i) with probability at least 1−e−C(R∧L), for all long-range edge esuch that dSR(O, e−)< R,\n|logwO,R,L (e)−logPO[ξ1=e]| ≤C0e−C1L+C2R,\n(ii) for all x∈ V, with probability at least 1−e−C(R∧L)conditional on G∖Gx, for all long-range\nedge eofGxsuch that dSR(x, e−)< R\n\f\flogwR,L(e|x)−logQι(e+),Ge[ξ1=e]\f\f≤C0e−C1L+C2R.\n54Proof.We prove the first bound in detail. As usual we forget about the conditionning by the\ntypes of O, η(O)which changes little in practice. Given k≥1, letAk=Ak(q0, α)be the set of\nquasi-trees for which every path of length kstarting from the root contains a proportion at least\nαof vertices xsuch that qEsc(x)≥q0. Let\nE0:={GO∈ AR∩ AL} ∩ {∀ x∈BSR(O, R) :qEsc(x)≥c0e−c1R}.\nWe claim the event Ec\n0occurs with exponentially small probability in R∧Lfor an appropriate\nchoice of parameters. First there exist q0, α > 0such that G/∈ AR∩ ALwith exponentially small\nprobability by Proposition 5.2. Next, Proposition 5.1 implies there exists c∈(0,1)such that for\na fixed x∈BSR(O, R),qEsc(x)< cδ4Rwith probability at most c2R. However there are O(∆R)\nvertices in this ball, hence the claim by union bound.\nNotice then that to have one of the two events e∈ξorXτL∈ Gerealized exclusively, the chain\nneeds to backtrack from level Lto level 0. On E0, this occurs with probability exponentially small\ninLby Lemma 7.2. Thus on E0\n|wO,R,L (e)−PO[ξ1=e]| ≤POh\nτ(R)\nSR∧τ(L)\nNB<∞i\n≤e−C′(R∧L)\nfor some constant C′>0. On the other hand, on E0\nwO,R,L (e)∧PO[ξ1=e]≥δRq0\nas the right-hand side is a lower bound on the probability to go from Otoeand then escape to\ninfinity, which forces both e∈ξandXτL∈ Ge. Using |logx−logy| ≤ |x−y|/(x∧y), we deduce\nthat on E0,\n|logwO,R,L (e)−logPO[ξ1=e]| ≤C0e−C1L+C2R\nfor some constants C0, C1, C2>0.\nThe second bound is proved similarly conditional on G∖Gx. The main difference is the addi-\ntional conditionning by either τL< τη(x)orτη(x)=∞. Since the differences implies backtracking,\nthese can be controlled using the same arguments as above and Proposition 5.1.\nLemma 7.6. There exist constants C, C 1, . . . , C 3>0andβ≥1such that the following holds.\nFor all ε >0there exist C0=C0(ε), Ch=Ch(ε)such that for all s, t≥0, for all R, L > 0, with\nprobability at least 1−ε−2(s+t)e−C(R∧L)conditional on the types of Oandη(O),\n|−logwXs,R,L(ξ(Xs···X s+t))−ht| ≤Ch√\nt+C0t e−C1L+C2R+C3RLβ+1.(55)\nProof.FixR, Lfor the rest of the proof. For notational simplicity, we drop subscripts R, Lfrom\nthe weights and omit writing the conditionning by ι(O), ι(η(O))but the probabilistic statements\nbelow should be interpreted conditional on these. Recall G(R)is the quasi-tree truncated at the\nR-boundary of small-range components. Note that that weights are positive only for edges in\nG(R).\nGiven t≥0, let\nNt:= max {k≥0|Tk≤t}.\n55We first argue there exists a constant C3>0such that for all s, t≥0ifX0···X s+tis included in\nG(R)then\n\f\f\flogwXs(ξ(Xs···X s+t))−logw(ξLNs+2···ξLNs+t|ξLNs+1)\f\f\f≤C0RL(TNs+2−TNs\n+TNs+t+1−TNs+t\u0001\n.(56)\nNote first that ξLNs+1···ξLNs+tis necessarily part of the path ξ(Xs···X s+t). Since weights are\nbelow one, we can easily lower bound\n−logwXs(ξ(Xs···X s+t))≥ −logw(ξLNs+2···ξLNs+t|ξLNs+1).\nOn the other hand the path ξ(Xs···X s+t)contains at most TNs+2−s≤TNs+2−TNsedges until\nit reaches ξLNs+2and similarly it contains at most TNs+t+1−TNs+tafter it leaves ξLNs+t. Then we\nbound the weights of these edges. If x∈ Vandeis a long-range edge such that dSR(x, e−)< R,\nδRlower bounds the probability the chain goes from xtoeand leaves by this edge. As small-range\ncomponents have size at least 2(H3), δRLlower bounds the probability that the chain goes from\nxtoe+and reaches long-range distance Lbefore coming back to e−. Consequently, for some\nconstant C3>0, we can always bound −logwx(e)≤C3RLfor all x, einG(R), for any realization\nof the quasi-tree. We deduce that\n−logwXs(ξ(Xs···X s+t)) + log wXs(ξLNs+2···ξLNs+t)≤C3RL(TNs+2−TNs\n+TNs+t+1−TNs+t\u0001\n.\nNext fix ε∈(0,1)and let s, t≥0. LetAbe the set of quasi-trees for which every long-range\nedge esuch that dSR(O, e−)< Rsatisfies the bounds of Lemma 7.5 for some C0, C1, C2>0.\nConsider the following events:\nE0:={G ∈ A} ∩\\\ns′≤s+t{GXs′∈ A} ,\nE1:={τ(R)\nSR> s+t}\nE2:={\f\fLNs+t−LNs−dt\f\f≤CLR√\nt}\nwith CLR=CLR(ε)>0to determine, and write E:=E0∩E1∩E2. By Lemmas 7.5 and 7.1,\nC0, C1, C2can be chosen so that PO[Ec\n0]≤(s+t)e−C4(R∧L)for some C4>0. By Lemma 7.3\nthere exists C5>0such that PO[Ec\n1]≤(s+t)e−C5R. Finally from the proof of Proposition 7.1,\nCLR(ε)can be taken so that PO[Ec\n2]≤ε. Hence PO[Ec]≤ε+ (s+t)e−C(R∧L), for some constant\nC >0.\nThen let l+:=⌈dt+CLR√\nt⌉. On the event E, using (54)\n\f\f\flogw(ξLNs+2···ξLNs+t|ξLNs+1)−logPh\nξLNs+2···ξLNs+t∈ξ′\f\f\fX, ξ′\nLNs+1=ξLNs+1i\f\f\f\n≤l+e−C1L+C2R.\nLetting h′be the constant of Lemma 7.4, we claim then that for h=h′/EQµ[T1]and some constant\nCh=Ch(a, ε)\n\f\f\f−logPh\nξLNs+2···ξLNs+t∈ξ′\f\f\fX, ξ′\nLNs+1=ξLNs+1i\n−ht\f\f\f≤Ch√\nt\n56with probability at least ε. This can be done using the same arguments as in the proof of\nProposition 7.1, combining the fluctations for Ns+t(49) established in the said proof with Lemma\n7.4.\nFinally, we are left with bounding the right hand side of (56). Using again the fluctations for\nNs+t(49) with the stretched exponential tail of regeneration times, there exists β≥1andC6>0\nsuch that\nP\u0002\nTNs+1−TNs+TNs+t+1−TNs+t> Lβ\f\fO, η(O)\u0003\n≤ε+√\ns+te−C6L.\nAll in all, up to changing the value of Cwe have thus proved that with probability at least\n1−2ε−(s+t)e−C(R∧L),\n|logwXs(ξ(Xs···X s+t))−ht| ≤Ch√\nt+C0l+e−C1L+C2R+C3RLβ+1.\nSince l+=Oε(t)this proves the result.\n8 First steps towards nice paths\nWe now come back to the finite setting. The goal of this section is to prove Lemmas 3.1, 3.2 and\nProposition 3.1, which will be used in the next section to justify the chain is likely to follow a nice\ntrajectory.\n8.1 The case of typical starting states\nWe start with the case of typical starting states, for which there is little left to prove.\nProof of (i) in Lemma 3.2. This is essentially a matter of checking definitions. For fixed x∈V,\nthe choice of s, t=O(logn)a, a≥1, Lemma 2.1 ensures the finite chain Xand the chain Xon\nthe quasi-tree can be coupled until time s+twith annealed probability 1−o(n−ε)for some ε >0.\nProvided the coupling succeeds, the property of backtracking or hitting the boundary of a small-\nrange ball coincides for both chains, while the regeneration times of Xobviously give regeneration\nedges for the finite chain. Thus from Lemma 2.1 and Lemma 7.3 we deduce\nPx[(Xs···Xs+t··· ∈ Γ(R, L, M ))]≤(s+t)e−C(R∧L∧Mα)+o(n−ε) (57)\nSince s, t= (log n)a, for any b >0the choice of R, L =O(log log n)with large enough implicit\nconstants depending on a, bandM= (log log n)2/αmakes the right hand side o((log n)−b). The\nfirst moment argument (2) implies (i) in Lemma 3.2.\nWe can repeat the same kind of arguments for point (i) in Proposition 3.1 however we obtain\nfor the time being a slightly weaker statement that the one expected. Because of the dependance\ninεin the annealed probability of (48) and (55), applying the first moment arguments will not\nestablish the concentration property with high probability but only 1−ε−o(1). To obtain a\nhigh probability result we need to apply a higher order Markov’s inequality, which is also the key\nargument to obtain the uniform statements.\n578.2 Bootstrapping annealed bounds with parallel chains\nArguments used so far above establish errors on annealed error in o(1), nay in ε+o(1). Following\nideas from [6, 7], one strategy to improve these error bounds consists in using higher order mo-\nments in Markov’s inequality, which leads to an argument of \"parallelizing chains\" on the same\nenvironment. Namely for all ε >0andk≥1, for any trajectorial event A\n∀x∈V:P[Px[A]> ε]≤Px[A]k\nεk=ε−kPx\"k\\\ni=1{X(i)∈A}#\n(58)\nand\nP\u0014\nmax\nx∈VPx[A]> ε\u0015\n≤1\nεkE\"X\nx∈VPx[A]k#\n≤n\nεkmax\nx∈VPx\"k\\\ni=1{X(i)∈A}#\n, (59)\nwhere X(1), . . . , X(k)arekversions of the chain Xgenerated independently conditional on the\nsame environment. These ktrajectories can be generated sequentially with the environment,\nsampling the environment only when exploring parts of the environment not already visited by\nthe previous chains. The idea will then be to argue that the chains are unlikely to follow each\nother, so they end up exploring disjoint parts of the environments. If the event Adepends only on\nwhat happens after the trajectories separate from each other (hence the consideration of shifted\ntrajectories), we can then expect the annealed probability to decouple and become P[A]k. If the\nevent Ahas probability below f(ε)for some function f, this can be sufficient to beat the ε−kor\nε−knfactor.\nAs an example of application, we can start proving point (i) in Proposition 3.1, which also\nillustrates some of the arguments used in the sequel.\nProof of (i) in Proposition 3.1. Letε > 0,t=O(logn)andx∈V. For CLR=CLR(ε)and\nCh=Ch(ε)>0to determine, consider the set of paths\nΓEnt:=(\np\f\f\f\f\f||ξ(p)| −d|p|| ≤CLRp\n|p|\n|−logwp0,R,L(ξ(p))−h|p|| ≤Chp\n|p|)\n(60)\nwhich can be paths either in G∗orG. The goal is thus to show Px[(X0)···Xt/∈ΓEnt] =o(1).\nConsider kversions X(1), . . . , X(k)of the chain Xgenerated independently conditional on the\nsame environment, with k≫1, k=O(log log n). Then using the same coupling as in Section 2.7\nthese can be coupled with kversions X(1), . . . ,X(k)of the chain X. For j∈[k]letG(j, t)denote\nthe union of the first jtrajectories up to time t, along with their long-range neighbourhoods up\nto depth L. Compared to the case of a single chain, the number of visited vertices gets multiplied\nbykat most, so the proof of Lemma 2.1 shows the chains X(1), . . . , X(k)can be coupled with\nhigh probability, along with their long-range neighbourhood up to length L=O(log log n), up to\ntime t=O(logn), to coincide with G(k, s+t). Now the definitions of the long-range distance\nand of weights ((15) and (53)), which are measurable with respect to this neighbourhood, were\nexactly made to coincide in the finite and quasi-tree settings. Thus if the coupling succeeds\n58(X0···Xt)∈ΓEntif and only in X0···X t∈ΓEnt. Letting for j∈[k]:\nBj:=j\\\ni=1n\nX(i)\n0···X(i)\nt/∈ΓEnto\n(58) and the coupling imply\nPx[X0···Xt/∈ΓEnt]≤ε−kP[Bk|ι(O) =x] +o(1),\nso it suffices to show PO[Bk|ι(O) =x] =o(εk). We consider in fact a slightly modified event.\nLetl:= (log log n)3and consider the set of quasi-trees Al=Al(α, q0)for which every length l\npath from the root contains a proportion at least αof vertices for which the probability to escape\nto infinity outside pis lower bounded by q0. Recalling Proposition 5.2 and Remark 5.2, Lemma 7.1\nimplies the existence of constants q0, α > 0andC > 0such that all vertices in G(j, t)have their\nsubquasi-tree in Alwith probability at least 1−te−Cl. Since t=O(logn)andl= (log log n)3\nthis probability is 1−o(1). Thus we can suppose that the event Bjalso contains the event that\nall vertices of G(j, t)have their subquasi-tree in Al. Secondly, consider the l-th regeneration time\nT(i)\nlof the i-th chain. By the stretched exponential tails of regeneration times 6.4 and Chernoff’s\nbound, there exist constants α∈(0,1]andC >0such that for all i≤k\nPOh\nT(i)\nl> t0\f\f\fι(O) =xi\n≤e−tα\n0+Cl\nThus considering t0:= (log log n)β, for β >3/αlarge enough makes the right hand side o(k−1).\nConsequently by union bound we obtain that POh\n∃i≤k:T(i)\nl> t0i\n=o(1), so we can also incor-\nporate to the events Bjthe event that T(i)\nl≤t0fori≤j.\nNext we claim that PO[B1|ι(O) =x]≤ε/2+o(1)andPO[Bj|ι(O) =x, B j−1]≤ε/2+o(1)\nfor all j∈[2, k]. Noting that\nPO[Bk|ι(O) =x] =PO[B1|ι(O) =x]kY\nj=2PO[Bj|ι(O) =x, B j−1],\nthe claims imply that P[Bk|ι(O) =x] = (ε/2 +o(1))k=o(εk)as desired.\nLet us prove the claims. The first claim is the consequence of Proposition 7.1 and Lemma\n7.6, provided the constants CLR, Chare chosen adequatly. To prove the second claim, let j≥2\nand suppose Bj−1holds. We argue that X(j)has quenched probability at least 1−o(1)to exit\nG(j−1, t)for the last time by time t0. Consider then the loop-erased chain (ξ(j)\nm)mobtained from\nX(j). Thanks to the first event we added to Bj−1, for any long-range path ζof length lcontained\ninG(j−1, t),\nPOh\nξ(j)\nl=ζl\f\f\fBj−1i\n≤e−Cl\nfor some constant C > 0. Since G(j−1, t)is made of the long-range neighbourhood of the first\nj−1trajectories, it contains at most kt∆R(L+1)long-range paths of length l. Thus by union\nbound\nPOh\nξ(j)\nl∈ G(j−1, s+t)\f\f\fBj−1i\n≤kt∆R(L+1)e−Cl=e−C(log log n)3+O(log log n)2=o(1).\nOn the other hand, the second event added to Bjimplies that T(j)\nl≤t0so the chain must in\nparticular have made the first lsteps of its loop-erased trace before time t0. Combined with what\n59precedes we deduce that conditional on Bj−1, the chain X(j)leaves the graph G(j−1, t)before\ntime t0with probability 1−o(1).\nLetL(j)denote now this last exit time. Conditional on L(j)andXL(j)=u, the quasi-tree\nthat contains the subsequent trajectory needs then to be generated according to the measure Qu\n(41). Using Remark 6.2 upper bounds under this law only differ by a constant factor from upper\nbounds with the usual law. We can thus apply Proposition 7.1 and Lemma 7.6 conditional on Lj\nto obtain that\nPOh\n(X(j)\nLj···X(j)\nLj+t)/∈ΓEnt\f\f\fBj−1, ι(O) =xi\n≤ε/2 +o(1).\nNow by what precedes dLR(X(j)\n0,X(j)\nLj)≤lwith probability 1−o(1)conditional on Bj−1, hence\ndLR(X(j)\n0,X(j)\nt)∈[dt−CLR√\nt,dt−CLR√\nt+l]\nand\n−logwX0,R,L(ξ(X(j)\n0···X(j)\nt))∈[ht−Ch√\nt, ht+Ch√\nt+lwmax],\nwhere wmaxdenotes an upper bound on all the weights contained in G(j−1, t). For any realization\nof the quasi-tree we can bound wmax≤O(RL)as in the proof of Lemma 7.6. All in all, since\nlwmax=o(√\nt), this ensures (X(j)\n0,X(j)\nt)∈ΓEntwith probability 1−ε/2−o(1), proving the\nclaim.\n8.3 From o(1)too(1/n)bounds\n8.3.1 Quasi-tree with a cycle\nThe rest of this section will now be devoted to the proof of the uniform, worst-case statements.\nThe main ideas are the same as those used above. From (59), we only need to push them yet\na little further to make error bounds on annealed probabilities o(1/n). From the proof of (i) in\nProposition 3.1 two arguments require a modification: the coupling with the a quasi-tree and the\nuse of escape probabilities.\nLet us focus on the first point. It seems quite obvious that the coupling with quasi-trees cannot\nbeimprovedbeyondtoa o(1/n)errorassomeverticesdonothaveaquasi-treelikeneighbourhood.\nHowever from the proof of Lemma 2.1 this error can be achieved allowing just one cycle. We are\nthus led to consider a modified quasi-tree model which contains at most one long-range cycle.\nLet(ui)l\ni=0,(vi)l\ni=0, l≥0be two sequences of vertices in Vsuch that ui+1∈B+\nSR(vi,∞)for all\ni∈[0, l−1]andB+\nSR(vl, R)∩B−\nSR(u0, R)̸=∅. For every iand vertex z∈B+\nSR(ui,∞)∖{ui, vi},\nadd an edge (z, η(z))and grow a one-sided quasi-tree rooted at η(z). We call the oriented graph\nG= (V,E)obtained a quasi-tree with a cycle ((ui, vi))l\ni=0. As a usual quasi-tree, it is given by\nmaps ι, ηsuch that ιidentifies vertices of Vwith vertices in V, while η:V → Vis an involution.\nHere ηis obtained by the corresponding maps in the quasi-trees outside the cycle whereas for the\ncycle we set η(ui) :=vi+1for all i= 0, . . . l−1. A root Ocan be chosen, which does not to be on\nthe cycle. The definition of the Markov chain X(8) extends directly to this setting.\nThe coupling of Section 2.7 gives a natural way to couple Xon a random realization of Gwith\nthe Markov chain X: the rejection scheme is used until the first occurence of a cycle, after which\ncycles are ignored in the construction of G. The stochastic comparison in the proof of Lemma 2.1\n60still holds, but using now that\nP[Z≥2]≤m4∆2R\nn2(61)\nifZis a binomial Bin(m, m ∆R/n), we deduce that for t=no(1), the chains XandXcan be\ncoupled so that the chains coincide and BLR(Xs, L)andBLR(Xs, L)are isomorphic for all s≤t\nwith probability 1−o(1/n).\nMore generally, let k≥1andX(1), . . . , X(k), resp. X(1), . . . ,X(k),kversions of the chain X,\nresp. Xgenerated independently conditional on the same environment. Letting G∗(k, t),G(k, t)\nbe the environments explored by these trajectories up to time talong with their L-long-range\nneighbourhoods, the same arguments as above implies that\nP\"\nG∗(k, t)andG(k, t)are isomorphic\n∀i∈[1, k],∀s≤t:X(i)\ns=ι(X(i)\ns)#\n= 1−o((kt)4/n7/4) =o(εk/n) (62)\nif for instance k=⌊logn/2 logε−1⌋.\n8.3.2 An essentially uniform bound on escape probabilities\nThe second argument that needs to be adapted is the use of escape probabilities, which can be\nexpected to of constant order only for typical states. Consider Gto be a random realization of a\nquasi-tree with a cycle as described above and Xthe associated Markov chain. Conditional on the\ncycle, the quasi-trees that are added to it are generated according to the model studied in Sections\n5 - 7, therefore the asymptotic analysis of the chain Xdirectly extends to that case, conditional\non the cycle. The chain ultimately leaves the cycle, after which it stays in a genuine quasi-tree.\nGiven a vertex xwhich is not on the cycle, let qEsc(x)denote the probability to escape in the\ncorresponding quasi-tree. If xis on the cycle, there is no quasi-tree at x, so let qEsc(x)denote the\nprobablity of escaping through one of the vertices that are in the same component as x. Using\nthat components of V1have size at least 3(H3), these escape probabilities can be bounded exactly\nas in Section 5, up to a change of constants, conditional on the cycle. Thus from Proposition 5.1,\nthere exists β≥1such that for all u∈ Vnot on the cycle,\nP\u0014\nqEsc(u)<1\n(logn)β\f\f\f\fG∖Gu\u0015\n=O(1/n2). (63)\nCombined with Lemma 7.1, we see that on a time scale t=O((log n)a), a≥1, the probability\nthat the chain Xmeets a vertex which does not satisfy the lower bound on escape probability\nabove is still o(1/n). This makes the chain sufficiently unlikely to meet such vertices that the chain\nXcannot reach such quasi-tree-like neighbourhoods either, thanks to the the coupling described\nabove. Informally, this allows us to proceed as if escape probabilities were everywhere lower\nbounded by 1/(logn)βand reason conditional on the environment, as we did in the reversible\ncase.\n8.4 Nice paths from arbitrary states\nA first application of these uniform lower bounds is the following.\nProof of Lemma 3.1. We prove the first point. Let L=CLlog log nfor a fixed constant CL>0,\nt≪n1/16andx∈V. By Remark 3.1 it suffices to consider the case of s=C(logn)afor some\n61a >0to determine. Using the coupling described above (with only one chain here, ie k= 1) we\ncan couple the trajectory of Xand its whole Llong-range neighbourhood with that of X. By (62)\nthe coupling succeeds up to time s+twith probability 1−o(1/n), hence for u∈V\nPu[∃s′∈[s, s+t] :BLR(Xs′, L)is not quasi-tree-like ]\n=PO[∃s′∈[s, s+t] :Xs′is at distance less than Lfrom the cycle |ι(O) =u] +o(1/n).\nNow let β≥1be as in (63) and consider the set\nEsc := {x∈ G | qEsc(x)≥(logn)−β}.\nWrite τEsccfor the hitting time of the complement set of Esc. The number of vertices in the L\nlong-range neighbourhood of the trajectory being bounded by O(s+t)∆R(L+1)=O(n1/16), union\nbound and (63) imply\nPO[τEscc≤s+t|ι(O) =u] =o(1/n).\nWe are being a bit sketchy here, but the argument can be formalized using the Ulam labelling.\nOn the other hand, when the chain is at a vertex of Escits long-range distance to the cycle has\na quenched probability at least (logn)−βto increase by 1indefinitely. Thus if we consider the\nsequence of times (Sk)k≥1at which this happens, for all x∈ V, s≥0,\nPx[τEscc> s+t, S1> s]≤e−s/(logn)β.\nConsequently by union bound and Strong Markov’s property\nPx[∃s′∈[s, s+t] :Xs′is at distance less than Lfrom the cycle , τEscc> s+t]\n≤Px[SL> s, τ Escc>s+t]≤Le−s/L(logn)β.\nSince s= (log n)a, choosing a > β +1makes these two bounds o(1/n). The first moment argument\n(59) concludes the proof of the first point in the general case.\nFinally point (ii) can be proved with similar arguments.\nIt remains to prove (ii) in Lemma 3.2 and Proposition 3.1. The argument is similar and uses\nthe same ideas as above.\nProof of (ii) in Lemma 3.2 and Proposition 3.1. Consider the concentration of the drift and en-\ntropy. Let x∈V,ε >0andk:=⌊logn/2 logε−1⌋. As for point (i) we combine the higher Markov\ninequalities (59) and the coupling with kchains on a quasi-tree with a cycle. As a cycle is allowed\nthe error due to the coupling is o(1/n)by (62) so we need only focus on the infinite setting: letting\nBj:=j\\\ni=1{X(i)\ns···X(i)\ns+t/∈ΓEnt}.\nforj∈[k], with ΓEntas in (60) it suffices to prove PO[Bk|ι(O) =x] =o(εk/n). Then we claim\nthatPO[B1|ι(O) =x]≤ε3/2+o(1)andPO[Bj|ι(O) =u, B j−1]≤ε3/2+o(1)for all j∈[2, k].\nWith our choice of kthis will imply PO[Bk|ι(O) =x] = (ε3/2 +o(1))k=o(εk/n)as needed.\nAs we mentionned already, results proved for genuine quasi-trees extend to the case where an\nadditional cycle is present. In particular, the first regeneration time and level, defined to occur\n62outsidethecycle, havestretchedexponentialorexponentialtailrespectively, conditionalonthecy-\ncle, while the remaining regenerations occurs on genuine quasi-trees. This is sufficient to establish\nthat Proposition 7.1 and Lemma 7.6 still hold in this case, implying that P[B1|ι(O) =u]≤ε3/2\nfor a good choice of Ch, CLR(ε).\nFor the second claim, we argue as for (i) by showing the chains are unlikely follow each other.\nLetβ≥1be as in (63), consider the set\nEsc := {x∈ G | qEsc(x)≥(logn)−β}.\nand let τEsccbe the first exit time by any of the chains. As we argued above in the proof of\nLemma 3.1, PO[τEscc≤s+t|ι(O) =u] =o(1/n). On the other hand, while on this set the\nchains have a quenched probability at least (logn)βto have their long-range distance increased by\n1permanently, hence for any j∈[k], letting (S(j)\nk)kbe the sequence of times at which this occurs\nfor the j-th chain,\nPOh\nS(j)\nl> s, τ Escc> s+ti\n≤le−C1s/l(logn)β(64)\nfor some constant C1>0. Taking a >1 + 2βins= (log n)sandl= (log n)β(log log n)3the right\nhand side is o(1).\nConsider now the loop-erased chain (ξ(j)\nm)mobtained from X(j). If all vertices visited by the\nchain are in Esc, and using that components of V1have size at least 3, the probability to follow a\ngiven path is exponentially small in the length. Since G(j−1, s+t)contains at most one long-range\ncycle this subgraph contains at most 2k(s+t)∆R(L+1)long-range paths, so by union bound\nPOh\nξ(j)\nl∈ G(j−1, s+t), τEscc> s+ti\n≤2k(s+t)∆R(L+1)e−C2l/(logn)β.\nfor some constant C2>0. For l= (log n)β(log log n)3this is o(1).\nCombining the two previous observations, we deduce that conditional on Bj−1, the last exit of\nG(j−1, s+t)by the trajectory (X(j)\nt′)t′≤s+toccurs before s, leaving at least tsubsequent steps.\nLetting L(j)denote this last exit time, we can reason conditional on Ljand use Proposition 7.1\nand Lemma 7.6 about shifted trajectories to deduce the claim.\nFinally,theproofofLemma3.2usesthesamearguments,consideringthesetofpaths Γ(R, L, M )\ninstead of ΓEnt.\n9 Approximation by nice paths: proof of Proposition 3.2\nWe move to the second part of the proof of Theorem 1.1. Most of this section is identical to the\nreversible case so we will skip some of the proofs that can be found in [11].\n9.1 Forward neighbourhood\nNice paths between xandywill have their first steps and last steps contained respectively in some\nquasi-tree-like neighbourhoods of xandy. We define here the forward neighbourhood of x.\nLetx∈V,l≥1an integer and wmin≥0. The forward graph K(x, l, w min)is designed\nessentially as a \"spanning quasi-tree\" of the ball BLR(x, l), obtained by exploring this ball algo-\nrithmically but giving priority to paths with large weights and truncating whenever cycles are\nencountered. This process will thus build iteratively a sequence (Km)τ\nm=0of subsets of BLR(x, l),\n63until it stops at a random time τto yield K(x, l, w min) :=Kτ. Unless the procedure is initiated\nat a vertex xwhose ball BLR(x, L)is not quasi-tree-like, Kmremains at all time quasi-tree-like.\nIn this case for every long-range edge e∈Emthere exists a unique long-range path ξ(e)from x\ntoecontained in Km. Define its cumulative weight as ˆw(e) :=wx(ξ(e)). Because weights require\nthe knowledge of (L−1)-neighbourhoods, the exploration queue will consist in subsets Emof\nlong-range edges for which the whole long-range (L−1)-neighbourhood is contained in Km, so\nthat cumulative weights can be computed from Kmonly. Finally, a constraint of minimal weights\nis added during the procedure, in order to keep the number of vertices explored as o(n).\nExplorationoftheforwardneighbourhood Theprocedureisinitiatedwith K0:=BLR(x, L).\nIfK0contains a long-range cycle, E0:=∅and the procedure stops. Otherwise let E0be the set\nof long-range edges at distance 0from x. Then for all m≥0the(m+ 1)-th step goes as follows:\n1. Among all long-range edges einEmat long-range depth at most lfrom xinKmand such\nthat ˆw(e)≥wmin, pick the edge em+1with maximal cumulative weight, using an arbitrary\nordering of the vertices to break ties. If there is no such edge, the procedure stops.\n2. Explore the depth- Lneighbourhood of em+1: for each descendant z∈∂Kmat long-range\ndistance L−1from em+1, reveal η(z). Thisexplorationphasestopsifarevealededgeviolates\nthe quasi-tree structure: this occurs if for some zthe small-range ball BSR(η(z), R)has a\nnon-empty intersection with Kmor one of the previously revealed balls.\n3. If the previous exploration phase stopped because of intersecting small-range balls, then\nEm+1:=Em∖{em+1}andKm+1:=Km. If it stopped because of an intersection with\nKm, letZmbe this intersection. Then Em+1is obtained by deleting from Emevery long-\nrange edge which has either a descendant or an ancestor in Zm, as well as the edge em+1,\nand set Km+1:=Km. Finally, if the exploration ended without a violation of the quasi-tree\nstructure, add the subsequent long-range edges of em+1toEm+1, whereas the newly revealed\nvertices are added to Km+1.\nWhen the procedure ends, the set Eτconsists by construction of edges at long-range distance\nl−Lfrom x, which contain no long-range cycles in their (L−1)-neighbourhood and whose\nweights are measurable with respect to K(x, l, w min). The two following lemmas were proved for\nthe reversible case.\nLemma 9.1 ([11, Lemma 8.1]) .Letκmdenote the number of long-range edges revealed during\nthe first msteps mandκ(x, l, w min) :=κτthe total number of long-range edges revealed during\nthe construction of K(x, l, w min). Suppose BLR(x, L)is quasi-tree-like so that τ≥1. There exists\na constant C >0such that for all m∈[1, τ]\nˆw(em)≤Cl∆RL\nκm. (65)\nIn particular\nκ(x, l, w min)≤Cl∆RL\nwmin. (66)\nLemma 9.2 ([11, Lemma 8.2]) .Ifm > τ, letcycle( em+1)be the event that the exploration of the\n(L+1)neighbourhood of the long-range edge em+1∈Emconsidered at the (m+1)-th step revealed\n64a cycle. Let ε∈(0,1)and consider the following process. If BLR(x, L)is not quasi-tree-like, let\nWm:= 1for all m≥0, otherwise set W0:= 0and for all m≥0define\nWm+1:=Wm+ ( ˆw(em+1)∧ε/2) 1m<τ 1cycle( em+1).\nThis is the total cumulative weight of edges that violated the quasi-tree structure at step m+ 1\nand that are below ε/2. Suppose l=O(logn),R, L =O(log log n)and that wmin≥e(log log n)3/n.\nThen for all s=s(n), with high probability, for all x∈V,\nWτ≤Ws+ε. (67)\n9.2 Nice paths: definition\nWe now define nice paths in order to prove Proposition 3.2. For the rest of this section set\nR:=CRlog log n, C L:=CLlog log n, M := (log log n)κ.\nwhere CR, CL>0andκ≥1are constants chosen large enough so that the conclusions of\nLemma 3.2 holds. Fix ϵ∈(0,1)for the rest of this section. In the sequel, we consider several\nconstants C0, . . . , C 5, defined in terms of the constants CLR(ε), Ch(ε)of Proposition 3.1, which\ncan in particular depend on ε. They are indexed in the reverse order in which they are fixed, so\nC0is chosen after C1, which is chosen after C2, etc. Let\ns:=\u0016logn\n10 log ∆\u0017\n∧\u0016logn\n10h\u0017\n, t :=\u0016logn\nh+C0p\nlogn\u0017\n,\nl1:=d(t−s)−C4√\nt, w min:=e−h(t−s)−Ch√\ntwmax:=e−ht+C1√\nt.(68)\nGiven x, y∈V, r∈N/2,L≤r≤Mandl3∈[ds−C5√s,ds+C5√s], consider the following\nthree-stage exploration of the environment:\n1. Explore K=K(x0, l1, wmin)as explained in the previous section. Let E:=Eτbe the set of\nlong-range edge remaining in the exploration queue at the end of the procedure, and consider\nthe set E′of boundary vertices at long-range distance l1from x0, whose image under ηis\nyet to determine.\n2. The backward neighbourhood is B=B(r, r+s, l3) :=B−\nP(y, r+s)∩B−\nLR(y, l3).\n3. Finally, reveal everything else.\nItwillbecrucialinthesequeltocontrolthenumbers N1, N2oflong-rangeedgesrevealedduring\nthe two first stages. By definition N1:=κ(x, l1, wmin). Observe that for any ε′<h\n10 log ∆∧1\n10,\nh(t−s)<(1−ε′) logn+O(√logn). Thus by (66),\nN1=O(l1∆RLeh(t−s)+Ch√\nt) =O(logn∆CRCL(log log n)2n1−ε′eC√logn)\nfor some constant C. The cardinality of Bis bounded by that of B−\nP(y, r+s), which is O(∆r+s) =\nO(n2/10)by choice of r≤Mands. All in all, for any ε′<h\n10 log ∆∧1\n10,\nN1=O(n1−ε′),\nN2=O(n1/10).(69)\n65LetFr,lbe the σ-algebra generated by the long-range edges revealed during the two first stages.\nSince Kis quasi-tree-like, a non-backtracking long-range path ξfrom xtoE′entirely contained in\nKmust cross a unique edge of E. Let ξEdenote this edge and define wE(ξ) :=wx,R,L(ξ1···ξE).\nThis is essentially the weight wx(ξ), but where last steps of the path were truncated to keep a\nweight that is Fr,l-measurable.\nInB, letF′be the set of boundary vertices which are at long-range distance exactly lfrom\ny, for which the shortest long-range path to yis unique and has a tree-like neighbourhood in\nBP(y, r+s), that is B∩BLR(z, L)contains no long-range cycle for all vertex zon this path.\nConsider now the set Fof long-range edges in Bthat are at long-range distance L−1from F′. If\nξis a non-backtracking long-range path from F′toy, letξFdenote the unique edge of Fcrossed\nbyξandξF+1the subsequent edge. Set wF(ξ) := wξ+\nF,R,L(ξF+1···ξ|ξ|−L+1), where ξ+\nFis the\nendpoint of ξFclosest from y. Here we truncate the path at both ends: the first steps to have a\nFr,l-measurable weight but also the last steps, as the trajectories considered afterwards end at y.\nDefinition 9.1. Given r∈N/2,r≤Mandl∈[ds−C5√logn,ds+C5√logn]letNt\nr,l(x, y)be\nthe set of length tpaths pbetween xandysuch that\n(i)p∈Γ(R, L, M )\n(ii)pcan be decomposed as the concatenation p=p1p2p3of three paths such that: p1is a\npath from xtoE′entirely contained in K(x, l1), whose endpoint is the only vertex of E′it\ncontains (which implies that it starts and ends with a long-range step),\n(iii)p2is a path between E′andF′which starts with a small-range step but ends with a long-\nrange step such that\nC3p\nlogn≤ |p2| ≤C2p\nlogn.\nfor some C2, C3. In addition, p2contains a regeneration edge outside K(x, l1, wmin)before\nstepML, and the endpoint of p2is the only vertex of B(r, r+s, l)it contains.\n(iv)p3is a path of length r+sfrom F′toyentirely contained in B(r, r+s, l), which starts and\nends with a small-range step and does not contains any regeneration edge in its first rsteps,\n(v)wE(ξ(p1))wF(ξ(p3))≤wmax.\nIn the sequel we consider\nPt\nr,l(x, y) :=X\np∈Nt\nr,l(x,y)P(p).\nThe complete set of nice paths is obtained by taking the union over parameters r, l, namely\nNt(x, y) :=[\nr≤M\nr∈N+1/2ds+C5√s[\nl=ds−C5√sNr,l(x, y).\nand the total probability of nice paths by Pt\nN(x, y) :=P\np∈Nt(x,y)P(p).\nConditions (i) and (ii) allow to relate the probability of following a nice path with the weight\nconstraint (v): thanks to the tree structure of K(x, l1), each vertex in E′has a unique ancester\nedge in E. Given e∈E, letE′(e)denote the set of vertices in E′with ancester eand recall ξ(e)\n66is the unique non-backtracking long-range path from xtoe. Similarly for f∈F, letF′(f)be the\nset of vertices of F′from which the unique non-backtracking long-range path to ygoes through f\nandξ(f)the unique non-backtracking long-range path from f+toy(which thus does not include\nf). Then for a fixed total long-range length l:\nX\np:ξ(p1)E=e\nξ(p3)F=f\n|ξ(p)|=lP(p)≤X\nξ:ξE=e\nξF=f,\n|ξ|=lPx[ξ(X0···Xτl)≤l−L+1=ξ,(X0···Xτl)∈Γ(R, L, M )]\n≤X\nξ:ξE=e\nξF=f,\n|ξ|=lwx,R,L(ξ).\nby Lemma 3.3. For each ξin the sum,\nwx,R,L(ξ) =wx,R,L(ξ1···ξE)wξ+\nE,R,L(ξE+1···ξF)wξ+\nF,R,L(ξF+1···ξl−L+1)\n=wE(ξ1···ξE)wξ+\nE,R,L(ξE+1···ξF)wF(ξF+1···ξl−L+1)\nObserve now that for fixed eandfthe first and third factors are fixed as well and determined by\nξ(p1), ξ(p3), so the sum is only over the steps ξE+1···ξF−1. Since weights sum up to 1, we can\nalso sum over land boundX\np:ξ(p1)=ξ1\nξ(p3)=ξ3P(p)≤wmax (70)\nthanks to the weight constraint (v). In words: the total probability of nice paths with prescribed\nlong-range edges in EandFis upper bounded by wmax.\n9.3 Nice paths are typical\nWe prove here points (i)-(ii) of Proposition 3.2. The argument is the same for both points and\nrelies on Lemmas 2.1, 3.1, 3.2 and Proposition 3.1. Let t= log n/h+C0(ε)√logn. To prove (i) we\ncan consider a fixed starting state x∈Vand then need to ensure that the tsteps of a trajectory\nare nice with quenched probability at least 1−ε, while for point (ii) we need to prove the last t\nsteps of a length s′+ttrajectory are nice, for s′= (log n)a, a≥1, this time independently of the\nstarting state. The proof thus only differs by which statement of the aforementionned Lemmas\nand Proposition we use. Let us for instance focus on point (ii). Consider a≥1such that the\nstatement of Lemmas 3.1, 3.2 and Proposition 3.1 hold. There are several properties to check.\nSince there are finitely many of them, once a property is shown to hold with probability 1−oP(1)\nor1−εwe can automatically assume it is satisfied when checking the remaining properties. Write\nt′:=s′+tto simplify notation.\nWe already proved in Lemma 3.2 that a failure of the requirement (Xs′···Xt′)∈Γ(R, L, M )\noccurs with probability oP(1). For fixed r, lthe very definition of the backward neighbourhood\nB(r, r+s, l)implies that it necessarily contains the last r−ssteps of the trajectory provided\nthey have the prescribed long-range length l. Summing over randl, the constraint that p2\nonly has its last endpoint in the backward neighbourhood while p3is contained in it amounts\nto conditionning by the last regeneration time occurring before s. In particular this requires the\nexistence of a regeneration time in the interval [t′−s−M, t′−s]but this is exactly ensured by\n67the fact that (Xs′···Xt′)∈Γ(R, L, M ). Assume now this property hold and let TF′denote the\nlast regeneration before time t′−s. The remaining obstructions to following a nice paths are:\n1. the first steps of (Xs′···Xt′)are not contained in K=K(Xs′, l, w min), which occurs if\nBLR(Xs′, L)is not quasi-tree-like. This occurs with probability oP(1)by Lemma 3.1.\n2. from Xs′, the chain leaves Kbefore it reaches long-range distance l1, which occurs if:\n•the loop-erased trace exits Kthrough the Llong-range neighbourhood of an edge e\nwhich satisfied ˆw(e)< w min: since cumulative weights along a path are non-increasing\nthis implies −logw(ξ(Xs′···Xt′−s))>−logwmin=h(t−s)+Ch√\ntwhich occurs with\nprobability less than εby Proposition 3.1.\n•the chain crosses an edge which violated the quasi-tree structure. Recall the process\nconsidered in Lemma 9.2. As mentionned above we can suppose BLR(Xs′, L)is quasi-\ntree-like. Then by Proposition 3.1, the total probability of paths with long-range length\nLandweightsabove ε/2isoP(1). Thusuptothis oP(1)errorthequantity Wκconsidered\nin the lemma exactly counts the probability to exit Kat an edge which violated the\ntree structure. Thanks to the choice of wmin(68) the lemma establishes that with high\nprobability, Wκ≤W⌊L/2⌋+εfor any value of Xs′. However W⌊L/2⌋= 0asBLR(Xs′, L)\nis quasi-tree-like.\n3. the path from Xt′−s−TF′toyis not unique or its long-range length is not in the interval\n[ds−C5√logn,ds+C5√logn]. If this path is not unique, this implies in particular that\nthe ball B(Xt′−s−M, t′−s−M)is not quasi-tree-like. Since s+M=O(s)Lemma 3.1 shows\nthis occurs with probability oP(1). For the long-range distance requirement, Proposition 3.1\nshows that ||ξ(Xt′−s. . . X t′)| −ds| ≤CLR√swith probability at least 1−ε−o(1). Then\nnote that the long-range distance traveled in the intervals [TF′, t′]and[t′−s, t′]differ by\nat most M, hence\f\f\f\fξ(Xt′−s−TF′···Xt′)\f\f−ds\f\f≤CLR√s+M≤C5√sfor a large enough\nconstant C5, using that M=o(s).\n4. No regeneration occured outside outside Kduring the first MLsteps of the intermediate\npath p2or the latter does not have length O(√\nt). Since the trajectory is in Γ(R, L, M )there\nmust be regeneration edge in less than Msteps after having first reached E′. By the non-\nbacktracking property, this regeneration edge is either outside Kor at long-range distance\nless than Lfrom E′, after which the chain is forced to exit K. Thus the L-th consecutive\nregeneration edge must be outside K. Using again that regeneration edges are spaced by\nat most Msteps, this makes in total MLsteps at most until a regeneration occurs outside\nK(x, l1).\nFor the length requirement observe that the long-range length is sub-additive. Since by\ndefinition a nice trajectory decomposes as the concatenation (Xs′···Xt′) =p1p2p3the sub-\nadditivity implies\n|ξ(Xs′···Xs′+t)| ≤ |ξ(p1)|+|ξ(p2)|+|ξ(p3)|.\nThe path ξ(p1)has length l1, whereas ξ(p3)has variable length but from the bounds on lin\nDefinition 9.1 and (68) we infer that their combined length is\n|ξ(p1)|+|ξ(p3)| ≤dt−(C4−C5)√\nt,\n68while the intermediate path obviously has length |ξ(p2)| ≤ |p2|. Choose C4≥C5+ 2CLR.\nHence if |p2|< C 3√\ntwith C3:=CLR, then |ξ(Xs′···Xs′+t)|<dt−CLR√\nt, which occurs\nwith probability at most oP(1)by Proposition 3.1. To prove the upper bound on |p2|, observe\n|p1|+s′coincides with the first hitting time τl1of long-range distance l1after step s′. From\nProposition 3.1 we can deduce the existence of C2>0such that τl1−s′≥t−s−C2√\ntwith\nprobability at least 1−ε. Since |p3| ≥s, we deduce that\n|p2|=t− |p1| − |p3| ≤C2√\nt.\n5.wE(ξ(Xs′···Xτl1))wF(ξ(Xt′−s−TF′···Xt′))> w max: let ξ:=ξ(Xs′···Xt). When deriving\n(70) we used that\nwx,R,L(ξ) =wE(ξ1···ξE)wξ+\nE,R,L(ξE+1···ξF)wF(ξF+1···ξl)\nwher ξlisthelastedgeof ξ. Nowbythenon-backtrackingpropertyofnicepaths ξ(Xs′···Xτl1)\nandξ(Xt′−s−TF′···Xt′)contain ξ1···ξEandξF+1···ξlrespectively so the goal is to show\nthat wE(ξ1···ξE)wF(ξF+1···ξl)> w maxwith probability at most ε. As we argued for the\nprevious point, Proposition 3.1 implies τl1−s′≥t1:=t−s−C2√\ntwith probability at least\n1−ε. Since cumulative weights are non-increasing along a path, we deduce that\nwE(ξ1···ξE)≤wx,R,L(ξ(Xs′···Xt1))≤e−t1h+Ch√t1\nwith probability at least 1−ε. Similarly we know that TF′≤t′−s, hence\nwE(ξ1···ξF)≥w(ξ(Xs′···Xt′−s))≥e−(t−s)h−Ch√t−s\nwith probability at least 1−ε−o(1). From these two bounds, we deduce\nwξ+\nE,R,L(ξE+1···ξF)≥e−Ch√t−s−Ch√t1−Ch√\nt≥e−C′√\nt\nfor some C′=C′(ε)>0. Thus if wE(ξ1···ξE)wF(ξF+1···ξl)> w maxwe obtain that\nwx,R,L(ξ)≥wmaxe−C′√\nt\nwhich has probability at most εif the constant C1inwmax(68) is taken sufficiently large.\n9.4 Concentration of nice paths\nThe proof of the concentration of Pt\nNis identical to the reversible case so we will only sketch\nthe arguments. Recall Fr,lis the σ-algebra generated by the long-range edges revealed during\nthe two first stages. Concentration is established for r, landx, yfixed then extended by union\nbound. It is based on the following concentration inequality for functions on the symmetric group,\nproved specifically for this problem. It is inspired from a concentration inequality by Chatterjee\n([9, Prop. 1.1], see also [8]) that was used to prove the cutoff of non-backtracking chains in [7]. It\ncorresponds to the degree d= 1case, but provides better constants.\nProposition 9.1 ([11, Corollary 1.1]) .Letϕbe a polynomial in the indeterminates (Xij)i,j∈[n],\nwith non-negative coefficients, of degree at most 1in each entry and total degree d. We write ∂ij\nfor the partial derivative in the (ij)-entry. Let M(ϕ)be the maximal coefficient of ϕand given\n69a matrix S= (Sij)i,j∈[n], letN(ϕ, S)be the number of monomials in ϕwhich are non-zero when\nevaluated at S. Finally consider Aϕ:=M(ϕ) max SN(ϕ, S)where the maximum is taken over all\npermutation matrices and A∇ϕ:= max i,j∈[n]A∂ijϕ. Then if Sis a uniform permutation matrix,\nfor all t≥0,\nP[|ϕ(S)−E[ϕ(S)]| ≥t]≤2 exp \n−t2\n2αϕ(4\n3E[ϕ] +2d+1(d−1)\n3nAϕ+t)!\n. (71)\nwith\nαϕ:= 6d2dA∇ϕ \nlog\u00124Aϕn\nA∇ϕ\u0013+\n+(2/n)(2−e−2/n)\n1−e−2/n!\nLet us clarify a bit this result which is arguably a bit heavy in notation. First note any function\non permutations can be represented as a polynomial in permutations matrices of degree at most\nn−1, although not uniquely. Furthermore, since permutation matrices have 0−1entries there\nis no loss of generality in supposing the degree is at most in each entry. The prohibitive term\nof the inequality are obviously the exponential terms in the degree, limiting the applications to\nsmall degrees that can nonetheless depend on n, and the term A∇ϕ. As the notation indicates,\nthis term is essentially a bound on the norm of the gradient of the function. In the case d= 1, the\nrandom variable ϕ(S)can be written as tr(B⊤S)for some matrix B, in which case Bis exactly the\ngradient of ϕand we have A∇ϕ= max i,jBij. All in all, Proposition 9.1 provides a concentration\ninequality for smooth low-degree functions on the symmetric group.\nIn our case, this result is applied conditional on Fr,l, so all the randomness comes from the\nintermediate path p2, whose length will thus give the degree of the function. Then derivating the\nfunction at some entry (i, j)comes down to focusing on the probability of nice paths which follow\na prescribed long-range edge. The term M(∂ijϕ)will thus correspond to the maximal probability\nof following one given nice path, which is wmaxby (70), while the boundedness of degrees allows\nto upper bound the number of nice paths which can use the long-rang edge (i, j). We thus obtain\nthe following bounds, which will be sufficient to apply the proposition.\nLemma 9.3 ([11, Lemma 8.3]) .With the notations of Proposition 9.1, for all r≤M, l≥0,\nconditional on Fr,lPt\nNr,l(x, y)can be realized as a function ϕonSn′of degree at most\nd:=C2p\nlogn′. (72)\nwhich satisfies\nαϕ=O\u0000\nd3(4∆)dwmaxlogn\u0001\n, A ϕ=O(d2∆dn2/10wmax)\nand A∇ϕ=O\u0000\nd2(2∆)dwmax\u0001\n.\nThe next step is to compute the expectation of nice paths. The following Lemma will be\nsufficient to deduce Proposition 3.2. Recall the measure Q(L)\nuwas defined in (18). To ease notation\nwewilldroptheexponent (L)inthesequel. Recallalsothat T(G,l)\n1, T(G,l)\n1denoteregenerationtimes\nwith horizon LinG∗andGrespectively, µis the invariant measure of the regeneration chain in the\nquasi-tree and νwas considered in Proposition 6.4. Below we write Qν+c=P\nu(ν(u) +c(u))Qu.\nLemma 9.4. There exists a measure conVsuch thatP\nv∈Vc(v) =oP(1)and\nEh\nPt\nNr,l(x, y)\f\f\fFr,li\n=(1−oP(1))\nEQµh\nT(G,∞)\n1iQν+ch\nXr+s=y,|ξ(X0···Xr+s)|=l, r < T(G,L)\n1≤Mi\n.\n70Proof of Lemma 9.4. The proof is almost identical to that of Lemma 8.4 in [11] for the reversible\ncase. It is essentially an application of Proposition 6.4 combined with the coupling on the quasi-\ntree.\nLetτE′be the hitting time of E′andUt′(K)the time spent in Kbefore the time t′≥0. By\ndefinition, a nice path requires that τE′≤t−sand the first part p1of the path is the trajectory\nuntil τE′. The second part of a nice path is the trajectory until hitting F′and is assured to spend\nless than m:=MLsteps in K. Therefore by strong Markov’s property, one can bound\nX\np∈Nr,lP(p)≤X\nt1+t2=t−(r+s)\nC3√logn≤t2≤C2√lognX\nu∈E′X\nv∈F′(Px[τE′=t1, Xt1=u] (73)\n×Ph\n∃k≥0 :T(G,L)\nk=t2, Xt2=v, Ut2(K)≤m\f\f\fX1/2=ui\nP\u0002\nXr+s=y, r < T 1≤M|X1/2=v, τη(v)>τL\u0003\u0001\n.\nIn each term of this sum, the first and third factor are Fr,l-measurable, so only the second factor\ngets averaged when taking conditional expectation. We claim this expectation satisfies\nX\nv∈F′\f\f\f\f\f\fPh\n∃k≥0 :T(G,L)\nk=t2, Xt2=v, Ut2(K)≤m\f\f\fX1/2=u,Fr,li\n−ν(v)\nEQµh\nT(G,∞)\n1i\f\f\f\f\f\f+oP(1).\nNote that ν/EQµh\nT(G,∞)\n1i\nis independent of uandt1, while the first factor in the sum considered\nabove can be summed to at most 1. We can also recognize the measure Qv(18) in the third factor.\nTherefore provided the claim holds one obtains that for some c= (cv)vsatisfyingP\nv∈Vc(v) =\noP(1),\nE\nX\np∈Nr,lP(p)\f\f\f\f\f\fFr,l\n≤X\nv∈F′\nν(v)\nEQµh\nT(G,∞)\n1i+c(v)\nQv[Xr+s=y, r < T 1≤M]\n≤X\nv∈Vν(v) +c(v)\nEQµh\nT(G,∞)\n1iQv[Xr+s=y, dLR(v, y) =l, r < T 1≤M],\nwhere in the second we implicitely changed the definition of c, using that EQµh\nT(G,∞)\n1i\n=O(1).\nThis proves an upper bound. To prove a matching lower bound, note that the inequality (73) is\nnot sharp if the trajectory is not in Γ(R, L, M )or there is no regeneration time in the interval\n[τl1, ML ]. This was shown to occur with probability oP(1)in Section 9.3, so the upper bound is\nalso a lower bound up to a oP(1)error.\nLet us prove the claim. Let u∈E′, v∈F′. The idea is to couple the chains XandX\non a quasi-tree to relate their regeneration times, in order to use Proposition 6.4. However the\nconditionning by Fr,lalready revealed some-long range edges, which requires in turn a similar\nconditionning in the quasi-tree. We argue that the only conditionning required is by BLR(O, L),\nthe ball of radius Lin a quasi-tree. Let u0be the ancestor of Eat long-range distance Lfrom\nu. By definition of K(x, l1), the Llong-range neighbourhood BLR(u0, L)contains no long-range\ncycle and thus is a possible realization of BLR(O, L)around the root of a quasi-tree. Consider\na quasi-tree Gwhich has this ball as the neighbourhood of its root and is completed with the\nstandard procedure (so without taking consideration of the long-range edges revealed in K∪B).\nUsing the coupling of Section 2.7, the chain Xstarted at ucan thus be coupled with the chain X\n71onG, started at the vertex of BLR(O, L)that identifies with u. This coupling fails after Xenters\nBor if it re-enters Kby another path that the one it used. Let τcoupdenote this new coupling\ntime.\nFrom (69), the probability of sampling an element of either KorBunder the uniform measure\nonVisO(1/nε′)for some ε′>0. Consequently, using the same comparison with a binomial\nwe used in the proof of Lemma 2.1, this implies that the chain Xre-enters Kor reaches Bin\nO(√logn)steps is O(logn/n2ε′) =o(1). Note that considering regeneration times requires the\nknowledge of the Lsteps ahead but this is O(log log n). Consequently it remains true that for\nt2+L=O(√logn),\nX\nv∈F′\f\f\fPh\n∃k≥0 :T(G,L)\nk=t2, Xt2=v, Ut2(K)≤m\f\f\fX1/2=u,Fr,li\n−Ph\n∃k≥0 :T(G,L)\nk=t2, ι(Xt2) =v, Ut2(K)≤m\f\f\fX1/2=x, B LR(O, L)i\f\f\f=o(1)\nwhere x∈BLR(O, L)is a vertex at long-range distance Lwith type u. Obviously, {T(G,∞)\nk, k≥\n1} ⊂ { T(G,L)\nk, k≥1}. This inclusion may be strict however, if the chain backtracks over a long-\nrange distance L. Lemma 7.3 shows this occurs before time O(√logn)with probability o(1).\nConsequently, with high probability T(G,L)\nk=T(G,∞)\nkfor all regeneration times that occur before\nt2, so we can exchange these random times in the equation above. Now since t2= Θ(√logn)and\nm2=O((log log n)2κ+2) =o(t2), Proposition 6.4 proves that\nX\nv∈V\f\f\f\f\f\fPh\n∃k≥0 :T(G,∞)\nk=t2, ι(Xt2) =v, Ut2(K)≤m\f\f\fX1/2=x, B LR(O, L)i\n−ν(v)\nEQµh\nT(G,∞)\n1i\f\f\f\f\f\f\n≤P\u0002\nUt2(K)> m| X1/2=x, B LR(O, L)\u0003\n/EQµh\nT(G,∞)\n1i\n+o(1).\nBy Lemma 6.4, the first termis stretchedexponentialin mand thus o(1). Usingtriangle inequality\nto combine the two previous bounds yields the claim.\nNow comes the last proof of this paper.\nProof of Proposition 3.2. Points (i)-(ii) were established proved in Section 9.3 so we now prove\n(iii) - (iv).\nFor (iii), suppose first x, y∈Vandr, lare fixed. Let ϕ:=Pt\nNr,l(x, y)as in Lemma 9.3 and\nz:=ε\n2E[ϕ] +ε\n2C6Mn√lognfor some C6>0. Note that wmax≤e−C√logn/nfor some constant C\nwhich tends to +∞as the constant C0in the definition of tgrows, while other factors of αϕ, A∇ϕ\nare of all of order at most eC′√logn. Thus for any choice of C=C(ε)>0Lemma 9.3 shows that\nαϕ,A∇ϕcan both be bounded by e−C√logn/n, provided the constant C0is sufficiently large, while\nd2(d−1)Aϕ/n=o(n−3/5). In particular we can choose Cso that A∇ϕ≤z. Since E[ϕ]≤2z/ε\nandz≥ε/(2C6Mn√logn), applying Proposition 9.1 yields\nP[|ϕ−E[ϕ]| ≥z]≤2 exp\u0012−C′ε2\nαϕMn√logn\u0013\n(74)\nforsome C′=C′(ε)>0. Uptoincreasingagainthevalueof C, wecanensurethat αϕMn√logn≤\n(logn)−2, hence\nP\u0014\n|ϕ−E[ϕ]|>ε\n2E[ϕ] +ε\n2C6Mn√logn\u0015\n≤2 exp\u0000\n−C′ε2(logn)2\u0001\n.\n72This is sufficent to take a union bound over x, y∈Vandr≤M, l∈[ds−C5√logn,ds+\nC5√logn]. Thus summing over r, lwe obtain that with high probability, for all x, y∈V,\n\f\f\f\f\f\fPt\nN(x, y)−X\nr,lEh\nPt\nNr,l(x, y)\f\f\fFr,li\f\f\f\f\f\f≤ε\n2X\nr,lEh\nPt\nNr,l(x, y)\f\f\fFr,li\n+ε\nn.\nLemma 9.4 gives an estimate of the conditional expectation which shows:\nPt\nN(x, y)≤1 +ε/2\nEQµh\nT(G,∞)\n1iMX\nr=0Qν+ch\nXr+s=y, r < T(G,L)\n1≤Mi\n+ε\nn\n= (1 + ε/2)Aˆπ(y) +(1 +ε/2)A\nEQνh\nT(G,L)\n1iMX\nr=0Qch\nXr+s=y, r < T(G,L)\n1≤Mi\n+ε\nn.(75)\nwith A:=EQνh\nT(G,L)\n1i\nEQµh\nT(G,∞)\n1i. Summing over y∈Vandr∈[0, M]in the second term yields (1 +\nε/2)AP\nv∈Vc(v)≤2Aεwith high probability. On the other hand, from Lemma 9.4 we also have\nthe lower bound\nPt\nN(x, y)≥(1−ε/2)(1−oP(1))Aˆπ(y)−ε/n,\nwhich by summing over y∈Vshows that A≤1 + 10 εwith high probability. Plugging this in the\nupper bound above yields the second statement of the proposition.\nLet us move to the proof of (iv). Define\nˆπ1(v) :=1\nEQνh\nT(G,L)\n1iMX\nr=0X\nu∈Sν(u)Quh\nXr+s=v, r < T 1≤M, τ(R)\nSR∧τ(L)\nNB> M +si\nfor all v∈Vand decompose ˆπ=: ˆπ1+ ˆπ2.\nWe start bounding ˆπ2(V). Summing over yin (75), the left hand side becomes the total\nprobability of following a nice path, which by the first part of the proposition is at least 1−ε\nwith high probability for a typical x∈V. Consequently Acan be lower bounded by a constant\nwith high probability, which implies that for some constant c >0,EQνh\nT(G,L)\n1i\n≥cwith high\nprobability. Thus by summing over v∈Vwe can bound\nˆπ2(V)≤c−1MQνh\nτ(R)\nSR∧τ(L)\nNB≤M+si\n+oP(1).\nThennoticethatthequenchedprobability P\u0002\nτη(u)> τL\f\fX1/2=u\u0003\ncanalwaysbelowerbounded\nbyqn:=δCLfor some constant C >0, hence\nˆπ2(V)≤c−1MX\nu∈Vν(u)Ph\nτ(R)\nSR∧τ(L)\nNB≤M+s, τη(u)> τL\f\f\fX1/2=ui\nP\u0002\nτη(u)> τL, X1/2=u\u0003 +oP(1)\n≤c−1q−1\nnMX\nu∈Vν(u)Ph\nτ(R)\nSR∧τ(L)\nNB≤M+s\f\f\fX1/2=ui\n+oP(1).\nAsL=O(log log n)one has q−1\nn=O((log n)−b)for some b >0. Next by Markov’s property for\nallu∈V\nPh\nτ(R)\nSR∧τ(L)\nNB≤M+s\f\f\fX1/2=ui\n=Ph\nτ(R)\nSR∧τ(L)\nNB≤M+s\f\f\fX0=u, X 1/2=ui\n=Puh\nτ(R)\nSR∧τ(L)\nNB≤M+s, X 1/2=ui\np(u, η(u))\n≤c′Puh\nτ(R)\nSR∧τ(L)\nNB≤M+si\n73for some constant c′>0as the entries of pare bounded (Assumption (H2)). For fixed u∈V,\nLemma 3.2 shows Puh\nτ(R)\nSR∧τ(L)\nNB≤M+si\n=oP((log n)−b) =oP(q−1\nn)asM+s=O(logn).\nLetting U∈Vbe a random variable independent from the environment, this can be rephrased\nas the fact that PUh\nτ(R)\nSR∧τ(L)\nNB≤M+si\n=oP(q−1\nn)conditional on U, but then this statement\nmust also hold unconditionally. As the measure νis deterministic we can take Uof law ν, after\nwhich by taking the conditional expectation with respect to the environment we get that\nEh\nPUh\nτ(R)\nSR∧τ(L)\nNB≤M+si\f\f\fηi\n=X\nu∈Vν(u)Puh\nτ(R)\nSR∧τ(L)\nNB≤M+si\n=oP(q−1\nn).\nAll in all, this proves ˆπ2(V) =oP(1).\nTo bound the ℓ2norm of ˆπ1, we can use what precedes to first bound\nˆπ1≤c−1Mmax\nr≤MQνh\nXr+s=·, τ(R)\nSR∧τ(L)\nNB> si\n≤c−1q−1\nnMmax\nr≤MX\nu∈Vν(u)Ph\nXr+s=·, τ(R)\nSR∧τ(L)\nNB> M +s\f\f\fX1/2=ui\n.\nConsequently\nE\"X\nv∈V˜π1(v)2#\n≤c−2q−2\nnM2max\nr≤MP\"\nXr+s=X′\nr+s, τ(R)\nSR∧τ(L)\nNB> s,\nτη(X1/2)> τL, τη(X′\n1/2)> τL#\nwhere X, X′are two independent chains conditional on the same environment, both started at\ntime 1/2under the law ν. Here the event τ(R)\nSR∧τ(L)\nNB> sapplies to both chains. Fix r≤M.\nObserve that if all long-range edges crossed by X′are at distance more than Rfrom the trajectory\nfollowed by the first chain X, then X′can reach Xr+sonly by deviating by a small-range distance\nRor by crossing the long-range edge (X′\n1/2, η(X′\n1/2))ifη(X′\n1/2)∈B−\nSR(X1/2. . . X r+s, R). The\nevent {τη(X′\n1/2)> τL}proscribes this last possibility to occur before the chain has crossed a long-\nrange distance L, so the chain must backtrack. Thus either of these two possibilities contradicts\nτ(R)\nSR∧τ(L)\nNB> s. Consequently the only possibility that the two chains meet is that the second\none crosses a long-range edge at small-range distance less than Rfrom the first chain. From (45)\ndSR(X1/2, X′\n1/2)≤Rwith probability O(∆R/n) =o(1)under ν. Then assuming this does not\nhold, under the annealed law the environment can be generated sequentially along the trajectories\nof the chains as explained in Section 2.7. The number of long-range edges met by X′up to time\nM+swhich fall in a Rsmall-range neighbourhood of Xis thus stochastically dominated by a\nbinomial Bin(M+s,∆R(M+s)/n). As qn=O((log)b), R=O(log log n),s=O(logn), M=\no(logn)we deduce that\nE\"X\nv∈Vˆπ2(v)#\n≤O \nC(logn)b′\nn!\nfor some b′>0, which gives the result by the first moment argument (2).\nReferences\n[1] Elie Aidékon. Large deviations for transient random walks in random environment on a gal-\nton–watson tree. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques , 46(1):159–\n189, 2010. doi:doi:10.1214/09-AIHP204 . 39, 41\n74[2] Zsuzsanna Baran, Jonathan Hermon, Anđela Šarković, and Perla Sousi. Phase transition for\nrandom walks on graphs with added weighted random matching. 2023. arXiv:2306.13077 .\n4, 5\n[3] Anna Ben-Hamou and Justin Salez. Cutoff for nonbacktracking random walks on sparse\nrandom graphs. 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The evolution of the mixing rate of a sim-\nple random walk on the giant component of a random graph. Random Structures\n& Algorithms , 33(1):68–86, 2008. URL: https://onlinelibrary.wiley.com/doi/abs/\n10.1002/rsa.20210 ,arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/rsa.\n20210,doi:https://doi.org/10.1002/rsa.20210 . 3\n[13] Geoffrey Grimett and Harry Kesten. Random Electrical Networks on Complete Graphs II:\nProofs. 08 2001. 25\n[14] Jonathan Hermon, Allan Sly, and Perla Sousi. Universality of cutoff for graphs with an added\nrandom matching. The Annals of Probability , 50(1), Jan 2022. URL: https://doi.org/10.\n1214%2F21-aop1532 ,doi:10.1214/21-aop1532 . 1, 4, 5, 7, 9, 23, 39\n75[15] Jonathan Hermon, Anđela Šarković, and Perla Sousi. Cutoff for random walk on random\ngraphs with a community structure. 2022. arXiv:2212.04469 . 4, 5\n[16] V. A. Kaimanovich and A. M. Vershik. Random walks on discrete groups: Boundary and\nentropy. 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Cambridge University Press, Cambridge, 2000. doi:10.1017/\nCBO9780511470967 . 27\n76"    },    {        "title": "2402.03422v1.The_Heterogeneous_Surface_of_Asteroid__16__Psyche.pdf",        "content": "manuscript submitted to JGR: Planets\nThe Heterogeneous Surface of Asteroid (16) Psyche\nSaverio Cambioni1, Katherine de Kleer2, Michael Shepard3\n1Department of Earth, Atmospheric & Planetary Sciences, Massachusetts Institute of Technology,\nCambridge, MA, USA\n2Division of Geological & Planetary Sciences, California Institute of Technology, Pasadena, CA, USA\n3Department of Environmental, Geographical & Geological Sciences, Bloomsburg University, Bloomsburg,\nPA, USA\nKey Points:\n•Longitude-latitude maps of the thermal inertia and dielectric constant of aster-\noid (16) Psyche derived from ALMA data.\n•The surface of Psyche is heterogeneous and shows signatures suggestive of both\nmetal- and silicate-rich materials.\n•A depression on Psyche with distinctive thermal-inertia properties suggests an evolved\nsurface processed by impacts.\nCorresponding author: Saverio Cambioni, cambioni@mit.edu\n–1–arXiv:2402.03422v1  [astro-ph.EP]  5 Feb 2024manuscript submitted to JGR: Planets\nAbstract\nMain-belt asteroid (16) Psyche is the largest M-type asteroid, a class of object classi-\ncally thought to be the metal cores of differentiated planetesimals and the parent bod-\nies of the iron meteorites. de Kleer, Cambioni, and Shepard (2021) presented new data\nfrom the Atacama Large Millimiter Array (ALMA), from which they derived a global\nbest-fit thermal inertia and dielectric constant for Psyche, proxies for regolith particle\nsize, porosity, and/or metal content, and observed thermal anomalies that could not be\nexplained by surface albedo variations only. Motivated by this, here we fit a model to\nthe same ALMA dataset that allows dielectric constant and thermal inertia to vary across\nthe surface. We find that Psyche has a heterogeneous surface in both dielectric constant\nand thermal inertia but, intriguingly, we do not observe a direct correlation between these\ntwo properties over the surface. We explain the heterogeneity in dielectric constant as\nbeing due to variations in the relative abundance of metal and silicates. Furthermore,\nwe observe that the lowlands of a large depression in Psyche’s shape have distinctly lower\nthermal inertia than the surrounding highlands. We propose that the latter could be ex-\nplained by a thin mantle of fine regolith, fractured bedrock, and/or implanted silicate-\nrich materials covering an otherwise metal-rich surface. All these scenarios are indica-\ntive of a collisionally evolved world.\nPlain Language Summary\nAsteroid (16) Psyche is the target of the eponymous NASA mission, which will as-\nsess whether the asteroid is the exposed core of an early pre-planet. The Atacama Large\n(Sub-)Millimeter Array located on the Chajnantor Plateau in Chile allowed the acqui-\nsition of temperature images of Psyche at a resolution of 30 km/pixel (the highest ever\nachieved from Earth). We analyze these images to map the metal content of the first mil-\nlimeters of the surface as function of longitude and latitude. We find that the relative\nabundance of metals and silicates varies across the surface. Additionally, we observe that\nthe lowlands of a large depression change temperature much faster than the surround-\ning highlands as Psyche rotates on its spin axis. Based on this, we propose that the low-\nlands could exhibit ponds of fine-grained materials, be highly fractured, and/or host silicate-\nrich materials implanted by impacts. All these scenarios are indicative of an evolved sur-\nface processed by impacts.\nKeywords : (16) Psyche; Asteroids – Surfaces; Remote Sensing – ALMA; Origin and\nEvolution\n1 Introduction\nThe upcoming NASA Psyche mission will explore Main Belt asteroid (16) Psyche\n(henceforth just “Psyche”), an M-type asteroid of about 220 km in size which could have\nbeen part of a differentiated planetesimal (Elkins-Tanton et al., 2022). Planetesimals are\nthought to be born 100 km in size and larger (e.g., Morbidelli et al., 2009). During the\nevolution of the solar system, some may have survived relatively intact, while others suf-\nfered collisions and created asteroid families (e.g., Delbo’ et al., 2017), and others may\nhave lost their mantle materials with no apparent associated family (Davis et al., 1999).\nPsyche could be a differentiated planetesimal that lost its mantle in a hit-and-run col-\nlision (e.g., Asphaug & Reufer, 2014), a rubble pile of material that was previously part\nof a differentiated body, or a primitive object of unusually metal-rich composition. An-\nswering the question ‘what is Psyche?’ is valuable for constraining the collisional evo-\nlution of the planetesimals and requires linking remote sensing observations of their sur-\nfaces to the properties of their interiors.\nThe M-type asteroids (Tholen, 1984) are a diverse group (Table 1), which is bro-\nken down in the Bus–DeMeo taxonomy into the Xc class, associated with enstatite chon-\n–2–manuscript submitted to JGR: PlanetsTable 1. Properties of some M-type asteroids\nAsteroid a [AU] D [km] Radar albedo Γ [tiu] Analog? 3- µm? References\n16 Psyche 2.92 222+4\n−1 0.34±0.08 280 ±100 see text yes S21; DK21; T17\n21 Lutetia 2.44 98 ±2 0.24±0.07 20–30 EC yes S15; C11; S10; R11\n22 Kalliope 2.91 168 ±3 0.18±0.05 125 ±125 IM+PYR yes S15; M12; F10; OB10\n55 Pandora 2.76 85 ±3 - - IM yes OB10\n69 Hesperia 2.98 . 136 ±14 0.45±0.12 - IM+PYR yes S15; H05; L15\n77 Frigga 2.67 61.4 ±0.2 0.14±0.04 - - - S15\n92 Undina 3.19 108 ±5 0.38±0.09 - IM+PYR yes S15; F11; R00\n97 Klotho 2.67 88 ±24 0.26±0.05 - EC, CH no S15; S10; OB10\n110 Lydia 2.73 88 ±3 0.37±0.10 135 ±65 IM+PYR yes S15; DT09; H05; OB10\n125 Liberatrix 2.74 48.4 ±0.5 - 71+26\n−24IM+PYR no ME21; H05; R00\n129 Antigone 2.87 128.7 ±0.6 0.36±0.09 - CB? yes S15; S10; OB10\n135 Hertha 2.43 71 ±3 0.18±0.05 - CH, EC yes S15; S10; OB10\n136 Austria 2.29 36.9 ±0.5 - - - yes L15\n201 Penelope 2.68 86 ±3 0.40±0.10 <50 IM+PYR yes S15; M05; H05; R00\n216 Kleopatra 2.79 119 ±3 0.43±0.10 >50 IM+PYR no S18; M05; OB10; L15\n224 Oceana 2.65 58.2 ±0.8 0.25±0.10 - EC, CH - S15; S10\n250 Bettina 3.15 121 ±2 - - IM+PYR - OB10; H07\n347 Pariana 2.61 48.6 ±0.1 0.36±0.09 - IM, CB - S15; S10\n382 Dodona 3.12 65.2 ±0.5 - 60+90\n−45IM+PYR - ME21; H11\n413 Edburga 2.58 25 ±5 0.35±0.09 100+60\n−60 IM, EC no S15; ME21; H07; R00\n441 Bathilde 2.81 66 ±2 0.20±0.05 180+20\n−60 - - S15; PG19\n497 Iva 2.85 40.9 ±0.3 0.24±0.08 70+19\n−25 SIM, CH no S15; ME21; OB10; OB10\n766 Moguntia 3.02 41.0 ±0.1 - - IM+OLI - H11\n785 Zwetana 2.57 49 ±1 0.26±0.07 <50 IM, CB no S15; M05; OB10; OB10\n798 Ruth 3.01 46 ±7 - - IM+OLI - H11\n1210 Morosovia 3.01 33.7 ±0.3 - - IM+OLI - H11\nTable 2. “a” is the semi-major axis of the asteroid in astronomical units (AU). D is the mean or effective diameter, latest WISE value from Masiero et al. (2011,\n2012, 2014, 2021) as listed in the Minor Planet Physical Properties Catalogue ( https://mp3c.oca.eu/ ), except for Psyche, (21) Lutetia, and (69) Hesperia and\n(798) Ruth, for which the values are from Shepard et al. (2021), Sierks et al. (2011), and Al´ ı-Lagoa et al. (2018), respectively. Γ is thermal inertia, that is, the\nresistance of surface material to change temperature. Uncertainties are 1-standard deviation. IM = iron meteorites; EC = enstatite chondrite; CB = high-iron car-\nbonaceous chondrites; CH = CH meteorites; SIM = stony-iron meteorites. The acronyms PYR and OLI indicate the detection of pyroxene- and olivine-bands in\nthe bodies’ spectra. References for radar albedo, thermal inertia, silicates and OH/H 2O band are listed using the following acronyms: C11: Coradini et al. (2011);\nDK21: de Kleer, Cambioni, and Shepard (2021); DT09: Delbo’ and Tanga (2009); F10: Fornasier et al. (2010); F11: Fornasier et al. (2011); H05: Hardersen et al.\n(2005a); H07: Hardersen et al. (2007); H11: Hardersen et al. (2011); L15: Landsman et al. (2015); M05: Mueller et al. (2005); M12: Marchis et al. (2012); ME21:\nMacLennan and Emery (2021); OB10: Ockert-Bell et al. (2010); PG19: Podlewska-Gaca et al. (2020) R11: Rivkin et al. (2011); S10: Shepard et al. (2010); S15:\nShepard et al. (2015); S18: Shepard et al. (2018); S21: Shepard et al. (2021); T17: Takir et al. (2017).\n–3–manuscript submitted to JGR: Planets\ndrites, and the Xk class, associated with mesosiderites (DeMeo et al., 2009). Neeley et\nal. (2014) compared the optical-near-infrared spectra for 29 M-type asteroids to mete-\norites and found that, while the best analog types are iron meteorites in most cases, 21%\nof the targets were better fit by enstatite chondrites, consistent with early studies (e.g.,\nChapman & Salisbury, 1973; Vernazza et al., 2009). Insights into the concentration of\nmetal in the near-surface of asteroids come from measurement of radar echoes, in which\na circularly polarized continuous wave signal is trasmitted to the asteroid and the echo\npower in the same (SC) and opposite (OC) senses of polarization is measured. The OC\nradar albedo (henceforth just “radar albedo”) is the ratio of the power received from tar-\nget asteroids relative to that which would be measured from a metallic sphere of the same\ncross-sectional area and at the same distance. Radar albedo increases with increasing\nmetal content of the near-surface material (e.g., Shepard et al., 2015). The average radar\nalbedo of the M-type group is 0.28 ±0.13 (Shepard et al., 2010; Shepard et al., 2021),\nwhich is much higher than that of S-type or C-type asteroids ( ∼0.14, e.g., Magri et al.,\n2007) and, as such, indicative of a higher metallic content of the surface material (Ostro\net al., 1985). Another proxy for surface metal content is the thermal inertia Γ, which is\nthe resistance of the surface material to change temperature during the diurnal illumi-\nnation cycle. Γ is a composite parameter function of the material thermal conductivity\nκ, specific heat capacity cpand bulk density ρof the surface (Γ =√κcpρ). Thermal\ninertia Γ is measured in units of J m−2K−1s−0.5, which will hereafter be referred to as\n“tiu” for “thermal inertia unit”. For a given particle size, metal-rich regolith tend to have\nhigher thermal inertia than silicate-rich ones (Matter et al., 2013); consistently, some M-\ntype asteroids appear to have higher thermal inertia than similar-sized C-type or S-type\nasteroids (e.g., Delbo et al., 2015). However, evidence for a correlation between thermal\ninertia and radar albedo among M-type asteroids is marginal to none (Table 1; see also\nLandsman et al., 2019; Elkins-Tanton et al., 2020).\nSpectroscopic observations of M-type asteroids also revealed the presence of sili-\ncate features in otherwise featureless iron-meteorite-like spectra (e.g., Fornasier et al.,\n2010; Hardersen et al., 2005b). This material may be what remains of stripped rocky man-\ntles, supporting the hypothesis that M-type asteroids are the cores of differentiated plan-\netesimals (e.g., Bell et al., 1989; Asphaug & Reufer, 2014; Scott et al., 2015). At the low\npressures of planetesimals’ magma oceans, olivine is expected to be the first mineral to\ncrystallize across a vast range of possible bulk compositions (Elkins-Tanton et al., 2013;\nElkins-Tanton & Weiss, 2017). Although the spectroscopic signature of olivine was ob-\nserved on M-type asteroids (766) Moguntia, (798) Ruth, and (1210) Morosovia (Hardersen\net al., 2011), the near-infrared spectra of many other M-type asteroids, such as Psyche,\n(22) Kalliope, (69) Hesperia, (110) Lydia, and (201) Penelope were found to show only\na weak absorption feature near 0.93 µmindicative of low-Fe, low-Ca pyroxene (B. E. Clark\net al., 2004; Hardersen et al., 2005a; Fornasier et al., 2010; Birlan et al., 2007; Sanchez\net al., 2017). Pyroxenes may form through reduction of olivine in a low oxygen fugac-\nity environment, such as that of the nebular region close to the Sun or in a forming plan-\netary core (Hardersen et al., 2005a; Elkins-Tanton et al., 2013). The dominance of one\nmineralogy over the other in the M-type population could be indicative of where the as-\nteroids accreted or how they differentiated (Hardersen et al., 2005a).\nThe third component detected on the surfaces of M-type asteroids is hydrated min-\nerals (e.g., Rivkin et al., 2000; Shepard et al., 2015; Takir et al., 2017). If hydrated min-\nerals are endogenic to M-types, this would suggest that they accreted beyond the water-\nice condensation line of the early solar system (Landsman et al., 2015; Elkins-Tanton et\nal., 2022). Another explanation is that the hydrated material is exogenic (Busarev, 2002;\nShepard et al., 2015; Avdellidou et al., 2018), consistent with predictions of low volatile\ncontent in silicate magmas on planetesimals (Weiss & Elkins-Tanton, 2013) and with space-\ncraft observations of implanted exogenic material on asteroids of sizes ranging from 500-\nkm-sized Vesta to sub-km-sized (101955) Bennu (Reddy et al., 2012; DellaGiustina et\nal., 2021). Another possible explanation is that space weathering of M-type asteroids’\n–4–manuscript submitted to JGR: Planets\nsurface material may produce hydrated minerals via interaction of solar wind protons\nwith oxygen-bearing minerals (Landsman et al., 2015), as was proposed for the Moon\n(e.g., R. N. Clark, 2009; Sunshine et al., 2009; Pieters et al., 2009); although, more lab-\noratory experiments are needed to test this proposal specifically for M-types.\n1.1 This work\nOn Psyche, variations in metal and silicate content have been proposed on the ba-\nsis of rotational variations of the surface at visible, infrared and radar wavelengths (Viikinkoski\net al., 2018; Ferrais et al., 2020; Hardersen et al., 2005b; Sanchez et al., 2017; Takir et\nal., 2017; Shepard et al., 2021). de Kleer, Cambioni, and Shepard (2021) observed Psy-\nche in thermal emission using the Atacama Large (Sub-)Millimeter Array (ALMA) at\na resolution of 30 km over 2/3 of its rotation and fit for the thermal inertia and dielec-\ntric constant (that is, the polarizability of the surface material, to be presented in Sec-\ntion 2.2) with a single value of each property over the entire surface. Their results showed\nthe presence of thermal anomalies which support the heterogeneity of surface compo-\nsition and could not be explained by surface albedo variations only. Motivated by this,\nhere we fit a model to the same ALMA dataset that allows thermal inertia and dielec-\ntric constant to vary across the surface. We describe the methodology in Section 2. In\nSection 3, we present the results as longitude-latitude maps of surface properties and test\ntheir robustness to model assumptions. In Section 4, we discuss possible metal-rich and\nmetal-poor areas on the basis of dielectric constant measurements and discuss a promi-\nnent feature in the thermal inertia map.\n2 Materials and Methods\n2.1 Data\nThe data analyzed in this paper were obtained at ALMA on 2019 June 19 between\n06:33 and 09:11 UT. The observations, data reduction, and imaging methods are described\nin de Kleer, Cambioni, and Shepard (2021) and are only briefly summarized here. Data\nwere obtained at a wavelength of 1.3 mm over 8 GHz of total bandwidth, all of which\nwas used in the analysis. The maximum baseline at the time of observation was 16.2 km,\nand Psyche was at a distance of 2.04 AU from Earth (and 2.78 AU from the Sun) result-\ning in a spatial resolution of ∼30 km on Psyche, or an angular resolution of ∼0.02”. The\nsub-observer and sub-solar latitudes on Psyche at the time of observation were -14◦and\n3◦respectively, and the solar phase angle was 17◦. Rotation curves of the total flux den-\nsity and peak brightness temperature for the ALMA observations are in Figure 2 of de\nKleer, Cambioni, and Shepard (2021). The data were reduced and calibrated by the ALMA\npipeline, and were further calibrated and imaged using the Common Astronomy Soft-\nware Applications (CASA) package (McMullin et al., 2007). The duration of each ALMA\nscan of Psyche was under 1 minute, and several scans were imaged together to improve\nthe signal-to-noise. Loss of resolution from smearing was avoided by keeping the observ-\ning interval of jointly-imaged scans below 6 minutes. Each final image has a signal-to-\nnoise of ∼45-50 and a noise level around 2.5 K.\n2.2 Choice of model parameters\nMeasurements of thermal emission of asteroids provide constraints on the parti-\ncle size, porosity, and/or metal content of the surface materials. Thermal observations\nfrom Earth have been typically acquired in infrared wavelengths and interpreted using\nthermophysical models (e.g., Delbo et al., 2015) for which the free parameters are the\nthermal inertia Γ (defined in Section 1) and the roughness of the surface f. The latter\nmeasures the root-mean-square slope of topography that has spatial scales below the res-\nolution of the shape model of the asteroid. A rougher surface has higher mean surface\n–5–manuscript submitted to JGR: Planets\nslope and stays warmer for longer periods of time than it would if it were smoother due\nto higher absorption of sunlight and more self-heating (e.g., Delbo et al., 2015; Rozitis,\n2017). Different approaches to model surface roughness exist in literature (Delbo et al.,\n2015). The thermophysical model we use here (to be presented in Section 2.3) allows for\nmodelling of surface roughness by carving one spherical craters onto each facet of the\nshape model and tune the root-mean-square slope fof the facets by either changing the\nopening angle or the areal fractional coverage of the craters, or both.\nMid-infrared observations do not currently resolve the thermal emission from the\nsurface of Psyche because the high required angular resolution ( ∼0.03”) is beyond the\ncapabilities of existing facilities. Alternatively, de Kleer, Cambioni, and Shepard (2021)\nused ALMA to acquire thermal emission maps of Psyche at a resolution of 30km/pixel.\nALMA is an astronomical interferometer composed of 66 antennas distributed in the Cha-\njnantor plateau in Chile. The antennas observed Psyche’s thermal emission at millim-\niter wavelengths and combined the signals interferometrically into a synthetic signal equiv-\nalent to that observed by a putative telescope with an effective diameter of 16 kilome-\nters. The foundational theory of how to analyze thermal emission of solar system bod-\nies at millimeter wavelengths can be found in numerous investigations of the Moon (e.g.,\nPiddington & Minnett, 1949; Muhleman, 1972; Ulich et al., 1974; Zheng et al., 2019),\ndwarf planet Ceres (W. J. Webster et al., 1988; J. Webster William J. & Johnston, 1989;\nRedman et al., 1998; Altenhoff et al., 1994; Li et al., 2020), asteroids (3) Juno (Altenhoff\net al., 1994; ALMA Partnership et al., 2015), (4) Vesta (e.g., J. Webster William J. &\nJohnston, 1989; Altenhoff et al., 1994), Psyche (de Kleer, Cambioni, & Shepard, 2021),\n(21) Lutetia (Gulkis et al., 2012) and (2867) Steins (Gulkis et al., 2010), and of icy so-\nlar system bodies (Muhleman & Berge, 1991; Lellouch et al., 2017; Trumbo et al., 2018;\nde Kleer, Butler, de Pater, et al., 2021). Discussion of issues related to physical prop-\nagation of thermal emission at different wavelengths can be found in Lagerros (1996);\nRedman et al. (1998); S. J. Keihm (1982); S. Keihm et al. (2013); Hayne et al. (2017),\nand in de Kleer, Butler, Cordiner, et al. (2021) for the specific case of solar system sci-\nence with ALMA.\nAt millimeter wavelengths, the thermal emission of the surface (or, equivalently,\nits brightness temperature) is sensitive to the complex dielectric constant ϵof the near\nsurface material and the temperature profile within the (sub)surface. The dielectric con-\nstant measures the polarizability of the surface material, has real part ϵ′and imaginary\npartϵ′′(as such, ϵ=ϵ′+iϵ′′) and is equal to ˜ n2, where ˜ n=n+iKis the complex re-\nfractive index. nis the ratio of the speed of light in a vacuum to the speed of light in\nthe material and Kis the extinction coefficient measuring the amount of attenuation when\nthe electromagnetic wave propagates through the material. The extinction coefficient de-\nfines the electrical skin depth δelec=λ/(4πK) within which the emission is reduced\nby 1/e assuming a non-conducting material for emission at wavelength λ. While this depth\nis small at infrared wavelengths, it becomes significant at longer wavelengths, hence the\nneed to integrate the thermal emission within the subsurface (e.g., Gulkis et al., 2010)\nfor a temperature profile as a function of thermal inertia Γ and surface roughness f. Since\n˜ndefines the reflection coefficients of the surface, the millimeter emissivity of the sur-\nfaceE, that is, the degree of emission relative to a blackbody, is a decreasing function\nof dielectric constant. For low-emissivity asteroids like Psyche (global E= 0.4 – 0.7, de\nKleer, Cambioni, & Shepard, 2021), thermal emission at millimiter wavelengths is par-\nticularly sensitive to metal content and porosity of the surface material (see their sec-\ntion 3.5.3).\n2.3 Thermal emission images\nTo interpret the ALMA observations, we generate thermal emission images at the\nsame viewing geometry and wavelength of the ALMA data with thermal inertia and di-\nelectric constant as free parameters. The thermophysical simulation setup and the method-\n–6–manuscript submitted to JGR: Planets\nology to create model images are the same of de Kleer, Cambioni, and Shepard (2021),\nwith a few adaptions as described in the following.\nWe use a well-established ThermoPhysical Model (TPM, Delbo et al., 2015) to solve\nthe non-dimensionalized 1-D heat conduction equation with thermal inertia Γ as free pa-\nrameter. We explore the following values of thermal inertia: Γ = (25, 75, 116, 135, 156,\n181, 210, 283, 442, 594, 1000) tiu. We assume surface roughness f= 0◦(that is, a smooth\nsurface) as this was found to best-fit the thermal emission data at both infrared (Matter\net al., 2013; Landsman et al., 2018) and mm-wavelengths (de Kleer, Cambioni, & Shep-\nard, 2021). We assume bolometric emissivity Ebol= 0.9, bond albedo A= 0.05, and\nasteroid topography as defined by the shape model by Shepard et al. (2021). We test the\nrobustness of the results against the assumption that the facets of the shape model are\nsmooth and the entire surface has the same Bond albedo in Section 3.4.\nNext, we compute the blackbody flux density for the facets of the shape model based\non the temperature output of the TPM. For each facet of the shape model, the TPM com-\nputes the temperature as a function of dimensionless depth parameter x’ = x/ δth, where\nx is the depth in physical units and δthis the diurnal thermal skin depth over which the\namplitude of the diurnal temperature wave is reduced by 1/e:\nδth=r\nP\nπΓ\nρcp(1)\nwith rotation period P= 4.195948 hours for Psyche (Shepard et al., 2021). For each value\nof Γ, we compute the corresponding δthassuming a bulk density ρ= 3500 kg/m3which\nwas derived from Psyche’s radar albedo of 0.34 ±0.08 by Shepard et al. (2021) and a\nvalue of specific heat cp= 370 J/kg K. The latter is an average value of possible me-\nteorite analogs for Psyche identified by Elkins-Tanton et al. (2020): mesosiderites ( cp=\n383±6 J/kg K), iron IIAB ( cp= 342 ±6 J/kg K) and iron IIIAB ( cp= 375 ±23 J/kg\nK) computed at ∼200 K (Consolmagno & Britt, 2013; Consolmagno et al., 2013). We\nconvert the dimensionless depth parameter x’ to physical units and calculate the tem-\nperature as a function of physical depth. We use Eq. 3 from de Kleer, Butler, de Pater,\net al. (2021) to integrate the blackbody flux density through the subsurface along the\nviewing path. The integration along the viewing path depends on the dielectric constant\nϵ. Because K2/ϵ′<0.006 for most meteorites (Campbell & Ulrichs, 1969), hereafter we\nmake the simplification that ϵ=ϵ′as in de Kleer, Cambioni, and Shepard (2021). We\nexplore the following values of ϵ: (3.0, 7.5, 12.0, 16.5, 17.5, 18.5, 19.5, 20.5, 21.5, 25.5,\n30.0, 55.0, 80.0). This array is more finely sampled around the best-fit ϵ= 18.5 found\nby de Kleer, Cambioni, and Shepard (2021). We discuss the range of applicability of the\nresults based on the assumed bulk density value in Section 4.1.\nNext, we correct the thermal emission from each facet for the millimeter emissiv-\nityEof the surface material. The emissivity as a function of emission angle is computed\nusing the Fresnel reflection coefficients for polarization parallel and perpendicular to the\ndirection of propagation, which in turn depend on the complex refractive index (see Eq.\n9 in de Kleer, Cambioni, & Shepard, 2021) and thus on the dielectric constant of the ma-\nterial (Section 2.2). We explore the same array of values of dielectric constant as described\nabove.\nNext, we obtain model thermal-emission images for Psyche by projecting the shape\nmodel by Shepard et al. (2021) onto the plane-of-sky in the International Celestial Ref-\nerence System (ICRS), which is that of the ALMA observations. We illustrate the pro-\njection of the surface areas onto the R.A. and decl. observed by ALMA in Figure 1. The\nvertices of the facets are arrays defined in the asteroid reference system, which has pos-\nitive ˆ xaxis along the direction of the prime meridian as defined by Shepard et al. (2021),\npositive ˆ zaxis in the direction of the rotation pole, and the third axis to complete the\n–7–manuscript submitted to JGR: Planets\northonormal frame. We rotate the vertices to ICRS as described on the DAMIT web-\nsite ( https://astro.troja.mff.cuni.cz/projects/damit/pages/documentation ):\nrICRS =Rx(θ3)Rz(θ1)Ry(90−θ2)Rz(ϕ0+2π\nP(t−t0))rast (2)\nwhere θ1= 36◦andθ2= -8◦are the longitude and latitude of the asteroid spin axis, ϕ0\n= 341.56◦is the initial rotation angle of the asteroid, tis the observation epoch, t0is\nJ2000, and θ3= 23.44◦is the Earth’s obliquity. The matrices Rx,RyandRzare de-\nfined, for a generic angle θ, as:\nRx(θ) =\n1 0 0\n0 cos θ−sinθ\n0 sin θcosθ\n, Ry(θ) =\ncosθ0 sin θ\n0 1 0\n−sinθ0 cos θ\n, Rz(θ) =\ncosθ−sinθ0\nsinθcosθ0\n0 0 1\n\n(3)\nNext, we interpolate the thermal emission values of the projected facets that are\nvisible to ALMA in R.A. and decl. to produce a 2D array whose pixels correspond to\nthose of the ALMA images. Finally, the model images are convolved with the ALMA\nbeam to model the observation conditions. The above procedure is used to build a look-\nup table of images for every combination of the values of thermal inertia Γ and dielec-\ntric constant ϵlisted above.\n2.4 Fitting Γ and ϵto the thermal-emission curves\nOur goal is to fit a value of thermal inertia and a value of dielectric constant to the\nALMA images for each surface area, and see whether this can explain the residuals from\nthe best-fit global model by de Kleer, Cambioni, and Shepard (2021), who determined\na global best-fit value of thermal inertia and dielectric constant for the entire surface to\nthe data. Here the term “surface area” indicates the ensemble of projected facets of the\nshape model that are imaged in different ALMA pixels as the asteroid rotates and the\nsurface area moves across the plane of sky. We apply the procedure of Section 2.3 to the\ndata to determine the R.A. and decl. of the projected facets that are visible at a given\nobserving epoch and associate thermal-emission values to such facets based on the cor-\nresponding ALMA pixels. The thermal emission curve Dof surface area is the variation\nin thermal emission as a function of time as the asteroid rotates and a facet moves across\nthe plane of sky. We repeat the same procedure to generate model thermal-emission curves\nM(Γ, ϵ) for each facet as a function of thermal inertia Γ and dielectric constant ϵ. We\nevaluate the goodness-of-fit of each modelled thermal emission curve M(Γ, ϵ) for a given\nthermal inertia Γ and dielectric constant ϵby evaluating its χ2\nr(reduced chi-squared) func-\ntion (e.g., Hanuˇ s et al., 2015):\nχ2\nr(Γ, ϵ) =1\nNobs−2NobsX\ni=1(Mi(Γ, ϵ)−Di)2\nσ2\nD, (4)\nwhere σDis the image background noise (Section 2.1) and Nobs≤22 is the number of\ntimes the projected facet was observed by ALMA. The set Sof values of thermal iner-\ntia and dielectric constant that best fit the thermal emission curve Dare found by tak-\ning the median of the (Γ , ϵ) whose χ2\nr(Γ, ϵ) values meet the following criterion:\n(Γ, ϵ)∈Sif χ2\nr(Γ, ϵ)< min χ2r×\u0012\n1 +r\n2\nNobs−2\u0013\n. (5)\nWe consider the median to be a more reliable estimator of the central tendency of\nSthan the minimum of the χ2\nr, which is not distinguishable from other solutions in S\n–8–manuscript submitted to JGR: Planets\nin virtue of Equation 5, the mean of S, which is more sensitive to outliers that the me-\ndian, and the mode of S, which is not sensitive to elements in the set different from the\nmost recurrent ones. We compute the uncertainty of the best-fit values of (Γ , ϵ) as the\nstandard deviation of the solutions in S.\n2.5 Mapping (Γ , ϵ) in longitude and latitude\nWe display the best-fit (Γ , ϵ) and uncertainties as function of asteroid-centric lat-\nitude αand latitude βusing the Mollweide (equal-area, pseudocylindrical) cartographic\nmap projection technique with the prime meridian (PM) at the center of the map. To\ntake into account the correlation between nearby pixels that characterizes the ALMA\nimages (that is, each ALMA beam contains about 45 pixels, de Kleer, Cambioni, & Shep-\nard, 2021), we bin the surface properties in longitude and latitude on a grid with elements\nof angular size 5◦×5◦and apply Gouraud shading (Gouraud, 1971). We hatch the re-\ngion between longitudes 60◦W and 120◦W to indicate that the solution for that region\ncould be affected by model artifacts. The region was observed at high emission angles\nin less than 1/3 of the ALMA observations and includes an equatorial area (longitude\n∼90◦W, latitudes ±30◦) where unmapped topography could be present (Shepard et\nal., 2021). We remove from the maps those areas that have a bad goodness of fit, that\nis,χ2\nr>10 (the optimal value is χ2\nr∼1), or Nobs≤3. The latter condition follows from\nEquations 4–5, where a surface area shall be visible a number of times at most equal to\nthe number of degrees of freedom plus 1 in order for the inverse problem to be well-conditioned.\nAt least two data points are needed to derive the best-fit value of Γ (which controls the\nvariation of temperature as a function of time), and another data point is needed to break\nthe degeneracy on the best-fit value of ϵ. However, if a surface area is observed just for\na few epochs near the terminator and then it rotates away from sight, its surface prop-\nerties may remain unconstrained even if Nobs≥3. To assess the minimum required num-\nber of Nobsto fit the data, we test the fitting procedure in Section 2.4 to retrieve the sur-\nface properties of synthetic Psyche’s images with added Gaussian noise based on σDfrom\nthe data. We confirm that constraining a solution for Nobs≥3 is possible everywhere\non Psyche despite for a few areas at the south pole, which we remove from the map.\n3 Results\n3.1 Comparison between the global solution and the local solution\nFigure 2a is the map of brightness temperature corresponding to the time-averaged\nthermal emission observed with ALMA, that is, the brightness temperature correspond-\ning to the observed thermal emission averaged across the different epochs when the lo-\ncation was visible. Figure 2b and 2c are maps of the time-averaged model brightness tem-\nperature and time-averaged residual brightness temperature, respectively, for a smooth\nsurface with Γ = 283 tiu and ϵ= 18.5 which is the global best-fit case from de Kleer,\nCambioni, and Shepard (2021). Figure 2d and 2e are the maps of the time-averaged model\nbrightness temperature and time-averaged residual brightness temperature obtained by\nfitting a value of Γ and ϵto the thermal emission curve of each area on Psyche (this study).\nHereafter, we will refer to the results in panels b and c as the “global solution” and those\nin panels d and e as the “local solution”. We find that the structures in the tempera-\nture residual map of the global solution (Figure 2c) are well explained by a variegation\nin thermal inertia and dielectric constant over the surface: the residuals for the local so-\nlution (Figure 2e) are randomly distributed over the surface with mean value and stan-\ndard deviation equal to -0.3 K and 1.3 K, respectively.\n–9–manuscript submitted to JGR: Planets\nFigure 1. As the asteroid rotates during the observation epochs, the same facet of longitude\nαand latitude βis seen at different right ascensions R.A. and declinations decl. in the ALMA\nimages. We use the knowledge of the (R.A., decl.) of the projected facets and the (R.A., decl.) of\nthe ALMA pixels to associate a thermal-emission curve to each facet. We then fit modelled ther-\nmal emission curves corresponding to different values of thermal inertia and dielectric constant to\nthe observed thermal emission curves.\n–10–manuscript submitted to JGR: Planets\nGlobal solution\nA single value of thermal inertia and \ndielectric constant for the entire surface Local solution\nA value of thermal inertia and dielectric \nconstant for each surface region a\nb\ncd\ne\nFigure 2. ALMA data (top panel), global solution (left panels) and local solution\n(right panels) of the thermal emission of Psyche a, Time-averaged brightness temperature\ncorresponding to ALMA data. b, Time-averaged brightness temperature corresponding to the\nbest-fit surface properties in de Kleer, Cambioni, and Shepard (2021). c, Time-averaged resid-\nual brightness temperature (that is, time-average of the model minus the data) for the model in\npanel b. d, Time-averaged brightness temperature corresponding to the best-fit properties for\neach surface area (this study). e, Time-averaged residual brightness temperature for the model\nin panel d. In panels d and e, we removed the areas that were either observed less than 3 times\nor have bad goodness of fit ( χ2\nr>10). The hatched region is where possible model artifacts may\naffect the quality of the solution (see Section 2.5). The meridians are spaced 60◦apart, so that\n“West” indicates PM −180◦and “East” indicates PM+180◦. The parallels are spaced 30◦apart.\nThe sub-observer point is at latitude −14◦, around which the resolution of the maps is estimated\nto be∼5◦.\n–11–manuscript submitted to JGR: Planets\nFigure 3. Maps of thermal inertia and dielectric constant of asteroid (16) Psyche\nderived from the ALMA thermal emission data. a, Best-fit thermal inertia. b, Uncertainty\nof the thermal inertia. c, Best-fit dielectric constant. d, Uncertainty of the dielectric constant.\ne, Goodness-of-fit in units of χ2\nr(Equation 4). f, Surface coverage (that is, how many times how\nmany times a surface area was observed by ALMA). In panel a, the dashed box indicates the\nBravo-Golf region discussed in Sections 3.3 and 4.2. In panel c we indicate 10 areas whose surface\nmaterial has median dielectric constant at least 1 standard deviation higher (squares, labelled\n1–5) or lower (circles, labelled 6–10) than the median global value of Psyche. For the significance\nof these comparisons and their interpretation, see Section 4.1. The names of surface area in panel\nf are from Shepard et al. (2021). 3-D animations of panels a and c are available as supporting\ninformation (videos S1 and S2, respectively).\n–12–manuscript submitted to JGR: Planets\nFigure 4. Most of Psyche’s surface has thermal inertia ∼150–300 J m−2K−1\ns−0.5and dielectric constant ∼15–25, consistent with the global values presented\nby de Kleer, Cambioni, and Shepard (2021). a, Thermal inertia. b, Dielectric constant. c,\nGoodness-of-fit. The black histograms display areal abundance of surface area with Γ, ϵorχ2\nrin\nthe bin. The values are those from the maps of Figure 3, weighted according to the correspond-\ning area on the best-fit ellipsoid by Shepard et al. (2021). The red curves are the corresponding\ncumulative distributions. For panel a and b, the bin edges corresponds to the midpoints between\nconsecutive values of Γ and ϵexplored with the thermophysical model (Section 2).\n–13–manuscript submitted to JGR: Planets\n3.2 Maps of thermal inertia and dielectric constant\nFigure 3 displays the maps of Psyche’s surface properties derived from the ALMA\ndata: thermal inertia Γ (best-fit value: panel a; uncertainty: panel b); dielectric constant\nϵ(best-fit value: panel c; uncertainty: panel d); the goodness-of-fit of the local solution\n(panel e); and the surface coverage achieved with ALMA (panel f), that is, how many\ntimes a surface area was observed by ALMA. The areal abundance of surface areas with\na certain value of thermal inertia, dielectric constant and goodness-of-fit is displayed in\nFigure 4, together with the corresponding cumulative distributions (red curves). Most\nareas have χ2\nr<5–6, which we consider satisfactory for these types of observations.\nWe find that the surface thermal inertia of Psyche ranges between ∼25 tiu and ∼600\ntiu, although most of the surface has a thermal inertia in the range 150–300 tiu (Figures\n3a and 4a). To the west of longitude 30◦W, we observe that the uncertainty of the ther-\nmal inertia measurements (Figure 3b) are comparable to the best-fit values of the ther-\nmal inertia, that is, the measurements are not well constrained, while the thermal in-\nertia measurements are better constrained in most of the Eastern hemisphere. This is\nprobably because the Western hemisphere was observed a fewer number of epochs (and\nhence times of day) than the Eastern hemisphere (Figure 3f). In areas with thermal in-\nertia Γ >300 tiu, the best-fit thermal inertia tend to have higher uncertainties because\nthe diurnal curves of the thermal emission become more and more alike as the surface\nthermal inertia increases.\nWe find that the dielectric constant of Psyche’s surface material ranges between\n∼8 and ∼60 (Figures 3b and 4b), although most of the surface has dielectric constant\nin the range 15–25, consistent with the global solution by de Kleer, Cambioni, and Shep-\nard (2021). Analogously to the thermal inertia measurements, the highest uncertainties\nfor the dielectric constant are observed in the Western hemisphere likely because of lower\nsurface coverage (Figure 3f). Overall, the measured dielectric constant values have much\nlower uncertainties than the thermal inertia values.\n3.3 Thermal signature of a large mass-deficit region\nIn the Eastern hemisphere (where the values of thermal inertia are better constrained),\nwe observe a prominent feature in the thermal inertia map extending between longitudes\n15◦W and 60◦E (that is, the region in the dashed white box in Figures 3a and 5a). The\nfeature is a low-thermal-inertia region surrounded by a ring of high-thermal inertia ar-\neas and corresponds to the Bravo and Golf regions in Shepard et al. (2021). We refer to\nthis region as Bravo-Golf hereafter. In Figure 5a, we plot the altitude map of Psyche com-\nputed by subtracting Psyche’s best-fit ellipsoid to its shape model. Figure 5b is the same\nmap of thermal inertia of Figure 3a with the altitude contours superimposed onto it. In\nFigure 5c, we plot both a thermal-inertia profile and an altitude profile (black and red\ncurves, respectively) taken along longitude 35◦E. We observe that the two profiles closely\nresemble one another, and that the thermal inertia of the lowlands is statistically dis-\ntinct from that of the highlands.\nTo take into account the uncertainties in the value of thermal inertia, we perform\na two-sided Spearman test to try the null hypothesis that a random distribution in ther-\nmal inertia and elevation in-between longitudes 15◦W and 60◦E and latitudes ±50◦could\nreproduce the observed feature. We perform the test 10,000 times, where at each trial\nwe draw sample from a Gaussian distribution with mean and standard deviation equal\nto the best-fit Γ and its uncertainty. We find that the correlation between altitude and\nthermal inertia in the region is statistically significant as the null hypothesis has a prob-\nability below 10−5in each trial (Figure 5d). We discuss the thermal-inertia feature in\nSection 4.2.\n–14–manuscript submitted to JGR: Planets\nFigure 5. The lowlands in the Bravo-Golf region (longitudes 15◦W to 60◦E) have\na lower thermal inertia than the surrounding highlands. a, Altitude map computed by\nsubtracting the best-fit ellipsoid of Psyche from its shape model by Shepard et al. (2021). The\ndashed white box indicates the Bravo-Golf region analyzed in Section 3.3 and discussed in Sec-\ntion 4.2. We also indicate the same 10 areas of Figure 3c whose surface material has median\ndielectric constant at least 1 standard deviation higher (squares, labelled 1–5) or lower (circles,\nlabelled 6–10) than the median global value of Psyche (Section 4.1). b, Altitude contours super-\nimposed on the map of best-fit thermal inertia of Figure 3a. c, Latitudinal profiles at longitude\n35◦E of thermal inertia and altitude (black and red curves, respectively). d, Results of the two-\nsided Spearman test of the null hypothesis that a random distribution of thermal-inertia- and\nelevation-values could produce the observed correlation of panel b when uncertainties in thermal\ninertia are considered, for values in-between longitudes 15◦W and 60◦E and latitudes ±50◦. The\nSpearman p-value is lower than 10−5in all the 10,000 trials, which confirms that the thermal\ninertia is correlated with altitude in the region.\n–15–manuscript submitted to JGR: Planets\n3.4 Robustness of the results against model assumptions\nWe test the robustness of the results to the assumed value of Bond albedo (Sec-\ntion 2.3). Variegations in albedo on Psyche were observed by Viikinkoski et al. (2018),\nwho assumed that the albedo could vary up to ±25% of the nominal value — a reason-\nable range given that the problem of extracting albedo variations from lightcurves is poorly\nconstrained. Using instantaneous thermal equilibrium, de Kleer, Cambioni, and Shep-\nard (2021) demonstrated that a variation of ±30% in Bond albedo is too small to ex-\nplain the magnitude of the time-averaged temperature residuals in Figure 2c. We con-\nfirm this by running additional TPM simulations for the global model of de Kleer, Cam-\nbioni, and Shepard (2021), but with 30% higher/lower albedo than the nominal case. While\nwe cannot exclude that the thermal anomalies are in part contributed by differences in\ntemperature driven by albedo variations, we conclude that the anomalies are mainly due\nto heterogeneity in the thermal inertia and dielectric constant of the surface materials.\nNext, we test the robustness of the results to the assumption of smooth surface (that\nis,f= 0◦, Section 2.3). We run additional simulations in which we carve on each facet\nan hemispherical crater with area equal to 60% of that of the facets (“rough model”).\nThis corresponds to a root-mean-square slope of f∼40◦whose effect on emissivity is\nmodelled as done in de Kleer, Cambioni, and Shepard (2021). We experiment with Γ =\n(116, 283, 442) tiu, which correspond to the lower quartile in Figure 4a, the best-fit global\nvalue from de Kleer, Cambioni, and Shepard (2021), and the upper quartile in Figure\n4a, respectively. Figure 6 is a map of those areas on Psyche where the rough model has\na better goodness of fit than the smooth model, that is, χ2\nr(rough )< χ2\nr(smooth ) (Eq.\n5). We find that χ2\nr(rough )>10 for most of the surface and that, in the few areas where\nχ2\nr(rough )< χ2\nr(smooth ), the difference between Γ( rough ) and Γ( smooth ) is within\nthe uncertainty of Γ( smooth ). The thermal-inertia solution in the Bravo-Golf region has\nχ2\nr(smooth )< χ2\nr(rough ) in most of the lowlands and χ2\nr(rough )>10 in the highlands,\nsuggesting that the thermal-inertia signature analyzed in Section 3.3 is unlikely to be\ndue to unmodelled roughness. We conclude that a smooth surface is a good assumption\nnot only globally, but also locally.\n4 Discussion and interpretation of the results\nIn Section 4.1, we interpret the variations of Psyche’s dielectric constant in terms\nof metal content of the uppermost few millimeters of regolith. In Section 4.2, we discuss\nthe thermal signature of the Bravo-Golf region (that is, the region delimited by the dashed\nwhite box in Figures 3a and 5a) and propose three scenarios that may explain it.\n4.1 Relative abundance of metals and silicates over the surface\nThe thermal inertia values of most of Psyche’s surface (Γ ∼150–300 tiu) is con-\nsistent with a metal-rich surface as previously noted by Matter et al. (2013) and de Kleer,\nCambioni, and Shepard (2021). Our results are also consistent with the low-emissivity\ncase from Landsman et al. (2018), who measured a thermal inertia of 100–200 tiu for an\nemissivity of 0.5 and 0.7, although their high-emissivity case finds a lower thermal in-\nertia of 5–25 tiu. The dielectric constant of most of Psyche’s surface ( ϵ∼15–25) is also\nconsistent with a metal-rich surface because it is higher than the typical range of Earth\nrocks ( ϵ∼2–10) and closer to that of meteorites with high metal content (e.g., Camp-\nbell & Ulrichs, 1969).\nIntriguingly, we do not observe a global correlation between thermal inertia and\ndielectric constant on Psyche. Interpreting this finding requires new laboratory measure-\nments of thermal conductivity of metal-rich particulates and their dielectric constants\nat ALMA-relevant frequencies. Numerous laboratory measurements of the dielectric con-\nstant of terrestrial rocks and meteorites exist in the range 20 MHz–35 GHz (Parkhomenko,\n–16–manuscript submitted to JGR: Planets\nFigure 6. Tests of robustness of the results. Relative difference between the thermal\ninertia Γ( rough ) obtained using the rough model and thermal inertia Γ( smooth ) obtained using\nthe smooth model, normalized to the standard deviation of the latter. The map plots only those\n(few) areas where the rough solution has better goodness of fit than the smooth solution (that is,\nχ2\nr(rough )< χ2\nr(smooth )<10).\n–17–manuscript submitted to JGR: Planets\n1967; Campbell & Ulrichs, 1969; Hickson et al., 2020). By contrast, laboratory data for\nrocks at millimeter wavelengths ( ∼100–300 GHz) are very limited (e.g., Brouet et al.,\n2014), and there are not data for powdered metals.\nWhile Psyche’s surface appears to be predominantly metal-rich, we also observe\nareas whose dielectric constant ranges between ∼8 to∼60. Assuming that the metal\non Psyche is in the form of metallic inclusions (de Kleer, Cambioni, & Shepard, 2021),\nϵ= 60 corresponds to porosity lower than 25% and a metal content >40%, while the\nporosity and metal content are unconstrained for ϵ∼5. These calculations assume no\nreduction in emissivity due to volume scattering (see Figure 8c in de Kleer, Cambioni,\nand Shepard (2021)). As such, the dielectric constant measurements (Figure 3c) suggest\nthat the relative abundance of metals and silicates of the first millimeters of regolith varies\nacross the surface.\nWe observe that the regions labelled as n. 1, 2, 3, 4 and 5 in Figure 3c and 5c and\ncentered at ∼(15◦N, 100◦W), (10◦N, 15◦W), (45◦S, 60◦E), (30◦N, 60◦E), and (40◦S, 120◦W)\nhave dielectric constant ∼4.9, 2.6, 2.6, 1.0, and 4.9 standard deviations higher than the\nmedian dielectric constant of Psyche, respectively. This is indicative of a locally higher\nmetal content of the first millimeters of the surface. Intriguingly, areas n. 1, 2, and 3 are\nin proximity of large depressions of Psyche’s shape (Figure 5a and Shepard et al., 2021)\nwhere ferrovolcanism, that is, eruption of metallic lava flows on the surface, could have\npreferentially occurred (Abrahams & Nimmo, 2019; Johnson et al., 2020). Abrahams and\nNimmo (2019) showed that molten iron may erupt on the surface of a pure iron-nickel\nremnant. If there is a silicate layer on top of the core, the iron melt may create intru-\nsive diapirs which can be successively uncovered through excavation by impacts (Raducan\net al., 2020). Johnson et al. (2020) showed that, if Psyche is a differentiated object with\na relatively thin silicate mantle and iron core, volatile-rich molten iron could be injected\ninto the mantle and erupt on the surface during its cooling. They do not provide spe-\ncific predictions on the eruption sites, but suggest that these are more favoured where\nthe crust/mantle are the thinnest, which could be associated with large depressions.\nThe areas labelled as n. 6, 7, 8, 9 and 10 in Figure 3c and 5c and centered at ∼\n(10◦S, 130◦W), (15◦S, 15◦E), (5◦S, 130◦E), (50◦S, 120◦E) and (50◦N, 50◦W) have di-\nelectric constant ∼1, 1, 1.5, 1.5, and 4.4 standard deviations lower than the median di-\nelectric constant of Psyche, respectively. This is indicative of a locally lower metal con-\ntent of the first millimeters of the surface. The surface materials at these locations could\nbe a mixture between metal- and silicate-rich materials. The presence of silicate mate-\nrials on Psyche in the form of orthopyroxenes was proposed by Sanchez et al. (2017) to\nexplain the detection of the 0.93- µmband in the visible near-infrared spectra of Psyche.\nAdditionally, Hardersen et al. (2005b), Fornasier et al. (2010) and Takir et al. (2017) de-\ntected the presence of the 3- µmband at longer wavelengths, which they associated with\nOH- or H 2O-bearing phases, possibly in the form of phyllosilicates (Takir & Emery, 2012).\nThe carbonaceous materials are likely exogenic (e.g., Shepard et al., 2015; Avdellidou\net al., 2018), consistent with experiments showing that hydrated projectiles may implant\nthe 3- µmband on metal-rich surfaces (Libourel et al., 2019). Exogeneous silicates are\ntherefore a likely contributor to the observed areas with low dielectric constant, but we\ncannot conclude that they are all predominantly carbonaceous, because dielectric con-\nstant is a poor discriminator of composition between different silicate-rich materials (Brecher\net al., 1975; Palmer et al., 2015). Area n. 10 has the lowest dielectric constant on Psy-\nche (ϵ∼8) and could be particularly rich in silicate materials within the first millime-\nters of the surface. However, we caution that model artifacts could be present at that\nlocation because the region was imaged in less than 1/3 of the ALMA observations and\nalways at high emission angles.\nAssuming that Psyche’s surface material is a mixture of rocks and iron sulfides/oxides\n(Eq. 15 in de Kleer, Cambioni, & Shepard, 2021), the range of dielectric constant on Psy-\nche corresponds to a local bulk density ranging from ρ∼1500 kg/m3(i.e., 100 wt.%\n–18–manuscript submitted to JGR: Planets\niron sulfides/oxides and 67% macroporosity) to 4500 kg/m3(i.e., 97 wt.% iron sulfides/oxides\nand 0% macroporosity). This density range encompasses the radar-derived ρ= 3500 kg/m3\n(which we assumed for Equation 1) and may also be consistent with the full density range\ninferred from the full range of measured radar albedos by Shepard et al. (2021). How-\never, we do not observe a spatial correlation between dielectric constant and radar albedo\non Psyche. For example, area n. 3 (India in Shepard et al., 2021) is a region of both high\ndielectric constant and elevated radar albedo, but area n. 9 (Hotel in Shepard et al., 2021)\nis a radar-bright area with low dielectric constant. Radar-bright regions with low ALMA\ndielectric constant could be metal-rich areas covered by a thin mantle of silicate-rich ma-\nterial or be covered in a silicate-rich regolith with embedded metal particles. We can-\nnot disentangle these two scenarios just based on dielectric constant and thermal iner-\ntia alone because our model assumes constant density with depth. In fact, the density\nof the uppermost millimeters of the surface (to which ALMA is sensitive to) can be dif-\nferent than that of the underlying first tens of centimeters, to which the radar is sensi-\ntive to. The latter is certainly true for the Moon (Hayne et al., 2017). Further research\nis therefore warranted to adjust radar densities to observations at ALMA wavelengths.\nThis task offers many challenges, including limitations of current techniques to convert\nradar albedo and dielectric constant into bulk density of the regolith, the different spa-\ntial resolutions of radar versus ALMA measurements, and the lack of measurements of\nthe dielectric properties of metal-rich materials in both solid and particulate form at ALMA\nwavelengths.\n4.2 Thermal inertia of the Bravo-Golf region\nHere we propose three non–mutually-exclusive scenarios to explain why the ther-\nmal inertia of lowlands of the Bravo-Golf region is systematically lower than that of the\nhighlands. The three scenarios are sketched in Figure 7.\nA first hypothesis is that the lowlands are metal-rich (thus radar-bright as observed\nby Shepard et al., 2021), but covered in a thin mantle of fine regolith, which lowers their\nthermal inertia with respect to the highlands where the regolith has coarser particle size.\nFor a regolith particulate, thermal conductivity increases with increasing particle size\nfor a given bulk density and specific heat capacity (e.g., Sakatani et al., 2018; Cambioni\net al., 2019). On airless worlds, impacts may induce accumulation of low-thermal-inertia\nfine regolith in places where the gravity field is the highest. The Bravo-Golf region could\nbe a place where fine material preferentially ponds because it is closer to the center of\nmass of Psyche than any other point on the surface. A similar argument was presented\nfor the bi-lobate asteroid (216) Kleopatra by Shepard et al. (2018) who, based on the\nanalysis of asteroid geopotential surface, suggested that loose material should preferen-\ntially migrate to Kleopatra’s neck region, consistent with radar observations. On Psy-\nche, we speculate that the large impacts which formed some of its depressions may have\nkicked-up fine regolith and/or generated seismic shaking which provoked accumulation\nof fines in the lowlands of the Bravo-Golf region. On asteroids greater than 100 km in\ndiameter, individual impacts are expected to induce only localized (regional) seismic ef-\nfects, but large impacts may still have widespread effects on the surface (Richardson et\nal., 2005).\nA second hypothesis is that the surface material covering the lowlands is more porous\nthan that of the highlands. Rock’s thermal inertia decreases with increasing rock poros-\nity (e.g., Grott et al., 2019; Cambioni et al., 2021), although measurements of meteorite\nthermal conductivity as function of porosity are primarily based on chondrites. Rock poros-\nity can be enhanced by impact-induced fractures. Laboratory experiments of hyperve-\nlocity impacts onto iron-nickel meteorite and ingots (Matsui & Schultz, 1984; Libourel\net al., 2019; Marchi et al., 2020) show that the fractures permeate the floor of the crater,\nwhile the rims acquire a rose-petal-shaped configuration and are not fractured, although\nresults could be affected by edge effects in the samples. If the fractures extend to the en-\n–19–manuscript submitted to JGR: Planets\nFigure 7. The three possible scenarios proposed to explain the thermal-inertia\ncontrast between the lowlands and the highlands in the Bravo-Golf region . The sce-\nnarios are described in Section 4.2.\ntire body, this may potentially explain the low density of some M-type asteroids (e.g.,\nPsyche and Kleopatra, Elkins-Tanton et al., 2020; Marchis et al., 2021). Alternatively,\nthe surface could be covered in highly porous boulders, whose thermal inertia could mimic\nthat of fine regolith (Rozitis et al., 2020; Grott et al., 2019; Okada et al., 2020; Cambioni\net al., 2021). This scenario, however, is unlikely for Psyche if its surface material has a\ncomposition analogous to that of iron meteorites (porosity <10%, Macke, 2010) or mesosiderites\n(porosity ∼12%, similar to the ∼14% of LL chondrites, but much lower than ∼30%\nof CM chondrites, Britt & Consolmagno, 2003).\nA third hypothesis is that the lowlands have a higher abundance of silicate-rich ma-\nterials than the highlands, consistent with having lower dielectric constant than some\nareas of the highlands (e.g., area 7 versus area 4 in Figure 3c). The silicate-rich mate-\nrials could be the residue of a silicate impactor that may have formed the Bravo-Golf\ndepression and whose material was mixed with the (metal-rich, thus radar-bright, Shep-\nard et al., 2021) subsurface material by subsequent smaller impacts. If that is the case,\nwe speculate that Psyche’s surface was behaving in a ductile way at the time of impact\nthat formed the Bravo-Golf depression, because projectile implantation efficiency is close\nto zero for most cooled impacts (Marchi et al., 2020), assuming a target composition as\nthat of iron meteorites. We nevertheless caution that impact contamination is unlikely\nto be limited to the crater floor (Libourel et al., 2019; Marchi et al., 2020) and the high-\nlands have lower albedo than the lowlands (Viikinkoski et al., 2018), in contrast with the\nfindings by Libourel et al. (2019) that silicate implantation lowers the albedo of the crater\nwith respect to exposed iron.\n–20–manuscript submitted to JGR: Planets\n5 Conclusion\nWe present a new analysis of the ALMA thermal emission data of asteroid (16) Psy-\nche published in de Kleer, Cambioni, and Shepard (2021) using a model which derive\na value of thermal inertia and dielectric constant for each region on the surface. We find\nthat most of Psyche’s surface has thermal inertia in-between 150–300 tiu and dielectric\nconstant 15–25.\nOur results support the metal-rich nature of Psyche’s surface materials, consistent\nwith previous studies (Matter et al., 2013). However, because our data spatially resolve\nthe surface, we also observe heterogeneity in the surface properties due to the presence\nof terrains whose thermal inertia range between ∼25 tiu and ∼600 tiu and dielectric con-\nstant between ∼8 and ∼60. However, we do not observe a direct correlation between ther-\nmal inertia and dielectric constant on Psyche’s surface. The array of detected dielectric\nconstant values is indicative of a mineralogy of the first millimiters of the surface rang-\ning from low-metal to high-metal content depending on porosity. Some metal-rich ar-\neas with high dielectric constant correspond to depressions where ferrovolcanism (Abrahams\n& Nimmo, 2019; Johnson et al., 2020) could have preferentially occurred. Other regions\nwith low dielectric constant could have higher abundance of silicate materials mixed with\nmetals. This is consistent with spectroscopic evidence of orthopyroxenes on Psyche (Sanchez\net al., 2017) and hydrated minerals (Takir et al., 2017) of likely exogeneous origin (Avdellidou\net al., 2018). In the thermal inertia map, we observe that the lowlands of a large depres-\nsion (dubbed Bravo-Golf and located between longitudes 15◦W and 60◦E) have statis-\ntically lower thermal inertia than the surrounding highlands. We propose that this could\nbe a signature of a thin mantle of fine regolith covering the lowlands; presence of frac-\ntures induced by impacts; and/or silicate materials implanted by impacts. All three sce-\nnarios are indicative of a collisionally evolved surface.\nIn conclusion, we provide evidence that Psyche is a metal-rich asteroid whose sur-\nface is heterogeneous, shows both metal and silicate materials, and appear evolved by\nimpacts. We look forward to the data from the NASA Psyche mission to get new insight\ninto this unusual and fascinating world.\nAcknowledgments\nThis paper makes use of the following ALMA data: ADS/JAO.ALMA#2018.1.01271.S.\nALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS\n(Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Re-\npublic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Obser-\nvatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Ob-\nservatory is a facility of the National Science Foundation operated under cooperative agree-\nment by Associated Universities, Inc. This research was funded in part by the Heising-\nSimons Foundation through grant 2019-1611. S.C. acknowledge funding through the Crosby\nfellowship of the Department of Earth, Atmospheric and Planetary Sciences of the Mas-\nsachusetts Institute of Technology. The authors thank M. Delbo and B. Weiss for insight-\nful discussions that improved this manuscript.\nConflict of Interest Statement\nThe authors declare no competing interests.\nData Availability Statement\nThe ALMA data are freely available at the ALMA data archive ( https://almascience\n.nrao.edu/alma-data ) under the project code: 2018.1.01271.S. The paper makes use\nof the ThermoPhysical Model by Delbo et al. (2015), which is publicly available at https://\nwww.oca.eu/en/marco-delbo in the section “Asteroid thermal models”, and the soft-\nware package PyVista by Sullivan and Kaszynski (2019). All the relevant equations and\nparameters for the reproducibility of the results are given in this paper and in de Kleer,\n–21–manuscript submitted to JGR: Planets\nCambioni, and Shepard (2021). Datasets for this research are available in these in-text\ndata citation references: results in Cambioni et al. (2022); shape model in Shepard et\nal. 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If the PQ symmetry breaking scale associated with the axion is below the\ninflationary reheating temperature, axion strings and domain walls populate the universe.\nMost of these strings and walls decay away into axion dark matter, but a small subset of the\nwalls will be self-enclosed surfaces that are not attached to any strings. These enclosed walls\ncan collapse in on themselves, compressing a large amount of energy into a small volume\nand potentially forming primordial black holes (PBHs). We study the number density and\ndynamics of these self-enclosed walls, taking into account their size distribution, Hubble\nexpansion, asphericities, and all stages of domain wall dynamics using a combination of semi-\nanalytic and numerical approaches. We find that axion models with a high axion decay\nconstant fa, such as those of interest in early matter-dominated cosmologies, yield a PBH\nabundance potentially observable by future gravitational lensing surveys. We note that the\nformalism developed here is also useful for predicting relic PBH abundances in other models\nthat exhibit unstable domain walls.arXiv:2402.03426v2  [hep-ph]  9 Mar 2024Contents\n1 Introduction 2\n2 Overview of Axion Cosmology and Topological Defects 4\n3 Closed Domain Wall Abundance 6\n3.1 Enclosed Domain Wall Number Density 6\n3.2 Domain Wall Shape Distribution 9\n3.3 Wall Formation Times 14\n4 Superhorizon Expansion of Domain Walls 17\n4.1 Simulation Setup 17\n4.2 Re-entry Radius and Collapse Time 20\n4.3 Re-entry and Collapse Tension 21\n5 Subhorizon Collapse of Domain Walls 23\n5.1 Compressibility of Perfectly Spherical Walls 23\n5.2 Compressibility of Realistic Walls with Asphericities 26\n6 Conditions for PBH Formation 29\n6.1 Compression into a Schwarzschild Radius 29\n6.2 Impediments to Collapse: Angular Momentum 31\n6.3 Impediments to Collapse: Collisions with Neighboring Defects 34\n7 PBH Abundance and Detection Prospects 35\n8 Conclusions 37\n9 Acknowledgements 40\nA Reheat Temperature from an Early Matter-Dominated Era 40\nB Superhorizon Evolution of Aspherical Domain Walls 43\nC Domain Wall Self-Gravitation 44\nD PBH Abundance in ALP Models 45\n– 1 –1 Introduction\nThe QCD axion (henceforth, just the axion1) is a highly motivated dark matter (DM) candi-\ndate [1–3] and may solve the Strong CP Problem in the Standard Model (SM). The axion is a\npseudoscalar particle which arises naturally as a pseudo-Nambu-Goldstone boson of a broken\nglobal U(1)PQsymmetry [4–11]. This PQ symmetry breaks at the PQ energy scale fadue to a\ncomplex scalar field Φ acquiring a vacuum expectation value. Two distinct axion cosmologies\ncan occur: If the PQ scale lies above the inflationary reheating scale, the axion field config-\nuration would have been homogeneous across the universe at early times [1–3, 12]. However,\nif the PQ scale lies below the inflationary reheating scale, the axion field configuration would\nhave varied from horizon to horizon [12, 13].\nIn this second scenario, the winding of the complex PQ field leads to the production of\ntopological line defects known as cosmic strings [14, 15]. These strings dynamically evolve\nuntil the universe cools to a temperature of TQCD≈102MeV, at which point the thermal\naxion mass grows larger than the Hubble scale. This creates dynamical brane-like structures\nknown as domain walls [13]. Most domain walls are appended to cosmic strings, but a few\nare entirely self-enclosed. These self-enclosed domain walls are the focus of this paper.\nIn a theory with only one classical axion field vacuum (often referred to as a theory where\nthedomain wall number NDWis equal to one), the string-wall network is not topologically\nstable and can decay [13]. For self-enclosed domain walls, this manifests itself as an inward\ncollapse: the wall surface rapidly accelerates inwards, converting its rest mass energy into\nkinetic energy, and compressing the energetic wall surface into a smaller and smaller volume.\nAt this stage, one of two things happens: The wall may break apart and begin oscillating,\nradiating away axion particles until the wall completely vanishes. Alternatively, if the wall\nis massive enough, sufficient energy may be forced towards the center of mass of the wall so\nthat a primordial black hole (PBH) forms [16]. In this paper, we study this PBH formation\nmechanism and calculate the PBH abundance predicted by axion models.\nA wide range of cosmological models predict the formation of PBHs in the early universe\n(see refs. [17–19] and references therein). Depending on the model, PBHs can take on a wide\nrange of masses, and they may make up some or all of the DM of the universe. PBHs are an\nespecially intriguing DM candidate in part because of their observability. If they have masses\nbelow 10−16M⊙, they emit Hawking radiation in the form of gamma rays, which has led to\nconstraints from gamma ray telescopes [20–23]. Heavier PBHs in the mass range 10−10M⊙\nto 1010M⊙have strong gravitational fields that deflect light from background stars as they\nmove across the sky via gravitational lensing. This has allowed multiple telescopes across the\nelectromagnetic spectrum to constrain this mass range [24–32], and near-future and current\nmissions will constrain these PBHs even further [33–35].\nIn this work, we present the first detailed study of PBH formation from self-enclosed\ndomain walls collapsing within an NDW= 1 axion cosmology where the PQ phase transition\noccurs after inflationary reheating. We carefully account for the number density and size\n1Not to be confused with axion-like particles (ALPs), which we consider in Appendix D.\n– 2 –distribution of self-enclosed domain walls around the QCD phase transition, and study all\nstages of dynamical wall evolution, starting with the walls expanding with the scale factor of\nthe universe, through their subhorizon collapse, and ending with their maximal compression.\nMoreover, we consider wall formation in both a standard cosmology and cosmologies with\nearly matter domination. These latter cosmologies are consistent with fa≳1011GeV [3, 36–\n39], which is preferred for PBH production. Note that since we are simply considering the\nQCD axion, a majority of the present-day DM abundance will naturally consist of axion\nparticles created via the misalignment mechanism and defect radiation. PBHs will necessarily\nmake up a sub-component of the DM since only a fraction of the axion field energy density\nexists in self-enclosed domain walls when the string-wall network collapses.\nIn related work, refs. [40, 41] discussed the collapse of axion domain walls into PBHs\nfor axion models where the number of degenerate vacuua NDWis greater than one. In\nthese models, the temperature at which domain walls form and the temperature at which\nstring-walls network begin to radiate and contract can be separated by several orders of\nmagnitude, allowing for more efficient PBH production (see also refs. [42–46], which emphasize\ngravitational wave production in similar scenarios). However, the stability of string-wall\nnetworks in these models also presents a problem: the domain walls overclose the universe [13,\n47]. This problem can be overcome by adding extra modifications to the axion potential to be\ncompatible with current cosmological constraints [48]. Ref. [49] proposed a model in which\naxion strings form during inflation, and experience a specific number of e-foldings such that\nthey re-enter the horizon right before BBN, allowing closed domain walls to be orders of\nmagnitude larger than the cosmological horizon at the QCD phase transition. Refs. [50, 51]\ndiscuss the potential of forming PBHs from axion domain wall collapse, but do not make a\nconcrete prediction for the fraction of DM composed of black holes fPBH. In our work, we\ncalculate the number density of enclosed walls and the effects of asphericities carefully and\nhence believe our manuscript to be the first to provide a reliable estimate of the present day\nPBH energy density Ω PBHin a fully post-inflationary scenario.\nIn Sec. 2, we briefly review axion cosmology, strings, and domain walls. We also discuss\nearly matter domination. In Sec. 3, we calculate the statistical properties of enclosed walls\nsuch as their size and number density using the methods of percolation theory (effectively, the\nKibble mechanism [14]) and random fields. In Sec. 4, we study the dynamics of domain walls\nbefore they enter the cosmic horizon. In Sec. 5, we study the evolution and compressibility\nof domain walls as they enter the horizon and collapse in isolation. In Sec. 6, we discuss how\nefficiently energy must be compressed in order for a domain wall to form a PBH, and we\ndiscuss how angular momentum and collisions with other defects suppress PBH formation in\npart of the axion parameter space. In Sec. 7, we compute Ω PBHandfPBH, and we discuss\nthese results in the context of current and future gravitational lensing surveys, which may be\nable to probe the PBH masses and abundances we predict. Finally, in Sec. 8, we provide a\nsummary and concluding remarks.\n– 3 –2 Overview of Axion Cosmology and Topological Defects\nAxion topological defects can be understood from the symmetries of the axion Lagrangian.\nAt high temperatures in the early universe, the UV physics of the axion is captured by the\nLagrangian of a complex scalar field Φ [7–11]\nLUV⊃ |∂µΦ|2−VPQ(Φ). (2.1)\nThe potential VPQis invariant under a global phase rotation of Φ (and a simultaneous phase\nrotation of the fields Φ couples to) and therefore respects a U(1) symmetry [4, 5]. As the\nuniverse cools, this U(1) symmetry is spontaneously broken and Φ acquires a vacuum expec-\ntation value ⟨Φ⟩=fPQ/√\n2eia/f PQ, with the axion, a, defined as the angular degree of freedom\nof Φ and fPQthe PQ breaking scale.2\nThe breaking of the U(1) symmetry implies a non-trivial vacuum, which results in the\nproduction of topological strings at the temperature T≈fPQ[53]. These axion strings are\nlines in physical space where the expectation value of Φ remains zero [14]. Outside the string\ncore of size δs∼m−1\nΦ∼f−1\nPQ, Φ rapidly approaches the true vacuum |⟨Φ⟩|=fPQ/√\n2, and\nawinds from 0 to 2 π×fPQaround the string [53]. The energy per unit length of the axion\nstring, µ, is roughly the energy density in the string core times the cross-section of the string\ncore, µ≈VPQ(Φ = 0) δ2\ns∼f4\nPQ×f−2\nPQ=f2\nPQ. A more realistic value is µ≈πf2\nPQln(fPQL),\nwhere the log arises from the integration of the U(1) potential roughly from the string core\nto the string separation length L[15].\nInitially, the abundance of strings formed at the PQ phase transition is set by the corre-\nlation length of the Φ field, as described by the Kibble-Zurek mechanism [14, 54–56]. The\ntangled axion strings self-intersect and approach an attractor solution with a density of ap-\nproximately one string per horizon [14, 57] as discussed in Sec. 3.3. The distance between\nstrings sets the correlation length of the axion field [14, 58] which in turn sets the typical\nscale of the axion domain walls that form much later around the QCD phase transition.\nThe origin of axion domain walls can be seen by integrating out the heavy radial mode of\nthe Φ field and considering the potential generated for aby strong dynamics near the QCD\nphase transition,\nLIR=1\n2∂µa ∂µa−VQCD(a), V QCD≃m2\na(T)f2\nPQ\nN2\nDW\u0014\n1−cos\u0012\nNDWa\nfPQ\u0013\u0015\n, (2.2)\nwhere fa≡fPQ/NDWis the axion decay constant, NDWis the domain wall number, Tis the\ntemperature of the SM gluons, and ma(T) is the temperature-dependent axion mass [15, 59].\nAt high temperatures far above the QCD phase transition, ma(T) is small so that VQCD\nis negligible. However, as the temperature of the universe drops and approaches TQCD≈\nΛQCD,ma(T) grows dramatically, and the explicit breaking of the U(1)PQsymmetry by\n2If Φ couples to other complex scalar fields that also acquire vacuum expectation values, such as in DFSZ\nmodels [7, 9, 11, 52], then abecomes a linear combination of the angular degrees of freedom of those fields.\n– 4 –VQCD becomes important. The resultant cosine potential possesses NDWunique minima\nfor axion field values θ≡a/fPQbetween 0 and 2 π(the minima at θ= 0 and 2 πmust\nbe identified). Since the axion field varies by this same amount as one winds around the\naxion string in physical space, any curve enclosing an axion string traces out these NDW\nunique vacuua. This explicit breaking of the U(1)PQsymmetry down to ZNDWresults in\nthe production of NDWdomain walls emanating from each string. These domain walls are\nsurfaces in physical space where NDWθ=πand where the axion field sharply interpolates\nbetween two adjacent minima in the potential.3The physical thickness of the wall, δ, is set\nby the inverse curvature of cosine potential, ma(T)−1, while the energy per unit area of the\nwall, σ, is set by the energy density in the wall surface times the thickness of the wall, which\nis given parametrically by σ≈VQCD(NDWθ≈π)δ∼m2\na(T)f2\na×δ∼ma(T)f2\na. A precise\nderivation of σgives σ= 8ma(T)f2\na[15, 59, 60].4\nTheNDWdomain walls which radially ‘fan’ out from each axion string are initially thick\nin the early universe when the horizon size is smaller than their thickness, δ. However, the\nwalls become dynamical and ‘form’ when δbecomes smaller than the horizon, t. Since the\nwall thickness δ≡ma(T)−1is a function of temperature, with ma(T) rapidly approaching\nits zero temperature value near the QCD phase transition, δtypically becomes less than t\naround this time too. The temperature-dependent wall thickness δ≡m−1\nais given by\nδ≡ma(T)−1≃ma(T= 0)−1×\n\n\u0012T\nTQCD\u0013n/2\nT≳TQCD,\n1 T≲TQCD.(2.3)\nFor the QCD axion, instanton and lattice computations suggest n≈7−8 and TQCD≃100\nMeV [61–63]. To parameterize the uncertainty in the power nas well as extend the results\nof this paper to other axion-like-particles (ALPs), we keep nandTQCD as free parameters\nfor the remainder of this paper unless explicitly referring to the QCD axion. As discussed\nin Secs. 5 and 6, the conditions for PBH formation of collapsing axion walls do not depend\nexplicitly on the value of TQCDorma(T= 0), just ratios of T/T QCDorma(T)/ma(T= 0) up\nto mild factors of the relativistic degrees of freedom in the thermal bath, g∗. For ALPs, we\ndefine TQCD≡p\nma(T= 0)fa. For the QCD axion, ma(T= 0) can be precisely computed\nin chiral perturbation theory in terms of the up and down quark masses, muandmd, as well\nas the pion mass, mπ, and decay constant, fπ[6, 64]\nma(T= 0)≃6µeV\u0012fa\n1012GeV\u0013−1\n(QCD axion) . (2.4)\nFor the remainder of this paper, we focus on the case where NDW= 1 which are axion\nmodels that possess one unique vacuum, such as the KSVZ axion [7, 10]. These domain walls\nare surfaces in space where the axion field is θ=πand separate regions of space where θgoes\n3ForNDW= 1, the wall is the axion field configuration that interpolates between θ= 0 and 2 π.\n4The axion potential is not exactly Sine-Gordon due to axion-pion mixing as pointed out by Sikivie [59].\n– 5 –to 0 and 2 πon either side of the wall (see Figs. 8, 13). Axion theories where NDW= 1 are not\ntopologically stable since no residual discrete symmetry remains after U(1)PQis broken by\nQCD. The instability of a string-wall system is physically manifested by the walls pulling the\nattached string network apart into wall-bounded string membranes, which quickly decay into\naxions. Theories where NDW>1, such as the DFSZ axion, possess more than one unique\nvacuum and are topologically stable due to the residual ZNDWsymmetry. A universe with\ntopologically stable walls present today is incompatible with our present-day cosmology since\nthe walls will come to dominate the energy density of the universe [13, 47]. It is possible to\navoid this ‘domain wall problem’ if inflation occurs after the axion strings form or if there\nexists additional explicit U(1)PQbreaking which can lift the degeneracy between the different\nvacuua, thereby generating a pressure difference on the walls and causing the string-wall\nsystem to collapse. See ref. [40] for formation of PBHs in such cosmologies.\nWe consider wall formation in either radiation-dominated (RD) or matter-dominated\n(MD) eras to allow for consistent cosmologies across a broad range of fa: for example, for\nsufficiently large fa, axions may be overproduced as DM from the misalignment mechanism\nor from axion strings bounded by walls in a RD era. In an early MD era, however, the entropy\ngenerated during reheating can dilute this background abundance of axions to the observed\nDM in our universe today. An early MD era has been motivated in several different contexts,\ne.g. long-lived relics that decay out of equilibrium [37, 65], and the potential cosmological\nimplications of early MD on axions have been widely studied (see refs. [36–39, 66]).\nIn the following section, we discuss the abundance of self-enclosed walls – that is, walls\nthat are not attached to any axion string, which can form by chance due to θrandomly varying\non distance scales of order the string separation length L. These enclosed walls collapse under\ntheir own self tension and can form PBHs as discussed in Secs. 4-6.\n3 Closed Domain Wall Abundance\nIn this section, we discuss the number density and size distribution of self-enclosed domain\nwalls that form around the QCD phase transition. First, we explain how the lack of correlation\nof the axion field on distances greater than the string separation, L, gives rise to enclosed\nwalls. In Secs. 3.1 and 3.2, we perform a series of Monte Carlo numerical simulations of\nboth discrete and continuous uniform random fields (with support on [0 ,2π)) to mimic the\nincoherence of the axion field in different patches of the universe and use these results to\nidentify enclosed walls and calculate their number density and size distribution.\n3.1 Enclosed Domain Wall Number Density\nAround the QCD phase transition, every axion string forms the boundary of a domain wall as\ndiscussed in Sec. 2. However, it is also possible to form domain walls which are not attached\nto any string; these walls are closed surfaces of constant θ=π. The formation probability\nand size of enclosed walls arise due to the loss of correlation of the axion field θ=a/faon\ndistance scales greater than the string separation, L. That is, because of the randomness of\n– 6 –Figure 1 . An overview of the distinct cosmological phases of PBH formation from axion domain\nwall collapse. Left: The axion string network after the PQ Phase Transition. Strings are indicated in\nblue. Most strings are infinite in extent, but some are closed loops [67, 68]. The curved purple arrows\nindicate the 2 πwinding of the axion field θaround the string as discussed in Sec. 2. Center : The\naxion string-wall network shortly after the QCD Phase Transition, where domain walls form. Note\nthat the strings, still indicated in blue, are now attached to domain walls. Importantly, there also\nexist domain walls which are not attached to strings, but which are rather completely self-enclosed as\ndiscussed in Sec. 3, and here indicated by a spheroid in the top middle of the panel. The purple arrows\nindicate that the self-enclosed domain wall eventually collapses in on itself, as discussed in Secs. 4-5,\nwhile the orange arrows indicate axion emission. This emission arises when domain walls pull apart\nthe infinite string-wall network, creating violent oscillations and causing the entire network to radiate\naway. Right : The present-day universe. The small black dots indicate axions created from defect\nradiation and the misalignment mechanism.6The large black sphere indicates a PBH created from the\ncollapse of the self-enclosed axion domain wall as discussed in Secs. 6-7. Note that the present-day\nDM abundance consists of both axion particles and leftover PBHs from domain wall collapse.\nθon scales greater than L, it is possible, by chance, to find a region of the universe where\nθ < π surrounded entirely by a region of the universe where θ > π or vice versa. An enclosed\ndomain wall is the boundary interpolating these two regions of the universe.\nThe probability of finding a rare patch of the universe with this axion field configuration\ncan be understood using the methods of percolation theory [69, 70], which is closely related\nto the Kibble-Zurek mechanism [14, 55]. The application of percolation theory to estimate\nthe distribution of both walls bounded by strings and enclosed walls was first put forth by\nSikivie [71, 72] (see also [67]). We first examine a discretized version of Sikivie’s algorithm\nto provide intuition for the formation of enclosed walls before extending the algorithm to the\ncontinuum limit, which provides both a cross-check to the discrete case and more information\non the geometry of closed walls.\nThe discretized percolation theory approach can be understood by the following algo-\n6Here, we represent axion particles as black dots, but because typical axions have sub-eV masses, axion\nDM is highly wave-like.\n– 7 –rithm: partition the universe into a series of lattice sites where the separation between each\nlattice center is the correlation length of the axion field, the typical size over which θvaries\nrandomly from patch to patch in the universe. For the axion field θ, the correlation length is\nthe string separation distance L(which is roughly the horizon size). Next, randomly assign\nto each site the label θ1, θ2orθ3corresponding to whether the mean θwithin that patch\nof the universe is between [ π, π/ 3), [π/3,5π/3), or [5 π/3, π), respectively. We assign θ1,2,3\nto a uniform interval between 0 and 2 πbecause the probability distribution of θin a given\ncorrelation volume (lattice volume) of the universe is uniformly distributed before the axion\nstarts oscillating near the QCD phase transition. As discussed in Sec. 3.3, the assignment\ndoes not have to be uniformly chosen on the interval [0 ,2π) after the axion starts oscillating.\nA planar cross-section of such a random lattice is shown in the right panel of Fig. 2.\nAdjacent θ1andθ3lattice sites are separated by a domain wall since θcrosses πbetween\nthem as indicated by the red lines on lattice boundaries. Similarly, a configuration of adjacent\nlattice sites that change monotonically from θ1→θ2→θ3in a clockwise (counterclockwise)\norientation indicate that they wind around an axion string going into (out of) their shared\nvertex, as shown by the white (black) circles.\nAs can be seen from the right panel of Fig. 2, every string is attached to a domain wall\nbut there exists the rare enclosed wall as shown by the lattice site whose entire boundary is\nred. Most enclosed walls have an extent of order the string separation L, though enclosed\nwalls that span more than one lattice site exist too, but are rarer. It is expected that these\nenclosed walls with size much larger than Lare exponentially suppressed in abundance due to\nthe decreasing probability of finding such a patch of the universe without any strings [71, 72].\nIndeed, an ansatz in percolation theory for the number density of clusters of a given size (in\nour case, closed domain walls of a given volume) is [67, 69]\nncDW(V) =AV−Bexp\u0002\n−CV2/3], (3.1)\nwhere Vis the volume enclosed by the domain wall (equivalent to the number of lattice sites\nit comprises), and A,BandCare constants to be determined numerically.\nUsing this algorithm, we run a series of simulations on a cubic lattice consisting of 253\nlattice sites, corresponding to roughly 253Hubble volumes. To mitigate boundary effects, we\nrecord only walls and strings two lattice sites away from the boundary of the lattice. Ensuring\nconvergence requires realizing the random axion field phases on the lattice many times. We\nrealize it 15,000 times, as illustrated in the left panel of Fig. 3. The right panel of Fig. 3\nshows the number density of enclosed walls as a function of their volume (number of lattice\nsites they enclose) in units of number per L3. The results for ncDWgive good agreement with\nEq. (3.1), with the fit\nA= 0.512, B = 0.552, C = 6.430, (3.2)\nwhich is shown by the black line in the right panel of Fig. 3.\nSince the mean curvature radius, R, of the walls scales with volume as V∝R3, Eqns. (3.1)-\n(3.2) suggest that the number density distribution of enclosed walls scales as dncDW/dlnR∝\n– 8 –Wall\nRegion   \nRegion   Region  String  \n(Into Page)✓1<latexit sha1_base64=\"qln7EHCGtxTJjsm0fvSdIrLp1oo=\">AAAB73icdVDJSgNBEO2JW4xb1KOXxiB4GqazmlvQi8cIZoFkCD2dnqRJz2J3jRCG/IQXD4p49Xe8+Td2FkFFHxQ83quiqp4XS6HBcT6szNr6xuZWdju3s7u3f5A/PGrrKFGMt1gkI9X1qOZShLwFAiTvxorTwJO8402u5n7nnistovAWpjF3AzoKhS8YBSN1+zDmQAdkkC84NqmQeolgxy6SarVeNqRESKlWw8R2FiigFZqD/Ht/GLEk4CEwSbXuEScGN6UKBJN8lusnmseUTeiI9wwNacC1my7uneEzowyxHylTIeCF+n0ipYHW08AznQGFsf7tzcW/vF4C/oWbijBOgIdsuchPJIYIz5/HQ6E4Azk1hDIlzK2YjamiDExEORPC16f4f9Iu2qRsV27KhcblKo4sOkGn6BwRVEMNdI2aqIUYkugBPaFn6856tF6s12VrxlrNHKMfsN4+ATUlkBo=</latexit>\n✓2\n<latexit sha1_base64=\"sN7Kx8fWiMQP+KfmrqbprBBl9/E=\">AAAB73icdVDJSgNBEO2JW4xb1KOXxiB4GqazmlvQi8cIZoFkCD2dnqRJz2J3jRCG/IQXD4p49Xe8+Td2FkFFHxQ83quiqp4XS6HBcT6szNr6xuZWdju3s7u3f5A/PGrrKFGMt1gkI9X1qOZShLwFAiTvxorTwJO8402u5n7nnistovAWpjF3AzoKhS8YBSN1+zDmQAfFQb7g2KRC6iWCHbtIqtV62ZASIaVaDRPbWaCAVmgO8u/9YcSSgIfAJNW6R5wY3JQqEEzyWa6faB5TNqEj3jM0pAHXbrq4d4bPjDLEfqRMhYAX6veJlAZaTwPPdAYUxvq3Nxf/8noJ+BduKsI4AR6y5SI/kRgiPH8eD4XiDOTUEMqUMLdiNqaKMjAR5UwIX5/i/0m7aJOyXbkpFxqXqziy6ASdonNEUA010DVqohZiSKIH9ISerTvr0XqxXpetGWs1c4x+wHr7BDapkBs=</latexit>\n✓3<latexit sha1_base64=\"F2W050S6gFzu51aLGhciCL3Lbho=\">AAAB73icdVDJSgNBEO2JW4xb1KOXxiB4GqazmlvQi8cIZoFkCD2dnqRJz2J3jRCG/IQXD4p49Xe8+Td2FkFFHxQ83quiqp4XS6HBcT6szNr6xuZWdju3s7u3f5A/PGrrKFGMt1gkI9X1qOZShLwFAiTvxorTwJO8402u5n7nnistovAWpjF3AzoKhS8YBSN1+zDmQAelQb7g2KRC6iWCHbtIqtV62ZASIaVaDRPbWaCAVmgO8u/9YcSSgIfAJNW6R5wY3JQqEEzyWa6faB5TNqEj3jM0pAHXbrq4d4bPjDLEfqRMhYAX6veJlAZaTwPPdAYUxvq3Nxf/8noJ+BduKsI4AR6y5SI/kRgiPH8eD4XiDOTUEMqUMLdiNqaKMjAR5UwIX5/i/0m7aJOyXbkpFxqXqziy6ASdonNEUA010DVqohZiSKIH9ISerTvr0XqxXpetGWs1c4x+wHr7BDgtkBw=</latexit>\n⇡\n3\n<latexit sha1_base64=\"8dmuKMoi7d0Tr+oPebpMtxVECwc=\">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0oseiF48V7Ac0oWy2m3bpZhN3N4US8ju8eFDEqz/Gm//GTZuDtj4YeLw3w8w8P+ZMadv+tkpr6xubW+Xtys7u3v5B9fCoo6JEEtomEY9kz8eKciZoWzPNaS+WFIc+p11/cpf73SmVikXiUc9i6oV4JFjACNZG8txAYpK6McvSy2xQrdl1ew60SpyC1KBAa1D9cocRSUIqNOFYqb5jx9pLsdSMcJpV3ETRGJMJHtG+oQKHVHnp/OgMnRlliIJImhIazdXfEykOlZqFvukMsR6rZS8X//P6iQ5uvJSJONFUkMWiIOFIRyhPAA2ZpETzmSGYSGZuRWSMTQ7a5FQxITjLL6+SzkXdadSvHhq15m0RRxlO4BTOwYFraMI9tKANBJ7gGV7hzZpaL9a79bFoLVnFzDH8gfX5AxubklQ=</latexit>\n5⇡\n3\n<latexit sha1_base64=\"4XK6LLdCnjXsBOCEPH/XEjMmius=\">AAAB9XicbVBNS8NAEJ3Ur1o/WvXoZbEInkqiLXosevFYwX5AE8tmu2mXbjZhd6OUkP/hxYMiXv0v3vw3btsctPXBwOO9GWbm+TFnStv2t1VYW9/Y3Cpul3Z29/bLlYPDjooSSWibRDySPR8rypmgbc00p71YUhz6nHb9yc3M7z5SqVgk7vU0pl6IR4IFjGBtpAc3kJikDTdmWXqRDSpVu2bPgVaJk5Mq5GgNKl/uMCJJSIUmHCvVd+xYeymWmhFOs5KbKBpjMsEj2jdU4JAqL51fnaFTowxREElTQqO5+nsixaFS09A3nSHWY7XszcT/vH6igysvZSJONBVksShIONIRmkWAhkxSovnUEEwkM7ciMsYmCG2CKpkQnOWXV0nnvObUa427erV5ncdRhGM4gTNw4BKacAstaAMBCc/wCm/Wk/VivVsfi9aClc8cwR9Ynz+U0ZKT</latexit>\n0<latexit sha1_base64=\"zYY7oUhfdNehIqni3NNg/SSxHQs=\">AAAB6HicdVDJSgNBEO1xjXGLevTSGARPw3RWcwt68ZiAWSAZQk+nJmnTs9DdI4QhX+DFgyJe/SRv/o2dRVDRBwWP96qoqufFgivtOB/W2vrG5tZ2Zie7u7d/cJg7Om6rKJEMWiwSkex6VIHgIbQ01wK6sQQaeAI63uR67nfuQSoehbd6GoMb0FHIfc6oNlLTGeTyjk3KpFYk2LELpFKplQwpElKsVjGxnQXyaIXGIPfeH0YsCSDUTFClesSJtZtSqTkTMMv2EwUxZRM6gp6hIQ1Aueni0Bk+N8oQ+5E0FWq8UL9PpDRQahp4pjOgeqx+e3PxL6+XaP/STXkYJxpCtlzkJwLrCM+/xkMugWkxNYQyyc2tmI2ppEybbLImhK9P8f+kXbBJyS43S/n61SqODDpFZ+gCEVRFdXSDGqiFGAL0gJ7Qs3VnPVov1uuydc1azZygH7DePgHgyI0C</latexit>\n2⇡\n<latexit sha1_base64=\"GfZWgzasYF1tx3c0c8u3tNTMa5A=\">AAAB7HicdVDLSgNBEOz1GeMr6tHLYBA8LTt5mlvQi8cIbhJIQpidTJIhs7PLzKwQlnyDFw+KePWDvPk3Th6CihY0FFXddHcFseDaeN6Hs7a+sbm1ndnJ7u7tHxzmjo6bOkoUZT6NRKTaAdFMcMl8w41g7VgxEgaCtYLJ9dxv3TOleSTvzDRmvZCMJB9ySoyV/ALqxryfy3suLuNaESPPLeBKpVaypIhxsVpF2PUWyMMKjX7uvTuIaBIyaaggWnewF5teSpThVLBZtptoFhM6ISPWsVSSkOleujh2hs6tMkDDSNmSBi3U7xMpCbWehoHtDIkZ69/eXPzL6yRmeNlLuYwTwyRdLhomApkIzT9HA64YNWJqCaGK21sRHRNFqLH5ZG0IX5+i/0mz4OKSW74t5etXqzgycApncAEYqlCHG2iADxQ4PMATPDvSeXRenNdl65qzmjmBH3DePgF+vY6B</latexit>\n⇡<latexit sha1_base64=\"pXaUd9tpIhVqDq6r7ZmIijF7ylY=\">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoseiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz00Et4v1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vktZF1atVL+9rlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwBSVY3V</latexit>\nFigure 2 . An overview of the lattice defect simulation. Left: The spatial region around a string.\nIn our lattice simulation, we divide the full 2 πof the U(1) circle into three equal circle segments\nθ∈ {θ1, θ2, θ3}so that the arrangement θ1→θ2→θ3corresponds to winding around a string in\nphysical space. A string pointing into (out of) the page corresponds to a clockwise (counterclockwise)\nwinding of θ. The red line emanating from the string represents a domain wall which separates θ1\nandθ3regions. Right : An x-ycross section of the defect lattice simulation. The size of each lattice\nsite is physically interpreted as the average string separation L. Each site is independently assigned\na random phase θi. From this random assignment, it is possible to identify the strings and walls:\nThe unfilled (filled) black dots correspond to strings pointing into (out of) the page. The red lines\nindicate domain walls since θmust cross πwhen traversing adjacent regions of θ1andθ3. Walls that\nend on strings cannot be completely self-enclosed. The square closed wall shown in top middle of this\ncross section can correspond to a fully self-enclosed domain wall, although it can also be attached to\na string further into or out of the page. The combinatorics of this simple lattice simulations allows us\nto estimate the number density of closed domain walls of different sizes at the QCD phase transition.\n−(R3)2/3=−R2[67]. In the following section we confirm this using the continuum approach.\n3.2 Domain Wall Shape Distribution\nThe simulations outlined in Sec. 3.1 are a course-grained snapshot of the actual smooth\nθfield configuration of the universe with the separation between each lattice site equal to\nthe correlation length, L, of the axion field at that instant of time. In essence, the lattice\nsimulations in the previous section corresponds to a smooth random field θ(x) with two-point\nfunction\n⟨θ(x1), θ(x2)⟩Lattice =(\n1|x1−x2| ≤L ,\n0|x1−x2|> L .(3.3)\nThat is, θis entirely correlated within one lattice cell and entirely uncorrelated outside. The\nreal universe is of course not discrete and the axion field can take continuous values anywhere\n– 9 –0 2500 5000 7500 10000 12500 15000\nNumber of Simulations10-910-810-710-610-510-410-310-2ncDW [1/L3]V=1L3\nV=2L3\nV=3L3\nV=4L3\n1.0 1.5 2.0 2.5 3.0 3.5 4.0\nV [L3]10-910-810-710-610-510-410-310-2\nncDW(V)=AV−B\nexp[−CV2/3]Figure 3 . Results from numerical string-wall network simulations. Left: The average number of\nclosed domain walls of specified size per lattice volume V(interpreted physically as the cubed string\nseparation length L3) as a function of the number of simulations performed. The blue, red, yellow, and\npurple curve correspond to volume one, two, three, and four lattice volumes, respectively. Note that\nall four closed domain wall densities have converged. Right : The average number of closed domain\nwalls of specified size per lattice volume ncDW as a function of the enclosed volume V. The colored\npoints correspond to values obtained from simulations for a given size wall. The solid black line shows\nthe best fit of Eq. (3.1) to the simulation results.\nfrom 0 to 2 π.\nA more physical continuum limit of the lattice simulations entails generating a continu-\nous uniform random field θ(x) with support from 0 to 2 πand with a continuous correlation\nfunction that is highly correlated for scales |x1−x2| ≪Land uncorrelated for |x1−x2| ≫L,\nanalogous to Eq. (3.3). Such a random field can be generated first by generating a Gaussian\nrandom field with the desired correlation function and then taking its phase, which itself is a\nuniform random field. Generating a Gaussian random field with an arbitrary correlation func-\ntion (equivalently, power spectrum) is computationally efficient, and we follow the approach\nof [73, 74] which involves generating white noise in real space (which has unity correlation\neverywhere), fast-Fourier transforming to momentum-space, multiplying by the desired power\nspectrum (the Fourier transform of ⟨θ(x1), θ(x2)⟩), and inverse fast-Fourier transforming back\nto real-space. It is possible to use the same correlation function of Eq. (3.3) in this method,\nbut because of its sharp derivative at |x1−x2|=L, its Fourier series is oscillatory and\nsomewhat difficult to sample accurately for the fast-Fourier transforms. Instead, we take a\ncorrelation function\n⟨θ(x1), θ(x2)⟩Continuum =e−|x1−x2|2/L2, (3.4)\n– 10 –Figure 4 .Left: A contour map showing the surface of constant θ=πfrom a continuous uniform\nrandom field (domain 0 ≤θ < 2π) generated using the two-point function Eq. (3.4). The field is\ngenerated randomly in a box with size 203string correlation volumes L3and represents a statistical\nconfiguration of the axion field in the universe near TQCD. Domain walls can be seen as the purple\nmesh separating regions of space. Right : Same as left, but with the infinite string-wall network\n(non-self-enclosed domain walls) removed for z >5Land with self-enclosed domain walls highlighted\nin gold. We use this continuum uniform random field approach to verify that the number density\nof self-enclosed domain walls agrees with the discrete lattice simulation. We also extract the shape\ndistribution of self-enclosed walls from this random field in order to accurately account for domain\nwall asphericities.\nto model the continuous fluctuations of the axion field.\nFigure 4 shows the surface of points where θ=πfrom one simulation run of a generated\n3D uniform random field with correlation function given by Eq. (3.4) on a box of dimensions\n(20L)3. There is one large infinite string-wall network, as shown by the purple membrane\n(walls) which end on strings. For distances much greater than the correlation length, L, the\nwalls appear Brownian-like due to the lack of correlation in the axion field. As cross-sectional\nslices of the image are removed, two enclosed walls embedded in the string-wall network come\ninto view as shown by the gold spheroidal objects in the right panel. To gather statistics\non the number density distribution and shapes of these enclosed walls and to compare with\nthe lattice results of Sec. 3.1, we generate 15000 realizations of the axion random field like\nFig. 4, corresponding to 15000 ×(20L)3= 1.2×108L3total correlation volumes (roughly\nHubble volumes), and record the total number of enclosed walls observed and their following\nproperties: volume, and the mean, maximum, and minimum distance from the wall center of\nmass to its surface (centroid distance).\nTallying over all simulation runs yields nearly 8 ×104enclosed walls. The histograms of\n– 11 –Fig. 5 show how these enclosed walls are distributed as a function of their volume and mean\ncentroid distance, which we define as their mean radius, R. As expected, larger walls are\nmuch rarer than smaller walls – in fact, exponentially so as the lattice approach indicated.\nThe black curves overlaying each histogram show an analytic fit of the enclosed wall count in\neach bin. Dividing by the total simulation volume and bin width gives the number density\ndistributions of walls, which are well described by\ndncDW\ndV≈1.5×10−3V−.55exp\u0002\n−7V0.8\u0003\n, (3.5)\ndncDW\ndR≈1.5×10−3R−.25exp\u0002\n−6.4R2\u0003\n, (3.6)\nwhere ncDW,VandRare in units of L−3,L3andL, respectively. We have checked that\nthese distribution functions are independent of the histogram bin width except when the bin\nwidth is so small that there are large gaps between bins. Moreover, the above fits are of\ncourse, not unique, but the variety of alternative parameterizations do not greatly differ until\nextrapolating far beyond the domain of the histograms of Fig. 5. Since most of the relevant\nparameter space for observable PBHs will not be in this regime, the final results of this paper\nare quite insensitive to these alternative parameterization. With these caveats in mind, the\nform of Eqns. (3.5) and (3.6) nevertheless is in quite good agreement with the lattice results\nof Sec. 3.1, and confirm the expectation that dlnncDW/dR∝ −R2. Moreover, integrating\nEq. (3.5) or (3.6) gives the total number density of enclosed walls of nc,DW≃1.0×10−3per\ncorrelation volume L3, which again agrees with the lattice results (see Fig. 3).\nLast, another important result from the continuum simulations is that walls are, as\nexpected, not perfectly spherical. Fig. 6 shows the distribution of the ratio of the max to\nmean centroid distance and the min to mean centroid distance of enclosed walls. These ratios\nare a proxy for how ellipsoidal the walls are. Note, the asphericity of walls hinder PBH\nformation due to asphericity growth during collapse as we discuss in Sec. 5.\nMost walls have moderate ellipticities, with their largest (shortest) axis about a factor\nof 1.4 (0.6) greater (smaller) than their mean radius. Perfectly spherical walls are rare.\nAlthough we show in Sec. 5.2 that the domain wall abundance is not overly sensitive to the\nexact distribution of ellipticities – as long as most are not perfectly spherical – we find that the\ndistributions are well described by extreme-value Frechet and Weibull distributions [75, 76]\nparameterized by the following probability density functions\nfMax/Mean(x)∝x−α−1exp\u0002\n−(x/β)−α\u0003\nα= 12.38, β= 1.37, (3.7)\nfMin/Mean(x)∝xα−1exp [−(x/β)α] α= 7.34, β = 0.63, (3.8)\nas shown by the black contours overlaying the histograms in Fig. 6\nIn the following sections, we return to these distributions, Eqns. (3.6)-(3.8), when com-\nputing the abundance of enclosed walls that form PBHs.\n– 12 –1101001000104105\n1101001000104105Figure 5 .Left: The number of self-enclosed domain walls found using the continuous random field\nrealizations with two-point function given by Eq. (3.4), here shown as a function of enclosed volume.\nThe histogram shows the data obtained from our random field realizations, while the solid black line\nshows the best fit analytic function, given by Eq. (3.5). We note that there is strong agreement\nbetween the fit obtained using the lattice or continuum approaches, illustrating that the number\ndensity is governed by a simple scaling dictated by percolation theory. Right : Same as left, but as\na function of the mean wall radius. The black line shows the best fit analytic solution, given by\nEq. (3.6). Note that the dip on the leftmost side of the histogram is due to the limited resolution of\nthe continuous random field realizations.\nFigure 6 .Left: The normalized distribution of the max to mean centroid distance ratio for self-\nenclosed domain walls generated using a continous uniform random field with a two-point function\ngiven by Eq. (3.4). The histogram shows the data obtained from our random field realizations, while\nthe solid black line shows the best fit extreme value function, given by Eq. (3.7). Right : Same as left,\nbut showing the min to mean centroid distance ratio and best fit, given by Eq. (3.8).\n– 13 –\nHorizon<latexit sha1_base64=\"q5OLdwZKWtTmPXLzwU6RYpOB0Rs=\">AAAB7nicdVDLSsNAFJ3UV62vqks3g0VwFZJabdwV3XRZwT6gDWUynbRDJzNhZiLU0I9w40IRt36PO//GSRtBRQ9cOJxzL/feE8SMKu04H1ZhZXVtfaO4Wdra3tndK+8fdJRIJCZtLJiQvQApwignbU01I71YEhQFjHSD6XXmd++IVFTwWz2LiR+hMachxUgbqdsUkt4LPixXHPvS8868OnRsZ4GMVF2vdgHdXKmAHK1h+X0wEjiJCNeYIaX6rhNrP0VSU8zIvDRIFIkRnqIx6RvKUUSUny7OncMTo4xgKKQpruFC/T6RokipWRSYzgjpifrtZeJfXj/RoeenlMeJJhwvF4UJg1rA7Hc4opJgzWaGICypuRXiCZIIa5NQyYTw9Sn8n3Sqtluzz2+qlcZVHkcRHIFjcApcUAcN0AQt0AYYTMEDeALPVmw9Wi/W67K1YOUzh+AHrLdP3s+P7w==</latexit>\nNetwork<latexit 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sha1_base64=\"d8Ij0zq/aoUxnKg5tRBBW2b/t3E=\">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8cW7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GdzO//cS1EbF6wEnC/YgOlQgFo2ilR+xnPR2RemPaL1fcqjsHWSVeTiqQo94vf/UGMUsjrpBJakzXcxP0M6pRMMmnpV5qeELZmA5511JFI278bH7xlJxZZUDCWNtSSObq74mMRsZMosB2RhRHZtmbif953RTDGz8TKkmRK7ZYFKaSYExm75OB0JyhnFhCmRb2VsJGVFOGNqSSDcFbfnmVtC6q3mX1qnFZqd3mcRThBE7hHDy4hhrcQx2awEDBM7zCm2OcF+fd+Vi0Fpx85hj+wPn8AUsSkK0=</latexit>\nGrowth of correlated domains Time Time Important times in defect evolution\n(ma(tosc)'3H(tosc))\n<latexit sha1_base64=\"RwGUWHPvdBLSTstDGDm5jpXffsw=\">AAACE3icbVC7TsMwFHV4lvIKMLJYVEgtQ5VAEYwVLB2LRB9SE0WO67ZWbSfYDlIV9R9Y+BUWBhBiZWHjb3DbDLTlSFc6Oude3XtPGDOqtOP8WCura+sbm7mt/PbO7t6+fXDYVFEiMWngiEWyHSJFGBWkoalmpB1LgnjISCsc3k781iORikbiXo9i4nPUF7RHMdJGCuyzIg9QUQepJzmMFB6XoKcoJw/wojYvlwK74JSdKeAycTNSABnqgf3tdSOccCI0ZkipjuvE2k+R1BQzMs57iSIxwkPUJx1DBeJE+en0pzE8NUoX9iJpSmg4Vf9OpIgrNeKh6eRID9SiNxH/8zqJ7l37KRVxoonAs0W9hEEdwUlAsEslwZqNDEFYUnMrxAMkEdYmxrwJwV18eZk0z8tupXx5VylUb7I4cuAYnIAicMEVqIIaqIMGwOAJvIA38G49W6/Wh/U5a12xspkjMAfr6xd2D5yj</latexit>\n(twall'ma(twall)\u00001)\n<latexit sha1_base64=\"VWe1Qr3q02D/Tfrd2fIzFWQU91w=\">AAACFXicbVDLSsNAFJ3UV62vqEs3g0VoQUsiFV0W3bisYB/QxDCZTtqhM0mcmSgl5Cfc+CtuXCjiVnDn3zh9LGzrgQuHc+7l3nv8mFGpLOvHyC0tr6yu5dcLG5tb2zvm7l5TRonApIEjFom2jyRhNCQNRRUj7VgQxH1GWv7gauS3HoiQNApv1TAmLke9kAYUI6UlzzwuKS91BIePiLEMOpJycg+5h2b08l16YmdlzyxaFWsMuEjsKSmCKeqe+e10I5xwEirMkJQd24qVmyKhKGYkKziJJDHCA9QjHU1DxIl00/FXGTzSShcGkdAVKjhW/06kiEs55L7u5Ej15bw3Ev/zOokKLtyUhnGiSIgni4KEQRXBUUSwSwXBig01QVhQfSvEfSQQVjrIgg7Bnn95kTRPK3a1cnZTLdYup3HkwQE4BCVgg3NQA9egDhoAgyfwAt7Au/FsvBofxuekNWdMZ/bBDIyvX2V8nlU=</latexit>\n(tµ/\u0000'µ/\u0000)\n<latexit sha1_base64=\"C/s/a4U2XuqbgIyknDEahKwsEKs=\">AAACC3icbVDLSgMxFM3UV62vUZdugkWomzojFV0W3bisYB/QGYZMmmlDk8yYZIQydO/GX3HjQhG3/oA7/8a0HVBbD1zu4Zx7Se4JE0aVdpwvq7C0vLK6VlwvbWxube/Yu3stFacSkyaOWSw7IVKEUUGammpGOokkiIeMtMPh1cRv3xOpaCxu9SghPkd9QSOKkTZSYB9WdJB5PD3xFO1zNIamc3IHf6TjwC47VWcKuEjcnJRBjkZgf3q9GKecCI0ZUqrrOon2MyQ1xYyMS16qSILwEPVJ11CBOFF+Nr1lDI+M0oNRLE0JDafq740McaVGPDSTHOmBmvcm4n9eN9XRhZ9RkaSaCDx7KEoZ1DGcBAN7VBKs2cgQhCU1f4V4gCTC2sRXMiG48ycvktZp1a1Vz25q5fplHkcRHIBDUAEuOAd1cA0aoAkweABP4AW8Wo/Ws/Vmvc9GC1a+sw/+wPr4BkfKmpI=</latexit>Figure 7 . The effect of time-evolution on the string-wall network. Strings form at tPQbeing highly\nBrownian on scales larger than the correlation length of Φ. Strings eventually reach scaling at tscaling ,\nat which time the string correlation length Lis equal to the cosmic horizon size (times potential log\ncorrections). Small scale structure on the strings continually damps as scaling ensures that the axion\nfield remains largely homogeneous on scales of order the cosmic horizon size. Later, at twall, domain\nwalls begin to form since the domain wall thickness δdecreases below the horizon size. At tosc, the\nmisalignment production of DM is most significant. Finally, at tµ/σ, the entire string-wall network\ncollapses as the wall tension overpowers the string tension, meaning domain wall production ceases.\nThe entire domain wall forming time interval is indicated by the blue band.\n3.3 Wall Formation Times\nThe percolation theory approach of Sec. 3.1 makes manifest that the formation and size of\nenclosed walls is just a geometric problem. Where is the physics? At what point in time is\nthe geometric argument valid?\nThe physics is encoded in the correlation length Lwhich sets the scale of the distribution\nfunctions. To mapLto aphysical length scale requires knowledge of the physics that controls\nthe spatial deviations of the axion field at a given point in time. One can apply the geometric\narguments above at any time there exists separated domains which are correlated over a\ngiven region L7. Indeed, the mean curvature of an axion topological defect iseffectively the\ncorrelation scale of the axion field [58].\nThus, by identifying the mean string curvature radius with the size of the correlated\ndomain of the axion field, we can find the density of newly formed enclosed walls at different\npoints in time in the evolution of the universe. Physically, walls are continually being produced\n7We thank Robert Brandenberger for emphasizing this point.\n– 14 –throughout cosmic time, with their characteristic size being larger at later times. However,\nwalls and the small scale structure of strings are also continually being destroyed, which is\nwhy the correlation length of the axion field grows with time. For example, the left panel\nof Fig. 7 shows a pictorial representation of the infinite axion string network in the scaling\nregime, which is an attractor solution of defect network evolution where the average number\nof strings per horizon, ξ, is maintained around one [14, 54]. At a given point in time,\nthe infinite string has mean curvature of size L=t/√ξ≃t(red circle) – definitionally\ncorresponding to the axion field correlation length. The superhorizon perturbations of the\nstring are frozen by causality (Hubble friction [53]), while the subhorizon deviations are\ndynamic. The relativistic motion of the string on subhorizon scales facilitates axion emission\nand string intercommutation which damps small scale axion structure and increases the size\nof correlated regions of the axion field. As a result, the axion field is effectively ‘remixed’\nevery Hubble time [57, 67], allowing the percolation method to be applied again with a new\n(and larger) interpretation of the correlation length.\nA self-enclosing wall can form at any time t0between the creation and destruction of the\nstring-wall network. The right panel of Fig. 7 shows some key times in the evolution history\nof the entire axion topological network beginning with the time time of string formation,\ntPQ. After the network reaches scaling at tscaling , the correlation length of the axion field is\nL=t/√ξ≃t. One can of course identify walls during this early period by denoting them as\nsurfaces of constant θ=π. However, the wall thickness m−1\nais much larger than the horizon\nso that the ‘walls’ are not dynamical at this point; defining a wall tension σis not meaningful\nnor do the ‘walls’ collapse.\nSince enclosed walls do not collapse until they become dynamical, we focus on formation\ntimes after this occurs. The walls become dynamical at time twallsatisfying\ntwall≃ma(twall)−1(Walls dynamical) . (3.9)\nSoon afterwards, the axion field begins oscillating towards its minimum at time tosc. This\noccurs when 3 H(tosc)≃ma(tosc), or equivalently, at time\ntosc≃3k ma(tosc)−1, k =\n\n1\n2RD era\n2\n3MD era(Axion oscillates) . (3.10)\nHere, kis the exponent in the scale factor vs time relation a(t)∝tk. Note that after tosc, the\naxion field is still correlated within a region of size L. It is just not uniform between 0 and 2 π\nsince the axion field oscillates about its minimum at ⟨θ⟩= 0. This can be simply captured in\nthe percolation theory approach by reducing the θ2region (see Fig. 2) at the rate at which\nthe amplitude of the axion field falls towards its minimum in time.\nPast tosc, the string-wall network continues to evolve in the scaling regime until the\ndynamics of the wall overpower that of the string. This change happens when the tension\nforce from a string segment, Fstring =µ, becomes weaker than the force of the domain wall\n– 15 –of curvature radius tattached to it, Fwall≈σt[57, 77, 78]. This occurs at time\ntµ/σ≃µ/σ≃π\n4ln(fa/TQCD)ma(tµ/σ)−1(Walls overpower strings) . (3.11)\nAfter time tµ/σ, the walls pull the string network apart and the entire system quickly decays\ninto axions and gravitational waves (see Fig. 12 of ref. [78] for how quickly walls overpower\ntheir string boundary once their curvature radius grows beyond µ/σ). Consequently, the latest\ntime an enclosed wall can form is shortly after tµ/σ. The PBH abundance will be dominated\nat the latest possible time of enclosed wall formation, which is approximately tµ/σ. This is\nbecause larger walls form later, and the larger the wall, the easier it is to form a PBH upon\ncollapse as discussed in Sec. 6.\nNote that if the axion mass is still increasing at twall, then the times twall≃tosc≈tµ/σ\nare all comparable. This is generally the case if the QCD axion begins oscillating in a\nRD era. Conversely, if the axion mass is already at its zero temperature mass at twall, then\ntwall≈tosc≈.035(ln( fa/TQCD)/36)−1tµ/σ. This is generally the case if the QCD axion begins\noscillating in a MD era and fa≳1015GeV or for ALPs with a (at late times) temperature-\nindependent mass.\nFinally, we note that recent numerical work [79–83] confirms this picture of scaling with\nξ→1 independent of initial conditions.8However, refs. [68, 87–93], which fully simulate the\nfield-theoretic Euler-Lagrange equations of motion for strings, find evidence of logarithmic\nviolations to this scaling. In these simulations, ξstill has an attractor solution (meaning\nthe simulations converge to the same value of ξindependently of the initial conditions used),\nbut once the attractor is reached, ξdoes not stay fixed, but increases logarithmically with\ncosmic time (see however [94]). Simulations find ξlogarithmically grows in cosmic time up to\nξ∼1.2±0.2 but cannot tell if this trend continues indefinitely due to lack of computational\nresources; if extrapolating this trend in the growth of ξthrough orders of magnitude in time\nall the way to the QCD phase transition, ξcould be as large as ∼15 at the time twall.\nIn the remaining sections, we assume the traditional value of ξ≃1, which does not rely\non an (as of the present) uncertain scaling violation coefficient. This choice is likely more\nrobust in an early matter-dominated cosmology, such as the one we are primarily interested in,\nwhere logarithmic scaling violations are expected to be suppressed [39] if they exist. However,\nwe discuss how a different ξat the QCD phase transition would affect the present-day PBH\nabundance in Sec. 8 and in Appendix D where we consider ALPs in a radiation-dominated\ncosmology. In the following two sections, we consider the dynamics of a single enclosed wall\nthat forms at some arbitrary time t0, with twall≤t0≤tµ/σ, as shown by the blue strip in\nthe right panel of Fig. 7. Sec. 4 deals with the initial stage of superhorizon expansion while\nSec. 5 deals with subhorizon collapse.\n8Friction between the axion strings and the background plasma is ignored in these references as well as in\nthe methods simulating the full Euler-Lagrange equation of motion. Friction can dramatically increase the\ntime it takes for strings to reach scaling, though they are expected to still reach scaling before TQCD [84–86].\n– 16 –4 Superhorizon Expansion of Domain Walls\nIn this section, we describe the dynamics of self-enclosed domain walls starting from when\nthe walls form up until the moment they enter the cosmic horizon. During this stage, the rest\nmass of the wall increases in two distinct ways: First, the Hubble expansion of the universe\nstretches the wall, giving it a larger surface area. Second, the wall tension grows due to\nthe axion mass continuing to ‘turn on’ as the universe cools. Both mechanisms increase the\nprobability of PBH formation. We numerically simulate the full Euler-Lagrange equations of\nmotion of an enclosed wall in a Friedmann-Lemaitre-Robertson-Walker (FLRW) background\nto accurately account for these phenomena. First, we describe how these simulations are\nset up and executed. We then use the simulations to extract the peak radius of the wall\nat horizon entry and the time it takes for the wall to enter the horizon and to collapse.\nThese horizon entry characteristics are crucial in determining whether a PBH can form, as\ndiscussed in Secs. 5-6. Moreover, we use this information in Sec. 7 to calculate the final energy\nstored in a wall and hence the mass of the resultant PBH. Finally, while the analysis in this\nsection assumes the walls are perfectly spherical during superhorizon expansion for simplicity,\nwe also simulate and discuss the superhorizon evolution of aspherical walls in Appendix B.\nAsphericities are most important during the final stage of subhorizon collapse, as discussed\nin Sec. 5.2.\n4.1 Simulation Setup\nIn Minkowski spacetime, it is energetically favorable for a self-enclosed domain wall to shrink.\nEven in the absence of an energy gap between the vacuum inside and outside of the domain\nwall, the intrinsic surface tension, σ, causes a curved wall surface to accelerate inward, leading\nit to rapidly approach the speed of light.\nHowever, as discussed in Sec. 3, self-enclosed domain walls larger than the cosmic horizon\ncan form for which the non-flatness of an FLRW spacetime is important. Instead of collapsing\nimmediately, superhorizon walls stretch with the scale factor, a(t), of the universe. This only\noccurs for a finite amount of time due to the horizon size growing at a faster rate than the\nscale factor. For a wall that forms at time t0with radius R0≳t0, one can equate the horizon\nsizetwith the increasingly stretched radius of the wall, R(t) =R0a(t)/a(t0), to estimate the\nsize of the wall at horizon re-entry, RRE≈R0(R0/t0)2(3)in a RD (MD) era. To make this\nrelation more exact, we study domain wall growth numerically.\nAs soon as the domain wall re-enters the horizon, it is briefly at rest with a radius RRE,9\nso that its total energy at horizon re-entry is\nERE≃4πσR2\nRE≈32πma(tRE)f2\naR2\nRE, (4.1)\nwhere σ≃8ma(T)f2\nais the tension of the domain wall. Eq. (4.1) shows that the total domain\nwall energy scales with the square of the wall radius, meaning up until horizon re-entry\n9For aspherical walls, there is not single time at which the wall is at rest, but there is a short time interval\nin which the entire wall goes from being completely superhorizon to collapsing inwards.\n– 17 –E(t)∝a2(t)ma(t). Eq. (4.1) also accounts for mapotentially not having reached its zero\ntemperature value at t=tRE. It is preferable for PBH formation for the domain walls to take\na long time to collapse so that mais as large as possible per Eq. (2.3).\nWe directly simulate the enclosed wall from formation until collapse. This is done by\nnumerically solving the Euler-Lagrange equations associated with the IR axion Lagrangian,\ngiven by Eq. (2.2), in an FLRW background. Using θ(x)≡a(x)/fa, the classical equation of\nmotion for the axion field is\n\u0014\n∂2\nt+ 3˙a\na∂t−1\na2∇2\u0015\nθ(x) +m2\na(T) sin(θ) = 0 , (4.2)\nwhere a=a(t) denotes the scale factor. The Hubble scale His related to the SM plasma\ntemperature via\nH(T) =\n\nr\n8π3g∗(T)\n90T2\nMPlRD Era ,\ns\n8π3g2∗(T)\n90g∗(TRH)T4\nMPlT2\nRHMD Era (Non-Adiabatic) ,(4.3)\nwhere g∗is the effective number of relativistic degrees of freedom in the SM plasma, MPl≃\n1.22×1019GeV is the Planck mass, and TRHis the reheat temperature in the early matter-\ndominated scenario [95].\nWe transform the above equation into conformal time via the relation dη=dt/a(t) and\ndefine η0≡r0≡t0with t0chosen to be the wall formation time arbitrarily lying somewhere\nin the interval ( twall, tµ/σ) as discussed in Sec. 3.3. Eq. (4.2) now reduces to the simple form\n∂2θ\n∂(η/η0)2− ∇2\nr/r0θ+ 2H∂θ\n∂(η/η0)+ (η/η0)2˜m2\na\u0012η\nη0\u0013\nsinθ= 0, (4.4)\nwhere ˜ mais the dimensionless axion mass, given by\n˜ma\u0012η\nη0\u0013\n=ma(T= 0)×η0×min\u0014\u0012TQCD\nT(η/η0)\u0013n/2\n,1\u0015\n, n≈6.68, (4.5)\nand where His given by\nH=1\nRdR\nd(η/η0)=\n\n1\n(η/η0)RD Era ,\n2\n(η/η0)MD Era .(4.6)\nWe numerically solve this equation starting from η/η0= 1 with initial field configuration\n∂θ\n∂(η/η0)\f\f\f\f\nη/η0=1= 0, (4.7)\nθ\u0012r\nr0\u0013\f\f\f\f\nη/η0=1= 4 arctan exp\u0002\n˜ma\u0012η\nη0= 1\u00131\nr0(r−R0)·ˆn\u0003\n, (4.8)\n– 18 –0 2 4 6 8 10\nr/r0−π/20π/2π3π/22π5π/2θη= 1η0\n0 2 4 6 8 10\nr/r0η= 2η0\n0 2 4 6 8 10\nr/r0η= 6η0Figure 8 . The axion field configuration θ=a/fafor a spherical domain wall of initial radius R0=\n4t0as a function of the comoving radial coordinate r/r0at three conformal time slices. Here, the\ndomain wall is assumed to have formed at t=twall.Left: The initial axion field configuration\natη=η0, corresponding to the moment the axion mass and Hubble scale cross. The initial field\nconfiguration is given by Eq. (4.7) and the wall is taken to be completely at rest. Middle : The axion\nfield configuration at η= 2η0. Note that the wall appears considerably thinner due to the increase in\nthe thermal axion mass, athough this visual effect is further enhanced due to the choice of comoving\ncoordinates (see the ( η/η0)2factor multiplying ˜ m2\nain Eq. (4.2)). The red arrows indicate the motion\nof the field configuration, with the wall continuing to become thinner, but the wall otherwise being\nat rest in comoving coordinates as it continue to expand with the scale factor. Right : The axion field\nconfiguration at η= 6η0. Here, the wall has again become dramatically thinner and has entered the\ncosmic horizon, causing rapid inward movement. The wall has also started radiating axions, which\ncan be seen as outward moving ripples in the field.\nwhere R0is a vector pointing to the wall surface along the direction of randˆnis an out-\nward pointing vector normal to the domain wall at the point R0. This field configuration\ncorresponds to a single self-enclosed domain wall. Note that it is simply a standard planar\nSine-Gordon kink soliton with asymptotic limits 0 and 2 π, but deformed into a sphere [96].\nThis quasi-planar solution is a good approximation to the initial wall profile in the regime\nwhere the local radius of curvature of the wall is much greater than the wall thickness δ,\nbeing the exact solution when neglecting the 2 /r∂rterm in ∇2\nr/r0of Eq. (4.4). Due to the\nsignificant thinning of axion domain walls as the axion mass ma(T) =δ−1grows, and due to\nthe initial comoving expansion of the wall, this is a good approximation for any wall of size\nLor larger shortly after creation. Fig. 8 shows three time slices of the simulated axion field\nconfiguration for a collapsing spherically symmetrical domain wall. The axion mass turn-on\nhas no significant effect on the total amount of time it takes for a domain wall collapse.10\nHowever, the mass turn-on can be important in increasing the total energy of the wall and in\ndecreasing its thickness, which makes the domain wall more easily compressible into a black\nhole, which we discuss in Sec. 5.\n10This can be seen in the Nambu-Goto picture, where adding a time-dependent mass term to the action\nadmits the exact same solution to the equations of motion as the action with no time-dependent mass term.\n– 19 –1.02.5 5.0 7.5 10.0 12.5 15.0 17.5\nη/η0010203040506070R [t0]\n1t02t03t04t05t06t07t08t0R0=9t0Radiation Domination\n1 5 10 15 20\nη/η002004006008001000R [t0]\n1t0\n−2t03t04t05t06t07t08t0R0=\n9t0Matter Domination1 925 49 81121 169 240 306t/t0\n1 125 1000 3375 8000t/t0Figure 9 .Left: Collapse of closed spherical domain walls assuming radiation domination. The colored\ncurves indicate the physical radius Rof different closed domain walls with initial radius R0as a function\nof conformal time ηand physical time t. The triangles on each curve indicate the moment in time\nthat the corresponding domain wall enters the horizon, which we define as when the physical velocity\nof the wall reverses sign and points inwards. Notice that walls with larger R0re-enter the horizon\nlater and hence attain a larger surface area than walls with small R0, giving them more energy to\nform black holes. Right : Same as left, but assuming matter domination. Note here that since a∝t2/3\nrather than t1/2like for radiation domination, the walls grow much larger before collapsing.\n4.2 Re-entry Radius and Collapse Time\nWith this noted, we simulate spherical domain wall collapse via Eq. (4.4) for a range of initial\nwall radii R0in order to infer the relation between initial wall size, peak wall radius, and the\namount of time it takes for a wall to enter the horizon. Here, we define horizon re-entry as the\nmoment when the physical velocity of the wall reverses sign. The left (right) panel of Fig. 9\nshows the evolution of the physical radius Rof the wall as a function of time in a RD (MD)\nera. Note the wall stretches to a larger peak radius in a MD era due to the faster expansion\nrate of the universe. Similarly, Fig. 10 shows a linear fit to the re-entry time ηREfor wall\ncollapse during both RD (left) and MD (right) eras. In conformal time η, this relation is\nηRE\nη0≃(\n1.06α+ 0.26 RD Era ,\n1.51α+ 0.38 MD Era .(α≡R0/t0) (4.9)\nThe linearity of this relation is principally due to the fact that the ratio between any distance\ntimes the scale factor aand the horizon size tis linear in η. We also extract the re-entry\nradius RREas a function of α≡R0/t0. It is\nRRE≃R0×(\nmax{0.75α2, α} RD Era ,\nmax{3.75α3, α} MD Era ,(α≡R0/t0) (4.10)\n– 20 –0 2 4 6 8 10\nR0 [t0]1357911ηRE/η0Radiation Domination\n0 2 4 6 8 10\nR0 [t0]13579111315ηRE/η0Matter DominationFigure 10 . The conformal horizon re-entry time ηRE/η0as a function of the initial spherical wall\nradius R0computed from the domain wall collapse simulation shown in Figure 9. Left: The conformal\ntime at horizon re-entry ηRE/η0assuming radiation domination. The gray points are the simulation\nresults and the black line is a linear fit to the simulation results. Note that the relation is highly linear\nin conformal time, with corresponding coefficients given by Eq. (4.9). The α= 1 point is excluded\nin this fit to account for the wall already being horizon sized at formation. Right : Same as left, but\nassuming matter domination. Note that in a matter-dominated cosmology, it takes longer for the walls\nto re-enter the horizon and ultimately collapse compared to a radiation dominated cosmology.\nwhere we take the maximum of the initial radius αand the interpolated power laws in order\nto account for the fact that the smallest walls ( α≲1) are already subhorizon sized upon\nformation. We show the power law fits to the simulation data in Fig. 11.\n4.3 Re-entry and Collapse Tension\nWe also use simulations to account for the change in the domain wall thickness as it expands\nand later contracts. By Eq. (2.3), between formation and re-entry, the axion wall thickness\nchanges by a factor\nδRE\nδ0=ma(t0)\nma(tRE)=min\u001a\u0010TQCD\nT0\u0011n/2\n,1\u001b\nmin\u001a\u0010TQCD\nTRE\u0011n/2\n,1\u001b=min\u001a\u0010TQCD\nT0\u0011n/2\n,1\u001b\nmin\u001a\u0010TQCD\nT0T0\nTRE\u0011n/2\n,1\u001b, (4.11)\nwhere T0(TRE) is the temperature of the universe at the wall formation (horizon re-entry)\ntime t0(tRE). Note that δRE/δ0lies between ( TRE/T0)n/2≤1 and 1, where the former case\ncorresponds to the axion not reaching its zero temperature value by the time the wall re-enters\nthe horizon, and the latter case corresponds to the axion mass reaching its zero-temperature\nvalue by the time the wall forms.\nThe wall formation temperature T0ranges from the temperature at which the wall thick-\nness becomes smaller than the horizon, Twall, to when the string-wall network stops scaling and\n– 21 –2 4 6 8 10\nR0 [t0]010203040506070RRE [t0]Radiation Domination\n∝R20\n2 4 6 8 10\nR0 [t0]0500100015002000250030003500RRE [t0]Matter Domination\n∝R30Figure 11 .Left: The physical domain wall horizon re-entry radius RREas a function of initial radius\nR0. The gray points are the simulation results and the black line is a linear fit to the simulation\nresults, given by Eq. (4.10). Right : Same as left, but assuming matter domination where a∝t2/3,\nhence the larger re-entry radii.\nis destroyed, Tµ/σ. As can be seen by the right-most equation of (4.11), the ratio of the for-\nmation temperature T0toTQCDand to the corresponding re-entry temperature, TRE, matters\nsignificantly for δRE/δ0. When the walls form in a RD era and fa≲1017GeV, the formation\ntemperatures Twall≃Tosc≃Tµ/σare nearly identical. The value for T0/TQCD≃Tosc/TQCD\nis given in the top line of Eq. (A.1) as a function of fa. The ratio T0/TREdepends only on α\nand the time-temperature relation of the cosmological era\nT0\nTRE≃\n\nηRE\nη0\u0012g∗,RE\ng∗,0\u00131/3\nRD Era ,\n\u0012ηRE\nη0\u00133/4\u0012g∗,RE\ng∗,0\u00131/4\nMD (Non-Adiabatic) Era .(4.12)\nwhere ηRE/η0is given in Eq. (4.9) and is a function of α. If the walls form in a MD era and\nfa≳1015GeV (as will be the case for the largest PBH signal), then the axion is already at\nits zero temperature value by the time the universe cools to Twall. In that case, Eq. (4.11) is\nsimply unity for any T0.\nFinally, the total collapse time ηcof the wall can be determined from these simulations\nas well. Note that while we delegate studying the compression of energy into a PBH to\na distinct set of simulations covered in Sec. 5, the simulations used in this section yield\nan accurate prediction for the amount of time it takes for a domain wall to be maximally\ncompressed. We show a linear fit to these collapse times in Fig. 12. This linear relation is\ngiven by\nηc\nη0≃(\n1.84α+ 0.39 RD Era ,\n2.12α+ 0.31 MD Era .(α≡R0/t0) (4.13)\n– 22 –0 2 4 6 8 10\nR0 [t0]14710131619ηc/η0Radiation Domination\n0 2 4 6 8 10\nR0 [t0]159131721ηc/η0Matter DominationFigure 12 . The normalized conformal time length ηc/η0at which closed spherical domain walls have\nfully collapsed as a function of their initial radius R0.Left: The gray data points are collapse times\nobtained from our wall simulations assuming radiation domination. The black line shows the best\nlinear fit to these data points. Right : Same as left, but assuming matter domination.\nThe results of Eqns. (4.9), (4.10) and (4.13) capture the superhorizon behavior of perfectly\nspherical walls. As (more realistic) aspherical walls expand, they deform due to the expansion\nof the universe. Sides of the wall with different displacement from the wall center of mass\nstretch by differing amounts and re-enter the horizon at different times. We run identical sim-\nulations to the spherical ones described above, but with ellipsoidal initial wall configurations\nand describe the results in Appendix B to account for this. We find that prior to horizon\nre-entry, deformation of walls with large-scale asphericities typically amounts to a decrease\nin the ellipsoid parameters ϵ1andϵ2of a few percent.\n5 Subhorizon Collapse of Domain Walls\nIn this section, we study the dynamics of self-enclosed domain walls from the moment they\nre-enter the cosmic horizon. We first discuss perfectly spherical walls in Sec. 5.1 before\nconsidering realistic walls with asphericities in Sec. 5.2. In both cases, we calculate the\nmaximum energy within a given radius to determine how efficient walls compression is.\n5.1 Compressibility of Perfectly Spherical Walls\nAt horizon re-entry, tRE, walls are at their turning point transitioning from expansion to col-\nlapse. For a perfectly spherical wall, there are only two parameters that control the dynamics\nof wall collapse: the wall thickness δ≡m−1\na(in the rest frame of the wall) and the radius\nat horizon re-entry, R(tRE) =RRE. In fact, when the wall is subhorizon, Hubble friction is\nnegligible and the equation of motion can be made scale invariant so that it depends only on\n– 23 –0123456\n0 510 15 20 25 30012340123456\n0 510 15 20 25 300510152020406080\n0 510 15 20 25 30123-20246\n0 510 15 20 25 3005101520Figure 13 . The time evolution of a subhorizon domain wall. Top: The axion field configuration\nevolving according to Eq. (5.1), with each panel from left to right corresponding to a later time. In\nthe first panel, the wall is at rest. In the second panel, the wall has started Lorentz contracting and\nmoving to the left. In the third panel, the wall is maximally compressed. In the fourth panel, the\nwall has passed through itself, radiating a large amount of energy in the form of axions. Bottom : The\naxion field energy density corresponding to the field configurations shown in the top panels. Notice\nthat the energy density is highly localized in the domain wall surface, being maximally compressed\nin the third panel, but having been split into several components propagating outwards in panel four\nafter a PBH did not successfully form.\nthe ratio δ/R, even for a temperature-dependent axion mass.11Eq. (4.2) then becomes\n(1 +κ)2∂2θ\n∂˜t2−∂2θ\n∂˜r2−2\n˜r∂θ\n∂˜r+ sin θ= 0 κ=\n\nn\n4RD era ,\nn\n3MD adiabatic era ,\nn\n8MD non-adiabatic era ,(5.1)\nwhere n≈6.68 is the power at which m2\naincreases with temperature (see Eq. (2.3)) and\nκ/n=−(1/2)(dlnT/dlnt) is related to the time-temperature relation at the QCD phase\ntransition.\nWe numerically compute the evolution of the collapsing axion wall after horizon re-entry\nfor a wide range of δRE/RRE, taking a starting axion field configuration of\nθ(tRE, r) = 4 tan−1exp\u0002\nδ−1\nRE(r−RRE)\u0003\n, ˙θ(tRE, r) = 0 , (5.2)\nwhich is the analytic solution to Eq. (5.1) for large RRE/δRE. We use the numerical solution\nforθ(t,x) to compute the energy density of the axion field, ρ(t,x), as\nρ(t,x) =f2\na\u00141\n2˙θ(t,x)2+1\n2|∇θ(t,x)|2+ma(t)2(1−cosθ(t,x))\u0015\n, (5.3)\n11This scale invariance can be seen by the change of variables {t, r} → { ˜t,˜r} ≡ { ma(t)t, ma(t)r}and\ndropping terms O((mat)−1) which are small now that the wall is subhorizon.\n– 24 –and the energy within a given enclosed radius renclas a function of time as\nE(t, rencl) =Z\n|x|<renclρ(t,x)d3x. (5.4)\nFig. 13 shows the evolution of θ(t, r) and ρ(t, r) for a fiducial RRE/δRE= 15. The evolution\ncan be understood as follows: Initially, the wall is at rest and separates a region where θ= 0\ninside the wall and θ= 2πoutside as shown by the blue contour on the top of the first\npanel. The thickness of the wall, which can be seen more clearly from the width of the\nGaussian-like peak of the energy density, is of order m−1\na, as shown by the orange contour\non the bottom of the first panel. Next, the wall begins to collapse, converting rest mass\ninto kinetic energy and picking up speed as shown by the second panel. At this stage of\ncollapse, the Lorentz factor, γ, can be estimated from the Nambu-Goto effective field theory,\nwhich predicts γ= (R0/R)2. As the wall becomes more and more relativistic, a deviation\nfrom the arctan profile occurs with a small peak developing on the wall as shown on the\ntop of the second panel. Here, the thin-wall limit breaks down since the curvature term,\n2∂˜rθ/˜r∼γδREθ/R, becomes comparable to the potential term, sin θ[97]. This occurs at\nthe critical radius R∗≈γ∗δRE≈RRE(δRE/RRE)1/3, Lorentz factor, γ∗≈(RRE/δRE)2/3, and\nLorentz contracted wall thickness δ∗≈δRE/γ∗. At the final stage of collapse when R < R ∗,\nthe kinetic term in Eq. (5.1) is much greater than the potential (sin θ) term. The axion\nequation of motion effectively becomes the wave equation. The wave equation solution for\nR < R ∗is a radially inward-moving wave with a peak growing linearly with time, but with a\nconstant width. The minimum radius that a spherical wall can be fully compressed to is thus\napproximately the Lorentz contracted wall thickness δ∗=δRE(δRE/RRE)2/3.12A maximally\ncompressed wall is shown on the third panel of Fig. 13. Finally, the violent collision of the wall\nwith itself at collapse (assuming it has not formed a PBH), induces strong axion emission,\nas shown by the wiggles on the rightmost panel of Fig. 13, which travel radially outward.\nThe remnant of the wall expands and collapses again, losing about half its energy to axion\nemission with each bounce.\nThe left panel of Fig. 14 shows the largest fractional energy, f◦≡Max{E(t, rencl)/E(t,∞)},\nthat a given radius encloses for different contours of the wall re-entry radius to thickness ra-\ntio,RRE/δREof a spherical wall. Fig. 14 demonstrates that an O(1) fraction of the energy\nin a spherical wall can be compressed into a radius of order the wall thickness. Moreover,\nthe larger the initial wall size (larger RRE/δRE), the smaller the volume the wall can be\ncompressed to, as shown by the contours shifting to the left with increasing RRE/δRE. The\ngreater compressibility of larger walls is due to their greater Lorentz contraction. For a fixed\nrencl≲δ, the fractional energy enclosed grows as ( RRE/δRE)2, which is proportional to the\n12Particles in the background plasma that the axion couples to can damp wall motion if they reflect off\nthe wall [14, 98]. This occurs when the particle momentum is less than the inverse Lorentz contracted wall\nthickness 1 /δ∗[59, 99–101]. The δ∗of walls that dominantly end up forming PBHs is typically less than\n(100ma)−1, so that only non-relativistic axions from misalignment are cold enough to reflect off the enclosed\nwalls, as is the usual case for the mildly relativistic string-wall network [59, 72] (see however, [102]). The\nfriction generated by the cold axion population is small, especially after tosc[59].\n– 25 –volume of the minimum Lorentz contracted wall thickness, δ3\n∗. The energy enclosed falls off\nrapidly ( ∝r3\nencl) for rencl≪δ∗, since the energy density, ρ, of the axion field at small rencl\nis roughly a sharp peak of thickness δ∗, so that for rencl≪δ∗,ρis approximately constant.\nEq. (5.4) then indicates that E(t, rencl)∝r3\nenclas seen in the simulations.\nIn summary, numerical simulations indicate that for a collapsing spherical wall, the max-\nimum fraction of its total energy enclosed within a given radius, rencl, is nearly unity for\nrencl≫δ∗, but rapidly drops as r3\nenclforrencl≪δ∗and is proportional to the cube of the\nminimum Lorentz contracted wall thickness δ3\n∗∝(RRE/δRE)2for small rencl. This is captured\nby the semi-analytic broken power law\nf◦≡MaxE(t, rencl)\nE(t,∞)≈\n \nA−1\u0010rencl\nδ\u0011−3\u0012RRE\nδRE\u0013−2!2/3\n+ 1\n−3/2\n(5.5)\n≃\n\nA\u0010rencl\nδ\u00113\u0012RRE\nδRE\u00132\nrencl≪δ∗,\n1 rencl≫δ∗,(5.6)\nwhere the constant A ≈ 1/34 is calibrated by the numerical simulations and δis the instan-\ntaneous rest frame wall thickness. The semi-analytic fit, Eq. (5.5), is shown by the dashed\ncontours in the left panel of Fig. 14.\n5.2 Compressibility of Realistic Walls with Asphericities\nIn this section, we extend the analysis of Sec. 5.1 to determine the condition for PBH formation\nfor realistic, non-spherical walls. The effect of asphericities on collapse was first discussed by\nWidrow [16] who analytically found that large scale aspherical perturbations on an enclosed\ndomain wall are slightly damped in the early phases of collapse but then grow dramatically\nduring the final moments (see similarly Appendix B of [103]). There are two important caveats\nto this analysis: first, asphericities were only shown to grow to linear order in perturbation\ntheory, that is, only valid in a regime when the perturbations are small compared to the size of\nthe wall. Second, the analysis of [16, 103] was done in the Nambu-Goto picture which breaks\ndown in the regime where the perturbations are expected to grow the most (at R < R ∗).\nTo overcome these limitations, we study the effect of asphericities numerically using the full\nEuler-Lagrange equation of motion for the axion field (4.2). While we do find asphericities\ngrow during collapse, this growth is not as severe as previously expected in the literature; the\nultimate effect is to moderately reduce the efficiency of PBH formation.\nLarge scale asphericities on the wall are more important than small scale asphericities\nfor two reasons: First, Hubble freezes perturbations on the wall larger than the horizon\nwhile perturbations smaller than the horizon are free to move which become damped from\noscillating and emitting axions or gravitational waves. After re-entering the horizon, the walls\nof typical curvature radius RRE≈tREwill mostly have perturbations of size O(RRE). Second,\nfrom the mode analysis of [16] in the Nambu-Goto picture, it was found that the largest scale\n– 26 –Figure 14 . Energy compression in domain wall collapse. Left: The peak energy fraction compressed\ninto a spherical area of radius Renclosed as a function of Renclosed in units wall thickness δfor spherical\ndomain wall collapse. The solid colored lines indicate energy fractions for walls with five different\nre-entry radii RRE. Note, the slight kinks at Renclosed ∼δare caused by the appearance of a small\noscillatory correction to ˙θin the final moments before collapse, where the analytic domain wall solution\nbecomes invalid. The dashed colored lines indicate the best analytic fits to Eq. (5.5). Right : The\nrelative suppression in energy that can be compressed into a sphere of radius Renclosed as a function\nofRenclosed assuming that the domain wall is aspherical along a single axis with ellipsoid parameter\nϵ. The four distinct colored lines correspond to walls with four different ellipsoid parameters. Note\nthat the energy fraction within the sphere f(ϵ) has been normalized to the fraction in the perfectly\nspherical ( ϵ= 1) case. The dashed colored lines indicate the best analytic fits to Eq. (5.9).\nperturbations grow the most. For these reasons, we only consider large scale asphericities and\napproximate them by modeling the enclosed wall as an ellipsoid.\nWe repeat the simulations and analysis of Sec. 5.1 with an initial wall configuration that\npossesses θandϕdependence\nθ(tRE, r, θ, ϕ ) = 4 tan−1exp\u0002\nδ−1\nRE(r−R◦(θ, ϕ))\u0003˙θ(tRE, r, θ, ϕ ) = 0 , (5.7)\nwhere R◦(θ, ϕ) is the radial profile of the ellipsoidal wall with semi-axes ( RRE, ϵ1RRE, ϵ2RRE),\nR◦(θ, ϕ) =RREq\n(cos2ϕ+ϵ−2\n1sin2ϕ) sin2θ+ϵ−2\n2cos2θ. (5.8)\nWithout loss of generality, we take ϵ1≥1 and ϵ2≤1 so that the curvature radius RREis\ndefined as the median semi-axis at horizon re-entry. The random field simulations of Sec. 3\ngive a typical ϵ1∼1.4 an ϵ2∼0.6 as shown by Fig. 6. Our goal is to numerically compute\nthe reduction factor for the enclosed energy within a given radius, rencl, for an ellipsoidal wall\nof initial curvature radius RREand ellipsoidal parameters ( ϵ1, ϵ2) compared to a perfectly\nspherical wall of the same curvature radius RRE(i.e. with ϵ1=ϵ2= 1).\n– 27 –The right panel of Fig. 14 shows this reduction in enclosed energy of an ellipsoidal wall\ncompared to a spherical wall for RRE/δRE= 20 with ϵ2= 1 and ϵ1allowed to vary (i.e. the\nwalls are more like a baguette than a pancake) as shown by the colored contours. As expected,\nthe more ϵ1deviates from unity, the larger the suppression in enclosed energy compared to a\nperfectly spherical wall. Moreover, the right panel of Fig. 14 indicates that most of the energy\nin an elliptical wall can still be compressed into a radius rencl≫few×δbecause the relative\nsuppression compared to a spherical wall of the same mean RREis negligible. However, for\nrencl≲few×δ, the eccentricity of the walls grows dramatically and the suppression in the\nenergy enclosed compared to a spherical wall is severe, becoming more suppressed, with larger\nRRE/δRE. By simulating a variety of RRE/δREandϵ1, we find a semi-analytic function for\nthe relative suppression of an ellipsoidal wall of eccentricity ϵ1>1 and ϵ2= 1 compared to a\nspherical wall ( ϵ1=ϵ2= 1) as shown by the dashed curves in the right panel of Fig. 14. This\nfit is given approximately by a broken power law\nf(ϵ1, ϵ2= 1)\nf◦≡f◦(ϵ1)≈\n1 + \nB−1|ϵ1−1|\u0012RRE\nδRE\u00131.83\nS\u0010rencl\nδ, ϵ1\u0011!3\n−1/3\n, (5.9)\nwhere B ≈ 5.4 and Sis a sigmoid function that goes to 1 for rencl/δ≪1 and to 0 for\nrencl/δ≫1. Parametrically, Sis given by\nS\u0010rencl\nδ, ϵ1\u0011\n≈ \n2\n1 + exp\u0002\nCrencl\nδ|ϵ1−1|−.55\u0003!1.5\n, (5.10)\nwhere C ≈0.33. The sigmoid function Sgives rise to a characteristic radius rbreak > δwhere\nforrencl≳rbreak, the maximum fractional energy enclosed within a radius renclis effectively\nunity and unchanged relative to a spherical wall of the same re-entry radius RRE; conversely,\nforrencl≪rbreak, the maximum fractional energy enclosed is much less than the spherical\nwall of the same RREdue to the large scale asphericities on the wall which grow strongly for\nrencl≪rbreak. That is, the relative suppression of an ellipsoidal wall compared to a spherical\nwall, Eq. (5.9) is approximately\nf(ϵ1, ϵ2= 1)\nf◦≈\n\n1 rencl≳rbreak,\nB(ϵ1−1)−1\u0012RRE\nδRE\u0013−1.83\nrencl≪rbreak,(5.11)\nwhere\nrbreak\nδ≈5\u0012ϵ1−1\n0.3\u00130.55 \n1 +1\n5ln\"\u0012RRE/δRE\n20\u00131.83\u0012ϵ1−1\n0.3\u0013#!\n. (5.12)\nThe results of Eqns. (5.11) and (5.12) have important implications for the formation of PBHs.\nSince most walls are not perfectly spherical (see the distribution of Fig. 6), and the suppression\nin enclosed energy is strong for rencl< rbreak, most walls only form PBHs if their Schwarzschild\n– 28 –radius is larger than rbreak, which is parameterically much larger than δ∗for perfectly spherical\nwalls. We discuss this quantitatively in the following section.\nFinally, realistic walls have ϵ2<1. The maximum radius the entire wall can be com-\npressed into, rbreak, does not significantly change because the semi-axis RREϵ2is less than the\nother two semi-axes RREandRREϵ1. However, for rencl≪rbreak, the asphericities along the\nsemi-axis associated with ϵ2can grow. Through simulations of collapsing walls with varying\nϵ1andϵ2, we find that the reduction factor in the energy enclosed within a radius renclcom-\npared to a spherical wall of the same RREroughly factorizes in terms of f◦(ϵ) of Eq. (5.9)\nfor each axis,\nf(ϵ1, ϵ2)\nf◦≈f◦(ϵ1)×f◦(ϵ2), (5.13)\nthough this is harder to numerically check for RRE/δRE≫10 due to the computational\nlimitations of simulating high resolution three-dimensional walls. Note for r > r break, (5.13)\nis effectively equivalent to (5.11).\nIn summary, the maximum fractional energy enclosed within a radius renclof a collapsing\nwall in terms of its horizon re-entry radius RREand ellipsoidal parameters ϵ1,ϵ2is given by\nf≡MaxE(t, rencl)\nE(t,∞)=f◦×f(ϵ1, ϵ2)\nf◦, (5.14)\nwhere f◦is given by Eq. (5.5) and f(ϵ1, ϵ2)/f◦is given by Eq. (5.13).\n6 Conditions for PBH Formation\nIn this section, we determine the conditions for PBH formation of collapsing domain walls.\nFirst, we discuss the conditions for the wall to be compressed to within a Schwarzschild\nradius using the results of Sec. 5. We find a minimum RRE/δREneeded to form a PBH,\nwhich increases with decreasing fa. This follows because walls with low famust compensate\nby having large radii to achieve the same mass compared to a wall with larger fa. Next, we\ncarefully treat potential impediments to PBH formation, such as angular momentum which\ncan cause some walls to form a centrifugal barrier or break apart prior to compressing to\nwithin a Schwarzschild radius, and collisions with other topological defects. We find that\nangular momentum is only an obstacle to PBH formation if the domain wall is very light,\nwhile collision with other defects has a mild suppressive effect across the whole wall mass\nrange. In Appendix C, we furthermore discuss why gravity does not impede PBH formation.\n6.1 Compression into a Schwarzschild Radius\nA black hole forms when renclis smaller than the corresponding Schwarzschild radius, Rs=\n2GEf , where E=E(t,∞) =Awallσis the total energy of the wall, σ≃8maf2\na= 8δ−1f2\nais\nthe wall tension, fis the fraction of the total energy within rencl(5.14), and Awallis the wall\n– 29 –surface area at horizon re-entry, given by Awall= 4πR2\nREI◦(ϵ1, ϵ2), where I◦(ϵ1, ϵ2)/ϵ1ϵ2≡\nRG(ϵ−2\n1;ϵ−2\n2; 1)∼1 is the Carlson G elliptic integral.13\nIt is helpful first to gain some intuition for the conditions imposed on the wall parameters\n(RRE/δRE, ϵ1, ϵ2) to successfully collapse within a Schwarzchild radius for simpler walls or for\nnaive assumptions about collapse before considering how realistic walls with asphericities\nmodify these conditions for PBH formation. First, assume that the wall is perfectly spherical\nand can compress most of its energy to within a radius of order its rest-frame thickness,\nδ≡1/ma; that is f∼1 for rencl∼δ. Setting rencl=Rsgives the condition on the minimum\nsize of the wall re-entry radius to its thickness\nRRE\nδRE=\u0012MPl\nfa\u0013\u0012δ\nδRE\u0013\u00121\n64π\u00131\n2\u0010PBH Formation Condition:\nNaive Assumption\u0011\n. (6.1)\nAlthough Eq. (6.1) is not correct for realistic walls, it captures some of the parameteric\ndependence of the wall on forming a PBH, namely that the larger fais, the smaller the wall\nneeds to be to collapse to within a Schwarzschild radius and form a PBH.\nNext, drop the assumption that f∼1 for rencl∼δ. As discussed in Sec. 5, because of the\nLorentz contraction of the wall thickness, a perfectly spherical wall can be compressed into a\nradius smaller than its rest-frame thickness. Setting rencl=Rs, but with f=f◦of Eq. (5.5),\ngives an improved condition on the minimum size of the wall re-entry radius to its thickness\nRRE\nδRE=\u0012MPl\nfa\u00133\n4\u0012δ\nδRE\u00133\n4\u00121\n64π1\nA1/31\nf(f−2/3−1)1/2\u00133\n8\u0010PBH Formation Condition:\nPerfectly Spherical Walls\u0011\n.\n(6.2)\nAlthough Eq. (6.2) is similar to the naive expectation of Eq. (6.1), there are a few important\ndifferences: First, the power in MPl/fais reduced from 1 to 3 /4 which arises from the Lorentz\ncontraction of the wall, making it easier to form a PBH by not needing a large RRE/δRE.\nSecond, the energy fraction fis a free parameter in Eq. (6.2) which can be chosen to minimize\nthe right-hand-side of Eq. (6.2). For f≳0.95, the right-hand side of (6.2) blows up because\nrenclis so large that it is difficult to satisfy the black hole formation condition of rencl=Rs.\nConversely, for f≲0.05, the right-hand side again blows up because while renclis small,\nthe energy fraction within renclfalls off so rapidly that Rsdecreases much faster than rencl,\nmaking it again difficult to satisfy rencl=Rs. For fbetween these two values, the right-hand\nside is nearly constant and saturated near its minimum value. The blue contour labeled\nϵ= 1.0 in Fig. 15 shows the minimum RRE/δREneeded for a PBH formation as a function of\nfafor a perfectly spherical wall (which occurs for an energy fraction f≈0.54).\nFinally, drop the assumption that the wall is perfectly spherical. The condition for PBH\nformation is still given by rencl=Rs, but now with fof Eq. (5.14) for aspherical walls.\nBecause of the complicated dependence of fonrencl, it is not possible to write an exact\n13The Carlson G elliptic integral is the integral giving the surface area of an ellipsoid with semi-major axes\nRREϵ1andRREϵ2relative to a sphere of radius RRE[104].\n– 30 –Figure 15 . The minimum wall re-entry radius RREin units of the re-entry wall thickness δREneeded\nto form a PBH, as a function of fa. The different colored lines show the radius needed for different\nwall eccentricity parameters, where one axis is allowed to vary in length while the other two axes are\nfixed at radius RRE. Note that for aspherical walls, the minimal radius needed for PBH formation\nis substantially larger than in the spherical case (the blue line). This is due to the fact that it is\nsubstantially harder to compress energy into a small volume if the wall is aspherical. This difficulty\ncan be overcome, however, by driving up the domain wall radius to the point where the Schwarzschild\nradius of the potential black hole formed is equal to the thickness of the wall.\nanalytic bound like Eqns. (6.1) and (6.2), but the equation can be solved numerically for the\nminimum RRE/δREto form a PBH as shown by the ϵ̸= 1 contours of Fig. 15. The ϵ= 1.001\n(orange) and ϵ= 1.01 (green) contours highlight that the walls must be spherical to a high\ndegree at horizon re-entry for the perfectly spherical wall approximation, Eq. (6.2), to hold.\nForϵ≳1, aspheriticies hinder compression into Rs. In this regime, an approximate equation\nfor the numerical solution of the minimum RRE/δREcan be understood by taking f∼1 for\nrencl∼rbreak of Eq. (5.12) due to the rapid fall-off in enclosed energy for radii smaller than\nrbreak. Solving the equation rencl=Rsperturbatively gives the following minimum size of the\nwall re-entry radius to form a PBH\nRRE\nδRE≈\u0012MPl\nfa\u0013\u0012δ\nδRE\u0013\u00121\n64πI◦(ϵ1, ϵ2)\u00131\n2\u0010rbreak\nδ\u00111\n2\u0010PBH Formation Condition:\nAspherical Walls\u0011\n.\n(6.3)\nHere, rbreak/δis given in Eq. (5.12) with RRE/δREin its natural logarithm argument evaluated\nat the logarithmic-independent piece of Eq. (6.3) (i.e. first order in perturbation theory).\n6.2 Impediments to Collapse: Angular Momentum\nA simple requirement for PBH formation is that the angular momentum of the wall, JDW,\nmust be less than the maximum angular momentum a black hole can possess (the so-called\n– 31 –extremal limit ) [105]\nJDW< GM2\nDW (Maximum Wall Angular Momentum) . (6.4)\nEnclosed axion domain walls may form with, or acquire, angular momentum that violate\ncondition (6.4). The intrinsic angular momentum of the wall can arise from the uncorrelated\nvelocity of the axion field upon formation while the acquired angular momentum can arise in\nthree main ways: (1) collisions with other defects, (2) fracturing into multiple walls during\ncollapse, and (3) tidal gravitational forces from background density inhomogeneities. We now\ndiscuss each scenario in turn.\nThe angular momentum of an enclosed wall is given by\nJDW=I\nR×dp=I\nR×vγσdA (6.5)\n=MDW¯RvRMS\u0012H\nR×dAγv\n¯RvRMSR\ndA\u0013\n, (6.6)\nwhere Ris a vector pointing from the wall center of mass to a given point on the wall surface\nanddp=vγσdA the momentum of a patch of wall of infinitesimal wall surface area dA,\nvelocity v, Lorentz factor γ, and tension σ=MDW/H\ndA.¯RandvRMSare the mean wall\nradius and root-mean-square velocity of the wall surface, respectively.\nIn the second line, we use the fact that the wall velocity is always normal to the wall\nsurface. Thus, v can be thought of as 1-dimensional random velocity field at a given ( θ, ϕ)\ncoordinate on the wall surface; it may be positive or negative depending on whether the\nmotion is outward or inward, respectively, at a given point on the surface.\nFor an enclosed wall that is not significantly distorted, Rand the surface normal vector\nare roughly aligned. Consequently, the dimensionless factor in parenthesis – which involves\nthe perpendicular projection of these two vectors – is typically much smaller than unity.\nMoreover, it vanishes identically if γv is constant everywhere on the wall surface since by\nStokes’ theorem, the integral can be transformed into an integral over the boundary of an\nenclosed surface, which is 0. For any random velocity field v( θ, ϕ) on the wall surface – that\nis, a periodic random field on the rectangle 0 ≤θ < π , 0≤ϕ <2π– which is neither highly\ncorrelated nor highly uncorrelated, we find the magnitude of the term in parenthesis is of\norder∼10−2.14For velocity fields v that are either highly correlated or highly uncorrelated,\nwhich correspond to the limit where the velocity field is the same everywhere on the wall or\ntotally random everywhere, this factor in parenthesis is smaller, as expected.\n14Closed string loops (including string-bounded walls) have a somewhat larger specific angular momen-\ntum [106, 107]. The reason for this is likely geometric: only motion perpendicular to a defect is physical [53].\nA segment of string can move anywhere in the plane transverse to its tangent vector, while a segment of wall\ncan only only move parallel to its normal vector. The latter constraint naturally reduces the perpendicular\nprojection of the lever arm momentum in Eq. (6.5).\n– 32 –After horizon re-entry, the magnitude of the wall angular momentum compared to an\nextremal black hole is therefore\nJDW\nGM2\nDW≲.01vRMSRRE\nGM DW=.02vRMSRRE\nRs, (6.7)\nwhere Rsis the Schwarzschild radius of the PBH.\nSince the continual growth and remixing of correlated domains of the axion field in the\nuniverse – which are the source of enclosed walls — are generated by the motion of the infinite\nstring-wall network, it is expected that vRMSinherits its value from the network at the time t0\nwhen the enclosed wall forms. Numerical simulations and analytic expectations give a typical\nspeed v∞∼0.4−0.6 of the string-wall network [108, 109]. By the time the wall re-enters\nthe horizon, this speed may decrease due to the redshifting of momentum with the Hubble\nexpansion [53]. Consequently, Eq. (6.7) indicates that the intrinsic angular momentum of an\nenclosed wall likely cannot stop a PBH from forming if the wall needs to compress by factor\nof∼a few hundred to fall within its Schwarzschild radius. From the minimum RRE/δneeded\nto form a PBH and the scale rbreak/δ≈Rs/δinto which an O(1) fraction of the wall energy\ncan be compressed (Eq. (5.12)), it can be seen that RRE/Rs≲102forfa≳1015GeV, which\ndesignates the region where Eq. (6.4) is satisfied.\nEnclosed walls that collide with the wall-bounded strings that surround them can acquire\nfurther angular momentum. Since the relative speeds between the two colliding defects is\ncomparable to v∞, this additional angular momentum is similar to Eq. (6.7) when the two\ncolliding defects have comparable masses. If the enclosed wall is more massive compared to the\nstring-bounded wall it collides with, which is typically the case, then the angular momentum\ntransfer can be suppressed further by the ratio of their masses. If the wall is highly distorted\nand collides with itself during collapse, the wall can break apart into multiple enclosed walls.\nBut by the same logic as before, each piece will have some angular momentum comparable\nto Eq. (6.7) . Moreover, the shape of the enclosed wall must be highly non-spherical for this\nfracturing to occur, which will not be the case unless the initial wall size is far greater than\nthe horizon at formation, at which point, the surface is more Brownian-like. However, the\nwalls that eventually become the majority of PBHs form with sizes only mildly greater than\nthe horizon (i.e. α≡R0/t0∼1−2). Walls with α≫1 are anyway far too suppressed in\nabundance.\nFinally, if the enclosed wall is treated as a rigid object, then the wall can acquire angu-\nlar momentum from gravitational tidal forces if it is aspherical and possesses a quadrupole\nmoment [110]. The angular momentum acquired by a rigid elliptical wall of mass MDW=\nMBH∼4πσt2\nREfrom such tidal forces corresponds to a spin parameter [110]\nJDW\nGM2\nDW∼\u0012MDW\n10−5M⊙\u0013−1\u0012tRE\ntQCD\u0013\u0012Σ\n5×10−5\u0013\n, (6.8)\nwhere Σ is the variance in the primordial background density perturbations and tQCD is\nthe time of the QCD phase transition which is the typical timescale at which the walls re-\nenter the horizon and collapse. Eq. (6.8) indicates that QCD axion domain walls with mass\n– 33 –MDW≳10−5M⊙and moderate eccentricities are sub-extremal and able to form PBHs if\nEq. (6.4) is satisfied. This is typically the case for any fa≳1014GeV as discussed in Sec. 7.\n6.3 Impediments to Collapse: Collisions with Neighboring Defects\nCollisions with the surrounding string-bounded walls can also damage or destroy the enclosed\nwalls via axion emission or wall puncturing which, if the resultant holes are larger than\nRµ/σ≈µ/σ, will expand and ‘eat’ the wall [13, 77, 111, 112]. How often does this occur?\nConsider an enclosed wall of initial mean physical radius R0=αt0during superhorizon\nexpansion. In the comoving frame, the wall radius is stationary at fixed coordinate radius\nr=R0and the comoving flux of surrounding defects impinging on the enclosed wall is\nΦ =Z\nnvcosθf(v)d3v=1\n4n¯v , (6.9)\nwhere ¯ v=R\nvf(v)d3v≈v∞×a(t0)/a(t) is the mean coordinate velocity of the enclosed wall\nrelative to the surrounding defects15andn=t−3×(a(t)/a(t0))3is the comoving number\ndensity of the other defects in the medium. The rate of collisions is then\nΓcoll≈4πr2Φ≈πα2t2\n01\nt3\u0012t\nt0\u00133k\nv∞\u0012t0\nt\u0013k\n, (6.10)\nwhere kis again the exponent in the relation a(t)∝tkand is equal to 1 /2 (2/3) in a RD\n(MD) era. It follows from (6.10) that the probability of an enclosed wall notcolliding with\nanother defect after a time tis\nP(Not colliding after time t)≡P\b\bcoll= exp\u0012\n−Zt\nt0Γcoll(t′)dt′\u0013\n= exp\"\n−πα2v∞\n2−2k \n1−\u0012t0\nt\u00132−2k!#\n. (6.11)\nEq. (6.11) demonstrates that the majority of large walls collide with other defects in the\nnetwork, leading to their destruction. The factor of α2in Eq. (6.11) can be understood as a\ngeometric factor in the sense that the larger the total surface area of the wall, the greater its\ncross-sectional area. The asymptotic behavior of P\b\bcollto the constant exp\u0002\n−πα2v∞/(2−2k)\u0003\ncan be understood by the increase in physical separation of defects with the expansion of the\nuniverse, which reduces the rate at which enclosed walls collide with other defects.\nNote that Eq. (6.11) is valid while the wall is superhorizon and while the infinite string-\nwall network remains in scaling. After tµ/σ, the infinite string-wall network collapses relativis-\ntically so that the network decays within roughly a Hubble time tµ/σ[113]. Thus Eq. (6.11)\nshould be evaluated at the minimum of t≈tREandt≈2tµ/σ. For example, for walls that\nform at time near network collapse tµ/σ,P\b\bcoll(α)≈exp\u0002\n−1.6α2v∞\u0003\nin a RD or MD era.\n15This follows because the coordinate velocity is dx/dt=vpeca(t0)/a(t) where vpecis the peculiar velocity\nof the surrounding defects which is the speed of the infinite string-wall network v∞∼0.5.\n– 34 –7 PBH Abundance and Detection Prospects\nIn this section, we combine the results for the number density of enclosed walls (Sec. 3) with\ntheir superhorizon growth (Sec. 4), subhorizon collapse (Sec. 5), and conditions for PBH\nformation (Sec. 6) to determine the relic abundance of PBHs fPBHas a function of fa, as\nwell as the mass associated with the PBHs produced. We compare the computed abundances\nwith the reach of current and future gravitational lensing surveys. We focus on the early MD\ncosmological scenario, which yields an especially favorable fPBH, but share analogous results\nfor axion-like particles in RD cosmologies in Appendix D.\nThe total energy density in PBHs at the moment of production ρPBHis parameterized\nby the integral\nρPBH=Z\ndα dϵ 1dϵ2MPBH(α, ϵ1, ϵ2)dnPBH(α, ϵ1, ϵ2)\ndαfmax/mean(ϵ1)fmin/mean(ϵ2),(7.1)\nwhere fmax/mean(ϵ1) and fmin/mean(ϵ2) are the normalized eccentricity probability distribution\nfunctions given by Eq. (3.7) and Eq. (3.8), respectively, MPBH(α, ϵ1, ϵ2) is the mass of a PBH\nformed from a closed domain wall of radius α=R0/t0and eccentricity parameters ϵ1andϵ2.\ndnPBH/dαis the number density distribution of PBHs, which we construct as\ndnPBH(α)\ndα≡dncDW(α)\ndα×P\b\bcoll(α)×Θ [α−αmin], (7.2)\nwhere dnPBH(α)/dαis the self-enclosed domain wall number density given by Eq. (3.6), P\b\bcoll\nis the probability a self-enclosed wall does not collide with a neighboring defect, given by\nEq. (6.11). Θ is a Heaviside function with αminbeing the solution to Eq. (6.3) taking into\naccount superhorizon expansion of Eq. (4.10). For ease of computation, we also decompose\nMPBHin the following way\nMPBH(α, ϵ1, ϵ2)≃4πR2\nRE(α)I◦(ϵ1, ϵ2)σf(α, ϵ1, ϵ2) (7.3)\n≃32πR2\nRE(α)I◦(ϵ1, ϵ2)f2\nama(tc)f(α, ϵ1, ϵ2). (7.4)\nHere, f(α, ϵ1, ϵ2)∼1 is the fraction of the total wall energy that compresses to within\na Schwarzschild radius as given in Eq. (5.14), RREis the wall re-entry radius as given in\nEq. (4.10), tcis the collapse time given by Eq. (4.9). We take the latest possible t0≃tµ/σ\nwhich is the moment that the string-wall network scaling relation ends and the entire string-\nwall network begins to contract, given by Eq. (3.11). I◦is the surface area of an ellipsoidal\ndomain wall with semi-major and semi-minor axis lengths RREϵ1andRREϵ2, respectively,\nnormalized to the corresponding spherical surface area. We evaluate this integral numerically.\nWith ρPBHcalculated, the final step is to compute the total fraction of DM consisting of\nPBHs fPBH. In QCD axion models, the relic axion particle energy density Ω a(not including\nPBHs) has two components: the misalignment energy density Ω misand the relic density\nof axions produced from defect radiation (primarily strings) Ω strings . At the moment of\nmisalignment, the energy density in such axions ρmisis given by\nρmis(tosc)≃1\n2f2\na⟨θ2\nmis⟩ma(Tosc)2, (7.5)\n– 35 –where ⟨θ2\nmis⟩ ≃π2/3 is the mean square misalignment angle and Toscis the temperature at\nwhich the the axion field starts to oscillate [12]. The moment of oscillation tosc, corresponding\ntoTosc, is found by solving 3 H(Tosc) =ma(Tosc). We detail how to compute the oscillation\ntime in the early matter-dominated scenario scenario in Appendix A.\nWe now compute Ω PBH/Ωmisas\nΩPBH\nΩmis=ρPBH(tc)\nρmis(tosc)\u0014a(tc)\na(tosc)\u00153ma(Tosc)\nma(T= 0), (7.6)\nwhere the [ a(tc)/a(tosc)]3factor accounts for dilution of misalignment DM up until the moment\nof PBH formation and ma(Tosc)/ma(T= 0) accounts for the increase in the axion mass of\nmisalignment DM from toscuntil today. In the matter-dominated scenario, [ a(tc)/a(tosc)]3=\n(tc/tosc)2while in the radiation dominated scenario, [ a(tc)/a(tosc)]3= (tc/tosc)3/2. After tµ/σ,\nwhere the whole string-wall network collapses, the relic axion DM particles and the PBHs\nboth scale like matter for the rest of cosmic history, hence the fraction Ω PBH/Ωmisstays fixed\nuntil the present day.\nFinally, fPBH, the fraction of present-day DM consisting of PBHs, is defined as\nfPBH=ΩPBH\nΩmis+ Ω strings + Ω PBH≃ΩPBH\nΩmis+ Ω strings, (7.7)\nwhere Ω strings can be obtained from simulations. Despite being extensively studied, the num-\nber density of axions from string radiation is uncertain. As discussed in Sec. 3.3, log correc-\ntions to the string density parameter ξcould be slightly larger than 1 .0 by TQCD, but these\nlog corrections are generally thought to be suppressed in an early MD cosmology [39] due to\nthe greater expansion rate of the universe compared to a RD cosmology. Fig. 16 shows fPBH\nas a function of fawhere the blue band indicates the uncertainty from string radiation. The\ngray solid regions show current constraints from EROS [28] and OGLE [27], while the dashed\ngray regions show constraints projected to be achieved with the release of Gaia Data Release\n4 [33] and constraints projected to be achieved with the Roman Space Telescope ’s Galactic\nBulge Time Domain Survey [34, 35]. Note that these constraints from Roman are somewhat\noptimistic and may fall up to an order of magnitude lower, depending on systematics [34].\nFig. 16 indicates that fPBHplateaus at very high valaues of fabecause in this part of\nthe parameter space, even small, aspherical walls are energetic enough to successfully form\nPBHs upon collapsing. As fais lowered below fa≲a few times 1016GeV, the walls become\nprogressively less energetic, requiring their initial radius R0to be increasingly larger. However,\nthe number density of enclosed walls with a very large initial radius is exponentially suppressed\nrelative to small walls per Eq. (3.6). This means that for fa≲a few times 1015GeV, fPBH\ndrops well below a range observable by gravitational lensing surveys in the foreseeable future.\nHowever, the projections suggest that near-future gravitational lensing surveys will be able\nto probe the very high faregion of the relevant fPBHparameter space.\nIn the most minimal early MD cosmology, where a single particle species dominates the\nenergy density of the universe at early times and eventually decays into SM radiation at TRH,\n– 36 –obtaining the correct axion DM abundance at large farequires low TRH. The region to the\nright of the orange vertical line of Fig. 16 indicates roughly where TRHfalls below the Big\nBang Nucleosynthesis (BBN) temperature, TBBN≃2 MeV [114] for Ω strings≲Ωmis. For\nTRH≲TBBN, which roughly corresponds to fa≳3×1015GeV [66], the predicted abundance\nof light elements begins to observably differ from their measured values [114]. Moreover,\nforfa≳1016GeV, TPQalso risks growing larger than the upper bound on the inflationary\nreheating temperature [115]. Hence, for the QCD axion, a consistent MD cosmology may\nlimit fPBHto lie on the exponentially sensitive tail of the distribution of large walls such that\nfPBH≲10−10– out of reach of near-future lensing surveys. ALPs, which can form domain\nwalls at times earlier than the QCD axion, can avoid this low TRHbound and can thus have\nconsistent cosmologies for fa>3×1015GeV so that fPBHis large and observable by next\ngeneration lensing surveys. We discuss this further in Appendix D.\n8 Conclusions\nSelf-enclosed domain walls in axion models provide an exciting new avenue for PBH produc-\ntion, and these PBHs may potentially be observable by future gravitational lensing surveys.\nWe study the number density and PBH formation efficiency of these closed domain walls\nin both radiation dominated and early matter-dominated cosmologies: the latter being of\nparticular interest since it allows high values of the axion decay constant fa.\nFirst, we study the number density of self-enclosed domain walls near the moment the\nstring-wall network collapses tµ/σusing a simple geometric argument: namely, that the axion\nfield must be completely uncorrelated on length scales larger than the string separation length\nLand highly correlated on length scales shorter than L. Applying this argument to a discrete\nlattice representation of the early universe yields the distribution of self-enclosed domain walls\nof various sizes. Expanding the argument to a continuous random field furthermore yields\nthe distribution of shapes of self-enclosed domain walls, which we take to be approximately\nspheroidal – this is a valid approximation for studying PBH formation from domain wall\ncollapse since the walls are smooth on scales ∼L, which is the scale at which the domain wall\nnumber density is peaked.\nWe then study the dynamics of these domain walls upon formation. At early times, the\nwalls may stretch with the expansion of the universe until they enter the cosmic horizon.\nNumerically solving the low energy axion equations of motion in an FLRW background yields\nthe radius of domain walls at the moment of horizon re-entry RREand the amount of (con-\nformal) time it takes for walls to re-enter the horizon ηREand to fully collapse ηc. Depending\non the axion decay constant fa, the wall may also become thinner and more energetic due to\na higher wall tension as the axion mass continues to ‘turn on’ during the dynamical stage of\ndomain wall evolution.\nNext, we simulate the subsequent stage of wall evolution: the inwards collapse, where\nthe expansion of the universe may safely be neglected. We numerically solve the low energy\naxion equation of motion in a Minkowski spacetime and compute how much energy can be\n– 37 –101510161017\nfa [GeV]10−1010−810−610−410−21fPBH\nTRH/similarequal2 MeV\nQCD Axion\nMD EraΩstrings<Ωmis\nΩstrings/similarequal10ΩmisRoman Projection10−11MPBH [M⊙]\nGaiaProjectionEROS\nOGLEFigure 16 . The projected DM PBH fraction in an early matter-dominated cosmology. The bot-\ntom axis shows the QCD axion model facorresponding to a given fPBHabundance. The top axis\nshows the PBH mass MPBHassociated with each fa. Note that the PBH spectrum for a given fais\nnearly monochromatic, hence the one-to-one map from fatoMPBH. The top and bottom solid black\nlines show the fPBHabundance from axion domain wall collapse in the case where the relic radiated\nstring axion abundance Ω strings is<Ωmisand approximately 10 Ω mis, respectively. Ω strings is entirely\npredicted by faand the subject of extensive study, however its exact value remains uncertain due to\nsimulation limitations. The two black lines correspond to edge cases for Ω strings , meaning fPBHmay lie\nanywhere in the blue band. The gray solid regions show the PBH parameter space currently excluded\nby the EROS microlensing survey [28] and by the recent OGLE-IV survey [32]. The gray dashed lines\nshow projections for the PBH parameter space that will be reachable by Gaia Data Release 4 [33] and\nbyRoman ’s Galactic Bulge Time Domain Survey [34, 35]. The dashed orange line shows where the\nreheat temperature TRHequals the BBN temperature TBBN≃2 MeV [114] in the most minimal early\nmatter-dominated cosmology when Ω strings≲Ωmis[66]. Regions to the right of this line may over-\nproduce axion DM so that consistent early matter-dominated cosmologies for the QCD axion require\nfPBHto be below ∼10−10, although this constraint is easily avoided in ALP models. Projections for\nALP models are shown in Figs. 20, 21.\n– 38 –compressed into a small volume about the wall center of mass. The numerical solutions reveal\nthat for a spherical domain wall, the fraction of total wall energy that can be compressed into\na sphere of radius raround the wall’s center of mass is roughly unity for r≳δ∗, but drops off\n∝r3forr≲δ∗where δ∗is the Lorentz contracted domain wall thickness. Importantly, the\nnumerical solutions also show that the energy compressibility of aspherical walls is further\npower law suppressed relative to the spherical case. Realistic walls with asphericities can\ncompress most of their energy into a radius r≈rbreak which is parametrically larger than the\nrest-frame wall thickness δ= 1/ma.\nWe use these results to compute the initial domain wall size and shape necessary to form\na PBH. For the faparameter space of interest, there is a critical minimal initial domain wall\nradius R0necessary for successful PBH formation. This radius coincides with the domain\nwall being large enough that its Schwarzschild radius is somewhat greater than the wall\nthickness Rs≈rbreak. We also account for angular momentum acquisition by the walls due\nto interactions with neighboring defects and gravitational attraction between domain walls\nand surrounding density perturbations. However, careful study reveals that these effects are\nmarginal for the wall and PBH masses of interest.\nFinally, we use all of these considerations to calculate the fraction of present-day DM that\nis composed of PBHs produced via axion domain walls collapse, fPBH, in an early matter-\ndominated cosmology, shown in Fig. 16. We find that the present-day PBH abundance is\nsufficiently high in early matter-dominated scenarios with fa≳1015GeV to potentially be\nobservable in the future. For fa≳1016GeV, the PBH abundance is large enough that\ncurrent or near-future astrometric and photometric gravitational lensing surveys like Gaia\nData Release 4 and OGLE could potentially observe a PBH signal. However, for this fa\nrange, the early matter-dominated reheat temperature TRHcomes into tension with the Big\nBang Nucleosynthesis temperature TBBNin the simplest cosmological scenario. For smaller\nvalues of fa, this tension can be completely avoided, however the corresponding fPBH is\nsignificantly smaller and will likely not be observable in the near future. Another way to\navoid the potential tension with BBN, while still maintaining a near-future observable fPBH,\nis to consider axion-like particles, which we do in Appendix D.\nIn all of these scenarios, we have assumed that the string density parameter ξis roughly\nunity, however this could turn out to be inaccurate – especially in the radiation dominated\nscenario discussed in Appendix D (see Sec. 3 for a discussion of ξ). Should that be the case,\nwith ξgrowing larger than roughly unity at the QCD phase transition, Ω PBHwould be sub-\nject to a further exponential suppression due to the self-enclosed wall number density scaling\nin Eqns. (3.5)-(3.6). This would cause there to be fewer ∼horizon-sized walls, meaning expo-\nnentially fewer PBHs in the parameter space that relies on superhorizon growth to increase\nthe domain wall mass and form PBHs. As can be seen from Fig. 16, this is dominantly the\nregion where fa≲1016GeV.\nAs a final note, we emphasize that the formalism developed here can be applied to\nother models that exhibit unstable domain walls approximately described by a Sine-Gordon\nequation of motion. We therefore hope to extend this work to other classes of models not\n– 39 –involving the QCD axion in the future.\n9 Acknowledgements\nWe thank Robert Brandenberger, Malte Buschmann, Cyril Creque-Sarbinowski, Chris Dessert,\nJoshua W. Foster, Anson Hook, Junwu Huang, John March-Russell, Javier Redondo, Ken\nVan Tilburg, Giovanni Villadoro, and Neal Weiner for helpful discussions, Huangyu Xiao for\nclarification on low TRHlimits in axion matter-dominated cosmologies, and Robert Branden-\nberger, Cyril Creque-Sarbinowski, Chris Dessert, and Ken Van Tilburg for helpful comments\non the manuscript. This work was supported in part through the NYU IT High Performance\nComputing resources, services, and staff expertise. DD is supported by the James Arthur\nPostdoctoral Fellowship and is grateful to the Galileo Galilei Institute and the CERN Theory\nDivision where part of this work was performed. MK was funded by New York University’s\nCarl P. Feinberg graduate fellowship in theoretical physics when this work was conducted.\nMK is also grateful for the hospitality of Perimeter Institute. Research at Perimeter Institute\nis supported in part by the Government of Canada through the Department of Innovation,\nScience and Economic Development Canada and by the Province of Ontario through the\nMinistry of Colleges and Universities. Both authors also thank the Center Computational\nAstrophysics at the Flatiron Institute for hosting us while we carried out part of this work.\nThe Center for Computational Astrophysics at the Flatiron Institute is supported by the\nSimons Foundation. This research was supported in part by the Heising-Simons Foundation,\nthe Simons Foundation, grant no. PHY-2210551 and grant no. PHY-2309135 to the Kavli\nInstitute for Theoretical Physics (KITP). We made use of the software packages NumPy [116],\nSciPy [117], and py-pde [118].\nA Reheat Temperature from an Early Matter-Dominated Era\nIn this section, we derive the necessary reheat temperature at the end of the early MD era,\nTRH, to give the correct axion DM abundance from the misalignment mechanism, Ω mis,MD,\nand from axion strings, Ω string ,MD. This value of TRHis self-consistently used to derive the\ntemperature, Tosc,MD, when the axion begins oscillating in an early MD era. This is because\nthe oscillation temperature Tosc,MDin a MD era depends on TRHwhich enters into Hubble.\nOverall, the value of Tosc,MD/TQCD is important since it controls whether the axion mass is\nstill growing when walls form, which in turn affects the condition for PBH formation (see\nSec. 6).\nThe wall formation temperature relative to TQCDis\nTosc\nTQCD≃\n\n\u0012MPl\nfa\u00132\n4+n\u001210\n8π3g∗(T0)\u00131\n4+n\nRD era ,\n\u0012MPl\nfa\u00132\n8+n\u0012TRH\nTQCD\u00134\n8+n\u001210\n8π3g∗(T0)g∗(TRH)\ng∗(T0)\u00131\n8+n\nMD (Non-Adiabatic) Era .\n(A.1)\n– 40 –Here, MPl≃1.22×1019GeV is the Planck constant and g∗(T) are the relativistic degrees of\nfreedom in the thermal bath at temperature T. In a RD era, Tosc/TQCDdepends dominantly\nonfa, and to a much weaker extent, g∗. In a MD era, Toscalso depends on the temperature\nat which the universe returns to being radiation dominated, TRH. The lower TRHis, the more\nthe background axion abundance is diluted [39, 66, 119].\nThe axion abundance from both misalignment and strings in a MD era is\nΩa,MD(fa) = Ω mis,RD(fa)Ya(Tosc,MD)\nYa(Tosc,RD)1\nD| {z }\nΩmis,MD\u0012\n1 +Ωstring ,MD\nΩmis,MD\u0013\n. (A.2)\nThe first term in Eq. (A.2), Ω mis,RD, is the energy density of axions, ρa, relative to the critical\ndensity of the universe, ρc, if the axions were generated solely by misalignment in the standard\nRD cosmology\nΩmis,RD=ρa(ttoday)\nρc(ttoday)≃1\n2⟨θ2\nmis⟩f2\nama(tosc,RD)maa(tosc,RD)3\na(ttoday )3\nρc(ttoday)(A.3)\n≃ΩDM\u0012fa\nfa,DM\u00136+n\n4+n\n. (A.4)\nHere, fa,DMis the decay constant that gives the correct DM abundance, Ω DM, in a RD era\nfrom misalignment. For the QCD axion, n≃7 so that [120]\nΩmis,RD≈ΩDM\u0012fa\n3×1011GeV\u00131.18\n(QCD axion) . (A.5)\nNote that in the post-inflation scenario considered in this paper, the misalignment angle is\nnot a free parameter, but is instead the horizon averaged value of ⟨θ2⟩. For ALPs, fa,DMcan\nbe much larger, near 1016GeV, since ALP mass can be much lighter than the QCD axion for\na given fa[121].\nThe second term in Eq. (A.2), Ya(Tosc,MD)/Ya(Tosc,RD), is the ratio of the redshift in-\nvariant axion yields when the axion begins oscillating at Tosc,MD(Tosc,RD), in a MD (RD)\nera\nYa(Tosc,MD)\nYa(Tosc,RD)=na(Tosc,MD)/s(Tosc,MD)\nna(Tosc,RD)/s(Tosc,RD)(A.6)\n=ma(Tosc,MD)\nma(Tosc,RD)\u0012Tosc,RD\nTosc,MD\u00133\u0012g∗s(Tosc,RD)\ng∗s(Tosc,MD)\u0013\n(A.7)\n=min\u001a\u0010TQCD\nTosc,MD\u0011n/2\n,1\u001b\nmin\u001a\u0010TQCD\nTosc,RD\u0011n/2\n,1\u001b\u0012Tosc,RD\nTosc,MD\u00133\u0012g∗s(Tosc,RD)\ng∗s(Tosc,MD)\u0013\n. (A.8)\nEach yield corresponds to the conserved comoving number of axions once the axion begins\noscillating. Since the temperature at which the axion begins oscillating in a MD era is\n– 41 –less than a RD era for the same fa, these yields are different. The case where n= 8 and\nTosc,MD, Tosc,RD> TQCDcorresponds to Eq. (69) in ref. [39].\nThe third term in Eq. (A.2), D, is the dilution of the axion abundance by the entropy\ngenerated at the end of the early MD era cosmology when the universe returns to radiation\ndomination. Specifically, the dilution is the increase in the entropy of the universe, S=\ng∗sa3T3, at the end of the MD era (at temperature TRH) compared to the entropy when the\naxion begins oscillating (at Tosc,MD),\nD=S(TRH)\nS(Tosc,MD)=g∗s(TRH)a(TRH)3T3\nRH\ng∗s(Tosc,MD)a(Tosc,MD)3T3\nosc,MD(A.9)\n=\n\ng∗s(TRH)\ng∗s(Tosc,MD)\"\u0012g∗(Tosc,MD)\ng∗(TRH)\u00132/3\u0012Tosc,MD\nTRH\u00138/3#3\u0012TRH\nTosc,MD\u00133\n(Tosc,MD< TNA)\n≃g∗(Tosc,MD)\ng∗(TRH)\u0012Tosc,MD\nTRH\u00135\ng∗s(TRH)\ng∗s(TNA)\"\u0012g∗(TNA)\ng∗(TRH)\u00132/3\u0012TNA\nTRH\u00138/3#3\u0012TRH\nTNA\u00133\n(Tosc,MD> TNA)\n≃g∗(TNA)\ng∗(TRH)\u0012TNA\nTRH\u00135\n≃TMD\nTRH\n(A.10)\nHere, we use the fact that the entropy of the universe during the MD era is unchanged\nuntil the energy density of the relativistic decay products of the field responsible for matter\ndomination overtakes the primordial radiation, which marks the start of the non-adiabatic\nregime of matter domination and occurs at temperature TNA≃(T4\nRHTMD)1/5where TMDis\nthe temperature at which the early MD era begins [122]. That is, between TMDandTNA,\nthe scale factor vs. temperature relation of the universe is the usual a∝g−1/3\n∗sT−1, while\nbetween TNAandTRH, the expansion is non-adiabatic with a∝g−2/3\n∗sT−8/3[95]. As can be\nseen from the two cases in Eq. (A.9), the amount of dilution depends on whether the axion\nstarts oscillating after or before the universe enters the non-adiabatic regime of the MD era.\nThe first case is equivalent to the reciprocal of Eq. (70) in ref. [39]. Note the largest possible\ndilution occurs when Tosc,MDoccurs during the non-adiabatic regime of the early MD era.\nFinally, the last term in Eq. (A.2) is the contribution to Ω afrom axion string decay.\nThe ratio of axion DM from strings compared to misalignment, Ω string/Ωmis, depends on\nthe number of strings per horizon, ξ, and how relativistic the axions emitted by the strings\nare. As discussed in Sec. 3.3, although ξis expected to be O(1), there is large uncertainty\nfrom simulations due to extrapolation of ξfrom∼1 toO(10) which could make Ω string ,RD∼\n10Ω mis,RDin a RD era. In a MD era, however, the faster expansion rate of the universe is\nexpected to make ξmuch smaller even with logarithmic extrapolation of simulations [39].\nThe left panel of Fig. 17 shows the necessary TRH/TQCDto give the observed axion DM\nabundance from axion misalignment and string decay (i.e. Ω a,MD= Ω DM) as a function of fa\n– 42 –Figure 17 .Left: The reheat temperature TRHover the QCD temperature TQCD as a function of\nfa. The blue band indicates the uncertainty in TRHdue to the string radiated axion energy density\nΩstring, which alters what TRHis needed to obtain the correct present-day axion DM abundance. The\norange shaded region indicates the part of parameter space where TRHis driven below the Big Bang\nNucleosynthesis temperature TBBN in the most minimal early matter-dominated cosmology. Right :\nThe axion field oscillation temperature Tosc,MDover the QCD temperature TQCD as a function of fa.\nThe oscillation temperature controls when a majority of misalignment axion DM is produced. The\nblue band again indicates uncertainty in Tosc,MDdue to Ω string not being accurately known. The black\ndotted line indicates the boundary between the region of parameter space where the axion has reached\nits zero temperature value at T=Tosc,MDand where the mass is still turning on at T=Tosc,MD.\nfor the QCD axion. The thickness of the blue line corresponds to when Ω string ,MD<Ωmis,MD\n(the top contour) to Ω string ,MD≈10Ω mis,MD(the bottom contour). In the orange shaded\nregion, the reheat temperature after the MD ends is below T≈4 MeV and is excluded by\nBBN [123]. This limits the largest faconsistent with the most minimal early MD cosmology\nto be fa∼a few times 1015GeV.\nSimilarly, the right panel of Fig. 17 shows the corresponding Tosc,MD/TQCD using the\ncorrect TRHfrom the left panel to give the correct axion DM abundance. For fa≳5×1014\nGeV, the axion is already at its zero temperature value by the time the earliest axion walls\nform, as shown by the dashed horizontal line. Consequently, for falarger than this, the\nrest-frame wall thickness of the walls is always unchanged in time.\nB Superhorizon Evolution of Aspherical Domain Walls\nHere, we expand our analysis of the superhorizon evolution of domain walls described in\nSec. 4 to aspherical self-enclosed domain walls. We use Eq. (5.8) to paramaterize the surface\nof the aspherical wall and Eq. (5.7) as our initial field configuration. We then study how\nthis field configuration evolves via Eq. (4.2), completely analogous to the spherical case in\nSec. 4. As argued in Sec. 3, we approximate any aspherical wall formed with a characteristic\nmean radius of curvature R0as an ellipsoid with semi-major axis R0ϵ1and semi-minor axis\nR0ϵ2, which parameterize the asphericity of the domain wall. We simulate the superhorizon\n– 43 –Figure 18 . The eccentricity evolution of ellipsoidal domain walls up until the moment of wall re-entry.\nLeft: The eccentricity parameter ϵas a function of real time tand conformal time ηforα= 5 walls\nassuming radiation domination. The distinct colored curves correspond to distinct walls with the same\nα≡R0/t0, but distinct ellipsoid parameters initialized at t/t0=η/η0= 1. Notice that walls with\nϵ <1 re-enter the horizon faster than walls with ϵ >1, and that walls tend to become less elliptical\nbefore re-entry. Right : The same as left, but with α= 6. Note that larger walls re-enter the horizon\nlater.\nevolution of a host of these ellipsoidal walls of different α≡R0/t0and different ϵ1andϵ2. The\nprimary takeaway from these simulations is that during the superhorizon stretching phase,\nthe elliptical walls become more spherical, with ( ϵ(t0)−ϵ(tRE))/ϵ(t0)≃0.1−0.2. Fig. 18\nshows several examples of how a single eccentricity parameter ϵ=ϵ1(with ϵ2= 1) evolves\nwith time up until horizon re-entry. Since the asphericity becomes mildly less severe, we\nsafely neglect this superhorizon correction. However, we emphasize that asphericities are of\nmonumental importance in subhorizon collapse dynamics, primarily due to how they prevent\nthe compression of energy. We account for this in details in Sec. 5 and Sec. 6.\nC Domain Wall Self-Gravitation\nIt was pointed out by refs. [124, 125] that domain walls are gravitationally repulsive. For\nspherical walls close to their collapse radius, where gravity will act the most strongly, we\nmay neglect the Hubble expansion of the universe and simply consider the axion wall field\nconfiguration on an asymptotically flat spacetime background. Here, this ‘repulsion’ is simply\nthe statement that an observer exterior to the wall must accelerate inward in order to stay\na fixed height above the wall surface. For a spherical axion domain wall, Birkhoff’s theorem\nimplies that the spacetime region exterior to the wall is described by a Schwarzschild met-\nric [126]. Inside the domain wall, the spacetime is entirely flat, so an observer inside the wall\nmust also accelerate inward to stay a fixed distance above the wall surface. We note that this\nis in contrast to static planar domain walls, where a genuine repulsion is experienced by an\n– 44 –observer near any side of the wall. Solving the Einstein field equations for a timelike spherical\nshell of equal energy density and tension σyields the equation [125]\n(A+B)¨R=−2AGM\nR2−2AB(A+B)\nR, (C.1)\nwhere Gis the gravitational constant, M= 4πσR2(1−2πσGR) is the mass of the wall as\nmeasured by an observer far away and\nA= (1 + ˙R2)1/2, B=\u0014\n1−2GM\nR+˙R2\u00151/2\n. (C.2)\nwhere the dot denotes a time derivative with respect to an observer instantaneously comoving\nwith the wall [125]. Note that gravity aids the domain wall collapse via the first term in\nEq. (C.1). However, the gravitational contribution is negligible compared to the pure tension\nterm unless R≲σR2\nRE/M2\nPl∼maf2\naR2\nRE/M2\nPl∼Rs, where Rsis the Schwarzschild radius.\nThus, the gravitational contraction is completely negligible until the last moments of collapse,\nat which point the wall will already have accelerated to nearly the speed of light. Therefore,\nthe attractive gravitational force will not yield a significant speedup or change in the wall\nmomentum in the collapse and we ignore it in the text. For visualization, Fig. 19 compares\nhow the acceleration due to gravity is largely subdominant to the wall self-tension except\nwhen the wall has already contracted very close to the Schwarzschild radius.\nD PBH Abundance in ALP Models\nHere we compute the PBH abundance in axion-like particle (ALP) models with temperature-\nindependent masses ( n= 0 per Eq. (2.3)). ALP models accommodate a more general set\nof cosmologies than QCD axions because faandmaare independent parameters. First, we\nconsider an ALP model in a RD cosmology. Unlike in the QCD axion case, ALPs with\nhigh values of faare still cosmologically viable if the ALP mass is sufficiently lowered. We\nfind that in a RD era, the PBH abundance in ALP models is nevertheless smaller than the\nPBH abundance in the QCD axion + MD cosmology considered in the main text, as shown\nin Fig. 20. This is mainly because of the slower expansion rate of the universe during a\nRD era, which means that domain walls do not get stretched as much before entering the\nhorizon. The time interval between distinct ALP DM/PBH production stages (formation,\nmisalignment, collapse) is also less favorable than in the MD case unless the ALP possesses\na temperature dependent mass like the QCD axion. Fig 20 shows the ratio of Ω PBH/Ωmis\nfor ALPs in a RD era, assuming that the ALPs are at their zero temperature mass before\nwall formation and the string density is at ξ≃1. The ratio here is true regardless if the\nALP misalignment abundance is the DM or if it is overproduced and later diluted since\nthe PBHs and misalignment axions are produced nearly concurrently so that Ω PBH/Ωmisis\nredshift invariant. Note that since string scaling in the RD scenario is more susceptible to\nlog corrections, the Ω PBHresult presented here may be subject to exponential corrections if\n– 45 –100101102103\nRRE/Rs10-610-510-410-310-210-1100Normalized AccelerationAcceleration due to gravityAcceleration due to tension Total accelerationFigure 19 . The instantaneous acceleration of a perfectly spherical domain wall normalized to its\npeak acceleration as a function of the wall re-entry radius to its Schwarzschild radius RRE/Rs. In\nred, the inward acceleration of the domain wall due to tension alone. In blue, the inward acceleration\nof the domain wall due to gravity alone. In black, the total inward acceleration of the domain wall.\nThe total energy of the wall is taken to be E= 4πf2\namaR2\nREwith fa= 1014GeV. Note that the wall\nacceleration due to gravity only becomes significant when the wall radius is of order the Schwarzschild\nradius. For axion domain walls, Rs≪RRE, meaning walls will typically have reached a contraction\nspeed near the speed of light by the time the wall radius is close to the Schwarzschild radius. This\nmakes the enhanced acceleration due to gravity negligible when computing the total collapse time.\nξis not O(1). This is in contrast to Fig. 16 (the main result of this work) where systematics\nare under better control due to the MD cosmology (although the physical value of ξhas still\nnot been fully determined, even though it’s believed to be close to unity [39]).\nALP domain wall formation in a MD cosmology is also a possible cosmological scenario.\nWe show the ALP MD fPBHin Fig. 21. This projection is nearly identical to Fig. 16, except\nthe ALP mass is now a free parameter, which in turn makes mPBHa free parameter. Note\nthat in contrast to Fig. 16, the ALP scenario easily avoids coming into tension with TBBN\nif the ALP mass is slightly offset from the QCD axion line. For high values of fa, where\nthe PBH yield is largest, one needs mALP> m QCD Axion . This, in turn, makes the PBHs\nproduced slightly lighter.\n– 46 –1/parenleftbigmALP\nmQCDAxion/parenrightbig×MPBH [M⊙]\n10161017\nfa [GeV]10-1010-810-610-410-2100ΩPBH/ΩALP\nAxion-Like Particle, RD Era\nZero Temp. Mass ReachedΩstrings/similarequal10Ωmis\nΩstrings<ΩmisFigure 20 . The projected fraction of ALP DM consisting of PBHs assuming radiation domination\nand a static ALP mass at wall formation. The bottom axis shows the ALP model facorresponding\nto a given fPBHabundance. The top axis shows the PBH mass MPBHassociated with each fa. Note\nthat the PBH spectrum for a given fais nearly monochromatic, hence the one-to-one map from fa\ntoMPBH. The top and bottom solid black lines show the fPBHabundance from ALP domain wall\ncollapse in the case where the relic radiated string axion abundance Ω strings is less than Ω misand\napproximately 10 Ω mis, respectively. Ω strings is entirely predicted by faand the subject of extensive\nstudy, however its exact value remains uncertain due to simulation limitations. The two black lines\ncorrespond to edge cases for Ω strings , meaning fPBHmay lie anywhere in the blue band. Note that this\nfigure assumes that the ALP has reached its zero temperature mass at wall formation. Also note that\nmais now a free parameter which must be picked to obtain a PBH mass corresponding to each fa\n(however, this mais fixed by cosmology if one assumes that all of DM comes from ALP misalignment).\n– 47 –101510161017\nfa [GeV]10−1010−810−610−410−21ΩPBH/ΩALP\nAxion-Like Particle, MD Era\nZero Temp. Mass ReachedΩstrings/similarequal10Ωmis\nΩstrings<Ωmis10−11/parenleftbigmALP\nmQCDAxion/parenrightbig×MPBH [M⊙]Figure 21 . The projected fraction of ALP DM consisting of PBHs assuming matter domination and\na static ALP mass at wall formation. The bottom axis shows the ALP model facorresponding to\na given fPBHabundance. The top axis shows the PBH mass MPBHassociated with each fa. Note\nthat the PBH spectrum for a given fais nearly monochromatic, hence the one-to-one map from fa\ntoMPBH. The top and bottom solid black lines show the fPBHabundance from ALP domain wall\ncollapse in the case where the relic radiated string axion abundance Ω strings is less than Ω misand\napproximately 10 Ω mis, respectively. Ω strings is entirely predicted by faand the subject of extensive\nstudy, however its exact value remains uncertain due to simulation limitations. The two black lines\ncorrespond to edge cases for Ω strings , meaning fPBHmay lie anywhere in the blue band. Note that this\nfigure assumes that the ALP has reached its zero temperature mass at wall formation. 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This\ndifficulty gets worse with increasing telescope size as the photon flux per di ffraction-limited resolution element remains constant but\nthe evolution timescale decreases linearly with the di ffraction-limited resolution.\nAims. We aim to improve magnetic field reconstruction at the di ffraction limit without averaging the observations in time or space,\nand without applying noise filtering.\nMethods. The magnetic field vector tends to evolve slower than the temperature, velocity and microturbulence. We exploit this by\nadding temporal regularisation terms for the magnetic field to the linear least-squares fitting used in the weak-field approximation, as\nwell as to the Levenberg-Marquardt algorithm used in inversions. The other model parameters are allowed to change in time without\nconstraints. We infer the chromospheric magnetic field from Ca ii854.2 nm observations using the weak field approximation and the\nphotospheric magnetic field from Fe i617.3 nm observations, both with and without temporal regularisation.\nResults. Temporal regularisation reduce the noise in the reconstructed maps of the magnetic field and provides a better coherency in\ntime in both the weak-field approximation and Milne-Eddington inversions.\nConclusions. Temporal regularisation markedly improves magnetic field determination from spatially and temporally resolved obser-\nvations.\nKey words. Sun: chromosphere – Radiative transfer – Polarisation – Sun: magnetic fields – Stars: atmospheres\n1. Introduction\nThe degree of polarisation in spectral lines induced by magnetic\nfields in the solar atmosphere can be higher than 10% in regions\nwith strong field (e.g. Martinez Pillet et al. 1990; Castellanos\nDurán et al. 2020), but more commonly it is much lower: lin-\near polarisation induced by the Zeeman and Hanle e ffect in lines\nformed in the chromosphere tends to be smaller than 0.1%, espe-\ncially in the quiet Sun where field strengths are low (e.g. Jur ˇcák\net al. 2018; Centeno et al. 2021). Observations of the full Stokes\nvector therefore require long integration times and /or spatial av-\neraging over areas larger than the di ffraction limit of the tele-\nscope in order to reach the signal-to-noise ratio (SNR) required\nto detect signals with a strength of <0.1% of the local continuum\nintensity (e.g. Centeno et al. 2010).\nA major goal of modern solar telescopes is to perform\nimaging spectropolarimetry at the di ffraction limit. This is re-\nflected in the calibration accuracy of the instruments. For exam-\nple, the CRISP instrument at the Swedish 1-m Solar Telescope\n(Scharmer et al. 2003) can currently reach a polarimetric accu-\nracy of∼10−3and the DKIST telescope can be calibrated to an\naccuracy of 5×10−4(Rimmele et al. 2020).\nBecause of the finite brightness of the Sun, this SNR is not\nreached instantaneously. For example, it takes 2 s to reach a SNR\nof 5×10−4at the di ffraction limit in the continuum adjacent to the\nCa ii854.2 nm line at a spectral resolution λ/∆λ=80,000 and\na total transmission of the telescope-instrument-camera chain\nSend o ffprint requests to : J. de la Cruz Rodríguez e-mail:\njaime@astro.su.seof 5%. Tuneable-filter-based instruments like CRISP or VTF\nat DKIST require data acquisition typically at 10 to 20 wave-\nlengths, depending on the width of the line and the desired\nwavelength coverage. This leads to a total acquisition time of\nroughly 20 to 40 s for a di ffraction-limited full-Stokes line scan.\nThis time is large compared to the timescale on which the\ndensity, temperature and velocity in the atmosphere change: the\nsound crossing time for a di ffraction-limited pixel of a 1 m tele-\nscope at 854 nm is 9 s assuming a sound speed of 7 km s−1,\nfor a 4-m telescope it is 2 s. The sound speed should be con-\nsidered as a lower limit, in the chromosphere the Alfvén speed\nis a better indicator of the evolution speed. It can be larger than\n100 km s−1, implying changes of the thermodynamic parameters\nin less than 0.1 s.\nThus, using di ffraction-limited instruments based on tune-\nable filters to reach the required high SNR inevitably leads to ob-\nserved Stokes profiles that average di fferent atmospheric states\nin density, temperature, and velocity. Using integral field spec-\ntropolarimeters such as microlensed hyperspectral imagers (van\nNoort et al. 2022) or image slicers (Dominguez-Tagle et al.\n2022) mitigate – but not fully eliminate – this e ffect, but at the\nprice of a strongly reduced field-of-view (FOV).\nThis mismatch of required integration times and evolution\ntimescales poses a severe problem for inferring chromospheric\nmagnetic fields, especially over the large FOVs of Fabry-Pérot-\ntype instruments. The equations that describe radiative transfer\nare non-linear. An inversion of a time-averaged line profile does\nin general not yield an inferred atmosphere that represents the\ntime-average of the underlying atmosphere (Carlsson & Stein\nArticle number, page 1 of 8arXiv:2402.03440v1  [astro-ph.SR]  5 Feb 2024A&A proofs: manuscript no. ms\n1995). Time-averaging of the Stokes vector might even result in\nan observed average Stokes vector that cannot be reproduced by\nany single atmosphere model.\nThe magnetic field in the chromosphere, where the mag-\nnetic pressure is larger than the gas pressure, tends to be rather\nsmooth and slow-varying over space. Morosin et al. (2020) re-\nalised that this property can be exploited to improve inferring\nmagnetic fields from noisy data using the weak-field approxima-\ntion (WFA). They extended the merit function used in the WFA-\nfit with a term that penalises di fferences in magnetic field in\nneighbouring pixels, and showed that this leads to substantially\nimproved inference of the longitudinal magnetic field based on\nthe Mg i517.3 nm, Na i589.6 nm, and Ca ii854.2 nm lines.\nThe magnetic field in the chromosphere generally only\nevolves over granular turnover timescales of several minutes\n(Kleint 2017; Ferrente et al. 2023), but the mass density, temper-\nature, and velocity evolve with a timescale of seconds on spa-\ntial scales of a few tens of kilometers. We propose to exploit this\ntemporal smoothness to improve magnetic field inferral from ob-\nservation taken at or close to the di ffraction limit.\nThe general idea is to perform inversions where, for the same\nspatial pixel, observations at consecutive timesteps are allowed\nto have arbitrary variations in the velocity, temperature and other\nvariables that are well-constrained by Stokes Iat relatively low\nSNR. The magnetic field, which is constrained by Stokes Q,U,\nandVand requires a high SNR to be constrained, should vary as\nlittle as possible while staying consistent with the data, given the\nnoise level.\nThis concept can be viewed as applying Tikhonov regularisa-\ntion (Tikhonov & Arsenin 1977; Willoughby 1979) to the mag-\nnetic field in the time domain. In the case of the WFA this can\nbe implemented as a straightforward extension of the method\npresented by Morosin et al. (2020). In the case of inferral meth-\nods based on explicit radiative transfer, whether in the Milne-\nEddington approximation, in LTE, or in full non-LTE, it can be\nimplemented as an extension of the method described in de la\nCruz Rodríguez (2019). The former method is linear and re-\nquires only a single matrix inversion to yield a solution; the latter\nmethod is non-linear and can be solved using the Levenberg-\nMarquardt algorithm.\nIn this paper we present implementations for both cases. As\na proof of concept we apply the temporally-regularised WFA to\nobservations of quiet Sun in the Ca ii854.2 nm line, and apply\ntemporally regularised Milne-Eddington inversions to observa-\ntions of quiet Sun in the Fe i617.3 nm line. We chose to illustrate\nthe non-linear case with the Milne-Eddington approximation ap-\nplied to a photospheric line because it is easy to implement and\nfast to invert. The formalism remains identical also for LTE or\nnon-LTE inversions of chromospheric lines.\nThe observations that we use here are obtained with a Fabry-\nPerot tuneable-filter instrument. The individual line positions\ncontained in the data are not taken simultaneously, but sequen-\ntially. We assume that one line scan comprises a timestep that\nwill be represented by one atmosphere model. As we argued, this\nis an approximation as the solar evolution timescale is smaller\nthan the scan time, and this can translate into errors in the in-\nferred model parameters (e.g., Felipe et al. 2018). We stress that\ntemporal regularisation can also be used for integral-field spec-\ntropolarimeters. They record all wavelengths simultaneously,\nand the integration time used in a timestep is mainly set by the\ndesired SNR, and this time is significantly shorter than would be\nneeded by an tuneable-filter instrument.2. Observations\nWe use two datasets, acquired with the CRISP instrument (see\nScharmer 2006; Scharmer et al. 2019; de Wijn et al. 2021) at the\nSwedish 1-m Solar Telescope. The datasets were reduced with\nthe SSTRED pipeline (de la Cruz Rodríguez et al. 2015; Löf-\ndahl et al. 2021), and processed with the MOMFBD algorithm\nto minimise atmospheric distortions (Löfdahl 2002; Van Noort\net al. 2005). The polarimetric calibration was individually per-\nformed for each pixel as described in van Noort & Rouppe van\nder V oort (2008).\nThe first dataset, recorded on 2020-07-14 starting at 08:40\nUT at disk center, consists of a four-line program with data\nacquired in the Ca ii854.2 nm, Fe i617.3 nm, H αand\nMg i517.3 nm lines. We only used the 854.2 nm data in this\nstudy. The line was sampled between ±30 pm from line cen-\nter in regular steps of 7.5 pm, with a total integration time\nper wavelength of 0 .91 s. The overall cadence, including all\nfour lines is 34 .2 s, whereas the total acquisition time for one\n854.2 nm scan was approximately 13 s. The upper row in Fig. 1\nillustrates the four Stokes parameters in the first time-step at\n+15 pm from line center. The target is a quiet-Sun region that\nharbours two small network patches with opposite polarity. We\nestimated the noise level based on the outermost wavelength\npoints. The standard deviations of the noise were found to be\nσQ,U,V=(3.9,2.6,2.7)×10−3.\nThe second dataset, recorded on 2021-05-26 starting at\n09:48 UT on coordinates ( X,Y)=(698,387) (µ=0.54),\nconsists of a patch of quiet Sun close to active region\nAR12824. We recorded data in the Ca ii854.2 nm and\nFe i617.3 nm lines, but we only used the 617.3 nm data\nin this study. The Fe i617.3 nm line was sampled at ∆λ=\n[−17.5,−10.5,−7,−3.5,0,3.5,7,10.5,14,17.5] pm from line\ncenter, with a total integration time per wavelength of 0 .28 s.\nThe selected FOV is illustrated in the bottom row of Fig. 1 in\nall four Stokes parameters at −7 pm from line center. The back-\nground noise as estimated from the continuum point has standard\ndeviationsσQ,U,V=(1.8,2.3,2.5)×10−3.\n3. Method\n3.1. The linear case: an application to the weak-field\napproximation\nWhen the dependence of the observable with the model parame-\nters is linear, spatial and temporal regularisation can be trivially\nimposed. The weak-field approximation (Landi Degl’Innocenti\n& Landi Degl’Innocenti 1973; Je fferies et al. 1989) allows writ-\ning analytical expressions for the Q,U, and Vprofiles that are\nlinear in B2\n⊥andB∥.\nWe extend the implementation of Morosin et al. (2020) by\nalso adding regularisation in the temporal dimension. In order\nto derive general expressions for the linear case, let us assume\nthat we can express our synthetic prediction siof the i−thdata\npoint in a pixel as si=ciP, where ciis a quantity that does\nnot depend on the model parameters (but that can change for\neach data point) and Pis a model parameter. For a given pixel\nat coordinates t,y,x, we can write the regularised merit function\nArticle number, page 2 of 8de la Cruz Rodríguez and Leenaarts: magnetic field determination with spatio-temporal regularisation\n0 20 40\nx [arcsec]02040y [arcsec]I\n0 20 40Q\n0 20 40U\n0 20 40V\n−0.010 −0.005 0.000 0.005 0.010\n(Q,U,V)/<I>\n0 3 6\nx [arcsec]0369y [arcsec]I\n0 3 6Q\n0 3 6U\n0 3 6V\n−0.02 −0.01 0.00 0.01 0.02\n(Q,U,V)/<I>\nFig. 1. Observations used in this study. Top: Stokes parameters in the Ca ii854.2 nm line at +15 pm from line center. Bottom: Stokes parameters\nin the Fe i617.3 nm line at−17.5 (I) and−7.0 pm ( Q,U,V) from line center. Stokes Q,UandVhave been normalised by the spatially-averaged\nStokes Iintensity at the same wavelength.\nχ2as:\nχ2=1\nNdataNdataX\ni=1oi\nt,y,x−ciPt,y,x\nσi2\n+αh\n(Pt,y,x−Pt,y,x−1)2\n+(Pt,y,x−Pt,y,x+1)2+(Pt,y,x−Pt,y−1,x)2+(Pt,y,x−Pt,y+1,x)2i\n+β\u0010\nPt,y,x−P0\u00112+γh\n(Pt,y,x−Pt−1,y,x)2+(Pt,y,x−Pt+1,y,x)2i\n,\n(1)\nwhere oiis the i−thobserved data point, αis a spatial regularisa-\ntion weight, βis a low-norm regularisation weight that penalises\ndeviations from a constant value P0andγis a temporal regular-\nisation weight. Taking the derivative of Eq. 1 with respect to all\nPt,y,x, and equating it to zero to find the minimum, while absorb-\ning the scaling factors Ndataandσiintooiandci, we can derive a\nsparse linear system of equations ( AP=B), where the di fferent\ntemporal and spatial locations are coupled by the non-diagonalregularisation terms:\n(2)X\nic2\ni+4α+βPt,y,x−α\u0010\nPt,y−1,x+Pt,y+1,x+Pt,y,x−1+Pt,y,x+1\u0011\n−γ\u0010\nPt−1,y,x+Pt+1,y,x\u0011\n=X\nicioi\nt,y,x+βP0.\nEquation 2 couples the parameters of the di fferent pixels ( Pt,y,x)\nthrough the regularisation weights ( α,β,γ) on the left-hand side.\nObtaining a solution requires solving a global problem for all\npixels simultaneously. If the regularisation weights are zero, the\nAmatrix in Eq. 2 is diagonal and we recover the unconstrained\nalgorithm. Figure 2 shows the structure of Afor a small FOV of\n4×4 pixels and 3 time steps. Compared to Fig. 1 of Morosin et al.\n(2020), this matrix has two extra o ff-diagonal bands originating\nfrom the temporal regularisation terms.\nIf we replace Pt,y,xwith Bt,y,x\n∥andP\niciwithP\niC∂Ii/∂λ iwe\ndirectly obtain the expression for the fully regularised WFA,\nArticle number, page 3 of 8A&A proofs: manuscript no. ms\n0 15 30 45\n0\n15\n30\n45\n−10−5 0 5 10\nFig. 2. Regularisation matrix for the linear case. Each row corresponds\nto the regularisation function for one pixel in space and time. For dis-\nplaying purposes, we have assumed an observation with dimensions\nnt=3,ny=4,nx=4 and regularisation weights α=1,β=1 and\nγ=2.5. The outermost bands (darker blue) correspond to the temporal\nregularisation terms whereas the inner light-blue terms originate from\nthe spatial regularisation.\nwith C=4.67×10−12λ2\n0geff,λ0the central wavelength of the\nline in nm and geffthe effective Landé factor of the transition.\nFigure 3 illustrates a comparison of four reconstructions of\nB∥from the Ca ii854.2 nm dataset using the unconstrained WFA\nalgorithm, the spatially regularised method of Morosin et al.\n(2020) , and two variants of our method that include temporal\nregularisation alone as well as both temporal and spatial reg-\nularisation. The reconstruction of the unconstrained algorithm\nhas a very noisy background, and besides the magnetic field in\nthe strong network patches, small-scale loop-like features can\nbe discerned in the FOV . The inclusion of spatial regularisa-\ntion greatly decreases the noise in the background, making the\nsmall-scale loop-like features much more visible compared to\nthe unconstrained case. Temporal regularisation alone also de-\ncreases the noise, perhaps yielding a slightly sharper model than\nthe spatially-regularised one, while also decreasing the temporal\nfluctuations of the background noise along the time series. The\ncombined action of temporal and spatial regularisation further\ndecreases the noise compared to the previous cases and yields a\nmodel with the highest SNR. The standard deviation of the noise\nin the magnetic field, as measured in the lower-right corner of the\nimage, is reduced from 18 G in the unconstrained case to 9 G in\nthe fully-regularized case. The improvement induced by spatial\nregularisation is particularly obvious in the movie showing the\nentire time-series. The upper panel in Fig. 4 further illustrates the\nreduction of the noise in the regularised reconstructions along a\nslice through the FOV .4. The non-linear case: application to\nMilne-Eddington inversions\nThe Levenberg-Marquardt algorithm (LM, Levenberg 1944;\nMarquardt 1963) is one of the most e fficient methods to recon-\nstruct the parameters of non-linear models from observations.\nRegularised LM implementations are used in di fferent fields of\nastrophysics in order to constrain the parameters of the model\nunder consideration (e.g., Piskunov & Kochukhov 2002; de la\nCruz Rodríguez et al. 2019). The regularisation is represented\nby a set of Npenpenalty functions rn(p) that are squared in order\nto haveℓ−2 norm regularisation. Following the notation of de la\nCruz Rodríguez et al. (2019), the merit function χ2can generally\nbe expressed as a function of the model parameter vector pand\nthe data points x:\nχ2(p,x)=1\nNdataNdataX\nk=1\"ok−s(p,xk)\nσk#2\n+NpenX\nn=1αnrn(p)2, (3)\nwhere the weights αnregulate the influence of the regularisa-\ntion terms in the merit function. We note that, unlike in the lin-\near case, the penalty functions are not independently defined for\neach pixel (see below).\nThe model corrections predicted by the regularised\nLevenberg-Marquardt algorithm can be derived by linearising\nEq. 3 and taking the derivative with respect to the model pa-\nrameters. The correction to the model parameters ( ∆p), in vector\nform, is given by a linear system of equations:\n(J·JT+L·LT)∆p=J·(o−s)−L·r, (4)\nwhere Jis the Jacobian matrix of the model and Lis the Ja-\ncobian matrix of the penalty functions. In this expression, all\nconstants and normalising factors are implicitly contained in the\ncorresponding vectors. For any given parameter pt,y,x, the regu-\nlarisation functions are defined as:\n(5)X\nr2\nt,y,x=αt(pt,y,x−pt−1,y,x)2+αs(pt,y,x−pt,y−1,x)2\n+αs(pt,y,x−pt,y,x−1)2+αl(pt,y,x−p0)2,\nwhere we have included spatial, temporal and low-norm regular-\nisation terms.\nIf the penalty functions have a linear dependence with the\nmodel parameters, like the ones used in this study, we can ex-\npress them as r=Lp+c, where the coupling is contained in the\nJacobian matrix L. Figure 5 illustrates the structure of Lfor a\nproblem with nt=3,ny=4,nx=4 and npar=2. The maximum\nnumber of penalty functions should be npen<4nxnyntas some\nfunctions are not defined at the edges of the problem. Figure 6\nillustrates the approximate Hessian matrix of the regularisation\nfunctions L·LT. The size of the approximate Hessian matrix is\nset by the total number of free parameters of the problem, which\nis also the size of one row of the LTmatrix (4×4×3×2=96).\nIn the latter, the outer dark blue bands originate from the tem-\nporal regularisation whereas the inner light blue bands contain\nthe spatial coupling terms. For npar=1, this matrix has identical\nform to the one shown in Fig. 2 for the linear case.\nIn this case we are only including penalty functions that com-\npare a parameter value with its neighbouring values at t−1,y−1\nandx−1. However, because the product L·LTis similar to a\ncorrelation of each penalty function with all others, the resulting\ncoupling matrix in the left-hand side has identical structure as\nthe one in the linear case where we also included explicit com-\nparisons with the values at t+1,y+1 and x+1.\nArticle number, page 4 of 8de la Cruz Rodríguez and Leenaarts: magnetic field determination with spatio-temporal regularisation\n010203040y [arcsec]\nUnconstrained\n Full reg.\n0 10 20 30 40\nx [arcsec]010203040y [arcsec]\nTemporal reg.\n0 10 20 30 40\nx [arcsec]Spatial reg.\n−60−40−200204060B/bardbl[G]\nFig. 3. Reconstructions of the line-of-sight component of the magnetic field ( B∥), inferred by applying the weak-field approximation to the\nCa ii854.2 nm data. Upper-left panel: reconstruction performed with the unconstrained algorithm. Lower-left panel: reconstruction performed\nwith a temporally-regularised algorithm. Lower-right panel: reconstruction performed with a spatially-regularised algorithm. Upper-right panel:\nreconstruction performed with full regularisation in space and time. A movie with the time series is available online. The red tickmarks indicate\nthe location of the cut shown in Fig. 4.\nIn a previous study, de la Cruz Rodríguez (2019) imple-\nmented a multi-resolution inversion code that included spa-\ntial regularisation and allowed to deal with the e ffects of the\ntelescope point-spread-function in Milne-Eddington inversions\n(PyMilne1). We have extended this code to include temporal reg-\nularisation terms. With these changes, the code can now invert a\ntime series of maps as a global problem. We inverted a time-\nseries of 14 line scans acquired in the Fe i617.3 nm line in a\nquiet-Sun patch. These observations are not ideal because they\nalso included scans in the Ca ii854.2 nm line, inducing a time\ngap between consecutive Fe iscans. Nevertheless, the cadence\nis sufficiently high to illustrate the advantages of the method. In\nthe following, we focus on the reconstruction of B⊥, which is a\nmore challenging problem than reconstructing B∥.\n1https://github.com/jaimedelacruz/pyMilneWe performed reconstructions with the unconstrained algo-\nrithm, including spatial regularisation only, temporal regulari-\nsation only and a case including low-norm, spatial and temporal\nregularisation altogether. Regularisation was only imposed in the\nthree components of the magnetic field, leaving all other param-\neters unconstrained. Therefore, the thermodynamical quantities\nof the model are allowed to change freely in time and space.\nFigure 7 compares reconstructions of the time-series of\nB⊥from the unconstrained algorithm and the three regularised\ncases. The improvements of the fully-regularised reconstruction\nare threefold:\n–so-called inversion noise, caused by pixels where the algo-\nrithm did not converge to the global minimum, is absent.\nThis e ffect is clearest in the patches of strong magnetic field,\nwhere randomly-located black pixels are present in the un-\nconstrained reconstruction. Examples of inversion noise are\nalso visible in the lower panel of Fig. 4 at y≈3.7′′and\nArticle number, page 5 of 8A&A proofs: manuscript no. ms\n0 10 20 30 40\ny [arcsec]−80−4004080B/bardbl[G]unconstrained\nfully-regularized\n0 2 4 6 8 10 12\ny [arcsec]0300600900B⊥[G]unconstrained\nfully-regularized\nFig. 4. Vertical cuts of the reconstructed magnetic field. The upper panel\nshows a reconstruction of B∥with the weak-field approximation from\nthe 854.2 nm dataset along a cut indicated with red markers in Fig 3. The\nblack curve shows the unconstrained reconstruction, the red curve the\nreconstruction with both spatial and temporal regularisation. A similar\nplot is shown in the lower panel for the reconstruction of B⊥from the\nMilne-Eddington inversion of the 617.3 nm dataset presented in Fig. 7.\ny≈4.35′′. Inversion noise can be removed from the uncon-\nstrained method by initialising the inversion several times\nwith di fferent starting model parameters or by performing\na two-cycle inversion with a smoothing step in between (as\nsuggested by Mili ´c & van Noort 2018). However, the noise\nis naturally addressed in all the regularised cases, where the\ncoupling between pixels in space and /or time removes it en-\ntirely.\n–a reduction of the background noise. Such a background\nbias-level can appear in the presence of noise for parameters\nthat are defined positive (see Martínez González et al. 2012).\nWe find that all forms of regularisation decrease this bias,\nbut the low-norm regularisation term (with P0=0) and the\ntemporal regularisation are particularly e ffective in this re-\ngard. The spatially-regularised time-series shows a stronger\nframe-to-frame fluctuation in the background level, which\nseems to be induced by varying atmospheric seeing condi-\ntions along the series. This e ffect is also visible in the lower\npanel of Fig. 4, where the red curve is regularly predicting\nmuch lower values of B⊥in regions located between stronger\nnetwork patches.\n–the time coherence of weak magnetic structures is enhanced.\nStructures with signal barely over the noise appear more\npersistent in time with temporal regularisation. Without this\nregularisation they appear intermittently, only in those time-\n0 20 40 60 80\n0\n50\n100\n150\n200\n250\n300LT\n−1.5−1.0−0.50.00.51.01.5Fig. 5. Transpose of the Jacobian matrix of the penalty functions for the\nnon-linear case with nt=3,ny=4,nx=4,npar=2 and regularisation\nweightsα=1,β=1 andγ=2.5. Each row corresponds to one penalty\nfunction and each column indicates which free parameters are a ffected.\nsteps where the noise happens to be low enough for the in-\nversion to pick up the structure.\nHowever, despite these improvements, the evolution of mag-\nnetic fields in the photosphere is anchored to the forces orig-\ninating from gas pressure gradients (plasma β≫1), and our\nassumption of having a slowly evolving magnetic field relative\nto the rest of parameters is not optimal. Compared to the chro-\nArticle number, page 6 of 8de la Cruz Rodríguez and Leenaarts: magnetic field determination with spatio-temporal regularisation\n0 30 60 90\n0\n30\n60\n90L LT\n−10−5 0 5 10\nFig. 6. Hessian approximation of the regularisation functions ( LTL).\nThis matrix was computed using Lfrom Fig. 5.\nmospheric case presented in §3.1, we had to limit the amount of\ntemporal regularisation to perform these inversions.\n5. Discussion & conclusions\nWe present a method to include Tikhonov regularisation in the\ntime dimension for linear and non-linear fits of model parame-\nters. We show that the method is suitable for improving determi-\nnations of the magnetic field in the solar chromosphere.\nCompared to other techniques that operate on the data to de-\ncrease noise, such as temporal and spatial rebinning, Principal\nComponent Analysis filtering (see, e.g., Martínez González et al.\n2008), Fourier filtering, or neural-network-based noise reduction\n(Díaz Baso et al. 2019), regularisation leaves the data untouched\nand operates only on the model parameters. This is an important\nadvantage because filtering and /or averaging the data likely af-\nfects the reconstruction of all parameters, whether that is desired\nor not.\nTemporal regularisation addresses the di fferent time scales\non which the magnetic field and the thermodynamical param-\neters evolve in the chromosphere as well as the di fferent SNR\nrequired to constrain them. Because the thermodynamic param-\neters evolve relatively fast but require a low SNR, one can use\nzero (as we do here) or weak regularisation. The magnetic field\nevolves more slowly but requires high SNR, so it is regularised\nmore strongly.\nChoosing adequate regularisation weights is crucial to ob-\ntain results that are not overly smooth. The weights must be\nchosen in such a way that the final quality of the fit is not af-\nfected, but just around that threshold when the quality of the fits\nstarts to be a ffected by the limitations of the regularisation terms\n(Kochukhov 2017). For observations with low SNR, this might\nmean that the regularisation is so strong it makes the magnetic\nfield almost constant in time. We emphasise that with properly\nchosen weights, regularisation does not impose that the magneticfield is constant over time. If the SNR is su fficiently high it still\nallows sharp gradients in time. However, in cases of interme-\ndiate SNR it will produce smoother solutions if strong gradients\nare not required to minimise χ2. Regularisation has an advantage\nover smoothing the model parameters after an unconstrained in-\nversion, because smoothing washes out variations regardless of\nthe impact that they have on χ2, but regularisation keeps sharp\ngradients in locations where they have a significant impact in χ2.\nMost observational programs with Fabry-Perot instruments\nthat observe multiple lines are done by scanning once through\neach line and repeating this cycle. Long integration times are\nneeded for chromospheric lines in order to reach su fficient SNR\nto measure polarisation. As mentioned in the introduction this\ntakes about 20 s, and the observed line profile is not produced by\na single atmosphere as the Sun has evolved over this time.\nTemporal regularisation allows for observing strategies that\nmitigate this. Instead of a single slow scan one can perform mul-\ntiple fast scans that each have su fficient SNR to constrain the\nthermodynamic parameters. These profiles now su ffer less from\nsolar evolution and therefore lead to improved reconstructions of\nthe thermodynamic parameters. Temporal regularisation of the\nmagnetic field couples the scans together and increases the \"ef-\nfective\" SNR su fficiently to determine the field. These fast scans\ncan be interleaved with observations in other lines (which them-\nselves can consist of multiple scans) as needed, as temporal reg-\nularisation does not require a constant cadence. If needed, the\ntemporal regularisation weight can be made smaller for scans\nthat are not performed back-to-back.\nAnother obvious application is data taken with integral-field\nspectropolarimeters. Here the assumptions behind temporal reg-\nularisation hold much better, because all spatial and spectral dat-\napoints are taken simultaneously, and the cadence is typically\nhigh (Rouppe van der V oort et al. 2023).\nData taken at the di ffraction limit of a telescope have a lim-\nited SNR if the exposure time is kept short enough to prevent so-\nlar evolution from blurring the images. Spatio-temporal regular-\nisation of the magnetic field is a way of increasing the e ffective\nSNR of the data while keeping exposure times short. For data\ntaken with large-aperture telescopes, such as DKIST (Rimmele\net al. 2020) and the planned EST (Quintero Noda et al. 2022),\nthis technique will be invaluable.\nThe version of pyMilne that includes temporal regularisation\nis available in the o fficial repository1. The regularised WFA code\nused here is also publicly available2.\nAcknowledgements. The authors of this paper are thankful to the referee for his\ncareful evaluation of the manuscript. The Institute for Solar Physics is supported\nby a grant for research infrastructures of national importance from the Swedish\nResearch Council (registration number 2021-00169). The Swedish 1-m Solar\nTelescope is operated on the island of La Palma by the Institute for Solar Physics\nof Stockholm University in the Spanish Observatorio del Roque de los Mucha-\nchos of the Instituto de Astrofísica de Canarias. JL acknowledges financial sup-\nport from the Swedish Research council (VR, project number 2022-03535). No\nanimals were harmed in the making of this manuscript. This project has been\nfunded by the European Union through the European Research Council (ERC)\nunder the Horizon Europe program (MAGHEAT, grant agreement 101088184).\nViews and opinions expressed are however those of the author(s) only and do\nnot necessarily reflect those of the European Union or the European Research\nCouncil. Neither the European Union nor the granting authority can be held re-\nsponsible for them.\nReferences\nCarlsson, M. & Stein, R. F. 1995, ApJ, 440, L29\n2https://github.com/jaimedelacruz/fullReg_wfa\nArticle number, page 7 of 8A&A proofs: manuscript no. ms\n0 2 4 6\nx [arcsec]0246810y [arcsec]\nUnconstrained\n0 2 4 6\nx [arcsec]Spatial reg.\n0 2 4 6\nx [arcsec]Temporal reg.\n0 2 4 6\nx [arcsec]Full reg.\n0 100 200 300 400 500 600B⊥[G]\nFig. 7. Reconstruction of the transverse component of the magnetic field vector from Milne-Eddington inversions of the Fe i617.3 nm. Left: Un-\nconstrained reconstruction where each pixel has been inverted six times with five di fferent random initialisations. Middle-left: reconstruction with\nspatial regularisation where all pixels where only inverted once. Middle-right: reconstruction with temporal regularisation. Right: reconstruction\nincluding temporal, spatial and low-norm regularisation terms. 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G., Sliepen, G., & de la Cruz Rodríguez, J. 2019,\nA&A, 626, A55\nTikhonov, A. N. & Arsenin, V . Y . 1977, Solutions of Ill-posed problems\n(W.H. Winston)\nvan Noort, M., Bischo ff, J., Kramer, A., Solanki, S. K., & Kiselman, D. 2022,\nA&A, 668, A149\nVan Noort, M., Rouppe Van Der V oort, L., & Löfdahl, M. G. 2005, Sol. Phys.,\n228, 191\nvan Noort, M. J. & Rouppe van der V oort, L. H. M. 2008, A&A, 489, 429\nWilloughby, R. A. 1979, SIAM Review, 21, 266\nArticle number, page 8 of 8"    },    {        "title": "2402.03482v1.On_time_fractional_partial_differential_equations_of_time_dependent_piecewise_constant_order.pdf",        "content": "arXiv:2402.03482v1  [math.AP]  5 Feb 2024ON TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS OF\nTIME-DEPENDENT PIECEWISE CONSTANT ORDER\nYAVAR KIAN△, MARI´AN SLODI ˇCKA⋆,´ERIC SOCCORSI✷, AND KAREL VAN BOCKSTAL♠\nAbstract. Thiscontribution considersthe time-fractionalsubdiffus ionwithatime-dependent variable-\norder fractional operator of order β(t). It is assumed that β(t) is a piecewise constant function\nwith a finite number of jumps. A proof technique based on the Fo urier method and results from\nconstant-order fractional subdiffusion equations has been designed. This novel approach results in the\nwell-posedness of the problem.\n1.Introduction\nThe idea of fractional calculus (FC) dates back to the 17th centur y when mathematicians like\nLeibniz and L’Hˆ opital pondered the meaning of differentiation and int egration of noninteger orders.\nSignificant progress was made in the 19th century thanks to the wo rk of mathematicians like Liouville\nand Riemann.\nFractional calculus has applications in various scientific and engineer ing fields, including physics,\nengineering, signal processing, finance, and more. It has proven to be a powerful tool for describing\nsystems with long-range memory, fractal phenomena, and non-lo cal behaviour. The fractional calculus\nframework provides a deeper understanding of complex phenomen a that integer-order calculus cannot\nadequately describe.\nMachado et al. [22] summarised the historical perspective on the m ajor developments in fractional\ncalculussincethe1970s. Pleasenotethatmostpapersstudiedcon stant-order(CO)fractionaloperators.\nThere exist various types of CO fractional derivatives (Caputo, R iemann-Liouville, Gr¨ unwald-Letnikov,\netc.), see, e.g. [25]. The Caputo and the Riemann-Liouville variations c an be expressed in terms of the\nRiemann-Liouville kernel g1−β:\ng1−β(t) =t−β\nΓ(1−β), t > 0,0< β <1,\nwhere Γ represents the Gamma function. This kernel obeys\n(−1)jg(j)\n1−β(t)≥0,∀t≥0,j= 0,1,2;g′\n1−β/ne}ationslash≡0,\nand, therefore, it is strongly positive definite by [23, Corollary 2.2]. T his creates a powerful position\nwhen establishing energy estimates for a solution.\nDate: February 7, 2024.\n1In the literature, several definitions of fractional derivatives an d integrals in variable-order (VO)\nFC can be found as generalisations of their CO counterparts, see e .g. [2, 6, 8, 10, 14, 15, 16, 20, 34, 35,\n41, 33]. One of the possibilities uses the following time-dependent Riem ann-Liouville kernel\n/parenleftbig\ng1−β(t)/parenrightbig\n(t) =t−β(t)\nΓ(1−β(t)), t > 0,0< β(t)<1.\nThen, the Caputo VO fractional derivative reads as follows\n/parenleftBig\n∂β(t)\ntu/parenrightBig\n(t) =/parenleftbig\ng1−β(t)∗∂tu/parenrightbig\n(t) =/integraldisplayt\n0(t−s)−β(t−s)\nΓ(1−β(t−s))∂tu(s) ds, t > 0,0< β(t)<1.\nAnother possibility for a slightly different VO derivative is [6, 34]\n/parenleftBig\n∂β(t)\ntu/parenrightBig\n(t) =1\nΓ(1−β(t))/integraldisplayt\n0(t−s)−β(t)∂tu(s) ds, t > 0,0< β(t)<1.\nThis definition will be considered in this paper. Polymers, plastics, rub ber and oil are all viscoelastic\nmaterials that are used in biology, medicine, chemical engineering and other fields. CO fractional\nviscoelasticity models do not account for the evolution of the micros tructure of viscoelastic materials\nduring deformation, which changes the mechanical properties of d eforming materials. On the other\nhand, in VO fractional models with a fractional time-derivative of tim e-dependent order, the variation\nof the fractional order is used for describing the change of the me chanical properties of the material\nunder variousloadingconditions. For this reason, they can accura telypredict the viscoelasticbehaviour\nof soft materials, see e.g., [11, 26, 32]. In these models, the fractio nal order can be any (0 ,1)-valued\nfunction of the time variable, but in practice, it is fitted to piecewise c onstant time domain data, see,\ne.g. [1]. This is quite reminiscent of time-fractional multi-state regime -switching option pricing models.\nIndeed, although the CO time-fractional Black-Scholesequation c an account for the nonlocal properties\nof the assets’ prices, see e.g. [3], it cannot describe changes in mar ket states. Hence, regime-switching\nVO time-fractional models have been proposed in [5, 28] to price opt ions. Since each market state is\ndescribed by a CO time fractional model, the fractional order of th e VO model is piecewise constant,\nsee, e.g. [5][Eq. (1)].\nWhen solvinga problemwith time-dependent fractionalderivatives, onecan distinguish twomajor\ncases:\nThe highest order of the time derivative is a constant: This covers parabolic and hyper-\nbolic situations; see [29, 37, 36, 38, 39, 40]. In this framework, the VO time derivatives can be\ninterpreted as Volterra operators of lower order. Consequently , they can be incorporated into\nthe right-hand side of the governing differential equation. The well- posedness of the setting can\nbe shown using a generalised Gr¨ onwall lemma, see [13, Lemma 7.1.1] or [4, Lemma 1]. We refer\nthe reader to [9, 30] for a discrete version of this lemma. Up to now, the most general article\naddressing this proof technique is [31], which also includes nonlinear fu nctions of VO fractional\nderivatives.\n2The highest order of the time derivative is time-dependent: Kochubei[18,19]developed\na general FC for\nDϕf(t) :=d\ndt(ϕ∗f)(t)−ϕ(t)f(0),\nwhich is based on the Laplace transform. The integral kernel has t o obey some conditions\nconstraining ϕto be completely monotone (cf. [12, 21]). This is very restrictive fo r the choice\nof the kernels and, therefore, less interesting. The authors of [ 12] present a few examples\nof practical relevance (exponential, Mittag-Leffler, and error fu nction-type transitions). Until\nnow, there is no general theory that covers this situation. One wo uld like to have the existence\nand uniqueness of a solution for β(t), which is a piecewise smooth function. This case remains\nan open problem.\nHighlights of the paper. We consider the time-fractional diffusion problem (1.5) with the time-\ndependent highest order β(t) of the fractional derivative in time. We assume that β(t) is a piecewise\nconstant function with a finite number of steps. In Section 3, we de sign a new proof technique to\nestablish the well-posedness of the problem (stated in Theorem 2.2) by using Fourier analysis and the\ninterpretation of a solution for CO FC in terms of the Mittag-Leffler f unction. We want to point out\nthatβ(t) does not need to be monotone, and the convolution kernel is not p ositive definite. Our proof\ntechnique is limited to a finite number of steps/jumps of β(t). Until now, we have not found a way to\ntreat a continuously varying β(t).\n1.1.Formulation of the problem. Let Ω⊂Rd,d∈N:={1,2,...}, be an open bounded domain\nwith Lipschitz boundary ∂Ω. We consider a general second-order linear differential operato rLdefined\nby\nL(u) =−div (A(x)∇u)+c(x)u, (1.1)\nwhere\nA(x) = (ai,j(x))i,j=1,...,d, aij∈L∞(Ω,R),AT=A, c∈L∞(Ω,R).\nSetH:=L2(Ω). We assume that there exist two constants α∈(0,∞) andc0∈[0,ακ2), where κis\nthe Ω-related constant of the classical Poincar´ e inequality /ba∇dbl∇u/ba∇dblHd≥κ/ba∇dblu/ba∇dblHfor allu∈V:=H1\n0(Ω),\nthe closure of C∞\n0(Ω) in the first order Sobolev space H1(Ω), such that\nd/summationdisplay\ni,j=1aij(x)ξiξj≥α|ξ|2,for a.e.x∈Ω and all ξ= (ξi)i=1,...,d∈Rd, (1.2)\nand\nc(x)≥ −c0for a.e.x∈Ω. (1.3)\n3Moreover,for T∈(0,∞) fixed and for β: (0,T)→(0,1), wedefine the variable-orderfractionalintegral\noperator 0Iβ(t)\nt, and the variable-order fractional Caputo operator∂β(t)\n∂tβ(t)as follows [6, 20, 34, 41]\n0Iβ(t)\ntu(t) :=1\nΓ(β(t))/integraldisplayt\n0u(s)\n(t−s)1−β(t)ds,\n∂β(t)\ntu(t) :=0I1−β(t)\ntu′(t) =1\nΓ(1−β(t))/integraldisplayt\n0u′(s)\n(t−s)β(t)ds. (1.4)\nInthiscontribution, weexaminethe existenceanduniquenessissue forthesolutiontothefollowing\ninitial-boundary value problem (IBVP)\n\n\n/parenleftBig\n∂β(t)\ntu/parenrightBig\n(x,t)+Lu(x,t) =f(x,t), (x,t)∈Ω×(0,T),\nu(x,t) = 0, (x,t)∈∂Ω×(0,T),\nu(x,0) =u0(x), x ∈Ω,(1.5)\nin the special case where the function βis piecewise constant. Here u0(resp.,f) is a suitable initial\ncondition (resp., source term) that will be made precise further.\n2.Results\n2.1.Definitions and notations. In what follows, the usual norm in HorHdis denoted by /ba∇dbl·/ba∇dbl.\nWe introduce the following bilinear form\nℓ(u,v) :=d/summationdisplay\ni,j=1/integraldisplay\nΩ/parenleftbig\nai,j(x)∂xiu(x)∂xjv(x)dx+c(x)u(x)v(x)/parenrightbig\ndx, u,v∈V.\nEvidently, ℓis continuous in V×V, as we have\n|ℓ(u,v)| ≤max\n1≤i,j≤d/ba∇dblai,j/ba∇dblL∞(Ω)/ba∇dbl∇u/ba∇dbl/ba∇dbl∇v/ba∇dbl+/ba∇dblc/ba∇dblL∞(Ω)/ba∇dblu/ba∇dbl/ba∇dblv/ba∇dbl\n≤C/ba∇dblu/ba∇dblH1(Ω)/ba∇dblv/ba∇dblH1(Ω), u,v∈V,\nwhereC= max 1≤i,j≤d/ba∇dblai,j/ba∇dblL∞(Ω)+/ba∇dblc/ba∇dblL∞(Ω). Further, it readily follows from (1.2)-(1.3) and from the\nPoincar´ e inequality that ℓisV-coercive:\nℓ(u,u)≥α/ba∇dbl∇u/ba∇dbl2−c0/ba∇dblu/ba∇dbl2≥(ακ2−c0)/ba∇dblu/ba∇dbl2\nH1(Ω), u∈V.\nTherefore, the linear operator Lis associated with ( ℓ,V,H) in the sense of [7, Chap. VI, Section\n3.2.5] is selfadjoint (and positive) in H, and acts on its dense domain D(L) ={u∈V:Lu∈H}\nasLu=Lu. Moreover, since Vis compactly embedded in H, the resolvent of Lis compact, and\nconsequently, the spectrum of Lis discrete. We denote by λn,n≥1, the eigenvalues of L, arranged in\nnon-decreasing order and repeated with the (finite) multiplicity (se e, e.g., [27, Theorem XIII.64]):\n0< λ1≤λ2≤ ···\n4and by{Xn, n≥1}an orthonormal basis in Hof eigenfunctions such that LXn=λnXn.\nWe recall that\nD(L) =/braceleftBigg\nv∈H:∞/summationdisplay\nn=1λ2\nn|/an}b∇acketle{tv,Xn/an}b∇acket∇i}ht|2<∞/bracerightBigg\n,\nis a Banach space with respect to the norm\n/ba∇dblv/ba∇dblD(L):=/parenleftBigg∞/summationdisplay\nn=1λ2\nn|/an}b∇acketle{tv,Xn/an}b∇acket∇i}ht|2/parenrightBigg1/2\n.\nIn the next definition, we formulate how the solution to (1.5) should b e understood. We set\nI:= (0,T) and we recall that the space W1,1(I,H) consists of functions u∈L1(I,H) satisfying\n∂tu∈L1(I,H).\nDefinition 2.1. Letu0∈D(L)and letf∈L1(I,H). Then, a solution to the IBVP (1.5)is any\nfunction\nu∈L1(I,D(L))∩W1,1(I,H)\nsatisfying the two following conditions simultaneously:\ni)∂β\ntu(x,t)+Lu(x,t) =f(x,t)for a.e.(x,t)∈Ω×I,\nii)limt↓0/ba∇dblu(·,t)−u0/ba∇dbl= 0.\n2.2.Main result. LetM∈N. Givenβj∈(0,1),j= 0,1,...,M−1, andt0:= 0< t1< t2< ... <\ntM:=T, we consider β:I→(0,1) such that\nβ(t) :=βj, t∈[tj,tj+1), j= 0,1,...,M−1.\nMoreover, we put Ij:= (tj,tj+1),j= 0,...,M−1. Then, the main result of this paper can be stated\nas follows.\nTheorem 2.2. Letu0∈D(L), letf∈W1,1/parenleftbig\n∪M−1\nj=0Ij,H/parenrightbig\n, and assume that for all j= 0,...,M−1,\n∃εj∈(0,1−βj)such that/vextenddouble/vextenddouble(·−tj)βj+εjf′/vextenddouble/vextenddouble\nL∞(Ij,H)<∞. (2.1)\nThen, the IBVP (1.5)admits a unique solution\nu∈ C0(I,D(L))∩W1,1(I,H)\nin the sense of Definition 2.1. Moreover, there exists a posit ive constant C, depending only on Ω,T\nand{(tj,βj,εj), j= 0,...,M−1}, such that\n/ba∇dblu/ba∇dblC0(I,D(L))+/ba∇dblu/ba∇dblW1,1(I,H)\n≤C\n/ba∇dblu0/ba∇dblD(L)+M−1/summationdisplay\nj=0/parenleftBig\n/ba∇dblf/ba∇dblW1,1(Ij,H)+/vextenddouble/vextenddouble(·−tj)βj+εjf′/vextenddouble/vextenddouble\nL∞(Ij,H)/parenrightBig\n. (2.2)\n52.3.Outline and comments. Making use of an auxiliary result stated in Proposition 3.1, of which\nthe proof is postponed to Section 4, we will prove the main result of t his article, Theorem 2.2, in\nSection 3.\nTheorem 2.2 claims existence of a unique solution uin the sense of Definition 2.1 to the IBVP\n(1.5) provided that the initial state u0∈D(L) and the source term f∈W1,1/parenleftbig\n∪M−1\nj=0Ij,H/parenrightbig\nsatisfies\nthe condition (2.1). According to it, the first-order time derivative f′(·,t) should not blow up faster as\nt↓tj,j= 0,...,M−1, than the power function ( t−tj)−κjfor some κj∈(0,1). Such a condition is a\nkey element to the proof of Theorem 2.2. The reason is as follows.\nFor allj= 0,...,M−1,u|Ijis characterised as a solution on the interval Ijto a time-fractional\nPDE of constant order βj, with source term fj. Namely, we have f0=f, whilefj−fis expressed in\nterms of the first-order time derivative of the functions u|Ik,k= 0,...,j−1, when j= 1,...,M−1,\nsee (3.3) and (3.7). In the peculiar case where j= 1 in (3.3), we have\nf′\n1(·,t) =f′(·,t)+β1\nΓ(1−β1)/integraldisplayt1\n0(t−s)−1−β1u′(·,s)ds, t∈I1,\nso we need a good control on u′\n|I0in order to guarantee that f′\n1(·,t)∈L1(I1,H). This is achieved by\nenforcing the condition (2.1) with j= 0 onf, see (4.15).\nFinally, we point out that the statement of Theorem 2.2 is no longer va lid when Mgoes to infinity.\nThis can be understood from the identity (3.8) and the estimate (3.1 1) below (where the notation vj\nstands for u|Ij), as the constant in the inequality (2.2) blows up when Mbecomes infinitely large.\n3.Proof of Theorem 2.2\nThe proof of Theorem 2.2 relies on the Fourier method, results of CO FC and an auxiliary result\n(Proposition 3.1). We split the proof into three parts, which can be f ound in the next subsections.\n3.1.The Fourier method. Using the Fourier method, we build the solution to the IBVP (1.5).\nNamely, assuming that uis a solution to (1.5) in the sense of Definition (2.1), we write\nu(·,t) =∞/summationdisplay\nn=1un(t)Xn, t∈I,\nwhereun(t) :=/an}b∇acketle{tu(·,t),Xn/an}b∇acket∇i}htH. Evidently, for each n∈N,unis the solution to the following fractional\ndifferential system (FDS)\n\n\n∂β\ntun+λnun=f0,n, t∈I,\nun(0) =u0,n,(3.1)\n6with source term f0,n(t) :=/an}b∇acketle{tf(·,t),Xn/an}b∇acket∇i}htHand initial state u0,n:=/an}b∇acketle{tu0,Xn/an}b∇acket∇i}htH. Therefore, the function\nvj,n:=un|Ij,j= 0,1,...,M−1,n∈N, solves\n\n\n/integraldisplayt\ntj(t−s)−βj\nΓ(1−βj)v′\nj,n(s)ds+λnvj,n(t) =fj,n(t), t∈Ij,\nvj,n(tj) =υj,n,(3.2)\nwhere\nfj,n(t) :=f0,n(t)−j−1/summationdisplay\nk=0/integraldisplaytk+1\ntk(t−s)−βj\nΓ(1−βj)v′\nk,n(s)ds, t∈Ij, (3.3)\nand\nυ0,n:=u0,n, υj,n:=vj−1,n(tj), j= 1,...,M−1. (3.4)\nHere and in the remaining part of this text, any finite sum over k= 0 toj−1 is taken equal to zero\nwhenj= 0.\nAs a consequence, for all j= 0,1,...,M−1 and all n∈N, we get by substituting t−tjfortin\n[17, Theorem 5.15], that the solution to (3.2) reads\nvj,n(t) :=υj,nEβj,1(−λn(t−tj)βj)+/integraldisplayt\ntj(t−s)−1+βjEβj,βj(−λn(t−s)βj)fj,n(s)ds, t∈Ij,(3.5)\nwhereEα,βdenotes the two-parameter Mittag-Leffler function defined by Eα,β(z) =/summationtext∞\nk=0zk\nΓ(αk+β), see\n[24, Eq. (1.56)]. By [24, Theorem 1.6], we have for α∈(0,2) andβ >0 that\n∃CE>0,∀z∈[0,∞),|Eα,β(−z)| ≤CE\n1+z≤CE. (3.6)\n3.2.Auxiliary result. With reference to (3.3) and (3.5), we set\nvj(·,t) :=∞/summationdisplay\nn=1vj,n(t)Xn, fj(·,t) :=∞/summationdisplay\nn=1fj,n(t)Xn, t∈Ij,forj= 0,1,...,M−1,(3.7)\nand we note from this that\nu|Ij=∞/summationdisplay\nn=1un|IjXn=∞/summationdisplay\nn=1vj,nXn=vj.\nThe proof of Theorem 2.2 essentially relies on the following technical r esult.\nProposition 3.1. Suppose that u0andfsatisfy the assumptions of Theorem 2.2. Then, putting\nFj:=/ba∇dblu0/ba∇dblD(L)+j/summationdisplay\nk=0/parenleftBig\n/ba∇dblf/ba∇dblW1,1(Ik,H)+/vextenddouble/vextenddouble(·−tk)βk+εkf′/vextenddouble/vextenddouble\nL∞(Ik,H)/parenrightBig\n, j= 0,...,M−1,(3.8)\nwe have for all j= 0,...,M−1:\n(i)fj∈W1,1(Ij,H)and the estimates\n/ba∇dblfj/ba∇dblW1,1(Ij,H)≤CjFj, (3.9)\n/vextenddouble/vextenddoublef′\nj(·,t)/vextenddouble/vextenddouble≤CjFj(t−tj)−βj−εj, t∈Ij. (3.10)\n7(ii)vj∈ C0(Ij,D(L))∩W1,1(Ij,H)and the estimates\n/ba∇dblvj/ba∇dblC0(Ij,D(L))+/ba∇dblvj/ba∇dblW1,1(Ij,H)≤CjFj, (3.11)\n/vextenddouble/vextenddoublev′\nj(·,t)/vextenddouble/vextenddouble≤CjFj(t−tj)−1+βj, t∈Ij. (3.12)\nHere and in the remaining part of this text, Cjdenotes a generic positive constant depending only on\n{(Ik,βk,εk), k= 0,...,j}, which may change from line to line.\nThe proof of Proposition 3.1 being quite tedious, we postpone it to Se ction 4.\nWe notice from Proposition 3.1 that υj∈D(L) for all j= 0,...,M−1 (this follows from the\nidentity υ0=u0and the assumption u0∈D(L) whenj= 0, and from Proposition 3.1(ii), namely\nfromvj∈ C0(Ij,D(L)), when j= 1,...,M−1, sincevj−1(·,tj) =υjin this case). Moreover, we have\nvj(·,tj) =υjfor allj= 0,...,M−1, by virtue of (3.5)-(3.7), and consequently\nvj−1(tj) =vj(tj), j= 1,...,M−1. (3.13)\nNow, since vj∈ C0(Ij,D(L)) for all j= 0,...,M−1, by Proposition 3.1(ii), (3.13) and the identity\nvj=u|Ijthen yield that\nu∈ C0(I,D(L)). (3.14)\n3.3.End of the proof of Theorem 2.2. Let us first prove that u′∈L1(I,H). To this end, taking\ninto account that u∈L1(I,H), according to (3.14) and the embedding C0(I,D(L))⊂L1(I,H), we\nfind for every ϕ∈ C∞\n0(I) and a.e. x∈Ω, that\n/an}b∇acketle{tu′(x,·),ϕ/an}b∇acket∇i}htC∞\n0(I)′×C∞\n0(I)=−/an}b∇acketle{tu(x,·),ϕ′/an}b∇acket∇i}htC∞\n0(I)′×C∞\n0(I)\n=−M−1/summationdisplay\nj=0/integraldisplaytj+1\ntjvj(x,t)ϕ′(t)dt\n=M−1/summationdisplay\nj=0/integraldisplaytj+1\ntjv′\nj(x,t)ϕ(t)dt−M−1/summationdisplay\nj=0(vj(x,tj+1)ϕ(tj+1)−vj(x,tj)ϕ(tj)).\nHere we used that vj∈W1,1(Ij,H) for allj= 0,...,M−1, as stated in Proposition 3.1(ii). Further,\nbearing in mind that vj∈ C0(Ij,H) and that ϕ(0) =ϕ(T) = 0, we see that\nM−1/summationdisplay\nj=0(vj(x,tj+1)ϕ(tj+1)−vj(x,tj)ϕ(tj)) = 0, x∈Ω.\nThus, we have\n/an}b∇acketle{tu′(x,·),ϕ/an}b∇acket∇i}htC∞\n0(I)′,C∞\n0(I)=M−1/summationdisplay\nj=0/integraldisplaytj+1\ntjv′\nj(x,t)ϕ(t)dt, x∈Ω,\nfor allϕ∈ C∞\n0(I), from where we get that\nu′(x,t) =M−1/summationdisplay\nj=0χIj(t)v′\nj(x,t), x∈Ω, t∈I, (3.15)\n8whereχIdenotes the characteristic function of I. As a consequence we have u′∈L1(I,H) from\nProposition 3.1(ii), and hence u∈ C0(I,D(L))∩W1,1(I,H) by (3.14). Next, for all j= 0,...,M−1\nwe have\n/integraldisplayt\ntj(t−s)−βj\nΓ(1−βj)v′\nj(x,s)ds+Lvj(x,t) =fj(x,t), x∈Ω, t∈Ij,\nfrom the first equation of (3.1) and (3.7), whence\n/integraldisplayt\n0(t−s)−βj\nΓ(1−βj)u′(x,s)ds+Lu(x,t) =f(x,t), x∈Ω, t∈Ij,\nby (3.3) and (3.15). Moreover, we have u(·,0) =u0from (3.5)-(3.7) and the identities u|I0=v0and\nυ0=u0. This proves that uis a solution to the IBVP (1.5) in the sense of Definition 2.1. There exist s\nat most one solution to (1.5) since the solution to (3.2) on Ij,j= 0,...,M−1, is unique according to\n[17, Theorem 4.3].\nFinally, since u(·,t) =/summationtextM−1\nj=0χIj(t)vj(·,t) for allt∈I, we have\n/ba∇dblu/ba∇dblC0(I,D(L)= max{/ba∇dblvj/ba∇dblC0(Ij,D(L), j= 0,...,M−1}, (3.16)\nand\n/ba∇dblu/ba∇dblW1,1(I,H)=M−1/summationdisplay\nj=0/ba∇dblvj/ba∇dblW1,1(Ij,H), (3.17)\naccording to (3.15), so we obtain (2.2) by combining (3.11) with (3.16) -(3.17).\n4.Proof of Proposition 3.1\nThe proof is by induction on j. Givenj∈ {1,...,M−1}, the induction hypothesis (IH) jis that\nthe claim ofProposition3.1(and in particularthe estimates (3.9) – (3.1 2)) holdswhen k= 0,1,...,j−1\nis substituted for jin its statement.\nAs the method of the proof is basically the same for j= 0 and for j∈ {1,...,M−1}, we start\nwith the inductive step, only pointing out in the initial step the specific s of the proof for j= 0 when\nneeded.\n4.1.Inductive step. Letj∈ {1,...,M−1}be fixed. Assuming (IH) j, we aim to prove the statement\nof Proposition 3.1.\n4.1.1.Proof of (i). We split the proof into two main steps.\n9Step 1.We start by showing that fj∈L1(Ij,H) satisfies the estimate\n/ba∇dblfj/ba∇dblL1(Ij,H)≤CjFj. (4.1)\nTo this purpose we refer to (3.3) and (3.7), and obtain that\nfj(·,t) =∞/summationdisplay\nn=1/parenleftBigg\nf0,n(t)−j−1/summationdisplay\nk=0/integraldisplaytk+1\ntk(t−s)−βj\nΓ(1−βj)v′\nk,n(s)ds/parenrightBigg\nXn, t∈Ij. (4.2)\nNext, for all N∈Nwe have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftBigg\nf0,n(t)−j−1/summationdisplay\nk=0/integraldisplaytk+1\ntk(t−s)−βj\nΓ(1−βj)v′\nk,n(s)ds/parenrightBigg\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1f0,n(t)Xn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble+1\nΓ(1−βj)j−1/summationdisplay\nk=0/integraldisplaytk+1\ntk(t−s)−βj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1v′\nk,n(s)Xn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleds\n≤ /ba∇dblf(·,t)/ba∇dbl+1\nΓ(1−βj)j−1/summationdisplay\nk=0/integraldisplaytk+1\ntk(t−s)−βj/ba∇dblv′\nk(s)/ba∇dblds,\nand (t−s)−βj≤(t−tk+1)−βjwhenever s≤tk+1≤tj< tandk= 0,...,j−1. This and (4.2) yield\nthat\n/ba∇dblfj(·,t)/ba∇dbl ≤ /ba∇dblf(·,t)/ba∇dbl+1\nΓ(1−βj)j−1/summationdisplay\nk=0(t−tk+1)−βj/ba∇dblv′\nk/ba∇dblL1(Ik,H), t∈Ij.\nThus, weget, by integratingwith respectto toverIjandusing that ( tj+1−tk+1)1−βj−(tj−tk+1)1−βj≤\nt1−βj\nj+1for allk= 0,...,j−1, that\n/ba∇dblfj/ba∇dblL1(Ij,H)≤ /ba∇dblf/ba∇dblL1(Ij,H)+t1−βj\nj+1\nΓ(2−βj)j−1/summationdisplay\nk=0/ba∇dblv′\nk/ba∇dblL1(Ik,H).\nNow, applying (IH) j, and more precisely (3.11) where k= 0,...,j−1 is substituted for j, we find that\n/ba∇dblfj/ba∇dblL1(Ij,H)≤ /ba∇dblf/ba∇dblL1(Ij,H)+t1−βj\nj+1\nΓ(2−βj)j−1/summationdisplay\nk=0CkFk,\nand (4.1) follows from this and the estimates Fk≤ Fj−1fork= 0,...,j−1.\nStep 2.Having established (4.1), we turn now to the proof of (3.10). This ca n be achieved by differen-\ntiating (4.2) with respect to t∈Ij: we obtain that\nf′\nj(·,t) =f′(·,t)+βj\nΓ(1−βj)j−1/summationdisplay\nk=0/integraldisplaytk+1\ntk(t−s)−1−βjv′\nk(·,s)ds,\nand hence that\n/vextenddouble/vextenddoublef′\nj(·,t)/vextenddouble/vextenddouble≤ /ba∇dblf′(·,t)/ba∇dbl+βj\nΓ(1−βj)j−1/summationdisplay\nk=0/integraldisplaytk+1\ntk(t−s)−1−βj/ba∇dblv′\nk(·,s)/ba∇dblds. (4.3)\n10The main difficulty lies in handling the term k=j−1 in the above summation. We distinguish in two\ncases. In the first one, when k= 0,...,j−2, we get upon using that ( t−s)−1−βj≤(tj−tj−1)−βj−1\nfor allt∈Ijands∈Ik, that\n/integraldisplaytk+1\ntk(t−s)−1−βj/ba∇dblv′\nk(·,s)/ba∇dblds=/integraldisplaytk+1\ntk(t−s)−1−βj(s−tk)−1+βk/vextenddouble/vextenddouble(s−tk)1−βkv′\nk(·,s)/vextenddouble/vextenddoubleds\n≤(tj−tj−1)−1−βj/parenleftbigg/integraldisplaytk+1\ntk(s−tk)−1+βkds/parenrightbigg/vextenddouble/vextenddouble(·−tk)1−βkv′\nk/vextenddouble/vextenddouble\nL∞(Ik,H)\n≤β−1\nk(tj−tj−1)−1−βj(tk+1−tk)βk/vextenddouble/vextenddouble(·−tk)1−βkv′\nk/vextenddouble/vextenddouble\nL∞(Ik,H).(4.4)\nIn thesecondcase, when k=j−1, wehave( t−s)−βj−εj≤(t−tj)−βj−εjand(t−s)−1+εj≤(tj−s)−1+εj\nfor allt∈Ijands∈Ij−1, whence\n/integraldisplaytj\ntj−1(t−s)−1−βj/vextenddouble/vextenddoublev′\nj−1(·,s)/vextenddouble/vextenddoubleds\n=/integraldisplaytj\ntj−1(t−s)−βj−εj(t−s)−1+εj(s−tj−1)−1+βj−1(s−tj−1)1−βj−1/vextenddouble/vextenddoublev′\nj−1(s)/vextenddouble/vextenddoubleds\n≤(t−tj)−βj−εj/parenleftBigg/integraldisplaytj\ntj−1(tj−s)−1+εj(s−tj−1)−1+βj−1ds/parenrightBigg\n/vextenddouble/vextenddouble(·−tj−1)1−βj−1v′\nj−1/vextenddouble/vextenddouble\nL∞(Ij−1,H).\nFurther, by performing the change of variable r=s−tj−1\ntj−tj−1in the integral of the above line, we see that\n/integraldisplaytj\ntj−1(tj−s)−1+εj(s−tj−1)−1+βj−1ds= (tj−tj−1)−1+βj−1+εj/parenleftbigg/integraldisplay1\n0(1−r)−1+εjr−1+βj−1dr/parenrightbigg\n= (tj−tj−1)−1+βj−1+εjB(βj−1,εj),\nwhereBdenotes the beta function\nB(r1,r2) :=/integraldisplay1\n0sr1−1(1−s)r2−1ds, r1∈(0,∞), r2∈(0,∞).\nAs a consequence, we have for all t∈Ij,\n/integraldisplaytj\ntj−1(t−s)−1−βj/vextenddouble/vextenddoublev′\nj−1(·,s)/vextenddouble/vextenddoubleds\n≤(tj−tj−1)−1+βj−1+εjB(βj−1,εj)/vextenddouble/vextenddouble(·−tj−1)1−βj−1v′\nj−1/vextenddouble/vextenddouble\nL∞(Ij−1,H)(t−tj)−βj−εj.\nPutting this together with (4.3)-(4.4), and applying (3.12) where k= 0,...,j−1 is substituted for j,\nin accordance with (IH) j, we find for a.e. t∈Ijthat the difference\n(t−tj)βj+εj/vextenddouble/vextenddoublef′\nj(·,t)/vextenddouble/vextenddouble−(t−tj)βj+εj/ba∇dblf′(·,t)/ba∇dbl\nis upper bounded by the constant\nβj\nΓ(1−βj)/parenleftBigg\n(tj+1−tj)βj+εj\n(tj−tj−1)1+βjj−2/summationdisplay\nk=0Ck(tk+1−tk)βk\nβkFk+Cj−1B(βj−1,εj)\n(tj−tj−1)1−βj−1−εjFj−1/parenrightBigg\n≤CjFj−1.\nNow, (3.10) follows readily from this and (2.1). Moreover, using (4.1) and recalling that βj+εj∈(0,1),\nwe get (3.9) by integrating (3.10) over Ij.\n11We turn now to showing (ii), but prior to that, we notice for further use from (3.9) and from the\ncontinuous embedding W1,1(Ij,H)⊂ C0(Ij,H), that\n/ba∇dblfj(·,t)/ba∇dbl ≤CjFj, t∈Ij. (4.5)\n4.1.2.Proof of (ii). The proof of the second claim, (ii), of Proposition 3.1 being quite lengt hy, we break\nit into two parts.\nStep 1.The first step is to show that vj∈ C0(Ij,D(L)) satisfies /ba∇dblvj/ba∇dblC0(Ij,D(L))≤CjFj. To do that, we\nrecall from (i) that the function t/mapsto→fj,n(t) =/an}b∇acketle{tfj(·,t),Xn/an}b∇acket∇i}ht ∈W1,1(Ij), for all n∈N. Then, bearing in\nmind that\nd\ndtEα,1(−λntα) =−λntα−1Eα,α(−λntα), t >0.\nand integrating by parts, we obtain for all t∈Ijthat\n/integraldisplayt\ntj(t−s)−1+βjEβj,βj(−λn(t−s)βj)fj,n(s)ds\n=1\nλn/parenleftBigg\nEβj,1(0)fj,n(t)−Eβj,1(−λn(t−tj)βj)fj,n(tj)+/integraldisplayt\ntjEβj,1(−λn(t−s)βj)f′\nj,n(s)ds/parenrightBigg\n.\nAs a consequence, we have for all N∈Nthat\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftBigg/integraldisplayt\ntj(t−s)−1+βjEβj,βj(−λn(t−s)βj)fj,n(s)ds/parenrightBigg\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nD(L)\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftbig\nEβj,1(0)fj,n(t)−Eβj,1(−λn(t−tj)βj)fj,n(tj)/parenrightbig\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftBigg/integraldisplayt\ntjEβj,1(−λn(t−s)βj)f′\nj,n(s)ds/parenrightBigg\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble, t∈Ij. (4.6)\nThe first term on the right-hand side of (4.6) is simply handled by (3.6) , as follows:\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftbig\nEβj,1(0)fj,n(t)−Eβj,1(−λn(t−tj)βj)fj,n(tj)/parenrightbig\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1Eβj,1(0)fj,n(t)Xn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1Eβj,1(−λn(t−tj)βj)fj,n(tj)Xn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤CE\n/parenleftBiggN/summationdisplay\nn=1|fj,n(t)|2/parenrightBigg1/2\n+/parenleftBiggN/summationdisplay\nn=1|fj,n(tj)|2/parenrightBigg1/2\n\n≤CE(/ba∇dblfj(·,t)/ba∇dbl+/ba∇dblfj(·,tj)/ba∇dbl). (4.7)\nAs for the second term, we get through standard computations t hat\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftBigg/integraldisplayt\ntjEβj,1(−λn(t−s)βj)f′\nj,n(s)ds/parenrightBigg\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/integraldisplayt\ntj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1Eβj,1(−λn(t−s)βj)f′\nj,n(s)Xn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleds\n12≤CE/integraldisplayt\ntj/parenleftBiggN/summationdisplay\nn=1/vextendsingle/vextendsinglef′\nj,n(s)/vextendsingle/vextendsingle2/parenrightBigg1/2\nds\n≤CE/integraldisplayt\ntj/vextenddouble/vextenddoublef′\nj(s)/vextenddouble/vextenddoubleds.\nThis and (4.6)-(4.7) then yield that for all t∈Ij,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nn=1/parenleftBigg/integraldisplayt\ntj(t−s)−1+βjEβj,βj(−λn(t−s)βj)fj,n(s)ds/parenrightBigg\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nD(L)\n≤CE/parenleftBig\n2/ba∇dblfj/ba∇dblC0(Ij,H)+/vextenddouble/vextenddoublef′\nj/vextenddouble/vextenddouble\nL1(Ij,H)/parenrightBig\n.(4.8)\nNow, taking into account that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nn=1υj,nEβj,1(−λn(t−tj)βj)Xn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nD(L)=/parenleftBigg∞/summationdisplay\nn=1λ2\nn|υj,n|2Eβj,1(−λn(t−tj)βj)2/parenrightBigg1\n2\n≤CE/parenleftBigg∞/summationdisplay\nn=1λ2\nn|υj,n|2/parenrightBigg1\n2\n≤CE/ba∇dblυj/ba∇dblD(L), t∈Ij, (4.9)\nand recalling (3.5)-(3.7), we deduce from (4.8) that\n/ba∇dblvj(·,t)/ba∇dblD(L)≤CE/parenleftBig\n/ba∇dblυj/ba∇dblD(L)+2/ba∇dblfj/ba∇dblC0(Ij,H)+/vextenddouble/vextenddoublef′\nj/vextenddouble/vextenddouble\nL1(Ij,H)/parenrightBig\n, t∈Ij. (4.10)\nNext, bearing in mind that υj=vj−1(·,tj) sincej≥1 here, we infer from (IH) j(namely, upon\nsubstituting j−1 forjin (3.11)) that\n/ba∇dblυj/ba∇dblD(L)≤Cj−1Fj−1. (4.11)\nBy inserting this, (3.9) and (4.5) into (4.10), we obtain that\n/ba∇dblvj(·,t)/ba∇dblD(L)≤CjFj, t∈Ij. (4.12)\nFurther, we see from (3.5) that for all fixed N∈N, the function t/mapsto→/summationtextN\nn=1vj,n(t)Xn∈ C0(Ij,D(L)).\nMoreover, with reference to (3.5)-(3.7), it is clear from (4.8)-(4.9 ) that the series/summationtext∞\nn=1vj,n(t)Xn\nconverges to vj(·,t) inD(L), uniformly in t∈Ij. Therefore, we have\nvj∈ C0(Ij,D(L)) (4.13)\nand the estimate\n/ba∇dblvj/ba∇dblC0(Ij,D(L))≤CjFj, (4.14)\naccording to (4.12).\n13Step 2.Having established (4.13)-(4.14), we turn now to proving the estima te (3.12). For this purpose,\nwe differentiate (3.5) with respect to t∈Ij, and obtain that\nv′\nj,n(t) = (fj,n(tj)−λnυj,n)(t−tj)−1+βjEβj,βj(−λn(t−tj)βj)\n+/integraldisplayt\ntj(t−s)−1+βjEβj,βj(−λn(t−s)βj)f′\nj,n(s)ds, n∈N. (4.15)\nNext, using (3.6), we get for all N∈Nand allt∈Ijthat\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftbig\nfj,n(tj)−λnυj,n)(t−tj)−1+βjEβj,βj(−λn(t−tj)βj/parenrightbig\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1fj,n(tj)(t−tj)−1+βjEβj,βj/parenleftbig\n−λn(t−tj)βj/parenrightbig\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1λnυj,n(t−tj)−1+βjEβj,βj/parenleftbig\n−λn(t−tj)βj/parenrightbig\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤CE(t−tj)−1+βj\n/parenleftBiggN/summationdisplay\nn=1|fj,n(tj)|2/parenrightBigg1/2\n+/parenleftBiggN/summationdisplay\nn=1λ2\nn|υj,n|2/parenrightBigg1/2\n\n≤CE(t−tj)−1+βj/parenleftBig\n/ba∇dblfj(·,tj)/ba∇dbl+/ba∇dblυj/ba∇dblD(L)/parenrightBig\nand that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1/parenleftBigg/integraldisplayt\ntj(t−s)−1+βjEβj,βj(−λn(t−s)βj)f′\nj,n(s)ds/parenrightBigg\nXn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤/integraldisplayt\ntj(t−s)−1+βj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nn=1Eβj,βj(−λn(t−s)βj)f′\nj,n(s)Xn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleds\n≤CE/integraldisplayt\ntj(t−s)−1+βj/parenleftBiggN/summationdisplay\nn=1/vextendsingle/vextendsinglef′\nj,n(s)/vextendsingle/vextendsingle2/parenrightBigg1/2\nds\n≤CE/integraldisplayt\ntj(t−s)−1+βj/vextenddouble/vextenddoublef′\nj(·,s)/vextenddouble/vextenddoubleds.\nTherefore, it follows from (3.5)-(3.7) and (4.15) that for all t∈Ij,\n/vextenddouble/vextenddoublev′\nj(·,t)/vextenddouble/vextenddouble≤CE/parenleftBigg\n(t−tj)−1+βj/parenleftBig\n/ba∇dblfj(·,tj)/ba∇dbl+/ba∇dblυj/ba∇dblD(L)/parenrightBig\n+/integraldisplayt\ntj(t−s)−1+βj/vextenddouble/vextenddoublef′\nj(·,s)/vextenddouble/vextenddoubleds/parenrightBigg\n.(4.16)\nMoreover, since βj+εj∈(0,1), we apply (3.10) and get for all t∈Ijthat\n/integraldisplayt\ntj(t−s)−1+βj/vextenddouble/vextenddoublef′\nj(·,s)/vextenddouble/vextenddoubleds\n≤/vextenddouble/vextenddouble(·−tj)βj+εjf′\nj/vextenddouble/vextenddouble\nL∞(Ij,H)/integraldisplayt\ntj(t−s)−1+βj(s−tj)−βj−εjds\n≤CjFjB(1−βj−εj,βj)(t−tj)−εj\n≤CjFjB(1−βj−εj,βj)(tj+1−tj)1−βj−εj(t−tj)−1+βj,\n14so we obtain (3.12) by plugging the above estimate, (4.5) and (4.11) in to (4.16). Finally, (3.11) follows\nstraightforwardly from (3.12) and (4.14).\n4.2.Base step. Firstly, since f0=f|I0, it is clear from the assumption f|I0∈W1,1(I0,H) and from\n(2.1) that the claims of (i) in Proposition 3.1, hold for j= 0.\nSecondly, taking j= 0 in the derivation of (4.10), we get that for all t∈I0,\n/ba∇dblv0(·,t)/ba∇dblD(L)≤CE/parenleftBig\n/ba∇dblu0/ba∇dblD(L)+2/ba∇dblf/ba∇dblC0(I0,H)+/ba∇dblf′/ba∇dblL1(I0,H)/parenrightBig\n. (4.17)\nSimilarly, since\n/integraldisplayt\n0(t−s)−1+β0/ba∇dblf′(·,s)/ba∇dblds≤ B(1−β0−ε0,β0)t1−β0−ε0\n1/vextenddouble/vextenddouble(·)β0+ε0f′/vextenddouble/vextenddouble\nL∞(Ij,H)t−1+β0,\nfor allt∈I0, we find upon mimicking the proof of (4.16) that\n/ba∇dblv′\n0(·,t)/ba∇dbl ≤CE/parenleftBig\n/ba∇dblu0/ba∇dblD(L)+/ba∇dblf(·,0)/ba∇dbl+B(1−β0−ε0,β0)t−1+β0\n1/vextenddouble/vextenddouble(·)β0+ε0f′/vextenddouble/vextenddouble\nL∞(I0,H)/parenrightBig\nt−ε0.(4.18)\nTherefore, the estimates (3.11)-(3.12) for j= 0 follow from (4.17)-(4.18) and the continuity of the\nembedding W1,1(I0,H)⊂ C0(I0,H).\nThis completes the proof of the proposition.\nAcknowledgments\nThe work of K. Van Bockstal was supported by the Methusalem pro gramme of Ghent University\nSpecial Research Fund (BOF) (Grant Number 01M01021).\nReferences\n[1] K. Adolfsson, M. Enelund, and S. Larsson. Adaptive discr etization of an integro-differential equation with a weakly\nsingular convolution kernel. Comput. Methods Appl. Mech. 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SIAM Journal on Numerical Analysis , 47(3):1760–1781, 2009.\n(△)Univ Rouen Normandie, CNRS, Normandie Univ, LMRS UMR 6085, F-76000 Rouen, France\nEmail address :yavar.kian@univ-rouen.fr\n(⋆)Department of Electronics and Information Systems, resear ch group of Numerical Analysis and\nMathematical Modeling (NaM2), Ghent University, Krijgslaan 281, S8, Gent 9000, Belgium\nEmail address :marian.slodicka@ugent.be\n(✷)Aix-Marseille Univ, Universit ´e de Toulon, CNRS, CPT, Marseille, France\nEmail address :eric.soccorsi@univ-amu.fr\n(♠)Ghent Analysis & PDE center, Department of Mathematics: Analy sis, Logic and Discrete Mathe-\nmatics, Ghent University, Krijgslaan 281, 9000 Ghent, Belgi um\nEmail address :karel.vanbockstal@ugent.be\n17"    },    {        "title": "2402.03497v1.An_Analytic_Solution_for_Kernel_Adaptive_Filtering.pdf",        "content": "AN ANALYTIC SOLUTION FOR KERNEL ADAPTIVE FILTERING\nBenjamin Colburn⋆, Luis G. Sanchez Giraldo†, Kan Li⋆, and Jose C. Principe⋆\n⋆University of Florida,†University of Kentucky\nABSTRACT\nConventional kernel adaptive filtering (KAF) uses a pre-\nscribed, positive definite, nonlinear function to define the\nReproducing Kernel Hilbert Space (RKHS), where the op-\ntimal solution for mean square error estimation is approxi-\nmated using search techniques. Instead, this paper proposes\nto embed the full statistics of the input data in the kernel\ndefinition, obtaining the first analytical solution for nonlin-\near regression and nonlinear adaptive filtering applications.\nWe call this solution the Functional Wiener Filter (FWF).\nConceptually, the methodology is an extension of Parzen’s\nwork on the autocorrelation RKHS to nonlinear functional\nspaces. We provide an extended functional Wiener equation,\nand present a solution to this equation in an explicit, finite\ndimensional, data-dependent RKHS. We further explain the\nnecessary requirements to compute the analytical solution in\nRKHS, which is beyond traditional methodologies based on\nthe kernel trick. The FWF analytic solution to the nonlinear\nminimum mean square error problem has better accuracy than\nother kernel-based algorithms in synthetic, stationary data. In\nreal world time series, it has comparable accuracy to KAF but\ndisplays constant complexity with respect to number of train-\ning samples. For evaluation, it is as computationally efficient\nas the Wiener solution (with a larger number of dimensions\nthan the linear case). We also show how the difference equa-\ntion learned by the FWF from data can be extracted leading to\nsystem identification applications, which extend the possible\napplications of the FWF beyond optimal nonlinear filtering.\n1. INTRODUCTION\nNorbert Wiener’s 1949 work on minimum mean square es-\ntimation (MMSE) initiated the theory of optimum filter-\ning [34]. The mathematics to solve integral equations, the\nWiener-Hopf method [36], were crucial to arrive at the opti-\nmal parameter function, however, the methodology is rather\ncomplex. In digital signal processing using finite impulse\nresponse filters, the Wiener solution coincides with least\nsquares, as proven by the Wiener-Khinchin theorem [35]. Un-\nfortunately, the solution still belongs to the span of the input\nThis material is based upon work supported by the Office of the Under\nSecretary of Defense for Research and Engineering under award numbers\nN00014-21-1-2295 and N00014-21-1-2345.data i.e., the corresponding filter is linear in the parameters\nand therefore it is not a universal functional approximator.\nIn the late 50’s, Emmanuel Parzen [22] presented an al-\nternative approach to solve the MMSE problem in a Repro-\nducing Kernel Hilbert Space (RKHS) defined by the autocor-\nrelation function of the data. The history of RKHS applica-\ntions started in physics (Fredholm equations), statistics, sig-\nnal processing, and machine learning [32, 13, 12, 8]. Here, it\nwill also be clear that the RKHS framework provides a natu-\nral link between stochastic processes and deterministic func-\ntional analysis.\nParzen’s essential idea is that there exists a congru-\nence map between the set of random variables spanned by\nthe random process {Xt, t∈T}with covariance function\nR(t, s) =E[XtXs]and the RKHS of vectors spanned by the\nset{R(·, t), t∈T}denoted as HR. Note that the kernel esti-\nmates the second-order statistics of the data (a data-dependent\nkernel). Parzen clearly states that this RKHS offers an elegant\nfunctional analysis framework for MMSE solutions such as\nregression [7], least squares estimation of random variables,\ndetection of signals in Gaussian noise [6], and others. Un-\nfortunately, the autocorrelation kernel is a linear kernel, so\nHRyields only linear solutions to the regression problem.\nParzen’s beautiful interpretation did not provide any practical\nimprovement, so it was forgotten in signal processing.\nThis paper takes Parzen’s work one step further by defin-\ning an analytic solution for the Wiener filter in a space non-\nlinearly related to the input space, which is novel as far as we\nknow. Using analogous techniques to Parzen’s work in [22]\nwe create a data-dependent kernel based on the covariance\nfunction of projected feature vectors in a space nonlinearly\nrelated to the input space. The end result is an analytic solu-\ntion to optimal nonlinear filters for strictly stationary signals.\nThe proposed approach differs significantly from current ker-\nnel adaptive filtering (KAF) [16] [20], where a weighted sum\nof functionals centered at individual samples approximate the\nMMSE using search techniques. While a closed-form solu-\ntion for a nonlinear filter is interesting in its own right, the idea\nof embedding the statistics of input data, its internal depen-\ndency structure, into the inner product structure of an RKHS\ndefined by a nonlinear kernel is perhaps the more general and\nimportant contribution of this work, with many applications\nin science and engineering.\nIn order to achieve our goals, full access to the structurearXiv:2402.03497v1  [eess.SP]  5 Feb 2024of the RKHS is necessary, which is beyond the conventional\n“kernel trick” [29]. Since the kernel in our case preserves the\nstatistical time structure of the input random process (or of the\nmultivariate random variable), one needs the flexibility of an\noperator which includes time beyond the simple inner prod-\nuct between two projected samples. For this reason, we ap-\nproximate the kernel by an explicit, finite dimensional RKHS.\nIn the case of the Gaussian kernel, the RKHS is defined by\nHermite-Gaussian functions of a certain dimension selected\nby the user, as suggested in [17]. The advantage is that the in-\nner product becomes an explicit quadratic form, which allows\nfor much more control of the projections. The disadvantage\nis that the explicit RKHS is finite dimensional i.e., no longer\nuniversal, but the dimensionality is under control of the user.\nThe previous results [17] show that excellent approximations\nfor adaptive filters are obtained with just 20−50dimensions.\nThe rest of the paper will continue as follows. In sec-\ntion II, we give a brief review of optimal linear filtering in\nan RKHS. Then in section III we will describe the extension\nof this idea to a space defined by a nonlinear kernel, which\ngives rise to the Functional Wiener Filter (FWF). In section\nIV the analytic FWF solution is described, followed by the\nalgorithmic solution applied to data samples. We also present\nan error analysis of the FWF. In section V we discuss how\nto recover a difference equation from the FWF showing that\nthe method has application beyond just optimal nonlinear fil-\ntering. We then compare the performance of the FWF with\nKAF and other nonlinear filtering methods (Gaussian Pro-\ncesses and Kernel Ridge Regression) and give experimental\nvalidation of FWF even for non-stationary time series. Sec-\ntion VI summarizes the paper and compares the methodology\nwith KAF. In the appendix we give an explanation of why\nprevious attempts in [3] and [4] were unsuccessful when com-\npared to this derivation of the FWF.\n2. REVIEW OF LINEAR FILTERING OF\nCONTINUOUS TIME SERIES IN RKHS\nLetXbe a non-empty set, and K(x, x′)a function defined on\nX × X that is non-negative definite. The Moore-Aronzsajn\ntheorem [1], guarantees that K(x, x′)defines uniquely a\nRKHS, HK, such that K(·, x)∈ H Kand for any g∈ H K,\nyields ⟨g, K(·, x)⟩HK=g(x). Therefore, an RKHS is a\nspecial Hilbert vector space associated with a kernel such\nthat it reproduces (via the inner product) in the space, i.e.,\n⟨K(·, x), K(·, x′)⟩HK=K(x, x′); or equivalently, a space\nwhere every point evaluation functional is bounded.\nA stochastic process {Xt,ω, t∈T}is broadly defined as\na collection of random variables(r.v) on a measurable sample\nspace (Ω, BΩ), indexed by a set T. Here, we restrict Xt,ω\nto random variables taking values in R, and T⊂R, which\nwe call a time-series, {Xt, t∈T}and omit the dependence\nonωfor simplicity. We can further simplify the notation and\nwrite{Xt, t∈T}as{Xt}. For a time-series with finite sec-ond order moments, let L2({Xt})denote the space of all real\nvalued random variables spanned by the time series i.e., this\nspace consists of all r.v. that are either linear combinations of\nfinite number of {Xt}or are limits of such linear combina-\ntions.\nThe time structure is quantified by the joint probability\ndensity p(Xt, Xs)of the pair of random variables, Xtand\nXs, or a random process sampled at two points in time tand\ns. Assuming a strictly stationary stochastic model for Xt,\nthe marginal density pXt(x)is the same for any t. Normally,\nthe joint density p(Xt, Xs)is quantified by its mean value,\ncalled the autocorrelation function. To simplify notation, let\nus define the time autocorrelation of the time series as:\nR(s, t) =E[XsXt]. (1)\nThe autocorrelation function on time sample pairs is pos-\nitive semi-definite, hence by Moore-Aronzsajn theorem [1] it\ndefines a RKHS space of functions on T×T, denoted HR.\nNotice that the inner product in HRdepends on the statistics\nof the data through Xt, while the functions in HRare deter-\nministic because of the E[·]operator.\nFor any pair of random variables A, A′∈L2({Xt}), define\ntheir inner product as\n⟨A, A′⟩=E[AA′] (2)\nand the induced norm of AinL2({Xt})by⟨A, A⟩=E[A2].\nObviously, this inner product coincides with the autocorrela-\ntion function if AisXsandA′isXt. However, notice that\nL2({Xt})is not an RKHS.\n2.1. Explicit Expression for MMSE\nOne of the important problems in time series analysis is the\nrepresentation of an unobservable r.v. Z. Let{Xt, t∈T}be\nan observable time series assumed stationary. The goal is to\ncreate a linear combination of the observable time series that\nhas the smallest mean squared distance to Z. By the Hilbert\nprojection theorem, there is a unique minimum norm projec-\ntion between the abstract Hilbert space Hand any subspace\nMofH. Then, there exists a unique vector A∗inM, given\nbyA∗=E∗[A|M], which is the orthogonal projection of a\nvector AinHtoM. For a family of vectors {Xt, t∈T}the\nprojection becomes\nA∗=E∗[A|Xt, t∈T]. (3)\nThen with A=Z, the optimal linear predictor is the\nunique r.v. in L2({Xt})that satisfies\nE[E∗[Z|Xt, t∈T]Xs] =E[ZXs] (4)\nfor any s∈T. This result gives immediately rise to the\nfamous Wiener equation. Indeed, if Tis a finite interval\nT={t:a≤t≤b}andw(t)a weighting function in\nL2, the integralZb\naX(t)w(t)dt (5)\nrepresents a r.v. in L2(Xt, t∈T), then the weighting\nfunction of the best linear predictor can be written as\nE∗[Z|Xt, t∈T]≜Zb\naX(t)w∗(t)dt (6)\nand must satisfy the generalized Wiener equation\nZb\naw∗(t)R(s, t)dt=ρz(s) (7)\nina≤s≤b, with R(s, t) =E[XsXt], andρz(s) =E[ZXs].\nThis result states that one can always find a representation\nfor the function ρz(s)in terms of the functions {R(s, t), t∈\nT}such that the MMSE linear predictor E∗[Z|Xt, t∈T]can\nbe written in terms of the corresponding linear operator on the\ntime series {Xt, t∈T}.\nParzen showed that the solution of the prediction problem\nmay be written as follows [22]. If Zis a r.v. with finite second\nmoments and ρz(s) =E[ZXs]then\nρz∈ HR\nE∗[Z|Xt, t∈T] =⟨ρz, X⟩HR.(8)\nThe equivalent MMSE solution in the data space be-\ncomes,\nY=E∗[Z|Xt, t∈T] =⟨ρz, X⟩HR\n=Z\nt∈TZ\ns∈TR−1(s, t)ρz(s)X(t)dsdt(9)\nwhich coincides with the Wiener solutionR\nt∈TX(t)w∗(t)dt.\nFurthermore, in digital signal processing the double integral\nbecomes a double sum and it can be computed in the input\nspace directly from samples as\nY(t) =⟨Xt, R−1ρz⟩L2=⟨Xt, ρz⟩HR\n=L−1X\nt=0L−1X\ns=0X(t)R−1(s, t)ρz(s).(10)\nNote that the autocorrelation matrix and the cross-correlation\nvector must be first estimated from a training set. Another\ninteresting fact is that the dimension of HRis the number of\ndelays, which is under user control. Moreover, in HRthis\nsolution is coordinate free, does not use the approximation\nerror, and directly uses the structure of HR. In fact, it is suffi-\ncient to compute the linear projection between ρz(s)with the\ninput data because the covariance kernel R(s, t)is implicit\nin the inner product of HR, unlike Wiener-Hopf method that\nrequires spectral factorization. This coordinate free property\nof RKHS solutions with the covariance kernel was first notedby Lo `eve [21] who suggested that, instead of finding a set of\nfunctional projections (e.g. Karhunen-Lo `eve Transform), it\nis sufficient to employ the statistics of {Xt}embedded in the\nstructure of the RKHS. Parzen [22] further states that for this\nreason “RKHS defined by the covariance kernel is the natural\nsetting in which to solve problems of statistical inference on\ntime series”.\nThe fundamental issues with Parzen approach are twofold:\nfirst, it does not elucidate efficient alternatives to implement\nthe conditional mean operator; second, for discrete time sig-\nnal processing, this approach is computationally more com-\nplex than the famous Wiener solution w∗=R−1ρ, where R\nis the autocorrelation matrix (the kernel R(s, t)evaluated at\na finite set of delays), and ρis the cross-correlation vector.\nWe conclude that the advantage of the RKHS theory is on\nthe mathematical tools of congruence and representation of\ntime series in RKHS, which opens the door to seek analyti-\ncal solutions for nonlinear regression. A key feature of the\nRKHS theory is that functionals defined in the RKHS are in-\ndependent of the kernel utilized, therefore we can expect that\na similar form to (9), which finds a minimum norm projection\nfor nonlinear regression, exists in a different RKHS. Hence,\nthe key goal for nonlinear extensions of Parzen’s approach is\nto concentrate on designing proper kernels and more general\nframeworks based on operators.\n3. THE NONLINEAR PREDICTION CASE\nOur goal now is to define a novel RKHS that preserves the\ncorrelation structure defined by the data, as done in HR, but\nalso maps the time series by a nonlinear kernel to achieve\nConvex Universal Learning Machine (CULM) properties\n[23]. There are several ways we could proceed. Perhaps the\nmost straight forward is to define an RKHS with a kernel that\nextends the autocorrelation function [38], called correntropy\n[18], which is also a positive definite function of two argu-\nments defined in a Gaussian RKHS. Unfortunately, this route\nlead to the need for heuristic approximations which yields\npoor generalization as mentioned in [4] and [3].\nAs an alternative, here we define and apply an operator\nthat analytically computes the minimum norm projection in\nan explicit finite dimensional RKHS that approximates the\nGaussian RKHS i.e.\nGσ(x,x′) :X × X → R≈ˆϕ(x)Tˆϕ(x′) (11)\nwhere X=Rp, with mapping ˆϕ:X → RD,Gσis the\nGaussian function with kernel size σ,pis the dimension of\nthe input data, and Dis the number of dimensions in the finite\nRKHS. The definition of a nonlinear, but finite dimensional\nRKHS makes this task much easier, while with a controlled\nerror.\nIn [17] the explicit mapping function based on the Tay-\nlor expansion of the Gaussian kernel is used to define a finitedimensional RKHS which approximates the feature space de-\nfined by the Gaussian kernel. A more in depth discussion\ncan be found in [5]. This is just one way of getting an ex-\nplicit feature space; another notable technique employs ran-\ndom Fourier features (RFF) [26], but it is a more computa-\ntionally expensive method than the Taylor expansion based\nmethod. The advantage of the explicit RKHS is that we can\nemploy directly linear operators in the feature space, which\nprovides much more flexibility when compared with the ker-\nnel trick. We will refer to the explicit finite rank feature space\nasHB, with kernel B(·,·). The explicit mapping function for\nvectors given in [5] (see also [40, 39]) is given by\nϕd,j(x) =e−∥x∥2\n2σ21\nσd√\nd!dY\ni=1xji, (12)\nwhere j∈[p]denumerates over all selections of dcoordinates\nofx(allowing repetitions and enumerating over different or-\nderings of the same coordinates). For instance, the set [2]3\nconsists of eight 3-tuples, namely, (1,1,1),(1,1,2),(1,2,1),\n(1,2,2),(2,1,1),(2,1,2),(2,2,1), and (2,2,2). The finite\nrank kernel Bis obtained by truncating the series to the first\nkterms\nB(x,x′) =D\nˆϕ(x),ˆϕ(x′)E\nHB\n=kX\nd=0X\nj∈[p]dϕd,j(x)ϕd,j(x′)\n=e−∥x∥2\n2σ2e−∥x′∥2\n2σ2kX\nd=0\u0000\nxTx′\u0001d\nσ2dd!.(13)\nIt’s important to note that the Gaussian kernel is obtained\nwhen k(the number of terms) approaches infinity. However,\ndue to the exponential decay of the Gaussian function’s eigen-\nvalues, a good finite rank approximation can often be obtained\nwith just a few terms in the expansion [17].\nNormally algorithms which employ RKHSs are written in\nterms of weighted sums of kernels, and the underlying eigen-\nvalues and eigenfunctions which form a basis for the RKHS\nare not considered in detail. When we define explicitly the\nbasis of the RKHS we are really treating the RKHS more like\na Hilbert space for which we define a basis. Later we will see\nthat it is necessary to work with individual eigenfunctions of\nthe RKHS if we want to define an analytical MMSE solution.\nTherefore the explicit kernel provides an elegant way of per-\nforming the operations necessary for an analytical solution.\n3.1. Retaining Time Structure in HB\nWe can retain the time structure of a stochastic process,\n{Xt, t∈T}, by applying the finite rank kernel B(·,·). This\nforms the span of the set of random elements, {B(X(t),·), t∈\nT}taking values in HB. We call this set HRB. We can form\ncollections of elements from HRBby selecting different timeinstances. The inclusion of this set of time indices allow for\nthe quantification of the time structure of the projection of a\ntime series {Xt}intoHRB. In this collection of elements,\nwe can define functionals based on both time and also across\nthe dimensions of HRB(i.e. space), which would require\nprojections defined based on samples using kernel trick.\nIn order to explain the added flexibility, we give an exam-\nple here. FIR time filters operate with a finite window of sig-\nnal samples of X(t). Letxt= [X(t), . . . , X (t−(L−1))]T\nbe a window of size L. This window projected into HRBhas\nthe following form:\nϕ(xt) =\n| . . . |\n| . . . |\nϕ(X(t)). . . ϕ (X(t−(L−1)))\n| . . . |\n| . . . |\n\nD×L(14)\nThe resulting collection of random elements is a rank 2 tensor\ninHRBcontaining elements indexed by time instances tand\nspatial indices of D. Note that many calculations involing\nHRBcan be simplified by vectorizing the tensor given in (14).\nThis vectorization is done by stacking each column on top of\none another to form a vector in RD·L. We will refer to this\nvectorized version as ϕ(xt).\nIn most practical cases the user will define both the win-\ndow size ( L) and the number of dimensions ( D) ofHB, which\nin turn defines the dimensions of elements in HRB. Note that\nsince we use an explicit mapping function, we have the abil-\nity to operate with specific time and space coordinates of ele-\nments in HRB. When computing the kernel trick in KAF, we\nonly have access to the columns of the matrix, which for the\nGaussian kernel are infinite dimensional.\nFor example, KLMS [19], which is a nonlinear filter, op-\nerates with weighted sums of G(xt,xi)where xtis the L-tap\nvectors including the current input and xiare training samples\n(also vectors of size L). The correntropy function, defined as\nV(t, s) =Et,s[G(X(t), X(s))]can also be computed di-\nrectly from samples. It is shown in [18] that Vis a similarity\nmetric that contains all the even moments of the joint proba-\nbility density function pt,s(xt, xs), which means that autocor-\nrentropy is a measure of statistical dependence [27]. Both ex-\namples show the power of RKHS theory, but they don’t make\nexplicit use of the eigenstructure of the RKHS. For instance,\nsuppose that we want to estimate the similarity between the\nfirst and second dimensions for a given coordinate system in\nthe RKHS. These types of functionals are cumbersome to deal\nwith using the kernel trick as each of these dimensions would\nhave to be expressed as the weighted sum of kernel functions.\nThis would quickly become computationally infeasible. In\nthe Appendix we explain why this type of functional is needed\nto derive closed-form MMSE solutions.3.2. Correntropy in HRB\nIn this section we will define an extension to the autocorrela-\ntion function R(s, t)in the space of random elements HRB.\nWe will start by defining a tensor based covariance function,\nUd1,d2(t, s)working on pairs of time and spatial dimensions,\nas follows:\nUd1,d2(t, s) =E(t,s)[ϕd1(Xt)ϕd2(Xs)] (15)\nThis function selects the dimensions d1andd2of the explicit\nmappings to the RKHS of the time series at times tandsand\nthen computes their correlation. Equation (15) describes how\nto calculate one element in the Utensor. Looping through all\nspace indices will yield the tensor U(t, s). We will refer to\n(15) as the moment-wise correntropy function, because it is a\ndecomposition of the correntropy function described in [18].\nNote that if the time series is strictly stationary, then we can\ncombine the pair of time indices into a single index, τ=t−s\nand rewrite Uas\nUd1,d2(τ) =Et,t−τ[ϕd1(X(t))ϕd2(X(t−τ))] (16)\nIf the process is also ergodic, we can estimate Udirectly from\ntraining samples (realizations of {Xt}) as\nˆUd1,d2(τ) =1\nNNX\nt=1ϕd1(X(t))ϕd2(X(t−τ)) (17)\nwhere Nis the number of training samples. If we fix Land\nDthen we can represent Uas aD×L×D×Lshaped ten-\nsor where (U)d1,t,d2,t−τ= (U)d2,t+τ,d1,t. While this tensor\nview of the moment-wise correntropy in HRBmakes clear the\nidea of measuring correntropy across both time and space, we\ncan create a rank two tensor representation of this function\nwhich is easier to work with both in terms of mathematical\nnotation and practical computation. This rank two tensor(or\nmatrix) is just the unfolded version of the Utensor which is\na(D·L)×(D·L)matrix. This unfolding is a mode-2 un-\nfolding of the the tensor Uas described in [25], and will be\ndenoted as UM. Note that if D→ ∞ , then the trace of UM\nis exactly equal to the correntropy function introduced in [18]\nand utilized in [2] and [41].\nTheorem 1. The moment-wise correntropy function of ran-\ndom vectors is positive semi-definite (PSD).\nProof. Consider a random vector, x, whose realizations take\nvalues in RL. Letϕ(x), be another random vector in RD·L\nwhich is the result of applying the vectorized version of the\nexplict mapping function given in (14). Then its covariance\nfunction, UM, is defined as UM=E[ϕ(x)ϕ(x)T]. Let zbe\nan arbitrary real valued random vector with dimension D·L.If we show that zTUMz≥0thenUMis PSD.\nzTUMz=zTE[ϕ(x)ϕ(x)T]z\n=E[zTϕ(x)ϕ(x)Tz]\n=E[(zTϕ(x))(zTϕ(x))T](18)\nSince the inner product Y=zTϕ(x)is a real scalar valued\nr.v., then\nE[(zTϕ(x))(zTϕ(x))T] =E[Y2]≥0. (19)\nTheorem 2. If a matrix UMis PSD so is its pseudo-inverse1\nProof. LetUMbe a PSD matrix. Then it has an eigende-\ncomposition UM=QTΛQ. Where Qcontains the orthonor-\nmal eigenvectors of U, and Λis a diagonal matrix with non-\nnegative real eigenvalues. Then, its pseudo-inverse can be\nobtained as follows\nU†\nM=QΛ†QT. (20)\nWhere the entries in the diagonal of Λ†are the reciprocals\nof non-negative real numbers defined as 1/xforx >0and\n0forx= 0, and are therefore also non-negative real num-\nbers. Then U†\nMhas only non-negative real eigenvalues and is\ntherefore also PSD.\nTheorem 1 proves that Uis PSD and therefore it can be\nused to define a RKHS. If we consider the vectorized version\nof the explicit mapping function, ϕ(xt), then the unfolded\nversion of Uis the moment-wise correntropy function of real\nvalued random vectors.\nFurthermore, we can define a new RKHS, HUwith the\nfollowing kernel,\nKU(x,y) =ϕ(x)UMϕ(y). (21)\nwhere x,y∈RLandϕ(x),ϕ(y)∈RD·Lare the vector\nrepresentations of functions in HRB.\nKUcontains information about the correlations of a time\nseries{Xt, t∈T}projected into HRB, and therefore defines\na Hilbert space with a data-dependent inner product. How-\never, its important to note that these correlations are functions\nnot just of {Xt}, but also of the eigenfunctions of HB. This\nis more than a measure of second order statistics (correlation)\nbecause it contains the even moments of the probability den-\nsity function of the input data (up to order D) [18].\nRather than defining KUby directly estimating its eigen-\ndecomposition, we describe the kernel as an operator over\nHRBwhere the eigenfunctions of HRBare already explicitly\nknown. Since the kernel defining HBis a truncation of the\nGaussian kernel, we can only claim universality in the limit.\nHowever, we show experimentally that for many applications\n1We mean the Moore-Penrose pseudo-inverse.this truncation will still yield very good results in terms of\nMSE.\nTo elaborate on how we access the space HU, in many\nworks such as [31] and [28] an integral operator is estimated\nby finding the leading eigenfunctions and values. Here we do\nnot take this approach. Instead, we use the fact that all separa-\nble finite dimensional Hilbert spaces defined on the same do-\nmain with the same number of dimensions are isomorphic to\nRM[14], where Mis the dimension of the space. Therefore,\nthe two Hilbert spaces, HRBandHUare isomorphic. We find\nthe isomorphism between these two spaces ( U1\n2) and then use\nthis operator to estimate the kernel function KUwithout ever\nexplicitly defining the eigendecomposition of KU. This type\nof argument can be extended to infinite dimensional spaces by\nsaying that the different separable Hilbert spaces with infinite\ndimension are isomorphic to ℓ2the space infinite sequences\nwith finite energy [14]. However, the computation of an op-\nerator like U1\n2in an infinite dimensional space is of course\nimpractical. The ability to explicitly define and apply opera-\ntors is an important strength of finite dimensional RKHSs.\nFig. 1 .Depiction of the relationship between HUandHRB. Note that\nwe never explicitly find the eigenbasis {ψm}M\nm=1, but we are able to access\nHUthrough the kernel function U. M=D·L.\n4. THE WIENER SOLUTION IN HU\nThe MMSE for the linear case is given in (4), where the pro-\njection of a desired signal into the linear span of an input sig-\nnal is found. In the nonlinear case we would like to project a\ndesired signal into the span of the composition of a time series\nand the feature map given by the kernel of HB,\nE[E∗[Z|ϕ(Xt, t∈T)]ϕ(Xs)] =E[Zϕ(Xs)]. (22)\nParzen succinctly states that if one can find a representa-\ntion of ρz(s)in terms of linear operations on the covariance\nkernel, then the MMSE linear predictor, E∗[Z|Xt, t∈T],\ncan be written in terms of the corresponding linear operations\non the time series. With (22) in mind, we can use this idea\nto extend the Wiener solution to HRB, which is nonlinearly\nrelated the input space.\nFirst, we define the cross-covariance function between a\nrandom process projected into HRBand some r.v. Zas\nρd(t) =Et[Zϕd(Xt)]. (23)where d= 1, . . . , D . In tasks such as time series pre-\ndiction if we assume strict stationarity and ergodicity then ρ\nbecomes a function of only lag ( τ) and dimension ( d).\nρd(τ) =Et−τ[Zϕd(Xt−τ)] (24)\nNotice that if τ= 0, . . . , L −1then looping through all val-\nues of τwill result in a rank two tensor (ρ)d,τ, which is a\nD×Ltensor containing a scalar weight corresponding to\neach eigenfunction of HRB. In other words it is a function\ninHRBwith a similar definition to equation 14.\nPractically if we assume Ztakes a scalar value for every\nt∈T, thenρcan be estimated from training data samples as\nˆρ=1\nTTX\nt=1Z(t)ϕ(xt) (25)\nNote that (25) is given in terms of the vector representa-\ntion of functions in HRB, but ˆρcan be unfolded to recover\nthe tensor (ρ)d,τ. Following Parzen’s approach and by anal-\nogy with (10), the MMSE solution in the data space is\nˆY(t) =⟨ϕ(xt),ρ⟩HU\n=D·LX\nn=1D·LX\nm=1ϕn(xt)U−1\nM(n, m)ρm(26)\nwhere n, m= 1, . . . ,(D·L)loop through all combinations of\ntime and space indices. Equation (26) is the FWF analytic so-\nlution in the data space and an analog to linear solution given\nin (10). Notice that it is constant complexity, unlike other\nkernel-based solutions whose complexity increases with re-\nspect to training set size. The FWF solution is linear in HRB,\nbut nonlinear in the input space. The main difference between\nthis nonlinear solution and its linear counterpart in HRis the\naddition of an extra spatial dimension gained when we substi-\ntute the input data X(t)withϕ(X(t)).\nWe can show the ramifications of adding this spatial di-\nmension to the Wiener equation given in (7). In order to do\nthis we would like to create an analog to equation (7) which\ndescribes the problem we implicitly solve when we compute\nequation (26).\nIts important to remember that while ϕ(xt)andρare ma-\ntrices of size D×L, they represent spatiotemporal functions.\nFor example, consider some real-valued rank 2 tensor Vof\nsizeD×L. Let Ωbe the domain of the eigenfunctions of\nHB. Then in HRBVrepresents the following function.\nv(t, x) =DX\nd=1(V)d,tϕd(x) (27)\nwhere x∈Ω,Vd,tpoints to a single scalar valued element of\nthe tensor V, and t= 0, . . . , L −1assuming discrete time.\nRemember that when we take the inner product in HRBweare really taking the sum of Lintegrals. For example, con-\nsider two real-valued D×Ltensors VandV′. These rep-\nresent two functions in HRB. Then the following statements\nare equivalent.\n⟨V, V′⟩HRB=L−1X\nt=0DX\nd=1(V)d,t(V′)d,t\n=L−1X\nt=0Z\nΩv(t, x)v′(t, x)dx(28)\nwhere v(t, x)andv′(t, x)are the spatiotemporal functions\ndefined over the domain T×Ωrepresented by the tensors\nVandV′.\nNow, we will describe the set of integral operators whose\nmatrix representations are stored in the Utensor. To simplify\nthe description we will assume that some fixed LandDhave\nbeen chosen. Let s, t= 0, . . . L −1andd1, d2= 1, . . . , D .\nThen let U={(U):,:,s,t, s, t = 0, . . . , L −1}, where the\ncolons U:,:,s,tdenote taking all possible values for d1andd2.\nEach element in Uis aD×Dmatrix and is the matrix rep-\nresentation of an operator which acts on elements in HB. We\ncan rewrite each element in Uas an integral operator.\nLst\nU(x, y) =DX\nd1=1DX\nd2=1(U)d1,d2,s,tϕd1(x)ϕd2(y). (29)\nwhere x, y∈Ωand(U)d1,d2,s,tis a scalar valued element of\nthe tensor (U):,:,s,t. Equation 29 gives the integral operator\nfor one element in U, so we can store all of these integral\noperators in another set, LU={Lst\nU, s, t= 0, . . . L −1}.\nThenUandLUare two sets holding different representations\nof equivalent operators.\nThen the equation that is solved implicitly in (26) is\nL−1X\nt=0\u0014Z\nΩw(t, x)Lst\nU(x, y)dx\u0015\n=ρ(s, y) (30)\nwhere w(t, x)andρ(s, y)are two spatiotemporal equa-\ntions defined over the domain T×Ω. When we take the in-\nverse of UMand multiply this by the vector representation\nofρ(s, y)we solve for w∗which is the vector representation\nof the spatiotemporal function, w∗(t, x), which solves (30).\nEquation (30) is a direct discrete time analog to equation (7).\nFor this reason we call it the functional Wiener equation.\nThe differences between equations (7) and (30) appear be-\ncause we are no longer working with scalar valued functions\nof time, X(t). Rather, we are working with collections of\nfunctions indexed by time, ϕ(xt). In the linear case we can\nthink of R(s, t)as a set of scalars indexed by two time indices.\nThis is sufficient in the linear case, as w(t)is a scalar valued\nfunction of time. In the case of the FWF, the ”weights” wofthe system are spatiotemporal functions i.e., for every time t,\nw(t, x)is a function over Ωnot a scalar. Therefore, in order\nto operate on this function, we need an integral operator (not\na single scalar). This is why a second integral is needed over\nΩwhich describes the application of an integral operator, Lst\nU,\nto the function w(t, x).\nThe extension of equation (30) for continuous time can be\nin principle written as\nZb\na\u0014Z\nΩw(t, x)Lst\nU(x, y)dx\u0015\n=ρ(s, y) (31)\nwhere a≤s≤b. However, the set LUwill become uncount-\nable.\n4.1. Calculation of the Function Wiener Filter\nEquation (26) shows clearly the extension of Parzen’s solution\nto a space non-linearly related to the input space. However,\nthis form of the solution is not as computationally efficient as\nit could be. In this section we describe a computationally ef-\nficient way to calculate the MMSE solution described above.\nWe are given some finite number of input signal samples,\nX(i), and desired signal Z(i)with the goal of mapping win-\ndows of X(i)to its corresponding Z(i). Let, X={xi, i∈\nI}be the set of all windows of time in X(i)of length L.|I|\nis the number of training windows in X(i). Each window of\ntime in this data set can be projected into HRBusing (14)\nyielding a D×Ltensor for each window in time. Then we\nvectorize these tensors by stacking the columns of this tensor\non top of one another to form a vector of length D·Lfor each\nwindow of time in the training set. Finally, we end up with a\ndata matrix, X={ϕ(xi), i∈I}, which is a (D·L)×Imatrix\nwhere each column represents the projection of a window in\ntime into HRB. Letzidenote the desired value corresponding\nto the input vector xi. We can store all of these desired values\nin a vector z(I×1). Note that we may use the Moore-Penrose\npseudo inverse in the case that UMis ill-conditioned.\nThen we can calculate the matrix form of U,\nˆUM=XTX\n|I|. (32)\nThe estimation of the cross-correlation vector is\nˆρ=Xz\n|I|. (33)\nWhere ˆρis just the weighted sum of the columns of X. Then\nwe can calculate the projection of Zintoϕ({Xi, i∈I})as\nw∗=U−1\nMˆρ (34)\nWhere w∗is a vector of size (D·L)which represents the\noptimal solution in HRB. Lastly, in order to make predictions\nwith this solution we just take the inner product between w∗andϕ(xt)inHRB. Since we are using an explicit space this\nis just the inner product between two vectors.\nˆY(t) =⟨ϕ(xt),w∗⟩HRB=⟨ϕ(xt),ˆρ⟩HU. (35)\n4.2. Theoretical Error Analysis\nWe claim that the FWF is a closed-form optimal nonlinear\nfilter in an approximate universal space, so where does error\nfrom the FWF come from? First, it is important to remember\nthat we can only calculate the FWF solution in a finite dimen-\nsional RKHS, so the space HRBand therefore HUis only\nuniversal as Dapproaches infinity. This deficiency of HUis\na source of error. In [5] error bounds for the approximation\nmade by HBto the universal Gaussian kernel is given. Sec-\nond, we still need to select the Gaussian kernel size, which in\nFWF only affects the precision of the results for finite number\nof samples. Third, since we are quantifying the joint distribu-\ntion between pairs of samples projected into an RKHS and\ntaking their mean, implicitly we are assuming that the system\nwe are modeling is strictly stationary. Any deviation from\nthis strictly stationary assumption can cause error. Fourth, of\ncourse there can be numerical error associated with condition-\ning of the UMmatrix.\nFig. 2 .|Training - Theoretical |MSE in log scale (Top),Theoretical\nMSE (left), and Test MSE(right) as a function of DandLfor prediction\nof Mackey-Glass time series. Note the change in scale in the top plot. Kernel\nsize is 0.25\nIn [22] a theoretical MMSE for the solution in HRis given\nwith the equation\nE[|Z−E[Z|Xt, t∈T]|2] =E[|Z|2]− ⟨ρz, ρz⟩HR.(36)\nWe can extend this theoretical MMSE to HUby measur-\ning the distance between the projection of ZintoHUandZ\nin a similar manner to equation (36)E[|Z−E[Z|HB(Xt, t∈T)]|2] =E[|Z|2]− ⟨ρ,ρ⟩HU.(37)\nThis equation gives the MMSE for the solution in the fi-\nnite dimensional RKHS HUeven before we start make pre-\ndictions with the solution. Since the projection of Z in HUis\northogonal we just need to estimate the difference between the\npower of the desired response and its projection in HU. This\nis markedly different from KAF and other filtering methods,\nwhere we have to evaluate the error in the training, which is\nalways complex and is a function of the dynamics of learning.\nTherefore, discounting numerical error we can say that all the\nsources of error mentioned above are combined in the inner\nproduct calculation of the space HU. This means improving\nthe error is an issue of designing correctly HUand therefore\nHRB. As an example, since the kernel size is a continuous\nvariable, (37) can be used to estimate the optimal kernel size\nfor the application under study.\nDue to the analytical solution, the MSE on the training\nset should be very close to the theoretical MMSE for a given\nchoice of the hyper-parameters: kernel size, L, and dimension\nD. Figure 2 shows the theoretical MMSE, the absolute dif-\nference between theoretical and calculated MSE in the train-\ning set, and the test set MSE as a function of parameters for\nprediction on the nonlinear chaotic Mackey-Glass time series\nwith a kernel size of 0.25. As can be expected the larger the\nnumber of lags and the dimension of the explicit RKHS, the\nsmaller is the error. This figure also shows that the training set\nMSE follows the theoretical error almost exactly for a large\nrange of hyper parameter values. Figure 2 also shows the test\nset MSE as function of DandL. We can observe that the test\nset performance for large delays and number of eigenfunc-\ntions flattens, which is explained by overtraining. Since the\nFWF is a linear model, techniques for model order estimation\n[33] can be applied to improve generalization, but we leave\nthis to future work.\n5. EXPERIMENTS\nThroughout these experiments we will compare the FWF so-\nlution derived above in two important applications: nonlinear\nmapping of one time series into another (as required in system\nidentification) and in nonlinear time series prediction. The\nmodels used for comparison are Gaussian Process Regression\n(GPR) [37], Kernel Least-Mean Squares (KLMS) [19], Ex-\ntended Kernel Recursive Least-Squares (KRLS) [20], Kernel\nRidge Regression (KRR) [9], and Augmented Space Linear\nModel (ASLM) [24], which is a linear model in the joint space\nof the data and desired response, improving slightly the per-\nformance of the Wiener solution.Fig. 3 .Depiction of the FWF in finite dimensional RKHS(top left), Visualization of the solution learned by the FWF on the strictly stationary task(bottom\nleft and right).\n5.1. Strictly Stationary System and Function Recovery\nIn this section we will apply the methods to a signal that we\nknow follows the strictly stationary assumptions made by the\nFWF. we will also probe the function learned by the FWF giv-\ning insight into how the FWF can learn functions that create\nthe input data, which is very important for applications in sys-\ntem identification. Finally, we will compare the performance\nof the FWF with the other methods on this task.\nFirst, we discuss the system used in this experiment. We\nfirst generate white Gaussian noise, X(n), constructed by\nrandomly sampling the distribution, N(0, π). We will use\nthis white noise as the input to the models. The model output\nD(n)is created by applying a nonlinear mapping to X(n),\nD(n) = 0.5tanh(X(n))2+ sin(X(n −1))3+ 0.5tanh(X(n −2))3\n+ 0.2sin(X(n −3))2+ 0.75tanh(X(n −4))2.\n(38)\nIts interesting to note that since HRBmaintains both time\nand space indices we can examine more closely what func-\ntions the FWF actually learns and recover a difference equa-\ntion describing the function learned by the FWF. In linear fil-\ntering, one can interpret the output of a linear system as ap-\nplying a series of transformations (in this case simple scalarmultiplications) to individual samples and then summing up\nthe result of these transformations. The difference in the FWF\nis that these transformations are no longer restricted to scalar\nmultiplications. Since the FWF works with collections of\nfunctions, rather than collections of scalars, the transforma-\ntions applied to each sample can be any transformation which\ncan be described as a linear functional in HB.\nIn order to make this more concrete consider a window of\ntime mapped to HRB,ϕ(xn). Each column of ϕ(xn)corre-\nsponds to the mapping of a scalar to a function in HB. We\ncan create a tensor representation of the vector w∗by folding\nthe vector into a D×Lrank two tensor which we will denote\nasw∗\nd,τ. Note that d= 1, . . . , D andτ= 0, . . . , L −1.\nw∗\nd,τ=\nw∗\n1w∗\nD+1. . . w∗\n(L−1)D+1\n...... . . ....\nw∗\nDw∗\n2D. . . w∗\nLD\n\nD×L(39)\nwhere w∗\nnis the nthelement in the vector w∗. In this form\nits clear to see that each column of w∗\nd,τrepresents a transfor-\nmation to be applied to each sample in a time window. Now\nequipped with this representation the extension of the differ-\nence equation given by the FWF isˆY(n) =L−1X\nτ=0⟨w∗\n:,τ, ϕ(X(n−τ))⟩HB\n=⟨w∗,ϕ(xn)⟩HRB.(40)\nHere w∗\n:,τis a column of the tensor depicted in (39). Each\ninner product in the sum over the lags can represent any linear\nfunctional in HB.\nThis idea is depicted visually in the top left quadrant of\nfigure 3. The right half of figure 3 shows the Lfunctions (one\nfor each value of τ) found by the FWF with input signal X(n)\nand desired signal D(n). The green histograms show the PDF\nof the training set (scaled for visualization). We observe that\nin regions covered in the training set the functions learned by\nthe FWF are vertically shifted versions of the true functions\ngiven in the difference equation. However, when these shifted\nversions are summed together they converge to the true differ-\nence equation given in (38). Its also interesting that the FWF\noutputs a flat line for values of τwhich are larger than the true\nmemory depth of the system.\nWe can think of these functions as “modes” akin to the\nmodes found in Koopman theory [15]. The solution given by\nthe FWF is just a sum over these modes. This interpretation\nof the functions learned by the FWF provides the ability to\nextract modes learned by the FWF and gives added explain-\nability to the method.\nNow we will compare the performance of the FWF with\nthe methods mentioned above on this task. Each method was\ntested with 5 different training and testing windows at each\nvalue of N. The average test set MSE is shown in figure 4\nwith error bars depicting the variance of test set MSE across\nthe different training and testing windows. For each method a\nsearch across a range of kernel sizes was performed, and the\nbest results for each method are shown. The FWF provides a\nclear performance boost over the other methods on this task.\nThe nonlinearity is not static, so the KAF have difficulties in\napproximating the system well. Nevertheless, this suggests\nthat if the strict stationary assumptions made by the FWF are\nmet, then its performance may be better than the other the\nnonlinear filtering methods. Moreover, we get immediately\ninformation of the time functions learned by the FWF at each\nlag. In later experiments we will see that if the signals do not\nfollow this assumption the FWF is still comparable in terms\nof performance to the other nonlinear methods.\n5.2. Lorenz Mapping\nThe Lorenz system is governed by a well known set of non-\nlinear equations introduced in [32]. In this experiment the\nmodels are given the Xcomponent of the Lorenz equations\nas input, and asked to predict the Zcomponent 5 samples in\nthe future. As in the previous experiment 5 different training\nand testing windows for each value of Nwere used. A grid\n100 500 1000 1500 2000\nTraining Samples (N)103\n102\n101\n100T est MSE\nStationary Signal: T est MSE, L = 5\nFWF, ks=1.5\nKLMS, ks=2.0\nKRLS, ks=3.0\nGPR\nKRR, ks=3.0\nASLM\nWFFig. 4 .Comparison of test set MSE as a function of number of training\nsamples on the strictly stationary task. L= 5,D= 30 .\nsearch for kernel size was done for each kernel method and\nthe best kernel size was selected. Figure 6 shows the result\nat the kernel size with the best MSE. The number of lags for\neach method was 20. The number of dimensions in HBwas\n50.\nWe see that FWF performance is comparable to that of\nKRLS and GPR, and better than KLMS, KRR and ASLM, de-\nspite the use of a finite dimensional RKHS and a strict station-\nary assumption, which is not fulfilled in these experiments.\nMoreover, at low numbers of samples the FWF tends to do\nbetter than KRLS or GPR, showing it is quite data efficient.\nIn figure 5 the average train and test MSE across 5-folds\nfrom the FWF is shown as a function of number of dimen-\nsions Dof the RKHS. We see that beyond 20 the number of\ndimensions has little effect on the performance of the FWF.\nThis could be expected since the eigenvalue spectrum of the\nGaussian kernel falls off quite fast. Interestingly, the test MSE\ndoes not increase with the dimension of the space, which can\nbe related by the orthogonality of the bases. This figure also\nshows that the model overfits because the test set performance\nis slightly higher than the training performance (the scale is\nlogarithmic).Fig. 5 .Train and Test MSE as a function of the number of dimensions (D)\ninHU. Number of lags is 20, kernel size is 0.75, and number of training\nsamples is 2000. Note the log scale for the MSE.\n100 500 1000 1500 2000\nTraining Samples (N)104\n103\n102\n101\n100T est MSE\nLorenz: T est MSE, L = 20\nFWF, ks=0.75\nKLMS, ks=0.75\nKRLS, ks=0.75\nGPR\nKRR, ks=1.0\nASLM\nFig. 6 .Comparison of test set MSE as a function of number of training\nsamples on Lorenz.\n5.3. Mackey-Glass Prediction\nIn this experiment the models are tasked with prediction of\nthe Mackey-Glass time series. Again, 5 different training and\ntesting windows for each value of N were used. A grid search\nfor kernel size was done for each kernel method, and the best\nresults are shown in figure 7. The number of lags for each\nmethod was 20. The dimension of HBwas 50. Again we\nsee for low number of samples the FWF is superior, however\ngiven enough training samples the FWF performs worse than\nKRLS and GPR. This may be related to the low amplitude\nripple in the data that is continuously changing and that rep-\nresents a non-stationarity.\n100 500 1000 1500 2000\nTraining Samples (N)104\n103\n102\nT est MSE\nMackeyGlass: T est MSE, L = 20\nFWF, ks=0.25\nKLMS, ks=0.5\nKRLS, ks=0.5\nGPR\nKRR, ks=0.5\nASLMFig. 7 .Comparison of test set MSE as a function of Number of training\nsamples for prediction of Mackey-Glass.\n5.4. Noisy Mackey-Glass Prediction\nIn this section we test the same methods as before on the task\nof prediction of the Mackey-Glass time series with a noisy\ninput signal. We add noise sampled from Gaussian distribu-\ntions with different standard deviations (0,0.01,0.05,0.1,0.2).\nAgain 5-fold cross validation is used with a test signal length\nof 300 examples. Figure 8 shows the comparative perfor-\nmance of these methods as the level of noise is increased.\nThe dimension of HBwas 30. We observe that, as expected,\nperformance degrades with higher noise levels, and improves\nacross all models with larger training sets. The FWF seems\nto be more sensitive to noise for smaller training sets, but fol-\nlows closely the KRLS and GPR for larger number of training\nsamples.\n0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200\nNoise Std104\n103\n102\n101\nT est MSE\nT est MSE vs Noise Std, L=15, N=500, D=30\nFWF, ks=0.25\nKLMS, ks=0.25\nKRLS, ks=0.5\nGPR\nASLM\nKRR, ks=0.5\n0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200\nNoise Std105\n104\n103\n102\nT est MSE\nT est MSE vs Noise Std, L=15, N=2000, D=30\nFWF, ks=0.25\nKLMS, ks=0.25\nKRLS, ks=0.5\nGPR\nASLM\nKRR, ks=0.5\nFig. 8 .Comparison of test set MSE as a function of standard deviation\nof noise for prediction of Mackey-Glass with 500 train examples (Top) and\n2000 training examples (bottom) .5.5. Sun Spot Prediction\nIn this section we test the same methods, but on the sun spot\ndata set. The goal in this task is to use the number of sun\nspots in the previous Ldays, to predict the number of sun\nspots which will occur on the next day. The data set was\nnormalized to norm 1 for all the methods. We show the MSE\nusing again 5-fold cross validation with a test signal size of\n300. In this case the FWF starts with an higher error than the\nothers for 250 samples, but converged to the performance of\nthe best methods after 500 samples.\n100 500 1000 1500 2000\nTraining Samples (N)0.20.40.60.81.01.2T est MSE\nSun Spot: T est MSE, L = 20\nFWF, ks=3.0\nKLMS, ks=2.0\nKRLS, ks=3.0\nGPR\nKRR, ks=3.0\nASLM\nFig. 9 .Comparison of test set MSE as a function of of number of training\nsamples for prediction of Sun Spot. Dimension of HBis 50.\nMethod Training Evaluation\nFWF O(NDL ) +O((DL)2N) O(2DL)\nKLMS O(N) O(N)\nKRLS O(N2) O(N)\nASLM O(L2N) +O(LN) O(L) +O(log(N))\nKRR O(N3) O(N3)\nGPR O(N3) O(N3)\nTable 1 .Computational Complexity Comparison: D (dimension of HB),\nN (number of training samples), L(number of lags)\n6. DISCUSSION\nIn summary, this work extended the idea of the Wiener filter\nto a space nonlinearly related to the input space, yielding a\nclosed-form solution to the MMSE problem. In order to find\nthis MMSE solution, we must solve the system of equations\nresulting from the functional Wiener equation given in (30).\nThe solution utilizes a formulation of the Wiener solution in\na linear RKHS described in [22]. The extension is done in\nan explicit finite dimensional RKHS, HRB, which is nonlin-\nearly related to the data space and utilizes a data dependent\nkernel in HU. The solution, to the functional Wiener Filter is\ngiven in (35), however we have to assume ergodicity in ourapproach. The first conclusion is that the FWF can be inter-\npreted as an FIR, but nonlinear transformations may be ap-\nplied to each sample. We can extract the transformations ap-\nplied by the FWF to each sample giving added explainablility\nto the model. We believe that further research is needed to test\nthe limitations of the FWF for system identification tasks. In\nexperiments, it is shown that the performance of the FWF is\ncomparable to that of other nonlinear regression methods such\nas KRLS and GPR, with gains in performance for small num-\nber of training samples and on a synthetic data set which we\nknow follows the assumptions made by the FWF. We summa-\nrize in Table I the computational complexity of all the meth-\nods employed in the tests. As it is quite clear the computation\ncomplexity of the FWF is constant and much smaller than any\nof the other approaches. Effectively, it is of the same com-\nplexity as the Wiener solution for a filter order of D·Las can\nbe expected. This simplifies tremendously the application of\nKAF for online applications for edge computing.\nAlthough the FWF belongs to the KAF family, its prin-\nciples are totally different and represent a fresh approach to\nexploit the linear structure of the Hilbert space for compu-\ntation. The linear model is very well understood in signal\nprocessing and control theory, but it can not compete with the\nversatility of nonlinear models, as the neural network revolu-\ntion clearly demonstrated. The price paid by nonlinear model-\ning is that adaptation of parameters is non-convex, and full of\nlocal minima which brings brittleness in application. RKHS\ntheory allows for universal approximations with convex opti-\nmal solutions named CULMS (convex universal learning ma-\nchines)[23]. However, a weak link in the KAF approach is\nthe linear (or worse) growth in the function model in RKHS,\nwhich can be traced to the definition of the RKHS domain\n(pairs of data samples), and the error associated with iterative\ngradient-based solutions. The FWF defines a kernel that pre-\nserves the time structure of the random process (or multivari-\nate random variable) i.e., the inner product becomes now data\ndependent. We submit that this idea of designing the kernel\nwith the statistics of the data is the best way for kernel design,\nan open problem in KAF.\nHow to operate effectively with data dependent kernels\nhas been largely absent in the literature, only specific applica-\ntions have been published for kernel regression [30] and for\ntheoretical analysis [11]. This is a true advance in this paper,\nthe development of a theory to extend nonlinear modeling in\nRKHS using a fixed number of parameters. We started with\nthe MMSE problem that is crucial in statistics and time series\nanalysis. Our solution is still limited to finite dimensional\nkernels, which we call explicit kernels because they truncate\nthe eigenspectrum of universal kernels such as the Gaussian\nor the Laplacian kernels. In these spaces the kernel opera-\ntors can be defined as quadratic forms, which allows much\nmore flexibility than the kernel trick, because we can oper-\nate with individual dimensions of the RKHS. We claim that\nthe MMSE solution is possible because we can estimate di-rectly from data the correntropy along each dimension of the\nspace(i.e moment-wise correntropy). Our previous attempt of\nutilizing the correntropy RKHS for MMSE, also a data de-\npendent kernel, failed because the kernel trick is only capable\nof operating on pairs of columns of the tensor operator. We\nhad to employ clustering on the training set samples to cre-\nate models that would approximate the optimal solution [4],\nwhich did not generalized well.\nThe FWF goes beyond this “tweak” and develops a prin-\ncipled approach to MMSE in an explicit kernel RKHS. An an-\nalytic solution was developed and although not universal, the\nperformance is on par with the state of the art KAF filters with\nmuch less computation, and added explainability. We expect\nthat this approach will be quite fruitful for other stochastic\nsignal processing applications. We summarize below the dif-\nferences between the traditional KAF approach and the new\ndata dependent approach proposed in the paper.\nFirst, the FWF is not a search based solution, it is a ana-\nlytical solution. Second, the error in the FWF comes from its\nassumption of stationarity and the truncation of the Gaussian\nkernel, not from the search method. Third, the KAF solu-\ntion utilizes the kernel trick and therefore can be calculated\nin an infinite dimensional space. While this is a theoretical\nadvantage, it produces functional models that increase with\nthe training data. The use of the Uoperator in this deriva-\ntion of the FWF solution prevents a direct generalization to\nan infinite dimensional RKHS, but also fixes the model size\nregardless of the number of training examples used.\nThere are other contributions of this work beyond the\nclosed-form solution. We develop a data-dependent RKHS in\nHUwhich contains information about the correlation struc-\nture intrinsic to the data used to construct KU. While we do\nnot give an eigendecomposition of this kernel directly, we\ndo describe how to take inner products in HU. Since this\nRKHS is data-dependent the “rules” described as statistical\naverages with respect to different dimensions of the input\nspace are contained within the eigenstructure of the space.\nThis in theory could lead to very computationally efficient\nkernels tuned to deal with data from a specific distribution.\nMore research needs to be conducted to see if this is indeed\nthe case. Other uses for data-dependent kernels have been\ndescribed, for example, [10] uses a cross-density kernel to\nquantify dependence between random processes. There may\nbe different RKHSs like HUwhich need to be explored.\nAppendices\nA. WHY DO WE NEED MOMENT-WISE\nCORRENTROPY?\nIn this work we have proposed a method to solve the Func-\ntional Wiener equation in (30) in a finite dimensional RKHS.We do this by extending the idea of a covariance function\nR(s, t), to a moment-wise correntropy function Ud1,d2(s, t).\nHowever, if one is familiar with the KAF literature this is not\nthe most obvious extension of R(s, t). The most obvious ex-\ntension to a nonlinear space would be the correntropy function\ndescribed in [18]. This correntropy function, V(s, t), paired\nwith the Gaussian kernel, Gσ, is defined as\nV(s, t) =E[Gσ(Xs, Xt)]\nGσ(x, y) =e(−||x−y||2\n2\n2σ2).(41)\nIn [18] it is shown that this function is PSD and therefore\ndefines a data-dependent RKHS which we will call HV. This\ndefinition of correntropy is the expected value between kernel\nevaluations centered at a time series indexed with two differ-\nent time indices. This means Vis a measure of the average\ncorrelation across all spatial dimensions of Gσand does not\ncontain the inter-dimensional correlations that the Ufunction\nin equation (15) describes. An attempt to use the correntropy\nfunction in (41) in order to define a data-dependent RKHS in\nwhich to solve the Wiener equation is made in [4] with lim-\nited success. The question is why is Vinsufficient to give an\nanalytical solution to a nonlinear filter?\nSeparable Hilbert spaces can be isomorphic only if they\nhave the same number of dimensions. A key crux of the argu-\nment used by Parzen is that the linear span of a time series in\nL2is congruent to HR, suggesting an isomorphism between\nthese two spaces exists. This congruence allows one to solve\nthe MMSE solution in HRwith the guarantee that this solu-\ntion is equivalent to the optimal solution in the linear span\nof a time series in L2.HVis a data-dependent kernel and\nmeasures correlations between a time series at different time\nindices, but it forms an Ldimensional RKHS, therefore it can\nnot be congruent to HG(the gaussian RKHS). There cannot\nbe a one-to-one inner product preserving mapping between\nRKHSs with different number of dimensions. Therefore, we\ncannot hope to give an analytical solution in HVequivalent\nto picking some functional in HGwhich is the representation\nof the desired mapping function. In fact, many different time\nseries are equivalent in terms of V, because Vaverages over\nall spatial dimensions. This means that permutations of the\nsame spectral coefficients will result in the Vfunction giving\nthe same scalar value.\nWhen we use HUwe form a RKHS with the same num-\nber of dimensions as HRBso we do not encounter this is-\nsue.HUandHRBshare the same congruent relationship that\nL2({Xt})andHRshare. The operator U1\n2defines an iso-\nmorphism between the two spaces. This is why Parzen’s ar-\nguments could be readily extended using HU, but are not so\neasily extended to HV.References\n[1] Nachman Aronszajn. “The theory of reproducing ker-\nnels and their applications”. In: Cambridge Philos.\nSoc. Proc. 39 (1943), pp. 133–153.\n[2] Liangjun Chen, Hua Qu, and Jihong Zhao. “Gen-\neralized correntropy induced loss function for deep\nlearning”. In: 2016 International Joint Conference on\nNeural Networks (IJCNN) . 2016, pp. 1428–1433. DOI:\n10.1109/IJCNN.2016.7727366 .\n[3] Benjamin Colburn, Luis G. Sanchez Giraldo, and\nJose C. Principe. 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MIT Press, 2004.\nURL:https : / / proceedings . neurips .\ncc / paper _ files / paper / 2004 / file /\n85353d3b2f39b9c9b5ee3576578c04b7-Paper.\npdf.\n[41] Songlin Zhao, BD Chen, and Jose Principe. “Kernel\nadaptive filtering with maximum correntropy crite-\nrion”. In: Proceedings of the International Joint Con-\nference on Neural Networks (July 2011), pp. 2012–\n2017. DOI:10.1109/IJCNN.2011.6033473 ."    },    {        "title": "2402.03504v1.The_JWST_Resolved_Stellar_Populations_Early_Release_Science_Program_V__DOLPHOT_Stellar_Photometry_for_NIRCam_and_NIRISS.pdf",        "content": "DRAFT VERSION FEBRUARY 7, 2024\nTypeset using L ATEXtwocolumn style in AASTeX631\nThe JWST Resolved Stellar Populations Early Release Science Program V .\nDOLPHOT Stellar Photometry for NIRCam and NIRISS\nDANIEL R. W EISZ ,1ANDREW E. D OLPHIN ,2, 3ALESSANDRO SAVINO ,1KRISTEN B. W. M CQUINN ,4MAXJ. B. N EWMAN ,4\nBENJAMIN F. W ILLIAMS ,5NITYA KALLIVAYALIL ,6JAYANDERSON ,7MARTHA L. B OYER ,7MATTEO CORRENTI ,8, 9\nMARLA C. G EHA,10KARIN M. S ANDSTROM ,11ANDREW A. C OLE,12JACK T. W ARFIELD ,6EVAN D. S KILLMAN ,13ROGER E. C OHEN ,4\nRACHAEL BEATON ,7, 14, 15ALESSANDRO BRESSAN ,16ALBERTO BOLATTO ,17, 18MICHAEL BOYLAN -KOLCHIN ,19\nALYSON M. B ROOKS ,4, 20JAMES S. B ULLOCK ,21CHARLIE CONROY ,22MICHAEL C. C OOPER ,21JULIANNE J. D ALCANTON ,5, 20\nAARON L. D OTTER ,23TOBIAS K. F RITZ ,24CHRISTOPHER T. G ARLING ,6MARIO GENNARO ,7, 25KAROLINE M. G ILBERT ,7, 25\nLEOGIRARDI ,26BENJAMIN D. J OHNSON ,22CLIFF JOHNSON ,27JASON KALIRAI ,28EVAN N. K IRBY ,29DUSTIN LANG ,30\nPAOLA MARIGO ,31HANNAH RICHSTEIN ,6EDWARD F. S CHLAFLY ,32ERIKJ. T OLLERUD ,7AND ANDREW WETZEL33\n1Department of Astronomy, University of California, Berkeley, CA 94720, USA\n2Raytheon, 1151 E. Hermans Rd., Tucson, AZ 85756\n3Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85719, USA\n4Department of Physics and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA\n5Department of Astronomy, University of Washington, Box 351580, U.W., Seattle, WA 98195-1580, USA\n6Department of Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA 22904, USA\n7Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n8INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078, Monteporzio Catone, Rome, Italy\n9ASI-Space Science Data Center, Via del Politecnico, I-00133, Rome, Italy\n10Department of Astronomy, Yale University, New Haven, CT 06520, USA\n11Department of Astronomy & Astrophysics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA\n12School of Natural Sciences, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia\n13University of Minnesota, Minnesota Institute for Astrophysics, School of Physics and Astronomy, 116 Church Street, S.E., Minneapolis, MN 55455, USA\n14Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA\n15The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA\n16SISSA, Via Bonomea 265, 34136 Trieste, Italy\n17Department of Astronomy, University of Maryland, College Park, MD 20742, USA\n18Joint Space-Science Institute, University of Maryland, College Park, MD 20742, USA\n19Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712-1205, USA\n20Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\n21Department of Physics and Astronomy, University of California, Irvine, CA 92697 USA\n22Center for Astrophysics — Harvard & Smithsonian, Cambridge, MA, 02138, USA\n23Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA\n24Department of Astronomy, University of Virginia, Charlottesville, 530 McCormick Road, VA 22904-4325, USA\n25The William H. Miller IIIDepartment of Physics & Astronomy, Bloomberg Center for Physics and Astronomy, Johns Hopkins University, 3400 N. Charles\nStreet, Baltimore, MD 21218, USA\n26Padova Astronomical Observatory, Vicolo dell’Osservatorio 5, Padova, Italy\n27Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, 1800\nSherman Avenue, Evanston, IL 60201, USA\n28John Hopkins Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723, USA\n29Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA\n30Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada\n31Department of Physics and Astronomy G. Galilei, University of Padova, Vicolo dell’Osservatorio 3, I-35122, Padova, Italy\n32Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA\n33Department of Physics and Astronomy, University of California, Davis, CA 95616, USA\n(Accepted February 2, 2024)\nSubmitted to ApJS\nCorresponding author: Daniel R. Weisz\ndan.weisz@berkeley.eduarXiv:2402.03504v1  [astro-ph.GA]  5 Feb 20242 W EISZ ET AL .\nABSTRACT\nWe present NIRCam and NIRISS modules for DOLPHOT, a widely-used crowded field stellar photometry\npackage. We describe details of the modules including pixel masking, astrometric alignment, star finding, pho-\ntometry, catalog creation, and artificial star tests (ASTs). We tested these modules using NIRCam and NIRISS\nimages of M92 (a Milky Way globular cluster), Draco II (an ultra-faint dwarf galaxy), and WLM (a star-forming\ndwarf galaxy). DOLPHOT’s photometry is highly precise and the color-magnitude diagrams are deeper and have\nbetter definition than anticipated during original program design in 2017. The primary systematic uncertainties\nin DOLPHOT’s photometry arise from mismatches in the model and observed point spread functions (PSFs) and\naperture corrections, each contributing ≲0.01mag to the photometric error budget. Version 1.2 of WebbPSF\nmodels, which include charge diffusion and interpixel capacitance effects, significantly reduced PSF-related un-\ncertainties. We also observed minor ( ≲0.05mag) chip-to-chip variations in NIRCam’s zero points, which will\nbe addressed by the JWST flux calibration program. Globular cluster observations are crucial for photometric\ncalibration. Temporal variations in the photometry are generally ≲0.01mag, although rare large misalignment\nevents can introduce errors up to 0.08 mag. We provide recommended DOLPHOT parameters, guidelines for\nphotometric reduction, and advice for improved observing strategies. Our ERS DOLPHOT data products are\navailable on MAST, complemented by comprehensive online documentation and tutorials for using DOLPHOT\nwith JWST imaging data.\nKeywords: James Webb Space Telescope (2291); Stellar photometry (1620); Hertzsprung Russell diagram (725)\n1.INTRODUCTION\nJWST has the potential to resolve millions of stars in thou-\nsands of galaxies out to large distances (e.g., D∼100Mpc).\nSuch data will enable new foundational science in a broad\nrange of areas such as the cosmic distance ladder and local\nH0measurements, reionization, globular cluster formation,\ndark matter, the stellar initial mass function, galaxy assem-\nbly, the effects of rare red stars that can effect the spectral\nenergy distributions (SEDs) of galaxies at all cosmic epochs,\nand much more (e.g., see discussion in Weisz et al. 2023).\nMuch of this science comes from observations of resolved\nstars in crowded fields. In crowded fields, neighboring stars\nhave overlapping point spread functions (PSFs), which can\nlead to confusion over the number of stars and their rela-\ntive contributions to the observed flux in a given pixel. Re-\ncovering accurate and precise photometry for large numbers\nof stars in the limit of modest-to-severe crowding is tech-\nnically daunting and requires highly optimized observations\nand sophisticated analysis tools (e.g., Dalcanton et al. 2012a;\nWilliams et al. 2014).\nFortunately, crowded field stellar photometry is a mature\nfield based on a rich history of development dating back\nnearly ∼50years. Early crowded field photometry routines\ncombined pioneering work on photoelectric detectors with\ninnovative approaches to simultaneously modeling the stellar\nlight profiles of adjacent stars, resulting in a number of codes\nin the 1980s that could photometer thousands of stars in a\nfield (e.g., Buonanno et al. 1979; Tody 1980; Stryker 1983;\nLupton & Gunn 1986; Penny & Dickens 1986; Schechteret al. 1993). A major achievement of this era was the creation\nof the legacy software package DAOPHOT (Stetson 1987).1\nContinued improvements in crowded field photometry\nwere catalyzed by the launch of the Hubble Space Telescope\n(HST). Via the Hubble Key Project aimed at measuring H0,\nseveral independent photometric routines were developed to\ngauge systematics in the photometry (e.g., Stetson 1994;\nand see discussion in Freedman et al. 2001). Similarly, the\nground-breaking sensitivity and precision of HST/WFPC2,\nalong with its notoriously undersampled PSF, motivated the\ndevelopment of specialized photometric and astrometric rou-\ntines aimed at dense stellar fields (e.g., Holtzman et al. 1995;\nLauer 1999; Anderson & King 2000; Dolphin 2000). More\nrecently, stellar surveys of crowded fields such as the Galac-\ntic plane and M31 have provided important gains in the speed\nand flexibility of crowded field codes (e.g., Dalcanton et al.\n2012a; Schlafly et al. 2018). In the context of nearby galax-\nies, the Panchromatic Hubble Andromeda Treasury (PHAT)\nsurvey provided substantial new additions (e.g., simultaneous\nmulti-camera, multi-wavelength crowded field photometry)\nto DOLPHOT (Dolphin 2000, 2016), a crowded field photo-\nmetric package that has produced photometry for millions of\nstars in hundreds of galaxies in the Local Group (LG) and\nLocal V olume (e.g., Holtzman et al. 2006; Rizzi et al. 2007;\nWeisz et al. 2008; Dalcanton et al. 2009; McQuinn et al.\n2010; Radburn-Smith et al. 2011; Dalcanton et al. 2012b,a;\nWilliams et al. 2014; Jang & Lee 2017; McQuinn et al. 2017;\nSkillman et al. 2017; Sabbi et al. 2018; Anand et al. 2021;\n1As discussed by Stetson (1987), DAOPHOT was a version of the photomet-\nric routine POORMAN written by Mould & Shortridge that was improved\nto handle a higher density of stars. Unfortunately, there is no bibliographic\nrecord for POORMAN.JWST R ESOLVED STELLAR POPULATIONS V 3\nJang et al. 2021; Williams et al. 2021; Lee et al. 2022; Savino\net al. 2022; Riess et al. 2023; Williams et al. 2023).\nA main goal of our JWST Resolved Stellar Populations\nEarly Release Science is to provide the astronomy commu-\nnity with an easy-to-use and efficient means for perform-\ning crowded field photometry on JWST imaging that will\nultimately help to realize JWST ’s full potential for the re-\nsolved Universe. Specifically, we have developed mod-\nules for DOLPHOT that are tailored to the characteristics\nof NIRCam and NIRISS, which are important imaging in-\nstruments for studies of resolved stellar populations with\nJWST . DOLPHOT is a well-tested, widely-used, and pub-\nlicly available package that already supports modules spe-\ncific to several HST cameras (WFPC2, ACS, WFC3/UVIS\nand IR), has been a testing ground for the Roman Space Tele-\nscope , and includes general purpose routines that can be used\non virtually any images of resolved stars. The addition of\nJWST modules will enable a wide array of JWST -specific and\ncross-facility (e.g., JWST andHST) science, some of which\nhas already been demonstrated using early versions of our\nJWST DOLPHOT modules (e.g., Chen et al. 2023; Lee et al.\n2023a,b; McQuinn et al. 2023; Riess et al. 2023; Van Dyk\net al. 2023; Warfield et al. 2023; Peltonen et al. 2024; Li et al.\n2024).\nIn this paper, we describe the NIRCam and NIRISS stel-\nlar photometry modules for DOLPHOT. DOLPHOT’s under-\nlying algorithms are already well-documented in the litera-\nture, along with rigorous tests of their accuracy in a variety\nof crowded and uncrowded fields (e.g., Radburn-Smith et al.\n2011; Williams et al. 2014, 2023). Accordingly, our focus is\non describing the details specific to the DOLPHOT NIRCam\nand NIRISS modules and providing examples of its appli-\ncation to the three Early Release Science (ERS) targets in\nthe Local Group: M92, Draco II, and WLM. As described\nin Weisz et al. (2023), these targets, and the associated ob-\nserving strategies, were carefully selected to benchmark the\ndevelopment of DOLPHOT in a variety of regimes that we\nanticipate will be common for resolved star science, and thus\nneed to be vetted for study with JWST . This paper is designed\nto describe the modules and provide examples of their appli-\ncation to the ERS data. As part of the ERS program, we have\ncreated an extensive set of deliverables including online doc-\numentation and data products that allow interested readers to\nreduce ERS data identically to what is done in this paper,\nas well as explore the various aspects of the data for their\nown purposes (e.g., to customize catalog culling criteria). Es-\nsential DOLPHOT input and output data associated with the\nphotometric reductions in this paper are hosted as high-level\nscience products on MAST2, while step-by-step guides for\nour DOLPHOT reductions can be found on our DOLPHOT\ndocumentation page3.\nThis paper is organized as follows. We summarize the ERS\nobservations in §2. In §3, we describe the DOLPHOT NIR-\n2https://archive.stsci.edu/hlsp/jwststars/\n3https://dolphot-jwst.readthedocs.ioCam and NIRISS modules and provide a general outline of\nhow to apply these new modules to JWST imaging in order\nto produce stellar catalogs. We illustrate the application of\nthese modules to ERS data in §4. In §5, we examine the time\nvariability of the PSF and compare the DOLPHOT SNR es-\ntimates with expectations from the JWST ETC. Finally, we\nsummarize the paper and highlight future areas for improve-\nment in §6.\n2.OBSERV ATIONS\nExtensive details of our ERS survey and observations are\nprovided in Weisz et al. (2023). Here, we briefly summarize\nthe observations and list their basic characteristics in Table 1.\nIn June and July, 2022 our program acquired NIRCam and\nNIRSS imaging of three LG targets: globular cluster M92,\nultra-faint dwarf galaxy Draco II, and LG star-forming dwarf\ngalaxy WLM. These targets were selected to satisfy a number\nof science and technical goals including the development and\ntesting of DOLPHOT in a variety of scenes (e.g., crowded\nand uncrowded fields, varying surface brightness, varying de-\ngrees of saturation, a representative set of wide and medium\nfilters). In all cases, the NIRCam fields were placed centrally\non each target with locations and orientations set to maxi-\nmize overlap with archival HST imaging and schedulability\nearly in the ERS window. The NIRISS fields were acquired\nin parallel. Table 1 summarizes basic characteristics of our\nNIRCam and NIRISS observations.\nIn the process of analyzing our M92 data, we found that the\nthird exposure of M92 appears to be “corrupt” in the sense\nthat although the 3rd exposure of M92 visually looks fine, it\nresults in remarkably poor photometry for both NIRCam and\nNIRISS, despite extensive efforts to fix it. We therefore have\nexcluded it from the DOLPHOT reductions in this paper. A\nlater analysis of the fine guidance sensor data revealed insta-\nbilities in the telescope only during this exposure. We discuss\ndetails of the third exposure in Appendix A.\nOur observations of WLM were designed to sample RR\nLyrae light curves. However, the default JWST reduction\npipeline is currently not capable of producing the time series\nimages for short period observations needed to extract flux as\na function of time. The default pipeline currently only pro-\nvides time series data when the time series observation (TSO)\nmode is used. We were unable to use TSO mode because it\nprohibits dithering. While it is possible to modify the JWST\npipeline to produce the time series images necessary for short\nperiod variable analysis, it is a topic beyond the scope of\nthis paper. Instead, a Cycle 2 archival proposal undertaken\nby some members of our team (AR-03248; PI Skillman) is\ndeveloping the documentation and tools needed to recover\nvariable star light curves from NIRCam and NIRISS imag-\ning with DOLPHOT. These will be made publicly available\nupon completion.\nFinally, the NIRISS observations of Draco II, were located\nat several half-light radii from the galaxy. As far as we can\ntell, the field is consistent with being blank, i.e., no obvious\ngalaxy member stars, and we do not analyze or discuss this\nfield in the paper.4 W EISZ ET AL .\nTable 1. A summary of our JWST Early Release Science (ERS) observations taken in 2022. Although we acquired 4 exposures for M92, the\n3rd exposure produces poor photometry due to larger-than-normal jitter in the telescope stability that occurred only during this exposure. We\nexamine this issue in Appendix A. The first entry for M92 in this table reflects all observations taken, while the second entry is without the 3rd\nexposure. More details on the exact observations (e.g., dither pattern, readout mode) are given in Weisz et al. (2023) and are available in our\npublic Phase II file in APT.\nTarget Date Camera Filter texp[s] Groups Integrations Dithers\nM92 June 20–21 NIRCam F090W/F277W 1245.465 6 1 4\nNIRCam F150W/F444W 1245.465 6 1 4\nNIRISS F090W 1245.465 7 1 4\nNIRISS F150W 1245.465 7 1 4\nM92 (no 3rd exp) June 20–21 NIRCam F090W/F277W 934.099 6 1 3\nNIRCam F150W/F444W 934.099 6 1 3\nNIRISS F090W 934.099 7 1 3\nNIRISS F150W 934.099 7 1 3\nDraco II July 3 NIRCam F090W/F480M 11810.447 7 4 4\nNIRCam F150W/F360M 5883.75 7 2 4\nNIRISS F090W 11123.294 9 7 4\nNIRISS F150W 5883.75 10 3 4\nWLM July 23–24 NIRCam F090W/F430M 30492.427 8 9 4\nNIRCam F150W/F250M 23706.788 8 7 4\nNIRISS F090W 26670.137 17 9 4\nNIRISS F150W 19841.551 19 6 4\n3.NIRCAM AND NIRISS PHOTOMETRY WITH\nDOLPHOT\nIn this section, we provide an overview of the new\nDOLPHOT NIRCam and NIRISS modules with application\nto imaging from our ERS program. The core workings of\nDOLPHOT, along with extensive tests of its functionality and\nreliability are well-documented in the literature (e.g., Dol-\nphin 2000; Dalcanton et al. 2012b,a; Williams et al. 2014;\nDolphin 2016; Williams et al. 2021). Here, we will not re-\nvisit these details. Instead, we focus on modifications made\nto DOLPHOT for incorporating NIRCam and NIRISS imag-\ning into its existing framework.\n3.1. Overview\nAs input, DOLPHOT takes a reference image, a list of sci-\nence images, and several dozen input parameters with user\ndefined values. DOLPHOT astrometically aligns all the sci-\nence images to a reference image. It then performs simul-\ntaneously, multi-wavelength photometry on all science im-\nages by fitting PSF models to the signal-to-noise ratio (SNR)\npeaks (and any neighboring SNR peaks in crowded fields)\nit detects in each of the science images. The result of this\nprocess is a set of photometric measurements for all detected\nobjects in all science images. Stellar catalogs are created by\nculling the main source catalog using criteria such as SNR,\nhow compact/extended sources are, etc., which we detail in\n§3.5.\nForJWST , the science images we use are the stage 2 cal\nfiles, which are calibrated single exposure science imagesproduced by the JWST pipeline. The references images are\nI2D files, which are resampled, stacked science images, akin\nto drizzled images with HST. It is possible to use cal as ref-\nerence images as well. The JWST pipeline also makes avail-\nablecrf images, which are like cal science images, but\nwith cosmic ray flags applied. Generally, we have found as-\ntrometric alignment of the crf images in DOLPHOT to be\nworse than cal images.\nThe reference image is used only to align each of the sci-\nence images. Our general recommendation for a reference\nimage is to select the deepest image available. All images\nused in this analysis were created by the standard STScI\npipeline and downloaded from MAST4.\nIn this paper, we use FITS images with the following\nJWST pipeline versioning information CALVER=1.11.4,\nCRDS VER=11.17.2, and CRDS CTX=jwst p1147.pmap.\nThis version includes updates to the chip-to-chip zero points,\nthe switch from Vega to Sirus as a reference star, and updated\nflat fields released in late 2023.\nDOLPHOT requires only a few pre-processing steps for all\nimages. These steps include masking bad pixels (e.g., cosmic\nrays, hot pixels) based on the data quality (DQ) flags, multi-\nplying by the camera specific pixel area masks, and making\ninitial estimates of the sky and the positions of bright stars\nfor alignment.\n4The specific observations used in his paper can be accessed via DOI:\n10.17909/71kb-ga31.JWST R ESOLVED STELLAR POPULATIONS V 5\nFollowing pre-processing, DOLPHOT aligns each science\nimage to the reference image. The quality of the alignment is\ndetermined by several factors including the number of bright\nstars available, depth of the reference image, the fidelity of\nthe provided WCS information, SNRs, relative orientations\n(e.g., images with large rotations may be harder to align),\nstars in common between the reference and science image\n(e.g., images taken in very different filters, such as ultra-\nviolet and IR, may be hard to align as they may have few\nsources in common).\nWith all images aligned, DOLPHOT searches for objects\nto photometer by iteratively identifying signal-to-noise ratio\npeaks in the stack of science images. DOLPHOT measures\nthe fluxes of each object by simultaneously fitting a PSF\nmodel, and a local background model, to the target object\nplus all neighboring objects within a user specified radius.\nUpon completion of photometry, DOLPHOT provides ex-\ntensive output including its position on the reference image\nand the flux and a number of quality assessment metrics (e.g.,\nχ2, shape of the star’s light profile relative to the PSF) for\neach star in each image. It also provides combined fluxes\nand magnitudes for each object from which stellar catalogs\nare usually constructed.\nCharacterizing uncertainties for crowded field stellar pho-\ntometry requires artificial star tests (ASTs). ASTs are syn-\nthetic stars with known positions and magnitudes that are\ninserted into real JWST science images and then recovered\nby DOLPHOT. It is well-established that the difference in\ninput and recovered flux for ASTs provides a more realis-\ntic accounting of photometric uncertainties than the Poisson\nnoise that is reported by the crowded field photometric pro-\ncess alone (e.g., Stetson & Harris 1988).\n3.2. Pre-processing steps\nPrior to running DOLPHOT, pre-processing is required in\norder to convert the data to a format suitable for PSF-fitting\nphotometry. For the case of NIRCAM and NIRISS data,\nsteps in this process are as follows (using the nircammask\nandnirissmask utilities, respectively):\n• Mask out bad or saturated pixels. At the time of this\nwriting, bad pixels on cal andcrf images are iden-\ntified by having an SCI array value of NaN; previous\nversions of the pipeline have used SCI array values of\nexactly 0. The mask utilities will correctly interpret\neither approach. Additionally, saturated pixels in cal\nandcrf images are identified by having a DQ array\nflag with a value of 2. Bad pixels on I2D images are\nidentified by having a WHT array value of exactly 0.\n• Convert from the default calibration of MJy/sr to DN\n(data number). This is performed by dividing all pixel\nvalues by the FITS keyword PHOTMJSR, and subse-\nquently multiplying by the exposure time (FITS key-word EFFEXPTM)5. An additional step for cal and\ncrf images is to multiply pixel values by the pixel\narea map (AREA array).\n• Readout noise and gain values are also saved into the\nFITS file. Details of the pedigree of that data are avail-\nable in the versions.txt files that are included\nwith the NIRCAM and NIRISS DOLPHOT modules.\nThe result of the pre-processing step is an image in units\nof DN, along with FITS keywords for gain and readout noise,\nallowing PSF-fitting photometry to run.\nAs of this writing, for the purpose of this ERS program,\nthere was no need to incorporate information from the Ad-\nvanced Scientific Data Format (ASDF) metadata provided for\nJWST images (Greenfield et al. 2015). ASDF has the ability\nto host more detailed metadata than the standard FITS for-\nmat, such as improved WCS information. However, at this\ntime all necessary information (astrometry information, pho-\ntometric calibrations, pixel areas, etc.) for DOLPHOT are\navailable via FITS header keywords. If that situation changes\nin the future, we will explore writing an ASDF reader for\nDOLPHOT.\n3.3. Alignment\nThe first step in DOLPHOT’s reduction process is to align\nall of the original-sampling ( cal orcrf) images to a com-\nmon reference frame. Normally, the common reference is\nanI2D file. In principle, if no dithering was used in the\nobservations, one of the cal orcrf images could be used\nas a reference image. In practice, typically the deepest I2D\nfile makes for the best reference image as it allows for the\nmost star matches with the science images leading to better\nastrometric alignment. We note that DOLPHOT does not re-\nsample or re-bin any images, which can lead to issues in con-\nservation of intensity, for example. Instead, DOLPHOT only\nperforms photometry on the original, non-drizzled, science\nimages provided by the JWST pipeline.\nA key difference between the JWST modules and the previ-\nously released DOLPHOT HST modules is that DOLPHOT\ndoes not apply distortion corrections based on DOLPHOT’s\ninternal model. With the HST modules, common practice\nis to use the astrometric data in the header but not the dis-\ntortion. Using HST with DOLPHOT’s internal distortion\nmodel requires the parameter setting UseWCS=1, for which\nDOLPHOT estimates only the shift, scale, and rotation of\neach science image relative to the reference. For JWST the\nastrometric data included in the FITS header is used for both\nalignment and application of distortion corrections, requiring\na setting of UseWCS=2. In this case, DOLPHOT estimates a\n5We note that as of this writing, the actual time the telescope spends col-\nlecting data slightly differs from the exposure time in the FITS keywords,\nsuch as EFFEXPTM. In practice, the ramp fitting procedure begins at the\nend of the first group. But currently, the FITS exposure time keywords are\nbased on when the first group starts. This effect is generally subtle, i.e., the\nimpact on SNR is typically ≲1%, but it is now factored into DOLPHOT\nfor accuracy and completeness.6 W EISZ ET AL .\nfull distortion solution, which is beyond the shift, scale, and\nrotation typically used by DOLPHOT on HST images. As\nof this writing, NIRCAM data are provided with 3rd order\nSIP polynomials, while NIRISS data are provided with 4th\norder polynomials. DOLPHOT currently processes up to 5th\norder polynomials, so it can handle additional fidelity in the\nastrometry data, should it become available in the pipeline or\nadded by offline astrometry tools (e.g., astrometry.net; Lang\net al. 2010).\n3.4. Star Detection and Photometry\nAs with previously available DOLPHOT modules, pho-\ntometry proceeds once the images are aligned. An initial pass\n(and usually multiple passes) detects peaks in the SNR map\nacross the reference image. Photometry is performed at each\npeak to attempt to identify a point source. As with all previ-\nous versions of DOLPHOT, the user can adjust parameters to\nalter the noise models, photometry modes (e.g., aperture vs.\nPSF-fitting), sky fitting method, etc. Our internal testing on\nthe ERS data resulted in a set of recommended DOLPHOT\nparameters, which are listed in Table 4. We discuss the pro-\ncess by which we determined these parameters in more detail\nin §3.8.\nA standard feature of DOLPHOT is to make adjustments to\nthe model PSFs to improve the fit quality and photometry. As\ndiscussed in Dolphin (2000), mismatches between the shape\nof the model and the true PSF contribute to the photometric\nerror budget, which can grow large for faint sources. As part\nof its normal operation, DOLPHOT measures a PSF resid-\nual image relative to the pre-calculated PSF model library.\nIt then makes adjustments to the model PSFs based on com-\nparisons to bright stars in the field to improve agreement be-\ntween the model and observed PSFs. This step is performed\nautomatically (i.e., PSFRes=1) by DOLPHOT unless the use\nof residual PSF images is turned off (i.e., PSFRes=0). These\nadjustments are made independently in each DOLPHOT sci-\nence image. The amplitude of the PSF adjustments provides\na means to quantify the systematic uncertainty floor on the\nphotometry for a given set of PSF models. In §3.6, we pro-\nvide the typical PSF adjustments made by DOLPHOT on\nERS data relative to the default WebbPSF models.\nLikewise, aperture corrections are normally computed au-\ntomatically (i.e., ApCor=1), unless they are turned off in\nDOLPHOT (i.e., ApCor=0). This calculation will estimate\nthe magnitude difference between instrumental PSF-fitted\nmagnitudes and magnitudes within a standard 10-pixel ra-\ndius, accounting for the WebbPSF-predicted encircled en-\nergy within that standard radius.\nFinally, zero points are applied to convert from instrumen-\ntal magnitudes (in DN/sec) to the VEGAMAG system. These\nzero points are in units of Jy for Vega (though Sirus is now\nthe standard reference star) so they require conversion back\nfrom DN/sec to Jy using the FITS header keywords PHOT-\nMJSR and PIXAR SR before the calibration is applied. Zero\npoints for NIRISS were provided relative to 1 DN/sec. Thus,\nthey are applied directly without additional conversion. Fi-\nnally, we note that DOLPHOT parameters NIRCAMvega andNIRISSvega can be set to zero to report in ABmag instead of\nVEGAMAG.\n3.5. Post-Processing & Catalog Creation\nOnce complete, DOLPHOT saves all photometry data to a\nsingle ASCII file containing overall fit metrics (positions of\nobjects in the coordinate system of the reference image, χ2,\nS/N, sharpness, roundness, crowding, and the object type:\nsingle pixel, point source, or extended source) for all ob-\njects identified. Photometry and the same quality assessment\ninformation is also provided for all combined exposures in\neach filter (e.g., all F090W images), as well as for all indi-\nvidual exposures (e.g., each F090W images, which can be\nused for time domain studies, for example). The photometric\ndata provided by filter and image include counts (DN), back-\nground, calibrated magnitude, and calibrated count rates,\nwhich can be useful if an upper limit is informative, such\nas in multi-wavelength SED fitting and time-domain studies\n(e.g., Gordon et al. 2016).\nAs the DOLPHOT output includes all sources identified on\nthe images, it is necessary to establish criteria for good de-\ntections (i.e., stars). As part of this ERS program, Warfield\net al. (2023) developed a set of criteria to identify stars us-\ning DOLPHOT reported quality parameters. We adopt this\nscheme for this paper and classify good stars as those that\nsatisfy all of the following criteria:\n•SNR F090W≥5\n•SNR F150W≥5\n•sharp2\nF090W≤0.01\n•sharp2\nF150W≤0.01\n•crowd F090W≤0.5\n•crowd F150W≤0.5\n•flagF090W≤3\n•flagF150W≤3\n• Object Type ≤2\nThe sharpness parameter is zero for a perfectly-fit star, pos-\nitive for a star that is too sharp (i.e., the flux is concentrated in\na small number of pixels, e.g., a cosmic ray), and negative for\na star that is too broad (perhaps a blend, cluster, or galaxy).\nOur choice of SNR ≥5is lower than the SNR ≥10threshold\nin Warfield et al. (2023). Warfield et al. (2023) focused on\noptimizing the other parameters for star-galaxy separation,\nand therefore adopted a more conservative SNR threshold.\nThe crowding parameter is in magnitudes. It reports how\nmuch brighter the star would have been measured had nearby\nstars not been fit simultaneously. For an isolated star, the\nvalue is zero. High crowding values are generally a sign of\npoorly-measured stars.\nError flags are defined as: 0 is a star that is recovered ex-\ntremely well; 1 is that the photometry aperture extends offJWST R ESOLVED STELLAR POPULATIONS V 7\nchip; 2 is that there are too many bad or saturated pixels; 4 is\nthe center of the star is saturated; 8 is an extreme case of one\nof the above. The DOLPHOT manual suggests using values\nof 3 or less in general or 2 or less for precision photometry.\nObject types are as follows: 1 is a good star; 2 is a star\ntoo faint for PSF determination; 3 is an elongated object; 4 is\nan object that is too sharp; 5 is an extended object. As rec-\nommended by the DOLPHOT manual, we only keep object\ntypes 1 and 2 in our stellar catalogs.\nAs the deepest and highest angular resolution images, we\nfound that applying these criteria to F090W and F150W pho-\ntometry had the largest impact on the catalog culling (e.g.,\nWarfield et al. 2023). For targeted science (e.g., luminous\nred stars at longer wavelengths) other criteria and/or applica-\ntion to other filters may produce more desirable results. Sim-\nilarly, cuts on single bands may be useful for particular sci-\nence cases beyond CMDs (e.g., stellar SED fitting; Gordon\net al. 2016). Finally, as discussed by Warfield et al. (2023),\nthese criteria were focused on purity rather than complete-\nness. This is motivated by the large number of background\ngalaxies present in our ERS imaging. Less stringent cuts,\nparticularly in sharpness, can produce deeper CMDs with\nless conservative completeness limits, albeit with a larger de-\ngree of non-stellar contamination. We illustrate the effects of\nour fiducial culling criteria on our ERS targets in §4. Read-\ners who wish to explore alternative culling criteria can down-\nload our catalogs from the MAST high levels science prod-\nucts page.\n3.6. Point Spread Function Models\nThe currently available NIRCAM and NIRISS modules in-\ncorporate model PSF libraries calculated using WebbPSF.\nFor this paper, the PSF models were generated using\nWebbPSF version 1.2.1, which adopts “in-flight” optical per-\nformance data (as opposed to pre-launch data). The align-\nment and stability of JWST , including the wavefront and PSF,\nare known to vary over time (e.g., McElwain et al. 2023),\nwhich has the potential to affect photometry. WebbPSF in-\ncorporates time-dependent optical path delay (OPD) maps to\ncapture changes to the wavefront and PSF over time, enabling\ncorrections for temporal changes to JWST .\nThe WebbPSF model PSF library we implement in\nDOLPHOT consists of distorted PSF models for all the avail-\nable NIRCam/NIRISS filters, oversampled by a factor of 5,\ncalculated over a 51×51physical pixel region. The models\nare calculated on a 5×5spatial grid for each of the detec-\ntor chips. The models were generated using OPD maps from\nJuly 24th 2022 (O2022072401-NRCA3 FP1-1.fits). We used\na G5V source spectrum from the Phoenix stellar library (e.g.,\nHusser et al. 2013) to generate the PSF, which was sampled\nat 21, 9, and 5 wavelengths for wide, medium, and narrow\nbands, respectively. v1.2.1 of WebbPSF includes the effects\nof charge diffusion and interpixel capacitance, which were\nnot incorporated into previous WebbPSF models. We found\nthat the inclusion of these effects dramatically improves the\nquality of DOLPHOT photometry, including reducing photo-\nmetric systematics by nearly an order-of-magnitude relativeto WebbPSF models without these effects. We discuss the the\ntotal photometric error budget in §5.3.\nAs described in §3.4, DOLPHOT makes adjustments to the\nmodel PSFs to provide improved matches to the data. We\nsummarize the effect of these PSF adjustments on the pho-\ntometry in Table 2. This table provides the mean fractional\ncentral pixel brightness in each filter (i.e., the fraction of total\nPSF light in the central pixel averaged across all PSF mod-\nels), the scatter in the central pixel fraction flux, and the mean\nPSF adjustment in the central pixel measured by DOLPHOT.\nWe computed these quantities across NIRCam and NIRISS\nimages from our ERS program, except for the 3rd exposure\nof M92.\nAs DOLPHOT provides an average PSF correction (i.e., by\ncomputing the PSF adjustments on a set of bright, high SNR\nstars in each science image), we can quantify the uncertainty\nin this correction. Table 2 provides a simple estimate of the\n1-σphotometric error created by application of the same PSF\nresidual image to all stars in a given science image. The PSF\nadjustments are computed separately for each science image\nfor each target. That is, DOLPHOT has improved the model\nPSFs by adjusting the central pixel on average. This is an\nimprovement over using the native WebbPSF models without\nany adjustment, but it does still yield an uncertainty floor.\nConsidering the central pixel only, as DOLPHOT does for its\nPSF correction (see Dolphin 2000), the magnitude error can\nbe estimated as\nσ=2.5σmodel δmean\nlog(10) µ2\nmodel(1)\nwhere µmodel is the mean model flux in the central pixel\nacross all model PSFs in a given filter, σmodel is the scatter in\nthe central pixel fluxes across all model PSFs in a given filter,\nandδmean is the mean flux adjustment made by DOLPHOT.\nWhile this estimate is reasonably accurate for highly concen-\ntrated PSFs, such as the two NIRISS filters in our data sam-\nple, it becomes increasingly conservative for broader PSFs.\nAs Table 2 shows, the typical PSF adjustments, and the\ncorresponding photometric uncertainties are small. For all\nNIRCam filters, a modest amount of light is concentrated in\nthe central pixel and the PSF adjustments are only typically a\nsmall fraction of the total flux. The resulting photometric un-\ncertainties introduced by the PSF models range from 0.001\nto 0.008 mag. For NIRISS, the PSFs are slightly sharper,\ni.e., more of the total light is centrally concentrated in the\nmodel PSFs, which leads to larger PSF adjustments. The\ncorresponding photometric uncertainties are ∼0.01mag in\nthe NIRISS F090W and F150W filters. Overall, the small\nadjustments needed by DOLPHOT to improve the WebbPSF\nmodels indicate that the models themselves are already quite\ngood. This was not the case with PSF models from previ-\nous versions of WebbPSF, all of which were systematically\ntoo sharp and the corresponding photometric errors, i.e., sys-\ntematics from the PSF models, were at times larger than the\nphoton counting noise.\nWe note that the values listed in Table 2 were computed\nfor SW and LW images independently. We found all PSF ad-8 W EISZ ET AL .\nFilter Nexp Central Pixel Central Pixel Central Pixel Photometric Error\nModel Mean Model σ Mean δ\n(% flux) (% flux) (% flux) (mag)\n(1) (2) (3) (4) (5) (6)\nNIRCAM F090W 88 24 .18 3 .98 −0.69 0 .005\nNIRCAM F150W 88 16 .63 1 .41 −0.25 0 .001\nNIRCAM F250M 8 20 .19 2 .07 −0.24 0 .001\nNIRCAM F277W 6 18 .41 1 .68 −1.0 0 .005\nNIRCAM F360M* 8 12 .98 0 .76 −1.57 0 .008\nNIRCAM F430M 8 10 .22 0 .44 0 .07 0 .000\nNIRCAM F444W 6 10 .05 0 .43 −0.34 0 .002\nNIRCAM F480M* 8 8 .48 0 .30 −0.41 0 .002\nNIRISS F090W 7 31 .47 5 .57 −2.31 0 .014\nNIRISS F150W 7 28 .53 4 .48 −1.69 0 .010\nTable 2. Central pixel PSF data for the input WebbPSF PSF models (mean and standard deviation) and mean adjustments as measured by\nDOLPHOT from application to all ERS NIRCam and NIRISS images, except the 3rd exposure for M92. Values in columns 3-5 are fractions\nof the total stellar flux. Column 6 shows approximate photometric error in magnitudes caused by the application of the same PSF residual to\nall stars (see Equation 1). Asterisks for NIRCAM F360M and F480M denote the filters only observed in the very sparse Draco II field; results\nfrom those filters have correspondingly higher uncertainties.\njustments to be marginally larger when running SW and LW\nimages simultaneously, though the qualitative finding that the\nPSF adjustments are quite small (i.e., ≲0.01mag) still holds.\n3.7. Artificial Star Tests\nWhile DOLPHOT provides an estimate of photometric un-\ncertainties based on the goodness of fit and the noise char-\nacteristics of the data, a much better characterization of the\nphotometric measurement uncertainties and selection func-\ntion are accomplished through artificial star tests (ASTs).\nThis long-established approach (e.g., Stetson 1987; Stetson\n& Harris 1988) represents the “gold standard” in the field of\nresolved stellar population photometry. It relies on the injec-\ntion of mock stellar sources into the raw images, which are\nthen recovered using the identical photomeric procedure used\nto construct the raw and stellar DOLPHOT catalogs. The\noutput of such simulated data tests can be used to quantify a\nnumber of aspects of data quality. A common example is that\nthe comparison between the input and output magnitudes of\nthe mock catalog, as well as the fraction of mock stars that\nare successfully detected, provides a self-consistent charac-\nterization of photometric errors, systematic uncertainties, and\nphotometric completeness as a function of spatial position in\nthe images and location on the CMD. Throughout this paper,\nwe focus on ASTs run only on the SW data, which illustrate\nthe main points of ASTs. The same procedures we describe\ncan be used to run ASTs in an arbitrary number of bands,\nthough the computational time can become quite expensive\nfor large numbers of photometric bands.\nThe first step in running ASTs is to create a suitable input\nstar catalog. For each target and camera, we created a list of\n≳4×105mock stars (see Table 5), with positions drawn\nfrom uniform spatial distributions on the NIRCam/NIRISSfootprints, aside from gaps between the chips and modules.\nFor each star, we assign input magnitudes such that they are\nuniformly distributed over the F090W vs. (F090W-F150W)\nCMD with 17≤F090W ≤31and−0.5≤F090W −F150W\n≤2.\nThe artificial stars are then injected into all science (i.e.,\ncal) images using the best PSF model and realistic noise ob-\ntained from the original reduction run. The stars are injected\none at the time, to avoid altering the crowding properties of\nthe original images. The star magnitude and position is then\nmeasured by DOLPHOT, as if it were a real source. Perform-\ning this operation for all the input stars, we obtain a catalog\nof output magnitudes, positions, and goodness-of-fit param-\neters. This list is then culled using the same quality criteria\napplied to the original photometric catalog (cf. § 3.5).\n3.8. Optimizing Photometric Parameters\nDOLPHOT is a flexible code in that it provides the user\nextensive control over details of the photometric reduction\n(e.g., PSF adjustments, sky fitting, image alignment methods,\nnoise models). This ensures that DOLPHOT can produce\nhigh-quality photometry for a wide variety of images (e.g.,\ncrowded vs. uncrowded, presence/absence of surface bright-\nness gradients, images from multiple filters with widely dif-\nferent characteristics). At the same time, this flexibility\nis characterized by a large number of parameters that can\nrequire tuning in order to produce “optimal” photometry.\nDOLPHOT parameters have been refined in the context of\nmajor HST programs over the past decade, culminating in a\nset of parameters recommended by the PHAT program that\nencompass a wide range of image properties (e.g., Williams\net al. 2014). While these parameters produce excellent HSTJWST R ESOLVED STELLAR POPULATIONS V 9\nphotometry, it is important that we investigate parameters\nthat may provide better photometry for JWST .\nAccordingly, we performed a large set of DOLPHOT runs\nto explore the effect of changing a select set of DOLPHOT\nparameters. We performed these runs on all three ERS tar-\ngets, to experiment with different stellar density regimes:\nhigh crowding for WLM, low crowding for M92, and an\nalmost empty field for Draco II. As the SW and LW chan-\nnels have different detector characteristics (e.g., plate scale),\nwe executed the experiments on the two channels indepen-\ndently. For each target and each channel, we explored differ-\nent values of FitSky (which sets the method for local sky\nmeasurement), RAper (which sets the size of the aperture in\nwhich photometry is performed), and Rchi (which sets the\nsize of the region over which the fit is evaluated). We set\nthe value of FitSky to either 2 (fit the sky inside the PSF\nregion but outside the photometry aperture) or 3 (fit the sky\nwithin the photometry aperture as a 2-parameter PSF fit). For\na given FitSky , we let RAper range between 1 and 5 pix-\nels, in discrete increments, and Rchi range between 0.5 and\nRAper , in increments of 0.5 pixels. In the FitSky =2 case,\nthe values of Rsky2 (which set the inner and outer radius\nfor sky computation) are also adjusted and set to {RAper +1;\n2.5(RAper +1)}. ForFitSky =3, we explored an additional\ngrid, defined by RAper values of 7, 10, and 13 and Rchi\nvalues of 1.5, 2, and 3. This exploration results in 69 pa-\nrameter permutations per channel, per target, totaling 414\nDOLPHOT runs, which consumed nearly five years of CPU\ntime.\nFor each of these runs, we used thousands of ASTs (see\n§ 3.7) as one metric for evaluating photometric performance.\nASTs were injected at various locations on the CMD (from\nvery bright, high-SNR parts down to the detection limit).\nWe then used these stars to quantify the mean scatter in\ncolor and magnitude, and the completeness fraction at each\nCMD location. Inspection of these metrics revealed that\nin large regions of the parameter space, DOLPHOT perfor-\nmance was poor. For example, most runs with Rchi >\n3.0and/or FitSky =3 produced obviously poor photome-\ntry (e.g., low number of stars, poor completeness, large scat-\nter). Adopting Fitsky =2, we identified a small region of\ntheRAper -Rchi parameter space ( 2≤RAper ≤4and\n1.5≤Rchi ≤2.5) where DOLPHOT provided the “best”\nphotometry. Within this parameter region, the photometric\nperformance was fairly comparable. Different permutations\nof these parameters produced slight trade-offs in complete-\nness versus photometric precision, though the differences\nwere generally at the few percent level or less. We also found\na slight trend with crowding, in that the optimal RAper value\nwould increase as the field became less crowded, but again,\nthis effect was small.\nGiven the similarity of the photometry over this param-\neter space, we decided to adopt a single set of parameters\nfor each instrument and channel. Part of the motivation be-\nhind this choice is to provide the community with easy-to-\nuse guidance for DOLPHOT reductions that also produces\nhigh-quality photometry. The recommended PHAT parame-Target Camera Object Density Stellar Density\n(N/ arcsec2) ( N/ arcsec2)\n(1) (2) (3) (4)\nM92 NIRCam 28.1 2.8\nNIRISS 7.4 0.2\nWLM NIRCAM 48.8 13.2\nNIRISS 8.9 0.7\nDraco II NIRCAM 14.7 0.03\nTable 3. The average densities of objects (Column 3) and stars\n(Column 4) from the DOLPHOT photometric reductions of our ERS\nfields.\nters all live within the “optimal” DOLPHOT parameters we\nhave identified for JWST . Therefore, we decided to adopt the\nPHAT set up as our JWST DOLPHOT parameters. Specifi-\ncally, we recommend the PHAT WFC3/IR parameters for the\nNIRCam SW channel and the ACS/WFC parameters for the\nNIRCam LW channel and NIRISS. The full parameter set is\nlisted in Table 4.\n4.DOLPHOT PHOTOMETRY OF EARLY RELEASE\nSCIENCE DATA\nIn this section, we present DOLPHOT photometric reduc-\ntions of our ERS NIRCam and NIRISS data. We used the\nprocedures described in §3 and the DOLPHOT parameters\nlisted in Table 4 for all targets. For each target and instru-\nment, we discuss examples that illustrate the results of our\nDOLPHOT runs. The full catalogs are available on MAST\nfor interested readers to download. Similarly, step-by-step\ndetails for our runs are available on our ERS documentation\npage. Comparisons between our reductions and stellar mod-\nels (i.e., to demonstrate the reasonability of the zero points\nand stellar models, which have not changed since our past\npublication) were already made in Weisz et al. (2023), and\nwe do not repeat those comparisons here. We do not include\nthe Draco II NIRISS field in our analysis as there were not\nenough bright stars in the field for astrometric alignment in\nDOLPHOT.\n4.1. M92\n4.1.1. NIRCam\nFigure 1 shows the the spatial distribution of objects (left)\nand stars (right) recovered by our application of DOLPHOT\nto the M92 NIRCam imaging. This M92 DOLPHOT reduc-\ntion excludes the 3rd exposure for reasons discussed in §2\nand in Appendix A.\nIn total, DOLPHOT finds ∼9.8×105sources in the NIR-\nCam field. This translates to a density of ∼28objects per\narcsec2(Table 3). The spatial distribution of all objects (left\npanel) shows that a sizable number of these objects are ob-\nvious artifacts and not M92 stars. Visually, the most obvi-\nous contaminants are sources associated with bright, satu-\nrated stars. These objects trace both the cores and diffrac-\ntion spikes. Though less obvious visually, there are a large10 W EISZ ET AL .\nA1A2A3A4B4B3B2B1\nFigure 1. The spatial distribution of objects and stars detected by DOLPHOT in our NIRCam imaging of M92, excluding the 3rd exposure.\nLeft: A density map ( ∼50×50pixel bins) of the ∼9.8×105objects reported in the raw DOLPHOT catalog. There is a density gradient\ntoward the center of the image and cluster center. Additional features include a high density of sources that trace saturated stars and a lower\ndensity of sources in the SW chip gaps. Right: The density map of stars that passed the catalog culling criteria listed in § 3.5. These criteria\nremoved the vast majority of obvious artifacts (e.g., corresponding to saturated stars, diffraction spikes) and reveal a clear stellar density gradient\nas is expected for a GC. Note that the individual SW chips are labeled.\n(a)\n (b)\n (c)\nFigure 2. NIRCam CMDs of M92 in select filter combinations. These CMDs exclude the 3rd exposure. Panels (a), (b), and (c) show the\nCMDs for the SW (F090W −F150W), LW (F277W −F444W), and an example SW/LW (F090W −F277W) filter combination. The left side\nof each plot shows CMDs of all objects detected, while the right panels are CMDs of stars that passed the culling criteria. In all cases, the\nphotometry is of excellent quality and the culling critria removes a large fraction of obvious non-stellar sources. There appear to be two MSs\nin the F090W −F277W CMD that are offset in color from one another. This is due to small zero point differences between the NIRCam\nchips/modules.\nnumber of background galaxies in the spatial plot of all ob-\njects. The SW interchip gaps are clearly visible in both spa-\ntial plots. All sources in these chip gap regions correspond\nto objects detected in LW filters only, as there is no SW cov-\nerage in the chip gaps owing to the different detector shapes\nand our choice not to fill the gaps by dithering. The gaps are\ncompletely empty in the star-only plot (right panel) becauseour nominal culling criteria require detections in the SW fil-\nters.\nThe right panel of Figure 1 shows the spatial distribution\nof stars in the M92 NIRCam field (i.e., objects that passed\nthe culling criteria defined in §3.5). The culled catalog con-\ntains 9.7×104stars, which is ∼10% of the total number\nof objects detected. Visually, the spatial distribution of stars\nqualitatively follows what is expected of a globular clusterJWST R ESOLVED STELLAR POPULATIONS V 11\nFigure 3. The completeness and photometric uncertainty for the\nM92 NIRCam F090W and F150W data computed from artificial\nstar tests. The top panel shows the completeness function for\neach filter. The bottom panels show the recovered minus the in-\nput magnitude difference for ASTs that pass the same culling crite-\nria as applied to the photometry. The 50% completeness limits are\nmF090W= 26 .4andmF150W= 25 .4. Both filters have a bias\nconsistent with zero.\n(e.g., King 1962): a higher concentration of stars in the cen-\nter, with a decrease in density as a function of increasing ra-\ndius. Many of the obvious artifacts have been removed such\nas objects associated with saturated stars and more extended\nbackground galaxies. As discussed in Warfield et al. (2023),\nthese culling criteria are designed with purity in mind, though\nthey are not perfect, as some objects associated with diffrac-\ntion spikes and compact background galaxies can still be mis-\ntaken for stars and included in the culled catalogs. Though\nthese only represent a small fraction of the bona fide M92\nstars, careful inspection is required if individual stars/objects\nare of interest (e.g., those that occupy sparsely populated re-\ngions of a CMD).\nFigure 2 shows an illustrative set of CMDs for all objects\n(left panels) and stars (right panels) in the M92 NIRCam field\nfor select SW and LW filter combinations. Panel (a) shows\nthe F090W −F150W CMDs. The effects of the culling crite-\nria are quite dramatic, particularly at faint magnitudes. The\nmajority of non-stellar sources are located at the very bottom\nof the CMD (i.e., low S/N) or in regions of the CMD not\ntypically occupied by stellar sources. The application of the\nculling criteria removes ∼90% of the detected objects, pro-\nducing the exquisitely deep CMD shown in the right panel.\nThe resulting stellar CMD shows a very tight lower main\nsequence as expected for a metal-poor GC. The bright end\nof the CMD begins at the main sequence turn off (MSTO),\nwhile fainter features such as the MS kink and bottom of the\nstellar sequence (i.e., M⋆∼0.1M⊙) are evident. These fea-\ntures are discussed in Weisz et al. (2023). The sparse col-\nlection of objects near the bottom of the SW CMD are some\ncombination of compact background galaxies that were not\npicked up by the culling criteria and a small number of white\ndwarfs (Nardiello et al. 2022). Brown dwarfs are likely tooDetector Parameter Value\nNIRCam/SW RAper 2\nNIRCam/SW Rchi 1.5\nNIRCam/SW Rsky2 “3 10”\nNIRCam/LW RAper 3\nNIRCam/LW Rchi 2.0\nNIRCam/LW Rsky2 “4 10”\nNIRISS RAper 3\nNIRISS Rchi 2.0\nNIRISS Rsky2 “4 10”\nAll FitSky 2\nAll PSFPhotIt 2\nAll PSFPhot 1\nAll SkipSky 1\nAll SkySig 2.25\nAll SecondPass 5\nAll SigFindMult 0.85\nAll MaxIT 25\nAll NoiseMult 0.1\nAll FSat 0.999\nAll FlagMask 4\nAll ApCor 1\nAll Force1 0\nAll PosStep 0.25\nAll RCombine 1.5\nAll SigPSF 5.0\nAll PSFres 1\nAll InterPSFlib 1\nAll UseWCS 2\nAll CombineChi 0\nTable 4. Recommended DOLPHOT input parameters based for\nNIRCam SW, LW, and NIRISS imaging based on the extensive pho-\ntometric testing described in §3.8. A detailed description of each\nparameter can be found in the DOLPHOT manual and on our ERS\ndocumentation webpage.\nfaint to be included in this CMD (e.g., Dieball et al. 2019).\nWe examined the CMDs as a function of SW chip and found\nthem to be in generally good agreement.\nPanel (b) shows the LW NIRCam CMD of M92. Appli-\ncation of the culling criteria, which is based only on the SW\ndata, removes many artifacts and produces one of the deepest\nmid-IR CMDs of a GC to date. The LW CMD includes the\nMSTO at the bright end, and extends only to the middle of\nthe MS kink at the faint end. Culling criteria tailored specif-\nically to the LW channels may be able to produce a slightly\ndeeper CMD with fewer points away from the MS. However,\na full exploration of filter dependent culling criteria is beyond12 W EISZ ET AL .\nA1A2A3A4B4B3B2B1\nFigure 4. The spatial distribution of objects and stars detected by DOLPHOT in our NIRISS imaging of M92. Left: A density map ( ∼50×50\npixel bins) of the ∼1.3×105objects reported in the raw DOLPHOT catalog. The spatial distribution of detected objects is uniform. Some\nartifacts (e.g., bright stars, claws; Rigby et al. 2022) are visible. Right: The density map of the ∼2.9×103stars that passed the catalog culling\ncriteria listed in §3.5. These criteria removed the vast majority of obvious artifacts (e.g., corresponding to saturated stars, diffraction spikes),\nleaving a sparse sampling of stars. This low density is expected, given that the field is located at ∼5 half-light radii.\nFigure 5. The NIRISS CMD of M92, excluding the 3rd exposure.\nThe left panel shows all objects detected, the right panel shows the\nstars that passed the culling criteria. Compared to the SW NIRCam\nCMD in Figure 2, the NIRISS CMDs cover a smaller dynamic range\nin luminosity, owing to saturation effects at the bright end and lower\nsensitivity at the faint end. The diagonal feature of stars at F090W ∼\n20corresponds to objects in image artifacts that were not removed\nby the applied culling criteria. The increased scatter in this CMD,\nrelative to the SW NIRCam CMD, is due to the combination of\nlower SNR as well as WebbPSF models that are not as well-matched\nto the observed PSFs.\nthe scope of this paper. For interested readers, this exercise\ncan be readily done with the public ERS catalogs we provide\non MAST.Target Camera NASTs F090W F150W\n50% Comp. 50% Comp.\n(mag) (mag)\n(1) (2) (3) (4) (5)\nM92 NIRCam 2871773 26.4 25.4\nNIRISS 1168289 28.1 27.4\nWLM NIRCAM 1573112 28.7 27.7\nNIRISS 1168465 29.3 28.5\nDraco II NIRCAM 408455 29.6 28.3\nTable 5. Summary of the SW artificial star tests for each of our ERS\ntargets. Column (3) lists the number of ASTs run; Columns (4) and\n(5) list the F090W or F150W magnitude corresponding to the 50%\ncompleteness limit.\nThe LW CMD shows some structure at the brightest mag-\nnitudes of the LW near the MSTO. The source is like related\nto the nuances of saturation and star locations with respect to\nthe center of a pixel. Savino et al. in prep. discusses these\neffects in more detail.\nPanel (c) shows the F090W −F277W CMD of the M92\nNIRCam field. As with the other example CMDs, the culling\ncriteria provide for the removal of many non-stellar sources.\nThe resulting CMD extends from the MSTO at the bright\nend to the bottom of the MS at the faint end. A close in-\nspection of this CMD shows a slight bifurcation in the MS,\nthat is most visible near the MSTO. The two MSs are offset\nby∼0.05mag. Closer inspection of our data shows that\nthe color of the sequences changes as a function of NIRCam\nchip/module. The offsets are most apparent in the SW −LW\nCMDs (e.g., it is also clear in the F150W −F444W CMD),\nand are far smaller in the SW-only CMDs, and certainly notJWST R ESOLVED STELLAR POPULATIONS V 13\nA1A2A3A4B4B3B2B1\nFigure 6. The spatial distribution of objects and stars detected by DOLPHOT in our NIRCam imaging of WLM. Left: A density map ( ∼50×50\npixel bins) of the ∼1.7×106objects reported in the raw DOLPHOT catalog. The spatial distribution of detected objects is fairly uniform, with\na slight gradient toward chip A1, which is positioned closest to the center of the galaxy. Additional features include a high density of sources\nthat trace saturated stars and a lower density of sources in the SW chip gaps. Right: The density map of the ∼4.6×105stars that passed\nthe catalog culling criteria listed in § 3.5. These criteria removed the vast majority of obvious artifacts (e.g., corresponding to saturated stars,\ndiffraction spikes). The density of stars increases toward chip A1, which is closest to the center of the galaxy.\n(a)\n (b)\n (c)\nFigure 7. NIRCam CMDs of WLM in select filter combinations. Panels (a), (b), and (c) show the CMDs for the SW (F090W −F150W),\nLW (F250M −F430M), and an example SW/LW (F090W −F250M) filter combination. The left side of each plot shows CMDs of all objects\ndetected, while the right sides are CMDs of stars that passed the culling criteria described in § 3.5. The SW CMD of WLM is the deepest\never constructed for a galaxy that is not within the viral radius of the MW. A number of stellar evolution sequences are quite tight (e.g., RGB,\nsub-giant branch, young MS), in line with the exquisite SNR. Panel (b) shows the LW-only CMD, which shows a bright AGB star sequence,\na bright RGB, and a well-populated RC. The increased scatter below the red clump is the result of culling criteria applied only to SW data.\nPanel (c) shows an example SW-LW CMD. As with panel (a), a number of clear sequences emerge. The CMD nearly reaches the oldest MSTO,\ndespite the LW filter being a medium band.\nas large as our team initially reported in Boyer et al. (2022).\nOur findings indicate that the spatial zero points need further\nrefinement, which is a goal of the JWST absolute flux cali-\nbration program (Gordon et al. 2021). Updates to the zero\npoints should produce even tighter sequences in M92.Figure 3 shows the SW completeness functions (top panel)\nand photometric bias and scatter (bottom panels) as deter-\nmined from ∼106ASTs inserted into the NIRcam images\nof M92. The shape of the completeness functions behaves\nas expected for a mostly uncrowded stellar field. The com-14 W EISZ ET AL .\npleteness is >50% for the entirety of the stellar sequence,\nreaching 50% at mF090W= 26 .4andmF150W= 25 .4.\nThe completeness gradually decreases until it reaches zero\nat F090W ∼28.1and F150W ∼27.6.\nThe bottom two panels show the difference between the re-\ncovered and input magnitudes for ASTs that pass the culling\ncriteria. The mean of both distributions is 0, which indicates\nno bias in the AST recovery. The scatter increases as a func-\ntion of magnitude in the expected manner for ASTs (e.g.,\nMonelli et al. 2010; Dalcanton et al. 2012a). The LW fil-\nter for M92, and the other targets in our program, has similar\nAST characteristics – no bias and scatter that behaves as ex-\npected.\nTwo other studies have previously published CMDs of\nM92 using JWST ERS imaging. Nardiello et al. (2022) and\nZiliotto et al. (2023) performed photometry on our M92 NIR-\nCam imaging using empirical PSFs, based on the method of\nAnderson & King (2000). In general, the CMDs produced by\ntheir programs and DOLPHOT are qualitatively similar, i.e.,\ndeep and high SNR.\nHowever, there are a number of subtle details in the reduc-\ntion procedures that affect interpretation of the results at the\nfew to several percent level of precision and accuracy. For\nexample, Ziliotto et al. (2023) use the 3rd exposure of M92,\nwhich adds extra noise (see Appendix A) and do not note the\nmodule dependent zero point offset in F277W/F444W that\nwas present prior to the zero point updates provided by STScI\nin Fall 2023. We found these to introduce non-negligible bias\nand scatter in the LW photometry. In principle, such scatter\ncould enhance multiple population effects they identify.\nSimilarly, the rapid publication timescale of Nardiello et al.\n(2022) meant much of the calibration work on JWST was in-\ncomplete or unavailable. Their results preceded post-launch\nSTScI zero points, stable filter curves, flat field updates, and\nusable DQ arrays. They circumvented many of these issues\nby anchoring their photometry to theoretical predictions from\nthe BaSTI stellar evolution models (Hidalgo et al. 2018).\nThis has the effect of producing qualitatively good-looking\nCMDs, but it also glosses over many of the calibration effects\nfor which we chose M92 as an ERS target. Our program was\ndesigned to obtain high S/N of a target with a simple stellar\npopulation in order to help diagnose potential shortcomings,\nsystematics, etc., as we have done in this paper.\n4.1.2. NIRISS\nFigure 4 shows the spatial distribution of 1.3×105objects\n(left) and 2.9×103stars (right) for our NIRISS M92 field.\nThe objects are essentially uniformly distributed across the\nfield. There are some visually obvious artifacts, including\na bright foreground star in the center of the field, as well\nas some claws and wisps due to scattered light from nearby\nbright objects.\nAs with NIRCam, the culling criteria identified ∼98% of\nthe objects as non-stellar artifacts. The resulting stellar field\nis sparsely populated, which is expected due to its location at\n∼5 half-light radii.Figure 5 shows the NIRISS CMDs of M92 for all objects\n(left) and stars (right). The majority of non-stellar objects\nrejected by the catalog culling are faint sources near the bot-\ntom of the CMD. The resulting stellar CMD in the left panel\nshows a clear lower MS of M92. The NIRISS CMD spans a\nsmaller dynamic range in luminosity compared to NIRCam.\nAt the bright end, the CMD reaches the top of the MS kink,\nnot the oldest MSTO. This is likely due to saturation effects\nowing to different detector characteristics. The NIRISS stel-\nlar CMD has stars nearly as faint as those in NIRCam, but\nthe scatter is visually much larger at the bottom of the MS.\nThis may be due to the lower throughput of NIRISS. Addi-\ntionally, as discussed in §3.4, the NIRISS WebbPSF models\nare marginally worse matches to the observed PSFs (i.e., the\nmodels have too much centrally concentrated light), which\nresults in larger PSF adjustments from DOLPHOT and a\nlarger systematic uncertainty floor of ∼0.01mag per fil-\nter, which is an order-of-magnitude larger than the PSF-based\nuncertainties for NIRCam. The net result is that the increased\nwidth of the MS in M92 is at least in part driven by the model\nPSFs.\nTable 5 includes summary statistics of the NIRISS M92\nSW ASTs. The recovered ASTs show no bias and minimal\nscatter, much like the NIRCam ASTs. The 50% complete-\nness limits of the NIRISS field are ∼2magnitudes fainter\n(mF090W= 28 .1,mF150W= 27 .4) than the NIRCam\nfield, which is the result of no crowding in the NIRISS field,\ncompared to modest crowding in the center of the NIRCam\nfield. The density of objects and stars in the NIRISS field is\n∼4−10times less than it is in the NIRCam field (Table 3).\n4.2. WLM\n4.2.1. NIRCam\nFigure 6 shows the spatial distribution of objects (left) and\nstars (right) from our DOLPHOT photometric reduction of\nWLM. In total, DOLPHOT finds ∼1.7×106objects in the\nWLM field, yielding a typical density of ∼49objects per\nsq. arcsec. Of these, ∼4.6×105(∼27%) pass the culling\ncriteria, yielding a stellar density of ∼13stars per sq. arcsec,\nwhich is the highest density of our ERS fields.\nThe WLM observations are oriented such that chips A1\nand A2 are closest to the center of the galaxy, while B1 and\nB2 are farthest away. This orientation produces a spatial gra-\ndient from A1 (highest density) to B1 (lowest density) that\nis clearly visible in both the object and stellar spatial maps.\nAs with M92, the object map shows several artifacts (fore-\nground stars, background galaxies) that are largely rejected\nby the culling criteria. The inter-module and inter-chip gaps\nare not populated due to our choice in dithers and culling cri-\nteria.\nFigure 7 shows select NIRCam SW and LW WLM CMDs.\nPanel (a) plots the F090W −F150W CMDs for all objects\n(left) and stars (right). The majority of sources removed by\nthe catalog culling are faint, low S/N objects. The result-\ning stellar CMD of WLM is the deepest ever obtained for a\ngalaxy outside the virial radius of the MW. It’s remarkable forJWST R ESOLVED STELLAR POPULATIONS V 15\nA1A2A3A4B4B3B2B1\nFigure 8. The spatial distribution of objects and stars detected by DOLPHOT in our NIRISS imaging of WLM. Left: A density map ( ∼50×50\npixel bins) of the ∼1.6×105objects reported in the raw DOLPHOT catalog. The spatial distribution of detected objects is fairly uniform.\n‘Holes’ in the spatial distribution mainly correspond to saturated pixels that were entirely masked by DOLPHOT. Right: The density map of\nthe∼1.3×104stars that passed the catalog culling criteria listed in § 3.5. There is a modest spatial gradient that increases in the direction of\nthe main body of the galaxy.\nits depth and precision. Many of the features in the SW CMD\nare similar to a previous analysis of WLM with HST/ACS\n(Albers et al. 2019) and are discussed in more depth in our\nteam’s SFH paper of WLM (McQuinn et al. 2023). Here, we\nbriefly summarize CMD features. At the bright end, we see\na clear young MS population indicting the presence of re-\ncent star formation. Slightly redder than the MS, is the blue\ncore helium burning sequence (BHeB). Part of this sequence\nfalls in the instability strip and these stars appear as Cepheids,\nsome of which have been targeted from the ground over the\nlast few decades (e.g., Sandage & Carlson 1985; Pietrzy ´nski\net al. 2007). For redder bright stars, we see a well-defined\npopulation of asymptotic giant branch (AGB) stars that are\nlocated above a clearly defined tip of the red giant branch\n(TRGB). This AGB star population is analyzed in detailed in\nBoyer et al. (2024). The RGB is narrow and well-populated.\nA mixture of AGB stars and red core helium burning stars\nare located at slightly bluer colors than the RGB. There is\na prominent, tight red clump (RC) at mF090W∼25along\nwith a clear horizontal branch. Vertically extending from the\nred clump is the red helium burning sequence, the brightest\nof which are considered red supergiants. The CMD extends\n∼2magnitudes below the oldest main sequence turnoff.\nPanel (b) shows the LW CMDs of the WLM NIRCam field.\nThe culling criteria have left a fair number of low S/N ob-\njects on the stellar CMD, indicating that improvements could\nbe made by including the LW filters in the culling. Read-\ners who wish to explore this are encouraged to download our\nphotometry from MAST.\nAs expected, the RGB in the stellar CMD is narrow in this\nfilter combination as they are both well into the Rayleigh-\nJeans tail of RGB stellar flux distributions. There is a promi-\nnent, bright AGB star population. The CMD begins to\nbroaden substantially below the red clump. Though the LWCMD doesn’t reach the oldest MSTO, it is nevertheless re-\nmarkable as it is the deepest medium band CMD of a galaxy\noutside the MW satellites.\nPanel (c) shows an illustrative SW −LW CMD. Here, the\nculling criteria do an acceptable job of removing non-stellar\nobjects, though including F250M-specific criteria should im-\nprove the faint end source classification. The CMD features\nare generally similar to those shown in panel (a) though the\nCMD is not as tight in general or quite as deep. Culling cri-\nteria tailored to F250M, along with ASTs, are necessary to\ndetermine if the oldest MSTO is brighter than the 50% com-\npleteness limits.\nThe SW AST results for WLM are summarized in Ta-\nble 5. The 50% completeness limits of F090W (28.7) and\nF150W (27.7) are just below the oldest MSTO. Because the\nculling criteria were designed for purity, and not complete-\nness, relaxing the sharpness will extend the completeness\nlimits fainter. Such decisions need to be driven by the sci-\nence. For example, star formation history measurements may\nbe able to tolerate a decrease in purity in exchange for fainter\ncompleteness limits (e.g., McQuinn et al. 2023).\n4.2.2. NIRISS\nFigure 8 shows the spatial distribution of objects (left) and\nstars (right) in the NIRISS field of WLM. The objects are\ngenerally distributed uniformly, with a handful of low den-\nsity regions owing mainly to saturated pixels. The culling\ncriteria removes ∼90% of the objects from the field, leav-\ning a sparse stellar distribution, which is consistent with this\nfield’s location in WLM’s stellar halo. There is a slight gra-\ndient in the field toward the upper left portion of the field,\nwhich is also the direction of the center of the galaxy. The\nobject and stellar densities are 8.9 and 0.7 per sq. arcsec, in-\ndicating that this is not a crowded field.16 W EISZ ET AL .\nFigure 9. The NIRISS CMD of WLM. The left panel shows all ob-\njects detected, the right panel shows the stars that passed the culling\ncriteria. The NIRISS CMD has a similar depth to the NIRCam\nCMD, albeit with increased scatter. In part, the scatter is due to\nthe lower sensitivity of NIRISS in these filters, along with the less\naccurate NIRISS WebbPSF models. As expected, due to its location\noutside the main body of the galaxy, the NIRISS CMD lacks a stel-\nlar population younger than a few Gyr and is much more sparsely\npopulated.\nFigure 9 shows the CMD of objects (left) and stars (right)\nfor the WLM NIRISS field. The culling criteria remove much\nof the contamination around the RGB and below the oldest\nMSTO, producing the deep and clean stellar CMD. The stel-\nlar CMD lacks stars younger than at least 1-3 Gyr, due to\nits location in the stellar halo. Otherwise, it exhibits many\nof the expected features of old and intermediate age popula-\ntions (e.g., RGB, RC). The CMD extends well-below the old-\nest MSTO, similar to the NIRCam CMD. The RGB, RC, and\nMSTO appear slightly broader on the NIRISS CMD com-\npared to NIRCam. This is unlikely to be due to a more com-\nplex stellar population, and instead may reflect differences in\nNIRISS and NIRCAM, such as lower throughput and slightly\nless accurate PSF models for NIRISS.\nTable 5 summarizes the AST results for the WLM NIRISS\nfield. As with the other field, the ASTs show no bias and lit-\ntle scatter, indicating that they are well-recovered. The 50%\ncompleteness limits are mF090W= 29 .3andmF150W=\n28.5, which are 0.6 and 0.8 mag deeper than the same filters\nin NIRCam. This is because the NIRISS field is located in\nthe much less crowded stellar halo.\n4.3. Draco II\nFigure 10 shows the spatial distribution of objects (left)\nand stars (right) for our NIRCam observations of Draco II.\nThe object density plot shows many familiar artifacts includ-\ning bright foreground stars, chip gaps, and background galax-\nies. Additionally, there are large, diffuse overdensities thatcover chips A3, B3, and B4. These features are also present\nin the images themselves and are the result of persistence.\nPrior to our observations, program 1022 spent ∼12hours\ntesting the fine guidance sensor’s ability to track moving ob-\njects near Mars, Jupiter, and Saturn. This resulted in back-\nto-back observations of some of the brightest objects in the\nUniverse, followed by one of the faintest objects in the Uni-\nverse.\nAs shown in the right panel, the culling criteria does an\nexcellent job of removing these overdensities, along with the\nother artifacts. The result is an extremely low density stel-\nlar field with just 0.03 stars per sq. arcsec, which is typical\nof an ultra-faint dwarf galaxy. Bagley et al. (2023) report\npersistence in the CEERS data from the same Solar System\nprogram and develop a routine to mask out pixels affected by\npersistence. In our case, this would result in ∼1/3 of our field\nnot being analyzed at all. While this provides a suitable solu-\ntion, another may be to specify that observations should not\nbe scheduled when extreme persistence may be a problem.\nFigure 11 shows select CMDs of Draco II. Panel (a) shows\nthe SW CMD in which application of the culling criteria pro-\nduces a deep, fairly clean CMD of Draco II. The CMD ex-\ntends from the oldest MSTO to beyond the bottom of the stel-\nlar sequence. The MS kink is clearly visible at F090W ∼23.\nThe large scatter at the bottom of the CMD is the result of\nconfusion between background galaxies, stars, and the gap\nbetween stars and brown dwarfs. Even with cuts designed\nfor purity, we are not able to readily discern between stars\nand compact galaxies at such faint magnitudes. This is the\ndeepest CMD (i.e., it reaches the lowest mass main sequence\nstar) ever constructed of a galaxy outside the MW.\nPanel (b) shows the LW of Draco II. The SW culling cri-\nteria drastically reduce the number of contaminants, leaving\na clear MS in the right hand panel. Further contamination,\nparticularly near the faint end, could possibly be removed\nby adding LW-specific culling criteria. The F360M-F480M\ncolor provides little leverage on temperature, resulting in a\nnearly vertical MS.\nPanel (c) shows an example SW and LW CMD (F090W-\nF360M). The SW culling criteria do a reasonable job remov-\ning contamination down to F090W ∼26, below which there\nis a noticeable increase in scatter. This scatter is likely due\nto the low SNR of the F360M data at such faint magnitudes\nas well as the lack of an LW-specific culling criteria. The\nstellar CMD shows a clear lower MS, including the MS kink.\nThe F360M filter was taken specifically for its metallicity\nsensitivity and analysis of this CMD could, in principle, pro-\nvide one of the tightest constraints on whether Draco IIis a\nbona fide UFD or GC, which remains an open question in\nthe literature (e.g., Baumgardt et al. 2022; Fu et al. 2023),\nby determining if its metallicity distribution function has a\nstatistically significant spread.\nTable 5 lists the AST properties for this NIRCam field.\nThe 50% completeness limits are mF090W= 29 .6and\nmF150W= 28.3, which are nearly at the bottom of the CMD.\nThe NIRISS field did not have enough bright stars to align\nproperly in DOLPHOT and we therefore do not discuss it.JWST R ESOLVED STELLAR POPULATIONS V 17\nA1A2A3A4B4B3B2B1\nFigure 10. The spatial distribution of objects and stars detected by DOLPHOT in our NIRCam imaging of Draco II. Left: A density map\n(∼50×50pixel bins) of the ∼5.1×105objects reported in the raw DOLPHOT catalog. The spatial distribution of objects shows a number\nof features, including saturated foreground stars and background galaxies. The other prominent features are the diffuse object overdensities\nthat are present in chips A3, B3, and B4. These are the result of persistence from the previous program (1022), which was staring at/near\nMars, Jupiter, and Saturn for ∼12 hours in order to test the fine guidance sensor. Right: The density map of the ∼1.1×103stars that passed\nthe catalog culling criteria listed in §3.5. These criteria removed the vast majority of obvious artifacts (e.g., corresponding to saturated stars,\ndiffraction spikes) as well as eliminated most of the contamination from persistence.\n(a)\n (b)\n (c)\nFigure 11. NIRCam CMDs of Draco II in select filter combinations. Panels (a), (b), and (c) show the CMDs for the SW (F090W −F150W),\nLW (F360M −F480M), and an example SW/LW (F090W −F360M) filter combination. The SW CMD of Draco II (panel a) is the deepest\never (i.e., it reaches the lowest stellar mass) constructed for a galaxy outside the MW itself. It includes a clearly defined MS Kink. Panel\n(b) shows the LW-only CMD, which is essentially vertical at a color of ∼0due to the lack of temperature sensitivity in the F360M −F480M\nfilter combination. Panel (c) shows an example SW-LW CMD (F090W −F360M), which extends below the MS kink before photometric scatter\nwashes out the stellar sequence. This depth is impressive for a medium-band LW filter. For both CMDs that include the LW filter, specific LW\nculling criteria may reduce some of the noise in the CMD, particularly at the faint end.\nWe note that although the culling criteria appear to do a\ngood job of removing the contamination due to persistence, it\nis unclear how many stars in Draco II were also removed. In\ngeneral, it may be advisable in proposal planning to request\nthat observations of resolved galaxies be scheduled such thatpersistence is unlikely to be an issue. If this had been WLM\ninstead of Draco II, it is possible that a significant number of\nstars may have been lost in the persistence-induced noise.\n5.DISCUSSION18 W EISZ ET AL .\nFigure 12. An illustration of improvements in DOLPHOT NIRCam photometry since the publication of our survey paper in early 2023\ndemonstrated by showing F090W-F150W CMDs of M92 from various DOLPHOT runs in the past year. Panel (a) shows the CMD from our\nERS survey paper Weisz et al. (2023) from January 2023. It was constructed using all exposures of M92, including the anomalous 3rd exposure,\nand used JWST calibrations and WebbPSF models from late 2022. Panel (b) shows photometry from the same time frame, only without the 3rd\nexposure. Panel (c) shows the CMD of M92 (no 3rd exposure) using the older WebbPSF models (v1.1) and calibration data, but with updated\nNIRCam zero points released in Fall 2023. Panel (d) shows the M92 CMD published in this paper (with no 3rd exposure), which includes\nupdated WebbPSF models and very recent calibrations (e.g., flat fields) and zeropoints. From left to right, the CMDs have noticeably less\nscatter, tighter stellar CMD sequences (e.g., MSTO, MS kink), more stars, and improved depth.\n5.1. Evolution of JWST DOLPHOT Photometry\nOur knowledge of JWST and its instruments has greatly\nimproved since our team’s ERS data was acquired in mid-\n2022. Among the improvements during this time are more\naccurate calibrations (e.g., flat field, zero points), better data\nquality masking, and more realistic model PSFs. During the\ncourse of our ERS program, we have continued to incorpo-\nrate these changes into DOLPHOT.\nFigure 12 illustrates the impact of these revisions on the\nSW CMD of M92. Panel (a) shows the SW CMD of\nM92 that was originally published in our ERS survey pa-\nper (Weisz et al. 2023), which includes all 4 exposures.\nPanel (b) was reduced with the same DOLPHOT config-\nuration as panel (a), but without the anomalous 3rd ex-\nposure. CMDs in both panels (a) and (b) were con-\nstructed using images produced by the the JWST pipeline\nversion with CALVER=1.9.3, CRDS VER=11.16.18, and\nCRDS CTX=jwst p1063.pmap, as well as PSF models using\nWebbPSF version 1.1.1.\nThe CMD in Panel (c) used the same setup as above,\nbut with different zeropoints. Specifically, they are fromCRDS CTX=jwst p1126.pmap, whichas released in Fall\n2023. These zero points were applied to the photometry after\nit was already run. A notable improvement in panel (c) was a\nreduction in chip-to-chip photometric offsets, which we ob-\nserved to range from 0.02 to 0.1 in all filters in all previous\nversion of our photometry (i.e., panel b).\nFinally, panel (d) shows the CMD presented in this paper,\nwith the most up-to-date PSFs and calibrations. Some of the\nupdates include flat fields, zero points, the switch from Vega\nto Sirius as a reference star, and WebbPSF models that model\ninterpixel capacitance and charge diffusion. WebbPSF v1.2\nwas released in late 2023.\nVisually, there is clear and dramatic improvement in these\nCMDs over time. The progression from panel (a) to (d) is\none of tighter sequences, less scatter, improved definition of\nthe MSTO, and greater depth. Our most recent CMD also\ncontains more stars than previous versions.\nFigure 13 shows an even more dramatic improvement over\nthe same time and parameter range. Our initial NIRISS\nCMD (panel a) exhibited a tremendous about of scatter due\nin large part to the effect of the 3rd exposure. The removal\nof the 3rd exposure (panel b), significantly reduced the scat-JWST R ESOLVED STELLAR POPULATIONS V 19\nFigure 13. Improvements in DOLPHOT NIRISS photometry, illus-\ntrated by CMDs of M92. Panels (a) and (b) are the NIRISS CMDs\nreleased as part of the ERS survey paper in January 2023 (Weisz\net al. 2023); with the CMD constructed from 4 exposures shown\nin panel (a) and the CMD constructed without the 3rd exposure in\npanel (b). Panel (c) shows the NIRISS CMD from this paper, which\nincludes update WebbPSF models and JWST NIRISS calibrations.\nIt does not include the 3rd exposure. The CMD in panel (c) has\nless scatter, tigther sequences, more stars, and greater depth than\nour previous NIRISS CMDs.\nter in the CMD. Panel (c) shows the current NIRISS CMD\nof M92, with no 3rd exposure. Relative to panel (b) it has\na tighter main sequences and contains more stars. These\nimprovements were almost entirely the result of improved\nWebbPSF models. Previous WebbPSF models had too much\nlight concentrated in the central pixel compared to observa-\ntions. The current NIRISS WebbPSF models are still slightly\ntoo sharp, but this only affects the photometry at the level of\n∼0.01mag, whereas the previous generation introduced a\nscatter of ∼0.09mag.\n5.2. Point Spread Function Time Variability\nSpace-based telescopes are typically characterized by a\nmuch higher degree of PSF stability than ground-based fa-\ncilities. However, even space telescopes exhibit some PSF\ntime dependence, due to, for example, thermal variations or\nsmall impacts. HST is known to exhibit such effects (e.g.,\noptical telescope assembly breathing; Hasan 1994) and they\nare generally small and stable enough to be corrected for by\nthe PSF model adjustments performed by DOLPHOT. JWST\nis expected to have similar temporal changes in the PSF (e.g.,\nsee Sec 6.2 of McElwain et al. 2023). Here, we undertake a\npreliminary characterization of the effects of temporal vari-\nations in the PSF on the DOLPHOT photometry of the ERS\ntargets.\nFigure 14 shows the time-series wavefront sensing mea-\nsurements for JWST ’s optical telescope element (OTE),\nalong with related encircled-energy variations in the NIR-\nCam F150W PSF, for the months of July (top plot) and\nSeptember (bottom plot) 2022 as generated by WebbPSF. Formost of the measured epochs, JWST ’s optical performance\nshows remarkable stability, with minimal deviations from\ncommissioning alignment. However, sporadic events can\noccur when the telescope drifts away from nominal perfor-\nmance. While corrections to the mirror segment positioning\nare rapidly issued to bring the telescope back to commission-\ning alignment, observations taken before the corrections are\napplied will likely present significant variations from nomi-\nnal PSF models.\nWithin the twelve months period spanning June 1st, 2022\nto May 31st, 2023, 14 such events occurred. Deviations from\nnominal performance lasted between two and seven days be-\nfore corrections were issued. The largest event recorded so\nfar occurred between July 11th 2022 and July 15th 2022 (top\nplot of Fig. 14), with changes to the encircled-energy of var-\nious filters changing more than 5% at a 10 pixel radius.\nHowever, the majority of the alignment anomalies were\nmuch smaller. The bottom panel of Fig. 14 shows a typi-\ncal example of such an event, which, in this case, occurred\nbetween September 6th 2022 and September 10th 2022. Re-\nsultant changes to the PSF were within mission stability re-\nquirements.\nAs the majority of these events are related to thermal set-\ntling of the spacecraft (e.g., McElwain et al. 2023), they pri-\nmarily occurred in the first six months of scientific opera-\ntions. In fact, since November 2022, only two such misalign-\nments have occurred, both of them with minimal deviations\nfrom nominal performance. The outlook for JWST ’s optical\nstability is therefore very promising.\nNevertheless, is important to quantify the impact\nof time-dependent PSFs on the photometry, espe-\ncially for datasets acquired early in Cycle 1. To do\nso, we computed two alternative PSF grids for NIR-\nCam, using OPD maps corresponding to the July\n“large event” ( R2022071502-NRCA3 FP1-1.fits ;\nJuly 15th, 2022) and to the September “small event”\n(R2022090902-NRCA3 FP1-1.fits ; September 9th,\n2022). We also calculated a third grid which corresponds\nto nominal alignment changes in the telescope two months\nafter our official PSFs OPD (“O2022092302-NRCA3 FP1-\n1.fits”; September 23rd, 2022). We term this last test the\n“no event” case. The no event case is meant to test how\nnormal operational variations in the telescope (e.g., mirror\nalignments, thermal effects) manifest in DOLPHOT PSF\nphotometry under the assumption that the PSF is computed\nat one epoch but applied to data taken at an epoch 2 months\nlater. We then re-ran DOLPHOT on our three ERS targets,\nusing these alternative grids, and compared the differences\nin the photometry.\nThe DOLPHOT-generated catalogs from the two epochs\nare spatially cross-matched by (a) only considering stars with\nSNR>50in both epochs and (b) requiring their spatial coor-\ndinates to match within 0.15pix. For the large event, we were\nonly able to adequately match sources with a much larger\nradius of 2pix. We discuss the implications of the spatial\nmatching radius below. We use these high SNR stars cross-\nmatched between the two epochs to assess differences in the20 W EISZ ET AL .\nphotometry. As a point of reference, we also compare them\nto the expected scatter from ASTs of images analyzed with\nPSFs at the same epoch, i.e., our nominal photometry.\nFor each test, we compare properties of the cross-matched\nDOLPHOT photometry to the nominal photometry, i.e., the\nphotometry presented in §4. For the nominal case, we use\nthe ASTs with SNR >50 to assess the bias and scatter, which\nserve as a reference point by which to assess the effects of\ntime variations on the photometry. To illustrate expectations\nfrom the ASTs, we show the expected bias and scatter for\nM92 in the F150W filter in the top panel of Figure 15. We\nsee the bias is smaller than the scatter and is consistent with\nzero. The amplitude of the scatter increases as expected for\nASTs.\nThe next 3 panels in Figure 15 show an example of how\nthe M92 NIRCam F150W photometry compares between our\nnominal catalog and the three types of events we consider.\nSpecifically, we plot the difference in F150W magnitudes be-\ntween the two sets and compute the mean and scatter for all\nstars with SNR >50.\nBecause we are comparing photometry of identical stars\nbetween the epochs, the expectation is that the difference in\nmagnitude should always be zero. The only variable in the\nreduction is the PSF library, thus any differences we find are\nsolely due to variations in the PSF.\nFor the no event case, we find no bias ( µ=−0.001) and\na very small scatter ( σ= 0.008). This indicates that while\nthe photometry is not identical between the epochs, the dif-\nferences are small and on order of systematics introduced by\nthe PSF models at the same epoch. These effects are also\nsmaller than the noise reported by the ASTs, shown in panel\n(a).\nThe small event case (panel c) tells a similar story. The\nbias and scatter are small, though the scatter is not entirely\nnegligible. Specifically, in this case, an accurate characteri-\nzation of the noise would require adding σ= 0.012mag in\nquadrature to other sources of noise.\nThe bottom panel of Figure 15 shows the results for a large\nevent. In this case, the mean difference in the photometry\nis small, but non-zero ( µ=−0.02mag), and the scatter\nσ= 0.07mag) is larger than in the no event and small event\ncases. The scatter is a factor of ∼2larger than the noise re-\nported by the ASTs and a factor of ∼3.5larger than the noise\nfrom the Poisson noise in the photometry (i.e., SNR >50\ntranslates to a photometric error of <0.02mag). In the large\nevent case, the time variations in the PSF are actually the\ndominant source of photometric uncertainty and would need\nto be included in any subsequent modeling of the data.\nBeyond the addition of significant photometric noise, we\nalso found that the large event made the astrometry less\nrobust. Specifically, in order to match stars between the\nlarge event and nominal catalogs, we had to expand the\npixel matching radius from 0.15 to 2 pix, over an order-of-\nmagnitude increase. Smaller search radii did not yield a rea-\nsonable number of matches. We found that increasing the\nsearch radius to 2 pix was necessary for matching catalogs for\nlarge events for any of our ERS targets. It may be possible tomitigate some of this mismatch by increasing the DOLPHOT\nparameter Rcombine However, this exploration is outside\nthe scope of the current paper.\nThough detailed testing of the astrometric performance of\nJWST is beyond the scope of this paper, we suggest that such\ninvestigations are warranted, given that some science cases\n(e.g., proper motions of globular clusters, nearby galaxies)\nforJWST require astrometric precision of ≪1pix (e.g., An-\nderson & King 2000; Sohn et al. 2012; van der Marel et al.\n2012; Kallivayalil et al. 2013).\nTable 6 summarizes the temporal variations in the\nDOLPHOT photometry of NIRCam observations for all ERS\ntargets. In general, the trends illustrated for M92 in Figure 15\nhold for the other targets and filters. For no and small events,\nthe mean differences from the nominal catalogs are <1%,\nwhile the scatter, particularly for the medium bands, can be\nas high as ∼8%. For F090W and F150W, the same general\ntrends hold across all targets.\nTo illustrate the effects of time variations on the PSF,\nFigure 16 shows the pixel-by-pixel flux ratios for NIRCam\nF150W WebbPSF models for the three scenarios considered.\nThe most obvious change in the flux is for the large event,\nwhich shows a significant change in the PSF. Changes in the\nsmall and no event scenarios are more subtle, but still clearly\npresent. Even in the case of no event, changes in the stability\nofJWST and minor re-alignments of the mirrors introduce\nsome changes in the PSF.\nImportantly, as discussed above, in the context of\nDOLPHOT, the PSF alterations generally do not introduce\na substantial bias into the photometry, but can add noise.\nSimilar conclusions are reported elsewhere in the literature.\nFor example, Nardiello et al. (2022) report a variation in\nthe NIRCam PSF of 3-4% (presumably in both the F090W\nand F150W filters) by analyzing variations in empirical PSFs\ncomputed from our M92 and WLM imaging. This appears to\nbe within a factor of ∼2of our findings with WebbPSF mod-\nels for the no event and small event scenarios.\nThrough their own DOLPHOT testing with NIRCam imag-\ning of nearby galaxies, Riess et al. (2023) suggest a charac-\nteristic uncertainty in the absolute photometry of NIRCam to\nbeσ= 0.03in each of F090W, F150W, and F277W. The\namplitude of this uncertainty is higher than the typical bias\nand in the range of the scatter we report in Table 6. We also\nnote that Libralato et al. (2023) undertake tests of NIRISS\nPSF stability in the context of proper motions of the LMC\nand report modest temporal variations in the PSF.\nFortunately, there are some mitigation strategies for mini-\nmizing any extra noise due to time dependent PSF variations.\nFirst, data that are acquired in a single visit or roughly at\nthe same time is unlikely to be significantly affected by the\nabove issues. Data taken closely spaced in time (hours, days)\nshould not be subject to significant PSF variations. For ex-\nample, data for each target in our program was collected in\n1-2 periods and a single epoch PSF grid works well. This\nis likely true for short period variables (e.g., RR Lyrae) in\nWLM.JWST R ESOLVED STELLAR POPULATIONS V 21\nWF Trending for 2022-07\nWF Sensing61.42022-07-01\n68.52022-07-04\n75.82022-07-06\n63.62022-07-08\n63.62022-07-10\n253.22022-07-12\n61.92022-07-15\n62.12022-07-17\n62.92022-07-19\n63.82022-07-22\n63.22022-07-24\n65.42022-07-26\n65.32022-07-28\n66.12022-07-30\nDrifts\n7.2\n 22.7\n 29.8\n 22.9\n 8.1\n 247.3\n 120.9\n 19.5\n 9.1\n 11.9\n 9.3\n 12.1\n 9.7\n 10.6\nCorrections\n45.8\n 215.9\n0255075100125150Wavefront Error\n[nm rms]\nObservatory WFE at NIRCam NRCA3\nT elescope WFE\nWavefront control target range\n2022-07-04 2022-07-11 2022-07-18 2022-07-25\nDate0.050\n0.025\n0.0000.0250.050Change in \nEncircled Energy\nNIRCam F150W\nMedian EE within 0.31 arcsec = 0.881\nMedian EE within 0.08 arcsec = 0.685EE within 0.31 arcsec (10 pix)\nEE within 0.08 arcsec (2.5 pix)\n±3% change (stability requirement)\nWF Trending for 2022-09\nWF Sensing60.62022-09-01\n60.72022-09-03\n62.52022-09-05\n85.12022-09-07\n64.32022-09-09\n66.52022-09-13\n69.02022-09-15\n60.02022-09-17\n61.02022-09-19\n60.42022-09-21\n61.12022-09-23\n61.32022-09-26\n61.32022-09-27\nDrifts\n8.1\n 6.8\n 16.3\n 60.9\n 21.6\n 14.1\n 10.6\n 10.0\n 10.4\n 11.5\n 12.8\n 9.1\n 7.2\nCorrections\n58.5\n 34.8\n0255075100125150Wavefront Error\n[nm rms]\nObservatory WFE at NIRCam NRCA3\nT elescope WFE\nWavefront control target range\n2022-09-05 2022-09-12 2022-09-19 2022-09-26\nDate0.04\n0.02\n0.000.020.04Change in \nEncircled Energy\nNIRCam F150W\nMedian EE within 0.31 arcsec = 0.881\nMedian EE within 0.08 arcsec = 0.686EE within 0.31 arcsec (10 pix)\nEE within 0.08 arcsec (2.5 pix)\n±3% change (stability requirement)\nFigure 14. Figures output by WebbPSF that show variations in the telescope alignment, wavefront, and encircled energy as a function of\ntime over the period. At select epochs, we compute the effects of small, large, and no perturbations to the telescope stability on DOLPHOT\nphotometry.\nSecond, the effect of the PSF changes can be well-\napproximated by the addition of Gaussian noise. In the\ncourse of scientific analysis, one could add a Gaussian noise\nmodel with no bias and a scatter equal to a value listed in\nTable 6 to capture this additional source of noise.\nThird, it is possible to run DOLPHOT on each epoch sep-\narately with PSFs customized to that epoch. Users can gen-\nerate their own PSFs (e.g., using WebbPSF and DOLPHOT\nutilities such as nircammakepsf ) to perform per epoch pho-\ntometry. A detailed example of custom PSF generation isshown on our ERS DOLPHOT documentation webpage. In\nsuch a case, one would perform per epoch photometry and\nthen cross-match the photometry from each epoch to gen-\nerate catalogs. The same process should be used for ASTs\ngenerated by this approach. As noted above, the large mis-\nalignment events can affect the spatial cross-matching of cat-\nalogs. In this case, it is important that care be taken when\nmerging catalogs taken at different epochs.\nFourth, one can use empirical PSFs generated at each\nepoch. Within the context of DOLPHOT, this can be done22 W EISZ ET AL .\nFigure 15. An illustration of the effects of temporal variations on\nthe photometry of M92. We show scatter from the SNR >50M92\nASTs in F150W in panel (a) as a point of comparison. The other\npanels show the no event case (panel b; i.e., normal temporal vari-\nations), a small misalignment event (panel c), and a large misalign-\nment event (panel d). Each of these panels shows the difference\nin F150W magnitudes of the same high SNR stars (SNR >50) for\nour fiducial photometry and photometry computed using WebbPSF\nmodels from an epoch corresponding to an event. For normal op-\neration (panel b) and small misalignment events (panel c) there is\nvery little differnce in the photometry. For the a large misalignment\nevent, a non-negligible amount of scatter can be present and should\nbe included in the error budget. Large events may also impact the\nastrometry. See § 5.2 and Table 6 for more details. Fortunately,\nsuch large events appear to be rare.\nby constructing one’s own empirical PSFs (e.g., using the\nmethod of Anderson & King (2000)) and, if put into the same\nformat as WebbPSF models, imported into DOLPHOT using\nits PSF ingestion utilities (e.g., nircammakepsf ). Compared\nto theoretical PSFs, empirical PSFs have the advantage of\ncapturing the observed state of the PSF at each epoch. How-\never, empirical PSFs also rely upon having suitable stars at\neach epoch from which to construct the PSF over the entire\nfield, an appropriate observational strategy (e.g., sufficient\ndither patterns in each filter), and adequate sampling of the\nwings of the PSF (as opposed to just the cores, which are\noften a main focus for astrometry).\nFinally, we emphasize again, that in general, JWST appears\nto have had remarkable stability outside the first few months\nof operation, and misalignment events should be rare.\nFigure 16. An illustration of temporal variations in an example\nF150W NIRCam PSF computed using WebbPSF. Each panel shows\nthe ratio of flux per pixel in each model PSF for a given misalign-\nment scenario (panel a: no event; panel b: small event; panel c;\nlarge event) relative to the nominal PSF. The changes to the PSF are\nsmall, but non-zero, for both the no event and small event scenarios.\nThe large event results in substantial change to the PSF. As summa-\nrized in Table 6, time and/or alignment variations to the PSF do not\nappear to introduce bias into the photometry, but they do add scatter\nof∼2−10% depending on the size of the event and wavelength.\nDespite having paid so much attention to the issue of tem-\nporal variation, we find these results to be encouraging, par-\nticularly at such an early point in JWST ’s lifetime. PSF tem-\nporal variations generally introduce no bias into the photom-\netry, while the scatter of a few percent is adequate for many\nscience applications. The main concerns are with (a) a large\nevent and (b) working with data taken over long time base-\nlines. In the case (a), we recommend timely analysis of the\nwavefront stability with WebbPSF and photometric reduc-\ntions in order to diagnose any issues. For example, a program\nwith high SNR requirements may find that the photometric\n(or possibly astrometric) uncertainties introduced by a large\nevent are much larger than the formal uncertainties reported\nby DOLPHOT. In this event, re-observation at a time when\nJWST has returned to stability may be warranted. In case\n(b), observations over long time baselines, additional careJWST R ESOLVED STELLAR POPULATIONS V 23\nNo Event Small Event Large Event\nGalaxy Filter Rtol N⋆ µ σ R tol N⋆ µ σ R tol N⋆ µ σ\n(pix) (mag) (mag) (pix) (mag) (mag) (pix) (mag) (mag)\n(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)\nM92 F090W 0.15 85778 -0.001 0.009 0.15 85255 0.008 0.015 2 81710 -0.026 0.092\nF150W 0.15 90510 0.000 0.008 0.15 89806 0.001 0.012 2 85914 -0.022 0.071\nF277W 0.15 82230 0.002 0.035 0.15 81748 0.002 0.043 2 78461 0.005 0.082\nF444W 0.15 71722 0.001 0.073 0.15 71477 -0.007 0.077 2 68813 -0.001 0.126\nWLM F090W 0.15 185296 -0.004 0.011 0.15 182921 0.002 0.018 2 173627 0.035 0.081\nF150W 0.15 124555 -0.002 0.010 0.15 123568 0.002 0.016 2 119583 0.012 0.060\nF250M 0.15 39531 0.000 0.031 0.15 39517 0.007 0.036 2 39198 0.033 0.058\nF430M 0.15 24713 -0.001 0.066 0.15 24724 0.015 0.081 2 24565 0.053 0.078\nDraco II F090W 0.15 291 0.000 0.011 0.15 281 0.012 0.014 2 273 -0.018 0.117\nF150W 0.15 284 0.000 0.008 0.15 281 0.007 0.011 2 270 -0.001 0.097\nF360M 0.15 186 0.004 0.027 0.15 186 0.018 0.072 2 176 0.058 0.039\nF480M 0.15 111 -0.014 0.064 0.15 111 0.015 0.150 2 101 0.091 0.031\nTable 6. The impact of time variations in the PSF on DOLPHOT photometry. For each target, we compare fiducial photometry with that gen-\nerated during various misalignment events (none, small, large) and compute summary statistics. Specifically, we compute the mean difference\n(µ) and standard deviation ( σ) for the same stars ( N⋆) in all observed filters. In general, we note that no biases are present and that the scatter\ngenerally remains small, for all but large misalignment events. For large events, which appear to be rare, the scatter is as large as ∼8% and\nmatching photometry across epochs requires a much larger pixel matching radius ( Rtol).\nmust be taken, as the normal noise reported by DOLPHOT,\nincluding ASTs, does not include the additional noise term\nintroduced by temporal variations to JWST . Overall, we em-\nphasize that users should evaluate the true noise of the data\nrelative to what is required for their science use case and plan\ntheir observations and analysis accordingly.\n5.3. Systematic Uncertainties in the Photometry\nThe analysis of our ERS data allows us to estimate the\namplitude for various systematic uncertainties inherent to\nDOLPHOT JWST photometry. It remains premature to dis-\ncuss absolute flux uncertainties until results from the JWST\ncalibration program (Gordon et al. 2021) have been officially\npublished.\nAs discussed in Dolphin (2000), the two main sources of\nsystematics are PSF adjustments and aperture corrections.\nWe discussed and calculated the amplitude of systematic\nuncertainties due to PSF adjustments in §3.6, and found\nthat all NIRCam values were ≤0.008 mag per filter and\n≤0.014mag per NIRISS filter.\nAperture corrections are required to account for a star’s\nflux that may fall outside the finite area of the PSF. To es-\ntimate aperture corrections, DOLPHOT uses a set of bright,\nisolated stars in each science image. The fluxes measured\nfrom aperture photometry is compared to PSF photometry to\nestablished the amplitude of the aperture correction. Aper-\nture corrections are then applied to each object in the science\nframe.\nIn our ERS data, the aperture corrections range from ∼\n0.1−0.15mag for all datasets. The 1- σuncertainties onthese aperture corrections are ≲0.003mag. This latter num-\nber, the uncertainty on the aperture correction, is formally the\nsystematic uncertainty.\nHowever, in principle, the aperture corrections should be\nuniform across the targets for each filter. Instead, we find\nvariations of ∼0.005mag in the aperture corrections for the\nsame filter and chip, but for different targets. Given the for-\nmal uncertainty above and this variation in aperture correc-\ntion, we suggest that an upper limit of 0.01 mag on the aper-\nture correction is a reasonable, if slightly conservative value.\nWe do note that uncertainties in the aperture corrections\nmay be larger if DOLPHOT cannot find a sufficient number\nof isolated stars in a given field from which to calculate the\naperture corrections (e.g., if the entire field is so crowded\nthat no or few aperture stars are available, such as the bulge\nof M31 (e.g., Dalcanton et al. 2012a; Rosenfield et al. 2012).\nTime variations in the PSF can also contribute to the sys-\ntematic error budget. For images taken over a short period\nof time, these uncertainties are negligibly small as the PSF\ndoesn’t vary. Over longer timelines, normal operation of the\ntelescope (which include, for example, thermal distortions,\ntilt events, micrometeorite impacts; McElwain et al. 2023)\nmay introduce random uncertainties in the PSF with an am-\nplitude comparable to the current PSF systematics. Small\nmis-alignment events could introduce noise at the few hun-\ndredths of a magnitude. Large alignment events could in-\ncrease this random noise up to ∼0.08mag. Details of these\neffects are discussed in §5.2. We emphasize that the time\nvariation uncertainty in the PSF should not be significant for\nmost use cases.24 W EISZ ET AL .\nThe total systematic uncertainty budget in the photome-\ntry also includes contributions from flat field uncertainties\nand the absolute flux calibration (i.e., the global and chip-to-\nchip zero points). Our ERS data are not adequate to capture\nmost of these effects. Flat field uncertainties usually mani-\nfest over larger areas of the detector than are sampled by our\nsmall dithers. Similarly, while our program proved valuable\nfor identifying chip-to-chip offsets in the zero points (Boyer\net al. 2022), these offsets now appear to be sufficiently small\n(i.e.,≲0.02mag) that our data only provide limited new in-\nformation. The ongoing JWST absolute flux calibration pro-\ngram will provide more insight (Gordon et al. 2021).\nOne test of the overall accounting of systematics is to ob-\nserve the same stars at different spatial positions on the de-\ntectors to check for consistency in the reported photome-\ntry. In principle, the same sources should have photometry\nwithin reported uncertainties no matter their spatial location\non the chips. Any inconsistencies would point to an addi-\ntional source of uncertainty. One such example occurred in\nthe PHAT survey. Multiple orientations of HST showed that\nphotometry of the same bright stars varied by 0.02-0.04 mag\nas a function of position. Ultimately, this revealed the need\nfor PSF interpolation (Dalcanton et al. 2012a; Williams et al.\n2014), a feature that is now the default in DOLPHOT. Sub-\nsequent DOLPHOT analysis in M31 and M33 have found\nspatial variations in the HST PSF and photometry to be a\nsubdominant issue (Williams et al. 2021, 2023).\n5.4. Comparing Predicted and Measured Signal-to-Noise\nThe observing strategy of our ERS program was planned\nwith v1.5.2 of the JWST ETC in 2017. The ETC was used to\ntranslate the maximum photometric uncertainty (or minimum\nSNR) threshold for each target into integration times for a\ngiven observing strategy (i.e., dithers, groups, etc).\nIt is instructive to assess how our recovered SNRs\nfrom DOLPHOT compared to the initial goals of the pro-\ngram. We provided a preliminary comparison between the\nDOLPHOT and ETC SNRs in Weisz et al. (2023); here,\nwe briefly recap the ETC calculations for this program.\nFor planning our observations with the ETC, we used a\nK5V star ( Teff= 4250 K,log(g) = 4 .5dex) from the\nPhoenix stellar models, foreground extinction from Schlafly\n& Finkbeiner (2011), the observational strategy listed in\nTable 1, and v1.5.2 of the ETC. To meet our program\ngoals, we required a SNR ≥10at F090W,F150W =26,25.8\nfor M92, F090W,F150W =28.5,28.3 for WLM, and\nF090W,F150W =27, 26.8 for Draco II. The main science\nmotivations were to reach a 0.1M⊙MS star in M92, the\noldest MSTO in WLM, and a 0.2M⊙MS star in Draco II,\neach of which enables a wide variety of science from stars\nbrighter than these limits, as discussed in Weisz et al. (2023).\nWe estimated the JWST Vega magnitudes for each of these\ngoals using the MIST stellar models (Choi et al. 2016), as\nthere were no near-IR data deep enough to directly constrain\nthe locations of these features empirically. We built in a small\nmargin, which is reflected in the above magnitude limits, in\nthe event that JWST under-performed expectations.From the photometry presented in §4, we compute the\nSNR for both SW filters at the target depths listed above.\nSpecifically, we consider a 0.1 mag bin in each filter centered\non the magnitude of interest and then compute the mean SNR\nin F090W and F150W for stars that pass the culling criteria.\nFrom our DOLPHOT photometry of M92, we find\nSNRs of 19.39 and 15.11 at F090W,F150W =26,25.8 mag.\nFor WLM, we find SNRs of 25.24 and 14.85 at\nF090W,F150W =28.5,28.3. For Draco II, we find SNRs of\n49.31 and 24.31 at F090W,F150W =27,26.8.\nIn all cases, DOLPHOT recovers higher SNRs than the\noriginal ETC calculations by factors of ∼1.5−5. This\nis a good finding for the science and technical aims of our\nprogram, as it ensures we meet all minimum requirements.\nMoreover, other resolved stellar populations studies that rely\non our program for exposure time guidance should be en-\ncouraged that they should meet their minimum requirements\nas well.\nGiven the high community demand for JWST time, it is im-\nportant that not all observations be too conservative in their\nestimate of exposure time. We thus review several possible\nreasons for the higher than expected SNRs from DOLPHOT\ncompared to initial expectations. First, is that JWST is over-\nperforming pre-launch expectations in terms of sensitivity\n(e.g., Rigby et al. 2022; McElwain et al. 2023), resulting in\nhigher SNRs for fixed integration time. Second, there have\nbeen improvements to JWST data products (e.g., updated\nflat fields, post-launch PSFs, revised zero points) that have\nhelped to improve DOLPHOT’s performance (i.e., more pre-\ncise photometry) since the publication of the survey paper.\nThird, are improvements to JWST ’s ETC. Improved knowl-\nedge of JWST ’s in-flight performance has been incorporated\nintoJWST ’s ETC, providing for more realistic SNR and ex-\nposure time estimates than were available in 2017.\nOne lingering issue with the ETC is its reliance on\nonly aperture photometry for determining expected SNR.\nDOLPHOT relies on PSF fitting, which, for faint sources,\nprovides for improved flux recovery over aperture photom-\netry. This is a well-known issue in the context of the HST\nETC , which employs aperture photometry to provide a nom-\ninal SNR estimate, as well as an algorithm similar to PSF\nfitting that provides an ‘optimal’ SNR. In general, the opti-\nmal SNRs from HST are∼1.5−2×larger than the SNRs\nbased on aperture photometry. Experienced members of our\nteam have found the HST optimal SNRs are much closer to\nwhat DOLPHOT reports. There are pathways for incorporat-\ning PSF-like SNR extraction in the JWST ETC, but they have\nyet to be implemented.\nOur team is in the process of undertaking a detailed evalua-\ntion of the current ETC (v3.0) relative to DOLPHOT’s SNRs.\nGiven the amount of detail in this comparison, we will pub-\nlish this assessment of the current JWST ETC as a standalone\npaper (Savino et al. in prep).\n6.CONCLUSIONS\n6.1. Summary of ResultsJWST R ESOLVED STELLAR POPULATIONS V 25\nWe have developed new NIRCam and NIRISS modules\nfor the widely used crowded field stellar photometry package\nDOLPHOT. We describe modifications made to DOLPHOT\nthat are tailored to NIRCam and NIRISS imaging and sum-\nmarize the process by which DOLPHOT is run on JWST\nimaging. We tested the fidelity of these modules on NIR-\nCam and NIRISS imaging of three targets (M92, Draco II,\nand WLM) taken as part of the JWST Resolved Stellar Pop-\nulations Early Release Science Program (Weisz et al. 2023).\nFrom this testing we find:\n• DOLPHOT produces excellent CMDs for each of the\nERS targets. The CMDs have tight features (e.g.,\nRGB, RC, MS) that are precise enough to reveal\npercent-level systematics in the data calibration (e.g.,\nwith current PSF models).\n• Stability with fine guidance sensor lock on the guide\nstars affected the 3rd exposure of M92. It is not suit-\nable for being used in our data reduction. All normal\nDOLPHOT astrometric and photometric diagnostics\nappear reasonable, but the resulting CMD is of poor\nquality. We provide more details in the Appendix.\n• Despite significant persistence in the imaging of\nDraco II due to previous calibration observations of\nSolar system targets, we find DOLPHOT produces pre-\ncise and complete photometry of Draco II.\n• We find that WebbPSF models that include effects of\ncharge diffusion and interpixel capacitance are well-\nmatched to stars in the ERS data. The dominant\nsystematic uncertainties in the DOLPHOT photome-\ntry are the PSF models and aperture corrections, both\nof which are limited to ≲0.01mag in each filter.\nThe availability of suitable stars for determining aper-\nture corrections may affect this error budget in very\ncrowded fields. A full accounting of photometric un-\ncertainties will require better knowledge of the flat\nfield uncertainties than our program can provide, as\nwell as results from the absolute flux calibration pro-\ngram (Gordon et al. 2021).\n• There are small-to-modest temporal variations in the\ntheoretical PSFs for NIRCam. We examined these\nvariations for three telescope alignment scenarios:\nsmall and large misalignments and no misalignments.\nThey generally do not bias the photometry, how-\never they introduce additional scatter ranging from\n≲0.01mag for normal telescope operation up to ∼\n0.09mag for a large misalignment event. We present\nmitigation strategies.\n• We show that our program provided higher SNR data\nthan was anticipated during program design in 2017.\nThis is likely due to a combination of better than ex-\npected performance of JWST , as well as differences\nin how the JWST ETC operates versus the noise com-\nputed by DOLPHOT. An upcoming paper by our team(Savino et al. in prep.) extensively explores perfor-\nmance of the ETC.\n• Images and photometric catalogs used in this paper\ncan be downladoed from MAST6. Step-by-step guides\nfor our DOLPHOT reductions can be found on our\nDOLPHOT documentation page7.\n6.2. Future Outlook\nJWST is performing as well or better than expected (e.g.,\nRigby et al. 2022; McElwain et al. 2023), which greatly en-\nhances the prospects for exploration of the local Universe.\nMuch of this will be built on precise and accurate stellar\nphotometry, often in crowded fields. Accordingly, there\nare several areas for which we anticipate improvements in\nDOLPHOT and the data products that it needs as input. Here,\nwe briefly summarize some of these issues.\n• The Advanced Scientific Data Format (ASDF; Green-\nfield et al. 2015) was introduced as a replacement for\nthe FITS file format. It is currently available for JWST\ndata alongside conventional FITS headers. At the time\nof this writing, all necessary information to process\nNIRCam and NIRISS images with DOLPHOT appears\navailable in the FITS header and the creation of an\nASDF reader for DOLPHOT is not yet necessary. Be-\ncause ASDF is capable of storing more detailed meta-\ndata than FITS (e.g., more information on distortions),\nit may be necessary in the future for DOLPHOT to read\ndata from the ASDF header. We will continue to mon-\nitor the metadata provided with JWST images and add\nan ASDF reader if it becomes necessary.\n• In principle, Frame 0 data provide access to pixels that\nmay be saturated in longer integrations. We attempted\nto incorporate Frame 0 into our ERS DOLPHOT re-\nductions, but learned from STScI that, as of this writ-\ning, the Frame 0 data are not being correctly processed\nby the JWST pipeline and therefore are not ready for\nscience use. Given the significant saturation issues\npresent in many of our ERS images, we welcome the\navailability of suitably calibrated Frame 0 data. Be-\ncause Frame 0 data are simply an image of shorter in-\ntegration time, no modifications to DOLPHOT should\nbe necessary for their use.\n• The utility of DQ arrays has improved since the ear-\nliest days of JWST . Nevertheless, continued improve-\nment to the DQ arrays (better flagging of bad pixels\nand artifacts such as claws and wisps) would help re-\nmove contaminants and provide improved photometry.\n• As illustrated with our Draco II data, persistence can\nbe a real challenge with JWST . We suggest that users\n6https://archive.stsci.edu/hlsp/jwststars/\n7https://dolphot-jwst.readthedocs.io26 W EISZ ET AL .\nconsider indicating in the Special Requirements sec-\ntion of their JWST proposals that their observations be\nscheduled to minimize persistence. It is also advis-\nable that users visually inspect their data and attempt\nan early DOLPHOT reduction to provide time to file\na WOPR in the event that persistence significantly af-\nfects the photometry.\n• The culling criteria we have adopted in this paper are\nbased on Warfield et al. (2023) and are designed for\npurity (i.e., to enable better star-galaxy separation) at\nthe expense of completeness. They are also only ap-\nplied to the deepest data, which is F090W and F150W.\nThese culling criteria may not be optimal for all filters\nand/or science cases and readers may need to explore\nother permutations.\n• An analysis of photometry of the same stars observed\non different detectors would be a valuable way to\ngauge the overall photometric error budget. Our ERS\ndata were all taken at the same orientation and with\nsmall dithers, and are not suitable for this type of test-\ning. The LMC calibration fields may be suitable. Oth-\nerwise, we suggest that data suitable for these experi-\nments would be of high value for establishing the over-\nall photometric error budget for resolved stellar popu-\nlations science.We thank the anonymous referee for a positive and construc-\ntive report. Our entire ERS team greatly thanks all the peo-\nple that have worked so hard over the years in order to make\nJWST such an amazing telescope. This work is based on\nobservations made with the NASA/ESA/CSA James Webb\nSpace Telescope. The data were obtained from the Mikulski\nArchive for Space Telescopes at the Space Telescope Science\nInstitute, which is operated by the Association of Universi-\nties for Research in Astronomy, Inc., under NASA contract\nNAS 5-03127 for JWST. These observations are associated\nwith program DD-ERS-1334. This program also benefits\nfrom recent DOLPHOT development work based on obser-\nvations made with the NASA/ESA Hubble Space Telescope\nobtained from the Space Telescope Science Institute, which\nis operated by the Association of Universities for Research in\nAstronomy, Inc., under NASA contract NAS 5–26555. These\nobservations are associated with program HST-GO-15902.\nFacilities: JWST(NIRCAM), JWST(NIRISS)\nSoftware: This research made use of routines and\nmodules from the following software packages: Astropy\n(Astropy Collaboration et al. 2013, 2018, 2022), DOLPHOT\n(Dolphin 2016), IPython (Perez & Granger 2007),\nMatplotlib (Hunter 2007), NumPy (van der Walt et al.\n2011) and SciPy (Virtanen et al. 2020)\nREFERENCES\nAlbers, S. M., Weisz, D. R., Cole, A. A., et al. 2019, MNRAS, 490,\n5538, doi: 10.1093/mnras/stz2903\nAnand, G. S., Rizzi, L., Tully, R. 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Panel (b): Culled CMD based on photometry using exposures 1,2,3, and 4. Panel (c): Culled CMD of only exposure 3.\nA visual comparison of panels (a) and (b) shows that the inclusion of the 3rd exposure results in more scatter in the combined CMD. This is\nmost evident near the MSTO, which is much broader in panel (b) than in panel (a), but is also present in virtually all other regions of the CMD.\nThe CMD of the 3rd exposure in panel (c) shows poor photometry despite no obvious diagnostic issues raised in DOLPHOT. This pattern is\npresent in the 3rd exposure of the NIRISS M92 imaging. The issue appears to be a small amount of increased jitter in the telescope, as\nrevealed by analysis of the fine guidance sensor data.\nA.ANOMALOUS 3RD EXPOSURE IN M92\nDuring preliminary exploration of our M92 imaging in Fall 2022, we noticed that photometry from the 3rd exposure appeared\nto be of significantly lower quality in both NIRCam and NIRISS. We inspected all the normal diagnostics within DOLPHOT (e.g.,\nastrometric alignment, DOLPHOT-generated warnings, image properties, header information) and found no obvious source of\nthe problem. Program co-I Jay Anderson verifed the poor photometry independently using JWST1PASS, a version of HST1PASS\nupdated for use with JWST (Anderson 2022). The issue only becomes apparent when examining the photometry (i.e., plotting a\nCMD).\nFigure 17 illustrates the poor quality of the photometry from the 3rd exposure. Panel (c) shows the NIRCam SW CMD of only\nthe 3rd exposure. The bright stars show unusually broad scatter and even the lower MS is broader than expected for a single\nexposure. Panels (a) and (b) show the combined CMDs when excluding the 3rd exposure (a) and including the 3rd exposure\n(b). It is clear visually that omitting the 3rd exposure produces a much tigher CMD, particularly at the bright end. While the 4\nexposure CMD (panel b) appears to extend slightly fainter than the 3 exposure CMD (panel a), the former has a slightly broader\nMS.\nAnalysis of fine guidance sensor (FGS) data for the 3rd dither revealed an anomalous amount of jitter during this exposure.\nCompared to the ∼0.5milliarcsecond rms jitter per axis of exposures 1, 2, and 4, the jitter on the 3rd exposure is 5 and 1630 W EISZ ET AL .\nmilliarcseconds on the x- and y-axis, respectively. This amount of jitter is less than the typical size of the JWST PSFs and is\ntherefore not noticeable by visual inspection of the images. Moreover, it did not affect the alignment statistics within DOLPHOT.\nHowever, it is sufficiently large that it does degrade the image quality to a point that is noticeable in the resulting CMD, as the\nPSF is slightly smeared out relative to the model.\nThis problem was only discovered by analyzing data associated with the FGS. If users notice lower than expected quality of\ntheir CMDs, we suggest checking the FGS data using a tool such as spelunker8.\n8https://github.com/GalagaBits/JWST-FGS-Spelunker"    },    {        "title": "2402.03568v1.Exploring_freeze_out_and_flow_using_exact_solutions_of_conformal_hydrodynamics.pdf",        "content": "Exploring freeze out and flow using exact solutions of conformal hydrodynamics\nOwen Bradley1and Christopher Plumberg1,∗\n1Natural Science Division, Pepperdine University, Malibu, CA 90263, USA\nExact solutions to the equations of hydrodynamics provide valuable benchmark tests for numerical\nhydrodynamic codes and also provide useful insights into the nature of hydrodynamic flow. In this\npaper, we introduce two novel, closely related exact solutions with non-trivial rapidity dependence\nwhich are generalizations of the well-known Gubser flow solution to conformal hydrodynamics. We\nthen use one of our solutions to explore the consequences of choosing between two different criteria\nfor implementing the freeze out process in fluid dynamical simulations of nuclear collisions: freeze\nout at constant temperature vs. freeze out at constant Knudsen number. We find that, employing\nour exact solution, the differences between these freeze out criteria are heavily influenced by the\npresence of strong collective flow. Our results highlight the importance of accurately describing the\nfreeze out process in collisions with large flow gradients, particularly in small systems.\nI. INTRODUCTION\nRelativistic fluid dynamics has proved to be an extremely powerful tool for the description of high-energy nuclear\ncollisions [ 1–5]. To date, hydrodynamic models have been used to quantitatively describe a host of experimental\nobservables [ 6–20]. With its ability now to make predictions which are accurate at the percent level [ 21–23], the fluid\ndynamical paradigm has arguably begun to enter the domain of precision physics.\nOne of the remaining systematic uncertainties affecting nuclear collision phenomenology involves the treatment of\nthe freeze out process which terminates the fluid dynamical evolution. The most common approach is to freeze out\nalong a hypersurface of fixed temperature [ 24,25] or energy density [ 26], where the latter allows a straightforward\ngeneralization to finite chemical potentials which are probed by nuclear collisions [ 27,28]. An alternative (although\nless widely used) criterion for imposing freeze out is to terminate at constant Knudsen number Kn[2]. The Knudsen\nnumber represents the competition between mechanisms which drive the system out of equilibrium (e.g., strong flow\ngradients) and those which relax the system toward local equilibrium (e.g., strong interactions or dissipative effects)\nand thus provides a measure of how well a system may be described fluid dynamically. Freeze out at constant Knudsen\nnumber can thus be viewed as a choice to decouple the system in terms of the degree to which the fluid dynamical\ndescription is justifiable at a given stage in the evolution.\nThe extent to which the use of different freeze-out criteria influences experimental observables has already received\nsome attention in the context of hydrodynamic models [ 25,29–33], including systems both with and without event-by-\nevent fluctuations. However, less work has been done on the differences between these criteria in small systems where\nlarge flow gradients drive fluid dynamics to its breaking point [ 5,31,34–36], and relatively little progress has been\nmade in understanding the quantitative implications of these differences for observables which probe specifically the\nspace-time geometry of nuclear collisions, such as Hanbury Brown – Twiss interferometry [37].\nIn this paper, we begin to address this gap by exploring the extent to which the shape of the freeze out hypersurface\nis impacted by the choice of freeze out criterion and the presence of collective flow in the system. In order to build\nintuition and obtain results which can be straightforwardly interpreted, we have chosen to answer this question in the\ncontext of an exact solution to the equations of relativistic fluid dynamics. The search for such exact solutions has a\nlong history. Known exact solutions include those obtained by Bjorken assuming longitudinal boost invariance [ 38] and\na later generalization by Gubser [ 39,40] to include expansion in the transverse plane as well. Exact solutions have also\nbeen constructed for rotating systems [ 41,42] and systems which possess non-trivial dependence on the space-time\nrapidity [43, 44], as well as for various generalizations including viscous effects [45].\nIn this study, we are interested in exploring the freeze out process using exact solutions which exhibit strong transverse\nand longitudinal flow simultaneously. To do this, we first construct two new, exact, Gubser-like solutions which contain\nnon-trivial transverse and longitudinal flow, and then use one of these solutions to compare the geometries of fixed\ntemperature and fixed Knudsen number hypersurfaces. The first solution (“Solution I\") represents a generalization of\nGubser’s ideal hydrodynamic solution and does not include viscosity, while the second solution (“Solution II\") is a\ngeneralization of Gubser’s viscous solution [ 39,40] (and thus allows us to define and study freeze out in terms of the\nKnudsen number). Both of our solutions make use of a trick, recently suggested in [ 44] and which we refer to here as a\n∗christopher.plumberg@pepperdine.eduarXiv:2402.03568v1  [nucl-th]  5 Feb 20242\n“temporal shift,\" to induce a non-trivial rapidity dependence into the collective flow of a known solution. In this paper\nwe focus our attention on the derivation of the solutions themselves, as well as their qualitative implications for the\ngeometry of the freeze out hypersurface. We defer a more thorough discussion of the quantitative consequences for\nnuclear collision phenomenology, such as HBT measurements, to a subsequent study.\nWe have organized the paper as follows. In Sec. II, we construct Solution I as a generalization of Gubser’s ideal flow\nsolution and show how to introduce rapidity dependence by performing a temporal shift. We also discuss a number\nof properties of Solution I and consider how its space-time dependence is influenced by different choices of the free\nparameters it contains. In Sec. III, we apply the same temporal shifting procedure to the viscous version of Gubser’s\noriginal solution, in order to obtain our Solution II which possesses both viscosity and non-trivial η-dependence.\nFinally, in Sec. IV, we apply our Solution II to the question of how the presence of both transverse and longitudinal\nflow influence the shape of the freeze out surface. Some conclusions are presented in Sec. V.\nII. SOLUTION I\nBefore presenting our derivation of Solution I, we first briefly review the original ideal solution obtained by Gubser\n[39, 40].\nA. The ideal Gubser solution\nRelativistic fluid dynamics is predicated on the covariant conservation of energy and momentum, expressed by\n∇µTµν= 0 (1)\nwhere ∇µrepresents a covariant derivative and Tµνis the energy-momentum tensor. This yields 4 (3 space, 1 time)\nequations of motion which dictate the evolution of the system. For most phenomenological applications, the initial\nstates of nuclear collisions and their subsequent evolution are sufficiently complicated that they can only be solved\nnumerically. However, in certain highly symmetric scenarios, it may be possible to find exact solutions with at least\nsome phenomenological relevance to nuclear collisions. The challenge is then to identify a suitable symmetry group\nwhich mimics the dynamics of nuclear collisions sufficiently closely.\nThe original solution obtained by Gubser was constructed in order to respect the symmetry group SO(3)q×\nSO(1,1)×Z2, where SO(1,1)represents boost-invariance along the beam axis, Z2represents a reflection symmetry\nunder η→ −η, and SO(3)qdenotes a conformal symmetry which includes azimuthal rotations around the beam axis\nas a subgroup and is designed in such a way as to generate a non-trivial flow profile in the radial direction. Since the\ngenerators of the SO(3)qandSO(1,1)commute, one thus obtains a solution to the hydrodynamic equations (1)which\nexhibits radial flow while also respecting boost invariance.\nThedS3⊗Rframe is parameterized by the target space coordinates ˆxµ= (ρ, θ, ϕ, η ), while the physical Milne\ncoordinates are xµ= (τ, r, ϕ, η ).1In both sets of coordinates, ϕrepresents the azimuthal angle around the beam axis\nandηrepresents the longitudinal rapidity, while in Milne coordinates, τandrrepresent the Bjorken proper time and\nradial coordinate, respectively. Then the dS3⊗Rcoordinates ρandθare related to Milne coordinates by\nsinhρ=q2\u0000\nτ2−r2\u0001\n−1\n2qτ(2)\ntanθ=2qr\n1 +q2(τ2−r2)(3)\nand thus offer an alternative parametrization of the (τ, r)directions. In terms of the dS3⊗Rcoordinates, boost\ninvariance can be expressed as independence of the rapidity η, while the SO(3)qsymmetry amounts to independence\nof the coordinates (θ, ϕ). The dS3⊗Rframe has the metric\nˆgµν= diag\u0000\n1,−cosh2ρ,−cosh2ρsin2θ,−1\u0001\n(4)\n1Here and below, hatted quantities (e.g., ˆX) are defined in the dS3⊗Rframe, while quantities with out hats (e.g., X) are defined in\nphysical Milne coordinates.3\nwhile in Milne coordinates the metric is\ngµν= diag\u0000\n1,−1,−r2,−τ2\u0001\n(5)\nTo obtain Gubser’s original solution, one assumes the fluid is stationary in the dS3⊗Rframe:\nˆuµ= (1,0,0,0) (6)\nBy transforming the trivial solution (6) back to Milne coordinates, Gubser obtained the solution [39, 40]\nuτ=τ∂ρ\n∂τ=1 +q2\u0000\nτ2+r2\u0001\nq\n1 + 2 q2(τ2+r2) +q4(τ2−r2)2(7)\nur=−τ∂ρ\n∂r=2q2rτq\n1 + 2 q2(τ2+r2) +q4(τ2−r2)2(8)\nwith a corresponding energy density ϵgiven by\nϵ=ˆϵ0\nτ4/3(2q)8/3\n[1 + 2 q2(τ2+r2) +q4(τ2−r2)2]4/3(9)\nEqs.(7)-(9)define Gubser’s ideal solution to conformal hydrodynamics. It contains two free parameters: q, which\nrepresents an inverse lengthscale characterizing of the flow profile, and ˆϵ0, which sets the scale for the energy density.\nB. Derivation of Solution I\nWe now want to generalize Gubser’s solution by relaxing the assumption of boost invariance. In order to break the\nboost invariance of the Gubser solution, we may introduce ρandηdependence without violating the SO(3)qsymmetry.\nExplicitly, we take the following ansatz for the flow (cf. [39, 40]):\nˆuµ= (cosh ξ(ρ, η),0,0,sinhξ(ρ, η)) (10)\nThis ansatz reduces to a stationary fluid (and therefore to Gubser’s solution) whenever ξ= 0. We emphasize that\nboost invariance is broken by the inclusion of η-dependence, but the SO(3)qsymmetry is unaffected. Note that (10)is\nproperly normalized using the dS3⊗Rmetric:\nˆuµˆuνˆgµν= 1 (11)\nThe Gubser solution requires a conformal equation of state. For simplicity, we assume in this study that ˆp=ˆϵ/3.\nIn this case, the ideal energy-momentum tensor ˆTµν=ˆϵˆuµˆuν+ˆp(ˆuµˆuν−ˆgµν)has the following non-vanishing\ncomponents:\nˆTρρ=ˆϵ\n3\u0010\n4 (ˆuρ)2−1\u0011\n(12)\nˆTρη=4ˆϵ\n3ˆuρˆuη(13)\n=ˆTηρ(14)\nˆTθθ=ˆϵ\n3\u0010\n4\u0000\nˆuθ\u00012+ sech2ρ\u0011\n=ˆϵ\n3sech2ρ (15)\nˆTϕϕ=ˆϵ\n3\u0010\n4\u0000\nˆuϕ\u00012+ sech2ρcsc2θ\u0011\n=ˆϵ\n3sech2ρcsc2θ (16)\nˆTηη=ˆϵ\n3\u0010\n4 (ˆuη)2+ 1\u0011\n(17)\nThis energy-momentum tensor must be covariantly conserved:\n∇µˆTµν=∂µˆTµν+ Γµ\nαµˆTαν+ Γν\nαµˆTαµ= 0 (18)4\nEvaluating the Christoffel symbols for the metric (4) yields the following non-vanishing components:\nΓρ\nθθ= cosh ρsinhρ, Γρ\nϕϕ= sin2θcoshρsinhρ (19)\nΓθ\nρθ= Γϕ\nρϕ= tanh ρ, Γθ\nϕϕ=−sinθcosθ, Γϕ\nθϕ= cot θ (20)\nThen (1) implies two independent equations in the ρandηdirections:\n0 =∂µˆTµρ+ Γµ\nαµˆTαρ+ Γρ\nαµˆTαµ\n=∂ρˆTρρ+∂ηˆTρη+ 2 tanh ρˆTρρ+ cosh ρsinhρ\u0010\nˆTθθ+ sin2θˆTϕϕ\u0011\n(21)\n0 =∂µˆTµη+ Γµ\nαµˆTαη+ Γη\nαµˆTαµ\n=∂ρˆTρη+∂ηˆTηη+ 2 tanh ρˆTρη(22)\nDefining ˆϵ= exp(4 T)and making use of Eq. (10), the conservation equations (21)-(22) simplify to\n−tanhρ=∂T\n∂ρ+ cosh(2 ξ)\u0012\ntanhρ+∂ξ\n∂η+ 2∂T\n∂ρ\u0013\n+ sinh(2 ξ)\u0012∂ξ\n∂ρ+ 2∂T\n∂η\u0013\n(23)\n0 =−∂T\n∂η+ sinh(2 ξ)\u0012\ntanhρ+∂ξ\n∂η+ 2∂T\n∂ρ\u0013\n+ cosh(2 ξ)\u0012∂ξ\n∂ρ+ 2∂T\n∂η\u0013\n(24)\nSo far no assumptions have been made. In order to find an exact solution to these equations, we now consider the\nspecial case where both the flow rapidity ξandTare independent of η.2In this case, they can depend only on ρ, so\nthat the equations simplify further to\n0 = 2 cosh2ξtanhρ+ (1 + 2 cosh (2 ξ))dT\ndρ+ sinh (2 ξ)dξ\ndρ(25)\n0 = sinh (2 ξ)\u0012\ntanhρ+ 2dT\ndρ\u0013\n+ cosh (2 ξ)dξ\ndρ(26)\nThe second equation can be solved to find Tin terms of ρandξ(ρ); the solution is\nT(ρ) =c1−1\n2ln cosh ρ−1\n4ln sinh (2 ξ(ρ)), (27)\nwith c1an arbitrary constant. Substituting this solution for Tback into the first equation yields\nsinh(2 ξ) tanh ρ−(2 + cosh(2 ξ))dξ\ndρ= 0 (28)\nThis equation is trivially satisfied when ξ= 0. In this case, one would have found the solution for Tto be\nT(ρ) = ˜c1−2\n3ln cosh ρ, (29)\nfor some constant ˜c1, implying that\nˆϵ∝(cosh ρ)−8/3, (30)\nwhich is simply Gubser’s original solution [ 40]. The ansatz (10)thus provides a generalization of Gubser flow which\nreduces to Gubser’s ideal solution when ξ= 0, as anticipated.\nTo solve Eq. (28)when ξ̸= 0, we recast it into a more manageable form using the changes of variables x=cosh2ρ\nandξ=1\n2lny(x):\ndy\ndx=y\u0000\ny2−1\u0001\nx(1 +y(y+ 4))(31)\n2Note that ηdependence may still be included below by applying a temporal shift, as we describe in Sec. IID.5\nIt has the solution\ny(x)≡˜y(c2x), (32)\nwhere\n˜y(w) = 1−w\n3+24/3w(w−9)\na(w)+ 22/3a(w), (33)\na(w) =h\n−w\u0010\n54 + w(2w−27) + 3p\n3 (108 −w2)\u0011i1/3\n, (34)\nandc2is another arbitrary constant. Our solution in the dS3⊗Rframe can therefore be written\nr= cosh2ρ=1 + 2 q2\u0000\nτ2+r2\u0001\n+q4\u0000\nτ2−r2\u00012\n4q2τ2(35)\nˆϵ(ρ) =2e4c1\nr\u0012˜y(c2r)\n˜y(c2r)2−1\u0013\n(36)\nξ(ρ) =±1\n2ln ˜y(c2r) (37)\nandc1,c2<0are undetermined constants. Note that the energy density ˆϵis conformal; the physical energy density\nisϵ=ˆϵ/τ4in Milne coordinates. Note also that the rapidity ξcan be chosen positive or negative by the reflection\nsymmetry which is apparent in the above equations. Here we choose only the positive ‘branch’ of ξfor the sake of\ndefiniteness.\nThe flow uµ(10) can then be mapped to Milne coordinates by the transformation rules [40]\nuτ(τ, r) =τ∂ρ\n∂τcoshξ (38)\nur(τ, r) =−τ∂ρ\n∂rcoshξ (39)\nuη(τ, r) =−1\nτsinhξ (40)\nWe thus arrive at an exact solution for the energy density ϵand the flow profile uµ, given in Eqs. (33)-(40). Boost\ninvariance has clearly not yet been broken, since the solution is still independent of η.\nC. Asymptotics\nOur exact solution (33)-(40)is singular in the limit c2→0−unless c1is chosen in an appropriate way (cf. (33)and\n(36)). To avoid this, we define\nc1=1\n12ln\u0000\n−2c2ˆϵ3\n0\u0001\n(41)\nwhere the parameter ˆϵ0is the same one used in Gubser’s solution [ 39]. This combination ensures that the energy\ndensity ˆϵis regular and reduces to Gubser’s original solution as c2→0−. One can see this most easily by expanding\nthe solution for ˆϵ(ρ)in a power series for small c2<0:\nˆϵ(ρ)\nˆϵ0=1\n(cosh ρ)8/3−α\n(cosh ρ)4/3+3α2\n4+O(α3(cosh ρ)4/3) (42)\nwhere we have also defined α≡(−c2)2/3/(3·21/3). The general term in this expansion is clearly of order\nαn(cosh ρ)4(n−2)/3, so that the series reduces to (30) when α3/4≪coshρ, i.e., as c2→0−.\nOn the other hand, for sufficiently large ρone finds that the ˜ydecreases inversely with its argument: ˜y(w)∝1/w.\nPlugging this behavior into (36), we find that the conformal energy density scales for large ρas\nˆϵ(ρ)∼r−2∼e−4ρ(43)6\n0 2 4 6 8 10\nρ1010\n108\n106\n104\n102\n100\nˆ/epsilon1\nˆ/epsilon10\n(a)c2 = -1\nc2 = -10−1\nc2 = -10−2\nc2 = -10−3\nc2 = -10−4\nGubser\ne−4ρ\n0 2 4 6 8 10\nρ02468\nξ\n(b)c2 = -1\nc2 = -10−1\nc2 = -10−2\nc2 = -10−3\nc2 = -10−4\nGubser\nξ = ξ∗\nFIG. 1. The conformal energy density ˆϵ(panel (a)) and the flow rapidity ξ(panel (b)) vs. the time-like coordinate ρ, for\ndifferent choices of the free parameter c2. Gubser’s original solution (corresponding to c2= 0is given by the thick (solid, black)\nline in both panels. The horizontal dashed black line in the righthand panel shows that ξ∗=ξ(ρ∗)≈1.188, independent of c2.\nSee text for discussion.\nOne therefore expects a regime in ρwhere the leading behavior of ˆϵ(ρ)transitions from that of Gubser’s solution (30)\nto a steeper exponential fall-off in ρ(43). The transition should happen in the neighborhood of some critical value\nρ=ρ∗where the sub-leading corrections in (42) become comparable to the leading term:\n1\n(cosh ρ∗)8/3≈α\n(cosh ρ∗)4/3−→ρ∗≈arccosh\u0010\nα−3/4\u0011\n= arccosh\u0012541/4\n√−c2\u0013\n(44)\nInterestingly, we note that the definition (44)implies a constant value for ξ(ρ∗)≡ξ∗≈1.188, since the c2dependence\ncancels in Eq. (37). Thus, while the precise “Gubser time\" ρ∗at which the sub-leading corrections become important\ndepends on the value of c2, it occurs at the same flow rapidity ξ∗, regardless of c2.\nThis behavior is illustrated in Fig. 1, which shows the conformal energy density ˆϵ(panel (a)) and the flow rapidity ξ\n(panel (b)) for several choices of c2. For ˆϵ, one indeed observes that for sufficiently small ρ, our solution is close to\nGubser’s original solution (solid black line), while for sufficiently large ρ, the solution instead scales like e−4ρ(black\ndotted line). Moreover, the point ρ∗where the transition occurs depends on c2roughly as implied by (44). For\nexample, for c2=−10−3, we find using Eq. (44)that the critical value ρ∗≈5.14, in good agreement with Fig. 1.\nSeveral other values of ρ∗are marked for the various c2shown.\nFig. 1(b) shows that ξis a monotonically increasing function for positive ρ, and by symmetry under η→ −ηis\nmonotonically decreasing for negative ρ(cf.(35)). As c2→0−, however, the solution again converges smoothly back\nto Gubser’s boost invariant solution. Note that, since ξ̸= 0whenever c2̸= 0,uηis always non-zero except when\nc2= 0.\nFinally, we emphasize that, aside from the reflection symmetry η→ −η, our unshifted solution preserves all of\nGubser’s original symmetries (including rotational invariance around the beam axis and boost invariance). In the next\nsubsection, we will show how to modify this solution to break boost invariance as well.\nD. The shifted solution\nThe new solution found above consists of a non-trivial solution for ϵanduµin Milne coordinates, which introduces a\nnon-vanishing uηwhile remaining independent of η. In this sense, the solution is analogous to an azimuthally invariant\nsystem which nevertheless rotates in the azimuthal direction. However, our original goal was to construct a solution\nwithout boost invariance. As pointed out in Ref. [ 44], additional solutions possessing non-trivial ηdependence can\nbe generated by simply translating a known solution by a constant amount in the Minkowski temporal coordinate:\nt→t′=t+t0. We call this transformation a “temporal shift.\" This is equivalent to a change of Milne coordinates7\ngiven by\nτ→τ′(τ, η, r ) =q\nτ2+ 2t0τcoshη+t2\n0 (45)\nη→η′(τ, η, r ) = arctanh\u0012τsinhη\nτcoshη+t0\u0013\n(46)\nr→r′(τ, η, r ) =r (47)\nNotice that the transformation on the rcoordinate is trivial, since the temporal shift affects only τandη, but we\ninclude a shifted r′as well for clarity below.\nBy definition, the energy density ϵtransforms as a scalar:3\nϵ(τ, r, η ) =ϵ′(τ′, r) (48)\nwhere here and below, τ′is defined as in Eq. (45). Since the flow velocity is a four-vector, it transforms as\nuτ′=∂τ′\n∂τuτ+∂τ′\n∂ηuη+∂τ′\n∂rur(49)\nuη′=∂η′\n∂τuτ+∂η′\n∂ηuη+∂η′\n∂rur(50)\nur′=∂r′\n∂τuτ+∂r′\n∂ηuη+∂r′\n∂rur(51)\nThanks to the triviality of the transformation in the rdirection, the radial component of the flow velocity transforms\nonly by shifting the coordinates in Eq. (39) appropriately:\nur(τ, r, η ) =ur′(τ′, r) (52)\nThe remaining components of the flow velocity are then found by solving (49)and(50)foruτanduη. The result is\n(with c1set as in Eq. (41)):\nr′= cosh2ρ′=1 + 2 q2\u0010\nτ′2+r2\u0011\n+q4\u0010\nτ′2−r2\u00112\n4q2τ′2(53)\nˆϵ(ρ′)\nˆϵ0=(−16c2)1/3\nr′\u0012˜y(c2r′)\n˜y(c2r′)2−1\u0013\n(54)\nξ(ρ′) =1\n2ln ˜y(c2r′) (55)\nuτ(τ, η, r ) =\u0012τ+t0coshη\nτ′\u0013\nuτ′(τ′, r)−(t0sinhη)uη′(τ′, r) (56)\nur(τ, η, r ) =ur′(τ′, r) (57)\nuη(τ, η, r ) =\u0012\n1 +t0\nτcoshη\u0013\nuη′(τ′, r)−t0\nτ\u0012sinhη\nτ′\u0013\nuτ′(τ′, r) (58)\nEqs.(53)-(58), together with (45), constitute our complete new Solution I which generalizes Gubser flow to include\nnon-boost-invariant longitudinal flow, where the unshifted flow components uτ′,ur′, and uη′have been defined in\n(38)-(40).\nE. Properties of Solution I\nWe now consider how our Solution I obtained above depends on the parameters c2andt0. For definiteness, all other\nparameters are fixed to their values as defined in Refs. [39, 40]. Specifically, we take\nq= (4.3fm)−1andˆϵ0= 880\n3Here and below, quantities marked with primes (′) correspond to the unshifted frame; unprimed quantities are defined in the shifted\nframe.8\n101\n100101102103/epsilon1 (GeV/fm3)t0 = 0.1 fm/c\nc2 = 0\n(a)r = 0 fm,  η = 0\nr = 0 fm,  η = 2\nr = 0 fm,  η = 4r = 5 fm,  η = 0\nr = 5 fm,  η = 2\nr = 5 fm,  η = 4r = 10 fm,  η = 0\nr = 10 fm,  η = 2\nr = 10 fm,  η = 4\nt0 = 0.1 fm/c\nc2 = -0.1\n(b)\n0.1 0.5 5 10\nτ (fm/c)101\n100101102103/epsilon1 (GeV/fm3)t0 = 0.5 fm/c\nc2 = 0\n(c)\n0.1 0.5 5 10\nτ (fm/c)t0 = 0.5 fm/c\nc2 = -0.1\n(d)\nFIG. 2. The physical energy density ϵas a function of τ, for various radius rand rapidity η. Each panel shows a different\ncombination of free parameters t0andc2:t0= 0.1fm/candc2= 0(panel (a)); t0= 0.1fm/candc2=−0.1(panel (b));\nt0= 0.5fm/candc2= 0(panel (c)); t0= 0.5fm/candc2=−0.1(panel (d)).\nwhich are chosen in order to have a reasonable description of central Au+Au collisions with√sNN= 200 AGeV.\nWe begin by considering how the physical energy density depends on c2andt0. This is shown in Figure 2. We\nobserve several qualitative behaviors. First, by comparing panels (a) and (b) with the panels (c) and (d), we see that\nincreasing t0has the effect of steepening the rapidity dependence of the shifted solution. This is intuitively plausible,\nsince setting t0= 0must lead back to an η-independent solution. A similar effect is observed by making c2non-zero:\nthe rapidity dependence is weakened as the value of c2is decreased. Notice further that the effects of modifying t0or\nc2affect ϵin different ways: increasing t0causes ϵto fall faster at late τ, while decreasing c2leads to a faster decrease\nofϵin the radial direction and only weakly affects the τdependence. We only show curves with η≥0since the energy\ndensity is always symmetric under η→ −η(cf. (54) and (45)).\nOur solution’s flow profile is shown in Figs. 3 and 4. In Fig. 3, we show the ratio ur/uτplotted against rapidity η,\nfor different combinations of r,τ,t0, and c2. We observe that increasing t0(cf. panels (a) and (b) with panels (c)\nand (d)) again reduces the solution’s width in η. Decreasing c2, however, manifestly breaks the reflection symmetry\nη→ −ηas anticipated above (cf. panels (a) and (c) with panels (b) and (d)): having chosen the ‘positive’ branch of ξ\nfor our solution in Eq. (37), we see that the resulting transverse flow is stronger for η <0. This is a consequence of the\nfact that a positive flow rapidity ξyields anegative rapidity component uη, by Eq. (40). Choosing c2<0therefore\nleads to an asymmetric flow in η, by (56) and (58).\nIn Fig. 4, we show the corresponding results for the longitudinal flow velocity, τuη/uτ. In panels (a) and (c), we see9\n0.00.20.40.60.81.0\nur\nuτt0 = 0.1 fm/c\nc2 = 0(a)τ = 0.2 fm/c,  r = 1 fm\nτ = 0.2 fm/c,  r = 5 fm\nτ = 0.2 fm/c,  r = 10 fmτ = 1 fm/c,  r = 1 fm\nτ = 1 fm/c,  r = 5 fm\nτ = 1 fm/c,  r = 10 fmτ = 5 fm/c,  r = 1 fm\nτ = 5 fm/c,  r = 5 fm\nτ = 5 fm/c,  r = 10 fm\nt0 = 0.1 fm/c\nc2 = -0.1(b)\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5\nη0.00.20.40.60.81.0\nur\nuτt0 = 0.5 fm/c\nc2 = 0(c)\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5\nηt0 = 0.5 fm/c\nc2 = -0.1(d)\nFIG. 3. The transverse velocity ur/uτas a function of η, for various τandr. Each panel shows a different combination of free\nparameters t0andc2:t0= 0.1fm/candc2= 0(panel (a)); t0= 0.1fm/candc2=−0.1(panel (b)); t0= 0.5fm/candc2= 0\n(panel (c)); t0= 0.5fm/candc2=−0.1(panel (d)). See text for discussion.\nthat the only ηdependence arises from the shift to t0̸= 0; since c2= 0, there is no additional rapidity dependence\nwhich arises from breaking the reflection symmetry. In this case τuη/uτdepends only on τandη, but not on r(cf.\nEqs.(56)and(58)). One also observes that increasing t0steepens the rapidity dependence, as was noted above; setting\nt0= 0would result in uη= 0for all τandr. A more nuanced picture of the flow emerges once we let c2<0(shown in\nFig. 4(b) and (d)). In this case, the location with the longitudinal flow vanishes shifts to η <0, where the size of the\nshift depending on both τandr. One therefore finds that the regions where the rapidity flow ( uη) isweakestare also\nthose where the radial flow ( ur) isstrongest . This is intuitively what one would expect for an incompressible fluid.\nNote, however, that this rough correspondence is not exact: even in the symmetric case ( c2= 0),ur/uτdoes not reach\na maximum at η= 0where the longitudinal flow vanishes. This is due to the complicated nature of Gubser flow and\nthe effects of translating in the temporal direction (cf. panels (a) and (b) in Figs. 3 and 4).\nOur solution therefore exhibits some curious properties: the energy density is always symmetric under η→ −η,\nwhile the flow profile is asymmetric whenever c2<0. This suggests that scenarios with c2<0may not bear much\nphenomenological relevance to the modeling of nuclear collisions, where one would expect to find both flow profiles\nand energy densities to be asymmetric in η[44]. Nonetheless, this solution may also be useful in generating extreme\nflow profiles which can be used to subject hydrodynamic codes to valuable performance tests. In the specific case that\nc2= 0, however, the temporal shift induces non-trivial ηdependence without violating the Z2symmetry in η, and\nthus amounts to a shifted, non-boost-invariant generalization of Gubser’s original, boost-invariant solution for ideal\nhydrodynamics. In the next section, we show how to apply the same ‘temporal shift’ trick to the viscous version of10\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nτuη\nuτt0 = 0.1 fm/c\nc2 = 0\n(a)τ = 0.2 fm/c,  r = 1 fm\nτ = 0.2 fm/c,  r = 5 fm\nτ = 0.2 fm/c,  r = 10 fmτ = 1 fm/c,  r = 1 fm\nτ = 1 fm/c,  r = 5 fm\nτ = 1 fm/c,  r = 10 fmτ = 5 fm/c,  r = 1 fm\nτ = 5 fm/c,  r = 5 fm\nτ = 5 fm/c,  r = 10 fm\nt0 = 0.1 fm/c\nc2 = -0.1\n(b)\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5\nη1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nτuη\nuτt0 = 0.5 fm/c\nc2 = 0\n(c)\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5\nηt0 = 0.5 fm/c\nc2 = -0.1\n(d)\nFIG. 4. The longitudinal velocity τuη/uτas a function of η, for various τandr. Each panel shows a different combination of\nfree parameters t0andc2:t0= 0.1fm/candc2= 0(panel (a)); t0= 0.1fm/candc2=−0.1(panel (b)); t0= 0.5fm/cand\nc2= 0(panel (c)); t0= 0.5fm/candc2=−0.1(panel (d)). See text for discussion.\nGubser’s solution, for which the shear viscosity is non-zero.\nIII. SOLUTION II\nThe solution derived above in Sec. II was obtained assuming only ideal hydrodynamics. However, Gubser [ 39]\noriginally derived a version of his boost-invariant solution to the Navier-Stokes equations with a finite specific shear\nviscosity ˜η/s̸= 0.4Gubser’s viscous solution is thus a generalization of our Solution I (with c2= 0) to a finite shear\nviscosity. In this section, we apply a temporal shift to introduce rapidity dependence also into this viscous solution.\nThis will allow us to define and examine the Knudsen number in Sec. IV.\nGubser’s viscous solution is most conveniently written in terms of the temperature TG, where the subscript ‘G’\n4Here and below, we denote the shear viscosity by ˜η, in order to distinguish it from the space-time rapidity η.11\ndenotes ‘Gubser.’ TGis related to the energy density by ϵ≡f∗T4\nG(where f∗≡11) and is defined by [39, 40]\nTG(τ, r) =ℏc\nf1/4\n∗τˆT(ρ(τ, r)) (59)\nˆT(ρ) =ˆT0\n(cosh ρ)2/3\u0012\n1 +H0\n9ˆT0(sinh ρ)3\n2F1\u00123\n2,7\n6,5\n2;−sinh2ρ\u0013\u0013\n, (60)\nwhere ρis defined by Eq. (2)andH0∝η/s. We set ˆT0= 5.55andH0= 0.33, corresponding to a choice of η/s= 0.134\n[39].\nThe flow uµ\nGin the viscous case is the same as that obtained in the ideal case (7)-(9). One easily checks that\n(uτ\nG)2−(ur\nG)2= 1. Gubser’s complete viscous solution for TGanduµ\nGis thus given by Eqs. (7), (8), and (59).\nTo obtain the shifted version of Gubser’s viscous solution (denoted by the subscript ‘sG’), we apply the same\ntransformation discussed in Sec. IID. Like the energy density ϵ, the temperature TGmust transform as a scalar. This\nimplies that Gubser’s viscous solution with rapidity dependence is given by\nTsG(τ, r, η ) =TG(τ′, r) (61)\nIn analogy with (57), the radial component of the shifted flow uµ\nsGis\nur\nsG(τ, r, η ) =ur′\nG(τ′, r) (62)\nThe remaining components of the flow velocity are then found by solving (49)-(50)for the shifted components as\nbefore, and recalling that uη′\nG= 0as a result of the boost invariance and reflection symmetry of the Gubser solution.\nThe result is\nuτ\nsG(τ, r, η ) =uτ′\nG(τ′, r) \nτ+t0coshηp\nτ2+ 2t0τcoshη+t2\n0!\n(63)\nuη\nsG(τ, r, η ) =−uτ′\nG(τ′, r) \nt0sinhη\nτp\nτ2+ 2t0τcoshη+t2\n0!\n(64)\nAgain, one easily confirms that the shifted flow still possesses the correct normalization:\n(uτ\nsG)2−(ur\nsG)2−(τuη\nsG)2= 1 (65)\nEqs.(61)-(64)thus provide a generalization of Gubser’s viscous solution which relaxes the assumption of boost\ninvariance by introducing a non-trivial dependence on the space-time rapidity η. It coincides with the first solution\nfound above in Secs. II only if one takes c2= 0and˜η/s= 0. In Sec. IV, we will consider several properties of\nthis generalized solution and their implications for the relationship between the freeze out hypersurface and flow in\nrelativistic nuclear collisions.\nIV. FLOW AND FREEZE OUT\nHaving introduced the two new exact solutions derived above, we turn finally to discuss how we can use them to\nbetter understand the role of flow in influencing the freeze out process. Since we wish to compare freeze out at constant\ntemperature with that at constant Knudsen number, we will focus our attention on Solution II.\nIn general, the criteria defining freeze out are normally formulated [ 2] in terms of quantities which depend locally\non the hydrodynamic state of the system and which reflect (directly or indirectly) the extent to which the system\nin a given region may be represented as a strongly coupled plasma of deconfined color charges. The two criteria we\nconsider here are: (i), freeze out at constant temperature; and (ii), freeze out at constant Knudsen number. Freeze\nout at constant temperature has historically been the more widely used criterion, and may be interpreted either as a\nphenomenological choice of where to ‘switch’ from a partonic to a hadronic description of the system [ 11,46] or as a\nway of characterizing the system’s passage through the confinement/deconfinement transition [ 20]. On the other hand,\nas noted earlier, freeze out at constant Knudsen number characterizes the partonic-hadronic transition in terms of the\ndegree to which a fluid dynamical description is applicable. Insofar as constant Thypersurfaces closely approximate\nthose of constant Knin realistic hydrodynamic simulations [ 31], the distinction between them is largely academic. Our12\ngoal here, however, is to explore in the context of an analytical solution whether this identification always holds, and if\nnot, under what kinds of conditions it may fail.\nThe Knudsen number Kn is defined by [2, 31]\nKn≡ℓmfp\nℓmacro(66)\nwhere ℓmfpis the mean free path of the fluid and ℓmacrois an appropriate macroscopic length scale compared to which\nℓmfpshould be sufficiently small in order to justify the applicability of fluid dynamics.\nIn order to compute the Knudsen number, we must estimate both lengthscales entering Eq. (66). In terms of the\nquantities employed here, ℓmfpcan be estimated by [2, 31]\nℓmfp∼5˜η\nsT\nwhere Tis the temperature, and ˜η/sis the ratio of the shear viscosity to entropy density. This choice of ℓmfpcorresponds\nto the value of the shear relaxation time τπusing the 14-moment approximation of the relativistic Boltzmann equation\nin the massless limit [ 47,48]. As noted above, since a finite ℓmfp̸= 0implies a nonvanishing shear viscosity ratio\n˜η/s̸= 0, we limit our discussion in this section to the shifted viscous Gubser solution obtained in Sec. III.\nMany choices are possible for the macroscopic lengthscale ℓmacro[31]; here, we define it in terms of the scalar\nexpansion rate θ,5which defines a corresponding lengthscale ℓmacro =Lθgiven by\n1\nLθ≡θ(τ, r, η ) =∂µuµ+ Γµ\nµαuα=1 +q2\u0010\nr2+ 5τ′2\u0011\nτ′q\n1 + 2 q2\u0000\nτ′2+r2\u0001\n+q4\u0000\nτ′2−r2\u00012(67)\nwhere τ′is defined in (45), and the non-vanishing Christoffel symbols in Milne coordinates are\nΓτ\nηη=τ, Γη\nητ= Γη\nτη=1\nτ, Γr\nϕϕ=r, Γϕ\nϕr= Γϕ\nrϕ=1\nr(68)\nLθthus characterizes the typical lengthscale over which the strength of the system flow changes appreciably. As\npointed out above and in [ 31], these scales should be much larger than the relevant microscopic scale which determines\nthe strength or frequency of interaction, i.e., the mean free path ℓmfp, in order for fluid dynamics to be applicable.\nThe Knudsen number may therefore be defined as\nKn≡Knθ≡ℓmfp\nLθ=5˜ηθ\nsT(69)\nFrom(69), it follows that that the Knudsen number is small enough ( Kn≪1) to warrant the use of fluid dynamics\nonly when either ˜η/s(i.e., out of equilibrium viscous corrections) or θ(i.e., the flow gradients) is sufficiently small. In\nthe former case, one may have a violently expanding system which nevertheless is describable using fluid dynamics, as\nlong as the mean free path is sufficiently short and the frequency of interactions sufficiently high. In the latter case, one\nmay have a relatively large mean free path but nevertheless maintain local equilibrium if the system evolves sufficiently\nslowly. Conversely, when either ˜η/sorθbecomes too large relative to the other, then the interactions become too\ninfrequent to maintain local equilibrium and thus fluid dynamics no longer provides a justifiable description of the\nsystem. It is at this point when freeze out in terms of Knshould be implemented.\nIn the remainder of this section, we consider two questions, using our Solution II as a concrete model:\n1.How much do the two freeze out criteria (constant temperature, constant Knudsen number) differ in the freeze\nout contours which they produce?\n2.To what extent are these differences influenced by the strength of collective flow and the size of its gradients\nthroughout the system?\n5Another possible choice is the spatial gradient of the energy density, ϵ, which is expected to be of the same order of magnitude as the\ndefinition we use here [31].13\n0 5 10 15051015\n0 5 10 15\n0.10.30.50.70.9\nFIG. 5. Pairs of freeze out contours using criteria of constant temperature (solid black lines) and constant Knudsen number Kn\n(dashed black lines), plotted on top of the transverse flow vr=ur/uτin the τ-rplane. Both panels take t0= 0.5fm/cand\neither η= 0(panel (a), where uη= 0) orη= 2(panel (b)). The Knudsen number contours correspond to Kn=0.50, 0.75, 1.00,\nand 1.25 (ordered by increasing τatr= 0), and the temperature contours have been adjusted to match the Knudsen number\ncontours at r= 0. A solid red contour at T= 130MeV is included for reference.\nTo address the first question, in Fig. 5 we plot contours of fixed Knudsen number (69)for Solution II (with\nt0= 0.5fm/c) as a function of τandr(displayed as solid, black lines), for η= 0(panel (a)) and η= 2(panel (b)).\nWe also show for reference an isothermal contour of T= 130MeV (solid, red line) corresponding roughly to Kn≈0.75,\naccording to the definition (69). The contours are displayed on top of a density plot of the radial velocity vr=ur/uτ\nwhich indicates where the transverse flow is the strongest. Four pairs of T/Kn(solid/dashed) contours are shown,\nwhere each pair is adjusted to coincide when r= 0. The Knudsen contours assume values of Kn=0.50, 0.75, 1.00, and\n1.25 (corresponding at r= 0respectively to τ≈2.2, 4.7, 7.3, and 10.3 fm /cand temperatures of T≈209MeV, 115\nMeV, 69 MeV, and 44 MeV).\nOne finds that, for η= 0,Kn<∼0.5at sufficiently small r<∼3fm and times which are neither too early nor too\nlate ( τ<∼2fm/c), and thus that both sets of contours (constant Tand constant Kn) demarcate similar regionsin\nthe(τ, r)-plane where the system may be treated fluid dynamically. Indeed, the contour of Kn= 0.5clearly agrees\nquantitatively with the corresponding contour of constant Tat sufficiently small radii rand late times τ. Similar\nstatements apply to the η= 2case, except that all freeze out contours are reached at earlier times and thus the regime\nwhere fluid dynamics is valid shrinks accordingly.\nFor larger Kn, Fig. 5 shows explicitly that the shapesof constant temperature contours can differ quite significantly,\nboth qualitatively and quantitatively, from contours of constant Knudsen number. We find that the contours of fixed\nTacquire a positive concavity at r= 0at sufficiently late times and low temperatures.6This leads to the formation of\na ‘wing-like’ structure at large radii which has been noted previously [ 34] and which has been shown to reflect the\nviolent collective expansion of the system [ 37,49]. This is because the interior of the fireball freezes out before the\nedges do, thanks to the rapid cooling produced by the strong collective flow.\nMoreover, one finds that, while the contours corresponding to Kn= 0.50and 0.75 agree qualitatively with one\nanother, significant discrepancies develop in the contour pairs with Kn=1.00 and 1.25 in the region where transverse\nflow becomes the strongest. This has a straightforward and intuitive interpretation: since Knessentially differs from T\n6In the ideal case, which is a reasonable approximation to the viscous solution at r= 0, one easily shows that the concavity of the\ntemperature contour changes sign at a critical proper time of τc≡1/(cq) = 4 .3fm/c, corresponding to an ideal temperature of\nT(τc) =ℏcqˆT0/f1/4\n∗≈140MeV. This seems to agree quite well with where the temperature contours in Fig. 5 become flat at r= 0.14\n0 5 10 150246810\n0 5 10 15020406080100120140\n0 5 10 150.00.20.40.60.8\nFIG. 6. Panel (a) displays thick contours in the (τ, r)-plane for Knθ= 0.75(solid blue), 1.00 (dashed purple), and 1.25\n(dot-dashed red). For each of these respective contours, we also plot the temperature T(panel (b)) and radial velocity ur/uτ\n(panel (c)) as a function of radius r, with the same styling of lines in each panel. For each thick contour representing Knθ, we\nalso plot a thin contour of the same color and linestyle to represent gKnθ.\nonly by a constant factor times θ, the differences between constant temperature and constant Knudsen contours must\narise from the expansion rate which pushes the constant Knθcontours to earlier times and smaller radii, compared\nwith constant Tcontours. Note that since uη= 0when η= 0(by Eq.(58)), in this case the discrepancies between the\ncontours are completely determined by ur/uτ, as well as the gradients of both transverse and longitudinal flow.\nIn response to our first question above, then, we conclude that different freeze out criteria such as decoupling at a\nfixed Tvs. a fixed Knmay indeed lead to dramatically different contours and consequently produce different geometries\nfor the freeze out process. In the present example, the largest differences occur at larger Knudsen numbers Kn>∼1.00\nand hence smaller temperatures T<∼100MeV. However, as we have already suggested and now demonstrate, the real\norigin of these discrepancies is the effect of powerful collective flow with large gradients which drives a rapid cooling of\nthe system and forces an earlier freeze out when it is sufficiently strong.\nIn order to facilitate a direct comparison between the contours of fixed Tand fixed Knand to show how their\ndifferences are influenced by collective flow, in Fig. 6 we plot the temperature Tand radial velocity vralong the\ncontours of constant Knthemselves, assuming t0= 0.5fm/c. In Fig. 6(a), we plot portions of the Knθas thick\n(τ, r)-contours at values of 0.75 (solid blue), 1.00 (dashed purple), and 1.25 (dot-dashed red), extending at late times\nfrom r= 0to approximately where dτ/dralong the contour diverges. Unlike the constant Tcontours in Fig. 5, the\nconstant Kncontours exhibit negative concavity at r= 0: freeze out occurs lastat the center of the fireball. In\nFig. 6(b) and (c), we plot Tandvralong these same contours as functions of rand find that Tis decidedly not\nconstant along the Kncontours. Instead, thanks to the rapid growth of transverse flow with rand its competition with\nthe temperature gradients in the fireball’s interior [ 25], one finds that Tfirstincreases from r= 0, before reaching a\npeak and then decreasing toward larger radii. By inspecting the curves in Fig. 6(b), we observe that the relative height\nof the peak above the central temperature is approximately 13%forKnθ= 0.75,37%forKnθ= 1.00, and 72%for\nKnθ= 1.25. Comparison with the curves in Fig. 6(c) then confirms that this relative difference increases with the\nchange in the maximum transverse velocity of the fluid, indicating the strong influence of the latter on the former. We\ncan therefore answer our second question by confirming that strong collective flow gradients can generate qualitatively\nsignificant discrepancies between different freeze out criteria.7\nThe tight connection between flow and the freeze out process may be further clarified by considering how the\nKnudsen contours would be affected by the omission of a portion of the flow. To see this, we construct an unphysical\n7Note that only very slight discrepancies between freeze out criteria were observed in Ref. [ 31] in the case of p+Pb collisions where\nthe same ‘wing-like’ structure was observed in the constant Tcontours, in apparent contrast to the results found here. In this regard,\nhowever, we emphasize several key differences between our exact solution and the calculations performed in [ 31]: (i), Gubser’s solution\n(and thus our Solution II) assumes Navier-Stokes hydrodynamics, for which the shear correction πµνis directly proportional to the shear\ntensor σµν, whereas [ 31] employed Israel-Stewart hydrodynamics [ 50], in which πµνevolves according to a relaxation equation; (ii), [ 31]\ndid not use a constant ˜η/sas assumed here; and (iii), it is unclear whether the Gubser solution’s flow gradients are really comparable to\nthose present in the p+Pb simulation shown in [ 31]. Some combination of these differences may account for the much larger sensitivity to\nfreeze out criterion seen here.15\n0 5 10 150246810\n0 5 10 15020406080100120\n0 5 10 150.00.20.40.60.8\nFIG. 7. Panel (a) displays contours in the (τ, r)-plane for Knθ= 0.75(solid blue), 1.00 (dashed purple), and 1.25 (dot-dashed\nred). For each of these respective contours, we also plot the temperature T(panel (b)) and radial velocity ur/uτ(panel (c)) as\na function of radius r, with the same styling of lines in each panel. Thick contours represent Knθatη= 0, while thin contours\nrepresent η= 2.\nflow which omits the ηcomponent from uµ:\n˜uµ= (˜uτ,˜ur,0,0)≡\u0012q\n1 + (ur)2, ur,0,0\u0013\n(70)\nwith urgiven by the shifted Gubser solution used in Eq. (67). We emphasize that Eq. (70)isnota solution of the\noriginal equations of motion (1), although it remains properly normalized by construction:\n˜uµ˜uµ= 1 (71)\nUsing ˜uµ, we compute a corresponding unphysical Knudsen number fKnθwhich may in turn be used to define modified\nfreeze-out contours. Specifically, we define\nfKnθ=˜η\nsT\u0000\n∂µ˜uµ+ Γµ\nµα˜uα\u0001\n(72)\nWe use the same definition of Tfor both KnθandfKnθas in Eq. (67). The corresponding contours are shown as\nthin lines in Fig. 6. We use three of the same values ( Kn= 0.75, 1.00, 1.25) of the Knudsen numbers as before. By\ncomparing the respective thick and thin line pairs we may then estimate the importance of the component uηfor the\nshape of the freeze out surface.\nWe first observe the effects of omitting uηon the shape of the Knθcontours themselves. By Eq. (58), we see that\nchoosing t0= 0fm/cwould have resulted in uη= 0and therefore no differences between the KnθandfKnθcontours.\nWith t0= 0.5fm/c, however, we see that the unphysical fKnθcontours shift to smaller τandrvalues relative to the\nKnθcontours. Omitting uηfrom the shifted viscous Gubser solution of Sec. III thus effectively enhances the flow\ngradients in the system and shrinks the region where fluid dynamics is applicable. As a result of the artificially shifted\ncontours, the unphysical flow also steepens the temperature and radial flow dependence along the contours. Looking at\nthe center panel of Fig. 6, we find that the discrepancies between the central ( r= 0) and peak temperatures along the\ncontours are slightly enhanced along the shifted fKnθcontours. The effects on vr/care more pronounced, producing up\nto a 6% reduction in the peak radial velocity for the Knθ= 0.75contour. Although we do not show it here, the effects\nof omitting uηare enhanced further as t0is increased. We conclude that one effect of setting t0>0is to ‘redirect’\nsome amount of the unshifted solution’s flow into the rapidity direction, such that artificially suppressing the resulting\nηcomponent of the flow generates spurious gradients which exacerbate the differences between freeze out criteria.\nBoth transverse and longitudinal flow therefore play essential roles in determining the quantitatively correct shape of\nthe freeze out surface, as well as the variations in the temperature and radial flow along it.\nThe rapidity dependence itself of the shifted solution can be seen in Fig. 7, where we compare contours of constant\nKnθforη= 0(thick lines) with η= 2(thin lines). We emphasize that the solution is independent of η(i.e., boost\ninvariant) when t0= 0. For t0>0, however, we see in Fig. 7(a) that increasing ηaway from zero shifts freeze out\nto earlier τ, without dramatically affecting the radial dependence. The radial velocity (Fig. 7(c)) likewise shifts to\nsmaller values without modifying the radial dependence significantly, producing a roughly 8% in the maximum vr/c16\natη= 2relative to mid-rapidity when Knθ= 0.75. In Fig. 7(b), we observe that, since both TandKnare Lorentz\nscalars, their relationship is unaffected by the shift in t, and thus the thick and thin lines coincide for all Knθcontours,\nregardless of the values of t0andη. Furthermore, increasing the magnitude of ηfurther shifts freeze out to still earlier\ntimes. Taken together, these results imply once again that an accurate description of the system’s shape and flow are\nboth essential for a quantitatively precise characterization of the freeze out process in nuclear collisions.\nOur shifted version of Gubser’s original solution thus illustrates the importance of accounting for both transverse\nand longitudinal flow (and, more precisely, their gradients) on the shape of the freeze out surface. In particular, the\nconstant temperature contours and constant Knudsen contours imply qualitatively similar geometries for the freeze out\nhypersurface when the flow is relatively weak. However, in the presence of strong collective flow, such as one might\nexpect to find in small collision systems [ 4,34,51], which freeze out criterion one adopts is likely to have a larger\nsystematic effect on experimental observables. The effects could be especially pronounced for observables which are\ndirectly sensitive to the geometry of the freeze out hypersurface, such as the Hanbury Brown – Twiss interferometric\nradii [37, 49]. We plan to explore the nature of this sensitivity in a subsequent study.\nV. CONCLUSIONS\nIn this work, we have developed two novel exact solutions conformal hydrodynamics. Both solutions apply a\nconstant shift of the Minkowski time coordinate (a “temporal shift\") to induce a non-trivial rapidity dependence\ninto an already known solution. In the first case (our “Solution I\"), we have generalized Gubser’s solution of ideal\nrelativistic hydrodynamics to allow for flow in the rapidity direction and combined this with a temporal shift to\ngenerate a solution with rapidity dependence as well. In the second case (“Solution II\"), we have applied a temporal\nshift to Gubser’s original solution of viscousrelativistic hydrodynamics, yielding a solution for which we can define\nout-of-equilibrium quantities like the Knudsen number. Our new solutions admit a number of interesting applications,\nincluding providing useful benchmark tests for 3+1D hydrodynamic codes, as well as allowing analytically tractable\ninsights into the nature of far-from-equilibrium dynamics in nuclear collisions.\nAs a concrete instance of this latter application, we have used the Solution II constructed here to clarify two related\nquestions: first, the amount by which alternative freeze out criteria may differ in the freeze out hypersurfaces which\nthey produce, and second, the extent to which these differences are influenced by the strength of collective flow and\nthe size of its gradients throughout the system.\nWe have answered the first question by considering the qualitative differences which emerge in our solution between\ncontours of fixed temperature Tand fixed Knudsen number Kn. We have found that large discrepancies tend to arise\nas the system falls progressively away from local equilibrium: in our solution, this tends to occur at temperatures\nT<∼100MeV. We anticipate that these differences should lead to systematic effects on the implementation of the\nfreeze out process and the evaluation of, e.g., Cooper-Frye integrals [52, 53] in more realistic numerical simulations.\nTo address the second question, we directly plotted the local temperature Tand radial velocity vralong contours of\nconstant Knusing our Solution II. This allowed us to quantify the typical magnitude of these effects and to explore\nthe role played by collective flow on the discrepancies. We have also considered how these results are affected by the\nartificial suppression of uηin the full solution, as a way of assessing their sensitivity to the flow itself. We found that\nour solution predicts a strong temperature variation (of up to 72%) along constant Kncontours in regions where\nthe flow is sufficiently strong. Even in regions where the TandKncontours were in qualitative agreement, we still\nobserved up to 13% variation in T(relative to Kn) in those regions where the bulk of particle production occurs [ 25].\nThis underscores the fact that systematic effects can and do arise on the basis of which freeze out criterion one adopts\nin hydrodynamic simulations.\nSince the discrepancies we observe between different freeze out criteria are somewhat more pronounced than those\nseen in previous studies (e.g., [ 31,32]), it is important to determine whether the results obtained here can be reproduced\nin the context of more realistic numerical simulations and to assess what, if any, are the systematic effects on predictions\nfor experimental observables. 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Lett. 115, 072501 (2015), arXiv:1503.07514 [hep-th]."    },    {        "title": "2402.03734v1.Magnon_mediated_spin_pumping_by_coupled_ferrimagnetic_garnets_heterostructure.pdf",        "content": "Magnon mediated spin pumping by coupled ferrimagnetic garnets\nheterostructure\nAnupama Swain*,1Kshitij Singh Rathore*,1Pushpendra Gupta,1Abhisek Mishra,1Gary Lee,2Jinho Lim,3Axel\nHoffmann,3Ramanathan Mahendiran,2and Subhankar Bedanta1, 4,a)\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), An OCC of Homi Bhabha National Institute (HBNI),\nJatni-752050, Odisha, India\n2)Department of Physics, 2 Science Drive 3, National University of Singapore, 117551,\nRepublic of Singapore\n3)Department of Materials Science and Engineering and Materials Research Laboratory,\nUniversity of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA\n4)Center for Interdisciplinary Sciences (CIS), NISER, An OCC of Homi Bhabha National Institute (HBNI),\nJatni-752050, Odisha, India\n*equal contribution\nABSTRACT\nSpin pumping has significant implications for spintronics, providing a mechanism to manipulate and trans-\nport spins for information processing. Understanding and harnessing spin currents through spin pumping is\ncritical for the development of efficient spintronic devices. The use of a magnetic insulator with low damping,\nenhances the signal-to-noise ratio in crucial experiments such as spin-torque ferromagnetic resonance (FMR)\nand spin pumping. A magnetic insulator coupled with a heavy metal or quantum material offers a more\nstraight forward model system, especially when investigating spin-charge interconversion processes to greater\naccuracy. This simplicity arises from the absence of unwanted effects caused by conduction electrons unlike\nin ferromagnetic metals. Here, we investigate the spin pumping in coupled ferrimagnetic (FiM) Y 3Fe5O12\n(YIG)/Tm 3Fe5O12(TmIG)bilayerscombinedwithheavy-metal(Pt)usingtheinversespinHalleffect(ISHE).\nIt is observed that magnon transmission occurs at both of the FiMs FMR positions. The enhancement of\nspin pumping voltage ( Vsp) in the FiM garnet heterostructures is attributed to the strong interfacial exchange\ncoupling between FiMs. The modulation of Vspis achieved by tuning the bilayer structure. Further, the spin\nmixing conductance for these coupled systems is found to be ≈1018m−2. Our findings describe a novel\ncoupled FiM system for the investigation of magnon coupling providing new prospects for magnonic devices.\nKeywords: Ferrimagnet, Spin Pumping, Magnons, Coupling, Thin Films.\nThe need for ultra-low power consumption devices has\ngiven rise to the field of spintronics. This field uses\nthe spin degree of freedom of electrons1,2. Spintronic-\nbased applications rely on the generation of spin cur-\nrents and their conversion into charge currents in mag-\nnetic heterostructures3–6. However, in this process, the\nscattering of conduction electrons in the magnetic layer\nresults in Joule heating. In this context, moving to-\nwards the magnon-based information processing in in-\nsulators will solve this issue as there is no physical move-\nment of the electrons7. Magnons are the quanta of spin\nwaves, defined as the collective excitations of magnetic\nmoments in magnetically ordered materials. Magnons\ncan propagate over distances ranging from micrometers\nin low damping metallic thin films to around cm in high\nquality magnetic insulators8,9. The magnonic spin cur-\nrent can be employed to carry, transport, and process\ninformation10,11, as well as generate a spin torque act-\ning on the local magnetic moment that can be exploited\nto drive magnetization dynamics12,13and magnetic do-\nmain walls14–16. A promising technique for detecting\nmagnonic spin currents is the spin pumping induced in-\nverse spin Hall effect (ISHE)10,17. Spin pumping refers\nto the transfer of spin angular momentum via magne-tization precession from the ferromagnetic material to\nthe adjacent spin sink layer18. These pure spin currents\nare transformed into conventional charge currents by the\nISHE, which allows for a convenient electrical detection\nof spin-wave based spin currents11. After the discovery\nof the spin-pumping effect and the ways for enhancement\nof spin current in ferrimagnetic insulator (yttrium iron\ngarnet, Y 3Fe5O12, YIG)/non-magnetic metal (platinum,\nPt), there was rapidly emerging interest in the investi-\ngation of these structures19,20. In these magnetic insu-\nlators movement of individual electron gets restricted,\nwhich helps to avoid Joule heating dissipation and there-\nfore benefits modern upcoming spintronic devices21.\nYIG is often considered as the best medium for spin\nwavepropagationbecauseofitsverysmallGilbertdamp-\ning coefficient ( 2×10−5for bulk YIG)22. Being an elec-\ntrical insulator, electron-mediated angular momentum\ntransfer can only occur at the interface between YIG and\na metallic layer. In that context, metals with large spin\norbit coupling (SOC) like Pt where a pure spin current\ncan be generated through spin Hall effect (SHE) have\nbeen used to excite or amplify propagating spin waves\nthrough loss compensation in YIG23–25. Recently, other\ninsulator garnets like Tm 3Fe5O12(TmIG), Gd 3Fe5O12arXiv:2402.03734v1  [cond-mat.mtrl-sci]  6 Feb 20242\n(GdIG) etc. have been investigated with a focus on low\nGilbert damping, interlayer coupling and spintronics ap-\nplications. Here we investigate the manipulation, genera-\ntionanddetectionofmagnon-basedspincurrentsinferri-\nmagnetic insulators capped with heavy metal Pt. These\ninsulators with Pt have been explored a lot for magnon\ntransport physics in effects like ISHE26, Spin Hall mag-\nnetoresistance (SMR)27and spin-Seebeck effect (SSE)28.\nThere are few results published on spin waves in coupled\nferrimagnetic layers29,30, however, the effect of coupling\non spin pumping in such FiM system is scarce in liter-\nature. In order to study the interface and the growth\neffects on spin pumping, this work presents a systematic\nstudy by considering YIG/Pt, TmIG/Pt and bilayers of\nYIG/TmIG with Pt. This study reveals the increase in\nthe spin pumping voltage in the bilayers which can be\nattributed to the interfacial exchange coupling between\nYIG and TmIG.\nHigh-quality garnet films were prepared on (111)\noriented GGG (GdGa 5O12)single crystal substrate by\npulsed laser deposition (PLD) technique using an ex-\ncimer laser (λ= 248 nm ). The garnet targets were com-\nmercially purchased from M/s. Testbourne, UK. The\nbase pressure of the chamber was 8×10−7mbar. The\nsubstrate temperature was maintained at 560◦Cin a\n9×10−4mbar of oxygen partial pressure during YIG\nlayer deposition. The laser fluence and repetition rate\nwere 1.8 J/cm2and8 Hz, respectively. After deposition,\nthe sample was annealed for 2 hat 800◦Cin 300 mbar\nof ambient oxygen environment and cooled at 10◦C/min\nrate. TheTmIGlayerwasgrownbymaintainingthesub-\nstrate temperature at 750◦C, laser fluence at 1 J/cm2,\nwith a repetition rate of 6 Hzin 0.26 mbar of oxygen par-\ntial pressure. Post deposition, the prepared TmIG film\nwas in-situ annealed at same growth temperature for 25\nmins at 100 mbar oxygen pressure followed by cooling\nat 5◦C/min. During deposition the substrate to target\ndistance was kept 5 cmfor both the films. The bilayer\nsamples were prepared by following the same growth con-\ndition as the individual single layers. For the bilayer\nsamples, each FiM layer was first deposited and then an-\nnealed in similar conditions as of its corresponding sin-\ngle reference layer samples and then the subsequent layer\nwas deposited followed by its post-annealing. The details\nof prepared sample structure are mentioned in Table I.\nThe thickness of the corresponding layer is mentioned\nin the brackets. The Pt layer has been prepared via dc\nmagnetron sputtering in a high vacuum multi-deposition\nchamber manufactured by Mantis Deposition Ltd., UK.\nThe growth quality and thickness of the prepared\nfilmswereinvestigatedby X-raydiffraction(XRD).High-\nresolution transmission electron microscopy (HR-TEM)\nhas been performed in this study to verify the epitaxy of\ndeposited films. The saturation magnetization value has\nbeen taken from the superconducting quantum interfer-\nencedevice(SQUID)magnetometrydata. Theferromag-\nnetic resonance (FMR) and spin pumping induced ISHE\nmeasurements were performed using a coplanar waveg-TABLE I. Details of the sample structure studied in this work\nSample Sample\nname structure\nS1 GGG(111)/YIG(100 nm)/Pt( (5 nm)\nS2 GGG(111)/TmIG(30 nm)/Pt(5 nm)\nS3GGG(111)/YIG(100 nm)/TmIG(30 nm)/Pt(5 nm)\nS4GGG(111)/TmIG(30nm)/YIG(100 nm)/Pt(5 nm)\nuide (CPW) based setup. The sample was placed up-\nside down on the CPW. FMR measurements were car-\nried out in the 3-12 GHz of frequency range with 25 mW\nmicrowave power. The ISHE measurements were carried\noutat7GHz rffrequencybyconnectingananovoltmeter\nat the opposite edges of the sample. The measurement\ndetails are mentioned in our previous work31.\nFIG. 1. XRD patterns of (a) S3 and (b) S4 samples (the\ncorresponding insets show the XRD pattern for the 2 θrange\n20◦-80◦). HRTEM image and SAED pattern for sample S3\nare shown in (c) and (d), respectively. The inset shows the\nmagnified part of YIG and TmIG interface.\nThe prepared samples were structurally characterized\nby XRD to confirm the phase and growth quality. The\nXRDpatternofS3andS4samplesareshowninFig. 1(a-\nb). The XRD pattern of S1 and S2 is given in the supple-\nmentary file. The YIG and TmIG lattice parameters are\nveryclosetothatofthesubstrateGGGwhichensuresthe\nepitaxial growth of the structure. The observed diffrac-\ntion peaks were indexed with the corresponding crystal\nplane (h k l) values, and it is evident from the analysis\nthatthegrowthofpreparedsamplesisalongthe(111)di-\nrection. The absence of any additional peaks other than\nthe peaks corresponding to YIG, TmIG, and GGG in\nthe patterns ensures the phase purity of the grown struc-\ntures (shown in the corresponding insets). In Fig. 1(a),\nthe peak corresponding to YIG layer is dominating over\nthe TmIG peak as the YIG thickness is high. By ana-\nlyzing the XRD peak at (444) reflection yields a cubic\nlattice parameter of YIG in S1 is 12.46 Å, which is com-3\nparable to 12.38 Åfor the bulk YIG32. Moreover, the\nlattice parameters of YIG and TmIG are estimated for\nall the samples. The YIG lattice parameter in S3 and\nS4 is 12.43 Åand 12.48 Å, respectively. Likewise, the lat-\ntice parameters of TmIG in S2, S3, and S4 are 12.45 Å,\n12.54 Å, and 12.60 Å, respectively. It is to be noted that,\nthe lattice parameter has changed in S3 and S4 samples\nfor both YIG and TmIG with respective to their single\nlayer i.e., S1 and S2.This indicates the presence of strain\nin the films which may have the impact on the physical\nproperties of the samples.\nThe interface is further explored by cross-sectional\nHRTEM for the sample S3. The zoomed-in image [shown\nin the inset of Fig. 1(c)] reveals the epitaxial growth\nof YIG and TmIG. These high-quality images provide\nclearevidenceofthewell-definedinterfaceandcrystalline\nstructure of the film showing in-plane lattice matching\nwith the substrate. The thicknesses of YIG and TmIG\nwere found to be 100 nm and 30 nm respectively. No-\ntably, no defects or misalignment in lattice planes were\nobserved in the HRTEM image shown in Fig. 1(c). Fig.\n1 (d) shows the selected area electron diffraction (SAED)\npattern of sample S3, which confirms the single crys-\ntalline nature. The validation provided by these HRTEM\nimages and SAED pattern is crucial evidence, confirm-\ning the successful fabrication of the thin films with sharp\ninterface.\nThe spin dynamics properties were investigated by\nFMR measurements at room temperature. Fig. 2 (a-\nd) show the FMR spectra of the prepared samples at\ndifferent frequencies ( f) ranging from 3- 12 GHz. Dis-\ntinct FMR peaks were observed for YIG and TmIG in\nall samples. This confirms the room temperature mag-\nnetic phase of the prepared samples. A shift in Hres\ncorresponding to YIG and TmIG in the S3 and S4 sam-\nples compared to S1 and S2 samples is observed. This\ncould be attributed to the interfacial exchange coupling\nbetween YIG and TmIG, similar to other multilayered\nsystems33.\nFurther, the FMR signals were fitted by Lorentzian\nfunction to obtain line width (∆H)and resonance field\n(Hres) at each frequency. Later, the damping analy-\nsis is carried out for all the samples by plotting fvs\nHresand∆Hvsf(shown in Fig. 3 ). In case of S1,\nthe damping value is estimated considering the uniform\n(n=0) mode. Here, for the bilayer samples, the damping\nanalysis is done for the respective resonance fields of YIG\nand TmIG.\nThe plotted data in Fig 3 (a) were fitted by the Kittel\nequation34,\nf=γ\n2πq\n(Hres+HK)(Hres+ 4πMeff+HK)(1)\nwhere, γ\u0000\n=gµB\nℏ) is gyromagnetic ratio ( gis Lande g-\nfactor, µBis Bohr magneton), HKis in-plane anisotropic\nfield), 4πMeff\u0010\n= 4πMs+2KS\nMstFM\u0011\nis the effective de-\nmagnetizing field (KS, Ms, and tFMare perpendicu-\nFIG. 2. FMR spectra of (a) S1 (b) S2 (c) S3 and (d) S4\nsamples at different frequencies.\nlarsurfaceanisotropyconstant, saturationmagnetization\nand thickness of the magnetic layer, respectively). Af-\nterwards, the Gilbert damping constant ( α) values were\nestimated by fitting the plot in Fig 3(b) and (c) by the\nequation\n∆H= ∆H0+4παf\nγ. (2)\nTheobtainedvaluesof g,Msandα(tabulatedinTable\nII)areingoodagreementwiththeexistingliterature26,35.\nAn increase in αvalue is observed for both YIG and\nTmIG in S3 and S4 samples compared to the S1andS2\nsamples.\nThe ISHE measurements were carried out by a FMR\nbased setup. The observed voltage signal corresponding\nto the FMR resonance is shown in Fig 4 (a-d) for the\nprepared samples. In general case, for ISHE two layers\nare required, one is the source for spin current i.e., the\nmagnetic layer and the other one is the spin sink i.e., the\nhigh spin orbit coupling (HS) material. In this study, the\nHS layer is the Pt layer where the spin current source is\nthe YIG/TmIG bilayer in the samples S3 and S4. In-\nterestingly, here we have observed ISHE voltage at two\ndistinct FMR resonance field positions corresponding to\nthe resonances and concomitant spin currents mainly ex-\ncited from different layers, i.e., YIG and TmIG layers in\nboth S3 and S4 samples as shown in Fig. 4.\nIn order to extract the spin pumping voltage from the\nrectifications such as anomalous Hall effect (AHE) and\nanisotropic magneto resistance (AMR), the voltage was\nmeasured at different value of the angle ( ϕ) at a step of\n5◦in the range of 0◦to360◦at a constant frequency of 7\nGHz. We deliberately used higher frequency to avoid the\n3 magnon splitting phenomenon which in general takes4\nFIG. 3. (a) fvsHresplot for all the samples. ∆Hvsfplot for (b) YIG and (c) TmIG for respective resonance in FMR signals\nwith corresponding fittings.\nplace at lower frequencies36. Here ϕis defined as the\nangle between the direction of Hand the perpendicular\ndirection to contacts for voltage measurement. The mea-\nsured voltage is fitted by the Lorentzian equation. The\nsymmetric ( Vsym) and antisymmetric ( Vasym) contribu-\ntions of the voltage were extracted. The obtained Vsym\nandVasymvalues were plotted as a function of angle ϕ\n(shown in Fig. 5). It is to be noted that the antisym-\nmetric component is almost negligible compared to the\nsymmetric component. This clearly indicates the domi-\nnance of the spin pumping induced ISHE in the samples.\nMoreover, to quantify the spin pumping voltage and\nother rectification effects the estimated VsymandVasym\nvalues in Fig. 5 were fitted by the given equations (3)\nand (4), respectively37.\nVsym=Vspcos3(ϕ) +VAHE cos(θ)cos(ϕ)\n+VAMR⊥\nsym cos(2ϕ)cos(ϕ)\n+VAMR||\nsym sin(2ϕ)cos(ϕ)(3)\nVasym =VAHE sin(θ)cos(ϕ)+\nVAMR⊥\nasym cos(2ϕ)cos(ϕ)+\nVAMR||\nasym sin(2ϕ)cos(ϕ)(4)\nwhere VspandVAHEare voltages due to spin pumping\nand the anomalous Hall effect. Furthermore, VAMR∥\nasym,sym\nandVAMR⊥\nasym,sym are the parallel and perpendicular com-\nponents of the AMR voltage, respectively. θis the angle\nbetween the electric and magnetic fields of the microwave\nwhich is 90◦.\nThe extracted value from fitting is tabulated in Table\nII. The V∥,⊥\nAMRcomponent is calculated by the following\nformula38\nV∥,⊥\nAMR =r\u0010\nVAMR∥,⊥\nsym\u00112\n+\u0010\nVAMR∥,⊥\nasym\u00112(5)\nFrom Table II, it is further confirmed quantitatively\nthat the spin pumping voltage Vspis the dominating\nFIG. 4. (a)FMR and ISHE spectra for (a) S1, (b) S2, (c) S3\nand S4 samples at 7 GHz.\ncontribution to the measured ISHE voltage. The power-\ndependent data (shown in supplementary file Fig. 2) fur-\nther ensure the spin pumping dominance in all the sam-\nples. It is observed that the Vsphas decreased by one\norder for YIG in S3 sample as compared to the S1 sam-\nple. This is expected as in S3 sample, the spin current\ngenerated at YIG layer has to pass through the TmIG\nlayer before converting to charge current at Pt layer. In\nthis process, there may be dissipation of the spin angular\nmomentumwhichledtothedecreaseinthespinpumping\nvoltage. The Vspisalmost20timeslargerforTmIGinS3\nas compared to the S2 sample. The observed enhanced\nVspfor TmIG in S3 and S4 as compared to the S2 sam-\nple can be ascribed to the interfacial exchange coupling.\nThe presence of interlayer exchange coupling in magnetic\nbilayersystemshasalreadybeenshowntochangetheam-\nplitudes of different ferromagnetic resonance modes due\nto the dynamic exchange field at the interface39,40. As\nobservedfromtheFMRdata, thecouplingofthemagnon\ncurrent at the interface of YIG and TmIG led to the en-\nhancement of the Vspfor TmIG. Interestingly, the Vspis\nmaximum for YIG in S4 sample. This may be ascribed\nto the interfacial exchange coupling between YIG and\nTmIG.5\nTABLE II. Fitted parameters\nSample S1(GGG/YIG/Pt) S2(GGG/TmIG/Pt) S3(GGG/YIG/TmIG/Pt) S4(GGG/TmIG/YIG/Pt)\nYIG TmIG YIG TmIG\nα×10−30.51±0.07 17.00±0.03 1.10±0.0317.00±0.04 2.60±0.01 30.00±0.38\nVsp(µV) 180.0±7.8 0.06±0.01 18.0±0.71.00±0.01220.00±0.52 0.16±0.12\nVasym\nAHE (µV) 17.0±1.5 0.010±0.004 0.94±0.070.16±0.07 3.20±0.82(0.40±0.02)×10−2\nV⊥\nAMR(µV) 12.0±1.0 0.04±0.01 11.0±0.80.58±0.01 1.10±0.16 0.17±0.15\nV∥\nAMR(µV) 0.62±0.04 (0.40±0.04)×10−21.40±0.380.110±0.002 1.40±0.14 0.050±0.006\ng↑↓\neff 4.78×10182.39×10171.56×10171.12×10171.24×10181.76×1018\nFIG. 5. Angular dependence of VsymandVasymwith corre-\nsponding fits for (a) S1, (b) YIG in S3, (c) TmIG in S3, and\n(d) YIG in S4.\nIn this context, in order to quantify the spin current\npropagation, effective spin mixing conductance (g↑↓\neff) is\nevaluated by the following expression using the obtained\ndamping constant value6.\ng↑↓\neff=4π∆αMstFM\ngµB(6)\nwhere MS,tFM, and ∆αare the saturation magneti-\nzation, thickness of magnetic layer and change in Gilbert\ndamping from bilayer to single layer films, respectively.\nThese g↑↓\neffvalues (of the order of 1018m−2) are well\nmatched with the existing literature26. Hence, the inter-\nfacial exchange coupling led to the enhancement of spin\npumping for YIG and TmIG layer41.\nWehavestudiedspinpumpingandISHEforgarnet/Pt\nsystems. The damping analysis exhibits an enhancement\nofαin the bilayer garnet sample. The measured ISHE\nvoltage for YIG/Pt layer is around 180 µV where the ma-\njor contribution is from spin pumping. A decrease in the\nISHE voltage is observed by one order i.e., 17 µV in S3\nwhich is attributed to the presence of TmIG layer play-\ning as a hindrance to the transfer of angular momentumfrom the YIG layer. Whereas, an increase in spin pump-\ning voltage for YIG in S4 and TmIG in S3 is observed as\ncompared to their respective single layers. This may be\nattributed to the interfacial exchange coupling between\nYIG and TmIG. Moreover, further study can be carried\nout to have an insight to interface exchange coupling and\nthe effects of spin pumping on magnon-magnon interac-\ntions.\nACKNOWLEDGEMENT\nS.B., A.S., K. S. R., P.G., and A.M. thank the De-\npartment of Atomic Energy, Department of Science and\nTechnology, Science and Engineering Research Board\n(Grant No. CRG/2021/001245), Government of In-\ndia, Chanakya Post-doctoral fellowship, i-Hub quan-\ntum technology foundation (Sanction Order No. I-\nHUB/PDF/2022-23/04) for providing financial support.\nWork from J. L. and A. H. with respect to the data\nanalysis and discussion, as well as manuscript prepara-\ntion has been supported by the U.S. Department of En-\nergy, Office of Science, Basic Energy Sciences, Materials\nSciences and Engineering Division through Contract No.\nDE-SC0022060.\nDATA AVAILABILITY\nThe data that support the findings of this study are\navailable from the corresponding author upon request.\nREFERENCES\naElectronic mail: sbedanta@niser.ac.in\n1S. D. Bader and S. Parkin, Annu. Rev. Condens. Matter Phys.\n1, 71 (2010).\n2I. Žutić, J. Fabian, and S. D. Sarma, Reviews of modern physics\n76, 323 (2004).\n3A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands,\nNature physics 11, 453 (2015).\n4B. B. Singh, K. Roy, P. Gupta, T. Seki, K. Takanashi, and S. Be-\ndanta, NPG Asia Materials 13, 9 (2021).\n5P. Gupta, B. B. Singh, K. Roy, A. Sarkar, M. Waschk,\nT. Brueckel, and S. 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Maier-Flaig,\nM. Althammer, H. Huebl, R. Gross, R. D. McMichael, M. D.\nStiles, et al., Physical review letters 120, 127201 (2018).\n41J. Liu, Y. Xiong, J. Liang, X. Wu, C. Liu, S. K. Cheung,\nZ. Ren, R. Liu, A. Christy, Z. Chen, et al., arXiv preprint\narXiv:2309.03116 (2023)."    },    {        "title": "2402.03831v1.Soliton_Management_for_ultrashort_pulse__dark_and_anti_dark_solitons_of_Fokas_Lenells_equation_with_a_damping_like_perturbation_and_a_gauge_equivalent_spin_system.pdf",        "content": "Soliton Management for ultrashort pulse: dark\nand anti-dark solitons of Fokas-Lenells equation\nwith a damping like perturbation and a gauge\nequivalent spin system\nRiki Dutta1, Gautam K. Saharia1†, Sagardeep Talukdar1†,\nSudipta Nandy1*†\n1Department of Physics, Cotton University, Panbazar, Guwahati,\n781001, Assam, India.\n*Corresponding author(s). E-mail(s):\nsudipta.nandy@cottonuniversity.ac.in;\nContributing authors: rikidutta96@gmail.com;\ngautamks0802@gmail.com; talukdarsagardeep@gmail.com;\n†These authors contributed equally to this work.\nAbstract\nWe investigate the propagation of an ultrashort optical pulse using Fokas-Lenells\nequation (FLE) under varying dispersion, nonlinear effects and perturbation.\nSuch a system can be said to be under soliton management (SM) scheme. At first,\nunder a gauge transformation, followed by shifting of variables, we transform\nFLE under SM into a simplified form, which is similar to an equation given\nby Davydova and Lashkin for plasma waves, we refer to this form as DLFLE.\nThen, we propose a bilinearization for DLFLE in a non-vanishing background by\nintroducing an auxiliary function which transforms DLFLE into three bilinear\nequations. We solve these equations and obtain dark and anti-dark one-soliton\nsolution (1SS) of DLFLE. From here, by reverse transformation of the solution,\nwe obtain the 1SS of FLE and explore the soliton behavior under different SM\nschemes. Thereafter, we obtain dark and anti-dark two-soliton solution (2SS) of\nDLFLE and determine the shift in phase of the individual solitons on interaction\nthrough asymptotic analysis. We then, obtain the 2SS of FLE and represent the\nsoliton graph for different SM scheme. Thereafter, we present the procedure to\ndetermine N-soliton solution (NSS) of DLFLE and FLE. Later, we introduce a\nLax pair for DLFLE and through a gauge transformation we convert the spectral\n1arXiv:2402.03831v1  [nlin.SI]  6 Feb 2024problem of our system into that of an equivalent spin system which is termed as\nLandau-Lifshitz (LL) system. LL equation (LLE) holds the potential to provide\ninformation about various nonlinear structures and properties of the system.\nKeywords: Fokas-Lenells Equation, Soliton Management, Soliton, Lax Pair,\nLandau-Lifshitz Equation\n1 Introduction\nIn optics, soliton is a nonlinear optical pulse that propagates through a waveguide\nwithout getting distorted and hence can acts as a good information carrier in digital\nand quantum communication devices. So the research on soliton has always been in\ngreat demand. For the soliton to exist the system must balance the dispersion and non-\nlinear effects. Nonlinear Schr¨ odinger equation (NLSE) is the most basic equation that\naddresses both of the two effects and solving it results in a soliton solution (Hasegawa\nand Tappert, 1973a,b; Malomed and Weinstein, 1996; Hosseini et al, 2023; Serkin\nand Hasegawa, 2000; Chakraborty et al, 2015). Beyond optical soliton, NLSE (popu-\nlarly known by the name Gross–Pitaevskii equation (Pethick and Smith, 2008)) also\ndescribes the propagation of matter-wave in Bose-Einstein condensates, where the dis-\npersion is represented by kinetic energy of the bosons and nonlinear effect arises due to\nintra and inter bosonic interactions. In optics, besides NLSE, there are other equations\ntaking into account some higher order effects. Some of these well-studied equations are\nKaup–Newell equation (KNE) (Jawad et al, 2019), Chen–Lee–Liu equation (CLLE)\n(Gonz´ alez-Gaxiola and Biswas, 2018; Yıldırım et al, 2020), Tiriki-Biswas equation\n(TBE) (Triki and Biswas, 2018), sasa-satsuma equation (SSE) (Liu et al, 2017; Hos-\nseini et al, 2022; Yildirim, 2019), etc. It may be interesting to point out that KNE,\nCLLE and TBE are also referred to as byproducts of derivative NLSE (DNLSE). How-\never, a high intensity light source can produce ultrashort (femtosecond) pulses. To\ndescribe such a pulse, one must account for the higher order effects namely, spatio-\ntemporal dispersion (STD) and nonlinear dispersion (ND) effect along with the group\nvelocity dispersion (GVD) and Kerr nonlinear effects (KNLE). Fokas-Lenells equation\n(FLE) (Lenells and Fokas, 2008; Lenells, 2009) is a nonlinear equation that addresses\nall these effects. The dimensionless form of FLE is presented as\niUt+a1Uxx−a2Uxt+b|U|2(U+i a2Ux) = 0 (1)\nwhere Uis the field function that describes the complex waveform of an ultrashort\npulse. The suffix xandtdenote the partial differentiations of Uwith respect to x\nandtrespectively. Utis the temporal evolution of the pulse, Uxx,Uxt,|U|2Uand\n|U|2Uxrepresents GVD, STD, KNLE and ND respectively. Eqn. (1) is relatively a new\nequation and the study on this is going on in full swing. Many analytical approaches\nhave been employed to derive soliton solutions of FLE (Biswas et al, 2018; Hosseini\net al, 2020; Triki and Wazwaz, 2017; Baronio et al, 2015; Chen et al, 2018; Cinar\net al, 2022; Onder et al, 2022; Ullah et al, 2023; Gomez S et al, 2022; El-Shiekh and\n2Hamdy, 2023; Gaballah et al, 2023; Krishnan et al, 2019; Matsuno, 2012b,a; Talukdar\net al, 2023). In general the works on FLE is done for constant dispersion and nonlinear\ncoefficients. Only a few papers have been published with variable coefficients in FLE\n(Kundu, 2010; L¨ u and Peng, 2013; Wang et al, 2017). In this manuscript, however, our\npurpose is to study a system of an ultrashort pulse having time-varying dispersion,\nnonlinear effects and under a damping like perturbation. Under the SM scheme, Eqn.\n(1) modified accordingly and rewritten as\niUt+a1D(t)Uxx−a2Uxt+bR(t)|U|2(U+i a2Ux) = Γ( t) (Ux−i n U ) (2)\nhere the time-variable coefficient parameters D(t) and R(t) represents variation in\ndispersion and nonlinear effects respectively and Γ( t) is the gain parameter. The form\nof Γ( t) in this manuscript is considered as Γ( t) =R(t)D′(t)−R′(t)D(t)\n2D(t)R(t)and the symbol\nprime (′) represents the derivative with respect to t. The first and second terms on\nthe right hand side of Eqn. (2) are for damping and gain (or loss) of the medium.\nFrom hereon throughout the manuscript wherever we mention FLE, we are referring\nto Eqn. (2). The two main objectives of our manuscript are 1.) at first, we introduce\na suitable bilinear scheme to obtain dark and anti-dark soliton solutions of Eqn. (2).\nThereafter, explore the soliton solution under different SM schemes and provide a\nsystematic procedure to evaluate higher order soliton solutions. 2.) later, we are going\nto transform an already gauge transformed form of Eqn. (2) which we call as DLFLE\n(Eqn. (5)), into an equation of equivalent spin system called LLE. To the best of our\nknowledge this work had not been published before for such a system.\nTo address the first objective, we at first, through a gauge transformation followed\nby change in variables, convert Eqn. (2) into a simplified form (DLFLE), then we\npropose a bilinear scheme and introduce an auxiliary function to make the bilinear\nprocess more convenient. Our proposed scheme transforms the higher order nonlinear\nequation into three bilinear forms. We obtain dark and anti-dark 1SS of DLFLE.\nNow, by reverse transforming the obtained 1SS, we get the 1SS of FLE. Then we\nmention the criteria upon which the nature of soliton becomes dark or anti-dark and\nalso provide the graphical representation under different SM schemes. We also obtain\n2SS of DLFLE and determine the shift in phase of the two individual solitons upon\ninteraction. Eventually, we obtain the 2SS of FLE. In our previous paper (Dutta\net al, 2023), we had implemented an analogous scheme to realize soliton solution of\nFLE (DLFLE form to be precise), but for constant coefficients ( D(t) =R(t) = 1) and\nnow we extend the scheme to realize the soliton of FLE under SM (Eqn. (2)).\nAgain, it is observed that various nonlinear equations shows gauge equivalence\nwith spin system equations (referred as LLE) (Takhtajan and Zakharov, 1979;\nKundu, 1984; Ghosh et al, 1999; Ghosh and Nandy, 1999), even DLFLE having\nD(t) =R(t) = 1 shows gauge equivalence LLE (Dutta et al, 2023). LLE can provide\ninteresting features about a system which are important from physical and mathe-\nmatical point of view (Lakshmanan, 2011; Guo and Ding, 2008). We now introduce\na Lax pair for DLFLE and from that to accomplish our second objective, we are\n3going to search for a gauge transformation which transforms the spectral problem of\nDLFLE into a spectral problem of LLE.\nThis manuscript is arranged as: in the following section we consider a gauge trans-\nformation followed by change in variables that will convert Eqn. (2) into DLFLE\n(Eqn. (5)). Then using Hirota bilinearization, we derive dark and anti-dark 1SS of\nDLFLE and eventually 1SS of FLE and present the graphs under various SM scheme.\nThen, derive the multi-soliton solutions and perform asymptotic analysis. In the third\nsection we propose a Lax pair for DLFLE and through a gauge transformation we\ntransform the system into a gauge equivalent LLE. The fourth section will conclude\nthe manuscript.\n2 Bilinearization of FLE with non-vanishing\nbackground\nThe transformation of Eqn. (2) with respect to the variables ( xandt) gives us the\nliberty to choose the constant coefficients ( a1,a2,bandn) as per our convenience since\nthe coefficients can be absorbed in the process of variable transformation. Assuming\nn=1\na2,m=a1\na2>0 and positive b, consider the gauge transformation\nU=rm\nbnei(nx+2mnt )u (3)\nfollowed by the transformation of variables\nξ= 2(x+mt), τ =−mn2\n2t (4)\nand for the time-varying parameters\nD(t)→D(τ), R (t)→R(τ),Γ(t)→Γ(τ)\nwe get the following equation\nuξτ−D(τ)u+ 2i R(τ)|u|2uξ= Γ(τ)ux (5)\nhereuis the field function corresponding to the new transformed system. For constant\ndispersion and nonlinear effects, i.e. constant coefficients ( D(t) =R(t) = 1), Eqn. (5)\nis the first negative hierarchy of DNLSE and is similar to an equation governing the\ndynamics of short-wavelength ion-cyclotron waves in plasma (Davydova and Lashkin,\n1991; Lashkin, 2021). Davydova and Lashkin first realized this and hence in case of\nconstant coefficients, Eqn. (5) can be termed as Davydova Lashkin FLE (DLFLE).\nIn this manuscript, we refer Eqn. (5) as DLFLE, even though D(t) and R(t) are not\nconstants. For a constant Γ( τ) = Γ, the right hand side of Eqn. (5) takes the form of\na perturbation representing linear damping (Berger and Perkins, 1976).\n4For a soliton solution of DLFLE, we assume a non-vanishing background condition\nu→ρ ei(κξ+ω(τ))asξ→ ±∞ . Under this situation, we expect dark and anti-dark\nsoliton solutions of Eqn. (5) through bilinearization.\nTo write Eqn. (5) in the bilinear form let us assume\nu=s\nD(τ)\nR(τ)g\nf(6)\nwhere gandfare two complex functions of ( ξ, τ). Consequently, Eqn. (5) becomes\n1\nf2(DξDτ−D(τ))g.f−g\nf3DξDτ(f.f) +2i|g|2\nf3f∗D(τ)Dξ(g.f) +gλf.f\nf3−λg.f\nf2+\ns|g|2\nf3−s|g|2f∗\nf3f∗= 0 (7)\nwhere Dξ,Dτare Hirota derivatives (Hirota and Satsuma, 1976) and are defined as\nDm\nξDn\nτg(ξ, τ).f(ξ, τ) = (∂\n∂ξ−∂\n∂ξ′)m(∂\n∂τ−∂\n∂τ′)ng(ξ, τ).f(ξ′, τ′)\f\f\f\f\f\n(ξ=ξ′)(τ=τ′)(8)\nNotice that the last two terms in Eqn. (7) contain an auxiliary function swhich is\nintroduced so that the multilinear Eqn. (7) can be cast into three bilinear equations\ngiven below (Eqns. (9) - (11)). Here, the constant λwill also be determined while\nsolving the three equations. The bilinear equations in terms of g,fandsare\n(DξDτ−D(τ)−λ)g.f= 0 (9)\n(DξDτ−λ)f.f=sg∗(10)\n2i D(τ)Dξ(g.f) =sf∗(11)\nNow to obtain soliton solution, the expansions of g,fandswith respect to an\narbitrary parameter ϵare as follows\ng=g0(1 +ϵ2g2+ϵ4g4+...), f = 1 + ϵ2f2+ϵ4f4+... (12)\ns=s0(1 +ϵ2s2+ϵ4s4+...) (13)\n2.1 Dark and Anti-Dark 1SS\nTo obtain the 1SS of DLFLE, we drop the terms of order greater than or equal to ϵ3\ning,fands. Thus, Eqn. (6) can be represented as\nu=s\nD(τ)\nR(τ)g0(1 +ϵ2g(1)\n2)\n1 +ϵ2f(1)\n2\f\f\f\nϵ=1(14)\n5Let us consider the expressions for g0,s0,g(1)\n2,s(1)\n2andf(1)\n2are as follows\ng0=ρ ei(κξ+ω(τ))(15)\ns0=ρsei(κξ+ω(τ))(16)\ng(1)\n2=K1eθ1+θ∗\n1 (17)\ns(1)\n2=M1eθ1+θ∗\n1 (18)\nf(1)\n2=T1eθ1+θ∗\n1 (19)\nwhere θ1=p1ξ+ Ω 1R\nD(τ)dτ.p1, Ω1,K1,M1,T1are complex parameters. Let us\nconsider, p1=p1r+i p1iandT1=T1r+i T1iwhere p1r,p1i,T1randT1iare real.\nOn substituting the above expressions into Eqns. (9) - (11) yield the following\nρs=−2κρ (20)\nλ= 2κρ2(21)\nω(τ) = (−1\nκ−2ρ2)Z\nD(τ)dτ (22)\nΩ1=h1\np1(23)\nM1=T2\n1\nK∗\n1(24)\nK1=γ1T∗\n1 (25)\nwhere h1is real and γ1is a complex of absolute value 1, and these two are represented\nas\nγ1=(p1+p∗\n1)T1−i(T1−T∗\n1)κ\n(p1+p∗\n1)T∗\n1−i(T1−T∗\n1)κ(26)\nh1=−|p1|2(−1 +γ1)2κρ2\n(p1+p∗\n1)2γ1(27)\nand the system obeys the constraint\np2\n1r|T1|2+ 2p1rκ(1 +κρ2)T1rT1i+κ2(1 +κρ2)T2\n1i= 0 (28)\nnow keeping either T1rorT1ifixed, we can calculate the other from Eqn. (28). And\nthis will also lead to a restriction |p1r| ≤p\nκ3ρ2(1 +κρ2). It is interesting to point\nout that in case of D(t) =R(t) = 1, the obtained expressions for g,falign with the\nresults already deduced for DLFLE having constant coefficients (Dutta et al, 2023),\nwhich is expected.\nAs already mentioned, the soliton obtained will always have a background of ρ,\nand the nature of soliton being dark (dip in the background) or anti-dark (bulge on\n6the background) depends upon the parameters p1r,κandT1i. For positive p1r, when\nκandT1ihave different signs we get a dark soliton and for the same sign of κandT1i\nthe soliton becomes anti-dark. For the case of negative p1rthe criteria become vice\nversa. This statement is supported by the expression of the amplitude ( A1) given as\nA1=ρ\f\f\fs\nD(τ)\nR(τ)T1+γ1|T1|\nT1+|T1|\f\f\f (29)\nFor the mentioned criteria of anti-dark soliton in the above paragraph, A1is\ngreater than ρand for the dark soliton criteria, A1is less than ρ. Now, the obtained\n1SS “ u” (Eqn. (14)) is a solution of DLFLE). By replacing ξandτin terms of xandt\nfrom Eqn. (4) and putting the expression of “ u” inside of “ U” in Eqn. (3), we get the\n1SS of FLE. One important and obvious detail we like to mention at this point that\nall the anti-dark 1SS of FLE presented in Figs. (1) - (6) have parameters p1r=−10,\nκ= 5,T1r=−6 and for all the dark 1SS graphs, we have p1r= 10, κ=−5,T1r= 6.\nT1ris calculated using Eqn. (28). And for all the graphs, we fix ρ= 2, m= 0.2,\nn= 4,b= 1,a2=1\nn,a1=m a 2.\nAgain the role of D(τ),R(τ) and Γ( τ) very much impact the structure and propa-\ngation of the solitons. The existence of Γ( τ) also affects the background of the system.\nDepending upon the relationship between D(τ) and R(τ), Γ(τ) either vanishes or\nexists. In the following subsections (2.1.1) - (2.1.5) we are going to drive into some of\nthe different SM scheme.\n2.1.1 D(t) =R(t) = 1 (or some other constants) and Γ( t) vanishes\nD(t) and R(t) are linearly dependent functions, hence Γ( t) = 0. The background in\nthis case remains unaffected since the gain parameter is zero. Fig. (1a) demonstrates\nthis condition for anti-dark 1SS in a 3D plot. Fig. (1b) demonstrates the same but for\ndark 1SS.\n2.1.2 D(t) = 1, R(t) = 1 + σcos(kt) and Γ( t) is non-vanishing\nD(t) and R(t) are linearly independent functions, hence Γ( t)̸= 0. The background in\nthis case becomes sinusoidal for the contribution of “cosine” function in R(t) and the\ngain parameter being non-zero. Figs. (2a) - (2b) demonstrate this condition. We keep\nthe value of σbelow 1 to avoid the scenario R(t)→0 and eventually u, U→ ∞ , which\ncan be problematic in graphical presentation. If D(t) and R(t) interchange their values\nthan the sinusoidal nature of the graph reserves (ups and downs get interchanged)\nas demonstrated in Fig. (2c). And for the same expressions of D(t) and R(t) (=\n“1 + σcos(kt)”), implies Γ( t) = 0, then for a small value of σ, the graph will become\nsimilar to that of Figs. (1a) and (1b).\n7(a) Anti-Dark 1SS\n (b) Dark 1SS\nFig. 1 : 3D plot representation of 1SS under SM scheme: D(t) =R(t) = 1. (a) repre-\nsents anti-dark soliton and (b) represents dark soliton.\n(a) Anti-Dark 1SS\n (b) Dark 1SS\n (c) Anti-Dark 1SS\nFig. 2 : 3D plot representation of 1SS under SM scheme: D(t) = 1, R(t) = 1+ σcos(kt)\nwith σ= 0.1 and k= 10π. (a) represents anti-dark soliton and (b) represents dark\nsoliton. (c) represents anti-dark soliton but under SM scheme D(t) = 1 + σcos(kt),\nR(t) = 1.\n2.1.3 D(t) = 1, R(t) = 1 + σe(−kt2)and Γ( t) is non-vanishing\nD(t) and R(t) are linearly independent functions, hence Γ( t)̸= 0. In this case the\nbackground has a depth for the Gaussian term “ σe(−kt2)” inR(t) and the gain param-\neter being non-zero. Figs. (3a) - (3b) demonstrate this condition. If D(t) and R(t)\ninterchange their values than the background will have a bulge instead of a depth\nas demonstrated in Fig. (3c). Again for the same expressions of D(t) and R(t) (=\n“1 + σe(−kt2)”), Γ( t) = 0, then for a small value of σ, the graph will become similar\nto that of Figs. (1a) and (1b).\n8(a) Anti-Dark 1SS\n (b) Dark 1SS\n (c) Anti-Dark 1SS\nFig. 3 : 3D plot representation of 1SS under SM scheme: D(t) = 1, R(t) = 1+ σe(−kt2)\nwith σ= 0.25 and k= 100. (a) represents anti-dark soliton and (b) represents dark\nsoliton. (c) represents anti-dark soliton but under SM scheme: D(t) = 1 + σe(−kt2),\nR(t) = 1.\n(a) Anti-Dark 1SS\n (b) Dark 1SS\nFig. 4 : 3D plot representation of 1SS under SM scheme: D(t) =R(t) =e(σt)cos(kt)\nwith σ= 0.25 and k=π. (a) represents anti-dark soliton and (b) represents dark\nsoliton.\n2.1.4 D(t) =R(t) =e(σt)cos(kt) and Γ( τ) vanishes\nD(τ) and R(τ) are linearly dependent functions, hence Γ( τ) = 0. In this case the\nbackground is unaffected. D(τ) and R(τ) are periodic functions and this makes the\npropagation of the soliton harmonic from mean position ( x= 0) along the time ( t)\naxis. Figs (4a) and (4b) demonstrate this case. The amplitude remains constant along\nthe propagation of the soliton.\n9(a) Anti-Dark 1SS\n (b) Dark 1SS\nFig. 5 : 3D plot representation of 1SS under SM scheme: D(t) =e(σt)cos(kt) and\nR(t) =cos(kt) with σ= 0.25 and k=π. (a) represents anti-dark soliton and (b)\nrepresents dark soliton.\n2.1.5 D(t) =e(σt)cos(kt) and R(t) =cos(kt) and Γ( t) is\nnon-vanishing\nD(τ) and R(τ) are linearly independent functions and the gain parameter is a non-\nzero constant, Γ( τ) =σ\n2. In this case the background has a constant gain (or lose) of\n“σ\n2” factor. Just like the previous case (subsection (2.1.4)), here also D(τ) and R(τ)\nare periodic functions and thus the propagation of soliton is harmonic from mean\nposition along the time axis.Figs (5a) and (5b) demonstrate this case. Amplitude\namplification is clearly visible from the graphs unlike the soliton in Figs. (4) where\nthe amplitude remains constant.\nNow, exploring the last two cases, i.e. subsections (2.1.4) and (2.1.5), we find that\nσcontrols the deviation of the soliton from the mean position and kcan control\nthe frequency of the soliton and can also keep in check the deviation. In the last\nsubsection (2.1.5) which is of a non-vanishing constant gain (Γ( t) =σ\n2),σcan increase\n(or decrease) the amplitude of the soliton and also affect the background of the system.\nComparing Figs. (6a) and (6b), we can see increasing σfrom 0 .1 to 0 .3, increases the\ndeviation of soliton from mean position. Again, increasing kfrom πto 2π, increases\nthe frequency of deviating from mean along taxis and also reduces the magnitude of\ndeviation for same time propagation as we can see comparing Figs. (6c) with (6b).\nThe same are represented as density plots in Figs. (6d) - (6f) for a more clear view.\nKeeping in mind the distance required for the soliton to travel, one can fix the gain\nparameter in such a way that the background change is bearable and the soliton\nalso gain some desirable amplification. With this knowledge while constructing such\na SM FLE system, the soliton structure and the propagation behavior can be tuned\n10(a) Anti-Dark 1SS\n (b) Dark 1SS\n (c) Anti-Dark 1SS\n(d) Anti-Dark 1SS\n (e) Dark 1SS\n (f) Anti-Dark 1SS\nFig. 6 : Graphical representation of 1SS under SM scheme: D(t) =e(σt)cos(kt) and\nR(t) =cos(kt). (a), (b) and (c) are 3D plot representation and (d), (e) and (f) are\ndensity plot representation. (a) and (d): anti-dark soliton with σ= 0.05 and k= 10π,\n(b) and (e): dark soliton with σ= 0.5 and k= 10πand (c) and (f): anti-dark soliton\nwith σ= 0.5 and k= 40π.\naccording to one’s requirements. This is a big advantage that the SM scheme provides\nto a FLE system.\n2.2 Dark and Anti-Dark 2SS\nIn this subsection, we attempt to obtain the 2SS of DLFLE and eventually 2SS of\nFLE. For this we drop the terms of order greater than or equal to ϵ5ing,fands.\nHence, Eqn. (6) for 2SS can be rewritten as\nu=s\nD(τ)\nR(τ)g0(1 +ϵ2g(2)\n2+ϵ4g(2)\n4)\n1 +ϵ2f(2)\n2+ϵ4f(2)\n4\f\f\f\nϵ=1(30)\nLet us consider the expressions for g(2)\n2,g(2)\n4,s(2)\n2,s(2)\n4,f(2)\n2andf(2)\n4in 2SS are as follows\ng(2)\n2=K1eθ1+θ∗\n1+K2eθ2+θ∗\n2 (31)\n11g(2)\n4=K12eθ1+θ∗\n1+θ2+θ∗\n2 (32)\ns(2)\n2=M1eθ1+θ∗\n1+M2eθ2+θ∗\n2 (33)\ns(2)\n4=M12eθ1+θ∗\n1+θ2+θ∗\n2 (34)\nf(2)\n2=T1eθ1+θ∗\n1+T2eθ2+θ∗\n2 (35)\nf(2)\n4=T12eθ1+θ∗\n1+θ2+θ∗\n2 (36)\nwhere θj=pjξ+ Ωjτ.pj, Ωj,Kj,Mj,Tj,K12,M12,T12are complex parameters\n(j= 1,2). We assume pj=pjr+i pjiandTj=Tjr+i Tjiwhere pjr,pji,Tjrand\nTjiare real. Now, substituting the above equations into Eqns. (9) - (11), yields the\nfollowing expressions\nΩj=hj\npj(37)\nMj=T2\nj\nK∗\nj(38)\nM12=T2\n12\nK∗\n12(39)\nKj=γjT∗\nj (40)\nK12=γ1γ2T∗\n12 (41)\nT12=c12T1T2 (42)\nthe expressions for γj’s and hj’s are similar to Eqns. (26) and (27) respectively as\nγj=(pj+p∗\nj)Tj−i(Tj−T∗\nj)κ\n(pj+p∗\nj)T∗\nj−i(Tj−T∗\nj)κ(43)\nhj=−|pj|2(−1 +γj)2κρ2\n(pj+p∗\nj)2γj(44)\nandc12is real, expressed as\nc12=(p1rT1rT2i−p2rT1iT2r)2+ (p1r−p2r)2T2\n1iT2\n2i\n(p1rT1rT2i−p2rT1iT2r)2+ (p1r+p2r)2T2\n1iT2\n2i(45)\nthe 2SS system is under two constraints (for j= 1,2)\np2\njr|Tj|2+ 2pjrκ(1 +κρ2)TjrTji+κ2(1 +κρ2)T2\nji= 0 (46)\n12forTjr’s and Tji’s to be real, there must be restriction of |pjr| ≤p\nκ3ρ2(1 +κρ2).\nThe amplitude ( Aj) of the jth soliton is given as\nAj=ρs\nD(τ)\nR(τ)\f\f\fTj+γj|Tj|\nTj+|Tj|\f\f\f (47)\nthis is similar to Eqn. (29), and thus we infer that the nature of the jth soliton to be\ndark or anti-dark is dependent upon the same criteria of κ,pjr,Tjias that in the 1SS\ncase.\n2.2.1 Asymptotic analysis of 2SS\nSoliton has a unique characteristics which is that soliton upon interaction with other\nsoliton, retains all of its characteristics like shape, size, etc. with a shift in its phase. To\nstudy the amplitudes and the phase shift of the individual solitons of 2SS, we are going\nto perform asymptotic analysis. The individual solitons of 2SS when asymptotically\napart, are essentially two separated 1SS. In the limit of before interaction τ→ −∞ ,\nfixing θ1to a finite value implies |eθ1|is finite and |eθ2| → ∞ . In this case, Eqn. (30)\ntakes the form\nu1−=ρ ei(κξ+ω(τ))s\nD(τ)\nR(τ)K2(1 +c12K1eθ1+θ∗\n1)\nT2(1 +c12T1eθ1+θ∗\n1)(48)\n⇒u1−=ρ ei(κξ+ω(τ))s\nD(τ)\nR(τ)eiϕ1(1 +K′\n1eθ1+θ∗\n1)\n(1 +T′\n1eθ1+θ∗\n1)(49)\nhereK′\n1=c12K1andT′\n1=c12T1. Eqn. (49) is simply an expression for 1SS with an\nadditional phase ϕ1. The fractional term “K2\nT2” in Eqn. (48) has a magnitude of 1 but\na phase difference of ϕ1which results the phase term “ eiϕ1” in Eqn. (49).\nIn the limit of after interaction τ→ ∞ and for a fixed finite value of θ1, we have a\nfinite|eθ1|and|eθ2| →0. This time, the corresponding asymptotic form of Eqn. (30)\nbecomes\nu1+=ρ ei(κξ+ω(τ))s\nD(τ)\nR(τ)(1 +K1eθ1+θ∗\n1)\n(1 +T1eθ1+θ∗\n1)(50)\nwe see Eqn. (50) is also in the form of 1SS just like Eqn. (49) with a difference in\nphase. This phase shift ϕ1is given as\nϕ1=tan−1h\n−2κ T2\ni2(κ Tr2Ti2+pr2|T2|2)\n(κ(Tr2−Ti2)Ti2+pr2|T2|2) (κ(Tr2+Ti2)Ti2+pr2|T2|2)i\n(51)\nThe amplitudes of both the Eqns. (49) and (50) are same and can be expressed\nthrough Eqn. (29). This case represents a purely elastic collision between two solitons\n13since each of them retains all their characteristics with a slight change in phase. We\ncan perform the same analysis by keeping θ2finite. This time also, the amplitude\nbefore and after interaction remains the same and the phase shift can be represented\nin a similar way as in Eqn. (51) but by replacing the components of p2andT2with the\nrespective components of p1andT1. With this, we can infer that our system DLFLE\nwith a non-vanishing background condition supports elastic soliton interaction.\nFinally, just like in 1SS case, here also we get the 2SS of FLE by putting Eqn. (30)\nin Eqn. (3). Figs. (7a) - (7d) demonstrate 2SS under SM scheme: D(t) =R(t) = 1\nand hence Γ( t) = 0. This implies that the background is unaffected. We can observe\nfrom the two density plots (Figs. (7c) - (7d)) that the individual solitons even upon\ninteraction with each other, propagates with the same characteristics as was before\ninteraction, but only a shift occurs in their respective phases.\nNow, for the SM scheme as demonstrated in subsection (2.1.5): D(t) =\ne(σt)cos(kt),R(t) =cos(kt) and hence Γ( t) =σ\n2, we compare two 1SS represented in\nFigs. (8a) and (8b) with 2SS represented in Fig. (8c). We notice that upon interac-\ntions, the individual solitons of 2SS in Fig. (8c) repeatedly alternating their phases,\nthereby maintaining their original trajectories as they would have without interactions\nfor 1SS case, as seen in Figs. (8a) and (8b). At a first glance of Fig. (8c), it might\nappear that the two individual solitons are not interacting with each other but in real-\nity they are interacting with repeated phase shifts that their trajectories appear to be\nof non interaction . Figs. (8d) - (8f) represent the same in density plot.\n2.3 Scheme for obtaining NSS\nIn this subsection, we represent the scheme to obtain NSS of DLFLE and FLE. For\nthis, we have to drop the terms greater than or equal to ϵ2N+1ing,fands. Eqn. (6)\nbecomes\nu=s\nD(τ)\nR(τ)g(N)\nf(N)(52)\nthe expansions of g(N)andf(N)are as follow\ng(N)=g0\u0010\n1 +NX\nj=1Kjeθj+θ∗\nj+1\n2NX\nj,l=1;j̸=lKjleθj+θ∗\nj+θl+θ∗\nl+...+K12...NeNP\nj=1θj+θ∗\nj\u0011\n(53)\nf(N)= 1 +NX\nj=1Tjeθj+θ∗\nj+1\n2NX\nj,l=1;j̸=lTjleθj+θ∗\nj+θl+θ∗\nl+...+T12...NeNP\nj=1θj+θ∗\nj\n(54)\n14(a) Anti-dark-Anti-dark 2SS\n (b) Dark-Dark 2SS\n(c) Anti-dark-Anti-dark 2SS\n (d) Dark-Dark 2SS\nFig. 7 : Graphical representation of 2SS under SM scheme: D(t) =R(t) = 1. (a) and\n(b) are 3D plot representation and (c) and (d) are density plot representation. (a) and\n(c) represent anti-dark-anti-dark soliton with p1r=−10,p2r=−20,κ= 5,T1r=−6,\nT2r=−9 and (b) and (d) represent dark-dark soliton with p1r= 10, p2r= 20, κ=−5,\nT1r= 6, T2r= 9. T1iandT2iare calculated using Eqn. (46). In all the four graphs,\nwe fix ρ= 2,m= 0.2,n= 4,b= 1,a2=1\nn,a1=m a 2.\nwhere θj=pjξ+ Ωjτ.pj, Ωj, parameters within g(N)(i.e.K′s) and within f(N)\n(i.e.T′s) are complex parameters (subscript j= 1, ... , N). These parameters can be\ncalculated using the bilinear Eqns. (9) - (11). Just like 2SS has two constraints (Eqn.\n(46)), the NSS consists of N number of constraints similar to that expressed in Eqn.\n(46) but for j= 1, ... , N. The individual jth soliton of NSS has the same amplitude\nexpression Ajfrom Eqn. (47) and obey the same criteria for dark and anti-dark soliton\n15(a) Anti-Dark 1SS\n (b) Anti-Dark 1SS\n (c) Anti-Dark 2SS\n(d) Anti-Dark 1SS\n (e) Anti-Dark 1SS\n (f) Anti-Dark 2SS\nFig. 8 : Graphical representation of 2SS under SM scheme: D(t) =e(σt)cos(kt) and\nR(t) =cos(kt) and Γ( t) =σ\n2with σ= 0.25 and k=π. (a), (b) and (c) are 3D\nplot representation and (d), (e) and (f) are density plot representation. (a) and (d)\nrepresent anti-dark 1SS with p1r=−10,T1r=−6,T1ifrom Eqn. (28), (b) and (e)\nrepresent anti-dark 1SS with p1r=−20,T1r=−9,T1ifrom Eqn. (28) and (c) and (f)\nrepresent anti-dark-anti-dark 2SS with p1r=−10,p2r=−20,T1r=−6,T2r=−9,\nT1iandT2ifrom Eqn. (46). In all the six graphs, we fix ρ= 2,κ= 5,m= 0.2,n= 4,\nb= 1,a2=1\nn,a1=m a 2.\nnature with respect to the parameters κ,pjr,Tjias already discussed for 1SS and\n2SS. And eventually to obtain NSS of FLE, we have to put Eqn. (52) into Eqn. (3).\n3 Gauge Connected Landau-Lifshitz equation\nIn this section we are going to establish the gauge connection between Eqn. (5) with\nits counterpart LLE. To approach this, we exploit the equivalence of the Lax pairs of\nthe two systems (DLFLE and LLE). Let us consider the Lax pair ( L,M) for Eqn. (5)\nΨξ=LΨ Ψ τ=MΨ (55)\n16here Ψ is the Jost function corresponding to the field function uin which ( L,M) are\ngiven by\nL=−iζ2\n2Σ +s\nR(τ)\nD(τ)ζqξ (56)\nM=i\n2ζ2D(τ)Σ−i\nζp\nD(τ)R(τ) Σq+iR(τ) Σq2(57)\nζis the spectral parameter, Σ and qare 2×2 matrices given as follow\nΣ =\u0012\n1 0\n0 -1\u0013\n, q =\u0012\n0 u\n−u∗0\u0013\nfrom Eqn. (55) follows the compatibility condition which is\nLτ−Mξ+ [L, M ] = 0 (58)\nEqn. (58) is also called zero curvature equation. Under a local gauge transformation\nwe write a matrix gsuch as g(ξ, τ, ζ 0) = Ψ( ξ, τ, ζ )|ζ=ζ0and the Jost function Ψ\ntransforms to Φ as Ψ →Φ(ξ, τ, ζ, ζ 0) =g(ξ, τ, ζ 0)−1Ψ(ξ, τ, ζ ). Φ is the Jost function\ncorresponding to the spin field function Sof the LL system. The new Lax pair ( L′,\nM′) associated with Φ written as\nΦξ=L′Φ Φ τ=M′Φ (59)\nand the relation between ( L′,M′) and ( L,M) expressed as\nL′=g−1Lg−g−1gξ (60)\nM′=g−1Mg−g−1gτ (61)\nwhere\ngξg−1=L|ζ=ζ0, g τg−1=M|ζ=ζ0 (62)\nthe compatibility condition Eqn. (58) for the new system rewritten as\nL′\nτ−M′\nξ+ [L′, M′] = 0 (63)\nEqn. (63) is corresponding to the spin field function ( S).Sis expressed as\nS=g−1Σg (64)\n17andS2=I. This leads to the expressions for the derivatives of Snamely Sξ,Sτand\nSτξas\nSξ=g−1[Σ, L0]g⇒Sξ= 2ζ0s\nR(τ)\nD(τ)g−1Σqξg (65)\nSτ=g−1[Σ, M 0]g⇒Sτ=−2i\nζ0p\nD(τ)R(τ)g−1qg (66)\nSτξ= 2iR(τ)(g−1qξqg−g−1q qξg)−2i\nζ0p\nD(τ)R(τ)g−1qξg (67)\nusing the above equations, the compatibility condition Eqn. (63) yields\nD(τ)Sξ+ζ2\n0\n2i[S, Sτξ] = 0 (68)\nThis is the gauge connected LLE for the Eqn. (5). One important thing to note is\nthat Eqn. (68) is consistent for the system of Eqns. (59), (60), (61) in any matrix S\nof arbitrary dimension as long as Ssatisfies S2=I.\n4 Conclusion\nWe have transformed FLE under SM into DLFLE and using Hirota bilinearization\nwe obtained 1SS and 2SS of DLFLE and under reverse transformation we obtained\n1SS and 2SS of FLE. Under different SM scheme, we have explored the soliton using\ngraphical representation. We have also mentioned that under certain SM scheme the\nsoliton can be amplified accompanied by a certain amount of affecting the background.\nThis information can be utilized to construct a desirable FLE system for ultrashort\npulse with customized properties. Thereafter, we have determined the shift in phase of\nthe individual solitons suffered upon interaction and also showed that the interaction\nwas elastic. We believe that the derived soliton solutions will be a useful information\ncarrier in digital and quantum communication where higher order effects like STD and\nND are taken into account along with GVD, KLNE. The proposed method may be\nextended to other nonlinear models such as coupled FLE under SM to obtain soliton\nsolutions in multi-mode waveguide. This will be one of our future objectives. 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Optical and Quantum\nElectronics 55(6):495\n21Wang ZQ, Wang X, Wang L, et al (2017) Higher-order peregrine combs and pere-\ngrine walls for the variable-coefficient lenells-fokas equation. Superlattices and\nMicrostructures 102:189–201\nYildirim Y (2019) Optical solitons to sasa–satsuma model with trial equation\napproach. Optik 184:70–74\nYıldırım Y, Biswas A, Asma M, et al (2020) Optical soliton perturbation with chen–\nlee–liu equation. Optik 220:165177\nStatements & Declarations\nFunding. R. Dutta and S. Talukdar receive fellowship from Department of Science\n& Technology, Govt. of India under INSPIRE programme. Corresponding fellowship\naward numbers DST/INSPIRE Fellowship/2020/IF200303 and DST/INSPIRE Fel-\nlowship/2020/IF200278. Other than this, the authors did not receive any fund support\nfrom any organization for the submitted work.\nCompeting Interests. The authors have no relevant financial or non-financial\ninterests to disclose.\nAuthors Contribution. Conceptualization: Riki Dutta, Sudipta Nandy; For-\nmal Analysis: Riki Dutta; Supervision: Sudipta Nandy; Validation: Sudipta Nandy,\nSagardeep Talukdar, Gautam K. Saharia; Writing - original draft: Riki Dutta; Writing\n- review & editing: All the authors.\n22"    },    {        "title": "2402.03839v1.Random_features_models__a_way_to_study_the_success_of_naive_imputation.pdf",        "content": "Random features models: a way to study the success\nof naive imputation\nAlexis Ayme1, Claire Boyer1,2, Aymeric Dieuleveut3, and Erwan Scornet1\n1Sorbonne Universit´ e and Universit´ e Paris Cit´ e, CNRS, Laboratoire de\nProbabilit´ es, Statistique et Mod´ elisation, F-75005 Paris, France\n2Institut Universitaire de France (IUF)\n3CMAP, Ecole Polytechnique\nAbstract\nConstant (naive) imputation is still widely used in practice as this is a first\neasy-to-use technique to deal with missing data. Yet, this simple method could be\nexpected to induce a large bias for prediction purposes, as the imputed input may\nstrongly differ from the true underlying data. However, recent works suggest that\nthis bias is low in the context of high-dimensional linear predictors when data is\nsupposed to be missing completely at random (MCAR). This paper completes the\npicture for linear predictors by confirming the intuition that the bias is negligible\nand that surprisingly naive imputation also remains relevant in very low dimension.\nTo this aim, we consider a unique underlying random features model, which offers\na rigorous framework for studying predictive performances, whilst the dimension\nof the observed features varies. Building on these theoretical results, we establish\nfinite-sample bounds on stochastic gradient (SGD) predictors applied to zero-\nimputed data, a strategy particularly well suited for large-scale learning. If the\nMCAR assumption appears to be strong, we show that similar favorable behaviors\noccur for more complex missing data scenarios.\n1 Introduction\nMissing data appear in most real-world datasets as they arise from merging different data\nsources, data collecting issues, self-censorship in surveys, just to name a few. Specific\nhandling techniques are required, as most machine learning algorithms do not natively\nhandle missing values. A common practice consists in imputing missing entries. The\nresulting complete dataset can then be analyzed using any machine learning algorithm.\nWhile there exists a variety of imputation strategies (single, multiple, conditional,\nmarginal imputation ; see, e.g., Bertsimas et al., 2018, for an overview), mean imputation\n1arXiv:2402.03839v1  [math.ST]  6 Feb 2024is definitely one of the most common practices. Such a procedure has been largely\ncriticized in the past as (single) mean imputation distorts data distributions by lowering\nvariances, which can lead to inconsistent parameter estimation. Indeed, a large part\nof the literature on missing values focuses on inference in parametric models, such as\nlinear (Little, 1992; Jones, 1996) or logistic models (Consentino and Claeskens, 2011;\nJiang et al., 2020). From an empirical perspective, benchmarks of imputation techniques\n(Wo´ znica and Biecek, 2020) indicate that simple imputation, such as the mean, induces\nreasonable predictive performances, compared to more complex imputation techniques\nsuch as MICE (Perez-Lebel et al., 2022).\nOn the contrary, a recent line of work (Josse et al., 2019) aims at studying the\npredictive performances of impute-then-regress strategies that work by first imputing\ndata (possibly with a very simple procedure) and then fitting a learning algorithm on the\nimputed dataset. Whereas mean imputation leads to inconsistent parameter estimation ,\nJosse et al. (2019) and Bertsimas et al. (2021) show that impute-then-regress procedures\nhappen to be consistent if the learning algorithm is universally consistent. Le Morvan\net al. (2021) generalize the consistency results of mean-imputation by Josse et al. (2019);\nBertsimas et al. (2021) and prove that for any universally consistent regression model,\nalmost all single imputation strategies can lead to consistent predictors. Therefore,\nthe impact of a specific imputation strategy has to be analyzed for specific regression\nmodels.\nWithout dispute, linear models are the most classic regression framework. However,\ntheir study becomes challenging in presence of missing values as they can require to\nbuild 2dnon-linear regression models (one for each missing data pattern), where dis\nthe number of input variables (Le Morvan et al., 2020; Ayme et al., 2022). In the\ncontext of linear models with missing inputs, Le Morvan et al. (2020) establish finite-\nsample generalization bounds for zero-imputation, showing that this strategy is generally\ninconsistent. However, assuming a low-rank structure on the input variables, Ayme et al.\n(2023) prove that zero-imputation prior to learning is consistent in a high-dimensional\nsetting. Note that the impact of zero-imputation with low-rank inputs has also been\nanalyzed by Agarwal et al. (2019) in the context of Principal Components Regression,\nwhere the same type of generalization bounds were established. In this paper, we want\nto go beyond the low-rank structure by considering a (possibly infinite) latent space,\nand using the random feature framework.\nRelated work - Random features First introduced by Rahimi and Recht (2007),\nrandom features are used in nonparametric regression to approximate, with a few features,\na kernel method where the final predictor belongs to an infinite-dimensional RKHS.\nRudi and Rosasco (2017); Carratino et al. (2018) obtain generalization upper bounds\nfor kernel regression learned with a small number of features, leading to computational\nefficiency. Random features are also used to describe a one-hidden-layer neural network\n(Bach, 2017).\n2Related work - high-dimensional linear models Linear models have been widely\nstudied in a fixed design, considering the input variables are fixed (see, e.g., Hastie et al.,\n2015, for an analysis in the high-dimensional case). Quite notably, few works analyze\n(high-dimensional) linear models in the random design setting, a necessary framework\nto assess the predictive performance of linear models on unseen data (Caponnetto and\nDe Vito, 2007; Hsu et al., 2012; Mourtada and Rosasco, 2022). These works mainly\nfocus on the statistical properties of the Empirical Risk Minimizer (ERM) with a\nridge regularization using uniform concentration bounds. On the other hand, Bach\nand Moulines (2013); Raskutti et al. (2014); Dieuleveut et al. (2017) directly control\nthe generalization error of predictors resulting of stochastic gradient strategies, while\nperforming a single pass on the dataset. The obtained bounds have therefore the\nadvantage of being dependent on the training algorithm involved.\nContributions In this paper, we analyze the impact of the classic imputation-by-zero\nprocedure on predictive performances, as a function of the input dimension. To this\naim, we consider a latent space from which an arbitrary number of input variables are\nbuilt. The output variable is assumed to result from a linear transformation of the\nlatent variable. Such a framework allows us to analyze how predictive performances vary\nwith the number of input variables, inside a common fixed model. Under this setting,\nwe assume that all entries of input variables are observed with probability ρ∈(0,1),\nwithin a MCAR scenario, and study the performance of a linear model trained on\nimputed-by-zero data.\n•We prove that when the input dimension dis negligible compared to that of the\nlatent space p, the Bayes risk of the zero-imputation strategy is negligible compared\nto that induced by missing data themselves. Therefore, naive imputation is on\npar with best strategies.\n•When d≫p, both above errors vanish, which highlights that neither the pres-\nence of missing data or the naive imputation procedure hinders the predictive\nperformances.\n•From a learning perspective, we use Stochastic Gradient Descent to learn param-\neters on zero-imputed data. We provide finite-sample generalization bounds in\ndifferent regimes, highlighting that the excess risk vanishes at 1 /√nfor very low\ndimensions ( d≪p) and high dimensions ( d >(1−ρ)√n/ρ).\n•Two different regimes arise from the finite dimension of the latent space. To move\nbeyond this disjunctive scenario, we consider a latent space of infinite dimension\nand analyze predictors built on dzero-imputed input variables. We prove that\nthe corresponding Bayes excess risk is controlled via the excess risk of a kernel\nridge procedure, with a penalization constant depending on ρandd. A finite-\nsample generalization bound on the SGD strategy applied on zero-imputed data\nis established and shows that zero-imputation is consistent in high-dimensional\nregimes ( d≫√n).\n3•The MCAR assumption considered throughout the paper, and often in the litera-\nture, can be attenuated at the cost of weaker theoretical results but which allows\nto show that naive imputation is relevant in high dimension even for non-trivial\nMissing Not At Random (MNAR) scenarios.\nNotations. Forn∈N, we denote [ n] ={1, . . . , n }. We use ≲to denote inequality\nup to a universal constant.\n2Imputation is adapted for very low and high-\ndimensional data\n2.1 Setting\nWe adopt the classical regression framework in which we want to predict the value of\nan output random variable Y∈Rgiven an input random variable X∈ X =Rdof\ndimension d. More precisely, our goal is to build a predictor ˆf:X →Rthat accurately\nestimates the regression function f⋆(also called Bayes predictor) defined as a minimizer\nof the quadratic risk\nR(f) :=E\u0002\n(Y−f(X))2\u0003\n, (1)\nover the class of measurable functions f:X →R. When fis linear, we simply denote\nR(θ) the risk of the linear function parameterized by θ, i.e., such that for all x∈Rd,\nf(x) =x⊤θ.\nRandom features Real datasets are often characterized by high correlations between\nvariables, or equivalently by a hidden low-rank structure (Johnstone, 2001; Udell and\nTownsend, 2019). The random feature framework (Rahimi and Recht, 2007) constitutes\na general and flexible approach for modeling such datasets. We, therefore, assume that\nthe inputs ( Xi)i∈[n], i.i.d. copies of X, actually result from a random feature (RF) model.\nFor pedagogical purposes, we start by restricting ourselves to finite-dimensional latent\nmodels.\nAssumption 1 (Gaussian random features) .The input variables (Xi)i∈[n]are assumed\nto be given by\nXi,j=Z⊤\niWj, fori∈[n]andj∈[d] (2)\nwhere the p-dimensional latent variables Z1, . . . , Z nare i.i.d. copies of Z∼ N(0, Ip), and\nwhere the p-dimensional random weights W1, . . . , W dare i.i.d. copies of Wuniformly\ndistributed on the sphere Sp−1, i.e., W∼ U(Sp−1).\nThe latent space in Assumption 1 corresponds to Rp. We have only access to n\nobservations ( Xi)i∈[n]of dimension dthat can be seen as random projections using d\ndirections of the latent features ( Zi)i∈[n]. The total amount of information contained in\n4the latent variables ( Zi)i∈[n]cannot be recovered in expectation by that contained in the\nobservations ( Xi)i∈[n]ifd < p . This no longer holds when d≫p, and the observations\n(Xi)i∈[n]can be therefore regarded as low-rank variables of rank p.\nLinear models with RF In the following, we present the model governing the\ndistribution of the output variable Y.\nAssumption 2 (Latent linear well specified model) .The target variable Yis assumed\nto follow a linear model w.r.t. the latent covariate Z, i.e.,\nY=Z⊤β⋆+ϵ, (3)\nwhere the model parameter is denoted by β⋆∈Rpand the noise ϵ∼ N(0, σ2)is assumed\nto be independent of Z.\nConsidering such a random feature setting is particularly convenient when studying\nthe influence of the input dimension don learning without modifying the underlying\nmodel (indeed, the distribution of Y–and Z– does not depend on the input dimension\nd). Note that a similar model (with fixed weight ( Wj)j) has already been introduced in\nHastie et al. (2022).\nMissing data Often one does not have access to the full input vector Xbut rather to a\nversion of Xcontaining missing entries. On the contrary, the output Yis always assumed\nto be observed. To encode such missing information on X, we introduce, the random\nvariable P∈ {0,1}d, referred to as the missing pattern (or actually an observation\nindicator), such that Pj= 1 if Xjis observed and Pj= 0 otherwise. Assuming that\nall variables are equally likely to be missing, we define ρ:=P(Pj= 1) for any j∈[d],\ni.e., 1−ρis the expected proportion of missing values for any feature. In this paper,\nwe thoroughly analyze the classical Missing Completely At Random (MCAR) setting,\nwhere the missing pattern Pand the complete observation ( X, Y) are independent.\nNote that we will extend some of our theoretical findings for the MCAR case to relaxed\nmissing scenarios in Section 4.\nAssumption 3 (MCAR pattern with independent components) .The complete observa-\ntion(X, Y)and the missing pattern Pare assumed to be independent, i.e., (X, Y)⊥ ⊥P\nand such that Pfollows a Bernoulli distribution B(ρ)⊗d, i.e., for any j∈[d],ρ=P(Pj=\n1), with 0< ρ≤1denoting the expected proportion of observed values.\nImputation Most machine learning algorithms are not designed to deal directly with\nmissing data. Therefore, we choose to impute the missing values (both in the training\nand test sets) by zero (or by the mean for non centered inputs). The imputed inputs in\nthe train and test sets are thus denoted, for all i, by\n˜Xi=Pi⊙Xi, (4)\n5where ⊙represented the component-wise product. The impact of missing data, and\ntheir handling by naive imputation, in this supervised learning task can be scrutinized\nthrough the evolution of the following key quantities:\n•The Bayes risk based on complete random features (without missing entries):\nR⋆(d) := inf\nfE\u0002\n(Y−f(X)2|W1, . . . , W d\u0003\n,\nwhere the infimum is taken over all measurable functions.\n•The Bayes risk given Xmiss= (˜X, P)∈Rd× {0,1}d:\nR⋆\nmiss(d) := inf\nfE\u0002\n(Y−f(Xmiss))2|W1, . . . , W d\u0003\n,\nwhere the infimum is taken over all measurable functions. It has been shown to\nbe attained for a pattern-by-pattern predictor (Le Morvan et al., 2020). The bias\nor deterministic error due to learning with missing inputs can be characterized as\n∆miss(d) :=E[R⋆\nmiss(d)−R⋆(d)].\n•The risk of the best linear predictor relying on zero-imputed inputs:\nR⋆\nimp(d) := inf\nθ∈RdEh\n(Y−˜X⊤θ)2|W1, . . . , W di\n.\nThe approximation error associated with this specific class of predictors, among\nthose handling missing inputs, is denoted by\n∆imp/miss(d) :=E\u0002\nR⋆\nimp(d)−R⋆\nmiss(d)\u0003\n.\nNote that these three risks decrease with the dimension and are ordered as follows,\nR⋆(d)≤R⋆\nmiss(d)≤R⋆\nimp(d). (5)\nIn what follows, we give a precise evaluation of R⋆(d) and R⋆\nmis(d) and provide bounds\nforR⋆\nimp(d) as well.\n2.2 Theoretical analysis\nOur goal is to dissect the systematic errors introduced by either the occurrence of missing\ninputs or their handling via naive zero imputation. To do so, we start by characterizing\nthe optimal risk over the class of linear predictors when working with complete inputs.\nProposition 2.1. Under Assumptions 1 and 2, the Bayes risk for linear predictors\nbased on complete random features is\nE[R⋆(d)] =\u001aσ2+p−d\np∥β⋆∥2\n2,when d < p,\nσ2when d≥p,\nwhere the expectation is taken over (Wj)j∈[d].\n6Proposition 2.1 highlights that learning with a number dof random features larger\nthan the latent dimension pis equivalent to learning directly with the latent covariate Z.\nBesides, when d < p , the Bayes predictor suffers from an increased risk, as learning is\nflawed by a lack of information in the (fully observed) inputs. This can be compensated\nby increasing the number dof random features, as the explained variance of Y, i.e.,\nEY2−ER⋆(d) =d\np∥β⋆∥2\n2,increases with dford≤p.\nProposition 2.2. Under Assumptions 1 to 3, the Bayes risk for predictors working\nwith missing data is given by\nE[R⋆\nmiss(d)] =(\nσ2+p−ρd\np∥β⋆∥2\n2 when d < p,\nσ2+E[(p−B)1B≤p]\np∥β⋆∥2\n2when d≥p,\nwhere the expectation is taken over the random weights (Wj)j∈[d]andB∼ B(d, ρ).\nTherefore,\n∆miss(d) =(\n(1−ρ)d\np∥β⋆∥2\n2 when d < p,\nE[(p−B)1B≤p]\np∥β⋆∥2\n2,when d≥p.\nTo our knowledge, this result is the first one to precisely evaluate the error induced\nby missing inputs when learning a linear latent model. More specifically, two regimes\nare identified. In the first regime d < p , i.e., when working with random features of\nlower dimension than that of the latent model, R⋆\nmisstakes the same form as R⋆, where\nthe input dimension dis replaced by ρd. This can be interpreted as the cost of learning\nwith ρdobserved features in expectation instead of the dinitial features.\nIn the second regime, when d≥p, the error due to missing data becomes more\nand more negligible as dincreases, as the redundancy of the random feature model is\nsufficient to retrieve the information contained in the latent covariate of lower dimension\np. Furthermore, if d≥(p+ 1)(1−ρ)e\nρ, we can bound ∆ miss(d) from above and below,\nρ\n2e(1−ρ)d−1∥β⋆∥2\n2≤∆miss(d)≤p\u0012dρ\np(1−ρ)\u0013p\n(1−ρ)d∥β⋆∥2\n2, (6)\nshowing that ∆ miss(d) decays exponentially fast with din the high-dimensional regime:\nthe impact of missing data on learning is therefore completely mitigated in high dimen-\nsion.\nTheorem 2.3. Under Assumptions 1 to 3, the Bayes risk for predictors based on\nzero-imputed random features satisfies\nE\u0002\nR⋆\nimp(d)\u0003\n−σ2≤\n\ninf\nk≤dn\np−ρk\np+(1−ρ)ρ(k−1)\np−ρ(k−1)−2k\npo\n∥β⋆∥2\n2ifd < p,\np\nρd+(1−ρ)p∥β⋆∥2\n2 ifd≥p.(7)\n7Thus, when d < p ,\n∆imp/miss(d)≤(1−ρ)ρ(d−1)\np−ρ(d−1)−2d\np∥β⋆∥2\n2 (8)\n=ρ(d−1)\np−ρ(d−1)−2∆miss(d). (9)\nAnd, when d≥p,\n(1−ρ)p\nρd+ (1−ρ)p∥β⋆∥2\n2≤∆imp/miss(d) + ∆ miss(d) (10)\n≤p\nρd+ (1−ρ)p∥β⋆∥2\n2. (11)\nTheorem 2.3 is the first result to provide a complete view of the impact of naive\nimputation on learning linear latent model. In particular, it sheds light on the following\nlow-dimensional behavior. When ρd≪p, the error due to naive imputation appears to\nbe negligible in comparison to the error ∆ miss(d) due to missing data. Low-dimensional\n(missing) random features are unlikely to be strongly correlated, thus making imputation\nbefore training competitive (compared to the best predictor based on missing values).\nThis is all the more true as the expected number of observed entries ρdis negligible\ncompared to p.\nIn high dimensions where d≫p, both errors ∆ imp/miss(d) and ∆ miss(d) vanish:\nneither the occurrence of missing data, nor their naive handling through imputation,\nhinder the learning task. This provides a refined and more rigorous analysis of this\nfavorable behavior already identified in Ayme et al. (2023). Remark that, contrary\nto ∆ miss(d) (Equation (6)), ∆ imp/miss(d) does not seem to decrease exponentially as d\nincreases, but only as 1 /d. Not that, according to (10), this rate is optimal.\nIllustration We illustrate the bounds obtained in Proposition 2.1, Proposition 2.2\nand Theorem 2.3 in Figure 1. In particular, we remark that the upper bounds (7)\nrepresented in (a) decrease with the number of features dbut is loose for dclose to and\nsmaller than p. Indeed, for this regime, features are not enough isotropic to say that\nimputation by the mean (here by 0) is relevant, and not enough correlated to exploit\nshared information between features. Figure (b) illustrates the shift point in pforRmis\nandRimp/misand the difficulties to learn with missing values (imputed or not) around\nd=p.\n2.3 Learning from imputed data with SGD\nWe leverage on our analysis of the Bayes risk of impute-then-regress strategy to propose\na learning algorithm based on an SGD strategy. Our algorithm is computationally\nefficient, as it requires only one pass over the dataset, and shown to have theoretical\nguarantees. Note that due to missing data, the model becomes miss-specified (see Ayme\net al., 2023), a challenging study case that can still be handled by SGD procedures.\n80 100 200 300 4000.00.20.40.60.81.0 d=pRimp (UB)\nRmis\nR\n0 100 200 300 4000.00.20.40.60.81.0\nd=pimp+mis (UB)\nmis\n(a) (b)\nFigure 1: Evolution of different risks w.r.t. the number dof random features, with\np= 100, ∥β⋆∥= 1,ρ= 0,8 and σ= 0.\nSGD estimator. An averaged stochastic gradient descent (SGD) on the imputed\ndataset ( ˜Xi)i∈[n]is performed to directly minimize the theoretical risk θ7−→Rimp(θ)\noverRd. The algorithm starts from θ0= 0 with a step-size γ >0, then follows the\nrecursion\nθt=h\nI−γ˜Xt˜X⊤\nti\nθt−1+γYt˜Xt, (12)\nand outputs after niterations the Polyak-Ruppert averaging ¯θ=1\nn+1Pn\nt=1θt, used to\nestimate θ⋆\nimp. This algorithm (performing one pass over the training data) is optimal in\nterms of computational complexity. Note that the choice of the step size should depend\nintimately on the input dimension, with a slight variation, according to whether the\nsetting is low or high-dimensional (see, Dieuleveut and Bach, 2016, for example).\nTheorem 2.4. Under Assumptions 1 to 3, for d < p −1, the SGD recursion with\nPolyak-Ruppert averaging and step size γ=1\ndsatisfy\nE[Rimp(¯θ)−R⋆\nmiss(d)]≲ρ(1−ρ)d(d−1)\np(p−ρ(d−1)−2)∥β⋆∥2\n2\n+d\nρnd\n(p−ρ(d−1)−2)∥β⋆∥2\n2+ρd\nn(σ2+∥β⋆∥2\n2). (13)\nFord≥p, the choice γ=1\nd√nleads to\nE[Rimp(¯θ)−R⋆(d)]≲1\nρp\nd∥β⋆∥2\n2+p\nρ2(1−ρ)√n∥β⋆∥2\n2+σ2+∥β⋆∥2\n2√n. (14)\nRegardless of the regime, these generalization upper bounds are composed of three\nterms. The first one encapsulates the (deterministic) error due to the imputation of\n9missing values. The last two are stochastic errors inherent to learning, corresponding\nrespectively to the initial condition and the variance of the SGD recursion.\nWhen d < p−1, the learning error\nd\nρnd\n(p−ρ(d−1)−2)∥β⋆∥2\n2+ρd\nn(σ2+∥β⋆∥2\n2)\ndecreases fast with the number nof observation. However, the error due to the imputation\nof missing data ρ(1−ρ)d(d−1)\np(p−(d−1)−2)∥β⋆∥2\n2becomes negligible only for extremely low-\ndimensional regimes ( d≪p) and remains significant when d≲p. Therefore, for such\na regime, even with a lot of observations, the imputation produces a large bias, and\nwe recommend using other methods natively capable of handling missing values (e.g.,\ndedicated tree-based methods, see Stekhoven and B¨ uhlmann, 2012).\nIn the regime d≫p, the error1\nρp\nd∥β⋆∥2\n2due to missing data and the zero-imputation\nprocedure is low. Besides, the learning errorp\nρ2(1−ρ)√n∥β⋆∥2\n2+σ2+∥β⋆∥2\n2√ndecreases at a\n(slow) rate 1 /√n. This slow rate of learning error is due to the fact that the covariance\nmatrix of the imputed data Σ imp=E[˜X˜X⊤] has a rank equal to dand eigenvalues lower\nbounded by ρ(1−ρ). Hence imputed data are not of low rank, even for d≫p. However,\nthe upper bound (14)becomes dimension-free for the regime d > ρ (1−ρ)√n. In this\ncase, the bias due to missing data and zero imputation is negligible compared to the\nlearning error. This gives a clear practical recommendation: if the observed rate is such\nthatρ(1−ρ)< d/√n, then zero-imputation does not deteriorate the learning procedure\nfordlarge enough.\nOverall, we have fully characterized the inherent error due to missing values in learning\nlinear latent models, and proposed an efficient predictor based on SGD strategies. This\nsection outlines that naive imputation thus remains a competitive and relevant technique\nnot only for high-dimensional models but also for extremely low-dimensional ones, a\nfavorable regime that was not identified so far in the literature. Note in passing that\nfor the latter, the error bound (13)of the SGD predictor enjoys both a fast rate and a\nnegligible approximation error, which can only be marginally improved. However, the\nanalysis conducted so far relies on strong assumptions (finite-dimensional latent model\nwith Gaussian random features and uniform weights, linear model). In the next section,\nwe propose an extension of our high-dimensional results to a more general framework.\n3 Extension to infinite-dimensional latent space\nIn this section, we analyze the influence of missing data in learning, when the random\nfeature model involves an infinite-dimensional latent space.\n3.1 The extended random feature framework\nConsider a latent space Z(taking the place of Rp), possibly of infinite dimension. We\ndenote by ( Zi)i∈[n]i.i.d. latent variables, distributed as a generic random variable Z∈ Z.\n10We only observe the variables ( Xi)i∈[n], i.i.d. copies of X∈ X=Rd, resulting from the\nfollowing transformation of the latent variables.\nAssumption 4 (General random features) .The input variables are assumed to be given\nby\nXij=ψ(Zi, Wj),for all i∈[n]andj∈[d], (15)\nwhere the weights W1, . . . , W d∈ W are i.i.d. drawn according to a distribution ν, and\nwhere ψ:Z × W → R. Furthermore, we assume that ψ(Zi, .)∈L2(ν), and there exists\nL >0such that E[ψ(Z, W )2|W]≤L2almost surely.\nAssumption 4 is an extension of Assumption 1 considering Z=W=Rp,ψ(Zi, Wj) =\nZ⊤\niWj,νthe uniform distribution on Sp−1andL2= 1. But, such a setting of general\nrandom features encompasses many more scenarios, and has been extensively studied\n(Rahimi and Recht, 2007).\nIn this framework, we aim to study the linear prediction of an output Ygiven X,\ni.e., to build a prediction function of the form g(X) =X⊤θwith θ∈Rd. Note that this\ntype of prediction can be also obtained as a function of the (latent) variable Z, indeed,\ng(X) =dX\nj=1θjψ(Z, W j) =:f(Z). (16)\nWe can therefore define the corresponding class of functions with input space Zas\nF(d)\nν:=n\nf:Z →R, f(z) =dX\nj=1θjψ(z, W j), θ∈Rdo\nNote that the class F(d)\nνis random because the weights ( Wj)j∈[m]themselves are random.\nWhen the number dof random features tends to infinity, we can define the set F(∞)\nνof\nfunctions which take the form:\nf(Z) =Z\nαf(w)ψ(Z, w)dν(w), (17)\nfor any αf∈L2(ν). The associated norm is given by\n∥f∥2\nν:= inf\nα∈L2(ν)Z\n|α(w)|2dν(w) s .t.∀z∈ Z, f(z) =Z\nα(w)ψ(z, w)dν(w).(18)\nThis norm corresponds to an RKHS norm, we refer the interested reader to Bach (2017)\nfor further details. We denote by R⋆(∞), the risk of the best predictor belonging to the\nclassF(∞)\nν, i.e., R⋆(∞) = inff∈F(∞)\nνE[(Y−f(Z))2].\nThe setting considered in this section includes, for instance, Fourier random features\n(Rahimi and Recht, 2007; Rudi and Rosasco, 2017).\n11Example 3.1 (Fourier random features) .Consider Z∈ Zwhere Zis a compact subset\nofRp,W= (A, B, C )∈ W =Rp×R× {− 1,1}and fix\nψ(Z, W ) = cos( A⊤Z+B) + 2C,\nwith A∼ N(0, I),B∼ U([0,2π]) and C∼ U({−1,1}). Note that Assumption 4 holds\nhere with L2= 3. The resulting function class F(∞)\nνdescribed by these random features\nis dense for ∥ · ∥∞in the space of continuous functions.\nAs shown in this example, with a proper choice of νandψ, the class F(∞)\nνcan\napproximate any function that makes the following assumption feasible.\nAssumption 5. The Bayes predictor f⋆(z) =E[Y|Z=z]belongs to F(∞)\nν.\nUnder Assumption 5, the model Y=f⋆(Z) +ϵ, is well defined, i.e., E[ϵ|Z] =\n0. Remark that Assumption 5 can be seen as a natural extension of linear model\nAssumption 2.\n3.2 Impact of missing data and imputation in the RF frame-\nwork\nThe general random features ( Xi)i∈[n]are assumed to be corrupted by MCAR entries,\nwhether during training and test phases. Our goal is to study the quantity R⋆\nimp(d).\nNote that (5) can be rewritten here as\nR⋆(∞)≤R⋆(d)≤R⋆\nmiss(d)≤R⋆\nimp(d).\nThus, introducing the quantity\n∆(∞)\nimp(d) :=ER⋆\nimp(d)−R⋆(∞)\n= ∆ miss(d) + ∆ imp/miss(d) +ER⋆(d)−R⋆(∞), (19)\nwe encapsulate (i) the error ER⋆(d)−R⋆(∞) due to learning from a finite number of\nrandom features, (ii) the error ∆ miss(d) due to learning with missing inputs and, (iii)\nthe approximation error ∆ imp/miss(d) due to the imputation by zero.\nTheorem 3.2. Under Assumptions 3 and 4,\n∆(∞)\nimp(d)≤inf\nf∈F(∞)\nν\u001a\nR(f)−R⋆(∞) +λimp\nd∥f∥2\nν\u001b\n,\nwithλimp=L2\nρ. In particular, under Assumption 5,\n∆(∞)\nimp(d)≤λimp\nd∥f⋆∥2\nν.\n12Theorem 3.2 provides an upper bound on ∆(∞)\nimp(d) for general random feature models.\nIn particular, the latter can be compared to a ridge bias when performing a kernel\nridge regression in F(∞)\nν, and choosing the penalization strength of the order of λimp/d.\nFurthermore, under Assumption 5 (well-specified model), this bias converges to zero\nwith a rate of ∥f⋆∥2\nν/(ρd). By applying this result to a finite-dimensional latent model\nunder Assumptions 1 and 2, and remarking that ∥f⋆∥2\nν=p∥β⋆∥2\n2, we recover the same\nratep/(ρd)∥β⋆∥2\n2exhibited in Theorem 2.3. According to Theorem 2.3, this rate cannot\nbe improved in general. More globally, missing data in RF models become harmless\nwhen learning with a large number of random features. It should be noted that when\nAssumption 5 does not hold anymore, one can still conclude that the bias ∆(∞)\nimptends\nto zero but at an arbitrarily slow rate. Regarding Assumption 4, it remains a mild\nrequirement; in particular, it does not require centered inputs. This underlines that there\nis no need to impute by the mean to obtain ∆(∞)\nimpconverging to 0 in high-dimensional\nregimes.\n3.3 SGD generalization upper bound\nIn this section, we assess the generalization performance of the SGD iterates when\nworking with an underlying general random feature model.\nAssumption 6. There exists ℓ >0such that, almost surely\nE[ψ(Z, W )2|W]≥ℓ2.\nThis assumption holds when features are renormalized (i.e., when ℓ=L= 1) or in\nthe case of random Fourier features (see Example 3.1) with ℓ= 1.\nAssumption 7. Assume that almost surely,\n|ψ(Z, W )|2≤κL2.\nThis assumption is satisfied in Example 3.1 with κ= 2 and L2= 3.\nTheorem 3.3. Under the general framework covered by Assumptions 4 to 7 with MCAR\ndata (Assumption 3), the SGD recursion with Polyak-Ruppert averaging and step size\nγ= 1/(κd√n)satisfy\nE[Rimp(¯θ)−R⋆(∞)]≲L2\nρd∥f⋆∥2\nν+L2\nℓ2L2\nρ2(1−ρ)√n∥f⋆∥2\nν+κL2EY2\n√n. (20)\nTheorem 3.3 outlines that, even for very general random features model (with\npossibly a latent space of infinite dimension), the impact of (i) the finite number\nof features, (ii) the missing data, and (iii) the imputation by 0, represented by the\nquantityL2\nρd∥f⋆∥2\nν, remains negligible in high dimension. Similarly to (14), the learning\nerrorL2\nℓ2L2\nρ2(1−ρ)√n∥f⋆∥2\nν+κL2EY2√ndecreases with a slow rate. More precisely, when d≫\n13L2\nℓ2ρ(1−ρ)√n, the upper bound is dimension free and the bias ∆(∞)\nimpdue to imputation\nbecomes completely negligible. Note that, for renormalized features ( L2=l2= 1),\nthe transition from a low-dimensional regime to a high-dimensional one is given by\nd=ρ(1−ρ)√n(as for Theorem 2.4), which is very easy to evaluate.\n4 Beyond the MCAR assumption\nTo go beyond the MCAR missing data framework used in the previous section, we now\nconsider missing not at random (MNAR) data, in which the missingness indicator of\nany variable can depend on the (possibly missing) value of the variable. In particular,\nwe assume that the missing patterns ( Pi)idepend on the latent features ( Zi)i, which\nresults in a MNAR scenario.\nAssumption 8. Suppose that PandYare independent. Furthermore, consider that\nthere exists an i.i.d. sequence (W′\nj)j∈[d]i.i.d. drawn according to some distribution µ\nsupported on a set W′and assume that P1, . . . , P d|Z, W′\n1, . . . , W′\ndare independent. We\nassume that the sequences (W′\nj)j∈[d]and(Wj)j∈[d]are independent and that\nP\u0000\nPj|Z, W′\nj\u0001\n=ϕ(Z, W′\nj),for all j∈[d],\nwhere ϕ:Z × W′→(0,1]is a continuous function.\nThe following result shows that the asymptotic property of R⋆\nimpin a MCAR setting\n(Theorem 2.3) remains valid in the MNAR setting of Assumption 8.\nTheorem 4.1. Under Assumption 8, consider one of the following settings:\n(i)(finite-dimensional latent space) Under Assumptions 1 and 2, assume the distri-\nbution of the missing mechanism to be given for W′= (W′\n0, W′)∈R×Rd, by\nϕ(Z, W′) = Φ( Z⊤W′+W′\n0)withΦa Lipschitz function. Additionally, 0is required\nto belong to the support of W′.\n(ii)(general latent space) Under Assumptions 4 and 5, assume in addition that Z\nis compact, f⋆continuous and F(ν)dense in the space of continuous functions\nequipped with the norm ∥ · ∥∞.\nThen, almost surely,\nlim\nd→+∞R⋆\nimp(d) =R⋆(∞).\nAs a consequence,\nlim\nd∆(∞)\nimp(d) = lim\nd∆miss(d) = lim\nd∆miss/imp(d) = 0 .\nThis result shows that the benign impact of missing data and imputation on predictive\nperformances in high dimension holds true outside the MCAR assumption, even for\nmissing scenarios (MNAR) often considered as more challenging. Let us consider two\nnon-trivial examples.\n14Example 4.2 (Gaussian random features with logistic model) .Consider the finite-\ndimensional latent model of Assumptions 1 and 2, where Z=RpandW′\nj= (W′\n0j, W′\nj)∈\nR×Rd, and assume that the conditional distribution of the missing patterns Pjis given\nby\nP\u0000\nPj|Z, W′\nj\u0001\n= Φ(W′\n0j+W′⊤\njZ) =1\n1 +eW′\n0j+W′⊤\njZ.\nIn this example, the features Xjare assumed to be missing according to a logistic model\non the latent variables Z. In this setting, we can show that Theorem 4.1 ( i) applies,\nsince in particular, 0 belongs the support of W′\nj. Note that, if W′\nj= 0 almost surely then\nthis model corresponds to a MCAR scenarios but with different proportion of missing\nvalues for each feature. The model is no longer MCAR as soon as random variable W′\nj\nis not exactly equal to 0.\nExample 4.3 (Fourier random features for any function ϕ).Let us consider the framework\nof Example 3.1 with a continuous function f⋆. Then Theorem 4.1 ( ii) applies for any\ncontinuous function ϕ(in particular, we can consider the logistic model of Example 4.2\nwithout any condition on W′).\nFor these two MNAR examples,\nlim\nd∆(∞)\nimp(d) = lim\nd∆miss(d) = lim\nd∆miss/imp(d) = 0 ,\nwhich means that missing values and imputations vanishes with the dimension.\n5 Conclusion\nThanks to the rigorous framework of random features models, we prove that naive\nimputation is relevant both in high- and low-dimensional regimes. In particular, the\nbias induced by imputation is negligible compared to the one induced by missing data,\ntherefore showing that zero-imputation strategies may lead to near-optimal predictors.\nFollowing this analysis, we prove that an SGD procedure trained on zero-imputed data\nreaches near-optimal rate of consistency in low-dimensional regimes, but still suffer from\nslow rates in high-dimensional ones. Obtaining fast rates for the latter setting is still an\nopen and interesting question. 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Why are big data matrices approximately low\nrank? SIAM Journal on Mathematics of Data Science , 1(1):144–160, 2019.\nDietrich Von Rosen. Moments for the inverted wishart distribution. Scandinavian\nJournal of Statistics , pages 97–109, 1988.\nKatarzyna Wo´ znica and Przemys law Biecek. Does imputation matter? benchmark for\npredictive models. arXiv preprint arXiv:2007.02837 , 2020.\n18A Notations\nFor two vectors (or matrices) a, b, we denote by a⊙bthe Hadamard product (or\ncomponent-wise product). [ n] ={1,2, ..., n}. For two symmetric matrices AandB,\nA⪯Bmeans that B−Ais positive semi-definite. The symbol ≲denotes the inequality\nup to a universal constant. Table 1 summarizes the notations used throughout the paper\nand appendix.\nTable 1: Notations\n∥u∥2\nM u⊤Muthe semi-norm induced by a positive matrix M\n∥M∥2\nFr The Frobenius norm of M\nTr(M) The sum of diagonal elements of M\nM+λ Abuse of notation for M+λIp\nM†The Moore-Penrose pseudoinverse of M\nSp The unit sphere of Rp\nSpan( uj, j∈[k]) The linear span induced by ( uj)j∈[k]\nP The mask\nW (W1, . . . , W d)⊤the matrix of weights\nR(θ) the risk of linear predictor θon complete data\nRimp(θ) the risk of linear predictor θon imputed data\nθ⋆Best linear predictor on complete data\nθ⋆\nimp Best linear predictor on imputed data\nΣ E[XX⊤|W]\nλj eigenvalues of Σ\nuj eigendirections of Σ\nρ Theoretical proportion of observed entries\nL2(ν) The set of two square νintegrable functions\nB Preliminary results - random matrices\nWe provide here a reminder on singular values decomposition and Moore-Penrose\npseudoinverse. We can found these results and more on linear algebra in Giraud (2021,\nappendix).\nTheorem B.1. Anyn×preal-valued matrix of rank rcan be decomposed as\nA=rX\nj=1σjujv⊤\nj,\nwhere\n•σ1≥ ··· ≥ σr>0,\n19•(σ1, . . . , σ r)are the nonzero eigenvalues of A⊤AandAA⊤, and\n•(u1, . . . , u r)and(v1, . . . , v r)are two orthonormal families of RnandRp, such that\nAA⊤uj=σ2\njujandA⊤Avj=σ2\njvj.\nFurthermore, the Moore-Penrose pseudo inverse defined as\nA†=rX\nj=1σ−1\njvju⊤\nj,\nsatisfied\n1.A†Ais the orthogonal projector on lines of A,\n2.AA†is the orthogonal projector on columns of A,\n3.(AO)†=O⊤A†for any orthogonal matrix O.\nFor any function f:R7−→R, and any positive matrix A∈Rd×dwith the following\nspectral decomposition A=Pd\nj=1λjvjv⊤\nj, we denote by f(A) the matrix corresponding\nto the spectral decomposition\nf(A) :=dX\nj=1f(λj)vjv⊤\nj.\nTheorem B.2 (Jensen inequality for random positive matrix, see Theorem 2.10 in\nCarlen (2010)) .LetAbe a random positive matrix. For all convex functions f, we have\nTr(f(EA))≤ETr(f(A)).\nProposition B.3. LetA=Pd\nj=1ZjZ⊤\njwithZ1, . . . , Z di.i.d. random vector of Rpwith\n∥Z1∥2\n2≤1almost surely and EZZ⊤=αIp, then, for all λ >0\np\ndα+λ≤ETr\u0000\n(A+λIp)−1\u0001\n≤(1 + 1 /λ)p\ndα+λ. (21)\nProof of Proposition B.3. This result is a direct application of Mourtada and Rosasco\n(2022, Lemma 2) considering ˆΣ =1\ndA\nLemma B.4. LetM∈Rp×pbe a random symmetric matrix, such that for all vectors\nu, v∈Sp−1,Law( u⊤Mu) = Law( v⊤Mv). Then, for all β∈Rp,\nE\u0002\nβ⊤Mβ\u0003\n=∥β∥2\n2ETr(M)\np.\nThis is in particular satisfied if, for any orthogonal matrix O,OMO⊤has the same law\nasM.\n20Proof. By assumption, for all u, v∈Sd−1,Eu⊤Mu=Ev⊤Mv. Thus, there exists α\nsuch that, for all v∈Sd,v⊤EMv=Ev⊤Mv=α, which entails that EM=αIby\ncharacterization of symmetric matrices. Therefore, ETr(M) =Tr(EM) =pα, and\nEM=ETr(M)\npI. Hence, for all β∈Rp\nE\u0002\nβ⊤Mβ\u0003\n=β⊤EMβ=∥β∥2\n2ETr(M)\np.\nThe last point easily follows, see for example Page Jr (1984, Proposition 2.14) for the\ncase of invariant distributions by orthogonal transforms.\nThe following result is inspired by the result of Cook and Forzani (2011), that is an\nadaptation of that of Von Rosen (1988).\nLemma B.5. For all 0< d < p −1, letW∈Rp×dsuch that columns of Ware i.i.d.\nand uniform over Sp−1, then\nE∥W†∥2\nFr=d\u0012\n1 +d−1\np−d−1\u0013\n.\nProof. Up to a polar coordinate change of variable, one can show that the distribution\nof the columns of Wcorresponds to that of normalized Gaussian vectors, i.e., for all\nj∈[d],\nWj=Gj\n∥Gj∥2,\nwhere ( Gj) are i.i.d. of law N(0, Ip). Note that the columns of Ware the rows of\nM=W⊤. As d < p ,\nMM†=Id.\nbecause the rows of Mare almost surely linearly independent. For all j∈[d], we let lj\nbe the j-th row of M, and cjthej-th column of M†. Therefore, for all k̸=j,l⊤\nkcj= 0,\nthen cj∈Span (lk, k̸=j)⊥. Note, that Span (lj, j∈[d]) =Span (cj, j∈[d]) by property\nof Moore-Penrose pseudoinverse (Theorem B.1). Thus, cjas the form,\ncj=θjPjlj,\nwhere Pjis the orthogonal projection on Span(lk, k̸=j)⊥. Besides, l⊤\njcj= 1 gives us\nthat\nθj=1\n∥Pjlj∥2\n2.\nThus,\n∥M†∥2\nFr=X\nj∥cj∥2\n2=X\nj1\n∥Pjlj∥2\n2(22)\n21Aslj=Wj=Gj\n∥Gj∥2, we can write\n1\n∥Pjlj∥2\n2=∥Gj∥2\n2\n∥PjGj∥2\n2\nUsing that ∥Gj∥2\n2=∥PjGj∥2\n2+∥(Ip−Pj)Gj∥2\n2, we have\n1\n∥Pjlj∥2\n2= 1 +∥(Ip−Pj)Gj∥2\n2\n∥PjGj∥2\n2.\nConditioning by ( Gk) with k̸=j, and using Cochran theorem ( Ip−Pj)Gj|Gk, k̸=jand\nPjGj|Gk, k̸=jare two independent standard normal vector of respective dimensions\np−(p−d+ 1) = d−1 and p−d+ 1. Thus,\nE\u00141\n∥Pjlj∥2\n2|Gk, k̸=j\u0015\n= 1 + E\u0002\n∥(Ip−Pj)Gj∥2\n2|Gk, k̸=j\u0003\nE\u00141\n∥PjGj∥2\n2|Gk, k̸=j\u0015\n(23)\n= 1 +d−1\np−d−1, (24)\nbecause E[∥(Ip−Pj)Gj∥2\n2|Gk, k̸=j]=d−1 and Eh\n1\n∥PjGj∥2\n2|Gk, k̸=ji\n=1\np−d−1as the\nexpectation of an inverse-chi-squared of parameter p−d+ 1>2 (with d < p−1). Then,\ntaking the expectation of (22) leads to the result,\nE∥M†∥2\nFr=d+d(d−1)\np−d−1.\nLemma B.6. LetA, B, V three symetrics non-negative matrix, if A⪯BthenA⊙V⪯\nB⊙V.\nProof. LetX∼ N(0, V) and θ∈Rd,\n22∥θ∥2\nA⊙V=θ⊤A⊙V θ\n=θ⊤\u0000\u0000\nEXX⊤\u0001\n⊙A\u0001\nθ\n=E\u0002\nθ⊤\u0000\u0000\nXX⊤\u0001\n⊙A\u0001\nθ\u0003\n=E\"X\ni,jθi\u0000\u0000\nXX⊤\u0001\n⊙A\u0001\ni,jθj#\n=E\"X\ni,jθiXiXjAi,jθj#\n=E\"X\ni,j(θiXi) (θjXj)Ai,j#\n=E\u0002\n∥X⊙θ∥2\nA\u0003\n≤E\u0002\n∥X⊙θ∥2\nB\u0003\n=∥θ∥2\nB⊙V\nC Proof of Section 2\nThe following result, established by Ayme et al. (2023), is used to derive an expression\nof ∆ missing + ∆ imp/miss.\nLemma C.1 (Proposition 3.1 of (Ayme et al., 2023)) .For all θ∈Rd,\nRimp(θ) =R(ρθ) +ρ(1−ρ)∥θ∥2\ndiag(Σ) . (25)\nRecalling that\n∆miss+ ∆ imp/miss=E\u0002\nR⋆\nimp(d)−R⋆(d)\u0003\n, (26)\nwe deduce from Lemma C.1 that\n∆miss+ ∆ imp/miss=Einf\nθ∈Rd\u001a\nR(θ)−R⋆(d) +1−ρ\nρ∥θ∥2\ndiag(Σ)\u001b\n. (27)\nAdditionally, when diag(Σ) = Ip(in particular for model (1)), by optimization, we\nobtain,\n∆miss+ ∆ imp/miss=λE∥θ⋆∥2\nΣ(λI+Σ)−1, (28)\nwith λ=1−ρ\nρ.\n23Lemma C.2. Under Assumption 3,\nER⋆\nmiss(d) =dX\nk=0P(B=k)ER⋆(k),\nwhere Bis a binomial random variable of parameters dandρ.\nProof. Using the decomposition of the Bayes predictor from Le Morvan et al. (2020),\nwe have\nR⋆\nmiss(d) =X\nm∈{0,1}dP(P=m)R⋆\nm, (29)\nwhere\nR⋆\nm= inf\nfE\u0002\n(Y−f(Xobs(m)))2|P=m, W 1, . . . , W d\u0003\n,\nis the local Bayes risk given ( P=m). Using MCAR assumption (Assumption 3), and\nGaussian assumption, according to Le Morvan et al. (2020), each local Bayes predictor\nare linear, thus\nR⋆\nm= inf\nθE\u0002\n(Y−θ⊤Xobs(m)))2|Wj, j∈obs(m)\u0003\n.\nAs (Wj) are i.i.d. (and independent of Y),R⋆\nmhas the same law as R⋆(|m|) where |m|\nis the number of observed components of m. Thus,\nER⋆\nmiss(d) =X\nm∈{0,1}dP(P=m)ER⋆(|m|)). (30)\nGrouping the missing patterns of the same size, we conclude that,\nER⋆\nmiss(d) =dX\nk=0P(B=k)ER⋆(k),\nwhere Bis a binomial law of parameters dandρ.\nC.1 Proof of Proposition 2.1\nBy definition,\nR⋆(d) =E\u0002\n(X⊤θ⋆−Y)2|W1, . . . , W d\u0003\n=E\u0002\n(X⊤θ⋆−Z⊤β⋆−ϵ)2|W1, . . . , W d\u0003\n(using (2))\n=σ2+E\u0002\n(X⊤θ⋆−Z⊤β⋆)2|W1, . . . , W d\u0003\n,\n24using that ϵis an independent noise of variance σ2. We have X⊤θ⋆=Z⊤P\njθ⋆\njWj.\nThen,\nR⋆(d) =σ2+E\n\n dX\nj=1θ⋆\njWj−β⋆)!⊤\nZ\n2\f\f\f\f\fW1, . . . , W d\n\n=σ2+\r\r\r\r\rβ⋆−dX\nj=1θ⋆\njWj\r\r\r\r\r2\n2,\nby isotropy of Z(Z∼ N(0, Ip) and thus EZZ⊤=I). Using thatPd\nj=1θ⋆\njWjbelongs to\nSpan (W1, . . . , W d), we get thatPd\nj=1θ⋆\njWj=Pdβ⋆where Pdis the orthogonal projection\non Span( W1, . . . , W d). Then,\nR⋆(d) =σ2+∥(I−Pd)β⋆∥2\n2=σ2+ (β⋆)⊤(I−Pd)β⋆.\nRemark that Pdis a random matrix (since W1, . . . , W dare random). Denoting by W\nthe matrix admitting W1, . . . , W das rows, the projection matrix can be rewritten as\nPd=W†W. Thus, for all orthogonal matrix O,OPdO⊤= (WO⊤)†WO⊤. The matrix\nWO⊤has rows O⊤W1, . . . , O⊤Wd, which is an i.i.d. sequence of random vectors on\nthe unit sphere (since O⊤is an orthogonal). Indeed, WO⊤andWhave the same\ndistribution, in consequence PandOPdOhave the same distribution too. Thus, by\nLemma B.4, if d < p ,\nER⋆(d) =σ2+1\np∥β⋆∥2\n2ETr(I−Pd)\n=σ2+p−d\np∥β⋆∥2\n2,\nusing that Tr(I−Pd) = rank(I−Pd) = p−d. Besides, if d≥p,Pd=Ip, and\nER⋆(d) =σ2.\nC.2 Proof of Proposition 2.2\nUsing Lemma C.2, we have\nER⋆\nmiss(d) =dX\nk=0P(B=k)ER⋆(k),\nwhere Bis a binomial random variable of parameters dandρ. Using Proposition 2.1,\nwe have\nE[R⋆(k)] =\u001aσ2+p−k\np∥β⋆∥2\n2,when k < p,\nσ2when k≥p.\n25Combining the two previous equalities, we obtain that\nER⋆\nmiss(d) =σ2+E[(p−B)1B≤p]\np∥β⋆∥2\n2.\nIn the case where d≤p,1B≤p= 1 almost surely, and we obtain\nER⋆\nmiss(d) =σ2+p−ρd\np∥β⋆∥2\n2.\nC.3 Proof of Theorem 2.3\nC.3.1 Preliminaries\nIn the rest of the proof, we denote by W= (W1, . . . , W d)⊤∈Rd×pthe weight matrix\nthat admits the weight vectors Wj∼ U(Sp−1) for rows. We call Σ = E\u0002\nXX⊤|W\u0003\nthe\ncovariance matrix of an input X∈Rdgiven the weight matrix W. Recall that the\nlatter, resulting from a random feature model, is such that X=WZ, forZ∈Rpthe\ncorresponding latent vector.\nLemma C.3. Under assumptions of Theorem 2.3,\n∆imp/miss+ ∆ miss=(λ∥β⋆∥2\n2\npETr ((Σ + λId)−1) ifd < p\nλ∥β⋆∥2\n2\npETr\u0000\n(W⊤W+λIp)−1\u0001\nifd≥p,\nwithλ=1−ρ\nρ.\nProof. One has for θ∈Rp,\nR(θ) =σ2+E\u0002\n(Z⊤β⋆−θ⊤X)2|W\u0003\n.\nUsing that X=WZ, we have\nR(θ) =σ2+E\u0014\u0010\u0000\nβ⋆−W⊤θ\u0001⊤Z\u00112\n|W\u0015\n=σ2+\r\rβ⋆−W⊤θ\r\r2\n2,\nby isotropy of Z. Since θ⋆minimizes the risk R,θ⋆minimizes the least-squares criterion\nabove. Therefore, in the case where p > d (the “design” W⊤being long), θ⋆is unique\nand given by θ⋆= (WW⊤)−1Wβ⋆. In the case where d < p (the design matrix W⊤\nbeing fat), there exists an infinite number of minimizers (all are solutions of the system\nβ⋆=W⊤θ), but one can look at the solution of minimal ℓ2-norm. Then, KKT conditions\nprovide the particular solution θ⋆=W(W⊤W)−1β⋆. In both cases, θ⋆can be written\nin the following unified way:\nθ⋆=\u0000\nW⊤\u0001†β⋆.\n26Futhermore,\nΣ =E\u0002\nXX⊤|W\u0003\n=E\u0002\nWZ(WZ)⊤|W\u0003\n=E\u0002\nWZZ⊤W⊤|W\u0003\n=WW⊤.\nThen, using (28),\n∆imp/miss+ ∆ miss=λE∥θ⋆∥Σ(Σ+ λI)−1\n=λE\r\r\r\u0000\nW⊤\u0001†β⋆\r\r\r2\nWW⊤(WW⊤+λI)−1\n=λE∥β⋆∥2\nW†WW⊤(WW⊤+λI)−1(W⊤)†\n=λ∥β⋆∥2\n2\npETr\u0010\nW†WW⊤(WW⊤+λI)−1\u0000\nW⊤\u0001†\u0011\n,\nusing Lemma B.4 remarking that, for all orthogonal matrix O∈Rp×p\nOW†WW⊤(WW⊤+λI)−1\u0000\nW⊤\u0001†O⊤=OW†WO⊤OW⊤(WO⊤OW⊤+λI)−1\u0000\nW⊤\u0001†O⊤\n= (WO⊤)†WO⊤(WO⊤)⊤(WO⊤(WO⊤)⊤+λI)−1\u0000\n(WO⊤)⊤\u0001†,\nby orthogonality of O(O⊤O=Ip). Then, OW†WW⊤(WW⊤+λI)−1\u0000\nW⊤\u0001†O⊤has\nthe same distribution as W†WW⊤(WW⊤+λI)−1\u0000\nW⊤\u0001†, since WO⊤dist=W.\nConsider the singular value decomposition (SVD) of W,\nW=rX\nj=1σjujv⊤\nj,\nwhere r=p∧dis the rank of W†, (uj) is an orthonormal basis of Rd,and ( vj) is an\northonormal basis of Rp. The SVD of its pseudo-inverse is therefore\nW†=rX\nj=1σ−1\njvju⊤\nj.\nThen, we obtain\nW†WW⊤(WW⊤+λI)−1\u0000\nW⊤\u0001†=rX\nj=11\nλ+σ2\njuju⊤\nj.\nThus,\nTr\u0010\nW†WW⊤(WW⊤+λI)−1\u0000\nW⊤\u0001†\u0011\n=rX\nj=11\nλ+σ2\nj.\nWe recognize ( λ+σ2\nj)j∈[r]as the eigenvalues of Σ + λId=WW⊤+λIdwhen d < p\nandrank(W) =d, or as the eigenvalues of W⊤W+λIpwhen d≥pandrank(W) =p.\nHence,\n∆imp/miss+ ∆ miss=(λ∥β⋆∥2\n2\npETr ((Σ + λId)−1) if d < p\nλ∥β⋆∥2\n2\npETr\u0000\n(W⊤W+λIp)−1\u0001\nifd≥p.\n27C.3.2 Proof of (7)(d < p−1)\n(First step) Decomposition of R⋆\nimp(d).Note that for x≥0 (to be chosen later),\none has\nR⋆\nimp(d)≤Rimp(xθ⋆)≤R(xρθ⋆) +ρ(1−ρ)∥xθ⋆∥2\n2=R(xρθ⋆) +ρ(1−ρ)x2∥θ⋆∥2\n2,\nusing Lemma C.1. Then,\nR⋆\nimp(d)−R⋆(d)≤R(xρθ⋆)−R⋆(d) +ρ(1−ρ)x2∥θ⋆∥2\n2.\nNote that,\nR(xρθ⋆)−R⋆(d) =∥xρθ⋆−θ⋆∥2\nΣ\n= (1−xρ)2∥θ⋆∥2\nΣ.\nThus, we have,\nR⋆\nimp(d)−R⋆(d)≤(1−xρ)2∥θ⋆∥2\nΣ+x2ρ(1−ρ)∥θ⋆∥2\n2. (31)\n(Second step) Calculus of E∥θ⋆∥2\n2.Since θ⋆= (W⊤)†β,\nE∥θ⋆∥2\n2=Eβ⊤(W†(W⊤)†)β.\nAgain, for an orthonormal matrix O,OW†(W⊤)†O⊤= (WO⊤)†((WO⊤)⊤)†has the\nsame law as that of W†(W⊤)†because WO⊤has the same law of W. Using Lemma B.4,\nE∥θ⋆∥2\n2=∥β∥2\n2ETr\u0000\n(W†)⊤W†\u0001\np\n=∥β∥2\n2E∥W†∥2\nFr\np,\nby definition of the Frobenius norm. Then, by Lemma B.5, we obtain\nE∥θ⋆∥2\n2=d\np\u0012\n1 +d\np−d−1\u0013\n∥β∥2\n2. (32)\n(Third step) Calculus of E∥θ⋆∥2\nΣ.We have by optimization result,\nEY2=∥θ⋆∥2\nΣ+R⋆(d).\nUsing that, EY2=σ2+∥β∥2\n2, and taking the expectation, we obtain\nσ2+∥β∥2\n2=E∥θ⋆∥2\nΣ+ER⋆(d).\nFurthermore, by Proposition 2.1,\nER⋆(d) =σ2+p−d\np∥β∥2\n2.\n28Thus, we obtain,\nE∥θ⋆∥2\nΣ=d\np∥β∥2\n2. (33)\n(Fourth step) Conclusion. Putting things together, one gets\nE\u0002\nR⋆\nimp(d)−R⋆(d)\u0003\n≤(1−xρ)2E∥θ⋆∥2\nΣ+x2ρ(1−ρ)E∥θ⋆∥2\n2\n=d\np∥β∥2\n2\u0012\n(1−xρ)2+x2(1−ρ)ρ\u0012\n1 +d−1\np−d−1\u0013\u0013\n.\nThe bound on the right hand side can be optimized with respect to x. It corresponds to a\nstrongly convex function of the form f:x7−→(1−ax)2+bx2. We have f′(x) =−2a(1−\nax) + 2bx, so that the only critical point is x⋆=a\na2+b, leading to minf=f(x⋆) =b\na2+b.\nTherefore,\nE\u0002\nR⋆\nimp(d)−R⋆(d)\u0003\n≤d\np∥β∥2\n2(1−ρ)ρ\u0010\n1 +d−1\np−d−1\u0011\nρ2+ (1−ρ)ρ\u0010\n1 +d−1\np−d−1\u0011\n=d\np∥β∥2\n2(1−ρ)\u0010\n1 +d−1\np−d−1\u0011\nρ+ (1−ρ)\u0010\n1 +d−1\np−d−1\u0011\n=d\np∥β∥2\n2(1−ρ) (p−2)\nρ(p−d−1) + (1 −ρ) (p−2)\n=d\np∥β∥2\n2(1−ρ) (p−2)\np−ρ(d−1)−2\n=d\np∥β∥2\n2\u0012\n(1−ρ) + (1 −ρ)(p−2)−(p−ρ(d−1)−2)\np−ρ(d−1)−2\u0013\n= (1−ρ)d\np∥β∥2\n2\u0012\n1 +ρ(d−1)\np−ρ(d−1)−2\u0013\n,\nwhich leads to the desired result. We obtain also (8)and(9)using equality obtain in\nProposition 2.2.\nC.3.3 Proof of upper and lower bounds (10) (11) (d≥p)\nUsing Lemma C.3, we have\n∆imp/miss+ ∆ miss=λ∥β⋆∥2\n2\npETr\u0000\n(W⊤W+λIp)−1\u0001\nRemark that W⊤W=Pd\nj=1WjW⊤\nj. Furthermore, note that when W1∼ U(Sp−1),\nfor any 1 ≤k≤p,W1= (W11, . . . , W 1k, . . . , W 1p)⊤has the same distribution as\n29(W11, . . . ,−W1k, . . . , W 1p)⊤. Therefore, for all 1 ≤k̸=k′≤p,E[W1kW1k′] =−E[W1kW1k′],\nleading to E[W1kW1k′] = 0. Furthermore E[W2\n11+. . .+W2\n1p] =E[W2\n11]+. . .+E[W2\n1p] = 1,\nso that by exchangeability, for all 1 ≤k≤p,E[W2\n1k] = 1/pand finally EW1W⊤\n1=1\npIp.\nApplying Proposition B.3, we obtain\nλ∥β⋆∥2\n2\npp\nd/p+λ≤∆imp/miss+ ∆ miss≤λ∥β⋆∥2\n2\np(1 + 1 /λ)p\nd/p+λ,\nand\n∥β⋆∥2\n2(1−ρ)p\nρd+ (1−ρ)p≤∆imp/miss+ ∆ miss≤ ∥β⋆∥2\n2p\nρd+ (1−ρ)p.\nD Proof of Section 3\nD.1 Proof of Theorem 3.2\nStart by writing\n∆(∞)\nimp=ER⋆\nimp(d)−R⋆(∞)\n= ∆ imp/miss+ ∆ miss+ER⋆(d)−R⋆(∞)\n=Einf\nθ∈Rd\u001a\nR(θ)−R⋆(∞) +1−ρ\nρ∥θ∥2\ndiag(Σ)\u001b\n,\nusing (27). Considering Assumption 4, diag(Σ) ⪯L2Id, that leads to\n∆(∞)\nimp≤Einf\nθ∈Rd\u001a\nR(θ)−R⋆(∞) +L21−ρ\nρ∥θ∥2\n2\u001b\n.\nFixing λ=L21−ρ\nρ, we aim at providing an upper bound for\n∆λ:=Einf\nθ∈Rd\b\nR(θ)−R⋆(∞) +λ∥θ∥2\n2\t\n.\nLetf∈ F(∞)\nνwith one representative α∈L2(ν). We set θ(α)∈Rdsuch that for all\nj∈[d],θ(α)\nj=1\ndα(Wj). We have,\n∆λ+R⋆(∞) =Einf\nθ∈Rd\b\nR(θ) +λE\u0002\n∥θ∥2\n2|W\u0003\t\n≤E\u0014\nR\u0000\nθ(α)\u0001\n+λE\u0014\r\r\rθ(α)\r\r\r2\n2|W\u0015\u0015\n=ER\u0010\nθ(α)\u0011\n+λE\u0014\r\r\rθ(α)\r\r\r2\n2\u0015\n30First term. Remark that by definition of random features (15),X⊤θ(α)=Pd\nj=1θ(α)\njψ(Z, W j) =\n1\ndPd\nj=1α(Wj)ψ(Z, W j). In consequence, Eh\nX⊤θ(α)|Zi\n=R\nα(W)ψ(Z, W )dν(W) =\nf(Z). Then\nER\u0010\nθ(α)\u0011\n=E\u0014\nE\u0014\u0010\nX⊤θ(α)−Y\u00112\n|W\u0015\u0015\n=E\u0014\nE\u0014\u0010\nX⊤θ(α)−Y\u00112\n|Z\u0015\u0015\nusing Fubini’s theorem\n=E\u0014\nE\u0014\u0010\nX⊤θ(α)−f(Z) +f(Z)−Y\u00112\n|Z\u0015\u0015\n=E\u0014\nE\u0014\u0010\nX⊤θ(α)−f(Z)\u00112\n+ (f(Z)−Y)2|Z\u0015\u0015\nusingEh\nX⊤θ(α)|Zi\n=f(Z)\n=Eh\nVh\nX⊤θ(α)|Zii\n+R(f).\nThen,\nVh\nX⊤θ(α)|Zi\n=V\"\n1\nddX\nj=1α(Wj)ψ(Z, W j)|Z#\n=1\ndVν[α(W)ψ(Z, W )|Z] ( Wj) being i.i.d\n≤1\ndEν\u0002\nα(W)2ψ(Z, W )2|Z\u0003\n.\nThen using Fubini’s theorem,\nVh\nX⊤θ(α)i\n≤1\ndE\u0002\nE\u0002\nα(W)2ψ(Z, W )2|W\u0003\u0003\n=1\ndE\u0002\nα(W)2E\u0002\nψ(Z, W )2|W\u0003\u0003\n.\nUnder Assumption 4,\nVh\nX⊤θ(α)i\n≤L2\ndE\u0002\nα(W)2\u0003\n.\nThus,\nER\u0010\nθ(α)\u0011\n≤L2\ndEν\u0002\nα(W)2\u0003\n+R(f).\nSecond term.\nE\u0014\r\r\rθ(α)\r\r\r2\n2\u0015\n=E\"dX\nj=1θ2\nj#\n=1\ndE\"\n1\nddX\nj=1α(Wj)2#\n=1\ndE\u0002\nα(W)2\u0003\n,\n31using that ( Wj)jare i.i.d..\nConclusion. Combining these two terms, we have\n∆λ+R⋆(∞)≤R(f) +λ+L2\ndEν\u0002\nα(W)2\u0003\n.\nThis result is valid for any fandα, thus\n∆(∞)\nimp+R⋆(∞)≤∆λ+R⋆(∞)≤inf\nf∈F(ν)\u001a\nR(f) +λ+L2\nd∥f∥2\nν\u001b\n.\nE Proof of error bounds for SGD estimators\nE.1 Proof of equation (13)of Theorem 2.4\nIn this section, we apply results from the SGD literature, in particular, Bach and\nMoulines (2013, Theorem 1), to our framework.\nTheorem E.1. In the framework of Section 2, for γ=1\n6d, and when d < n , we have\nRimp(¯θ)−R⋆\nimp(d)≲d\nn∥θ⋆\nimp∥2\n2+d\nnσ2\nimp, (34)\nwithσ2\nimp:=σ2+ρ−1(R⋆\nimp(d)−R⋆(d) +∥θ⋆∥2\nΣ).\nProof. The proof of this theorem consists of verifying that assumptions of Bach and\nMoulines (2013, Theorem 1) hold in our case. Assumptions (A1-5) are easily satisfied.\nLet us show that E\u0014\n˜X˜X⊤\r\r\r˜X\r\r\r2\n2|W\u0015\n⪯R2Σimp. Indeed,\nE\u0014\n˜X˜X⊤\r\r\r˜X\r\r\r2\n2|W\u0015\n⪯Eh\n˜X˜X⊤∥X∥2\n2|Wi\n,\nusing that\r\r\r˜X\r\r\r2\n2≤ ∥X∥2\n2, and 0 ⪯˜X˜X⊤. Then,\nEh\n˜X˜X⊤∥X∥2\n2|Wi\n=EEh\n˜X˜X⊤∥X∥2\n2|P,Wi\n=EE\u0002\nPP⊤⊙XX⊤∥X∥2\n2|P,W\u0003\n=E\u0002\nΣP⊙XX⊤∥X∥2\n2|W\u0003\n= Σ P⊙\u0000\nE\u0002\nXX⊤∥X∥2\n2|W\u0003\u0001\n.\nXis Gaussian vector, thus E\u0002\nXX⊤∥X∥2\n2\u0003\n⪯R2Σ with R2= 3d, and Lemma B.6\nlead to\nE\u0014\n˜X˜X⊤\r\r\r˜X\r\r\r2\n2\u0015\n⪯R2ΣP⊙Σ =R2Σimp.\n32Define ϵimp=Y−˜X⊤θ⋆\nimp=X⊤θ⋆+ϵ−˜X⊤θ⋆\nimp. First, we have ϵ2\nimp≤3\u0012\nϵ2+\u0010\n˜X⊤θ⋆\nimp\u00112\n+\u0000\nX⊤θ⋆\u00012\u0013\n,\nthen\nEh\nϵ2\nimp˜X˜X⊤i\n⪯3Eh\nϵ2˜X˜X⊤i\n+ 3E\u0014\u0010\n˜X⊤θ⋆\nimp\u00112˜X˜X⊤\u0015\n+ 3E\u0014\u0010\n˜X⊤θ⋆\u00112˜X˜X⊤\u0015\n.(35)\nUsing that ϵis an independent noise, Eh\nϵ2˜X˜X⊤i\n=σ2Σimp. Let u, vinRd, note that\nv⊤E\u0014\u0010\nu⊤˜X\u00112˜X˜X⊤\u0015\nv=E\u0014\u0010\nu⊤˜X\u00112\u0010\nv⊤˜X\u00112\u0015\n≤E\u0014\u0000\nu⊤X\u00012\u0010\nv⊤˜X\u00112\u0015\n≤ρEh\u0000\nu⊤X\u00012\u0000\nv⊤X\u00012i\n≤ρr\nEh\n(u⊤X)4i\nEh\n(v⊤X)4i\n,\nusing the Cauchy-Schwarz inequality. Then, by the kurtosis boundedness of Gaussian\nvectors, we have Eh\u0000\nu⊤X\u00014i\n≤3∥u∥4\nΣandEh\u0000\nv⊤X\u00014i\n≤3∥v∥4\nΣ. Then,\nv⊤E\u0014\u0010\nu⊤˜X\u00112˜X˜X⊤\u0015\nv= 3ρ∥u∥2\nΣ∥v∥2\nΣ\n≤3ρ−1∥u∥2\nΣ∥v∥2\nΣimp.\nThis shows that\nE\u0014\u0010\nu⊤˜X\u00112˜X˜X⊤\u0015\n⪯3ρ−1∥u∥2\nΣΣimp. (36)\nUsing similar arguments,\nEh\u0000\nu⊤X\u00012˜X˜X⊤i\n⪯3ρ−1∥u∥2\nΣΣimp. (37)\nThese two above equations can be used when uis equal to θ⋆\nimpandθ⋆, to transform\n(35) into\nEh\nϵ2\nimp˜X˜X⊤i\n⪯(3σ2+ 9ρ−1\r\rθ⋆\nimp\r\r2\nΣ+ 9ρ−1∥θ⋆∥2\nΣ)Σimp.\nRemarking, that\r\rθ⋆\nimp\r\r2\nΣ≤2\r\rθ⋆\nimp−θ⋆\r\r2\nΣ+ 2∥θ⋆∥2\nΣ= 2(R⋆\nimp(d)−R⋆(d) +∥θ⋆∥2\nΣ), we\nget\n3σ2+ 9ρ−1\r\rθ⋆\nimp\r\r2\nΣ+ 9ρ−1∥θ⋆∥2\nΣ≲σ2+ρ−1(R⋆\nimp(d)−R⋆(d) +∥θ⋆∥2\nΣ),\nleading to the desired results.\n33Lemma E.2. Under assumptions of Theorem 2.4. The norm of θ⋆\nimp, the best predictor\nworking with imputed by 0inputs, satisfies\nE∥θ⋆\nimp∥2≤(\nd\npρp−2\np−d−1∥β∥2\n2when d < p−1,\np\ndρ2(1−ρ)∥β∥2\n2,when d≥p−1.\nProof. Let’s begin by,\nρ(1−ρ)∥θ⋆\nimp∥2\n2≤R(ρθ⋆\nimp)−R(ρθ⋆) +ρ(1−ρ)∥θ⋆\nimp∥2\n2. (38)\nbecause R(θ⋆)≤R(ρθ⋆\nimp). Using, Lemma C.1, we obtain\nρ(1−ρ)E∥θ⋆\nimp∥2\n2≤∆miss+ ∆ imp/miss (39)\nFirst case: d < p−1.In this, case\n∆miss+ ∆ imp/miss≤(1−ρ)d\np∥β∥2\n2\u0012\n1 +ρ(d−1)\np−ρ(d−1)−2\u0013\n.\nWe obtain using (39),\nE∥θ⋆\nimp∥2\n2≤d\nρp∥β∥2\n2\u0012\n1 +ρ(d−1)\np−ρ(d−1)−2\u0013\n≤p−2\nρp∥β∥2\n2ρ(d−1)\np−ρ(d−1)−2.\nSecond case: d > p−1.In this case,\n∆miss+ ∆ imp/miss≤p\nρd+ (1−ρ)p∥β⋆∥2\n2≤p\nρd∥β⋆∥2\n2.\nWe obtain using (39),\nE∥θ⋆\nimp∥2\n2≤p\nρ2(1−ρ)d∥β⋆∥2\n2.\nProof of (13).\nRimp(¯θ)−R⋆(d) =Rimp(¯θ)−R⋆\nimp(d) +R⋆\nimp(d)−R⋆(d)\nUsing Theorem E.1 to bound the first term, we find\nRimp(¯θ)−R⋆(d)≲\u0012\n1 +d\nρn\u0013\n(R⋆\nimp(d)−R⋆(d)) +d\nn∥θ⋆\nimp∥2\n2+d\nn(σ2+∥θ⋆∥2\nΣ).\nThus, taking the expectation,\nE[Rimp(¯θ)−R⋆(d)]≲\u0012\n1 +d\nρn\u0013\n(∆imp/miss+ ∆ miss) +d\nnE∥θ⋆\nimp∥2\n2+d\nn(σ2+E∥θ⋆∥2\nΣ).\n34Note that E∥θ⋆\nimp∥2\n2≤1\n(1−ρ)ρ(∆imp/miss+ ∆ miss) (using (39)) and E∥θ⋆∥2\nΣ=d\np∥β⋆∥2\n2\n(using Proposition 2.1). Thus,\nE[Rimp(¯θ)−R⋆(d)]≲\u0012\n1 +d\nρn+d\n(1−ρ)ρn\u0013\n(∆imp/miss+ ∆ miss) +d\nn\u0012\nσ2+d\np∥β⋆∥2\n2\u0013\n≲\u0012\n1 +d\n(1−ρ)ρn\u0013\n(∆imp/miss+ ∆ miss) +d\nn\u0012\nσ2+d\np∥β⋆∥2\n2\u0013\n.\nThus,\nE[Rimp(¯θ)−R⋆\nmiss(d)]≲∆imp/miss+d\nρn+d\n(1−ρ)ρn(∆imp/miss+ ∆ miss) +d\nn\u0012\nσ2+d\np∥β⋆∥2\n2\u0013\n≲(1−ρ)d\np∥β∥2\n2ρ(d−1)\np−ρ(d−1)−2+d\n(1−ρ)ρnd\np∥β∥2\n2(1−ρ) (p−2)\np−ρ(d−1)−2+d\nn\u0012\nσ2+d\np∥β⋆∥2\n2\u0013\n≲(1−ρ)d\np∥β∥2\n2ρ(d−1)\np−ρ(d−1)−2+d\n(1−ρ)ρnd\np∥β∥2\n2(1−ρ) (p−2)\np−ρ(d−1)−2+d\nnσ2\n≲d2∥β∥2\n2\np(p−ρ(d−1)−2)\u0012\n(1−ρ)ρ+p−2\nρn\u0013\n+d\nnσ2.\nE.2 Proof of Equation (14)in Theorem 2.4 and proof of Theo-\nrem 3.3\nLet’s start with a result of Ayme et al. (2023) for the deterministic case (without random\nfeatures).\nAssumption 9. There exist σ >0andR > 0such that E[XX⊤∥X∥2\n2]⪯R2Σand\nE[ϵ2∥X∥2\n2]≤σ2R2, where ϵ=Y−X⊤θ⋆.\nTheorem E.3. (Ayme et al., 2023) Under Assumption 9, choosing a constant learning\nrateγ=1\nκTr(Σ)√nleads to\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−R(θ⋆)≲R2\n√n\r\rθ⋆\nimp\r\r2\n2+σ2+∥θ⋆∥2\nΣ√n+R⋆\nimp(d)−R⋆(d),\nwhere θ⋆(resp. θ⋆\nimp) is the best linear predictor for complete (resp. with imputed missing\nvalues) case.\nProof of (14) of Theorem 2.4. Xis Gaussian vector, thus E\u0002\nXX⊤∥X∥2\n2\u0003\n⪯R2Σ with\nR2= 2d. Furthermore, in the cass p≤d X⊤θ⋆=Z⊤β⋆andϵ=Y−X⊤thus the noise\nandXare independent and E[ϵ2∥X∥2\n2]≤σ2Tr(Σ). Then, Assumption 9 is satisfied\nwith κ= 3. Taking the expectation in Theorem E.3, we found\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−ER(θ⋆)≲Tr(Σ)√nE\r\rθ⋆\nimp\r\r2\n2+σ2+E∥θ⋆∥2\nΣ√n+ ∆ imp/miss+ ∆ miss,\n35Let’s recall that, R(θ⋆) =σ2almost-surely, ∥θ⋆∥2\nΣ=∥β∥2\n2almost surely and Tr(Σ) = d.\nThen,\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−σ2≲d√nE\r\rθ⋆\nimp\r\r2\n2+σ2+∥β∥2\n2√n+ ∆ imp/miss+ ∆ miss.\nThen applying (39),\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−σ2≲\u0012\n1 +d\nρ(1−ρ)√n\u0013\n(∆imp/miss+ ∆ miss) +σ2+∥β∥2\n2√n.\nApplying (11), we finally found,\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−σ2≲\u0012\n1 +d\nρ(1−ρ)√n\u0013p∥β⋆∥2\n2\nρd+ (1−ρ)p+σ2+∥β∥2\n2√n\n≲\u0012\n1 +d\nρ(1−ρ)√n\u0013p∥β⋆∥2\n2\nρd+ (1−ρ)p+σ2\n√n.\nProof of Theorem 3.3. Under Assumption 7, ∥X∥2\n2≤κL2dalmost surely, then E[XX⊤∥X∥2\n2]⪯\nκL2dΣ. And,\nE[ϵ2∥X∥2\n2]≤E[ϵ2]κL2d=R⋆(∞)κL2d.\nThus we can applied Theorem E.3, that gives us,\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−R(θ⋆)≲κL2d√n\r\rθ⋆\nimp\r\r2\n2+R⋆(∞)Lκ+∥θ⋆∥2\nΣ√n+R⋆\nimp(d)−R⋆(d).\nNote that,\n∆(∞)\nimp=ER⋆\nimp(d)−R⋆(∞)\n= ∆ imp/miss+ ∆ miss+ER⋆(d)−R⋆(∞).\nThus taking the expectation,\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−R⋆(∞)≲κL2d√nE\r\rθ⋆\nimp\r\r2\n2+R⋆(∞)κL2+∥θ⋆∥2\nΣ√n+ ∆(∞)\nimp.\nUnder Assumption 6, l2I⪯diag(Σ),\nℓ2ρ(1−ρ)E\r\rθ⋆\nimp\r\r2\n2≤ρ(1−ρ)E\r\rθ⋆\nimp\r\r2\ndiag(Σ)\n≤∆imp/miss+ ∆ miss\n≤∆(∞)\nimp.\n36Then,\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−R⋆(∞)≲κL2d\nℓ2ρ(1−ρ)√n∆(∞)\nimp+R⋆(∞)κ+∥θ⋆∥2\nΣ√n+ ∆(∞)\nimp\n≲\u0012\n1 +κL2d\nℓ2ρ(1−ρ)√n\u0013\n∆(∞)\nimp+R⋆(∞)κL2+∥θ⋆∥2\nΣ√n.\nRecall that using Theorem 3.2,\n∆(∞)\nimp≤λimp\nd∥f⋆∥2\nν=L2\nρd∥f⋆∥2\nν,\nand∥θ⋆∥2\nΣ≤EY2almost-surely. Thus\nE\u0002\nRimp\u0000¯θimp\u0001\u0003\n−R⋆(∞)≲\u0012\n1 +κL2d\nℓ2ρ(1−ρ)√n\u0013l2\nρd∥f⋆∥2\nν+R⋆(∞)κ+EY2\n√n\n≲\u0012\n1 +κL2d\nℓ2ρ(1−ρ)√n\u0013L2\nρd∥f⋆∥2\nν+ (1 + κL2)EY2\n√n.\nFProof of Theorem 4.1 (under MNAR assumption)\nFirst step (bias-variance) We denote by W′the matrix of W′\njLetθ∈Rp,\nRimp(θ) =EZE\u0014\u0010\nY−˜X⊤θ\u00112\n|Z,W,W′\u0015\n=EZE\u0014\u0010\nY−Eh\n˜X⊤θ|Z,W,W′i\u00112\n|Z,W,W′\u0015\n+EZVh\n˜X⊤θ|Z,W,W′i\n,\nusing bias-variance decomposition. Futhermore,\nEh\n˜X⊤θ|Z,W,W′i\n=dX\nj=1θjEh\n˜Xj|Z,W,W′i\n=dX\nj=1θjϕ(Z, W′\nj)ψ(Z, W j)\nand\nVh\n˜X⊤θ|Z,W,W′i\n=dX\nj=1θ2\njVh\n˜Xj|Z, W j, W′\nji\n=dX\nj=1θ2\njϕ(Z, W′\nj)(1−ϕ(Z, W′\nj))ψ(Z, W j)2.\n37Letα∈L2(µ⊗ν), and define θ(d)∈Rdsuch that θ(d)\nj=α(Wj, W′\nj)/d. We have\nR⋆\nimp(d)≤Rimp(θ(d))\n=EZ\n \nY−1\nddX\nj=1α(Wj, W′\nj)ϕ(Z, W′\nj)ψ(Z, W j)!2\n+EZ\"\n1\nd2dX\nj=1α(Wj, W′\nj)2ϕ(Z, W′\nj)(1−ϕ(Z, W′\nj))ψ(Z, W j)2#\n.\nConvergence of variance term Using that ϕ(Z, W′\nj)(1−ϕ(Z, W′\nj))≤1 almost-surely,\nwe have\nEZ\"\n1\nd2dX\nj=1α(Wj, W′\nj)ϕ(Z, W′\nj)(1−ϕ(Z, W′\nj))ψ(Z, W j)2#\n≤EZ\"\n1\nd2dX\nj=1α(Wj, W′\nj)2ψ(Z, W j)2#\n=1\nd2dX\nj=1α(Wj, W′\nj)2EZψ(Z, W j)2\n≤1\nd2dX\nj=1α(Wj, W′\nj)2L2.\nUsing that\u0000\nα(Wj, W′\nj)2\u0001\njare an i.i.d. sequences of integrable random variables, we\nobtain that\nlim\nd→+∞EZ\"\n1\nd2dX\nj=1α(Wj, W′\nj)ϕ(Z, W′\nj)(1−ϕ(Z, W′\nj))ψ(Z, W j)2#\n= 0,\nalmost-surely.\nConvergence of bias term α(Wj, W′\nj)ϕ(Z, W′\nj)ψ(Z, W j) is integrable because\n|α(Wj, W′\nj)ϕ(Z, W′\nj)| ≤ |α(Wj, W′\nj)|, and ψ(z, W j)∈L2(ν). Then, using Kolmogorov’s\nlaw and mapping continuous theorem, we obtain\nlim\nd→+∞EZ\n \nY−1\nddX\nj=1α(Wj, W′\nj)ϕ(Z, W′\nj)ψ(Z, W j)!2\n=EZ\"\u0012\nY−Z\nα(w, w′)ϕ(Z, w′)ψ(Z, w)dµ⊗ν(w, w′)\u00132#\n.\nThus we obtain,\nlim sup\nd→+∞R⋆\nimp(d)≤EZ\"\u0012\nY−Z\nα(w, w′)ϕ(Z, w′)ψ(Z, w)dµ⊗ν(w, w′)\u00132#\n.\nDenoting by G, the functions of the form\ng(Z) =Z\nα(w, w′)ϕ(Z, w′)ψ(Z, w)dµ⊗ν(w, w′),\n38we obtain that\nlim sup\nd→+∞R⋆\nimp(d)≤inf\ng∈GEZ\u0002\n(Y−g(Z))2\u0003\n.\nUsing Fubini theorem, functions of the form\ng(Z) =Z\nα(w)ψ(Z, w)dν(w)Z\nβ(w′)ϕ(Z, w′)dµ(w′)\n=f(Z)h(Z),\nare include in G. For the following, we denote by Hthe set of functions, of the form\nh(Z) =Z\nβ(w′)ϕ(Z, w′)dµ(w′),\nwith β∈L2(µ). Thus we obtain, the following bound,\nlim sup\nd→+∞R⋆\nimp(d)≤ inf\n(f,h)∈Fν×GEZ\u0002\n(Y−f(Z)h(Z))2\u0003\n. (40)\nProof for the case Zcompact, Fdense in continuous function, and\nf⋆continuous First, let’s show that the risk is continuous for the set of continuous\nprediction functions. Let f, gtwo continuous function on B∞(0, B) (ball for ∥∥∞), then\n|R(f)−R(g)|=\f\fE\u0002\n(f(Z)−f⋆(Z))2−(g(Z)−f⋆(Z))2\u0003\f\f\n=|E[(f(Z)−g(Z))(g(Z) +f(Z)−f⋆(Z))]|\n≤ ∥f−g∥∞(∥f∥∞+∥g∥∞+∥f⋆∥∞)\n≤ ∥f−g∥∞(2B+∥f⋆∥∞).\nThat concludes that risk is continuous. Then, we consider β= 1,thus h(Z) =R\nβ(w′)ϕ(Z, w′)dµ(w′)>0 almost-surely because ϕ(Z, w′)>0 almost surely. Thus\nconsidering in (40),f(Z) =f⋆(Z)/h(Z),fis continuous, we conclude using continuity\nof risk and f∈¯F.\nProof for Gaussian RF In this case ϕ(z,(w, w′\n0)) = Φ( z⊤w′+w′\n0), consider I=R\nΦ(w′\n0)dµ. We consider hϵ(z) =R\n∥w′∥≤ϵϕ(Z,(w′, w′\n0)dµ(w′, w0)/R\n∥w′∥≤ϵΦ(w′\n0)dµ(w′, w′\n0),\n|I−hϵ(Z)| ≤1R\n∥w′∥≤ϵΦ(w′\n0)dµ(w′)Z\n∥w′∥≤ϵ|ϕ(0,(0, w′\n0))−ϕ(Z,(w′, w0))|dµ(w′, w′\n0).\nUsing that that Φ is CLipschitz,\n|I−hϵ(Z)| ≤Cϵ∥Z∥.\nWe conclude using h=hϵfor a decreasing sequence of ϵandf=f⋆/Iin (40).\n39"    },    {        "title": "2402.03844v1.A_Unified_Model_for_Non_Fickian_Diffusion_and_Anomalous_Swelling_of_Glassy_Polymers.pdf",        "content": "A Unified Model for Non-Fickian Diffusion and Anomalous Swelling of Glassy\nPolymers\nPeihan Lyu,1Zhaoyu Ding,1Masao Doi,1, 2,∗and Xingkun Man1, 3,†\n1School of Physics, Beihang University, Beijing 100191, China\n2Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou 325000, China\n3Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China\nA sheet of glassy polymers placed in solvent shows swelling behaviors quite different from that\nof soft polymers (rubbers and gels). (1) Non-Fickian diffusion (called case II diffusion): As solvent\npermeates into the sample, a sharp front is created between the swollen part and the glassy part,\nand it moves towards the center at constant speed. (2) Non-monotonous swelling: The thickness\nof the sample first increases and then decreases towards the equilibrium value. Here, we propose\na unified theory for the swelling of glassy polymers. The theory regards the glassy polymers as a\ncontinuous mixture of viscoelastic network and solvent, where the relaxation time of polymer varies\nenormously depending on solvent concentration. We will show that this theory explains the above\ntwo characteristics of such process in a simple and unified framework. The theory predicts how the\npermeation speed of solvent and the characteristic times of the swelling process depend on material\nparameters and experimental conditions, which can be checked experimentally.\nWhen a thin sample of polymer is placed in a bath of\nsolvent, the solvent permeates into the polymer and in-\ncreases the volume of the sample. In soft polymers (rub-\nbers and gels), the thickness and the area of the sample\nincrease monotonically in time t, following the Fickian\ndiffusion law (in proportion to t1/2) and reach the equi-\nlibrium value. On the other hand, in glassy polymers,\nthe swelling behavior is completely different [1–6]. In the\nclassical experiment, Thomas and Windle [1] reported\nthe following anomalies in the swelling of PMMA(poly-\nmethyl-methaclylate) sheet in methanol (see Fig.1). (a)\nAt first, as the solvent permeates into the polymer, a\nsoft rubbery part is created at the surface of the polymer\nand moves towards the center. During this process, the\ninterface between the rubbery part and the glassy part\nremains sharp, and it moves at constant speed. At this\nstage, the sample thickness increases in time in propor-\ntion to t, while the sample width remains constant. (b)\nAfter some time, the two fronts created at both surfaces\nmeet at the center and the whole polymer becomes rub-\nbery. At this stage, the thickness decreases in time while\nthe width increases towards their equilibrium values.\nThe non-Fickian diffusion dynamics characterizing the\nfirst stage is observed for many glassy polymers, and is\ncalled case II diffusion [7–12]. The case II diffusion is\ncharacterized by the existence of sharp front of swollen\nregion that moves at constant speed. Such behavior is\nconsidered to be caused by the glass transition of poly-\nmers induced by solvent. Numerous theories have been\nproposed to describe the case II diffusion [13–17]. How-\never, there is still no consensus or clear understanding on\nhow the case II behavior is caused by the glass transition.\nAll previous theories have a common drawback that\nthey only discuss the one dimensional problem of diffu-\nsion in the first stage [18–26]. Such theory cannot discuss\nthe three dimensional problem of shape change in the\nsecond stage and how the first stage transforms to thesecond stage. To deal with the second stage, separate\ntheory is needed.\nIn this paper, we propose a three dimensional theory\nfor the swelling of glassy polymers. We regard the glassy\npolymer as a viscoelastic solid and discuss its swelling\nusing the same framework as that for the soft polymer\ngels [27–29]. The glass transition is taken into account by\nassuming that the kinetic parameters (solvent diffusion\nconstant and polymer relaxation time) change drastically\nat a certain solvent concentration. We shall show that\nthis simple model reproduces the two anomalies, the case\nII diffusion and the non-monotonous shape change with-\nout introducing any additional assumptions.\nWe consider a square sheet of glassy polymers that\nTimeFickian\nExpansion\n(iii)Case II\nDiffusion\n(ii) (i)(ii) (iii)\ntSolvent profile(a)\n(c)(i)\nTimeTimeL02h0\nInduction(b)\nrubbery glassy\nFIG. 1. (a) Schematic of the swelling of a square sheet of\nglassy polymers in a bath of solvent. The sheet initially has\nside length L0and thickness 2 h0(h0≪L0). As solvent per-\nmeates, sharp front is formed between outer rubbery part\n(blue) and inner glassy part (black). After some time, the\nentire sample becomes rubbery, relaxing towards the equilib-\nrium state. (b) Time evolution of thickness (top) and area\n(bottom) show three stages. (i) Induction stage (black): both\nthickness and area are nearly unchanged. (ii) Case II diffu-\nsion stage (red): thickness increases in time in proportion to\nt, while area nearly remains constant. (iii) Fickian expansion\nstage (blue): thickness decreases but area increases towards\nthe equilibrium value. (c) Time evolution of the solvent pro-\nfile along z-axis in the three stages.arXiv:2402.03844v1  [cond-mat.soft]  6 Feb 20242\nis initially at equilibrium with thickness 2 h0and side\nlength L0(h0≪L0), (see Fig.1). We take a Cartesian\ncoordinate such that the right and the left surfaces are\natz=±h0. Suppose that a point located at ron the\npolymer network in the dry state is displaced to r+u(r, t)\nat time t. We assume that the displacement gradient\nεαβ=∂uα/∂rβ(α, βstand for x, y, z ) is small, and that\nthe swelling takes place uniformly and isotropically in\nx-yplane. The displacement vector can be written as\nux(t) =α(t)x,uy(t) =α(t)y, and uz=u(z, t), where\nα(t) and u(z, t) are the unknowns to be obtained.\nFor small deformation, the polymer velocity vpatr\nis given by the time derivative of u(r, t), i.e. vp=˙u.\nThe solvent velocity vsis different from vpand the dif-\nference represents the diffusion (or permeation) of sol-\nvent in polymer network. We assume that the system is\nincompressible, and that initially there is no solvent in\nsample. This allows us to write the volume fraction of\nsolvent ϕsin terms of the volume expansion of polymer\nnetwork as ϕs=∇ ·u=∂u/∂z + 2α. Therefore, the\nstate of the sample at time tis completely characterized\nbyu(z, t) and α(t).\nTo determine the time evolution of u(z, t) and α(t),\nwe use Onsager principle, which states that their evolu-\ntion rate ˙ uand ˙αare determined by the minimum of the\nRayleighian function defined by ℜ=˙F+ Φ. Here, ˙Fis\nthe time derivative of free energy F, and Φ is the energy\ndissipation function of the system [30, 31].\nThe free energy Fincludes elastic energy of polymer\nnetwork and mixing energy of polymer and solvent. For\nsmall deformation, the free energy is given in the same\nform as in soft gels [28, 29]\nF\nL2\n0=Zh0\n0dz\"\nK\n2\u0012\n2α+∂u\n∂z−3εeq\u00132\n+2G\n3\u0012∂u\n∂z−α\u00132#\n(1)\nwhere εeqis the equilibrium strain, and KandGare\nthe material constants called osmotic bulk modulus and\nshear modulus, respectively.\nThe energy dissipation arises from two parts: (a) Φ dif\ndue to the friction between polymer and solvent. This\nis the same as that for soft gels Φ dif=R\ndrξ\n2(vp−vs)2,\nwhere ξis the friction coefficient. (b) Φ rhedue to the\nviscosity of polymer η. This term is negligible in soft\npolymers, but becomes very important in glassy poly-\nmers. In glassy polymers, the material transforms from\nhard glassy state to soft rubbery state across the glass\ntransition, and this is due to the viscoelastic effect of\nthe polymers. We therefore model the glassy polymer by\nthe standard Kelvin-Voigt model for viscoelastic solid.\nThe energy dissipation function of this model is given\nby Φ rhe=R\ndrη\n4P\nαβ\u0010\n˙εαβ+ ˙εβα−2\n3δαβP\nγ˙εγγ\u00112\n[32].\nHere ηis a material parameter called internal viscosity,\nwhich is defined by the rheological relaxation time τrhe\nasη=Gτrhe.\n(c)tmeet tfin tind(a)\n(b)\nFIG. 2. (a) Time evolution of the solvent volume fraction\nϕs(z, t) in the swelling process. (b) Zoomed-in view of (a)\nin the region near the surface at early time of permeation.\n(c) Time dependence of the front position hc(t), indicating\nthree characteristic times: (i) tind:hcremains unchanged\nat the surface, (ii) tmeet:hcmoves towards the center at a\nnearly constant speed, and (iii) tfin:hcbecomes equal to zero.\nAll times are scaled by τdif=h2\n0/Dr. Other parameters are\nξg/ξr= 5000, ηg/ηr= 100 and ηr/ξrh2\n0= 0.05.\nWe take the conventional view on the glass transi-\ntion that it is a kinetic transition, and assume that the\nkinetic parameters ξandηchange by orders of mag-\nnitude at some characteristic solvent volume fraction\nϕcri. Specifically, we assume that ξandηdepend on\nϕsasξ(ϕs) = ξr+1\n2(ξg−ξr)\u0010\n1 + tanhϕcri−ϕs\nϕw\u0011\nand\nη(ϕs) =ηr+1\n2(ηg−ηr)\u0010\n1 + tanhϕcri−ϕs\nϕw\u0011\n, where ξgand\nηg(ξrandηr) are the friction coefficient and viscosity in\nthe glassy state (rubbery state), and ϕwis the width of\nthe glass transition region. On the other hand, we assume\nthat the equilibrium quantities KandGare constants\nindependent of ϕs.\nFor the present problem, the energy dissipation func-\ntion is given by Φ = Φ dif+ Φ rhe, and is written as [32]\nΦ\nL2\n0=Zh0\n0dz\"\nξ(ϕs)\n2( ˙u+ 2 ˙αz)2+2η(ϕs)\n3\u0012∂˙u\n∂z−˙α\u00132#\n(2)\nThe Rayleighian of the system is therefore given by\nℜ= Φ + ˙F. Then, the time evolution of u(z, t) and α(t)\nare determined by the condition that δℜ/δ˙u=δℜ/δ˙α=\n0, which gives [32]\nξ( ˙u+ 2 ˙αz) = (K+4\n3G)∂2u\n∂z2+4\n3∂\n∂z\u0014\nη\u0012∂˙u\n∂z−˙α\u0013\u0015\n(3)\nG(u−αh0) +Zh0\n0η\u0012∂˙u\n∂z−˙α\u0013\ndz= 0 (4)3\nand the boundary condition at z=h0\n4\n3η\u0010\n˙ϕs−3 ˙α\u0011\n+K(ϕs−3εeq) +4\n3G(ϕs−3α) = 0 (5)\nEquations (3) and (4) are the set of equations which de-\nscribe the swelling process of glassy polymers.\nEquation (3) indicates that the evolution of sys-\ntem is determined by the balance of two terms, the\ndiffusion term ( K+4\n3G)∂ϕs\n∂zand the viscosity term\n4\n3∂\n∂zh\nη(˙ϕs−3 ˙α)i\n. The diffusion pushes solvent towards\nglassy region, but the polymer viscosity hinders such per-\nmeation. The balance of these two terms gives the sharp\nfront of solvent.\nThe boundary condition (5) is automatically derived\nfrom the variational calculus. It represents the condition\nthat the normal stress acting on the polymer network at\nz=h0is zero. This boundary condition is quite different\nfrom the usual Neuman condition for soft gels [28, 29],\nand becomes very important in subsequent discussion,\nin determining the swelling speed. The other boundary\ncondition is u(0, t) = 0 at z= 0.\nWe solved eqs. (3)-(5) numerically, and calculated the\ntime variation of the solvent volume fraction ϕs(z, t) =\n∂u/∂z + 2α, the thickness ∆ h(t) =u(h0, t) and the area\n2L2\n0α(t), separately.\nFigure 2(a) shows the time dependence of the solvent\nvolume fraction ϕs(z, t). All times are scaled by the dif-\nfusion time τdif=h2\n0/Dr, where Dr= (K+4\n3G)/ξris\nthe diffusivity of solvent in rubbery polymers [33]. It is\nseen that the characteristic behavior of the case II diffu-\nsion is reproduced by the present theory: a sharp front\nis formed between the glassy region and the rubbery re-\ngion, and the front moves at constant speed keeping its\nshape constant.\nSuch behaviors can be understood by looking at the\nsolvent profile at an early stage shown in Fig. 2(b). It\nis seen that the solvent permeates into the glassy region\nobeying the Fickian diffusion. The permeation speed is\ngiven by the collective diffusion constant in glassy state\nDg= (K+4\n3G)/ξg. Since ξgis very large, the pro-\nfile changes very slowly during this period. We call this\nperiod induction period. The induction period persists\nuntil ϕsat the surface becomes equal to ϕcri.\nOnce the solvent volume fraction at the surface exceeds\nϕcri, it increases quickly and creates a sharp front in the\nprofile. The plateau value behind the front is close to the\nequilibrium value 3 εeq, but is slightly less than that. This\nis because the center at z= 0 is still in glassy state, and\ntherefore the lateral expansion α(t) remains at the initial\nvalue α(0), and ϕs(z, t) is given by the minimum of the\nfree energy eq. (1) subject to this constraint. Therefore,\nthe front moves slowly at constant speed, leaving the con-\nstrained equilibrium part behind. This is the mechanism\nof case II diffusion.\nWe define the front position hc(t) as the point where\nϕs(z, t) is equal to ϕcri. Figure 2(c) shows the time depen-\n(a) Diffusion -dominated (b) Viscosity -dominated\ntmeet tfin\ntindtindFIG. 3. Time dependence of the two strains ∆ h(t)/h0(black\nsolid line) and α(t) (red solid line) for glassy polymers with\n(a) lower viscosity ( ηr/ξrh2\n0= 0.005) and (b) higher viscosity\n(ηr/ξrh2\n0= 0.05). The top figure shows schematic time evo-\nlution of the apparent shape of the sample. The last stage in\n(a) is diffusion-dominated and in (b) is viscosity-dominated.\nOther parameters are the same as in Fig. 2.\ndence of hc(t). It is seen that apart from the induction\nperiod, the front moves at almost constant speed.\nThe stage of case II diffusion ends when hc(t) becomes\nequal to 0, i.e., when the two fronts meet at the center.\nAt this point, the entire sample becomes rubbery, and\nthe rest process is governed by the relaxation of polymer\nin rubbery state. We call this last relaxation process as\nFickian expansion stage.\nFigure 3 shows the time dependence of the strain nor-\nmal to the surface ∆ h(t)/h0and that parallel to the sur-\nfaceα(t) for two cases: Fig. 3(a) for ηr/ξrh2\n0= 0.005\nand Fig. 3(b) for ηr/ξrh2\n0= 0.05. The difference be-\ntween the two cases is seen in the last stage: In Fig. 3(a),\nthe thickness first decreases and increases again towards\nthe equilibrium value. On the other hand in Fig. 3(b),\nthe thickness decreases monotonically to the equilibrium\nvalue. Such distinctive behaviors were reported in previ-\nous experiments [7], but no explanation has been given.\nOur model explains the reason as follows.\nWhen the two fronts meet at the center, the entire sys-\ntem becomes rubbery, but the system is not at equilib-\nrium since the lateral dimension has been constrained.\nThe free energy expression in eq. (1) indicates that\nfor the system to be at equilibrium, the volume strain\nεz+ 2α−3εeqand the shear strain εz−αhas to relax\nto zero. The relaxation of the volume strain is governed\nby the diffusion of solvent, which is characterized by the\ndiffusion time τdif. On the other hand, the relaxation of\nshear strain is governed by the viscoelasticity of polymer\nin the rubbery state, which is characterized by the rhe-\nological relaxation time τrhe=ηr/G. The ratio of the\ntwo, De=τrhe/τdif, is the Deborah number. If ηris\nsmall, the shear strain εz−α(t) quickly relaxes to zero,\nand the resulting isotropic strain εz=α(t) relaxes to the\nequilibrium value εeqwith diffusion time τdif. This gives\nthe behavior shown in Fig. 3(a). On the other hand, if4\nηris large, εzandα(t) relax to their equilibrium values\nindependently of each other with rheological relaxation\ntime τrhe. This explains the behavior shown in Fig. 3(b).\nFigure 4(a) and (b) show how the induction time tind\nand the front velocity vfrontdepend on material param-\neters. The induction time is the time at which case II\ndiffusion starts. An estimation for tindis obtained from\nthe boundary condition eq. (5). During the induction\nperiod, the entire sample is in glassy state. Therefore we\nmay set, η=ηg,α=ϕ0/3 (ϕ0being the initial value of\nϕs(z, t), which is assumed to be 0 in the present context,\nbut is introduced here for the purpose of later discus-\nsion) and ˙ α= 0. The boundary condition then reduces\nto4\n3ηg˙ϕs+K(ϕs−ϕeq)+4\n3G(ϕs−ϕ0) = 0. The solution of\nthis simple ODE for ϕs(t) gives the following expression\nfortind[32],\ntind≈4\n3ηg\nKϕcri−ϕ0\nϕeq(6)\nEquation (6) indicates that tindis essentially determined\nby the rheological relaxation time of glassy state τrhe=\nηg/Gsince KandGare usually of the same order.\nEquation (6) for the induction time is compared with\nthe result of numerical calculation in Fig. 4(a) and (c),\nwhere the numerical induction time is obtained as the\ntime at which ϕsat the surface reaches ϕcri. Fig. 4(a)\nconfirms the validity of the scaling relation tind∼ηg/K\nfor the case of ϕ0= 0, and Fig. 4(c) confirms the pre-\ndiction that tindincreases with the decrease of ϕ0in pro-\nportion to ϕcri−ϕ0. Both figures confirm the validity of\neq. (6).\nFor the case of ϕ0= 0, the front velocity vfrontcan be\nestimated by the following argument. For the outside sol-\nvent to penetrate the glassy polymer, it needs induction\ntime tindto change the polymer from glassy state to rub-\nbery state. As it is seen in Fig. 2(b), during this period,\nthe front of the Fickian precursor moves the distance\nh0\nw=p\nDgtind. Therefore the front velocity is estimated\nbyvfront≃h0\nw/tind=p\nDgK/η g. This justifies the scal-\ning relation for vfront shown in Fig. 4(b). The scaling\nrelation vfront∼p\nDgK/η gis consistent with the previ-\nous argument by Thomas and Windle model [14, 18, 34].\nOn the other hand, the above simple scaling argu-\nment cannot explain the ϕ0dependence of vfrontshown in\nFig. 4(d). The above argument gives the relation vfront∼\n(ϕcri−ϕ0)−1/2, while the result of the numerical calcula-\ntion shown in Fig. 4(d) indicates vfront∼(ϕcri−ϕ0)−1.\nWe numerically checked that the Fickian precursor\nlength hwis independent of ϕ0, and is equal to h0\nw, the\nprecursor length of solvent free polymer [32]. This in-\ndicates that hwis a material parameter determined by\nDgandτrhe. If this is accepted, vfront is estimated by\nh0\nw/tind, or\nvfront∼s\nDgK\nηg1\nϕcri−ϕ0(7)\n(c) (d)(a) (b)\ntfin\ntmeet\ntind1\n1-11FIG. 4. (a) The dependence of induction time tindon ma-\nterial parameters ηg/K(in units of τdif). The black-dashed\nline is analytical solution of eq. (6) for ϕ0= 0. (b) The de-\npendence of the front velocity vfrontonp\nDgK/η g(in units of\nh0/τdif) forϕ0= 0. (c) The dependence of induction time tind\n(black squares), case II diffusion time tmeet(red circle), and\nFickian expansion time tfin(blue triangles) on ϕcri−ϕ0. The\nblack-dashed line is obtained by eq. (6). (d) The dependence\nof the front velocity vfront(in units of h0/τdif) (red circle) on\nϕcri−ϕ0. All data points are numerically calculated from the\nfull model, while all solid lines are guiding lines.\nAs far as we know there are no experimental reports on\nhow the front velocity changes when the initial solvent\nvolume fraction is changed. Eq. (7) is a prediction of our\nmodel, and it may be checked experimentally.\nFigure 4(c) also shows other characteristic times de-\nfined in Fig. 2(c). tmeetis the duration time of the case\nII diffusion stage, i.e., the time for the two fronts to meet\neach other after they are created, and tfinis the relax-\nation time in the final Fickian expansion stage. tmeetis\nestimated by h0/vfront, and is therefore proportional to\nϕcri−ϕ0.tfincan be estimated by tfin∼h2\n0/Drthat is\nindependent of ϕ0.\nSuch characteristic times depend on the sample thick-\nness h0. One can guess that tindis independent of h0,\nand that tmeetandtfinare proportional to h0andh2\n0re-\nspectively. These relations are checked by the numerical\ncalculation [32].\nIn summary, we have proposed a unified theory ac-\ncounting for the diffisio-viscoelasticity coupling to study\nthe swelling of glassy polymers. The theory explains the\nnon-Fickian diffusion and the anomalous swelling behav-\nior observed over many decades in experiments. The\nmodel reveals three featured stages of the swelling: induc-\ntion stage, case II diffusion stage and Fickian expansion\nstage. The induction time is proportional to the rheo-\nlogical time, tind∼ηg/K. The case II diffusion time is5\ntmeet∼h0/p\nDgK/η g, and the Fickian expansion time\nistfin∼h2\n0/Dr. The theory shows that the front ve-\nlocity vfrontis constant in case II diffusion, and is given\nby the Fickian precursor length hwandtindbyhw/tind.\nThe theory predicts how the initial solvent volume frac-\ntionϕ0and the initial film thickness h0affect the three\ncharacteristic times and the front velocity, providing an\neffective way to control the swelling of glassy polymers.\nAll analytical results are confirmed by numerical results\nof the unified model. Our theory can be applied to a wide\nrange of problems where solvent diffusion is coupled with\nmechanical properties such as plastic deformation, mem-\nbrane filtration and fracture or healing.\nWe thank Hugues Chat´ e for useful discussions. 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Soft Matter Physics ; Oxford University Press,\n2013\n[31] Doi, M. Onsager principle in polymer dynamics. Prog.\nPolym. Sci. 2021 ,112, 101339\n[32] see supporting information\n[33] Tanaka, T.; Fillmore, D. J. Kinetics of swelling of gels.\nJ. Chem. Phys. 1979 ,70, 1214–12186\n[34] Hui, C.-Y.; Wu, K.-C.; Lasky, R. C.; Kramer, E. J. Case-\nII diffusion in polymers. II. Steady-state front motion. J.Appl. Phys. 1987 ,61, 5137–51497\nfor Table of Contents use only\nA Unified Model for Non-Fickian Diffusion and Anomalous Swelling of Glassy Polymers\nPeihan Lyu, Zhaoyu Ding, Masao Doi*, Xingkun Man*\nrubbery\nTimeglassy\nTime\nAreaThickness"    },    {        "title": "2402.03878v1.The_generalizations_of_Hamiltonian_in_oriented_graphs.pdf",        "content": "The generalizations of Hamiltonian in oriented graphs∗\nJia Zhou, Zhilan Wang, Jin Yan†\nSchool of Mathematics, Shandong University, Jinan 250100, China\nAbstract\nAn oriented graph is an orientation of a simple graph. In 2009, Keevash, K¨ uhn and\nOsthus proved that every sufficiently large oriented graph Dof order nwith (3 n−4)/8\nis Hamiltonian. Later, Kelly, K¨ uhn and Osthus showed that it is also pancyclic.\nInspired by this, we show that for any given constant tand positive integer partition\nn=n1+···+nt, ifDis an oriented graph on nvertices with minimum semidegree\nat least (3 n−4)/8, then it contains tdisjoint cycles of lengths n1, . . . , n t. Also, we\ndetermine the bounds on the semidegree of sufficiently large oriented graphs that are\nstrongly Hamiltonian-connected, k-ordered Hamiltonian and spanning k-linked.\nKeywords: Oriented graphs; semidegree; cycle-factor; Hamiltonian\nAMS subject classifications. 05C20, 05C38, 05C70\n1 Introduction\nA digraph DisHamiltonian ifDcontains a cycle which encounters every vertex only once.\nHamiltonian is one of the most central notions in graph theory, and has been intensively\nstudied by numerous researchers in recent decades. One of the sufficient conditions for\nHamiltonian in digraphs was established by Ghouila-Houri [6]. He confirmed that any di-\ngraph Donnvertices with δ0(D)≥n/2 is Hamiltonian, where the minimum semidegree\nδ0(D) = min {δ+(D), δ−(D)}. However, the situation is complicated when one considers\nHamiltonian of oriented graphs , which is a digraph that can be obtained from a (simple)\nundirected graph by orienting its edges. The question concerning the analogue of the exis-\ntence of the Hamiltonian cycle in oriented graphs was raised by Thomassen [16]. In 2009,\nKeevash, K¨ uhn and Osthus [8] gave an exact answer to the question as follows.\nTheorem 1.1 [8] There exists a number n0so that any oriented graph Donn≥n0vertices\nwithδ0(D)≥(3n−4)/8is Hamiltonian.\n∗The author’s work is supported by NNSF of China (No.12071260,11671232)\n†Corresponding author. E-mail adress: yanj@sdu.edu.cn\n1arXiv:2402.03878v1  [math.CO]  6 Feb 2024Note that the semidegree condition of Theorem 1.1 is tight, some relevant counterexam-\nples are indicated in [8]. Furthermore, Kelly, K¨ uhn and Osthus [11] showed that the same\ncondition as Theorem 1.1 guarantees that Dispancyclic , which contains a cycle of length l\nfor every l∈ {3,4, . . . , n }.\nIn this paper, we will discuss several natural problems related to the Hamiltonicity.\nNote that a cycle-factor of a digraph is a set of disjoint cycles that covers all vertices of\nit. Hence, the Hamiltonian cycle can be regarded as a cycle-factor with only one cycle. We\nare particularly interested in problems concerning cycle-factors in oriented graphs. In 2009,\nKeevash and Sudakov [9] stated that there exist constants c, C > 0 such that for sufficiently\nlarge oriented graph Donnvertices with minimum semidegree at least (1 /2−c)n, if\nn1, . . . , n tare numbers satisfyingPt\ni=1ni≤n−C, then Dcontains disjoint cycles of lengths\nn1, . . . , n t. In this paper, we prove the following theorem.\nTheorem 1.2 Given an integer t≥1, there exists number n0, so that any oriented graph\nDonn≥n0vertices with δ0(D)≥(3n−4)/8contains tdisjoint cycles of lengths n1, . . . , n t,\nwhere n=n1+···+ntis any positive integer partition of n.\nTheorem 1.2 states that there exists any cycle-factor with a limited cycle number instead\nof ”almost” cycle-factor in oriented graphs with the semidegree of Theorem 1.1. Note that\nthe semidegree of Theorem 1.2 is sharp for cycle-factor problem in oriented graphs, since it\nis tight for the existence of Hamiltonian cycle in oriented graphs.\nAlso, Wang, Wang and Yan [19] made a new progress for digraphs, they showed the\nfollowing result. For every digraph Dwith W⊆V(D), if|W| ≥2kanddD(x) +dD(y)≥\n3n−3 for all {x, y} ⊆W, then for any integer partition |W|=Pk\ni=1niwith ni≥2 for each\ni, there are kdisjoint cycles containing exactly n1, . . . , n kvertices of W, respectively. For\nthe cycle-factor problems in graphs, El-Zahar [5] conjectured that one may get any cycle-\nfactor with lengths n1, n2, . . . , n tin graphs of order nwith the minimum degree at leastPt\ni=1⌈ni/2⌉. This fascinating conjecture has been verified for t= 2 in [5]. In particular, it\nhas also been confirmed for sufficiently large graphs in [1]. And more relevant results can\nbe found in references [4, 17, 18].\nAnother natural generalization of Hamiltonianity is to require the existence of a Hamil-\ntonian path between any pair of vertices. We call Disstrongly Hamiltonian-connected if\nfor any two vertices xandy, there is a Hamiltonian path from xtoy. In addition, Berge\n[3] demonstrated that every digraph Donnvertices with δ0(D)≥(n+ 1)/2 is strongly\nHamiltonian-connected. Theorem 1.3 yields a result about strongly Hamiltonian-connected\nfor sufficiently large oriented graphs.\nTheorem 1.3 There is an integer n0such that every oriented graph Donn≥n0vertices\nwithδ0(D)≥3n/8 + 2 is strongly Hamiltonian-connected.\nIn addition, it also makes sense to seek a Hamiltonion cycle visiting several vertices in\na specific order. A digraph Disk-ordered Hamiltonian if for every sequence s1, . . . , s kof\n2distinct vertices of Dthere is a directed Hamiltonian cycle which encounters s1, . . . , s kin\nthis order. Also, we call a digraph Disk-linked if it contains a system of disjoint paths\nP1, P2, . . . , P ksuch that Piis a path from xitoyi, for every choice of distinct vertices\nx1, . . . , x k, y1, . . . , y k. If the union of these kdisjoint paths span D, we call Disspanning\nk-linked . Indeed, every k-linked digraph is k-ordered and every 2 k-ordered digraph is k-\nlinked. In 2008, K¨ uhn, Osthus, and Young [13] proved that every digraph Donn≥n0(k)\nvertices with δ0(D)≥ ⌈(n+k)/2⌉ −1 isk-ordered Hamiltonian. Here, we give a result\nabout k-ordered Hamiltonian and spanning k-linked in oriented graphs.\nTheorem 1.4 For any positive integer k, there is an integer n0such that for every oriented\ngraph Donn≥n0vertices, the following holds.\n(i)Ifδ0(D)≥3n/8 + 5 k/2−2, then Disk-ordered Hamiltonian.\n(ii)Ifδ0(D)≥3n/8 + 7 k/2−1, then Dis spanning k-linked.\nThe rest of the paper is organized as follows. The aim of Section 2 is to prepare some\nnotation and basic tools used in the paper. In Section 3, we show some lemmas which are\nuseful in the proof of Theorem 1.2. And then we give the proof of Theorem 1.2. The proof\nof Theorem 1.3 and Theorem 1.4 are shown in Section 4 and Section 5, respectively. Section\n6 mentions some related problems about the generalizations of Hamiltonianity in oriented\ngraphs.\n2 Preliminaries and tools\n2.1 Notation and definitions\nLetD= (V, A) be a digraph with vertex set Vand arc set A. We write |D|for the number of\nvertices in Dand denote a(D) to be the number of arcs in D. Given two vertices x, y∈V, we\nwrite xyfor the arc directed from xtoy. For a vertex xofD, we write N+\nD(x) ={y:xy∈A}\n(resp. N−\nD(x) ={y:yx∈A}) to be the out-neighbourhood (resp. in-neighbourhood ) and\nd+\nD(x) =|N+\nD(x)|(resp. d−\nD(x) =|N−\nD(x)|) to be the out-degree (resp. in-degree ) ofx.\nFurther, let δ+(D) and δ−(D) denote the minimum out-degree andthe minimum in-degree\nofD. Let minimum degree δ(D) = min {d+(v) +d−(v) :v∈V(D)}. Given a set X⊆V,\nthe subdigraph of Dinduced by Xis denoted by D[X] and let D−Xdenote the digraph\nobtained from Dby deleting Xand all arcs incident with X. Denote N+\nD(X) to be the\nunion of the out-neighbourhood of every vertex x∈X. For two subsets X, Y inV, denote\nA(X, Y) to be the set of arcs from XtoY, and define a(X, Y) =|A(X, Y)|. A matching in\na digraph is a set of disjoint arcs with no common endvertices. A matching Mis maximum\nifMcontains the maximum possible number of disjoint arcs. And in this paper, the length\nof a path is its vertices. We abbreviate the term ”vertex disjoint” as ”disjoint”.\nThroughout this paper, the notation 0 < β≪αis used to make clear that βcan be\nselected to be sufficiently small corresponding to αso that all calculations required in our\n3proof are valid. For a positive integer t, simply write {1, . . . , t}as [t]. For convenience, write\na±bto be an unspecified real number in the interval [ a−b, a+b]. Next, we mention the\nimportant definition as follows.\nDefinition 2.1 (robust (µ, τ)-outexpander )Given 0< µ≤τ <1, we say a digraph Ris\na robust (µ, τ)-outexpander if |N+\nR(S)| ≥ |S|+µ|R|for every S⊆V(R)satisfying τ|R| ≤\n|S| ≤(1−τ)|R|.\nIn fact, our proof use the technique of [8], which finds a Hamiltonian cycle in an oriented\ngraph. Therefore, we also continue use the symbol of [8]. Set cto be some constant, let µbe\na sufficiently small real number, and write α= (1/100−c√µ)n/4. We partition an oriented\ngraph Donnvertices into four parts D1, D2, D3, D4. For simplicity of notation, for a vertex\nxinDi, if the cardinality of N+(x)∩Djis at least α, then briefly write Di: (Dj)>α. Next,\nwe give the more definitions.\nDefinition 2.2 (A vertex is acceptable ) If a vertex satisfies one of the following 16 proper-\nties, then it is acceptable.\n•D1: (D2)>α(D4)>α, D1: (D1)>α(D4)>α, D1: (D1)>α(D2)>α, D1: (D1)>α\n>α,\n•D2: (D1)>α(D3)>α, D2: (D1)>α(D4)>α, D2: (D3)>α(D4)>α, D2: (D4)>α\n>α,\n•D3: (D2)>α(D4)>α, D3: (D2)>α(D3)>α, D3: (D3)>α(D4)>α, D3: (D3)>α\n>α,\n•D4: (D1)>α(D3)>α, D4: (D1)>α(D2)>α, D4: (D2)>α(D3)>α, D4: (D2)>α\n>α.\nDefinition 2.3 (A vertex xis circular ) If x∈Disuch that all but at most c√µ|Di+1|\nvertices in Di+1are out-neighbourhood of xand all but at most c√µ|Di−1|vertices in Di−1\nare in-neighbourhood of x, then the vertex xis circular.\nDefinition 2.4 (A path Pis circular ) A path Pis a circular path if every vertex of it is\ncircular.\nVisibly, the definition of circular is stronger than the definition of acceptable. Now we\ngive the characterization of the extremal family F. Here, the partition ( D1, D2, D3, D4) of\nDis ordered.\nDefinition 2.5 (Extremal family F) A family Fof oriented graphs is extremal (Fig. 1) if\nevery oriented graph D∈ F onnvertices and for a sufficiently small real µ, there exists a\npartition ( D1, D2, D3, D4) ofDsatisfying the following properties:\n• |Di|= (1/4±16µ)n, for i∈[4].\n•a(Di, Di+1)>(1−800µ)n2/16, for i∈[4] (modulo 4 );a(Di)>(1/2−250µ)n2/16,\nfori∈ {1,3};anda(Di, Dj)>(1/2−300µ)n2/16, for i, j∈ {2,4}satisfying i̸=j.\n•Every vertex is acceptable and the number of non-circular vertices is at most 100√µn.\n4Fig. 1: A member Dof extremal family Fof order n, where the number of arcs from Dito\nDi+1are close to n2/16, for i∈[4] (shown in bold); the number of arcs from D2toD4and\nthe number of arcs from D4toD2are close to n2/32 (shown in bold).\n2.2 Diregularity lemma and Blow-up lemma\nIn this subsection, we collect the information we need about Diregularity Lemma and Blow-\nup Lemma. The density of a bipartite graph G= (A, B) with vertex classes AandBis\ndefined to be dG(A, B) :=eG(A, B)\n|A||B|. We often write d(A, B) if this is unambiguous. Given\nε >0, we say that Gisε-regular if for all subsets X⊆AandY⊆Bwith|X|> ε|A|\nand|Y|> ε|B|we have that |d(X, Y)−d(A, B)|< ε. Given d∈[0,1] we say that Gis\n(ε, d)-super-regular if it is ε-regular and furthermore dG(a)≥(d−ε)|B|for all a∈Aand\ndG(b)≥(d−ε)|A|for all b∈B.\nIn 1978, Szemer´ edi proposed Regular Lemma on graphs, and later Alon and Shapira [2]\nextended it to the digraph version.\nLemma 2.1 [2](Diregularity Lemma )For every ε∈(0,1)and the number M′there\nare numbers M(ε, M′)andn0such that if Dis a digraph on n≥n0vertices, and d∈[0,1]\nis any real number, then there is a partition of the vertices of DintoV0, V1, . . . , V kand a\nspanning subdigraph D′ofDsuch that the following holds:\n• |V0| ≤εn,|V1|=···=|Vk|, where M′≤k≤M,\n•for each σ∈ {+,−},dσ\nD′(x)> dσ\nD(x)−(d+ε)nfor all vertices x∈D,\n•for all i= 1, . . . , k the digraph D′[Vi]is empty,\n•for all 1≤i, j≤kwith i̸=jthe bipartite digraph whose vertex classes are Viand\nVjand whose arcs are all of A(Vi, Vj)arcs in D′isε-regular and has density either 0or\ndensity at least d.\nGiven V1, . . . , V kand a digraph D′, the reduced digraph R′with parameters ( ε, d) is the\ndigraph whose vertex set is [ k] and in which ijis an arc if and only if the bipartite digraph\nwhose vertex classes are ViandVjand whose arcs are all the arcs from VitoVjinD′is\nε-regular and has density at least d. Note that R′is not necessarily an oriented graph even\nifDis. The next lemma shows that there is a reduced oriented graph R⊆R′which still\nalmost inherits the minimum semidegree and density of D.\n5Lemma 2.2 [10] For every ε∈(0,1)there exist numbers M′=M′(ε)andn0=n0(ε)\nsuch that the following holds. Let d∈[0,1]with ε≤d/2. Let Dbe an oriented graph\nof order n≥n0andR′the reduced digraph with parameters (ε, d)obtained by applying\nDiregularity Lemma to D. Then R′has a spanning oriented subgraph Rsuch that δ0(R)≥\n(δ0(D)/|D| −(3ε+d))|R|.\nTo find a subdigraph with the maximum degree to be bounded, the standard idea is to\nfind a special structure in the reduced oriented graph Rand restore it to the subdigraph in\nD. So Blow-up Lemma of Koml´ os, S´ ark¨ ozy and Szemer´ edi [12] is necessary. Notice that, in\nthe general case, we use the Blow-up Lemma on the underling graph of R. The following\nlemma states that dense regular pairs and complete bipartite graphs behave identically for\nthe problem of finding subgraphs with bounded degree in graphs.\nLemma 2.3 [12](Blow-up Lemma )Given a graph Fon[k]and positive numbers d,∆,\nthere is a positive real η0=η0(d,∆, k)such that the following holds for all positive numbers\nl1, . . . , l kand all 0< η≤η0. Let F′be the graph obtained from Fby replacing each vertex\ni∈Fwith a set Vioflinew vertices and joining all vertices in Vito all vertices in Vj\nwhenever ijis an edge of F. Let G′be a spanning subgraph of F′such that for every edge\nij∈Fthe graph (Vi, Vj)G′is(η, d)-super-regular. Then G′contains a copy of every subgraph\nHofF′with ∆(H)≤∆.\n3 Cycle-factor in oriented graphs\nBefore staring the proof of Theorem 1.2, we outline the main idea.\nSketch of proof of Theorem 1.2. Suppose that Dis an oriented graph as stated in\nTheorem 1.2. Assume that Theorem 1.2 is false, namely, Dcontains no some cycle-factor\nC. Using the asymptotic pancyclicity (Lemma 3.1) of D, we can find all disjoint short\ncycles in C. Applying Diregularity lemma for the remaining digraph, there is an reduced\noriented graph R. According to Lemmas 3.3-3.4, Dis not an extremal oriented graph in\nFas otherwise we can find the cycle-factor C. Further, this implies that Ris a robust\n(µ,1/3)-outexpander. Then the remaining cycles in Ccan be find by using the splitting\noperation (see Claim 3.2), a contradiction.\n3.1 Preliminary lemmas\nSuppose that Dis an oriented graph on nvertices with δ0(D)≥(3n−4)/8, where nis large\nenough. Here, we show a sequence of lemmas that are crucial to our proof.\nIn 1998, Shen [15] showed that any oriented graph Donnvertices with δ+(D)≥0.355n\ncontains a triangle. And we write l-cycle to be a cycle with length l. In [11], authors also\nshowed that every oriented graph Donnvertices with δ0(D)≥n/3 + 1 contains an l-cycle\n6for every l∈ {4, . . . , n/ 1010}. For the sake of presentation, we call a digraph is asymptotic\npancyclic if it has an l-cycle for every l∈ {3,4, . . . , n/ 1010}. Therefore, the following lemma\nis straightforward.\nLemma 3.1 ( Asymptotic Pancyclic )IfDis an oriented graph on nvertices with\nδ0(D)≥0.355n, then Dis asymptotic pancyclic.\nFact 3.1 IfRwith δ0(R)≥(3/8−3d)|R|is a robust (µ,1/3)-outexpander, then Ris also\na robust (µ, τ)-outexpander, where d≪µ≤τ≤1/100.\nProof Let|R|=k. By the definition of robust ( µ,1/3)-outexpander, we have that |N+\nR(S)| ≥\n|S|+µkfor every S⊆V(R) satisfying k/3≤ |S| ≤2k/3. For a subset SofV(R) satisfying\nτk≤ |S| ≤k/3, it is easy to check that |N+\nR(S)| ≥(3/8−3d)k≥k/3 +k/100≥ |S|+µk.\nAnd if 2 k/3≤ |S| ≤(1−τ)k, then |S|+|N−\nR(v)|> k, that is, S∩N−\nR(v)̸=∅, for any v∈R.\nAnd then N+\nR(S) =V(R). So\n|N+\nR(S)|=|R|= (1−τ)k+τk≥ |S|+τk≥ |S|+µk.\nThis implies that Ris a robust ( µ, τ)-outexpander, as desired. □\nLemma 3.2 [8] Given n−1/6\n0≪1/M′≪ε≪d≪µ≪τ≪η≤1. Let Dbe an oriented\ngraph on n≥n0vertices, and Lis a subset of V(D)with cardinality at most tε2n. Let R′be\nthe reduced digraph of D−Lwith parameters (ε, d). Suppose that Ris a spanning oriented\nsubgraph of R′as stated in Lemma 2.2.\n(i)IfRis a robust (µ, τ)-outexpander and δ0(D)≥2ηn, then Dis Hamiltonian.\n(ii)IfRis not a robust (µ,1/3)-outexpander and δ0(D)≥(3n−4)/8, then Dis a\nmember of family F.\nThe following lemma is come from Claim 2.5-2.7 in [8] (see pages 20-22 for details).\nLemma 3.3 [8] Let Dbe an oriented graph in Fwith a partition (D1, D2, D3, D4)and\norder n. Ifδ0(D)≥(3n−4)/8, then there exists an oriented graph D′which is obtained by\ncontracting a few short paths or resetting a few vertices from Dsatisfies |D2|=|D4|inD′.\nIn the proof of the next lemma, we often use an operation on contracting a path, which\ncan be summarized as an algorithm here.\nAlgorithm 1 Contracting paths\nInput: An oriented graph D∈ F with a partition ( D1, D2, D3, D4) and disjoint paths\nP1, P2, . . . , P k, where the initial and terminal vertices of each path are located in the\nsame part.\nOutput: A new oriented graph H∈ Fand distinct vertices p1, p2, . . . , p k.\n1:SetH0=D.\n2:forj= 1do\n73: while j̸=k+ 1do\n4: SetPj=j1j2. . . j rwhere j1, jr∈Dl, for some l∈[4].\n5: Construct the new oriented graph Hjby contracting the path Pjfrom Hj−1into\na new vertex pjwith N−\nHj(pj) =N−\nHj−1(i1)∩Dl−1andN+\nHj(pj) =N+\nHj−1(ir)∩Dl+1, and\nthen put pjinto the part Dl.\n6: Letj=j+ 1.\n7: end while\n8:end for\n9:SetH=Hk.\nLemma 3.4 states that there are some oriented graphs in extremal family Fcontains\nany cycle-factor as desired.\nLemma 3.4 Given a constant tand a positive real µsuch that µ≪1/(1010·t2), there\nexists a number n0so that every oriented graph D∈ F (Fig. 1) onn≥n0vertices with\na partition (D1, D2, D3, D4)and|D2|=|D4|contains tdisjoint cycles of lengths n1, . . . , n t,\nwhere n=n1+···+ntis any positive integer partition of n.\nProof On the contrary, suppose that Dhas no cycle-factor C={C1, C2, . . . , C t}with\n|V(Ci)|=nifor each i∈[t] and n=Pt\ni=1ni. For convenience, we represent a path using\na sequence of the parts corresponding to each vertex along the path. We call a path (resp.\ncycle) is circular, if the successive vertices of the path (resp. cycle) lie in successive classes.\nAnd let the index ibe always taken modulo 4, in this lemma. Without loss of generality,\nsuppose that n1≥nifori∈[t]. Also, assume that each of n2, n3, . . . , n l′is equal to 3 and\nni=\n\n= 0 i∈ {l′+ 1, l′+ 2, . . . , l 0};\n= 1 i∈ {l0+ 1, l0+ 2, . . . , l 1};\n= 2 i∈ {l1+ 1, l1+ 2, . . . , l 2};\n= 3 i∈ {l2+ 1, l2+ 2, . . . , l 3}.(mod 4)\nFirstly, we will find all the triangles which are needed. Constituting l′−1 disjoint\ntriangles such that every triangle has a vertex in D2, a vertex in D3and a vertex in D4.\nRecall that there is a few vertices are non-circular and a(D2, D4)≥(1/2−300µ), which\nimplies that there are l′−1 disjoint arcs from D2toD4with endvertices are all circularity.\nOwing to the definition of circular, we can obtain l′−1 disjoint triangles as desired. From\nnow on, define Sto be the set of l′−1 triangles. Let D′=D−S, we conclude that D′∈ F\nand|D2|=|D4|inD′, since l′is a constant.\nIn order to find cycles with given lengths, we will find l3−l0suitable disjoint paths. Select\nl1−l0disjoint arcs whose two endvertices are circular vertices in D1asl1−l0paths. And pick\nl2−l1disjoint paths such that every path is a circular path D3D3D3. And then find l3−l2\ndisjoint circular paths, where each path is shaped like D2D4D1D2. Note that we can ensure\n8that all of above paths are disjoint, since the numbers of arcs in D1, D3, from DitoDi+1and\nfrom D2toD4are large enough. These yield that a path system P={Pl0+1, Pl0+2, . . . , P l3}.\nNow, the new oriented graph H1and contraction vertices pl0+1, pl0+2, . . . , p l3are obtained\nby applying Algorithm 1 with the oriented graph D′and the path system P. It is easy to\ncheck that all contraction vertices are circular and |D2|=|D4|inH1. Set Z={pl0+1, pl0+2,\n. . . , p l3}. Removing the vertices in Zand arcs which adjacent with Zfrom H1, we can\nobtain a new oriented graph H2.\nActually, we need all vertices to be circular. Thereby, our purpose is to find some\ndisjoint short paths containing all non-circular vertices so that all vertices are circular after\ncontracting these paths. Assume v1, . . . , v rare non-circular vertices in H2. Next, we will\nexplain that the following disjoint paths Q1, Q2, . . . , Q rare paths that we want. For each vi,\nchoose a circular out-neighbour v+\niand a circular in-neighbour v−\niso that they are distinct.\nThen there exist disjoint paths Q1, Q2, . . . , Q rsuch that for each i∈[r], the following hold.\n•The path Qistarting at v−\niviv+\niand ending at a circular vertex which lies in the same\nclass as v−\niwith length at most 7.\n•The path Qiis a circular path.\nApplying Algorithm 1 with the oriented graph H2and the paths Q1, Q2, . . . , Q r, we\nobtain a new oriented graph H3and contraction vertices q1, q2, . . . , q r. For each j∈[r],\nsince the endvertices of Qiare circular vertices, vertex qjis circular, which is easy to check\nby the construction of qj. Now every vertex of H3is circular.\nWe construct a new oriented graph H4by adding the vertex set ZtoH3and arcs between\nZandV(H3) inA(H1). Hence |D2|=|D4|inH4and we still have that\n||Di| − |Dj|| ≤ | (1/4 + 16 µ)n−(1/4−16µ)n|+t+r≤300√µn. (1)\nRecall that a(D2, D4), a(D4, D2)>(1/2−300µ)n2/16 in D, which gives a matching M0of\nsize 2×104√µn, where 104√µnarcs are from D2toD4and 104√µnarcs are from D4to\nD2inH4.\nNext, we begin by finding disjoint cycles Cl′+1, Cl′+2, . . . , C t. Due to the definition of\ncircularity and µ≪1/(1010·t2), for any two circular vertices y∈Dj−2, x∈Dj, we actually\nget that\n|N+(y)∩N−(x)∩Dj−1| ≥n/4−6000√µn≥n/4−n/4t+ 100√µn+ 2×104√µn. (2)\nThis implies that there are at least n/4−n/4t+ 2×104√µncircular vertices in N+(y)∩\nN−(x)∩Dj−1. However, since n1≥nifor 1≤i≤t, we get n1≥n/t. Thus n−n1≤n−n/t.\nAlso, it is clear from (2) that there are l1+l2+l3disjoint cycles C′\nl′+1, . . . , C′\ntsatisfying the\nfollowing statements.\n•For each i∈[l′+ 1, t],C′\niwith length ⌊ni/4⌋ ×4 is a circular cycle which avoids the\nvertices in the matching M0.\n•For each i∈[l0+ 1, t],C′\nicontains the corresponding contraction vertex pi.\n9Consider each C′\ni, if we replace the contraction vertex piwith the path Pi, we can obtain a\ncycle of length niinD, fori∈[l0+1, t]. Set U=V(C′\nl′+1∪C′\nl′+2∪···∪ C′\nt) and H5:=H4−U.\nClearly, M0⊆H5. Moreover, |D2|=|D4|still holds in H5.\nRecall that Ddoes not have a cycle-factor C, then H5cannot be Hamiltonian. However,\nwe will claim that H5is Hamiltonian, which contradicts the hypothesis and proves the\nlemma. In order to apply Blow-up Lemma, we also need |D1|=|D2|=|D3|=|D4|. we\ncan do this by finding two suitable disjoint paths to contract. Without loss of generality,\nassume that |D1|<|D3|inH5. Let s:=|D3| − |D1|inH5, (1) implies that s <300√µn.\nDue to the structure of D, almost all pairs of vertices in D3are connected by an arc. Let\nH6=H5−V(M0). Thus we greedily find a path R1,3of the form\nD3D3D4D1D2···D3D3D4D1D2D3,\nwhere the fragment D3D3D4D1D2consists of circular vertices and it repeats stimes. So\nR1,3starts with an arc between two circular vertices in D3. Let H7be the oriented graph by\napplying Algorithm 1 with the oriented graph H6and the path R1,3. Then add the vertex\nsetV(M0) and arcs between V(M0) and V(H7) inA(H5), assuming there is no confusion,\nwe still call it H7. It follows that |D1|=|D3|,|D2|=|D4|inH7and all vertices of H7\nare still circular. Without loss of generality, suppose that we have |D2|>|D1|inH7. Let\ns:=|D2| − |D1|, (1) implies that s <300√µn. By using the arcs in the matching M0, we\ncan find a path R2,1of the form\nD2D4D1D2D3D4D2D3D4D1···D2D4D1D2D3D4D2D3D4D1D2,\nwhere the fragment D2D4D1D2D3D4D2D3D4D1appears repeatedly for stimes. It is easy\nto check that D7has such a path. Likewise, by contracting R2,1, we obtain an oriented\ngraph H8. InH8, we get that\n|D1|=|D2|=|D3|=|D4| ≥(n1−6r−5×300√µn−10×300√µn)/4\n≥(n1−104√µn)/4.\nNote that, all vertices are still circular in H8. Namely, for every vertex v∈Di, there are at\nleast (1 −ct√µ)|Di+1|out-neighbours in Di+1.\nLetF′be the 4-partite graph with vertex classes D1, D2, D3, D4inH8, where the bipartite\ngraphs induced by ( Di, Di+1) are all complete. Clearly F′is Hamiltonian. On the other\nhand, assume that Gis the underlying graph corresponding to the set of edges oriented\nfrom DitoDi+1inH8, fori∈[4]. Pick ηwith ct√µ≪η2. Since all vertices of H8are still\ncircular, for any subset X⊆DiandY⊆Di+1with|X| ≥η|Di|,|Y| ≥η|Di+1|, it follows\nthat\nd(X, Y)≥(|Y| −ct√µ|Di+1|)|X|\n|X||Y|≥1−ct√µ\nη.\nTherefore, each pair ( Di, Di+1) isη-regular pair in G. Further, each pair ( Di, Di+1) is (η,1)-\nsuper-regular pair as all vertices of H8are circular. Also, Gis simple. So we can apply\n10Lemma 2.3 with k= 4, ∆ = 2 to get a Hamiltonian cycle in G. Since the construction of\nG,H8is Hamiltonian. Recall that H8is obtained by H5contracting two paths, this implies\nthatH5is Hamiltonian. This contradiction completes the proof of the lemma. □\n3.2 Proof of Theorem 1.2\nDefine constants M′, ε, d, µ, τ, η, n 0satisfying\nn−1/6\n0≪1/M′≪ε2≪d≪µ≪τ≪η≪1/t. (3)\nSuppose that Dis an oriented graph on n≥n0vertices with δ0(D)≥(3n−4)/8. Assume\nthat Theorem 1.2 is false. Namely, Dcontains no cycle-factor Cwith cycles C1, C2, . . . , C t\nwhose orders are n1, n2, . . . , n t, respectively. Without loss of generality, assume that n1≥\nn2≥ ··· nj≥ε2n > n j+1≥ ··· ≥ nt. Let’s start by finding disjoint cycles whose lengths\nless than ε2n. This can be done by applying Lemma 3.1. These yield t−jdisjoint cycles\nCj+1, Cj+2, . . . , C twhere each cycle Cihas the length ni< ε2n. Define D∗to be the oriented\ngraph D−St\ni=j+1Ci. It is easy to check that the semidegree of D∗at least (3 /8−tε2)n.\nApply Diregularity Lemma (Lemma 2.1) with parameters ε2, d, M′for the digraph D∗to\nobtain a partition V0, V1, V2, . . . , V kand a reduced oriented graph Rby Lemma 2.2. Further,\nwe have δ0(R)≥(3/8−tε2−d−3ε)|R| ≥(3/8−3d)|R|, as (3).\nClaim 3.1 Ris a robust (µ, τ)-outexpander.\nProof Suppose that Ris not a robust ( µ,1/3)-outexpander. It follows from Lemma 3.2\nthatDis an extremal oriented graph. Owing to the semidegree of Dand Lemma 3.3, there\nexists an oriented graph obtained by contracting a few short paths or resetting a few vertices\nfrom Dsatisfies the hypothesis of Lemma 3.4. Namely, there is a cycle-factor with lengths\nn1, n2, . . . , n t, a contradiction. Thereby, Ris a robust ( µ,1/3)-outexpander. Combining\nwith Fact 3.1, Ris a robust ( µ, τ)-outexpander. □\nIn the following, we will apply Lemma 3.2 to find disjoint cycles with lengths n1, n2, . . . , n j.\nTo do this, we split V(D∗) into S1, S2, . . . , S jwith almost inherit the semidegree condition\nofD∗. In this process, ”Chernoff bound” is essential, which states that P(|X−EX|> a)<\ne−a2/(3·EX), where Xis the hypergeometric random variable and EXis the expectation of\nX. Set ξi=⌊ni\nk⌋ ·1\n|Vi|. Namely, ξi> ε2, for each i∈[j].\nClaim 3.2 There exists a partition of V(D∗)intojsets(S1, S2, . . . , S j)such that the fol-\nlowing properties hold.\n(i)|Si|=ni≥ε2nandδ0(D[Si])≥2η|Si|, for i∈[j].\n(ii)For each i∈[j], there is an oriented reduced graph Riwith parameters (ε2,5d/6)\ncorresponding to the partition Si,1,···, Si,kofSi. Moreover, every Riis isomorphic with R.\n11Proof (i). For each i∈[j], we will show that Si=S\nl∈[k]Si,l∪Vi,0, where Si,lis a subset of\nViwith order ξi|Vi|andVi,0is a subset of V0with order ni−ξik|Vi|, so|Si|=ni. For each\nVl, consider a random partition of VlintojsetsS1,l, S2,l, . . . , S j,l. By assigning every vertex\nx∈VltoSi,lwith probability ξiindependently. For every vertex v∈V(D∗), let the random\nvariable A+\nv,i(resp., A−\nv,i) calculate the number of out-neighbours (resp., in-neighbours) of v\ninSi,l. It is not hard to see that\nEA+\nv,i=d+\nVl(v)\n|Vl|· |Si,l|=ξid+\nVl(v).\nOwing to Chernoff bound, this yields that\nP(A+\nv,i−EA+\nv,i<−n2/3)< e−n4/3\n3·d+\nVl(v).\nThis implies that\nX\nv∈V(D∗)P(A+\nv,i−EA+\nv,i<−n2/3)< ne−n4/3\n3·d+\nVl(v).\nAnalogously we can get that\nX\nv∈V(D∗)P(A−\nv,i−EA−\nv,i<−n2/3)< ne−n4/3\n3·d+\nVl(v).\nSo with non-zero probability we obtain that dσ\nSi,l(v)≥ξidσ\nVl(v)−n2/3for every vertex\nv∈V(D∗), every i∈[k] and σ∈ {+,−}. For each i∈[j], arbitrarily pick ni−ξik|Vl|\nvertices from the set V0, and form the set Siwith the setSk\nl=1Si,l. For σ∈ {− ,+}, (3) and\n|Si|=ni≥ε2nyield that\nδσ(D∗[Si])≥(3/8−t·ε2)|Si| −n2/3≥(3/8−t·ε2−n−1/3/ε2)|Si|>2η|Si|,asn→ ∞ .\n(ii). For every edge l1l2∈E(R), it follows from the construction of RthatA[Vl1, Vl2] is\nε2-regular pair with density at least d. From the definition of regularity and |Si,l1| ≥ε|Vl1|,\n|Si,l2| ≥ε|Vl2|, it can be concluded that |d(Si,l1, Si,l2)−d(Vl1, Vl2)|< ε2, that is d(Si,l1, Si,l2)≥\nd−ε2≥5d/6. On the other hand, there is no arc from Si,l1toSi,l2whenever l1l2/∈E(R).\nHence, there is an oriented reduced graph Riwith parameters ( ε2,5d/6) corresponding to\nthe partition Si,1,···, Si,kofSi. Clearly, V(Ri) =V(R). These analyses make it obvious\nthatRiis isomorphic with R. □\nTogether with Lemma 3.2 ( i), each oriented graph D∗(Si) is Hamiltonian. Thereby, we\nfind disjoint cycles of lengths n1, n2, . . . , n t, a contradiction. This completes the proof. □\n124 Strongly Hamiltonian-connected in oriented graphs\nBefore proving Theorem 1.3, we show a key lemma to the proof of Theorem 1.3. Define\nβ= (1/100 + c√µ)n/4. When |D2|>|D4|, we call a vertex good if it is acceptable, and it\nbelongs to D4or has one of the properties D1: (D2)<β(D3)<β,D2: (D1)<β(D2)<β\n<β(D3)<β,\nD3: (D1)<β(D2)<β. Similarly, when |D2|<|D4|, we call a vertex good if it is acceptable,\nand the vertex in D2or has one of the properties D1: (D2)<β(D3)<β,D3: (D1)<β(D2)<βor\nD4: (D1)<β(D4)<β\n<β(D3)<β. Naturally, if a vertex is not good, we call it a badvertex. The\nidea of proving Lemma 4.1 is very similar to proving Theorem 1.2, so here we focus on the\nplacement of the vertex x. Here, we still denote α= (1/100−c√µ)n/4.\nLemma 4.1 Suppose Dis an oriented graph on n≥n0vertices with δ0(D)≥3n/8, where\nn0is some integer. The oriented graph D′obtained from Dby adding a vertex xsuch that\nthe semidegree of xinD′is at least 4α, then D′is Hamiltonian.\nProof Suppose, contrary to our lemma, that D′is not Hamiltonian. Define constants\nM′, ε, d, µ, τ, η, n 0satisfying n−1/6\n0≪1/M′≪ε≪d≪µ≪τ≪1/100.Applying\nDiregularity Lemma with parameters ε2, d, M′for the digraph D, this yields a partition\nV0, V1, . . . , V kand there is a corresponding reduced oriented graph Rwith δ0(R)≥(3/8−\n3d)|R|by Lemma 2.2. If Ris a robust ( µ,1/3)-outexpander, then Ris a robust ( µ, τ)-\noutexpander by Fact 3.1. Put xinto the exception set V0. Lemma 4.1 holds as Lemma 3.2\n(i), a contradiction. Otherwise, Ris not a robust ( µ,1/3)-outexpander, then D∈ Fwith a\npartition ( D1, D2, D3, D4) by Lemma 3.2 ( ii).\nClaim 4.1 There is a part Drsuch that we can put xintoDrto be an acceptable vertex,\nr∈[4].\nProof Owing to the semidegree of x, there exist two parts DiandDjsuch that |N+(x)∩\nDi|> α and|N−(x)∩Dj|> α,i, j∈[4]. Therefore, the choice of Drdepends on the\nfollowing circumstances:\n•When i∈ {1,2}, ifj∈ {1,4}then r= 1, otherwise, r= 4.\n•When i∈ {3,4}, ifj∈ {2,3}then r= 3, otherwise, r= 2.\nIt is easy to check that xis an acceptable vertex in Dr. □\nNow, all vertices in D′are acceptable. In addition, it is not difficult to find D′belongs\ntoF. Indeed, if |D2|=|D4|inD′by few arrangement, then D′satisfies the hypothesis of\nLemma 3.4. Thus, D′is Hamiltonian, a contradiction. Therefore, |D2| ̸=|D4|always hold\nin the following procedures as we can’t move more than 2( |D2|−|D4|) vertices. Without lose\nof generality, suppose that |D2|>|D4|inD′. In the following, we only need to prove that\nthere is a contradiction with the semidegree of D. Its proof is the same as Claims 2.5-2.7\nin [8], but for the sake of completeness of proof, we still give the complete proof here.\n13Claim 4.2 [8] All vertices in D′will become good by some arrangement. And then there is\na contradiction with the semidegree of D.\nProof Sets=|D2|−|D4|. Note that we put a vertex vintoD4to be an acceptable vertex,\nthen vis a good vertex by the definition of good vertex.\nSubclaim 1 There are at most s−1bad vertices in D1∪D3. Moreover, we can arrange\nthem to be good vertices by moving them to D4.\nProof Ifvis a bad vertex in D1, we can obtain that |N−(v)∩D2| ≥βor|N−(v)∩D3| ≥β.\nIt follows from vis an acceptable vertex that |N+(v)∩D1| ≥αor|N+(v)∩D2| ≥α. It is\neasy to check that we move the vertex vtoD4then vis an acceptable vertex. Furthermore,\nit is good. Similarly for the case v∈D3. Thus, if there are sbad vertices in D1∪D3,\nthen we can arrange them to be good vertices by moving them to D4. This yields that\n|D2|=|D4|, a contradiction. □\nSubclaim 2 All bad vertices in D2will become acceptable in D1∪D3by some arrangement.\nProof Suppose that vis not a good vertex in D2. Then vsatisfies at least one of properties\nD2: (D1)>β,D2: (D2)>β,D2: (D2)>βandD2: (D3)>β. By the acceptability of v, we get\nthat vhas a large in-neighbourhood in D1orD4and a large out-neighbourhood in D2or\nD3. Hence, if vhas properties D2: (D1)>βorD2: (D2)>β, then it is also an acceptable\nvertex in D1. So we move the vertex vto the part D1. And if vhas properties D2: (D2)>β\norD2: (D3)>β, then vis also an acceptable vertex in D3. In this case, we move the vertex\nvtoD3. By Subclaim 1, we can arrange vto be good in some part Dk. □\nNote that |D2| ̸=|D4|during the whole process. However, it may happen that |D2|−|D4|\ngoes from +1 to −1 if a vertex vis moved from D2toD4. In this case we can put vinto\nD1orD3to be an acceptable vertex, by Claim 2. This yields that |D2|=|D4|inD′, a\ncontradiction. Thereby all vertices are good by our arrangement and |D2|>|D4|inD′.\nNext, we will show that there is no arc from D3toD2and from D2∪D3toD1and in\nD2. Construct a spanning oriented graph H′ofD′whose arc set is A(H′) =A(D2, D1)∪\nA(D2)∪A(D3, D1)∪A(D3, D2). Let Mbe a maximum matching in H′. For each i∈[3],\nsuppose Li=V(M)∩Di. Ifa(M)≥s, we could extend sarcs of Mtosdisjoint paths\nsuch that the difference between |D2|and|D4|decreased by sby contracting these paths.\nThen the oriented graph D′′obtained by contracting these paths from D′has|D2|=|D4|.\nBy Lemma 3.4, D′′is Hamiltonian. Further, D′is Hamiltonian, a contradiction. Hence,\na(M)< s.\nThen we will claim that a(M) = 0. Assume to the contrary that a(M)≥1. The\nmaximality of Mand the definition of the good vertex imply\na(D3, D1)≤a(L3, D1) +a(D3, L1)\n≤(|L3|+|L1|)β≤a(M)|D1|/45.\n14Similarly, we also can obtain that\na(D2, D1)≤a(L2, D1) +a(D2, L1)< a(M)|D1|/45,\na(D3, D2)≤a(L3, D2) +a(D3, L2)< a(M)|D3|/45,\na(D2)≤ |L2| ·2β <2a(M)|D2|/45.\nHence, it follows that\nX\nv∈D1d−(v)≤|D1| ·(|D1−1|)\n2+ 2a(M)|D1|/45 +|D1| · |D4|,\nX\nv∈D2d(v)≤(|D1|+|D3|)· |D2|+ 4a(M)|D2|/45 +|D2| · |D4|,\nX\nv∈D3d+(v)≤|D3| ·(|D3−1|)\n2+ 2a(M)|D3|/45 +|D3| · |D4|.\nWithout loss of generality, assume that the vertex xis a good vertex in D1. By pigeonhole\nprinciple, there are three vertices u∈D1, v∈D2, w∈D3with u̸=xsuch that\nd−(u)<|D1|\n2+|D1| · |D4|\n|D1| −1+2a(M)|D1|\n45·(|D1| −1),\nd(v)<|D1|+|D3|+|D4|+ 4a(M)/45,\nd+(w)<|D3|/2 +|D4|+ 2a(M)/45.\nTogether with |D1|+|D2|+|D3|+|D4| ≤n, it yields that\n3n\n2≤d−(u) +d+(w) +d(v)\n<3\n2(n− |D2|+|D4|) + 2|D4|+|D1| · |D4|\n|D1| −1+6a(M)\n45+2a(M)|D1|\n45(|D1| −1)\n<3\n2(n− |D2|+|D4| · |D1|\n|D1| −1) +6a(M)\n45+2a(M)|D1|\n45(|D1| −1).(4)\n□\nRecall that |D2|−|D4|=s > a (M), so|D2|−|D4| · |D1|\n|D1| −1> a(M) by simple calculation.\nThus (4) gives that\n3\n2·a(M)<6a(M)\n45+2a(M)|D1|\n45(|D1| −1).\nThis is impossible, so a(M) = 0. Further, (4) also implies that |D2| −|D4| · |D1|\n|D1| −1<0.\nNamely, |D2| − |D4|= 0. It contradicts the assumption, and thus the proof is completed. □\nProof of Theorem 1.3. Suppose Dis an oriented graph on n≥n0vertices with\nδ0(D)≥3n/8 + 2. And assume that we need find a Hamiltonian path from utov. If\nN+\nD(u)∩N−\nD(v) is not an empty set, divide N+\nD(u)∩N−\nD(v) equally into two parts Nu, Nv.\n15Next, an auxiliary oriented graph His obtained by removing the vertices u, vfrom D, then\nadd a vertex wwith N+\nH(w) =N+\nD(u)\\NvandN−\nH(w) =N−\nD(v)\\Nu. Then semidegree of w\ninHis at least 3 n/16 and δ0(H−{w})≥3n/8, which is clear from δ0(D)≥3n/8+2. Thus\nHcontains a Hamiltonian cycle since Lemma 4.1. Restore the vertex winto the vertices\nu, v. It means that there is a Hamiltonian path from utovinD. The theorem is established\nby the arbitrariness of uandv. □\n5 Proof of Theorem 1.4.\nBefore proving Theorem 1.4, we need to introduce more concepts. Denote the distance of\ntwo vertices x, yto be the length of a shortest path from xtoy. In 2008, Lichiardopol [14]\nshowed that the diameter of an oriented graph Dof order nwith δ0(D)≥n/3 is at most 5.\nProof of Theorem 1.4. (i) Assume the sequence s1, . . . , s kof distinct vertices of D\nis the order which we need to encounter by a Hamiltonian cycle. We proof Theorem 1.4( i)\nby finding a path Pfrom s1toskwhose length is at most 4 k−3. For each j∈[k],\nthere exists a path Piform sitosi+1whose length is at most 5 and avoids the vertices\n{s1, s2, . . . , s i−1, si+2, . . . , s k} ∪Si−1\nj=1Pjsince δ0(D−Sk\nj=1Pj)≥3n/8 + 5 k/2−2−(4k−\n3)> n/ 3.Therefore, there is a path Pfrom s1toskand encounters each siin the order.\nApparently, the length of Pis at most 4 k−3. Let Hbe the oriented graph that remove the\nvertices P\\ {s1, sk}from D. Owing to δ0(D−P\\ {s1, sk})≥3n/8 + 5k/2−2−(4k−3)>\n3(n−4k−3)/8 + 2 and Theorem 1.3, there is a Hamiltonian path from sktos1inH. This\nimplies that Disk-ordered Hamiltonian.\n(ii) Assume X={x1, x2, . . . , x k},Y={y1, y2, . . . , y k}be 2kvertices to be linked. Since\nδ0(D−Sk\nj=1Pj)≥3n/8+7k/2−1−5k > n/ 3, there exists a path Pifrom xitoyiof length at\nmost 5 and avoiding the vertices ( X∪Y∪Si−1\nj=1Pj)\\{xi, yi}, for each i∈[k−1]. This yields\nthat the semidegree of D−Sk−1\nj=1Pjis at least 3 n/8+7k/2−1−5(k−1)≥3(n−5k+5)/8+2.\nTogether with Theorem 1.3, we get that D−Sk−1\nj=1Pjcontains a Hamiltonian path from xk\ntoyk. Hence Dis spanning k-linked. □\n6 Remark\nIn 1993, H¨ aggkvist [7] also made the following conjecture. Here, define δ∗(D) =δ(D) +\nδ+(D) +δ−(D).\nConjecture 6.1 Every oriented graph Dwithδ∗(D)>(3n−3)/2is Hamiltonian.\nIn [10], this conjecture was verified approximately; that is, if δ∗(D)≥(3/2 +o(1))n,\nthen the oriented graph Dis Hamiltonian. In addition, notice that in Theorem 1.2, we\nonly get any cycle-factor with constant cycles, which implies that there must exist a long\n16cycle whose length at least n/tin any cycle-factor. Sudakov [9] described an infinite class of\ntournaments Tof order nwith δ0(T)≥(n−1)/2−1 which are not have a perfect packing\nof cyclic triangles. Naturally, we put forward the following question.\nProblem 6.1 Whether there is any cycle-factor with tcycles in an oriented graph Don\nnvertices with δ0(D)≥(3n−4)/8(resp. δ∗(D)>(3n−3)/2), for any natural number\nt < n/ 3?\nFinally, we can also consider the linkage problem in oriented graphs. Indeed, it is easy\nto check that every oriented graph Donnvertices with δ0(D)≥n/3 + 5 k−5 isk-linked.\nOn the other hand we can construct an oriented graph Donnvertices with δ0(D)≥\nn/4 + 3 k/2−5/2, but not k-linked.\nLetDbe an oriented graph on vertex set V=B∪C∪X∪Y, where X={x1, . . . , x k},\nY={y1, . . . , y k}and|B|=|C|= (n−2k)/2 and whose arcs are listed in the following\nways.\n(1) The arcs in both XandYare oriented arbitrarily.\n(2) The arcs in both BandCare oriented so that D[B] and D[C] form a regular tourna-\nment.\n(3) All arcs are oriented from {xk, yk} ∪CtoB, from Bto (X∪Y)\\ {xk, yk}, from Cto\n{xk, yk}and from ( X∪Y)\\ {xk, yk}toC.\n(4) All arcs are oriented from XtoYexcept for arcs yixifor each i∈[k].\nFig. 2: An oriented graph Dwith δ0(D)≥n/4 + 3 k/2−5/2 that is not k-linked.\nFirst, we shall show that δ0(D)≥n/4+3k/2−5/2. It is easy to check that every vertex\nin{x1, . . . , x k−1, y1, . . . , y k−1}has out-neighbourhood Cand in-neighbourhood B. This gives\nthat its semidegree is at least ( n−2k)/2. Similarly, the semidegree of xk(resp. yk) is at\nleast ( n−2k)/2. Therefore, it remains to show that every vertex in B∪Chas semidegree\nat least n/4 + 3k/2−5/2. By the construction of (2)-(3), for every vertex x∈B∪C, there\nis\ndσ(x)≥(n−2k)/2−1\n2+ 2(k−1)≥n/4 + 3 k/2−5/2, σ∈ {+,−}.\n17But it is not difficult to see that Ddoes not contain an ( xk,yk)-path which avoids X∪Y.\nTherefore, we provide the following problem.\nProblem 6.2 Whether there is a constant csuch that every oriented graph Donnvertices\nwithδ0(D)≥n/4 +ckisk-linked?\nReferences\n[1] S. Abbasi, Spanning cycles in dense graphs , Ph.D. thesis, Rutgers University, 1999.\n[2] N. Alon and A. Shapira, Testing subgraphs in directed graphs , J. Comput. Syst. Sci. 69\n(2004), 354–382.\n[3] C. Berge, Graphs , North-Holland, Amsterdam, 1985. Second revised edition of part 1\nof the 1973 English version.\n[4] K. Corr´ adi and A. Hajnal, On the maximal number of independent circuits in a graph ,\nActa Math. Hung. 14(1963), 423–439.\n[5] M.H. El-Zahar, On circuits in graphs , Disc. Math. 50(1984), 227–230.\n[6] A. Ghouila-Houri, Une condition suffisante d’existence d’un circuit hamiltonien , C. R.\nMath. Acad. Sci. Paris 25(1960), 495–497.\n[7] R. H¨ aggkvist, Hamilton cycles in oriented graphs , Combin. Probab. Comput. 2(1993),\n25–32.\n[8] P. Keevash, D. K¨ uhn and D. Osthus, An exact minimum degree condition for Hamilton\ncycles in oriented graphs , J. Lond. Math. Soc. 79(2009), 144–166.\n[9] P. Keevash and B. Sudakov, Triangle packings and 1-factors in oriented graphs , J.\nCombin. Theory Ser. B 99(2009), 709–727.\n[10] L. Kelly, D. K¨ uhn and D. Osthus, A Dirac-type result on Hamilton cycles in oriented\ngraphs , Combin. Probab. Comput. 17(2008), 689–709.\n[11] L. Kelly, D. K¨ uhn and D. Osthus, Cycles of given length in oriented graphs , J. Combin.\nTheory Ser. B 100(2010), 251–264.\n[12] J. Koml´ os, G. N. S´ ark¨ ozy, and E. Szemer´ edi, Blow-up Lemma , Comb. 17(1997), 109–\n123.\n[13] D. K¨ uhn, D. Osthus and A. Young, k-ordered Hamilton cycles in digraphs , J. Combin.\nTheory 98(2008), 1165–1180.\n18[14] N. Lichiardopol, A new lower bound on the strong connectivity of an oriented graph.\nApplication to diameters with a particular case related to Caccetta H¨ aggkvist conjecture ,\nDiscrete Math. 303(2008), 5274–5279.\n[15] J. Shen, Directed triangles in digraphs , J. Combin. Theory Ser. B 74(1998), 405-407.\n[16] C. Thomassen, Long cycles in digraphs with constraints on the degrees , London Math.\nSoc. Lecture Notes 38(1979), 211–228.\n[17] H. Wang, Proof of the Erd˝ os-Faudree conjecture on quadrilaterals , Graphs Combin. 26\n(2010), 833–877.\n[18] H. Wang, Disjoint 5-cycles in a graph , Discuss. Math. Graph Theory 32(2012), 221–\n242.\n[19] H. Wang, Y. Wang, and J. Yan, Disjoint cycles in a digraph with partial degree , SIAM\nJ. Discrete Math 37(2023), 221–232.\n19"    },    {        "title": "2402.03901v1.Batch_Universal_Prediction.pdf",        "content": "arXiv:2402.03901v1  [cs.IT]  6 Feb 2024Batch Universal Prediction\nMarco Bondaschi and Michael Gastpar\nSchool of Computer and Communication Sciences\nEPFL\nSwitzerland\nEmail: {marco.bondaschi, michael.gastpar}@epfl.ch\nAbstract —Large language models (LLMs) have recently gained\nmuch popularity due to their surprising ability at generati ng\nhuman-like English sentences. LLMs are essentially predic tors,\nestimating the probability of a sequence of words given the p ast.\nTherefore, it is natural to evaluate their performance from a\nuniversal prediction perspective. In order to do that fairl y, we\nintroduce the notion of batch regret as a modification of the\nclassical average regret, and we study its asymptotical val ue for\nadd-constant predictors, in the case of memoryless sources and\nfirst-order Markov sources.\nI. I NTRODUCTION\nPrediction refers to the problem of estimating the next\nsymbols of a sequence given its past, and evaluating the\nconfidence of such an estimate. Such a problem appears in\na large number of research areas, such as information the-\nory, statistical decision theory, finance, and machine lear ning.\nSome knowledge about the probability distribution that mod els\nthe sequence to be predicted is clearly helpful. Unfortunat ely,\nin many practical applications such knowledge is partial or\nmissing. If this is the case, then one may wish to say\nsomething about the future of the sequence when the true\nmodel of the source that is producing the symbols is anyof\nthe models belonging to a certain class. This problem usuall y\ngoes under the name of universal prediction [1], and it found\napplications in a wide range of areas, such as compression\n[2], [3], gambling [4] and machine learning [5], [6]. With th e\nrecent rise in popularity of large language models (LLMs), t he\nproblem of universal prediction is more timely than ever. In\nfact, LLMs are essentially predictors: given an n-word input\nsequence, language models can output an estimated probabil ity\nfor the next ℓwords in an online fashion. Furthermore, LLMs\ncan be interpreted as universal predictors, in the sense that\nbefore training, the model does not assume any information\nabout the source generating the input data, and can therefor e\nbe used in principle for next-word prediction of data from an y\ndistribution, the model improving its prediction accuracy as\nmore and more input data are fed into the network.\nFormally, given a finite input alphabet X, for every sequence\nof symbols xi= (x1,x2,...,x i)∈ Xi, a predictor ˆpassigns\na probability ˆp(y|xi)for the(i+1) -th symbol to be equal to\ny∈ X given the past xi. In universal prediction, a predictor is\ngenerally required to perform well if the data is generated\naccording to any distribution in a given class P. In order\nto evaluate the quality of the estimates of a predictor, a\nloss function is used. In language models, this loss functionis usually the logarithmic (or cross-entropy) loss, which is\ndefined point-wise as\nL(ˆp,y|xi) =−logˆp(y|xi). (1)\nIn the case ℓsequential symbols have to be predicted, the\ncumulative loss equals\nL(ˆp,yℓ|xi) =−log ˆp(yℓ|xi),ˆp(yℓ|xi) =ℓ/productdisplay\nj=1ˆp(yj|xi,yj−1).\n(2)\nFor a given ground-truth distribution p∈ P , the regret is\ndefined as the difference between the loss of a candidate\npredictor ˆpand that of p, i.e.,\nR(ˆp,p,yl|xi) =L(ˆp,yℓ|xi)−L(p,yℓ|xi) = logp(yℓ)\nˆp(yℓ|xi).\n(3)\nThe average regret is defined as the expected regret over\nsequences distributed according to p,\nR(ˆp,p) =Ep[R(ˆp,p,Yℓ|Xi)] (4)\n=/summationdisplay\nxip(xi)/summationdisplay\nyℓp(yℓ)logp(yℓ)\nˆp(yℓ|xi). (5)\nThe maximal average regret is the maximum average regret\nover all distributions in P, i.e.,R(ˆp) = max p∈PR(ˆp,p).\nIn universal prediction literature, it is customary to cons ider\nthe regret for the prediction of an entire sequence of nsymbols\n[1], [7]. More precisely, a predictor is defined to output\nan estimated probability for every n-sequence yn, which is\ndenoted by ˆp(yn). The average regret is then Rn(ˆp,p) =/summationtext\nynp(yn)logp(yn)\nˆp(yn). This case has been studied extensively,\nin particular for the memoryless case, where Pis the class\nof distributions generating i.i.d. symbols. For this case, the\nasymptotical expression for Rn(ˆp,p)asn→ ∞ has been\nderived [8]. Furthermore, the add-1\n2predictor, also called\nKrichevsky-Trofimov predictor, has been shown to be almost\nasymptotically optimal, in the sense that its asymptotic re gret\nis larger than the optimal one only by a constant independent\nofn.\nHowever, the rise of LLMs and the importance of viewing\nthem from a universal prediction perspective requires a dif -\nferent paradigm than the one presented above. In fact, LLMs\nare trained and tested on batches of data. In particular, during\nthe training phase a LLM model is fed nbatches of data,\nindependent of each other, each of them made of ℓsamples.At the end of the training phase, the LLM performance is\nthen measured over a fresh test batch of ℓsamples. In order\nto be able to fairly evaluate LLMs from a universal predictio n\nviewpoint, we introduce a new form of regret, batch regret ,\ndefined as follows.\nDefinition 1: Letxn= (x(1),...,x(n))be a training\nsequence of nbatches, each made of ℓsamples, x(j)=\n(x(j)\n1,...,x(j)\nℓ). Letˆp(yℓ|xn)be a predictor that, given xn,\nestimates the probability of a fresh sequence yℓofℓsamples,\ngenerated independently of xn.Batch regret is then defined\nas\nR(ˆp,p)/defines/summationdisplay\nxnp(x(1))···p(x(n))/summationdisplay\nyℓp(yℓ)logp(yℓ)\nˆp(yℓ|xn).\n(6)\nThe focus of this paper is to study the asymptotical batch\nregret for add-constant predictors, in the memoryless and fi rst-\norder Markov cases.\nRemark. We note here that if n= 0, then we fall back into\nthe classic full-sequence setting described above and desc ribed\nin [1], [8]. If instead ℓ= 1, we fall into the next-symbol\nprediction studied by Krichevsky [9]. In the full-sequence\nsetting, the Krichevsky-Trofimov predictor is asymptotica lly\nthe best “add-constant” estimator. In the next-symbol sett ing,\ninstead, the best add-constant estimator requires a slight ly\nlarger constant β0= 0.50922...Our setting can be thought\nas unifying and generalizing those two perspectives.\nNotation. The regime of ℓandnthat is perhaps most\nimportant from a LLM perspective is the one where the batch\nlengthℓ=ℓ(n)is a function of nsuch that limn→∞ℓ(n) =∞\nandlimn→∞ℓ(n)\nn<∞. This is the regime of nandℓ\nconsidered everywhere in this paper, unless differently sp ec-\nified. The little-o notation f(n,ℓ) =o/parenleftbig1\nnℓ/parenrightbig\nmeans that\nlimn→∞nℓ(n)f(n,ℓ(n)) = 0 . All logarithms are considered\nto be in base e.\nA. Related work\nThe asymptotics of average regret for memoryless sources\nhas been fully characterized in [8]. Markov sources from a\nuniversal prediction perspective have been studied in [4], [10],\n[11] for the worst-case regret case and recently in [12], [13 ] for\nthe average regret case. A form of conditional average regre t,\nthat matches batch regret in the special case of memoryless\nsources, was introduced in [14], where it is studied for the\nlocation family of distributions. More general forms of reg ret,\ndefined in terms of Rényi divergence, have been studied in\n[15], [16] for the memoryless case.\nB. Overview\nThe remainder of the paper is organized as follows. In\nSection II we study the binary memoryless case, that is, we\nconsider Pto be the class of distributions generating i.i.d.\nbinary digits. In Section III we study the first-order Markov\ncase, where the next bit is generated according to a distribu tion\nthat solely depends on the previous bit. In this work we\nstick to binary alphabet. However, most of the results canbe generalized to arbitrary finite alphabets, although proo fs\nbecome more involved in the general case.\nII. M EMORYLESS SOURCES\nIn this section we focus on the following setting. Let\nX={0,1}be the binary alphabet, and let Pbe the class\nof memoryless sources, i.e., sources generating i.i.d. bin ary\ndigits with a given probability,\nP={pθ(xi) =θn1(1−θ)n0,θ∈[0,1],for anyi∈N+}\n(7)\nwheren1andn0are the number of ones and zeros in xi,\nrespectively. In this setting, the average regret is\nR(ˆp,θ) =t/summationdisplay\nt1=0/parenleftbiggt\nt1/parenrightbigg\nθt1(1−θ)t0\nℓ/summationdisplay\nℓ1=0/parenleftbiggℓ\nℓ1/parenrightbigg\nθℓ1(1−θ)ℓ0logθℓ1(1−θ)ℓ0\nˆp(yℓ|xt)(8)\nwheret=nℓ. We note here the performance of two naive\npredictors. The first predictor is an add-constant predicto r that\nestimates the probability of the new batch yℓignoring the past.\nSuch a predictor is defined, for a chosen parameter1\n2≤β≤1,\nby\nˆp(yℓ|xt) =ℓ/productdisplay\ni=1ˆp(yi|xt,yi−1), (9)\nwhere\nˆp(yi= 1|xt,yi−1) =ℓ(i−1)\n1+β\ni−1+2β(10)\nandℓ(i−1)\n1 is the number of ones in the sequence yi−1.\nFor such a predictor, the average regret simply becomes the\naverage regret in the full-sequence of length ℓdescribed\nbefore. In this case, β=1\n2achieves the lowest regret, which\nasymptotically equals R(ˆp1/2)≈1\n2logℓ+c. The second\nnaive predictor estimates the probability of a sequence onl y\nconsidering the training sequence xtand none of the past\nsymbols of yℓ. Such a predictor is\nˆpβ(yℓ|xt) =/parenleftbiggt1+β\nt+2β/parenrightbiggℓ1/parenleftbiggt0+β\nt+2β/parenrightbiggℓ0\n. (11)\nIn this case the regret would be equal to ℓtimes the Krichevsky\nnext-symbol regret. The best constant would then be β=β0\nand the asymptotic regret would be R(ˆp)≈β0ℓ\nt+o(1\nt) =β0\nn+\no(1\nnℓ). However, both these naive predictors are suboptimal.\nIn this section we study the add-constant predictor that tak es\ninto account both xtandyℓat the same time. The predictor\nis defined by\nˆpβ(yℓ|xt) =ℓ/productdisplay\ni=1ˆpβ(yi|xt,yi−1), (12)\nwhere\nˆpβ(yi= 1|xt,yi−1) =t1+ℓ(i−1)\n1+β\nt+i−1+2β(13)for a chosen1\n2≤β≤1. One can think of this predictor as a\nrefined version of the one in (11), where the added constant\nis updated every time a new symbol from yℓis revealed. The\npredictor in (12) can be rewritten using Gamma functions as\nˆpβ(yℓ|xt) =Γ(t1+ℓ1+β)Γ(t0+ℓ0+β)Γ(t+2β)\nΓ(t1+β)Γ(t0+β)Γ(t+ℓ+2β)(14)\nor again, using properties of the Gamma function, as\nˆpβ(yℓ|xt) =/integraldisplay1\n0wβ(θ|xt)pθ(yℓ)dθ. (15)\nHence, the predictor can also be interpreted as a mixture\npredictor , where the prior distribution on the parameter space\nwβdepends on the training sequence xt.\nFor this predictor we have the following results about the\nbatch regret. Theorem 1 concerns the regret in the interior o f\nthe simplex, while Theorem 2 concerns the boundary.\nTheorem 1: Letδ <1\n2andΘ = [δ,1−δ]. Let\nPδ={pθ(xi) =θn1(1−θ)n0,θ∈Θ,for anyi∈N+}(16)\nbe the class of distributions under consideration. Then,\nmax\nΘR(ˆpβ,θ) =1\n2logt+ℓ\nt+o/parenleftbigg1\nt/parenrightbigg\n(17)\n=1\n2log/parenleftbigg\n1+1\nn/parenrightbigg\n+o/parenleftbigg1\nnℓ/parenrightbigg\n(18)\nProof: The following proof follows and extends the one\nin [8]. We start by rewriting the regret as\nR(ˆpβ,θ) =z/summationdisplay\nz1=0/parenleftbiggz\nz1/parenrightbigg\nθz1(1−θ)z0logθz1(1−θ)z0\nΓ(z1+β)Γ(z0+β)\nΓ(z+2β)\n−t/summationdisplay\nt1=0/parenleftbiggt\nt1/parenrightbigg\nθt1(1−θ)t0logθt1(1−θ)t0\nΓ(t1+β)Γ(t0+β)\nΓ(t+2β)(19)\nwherez=t+ℓ. We analyze the first term in the equation. The\nsecond term can be analyzed similarly. We will extensively u se\nthe equality\nΓ(x) =√\n2πxx−1\n2e−xes(20)\nfor some s∈[0,1\n12x]. Using this equation in the first term of\nthe right-hand side of (19) we get\nz/summationdisplay\nz1=0/parenleftbiggz\nz1/parenrightbigg\nθz1(1−θ)z0logθz1(1−θ)z0\nΓ(z1+β)Γ(z0+β)\nΓ(z+2β)\n=z/summationdisplay\nz1=0/parenleftbiggz\nz1/parenrightbigg\nθz1(1−θ)z0log/braceleftBigg/radicalBigg\n(z1+β)(z0+β)\n2π(z+2β)\n·θz1(1−θ)z0/parenleftbiggz1+β\nz+2β/parenrightbigg−z1−β/parenleftbiggz0+β\nz+2β/parenrightbigg−z0−β\n·es(z)−s1(z1)−s2(z0)/bracerightBiggWe can split the last expression into three terms that will be\nanalyzed separately:\n−z/summationdisplay\nz1=0/parenleftbiggz\nz1/parenrightbigg\nθz1(1−θ)z0/parenleftbigg\nz1+β−1\n2/parenrightbigg\nlog(z1+β)\n−z/summationdisplay\nz0=0/parenleftbiggz\nz0/parenrightbigg\nθz1(1−θ)z0/parenleftbigg\nz0+β−1\n2/parenrightbigg\nlog(z0+β)(A)\n−1\n2log(2π)+/parenleftbigg\nz+2β−1\n2/parenrightbigg\nlog(z+2β) (B)\nz/summationdisplay\nz1=0/parenleftbiggz\nz1/parenrightbigg\nθz1(1−θ)z0(s(z)−s1(z1)−s0(z0)) (C)\nFor term (A), consider only the first half (the second follows\nby symmetry), with the positive sign for simplicity. We can\nsplit it into two further terms\nz/summationdisplay\nz1=0/parenleftbiggz\nz1/parenrightbigg\nθz1(1−θ)z0/parenleftbigg\nz1+β−1\n2/parenrightbigg\nlog/parenleftbigg\nz1+β−1\n2/parenrightbigg\n(A’)\nz/summationdisplay\nz1=0/parenleftbiggz\nz1/parenrightbigg\nθz1(1−θ)z0/parenleftbigg\nz1+β−1\n2/parenrightbigg\nlog/parenleftbigg\n1+1\n2(z1+β−1\n2)/parenrightbigg\n(A”)\nWe analyze term (A’), which equals E[(Z1+β−1\n2)log(Z1+\nβ−1\n2)]forZ1∼Binomial( z,θ). Using the bounds (valid\nfora≥0andb >0)\naloga≥blogb+(a−b)(1+log b)+(a−b)2\n2b−(a−b)3\n6b2\n(21)\nand\naloga≤blogb+(a−b)(1+log b)+(a−b)2\n2b\n−(a−b)3\n6b2+(a−b)4\n3z3(22)\nwitha=Z1+β−1\n2andb=zθ+β−1\n2, we have\nE/bracketleftbigg/parenleftbigg\nZ1+β−1\n2/parenrightbigg\nlog/parenleftbigg\nZ1+β−1\n2/parenrightbigg/bracketrightbigg\n≥/parenleftbigg\nzθ+β−1\n2/parenrightbigg\nlog/parenleftbigg\nzθ+β−1\n2/parenrightbigg\n+1−θ\n2−1\nzθ(23)\nand\nE/bracketleftbigg/parenleftbigg\nZ1+β−1\n2/parenrightbigg\nlog/parenleftbigg\nZ1+β−1\n2/parenrightbigg/bracketrightbigg\n≤/parenleftbigg\nzθ+β−1\n2/parenrightbigg\nlog/parenleftbigg\nzθ+β−1\n2/parenrightbigg\n+1−θ\n2+1\nzθ(24)forzlarge enough such that zθ≥2. For the term (A”) we\nhave\n1\n2−1\n2zθ≤E/bracketleftbigg/parenleftbigg\nZ1+β−1\n2/parenrightbigg\nlog/parenleftbigg\n1+1\n2(Z1+β−1\n2)/parenrightbigg/bracketrightbigg\n≤1\n2\n(25)\nwhich follows from the fact that for a≥0,\n1\n2−1\n2(a+1)≤alog/parenleftbigg\n1+1\n2a/parenrightbigg\n≤1\n2(26)\nand from the fact that E[1/(Z1+1)]≤1/zθ. The (straightfor-\nward) analysis of terms (B) and (C) is left in the Appendix.\nThe same analysis worked out for the first term of (19) can\nbe carried out for the second term as well, just with tin place\nofz. Putting the bounds together leads to the theorem.\nTheorem 2: Letθ∈ {0,1}. Then\nR(ˆpβ,θ) =βlogt+ℓ\nt+o/parenleftbigg1\nt/parenrightbigg\n(27)\n=βlog/parenleftbigg\n1+1\nn/parenrightbigg\n+o/parenleftbigg1\nnℓ/parenrightbigg\n(28)\nProof: See the Appendix.\nNote that Theorem 1 and 2 are not enough to prove the\nexact asymptotics of the batch regret in the case the class of\ndistributions is parametrized by the entire interval Θ = [0,1].\nHow the regret behaves in this case is open and left for future\nwork.\nIII. F IRST-ORDER MARKOV SOURCES\nIn this section we deal with the class of binary Markov\nsources of order 1, i.e., the class\nP=\n\npθ(xi) =p1(x1)i/productdisplay\nj=2p(xj|xj−1)\n\n. (29)\nFor notation purposes we define p1=p1(x1= 1) ,p=p(1|0),\nq=p(0|1)andθ= (p1,p,q)∈[0,1]3. While in traditional\nfull-sequence universal prediction or in the Krichevsky ne xt-\nsymbol prediction, the value of the initial distribution p1is\nasymptotically unimportant, in the batch prediction setti ng\ndescribed in this paper it gets a more prominent role. In fact ,\nthe training data is a collection of nfresh batches of ℓbits\neach, where each batch x(i)is generated independently of the\nothers according to a fixed Markov source, i.e.,\npθ(x(i)) =p1(x(i)\n1)ℓ/productdisplay\nj=2p(x(i)\nj|x(i)\nj−1). (30)\nDue to this different feature, depending on the regime of ℓand\nn, the starting distribution p1becomes fundamental. Hence, we\nmust introduce an estimator for the starting distribution p1as\nwell. For a predictor in the form\nˆp(yℓ|xn) = ˆp1(y1|xn)ˆp(yℓ\n2|xn,y1) (31)the batch regret equals\nR(ˆp,θ) =/summationdisplay\nxnpθ(xn)···pθ(x(n))/summationdisplay\nyℓpθ(yℓ)logpθ(yℓ)\nˆp(yℓ|xn)\n=/summationdisplay\nxnpθ(xn)/summationdisplay\nyp1(y)logp1(y)\nˆp1(y|xn)\n+/summationdisplay\nxnpθ(xn)/summationdisplay\nyp1(y)/summationdisplay\nyℓ\n2pθ(yℓ\n2|y1)logpθ(yℓ\n2|y1)\nˆpβ(yℓ\n2|xn,y)\n(32)\nwherepθ(xn) =/producttextn\ni=1pθ(x(i)). The last expression shows\nthat the regret can be seen as the sum of two terms: the first\nterm is the regret for the estimation of the initial distribu tion,\nwhile the second term is the regret for the estimation of the\ntransition probabilities of the Markov source. We now deriv e\nupper bounds for the two terms for Markov sources with pos-\nitive transition probabilities. Both bounds are asymptoti cally\nproportional to1\nn, so that neither of them can be neglected.\nTheorem 3: LetPbe the class of first-order binary Markov\nsources as in (29). The initial distribution regret\nR1(ˆp,θ) =/summationdisplay\nxnpθ(xn)/summationdisplay\nyp1(y)logp1(y)\nˆp1(y|xn)(33)\nis upper bounded by\nmin\nˆp1max\nθR1(ˆp,θ)≤β0\nn+o/parenleftbigg1\nn/parenrightbigg\n. (34)\nProof: Letx1= (x(1)\n1,x(1)\n1,...,x(n)\n1)be the sequence\nof bits in the first coordinate of each batch, and consider the\npredictor\nˆp1(y|xn) = ˆp1(y|x1) =t1+β0\nn+2β0(35)\nwheret1=/summationtextn\ni=1x(i)\n1is the number of ones in x1. Then,\nR1(ˆp,θ) =/summationdisplay\nx1p1(x1)/summationdisplay\nyp1(y)logp1(y)\nˆp1(y|x1). (36)\nThe last expression is precisely Krichevsky’s next-symbol\nregret for the add- β0predictor. Hence, Krichevsky’s bound\ncan be applied, leading to\nmax\nθR1(ˆp,θ)≤β0\nn+o/parenleftbigg1\nn/parenrightbigg\n. (37)\nFurthermore, if we limit the class of predictors to those\nthan only depend on the first coordinate of each batch,\nx1= (x(1)\n1,x(1)\n1,...,x(n)\n1), then the following lower bound\nonR1(ˆp,θ)follows directly from Krichevsky’s lower bound\non next-symbol prediction [9, Theorem 2].\nTheorem 4: LetˆPbe the class of predictors ˆp1(y|xn) =\nˆp1(y|x1)that only depend on x1. Then,\nmin\nˆp1∈ˆPmax\nθR1(ˆp,θ)≥1\n2n+o/parenleftbigg1\nn/parenrightbigg\n. (38)Remark. Theorem 3 suggests that one should use an add-\nβ0predictor with the first coordinate of the nbatches when\npredicting the initial distribution of a Markov source. How ever,\none can possibly achieve lower regret with a predictor that a lso\nuses the other coordinates of the batches to estimate the ini tial\ndistribution of the Markov source. In fact, information abo ut\np1also leaks to other coordinates. Note that the second symbol\nof each batch is distributed according to\nPr(X2= 1) =p1(1−p−q)+p. (39)\nBy inverting the last expression, one gets\np1=Pr(X2= 1)−p\n1−p−q(40)\nTherefore, one can combine estimators for Pr(X2= 1) ,\npandqto derive a predictor for p1. Obvious choices\nwould be to estimate Pr(X2= 1) with an add-constant\nestimator based on the counts in the second coordinate of\nthenbatchesxn, while add-constant predictors based on\nthe counts of transitions in the entire training set could be\nused to estimate pandq. One can derive similar predic-\ntors for p1from the other coordinates as well, since in\ngeneralPr(Xj= 1) = (p1−π1)(1−p−q)j−1+π1, where\nπ1=p\np+qis the probability assigned to 1by the stationary\ndistribution of the Markov source. It is therefore natural t o\naverage those ℓpredictors of p1to achieve lower regret than\nthe one obtained by only considering the first coordinate of\nthe batches.\nThe following is an upper bound for the regret on the\ntransition probabilities of the Markov source, provided th at\nall transition probabilities are bounded away from zero.\nTheorem 5: Let0< δ <1\n2and letPδbe the class of\nfirst-order binary Markov sources as in (29), such that p,q∈\n[δ,1−δ]. The transition probability regret\nRT(ˆp,θ) =/summationdisplay\nxnpθ(xn)/summationdisplay\nyℓpθ(yℓ)logpθ(yℓ\n2|y1)\nˆp(yℓ\n2|xn,y1)(41)\nis upper-bounded by\nmin\nˆpmax\npθ∈PδRT(ˆp,θ)≤1\n2n+o/parenleftbigg1\nnℓ/parenrightbigg\n. (42)\nProof: Consider the predictor\nˆp(yℓ\n2|y1) =/productdisplay\nh,k∈{0,1}2/parenleftbiggthk+1\n2\nth+1/parenrightbiggℓhk\n. (43)\nLetπbe the stationary distribution of the Markov source. Note\nthatE[Lhk] =ℓπ(h)p(k|h)+o(1\nℓ). Furthermore, we have\nE[Thk] =n/summationdisplay\ni=1E[T(i)\nhk] (44)\n=nℓπ(h)p(k|h)+o(n/ℓ) (45)Hence, we can write\nRT(ˆp,θ) =E\n/summationdisplay\nhkLhklogp(k|h)\nThk+1\n2\nTh+1\n (46)\n=/summationdisplay\nhkE[Lhk]E\nlogp(k|h)\nThk+1\n2\nTh+1\n (47)\n=ℓ/summationdisplay\nhπ(h)E\n/summationdisplay\nkp(k|h)logp(k|h)\nThk+1\n2\nTh+1\n+o/parenleftbigg1\nnℓ/parenrightbigg\n.\n(48)\nThe expectation in the last expression is the estimation risk\ndenoted by ˜ǫin [13], except for one key difference: here the\nexpectation is over nindependent batches of length ℓeach,\nwhile in [13] it is over one batch of length nℓ. However, the\nproof provided in [13, Section 10] also works in the batch cas e\npresented here. In fact, the key ingredient in the original p roof\nis the fact that ThandThkare highly concentrated around their\nmean. The same holds in the batch case, since ThandThkare\nsums of independent, highly concentrated random variables :\napplying [13, Lemma 19] and Hoeffding’s inequality to Thk\nandThgives the concentration bound Pr(|Th−nℓπ(h)|>\nt)/lessorsimilarexp(−t2/nℓ). Hence, one can prove that\nE\n/summationdisplay\nkp(k|h)logp(k|h)\nThk+1\n2\nTh+1\n≤1\n2nℓ+o/parenleftbigg1\nnℓ/parenrightbigg\n. (49)\nPutting this into (48) leads to the theorem.\nSimilarly as the bound on the regret for the initial distri-\nbution, also in this case a lower bound proportional to1\nncan\nbe derived if one limits the class of predictors to those in th e\nform (43). Using also the latter information can lead to a low er\nregret. A natural modified version of the add-constant predi ctor\nfor Markov sources that uses all available information take s\nthe form\nˆpβ(yℓ|xn) = ˆp1(y1|xn)ℓ/productdisplay\nj=2ˆpβ(yj|xn,yj−1), (50)\nwhere\nˆpβ(yj=k|xn,yj−2,yj−1=h) =thk+ℓ(j−1)\nhk+β\nth+ℓ(j−1)\nh+2β(51)\nwhere for h,k∈ {0,1}we define\nthk=n/summationdisplay\ni=1t(i)\nhk, t(i)\nhk=n. of consecutive hkinx(i)(52)\nth=n/summationdisplay\ni=1t(i)\nh, t(i)\nh=n. ofhinx(i)\\x(i)\nℓ(53)\nand where ℓ(j−1)\nhkis the number of consecutive hkinyj−1,\nwhileℓ(j−1)\nhis the number of hinyj−2. Furthermore,\nˆp1(y1|xt)is the predictor for the initial distribution p1. How-\never, we conjecture that one cannot find a predictor with a\nregret that decays asymptotically faster than1\nn, even if the\nmultiplying constant of the leading term may improve.REFERENCES\n[1] N. Merhav and M. Feder, “Universal prediction,” IEEE Trans. Inf.\nTheory , vol. 44, no. 6, pp. 2124–2147, 1998.\n[2] J. Ziv and A. Lempel, “A universal algorithm for sequenti al data\ncompression,” IEEE Trans. Inf. Theory , vol. 23, no. 3, pp. 337–343,\n1977.\n[3] F. Willems, Y . Shtarkov, and T. Tjalkens, “The context-t ree weighting\nmethod: basic properties,” IEEE Trans. Inf. Theory , vol. 41, no. 3, pp.\n653–664, 1995.\n[4] Q. Xie and A. Barron, “Asymptotic minimax regret for data compression,\ngambling, and prediction,” IEEE Trans. Inf. Theory , vol. 46, no. 2, pp.\n431–445, 2000.\n[5] Y . Fogel and M. Feder, “Universal learning of individual data,” in Proc.\n2019 IEEE Int. Symp. Inf. Theory (ISIT) , 2019, pp. 2289–2293.\n[6] F. E. Rosas, P. A. M. Mediano, and M. Gastpar, “Learning, c ompression,\nand leakage: Minimising classification error via meta-univ ersal compres-\nsion principles,” in Proc. 2020 IEEE Inf. Theory Workshop (ITW) , 2021.\n[7] P. D. Grünwald, The minimum description length principle . Cambridge,\nMA: MIT Press, 2007.\n[8] Q. Xie and A. Barron, “Minimax redundancy for the class of memoryless\nsources,” IEEE Transactions on Information Theory , vol. 43, no. 2, pp.\n646–657, 1997.[9] R. Krichevskiy, “Laplace’s law of succession and univer sal encoding,”\nIEEE Transactions on Information Theory , vol. 44, no. 1, pp. 296–303,\n1998.\n[10] B. Y . Ryabko, “Twice-universal coding,” Problems of information trans-\nmission , vol. 20, no. 3, pp. 173–177, 1984.\n[11] ——, “Prediction of random sequences and universal codi ng,” Problems\nof information transmission , vol. 24, no. 2, pp. 87–96, 1988.\n[12] M. Falahatgar, A. Orlitsky, V . Pichapati, and A. T. Sure sh, “Learning\nmarkov distributions: Does estimation trump compression? ” in 2016\nIEEE International Symposium on Information Theory (ISIT) . IEEE,\n2016, pp. 2689–2693.\n[13] Y . Hao, A. Orlitsky, and V . Pichapati, “On learning mark ov chains,”\nAdvances in Neural Information Processing Systems , vol. 31, 2018.\n[14] F. Liang and A. Barron, “Exact minimax strategies for pr edictive density\nestimation, data compression, and model selection,” IEEE Transactions\non Information Theory , vol. 50, no. 11, pp. 2708–2726, 2004.\n[15] S. Yagli, Y . Altu ˘g, and S. Verdú, “Minimax Rényi redundancy,” IEEE\nTrans. Inf. Theory , vol. 64, no. 5, pp. 3715–3733, 2018.\n[16] M. Bondaschi and M. Gastpar, “Alpha-nml universal pred ictors,” in 2022\nIEEE International Symposium on Information Theory (ISIT) . IEEE,\n2022, pp. 468–473.APPENDIX\nA. Analysis of terms (A) and (B) for the proof of Theorem 1\nFor term (B), we can use the bound\nx\n1+x≤log(1+x)≤x (54)\nvalid for x >−1, to easily get\n1\n2log(2π)−/parenleftbigg\nz+2β−1\n2/parenrightbigg\nlog(z+2β)\n≤1\n2log(2π)−zlogz−/parenleftbigg\n2β−1\n2/parenrightbigg\nlogz−2β+β\nz(55)\nand\n1\n2log(2π)−/parenleftbigg\nz+2β−1\n2/parenrightbigg\nlog(z+2β)\n≥1\n2log(2π)−zlogz−/parenleftbigg\n2β−1\n2/parenrightbigg\nlogz−2β−3β\nz.(56)\nFinally, for term (C) we have\nE[s(z)−s1(Z1)−s0(Z0)]≤1\n12z(57)\nand\nE[s(z)−s1(Z1)−s0(Z0)] (58)\n≥ −E/bracketleftbigg1\n12(Z1+β)/bracketrightbigg\n−E/bracketleftbigg1\n12(Z0+β)/bracketrightbigg\n(59)\n≥ −1\n12zθ−1\n12z(1−θ). (60)\nwhere in the last step we used E/bracketleftBig\n1\nZ1+1/bracketrightBig\n≤1\nzθ.\nB. Proof of Theorem 2\nWe prove the case θ= 1. The case θ= 0 follows in the\nsame way due to symmetry. We have\nR(ˆp,θ= 1) = logΓ(t+ℓ+2β)Γ(t+β)\nΓ(t+ℓ+β)Γ(t+2β)(61)\n= logΓ(t+ℓ+2β)\nΓ(t+ℓ+β)+logΓ(t+β)\nΓ(t+2β).(62)\nUsing the double inequality\nx\n(x+s)1−s≤Γ(x+s)\nΓ(x)≤xs(63)\nvalid for x >0and0< s <1, we get the upper bound\nR(ˆp,θ= 1)≤βlog(t+ℓ+β)+log(t+2β)1−β\nt+β(64)\n≤βlogt+ℓ\nt+β2\nt+ℓ+2β(1−β)\nt(65)\nand the lower bound\nR(ˆp,θ= 1)≥logt+ℓ+β\n(t+ℓ+2β)1−β−βlog(t+β)(66)\n≥βlog/parenleftbiggt+ℓ\nt/parenrightbigg\n−β\nt+ℓ−β2\nt. (67)\nThese two bounds together prove the theorem."    },    {        "title": "2402.03911v1.Stability_of_Rotating__Charged_Fluids__Generalization_of_the_Hoiland_Conditions_in_Newtonian_Non_conductive_Case.pdf",        "content": "Stability of Rotating, Charged Fluids: Generalization of the Høiland Conditions\nin Newtonian Non-conductive Case\nKris Schroven,∗Vladim´ ır Karas, and Jiˇ r´ ı Hor´ ak\nAstronomical Institute, Czech Academy of Sciences, Boˇ cn´ ı II 1401, 14100 Prague, Czech Republic\nAudrey Trova†and Eva Hackmann\nUniversity of Bremen, Center of Applied Space Technology and Microgravity (ZARM), 28359 Bremen, Germany\n(Dated: February 7, 2024)\nWe study the conditions for stability of electrically charged, non-conductive perfect fluid tori with\nrespect to linear perturbations. To this end we employ Lagrangian perturbation formalism and we\nassume a system where the fluid orbits a central body. Gravitational field of the latter is described\nin the Newtonian framework. We first formulate the criteria valid for a general, non-axisymmetric\nsituation, and then we concentrate on the axisymmetric model in more detail. In the latter case\nwe generalize the Høiland criterion of stability to non-vanishing electric charge and classify special\nexamples. Toroidal structures with constant angular momentum distribution are found to be linearly\nstable. Subsequently, like in the uncharged case, rotating charged fluids are found to be unstable\nwith respect to non-axisymmetric perturbations.\nI. INTRODUCTION\nEquilibrium configurations of a rotating fluid and the\nconditions for their stability play an important role in\nastrophysics [1, 2]. Previously, we discussed a special\ncase of electrically charged, non-conductive, energetically\nbound figures of fluid equilibria around black holes [3–\n9]. A possibility of (even small) non-vanishing electrical\ncharge greatly enlarges the parameter space of the sys-\ntem. The astrophysical motivation to explore the equi-\nlibrium structures of electrically charged, orbiting fluid\narises from the fact that combined action of irradiation\nand rapid rotation are expected to lead to electric cur-\nrents and charge separation. Unfortunately, no single\napproach is adequate to tackle in full generality the com-\nplex medium that emerges in these circumstances (how-\never, see e.g. reference [10] for discussion). We adopt\nthe fluid description in terms of magnetohydrodynami-\ncal scenario, nonetheless, even in this framework the in-\ntroduction of electromagnetic terms requires additional\nsimplifying assumptions about their microphysical origin\n[11, 12]. A globally neutral astrophysical system can de-\nvelop separated regions with non-zero net charge of the\nfluid which occur in vast range of systems from plane-\ntary neighborhood to accreting black holes. Numerical\nintegration of the equations of motion is probably the\nonly way to solve these structures under astrophysically\nrealistic circumstances, but it often fails to give full un-\nderstanding. On the other hand, analytical techniques\nare restricted to highly symmetrical configurations. Here\nwe adopt the latter approach and we confine ourselves to\naxially and symmetrical, stationary perfect fluids.\nA follow-up question arises: under which conditions\nthe equilibrium solutions are stable. In this paper we will\n∗k.schroven@t-online.de\n†audrey.trova@zarm.uni-bremen.deadopt the traditional approach to the stability analysis\nin the linear regime, which was applied previously to the\nneutral fluid and ideal magnetohydrodynamic equations\n(see e.g. [1, 13–16]). This resulted in the Høiland equa-\ntion. Here, we use the method of a linear Lagrangian\nperturbation theory to set up general stability equa-\ntions. We can show, that like in the uncharged case,\nrotating charged fluids are unstable with respect to non-\naxisymmetric perturbations. For axisymmetric perturba-\ntions we can formulate a corresponding generalized ver-\nsion of the Høiland criterion relevant to our specific setup\nof a charged, non-conductive fluid in an electromagnetic\nbackground field.\nThe Lagrangian perturbation approach will be intro-\nduced in section II. In section III we will introduce the\nfluid equations for a non-conductive, charged fluid im-\nmersed in an electromagnetic background field. The La-\ngrangian approach is then used in section IV to set up the\ngeneral linear stability equations for the charged fluid and\nthe instability of rotating, charged fluids with regards\nto non-axisymmetric perturbations is discussed briefly.\nThe generalized Høiland conditions for a charged, non-\nconductive fluid are derived in section V. These general-\nized conditions are afterwards discussed for selected spe-\ncial cases of model parameters, like angular momentum\ndistributions and background fields, which were discussed\nin previous works concerning the existence of bound,\ncharged fluid structures. A conclusion is given in sec-\ntion VII.\nII. LAGRANGIAN PERTURBATION\nAPPROACH\nWhile Eulerian perturbation theory describes the\nchange of fluid variables at a fixed point in space, the La-\ngrangian approach relates fluid variables in equilibrium\nto the perturbed ones by a Lagrangian displacement vec-\ntor field. This results in a description of the change ofarXiv:2402.03911v1  [gr-qc]  6 Feb 20242\nthe fluid variables with respect to a frame dragged along\nwith the displacement [13]. The Lagrangian formalism\nwas introduced by Lebovitz [17] and later on widely used\nto discuss the stability of (magnetohydrodynamic) flu-\nids in a Newtonian and general relativistic setup (e.g.\n[13, 14, 16, 18–20]).\nWe denote the Eulerian physical coordinate by x, and\nthe Lagrangian initial coordinate by r. In Lagrangian\nperturbation theory, the dynamical variable is the La-\ngrangian displacement field ξ(r, t), arising from devia-\ntions of a fluid line from its original course due to small\nperturbations of the fluid\nx(t) :=r+ξ(r, t). (1)\nIt is worth mentioning that ξ(r, t) = 0 leads to rbe the\nsame as the usual comoving coordinate at initial time\nt= 0.\nThe Lagrangian perturbation approach describes the\nperturbation of a fluid element with respect to a”La-\ngrangian frame“– a frame that is embedded in the fluid\nand dragged along the perturbation direction. The La-\ngrangian perturbation ∆ Qof an arbitrary tensor quan-\ntityQis then related to the Eulerian perturbation δQby\n[13]:\n∆Q=δQ+LξQ , (2)\nwhere the Lie derivative Lξis given by\nLξf=ξj∇jf, (3)\nfor scalar fields fand by\nLξui=ξj∇jui−uj∇jξi, (4)\nLξui=ξj∇jui+uj∇iξj. (5)\nfor contra- and co-variant vector fields ui,ui. Here, the\nEinstein summation notation is used.\nIn order to express the conservation of mass and equa-\ntion of state in terms of Lagrangian displacement follow-\ning [13] we consider\n∆ui=∂tξi, (6)\n∆ρ=−ρ∇iξi, (7)\nwhere uiis the fluid velocity and ρis the density. In\naddition, by utilizing the adiabatic assumption, as well\nas a polytropic equation of state, one finds that\n∆p=γp\nρ∆ρ and ∆ S= 0. (8)\nHere pis the pressure and γ= (∂lnp/∂lnρ)Sis the\nadiabatic index, while Sis the entropy of the fluid.A. Canonical energy and canonical displacements\nAn important fundamental property of the Lagrangian\nperturbation approach is gauge freedom related to the\ndisplacement vector field ξ. In fact, there is a class of\ntrivial displacements η, for which the corresponding Eu-\nlerian change in pressure, density, velocity and entropy of\nfluid vanishes. As a result, two canonical displacements\nξiandˆξicorrespond to the same physical perturbation\nif they differ only by a trivial displacement:\nξi=ˆξi+ηi. (9)\nHowever, there is a subtlety here. Even though ξandˆξ\ndescribe identical physical perturbations, the canonical\nenergy Ecis not preserved\nEc(ξi)̸=Ec(ˆξi+ηi). (10)\nThe canonical energy is defined in terms of the symplectic\nstructure W(ξ, ∂tξ) of a Lagrangian displacement and its\ntime derivative [13]\nEc:=1\n2W(ξ, ∂tξ) =1\n2H(ξ, ξ), (11)\nThe canonical energy plays a key role in the formula-\ntion of the stability criterion for a fluid configuration in\nthe following sense [21]:\nThe configuration is\n1. dynamically stable, if Ec(ξ)>0,\n2. dynamically unstable, if Ec(ξ) = 0,\n3. secularly unstable, if Ec(ξ)<0\nfor all initial conditions (in the last case, in particular,\nfor non-axisymmetric conditions).\nThis classification has a straightforward interpretation:\nif the canonical energy is positive for all possible config-\nurations, the perturbation can only die away over time,\nwhile Ecis decreasing. In the case that the canonical en-\nergy becomes negative for any possible configuration, the\nperturbations does not die away, as Eccan only decrease\nand the system is dynamically unstable.\nWe restrict the Lagrangian displacements to a subclass\nof so-called canonical displacements , which are orthogo-\nnal to all trivial displacements with respect to the sym-\nplectic product,\nW(ξ, η) = 0 ,∀η, (12)\nwhere ηis the trivial displacement [13]. Therefore, for\nthis subclass of canonical displacements the canonical en-\nergy is invariant.3\nIII. A CHARGED, NON-CONDUCTIVE FLUID\nIN THE PRESENCE OF AN\nELECTROMAGNETIC FIELD\nA charged, non-conductive fluid in the vicinity of an\nelectromagnetic background field is described by the fol-\nlowing set of equations; the Euler equation of a charged,\nnon-conductive fluid and the continuity equations for\nmass and charge, respectively\nDua\nDt+∇ap\nρ+∇aΦG−q\nρ\u0000\nFabub− ∇ aΦE\u0001\n= 0,(13)\n∂tρ+∇b(ρub) = 0 ,(14)\n∂tq+∇b(qub) = 0 .(15)\nHere, the external gravitational and electrostatic field\nare described by their potentials Φ Gand Φ E, respec-\ntively. Magnetic field is represented by the antisymmet-\nric Maxwell tensor Fab(Fab=∇aBb− ∇ bBa) and ρand\nqare mass and charge densities. Each term in Eq. (13)\nrepresents different force, that arises due to internal pres-\nsure gradients, gravity of the central compact body, and\nthe electromagnetic forces acting on the charged fluid\nmedium, respectively [4, 22]. Finally, we introduced the\ntotal time derivative\nD\nDt:=∂t+uj∇j (16)\nin Eq. (13). In the present discussion we neglect the fluid\nself-gravity and the electric current self-interaction. For\nsimplicity reasons we introduce the specific charge\nq\nρ:= ˆq . (17)\nFrom equations (14) and (15) we find that\nD\nDtˆq= 0, (18)\nwhich basically describes that the specific charge is pre-\nserved along the fluid course (i.e. the specific charge is a\nmaterial invariant).\nWithin the setup of our charged, non-conductive fluid\nwe will use the Lagrangian perturbation approach, de-\nscribed above. Next to the expressions for the Lagrangian\nperturbation of the fluid quantities pressure, mass den-\nsity, entropy and fluid velocity, already introduced in [13]\n(see Eqs. (6)–(8)), we receive for the Lagrangian pertur-\nbation of the specific charge\n∆ˆq= 0. (19)\nAdditionally, we derive the orthogonality condition be-\ntween the canonical and trivial displacements in our\ncharged setup, by closely following the procedure used\nin [13]:\nεijk∇iˆq∇j\u0010\n∆ˆξuk−ˆqFkbˆξb\u0011\n= 0, (20)where ∆ ˆξis the Lagrangian perturbation related to the\ndisplacement ˆξ. Our stability discussion will be restricted\nto the subclass of canonical displacements ˆξithat satisfy\nEq. (20).\nIV. STABILITY DISCUSSION WITH THE HELP\nOF THE CANONICAL ENERGY\nIt is our aim in this section to first derive the canon-\nical energy of the perturbed charged fluid system, and\nafterwards use it for further stability discussions, by fol-\nlowing the approach used in [13]. We are only inter-\nested in perturbing the fluid and not the external grav-\nitational or electromagnetic field. Therefore, we assume\nthat the Eulerian perturbations of these external fields\nvanish, δFab=δ∇aΦG=δ∇aΦE= 0. Furthermore,\nwe assume, that all discussed systems have a vanishing\ncharge and mass density at infinity. With these assump-\ntions, the Lagrangian perturbation of the Euler equation\n(13) is given by\n∆E=∂2\ntξa+ (2gabuc∇c−ˆqFab)∂tξb+\u0000\nub∇b\u00012ξa\n+1\nρ\u0000\n∇bξb∇ap− ∇ aξb∇bp− ∇ a(γp∇bξb)\u0001\n+ξb∇b∇aΦG+ ˆqξb∇b∇aΦE\n−ˆq\u0000\nFabuγ∇γξb+ξb∇bFaγuγ\u0001\n= 0.(21)\nWithout electric and magnetic fields, this result agrees\nwith the previous works. However even with them, the\nequation is of the form\nAab∂2\ntξb+Bab∂tξb+Cabξb= 0. (22)\nIfAab,BabandCabare hermitian, anti-hermitian and\nhermitian, respectively (see Appendix A for proof), Eq.\n(21) can be derived from a variational principle with the\naction potential\nI:=Z\nLdV\n=1\n2Z\n˙ξaAab˙ξb+˙ξaBabξb−ξaCabξbdV .(23)\nWe can now calculate the canonical energy Ecof the\nperturbed system, using the Lagrangian L. From (11)\nwe obtain\nEc=1\n2H(ξa, ξa) =Z\n˙ξa\u0012∂L\n∂˙ξa−L\u0013\ndV (24)\nBy using Eq. (21), the expression for the canonical en-\nergy in our set-up is explicitly given by4\nEc=1\n2Zn\nρ|˙ξ|2−ρ|uγ∇γξ|2+γp|∇aξa|2+ξa∗∇ap∇bξb+ξb∇bp∇aξa∗+ξa∗ξbρ[∇b∇ap+∇b∇aΦG]\n+ξa∗ξbρ\u0014\nˆq∇b∇aΦE−ˆq\n2(∇aFbγ+∇bFaγ)uγ\u0015\n−ρˆq\n2\u0010\nξa∗Fabuγ∇γξb+ξaFabuγ∇γξb∗\u0011\u001b\ndV ,(25)\nwhere we used ∇a(ρua) = 0 and assumed that the mass\ndensity, charge density and pressure of the fluid vanishes\nat the surface of the considered system.\nEquation (25) could be used to check the stability of\nthe normal modes of the system, where the normal mode\nunder discussion is represented by a specific ξa.\nA. Instability with regards to non-axisymmetric\nperturbations\nThe classical rotating, isentropic or nonisentropic,\nuncharged fluid is unstable with regards to non-\naxisymmetric perturbations [21]. Perturbation modes of\nnegative energy transport angular momentum and en-\nergy to infinity, while the rotating fluid is slowing down.\nThe energy transfer is inspired by variation of various\nmultipoles caused by the perturbations. This instability,\nhowever, is irrelevant for slowly rotating fluids [21].\nThe discussion by Friedman and Schutz [21] can be ex-\ntended to our case of a charged rotating fluid, leading to\nthe same instability. To check this, we use the following\nansatz for the non-axisymmetric perturbation of the fluid\nin cylindrical coordinates:\nξa=ζa(r, z)eimϕ, ˙ξa=˙ζa(r, z)eimϕ. (26)\nThe “amplitudes” ζa(r, z) and ˙ζa(r, z) can also depend\nonmbut should be bounded everywhere, independent of\nm. They are chosen such that ζϕvanishes and condition\n(20) is fulfilled\nζa=f∇aˆq , ˙ζa=2\nim∂a\u0000\nrΩζr−Fbϕζb\u0001\n,(27)\nwhere f=f(r, z) has to be a smooth function, which\nis constant along the fluid trajectories ( ∂t+Lu)f= 0.\nAccordingly, the divergence of ξabecomes\n∇aξa=∇aζa(r, z)eimϕ. (28)\nPlugging the ansatz into Eq. (25) leads to\nEc=K−Z\nρˆqΩζaζb(∇a∇bAϕ− ∇ a∇bΦE)dV\n−m2Z\nρΩ2|ζ|2dV .(29)\nHere, Kis a term that includes all terms from perturb-\ning the uncharged fluid with the given ansatz, which areindependent from m. According to the named assump-\ntions, both Kand the first integral in Eq. (29), which\ncontains the electromagnetic effects and is independent\nofmas well, are bounded from above. Therefore Ecwill\neventually become negative for sufficiently high frequen-\ncies in the non-axisymmetric perturbation.\nAs one can see from Eq. (29), the electromagnetic in-\nteractions of the fluid do not contribute any term that\nwould depend on the perturbation frequency m. Fol-\nlowing the discussion in Friedman and Schutz [21], we\nassume that a set of non-axisymmetric perturbations ex-\nist that do not die away in time, therefore leading to\ninstability.\nV. AXISYMMETRIC PERTURBATIONS: THE\nGENERALIZATION OF THE\nSOLBERG-HØILAND CONDITIONS\nWe saw that rotating, charged, non-conductive\nfluid structures will slow down over time under non-\naxisymmetric perturbations. We will now turn our focus\non the special case of axisymmetric perturbations, that\npreserve the angular momentum of the system. In the\nnon-charged fluid case, this stability discussion leads to\nthe Solberg-Høiland conditions or otherwise the Rayleigh\ncriterion in the special case of fluids with an overall con-\nstant entropy S.\nUnder certain conditions for the perturbations in the\nfluid, the first order variation δ1Etotof the total energy of\na charged fluid vanishes (the solution to the Euler equa-\ntion (13) has an extremal total energy only with respect\nto this class of perturbations), and the canonical energy\ncoincides with the second order variation δ2Etotof the\ntotal energy of the fluid. Since the variation δEtotof the\ntotal energy is a physical property of our configuration,\nEcwill loose its gauge dependence under this conditions,\nand a discussion of canonical and trivial displacements is\nnot necessary anymore. Perturbations of this class con-\nserve the fluid mass and charge, leave the system adia-\nbatic and further conserve the redefined fluid’s angular\nmomentum (∆ˆ uϕ= ∆( uϕ+ ˆqAϕ) = 0; see appendix B).\nA conservation of mass and charge is automatically sat-\nisfied under any perturbations, if the continuity equation\nfor mass and charge (see Eqs. (14)–(15)) are valid.\nIn this chapter we will focus the discussion on adia-\nbatic, axisymmetric perturbations that conserve the re-\ndefined angular momentum ˆ uϕof the fluid. In this case\nEccoincides with the second variation of the total en-5\nergy of the system and we can follow the approach used\nby Tassoul [1].\nA. Case of axial symmetry\nAxissymmetry (independence of the angular coordi-\nnate ϕcorresponding to that symmetry) leads to a con-\nservation constant. For an uncharged fluid, the conserved\nquantity is the angular momentum of the fluid L=uϕ.\nIn our case of a charged, non-conductive fluid, affected\nby an electromagnetic background field, the conserved\nquantity is a redefined angular momentum ˆL= ˆuϕ. It\ncan be directly derived (in e.g. cylindrical or spherical\ncoordinates) from the ϕ-component of the charged Euler\nequation (13), while using Eq. (18):\nD\nDtuϕ−ˆqFϕbub= 0, (30)\n⇒D\nDt(uϕ+ ˆqAϕ) =D\nDtˆuϕ= 0, (31)\nwhere we assumed, that the electromagnetic background\nfield is stationary.\nFurthermore, the magnetic vector potential can be re-\nstricted to its ϕ-component in our axisymmetric case (\nAa=Aφηa, while ηa= ˆea\nφcorresponds to the Killing\nvector for axial symmetry).\nIn order to use the conservation of ˆ uϕin the follow-\ning stability analysis, we will rewrite the charged Euler\nequation (13) in terms of\nˆuµ=uµ+ ˆqAµ. (32)\nIt becomes:\n\u0000\nˆub−ˆqAb\u0001\n∇b(ˆuµ−ˆqAµ)−ˆqFµb\u0000\nˆub−ˆqAb\u0001\n+∇µp\nρ+∇µΦG+ ˆq∇µΦE\n| {z }\nT2= 0,\n(33)\nˆub∇bˆuµ−ˆq\u0002\nAb∇bˆuµ+ ˆub∇bAµ\u0003\n+ ˆq2Ab∇bAµ\n−ˆq[∇µAb− ∇ bAµ] (ˆub−ˆqAb) +T2= 0,\n(34)\nˆub∇bˆuµ−ˆq\u0002\nAb∇bˆuµ+ ˆub∇µAb\u0003\n+ ˆq2Ab∇µAb+T2= 0.\n(35)In the case of circular fluid motion and a magnetic vec-\ntor field restricted to its ϕ-component, the second term\nin the equation above reduced to:\n\u0002\nAb∇bˆuµ+ ˆub∇µAb\u0003\n=gϕϕˆuϕAϕ\n,µ+gϕϕ,µˆuϕAϕ−2·\u00121\n2gϕϕ,µˆuϕAϕ\u0013\n,(36)\n=gϕϕˆuϕAϕ\n,µ. (37)\nFinally, we end up with the following new version of the\nEuler equation (13) in terms of ˆ ub:\nˆub∇bˆuµ−ˆq\u0000\nˆuϕAϕ\n,µ− ∇ µΦE\u0001\n+1\n2ˆq2(|Aϕ|2),µ+∇µp\nρ+∇µΦG= 0.(38)\nB. Rayleigh criterion for the charged,\nnon-conductive fluid\nWe are only interested in the perturbation of the fluid,\nwhich is why we do not consider any perturbation of the\nexternal gravitational or electromagnetic field. There-\nfore, δFab=δ∇aΦG=δ∇aΦE= 0. A linear pertur-\nbation of our new expression of the Euler equation (38)\nleads in the case of an initially unperturbed fluid in cir-\ncular motion to:6\nEc=δ2E= ∆ E=Z\nξaMabξbdV\n=Z\nξa\"\n∇bˆL2\n2g2φφ∂agφφ+∇ap\nρ\u0012∇bρ\nρ−∇bp\nγp\u0013\n− ∇ bˆq∇aΦE−1\n2∇bˆq2(|A|2),a+∇b(ˆqˆL)Aϕ\n,a#\nξbρdV\n+Z(δp)2\nγpdV\n|{z}\n≈0+Z\n∂Vξaδp dˆna\n|{z}\n=0.(39)\nwhere we assumed axisymmetric perturbations, which\npreserve the redefined angular momentum: ∆ ˆL= 0.\nFurthermore, we neglect the integral over ( δp)2. This\napproximation is valid, if the characteristic time of the\ndisturbance exceeds the travel time of a sound wave in\nthe perturbed area. This approximation basically ne-\nglects any possible p–modes, that could arise due to the\nperturbation [1].\nIn Eq. (39), the canonical energy is now expressed in\nform of a perturbation matrix Mab. As long as Mabis\npositive definite, Ecwill be positive for all ξa, and the\nsystem can be considered stable. The two new stabil-\nity conditions are therefore given by the conditions for a\nmatrix being positive definite. According to Tassoul [1],\nthese conditions are:\nTr(Mab) =Ma\na≥0, det (Mab)≥0. (40)\nThe pressure and density of a perfect fluid are con-\nnected to the entropy Sasα∇aS=−\u0010\n∇aρ\nρ−∇ap\nγp\u0011\n(see\ne.g. [1]). By using the found expression for Mabin (39),\nthe two stability conditions take the form:\nTr (Mab)\n=gab\" \nˆqAϕ\n,a−\u00121\ngφφ\u0013\n,aˆL!\n∇bˆL−α∇ap\nρ∇bS\n+\u0010\nEa−ˆq(|A|2),a+ˆLAϕ\n,a\u0011\n∇bˆqi\n≥0,\n(41)\ndet (Mab)\n=εabk1\nak2\nbεαβℓ1\nαℓ2\nβ+εabk1\nak3\nbεαβℓ1\nαℓ3\nβ\n+εabk2\nak3\nbεαβℓ2\nαℓ3\nβ≥0,(42)\nwith\nk1\na=ˆqAϕ\n,a−\u00121\ngφφ\u0013\n,aˆL , ℓ1\nb=∇bˆL , (43)\nk2\na=∇ap , ℓ2\nb=−α∇bS , (44)\nk3\na=Ea−ˆq(|A|2),a+ˆLAϕ\n,a, ℓ3\nb=∇bˆq . (45)\nIn the case of a vanishing electromagnetic field, the\nstability conditions (41) and (42) reduce to the Solberg-\nHøiland conditions. If aditionally the entropy is constant,\nFigure 1. Exemplary depiction of a gravitationally bound, ro-\ntating fluid structure (blue) immersed in a background mag-\nnetic field (grey arrows aligned with the vertical direction).\nThe central body (black circle) can be electrically charged,\nwhich adds an electric field to the system (not depicted here).\nThe fluid circulates in the azimuthal direction around the cen-\ntral body with a given velocity distribution.\nthe stability conditions further reduce to the Rayleigh\ncriterion:\n∇(ˆL2)≥0. (46)\nC. Specific cases\nIn this section we will apply the new found stability\nconditions for a non-conductive, charged fluid in an elec-\ntromagnetic background field to several special cases with\na given electromagnetic field and angular momentum dis-\ntribution. We assume a barotropic equation of state for\nall cases, which leads to a constant entropy distribu-\ntion throughout the fluid structure. All discussed special\ncases that follow were examined in previous works con-\ncerning the existence of bound charged fluid structures,\neither in a classical or general relativistic setup. This dis-\ncussion below is meant to evaluate, if the found bound\nstructures are stable concerning axisymmetric perturba-\ntions [3–7, 11, 12]. An exemplary depiction of such bound\nfluid structures is presented in Fig. 1.\na. Case I: Constant angular momentum, homoge-\nneous magnetic field. In this case, the cylindrical co-\nordinate system is used. In this coordinate system, the7\nvector potential for a homogeneous magnetic field is given\nby:\nAϕ=1\n2Bzr2, and Aϕ=1\n2Bz. (47)\nWe consider a magnetic field only, therefore setting Φ E=\n0. The two conditions for stable structures in case of\na constant angular momentum distribution and under\naxisymmetric perturbations reduce in this set up to:\nTr(Mab) = (ˆqBz)2+L\nr3Bz∂r(ˆqr2), (48)\ndet(Mab) = 0 . (49)\nAccording to this result, a charged fluid structure can\nbe considered stable to axisymmetric perturbations, if\nthe specific charge distribution – or more precisely ˆ qr2–\nincreases outwards in case of a repulsive Lorentz force or\ndecreases outwards in case of an attractive Lorentz force.\nThis is the direct generalization of the classical un-\ncharged Polish doughnut set up with a constant angu-\nlar momentum distribution. Since uncharged ideal fluid\nstructures are considered stable, if the specific angular\nmomentum in increasing outwards, a small fluid charge\ncan stabilize or destabilize the structure.\nb. Case II: Constant angular momentum, charged\ncentral object. In this case, the spherical coordinate sys-\ntem is used. In this coordinate system, the electric po-\ntential arising from the charge of the central object is\ngiven by:\nEa=\u0012Q\nr2,0,0\u0013\n. (50)\nWe consider no magnetic field, therefore setting Aa= 0.\nThe two conditions for stable structures for a constant\nangular momentum distribution and under axisymmetric\nperturbations reduce in this set up to:\nTr(Mab) =∂rˆqEr, (51)\ndet(Mab) = 0 . (52)\nAccording to this result, a charged fluid structure can be\nconsidered stable to axisymmetric perturbations, if the\ncharge increases (same charge as the BH) or decreases\n(opposite charge of the BH) outwards. This means, that\nin case of a repulsive electric force on the fluid, the repul-\nsive force should fall off slower than the attractive gravi-\ntational force and vice versa for the case of an attractive\nelectric force on the fluid.\nc. Case III: Constant angular momentum, charged\ncentral object immersed in a homogeneous magnetic field.\nConsidered is a homogeneous magnetic field and a central\nobject with charge. The two conditions for stable struc-\ntures for a constant angular momentum distribution andunder axisymmetric perturbations reduce to:\nTr(Mab) =∂aˆqEa+ (ˆqBz)2+LAϕgab∂agϕϕ\ng2\nϕϕ∂b(gϕϕˆq),\n(53)\ndet(Mab) =ˆqAϕˆL\ng2\nϕϕ\u0000\ngab∂agϕϕ∂bgϕϕEc∂cˆq\n−∂agϕϕEagcd∂cgϕϕ∂dˆq\u0001\n.(54)\nIn cylindrical coordinates, this leads to:\nTr(Mab) =∂aˆqEa+ (ˆqBz)2+L\nr3Bz∂r(ˆqr2),(55)\ndet(Mab) = ˆqBzEz∂zˆq2L\nr2+ (ˆqBz)2Ez∂zˆq . (56)\nGenerally one can deduce form this, that the fluid\nstructure is more likely to be stable, if only repulsive\nelectromagnetic forces occur (ˆ qEa>0 and ˆ qBzL > 0).\nAt the same time the charge density of the fluid structure\nshould increase outwards.\nd. Case IV: Constant angular velocity, homogeneous\nmagnetic field. In this case, the cylindrical coordinate\nsystem is used. In this coordinate system, the vector\npotential for a homogeneous magnetic field is given by:\nAϕ=1\n2Bzr2, and Aϕ=1\n2Bz. (57)\nWe consider a magnetic field only, therefore setting Φ E=\n0. The two conditions for stable structures in case of a\nconstant angular velocity distribution and under axisym-\nmetric perturbations reduce in this set up to:\nTr(Mab) = (2Ω + ˆ qBz)2+∂rˆqBzrΩ, (58)\ndet(Mab) = 0 . (59)\nAccording to this result, a charged fluid structure can be\nconsidered stable to axisymmetric perturbations, if the\ncharge increases (angular momentum of the fluid aligned\nto the magnetic field) or decreases (angular momentum\nof the fluid anti-aligned to the direction of the magnetic\nfield) outwards.\ne. Case V: Constant angular velocity, charged central\nobject. In this case, the spherical coordinate system is\nused. In this coordinate system, the electric potential\narising from the charge of the central object is given by:\nEa=\u0012Q\nr2,0,0\u0013\n. (60)\nWe consider no magnetic field, therefore setting Aa= 0.\nThe two conditions for stable structures for a constant\nangular velocity distribution and under axisymmetric\nperturbations reduce in this set up to:\nTr(Mab) = 4Ω2+∂rˆqErrΩ, (61)\ndet(Mab) = 2 rΩ2cosθ Er[rcosθ ∂rˆq−sinθ ∂θˆq].(62)8\n−2.00−1.75−1.50−1.25−1.00−0.75−0.50−0.25 0.00\nK10.000.250.500.751.001.251.501.752.00(/lscriptc//lscriptK,c)2\nSADDLE\nMINMAX\nSADDLEMAX\nSADDLE\nSADDLE\nMINTr(\nM)<0 Det(M)<0/lscript= constq= 0\nFamily IMq<0\nMq>0\n−2.50−2.25−2.00−1.75−1.50−1.25−1.00−0.75−0.50\nK10.00.20.40.60.81.0cosθcTr(\nM)<0\nDet(M)<0Mq<0\nFamily II\n0.8 1.0 1.2 1.4 1.6\n(r/rc) sinθ−0.4−0.20.00.20.4(r/rc) cosθ0.4\n0.80.010.01\n0.100.10\n0.6 0.8 1.0 1.2 1.4 1.6\n(r/rc) sinθ−0.8−0.6−0.4−0.20.00.20.40.60.8(r/rc) cosθ\nFigure 2. The figures of equilibrium and the corresponding distribution of pressure surfaces in charged slender tori immersed\nin the dipole magnetic field corresponding to two Families lying in the equatorial plane (I and II); their tentative existence was\noriginally discussed by Slan´ y et. al [4]. Top left panel: These tori correspond to ‘Family I’ solutions with the specific angular\nmomentum distribution ℓ=ℓc[(r/rc) sinθ]K1. The horizontal axis shows the parameter K1; the parameter ( ℓc/ℓK,c)2shown\non the vertical axis measures how much the rotation of the torus deviates from the Keplerian value at the torus center due\nto Lorentz force; the value ( ℓc/ℓK,c)2corresponds to uncharged tori, values greater/smaller than one correspond respectively\nto positively/negatively charged tori. The dashed areas denote regions in the parameter space, where the criteria for stability\nare not satisfied. The graph demonstrates that stable configurations do exist, however they are considerably restricted by\nthe stability criterion and limited to a small region of the parameter space. Bottom left panel: Example of the equi-pressure\nsurfaces of negatively charged tori of finite size, whose parameters correspond to the cross shown in the top panel. Even though\nthe center of the torus is stable, our criteria limit significantly possible sizes of the tori. Out of the three configurations shown\nby the thick black solid line, only the innermost one corresponds to an entirely stable torus. The corresponding panels in the\nright column have been constructed in the way analogous to the left column but for the case of Family II (off-equatorial tori).\nIn the latter case, we do not find any closed equi-pressure surface that would enclose a region of stability.\nAccording to this result, a charged fluid structure can be\nconsidered stable to axisymmetric perturbations, if the\ncharge increases (same charge as the BH) or decreases\n(opposite charge of the BH) outwards. Structures along\nthe equatorial plane seem stable, as long as the charge\ndoes not significantly change along the angle θ, or, if\nit does, decreases (same charge as the BH) or increases\n(opposite charge of the BH) towards the equatorial plane.\nf. Case VI: Constant angular velocity, charged central\nobject, homogeneous magnetic field. This time a homo-\ngeneous magnetic background field is assumed as well as\na charged central object.\nAssuming the following charge distribution is moti-vated by the former work [7]. In this paper bound\ncharged fluid structures were found with a charge dis-\ntribution of\nˆq=f(S), S =−ΦE+ ΩAϕ (63)\nin a general relativistic setting.\nThe two conditions for stable structures in case of a\nconstant angular velocity distribution and under axisym-9\nmetric perturbations reduce in this set up to:\nTr(Mab) =gab∂agϕϕ∂bgϕϕ\ngϕϕ\u0000\nΩ + ˆqAϕ\u00012\n+f′(S)gab(Ea+ Ω∂aAϕ) (Eb+ Ω∂bAϕ),\n(64)\ndet(Mab) =f′(S)\u0000\nΩ + ˆqAϕ\u00012\ngϕϕ\u0000\nεab∂agϕϕ(Eb+ Ω∂bAϕ)\u00012.\n(65)\nBoth conditions are fulfilled, if f′(S)≥0.\nThis setup was discussed in the general relativistic case\nby Kov´ aˇ r et al. [11]. The equatorial tori fulfill the stabil-\nity conditions in contrast to the chosen setup for the polar\nclouds. This could be a hint, that the found polar cloud\nsolutions are actually unstable with respect to axisym-\nmetric perturbations. A general relativistic discussion is\nneeded for a sound answer.\ng. Case VII: Magnetic dipole field Employing the\nspherical coordinates {r, θ, ϕ}, the background magnetic\nfield can be derived from the vector potential\nAϕ=M\nrsin2θ, and Aϕ=M\nr3. (66)\nIn addition, if the electric field vanish, Φ E= 0, the two\nconditions for stable equilibrium structures under the ax-\nisymmetric perturbations reduce to\nTr (Mab) =1\nr2sin2θ\"\u00122ℓ\nr+ ˆq∂rAϕ\u0013\n(∂rℓ+ ˆq∂rAϕ)\n+1\nr2(2ℓcotθ+ ˆq∂θAϕ) (∂θℓ+ ˆq∂θAϕ) +\n+ℓ\u0012\n∂rAϕ∂rˆq+1\nr2∂θAϕ∂θˆq\u0013#\n(67)\nand\ndet (Mab) =2ℓ2\nr4sin4θ\"\n1\nr∂θAϕ−(∂rAϕ) cotθ#\n×\"\n(∂rℓ+ ˆq∂rAϕ)∂θˆq+\n(∂θℓ+ ˆq∂θAϕ)∂rˆq#\n. (68)\nIn the case of a rigid-body rotation. Ω = ℓ/(rsinϑ)2=\nconst, a fluid equilibrium exists when the specific charge\ncan be expressed as function of the vector potential only,\nthat is when ˆ q=f(Aϕ). In that case, the stability con-\nditions (67) and (68) coincide with Eqs. (64) and (65)\ngiven above. Consequently, such configuration are there-\nfore stable as long as f′(Aϕ)>0.VI. DISCUSSION\nMore general equilibrium configurations, with specific\nangular momentum constant on cylinders rsinϑ= const\nhave been studied by Slan´ y et al [4]. In particular,\nthey consider angular momentum distribution given by a\npower-law functions of the form\nℓ=K2(rsinθ)K1+1(69)\nwith K1andK2being constant parameters ( K1=±1\ncorrespond to constant angular velocity and constant an-\ngular momentum, respectively; when K2=√\nGM and\nK1=−1/2, Eq. (69) describes Keplerian flows). It is\nalso assumed that the distribution of specific charge can\nbe expressed as\nˆq=Crλsin(θ)δ(70)\nwith C,λandδbeing yet another constants. The func-\ntions (69) and (70) describe equilibrium toroidal struc-\ntures only when the parameters K1,λandδsatisfy the\ncondition of integrability. In order to reduce a large num-\nber of free parameters, Slan´ y et al [4] consider four partic-\nular relations among the constants K1,λandδfor which\nthey are able to find bound fluid configurations. They\nreferred to as the family I, II, III, and IV, respectively.\nIn general, all four families allow equatorial tori, with\npressure maxima located in the equatorial plane. These\ntori are similar to thick-disks solutions of purely hydrody-\nnamic equations, modified by presence of Lorentz force.\nIn addition, the families II – IV permit also solutions\nwhere the Lorentz force pushes the pressure maxima out\nof the equatorial plane allowing off-equatorial “levitat-\ning” tori. As noted by Slan´ y et al [4], such configurations\nexist only for negative charge (i.e., when Mq <0).\nIn order to illustrate our results, we apply the stability\ncriteria on both types of the solutions. We first investi-\ngate stability properties of equatorial tori corresponding\nto Family-I solutions. The relation between the parame-\ntersλ,δandK1are (see Ref. [4])\nλ=−3\n2(K1−1), δ = 0. (71)\nThe equilibrium configurations are fully described by\nthree independent parameters: the exponent in the angu-\nlar momentum distribution K1, the charge parameter of\nthe torus Cand the maximal pressure pc(i.e. the value\nof the pressure at the torus center). The Lorentz force,\nproportional to C, determines a departure of the fluid\nangular momentum at the center of the torus ℓcfrom the\nlocal Keplerian value ℓK,cand thus also the value of the\nparameter K2. Here, we take the ratio ( ℓc/ℓK,c)2as a\nfree parameter, instead of C, because of its immediate\nphysical meaning.\nStarting with slender tori ( pc→0), in left-top panel\nof Fig. 2, we plot possible configurations in the param-\neter space of K1vs (ℓc/ℓK,c)2. The horizontal line la-\nbeled as q= 0 corresponds to uncharged hydrodynamic10\nslender tori, for which ℓc=ℓK,c. The blue-shaded\nand red-shaded areas below and above this line corre-\nspond to allowed combinations of these two parameters\nfor negatively- ( Mq < 0) and positively- ( Mq > 0)\ncharged tori, respectively. These regions are bounded\nby dashed black lines, where the topology of the central\npoint changes from maximum to saddle. Clearly only the\nformer one gives reasonable solutions.\nUsing expressions (67) and (68), we determine re-\ngions in the parameter space, where either Tr( Mab) or\ndet(Mab) is negative. These areas correspond to hatched\nregions in the plot. According to the discussion in previ-\nous sections, we know that even slender tori correspond-\ning to these parameters cannot be stable. Naturally, the\nsame applies also to bigger tori with finite thickness, as\ntheir equi-pressure surfaces surround the ones of the slen-\nder tori. In other words, a stability of slender tori is a\nnecessary condition for stability of tori with finite thick-\nness, whose maximal pressure points are at the same lo-\ncations. Therefore it follows that the only stable region\nin the parameter space left for possible stable configu-\nrations corresponds to negatively-charged tori with K1\nin the range [ −1,−0.5], where at least slender tori with\nnegligible cross-section are stable.\nWe next examine tori of finite thickness with parame-\nters corresponding to this region. The left bottom panel\nof Fig. 2 shows an example of the equi-pressure surfaces\nin the poloidal plane for the negatively charged torus\nwith angular momentum at the center ℓ2\nc= 0.922ℓ2\nK,cand\nK1=−0.6. The region of negative determinant det( Mab)\nis shown in the plot as a hatched area, the trace is posi-\ntive everywhere. We observe that although the center of\nthe torus lies in the stable area, our stability conditions\nput a serious constrain on the possible radial extend of\nthe torus.\nNext, we investigate stability of the off-equatorial\n(“levitating”) tori. We first concentrate on the Family-\nII solutions fully determined by three independent pa-\nrameters: the radial angular momentum distribution K1,\nheight of the torus maximal pressure point above the\nequatorial plane cos θcand the central pressure pc. The\nfamily II corresponds to the charge distributions of the\nform\nˆq=Cr3/2(sinθ)−3K1(72)\n(see Ref. [4]) and the force balance at the torus center\ndetermines values of the two remaining constants K2and\nC. Again we start by discussing stability of slender tori in\nthe limit pc→0. The center of the torus corresponds to\nthe pressure maximum when K1<−1/2, independently\nof the height of the torus above the equatorial plane.\nIn addition, both K1and cos θcdetermine the stability\nof the torus center. The regions where the trace and\ndeterminant of the matrix Mabare negative are shown\nin the top right panel of Fig. 2 as differently hatched\nareas. Clearly, no combination of K1and cos θcgives a\nstable configuration. Consequently, neither tori of finite\nthickness can be stable. In the bottom right panel ofFig. 2 this finding is illustrated by an example of the\nstructure of equi-pressure surfaces for K1=−2.25 and\nθc= 1.1. The regions of negative trace and determinant\nof the matrix Mabare shown in the plot by differently\nhatched areas, showing that even the maximal pressure\npoints of the toroidal structures are located well inside\nthe unstable areas.\nThese finding brinks up a natural question: Do stable\ncharged off-equatorial tori exist at all? To answer this\nquestion, we decide to go beyond the original analysis of\nSlan´ y et al. and let both K1andλto be independent pa-\nrameters. We examine stability of slender off-equatorial\ntori at arbitrary height (sin2θc≤1). The result is shown\nin Fig. 3. The blue-shaded area corresponds to param-\neters giving off-equatorial toroidal configurations. The\nfamilies II, III and IV introduced by Slan´ y et al. [4] cor-\nrespond to the three color lines indicated in the plot. We\nalso hatched the region where the trace or determinant\nof the stability matrix Mabis negative in full range of\nheight, 0 ≤θc≤π. Clearly, even with this more general\napproach, no combination of K1,λandθcmay give a\nstable slender torus and our question thus has a nega-\ntive answer. The same necessarily applies to tori of finite\nthickness, as for their stability, a stability of slender tori\nwith the same K1,λandθcis necessary condition.\nFinally, let us remark that the construction of off-\nequatorial tori can be still further generalized by allow-\ning for different powers of rand sin θcin the prescribed\nangular momentum distribution (69). It is likely that\nfor some particular values of these exponents a stable\ntoroidal configuration will exist. Deeper explorations of\nthis possibility is however behind the scope of this paper,\nwhere we mainly intent to illustrate power of our stabil-\nity criteria. We wish to return to this question in our\nfuture work.\nVII. CONCLUSIONS\nWe investigated the linear stability of charged,\nnon-conductive perfect fluid structures immersed in\nan electromagnetic background field concerning non-\naxisymmetric and axisymmetric perturbations in the\nNewtonian regime. In order to do so, we used the La-\ngrangian perturbation formalism introduced by Lebovitz\n[17].\nWe found that the charge distribution within a non-\nconductive fluid cannot stabilize the general instability of\nideal fluids with respect to non-axisymmetric perturba-\ntions, in agreement with the original argument by Fried-\nman and Schutz [21].\nIn the case of axisymmetric perturbations, we found a\ngeneralization of the Høiland conditions for stable fluid\nstructures for our case of charged, non-conductive fluids.\nThe generalized conditions were used to make specific\nstatements concerning the stability of seven special cases\nof electromagnetic background fields and angular mo-\nmentum distributions of a charged fluid structure. The11\n−4.0−3.5−3.0−2.5−2.0−1.5−1.0−0.5 0.0 0.5\nK10123456λ\nFamily IVFamily IIFamily IIITr(\nM)<0Det(M)<0\nFigure 3. Three families (II, III and IV) also allow for off-\nequatorial tori in the regions of K1–λparameter plane shown\nin the graph. Interestingly, none of these equilibrium configu-\nrations satisfy the stability criterion introduced in the present\npaper, as indicated by hatches covering the entire plane (see\nthe main text for details). Each colored line corresponds to\nthe range of K1values permitted by the equilibrium condition\nfor three distinct families.\nseven cases were chosen either by their simplicity and di-\nrect generalization of a classical non-charged fluid struc-\nture (Polish doughnut with constant angular momentum)\nor because these setups had been discussed concerning\nthe existence of bound charged fluid structures in previ-\nous works, and the question of stability of these struc-\ntures needed to be answered.\nCharged fluid structures with a constant angular mo-\nmentum distribution are found to be stable with respect\nto axisymmetric perturbations if the charge distribution\nwithin the structure increases outwards and gives rise to\na repulsive Lorentz force everywhere. We could show\nthat, while the found charged fluid structures discussed\nin [11] and some charged fluid structures of family I in\n[4] are stable in the case of equatorial tori, the solutions\nfor polar clouds appear to be unstable.\nStability could also not be confirmed for the case of\noff-equatorial tori that had been discussed in Slan´ y et al.\n[4] with the magnetic dipole background field. On the\nother hand, modifying the assumed profile of the radial\ndistribution of the charge density or the magnetic field\nstructure will likely lead to stable configurations.\nACKNOWLEDGMENTS\nWe wish to acknowledge the continued support from\nthe Czech Science Foundation EXPRO program (VK and\nJH, ref. 21-06825X) and the CTA-CZ research infras-\ntructure of the Czech Ministry of Education, Youth and\nSports (ref. LM2023047). AT and EH acknowledge the\nDeutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) funded under the project number\n420243324. Moreover, EH acknowledges the Deutsche\nForschungsgemeinschaft (DFG, German Research Foun-\ndation) under Germany’s Excellence Strategy - EXC-\n2123 QuantumFrontiers - 390837967. We further thank\nShokoufe Faraji, Jiˇ r´ ı Kov´ aˇ r and Petr Slan´ y for inspiring\ndiscussions.\nAppendix A: The (anti-)hermitian character of the\noperators in the perturbed charged Euler equation\nThis section is dedicated to show that the operators\nAab,BabandCab, occurring in Eq. (21), are hermitian,\nanti-hermitian, and hermitian respectively. Since this has\nalready been shown by Friedman and Schutz [13] in the\nuncharged fluid case, we will restrict the discussion here\nto those terms, which arise from the electromagnetic in-\nteraction of the charged fluid with the electromagnetic\nbackground filed.\nThe operators are given by:\nηaAabξb=gabηaξb, (A1)\nηaBabξb=ηa(2gabuc∇c−ˆqFab)ξb, (A2)\nηaCabξb=ηa\u0014\u0000\nub∇b\u00012ξa+1\nρ\u0000\n∇bξb∇ap− ∇ aξb∇bp\n−∇a(γp∇bξb)\u0001\n+ξb∇b∇aΦG+ ˆqξb∇b∇aΦE\n−ˆq\u0000\nFabuγ∇γξb+ξb∇bFaγuγ\u0001\u0003\n.\n(A3)\nSince Aabis hermitian, and the electromagnetic con-\ntribution in Babobviously anti-hermitian, ( Fabis anti-\nhermitian), only the electromagnetic terms in Cabneed\nto be checked for their symmetry. These terms are\nˆqηaξb∇b∇aΦE−ˆqηa\u0000\nFabuγ∇γξb+ξb∇bFaγuγ\u0001\n.(A4)\nSince covariant derivatives commute, the electric term\n(ˆq∇b∇aΦE) has to be hermitian. The magnetic contri-\nbution ˆ q(Fabuγ∇γ+∇bFaγuγ) can be rewritten as fol-\nlows\nqηa\u0000\nFabuγ∇γξb+ξb∇bFaγuγ\u0001\n=q\n2ηaξb[∇aFbc+∇bFac]uc\n−q\n2\u0002\nηaFabuc∇cξb+ξaFabuc∇cηb\u0003\n−1\n2∇c\u0000\nqucηaFabξb\u0001\n.(A5)\nThe last term under the volume integral over the sys-\ntem vanishes because we demand the charge density at\ninfinity to vanish, too. The two remaining terms are sym-\nmetric in exchanging ηandξ. It is therefore shown, that\nthe electromagnetic terms of Cabunder an integral over\nthe system volume are also hermitian.12\nAppendix B: Second variation of the total energy\nThe total energy of a charged, non-conductive fluid\nconfiguration is given by:\nEtot=T+U+WG+WE, (B1)\nwhere Tis the kinetic, WG=R1\n2ΦGρ dV the mass poten-\ntial,WE=R1\n2ˆqΦEρ dV the electric potential and Uthe\ninternal energy. The magnetic field does not contributeto the total energy of the system, since the Lorentz-force\nFabubis always perpendicular the the fluid motion. The\ngiven total energy therefore only has an additional elec-\ntric potential term WEcompared to the total energy of\nan uncharged, perfect fluid.\nLet us note that Friedman and Schutz [13] derived the\nvariation of the total energy until second order for an\nuncharged, adiabatic, perfect fluid under the premise of\nmass conservation:\nδEtot,nc=δ1Etot,nc+δ2Etot,nc+O(ξ3) (B2)\n=Z\u0014\nρξi\u0012\nuj∇jui+1\nρ∇ip+∇iΦG\u0013\n+ρuβ∆uβ\u0015\ndV (B3)\n+1\n2Z\nρ|∂tξ|2−ρ|uγ∇γξ|2+γp|∇aξa|2+ 2ξa∇ap∇bξb+ξaξbρ[∇b∇ap+∇b∇aΦG]dV+O(ξ3).(B4)\nIn the charged case, the variation of the electric potential\nuntil second order needs to be added to the equation\nabove. By taking into account, that the overall charge of\nthe system is conserved (see Eq. (15), and ∆ˆ q= 0) one\ngets analog to the variation of the gravitational potential:\nδWE=Z\nˆq∆\u00141\n2ΦEρ\u0015\ndV\n=Z\u0014\nˆqρξa∇aΦE+1\n2ˆqρξaξb∇a∇bΦE\u0015\ndV\n+O(ξ3).(B5)If the term ρuβ∆uβis rewritten in the form ρuβ∆ˆuβ\nin the charged case the variation of the total energy up\nto second order will take the form:\nδEtot=Z\u0014\nρξi\u0012\nuj∇jui+1\nρ∇ip+∇iΦG+ ˆq∇iΦE−ˆqFijuj\u0013\n+ρuβ∆ˆuβ\u0015\ndV\n+1\n2Z\nρ|∂tξ|2−ρ|uγ∇γξ|2+γp|∇aξa|2+ 2ξa∇ap∇bξb+ξaξbρ[∇b∇ap+∇b∇aΦG+ ˆq∇b∇aΦE]\n−ρˆqξa\u0000\nFabuγ∇γξb+ξb∇bFaγuγ\u0001\ndV+O(ξ3).(B6)\nThe first term in the first integral must vanish according\nto the Euler equation for charged fluids (see Eq. (13)).\nLike in the uncharged case, the first order variation be-\ncomes zero, and the second order variation coincides with\nthe canonical energy, if uβ∆ˆuβvanishes. In the case of a\nfluid in circular motion, this is the case, if the redefined\nangular momentum ˆ uϕ=uϕ+ ˆqAϕis conserved.\nEquation (B6) was derived by calculating the difference\nρˆquβ∆Aβbetween ρuβ∆uβandρuβ∆ˆuβto up to secondorder:\n∆Ai= ∆( gibAb) = ∆ gibAb+gib∆Ab+ ∆gib∆Ab,\n(B7)\nwhere\n∆gab=∇aξb+∇bξa+∇aξβ∇bξβ+O(ξ3) (B8)\nand\n∆Ai=\u0000\nδi\nj− ∇ jξi+∇jξβ∇βξi\u0001¯Aj−Ai+O(ξ3) (B9)13\n(see [13]). Here, ¯Aiis the perturbed vector field A′i,\nframedragged along ξ. In our case, the magnetic vector\npotential is not affected by the perturbations.\nTaylor expansion up to second order results in this case\nin:\n¯Ai=Ai(xa+ξa) =Ai+ξa∇aAi+1\n2ξaξb∇a∇bAi+O(ξ3).\n(B10)\nCalculating all terms up to second order results in:\n∆Ai=ξa∇aAi+Aa∇iξa+∇iξaξb∇bAa+1\n2ξaξb∇a∇bAi.\n(B11)This leads to\nZ\nρˆquβ∆AβdV\n=Z\nˆqρuβ\u0000\nξa∇aAβ+Aa∇βξa+∇βξaξb∇bAa\n+1\n2ξaξb∇a∇bAβ\u0013\ndV\n(B12)\n=Z\nρ ξaˆqFaβuβdV\n+1\n2Z\nρˆqξa\u0000\nFabuγ∇γξb+ξb∇bFaγuγ\u0001\ndV ,(B13)\nwhere we used ∇i(ρˆqui= 0) and assumed, that the\ncharge and mass density vanish at infinity. By adding\nthe variation of the electric potential to Eq. (B2) and\nreplacing ρuβ∆uβbyρuβ∆ˆuβ−ρˆquβ∆Aβfinally leads\nto Eq. (B6).\n[1] J.-L. Tassoul. Stellar Rotation . Cambridge University\nPress, 2000.\n[2] L. Mestel. Stellar rotation: a historical survey. In\nMichael J. Thompson and Jørgen Christensen-Dalsgaard,\neditors, Stellar Astrophysical Fluid Dynamics , pages 75–\n98. 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Oxford University Press, Oxford, England, Septem-\nber 2013."    },    {        "title": "2402.03915v1.Learning_Metrics_that_Maximise_Power_for_Accelerated_A_B_Tests.pdf",        "content": "Learning Metrics that Maximise Power for Accelerated A/B-Tests\nOlivier Jeunen\nShareChat\nEdinburgh, United KingdomAleksei Ustimenko\nShareChat\nLondon, United Kingdom\nABSTRACT\nOnline controlled experiments are a crucial tool to allow for con-\nfident decision-making in technology companies. A North Star\nmetric is defined (such as long-term revenue or user retention),\nand system variants that statistically significantly improve on this\nmetric in an A/B-test can be considered superior. North Star met-\nrics are typically delayed and insensitive. As a result, the cost of\nexperimentation is high: experiments need to run for a long time,\nand even then, type-II errors (i.e. false negatives ) are prevalent.\nWe propose to tackle this by learning metrics from short-term\nsignals that directly maximise the statistical power they harness\nwith respect to the North Star. We show that existing approaches\nare prone to overfitting, in that higher average metric sensitivity\ndoes not imply improved type-II errors, and propose to instead\nminimise the 𝑝-values a metric would have produced on a log of\npast experiments. We collect such datasets from two social media\napplications with over 160 million Monthly Active Users each, to-\ntalling over 153 A/B-pairs. Empirical results show that we are able\nto increase statistical power by up to 78% when using our learnt\nmetrics stand-alone, and by up to 210% when used in tandem with\nthe North Star. Alternatively, we can obtain constant statistical\npower at a sample size that is down to 12% of what the North Star\nrequires, significantly reducing the cost of experimentation.\n1 INTRODUCTION & MOTIVATION\nModern platforms on the web need to continuously make decisions\nabout their product and user experience, which are often central to\nthe business at hand. These decisions range from design and inter-\nface choices to back-end technology adoption and machine learning\nmodels that power personalisation. Online controlled experiments,\nthe modern web-based extension of Randomised Controlled Trials\n(RCTs) [ 29], provide an effective tool to allow for confident decision-\nmaking in this context [19] (bar some common pitfalls [13, 18]).\nA North Star metric is adopted, such as long-term revenue or\nuser retention, and system variants that statistically significantly\nimprove the North Star metric are considered superior to the tested\nalternative [ 7]. Proper use of statistical hypothesis testing tools\nsuch as Welch’s 𝑡-test [ 39], then allows us to define and measure\nstatistical significance in a mathematically rigorous manner.\nHowever effective this procedure is, it is far from efficient. Indeed,\nexperiments typically need to run for a long time, and statistically\nsignificant changes to the North Star are scarce. This can either\nbe due to false negatives (i.e. type-II error), or simply because the\nNorth Star is not moved by short-term experiments. In these cases,\nwe need to resort to second-tier metrics (e.g. various types of user\nengagement signals) to make decisions instead. These problems\nare common in industry, as evidenced by a wide breadth of related\nwork. A first line of research leverages control variates to reduce the\nvariance of the North Star metric, directly reducing type-II errors byincreasing sensitivity [ 2,3,8,10,26,40]. Another focuses on identi-\nfying second-tier “ proxy ” or “ surrogate ” metrics that are promising\nto consider instead of the North Star [ 28,34,38], or to predict long-\nterm effects from short-term data [ 1,9,33]. Finally, several works\nlearn metric combinations that maximise sensitivity [7, 16, 34].\nThis paper synthesises, generalises and extends several of the\naforementioned works into a general framework to learn A/B-\ntesting metrics that maximise the statistical power they harness.\nWe specifically extend the work of Kharitonov et al .[16] to appli-\ncations beyond web search, where the North Star can be delayed\nand insensitive. We highlight how their approach of maximising\nthe average 𝑧-score does not accurately reflect downstream metric\nutility in our case, in that it does not penalise disagreement with\nthe North Star sufficiently (i.e. type-III errors [ 15,24,35]). Indeed:\nwhilst this approach maximises the mean𝑧-score, it does not neces-\nsarily improve the median𝑧-score, and does not lead to improved\nstatistical power in the form of reduced type-II error as a result.\nAlternatively, optimising the learnt metric to minimise 𝑝-values\n—either directly or after applying a log-transformation— more eq-\nuitably divides gains over multiple experiments, leading to more\nstatistically significant results instead of a few extremely signifi-\ncant results. Furthermore, we emphasise that learnt metrics are not\nmeant to replace existing metrics, but rather to complement them.\nAs such, their evaluation should be done through multiple hypoth-\nesis testing (with appropriate corrections [ 31]) ifanyof the North\nStar, available vetted proxies and surrogates, orlearnt metrics are\nstatistically significant under the considered treatment variant. We\ncan then either adopt a conservative plug-in Bonferroni correction\nto temper type-I errors, or analyse synthetic A/A experiments to\nensure the final procedure matches the required confidence level.\nWe empirically validate these insights through two dataset of\npast logged A/B results from large-scale short-video platforms with\nover 160 million monthly active users each: ShareChat and Moj.\nExperimental results highlight that our learnt metrics provide sig-\nnificant value to the business: learnt metrics can increase statistical\npower by up to 78% over the North Star, and up to 210% when used\nin tandem. Alternatively, if we wish to retain constant statistical\npower as we do under the North Star, we can do so with down to\n12% of the original required sample size. This significantly reduces\nthe cost of online experimentation to the business. Our learnt met-\nrics are currently used for confident, high-velocity decision-making\nacross ShareChat and Moj business units.\n2 BACKGROUND & PROBLEM SETTING\nWe deal with online controlled experiments, where two system vari-\nants𝐴and𝐵are deployed to a properly randomised sub-population\nof users, adhering to best practices [13, 19].\nFor every system variant, for every experiment, we measure\nvarious metrics that describe how users interact with the platform.arXiv:2402.03915v1  [cs.LG]  6 Feb 2024Olivier Jeunen and Aleksei Ustimenko\nThese metrics include types of implicit engagement (e.g. video-\nplays and watch time), as well as explicit engagement (e.g. likes\nand shares) as well as longer-term retention or revenue signals. For\neach metric, we log empirical means, variances and covariances (of\nthe sample mean). For metrics 𝑚𝑖with 1≤𝑖≤𝑁, that is:\n𝝁=[𝜇1,...𝜇𝑁],and𝚺=𝜎1... 𝜎 1𝑖... 𝜎 1𝑁\n...............\n𝜎𝑖1... 𝜎𝑖... 𝜎𝑖𝑁\n...............\n𝜎𝑁1... 𝜎𝑁𝑖... 𝜎𝑁.(1)\nSuperscripts denote measurements pertaining to different variants\nin an experiment: e.g. 𝝁𝐴and𝝁𝐵.\n2.1 Statistical Significance Testing\nWe want to assess whether the mean of metric 𝑚𝑖is statistically\nsignificantly higher under variant 𝐴compared to variant 𝐵. To this\nend, we define a confidence level 𝛼(often𝛼≈0.05), correspond-\ning to the false-positive-rate we deem acceptable. Then, we apply\nWelch’s𝑡-test. The test statistic (also known as the 𝑧-score) for\nmetric𝑚𝑖and the given variants is given by:\n𝑧𝐴≻𝐵\n𝑖=𝜇𝐴\n𝑖−𝜇𝐵\n𝑖√︃\n𝜎𝐴\n𝑖+𝜎𝐵\n𝑖. (2)\nWe then transform this to a 𝑝-value for a two-tailed test as:\n𝑝𝐴≻𝐵\n𝑖=2·min(Φ(𝑧𝐴≻𝐵\n𝑖); 1−Φ(𝑧𝐴≻𝐵\n𝑖)). (3)\nHere,𝐴≻𝐵denotes a partial ordering between variants, implying\nthat𝐴is preferred over 𝐵.Φ(·)represents the cumulative distribu-\ntion function (CDF) for a standard Gaussian. For completeness, this\nCDF is given by:\nΦ(𝑧)=1√\n2𝜋∫𝑧\n−∞𝑒−𝑡2\n2d𝑡. (4)\nWhen𝑝𝐴≻𝐵\n𝑖<𝛼, we can confidently reject the null-hypothesis\nthat𝐴and𝐵are equivalent w.r.t. the mean of metric 𝑚𝑖. Note that\n𝑧-scores are signed , whereas two-tailed 𝑝-values are not. Indeed:\n𝑝𝐴≻𝐵\n𝑖≡𝑝𝐵≻𝐴\n𝑖, leaving room for faulty conclusions of directionality,\nknown as type-III errors [ 15,24,35]. We discuss these later in depth.\nA one-tailed 𝑝-value for the one-tailed null hypothesis 𝐴⊁𝐵\nis given by 1−Φ(𝑧𝐴≻𝐵\n𝑖), and rejected when <𝛼\n2. Throughout, we\nuse two-tailed 𝑝-values unless mentioned otherwise.\n2.1.1𝑝-value corrections. The above procedure is valid for a sin-\ngle metric, a single hypothesis, and importantly, a single decision.\nNevertheless, this is not how experiments run in practice. Without\nexplicit corrections on the 𝑝-values (or corresponding 𝑧-scores), vio-\nlations of these assumptions lead to inflated false-positive-rates. We\nconsider two common cases: a (conservative) multiple testing cor-\nrection when an experiment has several treatments, and a sequen-\ntial testing correction when experiments have no predetermined\nend-date or sample size at which to conclude. These corrections\nare applied as experiment-level corrections, to ensure that for any\nmetric𝑚𝑖and variants 𝐴,𝐵, the obtained 𝑝-values𝑝𝐴≻𝐵\n𝑖accurately\nreflect what they should reflect, yielding the specified coverage at\nvarying confidence levels 𝛼.Multiple comparisons. Often, launched experiments will have\nmultiple treatments deployed, leading to the infamous “ multiple\nhypothesis testing ” problem [ 31]. We apply a Bonferroni correction\nto deal with this. When there are 𝑇treatments, we consider a\ntreatment to be statistically significantly different from control\nwhen a two-tailed 𝑝<𝛼\n𝑇, instead of the original 𝑝<𝛼threshold.\nWe can equivalently apply this correction on 𝑧-scores instead,\nallowing us to directly compare 𝑧-scores across experiments with\nvarying numbers of treatments. Recall that the percentile point\nfunction is the inverse of the CDF. We obtain a one-tailed 𝑝-value\nas𝑝=1−Φ(𝑧), and we reject the null hypothesis when 𝑝<𝛼\n2.\nNow, instead, we reject when 𝑝<𝛼\n2𝑇. As such, computing corrected\n𝑧-scores as ¯𝑧=𝑧Φ−1(𝛼\n2)\nΦ−1(𝛼\n2𝑇)controls type-I errors effectively.\nAlways-Valid-Inference and peeking. A statistical test should only\nbe performed once, at the end of an experiment. When the treatment\neffect is large, this implies we may have been able to conclude\nthe experiment earlier. To this end, sequential hypothesis tests\nhave been proposed in the literature [ 37]. Modern versions make\nuse of Always-Valid-Inference (AVI) [ 11] to allow for continuous\npeeking at intermediate results and making decisions based on them,\nwhilst controlling type-I errors. Here, analogously, we can apply a\ncorrection on the 𝑧-scores as follows:\n¯𝑧=𝑧√︂\n(𝑁𝐴𝐵+𝜌)\n𝑁𝐴𝐵log\u0010𝑁𝐴𝐵+𝜌\n𝜌𝛼2\u0011, 𝜌 =10 000\nlog(log(𝑒\n𝛼2))−2 log(𝛼),\n(5)\nwhere𝑁𝐴𝐵is the total number of samples over both variants com-\nbined. For a detailed motivation, see Schmit and Miller [30].\nThese corrections are applied on a per-experiment level, both\nin the objective functions of methods introduced in the following\nSections and when evaluating the metrics that they produce.\n2.2 Learning Metrics that Maximise Sensitivity\nThe observation that we can learn parameters to maximise statis-\ntical sensitivity is not new. Yue et al .apply such ideas specifically\nfor interleaving experiments in web search [ 41]. Kharitonov et al .\nextend this to A/B-testing in web search, aiming to learn combina-\ntions of metrics that maximise the average 𝑧-score [ 16]. Deng and\nShi discuss “lessons learned” from applying similar techniques [ 7].\nWe introduce the approach presented by Kharitonov et al .[16], as\nour proposed improvements build on their foundations.\nWe consider new metrics as linear transformations on 𝝁:\n𝜔=𝝁𝑤⊺,where𝑤∈R1×𝑁. (6)\nThe advantage of restricting ourselves to linearity, it that we can\nwrite out the 𝑧-score of the new metric as a function of its weights:\n𝑧𝐴≻𝐵\n𝜔=𝝁𝐴𝑤⊺−𝝁𝐵𝑤⊺\n√︁\n𝑤𝚺𝐴𝑤⊺+𝑤𝚺𝐵𝑤⊺. (7)\nThese𝑧-scores can be used exactly as before to obtain 𝑝-values.\nAn intuitive property of the 𝑧-score, is that a relative 𝑧-score of\n𝑟=𝑧𝐴≻𝐵\n𝜔\n𝑧𝐴≻𝐵\n𝑖implies that 𝜔requires a factor 𝑟2fewer samples to\nachieve the same confidence level as 𝑚𝑖[4]. This can directly be\ntranslated to the cost of experimentation, as it allows us to run\nexperiments for shorter time-periods or on smaller sub-populations.Learning Metrics that Maximise Power for Accelerated A/B-Tests\nAs such, it comes naturally to frame the objective as learning\nthe weights 𝑤that maximise the 𝑧-score on the training data. This\ntraining data consists of a set of experiments with pairs of variants\nE={(𝐴,𝐵)𝑖}|E|\n𝑖=1. We consider three distinct relations between pairs\nof deployed system variants:\n(1)Known outcomes:(𝐴,𝐵)∈E+, where𝐴≻𝐵,\n(2)Unknown outcomes:(𝐴,𝐵)∈E?, where𝐴?𝐵,\n(3)A/Aoutcomes:(𝐴,𝐵)∈E≃, where𝐴≃𝐵.\nHere,𝐴≻𝐵implies that there is a known and vetted preference\nof variant𝐴over𝐵— typically because the North star or other\nguardrail metrics showed statistically significant improvements.\nWe denote inconclusive experiments as 𝐴?𝐵, implying statistically\ninsignificant outcomes on the North Star. In rare cases, the North\nstar might have gone up at the expense of important guardrail met-\nrics, rendering conclusions ambiguous. Finally, 𝐴≃𝐵represents\nA/A experiments, where we know the null hypothesis to hold true\n(by design). The first set of experiments is used to measure type-III\nerrors. Known and unknown outcomes are used to measure type-II\nerrors, and A/A experiments can inform us about type-I errors. This\ndataset of past A/B experiments is collected and labelled by hand,\nfrom natural experiments occurring on the platform over time.\n2.2.1 Optimising Metric Weights with a Geometric Heuristic. Note\nthat𝑧as a function of 𝑤isscale-free . That is, the direction of the\nweight vector 𝑤matters, but its scale does not. As Kharitonov et al .\nwrite [ 16], we can compute the optimal direction of 𝑤using the\nLagrange multipliers method, to obtain:\n𝑤★\n𝐴≻𝐵∝(𝚺𝐴+𝚺𝐵+𝜖𝐼)−1(𝝁𝐴−𝝁𝐵). (8)\nHere,𝜖∈Ris a small number to ensure that the matrix to be\ninverted is not singular. Kharitonov et al .fix this value at 𝜖=0.01\nand never adjust it throughout the paper. We wish to highlight\nthat technique is known as Ledoit-Wolf shrinkage [ 20,21], and\nthat it can have substantial influence on the obtained direction.\nIndeed: it acts as a regularisation term pushing the weights closer\nto𝑤=(𝝁𝐴−𝝁𝐵). This can be seen by observing that as 𝜖→inf, the\ninverse becomes1\n𝜖𝐼, and the solution hence becomes 𝑤=(𝝁𝐴−𝝁𝐵)\n𝜖.\nAs we only care about the direction we can ignore the denominator.\nTo ensure fair comparison, we also set 𝜖=0.01. Exploring the effects\nof Ledoit-Wolf shrinkage as a regularisation technique where 𝜖is a\nhyper-parameter, gives an interesting avenue for future work.\nIn order to include observations from multiple experiments into\na single set of learned weights, they propose to compute the optimal\ndirection per experiment, normalise, and average the weights:\n𝑤★\nE+=1\n|E+|∑︁\n(𝐴,𝐵)∈E+𝑤★\n𝐴≻𝐵\r\r𝑤★\n𝐴≻𝐵\r\r\n2. (9)\nWhilst this procedure provides no guarantees about the sensitivity\nof the obtained metric on the overall set of experiments, it is efficient\nto compute and provides a strong baseline method.\n2.2.2 Optimising Metric Weights via Gradient Ascent on 𝑧-scores. A\nmore principled approach is to cast the above as an optimisation\nproblem. The objective function for this optimisation problem con-\nsists of three parts. First, we wish to maximise the𝑧-score for allvariant pairs with known outcomes:\nL+\n𝑧(𝑤;E+)=1\n|E+|∑︁\n(𝐴,𝐵)∈E+𝑧𝐴≻𝐵\n𝜔.(10)\nSecond, we wish to maximise theabsolute𝑧-score for all variant\npairs with inconclusive outcomes under the North Star:\nL?\n𝑧(𝑤;E?)=1\n|E?|∑︁\n(𝐴,𝐵)∈E?\f\f\f𝑧𝐴≻𝐵\n𝜔\f\f\f. (11)\nThird, we wish to minimise theabsolute𝑧-score for all variant pairs\nthat are equivalent (i.e. A/A-pairs):\nL≃\n𝑧(𝑤;E≃)=1\n|E≃|∑︁\n(𝐴,𝐵)∈E≃\f\f\f𝑧𝐴≻𝐵\n𝜔\f\f\f. (12)\nThis gives rise to a combined objective as a weighted average:\nL𝑧(𝑤;E)=L+\n𝑧(𝑤;E+)+𝜆?L?\n𝑧(𝑤;E?)−𝜆≃L≃\n𝑧(𝑤;E≃).(13)\nKharitonov et al .demonstrate that, for a variety of different metrics\nin a web search engine, these approaches can exhibit improved\nsensitivity [ 16]. We apply this method to learn instantaneously\navailable proxies to a delayed North Star metric in general scenarios,\nand propose several extensions, detailed in the following Sections.\n3 METHODOLOGY & CONTRIBUTIONS\n3.1 Learning Metrics that Maximise Power\nWhen directly optimising 𝑧-scores, an implicit assumption is made\nthat the utility we obtain from increased 𝑧-scores is linear. This is\nseldom a truthful characterisation of reality, considering how we\nwish to actually usethese metrics downstream. We provide a toy\nexample in Table 1, reporting 𝑧-scores and one-tailed 𝑝-values for\ntwo experiments and three possible metrics 𝑚1,𝑚2,𝑚3, inspired\nby real data. In this toy example, we know that 𝐴≻𝐵, based on a\nhypothetical North Star metric. Nevertheless, as we do not know\nthis beforehand, we typically test for the null hypothesis 𝐴≃𝐵with\ntwo-tailed𝑝-values. In the table, this means that the outcome is\nstatistically significant if the reported one-tailed values are 𝑝<𝛼\n2\nor𝑝>1−𝛼\n2. We report the power of every metric at varying\nconfidence levels 𝛼∈{0.05,0.01}, reporting whether (i) the null\nhypothesis is correctly rejected ( 𝑝<𝛼\n2,✓), (ii) the outcome is\ninconclusive ( ?), i.e. a type-II error, or (iii) the null hypothesis is\nrejected, but for the wrong reason (𝑝>1−𝛼\n2,×).\nThis latter case is deeply problematic, as it signifies disagreement\nwith the North Star. Such errors have been described as type-III\nerrors in the statistical literature [ 15,24,35]. Naturally, we would\nrather have a metric that fails to reject the null than one that confi-\ndently declares a faulty variant to be superior. Indeed, Deng and Shi\nargue that both directionality andsensitivity are desirable attributes\nfor any metric [ 7]. Nevertheless, considering candidate metrics in\nTable 1, type-III errors are not sufficiently penalised by the aver-\nage𝑧-score: metric 𝑚3maximises this objective despite yielding\nstatistical power that is on par with a coin flip.\nDirectly maximising power might prove cumbersome, as it is\nessentially a discrete step-function w.r.t. the 𝑧-score, dependent\non the confidence level 𝛼. Instead, it comes natural to minimise\nthe one-tailed 𝑝-value reported in Table 1. Indeed, the 𝑝-value\ntransformation models diminishing returns for high𝑧-scores, whichOlivier Jeunen and Aleksei Ustimenko\n𝑧-score𝑝-value Power\n𝛼=0.05𝛼=0.01\n𝑚1Exp. 1 1.97 2.44e−2 ✓ ?\nExp. 2 1.97 2.44e−2 ✓ ?\nMean 1.97 2.44e−2 100% 0%\n𝑚2Exp. 1 1.90 2.87e−2 ? ?\nExp. 2 3.50 2.33e−4 ✓ ✓\nMean 2.70 1.45e−2 50% 50%\n𝑚3Exp. 1−2.58 9.95e−1× ×\nExp. 2 8.00 6.66e−16 ✓ ✓\nMean 5.29 4.98e−1 50% 50%\nTable 1: An example setting highlighting that maximising\naverage𝑧-scores (𝑚1) does not imply improved statistical\npower. Statistical power is dependent on the significance\nlevel𝛼,𝑝-values provide a good proxy to optimise for. We\nobserve similar trade-offs from real past A/B experiments.\nallows type-III errors to be sufficiently penalised. When considering\nthis objective, 𝑚3is clearly suboptimal whilst 𝑚2is preferred.\nNote that the change in objective would not affect the geomet-\nric heuristic as described in Section 2.2.1. As we simply apply a\nmonotonic transformation on the 𝑧-scores, the weight direction that\nmaximises the 𝑧-score equivalently minimises its 𝑝-value. When\nlearning via gradient descent, however, the 𝑝-value transformation\naffects how we aggregate and attribute gains over different input\nsamples. This allows us to stop focusing on increasing sensitiv-\nity for experiments that are already “sensitive enough”, and more\nequitably consider all experiments in the training data.\nThis change in objective provides an intuitive and efficient ex-\ntension to existing approaches, allowing us to directly optimise\nthe confidence we have to correctly reject the null-hypothesis. For\nknown outcomes, the loss function is given by:\nL+\n𝑝(𝑤;E+)=1\n|E+|∑︁\n(𝐴,𝐵)∈E+1−Φ(𝑧𝐴≻𝐵\n𝜔)\n=1\n|E+|∑︁\n(𝐴,𝐵)∈E+1−Φ \n𝝁𝐴𝑤⊺−𝝁𝐵𝑤⊺\n√︁\n𝑤𝚺𝐴𝑤⊺+𝑤𝚺𝐵𝑤⊺!\n,(14)\nand analogously extended to unknown L?𝑝and A/A-outcomes L≃𝑝.\nNevertheless, we wish to point out that we only want to maximise\n𝑝-values for A/A-outcomes if type-I error becomes problematic. As\nwe will show empirically in Section 4.3, this is not a problem we\nencounter. For this reason, we set 𝜆≃≡0.\nNote that whilst direct optimisation of 𝑝-values is an improve-\nment over myopic consideration of 𝑧-scores, there is another caveat:\nthe “worst-case” loss of a type-III error is bounded at 1, which does\nnot reflect our true utility function: metrics that disagree with the\nNorth Star are farless reliable than those that simply remain incon-\nclusive. As such, we also consider another variant of the objective,\nwhere ¯𝑝=−𝑝log(1−𝑝). Figure 1 provides visual intuition to clarify\nhow this monotonic transformation on the 𝑝-values more heavily\npenalises type-III errors, whilst retaining the optimum. From a the-\noretical perspective, this function provides a convex relaxation for\nminimising the number of type-III errors a metric produces. As a\nresult, we expect this surrogate to exhibit strong generalisation. We\nrefer to this objective as minimising the log𝑝-value.\n−3−2−1 0 1 2 3\nz-score0246Lossp-value\n−p(log(1−p))\np∈{0.025,0.975}Figure 1: Visualising our proposed optimisation objectives\nfor learnt metrics, as a function of their 𝑧-score.\nNote that one could envision further extensions here where\nthe confidence level 𝛼isdirectly incorporated into the objective\nfunction to maximise statistical power at a given confidence level 𝛼.\nNevertheless, we conjecture that their discrete nature might hamper\neffective optimisation and generalisation, compared to the strictly\nconvex and smooth surrogate we obtain from the log𝑝-value.\n3.2 Accelerated Convergence for Scale-Free\nObjectives via Spherical Regularisation\nThe objective functions we describe —either 𝑧-scores or(log)𝑝-\nvalues— are scale-free w.r.t. the weights that are being optimised.\nAs a result, out-of-the-box gradient-based optimisation techniques\nare not well-equipped to handle this efficiently.\nConsider a simple toy example where we have two observed\nmetrics for an experiment with a known preference 𝐴≻𝐵, and:\n𝝁𝐴=[1.0,1.0], 𝝁𝐵=[0.5,0.5], 𝚺𝐴=𝚺𝐵=𝐼. (15)\nFor this low-dimensional problem, we can visualise the 𝑧-score\nas a function of the metric weights in a contour plot, as shown in\nFigure 2a. Here, it becomes visually clear that whilst the direction of\nthe𝑤=[𝑤1,𝑤2]vector matters, its scale does not. The consequence\nis that the gradient vectors w.r.t. the objective on the right-hand plot\ncan lead to slow convergence, even in this concave objective. Indeed,\nfor poor initialisation in the bottom left quadrant (e.g. 𝑤=[−1,−2]),\nthe gradient direction is perpendicular w.r.t. the optima.\nRecent work makes a similar observation for discrete scale-free\nobjectives as they appear in ranking problems [ 36]. They propose to\nadopt projected gradient descent, normalising the gradients before\nevery update. Whilst effective, in our setting we would prefer to use\nout-of-the-box optimisation methods for practitioners’ ease-of-use.\nInstead, we introduce a simple regularisation term that represents\nthe distance between the scale of the 𝑤vector and a hyper-sphere:\nL∥𝑤∥=−𝛿\u0010\n𝑁−∥𝑤∥2\n2\u00112\n. (16)\nAll optima for this objective function are also optima to the origi-\nnal function—but the gradient field is more amenable to iterative\ngradient-based optimisation techniques. Figure 2b visualises how\nthis transforms the loss surface. Under this regularised objective,\nit is visually clear that gradient-based optimisation methods are\nlikely to exhibit faster convergence. Our empirical results confirm\nthis, for a variety of initialisation weights and learning objectives.\n4 EXPERIMENTS & DISCUSSION\nTo empirically validate the methods proposed in this work, we\nrequire a dataset containing logged metrics (sample means, their\nvariances and covariances), together with preference orderings overLearning Metrics that Maximise Power for Accelerated A/B-Tests\n−2 0 2\nw1−3−2−10123w2\n−0.495−0.475\n−0.450−0.400−0.350−0.300\n−0.250−0.200\n−0.150−0.100\n−0.0500.000\n0.0500.100\n0.1500.2000.250\n0.3000.3500.4000.450\n0.4750.495\n−2 0 2\nw1w2\n∇z\nw⋆\n−0.50.00.5\n(a) Original scale-free objective function.\n−2 0 2\nw1−3−2−10123w2\n−0.500−0.450−0.400−0.350\n−0.300−0.250−0.200\n−0.150−0.100−0.050\n0.0000.050\n0.1000.1500.200\n0.2500.300\n0.3500.400\n0.4000.4500.475\n0.495\n−2 0 2\nw1w2\n∇z\nw⋆−0.50.00.5\n (b) With added spherical regularisation ( 𝛿=5e−4).\nFigure 2: Directly maximising 𝑧-scores yields a scale-free objective that is not amenable to efficient optimisation with gradient\nascent ( left). Spherical regularisation retains all optima, whilst providing a gradient direction that is more aligned ( right ).\ncompeting system variants that were collected from real-world A/B-\ntests, ideally spanning large user-bases and several weeks.\nExisting work on this topic used a private dataset from Yandex fo-\ncused on web search experiments that ran between 2013–2014 [ 16].\nThey report type-I and -II errors for 8 metrics and a fixed 5% confi-\ndence level, over 118 A/B-tests and 472 A/A-tests.\nIn this work, we consider more general metrics that are relevant\nfor use-cases beyond web search (i.a. user retention and various\nengagement signals). Furthermore, we report type-I/II/III errors at\nvarying confidence levels, providing insights into the learnt metrics’\nbehaviour. For this, we leverage logs of past A/B-experiments on\ntwo large-scale short-video platforms with over 160 million monthly\nactive users each: ShareChat and Moj. The datasets consist of 153\nA/B-experiments total that ran in 2023, and over 25 000 A/A-pairs.\nIn total, we have access to roughly 100 metrics detailing various\ntypes of interactions with the platform, engagements, and delayed\nretention signals. Because our dataset is limited in size (a natural\nconsequence of the problem domain), we are bound to overfit when\nusing all available metrics as input features. As such, we limit\nourselves to 10 input metrics to learn from, and evaluate them w.r.t.\nthe delayed North Star. This feature selection step also ensures that\nour linear model consists of fewer parameters, which increases\npractitioners’ and business stakeholders’ trust in its output. We\nfocus on non-delayed signals, including activity metrics such as\nthe number of sessions and active days, and counters for positive\nand negative feedback engagements of various types. These are\nselected through an analysis of their type-I/II/III errors w.r.t. the\nNorth Star, as well as their 𝑧-scores: focusing on metrics with high\nsensitivity and limited disagreement. The research questions we\nwish to answer empirically using this data, are the following:\nRQ1 Do learnt metrics effectively improve on their objectives?\nRQ2 How do learnt metrics behave in terms of type-III errors?\nRQ3 How do learnt metrics’ type-I/II errors behave when considered\nas stand-alone evaluation metrics?\nRQ4 How do learnt metrics’ type-I/II errors behave when used in\nconjunction with the North Star and top proxy metrics?\nRQ5 How do learnt metrics influence required sample sizes when\nused in conjunction with the North Star and top proxy met-\nrics?\nRQ6 Do we observe accelerated convergence over varying objectives\nvia the proposed spherical regularisation technique?Objective 𝑧-score (↑) 𝑝-value (↓)\nMean Median Mean Median\nheuristic 7.31 3.07 1.88e−1 1.18e−3\n𝑧-score 7.55 2.67 2.33e−1 3.88e−3\n𝑝-value 5.22 3.08 4.32e−21.09e−3\nlog𝑝-value 4.33 3.17 5.19e−28.60e−4\nTable 2: Sensitivity results for learnt metrics, computed\nvia leave-one-out cross-validation on all experiments with\nknown outcomes. We observe that minimising the log𝑝-value\neffectively improves median sensitivity over alternatives.\nWe report results for the ShareChat platform in what follows,\nand provide further empirical results for Moj in Appendix A.\n4.1 Effectiveness of Learnt Metrics (RQ1)\nWe learn and evaluate metrics through leave-one-out cross-validation:\nfor every experiment, we train a model on all other experiments and\nevaluate the 𝑧-score (Eq. 7) and 𝑝-value (Eq. 3) the metric yields for\nthe held-out experiment. We report the mean and median 𝑧-scores\nand𝑝-values we obtain for all A/B-pairs with known outcomes (i.e.\nE+) in Table 2. Best performers for every column (either maximising\n𝑧-scores or minimising 𝑝-values) are highlighted in boldface . Em-\npirical observations match our theoretical expectations: whilst the\n𝑧-score objective does effectively maximise the average𝑧-score, it is\nthe worst performer for both mean and median 𝑝-values, and even\nthe median𝑧-score. Our proposed log𝑝-value objective effectively\nimproves both the median 𝑧-score and𝑝-value over alternatives.\n4.2 Agreement with the North Star (RQ2)\nFrom the obtained 𝑧-scores and 𝑝-values summarised in Table 2,\nwe can additionally derive (dis-)agreement with the North Star, for\nvarying confidence levels 𝛼. We visualise these results in Figure 3:\nif the obtained 𝑝-value under a learnt metric is lower than 𝛼, that\nmetric yields a statistically significant result (agreement). If the\nobtained𝑝-value for the alternative hypothesis (i.e. 𝐵≻𝐴when\nwe know𝐴≻𝐵) is lower than 𝛼, we have statistically significant\ndisagreement , or type-III error. This is a capital sin we wish to\navoid at all costs, as it severely diminishes the trust we can put in\nthe learnt metric. If the 𝑝-value reveals a statistically insignificant\nresult, we say the result is inconclusive, implying a type-II error. We\nobserve that both the 𝑧-score-maximising metric and the heuristicOlivier Jeunen and Aleksei Ustimenko\n0% 20% 40% 60% 80% 100%w⋆(p-value)w⋆(logp-value)w⋆(heuristic)w⋆(z-score)α= 1e−01\nAgreement Inconclusive ( type-II ) Disagreement ( type-III )0% 20% 40% 60% 80% 100%α= 5e−02\n0% 20% 40% 60% 80% 100%α= 1e−02\n0% 20% 40% 60% 80% 100%α= 1e−03\n0% 20% 40% 60% 80% 100%α= 1e−04\nFigure 3: Learnt metrics and their agreement with known North Star outcomes over varying confidence levels. A learnt metric\nthat maximises the average 𝑧-score exhibits significant type-III error, which can be alleviated by minimising 𝑝-values instead.\n10−410−310−210−1\nα10−510−410−310−210−1Type-I Error\nExpected (α) North Star Top-Proxy w⋆(p-value) w⋆(logp-value)10−410−310−210−1\nα0.50.60.70.8Type-II Error\nSensitivity & Power for Stand-alone Metrics\n(a) False-positive rate (type-I error) and false-negative rate (type-II error, 1−power) for stand-alone metrics.\n10−410−310−210−1\nα10−510−410−310−210−1Type-I Error\n10−410−310−210−1\nα0.30.40.50.60.70.8Type-II Error\nNorth Star + Top-Proxy or +w⋆(p−value) or +w⋆(logp−value)Sensitivity & Power for Sets of Metrics (a/f_ter Bonferroni correction)\n(b) False-positive rate (type-I error) and false-negative rate (type-II error, 1−power) for sets of metrics, after Bonferroni correction.\nFigure 4: When considering only learnt metrics, we improve type-II error significantly without hurting specificity. At 𝛼=0.05, we\nincrease statistical power by 67% ( upper plot ). In conjunction with the North Star and top proxy metrics, Bonferroni corrections\nare slightly conservative (type-I error <𝛼), and allow us to improve statistical power by 133% for 𝛼=0.05(lower plot ).\napproach fail to steer clear from type-III error. Optimising ( log)𝑝-\nvalues instead, alleviates this issue. For this reason, we only consider\nthese metrics for further evaluation. Indeed: an analysis of type-II\nerror is rendered meaningless when type-III errors are present.\nFigure 7a in Appendix A highlights that for the Moj platform as\nwell, type-III errors are common in the case of 𝑧-score-maximising\nor heuristic metrics. As such, we only consider ( log)𝑝-values to\nassess increases in statistical power and potential reductions to the\ncost of running online experiments.\n4.3 Power Increase from Learnt Metrics (RQ3–4)\nUntil now, we have leveraged experiments with known outcomes\nto assess sensitivity and agreement with the North Star. Now, we\nadditionally consider A/A-experiments ( E≃) and experiments with\nunknown outcomes (E?) to measure type-I and type-II error respec-\ntively. We measure this for the North Star, for the best-performing\nproxy metric that serves as input to the learnt metrics, and for learntmetrics that exhibit no empirical disagreement with the North Star.\nWe plot the type-I error (i.e. fraction of A/A-pairs in E≃that are\nstatistically significant at confidence level 𝛼) and the type-II error\n(i.e. fraction of A/B-pairs in {E+∪E?}that are statistically insignif-\nicant at confidence level 𝛼) for varying values of 𝛼in Figure 4a. We\nobserve that we are able to significantly reduce type-II errors com-\npared to the North Star (up to 78%), whilst keeping type-I errors at\nthe required level (i.e. 𝛼). However, we also observe that the type-II\nerror we obtain when using learnt metrics does not significantly\nimprove over the top proxy metric, when considered in isolation.\nNonetheless, this is not how evaluation metrics are used in prac-\ntice. Indeed, we track several metrics and can draw conclusions if\nanyof them are statistically significant. As such, metrics should be\nevaluated on their complementary sensitivity. That is, we compute\n𝑝-values for a set of metrics, apply a Bonferroni correction, and\nassess statistical significance. The statistical power that we obtain\nthrough this procedure is visualised in Figure 4b. We consider eitherLearning Metrics that Maximise Power for Accelerated A/B-Tests\n10−410−310−210−1\nα12345678Median Rel. z2\nNorth Star\n+ Top-Proxy\nor +w⋆(p−value)\nor +w⋆(logp−value)Relative Sample Size Decrease for Sets of Metrics (a/f_ter Bonferroni)\nFigure 5: When considering learnt metrics in conjunction\nwith the North Star and top proxy metrics, we require signif-\nicantly reduced sample sizes to obtain the same statistical\nconfidence level as we would get from the North Star.\nthe North Star in isolation, the North Star in conjunction with the\ntop-proxy, or a further combination with any learnt metric. Here,\nwe observe that the learnt metric provides a substantial increase in\nstatistical power: statistical power (i.e. 1−type-II error) is increased\nby up to 210% compared to the North Star alone, and 25–30% over\nthe North Star plus proxies. Furthermore, as the Bonferroni correc-\ntion is slightly conservative, we observe lower than expected type-I\nerror for higher confidence levels 𝛼. This implies that a more fine-\ngrained multiple testing correction can further improve statistical\npower. We empirically observe that this works as expected, but its\neffects are negligible in practice.\n4.4 Cost Reduction from Learnt Metrics (RQ5)\nSo far, we have shown that metrics learnt to minimise ( log)𝑝-values\nare effective at improving sensitivity (Table 2), whilst minimising\ntype-III error (Figure 3) and improving statistical power (Figure 4).\nOn one hand, powerful learnt metrics can lead to more con-\nfident decisions from statistically significant A/B-test outcomes.\nAnother view is that we could make the same amount of decisions\nbased on fewer data points, as we reach statistical significance with\nsmaller sample sizes. This implies a cost reduction, as we can run\nexperiments either on smaller portions of user traffic or for shorter\nperiods of time, directly impacting experimentation velocity.\nThis reduction in required sample size is equal to the square of\nthe relative 𝑧-score [ 4,16]. We visualise this quantity in Figure 5,\nfor varying confidence levels 𝛼, for the same Bonferroni-corrected\nprocedure as Figure 4b. To obtain a 𝑧-score for a setof metrics, we\nsimply take the maximal score and apply a Bonferroni correction\nto it as laid out in Section 2.1.1. Note that this procedure depends\non𝛼, explaining the slope in Figure 5. We observe that our learnt\nmetrics can achieve the same level of statistical confidence as the\nNorth Star with up to 8 times fewer samples, i.e. a reduction down\nto12.5%. This significantly reduces the cost of experimentation for\nthe business, further strengthening the case for our learnt metrics.\n4.5 Spherical Regularisation (RQ6)\nOur goal is to assess and quantify the effects of the proposed spheri-\ncal regularisation method in Section 3.2. We train models on all avail-\nable data with known outcomesE+, where we have a vetted prefer-\nence over variants 𝐴≻𝐵. We consider three weight initialisation\nstrategies to set 𝑤init, and normalise weights to ensure L∥𝑤init∥=0:(i)good initialisation at 𝑤init=1\n|E+|Í\n(𝐴,𝐵)∈E+𝝁𝐴−𝝁𝐵, (ii) con-\nstant initialisation at the all-one vector 𝑤init=®1, and (iii) bad\ninitialisation at 𝑤init=1\n|E+|Í\n(𝐴,𝐵)∈E+𝝁𝐵−𝝁𝐴. We train mod-\nels for all learning objectives we deal with in this paper: 𝑧-scores,\n𝑝-values, and log𝑝-values; whilst varying the strength of the spher-\nical regularisation term 𝛿. As discussed, this term does not affect\nthe optima, but simply transforms the loss to be more amenable to\ngradient-based optimisation methods. Thus, we expect convergence\nafter fewer training iterations. All models are trained until conver-\ngence with the adam optimiser [ 17], initialising the learning rate\nat5e−4and halving it every 1 000 steps where we do not observe\nimprovements. We use the radam variant to avoid convergence\nissues [ 22,27], and have validated that this choice does not sig-\nnificantly alter our obtained results and conclusions. We consider\na model converged if there are no improvements to the learning\nobjective after 10 000 steps. All methods are implemented using\nPython3.9 and PyTorch [25].\nFigure 6 visualises the evolution of the learning objective over\noptimisation steps, for all mentioned learning objectives, initialisa-\ntion strategies, and regularisation strengths. We observe that the\nmethod is robust, significantly improving convergence speed for all\nsettings, requiring up to 40% fewer iterations until convergence is\nreached. This positively influences the practical utility of the learnt\nmetric pipeline for researchers and practitioners.\n5 INSIGHTS FROM LEARNT METRICS\nIn this Section, we briefly discuss insights that arose through our\nempirical evaluation of all metrics: the North Star, classical sur-\nrogates and proxies, as well as learnt metrics. These insights are\nspecific to our platforms, but we believe they can contribute to a\ngeneral intuition and understanding of metrics for online content\nplatforms and broader application areas.\nRatio metrics are easily fooled. Often, important metrics can be\nframed as a ratio of the means (or sums) of two existing metrics [ 2,\n3]. Examples include click-through rate (i.e. clicks / impressions),\nvariants of user retention (i.e. retained users / active users), or\ngeneral engagement ratios (e.g. likes / video-plays). We observe that,\nwhilst these metrics can be important from a business perspective,\nthey typically exhibit significant type-III errors w.r.t. the North Star.\nIndeed, in the examples above both the numerator and denominator\nrepresent positive signals we wish to increase. Suppose an online\nexperiment increases the number of video-plays by 𝑌%, and the\noverall number of likes by 𝑋%<𝑌%. These two positive signals\nwill lead to a decreasing ratio, whilst we are likely to still prefer the\ntreatment w.r.t. the North Star if 𝑋and𝑌are substantially large.\nWe believe this is connected to common offline ranking evalua-\ntion metrics prevalent in the recommender systems field [ 12,32].\nIndeed, such metrics are cumulative in nature, optimising overall\nvalue instead of some notion of value-per-item [14].\nUser-level aggregations conquer general counters. In the previous\nexample, we describe general count metrics for the number of likes\nand the number of video-play events. User behaviour on online\nplatforms often follows a power-law distribution: a few “power\nusers” generate the majority of such events [ 5]. As a result, such\nmetrics are easily skewed, and they are not guaranteed to accuratelyOlivier Jeunen and Aleksei Ustimenko\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5\n×104−0.50.00.51.0Averagez-score×101\n winit=−µA−µB\n0.0 0.5 1.0 1.5 2.0\n×1040.900.951.00×101\n winit= 1\nδ= 0e+ 00\nδ= 1e−05\nδ= 1e−04\nδ= 1e−03\nδ= 5e−03Converged\nConverged\nConverged\nConverged\nConverged\n0.0 0.5 1.0 1.5 2.0 2.5\n×1040.850.900.951.00×101\n winit=µA−µB\n0 1 2 3 4\n×10410−1100Averagep-value\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\n×10410−1\n3×10−24×10−26×10−2\nδ= 0e+ 00\nδ= 1e−08\nδ= 5e−08\nδ= 1e−07\nδ= 1e−06Converged\nConverged\nConverged\nConverged\nConverged\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5\n×10410−1\n3×10−24×10−26×10−2\n0 1 2 3 4 5 6\nOptimisation steps×10410−210−1100101Average logp-value\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nOptimisation steps×10410−2\nδ= 0e+ 00\nδ= 1e−08\nδ= 1e−07\nδ= 1e−06\nδ= 5e−06Converged\nConverged\nConverged\nConverged\nConverged\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5\nOptimisation steps×10410−210−1\nFigure 6: Spherical regularisation significantly accelerates convergence for all considered objectives, up to 40%.\nreflect improvements for the full population of users. Aggregating\nsuch counters per users (e.g. count the number of days a user has at\nleast𝑋video-plays) instead of using raw event counters, provides\nstrong and sensitive proxies to the North Star.\nInterestingly, this framing is reminiscent of recall , as we effec-\ntively measure the coverage of users about whom we have positive\nsignals. Recall metrics are again strongly connected to offline eval-\nuation practices in recommender systems, especially in the first\nstage of two-stage systems common in industry [6, 23].\n6 CONCLUSIONS & OUTLOOK\nA/B-testing is a crucial tool for decision-making in online busi-\nnesses, and it has widely been adopted as a go-to approach to allow\nfor continuous system improvements. Notwithstanding their popu-\nlarity, online experiments are often expensive to perform. Indeed:\nmany experiments lead to statistically insignificant outcomes, pre-\nsenting an obstacle for confident decision-making. Experiments\nthat do lead to significant outcomes are costly too: by their very def-\ninition, a portion of user traffic interacts with a sub-optimal system\nvariant. As such, we want to maximise the number of decisions we\ncan make based on the experiments we run, and we want to min-\nimise the required sample size for statistically significant outcomes.\nIn this work, we achieve this by learning metrics that maximise\nthe statistical power they harness. We present novel learning ob-\njectives for such metrics, and provide a thorough evaluation of the\neffectiveness of our proposed approaches. Our learnt metrics are\ncurrently used for confident, high-velocity decision-making across\nShareChat and Moj business units.\nREFERENCES\n[1]Susan Athey, Raj Chetty, Guido W Imbens, and Hyunseung Kang. 2019. The\nSurrogate Index: Combining Short-Term Proxies to Estimate Long-Term Treatment\nEffects More Rapidly and Precisely . Working Paper 26463. 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ACM, 507–514. https://doi.org/10.1145/\n1835449.1835534Olivier Jeunen and Aleksei Ustimenko\n0% 20% 40% 60% 80% 100%w⋆(p-value)w⋆(logp-value)w⋆(heuristic)w⋆(z-score)α= 1e−01\nAgreement Inconclusive ( type-II ) Disagreement ( type-III )0% 20% 40% 60% 80% 100%α= 5e−02\n0% 20% 40% 60% 80% 100%α= 1e−02\n0% 20% 40% 60% 80% 100%α= 1e−03\n0% 20% 40% 60% 80% 100%α= 1e−04\n(a) Figure 3 reproduced with data from Moj.\n10−410−310−210−1\nα10−510−410−310−210−1Type-I Error\nExpected (α) North Star Top-Proxy w⋆(p-value) w⋆(logp-value)10−410−310−210−1\nα0.60.70.80.9Type-II Error\nSensitivity & Power for Stand-alone Metrics\n(b) Figure 4a reproduced with data from Moj.\n10−410−310−210−1\nα10−510−410−310−210−1Type-I Error\n10−410−310−210−1\nα0.30.40.50.60.70.80.9Type-II Error\nNorth Star + Top-Proxy or +w⋆(p−value) or +w⋆(logp−value)Sensitivity & Power for Sets of Metrics (a/f_ter Bonferroni correction)\n(c) Figure 4b reproduced with data from Moj.\n10−410−310−210−1\nα24681012Median Rel. z2\nNorth Star\n+ Top-Proxy\nor +w⋆(p−value)\nor +w⋆(logp−value)Relative Sample Size Decrease for Sets of Metrics (a/f_ter Bonferroni)\n(d) Figure 5 reproduced with data from Moj.\nFigure 7: Additional experimental results for Moj. Empirical observations match those for ShareChat.\nA ADDITIONAL EXPERIMENTAL RESULTS\nTo further empirically validate our theoretical insights w.r.t. the\nproposed methods, we repeat the experiments reported in Section 4\non data collected for the Moj platform, and reproduce Figures 3–5.\nResults are visualised in Figure 7 Observations match our expecta-\ntions, further strengthening trust in the replicability of our results.\nAll improvements in sensitivity and statistical power are a similar\norder of magnitude as those for ShareChat: learnt metrics thatminimise ( log)𝑝-values can substantially reduce type-II/III errors\nwithout affecting type-I errors. We observe an improvement over\nShareChat data in Figure 7d: learnt metrics for Moj exhibit a 12-\nfold reduction in the sample size that is required to attain constant\nstatistical confidence as to the North Star."    },    {        "title": "2402.03936v1.Commutativity_and_orthogonality_of_similarity_orbits_in_Banach_algebras.pdf",        "content": "arXiv:2402.03936v1  [math.FA]  6 Feb 2024COMMUTATIVITY AND ORTHOGONALITY OF SIMILARITY ORBITS IN\nBANACH ALGEBRAS\nMUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\nAbstract. For a semisimple unital Banach algebra AoverC, and elements a,b∈A,we show that\nthe similarity orbits, orb( a) and orb( b),over the principal component of the invertible group of A\ncommute precisely when there is at least one nonzero complex number not belonging to the spectrum\nof any product a′b′—where ( a′,b′)∈orb(a)×orb(b). In this case, the polynomially convex hull of\nthe spectra of the a′b′is constant. When orb( a) = orb( b),thenais central under the aforementioned\nassumption—and the result then generalizes part of an old th eorem due to J. Zem´ anek. We show\nfurther that the two classical characterizations of commut ative Banach algebras via the spectral radius\ncan be algebraically localized in the sense of ‘local’ impli es ‘global’. Thereafter, in Section 3, we give\na (somewhat weaker) localization of the above situation inv olving spectral perturbation on small\nneighborhoods in a similarity orbit. Finally, we apply the a bove results to algebraic elements and\nidempotents in particular, so that orthogonality of simila rity orbits of two idempotents is equivalent\nto a pair of spectral radius properties. To conclude with, a c ouple of localization theorems specific to\nidempotents and algebraic elements are presented. Similar statements to all of the above hold if a′b′\nis replaced by a′+b′,a′−b′, ora′+b′−a′b′.\n1 Introduction\nWe will assume throughout this paper that ( A,/bardbl·/bardbl) is a semisimple complex and unital Banach\nalgebra with unit denoted by 1. Furthermore, the group of invertible elements in Awill be denoted\nbyG(A), with the principal component containing 1denoted by G1(A) (or simply G1if there is no\nconfusion). We shall use σAto denote the spectrum\nσA(x) ={λ∈C:λ1−x/\\e}atio\\slash∈G(A)},\nandρAto denote the spectral radius\nρA(x) = sup{|λ|:λ∈σA(x)},\nof an element x∈A(and obviously drop the subscript if the context is clear as to which a lgebra is\nconcerned). The nonzero spectrum of x∈Awill be the set σ′(x) =σ(x)\\{0}, and the polynomially\nconvex hull of the spectrum of xwill be denoted using ˆ σ(x). Geometrically, ˆ σ(x) is the spectrum of\nxtogether with all of its holes filled in—which ensures that C\\ˆσ(x) is connected. Throughout this\npaper, we will use a◦bto denote the element a+b−ab, for any given a,b∈A. Notice then that\na◦b=1−(1−a)(1−b). The key ideas in our study will utilise and be concerned with the similar ity\norbit of an element a∈A, defined as:\norb(a):=/braceleftbig\nwaw−1:w∈G1/bracerightbig\n.\nNotice that, since G1is a group, the collection of all similarity orbits of elements of Aforms a partition\nofA. Due to the nature of the elements of the similarity orbit, Jacobson ’s Lemma is favourable in the\nDate: February 7, 2024.\n2020Mathematics Subject Classification. 46H05, 46H15, 47A10.\nKey words and phrases. Banach algebra, spectrum, spectral radius, similarity orb it, idempotent, algebraic element.\n12 MUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\nmatter of reducing the work required to prove certain facts in our upcoming results. As a reminder,\nJacobson’s Lemma says that the following is true for all a,b∈A:\nσ′(ab) =σ′(ba).\nBecause G(A) is a group, one can quickly show (without using the above equality) t hatσ(a) =\nσ(waw−1) for allw∈G1(A).\nIfEis the set of idempotents in a Banach algebra Aanda∈E, then J. Zem´ anek showed, amongst a\nlargenumberofotherresults, that the connectedcomponentof Econtaining aispreciselythe similarity\norbit ofa([5, Theorem 3.3]). Likewise, in 2016, E. Makai together with J. Zem´ anek in a collaborative\neffort had proven that if a polynomial pwhose roots all have multiplicity 1 is fixed, then the connected\ncomponents of the set E(p) ={x∈A:p(x) = 0}are precisely the similarity orbits of each of its\nmembers ([4, Theorem 3]). The connectedness of the similarity orbit s inE(p) is deeply investigated in\nthis latter paper, particularly connectedness via polynomial and po lygonal paths. Another important\nfact that will be used in a few instances is that ais central if and only if orb( a) ={a}. Hence, if\na′∈orb(a) is central, then a=a′since orb( a) = orb(a′) ={a′}.\nThe last major result in [5] provides ten statements which are equiv alent to an idempotent abeing\ncentral. In fact, these conditions are all intricately related to the similarity orbit and the spectrum of\na. To illustrate the various ideas that will henceforth be discussed, it is worthwhile to list some of the\nconditions given in [5, Theorem 4.3]. If a∈E, then the following are equivalent:\n(i)ais central;\n(ii)aE⊆E;\n(iii)aorb(a)⊆orb(a);\n(iv) sup\nr∈orb(a)ρ(ar)<∞;\n(v) sup\nr∈orb(a)ρ(r+a)<∞;\n(vi) sup\nr∈orb(a)ρ(r−a)<∞;\n(vii) sup\nr∈orb(a)ρ(a◦r)<∞;\n(viii)a◦E⊆E;\n(ix)a◦orb(a)⊆orb(a).\nIn the present paper, we not only show this list can be extended to m embers of E(p), but also a\nportion thereof is true for arbitrary Banach algebra elements. Fu rthermore, we had sought to obtain\nnew information concerning the interactions between the elements belonging to different similarity\norbits, say orb( a) and orb( b), inA. This is in stark contrast to the results in [5] where members of\norb(a) and only their interactions with awere studied. To clarify, compare the next question with\n(iv) in the above list: Suppose there is some ǫ >0 such that ρ(ar)≤ǫfor allr∈orb(b); what is the\nconnection between orb( a) and orb( b)? We answer this question via Corollary 2.5, which says that,\nunder the above premise, the similarity orbits must commute (with th e converse also being true). Keep\nin mind that there need not be anything extra assumed for the eleme nts; they need not necessarily be\nidempotents or even algebraic. Moreover, we also consider other a lgebraic variations, namely, r+a,\nr−a, anda◦r, asrvaries through orb( b)—and the same conclusions relating orb( a) and orb( b) are\ndeduced.\nThe improvements to the results due to Zem´ anek which we have in ou r current paper are corollaries\nto a more general theorem—namely—Theorem 2.2. In order to guar antee the commutation of the\nsimilarity orbits of aandb, we need only require that the union of all σ′(ar)—asrruns through\norb(b)—misses at least one nonzero complex number. We obtain the same s tatement for r+aandr−a\nand an ever-so-slightly weaker statement for a◦r. The details are thoroughly discussed in the theorem\ntogetherwith itsproof. Thisisthe highlightingresultofSection2. Ro undingoffthissectionisTheoremCOMMUTATIVITY AND ORTHOGONALITY OF SIMILARITY ORBITS IN BA NACH ALGEBRAS 3\n2.6, wherein we obtain an improvementupon two classicalcharacter izationsofcommutative semisimple\nBanach algebras related to subadditivity and submultiplicativity (res pectively) of the spectral radius.\nThe precise details of this will be expounded upon thoroughly prior to Theorem 2.6.\nInSection3, aweaklocalizationofTheorem2.2isproven. Insteadof rvaryingthroughoutorb( b), we\nconsider when rvaries in some open subset Nof orb(b)—but—we must add a few additional conditions\nin the premise. The first being that σ(ar)⊆C, for allr∈NwhereCis a fixed countable subset of\nC; the second assumption is that arhas a finite spectrum for some r∈orb(b). Then we can conclude\nthat the similarity orbits of aandbcommute. These additional assumptions are actually quite natural\nwhen one considers algebraic elements, as we do in the next section— wherein we apply the theory\nresulting from the prior sections in Section 4 to algebraic elements, a nd then to idempotents, so that\nnew facts concerning the similarity orbits of these types of element s are brought to light.\nIn this final section, supposing that aandbbelong to E(p), we expose some conditions under which\na′b′=µ1, for all ( a′,b′)∈orb(a)×orb(b)—andµ∈Cis fixed. Then, appropriately, if a,b∈E, it turns\nout that µ= 0, so that the similarity orbits of aandbare orthogonal. The pair of concluding results\nin this section discusses some localization aspects regarding idempot ents and algebraic elements in a\nsimilar vein to the main theorem of Section 3.\nMuch of the early work in this paper deals with representation theor y for Banach algebras. For a\nbrief idea of a typical argument using this, one can look at the proof of [3, Theorem 2.3] wherein a\ncharacterization of the Jacobson radical, rad( A), is given. A Banach algebra Ais said to be semisimple\nwhen rad( A) ={0}. We shall use the fact that rad( A) is precisely those members of Awhich belong to\nthe kernels of all continuous irreducible representations πofAthroughout our work (see [1, Theorem\n4.2.1]). Along with that, the next result due to A. Sinclair will be crucial to our proofs:\nTheorem 1.1 (Sinclair Density Theorem, [1, Corollary 4.2.6]) .Letπbe a continuous irreducible\nrepresentation of a Banach algebra Aon a Banach space X. If{ξ1,...,ξ n}and{η1,...,η n}are\nlinearly independent subsets of X, then there exists y∈G1(A)such that\nπ(y)ξi=ηifor each i∈ {1,...,n}.\nWe set forth a few additional conventions which will be the standard for the rest of this paper. Let\nL,M⊆C. To denote the number of distinct elements in L, we will simply write # L, and we also\ndefine the following sets:\nLM:={λµ:λ∈L,µ∈M},\nL+M:={λ+µ:λ∈L,µ∈M},\nαL:={αλ:λ∈L}forα∈C.\nFor the sake of brevity, we will let L2denote the set LMwhenever L=M. Ifa,b∈Acommute, then\nthe following spectral containments concerning their sum and prod uct are true:\nσ(a+b)⊆σ(a)+σ(b), σ(ab)⊆σ(a)σ(b).\nFor any metric space M, we will denote the open and closed balls with centre x∈ Mand radius ǫby\nBM(x,ǫ) andBM(x,ǫ), respectively.\n2 Commutativity of Similarity Orbits\nThe lemma below is the backbone of the proof of the main result—Theo rem 2.2—in that most of\nthe nuances regarding the representation theory used lie herein. The proof hinges on the fact that, if\nXis a Banach space and T∈ L(X), the Banach algebra of bounded linear operators on X, thenTis\na scalar multiple of the identity operator whenever every nonzero ξ∈Xis an eigenvector of T. Hence\nifTis non-central we must be able to find ξ∈Xsuch that {Tξ,ξ}is linearly independent in X.4 MUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\nLemma 2.1. Letπbe a continuous irreducible representation of a Banach alge braAon a Banach\nspaceX. Suppose that π(a)is non-central in L(X)andα∈C\\{0}. Then there exist η∈X\\{0}and\n{x,y,z} ⊆G1(A)such that {η,π(a)η}is a linearly independent set and\nπ(x)−1π(a)π(x)π(a)η=αη,\n/parenleftbig\nπ(y)−1π(a)π(y)+π(a)/parenrightbig\nη=αη,\n/parenleftbig\nπ(z)−1π(a)π(z)−π(a)/parenrightbig\nη=αη.\nMoreover, there exist γ∈X\\ {0}andw∈G1(A)such that {γ,π(a)γ}is linearly independent and\nπ/parenleftbig\n(w−1aw)◦a/parenrightbig\nγ= (1−α)γ.\nProof.If{π(a)η,η}is a linearly dependent set in Xfor allη∈X, then it can be shown that π(a)\nis a scalar multiple of the identity operator, implying that π(a) is central—which is obviously a con-\ntradiction. So let η/\\e}atio\\slash= 0 be such that {π(a)η,η}is a linearly independent set. We can certainly find\nsomeλ∈Csuch that λ2=α, and then {π(a)η,λη}is also linearly independent. By Sinclair’s Density\nTheorem we obtain some x∈G1(A) such that\nπ(x)π(a)η=λη,\nπ(x)(λη) =π(a)η.\nRewriting the second equality here gives λη=π(x)−1π(a)η. Therefore,\nπ/parenleftbig\nx−1axa/parenrightbig\nη=π/parenleftbig\nx−1a/parenrightbig\n(λη) =λ2η=αη.\nOne can easily show that {βπ(a)η−βαη,η}is linearly independent for all complex numbers β/\\e}atio\\slash= 0.\nThen recall {π(a)η,η}is linearly independent, so that an application of Sinclair’s Density Theo rem\nyields some yβ∈G1(A) such that\nπ(yβ)π(a)η=βπ(a)η−βαη\nπ(yβ)η=η.\nWith the second equality above we have that\nπ(a)π(yβ)η=π(a)η,\nand then the following is true:\nπ(yβ)−1[π(yβ)π(a)+π(a)π(yβ)]η=π(yβ)−1[(β+1)π(a)η−βαη],\nπ(yβ)−1[π(a)π(yβ)−π(yβ)π(a)]η=π(yβ)−1[(1−β)π(a)η+βαη].\nSinceπ(yβ)−1η=η, takingy=yβwhenβ=−1 gives us\n/parenleftbig\nπ(y)−1π(a)π(y)+π(a)/parenrightbig\nη=αη.\nOn the other hand if we set z=yβwhenβ= 1 we obtain\n/parenleftbig\nπ(z)−1π(a)π(z)−π(a)/parenrightbig\nη=αη.\nFinally, if we take a′=1−a, thenπ(a′) is non-central, and hence, by the first part, there is w∈G1(A)\nandγ∈X\\{0}such that {π(a′)γ,γ}is linearly independent and\nπ(w−1a′wa′)γ=αγ.\nHence it follows that {π(a)γ,γ}is linearly independent and\nπ((w−1aw)◦a)γ=π(1−(1−w−1aw)(1−a))γ\n=π(1−w−1a′wa′)γ=γ−αγ= (1−α)γ.\n/squareCOMMUTATIVITY AND ORTHOGONALITY OF SIMILARITY ORBITS IN BA NACH ALGEBRAS 5\nSince most of the technicalities were taken care of in the above lemma , the proof of Theorem 2.2\nis straightforward. The key point here is that the eigenvalues of an operator π(a) (for any continuous\nirreducible representation πofA) belong to σA(a). Considering the hypothesis of the lemma once\nmore, we observe that we have near-total freedom in our choice o f the complex number α(the only\nrestriction being that it is nonzero, with 1 also being excluded for the forma◦r). From this, conditions\n(1)–(4) become obvious.\nTheorem 2.2. LetAbe a semisimple Banach algebra and suppose that a,b∈A. Then every element\noforb(a)commutes with every element of orb(b)if and only if any one of the following holds:\n/uniondisplay\nr∈orb(b)σ′(ar)/\\e}atio\\slash=C\\{0} (1)\n/uniondisplay\nr∈orb(b)σ′(r+a)/\\e}atio\\slash=C\\{0} (2)\n/uniondisplay\nr∈orb(b)σ′(r−a)/\\e}atio\\slash=C\\{0} (3)\n/uniondisplay\nr∈orb(b)(σ′(a◦r)\\{1})/\\e}atio\\slash=C\\{0,1}. (4)\nProof.For the forward implication, we may use commutation of randato obtain\nσ(ar)⊆σ(a)σ(r), σ (r+a)⊆σ(a)+σ(r),\nσ(r−a)⊆σ(r)−σ(a), σ (a◦r)⊆σ(a)+σ(r)−σ(a)σ(r).\nRecall that σ(r) =σ(b) for allr∈orb(b), and hence our claim is true. With regards to the converse,\nwe will show commutation of aandbunder the assumption of (1)-(4) individually. So suppose that\nab/\\e}atio\\slash=ba, and let (1) hold. By semisimplicity there is a continuous irreducible rep resentation πofAon\nsome Banach space Xsuch that π(a) andπ(b) are not central. Let α∈C\\{0}such that α/\\e}atio\\slash∈σ(ar) for\nallr∈orb(b). Apply Lemma 2.1 to obtain some η/\\e}atio\\slash= 0 and y∈G1(A) such that {π(a)η,η}is linearly\nindependent and\nπ(y)−1π(a)π(y)π(a)η=αη.\nMoreover, fix γ/\\e}atio\\slash= 0 such that {π(b)γ,γ}is linearly independent, so that with Sinclair’s Density\nTheorem there must exist w∈G1(A) such that\nπ(w)π(a)η=π(b)γ,\nπ(w)η=γ.\nTherefore, one has that\nπ/parenleftbig\ny−1ayw−1bw/parenrightbig\nη=π/parenleftbig\ny−1ay/parenrightbig\nπ/parenleftbig\nw−1b/parenrightbig\nγ=π/parenleftbig\ny−1ay/parenrightbig\nπ(a)η=αη.\nNotice that this together with Jacobson’s Lemma forces α∈σ/parenleftbig\nayw−1bwy−1/parenrightbig\nwhich is a contradiction\nto our choice of α. Thus,aandbcommute.\nNow assume that (2) holds but ab/\\e}atio\\slash=baand letα∈C\\{0}such that α/\\e}atio\\slash∈σ(a+r) for allr∈orb(b).\nOnce more, there is a continuous irreducible representation πonAfor some Banach space Xsuch\nthatπ(a) andπ(b) are not central. With Lemma 2.1 we obtain some η/\\e}atio\\slash= 0 and y∈G1(A) such that\n{π(a)η,η}is linearly independent and\n/parenleftbig\nπ(y)−1π(a)π(y)+π(a)/parenrightbig\nη=αη.6 MUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\nAlso fixγ/\\e}atio\\slash= 0 such that {π(b)γ,γ}is a linearly independent set. By Sinclair’s Density Theorem there\nexistsw∈G1(A) such that\nπ(w)π(a)η=π(b)γ,\nπ(w)η=γ.\nHence it follows that\nπ/parenleftbig\ny−1ay+w−1bw/parenrightbig\nη=π/parenleftbig\ny−1ay/parenrightbig\nη+π/parenleftbig\nw−1bw/parenrightbig\nη=π/parenleftbig\ny−1ay/parenrightbig\nη+π/parenleftbig\nw−1b/parenrightbig\nγ\n=π/parenleftbig\ny−1ay/parenrightbig\nη+π(a)η=αη.\nNotice that this implies α∈σ/parenleftbig\na+yw−1bwy−1/parenrightbig\nwhich is a contradiction to our choice of α. Thus,a\nandbcommute.\nThe case for (3) is analogous to the one above. So assume that (4) holds but ab/\\e}atio\\slash=ba. Thenπ(a) and\nπ(b) are non-central for a continuous irreducible representation πofAon a Banach space X. Pick\nα′/\\e}atio\\slash∈C\\ {0,1}, so that 1 −α′/\\e}atio\\slash= 0. Therefore, Lemma 2.1 says that there exist ζ∈X\\ {0}and\nz∈G1(A) such that {π(a)ζ,ζ}is linearly independent and\nπ((z−1az)◦a)ζ=α′ζ.\nMoreover, let ω∈X\\{0}be chosen so that {π(b)ω,ω}is linearly independent in X; hence, Sinclair’s\nDensity Theorem gives rise to u∈G1(A) such that\nπ(u)π(a)ζ=π(b)ω,\nπ(u)ζ=ω.\nNow observe that we have the following:\nπ((zaz−1)◦(u−1bu))ζ=π(zaz−1)ζ+π(u−1bu)ζ−π(zaz−1u−1bu)ζ\n=π(zaz−1)ζ+π(a)ζ−π(zaz−1)π(a)ζ\n=π((zaz−1)◦a)ζ=α′ζ.\nThus, Jacobson’s Lemma says that α′∈σ(a◦(z−1u−1buz)). But because α′was arbitrary in C\\{0,1},\nwe now have a contradiction to (4). So aandbdo commute.\nFinally, observe that, for all the above arguments, we may use Jac obson’s Lemma to replace aandbby\narbitrary members coming from orb( a) and orb( b) respectively, so that the proof is now complete. /square\nUnder commutation of the similarity orbits, it follows from the commut ation of aandr(for each\nr∈orb(b)) thatσ(ar),σ(r+a),σ(r−a), andσ(a◦r) are all contained in respective fixed compact\nsubsets of C(specific to each of the four forms). The precise details of this is ac tually used in the proof\nbelow. Now by replacing σhere by ˆσ(the polynomially convex hull of the spectrum), we get not just\nset containment—but equality to some fixed compact subsets of th e complex plane.\nCorollary 2.3. LetAbe a semisimple Banach algebra and suppose that a,b∈A. If any one of the\nconditions (1)-(4)holds, then for each r∈orb(b)we have that\nˆσ(ar) = ˆσ(ab), ˆσ(r+a) = ˆσ(b+a),\nˆσ(r−a) = ˆσ(b−a), ˆσ(a◦r) = ˆσ(a◦b).\nProof.If any of (1)-(4) is true, then by Theorem 2.2 it follows that acommutes with any r∈orb(b).\nSuppose that r=ex1···exnbe−xn···e−x1is a fixed member of orb( b) and define the entire function\nf:C→orb(b) as follows:\nf(λ):=eλx1···eλxnbe−λxn···e−λx1.COMMUTATIVITY AND ORTHOGONALITY OF SIMILARITY ORBITS IN BA NACH ALGEBRAS 7\nSinceacommutes with f(λ) andσ(f(λ)) =σ(b), for all λ∈C, we have the following:\nσ(f(λ)+a)⊆σ(b)+σ(a), σ (f(λ)−a)⊆σ(b)−σ(a),\nσ(a◦f(λ))⊆σ(a)+σ(b)−σ(a)σ(b) σ(af(λ))⊆σ(a)σ(b).\nHenceLiouville’sSpectralTheorem([1, Theorem3.4.14])impliesthat ˆ σ(af(λ)), ˆσ(f(λ)+a), ˆσ(f(λ)−a),\nˆσ(a◦f(λ)) are all constant, so we have that\nˆσ(ar) = ˆσ(af(1)) = ˆσ(af(0)) = ˆσ(ab),\nˆσ(r+a) = ˆσ(f(1)+a) = ˆσ(f(0)+a) = ˆσ(b+a),\nˆσ(r−a) = ˆσ(f(1)−a) = ˆσ(f(0)−a) = ˆσ(b−a),\nˆσ(a◦r) = ˆσ(a◦f(1)) = ˆσ(a◦f(0)) = ˆσ(a◦b).\nSincerwas arbitrarily chosen from orb( b), we now have the result. /square\nOur next corollary considers the case where aandbhave the same similarity orbit. It turns out that\nacommutes with every member of Awhen we have that orb( a) = orb(b) and if one of (1)-(4) holds.\nCorollary 2.4. LetAbe a semisimple Banach algebra and suppose that a∈A. Thenais central if\nand only if any one of the following holds:\n/uniondisplay\nr∈orb(a)σ′(ar)/\\e}atio\\slash=C\\{0} (5)\n/uniondisplay\nr∈orb(a)σ′(r+a)/\\e}atio\\slash=C\\{0} (6)\n/uniondisplay\nr∈orb(a)σ′(r−a)/\\e}atio\\slash=C\\{0} (7)\n/uniondisplay\nr∈orb(a)(σ′(a◦r)\\{1})/\\e}atio\\slash=C\\{0,1}. (8)\nProof.Clearly,when aiscentral,wewillhavethatorb( a) ={a},sothen(5)-(8)arealltrue. Conversely,\nifanyoftheconditions(5)-(8)holds,thenbyTheorem2.2itfollowst hatacommuteswitheverymember\nof orb(a). If, to the contrary, ais not central, then there is a continuous irreducible representatio nπ\non a Banach space Xsuch that π(a) is non-central. Pick α∈C\\/parenleftbig\n{0}∪σ(a)2/parenrightbig\n. By Lemma 2.1 there\nexistw∈orb(a) and 0/\\e}atio\\slash=η∈Xsuch that\nπ(w−1awa)η=αη.\nHence,α∈σ(w−1awa). But because acommutes with w−1aw, we may conclude that σ(w−1awa)⊆\nσ(w−1aw)σ(a) =σ(a)2, and therefore α∈σ(a)2—which is clearly a contradiction. Therefore, ais\ncentral. /square\nThe following corollary, which is immediate from Theorem 2.2 and Corollar y 2.4, generalizes some\nof the statements given in [5, Theorem 4.3] to general Banach alge bra elements:\nCorollary 2.5. LetAbe a semisimple Banach algebra and suppose that a,b∈A. Then every element\noforb(a)commutes with every element of orb(b)if and only if any one of the following holds:\n(i) sup\nr∈orb(b)ρ(ar)<∞;\n(ii) sup\nr∈orb(b)ρ(r+a)<∞;\n(iii) sup\nr∈orb(b)ρ(r−a)<∞;8 MUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\n(iv) sup\nr∈orb(b)ρ(a◦r)<∞.\nIn particular, if orb(a) = orb(b), thenais central if and only if any one of the conditions (i)-(iv)holds.\nThe above corollary can be used to strengthen two well-known char acterizations of commutative\nsemisimple Banach algebras. It is known (see for instance [1, Corollar y 5.2.3] or the main theorem\nin [6]) that a semisimple Banach algebra Ais commutative if and only if any one of the following\nconditions hold:\n(i) there exists M >0 such that ρ(a+b)≤M(ρ(a)+ρ(b)) for all a,b∈A;\n(ii) there exists N >0 such that ρ(ab)≤Nρ(a)ρ(b) for alla,b∈A.\nObviously, if Ais commutative, then we can take M=N= 1. Observe that both MandNare\nindependent of all members a,b∈A, so the conditions are required to hold globally on A. The\nfollowing theorem localizes these results to similarity orbits—which, as a reminder, partition A.\nTheorem 2.6. LetAbe a semisimple Banach algebra. Then the following properti es are equivalent:\n(i)Ais commutative;\n(ii)for each similarity orbit orb(a)ofA, there exist Morb(a)>0anda′∈orb(a)such that ρ(a′+\nr)≤Morb(a)(ρ(a′)+ρ(r))for allr∈orb(a);\n(iii)for each similarity orbit orb(a)ofA, there exist Norb(a)>0anda′∈orb(a)such that ρ(a′r)≤\nNorb(a)ρ(a′)ρ(r)for allr∈orb(a).\nProof.IfAis commutative then (ii) and (iii) are trivially true. So suppose that (ii) h olds. Then, for\neachr∈orb(a), it follows that\nρ(a′+r)≤Morb(a)(ρ(a′)+ρ(r)) = 2Morb(a)ρ(a),\nwhence Corollary 2.5 implies that a′is central, and so, a′=a. Sincea∈Awas arbitrary, it follows\nthatAis commutative. If (iii) holds then the argument is analogous. /square\n3 Localization to Neighbourhoods in Similarity Orbits\nIfK1andK2are nonempty compact subsets of C, then the Hausdorff distance between these sets\nis defined as:\n∆(K1,K2):= max/braceleftbigg\nsup\nλ∈K1/parenleftbigg\ninf\nα∈K2|λ−α|/parenrightbigg\n,sup\nλ∈K2/parenleftbigg\ninf\nα∈K1|λ−α|/parenrightbigg/bracerightbigg\n.\nThe main result of this section is Theorem 3.1 and, for the sake of bre vity, we state it for only one type\nof operation, namely, multiplication. That is, the hypothesis and con clusion to the theorem stated\nbelow pertain only to the spectra of elements of the form raasrvaries through orb( b). The reader will\nno doubt find after perusing the proof of the theorem that there is really no need to limit the result\nto products of elements, but we may obtain identical results for ea ch of the remaining combinations of\nelements: r+a,r−aandr◦a. Refer to the remarks at the end of this section to view the tweaks\nnecessary to achieve this.\nTheorem 3.1. LetAbe a semisimple Banach algebra. Suppose that a,b∈AandCis a fixed countable\nsubset of C. If there exists a nonempty open subset Noforb(b)and a pair (a′,b′)∈orb(a)×orb(b)\nsuch that σ(ra)⊆Cfor allr∈Nand#σ(b′a′)<∞, then every member of orb(a)commutes with\nevery member of orb(b). Moreover, we have that σ(ra) =σ(ba)for allr∈orb(b).\nThe following two lemmas are a requisite for proving Theorem 3.1, and t he hypothesis thereof will\nbe assumed in the premise of each of these lemmas.\nLemma 3.2. σ(ra)is countable for all r∈orb(b).COMMUTATIVITY AND ORTHOGONALITY OF SIMILARITY ORBITS IN BA NACH ALGEBRAS 9\nProof.Because the map v/mapsto→vbv−1is continuous on G1, it follows that there exists an open ball\nB=BA(v0,ǫ) inG1such that v∈Bimpliesvbv−1∈Nand hence σ(vbv−1a)⊆Cfor allv∈B.\nSuppose that v0=ex1···exn. Now take y=ey1···eym∈G1to be arbitrary. By appropriately using\ne0=1, we may assume without loss of generality that m=n. Next we define the entire function\nf:C→G1to be\nf(λ):=e(1−λ)x1+λy1···e(1−λ)xn+λyn.\nThen if g(λ) =f(λ)b[f(λ)]−1a, we have via [1, Theorem 7.1.13] that λ/mapsto→σ(g(λ)) is an analytic\nmultifunction on C. Furthermore, the continuity of fsays that there exists δ >0 such that |λ|< δ\nimpliesf(λ)∈B, whence σ(g(λ))⊆Cwhenever |λ|< δ. But then the set of λ∈Csuch that σ(g(λ))\nis at most countable has nonzero capacity—in which case, [1, Theore m 7.2.8] forces σ(g(λ)) to be at\nmost countable for all λ∈C. In particular then, σ(g(1)) =σ(yby−1a) is at most countable. Because\ny∈G1was arbitrary, we now have the result. /square\nLemma 3.3. There exists an open ball BinG1such that σ(vbv−1a)is constant for all v∈B.\nProof.Denote by Kthe metric space consisting of all nonempty compact subsets of Cunder the\nHausdorff metric. Let T:G1→ Kbe the map v/mapsto→σ(vbv−1a). The open ball Bwill be the same one\nwhich we obtained in Lemma 3.2, that is, B=BA(v0,ǫ). By continuity of Tand the connectedness of\nB, it follows that T(B) is connected in K. To complete the proof, we need only show that T(B) is a\nsingleton. To this end, we will assume to the contrarythat there is s omev∈Bsuch that T(v)/\\e}atio\\slash=T(v0).\nNow there is an open ball D0=BT(B)(T(v0),δ0) inT(B) to which T(v) does not belong. Moreover, if\nthere does not exist any T(y) on the boundary ∂D0of the open ball then that would mean that T(B)\nis the union of two nonempty disjoint open sets, that is,\nT(B) =D0∪(T(B)\\D0),\nwhich is obviously a contradiction. So let T(y0)∈∂D0. Therefore, we can use the same reasoning to\nguarantee that for each real number δ∈(0,δ0], there is some yδ∈Bsuch that T(yδ)∈∂Dδ, where\nDδ=BT(B)(T(v0),δ) here. Recall that by the hypothesis, we have that T(v)⊆Cfor allv∈B, where\nCis a fixed countable subset of C. Because the spectrum is nonempty and compact, we have that if\nT(yδ)∈∂Dδthen the distance of T(yδ) toT(v0) is assumed at some complex numbers belonging to C.\nTherefore, to each δ∈(0,δ0] we can associated a pair ( αδ,βδ)∈T(yδ)×T(v0) such that |αδ−βδ|=δ.\nClearly,C×Cis countable whilst the interval (0 ,δ0] is uncountable, and so the preceding statement\nleads to the following pigeonhole argument: There must be distinct re al numbers, say δandγin (0,δ0]\nfor which the corresponding pairs ( αδ,βδ),(αγ,βγ)∈T(yδ)×T(v0) are the same. That is, αδ=αγ\nandβδ=βγin the above, in addition to the fact that\n|αδ−βδ|=δ,|αγ−βγ|=γ.\nNotice then that the contradiction δ=γis reached. /square\nProof of Theorem 3.1. In Lemma 3.3 we established that σ(vbv−1a) is constant for all v∈B, where\nB=BA(v0,ǫ) is an open ball in G1. Let us assume that, for each v∈B,\nσ(vbv−1a) ={αn}n∈N=M,\nwhereM⊆C. Suppose that y∈G1is arbitrary, and define the functions fandgas in Lemma 3.2,\nwithv0=ex1···exnandy=ey1···eyn. Then the map λ/mapsto→σ(g(λ)) is an analytic multifunction; with\nLemma 3.2 we have that σ(g(λ)) is countable for all λ∈Cand by our preceding remarks, there exists\nδ >0 such that |λ|< δimpliesσ(g(λ)) =M. Now for each n∈Nwe define the set\nZn:={λ∈C:αn∈σ(g(λ))},\nand we observe that BC(0,δ)⊆Znfor alln∈N—in which case, [1, Theorem 7.2.13] says that Zn=C\nfor eachn∈N. In particular then, αn∈σ(g(1)) =σ(yby−1a) for alln∈N. Asywas arbitrarily picked10 MUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\ninG1we can conclude that M⊆σ(yby−1a) for ally∈G1.\nBy the hypothesis and Jacobson’s Lemma, it follows that there is som ey0∈G1such that σ(y0by−1\n0a)\nis finite, and consequently, Mis finite. But recall we had that σ(vbv−1a) =Mfor allv∈B. So we may\nfix an arbitrary y∈G1as before and define the corresponding analytic function gaccording to Lemma\n3.2 to obtain existence of a δ >0 (via continuity of g) such that |λ|< δimplies that # σ(g(λ))<∞.\nHence the ScarcityTheorem (refer to [1, Theorem 3.4.25])now says that #σ(g(λ))≤#Mfor allλ∈C.\nBut by the first part, we have that σ(g(λ)) contains Mfor allλ∈Cand, in particular, M⊆σ(g(1)),\nso it follows that σ(g(1)) =M. Butywas arbitrary in G1so we conclude that σ(yby−1a) =Mfor all\ny∈G1. Finally, a simple application of Theorem 2.2 now gives the result. /square\nRemarks. We conclude this section with the following observations:\n(i) Notice that the condition in Theorem 3.1 assuming that there exist s a pair ( a′,b′)∈orb(a)×\norb(b) such that # σ(b′a′)<∞implies the following weaker statement:\n#\n/intersectiondisplay\nr∈orb(b)σ(ra)\n<∞.\nFurthermore, the above statement is sufficient to deduce the last part in the proof of Theorem\n3.1.\n(ii) If we go to the proof of Lemma 3.2 and replace g(λ) by one of the following entire functions:\nλ/mapsto→f(λ)b[f(λ)]−1+a, λ/mapsto→f(λ)b[f(λ)]−1−a, λ/mapsto→f(λ)b[f(λ)]−1◦a,\nthen we obtain Theorem 3.1 (in exactly the same manner) for the for msr+a,r−a,r◦a\nrespectively.\n4 Applications to Algebraic Elements\nIt will be interesting to consider the results highlighted in the preced ing section for elements in A\nwith more properties—specifically—algebraic elements. For this, it will be agreed upon that, for the\nremainder of this section, pwill be a polynomial of the form\np(λ):=n/productdisplay\ni=1(λ−λi),\nwhereK={λ1,...,λ n}is a fixed set of distinct complex numbers. The set of elements of Awhich are\nalgebraic under pwill be denoted by E(p). IfK={0,1}, then the elements of E(p) are idempotents.\nIfa∈E(p), then one can show that σ(a)⊆K. Something quite useful is the next result which\ndecomposes algebraic elements into linear combinations of idempoten ts:\nTheorem 4.1 ([4, Proposition 1]) .LetAbe a Banach algebra and suppose that a∈A. Thena∈E(p)\nif and only if there exists a set of idempotents {e1,...,e n}such that eiej= 0fori/\\e}atio\\slash=j,/summationtextn\ni=1ei=1,\nand\na=n/summationdisplay\ni=1λiei.\nIn fact, these idempotents are the Riesz idempotents obtained via the Holomorphic Functional\nCalculus. Hence if a,b∈E(p) commute and their decompositions are respectively given by:\na=n/summationdisplay\ni=1λiei, b=n/summationdisplay\ni=1λi˜ei,\nthenei˜ej= ˜ejeifor alli,j∈ {1,...,n}. Of course, E(p) is not necessarily connected; in fact if\na∈E(p), then the connected component of E(p) containing ais orb(a). This was established forCOMMUTATIVITY AND ORTHOGONALITY OF SIMILARITY ORBITS IN BA NACH ALGEBRAS 11\nidempotents in [5] and for algebraic elements in [4]. Before we directly a pply the theory obtained in\nSection 2, we establish that the product of two commuting algebraic elements is once again algebraic,\nvia the following proposition:\nProposition 4.2. LetAbe a semisimple Banach algebra and suppose that a,b∈E(p). If every\nelement of orb(a)commutes with every element of orb(b)thenorb(a)orb(b) = orb(b)orb(a)consists of\nalgebraic elements.\nProof.If we decompose a′∈orb(a) andb′∈orb(b) according to Theorem 4.1 as\na′=n/summationdisplay\ni=1λiei, b′=n/summationdisplay\ni=1λi˜ei,\nthen by direct calculation (and the aid of commutativity and orthogo nality of the Riesz idempotents)\nwe find that\na′b′=n/summationdisplay\ni=1n/summationdisplay\nj=1λiλjei˜ej,=/summationdisplay\nα∈K2αˆeα,\nwhere ˆeαˆeβ= 0 for all α/\\e}atio\\slash=βinK2, and/summationtext\nα∈K2ˆeα=1. Therefore, by Theorem 4.1, a′b′is algebraic\nunder the polynomial\nq(λ):=/productdisplay\nα∈K2(λ−α).\n/square\nIt turns out that when ais algebraic, we have a few additional equivalent statements to the lis t in\nCorollary 2.4. This then mostly completes generalizing [5, Theorem 4.3] to algebraic elements.\nTheorem 4.3. LetAbe a semisimple Banach algebra and suppose that a∈E(p). Then the following\nare equivalent:\n(i)ais central;\n(ii)there exists a polynomial q(λ)such that aE(p)⊆E(q);\n(iii)there exists a polynomial q(λ)such that aorb(a)⊆E(q);\n(iv)there exists a polynomial q(λ)such that a◦E(p)⊆E(q);\n(v)there exists a polynomial q(λ)such that a◦orb(a)⊆E(q).\nProof.Ifais central, then applying an identical argument as the one given in the proof for Proposition\n4.2 shows that (ii) and (iii) are true. Since a◦b=1−(1−a)(1−b), it follows that (i) is also sufficient\nfor (iv) and (v), for the subsequent reasons: Notice that if ais algebraic under the polynomial p, then\nso too is 1−a, under\nλ/mapsto→n/productdisplay\ni=1(1−λi−λ).\nTherefore, when b∈E(p), the element 1−bis algebraic as well and hence, via commutation of a\nandb(as well as the proof of Proposition 4.2), it follows that 1−(1−a)(1−b) is algebraic under a\npolynomial q(λ). A similar case can be made showing (i) ⇒(v).\nIf (ii) or (iii) is true, then obviously we must have that\nsup\nr∈orb(a)ρ(ar)<∞,\nwhence Corollary 2.5 implies that ais central. Similarly, if (iv) or (v) holds, then\nsup\nr∈orb(a)ρ(a◦r)<∞,\nso that again ais central. /square12 MUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\nFor the purposes of the next results, define the following:\nt1:= sup\nr∈orb(b)ρ(ar), t2:= sup\nr∈orb(b)ρ(a+r), κ:= inf/braceleftbig\n|λ|:λ∈K2/bracerightbig\n.\nTheorem 4.4. LetAbe a semisimple Banach algebra and suppose that a,b∈E(p). If we have that\n#/braceleftbig\nλ∈K2:|λ|=κ/bracerightbig\n= 1,\nthen there exist µ≥0andδ >0such that if t1< δ, then\norb(a)orb(b) = orb(b)orb(a) ={µ1}.\nIn particular, if a/\\e}atio\\slash∈G(A)then\norb(a)orb(b) = orb(b)orb(a) ={0}.\nProof.Chooseδto be the real number\ninf{|λ|:λ∈K2,|λ|> κ}.\nHence if t1< δ, Corollary 2.5 implies that a′b′=b′a′fora′∈orb(a) andb′∈orb(b). Furthermore, a′b′\nis algebraic and σ(a′b′)⊆K2. Letµbe the complex number in K2satisfying |µ|=κ. By Theorem\n4.1, ifa′b′/\\e}atio\\slash=µ1, it must be the case that σ(a′b′)∩K2/\\e}atio\\slash={µ}—implying that for some r∈orb(b)\n(depending on a′andb′) we will have that ρ(ar)≥δ, which is a contradiction.\nIfa /∈G(A)then0∈σ(a), sothat0 ∈K2andhence κ=µ= 0. Obviouslythen#/braceleftbig\nλ∈K2:|λ|=κ/bracerightbig\n=\n1 so by the first part the orthogonality of orb( a) and orb( b) is apparent. /square\nIn the special case for idempotents, we have that K2=K,δ= 1 and µ=κ= 0; that is, orb( a) and\norb(b) are orthogonal when t1<1.\nCorollary 4.5. LetAbe a semisimple Banach algebra and suppose that pandqare idempotents in\nA. Then the following statements are equivalent:\n(i) orb(p)orb(q) = orb(q)orb(p) ={0},\n(ii)t1= sup\nr∈orb(q)ρ(pr)<1,\n(iii)t2= sup\nr∈orb(q)ρ(p+r)<2.\nProof.Observe that from Theorem 4.4, we immediately have that\norb(p)orb(q) = orb(q)orb(p) ={0} ⇐⇒ t1<1.\nHence it suffices to show t2<2 is equivalent to t1<1. Ift2<2 then Corollary 2.5 implies that every\nelement of orb( p) commutes with every element of orb( q). Suppose to the contrary that t1/\\e}atio\\slash<1. By\ncommutation of pandr, the element pris an idempotent for all r∈orb(q). But not all such elements\nare 0, so fix r∈orb(q) such that σ(pr) ={0,1}. Using the following result from [2, Theorem 2.2],\n(9) λ/\\e}atio\\slash∈ {0,1} ⇒/bracketleftbig\nλ∈σ(p+r)⇐⇒(λ−1)2∈σ(pr)/bracketrightbig\n,\nand due to the fact that 2 /\\e}atio\\slash∈ {0,1}and (2−1)2∈σ(pr), it follows that 2 ∈σ(p+r)—which is\nobviously a contradiction to the hypothesis. Thus, t1<1. Conversely, suppose that t1<1. Then by\nthe first part it follows that orb( p) and orb( q) are orthogonal—in which case, p+ris an idempotent\nfor allr∈orb(q). Therefore, t2= 1<2. /square\nRemark. In the proof of Theorem 4.4, the chosen δis a sharp upper bound for t1. In Corollary 4.5, 1\nand 2 are sharp upper bounds for t1andt2, respectively. Of course, by Corollary 2.5, if t1ort2is finite\nthen the best we can do is infer that the elements of orb( p) commute with the elements of orb( q).COMMUTATIVITY AND ORTHOGONALITY OF SIMILARITY ORBITS IN BA NACH ALGEBRAS 13\nFor idempotents pandqinA, we have a very strong localization principle for similarity orbits. We\nneed not even know what σ(pr) contains, just that it is a singleton for all rin a small open subset of\norb(q)—in which case— pandqare the identity or their similarity orbits are orthogonal.\nCorollary 4.6. LetAbe a semisimple Banach algebra and suppose pandqare idempotents of Asuch\nthat#σ(pr) = 1for allrin an open subset Noforb(q). Then either p=q=1ororb(p)orb(q) =\norb(q)orb(p) ={0}.\nProof.We first claim that the hypothesis can be globalized in the following sens e: #σ(pr) = 1 for all\nrin orb(q). To see why this is so, fix a member ex1···exnqe−xn···e−x1inNand let\nr=ey1···eymqe−ym···e−y1\nbe arbitrary in orb( q). Without loss of generality, we may assume that m=n. Define the entire\nfunction f:C→Aby:\nf(λ):=eλx1+(1−λ)y1···eλxn+(1−λ)yn.\nHence by continuity of the map g:λ/mapsto→f(λ)q[f(λ)]−1, we also have that there is some disk B(1,δ) in\nCsuch that λ∈B(1,δ) implies g(λ)∈N— so that # σ(pg(λ)) = 1 for all λ∈B(1,δ). By the Scarcity\nTheorem, this must hold for all λ∈C, and in particular, 1 = # σ(pg(0)) = # σ(pr). Hence, # σ(pr) = 1\nfor allr∈orb(q).\nIf there is some r∈orb(q) such that 0 /\\e}atio\\slash∈σ(pr), thenpris invertible. Moreover, 0 = ( 1−p)primplies\nthatp=1. But then, r∈G(A) and hence r=q=p=1. Hence if p/\\e}atio\\slash=q, then\nσ(pwqw−1) ={0}for allw∈G1(A).\nNow observe that the value of t1is 0 in Corollary 4.5, so the components orb( p) and orb( q) are\northogonal, as advertised. /square\nRemarks. A couple of remarks related to the above corollary are in order.\n(i) The second part of Corollary 4.6 can be directly proven without th e use of Corollary 4.5 with\nsome basic representation theory. A brief outline of the proof is as follows: Begin by assuming\npq/\\e}atio\\slash= 0, to obtain a representation πon a Banach space Xsuch that π(pq)η/\\e}atio\\slash= 0, for some\nη∈X. Then set ξ=π(q)ηand argue as to why {π(p)ξ,π(q)ξ}is a linearly independent\nset. Thereafter a simple application of Sinclair’s Density Theorem to t his set will yield the\nnecessary contradiction.\n(ii) If in the hypothesis of the corollary we assumed that # σ(pr)≤n >1 for all r∈orb(q), then\nthe result would be invalid. A simple example illustrating the case for n= 2 can be obtained\nfromM2(C). Set\np=/bracketleftbigg1 0\n0 0/bracketrightbigg\n, q=/bracketleftbigg1 0\n0 1/bracketrightbigg\n.\nThen #σ(pr) = 2 for all r∈orb(q) butp/\\e}atio\\slash=qand orb( p)orb(q) = orb( p)/\\e}atio\\slash={0}. One may\nconstruct similar examples in Mn(C) for larger values of n.\nThe final result gives a localization principle for algebraic elements, an d, expectedly, its conclusion\nis stronger than that of Theorem 3.1, in that we have yet another c haracterization of central algebraic\nelements. Recall that in Theorem 3.1, we required in the premise that σ(b′a′) is finite for some pair\n(a′,b′)∈orb(a)×orb(b). Here we are looking at just orb( a), and obviously then σ(a2) is finite since\nσ(a) is finite.\nCorollary 4.7. LetAbe a semisimple Banach algebra and suppose that a∈Ais algebraic. Then a\nis central if and only if there exist a nonempty countable set C⊆Cand a nonempty open subset Nof\norb(a)such that σ(ra)⊆Cfor allr∈N.14 MUHAMMAD HASSEN, RUDI BRITS, AND FRANCOIS SCHULZ\nProof.Ifais central then orb( a) ={a}and hence Cmay be taken as the finite set σ/parenleftbig\na2/parenrightbig\nto obtain\nthe forward implication.\nFor the converse, we take the pair ( a,a)∈orb(a)×orb(a) in Theorem 3.1 to get that acommutes with\nall elements of its orbit, so that an identical argument as the one giv en in Corollary 2.4 implies the\ncentrality of a. /square\nReferences\n[1] B. Aupetit, A primer on spectral theory , Universitext (1979), Springer-Verlag, 1991, doi.\n[2] M. Barraa, E. H. Benabdi, and M. Boumazgour, On the spectra of products and linear combinations of idempo tents,\nAdv. Oper. Theory 6(2021), 1–9, doi.\n[3] R. Brits, On the multiplicative spectral characterization of the Jac obson radical , Quaest. Math. 31(2008), 179–188,\ndoi.\n[4] E. Makai and J. Zem´ anek, Nice connecting paths in connected components of sets of alg ebraic elements in a Banach\nalgebra, Czechoslov. Math. J. 66(2016), 821–828, doi.\n[5] J. Zem´ anek, Idempotents in Banach algebras , Bull. London Math. Soc. 11(1979), 177–183, doi.\n[6] J. Zem´ anek, Spectral radius characterizations of commutativity in Ban ach algebras , Studia Math. 61(1977), 257–268,\ndoi.\nUniversity of Johannesburg, Johannesburg, South Africa\nEmail address :mhassen@uj.ac.za\nUniversity of Johannesburg, Johannesburg, South Africa\nEmail address :rbrits@uj.ac.za\nUniversity of Johannesburg, Johannesburg, South Africa\nEmail address :francoiss@uj.ac.za"    },    {        "title": "2402.03960v1.Mechanical_Properties_of_Minerals_in_Lunar_and_HED_Meteorites_from_Nanoindentation_Testing__Implications_for_Space_Mining.pdf",        "content": "Mechanical Properties of Minerals in Lunar and HED Meteorites from Nanoindentation Testing: Implications for Space Mining   Eloy PEÑA-ASENSIO1,2*, Josep M. TRIGO-RODRIGUEZ2,3, Jordi SORT4,5, Jordi IBAÑEZ-INSA6, and Albert RIMOLA1  1 Departament de Química, Universitat Autònoma de Barcelona 08193 Bellaterra, Catalonia, Spain 2 Institut de Ciències de l’Espai (ICE, CSIC) Campus UAB, C/ de Can Magrans s/n, 08193 Cerdanyola del Vallès, Catalonia, Spain  3 Institut d’Estudis Espacials de Catalunya (IEEC) 08034 Barcelona, Catalonia, Spain 4 Departament de Física, Universitat Autònoma de Barcelona, E-08193 Cerdanyola del Vallès, Spain 5 Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluís Companys 23, E-08010 Barcelona, Spain 6 Geosciences Barcelona (GEO3BCN-CSIC), Lluís Solé i Sabarís s/n, E-08028 Barcelona, Spain *Corresponding author. E-mail: eloy.pena@uab.cat, eloy.peas@gmail.com   (Received …; revision accepted …)  Abstract- This study delivers an analysis of the mechanical and elemental properties of lunar meteorites DHOFAR 1084, JAH 838, NWA 11444, and HED meteorite NWA 6013. Utilizing microscale rock mechanics experiments, i.e., nanoindentation testing, this research reveals significant heterogeneity in both mechanical and elemental attributes across the mineral of the samples. Olivines, pyroxen, feldspar, and spinel demonstrate similar compositional and mechanical characteristics. Conversely, other silicate and oxide minerals display variations in the mechanical properties. Terrestrial olivines subjected to nanoindentation tests exhibit increased hardness and a higher Young's modulus compared to their lunar counterparts. A linear correlation is observed between the H/Er ratio and both plastic and elastic energies. Additionally, the alignment of mineral phases along a constant H/Er ratio suggests variations in local porosity. This study also highlights the need for further research focusing on porosity, phase insertions within the matrix, and structural orientations to refine our understanding of these mechanical characteristics. The findings have direct implications for in-situ resource utilization (ISRU) strategies and future state-of-the-art impact models. This comprehensive characterization serves as a foundational resource for future research efforts in space science and mining.  Keywords- Meteorites, Moon, Asteroid, ISRU, Mining, Mechanical Properties, Mineralogy  2  1. INTRODUCTION  In recent years, there has been a resurgence of interest in the exploration and colonization of the Moon and asteroids. With the increasing interest in lunar missions (Jia et al., 2018; NASA, 2020; Mitrofanov, 2021; Bhardwaj, 2021), it is important to understand the properties of the materials available on the lunar surface for in-situ resource utilization (ISRU) and manufacturing (Anand et al., 2012; Crawford, 2015; Naser, 2019; Duffard et al., 2021). The extension of space exploration endeavors to include near-Earth objects necessitates an enhanced understanding of both asteroid mining techniques and science (Ross, 2001; Anthony, 2018; Zacny, 2013; Andrews, 2015).  Lunar meteorites, also known as lunaites, are fragments of the Moon ejected by impacts that have subsequently fallen to Earth and provide a unique opportunity to study the composition and properties of lunar materials. Unlike the samples returned from the Apollo missions, which were collected by astronauts from limited locations, lunar meteorites are fragments of the Moon of unknown but, more importantly, random origin. These meteorites are excellent samples to study a wider range of lunar materials that were not screened during the astronaut extravehicular activities. Lunar meteorites have been found in various locations around the world and have been extensively studied using a variety of techniques, including petrographic analysis, X-ray diffraction, and mass spectrometry, among others. These studies have revealed a wealth of information about the mineralogy, chemistry, and geologic history of the Moon (Korotev, 2005). Similarly, despite the significant contributions of the Stardust (Brownlee et al., 2006), Hayabusa (Yano et al., 2006), Hayabusa2 (Watanabe et al., 2019), and OSIRIS-Rex (Lauretta et al., 2019) sample return missions, meteorites are the primary source of samples essential for inferring asteroidal and cometary properties. Indentation and tensile testing are common methods for measuring the mechanical properties of minerals, including hardness and Young’s modulus. However, due to the rarity of lunar meteorites, large-scale destructive experiments cannot be performed as they must be preserved for their scientific value. As a result, researchers have turned to microscale rock mechanics experiments as a nearly non-destructive method for characterizing the mechanical properties of meteorites (Moyano-Cambero et al., 2017; Tanbakouei et al., 2019; Wheeler, 2021; Zhang et al., 2022; Huang, 2023; Nie et al., 2023; Rabbi, 2023). The nanoindentation technique allows probing the physical properties of materials at the nanoscale and microscale. This is particularly important for the investigation of meteorites because: i) it can provide valuable insight into the properties of the constituent minerals, which do not need to be exactly equivalent to their terrestrial counterparts due to, for instance, compositional differences and different shock degrees; ii) knowledge of the mechanical properties of the minerals, combined with appropriate modeling of the behavior under different regimes (elastic, 3  plastic, etc.), may allow to infer the mechanical properties of asteroidal and lunar surface rocks in different scenarios relevant for ISRU; iii) from a more fundamental point of view, nanoindentation may be useful to gain information about how the shock history of the meteorites may affect their macro- and micro-structural properties and mechanical behavior; and iv) in composite rocks, how the individual mechanical properties of each constituent mineral correlate with the overall mechanical performance is of interest, for example, for mining prospections. In mining engineering and geomechanical analysis, differentiating between the mechanical properties of minerals and rocks is fundamental. Minerals' mechanical properties, while insightful, do not directly translate to rocks' macroscopic behaviors. This necessitates methods to extrapolate microscale (mineral-level) properties to macroscale (rock-level) phenomena. Xu et al. (2023) exemplify this by studying thermally induced microcracks in granite, analyzing mineral heterogeneity and thermal stress effects on granite's macroscale mechanical integrity using high-temperature microscopy and Accurate Grain-Based Modeling (AGBM). Conversely, Tang et al. (2023) employ microscale rock mechanics experiments and AGBM to determine asteroid rocks' Young's modulus, bridging mineral and interphase-level analyses to a comprehensive macroscale understanding. In relation to ISRU, it is important to emphasize that lunar materials have unique mechanical properties because of their composition and structure. Terrestrial rocks are composed primarily of silica- and silicate-based minerals such as quartz, feldspar, and mica, along with notable elemental constituents such as iron, aluminum, calcium, and magnesium. Lunar rocks have a similar silicate composition but exhibit distinct mineralogical variations (Papike, Taylor, & Simon, 1991). In addition, lunar rocks have a higher abundance of elements such as titanium, thorium, and rare earth elements. Besides compositional considerations, Earth’s materials have undergone a variety of recent geologic processes, including erosion, weathering, and tectonic activity. On the contrary, lunar material, exposed to the vacuum of space, has experienced limited influence from these geology processes. Nevertheless, lunar rocks have been subjected to the so-called space weathering: impact events caused by asteroids, meteoroids, and micrometeoroids, resulting in fractures and subsequent physical changes due to extreme temperature and pressure variations (Hapke, 2001).  Significant progress has been made in the development of lunar simulants, which are synthetic materials designed to mimic the properties of lunar materials (McKay et al., 1994; Kanamori et al., 1998; Zheng et al., 2009; Alberquilla et al., 2022). These simulants play a critical role in various scientific and engineering research activities related to lunar exploration and colonization. However, it remains essential to study lunar samples rather than relying solely on analogs, as they cannot perfectly replicate the precise characteristics and complexities of real materials. Lunar simulants often approximate certain properties such as composition, grain size, and density, but they may not 4  accurately capture the intricate details and variations found in real samples from the Moon. Therefore, it is necessary to study the mechanical properties of real samples to design appropriate manufacturing techniques and equipment for the lunar environment. By characterizing the mechanical behavior of lunar materials, we can develop suitable processes and tools for lunar exploration and colonization, and ultimately improve our understanding of the Moon's resources and potential for human exploration and habitation. Similarly, analyzing materials from asteroids follows the same principle, contributing to our comprehension of their utility in space exploration and science. In addition, the mechanical characterization of extraterrestrial material components is essential for future state-of-the-art impact modeling. Impact events are a ubiquitous process on the lunar and asteroid surfaces, and understanding the mechanics of these events is important for several fields, like geomorphological characterization of craters. While macroscopic mechanical properties are essential for understanding the behavior of materials under impacts and other extreme conditions, incorporating the micromechanical properties of the different phases of a material can provide a more accurate representation of the overall material's behavior, particularly when it is composed of several phases or minerals (Melosh, 1989; Davison, Collins, & Bland, 2016). Simulations that incorporate the micro- and nanomechanical properties of different phases can provide a more detailed representation of how macroscopic composite materials respond to impact and other extreme conditions. This can be particularly important for materials that are complex and heterogeneous, such as lunar meteorites, where the behavior of different phases can vary significantly due to differences in their composition, grain size, and orientation. In this study, we selected three lunar meteorites and one howardite–eucrite–diogenite (HED) meteorite for nanoindentation tests, aiming to generate valuable data pertinent to both lunar and asteroidal mining and science. In the next section, we will introduce the samples studied. Then, we will describe the results obtained for the main rock-forming minerals that have been tested. Finally, we will compare our results with previous data published in the literature. Our main goal is to exemplify the intrinsic value of a systematic study of meteorites using nanoindentation to infer the main mechanical properties of extraterrestrial minerals for ISRU.  2. SAMPLES AND EXPERIMENTAL METHODS  The thin section of the meteorites analyzed in this work (DHOFAR 1084, JAH 838, NWA 11444, and NWA 6013) belong to the Meteorite Collection of the Institute of Space Sciences (CSIC). The four samples are classified in the Meteoritical Bulletin Database, and their main characteristics are compiled in Table 1. 5   Table 1. Classification of the 4 samples analyzed in this work. Data extracted from the Meteoritical Bulletin Database. Note that NWA 6013 is not lunar.  Name Classification Fayalite (mol %) Ferrosilite (mol %) DHOFAR 1084 Feldespathic 43 30 JAH 838 Mingled regolith breccia 39.7 55.6 NWA 11444 Melt breccia 37.1 35.1 NWA 6013 Hartzburgitic diogenite 29.5 24  DHOFAR 1084 is a lunar meteorite found in 2001 in the Dhofar region of Oman, near Zufar. This meteorite has been classified as an average feldspathic lunar meteorite. Near this meteorite, DHOFAR 490 was also discovered, and both were isolated from other lunar rocks. However, it has still to be determined if they constitute a pairing. Fig. 1 shows the thin section obtained from DHOFAR 1084.  \n Figure 1. Mosaic of the thin section of lunar meteorite DHOFAR 1084. \n6  Jiddat al Harasis 838 (JAH 838) is a lunar meteorite found in Oman during a desert expedition in 2003, 28 km south of the Al Ghaftain. It is classified as a mingled regolith breccia and has a presence of mare and KREEPy material (potassium, rare-earth elements, and phosphorus), HASP (alumina–silica poor), and chondritic material., Fig. 2 shows the thin section of JAH 838.  \n Figure 2. Mosaic of the thin section of lunar meteorite JAH 838.  Northwest Africa 11444 (NWA 11444) is a lunar meteorite discovered in 2017 and collected in an unknown location in Mauritania. It is classified as a polymictic anorthositic breccia, which means that it is composed of fragments of anorthosite, a type of rock that is rich in the mineral plagioclase, fused together by the heat generated by a large impact event on the Moon’s surface. Fig. 3 shows the thin section obtained from NWA 11444.  \n7   Figure 3. Mosaic of the thin section of lunar meteorite NWA 11444.  Northwest Africa 6013 (NWA 6013) is a meteorite discovered in 2008 in Northwest Africa. It is classified as a Hartzburgitic diogenite, that is, HED achondrite, a type of ultramafic rock with a coarse-grained texture that contains subunits dominated by olivine and pyroxene with approximately equal abundances of 50 vol% each. Fig. 4 shows the thin section of NWA 6013.  \n Figure 4. Mosaic of the thin section of meteorite NWA 6013. \n8  A petrographic optical microscope, Zeiss Scope Axio, was used to examine the thin sections of the samples in both reflected and transmitted light modes at ×50, ×100, and ×250 magnification. To identify and name the various features and components, high-resolution mosaics were created, and a grid was superimposed on the mosaics. The creation of a grid mosaic was essential to facilitate the process of the identification of the Regions of Interest (ROIs) to perform the nanoindentation and mineral identification, as it allowed precise placement of the indenter tip on specific minerals and provided a visual reference for their location within the sample. This approach ensured accurate characterization of the mechanical properties of individual minerals within the lunar meteorites. We used scanning electron microscopy (SEM) with energy dispersive X-ray analysis (EDX) from the Catalan Institute of Nanoscience and Nanotechnology (ICN2), specifically the FEI Quanta 650 FEG in low vacuum backscattered electron mode (BSED), to examine the thin sections of the samples. An Inca 250 SSD Xmax20 EDS detector was used for elemental analysis. The EDS detector is equipped with Peltier cooling and has an active area of 20 mm2. Various magnifications were used to examine selected areas of the samples, from which we obtained the corresponding EDX spectra. This approach allowed a detailed analysis of the mineralogical composition of the selected ROIs and an overview of the elemental distribution within the sections. The information obtained from these analyses allowed us to identify the indented minerals. The mechanical properties of the minerals were obtained using the NHT2 Anton Paar nanoindentation instrument available at the Autonomous University of Barcelona (UAB), which uses a Berkovich pyramidal diamond tip. In the process of nanoindentation, controlled force is precisely applied to specific and localized areas of a sample using a diamond indenter. The indenter exerts pressure on the surface as the force is gradually increased up to a predetermined maximum value. Subsequently, after the loading phase, the force is systematically reduced back to zero, and the sample's surface retracts to some extent due to its elasticity. Throughout this entire load-unload cycle, the penetration depth is measured. By analyzing the load-depth curves derived from this process, valuable insights into deformation mechanisms (both elastic and plastic) and elastic recovery are obtained. To evaluate average mechanical properties, an array of 6-12 indentations was performed for each mineral with a maximum applied force of 25 mN. The thermal drift during nanoindentation was carefully kept below 0.05 nm/s. Precise corrections for contact area (calibrated with a fused silica sample), initial indentation depth, and instrument compliance were made (Fischer-Cripps & Nicholson 2004). Hardness (H) and reduced Young’s modulus (Er) values were obtained from the load-displacement curves following the method of Oliver & Pharr (1992) as depicted in Fig. 5.   9   Figure 5. Top: Load-displacement curve, showing the parameters used in the Oliver and Pharr method. Bottom: cross-section of an indentation. Adapted from Ovsik et al. (2021).  The Young’s modulus refers to a fundamental mechanical property that quantifies the stiffness or elasticity of a material., It measures the ability of a material to resist deformation under an applied force. On the other hand, the reduced Young's modulus (which is, in fact, the elastic property measured during nanoindentation) refers to a modified or adjusted version of Young's modulus, that considers convoluted contributions from the sample and the indenter tip (i.e., two-body contact):  \n10  !\"!=!#$\"\"−!#$#\"\"#\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(1)\t\tThe reduced modulus considers the elastic displacements occurring in both the specimen, characterized by Young’s modulus E and Poisson’s ratio ν, and the diamond indenter, which has Young’s modulus and Poisson ratio equal to Ei = 1,140 GPa and νi = 0.07, respectively. The hardness H is computed as: 𝐻=%$%&&                                                                  (1) \twhere Pmax is the maximum load applied and A is the projected contact area on the horizontal plane. The parameter H/Er merits consideration as it provides an indirect measure of wear resistance (Pellicer et al., 2021).  Elastic recovery values were obtained from the ratio between elastic and total indentation energies (We/Wt). We was determined from the area between the unloading curve and the displacement axis, whereas Wt was estimated from the area between the loading curve and the displacement axis. The plastic behavior was characterized analogously. The parameter Wp/Wt is often referred to as the plasticity index.  3. RESULTS  To gain valuable insights into the nature of lunar meteorites, we conducted a comprehensive analysis of their mechanical properties, identifying the main mineral of each tested region. Table 2 summarizes the mechanical properties obtained from nanoindentation testing for each of the studied lunar meteorites. We examined the atomic percentage composition of these lunar meteorites using SEM EDX measurements. Table 3 displays the atomic composition results, offering insights into the elemental composition of each indented mineral. Note that silicates that cannot be distinctly classified as pyroxenes, olivines, or feldspars, and which present ambiguous characteristics under SEM analysis, are categorized as 'other silicate'. Figure 6 presents four distinct examples of indentation tests conducted on various minerals found within each of the lunar meteorite samples, accompanied by their respective optical microscope images. The differences in the appearance of each indentation result from variations in illumination and reflection conditions. In the top row, corresponding to diopside/augite in the meteorite DHOFAR 1084, the load curves usually exhibit small and abrupt flat regions, potentially suggestive of porosity within the material or weak inclusions of phase within the matrix. Conversely, in the third row (pigeonite/enstatite in NWA 11444), several measurements had to be excluded due to the occurrence of sudden drops in the indentation curves, suggestive of cracking or pore collapse. Nonetheless, the 11  figure also shows the case of an outlier curve, which coincides with a fracture occurrence within the material.  Table 2. Compilation of the mechanical properties obtained for each mineral and meteorite from nanoindentation testing. The last column indicates the number of indentations performed at each test. Meteorite Cell Mineral Group H (GPa) Er (GPa) H/Er We/Wt Wp/Wt # DHO 1084 I11 Anorthite Feldspar 9.6±0.9 92±7 0.105±0.013 0.57±0.04 0.43±0.05 12 DHO 1084 F11 Anorthite (Oliv.) Feldspar 9.2±0.5 93±5 0.099±0.008 0.557±0.033 0.443±0.027 9 DHO 1084 E11 Anorthite Feldspar 8.8±0.2 99±7 0.090±0.007 0.50±0.05 0.50±0.06 6 DHO 1084 I12 Diopside/Augite Pyroxene 10.7±0.7 146±3 0.073±0.005 0.431±0.019 0.569±0.031 8 DHO 1084 H10   14.3±3.4 149±18 0.096±0.026 0.50±0.08 0.50±0.15 7 DHO 1084 F10 Other silicate Other 6.7±0.9 66±5 0.102±0.016 0.60±0.06 0.40±0.07 8 DHO 1084 A8 Titanomagnetite Spinel 10.6±0.4 154±2 0.0685±0.0025 0.392±0.021 0.608±0.035 6 JAH 838 L6 Anorthite Feldspar 9.3±0.5 90±5 0.103±0.008 0.583±0.032 0.417±0.023 8 JAH 838 L6 Forsterite Olivine 10.2±1.9 139±15 0.073±0.016 0.40±0.05 0.60±0.11 8 JAH 838 L6 Calcite Carbonate 2.8±0.2 74±8 0.038±0.005 0.27±0.04 0.73±0.07 8 JAH 838 L6 Forsterite Olivine 8.8±2.4 113±21 0.078±0.026 0.42±0.09 0.58±0.15 9 JAH 838 M3 Anorthite Feldspar 7.9±0.5 79±2 0.099±0.007 0.56±0.02 0.44±0.04 5 JAH 838 M3 Anorthite Feldspar 7.4±1 83±5 0.088±0.013 0.54±0.08 0.46±0.09 8 JAH 838 N12 Anorthite (Carb.) Feldspar 9.5±0.6 86±3 0.110±0.008 0.591±0.022 0.409±0.027 8 JAH 838 N12 Pigeonite Pyroxene 11±1 134±6 0.082±0.008 0.45±0.02 0.55±0.005 6 JAH 838 N12 Titanomagnetite Spinel 9.6±0.9 116±3 0.083±0.008 0.45±0.03 0.55±0.07 6 JAH 838 B9 Anorthite Feldspar 10.6±0.3 88±1 0.121±0.004 0.589±0.0016 0.411±0.019 8 JAH 838 G11 Other silicate Other 13.8±0.3 90±1 0.1525±0.0034 0.747±0.008 0.253±0.006 9 NWA 11444 D2 Pige./Enst. Pyroxene 10.2±0.6 155±5 0.066±0.005 0.403±0.021 0.597±0.028 6 NWA 11444 C3 Forsterite Olivine 11.7±1.1 156±13 0.075±0.010 0.41±0.04 0.59±0.07 7 NWA 11444 B6 Pigeonite Pyroxene 12.1±0.3 148±1 0.0817±0.0023 0.447±0.007 0.553±0.008 8 NWA 11444 C8 Diopside Pyroxene 12.2±0.7 162±5 0.075±0.005 0.4±0.3 0.6±0.4 6 NWA 11444 C8 Pige./Enst. Pyroxene 12.3±0.5 147±5 0.084±0.004 0.447±0.024 0.553±0.023 7 NWA 11444 C3 Calcite Carbonate 2.4±0.2 63±2 0.0377±0.0031 0.27±0.02 0.73±0.06 9 NWA 11444 C2 Ilmenite Oxide 9.5±1 134±12 0.071±0.010 0.41±0.04 0.59±0.06 9 NWA 11444 D2 Pige./Enst. Pyroxene 9.9±0.9 139±8 0.071±0.008 0.41±0.04 0.59±0.09 7 NWA 11444 D2 Pige./Enst. Pyroxene 10±0.6 135±6 0.074±0.006 0.44±0.03 0.56±0.04 8 NWA 11444 D2 Anorthite Feldspar 9±0.6 92±9 0.097±0.012 0.56±0.05 0.44±0.04 7 NWA 11444 C8 Forsterite Olivine 12.5±0.5 174±4 0.072±0.0031 0.402±0.012 0.598±0.019 9 NWA 11444 E4 Other silicate Other 4.9±1.3 38±5 0.13±0.04 0.67±0.08 0.33±0.01 7 NWA 6013 C16 Chromite Oxide 17.3±0.3 147±2 0.1178±0.0023 0.583±0.010 0.417±0.014 9 NWA 6013 C15 Troilite Sulfide 12.7±0.3 147±2 0.0864±0.0025 0.48±0.01 0.520±0.021 7 NWA 6013 F14 Labradorite Feldspar 10.6±0.3 88±1 0.120±0.004 0.601±0.013 0.399±0.016 9 NWA 6013 F14 Pigeonite Pyroxene 12.6±0.3 134±2 0.0946±0.0026 0.497±0.008 0.503±0.015 9 Table 3. Compilation of the atomic percentage composition from each nanoindentation performed. The last column indicates the number of measurements performed at each test. Meteorite Cell Mineral Group O (%) Na (%) Mg (%) Al (%) Si (%) S (%) Ca (%) Ti (%) Cr (%) Fe (%) Other (%) # DHO 1084 I11 Anorthite Feldspar 61.45±0.48 0.28±0.05 2.22±0.55 9.15±0.57 13.17±0.47 0.02±0.04 4.59±0.14 0.12±0.21  1.34±0.39 7.66±2.9 5 DHO 1084 I13 Anorthite (Oliv.) Feldspar 63.54±0.48 0.52±0.25 2.06±0.52 9.56±0.36 13.52±0.38 0.18±0.02 4.77±0.27   1.31±0.28 4.56±2.55 4 DHO 1084 I14 Anorthite Feldspar 64.15±0.6 0.24±0.03 2.61±1.11 8.84±1.19 13.92±0.16 0.18±0.02 4.76±0.66   1.92±0.95 3.39±4.72 3 DHO 1084 I15 Diopside/Augite Pyroxene 61.3±0.52  5.98±0.16 3.31±0.04 15.66±0.26  5.59±0.35 0.21±0.02 0.16±0.01 1.98±0.18 5.79±1.53 5 DHO 1084 I16   60.51±0.43  2.56±0.15 8.82±0.24 4.35±0.35  1.24±0.04 0.41±0.04 8.78±0.38 6.64±0.4 6.68±2.04 5 DHO 1084 I17 Other silicate Other 61.6±0.45 0.48±0.05 1.35±0.28 3.16±0.36 19.52±0.81  0.68±0.02 0.12±0.02  1.45±0.15 11.65±2.14 5 DHO 1084 I19 Titanomagnetite Spinel 63.36±0.72  1.45±0.09 0.98±0.02 4.53±0.29  0.75±0.08 11.75±0.21 0.09±0.03 11.94±0.22 5.17±1.66 4 JAH 838 I20 Anorthite Feldspar 58.5±0.44 0.18±0.04 0.56±0.15 9.96±0.2 11.41±0.02  5.29±0.07 0.5±0.1   13.6±1.03 3 JAH 838 I21 Forsterite Olivine 58.73±0.32  8.77±0.15 2.17±0.05 9.95±0.12  1.86±0.07   6.06±0.11 12.47±0.83 4 JAH 838 I22 Calcite Carbonate 57.6±0.16  1.05±0.06 2.49±0.06 3.45±0.14  17.07±0.16   0.69±0.06 17.66±0.64 5 JAH 838 I23 Forsterite Olivine 59.36±0.21  9.52±0.06 2.26±0.11 10.69±0.2  1.54±0.03   6.53±0.14 10.11±0.74 6 JAH 838 I24 Anorthite Feldspar 55.32±0.25 0.15±0.02 0.41±0.09 9.05±0.14 10.1±0.06  4.86±0.17   0.16±0 19.95±0.73 3 JAH 838 I25 Anorthite Feldspar 58.19±0.4 0.15±0.07 1.07±0.66 9.01±0.84 11.16±0.71  5.37±0.58 0.02±0.03  0.44±0.2 14.6±3.5 6 JAH 838 I26 Anorthite (Carb.) Feldspar 49.39±0.57 0.14±0.02 0.29±0.02 7.82±0.15 8.47±0.26  3.81±0.09   0.25±0.05 29.84±1.16 4 JAH 838 I27 Pigeonite Pyroxene 52.38±0.26  6.02±0.11 2.31±0.14 7.94±0.23  0.99±0.04   5.14±0.11 25.23±0.9 4 JAH 838 I28 Titanomagnetite Spinel 51.17±0.64  1.2±0.06 3.79±0.08 3.65±0.07  1.31±0.03 5.01±0.12 1.66±0.04 9.75±0.33 22.46±1.37 5 JAH 838 I29 Anorthite Feldspar 45±5.21  2.24±0.1 5.06±0.25 8.72±0.31  2.8±0.19   0.91±0.13 35.28±6.19 6 JAH 838 I30 Other silicate Other 60.21±0.11  0.21±0.01 0.99±0.04 20.18±0.43  1.19±0.04   0.11±0.01 17.11±0.64 3 NWA 11444 I31 Pige./Enst. Pyroxene 61.72±1.37  5.66±0.14 2.1±0.07 15.52±0.76  5.58±0.44 0.1±0.02 0.18±0.02 2.74±0.22 6.42±3.04 4 NWA 11444 I32 Forsterite Olivine 58.81±0.47  17.17±0.26 2.87±0.08 12.15±0.21 0.04±0.07 1.24±0.03   1.71±0.04 6±1.16 5 NWA 11444 I33 Pigeonite Pyroxene 57.54±1.03  5.66±0.19 2.55±0.05 13.86±0.34  1.96±0.14 0.08±0.01  4.73±0.18 13.64±1.94 4 NWA 11444 I34 Diopside Pyroxene 58.64±0.95  6.27±0.29 3.66±0.13 15.36±0.18  6.22±0.44 0.32±0.01 0.15±0.01 1.28±0.12 8.11±2.13 4 NWA 11444 I35 Pige./Enst. Pyroxene 58.28±0.24  13.97±0.27 2.92±0.13 11.93±0.18  1.26±0.07   3.7±0.11 7.95±0.99 4 NWA 11444 I36 Calcite Carbonate 57.73±1.63  0.77±0.17 2.46±0.09 2.89±0.16  13.81±0.72   0.22±0.02 22.13±2.79 4 NWA 11444 I37 Ilmenite Oxide 56.19±0.53  1.68±0.08 3.38±0.16 3.75±0.13  1.32±0.07 9.89±0.19  8.61±0.15 15.18±1.29 4 NWA 11444 I38 Pige./Enst. Pyroxene 61.67±0.23  6.46±0.15 3.26±0.12 15.73±0.21 0.12±0.09 2.42±0.22 0.08±0.04  5.16±0.11 5.09±1.18 5 NWA 11444 I39 Pige./Enst. Pyroxene 59.46±1.08  5.27±0.29 3.37±0.43 15.41±0.31  4.86±1.23 0.12±0.03 0.07±0.02 3.81±0.95 7.65±4.33 6 NWA 11444 I40 Anorthite Feldspar 61.16±1.44 0.17±0.11 1.31±0.83 9.86±1.69 12.26±1.08 0.04±0.06 5.71±0.54   0.73±0.32 8.77±6.08 4 NWA 11444 I41 Forsterite Olivine 58.9±0.48  10.81±0.11 2.48±0.04 12.15±0.02  1.02±0   7.97±0.06 6.68±0.72 3 NWA 11444 I42 Other silicate Other 66.62±0.19  0.5±0.05 1.5±0.03 26.48±0.06  0.85±0.02   0.16±0.02 3.9±0.37 4 NWA 6013 I43 Chromite Oxide 60.04 0.00 4.32 3.36 2.37 0.00 0.00 0.24 14.69 7.04 7.94 1 NWA 6013 I44 Troilite Sulfide 32.91 0.00 5.23 0.00 6.07 22.62 0.00 0.00 0.22 19.66 13.29 1 NWA 6013 I46 Labradorite Feldspar 55.76 1.04 1.11 7.64 11.90 0.00 3.26 0.00 0.00 0.64 18.65 1 NWA 6013 I47 Pigeonite Pyroxene 57.44 0.00 8.46 0.30 13.78 0.00 0.64 0.00 0.14 3.93 15.31 1  Figure 6. Representative nanoindentation curves and residual indentation imprints of different minerals in each meteorite. The median value is shown in red. Optical microscopic images of the indentation impressions are shown to the right of each curve. Indentations located on or near the edges of mineral phases have been systematically excluded. \n 14  Figure 7. Histograms of hardness (H), reduced Young’s modulus (Er), H/Er, and elastic recovery (We/Wt) for the meteorites examined in this study are presented in a single panel, with fixed frequency intervals of 5 to facilitate comparison.  Figure 7 shows that lunar meteorites feature a similar distribution of hardness across all nanoindentation tests, featuring a predominant concentration around 9 GPa with a small secondary cluster at approximately 3 GPa. In contrast, the HED meteorite exhibits a distinct pattern: none of the tested minerals fall within this lower-hardness secondary group and contains the hardest mineral. Concerning the reduced Young's Modulus, both DHOFAR 1084 and NWA 11444 show comparable distributions, with the former encompassing the range's minimum and maximum values. In terms of the H/Er ratio and elastic recovery, JAH 838 and NWA 11444 demonstrate similarities, although the latter spans the minimum and maximum values within the dataset. For all examined properties except hardness, the HED meteorite, NWA 6013, displays intermediate values and a distribution that diverges modestly from those of the lunar meteorites, albeit the number of samples is smaller. It is pertinent to \n 15 acknowledge that despite the presence of common mineralogical constituents in lunar meteorites, out of the four minerals identified within HED meteorites, three possess a unique mineralogy. The upper panel of Figure 8 portrays the average hardness of each mineral as a function of reduced Young's modulus, where each individual data point has been categorized by the respective meteorite and mineral group they belong to. Distinct mechanical properties emerge within each mineral group. Olivines, pyroxenes, feldspars, and spinels exhibit strikingly similar characteristics. Conversely, other silicate, and oxide minerals display significant variations in the mechanical properties. The carbonate group stands out as having the lowest hardness, while the oxides exhibit the highest hardness values.  The lower panel of Figure 8 presents the average hardness of each mineral as a function of reduced Young's modulus. In this case, the data points have been categorized based on the identified mineral type. The three indented silicates (others) exhibit distinct mechanical characteristics. A notable correlation is evident in their atomic compositions: as the other element percentage increases, both hardness and Young's modulus show a corresponding increase.  Most of the identified minerals share similar compositions within a given group, with slight variations observed in certain elements. Based on the H/Er ratio, the mechanical properties within each mineral group generally exhibit consistency, except for the oxide and other silicate. However, there is observed variability in the mechanical properties of individual minerals. For instance, the other silicates display Er of 38, 66, and 90 GPa, respectively, while forsterites show values of 113, 139, and 174 GPa. Conversely, minerals such as anorthites and pigeonites present a more uniform values in their mechanical properties. In indented regions where partial mixing with another mineral or a notable concentration of certain elements is detected, as seen in anorthite, diopside, and pigeonite, we observe no substantial variations in mechanical properties.    16  Figure 8. Mean hardness versus mean reduced Young’s modulus for all the nanoindentations performed. The top panel shows meteorite and group classification with mean H/Er depicted for each group. The bottom panel shows mineral classification with global minimum, maximum, and mean. Error bars are depicted in black.  \n 17  Figure 9. Scatter plot of H/Er ratio versus Wp/Wt with square markers. The linear fit is depicted, along with its parameters and the coefficient of determination, highlighting the linear relationship between these properties.   Figure 9 illustrated a distinct linear relationship between the H/Er and the normalized plastic energy (Wp/Wt), a trend that extends to the elastic component as well. This correlation is quantified through a linear regression analysis, the results of which are presented in the legend. The data points, represented by square markers, align closely with the fitted linear trend, signifying a strong empirical relationship. The coefficient of determination value (R²=0.95) is indicative of the strength of this correlation.  4. DISCUSSION   The comprehensive examination of the mechanical properties of the lunar meteorites DHOFAR 1084, JAH 838, and NWA 11444 provide insights into various aspects of lunar science and ISRU. These meteorites from our celestial neighbor offer a unique window into the lunar landscape that human missions will encounter.   DHOFAR 1084 has been used to unravel the geologic history of the Moon (Russell et al., 2004). Chemical analyses of this meteorite have revealed a substantial abundance of aluminum and other refractory elements, strongly suggesting that it originates from the lunar crust. Furthermore, the presence of impact glass and brecciated fragments within DHOFAR 1084 indicates that it was likely \n 18 formed through violent impact events—a common geological process that has left an indelible mark on the lunar surface over eons. The findings of this study underscore the significance of DHOFAR 1084 in shedding light on lunar geological processes.  JAH 838 has provided information on the chemical and isotopic composition of the Moon's interior. Researchers meticulously analyzed the isotopic ratios of oxygen, titanium, and other elements within this meteorite and discovered a striking resemblance to the samples brought back from the Apollo missions (Korotev, 2017). This congruence strongly confirms JAH 838's lunar origin. The chemical composition of JAH 838 also suggests its formation during the early stages of lunar history, a period marked by intense volcanic activity. JAH 838 is a fragmental breccia with mineral fragments of plagioclase, augite, pigeonite, poor-Ca pyroxene, and olivine set in a fine-grained matrix. Additionally, it contains lithic clasts of anorthosite, fragmental breccia, gabbro, norite, basalt, pyroxenite, and regolith (HASP spherule and chondritic lunar metal) embedded in a fine-grained dark gray matrix with metallic Ni-Fe, troilite, pyrrhotine, pentlandite, ilmenite, chromite, zirconolite, baddeleyite, and zircon (Bouvier et al., 2017).  NWA 11444 attests to the richness of lunar materials encapsulated within a fine-grain matrix (Gattacceca et al., 2019). The meteorite boasts an abundance of minerals and lithic clasts, featuring a diverse assortment of angular fragments, ranging from coarse-grained to aphanitic gabbros and basalts. Single-crystal types further diversify the composition of this meteorite, offering an idea of the lunar geological diversity. Gattacceca et al. (2019) delve into the remarkable attributes of NWA 11444, emphasizing its significance as a repository of lunar materials.  In turn, NWA 6013, the sole non-lunar meteorite under investigation in this study, is categorized as an olivine diogenite. These meteorites are believed to have originated at moderate depths within the mantle of the HED parent asteroid, which is thought to be Vesta. An interesting characteristic of NWA 6013 is the lack of clear boundaries between its subunits, imparting an overall metamorphic appearance to the rock. A noteworthy feature observed within NWA 6013 is the entrapment of olivine within pyroxene crystals, particularly in the pyroxene-rich subunits (Ruzicka et al., 2015). Additionally, the meteorite contains chromite grains, some of which attain sizes of several millimeters, further enriching its mineralogical diversity. Minor quantities of troilite and metallic iron are also present in the composition of this meteorite. In a broad sense, our present experiments indicate that NWA 6013 does not exhibit significantly distinct mechanical properties when compared to lunar meteorites.   Regarding the mean elemental composition measured in this study, each meteorite manifested a unique profile. DHOFAR 1084 was particularly rich in oxygen, aluminum, and titanium, whereas NWA 11444 excelled in magnesium and silicon concentrations. JAH 838 was noteworthy for its elevated calcium levels, which may have specialized chemical applications. NWA 6013, the sole HED  19 meteorite analyzed, exhibited a compelling elemental composition, notably featuring the highest iron content among the samples. Moreover, it presented increased concentrations of chromium and sulfur. Overall, a notable variability was observed in both mechanical and elemental properties across all meteorites, suggesting material heterogeneity. Such variability highlights the imperative for additional investigations to elucidate the microscale spatial distribution of these attributes.  The calcite identified in JAH 838 and NWA 11444 lunar meteorites is attributed to terrestrial weathering (Rubin, 1997). This mineral is commonly observed as a fine-grained substance filling fractures linked with metal alteration products. It also appears as a surface coating on sections of meteorites that were initially buried in soil and in minor quantities occluding intergranular pores within weathered meteorites (Al‐Kathiri et al., 2005). For instance, Huidobro et al. (2021) detected calcite in the lunar meteorite NWA 11273, which was discovered in the same region as NWA 11444.  The mineral chromite in the HED meteorite NWA 6013 demonstrates the highest mean hardness value at 17.3±0.3 GPa, indicating its superior resistance to deformation compared to the others. The particular case of olivine will be discussed in detail below. On the other end of the spectrum, the other mineral silicate in NWA 11444 has the lowest mean Er value at 38 GPa, implying more strain for a given constant stress. For comparison, the Chelyabinsk meteorite Er was determined to range from 69 to 78 GPa, as reported by Moyano-Cambero et al., (2017). This contrasts with the values obtained for samples from Itokawa, which ranged from 82 to 111 GPa (Tanbakouei et al., 2019). The Young's modulus values for ordinary chondrites, as reported by Yomogida & Matsui (1983), range from 10 to 140 GPa. In our study, the meteorite NWA 11444 exhibited the most variable Er, with a minimum value of 38 GPa and a maximum of 174 GPa. The mean Er, based on our indentation tests, was calculated to be 112±37 GPa.  The other mineral silicate (cell I30) in JAH 838 has the highest mean H/Er value of 0.1525, suggesting a relatively higher hardness compared to its elasticity. This might be indicative of a material with a higher wear resistance (Pellicer et al., 2021). Minerals from the oxide group, specifically chromite, shows varying mechanical properties, with chromite being the hardest, while other silicates have the highest H/Er ratio. The feldspar group, comprising anorthite and labradorite, generally shows moderate hardness and elasticity. While hardness and elasticity could be influenced by the mineralogical composition, they are not solely dictated by it.  While, in broad terms, minerals tend to exhibit analogous properties, the differences observed among them may arise from different factors, such as the presence of mixed second phases, phase insertions within the matrix, structural orientations, or porosity. Similarly, the varying degrees of shock evident in Figures 1-4 may also suggest differences in the structural fragility of these phases. It is  20 worth mentioning that the non-lunar meteorite examined (NWA 6013) for comparison in this study generally exhibits higher values of hardness, albeit with a smaller number of minerals tested.  A high H/Er ratio typically signifies a material that is relatively brittle and therefore easier to fracture, which is beneficial for mining operations. The meteorite JAH 838, rich in silicate minerals, exemplifies such a material, offering high hardness relative to its elasticity. For construction and manufacturing activities, materials with both high hardness and a high Young's Modulus are sought to ensure not only resistance to deformation but also good elastic recovery properties. The variability of the wear coefficient and its dependence on mechanical properties and abrasive geometry warrant further investigation for precise calibration in Moon ISRU and asteroid mining operations (Pintaude, 2013). Moreover, the material behavior during wear underscores the imperative for selecting mining tools composed of materials significantly harder than the target resources. Elastic effects also emerge as a critical factor in determining wear rates, suggesting that these considerations should be integrated into future predictive models to enhance the efficiency of resource extraction.   Mechanical properties like hardness and elasticity are often influenced by the material's mechanical history, including factors such as impacts, thermal history, and crystal structure. A meteorite subjected to high-impact events may have more refined crystal sizes due to shock-induced recrystallization, potentially enhancing its hardness and Young’s modulus. Similarly, a meteorite that has undergone thermal annealing may display altered mechanical properties due to the depletion of moderately volatile phases (Tartèse et al., 2021). Therefore, while the elemental composition provides some insights, the mechanical history of these meteorites likely plays a crucial role in defining their current mechanical properties. Further investigations, such as microstructural analyses or crystallographic studies, are essential to better understand these phenomena.  The linear correlation observed in our study between the H/Er ratio and normalized plastic energy resonates with the findings of Varea et al. (2012). Their research on nanostructured Cu-rich CuNi electrodeposited films demonstrated a similar linear relationship in mechanical properties under nanoindentation. The agreement of the results not only validates our findings but also reinforces the reliability of the H/Er ratio as a robust parameter for predicting mechanical behavior in different material compositions and structures.  The investigation of the mechanical properties of minerals in lunar meteorites, conducted through indentation and other experimental methods, holds significant relevance for future ISRU. A primary objective of such processes is the utilization of extraterrestrial resources, including minerals, to fabricate structures and products capable of supporting human activities on the lunar surface. To achieve this objective, a comprehensive understanding of the mechanical behavior of rocks to be employed in manufacturing processes is crucial. Exploring the mechanical properties of lunar  21 meteorites and their constituent minerals offers researchers valuable insights, enabling the extrapolation of these properties to the larger scale of lunar rocks, suitable for manufacturing. In this regard, Tang et al. (2022) emphasize the significant role of interphases between minerals and microcracks in determining rock mechanics. Such knowledge is fundamental for developing manufacturing processes and techniques adapted to lunar conditions and materials.  Insights garnered from studying the mechanical properties of minerals like olivine in lunar meteorites can inform the creation of manufacturing processes for producing robust structural materials and components, characterized by strength, durability, and resistance to wear. Additionally, the understanding of lunar materials can drive the development of novel alloys and composites optimized to thrive in the lunar environment and cater to the specific demands of lunar manufacturing. Enhanced comprehension of the mechanical properties of lunar materials facilitates multiple facets of future colonization activities, encompassing material characterization for quality control, coating design, and optimization, wear and fatigue assessment, and the development of new materials, among others.  Porosity, or the presence of voids within a material, plays a pivotal role in shaping its mechanical properties. Porosity can influence strength, stiffness, fracture toughness, and resistance to fatigue and wear. For instance, experimental measurements have demonstrated that an increase in porosity of approximately 20% results in a reduction of silicon's Young's modulus by nearly 50% (Magoariec & Danescu, 2009). Lunar regolith, the surface layer of loose material on the Moon, exhibits notable porosity, ranging from approximately 4% to 21%, with an average of around 12% (Wieczorek et al., 2013). This substantial porosity poses challenges when considering the direct use of lunar regolith as a construction material, as it may lack the necessary strength and stability. In Figure 8, varying mechanical properties across different phases are observed, yet align along a constant H/Er ratio. This alignment could be associated with changes in local porosity or density, indicating that as the porosity of a mineral increases, its properties diminish along the axis of a constant H/Er ratio (Luo & Stevens, 1999).  Kumamoto et al. (2017) reported that Young's modulus for terrestrial olivines ranges from 160 to 290 GPa. In contrast, our measurements for lunar olivines exhibit a range from 113 to 174 GPa. Correspondingly, hardness values for lunar olivines are lower, between 8.8 and 12.5 GPa, as opposed to terrestrial counterparts which range from 14 to 18 GPa. This disparity in mechanical properties compared to terrestrial olivines—irrespective of whether they are single crystals or polycrystalline aggregates—underscores the necessity for comprehensive characterization of lunar materials for ISRU, given their distinct behavior. The surface materials of the Moon have been subjected to numerous impacts, necessitating substantially more forceful impacts to eject these lunar rocks into  22 orbit. This could potentially result in lunar meteorites that are structurally weakened, exhibiting increased porosity or microfracturing. While porosity in olivine may predominantly arise from impact-induced microfracturing, we cannot discount the possibility of vaporization-induced voids in the crystalline structure during the shock wave transit.  To address this challenge, one approach involves the development of techniques for compacting and sintering lunar regolith to produce denser, stronger construction materials. Nanoindentation testing emerges as a valuable tool within this process, offering a non-destructive means of assessing the mechanical properties of compacted regolith samples, including hardness, elasticity, and fracture toughness.  In addition, the utilization of micromechanical properties derived from indentation testing holds immense potential for advancing our understanding of material behavior and enhancing the accuracy of impact models. By considering the complex responses of materials at the microscale, we can refine simulations, optimize designs, and better predict how materials perform under high-velocity impacts.  5. CONCLUSIONS   Lunar meteorites, originating from our celestial neighbor, the Moon, have captivated the scientific community for their intrinsic value in unraveling the mysteries of lunar geology and history. Ejected from the lunar surface through the tumultuous forces of high-velocity impacts, these enigmatic rocks make their journey to Earth, offering an unparalleled window into the composition, properties, and shock experiences of lunar materials. In contrast to the Apollo missions, where samples were meticulously collected from specific lunar locales, lunar meteorites represent fragments of the Moon with origins veiled in uncertainty and, in principle, randomness.  This study has provided a comprehensive analysis of the mechanical and elemental properties of lunar meteorites DHOFAR 1084, JAH 838, NWA 11444, and HED meteorite NWA 6013. The findings reveal significant heterogeneity in both mechanical and elemental attributes across the samples, underscoring the need for further investigation into the microscale spatial distribution of these properties.  The majority of the minerals identified in this study, particularly olivines, feldspars, pyroxenes, and spinels, demonstrate closely aligned compositional and mechanical characteristics. On the other hand, other silicate and oxide minerals exhibit important variations in their mechanical properties. The carbonate group distinguishes itself by manifesting the lowest hardness values, whereas the oxides are characterized by the highest hardness.   Forsterite in NWA 11444 has the highest reduced Young's Modulus value. In contrast, other silicate in the same meteorite shows the lowest mean Reduced Young's Modulus. The three indented  23 silicates (others) exhibit distinct mechanical characteristics, which not only have implications for their potential applications but also raise questions about the underlying factors influencing these properties.   Olivines from terrestrial origins, when analyzed using nanoindentation techniques, demonstrate greater hardness and a superior Young's modulus relative to olivines of lunar origin. A linear correlation is observed between the H/Er ratio and normalized plastic and elastic energies. Furthermore, various mineral phases display a constant H/Er ratio axis, which may indicate variations in local porosity. The need for further studies is evident, particularly focusing on aspects such as porosity, phase insertions within the matrix, and structural orientations, to provide a more comprehensive understanding of these mechanical characteristics.   By comprehending the mechanical characteristics of these compacted samples, researchers can gain valuable insights into optimizing compaction and sintering processes to yield more robust and durable building materials for lunar applications and asteroid mining. This advancement lays the groundwork for the creation of sustainable infrastructure, encompassing habitats, roadways, and other vital structures essential for long-term human presence on the Moon.  The use of microscale rock mechanics experiments to characterize the mechanical properties of minerals in meteorites provides valuable insight into the composition and behavior of materials on the lunar surface. The data obtained can be used to guide the development of extraterrestrial manufacturing techniques and space exploration technology, as well as contribute to next-generation impact modeling studies. The combination of mineral identification and nanoindentation testing provided a comprehensive characterization of the mechanical properties of the meteorites studied, which can inform forthcoming research efforts in space science and ISRU.  For future research, we aim to conduct a comparative analysis of the mechanical properties of lunar rocks, incorporating both Apollo mission return samples and analogous terrestrial minerals.  Acknowledgments- JMT-R and EP-A acknowledge financial support from the project PID2021-128062NB-I00 funded by MCIN/AEI/10.13039/501100011033. The lunar samples studied here were acquired in the framework of grant PGC2018-097374-B-I00 (P.I. JMT-R). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 865657) for the project “Quantum Chemistry on Interstellar Grains” (QUANTUMGRAIN). AR acknowledges financial support from the FEDER/Ministerio de Ciencia e Innovación – Agencia Estatal de Investigación (PID2021-126427NB-I00, PI: AR). Partial financial support from the Spanish Government (PID2020-116844RB-C21) and the Generalitat de Catalunya (2021-SGR-00651) is acknowledged. 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"    },    {        "title": "2402.03964v2.Almost_Perfect_Mutually_Unbiased_Bases_that_are_Sparse.pdf",        "content": "arXiv:2402.03964v2  [cs.DM]  14 Mar 2024Noname manuscript No.\n(will be inserted by the editor)\nAlmost Perfect Mutually Unbiased Bases that are Sparse\nAjeet Kumar ·Subhamoy Maitra ·Somjit Roy\nthe date of receipt and acceptance should be inserted later\nAbstract In dimension d, Mutually Unbiased Bases (MUBs) are a collection of orthono r-\nmal bases over Cdsuch that for any two vectors v1,v2belonging to different bases, the\ndot or scalar product |/a\\}⌊ra⌋k⌉tl⌉{tv1|v2/a\\}⌊ra⌋k⌉tri}ht|=1√\nd. The upper bound on the number of such bases is\nd+ 1. Construction methods to achieve this bound are known for cases when dis some\npower of prime. The situation is more restrictive in other ca ses and also when we con-\nsider the results over real rather than complex. Thus, certa in relaxations of this model\nare considered in literature and consequently Approximate MUBs (AMUB) are studied.\nThis enables one to construct potentially large number of su ch objects for Cdas well as\ninRd. In this regard, we propose the concept of Almost Perfect MUB s (APMUB), where\nwe restrict the absolute value of inner product |/a\\}⌊ra⌋k⌉tl⌉{tv1|v2/a\\}⌊ra⌋k⌉tri}ht|to be two-valued, one being 0\nand the other ≤1+O(d−λ)√\nd, such that λ >0 and the numerator 1 + O(d−λ)≤2. Each\nsuch vector constructed, has an important feature that larg e number of its components\nare zero and the non-zero components are of equal magnitude. Our techniques are based\non combinatorial structures related to Resolvable Block De signs (RBDs). We show that\nfor several composite dimensions d, one can construct O(√\nd) many APMUBs, in which\ncases the number of MUBs are significantly small. To be specifi c, this result works for d\nof the form ( q−e)(q+f), q,e,f∈N, with the conditions 0 ≤f≤efor constant e,fandq\nsome power of prime. We also show that such APMUBs provide set s of Bi-angular vectors\nwhich are of the order of O(d3\n2) in numbers, having high angular distances among them.\nFinally, as the MUBs are equivalent to a set of Hadamard matri ces, we show that the\nAPMUBs are so with the set of Weighing matrices.\nKeywords Almost Perfect Mutually Unbiased Bases, Combinatorial Des ign, Hadamard\nMatrices, Quantum Information Theory, Resolvable Block De sign, Weighing Matrix.\nMathematics Subject Classification (2010) 81P68\n1 Introduction\nMutually Unbiased Bases (MUBs) received serious attention in Quantum Information\nProcessing, as they are useful in different aspects of Quantu m Cryptology and Communi-\ncations like Quantum Key Distribution (QKD), Teleportatio n, Entanglement Swapping,\nA. Kumar\nApplied Statistics Unit, Indian Statistical Institute, Ko lkata, India, E-mail: ajeetk52@gmail.com\nS. Maitra\nApplied Statistics Unit, Indian Statistical Institute, Ko lkata, India, E-mail: subho@isical.ac.in\nS. Roy\nDepartment of Statistics, University of Calcutta, Kolkata , India, E-mail: somjit.roy2001@gmail.com2 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nDense Coding, Quantum Tomography, etc. (see [ 25] and the references therein). For any\nfinite dimensional Hilbert space Cd, the number of MUBs is bounded by d+1. However,\nin spite of intense research for several decades, these coul d be constructed only when d\nis some prime power [ 36,69,4,44,65]. For a specific dimension d, constructing a larger\nnumber of MUBs (upper bounded by d+ 1) is one of the most challenging problems in\nQuantum Information Theory.\nVarious mathematical tools have been used to construct MUBs , among which note-\nworthy being the use of finite fields [ 69,44] and maximal set of commuting bases [ 4].\nFor dimensions which are not power of primes, constructing l arge number of MUBs still\nremains elusive. This is the reason, various kinds of Approx imate MUBs have been con-\nstructed using character sums over Galois Rings or Galois Fi elds [60,45,66,16,63,51,70],\ncombinatorial design [ 48,47] and computational search [ 20].\nWhen MUBs are constructed over Rd, we get Real MUBs. They have interesting con-\nnections with Quadratic Forms [ 15], Association Schemes [ 49,23], Equi-angular Lines,\nEqui-angular Tight Frames over Rd[9], Representation of Groups [ 28], Mutually Unbiased\nReal Hadamard Matrices, Bi-angular vectors over Rd[33,43,8] and Codes [ 14]. As we\nhave elemented out earlier, large number of Real MUBs are non -existent for most of the\ndimensions [ 13]. In fact only for d= 4s,s >1, we have d/2 + 1 many MUBs, whereas\nfor most of the dimensions d, which are not perfect square, we have at best only 2 Real\nMUBs [13]. In view of this, various attempts have been made to constru ct Approximate\nReal MUBs (ARMUBs) which are available in literature [ 48,47,70].\nVarious efforts have been made to explore connections betwee n MUBs and geometrical\nobjects such as polytopes and projective planes [ 5,6,59,58,2]. Since the known meth-\nods for the construction of MUBs provides complete sets only whendis some power of\nprime, there are conjectures related to the existence of com plete sets of MUBs and finite\nprojective plane, which are also currently known to exist on ly for prime power orders.\nIfd=pn1\n1pn2\n2...pnss, then the lower bound on the number of MUBs is pnrr+ 1 where\npnrr= min{pn1\n1,pn2\n2,...,pnss}. Thus, constructing a large number of MUBs for any compos-\nite dimension has proven to be elusive even over Cd. In fact, the number of such bases is\nvery small when we consider the problem over the real vector s paceRd(see [13]). For dis-\ncussions on the basics and open problems related to MUBs and t he approximate version,\none may refer to our earlier works [ 48,47] and the references therein.\nTowards constructing the approximate MUBs over Cd, certain techniques were pro-\nposed in [ 60,45,16]. However, such techniques cannot be applied in constructi on of Real\nMUBs. In this direction, we could propose a construction in [ 48] for Approximate Real\nMUBs using Real Hadamard matrices. It has been shown in [ 48] that√\nd\n4+1 ARMUBs\nwith maximum value of the inner product as4√\ndcould be achieved for d= (4q)2, whereq\nis a prime. In [ 47], we have shown that such results can be generalized as well a s improved\nto a great extent using Resolvable Block Designs (RBDs). The earlier result of [ 48] could\nbe generalized in [ 47] ford=sq2, whereqis a prime power. Further, the parameters could\nbe improved too in [ 47]. It has been shown in [ 47] that ford=q(q+1), where qis a prime\npower and q≡3 mod 4, it is possible to construct ⌈√\nd⌉=q+ 1 many ARMUBs with\nmaximum value of the inner product upper bounded by2√\nd, between vectors belonging\nto different bases. Therefore, the improvement in the result of [47] is two-fold. First, the\nnumber of MUBs is greater and second, the maximum of the inner product values is lower\ncompared to [ 48]. As these were achieved by several kinds of combinatorial d esigns, we\nexplore this idea further. One important feature of all our c onstruction techniques [ 48,\n47] was that, all the components of the basis vectors so constru cted is either zero or of\nconstant magnitude. Hence taking the normalizingfactor ou tside the vectors would render\nthe numericalvalue of the components as zero or of unit magni tude,which is like Weighing\nmatrices [ 57] – a generalization of Hadamard matrices. Finally, note tha t the construction\nof [47] provides the vectors which are sparse and this property is i nherited in this paper\ntoo.Almost Perfect Mutually Unbiased Bases that are Sparse 3\nApproximate MUBs have been defined in various manners in the s tate of the art\nliterature. The cue has been taken from the two initial paper s [45,60]. Although the work\nof [45] is related to Approximate SIC POVM, the definition of “Appro ximate” has been\ncarriedovertoMUBs aswell.Variousmathematicalmeanings ofapproximationshavebeen\nused in relaxing the condition on absolute value of the inner product between two vectors\nsay|v1/a\\}⌊ra⌋k⌉tri}ht,|v2/a\\}⌊ra⌋k⌉tri}ht. For example, in these two papers [ 45,60], we get references to |/a\\}⌊ra⌋k⌉tl⌉{tv1|v2/a\\}⌊ra⌋k⌉tri}ht|\nbounded by1+o(1)√\nd,2+o(1)√\nd,O(log(d)√\nd),O(1\n4√\nd), andO(1√\nd) etc. Subsequent researchers\ninvestigating Approximate MUBs have also adopted similar m athematical definition [ 16,\n66,63,51,70].\nOne may note that each MUB in the space Cdconsists of dorthogonal unit vectors\nwhich, collectively, can be thought of as a unitary d×dmatrix. Two (or more) MUBs\nthus correspond to two (or more) unitary matrices, one of whi ch can always be mapped to\nthe identity Iof the space Cd, using a unitary transformation. For example, suppose we\nhavermany MUBs {M1,M2,M3,...,Mr}inCdwherer≤d+1 and also we can thought\nthem as arnumbers of d×dunitary matrices. If we multiply M−1\n1to each of the matrices\nat right, then one can obtain {I,M2M−1\n1,M3M−1\n1,...,MrM−1\n1}as the transformed set of\nMUBs. As the inverse of any unitary matrix is equal to its conj ugate transpose, to obtain\nMjM−1\ni, fori/\\⌉}atio\\slash=j, we are considering inner products of each row of the two matr ices in the\nset of MUBs. Thus, the modulus of each element of the product m atrix will be1√\nd. Taking\n1√\ndcommon, the modulus of each of the elements will be 1, i.e., we will have complex\nHadamard matrices. The result will be similar if we multiply the inverse from the left too.\nIn this regard, we now consider the Weighing matrices.\nDefinition 1 A square matrix of order dand weight wis called a (complex) weighing\nmatrix, denoted by W(w,d), if its elements belong to the set/braceleftBig\n0,exp(iθ)√w/bracerightBig\nwithθ∈R,\nand it satisfies W(w,d)†W(w,d) =I. If the elements are confined to the set/braceleftBig\n0,±1√w/bracerightBig\n, it\nbecomes a real weighing matrix.\nThe use of complex weighing matrices in quantum error correc ting codes has been\nexplored in [ 26]. Furthermore, the connection between real weighing matri ces and classical\ncodes are also investigated, as evident from the studies suc h as [22,40,41,3], which also\ndelve into applications involving spherical codes [ 54]. For more analysis, one can refer to\n[46] and the references therein. As we will proceed further, the definition of APMUBs will\nbe presented and we will show that the weighing matrices will related to APMUBs as the\nHadamard matrices relate to MUBs.\nLet us now summarize the contribution of this paper and outli ne its presentation.\n1.1 Organization and Contribution\nWe begin with Section 2to present a background of related combinatorial objects. T hen\ntowards the constructions, in Section 3, we show bounds on the values of certain parame-\nters, expressed in terms of the block size kand number of elements in the RBD, i.e., d. In\nthis regard, we define a combinatorial quantity A(d,k,µ), relevant to our analysis, which\ncan be of its own independent interest. Thereafter in Lemma 3we describe an interesting\nclass of RBDs which can be constructed from MOLS(s) yielding µ= 1 andr=N(s)+2.\nHereµis the maximum number of common elements between any pair of b locks from\ndifferent parallel classes, as we will explain in the followi ng background section. A con-\nstructive proof to obtain the same from MOLS(s) has also been given and in Lemma 4. We\nfurther show that the converse of Lemma 3is also true. The results are further explained\nwith illustrative examples.\nThen we consider the Almost Perfect MUBs (APMUBs) and some ge neric ideas of\nconstructioninSection 4.ThebasicmotivationanditsrelationshipwithBi-angular vectors4 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nare presented in Section 4.1. In Section 4.2, we analyze certain properties of the AMUBs\nwhich can be constructed using RBDs having blocks of constan t size. In this direction, we\nconsider RBD( X,A) with|X|=d=k·s= (q−e)(q+f) andAwith resolution r, where\neach block is of size ( q−e). We study the asymptotic behaviours of the parameters of\nAMUBs thus generated. It is also shown that our construction can provide APMUBs only\nwhenµ= 1, therefore putting strong constraints over the nature of the RBDs required for\nthis kind of constructions.\nSection5containsour algorithmstowardsconstructingRBDs that can be consequently\nused for the obtainingAPMUB’s withparametersthat could be achievedfor the first time.\nWe first show that whenever the dimension dis a composite number and can be expressed\nask·s,k≤s, such that β=/radicalbigs\nk≤2, one can construct N(s) + 1 many APMUBs,\nwhereN(s) is the number of MOLS( s). We refer to this as the MOLS Lower Bound\nConstruction for APMUBs. Since a composite number dcan be factored in multiple ways\nensuringβ≤2, there can be more than one MOLS Lower Bound Constructions f or a\ndimension d. It is to be noted that if s=q, some power of prime, then N(q) =q−1. Hence\nin such situations we get qmany APMUBs. The best known asymptotic bound for N(s)\nis given by N(s)→ O(s1\n14.8), which generally results into a small number of APMUBs. We\nimprove this significantly.\nIn this direction we show that when d= (q−e)(q+f),e,f∈Nande≥f, whereqis\nsome power of prime, we can obtain O(q) many APMUBs. That is, when e,fare constant,\nthen the number of such APMUBs is O(√\nd). Illustrative examples to describe the above\nconstruction have also been provided. This is explained in S ection5.2.\nWe conclude with the following two comments in Section 6towards future direction.\nFirst, the dimensions dfor which we obtain O(√\nd) many MUBs have the density of the\norder of the primes in the space of natural numbers. Secondly , the constructions of AP-\nMUBs directly translates to the construction of Bi-angular vectors. The other direction\nmay also be explored. It needs further disciplined effort to e xplore this part too.\n2 Background\nLet us start with the basic definition related to Mutually Unb iased Bases.\nDefinition 2 Consider two orthonormal bases,\nMl=/braceleftBig\n|ψl\n1/a\\}⌊ra⌋k⌉tri}ht,|ψl\n2/a\\}⌊ra⌋k⌉tri}ht,...,|ψl\nd/a\\}⌊ra⌋k⌉tri}ht/bracerightBig\nandMm=/braceleftbig\n|ψm\n1/a\\}⌊ra⌋k⌉tri}ht,|ψm\n2/a\\}⌊ra⌋k⌉tri}ht,...,|ψm\nd/a\\}⌊ra⌋k⌉tri}ht/bracerightbig\n,\nind-dimensional complex vector space, i.e., Cd. These two bases will be called Mutually\nUnbiased if we have/vextendsingle/vextendsingle/vextendsingle/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht/vextendsingle/vextendsingle/vextendsingle=1√\nd,∀i,j∈ {1,2,...,d}. (1)\nThe set M={M1,M2,...,Mr}consisting of such orthonormal bases will form MUBs of\nsizer, if every pair in the set is mutually unbiased.\nWhen the conditions among two different bases are relaxed suc h that/vextendsingle/vextendsingle/vextendsingle/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht/vextendsingle/vextendsingle/vextendsinglecan\ntake values other than1√\nd, then we consider the approximate version of this problem.\nThis is due to the fact that, for most of the dimensions, which are not power of some\nprimes, obtaining a large number of MUBs reaching the upper b ound is elusive. In this\ncontext we denote ∆for the set containing different values of/vextendsingle/vextendsingle/vextendsingle/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht/vextendsingle/vextendsingle/vextendsingleforl/\\⌉}atio\\slash=m. In this\ninitiative, extending the ideas of [ 47], we exploit the well known combinatorial object,\nthe Resolvable Block Design (RBD), towards construction of Approximate MUBs with\nimproved parameters. One may refer to the book [ 64] for more details on RBDs, and we\npresent certain definitions in this regard.Almost Perfect Mutually Unbiased Bases that are Sparse 5\nDefinition 3 A combinatorial block design is a pair ( X,A), whereXis a set of elements,\ncalled elements, and Ais a collection of non-empty subsets of X, called blocks. A combi-\nnatorial design is called simple, if there is no repeated blo ck inA.\nGenerally all the combinatorial designs are assumed to be si mple, i.e., they do not have\nany repeated blocks.\nDefinition 4 A combinatorial design ( X,A) is at−(d,k,λ) design if each block in Ais of\nsizek, and that any set of telements from X, appears as subset of exactly λblocks inA.\nNote that here t,d,kandλare positive integers with 1 <k<d.\n2.1 Resolvable Block Design (RBD)\nResolvable Block Design (RBD) is a special kind of Combinato rial design, where the set\nAcan be partitioned into parallel classes which are called re solutions of A. Initially RBD\nwas defined by R. C. Bose in the context of Balanced Incomplete Block Design [ 10,11,12].\nLater various generalizations could be achieved as explain ed in varied literatures [ 61,42,\n56,39]. For the purpose of our paper we consider the following simp le definition of RBD\nas presented in our previous paper [ 47].\nDefinition 5 Combinatorial design ( X,A), is called a Resolvable Block Design (RBD), if\nAcan be partitioned into r≥1 parallel classes, called resolutions. Where a parallel cl ass\nin design (X,A) is a subset of the disjoint blocks in Awhose union is X.\nThere is a special kind of RBD called Affine Resolvable BIBD (AR BIBD) [11,12,62]\n(see also [ 64, Chapter 5]). It is well known that whenever qis some power of a prime, one\ncan construct ( q2,q,1) ARBIBD. An Affine plane of order qis an example of this. Here\n|X|=q2, andAconsists of q(q+1) blocks, which can be resolved into q+1 many parallel\nclasses. Each parallel class consists of qmany blocks of constant size q. Most importantly,\nany pair of blocks from different parallelclasses has exactl y one element in common.Affine\nPlanes areknownonly when qis powerof a prime.For detail,one mayrefer to[ 64, Sections\n2.3, 5.2, 5.3, 6.4].\nLet us define the notation that we will use in this connection. For RBD(X,A), with\n|X|=d, we will indicate the elements (also called elements) of Xby simple numbering,\ni.e.,X={1,2,3,...,d}. Hererwill denote the number of parallel classes in RBD and\nparallel class will be represented by P1,P2,...,Pr. The blocks in the lthparallel class will\nbe represented by {bl\n1,bl\n2,...,bls}, indicating that the lthparallel class has smany blocks.\nSince in our entire analysis we will be using RBDs with consta nt block size, let us denote\nthe block size by k. Further, we denote the number of blocks in a parallel class o f RBD by\ns. Since in our analysis we are making of use of RBDs with consta nt block size, hence each\nparallel class will always have smany blocks and |X|=d=k·s. The notation bl\nijwould\nrepresent the jthelement of the ithblock in the lthparallel class. Further, the notation bl\ni\nwould represent ithblock oflthparallel class. Note that bl\nij∈X. In every block, we will\narrange the elements in increasing order, and we will follow this convention throughout\nthe paper, unless mentioned specifically. Thus bl\nij≤bl\ni,j+1,∀j. This will be important to\nrevisit when we convert the parallel classes into orthonorm al bases. Another important\nparameter for our construction is the value of the maximum nu mber of common elements\nbetween any pair of blocks from different parallel classes. W e denote this positive integer\nbyµ. Note that µ≥1 for any RBD, with r≥2. One may further refer to Lemma 2of\nSection3in this regard.\n2.2 Mutually Orthogonal Latin Square (MOLS)\nA Latin Square of order sis ans×sarray, and a cell of the array consists of a single\nelement from a set Y, such that |Y|=s. Every row of the Latin Square is a permutation6 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nof the elements of set Yand every column of the Latin square is also permutation of th e\nelements from the set Y. For more details one may refer to [ 64, Definition 6.1] as well as [ 1,\nExample 1.1].A pair of Latin Squares ( L1,L2) of same order and having entries from same\nsetY(or, a different set having same number of elements) is called Mutually Orthogonal,\nif in the ordered pair {(Y,Y)}={((L1)ij,(L2)ij)}, every pair x,y∈Yappears exactly\nonce. That is, if two of the Latin Squares are superimposed, a nd the resulting entries in\neach cell is written as ordered pairs, then every x,y∈Yappears exactly once in the cell.\nFurther, if there is a set of wmany Latin Squares, say {L1,L2,...,Lw}, each of order\ns, such that, every pair of Latin Squares is orthogonal, then t he set is called Mutually\nOrthogonal Latin Square of order s, which we denote as w-MOLS(s).\nLetN(s) denote the maximal value of wsuch that, there are wmany MOLS of order\ns[1,37], [64, Chapter 6]. While using the numericalvalues of N(s), in subsequent examples\nin this paper, we will use the currently known values of N(s) from [1, Table 3.87, page\n176]. Note that these are not always the actual values of N(s) (except when sis some\npower prime or of small order) as the exact value of N(s) is still an open question in most\nof the cases. It is known that, N(s)≤s−1∀s. When this bound is attained, we say that\nthere is a complete set of Mutually Orthogonal Latin Squares of orders. The construction\nfor complete sets of MOLS( s) is known when sis some power of prime [ 64, Section 6.4].\nWhensis not a power of prime, N(s) is much smaller than s−1. A table with the largest\nknown values for wis presented in [ 1] fors<10000.\nIt is known that there exists a constant n0, such that for all s≥n0, we have, N(s)≥\n1\n3s1\n91[21], which was later improved by Wilson [ 67] toN(s)≥s1\n17. Further, it was shown\nin [68, Section 4] that the exponent can be lower bounded by1\n14.8. One may note that\nN(s)→ ∞ass→ ∞in general, but for the finite cases only when sis some power of prime\nthenN(s) =s−1, else it is considerably small. In this regard, one may also note that the\nAffine Planes of order qare equivalent to ( q−1) MOLS(q) [64, Theorem 6.32]. For a brief\nsurvey on construction of MOLS, one may refer to [ 37]. Note that, in connection of RBD,\nwe usedsto indicate the number of blocks in a parallel class, whereas in the context of\nLatin Square, we use sto indicate the order of Latin Square.\n3 Some Important Technical Results\nLet us now consider a counting of combinatorial objects that is relevant to us.\nDefinition 6 LetT(d,k,µ), 0≤µ<k<d be the maximum number of subsets each of size\nk, that can be constructed from ddistinct objects, such that, between any two different\nsubsets there is a maximum of µobjects in common.\nFirst we should relate with the error correcting codes, as th is can be seen as the maximum\nnumber of codewords of the binary constant weight codes of le ngthdand weight kwith\nminimum distance 2( k−µ). For more details in this regard one may refer to [ 35, Theorem\n2.3.6], but we will only restrict here to some technical resu lts only. One may immediately\nnote that T(d,k,0) =/floorleftbigd\nk/floorrightbig, andT(d,k,k−1) =(d\nk). For arbitrary d,k,µ∈N, the following\nresult provides an estimate of T(d,k,µ).\nLemma 1 T(d,k,µ)≤/floorleftbigg(d\nµ+1)\n(k\nµ+1)/floorrightbigg\n=/floorleftBig\nd!(k−µ−1)!\nk!(d−µ−1)!/floorrightBig\n. The upper bound of/floorleftBig\nd!(k−µ−1)!\nk!(d−µ−1)!/floorrightBig\nis achieved\nwhenever (µ+1)−(d,k,1)design exists and in such cases T(d,k,µ)is the same as the number\nof blocks in (µ+1)−(d,k,1)design as in Definition 4.\nProofGiven a set of ddistinct elements, we like to construct the maximum number o f\nsubsets each of size k, such that any two subsets has µelements in common. Let us label\nthese blocks as {b1,b2,...br}, withr=T(d,k,µ).\nNow consider all the ( µ+1)-element subsets of ddistinct elements. They will be (d\nµ+1)\nin numbers. Now consider the blocks biandbj. Since there are a maximum of µelementsAlmost Perfect Mutually Unbiased Bases that are Sparse 7\nin common, any ( µ+1)-element subset of delements cannot exist, which occur in both\nthe blocks biandbj. Since each block biis of sizek, the number of ( µ+1)-element subsets\nwhich can be constructed by the kelements in biis(k\nµ+1). Let us denote this set by Si.\nSimilarlyfor block bjthe number of ( µ+1)-elementsubsets which can be constructed with\nitskelements is (k\nµ+1). Let us denote this set by Sj. We have already seen Si∩Sj=φ, else\nthere would be µ+1 element common between biandbj. Now since there are rsuch blocks\neach of size k, hence|S1|+|S2|+...|Sr|=r(k\nµ+1). This must be less than or equal to (d\nµ+1),\nwhich is the maximum possible ( µ+ 1)-element subsets that can be constructed from d\ndistinct elements. This implies r(k\nµ+1)≤(d\nµ+1)⇒ T(d,k,µ) =r≤/floorleftbigg\n(d\nµ+1)\n(k\nµ+1)/floorrightbigg\n=/floorleftBig\nd!(k−µ−1)!\nk!(d−µ−1)!/floorrightBig\n.\nTo see the second part of the lemma, note that t−(d,k,1) design is a design ( X,A)\nwhereAcontains the subsets of Xcalled blocks, such that |X|=dand each block contains\nexactlykelements. Every t-element subset of Xis contained in exactly one block. Hence\nthis implies that any two blocks of the design has maximum t−1 elements in common.\nThus it immediatelyfollowsthat if ( µ+1)−(d,k,1) design exists, then blocks of the design\nsatisfies the property of T(d,k,µ), and since number of blocks in ( µ+1)−(d,k,1) design\nis(d\nµ+1)\n(k\nµ+1)[64, Chapter 9, Theorem 9.4 and the following observation], whi ch is exactly the\nupper bound of T(d,k,µ) as proven above.\nThe upper bound of T(d,k,µ) is achieved whenever a ( µ+1)−(d,k,1) design exists,\nwhich is known for many values of 0 ≤µ<k<d . This implies thatthe bound for T(d,k,µ)\ngiven by the above result is tight. However, getting an exact value/expression of T(d,k,µ)\nappears to be an open and challenging problem.\nTo construct the AMUBs, our focus has been on RBDs with consta nt block size. In this\nconnection we now focus on few results that are relevant to ou r constructions of APMUBs\nin Section 5. Following is a lemma related to an RBD providing a bound for µandr\nin terms of the block size and the number of blocks in parallel classes where, as defined\npreviously, µis the maximum number of common elements between any pair of b locks\nfrom different parallel classes and ris the number of parallel classes.\nLemma 2 Consider an RBD (X,A)with|X|=d=k·swherek,s∈N, consisting of r >1\nparallel classes, each having blocks of size k. Thenµ≥ ⌈k\ns⌉, whereµis the maximum number\nof common elements between any pair of blocks from different pa rallel classes and r≤ T(d−\n1,k−1,µ−1). Further, if µ= 1, thenr≤ ⌊d−1\nk−1⌋=s+⌊s−1\nk−1⌋.\nProofSincer >1, consider any pair of parallel classes of RBD say ( Pl,Pm). Denote the\nblocks ofPlbe asbl\n1,bl\n2,...,bls. Since blocks are of constant size ⇒ |bl\ni|=kand the blocks\nbelonging to the same parallel class have no element in commo n,⇒bl\ni∩bl\nj=φ∀i,j=\n1,2,...,sandX=bl\n1∪bl\n2∪...∪bls,|X|=k·s. Similar relations will hold for blocks of any\nother parallel class.\nConsider any block of Pl, saybl\ni. Since,µ= maxm|bl\ni∩bm\nj|, we havebl\ni∩bm\nj≤µ,∀j=\n1,2,...,s. SinceX=bm\n1∪bm\n2∪...∪bms, hence,/summationtexts\nj=1|bl\ni∩bm\nj|=|bl\ni|=k. We also have/summationtexts\nm=1|bl\ni∩bm\nj| ≤/summationtexts\nm=1µ=µs⇒k≤µs. Sincek,s,µ∈N, we getµ≥ ⌈k\ns⌉. This implies\nminimum value of µ= 1 which is possible only if k≤sand on the other hand if k > s,\nthen minimum value of µ= 2. Thus, for µ= 1, we must have k≤s, i.e., number of blocks\nmust be greater than or equal to the block size of the RBD.\nTo obtain a bound on r, fix an element, say x∈X. Since the blocks of a parallel class\nare mutually disjoint and their union is X, each parallel class will have exactly one block\nwhich will contain x. Collect all the blocks that contain the element xand we will obtain\na set ofrsuch blocks. Denote this set by S. Now, remove xfrom every block in the set\nS. HenceSwill consist of blocks of size k−1, the maximum number of common elements\nbetween any two blocks in Swould be now ( µ−1) and the total number of elements\ncontained in these rblocks will be ≤(d−1). Therefore, we obtain r≤ T(d−1,k−1,µ−1).\nThus ifµ= 1 we have r≤ T(d−1,k−1,0) =⌊d−1\nk−1⌋=⌊s·k−1\nk−1⌋=s+⌊s−1\nk−1⌋.8 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nWe will see that, using our construction method for APMUBs, t he necessary condition on\nRBDs isµ= 1. Hence our effort will be to construct RBDs with µ= 1. For this to happen,\nk≤sandr≤s+⌊s−1\nk−1⌋.\n3.1 Results relating to MOLS\nWe will now consider a class of Resolvable Block Designs ( X,A) such that |X|=s2, which\ncan be constructed from a set of w-MOLS(s). HereAconsists of blocks having constant\nsizes, which can be resolved into w+2 many parallel classes, each having smany blocks,\nsuch that blocks from two different parallel classes have exa ctly one element in common.\nThrough the following construction, we explain a simple and direct way to convert a set\nofwmany MOLS( s) into such an RBD( X,A).\nConstruction 1 To construct an RBD having w+ 2number of parallel classes from w-\nMOLS(s)using the following steps.\n1. DefineMref, which is a s×sarray where each cell consists of one of the elements from\nX={1,2,...,s2}, as follows:\nMref=\n1 2 3 ... s\ns+1 s+2 s+3 ...2s\n2s+1 2 s+2 2 s+3...3s\n...............\n(s−1)s+1 (s−1)s+2 (s−1)s+3... s2\n,Lk=\nlk\n11lk\n12... lk\n1s\nlk\n21lk\n22... lk\n2s\n............\nlk\ns1lk\nn2... lk\nss\n\n2. Consider a Latin Square Lk, from the set of w-MOLS(s). Let(Lk)ij=lk\nijas indicated\nabove.\n3. Corresponding to the Latin Square Lk, construct a parallel class Pkconsisting of sdisjoint\nblocksbk\nt, each of size sas follows,\nbk\nt={(Mref)ij:lk\nij=t},wherei,j∈ {1,2,...,s}.\nEach row of the the Latin Square is a permutation of {1,2,...,s}. Hence, there will be a\npair(i,j)in each row for which lk\nij=t. Thus, the blocks Pk\ntwill have a total selements,\none from each row and column of Mref. Sincet={1,2,...,s}, there will be sblocks. Thus\nwe are essentially collecting all the elements of Mrefcorresponding to a particular symbol\ntofLkin one block bk\nt, and together the blocks bk\niwherei={1,2,...,s}form a parallel\nclassPk.\n4. Repeat the above step for all Latin Squares in the set of w-MOLS(s), thereby giving wmany\nparallel classes.\n5. Construct two more parallel classes, one using the horizo ntal rows of Mref, and other using\nthe vertical rows of Mrefas follows:\nP0=/braceleftBig\n(1,2,...,s),(s+1,s+2,...,2s),...((s−1)s+1,(s−1)s+2,...,s2)/bracerightBig\nP∞=/braceleftBig\n(1,s+1,...,(s−1)s+1),(2,s+2,...,(s−1)s+2),...,(s,2s,...,s2)/bracerightBig\n6. TheRBD(X,A)withX={1,2,3,...,s2}andA={P0,P∞,P1,P2,...,Pw}is the desired\noutcome.\nLemma 3 A set ofwmany MOLS (s)can be used to construct an RBD (X,A)such that |X|=\ns2, consisting of constant block sizes, each having selements, that can be resolved into w+2\nmany parallel classes. Here, any two blocks from different par allel classes will have exactly one\nelement in common.Almost Perfect Mutually Unbiased Bases that are Sparse 9\nProofWeclaimthat,anypairofblocksfromdifferentparallelclas ses{P0,P∞,P1,P2,...Pw}\nconstructed above has exactly one element in common. Consid er thetthandsthblocks of\nkthandmthparallel classes respectively. Then\nPk\nt∩Pm\ns={(Mref)ij:lk\nij=t}∩{(Mref)ij:lm\nij=s}.\nNow since LkandLmare the orthogonal Latin Squares, there will be exactly one p air\n(i,j) such that, ( Lk)ij=tand (Lm)ij=s. Hence exactly one element will be common\nbetween the blocks Pk\ntandPms.\nFrom the construction of P0andP∞, it is clear that any block has exactly one element\nin common. Since any block of Pk\ntis picking one element from each row and each column\nofMref, each block of Pk\ntwill have exactly one element in common with blocks of P0and\nP∞, which are collection of horizontal rows ( P0) and vertical rows ( P∞) ofMref.\nWe will now sketch the idea of the above method with a simple ex ample to convert a\n2-MOLS(5) into 4 parallel classes.\nExample 1 Let us consider the 2-MOLS(5) and Mrefas follows.\nL1=\n5 1 2 3 4\n1 2 3 4 5\n2 3 4 5 1\n3 4 5 1 2\n4 5 1 2 3\n,L2=\n5 1 2 3 4\n2 3 4 5 1\n4 5 1 2 3\n1 2 3 4 5\n3 4 5 1 2\n,Mref=\n1 2 3 4 5\n6 7 8 9 10\n11 12 13 14 15\n16 17 18 19 20\n21 22 23 24 25\n.\nWe use a 5 ×5 Reference Matrix Mrefconsistingof elements indicatedby {1,2,...,25}.\nWe put themina simplerowwiseincreasingsequence, whichis toensurethateachelement\noccur only once in the matrix Mref. Now corresponding to each MOLS L1andL2, we\nconstruct a parallel class, as per the Construction 1, thereby forming blocks of P1andP2\nby picking elements from Mref.\nP1={(2,6,15,19,23),(3,7,11,20,24),(4,8,12,16,25),(5,9,13,17,21),(10,14,18,22,1)},\nP2={(2,10,13,16,24),(3,6,14,17,25),(4,7,15,18,21),(5,8,11,19,22),(1,9,12,20,23)}.\nThe remaining two parallel classes will be constructed usin g horizontal and vertical ele-\nments ofMrefas follows:\nP0={(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)},\nP∞={(1,6,11,16,21),(2,7,12,17,22),(3,8,13,18,23),(4,9,14,19,24),(5,10,15,20,25)}.\nLet us now consider the construction in the other way, i.e., t he converse.\nConstruction 2 Consider an RBD (X,A), where|X|=s2andAconsist ofw+ 2numbers\nof parallel classes such that, each block of a parallel class is of a constant size sand any pair\nof blocks from different parallel classes have exactly one ele ment in common. Let us denote\nthe elements of Xby{1,2,...,s2}, parallel classes by {P0,P∞,P1,...,Pw}, and the blocks of\nPlbybl\ni. Since there are sblocks in each parallel class, therefore Pl={bl\n1,bl\n2,...,bls}. Lets\ndistinct symbols for construction of the Latin Squares be de noted byY={y1,y2,...,ys}. Now\nconstructw-MOLS(s)using RBD (X,A)as follows.\n1. UseP0andP∞to construct a reference matrix Mrefhaving elements from {1,2,...,s2}\nin the following manner:\nMref=\nb0\n1∩b∞\n1b0\n1∩b∞\n2... b0\n1∩b∞s\nb0\n2∩b∞\n1b0\n2∩b∞\n2... b0\n2∩b∞s\n............\nb0s∩b∞\n1b0s∩b∞\n2... b0s∩b∞s\n.\nHereMrefcontains all the elements of Xexactly once. As any two blocks from different\nparallel classes have exactly one element in common, if (Mref)ij= (Mref)lmthenb0\ni∩b∞\nj=\nb0\nl∩b∞m, and that implies all the blocks {b0\ni,b0\nl,b∞\nj,b∞m}have one element in common. Here,\nb0\niandb0\nlare the blocks in the Parallel class P0. Similarly, b∞\njandb∞mare the blocks of the\nparallel class P∞. Since blocks in a Parallel class are mutually disjoint, thi s is not possible,\ni.e.,(Mref)ij/\\⌉}atio\\slash= (Mref)lm.10 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\n2. Corresponding to the parallel class Pk, construct the Latin Squares Lkas follows\nif(Mref)ij∈bk\ntthen(Lk)ij=yt.\nThat is, we are substituting yt, wherever the element of the block bk\ntis appearing in Mrefto\nconstructLk. Note that Lkis a Latin Square. Since X=bk\n1∪bk\n2...bks, for every (Mref)ij\nthere will be one bk\ntsuch that (Mref)ij∈bk\ntand sincebk\ni∩bk\nj=φ,i,j∈ {1,2,...,s},\nfor each (Mref)ij, there will be a unique bk\ntsuch that (Mref)ij∈bk\nt. Now ifLkis not a\nLatin square, then there would be at least a pair of (i1,i2)corresponding to which (Lk)i1j=\n(Lk)i2jor a pair of (j1,j2)corresponding to which (Lk)ij1= (Lk)ijj. Consider (Lk)i1j=\n(Lk)i2j. This implies if x1= (Mref)i1j=b0\ni1∩b∞\nj∈bk\ntandx2= (Mref)i2j=b0\ni2∩b∞\nj∈\nbk\nt. Thusx1andx2∈b∞\nj. Hence |bk\nt∩b∞\nj| ≥ |{x1,x2}|= 2asx1= (Mref)i1j/\\⌉}atio\\slash=\n(Mref)i2j=x2. This is a contradiction as there is exactly one element comm on between\nthe blocks of different parallel classes, here P∞andPk. Similarly it can be argued that\n(Lk)ij1/\\⌉}atio\\slash= (Lk)ijjfor any pair of (j1,j2).\n3. Repeat the above step for each of the parallel class Pk, k= 1,2,...,w, thereby constructing\nthe set ofwLatin squares viz {L1,L2,...,Lw}.\nThe converse of Lemma 3is as follows.\nLemma 4 Given an RBD (X,A), where|X|=s2and consisting of w+2many parallel classes\nsuch that, each block of a parallel class is of a constant size sand any pair of blocks from different\nparallel classes have exactly one element in common. Then RB D(X,A)can be used to construct\nw-MOLS(s).\nProofWe claim that the set of Latin squares L1,L2,...,Lwas constructed above are\nMutually Orthogonal Latin Squares of order s.\nRecalling that a pair of Latin Squares L1andL2of same order and constructed\nfrom the entries from the same set Yis called mutually orthogonal, if the ordered pair\n((L1)ij,(L2)ij)∈ {(Y,Y)}appears exactly once.\nConsider the ordered pair, (( Lk)ij,(Lm)ij). Assume that, LkandLmare not Mutually\nOrthogonal Latin Squares, which implies that, there would b e at least one pair of Indices\n{(i,j),(u,v) : (i,j)/\\⌉}atio\\slash= (u,v)}such that, (( Lk)ij,(Lm)ij) = ((Lk)uv,(Lm)uv) = (yp,yq). Let\nyp= (Lk)ij∈bkpandyq= (Lm)ij∈bmq. This implies ( Mref)ij∈bkpandbmq. Similarly, yp=\n(Lk)uv∈bkpandyq= (Lm)uv∈bmqwhich implies ( Mref)uv∈bkpandbmq. However, then we\nwill have,bkp∩bmq={(Mref)ij,(Mref)uv}, but if (i,j)/\\⌉}atio\\slash= (u,v) then (Mref)ij/\\⌉}atio\\slash= (Mref)uv.\nThus,|bkp∩bmq| ≥2 which contradicts the fact that |bkp∩bmq|= 1. Hence we conclude that\nLkandLmare Mutually Orthogonal Latin Squares.\nWe now sketch the above method with an example to obtain a 2-MO LS(5) from 4\nparallel classes.\nExample 2 If we proceed with the same set of 4 parallel classes obtained as in Example\n1, it would naturally result into the same pair of MOLS(5), i.e .,L1andL2, with which\nwe have started Example 1. Therefore, we consider a different set of 4 parallel classes as\nfollows:\nP0={(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)},\nP∞={(1,6,11,16,21),(2,7,12,17,22),(3,8,13,18,23),(4,9,14,19,24),(5,10,15,20,25)},\nP1={(1,9,12,20,23),(2,10,13,16,24),(3,6,14,17,25),(4,7,15,18,21),(5,8,11,19,22)},\nP2={(1,7,13,19,25),(2,8,14,20,21),(3,9,15,16,22),(4,10,11,17,23),(5,6,12,18,24)}.\nNote that, P0andP∞are taken as above for convenience, providing the Mrefwith\nelements from X={1,2,...,25}. Observe that any pair of blocks from different parallel\nclasses have exactly one element in common. Now correspondi ng tos= 5, we simply useAlmost Perfect Mutually Unbiased Bases that are Sparse 11\nY={1,2,3,4,5}as five symbols to construct the Latin square. The remaining p arallel\nclassesP1andP2are used to construct L1andL2respectively, which are the required\n2-MOLS(5). Following the Construction 2above, we obtain Orthogonal Latin Squares L1\nandL2as follows:\nMref=\n1 2 3 4 5\n6 7 8 9 10\n11 12 13 14 15\n16 17 18 19 20\n21 22 23 24 25\n, L1=\n1 2 3 4 5\n3 4 5 1 2\n5 1 2 3 4\n2 3 4 5 1\n4 5 1 2 3\n, L2=\n1 2 3 4 5\n5 1 2 3 4\n4 5 1 2 3\n3 4 5 1 2\n2 3 4 5 1\n.\nBased on this we now proceed for our generic construction ide a in the next section.\n4 Definition of APMUBs and general construction ideas\nIn this section we first present the motivation of proposing s uch a combinatorial object\nand then proceed with the general construction ideas.\n4.1 Motivation for defining APMUBs and its Characteristics\nConsiderapairoforthonormalbases,say MlandMm.Iftheyare β-AMUBsthen |/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht| ≤\nβ√\nd, whereβis bounded by some constant. The definition of β-AMUBs does not rule out\n|/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|= 0. Let for n1pairs, the value of |/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|be 0, and for the remaining pairs\nn2, the value of |/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|be non-zero ( /\\⌉}atio\\slash= 0). If |/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|=βij√\nd(say), thenβij√\nd≤β√\nd.\nSince/summationtextd\nij|/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|2=d. Therefore,\nn1·0+d/summationdisplay\ni,j=1\n|/angbracketleftψl\ni|ψm\nj/angbracketright|/negationslash=0/parenleftbiggβij√\nd/parenrightbigg2\n=d⇒n2β2\nmin\nd≤d≤n2β2\nmax\nd,\nwhere,βmin= minijβijandβmax= maxijβijThis implies,\nn2β2\nmin\nd2≤1≤n2β2\nmax\nd2.\nNote that, n1+n2=d2⇒n1\nd2= 1−n2\nd2, then we have,\n1−1\nβ2\nmin≤n1\nd2≤1−1\nβ2max. (2)\nNote that,n1\nd2is the probability of randomly selecting two orthogonal vec tors from two\ndifferent bases. Therefore, if βmin=β=βmax, i.e.,∆={0,β√\nd}andβ= 1+O(d−λ),\nlambda > 0, then,n1\nd2=O(d−λ), λ >0. Hence, with these conditions on ∆andβ, the\nprobability of any pair randomly selected vectors from diffe rent orthonormal bases being\northogonal, tends to 0. Similarly, the probability that the angle between them isβ√\ndtends\nto 1 asymptotically. Since β→1, asdincreases, the bases of APMUBs would behave like\nMUBs in this sense.\nFurther note that, the vectors from a set of MUBs form a set of B i-angular vectors as\n∆={0,1√\nd}. Now if we restrict ∆={0,β√\nd}, then basis vectors of AMUBs would form set\nof Bi-angular vectors. Thus analysis of such AMUBs would als o shed light on the study of\nBi-angular vectors that has close connections with Weighin g Matrices, Error Correcting\nCodes, Orthogonal spreads, Frame theory, Association Sche mes etc. [ 14,7,8,34,52,32,17,\n33,43].\nWith this motivation, let us define APMUBs, which is the main f ocus of this paper.12 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nDefinition 7 The set M={M1,M2,...,Mr}will be called Almost Perfect MUBs (AP-\nMUBs) if∆=/braceleftBig\n0,β√\nd/bracerightBig\n, i.e.,the set containsjust twovalues, such that β= 1+O(d−λ)≤2,\nλ>0. When the bases are real, we call them Almost Perfect Real MU Bs (APRMUBs).\nIf the vectors are understood as states of a quantum system th en the absolute value of\nan inner product essentially indicates the overlap between these states. Hence randomly\npicking two quantum states corresponding to different basis of above set of APMUBs,\nthe overlap will have magnitude equal toβ√\ndwith probability almost 1. With negligible\nprobability it will be 0 which corresponds to the quantum sat es being orthogonal.\nThe sets of basis vectors of APMUB are Bi-angular as they are e ither orthogonal or\nhave constant absolute value of inner product, i.e., ∆={0,β√\nd}. Hence APMUBs form set\nof Bi-angular vectors with 0 being one of the value of inner pr oduct. Thus, for any pair of\nAPMUBs in Cd(orRd) we have the following lemma.\nLemma 5 Consider any pair of APMUBs in Cdwith∆={0,β√\nd}. Then any basis vector of\nan APMUB will be orthogonal to (1−1\nβ2)×dmany basis vectors of another APMUB and will\nbe at angleβ√\ndwith remainingd\nβ2many basis vectors.\nProofLetMlandMnbe any pair of APMUBs over Cdand let {|ψl\ni/a\\}⌊ra⌋k⌉tri}ht:i= 1,2,...,d}\nand{|ψm\nj/a\\}⌊ra⌋k⌉tri}ht:j= 1,2,...,d}be the corresponding basis vectors. Expressing |ψl\ni/a\\}⌊ra⌋k⌉tri}htas a linear\ncombination of {|ψm\nj/a\\}⌊ra⌋k⌉tri}ht:j= 1,2,...,d}, we get,\n|ψl\ni/a\\}⌊ra⌋k⌉tri}ht=αi1|ψm\n1/a\\}⌊ra⌋k⌉tri}ht+αi2|ψm\n2/a\\}⌊ra⌋k⌉tri}ht+...+αid|ψm\nd/a\\}⌊ra⌋k⌉tri}ht,\nwhereαij=/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht. Since the bases consist of unit vectors, i.e., /a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψl\ni/a\\}⌊ra⌋k⌉tri}ht= 1∀l,ihence,\n/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψl\ni/a\\}⌊ra⌋k⌉tri}ht=|αi1|2+|αi2|2+...+|αid|2=d/summationdisplay\nj=1|/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|2= 1.\nSince∆={0,β√\nd}, let us assume that |ψl\ni/a\\}⌊ra⌋k⌉tri}htis orthogonal, i.e., |/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|= 0 with t1many\nbasis vectors of Mm, and make an angle ofβ√\ndwith remaining t2=d−t1many basis\nvectors ofMm. Hence we have,\nd/summationdisplay\nj=1|/a\\}⌊ra⌋k⌉tl⌉{tψl\ni|ψm\nj/a\\}⌊ra⌋k⌉tri}ht|2= 1⇒t1·0+t2×/parenleftbigg\nβ√\nd/parenrightbigg2\n= 1⇒t1=/parenleftbigg\n1−1\nβ2/parenrightbigg\n×dandt2=d\nβ2.\nSince|ψl\ni/a\\}⌊ra⌋k⌉tri}htis arbitrary basis vector of Ml, hence the result.\nThe above result tells that β2can only be a rational number. Further, we have the\nfollowing corollary considering all the vectors in the set o f APMUBs.\nCorollary 1 Consider a set of rmany APMUBs on dimension dwith∆={0,β√\nd}, that will\nproducer×dvectors. In this set, each vector will have (d−1)+(r−1)(1−1\nβ2)dmany vectors\nas its orthogonal and (r−1)d\nβ2many vectors having the dot productβ√\nd.\nThe main contribution here is to show that one can construct O(√\nd) many APMUBs\nwith values of βslightly more than one. This says that our method can provide O(d3\n2)\nmany Bi-angular vectors where the dot product values are 0 an dβ√\nd. While there are\nconstructions of O(d2) Bi-angular vectors [ 53], but the angles we obtain with our methods\nare quite high than the existing constructions. Further we a chieve large sparsity and non-\nzero components of equal magnitude. This is related to coher ence property of unit norm\nvectors. This shows that the construction of APMUBs may prod uce interesting results in\nrelated domain. Further note that any set of Orthonormal Bas is vectors always form UnitAlmost Perfect Mutually Unbiased Bases that are Sparse 13\nNorm Tight Frames (UNTF). For a brief introduction on Frame t heory and its application\nin Hilbert space one may refer to [ 19,18]. Thus the basis vectors of the set of APMUBs\nalso constitute a Unit Norm TightFrames apart from being Bi- angular,whereas in general\nthe Bi-angular vectors constructed in [ 53] do not constitute tight frame. To see this, note\nthat since APMUBs are orthonormal basis vectors, hence any a rbitrary vector |u/a\\}⌊ra⌋k⌉tri}htcan\nbe uniquely expressed in terms of each of the APMUBs. Thus in t his context also the\nconstruction of APMUBs may be of independent interest.\n4.1.1 Connecting with Mutually Unbiased Weighing Matrices\nLet us now demonstrate how the construction of APMUBs bears i mplications to the exis-\ntence of Mutually Unbiased Weighing Matrices (MUWM).\nDefinition 8 LetW1andW2be a pair of weighing matrices of order dand weight w.\nIfW†\n1W2is again a weighing matrix with order dand weight w, then the pair is called\nmutually unbiased weighing matrices (MUWM). Moreover, let W={W1,W2,...,Wr}be\na set of weighing matrices such that every pair is mutually un biased. Then Wis referred\nto as a set of mutually unbiased weighing matrices. If Wconsists solely of real weighing\nmatrices, it is called a set of mutually unbiased real weighi ng matrices (MURWM).\nNote that, the MUWMs generalize mutually unbiased Hadamard matrices. The study\nof mutually unbiased weighing matrices of small orders has b een conducted in [ 8], where\ncomputer searches and some analyticalmethods were predomi nantly employed. Moreover,\nin[38],mutuallyunbiasedrealweighingmatriceshavebeen used t ostudythe binarycodes.\nIn this regard, we like to underline the following technical result.\nLemma 6 The existences of the following combinatorial objects are e quivalent:\n1.rmany APMUBs with ∆=/braceleftBig\n0,β√\nd/bracerightBig\n, and\n2.(r−1)many mutually unbiased weighing matrices of order dand weightd\nβ2.\nProof(1)⇒(2): Let {M1,M2,...,Mr}be a set of rmany APMUB. Choose any weighing\nmatrix, say M1and consider the set/braceleftBig\nM†\n1M1,M†\n1M2,...,M†\n1Mr/bracerightBig\n={I,W2,W3,...,Wr}.\nSinceMi’s are unitary matrices, Wiare also unitary. Moreover, since M1andMiare\nAPMUB, the elements of Wi=M†\n1Miare from the set/braceleftBig\n0,βexp(iθ)√\nd/bracerightBig\nwhereθ∈R. Hence,\nWiis a weighing matrices with weightd\nβ2.\nFurther,W†\niWj= (M†\n1Mi)†(M†\n1Mj) =M†\niM1M†\n1Mj=M†\niMj. SinceMiandMjare\nAPMUB, the elementsof M†\niMjare fromthe set/braceleftBig\n0,βexp(iθ)√\nd/bracerightBig\nwhereθ∈R, makingW†\niWj\na weighing matrices with weightd\nβ2. Hence,WiandWjare mutually unbiased weighing\nmatrices,foranypairof Wi,Wj.Thus{W2,W3,...,Wr}isasetof(r−1)mutuallyunbiased\nweighing matrices of weightd\nβ2.\n(2)⇒(1) The set of ( r−1) mutually unbiased weighing matrices along with identity\nmatrix (I), constitute the set of rmany APMUB.\nIn the lemma above, when we restrict the matrices to APRMUB, w e obtain a set\nof mutually unbiased real weighing matrices (MURWM). It’s a lso noteworthy that this\nlemmaparallelsthe connectionbetween MUBs and mutuallyun biasedHadamardmatrices\n(MUHM), where rMUBs are equivalent to ( r−1) MUHMs and vice versa.\n4.2 Our general construction ideas\nLet us now refer to the construction method of orthonormal ba ses using RBDs as given\nin [47, Section 3]. The constructionidea of [ 47] is generic in nature, where unitary matrices14 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\ncan be employed to construct orthogonal bases correspondin g to each parallel class of an\nRBD. However, here we will confine ourselves to the choice of H adamard Matrices (which\nare special kind of unitary matrices) for constructing orth onormal bases from parallel\nclasses of an RBD. This will enable us to bound βas per [47, Theorem 1], which is\nrequired for constructing APMUBs with good parameters. The order of the Hadamard\nmatrices used in this construction must be same as the block s ize of each parallel class.\nWith this method in place for constructing the set of orthono rmal bases from RBD, our\nwork here primarily focuses on constructing suitable RBDs, and consequently analyzing\nthe parameters ∆,βandǫof corresponding AMUBs constructed from them.\nWe focus on constructing RBDs, having constant block size. T he constant block size is\nessential if we want to use Hadamard matrices of same order (r eal or complex) for all the\nblocks of the RBD. This is required as it renders all the compo nents of the basis finally\nconstructed to be either zero or of a constant magnitude (1√\nk), which is the normalizing\nfactor of each basis vector. We will examine when they can sat isfy the conditions of AP-\nMUBs (APRMUBs). Our construction method results into vecto rs which are vary sparse\nwith the non-zero components of constant magnitude. This pr ovides large sets of both real\nas well as complex AMUBs for the dimensions where it is known t hat not more that two\nor three real MUBs exist.\nWe first present a generic result, which is dependent on the ex istence of suitable RBDs.\nThereafter, we explore the methods to construct such RBDs. W e further show that if an\nRBD satisfy µ= 1, then it will result into an APMUB. In fact, µis the most critical pa-\nrameter which we control in the constructionof RBDs. In all t he constructionsof AMUBs,\nthe number of elements in an RBD, i.e., |X|can be increased without bound whereas, the\nparameter µremainsconstant.All our constructionswillhave this prop erty,which justifies\nasymptotic analysis of the parameters for AMUBs thus constr ucted.\nTheorem 1 Consider an RBD (X,A)with|X|=d= (q−e)(q+f), withq,e,f∈N. If\nthe said RBD (X,A)consists of rparallel classes, each having blocks of size (q−e), and if\n0≤(e+f)≤/parenleftBig\nc2−µ2\nµc/parenrightBig\nd1\n2wherecis some constant, then one can construct rmanyβ-AMUBs\nin dimension d, whereβ=µ/radicalBig\nq+f\nq−e≤candǫ= 1−1\nq+f. Whenµ= 1andc= 2, we will\ngetrmany APMUBs with β= 1 +O(d−1\n2)≤2and∆={0,1\nq−e}. Further, if there exists\na real Hadamard matrix of order (q−e), we can construct rmany APRMUBs with the same\nparameters.\nProofWe have |X|= (q−e)(q+f) with each parallel class having the block size k= (q−e).\nHereµis the maximum number of elements that are common between two blocks from\ndifferent parallelclasses. Following[ 47, Theorem 1], this implies that, β=µ√\nd\nq−e=µ/radicalBig\nq+f\nq−e.\nFurther, using the relation d=q2+(f−e)q−ef, we obtain β=µ(√1+x2+x), where\nx=e+f\n2√\nd. From the definition of β-AMUBs,βmust be bounded for all values of d. Let this\nbound for βbec, thenµ(√1+x2+x)≤c⇒0≤(e+f)≤/parenleftBig\nc2−µ2\nµc/parenrightBig\nd1\n2. This inequality\ncan be restated in terms of qas 0≤(c2e+µ2f)≤(c2−µ2)q, which is the condition for β\nbeing bounded above by the constant c.\nIn order to see the asymptotic variation of βin terms of q, we consider the expansion\nof terms as follows:\nβ=µ/parenleftbigg\n1+e+f\n2q+(e+f)(3e−f)\n23q2+(e+f)(5e2−2ef+f2)\n24q3+.../parenrightbigg\n.(3)\nTo understand the asymptotic variation of βin terms of d, we again use the relation\nd=q2+(f−e)q−efto express qin terms of√\ndand thereafter, expanding the expression\nforβ=µ/radicalBig\nq+f\nq−e=µ(√1+x2+x), wherex=f+e\n2√\nd, in terms of negative power of√\ndandAlmost Perfect Mutually Unbiased Bases that are Sparse 15\nwe obtain\nβ=µ/parenleftbigg\n1+e+f\n2√\nd+(e+f)2\n23d−(e+f)4\n27d2+(e+f)6\n210d3−5(e+f)8\n215d4+.../parenrightbigg\n.(4)\nThus, for a given eandf, for large d(orq), asymptotically β=µ+O(1\nq) =µ+O(1√\nd).\nTherefore, if µ= 1, the construction yields APMUBs, provided β≤2⇒c= 2 as we have\nseen above that βis bounded by c. And in this situation we get 0 ≤(e+f)≤3\n2d1\n2.\nTo get the values of the set ∆, note that when µ= 1, there is maximum one element\ncommon between any pair of blocks from different parallel cla sses. And since Hadamard\nmatrices are used for constructing orthonormal bases, thus |/a\\}⌊ra⌋k⌉tl⌉{tu|v/a\\}⌊ra⌋k⌉tri}ht|=1\nq−ecorresponding\nto the situations when one element is common between the pair of blocks and |/a\\}⌊ra⌋k⌉tl⌉{tu|v/a\\}⌊ra⌋k⌉tri}ht|= 0\ncorresponding to situation, when no elements are common bet ween the pair of blocks.\nThus∆={0,1\nq−e}.\nTo calculate sparsity, note that for each vector constructe d from a block of size k, we\nwill have exactly kmany non-zero and d−kmany zero entries, hence\nǫ=d−k\nd= 1−k\nd= 1−q−e\n(q+f)(q−e)= 1−1\nq+f.\nIf a real Hadamard matrix of order ( q−e) exists, we can exploit it to obtain rmany\nreal approximate MUBs in Rd, with same values of the parameters β,∆andǫ.\nIf the constructionin [ 47] is to be used for APMUBs, then µshould be 1. Thus µ, which\nis the maximum number of elements common between any pair of b locks from different\nparallel classes, is the most critical parameter here. Furt her note that, µis always greater\nthan or equal to1, hence, a very limitedkinds of RBDs can be us ed to constructAPMUBs.\nAs per Lemma 2,µ≥ ⌈k\ns⌉. Thus, for µ= 1, an RBD having a constant block size must\nhavek≤s, i.e., the block size must not be greater than the number of bl ocks in the\nparallel class. In this connection, we have noted that an RBD constructed using MOLS\nhaveµ= 1. In fact, between any pair of blocks from different paralle l classes, in such an\nRBD, there is exactly one element in common.\nIn our above theorem, we have |X|=d= (q−e)(q+f), where number of elements\nin a block is ( q−e), i.e.,k= (q−e), and number of blocks in a parallel class is ( q+f),\nhences= (q+f). The reason we are expressing it like this will be clear in Th eorem3\nwhere we demonstrate the construction of such an RBD. Since eandfare bounded by\na positive integer, if e≥f, it will ensure that k≤s. Sinceβ=µ/radicalbigs\nk=µ/radicalBig\nq+f\nq−e, hence\nfor larged= (q−e)(q+f) we obtain β→µ, which is also evident from the asymptotic\nexpansion of βabove.\nFor|X|=d= (q−e)(q+f), we can have an RBD, where the block size is ( q+f), hence\nhaving (q−e) blocks in each parallel class. However, in such a situation ,µ >1 Lemma\n2, and hence we cannot get APMUBs. However, they can provide AM UBs as in [ 47]. The\nresult of [ 47, Theorem 4] is a particular case of this situation, with e= 0,f= 1 andµ= 2.\nIn this case, q+1 many parallel classes are there in an RBD, each having a con stant block\nsize of (q+1). In this case, β= 2/radicalBig\nq\nq+1= 2−O(1√\nd), i.e., though the maximum value of\nthe inner product was slightly less than2√\nd, but asymptotically βconverges to 2. Further\n∆is also not two-valued. Thus the construction did not satisf y the conditions needed for\nAPMUBs which is β= 1+O(d−λ), for some λ>0 and∆being the set consisting of just\ntwo elements with one being 0.\nThus, in order to obtain APMUBs, the RBDs in use must have µ= 1 and all the block\nsizes must be same. With this understanding, we will explore more suitable designs in the\nfollowing sections, so that the upper bound on the absolute i nner product values can be\nimproved than the results presented in [ 47] to obtain APMUBs.16 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\n5 Exact constructions of APMUBs through RBDs\nAs followed from previous section, an RBD having constant bl ock size must have µ= 1,\nin order to obtain APMUBs. In this section we explain the cons tructions of such RBDs.\nSince our focus is to build APMUBs in composite,we concentra teond=k·s, and consider\ntwo categories.\n–The first construction, being generic in nature, will work fo r any composite d=k·s=\n(s−e)swith 0≤e≤3\n2d1\n2. Here the number of APMUBs is at least N(s) + 1 when\ne>0 andN(s)+2 when e= 0.\n–The second one is considered when dcan be expressed as ( q−e)(q+f),0< f≤e,\nwhereqis some power of prime. Here the number of APMUBs is at least ⌊q−e\nf⌋+1.\nThese are presented in the Sections 5.1and5.2respectively as below. Thus here\nour approach to is to obtain large numbers of APMUBs, if the co mposite dimension d\ncan be expressed in some generic form. In the first category ou r starting point is w-\nMOLS(s) and in the second category we initiate with ( q2,q,1)-ARBIBD. In each case, we\nwill first demonstrate the construction with an example, the n outline the algorithm for\nthe respective construction and then provide the proof of co rrectness.\n5.1d=k·s= (s−e)s, 0≤e≤3\n2d1\n2\nLet us first demonstrate the method by explicitly constructi ng RBD(X,A) with|X|=\n2·5 = 10, i.e., here s= 5 andk= 2.\n1. To begin with, consider the following 4-MOLS(5) and the Mref:\nLS1=\n1 2 3 4 5\n5 1 2 3 4\n4 5 1 2 3\n3 4 5 1 2\n2 3 4 5 1\n,LS2=\n1 2 3 4 5\n4 5 1 2 3\n2 3 4 5 1\n5 1 2 3 4\n3 4 5 1 2\n,LS3=\n1 2 3 4 5\n3 4 5 1 2\n5 1 2 3 4\n2 3 4 5 1\n4 5 1 2 3\n,\nLS4=\n1 2 3 4 5\n2 3 4 5 1\n3 4 5 1 2\n4 5 1 2 3\n5 1 2 3 4\n,Mref=\n1 2 3 4 5\n6 7 8 9 10\n11 12 13 14 15\n16 17 18 19 20\n21 22 23 24 25\n.\n2. Using Construction 1, we construct RBD( ¯X,¯A), with|¯X|= 25 and ¯Ahaving 6 parallel\nclasses. We used Mrefas shown above to obtain the following RBD. We collect blocks\nfrom each parallel class, in a 5 ×5 matrix, where each row represents one block of the\nparallel class, and index the blocks as ¯bl\ni, wherelrepresents the index of parallel class\nandirepresents the block number within the parallel class.\n¯P1=\n¯b1\n5={1 7 13 19 25 }\n¯b1\n4={2 8 14 20 21 }\n¯b1\n3={3 9 15 16 22 }\n¯b1\n2={4 10 11 17 23 }\n¯b1\n1={5 6 12 18 24 }\n,¯P2=\n¯b2\n5={1 8 15 17 24 }\n¯b2\n4={2 9 11 18 25 }\n¯b2\n3={3 10 12 19 21 }\n¯b2\n2={4 6 13 20 22 }\n¯b2\n1={5 7 14 16 23 }\n,\n¯P3=\n¯b3\n5={1 9 12 20 23 }\n¯b3\n4={2 10 13 16 24 }\n¯b3\n3={3 6 14 17 25 }\n¯b3\n2={4 7 15 18 21 }\n¯b3\n1={5 8 11 19 22 }\n,¯P4=\n¯b4\n5={1 10 14 18 22 }\n¯b4\n4={2 6 15 19 23 }\n¯b4\n3={3 7 11 20 24 }\n¯b4\n2={4 8 12 16 25 }\n¯b4\n1={5 9 13 17 21 }\n,\n¯P∞=\n¯b5\n5={1 6 11 16 21 }\n¯b5\n4={2 7 12 17 22 }\n¯b5\n3={3 8 13 18 23 }\n¯b5\n2={4 9 14 19 24 }\n¯b5\n1={5 10 15 20 25 }\n,¯P0=\n¯b6\n5={1 2 3 4 5 }\n¯b6\n4={6 7 8 9 10 }\n¯b6\n3={11 12 13 14 15 }\n¯b6\n2={16 17 18 19 20 }\n¯b6\n1={21 22 23 24 25 }\n.\nHere¯A={¯P1∪¯P2∪¯P3∪¯P4∪¯P0∪¯P∞}. Note that, any pair of blocks from different\nparallel classes has exactly one element in common.\n3. Now we remove any 3 blocks from the parallel class ¯P0, say{¯b6\n1,¯b6\n2,¯b6\n3}.\n4. Next we remove the elements contained in this block from th e entire design ( ¯X,¯A).\n5. Then we discard ¯P0from the design. Here we get RBD( X,A) consisting of 5 parallel\nclasses, each having 5 blocks of size 2, where X={1,2,3,4,5,6,7,8,9,10}andA=\n{P1∪P2∪P3∪P4∪P∞}. Explicitly, we have,Almost Perfect Mutually Unbiased Bases that are Sparse 17\nP1=\nb1\n5={1 7}\nb1\n4={2 8}\nb1\n3={3 9}\nb1\n2={4 10}\nb1\n1={5 6}\n,P2=\nb2\n5={1 8}\nb2\n4={2 9}\nb2\n3={3 10}\nb2\n2={4 6}\nb2\n1={5 7}\n,P3=\nb3\n5={1 9}\nb3\n4={2 10}\nb3\n3={3 6}\nb3\n2={4 7}\nb3\n1={5 8}\n,\nP4=\nb4\n5={1 10}\nb4\n4={2 6}\nb4\n3={3 7}\nb4\n2={4 8}\nb4\n1={5 9}\n,P∞=\nb5\n5={1 6}\nb5\n4={2 7}\nb5\n3={3 8}\nb5\n2={4 9}\nb5\n1={5 10}\n.\nNote that, for this particular case using (10 ,2,1)-BIBD, one can construct an RBD\nwith 9 parallel classes each having 5 many blocks of constant block size 2. One must note\nthat this construction may not provide RBDs having maximum n umber of parallel classes,\nwithconstantblocksizeand µ= 1.Evenif weincludetheparallelclass P0=¯P0\\{¯b6\n1,¯b6\n2,¯b6\n3}\nin the RBD( X,A), it will not change the value of µwhich will remain equal to 1. However,\nthe block size of P0will be 5 then and hence RBD(X,A) will contain two different bl ock\nsizes. Therefore we discard the P0. Nevertheless we will see that even P0can be used to\nconstruct orthonormal basis which will be mutually unbiase d with all the orthonormal\nbasis constructed using {P1∪P2∪P3∪P4∪P∞}.\nThe technique is more formally explained for the general cas e in Construction 3below.\nConstruction 3 Letd=k·s= (s−e)s, with 0<e≤s.\n1. Using Construction 1, construct RBD(¯X,¯A), where ¯X={1,2,...,s2}. It will have\nr=N(s) + 2 many parallel classes, namely {¯P1,¯P2,...,¯Pw,¯P0,¯P∞}, each having s\nmany blocks of constant size s. Denoting blocks of the parallel class ¯Plwith¯bl\ni, for\ni= 1,2,...,s, we note that between any two blocks from different parallel c lasses,\nthere is exactly one element in common, i.e., |¯bl\ni∩¯bm\nj|= 1,∀l/\\⌉}atio\\slash=m.\n2. Pick a parallel class, say ¯P0. Removeemany blocks from it and denote as S={¯b0\n1∪\n¯b0\n2∪...∪¯b0e}.\n3. Removethe elementsin Sfrom¯Xand let us denote the new set with X, i.e.,X=¯X\\S.\nFurther, we remove the elements in Sfrom the parallelclasses {¯P2,¯P3,...,¯Pw,¯P∞}and\ndenote them by Pl, forl= 2,3,...,r, i.e.,Pl=¯Pl\\S. ThenA={P2,P3,...,Pw,P∞}\nwithPl={bl\n1,bl\n2,...,blq}, wherebl\ni=¯bl\ni\\S.\n4. Discard the parallel class ¯P0. The resulting RBD( X,A) is the required design. For\nconvenience, rename the elements from 1 to ( s−e)s.\nWe claim that the above design ( X,A) is an RBD, such that |X|= (s−e)sandA\nconsistofN(s)+1 manyparallelclasses,i.e., A={P2,P3,...,Pw,P∞}, eachhaving smany\nblocks, i.e., Pl={bl\n1,bl\n2,...,bls},l= 1,2,...,s, each of size ( s−e), i.e.,|bl\ni|= (s−e)∀i,l,\nsuch that blocks from different parallel classes have at most one element in common, i.e.,\n|bl\ni∩bm\nj| ≤1∀l/\\⌉}atio\\slash=m. We formalize this in the form of a lemma below.\nLemma 7 Letd= (s−e)sfors,e∈Nwith0<e≤s. Then one can construct an RBD (X,A),\nwith|X|=dhaving constant block size (s−e)withµ= 1, and having N(s)+1many parallel\nclasses, where N(s)is the number of MOLS (s).\nProofRefer to Construction 3above. In RBD( ¯X,¯A) any pair of blocks from different\nparallelclassesisofsize sandhasexactlyoneelementincommon,i.e., |¯bl\ni∩¯bm\nj|= 1∀l/\\⌉}atio\\slash=m.\nHence removalof the elements S={¯b1\n1∪¯b1\n2∪...∪¯b1e}from entire design will discard exactly\neelements from each block ¯bl\ni,l/\\⌉}atio\\slash= 1. Hence, the blocks bl\ni=¯bl\ni\\Swill be of constant size\n|bl\ni|=s−eand|bl\ni∩bm\nj| ≤1∀l/\\⌉}atio\\slash=m.\nNow we can use this RBD( X,A) to construct APMUBs in dimension d=|X|= (s−e)s\nfollowing Theorem 1.18 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nTheorem 2 Letd= (s−e)sfors,e∈Nwith0< e≤3\n2d1\n2. Then there exist N(s) + 1\nmany APMUBs with ∆={0,1\ns−e}andβ=/radicalBig\ns\ns−e= 1 +O(d−λ)≤2, whereλ=1\n2and\nsparsityǫ= 1−1\ns. Further, if there exists a real Hadamard matrix of order (s−e), then we\ncan construct N(s)+1many APRMUBs with the same parameters. For the case e= 0, there\nexistN(s)+2many MUBs, and if there exists a real Hadamard matrix of order s, then we can\nconstructN(s)+2many Real MUBs.\nProofIn order to show that we can produce such number of APMUBs, let us consider an\nRBD(X,A) with|X|= (s−e)shavingN(s)+1 parallelclasses of constantblock size ( s−e)\nsuch that between the blocks from different parallel classes , there is at most one element\nin common, and hence µ= 1. This follows from Lemma 7. Then using this RBD along\nwith a Hadamard matrix of order ( s−e), we can construct orthonormal bases following\nTheorem 1, with the values of ∆,β,ǫby substituting q=s, andf= 0. Further, the\ncondition that β≤2 for APMUB gives e≤3\n2d1\n2. In terms of sthis inequality becomes\ne≤3\n4sand in terms of k= (s−e), the inequality becomes s≤4kso thatβ≤2. The\nparameters of APMUBs are ∆={0,1\ns−e},β=/radicalBig\ns\ns−e= 1+O(d−1\n2)≤2 andǫ= 1−1\ns.\nSince the number of parallel classes is one more than number o f Mutually Orthogonal\nLatin Squares of Order s, we haver=N(s)+1.\nIn the situation when d=s2, i.e.,e= 0, using Construction 1above, we obtain\nRBD(X,A) havingN(s)+2 parallel classes, such that any pair of blocks have exact ly one\npoint in common. Now using this RBD( X,A), along with a Hadamard matrix of order\ns, we can construct orthonormal bases following Theorem 1which will provide N(s)+2\nmany MUBs. Hence for e= 0, the result follows directly. Construction 3is applied only\nwhene>0. Thus when d=s2, we getN(s)+2 MUBs, and when real Hadamard matrix\nof ordersis available, that can be used to construct N(s)+2 real MUBs.\nRemark 1 Note that we can construct an orthonormal basis correspondi ng to the parallel\nclassP0=¯P0\\S, again having ( s−e)selements. These basis vectors will be mutuallyunbi-\nasedwithalltheorthonormalbasesconstructedusingthepa rallelclasses P1,P2,...Pw,P∞.\nHowever, if we include it in the set of orthonormal bases then ∆={0,1√\nd,β√\nd}will have\nthree values, and the condition of APMUB will not be satisfied . Thus we ignore P0, even\nthough it provides an orthonormal basis which is mutually un biased with all the bases\nconstructed above.\nFor a composite s,N(s)→ O(s1\n14.8)≪s−1, which is an upper bound [ 1,64,68]. Thus in\ncasedcanbe expressedas d=k·s= (s−e)s,wheres,e∈Nwithk<s≤4k(or0<e≤3\n4s),\nwe can always construct N(s)+1 many APMUBs. We refer to this as Mutually Orthogonal\nLatin Square Lower Bound construction for APMUBs. For example, given d= 22·32·5·7,\n–this can be factored as d= 35·36, which will provide N(36)+1 = 9 many APMUBs\nwithβ= 1.01,\n–ord= 30·42 which will give N(42)+1 = 6 many APMUBs with β= 1.18,\n–ord= 28·45, which will give N(45)+1 = 7 many APMUB with β= 1.27.\nHere,N(36) = 8,N(42) = 5and N(45)= 6 arethepresentlyknownvaluesofthemaximum\nnumberofMOLS ofthese orders[ 1].Letus nowexplainthe significanceofourconstruction\nmethod through this example.\n–The number of complex MUBs for this d= 22·32·5·7 which can be constructed using\nprime power decomposition, and then taking tensor product, would be 22+ 1 = 5.\nThis is the lower bound and there is no better known result tha n this in the number of\nMUBs in this dimension. The value of βis 1 in this case as exact MUBs are referred.\n–Expressing d= 35·36, we have more number of APMUBs (9 many) than MUBs, with\nβ= 1.01.\n–Further, expressing d= 28·45, we can get 7 many APRMUBs with β= 1.27. This is\nbecause, we have real Hadamard matrix on the dimension 4 ·7 = 28.Almost Perfect Mutually Unbiased Bases that are Sparse 19\nOur Theorem 2can be compared with that of [ 68, Theorem 3], where the result could\nbe achieved using ( k,s)-nets. This, in turn, can be constructed from Mutually Orth ogonal\nLatin Squares of order s. Thus, for a square dimension d=s2, there would be N(s)+2\nmany MUBs. Moreover,the result in [ 47, Corollary3] points out that using RBDs, one can\nconstructq+1 many MUBs of dimension dwhend=q2. Note that N(q) =q−1 and thus\nthe number of MUBs from [ 47, Corollary 3] is same as that presented in [ 68, Theorem\n3]. The construction of [ 47, Corollary 3] had the advantage of using different Hadamard\nmatrices for the construction of each MUBs, whereas a single Hadamard matrix can be\nused for construction of MUBs in [ 68, Theorem 3].\nSince Theorem 2is based on RBDs as in [ 47, Corollary3], here alsodifferent Hadamard\nmatrices can be used for the construction of each MUBs. Hence , Theorem 2is a general-\nization of [ 68, Theorem 3] and [ 47, Corollary 3] as enumerated below.\n1. Ford=s2, Theorem 2reproduces the results of [ 68, Theorem 3] in terms of number\nof MUBs constructed.\n2. Ford=q2, Theorem 2reproduces the results of [ 47, Corollary 3], in terms of having\nthe advantage of using different Hadamardmatrices for the co nstruction of each MUBs\nand the number of MUBs constructed.\n3. As the additional contribution, for any composite d=k·s= (s−e)·s, with 0<e≤\n3\n2d1\n2, Theorem 2provides MOLS( s)+1 many APMUBs and one can also use different\nHadamard matrices for the construction of each basis.\nThe caseN(s) =s−1 corresponds to affine plane of order sand the corresponding\nRBD is called Affine Resolvable BIBD. When s=q, whereqis a prime power, we have well\nknown methods to construct Affine Resolvable ( q2,q,1)-BIBDs. Hence in such a situation\nwe will have q∼ O(√\nd) many APMUBs for composite dimensions which are not square.\nFor example,if d= 34·7 = 21·27,we have 29APMUBs with β= 1.13 or ford= 24·3 = 6·8\nwe obtain 9 APMUBs, with β= 1.15 whereas number of MUBs is 8 and 4 respectively in\nthese cases.\nNow let us explain the consequences for Approximate Real MUB s. Whenqis a prime\npower, and real Hadamard matrix of order ( q−e) exists, then we will obtain ( q+1)>√\nd\nmany APRMUBs, which provides large numbers of such objects o verRd. This is presented\nin the following result.\nCorollary 2 Letd= (q−e)q, whereqis a prime power and e∈N, with0<e≤3\n2d1\n2. Then\nthere existq+1many APMUBs with ∆={0,1\nq−e},β=/radicalBig\nq\nq−e= 1+O(d−λ), whereλ=1\n2\nandǫ= 1−1\nq. Further, if there exist real Hadamard matrices of order (q−e), then one can\nconstructq+1many Almost Perfect Real MUBs with same parameters.\nThe condition e≤3\n2d1\n2is because we require β≤2 for APMUBs. If e >3\n2d1\n2then\nβ >2, hence the constructed Approximate MUBs will not satisfy t he criteria for APMUBs\n[Definition 7]. Further examining the expression for βin Equation 4, for this particular\nconstruction with µ= 1,f= 0, we obtain the best possible APRMUBs when e= 1. That\nis, we have this situation when the dimensions are of the form d= (q−1)q. This we\nformally state in the following corollary.\nCorollary 3 Considerd= (q−1)q, such that qis a prime power and assume that a real\nHadamard matrix of order (q−1)exists. Then one can construct qmany Almost Perfect Real\nMUBs in dimension dwith∆=/braceleftBig\n0,1\nq−1/bracerightBig\n,β=/radicalBig\nq\nq−1= 1+O(d−1\n2), andǫ= 1−1\nq.\nFor example, when d= 20 and 156, there would be respectively 5 and 13 APRMUBs\nof the above type. One may note that we have ⌈√\nd⌉many APRMUBs in this case. If\nm=q−3\n2≡1 mod 4 and mis some prime power, then using the Paley Construction [ 55],\none can obtain Hadamard matrix of order 2( m+1) =q−1. Hence for any prime power\nq≡1 mod 4, ifq−3\n2is also some prime power and is equal to 1 mod 4, then the real20 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nHadamard matrix of order q−1 will necessarily exist through the Paley Construction.\nFor example, one can consider q= 13,29,53etc. For such q’s, the result will become\nindependent of the Hadamard Conjecture.\n5.2d=k·s= (q−e)(q+f),0<f≤eand 0<(e+f)≤3\n2d1\n2\nAs noted in Corollary 2that ifdcan be expresses as k·s= (q−e)qwithβ=/radicalbigs\nk=/radicalBig\nq\nq−e≤2, then there we can construct q=O(√\nd) many APMUBs. However, if dcan\nnot be expressed in this form then one can construct N(s) many APMUBs if dcan be\nexpressed as k·swithsa composite such that k≤s≤4k, i.e., for example in the cases\nd={2·32,2·11,2·3·7,22·3·7,...}etc. The condition k≤s≤4kensuresβ≤2 and\nifdcan be expressed as ( s−e)·sthen it is equivalent to 0 ≤e≤3\n2d1\n2. The best known\nlower bound for general sisN(s)∼s1\n14.8which is much less than s.\nIn order to obtain significantly larger number of APMUBs, for the dimensions that\ncannotbe expressedin the form d= (q−e)qwhereqis someprimepowerwith0 <e≤3\n2d1\n2,\nwe now consider the form of d= (q−e)(q+f), such that qis a prime power with e,f∈N.\nFirst we show that, in such a case if e≥f, then we can construct RBD( X,A) with\n|X|=dhaving constant block size ( q−e) such that Acan be partitioned into at least\nr=⌊q−(e−f)\nf⌋=⌊q−e\nf⌋+1 many parallel classes. Hence such an RBD( X,A) can be used\nto construct ⌊q−e\nf⌋+ 1 many orthonormal bases following [ 47, Theorem 1]. Further, if\n0<(e+f)≤3\n2d1\n2, thenβ≤2, and these orthonormal bases would be APMUBs, thus\nproviding us O(q) many APMUBs in such a situation.\nWeexplainthisconstructionintwoparts.Firstweconsider a(q2,q,1)AffineResolvable\nBIBD as the input. We call this RBD( ¯X,¯A) where |¯X|=q2and all the blocks of Ais of the\nsamesizeqandnumberofparallelclassesin Aisq+1.We use thistoconstructRBD( /tildewideX,/tildewideA),\nwhere|/tildewideX|= (q−e)(q+f) having same number of parallel class ( q+1), but the blocks are\nnotofthe samesize.The sizes ofthe blocksarefromset {(q−e),(q−e+1),...,(q−e+f),q}.\nIn the second part we use the RBD( /tildewideX,/tildewideA) as input and construct RBD( X,A) where\n|X|= (q−e)(q+f) such that each block in Ais of size (q−e). However, now the number\nof parallel class reduces to ⌊q−e\nf⌋+1.\nTo demonstrate the first part of the construction, we take |X|= (7−3)(7+1)= 4 ·8 =\n32, whereq= 7,e= 3 andf= 1. We use an Affine Resolvable (72,7,1)-BIBD which we\ncall RBD( ¯X,¯A). It will consist of eight parallel classes. Each parallel c lass would consist\nof seven blocks of constant size 7. We represent each paralle l class as a 7 ×7 matrix, where\neach rowrepresentone blockof the parallelclass.Hence the re wouldbe eightsuch matrices\nas below to represent the design.\n¯P1=\n¯b1\n7={1 2 3 4 5 6 7 }\n¯b1\n6={8 9 10 11 12 13 14 }\n¯b1\n5={15 16 17 18 19 20 21 }\n¯b1\n4={22 23 24 25 26 27 28 }\n¯b1\n3={29 30 31 32 33 34 35 }\n¯b1\n2={36 37 38 39 40 41 42 }\n¯b1\n1={43 44 45 46 47 48 49 }\n,¯P2=\n¯b2\n7={1 9 17 25 33 41 49 }\n¯b2\n6={2 10 18 26 34 42 43 }\n¯b2\n5={3 11 19 27 35 36 44 }\n¯b2\n4={4 12 20 28 29 37 45}\n¯b2\n3={5 13 21 22 30 38 46}\n¯b2\n2={6 14 15 23 31 39 47}\n¯b2\n1={7 8 16 24 32 40 48}\n,\n¯P3=\n¯b3\n7={1 10 19 28 30 39 48}\n¯b3\n6={2 11 20 22 31 40 49}\n¯b3\n5={3 12 21 23 32 41 43}\n¯b3\n4={4 13 15 24 33 42 44 }\n¯b3\n3={5 14 16 25 34 36 45 }\n¯b3\n2={6 8 17 26 35 37 46 }\n¯b3\n1={7 9 18 27 29 38 47}\n,¯P4=\n¯b4\n7={1 11 21 24 34 37 47 }\n¯b4\n6={2 12 15 25 35 38 48 }\n¯b4\n5={3 13 16 26 29 39 49}\n¯b4\n4={4 14 17 27 30 40 43}\n¯b4\n3={5 8 18 28 31 41 44}\n¯b4\n2={6 9 19 22 32 42 45}\n¯b4\n1={7 10 20 23 33 36 46 }\n,\n¯P5=\n¯b5\n7={1 12 16 27 31 42 46}\n¯b5\n6={2 13 17 28 32 36 47}\n¯b5\n5={3 14 18 22 33 37 48 }\n¯b5\n4={4 8 19 23 34 38 49 }\n¯b5\n3={5 9 20 24 35 39 43 }\n¯b5\n2={6 10 21 25 29 40 44}\n¯b5\n1={7 11 15 26 30 41 45}\n,¯P6=\n¯b6\n7={1 13 18 23 35 40 45 }\n¯b6\n6={2 14 19 24 29 41 46}\n¯b6\n5={3 8 20 25 30 42 47}\n¯b6\n4={4 9 21 26 31 36 48}\n¯b6\n3={5 10 15 27 32 37 49}\n¯b6\n2={6 11 16 28 33 38 43 }\n¯b6\n1={7 12 17 22 34 39 44 }\n,Almost Perfect Mutually Unbiased Bases that are Sparse 21\n¯P7=\n¯b7\n7={1 14 20 26 32 38 44}\n¯b7\n6={2 8 21 27 33 39 45 }\n¯b7\n5={3 9 15 28 34 40 46 }\n¯b7\n4={4 10 16 22 35 41 47 }\n¯b7\n3={5 11 17 23 29 42 48}\n¯b7\n2={6 12 18 24 30 36 49}\n¯b7\n1={7 13 19 25 31 37 43}\n,¯P8=\n¯b8\n7={1 8 15 22 29 36 43}\n¯b8\n6={2 9 16 23 30 37 44}\n¯b8\n5={3 10 17 24 31 38 45}\n¯b8\n4={4 11 18 25 32 39 46}\n¯b8\n3={5 12 19 26 33 40 47 }\n¯b8\n2={6 13 20 27 34 41 48 }\n¯b8\n1={7 14 21 28 35 42 49 }\n,\nIn order to construct RBD( /tildewideX,/tildewideA), where |/tildewideX|= (q−e)(q+f) = (7−3)(7+1) = 32 such\nthatµ= 1, we consider the following steps.\n1. Choose any h=e−f= 3−1 = 2 blocks from ¯P1. Let these blocks be ¯b1\n1and¯b1\n2. Let\nS1=¯b1\n1∪¯b1\n2={36,37,38,39,40,41,42,43,44,45,46,47,48,49}, as annotated in red in\nthe above matrices.\n2. Choose another f= 1 block from ¯P1. Let it be ¯b1\n3. Now choose any e= 3 elements from\nit. Let these be S2={33,34,35}. SetS=S1∪S2={33,34,35,36,37,38,39,40, 41,42,\n43,44,45,46,47,48,49}(indicated in red).\n3. Remove the elements of set Sfrom RBD( ¯X,¯A). Call the resulting combinatorialdesign\nRBD(/tildewideX,/tildewideA), where |/tildewideX|= 32and /tildewideA={/tildewideP1,/tildewideP2,/tildewideP3,/tildewideP4,/tildewideP5,/tildewideP6,/tildewideP7,/tildewideP8}, presentedas below.\n/tildewideP1=\n/tildewideb1\n7={1 2 3 4 5 6 7 }\n/tildewideb1\n6={8 9 10 11 12 13 14 }\n/tildewideb1\n5={15 16 17 18 19 20 21 }\n/tildewideb1\n4={22 23 24 25 26 27 28 }\n/tildewideb1\n3={29 30 31 32 }\n,/tildewideP2=\n/tildewideb2\n7={1 9 17 25 }\n/tildewideb2\n6={2 10 18 26 }\n/tildewideb2\n5={3 11 19 27 }\n/tildewideb2\n4={4 12 20 28 29 }\n/tildewideb2\n3={5 13 21 22 30 }\n/tildewideb2\n2={6 14 15 23 31 }\n/tildewideb2\n1={7 8 16 24 32 }\n,/tildewideP3=\n/tildewideb3\n7={1 10 19 28 30 }\n/tildewideb3\n6={2 11 20 22 31 }\n/tildewideb3\n5={3 12 21 23 32 }\n/tildewideb3\n4={4 13 15 24 }\n/tildewideb3\n3={5 14 16 25 }\n/tildewideb3\n2={6 8 17 26 }\n/tildewideb3\n1={7 9 18 27 29 }\n,\n/tildewideP4=\n/tildewideb4\n7={1 11 21 24 }\n/tildewideb4\n6={2 12 15 25 }\n/tildewideb4\n5={3 13 16 26 29 }\n/tildewideb4\n4={4 14 17 27 30 }\n/tildewideb4\n3={5 8 18 28 31 }\n/tildewideb4\n2={6 9 19 22 32 }\n/tildewideb4\n1={7 10 20 23 }\n,/tildewideP5=\n/tildewideb5\n7={1 12 16 27 31 }\n/tildewideb5\n6={2 13 17 28 32 }\n/tildewideb5\n5={3 14 18 22 }\n/tildewideb5\n4={4 8 19 23 }\n/tildewideb5\n3={5 9 20 24 }\n/tildewideb5\n2={6 10 21 25 29 }\n/tildewideb5\n1={7 11 15 26 30 }\n,/tildewideP6=\n/tildewideb6\n7={1 13 18 23 }\n/tildewideb6\n6={2 14 19 24 29 }\n/tildewideb6\n5={3 8 20 25 30 }\n/tildewideb6\n4={4 9 21 26 31 }\n/tildewideb6\n3={5 10 15 27 32 }\n/tildewideb6\n2={6 11 16 28 }\n/tildewideb6\n1={7 12 17 22 }\n,\n/tildewideP7=\n/tildewideb7\n7={1 14 20 26 32 }\n/tildewideb7\n6={2 8 21 27 }\n/tildewideb7\n5={3 9 15 28 }\n/tildewideb7\n4={4 10 16 22 }\n/tildewideb7\n3={5 11 17 23 29 }\n/tildewideb7\n2={6 12 18 24 30 }\n/tildewideb7\n1={7 13 19 25 31 }\n,/tildewideP8=\n/tildewideb8\n7={1 8 15 22 29 }\n/tildewideb8\n6={2 9 16 23 30 }\n/tildewideb8\n5={3 10 17 24 31 }\n/tildewideb8\n4={4 11 18 25 32 }\n/tildewideb8\n3={5 12 19 26 }\n/tildewideb8\n2={6 13 20 27 }\n¯b8\n1={7 14 21 28 }\n.\nNote that here all the blocks are not of the same sizes, but any two blocks from different\nparallel classes have at most 1 element in common, i.e., µ= 1. The blocks sizes are in the\nset{(q−3),(q−2),q}={4,5,7}.The number ofparallelclasses in /tildewideAremainsq+1 = 8. This\ntechnique is now more formally explained for the general cas e in Construction 4below.\nLetd=k·s= (q−e)(q+f), with 0<f≤e≤q. The steps for constructing the RBD( X,A)\nare as follows.\nConstruction 4 Givenq, a prime power, construct ( q2,q,1)-ARBIBD. Call this design\n(¯X,¯A) with¯X={1,2,...,q2}and|¯A|=q(q+1) many blocks, each block is of constant\nsizeq.It willhave r=q+1manyparallelclasses,callthem {¯P1,¯P2,...,¯Pq+1}, eachparallel\nclasshaving qmanyblocksofconstantsize q.Betweenanytwoblocksfromdifferentparallel\nclasses, exactly one element will be common, i.e., |¯bl\ni∩¯bm\nj|= 1,∀l/\\⌉}atio\\slash=m.\n1. Givene≥f, chooseh=e−f≥0 many blocks from ¯P1, which are {¯b1\n1,¯b1\n2,...,¯b1\nh}. Let\nS1=¯b1\n1∪¯b1\n2∪...∪¯b1\nh. Therefore, |S1|=h·q.\n2. From {¯b1\nh+1,¯b1\nh+2,...,¯b1\nh+f}blocksof ¯P1, choose any enumber of elements fromeach of\nthem. LetS2be the union of all these elements. Therefore, |S2|=e·f. LetS=S1∪S2.\n3. Remove the elements of set Sfrom RBD ( ¯X,¯A) and call the resulting design as\nRBD(/tildewideX,/tildewideA).\nWe claim that the above RBD( /tildewideX,/tildewideA) is such that |/tildewideX|= (q−e)(q+f) and/tildewideAconsists\nofq+1 many parallel classes having different block sizes, such t hat blocks from different\nparallel classes have at most one element in common, i.e., |bl\ni∩bm\nj| ≤1,∀l/\\⌉}atio\\slash=m. Hence\nµ= 1. We formalize this in the form of a lemma below.22 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nLemma 8 Letd= (q−e)(q+f)forf,e∈Nwith0< f≤e≤qand some prime power q.\nThen one can construct an RBD (/tildewideX,/tildewideA), with|X|=dhaving block sizes from the set of integers\n{(q−e),(q−e+1),...,(q−e+f),q}withµ= 1, and having r=q+1many parallel classes.\nProofRefer to Construction 4above. Here RBD( ¯X,¯A) is an ARBIBD with |¯X|=q2,\nhaving constant block size q. Note that any pair of blocks from different parallel classes\nhave exactly one element in common, i.e., |¯bl\ni∩¯bm\nj|= 1∀l/\\⌉}atio\\slash=m. The number of elements\nin the set |S|=|S1∪S2|=|S1|+|S2|= (e−f)q+ef <q2, which is a proper subset of ¯X.\nThese are removed from all the parallel classes of RBD( ¯X,¯A). Hence, the resulting design\nRBD(/tildewideX,/tildewideA) is such that |/tildewideX|=q2−(e−f)q+ef= (q−e)(q+f) =d, having same number\nof parallel classes as in ¯A. The number of element common between any two blocks from\ndifferent parallel classes would be at most 1, i.e., |/tildewidebl\ni∩/tildewidebm\nj| ≤1,∀l/\\⌉}atio\\slash=m.\nTo obtain the sizes of the blocks in RBD( /tildewideX,/tildewideA), note that S1contains all elements\nfromh= (e−f) number of blocks of ¯P1. HenceS1would have at least helements in\ncommon with all the blocks of remaining parallel classes. Th us, removal of the elements\ninS1from the parallel classes ¯Pl,l≥2 will remove at least helements from each block\nof¯Plwhich implies |¯bl\ni\\S1|=q−(e−f). Further S2containseelements from fmany\nblocks of ¯P1. Thus, the blocks in ¯P2,¯P3,...,¯Pq+1will have at most fmany elements in\ncommon with S2. Consequently, after removal of all the elements in S, from the parallel\nclass¯Pl, the block size |/tildewidebl\ni|, l≥2 will be maximum q−(e−f) = (q−h) and minimum\nq−(e−f)−f= (q−e). Further, the blocks in /tildewideP1will be of sizes q,(q−e) and have total\n(q−h) number of blocks.\nWe willnowshowhowone canuse fmanyblocksof theparallelclass /tildewideP1in RBD( /tildewideX,/tildewideA),\nas constructed above, to reshape any one of the parallel clas ses/tildewidePl, l/\\⌉}atio\\slash= 1 into a parallel\nclass having ( q+f) many blocks each of size ( q−e), which we denote by Pland the\nresulting combinatorial design by RBD( X,A). Since /tildewideP1haveq−hnumber of blocks, thus\n⌊q−h\nf⌋many parallel classes of ¯Acan be reshaped into a parallel class of A.\nLet us first demonstrate the construction by using RBD( /tildewideX,/tildewideA), as constructed in the\nexample earlier in this section, where |/tildewideX|= (7−3)(7+1) = 4 ·8 = 32, with q= 7,e= 3,\nf= 1 andh=e−f= 2. Note that /tildewideP1hasq−h= 5 blocks, and all the other parallel\nclasses have q= 7 blocks each. Let us denote the excess number of elements on each\nblock of /tildewidebl\nj, l≥2 than (q−e) byml\nj. Henceml\ni=|/tildewidebl\ni| −(q−e). Here for each block of\nthe parallel class /tildewidePl, l≥2, the value of ml\niis either 0 or 1. If the block |/tildewidebl\ni|= 5, then\nml\ni=|/tildewidebl\ni| −(q−e) = 5−(7−3) = 1 and similarly for block |/tildewidebl\ni|= 4,ml\ni= 0. Note that/summationtextq\ni=1ml\ni= (q−e)·f= (7−3)·1 = 4,∀l≥2. We modify this RBD( /tildewideX,/tildewideA) as follows.\n1. Sincef= 1, consider one block /tildewideb1\n3of/tildewideP1, which has 4 elements in it. Remove these\nelements from different blocks of /tildewideP2and add them as separate block in /tildewideP2. Denote the\nresulting parallel class as P2.\n2. Consider the parallel class /tildewideP3and the next f= 1 block of /tildewideP1, i.e., the block /tildewideb1\n4. Choose\nablockfrom /tildewideP3, say/tildewideb3\n1. Since|/tildewideb3\n1|= 5,i.e.,it hasone elementmorethan q−e= 4,hence\nmark one common element between /tildewideb1\n4and/tildewideb3\n1which is 27 in this case. Sequentially\nexecute this for all the blocks of /tildewideP1. This will mark the elements {27,23,22,28}on/tildewideb1\n4.\n3. Sincem3\nj= 0 or 1, the above step will mark exactly ( q−e) = 4 elements on /tildewideb1\n4. In\na situation, if ml\nihas more than one elements, then further iterations are requ ired to\nmark exactly ( q−e) elements on the blocks of /tildewideP1. Refer to Step 3 of Construction 5\nlater for the exact strategy in this regard.\n4. Now remove the elements marked on /tildewideb1\n4, i.e.,{27,23,22,28}from/tildewideP3and add them as\na separate block of /tildewideP3and denote the resulting parallel class as P3.\n5. Consider the next parallel class /tildewideP4and the next f= 1 block of /tildewideP1, i.e., the block /tildewideb1\n5.\nThen repeat the Steps 2, 3, 4 to obtain P4.\n6. Since the number of blocks in /tildewideP1= 5, in this way r=5\n1= 5 parallel classes, i.e.,\n/tildewidePl, l= 2,3,4,5,6 can be modified. Discard /tildewideP7and/tildewideP8. The resulting RBD( X,A) isAlmost Perfect Mutually Unbiased Bases that are Sparse 23\nsuch that |X|= 32 with Aconsisting of parallel classes {P2,P3,P4,P5,P6}as shown\nbelow.\nP2=\nb2\n8={29 30 31 32 }\nb2\n7={1 9 17 25 }\nb2\n6={2 10 18 26 }\nb2\n5={3 11 19 27 }\nb2\n4={4 12 20 28 }\nb2\n3={5 13 21 22 }\nb2\n2={6 14 15 23 }\nb2\n1={7 8 16 24 }\n,P3=\nb3\n8={22 23 27 28 }\nb3\n7={1 10 19 30 }\nb3\n6={2 11 20 31 }\nb3\n5={3 12 21 32 }\nb3\n4={4 13 15 24 }\nb3\n3={5 14 16 25 }\nb3\n2={6 8 17 26 }\nb3\n1={7 9 18 29 }\n,P4=\nb4\n8={16 17 18 19 }\nb4\n7={1 11 21 24 }\nb4\n6={2 12 15 25 }\nb4\n5={3 13 26 29 }\nb4\n4={4 14 27 30 }\nb4\n3={5 8 28 31 }\nb4\n2={6 9 22 32 }\nb4\n1={7 10 20 23 }\n,\nP5=\nb5\n8={10 11 12 13 }\nb5\n7={1 16 27 31 }\nb5\n6={2 17 28 32 }\nb5\n5={3 14 18 22 }\nb5\n4={4 8 19 23 }\nb5\n3={5 9 20 24 }\nb5\n2={6 21 25 29 }\nb5\n1={7 15 26 30 }\n,P6=\nb6\n8={2 3 4 5 }\nb6\n7={1 13 18 23 }\nb6\n6={14 19 24 29 }\nb6\n5={8 20 25 30 }\nb6\n4={9 21 26 31 }\nb6\n3={10 15 27 32 }\nb6\n2={6 11 16 28 }\nb6\n1={7 12 17 22 }\n,P7=\nb7\n8={5 12 19 26 }\nb7\n7={1 14 20 32 }\nb7\n6={2 8 21 27 }\nb7\n5={3 9 15 28 }\nb7\n4={4 10 16 22 }\nb7\n3={11 17 23 29 }\nb7\n2={6 18 24 30 }\nb7\n1={7 13 25 31 }\n.\nWe will now discuss about P7as presented above. From the two discarded parallel classes\n/tildewideP7and/tildewideP8, we find that one can use the block /tildewideb8\n3and remove the elements in it from /tildewideP7,\nand place them as a separate block of parallel class /tildewideP7, resulting into another parallel class\nhaving(q+f) = 8 many blocks each block having ( q−e) = 4 elements each. We denote this\nparallel class as P7. Certainly this is not unique as there are other possibiliti es to obtain\na parallel class using /tildewideP7and/tildewideP8. Thus here we actually obtain r= 6>/floorleftBig\nq−e\nf/floorrightBig\n+ 1 = 5,\nindicating that/floorleftBig\nq−e\nf/floorrightBig\n+1 is not a tight lower bound in this example.\nThe above construction is formally explained for the genera l case in Construction 5\nbelow. We consider RBD( /tildewideX,/tildewideA) constructed in 4with|/tildewideX|= (q−e)(q+f) as the input for\nfollowing construction. To begin with, compute ml\nj=|/tildewidebl\nj|−(q−e), l≥2 for each block of\n/tildewidePl, l≥2, which is the count of the excess number of elements on each b lock of/tildewidebl\nj, l≥2\nthan what is required, which is q−e. Note that/summationtextq\nj=1ml\nj=/summationtextq\nj=1/parenleftBig\n|/tildewidebl\nj|−(q−e)/parenrightBig\n=\n/summationtextq\nj=1|/tildewidebl\nj|−q(q−e) = (q−e)(q+f)−q(q−e) = (q−e)f,∀l≥2. Thus/summationtextq\nj=1ml\nj= (q−e)f\nis constant for all the parallel classes of RBD( /tildewideX,/tildewideA) except /tildewideP1, which consists of q−h\nblocks of sizes {(q−e),q}and are being used to modify the rnumber of other parallel\nclasses and will be discarded in the end.\nConstruction 5 Letd=k·s= (q−e)(q+f), with 0< f≤e≤qand we consider\nRBD(/tildewideX,/tildewideA) from Construction 4with|/tildewideX|= (q−e)(q+f) as the input.\n1. Consider fmanyblocksof /tildewideP1,whichhas( q−e)elements,i.e.,theblocks {/tildewideb1\nh+1,/tildewideb1\nh+2,...,\n/tildewideb1\nh+f}. Remove the elements in the blocks {/tildewideb1\nh+1,/tildewideb1\nh+2,...,/tildewideb1\nh+f}from different blocks\nof/tildewideP2and add the blocks {/tildewideb1\nh+1,/tildewideb1\nh+2,...,/tildewideb1\nh+f}as blocks of /tildewideP2. Name the resulting\nparallel class as P2.\n2. Considertheparallelclass /tildewideP3andnextfmanyblocksof /tildewideP1,i.e.,{/tildewideb1\n(h+f+1),/tildewideb1\n(h+f+2),...,\n/tildewideb1\n(h+2f)}, eachconsistingof qmanyelements.Callthissetof blocksas S1\n3.Selecta block\nfrom/tildewideP3, say/tildewideb3\n1. Corresponding to this block, mark m3\n1number of elements which are\ncommon with the blocks in set S1\n3. Then move them to the next block of /tildewideP3, namely /tildewideb3\n2,\nand markm3\n2number of elements common with the blocks in S1\n3. Sequentiallycontinue\nthis for all the blocks of /tildewideP3.\n3. Now consider /tildewideb1u,/tildewideb1v∈S1\n3, such that /tildewideb1uhas more elements marked than ( q−e) and/tildewideb1v\nhas less elements marked than ( q−e). Identify the blocks of /tildewideP3which have a marked\nelement common with /tildewideb1u, but has an unmarked element common with /tildewideb1v, say block /tildewideb3\nj.\nThen unmark this element in /tildewideb1u, and mark the common element between the block /tildewideb3\nj24 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\nand/tildewideb1von the block /tildewideb1v. Thus, the marked element on /tildewideb1uis removed and the marked on\n/tildewideb1vis added. Iterate this for all such blocks in S1\n3which has marked elements different\nfrom (q−e) and continue this till all the blocks in set S1\n3have exactly ( q−e) marked\nelement.\nLater in Lemma 9, we will show that such a block /tildewideb3\njwill always exist. Further this\nprocess will terminate in a finite number of steps as there are finite number of blocks\nand elements, and every iteration adds the marked element of a block having less\nelements marked than ( q−e) and removes the mark element of a block having more\nelements marked than ( q−e).\n4. Remove the elements marked on each of the blocks in the set S1\n3from the parallel class\n/tildewideP3and add the elements marked on the block, say /tildewideb1\nh+f+1as separate blocks in /tildewideP3.\nSimilarly add elements marked on the block /tildewideb1\nh+f+2as separate blocks in /tildewideP3and so\non for all the blocks in S1\n3. Call the resulting parallel class as P3.\n5. Then consider the next parallel class /tildewideP4and the next fmany blocks of /tildewideP1and repeat\nthe steps 2, 3 and 4 as mentioned above. Denote the resulting c lass asP4and continue\ntill the number of blocks in /tildewideP1becomes less than f.\n6. Since the number of blocks in /tildewideP1isq−h=q−(e−f), in this way r=/floorleftBig\nq−(e−f)\nf/floorrightBig\n=/floorleftBig\nq−e\nf/floorrightBig\n+ 1 many parallel classes /tildewidePrcan be modified. Then the RBD( X,A), where\nX={1,2,...,(q−e)(q+f)}andA={P2,P3,...,Pr+1}is the required design.\nNote that in Step 4 above, since each block in the set S1\n3hasq−eelements marked and\ntotal number of blocks is f, hence total number of elements removed from /tildewideP3are (q−e)·f.\nThus, we are basically using a set of fblocks from the parallel classes /tildewideP1to reshape one\nof the remaining parallel class, into blocks of size ( q−e), having ( q+f) blocks. This is\nachieved by identifying ( q−e) elements of one block from the parallel classes /tildewideP1, and in\ndifferent blocks of parallel class say /tildewidePl. Thereafter, we delete these elements from different\nblocks/tildewidePl, and add these elements as a separate block in /tildewidePl. Thus, the resulting parallel\nclass will consist of ( q+f) blocks, each having ( q−e) elements.\nWe claim that the above design ( X,A) is a RBD, such that |X|= (q−e)(q+f) andA\nconsists of r=/floorleftBig\nq−e\nf/floorrightBig\n+1 many parallel classes each having ( q+f) many blocks each of\nsize (q−e), such that the blocks from different parallel classes have a t most one element\nin common, i.e., |bl\ni∩bm\nj| ≤1∀i/\\⌉}atio\\slash=j. We formalize this in the following lemma.\nLemma 9 Letd= (q−e)(q+f), forf,e∈Nwith0<f≤e≤qandqsome power of prime.\nThen one can construct an RBD (X,A), with|X|=dhaving constant block size (q−e)with\nµ= 1, and having at least r=/floorleftBig\nq−e)\nf/floorrightBig\n+1many parallel classes.\nProofRefer to Construction 5above. Since in RBD( /tildewideX,/tildewideA) any pair of blocks from different\nparallel classes has at most one element in common, we have |¯bl\ni∩¯bm\nj| ≤1∀l/\\⌉}atio\\slash=m.\nSince no element of RBD( /tildewideX,/tildewideA) has been deleted or added to it to obtain RBD( X,A),\nhenced=|/tildewideX|=|X|= (q−e)(q+f). From the Step 6 of Construction 5, we obtain\nr=/floorleftBig\nq−e)\nf/floorrightBig\n+1.\nNow we show that Step 2 can be successfully executed. That is, from the set of fmany\nblocks of /tildewideP1, it will be possible to mark/summationtextq\ni=1ml\ni= (q−e)·f,∀l≥2. Note that each block\n/tildewideb1\nj, j >ehas exactly one element common with /tildewidebl\nj, j= 1,2,...,q, l ≥2. As 0≤ml\nj≤f,\ncorresponding to each block /tildewidebl\nj, there will alwaysbe ml\njelements on different blocks, which\nis totalfin number. Hence Step 2 can be successfully executed.\nSteps 3 and 4 are related to the construction where elements a re to be marked on\nblocks in the set S1\n3havingfmany blocks. These steps will finally result into ( q−e)\nelements being marked on each of these blocks. For this, note that/summationtextq\ni=1ml\ni= (q−e)f.\nThat means, if it is not possible to mark q−eelements on each of the blocks in the set S1\n3,\nthen on some block there will have more elements marked than q−eand on some blocksAlmost Perfect Mutually Unbiased Bases that are Sparse 25\nthere are less element marked than q−e. It is not possible that all the blocks have less\nthan (q−e) elements marked or all the blocks have more than ( q−e) elements marked as\nin that case/summationtextq\ni=1ml\ni<(q−e)for>(q−e)faccordingly.\nIn case there is a block /tildewideb1u,/tildewideb1v∈S1\n3, such the /tildewideb1uhas more elements marked than\n(q−e) and/tildewideb1vhas less element marked than ( q−e), then there will exist a block of /tildewideP3\nwhich have marked element common with /tildewideb1u, but non-marked element common with /tildewideb1v.\nSuppose there is no such block in /tildewideP3. Then the marked elements of block /tildewideb1u(which is\n> q−e) and the unmarked elements of /tildewideb1v(which is> e) would all lie on the different\nblocks of /tildewideP3. However, this would imply number of blocks in |/tildewideP3|>q−e+e=q, which is\na contradiction as |/tildewideP3|=q.\nFinally, we show that µ= 1. Note that the blocks which has been added in the parallel\nclasses/tildewideP2,/tildewideP3,...,/tildewidePr+1to construct the parallel classes P2,P3,...,Pr+1are respectively\npart of the blocks of /tildewideP1. Since any block of /tildewideP1has at most one element common with any\nother block of /tildewidePl, l≥2,µwill be 1 for RBD( X,A).\nNow we can use this RBD( X,A) to construct APMUBs in dimension d=|X|= (q−\ne)(q+f) having parameters as given by Theorem 1, which we formally state and prove in\nthe next theorem.\nTheorem 3 Letd= (q−e)(q+f), for some prime power q, where0<f≤eand0<(e+f)≤\n3\n2d1\n2,e,f∈N. Then there exist at least r=/floorleftBig\nq−e\nf/floorrightBig\n+1many APMUBs, with ∆={0,1\nq−e},\nβ=/radicalBig\nq+f\nq−e= 1−O(d−λ)≤2, whereλ=1\n2andǫ= 1−1\nq+f. Further, if there exists a Real\nHadamard matrix of order (q−e), then one can construct rmany Almost Perfect Real MUBs\nwith same parameters.\nProofIn order to show that we can produce such APMUBs, let us consid er an RBD( X,A)\nwith|X|= (q−e)(q+f) havingrparallel classes of constant block size ( q−e), such that\nbetween the blocks from different parallel classes, there is at most one element in common,\nand hence µ= 1.Then using this RBD along with a Hadamard matrix of order ( q−e), we\ncan construct orthonormal bases following Theorem 1. Further, the condition that β <2\nfor APMUB gives ( e+f)≤3\n2d1\n2. In terms of q, this inequality becomes 4 e+f≤3q. The\nparameters of APMUBs are ∆={0,1\nq−e},β=/radicalBig\nq+f\nq−e= 1 +O(d−1\n2)≤2 and sparsity\nǫ= 1−1\nq+f. Further when a real Hadamard matrix of order ( q−e) is available, the same\ncan be used in the construction of Approximate Real MUBs. Sin ce number of parallel\nclassesris at least ⌊q−e\nf⌋+1, hence we get at least these many APMUBs.\nRemark 2 Note thatq−e\nfisO(√\nd), whene,fare considered to be constants. That is, in\nsuch cases we obtain O(√\nd) many APMUBs for the dimension d.\nFollowingTheorem 3, wecangetatleast r=/floorleftBig\nq−e\nf/floorrightBig\n+1manyAPMUBs. Thiscan enableus\nto beat the Mutually Orthogonal Latin Square (MOLS) Lower Bound constr uctionfor APMUBs\n(Theorem 2), according to which we obtain N(q+f)+1 many APMUBs. Let us present\na few illustrative examples in this regard.\n–Ford= 60 = 6 ·10, the known value of N(10) is 2 hence MOLS Lower Bound construction\nprovides three APMUBs with βvalue of 1.29. On the other hand, if we use above\nconstruction method, by expressing d= (9−3)(9+1), we obtain9−(3−1)\n1= 7 many\nAPMUBs with β= 1.29. In this case the number of complex MUBs, that can be\nconstructed following prime factorization formula, is 3+1 = 4 only.\n–Ford= 24 = 4 ·6, withN(6) = 1, the MOLS Lower Bound construction generates 2\nAPRMUBs with β= 1.22. On the other hand, expressing d= 24 = (5 −1)(5+1) we\nobtain 5 APRMUBs with β= 1.22. The number of real MUBs for d= 24 is 2 [ 13] only.\nFurther, to illustrate the advantage of Construction 5over Construction 3, consider\nthe example of d= 22·32·5·7 and different ways of expressing it as product of two factors ,\nthat we considered earlier following Theorem 2.26 Ajeet Kumar ,Subhamoy Maitra and Somjit Roy\n–Expressing d= 30·42, theMOLS Lower Bound construction will provide N(42)+1 = 6\nmany APMUBs with β= 1.18, where as expressing d= (41−11)(41+ 1) and using\nTheorem 3will provide 31 many APMUBs with β= 1.18.\n–or expressing d= 28·45, theMOLS Lower Bound construction will provide N(45)+1= 7\nmany APRMUB with β= 1.27, where as expressing d= (43−15)(43+2) and using\nTheorem 3will provide 15 many APRMUB with β= 1.27.\nNote that, N(42) = 5 and N(45) = 6 are the presently known values of the maximum\nnumber of MOLS of these orders [ 1]. Here expressing d= 35·36, we cannot use Theorem\n3as it cannot be expressed as ( q−e)(q+f), withqsome power of prime and 0 ≤f≤e.\nThus with this factorization of d,MOLS Lower Bound construction providesN(36)+1 = 9\nAPMUBs with β= 1.01.\nNote that, r=/floorleftBig\nq−(e−f)\nf/floorrightBig\n, with condition 0 < f≤e, is maximum for a given qwhen\nf=e= 1. From the asymptotic expression of β, it is clear that for 0 < f≤ewe will\nobtainβclosest to 1 when e=f= 1. Thus for e=f= 1, we state the result of APMUB\nas a corollary below.\nCorollary 4 Letd=q2−1 = (q−1)(q+ 1)whereqis a prime power. Then one can\nconstructqmany APMUBs with ∆=/braceleftBig\n0,1\nq−1/bracerightBig\nandβ=/radicalBig\nq+1\nq−1with sparsity ǫ= 1−1\nq+1. If\na real Hadamard Matrix of order q−1exists, then we have qmany APRMUBs with the same\nparameters.\nOur observation made in connection with Hadamard matrix of o rder (q−1), constructed\nthrough Paley method [ 55] after Corollary 3is applicable here as well. We revisit that\nonce again here for better explanation. That is, if m=q−3\n2≡1 mod 4 and mis some\nprime power, then using the Paley Construction [ 55], we can construct Hadamard matrix\nof order 2( m+ 1) =q−1. Thus, for a prime power q≡1 mod 4, ifq−3\n2is also some\nprime power, as well as, is equivalent to 1 mod 4, then the real Hadamard matrix of order\nq−1 will exist through the Paley Construction. For all such q’s, there exist qAPRMUBs\ninRq2−1and the above corollary will become independent of the Hadam ard Conjecture.\nExamples of such qare 13,29,53etc.\n–Ford= (13−1)(13+1) = 23·3·7, we obtain 13 many APRMUBs with β= 1.080. In\nthis case number of real MUBs is only 2 and complex MUBs is 4.\n–Ford= (29−1)(29+1) = 23·3·5·7, we obtain 29 many APRMUBs with β= 1.035.\nIn this case also the number of real MUBs is only 2 and complex M UBs is 4.\nThe above examples clearly indicates that as dincreases,βapproaches closer to 1, hence\nwe obtain APRMUBs which are significantly close to the MUBs.\n5.3 Some problems that require further attention\nIt was pointed out in the example constructed above for RBD( X,A), that for |X|=\nd= 4·8 = (7−3)(7 + 1), one could construct more number of parallel classe s than\nr=/floorleftBig\nq−e\nf/floorrightBig\n+1 = 5 in this case, q= 7,e= 3,f= 1. From our experience of constructing\nRBD(X,A), for the situation when |X|can be expressed as ( q−e)(q+e) =q2−e2, i.e., for\nthe situation e=f >0, there appears to be always more than r=/floorleftBig\nq\nf/floorrightBig\n+1 many parallel\nclasses. In this situation it is possible to use other parall el classes, apart from the first\none of (q2,q,1)-ARBIBD, which enable us to obtain more parallel classes f or RBD(X,A)\nthanr=/floorleftBig\nq\nf/floorrightBig\n+1. A proof of this in the following form in a general setting m ight be an\ninteresting open problem.\nLetd= (q−e)(q+e), fore∈Nwith0≤e≤qandqa prime power. Then one\ncan construct an RBD (X,A), with|X|=dhaving constant block size (q−e)with maximum\nintersection number µ= 1, and having r≥q\n2many parallel classes.Almost Perfect Mutually Unbiased Bases that are Sparse 27\nFurther our efforts for the following form of composite dcould not result into number\nof APMUBs of the order of O(√\nd), whereqis a prime power.\n–Ford=q(q+f), RBD having block size q, withq+fmany blocks in each parallel\nclasses.\n–Ford= (q−e)(q+f),0<e<f, RBD having block size q−ewithq+fmany blocks\nin each parallel classes.\nFor the above forms of d, we could not construct more number of APMUBs than what is\ngiven by Mutually Orthogonal Latin Square Lower Bound construction . Further efforts in this\ndirection or new ideas may be required for this. We believe th at it should be possible to\nimprove the MOLS lower bound in such cases as well.\nNote that our constructionof APMUBs are very sparse and henc e the set of Bi-angular\nvectors are very sparse. The sparsity of each vector inner co nstructions, is ǫ= 1−1\ns≈\n1−1√\nd. Also the non zero components of the vectors are all of the sam e absolute value,\nwhich is1\ns−e≈1√\nd. Our extensive search of literature could not find any study o n bounds\non the cardinality of such kind of sparse vectors, each havin g same sparsity. Nevertheless,\nthere are bounds on the cardinality of flat equiangular lines . Here flat signifies that all the\ncomponent of the vectors are of same magnitude. In such situa tion the cardinality of set\nof equiangular lines in Cdis bounded by ( d2−d−1), Refer to [ 29, Lemma 2.2] which is\nless thand2, which is cardinality when the constrain of flatness is relax ed. We similarly\nbelieve that the cardinality of such Bi-angular set, with wi th such large sparsity would\nbe significantly less than those given in [ 24, Table I] and [ 14, Equations 3.9,5.9]]. Hence\nwe subsequently intend to study the bounds on the cardinalit y of the set of Bi-angular\nvectors, with large sparsity.\n6 Conclusion\nIn this paper we consider construction of APMUBs, which are s ignificantly good approx-\nimation of MUBs. In asymptotic sense, the APMUBs are almost a s good as the MUBs.\nThat is, for a dimension d, the value of the dot product between two vectors from differe nt\nbases will be very close to1√\nd, and in a few cases 0. In this paper we have formalized the\ndefinition of APMUBs and shown that for a good proportion of in tegers, we can construct\nO(√\nd) many APMUBs. Such a generic result is elusive in cases of per fect MUBs. Thus,\nfor all practical purposes in the domain of quantum informat ion,or related areas, our con-\nstruction ideas open up a larger possibility of obtaining re quired combinatorialstructures.\nHow dense are these values of dfor which we can construct such APMUBs? We note by\npreliminary calculations that for d < N, there are approximately3\n4N\nlogNmany such d’s,\nwhich can be expressed as ( q−e)qwith 0≤e <3\n4q. One may note that this is of the\norder of the density of the primes. As the main scope of this pa per is understanding the\ncombinatorialtechniques,we leave thisas a future researc heffort. Anotherimportantissue\nin this regard is that our constructions are directly relate d to the concept of Bi-angular\nvectors. We primarily note that when two vectors are randoml y selected from the set of\nsuch Bi-angular vectors, there exists a very large probabil ity that they will be making\nan angle ofβ√\nd. In fact as dincreases, the probability converges to certainty. This is the\nscenario that happens in our APMUB related constructions. W e leave this too for future\ninvestigation in a disciplined manner.\n7 Declarations: Funding and/or Competing interests\nNo funding was received to assist with the preparation of thi s manuscript. 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In this article we investigate the large-time\nbehaviorof the observationdomain that maximizes the observability constant overall possible measurable\nsubsets ofa given Lebesgue measure. We provethat it converges exponentially, as the time horizongoes to\ninfinity, toalimit set that wecharacterize. The mathematicaltechn ique isnew andreliesonaquantitative\nversion of the bathtub principle.\nKeywords: parabolic equations, observability, shape optimization, quantitative bathtub principle.\nAMS classification: 93B07, 35L05, 49K20, 42B37.\n1 Introduction\nLetd∈IN∗and let Ω be an open bounded smooth connected subset of IRd. Letq∈IN∗. We consider the\nevolution system\n∂ty+A0y= 0 (1)\nwhere−A0:D(A0)→L2(Ω,Cq) is a densely defined operator generating a strongly continu ous semi-\ngroup onL2(Ω,Cq). Given any y0∈D(A0), there exists a unique solution y∈C0([0,+∞),D(A0))∩\nC1((0,+∞),L2(Ω,Cq)) of (1) such that y(0,·) =y0(·).\nFor an evolution system like ( 1), it is often required in practice, for instance in engineer ing problems, to\nreconstruct initial or final data of solutions, based on part ial measurements performed on a subset ωof Ω\nover a a time horizon T. This inverse problem is feasible as soon as the system is obs ervable onωin time\nT, which is mathematically modelled by an observability inequality , as follows. For any measurable subset\nωof Ω and any T >0, the system ( 1) is said to be observable onωin timeTif there exists C >0 such that\nC/ba∇dbly(T,·)/ba∇dbl2\nL2(Ω,Cq)/lessorequalslant/integraldisplayT\n0/integraldisplay\nω|y(t,x)|2dxdt (2)\nfor every solution of ( 1) such that y(0,·)∈D(A0). In (2),|y|is the Hermitian norm of y∈Cq.\nMany results have been established in the literature regard ing observability properties for some classes of\npartial differential equations. Since, in this article, we ar e going to focus on the case of parabolic equations\n∗CEREMADE, UMR CNRS 7534, Universit´ e Paris-Dauphine, Univ ersit´ e PSL, Place du Mar´ echal De Lattre De Tassigny,\n75775 Paris cedex 16, France ( mazari@ceremade.dauphine.fr ).\n†Universit´ e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France (yannick.privat@univ-lorraine.fr ).\n‡Institut Universitaire de France (IUF).\n§Sorbonne Universit´ e, CNRS, Universit´ e Paris Cit´ e, Inri a, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France\n(emmanuel.trelat@sorbonne-universite.fr ).\n1(see Section 2.1for assumptions on A0), here we simply quote few of them. For the heat equation, tha t\nis, when −A0is the Dirichlet Laplacian, observability holds true for an yT >0 and any subset of positive\nLebesgue measure (see [ 18,25] for open subsets and see [ 3] for the extension to measurable subsets). The\nsame holds true for anomalous diffusion equations, that is, wh enA0is some power α>1/2 of the Dirichlet\nLaplacian (see [ 32]), and for the Stokes equation (see [ 16, Lemma 1] and [ 9]). We also refer to [ 45, Chapter\n9] for some general results establishing observability for parabolic systems.\nTheobservability constant is defined as the largest possible nonnegative constant for w hich the observ-\nability inequality ( 2) holds, i.e., by\nCT(1ω) = inf/braceleftBigg/integraltextT\n0/integraltext\nΩ1ω(x)|y(t,x)|2dxdt\n/ba∇dbly(T,·)/ba∇dbl2\nL2(Ω,Cq)/vextendsingle/vextendsingley0∈D(A0)\\{0}/bracerightBigg\n, (3)\nwhere 1ωdenotes the characteristic function of ω. Note that ( 1) is observable on ωin timeTif and only\nifCT(1ω)>0. The observability constant defined by ( 3) provides an account, in some sense, for the well-\nposedness of the above-mentioned inverse problem: the larg er the constant, the more favorable it is to solve\nthe inverse problem. This is why it is important, in practice , to try to choose the observation subset ωin\nan optimal way and, when it is possible, the horizon of time Tover which observations are performed. Of\ncourse, in practice, there are limitations on the choice of t he subsetω, starting with its Lebesgue measure.\nThis is why, in what follows, we will consider measurable sub sets of a given measure.\nLetL∈(0,1) be fixed. Given any T >0, we consider the optimal design problem\nδT= sup\n1ω∈ULCT(1ω) (4)\nconsisting of maximizing the observability constant over t he set\nUL={1ω∈L∞(Ω;{0,1})|ωis a measurable subset of Ω of Lebesgue measure |ω|=L|Ω|}.(5)\nThisshapeoptimization problemmodelsthebestshapeandpl acement ofsensorsfortheevolution system( 1)\nin timeT. If an optimal set ω⋆\nTexists, then it represents the best possible place to instal l some (adequately\nshaped) sensors.\nUnfortunately, the shape optimization problem ( 5) is very difficult to handle, not only because of the\ncomplexity of the definition of the observability constant ( 3), which is an infimum over all possible solutions\nof (1), but also because of the set of unknowns UL, which is very large and does not enjoy good compactness\nproperties. Actually, it is not even known if a maximizer ω⋆\nTexists for ( 4).\nTo remedy this lack of compactness, some constraints may be a dded. The search of an optimal set may\nbe restricted to a compact finite-dimensional set, like in [ 24,33,46,47], or by adding constraints on the\nBV norm, on the number of connected components, or anything y ielding compactness (see, e.g., [ 20]), or by\nfixing the initial data, like in [ 17,35,38]. However, we want to keep the maximal freedom on the choice o f\nthe observation subset and this is why we consider the genera l setULdefined by ( 5).\nThis choice comes at a price: the set ULis not even closed for the weak-star topology of L∞(Ω), thus\nmaking even the basic question of existence of an optimizer c hallenging. Such a question is well known in\nshape optimization, and this issue is often tackled by consi dering a relaxed version of the problem. Here,\nthe relaxation procedure consists of extending the definiti on ofCTto theL∞weak-star closure of UL, which\nis the set\nUL=/braceleftbigg\na∈L∞(Ω;[0,1])|/integraldisplay\nΩa(x)dx=L|Ω|/bracerightbigg\n2and to define, in accordance with ( 3),\nCT(a) = inf/braceleftBigg/integraltextT\n0/integraltext\nΩa(x)|y(t,x)|2dxdt\n/ba∇dbly(T,·)/ba∇dbl2\nL2(Ω,Cq)/vextendsingle/vextendsingley0∈D(A0)\\{0}/bracerightBigg\n∀a∈UL. (6)\nThen, we consider the relaxed shape optimization problem\n¯δT= sup\na∈ULCT(a) (PT)\nNow, since ULis compact for the L∞weak-star topology (see [ 20, Prop. 7.2.17]) and CTis upper semi-\ncontinuous (as an infimum of bounded linear functionals), it follows that Problem ( PT) has at least one\nsolutiona⋆\nT. Characterizing a⋆\nTis more intricate. In this article, we will study a⋆\nTasT→ ∞and also give\nsome results on its small-time asymptotics as T→0.\nOf course, we have δT/lessorequalslant¯δT=CT(a⋆\nT) but we do not know, in general, whether the inequality may be\nstrict or not. Anyhow, observe the following a priori counter-intuitive result: in some cases, the constant\ndensitya≡Lis not a maximizer (see [ 37, Prop. 2]).\nTo tackle the difficulty of optimizing the observability cons tant (3), or its relaxed version ( 6), the point of\nview adopted in the series of papers [ 36,37,39,40,41,42,43] consisted in considering a randomized version\nCT,rand(a) of the observability constant, defined by taking the infimum in (6) not on all but on almost all\ninitial data in an appropriate sense, thus yielding a more tr actable expression of the functional, namely,\nCT,rand(a) = inf\nj∈IN∗γj(T)/integraldisplay\nΩa(x)|φj(x)|2dx, (7)\nwhere(φj)j∈IN∗is a Hilbert basis of eigenfunctions of A0and(γj(T))j∈IN∗is a sequenceof positive coefficients\ndepending on the operator. We have CT,rand(a)/greaterorequalslantCT(a) for every a∈ULand the inequality may be strict\n[39, Theorem 1]. The randomized observability constant CT,rand(a) is in some sense “less pessimistic” than\nthe deterministic observability constant CT(a) which provides an account for the worst observation cases.\nFrom the harmonic analysis point of view, the constant CT,rand(a) reflects the independent observations\non each mode, with no interaction between them, while the con stantCT(a) also takes into account all\ninteractions between modes, through crossed terms in a spec tral expansion of solutions.\nIn the above-mentioned series of papers, the problem of maxi mizing the randomized (relaxed or not)\nobservability constant has been solved for various classes of evolution equations, showing strong differences\nbetween parabolic and hyperbolic cases. The results have re vealed a close relationship with asymptotic\nproperties of high frequency eigenfunctions (quantum ergo dicity properties). Roughly speaking, it has been\nshownthat, in hyperboliccases, all modescount andamaximi zer does notexistin general, whileinparabolic\ncases, thereexists auniquemaximizingdomainwhichismore over characterized andcomputablefromafinite\nnumber of modes only.\nThe question of maximizing the deterministic ( i.e.without randomization) observability constant CT(a)\nhas remained unsolved theoretically, although it has been i nvestigated numerically in 1D (see [ 4,34]).\nAlthough we do not solve this difficult problem, the present ar ticle takes us one step further by studying\nthe large-time behavior of the optimal observability const ant for parabolic evolution systems and of the\nassociated maximizers. It is interesting to note that the au thors of [ 1] studied the relaxed problem for two-\nphase heat equations and proved that the relaxed maximizers converge, as T→+∞, to relaxed maximizers\nof the corresponding stationary problem.\nIn Section 2, we state our main results concerning the large-time asympt otics of the relaxed maximizers.\nWe prove in Theorems 1and2that, in large time T, maximizing CT(a) overULis approximately equivalent\n3to maximizing the lowest eigenvalue σ1(a) of a matrix – when A0is selfadjoint and its lowest eigenvalue λ1\nis simple then σ1(a) =/integraltext\nΩa|φ1|2whereφ1is a normalized eigenfunction of −A0associated to λ1. The latter\nlimit problem has a unique solution a1=1ω∗, withω∗enjoying nice regularity properties. We prove that\nany maximizer a⋆\nT∈ULofCTconverges to 1ω∗inL1norm, exponentially as T→+∞at a rate given by the\nspectral gap of the operator A0. To determine the speed of convergence, we use an improved, q uantitative\nversion of the so-called bathtub principle , inspired by [ 31] and [8, Prop. 2.7]. The use of this technique is\none of the main novelties of the present paper.\nWe provide in Section 2.2several examples and we comment on the assumptions under whi ch our main\nresults are established. Following a relevant comment by Sy lvain Ervedoza during a presentation of one of\nus, we comment in Section 2.3on the strategy of proof that we develop, showing that, altho ugh the main\nresult is perfectly intuitive, a naive approach of proof is b ound to fail, and we have to resort to a more\nelaborate technique using kinds of quasi-maximizers.\nThe proofs of the main results are done in Section 3.\nIn the last Section 4, we comment on related problems: first, the optimal controll ability domain problem,\nwhich is dual to the one investigated in this paper; second, i n contrast to what is developed here, the small-\ntime asymptotics of the optimal observability problems. Fi nally we give some open problems.\n2 Large-time behavior of maximizers\n2.1 Main results\nThroughout the paper, given y= (y1,...,yq) andz= (z1,...,zq) inCq, we denote by |y|= (/summationtextq\ni=1|yi|2)1/2\nthe Hermitian norm and by y·¯z=/summationtextq\ni=1yi¯zithe Hermitian inner product.\nWe consider the following assumptions:\n(H1) There exists a Hilbert basis ( φj)j∈IN∗ofL2(Ω,Cq) consisting of (complex-valued) eigenfunctions of A0,\nassociated with eigenvalues ( λj)j∈IN∗of finite multiplicity such that\nRe(λ1)/lessorequalslant···/lessorequalslantRe(λj)/lessorequalslant··· →+∞asj→+∞.\n(H2) For every j∈IN∗, the eigenfunction φjis analytic in Ω.\nWe define\nJ1={j∈IN∗|Reλj= Reλ1},\np0= min{p∈IN∗|Reλp>Reλ1}= min(IN∗\\J1)/greaterorequalslant2.\nThe positive real number Re λp0−Reλ1is thespectral gap of the operator A0.\n(HL) (Bathtub property at level L) For every j∈J1, the function φjbelongs toC1(Ω). Given any real-valued\nfunction Φ on Ω written as\nΦ =/summationdisplay\np∈J1ηp/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈J1βj,pφj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nfor some (βj,p)j,p∈J1∈IRN1×P\\{0},{ηp}p∈J1∈[0,1]Psuch that/summationtext\np∈J1ηp= 1, the set ω={Φ/greaterorequalslantν}\nwhereν >0 is such that |ω|=L|Ω|, satisfies\ninf\n∂ω\\∂Ω|∇Φ|>0.\n4InSection 2.2, wewill discussthese assumptions, give various examples, andshow that Assumption( HL)\ncannot be weakened.\nThis is because of Assumptions ( H1) and (H2) that we have used, in this paper and in particular in\nits title, the word “parabolic”. However, the word may be a bi t of a misnomer because it often means\nthat the semigroup generated by −A0is analytic. However, there exist some analytic hypoellipt ic operators\nsatisfying all above assumptions, while the semigroup they generate is not analytic [ 11, Remark 4.1].\nUnder (H1), for anyy0∈D(A0), the solution yof (1) such that y(0,·) =y0can be expanded as\ny(t,x) =+∞/summationdisplay\nj=1aje−λjtφj(x),\nwhere\naj=/integraldisplay\nΩy0(x)φj(x)dx∀j∈IN∗.\nUsing the change of variable bj=aje−λjTand a homogeneity argument, we have\nCT(a) = inf/summationtext+∞\nj=1|bj|2=1/integraldisplayT\n0/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+∞/summationdisplay\nj=1bjeλjtφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndxdt. (8)\nBefore stating our main results, let us give some notations. For anyj∈IN∗, set\nγj(T) =\n\ne2Re(λj)T−1\n2Re(λj)if Re(λj)/\\e}atio\\slash= 0,\nT if Re(λj) = 0.\nWe define the Hermitian matrix\nM1(a) =/parenleftbigg/integraldisplay\nΩa(x)φi(x)·φj(x)ds/parenrightbigg\ni,j∈J1\nand we denote by σ1(a) the lowest eigenvalue of M1(a). When #J1= 1, i.e.,J1={1}(this is the case if A0\nis selfadjoint and λ1is simple), then σ1(a) =M1(a) =/integraltext\nΩa|φ1|2.\nTheorem 1. Under(H1), we have\n¯δT∼γ1(T) max\n1ω∈ULσ1(1ω)asT→+∞. (9)\nUnder(H1),(H2)and(HL), any solution a⋆\nT∈ULof Problem (PT)satisfies\n/vextenddouble/vextenddouble/vextenddoublea⋆\nT−argmax\na∈ULσ1(a)/vextenddouble/vextenddouble/vextenddouble\nL1(Ω)= O/parenleftBig\ne−(1−ξ)Re(λp0−λ1)T/2/parenrightBig\n∀ξ>0. (10)\nRemark 1. Note that ( 9) makes sense since\nmax\n1ω∈ULσ1(1ω) = max\na∈ULσ1(a)/greaterorequalslantσ1(L) =L>0.\nTheorem 1highlights the interest of the shape optimization problem\nmax\na∈ULσ1(a). (Pσ1)\n5Theorem 2. Under(H1)and(HL), the problem (Pσ1)has a unique1solutiona1=1ω∗∈ ULsatisfying\nthe following properties:\n•There exists µ∗>0such that\nω∗=/braceleftbigg\nx∈Ω|/summationdisplay\nk∈J1αk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈J1bk\njφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n>µ∗/bracerightbigg\n, (11)\nfor some (αk)k∈J1∈[0,1]#J1and(bk\nj)j∈J1∈C#J1such that\n/summationdisplay\nj∈J1αj=/summationdisplay\nj∈J1|bk\nj|2= 1∀k∈J1.\n•There exists K >0such that\nσ1(a1)/greaterorequalslantσ1(a)+K/ba∇dbla−a1/ba∇dbl2\nL1(Ω)∀a∈UL. (12)\n•If in addition (H2)holds, then ω∗is semi-analytic2. In particular, it has a finite number of connected\ncomponents.\nRemark 2. Concerning the randomized observability constant defined b y (7), it follows from some results\nin [39] that, forTis large enough,\nsup\na∈ULCT,rand(a) =γ1(T) max\n1ω∈ULσ1(1ω).\nTherefore, Theorem 1shows that, in large time, maximizing the deterministic obs ervability constant is\nalmost equivalent to maximizing the randomized observabil ity constant.\nRemark 3. The exponential rate of convergence ( 10) in Theorem 1is established thanks to a quantitative\nversion of the bathtub principle . The bathtub principle is an elementary inequality yieldin g the solutions of\nthe shape optimization problem\nsup\nω∈UL/integraldisplay\nωf\nwherefis a given real-valued integrable function on Ω. In recent ye ars, quantified versions of this principle\nwere derived and used in optimal control problems. We will bu ild on the version stated in [ 31, Proposition\n24]. We refer to Section 4.1for more background on this inequality but we merely state he re that, while it\nis now a known tool for optimal control theory, it is to the bes t of our knowledge the first time that such\nquantified inequalities are used in the study of observabili ty constants.\nRemark 4. In the theory of rearrangements, the quantitative estimate (12) can be seen as a strengthened\nHardy-Littlewood inequality; to the best of our knowledge, the first paper to investigate such strengthening\nwas [10]. However, the results of [ 10] do not apply here, as they are given in a different functional s etting\n1Here and in the sequel, it is understood that the optimal set i s unique within the class of all measurable subsets of Ω\nquotiented by the set of all measurable subsets of Ω of zero me asure.\n2A subset ωof a real analytic finite dimensional manifold Mis said to be semi-analytic if it can be written in terms of\nequalities and inequalities of analytic functions. Recall that semi-analytic subsets are Whitney stratifiable (see [ 19,21]) and\nenjoy local finiteness properties, such that: local finite pe rimeter, locally finite number of connected components, etc .\n6(roughly speaking, with Lp−Lq(1< p <∞,q=p/(p−1)) constraints rather than L∞−L1). The\nquantitative estimate ( 12) can take different forms. We refer to [ 6, Lemma 7.73], [ 31, Proposition 24], for\nsimilar spectral quantitative inequalities in a different co ntext and to [ 27, Theorem 1] wheresuch inequalities\nare used to study the stability of Vlasov-Poisson systems. T he proofs of the inequality in the three latter\nreferences revolve around the same idea, that of partial Sch warz rearrangements. Finally, we mention [ 8,\nProposition 2.7] where similar inequalities are obtained i n the context of stability of bang-bang optimal\ncontrols, with a more direct proof that we will follow.\n2.2 Examples and comments on the assumptions\nThe Dirichlet-Laplacian operator. Takeq= 1 and−A0to be the Dirichlet-Laplacian operator defined\non its domain D(A0) =H1\n0(Ω)∩H2(Ω). It follows from the spectral theorem [ 7, Section 6.4] that ( H1) is\nsatisfied. By hypoelliptic analyticity of the Laplacian, th e eigenfunctions of A0are analytic in Ω (and we\ntake them real-valued) and ( H2) is satisfied. Regarding ( HL), note that the first eigenvalue λ1(Ω) ofA0is\nsimple (hence, J1={1}) according to the Krein-Rutman theorem and the eigenfuncti onφ1is positive in\nΩ. By (11), we haveω∗={φ2\n1>µ∗}and, by analyticity of φ1,ω∗satisfies the interior sphere property (see\n[20]),∂ωis at a positive distance of ∂Ω and, according to the Hopf maximum principle,\ninf\n∂ω/ba∇dbl∇φ1/ba∇dbl>0.\nHence (HL) is satisfied.\nThe Robin-Laplacian operator. Takeq= 1,β >0 and−A0to be the Robin-Laplacian operator with\nboundarycondition ∂ny+βy= 0 on∂Ω. By similar arguments, ( H1) and(HL) aresatisfied. Theanalyticity\nproperty ( H2) follows for instance from [ 15, Section V.4].\nThe Dirichlet Stokes operator in the 2D unit disk. Consider the Stokes equation\n∂ty−△y+∇p= 0,divy= 0\nin Ω ={x∈IR2| /ba∇dblx/ba∇dbl<1}, the Euclidean unit disk of IR2, with Dirichlet boundary conditions. The Stokes\noperatorA0:D(A0)→His defined by A0=−P△, withD(A0) ={y∈V|A0y∈H}, withV={y∈\n(H1\n0(Ω))2|divy= 0},H={y∈(L2(Ω))2|divy= 0, y|∂Ω.n= 0}andP: (L2(Ω))2=H⊥⊕H⊥→Hthe\nLeray projection (see [ 5]). The first eigenfunction is given by\nφ0,1(r,θ) =−J′\n0(/radicalbig\nλ0,1r)√π/radicalbig\nλ0,1|J0(/radicalbig\nλ0,1)|/parenleftbigg−sinθ\ncosθ/parenrightbigg\n,\nin polar coordinates ( r,θ) (see [23,26]), whereJ0is the Bessel function of the first kind of order 0 and\nλ0,1=z2\n1,1withz1,1>0 is the first positive zero of J0. Using that φ0,1is radially-symmetric, it follows from\nproperties of the Bessel functions that ( H1), (H2) and (HL) are satisfied.\nSystem of coupled heat equations in the cube. SettingY= (y1,y2,y3)⊤, consider the system of\ncoupled 1D heat equations\n∂tY−△Y+AY= 0,\n7in Ω = (0,1)3, with Dirichlet boundary conditions, where △Y= (△y1,△y2,△y3)⊤, whereA∈M3(C) is a\n3×3 matrix with three distinct complex eigenvalues µ1,µ2,µ3and such that −π2<Reµ1= Reµ2<Reµ3.\nLetEi= Span(ui), withui∈C3, be the eigenspace of Aassociated to µi, fori∈ {1,2,3}. Setρ= Reµ1.\nThe above system is known to be observable in any time T >0 on any nonempty open subset ωof Ω.\nWe refer for instance to [ 2, Theorem 1.1] or [ 30, Proof of the first point of Theorem 1].\nHere, we have q= 3,A0=△+AId onD(A0) = (H2(Ω)∩H1\n0(Ω))3,J1={1,2},p0= 3 and\nγ1(T) = (e2ρT−1)/(2ρ). The assumptions ( H1) and (H2) are satisfied, and φ1(x) = sin(πx1)sin(πx2)u1\nandφ2(x) = sin(πx1)sin(πx2)u2. We define\nΦ(x) =ηsin2(πx1)sin2(πx2)|β11u1+β12u2|2+(1−η)sin2(πx1)sin2(πx2)|β21u1+β22u2|2,\nwhereη∈[0,1] andβij/\\e}atio\\slash= 0. Reasoning as before, and using that β11u1+β12u2/\\e}atio\\slash= 0 andβ21u1+β22u2/\\e}atio\\slash= 0\n(by linear independence of u1andu2), we get that |∇Φ|is uniformly bounded below on any nontrivial level\nset of Φ, and thus ( HL) is satisfied.\nOn the sharpness of (HL): periodic Laplacian in the 1D torus. Take Ω = (0 ,2π),q= 1 and\nA0=−d2/dx2with periodic boundary conditions y(0) =y(2π) andy′(0) =y′(2π). The first eigenvalue is\nλ1= 1 with eigenspace spanned by x/ma√sto→cosxandx/ma√sto→sinx. For every a∈UL, we have\nM1(a) =/parenleftBigg/integraltext2π\n0a(x)cos2xdx/integraltext2π\n0a(x)cosxsinxdx\n/integraltext2π\n0a(x)cosxsinxdx/integraltext2π\n0a(x)sin2xdx/parenrightBigg\n,\nand thus\nσ1(a) =1\n2/integraldisplay2π\n0a(x)dx−1\n2/radicalBigg/parenleftbigg/integraldisplay2π\n0a(x)cos(2x)dx/parenrightbigg2\n+/parenleftbigg/integraldisplay2π\n0a(x)sin(2x)dx/parenrightbigg2\n.\nTherefore, the optimal design problem ( Pσ1) (maximizing σ1overUL) is equivalent to\nmax\na∈UL/parenleftbigg/integraldisplay2π\n0a(x)cos(2x)dx/parenrightbigg2\n+/parenleftbigg/integraldisplay2π\n0a(x)sin(2x)dx/parenrightbigg2\n.\nIt is easy to see that the optimal value for this problem is 0, m eaning that the maximal value of σ1isLπ\n2.\nMoreover, the maximal value is reached for any function agiven by\na(x) =L\n2+η/parenleftbigg\nac\n1cosx+as\n1sinx+m/summationdisplay\nk=3(ac\nkcos(kx)+as\nksin(kx))/parenrightbigg\nfor anym∈IN\\{0,1,2}and any families of real numbers ( as\nk)1/lessorequalslantk/lessorequalslantmand (ac\nk)1/lessorequalslantk/lessorequalslantm, withηchosen small\nenough such that a∈UL. Hence, there exist maximizers in UL\\ULand the conclusion of Theorem 2is not\ntrue in that case. The main reason is that ( HL) is not satisfied here. Indeed, one has # J1= 2 and it is\npossible to choose adequately the coefficients defining Φ so th at Φ(x) = cos2x+sin2x= 1, and its normal\nderivative vanishes on ∂ω.\n2.3 Comments on the strategy of proof\nIn this section, we describe our strategy of proof, highligh ting the difficulties that will be encountered, on\nthe example of the Dirichlet heat equations. We take q= 1 and −A0equal to the Dirichlet-Laplacian on\n8D(A0) =H1\n0(Ω)∩H2(Ω). The operator A0is selfadjoint, of compact resolvent and then its spectrum i s real\nand the first eigenvalue λ1(Ω) is simple, and we take real-valued eigenfunctions.\nRecalling that ¯δTis the optimal value for Problem ( PT), defined by ( PT), considering particular choices\nof coefficients bjin (8), we have\n¯δT/lessorequalslantγ1(T) max\n1ω∈UL/integraldisplay\nΩa(x)φ1(x)2dx∀a∈UL (13)\n(see the beginning of Section 3.2for details).\nLet us now establish a lower bound for ¯δT. LetaT\n1∈ULbe the unique solution to the shape optimization\nproblem on the right-hand side of ( 13). We define\nET,a(b) =/integraldisplayT\n0/integraldisplay\nΩa(x)/parenleftbigg+∞/summationdisplay\nj=1bjeλjtφj(x)/parenrightbigg2\ndxdt\nand we set ε=|b1|2andcj=bj/√1−εforj/greaterorequalslant2, so that/summationtext+∞\nj=2|cj|2= 1. A straightforward computation\ngives\nET,a(b) =εAT,a(c)+(1−ε)BT,a(c)+2/radicalbig\nε(1−ε)DT,a(c)\nwhere\nAT,a(c) =γ1(T)/integraldisplay\nΩaφ2\n1,\nBT,a(c) =/integraldisplayT\n0/integraldisplay\nΩa(x)/parenleftbigg+∞/summationdisplay\nj=2cjeλjtφj(x)/parenrightbigg2\ndxdt,\nDT,a(c) =/integraldisplayT\n0/integraldisplay\nΩa(x)eλ1tφ1(x)+∞/summationdisplay\nj=2cjeλjtφj(x)dxdt=+∞/summationdisplay\nj=2cje(λj+λ1)T−1\nλ1+λj/integraldisplay\nΩaφ1φj.\nFrom the maximality of ¯δT, we have\n¯δT/greaterorequalslantinf/summationtext+∞\nj=2c2\nj=1min\nε∈[0,1]εAT,a(c)+(1−ε)BT,a(c)+2/radicalbig\nε(1−ε)DT,a(c)\nDue to the presence of exponential terms, we could expect to b e able to exploit this decomposition to\nprove that the first mode dominates the others when Tis sufficiently large. This does not work however,\nessentially because the modes j/greaterorequalslant2 in the decomposition can modify the observability constan t in large\ntime. To understand this difficulty, note that, by the Cauchy- Schwarz inequality,\n|DT,aT\n1(c)|/lessorequalslant/radicalBig\nAT,aT\n1(c)BT,aT\n1(c).\nTherefore\n¯δT/greaterorequalslantmin\nε∈[0,1]/parenleftBig\nεAT,aT\n1(c)+(1−ε)BT,aT\n1(c)−2/radicalbig\nε(1−ε)/radicalBig\nAT,aT\n1(c)BT,aT\n1(c)/parenrightBig\n= 0.\nThe latter equality follows from the fact that the minimum is equal to the first eigenvalue of the matrix\n\nAT,aT\n1(c)/radicalBig\nAT,aT\n1(c)BT,aT\n1(c)\n/radicalBig\nAT,aT\n1(c)BT,aT\n1(c)BT,aT\n1(c)\n.\n9Unfortunately, this estimate is not useful. The difficulty he re is due to the presence of crossed terms that\nare difficult to handle. To overcome it, we will consider a part icular path of “quasi-maximizers”, replacing\naT\n1by a convex combination of the expected maximizer in large ti me and the constant function equal to L.\nWe will choose appropriate convexity weights, to give an inc reasing importance to the term aT\n1asT→+∞.\n3 Proofs\n3.1 Proof of Theorem 2\nFirst of all, noting that, for every a∈UL,\nσ1(a) = min/summationtext\nj∈J1|bj|2=1/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈J1bjφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx\nis as minimum of linear continuous functionals, σ1is upper semicontinuous for the weak star topology of\nL∞. By compactness of ULfor this topology, it follows that ( Pσ1) has at least one solution a1.\nWe are going to prove that a1is unique and that a1=1ω∗for some subset ω∗(“bang-bang property”).\nBang-bang property. In order to prove that a1is actually the characteristic function of some measur-\nable subset, we exploit first-order optimality conditions. From [13, Theorem 1], since σ1(a) has a finite\nmultiplicity, the mapping UL∋a/ma√sto→σ1(a) is sub-differentiable at a1. Let (bk)k∈/tildewideJ1be an orthonormal basis\nof the eigenspace associated to σ1(a1), with/tildewideJ1⊂J1. We have\n∂σ1(a1) = co/braceleftBig\n∂jk(a1)|k∈/tildewideJ1/bracerightBig\nwithjk(a) =/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈/tildewideJ1bk\njφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx,\nwhere “co” means “convex hull”. We consider the tangent cone3Ta1to the set ULata1, and the indicator\nfunctionι[0,1]given by\nιUL(u) =/braceleftbigg\n0 ifu∈UL,\n+∞otherwise.\nFinally, we define the linear functional j0:UL∋a/ma√sto→/integraltext\nΩa. The optimization problem ( Pσ1) can be recast\nas\nmin\na∈L∞(Ω)\nj0(a)=L|Ω|/parenleftBig\n−σ1(a)+ιUL(a)/parenrightBig\n.\nIfa1is an optimal solution, according to the Lagrange multiplie r rule, there must exist µ∗∈IR such that\n0∈∂(−σ1)(a1)+µ∗∂j0(a1)+∂ιUL(a1).\nEquivalently, there exists ( αk)k∈/tildewideJ1∈IR#/tildewideJ1\n+such that/summationtext\nk∈/tildewideJ1αk= 1 and for every h∈L∞(Ω) such that\n0/lessorequalslanta1+ηh/lessorequalslant1 forη>0 small enough,\n−/summationdisplay\nk∈/tildewideJ1αk/a\\}b∇acketle{tdjk(a1),h/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tdj0(a1),h/a\\}b∇acket∇i}ht= 0⇐⇒/integraldisplay\nΩh(x)(Ψ(x)−µ∗)dx/greaterorequalslant0, (14)\n3The tangent cone Ta1is the set of functions h∈L∞(Ω) such that, for any sequence of positive real numbers εndecreasing\nto 0, there exists a sequence of functions hn∈L∞(Ω) converging to hasn→+∞, anda1+εnhn∈Ulfor every n∈IN (see\n[12] for example).\n10where the function Ψ : Ω →IR+is defined by\nΨ(x) =/summationdisplay\nk∈/tildewideJ1αk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈/tildewideJ1bk\njφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n.\nLet us now prove that a1is necessarily bang-bang , i.e., thata1=1ω∗for some Lebesgue measurable ω∗⊂Ω.\nBy contradiction, assume that |{0< a1<1}|>0. Letx0be a Lebesgue point of {0< a1<1}. There\nexists a sequence ( Gn)n∈INof measurable subsets with Gn⊂ {0<a1<1}such that\nlim\nε→0|Gn∩B(x0,ε)|\n|B(x0,ε|= 1, (15)\nwhereB(x0,ε) denotes the ball centered at x0with radius εin IRd. Settingh=1Gnand noting that\n0/lessorequalslanta1±ηh/lessorequalslant1 ifηis small enough, we infer from ( 14) that\n±/integraldisplay\nGn(Ψ(x)−µ∗)dx/greaterorequalslant0.\nDividing this inequality by |Gn|and letting Gnshrink to {x0}asn→+∞we infer that Ψ( x) =µ∗a.e. in\n{0<a1<1}. Since 0<|{0<a1<1}|/lessorequalslant|{Ψ =µ∗}|and Ψ is analytic, we must have Ψ = µ∗and∇Ψ = 0\neverywhere in Ω, which contradicts ( HL).\nWe thus infer that |{0<a1<1}|= 0 and hence a1=1ω∗for some Lebesgue measurable subset ω∗of\nΩ such that |ω∗|=L|Ω|.\nCharacterization of ω∗.Consider a Lebesgue point x0ofω∗. Let (Gn)n∈INbe a sequence of measurable\nsubsets such that Gn⊂ω∗and (15) holds. Setting h=−1Gnand noting that 0 /lessorequalslanta1+ηh/lessorequalslant1 ifηis small\nenough, we infer from ( 14) that/integraldisplay\nGn(Ψ(x)−µ∗)dx/greaterorequalslant0.\nDividing this inequality by |Gn|and letting Gnshrink to {x0}asn→+∞yields Ψ /greaterorequalslantµ∗a.e. inω∗.\nSimilarly, Ψ /lessorequalslantµ∗in (ω∗)c. We conclude that ω∗={Ψ>µ∗}by noting that {Ψ =µ∗}has zero Lebesgue\nmeasure.\nUniqueness and regularity of ω∗.Assume by contradiction that Problem ( Pσ1) has two distinct max-\nimizersa1anda2. As a consequence of our previous analysis, there exist ω1,ω2⊂Ω such that ai=1ωi\n(i= 1,2). By concavity of σ1, (a1+a2)/2 is also a solution of Problem ( Pσ1), which contradicts the\nbang-bang property of maximizers.\nFinally, the regularity property of ω∗follows from ( H2), which implies the analyticity of Ψ. Therefore,\nω∗is an open semi-analytic set.\nQuantitative estimate. At last, we prove the quantitative estimate ( 12); the argument we follow to\nestablish it is inspired and adapted from [ 8]. Observe that with the same function Ψ we have, for any\na∈UL,\nσ1(a)−σ1(a1)/lessorequalslant/integraldisplay\nΩ(a−a1)Ψ.\n11Let us prove that there exists C >0 such that\n∀a∈UL,/integraldisplay\nΩ(a−a1)Ψ/lessorequalslant−C/ba∇dbla−a1/ba∇dbl2\nL1(Ω),\nwhich can be seen as a quantitative version of the bathtub pri nciple.\nWe have proved that a1=1ω∗for someω∗⊂Ω such that |ω∗|=L|Ω|, and moreover that Ψ /greaterorequalslantµ∗inω∗\nand Ψ/lessorequalslantµ∗in Ω\\ω∗. Noting that for every a∈UL,/integraltext\nΩ(a−a1)Ψ =/integraltext\nΩ(a−a1)(Ψ−µ∗), thata−a1/lessorequalslant0 on\nω∗anda−a1/greaterorequalslant0 onω∗, we get that\n/integraldisplay\nΩ(a−a1)Ψ =−/integraldisplay\nΩ|a−a1|.|Ψ−µ∗|/lessorequalslant−/integraldisplay\nΩδ|a−a1|.|Ψ−µ∗|\nwhere, for a given δ >0, the set Ω δ={|Ψ−µ∗|/greaterorequalslantδ}is the complement of a tubular neighborhood of the\nlevet set {Ψ =µ∗}. By definition of Ω δ, it follows that\n/integraldisplay\nΩ(a−a1)Ψ/lessorequalslant−δ/ba∇dbla−a1/ba∇dblL1(Ωδ)=−δ(/ba∇dbla−a1/ba∇dblL1(Ω)−/ba∇dbla−a1/ba∇dblL1(Ω\\Ωδ))\n/lessorequalslant−δ(/ba∇dbla−a1/ba∇dblL1(Ω)−|Ω\\Ωδ|) (16)\nLemma 1. There exists M >0such that |Ω\\Ωδ|/lessorequalslantMδfor everyδ>0.\nProof.It suffices to prove this estimate for δ >0 small enough. We establish the existence of K >0 such\nthat ifs>0 is small enough then {Ψ =µ∗+s} ⊂Σµ∗+KδBwhere Σ µ∗={Ψ =µ∗}andBis the centered\nunit ball. Assuming such a Kexists, we infer that\nΩ\\Ωδ=δ/uniondisplay\ns=−δ{Ψ =µ∗+s} ⊂Σµ∗+KδB.\nNow, usingthedefinitionoftheperimeterofrectifiablecurv esviatheMinkowski content, weobtain |Ω\\Ωδ|/lessorequalslant\n2δPer(Σµ∗) forδsmall enough, whence the conclusion with M= 2KPer(Σµ∗). It thus remains to show\nthat\ndistH(Σµ∗,{Ψ =µ∗+s})/lessorequalslantMs\nfors>0 small enough, where dist His the Hausdorff distance between two compact sets.\nBy contradiction, assume that there exist two sequences ( xk)k∈INin Σµ∗and (yk)k∈INin Σµ∗+sk, where\n(sk)k∈INdenotes a sequence converging to 0, such that\n|xk−yk|= distH(Σµ∗+sk,Σµ∗) and lim\nk→+∞|Ψ(xk)−Ψ(yk)|\n|xk−yk|= 0.\nBy the mean value theorem, there exists zkbelonging to the segment [ xk,yk] such that\n|Ψ(xk)−Ψ(yk)|\n|xk−yk|=/angbracketleftbigg\n∇Ψ(zk),yk−xk\n|yk−xk|/angbracketrightbigg\n.\nSince Σ µ∗isC2, it satisfies the uniform interior ball property, and theref oreyk=xk+tkν(xk)+o(tk) fork\nlarge enough, with tk→0, whereν(xk) denotes the outward unit normal vector on Σ µ∗=∂{Ψ/lessorequalslantµ∗}. We\ninfer that /a\\}b∇acketle{t∇Ψ(zk),ν(xk)+o(1)/a\\}b∇acket∇i}ht →0 ask→+∞. This contradicts ( HL).\nGiven anya∈UL, we setδ=1\n2M/ba∇dbla−a1/ba∇dblL1(Ω). By (16) and by Lemma 1, we have\n/integraldisplay\nΩ(a−a1)Ψ/lessorequalslant−1\n2M/ba∇dbla−a1/ba∇dblL1(Ω)/parenleftbigg\n/ba∇dbla−a1/ba∇dblL1(Ω)−M\n2M/ba∇dbla−a1/ba∇dblL1(Ω)/parenrightbigg\n=−1\n4M/ba∇dbla−a1/ba∇dbl2\nL1(Ω).\nThe conclusion follows.\n123.2 Proof of Theorem 1\nConsidering particular choices of coefficients bjequal to 0 in IN∗\\J1in (8), we have\n¯δT/lessorequalslantγ1(T) sup\na∈ULmin/summationtext\nj∈J1|bj|2=1/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈J1bjφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx\n=γ1(T) max\n1ω∈ULmin/summationtext\nj∈J1|bj|2=1/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈J1bjφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx\n=γ1(T) max\n1ω∈ULσ1(1ω). (17)\nTo go from the first to the second line, we used Theorem 2. The rest of the proof is devoted to establishing\na lower bound on ¯δT. We define\nET,a(b) =/integraldisplayT\n0/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+∞/summationdisplay\nj=1bjeλjtφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndxdt.\nand we set ε=/summationtext\nj∈J1|bj|2(so that 1 −ε=/summationtext\nj∈IN∗\\J1|bj|2). We also define sequence ( cj)∈IN∗inℓ2(IR) by\ncj=/braceleftbiggbj/√εifj∈J1\nbj/√1−εotherwise\nso that /summationdisplay\nj∈J1|cj|2=/summationdisplay\nj∈IN∗\\J1|cj|2= 1.\nWe define the two subsets of ℓ2(C)\nΛJ1=/braceleftbigg\n(cj)∈C#J1,/summationdisplay\nj∈J1|cj|2= 1/bracerightbigg\nand Λ IN∗\\J1=/braceleftbigg\n(cj)∈ℓ2(IN∗\\J1),/summationdisplay\nj∈IN∗\\J1|cj|2= 1/bracerightbigg\n.\nA straightforward computation shows that\nET,a(b) =εAT,a(c)+(1−ε)BT,a(c)+2/radicalbig\nε(1−ε)DT,a(c),\nwhere\nAT,a(c) =γ1(T)/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈J1cjφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx\nBT,a(c) =/integraldisplayT\n0/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈IN∗\\J1cjeλjtφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndxdt\nDT,a(c) = Re/integraldisplay\nΩ/integraldisplayT\n0a(x)/summationdisplay\nk∈J1ckeλ1tφk(x)/summationdisplay\nj∈IN∗\\J1cjeλjtφj(x)dtdx\nConsequently,\nCT(a) = inf /summationtext\nj∈I N∗|bj|2=1ET,a(b)\n= inf\nε∈[0,1]inf\n(cj)∈ΛJ1/parenleftbigg\nεAT,a(c)+ inf\n(cj)∈ΛI N∗\\J1(1−ε)BT,a(c)+2/radicalbig\nε(1−ε)DT,a(c)/parenrightbigg\n13Moreover, note that\ninf\n(cj)∈ΛJ1AT,a(c) =γ1(T)σ1(a)\nLet us consider the unique solution a1=1ω∗of (Pσ1). From now on, with a slight abuse of notation, we\nwill use the notation AT,a1=γ1(T)σ1(a1).\nLetν∈(0,1) to be chosen later, and define aν=νa1+(1−ν)L∈UL(by convexity). By maximality\nof¯δT, we have\n¯δT/greaterorequalslantinf\nε∈[0,1]inf\n(cj)∈ΛI N∗\\J1(εAT,aν(c)+(1−ε)BT,aν(c)+2/radicalbig\nε(1−ε)DT,aν(c))\n/greaterorequalslantinf\nε∈[0,1]inf\n(cj)∈ΛI N∗\\J1(νεAT,a1+ν(1−ε)BT,a1(c)+(1−ν)(1−ε)BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c)).\nTo obtain this estimate, we have used that DT,aν(c) =νDT,a1(c), since\nDT,L(c) =LRe/summationdisplay\nj∈IN∗\\J1\nk∈J1cjcke(λj+λ1)T−1\nλ1+λj/integraldisplay\nΩφk(x)·φj(x)dx= 0\nby orthogonality of the functions φjinL2(Ω,Cq).\nWe have\ninf\n(cj)∈ΛI N∗\\J1(νεAT,a1+ν(1−ε)BT,a1(c)+(1−ν)(1−ε)BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c))\n=νεAT,a1+ inf\n(cj)∈ΛI N∗\\J1ν(1−ε)BT,a1(c)+(1−ν)(1−ε)BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c)\n/greaterorequalslantνεAT,a1+ inf\n(cj)∈ΛI N∗\\J1/parenleftBig\n(1−ν)(1−ε)BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c)/parenrightBig\n/greaterorequalslantνεAT,a1+(1−ν)(1−ε)Lγp0(T)\n2\n+ inf\n(cj)∈ΛI N∗\\J1/parenleftbigg(1−ν)(1−ε)\n2BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c)/parenrightbigg\n(18)\nby using that\nBT,L(c) =L/summationdisplay\nj∈IN∗\\J1|cj|2/integraldisplayT\n0e2Reλjtdt=L+∞/summationdisplay\nj=p0γj(T)|cj|2\nand that/summationtext+∞\nj=p0γj(T)|cj|2/greaterorequalslantγp0(T)/summationtext+∞\nj=p0|cj|2=γp0(T).\nLemma 2. LetT >0andµ∈(0,1)be given. We define\nεν,T,L=L2(1−ν)2γp0(T)\n16ν2γ1(T)+L2(1−ν)2γp0(T)∈(0,1). (19)\nFor everyε∈[0,εν,T,L], we have\ninf\n(cj)∈ΛI N∗\\J1/parenleftBig(1−ν)(1−ε)\n2BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c)/parenrightBig\n/greaterorequalslant0.\n14Proof of Lemma 2.By the Cauchy-Schwarz inequality, we have\n|DT,a1(c)|2=/vextendsingle/vextendsingle/vextendsingle/vextendsingleRe/integraldisplay\nΩ/integraldisplayT\n0a1(x)p0−1/summationdisplay\nk=1ckeλ1tφk(x)+∞/summationdisplay\nj=p0cjeλjtφj(x)dtdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslant/integraldisplay\nΩ/integraldisplayT\n0a1(x)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglep0−1/summationdisplay\nk=1ckeλ1tφk(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndtdx/integraldisplay\nΩ/integraldisplayT\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle+∞/summationdisplay\nj=p0cjeλjtφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndtdx\n/lessorequalslantp0−1/summationdisplay\nk=1γ1(T)|ck|2+∞/summationdisplay\nj=p0γj(T)|cj|2=γ1(T)+∞/summationdisplay\nj=p0γj(T)|cj|2.\nTo go from the second to the third line we used the fact that 0 /lessorequalslanta1(·)/lessorequalslant1. Therefore,\n(1−ν)(1−ε)\n2BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c)/greaterorequalslant(1−ν)(1−ε)\n2LXT(c)2−2ν/radicalbig\nε(1−ε)/radicalbig\nγ1(T)XT(c)\nwhere\nXT(c) =/parenleftbigg+∞/summationdisplay\nj=p0γj(T)|cj|2/parenrightbigg1/2\n.\nSinceXT(c)/greaterorequalslant/radicalbig\nγp0(T), the expected conclusion follows if\n/radicalBig\nγp0(T)/greaterorequalslant4ν√ε/radicalbig\nγ1(T)\nL(1−ν)√1−ε,\nwhich is satisfied for any ε∈[0,εν,T,L] whereεν,T,Lis defined by ( 19).\nLemma 3. One has\n¯δT/greaterorequalslantinf\nε∈[0,1]ψT,ν,L(ε), (20)\nwhere\nψT,ν,L(ε) =/braceleftBigg\nmin/parenleftBig\nνAT,a1,(1−ν)Lγp0(T)\n2/parenrightBig\nifε∈[0,εν,T,L]\n(1−ν)(1−ε)γp0(T) ifε∈(εν,T,L,1].(21)\nProof of Lemma 3.We combine Lemma 2and (18): first, observe that for any ε∈[0,1]\nνεAT,a1+(1−ν)(1−ε)Lγp0(T)\n2/greaterorequalslantmin/parenleftbigg\nνAT,a1,(1−ν)Lγp0(T)\n2/parenrightbigg\n.\nSecond, the expression of ψT,ν,Lon (εν,T,L,1] follows from ( 18), using that ET,a1(b)/greaterorequalslant0 for allb∈ℓ2(IR)\nand the fact that\nνεAT,a1+ν(1−ε)BT,a1(c)+(1−ν)(1−ε)BT,L(c)+2ν/radicalbig\nε(1−ε)DT,a1(c)/greaterorequalslant(1−ν)(1−ε)BT,L(c)\nfor allc∈ℓ2(IR).\nThe end of the proof consists in choosing νandεadequately to compute inf ε∈[0;1]ψT,ν,L(ε).\n15Lemma 4. There exists a family (νT)T>0converging to 1 as T→+∞such thatνT∈(0,1)for everyT >0,\nand\nlim\nT→+∞ενT,T,L= 1,lim\nT→+∞(1−νT)γp0(T)\nγ1(T)= +∞,lim\nT→+∞(1−νT)(1−ενT,T,L)γp0(T)\nγ1(T)= +∞.\nProof.Let (νT)T>0be a family of elements in [0 ,1] converging to 1 as T→+∞. LetθT= 1−νT. Since\nγp0(T)/γ1(T)∼e2TRe(λp0−λ1)asT→+∞, let us set θT=γ1(T))eηT/γp0(T), whereη∈IR∗\n+will be chosen\nappropriately. Note first that\n1−νT=θT∼e(η+2Re(λ1−λp0))TasT→+∞and(1−νT)γp0(T)\nγ1(T)=eηT.\nIn particular, this indicates that we should choose η∈(0,2Re(λp0−λ1)).\nRegarding the last equality of the lemma, note that\n1−ενT,T,L=16ν2\nTγ1(T)\n16ν2γ1(T)+L2(1−νT)2γp0(T)\nand thus\n(1−νT)(1−ενT,T,L)γp0(T)\nγ1(T)=16ν2\nT(1−νT)γp0(T)\n16ν2\nTγ1(T)+L2(1−νT)2γp0(T)=eηT\n1+L2e2ηT\n16ν2\nTγ1(T)\nγp0(T).\nSince\ne2ηTγ1(T)\nγp0(T)= O/parenleftBig\ne2T(η+λ1−λp0)/parenrightBig\n,\nchoosing any ηsuch thatη∈(Re(λp0−λ1),2Re(λp0−λ1)) yields the desired conclusion. Observe moreover\nthat, with this choice, we have\nενT,T,L=L2(1−νT)2γp0(T)\nL2(1−νT)2γp0(T)+16ν2\nTγ1(T)=L2e2ηTγ1(T)/γp0(T)\n16ν2\nT+L2e2ηTγ1(T)/γp0(T)= 1+O/parenleftbig\ne−2T(η−Re(λp0−λ1))/parenrightbig\nsinceη+Re(λ1−λp0)>0.\nConclusion: asymptotics of ¯δT.Let us consider ν=νTin (20), where (νT)T>0denotes the family\nconstructed in Lemma 4. One has\n¯δT/greaterorequalslantinf\nε∈[0,1]ψT,νT,L(ε)\nwhereψT,νT,Lis given by ( 21). From Lemma 4, there holds\nlim\nT→+∞(1−νT)γp0(T)L\n2γ1(T)= +∞and lim\nT→+∞(1−νT)(1−ενT,T,L)γp0(T)\nγ1(T)= +∞.\nFrom (20) we infer that\n¯δT\nγ1(T)/greaterorequalslantmin/parenleftbigg\nνTσ1(a1),(1−νT)γp0(T)L\n2γ1(T),(1−νT)(1−ενT,T,L)γp0(T)\nγ1(T)/parenrightbigg\n. (22)\nLettingTgo to +∞yields\nliminf\nT→+∞¯δT\nγ1(T)/greaterorequalslantσ1(a1).\nCombining this inequality with ( 17) finally gives ( 9).\n16Quantitative estimate on dist L1(Ω)/parenleftBig\na⋆\nT,argmax\na∈ULσ1(a)/parenrightBig\n.Since¯δT=CT(a⋆\nT) and since CT(a⋆\nT) is\ndefined as an infimum, considering particular choices of coeffi cientsbjequal to 0 in IN∗\\J1gives\nCT(a⋆\nT)/lessorequalslantγ1(T) min/summationtext\nj∈J1|bj|2/integraldisplay\nΩa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈J1bjφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx=γ1(T)σ1(a⋆\nT).\nNow, accordingto( 22), withtheparticularchoice ofparameter νTgiven byLemma 4, wealsohave CT(a⋆\nT)/greaterorequalslant\nγ1(T)νTσ1(a1) ifTis large enough, and therefore\nσ1(a⋆\nT)/greaterorequalslantCT(a⋆\nT)\nγ1(T)/greaterorequalslantνTσ1(a1).\nIn particular, this implies that 0 /lessorequalslantσ1(a1)−σ1(aT)/lessorequalslant(1−νT)σ1(a1). Using the constant Kgiven by\nTheorem 2we deduce that\nK/ba∇dbla⋆\nT−a1/ba∇dbl2\nL1(Ω)/lessorequalslant(1−νT)σ1(a1).\nBy construction of νTwe have\nνT∼e(η+2Re(λ1−λp0))TasT→+∞\nfor some fixed η∈(Re(λp0−λ1),2Re(λp0−λ1)), whence the conclusion.\n4 Conclusion and further comments\nIn this paper, we have described the large-time behavior of t he maximizers of the observability constant\nassociated with measurements of the solution of parabolic e quations on a subdomain.\nHereafter, we show by duality that our optimal observabilit y results can be applied as well to the optimal\nactuator (controllability) shapeand location problem. We then give a partial result regarding the small-time\nasymptotics, i.e., as T→0, of the optimal observability problem. We conclude with so me open questions.\n4.1 Controllability\nThe control system\n∂ty+A0y=u1ω (23)\nissaidtobeexactlynullcontrollableintime Tifeveryinitialdatum y(0,·)∈L2(Ω)canbesteeredto0intime\nTby means of an appropriate control function u∈L2((0,T)×Ω). It is well known that controllability and\nobservability are dual notions (see e.g.[14,44,45,48]), in the sense that the latter exact null controllability\nproperty is equivalent to the observability property of ( 1) onωin timeT, and the observability constant\nCT(1ω) coincides withthe inverse of theminimal L2control cost for the controllability problemfor ( 23). The\nHilbert Uniqueness Method (HUM, see [ 28,29,44,48]) provides a characterization unique control solving\nthe exact null controllability problem with a minimal L2norm. This control is referred to as the HUM\ncontrol and is characterized as follows. Define the HUM funct ionalJωby\nJω(φT) =1\n2/integraldisplayT\n0/integraldisplay\nω|φ(t,x)|2dxdt+/a\\}b∇acketle{tφT,y0/a\\}b∇acket∇i}htL2,\nwhereφis the solution of −∂tφ+A∗\n0φ= 0 such that φ(T,·) =φT. Under the observability assumption,\nthe functional Jωhas a unique minimizer and the HUM control uωsteeringy0to 0 in time Tisuω(t,x) =\n171ω(x)φ(t,x). The HUM operator Γ ω:L2(Ω)→L2((0,T)×Ω) is defined by Γ ω(y0) =uωand its operator\nnorm is given by\n/ba∇dblΓω/ba∇dbl= sup/braceleftBigg\n/ba∇dbluω/ba∇dblL2((0,T)×Ω)\n/ba∇dbly0/ba∇dblL2(Ω)|y0∈L2(Ω)\\{0)}/bracerightBigg\n.\nAdopting the convention 1 /0 = +∞, we have\n/ba∇dblΓω/ba∇dbl=1\nCT(1ω).\nWe refer to [ 36] for details. Thus\ninf\n1ω∈UL/ba∇dblΓω/ba∇dbl=/parenleftBig\nsup\n1ω∈ULCT(1ω)/parenrightBig−1\n.\nHence, the problem of maximizing the observability constan t is equivalent to the problem of minimizing the\noperator norm of Γ ω, which models the determination of a best control domain in s ome sense.\n4.2 Small-time asymptotics\nIn this section, ,we mention an issue similar to the one inves tigated in this article, which cannot be treated\nusing the same tools as in this paper: the limit of maximizers in small time, as T→0. We have the following\npartial result. It would be relevant to study how to obtain qu antitative estimates on the convergence of\n¯δT/T, as well as on the convergence of the maximizers.\nTheorem 3. Assume that q= 1and thatA0is the second-order elliptic differential operator\nA0=−d/summationdisplay\nj=1∂xj(gjk∂xk)+d/summationdisplay\nj=1bj∂xj+c\nwhere the functions gjk,bjandcare smooth on Ω. We assume that the matrix G= (gjk)1/lessorequalslantj,k/lessorequalslantdis real and\npositive definite in Ω, thatdet(G) = 1onΩand thatA0is symmetric for the Lebesgue measure on Ω. Then\n¯δT∼LTasT→0.\nProof.Using (8) and the fact that ( φj)j/greaterorequalslant1is a Hilbert basis, we first note that, taking a≡L,\nCT(L) = inf/summationtext+∞\nj=1|bj|2=1/integraldisplayT\n0/integraldisplay\nΩL/vextendsingle/vextendsingle/vextendsingle/vextendsingle+∞/summationdisplay\nj=1bjeλjtφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndxdt=Linf\nj∈IN∗γj(T) =Lγ1(T).\nSincea⋆\nTis a maximizer, we have ¯δT=CT(a⋆\nT)/greaterorequalslantCT(L) and thus\nliminf\nT→0¯δT\nT/greaterorequalslantLliminf\nT→0γ1(T)\nT=L.\nLet us prove the converse inequality. Actually, let us prove the stronger fact that\nlimsup\nT→0CT(a)\nT/lessorequalslantL∀a∈UL.\n18Using that CT(a) is defined by ( 8) as an infimum, taking bj= 1 and all other bkequal to 0, we have in\nparticular\nCT(a)/lessorequalslant/integraldisplayT\n0/integraldisplay\nΩa(x)eReλjt|φj(x)|2dxdt=γj(T)/integraldisplay\nΩa|φj|2∀j∈IN∗\nand thus, for every N∈IN∗,\nCT(a)/lessorequalslantinf/braceleftbiggN/summationdisplay\nj=1αjγj(T)/integraldisplay\nΩa|φj|2/vextendsingle/vextendsingle/vextendsingleα1,...,α N/greaterorequalslant0,N/summationdisplay\nj=1αj= 1/bracerightbigg\n.\nChoosing in particular αj=γj(T)−1//summationtextN\nk=1γk(T)−1for everyj∈ {1,...,N}, we infer that\n1\nNN/summationdisplay\nj=11\nγj(T)CT(a)/lessorequalslant/integraldisplay\nΩa1\nNN/summationdisplay\nj=1|φj|2(24)\nThe inequality ( 24) is valid for every a∈UL, everyN∈IN∗and everyT >0. FixingNand letting Ttend\nto 0, using that γj(T)→T, we obtain\nlimsup\nT→0CT(a)\nT/lessorequalslant/integraldisplay\nΩa1\nNN/summationdisplay\nj=1|φj|2∀N∈IN∗.\nTo conclude, we use the fact that the Ces` aro mean1\nN/summationtextN\nj=1|φj|2converges to the constant function1\n|Ω|as\nN→+∞, uniformly on any compact subset of the open set Ω for the C0topology and thus weakly in L2(Ω)\n(this follows from [ 22, Section 17.5, Theorem 17.5.7 and Corollary 17.5.8]). The e xpected result follows.\nNote that a similar argument was used in [ 40, Lemma 1].\nTheorem 3gives the asymptotic behavior of the optimal value ¯δT, but we do not know ho to study the\nbehavior of the maximizers a⋆\nTasT→0, as we did in the case T→+∞, in particular due to the fact that\nwe are not able to identify a limit problem as T→0. We let this question as an open problem.\n4.3 Other open questions\nOther open questions are in order:\n(i) A first one is to understand whether optimal profiles a⋆\nTare characteristic functions for any T >0.\nThisbang-bang property is usually a first step in deriving finer estimates an d geometric information.\n(ii) A second one is to establish quantitative estimates for the optimal observability constant: can we\nobtain an estimate akin to CT(a)−CT(a⋆\nT)/lessorequalslant−K/ba∇dbla−a⋆\nT/ba∇dbl2\nL1(Ω)? To answer this question, we believe\nthat a positive answer should first be obtained regarding the bang-bang property.\nAcknowledgment. We warmly thank Sylvain Ervedoza for his enlightening discu ssions on modal decom-\npositions and several ideas of examples introduced in Secti on2.2.\nThe authors acknowledge the support of the ANR project TRECO S, grant number ANR-20-CE40-0009. I.\nM-F was partially funded by a PSL Young Researcher Starting G rant 2023.\n19References\n[1] G. 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An introduction, Reprint of the 1995 edition.\n22"    },    {        "title": "2402.04035v1.Low_Distortion_Clustering_with_Ordinal_and_Limited_Cardinal_Information.pdf",        "content": "arXiv:2402.04035v1  [cs.GT]  6 Feb 2024Low-Distortion Clustering with Ordinal and Limited\nCardinal Information\nJakob Burkhardt∗Ioannis Caragiannis∗Karl Fehrs∗Matteo Russo†\nChris Schwiegelshohn∗Sudarshan Shyam∗\nAbstract\nMotivated by recent work in computational social choice, we extend the metric distortion\nframework to clustering problems. Given a set of nagents located in an underlying metric\nspace, our goal is to partition them into kclusters, optimizing some social cost objective. The\nmetric space is defined by a distance function dbetween the agent locations. Information about\ndis available only implicitly via nrankings, through which each agent ranks all other agents\nin terms of their distance from her. Still, even though no car dinal information (i.e., the exact\ndistance values) is available, we would like to evaluate clu stering algorithms in terms of social cost\nobjectives that are defined using d. This is done using the notion of distortion, which measures\nhow far from optimality a clustering can be, taking into acco unt all underlying metrics that are\nconsistent with the ordinal information available.\nUnfortunately, the most important clustering objectives ( e.g., those used in the well-known\nk-median and k-center problems) do not admit algorithms with finite distor tion. To sidestep\nthis disappointing fact, we follow two alternative approac hes: We first explore whether resource\naugmentation can be beneficial. We consider algorithms that use more than kclusters but\ncompare their social cost to that of the optimal k-clusterings. We show that using exponentially\n(in terms of k) many clusters, we can get low (constant or logarithmic) dis tortion for the k-center\nandk-median objectives. Interestingly, such an exponential bl owup is shown to be necessary.\nMore importantly, we explore whether limited cardinal info rmation can be used to obtain better\nresults. Somewhat surprisingly, for k-median and k-center, we show that a number of queries\nthat is polynomial in kand only logarithmic in n(i.e., only sublinear in the number of agents\nfor the most relevant scenarios in practice) is enough to get constant distortion.\n1 Introduction\nThe typical computational social choice problem consists o f optimizing a function over alternatives,\neach with a different associated cost or value. A classic exam ple is given by representative election.\nEach voter has a different representation score for every can didate, which we assume to correspond to\nthe distance in some underlying metric. Ideally, the repres entation minimizes the sum of distances of\neach voter to their closest representative. In the full info rmation setting, this corresponds to solving\nthe classic k-median problem. But this example already illustrates the di fficulty of implementing\nany voting mechanism: Even if the representation scores are assumed to be distances, they might\nbe unknown even to the participating voters. However, we may readily know if a voter prefers\nalternative aover alternative b.\n∗Department of Computer Science, Aarhus University, Åbogad e 34, 8200 Aarhus N, Denmark. Email: { jakob,\niannis ,karl,schwiegelshohn ,shyam }@cs.au.dk.\n†Department of Computer, Control, Management Engineering, Sapienza University of Rome, Via Ariosto 25,\n00185, Rome, Italy. Email: mrusso@diag.uniroma1.it .\n1Such examples have given rise to ordinal algorithms . An ordinal algorithm mainly allows for\ncomparisons between distances in the underlying metric. Th at is, given three points a,b,c, we are\nfreely given information whether d(a,b)≤d(a,c), but we are not given the exact numerical values\nofd(a,b)andd(a,c). The objective is to solve a given problem relying primarily on the ordinal\ninformation, while using as few (ideally zero) distance que ries as possible. The goodness of such an\nalgorithm is measured in terms of the quality of the computed solutionCcompared to the quality of\nthe optimal solution OPT that is given full information, com monly known as the metric distortion .\nFinding the median is arguably the most important problem in this field. Given a set of points X\nand a distance function d, the median mis defined to be the point minimizing the sum of distances.\nFollowing a long line of work [ 11,12,35,40,47,53], there now exists a deterministic algorithm with\noptimal metric distortion 3[39], which is also optimal [ 9,10]. Using randomization, Charikar et al.\n[24] recently achieved an important breakthrough, achieving a metric distortion of 2.753. The best\nknown lower bound is at least 2.1126[23].\nExtensions to more general clustering objectives such as (k,z)-clustering and facility location\nare comparatively much harder, see Anshelevich and Zhu [ 7], Caragiannis et al. [ 21]. In facility\nlocation, we ask for a set of centers Csuch that\n/summationdisplay\nx∈Xmin\nc∈Cd(x,c)+f·|C|\nis minimized, where fis the cost of opening a center. For (k,z)-clustering, we instead consider the\nobjective\nz/radicalBigg/summationdisplay\nx∈Xmin\nc∈Cd(x,c)z,\ni.e., the algorithm does not incur a cost for opening the cent ers, but instead has a budget of at\nmostkcenters that can be placed. Special cases include k-median where z= 1andk-center which\ncorresponds to z→∞.1\nUnfortunately, there are strong impossibility results for purely ordinal algorithms. Even for\n2-median, it is not possible to obtain an algorithm with bound ed metric distortion [ 7]. Therefore,\nresearch has begun to design algorithms that are given more p ower than purely ordinal information.\nIndeed, there has been some recent success in providing guar antees using only a constant number\nof queries per point, see Amanatidis et al. [ 5,4]. For clustering, recent work by Pulyassary [ 56] has\nshow that using at most polylog (n)distance queries per point, or n·polylog(n)queries overall, it\nis possible to achieve a constant factor approximation. The same work also showed that kqueries\nper point, or O(nk)queries overall are sufficient to achieve a constant factor ap proximation for\nk-median. Thus, we ask:\nQuestion 1.1. What is the minimum number of queries necessary for an algori thm to achieve\nconstant metric distortion for k-median, k-center, and facility location?\nWhile distance queries are a natural way of lending more powe r to the algorithm designer,\nobtaining the distances may be expensive as mentioned above . This leads to the question whether\nother models exist that allow the algorithm designer to boun d the metric distortion. A very natural\nway of doing so for clustering algorithms is by allowing the a lgorithm to return a (α,β)-bicriteria\napproximation. Such algorithms bound the clustering cost b y at most αtimes the cost of an optimal\nkclustering, while using βmany centers. We ask:\n1Sometimes the z√operation is omitted, as is the case for k-means corresponds to (k,2)-clustering. An α-\napproximation toz/radicalBig/summationtext\nx∈Xminc∈Cd(x,c)zimplies an O(αz)-approximation to/summationtext\nx∈Xminc∈Cd(x,c)z.\n2Question 1.2. What is the minimum value of βsuch that a bicriteria clustering algorithm using\nonly ordinal information has constant metric distortion?\n1.1 Our Results\nIn this paper we make substantial progress towards answerin g both questions. In the low-query\nsetting, we give two deterministic polynomial time algorit hms for k-center that, using at most\nO(k2)overall distance evaluations, obtain a 2-distortion and, using at most O(k)overall distance\nevaluations, obtain a 4-distortion. We also show that the latter result is optimal i n terms of the\nnumber of necessary queries, while the former is optimal for any polynomial time algorithm. For\n(k,z)-clustering, we obtain a randomized polynomial time algori thm that uses at most poly (k,logn)\noverall distance queries and achieves constant metric dist ortion. Note that all of these bounds are\nsublinear in the input size, that is assuming k≪n, we make o(1)queries per point.\nFinally, for facility location, there exists a simple adapt ation of the seminal Meyerson algorithm\n[52] that achieves a constant distortion using exactly one quer y per point or nqueries overall,\nsee also Section 4.1 of Pulyassary [ 56]. We show that no algorithm can achieve a constant factor\napproximation using less than Ω(n)queries, effectively closing the problem.\nIn the zero-query setting, we first show that there exists a (2,2k−1)-bicriteria algorithm for\nk-center. Moreover, this algorithm is optimal in the sense th at any algorithm achieving finite\ndistortion must use Ω(2k)centers. For (k,z)-clustering, we obtain two algorithms that solve all\n(k,z)-clustering objectives. The first succeeds with constant pr obability and achieves constant\ndistortion with (O(logn)k−1+o(1))many centers. The second requires (O(logn)k+o(1))and achieves\nO(1)distortion both in expectation and with high probability. W e complement this result by\nshowing that, for any constant factor distortion to k-median, Ω((2log∗n)k−1+2klogn)centers are\nnecessary even with a constant probability of success. For t he special case of 2-median, our bounds\nare optimal.\n1.2 Related Work\nOrdinal Preferences and Distortion The first paper to consider optimization problems using\nordinal information was probably Procaccia and Rosenschei n [55]. Subsequently, two main directions\nhave been established. Continuing to work with the model int roduced by Procaccia and Rosenschein ,\none line focuses mainly on maximizing welfare subject to nor malization assumptions, but without\nassuming any metric properties, see Amanatidis et al. [ 3,5,4], Caragiannis and Procaccia [ 20], Filos-\nRatsikas et al. [ 36]. The other line of work studies problem without the normali zation assumptions,\nbut assuming that the preferences are metric, i.e., they sat isfy the triangle inequality. Beyond\nclustering papers covered in the introduction, several oth er distortion problems have been studied\n[18,25,26,54]. While rare, it is also possible to achieve some results wit hout making either a\nnormalization or metric assumptions, see Abramowitz and An shelevich [ 1].\nClustering and Facility Location (k,z)-clustering is APX-hard in general metrics [ 30], though\nit is possible to obtain very accurate algorithms when makin g assumptions on either the metric [ 38,\n31] or the input [ 6,15,28]. Fork-center, Gonzalez [ 41] gave an optimal 2-approximation algorithm.\nFork-median, k-means and facility location, following a long line of resea rch [44,45,14,50,33,34],\nthe current state of the art is a 2.613approximation for k-median [ 42], a9approximation for k-\nmeans [ 2], and a 1.488approximation for facility location [ 49]. For general (k,z)-clustering, there\nare few claimed bounds, though most of the proofs for k-median and k-means go through while\nlosing aexp(z)approximation factor. Explicit results can be found in Cohe n-Addad et al. [ 29,32].\n32 Preliminaries\nLet(X,d)be a metric space where Xis a set of npoints and d:X×X→R≥0is a metric. The\ndistance between any two points x,y∈Xcan be accessed by a query of the form d(x,y). We assume\nthat such a query is associated with a cost. An algorithm is gi ven a budget and each query that\nthe algorithm makes consumes one unit of its budget. While qu erying the exact distance between\ntwo points is costly, our model assumes that, for every point ,ordinal information about its relative\ndistance to the other points is freely available. More speci fically, each point x∈Xprovides a\nrankingπx: [n]→Xthat is consistent withdin the sense that d(x,πx(i))≤d(x,πx(j))for every\ni,j∈[n],i < j . That is, points that are closer to xappear higher inx’s ranking. An ordinal\npreference profile Pis then just the collection of the points’ rankings, i.e., P={πx}x∈X. We write\nP(d)for the set of profiles where each point’s ranking is consiste nt with the distances d.\nIt is often convenient to restrict the ranking of a point to a c ertain subset of X. LetS⊆Xand\nm=|S|. The restriction of πxtoSis a function πx,S: [m]→Ssuch that, for any two y,y′∈S,y\nis ranked higher in πx,Sthany′if and only if yis ranked higher in πxthany′.\nThe ordinal preference profile provides a very rough sketch o f the underlying distance metric\nd. However, the relative distances expressed by the profile ca n enable an algorithm to allocate its\nbudget in a very economic way. Consider the following operat ion: For a set of points S⊆Xand a\npointx∈X, we define the distance of xtoSto bed(x,S) = min y∈Sd(x,y).\nGiven the ordinal information, the point z= argminy∈Sd(x,y)can readily be identified as x’s\nhighest ranked point among S. Hence, an algorithm can determine the distance of xtoSwith a\nsingle query d(x,z). Clearly, the same observation can be made about finding z= argmaxy∈Sd(x,y)\nand the distance d(x,z).\nWe intend to study the loss in outcome optimality if we restri ct an algorithmAto the ordinal\ninformation and a fixed query budget. We consider a variety of clustering problems where the goal\nis to find a solution that minimizes a given cost function φ. We denote byMthe set of all metric\nspaces. For a metric space (X,d)∈M and a profile P∈P(d), letA(P,d)be the solution (set of\ncenters) computed by algorithm A, and let C∗(d)be a solution (set of centers) of minimal cost. We\nsay that an algorithm Aachieves distortion Dwith constant (respectively high) probability, if\nsup\n(X,d)∈M\nP∈P(d)φ(A(P,d))\nφ(C∗(d))≤D\nwith probability at least 2/3(respectively probability at least 1−1/n). The expected distortion of\nAis given by the ratio\nsup\n(X,d)∈M\nP∈P(d)E[φ(A(P,d))]\nφ(C∗(d)).\nWe now state the definition of the (k,z)-clustering problem in the ordinal setting and introduce\na few standard terms that are commonly used in the context of c lustering problems.\nDefinition 2.1. In the ordinal(k,z)-clustering problem , we are given positive integers k,zand a\nsetXofnpoints that form a metric space (X,d)under distances d. Each point x∈Xreports a\nrankingπxthat is consistent with the distances d. LetP={πx}x∈X. For a subset S⊆Xof the\npoints, we denote the cost of a given solution C⊆Xby\nφC(S,d) =z/radicalBigg/summationdisplay\nx∈Sd(x,C)z.\n4The goal is to find a set Cofkpoints such that the cost function φC(X,d)is minimized. For\ncompactness, we drop the dependence on dand denote by φOPT(S)the cost of the optimal solution\non an arbitrary set of points S⊆X.\nGiven a solution Cto an ordinal (k,z)-clustering instance, we typically call the elements of\nCcenters .Cnaturally induces a partition of Xintokclusters{Ac}c∈Cwhere, for each c∈C,\nAc={x∈X:πx,C(1) =c}. We refer to the collection of these clusters as a clustering ofX.\nFinally, we define sampling probabilities for all (k,z)-clustering objectives.\nDefinition 2.2. Letzbe a positive integer, and let C⊆Xbe a set of centers. The sampling\nprobability of point c∈Xconditioned on having already selected a set of centers Cis\npz(c) :=P[cis added to C|C] =d(c,C)z\n/summationtext\nx∈Xd(x,C)z,\nand denote the induced distribution by D++\nz.\n3 Algorithms for k-Center\nWe present three algorithms for solving the ordinal k-center ((k,∞)-clustering) problem. Our algo-\nrithms are based on a greedy procedure by Gonzalez [ 41], which is known to yield a 2-approximation\nof thek-center problem. This procedure simply chooses an arbitrar y center to begin with and then,\nink−1iterations, chooses the center that is farthest away from th e already chosen centers (farthest-\nfirst traversal).\n3.12-Distortion Algorithms\nThe farthest-first traversal method lends itself well to be a dapted to the ordinal setting. Clearly,\ngiven a set of clusters, the farthest point from these cluste rs can be determined with one distance\nquery per cluster. For completeness, we give a pseudocode im plementation of the procedure in\nAppendix A.1. This immediately gives rise to the following result.\nTheorem 3.1. There exists a deterministic 2-distortion algorithm for k-center that makesk2−k\n2\ndistance queries.\nFor the zero-query regime, we extend the farthest-first trav ersal method such that, in every\niteration, the farthest point in every cluster is chosen. Since the algorithm and its analysis are\nstraightforward adaptions of Gonzalez [ 41], we merely state the result and give the details in Ap-\npendix A.2.\nTheorem 3.2. There exists a deterministic algorithm that, using only ord inal preferences, returns\na set of centers Cof size|C|= 2k−1, such that maxx∈Xd(x,C)≤2φOPT, whereφOPTis the cost\nof an optimal k-center clustering.\n3.24-Distortion Algorithm with O(k)Queries\nTo achieve a constant distortion via a linear (in k) number of queries, the idea is to perform a\n1\n2-approximate farthest-first traversal. Such a farthest-fir st traversal is robust with respect to the\ndistortion, losing only a factor of 2. Surprisingly, using ordinal information, we can execute a1\n2-\napproximate farthest-first-traversal with an optimal quer y bound. At a very high level, we keep\n5track of (center,farthest point) pairs for all clusters thr oughout the algorithm. However, we do not\nquery all the pairs. Instead we keep a track of an independent set of pairs to query which helps\nus bound the number of new pairs created, while ensuring that the distance of the unqueried pairs\nare at most twice the queried distances. We give the complete analysis here and the pseudocode in\nAppendix A.3.\nTheorem 3.3. There exists a deterministic 4-distortion algorithm to the optimal k-center clustering\nthat makes 2kqueries.\nThroughout the algorithm’s run, let Cbe the solution set and let Q⊆Cbe the so-called query\nset. Both CandQwill change over time, so we denote Cias the solution and Qias the query set\nafter the i-th iteration, for clarity of exposition. Moreover, for y∈Ci, letSy,ibe the set of points\nsuch that, for each of these points, yis the closest center among Ci, and let zi= argmax\nx∈Sy,id(y,x).\nNote that we query the distance d(y,zi), ifybelongs to the query set Qi.\nIn iteration i∈{0}∪[k−1]of the algorithm, we perform the following steps:\n1. Select the cluster Sy,i, fory∈Qisuch that d(y,zi)is maximized and add zitoCi, forming\nCi+1.\n2. Remove yfromQiand letRi+1:=Ci+1\\Qi(i.e.Ri+1always consists at least of yandzi).\n3. Add centers from Ri+1toQito obtain Qi+1as follows: Let u∈Ri+1.\n• If there exists a center p∈Qisuch that d(p,q)≥d(w,q), wherew= argmax\nx∈Su,i+1d(u,x)\nandq= argmax\nx∈Sp,i+1d(p,x), do not add utoQi.\n• If no such pexists, add utoQi.\nOnce all u’s have been discarded, we have obtained our new set Qi+1. All distances between\ncenters in Qi+1and the respective furthest points are queried. Note that we only have to\nquery novel pairs, i.e. already queried pairs do not require a new query.\nWe now prove several claims about the algorithm. The first two bound the number of queries.\nThe final two claims yield the desired bound on the distortion : In particular, we show that we\nselect, at each iteration, a point that is no closer than half the distance of the furthest point and\nthat such an approximate farthest-first traversal also yiel ds a constant distortion to the optimal\nk-center solution.\nInvariant 3.4. Ify∈Qiandzi= argmax\nx∈Sy,id(y,x)/∈Ci+1, thenargmax\nx∈Sy,id(y,x) = argmax\nx∈Sy,i+1d(y,x).\nProof. We prove this by induction, the base case of which is trivial a s initially we only have an\narbitrary center and its most distant point in S0andQ0.\nLet{w}=Ci+1\\Ciand letube the center of the cluster containing winCi. Consider any\ny∈Qi. Ifywas added to Qibeforeu, then we know d(zi,w)> d(y,zi), hencezi=zi+1. Ifywas\nadded to Qiafteru, thend(zi,w)> d(u,w). But since d(y,zi)≤d(u,w), we have d(y,zi)< d(zi,w)\nwhich also implies zi=zi+1.\nLemma 3.5. The total number of queries is at most 2k.\nProof. By Invariant 3.4, the only way a point can be removed from Qiis if it was added to Ci+1.\nTherefore, the number of queries made that lead to a deletion are exactly k. The remaining number\nof queries are upper bounded by at most k, and the claim follows.\n6This shows that the total number of queries made by the algori thm are bounded by O(k).\nWe now turn to the distortion factor. The following lemma sho ws that the algorithm executes a\n1\n2-farthest first traversal.\nLemma 3.6. Let{z}=Ci+1\\Ciand letz∈Sy,i. Then for any u∈Ciandw∈Su,i, we have\nd(y,z)≥1\n2·d(u,w).\nProof. We selected argmax\ny∈Qid(y,zi). Hence it suffices to compare d(y,zi)withd(u,w)foru /∈Qi.\nSinceu /∈Qi, we know that there exists some y′∈Qis.t.d(z′\ni,w)≤d(y′,z′\ni)≤d(y,zi). By the\ntriangle inequality. d(y′,z′\ni)≥d(y′,w)−d(z′\ni,w)≥d(u,w)−d(z′\ni,w). Rearranging, we have\nd(u,w)≤d(y′,z′\ni)+d(z′\ni,w)≤2d(y′,z′\ni)≤2d(y,zi),\nwhich concludes the proof.\nFinally, we show that an approximate farthest-first travers al yields a constant distortion to the\noptimalk-center solution.\nLemma 3.7. Suppose we iteratively select points such that, in every ite ration,d(z,Ci)≥α·\nargmax\nx∈Xd(x,Ci), forα∈(0,1]. Then, Ck−1yields a2\nα- distortion to the optimal k-center clus-\ntering:\nmax\nx∈Xmin\nu∈Ck−1d(x,u)≤2\nα·φOPT,\nwhereφOPTis the cost of an optimal k-center clustering.\nProof. LetC∗={A1,...Ak}be the optimal clustering. If Ck−1∩Ajis non-empty, for all Aj∈C∗,\nthe distortion is 2due to the triangle inequality. Otherwise, we let ibe the first iteration where we\nadded a second point x2from some cluster AjtoAiand letx1be the first point from Ajadded to\nC. Then for any u\nd(u,Ci)≤1\nα·d(x2,Ci)≤1\nα·d(x2,x1)≤2\nα·φOPT,\nwhich concludes the proof.\nCombining Lemmas 3.6,3.7, and 3.5then yields the theorem. Achieving a strictly smaller than\n4-distortion with a strictly subquadratic number of queries (or proving that it is imposs ible) is an\ninteresting open problem.\n4 Algorithms for (k,z)-Clustering\nIn this section, we present our algorithms for solving the (k,z)-clustering problems. The first makes\nuse of no queries and obtains a bi-criteria distortion guara ntee, seeking to trade off distortion with\nthe number of selected centers.\n4.1 Zero-Query Bi-Criteria Algorithm\nThe algorithm is based on distance sampling. The seminal k-means++ by Arthur and Vassilvitskii\n[13] iteratively selects points proportionate to the squared E uclidean distance of the current set of\ncenters. In this paper, we consider a generalization to (k,z)-clustering, where we sample points\nproportionate to their cost. In both cases, the expected cos t of the computed solution is with a\nfactor of O(logk)of that of an optimal k-means clustering2and this bound is tight even in the\n2The distribution has been analyzed repeatedly for the k-means problem. Similar statements for (k,z)-clustering\nare folklore, and we provide complete proofs for these probl ems in the appendix.\n7Euclidean plane. Improvements to this basic algorithm are a bundant in literature. Indeed, 2k\nrounds are already enough to achieve a O(1)bicriteria approximation, see Makarychev et al. [ 51]\nand Wei [ 58]. Alternatively, one may sample multiple points in each rou nd. This tends to yield a\nworse tradeoff between samples and cost, but combined with ot her algorithms, may yield a constant\napproximation [ 16,27,48,57,43].\nWhen adapting this procedure to the ordinal setting, the firs t challenge to overcome is that we\ndo not know pairwise distances. The key idea behind our algor ithm is to use ordinal information to\napproximate the sampling probabilities. We do this by over- sampling, i.e., we pick O(logn)points\nfor each point that the (k,z)++ algorithm picks. The main results of this section are the f ollowing\ntwo:\nTheorem 4.1. For the(2,z)clustering instance, Algorithm 1returnsO(logn)centers achieving a\nO(1)distortion with constant probability.\nInSection 5 , we show that this is optimal.\nTheorem 4.2. There exists a randomized algorithm achieving a O(1)-distortion for all (k,z)-\nclustering objectives simultaneously usingO(logn)k+o(1)centers, both on expectation and with high\nprobability.\nNow, we present the algorithm for Theorem 4.1and provide some intuition as to why it works.\nA similar reasoning can be extended to obtain the algorithm f or Theorem 4.2. Theorem 4.1and\nTheorem 4.2are formally proven in Appendix B. The algorithm for Theorem 4.2involves repeating\nAlgorithm 1to amplify success probability, and augmenting Theorem 3.3’sk-center algorithm to\nbound the worst case.\nFor the algorithm, we define some new notation. For any set of p ointsS∈Xand any point\nc /∈S, we define a partition of Scinto disjoint sets {Sc,0,Sc,1,...,S c,ℓ}whereℓ=⌊log|S|⌋. We\nconstruct the partition recursively starting from Sc,ℓ. Define Sc,ℓto be the singleton set containing\njust the farthest point in Sfromc. Next, for each 1< j < ℓ , defineSc,jto be the farthest 2ℓ−j\npoints from the set S\\{Sc,j+1∪Sc,j+2···∪Sc,ℓ}. Lastly, let Sc,1=S\\{Sc,2∪Sc,3···∪Sc,ℓ}.\nAlgorithm 1: (k,z)-clustering without queries\nInput: Point set X, ordinal information P={πp}p∈Aandk∈N\n1Initialize the set of centers C=∅\n2Sample a point cuniformly at random from X\n3LetC={c}\n4fori= 2tok−1do\n5Initialize Ci←∅foreach point cinCdo\n6 DefineS={x∈X:πx(c)≤πx(c′)∀c′∈C}, i.e.,Sis the set of points that belong\nto the cluster with center c, and let ℓ=⌊log|S|⌋\n7 Sample7logkpoints uniformly randomly from each of the sets {Sc,1,Sc,2,...,S c,ℓ}\n(defined above) and add them to CiC←C∪Ci\n8LetC←C∪C0whereC0is the output of Theorem 3.3’sk-center algorithm\n9returnC\nAnalysis: We now highlight a key property of Algorithm 1that shows us why it gives us a O(1)\ndistortion for the 2-median instance with constant probability. The following lemma show that,\n8Algorithm 1, in a sense, performs better than the (k,z)++ algorithm in each iteration. Formally,\nfor each point c∈X, we show that the probability that Algorithm 1picks the point in an iteration\nis at least the probability that the (k,z)++ algorithm picks the point.\nLemma 4.3. LetCbe the set of centers before at the beginning of line 3 of Algor ithm1in theith\niteration. For any point c∈Xafter line 8 of Algorithm 1, we have\nPAlg1[c∈Ci|C]≥pz(c).\nThe proof of the lemma is deferred to the appendix. Though Lem ma4.3gives us an idea as\nto why Algorithm 1indeed does well, it is important to note that statement, by i tself, does not\nimply the bounds in Theorem 4.1and Theorem 4.2. Specifically, Lemma 4.3does not imply that we\nperform better than the (k,z)++ algorithm. The reason is that Algorithm 1samples points from\ndifferent rings independently as opposed to the (k,z)++ algorithm. The analysis in Makarychev\net al. [ 51] and Bhattacharya et al. [ 17] points at the fickle nature of k-means++ algorithm and\nhow slightly perturbing it leads to a worse performance. To g et around this, we use over-sampling\nwithout making the asymptotic bicriteria approximation wo rse.\n4.2O(1)-Distortion Algorithm with O(k4log5n)Queries\nWe design an algorithm that achieves a constant distortion t o the cardinal objective with just a few\ncardinal queries. Formally, we show the following result:\nTheorem 4.4. There exists a randomized algorithm achieving an expected O(1)-distortion to the\noptimal(k,z)-clustering using O(k4log5n)queries.\nOur exposition mainly focuses on the k-median objective, for which z= 1, however, the proofs\nalmost seamlessly go through for other (k,z)clustering objectives. Due to space constraints, we\ngive a full proof and pseudocode for the algorithm in Appendix B and only highlight the key ideas\nhere. To this end, given a current set of centers C, we define an estimated cost for each of the rings\nin question, i.e.,\n/hatwiderφC(Si,j) =|Si,j|·min\nx∈Si,j−1d(x,ci).\nNote that to compute the above-estimated cost, we just need o ne query per ring (in each round).\nIndeed, we simply need to query the distance between point ciand the topmost point in ci’s\npreference list that belongs to Si,j−1. Since there are Trounds, the resulting number of queries is/summationtext\nt∈[T]t·log(|X|)≤T2logn. Now, we emulate thek-median++ algorithm by sampling a center c\nbelonging to ring Srjwith probability equal to\n/hatwidep(c) :=1\n|Srj|·/hatwiderφC(Srj)\n/summationtext\ni,j/hatwiderφC(Si,j).\nIt is not hard to see that the above is non-negative and summin g across all i,jwe obtain 1, thereby\nmaking the above a valid distribution, which, from now on, we will call D. Before discussing the\nmain algorithm in its full details, let us recall that, in the plaink-median++ algorithm, given a\ncurrent set of centers C, each new center cis sampled (adaptively) with probability\np(c) :=d(c,C)/summationtext\nx∈Xd(x,C).\nThis probability is proportional to how much they contribut e to the current overall cost. Recall that\nwe name the k-median++ induced distribution as D++(sincez= 1in this case). The following\nlemma relates the standard k-median++ distribution D++and the emulating distribution D.\n9Lemma 4.5. LetCbe the set of centers already chosen. For any point c∈Xsampled according to\ndistribution D\nPD[c∈Ci|C]≥1\n2·pz(c).\nAlgorithm. From this point onwards, our goal will be to show that Algorit hm6(whose formal\ndescription is deferred to Appendix B ) and achieves an O(1)distortion, as long as the number of\nroundsTis large enough. The high level idea is not to use a potential f unction that allows us to\nbound the cost of hit and not hit clusters, as is done in most k-means++ analyses. Instead, we show\nthat the cost decreases by a constant factor for a sufficient nu mber of samples, similar to Rozhon\n[57].\nUnfortunately, unlike these works, we cannot guarantee an u pper bound on the cost when\nrunning a sampling algorithm with the guarantee provided by Claim 4.5. Indeed, there is a non-\nzero probability that we hit the same clusters over and over a gain, which can lead to an arbitrarily\nhigh distortion.\nWe sidestep these issues with a careful initialization. For this we use the k-center solution\nresulting from Section 3. A sufficiently good k-center solution is within a factor O(n)of the cost\nof a(k,z)-clustering. Moreover, our k-center algorithms are deterministic, which modifies our\nprevious low probability event of having unbounded distort ion to a low probability event of having\nO(n)distortion.\nAnalysis. The proof proceeds as follows: First, let us consider the cur rent set of centers C(initial-\nized toC0, thek-center clustering output by the algorithm used to prove The orem 3.3). Then, we\nconsider the optimal clustering collection C∗={A1,...,A k}, and the union of uncovered clusters\nU, i.e., clusters not hit by C. We show that the probability that a given optimal cluster re mains\nuncovered after a fresh center is sampled is inversely expon entially related to its cost (normalized\nby the total cost). This is crucial because it helps us in show ing that the cost of uncovered points\nhas to drop by at least a constant factor at each new iteration of the algorithm, which is the second\nstep of our proof strategy. Lastly, we recall that the initia l clustering was a constant distortion to\nthe optimal k-center one, which means an O(n)-distortion to the optimal k-median clustering. This,\ncombined with the earlier considerations, leads to a consta nt distortion provided T∈O(klogn).\n5 Lower Bounds\nIn this section, we finally present our lower bounds. The lowe r bounds for k-center are simple and\noptimal. We, therefore, give the full proof in the main body. The lower bounds for k-median are\nsignificantly more complicated, but use a similar construct ion as the k-center lower bound.\nWe conclude this section by presenting a lower bound for the f acility location problem. The\nproofs of the latter two results are deferred to Appendix CandD, respectively.\nTheorem 5.1. For any fixed α, every bicriteria algorithm Afork-center that has distortion at\nmostαwith at least constant probability must return a solution of size at least Ω(2k). Moreover,\nany algorithm that has distortion at most αwith at least constant probability must make at least\nΩ(k)queries.\nWe remark that the distortion bound αhas no influence on the number of queries or the number\nof centers. That is, our lower bounds hold for arbitrary valu es ofα. This property together with\nthe observation that the cost of all (k,z)-clustering objectives are within a poly (n)factor implies\nthat the same bounds indeed hold for any (k,z)-clustering.\n10Proof. The hard instance is the same for the low query and zero query s etting. We start with an\nanalysis for the latter.\nThe hard instance: Our hard instance consists of 2k−1points. We begin by describing the\nordinal information and the underlying metric. Consider a c omplete binary tree Tof depth k−1.\nFor any two nodes p,q, we say that ais the common ancestor of pandqifais the minimum depth\nnode in the shortest path between pandqinT.\nThe interpretation of this tree is that the leaves are the poi nts and for any interior node a, the\nvalued(a)stored in adenotes the distances between all points p,qthat have aas the common\nancestor. Thus, we now require the following invariant to en sure that the tree encodes a metric.\nInvariant 5.2. If the subtree rooted at acontains the interior node b, thend(a)≥d(b).\nWe now specify the ordinal preferences, which we fix before determining the values d(a)of the\ninterior nodes. Let p,q,o be three leaves and let a(p,q),a(p,o), anda(q,o)be common ancestors\nof these pairs of nodes, respectively.\n• If the depth of a(p,q)is larger than the depth of a(p,o)anda(q,o)then the preference list of\npdetermines qto be closer to pthan too.\n• If the depth of a(p,q)anda(p,o)is equal then the relative ordering of qandoin the preference\nlist ofpis arbitrary (w.l.o.g., it may be chosen lexicographically ).\nWe now describe a hard input distribution that satisfies the i nvariant and is consistent with the\nordinal preferences. Select a random path Qbetween the root of Tand an arbitrary node rat\ndepthk−1. All nodes aalong that path receive the value d(a) =D. All remaining nodes receive\nthe value d(a) = 1.\nAnalysis: Note that, for any two trees sampled from the distribution, t he values assigned to the\ninterior nodes satisfy Invariant 5.2and thereby induce a metric on the set of leaf nodes. Since the\nordinal preferences are independent from these values, the two trees cannot be distinguished using\nthe ordinal information.\nWe now determine an optimal k-center solution C. For every interior node ainQ, the children\nofaform subtrees T(a,small)andT(a,large). The root bofT(a,small)satisfiesd(b) = 1 and the\nrootcofT(a,large)satisfiesd(c) =D. For the largest depth interior node ainQ, we introduce\nthe convention that T(a,large)contains the leaf r(i.e., the end point of Q).Cnow places exactly\none center on an arbitrary leaf of T(a,small)and one center on r. The cost of Cis therefore 1.\nNow consider any other solution C′. IfC′does not place a center on r, then the cost of C′isD.\nOtherwise, there must exist some a∈Qfor which T(a,small)does not receive a center. Hence,\nthe points in T(a,small)must be served by some center contained in T(a,large)or by a point not\ncontained in the subtree rooted at a. In both cases, the cost of these points is D.\nTo conclude, it now suffices to analyze the performance of the b est deterministic algorithm\nplacingKcenters against this hard input distribution. Since the alg orithm does not make any\nqueries and cannot determine Qbased on the ordinal information, its choice of centers is fix ed.\nThere are 2k−1many different nodes at depth k−1. Hence, the probability that Kincludes the\nleaf node risK/2k−1. Conversely, if K /∈Ω(2k)then the probability that Kdoes not include ris\nat least constant, which leads to a distortion of D.\nFinally, we remark on some generalizations of this lower bou nd. For the low query regime, an\nalgorithm needs to find the entire path Qor, equivalently, identify the leaf r. If it decides to not\ndo so then with probability at least1\n2it will have unbounded distortion of D. Again, consider the\n11performance of the best deterministic algorithm against th e input distribution. Given that Tis a\nbinary tree, at least one query queries is required to reduce the search space for Q(equivalently, for\nr) by a factor of1\n2in expectation. Hence, if the algorithm does not make Ω(k)queries, its distortion\nis unbounded.\nNext, we give a different lower bound for any bicriteria algor ithm for k-median. Specifically,\nwe show that any bicriteria algorithm for k-median requires Ω(2klogn)centers. For 2-median, this\nbecomes Ω(logn), which stands in contrast with 2-center, where we can obtain a true 2-distortion\nusing only ordinal information (see Theorem 3.2). We also show that there exists a slow growing\nfunction g(n)increasing in nfor every fixed ksuch that any bicriteria algorithm requires g(n)k\nmany queries. Moreover, g(n)may be lower bounded by 2log∗n, though somewhat higher bounds\nare likely possible using our construction. We conjecture t hat the true lower bound is (logn)k.\nTheorem 5.3. For any fixed α, every bicriteria algorithm Afork-median that has distortion\nless than αwith at least constant probability must return a solution of size at least Ω/parenleftBig\nlogn\nlogα·2k/parenrightBig\n.\nMoreover, any algorithm achieving a constant factor approx imation for k-median must make at least\nΩ(k+loglog n)queries.\nTheorem 5.4. For any fixed αand every fixed k, every bicriteria algorithm Afork-median that\nhas distortion less than αwith at least constant probability must return a solution of size at least\nΩ/parenleftBig/parenleftbig\n2log∗n/parenrightbigk−1/parenrightBig\n. The number of queries to achieve a constant distortion is at leastΩ(k·2log∗n).\nFinally, we return to the facility location problem. We are i nterested in lower-bounding the\nnumber of queries necessary to achieve any given distortion . Using no queries, it is not possible to\nobtain bounds on the distortion [ 8] beyond the trivial O(n)bound. Our lower bound essentially\nshows that Ω(n)queries are necessary to achieve constant distortion, maki ng the adaptation of\nMeyerson’s algorithm optimal.\nTheorem 5.5. For any fixed α, every algorithm Afor facility location that has distortion less than\nαwith at least constant probability must make Ω/parenleftbign\nα/parenrightbig\ndistance queries.\n6 Conclusion and Open Problems\nWe gave optimal algorithm for computing bicriteria approxi mations for kcenter both in terms of\nthe number of distance queries as well as number of additiona l centers in the purely ordinal setting.\nAdditionally, we gave optimal lower bounds for facility loc ation and substantially improved low\nquery and purely ordinal bicriteria algorithms for k-median.\nAside from closing the small remaining gaps left in our analy sis, several interesting open problems\npresent themselves. First, our bicriteria algorithm simul taenously achieves small distortion for all\n(k,z)-clustering. Another popular way to interpolate between k-median and k-center is ordered\nclustering Byrka et al. [ 19], Chakrabarty and Swamy [ 22]. Is it possible to achieve low distortion\nalgorithms for this problem as well?\nFurthermore, there exist many other clustering objectives , such as graph clustering. Which\ndistortion/query tradeoffs are possible for sparsest cut an d metric max cut?\nAcknowledgements\nIoannis Caragiannis and Sudarshan Shyam were partially sup ported by the Independent Research\nFund Denmark (DFF) under grant 2032-00185B. Matteo Russo is s upported by the ERC Advanced\n12Grant 788893 AMDROMA “Algorithmic and Mechanism Design Res earch in Online Markets”. Jakob\nBurkhardt and Chris Schwiegelshohn are partially supported by the Independent Research Fund\nDenmark (DFF) under a Sapere Aude Research Leader grant No 10 51-00106B.\nReferences\n[1] B. Abramowitz and E. Anshelevich. 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In Proceedings\nof the 29th Annual Conference on Neural Information Process ing Systems (NeurIPS) , pages\n604–612, 2016.\nA Omitted Content from Section 3\nA.12-Distortion Algorithm with O(k2)Queries\nAlgorithm 2is a straightforward adaption of the farthest-first travers al method by Gonzalez [ 41] to\nthe ordinal setting. Over k−1iterations, the algorithm performs/summationtextk−1\ni=1i=k2−k\n2distance queries in\ntotal. Furthermore, the approximation guarantee that Gonz alez [41] showed for their procedure in\nthe full information setting implies that Algorithm 2achieves a 2-distortion for the ordinal k-center\nproblem.\nAlgorithm 2: Ordinalk-center with k2queries\nInput:X,d,P,k\n1x←arbitrary point from X\n2C←{x}\n3for1...(k−1)do\n4Setδmax= 0\n5forc∈Cdo\n//define cluster with center c\n6 DefineAc={x∈X:πx,C(1) =c}\n//query distance from cto farthest point among Ac\n7 Letz= argmax x∈Acd(c,x)\n8 Queryδ=d(c,z)\n9 ifδ≥δmaxthen\n10 δmax=δ\n11 r←z\n12C←C∪{r}\n13returnC\nA.22-Distortion Algorithm with 2k−1Many Centers\nTheorem A.1. LetXbe a set of points in some metric space. There exists a determi nistic algo-\nrithm that, using only ordinal preferences, returns a set of centersCof size|C|= 2k−1, such that\nmaxx∈Xd(x,C)≤2φOPT, whereφOPTis the cost of an optimal k-center clustering.\nProof. LetCbe the solution returned by Algorithm 3. We first argue that Chas the right size.\nNote, that after initially having size 1, in each iteration iof the outer loop, the algorithm adds 2i−1\npoints to C, thus\n|C|= 1+k−2/summationdisplay\ni=02i= 1+2k−1−1 = 2k−1.\n17Algorithm 3: Ordinalk-center without queries\nInput:X,d,P,k\n1x←arbitrary point from X\n2C←{x}\n3for1...(k−1)do\n4T←∅ //new set of centers\n5forc∈Cdo\n//define cluster with center c\n6 DefineAc={x∈X:πx,C(1) =c}\n//add farthest point from camongActo solution\n7 Letz= argmax x∈Acd(c,x)\n8 T←T∪{z}\n9C←C∪T\n10returnC\nLetC∗be the optimal solution to the given ordinal k-center instance. C∗induces the clustering\nA∗={Aℓ}ℓ∈C∗. To prove that Algorithm 3has distortion at most 2, we consider two cases.\nIn the first case, we assume that for each cluster Aℓ∈A∗there is a point cℓ∈Csuch that also\ncℓ∈Aℓ. We say that the algorithm hitthe cluster Aℓwith center cℓ. For any center ℓ∈C∗of the\noptimal solution, consider now an arbitrary point x∈Aℓ. By the triangle inequality and optimality\nofC∗, we have that\nd(x,cℓ)≤d(x,ℓ)+d(ℓ,cℓ)≤2φOPT. (1)\nThus, for every point in Xthere is a point in Csuch that the distance between these points is at\nmost2φOPT.\nFor the other case, assume that there is a cluster A∈A∗such that the algorithm did not hit A\nwith a center, that is, C∩A=∅. Clearly, there must be at least one cluster in A∗such that the\nalgorithm hit the cluster with two centers. Let C′be the solution of the algorithm in the last iteration\nbefore a cluster in A∗was hit by a second center. Consider the point c= argmaxx∈Xd(x,C′)and\nnote that the algorithm selects cas the next center. Assume that c∈Aℓin the optimal solution.\nHence, there is another point c′∈C′such that also c′∈Aℓ. But then for all x∈X,\nd(x,C′)≤d(c,C′) = min\nz∈C′d(c,z)≤d(c,c′)≤2φOPT.\nHere, the first inequality stems from the definition of c, and the second inequality follows from the\nfact that c′∈C′. The last inequality is again due to the observation that two points in the same\noptimal cluster have distance at most 2φOPTfrom another, see Inequality ( 1). Hence, the cost of C′\nis already at most twice the cost of the optimal solution and a dding more centers to C′can never\nincrease the cost of the solution. This shows that the lemma a lso holds in this case and concludes\nthe proof.\nA.3 Pseudocode of 4-Distortion Algorithm with 2kQueries\nRecall that we defined Sy,ito be the set of points such that, for each of these points, yis the closest\ncenter among Cin thei-th iteration of Algorithm 4.\n18Algorithm 4: Ordinalk-center with 2kqueries\nInput:X,d,P,k\n1z←arbitrary point from X\n2C,Q←{z,πz(n)} //initialize with zand the farthest point from z\n3DefineSz,0={x∈X|xrankszhigher than πz(n)}andSπz(n),0=X\\Sz,0\n4fori= 1...(k−1)do\n5Setδmax= 0\n6fory∈Qdo\n//query distance from yto farthest point among Sy,i\n7 Letz= argmax\nx∈Sy,id(y,x)\n8 Queryδ=d(y,z)\n9 ifδ≥δmaxthen\n10 δmax=δ\n11 r←z\n12 v←y\n13C←C∪{r} //solution set in iteration i+1\n14Q←Q\\{v}\n15 DefineR←C\\Q\n16 foru∈Rdo\n17 add=true\n18 w= argmax\nx∈Su,i+1d(u,x)\n19 forp∈Qdo\n20 q= argmax\nx∈Sp,i+1d(p,x)\n21 ifd(p,q)≥d(w,q)then\n22 add=false\n23 ifaddthen\n24 Q←Q∪{u}\n25returnC\nB Omitted Content from Section 4\nB.1 Proof of Theorem 4.1and Theorem 4.2\nFirst, we give the proof of Lemma 4.3.\nProof. Let point cbelong to cluster Ainduced by the set of centers Cand in cluster A, letcbelong\ntojthring. The (k,z)++ distribution picks cwith probability φC(c)/φC(X), while Algorithm\n1pickscwith probability 1/|Si,j|. We show that 1/|Si,j| ≥φC(c)/φC(X). Towards this, we\nbound the value of φC(X)with respect to φC(c). Ascis in the jthring, we have that φC(X)≥\n19/summationtextl\nj′=j+1φC(Si,j′)+φC(c). Thus,\nφC(X)≥l/summationdisplay\nj′=j+1φC(Si,j′)+φC(c)\n≥l/summationdisplay\nj′=j+1φC(c)+φC(c)\n=\n1+l/summationdisplay\nj′=j+1|Sj′,i|\n·φC(c)\n= 2j·φC(c),\nwhere the second inequality holds because points in rings > jhave cost at least φC(c). To conclude,\nwe have\nφC(c)\nφC(X)≤φC(c)\n2j·φC(c)=1\n2j=1\n|Si,j|.\nAlgorithm 5: (k,z)-clustering without queries\nInput: Point set X, ordinal information P={πp}p∈Aandk∈N\n1Initialize the set of centers C=∅\n2for1...logndo\n3Sample a point cuniformly at random from A\n4LetC′={c}\n5fori= 2tok−1do\n6 Initialize Ci←φ//The centers to be added in this round\n7 foreach point cinCdo\n8 DefineS={x∈A:πx(c)≤πx(c′)∀c′∈C}, i.e.,Sis the set of points that\nbelong to the cluster with center c, and let l=⌊log|S|⌋\n9 SampleO(logk)points uniformly randomly from each of the sets\n{Sc,1,Sc,2,...,S c,ℓ}(defined above) and add them to Ci//As the size of S\nis bounded by n, we sample at most O(logk·logn)points in this\nstep\n10 C′←C′∪Ci\n11 C←C∪C′\n12LetC←C∪C0whereC0is the output of Theorem 3.3’sk-center algorithm\n13returnC\nWe will use two basic claims. The first relates the cost of vari ous(k,z)-clustering objectives.\nThe second gives a reduction from (α,β)bicriteria solutions to true O(α)-approximate solutions.\nBoth claims are arguably folklore and the experienced reader may skip their proofs.\nClaim B.1. Given a point set Xin some metric space. Then, any solution Swith distortion αto\nthe optimal k-center clustering on Xyields at most an α·n-distortion to the optimal (k,z)clustering\nonX.\n20Proof. Let the cost of the optimal k-center clustering instance be φOPT. Then the cost of Sfor the\nk-center problem is at most α·φOPT. The cost of Sfor the(k,z)clustering problem is at most\nz/radicalbig\n(n·αz·φz\nOPT)=z√n·α·φOPT.\nMoreover, the optimal solution for (k,z)clustering instance with cost φz,OPT will have k-center\ncost at most φz,OPT. Thus, Swill beα·n≥α·z√napproximation for the (k,z)clustering\ninstance.\nClaim B.2. Given a point set Xin some metric space. Let C′be an(α,β)bicriteria solution for\n(k,z)clustering. Interpret C′as the multiset where c∈C′is added for every point x∈Xassigned to\nc. Then any γ-approximate solution CforC′with respect to (k,z)clustering is an 4αγapproximate\nsolution for X.\nProof. We usecxto denote argmin\nc∈C′d(x,C′). Then\n/summationdisplay\nx∈Xdz(cx,C∗)≤2z−1/summationdisplay\np∈Xdz(cx,X)+dz(x,C∗)\n≤2z−1(α+1)·φz\nOPT.\nThis implies\n/summationdisplay\np∈Xdz(cx,S)≤γ·/summationdisplay\np∈Xdz(cx,C∗)\n≤γ·2z−1·(αz+1)·φz\nOPT.\nCombining, we then have\n/summationdisplay\nx∈Xdz(x,C)≤2z−1/summationdisplay\nx∈Xdz(cx,C)+dz(x,C′)\n≤2z−1/parenleftbig\nγz·2z−1·(αz+1)+αz/parenrightbig\n·φz\nOPT\n≤22z·γz·αz·φz\nOPT\nTaking the zthroot then yields a 4γαapproximation.\nWe remark that if we were optimizing the objective/summationtext\nx∈Xdz(x,C)instead ofz/radicalbig/summationtext\nx∈Xdz(x,C),\nthe claim changes to an 4z·α·γapproximation. Tighter bounds than claimed are possible, b ut\nsince we are only interested in O(1)distortion, the bounds we presented are sufficient for our nee ds.\nWe now show a generalized version of Lemma 3.2 in Arthur and Va ssilvitskii [ 13], holding for\nall(k,z)-clustering objectives in metric spaces. To that end, let us recall that\npz(c) :=d(c,C)z\n/summationtext\nx∈Xd(x,C)z,\nto be the probability we sample a point cconditioned on having selected a set of centers Calready.\nWe denote the induced distribution by D++\nz, for any z. Moreover, recall that C∗={A1,...,A k}is\nthe optimal clustering collection.\nClaim B.3. LetCbe the current set of centers and let Abe an optimal cluster from C∗. For some\nz,1≤k≤nandc∈A, letC′be the set of centers added to Csuch that P(c∈C′)≥pz(c). Then,\nED++\nz[φz\nC∪{c}(A)|c∈A∈C∗,C]≤2z+1·φz\nOPT(A),\nwhere the expectation is conditional on having selected set of centers Calready.\nED++\nz[φC∪{c}(A)|c∈A∈C∗,C]≤4·φOPT(A),\n21Proof. Every point a∈Awill contribute exactly min(d(a,C)z,d(a,C′)z)toφz\nC∪C′(A). Letcbe an\narbitrary point in AandC′be the set of centers Algorithm 1adds in current iteration. By Lemma\n4.3, we know that P[c∈C′]≥φz\nC(c)/φz\nC(X). As picking additional points only reduces the cost,\nwe have the following upper bound on the expected value of E[φz\nC∪C′(A)]\nE[φz\nC∪C′(A)]≤/summationdisplay\nc∈Aφz\nC(c)\nφz\nC(A)/summationdisplay\na∈Amin(d(a,C)z,d(a,a0)z)\nBy the triangle inequality, we have d(a0,C)≤d(a,C)+d(a,a0)and it follows from the power-\nmean inequality that,\nd(a0,C)z≤2z−1·(d(a,C)z+d(a,a0)z).\nAveraging over all a∈A, we get\nd(a0,C)z≤2z−1\n|A|/summationdisplay\na∈Ad(a,C)z+2z−1\n|A|/summationdisplay\na∈Ad(a,a0)z\n. Using this bound for d(a0,C)zand the fact that min(d(a,C)z,d(a,a0)z)≤d(a,C)z,d(a,a0)z, we\ngetE[φz\nC∪C′(A)]≤2z+1φz\nOPT(A).3\nFurther, by Jensen’s inequality, we have E[φC∪C′(A)]≤E[φz\nC∪C′(A)](1/z)≤4·φOPT(A).\nFrom the above proof, we get that any time we pick at least one p oint from a given cluster, say\nAi(from the optimal clustering), we get 4-approximation in expectation. It is important to note\nthat, we might pick multiple centers from the same cluster in some iteration.\nDefinition B.4 (Covered Optimal Cluster) .For alli∈[k], optimal cluster Aiis considered to be\ncovered ifφC(Ai)≤10·φC∗(Ai), and uncovered otherwise. For ease of notation, we use Uncovered\nto denote the set of points in uncovered clusters.\nWe prove that at the end of kiterations, the probability that we do not hit some uncovered\ncluster and we do not have an O(1)approximation is very small.\nLemma B.5. For some set of centers C, ifφC(X)≥20·φOPT,z(X), then choosing a point according\ntoDz++we hit an uncovered cluster with probability which is at leas t1/5.\nProof. Assuming φC(X)≥20·φOPT(X), first we prove that the uncovered clusters account for at\nleast1/2of the total cost. We have\n20zφz\nOPT(X)≤φz\nC(X)\n=/summationdisplay\ni∈Uncoveredφz\nC(Ai)+/summationdisplay\ni∈Coveredφz\nC(Ai)\n≤/summationdisplay\ni∈Uncoveredφz\nC(Ai)+10z·/summationdisplay\ni∈Coveredφz\nC∗(Ai)\n≤/summationdisplay\ni∈Uncoveredφz\nC(Ai)+10z·φz\nC∗(X).\nThus,PDz++[Anuncovered unhit cluster is hit ] =φC(Uncovered )\nφC(X)≥20z−10z\n20z≥1/2.\n3This last step is identical to the one in Arthur and Vassilvit skii [13].\n22But note that we sample points in different rings independentl y. Partition all the points from\nthe bad clusters into sets {X1,X2,...Xm}according to rings, i.e., group all points from the same\nring together. Then, the probability of picking at least one of these points is\n1−/productdisplay\ni∈[m]\n1−/summationdisplay\nx∈XiφC(x)\nφC(X)\n≥1\n2·/summationdisplay\nx∈UncoveredφC(x)\nφC(X)≥1\n2·1\n2=1\n4.\nThe proof of the above claim comes from the following argumen t. We give a sketch of the proof.\nFirst, using the AM-GM inequality, we have\n/productdisplay\ni∈[m]\n1−/summationdisplay\nx∈XiφC(x)\nφC(X)\n≤(1−p′)m,\nwherep′=1\nm·/summationtext\nx∈UncoveredφC(x)\nφC(X).\nMoreover, using the fact that ft(x) = (1−x/t)tis convex for any t≥1andft(1)≤1/e, we also\nhave\n1−1\n2·/summationdisplay\nx∈UncoveredφC(x)\nφC(X)≥/productdisplay\ni∈[m]\n1−/summationdisplay\nx∈XiφC(x)\nφC(X)\n.\nClaim B.3shows that sampling a single centercfromAaccording to distribution D++yields\nED++[φC∪{c}(Ai)]≤4·φC∗(Ai). Thus, by Markov’s Inequality, we have that\nPD++[φC∪{c}(A)≥5·φC∗(A)]≤4\n5.\nThis means that Aiwill be covered (Definition B.4) with a probability of at least1\n5, if we sample\naccording to D++. This is equivalent to saying that there exists A′⊆Athat has φC(A′)≥φC(A)\n5,\nand such that sampling a center c∈A′makesAcovered.\nThe probability that we hit an uncovered clusterAand make it a covered cluster is 4/5·1/4 = 1/5.\nWe conclude that in each iteration, we cover an uncovered cluster with probability 1/5.\nThe proof of Theorem 4.1follows directly from Lemma B.5. Algorithm 1can be viewed\nas repeating each step O(logk)times. Hence, the probability of failure in each iteration i s\n(4/5)O(logk)≤1/(2·k). A large enough constant for this to hold is c≥7. An upper bound\non the probability of failure in any one (at least one) of the kiterations is k·(1/(2·k)) = 1/2.\nHence, the algorithm succeeds with probability at least 1/2.\nFor the proof of Theorem 4.2, we use the analysis done so far and show that repeating the\nalgorithm logntimes amplifies the probability of success to give us a consta nt factor approximation.\nProof of Theorem 4.2.First, note that we augment the solution with the k-center approximate\nsolution which is an O(n)approximation with respect to any (k,z)objective. We prove that the\noutput of the sub-routine (Line 2-11) of Algorithm 5gives us a constant factor approximation with\nprobability 1−1/n. We know from Lemma B.5that if we sample one point, we always hit a\ncluster with probability at least 1/2. As we repeat the loop O(logk)times, we succeed (hit an\nuncovered cluster) with probability 1−1/2O(logk)= 1−1/(2·k). Asiranges from 1tok, the\nprobability of success (hitting all the uncovered clusters which are at most kin number) is at least\n1−k/(2·k) = 1/2and with constant probability we cover it. We use the union bound to bound\nthe probability of at least one failure, which is at most k/(2·k) = 1/2. As we repeat the whole\n23algorithm logntimes, the probability that we get a constant approximation in at least one of times\nis1−(1/2logn) = 1−1/n.\nNote that the worst-case approximation ratio of our cluster ing will be O(n)(because we augment\nthen-approximate k-center solution). Putting it all together, we get that the e xpected cost of the\nsolution is (1−1/n)·20·φC∗(X) + (1/n)·n·φC∗(X) =O(1)·φC∗(X)and that concludes the\nproof.\nB.2 Proof of Theorem 4.4\nAlgorithm 6: k-median with a O(k4log5n)queries\nInput: Point set X, ordinal information {πx}x∈Xandk∈N\n1Initialize C←C0from Theorem 3.3’sk-center algorithm\n2Sample a point cuniformly at random from A\n3C←C∪{c}\n4fort= 1toTdo\n5Samplec∈Sxjwith probability min/parenleftbigg\n1,T\n|Sxj|·/hatwiderφC(Sxj)/summationtext\ni,j/hatwiderφC(Sij)/parenrightbigg\n6C←C∪{c}\n7returnC\nClaim B.6. Given a set of current centers C, the probability that center cis sampled according to\ndistribution Dis\n/hatwidep(c)≥1\n2·p(c),\nwherep(c)is simply PD++[cis added to C], and similarly /hatwidep(c)isPD[cis added to C].\nProof. Let us begin by recalling that\np(c) =d(c,C)/summationtext\nx∈Xd(x,C)=d(c,C)\nφC(Ai)·φC(Ai)/summationdisplay\nx∈Xd(x,C)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=φC(X)\n/hatwidep(c) =1\n|Sxj|·/hatwiderφC(Sxj)\n/summationtext\ni,j/hatwiderφC(Sij).\nSince we also know that /hatwiderφC(Sxj) =|Sxj|·minq∈Sxj−1d(q,C)≥|Sxj|·d(c,C)for allc∈Sxj, then\nall we need to show is that/summationtext\ni,j/hatwiderφC(Sij)≤2·φC(X), in which case the claim holds. We have that\nfor alli,\n/hatwiderφC(Sij) =|Sij|·min\nq∈Sij−1d(q,C)\n≤2·|Sij−1|·min\nq∈Sij−1d(q,C)\n≤2·φC(Sij−1),\n24where the first inequality follows from the construction of t he rings Sij’s, and the second again by\nthe fact that /hatwiderφC(Sij)≥|Sij|·d(c,C). All in all, we have that\n/summationdisplay\ni,j/hatwiderφC(Sij)≤2·/summationdisplay\ni,jφC(Sij−1)≤2·φC(X),\nwhich concludes the proof.\nWe note that the proof may be readily adapted to squared dista nces or even distances with\narbitrary powers. Thus, the same analysis also works for (k,z)clustering. We also can improve the\nlower bound to /hatwidep(c)≥(1−ε)·p(c)for anyε, at the cost of increasing the number of queries by a\nfactorε−1. However, since this does not improve the analysis in any mea ningful way, we will use\nthe claim as stated.\nSince the cost of a (k,z)clustering is always between the cost of a (k,1)and a(k,∞)clustering,\nthe same claim also applies to (k,z)clustering in general.\nLemma B.7. LetCbe the current set of centers and let Abe some optimal but yet uncovered\ncluster from C∗. Then, the probability that Aremains uncovered after the addition of a new center\nis, at most\nP[Aremains uncovered ]≤exp/parenleftbigg\n−T·φC(A)\n10·φC(X)/parenrightbigg\n.\nProof. Claim B.3shows that sampling a single centercfromAaccording to distribution D++yields\nED++[φC∪{c}(Ai)]≤4·φC∗(Ai)(fork-median z= 1). Thus, by Markov’s Inequality, we have that\nPD++[φC∪{c}(Ai)≥5·φC∗(Ai)]≤4\n5.\nThis means that Aiwill be covered (Defn B.4) if we sample according to D++, with a probability of\nat least1\n5. This is equivalent to saying that there exists A′⊆Athat has φC(A′)≥φC(A)\n5, and such\nthat sampling a center c∈A′makesAcovered. Given that we sample according to distribution D\n(as opposed to D++) amplified (multiplicatively) Ttimes, we have that\nP[Aremains uncovered ]≤/productdisplay\nc∈A′(1−T·/hatwidep(c))\n≤/productdisplay\nc∈A′/parenleftbigg\n1−T·p(c)\n2/parenrightbigg\n≤exp/parenleftBigg\n−/summationdisplay\nc∈A′T·p(c)\n2/parenrightBigg\n= exp/parenleftbigg\n−T\n2·φC(A′)\nφC(X)/parenrightbigg\n≤exp/parenleftbigg\n−T·φC(A)\n10·φC(X)/parenrightbigg\n,\nwhere the second inequality comes from Claim 4.5, the third by 1+x≤exfor allx, and the last\nby recalling that φC(A′)≥φC(A)\n5.\nWe now suppose that we are not yet at the iteration where we hav e reached a constant distortion,\notherwise, we would already be done.\nLemma B.8. LetφOPTbe the cost of an optimal k-median clustering and let tbe such that φt(X)≥\n20·φOPT. Then,\nE[φCt+1(U)]≤1+exp/parenleftbig\n−T\n40k/parenrightbig\n2·φCt(U).\n25Proof. Before beginning, let us observe that the assumption φCt(X)≥20·φOPTimplies that\nφCt(U)≥φCt(X)\n2, as otherwise covered clusters would count for more than hal f the total cost of X,\ni.e.,φCt(X)<20·φOPT(by Definition B.4), which is a contradiction.\nLetUbe partitioned into the heavy collectionHt:=/braceleftBig\nA⊆U|φCt(A)≥φCt(U)\n2k/bracerightBig\n, and the light\ncollectionLtconsisting of all the remaining optimal clusters. By Lemma B.7 , we know that the\nprobability that a heavy optimal cluster Ais not hit in the t+1stiteration is bounded by\nexp/parenleftbigg\n−T·φCt(A)\n10·φCt(X)/parenrightbigg\n≤exp/parenleftbigg\n−T·φCt(U)\n20k·φCt(X)/parenrightbigg\n≤exp/parenleftbigg\n−T\n40k/parenrightbigg\n.\nThis means that heavy cluster Ais covered with at least the converse probability. In turn, t his\nimplies that the cost of uncovered clusters must decrease by at least the expected decrease of heavy\nclusterA’s cost, so that\nφCt(U)−E[φCt+1(U)]\n≥/parenleftbigg\n1−exp/parenleftbigg\n−T\n40k/parenrightbigg/parenrightbigg\n·/summationdisplay\nA∈HtφCt(A)\n=/parenleftbigg\n1−exp/parenleftbigg\n−T\n40k/parenrightbigg/parenrightbigg\n·/parenleftBigg\nφCt(U)−/summationdisplay\nA∈LtφCt(A)/parenrightBigg\n≥1−exp/parenleftbig\n−T\n40k/parenrightbig\n2·φCt(U),\nwhere the last inequality follows since light clusters have cost at most k·φCt(U)\n2k=φCt(U)\n2.\nWe now combine the above results to obtain the following theo rem (a restatement of Theorem\n4.4).\nTheorem B.9. Algorithm 6yields aO(1)-distortion (in expectation) to the optimal k-median clus-\ntering using O(k4log5n)queries.\nProof. Since Algorithm 5 uses Algorithm 4 as a subroutine, and the la tter outputs a 4-distortion to\nthe optimal k-center clustering, we have that E[φC0(U)]≤4n·φOPTby Claim B.1.\nBy Lemma B.8, we know that E[φCt+1(U)]≤20·φOPT+1+exp(−T\n40k)\n2·φCt(U), which means that\nby applying this expression repeatedly, we obtain\nE[φCT(U)]≤/parenleftBigg\n1+exp/parenleftbig\n−T\n40k/parenrightbig\n2/parenrightBiggT\n·4n·φOPT\n+20·φOPT·T−1/summationdisplay\nt=0/parenleftBigg\n1+exp/parenleftbig\n−T\n40k/parenrightbig\n2/parenrightBiggt\n≤/parenleftBigg/parenleftbiggn+1\n2n/parenrightbigg40klogn\n·4n+40·n\nn−1/parenrightBigg\n·φOPT\n≤42·φOPT,\n26where the first inequality holds by choosing T≥40klogn. Since, the number of iterations is also\nT, this means that opening 1600k2log2ncenters allows us to achieve\nE[φCT(X)]≤52·φOPT,\nwhich follows from E[φCT(X)]≤E[φCT(U)]+10·φOPT.\nUsing Claim B.2, and any arbitrary approximation algorithm, of which the be st currently know\nis a2.613approximation [ 42], we therefore obtain a 4·52·2.613<544distortion.\nC Lower Bounds for k-Median\nTheorem C.1. For any fixed α, every bicriteria algorithm Afork-median that has distortion\nless than αwith at least constant probability must return a solution of size at least Ω/parenleftBig\nlogn\nlogα·2k/parenrightBig\n.\nMoreover, any algorithm achieving a constant distortion fo rk-median must make at least Ω(k+\nloglogn)queries.\nProof. As with the proof for k-center above, we first describe the hard instance for the zer o-query\nregime and then remark on how to extend it. To simplify the cal culations, we prove the lower bound\nfor an input of size Θ(n), where we make the following two assumptions:\n• There is an integer n′such that n= 2k−2·n′.\n•n′andα+1are powers of 2.\nThe claim for general nandαcarries over with very minor details.\nThe hard instance: The first part of the instance is almost identical to that of k-center in the\nproof of Theorem 5.1. Indeed, since the distortion of k-center is unbounded, it is also unbounded\nfork-median as both costs are within a factor nof each other. Recall that our hard instance for\nk-center used a complete binary tree T. In our hard instance for k-median, we augment this tree\nby adding a hard instance for 2-median below each of its leaves.\nWe proceed to describe these 2-median instances. For a leaf node uinT, we refer to the 2-\nmedian instance below uasIu. For every leaf u,Iuconsists of n′=n/(2k−2)points. We group\nthe points in Iuinto bundles Bifori∈{0,...,logn′\nlog(α+1)}. Thei-th bundle has the property that\n|Bi|= (α+1)i.\nWe now introduce the ordinal preferences among the points in Iu, as well as between the points\nof different 2-median instances. Then, we describe the distribution over metrics consistent with said\npreferences.\n• Consider only points from a 2-median instance Iu. For any two points p,q∈Biand any point\no /∈Biwe have d(p,q)≤d(p,o),d(q,o). The remaining ordinal preferences among the points\ninIumay be chosen arbitrarily.\n• Letp∈Iu, and let q,obe two points such that either q∈Iv,v/\\e}atio\\slash=uoro∈Iv,v/\\e}atio\\slash=u.\nThen, whether pprefersqoveroor not, depends on the depths of the common ancestors\na(p,q),a(p,o)inT. We refer the reader to the description of the ordinal prefer ences in our\nhardk-center instance (see the proof of Theorem 5.1).\n27We now specify the hard input distribution over metrics that is consistent with these preferences.\nWe initialize Tas a binary tree of depth k−2and pick a leaf node runiformly at random. Let Q\nbe the path in Tfrom its root to r. We now assign values to each node in Tincluding its leaves .\nThese values are d(a) =Dfor every node athat lies on the path Q(whereDis some sufficiently\nlarge number) and d(a) =εotherwise (where ε >0is arbitrarily small).\nThe distance between any two points p,q∈Iu,u/\\e}atio\\slash=risε. For the 2-median instance Ir, we\nselect an ℓ∈{0,...,logn′\nlog(α+1)−1}uniformly at random. The distances now satisfy the followin g\nproperties:\n• For every pair of points p,qfrom a bundle Bjwithj≥ℓ, we setd(p,q) =ε.\n• For every pair of points p,qfrom a bundle Bjwithj < ℓ, we setd(p,q) = 1.\n• For every p∈Bjandq∈Bk,j/\\e}atio\\slash=k, we setd(p,q) = 1 ifk≤ℓ. Ifk,j > ℓ , we setd(p,q) =ε.\nSince the ordinal preferences were determined before sampl ingℓ, no algorithm using only ordinal\ninformation can determine any information about ℓ.\nAs before, the distance between any two points p∈Iu,q∈Iv,u/\\e}atio\\slash=vis given by the value that\nis stored at their common ancestor node a(p,q)inT(see the proof of Theorem 5.1).\nWe now consider the cost of an optimal solution C. As in our hard instance for k-center (Theorem\n5.1), the optimal solution must place at least one center in ever y subtree T(a,small)whereais\na node on the path Q. Otherwise, the solution has cost at least Dwhich can be arbitrarily high.\nConsider any subtree T(a,small)and note that its root bsatisfiesd(b) =ε. Hence, if the solution\nplaces a center on any leaf in T(a,small), then the contribution of the points in T(a,small)to the\ncost of the solution is negligible. Hence, the cost of any opt imal solution Cdepends only on the\ncost incurred for the points in Ir.\nWe claim that Cplaces a single center cℓinBℓand a single center in some bundle Bjwithj > ℓ.\nThe cost of the points in bundles Bjwithj > ℓ is nowε. The cost of a point pserved by cℓisε,\nifp∈Bℓand1ifp∈Bk,k < ℓ. Thus the overall cost is/summationtextℓ−1\ni=0αi=(α+1)ℓ−1\nα, ignoring negligible\ncontributions from the ε-valued distances.\nAny solution that does not intersect with a bundle Bj,j > ℓ costs at least (α+1)logn′\nlog(α+1)=n′.\nFinally, any solution that does not intersect with Bℓcosts at least (α+ 1)ℓ. Both of those terms\nare larger than(α+1)ℓ−1\nαby at least a factor αifn′is large enough so we can conclude that Cis\noptimal.\nAgain, it suffices to consider the performance of a determinis tic algorithm placing Kcenters\nagainst the hard input distribution. Since the ordinal info rmation offers no information on either\nQorℓ, the choice of centers C′is fixed. As in the proof of Theorem 5.1, the probability that C′\nincludes any point from IrisK/2k−2such that if K /∈Ω(2k)the distortion is Dwith at least\nconstant probability. Furthermore, the probability that C′intersects with Bℓis at mostlog(α+1)\nlogn′.\nThus,C′must consist of\nΩ/parenleftbigglogn′\nlog(α+1)·2k/parenrightbigg\n= Ω/parenleftbigglogn−k\nlog(α+1)·2k/parenrightbigg\ncenters to improve over an αdistortion. The first part of the theorem now follows by choos ingn′\nsuch that nis large enough compared to k.\nFor the low-query regime, the argument that we require at lea stΩ(k)queries is equivalent to\nthat of the k-center instance, being that the instances for the first k−2levels of the tree are identical.\nTheΩ(loglog n)query lower bounds follows from the fact that there are logn′many choices for the\nbundleBℓand every query can rule out half of the remaining possible ch oices.\n28Theorem C.2. For any fixed αand every fixed k, every bicriteria algorithm Afork-median that\nhas distortion less than αwith at least constant probability must return a solution of size at least\nΩ/parenleftBig/parenleftbig\n2log∗n/parenrightbigk−1/parenrightBig\n. The number of queries to achieve a constant distortion is at leastΩ(k·2log∗n).\nProof. Before beginning the proof, we require a bit of notation. For t wo non-negative integers a\nandb, we say thatab=/braceleftBigg\n1 ifa= 0\nba−1belse, i.e. the tetration bb..b\nwitha bs. We furthermore denote\nbyexpa\nb(x) =bb..bx\n, witha bs.\nThe hard instance: We assume that αis a sufficiently large non-negative integer. The instance\nconsists of a d-regular tree T, where the leaves contain the points, though this time the nu mber of\npoints in every leaf will typically be (far) greater than 1. The interior nodes of the tree will induce\ndistances between these nodes.\nThe tree is now described recursively as follows. Suppose th e tree has depth k−1. Then it\nhasdk−1many leaves. We number the leaves from 1todk−1. Fori=a·d+swithabeing a\nnon-negative integer and s∈{0,1,...,d−1}, we add αexpa\nd(s)points in leaf Li, for a sufficiently\nlarge constant d. Denote the number of points in leaf Libyni. Note that this satisfies the following\ninvariant, for αlarge enough:\nInvariant C.3. LetLibe a leaf. Then/summationtexti−1\nj=1nj≤α·ni.\nAs an immediate consequence, we also know that the total numb er of points is of the order\nαk−1d. Note that by definition of tetration, we have log∗(n) =kifαk−1d≤nαkd.\nWe define the common ancestor a(p,q)to be deepest interior node containing both pandq.\nNote that if pandqare contained in the same leaf, this ancestor is the leaf. We t hen have the key\nconstraint on the distances.\nInvariant C.4. Letp,q,o be points. If the common ancestor a(p,q)is deeper than the common\nancestors a(p,o)anda(q,o)thend(p,q)≤min(d(p,o),d(q,o)).\nThe preference lists are arbitrary, as long as they are consi stent with this constraint.\nWe now define the hard distribution. We choose a path from the r oot ofTto a random leaf\nLh. LetQbe the unique path. Let p∈Liandq∈Lj. Ifi < h thend(p,q) = 1. Now, let Q(p)\n(respectively Q(q)) be first interior node in Qin the path from Lito the root. If h≤i,jand\nQ(p) =Q(q) =a(p,q), thend(p,q) =ε, for a sufficiently small ε. Otherwise, d(p,q) = 1. Note that\nwe may break ties to enforce consistency with the preference lists.\nAnalysis: Since the preference lists were fixed before the outcome of th e random process, no\nalgorithm can determine the outcome of the process using onl y the ordinal information. Thus we\naim to show that the gap in the cost of an optimal k-median clustering is large for any two different\noutcomes of the random process. If this gap is large, then any ordinal algorithm must place centers\nin every leaf of the tree, i.e. in dk−1many centers. By choice of n, we have that αk−1d= Θ(n),\nwhich implies that we may choose d∈Ω(2log∗n), forαandkfixed. Thus all that remains is a\ncharacterization of the optimum.\nFor a fixed path Q= (a1,a2,...Lh), whereajare the interior nodes with a1being the root, let\nL(t)be the set of leaves Liwithi≥hand such that at=Q(p) =Q(q)forp∈Li∈L(t)and\nq∈Lj∈L(t). We place a center on an arbitrary point in one of the leaves L(t), for each t. Observe\nthat this places exactly kcenters by length of Q. Moreover, this choice is optimal, as the points on\nthe leaves Liwithi < h cost1in every solution and the remaining points cost ε, which is considered\n29negligible. Now, we consider an arbitrary other solution. By definition, there exists some set of\nleavesL(t)such that we do not place a center on any of the points in L(t). The number of points\nin the union of leaves in L(t)is at least nh. Thus, the cost of this solution must be at least nh·1.\nObserve that the cost of the optimum is at most/summationtexth−1\ni=1ni·1. Thus the approximation factor is of\nthe order αdue to Invariant C.3.\nAs with the preceding lower bounds, we must make at least logdmany queries at every depth\nof the tree to find the path Q. Thus the total number of queries is Ω(klogd)∈Ω(k·log∗n).\nD Facility Location with Uniform Opening Costs\nIn this section, we revisit the the seminal algorithm for onl ine facility location by Meyerson [ 52].\nFor worst case input orders, it is known to achieve an optimal O(logn/loglogn)approximation\n[37]. For random order inputs, it is known to achieve a 4-approximation [ 46], which we simulate.\nIn every iteration, the algorithm needs to determine the exa ct distance of point xto the set Cof\nalready opened facilities, see line 5in the description of Algorithm 7. This operation requires a\nsingle query given the ordinal information. Furthermore, t he operation is performed for every point\nexactly once. The following theorem summarizes this discus sion.\nAlgorithm 7: Meyerson’s algorithm for facility location\nInput:X,d,P,f\n1R←random permutation of X\n2C←{R(1)}\n3fort= 2...ndo\n4x←R(t)\n5p←min/braceleftBig\n1,d(x,C)\nf/bracerightBig\n6SetC=C∪{x}with probability p\n7returnC\nTheorem D.1 (See also Pulyassary [ 56]).Algorithm 7achieves constant expected distortion for the\nordinal facility location problem with uniform opening cos ts using one query per point.\nD.1 Lower Bounds for Facility Location\nTheorem D.2. For any fixed α, every algorithm Afor facility location that has distortion less than\nαwith at least constant probability must make Ω/parenleftbign\nα/parenrightbig\ndistance queries.\nProof. We assume that the opening costs per facility are 1. As before, we first describe the ordinal\npreferences and then sample a metric from a distribution con sistent with these preferences.\nThe hard instance: We group the points inton\nsclustersAi, each consisting of s∈Ω(α)points.\nAny two points from a cluster Aiprefer each other over any point from some other cluster Aj.\nMoreover, the distances between any two points p∈Aiandq∈Aj,i/\\e}atio\\slash=jare set to be∞(or a\nsufficiently large number if finite values are required). The p references inside the clusters, as well as\nacross clusters are arbitrary as long as every point p∈Aiprefers any point q∈Aiover any point\no∈Aj,j/\\e}atio\\slash=i.\n30The hard input distribution now consists of the following. W e select a cluster Aiuniformly at\nrandom, and toss a fair coin. With probability1\n2, all of the points in Aihave pairwise distance\nε. With probability1\n2, the pairwise distances in Aiare chosen to be a sufficiently larger number\nN≫n/s+s−1. For each cluster other than Ai, the distances between any two points within this\ncluster are ε.\nAnalysis In the case that the pairwise distances in Aiareε, the optimal solution consists of\nplacing exactly one facility in every cluster, leading to a c ost ofn/s(if we ignore the arbitrarily\nsmall connection costs). In the case that the pairwise dista nces areN, the optimal solution consists\nof placing exactly one facility in every cluster Aj,j/\\e}atio\\slash=iand placing a facility on every point in Ai,\nleading to an overall cost of n/s+s−1.\nTo distinguish between these two cases, the algorithm has to query at least one distance between\ntwo points in Ai. Suppose the algorithm makes Qqueries. If the algorithm fails to determine whether\nAihas pairwise distances Nor not, it must place a center on every point of a cluster it has not\nqueried, as otherwise the distortion is Nand therefore unbounded.\nBy Yao’s minimax principle, we may assume that the centers que ried by the algorithm are fixed\nuntilAiis detected. The probability that Aiis detected isQ·s\nn. Thus, if Aiis not detected, the\nalgorithm must place s−1additional facilities on each unqueried cluster, that is, (n/s−Q)·(s−1)\nadditional facilities in total. Hence, in the case that the d istances between the points in Aiareε\n(which occurs with probability1\n2), the algorithm incurs a distortion ofn/s+(n/s−Q)·(s−1)\nn/s∈Ω(α)for\nQ∈o(n/α). Here, we used that we chose the cluster size ssuch that s∈Ω(α). The claim now\nfollows by scaling sso that the distortion becomes exactly α.\n31"    },    {        "title": "2402.04044v1.Interlaboratory_comparison_of_particle_filtration_efficiency_testing_equipment.pdf",        "content": "Pre-print  (2024 ) \n \n \n  \nInterlaboratory  comparison of  particle  filtration \nefficiency testing equipment  \n \nTimothy A. Sipkens1, Ruth Perez  Calderon1, Richard G. Green1, Andrew Oldershaw1, Gregory J. \nSmallwood1 \n1 Metrology Research Centre, National Research Council Canada , Ottawa ON Canada  \n \nAbstract  \nThis work presents the results of two interlaboratory comparison s of particle filtration efficiency measurements performed by a \nnetwork of laboratories across  Canada  and Australia . Testing across multiple layers of a common verification material  \ndemonstrates a constant size -resolved quality factor  when layering uncharged material s. Size -resolved filtration curves also \nmatch expectations, with increasingly size -dependent curves  and a predictable increase in the PFE. Candidate reference materials \nwith controlled material properties were also tested across multiple laborato ries. Each set of materials sharing a common charge \nlevel show specific trends with the material basis weight.  Respirators showed more consistency between the laboratories  than \nthe other filters . However, a cross a majority of the tests, dark uncertainties, which are  otherwise unexplained  variability between \nlaboratories, are significant. This leaves room to improve the test  method  by developing  improved verification  procedures  and \nadditional reference materials .  \n \n1. Introduction  \nFace masks, respirators, and other related products are essential personal protective equipment , which  protect against a range \nof respiratory hazards , including  infectious pathogens , forest fire smoke , and occupation al hazards. Amongst the fundamental \nquantities characterizing these products is particle filtration efficiency  (PFE)  that quantif ies the ability of a filter to capture \nparticles in the air . The considerable attention given to PFE as a result of the COVID -19 pandemic highlighted the need for a \nbetter understanding of test results. Quantifying the uncertainty of test results, as well as the precision and bias of the associated \ntest methods , are key to removing doubt as to the transferability or traceability of t he test methods used to determine PFE. \nGiven potential challenges with understanding dark uncertainties reference materials supported by robust interlaboratory \ncomparisons , present a more viable pathway to establishing traceability.   \nInterlaboratory comparisons (ILCs) to quantify the reproducibility of measurements of respiratory protection  are not widely \navailable  and are often very limited in the range of materials and participating laboratories . Li et al. [1] considered two electret \nmaterials and one fiberglass sample , making size-resolved measurements across two laboratories.  Bourrous  et al. [2] performed \nan ILC with three laboratories using  filtration media making up community face coverings, with the goal of identifying optimal \nmaterials for use in such products. In that case, the aerosols varied between the three laboratories (NaCl, DEHS, and PSL), \nwhich were size -selected using an aerodynamic particle sizer. While impaction is the dominant mechanical filtration \nmechanism and is related to aerodynamic diameter, el ectrostatic filtration may differ between the different particle types and \ncould expand uncertainties  beyond simple interlaborato ry differences . Tessarolo et al. [3] presented a comparison of \nmeasurement of bacterial filtration efficiency (BFE) and pressure drop by six non -accredited laboratories in Italy. This study \nspanned three separate types of masks and provided some decomposition of the variability based on structure in the \nmeasurements . Fouqueau et al. [4] presented a comparison across  four laboratories examining the difference between  particle \nfiltration efficiency at 3 μm (two laboratories)  and bacterial filtration efficiency  (the remaining two laboratories) .  Page 2 of 14  Interlaboratory comparison for PFE  \n \n \nThis work  aims  to rigorously quantify differences between measurements of PFE across a series of independent \nlaboratories , specifically for the sodium chloride test typical ly used for testing respirators . It expands greatly on previous \ncomparisons , with a  substantially wider  range of filters and a larger number of laboratories spanning two countries , allowing \nfor a more robust estimate and understanding of interlaboratory variability in respiratory protection testing . Results are \nsupplemented with measurement s of pressure re sistance, enabling discussion of filter quality [5], and are examined in terms of \ntrends in certain  underlying material properties, including basis weight ( using qualified categories ), charge state, number of \nlayer s, and environmental conditioning (specific to a single respirator).  \n2. Methods  \n2.1 Experimental protocol  \nAll of the measurements throughout this work  were conducted using  a challenge aerosol consistent with  many standards for \nparticle filtration  testing of respirators  such as that used in NIOSH TEB -APR -STP-0059  [6] for certifying N95 respirators, GB \n2626 -2019  [7] for certifying KN95 respirators, and  CSA Z94.4.1:21  [8] for certifying CAN95 respirators. S odium chloride is \nnebulized to form a  challenge  aerosol  having a count median mobility diameter (CMD) of 75  nm ± 20 nm and a geometric \nstandard deviation  (GSD)  less than or equal to  1.86. In accordance with the cited test methods, the p articles are typically  dried, \ndiluted , and neutralized .  \nA range of i nstruments were permitted to measure aerosol concentrations and are separated into  three categories . Due to \nthe prevalence of the TSI 8130A, both in terms of its explicit mention in the TEB -APR -STP-0059 test method  [6] and in terms \nof its use by participating laboratories, the instrument receives its own category. The remaining instruments are categorized  \neither as photometer -based, where scattering is used to measure the quantity of particles, or count -based, where par ticles are \ncounted  (e.g., the Particle Filtration Efficiency Measurement System (PFEMS) , which was  developed by the National Research \nCouncil Canada [9, 10]). Note that the TSI 8130A  (TSI, USA)  is a photometer -based instrument , even though it has its own \ncategory. Other photometer -based instruments include the DustTrak  (TSI 8540/8543, USA) , PMFT 1000  (Palas , Germany ), \nand 100X  (ATI , USA ). Measurements with a photometer act as a close surrogate to the mass of particulate captured by a filter \n[11, 12]. As such, those laboratories employing count -based measure s were instructed to  convert  their measurements  to a mass -\nequivalent result for comparison against other data using an appropriate method  [11], with differences in these representations \nhaving been measured in the literature previously [1, 11, 13].  \nWithin this framework, measurements  were conducted across two interlaborat ory comparisons, summarized in Table 1. \nThe first ILC involved laboratories from Canada that took place in 2021, while the second ILC was a bilateral comparison \ninvolving laboratories from both Canada and Australia that took place in 2022. The distinction between the first and second \nILC is otherwise unimportant and receives little attention in the remainder of this  work. Rather, throughout, the measurements \nare separated into components , used to group a series of subcomponent s investigating a particular set of samples  (e.g., \nmeasurement of a fib erglass  material that is layered). In all cases, internal measurements made by the National Research \nCouncil Canada prior to sample distribution indicated uniformity amongst the samples sufficient to consider the distributed \nsamples as random ly selected  from the sample population .  \nLaboratory identifiers throughout this manuscript are anonymized  and are identified on a per -ILC basis  (i.e., lab 1 for the \nfirst ILC is not the same as lab 1 for the second ILC).  For the first ILC, laboratories are numbered  according to results from \nmeasurements for one layer of Green Line Paper  (prescribed calibration media documented in the TEB -APR -STP-0059 test \nmethod  [6]). For the second ILC, laboratories are sorted by the median of the  results from the  three sets of measurements  in \nthe fourth component, described in Sec. 2.1.2 .  \n Page 3 of 14  Interlaboratory comparison for PFE  \n \n \nTable 1. Summary of the various components of the two interlaborat ory comparisons presented in this work. The number of \nsubcomponents  refers to the number of different cases that were considered within  each component (e.g., measuring 1 to 5 layers for \nthe first component of the first ILC ).  \nILC Component  No. of \nsubcomponents  Sample type  Target parameters  No. of laboratories  \n1st 1 5 Green Line Paper (TSI 813010)  Δp, PFE, layering  16 \n1st 2 10 Candidate reference materials  Δp, PFE, matl. charge state  16 \n1st 3 2 Respirator  Δp, PFE, initial/minimum PFE  12 \n2nd 4 3 Respirator  Δp, PFE 15 \nΔp: pressure resistance ; P: penetration  \n2.1.1 First ILC \nThe first ILC had components driven by differences in the samples being tested, which spanned different materials, \nconstruction, and charge states . There are three components relevant here:  \nComponent 1.  This first component examined the PFE of layered Green Line Paper  (TSI 813010) , as supplied from a \nsingle lot by  the National Research Council Canada . Green Line Paper  is typically supplied for verification  of TSI 8130A \ninstruments [6]. The m aterial  is composed of uncharged fiberglass and exhibits a strongly -size dependent PFE  relative to many \nother materials . The filtration media was tested flat , i.e., without bending or cupping , and at a face velocity of 10 cm  s−1. Tests \ndiffered in terms of  layering multiple sheets of the filtration media, from 1 to 5 in increments of 1. This allows for some \nassessment of layering and allows for spanning multiple PFE levels , while holding the other properties of the material constan t.  \nComponent 2.  PFE was measured on a set of  candidate reference  material s composed of flat melt -blown polypropylene  \nsheets manufactured by Roswell Downhole Technologies and having  some  control  of the underlying material  properties . These \nmaterials  were also used in Sipkens et al. [11] and span a known range of PFE levels, basis weights, and charge states for a \nsimilar class of material. While the specifics of the material synthesis are outside of the scope of this work , the outcome is \nparticle filtration media with PFE values that span  40% – 99%, with a qualified level of charging  (none, half, and full) . The \nfirst six  subcomponents  are ordered according to the expected PFE, as measured by the National Research Council Canada \nprior to sample distribution.  This also correspond s to ordering the samples with respect  to increasing charge state, with the first \ntwo subcomponents  having  samples wi th no charge, the third subcomponent having a sample with h alf of the applied  charge, \nand the remainder of the  subcomponents having materials with a full charge applied . The final four  subcomponents  correspond \nto pairs of materials  each from the same lot (thus having the same basis weight and charge state)  that were tested both as -\nreceived (unconditioned) and environmentally conditioned according to the NIOSH TEB -APR -STP-0059  test method , namely \nfor 25 hr ± 1 hr at a temperature of 38 °C ± 2.5 °C and at a relative humidity of 85 % ± 5 % RH . Samples were  tested at a face \nvelocity of 10 cm  s−1.  \nComponent 3.  PFE was measured on formed respirators that were  supplied to each of laboratories from a single lot.  \nMeasurements were recorded both in terms of minimal loading (i.e., initial PFE) and the minimum PFE as the products are \nloaded, consistent with requirements in respirator testing. Samples were tested at a flow rate of 85 LPM.  \n2.1.2 Second ILC (bilateral)  \nThe second ILC had a focus on the test method prescribed in  CSA Z94.4.1:21  [8]:  \nComponent 4. For this component , PFE was measured on formed respirators , with measurements  reported at the minimum  \nloading possible , and were generally  measured  following the instructions i n CSA Z94.4.1:21  [8]. There was some variability \nin terms of the minimum amount of loading at which a measurement could be reported , given the range of  instruments  used in \nthis study . Conditioned samples were tested in both inhalation and exhalation configurations, while  unconditioned samples \nwere tested in the inhalation configuration , as described in CSA Z94.4.1:21 . Condition ing of samples is performed  following \nthe guidelines set out in CSA Z94.4.1:21 (which match the requirements in NIOSH TEB -APR -STP-0059). Samples were  tested Page 4 of 14  Interlaboratory comparison for PFE  \n \n \nas soon as possible  upon removal from the conditioning environment or placed and sealed in a gas -tight container at ambient \nconditions, and tested within 10 hours .  \n2.2 Log-penetration as a surrogate for PFE  \nResults for PFE are all reported in terms of the logarithm of penetration, with the equivalencies in terms of penetration and  \npercent PFE show n in Table 2, for reference. This treatment is expected to be more representative of the distribution of results \nas a consequence of how penetration scales with material thickness  and other underlying properties  [5] and the scaling of the \nnoise with penetration  [14], though this will be revisited in the results . This is also the approach  used in a number of academic \nworks (e.g., [15]) and is  cited in ASTM F2299  [16]. While uncertainties may not perfectly scale with the logarithm of \npenetration (as will be shown), this is a  far better approximation than linear scales with respect to PFE, where standard \ndeviations can easily extend above 100% for PFE , largely  making this an unacceptable metric . Such a treatment is also encoded \nin the filter  quality  factor [5], a metric balanc ing the penetration and pressure resistance that yields a quantitative measure of \nthe performance of a sample:  \n( )10 10\nF% log 1 log η 100 −\n− −==ppPQ\n  (1) \nwhere QF is the quality  factor , P is the penetration, Δp is the pressure resistance , and η is the PFE as a percent . This quantity \nhas been shown to be particularly useful in distinguishing between different  materials , e.g.,  [15, 17-21]. Note that  the quality \nfactor is often  stated in terms of the natural logarithm, such that this must be verified when comparing to other works.  The log 10 \nform is chosen  here due to convenience of the corresponding PFEs  for integer values , e.g., log 10P = −1 corresponds to 90% \nPFE.  It is also worth noting that the quality factor requires a consistent  face velocity and particle size, which further restricts \ncomparability between studies but makes the criterion useful if the test is identical across different studies.  It is expected that a \nmaterial , when layered will have a constant quality factor as the ratio of log-penetration a nd Δp does not change. This \nrelationship is linear on a log-penetration –Δp plot, with quality factor controlling the slope of the relationship.  \nPlots in this work typically reverse the penetration axis such that PFE still increases as one moves up the corresponding \npanels . Uncertainties in lo g-penetration will result in skewed uncertainties in terms of the  original  penetration and PFE. \nStandard deviations stated in terms of log-penetration will correspond to a geometric standard deviation in terms of the \npenetration itself. From simple error propagation, absolute uncertainties in PFE and  penetration  will be the same with opposite \nskew [14].  \n \nTable 2. Reference table showing the logarithm (base -10) of the penetration, the penetration as a fraction, and PFE as a percentage.  \nlog10P −3.000  −2.699  −2.301  −2.000  −1.699  −1.523  −1.301  −1.000  −0.301  0 \nPenetration, P/- 0.001  0.002  0.005  0.01 0.02 0.03 0.05 0.1 0.5 1 \nPFE, η /% 99.9 99.8 99.5 99 98 97 95 90 50 0 \n \n2.3 Statistical method : Hierarchical  random  effects  model  \nData is analyzed according to a random  effects  model . In this framework, effects are perceived as random variables , holding \nsome distribution. Then, the measured  value , yjk, is modeled  as a combination of effects  \nμ++=ij i ij ijyl \n  (2) \nwhere μi is the consensus value  for the ith test ; lij is the random  effect  for the jth laboratory  and represent s dark uncertainties ; \nand εij is the remaining error  within a given laboratory. The laboratory effect,  lij, is assumed to be normally -distributed and Page 5 of 14  Interlaboratory comparison for PFE  \n \n \nunbiased, lij ~ N(0, σl,i2), and is conditioned on the tests, allowing  for different degree s of lab -to-lab variation  for each test . \nHere, t he variance σl,i2 is used to represent dark, interlaboratory uncertainties  or biases  [22], which may not be evident in \nmeasurement by a single laboratory (i.e., are dark or hidden) and are representative of reproducibility. Errors , εij, are taken as \nnormally  distributed  with a variance from the  repeat measurements made by each laboratory  (i.e., Type A uncertainties or \nrepeatability) .  \nThe distribution for the effects and consensus value  were determined using Markov Chain Monte Carlo (MCMC). This \napproach generates a distribution for the interlaboratory variability,  lij, and the consensus  value . To minimize  the MCMC burn-\nin period, the  set of  lij were  initiated about the average  for each quantity  and μi about the global mean.  To restrict the solution \nspace and improve convergence with this Bayesian approach, we also apply priors  (encoding approximate information known \nbefore the statistical analysis)  to these quantities . The prior on μi was taken as exponential about the global average , maximizing \nthe information  entropy when only using the mean. The prior on the standard deviation of lij was taken as half Cauchy \ndistributions with a median of the difference between the global mean and the laboratory mean . The method used here makes \nuse of the Just Another Gibbs Sampler (JAGS) code (Hornik et al., 2003) , analogous to the approach used in Ref.  [23].  \nIn running the statistical analysis , outlier s are typically  not removed, so as  to capture the broader variability possible in \nthese measurements . This includes in determining the consensus value and the magnitude of the effects . However, when  \ndiscussing physical trends, outliers are identified  using the generalized extreme Studentized deviate  (GESD)  test, typically  \nusing  a threshold factor of 0.6. Exclusion of such measurements  would  act to underestimate the dark uncertainties .  \n3. Results  \n3.1 Green Line Paper and layering  \nFigure 1 demonstrates PFE measurements made by each laboratory, for 1 to 5 layers of Green Line Pape r. Results suggest that \nthe log -penetration transformation is likely reasonable , with roughly normally -distributed data following the transformation for \nmost  cases. Results still contain some  heteroskedastic ity in terms of  log-penetration, with a wider spread in the results for \nhigher PFE. It is worth noting, however, that plotting the results on a linear scale yields  far more heteroskedasticity , particularly \nwhen considering interlaboratory variability , further  affirming the choice to use log -penetration. The remaining \nheteroskedasticity  could be cause d by a number o f effects, including: (1)  the fact that these are integrated penetrations rather \nthan size -resolved penetrations , such that  these results are an additional step removed from the original filtration mechanisms  \nand thus not perfectly aligned with the model discussed by Hinds [5]; (2) a combination of Poisson and other detector noise in \nthe photometers  [14] that don’t scale with the filtration mechanisms; or (3) secondary interfacial effects between the layers  \n(though size -resolved measurements discussed subsequently suggest that this is unlikely) .  \n Page 6 of 14  Interlaboratory comparison for PFE  \n \n \n \n \nFigure 1. PFE results for an in increasing number of layers of Green Line Paper . Thick, coloured bars correspond to intralaborat ory \nuncertainties for each laboratory, while thin, black whiskers correspond to combined uncertainty with added interlaboratory \nuncertainties. Horizontal rules, from the center ou t, correspond to the consensus value  (μi) (solid lines) , intervals for uncertainties in the \nconsensus value  (σμ,i) (dashed lines) , and combined consensus and interlaborat ory uncertainties  ([σμ,i2 + σl,i2]1/2) (dotted lines) . All \nuncertainties correspond to k = 2.  \n \nDark uncertainties (related to reproducibility) contributed significantly to the overall uncertainties  across all of the \nsubcomponents (see also Figure 8 later in this work) . These contributions made up a larger proportion of the combined \nuncertainties at low PFE , in part  as the repeatability was higher  after the log -penetration transformation. Clear structure is \nvisible in the results, with laboratories that measured a lower PFE for a single layer also typically measuring lower  than the \nother laboratories  when the material was layered. This is further indicative of structured biases between the laboratories  for this \ntype of material.  \nThere is substantial scatter with respect to the different instrument types , though t he mass -based PFE computed from \ncount -based  measurements  tended to be lower. This observation  could be related to a combination of remaining differences \nbetween scattering - and mass -based filtration efficiency , where scattering -based filtration tends  to be higher [11]. Alternatively, \nthis could stem from  limitations in the existing  verification procedure  applied to the photometer instruments . The precise reason  \nrequir es further study , in particular to quantify the latter contribution.  \nFigure 2 plots log-penetration  against the pressure resistance for all of the laboratories , alongside isoline s of constant \nquality  factor , per Eq. (1). For each number of layers, there is a distinct central clustering of data (solid circles) , with some \nother results  that clearly  deviate from this trend. For this reason, outlier detection is applied to the data prior to plotting, with \nopen symbols corresponding to removed points. Deviant points typically  correspond to outliers in the pressure resistance \nmeasurement, which could be removed using an aggressive outlier treatment  (here, using a generalized extreme Studentized \ndeviate test with a threshold factor of 1 instead of the default 0.6 used for log-penetration ). This suggests that pressure resistance \nmeasurements could be improved, though this work does not address that specific issue.  Note that  the outlier s in terms of  log-\n-5.5\n-5.0\n-4.5\n-4.0\n-3.5\n-3.0\n-2.5\n-2.0\n-1.5\n-1.0\n-0.5\n0Log-penetration, log10(P)99.999\n99.99\n99.9\n99\n90\n0\nLaboratory identifier, first ILC1\n2\n34\n5\n6\n7\n8\n910\n11\n12\n13\n14\n1516\n17\n1\n2\n34\n5\n6\n7\n8\n910\n11\n12\n13\n14\n1516\n17\n1\n2\n34\n5\n6\n7\n8\n910\n11\n12\n13\n14\n1516\n17\n1\n2\n34\n5\n6\n7\n8\n910\n11\n12\n13\n14\n1516\n17\n1\n2\n34\n5\n6\n7\n8\n910\n11\n12\n13\n14\n1516\n17Particle filtration efficiency (PFE)/%TSI 8130A Photometer -based (except TSI 8130A) Count -based\nCombined (inter. and intra.)\nIntralaboratory Measurement mean, yijInstrument\n(a) (b) (c) (d) (e)μi\nμi+ 2σμ,i\nμi+ 2(σμ,i+ σl,i)1/2 2 2Page 7 of 14  Interlaboratory comparison for PFE  \n \n \npenetration are identified with crosses in Figure 1. Within the central cluster  of data , there is clear covariance between the \nmeasurements, with higher PFE for those results with a relatively high pressure drop. This is  largely  consistent with what we \nknow about mechanical filtration, wherein an increase in the  pressure drop is associated with  thicker or denser  materials , \ntranslating to  a higher filtration. Overlaid ellipses for these central data clusters  shown in Figure 2 correspond to k = 2 \ncovariance  following the outlier removal procedure .  \nLayering results in a trend  of increasing filtration  along a line of  similar  quality  factor . This is consistent with expectations , \nwhere mechanical filtration increases according to an exponential with  the thickness  of the same material  [5]. At the same time, \nthe pressure resistance of the materials increases roughly linearly.  There is some  reduction in the quality  factor  with an \nincreasing number of layers , decreasing from  roughly 5.5 to 4.5. This is more evident when plotting with respect to the quality \nfactor directly, Figure 2b. Plotting this way also shows that the spread in the quality factor reduce s with increasing pressure \ndrop and number of layers , consistent with observations for cotton fiber poplin weave in a lightweight flannel  by Zangmeister \net al. [19] albeit across different laboratories here . This is  in part  due to  averaging  of random variations in the underlying \nmaterial over multiple layer s. However, the standard error  does not scale with the square root of the number of layers, rather \nsaturating for more layers. This suggests a combination of the aforementioned effect  with increasing counting errors (see also \nFigure 3 and surrounding discussion). Overall, d espite the small decline in quality factor, this is a secondary effect, such that \nlayering can be reasonably approximated  using a constant quality factor, at least for this material.  \n \n \nFigure 2. Relationship between the (a) logarithm of the penetration  and (b) quality factor  (from log 10-penetration)  and the pressure \ndrop  when layering Green Line Paper . Points correspond to the mean for each laboratory and test. Grey, d ashed lines correspond to line \nof constant quality  factor, QF, using logarithm base -10, which are linear in this space. Ellipses correspond to the covariance in non -outlier \ndata ( k = 2). Solid symbols correspond to the central cluster, while open symbols correspond to the broader available data including \noutliers.  \n \nFigure 3 shows the size -resolved particle filtration for various numbers of layers of Green Line Paper measured using the \nPFEMS system [9, 10]. This system makes use of a pair of scanning mobility particle sizers (SMPSs), which classif y particles \naccording to their mobility diameter and then counts them to form distributions. Taking the ratio of particle count upstream \nand downstream of the sample allows for an assessment of the particle penetration  as a function of particle size. Increasing \nnoise in the size -resolved filtration as the PFE increases stems from a combination of low downstream counts and the log -\npenetration transformation, revealing visible single particle count or digital discretization noise in those signals. Noise at the \nedges of the curves representing the mobility diameter distributions correspond to cases where there was not a sufficient number \nof particles in the u pstream distribution to allow for a robust measurement of particle filtration, where error propagation predicts \na rapid expansion in measurement uncertainties [14].  \n-4.5\n-4.0\n-3.5\n-3.0\n-2.5\n-2.0\n-1.5\n-1.0\n-0.5log10(P)\n909999.999.99Particle filtration efficiency (PFE)/%\n3456 78\nLayersQF/kPa-1\n0 0.2 0.4 0.6 0.8 1.0 1.22345678Quality factor (log10), QF/kPa-1\nPressure drop, Δp/kPa0 0.2 0.4 0.6 0.8 1.0 1.2\nPressure drop, Δp/kPaLayersPage 8 of 14  Interlaboratory comparison for PFE  \n \n \nThe most penetrating particle size  (MPPS)  is relatively consistent across the different layers , again  consistent with previous \nobservations  [19, 24] and simple models  [5]. Further, the log -penetration at the MPPS appears to increase roughly linearly with \nthe number of layers, such that the curves  are evenly spread out vertically in the plot . This would correspond to a  constant \nquality factor regardless of the number of layers , again consistent with simple models [5]. Plotting the quality factor , Figure \n3b, shows that the size -resolved curves indeed collapse. In fact, a constant quality factor model seems to be more appropriate \nfor the size -resolved PFE  than for the PFE integrated over this size distribution  shown in Figure 2. This is not unexpected. The \nsize-resolved filtration is a more fundamental measurement in that it does not integrate over a range of filtration mechanism \nwith different size effects.  This observation  further affirms  that interfacial effects between the layers are secondary for these \nmaterials .  \n \n \nFigure 3. Size-resolved (a) particle filtration  and (b) quality factor for various layers of Green Line Paper measured using upstream and \ndownstream scanning mobility particle sizers. Upper axis gives an estimate of the corresponding aerodynamic diameter for the sodium \nchloride particles . In (a) the open circles correspond to the most penetrating particle size.  Dashed lines in (b) correspond to digitized \ndata from Li et al. [1], specifically  the 3M data  set at a face velocity of 10 cm s−1. (*) Note that quality factors are computed by assuming \n(1) the pressure drop at 10 cm s-1 is double that at 5 cm s−1 and (2) that the pressure scales linearly from the value for a single layer. As \nsuch and unlike the data from this work, lines only assess the effect of penetration on the quality factor.  \n \nQuality factors  measured here are also consistent with results from Li et al. [1], with the data for the fiberglass material as \nmeasured by 3M at a face velocity of 10 cm s−1 also shown in Figure 3b. Given that pressure resistance was only reported at 5 \ncm s−1 and only for a single layer, calculation of quality factors requires the assum ption  that the pressure resistance was a linear \nfunction of both the flow rate and number of layers. The resultant quality factors collapse onto a single curve, sharing a si milar \nMPPS and overall trend in the size -resolved quality factor  as the Green Line Paper  measured in this work . Note that assuming  \nthe pressure drop  is linear with the number of layers  would reduce variability in the quality factor  (given the functional form  \nof the quality factor) . Even so, the collapse of the data onto a single curve reinforces the sentiment that interfacial effects  \nbetween the layers  are insignificant for fiberglass . This results also appears to extend to the other fiberglass data sets reported \nby Li et al.  [1]. Applying the same treatment to t he data for flannel  from Zangmeister et al. [19] adds further  support  to this \nconclusion , this at substantially  lower quality factors  (a minimum of QF ~ 1.3  kPa−1, noting the difference in the base of the \nlog) and with some minor non -linearities in the pressure resistance  that prevented full collapse  (though these effects were small  \nat less than  10 %) . The electret materials considered by Li et al. [1], by contrast, did  demonstrate  decreasing quality factor  with \nan increased number of layers , suggesting that interfacial effects may be significant for electret  materials.  \n-4.0\n-3.5\n-3.0\n-2.5\n-2.0\n-1.5\n-1.0\n-0.5\n099.99\n99.9\n99.9\n90\n0log10(P)Particle filtration efficiency (PFE)/%\n100200 300 400 80 60 50 40 30\nMobility diameter/nmEstimated aerodynamic diameter for NaCl/nm\n100 200 80 60 50 40 30 20\n345678910(b) (a)\n100200 300 400 80 60 50 40 30\nMobility diameter/nmEstimated aerodynamic diameter for NaCl/nm\n100 200 80 60 50 40 30 20Quality factor (log10), QF/kPa−1\nLayersLi et al.*\nFiberglass, 3M, 10 cm s−1\nThis workPage 9 of 14  Interlaboratory comparison for PFE  \n \n \n3.2 Candidate reference material s \nPFE results for the candidate reference materials are shown in Figure 4, where d ark, interlaboratory uncertainties were  found \nto be significant across all  of the  material s tested and are domina nt in all cases . Much of the expansion of dark uncertainties is \nassociated with several measurements consistently above the central data cluster (these potential outliers are detected using \nGESD and are identified with crosses in Figure 4), which would add some skew in the underlying distribution of laboratory \neffects  and is not fully captured in the chosen analysis method.  These potential outliers expand the dark uncertainties by factor s \nof 1.4 – 2.6. Test 6 ( Figure 4f) is an exception, where there was sufficient spread in the data that no outliers were identified . \nThe central data cluster  after outlier removal  was reasonably described by a normal distribution.  \nAgain, substantial scatter was observed with respect to instrument type. However, in t his case, the count -based instruments \nare also interspersed with the other results. This is roughly consistent with previous observations across a broader range of \nmaterials spanning to lower PFEs [11].  \n \nFigure 4. PFE results for the candidate reference materials  with increasing consensus PFE , noting that the y-axes have different ranges \nacross the panels . Laboratories are reordered in terms of the median across all of the panels. As with Figure 1, thick, coloured bars \ncorrespond to intralaboratory uncertainties for each laboratory, while thin, black whiskers include  added interlaboratory uncertainties. \nHorizontal rules, from the center out, correspond to the consensus value  (solid lines) , intervals for uncertainties in the consensus value  \n(dashed lines) , and combined consensus and interlaboratory uncertaintie s (dotted lines) . All uncertainties correspond to k = 2. \n \nFigure 5 shows the relationship between pressure and log -penetration for these materia ls, following application of an \noutlier detection procedure , consistent with Figure 2. Results for each charge state generally follow lines of constant quality . \nUnlike Figure 2, the current results show this trend with increasing basis weight rather than layering of materials, showing that \nan analogous effect seems to apply between these two scenarios . This again suggest s that interfacial effects are secondary to \nbasis weight/layering  across a broader set of materials (though not necessarily all) .  \nCorrelation between the pressure drop and penetration measured for a single material by the laboratories is less pronounced \nthan in the Green Line Paper  (manifesting as round circles instead of titled ellipses in Figure 5). This could be due to the lower \npressure drop , resulting in larger coefficient in variation  in the pressure drop and more scatter in that dimension . Alternatively, \nthis may indicate that, unlike Green Line Paper, differences between samples of the material differ  more  in their makeup rather \n45\n40\n35\n30\n25\n20\n15\n10\n5\n065\n60\n55\n50\n45\n40\n35\n30\n2585\n80\n75\n70\n65\n60\n55\n50\n45\n4090\n80\n70\n5060\n4092949596\n9999.9\n909092\n809496\n9597\n8\n394\n5126\n17157\n1321\n101411\n168\n394\n5126\n17157\n1321\n101411\n168\n394\n5126\n17157\n1321\n101411\n168\n394\n5126\n17157\n1321\n101411\n168\n394\n5126\n17157\n1321\n101411\n168\n394\n5126\n17157\n1321\n101411\n16\nLaboratory identifier, first ILC1(η= 22%) 2(η= 45%) 3(η= 68%) 4(η= 85%) 5(η= 91%) 6(η= 98.2%)(a) (b) (c) (d) (e) (f)TSI 8130A Photometer -based (except TSI 8130A) Count -based\nCombined (inter. and intra.)\nIntralaboratory Measurement mean, yijInstrumentParticle filtration efficiency (PFE)/%Page 10 of 14  Interlaboratory comparison for PFE  \n \n \nthan simply having variability in t he basis weight  and/or thickness.  It is worth noting that the uncharged case in Figure 5 also \nexhibits substantial scatter in pressure drop, indicating that  the scattering in the pressure drop is independent of any charging \neffects , as expected . At the same time, i t is also worth noting that scatter in the log -penetration dimension increases at higher \ncharge states, which is likely an indication of the effect of variability in the applied charge expanding the uncertainties.  The \nprecise reason is difficult to discern and will be convolved with differences in the instruments and their calibration.  \n \n \n \nFigure 5. Relationship between the (a) logarithm of the penetration  and (b) quality factor and the pressure drop  across the candidate \nreference material . Results span different basis weights and charge states , though materials sharing a charge state had very similar \nquality across different basis weight categories and realizations. Select cases (2, 3, and 6) have original data overlaid on the plot , with \nopen symbols corresponding to outlier data . Other cases have ellipses representing k = 2 intervals on the covaria nce between the log -\npenetration and pressure drop.  \n \n3.3 Respirator tests  \nBoth the final component of the first ILC and the entirety of the second ILC  made use of respirators that were supplied to \nparticipants. Figure 6 shows the PFE results across the different measurements, which considered (a -b) the initial and minimum \n(while loading the material according to  TEB -APR -STP-0059  [6]) and (c -e) the effect of conditioning and mounting direction \n(i.e., inhalation versus exhalation).  \nFor the first ILC, repeatability within a laboratory was a good surrogate for the reproducibility across the laboratories in \nmost cases. In other words, for the reporting laboratories, measurements seem to be reproducible both in terms of the initial \nand minimum PFE. Further, those laboratories with the least amount of interlaboratory variability also tended to have \nmeasurements close to the consensus value.  \nFor the second ILC, an identical respirator was used across all three tests  and an initial PFE was reported. Little structure \nwas observed between measurements by the laborator ies, with a random ordering in terms of PFE across the different panels \n(i.e., on average, no lab consistently measured below the others). Dark uncertainties were more significant for these respira tors, \ndriven in large part by laboratories that had repeatable measurements  but are located at the periphery of the data (e.g., laboratory \n1 in Figure 1c).  \nThe use of an identical respirator necessitates that  the only difference between the cases should stem from the change in \nconditioning and/or mounting direction. No statistically significant differences were observed  in the consensus value  between \n2.711.517.8\n20 40 60 80 100 120-2.5\n-2.0\n-1.5\n-1.0\n-0.5\n050 QF/kPa-1\n20\n10\n5\n2\n126\n9\n105\n8\n7 4\n3\n1\nPressure drop, Δp/Pa2.711.517.8\n90\n099log10(P)Particle filtration efficiency (PFE)/%\nQuality factor (log10), QF/kPa−1(b) (a)\n20 40 60 80 100 120051015202530\nPressure drop, Δp/Pa6\n910 58\n7 4\n23\n1Low Medium -high Medium -low HighBasis weight\nNone Half FullCharge state\nConditioned UnconditionedConditioningPage 11 of 14  Interlaboratory comparison for PFE  \n \n \nthese different scenarios. Results were more varied for the exhalation case. This could indicate challenges associated with \nconsistent mounting the respirators  to face the other direction , which is a non -conventional  setup . Yet, despite this reduction in \nreproducibility, the consensus PFE was not affected by mounting direction.  \n \n \nFigure 6. PFE results for testing respirators across  (a-b) the final component of the first ILC and  (c-e) the second ILC . Vertical axis is \nidentical across all of the  panels. For each ILC, the laboratories are sorted according  to the first panel , that is (a) for the first ILC and (c) \nfor the second ILC . The laboratory identifiers are not the same across the two ILCs  (i.e., the laboratories in (a) and (b) are not the same \nas (c – e)). As with Figure 1, thick, coloured bars correspond to intralaboratory uncertainties for each laboratory, while thin, black whiskers \ncorrespond to added interlaboratory uncertainties. Horizontal rules, from the center out, correspond to the consensus value  (solid lines) , \nintervals for uncertainties in the consensus value  (dashed lines) , and combined consensus and interlaboratory uncertaintie s (dotted \nlines) . All uncertainties correspond to k = 2.   \n \n3.4 Summary of  uncertainty  and its decomposition  \nFigure 7 summarizes the combined uncertainties across all of the samples and as a function of PFE. Uncertainties trended \nsimilarly across all of the samples, increasing in a relative sense  (for log -penetration)  with increasing PFE but decreasing \nsignificantly in an absolute sense . Thus, while relative uncertainties ranged from 11 – 30 % of the nominal value of log -\npenetration, corresponding  95 % credible intervals  on the PFE  vary from  [10, 34 ] for the lowest PFE to [99.886, 99.994] for \nthe highest PFE . The respirators from the second ILC were a bit of an exception to the overall trends, with larger uncertainties  \nin particular for the exhalation case . This larger span was driven almost entire ly by a few potential outliers, as evident in Figure \n6d. Quality checks should be  recommended for these laboratories.   \nFigure 8 summarizes  the relative intralaboratory (repeatability), interlaboratory (dark, reproducibility), and consensus \nuncertainty contributions  to the overall variance. Uncertainties in the consensus value were minimal for all of the measurements. \nIn all but two respirator subcomponents,  the majority of the combined uncertainties corresponded to dar k, interlaboratory \nvariability . Much of the observed dark uncertainties stem from what might  be considered outlier laboratories. The removal of \n14\n3\n9\n4\n11\n13\n1\n10\n7\n5\n6\n15-5.0\n-4.5\n-4.0\n-3.5\n-3.0\n-2.5\n-2.0\n-1.5\n-1.0\n-0.5\n0\n14\n3\n9\n4\n11\n13\n1\n10\n7\n5\n6\n15\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15Log-penetration, log10(P)\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n1599.999\n99.99\n99.9\n99\n90\n0Particle filtration efficiency (PFE)/%\nLaboratory identifier, first ILC Laboratory identifier, second ILCInitial PFEMinimum PFE\n(w/ loading) Inhalation -conditioned Exhalation -conditioned Inhalation -unconditioned(a)(b)\n(c) (d) (e)TSI 8130A Photometer -based (except TSI 8130A) Count -based\nCombined (inter. and intra.)\nIntralaboratory Measurement mean, yijInstrumentPage 12 of 14  Interlaboratory comparison for PFE  \n \n \noutliers  by a GESD procedure  gives an indication of the degree to which these stray measurements  drive up dark uncertainties \nin the present analysis , with the leftover dark uncertainties shown as slightly darker , teal dashed border  bars in Figure 8. The \nrespirators are particularly prone to this effect . This is either  an indication  of the way overall uncertainties scale or  that better \ncontrols are required on laboratory measurements to either (1) update the distribution used to describe the measurements or ( 2) \nimprovements to quality control to ensure robust measurement across laboratories.  \n \n \nFigure 7. Consensus value with corresponding uncertainties ( k = 2) across all of the samples. Symbols and colors denote the different \nILC subcomponents.  \n \n  \nFigure 8. Absolute uncertainty and its decomposition for the PFE across all of the measurements. Lowest portions of the bars correspond  \nto dark, interlaboratory contributions, followed by intralaboratory variability, and finally uncertainties in the consensus v alue at the top. \nIntralaboratory contributions are averaged over the various laboratories. Note that candidate reference materials 8 and 10 we re \nconditioned version of materials 7 and 9, respectively.  \n-5.0\n-4.5\n-4.0\n-3.5\n-3.0\n-2.5\n-2.0\n-1.5\n-1.0\n-0.5\n099.999\n99.99\n99.9\n99\n90\n0959798\n50708099.599.799.899.9599.9799.9899.99599.99799.998Particle filtration efficiency (PFE)/%\nSampleLog-penetration, log10(P)Green Line Paper\nRespirator (second ILC)Candidate reference material\nRespirator (first ILC)\n3456779190939495979899.299.399.99\n99.9599.94\n99.8699.9399.98\n99.9799.994\n3255727974 7785898594.7 94.9\n84.3093.45 93.9198.6599.2699.5099.6899.886\n10\n00.20.40.6Absolute uncertainty in log -penetration/ -Dark, interlaboratory (reproducibility)\nDark (after outlier removal)Legend\nIntralaboratory (repeatability)\nConsensus uncertainty1 layer\n2 layers\n3 layers\n4 layers\n5 layers\n1 (η= 21%)\n2 (η= 43%)\n3 (η= 66%)\n4 (η= 83%)\n5 (η= 91%)\n6 (η= 98.2%)\n7 (η= 85%)\n8 (η= 85%)\n9 (η= 94%)\n10 (η= 94%)\nInitial\nMinimum\nInhalation -\nconditioned\nExhalation -\nconditioned\nInhalation -\nunconditioned\nGreen Line Paper Candidate reference material RespiratorsC C11.11.21.31.41.51.61.71.8Geometric standard deviation, GSDDarkPage 13 of 14  Interlaboratory comparison for PFE  \n \n \n4. Conclusions  \nThis work has quantified the interlaboratory variability of particle filtration measurements for respiratory protection, span ning \na range of materials for the conditions typical of testing respirators. Dark, interlaboratory variability was the dominant source \nof uncertainty in a vast majority  of measurements . Overall uncertainties in the measurements varied from 10 % – 30 % of the \nnominal value of the log -penetration . Further investigation is required to understand and minimize the effect of  both \nintralaboratory and interlaboratory  (dark)  differences in the measurements . Particular focus should be placed on the dark  \nuncertainties, including studies of the procedures used to verify instrument function  (e.g., used to constrain the challenge aerosol \nsize distribution  and number concentration as well as verifying the optical particle counters ) and the persistence of outlier s in \nthese measurements , which contribute significantly to the dark uncertainties .  \nThe uncharged Green Line Paper showed a linear relationship between the number of layers and log -penetration, consistent \nwith simple multiplicative penetration through each layer, without any indication that interfacial effects are significant. \nEvidence for this approach is relative ly clear in  the size -resolved particle filtration , with some minor secondary effects when \nthe particle filtration is integrated over a size distribution (for mass -based PFE) . Similar results were observed when a controlled \nlevel of charge was applied to the candidate reference material. As such, this simple model and the traditional quality measure \n[5] should be a reasonable surrogate in terms of predicting the effect of layering  and basis weight  when the structure of the \nmaterial does not otherwise change .  \nAcknowledgements  \nThis work was supported by the Pandemic Response P rogram  at the National Research Council Canada and the Public Health \nAgency of Canada. We would also like to acknowledge participation by  members of  the Canadian PPE Laboratory Network  \nand Australian laboratory network , including all the practitioners at the individual organizations that participated. The authors \nacknowledge the Canadian Association for Personal Protective Equipment Manufacturers  (CAPPEM)  for supplying some of \nthe materials.   \nReferences  \n[1] Li L, Zuo Z, Japuntich DA, Pui DYH. 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ACS nano. 2020;14:9188 -200. \n[20] Podgorski A, Bałazy A, Gradoń L. Application of nanofibers to improve the filtration efficiency of the most penetrating \naerosol particles in fibrous filters. Chemical Engineering Science. 2006;61:6804 -15. \n[21] Sankhyan S, Heinselman KN, Ciesielski PN, Barnes T, Himmel ME, Teed H, et al. Filtration Performance of Layering \nMasks and Face Coverings and the Reusability of Cotton Masks after Repeated Washing and Drying. Aerosol and Air \nQuality Research. 2021;21: 210117.  \n[22] Thompson M, Ellison SL. Dark uncertainty. Accreditation and Quality Assurance. 2011;16:483 -7. \n[23] Melanson JE, Thibeault M -P, Stocks BB, Leek DM, McRae G, Meija J. Purity assignment for peptide certified reference \nmaterials by combining qNMR and LC -MS/MS amino acid analysis results: application to angiotensin II. Anal Bioanal \nChem. 2018;410:6719 -31. \n[24] Leung WW -F, Hung C -H, Yuen P -T. Effect of face velocity, nanofiber packing density and thickness on filtration \nperformance of filters with nanofibers coated on a substrate. Separation and purification technology. 2010;71:30 -7. "    },    {        "title": "2402.04051v2.Analysis_of_Linear_Mode_Connectivity_via_Permutation_Based_Weight_Matching.pdf",        "content": "Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nAkira Ito1Masanori Yamada1Atsutoshi Kumagai1\nAbstract\nRecently, Ainsworth et al. (2023) showed that\nusing weight matching (WM) to minimize the\nL2distance in a permutation search of model pa-\nrameters effectively identifies permutations that\nsatisfy linear mode connectivity (LMC), in which\nthe loss along a linear path between two indepen-\ndently trained models with different seeds remains\nnearly constant. This paper provides a theoretical\nanalysis of LMC using WM, which is crucial for\nunderstanding stochastic gradient descent’s effec-\ntiveness and its application in areas like model\nmerging. We first experimentally and theoreti-\ncally show that permutations found by WM do\nnot significantly reduce the L2distance between\ntwo models and the occurrence of LMC is not\nmerely due to distance reduction by WM in it-\nself. We then provide theoretical insights show-\ning that permutations can change the directions\nof the singular vectors, but not the singular val-\nues, of the weight matrices in each layer. This\nfinding shows that permutations found by WM\nmainly align the directions of singular vectors as-\nsociated with large singular values across models.\nThis alignment brings the singular vectors with\nlarge singular values, which determine the model\nfunctionality, closer between pre-merged and post-\nmerged models, so that the post-merged model re-\ntains functionality similar to the pre-merged mod-\nels, making it easy to satisfy LMC. Finally, we\nanalyze the difference between WM and straight-\nthrough estimator (STE), a dataset-dependent per-\nmutation search method, and show that WM out-\nperforms STE, especially when merging three or\nmore models.\n1Nippon Telegraph and Telephone Corporation, 3–9–11 Midori-\ncho, Musashino-shi, Tokyo, Japan. Correspondence to: Akira Ito\n<akira.itoh@ntt.com >.\nPreprint. Under review.1. Introduction\nDeep learning has significantly advanced various fields, in-\ncluding image classification, speech recognition, and nat-\nural language processing (Vaswani et al., 2017; van den\nOord et al., 2016; Zhao et al., 2023). Large-scale neural\nnetworks (NNs), such as Transformers, are widely used in\nthese applications, and optimizing their parameters poses\na massive non-convex optimization problem. Remarkably,\nstochastic gradient descent (SGD), which is widely used\nfor training NNs, is known to find good solutions with high\nreproducibility despite its simplicity. One hypothesis for\nthis seemingly counterintuitive phenomenon is that the land-\nscape of the loss function may be much simpler than we\npreviously thought. Recently, Garipov et al. (2018) have\nshown that SGD solutions are connected by simple curves\nwhere the loss values remain consistently low. Moreover,\nEntezari et al. (2022) conjectured that Conjecture 1.1 holds,\nconsidering all possible permutation symmetries of NNs:\nConjecture 1.1 (Permutation invariance, informal) .Letθa\nandθbbe any two SGD solutions (model parameters). Then,\nwith high probability, there exists a permutation πsuch that\nthe barrier (defined in Definition 2.1) between θaandπ(θb)\nis sufficiently small.\nHere, the barrier represents the amount of increase in loss\nobserved when linearly interpolating between the two mod-\nels. Thus, a sufficiently small barrier implies that the model\nobtained by linear interpolation of the two models performs\nthe same as the models before interpolation. If the barrier be-\ntween two models is sufficiently small, we say linear model\nconnectivity (LMC) is satisfied between them (Frankle et al.,\n2020). Conjecture 1.1 suggests that most SGD solutions\nare part of the same loss basin by transferring each SGD\nsolution to the same loss basin through permutations. In-\ndeed, Ainsworth et al. (2023) demonstrated experimentally\nthat this conjecture is valid for various datasets and models\nby using weight matching (WM), in which permutations\nare identified that minimize the L2distance between the\nweights of the models.\nThe current theoretical analysis of why LMC occurs depends\non the feasibility of closely matching NN weights using\npermutations. Recent work by Zhou et al. (2023) has proven\nthat if the distance between the weights of two models can be\nsufficiently reduced through permutation, then LMC holds.\n1arXiv:2402.04051v2  [cs.LG]  19 Feb 2024Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nIntuitively, given two SGD solutions θaandθb, if one can\nfind a permutation πsuch that θa≈π(θb), it is reasonable\nto expect that the outputs of the model corresponding to the\ninterpolation between the two parameters θaandθbwill be\napproximately equal to the outputs of the models θaandθb.\nHowever, this paper reveals that this intuitive explanation is\nactually inaccurate and clarifies a more fundamental reason\nfor why LMC holds through finding permutations by WM.\nThe exploration of LMC principles is important not only to\nunderstand how SGD works in deep learning but also for\nits application in model merging (Ainsworth et al., 2023),\nwhere two independently trained models are combined. In\nparticular, the method of finding permutations using only\ntheL2distance is versatile, dataset-independent, and com-\nputationally efficient. Therefore, the analysis in this study\nholds considerable potential for various applications such as\nmodel merging, federated learning, and continual learning.\nThe contributions of this paper are threefold:\n1. Demonstrating that WM does not make the L2dis-\ntance small enough that the two models can be consid-\nered approximately equal. This paper experimentally\ndemonstrates that permutations found by WM do not signif-\nicantly reduce the L2distance between two models. Specif-\nically, even when distances are minimized using permuta-\ntions, the resulting distance between two models is not suf-\nficiently close to zero. Our results indicate that, even under\nLMC satisfaction, permutation reduces the model weight\ndistance by no more than 20%. Our findings, supported\nby analysis based on a Taylor approximation, indicate that\nreducing the L2distance by permutations is not in itself a\nreason to satisfy LMC.\n2. Revealing a reason for satisfying LMC by WM. We\nprovide both theoretical and experimental evidence show-\ning that WM satisfies LMC by aligning the directions of\nsingular vectors in the weights of each layer. First, we show\nthat the singular values of each layer in a trained model do\nnot vary significantly between models trained with differ-\nent initializations. This indicates that the differences in the\ntrained models arise not from the singular values, but from\nthe directions of the singular vectors of each layer. Fur-\nthermore, we show both theoretically and experimentally\nthat permutations found by WM align the directions of the\nsingular vectors between two models. In particular, when\nminimizing the L2distance, the directions of singular vec-\ntors with larger singular values are more likely to be aligned.\nBy this singular-vector alignment, the singular vectors with\nlarge singular values, which determine the functionality of\nthe model, become close between the pre-merged and post-\nmerged models, so that the post-merged model has similar\nfunctionality to the pre-merged models, making it easy to\nsatisfy LMC.3. Demonstrating the superiority of WM over the\nstraight-through estimator (STE) in merging three or\nmore models. To see the difference between WM and\nother permutation search methods independent of L2dis-\ntance, we examine straight-through estimator (STE), which\nfocuses on minimizing the barrier itself rather than the L2\ndistance. Our experiments reveal that permutations found\nby STE do not align the directions of singular vectors, high-\nlighting a fundamental difference from WM in achieving\nLMC. In addition, we demonstrate experimentally that this\ndifference has a significant impact on satisfying the LMC\nwhen merging three or more models.\n2. Background and Preliminaries\n2.1. Notation\nFor any natural number k∈N, let[k] ={1,2, . . . , k }. Bold\nuppercase variables represent tensors, including matrices\n(e.g.,X), and bold lowercase variables (e.g., x) represent\nvectors. For any tensor X, its vectorization is denoted by\nvec(X). Throughout this paper, |X|denotes the Frobenius\n(L2) norm of the tensor X.\n2.2. Permutation Invariance\nWe consider multilayer perceptrons (MLPs) f(x;θ)with\nLlayers for simplicity while our analyses in this paper can\nbe applied to any model architectures. Here, x∈Rdinis\nthe input to the NN, and θ∈Rdparam represents the model\nparameters, where din∈Nis the dimension of the input,\nanddparam∈Nis the dimension of the parameters. Let\nzℓbe the output of the ℓ-th layer (i.e., z0=x, and, for\nallℓ∈[L],zℓ=σ(Wℓzℓ−1+bℓ)). Here, σdenotes the\nactivation function, and Wℓandbℓrepresent the weight and\nbias of the ℓ-th layer, respectively. Note that in this MLP,\nwe have θ=\r\rL\nℓ=1(vec(Wℓ)∥bℓ), where ∥represents the\nconcatenation of vectors.\nNNs have permutation symmetries of weight space. Consid-\nering an NN with model parameters θ, for its ℓ-th interme-\ndiate layer,\nzℓ=P⊤Pzℓ=P⊤σ(PW ℓzℓ−1+Pbℓ)\nholds, where Pis a permutation matrix. Note that per-\nmutation matrices are orthogonal, so we have P⊤=P−1.\nTherefore, by permuting the input of the (ℓ+1)-st layer with\nP⊤, the model parameters can be changed without altering\nthe input-output relationship of the NN. Specifically, the\nnew weights and bias are given by W′\nℓ=PW ℓ,b′\nℓ=Pbℓ,\nW′\nℓ+1=Wℓ+1P⊤. Such permutations can be applied to\nall layers. In this paper, we denote the tuple of permutations\ncorresponding to each layer as π= (Pℓ)ℓ∈[L]. Moreover, if\na model θis given, the application of permutation πtoθis\ndenoted by π(θ).\n2Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\n2.3. Linear Mode Connectivity (LMC)\nLetθ∈Rdparam be a model and L(θ)denote the value\nof the loss function for model θ. Here, we define the loss\nbarrier between two given models θaandθbas follows:\nDefinition 2.1. For two given models θaandθb, their loss\nbarrier is defined as\nB(θa,θb) := max\nλ∈[0,1]\u0000\nL(λθa+ (1−λ)θb)\n−(λL(θa) + (1 −λ)L(θb))\u0001\n.\nIntuitively, the barrier represents the amount of increase in\nloss because of the linear interpolation of the two models.\nTwo models θaandθbare said to be linearly mode con-\nnected when the loss barrier between them is approximately\nzero.\n2.4. Permutation Selection\nEntezari et al. (2022) conjectured that for any SGD solutions\nθaandθb, there exists a permutation πsuch that LMC holds\nbetween θaandπ(θb)with high probability. Afterward,\nAinsworth et al. (2023) proposed WM and STE as methods\nfor finding such permutations. This subsection explains\nWM, the main focus of this paper. STE is discussed in\nSection 5.\nIn WM, we search for a permutation that minimizes the L2\ndistance between two models1:\narg min\nπ∥θa−π(θb)∥2\n= arg min\nπX\nℓ∈[L]∥W(a)\nℓ−PℓW(b)\nℓP⊤\nℓ−1∥2,(1)\nwhere, without loss of generality, let PL=IandP0=I,\nandIis an identity matrix. This minimization problem\nis known as the sum of the bilinear assignments problem,\nwhich is NP-hard (Koopmans & Beckmann, 1957; Sahni\n& Gonzalez, 1976; Ainsworth et al., 2023). Therefore,\nAinsworth et al. (2023) proposed a method to obtain an\napproximate solution by dividing the optimization for each\nlayer separately and solving them individually. The divided\noptimization problem can be reduced to a simpler problem\ncalled the linear assignment problem, which can be easily\nsolved using existing libraries like Scipy (Virtanen et al.,\n2020). Moreover, Pe ˜na et al. (2023) proposed solving Equa-\ntion (1) using Sinkhorn’s algorithm (Adams & Zemel, 2011)\nby considering it as an optimal transport problem. This\nmethod allows optimization of all layers at once, poten-\ntially finding better solutions. Therefore, this paper adopts\nSinkhorn’s algorithm-based method proposed in (Pe ˜na et al.,\n2023).\n1Although only the weights are considered here, the biases can\nalso be dealt with by concatenating the biases and the weights.3. Motivating Observations\nPrevious studies have suggested that the closeness of two\nparameters in terms of L2distance is critical for satisfying\nLMC. For example, Zhou et al. (2023) showed that LMC\nholds if a commutativity property is satisfied. This property\nholds if, for all layers ℓ,W(a)\nℓ−PℓW(b)\nℓP⊤\nℓ−1=0. Zhou\net al. (2023) argued in Section 5.2 that since WM finds the\npermutation that minimizes Equation (1), WM can be seen\nas searching for permutations that satisfy the commutativity\nproperty. In particular, there is a huge number of permu-\ntations because the total number of possible permutations\ngrows exponentially as the number of layers and the width\nincrease, and thus, some of them may sufficiently reduce\nthe distance between the two models. However, this section\nexplains from the perspective of a Taylor approximation\nthat this intuition is not always correct. That is, our results\ndemonstrate that even in cases where LMC is satisfied, the\npermutations found by WM do not necessarily bring the\nmodels as close as expected. The facts observed from the\nexperiments in this section motivate us to explore other\nreasons for satisfying LMC in the following sections.\n3.1. Closeness of Two Models in Terms of Taylor\nApproximation\nThis subsection demonstrates in terms of a second-order\nTaylor approximation that even with permutations found\nby WM, the distance between the two models is not close\nenough to be considered equal. Let θaandθbbe two SGD\nsolutions, and πbe a permutation found by WM to make\nπ(θb)close to the model θa. If the permutation brings\nθaandπ(θb)sufficiently close, we can expect the Taylor\napproximation to hold on the loss function. Conversely, if\nthe Taylor approximation does not hold, it means that θa\nandπ(θb)are far enough apart that the approximation fails.\nLetθc=λθa+ (1−λ)π(θb)be the merged model at the\nratioλ∈[0,1]. Ifθaandπ(θb)are sufficiently close, then\ntheir linear interpolation θcshould be close to both models.\nTherefore, the loss of the parameter θcshould be able to\nbe approximated by the Taylor approximation. In fact, the\nfollowing theorem holds if θaandπ(θb)are sufficiently\nclose:\nTheorem 3.1. The loss function L:U∋θ7→ L(θ)∈R\nis assumed to be a twice differentiable function on an open\nsetUoverRdparam. LetHaandHbbe the Hessian matrices\ncentered at the models θaandπ(θb), respectively. If, for\nanyλ∈(0,1),λθa+ (1−λ)θb∈Uholds, then we have\nB(θa, π(θb)) = max\nλλ(1−λ)\u0002\nβµ⊤∇(L(θa)−L(π(θb)))\n+1\n2β2µ⊤((1−λ)Ha+λHb)µ\u0003\n+O(β3),(2)\nwhere∇is the gradient with respect to the parameters, and\n3Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nTable 1. Results of WM and the estimated value of barrier using Taylor approximation when λ= 1/2\nDataset Network Barrier ( λ= 1/2) Taylor approx. L2dist. w/o WM L2dist. w/ WM\nCIFAR10VGG11 0.035±0.1 2 .956±0.35 799 .503±16.396 746 .465±19.576\nResNet20 0.167±0.035 7 .517±0.573 710 .762±16.261 661 .055±12.539\nFMNIST MLP −0.183±0.049 0 .928±0.175 121 .853±5.83 100 .041±4.71\nMNIST MLP −0.033±0.006 0 .036±0.03 81 .231±5.58 64 .751±4.795\nOis the Landau symbol, and µis the unit vector from θa\ntoπ(θb)(i.e.,µ= (π(θb)−θa)/∥π(θb)−θa∥), and β\nrepresents the L2distance between θaandπ(θb).\nWe prove this theorem in Appendix B.1. The theorem states\nthat if θaandπ(θb)are sufficiently close, the value of\nthe barrier can be predicted from the gradients and Hessian\nmatrices around each model. In particular, when the distance\nβis small, the barrier tends to be close to zero regardless of\nthe value of λ.\n3.2. Experimental Results\nWe conducted experiments to empirically verify whether\nthe Taylor approximation given by Equation (2) is a good\napproximation to estimate the barrier of two SGD solutions.\nTable 1 shows the mean and standard deviation results from\nfive trials of model merging (i.e., linear concatenation of the\nmodels (θa+π(θb))/2). Details about the datasets, network\ntraining procedures, and permutation search methods used\nin these experiments are described in Appendix C. The\ncolumn labeled “Barrier” shows the value of the barrier at\nλ= 1/2. We choose λ= 1/2because it is empirically\nknown that the midpoint between two models results in the\nhighest loss (Ainsworth et al., 2023; Pe ˜na et al., 2023). Also,\nthe approximation of Equation (2) at this point is shown in\nthe column labeled “Taylor approx.” A negative value in the\nBarrier column indicates that the loss of the merged model\nis lower than the models before merging. As can be seen\nfrom the results, except for the MNIST dataset, there is a\nsignificant difference between the actual values of the barrier\nand those estimated by the Taylor approximation. These\ndifferences are particularly large for VGG11 and ResNet20.\nThe table also shows the L2distance between the models θa\nandθbbefore and after applying the permutation (i.e., “w/o\nWM” and “w/ WM”). The results show that the L2distance\nchanges by only about 6% to 20% from the original distance\nbetween θaandθb. This suggests that WM does not bring\nthe models sufficiently close, at least not close enough for a\nsecond-order Taylor approximation to hold.\n4. Analysis of LMC based on WM\nThis section explains why the permutation found by WM\nmakes it possible to satisfy the LMC between the two mod-\nels. First, Section 4.1 shows that by performing singularvalue decomposition (SVD) on the weight of each layer\nof the NN, the L2distance between the two models can\nbe expressed as the difference between the singular values\nand the singular vectors of the two models. Next, we show\nexperimentally in Section 4.2 that the distributions of the\nsingular values are equal between independently trained\nmodels, indicating that the difference in weights between\nthe models arises only from the difference in the singular\nvectors. We then show theoretically in Section 4.3 that WM\nsearches for permutations that match the directions of the\nsingular vectors between the two models. In addition, we\nshow experimentally in Section 4.4 that the permutations\nobtained by WM cannot perfectly align the directions of all\nsingular vectors of the models, and only singular vectors\nwith large singular values can be aligned. Since the input-\noutput relationship of a linear map induced from a matrix\nis approximately determined by its singular vectors with\nlarge singular values, the permutations obtained by WM are\nexpected to play an important role in satisfying the LMC.\n4.1. Analysis of WM Using SVD\nThe basic idea in analyzing WM is to perform an SVD of\nthe weight in each layer. In WM, permutation matrices\nare searched for to minimize Equation (1). We denote the\nSVD of W(a)\nℓandW(b)\nℓbyW(a)\nℓ=U(a)\nℓS(a)\nℓ(V(a)\nℓ)⊤=P\niu(a)\nℓ,is(a)\nℓ,i(v(a)\nℓ,i)⊤andW(b)\nℓ=U(b)\nℓS(b)\nℓ(V(b)\nℓ)⊤=\nP\nju(b)\nℓ,js(b)\nℓ,j(v(b)\nℓ,j)⊤, respectively. Here, we assume that\nthe singular values are ordered in descending order (i.e., for\nallℓ∈[L],s(a)\nℓ,1≥s(a)\nℓ,2≥ ··· ≥ s(a)\nℓ,nands(b)\nℓ,1≥s(b)\nℓ,2≥\n··· ≥ s(b)\nℓ,n), where nis the number of singular values. Then,\nwe have\narg min\nπ∥θa−π(θb)∥2\n= arg min\nπX\nℓ∈[L]\r\r\rX\niu(a)\nℓ,is(a)\nℓ,i(v(a)\nℓ,i)⊤\n−X\njPℓu(b)\nℓ,js(b)\nℓ,j(Pℓ−1v(b)\nℓ,j)⊤\r\r\r2\n.(3)\nEquation (3) shows that the permutation matrices Pℓand\nPℓ−1are multiplied by the left and right singular vectors\nof the model θb, respectively. Since permutation matrices\nare orthogonal, the permutation matrices can change their\n4Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\n0200400\nIndex0510Singular valueMLP , MNIST\n0200400\nIndex0510Singular valueMLP , FMNIST\n0200004000060000\nIndex02040Singular valueVGG11, CIFAR10\n0100000200000\nIndex020Singular valueResNet20, CIFAR10\nFigure 1. Distribution of the singular values of the second layer.\nThe distribution of the singular values of all layers is shown in\nAppendix D.3.\ndirections, but not the norms, and therefore do not affect the\nmagnitudes of the singular values. If the singular values of\nboth models are almost equal (i.e., for all ℓandi,s(a)\nℓ,i=\ns(b)\nℓ,i) and we can find permutation matrices such that the\nleft and right singular vectors of the two models match,\nthen the two models θaandθbwill be exactly the same.\nIn other words, if one can match the singular values and\nsingular vectors between the two models, then the LMC\nclearly holds. Therefore, in the following, we will discuss\nthe differences in (1) singular values and (2) singular vectors\nof independently trained models.\n4.2.Difference Between Singular Values of Two Models\nFirst, we investigate the differences between the singular\nvalues of two independently trained models. To this end, ten\nmodels are trained independently under identical conditions\nexcept for the seed, and their singular values are compared.\nFigure 1 plots the singular values in the second intermediate\nlayer of independently trained models in descending order.\nThe evaluation results for all the layers are shown in Figure 6.\nThe singular values of ten independently trained models (i.e.,\ntrained with different seeds) are plotted in different colors.\nAs can be seen in the figures, in the intermediate layers, the\nsingular values are very close across all models.2Therefore,\nthe differences in singular values between the models are\nnot a major issue in achieving LMC.\n4.3. Role of Permutations in WM\nWe found that the distribution of singular values for each\nlayer is approximately equal among independently trained\n2As shown in Figure 6, there is some variation in the distribu-\ntion of singular values in the final layer, but this is not a problem\nin terms of the test accuracy as described in Appendix D.2.\n=0.3\nMLP , MNIST=0\n=0.3\nMLP , FMNIST=0\n=0.3\nVGG11, CIFAR10=0\n=0.3\nResNet20, CIFAR10=0\n0.00.20.4R(a,(b))\nw/ WM\nw/o WMFigure 2. Evaluation results of singular vector alignment of θaand\nπ(θb).\nmodels. Therefore, the differences between the models\nderive only from differences in the directions of the singular\nvectors.\nThen, we show that WM finds the permutation matrices that\nalign the singular vectors between two models. Specifically,\nthe following theorem holds:\nTheorem 4.1. Given the trained L-layer MLPs θaandθb,\nEquation (3)is equivalent to\narg max\nπ=(Pℓ)ℓX\nℓ,i,js(a)\nℓ,is(b)\nℓ,j(u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j).\n(4)\nThe proof of this theorem is shown in Ap-\npendix B.2. Focusing on the term for each layerP\ni,js(a)\nℓ,is(b)\nℓ,j(u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j)in Equa-\ntion (4), (u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)is the inner product between\nthe left singular vector u(a)\nℓ,iof the model θaand the\nleft singular vector u(b)\nℓ,jof the model θb, applied with\nthe permutation matrix Pℓ. The permutation matrix is\northogonal, so it only permutes the elements without\nchanging the norms of the left singular vectors. Therefore,\nthis inner product is maximized when the directions of the\ntwo left singular vectors are aligned by the permutation\nmatrix. Similarly, (v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j)is the inner product\nof the right singular vectors, which is also maximized when\nthe directions of the right singular vectors between two\nmodels are aligned by the permutation matrix Pℓ−1. Since\nthese permutation matrices PℓandPℓ−1are orthogonal,\nthey do not change the norms of the vectors and thus do not\naffect the singular values s(a)\nℓ,iands(b)\nℓ,j. Thus, Equation (4)\nis equivalent to finding permutation matrices that align the\ndirections of the left and right singular vectors for all layers\nbetween two models, especially those corresponding to\nlarge singular values.\nWe have analyzed the MLPs, but as shown in Appendix A,\nalmost the same holds for convolutional layers.\n5Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nw/ WM\nMLP , MNISTw/o WM w/ WM\nMLP , FMNISTw/o WM w/ WM\nVGG11, CIFAR10w/o WM w/ WM\nResNet20, CIFAR10w/o WM0.40.60.8\nR(a,(a+(b))/2)\nR((b),(a+(b))/2)\n(a) Evaluation results with the threshold γ= 0.\nw/ WM\nMLP , MNISTw/o WM w/ WM\nMLP , FMNISTw/o WM w/ WM\nVGG11, CIFAR10w/o WM w/ WM\nResNet20, CIFAR10w/o WM0.40.60.8\nR(a,(a+(b))/2)\nR((b),(a+(b))/2)\n (b) Evaluation results with the threshold γ= 0.3.\nFigure 3. Evaluation results of WM between θaandθb. The blue bars represent the evaluation results of Rbetween the model θaand the\nmerged model (θa+π(θb))/2, while the blue bars represent the evaluation results of Rbetween the model π(θb)and the merged model\n(θa+π(θb))/2.\n4.4. Experimental Validation of Singular-Vector\nAlignment\nAs already mentioned, the singular values are almost the\nsame among independently trained models. Furthermore,\nthe WM searches for permutations that minimize the dif-\nference between the singular vectors. If the WM can make\nthe difference in singular vectors small enough, then the\nL2distance between the two models also becomes almost\nzero, and the LMC is satisfied. However, as shown in Sec-\ntion 3, WM does not make the L2distance very small. In\nother words, WM cannot perfectly align the directions of all\nsingular vectors between the two models, and the number\nof singular vectors that can be changed in the direction by\nthe WM is thought to be limited. Meanwhile, Theorem 4.1\nshows that WM preferentially aligns the directions of singu-\nlar vectors with large singular values. As shown in Figure 1,\nthe singular vectors with large singular values are a small\nnumber of the total, so we expect that WM aligns the di-\nrections of only such dominant singular vectors. In this\nsubsection, we experimentally confirm that this expectation\nis correct.\nTo evaluate how well the singular vectors are aligned, we\ncalculate:\nR(θa, π(θb)) =P\nℓ,i,j(u(a)\nℓ,i)⊤Pℓu(b)\nℓ,j(v(a)\nℓ,i)⊤Pℓ−1v(b)\nℓ,jP\nℓnℓ,\n(5)\nwhere nℓis the number of singular values in the ℓ-th layer.\nNote that |R(θa, π(θb))| ≤1holds and, we have equality\nif for all ℓandi,u(a)\nℓ,i=Pℓu(b)\nℓ,iandv(a)\nℓ,i=Pℓ−1v(b)\nℓ,ihold\n(proof in Appendix B.5). Therefore, if R(θa, π(θb))is close\nto one, the directions of the singular vectors of the models\nare aligned.\nFigure 2 shows the results of evaluating Equation (5) be-\ntween θaandπ(θb). The vertical axis in the figure repre-\nsents the value of R. The red bars are the results when a\npermutation found by WM is applied to θb, and the blue\nbars are the results when no permutations are applied. The\nfigure shows the average and standard deviation of the re-\nsults obtained from five permutation searches using WM.In addition, a threshold γis introduced to examine whether\nthe singular vectors with the large singular values are prefer-\nentially aligned. For each model, we evaluate Equation (5)\nusing only those whose ratio to the largest singular value\nis greater than γ. Thus, γ= 0in the figure corresponds to\nthe results when all singular vectors are used, and γ= 0.3\ncorresponds to the results when only singular values whose\nratio to the largest singular value exceeds 0.3 are used.\nThe figure shows that the directions of the singular vectors\nare clearly more aligned with WM than without it. In par-\nticular, without WM, the value of Ris almost zero, thus\nindicating that the singular vectors are nearly orthogonal.\nThis would be related to the fact that the intermediate layer\nof the neural network is high-dimensional and random vec-\ntors on the high-dimensional space are orthogonal with high\nprobability (Vershynin, 2018).\nIn addition, focusing on the difference in γ, when the sin-\ngular vectors are aligned by using WM, the value of Ris\nclearly larger when γis 0.3. This means that singular vec-\ntors with larger singular values are aligned better. Although\nthe value of Ris not so large, especially around 0.2 at most\nfor VGG11 and ResNet20, this singular vector alignment\naffects merged models.\nFigure 3 shows the evaluation results of Equation (5) be-\ntween the merged (i.e., (θa+π(θb))/2) and the pre-merged\nmodels (i.e., θaandπ(θb)). If LMC holds, then the func-\ntionality of the post- and pre-merged models should be the\nsame, which indicates that the directions of the singular\nvectors between these models should be close. To investi-\ngate how well the directions of singular vectors with large\nvalues are aligned between the post-merged and pre-merged\nmodels, we also show the results for γ= 0.3in Figure 3(b).\nThe figures show that when γ= 0, the value of Rdoes\nnot change much depending on whether WM is used or not,\nwhereas when γ= 0.3, the value of Rchanges significantly\ndepending on whether WM is used or not. This result indi-\ncates that the directions of singular vectors with particularly\nlarge singular values are aligned before and after the merge.\nThe MLP results especially show that the value of Rex-\nceeds 0.7 by using WM. Since the input-output relationship\n6Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nMLP , MNIST MLP , FMNIST VGG11, CIFAR10 ResNet20, CIFAR100.00.20.4R(a,b(b))\nR(a,c(c))\nR(b(b),c(c))\nFigure 4. Evaluation results of Rbetween each pair of the models\nwithγ= 0.3.\n0 20 40 600204060MLP MNIST\n97.93\n0 50 100020406080100MLP FMNIST64.8366.9068.97\n71.0373.1075.17\n77.2479.3181.38\n83.45\n85.5287.5987.59\n89.66\n89.6689.66\n0 200 400 6000200400600ResNet20 CIFAR10\n83.45\n85.52\n87.5989.66\n91.72\n91.7291.72\n0 250 500 7500200400600VGG11 CIFAR10\n75.17\n77.2479.31\n81.3883.45\n83.4583.4585.52\n85.5285.52\n87.59\n87.5987.59\n89.66\n89.6689.66\n40.0046.2152.4158.6264.8371.0377.2483.4589.6695.86\n(a) Results of WM.\n0 25 50 750204060MLP MNIST\n91.7293.7995.86 97.93\n0 50 100020406080100MLP FMNIST\n60.6962.7664.8366.9068.9771.03\n73.1075.17\n77.2479.3181.3881.38\n83.4583.45\n85.52 85.52\n87.59\n87.5987.59\n89.6689.66\n0 200 400 6000200400600ResNet20 CIFAR10\n66.9068.9771.03\n71.0373.1073.1073.10\n75.17\n77.2479.3179.31\n81.3881.38\n83.4583.4583.45\n85.52\n85.5285.5287.59\n87.5987.59 89.66\n89.6689.66\n91.72\n91.7291.72\n0 250 500 7500200400600VGG11 CIFAR10\n50.3452.4154.4856.5558.6260.6962.7662.76\n64.8366.9068.9771.0373.1075.1777.2479.3181.38\n81.3883.45\n83.4583.4585.52\n85.5285.52\n87.59\n87.5987.59\n89.66\n89.6689.66\n40.0046.2152.4158.6264.8371.0377.2483.4589.6695.86\n(b) Results of STE.\nFigure 5. Accuracy landscape around θa,πb(θb)andπc(θc). The\nstar in the lower left represents θa, and the squares in the lower\nright and upper represent πb(θb), andπc(θc), respectively.\nof the linear map induced from the weight matrices of each\nlayer is determined by the dominant singular vector with\nlarge singular values3, the function of the merged model\nshould approach that of the model to be merged (i.e., θaand\nθb) by using WM, makes it easy to satisfy LMC.\nThe effect of changing model width on the singular-vector\nalignment by WM is discussed in Appendix D.3.\n5. Comparison with STE\nThis section discusses the relationship between STE, a more\ndirect method of finding permutation, and WM in terms of\nsingular vectors. STE uses a dataset to find permutation ma-\ntrices with a small barrier value. This section also explains\nthat the STE and WM are based on essentially different\nprinciples, and shows that the difference has a significant\nimpact on LMC among the three or more models.\n3In fact, several papers (Choudhary et al., 2020; Sainath et al.,\n2013; Prabhavalkar et al., 2016) have been proposed to reduce\nthe number of parameters without compromising performance by\nusing low-rank approximation of NNs.5.1. Straight-through Estimator (STE)\nAinsworth et al. (2023) proposed the STE, a permutation\nsearch method that finds a permutation πsuch that\narg min\nπL\u00121\n2(θa+π(θb))\u0013\n. (6)\nSince Equation (6) is difficult to solve directly, Ainsworth\net al. (2023) proposed a method to decompose this opti-\nmization into a forward path to find permutations and a\nbackward path to train the model obtained by the permu-\ntations found in the forward path, and to approximate the\nsolution by iterating this process. In addition, Pe ˜na et al.\n(2023) proposed a method to solve Equation (6) directly\nusing Sinkhorn’s algorithm. In this section, we adopt the\nlatter method based on Sinkhorn’s algorithm, which we call\nSTE in this paper. In the following subsections, we discuss\nthe difference between the permutations found by STE and\nWM.\n5.2. Experimental Results of Model Merging by STE\nThe results of model merging using STE are shown in Ta-\nble 2. It also shows the L2distance and the Rvalue between\nthe two models before and after the permutation, as well\nas Equation (2). Table 2 shows that despite the relatively\nsmall barrier value, the L2distance between the two models\nbefore and after permutation hardly changes compared to\nthe results with WM shown in Table 1. Since R(θa, π(θb))\nis nearly zero, the directions of the singular vectors between\nthe two models are likely not aligned at all. Therefore, the\nreason for satisfying LMC by STE should be completely\ndifferent from that of WM.\n5.3. Merging Three Models\nAs shown in the previous subsection, the permutation matri-\nces found by STE do not align the directions of the singular\nvectors of the models. This indicates that STE may find\npermutation matrices that reduce the loss of the merged\nmodel in a data-dependent manner (i.e., dependent on the\nloss landscape) rather than on the linear algebraic properties\nof the weight matrices of each layer. The difference between\nthe principles of STE and WM could result in a qualitative\ndifference in a model merging among three or more models.\nSuppose we have three SGD solutions θa,θb, andθc. Let\nπbandπcbe permutations that satisfy LMC between θa\nandπb(θb), andθaandπc(θc), respectively. If permuta-\ntions found by STE depend on the locality of the loss land-\nscape rather than the linear algebraic properties of the model\nweights, there is no guarantee that πb(θb)andπc(θc)are\nlinearly mode-connected. In contrast, permutations found\nby WM align the directions of the singular vectors of the\ntwo models. In other words, the singular vectors of πb(θb)\n7Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nTable 2. Results of model merging with STE.\nDataset Network Barrier ( λ= 1/2)L2dist. w/o STE L2dist. w/ STE R(θa, π(θb))(γ= 0.3)\nCIFAR10VGG11 0.06±0.042 799 .503±16.396 799 .779±16.177 0 .008±0.002\nResNet20 0.119±0.119 710 .762±16.261 711 .142±16.048 0 .005±0.004\nFMNIST MLP −0.342±0.066 121 .853±5.83 118 .316±5.453 0 .076±0.006\nMNIST MLP −0.037±0.008 81 .231±5.58 73 .994±5.58 0 .211±0.014\nandπc(θc)are also expected to be aligned. Thus, the LMC\nbetween πb(θb)andπc(θc)may not be satisfied in STE,\nwhile it is likely to be satisfied in WM.\nWe performed model merging experiments among three\nmodels to confirm the validity of the above discussion. First,\nFigure 4 presents the results of examining how well the\nsingular vectors are aligned in each model by WM. Since\nthe models θbandθcare matched to θathrough WM, it\nis expected that R(θa, πb(θa))andR(θa, πc(θc))would\nbe large. On the other hand, although θbandθcwere not\nexplicitly made to approach each other, R(πb(θb), πc(θc))\nis clearly greater than zero, indicating that the directions of\nthese two singular vectors are indirectly aligned by WM.\nFrom the result, the barrier between the models πb(θb)and\nπc(θc)is expected to be small. To confirm this, Figure 5\nshows the test accuracy landscape around θa,πb(θb)and\nπc(θc). The results of model merging at λ= 1/2for each\npair of models are also shown in Table 3. Table 3 shows the\nmean and standard deviation of model merging three times.\nAs can be seen from Figure 5, the barrier between πb(θb)\nandπc(θc)is smaller in WM than in STE. This means that\nthere is a significant difference between the principles of\npermutations obtained by WM and STE. Figure 5 also shows\nthat the landscape of test accuracy is flatter around the three\nmodels in WM than in STE. Therefore, WM is likely to be\nmore advantageous, especially for merging more than or\nequal to three models.\n6. Related Work\n(Linear) Mode Connectivity. Several studies (Garipov\net al., 2018; Draxler et al., 2018; Freeman & Bruna, 2017)\nhave found that different NN solutions can be connected\nby nonlinear paths with almost no increase in loss. Na-\ngarajan & Kolter (2019) first found that solutions can be\nconnected by linear paths with an almost constant loss value\nin the case of training models on MNIST with the same\nrandom initial values. Later, Frankle et al. (2020) showed\nexperimentally that LMC is not always satisfied between\ntwo SGD solutions, even with the same initial values, de-\npending on the datasets and model architectures. However,\nthey also showed that if a single model is trained for a cer-\ntain period and then two models are trained independently\nfrom this pre-trained model as a starting point, they are lin-\nearly mode-connected. Furthermore, Frankle et al. (2020)\nshowed the relationship between LMC and the lottery-tickethypothesis (Frankle & Carbin, 2019). Entezari et al. (2022)\nconjectured that LMC is satisfied with a high probability\nbetween two SGD solutions by taking into account the per-\nmutation symmetries in the hidden layers. Then, Ainsworth\net al. (2023) proposed a WM method by formulating the\nneuron alignment as a bipartite graph matching problem\nand solving it approximately. Afterward, Pe ˜na et al. (2023)\nproposed using Sinkhorn’s algorithm to solve the WM di-\nrectly. While several papers (Venturi et al., 2019; Nguyen\net al., 2019; Nguyen, 2019; Kuditipudi et al., 2019) have\ndiscussed non-linear mode connectivity, there is little theo-\nretical analysis in LMC. Zhou et al. (2023) introduced the\nconcept of layerwise linear feature connectivity (LLFC) and\nshowed that LLFC implies LMC. Also, Zhou et al. (2023)\nshowed that if the ReLU function is approximately linear\n(weak additivity for ReLU activation) and the commutativity\nproperty is satisfied, then LLFC and LMC are also satisfied.\nHowever, we show that the L2distance between the models\nafter permutation is not close enough to satisfy the commu-\ntativity property. This fact motivated us to investigate the\nrelationship between LMC and WM.\nModel Merging. LMC is a property related to merging\ntwo models, so its relevant topics include model merging\nand federated learning. McMahan et al. (2017); Kone ˇcn´y\net al. (2016) introduced the concept of federated learning,\nwhere a model is trained on divided datasets. Wang et al.\n(2020) proposed a method for federated learning by permut-\ning each component unit and then averaging the weights of\nthe models. Singh & Jaggi (2020) proposed a method for\nmerging models by performing a soft alignment of model\nweights using optimal transport, which is very similar to\nthe method proposed by Zhou et al. (2023). Wortsman et al.\n(2022) proposed a method to improve test accuracy with-\nout increasing inference cost, unlike ensemble methods, by\naveraging the weights of models fine-tuned with different\nhyperparameters.\n7. Conclusion\nThis paper analyzed why linear mode connectivity (LMC) is\nsatisfied by permutation search with weight matching (WM).\nFirst, we showed that WM does not reduce the distance be-\ntween the weights of two models as much as previously\nthought, and then we analyzed WM in terms of singular\nvalue decomposition. Our analysis showed that WM has the\n8Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\neffect of aligning the directions of singular vectors with large\nsingular values, which plays an important role in achiev-\ning LMC. Finally, we compared WM with straight-through\nestimator (STE), a dataset-dependent permutation search\nmethod, and showed that WM is superior at merging three\nor more models.\nPotential Broader Impact\nThis paper presents work whose goal is to advance the field\nof Machine Learning. There are many potential societal\nconsequences of our work, none which we feel must be\nspecifically highlighted here.\nReferences\nAdams, R. P. and Zemel, R. S. Ranking via sinkhorn propa-\ngation, 2011.\nAinsworth, S., Hayase, J., and Srinivasa, S. Git re-basin:\nMerging models modulo permutation symmetries. In The\nEleventh International Conference on Learning Represen-\ntations , 2023. 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(eds.), Pro-\nceedings of the 39th International Conference on Ma-\nchine Learning , volume 162 of Proceedings of Machine\nLearning Research , pp. 23965–23998. PMLR, 17–23 Jul\n2022. URL https://proceedings.mlr.press/\nv162/wortsman22a.html .\nZhao, W. X., Zhou, K., Li, J., Tang, T., Wang, X., Hou, Y .,\nMin, Y ., Zhang, B., Zhang, J., Dong, Z., Du, Y ., Yang, C.,\nChen, Y ., Chen, Z., Jiang, J., Ren, R., Li, Y ., Tang, X.,\nLiu, Z., Liu, P., Nie, J.-Y ., and Wen, J.-R. A survey of\nlarge language models, 2023.\nZhou, Z., Yang, Y ., Yang, X., Yan, J., and Hu, W. Going be-\nyond linear mode connectivity: The layerwise linear fea-\nture connectivity. In Thirty-seventh Conference on Neural\nInformation Processing Systems , 2023. URL https:\n//openreview.net/forum?id=vORUHrVEnH .\n11Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nA. Convolutional Layers\nThis section proves the theorem like Theorem 4.1 holds for convolutional neural networks (CNNs).\nA.1. Notation\nWe introduce the notation used in the following sections. Each element of a tensor is specified by a simple italic variable\nwith subscripts (e.g., for a third-order tensor X, itsi, j, k -th component is denoted by Xi,j,k). In this paper, we also use\nPython-like slice notation. For example, X1,:denotes the first row of the matrix X. For a complex matrix X, letX∗=X⊤\nbe its unitary transpose, where Xdenotes the complex conjugate of X.\nA.2. Matrix Representation of Convolutional Layer\nThis subsection introduces the matrix representation of a convolutional layer. Let X∈Rm×n×nandY∈Rm×n×nbe the\ninput and the output of the ℓ-th convolutional layer, respectively, where mdenotes the number of input and output channels\nandndenotes the size of height and width of the input. Although we assume that the numbers of output channels and input\nchannels are identical and the sizes of the height and width of the input are also identical for simplicity, our analysis can be\napplicable even when they are not identical. Let K∈Rn×n×m×mbe the kernel of the ℓ-th layer. Then we have\nYc,r,i=X\nd∈[m],p∈[n],q∈[n]Xd,r+p,i+qKp,q,c,d .\nHere, there exists a matrix Msuch that vec(Y) =Mvec(X)holds (Sedghi et al., 2019; Jain, 1989; Goodfellow et al.,\n2016), where\nM=\nB1,1B1,2. . .B1,m\nB2,1B2,2. . .B2,m\n............\nBm,1Bm,2. . .Bm,m\n. (7)\nHere for all c, d∈[m],Bc,dis a doubly circulant matrix defined by\nBc,d=\ncirc(K1,:,c,d) circ( K2,:,c,d). . . circ(Kn,:,c,d)\ncirc(Kn,:,c,d) circ( K1,:,c,d). . . circ(Kn−1,:,c,d)\n............\ncirc(K2,:,c,d) circ( K3,:,c,d). . . circ(K1,:,c,d)\n,\nwhere circis a function to generate a circulant matrix from a given vector. For example, given a vector a= (a1, a2, . . . , a 3),\nthe circulant matrix generated by ais given by circ(a) =\na1a2a3\na3a1a2\na2a3a1\n.\nA.3. Singular Value Decomposition and Weight Matching of Convolutional Layer\nSince Equation (7) represents the matrix form of the convolutional layer, we can reach a similar conclusion to Theorem 4.1\nby performing a singular value decomposition (SVD) on it. However, as this matrix is a huge matrix of size mn2×mn2,\ndirectly performing an SVD is not practical. Therefore, we decompose it into a more SVD-friendly form by applying a\nFourier transform. Using the imaginary unit as η=√−1, and setting ω=e−2πη/n, a one-dimensional Fourier transform\nmatrix Fis defined by Fi,j= (ω(i−1)(j−1))i,j.4A matrix corresponding to the two-dimensional Fourier transform can be\ndefined as Q= (F⊗F∗)/n. Here, ⊗denotes the Kronecker product, and F∗represents the unitary transpose of the matrix\nF. By using this two-dimensional Fourier transform matrix Q, the matrix Mcan be decomposed as follows:\nM= (Im⊗Q)∗L(Im⊗Q),\n4Usually, the alphabet letters iorjare used for the imaginary unit, but since they are used as indices here, we conveniently use η.\n12Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nwhere Imdenotes the identity matrix with size of m×m, and we have\nL=\nD1,1D1,2. . .D1,m\nD2,1D2,2. . .D2,m\n............\nDm,1Dm,2. . .Dm,m\n.\nHere, for all c, d∈[m],Dc,d=QBc,dQ∗is a complex diagonal matrix (Sedghi et al., 2019). Let G:,:,wbe a matrix formed\nby extracting the w-th diagonal element of each diagonal matrix Dc,dand arranging them (i.e., Gc,d,w = (Dc,d)w,w). Then,\nthe following theorem holds:\nTheorem A.1 (SVD of convolutional layer) .Letsw,i,uw,i, andvw,ibe the i-th singular value, left singular vector, and\nright singular vector of G:,:,w, respectively. Then, the matrix Mrepresenting the convolutional layer can be decomposed\nas follows:\nM=X\nw,i(uw,i⊗Qew)sw,i(vw,i⊗Qew)∗,\nwhere ewrepresents the orthonormal basis in Euclidean space Rn2, and sw,i,uw,i⊗Qew, andvw,i⊗Qeware the\nsingular value, left singular vector, and right singular vector of M, respectively.\nThe proof is shown in Appendix B.3. From this theorem, the following theorem can be proved:\nTheorem A.2. LetM(a)andM(b)be the matrix representations of convolutional layers of two CNNs. From Theorem A.1,\ntheir SVDs are given by M(a)=P\nw,i(u(a)\nw,i⊗Qew)s(a)\nw,i(v(a)\nw,i⊗Qew)∗andM(b)=P\nw,i(u(b)\nw,i⊗Qew)s(b)\nw,i(v(b)\nw,i⊗\nQew)∗, respectively. Then, the WM between M(a)andM(b)is equivalent to finding permutation matrices PℓandPℓ−1\nsuch that\narg max\nPℓ,Pℓ−1ℜX\nw,i,js(a)\nw,is(b)\nw,j(u(a)\nw,i)∗(Pℓu(b)\nw,j)(v(a)\nw,i)∗(Pℓ−1v(b)\nw,j),\nwhere ℜzis the real part of zfor a complex number z.\nThe proof is shown in Appendix B.4. Similar to the case of MLP (Theorem 4.1), Theorem A.2 indicates that WM has the\neffect of aligning the directions of the corresponding singular vectors in convolutional layers.\nB. Proofs\nB.1. Proof of Theorem 3.1\nProof. From the assumption and Taylor theorem centered at θa, we have\nL(θc) =L(θa) + (θc−θa)∇L(θa) +1\n2(θc−θa)⊤Ha(θc−θa) +O(∥θc−θa∥3)\n=L(θa) + (1 −λ)βµ∇L(θa) +1\n2(1−λ)2β2µ⊤Haµ+O(β3).\nSimilarly, using Taylor theorem centered at π(θb), we get\nL(θc) =L(π(θb)) + (θc−π(θb))∇L(π(θb)) +1\n2(θc−π(θb))⊤Ha(θc−π(θb)) +O(∥θc−π(θb)∥3)\n=L(π(θb))−λβµ∇L(π(θb)) +1\n2λ2β2µ⊤Haµ+O(β3).\nCombining these equations, the barrier can be obtained as\nB(θa, π(θb)) = max\nλ(L(θc)−λL(θa)−(1−λ)L(π(θb)))\n= max\nλ(λ(L(θc)− L(θa)) + (1 −λ)(L(θc)− L(π(θb))))\n= max\nλ\u0012\nβλ(1−λ)µ⊤(∇L(θa)− ∇L (π(θb))) +1\n2β2λ(1−λ)µ⊤((1−λ)Ha+λHb)µ\u0013\n+O(β3).\n13Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nB.2. Proof of Theorem 4.1\nProof. Consider the L2norm of the ℓ-th layer ∥W(a)\nℓ−PℓW(b)\nℓP⊤\nℓ−1∥2. Using the fact that the L2norm can be rewritten\nusing trace, we have\n∥W(a)\nℓ−PℓW(b)\nℓP⊤\nℓ−1∥2= tr\u0010\n(W(a)\nℓ−PℓW(b)\nℓP⊤\nℓ−1)(W(a)\nℓ−PℓW(b)\nℓP⊤\nℓ−1)⊤\u0011\n= tr\u0010\nW(a)\nℓ(W(a)\nℓ)⊤\u0011\n+ tr\u0010\nW(b)\nℓ(W(b)\nℓ)⊤\u0011\n−2 tr\u0010\nPℓW(b)\nℓP⊤\nℓ−1(W(a)\nℓ)⊤\u0011\n. (8)\nWe focus on the last term because only it depends on the permutation matrices. The SVDs of the weights W(a)\nℓandW(b)\nℓ\nare denoted by W(a)\nℓ=U(a)\nℓS(a)\nℓ(V(a)\nℓ)⊤=P\niu(a)\nℓ,is(a)\nℓ,i(v(a)\nℓ,i)⊤andW(b)\nℓ=U(b)\nℓS(b)\nℓ(V(b)\nℓ)⊤=P\nju(b)\nℓ,js(b)\nℓ,j(v(b)\nℓ,j)⊤,\nrespectively. Thus, the last term of Equation (8) can be rewritten as\n−2 tr\u0010\nPℓW(b)\nℓP⊤\nℓ−1(W(a)\nℓ)⊤\u0011\n=−2X\ni,js(a)\nℓ,is(b)\nℓ,j(u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j).\nTherefore, Equation (1) equals\narg min\nπ∥θa−π(θb)∥2= arg max\nπ=(Pℓ)ℓX\nℓ,i,js(a)\nℓ,is(b)\nℓ,j(u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j),\nwhich completes the proof.\nB.3. Proof of Theorem A.1\nProof. Note that the matrix Lcan be decomposed to L=P\nwG:,:,w⊗(ewe⊤\nw)by using the tensor G, where ewis the\northonormal basis in Eucrlidean space Rn2. Thus, the SVD of Lis given by\nL=X\nwX\ni(uw,i⊗Q)sw,i(vw,i⊗Q)∗⊗(ewe⊤\nw)\n=X\nwX\nisw,i(uw,iv∗\nw,i)⊗(ewe⊤\nw)\n=X\nwX\nisw,i(uw,i⊗ew)(vw,i⊗ew)∗.\nThus, the SVD of Mis also given by\nM=X\nwX\nisw,i(Im⊗Q)(uw,i⊗ew)(vw,i⊗ew)∗(Im⊗Q)∗=X\nwX\nisw,i(uw,i⊗Qew)(vw,i⊗Qew)∗,\nwhich completes the proof.\nB.4. Proof of Theorem A.2\nBefore proving Theorem A.2, we first prove the following lemma:\nLemma B.1. LetKandK′be kernels of a convolutional layer. We have ∥M−M′∥2=n2∥K−K′∥2, where Mand\nM′are the matrix representations of KandK′, respectively.\nProof. From the definition of MandM′, we have\nM=\nB1,1B1,2. . .B1,m\nB2,1B2,2. . .B2,m\n............\nBm,1Bm,2. . .Bm,m\n,M′=\nB′\n1,1B′\n1,2. . .B′\n1,m\nB′\n2,1B′\n2,2. . .B′\n2,m\n............\nB′\nm,1B′\nm,2. . .B′\nm,m\n,\nwhere for any c, d∈[m],Bc,dandB′\nc,ddenote the doubly circulant matrices obtained from the kernels KandK′.\nThus,∥M−M′∥2=P\nc,d∥Bc,d−B′\nc,d∥2=nP\nc,dP\ni∥circ(Ki,:,c,d)−circ(K′\ni,:,c,d)∥2=n2P\nc,dP\ni,j(Ki,j,c,d−\nK′\ni,j,c,d )2=n2∥K−K′∥2holds.\n14Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nProof of Theorem A.2. In convolutional layers, permutation matrices permute the input and output channels of the kernel.\nTherefore, the permutation matrices PℓandPℓ−1corresponding to the input and output are m×mmatrices. By using\nthese matrices, the permutation of the matrix representation of the convolutional layer of the model θbis denoted by\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤. Lemma B.1 indicates that finding the permutation matrices that minimize the L2distance\nbetween the two kernels is equivalent to minimizing ∥M(a)−(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤∥. Therefore, we have\n∥M(a)−(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤∥2\n= tr\u0010\u0010\nM(a)−(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011\u0010\nM(a)−(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011\n= tr\u0010\nM(a)(M(a))⊤\u0011\n+ tr\u0010\nM(b)(M(b))⊤\u0011\n−tr\u0010\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011\n−tr\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤(M(a))∗\u0011\n. (9)\nIn Equation (9), the permutation matrices PℓandPℓ−1are only related to the last two terms. Therefore, we focus only on\nthem. By using some properties of trace, we have\n−tr\u0010\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011\n−tr\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤(M(a))∗\u0011\n=−tr\u0010\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011\n−tr\u0012\u0010\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011∗\u0013\n=−tr\u0010\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011\n−tr\u0010\nM(a)\u0000\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0001∗\u0011\n=−2ℜtr\u0010\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011\n.\nHere, from Theorem A.1,\nM(a)=X\nw,i(u(a)\nw,i⊗Qew)s(a)\nw,i(v(a)\nw,i⊗Qew)∗=X\nw,is(a)\nw,i(u(a)\nw,i(v(a)\nw,i)∗⊗Qewe⊤\nwQ∗) =X\nw(C(a)\nw⊗Qewe⊤\nwQ∗),\n(10)\nand\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤=X\nw,is(b)\nw,i(Pℓ⊗Im)(u(b)\nw,i⊗Qew)(v(b)\nw,i⊗Qew)∗(Pℓ−1⊗Im)⊤\n=X\nw,is(b)\nw,i(Pℓu(b)\nw,i(Pℓ−1v(b)\nw,i)∗⊗Qewe⊤\nwQ∗)\n=X\nw(C(b)\nw⊗Qewe⊤\nwQ∗) (11)\nholds, where we let C(a)\nw=P\nis(a)\nw,iu(a)\nw,i(v(a)\nw,i)∗andC(b)\nw=P\nis(b)\nw,i(Pℓu(a)\nw,i)(Pℓ−1v(a)\nw,i)∗. From Equation (10) and\nEquation (11), we have\n−2ℜtr\u0010\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011∗\u0011\n=−2ℜtr X\nw(C(a)\nw⊗Qewe⊤\nwQ∗) X\nw′(C(b)\nw′⊗Qew′e⊤\nw′Q∗)!∗!\n=−2ℜtr\nX\nw,w′\u0010\nC(a)\nw(C(b)\nw′)∗⊗Qewe⊤\nwew′e⊤\nw′Q∗\u0011\n.\n15Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nUsing the fact that if w̸=w′, thene⊤\nwew′= 0, and otherwise, e⊤\nwew′= 1, we have\n−2ℜtr\u0012\nM(a)\u0010\n(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤\u0011⊤\u0013\n=−2ℜX\nwtr\u0010\nC(a)\nw(C(b)\nw)∗⊗Qewe⊤\nwQ∗\u0011\n=−2ℜX\nwtr\u0010\nC(a)\nw(C(b)\nw)∗\u0011\ntr\u0000\nQewe⊤\nwQ∗\u0001\n=−2ℜX\nwtr\u0010\nC(a)\nw(C(b)\nw)∗\u0011\n=−2ℜX\nwtr\n X\nis(a)\nw,iu(a)\nw,i(v(a)\nw,i)∗!\nX\njs(b)\nw,j(Pℓu(b)\nw,j)(Pℓ−1v(b)\nw,j)∗\n∗\n\n=−2ℜX\nwX\ni,js(a)\nw,is(b)\nw,j\u0010\n(u(a)\nw,i)∗(Pℓu(b)\nw,j)\u0011∗\u0010\n(v(a)\nw,i)∗(Pℓ−1v(b)\nw,j)\u0011\n.\nFrom the above, the minimization of ∥M(a)−(Pℓ⊗Im)M(b)(Pℓ−1⊗Im)⊤∥is equivalent to the maximization of\nℜP\nwP\ni,js(a)\nw,is(b)\nw,j\u0010\n(u(a)\nw,i)∗(Pℓu(b)\nw,j)\u0011∗\n((v(a)\nw,i)∗(Pℓ−1v(b)\nw,j)).\nB.5. Proof of |R(θa, π(θb))| ≤1\nThis subsection proves the following theorem:\nTheorem B.2. Letθaandθbbe the parameters of two MLPs with L-layers. For any permutation π= (Pℓ)ℓ∈[L], we have\n|R(θa, π(θb))|=\f\f\fP\nℓ,i,j(u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j)\f\f\f\nP\nℓnℓ≤1. (12)\nThe equality holds if, for all ℓ, i,u(a)\nℓ,i=Pℓu(b)\nℓ,iandv(a)\nℓ,i=Pℓ−1v(b)\nℓ,i.\nProof. If, for all ℓ, i,u(a)\nℓ,i=Pℓu(b)\nℓ,iandv(a)\nℓ,i=Pℓ−1v(b)\nℓ,i, the equality obviously holds. Thus, we prove that Equation (12)\nholds. Using the property of trace, we have\nX\nℓ,i,j(u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j) =X\nℓ,i,j(Pℓu(b)\nℓ,j)⊤(u(a)\nℓ,i)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j)\n=X\nℓtr\nX\ni(u(a)\nℓ,i)(v(a)\nℓ,i)⊤X\nj\u0010\n(Pℓu(b)\nℓ,j)(Pℓ−1v(b)\nℓ,j)⊤\u0011⊤\n.\nLetW′(a)\nℓ=P\ni(u(a)\nℓ,i)(v(a)\nℓ,i)⊤andW′(b)\nℓ=P\nj(Pℓu(b)\nℓ,j)(Pℓ−1v(b)\nℓ,j)⊤. Obviously, they are matrices with all singular\nvalues of 1, and thus by using von Neuman’s trace inequality (von Neumann, 1962), we have\nX\nℓ\f\f\ftr\u0010\nW′(b)\nℓ(W′(a)\nℓ)⊤\u0011\f\f\f≤X\nℓnℓ.\nTherefore, the triangle inequality yields that\nX\nℓ,i,j(u(a)\nℓ,i)⊤(Pℓu(b)\nℓ,j)(v(a)\nℓ,i)⊤(Pℓ−1v(b)\nℓ,j) =\f\f\f\f\fX\nℓtr\u0010\nW′(b)\nℓ(W′(a)\nℓ)⊤\u0011\f\f\f\f\f≤X\nℓ\f\f\ftr\u0010\nW′(b)\nℓ(W′(a)\nℓ)⊤\u0011\f\f\f≤X\nℓnℓ,\nwhich completes the proof.\n16Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\nTable 3. Evaluation results of barrier and R values between each pair of models\nAccuracy barrier Loss barrier Rvalue\nDataset Network (θa+πb(θb))/2 (θa+πc(θc))/2 (πb(θb) +πc(θc))/2 (θa+πb(θb))/2 (θa+πc(θc))/2 (πb(θb) +πc(θc))/2R(θa, πb(θb))R(θa, πc(θc))R(πb(θb), πc(θc))\nWMCIFAR10VGG11 0.094±0.158 0 .037±0.156 0 .141±0.141 8 .362±5.677 7 .555±4.978 10 .12±5.117 0 .199±0.042 0 .195±0.027 0 .1±0.015\nResNet20 0.135±0.026 0 .098±0.011 0 .294±0.098 3 .312±0.61 2 .995±0.064 7 .23±0.99 0 .23±0.016 0 .211±0.008 0 .102±0.005\nFMNIST MLP −0.211±0.029 −0.174±0.044 −0.174±0.051 1 .947±0.501 1 .703±0.289 4 .337±1.434 0 .335±0.003 0 .334±0.002 0 .209±0.003\nMNIST MLP −0.027±0.005 −0.034±0.003 −0.031±0.003 0 .173±0.04 0 .198±0.032 0 .475±0.069 0 .421±0.009 0 .425±0.006 0 .273±0.005\nSTECIFAR10VGG11 0.081±0.031 0 .099±0.042 2 .172±0.989 4 .86±0.815 5 .76±0.537 32 .013±8.193 0 .01±0.001 0 .01±0.001 0 .019±0.002\nResNet20 0.466±0.154 0 .446±0.138 1 .693±0.168 15 .005±3.765 13 .942±4.008 34 .483±2.426 0 .008±0.004 0 .007±0.003 0 .02±0.004\nFMNIST MLP −0.372±0.016 −0.343±0.055 0 .023±0.118 2 .667±0.248 2 .483±0.621 15 .97±1.724 0 .073±0.002 0 .081±0.009 0 .047±0.002\nMNIST MLP −0.037±0.011 −0.039±0.006 0 .017±0.014 0 .253±0.176 0 .358±0.198 2 .312±0.457 0 .201±0.019 0 .203±0.007 0 .119±0.015\nC. Experimental Setup\nWe describe the experimental setup for training neural networks to obtain SGD solutions. In this work, we apply Sinkhorn’s\nalgorithm for permutation based on WM and STE. Thus, we also detail the experimental setup for Sinkhorn’s algorithm.\nIn our paper, all experiments were conducted on a Linux workstation with two AMD EPYC 7543 32-Core processors, eight\nNVIDIA A30 GPUs, and 512 GB memory.\nC.1. Model Training\nMLP on MNIST and FMNIST. Following the settings in (Ainsworth et al., 2023), we train a Multi-Layer Perceptron\n(MLP) with three hidden layers, each comprising 512 units. The intermediate layers employ the ReLU function as their\nactivation function. For MNIST and FMNIST datasets, we optimize using the Adam algorithm with a learning rate of\n1×10−3.\nVGG11 and ResNet20 on CIFAR10. To accomplish Layer-wise Model Compression (LMC), we increase the widths of\nVGG11 and ResNet20 by factors of 4 and 16, respectively. As described in (Jordan et al., 2023), we repair the BatchNorm\nlayers in these models during model merging by using the training dataset. Optimization is conducted using Adam with a\nlearning rate of 1×10−3.\nC.2. Permutation Search\nFor permutation search in this study, we employ the method based on Sinkhorn’s algorithm as proposed by Pe ˜na et al.\n(2023). We utilize the implementation provided by the authors in their GitHub repository5. DistL2Loss and MidLoss are\nused as loss functions for permutation searches corresponding to WM and STE, respectively. Optimization for all models is\nperformed using Adam with a learning rate of 1, setting the maximum number of epochs to ten for DistL2Loss and five for\nMidLoss.\nD. Additional Results\nD.1. Experimental Results of STE and WM\nIn this section, additional experimental results for Section 5.3 are shown in Table 3 for the barrier values and Rvalues\nbetween each pair of models. The table shows the model-merging results with λ= 1/2, and the mean and standard deviation\nof five model merges. In the table, a negative value for the barrier indicates an improvement in performance due to the\nmerging. From the Table 3, we can see that the barrier between πb(θb)andπc(θc)is also smaller for WM than for STE.\nD.2. Distribution of Singular Values\nFigure 6 shows the distribution of the singular values of all the layers. The figure shows in the intermediate layers, the\nsingular values are very close across all models. Meanwhile, in the final layers, there is variability in the singular values.\nNote that this variability does not affect the accuracy values for the merged model. Let W(a)\nL=P\nis(a)\nL,iu(a)\nL,i(v(a))⊤\nL,iand\nW(b)\nL=P\nis(b)\nL,iu(b)\nL,i(v(b)\nL,i)⊤be the final layer weights of the two trained models. The figure shows that the difference\nbetween the singular values of the two models is approximately a constant multiple. In other words, there exists a constant\nαands(a)\nL,i≈αs(b)\nL,ifor all i. Then, if the singular vectors of the two weights are equal (i.e., v(a)\nL,i=v(b)\nL,iandu(a)\nL,i=u(b)\nL,i\n5https://github.com/fagp/sinkhorn-rebasin\n17Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\n05000510FC 1\n05000510FC 2\n050005FC 3\n0512FC 4\nIndexSingular value\n(a) MLP, MNIST.\n0500010FC 1\n05000510FC 2\n0500010FC 3\n0524FC 4\nIndexSingular value\n(b) MLP, FMNIST.\n02000468Conv 1\n0250005000002040Conv 2\n0200000204060Conv 3\n02500050000050100Conv 4\n010000050100Conv 5\n020000050100150Conv 6\n050000204060Conv 7\n050000255075Conv 8\n051.01.5FC 1\nIndexSingular value(c) VGG11, CIFAR10.\n025001015Conv 1\n0250000025Conv 2\n0250000020Conv 3\n0250000020Conv 4\n0250000020Conv 5\n0250000020Conv 6\n0250000050Conv 7\n0250000025Conv 8\n0100000050Conv 9\n0250000050Conv 10\n0100000050Conv 11\n0100000050Conv 12\n0100000050Conv 13\n01000000100Conv 14\n0100000050Conv 15\n050000050Conv 16\n0100000050Conv 17\n0500000100Conv 18\n0500000100Conv 19\n0500000100Conv 20\n0500000100Conv 21\n051.52.02.5FC 1\nIndexSingular value\n(d) ResNet20, CIFAR10.\nFigure 6. Distribution of the singular values of each layer.\n18Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\n0.0 0.5 1.0\nRatio0.00.51.0Singular value ratioVGG11\n× 1/8\n× 1/4\n× 1/2× 1\n× 2\n× 4\n0.0 0.5 1.0\nRatio0.00.51.0Singular value ratioResNet20\n× 1\n× 2\n× 4× 8\n× 16\nFigure 7. Distribution of singular values as the multiplier of the model width changes.\n1/8 1/4 1/2 1 2 4\nWidth multiplier0.00.10.20.3Value of RVGG11\nw/ WM\nw/o WM\n1 2 4 8 16\nWidth multiplier0.00.10.20.3Value of RResNet20\nw/ WM\nw/o WM\n(a) Evaluation results of Equation (5) for all the singular vectors.\n1/8 1/4 1/2 1 2 4\nWidth multiplier0.00.10.20.3Value of RVGG11\nw/ WM\nw/o WM\n1 2 4 8 16\nWidth multiplier0.00.10.20.3Value of RResNet20\nw/ WM\nw/o WM(b) Evaluation results of Equation (5) for the singular vectors with\nγ= 0.3.\nFigure 8. Relation between the model width and the difficulty in aligning the directions of singular vectors.\nfor all i), then W(a)\nL≈αW(b)\nLholds (indeed, as mentioned in Section 4.3, the permutation matrix brings the directions\nof the singular vectors close). Therefore, the weight of the final layer of the merged model at the ratio λ∈[0,1]is given\nbyλW(a)\nL+ (1−λ)W(b)\nL≈λW(a)\nL+ (1−λ)αW(a)\nL= (λ+ (1−λ)α)W(a)\nL. Thus, we can consider that the weight\nand activation function of the merged model are given by W(a)\nLand a softmax function with an inverse temperature of\n1/(λ+ (1−λ)α). Since the inverse temperature does not affect the accuracy value, the difference in the singular values of\nthe final layer would not matter in satisfying the LMC, at least in terms of accuracy value.\nD.3. Relationship with Model Width\nPrevious studies have demonstrated that the width of the model architecture affects the ease of achieving LMC. In this\nsection, we explain this reason from the following two facts: (i) the proportion of dominant singular values decreases as\nthe model width increases, and (ii) the WM preferentially aligns the directions of singular vectors corresponding to such\ndominant singular values.\nRelationship between Model Width and Distribution of Singular Values. As we mentioned, the proportion of relatively\nlarge singular values in all singular values decreases as the model width increases. To verify this, Figure 7 shows the\ndistribution of the singular values of all layers of VGG11 and ResNet20 trained on CIFAR10. Figure 7 shows the results\nof different model widths (i.e., dimensionality). The vertical axis represents the singular values divided by the maximum\nsingular value of each model, and the horizontal axis represents the ratio among all singular values (e.g., the point at 0.5 on\nthe horizontal axis represents the singular value in the middle of all values sorted in descending order). As can be seen,\nthe proportion of relatively large singular values decreases as the model width increases. Thus, the proportion of singular\nvectors that need to be aligned in the model decreases as width increases, suggesting that wider models make it easier to\nsatisfy LMC by WM.\nSingular-Vector Alignment We conduct an experiment to examine how well the directions of singular vectors are aligned\nas the model width is increased when the permutations found by the WM are applied. The results are shown in Figure 8. The\nfigures show the results of evaluating R(θa, π(θb))for the trained models θaandθbby searching permutations πso that θb\napproaches θaby WM. For comparison, the case where no permutations are applied (i.e., πis an identity map) is also shown.\nA threshold γwas also introduced to evaluate how well the directions of the singular vectors with large singular values are\n19Analysis of Linear Mode Connectivity via Permutation-Based Weight Matching\n1/8 1/4 1/2 1 2 4\nWidth multiplier0.40.50.60.7Value of RR(a,(a+(b))/2)\nw/ WM\nw/o WM\n1/8 1/4 1/2 1 2 4\nWidth multiplier0.40.50.60.7Value of RR((b),(a+(b))/2)\nw/ WM\nw/o WM\n(a) VGG11.\n1 2 4 8 16\nWidth multiplier0.40.50.60.7Value of RR(a,(a+(b))/2)\nw/ WM\nw/o WM\n1 2 4 8 16\nWidth multiplier0.40.50.60.7Value of RR((b),(a+(b))/2)\nw/ WM\nw/o WM (b) ResNet20.\nFigure 9. Relation between the model width and the difficulty in aligning the directions of singular vectors.\naligned. First, focusing on the results for Figure 8(a) with γ= 0, we can see that the value of Rdecreases even when the\nwidth increases and WM is used. On the other hand, the Figure 8(b) shows that the directions of the singular vectors with\nparticularly large singular values are aligned by permutation when the model width is increased. This indicates that even\nwith WM, it is difficult to perfectly align the directions of singular vectors between the two models, while increasing the\nwidth decreases the fraction of singular vectors with large singular values, and thus WM attempts to align the directions of\nsuch dominant singular vectors.\nTo investigate the impact of this alignment on the singular vectors of the merged model, we evaluate how well the singular\nvectors of the model are aligned before and after merging, and the results are shown in Figure 9. In the figures, the value of\nthe function Ris evaluated with γ= 0.3in order to focus on the singular vectors with large singular values. Figure 9 shows\nthat as the model width increases, the singular vectors’ directions with large singular values are aligned between the models\nbefore and after merging. Considering that the input-output relationship of each model layer is determined by the singular\nvector with the large singular value, this figure indicates that by increasing the model width, the function of the merged\nmodel approaches the models θaandθb. Therefore, increasing the width is expected to make the LMC more feasible.\nE. Limitations\nOur analysis in this paper has some limitations. In this paper, we have explained LMCs by WM in terms of SVD for\nthe weights of each layer, but this does not explain why all LMCs occur. For example, STE, which directly searches for\npermutations that reduce the barrier, does not align the directions of singular vectors, so the method of analysis in this paper\ncannot explain why STE can achieve LMC. In addition, the analysis in this paper can only be applied to image classification\ntasks using the same dataset. The applicability to different datasets and image generation tasks is a subject for future work.\n20"    },    {        "title": "2402.04066v1.On_the_Modelling_of_Ship_Wakes_in_S_Band_SAR_Images_and_an_Application_to_Ship_Identification.pdf",        "content": "ON THE MODELLING OF SHIP WAKES IN S-BAND SAR IMAGES\nAND AN APPLICATION TO SHIP IDENTIFICATION\nKamirul Kamirul∗, Odysseas Pappas, Igor G. Rizaev, Alin Achim\nVisual Information Laboratory\nUniversity of Bristol, UK\nABSTRACT\nWe present a novel ship wake simulation system for generating\nS-band Synthetic Aperture Radar (SAR) images, and demon-\nstrate the use of such imagery for the classification of ships\nbased on their wake signatures via a deep learning approach.\nShip wakes are modeled through the linear superposition of\nwind-induced sea elevation and the Kelvin wakes model of\na moving ship. Our SAR imaging simulation takes into ac-\ncount frequency-dependent radar parameters, i.e., the complex\ndielectric constant ( ε) and the relaxation rate ( µ) of seawa-\nter. The former was determined through the Debye model\nwhile the latter was estimated for S-band SAR based on pre-\nexisting values for the L, C, and X-bands. The results show\ngood agreement between simulated and real imagery upon vi-\nsual inspection. The results of implementing different training\nstrategies are also reported, showcasing a notable improve-\nment in accuracy of classifier achieved by integrating real and\nsimulated SAR images during the training.\nIndex Terms —Ship wakes, sea waves, SAR simulation,\nS-band, NovaSAR-1, vessel classification, sea modelling\n1. INTRODUCTION\nSynthetic Aperture Radar (SAR) technology exhibits weather-\nindependent performance and enables observations in any at-\nmospheric and illumination conditions [ 1]. SAR imaging\nis widely used in a variety of applications, including mar-\nitime surveillance, environmental monitoring, and disaster\nresponse [2, 3].\nShip wake detection in SAR images is gaining popularity\nas the ship wake signatures can serve as a way to estimate ship\ndynamics parameters and can also be utilized for more practi-\ncal applications, e.g., building a model for ship classification\nbased on the wakes [ 4,5]. This latter application unlocks the\ncapability to identify ship classes from SAR images, offering\nthe potential for advanced maritime surveillance capabilities.\nHowever, developing such classification models requires a\nhuge volume of training images, which can be challenging\n∗The authors would like to thank SSTL and Airbus D&S for\nthe provision of NovaSAR-1 data. Contact author e-mail address:\nkamirul.kamirul@bristol.ac.ukto obtain due to the limited availability of publicly accessible\ndatasets.\nThe majority of SAR wake simulation literature has primar-\nily focused on L-, C-, and X-band radar, as these are used by\nsome of the most widely utilised SAR satellite missions (such\nas TerraSAR-X, COSMO-SkyMed, ICEYE, ALOS-2, ERS-2,\nand SEASAT [ 6,7,8,9]). However, new satellite missions\nsuch as NovaSAR-1 and NISAR-S utilise S-band instruments\ndue to their advantages for maritime monitoring applications.\nMotivated by the above, our work focuses on simulating ship\nwakes specifically as imaged under a S-Band SAR system, and\nthen utilizes the resulted images for classification purposes\nin real scenarios. Therefore, the simulation results from this\nwork will not only supplement the existing modelling in terms\nof the frequency band used but also provide insights into the\ncharacteristics of ship wake signatures in the S-band scenario.\nShip wake simulation under S-Band SAR system have not\nbeen previously conducted largely due to limited information\non the frequency-dependent parameters used for the simula-\ntion, and the absence of real datasets for verifying simulation\nresults. Two essential frequency-dependent parameters, i.e.,\nthe complex dielectric constant ( ε) and relaxation rate ( µ) for\nS-Band were estimated and provided in this work. These pa-\nrameters were then used for calculating the normalized radar\ncross-section (NRCS) of sea surface and hydrodynamic modu-\nlation effect respectively. The complex dielectric constant ε\nfor S-band SAR was obtained using the Debye approach, as\nreported by Meissner and Wentz in [ 10,11], which is based\non an extensive experimental dataset for seawater. Meanwhile,\nthe value of µwas estimated from pre-existing values in the L,\nC, and X-bands, according to [6] and [12].\nTo demonstrate the utility of the simulated SAR images,\nwe employed the images to train a deep-learning model,\nAlexNet, for classifying real ships in SAR images acquired by\nNovaSAR-1. Different training strategies, including transfer\nlearning and mixing of simulated and limited numbers of real\nimages in the training dataset, have been conducted and evalu-\nated concerning their impact on classification performance.\nIn summary, our contributions are two-fold. Firstly, to the\nbest of our knowledge, we are the first to conduct ship wakes\nsimulation for S-Band SAR. Therefore, our simulation can\nserve as a theoretical foundation for future work. Secondly,arXiv:2402.04066v1  [eess.IV]  6 Feb 2024the training strategies outlined in our work provide a pathway\nfor utilizing simulated ship wake images to build a model for\nclassifying real ship wakes images.To demonstrate the utility\nof simulated SAR images, we employed the images to train a\ndeep-learning model, AlexNet, for classifying real ship images\nacquired by NovaSAR-1.\n2. MATERIALS AND METHODS\nThis work employs prior work on ship wake generation,\nwhich we have here extended to S-band SAR; a comprehen-\nsive explanation of the previous simulator can be found in\n[6] while code can be accessed through the University of\nBristol Research Data Repository at doi.org/10 .5523/\nbris .el0p94vgxjhi2224bx78actb4 .\n2.1. Sea Elevation Model\nIn this work, the sea elevation model is formulated as the\nsuperposition of the wind-induced sea surface and the Kelvin\nwakes generated by a moving ship. Wind-induced sea elevation\nat time tcan be expressed as [6, 12, 13]:\nZsea=X\niX\njAij\ncos (ki[xcosθi+ysinθj+ωt+σij]), (1)\nwhere kandωrespectively are the wavenumber and radial\nfrequency while σis the initial phase uniformly distributed\nbetween (0, 2 π).Aijrepresents the amplitude expressed as:\nAij=q\n2S(ki)D(kiθj)∆ki∆θj, (2)\nwhere S(k)andD(kiθj)are wave spectrum and angular\nspreading function [6].\nIn modelling the waves generated by moving ships, we\nused the Kelvin waves model, Zship, represented by Zilman et\nal. in [14]:\nZship=Vs\ng∂Φs\n∂x, (3)\nwithVsandΦsbeing ship velocity and an approximated form\nof fluid velocity potential. Finally, composite sea elevation\nmodel ( Z) due to existing wind and moving ship can be ob-\ntained as the superposition of ZseaandZship.\n2.2. Frequency-Dependent Parameters\nThere are two essential parameters that need to be determined\nto model the interaction between incoming waves and the sea\nsurface, i.e., complex dielectric constant ( ε) and the relaxation\nrate (µ).\nThe complex dielectric constant ( ε) has been calculated\nbased on the work of Meissner and Wentz in [ 10,11]. Therein,extensive experiments were conducted to determine εfor both\npure and sea water over a temperature range from -20 °C to\n+40°C and a frequency range up to 90 GHz. The dielectric\nconstant ( ε) of sea water with a specific salinity ( S) and tem-\nperature ( T), radiated under a specific wavelength ( λ), can be\ncalculated as:\nε(λ, S, T ) =\nεR+εS(S, T)−εR\n1 + [iλR(S, T)/λ]1−α−2iσ(S, T)λ\nc,(4)\nHere, λR,εS, and σare all dependent on SandT, while εR\nandαare set to 4.44 and 0.012, respectively, and crepresents\nthe speed of light. The detailed implementation of above equa-\ntion can be found on [ 10]. Assuming that S= 35 ppt [ 15],\nT= 21◦C[16], and NovaSAR-1 frequency is 3.2 GHz [ 17],\nthe corresponding dielectric constant for our simulation is\n69.63- i38.95.\nThe magnitude of µis poorly known experimentally, es-\npecially for S-Band case. In our experiment, the µvalue for\nS-band was interpolated from the known µvalues for L, C,\nand X-band, which were provided in [ 12,18]. Finally, the\nestimated value for µin the NovaSAR-1 band are µ= 0.05\ns-1forVw≤5 m/s and µ= 0.39s-1forVw>5 m/s.\n2.3. SAR Imaging of Sea Surface\nTo simulate SAR images of the modeled sea surface, we em-\nploy the two-scale model (TSM). This model is based on the\nresonant Bragg scattering theory describing the scattering of\nelectromagnetic waves from a rough surface [ 6,13]. The\nBragg scattering solution for normalized radar cross-section\n(NRCS) with VV and HH polarizations is expressed as [6]:\nσ0= 8πk4\necos4θlW(kBx, kBy)|T2\nc|, (5)\nwhere keis radar wavenumber. The θl,W, and Tcare respec-\ntively local incidence angle, wavenumber spectral density and\ncomplex scattering function. Then, the NRCS after taking into\naccount hydrodynamic and tilt modulations:\nσ(x, y) =σ0\u0014\n1 +Z\n(M(k)F(k)eikx+c.c)dk\u0015\n,(6)\nwhere F(k)is the 2D FFT of composite sea elevation model\nZ.M(k)is defined as follow [12, 19]:\nM(k) =−4.5k2\ny\n|k|ω−iµ\nω+iµ+4 cotθr\n1±sin2θriky, (7)\nwhere θris the nominal radar incident angle. The SAR image\nintensity can then be obtained using:\nI(x, y) =ZZ\n∂(yi−y)\nσ(x, y)\nP′\na(x, y)exp\n−π2\"\nxi−x−R\nVUr\nb#2\ndxdy. (8)∂(.)is the Dirac delta function of the impulse response\nfunction in range direction while R,H, and Vare the range\ndistance, platform height, and platform velocity, respectively.\nP′\nais degraded azimuth resolution which depends on number\nof incoherent looks and radar wavelength [ 6]. Finally, multi-\nplicative random noise is added to the final image to simulate\nspeckle, using the Weibull probability density function (PDF)\nas described in [20].\n2.4. Ship Classification\nIn this work, the AlexNet [ 21] model was trained to differen-\ntiate wakes produced by cargo and tanker ships. Due to the\nlimited availability of real ship wake images for testing, we\nutilized ship wake crops from five real NovaSAR-1 products\nand applied data augmentation by rotating, flipping, and vary-\ning noise amplitude. In total, this dataset consisted of 960 real\nimages and is hence denoted as NVReal dataset.\nThree different strategies for training the model have\nbeen conducted. Both Experiment I andExperiment II\nare based on a transfer learning approach. For this purpose,\na pre-trained model was produced using a large dataset,\nSynthwakeSAR , a synthetic dataset contains a total of\n46,080 simulated SAR ship wakes images [ 22].Synth-\nwakeSAR dataset is accessible at doi.org/10 .5523/\nbris .30kvuvmatwzij2mz1573zqumfx . Since the Syn-\nthwakeSAR dataset was originally generated for X-band SAR,\nwe reproduced the dataset to represent images acquired by an\nS-band SAR system (denoted as SynthwakeSAR-S dataset\nin this work) and used that for initial training. Afterwards, a\nsimulated NovaSAR-1 dataset ( NVSim ) was generated based\non parameters of real ship wake images from NovaSAR-1\n(NVReal ). This NVSim dataset was used to re-train the\npre-trained model via fine-tuning ( FT) for Experiment I and\nvia feature extraction ( FE) schemas for Experiment II .\nNVSim contains 18,000 images which were produced by\nassigning parameters within the range of the real parameters\nfor the real NovaSAR images in NVReal , as shown in the Table\n1 (denoted as Rsuperscripts). This step was to ensure that the\nsimulated images closely resemble their real counterparts so\nthat the classification model can learn their representation for\nrecognizing the real images. Individual ship parameters are\nprovided in Table 2.\nTable 1 . Sea state and imaging parameters for generating\nNVSim .\nParameters SymbolValue\nrangeNumber of\nsampling\nWind speed 10m U10 UR\n10±0.5m/s 3\nWind direction θw θR\nw±15◦10\nShip speed Vs VR\ns±0.5m/s 5\nShip direction θs θR\ns±30◦8\nIncident angle θi θR\ni±2◦3\nFig. 1 . Flowchart of experiments. The SynthwakeSAR-\nSrefers to reproduced version of SynthwakeSAR using\nNovaSAR-1 imaging properties while NVSim andNVReal\nare respectively simulated and real datasets of ship wakes.\nIn contrast to Experiment I andExperiment II ,Exper-\niment III did not employ any pre-trained network. In this\ncase, the model was directly trained using the NVSim dataset.\nThis experiment aims to assess the model’s performance in\nthe absence of a pre-existing dataset. Furthermore, we incor-\nporated a portion of images from NVReal and mixed them\nwith NVSim during the training stage for all experiments. The\namount of real images involved was varied from 10% to 30%.\nWe did not extend beyond this range as it would reduce the\nnumber of the remaining NVReal for testing the model. The\nflowchart of the experiments is given in Figure 1.\n3. RESULTS AND DISCUSSION\nIn this section, we present the visual comparison between the\nreal and simulated images. Moreover, the performance of the\nclassification model trained under different strategies is also\ndiscussed.\n3.1. Simulated NovaSAR-1 Images\nIn general, evaluating the accuracy of simulated images can be\nachieved through both qualitative and quantitative approaches.\nHowever, for our case, like-for-like comparison between simu-\nlated and real images is challenging because simulated images\nmay not be entirely identical but rather akin or resembling the\nreal ones. Therefore, measuring similarity between the images,\nfor example using Structural Similarity (SSIM) index or cor-\nrelation coefficient, would be inappropriate as these metrics\nwork on a pixel-by-pixel basis.\nConsidering the aforementioned limitation, we chose to\ncompare the images through visual inspection. As depicted\nin Figure 2, the simulated image closely resembles the real\none, except for the appearance of turbulent wake on the real\nimages. Furthermore, as the ship moves faster and parallel\nto the platform direction, the height of wind-induced ambient\nwaves increases, resulting in the transverse waves becoming\nscarcely visible. This phenomenon leaves only the prominentTable 2 . Ship parameters for generating NVSim .\nMMSI Class L(m)B(m)T(m)Fetch (km)UR\n10(m/s) θR\nw(◦)VR\ns(m/s) θR\ns(◦)θR\ni(◦)\n215071000 Tanker 183 32 9.30 80 4.13 343.50 6.73 351.50 29.35\n218433000 Cargo 126 23 6.90 25 6.10 359.50 7.45 182.50 29.31\n371720000 Tanker 229 103 9.50 80 3.33 332.50 7.56 355.50 23.96\n636013817 Tanker 227 150 9.70 80 4.85 4.85 6.89 156.50 27.87\n477665200 Cargo 266 150 10.50 80 6.20 341.50 8.44 181.50 28.23\nKelvin arms visible in both images, indicating a consistent\npattern or agreement between the two images. This behavior\nhas also been noted in [6, 23].\nFig. 2 . Comparison between simulated (A) and real (B) images\nof ship wakes for NovaSAR-1 SAR system. The parameters\nused are: L=126 m, B=23 m, T=6.9 m, θs= 182◦,Vs= 7.45\nm/s,θwind= 359 .5◦,Vw= 6.10m/s, and θi= 29.31◦.\n3.2. Classification Results and Performance\nIn this section, the use of simulated images for classifying real\nimages is demonstrated. Table 3 shows the classification accu-\nracy for each experimental setup under different proportions\nofNVReal included for training the model.\nTable 3 . Classification performance for all three experiments.\nNVReal Portion (%)Accuracy (%)\nExp. I Exp. II Exp. III\n0 60.41 50.86 51.56\n10 60.25 60.41 63.96\n20 40.02 64.14 59.97\n30 68.55 72.13 71.83\nBased on the experimental results, three key findings have\nbeen highlighted. Firstly, the model accuracy is generally low\nif no real images are introduced to the model during train-\ning. While it may be assumed that this result suggests a poor\nrepresentation of real images in the simulated dataset, it is\nnoteworthy that the accuracy across all experiments remains\nabove 50%. This implies that training the model solely us-\ningNVSim does provide actionable information and is not\nreduced to random guessing.Secondly, introducing a portion of real images in the train-\ning process can significantly boost the model performance.\nIt is found that adding 30% of the available NVReal data to\nNVSim in the training process improves the accuracy by al-\nmost 12% for Experiment II . This result indicates that it is\nstill possible to build a good classification model even when\nthere is limited real data available for training.\nLastly, fine-tuning a model pre-trained under synthetic\nship wake dataset (i.e, Experiment II ) is capable of producing\na higher accuracy of up to 72%. This result suggests that\ncombining representative simulated dataset and re-training\nall the network weights could provide benefits for building a\nclassification model in the absence of a real dataset for training.\nIt should be noted that our results are based on experiments\nconducted on five ship models due to the challenges in obtain-\ning real images along with their ship and sea states, as well\nas imaging parameters. Therefore, a more refined result is\nlikely to be obtained with an increased number of real datasets.\nThis would enhance the model’s ability to learn features from\na wider range of real images rather than relying extensively\non augmented images. Nevertheless, our results are presented\nto showcase the benefits of using simulated images in devel-\noping a deep learning model for classifying real ships in SAR\nimages.\n4. CONCLUSIONS\nIn this work, the first results on simulation of ship wake ac-\nquired under S-Band SAR system is reported. The simulation\nhas been conducted by employing a two-scale model, which\nrequires frequency-dependent parameters, i.e., the complex\ndielectric constant ( ε) and the relaxation rate ( µ) of seawa-\nter. The value of εwas determined using the Debye model,\nwhereas µwas estimated for the S-band based on pre-existing\nvalues in the L, C, and X-bands.\nBased on visual comparison, there is good agreement be-\ntween the simulated and real images. Additionally, we show-\ncased the utilization of simulated images to address a real-\nworld challenge, i.e., ship wake classification on NovaSAR-1\nimages. We have found that a classification model having\naccuracy of up to 72% can be attained by incorporating a very\nsmall proportion of real images along with simulated ones in\nthe training stage.5. REFERENCES\n[1]Y . S. Tsai, A. Dietz, N. Oppelt, and C. Kuenzer, “Re-\nmote Sensing of Snow Cover Using Spaceborne SAR: A\nReview,” Remote Sensing , vol. 11, no. 12, 2019.\n[2]S. Adeli, B. Salehi, M. 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Whittaker, B. Stern, N. Angli, M. Cohen, and\nR. Guida, “NovaSAR-S: A Low Cost Approach to SAR\nApplications,” in Proc. of 2013 Asia-Pacific Conf. on\nSynthetic Aperture Radar (APSAR) , 2013, pp. 84–87.\n[18] E. A. Caponi, D. R. Crawford, H. C. Yuen, and P. G.\nSaffman, “Modulation of Radar Backscatter from the\nOcean by a Variable Surface Current,” J. Geophys. Res. ,\nvol. 93, no. C10, pp. 12,249–12,263, Oct 1988.\n[19] D. R. Lyzenga, “Numerical Simulation of Synthetic\nAperture Radar Image Spectra for Ocean Waves,” IEEE\nTrans. Geosci. Remote Sens. , vol. GE-24, no. 6, pp. 863–\n872, 1986.\n[20] O. Karaku s ¸, E. E. Kuruo ˘glu, and A. Achim, “A Gen-\neralized Gaussian Extension to the Rician Distribution\nfor SAR Image Modeling,” IEEE Trans. Geosci. Remote\nSens. , vol. 60, pp. 1–15, 2022.\n[21] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Ima-\ngeNet Classification with Deep Convolutional Neural\nNetworks,” in Adv. Neural Inf. Process. Syst. 2012,\nvol. 25, pp. 1097–1105, Curran Associates, Inc.\n[22] I. G. Rizaev and A. Achim, “SynthWakeSAR: A Syn-\nthetic SAR Dataset for Deep Learning Classification of\nShips at Sea,” Remote Sensing , vol. 14, no. 16, 2022.\n[23] B. 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These effects can be combined\nto derive a gravitational analogue of the quantum/electric metrological triangle. The gravitational\nmetrological triangle may have applications in metrology and could be used to investigate the relation\nof the Planck constant to fundamental particle masses. This allows for quantum tests of the Weak\nEquivalence Principle. Moreover, the similarity of the gravitational and the quantum/electrical\nmetrological triangle can be used to test the universality of quantum mechanics.\nMETROLOGY, THE NEW SI, AND\nMETROLOGICAL TRIANGLES\nA major aspect of metrology is the definition and dis-\nsemination of physical units such as the SI base units sec-\nond, meter, kilogram, coulomb, kelvin, mol, and candela\n[1]. While once these units were based on specific real-\nizations and artefacts, they are now defined through the\nfixed numerical values of seven fundamental constants.\nBesides the second as a given number of oscillations of\nlight, emitted from a certain atomic transition in Cae-\nsium, those constants are the velocity of light c, the\nPlanck constant h, the electric charge e, the Boltzmann\nconstant kB, the Avogadro constant NA, and the lumi-\nnous efficacy Kcd[2], each playing central roles in various\nbranches of physics. It is clear that, by this choice, the\nMeter Convention established a deep relation between\nmetrology and the structure of physics and Special and\nGeneral Relativity, in particular [3].\nBefore the redefinition of the SI was put into practice in\n2019, it was needed to measure the defining constants to\nhighest precision [4–6]. In the case of the electron charge\neand the Planck constant hthis only became possible\nthanks to the quantum revolution in metrology, heralded\nby the discovery of the Josephson effect and the quan-\ntum Hall effect [7, 8]. Each of these effects gave experi-\nmental access to a fundamental measurand, the Joseph-\nson constant KJ= 2e/hand the von Klitzing constant\nRK=h/e2, respectively. Together with the electron\ncharge e, these constants form the quantum/electrical\nmetrological triangle [9, 10], see Fig. 1, where the value\nofecan be obtained from electron counting, e.g. us-\ning a single electron pump. This framework allowed for\ndifferent schemes to measure eandh, and thus offered\nsubstantial options for consistency checks improving our\nunderstanding of the underlying physical processes. Also\ntoday, after the SI redefinition, the quantum/electricalmetrological triangle remains a powerful tool for metrol-\nogy and fundamental physics.\nNowadays, the constituting effects of the triangle are\nused to realize electric standards, such as the units am-\npere, ohm, and volt which complement the realizations\nof the SI base units. For instance, an array of Joseph-\nson junctions is used to realize a voltage of ±10 V with\nonly nV uncertainty [11, 12]. That voltage could be used\nfurther to obtain a precise current from a quantum Hall\ndevice. The same device also can be used to compare re-\nsistors with the exact value of the von Klitzing constant\nRK= 25.8128...kΩ. Thus, the individual effects, but also\ntheir interplay can be exploited for scientific and techni-\ncal application. Based on these unit realizations the ef-\nfects can be exploited for the investigation of electromag-\nnetic fields, for instance by measuring the induced volt-\nage in a conductor or the magnetic flux in a SQUID. This\nalso allows for a precise and independent determination\nand detailed study of additional fundamental constants,\nsuch as the vacuum permittivity ϵ0and vacuum perme-\nability µ0, which should be related by ϵ0µ0= 1/c2. From\nthese, further quantities, e.g., the wave impedance of the\nFIG. 1. The metrological triangle for eas well as for h.arXiv:2402.04135v1  [gr-qc]  6 Feb 20242\nvacuum Z0=p\nµ0/ϵ0≈377 Ω and the fine structure\nconstant α=e2(4πϵ0ℏc)−1=Z0(2RK)−1≈1/137 can\nbe derived and measured in optical and atomic physics\nexperiments.\nIn this letter we will propose to follow the same line of\nthought to introduce a quantum/gravitational metrologi-\ncal triangle. For that we use the fact that Einstein’s Gen-\neral Relativity in its weak field limit resembles the math-\nematical structure of electrodynamics. We deduce the\ngravitational analogues of the Josephson and the quan-\ntum Hall effect, which together with the counting of el-\nementary particle masses, form a triangle that provides\na great potential to investigate the interaction of gravity\nand quantum particles in various experiments.\nTHE STRUCTURE OF WEAK FIELD EINSTEIN\nGRAVITY\nIn what follows, we will motivate that gravity, in its\nweak field limit, is described by similar equations as elec-\ntrodynamics while both are fundamentally different in\nnature. Einstein’s General Relativity is based on the Ein-\nstein field equation\nRµν−1\n2gµνR=8πG\nc4Tµν. (1)\nHere, RµνandRare the Ricci tensor and Ricci scalar,\nrespectively, and gµνis the space-time metric. Tµνis\nthe energy momentum tensor and Gthe Newton gravi-\ntational constant. A standard weak field approximation\nuses a decomposition of the metric into a Minkowskian\nbackground ηµνand a small variation hµν≪1\ngµν=ηµν+hµν. (2)\nIf we define the mass density ρ:=1\nc2Ttt, the mass flux\ndensity ji:=1\nc2Tti, and the gravitational scalar and vec-\ntor potentials\nϕ:=¯httc2, ai:=¯hti, (3)\nwhere ¯hµν=hµν−1\n2ηµνηρσhρσ[13], then we obtain the\nlinearized Einstein equations in harmonic gauge\n∇∇∇ ·BBBg= 0 ∇∇∇ ×EEEg+˙BBBg= 0 (4)\n∇∇∇ ·EEEg= 4πGρ ∇∇∇ ×BBBg−1\nc2˙EEEg=4πG\nc2jjj(5)\nwhere we defined the weak field gravitational field\nstrengths, that is, the gravitoelectric and the gravito-\nmagnetic field strengths\nEg=−∇ϕ−˙a,Bg=∇×a. (6)\nBoldface symbols are 3-vectors and the dot denotes the\ntime derivative. With that, (4) and (5) are the weak field\nEinstein equations, see e.g. [13, 14].We may push even further the analogy to the Maxwell\nequations by eliminating the velocity of light: Introduc-\ning another gravitational constant µg:=c2/Gallows us\nto define the gravitoelectric and gravitomagnetic excita-\ntionsDg:=1\nGEgandHg:=µgBgand to rewrite the\ninhomogeneous equations (5) as\n∇·Dg= 4πρ ∇×Hg−˙Dg= 4πj.(7)\nHaving formulated the weak field Einstein equations (4)\nand (7) without the speed of light we obtained on this\nlevel a pre-metric version of these equations [15]. The\nconstant 1 /Gnow plays the role of a gravitoelectric vac-\nuum permittivity, and 1 /µgthe role of a gravitomagnetic\nvacuum permeability. Ggives the strength of the gravi-\ntoelectic attraction between two masses and µggives the\nstrength of the gravitomagnetic interaction between spin-\nning masses. Accordingly, one can use test particles to\nmeasure the gravitoelectric attraction generated by given\nmasses as well as the gravitomagnetic effects generated by\nrotating masses such as the Lense-Thirring or the Schiff\neffect, from which one can determine the two constants\nGandµg. Both constants together give a velocity which\nusually is interpreted as the velocity of light c, that is,\nGµg=c2. However, in principle we should interpret this\ncas the speed of gravity, Gµg=c2\ng. In this way, that\nis, by measuring ϵ0andµ0from the motion of charged\nparticles and Gandµgfrom the motion of massive par-\nticles, we have formulated a genuine test of whether the\nspeed of gravity coincides with the speed of light. – As\nin the electromagnetic case we also may define a gravita-\ntional wave impedance of vacuum Z(g)\n0:=p\nG/µ gwhich\nis∼2.34·10−19m2/kg s. The unit m2/kg s is the me-\nchanical analogy to the unit Ω of the electrical resistance\nin the relation I=U/R, ifIdescribes a mass current in\nkg/s and Ua gravitational potential difference in m2/s2.\nHaving discussed the strong analogy between electro-\ndynamics and weak field gravity regarding the field equa-\ntions as well as the equations of motion of charged and\nmassive particles, we now ask for implications of this\nanalogy for massive quantum particles.\nA GRAVITATIONAL JOSEPHSON EFFECT\nThe analogy between electrostatic and Newtonian\ngravitational forces gives a motivation to formulate a\ngravitational analogue of the electric Josephson effect.\nFor a gravitational Josephson effect we first need a grav-\nitational Josephson junction. This can be realized using\nquantum condensates on a horizontal table [16, 17] with\na step of height h0and a barrier, giving a non-symmetric\ndouble well potential (DWP). In the following we con-\nsider neutral quantum particles.\nThe non-symmetric DWP we are considering is a box\nof width 2 b, a step of height h0atx= 0 and a barrier of3\nfinite height and of width 2 a, see Fig 2. The whole DWP\nis in a constant gravitational field. The step provides\na constant potential difference gh0between two regions\nwhere gis the constant gravitational acceleration. The\ntotal Hamiltonian of the system is given by H=H0+\nV(x) with\nH0=−ℏ2\n2m∆ +mgz+(\nV0for|x|< a\n0 elsewhere, (8)\ndescribing a symmetric DWP with a tunnel barrier of\nheight V0, and a small potential step h0making the DWP\nnon-symmetric\nV(x) =(\nmgh 0forx <−a\n0 elsewhere. (9)\nWe will solve this problem whereby the non-symmetric\npart will be treated as perturbation.\nWe have a stationary situation so that Hψ=Eψwith\nE >0. Since in the considered problem the ydependence\nis trivial we will omit this degree of freedom in the fol-\nlowing. Then the above problem is a problem in xandz.\nWe assume ψ(x, z) =ψx(x)ψz(z) and take ψx(x) =e±ikx\nwhere the kis allowed to differ in the regions I,II and in\nthe barrier region m. Inserting this separation ansatz in\nthe Schr¨ odinger equation we obtain the following three\nequations\n\u0012\nE−mgh 0−ℏ2k2\nI\n2m\u0013\nψzI=−ℏ2\n2mψ′′\nzI+mgz ψ zI\n\u0012\nE−V0−ℏ2k2\nm\n2m\u0013\nψzm=−ℏ2\n2mψ′′\nzm+mgz ψ zm(10)\n\u0012\nE−ℏ2k2\nII\n2m\u0013\nψzII=−ℏ2\n2mψ′′\nzII+mgz ψ zII,\nwhere the prime denotes a derivative with respect to z.\nSee also Fig. 2. A global solution to these equations is\ngiven by\nψz(z)∼Ai\u0010z\nℓ−ϵi\u0011\n(11)\nthat reduces to ψzI,ψzmandψzIIin the corresponding\ndomains. Here ℓ=\u0000\n2m2g/ℏ2\u0001−1/3is a characteristic\nlength scale and ϵiis a zero of the Airy function, see\ne.g. [16]. The energy and the momenta take the values\nE=ℏ2k2\n2m+mgℓϵ i,kII≡k,kI=p\nk2−2m2gh0/ℏ2and\nkm=p\n2mV0/ℏ2−k2≡κ.\nNow we have to incorporate the boundary conditions\nregarding the coordinate z:ψz(h0) = 0, and ψz(0) = 0.\nOne way to achieve this is to choose h0= (ϵi−ϵj)ℓfor\nany two zeros i > j of the Airy function. The boundary\nand jump conditions regarding the coordinate xleads to\nthe quantization of the energy. The corresponding en-\nergies and wave functions can be determined along the\nFIG. 2. The choice of the height h0. The potential is shown\nin gray, the Airy function is given in terms of their position\nprobability. The horizontal dashed line indicates the node of\nthe Airy function which coincides with the height h0of the\nleft region. 2 ais the width of the tunnel barrier V0and 2 b\nis the size of the box. (The z-axis shows the density of the\nwave function depending on zbut also shows the height of\nthe potential V0.)\nusual procedure, see the Supplement Materials. The non-\nsymmetric DWP potential then is implemented as per-\nturbation. This way we obtain the Josephson equations\niℏ˙wL=ELwL+KwR (12)\niℏ˙wR=ERwR+KwL, (13)\ncharacterizing the tunneling dynamics in a non-\nsymmetric DWP. Here, the wL,Rare the probabilities\nto find the particle in the region I or II. EL,Ris the en-\nergy of the corresponding states, and K=1\n2∆Ewith\n∆E=mgh 0.\nWe obtained equations which have exactly the same\nstructure as for the description of the Josephson effect in\nsuperconductivity. Thus, we obtain the same solutions,\nthat is, for a constant h0̸= 0 we get the AC Josephson\neffect with a current I=I0sin(νt) with ν=m\nhv0where\nv0=gh0. For a mass of 1 amu and h= 1 cm we get\nν= 104Hz and for h∼10µm we have ν∼10 Hz.\nBased on that we can define a gravitational Josephson\nconstant\nK(g)\nJ:=m\nhins\nm2=Hz\nm2\ns2. (14)\nFor neutrons, for instance, we get K(g)\nJ= 1.2667·106s\nm2.\nIn order to measure this effect one needs a collective\nquantum system as, e.g., a Bose-Einstein condensate. For4\nFIG. 3. Quantum particle SQUID for measuring rotations\nand gravitomagnetism.\na system of incoherent particles the phases are different\nand average out so that no net current will be observ-\nable. In fact, the Josephson constant for BECs in a non-\nsymmetric DWP has been measured in [18, 19].\nThis Josephson junction may be used to build up a\ngravitational SQUID, see Fig. 3(a). For any ring quan-\ntum particle current in a rotating frame using k=\nv+ω×xwe have\n2πn=I\nk·dx=m\nℏI\nvvv·dxxx+m\nℏΦ(g)≈m\nℏΦ(g)(15)\nfor small velocities. Based on that we can define a flux\nquantum Φ(g)\n0:=h\nmso that\nΦ(g)=nΦ(g)\n0. (16)\nThis is deeply related to the Sagnac effect. The rota-\ntionωmay consist of a kinematical rotation and of the\ngravitomagnetic rotation.\nIn analogy to the electromagnetic case the current in\nFig. 3(b) is given by\nItot= 2Icsinδcos \nπΦ(g)\nΦ(g)\n0!\n(17)\nwith the critical current Icand with\nδ=γa+πΦ(g)\nΦ(g)\n0(18)\nwhere γais the phase shift at the Josephson junction\na, Fig. 3(b). The typical resolution of the flux is ∼\n10−6Φ(g)\n0. For an area of Σ ∼10−2m2this implies a\nsensitivity of ω∼10−10Hz for measuring a rotation.\nThis is a gyroscope based on the Sagnac effect for matter\nwaves realized through a gravitational SQUID for BECs,\nsee also [19].\nA GRAVITATIONAL QUANTUM HALL EFFECT\nWith mass currents, e.g. neutrons, atoms, or BECs, on\na rotating surface a gravitational analogue of the quan-\ntum Hall effect can be realized. According to Fig. 4,in thin rotating plate the incoming current (red) will be\ndeviated to the right. This can be compensated by a\nsmall gravitational acceleration in y-direction induced by\na slight tilting of the x−y-plane in y-direction.\nThe Hamilton operator of a point mass in a rotating\nframe [20] is given by\nH=1\n2mp2+ω·L, (19)\nwhere Lis the angular momentum of the particle. If the\nrotation is around the z-axis and the particles move only\ninx-y-plane, see Fig. 4, then pz= 0 and the Hamilton\noperator reads\nH=1\n2mppp2−ωzypx (20)\n≈1\n2m(px−mωzy)2+1\n2mp2\ny, (21)\nwhere we neglected terms quadratic in the rotation. This\nHamiltonian has exactly the same structure as the Hamil-\nton operator for the quantum Hall effect in the Landau\ngauge. One can calculate the gravitational analogue of\nthe Landau levels and the mass current for the case that\nthe Landau levels are filled up to some level s. By tilt-\ning the plane one may also induce a small gravitational\nacceleration needed to compensate the rotation induced\nacceleration, see Fig. 4. This then gives a gravitational\nquantum Hall effect described by the transversal mass\ncurrent resistivity\nρxy=h\nm21\ns. (22)\nThis leads to the gravitational von Klitzing constant\nR(g)\nK:=h\nm2inm2\nkg s(23)\nfor the used masses. If we take for instance neutrons or\nhydrogen atoms, then R(g)\nK= 2.481·1018m2\nkg s.\nThe classical gravitational Hall effect adds the Cori-\nolis and gravitational acceleration a=g+v×Ωwith\nv=p/m, and requires a= 0,Ω= Ω zezandv⊥Ω.\nIn equilibrium we have vy= 0 so that ggg=vxΩzeeey.\nAccordingly, the induced acceleration can be compen-\nsated by the plane being slightly tilted in y-direction (for\nΩ = 0 .1 Hz and vx= 10 cm/s we need g= 1 cm/s2=\n10−3gEarth what requires a tilt angle of ∼1′).\nThis can also be described within the Drude model.\nThe equation of motion including a friction/damping\nterm for reaching an equilibrium ( τis a damping time)\nthen is a=g+v×Ω−v\nτ. In equilibrium a= 0 and with\nj=nmvwe get Ohm’s law g=ρjwith the resistivity\nρ=\u00121\nnmτ−Ωz\nnmΩz\nnm1\nnmτ\u0013\n, (24)\nfrom which we can read off the longitudinal and transver-\nsal (Hall) resistivity\nρxx=1\nnmτρxy=Ωz\nnm. (25)5\nFIG. 4. Setup for a gravitational quantum Hall effect. Neu-\ntrons or atoms are moving on a thin rotating plate with a\ncurrent I. The rotation induces an acceleration field orthog-\nonal to the motion of the particles. A compensation of the\ntransversal acceleration can be achieved by a small tilt of the\nplane around the x-axis what induces a small acceleration a.\nA GRAVITATIONAL METROLOGICAL\nTRIANGLE\nHaving discussed the gravitational Josephson effect\nand the gravitational quantum Hall effect, we can use\nthese results to construct the gravitational analogue of\nthe quantum/electrical metrological triangle. In total\nanalogy to the electric constants e,KJ, and RK, which\nobey the relations [9, 10]\ne=2\nKJRK, h =4\nK2\nJRK, (26)\nwe have found their gravitational counterparts, which\nsatisfy\nm=1\nK(g)\nJR(g)\nK, h =1\nK(g)2\nJR(g)\nK, (27)\nsee Fig. 5. Here, we find that the mass mappears on the\nsame level as 2 ein (26), where it represents the charge\nof a Cooper pair. In (27), however, all effects needed to\nrealize the corresponding experiments are purely gravi-\ntational and without electromagnetic interaction. At one\nhand, this could open a new window to quantum based\nmetrology. At the other hand, this relations give raise\nto a number of possible consistency checks, to test our\nunderstanding of quantum systems in gravity. Ignoring\nfor a moment, that we fixed hin the current SI, we could\nassume experiments, where a fundamental mass mis pre-\ncisely known. In this version of the gravitational metro-\nlogical triangle, we would have access to three measur-\nands\nh , m/K(g)\nJ, m2/R(g)\nK, (28)\nFIG. 5. The metrological triangle for mas well as for h.\nwhich have to coincide. Within the modern SI, where\nthe Planck constant his exactly known we, in turn, have\nthree methods to obtain the mass of a chosen reference\nparticle,\nm , K(g)\nJh ,q\nR(g)\nKh , (29)\nwhich allows similar insights as in the previous case.\nIn that scenario, however, we have an additional op-\ntion, since the fixed constant his the same as for\nquantum/electrical metrological triangle. Assuming the\nPlank constant to have the same value for the electro-\nmagnetic andthe gravitational interaction of quantum\nparticles, we can relate its identities from Eqs. (26) and\n(27) to obtain\nK2\nJRK= 4K(g)2\nJR(g)\nK. (30)\nThus, we find both triangles to be not independent, what\nallows to check whether the Planck constant his the\nsame in the gravitational and in the electrical context.\nThis might be interpreted as a test for the universality\nof quantum mechanics [21].\nSUMMARY AND DISCUSSION\nA gravitational analogue of the quantum/electrical\nmetrological triangle has been introduced. The new tri-\nangle is based on quantum effects of neutrons or atoms in\na gravitational field – in complete analogy to the electro-\nmagnetic interaction of charged quantum particles used\nin the quantum/electrical triangle. This is possible since\nthe mathematical structure of weak field gravity resem-\nbles the governing equations of electromagnetism. We\nshowed that two fundamental quantum effects can also\nbe realized with the gravitational interaction: the gravi-\ntational Josephson effect and the gravitational quantum\nHall effect. Together with particle counting in analogy to\nthe count of electrons in a single electron pump those can\nbe used to trace back for instance mass measurements to6\nthe values of the Planck constant hand a chosen atomic\nor fundamental particle mass m. Fundamental masses\ncan be determined to tremendous precision, for instance\nin particle trap experiments [6, 22]. With that a pure\nquantum realization of the kilogram – besides the Avo-\ngadro experiment [23] and the Kibble balance [24] – may\nbe possible, see also [25].\nIn addition to their metrological aspects, the gravita-\ntional Josephson effect and the gravitational quantum\nHall effect can also provide fundamental test of the be-\nhavior of quantum systems in gravity: They are suitable\nto construct tests of the universality of quantum mechan-\nics, i.e, whether quantum mechanics is valid and the same\nfor interactions of quantum particles with electromag-\nnetic and gravitational fields. According to [18, 19], the\ngravitational Josephson constant might be measurable\nwith an accuracy of 10−3. The gravitational quantum\nHall effect has not yet been experimentally realized.\nAs in the electromagnetic case one may use devices\nbased on the discussed effects for a better exploration of\nthe dynamics of matter and, thus, of the gravitational\nfields. However, the new approach will not help in im-\nproving the precision of measurements due to the weak-\nness of gravity on the laboratory scale. Owing to the\nWeak Equivalence Principle, the mass of a test particle\nplays no role in the dynamics of testparticles, and in the\nastrophysical context a mass Mappears in the combina-\ntionGM only.\nNevertheless, from the motion of test objects it is possi-\nble to measure the gravitational constants Gandµg. The\nNewton gravitational constant Gis given by the force\nbetween two massive bodies [6], and the gravitomagnetic\nconstant µgcan be determined from the Lense-Thirring\neffect [26]. Gcan be measured with 0.1% precision, and\nforµgone approaches 1%. In analogy to the electro-\nmagnetic case one may – in principle – measure Gwith\nclocks in a gravitational field given by a laboratory mass\nor measure the gravitomagnetic field with gravitational\nSQUIDs.\nA further issue is a hypothetical time dependence of\nthe constants. While the SI-defining constants are con-\nstant by definition, the measured constants like ϵ0,µ0,\norG(or parameters relevant for the standard model of\nelementary particles) may depend on time. While a time\ndependent Gwill not interfere with the constancy of the\ndefining constants a time-dependent ϵ0, for example, will\ninfluence the energy levels of atoms and, thus, the defini-\ntion of the second. However, since the second is based on\none transition only, nothing will change and everything\nstill is consistent. One may detect a hypothetical time-\ndependence of ϵ0– what is equivalent to a time depen-\ndence of the fine structure constant – through the com-\nparison of different energy transitions within one atom\nbut this does not influence the particular transition cho-\nsen for the definition of the second. In any case, any\nhypothetical time dependence of Gor the fine structureconstant has been experimentally excluded with highest\nprecision [27, 28].\nFrom Gandµgfurther quantities can be derived, e.g.,\nthe gravitational wave impedance Z(g)\n0:=p\nG/µ g∼\n2.34·10−19m2/kg s, which governs the propagation of\ngravitational waves in vacuum.\nHaving started our investigation at the level of Einstein\nequations, we also have to take the Weak Equivalence\nPrinciple as granted. That means that one has to be\nsure that the gravitational interaction acts universally,\nthat is, independent from the composition and weight\nof pointlike bodies. If we distinguish between inertial\nand gravitational mass, then inspection of the derivation\nof the gravitational Josephson and quantum Hall effects\nshows that the gravitational Josephson constant depends\non the gravitational mass K(g)\nJ=mg\nhand the gravita-\ntional von Klitzing constant on the product of inertial\nand gravitational mass R(g)\nK=h\nmimg. Then in the rela-\ntions (27) the left hand sides will be replaced by h→hmi\nmg\nandm→mi, respectively. This agrees with the general\nobservation that in all equations of motion of test mat-\nter the gravitational mass appears in the combination\nmg/mi. Thus, in principle the MTs also provide a pure\nquantum test of the Weak Equivalence Principle when\nrealized with different types of, e.g., atoms. The recently\npublished final result of the MICROSCOPE mission pro-\nvided a classical test of the Weak Equivalence Principle\nconfirming it at the order 10−15[29], see also [30].\nWe thank H. Abele, G. Czycholl, F. W. Hehl, T.\nMehlst¨ aubler, A. Pelster, G. Sch¨ afer, A. Surzhykov, and\nJ. Ullrich for fruitful discussions. We acknowledge the\nsupport by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) under Germany’s Excel-\nlence Strategy-EXC-2123 “QuantumFrontiers” – Grant\nNo. 390837967 and the CRC 1464 “Relativistic and\nQuantum-based Geodesy” (TerraQ).\n∗claus.laemmerzahl@zarm.uni-bremen.de; Also at Gauss-\nOlbers Space Technology Transfer Center, c/o ZARM,\nUniversity of Bremen, 28359 Bremen, Germany\n†sebastian.ulbricht@ptb.de\n[1] International Bureau of Weights and Measures (BIPM,\nThe International System of Units (SI), 9th Edition\n(2019).\n[2] I. M. Mills, P. J. Mohr, T. J. Quinn, B. N. Taylor, and\nE. R. 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Lett. 126,\n011102 (2021), arXiv:2010.06620 [physics.atom-ph].\n[29] P. Touboul et al. (MICROSCOPE), MICROSCOPE\nMission: Final Results of the Test of the Equiva-\nlence Principle, Phys. Rev. Lett. 129, 121102 (2022),\narXiv:2209.15487 [gr-qc].\n[30] V. V. Singh, J. M¨ uller, L. Biskupek, E. Hackmann, and\nC. L¨ ammerzahl, Equivalence of Active and Passive Grav-\nitational Mass Tested with Lunar Laser Ranging, Phys.\nRev. Lett. 131, 021401 (2023), arXiv:2212.09407 [gr-qc].\nSUPPLEMENTARY MATERIAL\nHere we derive the essential equations for the gravi-\ntational Josephson effect. For that we first determine\nthe energies and eigenstates for the DWP and then the\ndynamics of symmetric and antisymmetric states in a\nnon-symmetric DWP.\nThe symmetric double well potential\nWe take the symmetric DWP as given by (8). We\nemploy the boundary conditions in x-direction, that is,\nψx(±b) = 0 and the smoothness of the wave function\natx=±a. In what follows, we first consider the sym-\nmetric potential ( h0= 0) and treat the asymmetry as a\nperturbation afterwards. Then the wave functions are\nψx(x) =\n\nAsin (k(x+b)) for −b < x < −a\nAeκasin(k(b−a))\nseκa+e−κa\n×\u0000\neκ(x−a)+se−κ(x+a)\u0001\nfor|x|< a\nAssin (k(−x+b)) for a < x < b ,\n(31)\nwhere s=±1 stands for the symmetric/antisymmetric\nsolution, respectively. Ais a normalization factor and κ\nandkhave to obey the condition\nκtanhs(κa) =−kcot (k(b−a)), (32)8\nwhat leads to a discrete set of solutions for kandκ. For\nsimplicity, we consider the case of weak coupling between\nI and II, that is, κa≫1, and of low energies of the\nparticles,k\nκ≪1. We then obtain the momentum and\nenergy eigenvalues\nkn=1\nb−a\u0012\nnπ−k0\nn\nκ0n\u0010\n1±e−2κ0\nna\u0011\u0013\n(33)\nEi,n,±=E0\nn\u0012\n1−2\n(b−a)κ0n\u0010\n1±2e−2κ0\nna\u0011\u0013\n+mgℓϵ i.\n(34)\nwith κ0\nn=1\nℏp\n2m(V0−E0n),k0\nn=nπ\nb−a, and E0\nn=\n(ℏk0\nn)2/(2m) where nshould be not too large. We also\ndefine the average energy and the energy difference\nEi,n=1\n2(Ei,n,++Ei,n,−) (35)\n∆En=Ei,n,+−Ei,n,−=8E0\nn\n(b−a)κ0ne−2κ0\nna(36)\nof symmetric and anti-symmetric states of given nandi.\nThe gravitational Josephson equations for the\nnon-symmetric DWP\nHaving solved the problem for the symmetric DWP, we\nnow investigate the influence of the small potential step\ngh0. Application of the standard first order perturbation\ntheory to the small potential step (9) yields the corrected\nstates\n˜ψi,n,s(x, z) =ψi,n,s(x, z) +s\n2mgh 0\n∆Enψi,n,−s(x, z) (37)\nobeying to first order approximation\n[H0+V(x)]˜ψi,n,s(x, z) =\u0012\nEi,n,s+1\n2mgh 0\u0013\n˜ψi,n,s(x, z).\n(38)\nThus, the small potential step gh0leads to a mixing be-\ntween symmetric ( s= +1) and anti-symmetric ( s=−1)\nwave functions, while mixing between states of different\nnis suppressed by a factor ∆ En/Ei,n,s. In what follows\nwe drop the indices nandiassuming them to be fixed\nand show them only when needed.We now want to define left and right bin states by a\nsuperposition of eigen states of the non-symmetric DWP\n(37). With the ansatz\nψL= cos θ˜ψ++ sin θ˜ψ− (39)\nψR= sin θ˜ψ+−cosθ˜ψ− (40)\nwe minimizeR0\n−b|ψR|2dx+Rb\n0|ψL|2dxwhat is the prob-\nability to find a particle described by ψRin the left bin\nand vice versa. One finds, that the best fitting for left\nand right states are given by\nψL=1√\n2(ψ++ψ−) (41)\nψR=1√\n2(ψ+−ψ−). (42)\nThese are the same states as the optimal left and right\nstates for the symmetric DWP. However, in this case θ\nreads\nθ≈ −π\n4+mgh 0\n2∆En(43)\nand the action of the Hamiltonian is given by\n(H0+V)ψL=ELψL+KψR (44)\n(H0+V)ψR=ERψR+KψL (45)\nwith the energies and the coupling constants\nEi,n,R/L=Ei,n±1\n2∆Encos(2 θ)≈Ei,n±1\n2mgh 0(46)\nKn=−1\n2∆Ensin(2θ)≈1\n2∆En. (47)\nWe find in particular that the energy difference Ei,n,R−\nEi,n,L=mgh 0is related to the potential step (9) and\ndoes not depend on iorn.\nThe wave functions ψL/R(x) are no eigenstates of the\nHamilton operator. In order to check how the actual\nwave function decomposes into the left and right wave\nfunctions we make the ansatz\nΨ(t, x, z ) =1√\n2[wL(t)ψL(x, z) +wR(t)ψR(x, z)] (48)\nand determine the equations of motion for the pobabili-\ntieswL/R(t) from iℏ˙Ψ(x, t) = [H0+V(x)]Ψ(x, t)."    }]